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+{"text":"\\section{Introduction and Motivation} \nIn the background of any investigation into learning algorithms are no-free-lunch phenomena: roughly, the observation that assumption-free statistical learning is infeasible in general (see, e.g., \\cite[Ch. 5]{MLbook} for a formal statement). Common wisdom is that learning algorithms and architectures must adequately reflect non-trivial features of the data-generating distribution to gain inductive purchase. \n\nFor many purposes we need to move beyond passive observation, focusing instead on what \\emph{would} happen were we to \\emph{act} upon a given system. Even further, we sometimes desire to \\emph{explain} the behavior of a system, raising questions about what \\emph{would have} occurred had some aspects of a situation been different. Such questions depend not just on the data distribution; they depend on deeper features of underlying data-generating \\emph{processes} or \\emph{mechanisms}. It is thus generally acknowledged that stronger assumptions are  required if we want to draw \\emph{causal} conclusions from data \\citep{Spirtes2001,Pearl2009,Imbens2015,Peters,9363924}. \n\n\nWhether implicit or explicit, any approach to causal inference involves a space of candidate causal models, viz. data-generating processes. Indeed, a blunt way of incorporating inductive bias is simply to \\emph{omit} some class of possible causal hypotheses from consideration. Many (im)possibility results in the literature can accordingly be understood as pertaining to all models within a class. \nFor instance, if we can restrict attention to \\emph{Markovian} models that satisfy \\emph{faithfulness}, then we can \\emph{always} identify the structure of a model from experimental data (e.g., \\cite{Eberhardt2005,Spirtes2001}). If we can restrict attention to Markovian (continuous) models with linear functions and \\emph{non}-Gaussian noise, then \\emph{every} model can be learned even from purely observational data \\citep{JMLR:v7:shimizu06a}. As a negative example, in the larger class of (not necessarily Markovian) models, \\emph{no} model can \\emph{ever} be determined from observational data alone \\citep{Spirtes2001,BCII2020}.\n\nAt the same time, in many settings it is sensible to aim for results with ``nearly universal'' force. It is natural to ask, e.g., within the class of all Markovian models, how ``typical'' are those in which the faithfulness condition is violated? This might tell us, for instance, how typically we could expect failure of a method that depended on these assumptions. A well-known result shows that, fixing any particular causal dependence graph, such violations have \\emph{measure zero} for any smooth (e.g., Lebesgue) measure on the parameter space of distributions consistent with that graph \\citep{Meek1995}. In fact, the standard notion of statistical consistency itself, which underlies many possibility results in causal inference, requires omission of some purportedly ``negligible'' set of possible data streams \\citep{DiaconisFreedman,Spirtes2001}.\n\nThere are two standard mathematical approaches to making concepts like ``typical'' and ``negligible'' rigorous: \\emph{measure-theoretic} and \\emph{topological}. While the two approaches often agree, they capture slightly different intuitions \\citep{Oxtoby}.  One virtue of the measure-theoretic approach is its natural  probabilistic interpretation: intuitively, we are exceedingly \\emph{unlikely} to hit upon a set with measure zero. At the same time, the measure-theoretic approach is sometimes criticized in statistical settings for its alleged  dependence on a measure, and this has been argued to favor topological approaches (see, e.g.,  \\cite{BELOT2020159} on no-free-lunch theorems). The latter of course in turn demands an appropriate topology.\n\nIn the present work we show how to define a sequence of meaningful topologies on the space of causal models, each corresponding to a progressively coarser level of the so called \\emph{causal hierarchy} (\\cite{pearl2018book,BCII2020}; see Fig. \\ref{fig:hierarchy} for an abbreviated pictorial summary).\nWe aim to demonstrate that topologizing causal models in this way helps clarify the scope and limits of causal inference under different assumptions, as well as the potential empirical status of those very assumptions, in a highly general setting. \n\nOur starting point is a canonical topology on the space of Borel probability distributions called the \\emph{weak topology}. The weak topology is grounded in the fundamental notion of \\emph{weak convergence} of probability distributions \\citep{Billingsley} and is thereby closely related to problems of statistical inference (see, e.g., \\cite{DemboPeres}). Recent work has sharpened this correspondence, showing that open sets in the weak topology correspond exactly to the statistical hypotheses that can be naturally deemed  \\emph{verifiable} \\citep{Genin2018,GeninKelly}.\nWe extend the correspondence to higher levels of the causal hierarchy, including the most refined and expansive ``top'' level consisting of all (well-founded) causal models. Lower levels and  natural subspaces (e.g., corresponding to prominent causal assumption classes) emerge as coarsenings and projections of this largest space. As an illustration of the general approach, we prove a topological version of the causal hierarchy theorem from \\cite{BCII2020}. Rather than showing that collapse happens only in a measure zero set  as in \\cite{BCII2020}, our Theorem \\ref{thm:hierarchyFormal} show that collapse is \\emph{topologically meager}. Conceptually, this highlights a  different (but complementary) intuition:\nnot only is collapse  exceedingly unlikely in the sense of measure, meagerness implies that collapse could never be \\emph{statistically verified}. Correlatively, this implies that any causal assumption that would generally allow us to infer counterfactual probabilities from experimental (or ``interventional'') probabilities must itself be statistically unverifiable (Corollary \\ref{cor:hierarchy}). \n\nTo derive such a result we actually show something slightly stronger (see Lem.~\\ref{lem:separation}): even with respect to the subspace of models consistent with a fixed \\emph{temporal order} on variables, the causal hierarchy theorem holds. Merely knowing the temporal order of the variables is not enough to render collapse of the hierarchy a statistically verifiable proposition.\nFurthermore, we show that the \\emph{witness to collapse} can be taken as any of the well-known counterfactual ``probabilities of causation'' (see, e.g., \\cite{Pearl1999}): probabilities of necessity, sufficiency, necessity \\emph{and} sufficiency, enablement, or disablement. That is, none of these important quantities are fully determined by experimental data except in a meager set. \n\n\n\nIn \\S\\ref{scms} we give background on causal models, and in \\S\\ref{sec:causalhierarchy} we present a model-theoretic characterization of the causal hierarchy as a sequence of spaces. Topology is introduced in \\S\\ref{sec:weaktopology}, and the main results about collapse appear in \\S\\ref{sec:main}.\nFor the technical results, we include proof sketches in the main text to provide the core intuitions, relegating some of the details to an exhaustive technical appendix, which also includes additional supplementary material.\n\n\n\\section{Structural Causal Models} \\label{scms}\nA fundamental building block in the theory of causality is the \\emph{structural causal model} \\citep{Pearl1995,Spirtes2001,Pearl2009} or SCM, which formalizes the notion of a data-generating process. In addition to specifying data-generating distributions, these models also specify the generative mechanisms that produce them. For the purpose of causal inference and learning, SCMs provide a broad, fine-grained hypothesis space.\n\nThe notions in this section have their usual definition following, e.g., \\cite{Pearl2009}, but we have\nrecast them in the standard language of Borel probability spaces so as to handle the case of infinitely many variables rigorously.\nWe start with notation, basic assumptions, and some probability theory\n\n\n\\begin{notation*}\nThe signature (or range) of a variable $V$ is denoted $\\chi_V$. \nWhere $\\*S$ is a set of variables, let $\\chi_{\\*S} = \\bigtimes_{S \\in \\*S} \\chi_S$.\nGiven an indexed family of sets $\\{S_\\beta\\}_{\\beta \\in B}$ and elements $s_\\beta \\in S_\\beta$, let $(s_\\beta)_\\beta$ denote the tuple whose element at index $\\beta$ is $s_\\beta$, for all $\\beta$.\nFor $B' \\subset B$ write $\\pi_{B'} : \\bigtimes_{\\beta \\in B} S_\\beta \\to \\bigtimes_{\\beta \\in B'} S_{\\beta}$ for the \\emph{projection map} sending each $(s_\\beta)_{\\beta \\in B} \\mapsto (s_{\\beta'})_{\\beta' \\in B'}$; abbreviate $\\pi_{\\beta'} = \\pi_{\\{\\beta'\\}}$, where $\\beta' \\in B$. \n\\end{notation*}\n\nThe reader is referred to standard texts \\cite{johnkelley1975,Bogachev2007} for elaboration on the concepts used below.\n\n\\begin{definition}[Topology]\\label{def:basictopology}\nFor discrete spaces (like $\\chi_S$, for a single categorical variable $S$) we use the discrete topology and for product spaces (like $\\chi_{\\*S}$ for a set of variables $\\*S$) we use the product topology.\nNote that the so-called \\emph{cylinder sets} of the form $\\pi^{-1}_{\\*Y}(\\{\\*y\\})$ for finite subsets $\\*Y \\subset \\*S$ and $\\*y \\in \\chi_{\\*Y}$\nform a basis for the product topology on $\\chi_{\\*S}$. This cylinder set is a subset of $\\chi_{\\*S}$, and contains exactly those valuations agreeing with the value $\\pi_Y(\\*y)$ specified in $\\*y$ for $Y$, for every $Y \\in \\*Y$. Following standard statistical notation this cylinder is abbreviated as simply $\\*y$.\n\\end{definition}\n\\begin{definition}[Probability]\n\\label{def:probability}\nWhere $\\vartheta$ is a topological space write $\\mathcal{B}(\\vartheta)$ for its Borel $\\sigma$-algebra of measurable subsets.\nLet $\\mathfrak{P}(\\vartheta)$ be the set of probability measures on $\\mathcal{B}(\\vartheta)$.\nSpecifically, elements of $\\mathfrak{P}(\\vartheta)$ are functions $\\mu: \\mathcal{B}(\\vartheta) \\to [0, 1]$ assigning a probability to each measurable set such that $\\mu(\\vartheta) = 1$ and $\\mu\\big(\\bigcup_{i=1}^{\\infty}(S_i)\\big) = \\sum_{i=1}^{\\infty} \\mu(S_i)$ for each sequence $S_1, S_2, \\dots$ of pairwise disjoint sets from $\\mathcal{B}(\\vartheta)$.\nA map $f: \\vartheta_1 \\to \\vartheta_2$ is said to be \\emph{measurable} if $f^{-1}(S_2) \\in \\mathcal{B}(\\vartheta_1)$ for every $S_2 \\in \\mathcal{B}(\\vartheta_2)$.\n\\end{definition}\n\n\\begin{fact}[Lemma~1.9.4 \\cite{Bogachev2007}]\nA Borel probability measure is determined by its values on a basis\n\\end{fact}\n\n\\subsection{SCMs, Observational Distributions}\\label{ss:scml1}\n\nLet $\\*V$ be a set of \\emph{endogenous variables}.\nWe assume for simplicity every variable $V \\in \\*V$ is dichotomous\nwith $\\chi_V = \\{0, 1\\}$, although the results here generalize to any larger countable range.\nInfluences among endogenous variables are the main phenomena our formalism aims to capture.\nA well-founded\\footnote{See Appendix~\\ref{app:causalmodels} for additional background on orders and relations.} \\emph{direct influence} relation $\\rightarrow$ on $\\*V$ encapsulates the notion of one endogenous variable possibly influencing another. For each $V \\in \\*V$, we call $\\{V' \\in \\*V : V' \\rightarrow V\\} = \\mathbf{Pa}(V)$ the \\emph{parents} of $\\*V$.\nWe assume every set $\\mathbf{Pa}(V)$ is finite; this condition is called \\emph{local finiteness}. These two assumptions (well-foundedness and local finiteness) generalize the common \\emph{recursiveness} assumption to the infinitary setting, and have an alternative characterization in terms of ``temporal'' orderings:\n\\begin{fact}\\label{fact:omegalike}\nSay that a total order $\\prec$ on $\\*V$ is \\emph{$\\omega$-like} if every node has finitely many predecessors: for each $V \\in \\*V$, the set $\\{V' : V' \\prec V\\}$ is finite.\nThen the influence relation $\\rightarrow$ is extendible to an $\\omega$-like order iff $\\rightarrow$ is well-founded and locally finite.\n\\end{fact}\nIn addition to endogenous variables, causal models have \\emph{exogenous variables} $\\*U$.\nEach endogenous $V$ depends on a subset $\\*U(V)\\subset \\*U$ of ``exogenous parents'' and\nuncertainty enters via exogenous noise, that is, a distribution from $\\mathfrak{P}(\\chi_{\\*U})$.\nA \\emph{structural function} (or \\emph{mechanism}) for $V \\in \\*V$ is a measurable $f_{V} : \\chi_{\\mathbf{Pa}(V)} \\times \\chi_{\\*U(V)} \\to \\chi_{V}$\nmapping parental endogenous and exogenous valuations to values.\n\\begin{definition}\\label{def:scm:lit}\nA \\emph{structural causal model} is a tuple $\\mathcal{M} = \\langle \\*U, \\*V, \\{f_V\\}_{V \\in \\*V}, P \\rangle$ where $\\*U$ is a collection of exogenous variables, $\\*V$ is a collection of endogenous variables, $f_V$ is a structural function for each $V \\in \\*V$, and $P \\in \\mathfrak{P}(\\chi_{\\*U})$ is a probability measure on (the Borel $\\sigma$-algebra of) $\\chi_{\\*U}$.\n\\end{definition}\n\nAs is well known, recursiveness implies that each $\\*u \\in \\chi_{\\*U}$ induces a unique $\\*v \\in \\chi_{\\*V}$ that solves the simultaneous system of structural equations $\\{V = f_V\\}_V$:\n\\begin{proposition}\nAny SCM ${\\mathcal{M}}$ with well-founded, locally finite parent relation $\\rightarrow$ induces a unique measurable $m^{{\\mathcal{M}}} : \\chi_{\\*U} \\to \\chi_{\\*V}$ such that $f_V\\big( \\pi_{\\mathbf{Pa}(V)}(m^{{\\mathcal{M}}}(\\*u)), \\pi_{\\*U(V)}(\\*u) \\big) = \\pi_{V}\\big(m^{{\\mathcal{M}}}(\\*u)\\big)$ for all $\\*u \\in \\chi_{\\*U}$ and $V \\in \\*V$.\n\\end{proposition}\nMeasurability then entails that the exogenous noise $P$ induces a distribution on joint valuations of $\\*V$, called the \\emph{observational distribution},\nwhich characterizes passive observations of the system.\n\\begin{definition}\nThe observational distribution $p^{{\\mathcal{M}}} \\in \\mathfrak{P}(\\chi_{\\*V})$ is defined on open sets by $p^{{\\mathcal{M}}}(\\*y) = P\\big((m^{{\\mathcal{M}}})^{-1}(\\*y)\\big)$. Here recall that $\\*y$ represents a cylinder subset (Definition~\\ref{def:basictopology}) of $\\chi_{\\*V}$.\n\\end{definition}\n\n\n\n\n\n\n\\subsection{Interventions}\\label{ss:scml2}\nWhat makes SCMs distinctively causal is the way they accommodate statements about possible manipulations of a causal setup capturing, e.g., observations resulting from a controlled experimental trial. This is formalized in the following definition.\n\\begin{definition}\nAn \\emph{intervention} is a choice of a finite subset of variables $\\*W \\subset \\*V$ and $\\*w \\in \\chi_{\\*W}$. This intervention is written $\\*W \\coloneqq \\*w$, and we let $A$ be the set of all interventions.\nUnder this intervention, each $W \\in \\*W$ is held fixed to its value $\\pi_{W}(\\*w) \\in \\chi_W$ in $\\*w$ while the mechanism for any $V \\in \\*V \\setminus \\*W$ is left unchanged.\nSpecifically, where ${\\mathcal{M}}$ is as in Definition \\ref{def:scm:lit}, the manipulated model for $\\*W \\coloneqq \\*w$ is the model ${\\mathcal{M}}_{\\*W \\coloneqq \\*w} = \\langle \\*U, \\*V, \\{f^{\\*W \\coloneqq \\*w}_V\\}_{V \\in \\*V}, P\\rangle$ where\n\\begin{align*}\nf^{\\*W \\coloneqq \\*w}_V = \\begin{cases}\nf_V, & V \\notin \\*W\\\\\n\\text{constant func. mapping to } \\pi_V(\\*w), & V \\in \\*W.\n\\end{cases}\n\\end{align*}\n\\end{definition}\nThe \\emph{interventional} or \\emph{experimental distribution} $p^{\\mathcal{M}_{\\*W \\coloneqq \\*w}} \\in \\mathfrak{P}(\\chi_{\\*V})$ is just the observational distribution for the manipulated model ${\\mathcal{M}}_{\\*W \\coloneqq \\*w}$, and it encodes the probabilities for an experiment in which the variables $\\*W$ are fixed to the values $\\*w$.\n\\begin{remark}\nEmpty interventions $\\varnothing \\coloneqq ()$ are just passive observations, i.e., $p^{{\\mathcal{M}}_{\\varnothing \\coloneqq ()}} = p^{{\\mathcal{M}}}$.\n\\end{remark}\n\n\n\\subsection{Counterfactuals}\\label{ss:scml3}\nBy permitting multiple manipulated settings to share exogenous noise, not only the distribution arising from a single manipulation, but also joint distributions over multiple can be considered. These are often called \\emph{counterfactuals}. The set $\\mathfrak{P}(\\chi_{A \\times \\*V})$ encompasses the combined joint distributions over $\\*V$ for any combination of interventions from $A$. A basis for the space $\\chi_{A \\times \\*V}$ are the cylinder sets of the following form, for some sequence $(\\*X \\coloneqq \\*x, \\*Y), \\dots, (\\*W \\coloneqq \\*w, \\*Z)$ of pairs, where $ \\*Y, \\dots, \\*Z \\subset \\*V$ are finite, and $\\*X \\coloneqq \\*x, \\dots, \\*W \\coloneqq \\*w \\in A$ are interventions:\n\\begin{align*}\n    \\pi^{-1}_{\\{\\*X \\coloneqq \\*x\\}\\times \\*Y}(\\{\\*y\\}) \\cap \\dots \\cap \\pi^{-1}_{\\{\\*W \\coloneqq \\*w\\}\\times \\*Z}(\\{\\*z\\}).\n\\end{align*}\nWe will abbreviate this open set as ${\\*y}_{\\*x}, \\dots, {\\*z}_{\\*w}$, writing, e.g. simply $\\*x$ for the intervention $\\*X = \\*x$.\n\\begin{definition}\nGiven ${\\mathcal{M}}$, define a counterfactual distribution $p_{\\text{cf}}^{{\\mathcal{M}}} \\in \\mathfrak{P}(\\chi_{A \\times \\*V})$ on a basis as follows:\n\\begin{equation*}\n    p_{\\text{cf}}^{{\\mathcal{M}}}( \\*y_{\\*x}, \\dots, {\\*z}_{\\*w} ) = P\\big((m^{{\\mathcal{M}}_{\\*X \\coloneqq \\*x}})^{-1}(\\*y) \\cap  \\dots \\cap (m^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}})^{-1}(\\*z)\\big).\n\\end{equation*}\nHere, the letters $\\*y, \\dots, \\*z$ on the right-hand side abbreviate the respective cylinder sets (Definition~\\ref{def:basictopology}) $\\pi^{-1}_{\\*Y}(\\{\\*y\\}), \\dots, \\pi^{-1}_{\\*Z}(\\{\\*z\\})$.\n\\end{definition}\n\\begin{remark}\nMarginalizing $p^{{\\mathcal{M}}}_{\\text{cf}}$ to any single intervention $\\*W \\coloneqq \\*w$ yields $p^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}}$. If $\\chi_{\\*U}$ is finite, we obtain a familiar \\cite{Galles1998} sum formula\n$p_{\\text{cf}}^{{\\mathcal{M}}}(\\*y_{\\*x}, \\dots, {\\*z}_{\\*w}) = \\sum_{\\{\\*u \\mid m^{{\\mathcal{M}}_{\\*X \\coloneqq \\*x}}(\\*u) \\in \\*y, \\dots, m^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}}(\\*u) \\in \\*z\\}} P(\\*u)$.\n\\end{remark}\n\n\\begin{example} As a very simple example (drawn from \\cite{Pearl2009,BCII2020}), just to illustrate the previous definitions and notation, consider a scenario with two binary exogenous variables $\\mathbf{U} = \\{U_1,U_2\\}$ and two binary endogenous variables $\\mathbf{V} = \\{X,Y\\}$. Let $U_1,U_2$ both be uniformly distributed, and define $f_X:\\chi_{U_1} \\rightarrow \\chi_X$ to be the identity, and $f_Y:\\chi_X \\times \\chi_{U_2} \\rightarrow \\chi_Y$ by $f_Y(u,x) = ux + (1-u)(1-x)$. This fully defines an SCM ${\\mathcal{M}}$ with influence $X \\rightarrow Y$, and produces an observational distribution $p^{\\mathcal{M}}$ such that $p^{\\mathcal{M}}(x,y) = \\nicefrac{1}{4}$ for all four settings $X=x,Y=y$. \n\nThe space $A$ of interventions in this example includes the empty intervention and all combinations of $X:=x$ and $Y:=y$, with $x,y \\in \\{0,1\\}$. Notably, all interventional distributions here collapse to observational distributions, e.g., $p^{{\\mathcal{M}}_{X:= x}}(X,Y) = p^{\\mathcal{M}}(X,Y)$, for both values of $x$. Thus, ``experimental'' manipulations of this system reveal little interesting causal structure. The counterfactual distribution $p^{\\mathcal{M}}_{\\mathrm{cf}}$, however, does not trivialize. For instance, $p^{\\mathcal{M}}_{\\mathrm{cf}}((X:=1, Y=1),(X:=0,Y=0)) = \\nicefrac{1}{2}$. This term is known as the \\emph{probability of necessity and sufficiency} \\cite{Pearl1999}, which we can abbreviate by $p^{\\mathcal{M}}_{\\mathrm{cf}}(y_x,y'_{x'})$. Note that $p^{\\mathcal{M}}_{\\mathrm{cf}}(y_x,y'_{x'}) \\neq  p^{\\mathcal{M}}_{\\mathrm{cf}}(y_x)p^{\\mathcal{M}}_{\\mathrm{cf}}(y'_{x'}) = \\nicefrac{1}{4}$. Similarly, $p^{\\mathcal{M}}_{\\mathrm{cf}}(y'_x,y_{x'}) = \\nicefrac{1}{2}$.\n\n\\end{example}\n\n\n\n\n\\subsection{SCM classes}\nWe now define several subclasses of SCMs that we will use throughout the paper.\nNotably, we do not require their endogenous variable sets $\\*V$ to be finite. It is infinite in many applications, for instance, in time series models, or generative models defined by probabilistic programs (see, e.g., \\citep{II2019,Tavares}). Because the proofs call for slightly different methods, we deal with the infinite and finite cases separately. We make one additional assumption in the infinite case.\n\\begin{definition}\n$\\mu \\in \\mathfrak{P}(\\vartheta)$ is \\emph{atomless} if $\\mu(\\{t\\}) = 0$ for each $t \\in \\vartheta$;\n ${\\mathcal{M}}$ is atomless if $p^{{\\mathcal{M}}}_{\\text{cf}}$ is atomless.\n\\end{definition}\nIntuitively, an atomless distribution is one in which weight is always ``smeared'' out continuously and there are no point masses; infinitely many fair coin flips, for example, generate an atomless distribution as the probability of obtaining any given infinite sequence is zero.\n\\begin{definition}\nFor the remainder of the paper, fix a countable endogenous variable set $\\*V$. Define the following classes of SCMs:\n\\begin{align*}\n    \\mathfrak{M}_\\prec &= \\text{SCMs over } \\*V \\text{ whose influence relation is extendible to the } \\omega\\text{-like order } \\prec;\\\\\n    \\mathfrak{M}_X &= \\text{SCMs over } \\*V \\text{ in which the variable } X \\text{ has no parents: } \\mathbf{Pa}(X) = \\varnothing;\\\\\n    \\mathfrak{M} &= \\text{all SCMs over } \\*V = \\bigcup_{\\prec} \\mathfrak{M}_\\prec = \\bigcup_X \\mathfrak{M}_X.\n\\end{align*}\nIf $\\*V$ is infinite then all SCMs in the classes above are assumed to be atomless.\n\\end{definition}\n\n\\section{The Causal Hierarchy} \\label{sec:causalhierarchy}\n\nImplicit in \\S{}\\ref{scms}, and indeed in much of the literature on causal inference, is a hierarchy of causal expressivity. Following the metaphor offered in \\cite{pearl2018book}, it is natural to characterize three levels of the hierarchy as the \\emph{observational}, \\emph{interventional} (experimental), and \\emph{counterfactual} (explanatory). Drawing on recent work \\citep{BCII2020,ibelingicard2020} we make this characterization explicit. The levels will be defined in descending order of causal expressivity (the reverse of \\S\\ref{scms}). Fig.~\\ref{fig:hierarchy}(a) summarizes our definitions.\n\nHigher levels determine lower levels---counterfactuals determine interventionals, and the observational is just an (empty) interventional.\nThus movement ``downward'' in the causal hierarchy corresponds to a kind of projection.\nFor indexed $\\{S_\\beta\\}_{\\beta \\in B}$\nand $B' \\subset B$\nlet $\\varsigma_{B'} : \\mathfrak{P}(\\bigtimes_{\\beta \\in B} S_\\beta) \\to \\mathfrak{P}(\\bigtimes_{\\beta \\in B'} S_\\beta)$\nbe the \\emph{marginalization} map taking a joint distribution to its marginal on $B'$.\n\\begin{definition}\nDefine three composable \\emph{causal projections} $\\{\\varpi_i\\}_{1 \\le i \\le 3}$\nwith signatures and definitions\n\\begin{gather*}\n\\varpi_3 : \\mathfrak{M} \\to \\mathfrak{P}(\\chi_{A \\times \\*V}), \\quad \\varpi_2: \\mathfrak{P}(\\chi_{A \\times \\*V}) \\to  \\bigtimes_{\\alpha \\in A} \\mathfrak{P}(\\chi_{\\*V}), \\quad \\varpi_1: \\bigtimes_{\\alpha \\in A} \\mathfrak{P}(\\chi_{\\*V}) \\to \\mathfrak{P}(\\chi_{\\*V});\\\\\n    \\varpi_3: {\\mathcal{M}} \\mapsto p^{{\\mathcal{M}}}_{\\mathrm{cf}}, \\quad \\varpi_2: \\mu_3 \\mapsto \\big(\\varsigma_{\\{\\alpha\\} \\times \\*V}(\\mu_3)\\big)_{\\alpha \\in A}, \\quad \\varpi_1: (\\mu_\\alpha)_{\\alpha \\in A} \\mapsto  \\mu_{\\varnothing \\coloneqq ()} = \\pi_{\\varnothing \\coloneqq ()}\\big((\\mu_\\alpha)_\\alpha\\big).\n\\end{gather*}\nThe \\emph{causal hierarchy} consists of three sets $\\{\\mathfrak{S}_i\\}_{1 \\le i \\le 3}$ defined as images or projections of $\\mathfrak{M}$:\n\\begin{equation*}\n    \\mathfrak{S}_3 = \\varpi_3(\\mathfrak{M}), \\quad \\mathfrak{S}_2 = \\varpi_2(\\mathfrak{S}_3), \\quad \\mathfrak{S}_1 = \\varpi_1(\\mathfrak{S}_2).\n\\end{equation*}\n\\end{definition}\nThese are the three \\emph{Levels} of the hierarchy. The definitions cohere with those of \\S{}\\ref{scms} (and, e.g., \\cite{Pearl2009,BCII2020}):\n\\begin{fact}\nLet ${\\mathcal{M}} \\in \\mathfrak{M}$.\nThen\n$\\mu_3 = \\varpi_3({\\mathcal{M}}) \\in \\mathfrak{S}_3$ trivially coincides with its counterfactual distribution as defined in \\S{}\\ref{ss:scml3}, while\n$(\\mu_\\alpha)_\\alpha = \\varpi_2(\\mu_3) \\in \\mathfrak{S}_2$ coincides with the indexed family of all its interventional distributions (\\S{}\\ref{ss:scml2}), i.e., $\\pi_{\\*W \\coloneqq \\*w} \\big((\\mu_\\alpha)_\\alpha\\big) = p^{{\\mathcal{M}}_{\\*W \\coloneqq \\*w}}$ for each $\\*W \\coloneqq \\*w \\in A$.\nFinally $\\mu = \\varpi_1\\big((\\mu_\\alpha)_\\alpha\\big) \\in \\mathfrak{S}_1$ coincides with its observational distribution (\\S{}\\ref{ss:scml1}).\n\\end{fact}\nThus, e.g., $\\mathfrak{S}_3$ is the set of counterfactual distributions that are consistent with at least some SCM from $\\mathfrak{M}$. It is a fact that $\\mathfrak{S}_3 \\subsetneq \\mathfrak{P}(\\chi_{A \\times \\*V})$ and similarly not every interventional family belongs to $\\mathfrak{S}_2$; see\nAppendix \\ref{app:causalhierarchy} for explicit characterizations.\nAt the observational level, this is simple:\n\\begin{fact}\n$\\mathfrak{S}_1 = \\mathfrak{P}(\\chi_{\\*V})$ in the finite case. In the infinite case, $\\mathfrak{S}_1 = \\{ \\mu \\in \\mathfrak{P}(\\chi_{\\*V}): \\mu \\text{ is atomless}\\}$. \\label{prop:alternative}\n\\end{fact}\nWe will also use the subsets $\\{\\mathfrak{S}^\\prec_i\\}_{i}$ and $\\{\\mathfrak{S}^X_i\\}_{i}$, which are defined analogously but via projection from $\\mathfrak{M}_\\prec$ and $\\mathfrak{M}_X$ respectively.\n\n\n\n\n\n\\subsection{Problems of Causal Inference} \\label{subsection:probs}\nAs elucidated in \\cite{pearl2018book,BCII2020}, the causal hierarchy helps characterize many standard problems of causal inference, in as far as these problems typically involve ascending levels of the hierarchy. Some examples include: \n\\begin{enumerate}\n    \\item Classical identifiability: given observational data about some variables in $\\*V$, estimate a \\emph{causal effect} of setting variables $\\*X$ to values $\\*x$ \\citep{Pearl1995,Spirtes2001}. In the notation here, given information about $p^{\\mathcal{M}}(\\*V)$, can we determine $p^{\\mathcal{M}_{\\*X \\coloneqq \\*x}}(\\*Y)$?\n    \\item General identifiability: given a mix of observational data and limited  experimental data---that is, information about $p^{\\mathcal{M}}(\\*V)$ as well as some experimental distributions of the form $p^{\\mathcal{M}_{\\*W \\coloneqq \\*w}}(\\*V)$---determine $p^{\\mathcal{M}_{\\*X \\coloneqq \\*x}}(\\*Y)$ \\citep{tian-2002,lee2019general}.\n    \\item Structure learning: given observational data, and perhaps experimental data, infer properties of the underlying causal influence relation $\\rightarrow$ \\citep{Spirtes2001,Peters}.\n    \\item Counterfactual estimation: given a combination of observational and experimental data, infer a counterfactual quantity, such as probability of necessity  \\citep{robins-greenland}, or probability of necessity and sufficiency \\citep{Pearl1999,Tian2000} (see also \\S\\ref{section:pns} below). \n        \\item Global identifiability: given observational data drawn from $p^{\\mathcal{M}}(\\*V)$ infer the full counterfactual distribution $p_{\\text{cf}}^{{\\mathcal{M}}}(A \\times \\*V)$ \\citep{JMLR:v7:shimizu06a,drton2011}.\n\\end{enumerate} This is not an exhaustive list, and these problems are not all independent of one another. They are also all unsolvable in general. Problems 1, 2, and 3 involve ascending to Level 2 given information at Level 1 (and perhaps partial information at Level 2); problems 4 and 5 ask us to ascend to Level 3 given only Level 1 (and perhaps also Level 2) information. The upshot of the causal hierarchy theorem from \\cite{BCII2020} is that these steps are impossible without assumptions, formalizing the common wisdom, ``no causes in, no causes out'' \\cite{Cartwright}. \nTo understand the statement of the causal hierarchy theorem---and our topological version of it---we explain what it means for the hierarchy to collapse. \n\n\\subsection{Collapse of the Hierarchy}\\label{ss:collapse}\n\nIn the present setting a \\emph{collapse} of the hierarchy can be understood in terms of injectivity of the functions $\\varpi_i$.\n\nFor $i = 1, 2$ let $\\mathfrak{C}_i \\subset \\mathfrak{S}_i$ be the injective fibers of $\\varpi_i$, i.e.,\n$\\mathfrak{C}_i = \\{\\mu_i \\in \\mathfrak{S}_i : \\mu_{i+1} = \\mu_{i+1}' \\text{ whenever } \\varpi_i(\\mu_{i+1}) = \\varpi_i(\\mu'_{i+1}) = \\mu_i \\}$.\nEvery element $\\mu \\in \\mathfrak{C}_i$ is a witness to (global) collapse of the hierarchy: knowing $\\mu$ would be sufficient to determine the Level $i+1$ facts completely.\n\n\n\\begin{figure} \\centering \n\\subfigure [Causal Hierarchy] {\n   \\begin{tikzpicture}[framed]\n  \\node (m) at (0,0) {$\\mathfrak{M}$}; \n  \\node (ss3) at (1.5,0) {$\\mathfrak{S}_3$};\n  \\node (ss2) at (3,0) {$\\mathfrak{S}_2$};\n  \\node (ss1) at (4.5,0) {$\\mathfrak{S}_1$};\n  \n  \\path (m) edge[->]  (ss3);\n  \\node (l1) at (.7,.15) {\\small{}$\\varpi_3$};\n  \\path (ss3) edge[->]  (ss2);\n    \\node (l2) at (2.25,.15) {\\small{}$\\varpi_2$};\n  \\path (ss2) edge[->]  (ss1);\n    \\node (l3) at (3.75,.15) {\\small{}$\\varpi_1$};\n  \n  \\node (b1) at (-.5,0) {};\n  \\node (b2) at (5.5,0) {}; \n  \n  \\node (ns1) at (1.4,-1.25) {$\\mathfrak{S}^{X \\to Y}_{2}$};\n  \\node (b1) at (2.8,-1.25) {$\\dots$};\n  \\node (ns2) at (4.4,-1.25) {$\\mathfrak{S}^{X' \\to Y'}_{2}$};\n  \n    \\path (ss2) edge[->]  (ns1);\n   \\node (l3) at (1.6,-.55) {\\small{}$\\varpi^{X \\to Y}_{2}$};\n  \\path (ss2) edge[->]  (ns2);\n  \\node (l4) at (4.5,-.55) {\\small{}$\\varpi^{X' \\to Y'}_{2}$};\n   \\end{tikzpicture}\n   }\n\\subfigure [Collapse Set $\\mathfrak{C}_2$] {\n   \\begin{tikzpicture}\n\\draw (.7,1.2) ellipse (2.5cm and .5cm);\n\\draw [gray!60,fill=gray!20] (.15,0) ellipse (1.1cm and .25cm);\n\\draw (.7,0) ellipse (2cm and .4cm);\n\\node (s0) at (1.5,0) {$\\textcolor{blue}{\\bullet}$}; \n\\node (t0) at (1.2,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t1) at (1.55,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t2) at (1.9,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\path (t0) edge[->] (s0);\n\\path (t1) edge[->] (s0);\n\\path (t2) edge[->] (s0);\n\n\\node (s5) at (2.15,0) {$\\textcolor{violet}{\\bullet}$}; \n\\node (t6) at (2.3,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\node (t7) at (2.75,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\path (t6) edge[->] (s5);\n\\path (t7) edge[->] (s5);\n\n\n\\node (s1) at (.15,0) {$\\textcolor{orange}{\\bullet}$};\n\\node (s2) at (-.3,0) {$\\textcolor{green}{\\bullet}$};\n\\node (t3) at (.15,1.2) {$\\textcolor{orange}{\\bullet}$};\n\\node (t4) at (-.3,1.2) {$\\textcolor{green}{\\bullet}$};\n\\node (t5) at (-1,1.2) {$\\textcolor{red}{\\bullet}$};\n\\path (t3) edge[->] (s1);\n\\path (t4) edge[->] (s2);\n\\path (t5) edge[->,color=gray] (s2);\n\\node (x) at (-.675,.6) {$\\textcolor{red}{\\mathsf{X}}$};\n\\node (C2) at (.65,0) {$\\mathfrak{C}_2$};\n\n\\node (ss3) at (3.6,1.2) {$\\mathfrak{S}_3$};\n\\node (ss2) at (3.6,0) {$\\mathfrak{S}_2$};\n\n\\path (ss3) edge[->] (ss2);\n\n\\node (f) at (3.3,.6) {$\\varpi_2$};\n\n\\end{tikzpicture} \n}\n    \\caption{(a) $\\mathfrak{S}_3$ can be seen as a coarsening of $\\mathfrak{M}$, abstracting from irrelevant ``intensional'' details. $\\mathfrak{S}_2$ is obtained from $\\mathfrak{S}_3$ by marginalization (also a coarsening), while $\\mathfrak{S}_1$ is a projection of $\\mathfrak{S}_2$ via the ``empty'' intervention. Each map $\\varpi_i$, $i=1,2$, is continuous in the respective weak topology (Prop. \\ref{prop:causalprojectioncontinuous}). The projections $\\varpi^{X \\to Y}_{2}$ from $\\mathfrak{S}_2$ to the 2VE-spaces are likewise continuous and also open (Prop. \\ref{prop:causalprojectioncontinuous}). \\\\\n    (b) The shaded region, $\\mathfrak{C}_2 \\subset \\mathfrak{S}_2$, is the collapse set in which Level 2 facts determine all Level 3 facts: those points in $\\mathfrak{S}_2$ whose $\\varpi_2$-preimage in $\\mathfrak{S}_3$ is a singleton set. The main result of this paper is that $\\mathfrak{C}_2$ is \\emph{meager} in weak topology on $\\mathfrak{S}_2$ (Thm. \\ref{thm:hierarchyFormal}). This means $\\mathfrak{C}_2$ contains no open subset, which by Thm. \\ref{thm:l2learning} implies no part of $\\mathfrak{C}_2$ is statistically verifiable, even with infinitely many ideal experiments.}\\label{fig:hierarchy}\n\\end{figure}\n\n\\begin{comment} \n\\begin{figure}\\begin{center}\n   \\begin{tikzpicture}\n\\draw (.7,1.2) ellipse (2.5cm and .5cm);\n\\draw [gray!60,fill=gray!20] (.15,0) ellipse (1.1cm and .25cm);\n\\draw (.7,0) ellipse (2cm and .4cm);\n\\node (s0) at (1.5,0) {$\\textcolor{blue}{\\bullet}$}; \n\\node (t0) at (1.2,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t1) at (1.55,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\node (t2) at (1.9,1.2) {$\\textcolor{blue}{\\bullet}$};\n\\path (t0) edge[->] (s0);\n\\path (t1) edge[->] (s0);\n\\path (t2) edge[->] (s0);\n\n\\node (s5) at (2.15,0) {$\\textcolor{violet}{\\bullet}$}; \n\\node (t6) at (2.3,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\node (t7) at (2.75,1.2) {$\\textcolor{violet}{\\bullet}$};\n\\path (t6) edge[->] (s5);\n\\path (t7) edge[->] (s5);\n\n\n\\node (s1) at (.15,0) {$\\textcolor{orange}{\\bullet}$};\n\\node (s2) at (-.3,0) {$\\textcolor{green}{\\bullet}$};\n\\node (t3) at (.15,1.2) {$\\textcolor{orange}{\\bullet}$};\n\\node (t4) at (-.3,1.2) {$\\textcolor{green}{\\bullet}$};\n\\node (t5) at (-1,1.2) {$\\textcolor{red}{\\bullet}$};\n\\path (t3) edge[->] (s1);\n\\path (t4) edge[->] (s2);\n\\path (t5) edge[->,color=gray] (s2);\n\\node (x) at (-.65,.6) {$\\textcolor{red}{\\times}$};\n\\node (C2) at (.65,0) {$\\mathfrak{C}_2$};\n\n\\node (ss3) at (3.5,1.2) {$\\mathfrak{S}_3$};\n\\node (ss2) at (3.5,0) {$\\mathfrak{S}_2$};\n\n\\path (ss3) edge[->] (ss2);\n\n\\node (f) at (3.2,.6) {$\\varpi_2$};\n\n\\end{tikzpicture} \\end{center}\n    \\caption{Collapse of the hierarchy}\n    \\label{fig:my_label}\n\\end{figure}\n\\end{comment}\n\n\nA first observation is that $\\varpi_1$ is \\emph{never} injective. In other words, the distribution $p^{\\mathcal{M}}(\\*V)$ never determines all the interventional distributions $p^{\\mathcal{M}_{\\*X \\coloneqq \\*x}}(\\*Y)$. This is essentially a way of stating that correlation never implies causation absent assumptions. (See also \\cite[Thm. 1]{BCII2020}.) \n\\begin{proposition}\n$\\mathfrak{C}_1 = \\varnothing$. That is, Level 2 \\emph{never} collapses to Level 1 without assumptions. \\label{prop:collapse1}\n\\end{proposition}\n\nTo overcome this formidable inferential barrier, researchers often assume we are not working in the ``full'' space $\\mathfrak{M}$ of all causal models, but rather some proper subset embodying a range of causal assumptions. This may effectively eliminate counterexamples to collapse (cf. Fig. \\ref{fig:hierarchy}(b)). For problems of type 1 or 2 (from the list above in \\S\\ref{subsection:probs}) it is common to assume we are only dealing with models whose graph (direct influence relation) $\\rightarrow$ satisfies a fixed set of properties. For problems of type 3 it is common to assume that $p^\\mathcal{M}$ and $\\rightarrow$ relate in some way (for instance, through an assumption like \\emph{faithfulness} or \\emph{minimality} \\cite{Spirtes2001}). All of these problems become solvable with sufficiently strong assumptions about the form of the functions $\\{f_V\\}_V$ or the probability space $P$.\n\nIn some cases, the relevant causal assumptions are justified by appeal to background or expert knowledge. In other cases, however, an assumption will be justified by the fact that it rules out only a ``small'' or ``negligible'' or ``measure zero'' part of the full set $\\mathfrak{M}$ of possibilities. As emphasized by a number of authors \\cite{Freedman1997,Uhler,pmlr-v117-lin20a}, not all ``small'' subsets are the same, and it seems reasonable to demand further justification for eliminating one over another. We believe that the framework presented here can contribute to this positive project, but our immediate interest is in solidifying and clarifying limitative results about what cannot be done.\n\n\n\nThe issue of collapse becomes especially delicate when we turn to $\\mathfrak{C}_2$. When do interventional distributions fully determine counterfactual distributions? In contrast to Prop. \\ref{prop:collapse1} we have:\n\\begin{proposition} $\\mathfrak{C}_2 \\neq \\varnothing$. That is, there exists an SCM in which Level 3 collapses to Level 2. \n\\end{proposition}\n\\begin{proof}[Proof sketch] As a very simple example in the finite case, any fully deterministic SCM will result in collapse. This is because, if $(\\mu_{\\alpha})_{\\alpha \\in A}$ are all binary-valued then the measure $\\mu_3 \\in \\mathfrak{P}\\big(\\bigtimes_{\\alpha \\in A} \\chi_{\\*V}\\big)$ that produces the marginals $\\mu_{\\alpha}$ is completely determined: each $\\mu_{\\alpha}$ specifies an element of $\\chi_{\\*V}$, so $\\mu_3$ must assign unit probability to the tuple that matches $\\mu_{\\alpha}$ at the $\\alpha$ projection. \nIn the infinite case, any example must be non-deterministic by atomlessness, but collapse is still possible; see Example \\ref{example:collapse} in Appendix \\ref{app:causalhierarchy}.\n\\end{proof}\n\n\\subsection{Probabilities of Causation} \\label{section:pns}\n\nA handful of counterfactual quantitites over two given variables, collected below, have been particularly prominent in the literature (e.g., \\cite{Pearl1999}). Our main result will show that \\emph{any} of these six quantities (for any two fixed variables) is robust against collapse.\nBelow, fix two distinct variables $Y \\neq X \\in \\*V$ and distinct values $x \\neq x' \\in \\chi_X$, $y \\neq y' \\in \\chi_Y$.\n\\begin{definition}\\label{def:probcaus}\nThe \\emph{probabilities of causation} are the following quantities:\n\\begin{align*}\n    P(y_x, y'_{x'}): & \\text{ probability of necessity and sufficiency}\\\\\n    P(y'_x, y_{x'}): & \\text{ converse prob. of necessity and sufficiency}\\\\\n    P(y'_{x'} \\mid x, y): & \\text{ prob. of necessity} \\qquad P(y_x \\mid x', y'): \\text{ prob. of sufficiency}\\\\\n    P(y'_{x'} \\mid y): & \\text{ prob. of disablement} \\qquad P(y_x \\mid y'): \\text{ prob. of enablement}\n\\end{align*}\n\\end{definition}\nConsider, for example, the probability of necessity and sufficiency (PNS), which is the joint probability that $Y$ would take on value $y$ if $X$ is set by intervention to $x$, and $y'$ if $X$ is set to $x'$.\nPNS has been thoroughly studied \\citep{Pearl1999,Tian2000,avin:etal05}, in part due to its widespread relevance: from medical treatment to online advertising, we would like to assess which interventions are likely to be both \\emph{necessary} and \\emph{sufficient} for a given outcome.\nUsing the notation from \\S{}\\ref{ss:scml3}, PNS concerns the measure of sets\n$y_{x}, y'_{x'} = \\pi^{-1}_{(X \\coloneqq x, Y)}(\\{y\\}) \\cap \\pi^{-1}_{(X \\coloneqq x', Y)}(\\{y'\\})$.\n\n\nThe probabilities of causation are paradigmatically Level 3, and we will be interested in their manifestations at Level 2. In that direction we introduce a small part of $\\mathfrak{S}_2$, just enough to witness the behavior of $Y$ (and $X$) under the empty intervention and the two possible interventions on $X$:\n\\begin{definition}\nLet $A_X =\\{\\varnothing \\coloneqq (), X \\coloneqq 0, X \\coloneqq 1\\}$.\nDefine a small subspace $\\mathfrak{S}^{X \\to Y}_{2}\\subset \\bigtimes_{\\alpha \\in A_X} \\mathfrak{P}(\\chi_{\\{X,Y\\}})$ as the image of the map $\\varpi^{X\\to Y}_{2} = \\big(\\varsigma_{\\{X, Y\\}} \\times \\varsigma_{\\{X, Y\\}} \\times \\varsigma_{\\{X, Y\\}}\\big) \\circ \\pi_{A_X}$ (see Fig. \\ref{fig:hierarchy}(a)). Call $\\mathfrak{S}^{X \\to Y}_{2}$ a \\emph{two-variable effect} (2VE) \\emph{space}; fixing $X$, we have a 2VE-space for each $Y$.\n\\end{definition}\n\nIt is known in the literature that the probabilities of causation are not identifiable from the data $p(X,Y)$, $p(Y_{x})$, and $p(Y_{x'})$ (see, e.g., \\cite{avin:etal05} for PNS). As part of our proof of Theorem \\ref{thm:hierarchyFormal} below, we will strengthen this considerably to show them all to be \\emph{generically} unidentifiable, in a topological sense to be made precise.\n\n\n\n\n\n\n\n\\section{The Weak Topology} \\label{sec:weaktopology}\n\nWe now demonstrate how $\\mathfrak{S}_1,\\mathfrak{S}_2$ and $\\mathfrak{S}_3$ can be topologized.  \nIn general, given a space $\\vartheta$ and the set $\\mathfrak{S} = \\mathfrak{P}(\\vartheta)$ of Borel probability measures on $\\vartheta$, a natural topology on $\\mathfrak{S}$ can be defined as follows:\n\\begin{definition}\nFor a sequence $(\\mu_n)_n$ of measures in $\\mathfrak{S}$,\nwrite $(\\mu_n)_n \\Rightarrow \\mu$ and say it \\emph{converges weakly} \\citep[p.~7]{Billingsley} to $\\mu$ if $\\int_{\\vartheta} f \\, \\mathrm{d}\\mu_n \\to \\int_{\\vartheta} f \\, \\mathrm{d}\\mu$ for all bounded, continuous $f : \\vartheta \\to \\mathbb{R}$.\nThen the \\emph{weak topology} $\\tau^{\\mathrm{w}}$ on $\\mathfrak{S}$ is that with the following closed sets: $E \\subset \\mathfrak{S}$ is closed in $\\tau^{\\mathrm{w}}$ iff for any weakly convergent sequence $(\\mu_n)_n \\Rightarrow \\mu$ in which every $\\mu_n \\in E$, the limit point $\\mu$ is in $E$.\n\\end{definition}\nThere are several alternative characterizations of $\\tau^{\\mathrm{w}}$, which hold under very general conditions. For instance, it coincides with the topology induced by the so called L\\'{e}vy-Prohorov metric \\citep{Billingsley}. \nThe most useful characterization for our purposes is that it \ncan be generated by subbasic open sets of the form \\begin{equation}  \\{\\mu: \\mu(X)>r\\}\n\\label{subbasis-weak}\n\\end{equation} with $X$ ranging over basic clopens in $\\vartheta$ and $r$ over rationals (see, e.g., \\cite[Lemma A.5]{GeninKelly}).\n\nConceptually, the explication of $\\tau^{\\mathrm{w}}$ in terms of weak convergence strongly suggests a connection with statistical learning. We now make this connection precise, building on existing work \\cite{DemboPeres,Genin2018,GeninKelly}. \n\n\\subsection{Connection to Learning Theory}\nRoughly speaking, we will say a hypothesis $H \\subseteq \\mathfrak{S}$ is \\emph{statistically verifiable} if there is some error bound $\\epsilon$ and a sequence of statistical tests that converge on $H$ with error at most $\\epsilon$, when data are generated from $H$. More formally, a \\emph{test} is a function $\\lambda: \\vartheta^n \\rightarrow \\{\\mathsf{accept},\\mathsf{reject}\\}$, where $\\vartheta^n$ is the $n$-fold product of $\\vartheta$, viz. finite data streams from $\\vartheta$. The interest is in whether a ``null'' hypothesis can be rejected given data observed thus far. The \\emph{boundary} of a set $A\\subseteq \\vartheta$, written $\\mathsf{bd}(A)$, is the difference of its closure and its interior. Intuitively, a learner will not be able to decide whether to accept or reject on the boundary. Consequently it is assumed that $\\lambda$ is \\emph{feasible} in the sense that the boundary of its acceptance zone (in the product topology on $\\vartheta^n$) always has measure 0, i.e., $\\mu^n[\\mathsf{bd}(\\lambda^{-1}(\\mathsf{accept}))] = 0$  for every $\\mu \\in \\mathfrak{S}$, where $\\mu^n$ is the $n$-fold product measure of $\\mu$.\n\nSay a hypothesis $H \\subseteq \\mathfrak{S}$ is \\emph{verifiable} \\cite{Genin2018} if there is $\\epsilon>0$ and a sequence $(\\lambda_n)_{n\\in\\mathbb{N}}$ of feasible tests (of the complement of $H$ in $\\mathfrak{S}$, i.e., the ``null hypothesis'') such that \\begin{enumerate}\n   \\item $\\mu^n[\\lambda_n^{-1}(\\mathsf{reject})] \\leq \\epsilon$ for all $n$, whenever $\\mu \\notin H$;\n   \\item $\\underset{n\\rightarrow\\infty}{\\mbox{lim}}\\;\\mu^n[\\lambda_n^{-1}(\\mathsf{reject})] = 1$, whenever $\\mu \\in H$.\n\\end{enumerate} That is, to be verifiable we only require a sequence of tests that converges in probability to the true hypothesis in the limit of infinite data (requirement 2), while incurring (type 1) error only up to a given bound at finite stages (requirement 1). As an illustrative example, \\emph{conditional dependence} is verifiable \\cite{Genin2018}. This is a relatively lax notion of verifiability.\nFor instance, the hypothesis need not also be \\emph{refutable} (and thus ``decidable''). For our purposes this generality is a virtue: we want to show that certain hypotheses are not statistically verifiable by any method, even in this wide sense. The fundamental link between verifiability and the weak topology is the following, due to \\cite{Genin2018,GeninKelly}:\n\\begin{theorem} \\label{thm:genin} A set $H \\subseteq \\mathfrak{S}$ is verifiable if and only if it is open in the weak topology.\n\\end{theorem}\n\n\\subsection{Topologizing Causal Models}\nWe now reinterpret $\\tau^{\\mathrm{w}}$ at each level of the causal hierarchy: \n\\begin{definition}\nThe \\emph{weak causal topology} $\\tau^{\\mathrm{w}}_i$, $1 \\le i \\le 3$, is the subspace topology on $\\mathfrak{S}_i$, induced by\n\\begin{align*}\n\\text{if } i=3: \\tau^{\\mathrm{w}} \\text{ on } \\mathfrak{P}(\\chi_{A \\times \\*V}); \\quad\n\\text{if } i=2: \\text{product of }\\tau^{\\mathrm{w}} \\text{ on } \\bigtimes_{\\alpha \\in A} \\mathfrak{P}(\\chi_{\\*V}); \\quad\n\\text{if } i=1: \\tau^{\\mathrm{w}} \\text{ on } \\mathfrak{P}(\\chi_{\\*V}).\n\\end{align*}\n\\end{definition}\n\\begin{proposition}\\label{prop:causalprojectioncontinuous}\nIn the weak causal topologies,\n$\\{\\varpi_i\\}_{i = 1, 2}$ are continuous and all projections\n$\\varpi^{X \\to Y}_{2}$ are continuous and open.\n\\end{proposition}\nA significant observation is that the learning theoretic interpretation, originally intended for $\\tau^{\\mathrm{w}}_1$, naturally extends to $\\tau^{\\mathrm{w}}_2$. While data streams at Level 1 amount to passive observations of $\\*V$, data streams at Level 2 can be seen as sequences of experimental results, i.e., observations of ``potential outcomes'' $\\*Y_{\\*x}$. To make verifiability as easy as possible we assume a learner can observe a sample from all conceivable experiments at each step.  A learner is thus a function $\\lambda:\\mathcal{E}^n \\rightarrow \\{\\mathsf{accept},\\mathsf{reject}\\}$, where $\\mathcal{E}^n = ((\\chi_{\\*V})^n)_\\alpha$  is the set of potential experimental observations over $n$ trials (with $\\alpha$ indexing the experiments). Construing $\\mathcal{E}^n$ as a product space we can again speak of \\emph{feasibility} of $\\lambda$.  \n\nRecall that elements of $\\mathfrak{S}_2$ are tuples $(\\mu_\\alpha)_{\\alpha \\in A}$ of measures. Say  a hypothesis $H\\subseteq \\mathfrak{S}_2$ is \\emph{experimentally verifiable} if there is $\\epsilon>0$ and a sequence $(\\lambda_n)_{n \\in \\mathbb{N}}$ of feasible tests such that 1 and 2 above hold, replacing $\\mu^n[\\lambda_n^{-1}(\\mathsf{reject})]$ with $\\prod_{\\alpha} \\mu_\\alpha^n[(\\lambda_n^{-1}(\\mathsf{reject}))_\\alpha]$. That is, when experimental data are drawn from the interventional distributions $(\\mu_\\alpha)_{\\alpha \\in A} \\in H$, we require that the learner eventually converge on $H$ with bounded error at finite stages. We can then show (see Appendix \\ref{app:empirical}): \n\\begin{theorem} A set $H \\subseteq \\mathfrak{S}_2$ is experimentally verifiable if and only if it is open in $\\tau^{\\mathrm{w}}_2$. \\label{thm:l2learning}\n\\end{theorem}\nA similar result can be given for $(\\mathfrak{S}_3,\\tau^{\\mathrm{w}}_3)$, although it is less clear what the empirical content of this result would be.\nNote also that $\\tau^{\\mathrm{w}}_1,\\tau^{\\mathrm{w}}_2,\\tau^{\\mathrm{w}}_3$ give a sequence of increasingly fine topologies on the set of actual SCMs $\\mathfrak{M}$ by simply pulling back the projections. The point is that $\\tau^{\\mathrm{w}}_2$ is the finest that has clear empirical significance, while $\\tau^{\\mathrm{w}}_3$ is the finest in terms of relevance to the causal hierarchy.\n\n\n\n\\section{Collapse is Meager} \\label{sec:main}\nRecall that a set $X \\subseteq \\vartheta$ is \\emph{nowhere dense} if every open set contains an open  $Y$ with $X \\cap Y = \\varnothing$. A countable union of nowhere dense sets is said to be \\emph{meager} (or \\emph{of first category}). The complement of a meager set is \\emph{comeager}. Intuitively, a meager set is one that can be ``approximated'' by sets ``perforated with holes'' \\citep{Oxtoby}. Meagerness is notably preserved when taking the preimage under a map that is both continuous and open.\n\nAs discussed above, one intuition highlighted by the weak topology $\\tau^{\\mathrm{w}}$ is that open sets are the kinds of probabilistic propositions that could, in the limit of infinite data, be verified (Thms. \\ref{thm:genin},  \\ref{thm:l2learning}). Correlatively, meager sets in $\\tau^{\\mathrm{w}}$ are so negligible as to be unverifiable: as a meager set contains no non-empty open subsets (by the Baire Category Theorem \\cite{Oxtoby}), it is statistically unverifiable.\nWe will now show that the injective collapse set $\\mathfrak{C}_2$ from \\S{}\\ref{ss:collapse} is topologically meager.\n\nThe crux is to identify a ``good'' comeager 2VE-subspace where collapse \\emph{never} occurs (with separation witnessed by probabilities of causation).\nIn this subspace, the constraints circumscribing Level 3 have sufficient slack to make a tweak without thereby disturbing Level 2 (cf. Figure \\ref{fig:example:separation}).\nWe define the good set as the locus of a set of strict inequalities:\n\\begin{definition}\nA family $(\\mu_\\alpha)_{\\alpha \\in A_X} \\in \\mathfrak{S}^{X\\to Y}_{2}$ is \\emph{$Y$-good}\nif we have the following, abbreviating the members of $A_X$ as $(), x, x'$:\n\\begin{gather}\n0 < \\mu_x(y') - \\mu_{()}(x, y')\n < \\mu_{()}(x'),\\label{cns:ineq:1}\\\\\n0 < \\mu_{()}(x', y') < \\mu_{()}(x')\\label{cns:ineq:2}.\n\\end{gather}\n\\end{definition}\n\\begin{lemma}\n\\label{lem:goodcomeager}\nThe subspace of $Y$-good families is comeager in $\\mathfrak{S}^{X \\to Y}_{2}$.\n\\end{lemma}\n\\begin{proof}[Proof sketch]\nThe non-strict versions of \\eqref{cns:ineq:1}, \\eqref{cns:ineq:2} hold universally, so the complement of the good set is defined by equalities. This is closed and contains no nonempty open by the weak subbasis \\eqref{subbasis-weak}. \\end{proof}\nFigure~\\ref{fig:example:separation} presents the construction in a small, two-variable case, and Lemma~\\ref{lem:separation} below is proven by generalizing it to arbitrary $\\*V$.\nGuaranteeing agreement on every interventional distribution in the general case is subtle (Appendix~\\ref{app:main}):\nit has been observed that enlarging $\\*V$ can enable additional inferences (e.g., \\cite{9363924}), though the next result reflects a dependence on further assumptions.\n\n\\begin{lemma}\\label{lem:separation}\nSuppose $\\prec$ is an order in which $X$ comes first and $(\\mu_\\alpha)_{\\alpha \\in A} \\in \\mathfrak{S}^{\\prec}_2$ is such that $\\varpi^{X\\to Y}_{2}\\big((\\mu_\\alpha)_{\\alpha}\\big)$ is $Y$-good, and\nlet $\\varphi$ be PNS, the converse PNS, the probability of sufficiency, or the probability of enablement (Definition~\\ref{def:probcaus}).\nThen for any $\\mu_3 \\in \\mathfrak{S}^{\\prec}_3$ such that $\\varpi_2(\\mu_3) = (\\mu_\\alpha)_{\\alpha}$, there exists a $\\mu'_3 \\in \\mathfrak{S}^{\\prec}_3$ such that $\\mu_3$ and $\\mu'_3$ disagree on $\\varphi$.\n\\end{lemma}\n\\begin{figure} \\centering \n\\subfigure [$Y$-good Model] {\n\\begin{tabular}{ lllll } \n& & ${\\mathcal{M}}$ & & \\\\\n\\toprule\n$u$ & $P(u)$ & $X_u$ & $Y_{x, u}$ & $Y_{x', u}$\\\\\n\\midrule\n$u_0$ & $\\nicefrac{1}{2}$ & $x'$ & $y$ & $y$\\\\\n$u_1$ & $\\nicefrac{1}{2}$ & $x'$ & $y'$ & $y'$\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\subfigure [Example Separating Levels 2 and 3] {\n\\begin{tabular}{ lllll } \n& & ${\\mathcal{M}}'$ & & \\\\\n\\toprule\n$u$ & $P(u)$ & $X_u$ & $Y_{x, u}$ & $Y_{x', u}$\\\\\n\\midrule\n$u_0$ & $\\nicefrac{1}{2}- \\varepsilon$ & $x'$ & $y$ & $y$\\\\\n$u_1$ & $\\varepsilon$ & $x'$ & $y$ & $y'$\\\\\n$u_2$ & $\\nicefrac{1}{2}- \\varepsilon$ & $x'$ & $y'$ & $y'$\\\\\n$u_3$ & $\\varepsilon$ & $x'$ & $y'$ & $y$\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{(a): the structural functions and exogenous noise for a model ${\\mathcal{M}}$ with direct influence $X \\rightarrow Y$. This ${\\mathcal{M}}$ meets \\eqref{cns:ineq:1} and \\eqref{cns:ineq:2}, so we may apply Lemma~\\ref{lem:separation}, constructing the model $\\mathcal{M}'$ in (b), where $0 < \\varepsilon < \\nicefrac{1}{2}$. Note that $p^{{\\mathcal{M}}}_{\\text{cf}}(y_x, y'_{x'}) = 0$ while $p^{{\\mathcal{M}}'}_{\\text{cf}}(y_x, y'_{x'}) = \\varepsilon$, so that the two models disagree on a Level 3 PNS quantity; on the other hand, it is easy to check agreement on all of Level 2. Similarly, ${\\mathcal{M}}$ and ${\\mathcal{M}}'$ disagree on the converse PNS, probability of sufficiency, and probability of enablement (Definition~\\ref{def:probcaus}).}\n  \\label{fig:example:separation}\n\\end{figure}\nNote that by reversing the roles of $x$ and $x'$, we may obtain the same for the probability of necessity and probability of disablement.\nThe main theorem and its important learning-theoretic corollary are now straightforward.\n\\begin{theorem}[Topological Hierarchy] The set $\\mathfrak{C}_2$ of points where all Level 3 facts are identifiable from Level 2 is meager in $(\\mathfrak{S}_2,\\tau^{\\mathrm{w}}_2)$. \\label{thm:hierarchyFormal}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathfrak{D}^{X, Y}_2 \\subset \\mathfrak{S}_2^X$ be the preimage under $\\varpi^{X \\to Y}_{2}$ of the set of $Y$-good tuples in $\\mathfrak{S}^{X \\to Y}_{2}$.\nLemma~\\ref{lem:separation} implies that $\\mathfrak{C}_2 \\cap \\mathfrak{S}_2^{X}$ is contained in $\\mathfrak{S}_2^{X} \\setminus \\mathfrak{D}^{X, Y}_2$, for \\emph{any} $Y \\neq X$.\nMeanwhile, since $\\varpi^{X \\to Y}_{2}$ is continuous and open, Lemma~\\ref{lem:goodcomeager} implies that $\\mathfrak{S}_2^{X} \\setminus \\mathfrak{D}^{X, Y}_2$ is meager in $\\mathfrak{S}_2^{X}$, and thereby also in $\\mathfrak{S}_2$.\nThus $\\mathfrak{C}_2 = \\bigcup_{X \\in \\*V} \\mathfrak{C}_2 \\cap \\mathfrak{S}_2^{X}$ is a countable union of meager sets, and hence meager.\n\\end{proof}\n\\begin{corollary} \\label{cor:hierarchy} No causal hypothesis licensing arbitrary counterfactual inferences (and specifically those of the probabilities of causation) from observational and experimental data is itself statistically (even experimentally) verifiable.\n\\end{corollary}\n\n\\section{Conclusion}\\label{section:conclusion}\n\nWe introduced a general framework for topologizing spaces of causal models, including the space of all (discrete, well-founded) causal models. As an illustration of the framework we characterized levels of the causal hierarchy  topologically, and proved a topological version of the causal hierarchy theorem from \\cite{BCII2020}. While the latter shows that collapse of the hierarchy (specifically of Level 3 to Level 2) is \\emph{exceedingly unlikely} in the sense of (Lebesgue) measure, we offer a complementary result: any condition guaranteeing that we could infer arbitrary Level 3 information from purely Level 2 information must be \\emph{statistically  unverifiable}, even by experimental means. Both results capture an important sense in which collapse is ``negligible'' in the space of all possible models. As an added benefit, the topological approach extends seamlessly to the setting of infinitely many variables.\n\nThere are many natural extensions of these results. For instance, we have begun work on a version for continuous endogenous variables. Also of interest are subspaces embodying familiar causal assumptions or other well-studied coarsenings of SCMs (see, e.g., \\cite{pmlr-v117-lin20a} on Bayesian networks, or \\cite{GeninMayoWilson,Genin2021} on linear non-Gaussian models), which often render important inference problems solvable, though sometimes only ``generically'' so.\nIn the opposite direction, we expect analogous hierarchy theorems to hold for extensions of the SCM concept, e.g., that dropping the well-foundedness or recursiveness requirements \\cite{bongers2021foundations}.\nAs emphasized by \\cite{BCII2020}, a causal hierarchy theorem should not be construed as a purely limitative result, but rather as further motivation for understanding the whole range of causal-inductive assumptions, how they relate, and what they afford. We submit that the topological constructions presented here can help clarify and systematize this broader landscape.\n\n\n\\subsubsection*{Acknowledgments} \nThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-16565. We are very grateful to the five anonymous NeurIPS reviewers for insightful and detailed comments and questions that led to significant improvements in the paper. We would also like to thank Jimmy Koppel, Krzysztof Mierzewski, Francesca Zaffora Blando, and especially Kasey Genin for helpful feedback on earlier versions. Finally, we are indebted to Saminul Haque for identifying a gap in the published version of the paper, which has been corrected in the present arXiv version (see in particular the augmented statement of Prop.~\\ref{prop:causalprojectioncontinuous}). \n\n\n\\medskip\n\n{\n\\small\n\n\\bibliographystyle{abbrvnat}\n\n\\section{Structural Causal Models (\\S\\ref{scms})} \\label{app:causalmodels}\n\\subsection{Background on Relations and Orders}\n\n\\begin{definition}\nLet $C$ be a set.\nThen a subset $R \\subset C \\times C$ is called a \\emph{binary relation} on $C$. We write $cRc'$ if $(c, c') \\in R$.\nThe binary relation $R$ is \\emph{well-founded} if every nonempty subset $D \\subset C$ has a minimal element with respect to $R$, i.e., if for every nonempty $D \\subset C$, there is some $d \\in D$, such that there is no $d' \\in D$ such that $d' R d$.\nThe binary relation $\\left.\\prec\\right. \\subset C \\times C$ is a (strict) \\emph{total order} if it is irreflexive, transitive, and \\emph{connected}: either $c \\prec c'$ or $c' \\prec c$ for all $c \\neq c' \\in C$.\n\\end{definition}\n\\begin{example}\nThe edges of a dag form a well-founded binary relation on its nodes. If $\\*V = \\{V_n\\}_{n \\ge 0}$, then the binary relation $\\rightarrow$ defined by $V_m \\rightarrow V_n$ iff either $0 < m < n$ or $n = 0 < m$ is well-founded but not extendible to an $\\omega$-like total order (see Fact~\\ref{fact:omegalike}) and not locally finite: $V_0$ has infinitely many predecessors $V_1, V_2, \\dots$\n\\end{example}\n\n\\subsection{Proofs}\n\n\\begin{proof}[Proof of Proposition~1.]\nWe assume without loss that $\\*U(V) = \\*U$ for every $V \\in \\*V$.\nFor each $\\*u \\in \\chi_{\\*U}$,\nwell-founded induction along $\\rightarrow$ shows unique existence of a $m^{{\\mathcal{M}}}(\\*u) \\in \\chi_{\\*V}$\nsolving $f_{V}\\big(\\pi_{\\mathbf{Pa}(V)}(m^{{\\mathcal{M}}}(\\*u)), \\*u\\big) = \\pi_V(m^{{\\mathcal{M}}}(\\*u))$ for each $V$.\nWe claim the resulting function $m^{{\\mathcal{M}}}$ is measurable.\nOne has a clopen basis of cylinders, so it suffices to show each preimage $(m^{{\\mathcal{M}}})^{-1}(v)$ is measurable. Recall that here $v$ denotes the cylinder set $\\pi^{-1}_V(\\{v\\}) \\in \\mathcal{B}(\\chi_{\\*V})$, for $v \\in \\chi_V$.\nOnce again this can be established inductively. Note that\n\\begin{align*}\n(m^{{\\mathcal{M}}})^{-1}(v) =\n\\bigcup_{\\mathbf{p} \\in \\chi_{\\mathbf{Pa}(V)}}\\Big[ (m^{{\\mathcal{M}}})^{-1}(\\*p) \\cap \\pi_{\\*U}\\big(f_V^{-1}(\\{v\\}) \\cap (\\{\\*p\\} \\times \\chi_{\\*U})\\big) \\Big].\n\\end{align*}\nwhich is a finite union (by local finiteness) of measurable sets (by the inductive hypothesis) and therefore measurable.\nThus for any ${\\mathcal{M}}$\nthe pushforward $p^{{\\mathcal{M}}} = m^{{\\mathcal{M}}}_*(P)$ is a measure on $\\mathcal{B}(\\chi_{\\*V})$ and gives the observational distribution (Definition~4).\n\\end{proof}\n\n\\begin{proof}[Remark on Definition~6]\nTo see that $p^{{\\mathcal{M}}}_{\\mathrm{cf}}$ thus defined is a measure,\nnote that $p^{{\\mathcal{M}}}_{\\mathrm{cf}} = p^{{\\mathcal{M}}_A}$ and apply Proposition~1, where the model ${\\mathcal{M}}_A$ is defined in Definition~\\ref{def:counterfactualmodel}. This is similar in spirit to the construction of ``twinned networks'' \\citep{BalkePearl} or ``single-world intervention graphs'' \\citep{Richardson}.\n\\end{proof}\n\n\\begin{definition}\\label{def:counterfactualmodel}\nGiven ${\\mathcal{M}}$ as in Def.~\\ref{def:scm:lit}\nand a collection of interventions $A$\nform the following \\emph{counterfactual model} ${\\mathcal{M}}_A = \\langle \\*U, A \\times \\*V, \\{f_{(\\alpha, V)}\\}_{(\\alpha, V)}, P\\rangle$, over endogenous variables $A \\times \\*V$. The counterfactual model has the influence relation $\\rightarrow'$, defined as follows.\nWhere $\\alpha', \\alpha \\in A$ let $(\\alpha', V') \\rightarrow' (\\alpha, V)$ iff $\\alpha' = \\alpha$ and $V' \\rightarrow V$.\nThe exogenous space $\\*U$ and noise distribution $P$ of $\\mathcal{M}_A$ are the same as those of $\\mathcal{M}$,\nthe exogenous parents sets $\\{\\*U(V)\\}_V$ are also identical,\nand the functions are $\\{f_{(\\alpha, V)}\\}_{(\\alpha, V)}$ defined as follows.\nFor any $\\*W \\coloneqq \\*w \\in A$, $V \\in \\*V$, $\\mathbf{p} \\in \\chi_{\\mathbf{Pa}(V)}$, and $\\*u \\in \\chi_{\\*U(V)}$ let\n\\begin{align*}\nf_{(\\*W \\coloneqq \\*w, V)}\\big((\\*W \\coloneqq \\*w, \\mathbf{p}), \\*u\\big) = \\begin{cases}\n\\pi_V(\\*w), & V \\in \\*W\\\\\nf_V(\\mathbf{p}, \\*u), & V \\notin \\*W\n\\end{cases}.\n\\end{align*}\n\\end{definition}\n\n\n\\section{Proofs from \\S\\ref{sec:causalhierarchy}} \\label{app:causalhierarchy}\n\\begin{proof}[Remark on exact characterizations of $\\mathfrak{S}_3$, $\\mathfrak{S}_2$]\nRich probabilistic languages interpreted over $\\mathfrak{S}_3$ and $\\mathfrak{S}_2$ were axiomatized in \\cite{ibelingicard2020}.\nThis axiomatization, along with the atomless restriction, gives an exact characterization for the hierarchy sets.\nStandard form, defined below, gives an alternative characterization exhibiting each $\\mathfrak{S}_3^\\prec$ as a particular atomless probability space (Corollary~\\ref{cor:standardform}).\nFor $\\mathfrak{S}^{X \\to Y}_{2}$ (or $\\mathfrak{S}_2$ in the two-variable case) we need the characterization for the proof of the hierarchy separation result, so it is given explicitly as Lemma~\\ref{prop:p2:characterization:binary} in the section below on 2VE-spaces.\n\\end{proof}\n\\subsection{Standard Form}\\label{ss:app:standardform}\nFix $\\prec$. Note that the map $\\varpi_3$ restricted to $\\mathfrak{M}_\\prec$ does \\emph{not} inject into $\\mathfrak{S}_3^\\prec$, as any trivial reparametrizations of exogenous noise are distinguished in $\\mathfrak{M}_\\prec$.\nIt is therefore useful to identify a ``standard'' subclass $\\mathfrak{M}_\\prec^{\\mathrm{std}}$ on which $\\varpi_3$ is injective with image $\\mathfrak{S}_3^\\prec$, and in which we lose no expressivity.\n\\begin{notation*}\nLet $\\mathbf{Pred}(V) = \\{V' : V' \\prec V\\}$ and denote a \\emph{deterministic} mechanism for $V$ mapping a valuation of its predecessors to a value as $\\texttt{f}_V \\in \\chi_{\\mathbf{Pred}(V)} \\to \\chi_V$. Write an entire collection of such mechanisms, one for each variable, as $\\texttt{\\textbf{f}} = \\{\\texttt{f}_V\\}_{V}$.\nA set $\\*B \\subset \\*V$ is \\emph{ancestrally closed} if $\\mathbf{B} = \\bigcup_{V \\in \\*B} \\mathbf{Pred}(V)$.\nFor any ancestrally closed $\\*B$\nlet $\\xi(\\*B) = \\big\\{(V, \\*p): V \\in \\*B, \\*p \\in \\chi_{\\mathbf{Pred}(V)}\\big\\}$. Note that $\\texttt{\\textbf{F}}(\\*B) = \\bigtimes_{(V, \\mathbf{p}) \\in \\xi(\\*B)} \\chi_V$ encodes the set of all possible such collections of deterministic mechanisms, and we write, e.g., $\\texttt{f} \\in \\texttt{F}(\\*B)$.\nAbbreviate $\\xi(\\*V)$, $\\texttt{F}(\\*V)$ for the entire endogenous variable set $\\*V$ as $\\xi$, $\\texttt{F}$ respectively. We also use $\\texttt{f}$ to abbreviate the set\n\\begin{align}\n\\label{eq:probabilityofmechanisms}\n\\bigcap_{\\substack{V \\in \\*B\\\\ \\*p \\in \\chi_{\\mathbf{Pred}(V)}}} \\pi^{-1}_{(\\mathbf{Pred}(V) \\coloneqq \\*p, V)}(\\{\\texttt{f}(\\*p)\\}) \\in \\mathcal{B}(\\chi_{A \\times \\*V})\n\\end{align}\nso we can write, e.g., $p^{{\\mathcal{M}}}_{\\mathrm{cf}}(\\texttt{f})$ for the probability  in ${\\mathcal{M}}$ that the effective mechanisms $\\texttt{f}$ have been selected (by exogenous factors) for the variables $\\*B$.\n\\end{notation*}\n\\begin{definition}\\label{def:standardform}\n\nThe SCM ${\\mathcal{M}} = \\langle \\*U, \\*V, \\{f_V\\}_V, P\\rangle$ of Def.~\\ref{def:scm:lit} is in \\emph{standard form} over $\\prec$, and we write ${\\mathcal{M}} \\in \\mathfrak{M}_\\prec^{\\mathrm{std}}$, if we have that $\\left.\\rightarrow\\right. = \\left.\\prec\\right.$ for its influence relation, $\\*U = \\{U\\}$ for a single exogenous variable $U$ with $\\chi_U = \\texttt{{F}}$, $P \\in \\mathfrak{P}(\\texttt{F})$ for its exogenous noise space, and for every $V$, we have that $\\*U(V) = \\*U = \\{U\\}$ and the mechanism $f_V$ takes $\\mathbf{p}, (\\{ \\texttt{f}_V\\}_V) \\mapsto \\texttt{f}_V(\\mathbf{p})$ for each $\\*p \\in \\chi_{\\mathbf{Pred}(V)}$ and joint collection of deterministic functions $\\{ \\texttt{f}_V\\}_V \\in \\texttt{F} = \\chi_{U}$.\n\\end{definition}\nEach unit $\\*u$ in a standard form model amounts to a collection $\\{ \\texttt{f}_V\\}_V$ of deterministic mechanisms, and each variable is determined by a mechanism specified by the ``selector'' endogenous variable $U$.\n\\begin{lemma}\nLet ${\\mathcal{M}} \\in \\mathfrak{M}_\\prec$. Then there exists ${\\mathcal{M}}^{\\mathrm{std}} \\in \\mathfrak{M}_\\prec^{\\mathrm{std}}$ such that $\\varpi_3({\\mathcal{M}}) = \\varpi_3({\\mathcal{M}}^{\\mathrm{std}})$.\n\\end{lemma}\n\\begin{proof}\nTo give ${\\mathcal{M}}^{\\mathrm{std}}$ define a measure $P \\in \\mathfrak{P}(\\texttt{F})$ as in Def.~\\ref{def:standardform} on a basis of cylinder sets by the counterfactual in ${\\mathcal{M}}$\n\\begin{multline}\nP\\big(\\pi^{-1}_{(V_1, \\*p_1)}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{(V_n, \\*p_n)}(\\{v_n\\})\\big)\\\\\n= p_{\\mathrm{cf}}^{{\\mathcal{M}}}\\big( \\pi^{-1}_{(\\mathbf{Pred}(V_1) \\coloneqq \\*p_1, V_1)}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{(\\mathbf{Pred}(V_n) \\coloneqq \\*p_n, V_n)}(\\{v_n\\}) \\big). \\label{eq:standardformdefinition}\n\\end{multline}\nTo show that $\\varpi_3({\\mathcal{M}}) = \\varpi_3({\\mathcal{M}}^{\\mathrm{std}})$ it suffices to show that any two models agreeing on all counterfactuals of the form \\eqref{eq:standardformdefinition} must agree on all counterfactuals in $A$.\nSuppose $\\alpha_i \\in A$, $V_i \\in \\*V$, $v_i \\in \\chi_{V_i}$ for $i = 1, \\dots, n$.\nLet $\\*{B} = \\bigcup_{i} \\mathbf{Pred}(V_i)$ and given $\\texttt{f} = \\{\\texttt{f}_V\\}_V$,\ndefine $\\texttt{f}^{\\*W \\coloneqq \\*w}_V$ to be a constant function mapping to $\\pi_V(\\*w)$ if $V \\in \\*W$ and $\\texttt{f}^{\\*W \\coloneqq \\*w}_V = \\texttt{f}_V$ otherwise.\nWrite $\\texttt{f} \\models V = v$ if $\\pi_V(\\*v) = v$ for that $\\*v \\in \\chi_{\\*V}$ such that $\\texttt{f}_{V}\\big(\\pi_{\\mathbf{Pred}(V)}(\\*v)\\big) = \\pi_{V}(\\*v)$ for all $V$.\nFinally, \nnote that\n\\begin{align*}\n \\bigcap_{i=1}^n \\pi^{-1}_{(\\alpha_i, V_i)}(\\{v_i\\})  = \\bigsqcup_{\\substack{\\{\\texttt{\\textbf{f}}_{V}\\}_{V \\in \\*B} \\in \\texttt{\\textbf{F}}(\\*B) \\\\\\{\\texttt{\\textbf{f}}^{\\alpha_i}_{V}\\}_{V \\in \\*B} \\models V_i = v_i \\\\ \\text{for each } i }} \\{\\texttt{f}_V\\}_{V \\in \\*B}\n\\end{align*}\nwhere each set in the finite disjoint union is of the form \\eqref{eq:probabilityofmechanisms}.\nThus the measure of the left-hand side can be written as a sum of measures of such sets, which use only counterfactuals of the form \\eqref{eq:standardformdefinition},\nshowing agreement of the measures (by Fact~1).\n\\end{proof}\n\\begin{corollary}\\label{cor:standardform}\n$\\mathfrak{S}_3^\\prec$ bijects with the set of atomless measures in $\\mathfrak{P}(\\texttt{F})$, which we denote $\\mathfrak{S}^\\prec_{\\mathrm{std}}$.\nWe write the map as $\\varpi^\\prec_{\\mathrm{std}} : \\mathfrak{S}_3^\\prec \\to \\mathfrak{S}^\\prec_{\\mathrm{std}}$.\n\\qed\n\\end{corollary}\nWhere the order $\\prec$ is clear, the above result permits us to abuse notation, using e.g. $\\mu$ to denote either an element of $\\mathfrak{S}_3^\\prec$ or its associated point $\\varpi^\\prec_{\\mathrm{std}}(\\mu)$ in $\\mathfrak{S}^\\prec_{\\mathrm{std}}$.\nWe will henceforth indulge in such abuse.\n\n\\begin{proof}[Proof of Fact~\\ref{prop:alternative}]\nThe follows easily from Lem.~\\ref{lem:acausals1canonical} below, adapted from \\citet[Thm.~1]{suppes:zan81}.\nThis shows that every atomless distribution is generated by some SCM; furthermore, it can  chosen so as to exhibit no causal effects whatsoever.\n\\end{proof}\n\\begin{definition}\nSay that $\\nu \\in \\mathfrak{P}\\big(\\texttt{\\textbf{F}}(\\*V)\\big)$ is \\emph{acausal} if $\\nu(\\pi^{-1}_{(V, \\*p)}(\\{v_1\\}) \\cap \\pi^{-1}_{(V, {\\*p}')}(\\{v_2\\})\\big) = 0$\nfor every $(V, \\*p), (V, \\*p') \\in \\xi$ and $v_1 \\neq v_2 \\in \\chi_V$.\n\n\\end{definition}\n\\begin{lemma}\\label{lem:acausals1canonical}\nLet $\\mu \\in \\mathfrak{P}(\\chi_{\\*V})$ be atomless. Then there is a ${\\mathcal{M}} \\in \\mathfrak{M}^{\\mathrm{std}}_\\prec$ (see Def.~\\ref{def:standardform}) with an acausal noise distribution such that $\\mu = (\\varpi_1 \\circ \\varpi_2 \\circ \\varpi_3)({\\mathcal{M}})$. \n\\end{lemma}\n\\begin{proof}\nConsider $\\nu \\in \\mathfrak{P}\\big(\\texttt{\\textbf{F}}(\\*V)\\big) = \\mathfrak{P}\\big(\\bigtimes_{(V, \\*p)} \\chi_V \\big)$ determined on a basis as follows:\n$\\nu\\big( \\pi^{-1}_{(V_1, \\mathbf{p}_1)}(\\{v_1\\})\\cap \\dots \\cap \\pi^{-1}_{(V_n, \\*p_n)}(\\{v_n\\}) \\big) = \\mu\\big( \\pi^{-1}_{V_1}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{V_n}(\\{v_n\\}) \\big)$.\nThis is clearly acausal and atomless.\n\\end{proof}\n\n\\subsection{Proofs from \\S{}{3.2}}\n\n\\begin{proof}[Proof of Prop. \\ref{prop:collapse1} (Collapse set $\\mathfrak{C}_1$ is empty)]\nLet $\\mu \\in \\mathfrak{S}_1$ and $\\nu \\in \\mathfrak{S}^\\prec_{\\mathrm{std}}$ with $(\\varpi_1 \\circ \\varpi_2 \\circ \\varpi_{\\mathrm{std}}^{-1})(\\nu) = \\mu$.\nBy Lemma \\ref{lem:acausals1canonical} we may assume $\\nu$ is acausal.\nLet $X$ be the first, and $Y$ the second variable with respect to $\\prec$.\nNote there are $x^*$, $y^*$ such that $\\mu(\\pi^{-1}_X(\\{x^*\\}) \\cap \\pi^{-1}_Y(\\{y^*\\})) > 0$;\nlet $x^\\dagger \\neq x^*$, $y^\\dagger \\neq y^*$.\nConsider $\\nu'$ defined as follows where $\\digamma_3$ stands for any set of the form\n$\\pi^{-1}_{(V_1, {\\*p}_1)}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{(V_n, {\\*p}_n)}(\\{v_n\\}) \\subset \\texttt{\\textbf{F}}(\\*V)$, for $V_i \\in \\*V$, $\\*p_i \\in \\chi_{\\mathbf{P}(V_i)}$, $v_i \\in \\chi_{V_i}$,\nand $\\digamma_1$ is the corresponding $\\pi^{-1}_{V_1}(\\{v_1\\}) \\cap \\dots \\cap \\pi^{-1}_{V_n}(\\{v_n\\}) \\subset \\chi_{\\*V}$.\n\\begin{multline*}\n \\nu'\\big( \\pi^{-1}_{(X, ())}(\\{x\\}) \\cap \\pi^{-1}_{(Y, (x^*))}(\\{y_*\\}) \\cap \\pi^{-1}_{(Y, (x^{\\dagger}))}(\\{y_\\dagger\\}) \\cap \\digamma_3 \\big) =\\\\\n \\begin{cases}\n \\mu\\big(\\pi^{-1}_X(\\{x^*\\}) \\cap \\pi^{-1}_Y(\\{y^*\\}) \\cap \\digamma_1 \\big), & x = x^*, y_* = y^* \\neq y_\\dagger\\\\% = y^{\\dagger}\\\\\n 0, & x = x^*, y_* = y^{\\dagger} \\neq y_\\dagger\\\\% = y^*\\\\\n 0, & x = x^*, y_* = y_{\\dagger} = y^*\\\\\n \\mu\\big(\\pi^{-1}_X(\\{x^*\\}) \\cap \\pi^{-1}_Y(\\{y^\\dagger\\}) \\cap \\digamma_1 \\big), & x = x^*, y_* = y_{\\dagger} = y^\\dagger\\\\\n \\mu\\big(\\pi^{-1}_X(\\{x^\\dagger\\}) \\cap \\pi^{-1}_Y(\\{y\\}) \\cap \\digamma_1 \\big), & x = x^\\dagger\n \\end{cases}\n\n\n\\end{multline*}\nWe claim that $\\mu = \\mu'$ where $\\mu' = (\\varpi_1 \\circ \\varpi_2 )(\\nu')$; it suffices to show agreement on sets of the form $\\pi^{-1}_X(\\{x\\}) \\cap \\pi^{-1}_Y(\\{y\\}) \\cap \\digamma_1$. If $x = x^\\dagger$ then the last case above occurs; if $x = x^*$ and $y = y^\\dagger$ then we are in the fourth case; if $x = x^*$ and $y = y^*$ then exclusively the first case applies. In all cases the measures agree.\nLet $(\\nu_\\alpha)_\\alpha = \\varpi_2(\\nu)$ and $(\\nu'_\\alpha)_\\alpha = \\varpi_2(\\nu')$\nbe the Level 2 projections of $\\nu$, $\\nu'$ respectively.\nNote that $\\nu_{X \\coloneqq x^\\dagger}(y^\\dagger) < \\nu'_{X \\coloneqq x^\\dagger}(y^\\dagger)$.\nThis shows that the standard-form measures $\\nu$, $\\nu'$ project down to different points in $\\mathfrak{S}_2$ (in particular differing on the $Y$-marginal at the index corresponding to the intervention $X \\coloneqq x^\\dagger$) while projecting to the same point in $\\mathfrak{S}_1$.\nThus $\\mu \\notin \\mathfrak{C}_1$ and since $\\mu$ was arbitrary, $\\mathfrak{C}_1 = \\varnothing$.\n\\end{proof}\n\n\n\n\\begin{example}[Collapse set $\\mathfrak{C}_2$ is nonempty]\nWe present a $\\mu \\in \\mathfrak{S}_{\\mathrm{std}}^{\\prec}$ for which $\\varpi_2(\\mu) \\in \\mathfrak{C}_2$.\nLet ${\\*S}_n \\subset \\*V$ be the ancestrally closed (\\S\\ref{ss:app:standardform}) set of the $n$ least variables with respect to $\\prec$ and $X$ be the first variable with respect to $\\prec$; thus, e.g., ${\\*S}_1 = \\{X\\}$.\nWhere $\\texttt{\\textbf{f}} = \\{\\texttt{f}_V\\}_{V \\in {\\*S}_n} \\in \\texttt{F}(\\textbf{S}_n)$, define $\\mu(\\texttt{\\textbf{f}}) = 0$ if there is any $V \\in {\\*S}_n \\setminus \\{X\\}$, $\\*p \\neq (0, \\dots, 0) \\in \\chi_{\\mathbf{Pred}(V)}$ such that $\\texttt{f}_V(\\*p) = 0$, and otherwise define $\\mu(\\texttt{\\textbf{f}}) = 1\/2^n$.\nNote that this example is \\emph{monotonic} in the sense of \\cite{Angrist,Pearl1999}.\n\nWe claim $\\mu' = \\mu$ for any $\\mu' \\in \\mathfrak{S}_{\\mathrm{std}}^\\prec$ projecting to the same Level 2, i.e., such that $\\varpi_2(\\mu') = \\varpi_2(\\mu)$; note that it suffices to consider only candidate counterexamples with order $\\prec$ since $\\varpi_2(\\mu) \\notin \\mathfrak{S}_2^{\\prec'}$ for any $\\left.\\prec'\\right. \\neq \\left.\\prec\\right.$.\nIt suffices to show that $\\mu(\\texttt{f}) = \\mu'(\\texttt{f})$ for any $n$ and $\\texttt{\\textbf{f}} = \\{\\texttt{f}_V\\}_{V \\in {\\*S}_n}$; recall that in the measures, $\\texttt{f}$ denotes a set of the form \\eqref{eq:probabilityofmechanisms}.\nLet $(\\mu_\\alpha)_\\alpha = \\varpi_2(\\mu) \\in \\mathfrak{S}_2^\\prec$ and $(\\mu'_\\alpha)_\\alpha = \\varpi_2(\\mu')$, with $(\\mu_\\alpha)_\\alpha = (\\mu'_\\alpha)_\\alpha$.\nSince $\\mu'_{\\mathbf{Pred}(V) \\coloneqq \\*p}(\\pi^{-1}_V(\\{1\\})) = 1$ for any $V \\in {\\*S}_n \\setminus \\{X\\}$, $\\*p \\neq (0, \\dots, 0)$, probability bounds show $\\mu'(\\texttt{f})$ vanishes unless $\\texttt{f}_V(\\*p) = 1$ for each such $\\*p$, in which case\n\\begin{equation}\\label{eq:toreducetol2}\n\\mu'(\\texttt{f})=\n\\mu'\\Big(\n\\bigcap_{i=1}^n \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{v_i\\})\\Big)\n\\end{equation}\nfor some $v_i \\in \\chi_{V_i}$,\nwhere we have labeled the elements of ${\\*S}_n$ as $V_1, \\dots, V_n$,\nwith $V_1 \\prec \\dots \\prec V_n$.\nWe claim this is reducible---again using probabilistic reasoning alone---to a linear combination of quantities fixed by $(\\mu'_\\alpha)_\\alpha$, the Level 2 projection of $\\mu'$, which is the same as the projection $(\\mu_\\alpha)_\\alpha$ of $\\mu$.\nThis can be seen by an induction on the number $m = \\left|M\\right|$ where $M = \\{i : v_i = 1 \\}$: note\n\\eqref{eq:toreducetol2} becomes\n\\begin{multline*\n\\mu'\\Big(\n\\bigcap_{\\substack{i \\notin M}} \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{0\\})\\Big)\\\\\n- \\sum_{M' \\subsetneq M}\\mu'\\Big(\n\\bigcap_{\\substack{i \\notin M'}} \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{0\\})\n\\cap\n\\bigcap_{\\substack{i \\in M'}} \\pi^{-1}_{(V_i, \\{V_1, \\dots, V_{i-1}\\} \\coloneqq (0, \\dots, 0))}(\\{1\\})\n\\Big)\n\\end{multline*}\nand the inductive hypothesis implies each summand  can be written in the sought form\nwhile the first term becomes\n$\\mu'\\big(\\bigcap_{i\\notin M} \\pi^{-1}_{(V_i, ())}(\\{0\\})\\big) = \\mu'_{()}\\big(\\bigcap_{i \\notin M}\\pi^{-1}_{V_1}(\\{0\\})\\big) = \\mu_{()}\\big(\\bigcap_{i \\notin M}\\pi^{-1}_{V_1}(\\{0\\})\\big)$.\nHere $()$ abbreviates the empty intervention $\\varnothing \\coloneqq ()$.\nThus any Level 3 quantity reduces to Level 2, on which the two measures agree by hypothesis.\n\\label{example:collapse}\n\\end{example}\n\n\\subsection{Remarks on \\S{}{3.3}}\\label{ss:app:3.3}\n\n\\begin{lemma}\n\\label{prop:p2:characterization:binary}\n  Let $(\\mu_{\\alpha})_{\\alpha} \\in \\bigtimes_{\\alpha \\in A^{X \\to Y}_{2}} \\mathfrak{P}(\\chi_{X, Y})$.\n  Then $(\\mu_\\alpha)_{\\alpha} \\in \\mathfrak{S}^{X \\to Y}_{2}$ iff\n    \\begin{align}\\label{eq:snsr}\n    \\mu_{X \\coloneqq x}( x ) = 1\n    \\end{align}\n    for every $x \\in \\chi_X$\n    and \\begin{align}\\label{eq:snscm}\n    \\mu_{X \\coloneqq x}(y)\n    \\ge \\mu_{()}(x, y)\n    \\end{align}\n    for every $x \\in \\chi_{X}$, $y \\in \\chi_Y$. Here $x, y$ abbreviates the basic set $\\pi_X^{-1}(\\{x\\}) \\cap \\pi_Y^{-1}(\\{y\\})$.\n\\end{lemma}\n\\begin{proof\nIt is easy to see that \\eqref{eq:snsr}, \\eqref{eq:snscm} hold for any $(\\mu_\\alpha)_\\alpha$.\nFor the converse,\nconsider the two-variable model over endogenous $\\*Z = \\{X, Y\\}$ with $X \\prec Y$; note that $|\\texttt{\\textbf{F}}(\\*Z)| = 8$.\nA result of\n\\citet{tian:etal06} gives that this model is characterized exactly by \\eqref{eq:snsr}, \\eqref{eq:snscm} so for any such $(\\mu_\\alpha)_\\alpha$ there is a distribution on $\\texttt{\\textbf{F}}(\\*Z)$ such that this model induces $(\\mu_\\alpha)_\\alpha$.\nIt is straightforward to extend this distribution to an atomless measure on $\\texttt{\\textbf{F}}(\\*V)$.\n\\end{proof}\n\n\n\\section{Proofs from \\S\\ref{sec:weaktopology}} \\label{app:empirical}\n\n\\begin{proof}[Proof of Prop.~\\ref{prop:causalprojectioncontinuous}]\nThe continuity of any of the maps amounts to the continuity of projections in product spaces and marginalizations in weak convergence spaces. The latter follows easily from results in \\S{}3.1.3 of \\cite{Genin2018} or \\cite{Billingsley}.\n\nAs for the openness of any $\\varpi_2^{X \\to Y}$, note we can write any $S \\subset \\mathfrak{S}_2$ as $S = \\bigcup_{\\prec} S \\cap \\mathfrak{S}_2^\\prec$ where $\\prec$ in the union ranges over all total orders of $\\*V$. It thus suffices to show that for any $\\prec$ the image of any open $S \\subset \\mathfrak{S}_2^\\prec$ is open.\nDefine the map $\\varpi_2^{\\prec}: \\mathfrak{S}_{\\mathrm{std}}^\\prec \\to \\mathfrak{S}_2^\\prec$ and the map $\\varpi_2^{X \\to Y, \\prec}: \\mathfrak{S}_2^\\prec \\to \\mathfrak{S}_2^{X \\to Y}$ as restrictions of $\\varpi_2$ and $\\varpi_2^{X \\to Y}$ respectively. Evidently, $\\varpi_2^{\\prec}$ is continuous so it suffices to show that $\\varpi_2^{X \\to Y, \\prec} \\circ \\varpi_2^{\\prec}$ is open.\n\nFor any $n \\ge 1$ let $\\*S_{n, \\prec}$ be the initial segment of the first $n$ variables in $\\*V$ when ordered according to $\\prec$, as in Ex.~\\ref{example:collapse}, and define sets $\\mathfrak{S}_{\\mathrm{std}}^{n, \\prec}$, $\\mathfrak{S}_2^{n, \\prec}$ analogously to $\\mathfrak{S}_{\\mathrm{std}}^\\prec$, $\\mathfrak{S}_2^{\\prec}$ but over the set of variables $\\*{S}_{n, \\prec}$.\nDefine maps $\\varpi_{2}^{n, \\prec} : \\mathfrak{S}_{\\mathrm{std}}^{n, \\prec} \\to \\mathfrak{S}_2^{n, \\prec}$ and $\\varpi_2^{X \\to Y, n, \\prec} :\\mathfrak{S}_2^{n, \\prec} \\to \\mathfrak{S}_2^{X \\to Y}$, where $X, Y \\in \\*{S}_{n, \\prec}$, in a similar fashion.\nDefine also a map $\\varpi_{\\mathrm{std}}^{n, \\prec}: \\mathfrak{S}_{\\mathrm{std}}^{\\prec} \\to \\mathfrak{S}_{\\mathrm{std}}^{n, \\prec}$ as a marginalization taking a distribution over mechanisms (recall \\S{}\\ref{ss:app:standardform}) determining all of $\\*V$ to a distribution over deterministic mechanisms for $\\*S_{n, \\prec}$.\nLet $S \\subset \\mathfrak{S}_{\\mathrm{std}}^\\prec$ be an arbitrary basic open set and let $n$ be least such that $\\*{S}_{n, \\prec}$ contains $X$, $Y$, and every variable whose structural equation appears as a cylinder in the finite intersection defining $S$.\nThen note that $\\big(\\varpi_2^{X \\to Y, \\prec} \\circ \\varpi_2^{\\prec}\\big)(S) = \\big(\\varpi_2^{X \\to Y, n, \\prec} \\circ \\varpi_2^{n, \\prec} \\circ \\varpi_{\\mathrm{std}}^{n, \\prec}\\big)(S)$ and $\\varpi_{\\mathrm{std}}^{n, \\prec}(S)$ is certainly open, so it suffices to show that $\\varpi_2^{X \\to Y, n, \\prec} \\circ \\varpi_2^{n, \\prec} :  \\mathfrak{S}_{\\mathrm{std}}^{n, \\prec} \\to \\mathfrak{S}_2^{X \\to Y}$ is open for any $n$.\n\nTo see this, note that $\\mathfrak{S}_2^{X \\to Y}$ and $\\mathfrak{S}_{\\mathrm{std}}^{n, \\prec}$ in the weak topology are homeomorphic to (subsets of products of) probability simplices in appropriate Euclidean spaces $\\mathbb{R}^m$ with the standard topology, as they are distributions over a discrete space.\nThe latter in fact is exactly a probability simplex while $\\mathfrak{S}_2^{X \\to Y}$ is polyhedral by Lemma~\\ref{prop:p2:characterization:binary}, and $\\varpi_2^{X \\to Y, n, \\prec} \\circ \\varpi_2^{n, \\prec}$ is the restriction of a surjective linear mapping under the aforementioned homeomorphism.\nThis must be open by \\citet[Corollary~8]{MIDOLO20091186}.\n\\end{proof}\n\n\\begin{proof}[Proof of Thm. \\ref{thm:l2learning}]\nWe show how Theorem~3.2.1 of \\cite{Genin2018} can be applied to derive the result. Specifically, let $\\Omega = \\bigtimes_\\alpha \\chi_{\\*V}$. Let $\\mathcal{I}$ be the usual clopen basis, and let $W$ be the set of Borel measures $\\mu \\in \\mathfrak{P}(\\Omega)$ that factor as a product $\\mu = \\times_\\alpha \\mu_\\alpha$ where each $\\mu_\\alpha \\in \\mathfrak{S}_1$ and $(\\mu_\\alpha)_\\alpha \\in \\mathfrak{S}_2$. This choice of $W$ corresponds exactly to our notion of experimental verifiability.\n\nIt remains to check that a set is open in $W$ iff the associated set is open in $\\mathfrak{S}_2$ (homeomorphism).\nIt suffices to show their convergence notions agree. Suppose $(\\nu_n)_n$ is a sequence, each $\\nu_n \\in W$, converging to $\\nu = \\times_\\alpha \\mu_\\alpha \\in W$. We have for each $n$ that $\\nu_n = \\times_\\alpha \\mu_{n, \\alpha}$ such that $(\\mu_{n,\\alpha})_\\alpha \\in \\mathfrak{S}_2$. By Theorem~3.1.4 in \\cite{Genin2018}, which is straightforwardly generalized to the infinite product, for each fixed $\\alpha$ we have $(\\mu_{n, \\alpha})_n \\Rightarrow \\mu_\\alpha$. This is exactly pointwise convergence in the product space $\\mathfrak{S}_2$, and the same argument in reverse works for the converse.\n\\end{proof}\n\n\\section{Proofs from \\S\\ref{sec:main}} \\label{app:main}\n\nWe will use the following result to categorize sets in the weak topology.\n\\begin{lemma}\\label{prop:probiscontinuous}\n  If $X \\subset \\vartheta$ is a basic clopen,\n  the map $p_X : (\\mathfrak{S}, \\tau^{\\mathrm{w}}) \\to ([0, 1], \\tau)$ sending $\\mu \\mapsto \\mu(X)$ is continuous and open (in its image), where $\\tau$ is as usual on $[0, 1] \\subset \\mathbb{R}$.\n\\end{lemma}\n\\begin{proof}\nContinuous: the preimage of the basic open $(r_1, r_2) \\cap p_X(\\mathfrak{S})$ where $r_1, r_2 \\in \\mathbb{Q}$ is $\\{ \\mu: \\mu(X) > r_1 \\} \\cap \\{ \\mu : \\mu(X) < r_2 \\} = \\{ \\mu: \\mu(X) > r_1 \\} \\cap \\{ \\mu : \\mu(\\vartheta \\setminus X) > 1- r_2 \\}$, a finite intersection of the subbasic sets \\eqref{subbasis-weak} from \\S{}\\ref{sec:weaktopology}.\nSee also \\citet[Corollary~17.21]{Kechris1995}.\n\nOpen: if $X = \\varnothing$ or $\\vartheta$, then $p_X(\\mathfrak{S}) = \\{0\\}$ or $\\{1\\}$ resp., both open in themselves.\nElse $p_X(\\mathfrak{S}) = [0, 1]$;\nwe show any $Z = p_X\\big(\\bigcap_{i = 1}^n \\{\\mu: \\mu(X_i)>r_i\\} \\big)$ is open.\nConsider a mutually disjoint, covering $\\mathcal{D} = \\big\\{ \\bigcap_{i=0}^n Y_i : Y_0 \\in \\{X, \\vartheta \\setminus X\\},\\text{ each } Y_i \\in \\{X_i, \\vartheta \\setminus X_i\\}\\big\\}$ \nand space $\\Delta = \\{(\\mu(D))_{D \\in \\mathcal{D}} : \\mu \\in \\mathfrak{S}\\} \\subset \\mathbb{R}^{2^{n+1}}$.\nJust as in the Lemma, we have $\\textsf{p}_{S} : \\Delta \\to [0, 1]$, for each $S \\subset \\mathcal{D}$ taking $(\\mu(D))_D \\mapsto \\sum_{D \\in S} \\mu(D)$.\nNote $Z = \\textsf{p}_{\\{D: D\\cap X \\neq \\varnothing\\}}\\big( \\bigcap_{i=1}^n\\textsf{p}^{-1}_{\\{D: D\\cap X_i \\neq \\varnothing\\}}((r_i, 1])  \\big)$ so it suffices to show $\\textsf{p}_S$ is continuous and open; this is straightforward (see the end of the proof of Prop.~\\ref{prop:causalprojectioncontinuous}).\n\\end{proof}\n\n\\begin{proof}[Full proof of Lem.~\\ref{lem:goodcomeager}]\nWe show a stronger result, namely  that the complement of the good set is nowhere dense.\nBy rearrangement and laws of probability we find that the second inequality in \\eqref{cns:ineq:1} is equivalent to\n\\begin{align*}\n\\mu_x(y') \n &< \\mu_{()}(x') + \\mu_{()}(x, y')\\\\\n 1- \\mu_x(y) &< \\underbrace{\\mu_{()}(x') + \\mu_{()}(x)}_1 - \\mu_{()}(x, y)\\\\\n \\mu_x(y) &> \\mu_{()}(x, y).\n\\end{align*}\nLemma~\\ref{prop:p2:characterization:binary} then entails the non-strict analogues of all four inequalities in \\eqref{cns:ineq:1}, \\eqref{cns:ineq:2} are met for any $(\\mu_\\alpha)_\\alpha \\in \\mathfrak{S}^{X \\to Y}_{2}$, so we show that converting each to an equality yields a nowhere dense set, whose finite union is also nowhere dense.\nNote that we have a continuous, open (again, refer to the end of the proof of Prop.~\\ref{prop:causalprojectioncontinuous}), and surjective observational projection $\\pi_{()}: \\mathfrak{S}^{X \\to Y}_{2} \\to \\mathfrak{P}\\big(\\chi_{\\{X, Y\\}}\\big)$,\nand the first inequality in \\eqref{cns:ineq:2} is met iff $(\\mu_\\alpha)_\\alpha \\in \\big(p_{x', y'} \\circ \\pi_{()}\\big)^{-1}(\\{0\\})$ where $p_{x', y'}$ is the map from Lemma~\\ref{prop:probiscontinuous} and $x', y'$ denotes the set $\\pi_X^{-1}(\\{x'\\}) \\cap \\pi_Y^{-1}(\\{y'\\}) \\subset \\chi_{\\{X, Y\\}}$.\nThis is nowhere dense as it is the preimage of the nowhere dense set $\\{0\\}\\subset [0, 1]$ under a map which is continuous and open by Lemma~\\ref{prop:probiscontinuous}. The second inequality of \\eqref{cns:ineq:2} is wholly analogous after rearrangement.\n\nAs for \\eqref{cns:ineq:1},\ndefine a function $d: \\mathfrak{S}^{X \\to Y}_2 \\to [0, 1]$ taking $(\\mu_\\alpha)_\\alpha \\mapsto \\mu_{X \\coloneqq x}(y') - \\mu_{()}(x, y')$; this function $d$ is continuous by Lemma~\\ref{prop:probiscontinuous} and the continuity of addition and projection, and is once again open. Note\nthat the first inequality of \\eqref{cns:ineq:1} holds iff $d((\\mu_\\alpha)_\\alpha) = 0$.\nFor any $\\mu \\in \\mathfrak{S}_3^{X}$ such that $(\\varpi^{X \\to Y}_{2}\\circ \\varpi_2)(\\mu) = (\\mu_\\alpha)_\\alpha$, note that $d((\\mu_\\alpha)_\\alpha) = \\mu\\big(x', y'_x\\big)$ where $x', y'_x$ abbreviates the basic set $\\pi_{((), X)}^{-1}(\\{x'\\}) \\cap \\pi_{(X \\coloneqq x, Y)}^{-1}(\\{y'\\}) \\in \\mathcal{B}(\\chi_{A \\times \\*V})$.\nThus $d$ is surjective, so that $d^{-1}(\\{0\\})$ is nowhere dense since $\\{0\\} \\subset [0, 1]$ is nowhere dense.\nThe second inequality in \\eqref{cns:ineq:1} is again totally analogous.\n\\end{proof}\n\n\\begin{proof}[Proof of Lem.~\\ref{lem:separation}]\nAbbreviate $\\mu_3$ as $\\mu$, and without loss take $\\mu \\in \\mathfrak{S}^\\prec_{\\mathrm{std}}$.\nNote that \\eqref{cns:ineq:1}, \\eqref{cns:ineq:2} entail\n\\begin{equation*}\n0 < \\mu(x', y'_x) < \\mu(x'), \\quad 0 < \\mu(x', y'_{x'}) < \\mu(x').\n\\end{equation*}\nand therefore\n\\begin{align*}\n0 < \\mu\\big(\\pi^{-1}_{((), X)}(\\{x'\\}) \\cap \\pi^{-1}_{(x^*, Y)}(\\{1\\})\\big) < \\mu\\big(\\pi^{-1}_{((), X)}(\\{x'\\})\\big)\n\\end{align*}\nfor each $x^* \\in \\chi_X = \\{0, 1\\}$.\nIn turn this \nentails that there are some values $y_{0}, y_{1} \\in \\{ 0, 1 \\}$ such that\n$\\mu(\\Omega_1) > 0$, $\\mu(\\Omega_2) > 0$\nwhere the disjoint sets $\\{\\Omega_i\\}_i$ are defined as\n\\begin{align*}\n\\Omega_1 &= \\pi^{-1}_{((), X)}(\\{x'\\}) \\cap \\pi^{-1}_{(X \\coloneqq 0, Y)} (\\{y_0\\}) \\cap \\pi^{-1}_{(X \\coloneqq 1, Y)} (\\{y_1\\})\\\\\n\\Omega_2 &= \\pi^{-1}_{((), X)}(\\{x'\\}) \\cap \\pi^{-1}_{(X \\coloneqq 0, Y)} (\\{y^\\dagger_0\\}) \\cap \\pi^{-1}_{(X \\coloneqq 1, Y)} (\\{y^\\dagger_1\\})\n\\end{align*}\nwhere in the second line, $y_0^\\dagger = 1-y_0$ and $y_1^\\dagger = 1 - y_1$.\nNote that for $i = 1, 2$ we have conditional measures $\\mu_i(S_i) = \\frac{\\mu(S_i)}{\\mu(\\Omega_i)}$ for $S_i \\in \\mathcal{B}(\\Omega_i)$; further, $\\Omega_i$ is Polish, since each is clopen.\nThis implies $\\Omega_i$ is a standard atomless (since $\\mu$ is) probability space under $\\mu_i$.\nBy \\citet[Thm.~17.41]{Kechris1995}, there are Borel isomorphisms $f_i : \\Omega_i \\hookdoubleheadrightarrow [0, 1]$ pushing $\\mu_i$ forward to Lebesgue measure $\\lambda$, i.e., $\\mu_i(f_i^{-1}(B)) = \\lambda(B)$ for $B \\in \\mathcal{B}([0, 1])$.\nThus $g = f_2^{-1} \\circ f_1 : \\Omega_1  \\hookdoubleheadrightarrow \\Omega_2$ is $\\mu_i$-preserving: for $X_1 \\in \\mathcal{B}(\\Omega_1)$,\n\\begin{align}\n\\mu(g(X_1)) = \\frac{\\mu(\\Omega_2)}{\\mu(\\Omega_1)} \\mu(X_1).\\label{eqn:NOCBKNantKPlTpiViI0=}\n\\end{align}\n\nConsider $\\mu' = \\varpi_3(\\mathcal{M}')$ for a new $\\mathcal{M}' \\in \\mathfrak{M}_\\prec$, given as follows. Its exogenous valuation space is $\\chi_{\\*U} = \\Omega'$ where we define the sample space $\\Omega' = \\texttt{\\textbf{F}}(\\*V) \\times \\{ \\mathrm{T}, \\mathrm{H} \\}$; that is, a new exogenous variable representing a coin flip is added to some representation of the choice of deterministic standard form mechanisms.\nFix constants $\\varepsilon_1, \\varepsilon_2 \\in (0, 1)$ with $\\varepsilon_1 \\cdot \\mu(\\Omega_1) = \\varepsilon_2 \\cdot \\mu(\\Omega_2)$ and define its exogenous noise distribution $P$ by\n\\begin{equation}\\label{eq:modprobz}\nP(X \\times \\{\\mathrm{S}\\}) = \n\\begin{cases}\n(1-\\varepsilon_1)\\cdot \\mu(X), & X \\subset \\Omega_1, \\mathrm{S} = \\mathrm{T}\\\\\n\\varepsilon_1 \\cdot \\mu(X), & X \\subset \\Omega_1, \\mathrm{S} = \\mathrm{H}\\\\\n(1-\\varepsilon_2)\\cdot\\mu(X), & X \\subset \\Omega_2, \\mathrm{S} = \\mathrm{T}\\\\\n\\varepsilon_2 \\cdot \\mu(X), & X \\subset \\Omega_2, \\mathrm{S} = \\mathrm{H}\\\\\n\\mu(X), & X \\subset \\texttt{\\textbf{F}}(\\*V) \\setminus (\\Omega_1 \\cup \\Omega_2), \\mathrm{S} = \\mathrm{T}\\\\\n0, & X \\subset \\texttt{\\textbf{F}}(\\*V) \\setminus (\\Omega_1 \\cup \\Omega_2), \\mathrm{S} = \\mathrm{H}\n\\end{cases}.\n\\end{equation}\nWhere $\\texttt{\\textbf{f}}  \\in \\texttt{\\textbf{F}}(\\*V)$ and $V \\in \\*V$ write $\\texttt{f}_V$ for the deterministic mechanism (of signature $\\chi_{\\mathbf{Pred}(V)} \\to \\chi_V$) for $V$ in $\\texttt{f}$. (Note that each $\\texttt{f}$ is just an indexed collection of such mechanisms $\\texttt{f}_V$.)\nThe function $f'_V$ in $\\mathcal{M}'$ is defined at the initial variable $X$ as\n$f'_X(\\texttt{\\textbf{f}}, \\mathrm{S}) = \\texttt{\\textbf{f}}_X$ for both values of $\\mathrm{S}$,\nand for $V \\neq X$ is defined as follows, where $\\*p \\in \\mathbf{Pred}(V)$:\n\\begin{equation} \\label{eqn:modseqz}\n{f}'_V\\big(\\*p, (\\texttt{\\textbf{f}}, \\mathrm{S})\\big) =\n \\begin{cases}\n      (g(\\texttt{\\textbf{f}}))_V(\\*p), & \\texttt{\\textbf{f}}  \\in\\Omega_1, \\mathrm{S} = \\mathrm{H},\\, \\pi_X(\\*p) = x\\\\\n      (g^{-1}(\\texttt{\\textbf{f}}))_V(\\*p), & \\texttt{\\textbf{f}}  \\in\\Omega_2, \\mathrm{S} = \\mathrm{H},\\, \\pi_X(\\*p) = x\\\\\n      \\texttt{\\textbf{f}}_V(\\*p), &\\textnormal{otherwise}\n                                                 \\end{cases}.\n                                                 \\end{equation}\n\nWe claim that $\\varpi_2(\\mu') = \\varpi_2(\\mu)$.\nIt suffices to show for any $\\*Z \\coloneqq \\*z \\in A$ and ${\\mathbf{w}} \\in \\chi_{{\\*W}}$, $\\*W$ finite,\nwe have\n\\begin{equation}\n    \\mu(\\theta) = \\mu'(\\theta), \\text{ where }\\theta = \\bigcap_{W \\in {\\*W}}\\pi^{-1}_{(\\*Z \\coloneqq \\*z, W)}(\\{\\pi_W(\\*w)\\}).\\label{eqn:IOXCVOIXCJVSOIJDklfsjdlfksdj}\n\\end{equation}\nAssume $\\pi_Z(\\*w) = \\pi_Z(\\*z)$ for every $Z \\in \\*Z \\cap {\\*W}$, since both sides of \\eqref{eqn:IOXCVOIXCJVSOIJDklfsjdlfksdj} trivially vanish otherwise.\nWhere $\\texttt{\\textbf{f}} \\in \\texttt{\\textbf{F}}(\\*V)$\nwrite, e.g., $\\texttt{\\textbf{f}} \\models \\theta$ if $m^{{\\mathcal{M}}_A}(\\texttt{\\textbf{f}}) \\in \\theta$, where ${\\mathcal{M}}$ is a standard form model (Def.~\\ref{def:standardform}); for\n$\\omega' \\in \\Omega'$ write $\\omega' \\models' \\theta$\nif $m^{{\\mathcal{M}}'_A}(\\omega') \\in \\theta$. By the last two cases of \\eqref{eqn:modseqz} we have\n\\begin{align}\n\\mu'(\\theta)\n&= \\sum_{\\mathrm{S} =  \\mathrm{T}, \\mathrm{H}}P\\big( \\{ (\\texttt{\\textbf{f}}, \\mathrm{S}) \\in \\Omega' : (\\texttt{\\textbf{f}}, \\mathrm{S}) \\models' \\theta \\} \\big)\\nonumber\\\\\n&= \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\texttt{\\textbf{F}}(\\*V) \\setminus (\\Omega_1 \\cup \\Omega_2) : \\texttt{\\textbf{f}} \\models \\theta \\}\\big) +\n\\sum_{\\substack{\\mathrm{S} = \\mathrm{T}, \\mathrm{H}\\\\ i = 1, 2}} P\\big( \\{ (\\texttt{\\textbf{f}}, \\mathrm{S}) \\in \\Omega' : \\texttt{\\textbf{f}} \\in \\Omega_i, (\\texttt{\\textbf{f}}, \\mathrm{S}) \\models' \\theta \\} \\big).\\label{eqn:CXJSDFJKSDKFJKSDVC}\n\\end{align}\nApplying the first four cases of \\eqref{eq:modprobz} and the third case of \\eqref{eqn:modseqz}, the second term of \\eqref{eqn:CXJSDFJKSDKFJKSDVC} becomes\n\\begin{equation}\n\\sum_i \\Big[\\varepsilon_i \\cdot \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\theta \\} \\big) +\n \\left(1- \\varepsilon_i\\right) \\cdot\\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\theta \\} \\big)\\Big] .\\label{eqn:ckxljvXIOCJVOX}\n\\end{equation}\nEither $X \\in \\*Z$ and $\\pi_X(\\*z) = x$, or not.\nIn the former case: defining $X_i = \\{\\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\theta\\}$ for each $i = 1, 2$,\nthe first two cases of \\eqref{eqn:modseqz} yield that\n\\begin{align}\n\\{ \\texttt{\\textbf{f}} \\in \\Omega_1 : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\theta \\} &= \\{\\texttt{\\textbf{f}} \\in \\Omega_1 : g(\\texttt{\\textbf{f}}) \\models \\theta\\}\n= g^{-1}(X_2 )\\nonumber\\\\\n\\{ \\texttt{\\textbf{f}} \\in \\Omega_2 : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\theta \\} &= \\{\\texttt{\\textbf{f}} \\in \\Omega_2 : g^{-1}(\\texttt{\\textbf{f}}) \\models \\theta\\}\n= g(X_1).\\label{eqn:yDuRPR\/5r83+sSdJ2Iw=}\n\\end{align}\nApplying \\eqref{eqn:yDuRPR\/5r83+sSdJ2Iw=} and \\eqref{eqn:NOCBKNantKPlTpiViI0=}, \\eqref{eqn:ckxljvXIOCJVOX} becomes\n\\begin{gather}\n\\varepsilon_1 \\cdot \\frac{\\mu(\\Omega_1)}{\\mu(\\Omega_2)} \\cdot \\mu\\big( X_2 \\big)\n + \\left(1- \\varepsilon_1\\right) \\cdot\\mu\\big( X_1 \\big)\n  + \\varepsilon_2 \\cdot\\frac{\\mu(\\Omega_2)}{\\mu(\\Omega_1)} \\cdot \\mu\\big( X_1\\big)\n + \\left(1- \\varepsilon_2\\right) \\cdot\\mu\\big( X_2 \\big)\\nonumber\\\\\n = \\mu(X_1) + \\mu(X_2),\\label{eqn:CxoFLyrcYfY8on7xY\/A=}\n\\end{gather}\nthe final cancellation by choice of $\\varepsilon_1, \\varepsilon_2$.\nIn the latter case: since $m^{{\\mathcal{M}}}(\\texttt{\\textbf{f}}) \\in \\pi^{-1}_X(\\{x'\\})$ for any $\\texttt{\\textbf{f}} \\in \\Omega_1 \\cup \\Omega_2$, the third case of \\eqref{eqn:modseqz} gives $\\{ \\texttt{\\textbf{f}} \\in \\Omega_i : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models'\\theta\\} = X_i$. Thus \\eqref{eqn:ckxljvXIOCJVOX} becomes \\eqref{eqn:CxoFLyrcYfY8on7xY\/A=} in either case.\nPutting in \\eqref{eqn:CxoFLyrcYfY8on7xY\/A=} as the second term in \\eqref{eqn:CXJSDFJKSDKFJKSDVC}, we find $\\mu(\\theta) = \\mu'(\\theta)$.\n\nNow we claim $\\mu(\\zeta) \\neq \\mu'(\\zeta)$\nfor $\\zeta = \\zeta_0 \\cap \\zeta_1$ where\n$\\zeta_1 = \\pi^{-1}_{(X \\coloneqq 1, Y)}(\\{y_1\\})$ and $\\zeta_0 = \\pi^{-1}_{(X \\coloneqq 0, Y)}(\\{y_0\\})$.\nWe have\n\\begin{align}\n\\mu'(\\zeta) = &\\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega \\setminus (\\Omega_1 \\cup \\Omega_2) : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\\nonumber\\\\\n&+ \\sum_{i=1,2} \\Big[\\varepsilon_i \\cdot \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : (\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\zeta \\} \\big) +\n\\left(1- \\varepsilon_i\\right) \\cdot\\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\\Big].\\label{eqn:XT26+J4OsmYVQhi6+dY=}\n\\end{align}\nFirst suppose that $x = 0$.\nIf $\\texttt{\\textbf{f}} \\in \\Omega_1$, then\nnote that $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\zeta_0$ iff $g(\\texttt{\\textbf{f}}) \\models \\zeta_0$, but this is never so, since $g(\\texttt{\\textbf{f}}) \\in \\Omega_2$.\nIf $\\texttt{\\textbf{f}} \\in \\Omega_2$,\nthen\n$(\\texttt{\\textbf{f}}, \\mathrm{H}) \\models' \\zeta_1$ iff $\\texttt{\\textbf{f}} \\models \\zeta_1$, which is never so again by choice of $\\Omega_2$.\nIf $x = 1$ then we find that $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\not\\models \\zeta_1$ (if $\\texttt{\\textbf{f}} \\in \\Omega_1$) and $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\not\\models \\zeta_0$ (if $\\texttt{\\textbf{f}} \\in \\Omega_2$).\nThus $(\\texttt{\\textbf{f}}, \\mathrm{H}) \\not\\vDash ' \\zeta$ for any $\\texttt{\\textbf{f}} \\in \\Omega_1 \\cup \\Omega_2$ and \\eqref{eqn:XT26+J4OsmYVQhi6+dY=} becomes\n\\begin{equation*}\n \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\n - \\sum_{i=1,2} \\varepsilon_i \\cdot \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega_i : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big)\n = \\mu\\big( \\{ \\texttt{\\textbf{f}} \\in \\Omega : \\texttt{\\textbf{f}} \\models \\zeta \\} \\big) - \\varepsilon_1 \\cdot\\mu(\\Omega_1) < \\mu(\\zeta).\n\\end{equation*}\n\nIt is straightforward to check (via casework on the values $y_0$, $y_1$) that $\\mu$ and $\\mu'$ disagree also on the PNS: $\\mu(y_x, y'_{x'}) \\neq \\mu'(y_x, y'_{x'})$ as well as its converse.\nAs for the probability of sufficiency (Definition \\ref{def:probcaus}), note that\n\\begin{align*}\n    P(y_x \\mid x', y') = \\frac{P(y_x, x', y'_{x'}) + \\overbrace{P(y_x, y'_x, x', x)}^0}{P(x', y')}\n\\end{align*}\nand it is again easily seen (given the definition of the $\\Omega_i$) that $\\mu(y_x, x', y'_{x'}) \\neq \\mu'(y_x, x', y'_{x'})$ while the two measures agree on the denominator; similar reasoning shows disagreement on the probability of enablement, since\n\\begin{equation*}\n    P(y_x \\mid y') = \\frac{P(y_x, y'_{x'}, x') + \\overbrace{P(y_x, y'_{x}, x)}^0}{P(y')}. \\qedhere\n\\end{equation*}\n   \n   \n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nNon-equilibrium quantum many-body systems are nowadays routinely probed in experiments with ultracold atoms with unprecedented control over their parameters such as particle number, interaction strength, and external potentials. A plethora of non-equilibrium systems has been realized to address a wide variety of physics questions, including Anderson localization of Bose-Einstein condensates (BECs) \\cite{clement_suppression_2005, fort_effect_2005, schulte_routes_2005, billy_direct_2008, roati_anderson_2008}, many-body localization in disordered lattices \\cite{schreiber_observation_2015, lukin_probing_2019}, pre-thermalization of one-dimensional (1D) BECs \\cite{gring_relaxation_2012}, and quench-dynamics of spin-model systems \\cite{bernien_probing_2017}, to name just a few.\\\\ \nAt the same time, the theoretical understanding of non-equilibrium quantum many-body systems still lags behind that obtained for stationary or equilibrium systems. In particular, the role of ``classical'' chaos on non-equilibrium quantum many-body systems is currently subject of intense scrutiny, see e.g.~\\cite{ho_periodic_2019, hallam_lyapunov_2019, lewis-swan_unifying_2019, xu_does_2020}. For a bosonic quantum many-body system, the mean-field limit can be viewed as the classical limit in which the particle creation and annihilation operators lose their quantum properties and start to act as classical fields. The mean-field limit, typically involves non-linear (partial) differential equations, and can exhibit chaos with exponential separation in Hilbert space characterized by a positive Lyapunov exponent $\\lambda$ \\cite{brezinova_wave_2011, cassidy_threshold_2009}. One of the open questions is, whether wave chaos, more specifically the positive $\\lambda$, imprints itself onto the dynamics of a quantum many-body system even at finite particle numbers and how such an imprint could be measured. Recently, out-of-time-order correlators have been suggested as suitable probes for a positive $\\lambda$ (see e.g.~\\cite{Kitaev2014,sekino_fast_2008,shenker_black_2014,maldacena_bound_2016,lewis-swan_unifying_2019, xu_does_2020}). \\\\\nIn this paper, we find an imprint of chaos on a different observable within a paradigmatic bosonic system: A quasi 1D BEC initially trapped harmonically and then released to expand in a shallow disordered or periodic potential. We show that the fraction of non-condensed particles increases exponentially over time and that the associated rate is given by the Lyapunov exponent $\\lambda$ obtained from mean-field chaos. The depletion occurs on time scales during which most of the initial interaction energy is converted into kinetic energy, and comes to a halt at times close to the so-called scrambling time (or Ehrenfest time) \\cite{rammensee_many-body_2018, tomsovic_post-ehrenfest_2018, maldacena_bound_2016}. We observe chaos-induced depletion both in shallow disordered as well as periodic potentials showing that the effect is quite general and does not rely on the intrinsic randomness of a disordered landscape.\\\\\nFinally, we demonstrate that the condensate depletion and thus the Lyapunov exponent $\\lambda$ is accessible experimentally through the analysis of fluctuations of the total particle density in momentum space. While the condensed part is coherent and leads to interference fringes in the total density, the non-condensed part is incoherent and piles up over time as a non-fluctuating background. Analyzing the interference fringes after time of flight would thus allow to extract experimentally the fraction of non-condensed particles as a function of time and compare to the theoretically obtained $\\lambda$. Condensate depletion thus offers itself as an experimentally accessible probe to investigate the role of (mean-field or classical) chaos in non-equilibrium quantum matter.\\\\\nWhile our findings are generally applicable to bosonic systems that exhibit mean-field chaos, we pick one specific system already realized experimentally \\cite{billy_direct_2008} to obtain numerical results. The initially harmonically trapped quasi 1D BEC of $N$ $^{87}$Rb atoms is released at $t=0$ to expand in a shallow potential. As units we use $\\hbar=m=\\omega_0=1$, with $\\omega_0$ being the frequency of the initial longitudinal harmonic trap which amounts to a time unit of $t_0\\approx30$ms and a space unit of $x_0\\approx4.6\\mu$m. For the number of atoms, we take $N=1.2\\times 10^4$ following \\cite{billy_direct_2008}, as well as larger values, i.e.~$N=1.2\\times10^5$ and $N=1.2\\times10^6$ to investigate the effect of varying $N$.\\\\\nDescribing the quasi-1D system on a mean-field level the Gross-Pitaevskii equation (GPE) takes the form\n\\begin{equation}\ni\\frac{\\partial \\psi(x,t)}{\\partial t} = \n\\left(-\\frac{1}{2}\\frac{\\partial^2}{\\partial x^2}\n+V(x)+g|\\psi(x,t)|^2\\right)\\psi(x,t),\n\\label{eq:gpe}\n\\end{equation}\nwhere the nonlinearity is $g\\approx400$ with the above parameters and the normalization $\\int dx|\\psi(x,t)|^2=1$. The potential $V(x)$ corresponds to the harmonic potential at $t=0$, and to the periodic or disordered speckle potential at $t>0$ with amplitude much smaller than the mean energy per particle $e$. This system exhibits chaos on the mean-field level \\cite{brezinova_wave_2011}: Two wave functions, $\\psi_a(x,0)$ and $\\psi_{b}(x,0)$, respectively, initially very close to each other in Hilbert space as measured by a distance norm, separate exponentially in time until quasi-orthogonality is reached, see Fig.~\\ref{fig:N6}.\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{N6_grid_bog_comp_log.pdf}\n\t\\caption{$d^{(2)}$ (Eq.~\\ref{eq:d2}) for two initial conditions obtained by linear distortion of the mean-field ground state (black dashed-dotted line), distance function $\\bar d^{(2)}$ (Eq.~\\ref{eq:bar_d2}) averaged over the stochastic ensemble (triangles), and fraction of incoherent particles $n_\\text{incoh}$ (squares). Data shown at integer values of $t$ (except for $d^{(2)}$), the lines serve as guides for the eye. The red dashed lines mark an exponential increase with $\\lambda=1.44t_0^{-1}$. Number of particles is $N=1.2\\times 10^6$, periodic potential $V(x)=V_P\\cos{(k_Px)}$ used with $V_P=0.3e$, $k_P=\\pi\/3\\xi$ and $\\xi$ the healing length.}\n\t\\label{fig:N6}\n\\end{figure}\nThe rate of the exponential growth is given by the Lyapunov exponent $\\lambda$. For the distance norm we take\n\\begin{align}\nd^{(2)}_{a,b}(t) &= \\frac{1}{2}\\int dx |\\psi_a(x,t)-\\psi_{b}(x,t)|^2 .\n\\label{eq:d2}\n\\end{align}\nThe Lyapunov exponent $\\lambda$ shows systematic trends as a function of the parameters of the system: It vanishes in absence of inter-particle interactions for arbitrary potentials, as well as in presence of inter-particle interactions in free space (i.e.~without any potential). At fixed period of the periodic potential $k_P$, or fixed correlation length $\\sigma$ of the speckle potential, it increases both with nonlinearity and the potential amplitude \\cite{brezinova_wave_2011}, see the supplemental material (SM).\\\\\nTo find imprints of chaos on measurable observables of the quantum many-body system with finite $N$ a theory beyond mean-field has to be applied. The multi-configurational time-dependent Hartree method for bosons (see, e.g.~\\cite{alon_multiconfigurational_2008,lode_colloquium_2020}), while being in principle exact for a sufficient number of orbitals, suffers from the exponentially growing configuration space. For the particle numbers considered, only two orbitals can be afforded numerically \\cite{brezinova_wave_2012}. As more than two orbitals are populated during the propagation, the MCTHB method entails a large and not easy to quantify error. We, therefore, resort to the truncated Wigner approximation (TWA), see e.g.~\\cite{Steel1998,Sinatra2002,blakie_dynamics_2008,dujardin_breakdown_2016} which employs the Wigner representation $W$ for (in general) a many-body density matrix $\\hat \\rho$\n\\begin{align}\nW(&\\psi_1,\\ldots,\\psi_M,\\psi_1^* \\ldots,\\psi_M^*) = \\frac{1}{\\pi^{2M}}\\nonumber \\\\\n&\\times \\int dz^{2M}\\Tr\\left[\\hat \\rho e^{i\\sum_j\\left( z_j^*\\hat\\psi_j^\\dagger\n+iz_j\\hat{\\psi_j}\\right)}\n\\right]\ne^{-i\\sum_j\\left(z_j^*\\psi_j^*-iz_j\\psi_j\\right)}.\n\\end{align}\n$W$ can be viewed as a phase-space representation of the quantum many-body state. $M$ is the total number of modes in which particles can be created or annihilated and $\\hat \\psi_j^\\dagger$ and $\\hat \\psi_j$ are the corresponding creation and annihilation operators, respectively. In general, particles can be created or annihilated in an arbitrary single particle mode denoted by $j$. We choose $j$ to represent a specific point in space assuming for simplicity an equidistant spatial discretization. We have made sure, however, that the spatial grid is fine enough, i.e.~the distance between grid points $dx<1\/k_\\text{max}$ with $k_\\text{max}$ being the largest relevant momentum in the system, such that we are still in the continuum limit.\\\\ \nHaving $W$ as a function of time at disposal would allow to evaluate all expectation values of symmetrized products of creation and annihilation operators. The exact equation of motion for $W$ can be obtained using von Neumann's equation of motion for $\\hat\\rho$. However, it proves to be intractable, such that approximations have to be invoked. Within the TWA \\cite{Steel1998, Sinatra2002, blakie_dynamics_2008}, the \ntime evolution of $W$ is sampled stochastically with an ensemble of trajectories obeying the GPE, Eq.~\\ref{eq:gpe}. (The only modification comes from the fact that we have to discretize space such that the second derivative in Eq.~\\ref{eq:gpe} has to be replaced by its second-order finite difference approximation.) It has been shown \\cite{schlagheck_enhancement_2019, tomsovic_post-ehrenfest_2018, dujardin_describing_2015} that this approximation amounts to neglecting non-classical trajectories as well as interferences between distinct trajectories in many-body Hilbert space. The question then arises at which times do these neglected effects start to play a role and become non-negligible. For single- or few-particle systems, sampling the time evolution with classical trajectories is accurate up to the point where an initially maximally localized state has spread over the whole system. This time is called the Ehrenfest time $\\tau_E$ \\cite{ehrenfest_bemerkung_1927} which is, in presence of classical chaos, inversely proportional to $\\lambda$ and grows logarithmically with $1\/\\hbar$. This concept can be extended into the many-body regime for bosonic systems with $\\hbar$ being replaced by the effective Planck constant $\\hbar_\\text{eff}\\simeq 1\/N$. Following the lines of \\cite{rammensee_many-body_2018, tomsovic_post-ehrenfest_2018} we thus assume that our results are accurate up to the time $\\tau_E=\\frac{1}{\\lambda}\\log{N}$.\\\\\nThe initial conditions within the stochastic ensemble of trajectories are constructed such as to correctly sample the phase-space distribution of the underlying initial quantum state, which in our case is a BEC at zero temperature. The stochasticity of the ensemble comes solely from the sampling of this initial state since Eq.~\\ref{eq:gpe} is completely deterministic. We follow \\cite{Sinatra2002, blakie_dynamics_2008, Steel1998} and construct the initial wave functions by adding to the mean-field ground state in the harmonic trap vacuum fluctuations in form of Gaussian noise (see the SM).\\\\\nThe most relevant observables in our case will be the coherent part of the particle density given by $\\rho_\\text{coh}(x_j,t) = |\\langle \\hat \\psi_j(t) \\rangle|^2$, as well as the one-particle reduced density matrix (1RDM) $D_{ij}(t)=\\langle \\hat\\psi_i^\\dagger(t)\\hat\\psi_j(t)\\rangle$. The term ``coherent\" in defining $\\rho_\\text{coh}(x_j,t)$ points to the fact that only a macroscopically occupied state with a spatially non-random phase will survive the averaging. $\\rho_\\text{coh}(x_j,t)$ can therefore be associated with the density of condensed particles. Alternatively \\cite{penrose_bose-einstein_1956,leggett_bose-einstein_2001}, the condensate state is defined through a macroscopic occupation of one eigenstate of the 1RDM. We show in the SM that these two definitions of the condensate give practically identical results for the depletion over time such that we use throughout the remainder of the paper the term coherent synonymously to condensed.\\\\\nWithin the stochastic ensemble of trajectories, expectation values can be calculated as\n$\\langle \\hat\\psi_j(t)\\rangle = \\frac{1}{N_s}\\sum_{s=1}^{N_s}\\psi_s(x_j,t)\n$,\nwith $N_s$ ($\\gg1$) being the number of Gross-Pitaevskii trajectories $\\psi_{s}(x_j,t)$ within the ensemble. To calculate the 1RDM, one has to rewrite $\\langle \\hat\\psi_i^\\dagger\\hat\\psi_j + \\hat \\psi_j\\hat\\psi_i^\\dagger\\rangle = \n2\\langle \\hat\\psi_i^\\dagger\\hat\\psi_j \\rangle + \\frac{1}{Ndx}\\delta_{ij}$ using the commutator relation $[\\hat\\psi_j,\\hat\\psi_i^\\dagger] = \\frac{1}{Ndx}\\delta_{ij}$.\nThe term $\\delta_{ij}\/dx$ is the discrete version of the $\\delta$-function for a continuous system, and \nthe factor $1\/N$ comes from our normalization of the wave functions of Eq.~\\ref{eq:gpe} to one, or equivalently, the creation and annihilation operators to $1\/N$.\nThe 1RDM is then given by\n$D_{ij}(t) = \\frac{1}{N_s}\\sum_{s=1}^{N_s}\\psi^{*}_s(x_i,t)\\psi_s(x_j,t) - \\frac{1}{2Ndx}\\delta_{ij}\n$,\nand the total particle density is $\\rho_\\text{total}(x_j,t) = D_{jj}(t)$. The fraction of coherent particles is determined by \n$n_\\text{coh}(t)=\\sum_j dx\\rho_\\text{coh}(x_j,t)$. Accordingly, the fraction of incoherent particles is $n_\\text{incoh}(t) = 1-n_\\text{coh}(t)$. The crucial observation now is that\n\\begin{align}\nn_\\text{incoh}(t) = \\frac{1}{N_s^2}\\sum_{s,r}d^{(2)}_{s,r}(t)\n-\\frac{L}{2Ndx} = \\bar d^{(2)}(t),\n\\label{eq:bar_d2}\n\\end{align}\nwhich we obtain using Eq.~\\ref{eq:d2}, taking into account that the norm of the wave functions within the ensemble is $1\/N_s\\sum_s\\sum_j dx |\\psi_s(x_j)|^2 = 1 + L\/2Ndx$ with $L$ the length of the system. (Note that the term $L\/dx$ counts the number of single-particle modes to which vacuum fluctuations have been added.) For a detailed derivation, see the SM. The right-hand side of Eq.~\\ref{eq:bar_d2} is (apart from a constant term) the arithmetic mean over the distance function between all pairs of mean-field trajectories, and we denote it with $\\bar d^{(2)}(t)$.\\\\\nWe have explicitly verified the equality of Eq.~\\ref{eq:bar_d2} numerically by independently calculating the arithmetic mean of the distance function, $\\bar d^{(2)}(t)$, and comparing it to $n_\\text{incoh}(t)$, see Fig.~\\ref{fig:N6}.\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{N_number_comp_combined.pdf}\n\t\\caption{Left column (a) and (b) for the periodic potential with $V_P=0.3e$ and $k_P=\\pi\/3\\xi$: (a) Fraction of incoherent particles $n_\\text{incoh}$ for different particle numbers $N$ ($N$ being $1.2$ times the number near each curve), and $d^{(2)}$ for the linearly distorted initial conditions. Red dashed lines correspond to an exponential increase with $\\lambda=1.44 t_0^{-1}$. (b) Linear plot of (a) including $n^\\text{total}_\\text{low-env}$ extracted from the total density only, see Fig.~\\ref{fig:incoh_den}. The Ehrenfest time $\\tau_E=1\/\\lambda \\ln{N}$ is marked for each curve. Right column (c) and (d) same as left column but for one realization of speckle disorder with $V_D=0.3e$ and correlation length $\\sigma =0.57\\xi$. In (c) the Lyapunov exponent is $\\lambda =1.43 t_0^{-1}$. Data is shown at integer values of $t$ (except for $d^{(2)}$), the lines serve as guides for the eye.}\n\t\\label{fig:N_numb}\n\\end{figure}\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{incoherent_density_k_2.pdf}\n\t\\caption{Total density $\\tilde\\rho_\\text{total}(k,t)$ (solid), as well as coherent $\\tilde\\rho_\\text{coh}(k,t)$ (filled), and incoherent part $\\tilde\\rho_\\text{incoh}(k,t)$ (dashed) for $N=1.2\\times10^6$ at (a) $t=5t_0$ and (b) $t=9t_0$. The orange dots mark the lower envelope of the strong fluctuations of $\\tilde\\rho_\\text{total}(k,t)$, and can be used as an accurate estimate for $\\tilde\\rho_\\text{incoh}(k,t)$. Same potential as in Fig.~\\ref{fig:N6}.}\n\t\\label{fig:incoh_den}\n\\end{figure}\nThe observed rate of exponential growth does not depend on the specific choice of the two close initial conditions such that we can clearly associate it with a Lyapunov exponent $\\lambda$. The equality between $\\bar d^{(2)}(t)$ and $n_\\text{incoh}(t)$ proves that, if the mean-field limit is chaotic, the fraction of incoherent particles will grow exponentially with a rate given exactly by the mean-field $\\lambda$. The exponentially fast depletion is thus chaos-induced, or seen from another perspective, measures mean-field chaos. Importantly, the exponential increase happens on shorter time scales than $\\tau_E$, i.e.~before effects neglected within the TWA start to play a role.\\\\\nWhile Fig.~\\ref{fig:N6} depicts the exponential increase for $N=1.2\\times10^6$ particles, we see the same exponential increase, i.e.~the same $\\lambda$, also for smaller particle numbers, see Fig.~\\ref{fig:N_numb}. We have varied $N$ while keeping the nonlinearity $g\\propto a_sN$ constant, which amounts to increasing the scattering length $a_s$ by the same factor $N$ is decreased, which preserves the classical phase space. Indeed, with decreasing $N$ and increasing $a_s$, the BEC naturally shows larger initial depletion, but upon expansion in the periodic potential, the same $\\lambda$ emerges.\\\\ \nWe now turn to the question of how the present chaos-induced depletion could be observed in an experiment. We analyze the total particle density in momentum space $\\tilde\\rho_\\text{total}(k,t)$ which is accessible in experiments through time-of-flight measurements, see e.g.~\\cite{gericke_high-resolution_2008, erne_universal_2018}. During the expansion of the BEC, matter waves start to scatter at the potential landscape preserving initially their phase coherence. This scattering creates fluctuations in momentum space with increasingly higher frequencies as waves originating from points increasingly farther apart in real space coherently interfere. Ultimately, the density exhibits strong fluctuations reaching down to almost zero density, provided that inelastic scattering has been negligible up until this point in time, Fig.~\\ref{fig:incoh_den} (a). During inelastic scattering particles lose energy, phase information, and with it, the ability to create interference fringes. These particles constitute the incoherent part of the density which piles up in form of an almost non-fluctuating background. Using a simple algorithm that determines the lower envelope of the fluctuations in the total density, we obtain a functional form very close to $\\tilde\\rho_\\text{incoh}(k,t)$, see Fig.~\\ref{fig:incoh_den} (b). Interpolating between the points of the lower envelope and integrating, we obtain $n^\\text{total}_\\text{low-env}$, which follows $n_\\text{incoh}$ closely, see Fig.~\\ref{fig:N_numb} (b). We emphasize that $n^\\text{total}_\\text{low-env}$ is extracted from the total density only. From Fig.~\\ref{fig:N_numb} (b) it is obvious that the extraction mechanism will work best for high particle numbers with a small scattering length (e.g., for $N=1.2\\times 10^6$ two orders of magnitude of exponential growth can be resolved). For smaller $N$ and correspondingly larger scattering lengths $a_s$ the incoherent density starts to pile up before coherent scattering produces sufficiently strong fluctuations in the coherent part of the density. Therefore, the close association of a non-fluctuating density with $\\tilde\\rho_\\text{incoh}(k,t)$ is broken initially. It becomes, however, more and more accurate over time such that, in the experiment, one could observe the behavior of $\\tilde\\rho_\\text{incoh}(k,t)$ also beyond $\\tau_E$, where interferences of many-body trajectories not included within the TWA become relevant.\\\\\nIn order to measure the incoherent fraction of the total density in an experiment it is pivotal to resolve the deep minima of the fluctuations. Half of the distance between two minima is $\\Delta k\\gtrsim 0.04 x_0^{-1}$. Assuming a linear pixel size of a CCD camera of $2\\mu$m the fluctuations could be resolved after about $300$ms time of flight. The peak amplitude of the fluctuations is $\\tilde\\rho_\\text{total}(k)\\gtrsim0.03 x_0$ leading to $\\tilde\\rho_\\text{total}(k)\\Delta k = 1.2\\times10^{-3}$ such that the number of particles within each hump is greater than $100$ for $N=1.2\\times10^5$ and $N=1.2\\times10^6$. Despite the large time of flight necessary, we believe that the here proposed extraction could be realized in state-of-the-art BEC experiments.\\\\\nFor the disorder potential, we mostly see the same behavior as for the periodic potential, see Fig.~\\ref{fig:N_numb} (c) and (d): \n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{incoherent_density_k_disorder.pdf}\n\t\\caption{Particle density in momentum space at $t=8t_0$ for the speckle disorder with $V_D=0.3e$ and $\\sigma =0.57\\xi$ averaged over $10$ realizations (additional smoothing of the curves has been applied). The vertical lines mark the Landau velocity approximated by $k_L=\\sqrt{\\mu(t)}$ with $ \\mu(t)$ the chemical potential at time $t$.}\n\t\\label{fig:incoh_den_dis}\n\\end{figure}\n$n_\\text{incoh}$ grows exponentially with $\\lambda$ independent of the particle number $N$. Note that we did not perform any averages over disorder realizations here. As to the extraction of the incoherent part of the density from $\\tilde \\rho_\\text{total}(k,t)$ there is one point worth mentioning. Due to the broad spectrum of frequencies the speckle disorder offers, within few time steps, slow particles start to be scattered coherently and intertwine with particles that have lost their coherence through inelastic (i.e.~incoherent) scattering near $k=0$. The result is a local maximum in $\\tilde \\rho_\\text{total}(k,t)$ near $k=0$, and local minima near the Landau velocity $\\pm k_L$ due to inelastic scattering out of this momentum, see Fig.~\\ref{fig:incoh_den_dis}. Since, however, slow particles scatter from positions in space close to each other, this scattering produces fluctuations with low frequencies as compared to the fluctuations observed for larger $k$. It is, therefore, impossible to identify $\\tilde \\rho_\\text{incoh}(k,t)$ near $k=0$ based on the fluctuations of the total density, initially. At later times the local maximum near $k=0$ consists of incoherent particles only such that $n^\\text{total}_\\text{low-env}$ again accurately predicts the value of $n_\\text{incoh}$, see Fig.~\\ref{fig:N_numb}. For $N=1.2\\times10^6$ the agreement between $n^\\text{total}_\\text{low-env}$ and $n_\\text{incoh}$ is accurate only after $t\\gtrsim12t_0$ such that we refrained from plotting it.\\\\\nIn conclusion, we have shown that a BEC expanding in a shallow periodic or disordered potential is subject to an exponentially growing depletion, and that the depletion is characterized by the ``classical\" (mean-field) Lyapunov exponent $\\lambda$. We have thus found a new observable that allows to identify the finger-print of classical chaos on the non-equilibrium many-body dynamics of a quantum system with a finite number of particles. In addition, we have shown how our results could be measured in an experiment by analyzing the visibility of the fluctuations of the particle density after time of flight. This opens up the possibility to verify our predictions experimentally for a real many-body system.\\\\\n\nWe thank Joachim Burgd\u00f6rfer, David Gu\u00e9ry-Odelin, Dana Orsolits, Thorsten Schumm, and Juan-Diego Urbina for helpful discussions. This work has been supported by the WWTF grant MA14-002. S. D. acknowledges support by the International Max Plank Research School of Advanced Photon Science (IMPRS-APS). Calculations\nwere performed on the Vienna Scientific Cluster (VSC3). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\nMost of efficient cetacean localisation systems are based on the Time Delay Of Arrival (TDOA) estimation from detected\\footnote{As click\/whistles detector, matching filter is often prefered} animal's click\/whistles signals \\cite{nosalGT,fredericbenard_herveglotin_2009}. Long-base hydrophones'array is involving several fixed, efficient but expensive hydrophones \\cite{3Dtracking} while short-base version is requiring a precise array's self-localization to deliver accurate results. Recently (see \\cite{IFA_DCL}), based on Leroy's attenuation model versus frequencies \\cite{Leroy}, a range estimator have been proposed. This approach is working on the detected most powerful pulse inside the click signal and is delivering a rough range' estimate robust to head orientation variation of the animal. Our purpose is to use i) these hydrophone' array measurements recorded in diversified sea conditions and ii) the associated ground-truth trajectories of spermwhale (obtained by precise TDAO and\/or Dtag systems) to regress both position and azimuth of the animal from a third-party hydrophone\\footnote{We assume that the velocity vector is colinear with the head's angle.} (typically onboard, standalone and cheap model).\n\nWe claim, as in computer-vision field, that BoF approach can be successfully applied to extract a global and invariant representation of click's signals. Basically, the pipeline of BoF approach is composed of three parts: i) a local features extractor, ii) a local feature encoder (given a dictionary pre-trained on data) and iii) a pooler aggregating local representations into a more robust global one. Several choice for encoding local patches have been developed in recent years: from hard-assignment to the closest dictionary basis (trained for example by $K$means algorithm) to a sparse local patch reconstruction (involving for example Orthognal Maching Pursuit (OMP) or LASSO algorithms).\n\n\\section{Global feature extraction by spare coding}\n\n\n\\subsection{Local patch extraction}\n\nLet's denote by $\\g{C}\\triangleq\\{\\g{C}^j\\}$, $j=1,\\ldots,H$ the collection of detected clicks associated with the $j^{th}$ hydrophone of the array composed by $H$ hydrophones. Each matrix $\\g{C}^j$ is defined by $\\g{C}^j\\triangleq\\{\\g{c}_i^j\\}$, $i=1,\\ldots,N^j$ where $\\g{c}_i^j\\in\\mathds{R}^n$ is the $i^{th}$ click of the $j^{th}$ hydrophone. For our \\textit{Bahamas2} dataset \\cite{3Dtracking}, we choose typically $n=2000$ samples surrounding the detected click. The total number of available clicks is equal to $N=\\sum\\limits_{i=1}^{H}N^j$.\n\nAs local features, we extract simply some local signal patches of $p\\leq n$ samples (typically $p=128$) and denoted by $\\g{z}_{i,l}^j\\in\\mathbb{R}^p$. Furthermore all $\\g{z}_{i,l}^j$ are $\\ell_2$ normalized. For each $\\g{c}_{i}^j$, a total of $L$ local patches $\\g{Z}_{i}^j\\triangleq\\{\\g{z}_{i,l}^j\\}$, $l=1,\\ldots,L$ equally spaced of $\\lceil\\frac{n}{L}\\rceil$ samples are retrieved (see Fig.~\\ref{patch_extraction}). All local patches associated with the $j^{th}$ hydrophone is denoted by $\\g{Z}^{j}\\triangleq\\{\\g{Z}_i^{j}\\}$, $i=1,\\ldots,N^j$ while $\\g{Z}\\triangleq\\{\\g{Z}^j\\}$ is denoting all the local patches matrix for all hydrophones. A final post-processing consists in uncorrelate local features by PCA training and projection with $p'\\leq p$ dimensions.\n\n\\begin{figure*}[!ht]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[height=5cm,width=7.5cm]{ex_click.pdf} &\n\\includegraphics[height=5cm,width=7.5cm]{local_features.pdf}\n\\end{tabular}\n\\caption{Left: Example of detected click with $n=2000$. Right: extracted local features with $p=128$, $L=1000$ (one local feature per column).}\n\\label{patch_extraction}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Local feature encoding by sparse coding}\n\nIn order to obtain a global robust representation of $\\g{c}\\subset\\g{C}$, each associated local patch $\\g{z}\\subset\\g{Z}$ are first linearly encoded \\textit{via} the vector $\\g{\\alpha}\\in\\mathbb{R}^k$ such as $\\g{z}\\approx\\g{D}\\g{\\alpha}$ where $\\g{D}\\triangleq [\\g{d}_1,\\ldots,\\g{d}_k]\\in\\mathbb{R}^{p\\times k}$ is a pre-trained dictionary matrix whose column vectors respect the constraint $\\g{d}_j^T\\g{d}_j=1$. In a first attempt to solve this linear problem, $\\g{\\alpha}$ can be the solution of the Ordinary Least Square (OLS) problem:\n\\begin{equation}\\label{1}\nl_{OLS}(\\g{\\alpha}|\\g{z};\\g{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k}\\left\\lbrace\\frac{1}{2}\\Vert \\g{z} - \\g{D}\\g{\\alpha}\\Vert_2^2 \\right\\rbrace.\n\\end{equation}\nOLS formulation can be extended to include regularization term avoiding overfitting. We obtain the ridge regression (RID) formulation:\n\\begin{equation}\nl_{RID}(\\g{\\alpha}|\\mathbf{z};\\mathbf{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k}\\left\\lbrace\\frac{1}{2}\\Vert \\g{z} - \\g{D}\\g{\\alpha}\\Vert_2^2 + \\beta \\Vert \\g{\\alpha} \\Vert_2^2 \\right\\rbrace.\n\\end{equation}\nThis problem have an analytic  solution $\\g{\\alpha} = (\\g{D}^T\\g{D} + \\beta\\g{I}_k)^{-1}\\g{D}^T\\g{z}$. Thanks to semi-positivity of $\\g{D}^T\\g{D} + \\beta\\g{I}_k$, we can use a cholesky factor on this matrix to solve efficiently this linear system. In order to decrease reconstruction error and to have a sparse solution, this problem can be reformuled as a constrained Quadratic Problem (QP):\n\\begin{equation}\nl_{SC}(\\g{\\alpha}|\\g{z};\\g{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k} \\frac{1}{2}\\Vert \\g{z} - \\g{D} \\g{\\alpha} \\Vert_2^2 \\ s.t. \\ \\ \\Vert\\boldsymbol{\\alpha}\\Vert_1 = 1.\n\\end{equation}\nTo solve this problem, we can use a QP solver involving high combinatorial computation to find the solution. Under RIP assumptions \\cite{Tibshirani94regressionshrinkage}, a greedy approach can be used efficiently to solve and eq. 3 and this latter can be rewritten as:\n\\begin{equation}\\label{2}\nl_{SC}(\\g{\\alpha}|\\g{z};\\g{D}) \\triangleq \\min_{\\g{\\alpha} \\in \\mathbb{R}^k} \\frac{1}{2}\\Vert \\g{z} - \\g{D} \\g{\\alpha} \\Vert_2^2 + \\lambda\\Vert\\g{\\alpha}\\Vert_1,\n\\end{equation}\nwhere $\\lambda$ is a regularization parameter which controls the level of sparsity. This problem is also known as basis pursuit \\cite{Chen98atomicdecomposition} or the Lasso \\cite{Tibshirani94regressionshrinkage}. To solve this problem, we can use the popular Least angle regression (LARS) algorithm.\n\n\\subsection{Pooling local codes}\n\nThe objective of pooling \\cite{Boureau10atheoretical,Feng11} is to transform the joint feature representation into a new, more usable one that preserves important information while discarding irrelevant detail.\nFor each click signal, we usually compute $L$ codes denoted $\\g{V} \\triangleq \\left\\lbrace\\g{\\alpha}_i\\right\\rbrace$, $i = 1,\\ldots,L$.\nLet define $\\g{v}^{j}\\in\\mathbb{R}^L$, $j=1,\\ldots,k$ as the $j^{th}$ row vector of $\\g{V}$. It is essential to use feature pooling to map the response vector $\\g{v}^{j}$ into a statistic value $f(\\g{v}^{j})$ from some spatial pooling operation $f$. We use $\\g{v}^{j}$, the response vector, to summarize the joint distribution of the $j^{th}$ compounds of local features over the region of interest (ROI). We will consider the $\\ell_{\\mu}$-norm pooling and defined by:\n\\begin{equation}\nf_n(\\g{v};\\mu) = \\left(\\sum_{m=1}^L |v_m|^{\\mu}\\right)^{\\frac{1}{\\mu}} \\ \\ s.t. \\ \\mu \\neq 0.\n\\end{equation}\nThe parameter $\\mu$ determines the selection policy for locations. When $\\mu = 1$, $\\ell_{\\mu}$-norm pooling is equivalent to sum-pooling and aggregates the responses over the entire region uniformly. When $\\mu$ increases, $\\ell_{\\mu}$-norm pooling approaches max-pooling. We can note the value of $\\mu$ tunes the pooling operation to transit from sum-pooling to max-pooling.\n\n\n\\subsection{Pooling codes over a temporal pyramid}\nIn computer vision, Spatial Pyramid Matching (SPM) is a technic (introduced by \\cite{Lazebnik2006}) which improves classification accuracy by performing a more robust local analysis. We will adopt the same strategy in order to pool sparse codes over a temporal pyramid (TP) dividing each click signal into ROI of different sizes and locations. Our TP is defined by the matrix $\\g{\\Lambda}$ of size $(P \\times 3)$ \\cite{sebastienparis_xanaduhalkias_herveglotin_2013}:\n\\begin{equation}\n\\g{\\Lambda} = [\\g{a}, \\g{b}, \\g{\\Omega}],\n\\end{equation}\nwhere $\\g{a}$, $\\g{b}$, $\\g{\\Omega}$ are 3 $(P \\times 1)$ vectors representing subdivision ratio, overlapping ratio and weights respectively. $P$ designs the number of layers in the pyramid. Each row of $\\g{\\Lambda}$ represents a temporal layer of the pyramid, \\textit{i.e.} indicates how do divide the entire signal into sub-regions possibly overlapping. For the $i^{th}$ layer, the click signal is divided into $D_i=\\lfloor\\frac{1-a_i}{b_i}+1\\rfloor$ ROIs where $a_i$, $b_i$ are the $i^{th}$ elements of vector $\\g{a}$, $\\g{b}$ respectively. For the entiere TP, we obtain a total of $D=\\sum\\limits_{i=1}^{P}D_i$ ROIs. Each click signal $\\g{c}$ $(n \\times 1)$ is divided into temporal ROI $\\g{R}_{i,j}$, $i=1,\\ldots,P$, $j=1,\\ldots,D_i$ of size $(\\lfloor a_i.n\\rfloor \\times 1)$. All ROIs of the $i^{th}$ layer have the same weight $\\Omega_i$. For the $i^{th}$ layer, ROIs are shifted by $\\lfloor b_i.n\\rfloor$ samples. A TP with $\\g{\\Lambda} = \\left[\\begin{array}{ccc}\n1 & 1 & 1  \\\\ \\frac{1}{2} & \\frac{1}{4} & 1  \\end{array}\\right]$ is designing a 2-layers pyramid with $D=1+4$ ROIs, the entiere signal for the first layer and $4$ half-windows of $\\frac{n}{2}$ samples with $25\\%$ of overlapping for the second layer.\nAt the end of pooling stage over $\\g{\\Lambda}$, the global feature $\\g{x}\\in\\mathbb{R}^d$, $d=D.k$ is defined by the weighted concatenation (by factor $\\Omega_i)$ of $L$ pooled codes associated with $\\g{c}$.\n\n\n\n\\subsection{Dictionary learning}\n\nTo encode each local features by sparse coding (see eq.~\\ref{2}), a dictionary $\\g{D}$ is trained offline with an important collection of $M\\leq N.L$ local features as input. One would minimize the regularized empirical risk $\\mathcal{R}_M$:\n\\begin{equation}\n\\begin{array}{c}\n\\mathcal{R}_M(\\g{V},\\g{D}) \\triangleq \\displaystyle\\frac{1}{M}\\sum\\limits_{i=1}^M \\frac{1}{2}\\Vert \\g{z}_i - \\g{D} \\g{\\alpha}_i \\Vert_2^2 + \\lambda\\Vert\\g{\\alpha}_i\\Vert_1\n\\\\\n\\\\\n\\ s.t. \\ \\g{d}_j^T\\g{d}_j=1.\n\\end{array}\n\\end{equation}\nUnfortunatly, this problem is not jointly convex but can be optimized by alternating method:\n\\begin{equation}\n\\mathcal{R}_M(\\g{V}|\\g{\\hat{D}}) \\triangleq \\frac{1}{M}\\sum\\limits_{i=1}^M \\frac{1}{2}\\Vert \\g{z}_i - \\g{\\hat{D}} \\g{\\alpha}_i \\Vert_2^2 + \\lambda\\Vert\\g{\\alpha}_i\\Vert_1,\n\\end{equation}\nwhich can be solved in parallel by LASSO\/LARS and then:\n\\begin{equation}\n\\mathcal{R}_M(\\g{D}|\\g{\\hat{V}}) \\triangleq \\frac{1}{M}\\sum\\limits_{i=1}^M \\frac{1}{2}\\Vert \\g{z}_i - \\g{D} \\g{\\hat{\\alpha}}_i \\Vert_2^2 \\ \\ s.t. \\ \\g{d}_j^T\\g{d}_j=1.\\label{eq_dico}\n\\end{equation}\nEq.~\\ref{eq_dico} have an analytic solution involving a large matrix $(k\\times k)$ inversion and a large memory occupation for storing the matrix $\\g{V}$ $(k\\times M)$. Since $M$ is potentially very large (up to 1 million), an online method to update dictionary learning is prefered \\cite{Mairal_2009}. Figure \\ref{baha_click_range_dico} depicts 3 dictionary basis vectors learned \\textit{via} sparse coding. As depicted, some elements reprensents more impulsive responses while some more harmonic responses.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_range_dico.pdf}\n\\caption{Example of trained dictionary basis with sparse coding.}\n\\label{baha_click_range_dico}\n\\end{center}\n\\end{figure}\n\n\n\\section{Range and azimuth logistic regression from global features}\n\nAfter the pooling stage, we extracted unsupervisly $N$ global features $\\g{X}\\triangleq\\{x_i\\}\\in\\mathbb{R}^{d\\times N}$. We propose to regress \\textit{via} logistic regression both range $r$ and azimuth $az$ (in $x-y$ plan, when animal reach surface to breath) from the animal trajectory groundtruth denoted $\\g{y}$. For the current train\/test splitsets of the data, such as $\\g{X}=\\g{X}_{train}\\bigcup\\g{X}_{test}$, $\\g{y}=\\g{y}_{train}\\bigcup\\g{y}_{test}$ and $N=N_{train}+N_{test}$, $\\forall$ $\\{\\g{x}_i,y_i\\}\\in\\g{X}_{train}\\times \\g{y}_{train}$, we minimize:\n\\begin{equation}\n\\widehat{\\g{w}}_{\\theta}=\\arg\\min\\limits_{\\g{w}_{\\theta}}\\left\\{\\frac{1}{2}\\g{w}_{\\theta}^T\\g{w}_{\\theta} + C\\sum\\limits_{i=1}^{N_{train}}\\log(1+e^{-y_i\\g{w}_{\\theta}^T\\g{x}_i})\\right\\},\\label{logistic_regression}\n\\end{equation}\nwhere $y_i$ denotes $r_i$ and $az_i$ for $\\theta=r$ and $\\theta=az$ respectively. Eq.~\\ref{logistic_regression} can be efficiently solved for example with Liblinear software \\cite{liblinear2008}. In the test part, range and azimuth for any $\\g{x}_i\\in\\g{X}_{test}$ are recontructed linearly by $\\widehat{r}_i=\\widehat{\\g{w}}_r^T\\g{x}_i$ and by $\\widehat{az}_i=\\widehat{\\g{w}}_{az}^T\\g{x}_i$ respectively.\n\n\\section{Experimental results}\n\n\\subsection{bahamas2 dataset}\n\nThis dataset \\cite{3Dtracking} contains a total of $N=6134$ detected clicks for $H=5$ different hydrophones (named $H^7$, $H^8$, $H^9$, $H^{10}$ and $H^{11}$ and with $N^7=1205$, $N^8=1238$, $N^9=1241$, $N^{10}=1261$ and $N^{11}=1189$ respectively).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_traj_and_hydros.pdf}\n\\caption{The 2D trajectory (in $x\u2212y$ plan) of the single sperm whale observed during $25$ min and corresponding hydrophone's positions.}\n\\label{true_trajectory}\n\\end{center}\n\\end{figure}\nTo extract local features, we chose $n=2000$, $p=128$ and $L=1000$ (tuned by model selection). For both the dictionary learning and the local features encoding, we chose $\\lambda=0.2$ and fixed $15$ iterations to train dictionary on a subset of $M=400.000$ local features drawn uniformaly. We performed\n$K=10$ cross-validation where training sets reprensented $70\\%$ of the total of extracted global features, the rest for the testing sets. Logistic regression parameter $C$ is tuned by model selection. We compute the average root mean square error (ARMSE) of range\/azimuth estimates per hydrophone: $ARMSE(l)=\\frac{1}{K}\\sum\\limits_{i=1}^{K}\\sqrt{\\sum\\limits_{j=1}^{N_{test}^l}(y_{i,j}^l-\\widehat{y}_{i,j}^l)^2}$ where $y_{i,j}^l$, $\\widehat{y}_{i,j}^l$\n and $N_{test}^l$ represent the ground truth, the estimate and the number of test samples for the $l^{th}$ hydrophone respectively. The global ARMSE is then calculated by $\\overline{ARMSE}=\\frac{1}{H}\\sum\\limits_{l=1}^{H}ARMSE(l)$.\n\n \\subsection{$\\ell_{\\mu}$-norm pooling case study}\n\n For prilimary results, we investigate the influence of the $\\mu$ parameter during the pooling stage. We fix the number of dictionary basis to $k=128$ and the temporal pyramid equal to $\\g{\\Lambda}_1=\\left[1,1,1\\right]$, \\textit{i.e.} we pool sparse codes on whole the temporal click signal.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_range_nu_pooling.pdf}\n\\caption{$\\overline{ARMSE}$ vs. $\\mu$ for range estimation.}\n\\label{baha_click_range_nu_pooling}\n\\end{center}\n\\end{figure}\nA value of $\\mu=\\{3,4\\}$ seems to be a good choice for this pooling procedure. For $\\mu\\geq20$, results are similar to those obtained by max-pooling. For azimuth, we observe also the same range of $\\mu$ values.\n\n \\subsection{Range and azimuth regression results}\n\n Here, we fixed the value of $\\mu=3$ and we varied the number of dictionary basis $k$ from $128$ to $4096$ elements. We also investigated the influence of the temporal pyramid and we give results for two particulary choices: $\\g{\\Lambda}_1=\\left[1,1,1\\right]$ and $\\g{\\Lambda}_2=\\small\\left[\\begin{array}{ccc}1 & 1 &1\\\\ \\frac{1}{3} & \\frac{1}{3}&1\\end{array}\\right]$. For $\\g{\\Lambda}_2$, the sparse are first pooled over all the signal then pooled over 3 non-overlapping windows for a total of $1+3=4$ ROIs.\n In order to compare results of our presented method, we also give results for an hand-craft feature \\cite{IFA_DCL} specialized for spermwhales and based on the spectrum of the most energetic pulse d\u00e9tected inside the click. This specialized feature, denoted \\textit{Spectrum feature}, is a 128 points vector.\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_range.pdf}\n\\caption{$\\overline{ARMSE}$ vs. $k$ for range estimation with $\\mu = 3$.}\n\\label{baha_click_range}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[height=5.5cm,width=7.5cm]{baha_click_azimuth.pdf}\n\\caption{$\\overline{ARMSE}$ vs. $k$ for azimuth estimation with $\\mu = 3$.}\n\\label{baha_click_azimuth}\n\\end{center}\n\\end{figure}\n\nFor both range and azimuth estimate, from $k=2048$, our method outperforms results of the \\textit{Spectrum feature} and particulary for azimuth estimate. Using a temporal pyramid for pooling permits also to improve slightly results.\n\n\\section{Conclusions and perspectives}\n\nWe introduced in the paper, for spermwhale localization, a BoF approach \\textit{via} sparse coding delivering rough estimates of range and azimuth of the animal, specificaly towarded for mono-hydrophone configuration. Our proposed method works directly on the click signal without any prior pulses detection\/analysis while being robust to signal transformation issue by the propagation. Coupled with non-linear filtering such as particle filtering \\cite{Arulampalam02}, accurate animal position estimation could be perform even in mono-hydrophone configuration. Applications for anti-collision system and whale whatching are targeted with this work.\n\nAs perspective, we plan to investigate other local features such as spectral features, MFCC \\cite{Davis80,Rabiner93}, Scattering transform features \\cite{AndenM11}. These latter can be considered as a hand-craft first layer of a deep learning architecture with 2 layers.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\n\n\nThis note is a development based on a recent paper \\cite{Kallosh:2021ors} where we have performed a covariant (Lagrangian) quantization of gravity in a black hole background in the Regge-Wheeler  set up \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir}.  The gauge-fixing condition in  \\cite{Kallosh:2021ors} includes the Regge-Wheeler gauge for ${\\ell }\\geq 2$ modes, and a certain background covariant gauge  for ${\\ell }<2$  modes, where Regge-Wheeler gauge is not valid. We will refer to the  gauge in  \\cite{Kallosh:2021ors} covering all ${\\ell }$ modes, as a `generalized  Regge-Wheeler gauge'.\n\n\n\n\n\n\nThe Feynman path integral for gravity, viewed as quantum field theory (QFT), is  defined by De Witt-Faddeev-Popov  \\cite{DeWitt:1967ub,Faddeev:1967fc} and takes a form, in absence of sources \n\\begin{equation}\n \\int  D h\n \\,J_\\chi ( g,h)  \\delta \\big (\\chi_\\alpha ( g,h)\\big ) e^{\\mathrm{i} S ( g+h) } \\, .\n\\label{pi}\\end{equation}\nHere we integrate over the perturbations $h$ in the background metric $g$. The gauge-fixing conditions are $\\chi_\\alpha(g, h)=0$.  The Jacobian designed to make this path integral independent on the choice of the gauge-fixing conditions   can be presented with the help of the Faddeev-Popov (FP) ghosts \\cite{Faddeev:1967fc} \\begin{equation}\nJ_\\chi = \\int D\\bar C^\\alpha DC_\\beta \\, e^{\\mathrm{i} \\int \\mathrm{d}^4 x \\, \\bar C^\\alpha(x)  Q_{\\alpha} {}^{ \\beta}  (g, h)C_\\beta(x) }\\, .\n\\end{equation}\nThe differential operator in the ghost action is defined by the gauge variation of the gauge-fixing finctions\n$\n\\delta \\chi_\\alpha = Q_{\\alpha}{}^{\\beta} \\xi_\\beta\n$. For the  choice of the gauge-fixing functions made in \\cite{Kallosh:2021ors}, which in addition to Regge-Wheeler gauge for $l\\geq 2$ modes includes gauges for $l<2$ modes, a generalized  Regge-Wheeler gauge, we have found that the ghost actions do not have time derivatives  in  Schwarzschild coordinates. We therefore predicted that in the a generalized  Regge-Wheeler gauge  \\cite{Kallosh:2021ors} the canonical Hamiltonian according to the rules for gauge theories \n\\cite{Faddeev:1969su,Fradkin:1970pn,Faddeev:1973zb} is expected to be unitary.\n\nIn this note we will  present the quadratic in gravitational perturbations $h$ part of the gravity Hamiltonian in the spherical harmonic basis. Before doing this we will perform the standard counting of physical degrees of freedom in this case. The structure of the Hamiltonian will confirm this counting.\n\nThe standard counting of physical degrees of freedom in gauge theories in the QFT context of the Feynman path integral is the same in either Lagrangian or Hamiltonian quantization, and it is also gauge independent, if performed correctly. \nThe general counting formula is formulated for the number of gauge field components equal to $n+k$ in case of $k$ gauge symmetries.  The total number of physical degrees of freedom is\n\\begin{equation}  n-k\n\\label{number}\\end{equation}\nThis final counting formula in QFT is valid for any choice of  gauge-fixing, but  the procedure is different for unitary and pseudo-unitary gauges. \nFor example in 4D the metric has $n+k=10$   components and there are $k=4$ gauge symmetries, the counting is $n-k=(10- 4)-4=2$.\n\n\n\nIn QFT\nin the class of unitary gauges Hamiltonians have manifestly ghost-free underlying Hilbert spaces. There are $(p^*, q^*)$ variables in Faddeev's theorem \\cite{Faddeev:1969su} as described in \\cite{Kallosh:2021ors}.\nThis means that all $n-k$ physical states have positive definite metric. The S-matrix is unitary.\n\\begin{equation}\n\\# \\, {\\rm degrees \\, of\\, freedom}_{\\rm unitary \\, H} \\,  =  n-k\n\\label{numberU}\\end{equation}\nMeanwhile, \nin other gauges, for example, 4D Lorentz covariant gauges in gravity,  the Hamiltonians are ``pseudo-unitary'' with underlying state spaces with negative-norm ghost degrees of freedom \\cite{Fradkin:1977hw,Batalin:1977pb}. In such case the counting goes as follows: there are $n+k$ states with positive norm and $2k$ states with negative norm presented by FP anti-commuting ghosts, so the total counting, with account of negative norm states,  is the same as in unitary gauges\n\\begin{equation}\n\\# \\, {\\rm degrees \\, of\\, freedom}_{\\rm pseudo-unitary \\, H}  =  n+k -2k  \\Rightarrow n-k\n\\end{equation}\n The S-matrix is pseudo-unitary in a space of states with the indefinite metric.\n\n\n\n\n\n \nWe will see that the quadratic in $h$ part of the gravity Hamiltonian in spherical harmonic basis does support  this counting. In the class of gauges used in  \\cite{Kallosh:2021ors} the canonical Hamiltonian is unitary, as predicted there.\n\nThe complete form of the Hamiltonian to all orders of $h$  is beyond the scope of this paper. However, in \\cite{Kallosh:2021ors} we have argued that the non-linear couplings of ghosts to all orders in $h$ are free of time derivatives on ghosts. Therefore one would expect that the non-linear in $h$ terms in $H$ will be consistent with the unitarity of the Hamiltonian which will be deduced in this note  at the level quadratic in $h$.\n\n The corresponding part of the action $S( g+h)$, quadratic in $h$, is of the form\n\\begin{equation}\nS= {1\\over 2} \\int h_{\\mu\\nu} S^{ \\mu\\nu  \\lambda \\delta} (g) h_{\\lambda \\delta}\n\\label{quad} \\end{equation}\nHere $S^{ \\mu\\nu  \\lambda \\delta} (g)$ is a differential operator depending on the background metric $g$. The left hand side of equations of motion ${\\delta S\\over \\delta h_{\\mu\\nu} }=0$ linear in $h$ takes the form\n\\begin{equation}\nQ^{\\mu\\nu} \\equiv {\\delta S\\over \\delta h_{\\mu\\nu} }= S^{ \\mu\\nu  \\lambda \\delta} (g) h_{\\lambda \\delta}\n\\label{EOM}\\end{equation}\nOne can restore the action in eq. \\rf{quad} from the information available in eq. \\rf{EOM}.\n\nIn the spherical harmonic basis the 4D spacetime  is split into $\\mathcal{M} = \\mathcal{M}_2\\times\\mathbb{S}^2$  with coordinates $(x^{a},\\theta^A)$, $a=1,2$.  The 4D perturbations $h_{\\mu\\nu} $ are  represented by  2D fields for each $({\\ell }, m)$ \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir}. \nThe corresponding equations are known and we will use them as derived in \\cite{Martel:2005ir} in Schwarzschild coordinates. Once the quadratic Lagrangian is known, it is possible to derive the relevant quadratic in $h$ Hamiltonian.\nFor ${\\ell }\\geq 2$ modes the corresponding quadratic Hamiltonian was constructed by Moncrief in \\cite{Moncrief:1974am} where also the relevant Regge-Wheeler \\cite{Regge:1957td} and Zerilli \\cite{Zerilli:1971wd} equations were re-derived in the form of Hamiltonian equations of motion. The Hamiltonian was derived in \\cite{Moncrief:1974am} in absence of source terms.\nFor ${\\ell }<2$ the Hamiltonian was not studied, to the best of our   knowledge. In \\cite{Moncrief:1974am} it was explained that  the  attention was restricted to modes with $l\\geq 2$ since the modes with $l<2$ are nonradiative and require a special treatment.\n\n\nHere we will use the known field equations \\rf{EOM} in the form given in  \\cite{Martel:2005ir} in Schwarzschild coordinates, which allow to derive the Lagrangian in \\rf{quad}. From the quadratic Lagrangian we derive  a canonical  quadratic part of the Hamiltonian, with account of the algebraic constraints in our gauges.  We will conclude that there are no physical degrees of freedom suitable for quantization  at ${\\ell }<2$. Our definition of quantized degrees of freedom involves the QFT quantization conditions in 2D space of the form\n\\begin{equation}\n[q(r, t) , p(r', t) ]= i \\delta (r-r')\n\\label{quant}\\end{equation}\nThe classical field equations for low multipoles  in presence of sources are known to have non-trivial solutions. For example for the monopoles ${\\ell }=m=0$   there are solutions  like \n$\nh^{00}_{tt} \\sim {\\delta M\\over r}\n$, \nthey are known to affect  the  the black hole mass. However,  there are no solutions of the constraint equations compatible with the quantization condition \\rf{quant} for ${\\ell }<2$.\n\nAll our results are valid for any mass $M$ of the Schwarzschild black hole, and the limit to $M = 0$ is continuous. This means that they apply not only to the quantization in the black hole background, but also to the unitary quantization of the gravitational field in the Minkowski space background in spherical coordinates.\n\n\n\n\n\\section{ Counting Gravity  Physical Degrees of Freedom in the Spherical Harmonic Basis}\nThe ansatz of Regge-Wheeler for the metric perturbations $h_{\\mu\\nu}$  with  spherical harmonics of definite parity  is given in  \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir}. In our \n recent paper \\cite{Kallosh:2021ors} it was adapted for the purpose of quantization following  the formalism and notations in \\cite{Martel:2005ir}. In particular, we have presented the gauge symmetry transformations to all orders in $h$. The background metric in Schwarzschild coordinates is\n \\begin{equation}\ng_{\\mu\\nu}dx^\\mu dx^\\nu=\t-f(r)\\,\\mathrm{d}t^2 + \\frac{\\mathrm{d}r^2}{f(r)} + r^2(x)\\,\\mathrm{d}\\Omega^2_2\\, ,  \\qquad f(r)  = 1-\\frac{2GM}{r}\n\t\\label{SchldBackgroundGauge}\n\\end{equation} \n\n\\\n\n The 2D fields representing all components of $h_{\\mu\\nu}$ in 4D include the following  \n\\begin{eqnarray} \\label{RWA}\n&& h_{ab}^{\\ell m(+)}, \\quad j_{a}^{\\ell m(+)}, \\quad K^{\\ell m(+)}, \\quad G^{\\ell m(+)}\\,  \\hskip 3 cm {\\ell } >1  , \\quad  {\\rm even}\\\\\n&&  h_{a}^{\\ell m(-)}, \\quad h_2^{\\ell m(-)}\\,    \\hskip 6.5 cm {\\ell } > 1 , \\quad  {\\rm odd} \\\\\n&&h_{ab}^{1 m(+)}, \\quad  j_{a}^{1 m(+)},  \\quad K^{1m(+)}   \\qquad \\hskip 4 cm {\\ell }=1, \\quad  {\\rm even}\n\\label{ansatzDe} \\\\\n&&h_{a}^{1 m(-)}  \\hskip 8.2 cm  {\\ell }=1, \\quad   {\\rm odd}\\\\\n&&h_{ab}^{0 0(+)},  \\quad K^{00(+)}  \\hskip 6.6 cm  {\\ell }=0,  \\quad  {\\rm even}\n\\label{ansatzM} \n \\end{eqnarray}\nThe gauge symmetries are also expanded in spherical harmonics. In the form   given in our recent paper \\cite{Kallosh:2021ors} these are\n \\begin{eqnarray}\n\\xi^{{\\ell } > 1 }  \\quad  {\\rm even}  \\qquad   &&\\Rightarrow \\qquad \\{ \\xi_{a}^{\\ell m(+)}, \\xi^{\\ell m(+)} \\}\n\\label{sym}\\\\\n\\xi^{{\\ell } > 1}  \\quad  {\\rm odd}   \\qquad   &&\\Rightarrow \\qquad \\{  \\xi^{\\ell m(-)} \\}\n\\label{sym}\\\\\n\\xi^{{\\ell }=1}  \\quad  {\\rm even}  \\qquad   &&\\Rightarrow \\qquad  \\{ \\xi_{a}^{1 m(+)}, \\xi^{1 m(+)} \\} \n\\label{symDe} \\\\\n\\xi^{{\\ell }=1}  \\quad  {\\rm odd}  \\qquad  && \\Rightarrow \\qquad  \\{  \\xi^{1 m(-)}  \\} \\\\\n\\xi^{{\\ell }=0}   \\quad  {\\rm even}   \\qquad   &&\\Rightarrow \\qquad  \\{  \\xi_{a}^{0 0(+)} \\}\n\\label{symM} \n \\end{eqnarray}\nThe gauge symmetry parameters $\\xi_{a}^{\\ell m (+)}$, $\\xi^{\\ell m (+)}$, $\\xi^{\\ell m (-)}$  \ncan be regarded as scalar and vector fields on $\\mathcal{M}_2$.\n\nThe counting of physical degrees of freedom in these 5 sectors is\n\\begin{enumerate}\n  \\item ${\\ell } >1   \\quad  {\\rm even}:  \\quad  \\, n+k = 7, \\quad k= 3\\, \\quad \\Rightarrow \\quad n+k -2k = 7- 2\\cdot 3=1$\n  \\item ${\\ell } >1   \\, \\, \\quad  {\\rm odd}:  \\quad  \\, n+k = 3, \\quad k= 1\\, \\quad \\Rightarrow \\quad n+k -2k= 3- 2\\cdot 1=1$\n  \\item ${\\ell } =1    \\quad  {\\rm even}:  \\quad  \\, n+k = 6, \\quad k= 3\\, \\quad \\Rightarrow \\quad n+k -2k= 6- 2\\cdot 3=0$\n  \\item ${\\ell } =1   \\, \\,  \\quad  {\\rm odd}:  \\quad  \\, n+k = 2, \\quad k= 1\\, \\quad \\Rightarrow \\quad n+k -2k= 2- 2\\cdot 1=0$ \n  \\item ${\\ell } =0    \\quad  {\\rm even}:  \\quad  \\, n+k = 4, \\quad k= 2\\, \\quad \\Rightarrow \\quad n+k -2k= 4- 2\\cdot 2=0$\n\\end{enumerate}\nThus we find that in $l\\geq 2$ sector there is one even and one odd physical degree of freedom for each $({\\ell },m)$. There are no degrees of freedom for any of ${\\ell }<2$.\n\n\\section{Quadratic Lagrangian\/Hamiltonian  for ${\\ell }\\geq 2 $ Modes}\n\\subsection{${\\ell }\\geq 2 $ even} \n There are 7 fields here, $h_{ab}^{\\ell m(+)}, \\quad j_{a}^{\\ell m(+)}, \\quad K^{\\ell m(+)}, \\quad G^{\\ell m(+)}$.\n There are 7 equations of motion for these fields.   Now we can add the 3 Regge-Wheeler gauge-fixing conditions\n \\begin{equation}\n G=j_a=0\n \\end{equation} \nThe remaining 4 fields are $h_{ab}^{\\ell m(+)},  K^{\\ell m(+)}$. We expect to identify 3 constraints which will leave us with just one canonical degree of freedom. These equations are according to \\cite{Martel:2005ir}\n \\begin{eqnarray} \\label{MarP}\nQ^{tt} &=& -\\frac{\\partial^2}{\\partial r^2} {K} \n- \\frac{3r-5M}{r^2 f} \\frac{\\partial}{\\partial r} {K} \n+ \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{rr}\n+ \\frac{(\\lambda+2)r + 4M}{2r^3} {h}_{rr} \n+ \\frac{\\mu}{2r^2 f} {K}, \\\\ \nQ^{tr} &=& \\frac{\\partial^2}{\\partial t \\partial r} {K} \n+ \\frac{r-3M}{r^2 f} \\frac{\\partial}{\\partial t} {K} \n- \\frac{f}{r} \\frac{\\partial}{\\partial t} {h}_{rr} \n- \\frac{\\lambda}{2r^2} {h}_{tr}, \\cr\nQ^{rr} &=& -\\frac{\\partial^2}{\\partial t^2} {K} \n+ \\frac{(r-M)f}{r^2} \\frac{\\partial}{\\partial r} {K} \n+ \\frac{2f}{r} \\frac{\\partial}{\\partial t} {h}_{tr} \n- \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{tt}  \n+ \\frac{\\lambda r + 4M}{2r^3} {h}_{tt} \n- \\frac{f^2}{r^2} {h}_{rr} \n- \\frac{\\mu f}{2r^2} {K}, \\cr \nQ^\\flat &=& -\\frac{\\partial^2}{\\partial t^2} {h}_{rr} \n+ 2 \\frac{\\partial^2}{\\partial t \\partial r} {h}_{tr} \n- \\frac{\\partial^2}{\\partial r^2} \\tilde{h}_{tt} \n- \\frac{1}{f} \\frac{\\partial^2}{\\partial t^2} {K} \n+ f \\frac{\\partial^2}{\\partial r^2} \\tilde{K} \n+ \\frac{2(r-M)}{r^2 f} \\frac{\\partial}{\\partial t} {h}_{tr} \n- \\frac{r-3M}{r^2 f} \\frac{\\partial}{\\partial r} {h}_{tt} \n\\cr & & \\mbox{} \n- \\frac{(r-M)f}{r^2} \\frac{\\partial}{\\partial r} {h}_{rr} \n+ \\frac{2(r-M)}{r^2} \\frac{\\partial}{\\partial r} {K} \n+ \\frac{\\lambda r^2-2(2+\\lambda)Mr+4M^2}{2r^4 f^2}{h}_{tt}   \n- \\frac{\\lambda r^2-2\\mu Mr-4M^2}{2r^4} {h}_{rr}\\nonumber \n\\end{eqnarray} \nHere \n\\begin{equation}\n\\lambda = {\\ell }({\\ell }+1) \\qquad \\mu = ({\\ell }-1)({\\ell }+2)\n\\end{equation}\nThe quadratic in $h$ Lagrangian can be restored from these  equations  as explained in eqs. \\rf{quad}, \\rf{EOM}. One can proceed by defining for each of the 4  fields their canonical momenta. For example, there is no time derivative on $h_{tt}$ in the action, therefore $p_{tt}=0$, the other 3 coordinates in the form of ${\\cal L} (q, \\dot q)$ do have time derivatives, however, two more combinations of $q$'s and $p$'s are constrained. Only one independent canonical degree of freedom out of 4  is left.\n\nThe Hamiltonian of the related system starting with the  Arnowitt, Deser, Misner construction  was derived in \\cite{Moncrief:1974am}. We skip the details of the derivation here starting with the field equations \n\\rf{MarP} since the answer for the corresponding Lagrangian can be also reconstructed from the \n Zerilli-Moncrief function \\cite{Zerilli:1971wd,Moncrief:1974am} which in Regge-Wheeler gauge is\n\\[\n\\Psi_{\\rm even} = \\frac{2r}{{\\ell }({\\ell }+1)} \\biggl[ {K}   \n+ \\frac{2f}{\\Lambda} \\biggl( f {h}_{rr} \n- r  {K}_{,r} \\biggr) \\biggr], \\qquad l\\geq 2\n\\]\nwhere $\\Lambda = ({\\ell }-1)({\\ell }+2)  + 6M\/r$. The equation of motion in the form of the Zerilli-Moncrief function $\\Psi^{lm}_{even}$ as given in \\cite{Martel:2005ir} is\n\\begin{equation}\n(\\Box - V_{\\rm even})  \\Psi_{\\rm even}=S_{\\rm even}\n\\label{ZM}\\end{equation}\nwhere $\\Box= g^{ab} {\\cal D}_a {\\cal D}_b$ is the Laplacian operator on ${\\cal M}_2$, $V_{\\rm even}$ depends on $r$ as well as on  $M$ and on $l$,   and $S_{\\rm even}$ is the contribution from sources.  We refer to details given in \\cite{Martel:2005ir}, where also the relation between Zerilli-Moncrief function and the  original Regge-Wheeler function is explained.\nEquation  \\rf{ZM} can be derived from the Lagrangian of the form \\rf{quad}\n\\begin{equation}\n{\\cal L} = \\sum_{{\\ell } \\geq 2 ,  m} \\Big [  {1\\over 2} \\Psi_{\\rm even}( \\Box - V_{\\rm even})\\Psi_{\\rm even}  - \n \\Psi_{\\rm even} S_{\\rm even}\\Big ]\\end{equation}\n This can be rewritten in the form producing a quadratic part of the Hamiltonian. With $\\Psi_{\\rm even}\\equiv Q_{\\rm even}$ and its canonically conjugate $P_{\\rm even}$ and,  in absence of sources\n\\begin{equation}\n H_{{\\ell } \\geq 2, \\rm even}= {1\\over 2} \\sum_{{\\ell }\\geq 2, m} \n \\int \\Big [dr  f(  P^{{\\ell }, m } )^{2 }_{\\rm even}+  f (Q_{, r}^{{\\ell }, m })_{\\rm even}^2 +   V_{\\rm even}\n (Q^{{\\ell }, m })_{\\rm even}^2\\Big ]\n\\label{Heven} \\end{equation}\nwhere\n\\begin{equation} \nV_{\\rm even} = \\frac{1}{\\Lambda^2} \\biggl[ \\mu^2 \\biggl(\n  \\frac{\\mu+2}{r^2} + \\frac{6M}{r^3} \\biggr) \n+ \\frac{36M^2}{r^4} \\biggl(\\mu + \\frac{2M}{r} \\biggr) \\biggr] \n\\label{4.26}\n\\end{equation}\nThis is an example of the Faddeev's theorem  \\cite{Faddeev:1969su}, which we described in   \\cite{Kallosh:2021ors}, where starting from the original constrained  variables $(p_i, q^i)$  with constraints $\\phi^\\alpha(p,q)$ one can perform a canonical transformation with $p'_\\alpha =\\chi_\\alpha (p,q) =0$ and  $ q^{'\\alpha} = q^{'\\alpha} (p^*, q^*)$ \nso that the independent set of canonical variables is $(p^*, q^*)$. In this particular case we find just one set of $(p^*, q^*)$, which are the \n Zerilli-Moncrief function $\\Psi$ of the original variables,  and its canonical conjugate.\n\n \n\\subsection{${\\ell }\\geq 2 $ odd} \n\nThere are 3 fields in this sector:  $h_{a}^{\\ell m(-)}, \\quad h_2^{\\ell m(-)}$. In the RW gauge\n\\begin{equation}\nh_2^{\\ell m(-)}=0\\, .\n\\end{equation}\nEquations of motion for the remaining two fields are\n\\begin{eqnarray*} \nP^t &=& - \\frac{\\partial^2}{\\partial t \\partial r} {h}_r \n+ \\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r} \\frac{\\partial}{\\partial t} {h}_r \\\n- \\frac{\\lambda r - 4M}{r^3 f} {h}_t, \\\\ \nP^r &=& \\frac{\\partial^2}{\\partial t^2} {h}_r \n- \\frac{\\partial^2}{\\partial t \\partial r} {h}_t \n+ \\frac{2}{r} \\frac{\\partial}{\\partial t} {h}_t \n+ \\frac{\\mu f}{r^2} {h}_r,\n\\end{eqnarray*} \nRestoring the quadratic Lagrangian and using partial integration one can identify one  field which enters into Lagrangian without a time derivative, this is $h_t$.  \n\\begin{equation} \n{\\cal L} = h_t \\Big (- \\frac{\\partial^2}{\\partial t \\partial r} {h}_r \n+ {1\\over 2} \\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r} \\frac{\\partial}{\\partial t} {h}_r \\\n- {1\\over 2} \\frac{\\lambda r - 4M}{r^3 f} {h}_t \\Big ) + {1\\over 2} h_r \\Big (\\frac{\\partial^2}{\\partial t^2} {h}_r \n+ \\frac{\\mu f}{r^2} {h}_r\\Big ) \n\\end{equation}\nThus we find\n\\begin{eqnarray}\n&&p_t=0\\\\\n&&p_r= h_{t, r} - h_{r, t} -{2\\over r} h_t\n\\end{eqnarray}\nand there is a constraint for $h_t$ algebraically related to $p_r$  \n\\begin{equation}\n \\Big ( \\partial_r + \\frac{2}{r}  \\Big ) \\Big (p_r + h_{t, r} -{2\\over r} h_t\n\\Big ) \n-  {h}_{t,rr} \n+  \\frac{\\lambda r - 4M}{r^3 f} {h}_t =0\n\\end{equation}\nTherefore there is one independent degree of freedom  $(h_r, p_r)$. These are Faddeev's $(p^*, q^*)$ variables, exactly one set in agreement with the counting give above. One can write the corresponding Hamiltonian $H(h_r, p_r)$ and the field equations.\n\nOn the other hand, the Hamiltonian for this system was already derived in \\cite{Moncrief:1974am} in  the framework of the Arnowitt, Deser, Misner construction.  The field equations were derived in \\cite{Cunningham:1978zfa}, where the corresponding Cunningham-Price-Moncrief function was introduced. In notation of  \\cite{Martel:2005ir}, this function is\n\\begin{equation}\n\\Psi_{\\rm odd}^{lm}  = \\frac{2r}{({\\ell }-1) ({\\ell }+2) } \\biggl( \n {h}_{t , r} ^{{\\ell }m} - \n {h}_{r,t}^{{\\ell }m}  - \\frac{2}{r} {h}_t^{{\\ell }m}  \n\\biggr). \n\\end{equation}\nThis function in terms of canonical variables above depends on $(p_r, h_r)$.\nAs in the even case  discussed  above we are lead to a single field   equation for the Cunningham-Price-Moncrief  function\n\\begin{equation}\n(\\Box - V_{\\rm odd})  \\Psi_{\\rm odd}=S_{\\rm odd}\n\\label{odd}\\end{equation}\nHere the expressions for $V_{\\rm odd}$ and $  S_{\\rm odd}$ are given in \\cite{Martel:2005ir}, where also \nthe relation between Cunningham-Price-Moncrief  function  and the original Regge-Wheeler function is explained.\nWith $\\Psi_{\\rm odd}\\equiv Q_{\\rm odd}$ the Hamiltonian is\n\n \\begin{equation}\n H_{{\\ell } \\geq 2, \\rm odd}= {1\\over 2} \\sum_{{\\ell }\\geq 2, m} \n \\int \\Big [dr  f(  P^{{\\ell }, m } )^{2 }_{\\rm odd}+  f (Q_{, r}^{{\\ell }, m })_{\\rm odd}^2 +  \\Big ( \\frac{{\\ell }({\\ell }+1) }{r^2} - {6M \\over r^3}\\Big ) \n (Q^{{\\ell }, m })_{\\rm odd}^2\\Big ]\n \\label{Hodd}\\end{equation}\n  \n\\section{Quadratic Lagrangian\/Hamiltonian  for ${\\ell }<2$ Modes}\n\\subsection{${\\ell }=1$ even} \nOur 6 fields are $h_{ab}^{1 m(+)},   j_{a}^{1 m(+)},   K^{1m(+)}$.  We take a gauge-fixing condition \\cite{Kallosh:2021ors}\n\\begin{equation}\nj_{a}^{1 m(+)}= K^{1m(+)}=0\n\\end{equation}\nThe remaining filelds $h_{ab}^{1 m(+)}$ in this gauge satisfy the field equations\n\\begin{eqnarray*} \nQ^{tt} &=& \n \\frac{f}{r} \\frac{\\partial}{\\partial r}{h}_{rr}\n+ \\frac{2(r + M)}{r^3} {h}_{rr}, \\\\ \nQ^{tr} &=& \n- \\frac{f}{r} \\frac{\\partial}{\\partial t} {h}_{rr} \n- \\frac{1}{r^2} {h}_{tr}, \\\\\nQ^{rr} &=& \n \\frac{2f}{r} \\frac{\\partial}{\\partial t} {h}_{tr} \n- \\frac{f}{r} \\frac{\\partial}{\\partial r}{h}_{tt}  \n+ \\frac{ r + 2M}{r^3} {h}_{tt} \n- \\frac{f^2}{r^2}{h}_{rr} \n\\end{eqnarray*} \nWe can therefore reconstruct the Lagrangian of the form \\rf{quad} which will produce these equations.\n\\begin{equation}\n{\\cal L} = h_{tt} Q^{tt} +  \\Big (\\frac{\\partial}{\\partial t} h_{tr}\\Big)   \\frac{ 2f}{r}  {h}_{rr} \n-   h_{tr} \\frac{1}{2r^2} {h}_{tr} -h_{rr} \\frac{f^2}{ 2r^2}{h}_{rr} \n\\end{equation}\nWe now define $q\\equiv  h_{tr}, \\, p \\equiv  \\frac{ 2f}{r}  {h}_{rr}$ and $h_{tt} \\equiv \\lambda$\n\\begin{equation}\n{\\cal L} =  \\dot q p + \\lambda Q^{tt} (p, \\partial_r p)  \n-   q^2 \\frac{1}{2r^2} -{1\\over 8} p^2\n\\end{equation}\nWe integrate out the Lagrange multiplier and find\n\\begin{equation}\n{\\cal L} =  \\dot q p \n-   q^2 \\frac{1}{ 2r^2} -{1\\over 8} p^2\n\\end{equation}\nwhere \n\\begin{equation}\n \\frac{f}{r} \\frac{\\partial}{\\partial r}{rp\\over 2f}\n+ \\frac{2(r + M)}{r^3} {rp\\over 2f}=0  \\qquad \\Rightarrow \\qquad  p_{,r} + F(r) p=0\n\\label{Cp}\\end{equation}\nThe algebraic constraint which $p$ has to satisfy contradicts the  commutation relation which have to be imposed for quantization, as shown in eq. \\rf{quant}.\nThere is no solution of the algebraic constraint \\rf{Cp} for the canonical momentum $p(t,r)$ which would be consistent with the quantization condition, only $p=0$ is a consistent one. We conclude there that there are no physical degrees of freedom left in this sector,\n\\begin{equation}\nH_{{\\ell }=1, \\rm  even} =0\n\\end{equation}\n This is in agreement with the counting we presented above.\n\n\n\\subsection{${\\ell }=1$ odd} \n\nThere are 2 fields: $h_{a}^{1 m(-)}$. We take a gauge-fixing condition $h_{r}^{1 m(-)}=0$ \\cite{Kallosh:2021ors}. In this gauge the remaining field equation is\n\\begin{equation}\nP^t =  \\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r^2 } {h}_t\n\\end{equation}\nThe Lagrangian which will generate this equation is\n\\begin{equation}\n{\\cal L}={1\\over 2} h_t \n\\Big (\\frac{\\partial^2}{\\partial r^2} {h}_t \n- \\frac{2}{r^2} {h}_t \\Big  )\n\\end{equation}\nThere is one field here where the Lagrangian $ {\\cal L}(q)$ does not have time derivative of this field, therefore $p={\\delta {\\cal L}\\over \\dot h_t}=0 $. There are no canonical variables here and the Hamiltonian vanishes \n\\begin{equation}\nH_{{\\ell }=1, \\rm  odd} =0\n\\end{equation}\nThis is in agreement with the counting we presented above.\n\n\n\\subsection{${\\ell }=0$ even} \n\nThere are 4 fields here: $h_{ab}^{0 0(+)},  \\quad K^{00(+)}$. We take a gauge-fixing conditions $K=h_{tr}=0$ \\cite{Kallosh:2021ors}.\nThe remaining field equations are\n\\begin{eqnarray*} \nQ^{tt} &=&  \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{rr}\n+ \\frac{r + 2M}{r^3} {h}_{rr} \n, \\\\ \nQ^{rr} &=&  \n- \\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{tt}  \n+ \\frac{ 2M}{r^3} {h}_{tt} \n- \\frac{f^2}{r^2} {h}_{rr}\n\\end{eqnarray*} \nThe Lagrangian which will generate these equations is\n\\begin{equation}\n{\\cal L}=h_{tt} \\Big (\\frac{f}{r} \\frac{\\partial}{\\partial r} {h}_{rr}\n+ \\frac{(r + 2M)}{r^3} {h}_{rr} \\Big ) -{f^2\\over 2r^2} h_{rr}^2\n\\end{equation}\nThere are 2 fields, $q^1, q^2$, but there are no time derivatives in the Lagrangian, $p_1=p_2=0$, no canonical variables and the Hamiltonian vanishes\n\\begin{equation}\nH_{{\\ell }=0} =0\n\\end{equation}\n This is again in agreement with the counting we presented above.\n\n\n\n\\section{A special role of ${\\ell }=0,1$ in quantization of gravity}\n\nIs there any relation between the well known fact about the absence of radiation from monopoles and dipoles in gravity and the fact we observed here, that there are no quantum physical degrees of freedom in monopoles and dipoles when gravity is quantized in spherical harmonics basis? The answer is yes, and it has to do with the tensor nature of gravity, so that radiation starts with quadrupoles ${\\ell }\\geq 2$. \n\nRegge-Wheeler ansatz for ${\\ell }\\geq 2$ has 10 functions depending on coordinates of ${\\cal M}_2$ listed in eqs. \\rf{RWA}-\\rf{ansatzM}. Here we show them in the matrix form contracted with spherical functions.\n\\begin{equation}\nh_{\\mu\\nu}^{{\\ell }>1} =\\begin{pmatrix}\n h^{{\\ell }m}_{ab} Y^{{\\ell }m}   &   &  &  &  {\\color {blue}j^{{\\ell }m}_a Y_B^{{\\ell }m}} \\\\\n\\cr  \n {\\color {blue} j^{{\\ell }m}_a Y_B^{{\\ell }m} } &   &  &  & r^2 K^{{\\ell }m} \\Omega_{AB} Y^{{\\ell }m} + {\\color {red} G^{{\\ell }m} Y_{AB}^{{\\ell }m}}\\end{pmatrix}^{(+)} + \\begin{pmatrix}\n0   &   &  &  &  {\\color {blue} h^{{\\ell }m}_a X_B^{{\\ell }m}} \\\\\n\\cr  \n {\\color {blue} h^{{\\ell }m}_a X_B^{{\\ell }m}}  &   &  &  & {\\color {red} h^{{\\ell }m}_2 X_{AB}^{{\\ell }m}} \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nThe number gauge symmetries  in all cases with ${\\ell } >0$ is the same since $\\xi_\\mu$ is a vector\n\\begin{equation}\n\\xi_{\\mu}^{{\\ell }>0} =\\begin{pmatrix}\n\\xi^{{\\ell }m}_{a} Y^{{\\ell }m}   \\\\\n\\cr  \n {\\color {blue}\\xi^{{\\ell }m} Y_A^{{\\ell }m}}  \\end{pmatrix}^{(+)}  \\, + \\begin{pmatrix}\n0    \\\\\n\\cr  \n {\\color {blue}\\xi^{{\\ell }m}_a X_B^{{\\ell }m} }  \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nTherefore we find that  instead of 10 fields (even and odd) as for ${\\ell }\\geq 2$ we have 8 fields (even and odd)  for ${\\ell }=1$, no fields  in red\n\\begin{equation}\nh_{\\mu\\nu}^{{\\ell }=1} =\\begin{pmatrix}\n h^{{\\ell }m}_{ab} Y^{{\\ell }m}   &   &  &  & j^{{\\ell }m}_a Y_B^{{\\ell }m} \\\\\n\\cr  \n j^{{\\ell }m}_a Y_B^{{\\ell }m}  &   &  &  & r^2 K^{{\\ell }m} \\Omega_{AB} Y^{{\\ell }m} \\end{pmatrix}^{(+)} + \\begin{pmatrix}\n0   &   &  &  &h^{{\\ell }m}_a X_B^{{\\ell }m} \\\\\n\\cr  \nh^{{\\ell }m}_a X_B^{{\\ell }m}  &   &  &  & 0 \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nTherefore from 10-2 =8 states we subtract a double set of 4 symmetries, and find no degrees of freedom for ${\\ell }=1$ since  8-8=0.\n\nAt  ${\\ell}=0$  $Y_{AB}^{00}= X_{AB}^{00}=0$, the terms in red are absent, but also $Y_{A}^{00}= X_{A}^{00}=0$, all blue terms are absent. \n\\begin{equation}\nh_{\\mu\\nu}^{{\\ell }=0} =\\begin{pmatrix}\n h^{{\\ell }m}_{ab} Y^{{\\ell }m}   &   &  &  & 0 \\\\\n\\cr  \n0  &   &  &  & r^2 K^{{\\ell }m} \\Omega_{AB} Y^{{\\ell }m} \\end{pmatrix}^{(+)} + \\begin{pmatrix}\n0   &   &  &  &0 \\\\\n\\cr  \n0  &   &  &  & 0 \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\n\\begin{equation}\n\\xi_{\\mu}^{{\\ell }=0} =\\begin{pmatrix}\n\\xi^{{\\ell }m}_{a} Y^{{\\ell }m}   \\\\\n\\cr  \n0  \\end{pmatrix}^{(+)}  \\, + \\begin{pmatrix}\n0    \\\\\n\\cr  \n0 \\end{pmatrix}^{(-)} \\, .\n\\end{equation}\nWe are left with 4 fields and  2 gauge symmetries, there are no degrees of freedom for ${\\ell }=0$: 4-4=0.\n\n\n\\section{Quantization of Gravity in Spherical Harmonics Basis in the Flat Background}\nThe procedure of Lagrangian quantization performed in \\cite{Kallosh:2021ors} as well as the values of the unitary quadratic Hamiltonians presented in this paper, have a smooth limit from the Schwarzschild background to a flat one. In Schwarzschild  coordinates this means that the limit $M \\rightarrow 0$ is regular.\n\n In particular,   Zerilli-Moncrief function for ${\\ell }\\geq 2$ in  Regge-Wheeler gauge in the limit  $M \\rightarrow 0$  is \n\\begin{equation}\n\\Psi_{\\rm even}^{{\\ell }m} = \\frac{2r}{{\\ell }({\\ell }+1)} \\biggl[ {K}   \n+ \\frac{2}{({\\ell }-1)({\\ell }+2)} \\biggl( {h}_{rr} \n- r  {K}_{,r} \\biggr) \\biggr], \\qquad {\\ell }\\geq 2\n\\end{equation}\nThe Cunningham-Price-Moncrief function is\n\\begin{equation}\n\\Psi_{\\rm odd}^{{\\ell } m}  = \\frac{2r}{({\\ell }-1) ({\\ell }+2) } \\biggl( \n {h}_{t , r} ^{{\\ell }m} - \n {h}_{r,t}^{{\\ell }m}  - \\frac{2}{r} {h}_t^{{\\ell }m}  \n\\biggr \n), \n \\qquad {\\ell }\\geq 2 \\end{equation}\n The quadrartic part of the Hamiltonian in both cases  is\n \\begin{equation}\n H_{\\rm even\/odd}= {1\\over 2} \\sum_{{\\ell }\\geq 2, m} \n \\int \\Big [dr ( P^{{\\ell }, m } )^{2 }_{\\rm even\/odd}+  (Q_{, r}^{{\\ell }, m })_{\\rm even\/odd}^2 +   \\frac{{\\ell }({\\ell }+1) }{r^2} \n (Q^{{\\ell }, m })_{\\rm even\/odd}^2\\Big ]\n\\label{HamFlat} \\end{equation}\nHere $Q_{\\rm even\/odd} = \\Psi _{\\rm even\/odd}$ and $P_{\\rm even\/odd}$ is the corresponding canonical conjugate. At the quadratic level these are the only 2 physical states which appear in the unitary Hamiltonian.\n\nThe higher order terms in the each of the quantized actions, at the black hole background and in the flat background still have to be constructed.\n\n\\section{A comment on Regge-Wheeler and  Teukolsky formalism and gravity waves}\n\n\nThe Cunningham-Price-Moncrief (CPM) master function and the Zerilli-Moncrief (ZM) master function, which were identified here as canonical variables  in the gravity,  appear to play some role also in a more interesting case of the Kerr black holes. Namely, as pointed out in a  review \\cite{Pound:2021qin}, there is a relation via Chandrasekhar transformation between these functions and Teukolsky radial function. Note that \nTeukolsky equations for the Weyl tensor components  use the expansion in terms of the spin-weighted spheroidal harmonics. Such and expansion for the metric starts with ${\\ell }=2$.\n\nThere is also an interesting relation between the metric perturbation far from the source and our canonical variables in the generalized Regge-Wheeler gauge. Namely, according to  \\cite{Pound:2021qin} the gravitational wave strain can be determined directly from CPM and ZM functions of the metric. Using the \n Chandrasekhar transformation between these functions and Teukolsky radial function, and some properties of $\\psi_4= C_{n\\bar m n\\bar m}$ the gravitational strain was given as\n\n\n\\begin{equation}\nr(h_+ -i h_x) =\\sum_{{\\ell }\\geq 2} \\sum_{ |m|  \\leq {\\ell }} {D\\over 2} \\Big (\\Psi_{\\rm even}^{{\\ell }m}- i \\Psi_{\\rm even}^{{\\ell }m}\\Big ) \\, \\hskip 1 mm   {}_{- 2}  Y_{{\\ell }, m } (\\theta, \\phi)\n\\end{equation}\nwhere ${}_{- 2}  Y_{{\\ell }, m } (\\theta, \\phi)$ is the the spin-weighted spheroidal harmonic.\nThat equality holds in the limit $r  \\rightarrow  \\infty $  (at fixed $u=t-r_*$).\nHere the constant\n\\begin{equation}\nD= \\sqrt {({\\ell }-1) ({\\ell }+1) ({\\ell }+1)}\n\\end{equation}\nis the Schwarzschild limit of the constant that appears in the Teukolsky-Starobinsky identities. Clearly, the cases ${\\ell }=0,1$ drop from the formula for the gravitational waves. This is in agreement with the  fact established in this paper that these modes have no physical degrees of freedom.\n\n\n\n\\section{Summary} \n\n\n\nIn this note we have counted the number of physical quantized degrees of freedom  of Einstein gravity in spherical harmonic basis using the standard formula:  this number is given by $n-k$,  where  $n+k$ is the number of components of gauge fields and the  gauge theory has $k$ gauge symmetries. For example, in 4D the graviton has  $n+k= 6+4= 10$ components and there are $k=4$  gauge symmetries. The  number of physical degrees of freedom is $n-k= (n+k) - 2k= 10-8=2$.\n\n\nIn spherical harmonic basis we have found that for each ${\\ell }, m$ in ${\\ell } \\geq 2$ sector there is one degree of freedom for even parity states and one degree of freedom for odd parity states. In ${\\ell }<2$ sector of gravity we have found that there are no physical degrees of freedom.\n\nTo construct the Hamiltonian we start with the Regge-Wheeler formulation \\cite{Regge:1957td,Zerilli:1971wd,Martel:2005ir} of   Einstein gravity in spherical harmonic basis in the background of a Schwarzschild black hole. The part of the action $S(g+h)$ quadratic in perturbations $h_{\\mu\\nu}$ in eq. \\rf{quad} can be presented in  spherical harmonic basis using the explicit form of equations of motion linear in perturbations, as shown in eq. \\rf{EOM}. We take these explicit expressions $Q^{\\mu\\nu} = {\\delta S (g, h) \\over \\delta h_{\\mu\\nu}} $, which are linear in $h_{\\mu\\nu}$, \nfrom \\cite{Martel:2005ir}, and reconstruct  the part of the action $S(g+h)$ quadratic in perturbations $h_{\\mu\\nu}$. We impose the generalized Regge-Wheeler gauge\n \\cite{Kallosh:2021ors}. The action quadratic in fields we take in Schwarzschild  coordinates and proceed with canonical quantization, defining canonical momenta and constraints. \n\nFor ${\\ell } \\geq 2$ fields the procedure leads to one independent degree of freedom for even and one for odd modes in each case with ${\\ell }, m$, in agreement with the counting of physical degrees of freedom. We conclude that up to a canonical transformation such a Hamiltonian is equivalent to the one presented in \\cite{Moncrief:1974am} where the corresponding canonical variables are Zerilli-Moncrief function \\cite{Zerilli:1971wd,Moncrief:1974am} for even modes and a  Cunningham-Price-Moncrief function \\cite{Cunningham:1978zfa} for odd modes. In \\cite{Moncrief:1974am} the modes with ${\\ell }<2$ were not studied.\n\nWe apply our method also for  ${\\ell }<2$ modes. In each sector for ${\\ell }=1$, even and odd case and for \n${\\ell }=0$ we first reproduce the action from the explicit expressions $Q^{\\mu\\nu} = {\\delta S (g, h) \\over \\delta h_{\\mu\\nu}} $  linear in $h_{\\mu\\nu}$. We use the gauge-fixing condition for low multipoles in  \\cite{Kallosh:2021ors} and identify the canonical variables and constraints. In each case the conclusion is that there are no independent unconstrained canonical variables suitable for the quantized Hamiltonian. This is again in agreement with the counting of degrees of freedom performed earlier. \n\n\nThe original goal of this investigation was to develop a consistent method of quantization of gravitational field in the background  of a Schwarzschild black hole  \\cite{Kallosh:2021ors}.  However, we found that in Schwarzschild  coordinates the limit $M \\rightarrow 0$ is regular, and therefore the quantization procedure is valid in the Minkowski background as well. In this paper we found the  Hamiltonian describing unitary evolution of gravitational perturbations in spherical coordinates, which equally well applies to quantization of gravity in Minkowski  background as well as in the Schwarzschild black hole background. The choice of the generalized Regge-Wheeler gauge in \\cite{Kallosh:2021ors} where the gravity Hamiltonian is unitary requires to use the spherical harmonic basis for the metric perturbations. This {\\it unitary gauge} is a Regge-Wheeler gauge $G^{\\ell m(+)}=j_a^{\\ell m(+)}=h_2^{\\ell m(-)}=0$ for ${\\ell } \\geq 2$. For ${\\ell } =1$ it is $j_{a}^{1 m(+)}= K^{1m(+)}=h_{r}^{1 m(-)}=0$ and for ${\\ell } =0$ it is $K^{00}=h_{tr}^{00}=0$.\n\nIn this generalized Regge-Wheeler gauge, the quadratic part of the Hamiltonian for ${\\ell }<2$ modes is vanishing, whereas for ${\\ell }\\geq 2$ it is given in eqs. \\rf{Heven}, \\rf{Hodd} in the black hole background and in eq. \\rf{HamFlat} in Minkowski background. \n\n\\section*{Acknowledgement}\nI am   grateful to  A. Barvinsky,  E. Coleman, A. Linde,  E. Poisson,  A. Rahman, P. Stamp,  A. Starobinsky, A. Vainshtein,  A. Van Proeyen  and I. Volovich  for stimulating and helpful discussions.  \nI am supported by the SITP, by the US National Science Foundation Grant PHY-2014215 and  by the  Simons Foundation Origins of the Universe program (Modern Inflationary Cosmology collaboration).  \n \n\n\n\\\n\n\\\n\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe \\LaTeXe{} document class {\\tt appolb.cls} should be used \nby starting the file with\\\\\n{\\tt\\verb|\\documentclass{appolb}|}\n\nOur main goal is to let the authors see how the text and equations\nfit to our page layout --- the text column size is 126 mm\n$\\times$ 190 mm. The style is very similar to the original Latex\n{\\tt article}, \\ie most of the commands are used in the same way although\nsome of them result in a different text formatting.\nThere are also some new commands, which are described below. \n\n\\section{Options}\nOptional parameters to the {\\tt appolb} class can be given, as usually, in square\nbrackets, \\eg\\\\\n{\\tt\\verb|\\documentclass[letterpaper,draft]{appolb}|}\\\\\nDefault options are: {\\tt a4paper,final}.\n\n{\\parindent=0pt\\obeylines\nAvailable options:\n{\\tt draft} or {\\tt final} --- show or hide the overfull rule\n{\\tt letterpaper} or {\\tt a4paper} --- select paper size\n}\n\n\\section{Commands}\n\\parindent=0pt\n{\\tt\\verb|\\eqsec|}\n\nCall this macro before the first {\\tt\\verb|\\section|}\ncommand if you want equations numbered as \n(SectionNumber.EqNumber).\nYou can uncomment line \\the\\eLiNe\\ of this file \n({\\tt\\jobname.tex}) to see the effect.\n\n\\subsection{Shortcuts}\n{\\obeylines\n{\\tt \\verb|\\ie|} gives: \\ie\n{\\tt \\verb|\\eg|} gives: \\eg\n{\\tt \\verb|\\cf|} gives: \\cf\n}\nThe macros provide appropriate spacing\nwithout the need for any curly braces \\{\\}.\n\n\\subsection{Math mode operators}\n{\\tt \\verb|\\Tr|} gives: $\\Tr$\n\n{\\tt \\verb|\\e|} gives: $\\e$ --- straight `e' in math mode.\n\n\\subsection{{\\tt eqletters} environment}\n\nEnumarate equations with\na number and a lower-case letter, \\eg\n\\begin{eqletters}\n\\label{myeq}\n\\begin{eqnarray}\nA_1 &=& F(1)\\,,\n\\label{me1}\n\\\\\nA_2 &=& F(2)\\,.\n\\label{me2}\n\\end{eqnarray}\nAs long as the {\\tt eqletters} environment is active all equations are\nnumbered with letters, \\eg\n\\begin{equation}\nL = \\Half a = \\half A\n\\end{equation}\n\\end{eqletters}\n\nEquations (\\ref{me1}) and (\\ref{me2}) can be referenced as Eqs. (\\ref{myeq}).\nThe {\\tt \\verb|\\label|} statement used to generate the latter reference\nmust be placed outside any {\\tt eqnarray} or {\\tt equation} environment.\n\n\\end{document}\n\n\n\\section{Introduction}\n\nThese lectures concern the properties of strongly interacting matter at high energy density.  Such matter occurs in a  number of contexts.  The high density partonic matter that controls the early stages of hadronic collisions at very high energies is largely made of very coherent gluonic fields. In a single hadron, such matter forms the small x part of a wavefunction, a Color Glass Condensate.  After a collision of two hadrons, this matter almost instantaneously is transformed into longitudinal color electric and color magnetic fields.  The ensemble of these fields in their early time evolution is called the Glasma. The decay products of these fields thermalize and form a high temperature gas of quarks and gluons, the Quark Gluon Plasma.  In collisions at lower energy, and perhaps in naturally occurring objects such as neutron stars, there is high baryon density matter at low temperature.  This is Quarkyonic matter.\n\nThere is a very well developed literature concerning these various forms of matter.  It is not the purpose of these lectures to provide a comprehensive review.  I will concentrate on motivating and describing such matter from simple intuitive physical pictures and from simple structural aspects of QCD.  I will attempt at various places to relate what is conjectured or understood about such matter to experimental results from accelerator experiments.\n\n\n\n\\section{Lecture I: The Color Glass Condensate and the Glasma}\n\nThe parton distributions of gluons, valence quarks and sea quarks can be measured for some momentum scale less than a resolution scale $Q$ as a function of their fractional momentum $x$ of a high energy hadron.  The lowest value of $x$ accessible for a fixed hadron energy $E$ is typically\n$x_{min} \\sim \\Lambda_{QCD}\/E_{hadron}$.  The small x limit is therefore the high energy limit.\n\nIt is remarkable that as $x$ is decreased, as we go to the high energy limit, that the gluon density dominates the constituents of a hadron for $x \\le 10^{-1}$.  The various distributions are shown as a function of $x$ in Fig. \\ref{gluondominance}.  The gluon density rises like a power of $x$ like $x^{-\\delta}$, $\\delta \\sim .2-.3$\nat accessible energies\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=gluondominance.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The parton distribution as a function of $x$.}\n     \\label{gluondominance}\n\\end{figure}\nThe area of a hadron grows slowly with energies.  Cross sections grow roughly as $ln^2(1\/x)$ for small x.\nThis means that the rapidly growing gluon distribution results in a high density system of gluons.  At high density, the gluons have small separation and by asymptotic freedom, the intrinsic strength of their interaction must be weak.\n\nA small intrinsic interaction strength does not mean that interactions are weak.  Consider gravity:  The interactions between single protons is very weak, but the force of gravity is long range, and the protons in the earth act coherently, that is always with the same sign.  This results in a large force of gravity.  This can also happen for the gluons inside a hadron, if their interactions are coherent.\n\nTo understand how this might happen, suppose we consider gluons of a fixed size $r_0 \\sim 1\/p_T$ where\n$p_T$ is its transverse momentum.  We assume that at high energy, the gluons have been Lorentz contracted into a thin sheet, so we need only consider the distribution of gluons in the transverse plane.  If\nwe start with a low density of gluons at some energy, and then evolve to higher energy, the density of gluons increases.  When the density is of order one gluon per size of the gluon, the interaction remains weak because of asymptotic freedom.  When the density is of order $1\/\\alpha_S$, the coherent interactions are strong, and adding another gluon to the system is resisted by a force of order $1$.  The gluons act as hard spheres. One can add no more gluons to the system of this size.  It is however possible to add in smaller gluons, in the space between the closely packed gluons of size $r_0$.  This is shown in Fig. \\ref{saturation}\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=saturation.pdf, width=0.60\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  Increasing the gluon density in a saturated hadron when going to higher energy.}\n     \\label{saturation}\n\\end{figure}\n\nThe physical picture we derive means that below a certain momentum scale, the saturation scale $Q_{sat}$,\nthe gluon density is saturated and above this scale it is diffuse.  The saturation momentum scale grows with energy and need not itself saturate\\cite{Gribov:1984tu}-\\cite{McLerran:1993ka}.\n\nThe high phase space density of gluons, $dN\/dyd^2p_Td^2r_T \\sim 1\/\\alpha_S$ suggests that one can describe the gluons as a classical field.  A phase space density has a quantum mechanical interpretation\nas density of occupation of quantum mechanical states.  When the occupation number is large, one is in the classical limit.\n\nOne can imagine this high density gluon field generated from higher momentum partons.  We introduce the idea of sources corresponding to high $x$ partons and fields as low $x$ partons.  Because the high $x$ parton sources are fast moving, their evolution in time is Lorentz time dilated.  The gluon field produced by these sources is therefore static and evolves slowly compared to its natural time scale of evolution.  This ultimately means that the different configurations of sources are summed over incoherently, as in a spin glass. \n\nWe call this high energy density configuration of colored fields a Color Glass Condensate.  The word color is because the gluons that make it are colored.  The word condensate is used because the phase space density of gluons is large, and because this density is generated spontaneously.  The word glass is used because the typical time scale of evolution of the classical fields is short compared to the Lorentz time dilated scales associated with the sources of color.\n\nThere is an elaborate literature on the Color Glass Condensate and an excellent review is by Iancu and Venugopalan\\cite{Iancu:2003xm}.  Evolution of the CGC to small values of x is understood, as well as many relationships between deep inelastic scattering, deep inelastic diffraction and high energy nucleus-nucleus,\nproton-nucleus and proton-proton scattering.  The CGC is a universal form of matter in the high energy limit.  The theoretical ideas underlying the CGC are largely unchallenged as a description of the high energy limit of QCD, but the issue of when the approximation appropriate for the high energy limit are valid remains contentious. \n\nIn the description of high energy hadron hadron collisions, we consider the collision of two sheets of CGC as shown in Fig. \\ref{sheets}.  The color electric and color magnetic fields of the CGC are visualized as sheets of Lenard-Wiechart potentials.  These are classical gluon fields whose polarization and color are random, with an intensity distribution determined by the underlying theory of the CGC.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=sheets.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The collision of two sheets of CGC.}\n     \\label{sheets}\n\\end{figure}\n\nUpon collision of these sheets, the sheets become charged with color magnetic and color electric charge distributions of equal magnitude but opposite sign locally in the transverse plane of the sheets\\cite{Kovner:1995ja}-\\cite{Lappi:2006fp}. In the high\nenergy limit sources of color electric and color magnetic field must be treated on an equal footing because of the self duality of QCD.  This induced charge density produces longitudinal color electric and color magnetic fields between the two sheets.  These fields are longitudinally boost invariant and therefore have the correct structure to account for Bjorken's initial conditions in heavy ion collisions\\cite{Bjorken:1982qr}.  The typical transverse length scale over which the flux tubes vary is $1\/Q_{sat}$. The initial density of produced gluons is on dimensional grounds\n\\begin{equation}\n  {1 \\over {\\pi R^2}} {{dN} \\over {dy}} \\sim {{Q_{sat}^2} \\over {\\alpha_S}}\n\\end{equation}\nBecause there are both color electric and color magnetic fields, there is a topological charge density of maximal strength induced $FF^D \\sim Q_{sat}^2\/\\alpha_S$  \n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=glasma.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  Glasma flux tube produces after the collision.}\n     \\label{glasma}\n\\end{figure}\n\nThe decay of products of the Glasma is what presumably makes a thermalized Quark Gluon Plasma.  It is not clear how this thermalization takes place.  It is quite likely that in the decay of these fields, a turbulent fluid arises, and perhaps this fluid can generate an expansion dynamics similar to that of a thermalized QGP for at least some time\\cite{Dusling:2010rm}.\n\n\\subsection{The CGC and Electron-Hadron Scattering}\n\nIf the only momentum scale that controls high energy scattering is the saturation momentum,\nthen there will be scaling\\cite{Stasto:2000er}.  In particular, the cross section for deep inelastic scattering will be a function\n\\begin{equation}\n\\sigma_{\\gamma^*p} \\sim F(Q^2\/Q_{sat}^2)\n\\end{equation}\nrather than a function of $Q^2$ and $x$ independently.  The x dependence of the saturation momentum may be determined empirically as $Q_{sat}^2 \\sim Q_0^2\/x^\\delta$ where $\\delta = 0.2-0.3$, which is consistent with analysis of evolution equations\\cite{Balitsky:1995ub}-\\cite{Mueller:2002zm}.\nThe scaling relationship can be derived from the classical theory for $Q^2 \\le Q_{sat}^2$.  It can further be shown to extend over a much larger range of $Q^2$\\cite{Iancu:2002tr}.  For large values of $Q^2$ this scaling is a consequence of the linear evolution equations, but the global structure is determined by the physics of saturation.  Such a simple scaling relationship describes deep inelastic scattering data for $ x \\le 10^{-2}$.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=geometric.pdf, width=0.70\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The geometric scaling in deep inelastic scattering.}\n     \\label{geometric}\n\\end{figure}\n\nUsing evolution equations for the CGC including the effects of running coupling constant\\cite{Albacete:2007yr}-\\cite{Balitsky:2008zza}, one can compute\ndeep inelastic scattering structure functions at small x\\cite{Albacete:2009fh}.  This involves very few parameters, and provides comprehensive description of deep inelastic scattering data at $x \\le 10^{-2}$.\nThe description of $F_2$ in deep inelastic scattering is shown in Fig. \\ref{f2}.  It should be noted that in\nthe CGC description of deep inelastic scattering, the gluon distribution function is the Fock space distribution of gluons inside a hadron.  It can never become negative.  In the description of the $F_2$ data,\nthe gluon distribution function is not becoming small at small $Q^2$ as is the case in some linear evolution fits.  This is intuitively reasonable since we have no reason to expect that the Fock space distribtuion of gluons in a hadron should become small at small $Q^2$.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=f2.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The CGC description of F2 data in deep inelastic scattering.}\n     \\label{f2}\n\\end{figure}\n\nThe Color Glass Condensate description may also be applied to diffractive deep inelastic scattering,\nand with the same parameters that describe deep inelastic scattering does an excellent job of describing\nthe data.  In addition, there are measurements of the longitudinal structure function, a quantity directly proportional to the gluon density.  Conventional descriptions that use linear DGLAP evolution equations are somewhat challenged by this data, but the CGC description naturally fits the data.  \n\nTo summarize, the CGC description of deep inelastic scattering at small x naturally describes $F_2$, $F_L$\nand diffractive data. It is a successful phenomenology  Why is the CGC therefore not accepted as the standard description? The problem is that the linear evolution DGLAP descriptions describe $F_2$ adequately, except in the region where the perturbative computations most probably breaks down.  They do not do a very good job on the low $Q^2$ $F_L$ data, but this is where there is a fair uncertainty in the data.\nThe diffractive data is naturally described in the CGC framework, but there are other successful models.\nUltimately, there is no consensus within the deep inelastic scattering community that the CGC is needed in order to describe the data.\n\n\\subsection{The CGC and Heavy Ion Collisions}\n\n\\subsubsection{Multiplcities in RHIC Nulcear Collsions}\n\nOne of the early successes of the CGC was the description of multiplicity distributions in deep inelastic scattering\\cite{Kharzeev:2000ph}-\\cite{Kharzeev:2001yq}.  Recall that the phase space distribution of gluons up to the saturation momentum is of order $Q_{sat}^2\/\\alpha_S(Q_{sat})$.  We will assume that the distribution of\ninitially produced gluons is proportional to this distribution of gluons in the hadron wavefunctions of the colliding nuclei and further,that the multiplicity of produced gluon is proportional to the final\nstate distribution of pions.  We get\n\\begin{equation}\n{1 \\over \\sigma}~  {{dN} \\over {dy}} \\sim  {1 \\over{ \\alpha_S(Q_{sat})}}Q_{sat}^2 \\sim A^{1\/3} x^{-\\delta}\n\\end{equation}  \nHere $\\sigma$ is the area of overlap of the two nuclei in the collision and A the number of nucleons that participate in the collision.  $\\sigma ~Q_{sat}^2 \\sim A$\nat low energies assumes no shadowing of nucleon parton distributions and is consistent with\ninformation concerning deep inelastic scattering on nuclear targets.  In the collisions of nuclei one can directly measure the number of nucleonic participants in the collisions, a number that varies with the centrality of the collision. One can then compare the central region multiplicity with the number of participants so determined.  Such a comparison is shown in\\cite{Adcox:2004mh}-\\cite{Back:2004je} Fig. \\ref{aa} .\nThe saturation description of Kharzeev and Nardi provides a good description of the centrality dependence of the collisions\\cite{Kharzeev:2000ph}.  It also does well with the energy dependence.  Refinements of this description can provide a good description of the rapidity distribution of produced particles\\cite{Kharzeev:2001yq}.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=aa.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The multiplicity as a function of the number of nucleon participants in heavy ion collisions.}\n     \\label{aa}\n\\end{figure}\n\n\\subsubsection{Limiting Fragmentation}\n\nA general feature of high energy hadronic scattering is limiting fragmentation.  If one measures the distribution of particles as a function of rapidity up to some fixed rapidity from the rapidity of one of the colliding particles, then the distribution is independent of collision energy.  The region over which this scaling occurs\nincreases as the energy of the colliding particles increases.  Such scaling is shown in Fig. \\ref{limfrag}.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=limfrag.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  Limiting fragmentation in RHIC nuclear  collisions.}\n     \\label{limfrag}\n\\end{figure}\nSuch limiting fragmentation is natural in the CGC approach.  For example in Fig. \\ref{limfrag},\nwe see that the region of limiting fragmentation increases as beam energy increases\\cite{Back:2004je}.  If we think of the region where there is limiting fragmentation as sources for fields at small more central rapidities, then we see that going to higher energies corresponds to treating a larger region as sources.  In a renormalization group language, this simply means that one is integrating out fluctuations at less central rapidities, to generate an effective theory for the particles at more central rapidity.  A quantitative description of limiting fragmentation within the theory of the  CGC is found in Ref.   \\cite{Gelis:2006tb}.\n\n\\subsubsection{Single Particle Distributions in dAu Collisions}\n\nSome of the early predictions of the CGC were generic features of the single particle inclusive distributions\nseen in hadron-nucleus collisions.  There are two competing effects.  The first is multiple scattering of a hadron as it traverses a nucleus.  This effect is included n the CGC gluon distributions as an enhancment\nof the gluon distribution for  $p_T$ at transverse momentum of the order of the saturation momentum,\nwith a corresponding depletion at smaller momentum.  There is little effect at high $p_T$.  The other effect is that in the evolution of the gluon distribution to small $x$, the saturation momentum acts as a cutoff in the\nbremstrahlung like integrals that generate such small x gluons.  Nuclei have a larger saturation momentum\nthan do hadrons, so the small x gluon distribution for nuclei will be suppressed relative to that for hadrons.  Put another way, this effect will generate a suppression for more central collisions.  The sum of these effects is shown in Fig. \\ref{dA}\\cite{Baier:2003hr}-\\cite{Iancu:2004bx}. \nThe different curves correspond to different rapidities of the produced particle,\nbeginning with the top curve being near the fragmentation region of the nucleus.  As one evolve further in rapidity, the enhancement at intermediate transverse momentum disappears and is replaced by a smooth curve with an overall suppression of produced particles.\n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=dA.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The ratio of particles emitted in dA  and AA collisions to that in proton due to CGC effects.}\n     \\label{dA}\n\\end{figure}\n\nThe pattern of suppression suggested by the Color Glass Condensate was first seen in dAu collisions\nin the Brahms collaboration\\cite{Arsene:2004fa}, and later confirmed by the other experiments \\cite{Adcox:2004mh},\\cite{Back:2004je},\\cite{Adams:2005dq}.  The Brahms experiments demonstrated\nthat in the nuclear target fragmentation region that at intermediate $p_T$ there was en enhancement in $R_{dA}$\nas a function of centrality, but in the deuteron fragmentation region, there was a depletion as a function of centrality.  The CGC provided the only model that predicted such an effect, and it remains the only \ntheory that can quantitatively explain the suppression seen in the deuteron fragmentation region.\n\n\\subsubsection{Heavy Quark and $J\\Psi$ Production}\n\nIf the saturation momentum is small compared to a quark mass, it can be treated as very heavy.  It should have perturbative incoherent production cross sections.  If the saturation momentum is large compared to a quark\nmass, the quarks should be thought of as light mass.  Cross sections for production should be coherent,\nand for example in $pA$ collisions, scale as $A^{2\/3}$.   In the deuteron fragmentation region of dAu collisions we would expect suppression of heavy quark and charmonium cross sections relative to the nuclear fragmentation region. \n\\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=jpsi.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The $J\/\\Psi$ production cross section as a function of centrality and rapidity.}\n     \\label{jpsi}\n\\end{figure}\n In Fig. \\ref{jpsi}, the ratio of central to peripheral cross sections for $J\/\\Psi$ production is shown as a function of centrality and rapidity.  Note the strong suppression in the forward region for central collisions, as expected from the CGC.  Precise computations are difficult for the charm quark since its mass is close to the saturation momentum.  Such computations are in agreement with the data at forward rapidity\\cite{Kharzeev:2008nw}-\\cite{Kopeliovich:2010nw}.\n \n \\subsubsection{Two Particle Correlations}\n \n The Glasma flux tubes induced by the collision of two hadrons will generate long range correlations in rapidity.  In heavy ion collisions, this may be seen in forward backward correlations, as measured in STAR.  The correlation increases in strength with higher energy collisions or more central collisions.  This is expected in the CGC-Glasma description because for more central collisions the saturation momentum\n is bigger, so that the system is more correlated.  (The coupling becoming weaker means the system is more\n classical, and therefore the leading order contribution associated with Glasma flux tubes becomes\n relatively more important.)  Such forward-backward correlations are shown in Fig. \\ref{fb} as a function of rapidity and centrality\\cite{:2009dqa}-\\cite{Lappi:2009vb}.  The value of the correlation coefficient b\n can be shown to be bounded $b \\le 1\/2$.\n \\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=fb.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small  The strength of forward backward correlations as a function of rapidity and centrality.}\n     \\label{fb}\n\\end{figure}\n\nSuch two particle correlations in the Glasma can generate ridge like structures seen in two particle correlation experiments in azimuthal angle and rapidity\\cite{Shuryak:2007fu}-\\cite{Dumitru:2008wn}.  The long range rapidity correlation is intrinsic to the Glasma.  The angular correlation might be generated by flow effects at later times in the collision, by opacity and trigger bias effects, or by\nintrinsic angular correlations associated with the decay of Glasma flux tubes\\cite{Gavin:2008ev}-\\cite{Dumitru:2010iy}.\n\n\\subsubsection{The Negative Binomial Distribution}\n\nThe decay of a single Glasma flux tube generates a negative binomial distribution of produced particles\\cite{Gelis:2009wh}.\nA sum of negative binomial distributions is again a negative binomial distribution.  Such oa form of the distribution describes the RHIC data well.  It is difficult with the heavy ion data to isolate those effects due to an intrinsic negative binomial distribution and those due to impact parameter.  It is possible to isolate the effects of impact parameter, but it demands a high statistics study.  \n\n\\subsubsection{Two Particle Azimuthal Angular Correlations in dA Collisions}\n\nThe CGC will de-correlate forward-backward angular correlations when the the transverse momentum of produced particles is of order the saturation momentum\\cite{Kharzeev:2004bw}-\\cite{Albacete:2010rh}.\nThis is because near the produced particles get momentum from the CGC and\ntherefore are not back-to-back correlated.  In dAu collisions such an effect will be largest at forward rapidities near the fragmentation region of the deuteron, since this corresponds to the smallest values of x for the nuclear target.  This kinematic region is least affected by multiple scattering on the nucleus.  This effect has been seen by the STAR and PHENIX collaborations\\cite{Braidot:2010ig}-\\cite{Meredith:2009fp}.  There is a good quantitative description by Tuchun and by Albacete and Marquet,\\cite{Tuchin:2009nf}-\\cite{Albacete:2010rh}  as shown in the figure \\ref{dafb}\n \\begin{figure}[!htb]\n\\begin{center}\n  \\mbox{{\\epsfig{figure=dafb.pdf, width=0.90\\textwidth}}}\n   \\end{center}\n\\caption[*]{ \\small Forward rapidity, forward backward angular correlations in dAu collisions\nas a function of centrality.}\n     \\label{dafb}\n\\end{figure}\n\n\\subsection{Concluding Comments on the CGC and the Glasma}\n\nThere is now a wide variety or experimental data largely consistent with the CGC and Glasma based description.  There is a well developed theoretical framework that provides a robust phenomenology\nof both electro-hadron scattering and hadron scattering,  There are new areas that are developing that I have not had time to discuss.  One is the possibility to see effects of topological charge change in heavy ion collisions, the Chiral Magnetic Effect\\cite{Kharzeev:2007jp}.  Another area is pp collisions at the LHC, where some work concerning recent experimental data was developed at this school\\cite{McLerran:2010ex}.  \n\n\n\\section{Lecture II: Matter at High Temperature: The Quark Gluon Plasma}\n\n\\section{Matter at Finite Temperature}\n\n\\subsection{Introdcution}\n\nIn this lecture I will describe the properties of matter at high temperature.  The discussion here will be theoretical.  There is a wide literature on the phenomenology of the Quark Gluon Plasma and its possible\ndescription of heavy ion collisions at RHIC energies.  The interested reader is referred to that literature.\nI will here develop the ideas of decofinement, chiral symmetry restoration based in part on a simple description using the large number of colors limit of QCD.\n\n\\subsection{Confinement}\n\nThe partition function is\n\\begin{equation}\n Z = Tr~e^{-\\beta H + \\beta \\mu_B N_B}\n\\end{equation}\nwhere the temperature is $T = 1\/\\beta$ and $N_B$ is the baryon number and $\\mu_B$ is the baryon number chemical potential.  Operator expectation values are\n\\begin{equation}\n <O> = {{Tr~ O ~e^{-\\beta H + \\beta \\mu_B N_B}} \\over Z}\n\\end{equation}\n Under the substitution $e^{-\\beta H} \\rightarrow e^{-itH}$, the partition function becomes the time\n evolution operator of QCD.  Therefore, if we change $t \\rightarrow it$,and redefine zeroth\n components of fields by  appropriate factors of i, and introduce Euclidean gamma matrices with anti-commutation relations\n \\begin{equation}\n   \\{ \\gamma^\\mu, \\gamma^\\nu \\} = -2 \\delta^{\\mu \\nu}\n \\end{equation}\n  then for QCD, the partition function has the path integral representation \n  \\begin{equation}\n  Z = \\int~[dA] [d\\overline \\psi ] [d\\psi] exp\\left\\{ -\\int_0^\\beta~ d^4x~\\left( {1 \\over 4 }F^2 \n  +\\overline \\psi \\left[ {1 \\over i} \\gamma \\cdot D + m+ i \\mu_Q \\gamma^0 \\psi \\right]\\right) \\right\\}\n  \\end{equation}\nHere the fermion field is a quark field so that the baryon number chemical potential is\n\\begin{equation}\n  \\mu_Q = {1 \\over N_c} \\mu_B\n\\end{equation}\nThis path integral is in Euclidean space and is computable using Monte Carlo methods when\nthe quark chemical potential vanishes.  If the quark chemical potential is non-zero, various contributions appear with different sign, and the Monte Carlo integrations are poorly convergent.  Boundary conditions\non the fields must be specified on account of the finite length of the integration in time.  They are periodic for Bosons and anti-periodic for Fermions, and follow from the trace in the definition of the partition function.\n\nA straightforward way to probe the confining properties of the QCD matter is to introduce a heavy\ntest quark.  If the free energy of the heavy test quark is infinite, then there is confinement,\nand if it is finite there is deconfinement.  We shall see below that the free energy of an quark added to the system is\n\\begin{equation}\n  e^{-\\beta F_q} = <L>\n  \\label{L}\n\\end{equation}\nwhere\n \\begin{equation}\n    L(\\vec{x}) = {1 \\over N_c} Tr~ P~ e^{i \\int ~dt~ A^0(\\vec{x},t)}\n \\end{equation}\nSo confinement means $<L> = 0$ and deconfinement means that $<L>$ is finite.  The path ordered phase integration which defines the line operator $L$ is shown in Fig. \\ref{line}.  Such a path ordered phase  is called a Polyakov loop or Wilson line.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.50\\textwidth]{line.jpg}\n        \\end{center}\n        \\caption{The contour in the t plane which defines the Polyakov loop.  The space is closed in time because of the periodic boundary conditions imposed by the definition of the partition function.}\n\\label{line}\n\\end{figure}\n\nIt is possible to prove that the free energy of a heavy static quark added to the system is given by Eqn. \\ref{L}\nusing the effective action for a very heavy quark:\n\\begin{equation}\n S_{HQ} = \\int ~dt ~\\overline \\psi (\\vec{x},t) ~{1 \\over i}\\gamma^0 D^0~ \n\\psi (\\vec{x},t).\n\\end{equation}\nThe Yang-Mills action is invariant under gauge transformations that are periodic up to an element of the center of the gauge group.  The center of the gauge group is a set of diagonal matrices matrix $Z_p = e^{2\\pi i p\/N} \\overline I$ where $\\overline I$ is an identity matrix.\nThe quark contribution to the action is not invariant, and  $L \\rightarrow Z_p L$ under this transformation.  In a theory with only dynamical gluons, the energy of a system of $n$ quarks minus antiquarks is invariant under the center symmetry transformation only if $n$ is an integer multiple of $N$.  Therefore, when the center symmetry is realized, the only states of finite free energy\nare baryons plus color singlet mesons.\n\nThe realization of the center symmetry, $L \\rightarrow Z_p L$\nis equivalent to confinement.  This symmetry is like the global rotational symmetry of a spin system, and it may be either realized or broken.   At large separations, the correlation of a line and its adjoint, corresponding to a quark-antiquark pair is\n\\begin{equation}\n        lim_{r \\rightarrow \\infty} <L(r) L^\\dagger (0)> = Ce^{-\\kappa r} + <L(0)><L^\\dagger (0)>\n\\end{equation}        \nsince upon subtracting a mean field term, correlation functions should vanish exponentially.  Since\n\\begin{equation}\n   e^{-\\beta F_{q \\overline q} (r)} = <L(r) L^\\dagger (0)>\n\\end{equation}\nwe see that in the confined phase, where $<L> = 0$, the potential is linear, but in the unconfined phase,\nwhere $<L>$ is non-zero, the potential goes to a constant at large separations.\n\nThe analogy with a spin system is useful.  For the spin system corresponding to QCD\nwithout dynamical quarks,\nthe partition function  can be written as\n\\begin{equation}\n Z = \\int~ [dA]~e^{- {1\\over g^2} S[A]}\n\\end{equation}\nThe effective temperature of the spin system associated with the gluon fields is $T_{eff} \\sim g^2$.\nBy asymptotic freedom of the strong interactions, as real temperature gets larger, the effective temperature gets smaller.  So at large real temperature (small effective temperature) we expect an ordered system, where the $Z_N$ symmetry is broken, and there is deconfinement.  For small real temperature corresponding to large effective temperature, there is disorder or confinement.\n\nThe presence of dynamical fermions breaks the $Z_N$ symmetry.  This is analogous to placing a spin system in an external magnetic field.  There is no longer any symmetry associated with confinement, and the phase transition can disappear.  This is what is believed to happen in QCD for physical masses of quarks.  What was a first order phase transition for the theory in the absence of quarks becomes  a continuous change in the properties of the matter for the theory with quarks.\n\nAnother way to think about the confinement-decofinement transition is a change in the number of degrees of freedom.  At low temperatures, there are light meson degrees of freedom.  Since these\nare confined, the number of degrees of freedom is of order one in the number of colors.  In the unconfined world, there are $2(N_c^2-1)$ gluons, and $4N_cN_f$ fermions where $N_f$ is the number of light mass fermion families.  The energy density scaled by $T^4$ is a dimensionless number and directly proportional to the number of degrees of freedom.  We expect it to have the property shown in Fig. \\ref{et4} for pure QCD in the absence of quarks.  The discontinuity at the deconfinement temperature, $T_d$ is the latent heat of the phase transition.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.60\\textwidth]{et4.jpg}\n        \\end{center}\n        \\caption{The energy density scaled by $T^4$ for QCD in the absence of dynamical quarks.}\n\\label{et4}\n\\end{figure}\n\nThe energy density can be computed using lattice Monte Carlo methods.  The result of such computation is shown in Fig. \\ref{et4lat}.  The discontinuity present for the theory with no quarks becomes a rapid cross over when dynamical quarks are present.\n\nThe large $N_c$ limit gives some insight into the properties of high temperature \nmatter\\cite{'tHooft:1973jz}-\\cite{Thorn:1980iv}. As $N_c \\rightarrow \\infty$, the energy density itself is an order parameter for the decofinement phase transition.  Viewed from the hadronic world, there is an amount of energy density $\\sim N_c^2$ which must be inserted \nto surpass the transition temperature.  At infinite $N_c$ this cannot happen, as this involves an infinite amount of energy.  There is a Hagedorn limiting temperature, which for finite $N_c$ would have been the deconfinement temperature.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.60\\textwidth]{et4lat.jpg}\n        \\end{center}\n        \\caption{The energy density scaled by $T^4$ measured in QCD from lattice Monte Carlo simulation.  Here there are quarks with realistic masses.}\n\\label{et4lat}\n\\end{figure}\n\nThe Hagedorn limiting temperature can be understood from the viewpoint of the hadronic world as arising from an exponentially growing density of states.  In a few paragraphs, we will argue that mesons and glueballs are very weakly interacting in the limit of large $N_c$.  Therefore, the partition function is\n\\begin{equation}\n  Z = \\int~ dm~\\rho (m) e^{-m\/T}\n \\end{equation}\n Taking $\\rho(m) \\sim m^\\alpha e^{\\kappa m}$, so that \n \\begin{equation}\n   <m> \\sim {1 \\over {1\/T-\\kappa}}\n \\end{equation} \ndiverges when $T \\rightarrow 1\/\\kappa$\n\n\\subsection{A Brief Review of the Large $N_c$ Limit}\n\nThe large $N_c$ limit for an interacting theory takes $N_c \\rightarrow \\infty $ with the 't Hooft coupling\n$g^2_{'t Hooft} = g^2 N_c$ finite.  This approximation has the advantage that the interactions among quarks and gluons simplify.  For example, at finite temperature, the disappearance of confinement\nis associated with Debye screening by gluon loops, as shown in Fig. \\ref{loop}a.  This diagram generates a screening mass of order $M^2_{screening} \\sim g^2_{'t Hooft} T^2$.   On the other hand the quark loop contribution is smaller by a power of $N_c$ and vanishes in the large $N_c$ limit.\n\\begin{figure}[htbp]\n\\begin{center}\n\\begin{tabular} {l l l}\n\\includegraphics[width=0.50\\textwidth] {glueloop.jpg}  & &\n\\includegraphics[width=0.46\\textwidth] {quarkloop.jpg} \\\\\n& & \\\\\na & & b \\\\\n\\end{tabular}\n\\end{center}\n\\caption{a:  The gluon loop contribution to the heavy quark potential.  b:  The quark loop contribution to the potential}\n\\label{loop}\n\\end{figure}\n\nTo understand interactions, consider Fig. \\ref{int}a.  This corresponds to a mesonic current-current interaction through quarks.  In powers of $N_c$, it is of order $N_c$.  Gluon interactions will not change this overall factor.  The three current interaction is also of order $N_c$ as shown in Fig. \\ref{int}b.  The three meson vertex, $G$ which remains after amputating the external lines, is therefore of order $1\/\\sqrt{N_c}$.  A similar argument shows that the four meson interaction is of order $1\/N_c$.\nUsing the same arguments, one can show that the 3 glueball vertex is of order $1\/N_c$ and the four glueball interaction of order\n$1\/N_c^2$.\n\nThese arguments show that QCD at large $N_c$ becomes a theory of non-interacting mesons and glueballs.  There are an infinite number of such states because excitations can never decay.  In fact, the spectrum of mesons seen in nature does look to a fair approximation like non-interacting particles.  Widths of resonances are typically of order $200~ MeV$, for resonances with masses up to several $GeV$.\n\\begin{figure}[htbp]\n\\begin{center}\n\na  \\includegraphics[width=0.75\\textwidth ] {loop1.jpg}  \\\\\n ~ \\\\\n~~b ~~~ \\includegraphics[width=0.75\\textwidth ] {loop2.jpg} \\\\\n\n\\end{center}\n\\caption{a:  The quark loop corresponding to a current-current interaction.  b: A quark loop corresponding to a three current interaction.}\n\\label{int}\n\\end{figure}\n\\subsection{Mass Generation and Chiral Symmetry Breaking}\n\nQCD in the limit of zero quark masses has a $U(1) \\times SU_L(2) \\times SU_R(2)$ symmetry.  (The $U_5(1)$ symmetry is explicitly broken due to the axial anomaly.)  Since the pion field, $\\overline \\psi \\tau^a \\gamma_5 \\psi$ is generated by an $SU_{L-R}(2)$ transformation of the sigma field, $\\overline \\psi \\psi$, the energy (or potential) in the space of the pion-sigma field is degenerate under this transformation.\nIn nature, pions have anomalously low masses.  This is believed to be a consequence of chiral symmetry breaking, where the $\\sigma $ field acquires an expectation value, and the pion fields are Goldstone bosons associated with the degeneracy of the potential under the chiral rotations.\n\nSuch symmetry breaking can occur if the energy of a particle-antiparticle pair is less than zero, as shown in Fig. \\ref{hole}.  On the left of this figure is the naive vacuum where the negative energy states associated with quark are filled.  The right hand side of the figure corresponds to a particle hole excitation, corresponding to a sigma meson.  Remember that a hole in the negative energy sea corresponds to an antiparticle with the opposite momentum and energy. If the $\\sigma$ meson excitation has negative energy, the system is unstable with respect to forming a condensate of these mesons.\n\\begin{figure}[ht]\n        \\begin{center}\n        \\includegraphics[width=0.40\\textwidth]{hole.jpg}\n        \\end{center}\n        \\caption{The energy levels of the Dirac equation.  Unfilled states are open circles and filled states are solid circles.  For the free Dirac equation, negative energy states are filled and positive energy states are unoccupied, as shown on the left hand side.  A mesonic excitation corresponding to a particle hole pair is shown on the right hand side.}\n\\label{hole}\n\\end{figure}\n\nAt sufficiently high temperature, the chiral condensate might melt.  Indeed this occurs\\cite{Karsch:2001cy} .For QCD,\nthe chiral and deconfinement phase transition occur at the same temperature.  At a temperature of about $170 - 200~ MeV$, both the linear potential disappears and chiral symmetry is restored.  It is difficult to make a precise statement about the indentification of the chiral and deconfinement phase transitions,\nsince as argued above, for QCD with quarks, there is not a real phase transition associated with deconfinement\\cite{Bazavov:2009zn}-\\cite{Borsanyi:2010cj}.\nAlso, when quarks have finite masses, as they do in nature, chiral symmetry is not an exact symmetry, and there need be no strict phase transition associated with its restoration.  Nevertheless, the cross over is quite rapid, and there are rapid changes in the both the potential and the sigma condensate $<\\overline \\psi \\psi >$ at temperatures which are in a narrow range.\n\n\\section{Lecture III: Matter at High Baryon Number Density: Quarkyonic Matter}\n\nI now turn to a discussion of the phase diagram of QCD at finite baryon number density.\n\nIn the large $N_c$ limit of QCD, the nucleon mass is of order $N_c$\\cite{'tHooft:1973jz}-\\cite{Witten:1979kh}.  This means that in the confined phase of hadronic matter, for baryon chemical potential $\\mu_B \\le M_N$, the baryon number density is\nessentially zero:\n\\begin{equation}\n   <N_B> \\sim e^{(\\mu_B-M_N)\/T} \\sim e^{-N_c}\n\\end{equation}   \nFor temperatures above the de-confinement phase transition the baryon number is non-zero since there the baryon number density is controlled by $e^{-M_q\/T} \\sim 1$, and quark masses are independent of\n$N_c$.  For sufficiently large chemical potential the baryon number density can be nonzero also. The Hadronic Matter phase of QCD is characterized in large $N_c$ by zero baryon number density, but at higher density there is a new phase.\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.30]{phasediagram}\t       \n\\caption[]{The revised phase diagram of QCD}\n\\label{phasediagram}\n\\end{figure}\n\nIn the large $N_c$ limit, fermion loops are suppressed by a factor of $1\/N_c$.  Therefore the contribution to Debye screening from quarks cannot affect the quark potential until\n\\begin{equation}\n  M_{Debye}^2 \\sim \\alpha_{t'Hooft}~ \\mu_{quark}^2\/N_c \\sim \\Lambda^2_{QCD}\n\\end{equation}\nHere the quark chemical potential is $\\mu_B = N_c \\mu_{quark}$.  The relationship involving the Debye mass means there is a region parametrically large  chemical potential $M_N \\le \\mu_B \\le \\sqrt{N_c}M_N$ where matter is confined, and has finite baryon number.  This matter is different than either the Hadronic Matter or the De-Confined Phases.  It is called Quarkyonic because it exists at densities parametrically large compared to the QCD scale, where quark degrees of freedom are important,\nbut it is also confined so the degrees of freedom may be thought of also as those of confined \nbaryons\\cite{McLerran:2007qj}-\\cite{Hidaka:2008yy}.\n\nThe width of the transition region between the Hadronic phase and the Quarkyonic phase is estimated\nby requiring that the baryon number density become of order $N_B\/V \\sim k_{Fermi}^3 \\sim \\Lambda_{QCD}^3$.  Recall that the baryon chemical potential is $\\mu_B \\sim M_N + k_f^2\/2M_N$ for small $k_F$, so that the width of the transition in $\\mu_B$ is very narrow, of order $1\/N_c$. This is $\\delta \\mu_{qaurk} \\sim  1\/N_c^2$ when expressed in terms of $\\mu_{quark}$ which is the finite variable in the large $N_c$ limit.\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.60]{line_pbm}\t       \n\\caption[]{Chemical potentials and temperatures at decoupling.}\n\\label{line}\n\\end{figure}\n\n\nThe transition from Hadronic Matter to that of the Quark Gluon Plasma may be thought of as a change in the number of degrees of freedom of matter.  Hadronic Matter at low temperatures has  3 pion degrees of freedom.  The quark gluon plasma has of order $2(N_c^2-1)$ degrees of freedom corresponding to gluons and $4 N_c$ degrees of freedom for each light mass quark.  The change in degrees of freedom is of order $N_c^2$ in the large $N_c$ limit.  At very high baryon number densities, the quarks in the Fermi sea interact at short distances, and although strictly speaking are confined, behave like free quarks.  The number of degrees of freedom is therefore of order $N_c$.  Each phase has different numbers of degrees of freedom, and is presumably separated from the other by a rapid crossover.\n \\begin{figure}\n\\centering\n\\includegraphics[scale=0.60]{horn}\t       \n\\caption[]{Ratios of abundances of various particles .}\n\\label{horn}\n\\end{figure}\nQuarkyonic matter is confined and therefore thermal excitations such as mesons, glueballs, and Fermi surface excitations must be thought of as confined.  The quarks in the Fermi sea are effectively weakly interacting since their interactions take place at short distances.  So in some sense, the matter is ``de-confined\" quarks in the Fermi sea with confined glueball, mesons and Fermi surface excitations\\cite{Castorina:2010gy}.\n\nIn Hadronic Matter, chiral symmetry is broken and in Deconfined Matter it is broken.  In Quarkyonic Matter chiral symmetry is broken by the formation of charge density waves from binding of quark and quark hole excitations near the Fermi surface\\cite{Deryagin:1992rw}.  In order that the quark hole have small relative momentum to the quark, the quark hole must have momentum opposite to that of the quark.  This means the quark-quark hole excitation has total net momentum, and therefore the finite wavelength of the corresponding bound state leads to a breaking of translational invariance.  The chiral condensate turns out to be a chiral spiral where the chiral condensate rotates between different Goldstone boson\nas one moves through the condensate\\cite{Kojo:2009ha}.  Such condensation may lead to novel crystalline structures\\cite{tsvelik}.\n\nA figure of the hypothetical phase diagram of QCD is shown in Fig. \\ref{phasediagram} for $N_c = 3$.  Also shown is the weak liquid-gas phase transition, and the phase associated with color superconductivity.  Although the color superconducting phase cannot coexist with quarkyonic matter in infinite $N_c$, for finite $N_c$ there is such possibility.  The lines on this phase diagram might correspond to true phase transitions or rapid cross overs.  The confinement-deconfinement transition is known to be a cross over.  In the FPP-NJL model\\cite{Fukushima:2003fw}-\\cite{Pisarski:2000eq}, the Hadronic-Quarkyonic transition is first order\\cite{McLerran:2008ua},  but nothing is known from lattice computations.  If as we conjecture, there is region where chiral symmetry is broken by translationally non-invariant modes, then this region must be surrounded by a line of phase transitions.  I call this region Happy Island becuase it is an island of matter in the $\\mu_B-T$ plane.\n\nA remarkable feature of this plot is the triple point where the Hadronic Matter, Deconfined Matter and\nQuarkyonic Matter all meet\\cite{Andronic:2009gj}.  This triple point is reminiscent of the triple point for the liquid, gas and vapor phases of water. \n\nSince we expect a rapid change in the number of degrees of freedom across the transitions between\nthese forms of matter, an expanding system crossing such a transition will undergo much dilution would undergo much dilution at a fixed value of temperature or baryon chemical potential\\cite{BraunMunzinger:1994xr}-\\cite{Heinz:1999kb}.  One might expect in heavy ions to see decoupling of particle number changing processes at this transition, and the abundances of produced particles will be characteristic of the transition.  In Fig. \\ref{line}\n\nIn the Fig. \\ref{line}, the expectations for the confinement-deconfinement transition are shown with the dotted red line.  It is roughly constant with the baryon chemical potential, and the constant value of temperature is taken from lattice estimates.  The dark dashed curve represents $\\mu_B -T = cons \\times M_N$, corresponding to a simple model for the Quarrkyonic transition.  Such a very simple description does remarkably well.  \n\nA triple point is suggested at a baryon chemical potential near 400 MeV, and temperature near 160 MeV.  This corresponds to a center of mass energy for Pb-Pb collisions of 9-10 GeV.  This is near where there are anomalies in the abundances of rations of particles\\cite{Gazdzicki:1998vd}, as shown in Fig. \\ref{horn}.\nShown are fits using statistical models of abundances of particles using chemical potentials and temperature extracted from experimental data.  The sharp peak reflects the change in behavior as one proceeds along the dashed line of Fig. \\ref{line} corresponding to the Quarkyonic transition and joins to the dotted red line \nof the deconfinement transition\n\nIt is remarkable that the value of beam energy where this occurs corresponds to the hypothetical triple point of Fig. \\ref{line}, and that this is the density where the energy density stored in baryons becomes equal to that stored in mesons, Fig. \\ref{baryons},\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.60]{baryons}\t       \n\\caption[]{Energy density stored in baryons compared to that stored in mesons.}\n\\label{baryons}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nI gratefully acknowledge the organizers of the 50'th Crakow School of Theoretical Physics, in particular,\nMichal Praszalowicz, for making this wonderful and extraordinary meeting.\nThe research of  L. McLerran is supported under DOE Contract No. DE-AC02-98CH10886.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nHigh-energy neutrino astronomy has finally begun. \nThe detection of PeV-energy neutrinos~\\cite{icecubePeV2013} and the follow-up\nanalyses~\\cite{icecubeHESE2013} by the IceCube Collaboration revealed the existence\nof astrophysical ``on-source'' neutrinos at energies ranging from TeV to PeV.\nThese neutrinos are expected to be produced by the interactions\nof ultrahigh-energy cosmic-ray (UHECR) protons via $pp$ collisions\nor $\\gamma p$ collisions. The bulk intensity of these neutrinos,\n$E_\\nu^2 \\phi_{\\nu_e+\\nu_\\mu+\\nu_\\tau}\\simeq \n3.6\\times 10^{-8} {\\rm GeV} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$,\nprovides an important clue to understanding the general characteristics\nof UHECR sources through the connection between the observed cosmic-ray\nand neutrino intensities. \n\nIn the even higher-energy region from EeV to 100 EeV (EeV $=10^9 {\\rm GeV}$),\nthe highest-energy cosmic-ray (HECR) protons generate EeV-energy neutrinos\nvia interactions with cosmic microwave background (CMB) photons~\\cite{GZK} and extragalactic background light (EBL)\nduring their propagation in intergalactic space. The intensity of these\n``GZK cosmogenic'' neutrinos~\\cite{BZ} averaged over the sky is a consequence\nof the integral of the HECR emission over cosmic time, as neutrinos are\nstrongly penetrating particles that can travel cosmological\ndistances. It is, therefore, an observational probe to trace\nthe HECR source evolution. In particular, the cosmogenic neutrino intensity\nfrom 100 PeV to 10 EeV is highly sensitive to the evolution of the HECR emission rate\nand less dependent on other uncertain factors such as the highest energy\nof accelerated cosmic rays at their sources. As this energy range coincides with\nthe central region covered by the IceCube ultrahigh-energy neutrino searches,\nthe flux sensitivity achieved by IceCube has started to constrain a sizable\nparameter space of HECR source evolution, revealing the general\ncharacteristics of UHECR and HECR sources independent of the cosmic-ray acceleration model.\n\nIn this article, we review the new knowledge of UHECR\/HECR sources\nprovided by neutrino observations by IceCube. \nThe standard $\\Lambda$CDM cosmology with $H_0 = 73.5$ km s$^{-1}$ Mpc$^{-1}$, $\\Omega_{\\rm M} = 0.3$, \nand $\\Omega_{\\Lambda}=0.7$ is assumed throughout this article.\n\n\\section{The cosmic neutrino spectrum: Overview}\n\\label{sec:overview}\n\n\\begin{figure}\n\\includegraphics[width=0.45\\textwidth]{.\/NeutrinoFluxOverviewWzCaption.pdf}\n\\caption{Illustrative display of the differential fluxes of diffuse neutrinos having various origins. \nSupernova relic neutrinos are expected to appear in the MeV sky. Atmospheric\nneutrinos originating in extensive cosmic-ray airshowers dominate the background\nfor high-energy cosmic neutrino detection. Astrophysical neutrinos produced\n{\\it in situ} at cosmic-ray sources emerge at TeV and PeV energies, which have now been\ndetected by IceCube. GZK cosmogenic neutrinos, whose flux has yet to be measured, fill the highest-energy universe. Shown here is the possible range of fluxes\nfrom various source evolution models. The proton-dominated HECR composition is assumed.}\n\\label{fig:neut_fluxes}\n\\end{figure}\n\nFigure~\\ref{fig:neut_fluxes} displays the spectrum of neutrinos coming from\nall over the sky, {\\it i.e.}, diffuse neutrino fluxes. The massive background\nof atmospheric neutrinos ranging in energy over many orders of magnitude\nhad masked the astrophysical neutrinos until the IceCube observatory finally\nrevealed their existence. The spectrum shown here, taken from a model~\\cite{YoshidaTakami2014},\nhas a cutoff feature at $\\sim{\\rm PeV}$, but it is not clear yet whether the IceCube neutrino\nfluxes have a spectral cutoff. It is not even obvious that the spectrum can be well\ndescribed by a single power law formula. An analysis with enhanced sensitivity\nin the 10 TeV region seems to exhibit a trend toward a softer spectrum~\\cite{MESE} than\nthe up-going diffuse muon neutrino analysis, which is sensitive\nat energies above $\\sim 100\\ {\\rm TeV}$~\\cite{diffuse_nu}.\nOn-going efforts in the IceCube Collaboration will ultimately resolve these \nissues. Nevertheless, the intensity, \n$E_\\nu^2 \\phi_{\\nu_e+\\nu_\\mu+\\nu_\\tau}\\simeq \n3.6\\times 10^{-8} {\\rm GeV} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$, \nhas been well determined within a factor of two, and the implications\nfor the origin of UHECRs based on the intensity are not affected by these\ndetails of the spectral structure. If the neutrino emitters are also\nsources of the cosmic rays we are observing (which is very likely but not an undeniable assumption),\nwe can associate the neutrino flux with their parent cosmic-ray proton flux, \nand its comparison to the {\\it observed} cosmic-ray spectrum places\nsome constraints on the source characteristics.\n\nThe GZK cosmogenic neutrinos are expected to emerge in the 100 PeV--EeV sky.\nTheir intensity at the highest-energy end ($\\sim50-100\\ {\\rm EeV}$) depends mainly on\nthe maximal accelerated energy of cosmic rays at their sources\nand is not relevant to the ultrahigh-energy neutrino search by IceCube,\nas it is most sensitive at energies below 10 EeV.\nThe intensity at the lowest-energy tail ($\\sim10-100\\ {\\rm PeV}$)\nis determined by the EBL density\nand its evolution, which is EBL-model-dependent and could vary\nthe flux by a factor of $\\sim5$ at $\\sim10\\ {\\rm PeV}$.\nThe EeV-energy intensity is decided primarily by the HECR source evolution\nin redshift space.\nThe integral intensity of cosmogenic neutrinos above 100 PeV ranges from $10^{-17}$ to \n$\\sim3\\times 10^{-16} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$\ndepending on these factors. The IceCube detection exposure\nfor UHE neutrinos has now reached $\\sim3\\times 10^{16} {\\rm cm^{2}} \\sec {\\rm sr}$,\nand one can see that the IceCube sensitivity enables access to\na significant parameter space of the cosmogenic neutrino production models.\n\n\\section{The constraints on PeV- and EeV-energy UHECR sources}\n\\label{sec:PeV-EeV}\n\nThe flux of astrophysical neutrinos produced by UHECR protons at\ntheir sources is related to their parent cosmic-ray intensity\nvia the proton-to-neutrino conversion efficiency. The efficiency is usually\nparameterized in the form of the optical depth of the proton interactions.\nFor neutrinos produced through photomeson production ($\\gamma p$),\ntheir diffuse flux integrating emitted neutrinos over all the sources\nof UHECR protons with a spectrum in the source\nframe, $\\sim \\kappa_{\\rm CR}(E_{\\rm CR}\/E_0)^{-\\alpha}$ (where $E_0$ is the reference energy,\nwhich is conveniently set to $\\sim10$ PeV), is described as~\\cite{YoshidaTakami2014}\n\\begin{widetext}\n\\begin{eqnarray}\n\\phi_{\\nu_e + \\nu_\\mu + \\nu_\\tau}(E_\\nu) &\\simeq& \n\\frac{2 n_0 \\kappa_{\\rm CR}}{\\alpha^2} \\frac{c}{H_0} \n\\frac{s_{\\rm R}}{\\sqrt{(s_{\\rm R} + m_{\\pi}^2 - m_p^2)^2 - 4 s_{\\rm R} m_{\\pi}^2}} \n\\frac{3}{1 - r_{\\pi}} \n\\frac{(1 - e^{-\\tau_0})}{2 (m - \\alpha) - 1} \\Omega_{\\rm M}^{- \\frac{m - \\alpha + 1}{3}} \\nonumber \\\\\n&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\times \n\\left[ \\left\\{ \\Omega_{\\rm M} (1 + z_{\\rm max})^3 + \\Omega_{\\Lambda} \\right\\}^{\\frac{m - \\alpha}{3} - \\frac{1}{6}} - 1 \\right] \n\\left( \\frac{E_{\\nu}}{E_0 x_{\\rm R}^+ (1 - r_{\\pi})} \\right)^{-\\alpha}. \n\\label{eq:onsource_simple}\n\\end{eqnarray}\n\\end{widetext}\nHere $n_0$ is the source number density at the present epoch,\nand the source evolution is parameterized as $\\psi(z)=(1+z)^m$\nextending to the maximal redshift $z_{\\rm max}$\nsuch that the parameter $m$ represents the scale of the cosmological\nevolution often used in the literature. Further, $s_R\\ (\\simeq 1.5 {\\rm GeV^2})$ \nis the squared collision energy at the $\\Delta$ resonance of photopion production,\n$r_{\\pi} \\equiv m_{\\mu}^2 \/ m_{\\pi}^2 \\simeq 0.57$ is the muon-to-pion mass-squared ratio,\n$m_p$ is the proton mass, and $x_{\\rm R}^+\\ (\\simeq 0.36)$ \nis the kinematically maximal bound \nof the relative energy of emitted pions normalized by the parent cosmic-ray energy.\n$\\tau_0$, the optical depth of $\\gamma p$ interactions at the reference energy $E_0$,\nlinks the neutrino flux to the parent cosmic-ray intensity determined by $\\kappa_{\\rm CR}$.\n\nThe cosmic-ray flux integrating a UHECR spectrum over all\nthe sources in the redshift space, which corresponds to the UHECR spectrum\nwe {\\it observe}, is given by\n\\begin{eqnarray}\n\\phi_{\\rm CR}(E_{\\rm CR}) &=& \nn_0 c \\kappa_{\\rm CR} \\int_0^{z_{\\rm max}} dz \\nonumber \\\\\n&& (1 + z)^{1 - \\alpha} \\psi(z) \\left| \\frac{dt}{dz} \\right| \ne^{- \\tau_0} \n\\left( \\frac{E_{\\rm CR}}{E_0} \\right)^{-\\alpha},\\nonumber\\\\\n&&\n\\end{eqnarray}\nneglecting intergalactic magnetic fields and the energy loss\nin the CMB field during UHECR propagation.\nIntroducing some analytical approximations leads to the following simple formula:\n\\begin{eqnarray}\n\\phi_{\\rm CR}(E_{\\rm CR}) &\\simeq& \n2n_0 \\kappa_{\\rm CR} \\frac{H_0}{c} e^{-\\tau_0}\\left( \\frac{E_{\\rm CR}}{E_0} \\right)^{-\\alpha}\\nonumber\\\\\n&& \\frac{1}{2(m-\\alpha)-1} \\Omega_{\\rm M}^{- \\frac{m-\\alpha+1}{3}}\\nonumber\\\\\n&&\\left[ \\left\\{ \\Omega_{\\rm M} (1 + z_{\\rm max})^3 + \\Omega_{\\Lambda} \\right\\}^{\\frac{m-\\alpha}{3} - \\frac{1}{6}} - 1 \\right].\\nonumber\\\\\n&& \n\\label{eq:UHECR_flux_approx}\n\\end{eqnarray}\n\nComparing this formula to Equation~(\\ref{eq:onsource_simple}),\none can find that the source evolution effect represented by the evolution\nparameter $m$ is canceled in the ratio of the neutrino flux\nto the parent UHECR flux. This is because both the secondary produced neutrinos\nand emitted UHECRs originate in the same sources with the same evolution history.\nConsequently, the optical depth $\\tau_0$ is a deciding factor\nin the relation between these two fluxes. The TeV--PeV neutrino observation\nby IceCube that determines the neutrino flux $\\phi_\\nu$ thus\nassociates the UHECR source optical depth with the UHECR flux.\n\n\\begin{figure}[tb]\n  \\includegraphics[width=0.4\\textwidth]{.\/CRFluxWzDumpOpticalDepthContourHESE_SFR.pdf}\n  \\caption{Constraints on the optical depth of UHECR sources \nfor PeV-energy neutrino production and the energy flux of \nextragalactic UHECRs, $E_{\\rm CR}^2\\phi_{\\rm CR}$, at an energy of 10 PeV~\\cite{YoshidaTakami2014}. \nThe regions between the two blue solid curves ($\\alpha=2.5$), \ngreen dashed curves ($\\alpha=2.7$), and light blue dot-dashed curves ($\\alpha=2.3$) \nare allowed by the present IceCube observations~\\cite{icecubePeV2013,icecubeHESE2013}.\nThe unshaded region highlights the allowed region for $\\alpha=2.5$ \ntaking into account the observed intensity of UHECRs measured by the IceTop experiment~\\cite{icetop2013}.}\n\\label{fig:constraints_10PeV} \n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=0.4\\textwidth]{.\/CRFluxWzDumpOpticalDepthContourHESE1EeV_SFR.pdf}\n  \\caption{Same as Figure~\\ref{fig:constraints_10PeV}, but constraints against the cosmic-ray flux\n    at an energy of 1 EeV, showing the constraints when the UHECR spectrum from PeV neutrino\n    sources extends to higher energies.}\n\\label{fig:constraints_1EeV} \n\\end{figure}\n\nFigure~\\ref{fig:constraints_10PeV}\ndisplays the relations between the optical depth and the UHECR flux for several values of $\\alpha$,\nall of which are consistent with the IceCube observation\nat the present statistics~\\cite{icecubeHESE2013}. Star-formation-like source evolution\nis assumed, but the other assumptions regarding the evolution \nwould not change the main results, as explained above.\nThe smaller optical depth, implying a lower neutrino production efficiency, would require\nmore UHECR protons to be compatible with the neutrino intensity measured by IceCube.\nThe optical depth of the UHECR sources must be larger than 0.01,\nas the parent UHECR flux would exceed the observed cosmic-ray flux otherwise.\nThe proton flux from the neutrino sources contributes more than at least a few percent\nof all the UHECRs in the 10-PeV energy range. The magnetic horizon effect\nwould not change these constraints unless the sources are very rare, for example, \nif their number density is much smaller than $\\sim10^{-6} {\\rm Mpc}^{-3}$~\\cite{YoshidaTakami2014}.\nNote that the lower bound of the source density set by the small-scale UHECR anisotropy study\nconducted by the Auger observatory~\\cite{auger_density} is $6\\times 10^{-6} {\\rm Mpc}^{-3}$.\n\nGamma-ray bursters (GRBs) are strong candidates for UHECR acceleration sites\nand therefore high-energy neutrino production sites. \nInternal shocks are the most popular sites to produce\nhigh-energy neutrinos. An optical depth of $0.1-10^{-2}$ can be achieved,\ndepending on the dissipation radius, which satisfies the optical depth condition\nshown in Figure~\\ref{fig:constraints_10PeV}.\nHowever, their energetics may be problematic.\nThe typical gamma-ray energy output of a\nregular GRB is $10^{52}$ erg in gamma rays, and the local\noccurrence rate of long-duration GRBs is $\\sim1 {\\rm Gpc^{-3}}\\ {\\rm yr^{-1}}$.\nThese data indicate that the luminosity of local cosmic rays generated from GRBs is\n$10^{44}(\\eta_p\/10) {\\rm erg}\\ {\\rm Mpc^{-3}}\\ {\\rm yr^{-1}}$,\nwhere $\\eta_p$ is the ratio of\nthe UHECR output and $\\gamma$-ray output, which is known as the\nbaryon loading factor. This luminosity is 2 orders\nof magnitude smaller than that of UHECRs at 10 PeV and\nthus is too low to meet the requirement shown in Figure~\\ref{fig:constraints_10PeV}\nthat the UHECR flux from the sources must account for $\\geq0.1$ of the total UHECR flux.\nWe conclude that \nGRBs are unlikely to be major sources of both PeV-energy UHECRs and neutrinos.\n\nAmong known astronomical objects,\nonly flat-spectrum radio quasars (FSRQs)\ncan realize a large $\\gamma p$ optical depth ($\\geq0.01$)\nand large energetics. The typical $\\gamma$-ray luminosity density\nof FSRQs is $\\sim10^{46} {\\rm erg}\\ {\\rm Mpc^{-3}}\\ {\\rm yr^{-1}}$\nin our local universe. This is comparable to the local density\nof UHECRs at $\\sim$10 PeV.\n\nFigure~\\ref{fig:constraints_1EeV} shows the constraints on the $\\gamma p$ optical depth\nand UHECR flux when the energy spectrum of UHECR protons emitted from the neutrino sources\nextends to much higher energies than the observed neutrino energies.\nThe allowed regions in the parameter space become much smaller than those\nfor the constraints for $E_0 = 10$ PeV because the spectrum of the observed cosmic rays\nis steeper than the observed neutrino spectrum.\nNote that the optical depth constrained here is not at the EeV level, but at the PeV level,\nbecause PeV-energy protons are responsible for the neutrinos detected by IceCube.\nThe constraints suggest that the optical depth of protons for PeV-energy\nneutrinos is rather high, $\\tau_0\\geq 0.2$, and also that a major\nfraction of UHECRs in the EeV region is extragalactic\nprotons. This supports the ``dip'' transition model~\\cite{dip_model} of UHECR protons, \nwhere the ankle structure of the cosmic-ray spectrum, which\nappears at 3 to 10 EeV, is caused by the energy loss of\nextragalactic UHECR protons by Bethe--Heitler pair production with\nCMB photons. This model predicts a high GZK cosmogenic neutrino flux \nat 10--100 PeV.\nHowever, as we see in the next section,\nthe null detection of cosmogenic neutrino candidates in the IceCube seven year dataset\nexcludes the dip transition scenario if HECRs are proton-dominated.\nThis suggests that the source of neutrinos\nseen by IceCube is not the main source of cosmic rays at EeV energies or higher.\nNo known class of astronomical objects can \nmeet the stringent requirement of the $\\gamma p$ optical depth, $\\tau_0\\geq 0.1$,\nand the UHECR energetics. Only bright FSRQs such as those with \n$L_\\gamma\\sim 10^{50} {\\rm erg}\\ \\sec^{-1}$ {\\it could} realize\n$\\tau_0\\sim 0.1$, but such FSRQs are too rare to energetically reproduce\nthe observed UHECR flux at 1 EeV.\n\nAccording to these considerations, none of the known extragalactic objects\ncan be found to function as an origin of both PeV-energy\nneutrinos and HECRs. \n\n\\section{The constraints on the highest-energy cosmic-ray sources}\n\\label{sec:EeV}\n\n\\begin{figure*}\n  \\includegraphics[width=0.4\\textwidth]{.\/GZK2DHistogram.pdf}\n  \\includegraphics[width=0.4\\textwidth]{.\/AstrophysE2_58_2DHistogram.pdf}\n  \\caption{Expected event distributions on the plane of the energy proxy\nand cosine of the reconstructed zenith angle seen by IceCube. \nSimulation of the GZK cosmogenic model~\\cite{Ahlers2010}\n(left panel) and astrophysical neutrino models with a spectrum following $E_\\nu^{-2.5}$\nare shown with the intensity measured by IceCube~\\cite{IceCubeHESE2014}. The $z$ axis displays the number of events seen by the IceCube extremely high-energy analysis\nbased on the seven year data.}\n\\label{fig:energy_cosZenith}\n\\end{figure*}\n\nThe analysis of seven years of IceCube data obtained in the search for ultrahigh-energy\nneutrinos (energies larger than 10 PeV) has been reported~\\cite{EHE2016}.\nThe analysis was optimized in particular for neutrinos with energies above 100 PeV.\nThe exposure has reached $\\sim10^{17}{\\rm cm^2} \\sec\\ {\\rm sr}$ at 1 EeV,\nwhich makes it possible to probe an important region of the parameter space \nin the GZK cosmogenic neutrino models. \nTwo events with estimated deposited energies of\n2.6 and 0.77 PeV, respectively, were identified in this analysis, but no events were found\nin the higher-energy region. This observation presents a serious challenge\nto the standard baseline candidates of HECR sources discussed in the literature.\n\nAn IceCube simulation was used\nto predict the number of events IceCube would detect on the plane of the reconstructed\nenergy and zenith angle for each model of ultrahigh-energy neutrinos (including GZK cosmogenic neutrinos). \nFigure~\\ref{fig:energy_cosZenith} shows examples\nfrom the models. The resolution of the reconstructed deposited energies of energetic\nevents is less than excellent owing to the stochastic nature of the muon energy loss profile\nat PeV energies. Furthermore, IceCube's ability to associate the estimated energy deposit\nof an event with its parent neutrino energy is rather limited because only a small fraction\nof the neutrino energy is converted to the visible form ({\\it i.e.}, the deposited muon energy)\nby the IceCube detectors. Nevertheless, Figure~\\ref{fig:energy_cosZenith} exhibits\nclear differences between two different models. The GZK model yields an event distribution\nwith an energy peak higher than the softer astrophysical neutrino models.\nThe events from the GZK model are also distributed more sharply in the horizontal\ndirection {\\it i.e.}, $\\cos({\\rm zenith})=0$. This is because neutrinos with\nenergies at the EeV level experience strong absorption effects \nas they propagate through the Earth. These features, as well as\nthe total event rate (which is equivalent to the normalization of the event distributions\nshown in Figure~\\ref{fig:energy_cosZenith}), makes it possible to determine which models\nare compatible with the observation. The binned Poisson log-likelihood ratio\ntest was performed. The simulated event distributions \non the energy--zenith angle plane, such as those shown in Figure~\\ref{fig:energy_cosZenith},\ngive the expected number of events in each bin of the energy proxy and\ncosine of the zenith, which were used to construct the binned Poisson likelihood. A log-likelihood ratio was\nused as a test statistic, and an ensemble of pseudo-experiments to derive\nthe test statistical distribution was used to calculate the p-values.\n\n\\begin{figure}\n  \\includegraphics[width=0.4\\textwidth]{.\/GZKFluxConstraints.pdf}\n  \\caption{Energy spectra of various cosmogenic neutrino\n    models~\\cite{Ahlers2010, Kotera2010,Aloisio2015, Yoshida1993}. The spectra shown\nby red (brown) curves are rejected (disfavored), with p-values less than 10\\% (32\\%).\nAll these models assume a proton-dominated HECR composition.}\n\\label{fig:gzk_flux}\n\\end{figure}\n\nThe hypothesis that the two observed {\\it PeV-ish} events are of GZK cosmogenic origin\nis rejected, with a p-value of 0.3\\% (which implies that it is incompatible \nwith the event distribution shown in the left panel of Figure~\\ref{fig:energy_cosZenith})\nbut is consistent with a generic astrophysical power-law flux such\nas $E_\\nu^{-2}$ or $E_\\nu^{-2.5}$~\\cite{EHE2016}. \nThis result makes it possible to set an upper limit\non the ultrahigh-energy neutrino flux extending above 10 PeV and thus to also test\nthe GZK cosmogenic neutrino models using the binned Poisson log-likelihood ratio method.\n\nThe various cosmogenic neutrino energy spectra are displayed in\nFigure~\\ref{fig:gzk_flux}. Many of them are rejected or disfavored by the IceCube\nobservation. Regardless of where the HECR sources are\nand how they accelerate cosmic rays, the emitted HECR protons {\\it must} produce\nsecondary neutrinos by the GZK mechanism as they travel through space.\nIn this sense, any consequences of these bounds on GZK neutrinos\nare considered as robust and model-independent arguments.\n\nWe summarize the findings below.\n\n\\begin{itemize}\n\\item Cosmogenic models with the maximal flux allowed\n  by the Fermi-LAT measurement~\\cite{FermiDiffuse} of \n  the diffuse extragalactic $\\gamma-$ray background are rejected.\n  This finding implies that the present limits imposed by the neutrino observation are\n  at least as stringent as those imposed by the $\\gamma-$ray observation.\n\n\\item HECR source evolution comparable to the star formation rate (SFR) is beginning to\nbe constrained. Sources evolving more strongly than the SFR, such as FSRQs and GRBs, are unlikely\nto be HECR sources; otherwise, IceCube would have detected cosmogenic-neutrino-induced events already.\n\n\\item Any GZK cosmogenic type of energy spectrum must have an intensity below\n$E_\\nu^2\\phi_{\\nu_e + \\nu_\\mu + \\nu_\\tau}(E_\\nu) = 3\\times 10^{-9} \n{\\rm GeV}\\ {\\rm cm}^{-2}\\ \\sec^{-1}\\ {\\rm sr}^{-1}$ at 100 PeV. This limit rejects the dip transition\nmodel of UHECRs.\n\n\\end{itemize}\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{.\/Plot_2DConstraint_m_zmax_0909.pdf}\n  \\caption{Constraints on HECR source evolution parameters. The emission rate per co-moving volume\n    is parameterized as $\\psi(z)=(1+z)^m$ with redshifts up to $Z_{\\rm max}$. GZK cosmogenic neutrino\n    fluxes for various $m$ and $Z_{\\rm max}$ values are calculated by the approximated analytical\n    formulation~\\cite{YoshidaIshihara2012} and used for the likelihood calculation to derive\n    the confidence levels. The boxes indicate approximate parameter regions for the SFR~\\cite{Beacom} and\n    FR-II-A~\\cite{FR2-A} and -B~\\cite{FR2-B} radio galaxies.}\n\\label{fig:evolution}\n\\end{figure}\n\nMore generic constraints obtained by the IceCube Collaboration~\\cite{EHE2016} by scanning the parameter space for the\nsource evolution function, $\\psi(z)=(1+z)^m$,\nextending to the maximal redshift $Z_{\\rm max}$ \nare shown in Figure~\\ref{fig:evolution}.\nThe parameterized analytical formula for the cosmogenic fluxes~\\cite{YoshidaIshihara2012} is used here.\nBecause only the CMB is assumed as the target photon field in the parameterization,\nthe limits are systematically weaker than those on the models that include EBLs.\nApproximate regions for the SFR and the evolution\nof Fanaroff--Riley type II (FR-II) galaxies are also shown for comparison.\nNote that neutrinos yielded at redshifts larger than 2\nrepresent only a minor portion of the total cosmogenic fluxes owing to redshift\ndilution~\\cite{Kotera2010, YoshidaIshihara2012}.\nThis is especially true for the cosmogenic neutrino component created\nby interactions with the CMB (not the EBL). Considering this fact,\ntogether with the estimation that\nthe luminosity function of FR-II-type AGNs or FSRQs falls off rapidly at redshifts beyond\n$z\\simeq 2$, and that the evolution of the SFR becomes more or less constant or falls off\nat redshifts beyond $z\\sim2.5$, the boxes representing SFR and FR-II evolution\nin Figure~\\ref{fig:evolution} approximate well their representation by\nthe generic evolution function $\\psi(z)=(1+z)^m$ used in the plot.\nOne can find that HECR source evolution stronger than the SFR is unlikely.\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{.\/AstroMuraseAGNConstraints.pdf}\n  \\caption{Constraints on the fluxes of astrophysical neutrinos produced\nin the inner jets of radio-loud AGNs~\\cite{murase2014}. Two bounds for\nthe UHECR spectral index, $\\alpha=2.0$ (thin) and $2.3$ (thick),\nare shown.}\n\\label{fig:agn_nu}\n\\end{figure}\n\nAll the constraints on the HECR origins described so far \nrely on one critical assumption, that HECRs are proton-dominated.\nIf HECRs are of mixed- or heavy-nuclei composition, the resultant GZK cosmogenic\nflux is lower than the proton UHECR case by more than an order of magnitude,\nand the present IceCube detection sensitivity cannot reach this low intensity.\nIt is expected, however, that neutrinos with energies from the PeV level to the EeV level and beyond\nmay be produced {\\it in situ} at the HECR acceleration site. \nThe AGN neutrino models are good examples. A recent theoretical study\nof ultrahigh-energy neutrino generation in the inner jets \nof radio-loud AGNs~\\cite{murase2014} found that, \ntaking into account the blazar sequence,\nFSRQs can emit PeV--EeV neutrinos, and BL Lac objects\ncan be HECR (heavy) {\\it nuclei} sources~\\cite{murase2012}.\nThe predicted PeV--EeV neutrino intensity is proportional\nto the baryon loading factor, that is, the ratio of the UHECR luminosity\nto the electromagnetic radiation luminosity $L_{\\rm CR}\/L_\\gamma$.\nThe null detection of 100 PeV--EeV neutrinos by IceCube thus\nbounds this factor.\n\nFigure~\\ref{fig:agn_nu} shows the present bound\non the fluxes of neutrinos from radio-loud AGNs\nby IceCube~\\cite{EHE2016}.\nThe observed HECR generation rate, $\\sim$10 EeV \n($10^{44}\\ {\\rm erg}\\ {\\rm Mpc}^{-3}\\ {\\rm yr}^{-1}$), requires loading factors\nof around 3 and 100 for UHECR spectral indices of $\\alpha = 2$ and 2.3, respectively.\nThe present constraints are comparable to or slightly below the values\nrequired for radio-loud AGN inner jets to be responsible for\nthe majority of UHECRs\/HECRs. \nThe neutrino observation has started to exclude a sizable parameter\nspace in the models of AGNs as an origin of HECRs even if\nHECRs are composed of heavy nuclei, although this is a model-dependent\nargument.\n\nFast-spinning newborn pulsars are also proposed as candidate sources\nof HECRs~\\cite{olinto1997}. This proposal predicts a heavy-nuclei-dominated composition\nat the highest energies and thus would yield GZK neutrinos too rare\nto be detected. However, in this model, the accelerated particles traveling through\nthe expanding supernova ejecta surrounding the star\nproduce neutrinos with energies of 100 PeV to $\\sim$EeV. The predicted\ndiffuse neutrino flux from fast pulsars is \n$E_\\nu^2 \\phi_{\\nu_e+\\nu_\\mu+\\nu_\\tau}\\simeq \n1.1\\times 10^{-8} {\\rm GeV} {\\rm cm^{-2}} \\sec^{-1} {\\rm sr^{-1}}$,\ndepending on the source emission evolution, and\nis accessible at the IceCube detection sensitivity~\\cite{fang2014}.\n\nA binned Poisson log-likelihood test of this model was performed by\nthe IceCube Collaboration~\\cite{EHE2016}. The model is rejected\nif the evolution of the source emission history traces the standard SFR, although\nit is not ruled out if the emission rate evolves more slowly than the SFR.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe detection of TeV--PeV neutrinos and the null detection\nof EeV neutrinos by IceCube have yielded many insights on the origin of UHECRs, which cannot be probed by other cosmic messengers.\nThe most popular candidates for UHECR\/HECR sources, GRBs and radio-loud AGNs,\nhave now faced serious challenges from recent results in neutrino astronomy.\nGRBs and AGNs can still contribute to the observed bulk\nof the highest-energy cosmic rays but are unlikely to be their {\\it dominant} sources.\nAstronomical objects tracing the standard SFR or evolving much more slowly\nare needed to explain the observation without fine-tuning.\nIf the highest-energy cosmic rays are not proton-dominated,\nthese constraints are certainly relaxed. The model-dependent tests\ndescribed here, however, have already placed limits on some of\nthe parameter space of the AGN\/pulsar scenarios.\n\nThere is a loophole: a hypothesis that any high-energy neutrino emission\ndoes not involve cosmic-ray emission. A good example is the GRB choked jet\nmodel~\\cite{chokedGRB}. In dense environments, the optical depth $\\tau_0\\gg 1$,\nwhich implies that all the proton\u5c08 energy is\nconverted into neutrinos; {\\it i.e.}, the observed UHECRs and\nneutrinos are not directly connected.\n\nHow can we identify UHECR\/HECR sources that evolve\nat the usual SFR, or even more slowly?\nReal-time multi-messenger observation\ntriggered by high-energy neutrinos is a possible answer. IceCube has launched\nthe Gamma-ray Coordinates Network-based alert delivery system~\\cite{icecubeGCN}.\nThe search algorithms for Extremely High-Energy  neutrinos~\\cite{icecubePeV2013, EHE2016}\nand High-Energy Starting Events~\\cite{icecubeHESE2013, IceCubeHESE2014},\nthe analysis channels that discovered the high-energy cosmic neutrinos, are now running\nin real time at IceCube's South Pole data servers.\nOnce a high-energy-neutrino-induced event is detected, an alert is sent immediately\nto trigger follow-up observations by other astronomical instruments.\nIf UHECR\/HECR sources are transient neutrino sources,\nwe may be able to identify them by follow-up detection with optical\/X-ray\/$\\gamma$-ray\/radio\ntelescopes. This is probably a promising way to approach identification of\nthe yet-unknown origins of high-energy cosmic rays.\n\n\\section*{Acknowledgments}\nI am grateful to the CRIS 2016 organizers for\ntheir warm hospitality.\nI acknowledge my colleagues in the IceCube Collaboration\nfor useful discussions and suggestions.\nI also appreciate the input of Kunihito Ioka, Kumiko Kotera, Kohta Murase, \nand Hajime Takami on the theoretical arguments.\nSpecial thanks go to Aya Ishihara, who has worked together\non the analyses of extremely high-energy neutrinos \nfor many years. This work is supported by JSPS\nGrants-in-Aid for Scientific Research (Project \\#25105005 and \\#25220706).\n\n\n\n\n\\nocite{*}\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Sun is the best known star to astronomers, and is commonly used as a template in the study of other similar objects. Yet, there are still some of its aspects that are not well understood and that are crucial for a better understanding of how stars, and consequently how planetary systems and life evolve: how do the more complex physical parameters of a Sun-like star, such as rotation and magnetic activity, change with time? Is the Sun unique or typical (i.e., an average Sun-like star)? If the Sun is common, it would mean that life does not require a special star for it to flourish, eliminating the need to evoke an anthropic reasoning to explain it.\n\nIn an effort to assess how typical the Sun is, \\citet{2008ApJ...684..691R} compared 11 of its physical parameters with nearby stars, and concluded that the Sun is, in general, typical. Although they found it to be a slow-rotator against 276 F8 -- K2 (within $\\pm 0.1$ M$_\\odot$) nearby stars, this result may be rendered inconclusive owing to unnacounted noise caused by different masses and ages in their sample. Other studies have suggested that the Sun rotates either unusually slow \\citep{1979PASP...91..737S, 2015A&A...582A..85L} or regularly for its age \\citep{1983ApJS...53....1S, 1985AJ.....90.2103S, 1984ApJ...281..719G, 1998SSRv...85..419G, 2003ApJ...586..464B}, but none of them comprised stars that are very similar to the Sun, therefore preventing a reliable comparison. In fact, with \\textit{Kepler} and \\textit{CoRoT}, it is now possible to obtain precise measurements of rotation periods, masses and ages of stars in a very homogeneous way \\citep[e.g.,][]{2015EPJWC.10106016C, 2012A&A...548L...1D, 2014ApJS..210....1C}, but they generally lack high precision stellar parameters, which are accessible through spectroscopy. The challenging nature of these observations limited ground-based efforts to smaller, but key stellar samples \\citep[e.g.,][]{2003A&A...397..147P, 2012AN....333..663S}.\n\nThe rotational evolution of a star plays a crucial role in stellar interior physics and habitability. Previous studies proposed that rotation can produce extra mixing that is responsible for depleting the light elements Li and Be in their atmospheres \\citep{1989ApJ...338..424P, 1994A&A...283..155C, 2015A&A...576L..10T}, which could explain the disconnection between meteoritic and solar abundances of Li \\citep{2010A&A...519A..87B}. Moreover, rotation is highly correlated with magnetic activity \\citep[e.g.,][]{1984ApJ...279..763N, 1993ApJS...85..315S, 1995ApJ...438..269B, 2008ApJ...687.1264M}, and this trend is key to understand how planetary systems and life evolve in face of varying magnetic activity and energy outputs by solar-like stars during the main sequence \\citep{2009IAUS..258..395G, 2005ApJ...622..680R, 2016ApJ...820L..15D}.\n\nA theoretical treatment of rotational evolution from first principles is missing, so we often rely on empirical studies to infer about it. One of the pioneer efforts in this endeavor produced the well known Skumanich relation $v \\propto t^{-1\/2}$, where $v$ is the rotational velocity and $t$ is the stellar age \\citep{1972ApJ...171..565S}, which describes the rotational evolution of solar-type stars in the main sequence, and can be derived from the loss of angular momentum due to magnetized stellar winds \\citep[e.g.,][]{1988ApJ...333..236K, 1992ASPC...26..416C, 2003ApJ...586..464B, 2013A&A...556A..36G}. This relation sparked the development of gyrochronology, which consists in estimating stellar ages based on their rotation, and it was shown to provide a stellar clock as good as chromospheric ages \\citep{2007ApJ...669.1167B}. However, in Skumanich-like relations, the Sun generally falls on the curve (or plane, if we consider dependence on mass) defined by the rotational braking law by design. Thus it is of utmost importance to assess how common the Sun is in order to correctly calibrate it.\n\nSubsequent studies have proposed modifications to this paradigm of rotation and chromospheric activity evolution \\citep[e.g.,][]{1991ApJ...375..722S, 2004A&A...426.1021P}, exploring rotational braking laws of the form $v \\propto t^{-b}$. The formalism by \\citet{1988ApJ...333..236K} shows that this index $b$ can be related to the geometry of the stellar magnetic field, and that Skumanich's index ($b = 1\/2$) corresponds to a geometry that is slightly more complex than a simple radial field. It also dictates the dependence of the angular momentum on the rotation rate, and in practice, it determines how early the effects of braking are felt by a model. Such prescriptions for rotational evolution have a general agreement for young ages up to the solar age \\citep[see][and references therein]{2016arXiv160507125S, 2016A&A...587A.105A}, but the evolution for older ages still poses an open question. In particular, \\citet{2016Natur.529..181V} suggested that stars undergo a weakened magnetic braking after they reach a critical value of the Rossby number, thus explaining the stagnation trend observed on the rotational periods of older Kepler stars.\n\nIn order to assess how typical the Sun is in its rotation, our study aims to verify if it follows the rotational evolution of stars that are very similar to it, an objective that is achieved by precisely measuring their rotational velocities and ages. We take advantadge of an unprecedented large sample of solar twins \\citep{2014A&A...572A..48R} using high signal-to-noise ($S\/N > 500$) and high resolution ($R > 10^5$) spectra, which provides us with precise stellar parameters and is essential for the analysis that we perform (see Fig. \\ref{widths} for an illustration of the subtle effects of rotation in stellar spectra of Sun-like stars).\n\n\n\\section{Working sample}\n\nOur sample consists of bright solar twins in the Southern Hemisphere, which were mostly observed in our HARPS Large Program (ID: 188.C-0265) that aimed to search for planetary systems around stars very similar to the Sun \\citep[][Papers I, II and III, respectively, of the series The Solar Twin Planet Search]{2014A&A...572A..48R, 2015A&A...581A..34B, 2016A&A...590A..32T}. These stars are loosely defined as those that have T$\\mathrm{_{eff}}$, $\\log{g}$ and [Fe\/H] inside the intervals $\\pm 100$ K, $\\pm 0.1$ [cgs] and $0.1$ dex, respectively, around the solar values. It has been shown that these limits guarantee $\\sim$0.01 dex precision in the relative abundances derived using standard model atmosphere methods amd that the systematic uncertainties of that analysis are negligible within those ranges \\citep{2014ApJ...795...23B, 2015A&A...583A.135B, 2015A&A...582A..17S, 2016A&A...589A..17Y}. In total, we obtained high precision spectra for 73 stars and used data from 9 more targets observed in other programs, all of them overlapping the sample of 88 stars from Paper I. We used the spectrum of the Sun (reflected light from the Vesta asteroid) from the ESO program 088.C-0323, which was obtained with the same instrument and configuration as the solar twins.\n\nThe ages of the solar twin sample span between $0-10$ Gyr and are presented in the online material (Table \\ref{params}). They were obtained by \\citet{2016A&A...590A..32T} using Yonsei-Yale isochrones \\citep{2001ApJS..136..417Y} and probability distribution functions as described in \\citet{2013ApJ...764...78R,2014A&A...572A..48R}. Uncertainties are assumed to be symmetric. These ages are in excellent agreement with the ones obtained in Paper I, with a mean difference of $-0.1 \\pm 0.2$ Gyr (see footnote 5 in Paper III). We adopted 4.56 Gyr for the solar age \\citep{1995RvMP...67..781B}. The other stellar parameters ($T\\rm_{eff}$, $\\log{g}$, [Fe\/H] and microturbulence velocities $v\\rm_t$) were obtained by \\citet{2014A&A...572A..48R}. The stellar parameters of HIP 68468 and HIP 108158 were updated by \\citet{2016A&A...590A..32T}.\n\nOur targets were observed at the HARPS spectrograph\\footnote{\\footnotesize{The initial plan was to use the observations from the MIKE spectrograph, as described by the Paper I. However, we decided to use the HARPS spectra due to its higher spectral resolving power.}} \\citep{2003Msngr.114...20M} which is fed by ESO's 3.6 m telescope at La Silla Observatory. When available publicly, we also included all observations from other programs in our analysis in order to increase the signal to noise ratio ($S\/N$) of our spectra. However, we did not use observations for 18 Sco (HIP 79672) from May 2009\\footnote{\\footnotesize{These observations have instrumental artifacts.}} and we did not include observations post-HARPS upgrade (June 2015) when combining the spectra\\footnote{\\footnotesize{The spectra had a different shape in the red side, and since there were few observations, we chose not to use them to eliminate eventual problems with combination and normalization.}}.\n\nThe wavelength coverage for the observations ranged from 3780 to 6910 \\AA, with a spectral resolving power of $R = \\lambda\/\\Delta\\lambda = 115000$. Data reduction was performed automatically with the HARPS Data Reduction Software (DRS). Each spectrum was divided in two halves, corresponding to the mosaic of two detectors (one optimized for the blue and other for the red wavelengths). In this study we only worked with the red part (from 5330 to 6910 \\AA) due to its higher $S\/N$ and the presence of cleaner lines. The correction for radial velocities was performed with the task \\texttt{dopcor} from IRAF\\footnote{\\footnotesize{IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.}}, using the values obtained from the pipeline's cross-correlation function (CCF) data. The different observations were combined with IRAF's \\texttt{scombine}. The resulting average (of the sample) signal to noise ratio was 500 around 6070 \\AA. The red regions of the spectra were normalized with $\\sim$30th order polynomial fits to the upper envelopes of the entire red range, using the task \\texttt{continuum} on IRAF. We made sure that the continuum of the stars were consistent with the Sun's. Additionally, we verified that errors in the continuum determination introduce uncertainties in $v \\sin{i}$ lower than $0.1$ km s$^{-1}$.\n\n   \\begin{figure}[!t]\n   \\centering\n   \\includegraphics[width=\\hsize]{line_comp.pdf}\n   \\caption{Comparison of the spectral line broadening between two solar twins with different projected rotational velocities. The wider line correspond to HIP 19911, with $v \\sin i \\approx 4.1$ km s$^{-1}$, and the narrower one comes from HIP 8507, with $v \\sin i \\approx 0.8$ km s$^{-1}$.}\n   \\label{widths}\n   \\end{figure}\n\n\n\\section{Methods}\\label{methods}\n\nWe analyze five spectral lines, four due to Fe I and one to Ni I (see Table \\ref{lines}; equivalent widths were measured using the task \\texttt{splot} in IRAF.), that were selected for having low level of contamination by blending lines. The rotational velocity of a star can be measured by estimating the spectral line broadening that is due to rotation. The rotation axes of the stars are randomly oriented, thus the spectroscopic measurements of rotational velocity are a function of the inclination angle ($v \\sin{i}$).\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Line list used in the projected stellar rotation measurements.}\n\\begin{tabular}{cccccc}\n\\hline \\hline\\\\[-2ex]\nWavelength & Z & Exc. pot. & $\\log{(gf)}$ & $v_{\\mathrm{macro}}^\\odot$ & EW$^\\odot$\\\\\n(\\AA) &  & (eV) & & (km s$^{-1}$) & (\\AA)\\\\ \\hline\n6027.050 & 26 & 4.076 & -1.09  & 3.0 &\t0.064 \\\\\n6151.618 & 26 & 2.176 & -3.30 & 3.2 &\t0.051 \\\\\n6165.360 & 26 & 4.143 & -1.46  & 3.1 &\t0.045 \\\\\n6705.102 & 26 & 4.607  & -0.98  & 3.6 &\t0.047 \\\\\n6767.772 & 28 & 1.826 & -2.17  & 2.9 &\t0.079 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{EW are the equivalent widths and $v_{\\mathrm{macro}}$ are the macroturbulence velocities measured as in Sect. \\ref{vmacro_det}.}\n\\label{lines}\n\\end{center}\n\\end{table}\n\nWe estimate $v \\sin{i}$ for our sample of solar twins using the 2014 version of MOOG Synth \\citep{1973PhDT.......180S}, adopting stellar atmosphere models by \\citet{2004astro.ph..5087C}, with interpolations between models performed automatically by the Python package qoyllur-quipu\\footnote{\\footnotesize{Available at \\url{https:\/\/github.com\/astroChasqui\/q2}}} \\citep[see][]{2014A&A...572A..48R}. The instrumental broadening is taken into account by the spectral synthesis. We used the stellar parameters from \\citet{2016A&A...590A..32T} and microturbulence velocities from \\citet{2014A&A...572A..48R}. Macroturbulence velocities ($v_{\\mathrm{macro}}$) were calculated by scaling the solar values, line by line (see Sect. \\ref{vmacro_det}). Estimation of the rotational velocities was performed with our own algorithm\\footnote{\\footnotesize{Available at \\url{https:\/\/github.com\/RogueAstro\/PoWeRS}}} that makes automatic measurements for all spectral lines for each star. We applied fine tuning corrections by eye for the non-satisfactory automatic line profile fittings, and quote $v \\sin{i}$ as the mean of the values measured for the five lines. See Sects. \\ref{vmacro_det} and \\ref{vsini_det} for a detailed description on rotational velocities estimation as well as their uncertainties. Fig. \\ref{sun_fit} shows an example of spectral line fitting for one feature in the Sun.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\hsize]{sun_line.pdf}\n\\caption{Example of line profile fitting for the Fe I feature at 6151.62 \\AA\\ in the spectrum of the Sun. The continuous curve is the synthetic spectrum, and the open circles are the observed data.}\n\\label{sun_fit}\n\\end{figure}\n\n\\subsection{Macroturbulence velocities}\\label{vmacro_det}\n\nWe tested the possibility of measuring $v_{\\mathrm{macro}}$ (radial-tangential profile) simultaneously with $v \\sin{i}$, but even when using the extremely high-resolution spectra of HARPS, it is difficult to disentangle these two spectral line broadening processes, which is probably due to the low values of these velocities. Macroturbulence has a stronger effect on the wings of the spectral lines, but our selection of clean lines still has some contamination that requires this high-precision work to be done by eye. Some stars show more contamination than others, complicating the disentaglement. Fortunately, the variation of macroturbulence with effective temperature and luminosity is smooth \\citep{2005oasp.book.....G}, so that precise values of $v_{\\mathrm{macro}}$ could be obtained by a calibration. Thus we adopted a relation that fixes macroturbulence velocities in order to measure $v \\sin{i}$ with high precision using an automatic code, which provides the additional benefits of reproducibility and lower subjectivity.\n\nThe macroturbulence velocity is known to vary for different spectral lines \\citep{2005oasp.book.....G}, so for our high-precision analysis, we do not adopt a single value for each star. Instead, we measure the $v_{\\mathrm{macro}}$ for the Sun in each of the spectral lines from Table \\ref{lines}, and use these values to scale the $v_{\\mathrm{macro}}$ for all stars in our sample using the following equation\\footnote{\\footnotesize{In the future, it should be possible to calibrate macroturbulence velocities using 3D hydrodynamical stellar atmosphere models \\citep[e.g.,][]{2013A&A...557A..26M} by using predicted 3D line profiles (without rotational broadening) as observations and determine which value of $v_{\\rm{macro}}$ is needed to reproduce them with 1D model atmospheres.}}:\n\n\\begin{eqnarray}\nv_{\\mathrm{macro},\\lambda}^{*} = v_{\\mathrm{macro},\\lambda}^{\\odot} - 0.00707\\ T_{\\mathrm{eff}} + 9.2422 \\times 10^{-7}\\ T_{\\mathrm{eff}}^2 \\nonumber \\\\ + 10.0 + k_1 \\left(\\log{g} - 4.44\\right) + k_2 \\\\\n\\equiv f(T_{\\mathrm{eff}}) + k_1\\left(\\log{g} - 4.44\\right) + k_2 \\nonumber\n\\label{vmacro_eq}\n\\end{eqnarray}where $v\\rm_{macro,\\lambda}^{\\odot}$ is the macroturbulence velocity of the Sun for a given spectral line, $T_{\\mathrm{eff}}$ and $\\log{g}$ are, respectively, the effective temperature and gravity of a given star, $k_1$ is a proportionality factor for $\\log{g}$ and $k_2$ is a small correction constant.\n\nThis formula is partly based on the relation derived by \\citet{2012A&A...543A..29M} (Eq. E.1 in their paper) from the trend of macroturbulence with effective temperature in solar-type stars described by \\citet{2005oasp.book.....G}. The $\\log{g}$-dependent term (a proxy for luminosity) comes from the empirical relation derived by \\citet{2014MNRAS.444.3592D} (Eq. 8 in their paper), and is based on spectroscopic measurements of $v_{\\mathrm{macro}}$ of \\textit{Kepler} stars, which were disentangled from $v \\sin{i}$ using asteroseismic estimates of the projected rotational velocities. Doyle et al. obtained a value for the proportionality factor $k_1$ of -$2.0$. However their uncertainties on $v_{\\mathrm{macro}}$ were of the order of 1.0 km s$^{-1}$. Thus, we decided to derive our own values of $k_1$ and $k_2$ by simultaneously measuring $v_{\\mathrm{macro}}$ and $v \\sin{i}$ of a sub-sample of solar twins.\n\nThis sub-sample was chosen to contain only single stars or visual binaries mostly in the extremes of $\\log{g}$ ($4.25$ -- $4.52$) in our entire sample. We assume these values to have a linear relationship with $v_{\\mathrm{macro}}$ inside this short interval of $\\log{g}$. We used as a first guess the values of $v \\sin{i}$ and $v_{\\mathrm{macro}}$ from a previous, cruder estimation we made, and performed line profile fits by eye using MOOG Synth. The velocities in Table \\ref{vmacro_stars} are the median of the values measured for each line and their standard error. Note that these $v \\sin{i}$ are not consistently measured in the same way that the final results are. The rotational velocity broadening was calculated by our own code (see Sect. \\ref{vsini_det} for details). By performing a linear fit in the $v_{\\mathrm{macro}} - f(T_{\\mathrm{eff}})$ vs. $\\log{g}-4.44$ relation ($f$ comprises all the $T_{\\mathrm{eff}}$-dependent terms, the macroturbulence velocity of the Sun and the known constant on Eq. \\ref{vmacro_eq}), we obtain that $k_1 = -1.81 \\pm 0.26$ and $k_2 = -0.05 \\pm 0.03$ (see Fig. \\ref{vmacro_logg}). For the stars farthest from the Sun in $\\log{g}$ from our sample, these values of $k_1$ and $k_2$ would amount to differences of up to $\\pm 0.4$ km s$^{-1}$ in their macroturbulence velocities, therefore it is essential to consider the luminosity effect on $v\\rm_{macro}$ for accurate $v \\sin{i}$ determinations.\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Simultaneous measurements of rotational and macroturbulence velocities of stars in the extremes of $\\log{g}$ from our sample of solar twins.}\n\\begin{tabular}{lcccc}\n\\hline \\hline\\\\[-2ex]\nStar & $v \\sin{i}$ & $v_{\\mathrm{macro}}$ & $T\\rm_{eff}$ & $\\log{g}$ \\\\\n & (km s$^{-1}$) & (km s$^{-1}$) & & \\\\ \\hline\nHIP 115577 & $0.95 \\pm 0.05$ & $3.35 \\pm 0.09$ & 5699 & 4.25 \\\\\nHIP 65708 & $1.20 \\pm 0.09$ & $3.55 \\pm 0.08$ & 5755 & 4.25 \\\\\nHIP 74432 & $1.40 \\pm 0.03$ & $3.35 \\pm 0.08$ & 5684 & 4.25 \\\\\nHIP 118115 & $1.40 \\pm 0.10$ & $3.43 \\pm 0.09$ & 5808 & 4.28 \\\\\nHIP 68468 & $1.75 \\pm 0.07$ & $3.70 \\pm 0.08$ & 5857 & 4.32 \\\\\nHIP 41317 & $1.55 \\pm 0.03$ & $3.10 \\pm 0.06$ & 5700 & 4.38 \\\\\nSun & $1.75 \\pm 0.07$ & $3.30 \\pm 0.06$ & 5777 & 4.44 \\\\\nHIP 105184 & $2.50 \\pm 0.03$ & $3.21 \\pm 0.08$ & 5833 & 4.50 \\\\\nHIP 10175 & $1.55 \\pm 0.06$ & $3.05 \\pm 0.08$ & 5738 & 4.51 \\\\\nHIP 114615 & $2.20 \\pm 0.03$ & $3.25 \\pm 0.08$ & 5816 & 4.52 \\\\\nHIP 3203 & $3.90 \\pm 0.03$ & $3.40 \\pm 0.10$ & 5850 & 4.52 \\\\\n\\hline\n\\end{tabular}\n\\label{vmacro_stars}\n\\end{center}\n\\end{table}\n\n   \\begin{figure}[!t]\n   \\centering\n   \\includegraphics[width=\\hsize]{vmacro_logg.pdf}\n   \\caption{Linear relation between $v_{\\mathrm{macro}}$ and $\\log{g}$ (a proxy for luminosity) for the stars on Table \\ref{vmacro_stars}. See the definition of $f(T_{\\mathrm{eff}})$ in Sect. \\ref{vmacro_det}. The orange continuous line represents our determination of a proportionality coefficient of -1.81 and a vertical shift of -0.05 km s$^{-1}$. The black dashed line is the coefficient found by \\citet{2014MNRAS.444.3592D}. The light grey region is a composition of 200 curves with parameters drawn from a multivariate gaussian distribution. The Sun is located at the origin.}\n   \\label{vmacro_logg}\n   \\end{figure}\n\nTo obtain the macroturbulence velocities for the Sun to use in Eq. \\ref{vmacro_eq}, we forced the rotational velocity of the Sun to 1.9 km s$^{-1}$ \\citep{1970SoPh...12...23H}, and then estimated values of $v\\rm_{macro, \\lambda}^{\\odot}$ by fitting each line profile using MOOG Synth, and the results are shown in Table \\ref{lines}. We estimate the error in determining $v\\rm_{macro,\\lambda}^{\\odot}$ to be $\\pm 0.1$ km s$^{-1}$. Since Eq. \\ref{vmacro_eq} is an additive scaling, the error for $v\\rm_{macro}$ of all stars is the same as in the Sun\\footnote{\\footnotesize{The uncertainties in stellar parameters have contributions that are negligible compared to the ones introduced by the error in $v_{\\mathrm{macro}}$.}}.\n\n\\subsection{Rotational velocities}\\label{vsini_det}\n\nOur code takes as input the list of stars and their parameters (effective temperature, surface gravity, metallicity and microturbulence velocities obtained on Paper I), their spectra and the spectral line list in MOOG-readable format. For each line in a given star, the code automatically corrects the spectral line shift and the continuum. The first is done by fitting a second order polynomial to the kernel of a line and estimating what distance the observed line center is from the laboratory value. Usually, the spectral line shift corrections were of the order of $10^{-2}$ \\AA, corresponding to 0.5 km s$^{-1}$ in the wavelength range we worked on. This is a reasonable shift that likely arises from a combination of granulation and gravitational redshift effects, which are of similar magnitude. The continuum correction for each line is defined as the value of a multiplicative factor that sets the highest flux inside a radius of 2.5 \\AA\\ around the line center to 1.0. The multiplicative factor usually has a value inside the range $1.000 \\pm 0.002$.\n\nThe code starts with a range of $v \\sin{i}$ and abundances and optimizes these two parameters through a series of iterations that measure the least squares difference between the observed line and the synthetic line (generated with MOOG synth). Convergence is achieved when the difference between the best solution and the previous one, for both $v \\sin{i}$ and abundance, is less than 1\\%. Additionally, the code also forces at least 10 iterations in order to avoid falling into local minima.\n\nOne of the main limitations of MOOG Synth for our analysis is that it has a \"quantized\" behavior for $v \\sin{i}$: the changes in the synthetic spectra occur most strongly in steps of 0.5 km s$^{-1}$. This behavior is not observed in varying the macroturbulence velocities. Therefore, we had to incorporate a rotational broadening routine in our code that was separated from MOOG. We used the Eq. 18.14 from \\citet{2005oasp.book.....G}, in velocity space, to compute the rotational profile\\footnote{\\footnotesize{This is the same recipe adopted by the radiative transfer code MOOG.}}:\n\n\\begin{equation}\nG(v) = \\frac{2(1-\\epsilon)\\left[ 1-(v\/v\\rm_L)^2 \\right]^{1\/2} + \\frac{1}{2} \\pi \\epsilon \\left[ 1-(v\/v\\rm_L)^2 \\right]}{\\pi v\\rm_L (1-\\epsilon\/3)}\\mathrm{,}\n\\end{equation}where $v\\rm_L$ is the projected rotational velocity and $\\epsilon$ is the limb darkening coefficient (for which we adopt the value 0.6). The rotational profile $G(v)$ is then convolved with MOOG's synthetic profiles (which were generated with $v \\sin{i}$ = 0).\n\nThe total uncertainties in rotational velocities are obtained from the quadratic sum of the standard error of the five measurements and an uncertainty of 0.1 km s$^{-1}$ introduced by the error in macroturbulence velocities. Systematic errors in the calculation of $v\\rm_{macro,\\lambda}$ for the stars do not significantly contribute to the $v \\sin{i}$ uncertainties.\n\nSome of the stars in the sample show very low rotational velocities, most probably due to the effect of projection (see left panel of Fig. \\ref{sku}). The achieved precision is validated by comparison with the values of the full-width at half maximum (FWHM) measured by the cross-correlation function (CCF) from the data reduction pipeline, with the effects of macroturbulence subtracted (see Fig. \\ref{ccf}). The spectroscopic binary star HIP 103983 has an unusually high $v \\sin{i}$ when compared to the CCF FWHM, and a verification of its spectral line profiles reveals the presence of distortions that are the most probably caused by mis-measurement of rotational velocity (contamination of the combined spectrum by a companion -- observations range from October 2011 to August 2012). We obtained a curve fit for the $v \\sin{i}$ vs. CFF FWHM (km s$^{-1}$) using a similar relation as used by \\citet{2001A&A...375..851M, 2004A&A...426.1021P, 2007A&A...475.1003H}, which resulted in the following calibration: $v \\sin{i} = \\sqrt{(0.73 \\pm 0.02) \\left[\\mathrm{FWHM}^2 - v_{\\mathrm{macro}}^2 - (5.97 \\pm 0.01)^2\\right]}$ km s$^{-1}$ \\citep[estimation performed with the MCMC code \\texttt{emcee}\\footnote{\\footnotesize{Available at \\url{http:\/\/dan.iel.fm\/emcee\/current\/}}}][]{2013PASP..125..306F}. The scatter between the measured $v \\sin{i}$ and the ones estimated from CCF is $\\sigma = 0.20$ km s$^{-1}$ (excluding the outlier HIP 103983). The typical uncertainty in the rotational velocities we obtain with our method -- line profile fitting with extreme high resolution spectra -- is 0.12 km s$^{-1}$, which implies that the average error of the CCF FWHM $v \\sin{i}$ scaling is 0.16 km s$^{-1}$, which could be significantly higher if the broadening by $v_\\mathrm{macro}$ is not accounted for.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\hsize]{ccf_vs_moog.pdf}\n\\caption{Comparison between our estimated values of $v \\sin{i}$ (y-axis) and the ones inferred from the cross-correlation funcion FWHM (x-axis). The spread around the 1:1 relation (black line) is $\\sigma = 0.20$ km s$^{-1}$.}\n\\label{ccf}\n\\end{figure}\n\n\n\\section{Binary stars}\n\nWe identified 16 spectroscopic binaries (SB) in our sample of 82 solar twins by analyzing their radial velocities; some of these stars are reported as binaries by \\citet{2014AJ....147...86T, 2014AJ....147...87T, 2001AJ....121.3224M, 2015ApJ...802...37B}. We did not find previous reports of multiplicity for the stars HIP 30037, HIP 62039 and HIP 64673 in the literature. Our analysis of variation in the HARPS radial velocities suggest that the first two are probable SBs, while the latter is a candidate. No binary shows a double-lined spectrum, but HIP 103983 has distortions that could be from contamination by a companion. The star HIP 64150 is a Sirius-like system with a directly observed white dwarf companion \\citep{2013ApJ...774....1C,2014ApJ...783L..25M}. The sample from Paper I contains another SB, HIP 109110, for which we could not reliably determine the $v \\sin{i}$ due to strong contamination in the spectra, possibly caused by a relatively bright companion. Thus, we did not include this star in our sample.\n\n\\begin{figure*}[!ht]\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.48\\textwidth]{all_stars.pdf} & \\includegraphics[width=0.48\\textwidth]{sku_final.pdf} \\\\\n\\end{tabular}\n\\caption{Projected rotational velocity of solar twins in function of their age. The Sun is represented by the symbol $\\odot$. Left panel: all stars of our sample; the orange triangles are spectroscopic binaries, blue circles are the \\textit{selected sample} and the blue dots are the remaining non-spectroscopic binaries. Right panel: the rotational braking law; the purple continous curve is our relation inferred from fitting the \\textit{selected sample} (blue circles) of solar twins with the form $v \\sin{i} = v_\\mathrm{f} + m\\ t^{-b}$, where $t$ is the stellar age, and the fit parameters are $v_\\mathrm{f} = 1.224 \\pm 0.447$, $m = 1.932 \\pm 0.431$ and $b = 0.622 \\pm 0.354$, with $v_\\mathrm{f}$ and $b$ highly and positively correlated. The light grey region is composed of 300 curves that are created with parameters drawn from a multivariate gaussian distribution defined by the mean values of the fit parameters and their covariance matrix. Skumanich's law (red $\\times$ symbols, calibrated for $v^\\odot_{\\mathrm{rot}} = 1.9$ km s$^{-1}$) and the rotational braking curves proposed by \\citet[][black dashed curve, smoothed]{2014ApJ...790L..23D} and \\citet[][black dot-dashed curve]{2004A&A...426.1021P} are plotted for comparison.}\n\\label{sku}\n\\end{figure*}\n\nOf these 16 spectroscopic binaries, at least four of them (HIP 19911, 43297, 67620 and 73241) show unusually high $v \\sin{i}$ (see the left panel of Fig. \\ref{sku}). These stars also present other anormalities, such as their [Y\/Mg] abundances \\citep{2016A&A...590A..32T} and magnetic activity \\citep{2014A&A...572A..48R, F16inprep}. The solar twin blue straggler HIP 10725 \\citep{2015A&A...584A.116S}, which is not included in our sample, also shows a high $v \\sin{i}$ for its age. We find that five of the binaries have rotational velocities below the expected for Sun-like stars, but this is most likely an effect of projection of the stars' rotational axes. For the remaining binaries, which follow the rotational braking law, it is again difficult to disentangle this behavior from the $\\sin{i}$, and a statistical analysis is precluded by the low numbers involved. Tidal interactions between companions that could potentially enhance rotation depend on binary separation, which is unknown for most of these stars. They should be regular rotators, since they do not show anormalities in chromospheric activity \\citep{F16inprep} or [Y\/Mg] abundances \\citep{2016A&A...590A..32T}.\n\nBased on the information that at least 25\\% of the spectroscopic binaries in our sample show higher rotational velocities than expected for single stars, we conclude that stellar multiplicity is an important enhancer of rotation in Sun-like stars. Blue stragglers are expected to have a strong enhancement on rotation due to injection of angular momentum from the donor companion.\n\n\n\\section{The rotational braking law}\\label{br_law}\n\nIn order to correctly constrain the rotational braking, we removed from this analysis all the spectroscopic binaries. The non-SB HIP 29525 displays a $v \\sin{i}$ much higher than expected ($3.85 \\pm 0.13$ km s$^{-1}$), but it is likely that this is due to an overestimated isochronal age ($2.83 \\pm 1.06$ Gyr). Because it is a clear outlier in our results, we decided to not include HIP 29525 in the rotational braking determination. \\citet{2010A&A...521A..12M} found X-ray and chromospheric ages of 0.55 and 0.17 Gyr, respectively, for HIP 29525. We then divided the remaining 65 stars and the Sun in bins of 2 Gyr, and removed from this sample all the stars which were below the 70th percentile of $v \\sin{i}$ in each bin\\footnote{\\footnotesize{By doing a simple simulation with angles $i$ drawn from a flat distribution between 0 and $\\pi\/2$, we verify that 30\\% of the stars should have $\\sin{i}$ above 0.9.}}. This allowed us to select the stars that had the highest chance of having $\\sin{i}$ above $0.9$. In total, 21 solar twins and the Sun compose what we hereafter reference as the \\textit{selected sample}. Albeit this sub-sample is smaller, it has the advantage of mostly removing uncertainties on the inclination angle of the stellar rotation axes\\footnote{\\footnotesize{This procedure can also allow for unusually fast-rotating stars (although rare) with $\\sin {i}$ below 0.9 to leak into our sample.}}. We stress that the only reason we can select the most probable edge-on rotating stars ($i = \\pi\/2$) is because we have a large sample of solar twins in the first place.\n\nWe then proceeded to fit a general curve to the \\textit{selected sample} (see Fig. \\ref{sku}) using the method of orthogonal distance regression \\citep[ODR,][]{boggs1990orthogonal}, which takes into account the uncertainties on both $v \\sin{i}$ and ages. This curve is a power law plus constant of the form $v = v_\\mathrm{f} + m\\ t^{-b}$ \\citep[the same chromospheric activity and $v \\sin{i}$ vs. age relation used by][]{2004A&A...426.1021P, 2009IAUS..258..395G}, with $v$ (rotational velocity) and $v\\rm_f$ (asymptotic velocity) in km s$^{-1}$ and $t$ (age) in Gyr.\n\nWe find that the best fit parameters are $v_\\mathrm{f} = 1.224 \\pm 0.447$, $m = 1.932 \\pm 0.431$ and $b = 0.622 \\pm 0.354$ (see right panel of Fig. \\ref{sku}). These large uncertainties are likely due to: i) the strong correlation between $v_\\mathrm{f}$ and $b$; and ii) the relatively limited number of datapoints between 1 and 4 Gyr, where the parameter is most effective in changing the values of $v$. This limitation is also present in past studies \\citep[e.g.,][]{2016Natur.529..181V, 2003ApJ...586..464B, 2004A&A...426.1021P, 2008ApJ...687.1264M, 2014A&A...572A..34G, 2016A&A...587A.105A}. On the other hand, our sample is the largest comprising solar twins, and therefore should produce more reliable results. With more datapoints, we could be able to use 1 Gyr bins instead of 2 Gyr in order to select the fastest rotating stars, which would result in a better sub-sample for constraining the rotational evolution for young stars.\n\nThe relation we obtain is in contrast with some previous studies on modelling the rotational braking \\citep{2001ApJ...561.1095B, 2003ApJ...586..464B, 2015A&A...584A..30L} which either found or assumed that the Skumanich's law explains well the rotational braking of Sun-like stars. The conclusions by \\citet{2016Natur.529..181V} limit the range of validation up to approximately the solar age (4 Gyr) for stars with solar mass. When we enforce the Skumanich's power law index $b = 1\/2$, we obtain a worse fit between the ages 2 and 4 Gyr (and, not surprisingly, also after the solar age).\n\nOur data and the rotational braking law that results from them show that the Sun is a normal star regarding its rotational velocity when compared to solar twins. However, they do not agree with a regular Skumanich's law \\citep[][red $\\times$ symbols in Fig. \\ref{sku}]{2007ApJ...669.1167B}. We find a better agreement with the model proposed by \\citet[][black dashed curve in Fig. \\ref{sku}]{2014ApJ...790L..23D}, especially for stars older than 2 Gyr. This model is thoroughly described in Appendix A of \\citet{2012A&A...548L...1D}. In summary, it uses an updated treatment of the instabilities relevant to the transport of angular momentum according to \\citet{1992A&A...265..115Z} and \\citet{1997A&A...317..749T}, with an initial angular momentum for the Sun $J_0 = 1.63 \\times 10^{50}$ g cm$^2$ s$^{-1}$. Its corresponding rotational braking curve is computed using the output radii of the model, which vary from $\\sim$1 R$_\\odot$ at the current solar age to 1.57 R$_\\odot$ at the age of 11 Gyr, and it changes significantly if we use a constant radius $R = 1$ R$_\\odot$, resulting in a more Skumanich-like rotational braking.\n\nOur result agrees with the chromospheric activity vs. age behavior for solar twins obtained by \\citet{2014A&A...572A..48R}, in which a steep decay of the $R'\\rm_{HK}$ index during the first 4 Gyr was deduced (see Fig. 11 in their paper). The study by \\citet{2004A&A...426.1021P} also suggests a steeper power-law index ($b = 1.47$) than Skumanich's ($b_\\mathrm{S} = 1\/2$) in the rotational braking law derived from young open clusters, the Sun and M 67. However, as seen in Fig. \\ref{sku}, their relation significantly overestimates the rotational velocities of stars, especially for those older than 2 Gyr. This is most probably caused by other line broadening processes, mainly the macroturbulence, which were not considered in that study. As we saw in Sect. \\ref{vmacro_det}, those introduce important effects that are sometimes larger than the rotational broadening. Moreover, a CCF-only analysis tends to produce more spread in the $v \\sin{i}$ than the more detailed analysis we used.\n\nThe rotational braking law we obtain produces a similar outcome to that achieved by \\citet{2016Natur.529..181V} for stars older than the Sun (a weaker rotational braking law after solar age than previously suggested). Our data also requires a different power law index than Skumanich's index for stars younger than the Sun, one that accounts for an earlier decay of rotational velocities up to 2 Gyr.\n\nThe main sequence spin-down model by \\citet{1988ApJ...333..236K} states that, for constant moment of inertia and radius during the main sequence, we would have\n\n\\begin{equation}\n v_{\\mathrm{eq}} \\propto t^{-3\/(4an)} \\mathrm{,}\n \\label{kawaler}\n\\end{equation}where $v_{\\mathrm{eq}}$ is the rotational velocity at the equator and $a$ and $n$ are parameters that measure the dependence on rotation rate and radius, respectively (see Eqs. 7, 8 and 12 in their paper). If we assume a dipole geometry for the stellar magnetic field ($B_\\mathrm{r} \\propto B_0 r^3$), then $n = 3\/7$. Furthermore, assuming that $a = 1$, then Eq. \\ref{kawaler} results in $v_{\\mathrm{eq}} \\propto t^{-7\/4} = t^{-1.75}$. Skumanich's law ($v_{\\mathrm{eq}} \\propto t^{-0.5}$) is recovered for $n = 3\/2$, which is close to the case of a purely radial field ($n = 2$, $v_{\\mathrm{eq}} \\propto t^{-0.38}$). A more extensive exploration of the configuration and evolution of magnetic fields of solar twins is outside the scope of this paper, but our results suggest that the rotational rotational braking we observe on this sample of solar twins stems from a magnetic field with an intermediate geometry between dipole and purely radial.\n\n\n\\section{Conclusions}\n\nWe analyzed the rotational velocities of 82 bright solar twins in the Southern Hemisphere and the Sun using extremely high resolution spectra. Radial velocities revealed that our sample contained 16 spectroscopic binaries, three of which (HIP 30037, 62039, 64673) were not listed as so in the literature. At least five of these stars show an enhancement on their measured $v \\sin{i}$, which is probably caused by interaction with their close-by companions. They also present other anomalies in chemical abundances and chromospheric activities. We did not clearly identify non-spectroscopic binary stars with unusually high rotational velocities for their age.\n\nIn order to better constrain the rotational evolution of the solar twins, we selected a subsample of stars with higher chances of having their rotational axis inclination close to $\\pi\/2$ (almost edge-on). We opted to use carefully measured isochronal ages for these stars because it is the most reliable method available for this sample. We finally conclude that the Sun seems to be a common rotator, within our uncertainties, when compared to solar twins, therefore it can be used to calibrate stellar models.\n\nMoreover, we have found that Skumanich's law does not describe well the rotation evolution for solar twins observed in our data, a discrepancy that is stronger after the solar age. Therefore, we propose a new rotational braking law that supports the weakened braking after the age of the Sun, and comes with a earlier decay in rotational velocities up to 2 Gyr than the classical Skumanich's law. Interestingly, it also reveals an evolution that is more similar to the magnetic activity evolution observed in Sun-like stars, which sees a steep decay in the first 3 Gyr and flattens near the solar age. Additionally, we suggest that more high-precision spectroscopic observations of solar twins younger and much older than the Sun could help us better constrain the rotational evolution of solar-like stars.\n\\begin{acknowledgements}\n      LdS thanks CAPES and FAPESP, grants no. 2014\/26908-1 and 2016\/01684-9 for support. JM thanks for support by FAPESP (2012\/24392-2). LS acknowledges support by FAPESP (2014\/15706-9). We also would like to thank the anonymous referee for the valuable comments that significantly improved this manuscript.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\setcounter{footnote}{0}\n\n\nMore than a decade ago, matrix model was expected to realize the non-perturbative definition of string field theories through the double scaling limit\\cite{DS}.\nIn the matrix models, the interaction of the spin cluster domain wall (I-K type interaction) plays an important role to obtain the non-critical string field theory\\cite{IK}.\nHermitian matrix models formulate the Dynamical Triangulation of the discrete 2D surface, in which orientable strings propagate or interact\\cite{JR}\\cite{AJ}\\cite{Mog}. \nThe Loop gas model is a description of the non-critical string field theory, in which every string is located in 1-dimensional discrete target-space point $x$ and interacts with another one in the same point or neighboring points in each time evolution\\cite{KK}.\nIt is formulated by the matrix possessing the additional index $x$\\cite{Kos}\\cite{EKN}.\nOn the other hand, it is well known that the stochastic quantization of the matrix models is effective to deduce the string field theories, and the stochastic time plays the role of the geodesic distance on the 2D random surfaces\\cite{JR}\\cite{Nak}.\nHowever, one of the problems in the Dynamical Triangulation is that the probability of splitting interaction becomes too large to construct the realistic space-time.\nThis problem becomes more severe in higher dimension.\nEven in 2D model, the whole surface of the world-sheet is covered with many projections of infinitesimal baby universe.\n\nThe Causal Dynamical Triangulation (CDT) model is proposed to improve the above problem\\cite{AL}.\nIn this model, the triangulation is severely restricted because of the time-foliation structure.\nThe most characteristic feature of the CDT model is that the causality forbid the splitting and merging interaction.\nThere appears no baby universe and a string propagator becomes a torus with smooth surface.\nThe CDT model is generalized to include only the splitting interaction but not the merging interaction.\nIt is the Generalized Causal Dynamical Triangulation (GCDT) model and baby universes make the world surface not be smooth\\cite{ALWZ}.\nA string field theory is constructed from the CDT and its Schwinger-Dyson (S-D) equations are investigated\\cite{ALWWZ1}.\nIt is also formulated by a matrix model further\\cite{ALWWZ2}.\nRecently, the GCDT model is extended to include additional I-K type interaction and its matrix model formulation is also proposed\\cite{FSW}.\nThe S-D equation of this model has features of the non-critical string field theory.\n\nIn this note, we construct the CDT model from the matrix model for the loop gas model.\nWe assign the matrix an additional discrete index, which is interpreted as discrete time or geodesic distance.\nIn the original loop gas model, it is interpreted as space.\nThen, we apply the stochastic quantization method to this model in order to realize a string field theory of the GCDT model.\nThe main difference from other matrix model formulation is that the stochastic time does not have relation to the geodesic distance in our model.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics [width=100mm, height=20mm]{CDT1.eps}\n\\caption{A configulation of the CDT triangulation of the 1-step propagation}\n\\label{fig:CDT1}\n\\end{center}\n\\end{figure}\n\nAt the beginning, we briefly review some fundamental nature about the CDT in the 2D space-time.\nIn this model, a torus of loop propagation is sliced to many rings with small width $a$, the minimal discrete time.\nEach ring corresponds to the 1-step time propagation of a loop, from an edge to another edge.\nLoops of the edges are composed of the links with length $a$ so that the length of loop is also discretized.\nA ring is constructed with triangles, one of whose three edges has to be a component of the loop on one side and the other two edges are connected to other triangles.\nTherefore the 1-step propagation from a loop with the length $k$ to another one with the length $m$ is composed of $k+m$ triangles, with $k$ upward ones and $m$ downward ones (Fig.\\ref{fig:CDT1}).\nSince the number of the configuration corresponds to the amplitude of this 1-step propagation, we define the 1-step \"two-loop function\" as\n\\bea\n\\label{eq:amplitude}\nG^{(0)} (k,m; 1) \\equiv {g^{k+m} \\over k+m}~_{k+m}{\\rm C}_k ,\n\\ena\nwhere $g$ is associated with each triangle and $_{k+m}{\\rm C}_k$ expresses the binomial coefficient.\nIt is rather convenient to define the 1-step marked propagator of the loop with length $k$ as $G^{(1)} (k,m;1) \\equiv k G^{(0)}(k,m;1)$.\nWe can construct $t$-step unmarked (and marked) propagators by piling up the 1-step unmarked (and marked) propagators\n\\bea\n\\label{eq:tstep}\nG^{(0)} (n,m;t) & = & \\sum_{k=1}^{\\infty} G^{(0)}(n,k;t-1) k G^{(0)} (k,m;1), \\nonumber \\\\\nG^{(1)} (n,m;t) & = & \\sum_{k=1}^{\\infty} G^{(1)}(n,k;t-1) G^{(1)} (k,m;1),\n\\ena\nrespectively.\nThe disc amplitude $W(n)$ is the summation of the amplitudes such that the loop with the length $n$ becomes to zero in some future time, and it is expressed as\n\\bea\n\\label{eq:discamp}\nW(n) \\equiv \\sum_{t}^{\\infty} G^{(1)} (n,0;t).\n\\ena\nThen, we expect the superposing relation,\n\\bea\n\\label{eq:discsum}\nW(n) \\equiv \\sum_{k=1}^{\\infty} G^{(1)} (n,k;1) W(k)+G^{(1)}(n,0;1).\n\\ena\n\nOriginally, the CDT model does not contain splitting interaction nor merging interaction, because these processes violate the causality.\nThis means that a saddle point on the world-sheet causes to two distinct light cones.\nHowever, we can include the splitting interaction if we impose the condition such that any separated baby loop shrinks to length zero and the mother loop propagates without interacting with it.\nIt is the GCDT model, in which the \"causality\" in a broad sense is respected.\n \nWe propose a matrix model of the modified version of the loop gas model, with a fundamental matrix $(M_{tt'})_{ij}$, where the indices $i,~j$ run from 1 to $N$.\nThe $N \\times N$ matrix $M_{tt'}$ corresponds to a link variable which connects two sites on the discrete times $t$ and $t'$ with the direction from $t$ to $t'$.\nThen we start with the action of $U(N)$ gauge invariant form,  \n\\bea\n\\label{eq:actionM}\nS[M] = -g\\sqrt{N} {\\rm tr} \\sum_t M_{tt'} + {1 \\over 2}{\\rm tr} \\sum_{t,t'} M_{tt'} M_{t't} - {g \\over 3\\sqrt{N}} {\\rm tr} \\sum_{t,t',t''} M_{tt'} M_{t't''} M_{t''t},\n\\ena\nwith the partition function $Z = \\int {\\cal D} M e^{-S[M]}$.\n$M_{tt} \\equiv A_t$ is a hermitian matrix, which corresponds to a link of discrete string element soaked in one time $t$.\n$M_{t,t+1} \\equiv B_t$ and $M_{t+1,t} \\equiv B^{\\dagger}_t$ are associated with a link connecting sites on the nearest neighboring times $t$ and $t+1$.\nOtherwise $M_{tt'} = 0$ (for $t' \\neq t, t \\pm 1$).\nHence we can rewrite the partition function as $Z = \\int {\\cal D} A {\\cal D} B {\\cal D} B^{\\dagger} e^{-S[A,B,B^{\\dagger} ] } $ with the action \n\\bea\n\\label{eq:actionAB}\nS[A, B, B^{\\dagger}] &=& -g\\sqrt{N} {\\rm tr} \\sum_t A_{t} + {1 \\over 2}{\\rm tr} \\sum_t A_t^2 + {\\rm tr} \\sum_t B_t B_t^{\\dagger}  \\nonumber \\\\\n & & - {g \\over 3\\sqrt{N}} {\\rm tr} \\sum_t A_t^3 - {g \\over \\sqrt{N}} {\\rm tr} \\sum_t (A_t B_t B_t^{\\dagger} + A_{t+1} B_t^{\\dagger} B_t).\n\\ena\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics [width=100mm, height=35mm]{CDT2.eps}\n\\caption{The triangulation assignment for the cubic terms and the square term}\n\\label{fig:CDT2}\n\\end{center}\n\\end{figure}\nTwo square terms in the first line play the role of gluing the links of two triangles, and three cubic terms in the second line express the triangles which are the elements of surface (Fig.\\ref{fig:CDT2}).\nWhile the last two terms composed of $A, B$ and $B^{\\dagger}$ correspond to the triangles on the surfaces of string propagation ring, the cubic term of $A$ produces a triangle soaked in one time-slice, which does not exist in the CDT model.\nThrough integrating out the non-hermitian matrices $B_t$ and $B^{\\dagger}_t$, we obtain the effective theory only with equi-temporal link matrices $A_t$.\nThe partition function becomes $Z \\equiv \\int {\\cal D} A e^{-S_{\\rm eff} [A]}$, where\n\\begin{eqnarray*}\nS_{\\rm eff} [A] = {\\rm tr} \\sum_t \\left[ -g \\sqrt{N} A_t + {1 \\over 2} A_t^2  - {g \\over 3\\sqrt{N}} A_t ^3 + {\\rm log} \\Bigl\\{ {\\bf 1} - {g \\over \\sqrt{N}}  \\left( A_t {\\bf 1}_{t+1} + {\\bf 1}_t A_{t+1}  \\right) \\Bigr\\} \\right].\n\\end{eqnarray*}\nIf we define a loop variable $\\phi _t (n) \\equiv {1 \\over N}{\\rm tr}({A_t \\over \\sqrt{N}})^n$ as the discrete closed string of length $n$ in time $t$, the effective action of the loop variables is written as\n\\bea\n\\label{eq:actioneff}\nS_{\\rm eff} [\\phi , A] & = & - N^2 \\sum_t \\left[ {g \\over N} {\\rm tr}  {A_t \\over \\sqrt{N}} - {1 \\over 2N}{\\rm tr}  \\left( {A_t \\over \\sqrt{N}} \\right) ^2  + { g \\over 3N} {\\rm tr}  \\left( {A_t \\over \\sqrt{N}} \\right) ^3 \\right. \\nonumber \\\\\n & &  \\left. +  \\sum_{k=0}^{\\infty} \\sum_{m=0}^{\\infty} G^{(0)}(k,m;1) \\phi _t (k) \\phi _{t+1}(m) \\right],\n\\ena\nwhere $G^{(0)}(k,m;1)$ is the two-loop function of the 1-step time appeared in the CDT model.\nThe last term of eq.(\\ref{eq:actioneff}) is expressed graphically in Fig.\\ref{fig:CDT1}.\nWe may construct the $t$-step propagator as \n\\begin{eqnarray*}\nnmG^{(0)}(n,m;t)  = \\langle \\phi_0 (n) \\phi _t (m) \\rangle  = {1 \\over Z} \\int {\\cal D} A \\phi _0 (n) \\phi _t (m) e^{-S_{\\rm eff}[A]}, \n\\end{eqnarray*}\nThis expresses the sum over all ways of connecting $\\phi _0 (n)$ with $\\phi _t (m)$ by $t$ times piling of 1-step two-loop functions.\nThus we realize the matrix model formulation of the CDT model.\n\nNow, we extend this model to the GCDT with loop interactions by applying the stochastic quantization method to the above model.\nThe Langevin equation for a matrix variable and white noise correlation\n\\bea\n\\label{eq:langevinA}\n\\v (A_t)_{ij} = - {{\\partial S_{\\rm eff}} \\over {\\partial (A_t)_{ji}}} \\v \\t + \\v (\\xi _t)_{ij}, ~~~~~~\n\\langle \\v (\\xi _t)_{ij} \\v (\\xi _t')_{kl} \\rangle _\\xi = 2\\v \\t \\d _{tt'} \\d _{il} \\d _{jk},\n\\ena\ndescribe the evolution of the matrices on the step of the unit stochastic time $\\v \\t$.\nThey generate the Langevin equation for a loop variable \n\\bea\n\\label{eq:langevin}\n\\v \\phi _t (n) &=& \\v \\t n \\left[ g \\phi _t (n-1) - \\phi _t (n) + g \\phi _t (n+1) \n + \\sum_{k =0}^{n-2} \\phi _t ( k ) \\phi_t (n - k - 2) \\right. \\nonumber \\\\\n & & + \\sum_{k=1}^{\\infty} \\sum_{m=0}^{\\infty} G^{(1)} (k,m;1) \\phi _t (n+k-2) \\phi _{t+1} (m)  \\nonumber \\\\\n & & \\left. + \\sum_{k=1}^{\\infty} \\sum_{\\ell =0}^{\\infty} G^{(2)} (\\ell ,k;1) \\phi _t (n+k-2) \\phi _{t-1} (\\ell) \\right] + \\v \\zeta _t (n),\n\\ena\nwhere $G^{(1)} (k,m;1) \\equiv kG^{(0)} (k,m,t=1) $ and $G^{(2)} (k,m;1) \\equiv mG^{(0)} (k,m,t=1) $ are 1-step marked propagators with a mark on the entrance loop and the exit loop, respectively.\n$\\v \\zeta _t (n) \\equiv N^{-1-{n \\over 2}} n {\\rm tr} ( \\v \\xi _t A_t^{n-1} )$ is the constructive noise term which satisfies the correlation\n\\bea\n\\label{eq:noisecorrelation}\n\\langle \\v \\zeta _t (n) \\v \\zeta _{t'} (m) \\rangle _\\xi = 2 \\v \\t \\d _{tt'} {1 \\over N^2} nm \\langle \\phi _t (n+m-2) \\rangle _\\xi .\n\\ena\nAny observable $O(\\phi)$ composed of loop variables is deformed, under the stochastic time 1-step progress $\\v \\t$, following the variation of $\\phi _t (n)$ with the Langevin equation (\\ref{eq:langevin}) and the noise correlation (\\ref{eq:noisecorrelation}).\nThe generator of this $\\v \\t$ evolution corresponds to the Fokker-Planck (F-P) Hamiltonian $H_{\\rm FP}$,\n\\bea\n\\label{eq:FPHdef}\n\\langle \\v O(\\phi ) \\rangle _\\xi &=& \\langle \\sum_m \\v \\phi _t (m) {\\partial \\over \\partial \\phi _t (m) } O(\\phi ) + {1 \\over 2} \\sum_{m,n} \\v \\phi _t (m) \\v \\phi _t (n) {\\partial ^2 \\over \\partial \\phi _t (m) \\partial \\phi _t (n)} O(\\phi ) \\rangle _\\xi \\nonumber \\\\\n & & +{\\rm O}(\\v \\t^{3 \\over 2}) \\nonumber \\\\\n & \\equiv & - \\v \\t \\langle H_{\\rm FP} O(\\phi ) \\rangle _{\\xi}.\n\\ena\nWe interpret $\\phi _t (n)$ and $\\pi _t (n) \\equiv {\\partial \\over \\partial \\phi _t (n) }$ as the operators for creation and annihilation of the loop with length $n$ at time $t$, respectively.\nThey fulfill the following commutation relation:\n\\bea\n\\label{eq:commutation}\n[\\pi _t (n), \\phi _{t'} (m) ] = \\d _{tt'} \\d _{nm}.\n\\ena\nThen the F-P Hamiltonian is expressed as\n\\bea\n\\label{eq:FP1}\nH_{\\rm FP} = -{1 \\over N^2} \\sum_t \\sum_{n=1}^{\\infty} n L_t (n-2) \\pi _t (n),\n\\ena\nwhere $L_t (n)$ is defined by\n\\bea\n\\label{eq:generator}\nL_t (n) &=& -N^2 \\Biggl[ g \\phi _t (n+1) - \\phi _t (n+2) + g \\phi _t (n+3) \\Biggr. \\nonumber \\\\\n& & +\\sum_{k=1}^{\\infty} \\sum_{m=0}^{\\infty} G^{(1)} (k,m;1) \\phi _t (n+k) \\phi _{t+1} (m) \n + \\sum_{k=1}^{\\infty} \\sum_{\\ell =0}^{\\infty} G^{(2)} (\\ell ,k;1) \\phi _t (n+k) \\phi _{t-1} (\\ell) \\nonumber \\\\\n& & \\left. + \\sum_{k=0}^n \\phi _t (k) \\phi _t (n-k) + {1 \\over N^2} \\sum_{k=1}^{\\infty} k \\phi _t (n+k) \\pi _t (k) \\right],\n\\ena\nand it satisfies the Virasoro algebra\n\\bea\n\\label{eq:virasoro}\n[L _t (n), L _{t'} (m) ] = (n-m)\\d _{tt'} L _t (n+m).\n\\ena\n\nIn order to take the continuum limit we introduce the minimum length of this matrix model $a$.\nThe continuum limit is realized by taking $a$ to zero simultaneously with $N$ to infinity.\nIt is called the double scaling limit.\nAccording to the CDT model, we set the continuum loop length as $ L \\equiv an $ and time as $T \\equiv at $\\cite{AL}.\nWe also define the cosmological constant $\\Lambda$ by ${1 \\over 2}e^{-{1 \\over 2}a^2 \\Lambda} = g$.\nHere we introduce two scaling parameters $D$ and $D_N$.\nSince the commutation relation of the continuum field operators satisfy,\n\\bea\n\\label{eq:commutator}\n[\\Pi  (L;T), \\Phi  (L';T') ] = \\d (T-T') \\d (L-L'),\n\\ena\nthe scaling of the loop field operators can be described as $\\Phi  (L;T) \\equiv a^{-{1 \\over 2}D} \\phi _t (n) $ and $\\Pi (L;T) \\equiv a^{{1 \\over 2}D-2} \\pi _t (n)$ by using $D$.\nTo keep the effect of the first term of the last line in eq.(\\ref{eq:generator}), the splitting interaction, we fix the scaling of the infinitesimal stochastic time as\n\\bea\n\\label{eq:stochastic}\nd \\t \\equiv a^{{1 \\over 2}D-2} \\v \\t.\n\\ena\nHence the existence of the continuum stochastic time requires $D>4$.\nThe terms in the second line of eq.(\\ref{eq:generator}) express the characteristic interaction of this model, which corresponds to the I-K type interaction.\nThus we maintain these terms by redefining the 1-step propagator as $\\tilde{G}^{(1)}(L_1, L_2 ; a) \\equiv a^{-1} G^{(1)} (k,m;1)$, which gives the expression using the continuum lengths.\nWe define the string coupling constant as $G_{\\rm st} \\equiv {1 \\over N^2} a^{D_N}$ with $D_N$.\nThen we obtain the continuum limit of the F-P Hamiltonian ${\\cal H}_{\\rm FP}$ by $\\v \\t H_{\\rm FP} \\equiv d \\t {\\cal H}_{\\rm FP}$, and it is written as\n\\bea\n\\label{eq:FPhamiltonian}\n{\\cal H}_{\\rm FP}  & = & \\int dT \\int_0^{\\infty} dL L \\left[ a^{-{1 \\over 2}D+3} {1 \\over 2} \\left( {{\\partial ^2} \\over {\\partial L ^2}} - \\Lambda \\right) \\Phi (L;T)  \\right. \\nonumber \\\\\n & & +  \\int _0^{\\infty} dL_1 \\int _0^{\\infty} dL_2 \\tilde G ^{(1)} (L_1, L_2; a) \\Phi (L+L_1;T) \\Phi (L_2; T+a)  \\nonumber \\\\\n & & +  \\int _0^{\\infty} dL_1 \\int _0^{\\infty} dL_2 \\tilde G ^{(2)} (L_2, L_1; a) \\Phi (L+L_1;T) \\Phi (L_2; T-a)  \\nonumber \\\\\n & & + \\int _0^L dL_1 \\Phi (L_1 ; T) \\Phi (L - L_1 ;T) \\nonumber \\\\\n & & \\left.  + a^{-D_N -D+1} G_{\\rm st} \\int _0^{\\infty} dL_1 L_1 \\Phi (L+L_1; T) \\Pi (L_1; T) \\right]  \\Pi(L ; T).\n\\ena\nThe first term on the r.h.s is the potential term, which means the propagation of a loop in an equi-temporal slice.\nWe have to remember that any propagation of the loop in one equi-temporal slice is not contained in the GCDT model.\nHence we expect this term to scale out in the continuum limit.\nThis fact requires $D<6$.\nIt should be noted that, thanks to the scaling of the cosmological constant, after summing up the first three terms in eq.(\\ref{eq:generator}) the scaling of the propagation terms are enhanced two order higher compared with that of the original terms.\nThis enhancement makes the above restriction $D<6$ for $D$ consistent with another restriction $D>4$ from eq.(\\ref{eq:stochastic}).\n\nThe next two terms are I-K type interactions which are similar terms appeared in the non-critical string field theory model (Fig.\\ref{fig:CDT3}(b))\\cite{IK}.\nThe second and the third terms cause the annihilation of a loop with the length $L$ and creation of a loop with the length $L+L_1$ at the same time $T$.\nThey also create another loop with the length $L_2$ at the infinitesimal future time $T+a$ and infinitesimal past time $T-a$, respectively.\nThen the lengths $L_1$ and $L_2$ are connected by the infinitesimal 1-step marked propagator.\nThe fourth term is the splitting interaction, which annihilates a loop with length $L$ and create two loops with the sum of their lengths $L$, simultaneously.\nThe last term expresses the merging interaction, which annihilates two loops and create one loop whose length is equal to the sum of the two annihilated loops.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics [width=100mm, height=25mm]{CDT3.eps}\n\\caption{(a) Splitting interaction and (b) I-K type interaction. The entrance loop propagates upward to the exit loop as the mother universe, while the baby universe must disappear to the vacuum before long. On the surface of propagating world-sheet, many baby universes of two types are attached as quantum effect.}\n\\label{fig:CDT3}\n\\end{center}\n\\end{figure}\n\nWhile the splitting interaction is permitted in the GCDT model, the merging interaction should be forbidden because of the \"causality\" in a broad sense.\nFor this purpose we restrict the scaling parameter $D_N$ to satisfy\n\\bea\n\\label{eq:DN}\n D_N < -D+1.\n\\ena\nCombining this condition and $4<D<6$, the value of $D_N$ can be fixed.\nIf $D_N<-5$ the causality is respected, while if $D_N>-3$ it is violated.\nWhen $-5 \\le D_N \\le -3$, the condition of eq.(\\ref{eq:DN}) must be taken into account.\n\nThe interpretation of the stochastic process is following.\nIf the stochastic process is not included, we have the exact CDT model.\nEach stochastic time step $\\v \\t$ produces one baby universe in either of two ways, that is the ordinary splitting interaction and the I-K type interaction (Fig.\\ref{fig:CDT3}).\nWhile infinitesimal time or geodesic distance scales as $dT = a$, infinitesimal stochastic time scales by eq.(\\ref{eq:stochastic}) with $4<D<6$.\nThis may be expressed as $d\\t = a^{D_{\\t}} \\v \\t $, with $0<D_{\\t} <1 $.\nThis expression shows that $d \\t$ goes to zero more slowly than $dT$.\nFrom this we give a conjecture that the stochastic process, or the quantum effect occurs less frequently than the ordinary propagation in time.\nTherefore we can expect the processes creating baby universes may not become dominant in the propagation.\n\nIt is interesting to suppose the change of the scaling parameter $D$.\nThe GCDT model is realized for $4<D<6$, and then the world-sheet might have reasonable number of baby universes of two types on its surface.\nWhen $D>6$, the loop propagator contains too many stochastic processes, dominated by the propagation in the equi-temporal slice.\nThe string interactions are suppressed except for the merging interaction, whose probability depends on the scaling parameter $D_N$.\nHence, we can imagine that the loop propagator has too much length-fluctuation in the time evolution.\nIn $D=6$, each equi-temporal slice contains fertile stochastic processes including propagation, as well as splitting and I-K type interactions.\n\\footnote{In the S-D equation, as we will discuss later, an additional potential term concerning propagation $L ({\\partial ^2 \\over \\partial L^2}-\\Lambda ){W}(L)$, that is characteristic in the CDT model, survives in the scaling limit.}\\\nHowever, it is not desirable from the viewpoint of our model.\n\nFinally, we derive the S-D equation from the continuum version of the Langevin equation.\nThe disc amplitude is the expectation value of a loop variable, which propagates and shrink to nothing eventually.\nIt is expressed in the continuum expression as\n\\begin{eqnarray*}\nW(L) \\equiv \\langle \\Phi (L;T) \\rangle \\equiv \\int _0^{\\infty}  dT \\tilde{G}^{(1)} (L,0;T).\n\\end{eqnarray*}\nWith the help of eq.(\\ref{eq:discsum}), at the level of the expectation value we can expect the relation \n\\bea\n\\label{eq:disc}\n\\langle \\int _0^{\\infty} dL_2 \\tilde{G}^{(1)}(L_1,L_2;a) \\Phi (L_2; T+a) \\rangle = W(L_1).\n\\ena\nFrom this the S-D equation is derived as follows,\n\\bea\n\\label{eq:SD}\n\\int _0^L dL_1 W(L_1) W(L-L_1) +2 \\int _0^{\\infty} dL_1 W(L+L_1)W(L_1) =0. \n\\ena\nIn terms of the Laplace-transformed variable $\\tilde{W}(z) \\equiv \\int _0^{\\infty} dL e^{-Lz} W(L) $, eq.(\\ref{eq:SD}) is transformed to the expression in the Laplace space.\nWith the exchange of $z \\rightarrow -z$, we obtain another equation for $\\tilde{W}(-z)$.\nThen using above two equations, we obtain the S-D equation as\n\\bea\n\\label{eq:LSD}\n\\tilde{W}(z)^2 +2 \\tilde{W}(z) \\tilde{W}(-z) +\\tilde{W}(-z)^2 = \\mu ,\n\\ena\nwhere $\\mu$ is some constant.\nThis S-D equation expresses the same form as the one of the non-critical closed string field theory except for the coefficient of the second term of the l.h.s..\nIt is replaced with a coefficient $2{\\rm cos} \\pi p_0 $, where the background momentum $p_0 ={1 \\over m}$  corresponts to the central charge $c=1-{6 \\over m(m+1)}$.\nThe last term of the effective action eq.(\\ref{eq:actioneff}) is rewritten as $\\sum_{t,t'}  C_{tt'} \\sum_{k,m}^{\\infty} G^{(0)}(k,m;1) \\phi _t (k) \\phi _{t'}(m) $ with an adjacency matrix $C_{tt'} \\equiv \\d _{t',t+1} + \\d _{t',t-1} $.\nIf we adopt a twisted adjacency matrix $C^{(p_0)}_{tt'}= e^{i\\pi p_0}\\d _{t',t+1} + e^{-i\\pi p_0} \\d _{t',t-1}  $ as ref.\\cite{Kos}, instead of our choice $C_{tt'} $, we can express the loop bilinear term of the effective action eq.(\\ref{eq:actioneff}) as\n\\begin{eqnarray*}\n 2 {\\rm cos}\\pi p_0 \\sum _{t} \\sum_{k=0}^{\\infty} \\sum_{m=0}^{\\infty} G^{(0)}(k,m;1) \\phi _t (k) \\phi _{t+1}(m).\n\\end{eqnarray*}\nWe obtain the S-D equation of the non-critical string field theory exactly which has the right coefficient for the second term on the l.h.s. of eq.(\\ref{eq:LSD}).\n\nIn conclusion, we have proposed the matrix model formulation to construct the 2D GCDT model.\nUsing the stochastic quantization approach and taking a continuum limit, we obtain the non-critical string field theory.\nThe scaling parameter $D$ and $D_N$ on the double scaling limit are fixed as eq.(\\ref{eq:DN}) with $4<D<6$ by requiring the consistency to the CDT model.\nWe present one comment about a difference between the GCDT model and the discrete version of loop field theory formulated by our matrix model.\nIn our effective action eq.(\\ref{eq:actioneff}) the sum of the 1-step propagation terms contains the propagation whose length of the exit loop or the entrance loop equal zero.\nIt is not contained in the original CDT model\\cite{AL}.\nWe can cancel the $G^{(0)}(k,0;1)$ and $G^{(0)}(0,m;1)$ terms in the effective action by adding two fermionic degrees of freedom to the matrix model action.\nIt is done by assigning the Grassmann variables $C_t^{\\a} , \\overline{C}_t^{\\a} (\\a =1,2)$ to the link matrix connecting the sites in the same time slices, or $M_{tt} = A_t, C_t^{\\a}, \\overline{C}_t^{\\a}$.\n\\footnote{The fermionic matrices $C_t^{\\a}, \\overline{C}_t^{\\a}$ satisfy $(\\overline{C}_t^{\\a})_{ij}^* = - (C_t^{\\a})_{ji}$.}\\\nThen the fundamental action is modified as $S_1 \\equiv S - {\\rm tr} \\sum _t \\overline{C}_t C_t + {g \\over \\sqrt{N}}{\\rm tr} \\sum _t \\overline{C}_t A_t C_t $, where $S$ is the action of eq.(\\ref{eq:actionAB}).\nWhen we integrate out the fermionic degrees of freedom in $Z = \\int {\\cal D} A {\\cal D} B {\\cal D} B^{\\dagger} {\\cal D} C {\\cal D} {\\overline C} e^{-S_1 } $, the effective action has the additional term, $-N {\\rm tr} \\sum _t \\sum _{n=1}^{\\infty} {1 \\over n} ( {g \\over \\sqrt{N}} )^n A_t^n$, which cancels the terms concerning to the loop-to-vacuum 1-step processes.\nTherefore, we need not worry about whether the summation of the loop length begins from $n=0$ or $n=1$.\n\n\\section*{Acknowledgments}\nThe author would like to thank D. Suematsu for valuable comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe base $\\Lambda$CDM (Lambda cold dark matter) model is the simplest cosmological model that describes most of the data with remarkable accuracy~\\cite{Riess:1998cb,SupernovaCosmologyProject:1998vns,Planck:2018vyg,Alam:2020sor,DES:2021wwk}. However, even if we set aside its persistent theoretical issues associated with the cosmological constant $\\Lambda$~\\cite{Weinberg:1988cp,Sahni:1999gb,Peebles:2002gy,Padmanabhan:2002ji}, with the increase in the diversity and precision of observational measurements and also advances in data analysis and statistical methods, it has become increasingly clear that a more realistic alternative model is required to replace the six-parameter base $\\Lambda$CDM model as the new standard model of cosmology, and should phenomenologically exhibit non-trivial, if not significant, deviations from it~\\cite{DiValentino:2020vhf,DiValentino:2020zio,DiValentino:2020vvd,DiValentino:2020srs,DiValentino:2021izs,Perivolaropoulos:2021jda,Abdalla:2022yfr}. These deviations, encoded in modifications on top of the Hubble function of the standard cosmological model $H_{\\Lambda\\rm CDM}(z)$, when associated with dark energy (DE), as an effective or actual source, suggest features that are difficult to obtain within the canonical\/simple extensions of $\\Lambda$CDM and are hard to deal with within the established fundamental theories of physics; e.g., DE models that yield a density that attains negative values in the past which present at least one pole in their equation of state (EoS) parameters~\\cite{Delubac:2014aqe,Aubourg:2014yra,Sahni:2014ooa,DiValentino:2017rcr,Mortsell:2018mfj,Poulin:2018zxs,Capozziello:2018jya,Wang:2018fng,Akarsu:2019ygx,Dutta:2018vmq,Banihashemi:2018oxo,Banihashemi:2018has,Farhang:2020sij,Banihashemi:2020wtb,Visinelli:2019qqu,Akarsu:2019hmw,Ye:2020btb,Ye:2020oix,Perez:2020cwa,Calderon:2020hoc,Paliathanasis:2020sfe,Bonilla:2020wbn,Vazquez:2012ag,Akarsu:2020yqa,LinaresCedeno:2021aqk,Zhou:2021xov,LinaresCedeno:2020uxx,DiValentino:2020naf,Akarsu:2021fol,Ozulker:2022slu,DiGennaro:2022ykp}, and\/or present non-trivial characteristics such as an oscillatory EoS parameter that can even cross below the phantom divide line \\cite{Zhao:2017cud,Escamilla:2021uoj,Hojjati:2009ab,Xia:2006rr,Pace:2011kb,Cicoli:2018kdo,Ruchika:2020avj,Heisenberg:2022gqk,Tamayo:2019gqj}, and\/or an oscillatory density \\cite{Colgain:2021pmf,Pogosian:2021mcs,Raveri:2021dbu,Wang:2018fng,Escamilla:2021uoj,Bernardo:2021cxi,Tamayo:2019gqj}.\n\nWhile such deviations are highly non-trivial, the reason they appear in observational studies, as we will show, may be fairly simple. To begin with, let us describe the deviation of any alternative cosmological model from the $\\Lambda$CDM model by ${\\Delta H(z)\\equiv H(z)-H_{\\Lambda\\rm CDM}(z)}$; within the framework of spatially flat Robertson-Walker (RW) metric, fixing the comoving angular diameter distance $D_M(z)=c\\int_0^{z}\\dd{z'}H^{-1}(z')$ of the alternative model to that of $\\Lambda$CDM at any redshift $z=z_{\\rm s}$, requires that $\\int_0^{z_{\\rm s}}\\dd{z'}H^{-1}_{\\Lambda \\rm CDM}(z')= \\int_0^{z_{\\rm s}}\\dd{z'}H^{-1}(z')$. This is satisfied only if there exist at least two redshifts $z_1$ and $z_2$ in the interval $(0,z_{\\rm s})$ for which $\\Delta H(z_1)\/\\Delta H(z_2)<0$ unless $\\Delta H(z<z_{\\rm s})$ vanishes everywhere. Thus, a negative $\\Delta H(z)$ at any redshift (e.g., a dip as preferred by the Ly-$\\alpha$ data \\cite{Sahni:2014ooa,Aubourg:2014yra,Planck:2018vyg}) should be compensated by at least one positive $\\Delta H(z)$ somewhere else (e.g., the bump found in DE density reconstructions at $1.5\\lesssim z\\lesssim2$ \\cite{Wang:2018fng,Escamilla:2021uoj,Bernardo:2021cxi}). This compensatory behaviour implies an oscillation (not necessarily periodic) on top of $H_{\\Lambda\\rm CDM}(z)$ as suggested in the above mentioned observational analyses---which is in line with such behavior being favored by the baryon acoustic oscillations (BAO) data. In particular, we can choose $z_{\\rm s}$ to be the redshift of last scattering $z_*$ as $D_M(z_*)$ is strictly constrained by cosmic microwave background (CMB) observations almost model-independently for a given pre-recombination expansion history. This allows for oscillations up to $z_*$ with no constraints on their characteristics as long as they compensate each other so that the $D_M(z)$ integral is satisfied (they can be very frequent or just a single oscillation spread throughout the whole interval with an arbitrary shape etc.); however, it is conceivable that as cosmological observations other than CMB mostly probe redshifts $z\\lesssim3$, the shape and the place of the oscillations will be constrained (possibly amplified due to overfitting) by these local data.\n\n\nIn addition to the oscillations that arise from fixing the pre-recombination universe and hence $D_M(z_*)$ to that of $\\Lambda$CDM, if one also respects the success of $\\Lambda$CDM in the late universe ($z\\sim0$), we show that, the deviations from the Hubble radius, $H^{-1}(z)$, of $\\Lambda$CDM are described by localized oscillatory functions, namely, \\textit{wavelets}, and, these wavelets should satisfy the \\textit{admissibility condition}. Admissible wavelets are oscillatory functions with a vanishing integral over their whole range, and either have compact support or vanish approximately outside of a compact set of their parameters \\cite{Chui:1992}. They can generically be obtained from derivatives of probability distributions but are by no means limited to this method. In cosmology, the wavelets have been used in various contexts. The wavelet transforms have been used in analyzing the CMB signals \\cite{Sanz:1999xu,Tenorio:1999mf,McEwen:2007ni}, and analyzing the large-scale structure of the universe (to capture its non-Gaussian information content) \\cite{Einasto:2010bi,Allys:2020vld,Valogiannis:2021chp,Eickenberg:2022qvy,Wang:2021zse}; also, see Ref.~\\cite{Valogiannis:2022xwu} and references therein for some applications of wavelets in cosmology and astrophysics. Wavelets have also been considered for investigating possible oscillatory deviations in the DE EoS parameter from minus unity describing the cosmological constant~\\cite{Hojjati:2009ab}; however, note that, their approach of characterizing the oscillations of the EoS parameter with wavelets is fundamentally different from the central idea in this paper that deviations from $H^{-1}_{\\Lambda \\rm CDM}(z)$ should be described by admissible wavelets. Such deviations in the Hubble radius may also be described by wavelet oscillations in the EoS parameter, but, they may also correspond to much more violent behaviours with singularities, or even correspond to a cosmological constant if the wiggles in the Hubble radius are not attributed to the DE. Here, we show that models whose deviations from $\\Lambda$CDM are described by admissible wavelets on top of $H^{-1}_{\\Lambda \\rm CDM}(z)$ constitute a family of cosmological models that are in excellent agreement with the CMB measurements; and, discuss how even the simplest wavelets can lead to non-trivial behaviours in the Hubble parameter describing the BAO measurements without excessive number of free parameters. We also discuss that the same deviations from $H^{-1}_{\\Lambda\\rm CDM}(z)$ can originate from many different extensions in fundamental physics: modified theories of gravity, dynamical or non-minimally interacting DE etc.; and, we expose through examples, how the behaviour of some functions relevant to the source phenomena can be even more non-trivial. For example, if the deviations from $H^{-1}_{\\Lambda\\rm CDM}(z)$ are attributed to DE, the oscillations of the wavelet may cause the DE density to oscillate with a large enough amplitude so that the density attains negative values, resulting in divergences in its EoS parameter. Note that, while the same background dynamics may originate from different extensions, it may be possible to differentiate between these scenarios as we show by comparing the implications of attributing the deviations to the gravitational ``constant\", with attributing them to the DE.\n\n\\section{Wavelets on top of the standard cosmological model's Hubble radius}\n\nWe begin with the fact that the angular scale of the sound horizon at last scattering,\n\\begin{equation}\n\\theta_*=\\frac{r_*}{D_M(z_*)},\\label{eq:theta}\n\\end{equation}\nis measured almost model-independently, e.g., $100\\theta_*=1.04110\\pm 0.00031$ ($\\Lambda$CDM Planck18), with a precision of 0.03\\%, where $r_*$ is the comoving sound horizon at last scattering, and $D_M(z_*)$ is the comoving angular diameter distance out to last scattering~\\cite{Planck:2018vyg}. Then, fixing the pre-recombination physics to that of the standard cosmological model, i.e., $\\Lambda$CDM,\n\\begin{equation}\\label{eq:soundhorizon}\nr_*=\\int_{z_*}^\\infty \\frac{c_{\\rm s}(z)}{H_{\\Lambda \\rm CDM}(z)}\\dd{z}\n\\end{equation}\nis also determined, viz., $r_*=144.43\\pm 0.26\\,\\rm Mpc$ ($\\Lambda$CDM Planck18). Here $c_{\\rm s}(z)$ is the sound speed in the plasma and $z_{*}\\approx 1090$ is the redshift of last scattering (redshift for which the optical depth to Thomson scattering reaches unity) and $H_{\\Lambda \\rm CDM}(z)$ is the Hubble parameter of the standard cosmological model: \n\\begin{equation}\n3H^2_{\\Lambda \\rm CDM}(z)=\\rho_{\\rm m0}(1+z)^3+\\rho_{\\rm r0}(1+z)^4+\\rho_{\\Lambda},\n\\end{equation}\nwhere $\\rho_{\\rm m0}$, $\\rho_{\\rm r0}$, and $\\rho_{\\Lambda}$\nare the present-day energy densities corresponding to those of the pressureless matter (m), the radiation (r), and the cosmological constant ($\\Lambda$)---or, equivalently, the usual vacuum energy of the quantum field theory. We work, for convenience, in units for which the Newton's constant $G_{\\rm N}=1\/8\\pi$ and the speed of light $c=1$ unless they are shown explicitly. Here and in what follows, the subscript $0$ denotes the present-day ($z=0$) value of any quantity. While the values of these energy densities are subject to observational constraints, for the rest of this paper, we will assume them to be fixed (but unknown) values for which\n\\begin{equation}\nD_{M}(z_*)=c\\int_0^{z_*}\\frac{\\dd{z}}{H_{\\Lambda\\rm CDM}(z)}\n\\end{equation}\nis consistent with~\\cref{eq:theta,eq:soundhorizon}, and $\\rho_{\\rm m0}$ is compatible with the positions and relative heights of the peaks in the CMB power spectrum and $\\rho_{\\rm r0}$ is compatible with the observed CMB monopole temperature and standard model of particle physics. This ensures the basic consistency of $\\Lambda$CDM with the CMB data at the background level.\n\nAssuming $H_{\\Lambda \\rm CDM}(z)$ accurately describes the pre-recombination universe (hence $r_*$ is known), for a universe described by the spatially flat RW metric, the comoving angular diameter distance to $z_*$,\n\\begin{equation}\n      D_{M}(z_*)=c\\int_0^{z_*}\\frac{\\dd{z}}{H(z)},\\label{eq:DM}\n\\end{equation}\n of any model described by the Hubble parameter $H(z)$, is strictly constrained almost model-independently through the measurement of $\\theta_*$. Thus, for any $H(z)>0$ (expanding universe) deviating from $H_{\\Lambda \\rm CDM}(z)$, strict observational constraints from CMB still require\n\\begin{equation}\n\\int_0^{z_*}\\frac{\\dd{z}}{H_{\\Lambda \\rm CDM}(z)}\\approx \\int_0^{z_*}\\frac{\\dd{z}}{H(z)}, \\label{eq:approx}\n\\end{equation}\ncf., $D_M(z_*)=13872.83\\pm 25.31\\,\\rm Mpc$ ($\\Lambda$CDM Planck18).\nFor simplicity, we will assume\nthe approximation in Eq.~\\eqref{eq:approx} to be exact, and comment on the approximate case when necessary. Now, we define the deviation of a cosmological model from $\\Lambda$CDM in terms of its Hubble radius, $H(z)^{-1}$, as follows:\n\\begin{equation}\n\\psi(z)\\equiv\\frac{1}{H(z)}-\\frac{1}{H_{\\Lambda \\rm CDM}(z)}.\\label{eqn:devdef}\n\\end{equation}\nThen, we have\n\\begin{equation}\n D_{M}(z_*)=c\\int_0^{z_*}\\dd{z}\\qty[\\frac{1}{H_{\\Lambda \\rm CDM}(z)}+\\psi(z)],\\label{eq:exact}\n\\end{equation}\nand consequently, the exact version of Eq.~\\eqref{eq:approx} implies\n\\begin{equation}\n\\Psi(z_*)\\equiv\\int_0^{z_*}\\psi(z) \\dd{z}=0.\\label{eq:int}\n\\end{equation}\nOur assumption that the pre-recombination universe is accurately described by $H_{\\Lambda \\rm CDM}(z)$, viz., $H(z\\geq z_*)=H_{\\Lambda \\rm CDM}(z\\geq z_*)$, implies another condition on $\\psi(z)$, that is, \n\\begin{equation}\n\\psi(z\\geq z_*)=0. \\label{eq:prereccond}\n\\end{equation}\n\nThis mathematical framework allows one to naturally classify a family of $H(z)$ functions which can deviate, even significantly, from $H_{\\Lambda \\rm CDM}(z)$, but still have the same $D_M(z_*)$ the $\\Lambda$CDM model has, ensuring basic consistency with the CMB measurements at the background level (one might want to also consider the constraints on $\\rho_{\\rm m0}$ and $\\rho_{\\rm r0}$ from CMB). This family is described by \n\\begin{equation}\nH(z)=\\frac{H_{\\Lambda \\rm CDM}(z)}{1+\\psi(z)H_{\\Lambda \\rm CDM}(z)},\\label{eq:hz}\n\\end{equation}\nwhere $\\psi(z)$ satisfies the conditions introduced in~\\cref{eq:int,eq:prereccond}. We notice from this equation that introduction of the condition $-H^{-1}_{\\Lambda \\rm CDM}(z)<\\psi(z)<\\infty$ ensures that, in the past ($z>0$), $H(z)$ never diverges (except at the Big Bang) and the universe has always been expanding. Also, on top of all these conditions, let us demand \n\\begin{equation}\n\\psi(z=0)=0\\label{eq:present}\n\\end{equation}\nsince we know the universe at $z\\sim0$ is well described by the standard $\\Lambda$CDM model \\cite{Planck:2018vyg,Alam:2020sor,DES:2021wwk}.\n\nWe notice that \\cref{eq:int,eq:prereccond,eq:present} describe characteristic properties of functions that are known as \\textit{wavelets} where \\cref{eq:int} is true for wavelets that satisfy the \\textit{admissibility condition} \\cite{Chui:1992}. Wavelets are oscillatory (not necessarily periodic) functions that are well-localized, i.e., they have compact support or they vanish approximately outside of a compact set of its parameters. With such boundary conditions that the function should absolutely or approximately vanish outside of certain bounds, \\cref{eq:int} requires that the function oscillates at least once if it does not vanish everywhere; because, say $\\psi(z)<0$ for a certain value of $z$, this integral can vanish only if $\\psi(z)>0$ at another value of $z$, hence the oscillation. Note that, for a continuous $\\psi(z)$, this argument also implies that there exists at least one value of $z$ in the interval $(0,z_*)$ for which $\\psi=0$; this corresponds to the Rolle's theorem, which in our particular case states that the conditions $\\Psi(0)=0$ and $\\Psi(z_*)=0$ imply the existence of a $z_p\\in(0,z_*)$ for which $\\psi(z_p)=0$. \\textit{Thus, the deviations from the standard $\\Lambda$CDM model's Hubble radius, $\\psi(z)$, must be described by admissible wavelets, i.e., must have a wiggly (wave-like) behaviour characterized by the conditions given in \\cref{eq:int,eq:prereccond,eq:present}.}\n\nWe proceed with showing explicitly that the characteristics of $\\psi(z)$ described above, corresponds to a wiggly behaviour for $H(z)$ with respect to $H_{\\Lambda \\rm CDM}(z)$ \\textit{in a particular way}, namely, not necessarily wavelet type but such that $\\psi(z)$ is an admissible wavelet; to see this, we define a unitless parameter $\\delta(z)$, namely, the fractional deviation from $H_{\\Lambda\\rm CDM}(z)$, as follows;\n\\begin{equation}\n\\label{eqn:deltaH}\n\\delta(z)\\equiv \\frac{H(z)-H_{\\Lambda \\rm CDM}(z)}{H_{\\Lambda \\rm CDM}(z)}=-\\frac{\\psi(z)H_{\\Lambda \\rm CDM}(z)}{1+\\psi(z)H_{\\Lambda \\rm CDM}(z)}.\n\\end{equation}\nWe see that if we demand an ever-expanding universe $H(z)>0$, we should set $\\delta(z)>-1$. And, in what follows, unless otherwise is stated, we continue our discussions with the assumption that $\\delta(z)>-1$. For small deviations from $\\Lambda$CDM, i.e., $|\\delta(z)|\\ll1$, we can also write\n\\begin{equation}\n\\label{eqn:sdevH}\n\\delta(z)\\approx-\\psi(z)H_{\\Lambda \\rm CDM}(z).\n\\end{equation}\nThe small deviation region is quite important to study; because, despite its shortcomings, $\\Lambda$CDM is still the simplest model to explain the cosmological observations with remarkable accuracy. Particularly, in the late universe, the small deviation approximation is robustly imposed by many cosmological probes that require $|\\delta(z)|\\ll1$ for ${z\\lesssim2.5}$; even the largest discrepancies between the $H_{\\Lambda \\rm CDM}(z)$ of the Planck 2018 $\\Lambda$CDM~\\cite{Planck:2018vyg} and observed $H(z)$ values, viz., $H_0=73.04 \\pm 1.04$ km s${}^{-1}$ Mpc${}^{-1}$ (the SH0ES $H_0$ measurement~\\cite{Riess:2021jrx}) and $H(2.33)=224\\pm8{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ (the Ly-$\\alpha$-quasar data~\\cite{eBOSS:2020yzd}) correspond to $\\delta_0\\sim0.08$ and $\\delta(z=2.33)\\sim-0.05$, respectively. The form of~\\cref{eqn:sdevH} makes it even easier to see that $H(z)$ will have wiggles; since $H_{\\Lambda \\rm CDM}(z)$ is a monotonically varying function of $z$ and strictly positive, when $\\psi(z)$ changes sign (as it must at least once), this sign change (around which the small deviation condition is clearly satisfied) is directly reflected on $\\delta(z)$, producing a wiggle. Furthermore, respecting the successes of the $\\Lambda$CDM model, one may even wish to impose $|\\delta(z)|\\ll1$ at all times. In this case, since $H_{\\Lambda \\rm CDM}(z)$ monotonically grows with increasing redshift, one should have $\\psi(z)\\to0$ fast enough with $z\\to z_*$, such that the small deviation condition $\\abs{\\psi(z)H_{\\Lambda \\rm CDM}(z)}\\ll1$ is not broken.\n\nHaving said that, note the interesting extra behaviours apparent from the full form of $\\delta(z)$ in~\\cref{eqn:deltaH}: first, as mentioned before, ${\\psi(z)=-H^{-1}_{\\Lambda \\rm CDM}(z)}$ results in a singular $H(z)$ function and is not allowed for finite $z$ values; second, while the previous condition might seem as it requires either one of the confinements ${\\psi(z)>-H^{-1}_{\\Lambda \\rm CDM}(z)}$ or ${\\psi(z)<-H^{-1}_{\\Lambda \\rm CDM}(z)}$ at all times, in principle, $\\psi(z)$ can be discontinuous and is not necessarily confined to one of these regions; third, \\cref{eqn:sdevH} indicates that $\\psi(z)<0$ corresponds to $\\delta(z)>0$, yet, for a region in which ${\\psi(z)<-H^{-1}_{\\Lambda \\rm CDM}(z)}$, we have $\\delta(z)<0$ despite having $\\psi(z)<0$, but, looking at~\\cref{eqn:devdef}, such a region also corresponds to an extreme case with $H(z)<0$ and the universe would have gone through a contracting phase.\n\n\nFinally, it is worth noting that due to the wiggly behaviour of the wavelets, similar to $H(z)$, the other important kinematical parameters in cosmology, the deceleration parameter $q=-1+\\frac{{\\rm d}}{{\\rm d}t}\\left[H^{-1}(z)\\right]$ and the jerk parameter $j=\\frac{{\\rm d}^3a\/{\\rm d}t^3}{aH^3(z)}$ (which is simply $j_{\\Lambda\\rm CDM}=1$ for $\\Lambda$CDM) will also exhibit wiggly behaviors; the deceleration parameter will oscillate around its usual evolution in $\\Lambda$CDM, $q_{\\Lambda\\rm CDM}(z)$, as can be immediately seen from\n\\begin{equation}\n    q(z)=q_{\\Lambda\\rm CDM}(z)+\\frac{{\\rm d}\\psi(z)}{{\\rm d}t},\n\\end{equation}\nobtained by using \\cref{eqn:devdef} in the definition of $q(z)$. And, the jerk parameter will oscillate around its constant $\\Lambda$CDM value of unity. These behaviours are reminiscent of the non-parametric reconstructions in Refs.~\\cite{Mukherjee:2020vkx,Mukherjee:2020ytg}.\n\n\n\\section{Wiggles in dark energy density descended from the wavelets}\n\nIn the late universe where dust and DE are the only relevant components, we can treat $H(z)$ as an extension of ${H_{\\Lambda \\rm CDM}}(z)$ with the same matter density parameter $\\rho_{\\rm m}(z)$ but with a minimally interacting dynamical DE that explains the deviation of $\\delta(z)$ from zero; hereby, can write the DE density as $\\rho_{\\rm DE}(z)\\equiv3H^2(z)-\\rho_{\\rm m}(z)$, viz.,\n\\begin{equation}\n\\begin{aligned}\n\\label{eqn:drho}\n\\rho_{\\rm DE}(z)&=3H^2_{\\Lambda \\rm CDM}(z)[1+\\delta(z)]^2-\\rho_{\\rm m0}\\qty(1+z)^3\\\\\n&=\\rho_{\\rm DE0}+3H^2_{\\Lambda \\rm CDM}(z)\\delta(z)[2+\\delta(z)],\n\\end{aligned}\n\\end{equation}\nfrom which we can write the deviation of the DE density from $\\Lambda$, i.e., ${\\Delta\\rho_{\\rm DE}(z)\\equiv\\rho_{\\rm DE}(z)-\\rho_{\\Lambda}}$ (where we have ${\\rho_\\Lambda=\\rho_{\\rm DE0}}$), as follows:\n\\begin{equation}\n\\Delta\\rho_{\\rm DE}(z)=3H^2_{\\Lambda \\rm CDM}(z)\\delta(z)[2+\\delta(z)].\\label{eq:deltarho}\n\\end{equation}\nFor small deviations from $\\Lambda$CDM, these read\n\\begin{align}\n \\rho_{\\rm DE}(z)&\\approx \\rho_{\\rm DE0}+6\\delta(z)H^2_{\\Lambda \\rm CDM}(z),\\\\\n \\Delta\\rho_{\\rm DE}(z)&\\approx6\\delta(z)H^2_{\\Lambda \\rm CDM}(z),\n\\end{align}\ncorrespondingly. Thus, because $\\delta(z)$ is oscillatory around zero, $\\Delta\\rho_{\\rm DE}(z)$ will also be oscillatory around zero and its small oscillations will be scaled by $6H^2_{\\Lambda \\rm CDM}(z)$; in other words, the wiggles in $H(z)$ are implied by wiggles in $\\rho_{\\rm DE}(z)$ scaled by $6 H^2_{\\Lambda \\rm CDM}(z)$. That is, observational fitting\/non-parametric reconstruction procedures predicting wiggles in $H(z)$ will predict corresponding wiggles in $\\rho_{\\rm DE}(z)$ reconstructions.\n\nEven if our assumption that the pre-recombination universe is not modified with respect to the standard cosmology [implying~\\cref{eq:prereccond}], is taken to be approximate, for $z>z_*$, the fluctuations in the DE density should be much smaller than the matter energy density, ${\\abs{\\Delta\\rho_{\\rm DE}(z)\/\\rho_{\\rm m}(z)}\\ll1}$, in the matter dominated epoch, and much smaller than the radiation energy density, ${\\abs{\\Delta\\rho_{\\rm DE}(z)\/\\rho_{\\rm r}(z)}\\ll1}$, in the radiation dominated epoch. Since for both of these epochs the relevant energy densities can be well-approximated by the critical energy density of $\\Lambda$CDM $\\rho_{\\rm c}(z)\\equiv3H^2_{\\Lambda\\rm CDM}(z)$, in this approximate case for $z>z_*$, instead of $\\Delta \\rho_{\\rm DE}(z)=0$, we can write the more relaxed condition\n\\begin{equation}\n \\abs{\\frac{\\Delta\\rho_{\\rm DE}(z)}{\\rho_{\\rm c}(z)}}=\\abs{\\delta(z)[2+\\delta(z)]}\\ll1. \\label{eq:pert}\n\\end{equation}\nThis is satisfied for both $\\delta(z)\\sim 0$ (small deviation from $\\Lambda$CDM), and $\\delta(z)\\sim-2$ (corresponds to contracting universe), but only the former is of interest to us. Since~\\cref{eq:pert} requires small $|\\delta(z)|$ to be satisfied, it can be rewritten as\n\\begin{equation}\n\\abs{\\frac{\\Delta\\rho_{\\rm DE}(z)}{\\rho_{\\rm c}(z)}}\\approx2\\abs{\\delta(z)}\\approx2\\abs{-\\psi(z)H_{\\Lambda \\rm CDM}(z)}\\ll1, \\label{eq:rapidity}\n\\end{equation}\nfrom which we immediately see that $\\psi(z)$ should vanish rapid enough with increasing $z$ at large redshifts so that our assumption of almost unmodified pre-recombination physics holds.\n\nWe calculate from Eq.~\\eqref{eqn:drho} that the DE density passes below zero, $\\rho_{\\rm DE}(z)<0$, for\n\\begin{equation}\n\\delta(z)<-1+\\sqrt{1-\\frac{\\rho_{\\rm DE0}}{3H^2_{\\Lambda \\rm CDM}(z)}},\n\\end{equation}\nwhich can also be written as follows:\n\\begin{equation}\n\\delta(z)<-1+\\sqrt{1-\\frac{\\Omega_{\\rm DE0}}{\\Omega_{\\rm DE0}+(1-\\Omega_{\\rm DE0})(1+z)^3}}.\n\\end{equation}\nAccordingly, using Planck 2018 best fit $\\Lambda$CDM values $\\Omega_{\\rm m0}=0.3158$ and $H_0=67.32{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$~\\cite{Planck:2018vyg}, it turns out that $\\delta (2.33)<-0.028$, i.e., \n$\\Delta H(2.33)\\equiv{H(2.33)-H_{\\Lambda\\rm CDM}(2.33)<-6.65{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}}$ (corresponding to $H(2.33)\\lesssim230.536{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$), requires the DE density to yield negative values. Note that $H(2.33)=228\\pm7{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ from the Ly-$\\alpha$-Ly$\\alpha$ and $H(2.33)=224\\pm8{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ from the Ly-$\\alpha$-quasar data~\\cite{eBOSS:2020yzd}. \\textit{Thus, considering their mean values, fitting these data perfectly, requires a negative DE density at $z\\sim 2.3$.} In this sense, the function $\\delta(z)$ can also be used as a diagnostic to test for a negative DE density if all modifications to Planck $\\Lambda$CDM are attributed to DE.\n\nLastly, the continuity equation for the DE, viz., $\\dot\\rho_{\\rm DE}(z)+3H(z)[\\rho_{\\rm DE}(z)+p_{\\rm DE}(z)]=0$, implies $\\varrho_{\\rm DE}(z)=\\frac{1+z}{3} \\rho_{\\rm DE}'(z)$ for the inertial mass density, $\\varrho_{\\rm DE}(z)\\equiv\\rho_{\\rm DE}(z)+p_{\\rm DE}(z)$, and $w_{\\rm DE}(z)=-1+\\frac{1+z}{3} \\rho_{\\rm DE}'(z)\/\\rho_{\\rm DE}(z)$ for the corresponding EoS parameter $w_{\\rm DE}(z)\\equiv p_{\\rm DE}(z)\/\\rho_{\\rm DE}(z)$, where $\\rho_{\\rm DE}(z)$ is the DE density as defined in~\\eqref{eqn:drho}, $p_{\\rm DE}(z)$ is its pressure, and $'\\equiv\\dv{z}$. Accordingly, we have\n\\begin{equation}\n\\begin{aligned}\n    \\varrho_{\\rm DE}(z)&=2(1+z)H_{\\Lambda\\rm CDM}^2\\qty[\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta(\\delta+2)+\\delta'(\\delta+1)]\\\\\n    &\\approx2(1+z)H^2_{\\Lambda\\rm CDM}\\qty[2\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta+\\delta'],\n\\end{aligned}\n\\end{equation}\nfor the DE inertial mass density, and\n\\begin{equation}\n\\begin{aligned}\\label{eq:eos}\n    w_{\\rm DE}(z)&=-1+\\frac{2(1+z)\\qty[\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta(\\delta+2)+\\delta'(\\delta+1)]}{3\\qty[\\frac{\\rho_{\\rm DE0}}{\\rho_{\\rm c}}+\\delta(2+\\delta)]}\\\\\n    &\\approx-1+\\frac{2(1+z)\\qty[2\\frac{H'_{\\Lambda\\rm CDM}}{H_{\\Lambda\\rm CDM}}\\delta+\\delta']}{3\\qty[\\frac{\\rho_{\\rm DE0}}{\\rho_{\\rm c}}+2\\delta]},\n\\end{aligned}\n\\end{equation}\nfor the corresponding DE EoS parameter; in these two equations, the second lines are for small deviations from $\\Lambda$CDM. Notice that, in the exact form of~\\cref{eq:eos}, $w_{\\rm DE}(z)$ blows up if $\\rho_{\\rm DE0}\/\\rho_{\\rm c}(z)=-\\delta(z)[2+\\delta(z)]$ is satisfied for a redshift $z_{\\rm v}$. Comparing with \\cref{eqn:drho}, we see that this condition is equivalent to $\\rho_{\\rm DE}(z_{\\rm v})=0$; indeed, if the DE submits to the continuity equation as it does in this case, a vanishing energy density necessitates such a singularity~\\cite{Ozulker:2022slu}. Such infinities in the EoS parameter are not problematic from the fundamental physics point of view, instead, hints that the DE density is perhaps an effective one originating from a modified gravity model.\n\n\\section{Wiggles in Newton's ``constant\" descended from the wavelets}\n\nAlternatively, we can attribute the deviation of $H(z)$ from $H_{\\Lambda \\rm CDM}(z)$ to the deviations in the gravitational coupling strength, $G_{\\rm eff}(z)$, from the Newton's gravitational constant $G_{\\rm N}$ measured locally. We have, as usual,\n\\begin{equation}\n3H^2_{\\Lambda \\rm CDM}(z)=8\\pi G_{\\rm N}\\qty[\\rho_{\\rm m0}(1+z)^3+\\rho_{\\rm r0}(1+z)^4+\\rho_\\Lambda],\n\\end{equation}\nwhere the constant value $\\rho_\\Lambda$ is either the usual vacuum energy density or $\\rho_\\Lambda=\\frac{\\Lambda}{8\\pi G_{\\rm N}}$. We can write the Hubble parameter of the new model as\n\\begin{equation}\n3H^2(z)=8\\pi G_{\\rm eff}(z)\\qty[\\rho_{\\rm m0}(1+z)^3+\\rho_{\\rm r0}(1+z)^4+\\rho_\\Lambda],\n\\end{equation}\nfrom which, using the definition in~\\cref{eqn:deltaH},\n\\begin{equation}\n\\begin{aligned}\n\\label{eqn:Geff}\nG_{\\rm eff}(z)=\\qty[1+\\delta(z)]^{2}G_{\\rm N}\n\\end{aligned}\n\\end{equation}\ndirectly follows. Note that $G_{\\rm eff}(z)$ is also a wiggly function led by the wiggles of $\\delta(z)$, but, $G_{\\rm eff}(z)$ equals $G_{\\rm N}$ when $\\psi(z)=0$, and thereby, $G_{\\rm eff}(z=0)=G_{\\rm eff}(z>z_*)=G_{\\rm N}$ from~\\cref{eq:prereccond,eq:present}. And,\nfor small deviations from $\\Lambda$CDM, \\cref{eqn:Geff} reads\n\\begin{equation}\n\\begin{aligned}\nG_{\\rm eff}(z)\\approx [1+2\\delta(z)]G_{\\rm N}.\n\\end{aligned}\n\\end{equation}\nNote that, if we are to treat $\\rho_\\Lambda$ as the effective energy density of the cosmological ``constant\", i.e. ${\\Tilde{\\Lambda}(z)=8\\pi G_{\\rm eff}(z) \\rho_{\\Lambda}}$, this new cosmological term $\\Tilde{\\Lambda}(z)$ is not a constant anymore.\n\nIt is crucial to note that, while attributing the wiggles to the DE density or $G_{\\rm eff}(z)$ is indistinguishable in their background dynamics, this is not so for all physical observables. Particularly, a direct effect of the dynamical gravitational coupling strength would be observable, for instance, as this would promote the absolute magnitude $M_B=\\rm const$ of type Ia supernovae (SNIa) to a quantity varying with the redshift $M_B=M_B(z)$. Such an effect in the very late universe ($z\\lesssim0.1$) was recently suggested and investigated in a series of papers to address the so-called $M_B$ (and $H_0$) tension~\\cite{Alestas:2020zol,Marra:2021fvf,Perivolaropoulos:2021bds,Alestas:2022xxm,Perivolaropoulos:2022vql,Perivolaropoulos:2022txg}. Also, the idea that the supernovae absolute magnitudes are constant with redshift, has been questioned by observations and the question of whether or not this idea is valid has recently gained interest \\cite{Benisty:2022psx,DiValentino:2020evt,Rose:2020shp,Kang:2019azh,Kim:2019npy,Tutusaus:2017ibk,Linden:2009vh,Ferramacho:2008ap}. A possible variation of the $M_B(z)$ and equivalently of the SNIa luminosity $L(z)\\propto 10^{-\\frac{2}{5}M_B(z)}$ could be due to a variation of the Newton's ``constant\". Since the SNIa luminosity is proportional to the Chandrasekhar mass, which, in this case, is no longer a constant equal to $1.4\\,M_{\\odot}$, but a quantity that varies with $G_{\\rm eff}(z)$, we have $L(z)\\propto M_{\\rm Chandra}(z)$, so that\n$L(z)\\propto G_{\\rm eff}^{-3\/2}(z)$, which in turn leads, in this approach, to\n\\begin{equation}\n\\begin{aligned}\nM_{B}(z)-M_{B,G_{\\rm N}}=\\frac{15}{4}\\log \\frac{G_{\\rm eff}(z)}{G_{\\rm N}}=\\frac{15}{2}\\log[1+\\delta(z)],\\label{eq:mbwig}\n\\end{aligned}\n\\end{equation}\nwhere $M_{B,G_{\\rm N}}$ denotes the SNIa absolute magnitude when $G_{\\rm eff}(z)=G_{\\rm N}$, which satisfies ${M_{B,G_{\\rm N}}=M_{B,0}}$ due to~\\cref{eq:present}.\nThus, attributing wiggles to $G_{\\rm eff}(z)$ will have consequences not only on the expansion of the universe, but also on the absolute magnitudes of SNIa at different redshifts; and, as \\cref{eq:mbwig} shows, the wiggles of $G_{\\rm eff}(z)$ are directly manifested in the SNIa absolute magnitudes as a wiggly $M_B(z)$ reminiscent of the findings of Ref.~\\cite{Benisty:2022psx}. Investigating how this dual modification to the standard cosmology affects the cosmological parameter estimates from SNIa data and furthermore the so-called $M_B$ tension~\\cite{Efstathiou:2021ocp,Camarena:2021jlr}, is beyond the scope of this paper, and deserves a separate study.\n\n\\begin{figure*}[ht!]\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{wavelet.pdf}\n    \\includegraphics[width=0.46\\textwidth]{sin.pdf}\n      \\includegraphics[width=0.46\\textwidth]{hdot.pdf}\n      \\includegraphics[width=0.46\\textwidth]{sindm.pdf}\n    \\caption{The deviations from the $\\Lambda$CDM model in terms of some kinematical parameters for some wavelet examples of $\\psi(z)$ given in the top left panel where $\\Bar{\\alpha}$ and $\\alpha$ are in units of ${\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$, $\\Bar{\\beta}$ and $\\beta$ are unitless, $\\Bar{z}_\\dagger$ and $z_\\dagger$ are redshifts anchoring the wavelets. The dashed line corresponds to no deviation, i.e., $\\Lambda$CDM itself. The blue bars correspond to the TRGB $H_0$ measurement and various BAO measurements. See \\cref{sec:ex} for details.}\n    \\label{fig:kinematics}\n\\end{figure*}\n\n\\section{Employing some simplest wavelets}\n\\label{sec:ex}\n\n\n\n\\begin{figure*}[ht!]\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{dens.pdf}\n    \\includegraphics[width=0.45\\textwidth]{omegade.pdf}\n    \\includegraphics[width=0.46\\textwidth]{eoswig.pdf}\n    \\includegraphics[width=0.46\\textwidth]{inertial.pdf}\n    \\caption{The deviations from the cosmological constant, if the wavelet examples of $\\psi(z)$ are attributed to a dynamical DE, i.e., the wiggles in $H(z)$ are produced solely by a dynamical DE. The plots are matched by color to those in \\cref{fig:kinematics}.}\n    \\label{fig:dens}\n\\end{figure*}\n\n\\begin{figure}[t!]\n    \\begin{flushright}\n    \\includegraphics[width=0.45\\textwidth]{geff.pdf}\n    \\includegraphics[width=0.467\\textwidth]{mb.pdf}\n    \\end{flushright}\n    \\caption{ The deviation from $G_{\\rm N}$ if wiggles are produced solely by a varying Newton's ``constant\"; we also show the variation in the absolute magnitude $M_B$ of supernovae assuming the unmodified value to be the mean value of the measurement in Ref.~\\cite{Camarena:2021jlr}. The variation of $G_{\\rm eff}$ is less than $\\sim10\\%$ at all times, and there is practically no variation for $z\\sim0$ and $z\\gg0$.\n    The plots are matched by color to those in \\cref{fig:kinematics}.}\n    \\label{fig:ex}\n\\end{figure}\n\nWavelets constitute a wide family of functions that may or may not be smooth. They exhibit an oscillatory (not necessarily periodic) behaviour over a compact set of their parameters, and either vanish or quickly decay outside of this set. Even the superposition of arbitrarily many wavelets would describe another one. Here, we will consider some simplest examples: one discontinuous, namely, the Haar mother wavelet (\\cref{sec:haar}); and other smooth wavelets, namely, the Hermitian wavelets (\\cref{sec:hermitian}) that are acquired from the derivative\/s of a Gaussian distribution function. These examples have no inherent superiority to other possible wavelets; we provide them only because of their simplicity and to give a taste of how wavelets behave and their cosmological consequences.\n\nThese example wavelets, and their corresponding cosmologically relevant functions are plotted in~\\cref{fig:dens,fig:ex,fig:kinematics} for various values of their free parameters; matching colors in different figures indicate the same wavelet with the same choice of parameters. The dashed line corresponds to a vanishing wavelet, i.e., to the reference $\\Lambda$CDM model described with $H_{\\Lambda\\rm CDM}(z)$; for the figures, we neglected radiation for $z<z_*$, and used the mean values of the Planck 2018 TT,TE,EE+lowE+lensing results~\\cite{Planck:2018vyg} for $\\rho_{\\rm m0}\/H_0^2$, $r_*$, $\\theta_*$, and $z_*$. \\cref{fig:kinematics} shows the wavelets themselves scaled by $10^{3}$ on the top left panel, and the rest of the panels show some cosmological kinematics. The top right panel shows the comoving Hubble parameter, viz., the expansion speed $\\dot{a}=H(z)\/(1+z)$, where $a$ is the scale factor of the spatially flat RW metric, and dot denotes derivative with respect to cosmic time $t$. The lower right panel shows $D_M^{-1}(z)$ scaled by $c\\ln(1+z)$. The data points in these right panels are the TRGB ${H_0=69.8\\pm0.8\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$ measurement~\\cite{Freedman:2019jwv}, and the BAO measurements (see Ref.~\\cite{eBOSS:2020yzd} and references therein): BOSS DR12 consensus Galaxy (from $z_{\\rm eff}=0.38,\\,0.51$), eBOSS DR16 LRG (from $z_{\\rm eff}=0.70$), eBOSS DR16 Quasar (from $z_{\\rm eff}=1.48$), eBOSS DR16 Ly-$\\alpha$-Ly-$\\alpha$ (from $z_{\\rm eff}=2.33$), and eBOSS DR16 Ly-$\\alpha$-quasar (from $z_{\\rm eff}=2.33$ but shifted to $z=2.35$ in the figures for visual clarity). The lower left panel shows the derivative of the Hubble function with respect to the cosmic time normalized by $3H^2$, a crossing of the zero would indicate a non-monotonic behaviour in the Hubble function as suggested in~\\cite{Aubourg:2014yra,Akarsu:2019hmw,Akarsu:2021fol}. \\cref{fig:dens} shows various plots related to the DE dynamics when the wiggles are attributed to the DE. The top left panel shows the DE densities normalized by the present-day  critical energy density $\\rho_{\\rm c0}$, the top right panel shows the DE density parameters, $\\Omega_{\\rm DE}(z)\\equiv \\rho_{\\rm DE}(z)\/3H^2(z)$, the lower left panel shows the corresponding DE EoS parameters, and the lower right panel shows the DE inertial mass densities, $\\varrho_{\\rm DE}(z)\\equiv\\rho_{\\rm DE}(z)+p_{\\rm DE}(z)$, normalized by $\\rho_{\\rm c0}$. Finally, \\cref{fig:ex} shows the corresponding results when wiggles in $H(z)$ are attributed to $G_{\\rm eff}(z)$. The top panel shows $G_{\\rm eff}(z)$ normalized by $G_{\\rm N}$, say $G_{\\rm eff}(z=0)$, and the lower panel shows the variation of the absolute magnitude $M_B(z)$ for the mean value of the measurement $M_{B,G_{\\rm N}}=M_{B}(z=0)=-19.244\\pm0.037$ mag inferred in Ref.~\\cite{Camarena:2021jlr} (using Pantheon SnIa data set~\\cite{Pan-STARRS1:2017jku} along with Cepheid stars at $z<0.01$ for their calibration).\n\n\\subsection{Haar wavelet}\n\\label{sec:haar}\nTo begin with, we consider the simplest wavelet, the Haar mother wavelet \\cite{Chui:1992}; namely, $\\psi_{\\rm h}(z)=1$ for ${0\\leq z< 1\/2}$, ${\\psi_{\\rm h}(z)=-1}$ for $1\/2\\leq z<1$, and zero everywhere else; so that $\\Psi_{\\rm h}(z_*)=0$.  Shifting and scaling $\\psi_{\\rm h}$ with three parameters, i.e., defining \n\\begin{equation}\n   \\psi_{\\rm H}(z)\\equiv \\Bar{\\alpha} \\psi_{\\rm h}\\qty[\\Bar{\\beta}\\qty(z-\\Bar{z}_\\dagger)+\\frac{1}{2}],\n\\end{equation}\nwe can produce discontinuous wiggles on $H(z)$. For a good description of the Ly-$\\alpha$ data, we choose ${\\Bar{\\alpha}=0.00015}{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$, ${\\Bar{z}_\\dagger=2}$, and ${\\Bar{\\beta}=\\frac{1}{2}}$ for our example; see the green lines in the figures. It is clear that, for these values of its parameters, $\\psi_{\\rm H}(z)$ satisfies all the conditions we imposed on $\\psi(z)$, the major ones being~\\cref{eq:int,eq:prereccond,eq:present}. In the figures, $\\delta(z)<0$ for the interval $z\\in[2,3)$ leads to a dip in $H(z)$ for this interval that is in excellent agreement with the Ly-$\\alpha$ data. This is compensated by $\\delta(z)>0$ for the interval $z\\in[1,2)$ so that~\\cref{eq:int} is satisfied; this region constitutes a bump on $H(z)$. This bump presents itself in other functions such as $\\rho_{\\rm DE}(z)$ and $\\varrho_{\\rm DE}(z)$ [or $G_{\\rm eff}(z)$]; and, it is reminiscent of those that are found in non-parametric DE density reconstructions~\\cite{Escamilla:2021uoj,Bernardo:2021cxi} from observational data. For our particular example, we see in the figures that the bump results in slight disagreement with the eBOSS DR16 Quasar data at $z_{\\rm eff}=1.48$ for both $H(z)$ and $D_{M}(z)$. This can be mitigated by a different choice of parameters or more interestingly by adding more wiggles, for example, by superposing multiple Haar wavelets; however, this superposition would increase the number of free parameters. In the next subsection, we will increase the number of wiggles without increasing the number of free parameters. Note that, the $\\Dot{H}(z)\/3H^2(z)$ plot of the Haar example appears to never cross the zero line, implying monotonic behaviour for $H(z)$; however, this is not true. The discontinuities of the $H(z)$ function at $z=1,2,3$ results in spikes (Dirac delta distributions) that are not shown in~\\cref{fig:kinematics} for the $\\dot{H}(z)$ function at these redshifts, resulting in two crossings of the zero line at $z=1,2$ and a non-monotonic $H(z)$ that increases instantaneously in time at $z=2$. Similar spikes also exist for the $w_{\\rm DE}(z)$ and $\\varrho_{\\rm DE}(z)$ functions if the wiggles are attributed to the DE, but again are not shown in \\cref{fig:dens}. Additionally, if the deformations of the Hubble function described by $\\delta(z)$ is attributed to the DE density, the $w_{\\rm DE}(z)$ has a discontinuity at $z\\sim 2.2$ (as suggested in~\\cite{Akarsu:2019hmw,Akarsu:2021fol}) in addition to the obvious ones at $z={1,2,3}$. This discontinuity (present as a singularity) happens exactly at the redshift that $\\rho_{\\rm DE}(z)$ crosses from negative to positive values, and is characteristic of energy densities that have vanishing values in time and not problematic from the point of view of fundamental physics as discussed below~\\cref{eq:eos}. Of course, the discontinuities at $z={1,2,3}$ are not very compelling physically, but the Haar wavelet is the simplest example and shows what we should expect from the form of $H(z)$ for a minimal wavelet type deviation of the Hubble radius, $H(z)^{-1}$, from that of the standard cosmological model, $H_{\\Lambda \\rm CDM}(z)^{-1}$. A good alternative to the Haar wavelet can be the Beta wavelet~\\cite{betawave} derived from the derivative of the Beta distribution ${P_\\beta(z|\\gamma,\\lambda)\\equiv1\/B(\\gamma,\\lambda)z^{\\gamma-1}(1-z)^{\\lambda-1}}$ where $B(\\gamma,\\lambda)\\equiv\\int_0^1 k^{\\gamma-1}(1-k)^{\\lambda-1}\\dd{k}$ is the Euler beta function, $0\\leq z\\leq1$, and $1\\leq\\gamma,\\,\\lambda\\leq\\infty$. Beta wavelets are in some sense softened Haar wavelets as both have compact support and are unicycle (i.e., they have just one bump and one dip), however, unlike the Haar wavelet, the Beta wavelet is continuous~\\cite{betawave}. Thus, to describe more wiggles, one would need to superpose multiple Beta wavelets just like in case of the Haar wavelet, increasing the number of free parameters. While the Beta wavelets can satisfy \\cref{eq:int,eq:prereccond,eq:present} exactly without compromising continuity, they do not have a closed-form expression and are mathematically less tractable; thus, for simplicity, we will proceed with Hermitian wavelets that are also continuous\\footnote{One may also wish the wavelet satisfying~\\cref{eq:int,eq:prereccond,eq:present} exactly to have the stronger property of being smooth. However, since these conditions require that every derivative of the wavelet vanish outside of the interval $[0,z_*]$, but not inside, such a wavelet cannot be analytic. Non-analytic smooth functions can be constructed piecewise similar to splines but the pieces are not necessarily polynomial. These kind of functions are not compelling for the demonstrative purposes of this paper but may turn out to be useful in observational analyses.} and simpler, and satisfy~\\cref{eq:int,eq:prereccond,eq:present} to high precision.\n\n\n\\subsection{Hermitian wavelets}\n\\label{sec:hermitian}\nThe discontinuous features of the Haar wavelet can be considered as an approximate description of a rapidly varying smooth function which would be physically more relevant. A simple family of smooth wavelets can be acquired from the derivative\/s of a Gaussian distribution (cf., the Hermitian wavelets~\\cite{hermitianwave}). To do so, we consider the Gaussian distribution defined as follows;\n\\begin{equation}\n\\label{eqn:G0}\n{\\psi_{\\rm G0}(z)=-\\frac{\\alpha}{2\\beta}e^{-\\beta(z-z_\\dagger)^2}},\n\\end{equation}\nwhere $\\alpha$, $\\beta>0$, and $z_\\dagger>0$ are the three free parameters that will set, respectively, the amplitude, support, and center of the wiggles. The real part of the $n^{\\rm th}$ Hermitian wavelet can be obtained from the $n^{\\rm th}$ derivative of a Gaussian distribution ${\\psi_{\\text{G}n}(z)\\equiv\\dv[n]{\\psi_{\\rm G0}(z)}{z}}$; accordingly, utilizing~\\cref{eqn:G0} we obtain\n\\begin{equation}\n\\begin{aligned}\n\\psi_{\\rm G1}(z)=&-2\\beta(z-z_{\\dagger}) \\psi_{\\rm G0}(z),\\\\\n\\psi_{\\rm G2}(z)=&4\\beta \\qty[\\beta(z-z_{\\dagger})^2-\\frac{1}{2}]\n\\psi_{\\rm G0}(z), \\\\\n\\psi_{\\rm G3}(z)=&-8\\beta^2 \n\\qty[\\beta(z-z_{\\dagger})^3-\\frac{3}{2}(z-z_\\dagger)]\\psi_{\\rm G0}(z), \\\\\n\\psi_{\\rm G4}(z)=&16\\beta^2\\qty[\\frac{3}{4}+(z-z_{\\dagger})^4\\beta^2-3\\beta(z-z_{\\dagger})^2]\\psi_{\\rm G0}(z),\n\\end{aligned}\n\\end{equation}\netc., where only up to $4^{\\rm th}$ derivative are written explicitly. $\\psi_{\\rm G1}(z)$ and $\\psi_{\\rm G2}(z)$ are well-known wavelets and the latter is also known as the Ricker (Mexican hat) wavelet. $\\psi_{\\text{G}n}(z)$ are quasi-periodic functions, i.e., the redshift difference between consecutive peaks (whose amplitudes may differ) of the wave varies. We note that $\\psi_{\\rm G0}(z)$ itself is responsible for the fast damping of the wavelet function $\\psi_{\\text{G}n}(z)$ as $z$ moves away from $z_\\dagger$ and that $n^{\\rm th}$ derivative of $\\psi_{\\rm G0}(z)$ brings an $n^{\\rm th}$ degree polynomial as a factor to itself, which in turn implies that $n$ stands also for the number of nodes of the $\\psi_{\\text{G}n}(z)$ function, i.e., the number of times the function crosses zero. These $n$ nodes correspond to $n+1$ wiggles [total of $n+1$ dips and bumps of $\\psi(z)$]; the bumps of $\\psi(z)$ manifest themselves as dips, and dips of $\\psi(z)$ manifest themselves as bumps in $\\delta(z)$ and equivalently $H(z)$, cf., Eq.~\\eqref{eqn:sdevH}. These manifestations directly translate to wiggles on either $\\rho_{\\rm DE}(z)$ or $G_{\\rm eff}(z)$ depending on which function we attribute them to. The wiggly structure in these functions resemble the wiggles in their respective functions that are acquired from observational analyses utilizing parametric or non-parametric reconstructions~\\cite{Escamilla:2021uoj,Bernardo:2021cxi}. Wiggles acquired in observational reconstructions are no surprise even if the data set does not contain CMB, because, wiggles are necessary for $H(z)$ to fit the measurements of the Hubble parameter from the BAO data without spoiling the success of $\\Lambda$CDM in fitting the $D_M(z)$ values measured from the same BAO data (see Fig.~\\ref{fig:kinematics}); and the logic we used to show the necessity of bumps still apply when $z_*$ is swapped for the effective redshift of a BAO measurement. The necessity of the wiggles only when low redshift data ($z<3$) is considered, is the subject of an upcoming work. \n\nCoincidentally, the first derivative of the Gaussian distribution \\eqref{eqn:G0}, i.e., $\\psi_{\\rm G1}(z)$, can be used to roughly approximate the Haar wavelet smoothly. For $\\psi_{\\rm G1}(z)$, we pick $\\alpha=0.0005{\\rm \\,km\\, s^{-1}\\, Mpc^{-1}}$, $z_\\dagger=2$, and $\\beta=2$, so that the wavelet approximates our previous Haar example. For rest of the examples, $\\psi_{\\rm G2}(z)$, $\\psi_{\\rm G3}(z)$, and $\\psi_{\\rm G4}(z)$, the values of the parameters are shown on the top left panel of~\\cref{fig:kinematics} and the increased number of wiggles for higher derivatives are clearly seen. The top right, and bottom right panels show how increasing the number of wiggles can provide a better description of the BAO data. Unlike the Haar and $\\psi_{\\rm G1}(z)$ examples, $\\psi_{\\rm G2}(z)$ and $\\psi_{\\rm G4}(z)$ examples better describe also the eBOSS DR16 Quasar data at $z_{\\rm eff}=1.48$ while retaining better agreement with the Ly-$\\alpha$ BAO data at $z_{\\rm eff}=2.33$; the $\\psi_{\\rm G4}(z)$ example even complies with the trend of the Galaxy BAO data (at $z_{\\rm eff}=0.38,\\,0.51,\\,0.70$) $H(z)\/(1+z)$ measurements that increase with redshift, which cannot be achieved within $\\Lambda$CDM (even though $\\Lambda$CDM is not in strong tension with any of these data points). Still, we emphasize that these wavelets are just illustrative examples and better wavelets can be looked for. Again, attributing the wiggles to the DE, the DE density also wiggles smoothly; however, for the $\\psi_{\\rm G1}(z)$ and $\\psi_{\\rm G2}(z)$ examples, two safe\/expected singularities are again present in $w_{\\rm DE}(z)$ at the redshifts that the DE density vanishes.\n\n\nNote that \\cref{eq:int,eq:prereccond,eq:present} are satisfied exactly only for admissible wavelets with compact support in the redshift interval $[0,z_*]$; thus, unlike the Haar wavelet, $\\psi_{\\text{G}n}(z)$ does not satisfy~\\cref{eq:int,eq:prereccond,eq:present} exactly, but rather approximately\\footnote{We emphasize that the Haar and Hermitian wavelets are just convenient examples we used to demonstrate various aspects of the wavelet framework. The previously mentioned Beta wavelets can satisfy these conditions exactly without compromising continuity (at the cost of simplicity due to their lack of closed-form expression); and working with wavelets generated by higher order derivatives of the Beta distribution, it should be possible to increase the number of wiggles without increasing the number of free parameters, but to our knowledge, there is no established literature on wavelets derived from their higher order derivatives. Another possibility is constructing wiggles out of splines that are piecewise polynomials which can have compact support, but these are likely to suffer from excessive number of free parameters. Also, a middle ground exists where some of the conditions are satisfied exactly and some approximately. For example, the $n^{\\rm th}$ Poisson wavelet, viz., $\\psi_{{\\rm P}n}(z)\\equiv\\frac{z-n}{n!}z^{n-1}e^{-z}$ for $z\\geq0$ and vanishing everywhere else, satisfies \\cref{eq:present} exactly but the other two equations approximately for $n>1$.} (yet, beyond a level that cannot be resolved by observation). These three conditions were imposed on $\\psi(z)$ through arguments relying on the robustness of certain observations; however, no matter how robust and model independent they are, the uncertainties of the measurements itself require only that \\cref{eq:int,eq:prereccond,eq:present} hold approximately. Reassuringly, for large redshifts, ${\\psi_{\\text{G}n}(z)H_{\\Lambda\\rm CDM}(z)\\propto z^{n+\\frac{3}{2}}e^{-\\beta z^2}}$ for matter dominated and $\\propto z^{n+2}e^{-\\beta z^2}$ for radiation dominated universes; both of these functions rapidly decay by virtue of the exponential term which eventually decays faster than any polynomial growth, ensuring $\\delta(z)\\to0$ at large redshifts, see \\cref{eqn:deltaH}. A similar argument can be made for $\\Delta\\rho_{\\rm DE}(z)\\to0$ through \\cref{eq:deltarho} at large redshifts. Finally, to demonstrate how successfully the $\\psi_{\\text{G}n}(z)$ examples approximate the conditions given in \\cref{eq:int,eq:prereccond,eq:present}, we examine our $\\psi_{\\rm G3}(z)$ example as it is the one that violates these conditions most strongly. The values we pick in our $\\psi_{\\rm G3}(z)$ example correspond to: $\\psi_{\\rm G3}(0)=(41.75\\times10^{-6})\\,{\\rm \\,km^{-1}\\, s\\, Mpc}$, which can be compared with ${H^{-1}_{\\Lambda\\rm CDM}(0)=(14.78\\times10^{-3})\\,{\\rm \\,km^{-1}\\, s\\, Mpc}}$ from Planck 2018 and corresponds to ${\\delta(0)\\sim3\\times10^{-3}}$; $\\psi_{\\rm G3}(z_*)\\sim 10^{-10^6}{\\rm \\,km^{-1}\\, s\\, Mpc}$ which can be compared with ${H^{-1}_{\\Lambda\\rm CDM}(z_*)=7.3\\times10^{-7}{\\rm \\,km^{-1}\\, s\\, Mpc}}$ from Planck 2018 and corresponds to ${\\delta(z_*)\\sim10^{-10^6}}$; and, ${c\\times\\Psi_{\\rm G3}(z_*)=-5.46\\,\\rm Mpc}$ which is extremely well within the $1\\sigma$ uncertainty of $D_M(z_*)=13872.83\\pm 25.31\\,\\rm Mpc$ measured in Planck 2018.\n\n\n\n\\section{Conclusion}\nIt is well-known that the comoving angular diameter distance to last scattering, $D_M(z_*)$, is strictly constrained by observations almost model-independently. Therefore, in a viable cosmological model, this distance should be the same with the one measured by assuming $\\Lambda$CDM, so that consistency with CMB data is ensured at the background level. We have shown that, assuming the pre-recombination and present-day universes are well described by $\\Lambda$CDM, this is satisfied only if the deviation of any model from $\\Lambda$CDM described by the function $\\psi(z)=H(z)^{-1}-H(z)_{\\Lambda\\rm CDM}^{-1}$, which is the deviation from the standard $\\Lambda$CDM model's Hubble radius, should be, or well approximated by, an admissible wavelet. \\textit{In other words, in a viable alternative cosmological model that leaves the pre-recombination and present-day universes as they are in the standard cosmological model, the modifications cannot be arbitrary but should satisfy (exactly or approximately at a precision level that can be absorbed within the precision of the available observational data) a Hubble radius function whose deviation from the one in the standard cosmological model is a member of the set of admissible wavelets.}\n\nThe admissible wavelets describing $\\psi(z)$ can be converted to modifications in various cosmological kinematic functions such as the Hubble and comoving Hubble parameters, $H(z)$ and $H(z)\/(1+z)$ as shown in \\cref{fig:kinematics}, as well as, the deceleration and jerk parameters, $q(z)$ and $j(z)$. The wiggly nature of wavelets describing $\\psi(z)$ leads to wiggles in these functions, but none of them are necessarily wavelets, moreover, even the ones that arise from the simplest wavelets have non-trivial behaviour that is highly unlikely to be constructed\/introduced by hand in the first place. Accordingly, while requiring $\\psi(z)$ to be an admissible wavelet ensures consistency with the CMB at the background level, the wiggly nature of the kinematic functions can be immensely effective in fitting the multitude of BAO data which have no clear common trend compared to $\\Lambda$CDM. Also, as the wavelets we used as examples show, the number of wiggles in $\\psi(z)$, hence also in cosmological kinematics, can be varied and then quite featured kinematics well fitting the observational data can be achieved without further increasing the number of free parameters; e.g., one may introduce any number of wiggles by taking a sufficient number of derivatives of the Gaussian distribution and have only three extra free parameters. These non-trivial modifications we have found in the cosmological kinematics can then be attributed to different physical origins. As the first examples that come to mind, we have attributed them either to a dynamical DE, viz., $\\rho_{\\rm DE}(z)$, or to a dynamical gravitational coupling strength, viz., $G_{\\rm eff}(z)$, and briefly discussed how these different approaches are, in principle, observationally distinguishable, even though they give rise to the same background kinematics, see \\cref{fig:dens,fig:ex}. We demonstrated also that the dynamics of the DE, or the gravitational ``constant'', led by the simplest wavelets, are even more non-tirivial compared to the kinematics; for instance, the DE density can change sign in the past, accompanied by singularities in its EoS parameter.\n\nA wiggly structure may be described as consecutive bumps and dips on a function. By using the simplest admissible wavelets, we encountered a common pattern that our toy examples, which well describe the BAO data, present a bump in the Hubble parameter (which can be attributed to a bump in the DE density) at $1.5\\lesssim z\\lesssim2$ just as found in various observational reconstructions \\cite{Wang:2018fng,Escamilla:2021uoj,Bernardo:2021cxi}. The existence of bumps is a natural outcome of our findings, because the dips in $H(z)$ required for better description of the data, e.g., at $z\\sim2.3$ relevant to the Ly-$\\alpha$ data, should be compensated by bumps elsewhere so that the comoving angular diameter distance to last scattering remains unaltered. This should raise serious concerns that the bumpy features in the non-parametric $H(z)$ and\/or $\\rho_{\\rm DE}(z)$ reconstructions may be fake and caused by overfitting to the BAO data; since various BAO data call for dips in the Hubble parameter, there will be compensatory bumps where there are no data points to oppose them. Although the redshift range devoid of data where these bumps may be present is arbitrary and can extend to very high redshifts (e.g., a plateau with a small amplitude over a large redshift range compensating a tight dip at $z\\lesssim3$), most observational analyses reconstruct the cosmological functions up to $z\\sim3$ where the most suitable redshift range for a fake bump appears to be at $1.5\\lesssim z\\lesssim2$. It is worth noting here that the wiggles in the DE density are not expected to be representative of an Effective Field Theory, more concretely any minimally coupled scalar model \\cite{Colgain:2021pmf}, and thus it is conceivable that the introduction of theoretical priors should smooth out the wiggles in the DE density \\cite{Pogosian:2021mcs,Raveri:2021dbu}. This may be implying that the origin of the wiggles in $H(z)$ must be sought in modified gravity theories. However, it may also be too hasty to completely ignore the possibility of finding highly wiggly (may be discreet) DE densities; see, for instance, the so-called Everpresent $\\Lambda$ model, which suggests the observed $\\Lambda$ fluctuates between positive and negative values with a magnitude comparable to the cosmological critical energy density about a vanishing mean, $\\braket{\\Lambda}=0$, in any epoch of the Universe, in accordance with a long-standing heuristic prediction of the causal set approach to quantum gravity \\cite{Ahmed:2002mj,Zwane:2017xbg,Surya:2019ndm}.\n\nUp until now we have avoided discussing the $H_0$ tension and assumed that any alternative cosmological model would not deviate from $\\Lambda$CDM at $z\\sim0$, based on the observational argument that $\\Lambda$CDM describes local observational data well and is also supported by non-parametric reconstructions. However, this no deviation condition, cf., \\cref{eq:present}, is stricter than necessary, as observational evidence suggests that it is essentially the functional form of ${3H^2_{\\Lambda\\rm CDM}(z)=\\rho_{\\rm m,0}(1+z)^3+\\rho_{\\Lambda}}$ that is favored by local data. This suggests that, the reference model from which the deviations are defined, can be taken to be any model that is compatible with CMB data while agreeing with the functional form of $\\Lambda$CDM exactly or approximately in the vicinity of the present-time of the universe, instead of the exact $\\Lambda$CDM model itself. Such models can be compatible with both CMB and local $H_0$ measurements at the same time, see e.g., Ref.~\\cite{Akarsu:2019hmw,Akarsu:2021fol}. Even the requirement of this functional form can be relaxed and the well-known CPL parametrization and $w$CDM model can be used for the reference model, in which case $\\psi(z)$ being an admissible wavelet is not a necessary condition but an analytically compelling case. However, it is possible that strict observational constraints from BAO data, prevent these models from occupying the part of their parameter space that allows them to simultaneously fit the CMB and $H_0$ measurements. If these models are taken to be the reference model, the $H_0$ tension may also be resolved within our wavelet framework; more importantly, if the observational success of these models were held back by the BAO data, the use of wavelets may resurrect them by letting them fit the BAO data without compromising their successful description of the CMB and $H_0$ observations.\n\nIn our discussions we basically allowed wavelets to have quite a bit of freedom, apart from requiring them to be admissible and vanish outside of the interval $z=[0,z_*]$, see \\cref{eq:int,eq:prereccond,eq:present}. However, it can also be very useful to focus on various subsets of these wavelets. Namely, using arguments based on the history of the expansion of the universe and\/or fundamental physics (also, these two can be related in a certain way through the putative theory of gravity), we can impose more conditions on them, thereby narrow down the extent of the family of cosmological models satisfying our conditions. For example, as we have already discussed to some extent, with regard to the kinematics of the universe, one may demand an ever expanding universe ($H(z)>0$) and\/or a monotonically decreasing Hubble parameter ($\\dot{H}(z)<0$) from beginning to the present, or, with regards to dynamics of the DE (supposing that GR is valid and the deviations are attributed to a dynamical DE fluid), one may demand a non-negative DE density ($\\rho_{\\rm DE}(z)\\geq0$) at all times, or a non-negative DE inertial mass density corresponding to the null energy condition ($\\varrho_{\\rm DE}(z)\\geq0$) at all times, or at least be cautious so that no instability problems are encountered. Indeed, DE fluids that leads to our example admissible wavelets, seem to easily violate the conventional energy conditions; namely, the EoS parameter crosses below minus unity and\/or plus unity and even exhibits a pole\/s in some cases; the DE inertial mass density, and even the DE density itself in some cases, cross below zero. Such violations are generally known to indicate possible instability issues in the DE fluid. One way out in this case, as we mentioned earlier, would be the possibility of deriving such dark energies from modified gravity theories as effective sources without causing some other instability problems. Employing the Parameterized Post-Friedmann (PPF) \\cite{Hu:2008zd, Fang:2008sn} approach may also provide us with another way out, namely, the PPF discussed in  \\cite{Hu:2008zd, Fang:2008sn} may be used to placate the violent behaviors of the DE source, particularly to solve the instability issues related to the DE EoS parameter or make them less severe by pulling it towards the safer interval $[-1,1]$. This approach that replaces the condition of DE pressure perturbation with a smooth transition scale will help us understand the momentum density of the DE and other components on the large scale structure. We leave advantages of considering such reconstruction methods in relevance with the family of the DE models introduced in this paper for future consideration.\n\nTo conclude with, the wavelet framework presented in this paper seems to have the potential to be a good guide to find new cosmological models, alternative to the base $\\Lambda$CDM model, that are consistent with the observational data and to analyze existing ones, but further observational and theoretical studies are required to uncover the full scope of the implications and applications of this framework.\n\n\\begin{acknowledgments}\n The authors thank to Bum-Hoon Lee and Kazuya Koyama for useful insights and discussions. \\\"{O}.A. acknowledges the support by the Turkish Academy of Sciences in the scheme of the Outstanding Young Scientist Award  (T\\\"{U}BA-GEB\\.{I}P). E.\\'O.C. was supported by the National Research Foundation of Korea grant funded by the Korea government (MSIT) (NRF-2020R1A2C1102899). E.\\\"{O}.~acknowledges the support by The Scientific and Technological Research Council of Turkey (T\\\"{U}B\\.{I}TAK) in scheme of 2211\/A National PhD Scholarship Program.  S.T. and L.Y. were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education through the Center for Quantum Spacetime (CQUeST) of Sogang University (NRF-2020R1A6A1A03047877).\n \\end{acknowledgments}\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\nIn recent years, supersymmetric quantum mechanics (SUSY QM) has provided a\ndeeper understanding of the exact solvability of several well known\npotentials in 1-dimensional QM. In particular, using the ideas of shape\ninvariance (SI), it provides a procedure for getting the spectrum, the\neigenfunctions and the S-matrix (i.e.  the reflection and transmission\ncoefficients) algebraically [1].  There also exist interesting connections\nbetween SUSY QM and soliton solutions. Despite these (and more)\ninteresting developments in SUSY QM for one particle system in one\ndimension, so far, not many of these results could be extended either for\n$N$-particle systems in one dimension or for one particle systems in more\nthan one dimension.\n\nIn recent times there is a revival of interest in the  $N$-body problems\nin one dimension with inverse square interaction which were introduced and\nstudied by Calogero [2] and developed by Sutherland [3] and others [4].\nThese models have several interesting features, like exact solvability,\nclassical and quantum integrability and also have interesting applications\nin several branches of physics [5,6]. Apart from the well known\ntranslational invariant inverse square interaction models, referred to as\n$A_{N - 1}$ Calogero-Sutherland Model (CSM), there also exist\ngeneralizations of this model, but without  the translational invariance,\nreferred to as $BC_{N}$, $B_N$, $D_N$ models. These nomenclatures refer to\nthe relationship of these models to the root system of the classical Lie\ngroup. It might be added here that these models also share with $A_{N -\n1}$ CSM, features like exact solvability, and integrability and have also\nfound application in certain physical systems.\n\nThe purpose of this note is to enquire if the ideas of one dimensional\nSUSY QM could be extended to the $N$-particle case. In particular, whether\nthe spectrum of the celebrated Calogero and other models could be obtained\nalgebraically by using the ideas of SI and SUSY QM.  The first step in\nthat direction was taken recently by Efthimiou and Spector [7] who showed\nthat the well known Calogero model (also termed as $A_{N-1}$ CSM) exhibits\nSI. However, they were unable to obtain the spectrum algebraically. This\nis because using SUSY they were unable to relate the eigenspectra of the\ntwo SUSY partner potentials.  In this paper we demonstrate that using SUSY\nQM, SI and exchange operator formalism [8], the spectrum of the rational\n$A_{N-1}$ CSM, and also of all its generalizations like $B_N$, $D_N$ and\n$BC_N$ can be obtained algebraically.  It is worth mentioning that the SI\nin our case is somewhat different from that of Efthimiou and Spector [7].\nSo far as we are aware off, this is the first instance when an\n$N$-particle quantum system has been solved using the techniques of SUSY\nQM and SI.\n\n\n\nThe plan of the paper is the following. We briefly review the ideas of one\ndimensional SUSY QM in Sec. II with the main emphasis on solvability using\nSI.. In Sec. II.A, we apply these ideas to the Calogero model, i.e., the\nrational $A_{N-1}$ model.  We show that the spectrum of such a model can\nbe derived using the ideas of SUSY QM, SI and the exchange operator\nformalism[8]. We also briefly discuss as to how to obtain the\ncorresponding eigen-functions. In Sec. II.B, we treat the rational $BC_N$\nmodel, a translationally non-invariant system, in the same spirit.  The\nfull spectrum is obtained and the method for obtaining the exact\neigen-functions is explicitly spelled out. It is also shown in this\nsection that the $BC_N$ model posses SI even if the exchange operator\nformalism is not employed. This is a generalization of Efthimiou et al's\nwork [7] on $A_{N-1}$ model to the $BC_{N}$ case.  Finally, in Sec. III,\nwe summarize our results and discuss the possible directions to be\nfollowed in order to have a viable formalism of many-body SUSY QM.  We\nalso point out the difficulties involved in extending these results to the\ntrigonometric case.  In Appendix we show that the $BC_N$ trigonometric\nmodel is also shape invariant.\n\n\\section{SUSY, SI and Solvability}\n\nIt may be worthwhile to first mention the key steps involved in obtaining\nthe eigen-spectrum of a one body problem by using the concepts of SUSY QM\nand SI.  One usually defines the SUSY partner potentials $H_1$ and $H_2$\nby\n\\be\\label{1.1} \nH_1= A^{\\dag} A \\, , \\ H_2 = A A^{\\dagger}~~,\n\\ee\n\\noindent where ($\\hbar = 2m =1$)\n\\be\\label{1.2}\nA = {d \\over dx} + W(x)~,~ \nA^{\\dag} = -{d \\over dx} + W(x)~.\n\\ee\nIn the case of unbroken SUSY, the ground state wave function is given in \nterms of the superpotential $W(x)$ by, \n\\be\\label{1.3} \n\\psi_0 (x) \\propto e^{-\\int^{x} W(y)dy}~~,\n\\ee\n\\noindent while the  \nenergy eigenvalues and the wave functions of $H_1$ and $H_2$\nare related by, \n$(n=0,1,2,...)$\n\\be\\label{1.4}\nE_n^{(2)} = E_{n+1}^{(1)}~~, \\hspace{.2in} E_0^{(1)} = 0~~,\n\\ee\n\\be \\label{1.5}\n\\psi_n^{(2)} = [E_{n+1}^{(1)}]^{-1\/2} A \\psi_{n+1}^{(1)}~~, \\ \\ \\ \\\n\\psi_{n+1}^{(1)} = [E_{n}^{(2)}]^{-1\/2} A^{\\dag}  \\psi_{n}^{(2)}~~.\n\\ee\n\nLet us now explain precisely what one means by SI.\nIf the pair of SUSY partner Hamiltonians $H_1,H_2$\ndefined above are similar in shape and differ only in the\nparameters that appear in them, then they are said to be SI.\nMore precisely, if the partner Hamiltonians $H_{1,2}(x;a_1)$ satisfy the\ncondition,\n\\be \\label{1.6}\nH_2(x;a_1) = H_1(x;a_2) + R(a_1),\n\\ee\nwhere $a_1$ is a set of parameters, $a_2$ is a function of $a_1$ (say\n$a_2=f(a_1)$) and the remainder $R(a_1)$ is independent of $x$, then\n$H_{1}(x;a_1)$ and $H_{2}(x;a_1)$ are said to be SI. \nThe property of SI permits an immediate analytic determination of the\nenergy eigenvalues, eigenfunctions and the scattering matrix [1]. In\nparticular the eigenvalues and the eigenfunctions of $H_1$ are given by\n($n = 1,2,...$)\n\\be \\label{1.7}\nE^{(1)}_n (a_1) = \\sum^n_{k=1} R(a_k)~, \\hspace {.2in}  E^{(1)}_0 (a_1)=0~~,\n\\ee\n\\be\\label{1.8}\n\\psi^{(1)}_n (x;a_1) \\propto A^{\\dag}(x;a_1)A^{\\dag}(x;a_2)....A^{\\dag}(x;a_n)\n\\psi^{(1)}_0 (x;a_{n+1})~,\n\\ee\n\\be\\label{1.9}\n\\psi^{(1)}_0 (x;a_1) \\propto e^{-\\int^{x} W(y;a_1) dy}~~.\n\\ee\n\t\n\\subsection{Rational $A_{N-1}$ Calogero Model}\n\nWe now apply the exchange operator formulation to the rational $A_{N-1}$ CSM.\nThe Hamiltonian of the rational $A_{N-1}$ CSM is given by \n\\be\\label{9}\nH_{CSM} =  {\\sum_{i}{\\frac{1}{2}} {p_{i}^2}}  + l(l \\mp 1) \\sum_{i < j} \n{(x_{i} - x_{j})}^{-2} + \\omega {\\sum_{i} {\\frac{1}{2}} {{x_{i}^2}}} \\, .\n\\ee\n\\noindent \nThe sign $\\mp$ in (\\ref{9})\nrefers to the fact that $H$ acts on completely anti-symmetric or symmetric\nfunctions, respectively.\nLet us now define an operator $D_i$\n\\be\\label{10}\nD_{i} = -i \\partial_{i} + il \\sum_{j}^\\prime {(x_i -x_j)}^{-1} M_{ij} ,\n\\ee\n\\noindent known as the Dunkl operator in the literature. Hereafter\n$\\prime$ means $i=j$ is excluded in the summation. The exchange operator\n$M_{ij}$\nhave the following properties [8]\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\nM_{ij}^2=1~, \\ \\ \\ M_{ij}^\\dagger=M_{ij}~,\n\\ \\ \\ M_{ij} \\psi^{\\pm} = \\pm \\psi^{\\pm}~,\\nonumber \\\\\nM_{ij} D_{i} = D_{j} M_{ij}~, \\ \\\nM_{ij} D_k = D_k M_{ij}~, \\ \\ k \\neq i,j~~,\\nonumber \\\\\nM_{ijk}=M_{ij} M_{jk}~, \\ \\ M_{ijk}=M_{kij}=M_{jki}~,\n\\eea\n\\noindent where $\\psi^{\\pm}$ is a(an) symmetric(antisymmetric) function.\nNote that the Dunkl operator is hermitian by construction\nand $[D_i, D_j]=0$. If we now define\n\\be\\label{11}\na_i = D_i - i \\omega x_i \\, , \\ \n{a_i}^\\dagger = D_i +i \\omega x_i \\, ,\n\\ee\nthen it is easy to see that\n\\be\\label{12}\n{[ a_i, {a_j}^\\dagger]} = 2 w {\\delta_{ij}}(1 + l{\\sum_{k}^{\\prime}M_{ik}})\n\t     - 2 (1-{\\delta_{ij}}) l w M_{ij} \\, .\n\\ee\n\\noindent Let us now consider the SUSY partner potentials \n$H$ and $\\tilde {H}$ defined by\n\\be\\label{13}\nH = \\frac{1}{2}\\sum_{i}{a_i}^\\dagger a_i~,\\ \\ \\ \n\\tilde {H} =\\frac{1}{2} \\sum_{i}a_i {{a_i}^\\dagger} \\, . \n\\ee\n\\noindent Using eqs. (\\ref{10}) and (\\ref{11}) it is easily shown that \n\\be\\label{14}\nH_{CSM} = H + E_0^{CSM}~, \\ \\ \\\nE_0^{CSM} = [\\frac{N}{2} \\mp \\frac{l}{2} N (N-1)] \\omega~..\n\\ee\n\\noindent Thus, by construction, the ground state energy of $H$ is zero.\n\nUsing eqs. (\\ref{10}) and (\\ref{11}) it is easily shown that if  \n$\\psi$ is the eigenstate of $H$ with eigenvalue $E (>0)$, then \n$A_1 \\psi$ is the eigenstate of $\\tilde H$ with eigenvalue $E+ \\delta_1$ i.e. \n\\be\\label{16}\n\\tilde{H} (A_1 \\psi)=[E+ \\delta_1](A_1 \\psi),\n\\ee\nwhere,\n\\be\\label{17}\nA_1 ={\\sum_{i}} a_{i}~, \\ \\ \\\n\\delta_1 = [(N-1) \\pm l  N(N-1)] \\omega~~.\n\\ee\n\\noindent \nSimilarly, if $\\tilde {\\psi}$ is the eigenfunction of $\\tilde H$ with \neigenvalue $\\tilde E$, then $A_1^{\\dag}\\psi$ is the eigenfunction of $H$ \nwith eigenvalue $\\tilde{E} - \\delta_1$ i.e.\n\\be\\label{19}\nH({{A}_1^\\dagger}\\tilde {\\psi}) =[\\tilde {E}-\\delta_1] ({{A}_1^\\dagger} \n\\tilde{\\psi}).\n\\ee\n\\noindent This proves one to one correspondence between the non-zero\nenergy eigen values of $H$ and $\\tilde {H}$. \nThus it follows from here that the energy eigenvalues and eigenfunctions \nof the two partner Hamiltonians $H$ and $\\tilde H$ \nare related by \n\\be\\label{20}\n\\tilde{E}_n = E_{n+1} +\\delta_1 \\, ,~~ E_0 = 0 \\, , \\ \\ n = 0,1,2,...\n\\ee \n\\be\\label{21}\n\\tilde{\\psi}_n = {A_1 \\psi_{n+1} \\over \\sqrt {E_{n+1} +\\delta_1}}~~,\n~\\psi_{n+1} = {A_1^{\\dag} \\tilde{\\psi}_n \\over \\sqrt{E_{n+1}}}~~.\n\\ee\n\\noindent Note that $\\delta_1$ vanishes for $N=1$ and we\nrecover the usual results of SUSY QM with one degree of freedom.\nIt is worth noting that unlike the case of one dimensional QM, \nin this case the (positive) energy levels of $H$ and $\\tilde{H}$ \nare not degenerate. \n\nUsing eqs.. (\\ref{10}) and (\\ref{11}) it is also easily shown that $H$ and \n$\\tilde{H}$ satisfy the shape invariance condition \n\\be\\label{23}\n\\tilde{H} (\\{x_i\\} ,l)  = H (\\{x_i\\}, l) +R(l),\n\\ee\n\\noindent where\n\\be\\label{24}\nR(l) = [ N \\pm l N(N-1)] \\omega = \\omega + \\delta_1~~.\n\\ee\n\\noindent \nAs a result, using the formalism of SUSY QM, and the relation between\n$E_{n+1}$ and $\\tilde{E_n}$ as given by eq. (\\ref{20}), \nthe spectrum of H is given by\n\\be \\label{25}\n{E_{n}} =\\sum_{i}R(l_{i}) - n\\delta_1~.\n\\ee\n\\noindent Note that in this particular case all $l_i$ are \nidentical so that using $\\delta_1$ and $R(l)$ as given by eqs. (\\ref{17}) \nand (\\ref{24}), the spectrum turns out to be\n\\be\\label{26}\nE_{n} = n(R-\\delta_1)\n\t= n \\omega ~.\n\\ee\n\\noindent \nUsing eq. (\\ref{14}) we then get the correct spectrum of the Calogero $A_{N-1}$\nmodel.\n\nLet us now discuss as to how to obtain the eigenfunctions of CSM using the\nformalism of SUSY QM.\nWe have seen that $A_1$ and $A_1^\\dagger$ relate the non-zero eigen states\nof the partner Hamiltonians $H$ and $\\tilde{H}$. Once a particular state of\n$H (\\tilde{H})$ with non-zero eigen value is known, the use of eq.\n(\\ref{21}) enables us to find the corresponding state of \n$\\tilde{H}\n(H)$. In particular, using eq. (\\ref{1.8}) and the fact that in this case \nall the $l_i$ \nare identical, it follows that all\nthe eigen-functions can be obtained from the ground state wave\nfunctions $\\psi_0$ as, $\\psi_n=(A_1^\\dagger)^n \\psi_0$. Note that this\nis justified from the operator algebra also, since $A_1$ and $A_1^\\dagger$\ncan be identified as the annhilation and the creation operator respectively.\nIn particular, one can show\nusing eqs. (\\ref{16}), (\\ref{19}), (\\ref{23}) and (\\ref{24}) that\n$[H,A_1]=-A_1$ and $[H,A_1^\\dagger]=A_1$. \n\nThis procedure for obtaining the\neigen-functions is similar to that of Isikov {\\it et al.} [9].\nTo see this, define a set of operators,\n\\be\\label{27}\nA_n =\\sum_{i=1}^N a_i^n, \\ \\ \\ \\ n \\leq N, \\\n\\ee\n\\noindent which are symmetric in the particle indices. These operators\nsatisfy relations which are analogous to those given by eqs. (\\ref{16}) and \n(\\ref{19}) for any $n$\n(see the next paragraph).\nIt is easily checked that $[H,A_n]=-n A_n$ and\n$[H,A_n^\\dagger]=n A_n^\\dagger$. Following [9], \nthe $k$-th eigen-state is given by,\n\\be\\label{28}\n\\psi_{\\{n_i\\}} = \\prod_{i=1}^N \\left ( {A}^\\dagger_i \\right )^{n_i} \\psi_0,\n\\ \\ \\ \\ a_i \\psi_0 =0, \\ \\ \\ \\ k=\\sum_{i=1}^N n_i ~.\n\\ee\n\\noindent Note that $\\psi_{\\{n_i\\}}$ incorporates all the degenerate states\ncorresponding\nto a particular value of $k$ and all the corresponding states of $\\tilde{H}$\ncan be obtained by applying the same $A_1$ on $\\psi_{\\{n_i\\}}$.\n\n\nLet us now ask the question whether or not $A_1$ is the only operator which\nrelates the states with nonzero eigen values of the partner Hamiltonians. The\nanswer obviously is negative and in fact, any operator which is symmetric\nin the particle\nindices can be used to relate the non-zero eigenstates of the partner\nHamiltonians. However, none of these operators are useful\nin deriving the full spectrum of the $A_{N-1}$ CSM model.\nFor example, if $\\psi$ is an\neigen-function of $H$ with\nnon-zero energy eigen-value $E$, then,\n\\be\\label{29}\n\\tilde{H} ( A_n \\psi ) = \\left [ E + \\delta_n \\right ] ( A_n \\psi )~~,\n \\ \\ \\ \\ \\delta_n = [(N-n)  \\pm l N (N-1)] \\omega~.\n\\ee\n\\noindent Note that the above equation is valid only if $\\psi$ is at least\nthe $n$-th excited state, since $A_n(A_n^\\dagger)$ anhilates(creates)\n$n$ states.\nSimilarly, one can show that any state $\\tilde{\\psi}$ of $\\tilde{H}$,\nwhich represents at least the $(n-1)$-th excited state with energy\neigen value $\\tilde{E}$, is related to a state\n${A_n^\\dagger} \\tilde{\\psi}$ of $H$ with the eigen value \n$\\left ( \\tilde{E} - \\delta_n \\right )$. This again proves one to one\ncorrespondence between the $n$-th excited state of $H$ and the $(n-1)$-th\nexcited state of $\\tilde{H}$. As a result, the use of SI gives only\nthe spectrum beginning with the $n$'th excited state of $H$ and not the \nfull spectrum. \n\nIt is worth pointing out that for the $B_N$ type models, however, \nthe symmetry arguments force us to replace $A_1$ by $A_2$ in order\nto derive the full spectrum using SUSY QM.\nThis is discussed below in detail.\n\n\\subsection{Rational $BC_N$ Calogero Model}\n\n\nThe Hamiltonian for $BC_{N}$ Calogero model is given by\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\\label{2.1}\nH_{BC_{N}} & = & {\\frac{1}{2}}[{\\sum_{i} {p_{i}^2}}  + \n l(l \\mp 1) \\sum_{i,j}^{'} \\left [(x_{i} - x_{j}) ^{-2} +\n{(x_{i} +x_{j})} ^{-2} \\right ]\\nonumber \\\\\n & + & (l_{1}-1)l_{1} {\\sum_{i}x_{i}}^{-2}\n + \\frac{(l_2)(l_2-1)}{2} \\sum_i { x_i}^{-2}+\n{\\frac{\\omega}{2}}  {\\sum_{i} x_i^2}] ~.\n\\eea\n\\noindent The sign $\\mp$ in front of the second term implies\nthat $H$ is restricted\nto act on the space of anti-symmetric (symmetric) wave-functions only.\nThis model reduces to CSM of $B_N$, $C_N$ and $D_N$\ntype in the limit $l_2=0$, $l_1=0$ and $l_2=l_1=0$, respectively.\nWithout loss of generality, in this section, we therefore only study \nthe $B_N$ type model,\ni.e. $l_2=0$. The other cases are easily obtained from here.\n\nIt is interesting to observe that this system also shares  \nthe property of SI\nas found in [7] in the case of the $A_{N-1}$ model. \nSuperpotential corresponding to this model is \n\\be\nW_{i} =\\frac{\\partial G(x_{1}...x_{N})}{ \\partial x_{i}}=\n\\frac{\\partial (ln \\psi_0)}{\\partial x_i}~,\n\\ee\n\\noindent where $\\psi_0$ is the ground state wave function of $H_{B_{N}}$\nand $G$ is given by \n\\be\nG = +l_{1} \\sum_{i} \\ln (x_{i}) +l\\sum_{i > j}\\ln (x_{i} - \nx_{j}){(x_{i} +x_{j})} -\\frac{\\omega}{2} \\sum_i x_i^2~. \n\\ee\n\\noindent Thus, the superpotential takes the form \n\\be\nW_i =+l \\sum_j^\\prime \\left [ (x_i - x_j)^{-1} +(x_i +x_j)^{-1} \\right ]+\nl_{1} x_i^{-1} - \\omega x_i~.\n\\ee\n\\noindent Following [7], define \n${\\cal A}_{i} ({\\cal A}_{i}^\\dagger)=\\pm \n\\partial_{i}+W_i $,from which Hamiltonian (\\ref{2.1}) with $l_2=0$\ncan be expressed as \n\\be\nH^{B_N}=\\sum_i {{\\cal A}_{i}}^\\dagger {\\cal A}_i \\ \\\n-\\left [ \\frac{N}{2} -l N (N-1) -l_{1} N \\right] \\omega~, \\ \\\n\\ee\n\\noindent\nShape invariance follows due to the\nidentity,\n\\be\n\\sum_{i}{\\cal{A}}_{i}{\\cal{A}}_{i}^\\dagger(l,l_{1}) =\n\\sum_i {\\cal{A}}_{i}^\\dagger\n{\\cal{A}}_{i}(l+1,l_{1} +1)~.\n\\ee\n\\noindent\nShape invariance as observed in [7] for $A_{N-1}$ CSM, is\npresent not only\nin the rational $B_N, D_N, BC_N$ models, but also in their trigonometric\ncounterparts. For trigonometric $BC_N$ models this is shown  \nin the Appendix .\n\nAs in the $A_{N-1}$ case, the SI condition does not help us in obtaining\nthe spectrum of the rational $BC_N$ models\nunless we employ the\nexchange operator formalism. Further, \nthe Hamiltonian in eq. (\\ref{2.1}) can also be cast in a diagonal form using\nexchange\noperator method. This however requires including\na reflection operator $(t_{i})$ where $t_{i}$ commutes with $x_{j}$ and \nanti-commutes\nwith $x_{i}$. The Dunkl derivative operator ( analogous to the $A_{N-1}$ case) \nis given by\n\\be\\label{2.2}\n{\\cal{D}}_{i}\n= -i \\partial_{i} + il \\sum_{j}^\\prime \\left [ {(x_i -x_j)}^{-1} M_{ij} +\n{(x_i +x_j)}^{-1} \\tilde {M_{ij}} \\right ] +i l_1 x_{i} ^{-1},\\ \\ \\\n\\tilde{M_{ij}} = t_{i}t_{j}M_{ij}~.\n\\ee\n\\noindent\nThe reflection operator $t_i$ satisfies the following relations \n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\n t_i^2=1~, \\ \\ t_i \\psi(x_1, \\dots, x_i, \\dots, x_N)=\n\\psi(x_1, \\dots, -x_i, \\dots, x_N)~,\\nonumber \\\\\nM_{ij} t_i = t_j M_{ij}~,\\ \\ \n\\tilde{M}_{ij}^\\dagger = \\tilde{M}_{ij}~,\\ \\\nt_i {\\cal{D}}_i = - {\\cal{D}}_{i} t_i~,\\ \\\nt_i {\\cal{D}}_j = {\\cal{D}}_j t_i~,\\ \\ j \\neq i,\\nonumber \\\\\n\\tilde {M_{ij}} {\\cal{D}}_i = -{\\cal{D}}_{i} \\tilde{M_{ij}} \\, .\n\\eea\n\\noindent It follows from eq. (\\ref{2.2}) that $[{\\cal{D}}_i,\n{\\cal{D}}_j] = 0$ and \n\\be\n[x_i,{\\cal{D}}_j]=i\\delta_{ij} \\left ( 1 + l \\sum_{k}^\\prime (M_{ik}+\n\\tilde{M_{ik}})+2l_1t_i \\right )\n-i(1-\\delta_{ij})l(M_{ij} -\\tilde{M_{ij}}) ~.\n\\ee\n\\noindent\nDefining, ${\\hat{a}}_i$  and ${\\hat{a}}_i^\\dagger$ \nwith the same defintion as in the previous case (see eq. (\\ref{11}))\nand using the above equations,\none finds\n\\be\n[{\\hat{a}}_i, {\\hat{a}}_j^\\dagger] = 2 \\omega \\delta_{ij} \\left (1+\nl\\sum_{k}^{\\prime} ( M_{ik}+\n\\tilde{M_{ik}}) +2l_{1}t_{i} \\right )\n- 2(1- \\delta_{ij}) l \\omega (M_{ij} - \\tilde{M_{ij}})~.      \n\\ee\n\n\\noindent As before, the SUSY partner Hamiltonians ${\\cal{H}}$\nand $\\tilde {{\\cal{H}}}$ for\nthe $B_{N}$ case \nare defined as \n\\be\n{\\cal{H}} = \\frac{1}{2}\\sum_{i}{\\hat{a}}_i^\\dagger {\\hat{a}}_i \\, ~, \\ \\ \n\\tilde{{\\cal{H}}} = \\frac{1}{2}\\sum_{i}{\\hat{a}}_i {\\hat{a}}_i^\\dagger \\, ~.\n\\ee\n\\noindent \nIt can be seen that,\n\\be\nH_{B_{N}} ={\\cal{H}} + E_{0}^{B_{N}}~, \\ \\ \\\nE_{0}^{B_{N}} = [\\frac{N}{2} \\mp \\frac{l}{2} N (N-1)+l_1 N] \\omega~.\n\\ee\n\\noindent\nThe operator which brings in a correspondence between the \neigenstates $\\psi$\nand $\\tilde{\\psi}$ are respectively\n\\be\n\\hat{A}_{2} = \\sum_{i} {\\hat{a}}_{i}^2 \\, ~, \\ \\\n\\hat{A}_{2}^\\dagger = \\sum_i (\\hat{a}_i^\\dagger)^2 \\, ~.\n\\ee\n\\noindent One can show that if ${{\\psi}} (\\tilde {{{\\psi}}})$ is the\neigenfunction of \n${\\cal{H}} (\\tilde{{\\cal {H}}})$ with eigenvalue ${\\cal{E}}\n(\\tilde {{\\cal{E}}})$ then \n\\be\n{\\cal{H}}(\\hat{A_{2}}^\\dagger {\\psi}) =\n(\\tilde{{\\cal{E}}}-\\hat{\\delta}_{2})(\\hat{A_{2}}^\\dagger \n\\tilde {\\psi})~,\\ \\\n\\tilde {{\\cal{H}}}(\\hat{A}_{2}\\psi) =\n({\\cal{E}}+\\hat{\\delta}_{2})(\\hat{A}_{2}\\psi)~~.\n\\ee\n\\noindent where,\n\\be\n\\hat{\\delta}_{2} = [N-2 \\pm 2lN(N-1) +2l_{1}N ] \\omega~.\n\\ee\n\\noindent Now the question is why we\nshould take $\\hat{A}_2$ instead of $\\hat{A}_1$ (note that \n$\\hat{A}_n=\\sum_i \\hat{a}_i^n$)?\nThe point is, unlike the $A_{N-1}$ case, the $BC_N$ Hamiltonian has the \nreflection \nsymmetry, $x_{i}{\\rightarrow}-x_{i}$. Such a symmetry on the wave-functions \nis ensured only if one uses\n$\\hat{A}_2$ and not $\\hat{A}_1$. \n\n\\noindent Following the treatment in the $A_{N-1}$ case, it is easy to show\nthat $H$ and $\\tilde {H}$ of the $B_N$ model also satisfy the SI condition i.e.\n\\be\n\\tilde {H} (\\{x_i\\},l,l_1) = H (\\{x_i\\},l,l_1) + R_2 (l,l_1)\n\\ee\nwhere\n\\be\nR_{2}(l,l_1) =[N  \\pm 2lN(N-1) +2l_{1}N] \\omega~~.\n\\ee \nSince in this case also all the $l_i$ are identical, hence it is easy to\nsee that the spectrum is given by\n\\be\nE_{n}=n(R_{2}-\\hat{\\delta}_{2})\n     =2n\\omega~.\n\\ee     \n\\noindent  Note that now the spectrum is given by $2n\\omega$, instead\nof $n\\omega$ as\nin the case of $\\hat{A}_{N-1}$. This spectrum was also obtained earlier in \n[10], but by different method.\n\nThus we have shown that for the N-body Calogero models,\nthe spectrum can also be obtained by using the ideas of SQM, SI and exchange \noperator formalism.\n\n\n\n\\section{Summary \\& Discussions }\n\nIn this paper, we have generalized the ideas of SUSY QM with one degree\nof freedom to the rational-CSM,\nwhich is a many-body problem.  In particular, we have shown that the exchange\noperator formalism is suitable for relating the non-zero eigen states\nof the partner Hamiltonians of CSM. The shape invariance in this formalism\nbecomes\ntrivial compared to the case discussed in [7]. In fact, the\npotentials\nof the partner Hamiltonians differ by a constant and\nthis is reminiscent of the usual harmonic oscillator case. \nAs a result, the operator method employed in [9] for solving\nthe rational-CSM algebraically and the SUSY method described here are \nnot very different from each other.\n\nOne of the nontrivial check of the applicability of the SUSY QM and the SI\nideas to the many-body problems lies in solving the trigonometric CSM,\nsince unlike the oscillator case, in this case  the energy spectrum is not\nlinear in the radial quantum number.  Unfortunately, the generalized\nmomentum operator $D_i$ for all types of models, rational as well as\ntrigonometric CSM associated with the root structure of $A_n$, $B_n$,\n$D_n$ and $BC_n$, are hermitian by construction. So, we can not talk of\npartner Hamiltonians in terms of $D_i$ alone. We can define the usual\ncreation and the anhilation operators, $a_i^\\dagger$ and $a_i$, in case we\nare dealing with the rational-CSM and construct partner Hamiltonians.\nUnfortunately, this can not be done for the trigonometric models. On the\nother hand, as described in [7], we can indeed introduce partner\nHamiltonians for $A_n$ type of trignometric models provided the exchange\noperator formalism has not been used. The SI is present in this formalism\nalso but the task of relating the eigenspectrum of the partner\nHamiltonians is unknown as yet.  We have shown in this paper that the SI\nis also present in the most general $BC_N$ type of trignometric models.\nHowever, the problem again lies in our inability to relate the the\nspectrum of the partner Hamiltonians.\n\n\n\n \n\n\\begin{appendix}\n\n\\section{SI in Trigonometric $BC_N$ CSM model}\n\nIn this Appendix, we present the SI conditions for the trigonometric\n$BC_N$ models.\nThe trigonometric $BC_N$ Hamiltonian is given by,\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray} \nH_{BC_N} &=& \n- \\sum_i \\partial_i^2 + l (l - 1) \\sum_{i,j}^{\\prime} \\left [\n\\frac{1}{\\sin^2(x_i - x_j)} + \\frac{1}{\\sin^2(x_i + x_j)} \\right ]\\nonumber \\\\\n&& + \\sum_i \\frac{l_1 (l_1\n- 1)} {\\sin^2 x_i}\n+ l_2 (l_2 - 1) \\sum_i \\frac{1}{\\sin^2 2x_i}~.\n\\eea\n\\noindent This model reduces to $B_N$, $C_N$ and $D_N$ for\n(a) $l_2=0$, (b) $l_1=0$,\n(c) $l_1=l_2=0$, respectively. We define a superpotential\nof the form,\n\\be\nW_i = l \\sum_j \\left [ \\cot(x_i - x_j) + \\cot(x_i + x_j) \\right] + l_1 \\cot\nx_i + l_2 \\cot 2x_i~.\n\\ee\n\\noindent \nUsing this expression of $W_i$\nin the definition of the creation and the anhilation operators as defined\nin (\\ref{1.1}), one can construct partner Hamiltonians which are equivalent \nto $H_{BC_N}$ up to an overall constant. The SI condition for these partner \nHamiltonians is\n\\be \nH_2^{BC_N} (\\{x_i\\}, l , l_1 , l_2) = \nH_1^{BC_N} (\\{x_i\\}, l^\\prime , l_1^\\prime , l_2^\\prime) +\nR^{BC_N}~,\n\\ee\n\\noindent where\n\\begin{eqnarray}}\\newcommand{\\eea}{\\end{eqnarray}\nR^{BC_N} & = & \n2 ( l^\\prime l_1^\\prime - l l_1) N (N - 1) + \\frac{4}{3} N (N - 1) (N - 2)\n(l^{\\prime 2} - l^2) \n+ 4 N (l_2^\\prime l_1^\\prime - l_2 l_1)\\nonumber \\\\\n&& + 4 N (N - 1)\n(l_2^\\prime l^\\prime - l_2 l) + 4 N (l_2^{\\prime 2} - l_2)\n+ N (l_1^{\\prime 2} - l_1) + 2 N (N - 1) (l^{\\prime 2} - l^2)\n\\eea\n\\noindent and\n\\be \nl^\\prime=l-1~, \\ \\ l_1^\\prime = l_1 - 1 ~,\\ \\ l_2^\\prime=l_2 - 1~.\n\\ee\n\\noindent The SI condition for $B_N$, $C_N$ and $D_N$ can be obtained\nfrom the above equations by taking appropriate limits. In particular,\nby putting $l_2 = 0$, $l_1 = 0$ or $l_2 = l_1 = 0$, we obtain the corresponding\nresults for the $B_N$, $C_N$ and $D_N$ type models respectively.\\\\\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Exponential lower bound for planar graphs}\\label{sec:main}\nIn this section we present the main result of the paper.\nWe provide a construction that proves that there are planar graphs with $k$ terminals whose mimicking networks are of size $\\Omega(2^k)$. \n\nIn order to present the desired graph, for the sake of simplicity, we describe its dual graph $(\\du{G},\\du{c})$. We let $\\du{\\ensuremath{Q}}=\\{\\du{f_n},\\du{f_s},\\du{f_1},\\du{f_2},\\dots,\\du{f_{k-2}} \\}$ be the set of faces in $\\du{G}$ corresponding to terminals in the primal graph $\\duu{G}$.%\n\\footnote{Since the argument mostly operates on the dual graph, for notational simplicity,\n  we use regular symbols  for objects in the dual graph, e.g., $G$, $c$, $f_i$,\n  while starred symbols refer to the dual of the dual graph, that is, the primal graph.}\nThere are two special terminal faces $\\du{f_n}$ and $\\du{f_s}$, referred to as the north face and the south face. The remaining faces of $\\du{\\ensuremath{Q}}$ are referred to as equator faces.\n\nA set $\\du{\\ensuremath{S}} \\subset \\du{\\ensuremath{Q}}$ is \\emph{important} if $\\du{f_n} \\in \\du{\\ensuremath{S}}$ and $\\du{f_s} \\notin \\du{\\ensuremath{S}}$. Note that there are $2^{k-2}$ important sets; in what follows we care only\nabout minimum cuts in the primal graph for separations between important sets and their complements.\nFor an important set $\\du{\\ensuremath{S}}$, \nwe define its \\emph{signature} as a bit vector $\\sign{\\du{\\ensuremath{S}}} \\in \\bitv{|\\du{\\ensuremath{Q}}|-2}$ whose $i$'th position is defined as $\\sign{\\du{\\ensuremath{S}}}[i]= 1 \\text{ iff } \\du{f_{i}} \\in \\du{\\ensuremath{S}}$. \nGraph $\\du{G}$ will be composed of $2^{k-2}$ cycles referred to as important cycles, each corresponding to an important subset $\\du{\\ensuremath{S}} \\subset \\du{\\ensuremath{Q}}$.\nA cycle corresponding to $\\du{\\ensuremath{S}}$ is referred to as $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ and it separates $\\du{\\ensuremath{S}}$ from $\\overline{\\du{\\ensuremath{S}}}$.\nTopologically, we draw the equator faces on a straight horizontal line that we call the equator. We put the north face $\\du{f_n}$ above the equator and the south face $\\du{f_s}$ below the equator. For any important $\\du{\\ensuremath{S}} \\subset \\du{\\ensuremath{Q}}$, in the plane drawing of $\\du{G}$ the corresponding cycle $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ is a curve that goes to the south of $\\du{f_i}$ if $\\du{f_i} \\in \\du{\\ensuremath{S}}$ and otherwise to the north of $\\du{f_i}$. We formally define important cycles later on, see Definition~\\ref{def:impcyc}.\n\nWe now describe in detail the construction of $\\du{G}$.\nWe start with a graph $H$ that is almost a tree, and then embed $H$ in the plane\nwith a number of edge crossings, introducing a new vertex on every edge crossing.\nThe graph $H$ consists of a complete binary tree of height $k-2$ with root $v$ and an extra vertex\n$w$ that is adjacent to the root $v$ and every one of the $2^{k-2}$ leaves of the tree.\nIn what follows, the vertices of $H$ are called \\emph{branching vertices}, contrary\nto \\emph{crossing vertices} that will be introduced at edge crossings\nin the plane embedding of $H$.\n\nTo describe the plane embedding of $H$, we need to introduce some notation of the vertices\nof $H$.\nThe starting point of our construction is the edge $\\du{e}=\\{ \\du{w}, \\du{v} \\}$.\nVertex $\\du{v}$ is the first branching vertex and also the root of $H$.\nIn vertex $\\du{v}$, edge $\\du{e}$ branches into $\\du{e_0}=\\{\\du{v},\\du{v_0}\\}$ and $\\du{e_1}=\\{\\du{v},\\du{v_1} \\}$. Now $\\du{v_0}$ and $\\du{v_1}$ are also branching vertices.\nThe branching vertices are partitioned into layers $L_0,\\ldots,L_{k-2}$. Vertex $\\du{v}$ is in layer $L_0=\\{ \\du{v} \\}$, while $\\du{v_0}$ and $\\du{v_1}$ are in layer $L_1=\\{ \\du{v_0}, \\du{v_1} \\}$.  Similarly, we partition edges into layers $\\mathcal{E}^H_0,\\ldots \\mathcal{E}^H_{k-1}$. So far we have $\\mathcal{E}^H_0=\\{ \\du{e} \\}$ and $\\mathcal{E}^H_1=\\{ \\du{e_0}, \\du{e_1} \\}$. \n\nThe construction continues as follows. For any layer $L_i, i \\in \\{1, \\ldots , k-3 \\}$, all the branching vertices of $L_i=\\{ \\du{v_{00 \\ldots 0}} \\ldots \\du{v_{11 \\ldots 1}} \\}$ are of degree $3$. In a vertex $\\du{v_a} \\in L_i$, $a \\in \\bitv{i}$, edge $\\du{e_a} \\in \\mathcal{E}^H_i$ branches into edges $\\du{e_{0a}}=\\{ \\du{v_a}, \\du{v_{0a}} \\},\\du{e_{1a}}=\\{ \\du{v_a}, \\du{v_{1a}} \\} \\in \\mathcal{E}^H_{i+1}$, where $\\du{v_{0a}},\\du{v_{1a}} \\in L_{i+1}$. We emphasize here that the new bit in the index is added \\emph{as the first symbol}. \nEvery next layer is twice the size of the previous one, hence $|L_i|=|\\mathcal{E}^H_i|=2^i$. Finally the vertices of $L_{k-2}$ are all of degree $2$. Each of them is connected to a vertex in $L_{k-3}$ via an edge in $\\mathcal{E}^H_{k-2}$ and to the vertex $w$ via an edge in $\\mathcal{E}^H_{k-1}$.\n\nWe now describe the drawing of $H$, that we later make planar by adding crossing vertices, in order to obtain the graph $G$.\nAs we mentioned before, we want to draw equator faces $\\du{f_1}, \\ldots \\du{f_{k-2}}$ in that order from left to right on a horizontal line (referred to as an equator). Consider equator face $\\du{f_i}$ and vertex layer $L_i$ for some $i>0$. Imagine a vertical line through $\\du{f_i}$ perpendicular to the equator, and let us refer to it as an $i$'th meridian. We align the vertices of $L_i$ along the $i$'th meridian, from the north to the south. We start with the vertex of $L_i$ with the (lexicographically) lowest index, and continue drawing vertices of $L_i$ more and more to the south while the indices increase. Moreover, the first half of $L_i$ is drawn to the north of $\\du{f_i}$, and the second half to the south of $\\du{f_i}$.\nEvery edge of $H$, except for $e$, is drawn as a straight line segment connecting its endpoints.\nThe edge $\\du{e}$ is a curve encapsulating the north face $\\du{f_n}$ and separating it from $\\du{f_s}$-the outer face of $\\du{G}$. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\input{figures\/dual}\n\\caption{The graph $\\du{G}$.\\label{dual}}\n\\end{center}\n\\end{figure}\n\nThe crossing vertices are added whenever the line segments cross. This way the edges of $H$\nare subdivided and the resulting graph is denoted by $\\du{G}$.\nThis completes the description of the structure and the planar drawing of $\\du{G}$.\nWe refer to Figure~\\ref{dual} for an illustration of the graph $G$.\nThe set $\\mathcal{E}_i$ consists of all edges of $G$ that are parts of the (subdivided) edges of $\\mathcal{E}^H_i$ from $H$, see Figure~\\ref{subdivide}.\nWe are also ready to define important cycles formally.\n\n\\begin{figure}[t]\n\\centering\n\\input{figures\/subdivide}\n\\caption{The layer $\\mathcal{E}_{i+1}$.\n  The vertex and edge names are black, their weights are blue.\\label{subdivide}}\n\\end{figure}\n\n\n\\begin{definition}\\label{def:impcyc}\nLet $\\du{\\ensuremath{S}} \\subset \\du{\\ensuremath{Q}}$ be important.\nLet $\\pi$ be a unique path in the binary tree $H-\\{w\\}$ from the root\n$\\du{v}$ to $\\du{v_{\\rev{\\sign{\\du{\\ensuremath{S}}}}}}$, \nwhere $\\rev{\\cdot}$ operator reverses the bit vector.\nLet $\\pi'$ be the path in $G$ corresponding to $\\pi$.\nThe important cycle $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ is composed of $\\du{e}$, $\\pi'$, and an edge in $\\mathcal{E}_{k-1}$ adjacent to $\\du{v_{\\rev{\\sign{\\du{\\ensuremath{S}}}}}}$.  \n\\end{definition}\n\nWe now move on to describing how weights are assigned to the edges of $\\du{G}$. \nThe costs of the edges in $\\du{G}$ admit $k-1$ values: $c_1, c_2, \\ldots c_{k-2}$, and $C$. Let $c_{k-2}=1$. For $i \\in \\{1 \\dots k-3 \\}$ let $c_i= \\sum_{j=i+1}^{k-2}|\\mathcal{E}_{j}|c_{j}$.\nLet $C=\\sum_{j=1}^{k-2} |\\mathcal{E}_i|c_i$. Let us consider an arbitrary edge $\\du{e_{ba}}=\\{ \\du{v_{a}}, \\du{v_{ba}} \\}$ for some $a \\in \\bitv{i}, i \\in \\{ 0 \\ldots k-3 \\}, b \\in \\{ 0,1 \\}$ (see Figure~\\ref{subdivide} for an illustration). As we mentioned before, $\\du{e_{ba}}$ is subdivided by crossing vertices into a number of edges. If $b=0$, then edge $\\du{e_{ba}}$ is subdivided by\\footnote{For a bit vector $a$, $\\dec{a}$ denotes the integral value of $a$ read as a number in binary.} $\\dec{a}$ crossing vertices into $\\dec{a}+1$ edges: $\\du{e^1_{ba}}=\\{ \\du{v_a}, \\du{x^1_{ba}} \\}, \\du{e^2_{ba}}=\\{ \\du{x^1_{ba}},\\du{x^2_{ba}} \\} \\ldots \\du{e^{\\dec{a}+1}_{ba}}=\\{ \\du{x^{\\dec{a}}_{ba}}, \\du{v_{ba}} \\}$. Among those edges $\\du{e^{\\dec{a}+1}_{ba}}$ is assigned cost $C$, and the remaining edges subdividing $\\du{e_{ba}}$ are assigned cost $c_i$. Analogically, if $b=1$, then edge $\\du{e_{ba}}$ is subdivided by $2^i-1-\\dec{a}$ crossing vertices into $2^i-\\dec{a}$ edges: $\\du{e^1_{ba}}=\\{ \\du{v_a}, \\du{x^1_{ba}} \\}, \\du{e^2_{ba}}=\\{ \\du{x^1_{ba}},\\du{x^2_{ba}} \\} \\ldots \\du{e^{2^i-\\dec{a}}_{ba}}=\\{ \\du{x^{2^i-1-\\dec{a}}_{ba}}, \\du{v_{ba}} \\}$. Again, we let edge $\\du{e^{2^i-\\dec{a}}_{ba}}$ have cost $C$, and the remaining edges subdividing $\\du{e_{ba}}$ are assigned cost $c_i$.\nFinally, all the edges connecting the vertices of the last layer with $w$ have weight $c_{k-2} = 1$.\nThe cost assignment within an edge layer is presented in Figure~\\ref{subdivide}. \n\nThis finishes the description of the dual graph $G$. We now consider the primal graph $\\duu{G}$ with the set of terminals $\\duu{\\ensuremath{Q}}$ consisting of the\n$k$ vertices of $\\duu{G}$ corresponding to the faces $\\ensuremath{Q}$ of $G$. In the remainder of this section we show that there is a cost function on the edges of $\\duu{G}$, under which any mimicking network for $\\duu{G}$\ncontains at least $2^{k-2}$ edges. This cost function is in fact a small perturbation of the edge costs implied by the dual graph $G$.\n\nIn order to accomplish this,\nwe use the framework introduced in~\\cite{KrauthgamerR13}. In what follows,\n   $\\mincut{G}{c}{S}{S'}$ stands for the minimum cut separating $S$ from $S'$ in a graph $G$ with cost function $c$. Below we provide the definition of the cutset-edge incidence matrix and the Main Technical Lemma from~\\cite{KrauthgamerR13}. \n\n\\begin{definition}[Incidence matrix between cutsets and edges] Let $(G,c)$ be a $k$-terminal network, and fix an enumeration $S_1, \\ldots S_m$ of all $2^{k-1}-1$ distinct and nontrivial bipartitions $Q=S_i \\cup \\overline{S}_i$. The cutset-edge incidence matrix of $(G,c)$ is the matrix $A_{G,c} \\in \\{ 0,1 \\}^{m \\times E(G)}$ given by\n$$\n(A_{G,c})_{i,e}=\n\\begin{cases}\n1 \\text{ if } e \\in \\mincut{G}{c}{S_i}{\\overline{S}_i}\\\\\n0 \\text{ otherwise.}\n\\end{cases}\n$$\n\\end{definition}\n\n\\begin{lemma}[Main Technical Lemma of \\cite{KrauthgamerR13}]\\label{lem:mtl}\nLet $(G,c)$ be a $k$-terminal network. Let $A_{G,c}$ be its cutset-edge incidence matrix, and assume that for all $S \\subset Q$ the minimum $S$-separating cut of $G$ is unique. Then there is for $G$ an edge cost function $\\tilde{c}: E(G) \\mapsto \\mathbb{R}^+$, under which every mimicking network $(G',c')$ satisfies $|E(G')| \\geq \\rank{A_{G,c}}$. \n\\end{lemma}\n\nRecall that $\\duu{G}$ is the dual graph to the graph $\\du{G}$ that we constructed.\nBy slightly abusing the notation, we will use the cost function $c$ defined on the dual edges\nalso on the corresponding primal edges.\nLet $\\duu{\\ensuremath{Q}}=\\{ \\duu{f_n}, \\duu{f_s}, \\duu{f_1}, \\ldots \\duu{f_{k-2}} \\}$ be the set of terminals in $\\duu{G}$ corresponding to $\\du{f_n}, \\du{f_s}, \\du{f_1}, \\ldots \\du{f_{k-2}}$ respectively. We want to apply Lemma~\\ref{lem:mtl} to $\\duu{G}$ and $\\duu{\\ensuremath{Q}}$. For that we need to show that the cuts in $\\duu{G}$ corresponding to important sets are unique and that $\\rank{A_{\\duu{G},c}}$ is high.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\input{figures\/primal}\n\\caption{Primal graph $G^\\ast$.\\label{primal}}\n\\end{center}\n\\end{figure}\n\nAs an intermediate step let us argue that the following holds.\n\\begin{claim}\\label{clm:kc}\nThere are $k$ edge disjoint simple paths in $\\duu{G}$ from $\\duu{f_n}$ to $\\duu{f_s}$: $\\pi_0, \\pi_1, \\ldots, \\pi_{k-2}, \\pi_{k-1}$. Each $\\pi_i$ is composed entirely of edges dual to the edges of $\\mathcal{E}_i$ whose cost equals $C$. For $i \\in \\{ 1 \\ldots k-2 \\}$, $\\pi_i$ contains vertex $\\duu{f_i}$. Let $\\pi_i^n$ be the prefix of $\\pi_i$ from $\\duu{f_n}$ to $\\duu{f_i}$ and $\\pi_i^s$ be the suffix from $\\duu{f_i}$ to $\\duu{f_s}$. The number of edges on $\\pi_i$ is $2^i$, and the number of edges on $\\pi_i^n$ and $\\pi_i^s$ is $2^{i-1}$. \n\\end{claim}\n\\begin{proof}The primal graph $\\duu{G}$ together with paths $\\pi_0, \\pi_1 \\ldots \\pi_{k-2},\\pi_{k-1}$ is pictured in Figure~\\ref{primal}. The paths $\\pi_{k-2},\\pi_{k-1}$ visit the same vertices in the same manner, so for the sake of clarity only one of these paths is shown in the picture. This proof contains a detailed description of these paths and how they emerge from in the dual graph $\\du{G}$.\n\nConsider a layer $L_i$. Recall that for any $ba \\in \\bitv{i}$ edge $\\du{e_{ba}}$ of the almost tree is subdivided in $\\du{G}$, and all the resulting edges are in $\\mathcal{E}_i$. If $b=0$, then edge $\\du{e_{ba}}$ is subdivided by $\\dec{a}$ crossing vertices into $\\dec{a}+1$ edges: $\\du{e^1_{ba}}=\\{ \\du{v_a}, \\du{x^1_{ba}} \\}, \\du{e^2_{ba}}=\\{ \\du{x^1_{ba}},\\du{x^2_{ba}} \\} \\ldots \\du{e^{\\dec{a}+1}_{ba}}=\\{ \\du{x^{\\dec{a}}_{ba}}, \\du{v_{ba}} \\}$, where $\\du{c}(\\du{e^{\\dec{a}+1}_{ba}})=C$. Analogically, if $b=1$, then edge $\\du{e_{ba}}$ is subdivided by $2^i-1-\\dec{a}$ crossing vertices into $2^i-\\dec{a}$ edges: $\\du{e^1_{ba}}=\\{ \\du{v_a}, \\du{x^1_{ba}} \\}, \\du{e^2_{ba}}=\\{ \\du{x^1_{ba}},\\du{x^2_{ba}} \\} \\ldots \\du{e^{2^i-\\dec{a}}_{ba}}=\\{ \\du{x^{2^i-1-\\dec{a}}_{ba}}, \\du{v_{ba}} \\}$. Again, $\\du{c}(\\du{e^{2^i-\\dec{a}}_{ba}})=C$. Consider the edges of $\\mathcal{E}_i$ incident to vertices in $L_i$. If we order these edges lexicographically by their lower index, then each consecutive pair of edges shares a common face. Moreover, the first edge $\\du{e^1_{00\\ldots0}}$ is incident to $\\du{f_n}$ and the last edge $\\du{e^1_{11\\ldots1}}$ is incident to $\\du{f_s}$. This gives a path $\\pi_i$ from $f_n$ to $f_s$ through $f_i$ in the primal graph where all the edges on $\\pi_i$ have cost $C$. Path $\\pi_{k-1}$ is given by the edges of $\\mathcal{E}_{k-1}$ in a similar fashion and path $\\pi_0$ is composed of a single edge dual to $\\du{e}$. \n\\end{proof}\n\nWe move on to proving that the condition in Lemma~\\ref{lem:mtl} holds.\nWe extend the notion of important sets $\\ensuremath{S} \\subseteq \\ensuremath{Q}$ to sets $\\duu{\\ensuremath{S}} \\subseteq \\duu{\\ensuremath{Q}}$\nin the natural manner.\n\\begin{lemma}\\label{lem:uniquecuts}\nFor every important $\\duu{\\ensuremath{S}} \\subset \\duu{\\ensuremath{Q}}$, the minimum cut separating $\\duu{\\ensuremath{S}}$ from $\\overline{\\duu{\\ensuremath{S}}}$ is unique and corresponds to cycle $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ in $\\du{G}$.   \n\\end{lemma}\n\n\\begin{proof}\nLet $\\ensuremath{\\mathcal{C}}$ be the set of edges of $G$ corresponding to some\nminimum cut between $\\duu{\\ensuremath{S}}$ and $\\duu{\\overline{\\ensuremath{S}}}$ in $\\duu{G}$.\nLet $\\ensuremath{S} \\subseteq \\ensuremath{Q}$ be the set of faces of $G$ corresponding to the set $\\duu{\\ensuremath{S}}$.\nWe start by observing that the edges of $\\duu{G}$ corresponding to $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$\nform a cut between $\\duu{\\ensuremath{S}}$ and $\\duu{\\overline{\\ensuremath{S}}}$. Consequently,\n     the total weight of edges of $\\ensuremath{\\mathcal{C}}$ is at most the total weight of the edges of\n     $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$.\n\nBy Claim~\\ref{clm:kc}, $\\ensuremath{\\mathcal{C}}$ contains at least $k$ edges of cost $C$, at least one edge of cost $C$ per edge layer (it needs to hit an edge in every path $\\pi_0 , \\ldots \\pi_{k-1}$). Note that $\\ensuremath{\\mathcal{C}}_{\\sign{ \\du{\\ensuremath{S}} }}$ contains exactly $k$ edges of cost $C$. We assign the weights in a way that $C$ is larger than all other edges in the graph taken together.\nThis implies that $\\ensuremath{\\mathcal{C}}$ contains exactly one edge of cost $C$ in every edge layer $\\mathcal{E}_i$.\nIn particular, $\\ensuremath{\\mathcal{C}}$ contains the edge $e = \\{ v,w \\}$.\n\nFurthermore, the fact that $\\duu{f_i}$ lies on $\\pi_i$ implies that\nthe edge of weight $C$ in $\\mathcal{E}_i \\cap \\ensuremath{\\mathcal{C}}$ lies on $\\pi_i^n$ if $\\duu{f_i} \\notin \\ensuremath{S}$\nand lies on $\\pi_i^s$ otherwise.\nConsequently, in $\\duu{G}-\\ensuremath{\\mathcal{C}}$ there is one connected component containing all vertices\nof $\\duu{\\ensuremath{S}}$ and one connected component containing all vertices of $\\overline{\\duu{\\ensuremath{S}}}$.\nBy the minimality of $\\ensuremath{\\mathcal{C}}$, we infer that $\\duu{G}-\\ensuremath{\\mathcal{C}}$ contains \nno other connected components apart from the aforementioned two components.\nBy planarity, since any minimum cut in a planar graph corresponds to a collection of cycles\nin its dual, this implies that $\\ensuremath{\\mathcal{C}}$ is a single cycle in $G$.\n\nLet $e_i$ be the unique edge of $\\mathcal{E}_i \\cap \\ensuremath{\\mathcal{C}}$ of weight $C$\nand let $e_i'$ be the unique edge of $\\mathcal{E}_i \\cap \\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ of weight $C$.\nWe inductively prove that $e_i = e_i'$ and\nthat the subpath of $\\ensuremath{\\mathcal{C}}$ between $e_i$ and $e_{i+1}$ is the same as on\n$\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$.\nFor the base of the induction, note that $e_0 = e_0' = e$.\n\nConsider an index $i > 0$ and the face $\\du{f_i}$. If $\\du{f_i} \\in \\du{\\ensuremath{S}}$, i.e., $\\du{f_i}$ belongs to the north side, then $e_i$ lies south of $f_i$, that is, lies on $\\pi_i^s$.\nOtherwise, if $f_i \\notin \\ensuremath{S}$, then $e_i$ lies north of $f_i$, that is, lies on $\\pi_i^n$.\n\nLet $v_a$ and $v_{ba}$ be the vertices of $\\ensuremath{\\mathcal{C}}_{\\sign{\\ensuremath{S}}}$ that lie \non $L_{i-1}$ and $L_i$, respectively. By the inductive assumption, $v_a$ is an endpoint\nof $e_{i-1}' = e_{i-1}$ that lies on $\\ensuremath{\\mathcal{C}}$.\nLet $e_i = xv_{bc}$, where $v_{bc} \\in L_i$ and let $e_i' = x'v_{ba}$.\nSince $\\ensuremath{\\mathcal{C}}$ is a cycle in $G$ that contains exactly one edge on each path $\\pi_i$,\nwe infer that $\\ensuremath{\\mathcal{C}}$ contains a path between $v_a$ and $v_{bc}$ that consists of\n$e_i$ and a number of edges of $\\mathcal{E}_i$ of weight $c_i$.\nA direct check shows that the subpath from $v_a$ to $v_{ba}$ on $\\ensuremath{\\mathcal{C}}_{\\sign{\\ensuremath{S}}}$\nis the unique such path with minimum number of edges of weight $c_i$.\nSince the weight $c_i$ is larger than the total weight of all edges of smaller weight,\nfrom the minimality of $\\ensuremath{\\mathcal{C}}$ we infer that $v_{ba} = v_{bc}$ and $\\ensuremath{\\mathcal{C}}$\nand $\\ensuremath{\\mathcal{C}}_{\\sign{\\ensuremath{S}}}$ coincide on the path from $v_a$ to $b_{ba}$.\n\nConsequently, $\\ensuremath{\\mathcal{C}}$ and $\\ensuremath{\\mathcal{C}}_{\\sign{\\ensuremath{S}}}$ coincide on the path from the edge $e=vw$\nto the vertex $v_{\\rev{\\sign{\\ensuremath{S}}}} \\in L_{k-2}$. From the minimality of $\\ensuremath{\\mathcal{C}}$\nwe infer that also the edge $\\{w,v_{\\rev{\\sign{\\ensuremath{S}}}} \\}$ lies on the cycle $\\ensuremath{\\mathcal{C}}$ and, hence,\n   $\\ensuremath{\\mathcal{C}} = \\ensuremath{\\mathcal{C}}_{\\sign{\\ensuremath{S}}}$. This completes the proof.\n\\end{proof}\n\n\n\\begin{claim}\\label{clm:rank}\n$\\rank{A_{G,c}} \\geq 2^{k-2}$. \n\\end{claim}\n\\begin{proof}\nRecall Definition~\\ref{def:impcyc} and the fact that $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ is defined for every important $\\ensuremath{S} \\subseteq \\ensuremath{Q}$.\nThis means that the only edge in $\\mathcal{E}_{k-1}$ that belongs to $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ is the edge adjacent to $\\du{v_{\\rev{\\sign{\\du{\\ensuremath{S}}}}}}$. Let us consider the part of adjacency matrix where rows correspond to the cuts corresponding to $\\ensuremath{\\mathcal{C}}_{\\sign{\\du{\\ensuremath{S}}}}$ for important $\\ensuremath{S} \\subset \\ensuremath{Q}$ and where columns correspond to the edges in $\\mathcal{E}_{k-1}$ of weight $C$. Let us order the cuts according to $\\rev{\\sign{\\du{\\ensuremath{S}}}}$ and the edges by the index  of the adjacent vertex in $L_{k-2}$ (lexicographically). Then this part of $A_{G,c}$ is an identity matrix. Hence, $\\rank{A_{G,c}} \\geq 2^{k-2}$. \n\\end{proof}\n\nLemma \\ref{lem:uniquecuts} and Claim~\\ref{clm:rank} provide the conditions necessary for Lemma~\\ref{lem:mtl} to apply. This proves our main result stated in Theorem~\\ref{thm:main}. \n\n\\section{Doubly exponential example}\\label{sec:side}\n\n\\begin{figure}[tb]\n\\centering\n\\input{figures\/double-exp}\n\\caption{Illustration of the construction. The two panels correspond to two cases in the proof, either $u_{S_0} \\in Z$ (top panel) or $u_{S_0} \\notin Z$ (bottom panel).}\\label{fig:double-exp}\n\\end{figure}\n\nIn this section we show an example graph for which the compression technique introduced by Hagerup et al~\\cite{HagerupKNR98} does indeed produce a mimicking network on\nroughly $2^{\\binom{k-1}{\\lfloor (k-1)\/2 \\rfloor}}$ vertices.\nOur example relies on doubly exponential edge costs. Note that an example with single exponential costs can be compressed into a mimicking network of size single exponential in $k$ using the techniques of~\\cite{KratschW12}.\n\nBefore we go on, let us recall the technique of Hagerup et al~\\cite{HagerupKNR98}. Let $G$ be a weighted graph and $Q$ be the set of terminals. Observe that a minimum cut separating $S \\subset Q$ from $\\overline{S}=Q \\setminus S$, when removed from $G$, divides the vertices of $G$ into two sides: the side of $S$ and the side of $\\overline{S}$. The side is defined for each vertex, as all connected components obtained by removing the minimum cut contain a terminal. Now if two vertices $u$ and $v$\nare on the same side of the minimum cut between $S$ and $\\overline{S}$ for every $S \\subset Q$, then they can be merged without changing the size of any minimum $S$-separating cut. As a result there is at most $2^{2^k}$ vertices in the graph;\nas observed by~\\cite{ChambersE13,KhanR14}, this bound can be improved to roughly $2^{\\binom{k-1}{\\lfloor (k-1)\/2 \\rfloor}}$. After this brief introduction we move on to describing our example.\n\nOur construction builds up on the example provided in~\\cite{KrauthgamerR13} in the proof of Theorem 1.2.\nAs stated in Theorem~\\ref{thm:side} of this paper, our construction works for parameter $k$ equal to $6$ modulo $8$.\nLet $k = 2r+2$, that is, $r$ is equal to $2$ modulo $4$.\nThese remainder assumptions give the following observation via standard calculations.\n\\begin{lemma}\\label{lem:ell-even}\nThe integer $\\ell := \\binom{2r+1}{r}$ is even.\n\\end{lemma}\n\\begin{proof}\nRecall that $r$ equals $2$ modulo $4$.\nSince $\\binom{2r+1}{r} = \\frac{(2r+1)!}{r!(r+1)!}$, while the largest power of $2$ that divides $a!$ equals $\\sum_{i=1}^\\infty \\lfloor \\frac{a}{2^i} \\rfloor$, we have that\nthe largest power of $2$ that divides $\\binom{2r+1}{r}$ equals:\n\\begin{align*}\n& \\sum_{i=1}^\\infty \\left\\lfloor \\frac{2r+1}{2^i} \\right\\rfloor - \\sum_{i=1}^\\infty \\left\\lfloor \\frac{r}{2^i} \\right\\rfloor - \\sum_{i=1}^\\infty \\left\\lfloor \\frac{r+1}{2^i} \\right\\rfloor \n = r + \\sum_{i=1}^\\infty \\left\\lfloor \\frac{r}{2^i} \\right\\rfloor - 2 \\sum_{i=1}^\\infty \\left\\lfloor \\frac{r}{2^i} \\right\\rfloor  \\\\\n&\\quad = r - \\sum_{i=1}^\\infty \\left\\lfloor \\frac{r}{2^i} \\right\\rfloor \n = r - \\frac{r}{2} - \\frac{r-2}{4} - \\sum_{i=1}^\\infty \\left\\lfloor \\frac{r}{4 \\cdot 2^i} \\right\\rfloor \n \\geq  \\frac{1}{2} + \\frac{r}{4} - \\sum_{i=1}^\\infty \\frac{r}{4 \\cdot 2^i}\n = \\frac{1}{2}.\n\\end{align*}\nIn particular, it is positive. This finishes the proof of the lemma.\n\\end{proof}\n\nWe start our construction with a complete bipartite graph $G_0 = (Q_0, U, E)$, where one side of the graph consists of $2r+1 = k-1$ terminals $Q_0$, and the other side of the graph consists of $\\ell = \\binom{2r+1}{r}$ non-terminals\n$U = \\{u_S~|~S \\in \\binom{Q_0}{r}\\}$. That is, the vertices $u_S \\in U$ are indexed by subsets of $Q_0$ of size $r$.\nThe cost of edges is defined as follows. Let $\\alpha$ be a large constant that we define later on.\nEvery non-terminal $u_S$ is connected by edges of cost $\\alpha$ to every terminal $q \\in Q_0 \\setminus S$ and by edges of cost $(1+\\frac{1}{r} + \\frac{1}{r^2})\\alpha$ to every terminal $q \\in S$.\nTo construct the whole graph $G$, we extend $G_0$ with a last terminal $x$ (i.e., the terminal set is $Q = Q_0 \\cup \\{x\\}$)\n and build a third layer of $m = \\binom{\\ell}{\\ell\/2}$ non-terminal vertices $W = \\{w_Z~|~Z \\in \\binom{U}{\\ell\/2}\\}$. That is, the vertices $w_Z \\in W$ are indexed by subsets of $U$ of size $\\ell\/2$.\nThere is a complete bipartite graph between $U$ and $W$ and every vertex of $W$ is adjacent to $x$.\nThe cost of edges is defined as follows. An edge $u_S w_Z$ is of cost $1$ if $u_S \\in Z$, and of cost $0$ otherwise. Every edge of the form $xw_Z$ is of cost $\\ell\/2 - 1$.\nThis finishes the description of the construction. For the reference see the top picture in Figure~\\ref{fig:double-exp}.\n\nWe say that a set $S \\subseteq Q$ is \\emph{important} if $x \\in S$ and $|S| = r+1$. Note that there are $\\ell = \\binom{2r+1}{r} = \\binom{k-1}{\\lfloor (k-1)\/2 \\rfloor}$ important sets.\nWe observe the following.\n\\begin{lemma}\\label{lem:dblexp}\nLet $S \\subset Q$ be important and let $S_0 = S \\setminus \\{x\\} = S \\cap Q_0$. For $\\alpha > r^2 \\ell|W|$,\n    the vertex $w_Z$ is on the $S$ side of the minimum cut between $S$ and $Q \\setminus S$\n    if and only if $u_{S_0} \\in Z$. \n\\end{lemma}\n\n\\begin{proof}\nFirst, note that if $\\alpha > r^2 \\ell |W|$, then the total cost of all the edges incident to vertices of $W$ is less than $\\frac{1}{r^2} \\alpha$.\nIntuitively, this means that cost of the cut inflicted by the edges of $G_0$ is of absolutely higher importance than the ones incident with $W$.\n\nConsider an important set $S \\subseteq Q$ and let $S_0 = S \\setminus \\{x\\} = S \\cap Q_0$.\n\nLet $u_{S'} \\in U$. The \\emph{balance} of the vertex $u_{S'}$, denoted henceforth $\\beta(u_{S'})$, is the difference of the cost of edges\nconnecting $u_{S'}$ with $S_0$ and the ones connecting $u_{S'}$ and $Q_0 \\setminus S$. Note that we have\n$$\\beta(u_{S_0}) = r \\cdot \\left(1+\\frac{1}{r}+\\frac{1}{r^2}\\right) \\alpha - \\left(r+1\\right) \\cdot \\alpha = \\frac{1}{r} \\alpha.$$\nOn the other hand, for $S' \\neq S_0$, the balance of $u_{S'}$ can be estimated as follows:\n$$\\beta(u_{S'}) \\leq (r-1) \\cdot \\left(1+\\frac{1}{r}+\\frac{1}{r^2}\\right) \\alpha + \\alpha - r \\cdot \\alpha - \\left(1+\\frac{1}{r}+\\frac{1}{r^2}\\right) \\alpha \n              = -\\frac{r+2}{r^2} \\alpha < -\\frac{1}{r^2} \\alpha.$$\nConsequently, as $\\frac{1}{r^2} \\alpha$ is larger than the cost of all edges incident with $W$,\nin a minimum cut separating $S$ from $Q \\setminus S$, the vertex $u_{S_0}$ picks the $S$ side, while every vertex $u_{S'}$ for $S' \\neq S_0$ picks the $Q \\setminus S$ side.\n\nConsider now a vertex $w_Z \\in W$ and consider two cases: either $u_{S_0} \\in Z$ or $u_{S_0} \\notin Z$; see also Figure~\\ref{fig:double-exp}.\n\n\\myparagraph{Case 1: $u_{S_0} \\in Z$. } As argued above, all vertices of $U$ choose their side according to what is best in $G_0$, so $u_{S_0}$ is the only vertex in $U$ on the $S$ side.\nTo join the $S$ side, $w_Z$ has to cut $\\ell\/2-1$ edges $u_{S'} w_Z$ of cost $1$ each, inflicting a total cost of $\\ell\/2-1$;\nnote that it does not need to cut the edge $u_{S_0}w_Z$, which is of cost $1$ as $u_{S_0} \\in Z$.\nTo join the $Q \\setminus S$ side, $w_Z$ needs to cut $xw_Z$ of cost $\\ell\/2-1$\nand $u_{S_0}w_Z$ of cost $1$, inflicting a total cost of $\\ell\/2$.\nConsequently, $w_Z$ joins the $S$ side.\n\n\\myparagraph{Case 2: $u_S \\notin Z$. } Again all vertices of $U$ choose their side according to what is best in $G$, so $u_{S_0}$ is the only vertex in $U$ on the $S$ side.\nTo join the $S$ side, $w_Z$ has to cut $\\ell\/2$ edges $u_{S'}w_Z$ of cost $1$ each, inflicting a total\ncost of $\\ell\/2$.\nTo join the $Q \\setminus S$ side, $w_Z$ has to cut one edge of positive cost, namely the edge $xw_Z$ of cost $\\ell\/2-1$.\nConsequently, $w_Z$ joins the $Q \\setminus S$ side.\n\nThis finishes the proof of the lemma.\n\\end{proof}\nLemma~\\ref{lem:dblexp} shows that $G$ cannot be compressed using the technique presented in~\\cite{HagerupKNR98}.\nTo see that let us fix two vertices $w_Z$ and $w_{Z'}$ in $W$,\nand let $u_S \\in Z \\setminus Z'$.\nThen, Lemma~\\ref{lem:dblexp} shows that $w_Z$ and $w_{Z'}$ lie on different\nsides of the minimum cut between $S$ and $Q \\setminus S$.\nThus, $w_Z$ and $w_{Z'}$ cannot be merged.\nSimilar but simpler arguments show that no other pair of vertices in $G$ can be merged.\nTo finish the proof of Theorem~\\ref{thm:side}, observe that\n$$|W| = \\binom{\\ell}{\\ell\/2} = \\Omega\\left(2^{\\ell}\/\\sqrt{\\ell}\\right) = \\Omega\\left(2^{\\binom{k-1}{\\lfloor (k-1)\/2 \\rfloor} - k\/2}\\right).$$\n\n\n\n\n\n\\section{Introduction}\\label{sec:intro}\nOne of the most popular paradigms when designing effective algorithms is preprocessing.\nThese days in many applications, in particular mobile ones, even though fast running time is desired, the memory usage is the main limitation. The preprocessing needed for such applications is to reduce the size of the input data prior to some resource-demanding computations,\n without (significantly) changing the answer to the problem being solved.\nIn this work we focus on this kind of preprocessing, known also as graph compression, for flows and cuts.\nThe input graph needs to be compressed while preserving its essential flow and cut properties.\n\nCentral to our work is the concept of a \\emph{mimicking network}, introduced by \nHagerup, Katajainen, Nishimura, and Ragde~\\cite{HagerupKNR98}.\nLet $G$ be an edge-weighted graph with a set $Q \\subseteq V(G)$ of $k$ terminals.\nFor a partition $Q = S \\uplus \\bar{S}$, \na minimum cut between $S$ and $\\bar{S}$ is called a \\emph{minimum $S$-separating cut}. \nA \\emph{mimicking network} is an edge-weighted graph\n$G'$ with $Q \\subseteq V(G')$ such that the weights of minimum $S$-separating cuts\nare equal in $G$ and $G'$ for every partition $Q = S \\uplus \\bar{S}$.\nHagerup et al~\\cite{HagerupKNR98} \nobserved the following simple preprocessing step: if two vertices $u$ and $v$\nare always on the same side of the minimum cut between $S$ and $\\bar{S}$ for every choice\nof the partition $Q = S \\uplus \\bar{S}$, then they can be merged without changing the size\nof any minimum $S$-separating cut. \nThis procedure always\nleads to a mimicking network with at most $2^{2^k}$ vertices. \n\nThe above upper bound can be improved to a still double-exponential bound\nof roughly $2^{\\binom{k-1}{\\lfloor (k-1)\/2 \\rfloor}}$, as observed both by \nKhan and Raghavendra~\\cite{KhanR14} and by Chambers and Eppstein~\\cite{ChambersE13}.\nIn 2013, Krauthgamer and Rika~\\cite{KrauthgamerR13} observed that the aforementioned preprocessing\nstep can be adjusted to yield a mimicking network of size $\\mathcal{O}(k^2 2^{2k})$ for planar graphs.\nFurthermore, they introduced a framework for proving lower bounds, and showed that\nthere are (non-planar) graphs, for which any mimicking network \nhas $2^{\\Omega(k)}$ edges; a slightly stronger lower bound \nof $2^{(k-1)\/2}$ has been shown by Khan and Raghavendra~\\cite{KhanR14}.\nOn the other hand, for planar graphs the lower bound of~\\cite{KrauthgamerR13} is $\\Omega(k^2)$. \nFurthermore, the planar graph lower bound applies even in the special case when all the terminals\nlie on the same face.\n\nVery recently, two improvements upon these results for planar graphs have been announced. \nIn a sequel paper, Krauthgamer and Rika~\\cite{KrauthgamerR17} improve the \npolynomial factor in the upper bound for planar graphs to $\\mathcal{O}(k 2^{2k})$ and show that the exponential dependency\nactually adheres only to the \\emph{number of faces containing terminals}: if\nthe terminals lie on $\\gamma$ faces, one can obtain a mimicking network\nof size $\\mathcal{O}(\\gamma 2^{2\\gamma} k^4)$. \nIn a different work, Goranci, Henzinger, and Peng~\\cite{GoranciHP17} showed a tight $\\mathcal{O}(k^2)$ upper bound\nfor mimicking networks for planar graphs with all terminals on a single face.\n\n\\myparagraph{Our results.}\nWe complement these results by showing an exponential lower bound for mimicking networks in planar graphs.\n\\begin{theorem}\\label{thm:main}\nFor every integer $k \\geq 3$,\nthere exists a planar graph $G$ with a set $Q$ of $k$ terminals\nand edge cost function under which every mimicking network for $G$ has\nat least $2^{k-2}$ edges.\n\\end{theorem}\nThis nearly matches the upper bound of $\\mathcal{O}(k2^{2k})$ of  Krauthgamer and Rika~\\cite{KrauthgamerR17}\nand is in sharp contrast with the polynomial bounds when the terminals lie on a constant\nnumber of faces~\\cite{GoranciHP17,KrauthgamerR17}.\nNote that it also nearly matches the improved bound of $\\mathcal{O}(\\gamma 2^{2\\gamma} k^4)$ for terminals on $\\gamma$ faces~\\cite{KrauthgamerR17},\nas $k$ terminals lie on at most $k$ faces.\n\nAs a side result, we also show a hard instance for mimicking networks in general graphs.\n\\begin{theorem}\\label{thm:side}\nFor every integer $k \\geq 1$ that is equal to $6$ modulo $8$, there\nexists a graph $G$ with a set $Q$ of $k$ terminals \nand $\\Omega(2^{\\binom{k-1}{\\lfloor (k-1)\/2 \\rfloor} - k\/2})$ vertices,\n    such that no two vertices can be identified without strictly increasing the size of some minimum $S$-separating cut.\n\\end{theorem}\nThe example of Theorem~\\ref{thm:side}, obtained by iterating the construction of Krauthgamer and Rika~\\cite{KrauthgamerR13},\nshows that the doubly exponential bound\nis natural for the preprocessing step of Hagerup et al~\\cite{HagerupKNR98}, and\none needs different techniques to improve upon it.\nNote that the bound of Theorem~\\ref{thm:side} is very close to the upper bound given\nby~\\cite{ChambersE13,KhanR14}.\n\n\\myparagraph{Related work.}\nApart from the aforementioned work on mimicking\nnetworks~\\cite{GoranciHP17,HagerupKNR98,KhanR14,KrauthgamerR13,KrauthgamerR17},\nthere has been substantial work on preserving cuts and flows approximately,\nsee e.g.~\\cite{robi-apx,robi-old,mm-sparsifiers}.\nIf one wants to construct mimicking networks for vertex cuts in\nunweighted graphs with deletable terminals (or with small integral\nweights), the representative sets approach of Kratsch and Wahlstr\\\"{o}m~\\cite{KratschW12}\nprovides a mimicking network with $\\mathcal{O}(k^3)$ vertices, improving upon a previous\nquasipolynomial bound of Chuzhoy~\\cite{Chuzhoy12}.\n\n\n\\medskip\n\nWe prove Theorem~\\ref{thm:main} in Section~\\ref{sec:main}\nand show the example of Theorem~\\ref{thm:side} in Section~\\ref{sec:side}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\nAbout $100$ young massive stars are located in the central half-parsec of the Milky-Way's nuclear star cluster, with a supermassive black hole at is center. The proper motions of these stars indicate an average sense of clockwise rotation  around the black hole \\citep{2000MNRAS.317..348G}. More detailed 3-dimensional analyses show that a significant fraction ($\\sim 30-60 \\%$) of these stars are moving in a thin clockwise disc \\citep{2003ApJ...590L..33L, 2003ApJ...594..812G, 2006ApJ...643.1011P, 2006ApJ...648..405B, 2009ApJ...690.1463L,  2009ApJ...697.1741B, 2014ApJ...783..131Y, 2022ApJ...932L...6V}. The other stars clearly do not belong to the disc, with several groups disagreeing about the interpretation of their precise geometry. \\cite{2003ApJ...594..812G}, \\cite{2006ApJ...643.1011P}, \\cite{2009ApJ...697.1741B}, and \\cite{2022ApJ...932L...6V}   argue that the other stars form another disc-like structure, albeit with somewhat diffuse clustering of the stars' orbital planes, possibly connected by a strong warp to the clockwise disc.  \\cite{2022ApJ...932L...6V} goes further and claims that several diffuse disc-like structures can be identified. By contrast, \\cite{2009ApJ...690.1463L} and \\cite{2014ApJ...783..131Y} argue that all these secondary disc-like structures are not statistically significant. These groups also disagree on the precise fraction of the young stars that belong to the clockwise disc.\n\nIt is appealing to consider this kinematic data as a result of a partial disruption of an initially coherent stellar disc. There is strong evidence that the young stars were formed {\\it in situ} \\citep{2005MNRAS.364L..23N, 2006ApJ...643.1011P, 2009ApJ...690.1463L}, and by far the most natural scenario for this is the star formation inside a gravitationally unstable gas disc \\citep{2003ApJ...590L..33L, 2008Sci...321.1060B} . \\cite{2009A&A...496..695S} suggested that a torque from the circumnuclear gas ring located at a distance $\\sim 2$pc from the black hole, could exert a disruptive torque on the disc, but their analysis did not take the disc's self-gravity into account. A direct $N$-body simulations of this process by \\cite{2016ApJ...818...29T} showed that it was not efficient in disrupting the disc. \\cite{2011MNRAS.412..187K, 2015MNRAS.448.3265K, 2022MNRAS.tmp.2848P} explored whether the disc could be disrupted by stochastic torques from vector resonant relaxation (VRR), but found that in order for this to happen in $5\\times 10^6$ years, the masses of the background stars in the cluster had to be unrealistically high, over $100M_{\\odot}$.\n\nWe think that the key to the puzzle of the disrupted disc is in its gravitational interaction with the rotating nuclear star cluster inside which it resides. More specifically, we show that the effect called ``resonant friction'' \ncan produce a very strong torque on the disc that tends to align the disc's rotation with that of the cluster, and in the process can partially disrupt \nthe disc in less that $5\\times 10^6$ years. The Milky-way's nuclear cluster is rotating neither clockwise nor counterclockwise relative to the line of sight; instead its rotational axis is close to that of the Galaxy, and thus the cluster's rotation is misaligned with that of the disc \\citep{2014A&A...570A...2F}. In Section 2 we demonstrate using numerical simulations that this configuration naturally leads to the disc disruption, and  that the expected orbital distribution is qualitatively similar to the one that is observed in the Galactic Center. Typically we see an inner disc, sometimes warped or broken up into rings, co-existing with a more diffuse disoriented or weakly clustered orbits of the outer stars, with the inner disc containing $\\sim 50\\%$ of the total stars. The reader uninterested in the details of this paper should just look at Figure 1: it contains the paper's most interesting results. In this section we also explain how we choose the range of rotational parameters of the cluster based on existing observations of the cluster's mean radial velocities. In Section 3 we back up the numerical results by obtaining  an analytical estimate of  the Resonant Friction timescale inside a slowly-rotating cluster. In Section 4 we speculate that resonant friction would reorient gaseous accretion discs that are  formed inside a nuclear star cluster from radially infalling, tidally captured gas clouds. We discuss what impact this would have on spins of supermassive black holes. We present our numerical algorithm in the Appendix.\n\\begin{figure*}\n\\centering\n   \n   \\includegraphics[width=.450\\textwidth]{norotation.jpg}\n   \\includegraphics[width=.450\\textwidth]{rotation0.1octupole.jpg} \n   \\includegraphics[width=.450\\textwidth]{rotation0.2octupole.jpg}\n   \\includegraphics[width=.450\\textwidth]{rotation0.3octupole.jpg}\n   \\includegraphics[width=.45\\textwidth]{rotation0.4octupole.jpg}\n   \\includegraphics[width=.45\\textwidth]{rotation0.5octupole.jpg}\n \n \\caption{{\\bf Evolution of the stellar disc.} The background cluster is initially rotating along the $z$ axis with the dimensionless rotation rate $\\tilde{\\gamma}$ defined in the text and specified at the top of each subfigure. The counter-rotating disc is injected at $t=0$. The figure shows some randomly chosen examples of the time evolution of $n_z$, where $\\vec{n}$ is the unit vector directed along the star's angular momentum,  for all the stars in the disc. The red lines are marking the inner half of the disc's stars and the green lines are marking those the outer half. As the rotation rate increases, we observe the disruption of the outer parts of the disc and tendency for the inner disc to align with the cluster. At high rotation rate, the disc gets warped or  broken into transient rings. These effects are very important at $5$Mys, the likely age of the young stellar population in the Galactic Center.\n\n \\small\n }\n\\end{figure*}\n\\newline\\newline\n\\section{The stellar disc inside a rotating cluster}\n\\subsection{The results of numerical experiments}\nWe model only the inner part of the central cluster, which in our simulations consists of $1.2\\times 10^5$ solar-mass stars, located inside the central $0.4$pc with the spacial number density of stars given by the Peebles-Young distribution,  $n(r)\\propto r^{-1.5}$. The cluster, if it were extended out to $1$pc, would contain about half a million solar masses. The numbers are  based on \\cite{2007A&A...469..125S}, but it is questionable how faithfully they represent the distribution of dynamical scatterers in the Galactic Center, since (a) there seems to be a hole in the stellar distribution in the central $\\sim 0.2$pc \\citep{2009A&A...499..483B, 2009ApJ...703.1323D, 2010ApJ...708..834B}, and (b) stellar-mass black holes are expeceted to segregate inside central $0.1$pc \\citep{2000ApJ...545..847M, 2009ApJ...697.1861A}. \n\n\nDespite these uncertainties, we believe that  the numbers are correct to within an order of magnitude. Our young massive disc consists of $105$  stars  of $80 M_\\odot$ each. It is the total mass of the disc that plays key role, and we get similar results if we increase the number of stars and decrease their masses, while keeping the total mass constant. The disc's surface density is $\\Sigma\\propto r^{-2}$ \\citep{2006ApJ...643.1011P} and its outer edge is at $0.4$pc.\n\nIn our numerical experiments, we sacrifice the precision of individual interactions between stellar orbits in favour of extremely rapid speed and ease with which we can explore physical effects. The fastest dynamics responsible for the time evolution in the orientation of  orbital planes, is captured when the precessing orbits are replaced with the annuli that they trace; this type of relaxation is called ``vector resonant relaxation\" \\citep{1996NewA....1..149R}. We however, replace each annulus with a circular ring of comparable radius, and  keep only quadrupolar and octupolar terms in the interaction potential between the rings. Furthermore, instead of considering a continuum of the ring radii, we choose a modest number of discreet radius values, i.e., ``the bins\" ($10$ for the computations shown in Figure 1). Our computational vectorization benefits from each bin having an equal number of stars, including the stars that belong to the disc; there is no restriction on the stars' individual masses. The requirement that the overall distribution (approximately) represents $n(r)\\propto r^{-1.5}$  determines the value of the radius assigned to each bin. The main advantage of the binning is that it allows us to perform $\\sim N_{s}$ instead of $\\sim N_s^2$ operations per each step. This point, together with other details of our numerical procedure, is explained in the Appendix. The commented Matlab code is available upon request.\n\nEach simulation runs models $3\\times 10^7$ years of evolution, on a usual desktop computer. It takes several minutes to complete if only quadrupolar terms are taken into account, and several hour if the octupole terms are included\nWe use the \"octupolar\" simulations for the production runs, and the ``quadrupolar'' ones as a super-fast way to explore the parameter space. The only significant difference is that the inner half of the disc tends to remain somewhat more coherent if the octupolar terms are included. This allows us to quickly change the parameters of the system, and explore the convergence and stability of the results. The results of the numerical simulations are illustrated in Figure 1. The disc is injected in counter-rotation with respect to the cluster, so that the $\\vec{n}_i=\\vec{l}_i\/l_i$ are clustered in the southern hemisphere relative to the cluster rotation; here $\\vec{l}_i$ is the angular momentum of the $i$'th star. If the cluster is non-rotating, the disc stays coherent even though its bending modes are excited, consistent with the findings of \\cite{2011MNRAS.412..187K,2015MNRAS.448.3265K} and, most convincingly, with the direct simulations of \\cite{2022MNRAS.tmp.2848P}. For greater rotation rates, we see that within several million years, the  orbits of the inner half of the stars (represented by the red lines) are dragged towards corotation with the cluster, while the outer stars get dispersed. For more rapidly rotating clusters we observe the disc being warped or split into distinctly oriented rings, which are seen as coherent groups of the $\\vec{n}$-vectors. This complexity is consistent with the phenomenology seen in the galactic center.\n\nWhat is the rotation rate of the cluster?\nIn the next subsection we mathematically define the dimensionless rotation rate, and show how to estimate it from the radial velocity data. Using existing literature, we obtain a  rough estimate of $\\tilde{\\gamma}\\sim 0.3$.\n\n\\subsection{The rotation rate of the relaxed cluster}\n\n Inside the gravitational radius of influence of a supermassive black hole, the orbit-averaged torques between the stars moving on slowly-evolving elliptical orbits, drive a fast stochastic evolution of the stars' inclinations and eccentricities \\citep{1996NewA....1..149R, 2018ApJ...860L..23B}. This is known as Resonant Relaxation. Importantly, the orbit-averaged dynamics leaves the semimajor axes unchanged. As argued in the original discovery paper, the relaxed state should be in a statistical  equilibrium, with the probability distribution function  proportional to the appropriate Boltzmann weight:\n\\begin{eqnarray}\n    P(m, a, l, l_z,\\omega,\\Omega)&=&N(m, a)\\times\\nonumber\\\\\n    & &\\exp\\left[-m\\left(\\beta\\epsilon-\\vec{\\gamma}\\cdot\\vec{l}~\\right)\\right].\n    \\label{Prob}\n\\end{eqnarray}\nHere $m$ is the mass and $a$ is the semimajor axis. On the left-hand side of the equation, the two Delaunay actions $l, l_z$ are the magnitude and the z-component of the specific angular momentum vector $\\vec{l}$, and the two Delaunay angles $\\omega, \\Omega$ are the argument of periapsis and the longitude of ascending node.  On the right-hand side $N(m,a)$ is the normalization factor that will play no role in the following discussion,  and $\\epsilon$ is the specific orbit-averaged potential energy of the star's interaction with the other stars in the cluster. The inverse temperature $\\beta$ can be either positive, zero or negative, and $\\vec{\\gamma}$ is the rotational vector of the cluster; $\\vec{\\Omega}_{\\rm cl}\\equiv \\vec{\\gamma}\/\\beta$ is known as a thermodynamic angular velocity of the cluster, and it coincides with the angular velocity of the cluster's precession if the cluster is lopsided (e.g., Gruzinov et al.~2020). We can define the dimensionless rotation rate of the cluster as follows:\n\\begin{equation}\n    \\tilde{\\gamma}=m_0 l_0 \\gamma,\n    \\label{tildegamma}\n\\end{equation}\nwhere $m_0$ is the characteristic mass of a star in the cluster and $l_0$ is the characteristic specific angular momentum of a stellar orbit in the cluster. We choose $m_0=M_\\odot$ and $l_0=\\sqrt{GMr_0}$, where $M$ is the mass of the supermassive black hole and $r=0.1$pc, the characteristic radius of the stellar disc. It is this rotation parameter that labels the plots in Figure 1.\n\nA number of authors have explored thermodynamic equilibria of this form \\citep{1996NewA....1..149R,2014JPhA...47C2001T, 2015MNRAS.448.3265K, 2017ApJ...842...90R, 2018PhRvL.121j1101S, 2019PhRvL.123b1103T, 2020MNRAS.493.2632T, 2020ApJ...905...11G, 2022MNRAS.514.3452M, 2022arXiv220207665M}.\nWe now show that for a slowly-rotating edge-on cluster, Eq.~(\\ref{Prob}) together with some reasonably natural assumptions leads to a simple relationship\nbetween the mean radial velocity and its dispersion, both measured from, e.g., pixel-integrated spectroscopy. This relationship can be used to estimate the rotational velocity of the relaxed cluster (or infer it, if the data is detailed enough).\n\nConsider a cluster whose axis of rotation $z$ lies in the plane of the sky; thus $\\vec{\\gamma}=\\gamma \\hat{z}$. We shall assume that the cluster has a rotational symmetry about this axis. The Milky Way nuclear star cluster's rotation axis is perpendicular to the galaxy \\citep{2008A&A...492..419T,2014A&A...570A...2F}, and the cluster appears to be axially symmetric in its inner parts despite some inferred triaxiality in its outer parts \\citep{2017MNRAS.466.4040F}.  Let the $x$-axis be directed away from the observer along line of sight to the cluster, and let the origin of the $x, y, z$ coordinate system coincide with the center of the cluster. Each line of sight is characterized by coordinates $y,z$. The mean line-of-sight radial velocity is given by\n\\begin{equation}\n    \\langle v_r(y,z)\\rangle=\\int  P(\\vec{r}, \\vec{v})~v_x~dv_x~dv_y~dv_z~dx,\n    \\label{radvel}\n\\end{equation}\nwhere $P(\\vec{r}, \\vec{v})$ is the probability distribution function for a star moving with velocity $\\vec{v}$ to be located at position $\\vec{r}$. The latter can be obtained by using Eq.~(\\ref{Prob}) and expressing all the Delaunay variables in terms of $\\vec{r}, \\vec{v}$ (this is true because the Jacobian of a canonical transformation equals $1$).  Consider a reflection \n\\begin{eqnarray}\n    x&\\rightarrow&-x\\nonumber\\\\\n    v_x&\\rightarrow&-v_x\\label{symmetry}\n\\end{eqnarray}\nabout the plane of the sky. This leaves $a,l,\\epsilon$ unchanged, but flips the sign of $\\vec{\\gamma} \\cdot \\vec{l}$. Therefore,\n\\begin{eqnarray}\nP(-x,y,z,-v_x, v_y, v_z)&=&\\exp\\left[-2 m\\gamma(y v_x-x v_y)\\right]\\times\\nonumber\\\\\n  & &P(x,y,z,v_x,v_y, v_z).\\label{flip1}\n\\end{eqnarray}\nWe assume that the cluster is rotating slowly, with $m \\gamma l\\ll 1$\nWe therefore expand $\\exp\\left[-2 m\\gamma(y v_x-x v_y)\\right]\\simeq 1-2 m\\gamma(y v_x-x v_y)$. Multiplying the above equation by $v_x$ and integrating over the velocities and over $x$, we obtain the following relation:\n\\begin{equation}\n    \\langle v_r \\rangle -m \\gamma y \\langle v_r \\rangle^2=m\\gamma \\left(y\\sigma_r^2-\\langle x T_{xy}\\rangle\\right).\n    \\label{relation1}\n\\end{equation}\nHere $\\sigma$ is the radial velocity dispersion, $T_{ij}=\\overline{v_i v_j}$ is the velocity tensor, and $\\langle \\rangle$ stands for the average along the line of sight. Solving for ${\\gamma}$ and using Eq.~(\\ref{tildegamma}), and assuming that $m=m_0$ is the typical mass of the observed stars, we get\n\\begin{equation}\n    \\tilde{\\gamma}={l_0 \\langle v_r\\rangle\\over y(\\sigma_r^2+\\langle v_r\\rangle^2)-\\langle x T_{xy}\\rangle}.\n\\end{equation}\nIn the above equation everything is measurable except $\\langle x T_{xy}\\rangle$, since even if $v_y$ for individual stars could be measured, we would have no information about  $x$. Moreover, this term is in fact non-zero for anisotropic velocity ellipsoid and could be similar to $y \\sigma_r^2$ in magnitude. Note however, $T_{xy}=0$ for an isotropic velocity distribution, which is expected to be produced by scalar resonant relaxation near the black hole\\footnote{The prediction from the Resonant Relaxation is that the closer to the black hole, the more isotropic is the velocity ellipsoid.}. We shall assume this; in principle this assumption could be tested for consistency by checking that so-measured $\\tilde{\\gamma}$ is pixel-independent\\footnote{\\cite{2017MNRAS.466.4040F} performed orbit-modelling of the whole nuclear cluster out to $\\sim 8$pc, and found that the anisotropy of the velocity ellipsoid is significant in the intermediate range of radii around $\\sim 1$pc, but smaller at greater or much smaller radii. The inference is clearly not very precise in the regions where the stellar disc is located.}. We expect the result thus obtained to be correct to within a factor of $\\sim 2$.\n\nThe data in \\cite{2014A&A...570A...2F} is quite noisy at distances of interest. From Figure $11$ of that paper, the inner bin at $10$ arcsec (which corresponds to $\\sim 0.4$pc) has $\\langle v_r\\rangle\\sim 25 $km\/sec and $\\sigma\\sim 85$km\/sec. Plugging the numbers in the above equation, we get \n\n\\begin{equation}\\tilde{\\gamma}\\sim 0.3\\end{equation}. \n\nWe emphasize that this number is only an order of magnitude estimate and therefore we explore a range of values, as indicated in Figure $1$.\n\n\n \n\n\n\n\n\n \\section{Analytical estimate of Resonant Friction timescale}\nResonant friction was first discussed as a phenomenon by \\cite{1996NewA....1..149R}, as a dissipative counterpart to the stochastic resonant relaxation. Both are necessary for the thermal equilibrium to be established. The origin of the friction can be understood as follows. Consider the probability distribution in Eq.~(\\ref{Prob}) for high masses, $m\\gg m_0$. A star with such mass will tend to be near the orbit that minimizes the Jacoby constant $\\epsilon-\\vec{l}\\cdot\\vec{\\gamma}\/\\beta$. \\cite{2020ApJ...905...11G} studied such orbits and showed that they are stationary in the frame of reference that is rotating with the angular velocity $\\vec{\\gamma}\/\\beta$, and are typically aligned with the cluster's rotation. This occurs despite stochastic Vector Resonant Relaxation torques that are perturbing the massive orbit, because Resonant Friction {\\it drives} massive objects in a cluster towards these special orbits. The existence of Resonant Friction is thus \nclosely connected to the existence of thermodynamical equilibrium.\n\nResonant Friction explains why in numerical simulations of an Intermediate-Mass Black Hole inspiraling through a nuclear cluster, the former's orbit rapidly orients itself with the cluster's rotation. \\cite{2012ApJ...754...42M} demonstrated explicitly in numerical experiments that this reorientation occurs because of secular torques, but they did not make the connection to thermodynamics (the community has been somewhat reluctant to accept Madigan \\& Levin's arguments, instead attributing the reorientation to the $2$-body scattering processes).\n\nIt is the balance between the fluctuations and the dissipation that establishes the Boltzmann distribution. Below we use this fact to estimate the dissipation timescale in a rotating cluster.\nFirst, we observe that the $z$-component of the angular momentum of an object inside the cluster experiences a. stochastic walk, due\nto random torques from other orbits, and b. systematic drift upwards, which tends to align the orbit with the\ncluster rotation. The corresponding evolution equation\nfor the $l_z$-distribution $f(m,l_z,t)$ can be written as\n\\begin{equation}\n    {\\partial f\\over \\partial t}=-{\\partial\\over \\partial l_z}\\left[F_{\\rm drift}+F_{\\rm stochastic}\\right].\n   \n    \\label{evolution}\n\\end{equation}\n\n\n\nHere $m$ is the mass of a star, and $l_z=L_z\/m$ where $L_z$ is the $z$-component of its angular momentum, and $F_{\\rm drift}$ and $F_{\\rm stochastic}$ are the fluxes in $l_z$-space due to the resonant friction and stochastic resonant relaxation, respectively.\nThe Fokker-Planck form of the fluxes is given by\n\\begin{eqnarray}\n   F_{\\rm drift}(m,l_z)&=&V(m,l_z)~f(m,l_z)\\nonumber\\\\\n   F_{\\rm stochastic}(m,l_z)&=&-{\\partial\\over\\partial l_z}\\left[ D(l_z) f(m,l_z)\\right]\\label{fluxes}\n\\end{eqnarray}\nThe drift velocity $V(m,l_z)$ is mass-dependent, while the stochastic diffusion coefficient $D(l_z)$ is mass-independent because of the equivalence principle (the torque per mass on the orbit from the other stars depends only on the orbit, and not on the mass of the star).\nMoreover, since flipping the direction of the angular momentum $\\vec{l}\\rightarrow-\\vec{l}$ does not charge the period-averaged orbit,\nwe must have $D(l_z)=D(-l_z)$.\n\nFor slowly rotating, nearly spherically symmetric clusters $\\epsilon$ depends only weakly on the orbit's orientation. Therefore,  the equilibrium distribution is given by\n\\begin{equation}\n    f(m,l_z)=f_0(m)\\exp(m\\gamma l_z),\n    \\label{equilibrium}\n\\end{equation}\nwhere $\\vec{\\gamma}=\\gamma~\\hat{z}$ is the rotational vector of the cluster. In equilibrium, $F_{\\rm drift}+F_{\\rm stochastic}=0$. Therefore,\nsubstituting Eq.~(\\ref{equilibrium}) into Eq~(\\ref{fluxes}), we obtain the relationship for the  drift velocity\n\\begin{equation}\n    V(m,l_z)=V(0,l_z)+m\\gamma D(l_z).\n\\end{equation}\nThe second term of the right-hand side is the resonant-friction induced\npart of the drift velocity. The first term\n\\begin{equation}\n    V(0,l_z)=D^{\\prime}(l_z)\n\\end{equation}\nis the drift velocity of the zero-mass particles, required to enforce their\nfully isotropic equilibrium distribution. The parity symmetry of $D(l_z)$ results in $V(0,0)=0$, and thus\n\\begin{equation}\n    V(m,0)=m\\gamma D(0),\n\\end{equation}\nwhich establishes a key relationship between $V$, $\\gamma$, and $m$.\nIt allows us to relate the frictional timescale $t_{\\rm fr}$ to that of the vector resonant relaxation, $t_{\\rm VRR}$. We have\n\\begin{eqnarray}\n   t_{\\rm fr}&\\sim & l_0\/V\\sim l\/(m\\gamma D),\\nonumber\\\\\n   t_{\\rm VRR}&\\sim& l_0^2\/D,\\label{timescales}\n\\end{eqnarray}\nwhere like in the previous section, $l_0$ is the characteristic specific angular momentum of an orbit in the cluster. Thus we have\n\\begin{equation}\n    t_{\\rm fr}\\sim \\tilde{\\gamma}^{-1} {m_0\\over m} t_{\\rm VRR},\n    \\label{tfr1}\n\\end{equation}\nwhere $m_0$ is the mass of a typical star in the cluster and like before, the the dimensionless rotation parameter is given by\n\\begin{equation}\n    \\tilde{\\gamma}=m_0\\gamma l_0.\n\\end{equation}\n\nSo what is $t_{\\rm VRR}$? Following the arguments of \\cite{1996NewA....1..149R}, one can show that\n\\begin{equation}\n    t_{\\rm VRR}\\sim {M_{\\rm BH}^2\\over M_* m_0} {P^2\\over t_{\\rm coh}},\n\\end{equation}\nwhere $M_{\\rm BH}$  is the mass of the central black hole, $M_*$ is the mass of the stellar cluster, $P$ is the characteristic orbital period, and  $t_{\\rm coh}$ is the characteristic coherence timescale for the fluctuating torque. \\cite{1996NewA....1..149R} took $t_{\\rm coh}\\sim t_{\\rm VRR}$, arguing that VRR is the main mechanism for the change in orientation of the orbits. Thus they obtain\n\\begin{equation}\n    t_{\\rm VRR}\\sim {M_{\\rm BH}\\over \\sqrt{M_* m_0}}P.\n    \\label{tVRR1}\n\\end{equation}\nSubstituting this into Eq.~(\\ref{tfr1}), we obtain\n\\begin{equation}\n    t_{\\rm fr}\\sim {M_{\\rm BH}~P\\over \\tilde{\\gamma}\\sqrt{M_* m_0}}.\n    \\label{tfr2}\n\\end{equation}\nHowever, this argument has a serious limitation. If a cluster is rotating, its stellar distribution is flattened, with the ellipticity $\\sim \\tilde{\\gamma}^2$. This ellipticity drives precession of the stellar orbits with characteristic timescale\n\\begin{equation}\n    t_{\\rm prec}\\sim {M_{\\rm BH}\\over M_*}\\tilde{\\gamma}^{-2}P.\n\\end{equation}\nThis timescale becomes comparable the one given in Eq.~(\\ref{tVRR1}) for $\\tilde{\\gamma}\\sim N_s^{-1\/4}\\sim 0.1$, where $N_s=M_*\/m_0$ is the number of stars in the cluster. Therefore for $\\tilde{\\gamma}\\gg N_s^{-1\/4}$, we should consider $t_{\\rm coh}\\sim t_{\\rm prec}$. In that case,   we get\n\\begin{equation}\nt_{\\rm VRR}\\sim {M_{\\rm BH}\\over m_0}\\tilde{\\gamma}^2 P.\n\\end{equation}\nSubstituting this into Eq.~(\\ref{tfr1}), we get\n\\begin{equation}\nt_{\\rm fr}\\sim \\tilde{\\gamma} {M\\over m} P.\n\\label{tfr3}\n\\end{equation}\nNote that this does not depend on the cluster mass $M_*$. To sum up: if $\\tilde{\\gamma}\\ll N_s^{-1\/4}$, one should use Eq.~(\\ref{tfr2}); in the opposite case, one should use Eq.~(\\ref{tfr3}). Clearly, these expressions are very approximate, and one needs more precise arguments which are beyond the scope of this paper,\nto obtain more reliable expressions with more precisely stated domains of validity. \n\nHaving derived the Resonant Friction timescale for a stellar orbit of mass $m$, we note that the effect should be there for any massive object inside the cluster that is gravitationally coherent. Thus it should apply to a disc of stars, for as long as the orbital planes of the stellar orbits remain clustered. We thus apply the expression above to the whole  stellar disc. Taking $m=M_{\\rm disc}=8000 M_\\odot$, $M_{\\rm BH}=4\\times 10^6 M_\\odot$, $\\tilde{\\gamma}=0.3$, and $P=1500$yr (corresponding to a circular orbit at $0.1$pc), we get \n$t_{\\rm fr}\\sim 2.5$Myrs. This is consistent with the results of our numerical experiments described in Section 2.\n\n\n\n\\section{Alignment of accretion discs with the nuclear cluster rotation}\n\\begin{figure*}[t]\n   \\centering\n    \\epsfxsize=16cm \n    \\epsfbox{scenario2.jpeg}\n \n \\caption{\n\n \\small\nPictorial description of the scenario described here. An infalling cloud of gas is tidally disrupted and its material forms an accretion disc around the SMBH. The initial orientation of the disc is random, but the Resonant Dynamical Friction drives the disc's angular momentum $\\vec{L}_{\\rm disc}$ into alignment with that of the cluster, $\\vec{L}_{\\rm cluster}$.  }\n   \\end{figure*} \n \n\\subsection{General remarks}\nA classic argument by \\cite{1982MNRAS.200..115S}, \nand its more recent elaborations \n\\citep{2002MNRAS.335..965Y, 2012AdAst2012E...7K}, \nstrongly suggest that supermassive black holes \n(SMBH) \nacquire most of their mass by accreting gas from thin discs in galactic nuclei. If the orientation of an accretion disc\nrelative to the black hole could be maintained, this mode of accretion would drive the black hole to a  very high spin, with dimensionless spin parameter $\\alpha>0.9$  \n\\citep{1973blho.conf..343N}. \nThe discovery of \n\\cite{1975ApJ...195L..65B} \nthat gravito-magnetic forces drive an accretion\ndisc into alignment with the equatorial plane near the black hole, would suggest that we could expect essentially all supermassive black holes to be rapidly spinning. Measurements  of  x-ray  and  radio  emission  from accreting supermassive black holes in galactic nuclei indicate that they are indeed rotating rapidly \\citep{2020arXiv200808588J,2019ApJ...886...37D}. The spin energy of SMBHs is thought to be responsible for powering relativistic jets emanating from galactic nuclei, and  thus SMBH spin is the key agent of feedback during the formation of elliptical galaxies. \n\nHowever, much remains to be understood about the spins of SMBHs. The measurements of the black hole spins rely on rather complex models of accretion discs and jets, and thus may contain systematic uncertainties. On the theoretical side,\n\\cite{2006MNRAS.373L..90K} \nargued that the accretion is likely to be driven by randomly-oriented infall episodes; \nthis is supported observationally by the fact that many radio jets seem to be randomly oriented relative to their host spiral galaxies. In this stochastic-accretion picture, one expects  nearly half of the transient accretion discs to be counter-rotating relative to the black hole. \nImportantly, a counter-rotating innermost stable orbit has a greater radius than a co-rotating innermost stable orbit, and therefore the counter-rotating discs have a larger lever arm. \\cite{2008MNRAS.385.1621K} argued that as a result  of this, the supermassive black holes are spun down, on average, to low spins of $\\alpha\\sim 0.3$. A more elaborated model of \\cite{2018MNRAS.477.3807F} makes a distinction between lower-mass black holes of $M_{\\rm BH}\\lesssim 10^7M_\\odot$ and the more massive ones. These authors argue that despite of the stochastic feeding, the spin directions of the lower mass black holes are expected to remain adiabatically aligned with the\nangular momentum directions of their accretion discs, while the higher-mass black holes might experience both spin-ups and spin-downs in equal measure.  \n\nLISA will provide exquisitely precise measurements of the supermassive black hole spins, by measuring the gravitational waves from mergers of  stellar-mass and intermediate-mass black holes with SMBHs. It is thus imperative to develop reliable theoretical predictions for the SMBH spins.\nBelow we argue that a key piece of physics is missing in the current theoretical analyses, namely the stabilizing effect of rotating black hole clusters on the orientation of the accretion disc. \nIndeed if the stellar disc in our Galactic Center gets reoriented into alignment with the cluster, why would not an accretion disc in a galactic nucleus which is more active than ours? \n\n\n\\subsection{Resonant friction on accretion discs}\nThe scenario we consider is depicted in Figure 2. In its first stage, a cloud of gas, with the mass of $\\sim 10^4M_\\odot$ falls in, gets tidally disrupted and forms a gaseous accretion disc. \nIf the cluster is  flattened due to its rotation, it will  exert a gravitational torque on the\ndisc, causing it to precess. The differential precession could distort the disc. The hydrodynamics of distorted discs could be complicated, but it is reasonable to expect that after much dissipation from shocks etc., the disc would reassemble in the equatorial plane of the cluster. Naively it would seem that the disc could equally likely be co-rotating and counter-rotating with the cluster.\n\nHowever, the resonant friction is expected to break the symmetry between co- and counter-rotation, and drive the disc's and the cluster's angular momenta into alignment. \nResonant friction should act on any massive object inside the cluster, in particular it should affect the accretion disc, so long as the disc remains coherent and creates a gravitational perturbation that affects the cluster. Whether the disc remains coherent long enough to experience the effect is the key question.\nIn absence of a full hydrodynamical treatment, our intuition can be guided by simulations of stellar disc evolution inside a rotating cluster, such as the one illustrated in Figure $1$. \nThe initial disc remains coherent enough, until its overall orbital angular momentum flips into co-rotation with the cluster.  If instead of the stellar disc we introduced a gaseous disc of the same mass, it would\nfirstly flip into co-rotation with the cluster, and then settle into the equatorial plane of the cluster due to the hydrodynamic torques that are caused by the disc's differential precession.\n\nIf this picture were correct (and it obviously needs to be tested by hydrodynamic simulations!), then the rotating clusters could serve as\na stabilizing flywheel for the material that accretes onto the SMBH. It would imply that\n\\begin{itemize}\n    \\item The SMBH spin direction is aligned with that of its host cluster\n    \\item The SMBH spin magnitude could reach a very high value, unless there exists an as yet undetermined mechanism for the SMBH spindown (such as its super-radiant coupling to a scalar field or a cosmic string)\n    \\item Since the resonant friction from cluster rotation will affect inspiraling Intermediate-Mass \n    Black Holes, one expects a non-trivial alignment between the inspiraling orbit and the SMBH spin. Such an alignment will be  detectable by LISA.\n\\end{itemize}\n\nIn proposing hydrodynamic numerical experiments, we need to ascertain that the scenario is reasonable by comparing the resonant friction timescale with that of the dissolution of the disc due to a differential precession. \nAs argued in the previous section, the timescale for the friction to align the disc is\n\\begin{equation}\n    t_{\\rm fr}\\sim  \\tilde{\\gamma} ~{M_{\\rm BH}\\over M_{\\rm disc}}~P,\n\\end{equation}\nthis equation is valid so long as $\\tilde{\\gamma}>N_s^{-1\/4}$, where $N_s$ is the number of stars in the cluster. Note that this timescale is comparable to the period of a bending mode of the disc due to its self-gravity, $t_{\\rm bend}\\sim (M_{\\rm BH}\/M_{\\rm disc}) P$. Therefore the reorientation of the disc due to Resonant Friction is expected to be accompanied by  excitation of the bending motion of the disc. This explains the excitation of the\ndisc warps,  and the disc's breaking into several rings in some of our simulations.\n\nNote also that the mass of the cluster, assumed to satisfy $M_{\\rm disc}<M_{*}<M_{\\rm BH}$, does not enter into the expression above; this is because the cluster contributes to the timescale in two distinct ways, and the contributions cancel each other. On the one hand, the more stars are affected by the disc, the stronger is their gravitational back-reaction on the disc. On the other hand, the rate of precession of the orbital angular momenta of the cluster\nstars is  $\\propto M_*$, and a fast rate of precession is suppressing the cluster's gravitational response to the disc. The rate of differential precession is given by\n\\begin{equation}\n    t_{\\rm prec}\\sim \\tilde{\\gamma}^{-2} ~ (M_{\\rm BH}\/M_{*})~P.\n\\end{equation}\nComparing the two timescales,\n\n\\begin{equation}\n    {t_{\\rm fr}\\over t_{\\rm prec}}\\sim 0.5 \\left({\\tilde{\\gamma}\\over 0.3}\\right)^3~{M_{*}\\over 15 M_{\\rm disc}}.\n\\end{equation}\n\\newline\nIn the numerical example above, we used the parameters relevant for the Galactic Center stellar disc, with $M_*$ being the mass of the stars within the disc's outer radius.\nThe estimate suggests that for a realistic parameter range the disc alignment is promising. Neglected in this estimate is are the effects of Reynolds stresses,  which are expected to enhance the disc's coherence and thus increase the robustness of alignment. We thus conclude that the idea deserves further theoretical investigation,  with numerical experiments that involve both stellar and gas dynamics. \n\n\n\n\n\\section{Discussion}\nRotating nuclear star clusters in galactic nuclei are giant cosmic centrifuges. Any heavy object inside the cluster (an IMBH or a massive stellar or gas disc) feels a very strong Resonant Friction torque that tends to align its orbit with the cluster's rotation. This torque can (partially) disrupt an impulsively injected disc, and we have given strong arguments that this is what happened with the young stellar disc at the center of the Milky Way. We speculate that this process is also efficient for stochastically injected gaseous accretion discs. If future numerical experiments confirm our speculation, then there would be  reason to believe that the direction of spin of the supermassive  black hole could be anchored to that of its host cluster. Contrary to the arguments of \\cite{2008MNRAS.385.1621K}, the episodes of successive accretion would then allow the black hole to acquire high rotation rate. We must note, however, that the SgrA* shadow observations by the EHT collaborations, disfavour the edge-on spin of the supermassive black hole \\citep{2022ApJ...930L..12E}. These interesting, albeit model-dependent measurements need to be considered as the models of the disc-cluster interaction become more reliable.\n\nThe result for the Milky Way's young stellar disc has special significance. Its partially disrupted state has been a puzzle since its discovery, and several authors including the current one appealed to massive perturbers to produce the necessary gravitational violence \\citep{2005ApJ...635..341L, 2020ApJ...905..169Z, 2021ApJ...919..140S}. This paper demonstrated that hypothetical massive perturbers are unnecessary for explaining the observations.\n\nThis work has several technical shortcomings that will need to be improved upon in future work. In our simulations we considered only low-order multipole interactions, which are likely adequate for the study of the long-range torques responsible for resonant friction, but perhaps underestimate the stochastic torques experienced by the stars, which may negatively impact the coherent parts of the inner disc. Definitive confirmation should come from a direct (and extremely expensive) $N$=body simulations, similar to the ones performed in \\cite{2022MNRAS.tmp.2848P}, but modified to include  rotation of the host cluster. Resonant friction in rotating clusters deserves a more rigorous analytical treatment than the scaling arguments given here. Gas dynamics of accretion discs warped by resonant friction torques is an interesting topic of future study.\n\nWe thank Andrei Gruzinov and Scott Tremaine for insightful discussions, and for raising important conceptual and practical points after reading the first draft of this paper. We thank Sophie Koudmani for explaining the current theoretical models for the black-hole spin evolution, and Sarah Levin for advice on the prose. This research is supported by the Simons Investigator Grant 827103.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCharacteristic classes are cohomology classes associated to isomorphic classes of  vector bundles \\cite{Hirzebruch,Milnor,Fulton.Intersection}. They measure how a vector bundle is twisted or non-trivial.   Characteristic classes of a nonsingular variety are defined via its tangent bundle. \n\n\nCharacteristic classes are instrumental in many questions of geometry and theoretical physics. In string theory  and in  supergravity theories, characteristic classes appear in discussions of anomaly cancellations \\cite{Gaume,AlvarezGaume:1983ig} and tadpole cancellations \\cite{Sethi:1996es,CDE,Denef:2008wq}, in the computations of the index of supersymmetry operators, and in the definition of the charges of D-branes \\cite{Minasian:1997mm,AE1} and orientifold planes \\cite{Scrucca}.  \n\n\n\nThe aim of this paper is to compute the characteristic numbers of elliptic fibrations  that are crepant resolutions of singular Weierstrass models given by the output of Tate's algorithm \\cite{Tate}. Such elliptic fibrations are called  $G$-models.\n\n\n\\begin{defn}[Elliptic fibrations]\n A variety is said to be an elliptic $n$-fold if it is endowed with a proper surjective morphism $\\varphi:Y\\to B$ to a variety of dimension $n-1$  such that the generic fiber of $\\varphi$ is a smooth projective curve of genus one, and $\\varphi$ has a rational section. \n\\end{defn}\n\n \\begin{defn}[$G$-models]\n Let  $G$ be a  simple, simply-connected compact complex Lie group with Lie algebra $\\mathfrak{g}$.  A $G$-model is an elliptic fibration $\\varphi:Y\\to B$ with a  discriminant locus  containing an irreducible component $S$ such  that \n \\begin{enumerate}\n \\item the generic fiber over any other component of the discriminant is irreducible (that is, of Kodaira type I$_1$ or II), \n \\item the fiber over  the generic point of $S$ has a dual graph that becomes of the same type \n as the Dynkin diagram of the Langlands dual of $\\mathfrak{g}$ after removing the node corresponding to the component touching the section of the elliptic fibration. \n \\end{enumerate}\n \\end{defn}\n \n\n\n$G$-models are used to geometrically engineer gauge theories in compactifications of M-theory and F-theory \\cite{Vafa:1996xn,Morrison:1996na,Bershadsky:1996nh,Katz:2011qp}.\nThey also play an important role in studying superconformal gauge theories (for a review see \\cite{Heckman:2018jxk} and reference therein). \n $G$-models are typically defined by crepant resolutions of singular Weierstrass models given by Tate's algorithm and characterizing  a specific (decorated) Kodaira fiber \\cite{Andreas:2009uf,G2,Euler,EK1,SUG,EKY,EKY2,ES,ESY1,ESY2,EY,Lawrie,Mayrhofer:2014opa}. \n\n\nWe will restrict ourselves to the case of elliptic fibrations with a unique divisor over which the generic fiber is reducible. We also consider only elliptic fibration of complex dimension four with a trivial Mordell-Weil group. \nThat corresponds to groups $G$ that are simple and simply-connected. We note that our techniques apply without subtleties to the cases with non-trivial Mordell--Weil groups or to cases where $G$ is semi-simple. \nThe elliptic fibrations that we analyze are all given explicitly by crepant resolutions of singular Weierstrass models as discussed in section \\ref{sec:GModels}. \n\n\nEuler characteristics of $G$-models have been computed recently in \\cite{Euler}. We continue the work started in \\cite{Euler} by providing additional characteristic numbers  of $G$-models.\nIn our analysis,  we do not impose the Calabi-Yau condition since it will reduce the problem  to a computation of Euler characteristics, which is already addressed in \\cite{Euler}. \nThe Calabi-Yau fourfold case is reviewed  in section \\ref{sec:CY4}.  We also point out integrality issues for the invariant $\\chi_1(Y)$ in the  cases of SU($5$), SU($6$), and E$_6$ in section \\ref{Sec:Oddities}. \nWe  compute the characteristic numbers by explicitly performing pushforwards after correcting the Chern classes to take into account the effect of the sequence of blowups necessary to define the crepant resolution. \n\nThe theory of characteristic classes was founded in the 1930s and 1940s by  Whitney,  Stiefel, Pontryagin, and Chern \\cite{Hirzebruch,Milnor}. \nTo these days,  the most famous characteristic numbers are the Stiefel-Whitney numbers, the Pontryagin numbers, the Chern numbers, together with the Euler characteristic, which is the oldest topological invariant. \n The theory of characteristic classes relies deeply on sheaf theory as developed by  Kodaira, Spencer, and Serre; and historically, also  on Thom's theory of cobordism.  \nGrothendieck introduced an axiomatic definition of Chern classes in the Chow ring of a variety using projective bundles.   \nThis establishes characteristic classes as familiar objects in intersection theory \\cite[Chap 3]{Fulton.Intersection}. \nIn a sense, any natural transformation from the complex vector bundles to the cohomology ring is a polynomial in the Chern numbers. In his seminal book, Hirzebruch expresses all characteristic classes in terms of Chern classes \\cite{Hirzebruch}.\n\n\\subsection{List of characteristic numbers}\n\nWe compute  the following six types of rational Chern and Pontryagin numbers for each $G$-models:\n\\begin{enumerate}\n\\item The Chern numbers \n\\begin{equation}\n\\text{$\\int_Y c_1(TY)^4$, $\\int_Y c_1(TY)^2 c_2 (TY)$, $\\int_Y c_1(TY) c_3 (TY)$, $\\int_Yc_2^2(TY)$,  and $\\int_Y c_4 (TY).$}\n\\end{equation}\n\\item The holomorphic genera  $\\chi_p(Y) = \\sum_{q=0}^n (-1)^q h^{p, q}(Y)$ \\cite{Klemm:1996ts}:\n  \\begin{equation}\n  \\begin{aligned}\n  \\chi_0(Y)&=\\int_Y \\mathrm{Td}(TY)=\\frac{1}{720}\\int_Y (-c_4 + c_1 c_3 +3 c_2^2 + 4 c_1^2 c_2 -c_1^4),\\\\\n  \\chi_1(Y) & = \\frac{1}{180}\\int_Y (-31c_4 -14 c_1 c_3 +3 c_2^2 + 4 c_1^2 c_2 -c_1^4),\\\\\n  \\chi_2(Y) & = \\frac{1}{120}\\int_Y (79c_4 -19 c_1 c_3 +3 c_2^2 + 4 c_1^2 c_2 -c_1^4).\n  \\end{aligned}\n  \\end{equation}\n  The holomorphic Euler characteristic $\\chi_0$ is a birational   \\cite[Theorem II.8.19]{Hartshorne} and diffeomorphism invariant. \n  The other holomorphic genera are diffeomorphism invariants and invariant under crepant birational maps. \n\n\\item The  Pontryagin numbers of a fourfold are the numbers $\\int_Y p_2(TY)$ and $\\int_Y p_1^2(TY)$, where the Pontryagin classes $p_1(TY)$ and $p_2(TY)$ are defined as follows:\n \\begin{equation}\n \\begin{aligned}\n p_1(TY) &=c_1^2(TY) -2c_2(TY), \\\\\n  p_2(TY) &= c_2^2(TY)-2 c_1(TY) c_3(TY)+2c_4(TY).\n \\end{aligned}\n \\end{equation}\n\\item The Hirzebruch signature of a fourfold, \n \\begin{equation}\n\\sigma(Y)=\\frac{1}{45}\\int_Y \\Big(7p_2(TY) -p_1^2(TY)\\Big)=\\frac{1}{45}\\int_Y (-c_1^4+4 c_1^2 c_2+3 c_2^2-14 c_1 c_3+14 c_4).\n\\end{equation}\nThe signature is the degree of the Hirzebruch $L$-genus. The $L$-genus  is always an integer  \\cite[Corrolary 19.5, p. 226]{Milnor} and depends only on the oriented homotopy type of the variety \\cite[Corrolary 19.6, p. 226]{Milnor}.\n\\item The $\\hat{\\text{A}}$-genus of a fourfold,\n\\begin{equation}\n\\begin{aligned}\n\\int_Y \\hat{\\text{A}}_2(TY)&=\\frac{1}{5760}\\int_Y\\Big(7p_1^2(TY)-4p_2(TY)\\Big)\\\\\n&=\\frac{1}{5760}\\int_Y\\Big( {7 c_1^4-28 c_1^2 c_2+8 c_1 c_3+24 c_2^2-8 c_4} \\Big).\n\\end{aligned}\n \\end{equation}\nBy the Atiyah-Singer theorem, if the fourfold $Y$ is a spin manifold,  the degree of  $ \\hat{\\text{A}}_2$ gives the index of the Dirac operator on  $Y$. We will see that the $\\hat{\\text{A}}$-genus is independent of our choice of a crepant resolution and is also independent of $G$. \n\\item \n We also compute the following form that plays an important role in many questions of anomaly cancellations, it  detects the appearance of a non-vanishing contribution to the one-point function for the two, three, and four forms in type IIA, M, and F-theory \\cite{Vafa:1995fj,Sethi:1996es}:\n\\begin{equation}\nX_8(Y)=\\frac{1}{192 }\\int_Y \\Big( p_1^2(TY) -4p_2(TY)\\Big).\n\\end{equation}\nIn string theory, $X_8(Y)$ typically appears as a curvature invariant. As it is  expressed by Pontryagin numbers, $X_8(Y)$ is an oriented  diffeomorphism invariants and is also invariants under crepant birational maps. \n Our computations of $X_8(Y)$ has some overlaps with  \\cite{Lawrie:2018jut}, see Appendix \\ref{AD} for more information. \n\\end{enumerate}\n\n\\subsection{Chern numbers of fourfolds are $K$-equivalence invariants}\n\nCrepant resolutions are in a sense the mildest form of desingularizations, since they do not modify a singular  variety away from its singular locus and  preserves the canonical class. \nCrepant resolutions are also relative minimal models over the underlying singular variety. \nWhen they exist, crepant resolutions are not necessarily unique for varieties of dimension three or higher. For $G$-models, the number of flops can be pretty big \\cite{EJJN1,EJJN2,Hayashi:2014kca}. \nWhen two varieties are crepant resolutions of the same underlying singular variety, a natural question to ask is if their characteristic numbers are the same. \n For instance, the Betti numbers and  the Hodge numbers are invariants under crepant birational maps \\cite{Batyrev.Betti,Kontsevich.Orsay}.\n\nA key point that makes this paper possible is the fact that  it is enough to know a single crepant resolution to compute the Chern and Pontryagin numbers of a given $G$-model. \nThis is because the Chern numbers of a fourfold are invariants under crepant birational maps as proven in Theorem \\ref{Thm:TheInvariance} using  a result of  Aluffi \\cite[p. 3368]{Aluffi.IMRN}  and the birational invariance of the Todd genus.  \nWe would like to point out that such an invariance for the Chern numbers should not be taken for granted as it is not generally true that Chern and Pontryagin numbers are invariant under crepant birational maps. \nThe first counter-example appears in dimension five (see Example \\ref{GoMAEx1}). This fact motivated our choice to present the results only for fourfolds.\nFor a projective five-fold, the Chern numbers are \n$$\n\\int c_1^5, \\quad \\int c_1^3 c_2 , \\quad \\int c_1^2 c_3, \\quad \\int c_1 c_4, \\quad \\int c_5, \\quad \\int c_1 c_2^2, \\quad \\int c_2 c_3,\n$$\nwhere only the first five are invariant under K-equivalence  by Aluffi's theorem (see Theorems \\ref{Thm:Al} and \\ref{Thm:TheInvariance}). In fact, \nGoresky and MacPherson gave the following example of a five-dimensional Schubert\nvariety with two different small resolutions with  the same Chern\nnumbers with the exception of $\\int c_2 c_3$.  For more information on $K$-invariance, see \\cite{Wang03,Wang2002}.\n\n\\begin{exmp}[Goresky and MacPherson, {\\cite[Example 2, page 221]{GoMa}}]\\label{GoMAEx1}\nLet $X$ be the Schubert variety in the Grassmannian $G_2(\\mathbb{C}^4)$ consisting of all complex two-planes $V\\subset \\mathbb{C}^4$ such that $\\dim( V \\cap\\mathbb{C}^2)\\geq 1$. \nThis variety $X$ has a singularity at the point $V=\\mathbb{C}^2$ , and it has a small resolution $\\varphi_1: \\widetilde{X}_1\\to X$ where $\\widetilde{X}_1$ consists of all  $(1,2)$- flags $V^1\\subset V^2\\subset\\mathbb{C}^4$  such that $V^1\\subset\\mathbb{C}^2$. \nIt has a second small resolution $\\varphi_2: \\widetilde{X}_1\\to X$ which consists of all $(2,3)$ flags $V^2\\subset V^3\\subset\\mathbb{C}^4$  such that $\\mathbb{C}^2\\subset V^3$. Although $\\widetilde{X}_1$ and $\\widetilde{X}_2$ are homeomorphic, by a computation of Verdier, they do not have the same Chern classes.  \n\\end{exmp}\n\n\n\\begin{exmp}[Goresky and MacPherson, {\\cite[Example 2, page 222]{GoMa}}]\nLet $X$ be a Schubert variety $X=\\{V\\in G_2(\\mathbb{C}^5|\\dim(V\\cap \\mathbb{C}^3)\\leq 1\\}$.  \nLet $\\widetilde{X}_1$ be the variety of partial flags $V_1\\subset V_2\\subset \\mathbb{C}^5$ \nsuch that $V_1\\subset \\mathbb{C}^2$. Let $\\tilde{X}_2$ be the variety of \npartial flags $V_2\\subset V_4\\subset\\mathbb{C}^4$ such that \n$\\mathbb{C}^3\\subset V_4$. Both $\\widetilde{X}_1$ and $\\widetilde{X}_2$ are small resolutions \nof $X$ but their cohomology rings are not even abstractly isomorphic. \n\\end{exmp}\n\n\n\nThe following theorem of Libgober and Wood was known to Hirzebruch in the case of fourfolds.\n\\begin{thm}[Libgober and Wood, {\\cite{LW}, Theorem 3}]\nFor a compact complex manifold $X$, the Chern number $\\int_X c_1c_{n-1} $ is determined by the holomorphic genera and hence by the Hodge numbers. \n\\end{thm}\n\n\n\n\nEven though Chern numbers other than the Euler characteristic are not topological invariants, some are invariant under crepant birational maps as proven by Aluffi.\n\n\\begin{thm}[Aluffi, { \\cite[page 3368]{Aluffi.IMRN}}]\\label{Thm:Al}\nFor two nonsingular $n$-dimensional complete \tvarieties $X$ and $Y$ connected by a crepant birational map\n$$\n\\int_X c_1(TX)^i c_{n-i}(TX)=\\int_Y c_1(TY)^i c_{n-i}(TY), \\quad i=0, 1, \\dots, n.\n$$\n\\end{thm}\nThe following theorem is also used in this paper.\n\\begin{thm}[Aluffi, { \\cite[Corollary 1.2]{Aluffi.IMRN}}]\nLet $f:\\widetilde{X}\\to X$ be a crepant resolution. Then the class $$f_*\\Big( c(T\\widetilde{X})\\cap[\\widetilde{X}]\\Big),$$\n in $(A_* X)_{\\mathbb{Q}}$ is independent of $X$. \n\\end{thm}\n\n\n\nThe following theorem assets that the Chern numbers of fourfolds are  $K$-equivalence  invariants.  The proof is a simple application of  Theorem \\ref{Thm:Al} of Aluffi. \n\\begin{thm}\\label{Thm:TheInvariance}\nThe Chern and Pontryagin numbers of an algebraic variety of complex dimension four are $K$-equivalence invariants.\n\\end{thm}\n\\begin{proof}\nThe Chern numbers of a fourfold are $\\int_Y c_1(TY)^4$, $\\int_Y  c_1(TY)^2 c_2 (TY)$, $\\int_Y  c_1(TY) c_3 (TY)$, $\\int_Y c_2^2(TY)$,  and $\\int_Y c_4 (TY)$.  \nHirzebruch showed that  $\\int_Y  c_1(TY)c_3(TY)$ can be expressed by Hodge numbers. Moreover, from a result of Aluffi, we know that the Chern numbers $\\int_Y  c_1^i(TY)c_{n-i}(TY)$ are $K$-equivalence invariants.  The Chern number  $\\int_Y  c_2^2(TY)$ is also a $K$-equivalence invariant as it can be expressed as a linear combination of the holomorphic Euler characteristic (which is a birational invariant) and the $\\int_Y  c_1^i(TY)c_{n-i}(TY)$ for $i=0,1,2,3,4$. It follows that all Chern numbers of a fourfold are $K$-equivalence invariants.  The same is true for Pontryagin numbers since they are linear combinations of Chern numbers.\n\\end{proof}\n\n Theorem \\ref{Thm:TheInvariance} allows us to compute Chern numbers of $G$-models in the crepant resolution of our choice since  it is independent of a choice of a crepant resolution. \n\n\nThe Chern number $\\int_Yc_1(TY)^2 c_2(TY)$, the $\\hat{A}$-genus, and the Todd-genus (the holomorphic Euler characteristic) are invariants of the choice of $G$. \nThey can all be expressed as invariants of the divisor $W$ defined by the vanishing locus of a smooth section of $\\mathscr{L}$. \n\n\\subsection{Independence of Chern numbers}\nBy expressing $\\chi_0$ and $\\hat{A}$ in terms of Chern numbers and using the identity $\\int_Y c_1^4=0$, which holds for any crepant resolution of a Weierstrass model (see Theorem \\ref{thm.1.4}), we get the following expressions of $c_1^2 c_2$ and $c_1 c_3$:\n\\begin{align}\n\\int_Y c_1^2c_2=96(\\chi_0(Y)-\\hat{A}(Y)),\\quad \n\\int_Y c_1 c_3 = 384\\hat{A}(Y)+ 336\\chi_0(Y)+\\chi(Y)- 3 \\int_Y c_2^2.\n\\end{align}\nFor an elliptic fibration that is a crepant resolution of a Weierstrass model, this shows that $\\int_Y c_1^2 c_2$ gives the same value as a smooth Weierstrass model with the same  fundamental line bundle $\\mathscr{L}$. \nIt is therefore enough to compute only $\\int_Y c_2^2$ and the Euler characteristic $\\chi(Y)=\\int_Y c_4$ to know all the Chern numbers of a crepant resolution of a Weierstrass model. \n\n\n\\section{Strategy and roadmaps of  results}\n\n\n\n\n\n\n\nThe data at the heart of our computations are the  lists of blowups that give a  crepant resolution for each of these $G$-models. \nWe can summarize our strategy as follows \n\\begin{enumerate}\\item \nThe  $G$-models considered in this paper are defined by crepant resolutions of Weierstrass models given  in section \\ref{sec:Tate}. \n\\item They crepant resolutions are obtained by  sequence of blowups given in Table  \\ref{tab:blowupcenters}. \n\\item The Chern numbers and Pontryagin numbers of the $G$-models are listed respectively in Table \n\\ref{Table.ChernNumbers} and  Table \\ref{Table.Pontryagin}. \n\\item The holomorphic genera are listed in Table \\ref{Table.HolomorphicEC}. \n\\item The invariant $X_8(Y)$, the signature $\\sigma(Y)$, and the $\\mathnormal{\\hat{A}}$-genus are given in Table \\ref{Table.Pontryagin2}. \n \\end{enumerate}\n   Let  $\\pi:X_0\\to B$ be the projective bundle in which the Weierstrass model $Y_0$ is defined.  \n   We define a crepant resolution $f:Y\\to Y_0$ by  a composition of $k$ blowups   $f_i: X_{i}\\to X_{i-1}$  $i=1,\\cdots, k$ with smooth centers  that are transverse complete  intersections in $X_i$. \n  Using Theorem \\ref{Thm:AluffiCBU} for each $f_i$, we  compute the Chern classes  of each $X_i$ and by adjunction. We then determine the Chern classes of $Y$ by adjunction. \n \nFor  an element $Q$ of the Chow ring $A^*(Y)$, we compute $\\int_Y Q$ as a function of the topology of the base as follows. \nUsing Theorem  \\ref{Thm:Push}, we then express the Chern numbers in terms of the Chow ring of the original ambient space $X_0$ in which the Weierstrass model is defined. \nNext, we pushforward these to the base $B$ of the elliptic fibration using Theorem \\ref{Thm:PushH}. Since we fix the base to be a threefold, we can also simply use Lemmas \\ref{lem:Push2}--\\ref{lem:PushH} for our pushforwards. In summary we get,\n\\begin{equation}\\int_Y Q=\\int_{X_k} Q\\cdot [Y]=\\int_{B} \\pi_*  f_{1*} f_{2*} \\cdots f_{k*} \\Big(Q\\cdot [Y]\\Big).\n\\end{equation}\n\n\n\nThe following theorem gives the behaviors of intersection numbers involving Chern classes and Pontryagin classes of dimension too small to give Chern or Pontryagin numbers. \nTo give a number, they must be multiplied by an element of the Chow ring of appropriate dimension. \n\\begin{thm}\\label{thm.1.4}\nLet  $\\varphi:Y\\to B$ be an  elliptic fibration\ngiven by the crepant resolution of a singular Weierstrass model of dimension $n$. Then,\n\\begin{align}\n& \\int_Y c_1^i(TY)\\cdot  \\alpha &&=\\int_B (c_1-L)^i\\cdot \\varphi_* \\alpha, \\quad &&\\alpha \\in A^*(Y),\\label{Thm21.1}  \\\\\n& \\int_Y c_1^i(TY)\\cdot  \\varphi^* \\beta &&=\\int_B (c_1-L)^i \\beta,  &&\\beta \\in A^*(B), \\label{Thm21.2} \\\\\n& \\int_Y c_1^n(TY) &&=\\int_B (c_1-L)^n=0, \\label{Thm21.3}\\\\\n& \\int_Y c_3(TY)\\cdot  \\varphi^* \\beta  &&=\\int_B \\varphi_* (c_3 (TY) [Y]) \\cdot \\beta, \\quad &&\\beta \\in A^*(B), \\label{Thm21.4}\\\\\n& \\int_Y c_2(TY)\\cdot  \\varphi^* \\beta  &&=12\\int_B L \\cdot \\beta, \\quad &&\\beta \\in A^*(B), \\label{Thm21.5}\\\\\n& \\int_Y p_1(TY)\\cdot  \\varphi^* \\beta  &&=\n \\int_B(c_1-L)^2 \\beta-24\\int_B L\\cdot \\beta \\quad &&\\beta \\in A^*(B)\\label{Thm21.6}\n\\end{align}\nwhere $ \\varphi_* (c_3 (TY) [Y])$ is given in Table \\ref{Table:ChernNumbers} for the $G$-models considered in the paper and is invariant of a choice of a crepant resolution. \n\\end{thm}\n\n\\begin{proof}\nSince $Y$ is a crepant resolution of a Weierstrass model, we have that the first Chern class of $Y$ is the pullback $c_1(TY)=\\varphi^*(c_1-L)$.\nEquation \\eqref{Thm21.1} is therefore a direct consequence of the projection formula and the invariance of the degree under a proper map: \n$$\n\\int_Y \\varphi^* (c_1-L)^i \\alpha =\\int_B \\varphi_* (\\varphi^* (c_1-L)^i \\alpha)= \\int_B (c_1-L)^i \\varphi_*\\alpha.\n$$ \nEquations \\eqref{Thm21.2} and \\eqref{Thm21.3} are direct specializations of equation \\eqref{Thm21.1}. \nIn particular, if $Y$ is an $n$-fold, we have $\\int_Y \\varphi^* (c_1-L)^n=\\int_B (c_1-L)^n=0$. \n Equation \\eqref{Thm21.4} is also a direct consequence of the projection formula.\n Equation \\eqref{Thm21.5} follows from Theorem \\ref{Thm.C2} and the fact that $\\int_{Y_0}\\pi_* c_2 = 12L$ for a smooth Weierstrass model $\\pi: Y_0\\to B$. \n  Theorem \\ref{Thm.C2}, asserts that  for any crepant blowup centered at a transverse complete intersection of  smooth divisors, we have $f_*(c_2(TY)\\cdot[Y])=c_2(TY_0)\\cdot [Y_0]$. \n We note that  these are exactly the types of blowups we use in  section \\ref{sec:GModels}. \n Equation \\eqref{Thm21.6}  is derived by linear combination using $p_1=c_1^2-2c_2$ and equations \\eqref{Thm21.2} and \\eqref{Thm21.5}.\n  \\end{proof}\n  \n  \\begin{table}[htb]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.8\n\\scalebox{1}{$\n\\begin{array}{|c|c|}\n\\hline \n\\text{Algebra} & \\mu_G =\\int_Yc_2^2(TY)-\\int_{Y_0}c_2^2(TY_0) \\\\\\hline\n\\text{A}_1 & -2 \\int_BS (7 L-S)^2 \\\\\n\\hline\n\\text{A}_2, \\ \\text{C}_2, \\ \\text{G}_2 &-8\\int_B S \\left(19 L^2-8 L S+S^2\\right) \\\\\n\\hline\n\\text{A}_3, \\ \\text{B}_3 & -4 \\int_BS \\left(50 L^2-28 L S+5 S^2\\right) \\\\\n\\hline\n\\text{D}_4, \\ \\text{F}_4 & -8 \\int_BS \\left(27 L^2-16 L S+3 S^2\\right) \\\\\n\\hline \n\\text{A}_4 & -5 \\int_BS \\left(50 L^2-35 L S+8 S^2\\right) \\\\\n\\hline \n\\text{D}_5 & -4 \\int_BS \\left(63 L^2-44 L S+10 S^2\\right) \\\\\n\\hline \n\\text{A}_5 & - \\int_BS \\left(298 L^2-251 L S+70\\right) \\\\\n\\hline \n\\text{A}_6 & -2 \\int_BS \\left(174 L^2-171 L S+56 S^2\\right) \\\\\n\\hline \n\\text{E}_6 & -3 \\int_BS \\left(86 L^2-61 L S+14 S^2\\right) \\\\\n\\hline \n\\text{E}_7 &\\int_B \\left(135 L^2-100 L S+24 S^2\\right) \\\\\n\\hline \n\\text{E}_8 &\\int_B \\left(8 L^2-7 L S+2 S^2\\right) \\\\\n\\hline \n\\end{array}\n$}\n\\end{center}\n\\caption{$\\mu_G$ for all the G-models. $Y$ is the crepant resolution of a singular Weierstrass model corresponding to a $G$-model, and $Y_0$ is a smooth Weierstrass model over the same base with the same fundamental line bundle $\\mathscr{L}$. \\label{Table.muG} }\n\\end{table}\n\n  \\begin{cor}\\label{Cor.c2}\n  Let  $\\varphi:Y\\to B$ be an  elliptic fibration\ngiven by the crepant resolution of a singular Weierstrass model of dimension $n$ with fundamental line bundle $\\mathscr{L}$. Then,\n\\begin{equation}\n  \\int_Y c_1(TY)^{n-1}c_2(TY) = 12\\int_B (c_1-L)^{n-1} L.\n \\end{equation}\n  \\end{cor}\nThe following theorem shows how the dependence of the Pontryagin numbers on the Kodaira type (or better, on the group $G$) is controlled by the Chern number $\\int_Y c_2^2(TY)$. \n\n\\begin{thm}\\label{thmP}\nFor a $G$-model,\ndefined by  a crepant resolution of one of the  Weierstrass model given  in section \\ref{sec:Tate} and resolved by the sequence of blowups given in Table  \\ref{tab:blowupcenters}, we have \n\\begin{equation}\n\\begin{aligned}\n\\int_Y c_2^2(TY) &=\\    \\   24\\int_B L (c_2-c_1 L+6 L^2)+\\mu_G, \\\\\n\\int_Y p_2 (TY)&=-24 \\int_B L (2 c_2-c_1^2+36 L^2)+7\\mu_G,\\\\\n\\int_Y  p_1^2(TY)&=\\  \\   48\\int_B L (2 c_2-c_1^2+11 L^2)+4\\mu_G,\n \\end{aligned}\n\\end{equation}\nwhere $\\mu_G= \\int_B S (\\alpha_0 L^2+ \\alpha_1 LS+ \\alpha_2 S^2)$ is the contribution from the singularities induced by the Kodaira type over $S$.  The different values of $\\mu_G$ are listed in Table \\ref{Table.muG}.\nThe correction for $\\lambda_2\\int_Y p_2 + \\lambda_1 \\int_Y p_1^2$ is then $(7\\lambda_2+ 4\\lambda_1)\\mu_G$.\nIn particular, \n\\begin{align}\n\\begin{aligned}\\label{Eq.p}\n\\int_Y\\Big( p_1^2(TY)-4p_2(TY)\\Big) &=48\\int_B L \\left(c_1^2-2 c_2-61 L^2\\right)-24 \\mu_G, \\\\\n\\int_Y\\Big(7p_2(TY)-p_1^2(TY)\\Big) &=120\\int_B L \\left(2 c_2-c_1^2+46 L^2\\right)+45\\mu_G,\\\\\n\\int_Y\\Big(7p_1^2(TY)-4p_2(TY)\\Big) & = 240 \\int_BL \\left(2 c_2-c_1^2+L^2\\right).\n\\end{aligned}\n\\end{align}\n\\end{thm}\n\\begin{proof}\n This follows directly from an inspection of Table \\ref{Table.ChernNumbers},  Table \\ref{Table.Pontryagin}, and Table \\ref{Table.Pontryagin2}. \n\\end{proof}\n\n\n\n\n\\section{Geometric discussion}\n\\subsection{The Calabi-Yau fourfold case}\\label{sec:CY4}\nIn the case of Calabi-Yau fourfolds, knowing the Euler characteristic is enough to also compute other invariants such as the Chern number $c_2(TY)^2$ is a function of the Euler characteristic. \n\\begin{thm}\\label{Thm2.CY4}\n The Chern numbers and Pontryagin numbers of a  Calabi-Yau fourfold are topological invariants depending only on its Euler characteristic. \n\\end{thm}\n\\begin{proof}\nThe only non-zero Chern numbers of a Calabi-Yau  fourfold are $\\int_Y c_2^2(TY)$ and $\\int_Y c_4(TY)=\\chi(Y)$. These two Chern numbers  are related linearly as  \n$$\\int_Y c_2^2(TY)=480+\\frac{1}{3}\\chi(Y).$$ Thus, for a Calabi-Yau fourfold, all Chern numbers and Pontryagin numbers are topological invariants as they are functions of the Euler characteristic of $Y$. \n\\end{proof}\n \\noindent For more information see section \\ref{sec:CY4}. \nIt follows that in the Calabi-Yau fourfold case, all Chern and Pontryagin numbers  can be read from the computation of the Euler characteristic of $G$-models in \\cite{Euler} as  we have \\cite{Klemm:1996ts,Sethi:1996es}\n  $$\n  \\begin{aligned}\n\\int_Y  c_2^2(TY) &=480+\\frac{1}{3}\\chi(Y),\\quad\n  \\sigma =32+\\frac{1}{3}\\chi(Y),\\\\\n  \\chi_0&=2 ,\\quad\n  \\chi_1  =8-\\frac{1}{6} \\chi(Y),\\quad\n  \\chi_2  =12+ \\frac{2}{3}\\chi(Y),\\\\\n  X_8&=-\\frac{1}{24}\\chi(Y),\\quad\n \\frac{1}{5760} \\int_Y \\hat{\\text{A}}_2 = 2.\n   \\end{aligned}\n  $$\nFor $G$-models, the Euler characteristic of a Calabi-Yau fourfold is given in Table 10 of \\cite{Euler}, which we reproduce here for completeness. We notice that the same table can be obtained from the Euler characteristic $c_4(TY)$ after putting $L=c_1$, which is the condition that ensure that the canonical class of $Y$ is trivial. \n\n\\begin{table}[htb]\n\\begin{center}\n\\scalebox{.98}{$ \n \\begin{array}{|c|c |}\n \\hline\n \\text{Models}  & \\text{ $\\chi(Y_4)$,   Euler characteristic} \\\\\n \\hline\n\\text{Smooth Weierstrass} &  12 \\int_B(c_1 c_2 + 30 c_1^3) \\\\\n \\text{SU}(2) & 6 \\int_B (2 c_1 c_2 + 60 c_1^3 - 49 c_1^2 S + 14 c_1 S^2 - S^3)\\\\\n \\text{SU}(3)\\ \\text{or}\\   \\text{USp}(4) \\ \\text{or}\\  G_2 & 12 \\int_B (c_1 c_2 + 30 c_1^3 - 38 c_1^2 S + 16 c_1 S^2 - 2 S^3)\\\\\n\\text{SU}(4)\\ \\text{or}\\  \\text{Spin}(7)&  12 \\int_B(3 c_1 c_2 + 30 c_1^3 - 50 c_1^2 S + 28 c_1 S^2 - 5 S^3))\\\\\n\\text{Spin}(8) \\ \\text{or} \\  F_4& 12\\int_B  (c_1 c_2 +30 c_1^3 - 54 c_1^2 S + 32 c_1 S^2 - 6 S^3) \\\\\n\\text{SU}(5)&  3\\int_B (4 c_1 c_2+120 c_1^3-250 c_1^2 S+175 c_1 S^2-40 S^3)\\\\\n\\text{SU}(6)& 3\\int_B (4c_1 c_2+120c_1^3 -298 c_1^2S+251 c_1 S^2-70 S^3)\\\\\n\\text{SU}(7)& 6\\int_B (2c_1 c_2+60c_1^3-174 c_1^2S+171 c_1 S^2-56 S^3)\\\\\n\\text{Spin}(10)& 12\\int_B (c_1c_2+30c_1^3-63 c_1^2 S+44 c_1 S^2-10 S^3) \\\\\n \\text{E}_6& 3\\int_B  (4 c_1 c_2+120 c_1^3  - 258 c_1^2 S + 183 c_1 S^2 - 42 S^3)\\\\\n \\text{E}_7& 6\\int_B ( 2 c_1 c_2+60 c_1^3 - 135 c_1^2 S + 100 c_1 S^2 - 24 S^3)\\\\\n\\text{E}_8&  12\\int_B  ( c_1 c_2+ 30 c_1^3 - 80 c_1^2 S + 70 c_1 S^2 - 20 S^3)\\\\\n\\hline\n\\end{array}\n$}\n\\caption{Euler characteristic for Calabi-Yau elliptic fourfolds where $c_1=L$.}\n\\label{table:CY4}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\subsection{A geometric interpretation of the $\\hat{A}$-genus and the Todd-genus of  a $G$-model. }\n\nIn this section, we will see that the $\\hat{A}$-genus of a $G$-model $Y$ can be understood as the $A$-genus of the surface $W$ with normal bundle $\\mathscr{L}$ in the base $B$. \nWe call $W$ the {\\em Weierstrass divisor}. \n\n\nThe $\\hat{\\text{A}}$-genus of a $G$-model is \n$$\n\\int_Y \\hat{A}(Y)=\\frac{1}{5760}\\int_Y(7p_1^2-4p_2)= \\frac{1}{24}\\int_B L(-c_1^2+2c_2+L^2)=\\int_{W} \\hat{A}_1(W).\n$$\nWe compute the Chern classes of $W$ by the adjunction formula: \n$$\nc(TW)=\\frac{c(TB)}{1+L}=1+(c_1-L)+(c_2-c_1 L +L^2).\n$$\nIt is then direct to determine the first Pontryagin class of $W$:\n$$\np_1(TW)=c_1(TW)^2-2c_2 (TW)= (c_1-L)^2 -2 (c_2-c_1 L +L^2)=c_1^2-2c_2-L^2.\n$$\nSince $W$ is a surface, it has a unique  Pontryagin number, namely $\\int_W p_1(TW)$:\n$$\n\\int_W p_1(TW)=\\int_B L(c_1^2-2c_2-L^2)=-24 \\int_W \\hat{A}(W).\n$$\nThe last equality uses the fact that  $\\hat{A}_1=-p_1\/24$.\nIf $W$ is a spin manifold, $\\int_{W} \\hat{A}_1(W)$ is an integer number and so is $\\hat{A}(Y)$. \nWe  note that $Y$   has a vanishing canonical class if and only if  $W$ does too. In other words, $Y$ is  Calabi-Yau if and only if  $W$ is a K3 surface. \nIn that case, $\\hat{A}(Y)=\\hat{A}(W)=2$.\n\n A direct computation shows that the arithmetic genus of $Y$ is the same as the arithmetic genus of the surface $W$.\n\\begin{align}\n\\chi(W, \\mathscr{O}_W)&=\\int_B (1-e^{-L}) \\mathrm{Td}(TB)=\n\\frac{1}{12}\\int_W\\Big( c_1(TW)^2+c_2(TW)\\Big)\\\\\n&=\\frac{1}{12}\\int_B L(c_1^2-3 c_1 L+c_2+2 L^2)=\\chi_0(Y).\n\\end{align}\n\n\n\n\\subsection{Integrability problems for SU($5$) and E$_6$ Calabi-Yau fourfolds.}\\label{Sec:Oddities}\nA Calabi-Yau fourfold $Y$ has a  trivial canonical class and therefore satisfies the following relation \n\\begin{equation}\n\\chi_1(Y):=\\chi(\\Omega^1_Y)=\\sum_{p=0}^4 (-)^p h^{1,p}(Y)=8-\\frac{1}{6}\\chi(Y).\n\\end{equation}\n Since $\\chi_1(Y)$ is obtained by adding and subtracting Hodge numbers, it is an integer number. Thus, the Euler characteristic $\\chi(Y)$ of a Calabi-Yau fourfold  should be a multiple of $6$. \nBy a direct inspection of Table \\ref{table:CY4}, we observe that this is indeed the case for each $G$-models without any additional condition with the exception of SU($5$), SU($6$), and E$_6$. \n\\begin{lem} Let $S$ be the smooth divisor supporting a generic fiber of  Kodaira type I$_5^s$, I$_6^s$ or IV$^{* \\text{ns}}$ in the \\textnormal{SU($5$)}, \\textnormal{SU($6$)} or \\textnormal{E$_6$} model.  The \nThe Euler characteristics of the \\textnormal{SU($5$)}-model , \\textnormal{SU($6$)}, and the \\textnormal{E$_6$}-model are  divisible by $6$ in the Calabi-Yau case if and only if  \n\\begin{equation}\n\\int_B c_1(TB) S^2\\quad \\text{is an even integer number}.\n\\end{equation}\n\\end{lem}\nThis condition is clearly not always realized as seen in Example  \\ref{Examp.no}. If $c_1(TB)$ or $S$ is an even class, or if $B$ is a flat fibration with fiber $S$,  the condition is automatically satisfied. \nA  geometric situation naturally realizing the condition that $\\int_B c_1(TB) S^2$ is even is presented in Example \\ref{Examp.Curve}.\n\\begin{exmp}\\label{Examp.no}\nIf $B$ is a smooth quadric threefold in $\\mathbb{P}^4$ and $S$ is the intersection of  a hyperplane of $\\mathbb{P}^4$ with $B$, then,  $\\int_B c_1(TB) S^2=(5-2) 1^2=3$. \nNote that a smooth quadric threefold is Fano. Thus we can construct a Weierstrass model with $\\mathscr{L}$ the anticanonical line bundle of $B$. \n\\end{exmp}\n\n\n\\begin{exmp}\\label{Examp.Curve}\n The line bundle $\\mathscr{O}_B(S)$ has two distinct sections whose zero loci are smooth transverse divisors in $B$, then we can compute their complete intersection, which gives a smooth curve in $B$. \nBy the adjunction formula, the first Chern class of such a curve would be \n$\nc_1(TB)-2 S$. \nIt follows that its Euler characteristic is \n$\n2-2g=\\int_B ( c_1 -2S)S^2.\n$  \nIn such a case, we see that $\\int_S c_1 S^2=2(1-g+\\int_B S^3)$ is an even number. \n\\end{exmp}\n\n\n\\section{Invariance of characteristic numbers}\n\n  In contrast to Chern numbers, Pontryagin numbers are oriented diffeomorphism invariants. \n  There are even oriented homeomorphic invariants as proven by Novikov in the 1960s \\cite{Novikov}. \nIt follows that characteristic classes that are expressed in terms of Pontryagin classes and the Euler characteristic are both oriented diffeomorphic invariants and oriented homeomorphic invariants. \n   In contrast to the Euler characteristic,  Pontryagin numbers change their signs when the variety changes its orientation. \nThus, if any Pontryagin class of a variety is non-zero, the variety cannot possess any orientation reversing diffeomorphism (or homeomorphism). \n\n\n\nIn 1954, Hirzebruch asked in \\cite{Hirzebruch.Q} which linear combinations of Chern numbers \nof a nonsingular projective variety are topological invariants. \nSince  a complex projective \nvariety comes with a special choice of orientation, it is natural to restrict Hirzebruch question  to oriented homeomorphism or an oriented diffeomorphism. \nWith that restriction, the question was answered by  Kotschick in the following theorem. \n\n\n\\begin{thm}[Kotschick, {\\cite{Kotschick1}}]\n\\begin{enumerate}\n\\item A rational linear combination of Chern numbers is an\noriented diffeomorphism invariant of a smooth complex projective\nvariety if and only if it is a linear combination of the Euler and\nPontryagin numbers.\n\\item In complex dimension $n\\geq 3$, a rational linear combination of Chern numbers is a\ndiffeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of\nthe Euler characteristic. In complex dimension two, both Chern numbers $\\int c_1^2$ and $\\int c_2$ are oriented diffeomorphism invariant.  \n\\end{enumerate}\n\\end{thm}\nThe statement about surfaces is a consequence of Seiberg-Witten's theory \\cite{Kotschick2,Kotschick3}. \n \nThere are also invariants such as the Euler characteristic, the holomorphic genus, the holomorphic genera, and the signature, which are linear combination of Hodge numbers. \nHodge numbers are known to be the same for varieties in the same $K$-equivalence class. The following theorem characterizes which linear combinations of Chern numbers are also  linear combinations  of Hodge numbers. \n\n\n\n\\begin{lem}[Kotschick, {\\cite{Kotschick1}}]\nA rational linear combination of Chern numbers of a smooth complex projective variety\nis determined by the Hodge numbers if and only if it is a linear combination of the holomorphic genera  \n $\\chi_p$.\n\\end{lem}\n\n\nThe Euler characteristic is a homotopy invariant while the signature is an oriented homotopy invariant. The following theorem of Khan characterizes linear combination of Chern numbers that are oriented homotopy invariants. \n\n\\begin{thm}[Khan, {\\cite{Khan}}]\nA linear rational combination of Chern numbers is an oriented homotopy invariant, for almost-complex manifolds, if and only if it is a rational linear combination of the Euler characteristic and the signature of the manifold. \n\\end{thm}\n\n\n\n\n\n\\section{Pushforward of blowups and projective bundles}\n\\label{sec:push}\n\\subsection{A brief history of pushforward in string theory}\nIntersection theory in algebraic geometry as we know it today is presented in the seminal book of Fulton \\cite{Fulton.Intersection}. \nChern classes are defined from Segre classes, which satisfy stronger functorial properties. \nThe first formula computing  the  Chern classes  of the tangent bundle of the blow-up of\na nonsingular variety along a nonsingular center  was conjectured by Todd and Segre, and proven by  Porteous \\cite{Porteous}. A generalization of Porteous to singular varieties was obtained by Aluffi   \\cite{Aluffi_CBU}, who also provided user friendly descriptions of Porteous formula in the case of blow-ups whose centers are smooth complete intersections. Aluffi formula are center to this paper, and are reviewed in section \\ref{sec:push}.\n\nThe pushforward allows to compute intersection numbers of a fibration in terms of the Chow ring  of its base. \nThe first use of implicit pushforward methods in string theory is the computation of the Euler characteristic of a Weierstrass model by Sethi, Vafa and Witten for Calabi-Yau threefolds and fourfolds in terms of the Chow ring of its base \\cite{Sethi:1996es}.  \nUsing efficient pushforward techniques, generating functions for the Euler characteristic of a smooth Weierstrass model was derived by Aluffi and Esole \\cite{AE1}.\nIntersection theory is instrumental in defining the induced D3-charges in presence of an orientifold  \\cite{CDE}, which is to constrain to satisfy a ``tadpole condition''. \nSee \\cite{AE2,EFY,EKY} for additional examples of topological relations in the Chow ring motivated by string dualities and tapdole cancellations. \n\nThe first application of Porteous formula in string theory is  \\cite{Andreas:1999ng}. Aluffi's formula was used to compute the Euler characteristic of an SU(5) model  \\cite{EY} in   \\cite{Marsano} and in the study of anomaly cancellations in six dimensional gauge theories in  \\cite{EK1,EKY,EKY2,ES}. \nPowerful methods to compute pushforwards in projective bundles using the functorial properties of the Segre class were obtained in \\cite{FH2}, extended a point of view  used in \\cite{AE1,AE2}. Recently in \\cite{Euler}, we introduced new theorems that streamline the computation of pushforwards of blowups as simple algebraic manipulations involving  rational polynomials. These techniques allow a simple computations of pushforward of any analytic expression of the exceptional divisors of the blowup of a complete intersections of smooth divisors. \nAll these techniques are illustrated in  \\cite{Euler} by computing the Euler characteristics of crepant resolutions of Weierstrass models. \n\n\\begin{thm}[Aluffi-Esole, {\\cite{AE1}}]\nFor a smooth Weierstrass model, we have \n\\begin{equation}\\label{Eq.AE1}\n \\varphi_*c(Y)= 12\\ \\frac{L}{1+6L} c(B).\n \\end{equation}\n\\end{thm}\n\n\\begin{thm}[Esole--Jefferson--Kang,{\\cite{Euler}}]\nThe \\textnormal{SU($2$)}-model over a base of arbitrary dimension, gives \n \\begin{equation}\\label{Eq.Euler}\n \\varphi_*c(Y)= \\frac{2 L+3 L S-S^2}{(1+S)(1+6L-2S)} c(B),\n \\end{equation}\n where $S$ is the divisor supporting the fiber of type \\textnormal{ I$_2$} or \\textnormal{III}. \n \\end{thm}\n\n\n\n\\begin{thm}[Esole--Fullwood--Yau, {\\cite[Theorem 1.4]{EFY}}]\nLet $\\varphi: Y\\to B$ be an elliptic fibration defined by the \ncomplete intersection of two divisors of class $\\mathscr{O}(2)\\otimes\\pi^* \\mathscr{L}^{\\otimes 2}$ in the projective bundle \n$\\pi:\\mathbb{P}\\Big(\\mathscr{O}_B\\oplus \\mathscr{L}\\oplus \\mathscr{L}\\oplus\\mathscr{L}\\Big)\\to B$.  \nThen \n\\begin{equation}\\label{Eq.EFY}\n\\varphi_* c(Y)= \\frac{4L(3+5L)}{(1+2L)^2} c(B),\n\\end{equation}\nwhere  $L=c_1(\\mathscr{L})$. \n\\end{thm}\n\n\\begin{thm}[Esole--Kang--Yau, {\\cite{EKY}}]\nConsider a projective variety $B$ endowed with two line bundles $\\mathscr{L}$ and $\\mathscr{D}$. \nLet $\\varphi:Y\\to B$ be a smooth elliptic fibration defined as the  zero scheme  of a section of the  line bundle $\\mathscr{O}(3)\\otimes\\pi^* \\mathscr{L}^{\\otimes 2}\\otimes \\pi^*\\mathscr{D}$ in the projective bundle \n$\\pi:\\mathbb{P}\\Big(\\mathscr{O}_B\\oplus \\mathscr{L}\\oplus \\mathscr{D}\\Big)\\to B$. Then \n\\begin{equation}\\label{Eq.EKY}\n\\varphi_* c(Y)= 6 \\frac{(2 L + 2 L^2 - L D + D^2)}{(1 + 2 L - 2 D) (1 + 2 L + D)} c(B),\n\\end{equation}\nwhere  $D=c_1(\\mathscr{D})$ and $L=c_1(\\mathscr{L})$. \n\\end{thm}\n\n\n\n\n\n\\subsection{Definitions and notations}\nThroughout this paper, we work over the field of complex numbers. \nA variety is a reduced and irreducible algebraic scheme. \n We denote the vanishing locus of the sections $f_1, \\ldots, f_n$ by $V(f_1, \\ldots, f_n)$. \n  The tangent bundle of a variety $X$ is denoted by $TX$ and the  normal bundle of a subvariety $Z$ of a  variety $X$ is denoted by  $N_Z X$. \n Let $\\mathscr{V}\\rightarrow B$ be a vector bundle over a variety $B$. We denote the by $\\mathbb{P}(\\mathscr{V})$ the projective bundle of lines in  $\\mathscr{V}$.\nWe use Weierstrass models defined with respect to the projective bundle $\\pi : X_0 = \\mathbb P[\\mathscr O_B \\oplus \\mathscr L^{\\otimes 2} \\oplus \\mathscr L^{\\otimes 3} ] \\rightarrow B$  where $\\mathscr{L}$ is a line bundle of $B$. \nWe denote the pullback of $\\mathscr{L}$ with respect to $\\pi$ by $\\pi^* \\mathscr{L}$. \nWe denote by $\\mathscr{O}_{X_0} (1)$  the  canonical line bundle on $X_0$, i.e., the  dual of the tautological line bundle of $X_0$  (see \\cite[Appendix B.5]{Fulton.Intersection}).\nThe first Chern class of  $\\mathscr{O}_{X_0} (1)$ is denoted $H$ and the first Chern class of $\\mathscr{L}$ is denoted $L$. \n The Weierstrass model $\\varphi : Y_0 \\rightarrow B$ is defined as  the zero-scheme of a section of $\\mathscr{O}_{X_0} (3) \\otimes \\pi^* \\mathscr L^{\\otimes 6}$. \n The Chow group $A_*(X)$ of a nonsingular variety $X$ is the group of divisors modulo rational equivalence \\cite[Chap. 1,\\S 1.3]{Fulton.Intersection}.\nWe use $[V]$ to refer to the class of a subvariety $V$ in $A_*(X)$.\n Given a class $\\alpha \\in A_*(X)$, the degree of $\\alpha$ is denoted $\\int_X \\alpha$ (or simply $\\int \\alpha$ if $X$ is clear from the context.) Only the zero component of $\\alpha$ is relevant in computing $\\int_X \\alpha$---see \\cite[Definition 1.4, p. 13]{Fulton.Intersection}. We use $c(X)=c(TX)\\cap [X]$ to refer to the total homological Chern class of a nonsingular variety $X$, and likewise we use $c_i(TX)$ to denote the $i$th Chern class of the tangent bundle $TX$.  \n  Given   two varieties $X, Y$ and a proper morphism $f: X \\rightarrow Y$, the  proper pushforward associated to \n$f$ is denoted $f_*$.  If $g: X\\rightarrow Y$ is a flat morphism, the pullback of $g$ is denoted $g^*$ and by  definition $g^*[V]=[g^{-1} (V)]$, see  \\cite[Chap 1, \\S 1.7]{Fulton.Intersection}.\n\\begin{defn}[Pushforward,  {\\cite[Chap. 1, p. 11]{Fulton.Intersection}}] \n Let $f: X\\longrightarrow Y$ be a proper morphism. \nLet $V$ be a subvariety of $X$, the image $W=f(V)$ a subvariety of $Y$, and the function field $R(V)$ an extension of the function field $R(W)$. \nThe pushforward $f_* : A_*(X)\\to A_*(Y)$ is defined as follows\n$$\nf_* [V]= \\begin{cases}\n0  &  \\text{if} \\quad \\dim V\\neq \\dim W,\\\\\n[R(V):R(W)] \\   [V_2]  & \\text{if} \\quad \\dim V= \\dim W,\n\\end{cases}\n$$\nwhere $[R(V):R(W)]$ is the degree of the field extension $R(V)\/R(W)$. \n\\end{defn}\n\\begin{defn}[Degree, {\\cite[Chap. 1, p. 13]{Fulton.Intersection}}]\nThe degree of a class $\\alpha$ of $A_*(X)$ is denoted by  $\\int_X \\alpha$ (or simply $\\int \\alpha$ if there is no ambiguity in the choice of $X$), and is defined to be the degree of its component in $A_0(X)$.\n\\end{defn}\nThe total homological Chern class $c(X)$ of any nonsingular variety $X$ of dimension $d$ is defined as\n\\begin{equation*}\nc(X)=c(TX)\\cap [X],\n\\end{equation*}\nwhere $TX$ is the tangent bundle of $X$ and $[X]$ is the class of $X$ in the Chow ring. The degree of $c(X)$ is the topological Euler characteristic of $X$: \n\\begin{equation*}\n\\chi(X)=\\int_X c(X).\n\\end{equation*}\nThe following Lemma gives an important functorial property of the degree.\n\\begin{lem}[{\\cite[Chap. 1, p. 13]{Fulton.Intersection}}]  \\label{lem:Push} Let $f:X\\longrightarrow Y$  be a proper map between varieties. \n For any class $\\alpha$ in the Chow ring $A_*(X)$ of $X$:\n$$\\int_X \\alpha=\\int_Y f_* \\alpha.$$\n\\end{lem}\n Lemma \\ref{lem:Push} means that an intersection number in $X$ can be computed in $Y$ through a pushforward of a proper map $f:X\\longrightarrow Y$. \n This simple fact  has far-reaching consequences as it allows us to express the topological invariants of an elliptic fibration in terms of those of the base. \n\n\n\nLet $X$ be a projective variety with at worst canonical Gorenstein singularities. \nWe denote the canonical class by $K_X$. \n\\begin{defn}\nA birational projective  morphism $\\rho:Y\\longrightarrow X$ is called a \\emph{crepant desingularization} of $X$ if $Y$ is smooth and \n$K_Y=\\rho^* K_X$. \n\\end{defn}\n\n\\begin{defn}\nA resolution of singularities of a variety $Y$ is a proper surjective birational morphism $\\varphi:\\widetilde{Y}\\longrightarrow Y$  such that  \n$\\widetilde{Y}$ is nonsingular\nand  $\\varphi$ is an isomorphism away  from the singular  locus of $Y$. In other words, $\\widetilde{Y}$ is nonsingular and  if $U$ is the singular locus of $Y$, $\\varphi$ maps $\\varphi^{-1}(Y\\setminus U)$  isomorphically  onto $Y\\setminus U$.  \n A \\emph{crepant resolution of singularities}  is a resolution of singularities such that  $K_Y=f^* K_X$. \n\\end{defn}\n\n\n\n\\subsection{Pushforward formulas}\nWhen pushing forward blowups of  a projective bundle $\\pi: X_0=\\mathbb{P}[\\mathscr{O}_B\\oplus\\mathscr{L}^{\\otimes 2} \\oplus \\mathscr{L}^{\\otimes 3}]\\longrightarrow B$, the key ingredients are the following three theorems. \nThe first one is a theorem of Aluffi which gives the Chern class after a blowup along a local complete intersection. \nThe second theorem is a pushforward theorem that provides a user-friendly method to compute invariant of the blowup space in terms of the original space. \nThe last theorem is a direct consequence of functorial properties of the Segre class and gives a simple method to pushforward analytic expressions in the Chow ring of the projective bundle $X_0$ to  the Chow ring of its base. \nWe follow mostly \\cite{Euler}. \n\n\n\\begin{thm}[Aluffi, {\n{\\cite[Lemma 1.3]{Aluffi_CBU}}}]\n\\label{Thm:AluffiCBU}\nLet $Z\\subset X$ be the  complete intersection  of $d$ nonsingular hypersurfaces $Z_1$, \\ldots, $Z_d$ meeting transversally in $X$.  Let  $f: \\widetilde{X}\\longrightarrow X$ be the blowup of $X$ centered at $Z$. We denote the exceptional divisor of $f$  by $E$. The total Chern class of $\\widetilde{X}$ is then:\n\\begin{equation}\nc( T{\\widetilde{X}})=(1+E) \\left(\\prod_{i=1}^d  \\frac{1+f^* Z_i-E}{1+ f^* Z_i}\\right)  f^* c(TX).\n\\end{equation}\n\\end{thm}\n\n\\begin{lem}[See {\\cite{Aluffi_CBU,Euler}}] \\label{Thm:PushE}\n\\label{lem:symhom}\nLet  $f: \\widetilde{X}\\longrightarrow X$ be the blowup of $X$ centered at $Z$. We denote the exceptional divisor of $f$  by $E$. Then \n\\begin{equation}\n f_* E^n=\n (-1)^{d+1} h_{n-d} (Z_1, \\ldots, Z_d) Z_1\\cdots Z_d,\\nonumber\n\\end{equation}\nwhere $h_i(x_1, \\ldots, x_k)$ is the complete homogeneous symmetric polynomial of degree $i$ in $(x_1, \\ldots, x_k)$ with the convention that $h_i$ is identically zero for $i<0$ and $h_0=1$.  \n\\end{lem}\n\n\n\\begin{thm}[Esole--Jefferson--Kang,  see  {\\cite{Euler}}] \\label{Thm:Push}\n    Let the nonsingular variety $Z\\subset X$ be a complete intersection of $d$ nonsingular hypersurfaces $Z_1$, \\ldots, $Z_d$ meeting transversally in $X$. Let $E$ be the class of the exceptional divisor of the blowup $f:\\widetilde{X}\\longrightarrow X$ centered \nat $Z$.\n Let $\\widetilde{Q}(t)=\\sum_a f^* Q_a t^a$ be a formal power series with $Q_a\\in A_*(X)$.\n We define the associated formal power series  ${Q}(t)=\\sum_a Q_a t^a$, whose coefficients pullback to the coefficients of $\\widetilde{Q}(t)$. \n Then the pushforward $f_*\\widetilde{Q}(E)$ is\n $$\n  f_*  \\widetilde{Q}(E) =  \\sum_{\\ell=1}^d {Q}(Z_\\ell) M_\\ell, \\quad \\text{where} \\quad  M_\\ell=\\prod_{\\substack{m=1\\\\\n m\\neq \\ell}}^d  \\frac{Z_m}{ Z_m-Z_\\ell }.\n $$ \n\\end{thm}\n\n\n\\begin{thm}[{See  \\cite{Euler} }]\\label{Thm:PushH}\nLet $\\mathscr{L}$ be a line bundle over a variety $B$ and $\\pi: X_0=\\mathbb{P}[\\mathscr{O}_B\\oplus\\mathscr{L}^{\\otimes 2} \\oplus \\mathscr{L}^{\\otimes 3}]\\longrightarrow B$ a projective bundle over $B$. \n Let $\\widetilde{Q}(t)=\\sum_a \\pi^* Q_a t^a$ be a formal power series in  $t$ such that $Q_a\\in A_*(B)$. Define the auxiliary power series $Q(t)=\\sum_a Q_a t^a$. \nThen \n$$\n\\pi_* \\widetilde{Q}(H)=-2\\left. \\frac{{Q}(H)}{H^2}\\right|_{H=-2L}+3\\left. \\frac{{Q}(H)}{H^2}\\right|_{H=-3L}  +\\frac{Q(0)}{6 L^2},\n$$\n where  $L=c_1(\\mathscr{L})$ and $H=c_1(\\mathscr{O}_{X_0}(1))$ is the first Chern class of the dual of the tautological line bundle of  $ \\pi:X_0=\\mathbb{P}(\\mathscr{O}_B \\oplus\\mathscr{L}^{\\otimes 2} \\oplus\\mathscr{L}^{\\otimes 3})\\rightarrow B$.\n\\end{thm}\n\nSince all the blowups used in this paper have  centers that are complete intersections of two or three smooth divisors, the following two Lemmas are all that is needed to compute pushforwards under such blowups when the base is a threefold. \nThey are direct consequences of Lemma \\ref{Thm:PushE}.\n\\begin{lem}\\label{lem:Push2}\nFor a blowup  $f:\\widetilde{X}\\longrightarrow X$ with center a transverse intersection of two divisors of class $Z_1$ and $Z_2$, we have \n\\begin{align}\\nonumber\n& f_* E=0, \\qquad f_* E^2=-Z_1 Z_2, \\qquad \\   \\   f_*E^3= -(Z_1 +Z_2)Z_1 Z_2, \\\\\n & f_* E^4=-(Z_1^2+Z_2^2+ Z_1 Z_2) Z_1 Z_2, \n\\quad f_* E^5= -(Z_1+Z_2)(Z_1^2+Z_2^2)Z_1 Z_2 .\n\\end{align}\n\\end{lem}\n\n\\begin{lem}\\label{lem:Push3}\nFor a blowup  $f:\\widetilde{X}\\longrightarrow X$ with center a transverse intersection of three divisors of class $Z_1$, $Z_2$, and $Z_3$, we have \n\\begin{align}\\nonumber\n& f_* E=0, \\qquad f_* E^2=0, \\qquad  f_*E^3= Z_1 Z_2 Z_3, \\\\\n&  f_* E^4=(Z_1+Z_2) Z_1 Z_2 Z_3, \\quad   f_* E^5=  (Z_1^2+Z_2^2+Z_3^2+Z_1 Z_2+Z_1 Z_3+Z_2 Z_3)Z_1 Z_2 Z_3.\n\\end{align}\n\\end{lem}\n\\begin{lem}[\\cite{Euler,Fullwood:SVW}] \\label{lem:PushH}\nGiven the projective bundle $ \\pi:X_0=\\mathbb{P}(\\mathscr{O}_B \\oplus\\mathscr{L}^{\\otimes 2} \\oplus\\mathscr{L}^{\\otimes 3})\\rightarrow B$, denoting the first Chern class of $\\mathscr{L}$ by $L$, we have:\n$$\n\\begin{aligned}\n\\pi_* 1 &= 0, \\quad \\pi_*H= 0, \\quad \\pi_*H^2= 1 , \\quad  \\pi_* H^3 = -5 L, \\quad\\pi_*  H^4= 19 L^2, \\quad \\pi_* H^5= -65 L^3,\\\\\n\\pi_* H^{k} &=\\left[(-2)^{k-1} -(-3)^{k-1} \\right] L^{k-2} \\quad n\\geq 1 \\nonumber.\n\\end{aligned}\n$$\n\\end{lem}\nUsing Lemma \\ref{lem:PushH}, it is then direct to show that \n\\begin{equation}\n\\pi_* \\Big(H^k (3H+6 L)\\Big)= -(-3)^k  L^{k-1}, \\quad k\\geq 1\n\\end{equation}\n\\subsection{Example: the second Chern class under a crepant birational map}\nWe now look at the behavior of the second Chern class of a divisor $V$ under a crepant blowup whose center is a complete intersection of $n$ smooth  divisors intersecting transversally in a smooth  ambient space $X$.\n\n\\begin{thm}\\label{Thm.C2}\nConsider a smooth variety $X$ and a blowup $f:\\widetilde{X}\\to X$ with center  the complete intersection of $n\\geq 2$ smooth divisors $Z_i$ intersecting transversally. We denote  the exceptional divisor by $E$. Consider a divisor  $V$ in $X$. If we ask for the restriction of the blowup to $V$ to be crepant, then the class of $V$ is such that its  proper transform  is  $\\widetilde{V}=f^* V-(n-1)E$. It follows that \n$$\nf_* \\Big(c_2 (T\\widetilde{V})\\cdot  \\widetilde{V} \\Big) =c_2(TV) \\cdot V.\n$$\n\\end{thm}\n\\begin{proof}\nUsing Aluffi's formula (Theorem \\ref{Thm:AluffiCBU}) and the adjunction formula, we have \n$$\nc(T\\widetilde{V})=(1+E)\\Big({\\prod_{i=1}^n \\frac{(1 +f^* Z_i -E)}{(1+f^*Z_i)}}\\Big) \\frac{f^* c(TX)}{ (1+f^* V-(n-1) E)}.\n$$ \nIn the case $n-2$, by a direct expansion: \n$$\nc_1(T\\widetilde{V})=f^*c_1(TX)-f^*V=f^* c_1 (TV), \\quad c_2(T\\widetilde{V})= f^* c_2(Y)-V(Y-Z_1-Z_2) -V^2,\n$$\nwhere   $c_2(TV)=c_2(TX)-c_1 V+ V^2$. It follows that \n$$f_*c_2 (T\\widetilde{V})\\cap \\widetilde{V}=f_* \\Big[f^* c_2(TV)\\cap [V] +f^*Vf^*( Z_1 + Z_2) E - f^*(Z1+Z2)E^2 + E^3]$$\nBy applying Lemma \\ref{lem:Push2}, we find \n$$f_*c_2 (T\\widetilde{V})\\cap (\\widetilde{V})= c_2(TV)\\cap [V].$$\n\nIn the case $n=3$, there is no contribution of order $E^3$ in $f_*c_2 (T\\widetilde{V})\\cap (\\widetilde{V})$ and by Lemma \\ref{lem:Push3} we have $ f_* E=f_* E^2=0$. \n\n$$\n\\begin{aligned}\nf_* \\Big(c_2 (T\\widetilde{V})\\cdot  \\widetilde{V} \\Big) &=f^*[c_2(V) \\cdot V]\n\\\\ & \\quad \\  + (2 c_1 V-2 c_2-4 V^2+V Z_1+V Z_2+V Z_3) E+(4 V-2 Z_1-2 Z_2-2 Z_3)E^2.\n\\end{aligned}\n$$\nIn the case $n\\geq 4$, the theorem is trivial because  \n$c_2 (T\\widetilde{V})\\cdot  \\widetilde{V}$ is at most cubic in $E$ and  $f^* E^i=0$ for  $i\\leq 3$ by Lemma \\ref{Thm:PushE}. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Example: Todd class of a flat fibration of genus-$g$ curves. }\\label{sec:Toddg}\n\\begin{thm}[Esole-Fullwood-Yau, {\\cite[Theorem A.1]{EFY}}]\nLet $\\varphi:X\\to B$ be a proper and flat morphism between smooth projective varieties such that the generic fiber of $\\varphi$ is a smooth curve of genus $g$. Then \n\\begin{equation}\n\\varphi_* Td (X)=  \\Big(1-ch(\\varphi_* \\omega_{X\/B}^\\vee) \\Big) Td (B),\n\\end{equation}\nwhere $\\omega_{X\/B}$ is the relative dualizing sheaf of the fibration. \n\\end{thm}\nWhen the variety $Y$ is smooth, $\\omega_{X\/B}=\\omega_Y\\otimes (\\varphi^*\\omega_B)^\\vee$. \nIn particular, in the case of an elliptic fibration, we get \n\\begin{equation}\n\\varphi_* Td (Y)=  (1-e^{-L}) Td (B),\n\\end{equation}\nwhere $L=c_1(\\mathscr{L})$ and $\\mathscr{L}$ is the fundamental line bundle of the Weierstrass model. \n\nThe previous theorem  shows that the holomorphic Euler characteristic depends only on the base and the line bundle $\\mathscr{L}$. \nIn particular, for a $G$-model, it does not depend on the Kodaira type. %\n\n\nthe holomorphic arithmetic genus of $Y$ is the same as $\\chi(W,\\mathscr{O}_W)$ where $W$ is the Weierstrass divisor defined as the zero locus of a smooth section of $\\mathscr{L}$. See \n\\cite[Appendix A]{EFY}.\n\\begin{equation}\n\\chi_0(Y)=\\int_Y \\varphi_* Td (Y)= \\chi(W,\\mathscr{O}_W).\n\\end{equation}\n\n\n\\subsection{Example: Smooth Weierstrass model}\nThe Euler characteristic of a smooth Weierstrass model $Y_0\\to B$ with fundamental line bundle $\\mathscr{L}$ is \n\\begin{equation}\n\\chi(Y_0)=12L\\int_B \\frac{1}{1+6L} c(TB),\\quad \\text{where}\\   L=c_1(\\mathscr{L}). \n\\end{equation}\n\nFor an elliptic fourfold given by a smooth Weierstrass model:\n\\begin{align}\n\\chi_0(Y_0) &= \\frac{1}{12}\\int_B L (c_1^2+c_2-3 c_1 L+2 L^2),\\quad\n\\chi_1(Y_0) =\n-\\frac{1}{3}\\int_B L (2 c_1^2+5 c_2-54 c_1 L+232 L^2),\\\\\n\\chi_2(Y_0) &=  \n-\\frac{1}{2} \\int_B L (3 c_1^2-17 c_2+71 c_1 L-554 L^2).\n\\end{align}\nUsing the adjunction formula, we have\n\\begin{equation}\nc(TY_0)= \\frac{(1+H)(1+H+2L)(1+H+3L)}{(1+3H+6L)} (1+c_1(TB)+c_2(TB)+c_3(TB))\n\\end{equation}\n We abuse notation and do not write the pullback. By expanding, we get: \n\\begin{align}\\label{Eq.ExpC}\n\\begin{aligned}\nc_1(TY_0)&= (c_1-L),\\quad c_2(TY)=c_2-c_1 L+13 H L+12 L^2+3 H^2, \\\\\nc_3(TY_0)&=-72 L^3+12 c_1 L^2-c_2 L+c_3+H (13 c_1 L-108 L^2)+H^2 (3 c_1-52 L)-8 H^3\\\\\nc_4(TY_0)&=(-72 c_1 L^3 + 12 c_2 L^2 - c_3 L + 432 L^4)+H (-108 c_1 L^2 + 13 c_2 L + 864 L^3)\\\\\n& \\quad + (-52 c_1 L+3 c_2+636 L^2)H^2+(204 L - 8 c_1)H^3 +24 H^4.\n\\end{aligned}\n\\end{align}\n\\begin{thm}[Chern and Pontryagin numbers of a smooth Weierstrass fourfold]\\label{Thm.CP.4}\nLet $B$ be a projective threefold and $\\varphi:Y_0\\to B$ be a smooth Weierstrass model with fundamental line bundle $\\mathscr{L}$. Denoting the first Chern class of $\\mathscr{L}$ by $L$  \nand writing the $i$th Chern class of the base $B$ simply as  $c_i$, \nthe Chern and Pontryagin numbers of $Y_0$ are\n\\begin{align}\n \\int_{Y_0} c_1^4(TY_0)&=0, \\\\\n \\int_{Y_0} c_1^2(TY_0) c_2(TY_0) &=12 \\int_BL (c_1-L)^2, \\\\\n \\int_{Y_0} c_2^2(TY_0) &=24  \\int_BL (6 L^2-c_1 L+c_2),\\\\\n \\int_{Y_0}c_1(TY_0) c_3(TY) &= 12  \\int_BL (c_1-6 L) (c_1-L),\\\\\n\\int_{Y_0} c_4(TY_0) &=12 \\int_B L (36 L^2-6 L c_1+c_2),\n\\end{align}\n\\begin{align}\n \\int_{Y_0}p_2 (TY_0)&=-24  \\int_BL (-c_1^2+2 c_2+36 L^2),\\\\\n \\int_{Y_0} p_1^2(TY_0) &=-48 \\int_B L (c_1^2-2 c_2-11 L^2).\n\\end{align}\n\\end{thm}\n\\begin{proof}\nFor each entries, we use \\eqref{Eq.ExpC} and compute \n$$\\int_{Y_0} A= \\int_{X_0} A \\cdot (3 H + 6L)=3\\int_B \\pi_*\\Big[A \\cdot  (H+2L)\\Big].$$ \nThe pushforward is then evaluated using Theorem \\ref{Thm:PushH} or Lemma \\ref{lem:PushH}. \n\nFor instance, $\\int_{Y_0} c_4(TY_0)=\\int_{X_0} c_4(TY_0)(3H+6L)$. Since terms independent of $H$ or linear in $H$ will not contribute (see Lemma \\ref{lem:PushH}), we have \n\\begin{align}\n\\int_{Y_0} c_4(TY_0)&=\\int_{X_0} c_4(TY_0)(3H+6L) \\nonumber \\\\ \n&=\\int_B \\pi_* \\Big(H^3 (-204 c_1 L + 9 c_2 + 3132 L^2) + \n H^2 (-636 c_1 L^2 + 57 c_2 L + 6408 L^3) + \\nonumber \\\\\n &\\quad\\quad  H^4 (756 L - 24 c_1) + 72 H^5\\Big)\\nonumber\\\\\n &=12 \\int_B L (36 L^2-6 L c_1+c_2)\\nonumber\n \\end{align}\n Using  Lemma \\ref{lem:PushH}, we implement the pushforward with the substitution $H^2\\to 1$, $H^3\\to -5L$, $H^4\\to 19 L^2$, $H^5\\to -65 L^3$, \n which gives the final answer. \n\\end{proof}\n As a direct application of Theorem \\ref{Thm.CP.4}, we compute the following invariants for a smooth Weierstrass model. \n \\begin{thm}[$L$-genus, $\\hat{A}$-genus, and $X_8$ invariant of a smooth Weierstrass model]\n Let $B$ be a projective threefold and $\\varphi:Y_0\\to B$ be a smooth Weierstrass model with fundamental line bundle $\\mathscr{L}$. Denoting the first Chern class of $\\mathscr{L}$ by $L$ and the $i$th Chern class of the base $B$ simply by  $c_i$, then, we have\n\\begin{align}\n45\\sigma &=\\int_{Y_0}\\Big(7p_2(TY_0)-p_1^2(TY_0)\\Big)\\ \\   =120\\int_B  L(-c_1^2+2 c_2+46 L^2),\\\\\n5760\\int_{Y_0}\\hat{A}_2&=\\int_{Y_0}\\Big(7p_1^2(TY_0)-4 p_2(TY_0)\\Big)=240\\int_B  L(-c_1^2+2 c_2+L^2),\\\\\n192X_8&=\\int_{Y_0}\\Big(p_1^2(TY_0)-4p_2(TY_0)\\Big)\\ \\  =48\\int_B  L (c_1^2-2 c_2-61 L^2).\n\\end{align}\n\\end{thm}\n\\begin{proof}\nTrue by linearity from the quantities computed in Theorem \\ref{Thm.CP.4}. \n\\end{proof}\n\n\n\n\n\\subsection{Example: the SU($2$)-model}\nIn this section, we discuss in detail the computation of Chern numbers  of the SU(2)-model. The result is independent of a choice of a possible Kodaira fiber realizing the Dynkin diagram of type A$_1$ because  type I$_2$ and III are resolved by the same blowup. The Weierstrass equations defining SU(2)-models are given in section \\ref{sec:Tate}. The Weierstrass equation is defined in the ambient space as the projective bundle $X_0$, where the projection map is $\\pi: X_0=\\mathbb{P}_B[\\mathscr{O}_B\\oplus \\mathscr{L}^{\\otimes 2}\\oplus \\mathscr{L}^{\\otimes 3}]\\to B$ and the defining equation is a section of $\\mathscr{O}(3)\\otimes\\mathscr{L}^{\\otimes 6}$.\nHence, we find \n\\begin{align}\nc(TX_0)&=(1+H)(1+H+3\\pi^* L)(1+H+2\\pi^* L) \\pi^*c(TB),\\nonumber\\\\\nc(TY_0)&=\\frac{c(X_0)}{1+3H+6\\pi^* L},\\nonumber\n\\end{align}\n where  $L=c_1(\\mathscr{L})$ and $H=c_1(\\mathscr{O}_{X_0}(1))$ is the first Chern class of the dual of the tautological line bundle of  $ \\pi:X_0=\\mathbb{P}(\\mathscr{O}_B \\oplus\\mathscr{L}^{\\otimes 2} \\oplus\\mathscr{L}^{\\otimes 3})\\rightarrow B$. \n We denote by $S=V(s)$ the Cartier divisor supporting the fiber I$_2$ or III with dual graph of Dynkin type $\\widetilde{A}_1$. \n\nThe singular elliptic fibration of an SU(2)-model is resolved by a unique blowup with the center $(x,y,s)$, which we denote as $f: X_1 \\longrightarrow X_0$ with the exceptional divisor $E_1$.\nThe center is a complete intersection of hypersurfaces $V(x)$, $V(y)$, and $V(s)$, whose classes are respectively \n\\begin{equation}\\nonumber\n Z_1= 2\\pi^*L+H, \\quad Z_2= 3\\pi^* L+H, \\quad Z_3=\\pi^* S.\n\\end{equation}\nThe proper transform  of the elliptic fibration $Y_0$ is denoted as $Y$, and is obtained from the total transform of $Y$ by removing $2E_1$. It follows that the class of $Y$ in $X_1$ is \n\\begin{align}\\nonumber\n[Y]=[f^*(3H+6 \\pi^* L)-2 E_1]\\cap[X_1].\n\\end{align}\nUsing Theorem \\ref{Thm:AluffiCBU}, we have the following Chern class for $X_1$\n\\begin{align} \\nonumber\nc(TX_1)&=(1+E_1)\\frac{(1+f^* Z_1-E_1)(1+f^* Z_2 -E_1)(1+f^* Z_3-E_1)}{(1+f^* Z_1)(1+f^* Z_2)(1+f^* Z_3)} f^* c(TX_0).\n\\end{align}\nThe adjunction formula gives\n\\begin{align} \\nonumber\nc(TY)&=\\frac{(1+E_1)(1+f^* Z_1-E_1)(1+f^* Z_2 -E_1)(1+f^* Z_3-E_1)}{(1+3f^*H+6f^*\\pi^* L-2E_1)(1+f^* Z_1)(1+f^* Z_2)(1+f^* Z_3)} f^* c(TX_0).\n\\end{align}\nConcretely, we replace $c(TB)$ by the Chern polynomial $c_t(TB)=1+c_1t + c_2t^2 + c_3 t^3$, $L$ by $ Lt$, and $S$ by $ St$.  Then, $c_i(TY)$ is given by the coefficient of $t^i$ in the Taylor expansion centered at $t=0$.\n From the expansion, we get the following expression (to ease the notation, we do not write the pullbacks. Thus, by $c_i$, $L$, $S$, and $H$ we mean $f^* \\pi^* c_i$, $f^*\\pi^* L$, $f^*\\pi^* S$, and $f^*H$ respectively).\n\\begin{align}\nc_1(TY) &= (c_1-L),\\\\\nc_2(TY) &= (c_2-c_1 L+12 L^2)+3 H^2+13 H  L+E_1(-4  H-7   L+S),\\\\\nc_3(TY) &= 3 c_1 H^2+13 c_1 H L+12 c_1 L^2-c_2 L+c_3-8 H^3-52 H^2 L-108 H L^2-72 L^3\\nonumber\\\\\n&\\  \\   +E_1 (-4 c_1 H-7 c_1 L+c_1 S+16 H^2+66 H L+66 L^2-L S-S^2)\\\\\n&\\  \\   +E_1^2 (-10 H-19 L+S)+2 E_1^3,\\nonumber\\\\\nc_4(TY) & = (-8 c_1 H^3-52  H^2c_1 L-108 c_1 H L^2-72 c_1 L^3+3 c_2 H^2+13 c_2 H L\\nonumber \\\\\n&\\  \\  +12 c_2 L^2-c_3 L+24 H^4+204 H^3 L+636 H^2 L^2+864 H L^3+432 L^4)\\nonumber\\\\\n&\\  \\   + E_1(16 c_1 H^2+66 c_1 H L+66 c_1 L^2-c_1 L S-c_1 S^2-4 c_2 H-7 c_2 L+c_2 S  \\\\\n&\\   \\  -64 H^3-398 H^2 L+3 H^2 S-810 H L^2+13 H L S-540 L^3+12 L^2 S+L S^2+S^3)\\nonumber\\\\\n&\\  \\ + E_1^2 (-10 c_1 H-19 c_1 L+c_1 S+59 H^2+239 H L-4 H S+240 L^2-8 L S-S^2) \\nonumber\\\\\n&\\   \\  + E_1^3( 2 c_1-22 H-45 L+S)+3 E_1^4.\\nonumber\n\\end{align}\nWe can now compute $\\int_Y c_2^2(TY)$ and apply the pushforward formula of Lemma \\ref{lem:Push3} \n\\begin{align}\n\\begin{split}\n\\int_{Y} c_2^2(TY)\n=& \\int_{X_1}  c_2^2(TY) [Y]=\\int_{X_1}  c_2^2(TY) (f^*(3H+6 \\pi^* L)-2 E_1)\\\\\n =&\\int_{X_0} f_*\\Big( c_2^2(TY) (f^*(3H+6 \\pi^* L)-2 E_1)\\Big) \\\\\n  =&\\int_{B}\\pi_* f_*\\Big( c_2^2(TY) (f^*(3H+6 \\pi^* L)-2 E_1)\\Big) \\\\\n=&24\\int_B L (c_2-c_1L+6 L^2)-2\\int_BS(7L-S)^2.\n\\end{split}\n\\end{align}\n In the second line, we use $\\int_Y A=\\int_{X_1} A\\cap [Y]$. In the third line, we use the functorial property $\\int_{X_1}A=\\int_{X_0} f_* A$ for a proper map $f:X_1\\to X_0$. \nIn the fourth line, we use again the functional property of the degree, but this time, for the proper map $\\pi: X_0\\to B$. \nIn the last line, we use Lemma \\ref{lem:Push3} to compute the pushforward of $f$ to $X_0$ and then  Lemma \\ref{lem:PushH} to compute of $\\pi$ to the base $B$. \n\n\nUsing the same logic, we now  compute $\\int_Y c_3(TY) f^*\\pi^*\\alpha$. \n\\begin{align}\n\\begin{split}\n\\int_Yc_3(TY) f^*\\pi^*\\alpha &=\\int_{X_1}c_3(TY)  (f^*(3H+6 \\pi^* L)-2 E_1) f^*\\pi^*\\alpha\\\\\n&=\\int_{X_0} f_* \\Big(c_3(TY) (3H+6L-2E_1)\\Big) \\pi^* \\alpha\\\\\n&=\\int_{B}\\pi_*f_* \\Big(c_3(TY) (3H+6L-2E_1)\\Big) \\alpha\\\\\n&=\\int_B 12L(c_1-6L)\\alpha +6\\int_B (5L-S) S \\alpha.\n\\end{split}\n\\end{align}\n In the first line, we use $\\int_Y A=\\int_{X_1} A\\cap [Y]$.\nThe second and third lines are projection formula for $f$ and $\\pi$.\nIn the last line, we use Lemma \\ref{lem:Push3} to compute the pushforward of $f$ to $X_0$ and then  Lemma \\ref{lem:PushH} to compute of $\\pi$ to the base $B$. \n\n\\section{$G$-models}\\label{sec:GModels}\n\n\n\\subsection{Tate's forms for $G$-models}\\label{sec:Tate}\nThe variable $s$ is a section of the line bundle $\\mathscr{O}_B(S)$. In other words, the zero locus locus $V(s)$ is the divisor $S$ supporting the singular fiber.\n\\begin{align}\n\\label{eqn:SO}\n&\\text{I}_2^{\\text{s}} &&\\text{SU(2)}&:&\\quad \t&&y^2z+a_1 x y z + a_{3,1} s y z =x^3 + a_{2,1} s  x^2z + a_{4,1}s x z^2+a_{6,2} s^2 z^3,\\\\\n&\\text{I}_{2n}^{\\text{ns}} &&\\text{USp($2n$)}&:&\\quad \t&&y^2z =x^3 + a_2 x^2z + a_{4,n} s^n x z^2+a_{6,2n} s^{2n} z^3,\\\\\n&\\text{I}_{2n+1}^{\\text{ns}} &&\\text{USp($2n$)}&:&\\quad \t&&y^2z =x^3 + a_2 x^2z + a_{4,n+1} s^{n+1} x z^2+ a_{6,2n+1} s^{2n+1} z^3,\\\\\n&\\text{I}_{2n}^{\\text{s}} &&\\text{SU($2n$)}&:&\\quad \t&&y^2z+ a_1 x y z =x^3 + a_{2,1} s x^2z + a_{4,n} s^n x z^2+a_{6,2n} s^{2n} z^3,\\\\\n&\\text{I}_{2n+1}^{\\text{s}} &&\\text{SU($2n+1$)}&:&\\quad \t&&y^2z+a_1 x y z+a_{3, n} s^n y z^2   =x^3 + a_{2,1}s x^2z + a_{4,n+1} s^{n+1} x z^2+a_{6,2n+1} s^{2n+1} z^3,\\\\\n&\\text{I}_0^{*\\text{ss}}\\quad &&\\text{Spin(7)}&:& \\quad && y^2z= x^3 + a_{2,1} s x^2 z+ a_{4,2} s^2 x z^2+ a_{6,4} s^{4} z^3,\\\\\n&\\text{I}_0^{*\\text{s}}\\quad &&\\text{Spin(8)}&:& \\quad && y^2z= (x-x_1sz)(x-x_2 sz)(x-x_3 sz)+  s^2 r x^2z + s^3 q x z^2+ s^4 t z^3, \\\\\n&\\text{III}&&\\text{SU(2)}&:&\\quad &&y^2z  =x^3  + s a_{4,1} xz^2 + s^2 a_{6,2} z^3,\\\\\n&\\text{IV}^{\\text{ns}} &&\\text{SU(2)}&:&\\quad  &&y^2z  =x^3 +s^2 a_{4,2} xz^2 + s^2 a_{6,2} z^3,\\\\\n&\\text{IV}^{\\text{s}}\\quad  &&\\text{SU(3)}&:& \\quad &&  y^2z + a_{3,1} s y z^2 =x^3  + s^2 a_{4,2} xz^2 + s^3a_{6,3} z^3,\\\\\n\t&\\text{I}_0^{*\\text{ns}}\\quad  &&\\text{G}_2&:& \\quad &&y^2z=x^3+s^{2} a_{4,2}xz^2 +s^3 a_{6,3}z^3,\\\\\n&\\text{IV}^{*\\text{ns}} \\quad &&\\text{F}_4&:&\\quad \t&&y^2z =x^3 + s^{3} a_{4,3}xz^2 +s^4 a_{6,4}  z^3,\\\\\n&\\text{IV}^{*\\text{s}} \\quad\\ &&\\text{E}_6&:&\\quad &&y^2z+a_{3,2} s^2 y z^2   = x^3 +s^{3}  a_{4,3} x z^2 +  s^5 a_{6,5}z^3,\\\\\n&\\text{III}^{*} \\quad\\   &&\\text{E}_7&:&\\quad  &&y^2z =x^3 +s^3 a_{4,3}  xz+s^{5} a_{6,5} z^3, \\\\\n&\\text{II}^{*} \\quad\\  &&\\text{E}_8&:& \\quad &&y^2z=x^3+s^4 a_{4,4} xz^2 +s^5 a_{6,5} z^3.\n\\end{align}\n\n\\begin{table}[hbt]\n\\begin{center}\n\\scalebox{0.88}{$\t\t\t\n\\begin{array}{|c | c | c |} \n\\hline\n\\text{Algebra} & \\text{Dynkin diagram} & \\text{Kodaira type}\n\\\\\\hline \n\t\t\n &\t\t\t\\scalebox{.97}{$\\begin{array}{c}\\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}, fill=black] (1) at (0,0){};\n\t\t\t\\end{tikzpicture}\\end{array}\n\t\t$} \t\t\n\t\t\t\t& \\text{I}_1, \\quad \\text{II}\\\\\\hline\n\n\t\t\t\\text{SU}(2)\n &\t\t\t\\scalebox{.97}{$\\begin{array}{c}\\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}, fill=black] (1) at (0,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (2) at (1.3,0){};\n\t\t\t\t\\draw[thick] (0.15,0.1) --++ (.95,0);\n\t\t\t\t\\draw[thick] (0.15,-0.09) --++ (.95,0);\n\t\t\t\t\n\t\t\t\\end{tikzpicture}\\end{array}\n\t\t$} \t\t& \\text{I}_2^\\text{s}, \\quad \\text{I}_2^{\\text{ns}}, \\quad\\text{I}_3^{\\text{ns}},\\quad \\text{III},\\quad \\text{IV}^{\\text{ns}}\n\t\t\t\t\\\\\\hline\n{ \\text{SU}(\\ell) \\ (\\ell\\geq 3)}\n& \\scalebox{.97}{$\\begin{array}{c} \\begin{tikzpicture}\n\t\t\t\t\\node[draw,circle,thick,fill=black,scale=1.25,label=above:{1}] (0) at (90:1.1){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (1) at (-2,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{1}] (2) at (-1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{1}] (3) at (1,0){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (4) at (2,0){};\t\n\t\t\t\t\\draw[thick]   (2)--(1)--(0)--(4)--(3);\n\\draw[ultra thick, loosely dotted] (2) to (3) {};\n\n\t\t\t\\end{tikzpicture}\\end{array}$}            &{ \\text{I}_{\\ell}^\\text{s}   }    \\\\\\hline\n\n\n\t\t\t\t\t\t\n\t\t\t\t\t\t \n\t\t\t\t\t\t \n\t\t\t\t\t\n{\\text{Spin}(2(4+\\ell )) \\ (\\ell\\geq 0)}\n\t\t\t\t\t\t& \n\t\t\t\t\t\t\\scalebox{.97}{$\\begin{array}{c} \\begin{tikzpicture}\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black,label=above:{1}] (1) at (-.1,.7){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (2) at (-.1,-.7){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (3) at (1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (4) at (2.1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (5) at (3.3,0){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (6) at (4.6,.7){};\t\n\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (7) at (4.6,-.7){};\t\n\t\t\t\t\\draw[thick] (1) to (3) to (4);\n\t\t\t\t\\draw[thick] (2) to (3);\n\t\t\t\t\\draw[ultra thick, loosely dotted] (4) to (5) {};\n\t\t\t\t\\draw[thick] (5) to (6); \n\t\t\t\t\\draw[thick] (5) to (7);\n\t\t\t\\end{tikzpicture}\\end{array}$}&  \\text{I}_{\\ell}^{*\\text{s}}    \n\t\t\t\t\t\t\t\t\t\t\t\t  \\\\\\hline\n\t\t\t\t\t\t\t\t\t\t\t\t  \n\\text{E}_{6} \n&\t\t\t\\scalebox{.92}{$\\begin{array}{c}\\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black,label=below:{1}] (0) at (0,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{2}] (1) at (1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{3}] (2) at (1*2,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{2}] (3) at (1*3,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{1}] (4) at (1*4,0){};\n\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=left:{2}] (5) at (1*2,1*1){};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (6) at (1*2,1*2){};\n\t\t\t\t\\draw[thick] (0)--(1)--(2)--(3)--(4);\n\t\t\t\t\\draw[thick] (2)--(5)--(6);\n\n\t\t\t\\end{tikzpicture}\\end{array}$}  &  \\text{IV}^{*\\text{s}}   \n\t\t\n\t\t\t\\\\\\hline\n\\text{E}_{7}\t\n\t\t\t\t\t\t\t\t& \\scalebox{.94}{$\\begin{array}{c}\n \\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black, label=below:{1}] (0) at (0,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{2}] (1) at (1,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{3}] (2) at (1*2,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{4}] (3) at (1*3,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{3}] (4) at (1*4,0){ };\n\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{2}] (5) at (5,0){ };\n\t\t\t\t\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{1}] (6) at (6,0){ };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (7) at (3,1){ };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\\draw[thick] (0)--(1)--(2)--(3)--(4)--(5)--(6);\n\t\t\t\t\\draw[thick] (3)--(7);\n\n\t\t\t\\end{tikzpicture}\n\t\t\t\\end{array}$}\t\t&  \\text{III}^{*}    \\\\\\hline\t  \n\\text{E}_{8}\t\t\t\t\t\t\t  \n\t\t\t\t\t\t&\n\t\t\t\t\t\t \\scalebox{.94}{\n $\\begin{array}{c}\n \\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black,label=below:{1}] (0) at (0,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{2}] (1) at (1*1,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{3}] (2) at (1*2,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{4}] (3) at (1*3,0){ };\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{5}] (4) at (1*4,0){ };\n\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{6}] (5) at (1*5,0){ };\n\t\t\t\t\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{4}] (6) at (1*6,0){ };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{3}] (8) at (1*5,1*1){ };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=below:{2}] (7) at (1*7,0){ };\n\t\t\t\t\t\t\t\t\n\t\t\t\t\\draw[thick] (0)--(1)--(2)--(3)--(4)--(5)--(6)--(7);\n\t\t\t\t\\draw[thick] (5)--(8);\n\n\t\t\t\\end{tikzpicture}\n\t\t\t\\end{array}$}  &  {\\text{II}^{*} }  \\\\\\hline\n\t\t\t\t\t\t\t\t\t\t\t\t  \n\t\t\t\t\t\t\t\t\t\t\t\t   \\begin{array}{c}\n\t\t\t\t\t\t\t\t\t\t\t\t  \\\\\n{\\text{Spin}(2(3+\\ell)+1) \\ (\\ell\\geq 0)}   \\\\\n\t\t\t\t\t\t \\\\\n\t\t\t\t\t\t \\end{array}\n\t\t\t\t\t\t &\\scalebox{.97}{$\\begin{array}{c} \\begin{tikzpicture}\n\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black,label=above:{1}] (1) at (-.1,.7){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (2) at (-.1,-.7){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (3) at (1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (4) at (2.1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (5) at (3.3,0){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (6) at (4.6,0){};\t\n\t\t\t\t\\draw[thick] (1) to (3) to (4);\n\t\t\t\t\\draw[thick] (2) to (3);\n\t\t\t\t\\draw[ultra thick, loosely dotted] (4) to (5) {};\n\t\t\t\t\\draw[thick] (3.5,-0.05) --++ (.9,0){};\n\t\t\t\t\\draw[thick] (3.5,+0.05) --++ (.9,0){};\n\t\t\t\t\\draw[thick]\n\t\t\t\t\t(3.9,0) --++ (60:.25)\n\t\t\t\t\t(3.9,0) --++ (-60:.25);\n\t\t\t\\end{tikzpicture}\\end{array}$}\t\t& {\\begin{cases} \\text{I}_{0}^{*\\text{ss}}  \\quad \\text{for} \\quad \\ell= 0 \\\\  \t \\text{I}_{\\ell}^{*\\text{ns}}\\quad  \\text{for}\\quad \\ell\\geq 1\\end{cases} }\n\t\t\t\\\\\\hline\n\t\t\t\n{\\text{USp}(2(2+\\ell)) \\ (\\ell\\geq 0)}\n\t\t\t\t\t\t   &\n\t\t\t\\scalebox{.97}{$\\begin{array}{c} \\begin{tikzpicture}\n\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black,label=above:{1}] (1) at (-.2,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (3) at (.8,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (4) at (1.8,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (5) at (2.8,0){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (6) at (3.8,0){};\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (7) at (4.8,0){};\t\n\t\t\t\t\\draw[thick]   (3) to (4);\n\t\t\t\t\\draw[thick] (5) to (6);\n\t\t\t\t\\draw[ultra thick, loosely dotted] (4) to (5) {};\n\t\t\t\t\\draw[thick] (4.,-0.05) --++ (.6,0){};\n\t\t\t\t\\draw[thick] (4.,+0.05) --++ (.6,0){};\n\t\t\t\t\t\t\t\t\\draw[thick] (-.2,-0.05) --++ (.8,0){};\n\t\t\t\t\\draw[thick] (-.2,+0.05) --++ (.8,0){};\n\n\t\t\t\t\\draw[thick]\n\t\t\t\t\t(4.4,0) --++ (-120:.25)\n\t\t\t\t\t(4.4,0) --++ (120:.25);\n\t\t\t\t\t\\draw[thick]\n\t\t\t\t\t(0.2,0) --++ (-60:.25)\n\t\t\t\t\t(0.2,0) --++ (60:.25);\n\t\t\t\\end{tikzpicture}\\end{array}$}\n& {\\text{I}_{4+2\\ell}^{\\text{ns}},\\quad    \\text{I}_{5+2\\ell}^{\\text{ns}}}\n\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\\\\\\hline\n\\text{F}_4\n&\t\t\t\\scalebox{.97}{$\\begin{array}{c}\\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,fill=black,label=above:{1}] (1) at (0,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (2) at (1,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{3}] (3) at (2,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (4) at (3,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (5) at (4,0){};\n\t\t\t\t\\draw[thick] (1) to (2) to (3);\n\t\t\t\t\\draw[thick]  (4) to (5);\n\t\t\t\t\\draw[thick] (2.2,0.05) --++ (.6,0);\n\t\t\t\t\\draw[thick] (2.2,-0.05) --++ (.6,0);\n\t\t\t\t\\draw[thick]\n\t\t\t\t\t(2.4,0) --++ (60:.25)\n\t\t\t\t\t(2.4,0) --++ (-60:.25);\n\t\t\t\\end{tikzpicture}\\end{array}$} &  \\text{IV}^{*\\text{ns}}\n\t\t\t\t\n\t\t\t\t\\\\\\hline\n\n\t\t\t\n\\text{G}_2\n &\t\t\t\\scalebox{.95}{$\\begin{array}{c}\\begin{tikzpicture}\n\t\t\t\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}, fill=black] (1) at (0,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{2}] (2) at (1.3,0){};\n\t\t\t\t\\node[draw,circle,thick,scale=1.25,label=above:{1}] (3) at (2.6,0){};\n\t\t\t\t\\draw[thick] (1) to (2);\n\t\t\t\t\\draw[thick] (1.5,0.09) --++ (.9,0);\n\t\t\t\t\\draw[thick] (1.5,-0.09) --++ (.9,0);\n\t\t\t\t\\draw[thick] (1.5,0) --++ (.9,0);\n\t\t\t\t\\draw[thick]\n\t\t\t\t\t(1.9,0) --++ (60:.25)\n\t\t\t\t\t(1.9,0) --++ (-60:.25);\n\t\t\t\\end{tikzpicture}\\end{array}\n\t\t$} \t\t&  \\text{I}_0^{*\\text{ns}}\n\n\t\t\t\t\\\\\\hline\\end{array}$}\n\t\\end{center}\n\t\\caption{\n{ Affine Dynkin diagrams appearing as dual graphs of decorated Kodaira fibers. \n } \\label{Table:Affine}\n}\n\\end{table}\n\\clearpage\n\\subsection{Crepant resolutions}\n\n\\begin{table}[h!]\n\\centering\n\\renewcommand{\\arraystretch}{1.2\n$\n\\scalebox{.91}{$\n\t\\begin{array}{|c|c|c|}\n\t\\hline\n\t\\text{Group} & \\text{Fiber Type} & \\text{Crepant Resolution} \\\\\\hline\n\t\t\\text{SU}(2) & \n\t\t\\begin{matrix}\n\t\t\\text{I}_2^\\text{s},     \\text{I}^{\\text{ns}}_2\\\\\n\t\t {\\text{I}^{\\text{ns}}_3},  \\text{III} \\\\   \\text{IV}^{\\text{ns}} \n\t\t \\end{matrix} & \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\t\n \\end{tikzpicture}\\end{array} \\\\\\hline\n \n\t\t\\begin{matrix} \\text{SU}(3) \\\\ \\text{USp}(4) \\\\ \\text{G}_2 \\end{matrix} & \\begin{matrix} \\text{I}_3^\\text{s} , { \\text{IV}^\\text{s}} \\\\ \\text{I}_4^\\text{ns} \\\\ \\text{I}_0^{* \\text{ns}} \\end{matrix} & \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\t\t\n \\end{tikzpicture}\\end{array}  \\\\\\hline\n\t\\begin{matrix}\t\\text{SU}(4) \\\\ \\text{Spin}(7) \\end{matrix} &\\begin{matrix}  \\text{I}_4^\\text{s} \\\\ \\text{I}_0^{* \\text{ss}} \\end{matrix} & \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n \\end{tikzpicture}\\end{array}  \\\\\\hline\n\t\t\\text{Spin}(8) &    \\text{I}^{*\\text{s}}_0 &  \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (8,0){$X_3$};\n\t\\node(X4) at (11,0){$X_4$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x-x_i s z,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(x-x_j s z,e_2|e_4)$} (X3);\n \\end{tikzpicture}\\end{array} \\\\\\hline\n\t\t\\text{F}_4 &  \\text{IV}^{*\\text{ns}} &  \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (8,0){$X_3$};\n\t\\node(X4) at (11,0){$X_4$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(e_3,e_2|e_4)$} (X3);\n \\end{tikzpicture}\\end{array}\\\\\\hline\n \t\t\\text{SU}(5) &  \\text{I}_5^{\\text{s}} &  \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (8,0){$X_3$};\n\t\\node(X4) at (11,0){$X_4$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(x,y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(y,e_1|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(y,e_2|e_4)$} (X3);\n \\end{tikzpicture}\\end{array}\\\\\\hline\n \t\t\\text{Spin}(10) &  \\text{I}_1^{*\\text{s}} &  \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\node(X4) at (10,0){$X_4$};\n\t\\node(X5) at (12.5,0){$X_5$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(y,e_3|e_4)$} (X3);\n\t\\draw[big arrow] (X5) -- node[above,midway]{$(e_2,e_3|e_5)$} (X4);\n \\end{tikzpicture}\\end{array}\\\\\\hline\n \\text{SU}(6)  & \\text{I}_6^{\\text{s}} &   \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\node(X4) at (10,0){$X_4$};\n\t\\node(X5) at (12.5,0){$X_5$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(y,e_3|e_4)$} (X3);\n\t\\draw[big arrow] (X5) -- node[above,midway]{$(x,e_4|e_5)$} (X4);\n \\end{tikzpicture}\\end{array} \\\\\\hline\n \\text{SU}(7)  & \\text{I}_7^{\\text{s}} &   \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\node(X4) at (10,0){$X_4$};\n\t\\node(X5) at (10,-1){$X_5$};\n\t\\node(X6) at (7.5,-1){$X_6$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(y,e_3|e_4)$} (X3);\n\t\\draw[big arrow] (X5) -- node[right,midway]{$(x,e_4|e_5)$} (X4);\n\t\\draw[big arrow] (X6) -- node[above,midway]{$(y,e_5|e_6)$} (X5);\n \\end{tikzpicture}\\end{array} \\\\\\hline\n \\text{E}_6  & \\text{IV}^{*\\text{s}} &   \\begin{array}{c} \\begin{tikzpicture}\n\t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\node(X4) at (10,0){$X_4$};\n\t\\node(X5) at (10,-1){$X_5$};\n\t\\node(X6) at (7.5,-1){$X_6$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(e_2,e_3|e_4)$} (X3);\n\t\\draw[big arrow] (X5) -- node[right,midway]{$(y,e_3|e_5)$} (X4);\n\t\\draw[big arrow] (X6) -- node[above,midway]{$(y,e_4|e_6)$} (X5);\n \\end{tikzpicture}\\end{array} \\\\\\hline\n %\n \\text{E}_7 &\\text{III}^* &  \\begin{array}{c}\\begin{tikzpicture}\n \t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\node(X4) at (10,0){$X_4$};\n\t\\node(X5) at (10,-1){$X_5$};\n\t\\node(X6) at (7.5,-1){$X_6$};\n\t\\node(X7) at (5,-1){$X_7$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(y,e_3|e_4)$} (X3);\n\t\\draw[big arrow] (X5) -- node[right,midway]{$(e_2,e_3|e_5)$} (X4);\n\t\\draw[big arrow] (X6) -- node[above,sloped,midway]{$(e_2,e_4|e_6)$} (X5);\n\t\\draw[big arrow] (X7) -- node[above,sloped,midway]{$(e_4,e_5|e_7)$} (X6);\n \\end{tikzpicture}\t\\end{array}\\\\\\hline\n \\text{E}_8 & \\text{II}^*  & \\begin{array}{c}\\begin{tikzpicture}\n \t\\node(X0) at (0,0){$X_0$};\n\t\\node(X1) at (2.5,0){$X_1$};\n\t\\node(X2) at (5,0){$X_2$};\n\t\\node(X3) at (7.5,0){$X_3$};\n\t\\node(X4) at (10,0){$X_4$};\n\t\\node(X5) at (10,-1){$X_5$};\n\t\\node(X6) at (7.5,-1){$X_6$};\n\t\\node(X7) at (5,-1){$X_7$};\n\t\\node(X8) at (2.5,-1){$X_8$};\n\t\\draw[big arrow] (X1) -- node[above,midway]{$(x,y,s|e_1)$} (X0);\t\n\t\\draw[big arrow] (X2) -- node[above,midway]{$(y,e_1|e_2)$} (X1);\n\t\\draw[big arrow] (X3) -- node[above,midway]{$(x,e_2|e_3)$} (X2);\t\t\n\t\\draw[big arrow] (X4) -- node[above,midway]{$(y,e_3|e_4)$} (X3);\n\t\\draw[big arrow] (X5) -- node[right,midway]{$(e_2,e_3|e_5)$} (X4);\n\t\\draw[big arrow] (X6) -- node[above,midway]{$(e_4,e_5|e_6)$} (X5);\n\t\\draw[big arrow] (X7) -- node[above,midway]{$(e_2,e_4,e_6|e_7)$} (X6);\n\t\\draw[big arrow] (X8) -- node[above,midway]{$(e_4,e_7|e_8)$} (X7);\n \\end{tikzpicture}\t\\end{array} \\\\\\hline\n\t\\end{array}\n\t$}\n\t$\n\\caption{The blowup centers of the crepant resolutions. The variable $s$ is a section of the line bundle $\\mathscr{O}_B(S)$. In other words, the zero locus locus $V(s)$ is the divisor $S$ supporting the singular fiber  given in the second column. }\n\\label{tab:blowupcenters}\n\\end{table}\t\n\\clearpage \n\n\\subsection{Tables of results}  \n\n\n\\begin{table}[h!]\n\\begin{center}\n\\renewcommand{\\arraystretch}{2\n\\scalebox{1}{ $\n\\begin{array}{|c|c|c|c|}\n\\hline \n\\text{Algebra} & \\text{G} &\\text{Kodaira} & \\varphi_* \\Big(c_3(TY)[Y]\\Big) \\\\\n\\hline\n\\text{A}_1 & \\text{SU}(2) & \\displaystyle \n\\begin{matrix}\n\t\t\\text{I}_2^\\text{s},     \\text{I}^{\\text{ns}}_2,  {\\text{I}^{\\text{ns}}_3}\\\\\n\t\t  \\text{III}, \\text{IV}^{\\text{ns}} \n\\end{matrix}\n&  12 L (c_1-6 L)+30 L S-6 S^2  \\\\\n\\hline\n\\begin{array}{c}\n\\text{A}_2 \\\\\n\\text{C}_2 \\\\\n\\text{G}_2 \n\\end{array}& \\begin{array} {c}\n\\text{SU}(3) \\\\\n\\text{USp}(4) \\\\\n\\text{G}_2\n\\end{array}& \\begin{array} {c}\n\\text{I}^{\\text{s}}_3, \\ { \\text{IV}^{\\text{s}}} \\\\\n\\text{I}_4^{\\text{ns}} \\\\\n\\text{I}^{* \\text{ns}}_0\n\\end{array}  & 12L (c_1-6 L)+48 L S-12S^2  \\\\\n\\hline\n\\begin{array}{c}{ \n\\text{A}_3} \\\\\n\\text{B}_3 \\\\\n\\end{array}& \\begin{array} {c}\n\\text{SU}(4) \\\\\n\\text{Spin}(7) \\\\\n\\end{array} & \\begin{array} {c}\n\\text{I}^{\\text{s}}_4 \\\\\n\\text{I}^{* \\text{ss}}_0\n\\end{array}      & 12 L (c_1-6 L)+64 L S-20 S^2  \\\\\n\\hline\n\\begin{array}{c}\n\\text{D}_4 \\\\\n\\text{F}_4 \n\\end{array} & \\begin{array} {c}\n\\text{Spin}(8) \\\\\n\\text{F}_4\n\\end{array} & \\begin{array} {c}\n\\text{I}^{*\\text{s}}_0 \\\\\n \\text{IV}^{*\\text{ns}}\n\\end{array}      & 12L (c_1-6 L)+72 L S-24 S^2 \\\\\n\\hline \n\\text{A}_4 &\\text{SU}(5) &\\text{I}_5^\\text{s} & 12 L (c_1-6 L)+80 L S-30 S^2  \\\\\n\\hline \n\\text{D}_5 &\\text{Spin}(10) &\\text{I}_1^{*\\text{s}}& 12 L (c_1-6 L)+84 L S-32 S^2  \\\\\n\\hline \n\\text{A}_5 &\\text{SU}(6) &\\text{I}_6^\\text{s} & 12 L (c_1-6 L)+96 L S - 42 S^2  \\\\\n\\hline \n\\text{A}_6 &\\text{SU}(7) &\\text{I}_7^\\text{s} & 12 L (c_1-6 L)+112 L S - 56 S^2  \\\\\n\\hline \n\\text{E}_6 & \\text{E}_6 &  \\text{IV}^{*\\text{s}}  & 12 L (c_1-6 L)+90 L S-36 S^2  \\\\\n\\hline \n\\text{E}_7 & \\text{E}_7 & \\text{III}^{*} & 12 L (c_1-6 L)+98 L S-42 S^2  \\\\\n\\hline \n\\text{E}_8 & \\text{E}_8 &  \\text{II}^{*}      & 12L (c_1-6 L)+120 L S-60 S^2  \\\\\n\\hline \n\\end{array} \n$ }\n\\end{center}\n\\caption{Chern numbers after pushforwards to the base. The divisor $S$ is the one supporting the reducible Kodaira fiber corresponding to the type of the Lie algebra  $\\mathfrak{g}$. By definition, $L=c_1(\\mathscr{L})$ and  $c_i$ denotes the $i$th Chern class of the base of the fibration.}\n \\label{Table:ChernNumbers}\n\\end{table}\n\n \\clearpage \n\\begin{table}[htb]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5\n\\scalebox{.72}{$\n\\begin{array}{|c|c|c|c|c|c|}\n\\hline \n\\text{Group} & c_1^4 (TY) & c_1^2 (TY) c_2(TY) & c_2^2 (TY) & c_1(TY) c_3(TY) & c_4(TY) \\\\\\hline\n\\text{SU}(2) & 0 & 12 L (c_1-L)^2 & \\begin{array} {c}\n24 L (L (6 L-c_1)+c_2) \\\\\n-2 S (7 L-S)^2\n\\end{array} & \\begin{array} {c}\n12 L (c_1-6 L) (c_1-L) \\\\\n+6 S (c_1-L) (5 L-S)\n\\end{array} & \\begin{array} {c}\n12 L (6 L (6 L-c_1)+c_2) \\\\\n+6 L S (5 c_1-54 L) \\\\\n-6 S^2 (c_1-15 L)-6 S^3 \n\\end{array} \\\\\n\\hline\n\\begin{array} {c}\n\\text{SU}(3) \\\\\n\\text{USp}(4) \\\\\n\\text{G}_2 \n\\end{array} & 0 & 12 L (c_1-L)^2 & \\begin{array} {c}\n24 L (L (6 L-c_1)+c_2) \\\\\n-8 S \\left(19 L^2-8 L S+S^2\\right)\n\\end{array} & \\begin{array} {c}\n12 L (c_1-6 L) (c_1-L) \\\\\n+12 S (c_1-L) (4 L-S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+24 L S (2 c_1-21 L) \\\\\n-12 S^2 (c_1-17 L)-24 S^3\n\\end{array} \\\\\n\\hline\n\\begin{array} {c}\n\\text{SU}(4) \\\\\n\\text{Spin}(7)\n\\end{array} & 0 & 12 L (c_1-L)^2 & \\begin{array} {c}\n24 L (L (6 L-c_1)+c_2) \\\\\n-4 S \\left(50 L^2-28 L S+5 S^2\\right)\n\\end{array} & \\begin{array} {c}\n12 L (c_1-L) (c_1-6 L) \\\\\n+4 S (c_1-L) (16 L-5 S)\n\\end{array} & \\begin{array} {c}\n12 L (6 L (6 L-c_1)+c_2) \\\\\n+8 L S (8 c_1-83 L) \\\\\n+4 S^2 (89 L-5 c_1)-60 S^3\n\\end{array} \\\\\n\\hline\n\\begin{array} {c} \\text{Spin}(8) \\\\ \\text{F}_4 \\end{array} & 0 & 12 L (c_1-L)^2 & \\begin{array} {c} \n24 L (L (6 L-c_1)+c_2) \\\\\n-8 S \\left(27 L^2-16 L S+3 S^2\\right)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+24 S (c_1-L) (3 L-S)\n\\end{array} & \\begin{array} {c}\n12 L (6 L (6 L-c_1)+c_2) \\\\\n+72 L S (c_1-10 L) \\\\\n-24 S^2 (c_1-17 L)-72 S^3\n\\end{array} \\\\\n\\hline \n\\text{SU}(5) & 0 & 12 L (c_1-L)^2 & \\begin{array} {c} \n24 L (L (6 L-c_1)+c_2) \\\\\n-5 S \\left(50 L^2-35 L S+8 S^2\\right)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+10 S (c_1-L) (8 L-3 S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+10 L S (8 c_1-83 L) \\\\\n+15 S^2 (37 L-2 c_1)-120 S^3 \\end{array} \\\\\n\\hline \n\\text{Spin}(10) & 0 & 12 L (c_1-L)^2 & \\begin{array} {c}  \n24 L (L (6 L-c_1)+c_2) \\\\\n-4 S \\left(63 L^2-44 L S+10 S^2\\right)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+4 S (c_1-L) (21 L-8 S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+84 L S (c_1-10 L) \\\\\n-16 S^2 (2 c_1-35 L)-120 S^3\n\\end{array} \\\\\n\\hline \n\\text{SU}(6) & 0 & 12 L (c_1-L)^2 & \\begin{array} {c} \n24 L (L (6 L-c_1)+c_2) \\\\\n-S (298 L^2-251 L S+70 S^2)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+6 S (c_1-L) (16 L-7 S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+6 L S (16 c_1 - 165 L) \\\\\n-3 S^2 (14 c_1-265 L)-210 S^3\n\\end{array} \\\\\n\\hline \n\\text{SU}(7) & 0 & 12 L (c_1-L)^2 & \\begin{array} {c} \n24 L (L (6 L-c_1)+c_2) \\\\\n-2 S (174 L^2-171 L S+56 S^2)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+56 S (c_1-L) (2 L-S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+4 L S (28 c_1 - 289 L) \\\\\n-2 S^2 (28 c_1-541 L)-336 S^3\n\\end{array} \\\\\n\\hline \n\\text{E}_6 & 0 & 12 L (c_1-L)^2 & \\begin{array} {c} \n24 L (L (6 L-c_1)+c_2) \\\\\n-3 S \\left(86 L^2-61 L S+14 S^2\\right)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+18 S (c_1-L) (5 L-2 S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+18 L S (5 c_1-48 L) \\\\\n-9 S^2 (4 c_1-65 L)-126 S^3\n\\end{array} \\\\\n\\hline \n\\text{E}_7 & 0 & 12 L (c_1-L)^2 & \\begin{array} {c}  \n24 L (L (6 L-c_1)+c_2) \\\\\n-2 S \\left(135 L^2-100 L S+24 S^2\\right)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+14 S (c_1-L) (7 L-3 S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+2 L S (49 c_1-454 L) \\\\\n-6 S^2 (7 c_1-107 L)-144 S^3\n\\end{array} \\\\\n\\hline \n\\text{E}_8 & 0 & 12 L (c_1-L)^2 & \\begin{array} {c}  \n24 L (L (6 L-c_1)+c_2) \\\\\n-40 S \\left(8 L^2-7 L S+2 S^2\\right)\n\\end{array} & \\begin{array} {c} \n12 L (c_1-L) (c_1-6 L) \\\\\n+60 S (c_1-L) (2 L-S)\n\\end{array} & \\begin{array} {c} \n12 L (6 L (6 L-c_1)+c_2) \\\\\n+120 L S (c_1-9 L) \\\\\n-60 S^2 (c_1-15 L)-240 S^3\n\\end{array} \\\\\n\\hline \n\\end{array}\n$}\n\\end{center}\n\\caption{Chern numbers of elliptically fibered fourfolds obtained from crepant resolutions of Tate's models. The divisor $S$ is supporting the reducible Kodaira fiber corresponding to the type of the Lie algebra. We abuse notation and omit the degree $\\int$ in the entries of the table. By definition, $L=c_1(\\mathscr{L})$ and  $c_i$ denotes the $i$th Chern class of the base of the fibration.}\n\\label{Table.ChernNumbers}\n\\end{table}\n\\clearpage\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\renewcommand{\\arraystretch}{2.1\n\\scalebox{.66}{$\n\\begin{array}{|c|c|c|c|}\n\\hline \n\\text{Group} & \\chi_0 & \\chi_1 & \\chi_2 \\\\\\hline\n\\text{SU}(2) & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+\\frac{1}{2} S^2 (3 c_1-31 L)-\\frac{1}{2} L S (15 c_1-113 L)+S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n+S^2 (59 L-3 c_1)+L S (15 c_1-211 L)-4 S^3\n\\end{array} \\\\ \n\\hline\n\\begin{array} {c} \\text{SU}(3) \\\\ \\text{USp}(4) \\\\ \\text{G}_2 \\end{array} & \\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right) & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+S^2 (3 c_1-35 L)-4 L S (3 c_1-22 L)+4 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n-2 S^2 (3 c_1-67 L)+8 L S (3 c_1-41 L)-16 S^3 \\end{array} \\\\ \n\\hline\n\\begin{array} {c} \\text{SU}(4) \\\\ \\text{Spin}(7) \\end{array} & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+S^2 (5 c_1-61 L)-4 L S (4 c_1-29 L)+10 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n-2 S^2 (5 c_1-117 L)+16 L S (2 c_1-27 L)-40 S^3\n\\end{array} \\\\ \n\\hline\n\\begin{array} {c} \\text{Spin}(8)\\\\ \\text{F}_4 \\end{array} & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+2 S^2 (3 c_1-35 L)-18 L S (c_1-7 L)+12 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n-4 S^2 (3 c_1-67 L)+36 L S (c_1-13 L)-48 S^3\n\\end{array} \\\\ \n\\hline \n\\text{SU}(5) & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+\\frac{5}{2} S^2 (3 c_1-38 L)-5 L S (4 c_1-29 L)+20 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n-5 S^2 (3 c_1-73 L)+20 L S (2 c_1-27 L)-80 S^3\n\\end{array} \\\\ \n\\hline \n\\text{Spin}(10) & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+8 S^2 (c_1-12 L)-21 L S (c_1-7 L)+20 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+8 S^2 (c_1-12 L)-21 L S (c_1-7 L)+20 S^3\n\\end{array} \\\\ \n\\hline \n\\text{SU}(6) & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+\\frac{1}{2} S^2 (21 c_1 - 272 L) - L S (24 c_1 - 173 L) + 35 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n+S^2 (523 L - 21 c_1) + 4 L S (12 c_1 - 161 L) - 140 S^3\n\\end{array} \\\\ \n\\hline \n\\text{SU}(7) & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n\\left(+S^2\\right) (14 c_1-185 L)-2 L S (14 c_1-101 L)+56 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n-4 S^2 (7 c_1-178 L)+8 L S (7 c_1-94 L)-224 S^3\n\\end{array} \\\\ \n\\hline \n\\text{E}_6 & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+\\frac{3}{2} S^2 (6 c_1-67 L)-\\frac{3}{2} L S (15 c_1-101 L)+21 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n-6 S^2 (3 c_1-64 L)+3 L S (15 c_1-187 L)-84 S^3\n\\end{array} \\\\  \n\\hline \n\\text{E}_7 & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+\\frac{1}{2} S^2 (21 c_1-221 L)-\\frac{1}{2} L S (49 c_1-319 L)+24 S^3\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{2} L \\left(3 c_1^2+71 c_1 L-17 c_2-554 L^2\\right)\\\\\n+S^2 (421 L-21 c_1)+L S (49 c_1-589 L)-96 S^3\n\\end{array} \\\\ \n\\hline \n\\text{E}_8 & \\begin{array} {c} \n\\frac{1}{12} L \\left(c_1^2-3 c_1 L+c_2+2 L^2\\right)\n\\end{array} & \\begin{array} {c} \n-\\frac{1}{3} L \\left(2 c_1^2-54 c_1 L+5 c_2+232 L^2\\right)\\\\\n+5 S^2 (3 c_1-31 L)+10 L S (19 L-3 c_1)+40 S^3\n\\end{array} & \\begin{array} {c} \n\\frac{1}{2} L \\left(-3 c_1^2-71 c_1 L+17 c_2+554 L^2\\right)\\\\\n+10 S^2 (59 L-3 c_1)+20 L S (3 c_1-35 L)-160 S^3\n\\end{array} \\\\ \n\\hline \n\\end{array}\n$}\n\\end{center}\n\\caption{Holomorphic genera. The divisor $S$ is the one supporting the reducible Kodaira fiber corresponding to the type of the Lie algebra  $\\mathfrak{g}$. To ease the notation, we abuse notation and omit the degree $\\int$ in the entries of the table. By definition, $L=c_1(\\mathscr{L})$ and  $c_i$ denotes the $i$th Chern class of the base of the fibration. The holomorphic Euler characteristic $\\chi_0(Y)$ is equal to $\\chi_0(W,\\mathscr{O}_W)$ where $W$ is the divisor defined by $\\mathscr{L}$ in \nthe base (see section \\ref{sec:Toddg}).}\n\\label{Table.HolomorphicEC}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\renewcommand{\\arraystretch}{2.3\n\\scalebox{.78}{$\n\\begin{array}{|c|l|l|}\n\\hline \n\\text{Group}  & \\hspace{4cm}\\int_Y p_2(TY) & \\hspace{4cm} \\int_Y p_1^2(TY)  \\\\\\hline\n\\text{SU}(2)  &  \n-24 L (-c_1^2+2 c_2+36 L^2) -14 S (7 L-S)^2\n & -48 L (c_1^2-2 c_2-11 L^2)-8 S (7 L-S)^2\n \\\\\n\\hline\n\\text{SU}(3), \\ \\text{USp}(4) , \\  \\text{G}_2   &  \n-24 L (-c_1^2+2 c_2+36 L^2) -56 S (19 L^2-8 L S+S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-32 S (19 L^2-8 L S+S^2)\n\\\\\n\\hline\n\\text{SU}(4),\\ \\text{Spin}(7)  &\n-24 L (-c_1^2+2 c_2+36 L^2) -28 S (50 L^2-28 L S+5 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-16 S (50 L^2-28 L S+5 S^2)\n \\\\\n\\hline\n \\text{Spin}(8), \\ \\text{F}_4  &  \n-24 L (-c_1^2+2 c_2+36 L^2) -56 S (27 L^2-16 L S+3 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-32 S (27 L^2-16 L S+3 S^2)\n \\\\\n\\hline \n\\text{SU}(5) &\n-24 L (-c_1^2+2 c_2+36 L^2) -35 S (50 L^2-35 L S+8 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-20 S (50 L^2-35 L S+8 S^2)\n  \\\\\n\\hline \n \\text{Spin}(10) &-24 L (-c_1^2+2 c_2+36 L^2) -28 S (63 L^2-44 L S+10 S^2)\n &  -48 L (c_1^2-2 c_2-11 L^2)-16 S (63 L^2 - 44 L S + 10 S^2) \n \\\\\n\\hline \n\\text{SU}(6) &\n-24 L (-c_1^2+2 c_2+36 L^2)-7 S (298 L^2-251 L S+70 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-4 S (298 L^2-251 L S+70 S^2)\n  \\\\\n\\hline \n\\text{SU}(7) &\n-24 L (-c_1^2+2 c_2+36 L^2)-14 S (174 L^2-171 L S+56 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-8 S (174 L^2-171 L S+56 S^2)\n  \\\\\n\\hline \n\\text{E}_6 & \n-24 L (-c_1^2+2 c_2+36 L^2) -21 S (86 L^2-61 L S+14 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-12 S (86 L^2-61 L S+14 S^2)\\\\\n\\hline \n\\text{E}_7 & \n-24 L (-c_1^2+2 c_2+36 L^2) -14 S (135 L^2-100 L S+24 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-8 S (135 L^2-100 L S+24 S^2)\n  \\\\\n\\hline \n\\text{E}_8 &  \n-24 L (-c_1^2+2 c_2+36 L^2) -280 S (8 L^2-7 L S+2 S^2)\n & -48 L (c_1^2-2 c_2-11 L^2)-160 S (8 L^2-7 L S+2 S^2)\n \\\\\n\\hline \n\\end{array}\n$}\n\\end{center}\n\\caption{Pontryagin numbers. The divisor $S$ is the one supporting the reducible Kodaira fiber corresponding to the type of the Lie algebra  $\\mathfrak{g}$. To ease the notation, we abuse notation and omit the degree $\\int$ in the entries of the table. By definition, $L=c_1(\\mathscr{L})$ and  $c_i$ denotes the $i$th Chern class of the base of the fibration. } \n\\label{Table.Pontryagin}\n\\end{table}\n\n\\clearpage\n\\begin{table}[htb]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.6\n\\scalebox{.9}{$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline \n\\text{Group} & 192 X_8=\\int_Y( p_1^2-4p_2) &45 \\sigma=45\\int_Y L_2= \\int_Y(7p_2-p_1^2) & 5760\\int_Y \\hat{\\text{A}}_2=\\int_Y(7 p_1^2-4 p_2) \\\\\\hline\n\\text{SU}(2) & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+48 S (7 L-S)^2\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-90 S (7 L-S)^2\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline\n\\begin{array} {c} \\text{SU}(3) \\\\ \\text{USp}(4) \\\\  \\text{G}_2 \\end{array} & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+192S (19 L^2-8 L S+S^2)\n\\end{array} & \\begin{array} {c} \n120L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-360 S (19 L^2-8 L S+S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline\n\\begin{array} {c} \\text{SU}(4)\\\\ \\text{Spin}(7) \\end{array} & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+96 S(50 L^2-28 L S+5 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-180  S(50 L^2 -28 L S+5 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline\n\\begin{array} {c} \\text{Spin}(8) \\\\ \\text{F}_4 \\end{array} & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+192 S(27 L^2 -16 L S+3 S^2)\n\\end{array} & \\begin{array} {c} \n120L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-360 S (27 L^2-16 L S+3 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{SU}(5) & \\begin{array} {c} \n48L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+120 S(50 L^2-35 L S+8 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-225 S (50 L^2 -35 L S+8 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{Spin}(10) & \\begin{array} {c} \n48L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+96 S(63 L^2-44 L S+10 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-180 S(63 L^2-44 L S+10 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{SU}(6) & \\begin{array} {c} \n48L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+24 S (298 L^2-251 L S+70 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-45 S (298 L^2-251 L S+70 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{SU}(7) & \\begin{array} {c} \n48L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+48 S (174 L^2-171 L S+56 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-90 S (174 L^2-171 L S+56 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{E}_6 & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+72  S(86 L^2 -61 L S+14 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-135 S(86 L^2 -61 L S+14 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{E}_7 & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+48 S(135 L^2 -100 L S+24 S^2)\n\\end{array} & \\begin{array} {c} \n120 L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-90 S (135 L^2 -100 L S+24 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\text{E}_8 & \\begin{array} {c} \n48 L \\left(c_1^2-2 c_2-61 L^2\\right) \\\\\n+960 S(8 L^2 -7 L S+2 S^2)\n\\end{array} & \\begin{array} {c} \n120L \\left(-c_1^2+2 c_2+46 L^2\\right) \\\\\n-1800  S(8 L^2 -7 L S+2 S^2)\n\\end{array} & \\begin{array} {c} \n240 L \\left(-c_1^2+2 c_2+L^2\\right)\n\\end{array} \\\\\n\\hline \n\\end{array}\n$}\n\\end{center}\n\\caption{Characteristic invariants: the anomaly invariant $X_8$, the signature $\\sigma$, and the index of the $\\hat{\\text{A}}$-genus. The divisor $S$ is the one supporting the reducible Kodaira fiber corresponding to the type of the Lie algebra  $\\mathfrak{g}$. To ease the notation, we abuse notation and omit the degree $\\int$ in the entries of the table. By definition, $L=c_1(\\mathscr{L})$ and  $c_i$ denotes the $i$th Chern class of the base of the fibration.}\n\\label{Table.Pontryagin2}\n\\end{table}\n\\clearpage\n\n\n\n\\clearpage\n\n\n\n\n\\section*{Acknowledgements}\nThe authors are grateful to  Paolo Aluffi, Patrick Jefferson,  Sungkyung Kang, Chin-Lung Wang,  and Shing-Tung Yau for discussions.  \nM.K. would like to thank Simons workshop 2018, Strings 2018, and String-math 2018 for their hospitality. \nM.E. is supported in part by the National Science Foundation (NSF) grant DMS-1701635  ``Elliptic Fibrations and String Theory''.\nM.J.K. would like to acknowledge a partial support from NSF grant PHY-1352084. \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\mathbb{Z}_p$ and $\\mathbb{Z}_{p^2}$ be the ring of integers modulo $p$ and $p^2$, respectively, where $p>2$ is prime.\nFor a positive integer $m$, $\\mathbb{Z}_p^m$ and $\\mathbb{Z}_{p^2}^m$ are the ring extension.\nThe linear codes over $\\mathbb{Z}_p$ and $\\mathbb{Z}_{p^2}$ are subgroups of $\\mathbb{Z}_p^m$ and $\\mathbb{Z}_{p^2}^m$, respectively.\n\nThe classical Gray map from $\\mathbb{Z}_{p^{k+1}}$ to $\\mathbb{Z}_p^{p^k}$, where $p$ is a prime and $k\\geq 1$, is given in \\cite{LS}.\nHere, let $k=1$, the Gray map $\\phi$ from $\\mathbb{Z}_{p^2}$ to $\\mathbb{Z}_p^{p}$ is as follows.\n\n$$\\begin{array}{ccc}\n\\mathbb{Z}_{p^2} & \\longrightarrow & \\mathbb{Z}_p^p\\\\\n\\theta & \\longmapsto & \\phi(\\theta)\n\\end{array},$$\n$\\phi(\\theta)=\\theta''(1,1,\\ldots,1)+ \\theta' (0,1,\\ldots,p-1)$, where\n$\\theta = \\theta'' p + \\theta',$ and $\\theta',\\theta''\\in\\{0,\\ldots,p-1\\}.$\n\nThe homogeneous weight  \\cite{LS} $wt_{hom}$ on $\\mathbb{Z}_{p^2}$ is $$\nwt_{hom}(x)=\\left\\{\n\\begin{array}{ll}\np,& \\text{ if } x\\in p\\mathbb{Z}_{p^2}\\backslash\\{0\\}, \\\\\np-1,& \\text{ if } x\\notin p\\mathbb{Z}_{p^2}, \\\\\n0,& \\text{ if } x=0.\n\\end{array}\n\\right.\n$$\nAnd the homogeneous distance is $d_{hom}=wt_{hom}(\\mathbf{a}-\\mathbf{b})$, where $\\mathbf{a},\\mathbf{b}\\in \\mathbb{Z}_{p^2}^n$.\nThen, $\\phi$ is an isometry from $(\\mathbb{Z}_{p^2}^n, d_{hom})$ to $(\\mathbb{Z}_p^{pn}, d_H)$, where $d_H$ is the Hamming distance on $\\mathbb{Z}_p^{p}$.\n\nIf $\\mathcal{C}$ is a linear code over $\\mathbb{Z}_{p^2}$, then the code $C=\\phi(\\mathcal{C})$ is called a \\emph{$\\mathbb{Z}_{p^2}$-linear} code.\nThe dual of a linear code $\\mathcal{C}$ of length $n$ over $\\mathbb{Z}_{p^2}$, denoted by $\\mathcal{C}^{\\bot}$, is defined as\n$$\\mathcal{C}^\\bot=\\{\\mathbf{x}\\in \\mathbb{Z}_{p^2}^n:\\langle\\mathbf{x},\\mathbf{y}\\rangle=0 ~\\rm {for ~all}~ \\textbf{y}\\in \\mathcal{C}\\},$$\nwhere $\\langle,\\rangle$ denotes the usual Euclidean inner product.\nThe code $C_\\bot=\\phi(\\mathcal{C}^{\\bot})$ is called the \\emph{$\\mathbb{Z}_{p^2}$-dual code} of $C=\\phi(\\mathcal{C})$.\n\n\nIn 1973, Delsarte first defined the additive codes in terms of association schemes \\cite{DP}, it is a subgroup of the underlying abelian group.\nBorges et al. \\cite{BFPR10} studied the standard generator matrix and the duality of $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes. Since then, a lot of work has been devoted to characterizing $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes. Dougherty et al. \\cite{DLY16} constructed one weight $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes and analyzed their parameters. Benbelkacem et al. \\cite{BBDF20} studied $\\mathbb{Z}_2\\mathbb{Z}_4$-additive complementary dual codes and their Gray images. In fact, these codes can be viewed as a generalization of the linear complementary dual (LCD for short) codes \\cite{M92} over finite fields. Joaquin et al. \\cite{BBCM} introduced a decoding method of the $\\mathbb{Z}_2\\mathbb{Z}_4$-linear codes.\nMore structure properties of $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes can be found in \\cite{BBDF11,JTCR2}. Moreover, the additive codes over different mixed alphabet have also been intensely studied, for example $\\mathbb{Z}_2\\mathbb{Z}_2[u]$-additive codes \\cite{BC2}, $\\mathbb{Z}_2\\mathbb{Z}_{2^s}$-additive codes \\cite{AS13}, $\\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}$-additive codes \\cite{AS15}, $\\mathbb{Z}_2\\mathbb{Z}_2[u,v]$-additive codes \\cite{SW}, $\\mathbb{Z}_p\\mathbb{Z}_{p^k}$-additive codes \\cite{SWD} and $\\mathbb{Z}_p(\\mathbb{Z}_p+u\\mathbb{Z}_p)$-additive codes \\cite{WS}, and so on. It is worth mentioning that $\\mathbb{Z}_2\\mathbb{Z}_4$-additive cyclic codes form an important family of $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes, many optimal binary codes can be obtained from the images of this family of codes. More details of $\\mathbb{Z}_2\\mathbb{Z}_4$-additive cyclic codes can be found in \\cite{BC,JCR,JTCR,JTCR2,YZ20}.\n\nIn \\cite{CFC}, Fern\\'{a}ndez et al. studied the rank and kernel of $\\mathbb{Z}_2\\mathbb{Z}_4$-linear codes, where $\\mathbb{Z}_2\\mathbb{Z}_4$-linear codes are $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes under the generalized Grap map $\\Phi$.\nThe authors also studied the rank and kernel of $\\mathbb{Z}_2\\mathbb{Z}_4$-additive cyclic codes in \\cite{JTCR2}.\nIt is a natural problem: can we study the rank and kernel of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes in terms of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive codes?\nIn this paper, we succeed in considering this general case for the kernel, and we also generalize the rank over $\\mathbb{Z}_3\\mathbb{Z}_9$.\n\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code of length $\\alpha+\\beta$ defined by a subgroup of $\\mathbb{Z}_p^\\alpha\\times\\mathbb{Z}_{p^2}^\\beta$. Let $n=\\alpha+p\\beta$ and\n$\\Phi:\\mathbb{Z}_p^\\alpha\\times\\mathbb{Z}_{p^2}^\\beta\\rightarrow \\mathbb{Z}_p^n$ is an extension of the Gray map, which\n$$\\Phi(\\mathbf{x},\\mathbf{y})=(\\mathbf{x},\\phi(y_1),\\ldots,\\phi(y_\\beta)),$$\nfor any $\\mathbf{x}\\in \\mathbb{Z}_p^\\alpha$, and $\\mathbf{y}=(y_1,\\ldots,y_\\beta)\\in \\mathbb{Z}_{p^2}^\\beta$. The code $C=\\Phi(\\mathcal{C})$ is called a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$\\emph{-linear code}.\nThere are two parameters of this nonlinear code: the rank and the dimension of the kernel.\nWe denote $\\langle C\\rangle$ the linear span of the codewords of $C$. The dimension of $\\langle C\\rangle$ is called the \\emph{rank} of the code $C$, denoted by $rank(C)$.\nThe \\emph{kernel} of the code $C$, which is denoted as $K(C)$, is defined as:\n$$K(C)=\\{\\mathbf{x}\\in\\mathbb{Z}_p^{n}|C+\\mathbf{x}=C\\},$$\nwhere $C+\\mathbf{x}$ means $\\mathbf{x}$ adds all codewords in $C$.\nWe will denote the dimension of the kernel of $C$ by \\emph{$ker(C)$}.\n\nThe rank and dimension of the kernel have been studied for some families of $\\mathbb{Z}_2\\mathbb{Z}_4$-linear codes \\cite{BPR,FPV,PRV}.\nThese two parameters are helpful to the classification of $\\mathbb{Z}_2\\mathbb{Z}_4$-linear codes.\nTherefore, we try to generalize them to the more general case over $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$.\n\n\\par\nThe paper is organized as follows.\nIn Section \\ref{sec:2}, we give some properties about $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive codes and $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes.\nIn Section \\ref{sec:3}, we find all values of the rank for $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes, and we construct a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code for each value.\nIn Section \\ref{sec:4}, we determine all values of the dimension of the kernel for $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes.\nWe also construct all $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes for the values of the dimension of the kernel.\nIn Section \\ref{sec:5}, pairs of rank and the dimension of kernel of $\\mathbb{Z}_3\\mathbb{Z}_{9}$-additive codes are studied.\nFor each fixed value of the dimension of the kernel, the range of rank is given.\nMoreover, the construction method of the $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes with pairs of rank and the dimension of the kernel is provided.\nIn Section \\ref{sec:6}, we conclude the paper.\n\n\\section{Preliminaries} \\label{sec:2}\n\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code. Then $\\mathcal{C}$ is isomorphic to an abelian structure $\\mathbb{Z}_p^\\gamma \\times \\mathbb{Z}_{p^2}^\\delta$ since it is a subgroup of $\\mathbb{Z}_p^\\alpha \\times \\mathbb{Z}_{p^2}^\\beta$.\nThe \\emph{order} of a codeword $\\mathbf{c}$ means the minimal positive integer $a$ such that $a\\cdot\\mathbf{c}=\\mathbf{0}$.\nTherefore, the size of $\\mathcal{C}$ is $p^{\\gamma+2\\delta}$, and the number of codewords of order $p$ in $\\mathcal{C}$ is $p^{\\gamma+\\delta}$.\n\nLet $X$ be the set of $\\mathbb{Z}_p$ coordinate positions, and $Y$ be the set of $\\mathbb{Z}_{p^2}$ coordinate positions, so $|X|=\\alpha$ and $|Y|=\\beta$.\nIn general, it is notice that the first $\\alpha$ positions corresponds to the set $X$ and the last $\\beta$ positions corresponds to the set $Y$.\nWe denote $\\mathcal{C}_X$ and $\\mathcal{C}_Y$ are the punctured codes of $\\mathcal{C}$ by deleting the coordinates outside $X$ and $Y$, respectively.\nLet $\\mathcal{C}_p$ be the subcode consisting of all codewords of order $p$ in $\\mathcal{C}$. Let $\\kappa$ be the dimension of the linear code $(\\mathcal{C}_p)_X$ over $\\mathbb{Z}_p$.\nIf $\\alpha=0$, then $\\kappa=0$.\nConsidering all these parameters, we will say that $\\mathcal{C}$ or $C=\\Phi(\\mathcal{C})$ is of type $(\\alpha, \\beta; \\gamma, \\delta; \\kappa)$.\n\\par\nFor a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code, every codeword $\\mathbf{c}$ can be uniquely expressible in the form\n$$\\mathbf{c}=\\sum_{i=1}^\\gamma \\lambda_i\\mathbf{u}_i+\\sum_{j=1}^\\delta \\nu_j\\mathbf{v}_j,$$\nwhere $\\lambda_i \\in \\mathbb{Z}_p$ for $1\\leq i \\leq \\gamma$, $\\nu_j \\in \\mathbb{Z}_{p^2}$ for $1 \\leq j \\leq \\delta$ and $\\mathbf{u}_i$, $\\mathbf{v}_j$ are vectors in $\\mathbb{Z}_p^\\alpha \\times \\mathbb{Z}_{p^2}^\\beta$ of order $p$ and $p^2$, respectively.\nThen, we get the generator matrix $\\mathcal{G}$ for the code $\\mathcal{C}$ by vectors $\\mathbf{u}_i$ and $\\mathbf{v}_j$.\n$$\\mathcal{G}=\\left( \\begin{array}{c}\\mathbf{u}_1\\\\ \\vdots \\\\\\mathbf{u}_\\gamma \\\\ \\mathbf{v}_1 \\\\\\vdots \\\\\\mathbf{v}_\\delta \\end{array}\\right)=\\left( \\begin{array}{c|c}B_1 & pB_3 \\\\\n \\hline B_2 & Q\\end{array}\\right),$$\nwhere $B_1$, $B_2$ and $B_3$ are matrices over $\\mathbb{Z}_p$ of size $\\gamma \\times \\alpha, \\delta \\times \\alpha$ and $\\gamma \\times \\beta$, respectively; and $Q$ is a matrix over $\\mathbb{Z}_{p^2}$ of size $\\delta\\times \\beta$.\n\nRecall that in coding theory, two linear codes $C_1$ and $C_2$ of length $n$ are \\emph{permutation equivalent}\nif there exists a coordinate permutation $\\pi$ such that $C_2=\\{\\pi(c)|c\\in C_1\\}$.\nThe permutation equivalent of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive codes can be defined similarly.\nThen we have the following theorem.\n\n\\begin{theorem}{\\rm\\cite{AS15}} \\label{th:1}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$. Then, $\\mathcal{C}$ is permutation equivalent to a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code $\\mathcal{C}^\\prime$ with generator matrix of the form:\n$$\\mathcal{G}_S=\\left(\\begin{array}{cc|ccc}I_\\kappa & T' & pT_2 & \\mathbf{0} & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{0} & pT_1 & pI_{\\gamma-\\kappa} & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S & R & I_\\delta\n\\end{array}\\right),$$\nwhere $I_\\delta$ is the identity matrix of size $\\delta\\times \\delta$; $T', S', T_1, T_2$ and $R$ are matrices over $\\mathbb{Z}_p$; and $S$ is a matrix over $\\mathbb{Z}_{p^2}$.\n\\end{theorem}\n\\begin{proof}\nStraightforward by setting $r=1,s=2$ of $\\mathbb{Z}_{p^r}\\mathbb{Z}_{p^s}$-linear codes in \\cite{AS15}.\n\\end{proof}\n\nFrom Theorem \\ref{th:1}, there is a $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-additive code $\\mathcal{C}$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ if and only if\n\\begin{equation}\\label{eq:1}\n\\begin{array}{c}\n\\alpha,\\beta,\\gamma,\\delta,\\kappa\\geq 0, \\alpha+\\beta>0,\\\\\n0<\\delta+\\gamma \\leq \\beta+\\kappa \\text{\\rm{ and }} \\kappa\\leq \\rm{min}(\\alpha,\\gamma).\n\\end{array}\n\\end{equation}\n\nThe definition of the duality for $\\mathbb{Z}_p\\mathbb{Z}_{p^k}$-additive codes is shown in \\cite{SWD}, which includes $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive codes.\nThe \\emph{inner product} between $(\\mathbf{v}_1|\\mathbf{w}_1)$ and $(\\mathbf{v}_2|\\mathbf{w}_2)$ in $\\mathbb{Z}_p^\\alpha \\times \\mathbb{Z}_{p^2}^\\beta$ can be written as follows:\n$$\\langle(\\mathbf{v}_1|\\mathbf{w}_1),(\\mathbf{v}_2|\\mathbf{w}_2)\\rangle=p\\langle \\mathbf{v}_1,\\mathbf{v}_2\\rangle + \\langle \\mathbf{w}_1,\\mathbf{w}_2\\rangle \\in \\mathbb{Z}_{p^2}.$$\nNote that the result of the inner product $\\langle \\mathbf{v}_1,\\mathbf{v}_2\\rangle$ is from $\\mathbb{Z}_p$,\nand multiplication of its value by $p$ should be formally understood as the natural homomorphism from $\\mathbb{Z}_p$ into $\\mathbb{Z}_{p^2}$,\nthat is, $p\\langle \\mathbf{v}_1,\\mathbf{v}_2\\rangle \\in p\\mathbb{Z}_{p^2} \\subseteq \\mathbb{Z}_{p^2}$. More detail are suggested to see \\cite{SWD}.\n\nThe \\emph{dual code} $\\mathcal{C}^\\bot$ of a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code $\\mathcal{C}$ is defined in the standard way by\n\\begin{eqnarray*}\n \\mathcal{C}^\\bot=\\big\\{(\\mathbf{x}|\\mathbf{y})\n {}\\in \\mathbb{Z}_p^\\alpha \\times \\mathbb{Z}_{p^2}^\\beta:\n {\n \\langle(\\mathbf{x}|\\mathbf{y}),(\\mathbf{v}|\\mathbf{w})\\rangle=0 ~\\rm {for ~all}~ (\\mathbf{v}|\\mathbf{w})\\in \\mathcal{C}\\big\\}.}\n\\end{eqnarray*}\nReadily, the dual code $\\mathcal{C}^\\bot$ is also a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code \\cite{AS15}.\nIn particular, it is shown that the type of the code $\\mathcal{C}^\\bot$ is $(\\alpha, \\beta; \\bar{\\gamma}, \\bar{\\delta}; \\bar{\\kappa})$, where\n$$\\begin{array}{l}\\bar{\\gamma}=\\alpha+\\gamma-2\\kappa, \\\\\n\\bar{\\delta}=\\beta-\\gamma-\\delta+\\kappa, \\\\\n\\bar{\\kappa}=\\alpha-\\kappa.\n\\end{array}\n$$\nThe code $\\Phi(\\mathcal{C}^\\bot)$ is denoted by $C_\\bot$ and called the \\emph{$\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-dual code} of $C$.\n\n\\begin{lemma}\\label{l:2}\nFor all $\\mathbf{u},\\mathbf{v}\\in \\mathbb{Z}_p^\\alpha \\times \\mathbb{Z}_{p^2}^\\beta$,\n$\\mathbf{u}=(u_1,\\ldots,u_{\\alpha+\\beta})$,\n$\\mathbf{v}=(v_1,\\ldots,v_{\\alpha+\\beta})$,\nwe have\n$$\n\\Phi(\\mathbf{u}+\\mathbf{v})=\\Phi(\\mathbf{u})+\\Phi(\\mathbf{v})+\\Phi(pP(\\mathbf{u},\\mathbf{v})),\n$$\nwhere $P(\\mathbf{u},\\mathbf{v})=(0,\\ldots,0,P(u_{\\alpha+1},v_{\\alpha+1}),\\ldots,P(u_{\\alpha+\\beta},v_{\\alpha+\\beta}))$ and\n$$\nP(u_i,v_i)=\nP(u'_i,v'_i)=\n\\begin{cases}\n 1 &\n \\mbox{if $u'_i+v'_i \\ge p$}\n \\\\ 0 &\\mbox{otherwise},\n             \\end{cases}\n\\qquad u_i=u''_i p + u'_i, \\quad  v_i=v''_i p + v'_i.\n$$\n\\end{lemma}\n\\begin{proof}\nIt is sufficient to prove the claim for one\n$\\mathbb{Z}_{p^2}$ coordinate.\nAssume\n $w_i\n= u_i + v_i$ (the addition is modulo $p^2$),\nwhere\n$w_i=w''_i p + w'_i$,\n$u_i=u''_i p + u'_i$, and\n$v_i=v''_i p + v'_i$.\nSince\n$w_i\n= u_i + v_i = (u_i''+v_i'')p + (u_i'+v_i') \\bmod p^2 $,\nwe have $w'_i = u'_i+v'_i$ and $w''_i = u''_i+v''_i \\bmod p$\nif $u'_i+v'_i < p$. If $u'_i+v'_i \\ge p$,\nthe formula is different:\n$w'_i = u'_i+v'_i \\bmod p$ and $w''_i = u''_i+v''_i + 1 \\bmod p$.\nUtilizing the definition of $P$, we get\n\\begin{equation}\\label{eq:'''}\nw'_i = u'_i+v'_i \\bmod\\ p \\qquad\\mbox{and}\\quad\nw''_i = u''_i+v''_i+P(u'_i,v'_i) \\bmod\\ p.\n\\end{equation}\nNow, from the definition of $\\phi$ and \\eqref{eq:'''}, it is straightforward\n$ \\phi( pP(u'_i,v'_i)) = P(u'_i,v'_i) (1,1,\\ldots,1),$\nand we find\n\\begin{multline}\\label{eq:phiu+v}\n \\phi(u_i + v_i) =\n ( u''_i+v''_i+P(u'_i,v'_i) ) (1,1,\\ldots,1)\n+ (u'_i+v'_i) (0,1,\\ldots,p-1)\\\\\n= \\phi(u_i) + \\phi(v_i) + \\phi( pP(u'_i,v'_i)).\n\\end{multline}\n Applying \\eqref{eq:phiu+v} to each coordinate from\n $\\alpha+1$ to $\\alpha+\\beta$ completes the proof.\n\\end{proof}\n\\begin{remark}\n The function $P$ can be treated as\n a $\\{0,1,\\ldots,p-1\\}$-valued function in\n two $\\{0,1,\\ldots,p-1\\}$-valued arguments\n $u'_i$, $v'_i$.\n As any such function, it can be represented\n as a polynomial of degree at most $p-1$ in each variable. For example $P(u'_i,v'_i) = u'_i v'_i$\n for $p=2$ and\n $P(u'_i,v'_i) = 2u'_i v'_i(1 + u'_i + v'_i) $\n for $p=3$. We also note that substituting\n $u_i$ and $v_i$ instead of $u'_i$ and $v'_i$\n in this polynomial does not change the value\n of $p P(u_i,v_i)$; so, for example,\n for $p=3$, it is safe to write\n $P(u_i,v_i) = 2u_i v_i(1 + u_i + v_i) $.\n For the rigor of the paper, we still use a new function $P'(\\mathbf{u},\\mathbf{v})$,\n where $P(\\mathbf{u},\\mathbf{v})=(p-1)P'(\\mathbf{u},\\mathbf{v})$.\n Note that $\\Phi(pP(\\mathbf{u},\\mathbf{v}))=(p-1)\\Phi(pP'(\\mathbf{u},\\mathbf{v}))$.\n\\end{remark}\n\n\\begin{lemma}\\label{l:4}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-additive code. The $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-linear code $C=\\Phi(\\mathcal{C})$ is linear if and only if\n$pP'(\\mathbf{u},\\mathbf{v})\\in \\mathcal{C}$ for all $\\mathbf{u},\\mathbf{v}\\in \\mathcal{C}$.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $C$ is linear. Because $\\mathcal{C}$ is additive, for any $\\mathbf{u},\\mathbf{v}\\in \\mathcal{C}$, we have $\\mathbf{u}+\\mathbf{v}\\in \\mathcal{C}$.\nThen, $\\Phi(\\mathbf{u}),\\Phi(\\mathbf{v}),\\Phi(\\mathbf{u+v})\\in C$.\nSince $C$ is linear, $\\Phi(pP'(\\mathbf{u},\\mathbf{v}))\\in C$ by Lemma \\ref{l:2}.\nTherefore, $pP'(\\mathbf{u},\\mathbf{v})\\in \\mathcal{C}$. Conversely, assume that $pP'(\\mathbf{u},\\mathbf{v})\\in \\mathcal{C}$ for all $\\mathbf{u},\\mathbf{v}\\in \\mathcal{C}$. Let $\\mathbf{x},\\mathbf{y}\\in C$, then there are $\\mathbf{u'},\\mathbf{v'}\\in \\mathcal{C}$ such that $\\mathbf{x}=\\phi(\\mathbf{u'}),\\mathbf{y}=\\Phi(\\mathbf{v'})$.\nThus $pP'(\\mathbf{u'},\\mathbf{v'})\\in \\mathcal{C}$.\nSince $\\mathcal{C}$ is additive, $\\mathbf{u'}+\\mathbf{v'}+pP'(\\mathbf{u'},\\mathbf{v'})\\in \\mathcal{C}$.\nTherefore, $\\Phi(\\mathbf{u'}+\\mathbf{v'}+pP'(\\mathbf{u'},\\mathbf{v'}))\\in C$.\nThen $$\\begin{aligned}\n\\Phi(\\mathbf{u'}+\\mathbf{v'}+pP'(\\mathbf{u'},\\mathbf{v'}))\n&=\\Phi(\\mathbf{u'}+\\mathbf{v'})+\\Phi(pP'(\\mathbf{u'},\\mathbf{v'}))\\\\\n&=\\Phi(\\mathbf{u'})+\\Phi(\\mathbf{v'})+\\Phi(pP(\\mathbf{u'},\\mathbf{v'}))+\\Phi(pP'(\\mathbf{u'},\\mathbf{v'}))\\\\\n&=\\Phi(\\mathbf{u'})+\\Phi(\\mathbf{v'})+(p-1)\\Phi(pP'(\\mathbf{u'},\\mathbf{v'}))+\\Phi(pP'(\\mathbf{u'},\\mathbf{v'}))\\\\\n&=\\Phi(\\mathbf{u'})+\\Phi(\\mathbf{v'}).\n\\end{aligned}$$\nHence, $\\Phi(\\mathbf{u'})+\\Phi(\\mathbf{v'})=\\mathbf{x}+\\mathbf{y} \\in C$.\n\\end{proof}\n\n\\section{Rank of $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-additive codes}\\label{sec:3}\n\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-additive code of type $(\\alpha, \\beta;\\gamma,\\delta;\\kappa)$ and $C=\\Phi(\\mathcal{C})$ of length $\\alpha+p\\beta$.\nWe have known the definition of $rank(C)$ before. For general prime $p$, it is very difficult to discuss all values of $r=rank(C)$ clearly.\nTherefore, in this section, we only consider $p=3$, then give the range of values $r=rank(C)$ and prove that there is a $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear code of type $(\\alpha, \\beta;\\gamma,\\delta;\\kappa)$ with $r=rank(\\mathcal{C})$ for any positive integer $r$.\n\nIf $p=3$, for all $\\mathbf{u},\\mathbf{v}\\in \\mathbb{Z}_3^\\alpha \\times \\mathbb{Z}_{9}^\\beta$,\nit is easy to check that $\\Phi(\\mathbf{u}+\\mathbf{v})=\\Phi(\\mathbf{u})+\\Phi(\\mathbf{v})+2\\Phi(3(\\mathbf{u}\\ast \\mathbf{v}+\\mathbf{u}\\ast \\mathbf{u}\\ast \\mathbf{v}+\\mathbf{u}\\ast \\mathbf{v}\\ast \\mathbf{v}))$ by Lemma \\ref{l:2}, where $\\ast$ denotes the componentwise multiplication.\nThen we have the following theorem.\n\n\\begin{theorem}\\label{th:5}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-additive code of type $(\\alpha, \\beta;\\gamma,\\delta;\\kappa)$ which satisfies (\\ref{eq:1}),\n$C=\\Phi(\\mathcal{C})$ be the corresponding $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code of length $n=\\alpha+3\\beta$.\n\n\\begin{itemize}\n\\item[(i)] Let $\\mathcal{G}$ be the generator matrix of $\\mathcal{C}$,\nand let $\\{\\mathbf{u}_i\\}_{i=1}^\\gamma$, $\\{\\mathbf{v}_j\\}_{j=1}^\\delta$ be the rows of order $3$ and $9$ in $\\mathcal{G}$, respectively.\nThen $\\langle C\\rangle$ is generated by $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l\\leq k\\leq \\delta}$\nand $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x\\leq y\\leq z\\leq\\delta}$.\n\\item[$(ii)$] $rank(C)\\in \\bigg\\{\\gamma+2\\delta,...,\\rm{min}\\bigg(\\beta+\\gamma+\\kappa,\\gamma+\\delta+\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}\\bigg)\\bigg\\}$.\nLet $rank(C)=r=\\gamma+2\\delta+\\overline{r}$. Then $\\overline{r}\\in\\bigg\\{0,1,...,\\rm{min}\\bigg(\\beta-(\\gamma-\\kappa)-\\delta,\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}-\\delta\\bigg)\\bigg\\}$.\n\\item[$(iii)$] The linear code $\\langle C\\rangle$ over $\\mathbb{Z}_3$ is $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\n$(i)$\nLet $\\mathbf{c}\\in \\mathcal{C}$, without loss of generality, $\\mathbf{c}$ can be expressed as $\\mathbf{c}=\\Sigma_{j=1}^\\zeta \\mathbf{v}_j+\\omega$,\nwhere $\\zeta\\leq\\delta$ and $\\omega$ is a codeword in $\\mathcal{C}$ of order $3$.\nBy Lemma \\ref{l:2}, we have $\\Phi(\\mathbf{c})=\\Phi(\\Sigma_{j=1}^\\zeta \\mathbf{v}_j)+\\Phi(\\omega)$,\nwhere $\\Phi(\\omega)$ is a linear combination of $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$.\nNote that\n$\\Phi(\\Sigma_{j=1}^\\zeta \\mathbf{v}_j)=\\Sigma_{j=1}^\\zeta\\Phi(\\mathbf{v}_j)+\\Sigma_{1\\leq l \\leq k\\leq \\zeta}(\\tilde{a}\\Phi(3(\\mathbf{v}_k\\ast v_l))+\\Sigma_{1\\leq x \\leq y\\leq z\\leq \\zeta}(\\tilde{b}\\Phi(3(\\mathbf{v}_x\\ast \\mathbf{v}_y\\ast \\mathbf{v}_z))$, $\\tilde{a},\\tilde{b}\\in \\mathbb{Z}_3$.\nSince $\\Phi(3\\mathbf{v}_j)=\\Phi(3\\mathbf{v}_j\\ast\\mathbf{v}_j\\ast\\mathbf{v}_j)$ by Lemma \\ref{l:2},\n$\\Phi(\\mathbf{c})$ is generated by $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$,\n$\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l\\leq k\\leq \\delta}$\nand $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x\\leq y\\leq z\\leq\\delta}$.\n\n$(ii)$ The bound $\\gamma+\\delta+\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}$ is straightforward by (i),\nand the bound $\\beta+\\gamma+\\kappa$ is trivial by the form of $\\mathcal{G}_S$ in Theorem \\ref{th:1}.\n\n$(iii)$ Let $\\mathcal{S}_\\mathcal{C}$ be a $\\mathbb{Z}_3\\mathbb{Z}_9$-additive code generated by\n$\\{\\mathbf{u}_i\\}_{i=1}^\\gamma$, $\\{\\mathbf{v}_j\\}_{j=1}^\\delta$, $\\{3\\mathbf{v}_k*\\mathbf{v}_l\\}_{1\\leq l\\leq k\\leq \\delta}$\nand $\\{3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z\\}_{1\\leq x\\leq y\\leq z\\leq\\delta}$,\nthen $\\mathcal{S}_\\mathcal{C}$ is of type $(\\alpha,\\beta;\\gamma+\\overline{r},\\delta;\\kappa)$ and $\\Phi(\\mathcal{S}_\\mathcal{C})=\\langle C\\rangle$.\n\\end{proof}\nFor convenience, we use $\\mathbf{v}_j^2$ to denote $\\mathbf{v}_j*\\mathbf{v}_j$, and write the vectors in Theorem \\ref{th:5} (i) in the following way:\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$.\nThe number $\\gamma+\\delta+\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}$ can be represented as\n$\\gamma+2\\delta+\\dbinom{\\delta}{2}+\\delta+2\\dbinom{\\delta}{2}+\\dbinom{\\delta}{3}$=\n$\\gamma+3\\delta+3\\dbinom{\\delta}{2}+\\dbinom{\\delta}{3}$.\n\nThe following theorem provides a method to construct a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code for each value of $rank(C)$ which is shown in Theorem \\ref{th:5}.\n\\begin{theorem}\\label{th:7}\nLet $\\alpha,\\beta,\\gamma,\\delta,\\kappa$ be positive integers satisfying (\\ref{eq:1}).\nThen, there is a $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear code $C$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with $rank(C)=r$ if and only if\n$$r\\in \\bigg\\{\\gamma+2\\delta,\\ldots,\\rm{min}\\bigg(\\beta+\\delta+\\kappa,\\gamma+\\delta+\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}\\bigg)\\bigg\\}.$$\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-additive code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with generator matrix\n$$\\mathcal{G}=\\left(\\begin{array}{cc|ccc}I_\\kappa & T' & \\mathbf{0} & \\mathbf{0} & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{0} & 3T_1 & 3I_{\\gamma-\\kappa} & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S_r & \\mathbf{0} & I_\\delta\n\\end{array}\\right),$$\nwhere $S_r$ is a matrix over $\\mathbb{Z}_{9}$ of size $\\delta\\times(\\beta-(\\gamma-\\kappa)-\\delta)$, and $C=\\Phi(\\mathcal{C})$ is a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code.\nLet $\\{\\mathbf{u}_i\\}_{i=1}^\\gamma$ and $\\{\\mathbf{v}_j\\}_{j=1}^\\delta$ be the row vectors of order $3$ and $9$ in $\\mathcal{G}$, respectively.\n\nThe necessary condition follows from Theorem \\ref{th:5}.\nFor the sufficiency of the theorem, we should construct a $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear code $C$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with $rank(C)=r$.\n\nLet $\\mathbf{h}_k,1\\leq k\\leq \\delta$ be the column vector of length $\\delta$, with a $1$ in the $k$-th coordinate and zeros elsewhere.\nDefine five matrices $\\bar{A}_{\\delta \\times \\delta}$, $\\bar{B},\\bar{C},\\bar{D}$ of size $\\delta \\times \\dbinom{\\delta}{2}$ and $\\bar{E}$ of size $\\delta \\times \\dbinom{\\delta}{3}$ over $\\mathbb{Z}_9$ as follows:\n\\begin{itemize}\n\\item $\\bar{A}_{\\delta \\times \\delta}=2I_\\delta$;\n\\item $\\bar{B}=(\\mathbf{b}_j)$, where $\\mathbf{b}_j=\\mathbf{h}_k+\\mathbf{h}_l$ for $1\\leq k< l\\leq \\delta, j\\in\\bigg\\{1,2,\\ldots, \\dbinom{\\delta}{2}\\bigg\\}$,\nand $\\mathbf{b}_{j_1}\\neq \\mathbf{b}_{j_2}$ if $j_1 \\neq j_2$;\n\\item $\\bar{C}=2\\bar{B}$;\n\\item $\\bar{D}=(\\mathbf{d}_j)$, where $\\mathbf{d}_j=\\mathbf{h}_k+2\\mathbf{h}_l$ (or equivalently, $\\mathbf{d}_j=2\\mathbf{h}_k+\\mathbf{h}_l$) for $1\\leq k< l\\leq \\delta, j\\in\\bigg\\{1,2,\\ldots, \\dbinom{\\delta}{2}\\bigg\\}$,\nand $\\mathbf{d}_{j_1}\\neq \\mathbf{d}_{j_2}$ if $j_1 \\neq j_2$;\n\\item $\\bar{E}=(\\mathbf{e}_j)$, where $\\mathbf{e}_j=\\mathbf{h}_x+\\mathbf{h}_y+\\mathbf{h}_z$ for $1\\leq x< y<z\\leq \\delta, j\\in\\bigg\\{1,2,\\ldots, \\dbinom{\\delta}{3}\\bigg\\}$,\nand $\\mathbf{e}_{j_1}\\neq \\mathbf{e}_{j_2}$ if $j_1 \\neq j_2$.\n\\end{itemize}\n\nSince the bound $\\beta+\\delta+\\kappa$ is trivial, it is enough to consider\n$\\rm{min}\\bigg(\\beta+\\delta+\\kappa,\\gamma+\\delta+\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}\\bigg)=\\gamma+\\delta+\\dbinom{\\delta+1}{2}+\\dbinom{\\delta+2}{3}$,\nthat is,\n$\\overline{r}\\in \\bigg\\{0,1,\\ldots,\\delta+3\\dbinom{\\delta}{2}+\\dbinom{\\delta}{3}\\bigg\\}$.\nWe construct $S_r$ over $\\mathbb{Z}_{9}$ in the following way such that we can get a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code.\n\n$(1)$ If $\\overline{r}\\in \\{0,1,\\ldots,\\delta\\}$,\nthen choose $\\overline{r}$ different column vectors from the matrix $\\bar{A}$ as $\\overline{r}$ columns in $S_r$, and the remaining columns are all-zero vectors.\nThat is, $S_r=(\\bar{\\bar{A}}|\\mathbf{0})$, where $\\bar{\\bar{A}}$ is a matrix consisting of $\\overline{r}$ vectors from $\\bar{A}$.\nIn this case, $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$\nare all zero vectors. Considering $\\gamma+3\\delta$ vectors $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$, we claim that there are $\\gamma+2\\delta+\\overline{r}$ vectors from them are linear independent. Namely, $rank(C)=\\gamma+2\\delta+\\overline{r}$. In fact, $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$ are linear independent\nby the form of $\\mathcal{G}$,\nand there are $\\overline{r}$ vectors in $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$ that are linear independent to all these $\\gamma+2\\delta$ vectors. Without loss of generality, assume $S'=\\mathbf{0}$ in $\\mathcal{G}$.\nNote that $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$ are linear independent. In all these $\\delta$ vectors $\\mathbf{v}_j$,\nthere are exactly $\\overline{r}$ vectors with the form\n$$\\mathbf{v}_{j'}=(0,\\ldots,0,2,0,\\ldots,0,1,0,\\ldots,0),$$ while other $\\delta-\\overline{r}$ vectors are with the form $$\\mathbf{v}_{j''}=(0,\\ldots,0,0,0,\\ldots,0,1,0,\\ldots,0).$$\nAs to these $\\overline{r}$ vectors, we have $$3\\mathbf{v}_{j'}=(0,\\ldots,0,6,0,\\ldots,0,3,0,\\ldots,0), 3\\mathbf{v}_{j'}^2=(0,\\ldots,0,3,0,\\ldots,0,3,0,\\ldots,0),$$\nthus $\\mathbf{v}_j,3\\mathbf{v}_j$ and $3\\mathbf{v}_j^2$ are linear independent under the Gray map $\\Phi$.\nHowever, other $\\delta-\\overline{r}$ vectors are linear independent.\n\n$(2)$ If $\\overline{r}\\in \\bigg\\{\\delta+1,\\ldots,\\delta+\\dbinom{\\delta}{2}\\bigg\\}$,\nthen choosing all $\\delta$ columns of $\\bar{A}$ and $\\overline{r}-\\delta$ columns from $\\bar{B}$\nas $\\overline{r}$ columns of $S_r$, and the remaining columns of $S_r$ are all-zero vectors. Namely,\n$$S_r=(\\overline{A}|\\bar{\\bar{B}}|\\mathbf{0}),$$ where $\\bar{\\bar{B}}$ is a matrix consisting of $\\overline{r}-\\delta$ vectors.\nIn this case, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$ are all zero vectors, and\n$\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$\nare the same as $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$ because of the form of $\\bar{B}$.\nFrom Case (1), $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$\nare linear independent.\n\nNext, it also need to prove that there are $\\overline{r}-\\delta$ linear independent vectors in $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$ such that\nthey are also linear independent to $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$. By the definition of $\\bar{B}$, for fixed $1\\leq k< l\\leq \\delta$, if $\\mathbf{h}_k+\\mathbf{h}_l$ is in $S_r$,\nwe denote $\\mathbf{h}_k+\\mathbf{h}_l$ in the $r'$-th column,\nthen the $r'$-th coordinate of $3(\\mathbf{v}_k*\\mathbf{v}_l)$ is $3$ and other coordinates are all zeros.\nTherefore, each $\\Phi(3(\\mathbf{v}_k*\\mathbf{v}_l))$ is independent to vectors\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$\nand $\\{\\Phi(3(\\mathbf{v}_s*\\mathbf{v}_t))\\},\\{s,t\\}\\neq\\{k,l\\}$.\n\n$(3)$ If $\\overline{r}\\in \\bigg\\{\\delta+\\dbinom{\\delta}{2}+1,\\ldots,\\delta+2\\dbinom{\\delta}{2}\\bigg\\}$,\nthen choosing all $\\delta+\\dbinom{\\delta}{2}$ columns from the matrices $\\bar{A},\\bar{B}$, and choosing $\\overline{r}-\\delta-\\dbinom{\\delta}{2}$ columns from the matrix $\\bar{C}$\nas $\\overline{r}$ columns of $S_r$, and the remaining columns of $S_r$ are all-zero vectors. Namely,\n$$S_r=(\\overline{A}|\\overline{B}|\\bar{\\bar{C}}|\\mathbf{0}),$$ where $\\bar{\\bar{C}}$ is a matrix consisting of $\\overline{r}-\\delta-\\dbinom{\\delta}{2}$ vectors.\n\nIn this case, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$ are all zero vectors, and\n$\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$ are the same as $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$\nby the form of the matrices $\\bar{B},\\bar{C}$.\nSimilar to Case $(2)$, by the definition of $\\bar{C}$ and the form of $S_r$, all vectors\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$\nare linear independent, and there are $\\overline{r}-\\delta-\\dbinom{\\delta}{2}$ linear independent vectors in $\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$\nthat are linear independent to all these vectors.\n\n$(4)$ If $\\overline{r}\\in \\bigg\\{\\delta+2\\dbinom{\\delta}{2}+1,\\ldots,\\delta+3\\dbinom{\\delta}{2}\\bigg\\}$,\nthen choosing all $\\delta+2\\dbinom{\\delta}{2}$ columns of matrices $\\bar{A},\\bar{B},\\bar{C}$, and $\\overline{r}-\\delta-2\\dbinom{\\delta}{2}$ column vectors from $\\bar{D}$\nas $\\overline{r}$ columns of $S_r$, and the remaining columns of $S_r$ are all-zero vectors.\n$$S_r=(\\overline{A}|\\overline{B}|\\overline{C}|\\bar{\\bar{D}}|\\mathbf{0}),$$ where $\\bar{\\bar{D}}$ is a matrix consisted of $\\overline{r}-\\delta-2\\dbinom{\\delta}{2}$ vectors.\n\nIn this case, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$ are still all zero vectors,\nbut $\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$ are not the same as $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$\nby the form of $\\bar{D}$.\nSimilar to cases $(2)$ and $(3)$, all vectors\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$\nare linear independent,\nand there are $\\overline{r}-\\delta-2\\dbinom{\\delta}{2}$ linear independent vectors in $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$\nthat are linear independent to all these vectors.\n\n$(5)$ If $\\overline{r}\\in \\bigg\\{\\delta+3\\dbinom{\\delta}{2}+1,\\ldots,\\delta+3\\dbinom{\\delta}{2}+\\dbinom{\\delta}{3}\\bigg\\}$,\nthen choosing all $\\delta+3\\dbinom{\\delta}{2}$ columns of the matrices $\\bar{A},\\bar{B},\\bar{C},\\bar{D}$, and $\\overline{r}-\\delta-3\\dbinom{\\delta}{2}$ columns from the matrix $\\bar{E}$\nas $\\overline{r}$ columns of $S_r$, and the remaining columns of $S_r$ are all-zero vectors. Namely,\n$$S_r=(\\overline{A}|\\overline{B}|\\overline{C}|\\overline{D}|\\bar{\\bar{E}}|\\mathbf{0}),$$ where $\\bar{\\bar{E}}$ is a matrix consisting of $\\overline{r}-\\delta-3\\dbinom{\\delta}{2}$ vectors.\n\nIn this case, $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$ are not all zero vectors,\nby the definition of the matrix $E$ and similar to the discussion of Case (2), all vectors\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_j^2)\\}_{j=1}^\\delta$, $\\{\\Phi(3\\mathbf{v}_k*\\mathbf{v}_l)\\}_{1\\leq l< k\\leq \\delta}$, $\\{\\Phi(3\\mathbf{v}_x^2*\\mathbf{v}_y)\\}_{1\\leq x< y\\leq \\delta}$,\n$\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y^2)\\}_{1\\leq x< y\\leq \\delta}$\nare linear independent,\nand there are $\\overline{r}-\\delta-3\\dbinom{\\delta}{2}$ linear independent vectors in $\\{\\Phi(3\\mathbf{v}_x*\\mathbf{v}_y*\\mathbf{v}_z)\\}_{1\\leq x< y<z\\leq \\delta}$\nthat are linear independent to all these vectors.\n\nHence, using this construction, we can get a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code for each value of $r$.\n\\end{proof}\n\nNow we give an example to illustrate the main result of Theorem \\ref{th:7}. It has been checked by the MAGMA program.\n\\begin{exmpl}\nConsider a $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear code $C$ of type $(\\alpha,\\beta;2,4;1)$, the generator matrix of $\\mathcal{C}=\\Phi^{-1}(C)$ is\n$$\\mathcal{G}=\\left(\\begin{array}{cc|ccc}1 & T' & \\mathbf{0} & 0 & \\mathbf{0}\\\\\n0 & \\mathbf{0} & 3T_1 & 3 & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S_r & \\mathbf{0} & I_4\n\\end{array}\\right),$$\nwhere $r\\in\\{10,11,\\ldots,\\min(\\beta+5,36)\\}$ and\n$\\overline{r}\\in\\{0,1,\\ldots,\\min(\\beta-5,26)\\}$.\n\nAccording to Theorem \\ref{th:7}, let\n\n\\begin{center}\n$\\bar{A}=\\begin{pmatrix} 2 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0\\\\ 0 & 0 & 2 & 0\\\\ 0 & 0 & 0 & 2 \\end{pmatrix}$,\n$\\bar{B}=\\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 1 & 1 & 0 \\\\ 0 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 1 & 1 & 0 & 1 \\end{pmatrix}$,\n$\\bar{C}=\\begin{pmatrix} 2 & 2 & 2 & 0 & 0 & 0 \\\\ 2 & 0 & 0 & 2 & 2 & 0 \\\\ 0 & 2 & 0 & 0 & 2 & 2 \\\\ 0 & 0 & 2 & 2 & 0 & 2 \\end{pmatrix},$\n\n$\\bar{D}=\\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\\\ 2 & 0 & 0 & 1 & 1 & 0 \\\\ 0 & 2 & 0 & 0 & 2 & 1 \\\\ 0 & 0 & 2 & 2 & 0 & 2 \\end{pmatrix},$\n$\\bar{E}=\\begin{pmatrix} 1 & 1 & 1 & 0 \\\\ 1 & 1 & 0 & 1\\\\ 1 & 0 & 1 & 1\\\\ 0 & 1 & 1 & 1 \\end{pmatrix}$,\n\\end{center}\nand $\\mathbf{a}_{j_1},\\mathbf{b}_{j_2},\\mathbf{c}_{j_3},\\mathbf{d}_{j_4},\\mathbf{e}_{j_5}$ be the column vectors in $\\bar{A},\\bar{B},\\bar{C},\\bar{D},\\bar{E}$, respectively,\nwhere $j_1,j_5\\in\\{1,2,3,4\\}$, $j_2,j_3,j_4\\in\\{1,2,3,4,5,6\\}$.\nIf $\\beta=10$, then $r\\in\\{10,11,12,13,14,15\\}$.\nLet $S_{10}=\\mathbf{0}$, then we have\n$$S_{11}=(a_1|\\mathbf{0}),\nS_{12}=(a_1,a_2|\\mathbf{0}),\nS_{13}=(a_1,a_2,a_3|\\mathbf{0}),$$\n$$S_{14}=(a_1,a_2,a_3,a_4|\\mathbf{0}),\nS_{15}=(\\bar{A}|b_1|\\mathbf{0}).$$\nThen we can get $6$ $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear codes.\nThat is, for any fixed $\\beta$, we can construct $S_r$ by choosing some columns in order from matrices\n$\\bar{A},\\bar{B},\\bar{C},\\bar{D},\\bar{E}$ and other columns are zero vectors.\n\\end{exmpl}\n\n\\begin{remark}\nThe construction of $S_r$ in the proof of Theorem \\ref{th:7} is not unique.\nIn fact, $S_r$ can be construceted by choosing any $\\overline{r}$ different columns from the matrix $M=(\\bar{A}|\\bar{B}|\\bar{C}|\\bar{D}|\\bar{E})$ with size $\\delta\\times \\bigg(\\delta+3\\dbinom{\\delta}{2}+\\dbinom{\\delta}{3}\\bigg)$.\n\\end{remark}\n\n\n\\section{Kernel dimension of $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-additive codes}\\label{sec:4}\n\nIn this section, we will explore the dimension of the kernel of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes $C=\\Phi(\\mathcal{C})$.\n\n\\begin{lemma}\\label{l:9}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-additive code and $C=\\Phi(\\mathcal{C})$. Then,\n$$K(C)=\\{\\Phi(\\mathbf{u})|\\mathbf{u}\\in\\mathcal{C}\\text{ \\rm{and} } pP'(\\mathbf{u},\\mathbf{v})\\in \\mathcal{C}, \\forall \\mathbf{v}\\in\\mathcal{C}\\}.$$\n\\end{lemma}\n\\begin{proof}\nThe result follows from Lemma \\ref{l:4} and the definition of $K(C)$.\n\\end{proof}\n\nIt's obvious that all codewords of order $p$ in $\\mathcal{C}$ under $\\Phi$ belong to $K(C)$,\nand $\\Phi(\\mathbf{u})\\notin K(C)$ if and only if there is $ \\mathbf{v}\\in \\mathcal{C}$ such that $pP'(\\mathbf{u},\\mathbf{v})\\notin \\mathcal{C}$.\n\n\n\\begin{lemma}\\label{l:10}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-additive code and $C=\\Phi(\\mathcal{C})$.\nIf $\\Phi(\\mathbf{x}_1)+\\Phi(\\mathbf{x}_2)+\\ldots+\\Phi(\\mathbf{x}_n)\\in K(C)$ for $\\mathbf{x}_1,\\mathbf{x}_2,\\ldots,\\mathbf{x}_n\\in \\mathcal{C}$,\nthen $\\Phi(\\mathbf{x}_1+\\mathbf{x}_2+\\ldots+\\mathbf{x}_n)\\in K(C)$.\n\\end{lemma}\n\\begin{proof}\nIt is enough to prove that $n=2$. The general $n$ can be obtained by induction.\nBy Lemma \\ref{l:2}, we have\n\\begin{eqnarray*}\n  \\Phi(\\mathbf{x}_1+\\mathbf{x}_2+pP'(\\mathbf{x}_1,\\mathbf{x}_2)) &=& \\Phi(\\mathbf{x}_1+\\mathbf{x}_2)+\\Phi(pP'(\\mathbf{x}_1+\\mathbf{x}_2)) \\\\\n   &=& \\Phi(\\mathbf{x}_1)+\\Phi(\\mathbf{x}_2)+p\\Phi(pP'(\\mathbf{x}_1,\\mathbf{x}_2))=\\Phi(\\mathbf{x}_1)+\\Phi(\\mathbf{x}_2).\n\\end{eqnarray*}\nBy Lemma \\ref{l:9}, if $\\Phi(\\mathbf{x}_1+\\mathbf{x}_2+pP'(\\mathbf{x}_1,\\mathbf{x}_2))\\in K(C)$, then for all $\\mathbf{v}\\in \\mathcal{C}$, we have $pP'((\\mathbf{x}_1+\\mathbf{x}_2+pP'(\\mathbf{x}_1,\\mathbf{x}_2)),\\mathbf{v})=pP'((\\mathbf{x}_1+\\mathbf{x}_2),\\mathbf{v})\\in \\mathcal{C}$, that is,\n$\\Phi(\\mathbf{x}_1+\\mathbf{x}_2)\\in K(C)$.\n\\end{proof}\n\nFrom this lemma, we can conclude that for $\\mathbf{x}\\in \\mathcal{C}$ and $a\\in \\mathbb{Z}_{p^2}$, if $a\\Phi(\\mathbf{x})\\in K(C)$, then $\\Phi(a\\mathbf{x})\\in K(C)$,\nbut the inverse does not have to be true unless the order of $\\mathbf{x}$ is $p$.\n\n\\begin{proposition}\\label{prop:11}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$, with generator matrix $\\mathcal{G}$, and $C=\\Phi(\\mathcal{C})$ with $ker(C)=\\gamma+2\\delta-\\overline{k}$, where $\\overline{k}\\in \\{1,2,\\ldots,\\delta\\}$.\nThen, there exists a set $\\{\\mathbf{v}_1,\\mathbf{v}_2,\\ldots,\\mathbf{v}_{\\overline{k}}\\}$ of row vectors of order $p^2$ in $\\mathcal{G}$ and $\\Phi(\\mathbf{v}_i)\\notin K(C)$, such that for all $a_i \\in \\mathbb{Z}_p$,\n$$C=\\bigcup_{a_i\\in \\mathbb{Z}_p}\\bigg(K(C)+\\Phi\\bigg(\\sum_{i=1}^{\\overline{k}}a_i\\mathbf{v}_i\\bigg)\\bigg).$$\n\\end{proposition}\n\n\\begin{proof}\nDenote $\\tilde{\\mathbf{v}}=\\sum_{i=1}^{\\overline{k}}a_i\\mathbf{v}_i, a_i\\in \\mathbb{Z}_p$.\nIt is obvious that $C$ can be written as the union of some cosets of $K(C)$.\nSince $|C|=p^{\\gamma+2\\delta}$ and $ker(C)=\\gamma+2\\delta-\\overline{k}$, we know that there are $p^{\\overline{k}}$ such cosets.\nFirstly, if $\\tilde{\\mathbf{v}}\\neq 0$, then $\\Phi(\\tilde{\\mathbf{v}})\\notin K(C)$.\nLet $\\mathbf{u}_1,\\ldots,\\mathbf{u}_\\gamma,\\mathbf{v}_1,\\ldots,\\mathbf{v}_\\delta$ be $\\gamma$ and $\\delta$ rows in $\\mathcal{G}$ of order $p$ and $p^2$, respectively.\nBy Lemma \\ref{l:9}, all codewords of order $p$ in $\\mathcal{C}$ under $\\Phi$ belong to $K(C)$.\nSo, there are $p^{\\gamma+\\delta}$ codewords of order $p$ in $K(C)$ generated by $\\gamma+\\delta$ codewords,\nand there are $\\delta-\\overline{k}$ codewords $\\omega_i,i\\in\\{1,2,\\ldots,\\delta-\\overline{k}\\}$ of order $p^2$ such that $\\Phi(\\omega_i)\\in K(C)$.\nIt is obvious that $\\Phi(\\mathbf{u}_1),\\ldots,\\Phi(\\mathbf{u}_\\gamma),\\Phi(p\\mathbf{v}_1),\\ldots,\\Phi(p\\mathbf{v}_\\delta)$\n,$\\Phi(\\omega_1),\\ldots,\\Phi(\\omega_{\\delta-\\overline{k}})$ are linear independent,\nand the code $\\mathcal{C}$ can also be generated by $\\mathbf{u}_1,\\ldots,\\mathbf{u}_\\gamma,\\omega_1,\\ldots,\\omega_{\\delta-\\overline{k}},\\mathbf{v}_{i_1},\\ldots,\\mathbf{v}_{i_{\\overline{k}}}$,\nwhere $\\{i_1,i_2,\\ldots,i_{\\overline{k}}\\}\\subseteq \\{1,2,\\ldots,\\delta\\}$.\nWithout loss of generality, assume that $\\mathbf{v}_{i_1},\\ldots,\\mathbf{v}_{i_{\\overline{k}}}$ are the $\\overline{k}$ row vectors $\\mathbf{v}_{1},\\ldots,\\mathbf{v}_{\\overline{k}}$ in $\\mathcal{G}$.\nIf $\\Phi(\\tilde{\\mathbf{v}})\\in K(C)$, then $\\Phi(\\mathbf{u}_1),\\ldots,\\Phi(\\mathbf{u}_\\gamma),\\Phi(p\\mathbf{v}_1),\\ldots,\\Phi(p\\mathbf{v}_\\delta)$\n,$\\Phi(\\omega_1),\\ldots,\\Phi(\\omega_{\\delta-\\overline{k}})$ and $\\Phi(\\tilde{\\mathbf{v}})$ are linearly independent, which contradicts with the size of $K(C)$.\nThat is, $\\Phi(\\tilde{\\mathbf{v}})\\in K(C)$ implies that $\\tilde{\\mathbf{v}}=0.$\nThen, it's sufficient to prove that $p^{\\overline{k}}-1$ nonzero vectors $\\Phi(\\sum_{i=1}^{\\overline{k}}a_i\\mathbf{v}_i)$ are from different cosets, where $a_i\\in \\mathbb{Z}_p$.\nIf $\\tilde{\\mathbf{v}}_1\\neq \\tilde{\\mathbf{v}}_2$, then $\\Phi(\\tilde{\\mathbf{v}}_1)\\neq \\Phi(\\tilde{\\mathbf{v}}_2)$ because $\\Phi$ is injective.\nSuppose that $\\Phi(\\tilde{\\mathbf{v}}_1)$ and $\\Phi(\\tilde{\\mathbf{v}}_2)$ are in the same coset, that is, $\\Phi(\\tilde{\\mathbf{v}}_1)=\\Phi(\\tilde{\\mathbf{v}}_2)+K(C)$.\nHence, $\\Phi(\\tilde{\\mathbf{v}}_1)-\\Phi(\\tilde{\\mathbf{v}}_2)\\in K(C)$, and then $\\Phi(\\tilde{\\mathbf{v}}_1)+(p^2-1)\\Phi(\\tilde{\\mathbf{v}}_2)\\in K(C)$.\nThus $\\Phi(\\tilde{\\mathbf{v}}_1+(p^2-1)\\tilde{\\mathbf{v}}_2)\\in K(C)$ by Lemma \\ref{l:10}.\nThus, $\\tilde{\\mathbf{v}}_1+(p^2-1)\\tilde{\\mathbf{v}}_2=\\tilde{\\mathbf{v}}_1-\\tilde{\\mathbf{v}}_2=0$, and then $\\tilde{\\mathbf{v}}_1=\\tilde{\\mathbf{v}}_2.$\n\\end{proof}\n\nNote that if $C$ is a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear code, then $K(C)$ is a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear subcode of $C$ by Lemma \\ref{l:10}.\nWe define the \\emph{kernel} of a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code $\\mathcal{C}$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ as $\\mathcal{K}(\\mathcal{C})=\\Phi^{-1}(K(C))$. According to Lemma \\ref{l:9}, $\\mathcal{K}(\\mathcal{C})=\\{\\mathbf{u}\\in \\mathcal{C}|pP'(\\mathbf{u},\\mathbf{v})\\in \\mathcal{C} \\text{ for all }\\mathbf{v}\\in \\mathcal{C}\\}$.\nIt's easy to check that $\\mathcal{K}(\\mathcal{C})$ is a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive subcode of $\\mathcal{C}$ of type $(\\alpha,\\beta;\\gamma+\\overline{k},\\delta-\\overline{k};\\kappa)$.\n\nBy Proposition \\ref{prop:11}, if $\\mathcal{C}$ is a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code with generator matrix $\\mathcal{G}$,\nthen there exists a set $\\{\\mathbf{v}_1,\\mathbf{v}_2,\\ldots,\\mathbf{v}_{\\overline{k}}\\}$ of row vectors of order $p^2$ in $\\mathcal{G}$, such that for all $a_i \\in \\mathbb{Z}_p$,\n$$\\mathcal{C}=\\bigcup_{a_i\\in \\mathbb{Z}_p}\\bigg(\\mathcal{K}(\\mathcal{C})+\\sum_{i=1}^{\\overline{k}}a_i\\mathbf{v}_i\\bigg).$$\n\n\\begin{lemma}\\label{l:12}\nLet $C$ be a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ and length $n=\\alpha+p\\beta$.\nThen, $\\ker(C)\\in\\{\\gamma+\\delta,\\gamma+\\delta+1,\\ldots,\\gamma+2\\delta\\}$.\n\\end{lemma}\n\\begin{proof}\nBecause all codewords of $\\mathcal{C}$ with order $p$ under $\\Phi$ belong to $K(C)$, thus $ker(C)\\geq\\gamma+\\delta$.\nIf $C$ is linear, then $ker(C)=\\gamma+2\\delta$. This completes the proof.\n\\end{proof}\n\n\\begin{theorem}\\label{Th:13}\nLet $\\alpha,\\beta,\\gamma,\\delta,\\kappa$ be integers satisfying (\\ref{eq:1}), then we have the following two assertions:\n\\begin{enumerate}\n  \\item [\\rm (i)] Let $C$ be a $\\mathbb{Z}_p \\mathbb{Z}_{p^2}$-linear code $C$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$,\nthen $ker(C)=\\gamma+2\\delta-\\overline{k}$, where $\\overline{k} \\in\\{0,1,\\ldots,\\delta\\}$.\n  \\item [\\rm (ii)] If $\\overline{k} \\in\\{0,1,\\ldots,\\delta\\}$,\n then there exists a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear code $C$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with $ker(C)=\\gamma+2\\delta-\\overline{k}$.\n  \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with generator matrix\n$$\\mathcal{G}=\\left(\\begin{array}{cc|ccc}I_\\kappa & T' & \\mathbf{0} & \\mathbf{0} & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{0} & \\mathbf{0} & pI_{\\gamma-\\kappa} & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S_k & \\mathbf{0} & I_\\delta\n\\end{array}\\right),$$\nwhere $S_k\\in \\mathbb{Z}_{p^2}^{\\delta\\times s}$ for $s=\\beta-(\\gamma-\\kappa)-\\delta$, and $\\{\\mathbf{u}_i\\}_{i=1}^\\gamma$, $\\{\\mathbf{v}_j\\}_{j=1}^\\delta$ are the row vectors of the matrix $\\mathcal{G}$ with order $p$ and $p^2$, respectively.\nLet $\\{\\mu_i\\}_{i=1}^{\\gamma+\\delta}$ be the rows of $\\mathcal{G}$.\nIn fact, $\\mu_i=\\mathbf{u}_i$ for $i\\in \\{1,\\ldots,\\gamma\\}$ and $\\mu_i=\\mathbf{v}_j$ for $i\\in\\{\\gamma+1,\\ldots,\\gamma+\\delta\\}$ and $j\\in \\{1,\\ldots,\\delta\\}$.\nLet $C=\\Phi(\\mathcal{C})$ with $ker(C)=k=\\gamma+2\\delta-\\overline{k}$. Then (i) follows from Lemma \\ref{l:12}.\nFor (ii), it is sufficient to prove that if the conditions in this theorem hold,\nthen we can construct a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code $\\mathcal{C}$ such that $C=\\Phi(\\mathcal{C})$ with $ker(C)=\\gamma+2\\delta-\\overline{k}$.\nAccording to the structure of the matrix $\\mathcal{G}$, we can construct $\\mathcal{C}$ by constructing $S_k$ as follows.\n\n(1) If $\\overline{k}=0$, the result is trivial.\n\n(2) If $\\overline{k}\\geq 1$, setting all elements in the first $\\overline{k}$ rows of $S_k$ is $p-1$, and zeros elsewhere.\n\n$$S_k=\\begin{array}{@{}c@{}c@{}c@{}c@{}c@{}c@{}}\n  \\left(\\begin{array}{c} p-1 \\\\ \\vdots \\\\ p-1\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\\right.\n& \\begin{array}{c} p-1 \\\\ \\vdots \\\\ p-1\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\n& \\begin{array}{c} \\cdots \\\\ \\vdots \\\\ \\cdots\\\\ \\cdots \\\\ \\vdots \\\\ \\cdots\\end{array}\n& \\begin{array}{c} p-1 \\\\ \\vdots \\\\ p-1\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\n& \\left.\\begin{array}{c} p-1 \\\\ \\vdots \\\\ p-1\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\\right)\n& \\begin{array}{c} 1 \\\\ \\vdots \\\\ \\overline{k}\\\\ \\overline{k}+1 \\\\ \\vdots \\\\ \\delta \\end{array}\n\\end{array}$$\nThen we will show $ker(C)=\\gamma+2\\delta-\\overline{k}$.\nFor $\\mathbf{c}\\in\\mathcal{C}$, let $ord(\\mathbf{c})$ mean the order of the codeword $\\mathbf{c}$, now consider the following two sets $A$ and $B$,\n$$A=\\{\\mathbf{c}|ord(\\mathbf{c})=p\\}$$ and\n\n$$\\begin{aligned}\nB=\\big\\{&\\mathbf{c}=\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i+\\sum_{j=1}^{\\gamma+\\overline{k}} b_j\\mu_j\\mid a_i\\in \\mathbb{Z}_{p^2} \\text{ and }\\exists a_i \\in\\mathbb{Z}_{p^2}\\backslash p\\mathbb{Z}_{p^2};\\\\\n&b_j\\in \\mathbb{Z}_{p} \\text{ for } j\\in \\{1,\\ldots,\\gamma\\} \\text{ and } b_j\\in p\\mathbb{Z}_{p^2} \\text{ for } j\\in \\{\\gamma+1,\\ldots,\\gamma+\\overline{k}\\}\\big\\}.\n\\end{aligned}$$\n\nIt is easy to check that the order of $\\mathbf{c}\\in B$ is $p^2$.\nThe size of $A$ is $p^{\\gamma+\\delta}$ and $A\\subseteq K(C)$ by Lemma \\ref{l:9}.\nThe size of $B$ is $p^{\\gamma+\\overline{k}}({p^{2\\delta-2\\overline{k}}-p^{\\delta-\\overline{k}}})=p^{\\gamma+2\\delta-\\overline{k}}-p^{\\gamma+\\delta}$.\nLet $T=A\\cup B$, therefore, the size of $T$ is $p^{\\gamma+2\\delta-\\overline{k}}$.\nBy the definition of $\\mathcal{C}$, it can be rewritten as\n$$\\begin{aligned}\n\\mathcal{C}=\\big\\{\\mathbf{c}^\\prime=\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a^\\prime_i \\mu_i+\\sum_{j=1}^{\\gamma+\\overline{k}} b^\\prime_j\\mu_j|& a_i^\\prime\\in \\mathbb{Z}_{p^2}; b_j^\\prime\\in \\mathbb{Z}_{p} \\text{ for } j\\in \\{1,\\ldots,\\gamma\\} \\text{ and }\\\\\n&b_j^\\prime\\in \\mathbb{Z}_{p^2} \\text{ for } j\\in \\{\\gamma+1,\\ldots,\\gamma+\\overline{k}\\}\\big\\}.\n\\end{aligned}$$\nFor simplicity, let $\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i,\\sum_{j=1}^{\\gamma+\\overline{k}} b_j\\mu_j,\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a^\\prime_i \\mu_i,\\sum_{j=1}^{\\gamma+\\overline{k}} b^\\prime_j\\mu_j$ be $\\mathbf{c}_1,\\mathbf{c}_2,\\mathbf{c}_1^\\prime,\\mathbf{c}_2^\\prime$, respectively.\nIf $\\mathbf{c}=\\mathbf{c}_1+\\mathbf{c}_2\\in B$, then for arbitrary $\\mathbf{c}^\\prime \\in \\mathcal{C}$,\nwe have $pP'(\\mathbf{c},\\mathbf{c}^\\prime)=pP'(\\mathbf{c}_1, \\mathbf{c}_1^\\prime)=\\mathbf{0}\\in \\mathcal{C}$ or the linear combination of $p(p-1)^{-1}\\mu_i \\in \\mathcal{C}$,\nwhere $i\\in \\{\\gamma+\\overline{k}+1, \\ldots, \\gamma+\\delta\\}$.\nTherefore, $B\\subseteq K(C)$ by Lemma \\ref{l:9}. Thus $T\\subseteq K(C)$.\n\nNext we prove that $K(C)= T$. Let $D=\\mathcal{C}\\backslash T$, and $D$ can be written as follows.\n$$\\begin{aligned}\nD=\\big\\{\\mathbf{d}=\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i+\\sum_{j=1}^{\\gamma+\\overline{k}} b_j\\mu_j\\mid &a_i \\in\\mathbb{Z}_{p^2},\nb_j\\in \\mathbb{Z}_{p} \\text{ for } j\\in \\{1,\\ldots,\\gamma\\} \\text{ and }\\\\\n& \\exists b_j\\in \\mathbb{Z}_{p^2}\\backslash p\\mathbb{Z}_{p^2} \\text{ for } j\\in \\{\\gamma+1,\\ldots,\\gamma+\\overline{k}\\}\\big\\}.\n\\end{aligned}$$\nWe only consider $s=1$ since all columns of $S_k$ are the same.\nFor any $\\mathbf{d}\\in D$, if there exists $ b_{j_1}\\in \\mathbb{Z}_{p^2}\\backslash p\\mathbb{Z}_{p^2},j_1\\in \\{\\gamma+1,\\ldots,\\gamma+\\overline{k}\\}$, then let $\\mathbf{c}^\\prime=(p-1)\\mu_{j_1}$,\nwe have $pP'(\\mathbf{d},\\mathbf{c}^\\prime$)=$(\\mathbf{0}|0,0,\\ldots,0,p(p-1)^{-1},0,\\ldots,0)$ or $(\\mathbf{0}|p(p-1)^{-1},0,\\ldots,0,p(p-1)^{-1},0,\\ldots,0)$,\nthat is, the value of the $(\\alpha+1)$-th coordinate is $0$ or $p(p-1)^{-1}$ and the value of the $(\\alpha+\\beta-\\delta+j_1-\\gamma)$-th coordinate is $p(p-1)^{-1}$.\nBoth of these two vectors are not in $\\mathcal{C}$ because $(\\mathbf{0}|p,0,\\ldots,0,p(p-1)^{-1},0,\\ldots,0)\\in \\mathcal{C}$.\nThus there exists a $\\mathbf{c}^\\prime=p(p-1)^{-1}\\mu_{j_1}\\in\\mathcal{C}$ such that $pP'(\\mathbf{d},\\mathbf{c}^\\prime)\\notin \\mathcal{C}$  for all $\\mathbf{d}\\in D$.\nTherefore, $D \\cap K(C)=\\emptyset$, that is, $(\\mathcal{C}\\backslash T) \\cap K(C)=\\emptyset$.\nThen $T=K(C)$ since $T\\subseteq K(C)$.\nThe assertion (ii) follows.\n\\end{proof}\n\\begin{exmpl}\nLet $p=3$, and $C$ be a $\\mathbb{Z}_3\\mathbb{Z}_9$-linear code of type $(\\alpha,9;2,5;1)$ with $\\ker(C)=12-\\overline{k}$. Let $\\mathcal{C}=\\Phi^{-1}(C)$.\nBy Theorem \\ref{Th:13} (i), $\\overline{k}\\in\\{0,1,2,3,4,5\\}$ and $\\ker(C)\\in\\{12,11,10,9,8,7\\}$.\nTaking the generator matrix of $\\mathcal{C}$ as follows:\n$$\\mathcal{G}_S=\\left(\\begin{array}{cc|ccc}\n1 & T' & \\mathbf{0} & 0 & \\mathbf{0}\\\\\n0 & \\mathbf{0} &\\mathbf{0} & 3 & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S_k & \\mathbf{0} & I_5\n\\end{array}\\right).$$\nBy Theorem \\ref{Th:13} (ii), $S_{12}=(\\mathbf{0})$, and $S_{11},S_{10},S_{9},S_{8},S_{7}$ are constructed as follows:\n$$\nS_{11}=\\left(\\begin{array}{ccc}\n2 & 2 & 2\\\\\n0 & 0 & 0\\\\\n0 & 0 & 0\\\\\n0 & 0 & 0\\\\\n0 & 0 & 0\n\\end{array}\n\\right),\nS_{10}=\\left(\\begin{array}{ccc}\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n0 & 0 & 0\\\\\n0 & 0 & 0\\\\\n0 & 0 & 0\n\\end{array}\n\\right),\nS_{9}=\\left(\\begin{array}{ccc}\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n0 & 0 & 0\\\\\n0 & 0 & 0\n\\end{array}\n\\right),$$\n$$\nS_{8}=\\left(\\begin{array}{ccc}\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n0 & 0 & 0\n\\end{array}\n\\right),\nS_{7}=\\left(\\begin{array}{ccc}\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n2 & 2 & 2\\\\\n2 & 2 & 2\n\\end{array}\n\\right).\n$$\n\\end{exmpl}\n\\begin{remark}\\label{rem:15}\nConsidering the construction in Theorem \\ref{Th:13}, it is not the only one.\nFor example, let the matrix\n$$S_k=\\begin{array}{@{}c@{}c@{}c@{}c@{}c@{}c@{}}\n  \\left(\\begin{array}{c} p-1 \\\\ \\vdots \\\\ p-1\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\\right.\n& \\begin{array}{c}  0 \\\\ \\vdots \\\\ 0\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\n& \\begin{array}{c} \\cdots \\\\ \\vdots \\\\ \\cdots\\\\ \\cdots \\\\ \\vdots \\\\ \\cdots\\end{array}\n& \\begin{array}{c} 0 \\\\ \\vdots \\\\ 0\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\n& \\left.\\begin{array}{c} 0 \\\\ \\vdots \\\\ 0\\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array}\\right)\n& \\begin{array}{c} 1 \\\\ \\vdots \\\\ \\overline{k}\\\\ \\overline{k}+1 \\\\ \\vdots \\\\ \\delta \\end{array}\n\\end{array}$$\ncan also hold, that is, it is enough that there is one column vector which the first $\\overline{k}$ coordinates are $2$ and the last $\\delta-\\overline{k}$ are $0$ in $S_k$.\nOther columns just hold the last $\\delta-\\overline{k}$ are $0$, other coordinates could be any elements in $\\mathbb{Z}_9$.\n\\end{remark}\n\n\n\n\n\\section{Pairs of rank and kernel dimension of $\\mathbb{Z}_3\\mathbb{Z}_{9}$-additive codes}\\label{sec:5}\n\nWe give the range of the rank of the $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes if its dimension of the kernel is fixed.\nOnce a possible pair of values $(r,k)$ is given,\nthe $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code $C$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with $r=rank(C)$ and $k=ker(C)$ can also be constructed.\n\\begin{lemma}\\label{l:15}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-additive code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$,\nand $C=\\Phi(\\mathcal{C})$ is the corresponding $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code.\nIf $rank(C)=\\gamma+2\\delta+\\overline{r}$ and $ker(C)=\\gamma+2\\delta-\\overline{k}$, where $\\overline{k}\\in\\{1,\\ldots,\\delta\\}$,\nthen $$1\\leq\\overline{r}\\leq \\dbinom{\\overline{k}}{2}+\\dbinom{\\overline{k}+2}{3}.$$\nMoreover, $\\overline{r}=0$ if $\\overline{k}=0$.\n\\end{lemma}\n\n\\begin{proof}If $\\overline{k}=0$, then $C$ is linear and $\\overline{r}=0$.\nIt is clear that $\\overline{r}\\geq 1$ if $\\overline{k}\\geq 1$.\nLet $\\{\\mathbf{u}_i\\}_{i=1}^\\gamma$ and $\\{\\mathbf{v}_j\\}_{j=1}^\\delta$ be the row vectors of the matrix $\\mathcal{G}$ with order $p$ and $p^2$, respectively.\nBy Proposition \\ref{prop:11}, there exists a set $\\{\\mathbf{v}_1,\\mathbf{v}_2,\\ldots,\\mathbf{v}_{\\overline{k}}\\}$ of row vectors of order $p^2$ in the generator matrix $\\mathcal{G}$,\nsuch that for all $a_i \\in \\mathbb{Z}_p$,\n$C=\\bigcup_{a_i\\in \\mathbb{Z}_p}\\bigg(K(C)+\\Phi\\bigg(\\sum_{i=1}^{\\overline{k}}a_i\\mathbf{v}_i\\bigg)\\bigg).$\nAccording to Lemma \\ref{l:2},\n$\\Phi\\bigg(\\sum_{i=1}^{\\overline{k}}a_i\\mathbf{v}_i\\bigg)=\\Sigma_{j=1}^{\\overline{k}}\\Phi(\\mathbf{v}_j)+\\Sigma_{1\\leq l \\leq k\\leq \\overline{k}}(\\tilde{a}\\Phi(3(\\mathbf{v}_k\\ast v_l))+\\Sigma_{1\\leq x \\leq y\\leq z\\leq \\overline{k}}(\\tilde{b}\\Phi(3(\\mathbf{v}_x\\ast \\mathbf{v}_y\\ast \\mathbf{v}_z))$, $\\tilde{a},\\tilde{b}\\in \\mathbb{Z}_3$.\n\nFrom the proof of Theorem \\ref{Th:13} (ii), $K(C)=T=A\\cup B$.\nThe set $A$ can be generated by $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$ and $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$.\n$$\\begin{aligned}\nB=\\big\\{&\\mathbf{c}=\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i+\\sum_{j=1}^{\\gamma+\\overline{k}} b_j\\mu_j\\mid a_i\\in \\mathbb{Z}_{9} \\text{ and }\\exists a_i \\in\\mathbb{Z}_{9}\\backslash 3\\mathbb{Z}_{9};\\\\\n&b_j\\in \\mathbb{Z}_{3} \\text{ for } j\\in \\{1,\\ldots,\\gamma\\} \\text{ and } b_j\\in 3\\mathbb{Z}_{9} \\text{ for } j\\in \\{\\gamma+1,\\ldots,\\gamma+\\overline{k}\\}\\big\\}.\n\\end{aligned}$$\nFor any vector $\\mathbf{c} \\in B$, $\\Phi(\\mathbf{c})=\\Phi\\bigg(\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i+\\sum_{j=1}^{\\gamma+\\overline{k}}b_j\\mu_j\\bigg)\n=\\Phi\\bigg(\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i\\bigg)+\\Phi\\bigg(\\sum_{j=1}^{\\gamma+\\overline{k}}b_j\\mu_j\\bigg)$ by Lemma \\ref{l:2}.\nThe order of $\\sum_{j=1}^{\\gamma+\\overline{k}}b_j\\mu_j$ is $3$, thus $\\Phi\\bigg(\\sum_{j=1}^{\\gamma+\\overline{k}}b_j\\mu_j\\bigg)$\ncan also be generated by $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$ and $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$.\nBy the definition of $\\mu_j$, now we only consider  $\\Phi\\bigg(\\sum_{i=\\gamma+\\overline{k}+1}^{\\gamma+\\delta} a_i\\mu_i\\bigg)=\\Phi\\bigg(\\sum_{j=\\overline{k}+1}^{\\delta} a_j\\mathbf{v}_j\\bigg), a_j \\in\\mathbb{Z}_9$.\n\nAt first, we will prove that if $k\\in\\{\\overline{k}+1,\\ldots,\\delta\\}$,\nthen for all $\\mathbf{v}\\in\\mathcal{C}$,\n$\\Phi(3(\\mathbf{v}_k*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}_k*\\mathbf{v}))$\nis a linear combination of $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$ and $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$.\nThe condition\n$k\\in\\{\\overline{k}+1,\\ldots,\\delta\\}$ implies that $\\Phi(\\mathbf{v}_k)\\in K(C)$ from the proof of Proposition \\ref{prop:11}, that is, for all $\\mathbf{v}\\in \\mathcal{C}$,\n$3(\\mathbf{v}_k*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}_k*\\mathbf{v})\\in \\mathcal{C}$ by Lemma \\ref{l:9}.\nNote that all vectors $\\{\\mathbf{u}_i\\}_{i=1}^\\gamma$, $\\{3\\mathbf{v}_j\\}_{j=1}^\\delta$ and\n$\\{3(\\mathbf{v}_k*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}_k*\\mathbf{v})|k\\in\\{\\overline{k}+1,\\ldots,\\delta\\}\\}$ are of order $3$,\nthus $\\Phi(3(\\mathbf{v}_k*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}*\\mathbf{v}+\\mathbf{v}_k*\\mathbf{v}_k*\\mathbf{v}))$ can be represented linearly by $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$ and $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$ for $k\\in\\{\\overline{k}+1,\\ldots,\\delta\\}$.\n\nWithout loss of generality, let $\\delta-\\overline{k}=2$, and $j_1=\\overline{k}+1, j_2=\\overline{k}+2$.\nThen $\\Phi(a_{j_1}\\mathbf{v}_{j_1}+a_{j_2}\\mathbf{v}_{j_2})=\\Phi(a_{j_1}\\mathbf{v}_{j_1})+\\Phi(a_{j_2}\\mathbf{v}_{j_2})+2\\Phi(3(\\mathbf{v}_{j_1}*\\mathbf{v}_{j_2}+\\mathbf{v}_{j_1}*\\mathbf{v}_{j_1}*\\mathbf{v}_{j_2}\n+\\mathbf{v}_{j_1}*\\mathbf{v}_{j_2}*\\mathbf{v}_{j_2}))$.\nWe know $\\Phi(3(\\mathbf{v}_{j_1}*\\mathbf{v}_{j_2}+\\mathbf{v}_{j_1}*\\mathbf{v}_{j_1}*\\mathbf{v}_{j_2}\n+\\mathbf{v}_{j_1}*\\mathbf{v}_{j_2}*\\mathbf{v}_{j_2}))$ is a linear combination of $\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$ and $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$.\nSince $a_{j_1},a_{j_2}\\in \\mathbb{Z}_9$, it is easy to check that $\\Phi(a_{j_1}\\mathbf{v}_{j_1}),\\Phi(a_{j_2}\\mathbf{v}_{j_2})$ can also be generated by\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$ and $\\{\\Phi(3\\mathbf{v}_j)\\}_{j=1}^\\delta$. Consequently, $\\langle C\\rangle$ can be generated by\n$\\{\\Phi(\\mathbf{u}_i)\\}_{i=1}^\\gamma$, $\\{\\Phi(\\mathbf{v}_j)\\}_{j=1}^\\delta$, $\\Sigma_{1\\leq l \\leq k\\leq \\overline{k}}\\Phi(3(\\mathbf{v}_k\\ast v_l))$ and\n$\\Sigma_{1\\leq x \\leq y\\leq z\\leq \\overline{k}}\\Phi(3(\\mathbf{v}_x\\ast \\mathbf{v}_y\\ast \\mathbf{v}_z))$.\nHence $\\overline{r}\\leq \\dbinom{\\overline{k}}{2}+\\dbinom{\\overline{k}+2}{3}.$\n\\end{proof}\n\nAccording to Lemma \\ref{l:15}, $rank(C)\\in\\bigg\\{\\gamma+2\\delta,\\ldots,\\gamma+2\\delta+\\dbinom{\\overline{k}}{2}+\\dbinom{\\overline{k}+2}{3}\\bigg\\}$,\nand we have\n$$rank(C)\\in \\bigg\\{\\gamma+2\\delta,...,\\rm{min}\\bigg(\\beta+\\gamma+\\kappa,\\gamma+2\\delta+\\dbinom{\\overline{k}}{2}+\\dbinom{\\overline{k}+2}{3}\\bigg)\\bigg\\}.$$\n\n\\begin{theorem}\\label{th:16}\nLet $\\alpha,\\beta,\\gamma,\\delta,\\kappa$ be positive integers satisfying (\\ref{eq:1}).\nThen, there is a $\\mathbb{Z}_3 \\mathbb{Z}_{9}$-linear code $C$ of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with $ker(C)=\\gamma+2\\delta-\\overline{k}$ and $rank(C)=\\gamma+2\\delta+\\overline{r}$ if and only if\n$\\overline{k}\\in\\{1,\\ldots,\\delta\\}$ and\n$\\overline{r}\\in \\bigg\\{1,\\ldots,\\rm{min}\\bigg(\\beta-(\\gamma-\\kappa)-\\delta,\\dbinom{\\overline{k}}{2}+\\dbinom{\\overline{k}+2}{3}\\bigg)\\bigg\\}$\nor $\\overline{k}=\\overline{r}=0$.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathcal{C}$ be a $\\mathbb{Z}_3\\mathbb{Z}_{9}$-additive code of type $(\\alpha,\\beta;\\gamma,\\delta;\\kappa)$ with generator matrix\n$$\\mathcal{G}=\\left(\\begin{array}{cc|ccc}I_\\kappa & T' & \\mathbf{0} & \\mathbf{0} & \\mathbf{0}\\\\\n\\mathbf{0} & \\mathbf{0} & \\mathbf{0} & 3I_{\\gamma-\\kappa} & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S_{r,k} & \\mathbf{0} & I_\\delta\n\\end{array}\\right),$$\nwhere $S_{r,k}$ is a matrix over $\\mathbb{Z}_{9}$ of size $\\delta\\times (\\beta-(\\gamma-\\kappa)-\\delta)$,\nand $C=\\Phi(\\mathcal{C})$ is its corresponding $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear code.\nThe necessary condition is clear by Lemma \\ref{l:15}.\nThen we will show the sufficient conditions.\n\nIf $\\overline{k}=0$, then $C$ is a linear code and $\\overline{r}=0$.\nIf $\\overline{k}\\geq 1$ and $\\overline{r}\\geq 1$,\nlet $\\mathbf{h'}_k,1\\leq k\\leq \\delta$ be the column vector of length $\\delta$, with a $1$ in the $k$-th coordinate and zeros elsewhere,\nsimilar to those matrices\n$\\bar{A},\\bar{B},\\bar{C},\\bar{D},\\bar{E}$ in the proof of Theorem \\ref{th:7},\ndefine matrices $\\bar{A'}_{\\overline{k} \\times \\overline{k}}$, $\\bar{B'},\\bar{C'},\\bar{D'}$ of size $\\overline{k} \\times \\dbinom{\\overline{k}}{2}$\nand $\\bar{E'}$ of size $\\overline{k} \\times \\dbinom{\\overline{k}}{3}$ over $\\mathbb{Z}_9$ as follows:\n\\begin{itemize}\n\\item $\\bar{A'}_{\\overline{k}\\times \\overline{k}}=2I_{\\overline{k}}$;\n\\item $\\bar{B'}=(\\mathbf{b'}_j)$, where $\\mathbf{b'}_j=\\mathbf{h'}_k+\\mathbf{h'}_l$ for $1\\leq k< l\\leq \\overline{k} , j\\in\\bigg\\{1,2,\\ldots, \\dbinom{\\overline{k}}{2}\\bigg\\}$,\nand $\\mathbf{b'}_{j_1}\\neq \\mathbf{b'}_{j_2}$ if $j_1 \\neq j_2$;\n\\item $\\bar{C'}=(\\mathbf{c'}_j)$, where $\\mathbf{c'}_1=(2,2,\\ldots,2)^T$ and $\\mathbf{c'}_j=2\\mathbf{b'}_j$ for $j\\in\\bigg\\{2,3,\\ldots, \\dbinom{\\overline{k}}{2}\\bigg\\}$;\n\\item $\\bar{D'}=(\\mathbf{d'}_j)$, where $\\mathbf{d'}_j=\\mathbf{h'}_k+2\\mathbf{h'}_l$ (or equivalently, $\\mathbf{d'}_j=2\\mathbf{h'}_k+\\mathbf{h'}_l$) for $1\\leq k< l\\leq \\overline{k} , j\\in\\bigg\\{1,2,\\ldots, \\dbinom{\\overline{k}}{2}\\bigg\\}$, and $\\mathbf{d'}_{j_1}\\neq \\mathbf{d'}_{j_2}$ if $j_1 \\neq j_2$;\n\\item $\\bar{E'}=(\\mathbf{e'}_j)$, where $\\mathbf{e'}_j=\\mathbf{h'}_x+\\mathbf{h'}_y+\\mathbf{h'}_z$ for $1\\leq x< y<z\\leq \\overline{k}, j\\in\\bigg\\{1,2,\\ldots, \\dbinom{\\overline{k}}{3}\\bigg\\}$,\nand $\\mathbf{e'}_{j_1}\\neq \\mathbf{e'}_{j_2}$ if $j_1 \\neq j_2$.\n\\end{itemize}\n\nThen $S_{r,k}$ can be constructed in the following way.\nDivide $S_{r,k}$ into two matrices $S_{1},S_{2}$, where $S_1$ is of size $\\overline{k}\\times(\\beta-(\\gamma-\\kappa)-\\delta)$,\n$S_2$ is of size $(\\delta-\\overline{k})\\times(\\beta-(\\gamma-\\kappa)-\\delta)$ such that\n$S_{r,k}=\\begin{pmatrix}S_1\\\\S_2\\end{pmatrix}$, and let $S_2=(\\mathbf{0})$.\nLet the first column of $S_1$ be $\\mathbf{c'}_1=(2,2,\\ldots,2)^T$,\nwhich guarantee $ker(C)=\\gamma+2\\delta-\\overline{k}$ by Theorem \\ref{Th:13} and Remark \\ref{rem:15}.\nAs to the remaining columns of $S_1$, similar to the proof of Theorem \\ref{th:7},\none can select $\\overline{r}$ columns from matrices $\\bar{A'},\\bar{B'},\\bar{C'},\\bar{D'},\\bar{E'}$ in turn.\nSince $\\mathbf{c'}_1$ is the first column, one can select the other $\\dbinom{\\overline{k}}{2}-1$ columns from $\\bar{C'}$.\nBy using the same argument as the proof of Theorem \\ref{th:7}, $rank(C)=\\gamma+2\\delta+\\overline{r}$. This completes the proof.\n\\end{proof}\n\n\\begin{exmpl}\nConsidering a $\\mathbb{Z}_3 \\mathbb{Z}_9$-linear code $C$ of type $(\\alpha,\\beta;2,6;1)$.\nFrom Theorem \\ref{th:16}, $\\overline{k}\\in\\{0,1,2,3,4,5,6\\}$ and $\\overline{r}\\in\\bigg\\{0,1,\\ldots,\\rm{min}\\bigg(\\beta-7,\\dbinom{\\overline{k}}{2}+\\dbinom{\\overline{k}+2}{3}\\bigg)\\bigg\\}$.\nLet $\\beta=14$, the possible dimensions of the kernel $k\\in\\{8,9,10,11,12,13,14\\}$,\nand the possible ranks $r\\in\\{14,15,16,17,18,19,20,21\\}$.\nThe values of $(r,k)$ are shown in Table \\ref{tab:1}.\n\\begin{table}[!htbp]\n\\centering\n\\caption{The values of $(r,k)$}\\label{tab:1}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\diagbox{$k$}{$r$}  & 14        & 15        & 16        & 17        & 18        & 19        & 20        & 21          \\\\ \\hline\n14                  & $\\bullet$ &           &           &           &           &           &           &             \\\\ \\hline\n13                  &           & $\\bullet$ &           &           &           &           &           &             \\\\ \\hline\n12                  &           & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ &           &             \\\\ \\hline\n11                  &           & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$   \\\\ \\hline\n10                  &           & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$   \\\\ \\hline\n9                   &           & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$   \\\\ \\hline\n8                   &           & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$ & $\\bullet$   \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nThe generator matrix of $\\mathcal{C}=\\Phi^{-1}(C)$ is\n$$\\mathcal{G}=\\left(\\begin{array}{cc|ccc}1 & T' & \\mathbf{0} & 0 & \\mathbf{0}\\\\\n0 & \\mathbf{0} & 3T_1 & 3 & \\mathbf{0}\\\\\n\\hline\n\\mathbf{0} & S' & S_{r,k} & \\mathbf{0} & I_4\n\\end{array}\\right).$$\n\nWhen $\\overline{k}=4$, the related matrices are as follows, others can be obtained similarly and be omitted.\n\\begin{center}\n$\\bar{A'}=\\begin{pmatrix} 2 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0\\\\ 0 & 0 & 2 & 0\\\\ 0 & 0 & 0 & 2 \\end{pmatrix}$,\n$\\bar{B'}=\\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 1 & 1 & 0 \\\\ 0 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 1 & 1 & 0 & 1 \\end{pmatrix}$,\n$\\bar{C'}=\\begin{pmatrix} 2 & 2 & 2 & 0 & 0 & 0 \\\\ 2 & 0 & 0 & 2 & 2 & 0 \\\\ 2 & 2 & 0 & 0 & 2 & 2 \\\\ 2 & 0 & 2 & 2 & 0 & 2 \\end{pmatrix},$\n\n$\\bar{D'}=\\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\\\ 2 & 0 & 0 & 1 & 1 & 0 \\\\ 0 & 2 & 0 & 0 & 2 & 1 \\\\ 0 & 0 & 2 & 2 & 0 & 2 \\end{pmatrix},$\n$\\bar{E'}=\\begin{pmatrix} 1 & 1 & 1 & 0 \\\\ 1 & 1 & 0 & 1\\\\ 1 & 0 & 1 & 1\\\\ 0 & 1 & 1 & 1 \\end{pmatrix}$.\n\n$S_{15,10}=\\left(\n\\begin{array}{@{}c@{}|c}  \\begin{array}{c}2\\\\2\\\\2\\\\2\\\\0\\\\0\\end{array}  & \\mathbf{0}\\end{array}\n\\right),$\n$S_{16,10}=\\left(\n\\begin{array}{@{}c@{}|c}  \\begin{array}{cc}2 & 2\\\\2& 0\\\\2& 0\\\\2& 0\\\\0& 0\\\\0& 0\\end{array}  & \\mathbf{0}\\end{array}\n\\right),$\n$S_{17,10}=\\left(\n\\begin{array}{@{}c@{}|c}  \\begin{array}{ccc}2 & 2 & 0\\\\2& 0&2\\\\2& 0& 0\\\\2& 0& 0\\\\0& 0& 0\\\\0& 0& 0\\end{array}  & \\mathbf{0}\\end{array}\n\\right),$\n\n$S_{18,10}=\\left(\n\\begin{array}{@{}c@{}|c}  \\begin{array}{cccc}2 & 2 & 0&0\\\\2& 0&2&0\\\\2& 0& 0&2\\\\2& 0& 0&0\\\\0& 0& 0&0\\\\0& 0& 0&0\\end{array}  & \\mathbf{0}\\end{array}\n\\right),$\n$S_{19,10}=\\left(\n\\begin{array}{@{}c@{}|c}  \\begin{array}{ccccc}2 & 2 & 0&0&0\\\\2& 0&2&0&0\\\\2& 0& 0&2&0\\\\2& 0& 0&0&2\\\\0& 0& 0&0&0\\\\0& 0& 0&0&0\\end{array}  & \\mathbf{0}\\end{array}\n\\right),$\n\n$S_{20,10}=\\left(\n\\begin{array}{@{}c@{}|c}  \\begin{array}{cccccc}2 & 2 & 0&0&0&1\\\\2& 0&2&0&0&1\\\\2& 0& 0&2&0&0\\\\2& 0& 0&0&2&0\\\\0& 0& 0&0&0&0\\\\0& 0& 0&0&0&0\\end{array}  & \\mathbf{0}\\end{array}\n\\right),$\n$S_{21,10}=\\left(\n\\begin{array}{ccccccc}2 & 2 & 0&0&0&1&1\\\\2& 0&2&0&0&1&0\\\\2& 0& 0&2&0&0&1\\\\2& 0& 0&0&2&0&0\\\\0& 0& 0&0&0&0&0\\\\0& 0& 0&0&0&0&0\\end{array}\n\\right).$\n\\end{center}\n\n\\end{exmpl}\n\n\n\\section{Conclusion}\\label{sec:6}\n\nIn this paper, we studied the kernel of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes and the rank of $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes.\nUsing combinatorial enumeration techniques, we gave the lower and upper bounds of the rank of $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes and dimension of the kernel of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes, respectively.\nMoreover, we have constructed $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear codes for each value of the dimension of the kernel,\nand constructed $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes for each value of the rank.\nFinally, we also constructed $\\mathbb{Z}_3\\mathbb{Z}_{9}$-linear codes for pairs of values of rank and the dimension of the kernel.\n\n\\section{Acknowledgement}\n\nThe authors are grateful to Professor Denis Krotov's helpful discussion.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{Intro}\n\nQuantum theory is first and foremost a calculus for computing the probabilities of outcomes of measurements made on physical systems.  Therefore, the generic problem in quantum theory is one of probabilistic inference, i.e. given a specified class of quantum states, compute the predicted probabilities of measurement outcomes and their correlations.  For example, computing the correlation functions of a system in the ground state of a Hamiltonian, or computing the probabilities for the possible measurement outcomes after implementing a quantum circuit, are problems of this general type.  Such quantum inferences present a formidable computational challenge as the number of subsystems becomes large, since the number of parameters needed to specify a quantum state grows exponentially with the number of subsystems, and the formulas for quantities of interest typically also involve an exponentially large number of terms. \n\nA similar problem arises in classical probabilistic inference, since the number of terms required to specify a general probability distribution also grows exponentially with the number of random variables involved.  A variety of algorithms for classical probabilistic inference have been discovered, of which Belief Propagation algorithms on Graphical Models are amongst the most powerful.  Such algorithms are particularly interesting for two reasons.  Firstly, they are highly parallelizable in the sense that they can be implemented by associating each random variable with a separate processor.  Messages are received and sent by the processors along the links of a network corresponding to the edges of a graph and, importantly, the order in which the messages arrive does not matter.  Secondly, Belief Propagation performs remarkably well as a heuristic algorithm, even in cases where it is not guaranteed to converge to the exact solution.  Important examples include the near optimal decoding of low density \\cite{Gal63a} and turbo \\cite{BGT93a} error correction codes, spin glass models \\cite{MP01a}, and random satisfiability problems \\cite{MPZ02a}. Understanding the reasons for this is currently an active area of research, but it is understood \\cite{Yed01a} to be related to a hierarchy of approximation schemes commonly used in statistical physics.  \n\nDue to the similarity between the classical and quantum problems, one might hope to leverage the power of Belief Propagation in the quantum case also, especially since quantum theory can be regarded as a noncommutative generalization of classical probability theory.  This is indeed the case, and in this paper we develop the necessary theory of Quantum Belief Propagation and its associated Graphical Models.\n\nThis paper should be of interest to researchers in Graphical Models and Belief Propagation, as well as to researchers in quantum theory, particularly in quantum information and the simulation of quantum many-body systems.  As such, it is intended to be as self-contained as possible, although we do assume familiarity with the basic formalism of quantum theory on finite dimensional Hilbert spaces, including the theory of density matrices, generalized measurements and completely positive maps, as used in quantum information theory.  These are covered in detail in the textbook of Nielsen and Chuang~\\cite{NC00a}, as well as in Preskill's lecture notes~\\cite{Pre99b}.  For further background on classical Graphical Models and Belief Propagation, we suggest the texts of Lauritzen~\\cite{Lau96a}, MacKay~\\cite{Mac03a}, and Neapolitan~\\cite{Nea90a,Nea04a}, as well as the review articles by Yedida et al.~\\cite{Yed01a,YFW02a} and Aji and McEliece~\\cite{AM00a}.\n\nThe remainder of this paper is structured as follows.  In \\S\\ref{Problem}, the generic classical and quantum probabilistic inference problems are defined.  In \\S\\ref{Cond}, we review the notions of classical and quantum conditional independence, which are crucial for the development of Graphical Models and Belief Propagation algorithms.  \\S\\ref{Cond:CMI} outlines the entropic approach to conditional independence based on the vanishing of conditional mutual information and the associated constraints on conditional and mutual probability distributions.  This entropic approach has a straightforward quantum generalization based on the equality conditions for strong-subadditivity, which is described in \\S\\ref{Cond:QCMI}.  \\S\\ref{Cond:CDO} introduces the quantum conditional and mutual density operators, which are analogous to classical conditional and mutual probability distributions, and \\S\\ref{Cond:QCI} explains how quantum conditional independence can be characterized directly in terms of them.    \n\nIn \\S\\ref{Graph}, we develop the theory of quantum Graphical Models.    \\S\\ref{Graph:MN} reviews the definition of classical Markov Networks and the Hammersley-Clifford theorem, which gives an explicit representation of the probability distributions supported on them.  Motivated by this, \\S\\ref{Graph:QGS} defines the class of quantum $n$-Bifactor Networks, which are the most general class of networks on which our Belief Propagation algorithms operate. \\S\\ref{Graph:DMG} reviews the theory of dependency models and graphoids, which are abstractions of the conditional independence relation, and a quantum graphoid is defined based on quantum conditional independence. \\S\\ref{Graph:HC} uses the quantum graphoid to define quantum Markov Networks and gives some partial characterization theorems, along the lines of the classical Hammersley-Clifford theorem, which connect quantum Markov Networks to $n$-Bifactor Networks.  \\S\\ref{Graph:OM} briefly discusses quantum generalizations of two other classical Graphical Models: Factor Graphs and Bayesian Networks.  Figure \\ref{fig:worldview} sketches the relation between some of these Graphical Models, and summarizes the Quantum Belief Propagation algorithm's domain of convergence.  \n\n\\S\\ref{QBP} discusses the Quantum Belief Propagation algorithms.  In \\S\\ref{QBP:Desc}, QBP algorithms are described for $n$-Bifactor Networks.  In \\S\\ref{QBP:Conv}, QPB is shown to converge for $1$-Bifactor Networks on trees and for general Bifactor Networks on trees that are also Quantum Markov Networks.  \\S\\ref{Heur}  discusses some methods for using QBP as a heuristic algorithm in cases where it is not known to converge.  These are coarse graining \\S\\ref{Heur:CG}, sliding window QBP \\S\\ref{Heur:SW} and the method of replicas \\S\\ref{Heur:Rep}.  \n\n\\S\\ref{App} presents two applications of QBP: to decoding quantum error correcting codes in \\S\\ref{App:QEC} and to simulating many-body quantum systems in \\S\\ref{App:Stat}.  In particular,  \\S\\ref{App:Stat} explains how projected entangled-pair states, which have been successfully used in statistical physics as approximations to the ground states of a wide class of Hamiltonians, can be incorporated into the framework of Bifactor Networks. \n\nTo conclude, \\S\\ref{Relate} discusses the relationship to other quantum generalizations of Graphical Models and Belief Propagation that have been proposed and \\S\\ref{Conc} describes open questions and future research directions suggested by this work. \n\n\n\nNote that a slightly unconventional notation for probability distributions on sets of random variables and for quantum states on tensor products of quantum systems is used throughout.  This is very convenient for describing Graphical Models and is reviewed in appendix \\ref{ProbNot}.  \n\n\\section{Classical and Quantum Probabilistic Inference}\n\n\\label{Problem}\n\nClassical Graphical Models are designed to be used as tools for making probabilistic inferences amongst large numbers of correlated random variables.  Consider a set random variables, $V = \\{v_1, v_2, \\ldots, v_N\\}$, each of which takes a finite number of integer values $\\{1,2,\\ldots d\\}$.  To specify a general probability distribution, $P(V)$, over the variables requires $O(d^N)$ parameters.  On learning that some subset of the variables $U \\subseteq V$ take particular values, denoted $\\tilde{U} = \\{u = j_u\\}_{u \\in U}$, an important task is to update the probability for some other disjoint subset of variables $W \\subseteq V$ via Bayes rule\n\\begin{equation}\n\\label{Intro:Bayes}\nP(W|\\tilde{U}) = \\frac{P(\\tilde{U}\\cup W)}{P(\\tilde{U})} = \\frac{\\sum_{V - (U \\cup W)}P(\\tilde{U} \\cup (V - U))}{\\sum_{V - U} P(\\tilde{U}\\cup (V - \\tilde{U}))}.\n\\end{equation}\nThis immediately raises two problems.  Firstly, the number of parameters needed to specify the input to the computation, i.e. the probability distribution itself, is exponential in $N$.  We would like to specify a well-defined computational problem in which $N$ measures the input size.  Therefore, it is not feasible to consider the full set of probability distributions over $N$ variables, and attention must be restricted to families of distributions that can be specified with a number of parameters that grows only polynomially in $N$.  Secondly, assuming that the sizes of $U$ and $W$ are held constant as $N$ increases, eq.~\\eqref{Intro:Bayes} involves sums over a number of terms that is exponential in $N$.  Thus, a straightforward evaluation of the formula would not give an efficient algorithm.  The restriction on the class of probability distributions must somehow be used to find an alternative method of computation that is efficient.\n\nClassical Graphical Models are designed to provide an efficient representation of classes of probability distributions and Belief Propagation algorithms are designed to solve the corresponding inference problem.\n\nIn quantum theory, the random variables are replaced by a set of $N$ quantum systems $V = \\{v_1,v_2,\\ldots,v_N\\}$, each associated with a Hilbert space of dimension $d$.   Again, it takes an exponential in $N$ number of parameters to specify a general density operator $\\rho_V$.  The analog of the inference in eq.~\\eqref{Intro:Bayes} is to perform a positive operator valued measure (POVM) $\\left \\{ E_U^{(j)}\\right \\}$ on a subsystem $U \\subseteq V$ and, on obtaining outcome $j$, update the state of some disjoint subsystem $W \\subseteq V$ according to\n\\begin{equation}\n\\label{Intro:QBayes}\n\\rho_{U| E_U^{(j)}} = \\frac{\\PTr{U}{E_U^{(j)}  \\rho_{U \\cup W}}}{\\Tr{E_U^{(j)}\\rho_{U}}} = \\frac{\\PTr{V - W}{E_U^{(j)}  \\rho_{V}}}{\\Tr{E_U^{(j)} \\rho_{V}}}.\n\\end{equation}\n\nIt should be noted that this quantum problem reduces to the classical case when all the operators involved commute and are diagonal in a product basis of the systems in $V$.  In this sense eq.~\\eqref{Intro:QBayes} is a noncommutative generalization of eq.~\\eqref{Intro:Bayes} and this correspondence provides the guiding principle that we use to generalize the classical theory. \n\nThe quantum problem raises the same sort of issues as in the classical case, since it takes an exponential in $N$ number of parameters to specify a state on $N$ subsystems and the trace and partial trace in eq.~\\eqref{Intro:QBayes} involve sums over an exponential number of terms.  In quantum many-body theory, physical considerations are often used to motivate solutions to the representation problem, e.g. we may restrict attention to the ground or Gibbs states of some class of efficiently specifiable Hamiltonians.  In this paper, we take a different approach and instead generalize the sort of constraints that are used in defining classical Graphical Models.  The reasons for this are twofold.  Firstly, with the advent of quantum information science, it is relevant to solve instances of eq.~\\eqref{Intro:QBayes} that are of broader scope than those typically considered in statistical physics.  For example, we may be interested in states that are the output of a class of polynomial quantum circuits, or in the code states of a quantum error correction code.  The most natural way to phrase such constraints is not always in terms of Hamiltonians, although it may be possible to do so.  Secondly, by focussing on constraints with a clear probabilistic and information theoretic meaning, the connection between the classical and quantum problems is elucidated and the results of the vast literature on the classical inference problem can be called into play.  \n\n\n\n\\section{Conditional Independence}\n\n\\label{Cond}\n\nThe formal construction of classical Graphical Models is based on the idea of placing conditional independence constraints on sets of random variables.  In this section, the relevant classical definitions are reviewed and their quantum generalizations are introduced.  In \\S\\ref{Cond:CMI}, the entropic approach to conditional independence is outlined and the corresponding constraints on conditional and mutual probability distributions are reviewed.  In \\S\\ref{Cond:QCMI}, the entropic definition is straightforwardly generalized to the quantum case by replacing the Shannon entropy with the von Neumann entropy.  In order to provide constraints on density operators that are analogous to those for classical conditional and mutual probability distributions, conditional and mutual density operators are defined in \\S\\ref{Cond:CDO} and quantum conditional independence is expressed in terms of them in \\S\\ref{Cond:QCI}. \n\n\\subsection{Classical Conditional Independence}\n\n\\label{Cond:CMI}\n\nFor a set $V$ of classical random variables with joint distribution $P(V)$, the marginal distribution for any $U \\subseteq V$ is defined as $P(U) = \\sum_{V-U} P(V)$\nand for any two disjoint sets $U,W \\subseteq V$, the conditional distribution of $U$ given $W$ is defined as\n\\begin{equation}\nP(U|W) = \\frac{P(U \\cup W)}{P(W)}\n\\end{equation}\nThe Shannon entropy of any $U \\subseteq V$ is defined as\n\\begin{equation}\nH(U) = -\\sum_{U} P(U) \\log_2 P(U)\n\\end{equation}\nFor disjoint $U,W \\subseteq V$, the conditional entropy of $U$ given $W$ is defined as\n\\begin{equation}\n\\label{Cond:SECond}\nH(U|W) = - \\sum_{U \\cup W} P(U \\cup W) \\log_2 P(U|W),\n\\end{equation}\nand satisfies the identity\n\\begin{equation}\n\\label{Cond:SEChain}\nH(U|W) = H(U \\cup W) -  H(W).\n\\end{equation}\nThe mutual information between $U$ and $W$ is defined to be\n\\begin{align}\nH(U:W) & = H(U) - H(U|W) \\label{Cond:SMut} \\\\\n& = H(U) + H(W) - H(U \\cup W). \\label{Cond:SMutJ}\n\\end{align}\nNote that $H(U:W) = 0$ iff $P(U \\cup W) = P(U)P(W)$.\nFor three disjoint sets $U,W,X \\subseteq V$, the conditional mutual information between $U$ and $W$, given $X$ is defined to be\n\\begin{align}\nH(U:W|X) & = H(U|X) - H(U|W \\cup X) \\\\\n& = H(U \\cup X) + H(W \\cup X) - H(X)  - H(U \\cup W \\cup X). \\label{Cond:SCMutJ}\n\\end{align}\nThe condition $H(U:W|X) = 0$ is known as \\emph{conditional independence} of $U$ and $W$ given $X$ and it is equivalent to any of the following conditions\n\\begin{align}\nP(U|W \\cup X) & = P(U|X) \\label{Cond:CI1} \\\\\nP(W|U \\cup X) & = P(W|X) \\label{Cond:CI2}\\\\\nP(U\\cup W|X) & = P(U|X)P(W|X) \\label{Cond:CI3} \\\\\nP(U \\cup W \\cup X) & = P(U|X)P(W|X)P(X). \\label{Cond:CI4}\n\\end{align}\n\n\\begin{Exa}\n\\label{Cond:MC}\nConsider a Markov chain consisting of three random variables $u - x - w$.  The defining condition for such a process is that $u$ and $w$ are conditionally independent given $x$.  Thus, eq. \\eqref{Cond:CI4} immediately implies that the joint probability distribution has the form \n\\begin{equation}\n\\label{Cond:CCause}\nP(u,x,w) = P(u|x)P(w|x)P(x).\n\\end{equation}\nIn general, a joint distribution of three variables can be written as $P(u,x,w) = P(w|u,x)P(x|u)P(u) = P(u|x,w)P(x|w)P(w)$ and so eqs. (\\ref{Cond:CI1}) and (\\ref{Cond:CI2}) imply that $P(u,x,w)$ can also be written as\n\\begin{align}\nP(u,x,w) & = P(w|x)P(x|u)P(u) \\label{Cond:Forward} \\\\\nP(u,x,w) & = P(u|x)P(x|w)P(w). \\label{Cond:Backwards}\n\\end{align}\nThe three equivalent decompositions given in eqs. (\\ref{Cond:CCause} - \\ref{Cond:Backwards}) are suggestive of three different types of causal scenario that might give rise to such a Markov chain: \n\\begin{align*}\n(\\ref{Cond:CCause}) \\,\\, \\text{suggests} \\,\\, x \\,\\, & \\text{is a common cause of} \\,\\, u \\,\\, \\text{and}\\,\\,  w:   \\qquad u \\leftarrow x \\rightarrow w \\\\\n(\\ref{Cond:Forward}) \\,\\, \\text{suggests} \\,\\, u \\,\\, & \\text{causes}\\,\\, x \\,\\, \\text{and then}\\,\\, x \\,\\, \\text{causes}\\,\\, w:   \\qquad u \\rightarrow x \\rightarrow w \\\\  \n(\\ref{Cond:Backwards}) \\,\\, \\text{suggests} \\,\\, w \\,\\, & \\text{causes}\\,\\, x \\,\\,\\text{and then}\\,\\, x \\,\\, \\text{causes}\\,\\, u:    \\qquad u \\leftarrow x \\leftarrow w.\n\\end{align*}\nThe common feature of these three scenarios is that in each case all the correlations between $u$ and $w$ are mediated by $x$.  Ultimately, conditional independence captures this common feature rather than implying any specific causal scenario.\n\\end{Exa}\n\nThe example shows that care should be taken when interpreting a decomposition of a joint probability distribution into conditional and marginal distributions.  Conditional independence is about the structure of correlations between random variables rather than their specific causal relations.  For this reason it is often useful to replace conditional probabilities with an object that is more closely connected with correlation. \n\nThe \\emph{mutual probability distribution} of disjoint $U, W \\subseteq V$ is given by\n\\begin{equation}\n\\label{Cond:MPD}\nP(U:W) = \\frac{P(U\\cup W)}{P(U)P(W)} = \\frac{P(U|W)}{P(U)}.\n\\end{equation}\nAs the name implies, this is related to the mutual information and it is easy to check that eq. \\eqref{Cond:SMut} can be rewritten as\n\\begin{equation}\n\\label{Cond:SMIMutForm}\nH(U:V) = \\sum_{U \\cup W} P(U,W) \\log_2 P(U:W).\n\\end{equation}\n\nThe conditional independence conditions eqs. (\\ref{Cond:CI1}-\\ref{Cond:CI4}) can be re-expressed in terms of mutual distributions as\n\\begin{align}\nP(U:W \\cup X) & = P(U:X) \\label{Cond:MI1} \\\\\nP(W:U \\cup X) & = P(W:X) \\label{Cond:MI2}\\\\\nP(U\\cup W:X) & = P(U:X)P(W:X) \\label{Cond:MI3} \\\\\nP(U \\cup W \\cup X) & = P(U:X)P(W:X)P(X)P(U)P(W). \\label{Cond:MI4}\n\\end{align}\n\n\\begin{Exa}\nReturning to the Markov chain of example \\ref{Cond:MC}, the decompositions eqs. (\\ref{Cond:CCause}-\\ref{Cond:Backwards}) can all be rewritten in terms of mutual distributions by replacing each conditional probability with the product of a marginal and a mutual distribution using the relation $P(U|W) = P(U:W)P(U)$.  All three decompositions reduce to the same expression:\n\\begin{equation}\n\\label{Cond:MCMDecomp}\nP(u,x,w) = P(u)P(x)P(w)P(u:x)P(x:w).\n\\end{equation}\nThis decomposition clearly shows that all correlations between $u$ and $w$ are mediated by $x$ and avoids the causal ambiguities that are implicit in the use of conditional probabilities.\n\\end{Exa}\n\n\\subsection{Quantum Conditional Independence}\n\n\\label{Cond:QCMI}\n\nTurning now to the quantum case, if $V$ is a set of subsystems then the joint state is a density operator $\\rho_V$.  For $U \\subseteq V$, the analog of a marginal distribution is the reduced state obtained by taking a partial trace over $V - U$, i.e. $\\rho_U = \\PTr{V-U}{\\rho_V}$.  The Shannon entropy is replaced by the von Neumann entropy, defined as\n\\begin{equation}\nS(U) = -\\Tr{\\rho_U \\log_2 \\rho_U}.\n\\end{equation}\nQuantum analogs of conditional and mutual probability distributions are not commonly discussed in the literature, but they are needed to obtain decompositions of the joint density operator analogous to eqs. (\\ref{Cond:CI1}-\\ref{Cond:CI4}) and eqs. (\\ref{Cond:MI1}-\\ref{Cond:MI4}), so they are introduced in the next section.  For now, note that the quantum conditional entropy, mutual information and conditional mutual information can already be defined by simply replacing $H$ with $S$ in the expressions \\eqref{Cond:SEChain}, \\eqref{Cond:SMutJ} and \\eqref{Cond:SCMutJ}, since these expressions only involve joint and marginal probability distributions.  \n\nBy comparison with the classical case, it is natural to consider $S(U:W|X) = 0$ as a definition of \\emph{quantum conditional independence}.  In fact, the inequality $S(U:W|X) \\geq 0$ always holds and is known as strong subadditivity, so quantum conditional independence is simply the equality condition for strong subadditivity.  This equality condition has been investigated extensively and has been shown \\cite{HJPW03a} to be equivalent to the existence a decomposition of the Hilbert space $\\mathcal{H}_X$ of the form\n\\begin{equation}\n\\mathcal{H}_X  = \\bigoplus_{j = 1}^d \\left ( \\mathcal{H}_{X_j^L} \\otimes \\mathcal{H}_{X_j^R} \\right ),\n\\label{eq:decomp_X}\n\\end{equation}\n(the superscripts $L$ and $R$ indicate the left and right sector of the tensor product) such that the joint density operator $\\rho_{U\\cup W \\cup X}$ can be written as\n\\begin{equation}\n\\label{Cond:Hayden}\n\\rho_{U \\cup W \\cup X} = \\sum^d_{j = 1} p_j \\sigma_{U X_j^L} \\otimes \\tau_{X_j^R W},\n\\end{equation}\nwhere $0 \\leq p_j \\leq 1$, $\\sum_{j = 1}^d p_j = 1$, and $ \\sigma_{U X_j^L}$ and $\\tau_{X_j^R W}$ are density operators on $\\mathcal{H}_{U} \\otimes \\mathcal{H}_{X_j^L}$ and $\\mathcal{H}_{X_j^R} \\otimes \\mathcal{H}_{W}$ respectively.\n\nLess explicit formulations of the equality condition have also been found \\cite{Rus02b}, such as the operator equality\n\\begin{equation}\n\\label{Cond:Ruskai}\n\\log \\rho_{UWX} + \\log \\rho_X = \\log \\rho_{UX} + \\log \\rho_{WX},\n\\end{equation}\nwhere the logarithms are restricted to the supports of the operators.\n\n\\subsection{Conditional and Mutual Density Operators}\n\n\\label{Cond:CDO}\n\nQuantum conditional independence can be expressed in a form closer to the classical conditions eqs.~(\\ref{Cond:CI1}-\\ref{Cond:CI4}) and (\\ref{Cond:MI1}-\\ref{Cond:MI4}) by introducing definitions of \\emph{conditional and mutual density operators}.  For this purpose, it is convenient to define a family of products for pairs of operators $A, B$ as follows. \n\\begin{equation}\nA \\pns{n} B = \\left ( A^{\\frac{1}{2n}} B^{\\frac{1}{n}} A^{\\frac{1}{2n}} \\right )^n\n\\end{equation}\nAn important property of the $\\pns{n}$ products is that if $A$ and $B$ are both positive operators then $A \\pns{n} B$ is also positive.  In what follows, the most frequently used of these products are $A \\ensuremath{\\star} B = A \\pns{1} B$ and\n\\begin{equation}\nA \\ensuremath{\\odot} B = \\lim_{n \\rightarrow \\infty} \\left ( A \\pns{n} B \\right ).\n\\end{equation}\nNote that whilst $\\ensuremath{\\odot}$ is commutative and associative, $\\pns{n}$ is neither in general, so particular attention must be paid to the ordering of operators.  \n\nThe product $\\ensuremath{\\odot}$ was previously introduced in \\cite{War05a}, in the context of a Bayesian calculus for quantum theory, and it satisfies the formula\n\\begin{equation}\n\\label{Cond:LogDef}\nA \\ensuremath{\\odot} B = \\exp \\left ( \\log A + \\log B \\right ),\n\\end{equation}\nwhenever $A$ and $B$ are strictly positive.  If $A$ and $B$ are semi-positive, then eq.~\\eqref{Cond:LogDef} may be extended by restricting the action of the logarithm to the supports of the operators.\n\nThe $\\pns{n}$ products can be used to define a family of conditional density operators.  Let $V$ be a set of quantum systems in a state $\\rho_V$ and let $U,W \\subseteq V$ be disjoint. Define\n\\begin{equation}\n\\rho\\ns{n}_{U|W} = \\rho_W^{-1} \\pns{n} \\rho_{U\\cup W}, \n\\end{equation}\nwhere $^{-1}$ denotes the Moore-Penrose pseudoinverse\\footnote{In the present case this means that $\\rho_W^{-1}$ is the inverse of $\\rho_W$ when restricted to the support of $\\rho_W$ and has the same null space as $\\rho_W$.}.  Note that if $W = \\emptyset$, so that $\\mathcal{H}_W =\\mathbb{C}$ is the trivial Hilbert space, then $\\rho\\ns{n}_{U|W} = \\rho_U$.  The conditional density operators used most frequently in this paper are $\\rho\\1_{U|W} = \\rho\\ns{1}_{U|W}$ and $\\rho\\ensuremath{^{(\\infty)}}_{U|W} = \\rho_W^{-1}  \\ensuremath{\\odot} \\rho_{U\\cup W}$.\n\nThe operator $\\rho\\ensuremath{^{(\\infty)}}_{U|W}$ was originally introduced \\cite{CA97a} because it allows the quantum conditional entropy to be expressed via a formula analogous to eq.~\\eqref{Cond:SECond}\n\\begin{equation}\nS(U|W) = -\\Tr{\\rho_{U\\cup W} \\log_2 \\rho\\ensuremath{^{(\\infty)}}_{U|W}}.\n\\end{equation}\nThe operator $\\rho\\1_{U|W}$ was introduced in \\cite{Lei06a, Lei06b, AKMS06a} and also exhibits strong analogies with classical conditional probability. \n\nThe corresponding family of \\emph{mutual density operators} is defined similarly via\n\\begin{equation}\n\\label{Cond:MDODef}\n\\rho^{(n)}_{U:W} = \\left ( \\rho_U^{-1} \\otimes \\rho_W^{-1} \\right) \\ensuremath{\\star}^{(n)} \\rho_{U\\cup W} = \\rho_U^{-1} \\ensuremath{\\star}^{(n)} \\rho^{(n)}_{U|W},\n\\end{equation}\nwith $\\rho\\ensuremath{^{(\\infty)}}_{U:W}$ and $\\rho\\1_{U:W}$ defined in the obvious way.  \n\nThe operator $\\rho\\ensuremath{^{(\\infty)}}_{U:W}$ was introduced \\cite{CA97a} in order to express the quantum mutual information via a formula analogous to eq.~\\eqref{Cond:SMIMutForm}\n\\begin{equation}\nS(U:W) = -\\Tr{\\rho_{U \\cup W} \\log_2 \\rho\\ensuremath{^{(\\infty)}}_{U:W}}.\n\\end{equation}\n\n\\subsection{Constraints on Conditional and Mutual Density Operators}\n\n\\label{Cond:QCI}\n\nIn this section, quantum conditional independence is shown to be equivalent to constraints on conditional and mutual density operators analogous to eqs.~(\\ref{Cond:CI1}-\\ref{Cond:CI4}) and eqs.~(\\ref{Cond:MI1}-\\ref{Cond:MI4}). \n\n\\begin{The}\nIf $S(U:W|X) = 0$ then the following conditions hold:\n\\begin{align}\n\\rho\\ns{n}_{U|X\\cup W} & = \\rho\\ns{n}_{U|X} \\otimes P_W \\label{Cond:CIo1}\\\\\n\\rho\\ns{n}_{W|X \\cup U} & = \\rho\\ns{n}_{W|X} \\otimes P_U \\label{Cond:CIo2}\\\\\n\\rho\\ns{n}_{U \\cup W |X} & = \\rho\\ns{n}_{U|X} \\rho\\ns{n}_{W|X}\\label{Cond:CIo3}\\\\\n\\rho_{U\\cup W \\cup X} & = \\rho_X \\pns{n} \\left ( \\rho\\ns{n}_{U|X} \\rho\\ns{n}_{W|X} \\right ), \\label{Cond:CIo4} \n\\end{align}\nwhere $P_W$ is the projector onto the support of $\\rho_W$ and $P_U$ is the projector onto the support of $\\rho_U$.\n\\end{The}\n\\begin{proof}\nThese conditions are a direct consequence of the decomposition given in eq. \\eqref{Cond:Hayden}.  Since each $\\mathcal{H}_{X_j^L}$ is a factor in a direct sum decomposition of $\\mathcal{H}_X$, it follows that the operators $\\sigma_{UX_j^L}$ have disjoint support.  Similarly, the operators $\\tau_{W X_j^R}$ have disjoint support.  Hence, to prove eq. \\eqref{Cond:CIo1} note that\n\\begin{equation}\n\\rho_{W \\cup X} = \\sum^d_{j = 1} p_j \\sigma_{X_j^L} \\otimes \\tau_{X_j^R W},\n\\end{equation} \nand hence\n\\begin{align}\n\\rho\\ns{n}_{U|W \\cup X}  & = \\rho_{WX}^{-1} \\pns{n} \\rho_{UWX} \\\\\n& = \\sum^d_{j = 1} \\left ( \\sigma_{X_j^L}^{-1} \\pns{n} \\sigma_{U X_j^L} \\right ) \\otimes \\left ( \\tau_{X_j^R W}^{-1} \\pns{n} \\tau_{X_j^R W} \\right ) \\\\\n& = \\sum^d_{j = 1} \\sigma\\ns{n}_{U |X_j^L} \\otimes P_{X_j^R W} \\\\\n& = \\rho\\ns{n}_{U|X} \\otimes P_W,\n\\end{align}\nwhere $P_{X_j^R W}$ is the projector onto the support of $\\tau_{X_j^R W}$.\n\nEqs. \\eqref{Cond:CIo2} and \\eqref{Cond:CIo3} are proved similarly, with the proviso that the decomposition given in eq. \\eqref{Cond:Hayden} implies that $\\rho\\ns{n}_{U|X}$ and $\\rho\\ns{n}_{W|X}$ commute, which is necessary to prove eq. \\eqref{Cond:CIo3}.  Finally, \\eqref{Cond:CIo4} is equivalent to \\eqref{Cond:CIo3} via the definition a conditional density operator.\n\\end{proof}\n\nIt is straightforward to adapt the proof in order to arrive at analogous decompositions in terms of mutual density operators.\n\\begin{The}\nIf $S(U:W|X) = 0$ then the following conditions hold:\n\\begin{align}\n\\rho\\ns{n}_{U:X\\cup W} & = \\rho\\ns{n}_{U:X} \\otimes P_W \\label{Cond:MIo1}\\\\\n\\rho\\ns{n}_{W:X \\cup U} & = \\rho\\ns{n}_{W:X} \\otimes P_U \\label{Cond:MIo2}\\\\\n\\rho\\ns{n}_{U \\cup W :X} & = \\rho\\ns{n}_{U:X} \\rho\\ns{n}_{W:X}\\label{Cond:MIo3}\\\\\n\\rho_{U\\cup W \\cup X} & = \\left ( \\rho_U \\otimes \\rho_W \\otimes \\rho_X \\right ) \\pns{n} \\left ( \\rho\\ns{n}_{U:X} \\rho\\ns{n}_{W:X} \\right ), \\label{Cond:MIo4}\n\\end{align}\n\\end{The}\n\n\nIt remains to determine whether any converse implications hold, i.e. which of the conditions eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo4}) and (\\ref{Cond:MIo1}-\\ref{Cond:MIo4}) imply that $S(U:W|X) = 0$.  For this purpose, it is only necessary to consider eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo3}) because eqs. (\\ref{Cond:MIo1}-\\ref{Cond:MIo4}) are equivalent to eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo4}) via the definition of a mutual density operator and eq. \\eqref{Cond:CIo4} is equivalent to eq. \\eqref{Cond:CIo3} via the definition of a conditional density operator.  In general, the situation appears to be more complicated than in the classical case and we are only able to obtain tight converse results for the cases $n \\rightarrow \\infty$ and $n = 1$.\n\n\\begin{The}\n\\label{Cond:RuskaiThe}\nIn the limit, $n \\rightarrow \\infty$, all the converse implications hold, i.e. any of the conditions (\\ref{Cond:CIo1}-\\ref{Cond:CIo3}) imply that $S(U:W|X) = 0$. \n\\end{The} \n\\begin{proof}\nThese results are simple consequences of the equality condition given in eq. \\eqref{Cond:Ruskai}.  For eq. \\eqref{Cond:CIo1} we have\n\\begin{equation}\n\\rho_{W \\cup X}^{-1} \\ensuremath{\\odot} \\rho_{U \\cup W \\cup X} = \\rho_{X}^{-1} \\ensuremath{\\odot} \\rho_{U \\cup X}.\n\\end{equation}\nUsing eq. \\eqref{Cond:LogDef} gives\n\\begin{equation}\n\\exp \\left ( \\log \\rho_{U\\cup W \\cup X} - \\log \\rho_{W \\cup X }\\right ) = \\exp \\left ( \\rho_{U \\cup X} - \\rho_X \\right ).\n\\end{equation}\nTaking logarithms and rearranging gives eq. \\eqref{Cond:Ruskai}.  The proofs for eqs. \\eqref{Cond:CIo2} and \\eqref{Cond:CIo3} follow by similar arguments.\n\\end{proof}\n\nFor the $n=1$ case, eqs. \\eqref{Cond:CIo1} and \\eqref{Cond:CIo2} imply converse results.  \n\n\\begin{The}\nIf $\\rho\\1_{U|X\\cup W}  = \\rho\\1_{U|X}$ or $\\rho\\1_{W|X \\cup U} = \\rho\\1_{W|X}$ then $S(U:W|X) = 0$. \n\\end{The}\n\\begin{proof}\nAs explained in \\cite{HJPW03a}, Uhlman's theorem \\cite{Uhl77a}, implies that $S(U:W|X) = 0$ iff there exists a trace preserving, completely positive map $\\mathcal{E}_{U\\cup X \\cup W|U \\cup X}: \\mathfrak{L}(\\mathcal{H}_{U} \\otimes \\mathcal{H}_X) \\rightarrow \\mathfrak{L}(\\mathcal{H}_{U} \\otimes \\mathcal{H}_X \\otimes \\mathcal{H}_W)$, such that both\n\\begin{align}\n\\mathcal{E}_{U \\cup X \\cup W|U \\cup X}(\\rho_U \\otimes \\rho_X) & =  \\rho_{U} \\otimes \\rho_{X \\cup W} \\\\\n\\mathcal{E}_{U \\cup X \\cup W|U \\cup X}(\\rho_{U \\cup X}) & =  \\rho_{U \\cup X \\cup W}\n\\end{align}\nhold simultaneously.  In the present case, this can be achieved via a map of the form $\\mathcal{E}_{U \\cup X \\cup W|U \\cup X} = \\mathcal{I}_U \\otimes \\mathcal{F}_{X \\cup W|X}$, where $\\mathcal{I}_U$ is the identity superoperator on $\\mathfrak{L}(\\mathcal{H}_U)$ and $\\mathcal{F}_{X\\cup W|X}: \\mathfrak{L}(\\mathcal{H}_X) \\rightarrow \\mathfrak{L}(\\mathcal{H}_X \\otimes \\mathcal{H}_W)$ is a trace preserving completely positive map.  $\\mathcal{F}_{X \\cup W|X}$ is defined via a Kraus representation $\\mathcal{F}_{X \\cup W|X}(\\sigma_X) = \\sum_j M^{(j)}_{X \\cup W|X} \\sigma_X M^{(j)\\dagger}_{X \\cup W|X}$, where\n\\begin{equation}\nM^{(j)}_{X \\cup W|X} = \\rho_{X \\cup W}^{\\frac{1}{2}} \\Ket{j}_W \\rho_{X}^{-\\frac{1}{2}},\n\\end{equation}\nand $\\Ket{j}_W$ are basis vectors for $\\mathcal{H}_W$.\n\nIt is straightforward to check that $\\sum_j M^{(j)\\dagger}_{X \\cup W|X} M^{(j)}_{X \\cup W|X} = P_X$, where $P_X$ is the projector onto the support of $\\rho_X$.  This can easily be extended to be a trace preserving map by adding an extra Kraus operator that has support only in the subspace orthogonal to the support of $\\rho_X$, but this can be omitted for the present purpose since it doesn't change the action of $\\mathcal{E}_{U \\cup X \\cup W|U \\cup X}$ on $\\rho_U \\otimes \\rho_X$ or $\\rho_{U \\cup X}$.  It is straightforward to check that $\\mathcal{F}_{X \\cup W|X} ( \\rho_X ) = \\rho_{X \\cup W}$, so the first condition is satisfied.  The action on $\\rho_{U \\cup X}$ is given by\n\\begin{align}\n\\mathcal{I}_U \\otimes \\mathcal{F}_{X \\cup W|X} (\\rho_{U \\cup X}) & = \\sum_j I_U \\otimes M^{(j)}_{X \\cup W|X} \\rho_{U \\cup X} I_U \\otimes M^{(j)}_{X \\cup W|X} \\\\\n& = \\rho_{X \\cup W}^{\\frac{1}{2}} \\sum_j \\Ket{j}_W \\Bra{j}_W \\rho_{X}^{-\\frac{1}{2}} \\rho_{U \\cup X} \\rho_{X}^{-\\frac{1}{2}} \\rho_{X \\cup W}^{\\frac{1}{2}}  \\\\\n& = \\rho_{X \\cup W}^{\\frac{1}{2}} \\rho\\1_{U|X} \\rho_{X \\cup W}^{\\frac{1}{2}}\n\\end{align}\nBy assumption, $\\rho\\1_{U|X\\cup W}  = \\rho\\1_{U|X}$, so it follows that $\\rho_{U \\cup X \\cup W} = \\rho_{X \\cup W}^{\\frac{1}{2}} \\rho\\1_{U|X} \\rho_{X \\cup W}^{\\frac{1}{2}}$, as required.  The result for  $\\rho\\1_{W|X \\cup U} = \\rho\\1_{W|X}$ follows by symmetry.\n\\end{proof}\n\nFor $n<\\infty$, it is not true that \\eqref{Cond:CIo3} implies conditional independence, even in the case $n=1$. This is illustrated by the following counterexample. \n\n\\begin{Exa}\n\\label{ex:notMarkov}\nLet $U$ and $W$ be single qubits, and $X$ be composed of  two qubits labeled $X^L$ and $X^R$. For $\\epsilon >0$, consider the normalized state\n\\begin{equation}\n\\rho_{U\\cup X \\cup W} = \\frac{4}{(1-\\epsilon)^{\\frac 1n} + 3(\\epsilon\/3)^{\\frac 1n}}\n\\rho_X \\pns{n} \\Big(P^-_{U\\cup X^L} \\otimes P^-_{W\\cup X^R}\\Big),\n\\end{equation}\nwhere\n\\begin{equation}\n\\rho_X = (1-\\epsilon) P^-_{X^L\\cup X^R} + \\frac{\\epsilon}{3} P^+_{X^L\\cup X^R}\n\\end{equation}\nand where $P_{A\\cup B}^\\pm$ denote the projector onto the symmetric and anti-symmetric subspaces of $\\mathcal{H}_A \\otimes \\mathcal{H}_B$. The conditional states are \n\\begin{align}\n\\rho_{U| X}^{(n)} &= \\frac{2}{\\sqrt{(1-\\epsilon)^{\\frac 1n} + 3(\\epsilon\/3)^{\\frac 1n}}} P^-_{U\\cup X^L} \\otimes I_{X^R} \\ \\mathrm{and} \\\\ \n\\rho_{W| X}^{(n)} &= \\frac{2}{\\sqrt{(1-\\epsilon)^{\\frac 1n} + 3(\\epsilon\/3)^{\\frac 1n}}} I_{X^L} \\otimes P^-_{W\\cup X^R}.\n\\end{align}   \nBy construction, condition \\eqref{Cond:CIo4}  is easily verified $\\rho_{U\\cup X\\cup W} = \\rho_X \\pns{n} (\\rho_{U| X}^{(n)} \\rho_{W| X}^{(n)})$. In the limit $\\epsilon \\rightarrow 0$, the state $\\rho_{U\\cup X \\cup W} \\rightarrow P^-_{U\\cup W} \\otimes P^-_{X^L\\cup X^R}$, which has $S(U:W|X) = 2$. By continuity, we claim that  for all $n<\\infty$, there exists an $\\epsilon > 0$ such that $\\rho_{U\\cup X\\cup W}$ is a density operator that does not saturate strong subadditivity. \n\\end{Exa}\n\nThe preceding example shows that some of the conditions given in eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo4}) are not sufficient to imply quantum conditional independence on their own.   Therefore, additional constraints need to be imposed in order to obtain converse results.  Two alternative approaches are considered here, one based on additional commutation conditions that hold for conditionally independent states and one based on the algebraic structure of such states. The approach based on commutation conditions is perhaps more elegant, but the algebraic conditions are also relevant because they are used in theorem \\ref{Graph:THCC} in \\S\\ref{Graph:HC} to provide a characterization result for quantum Markov Networks on trees.  The following sequence of results provides the approach based on commutation conditions.\n\n\\begin{The}\n\\label{Cond:EqThe}\nFor a fixed $n$, if $\\rho_X^{-\\frac{1}{2n}}\\rho_{U\\cup X}^{\\frac{1}{2n}}$ and its adjoint commute with $\\rho_X^{-\\frac{1}{2n}}\\rho_{W \\cup X}^{\\frac{1}{2n}}$, then the conditions given in eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo4}) are all equivalent.\n\\end{The}\n\\begin{proof}\nWe start by showing that $\\rho\\ns{n}_{U|W\\cup X} = \\rho\\ns{n}_{U|X}$ is equivalent to $\\rho\\ns{n}_{W|U \\cup X} = \\rho\\ns{n}_{W|X}$.  The first of these can be written explicitly in terms of joint and reduced density operators as\n\\begin{equation}\n\\rho_{W \\cup X}^{-\\frac{1}{2n}} \\rho_{U \\cup W \\cup X}^{\\frac{1}{n}} \\rho_{W \\cup X}^{-\\frac{1}{2n}} = \\rho_{X}^{-\\frac{1}{2n}} \\rho_{U \\cup X}^{\\frac{1}{n}} \\rho_{X}^{-\\frac{1}{2n}}.\n\\end{equation}\nLeft and right multiplying by $\\rho_{W\\cup X}^{\\frac{1}{2n}}$ gives\n\\begin{equation}\n\\label{Cond:WOIF1}\n\\rho_{U\\cup W\\cup X}^{\\frac{1}{n}} = \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_{X}^{-\\frac{1}{2n}} \\rho_{U \\cup X}^{\\frac{1}{n}} \\rho_{X}^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}}.\n\\end{equation}\nNow, define $T = \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}}$ so that $\\rho_{U\\cup W\\cup X}^{\\frac{1}{n}} = TT^\\dagger$.  In a similar fashion, $\\rho\\ns{n}_{W|U\\cup X} = \\rho_{W|X}$ can be shown to be equivalent to $\\rho_{U\\cup W\\cup X}^{\\frac{1}{n}} = T^\\dagger T$.  \n\nNow,\n\\begin{align}\nT^\\dagger & = \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{- \\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\\\\n& =  \\rho_X^{\\frac{1}{2n}} \\rho_X^{- \\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{- \\frac{1}{2n}} \\rho_{W \\cup X}^{\\frac{1}{2n}} \\\\\n& =  \\rho_X^{\\frac{1}{2n}} \\rho_X^{- \\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_X^{- \\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\label{Cond:CommUse1} \\\\\n& = \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_X^{- \\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\\\\n& = T,\n\\end{align}\nwhere the assumption that $\\rho_X^{- \\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}}$ commutes with $\\rho_X^{- \\frac{1}{2n}} \\rho_{W\\cup X}$ has been used to derive eq. \\eqref{Cond:CommUse1}.  Hence, $T$ is Hermitian and the two conditions are equivalent.\n\nFor the remaining condition note that $\\rho\\ns{n}_{U\\cup W|X} = \\rho\\ns{n}_{U|X} \\rho\\ns{n}_{W|X}$ is equivalent to\n\\begin{align}\n\\rho_{U\\cup W\\cup X}^{\\frac{1}{n}} & = \\rho_{U\\cup X}^{\\frac{1}{n}} \\rho_X^{-\\frac{1}{n}} \\rho_{W\\cup X}^{\\frac{1}{n}} \\\\\n& = \\rho_{U\\cup X}^{\\frac{1}{2n}}  \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \n\\end{align}\nThe commutativity of $\\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}}$ and $\\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}}$ then gives\n\\begin{align}\n\\rho_{U\\cup W\\cup X}^{\\frac{1}{n}} & = \\rho_{U\\cup X}^{\\frac{1}{2n}}  \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\\\\n& =  \\rho_X^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}}  \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}},\n\\end{align}\nand the commutativity of $ \\rho_X^{-\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}}$ and $\\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}}$ gives\n\\begin{align}\n\\rho_{U\\cup W\\cup X}^{\\frac{1}{n}} & =  \\rho_X^{+\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{W \\cup X}^{\\frac{1}{2n}}  \\rho_X^{-\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}} \\\\\n& =  \\rho_{W\\cup X}^{\\frac{1}{2n}}  \\rho_X^{-\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{n}}  \\rho_X^{-\\frac{1}{n}}  \\rho_{W\\cup X}^{\\frac{1}{2n}},\n\\end{align}\nwhich is equivalent to $\\rho\\ns{n}_{U|W\\cup X} = \\rho\\ns{n}_{U|X}$.\n\\end{proof}\n\nTheorem \\ref{Cond:EqThe} relates the conditions eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo3}) for a fixed value of $n$, but the conditions for different values of $n$ can also be related via the following corollary.\n\n\\begin{Cor}\n\\label{Cond:InductCor}\nFor fixed $n$, if  $\\rho_X^{-\\frac{1}{2n}}\\rho_{U\\cup X}^{\\frac{1}{2n}}$ and its adjoint commute with $\\rho_X^{-\\frac{1}{2n}}\\rho_{W\\cup X}^{\\frac{1}{2n}}$, then $\\rho\\ns{n}_{U|W\\cup X} = \\rho\\ns{n}_{U|X}$ implies $\\rho\\ns{2n}_{U\\cup W|X} = \\rho\\ns{2n}_{U|X}\\rho\\ns{2n}_{W|X}$.\n\\end{Cor}\n\\begin{proof}\nIn the preceding proof it was shown that $\\rho\\ns{n}_{U|W\\cup X} = \\rho\\ns{n}_{U|X}$ is equivalent to $\\rho_{U\\cup W \\cup X}^{\\frac{1}{n}} = TT^{\\dagger}$, where $T = \\rho_{W \\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{U\\cup X}^{\\frac{1}{2n}}$, and that the commutativity conditions imply that $T$ is Hermitian.  Therefore, $\\rho_{U \\cup W\\cup X}^{\\frac{1}{n}} = \\left ( T^{\\dagger} \\right )^2$, which implies $\\rho_{U\\cup W\\cup X}^{\\frac{1}{2n}} =  T^{\\dagger} = \\rho_{U\\cup X}^{\\frac{1}{2n}} \\rho_X^{-\\frac{1}{2n}} \\rho_{W\\cup X}^{\\frac{1}{2n}}$.  The latter is straightforwardly equivalent to $\\rho\\ns{2n}_{U\\cup W|X} = \\rho\\ns{2n}_{U|X}\\rho\\ns{2n}_{W|X}$\n\\end{proof}\n\nPutting these results together leads to a set necessary and sufficient condition for conditional independence.\n\n\\begin{Cor}\nIf $\\rho_X^{-\\frac{1}{2n}}\\rho_{U\\cup X}^{\\frac{1}{2n}}$ and its adjoint commute with $\\rho_X^{-\\frac{1}{2n}}\\rho_{W\\cup X}^{\\frac{1}{2n}}$ for every $n$, then any of the conditions given in eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo3}) imply that $S(U:W|X) = 0$.\n\\end{Cor}\n\\begin{proof}\nUnder these commutativity conditions, theorem \\ref{Cond:EqThe} implies that eqs. (\\ref{Cond:CIo1}-\\ref{Cond:CIo3}) are equivalent for any fixed $m$ and corollary \\ref{Cond:InductCor} shows that $\\rho\\ns{2m}_{U\\cup W|X} = \\rho\\ns{2m}_{U|X}\\rho\\ns{2m}_{W|X}$ can be derived from $\\rho\\ns{m}_{U|W \\cup X} = \\rho\\ns{m}_{U|X}$.  By applying theorem \\ref{Cond:EqThe} with $n=2m$, it follows that $\\rho\\ns{m}_{U|W\\cup X} = \\rho\\ns{m}_{U|X}$ implies $\\rho\\ns{2m}_{U|W\\cup X} = \\rho\\ns{2m}_{U|X}$.  By induction, this implies that $\\rho\\ns{2^s m}_{U|W\\cup X} = \\rho\\ns{2^s m}_{U|X}$ for any positive integer $s$.  Taking the limit $s \\rightarrow \\infty$ gives $\\rho\\ensuremath{^{(\\infty)}}_{U|W\\cup X} = \\rho\\ensuremath{^{(\\infty)}}_{U|X}$, which implies $S(U:W|X) = 0$ by theorem \\ref{Cond:RuskaiThe}.\n\\end{proof}\n\nWe now turn to the algebraic approach to proving converse results.  Firstly, note that eq.~\\eqref{Cond:CIo3} implies that $\\rho\\ns{n}_{U|X}$ and $\\rho\\ns{n}_{W|X}$ commute, since $\\rho\\ns{n}_{U \\cup W|X}$ is Hermitian.  It can be shown that whenever two operators $A_{U\\cup X}\\otimes I_W$ and $I_U\\otimes B_{W \\cup X}$ commute there exists a decomposition of $\\mathcal{H}_X$ as in eq.~\\eqref{eq:decomp_X} such that\n\\begin{align}\nA_{U  X} &= \\sum^d_{j = 1} a_{U X_j^L} \\otimes I_{X^R_j} \\ \\mathrm{and} \\label{Cond:Decomp1} \\\\\nB_{W X} &= \\sum^d_{j = 1}  I_{X^L_j}\\otimes b_{ X_j^RW}, \\label{Cond:Decomp2}\n\\end{align}\nso eq. \\eqref{Cond:CIo3} implies that $\\rho\\ns{n}_{U|X}$ and $\\rho\\ns{n}_{W|X}$ have this structure, as would be expected if the joint state is conditionally independent and hence satisfies eq.~\\eqref{Cond:Hayden}.  However, eq. \\eqref{Cond:Hayden} implies an additional constraint that has not been used so far, namely that $\\rho_X$ also respects the same tensor product structure on $\\mathcal{H}_X$, i.e. $\\rho_X$ is of the form\n\\begin{equation}\n\\rho_X = \\sum_{j  = 1}^d p_j \\sigma_{X_j^L} \\otimes \\tau_{X_j^R}.\n\\end{equation}\nMore generally, we will say that an operator $C_X$ is {\\em decomposable with respect to} the pair of commuting operators $A_{U\\cup X}$ and $B_{W \\cup X}$ if it has the same algebraic structure on $\\mathcal{H}_X$, i.e. if\n\\begin{equation}\nC_{X} = \\sum^d_{j = 1} c_{X_j^R} \\otimes c_{X_j^R}.\n\\label{def:decomposable}\n\\end{equation}\nfor some factorization of $\\mathcal{H}_X$, such that eqs. \\eqref{Cond:Decomp1} and \\eqref{Cond:Decomp2} hold.\nImposing the commutativity of $\\rho\\ns{n}_{U|X}$ and $\\rho\\ns{n}_{W|X}$, along with the decomposability of $\\rho_X$ with respect to  $\\rho\\ns{n}_{U|X}$ and $\\rho\\ns{n}_{W|X}$ as additional constraints is enough to straightforwardly show that any of eqs.~(\\ref{Cond:CIo1}-\\ref{Cond:CIo4}) imply conditional independence for all values of $n$. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Graphical Models}\n\n\\label{Graph}\n\nIn this section, quantum conditional independence is used to define quantum Graphical Models that generalize their classical counterparts.  The main focus is on quantum Markov Networks and $n$-Bifactor Networks, since these allow for the simplest formulation of the Belief Propagation algorithms to be described in \\S\\ref{QBP}.  \\S\\ref{Graph:MN} reviews the definition of classical Markov Networks and the Hammersley-Clifford theorem, which gives an explicit representation for the probability distributions associated with classical Markov Networks.  Motivated by this, \\S\\ref{Graph:QGS} defines the class of quantum $n$-Bifactor Networks, which are the most general class of networks on which our Belief Propagation algorithms operate.  \\S\\ref{Graph:DMG} reviews the theory of dependency models and graphoids, which is useful for proving theorems about Graphical Models, and shows that quantum conditional independence can be used to define a graphoid. \\S\\ref{Graph:HC} defines quantum Markov Networks and gives some partial characterization results for the associated quantum states, along similar lines to the Hammersley-Clifford theorem.  Most of these definitions and characterization results are summarized on Fig. \\ref{fig:worldview}.\n\nThe remaining two subsections briefly outline two other quantum Graphical Models: Quantum Factor Graphs in \\S\\ref{Graph:FG} and Quantum Bayesian Networks in \\S\\ref{Graph:BN}.  These structures are equivalent from the point of view of the efficiency of Belief Propagation algorithms, since it is always possible to convert them into $n$-Bifactor Networks and vice-versa with only a linear overhead in graph size. An explicit method for converting a quantum factor graph into a quantum $1$-Bifactor Network is given because factor graphs are used in the application to quantum error correction developed in \\S\\ref{App:QEC}.\n\n\\subsection{Classical Markov Networks}\n\n\\label{Graph:MN}\n\n\\begin{figure}\n\\center\\includegraphics{MN}\n\\caption{The equalities $H(a:d\\cup e\\cup f | b \\cup c) = 0$, $H(f:a\\cup b \\cup c \\cup d | e) = 0$, and $H(a\\cup b : e \\cup f | c \\cup d) = 0$ are examples of constraints that are satisfied when $(G,P(V))$ is a Markov Network.}\n\\label{fig:MN}\n\\end{figure}\n\nLet $G = (V,E)$ be an undirected graph and suppose that each vertex $v \\in V$ is associated with a random variable, also denoted $v$.  Let $P(V)$ be the joint distribution of the variables.  $(G,P(V))$ is a \\emph{Classical Markov Network} if for all $U \\subseteq V$, $H(U:V - (n(U) \\cup U)|n(U)) = 0$, where $n(U)$ is the set of nearest neighbors of $U$ in $G$ (see Fig.~\\ref{fig:MN}).  Further, if $P(V)$ is strictly positive for all possible valuations of the variables, then $(G,P(V))$ is called a \\emph{Positive Classical Markov Network}.  For such positive networks there is a powerful characterization theorem \\cite{Gri73a,Bes74a}.\n\\begin{The}[Hammersley-Clifford \\cite{HC71a}]\n\\label{thm:HC}\n$(G,P(V))$ is a positive classical Markov network iff it can be written as\n\\begin{equation}\n\\label{Graph:HCE}\nP(V) = \\frac{1}{Z} \\prod_{C \\in \\mathfrak{C}} \\psi(C),\n\\end{equation}\nwhere $\\mathfrak{C}$ is the set of cliques of $G$, $\\psi(C)$ is a positive function defined on the random variables in $C$ and $Z$ is a normalization factor.\n\\end{The}\nA set of vertices $C \\subseteq V$ in a graph is a clique if $\\forall u,v \\in C$, $u \\neq v \\rightarrow (u,v) \\in E$, i.e. every vertex in $C$ is connected to every other vertex in $C$ by an edge.  Note that the decomposition in eq.~\\eqref{Graph:HCE} is generally not unique, even up to normalization.  A distribution of the form of eq.~\\eqref{Graph:HCE} is said to factorize with respect to the graph $G$.\n\nMarkov chains are a special case of Markov Networks in which the graph is a chain.  These are included in the slightly more general class of networks where the graph is a tree.  For trees the only cliques are the individual vertices and the pairs of vertices that are connected by an edge, and the associated probability distributions have a representation in terms of marginal and mutual probability distributions of the form\n\\begin{equation}\n\\label{Graph:HCMut}\nP(V) = \\prod_{v \\in V} P(v) \\prod_{(u,v) \\in E} P(u:v),\n\\end{equation}\nwhich generalizes the decomposition for three variable Markov chain given in eq. \\eqref{Cond:MCMDecomp}.  For more general networks wherein the graph has cycles, there is no Hammersley-Clifford decomposition in which the functions $\\psi(C)$ are marginal and mutual probability distributions.\n\nThe Hammersley-Clifford decomposition can be put in a form more familiar to physicists by introducing a positive constant $\\beta$ and defining the functions $H(C) = - \\beta^{-1} \\log \\psi(C)$, which are always well defined since $\\psi(C)$ is positive.  Then eq.~\\eqref{Graph:HCE} can be written as\n\\begin{equation}\nP(V) = \\frac{1}{Z} \\exp \\left ( -\\beta \\sum_{C \\in \\mathfrak{C}} H(C) \\right ),\n\\end{equation}\nwhich is a Gibbs state for a system with a Hamiltonian $\\sum_{C \\in \\mathfrak{C}} H(C)$ and partition function $Z$.  This is a generalization of the lattice models studied in statistical physics  to arbitrary graphs.  Indeed, if $G$ is a lattice, then, as for trees, the only cliques are the individual vertices and pairs of vertices connected by an edge, so for lattices the edges represent local nearest-neighbor interactions.  \n\nIn many applications,  such as in statistical physics, the functions $\\psi(C)$ are often constants for cliques containing three or more vertices even in the case where the graph has cliques with more than two vertices.  In this case, we again have that the only nontrivial functions are defined on the vertices and edges of the graph, so the state can be written as\n\\begin{equation}\n\\label{Graph:CGS}\nP(V) = \\frac{1}{Z} \\prod_{v \\in V} \\psi(v) \\prod_{(u,v) \\in E} \\psi(u:v).\n\\end{equation}\nHere, the edge functions are denoted $\\psi(u:v)$ because of the close parallel with eq.~\\eqref{Graph:HCMut}, but they are general positive functions rather than mutual distributions.  We adopt the terminology \\emph{bifactor distribution} to describe distributions of the form of eq.~\\eqref{Graph:CGS} and \\emph{Bifactor Network} for the pair $(G,P(V))$.  For example, the distribution associated with a local nearest-neighbor model on an arbitrary graph, such as the spin-glasses studied in statistical physics, would be a bifactor distribution.  \n\n\\subsection{Quantum Bifactor Networks}\n\n\\label{Graph:QGS}\n\nA proper generalization of Markov Networks to quantum theory involves the replacement of random variables with quantum systems and the replacement of classical conditional independence with its quantum counterpart.  This theory is developed in the following sections, but it is convenient to first introduce a class of states that parallels the classical bifactor distributions of eq.~\\eqref{Graph:CGS}.\n\nLet $G = (V,E)$ be a graph, let each vertex $v \\in V$ be associated to a quantum system with Hilbert space $\\mathcal{H}_v$.  Let $\\mathcal{H}_V = \\bigotimes_{v \\in V} \\mathcal{H}_v$ and consider the class of states $\\rho_V$ that can be expressed as\n\\begin{equation}\n\\label{Graph:QGSEPre}\n\\rho_V = \\frac 1 Z \\left(\\bigotimes_{u\\in V} \\mu_u\\right) \\pns{n} \\left(\\left (\\pns{n} \\right )_{(v,w) \\in E} \\nu_{v:w}\\right),\n\\end{equation}\nwhere $Z$ is normalization constant,  the $\\mu_u$'s are operators on $\\mathcal{H}_u$ and the $\\nu_{v:w} = \\nu_{w:v}$ are operators on $\\mathcal{H}_v\\otimes\\mathcal{H}_w$.  As stated, this expression is ambiguous because the $\\pns{n}$ product is neither commutative or associative apart from in the limit $n \\rightarrow \\infty$.  To avoid this ambiguity we impose the additional constraint that $[\\nu_{u:v},\\nu_{w:x}]=0$ for finite $n$, in which case the expression $\\left ( \\pns{n} \\right )_{(v,w) \\in E} \\nu_{v:w}$ reduces to $\\prod_{(v,w) \\in E} \\nu_{v:w}$. The state $\\rho_V$ is an \\emph{$n$-bifactor state} if it can be written as \n\\begin{equation}\n\\label{Graph:QGSE}\n\\rho_V = \\frac 1 Z \\left(\\bigotimes_{u\\in V} \\mu_u\\right) \\pns{n} \\left(\\prod_{(v,w) \\in E} \\nu_{v:w}\\right),\n\\end{equation}\nwith $[\\nu_{u:v},\\nu_{w:x}]=0$, and it is an \\emph{$\\infty$-bifactor state} if it can be written as\n\\begin{equation}\n\\label{Graph:QGSEInf}\n\\rho_V = \\frac 1 Z \\left(\\bigotimes_{u\\in V} \\mu_u\\right) \\ensuremath{\\odot} \\left(\\ensuremath{\\odot}_{(v,w) \\in E} \\nu_{v:w}\\right),\n\\end{equation}\nwith no commutativity constraint on the $\\nu_{v:w}$.\nThe pair $(G,\\rho_V)$ is referred to as a quantum \\emph{$n$-Bifactor Network}, or \\emph{$\\infty$-Bifactor Network}, respectively.\n\nIt turns out that not every quantum Bifactor Network is a quantum Markov Network, but the quantum generalizations of Belief Propagation algorithms to be developed in \\S\\ref{QBP} can be formulated for any Bifactor Network.  Therefore, readers who are mainly interested in algorithms and applications rather than proofs can skip to \\S\\ref{QBP}, perhaps pausing to read \\S\\ref{Graph:FG} on the way in order to understand the application to quantum error correction.  \n\nThe next goal is to formulate the theory of quantum Markov Networks and provide characterization theorems analogous to the Hammersley-Clifford theorem.  In order to do so it is convenient to first introduce the theory of dependency models and graphoids, which is useful for proving theorems about Graphical Models.\n\n\\subsection{Dependency Models and Graphoids}\n\n\\label{Graph:DMG}\n\nGraphs and conditional independence relations share a number of important properties that are responsible for the structure of Graphical Models.  These properties are also shared by a number of other mathematical structures and they can be abstracted into structures known as dependency models and graphoids, which were introduced by Gieger, Verma, and Pearl \\cite{VP90a, GVP90a}.  Here, the theory is briefly reviewed and quantum conditional independence is shown to also give rise to a graphoid.\n\nA \\emph{dependency model} $M$ over a finite set $V$ is a tripartite relation over disjoint subsets of $V$.  The statement that $(U,W,X) \\in M$ will be denoted $I(U,W|X)$, with a possible subscript on the $I$ to denote the type of dependency model.  $I(U,W|X)$ should be taken to mean that ``$U$ and $W$ only interact via $X$'', or that ``$U$ and $W$ are independent given $X$''.   \n\n\\begin{Exa}\nAn \\emph{Undirected Graph Dependency Model} $I_G$ is defined in terms of an undirected graph $G$.  Let $V$ be the set of vertices of $G$ and then let $I_G(U,W|X)$ if every path from a vertex in $U$ to a vertex in $W$ passes through a vertex in $X$.  $I_G$ is often called the \\emph{Global Markov Property}.\n\\end{Exa}\n\n\\begin{Exa}\nA \\emph{Probabilistic Dependency Model} $I_P$ is defined in terms of a probability distribution $P(V)$ over a set $V$ of random variables. $I_P(U,W|X)$ is true if $U$ and $W$ are conditionally independent given $X$.  \n\\end{Exa}\n\n\\begin{Exa}\nA \\emph{Quantum Dependency Model} $I_\\rho$ is defined in terms of a density operator $\\rho_V$ acting on the tensor product of Hilbert spaces labeled by elements of a set $V$.  $I_\\rho(U,W|X)$ is true if $U$ and $W$ are quantum conditionally independent given $X$.\n\\end{Exa}\n\nA \\emph{graphoid} is a dependency model that for all disjoint $U,W,X,Y \\subseteq V$satisfies the following axioms:\n\\begin{align}\n\\text{Symmetry:}\\quad \t\t& I(U,W|X) \\Rightarrow I(W,U|X) \\\\\n\\text{Decomposition:}\\quad \t& I(U,W\\cup Y|X) \\Rightarrow I(U,W|X) \\\\\n\\text{Weak Union:}\\quad \t\t& I(U,W \\cup Y|X) \\Rightarrow I(U,W|X \\cup Y) \\\\\n\\text{Contraction:}\\quad \t\t& I(U,W|X) \\,\\, \\text{and} \\,\\, I(U,Y|X \\cup W) \\Rightarrow I(U,W \\cup Y|X).\n\\end{align}\nA \\emph{positive graphoid} is a graphoid that also satisfies the additional axiom\n\\begin{align}\n\\text{Intersection:}\\quad\t\t& I(U,W |X \\cup Y) \\,\\, \\text{and} \\,\\, I(U,Y|W \\cup X) \\Rightarrow I(U,W \\cup Y|X).\n\\end{align}\n\n\\begin{The}\nThe quantum dependency model is a graphoid.\n\\end{The}\n\\begin{proof}\nSymmetry is immediate because $S(U:W|X)$ is invariant under exchange of $U$ and $W$.  Decomposition and Weak Union follow from the strong subadditivity inequality.  Specifically, for $A,B,C \\subseteq V$, strong subadditivity asserts that $S(A:B|C) \\geq 0$, or in terms of von Neumann entropies\n\\begin{equation}\n\\label{Cond:SS}\nS(A\\cup C) + S(B\\cup C) - S(C) - S(A\\cup B \\cup C) \\geq 0.\n\\end{equation}\nDecomposition asserts that if $S(U:W \\cup Y|X) = 0$ then $S(U:W|X) = 0$.  This is true if $S(U:W \\cup Y|X) - S(U:W|X) \\geq 0$, since $S(U:W|X)$ is guaranteed to be positive by strong subadditivity.  Expanding  $S(U:W \\cup Y|X) - S(U:W|X)$ and canceling terms gives\n\\begin{equation}\n\\begin{split}\nS(U:W \\cup Y|X) - S(U:W|X) = & S(U \\cup W \\cup X) + S(W \\cup X \\cup Y) \\\\\n& - S(W \\cup X) - S(U \\cup W \\cup X \\cup Y),\n\\end{split}\n\\end{equation}\nbut the right hand side is positive by eq.~\\eqref{Cond:SS} with $A= U, B = Y, C = W\\cup X$.\n\nWeak Union is proved via a similar argument applied to $S(U:W \\cup Y|X) - S(U:W|X \\cup Y)$.  It follows from eq.~\\eqref{Cond:SS} by taking $A = U, B = Y, C = X$.  Finally, contraction follows from noting that $S(U:W|X) + S(U:Y|X \\cup W) = S(U:W \\cup Y|X)$, which is straightforward to show by expanding in terms of von Neumann entropies.\n\\end{proof}\n\nThe well-known analogous result for classical probability distributions follows immediately because classical probability distributions can be represented by density matrices that are diagonal in an orthonormal product basis, and for such states the von Neumann entropies of subsystems are equal to the Shannon entropies of the corresponding marginal distributions.  Additionally, if $P(V)$ is positive for all possible valuations of the variables then the associated dependency model is actually a positive graphoid.  The analogous quantum property would be to require that $\\rho_V$ is a strictly positive operator, i.e. it is of full rank, but we have not been able to prove that this property implies intersection.\n\nThe undirected graph dependency model is also a positive graphoid.  The proof is straightforward, so it is not given here.  The following theorem is important for the theory of Markov networks.\n\\begin{The}[Lauritzen \\cite{Lau06a}]\n\\label{Cond:UDGraph}\nThe undirected graph dependency model is equivalent to the dependency model obtained by setting \n$I\\left (U,V - (U\\cup n(U))|n(U) \\right )$ for all $U \\subseteq V$, where $n(U)$ is the set of nearest neighbors of $U$, and demanding closure under the positive graphoid axioms.\n\\end{The}\n\nThe condition $I\\left (U,V - (U\\cup n(U))|n(U) \\right )$ defines the \\emph{Local Markov Property} on a graph.  Note that although its closure under the positive graphoid axioms is equivalent to the Global Markov Property, this is not the case for a graphoid that doesn't satisfy intersection \\cite{Lau06a}.\n\n\\subsection{Quantum Markov Networks}\n\n\\label{Graph:HC}\n\nUsing the terminology of the previous section, the definition of a classical Markov Network can be conveniently reformulated as a pair $(G,P(V))$, where $G=(V,E)$ is an undirected graph and $P(V)$ is a probability distribution over random variables represented by the vertices, such that the graphoid $I_P$ satisfies the local Markov property with respect to the graph $G$.  The definition of a quantum Markov network can now be obtained by replacing the probabilistic dependency model with a quantum dependency model.\n\nLet $G = (V,E)$ be an undirected graph and suppose that each vertex $v \\in V$ is associated with a quantum system, also denoted $v$, with Hilbert space $\\mathcal{H}_{v}$.  Let $\\rho_V$ be a state on $\\mathcal{H}_V = \\bigotimes_{v \\in V} \\mathcal{H}_v$.  $(G,\\rho_V)$ is a \\emph{Quantum Markov Network} if the graphoid $I_\\rho$ satisfies the local Markov property with respect to the graph $G$.  Further, if $\\rho_V$ is of full rank, then $(G,\\rho_V)$ is called a \\emph{Positive Quantum Markov Network}.  Note that unlike in the classical case, we cannot conclude that the global Markov property holds for positive quantum Markov networks because the intersection axiom has not been proved.\n\nThe remainder of this section provides some partial characterization results for quantum Markov networks, along the lines of the Hammersley-Clifford theorem.  The most generally applicable of these results makes use of the $\\ensuremath{\\odot}$ product.\n\\begin{The}\n\\label{Graph:QHC}\nLet $G = (V,E)$ be an undirected graph and let $\\mathfrak{C}$ be the set of cliques of $G$.  If $(G,\\rho_V)$ is a positive quantum Markov network then there exist positive operators $\\sigma_C$ acting on the cliques of $G$, i.e. $C \\in \\mathfrak{C}$, such that\n\\begin{equation}\n\\label{Graph:QHCDecomp}\n\\rho_V = \\bigodot_{C \\in \\mathfrak{C}} \\sigma_C .\n\\end{equation}\n\\end{The}\nThis theorem is analogous to one direction of the Hammersley-Clifford theorem and the proof is very similar to a standard proof for the classical case \\cite{Pol04a}, but is somewhat involved so it is given in appendix \\ref{Proof}.  However, unlike the classical case, the converse does not hold, i.e. there are states of the form eq. \\eqref{Graph:QHCDecomp} that do not satisfy the local Markov property as illustrated by the following example.  \n\n\\begin{Exa}\n\\label{ex:heisenberg}\nConsider a chain of 3 qubits $A$, $B$, and $C$ coupled through an anti-ferromagnetic Heisenberg interaction $H = \\sigma^x_A\\sigma^x_BI_C + \\sigma^y_A\\sigma^y_BI_C + \\sigma^z_A\\sigma^z_BI_C + I_A\\sigma^x_B\\sigma^x_C + I_A\\sigma^y_B\\sigma^y_C + I_A\\sigma^z_B\\sigma^z_C$ where $\\sigma^x$, $\\sigma^y$, and $\\sigma^z$ denote the Pauli operators\n\\begin{equation}\n\\sigma^x= \n\\left( \\begin{array}{cc}\n0 & 1 \\\\\n1 & 0\n\\end{array} \\right),\\ \\ \n\\sigma^z= \n\\left( \\begin{array}{cc}\n1 & 0 \\\\\n0 & -1\n\\end{array} \\right), \\ \\ \n\\mathrm{and}\\ \\ \n\\sigma^y= \\sigma^z \\sigma^x.\n\\end{equation}\nThe Gibbs state $\\rho_{A\\cup B \\cup C}(\\beta) = \\frac{1}{Z(\\beta)} \\exp(-\\beta H)$ has the form eq. \\eqref{Graph:QHCDecomp}, but for any finite $\\beta$ it has a non-zero mutual information between $A$ and $C$ conditioned on $B$ as shown on Fig.~\\ref{Heisenberg}.\n\\begin{figure}[h!]\n\\center \\includegraphics[height=2.5in]{Heis}\n\\caption{Conditional mutual information for a 3-vertex anti-ferromagnetic Heisenberg spin-$\\frac 12$ chain as a function of inverse temperature $\\beta$.}\n\\label{Heisenberg}\n\\end{figure}\n\\end{Exa}\n\nFor trees, a decomposition into reduced and mutual density operators analogous to eq. \\eqref{Graph:HCMut} is possible.  For this, we need the following lemma.\n\n\\begin{Lem}\n\\label{Graph:Remove}\nLet $G= (V,E)$ be a graph, let $(G,\\rho_V)$ be a quantum Markov network and let $u \\in V$.  Let $G' = (V',E')$ be the graph obtained by removing $u$ from $V$ and removing all edges that connect $u$ to any other vertex from the graph.  Let $G'' = (V',E'')$ be the graph obtained by adding to $G'$ an edge between every pair of distinct neighbors of $u$ in the original graph $G$.  Let $\\rho_{V'} = \\PTr{u}{\\rho_{V'}}$.  Then $(G'',\\rho_{V'})$ is a quantum Markov network.\n\\end{Lem}\n\\begin{proof}\nFor $U \\subset V$, let $U_u = U-u$ if $u \\in U$ and $U_u = U$ otherwise, and denote $n_G(U_u)$ and $n_{G''}(U_u)$ the neighbors of $U_u$ in the graphs $G$ and $G''$ respectively.  It must be shown that $I_{\\rho_V}(U, V-(U \\cup n_G(U) | n_G(U))$ for all $U \\subset V$ implies $I_{\\rho_{V'}}(U_u, V'-(U_u \\cup n_{G''}(U_u) | n_{G''}(U_u))$ for every $U_u \\subset V'$. By symmetry, we can assume without loss of generality that $u \\in U$. There are two different cases to consider:\n\n\\noindent{\\bf Case I:} $n_G(u) \\cap U \\neq \\emptyset$.\\\\\nThis implies that $n_{G''}(U_u) = n_G(U)$ and so $V' - (U_u \\cup n_{G''}(U_u)) = V - (U \\cup n_G(U))$. We conclude that $I_{\\rho_{V'}}(U_u, V'-(U_u \\cup n_{G''}(U_u) | n_{G''}(U_u))$ is equivalent to $I_{\\rho_V}(U - u, V-(U \\cup n_G(U) | n_G(U))$, and the result follows from decomposition.\n \n\\noindent{\\bf Case II:} $n_G(u) \\cap U = \\emptyset$.\\\\\nThis implies that $n_{G''}(U_u) = n_G(U_u)$. Consider the local Markov property on the original graph $G$ applied to $U_u$: $I_{\\rho_{V}}(U_u, V-(U_u \\cup n_{G}(U_u) | n_{G}(U_u))$ which is equivalent to  $I_{\\rho_{V}}(U_u, u \\cup V'-(U_u \\cup n_{G''}(U_u) | n_{G''}(U_u))$, and the result follows from decomposition.  \n\\end{proof}\n\n\\begin{The}\nLet $G= (V,E)$ be a tree.  If $(G,\\rho_V)$ is a positive quantum Markov network then it can be written as\n\\begin{equation}\n\\rho_V = \\left (\\bigotimes_{v \\in V} \\rho_v \\right ) \\pns{n} \\left ( \\prod_{(v,u) \\in E} \\rho\\ns{n}_{v:u}   \\right ).\n\\label{eq:standard_graph_state}\n\\end{equation}\n\\label{The:mutual}\n\\end{The}\n\n\\begin{proof}\nThe proof is by induction on the number of vertices in the tree.  It is clearly true for a single vertex, so consider a tree $G=(V,E)$ with $N$ vertices and choose a leaf vertex $u \\in V$.  Construct the quantum Markov network $(G'',\\rho_{V'})$ as in lemma \\ref{Graph:Remove}.  Since $u$ is a leaf it only has one neighbor in $G$, denoted $w$, so the only difference between $G$ and $G''$ is that $u$ and the single edge connecting $u$ to the rest of the graph have been removed.  By the inductive assumption, $\\rho_{V'}$ has a  decomposition of the form\n\\begin{equation}\n\\label{Graph:InductAssump}\n\\rho_{V'} = \\left (\\bigotimes_{v \\in V'} \\rho_v \\right ) \\pns{n} \\left ( \\prod_{(v,x) \\in E''} \\rho\\ns{n}_{v:u}   \\right ).\n\\end{equation} \nGenerally, $\\rho_{V} = \\rho_{V' \\cup \\{u\\}} = \\rho_{V'} \\pns{n} \\rho\\ns{n}_{u|V'}$.  The local Markov property implies that $I_\\rho (u,V' - w|w)$, so that $\\rho_{u|V'} = \\rho_{u|w}$, which in turn can be written as $\\rho_{u|w} = \\rho_u \\pns{n} \\rho_{u:w}$, so \n\\begin{equation}\n\\rho_V = \\rho_{V'} \\pns{n} \\left ( \\rho_u \\pns{n} \\rho_{u:w}\\right  ).\n\\end{equation}  \nEvery term in eq.~\\eqref{Graph:InductAssump} commutes with $\\rho_u$, because they are defined on different tensor product factors.  Also, $\\rho_{u:w}$ commutes with all the other mutual density operators either because they act on different tensor product factors or because the fact that $w$ is the only neighbor of $u$ implies that $u$ is quantum conditionally independent of any other subsystem given $w$.\n\\end{proof}\n\nIn the classical case, the Hammersley-Clifford decomposition is not necessarily unique, and when the graph is a tree the decomposition into marginal and mutual distributions is only one possibility. Similarly, a state $\\rho_V$ might have a decomposition of the form of eq.~\\eqref{eq:standard_graph_state} but with more general operators in place of the mutual and marginal states. This provides another motivation for the definition of an $n$-bifactor state that was given in eq.~\\eqref{Graph:QGSE}.   As mentioned in \\S\\ref{Graph:QGS}, not all $n$-bifactor states are quantum Markov networks, but a subset of them are, as shown by the following theorem.\n\n\\begin{The}\n\\label{Graph:THCC}\nLet $G = (V,E)$ be a tree with each vertex $v \\in V$ associated to a quantum system with Hilbert space $\\mathcal{H}_v$.  Let $\\mathcal{H}_V = \\bigotimes_{v \\in V} \\mathcal{H}_v$ and let $\\rho_V$ be an $n$-bifactor state on $\\mathcal{H}_V$.  If $\\mu_v$ is decomposable with respect to all pairs $\\nu_{u:v}$ and $\\nu_{w:v}$, then $(G, \\rho_V)$ is a quantum Markov network.\n\\end{The}\n\nThe notion of decomposability used in the statement of this theorem is defined at eq. \\eqref{def:decomposable}. The proof is straightforward and we leave it as an exercise. \n\n\\subsection{Other Graphical Models}\n\n\\label{Graph:OM}\n\nIn this section quantum generalizations of two other Graphical Models are described: Factor Graphs and Bayesian Networks.  Generally, the choice of which model to use depends on the application and Belief Propagation algorithms have been developed for all of them in the classical case.  For example, Factor Graphs arise naturally in the theory of error correcting codes, Bayesian Networks are commonly used to model causal reasoning in artificial intelligence, and Markov Networks are useful in statistical physics. However, it is now understood that the classical versions of these three models are interconvertable, and that upon such conversion the different Belief Propagation algorithms are all equivalent in complexity \\cite{AM00a, YFW02a, KFL01a}.  Some similar results also hold for the quantum case, as we illustrate by showing how a quantum factor graph can be converted into a $1$-Bifactor Network. This construction is used in the application to quantum error correction described in \\S\\ref{App:QEC}.  \n\n\\subsubsection{Quantum Factor Graphs}\n\n\\label{Graph:FG}\n\n\\begin{figure}\n\\center\\includegraphics{FG}\n\\caption{Factor graph representation of the state $(\\Ket{000} + \\Ket{111})_{uvw}$, with $\\mu_u = \\mu_v = \\mu_w = I$ and $X_a = (I+\\sigma^z_u\\otimes \\sigma^z_v)$, $X_b = (I+\\sigma^x_u\\otimes \\sigma^x_v\\otimes\\sigma^x_w)$, and $X_c = (I+\\sigma^z_v\\otimes\\sigma^z_w)$.}\n\\label{fig:FG}\n\\end{figure}\n\nA \\emph{quantum factor graph} consists of a pair $(G, \\rho_V)$, where $G = (U,E)$ is a bipartite graph and $\\rho_V$ is a quantum state.  A bipartite graph is an undirected graph for which the set of vertices can be partitioned into two disjoint sets, $V$ and $F$, such that $(v,f) \\in E$ only if $v \\in V$ and $f \\in F$. The vertices in $V$ are referred to as ``variable nodes\" and those in $F$ as ``function nodes\".  Each variable node $v$ is associated with a quantum system, also labeled $v$, with a Hilbert space $\\mathcal{H}_v$, and $\\rho_V$ is a state on $\\bigotimes_{v \\in V} \\mathcal{H}_v$. The Hilbert space associated to a function node $f$ is the tensor product of the Hilbert spaces of the adjacent variable nodes\\footnote{The following equality is not just meant in the sense of an isomorphism, they are the same Hilbert spaces.}: $\\mathcal{H}_f = \\bigotimes_{v \\in n(f)} \\mathcal{H}_v$. The state associated with a factor graph is of the form\n\\begin{equation}\n\\label{Graph:FGE}\n\\rho_V = \\frac 1Z \\prod_{f \\in F}  X_f \\ensuremath{\\star} \\bigotimes_{v \\in V} \\mu_v \n\\end{equation}\nwhere $\\mu_v$ is an operator on $\\mathcal{H}_v$, $X_f$ is an operator on $\\mathcal{H}_f$ and $[X_f,X_g] = 0$. \n\nFor example, such a state would be obtained after performing a sequence of projective von Neumann measurements on a product state of the variable nodes (see Fig.~\\ref{fig:FG}). More precisely, for each $f \\in F$, let $\\{P_f^j\\}$ be a complete set of orthogonal projectors, and let $\\bigotimes _{v \\in V} \\mu_v$ be the initial state of $V$. When the projective measurements $\\{P_f^j\\}$ are performed at each function node and commuting outcomes $P_f^j = X_f$ are obtained, the post-measurement state is of the form of eq. \\eqref{Graph:FGE}. Similarly, factor graph states could be obtained from more general POVM measurements $\\{E_f^j\\}$, provided the state update rule $\\rho_V \\rightarrow \\frac{(E_f^j)^{\\frac{1}{2}} \\rho_V (E_f^j)^{\\frac{1}{2}}}{\\Tr{E_f^j \\rho_V}}$ is used. In that case, the $X_f$ could be any positive operator rather than being restricted to projectors as in the case of a von Neumann measurement. \n\nTo convert a factor graph into a $1$-Bifactor Network, we need to treat the function nodes as distinct quantum systems, and so endow them with their own Hilbert spaces $\\mathcal{H}_f = \\bigotimes_{v \\in n(f)} \\mathcal{H}_{R_v^f}$ where $ \\mathcal{H}_{R_v^f}$ is isomorphic to $ \\mathcal{H}_{v}$. The system $R_v^f$ is called a reference system for $v$ in $f$. Then, the state of the function nodes can be written on the graph $G = (U,E)$, where $U = V\\cup F$, $\\rho_V = \\PTr{F}{\\rho_{U}}$ and\n\\begin{equation}\n\\rho_U = \\frac 1Z  \\bigotimes_{u \\in U} \\mu_u \\ensuremath{\\star} \\prod_{(v,f) \\in E} \\nu_{v:f},\n\\end{equation}\nwhere for $u \\in F$, $\\mu_u = X_u^T$, $\\nu_{v:f} = d_v\\kb{\\Phi}{\\Phi}_{v\\cup R_v^f} \\otimes I_{f-R^f_v}$ and $\\Ket\\Phi_{v\\cup R_v^f} = \\frac{1}{\\sqrt d_v} \\sum_{j=1}^{d_v} \\Ket{j}_v\\Ket{j}_{R_v^f}$ denotes the maximally entangled state between $v$ and its reference $R_v^f$.  \n\n\\subsubsection{Quantum Bayesian Networks}\n\n\\label{Graph:BN}\n\n\\begin{figure}\n\\center\\includegraphics{BN}\n\\caption{This directed acyclic graph has two distinct ancestral orderings: $(a,b,c,d)$ and $(a,c,b,d)$. The equalities $S(d:a|b\\cup c) = 0$ and $S(b:d\\cup d | a) = 0$ are examples of constraints that are satisfied when $(G,\\rho_V)$ is a Quantum Bayesian Network.}\n\\label{fig:BN}\n\\end{figure}\n\nApart from Markov Networks, there are other Graphical Models that make use of the theory of dependency models and graphoids.  Bayesian Networks provide an example, and they are commonly applied in expert systems to model causal reasoning \\cite{Nea90a,Nea04a}.  The basic idea is to replace the undirected graph of a Markov network with a Directed Acyclic Graph (DAG), wherein the directed edges represent direct cause-effect relationships.  The quantum graphoid can be used to give a straightforward generalization of the classical networks, which we only treat briefly here.  To describe the generalization, a few definitions and facts about DAGs are required.\n\nFor a vertex $v$ in a DAG $G = (V,E)$, let $m(v)$ denote the parents of $v$, i.e. $m(v) = \\{u \\in V|(u,v) \\in E\\}$.  The set of ancestors of $v$ is denoted $a(v)$ and consists of those vertices $u$ for which there exists a path in the graph starting at $u$ and ending at $v$.  Conversely, the set of descendants of $v$ is denoted $d(v)$ and consists of those vertices $u$ for which there exists a path in the graph starting at $v$ and ending at $u$.  The set of parents of a subset $U \\subseteq V$ of vertices is defined as $m(U) = \\cup_{u \\in U} m(u) - U$ and similarly $a(U) = \\cup_{u \\in U} a(u) - U$ and $d(U) = \\cup_{u \\in U} d(u) - U$.  The set of nondescendants of a subset $U \\subseteq V$ of vertices is defined to be $nd(U) = V - (d(U)\\cup U)$.  Note that the vertices in $U$ are not considered to be nondescendants of $U$ for technical convenience.  Finally, every DAG has at least one ancestral ordering of its vertices $(v_1,v_2,\\ldots,v_n)$, such that if $v_j \\in a(v_k)$ then $j < k$ (see Fig. \\ref{fig:BN}).\n\nA \\emph{Quantum Bayesian Network} is a pair $(G,\\rho_V)$, where $G = (V,E)$ is a DAG, each vertex $v \\in V$ is associated with a quantum system, also denoted $v$, with Hilbert space $\\mathcal{H}_v$, and $\\rho_V$ is a quantum state on $\\mathcal{H}_V = \\bigotimes_{v \\in V} \\mathcal{H}_v$.  The state $\\rho_V$ satisfies the conditional independence constraints $I_\\rho (U, nd(U)-m(U)|m(U) )$ for all subsets $U \\subseteq V$.\n\nThe definition of a classical Bayesian Network is obtained by replacing the quantum systems with classical random variables.  It can be shown that $(G,P(V))$ is a classical Bayesian Network iff $P(V) = \\prod_{v \\in V} P(v|m(v))$, and a partial quantum generalization of this can be obtained using the conditional density operator.\n\nDue to the nonassociativity of the $\\pns{n}$ products, expressions like $A \\pns{n} B \\pns{n} C$ are ambiguous.  It is convenient to adopt the convention that they are evaluated left-to-right, so that $A \\pns{n} B \\pns{n} C = \\left ( A \\pns{n} B \\right ) \\pns{n} C$.  Similarly, we adopt the convention that\n\\begin{equation}\n\\left ( \\pns{n} \\right )_{j=1}^N A_j = \\left ( \\left ( \\left ( A_1 \\pns{n} A_2 \\right ) \\pns{n} A_3 \\right ) \\ldots \\right ) \\pns{n} A_N.\n\\end{equation}\n\n\\begin{The}\nIf $(G,\\rho_V)$ is a Quantum Bayesian Network and $(v_1,v_2,\\ldots,v_N)$ is an ancestral ordering of $V$ then\n\\begin{equation}\n\\rho_V = \\left ( \\pns{n} \\right )_{j=1}^N \\rho\\ns{n}_{v_j|m(v_j)}.\n\\end{equation}\n\\end{The}\n\\begin{proof}\nFor any ordering $(v_1,v_2,\\ldots,v_N)$ of the vertices, an arbitrary state can always be written as\n\\begin{equation}\n\\rho_V = \\left ( \\pns{n} \\right )_{j=1}^{N} \\rho\\ns{n}_{v_{j}|v_{j - 1} v_{j - 2} \\ldots v_1}.\n\\end{equation}\nThis is a quantum generalization of the chain rule for conditional probabilities, which follows straightforwardly from the definition of conditional density operators.  If $(v_1,v_2,\\ldots,v_N)$ is in fact an ancestral ordering, then $\\{v_{j-1}, v_{j - 2}, \\ldots, v_1\\} \\subseteq nd(v_{j})$, so $I_\\rho (v_{j}, nd(v_{j})|m(v_{j}))$ implies that $\\rho\\ns{n}_{v_j|v_{j-1} v_{j-2} \\ldots v_1} = \\rho\\ns{n}_{v_j|m(v_j)}$.\n\\end{proof}\n\n\n\\section{Quantum Belief Propagation}\n\n\\label{QBP}\n\nIn this section, we discuss algorithms for solving the inference problem that we started with in \\S\\ref{Problem} for the case of $n$-Bifactor Networks.  In fact, we start with the seemingly simpler problem of computing the reduced density operators of the state on the vertices and on pairs of vertices connected by an edge, and then present a simple modification of the algorithm to solve the inference problem for local measurements.  \n\nRecall that $n$-bifactor states are of the form\n\\begin{equation}\n\\label{Graph:QGSERem}\n\\rho_V = \\frac 1 Z \\left(\\bigotimes_{u\\in V} \\mu_u\\right) \\pns{n} \\left(\\prod_{(v,w) \\in E} \\nu_{v:w}\\right),\n\\end{equation}\nand that the operators associated with vertices and edges do not have to be straightforwardly related to the reduced and mutual density operators.  Therefore, it is not clear a priori that even the simpler task can be done efficiently.  {\\em Quantum Belief Propagation} (QBP) algorithms are designed to solve this problem by exploiting the special structure of $n$-bifactor states.  Since the class of states under consideration is different for each value of $n$, there is not one but a family of algorithms. The algorithm that is designed to solve inference problems on $n$-Bifactor Networks is denoted QBP$^{(n)}$.  \n\nTo avoid cumbersome notation, focus will be given to $n$-bifactor states with $n < \\infty$. Recall that the operators $\\nu_{u:v}$ defining these states mutually commute. This is not true of $\\infty$-bifactor states. Nevertheless, a Belief Propagation algorithm for $\\infty$-bifactor states can be readily defined from the finite $n$ one, by replacing {\\em all} products appearing in eqs.~(\\ref{message}-\\ref{belief2}) by the $\\odot$ product. Under this modification, the convergence Theorem \\ref{thm:QBP} applies to $\\infty$-Bifactor Networks, and its proof only requires straightforward modifications.\n\nThe remainder of this section is structured as follows.  \\S\\ref{QBP:Desc} gives a description of the QBP algorithms and \\S\\ref{QBP:Conv} shows that QBP$\\ns{n}$ converges on trees if the $n$-Bifactor Network is also a quantum Markov Network and that QBP$\\ns{1}$ converges on trees in general.  In both cases, the algorithm converges in a time that scales linearly with the diameter of the tree.  Finally, \\S\\ref{QBP:Inf} explains how to modify the algorithm to solve inference problems for local measurements.\n\n\\subsection{Description of the Algorithm}\n\n\\label{QBP:Desc}\n\nTo describe the operation of the QBP algorithms, it is helpful to imagine that the graph $G$ represents a network of computers with a processor situated at each vertex.  The algorithm could equally well be implemented on a single processor, in which case the network is just a convenient fiction.  Pairs of processors are connected by a communication channel if there is an edge between the corresponding vertices.  The processor at vertex $u$ has a memory that stores the value of $\\mu_u$ as well as the value of $\\nu_{u:v}$ for each vertex $v$ that is adjacent to $u$ in the graph.  The task assigned to each processor is to compute the local reduced state $\\rho_u$ and the joint states $\\rho_{u\\cup v}$\\footnote{Of course, it would be sufficient to only have one processor compute $\\rho_{u\\cup v}$ for each edge.}.  At each time step $t$, the processor at $u$ updates its ``beliefs'' about $\\rho_u$ and $\\rho_{u\\cup v}$ via an iterative formula.  These beliefs are denoted $b_u\\ns{n}(t)$ and $b_{uv}\\ns{n}(t)$, and are supposed to be approximations to the true reduced states $\\rho_v$ and $\\rho_{u\\cup v}$ based on the information available to the processor at time step $t$.  Since the reduced states may depend on information stored at other vertices, the processors pass operator valued messages $m\\ns{n}_{u \\rightarrow v}(t)$ along the edges at each time step in order to help their neighbors.  The message  $m\\ns{n}_{u \\rightarrow v}(t)$ is an operator on $\\mathcal{H}_v$ and is initialized to the identity operator  $m\\ns{n}_{u\\rightarrow v}(0) = I_v$ at $t = 0$.  For $t > 0$ it is computed via the iterative formula\n\\begin{equation}\nm^{(n)}_{u \\rightarrow v}(t) = \\frac 1Y \\PTr{u}{\\mu_u \\pns{n} \\Bigg[ \\Big\\{\\prod_{v' \\in n(u)-v} m^{(n)}_{v' \\rightarrow u}(t-1)\\Big\\} \\pns{n} \\nu_{u:v} \\Bigg]}.\n\\label{message}\n\\end{equation}\nHere, $Y$ is an arbitrary normalization factor that should be chosen to prevent the the matrix elements of $m\\ns{n}_{u \\rightarrow v}(t)$ becoming increasingly small as the algorithm proceeds.  It is convenient to choose $Y$ such that $\\PTr{v}{m\\ns{n}_{u \\rightarrow v}(t)} = 1$. \n\n\nThe beliefs about the local density operator $\\rho_u$ at time $t$ are given by the simple formula\n\\begin{equation}\nb^{(n)}_u(t)  = \\frac 1{Y'} \\mu_u \\pns{n} \\prod_{v' \\in n(u)} m^{(n)}_{v' \\rightarrow u}(t), \\label{belief1}\n\\end{equation}\nwhere $Y'$ is again a normalization factor that should be chosen to make $\\PTr{u}{b^{(n)}_u(t)} = 1$.\nOn the other hand, the beliefs about $\\rho_{u\\cup v}$ also depend on the messages received by the processor at $v$, so we have to imagine that each vertex shares its messages with its neighbors.  Having done so, the beliefs about $\\rho_{u\\cup v}$ are computed via\n\\begin{equation}\nb^{(n)}_{uv}(t) = \\frac 1{Y''} (\\mu_u \\mu_v) \\pns{n} \\Bigg[\\Big\\{\\prod_{w \\in n(u)-v} m^{(n)}_{w \\rightarrow u}(t) \\prod_{w' \\in n(v)-u} m^{(n)}_{w' \\rightarrow v}(t)\\Big\\} \\pns{n} \\nu_{u:v} \\Bigg] \\label{belief2},\n\\end{equation}\nwhere $Y''$ is again a normalization factor.\n\nThe beliefs obtained from the QBP$^{(n)}$ algorithm on input $\\{\\mu_u\\}_{u\\in V}$ and $\\{\\nu_{u:v}\\}_{(u,v) \\in E}$ after $t$ time steps are denoted $[b^{(n)}_{u}(t), b^{(n)}_{uv}(t) ] = \\mathrm{QBP}_t^{(n)}(\\mu_u,\\nu_{u:v})$.  The goal of the next section is to provide conditions under which the beliefs represent the exact solution to the inference problem, i.e. to find states and values of $t$ such that $\\mathrm{QBP}_t\\ns{n}(\\mu_u,\\nu_{u:v}) = [\\rho_u,\\rho_{u\\cup v}]$. \n\\subsection{Convergence on Trees}\n\n\\label{QBP:Conv}\n\nAt time $t$, the beliefs $b^{(n)}_{u}(t)$ and $b^{(n)}_{uv}(t)$ represent estimates of the reduced states $\\rho_u$ and $\\rho_{u\\cup v}$ of the input $n$-bifactor state $\\rho_V$. \nNote that when the $\\mu_u$ and the $\\nu_{u:v}$ all commute with one another and are diagonal in local basis, the QBP$^{(n)}$ algorithms all coincide for different $n$ (including $n=\\infty$) and correspond to the well known classical Belief Propagation algorithm. This algorithm always converges on trees in a time that scales like the diameter of the tree. Its convergence on general graphs is not fully understood and constitutes an active area of research \\cite{Yed01a, YFW02a}. In the quantum setting, the $\\mu_u$ and the $\\nu_{u:v}$ do not commute in general, but for finite $n$, the $\\nu_{u:v}$ commute with each other by assumption. This has straightforward consequence that will be of use later.\n\n\\begin{Prop}\n\\label{prop:commute}\nFor all $u,v \\in V$, $x \\in n(u)$, and $w \\in n(v)$, the following commutation relations hold $[\\nu_{u:v},m^{(n)}_{x\\rightarrow u}(t)] = 0$ and $[m^{(n)}_{w\\rightarrow v}(t),m^{(n)}_{x\\rightarrow u}(t)] = 0$ .\n\\end{Prop}\n\nBefore proving the convergence of Quantum Belief Propagation, the following classical example can help build intuition of its workings, and also serves to outline the crucial steps in proving convergence.\n\n\\begin{figure}\n\\center\\includegraphics{belief}\n\\caption{Belief $b_{uv}$ is a function of $\\mu_u$, $\\mu_v$, $\\nu_{u:v}$, and the incoming messages at vertices $u$ and $v$, except $m_{u\\rightarrow v}$ and $m_{v\\rightarrow u}$.}\n\\label{fig:belief}\n\\end{figure}\n\n\\begin{Exa}\n\\label{ex:classical}\nConsider the function $P$ of $N$ discrete variables $x_j \\in \\{1,2,\\ldots,d\\}$ \n\\begin{equation}\nP(x_1,x_2,\\ldots,x_N) = \\psi(x_1,x_2)\\psi(x_2,x_3)\\ldots\\psi(x_{N-1},x_N)\n\\label{eq:classic_ex}\n\\end{equation}\nwhich could be for instance a classical bifactor distribution on a chain with $N$ sites. To evaluate the marginal function $P(x_N) = \\sum_{x_1,x_2,\\ldots,x_{N-1}} P(x_1,x_2,\\ldots,x_N)$, one can proceed directly and carry the sum over $d^N$ terms. A more efficient solution is obtained by invoking the distributive law to reorder the various sums and products into\n\\begin{equation*}\nP(x_N) = \\sum_{x_{N-1}}\\Big(\\psi(x_{N-1},x_N)\\Big( \\ldots \\Big(\\sum_{x_2} \\psi(x_2,x_3)\\Big(\\sum_{x_1} \\psi(x_1,x_2) \\Big)\\Big) \\ldots\\Big) \\Big),\n\\end{equation*}\nand performing the sums sequentially, starting with $\\sum_{x_1}$, then $\\sum_{x_2}$, and so on\n\\begin{eqnarray*}\nP(x_N) &=& \\sum_{x_{N-1}}\\Big(\\psi(x_{N-1},x_N)\\Big( \\ldots \\Big(\\sum_{x_2} \\psi(x_2,x_3)M_{1\\rightarrow 2}(x_2)\\Big) \\ldots\\Big) \\Big) \\\\\n&=& \\sum_{x_{N-1}}\\Big(\\psi(x_{N-1},x_N)\\Big( \\ldots M_{2\\rightarrow 3}(x_3) \\ldots\\Big) \\Big) \\\\\n&&\\vdots \\\\\n&=& \\sum_{x_{N-1}} \\psi(x_{N-1}:x_N) M_{N-2 \\rightarrow N-1}(x_{N-1}) \n\\end{eqnarray*}\nwhere the ``messages\" are defined recursively $M_{j \\rightarrow j+1}(x_{j+1}) = \\sum_{x_{j}} \\psi(x_j:x_{j+1}) M_{j-1 \\rightarrow j}(x_j)$, with $M_{1 \\rightarrow 2} = \\sum_{x_1} \\psi(x_1:x_2)$. Each of these steps involves the sum of $d^2$ terms, so $P(x_N)$ can be computed with order $Nd^2$ operations. \n\\end{Exa}\n\nThis example differs from the Belief Propagation algorithm described in the previous section in three important aspects. Firstly, it relied on the distributive law, which does not hold in general for the $\\pns{n}$ product, i.e. $\\PTr{u}{X_{uv} \\pns{n} Y_{vw}} \\neq \\PTr{u}{X_{uv}} \\pns{n} Y_{vw}$ in general. This will motivate Theorems \\ref{thm:commute} and \\ref{thm:commute1}, that establish necessary conditions for the validity of the distributive law. Secondly, the graph in that example is a chain, whereas Belief Propagation operates on any graph. However, Belief Propagation is only guaranteed to converge on trees, and the above example generalizes straightforwardly to such graphs. Thirdly, the messages in the example must be computed in a prescribed order: $M_{i-1\\rightarrow i}$ is required to compute $M_{i \\rightarrow i+1}$. This last point is important and deserves an extensive explanation.  \n\nSuppose that instead of computing the messages $M_{i\\rightarrow i+1}$ sequentially, messages at each vertex were computed at every time step, following the rule $m_{i \\rightarrow i\\pm1}(t,x_{i\\pm1}) = \\sum_{x_i} m_{i\\mp1 \\rightarrow i}(t-1,x_i)\\psi(x_i:x_{i\\pm1})$, as in eq. \\eqref{message}, with the initialization $m_{i\\pm1 \\rightarrow i}(0,x_i) = 1$. Then, one can easily verify that for $t\\geq i$, $m_{i \\rightarrow i+1}(t,x_{i+1}) = M_{i \\rightarrow i+1}(x_{i+1})$. In other words, the messages $m_{i\\rightarrow i+1}$ become time independent after a time equal to the distance between vertex $i$ the beginning of the chain. This observation can in fact be generalized as follows.\n\n\\begin{figure}\n\\center\\includegraphics{depth}\n\\caption{For $(u,v) \\in E$, the graph $G_v^u$ is obtained from $G$ by considering $u$ as the root and removing the subtree associated to vertex $v$. In this example, $depth(G_v^u) = 2$.}\n\\label{fig:depth}\n\\end{figure} \n\n\\begin{Lem}\nWhen $G$ is a tree, the QBP$^{(n)}$ messages $m^{(n)}_{u\\rightarrow v}(t)$ are time independent for $t> depth(G_v^u)$, where $G_v^u$ is the tree obtained from $G$ by choosing $u$ as the root, and removing the subtree associated to $v$ (see Fig. \\ref{fig:depth}).\n\\end{Lem}\n\n\\begin{proof}\nThe proof is by induction. If $u$ is a leaf, it has a unique neighbor $n(u)$ and $m_{u \\rightarrow n(u)}^{(n)}(t) = \\PTr{u}{\\mu_u \\pns{n} \\nu_{u:n(u)}}$ which is time independent. If $u$ is not a leaf, it has two neighbors $L(u)$ and $R(u)$. Clearly, if $m_{L(u)\\rightarrow u}^{(n)}(t)$ is time independent for $t \\geq t^*$, then $m_{u\\rightarrow R(u)}^{(n)}(t) = \\PTr{u}{\\mu_u \\pns{n}\\big[m_{L(u)\\rightarrow u}^{(n)}(t-1) \\pns{n} \\nu_{u:R(u)}\\big]}$ is time independent for $t \\geq t^*+1$. \n\\end{proof}\n\nWhen operated on a tree, all beliefs computed by QBP algorithm converge to a steady state after a time equal to the diameter of the tree. Note that when the graph contains loop, the beliefs do not necessarily reach a steady state. It remains to be shown that on trees, this steady state is the correct solution. For this, we need a technical result that requires some new notation. Let $U$ and $W$ be two non-intersecting subsets of $V$. Define the two subsets of edges $E_U = \\{ (u,w) \\in E : u,w \\in U\\}$ and $E_{U:W}= \\{ (u,w) \\in E : u \\in U\\ {\\mathrm and}\\ w \\in W\\}$.  Let $\\Gamma_U = \\bigotimes_{u \\in U} \\mu_u$ and for any $F \\subset E$, let $\\Lambda_{F} = \\prod_{(u,w) \\in F} \\nu_{v:w}$. \n\n\\begin{The}\\label{thm:commute}\nLet $(G, \\rho_V)$ be an $n$-Bifactor Network with graph $G = (V,E)$. Let $U,W,X$ be non-intersecting subsets of $V$ such that $U\\cup W \\cup X =V$. When $S(U:X|W) = 0$, the following diagram is commutative.\n\\begin{equation}\\begin{CD}\n    \\Gamma_{U\\cup W} \\pns{n} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}})                 \n    @>{\\text{Tr}_U}>>           \n     \\PTr{U}{\\Gamma_{U\\cup W} \\pns{n} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}})}\\\\\n    @VV{\\Gamma_X \\pns{n}(\\cdot \\Lambda_{E_X})}V                      \n    @VV{\\Gamma_X \\pns{n}(\\cdot \\Lambda_{E_X})}V     \\\\\n    \\rho_V = \\Gamma_V \\pns{n} \\Lambda_{E_V} \n    @>{\\text{Tr}_U}>> \n    \\PTr{U}{\\rho_V}\n\\end{CD}\\label{eq:commute_diag}\\end{equation}\n\\end{The}\n\n\\begin{proof}\nThe down-right path is the simplest. The first equality follows from the fact that $\\Lambda_{E_X}$ commutes with $\\Gamma_{U\\cup W}$ and all other $\\Lambda_E$'s, and the definition $\\rho_V = (\\Gamma_{U \\cup W} \\otimes \\Gamma_X )  \\pns{n} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}} \\Lambda_{E_X})$. The second equality is just a definition. The right-down path uses the representation of states that saturate strong subadditivity eq.~\\eqref{Cond:Hayden}, which implies that $\\rho_V$ has a decomposition of the form $\\rho_V = \\sum^d_{j = 1} p_j \\sigma_{U W_j^{(1)}} \\otimes \\tau_{W_j^{(2)} X}$. First observe that \n\\begin{align}\n\\Gamma_{U \\cup W}  \\star^{(n)} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}} ) \n&= (\\Gamma_X^{-1}\\pns{n} \\rho_V) \\Lambda_{E_X}^{-1} \\\\\n&= \\Big(\\Gamma_X^{-1}\\pns{n} \\sum^d_{j = 1} p_j \\sigma_{U W_j^{(1)}} \\otimes \\tau_{W_j^{(2)} X}\\Big)\\Lambda_{E_X}^{-1} \\\\\n&=  \\sum^d_{j = 1} p_j \\sigma_{U W_j^{(1)}} \\otimes \\Big[\\big(\\Gamma_X^{-1}\\pns{n} \\tau_{W_j^{(2)} X}\\big)\\Lambda_{E_X}^{-1}\\Big].\n\\end{align}\nIt follows that\n\\begin{align*}\n\\Gamma_X\\pns{n}\\left[\\PTr{U}{\\Gamma_{U \\cup W}  \\star^{(n)} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}} )}\\Lambda_{E_X} \\right] \n&=  \\sum^d_{j = 1} p_j \\sigma_{W_j^{(1)}} \\otimes \\tau_{W_j^{(2)} X}\\\\\n&= \\PTr{U}{\\rho_V}.\n\\end{align*}\n\\end{proof}\n\nSpecializing to the case $n=1$ enables a stronger result to be derived that does not require  independence assumptions.\n\n\\begin{The}\\label{thm:commute1}\nLet $(G,\\rho_V)$ be a 1-Bifactor Network with graph $G = (V,E)$. Let $U,W,X$ be non-intersecting subsets of $V$ such that $U\\cup W \\cup X =V$. The following diagram is commutative.\n\\begin{equation}\\begin{CD}\n    \\Gamma_{U} \\ensuremath{\\star} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}})                 \n    @>{\\text{Tr}_U}>>           \n     \\PTr{U}{\\Gamma_{U} \\ensuremath{\\star} (\\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}})}\\\\\n    @VV{\\Gamma_{W\\cup X} \\ensuremath{\\star}(\\cdot \\Lambda_{E_X})}V                      \n    @VV{\\Gamma_{W\\cup X} \\ensuremath{\\star}(\\cdot \\Lambda_{E_X})}V     \\\\\n    \\rho_V = \\Gamma_V \\ensuremath{\\star} \\Lambda_{E_V} \n    @>{\\text{Tr}_U}>> \n    \\PTr{U}{\\rho_V}\n\\end{CD}\\end{equation}\n\\end{The}\n\n\\begin{proof}\nThe theorem follows simply from the cyclic property of the partial trace:\n\\begin{align}\n\\PTr{U}{\\rho_V}\n&= \\PTr{U}{[\\Gamma_U^\\frac 12 \\otimes \\Gamma_{W\\cup X}^\\frac 12] \\Lambda_{E} [\\Gamma_U^\\frac 12 \\otimes \\Gamma_{W\\cup X}^\\frac 12]} \\\\\n&=  \\Gamma_{W\\cup X}^\\frac 12 \\PTr{U}{ \\Gamma_U \\Lambda_{E}} \\otimes \\Gamma_{W\\cup X}^\\frac 12 \\\\\n&=  \\Gamma_{W\\cup X}^\\frac 12 \\PTr{U}{ \\Gamma_U \\Lambda_{E_{U\\cup W}} \\Lambda_{E_{U\\cup W:X}}} \\Lambda_{E_X} \\otimes \\Gamma_{W\\cup X}^\\frac 12.\n\\end{align}\n\\end{proof}\n\nWe are now positioned to state and prove the main result of this section. \n\n\\begin{The}\n\\label{thm:QBP}\nLet $(G,\\rho_V)$ be an $n$-Bifactor Network with graph $G = (V,E)$, and let $[b^{(n)}_{u}(t), b^{(n)}_{uv}(t) ] = \\mathrm{QBP}_t^{(n)}(\\mu_u,\\nu_{u:v})$. If $(G,\\rho_V)$ is a quantum Markov network and $G$ is a tree, then for all $t \\geq diameter(G)$, $b^{(n)}_u(t) = \\rho_u$ and  $b^{(n)}_{uv}(t) = \\rho_{u\\cup v}$.\n\\end{The}\n\n\\begin{proof}\nFirst, observe that $b^{(n)}_u(t) = \\PTr{v}{b^{(n)}_{uv}(t)}$, so it is sufficient to prove that $b^{(n)}_{uv}(t) = \\rho_{u\\cup v}$. Consider $u \\cup v$ to be the root of the tree. We proceed by induction, repeatedly tracing out leaves from the bifactor state except $u$ and $v$ until we are left with only vertices $u$ and $v$. Set $G(0) = G$ and let $G(t) = (V(t),E(t))$ be the tree left after $t$ such rounds of removing leaves. Denote the leaves of $G(t)$ apart from $u$ and $v$ by $l(t)$, the children of $x$ by $c(x)$, and the unique parent of $x$ by $m(x)$. At $t=0$, consider tracing out a leaf $w$ of $G$\n\\begin{align}\n\\PTr{w}{\\rho_V}\n&= \\PTr{u}{(\\mu_w\\otimes \\Gamma_{V-w})\\pns{n} (\\nu_{w:m(w)} \\Lambda_{E_{V-w}})}\\\\\n&= \\Gamma_{V-w} \\pns{n}\\left[\\PTr{w}{\\mu_w \\pns{n} \\nu_{w:m(w)}}\\Lambda_{E_{V-w}}\\right] \\\\\n&= \\Gamma_{V-w} \\pns{n}\\left[m_{w\\rightarrow m(w)}^{(n)}(1)\\Lambda_{E_{V-w}}\\right]\n\\end{align}\nwhere we have used Theorem~\\ref{thm:commute} going from the first to the second line.  Since this holds for all leaves, we conclude that \n\\begin{equation}\n\\PTr{l(0)}{\\rho_{V}} = \\Gamma_{V(1)} \\pns{n} \\left(\\prod_{x \\in l(0)} \\prod_{y \\in c(x)} m_{y\\rightarrow x}^{(n)}(1)\\Lambda_{V(1)}\\right).\n\\end{equation}\nWe thus make the inductive assumption that\n\\begin{equation}\n\\label{Graph:IndAss}\n\\rho_{V(t)} = \\Gamma_{V(t)} \\pns{n} \\left(\\prod_{x \\in l(t)} \\prod_{y \\in c(x)} m_{y\\rightarrow x}^{(n)}(t)\\Lambda_{V(t)}\\right).\n\\end{equation}\nIt follows that\n\\begin{align}\n\\rho_{V(t+1)} \n&= \\PTr{l(t)}{\\rho_{V(t)}} \\\\\n&= \\PTr{l(t)}{\\Gamma_{V(t)} \\pns{n} \\left[\\prod_{x \\in l(t)} \\prod_{y \\in c(x)} m_{y\\rightarrow x}^{(n)}(t)\\Lambda_{V(t)}\\right]} \\\\\n&= \\PTr{l(t)}{\\Gamma_{V(t+1)} \\pns{n} \\left[\\prod_{x \\in l(t)}  \\mu_x \\pns{n}\\Big( \\prod_{y \\in c(x)} m_{y\\rightarrow x}^{(n)}(t) \\nu_{x:m(x)} \\Lambda_{V(t+1)}\\Big)\\right]} \\\\\n&= \\Gamma_{V(t+1)} \\pns{n} \\left[\\prod_{x \\in l(t)} \\PTr{x}{ \\mu_x \\pns{n}\\Big( \\prod_{y \\in c(x)} m_{y\\rightarrow x}^{(n)}(t) \\nu_{x:m(x)}\\Big)} \\Lambda_{V(t+1)}\\right] \\\\\n&= \\Gamma_{V(t+1)} \\pns{n} \\left[\\prod_{x \\in l(t)} m_{x\\rightarrow m(x)}^{(n)}(t+1) \\Lambda_{V(t+1)}\\right] \\\\\n&= \\Gamma_{V(t+1)} \\pns{n} \\left[\\prod_{x \\in l(t+1)} \\prod_{y \\in c(x)} m_{y\\rightarrow x}^{(n)}(t+1)\\Lambda_{V(t+1)}\\right]\n\\end{align}\nalso assumes the same form, so eq.~\\eqref{Graph:IndAss} follows by induction.  We have again used Theorem~\\ref{thm:commute} in going from the third to the fourth line.  When $V(t)$ contains only $u$ and $v$ then this reduces to $\\rho_{u\\cup v} = b\\ns{n}_{uv}(t)$, which is what we set out to prove. \n\\end{proof}\n\nOnce again, specializing to the case $n=1$ enables a stronger result to be derived that does not rely on independence assumptions.\n\n\\begin{Cor}\n\\label{cor:QBP1}\nLet $(G,\\rho_V)$ be an $1$-Bifactor Network with graph $G = (V,E)$, and let $[b_{u}(t), b_{uv}(t) ] = \\mathrm{QBP}^{(1)}_t(\\mu_u,\\nu_{u:v})$. If $G$ is a tree, then for all $t \\geq diameter(G)$, $b_u(t) = \\rho_u$ and  $b_{uv}(t) = \\rho_{u\\cup v}$.\n\\end{Cor}\n\n\\begin{proof} This Corollary is a consequence of Theorem~\\ref{thm:commute1} and the fact that the proof of Theorem~\\ref{thm:QBP} only relies on the commutativity of the diagram eq.~\\eqref{eq:commute_diag}.\n\\end{proof}\n\nThis last result gives us additional information about the structure of correlations in 1-bifactor states that is captured by the following corollary.\n\n\\begin{Cor}\nLet $(G,\\rho_V)$ be an $1$-Bifactor Network on graph $G = (V,E)$. If $G$ is a tree, then the mutual density operators commute: $[\\rho_{u:v},\\rho_{w:x}] = 0$ for all $(u,v)$ and $(w,x) \\in E$.\n\\end{Cor}\n\n\\begin{proof}\nThe only non trivial case is $[\\rho_{u:v},\\rho_{v:w}]$ with $u \\neq w$. Let $[b_{u}(t), b_{uv}(t) ] = \\mathrm{QBP}^{(1)}_t(\\mu_u,\\nu_{u:v})$ and denote\n\\begin{equation}\nA_{u-v}(t) = \\prod_{w\\in n(u)-v} m_{w\\rightarrow u}(t).\n\\end{equation}\nObserve that $A_{u-v}(t)$ is an operator on $\\mathcal{H}_u$, and by Proposition~\\ref{prop:commute}, $[A_{u-v}(t),\\nu_{u:w}] = 0$ for all $u$, $v$, and $w\\in V$. From Theorem~\\ref{thm:QBP}, we have for $t \\geq diameter(G)$\n\\begin{equation}\n[\\rho_{u:v},\\rho_{v:w}] = \n[ A_{u-v}(t) A_{v-u}(t) \\nu_{u:v}, A_{v-w}(t) A_{w-v}(t) \\nu_{v:w} ] = 0.\n\\end{equation}\n\\end{proof}\n\nCorollary~\\ref{cor:QBP1} shows that for 1-bifactor states on trees, QBP$^{(1)}$ enables an efficient evaluation of  the one-vertex and two-vertex reduced density operators $\\rho_{u}$ for all $u \\in V$ and $\\rho_{u\\cup v}$ for all $(u,v) \\in E$. Can this result be generalized to arbitrary bifactor states? This question is of interest since, as we will detail in \\S\\ref{App:Stat}, the Gibbs states used in statistical physics are $\\infty$-bifactor states. However, it is known that approximating the ground state energy of a two-local Hamiltonian on a chain is QMA-complete \\cite{AGK07a,Ira07a}\\footnote{QMA stands for Quantum Merlin and Arthur and it is the natural quantum generalization of the classical complexity class NP. So to the best of our knowledge, solving a QMA-complete problem would require an exponential amount of time even on a quantum computer.}. Knowledge of $\\rho_{u\\cup v}$ leads to an efficient evaluation of the energy. Therefore, without any independence assumptions, it is unlikely that an efficient QBP algorithm for $n$-Bifactor Networks will converge to the correct marginals for $n>1$. This contrasts with classical BP that always converges to the exact solution on trees. However, \\S\\ref{Heur:Rep} gives a QBP algorithm that solves the inference problem for any $n$-bifactor state on a tree in a time that scales exponentially with $n$. \n\n\\subsection{Solving Inference Problems}\n\n\\label{QBP:Inf}\n\nWe close this section with a discussion of how QBP algorithm can solve inference problems when local measurements are executed on a bifactor state. In other words, for an outcome of a local measurement on a subsystem $U$ described by a POVM element $E^{(j)}_U = \\bigotimes_{u \\in U} E_u^{(j)}$, we are interested in evaluating the  marginal states $\\rho_{u|E^{(j)}_U}$ and $\\rho_{u\\cup v|E^{(j)}_U}$ conditioned on the outcome, where\n\\begin{align}\n\\rho_{u|E^{(j)}_U} & = \\frac 1 Y \\PTr{V - u}{(E^{(j)}_U)^{\\frac{1}{2}} \\rho_V (E^{(j)}_U)^{\\frac{1}{2}}} \\\\\n\\rho_{u\\cup v|E^{(j)}_U} & = \\frac 1 Y \\PTr{V - \\{u,v\\}}{(E^{(j)}_U)^{\\frac{1}{2}} \\rho_V (E^{(j)}_U)^{\\frac{1}{2}}},\n\\end{align}\nand $Y$ is a normalization factor.\nFor $u,v \\notin U$, this amounts to a local modification of the bifactor state that accounts for the action of the measurement, the QBP algorithm being otherwise unaltered. We focus on 1-Bifactor Networks and return to the general case at the end of this section.\n\n\\begin{The}\n\\label{thm:update}\nLet $(G,\\rho_V)$ be a 1-Bifactor Network with $G = (V,E)$ a tree. For $U \\subset V$, let $\\{E^{(j)}_U\\} = \\Big\\{\\bigotimes_{u \\in U} E_u^{(j)}\\Big\\}$ be a POVM on the subsystem $U$ and let $W = V-U$. Define $\\mu_u^{(j)}  = \\mu_u \\ensuremath{\\star} E_u^{(j)} $ for $u\\in U$ and $\\mu_u^{(j)} = \\mu_u$ for $u \\in W$. Let $[b_{uv}(t), b_{uv}(t) ] = \\mathrm{QBP}^{(1)}(\\mu_u^{(j)},\\nu_{u:v})$. Then for all $t \\geq diameter(G)$, $b_u(t) = \\rho_{u|E_U^{(j)}}$ for all $u \\in W$ and  $b_{u\\cup v}(t) = \\rho_{uv|E_U^{(j)}}$ for all $(u,v) \\in E_W$.\n\\end{The}  \n\n\\begin{proof}\nThe reduced state on $W$ conditioned on the measurement outcome $E_U^{(j)}$ is given by\n\\begin{align}\n\\rho_{W|E_U^{(j)}} & = \\frac 1Y \\PTr{U}{(E_U^{(j)})^{\\frac{1}{2}} \\rho_V (E_U^{(j)})^{\\frac{1}{2}}} \\\\\n&= \\frac 1Y \\prod_{\\substack{v \\in W \\\\ u\\in U}} \\prod_{(w,x) \\in E} \\mu_v^\\frac 12  \\PTr{U}{(E_u^{(j)})^{\\frac{1}{2}} \\mu_u^\\frac 12 \\nu_{w:x} \\mu_u^\\frac 12 (E_u^{(j)})^{\\frac{1}{2}}}\\mu_v^\\frac 12 \\\\\n&= \\frac 1Y \\prod_{\\substack{v \\in W \\\\ u\\in U}} \\prod_{(w,x) \\in E} \\mu_v^\\frac 12  \\PTr{U}{ \\nu_{w:x} \\mu_u^\\frac 12 E_u^{(j)} \\mu_u^\\frac 12 }\\mu_v^\\frac 12 \\\\\n&= \\frac 1Y \\prod_{\\substack{v \\in W \\\\ u\\in U}} \\prod_{(w,x) \\in E} \\Big(\\mu_v^{(j)}\\Big)^\\frac 12  \\PTr{U}{ \\Big(\\mu_u^{(j)}\\Big)^\\frac 12 \\nu_{w:x} \\Big(\\mu_u^{(j)}\\Big)^\\frac 12 }\\Big(\\mu_v^{(j)}\\Big)^\\frac 12 .\n\\end{align}\nThe result thus follows from Corollary~\\ref{cor:QBP1}.\n\\end{proof}\n\n\nThe result of Theorem~\\ref{thm:update} can easily be extended to compute the conditional marginal state $\\rho_{u|E^{(j)}_U}$ and $\\rho_{u\\cup v|E^{(j)}_U}$ for any $u$ and $v$, not just those in $W = V-U$. This is achieved by altering the beliefs as follows\n\\begin{equation}\nb_u(t)   = \\frac 1ZE_u^{(j)} \\ensuremath{\\star} \\mu_u \\ensuremath{\\star}  \\prod_{v' \\in n(u)} m_{v' \\rightarrow u}(t)\n\\end{equation}\nfor $u \\in U$, \n\\begin{equation}\nb_{uv}(t) = \\frac 1Z E_{uv}^{(j)} \\ensuremath{\\star} (\\mu_u \\mu_v)\\ensuremath{\\star}  \\Bigg[\\prod_{w \\in n(u)-v} m_{w \\rightarrow u}(t) \\prod_{w' \\in n(v)-u} m_{w' \\rightarrow v}(t) \\ensuremath{\\star} \\nu_{u:v} \\Bigg]\n\\end{equation}\nwith $E_{uv}^{(j)} = E_{u}^{(j)} \\otimes I_v$ when $u\\in U$ and $v \\in W$ and $E_{uv}^{(j)} = E_{u}^{(j)} \\otimes E_{u}^{(j)} $ when $u,v \\in U$. The proof is straightforward and we omit it.\n\nTheorem~\\ref{thm:update} shows how QBP leads to an efficient algorithm for solving inference problems on 1-bifactor states on trees with local measurements. This immediately implies an efficient algorithm for general $n$-bifactor states when $(G,\\rho_V)$ is a quantum Markov network. Indeed, Theorem~\\ref{thm:QBP} demonstrates that in that case the QBP$^{(n)}$ algorithm can be used to efficiently compute the marginal density operators $\\rho_{u\\cup v}$ for all $(u,v) \\in E$. From these, one can straightforwardly obtain the marginal operators $\\rho_u$ for all $u \\in V$ and mutual operators $\\rho_{u:v}$ for all $(u,v) \\in E$. Theorem~\\ref{The:mutual} states that $\\rho_V$ can be represented as a 1-bifactor state in terms of its marginal and mutual operators. The inference problem can then be solved using the QBP$^{(1)}$ algorithm as explained above. \n\n\n\\section{Heuristic Methods}\n\n\\label{Heur}\n\nThe previous section provided conditions under which QBP algorithms give exact solutions to inference problems on $n$-Bifactor Networks. Namely, the underlying graph must be a tree, and the state must be either a quantum Markov network or a 1-bifactor state. When these conditions are not met, QBP algorithms may still be used as heuristic methods to obtain approximate solutions to the inference problem, although in general these approximations will be uncontrolled. \n\nTo draw a parallel, classical Belief Propagation algorithms have found applications in numerous distinct scientific fields where they are sometimes known under different name: Gallager decoding, Viterbi's algorithm, sum-product, and iterative turbo decoding in information theory; cavity method and the Bethe-Peierls approximation in statistical physics; junction-tree and Shafer-Shenoy algorithm in machine learning to name a few. In many of these examples, BP algorithms exhibit good performance on graphs with loops, even though the algorithm does not converge to the exact solution on such graphs. In fact,  ``Loopy Belief Propagation\" is often the best known heuristic method to find approximate solutions to hard problems. Important examples include the near-Shannon capacity achieving turbo-codes and low density parity check codes. On the other hand, there are known examples for which loopy BP fail to converge and their general realm of applicability is not yet fully understood. \n\nAs in the classical case, one can expect loopy QBP to give reasonable approximations in some circumstances, for instance when the size of typical loops is very large. Intuitively, one expects a local algorithm to be relatively insensitive to the large scale structure of the underlying graph. However, quantum inference problems also pose a new challenge. Quite apart from issues regarding the graph's topology, an $n$-bifactor state with $n>1$ may not obey the independence conditions required to ensure the convergence of QBP. The goal of this section is to suggest three techniques that are expected to improve the performance of QBP in such circumstances. \n\n\\subsection{Coarse-graining}\n\n\\label{Heur:CG}\n\nBy definition, a quantum Markov network has the property that the correlations from one vertex to the rest of the graph are screened off by its neighbors. When this property fails, QBP will not in general produce the correct solution to an inference problem. Coarse graining is a simple way of modifying a graph in such a way that the state may be a closer to forming a quantum Markov network with respect to the new graph than it was with respect to the original graph. \n\nA coarse graining of a graph $G = (V,E)$ is a graph  $\\tilde G = (\\tilde V, \\tilde E)$, where $\\tilde V$ is a partition of $V$ into disjoint subsets of and $(U,W) \\in \\tilde{E}$ if there is an edge connecting a vertex in $U$ to a vertex in $W$ in $G$.  The coarse-grainings that are of most interest are those that partition $V$ into connected sets of vertices (see Fig.~\\ref{CG} for example).  It is an elementary exercise to show that if $(G,\\rho_V)$ is an $n$-Bifactor Network, then $(\\tilde G, \\rho_{\\tilde{V}})$ is an $n$-Bifactor Network for any coarse graining $\\tilde G$. The intuition for why coarse graining might get us closer to a Markov network is that it effectively ``thickens\" the neighborhood of each vertex, which may then be more efficient at screening off correlations. This intuition is illustrated in Fig.~\\ref{CG} and is supported by the fact that Markov networks are fixed points of the coarse graining procedure, i.e. if $\\tilde G$ is a coarse-graining of $G$, then $(\\tilde G, \\rho_{\\tilde{V}})$ is a quantum Markov network whenever $(G,\\rho_V)$ is a Markov network.\n\n\\begin{figure}[h!]\n\\center \\includegraphics[height=1.6in]{CG}\n\\caption{Example of a coarse-grained graph. Figure a) shows in light gray the neighborhood of the darkened vertex in the original graph. In b) the dashed ellipses represent coarse-grained vertices. The neighborhood of the darkened coarse-grained vertex is represented by the light gray set.}\n\\label{CG}\n\\end{figure}\n\nAlso note that every graph $G$ can be turned into a tree by a suitable coarse graining. When the obtained Bifactor Network is a Markov Network or when $n=1$, QBP is then guaranteed to converge to the exact solution. The Hilbert space dimension at the vertices of the coarse-grained graph is bounded by an exponential in the tree-width of $G$, so this technique is efficient only for graph of $O(\\log(N))$ tree-width. \n\n\\subsection{Sliding window QBP}\n\n\\label{Heur:SW}\n\nSliding window QBP is similar in spirit to coarse-graining but is mainly suitable for chains (although the idea is easily generalized to arbitrary trees of low degree). Consider an $n$-bifactor state $\\rho_V$ on a one dimensional lattice $G = (V,E)$ with $V = \\{v_1,v_2,\\ldots,v_N\\}$ and $E = \\{(v_j,v_{j+1})\\}_{j=1,\\ldots ,N-1}$. When $(G,\\rho_V)$ is not a quantum Markov Network, the diagram of eq.~\\eqref{eq:commute_diag} will generally fail to be commutative. The commutativity of this diagram is essential for the success of QBP, as for instance it implies\n\\begin{align}\n\\PTr{v_1}{(\\mu_{v_1} \\otimes \\mu_{v_2}) \\pns{n} (\\nu_{v_1:v_2} \\nu_{v_2:v_3})} &= \\mu_{v_2} \\pns{n}\\left[ \\PTr{v_1}{\\mu_{v_1} \\pns{n} \\nu_{v_1:v_2}} \\pns{n} \\nu_{v_2:v_3}\\right] \\\\\n&= \\mu_{v_2} \\pns{n} \\big[ m_{v_1\\rightarrow v_2} \\pns{n} \\nu_{v_2:v_3} \\big].\n\\end{align}\nThus, the Hilbert space of vertex $v_1$ is traced out before operators on vertex $v_3$ are brought into the picture. This enables the algorithm to progress along the lattice by evaluating a cumulative operator of constant dimension (i.e. the messages), much in the spirit of the transfer matrix of statistical physics. Without the Markov property, this is generally not possible. \n\nHowever, when vertices separated by a distance $\\ell$ are conditionally independent given the vertices between them, sliding window QBP can be operated efficiently to produce the exact solution of the inference problem. This works by defining new message operators\n\\begin{equation}\n\\tilde m_{v_{j +\\ell-1}\\rightarrow v_{j+\\ell}} = \\PTr{\\{v_1,v_2, \\ldots ,v_j\\}}{\\left[\\bigotimes_{k = 1}^{\\ell + j-1} \\mu_{v_k}\\right] \n\\pns{n} \\left[\\prod_{k = 1}^{\\ell+j-1} \\nu_{v_k:v_{k+1}}\\right]}\n\\end{equation}\nwhich act on $\\mathcal{H}_{v_{j+1}} \\otimes \\mathcal{H}_{v_{j+2}} \\otimes \\ldots \\mathcal{H}_{v_{j+\\ell}}$. When\n\\begin{equation}\nS(v_j:v_{j+\\ell}|\\{v_{j+1}, v_{j+2}, \\ldots , v_{j+\\ell-1}\\}) = 0\n\\label{correlation-length}\n\\end{equation}\nfor all $v_j \\in V$, we have the equality\n\\begin{equation}\n\\tilde m_{v_{j+\\ell} \\rightarrow v_{j+\\ell+1}} = \\PTr{v_{j+1}}{\\mu_{v_{j+\\ell}}\\pns{n}\\big[\\tilde m_{v_{j+\\ell-1}\\rightarrow v_{j+\\ell}} \\pns{n} \\nu_{v_{j+\\ell}:v_{j+\\ell+1}}\\big]},\n\\end{equation}\nso inference problems can be solved exactly with operators whose dimension grow exponentially with the $\\ell$ rather than the lattice size $N$.   In particular, this method can be applied to spin-systems that have a finite correlation length because then eq.~\\eqref{correlation-length} can be expected to hold approximately for some finite $\\ell$.\n\n\\subsection{Replicas}\n\n\\label{Heur:Rep}\n\nThe method of replicas maps $n$-bifactor states to 1-bifactor states on which QBP$^{(1)}$ can be implemented without concerns for independence. This is achieved by replacing the systems $v$ on each vertex of the graph $G$ by $n$ replicas, so that the Hilbert space associated to vertex $v$ becomes $\\mathcal{H}_v^{\\otimes n}$. As a consequence, the algorithm suffers an overhead exponential in $n$. The name ``replica\" is borrowed from the analogous technique used in the study of classical quenched disordered systems. The validity of this technique is based on the following observation. \n\n\\begin{Prop}\\label{prop:product}\nLet $\\{\\mathcal{H}_j\\}_{j=1,\\ldots,n}$ be isomorphic Hilbert spaces. Let $T^{(n)}$ be the operator that cyclicly permutes these $n$ systems. Let $A_1$ be an arbitrary operator on $\\mathcal{H}_1$, and define $A_j = (T^{(n)})^{j-1} A_1 (T^{(n)\\dagger})^{j-1}$ to be the corresponding operators on $\\mathcal{H}_j$. Then for any set of operators $\\{A_1^{(k)}\\}$ on $\\mathcal{H}_1$, the following equality holds\n\\begin{equation}\nA_1^{(1)}A_1^{(2)}\\ldots A_1^{(n)} = \\PTr{2,3,\\ldots, n}{[A_1^{(1)}\\otimes A_2^{(2)}\\otimes\\ldots\\otimes A_n^{(n)}]T^{(n)}}.\n\\end{equation}\n\\end{Prop}\n\nWe are now in a position to formalize the replica method. \n\n\\begin{The} Let $(G,\\rho_V)$ be an $n$-Bifactor Network, with operators $\\mu_u$ and $\\nu_{u:v}$. Then, $\\rho_V$ is locally isomorphic to a 1-bifactor state with Hilbert spaces comprising $n$ replicas of the original system $\\mathcal{H}_u' = \\mathcal{H}_{u_1} \\otimes \\mathcal{H}_{u_2} \\otimes \\ldots \\otimes \\mathcal{H}_{u_n}$ for all $u \\in V$. The partial isomorphism at vertex $u$ is given by $\\PTr{u_2,u_3,\\ldots,u_n}{(T^{(n)\\dagger}_u)^\\frac 12 \\cdot (T^{(n)}_u)^\\frac 12}$.  More precisely, we claim that\n\\begin{equation}\n\\rho_V = \\PTr{\\{u_2,u_3,\\ldots, u_n\\}_{u \\in V}}{U^\\dagger \\left(\\bigotimes_{u\\in V} \\tilde \\mu_u\\right) \\ensuremath{\\star} \\left(\\prod_{(v,w) \\in E} \\tilde\\nu_{v:w}\\right) U}\n\\end{equation}\nwhere\n\\begin{align}\n\\tilde \\mu_u &= \\left(\\mu_u^{\\frac{1}{n}} \\right)^{\\otimes n} \\left(T_u^{(n)}\\right) \\\\\n\\tilde \\nu_{u:v} &= \\left(\\nu_{u:v}^{\\frac 1n}\\right)^{\\otimes n}  \\\\\nU &= \\bigotimes_{u \\in V} (T^{(n)}_u)^\\frac 12\n\\end{align}\nare operators on $\\mathcal{H}_u'$\n\\end{The}\n\n\\begin{proof}\nFirst, note that $T_u^{(n)}$ commutes with $\\left(\\mu_u^{\\frac{1}{n}} \\right)^{\\otimes n}$, so  $\\tilde \\mu_u^\\frac 12 = \\left(\\mu_u^{\\frac{1}{2n}} \\right)^{\\otimes n} \\left(T_u^{(n)}\\right)^\\frac 12 = \\left(T_u^{(n)}\\right)^\\frac 12 \\left(\\mu_u^{\\frac{1}{2n}} \\right)^{\\otimes n}$. Thus\n\\begin{align}\n& \\PTr{\\{u_2,u_3,\\ldots, u_n\\}_{u \\in V}}{U^\\dagger \\left(\\bigotimes_{u\\in V} \\tilde \\mu_u\\right) \\ensuremath{\\star} \\left(\\prod_{(v,w) \\in E} \\tilde\\nu_{v:w}\\right)U} \\\\\n&= \\PTr{\\{u_2,u_3,\\ldots, u_n\\}_{u \\in V}}{U^\\dagger \\left(\\bigotimes_{u\\in V} T_u^{(n)} \\left(\\mu_u^{\\frac 1n} \\right)^{\\otimes n}\\right) \\ensuremath{\\star} \\left(\\prod_{(v,w) \\in E}  \\left(\\nu_{u:v}^{\\frac 1n}\\right)^{\\otimes n} \\right)U} \\\\\n&= \\PTr{\\{u_2,u_3,\\ldots, u_n\\}_{u \\in V}}{\\left(\\bigotimes_{u\\in V} \\left(\\mu_u^{\\frac 1n} \\right)^{\\otimes n}\\right) \\ensuremath{\\star} \\left(\\prod_{(v,w) \\in E}  \\left(\\nu_{u:v}^{\\frac 1n}\\right)^{\\otimes n} \\right)  \\bigotimes_{u \\in V} T_u^{(n)} } \\\\\n&= \\PTr{\\{u_2,u_3,\\ldots, u_n\\}_{u \\in V}}{\\left[\\left(\\bigotimes_{u\\in V} \\mu_u^{\\frac 1n} \\right) \\ensuremath{\\star} \\left(\\prod_{(v,w) \\in E}  \\nu_{u:v}^{\\frac 1n} \\right)\\right]^{\\otimes n}  \\bigotimes_{u \\in V} T_u^{(n)} } \\\\\n&= \\left[\\left(\\bigotimes_{u\\in V} \\mu_u^{\\frac 1n} \\right) \\ensuremath{\\star} \\left(\\prod_{(v,w) \\in E}  \\nu_{u:v}^{\\frac 1n} \\right)\\right]^n = \\rho_V\n\\end{align}\nwhere we used Proposition~\\ref{prop:product} to obtain the last line. \n\\end{proof}\n\nSince the dimension of the Hilbert at each vertex grows exponentially with $n$, the QBP$^{(1)}$ algorithm used to solve the corresponding inference problem suffers an exponential overhead. One can make a replica symmetry ansatz, assuming that the state is symmetric under exchange of replica systems at any given vertex. Since the symmetric subspace of $\\mathcal{H}_v^{\\otimes n}$ grows polynomially\\footnote{More precisely, it grows as $\\binom{n+d-1}{n} \\approx n^{d-1}$.} with $n$, QBP algorithm can be executed efficiently. The validity of this ansatz cannot be verified in general, but it may serve as a good heuristic method. \n\n\n\\section{Applications}\n\n\\label{App}\n\nThis section explains in some detail how QBP can be used as a heuristic algorithm to find approximate solutions to important problems in quantum error correction and the simulation of many-body quantum systems. The focus will be on the reduction of well established problems to inference problems on $n$-Bifactor Networks. One can make use of the techniques discussed in the previous section whenever the resulting Graphical Model does not meet the requirements to ensure convergence of QBP, or when these conditions cannot be verified efficiently.\n\n\\subsection{Quantum Error Correction}\n\n\\label{App:QEC}\n\nMaximum-likelihood decoding is an important task in quantum error correction (QEC). As in classical error correction, this problem reduces to the evaluation of marginals on a factor graph, also called Tanner graph in this context. More precisely, for independent error models, the quantum channel conditioned on error syndrome is a 1-bifactor state. As a consequence, qubit-wise maximum likelihood decoding of a QEC stabilizer code reduces to an inference problem on a 1-Bifactor Network. Thus, there is no independence condition that needs to be verified, although the graph will generally contain loops. Before demonstrating this reduction, a brief summary of stabilizer QEC is in order, see \\cite{Got97a} for more details. For details on the use of Belief Propagation for the decoding of classical error correction codes, the reader is referred to the text of MacKay \\cite{Mac03a} and forthcoming book of Richardson and Urbanke \\cite{RU05a}.  \n\nConsider a collection of $N$ two-dimensional quantum systems (qubits) $V = \\{u\\}_{u=1,\\ldots,N}$ with $\\mathcal{H}_u = \\mathbb{C}^2$. A QEC code is a subspace $\\mathcal{C} \\in \\mathcal{H}_V$ that is the $+1$ eigensubspace of a collection of commuting operators $S_j$, $j=1,\\ldots N-K$, called stabilizer generators. Each stabilizer generator is a tensor product of Pauli operators on a subset $U_j$ of $V$:\n\\begin{equation}\nS_j = \\bigotimes_{u \\in U_j} \\sigma^{\\alpha^u_j}_u\n\\end{equation}\nwhere $\\alpha_j^u \\in \\{x,y,z\\}$. When the stabilizer generators are multiplicatively independent, the code encodes $K$ qubits, i.e. $\\mathcal{C}$ has dimension $2^K$. For each $j=1,\\ldots N-K$, define the two projectors $P_j^\\pm = (I \\pm S_j)\/2$. The code space is therefore defined as $\\mathcal C = (\\prod_j P_j^+) \\mathcal H_V$. \n\nError correction consists of three steps. First, the system $V$ is prepared in a code state $\\rho_V$ supported on $\\mathcal C$, in such a way that $P_j^+ \\rho_V P_j^+ = \\rho_V$ for all $j$. The state is then subjected to the channel $\\rho_V \\rightarrow \\mathcal E_{V|V}(\\rho_V)$. Second, each stabilizer generator $S_j$ is measured, yielding an outcome $s_j = \\pm$ with probability $\\Tr{P^\\pm_j \\mathcal{E}_{V|V}(\\rho_V)}$. The collection of all $N-K$ measurement outcomes $s_j$, called the error syndrome, is denoted  ${\\bf s} = (s_1,s_2,\\ldots s_{N-K})\\in \\{-,+\\}^{N-K}$. Third, the channel $\\mathcal E_{V|V}$ is updated conditioned the error syndrome $\\bf s$. Based on this updated channel, the optimal recovery is computed and implemented. \n\nThe computationally difficult step in the above protocol consists in conditioning the channel on the error syndrome. To understand this problem, it is useful to express the channel in a Kraus form $\\mathcal{E}_{V|V}(\\rho_V) = \\sum_k M^{(k)}_{V|V} \\rho_V M^{(k)\\dagger}_{V|V}$ where $\\{M^{(k)}\\}$ are operators on $\\mathcal{H}_V$. When $s_j = +$, we learn that the error that has affected the state commutes with $S_j$, while $s_j = -$ indicates that the error anti-commutes with $S_j$. To update the channel conditioned on the error syndrome $s_j = +$ say, we first decompose each Kraus operator $M^{(k)}_{V|V}$ as the sum of an operator that commutes with $S_j$ and an operator that does not commute with $S_j$: $M^{(k)}_{V|V} = M^{(k)+}_{V|V} + M^{(k)\\prime}_{V|V}$ where $ M^{(k)+}_{V|V} = P_j^+  M^{(k)}_{V|V} P_j^+$ and $ M^{(k)\\prime}_{V|V} =  M^{(k)}_{V|V} -  M^{(k)+}_{V|V}$. The updated channel is obtained by throwing away the primed component  $ M^{(k)\\prime}_{V|V}$ of each Kraus operator, and renormalizing.  \n\nIn what follows, we demonstrate how the conditional channel can be expressed as a factor graph. This is most easily done using the Jamio\\l kowski representation of quantum channels. For each quantum system $v$, let $R_v$ denote a reference for $v$, with Hilbert space $\\mathcal{H}_{R_v} \\simeq \\mathcal H_v$. Define the maximally entangled state between system $v$ and its reference by $\\Ket\\Phi_{vR_v} =  \\frac{1}{\\sqrt d} \\sum_j \\Ket{j}_v\\Ket{j}_{R_v}$. Then, the Jamio\\l kowski representation of a channel $\\mathcal E_{V|V}$ is a density operator $\\rho_{\\overline V}$ on $\\mathcal H_{\\overline V} = \\mathcal H_V \\otimes \\mathcal H_{R_V}$ given by $\\rho_{\\overline V} = (\\mathcal E_{V|V} \\otimes \\mathcal I_{R_V|R_V})(\\kb{\\Phi}{\\Phi}_{VR_V})$, where $\\mathcal I$ denotes the identity channel. For independent error models considered here, $\\rho_{\\overline V} = \\bigotimes_{u \\in V} \\rho_{\\overline u}$.\n\nFor each stabilizer generator $S_j$, denote $\\overline{S}_j = \\bigotimes_{u \\in U_j} \\sigma^{\\alpha_j^u}_u \\otimes \\sigma^{\\alpha_j^u}_{R_u}$, and construct the associated projectors $\\overline{P}_j^\\pm = (I \\pm \\overline{S}_j)\/2$. An important property of these operator is that they fix the maximally entangled state $\\overline{S}_j \\Ket{\\Phi}_{VR_V} = \\overline{P}_j^+ \\Ket{\\Phi}_{VR_V} = \\Ket{\\Phi}_{VR_V}$. Let $E$ be an operator on $V$. If $E$ commutes with $S_j$, we have $\\overline{P}_j^+ (E\\otimes I_{R_V}) \\Ket{\\Phi}_{VR_V} = (E\\otimes I_{R_V}) \\Ket{\\Phi}_{VR_V}$ and $\\overline{P}_j^- (E\\otimes I_{R_V}) \\Ket{\\Phi}_{VR_V} = 0$, while if $E$ anti-commutes with $S_j$, the same identities hold with $\\overline P_j^+$ and $\\overline P_j^-$ exchanged. It follows from this observation that conditioned on the error syndrome $\\bf s$, the channel is described by the Jamio\\l kowski matrix\n\\begin{equation}\n\\rho_{\\overline V|{\\bf s}} = \\frac 1Z \\prod_{j} \\overline P_j^{s_j} \\ensuremath{\\star} \\bigotimes_{v \\in V} \\rho_{\\overline v}, \n\\end{equation}\nthat is a quantum factor graph. \n\nThere are a number of relevant quantities that can be evaluated from this factor graph. For instance, one can efficiently evaluate the conditional channel on any constant size set of qubits $W \\subset V$ vial partial trace. This is useful in iterative decoding schemes such as those used for quantum turbo-codes \\cite{OPT07a} and low density parity check codes \\cite{COT05a}.  In those cases, the conditional channel on $W$ can only be evaluated approximately since it requires loopy QBP. The factor graph also enables exact evaluation of the logical error in a concatenated block coding scheme \\cite{Pou06b} such as used in fault-tolerant protocols. \n\n\\subsection{Simulation of Many-Body Quantum Systems}\n\n\\label{App:Stat}\n\nIn statistical physics, the state of a many-body quantum system $V$ is a Gibbs state $ \\rho_V  = \\frac 1Z \\exp(-\\beta H)$ for some Hamiltonian $H$, where $\\beta = 1\/T$ is the inverse temperature. Typically, $H$ is the sum of single and two-body interactions $H = \\sum_{u \\in V} H_u + \\sum_{(u,w) \\in E} H_{uv}$ on some graph $G = (V,E)$. Understanding the correlations present in these states is a great challenge in theoretical physics. In this section, we describe how QBP can serve as an heuristic method to accomplish this task approximately. For an account of the use of Belief Propagation in classical statistical mechanical systems, we refer the reader to the text of M\\'ezard and Montanari \\cite{MM07a}.\n\nDefining $\\mu_u = \\exp(-\\beta H_u)$ and $\\nu_{v:w} = \\exp(-\\beta H_{vw})$ gives an expression for $\\rho_V$ of the form of eq.~\\eqref{Graph:QGSEInf}:\n\\begin{equation}\n\\rho_V = \\left(\\bigotimes_{v \\in V} \\mu_v\\right) \\odot \\left(\\bigodot_{(v,w) \\in E} \\nu_{v:w}\\right)\n\\end{equation}\nThus, $\\rho_V$ is an $\\infty$-bifactor state. As mentioned in \\S\\ref{QBP}, a QBP$^{(\\infty)}$ algorithm can easily be formulated for this type of bifactor state, and still converge to the exact solutions of the corresponding inference problem when $\\rho_V$ is a quantum Markov network and $G$ is a tree. This requires replacing all matrix products $\\prod$ by the commutative product $\\odot$ in the defining equations of QBP$^{(\\infty)}$ eqs.~(\\ref{message}-\\ref{belief2}). The proof of  convergence Theorem \\ref{thm:QBP} under these more general conditions follows essentially the same reasoning.  \n\nTo obtain a bifactor state that satisfies the commutation condition $[\\nu_{u:v},\\nu_{w:x}]=0$, it is possible to coarse-grain $G$ in a way that the resulting interaction between coarse-grained neighbors commute. Consider for instance a one dimensional chain $G = (V,E)$ with $V = \\{u\\}_{u=1,\\ldots,N}$ and $E = \\{(u,u+1)\\}_{u=1,\\ldots,N-1}$. We can construct a coarse-grained graph $\\tilde G$ by identifying all vertices $2 u-1$ and $2 u$ for $u=1,\\ldots,\\lfloor\\frac N 2\\rfloor$. The state $\\rho_V$ is then an $\\infty$-bifactor state on $\\tilde G$, with operators\n\\begin{align}\n\\tilde\\mu_u &= \\mu_{2u-1} \\odot \\mu_{2u} \\odot \\nu_{2u-1:2u} \\\\\n\\tilde\\nu_{u:u+1} &= \\nu_{2u:2u+1},\n\\end{align}\nsatisfying $[\\tilde{\\nu}_{u:u+1},\\tilde{\\nu}_{v:v+1}]$\nThus, $\\infty$-bifactor states are commonplace in quantum many-body physics. Unfortunately, the convergence of the QBP algorithm in this case requires the state to be a quantum Markov network, which cannot be tested directly in general. As we will now explain, it is often possible to reasonably approximate a Gibbs state by an $n$-bifactor with finite $n$, and sometime even $n=1$.\n\nA simple way to obtain an $n$-bifactor state is to approximate $\\odot$ by $\\pns{n}$ for some large value of $n$. In the context of many-body physics, this is called a Trotter-Suzuki decomposition of the Gibbs state, and becomes more accurate as the ratio $\\beta\/n$ decreases. The QBP$^{(n)}$ algorithm can then be operated on this $n$-bifactor state, but its convergence again requires some independence condition that cannot be verified systematically. Alternatively, one can use the replica method described in section \\ref{Heur:Rep} and solve the inference problem exactly with QBP$^{(1)}$, but with an increase in complexity exponential in $n$. The replica method is then reminiscent of the well known correspondence between quantum statistical mechanics in $d$ dimensions and classical statistical mechanics in $d+1$ dimensions, where the extra dimension represents inverse temperature. \n\nThe 1-bifactor states also capture the correlations of some non-trivial quantum many-body systems. {\\em Valence bond solid} (VBS) states were introduced in Ref. \\cite{AKLT87a,AKLT88a} as exact ground states (i.e. $T=0$ Gibbs states) of spin systems with interesting properties. Recent work has generalized these constructions to {\\em matrix product states} (MPS) in one-dimension \\cite{FNW92a,Vid04a,Vid06a}, and {\\em projected entangled-pair states} (PEPS) for higher dimensions \\cite{VC04a,SDV06a}. These form an important class of states for the description of quantum many-body systems. For instance, {\\em density matrix renormalization group} (DMRG) \\cite{Whi92a} --- one of the most successful method for the numerical study of spin chains --- is now understood as a variational method over MPS \\cite{OR95a,DMNS98a,VPC04b}. All these states are instances of 1-bifactor states.\n\n\\begin{figure}[h!]\n\\center \\includegraphics[height=1.6in]{VBS}\n\\caption{Projected entangled pair state on a two-dimensional square lattice. The vertices are associated to dashed circles. Each $\\bullet$---$\\bullet$ represents a maximally entangled state of $D$ dimension shared between neighboring vertices. A partial isometry $A_u: (\\mathbb{C}^D)^{c_u} \\rightarrow \\mathbb{C}^d$ is applied at each vertex, where $c_u$ is the degree of vertex $u$. }\n\\label{VBS}\n\\end{figure}\n\nFor sake of simplicity, we will demonstrate this claim for one-dimensional MPS, but the same argument holds for higher dimensions. The MPS $\\Ket\\Psi$ is a pure state of a collection of $N$ $d$-dimensional quantum systems displayed on a one dimensional lattice. Each vertex $u$ is assigned two ``virtual particles\" $L_u$ and $R_u$, where $L$ and $R$ stand for left and right (see Fig.~\\ref{VBS} for a illustration of this construction in two-dimensions). Each of these particles are associated a Hilbert space $\\mathcal{H}_{L_u} = \\mathcal{H}_{R_u} = \\mathbb{C}^D$. Initially, the right particle of vertex $u$ is in a maximally entangled state with the left particle of vertex $u+1$; $\\Ket{\\Phi}_{R_u\\cup L_{u+1}} = \\frac{1}{\\sqrt D} \\sum_{\\alpha=1}^D \\Ket{\\alpha}_{R_u}\\Ket{\\alpha}_{L_{u+1}}$ where $\\Ket\\alpha$ are orthogonal basis vectors for $\\mathbb{C}^D$. (The lattice can be closed to form a circle, in which case we identify $N+1 = 1$.) The initial state is therefore $\\Ket{\\Phi_0} = \\bigotimes_u \\Ket{\\Phi}_{R_u\\cup L_{u+1}}$. \n\nTo obtain the MPS, apply an operator $A_u: \\mathcal{H}_{L_u}\\otimes \\mathcal{H}_{R_u} \\rightarrow  \\mathbb{C}^d$\n\\begin{equation}\nA_u = \\sum_{j=1}^d \\sum_{\\alpha,\\beta = 1}^D A_u^{j,\\alpha,\\beta} \\kb{j}{\\alpha,\\beta}\n\\end{equation}\nto each vertex of the lattice. The vectors $\\Ket j$ form an orthogonal basis for $\\mathbb{C}^d$.  The resulting state is\n\\begin{equation}\n\\Ket{\\Psi} = \\bigotimes_{u=1}^N A_u \\Ket{\\Phi_0} \n\\propto \\sum_{j_1,j_2,\\ldots,j_N=1}^d \\Tr{B_1^{j_1}B_2^{j_2}\\ldots B_N^{j_N}} \\Ket{j_1,j_2,\\ldots,j_N}\n\\label{eq:MPS}\n\\end{equation}\nwhere the matrices $B_u^j$ are the submatrices of $A_u$ with matrix elements $(B_u^j)_{(\\alpha,\\beta)} = A_u^{j,\\alpha,\\beta}$. \n\n\nFor the corresponding 1-bifactor state, the underlying graph $G = (V,E)$ is also a one dimensional lattice $V = \\{1,2,\\ldots,N\\}$ and $E = \\{(1,2),(2,3),\\ldots ,(N-1,N)\\}$.  The Hilbert space associated to vertex $u$ is $\\mathcal{H}_u = \\mathbb{C}^D \\otimes\\mathbb{C}^D$. As above, it is convenient to imagine that each vertex $u$ is composed of two $D$-dimensional subsystems $L_u$ and $R_u$. Then, up to a local isometry, the MPS of eq.~\\eqref{eq:MPS} can be expressed as a 1-bifactor state eq.~\\eqref{Graph:QGSE} with\n\\begin{equation}\n\\mu_u = A_u^\\dagger A_u \\quad \\mathrm{and}\\quad \\nu_{u:v} = \\kb{\\Phi}{\\Phi}_{R_u\\cup L_v}.\n\\end{equation}\nMoreover, the operators $\\nu_{u:v}$ mutually commute. To see the relation with eq.~\\eqref{eq:MPS}, note that the operators $A^u$ can be polar decomposed $A_u = U_u \\sqrt{A_u^\\dagger A_u} = U_u \\mu_u^\\frac 12$. \\footnote{Note that $\\mu_u$ has rank $\\leq d$. This can be seen straightforwardly by writing $\\mu_u = \\sum_{j=1}^d A_u^{*j,\\alpha,\\beta}A_u^{j,\\gamma,\\delta} = \\sum_{j=1}^d \\kb{h_u^j}{h_u^j}$ where $|h_u^j\\rangle = \\sum_{\\alpha,\\beta} A_u^{j,\\alpha,\\beta} \\Ket{\\alpha,\\beta} \\in \\mathcal{H}_u$.} The matrix $U_u$ is a partial isometry $\\mathcal{H}_u \\rightarrow \\mathbb{C}^d$ and\n\\begin{align}\n\\kb\\Psi\\Psi &=  \\frac 1Z \\left(\\prod_{u \\in V} A_u\\right) \\kb{\\Phi_0}{\\Phi_0}\\left(\\prod_{u \\in V} A^\\dagger_u\\right) \\\\\n&= \\frac 1Z \\left(\\prod_{u \\in V} U_u\\mu_u^\\frac 12 \\right) \\left(\\prod_{(v,w) \\in E} \\nu_{u:v}\\right) \\left(\\prod_{u \\in V} \\mu_u^\\frac 12 U^\\dagger_u \\right) \\\\\n&= \\frac 1Z \\left(\\prod_{u \\in V} U_u \\right) \\left(\\bigotimes_{u \\in V} \\mu_u \\right) \\ensuremath{\\star} \\left( \\prod_{(v,w) \\in E} \\nu_{u:v}\\right) \\left(\\prod_{u \\in V}  U^\\dagger_u \\right)\n\\end{align}\nas claimed. \n\nBifactor states are thus relevant to the description of quantum many-body systems. QBP can sometimes be used to efficiently compute correlation functions, but in general for spatial dimension larger than one, its convergence is not guaranteed. This is mainly due to the presence of small loops in the underlying graph. Partial solutions have been proposed to overcome this difficulty \\cite{VC04a}, and it is conceivable that techniques from loopy Belief Propagation and its generalizations \\cite{YFW02a} will improve these algorithms. As in the classical case however, QBP may be more appropriate for the study of quantum systems on irregular sparse graphs, such as those encountered in classical spin glasses. \n\nFinally, it should be noted that the Markov conditions required to certify the convergence of QBP --- or the associated coarse-grained Markov conditions as explained in the previous section --- are weaker than those typically studied in statistical physics, namely the vanishing of connected correlation functions beyond some length scale. For pure quantum states, the two notions coincide and are equivalent to the absence of long-range entanglement.  At finite temperature however, the state is mixed and the vanishing of mutual information between vertices $u$ and $u+\\ell$ conditioned on vertices $u+1,\\ldots,u+\\ell-1$  eq. \\eqref{correlation-length} does not imply the absence of connected correlations $\\langle A_uA_{u+\\ell}\\rangle = \\Tr{\\rho_V A_u A_{u+\\ell}} - \\Tr{\\rho_V A_u}\\Tr{\\rho_VA_{u+\\ell}}$.\n\n\\section{Related Work}\n\n\\label{Relate}\n\nIn this section, our approach to quantum Graphical Models and Belief Propagation is compared to other proposals that have appeared in the literature.  Firstly, Tucci has developed an approach to quantum Bayesian Networks \\cite{Tuc95a}, Markov Networks \\cite{Tuc07a} and Belief Propagation \\cite{Tuc98a} based on a different analogy between quantum theory and classical probability, namely the idea that probabilities should be replaced by complex valued amplitudes.  Tucci's models require that these amplitudes should factorize according to conditions similar to those used in classical Graphical Models. One disadvantage of this is that the definition requires a fixed basis to be chosen for the system at each vertex of the graph, and the factorization condition for Bayesian Networks is not preserved under changes of this basis.  In contrast, our definition of quantum conditional independence is based on an explicitly basis independent quantity, so it does not have this problem.  Another difficulty with using amplitudes is that they are only well-defined for pure states, so that mixed states have to be represented as purifications on larger networks.  In our approach, density operators are taken as primary, so mixed states can be represented without purification.  On the other hand, the Tucci's definitions can easily accommodate unitary time evolution, whereas we do not have a general treatment of dynamics in our approach at the present time.  A related definition of quantum Markov Networks, also based on amplitudes but without a development of the corresponding Belief Propagation algorithm, has been proposed by La Mura and Swiatczak \\cite{LMS07a}, to which similar comments apply.\n\nThere has also been work on Quantum Markov networks within the quantum probability literature \\cite{Lei01a,AF03a,AF03b}, although Belief Propagation has not been investigated in this literature.  This is closer to the spirit of the present work, in the sense that it is based on the generalization of classical probability to a noncommutative, operator-valued probability theory.  These works are primarily concerned with defining the Markov condition in such a way that it can be applied to systems with an infinite number of degrees of freedom, and hence an operator algebraic formalism is used.  This is important for applications to statistical physics because the thermodynamic limit can be formally defined as the limit of an infinite number of systems, but it is not so important for numerical simulations, since these necessarily operate with a finite number of discretized degrees of freedom.  Also conditional independence is defined in a different way via quantum conditional expectations, rather than the approach based on conditional mutual information and conditional density operators used in the present work.  Nevertheless, it seems likely that there are connections to our approach that should to be investigated in future work. \n\nLastly, during the final stage of preparation of this manuscript, two related papers have appeared on the physics archive. An article by Laumann, Scardicchio and Sondhi \\cite{LSS07a} used a QBP-like to solve quantum models on sparse graphs.  Hastings \\cite{Has07b} proposed a QBP algorithm for the simulation of quantum many-body systems based on ideas similar to the ones presented here. The connection between the two approaches, and in particular the application of the Lieb-Robinson bound \\cite{LR72a} to conditional mutual information, is worthy of further investigation. \n\n\n\\section{Conclusion}\n\n\\label{Conc}\n\n\\begin{figure}\n\\center\\includegraphics[width=11cm]{veine}\n\\caption{Relation between Markov Networks, Bifactor Networks, and 1-Bifactor Networks in a) quantum theory and b) classical probability theory. The hashed regions indicate the domain of convergence of the associated Belief Propagation algorithms. Figure a). Convergence of Belief Propagation on trees for Markov Networks is Theorem \\ref{thm:QBP} and for 1-Bifactor states is Corollary \\ref{cor:QBP1}. That all Markov Networks on trees are Bifactor states is Theorem \\ref{The:mutual}. The existence of Bifactor Networks on trees that are not Markov Networks is given by Example \\ref{ex:notMarkov} for $n<\\infty$ and the Heisenberg anti-ferromagnetic spin chain of Example \\ref{ex:heisenberg} for $n = \\infty$. Markov Networks on trees with cliques of size $>2$ are generally not Bifactor Networks, c.f. Theorem \\ref{Graph:QHC}. Figure b). That all classical Bifactor Networks are Markov Networks is the Hammersley-Clifford Theorem \\ref{thm:HC}, and convergence of Belief Propagation on trees follows from Theorem \\ref{thm:QBP}. }\n\\label{fig:worldview}\n\\end{figure}\n\nIn this paper, we have presented quantum Graphical Models and Belief Propagation based on the idea that quantum theory is a noncommutative, operator-valued, generalization of probability theory.  Our main results are summarized on Fig. \\ref{fig:worldview}.  We expect these methods to have significant applications in quantum error correction and the simulation of many-body quantum systems.  We are currently in the process of implementing these algorithm numerically in both of these contexts. Belief Propagation based decoding of several types of quantum error correction codes has already been implemented quite successfully, e.g. on concatenated block codes \\cite{Pou06b}, turbo codes \\cite{OPT07a}, and  sparse codes \\cite{COT05a}. However, for the noise models considered there, the corresponding bifactor states only involve commuting operators and thus the corresponding inference problem could be solved by means of a classical Belief Propagation algorithm.  We conclude with several open questions suggested by this work. \n\nIn the context of many-body physics, it would be interesting to relate the class of solutions obtained by QBP to other approximation schemes used in statistical physics, much in the spirit of the work of Yedidia \\cite{Yed01a} in the classical setting. A related problem would be to understand how the different classes of bifactor states relate to each other. We suspect that when the Hilbert space dimension at each vertex of the graph is held fixed, the $n$-bifactor states on that graph form a subset of the $m$-bifactor states when $n<m$. If that conjecture were true, it might lead to a family of approximation schemes converging to the correct solution. It would also reveal an interesting discrepancy between the classical and quantum settings. Classically, the problem of computing correlation functions in a disordered many-body system and the problem of decoding an error correction code are equivalent. If our conjecture holds true, in the quantum case the latter is simpler than the former.\n\nWhilst our definition of a quantum Markov Network is well motivated as a direct analog of a classical Markov Network, it does not seem to represent the most general class of states to which our Belief Propagation algorithms are applicable.  In particular, in \\S\\ref{QBP:Conv} it was shown that QBP converges on trees for arbitrary bifactor states defined with respect to the $\\ensuremath{\\star}$ product.  One reason for this discrepancy might be that the quantum conditional independence condition, $I_\\rho (U,W|X)$, only allows classical correlations to be mediated between $U$ and $W$ via $X$, i.e. $\\rho_{U\\cup W}$ is always separable, whereas the classical condition $I_P(U,W|X)$ is compatible with an arbitrary distribution $P(U \\cup W)$.  This suggests that quantum conditional independence could be relaxed to a condition that allows quantum correlations, i.e. entanglement, to be mediated by $X$, whilst still preserving the validity of Belief Propagation.  It would be interesting to find a condition like this that also satisfies the graphoid axioms, so that it could naturally be represented on a graph.\n\nNevertheless, quite apart from their application in Belief Propagation algorithms, the mathematical structures investigated in this work should be of interest in other areas of quantum information and computation.  Firstly, the characterizations of quantum conditional independence in terms of conditional density operators given in \\S\\ref{Cond:CDO} should be useful, and indeed are currently being applied to the problem of pooling quantum states \\cite{LS07a}.  Another interesting area of investigation would be the computational complexity of inference on quantum Markov Networks.  In the classical case, it is fairly straightforward to find families of Markov Networks that encode instances of NP complete problems, such as satisfiability or graph colorability.  Therefore, one would expect to be able to encode problems that are similarly hard for quantum computers, i.e. complete for the complexity class QMA, as inference problems on quantum Markov Networks.  This should be closely related to the quantum marginals problem, which has recently be proved to be QMA-complete \\cite{AGK07a,Ira07a}.\n\nFinally, this work leaves open the question of fully characterizing quantum Markov Networks.  The most generally applicable result given here is theorem \\ref{Graph:QHC}, which is a direct analog of one direction of the classical Hammersely Clifford theorem using the $\\ensuremath{\\odot}$ product.  A full characterization would provide a converse to this theorem, i.e. a set of conditions on the operators in eq.~\\eqref{Graph:QHCDecomp}, satisfied by the construction used in the proof, such that all states of this form are guaranteed to satisfy the Markov condition.  Analogous theorems for the $\\pns{n}$ products would also be useful. This work also leaves open the question of intersection for quantum conditional mutual information, i.e. whether $S(U:W|X\\cup Y) = 0$ and $S(U:Y|W\\cup X) = 0$ imply $S(U:W\\cup Y|X) = 0$ for strictly positive states. This result would imply that positive quantum Markov networks obey global Markov properties.  \n\n\\section*{Acknowledgments}\n\nML would like to thank Rob Spekkens for useful discussions about quantum conditional independence. DP is grateful to Harold Ollivier for many stimulating discussion on Belief Propagation.   \n\nAt IQC, ML was supported in part by MITACS and ORDCF.  Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.  ML was also supported in part by grant RFP1-06-006 from The Foundational Questions Institute (fqxi.org). \n\nDP is supported in part by the Gordon and Betty Moore Foundation through Caltech's Center for the Physics of Information, by the National Science Foundation under Grant No. PHY-0456720, and by the Natural Sciences and Engineering Research Council of Canada.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\columnwidth]{Fig1_Motivation.pdf}\n\\caption{\\small\nThe statistics of common neighbors between different models.\nCNN-CNN and ViT-ViT curves denote the average number of common neighbors in the k nearest neighbors of each instance.\nThe features are extracted by two homogeneous networks trained with different initialization.\nSimilarly, CNN-ViT represents the common neighbors between CNN and ViT.\nFurthermore, Upbound refers to the maximum number of neighbors to consider.\nThese models are trained on the CUHK-SYSU dataset in a supervised manner and cluster on the Market1501 dataset.\nCompared to the CNN-CNN and ViT-ViT, the CNN-ViT contains fewer common neighbors, and the ways they distinguish two individuals are more different than homogeneous networks.\nIt demonstrates that heterogeneous networks address the task in different patterns.\n}\n\\label{fig1:Motivation}\n\\end{figure}\n\n\\IEEEPARstart{P}{erson} re-identification (Re-ID)~\\cite{series\/acvpr\/GongCLH14} aims at matching individual pedestrian images from images captured by different cameras according to identity.\n\\IEEEpubidadjcol\nThis task is challenging because the variations of viewpoints, body poses, illuminations, and backgrounds will influence a person's appearance.\nRecently, supervised person Re-ID methods~\\cite{journals\/ijcv\/LiZG20, journals\/pami\/SunZLYTW21, journals\/ijcv\/YinWZ20, conf\/eccv\/SunZYTW18, conf\/cvpr\/ZhangLZJ020, conf\/cvpr\/WangZGL18, conf\/iccv\/WuZL19, conf\/cvpr\/YuZ20, conf\/eccv\/ZhongZLY18, conf\/iccv\/HanYZTZGS19} made impressive progress.\nHowever, as the number of images increases and the ensuing scene changes, regular supervised learning approaches are losing their ability to adapt to complex scenarios.\nThe performance of person re-ID models trained on existing datasets will evidently suffer for person images from a new video surveillance system due to the domain gap.\nTo avoid time-consuming annotations on the new dataset, unsupervised domain adaptation (UDA) is proposed to adapt the model trained on the labeled source-domain dataset to the unlabeled target-domain dataset.\n\nGenerating trusted identity information on the target domain is seen as the core of the UDA task.\nSome UDA Re-ID methods~\\cite{conf\/cvpr\/Deng0YK0J18, conf\/iccv\/LiLLW19, conf\/cvpr\/WeiZ0018, conf\/eccv\/ZhongZLY18, conf\/eccv\/ZouYYKK20} directly apply GANs~\\cite{journals\/corr\/GoodfellowPMXWOCB14} to transfer the style of pedestrian images from the source domain to the target while keeping the identities to train the model.\nHowever, the complexity of the human form and the limited number of instance in a Re-ID dataset limit the quality of generated images.\nAfter abandoning the image generation, some methods~\\cite{conf\/aaai\/ChangYXH19, conf\/cvpr\/WangZGL18} introduce the attribute to bridge the domain gap.\nThese methods introduce additional annotation information which defeats the purpose of the UDA Re-ID task.\nLimited by the missing label on the target domain, others~\\cite{conf\/iccv\/QiWHZSG19, conf\/cvpr\/Zhong0LL019, conf\/cvpr\/YangZLCLLS21, journals\/tip\/DaiCWHYTLJ22} align the distributions of target and source domains while only learning classifying on the source.\nTo better adapt the distribution of the target domain and train with the target-domain identity knowledge, various methods~\\cite{conf\/aaai\/LinD00019, conf\/iccv\/ZhangCSY19, conf\/cvpr\/BaiWW0D21} apply a clustering algorithm in the target domain to generate the pseudo labels for training in a supervised manner. \nOne of the keys to improving performance is alleviating the influence of noisy labels. \nIn this context, many methods~\\cite{conf\/iclr\/GeCL20, conf\/cvpr\/ZhengLH0LZ21, conf\/eccv\/ZhaiYLJJ020} based on clustering algorithms are proposed to rule out the harmony from the noisy labels by introducing more than one framework to predict pseudo labels.\nThey aim to generate knowledge with specific differences in samples and exchange the knowledge among the networks to enhance their ability.\nDespite encouraging progress, the benefits from the knowledge mined by homogeneous networks are limited.\nAs shown in Fig.~\\ref{fig1:Motivation} CNN-CNN and ViT-ViT, these homogeneous networks with similar structures identify pedestrians in a comparable manner, and the relation among the instances are similar.\nIt suggests they use similar patterns to extract pedestrian features, and networks may converge to equal each other.\nFurthermore, this mode of operation makes it possible for the networks to make the same mistakes and not be able to correct them.\nSuch a design limits the knowledge models can learn from the training set and makes it possible for the networks to repeat mistakes without being able to remedy them.\nAs a result, mining the information from different subspaces is required to broaden the scope of knowledge and generate reliable pseudo labels.\n\nTo tackle this problem, heterogeneous networks, as shown in Fig.~\\ref{fig1:Motivation} CNN-ViT, can discover the information from multiple subspaces and have more extensive latent knowledge.\nWe propose Dual-level Asymmetric Mutual Learning (DAML), a novel unsupervised domain adaption method for person Re-ID that broadens the scope of knowledge for the network by exploiting information from two different subspaces and selectively transferring information between heterogeneous networks.\nThe proposed DAML consists of a CNN that focuses on identity learning as a teacher network and a ViT that concentrates on adapting knowledge from the target domain as a student network for embedding samples into different subspaces and setting the constraints among the classifiers for asymmetric mutual learning.\n\nIn particular, the CNN that works as a teacher will train on both source and target datasets under the supervision of ground-truth source-domain labels and pseudo-target-domain labels.\nThe former can provide reliable identity information for extracting discriminative feature representation, while the latter will assist the network in adapting the  distribution of the target domain.\nHowever, learning from the source domain will harm the distribution that the network adapted limiting the performance.\nTo avoid this disadvantage, the ViT that works as a student only trains with the guidance of pseudo-target-domain labels and learns the knowledge from the teacher.\nIn the pseudo label generation stage, the relationship between two samples is weighted according to their teacher and student features similarity.\nMoreover, this process wholly exchanges the knowledge learned from two different subspaces.\nAfter predicting the identities of input images, the asymmetric constraints between two heterogeneous networks selectively exchange the knowledge.\nThe student learns the identity knowledge from the teacher network under the constraints from the target-domain samples.\nFurthermore, for the student can benefit more from the teacher and better utilize the ground-truth labels, the source-domain identity knowledge learned by the teacher is transferred to the target domain with the constraints based on source-domain samples.\nIn summary, the DAML employs diverse subspaces to generate reliable pseudo label in the target domain and help student adopt ground-truth knowledge in the source domain.\n\nOur main contributions are summarized below:\n\n\\begin{itemize}\n    %\n    \\item We address the diverse subspaces learning and target-domain identity learning for unsupervised domain adaptation person Re-ID with proposed Dual-level Asymmetric Mutual Learning (DAML). \n    %\n    The former has rarely been studied in the existing research, while the latter is crucial for retrieving person in the target domain.\n    %\n    \\item We propose a novel Dual-level Asymmetric Mutual Learning (DAML) method for unsupervised domain adaptation person Re-ID. \n    %\n    The asymmetric knowledge learning between the teacher and the student helps them play their roles better.\n    %\n    \\item To learn from diverse subspaces, the proposed DAML introduces two heterogeneous networks to mine valuable information from different subspaces and selectively exchange the information between them.\n    %\n    \\item To better utilize the knowledge mined by heterogeneous networks and ensure the networks orient to the task, the proposed DAML smoothly update the classifiers in a hard distillation manner and exchange knowledge during training in a soft distillation manner.\n    %\n\\end{itemize}\n\n\\section{Related Work}\n\n\\subsection{Unsupervised Domain Adaptation Person Re-ID}\nUnsupervised Domain Adaptation Person Re-ID has attracted increasing attention in recent years due to its effectiveness in reducing manual annotation costs.\nThere are two main categories of methods are proposed to address this issue.\nFirstly, GAN-based methods aim to transfer samples from the source domain to the target domain without altering their identities. \nSPGAN~\\cite{conf\/cvpr\/Deng0YK0J18} and PDA-Net~\\cite{conf\/iccv\/LiLLW19} transfer images directly from the source domain to the target domain while maintaining the original identity knowledge.\nThe generated images have a similar style to the target-domain images and are used to train the model under the supervision of their original labels in the source domain.\nTo produce generated images that are more realistic and have more detail, DG-Net~\\cite{conf\/cvpr\/ZhengYY00K19} and DG-Net++~\\cite{conf\/eccv\/ZouYYKK20} introduce disentanglement for the generation stage.\nBut the generation is expensive and the style of generated images may not well fit the target domain.\nRather than transfer images from the source domain to the target domain, HHL~\\cite{conf\/eccv\/ZhongZLY18} transfers target-domain images among the cameras to generate images that have the same identity but contain the difference at the same time.\nSecondly, the clustering-based methods clustering based methods do not require expensive GAN networks for generation and have achieved state-of-the-art performance to date.\nTo reduce the impact of noisy label, MMT~\\cite{conf\/iclr\/GeCL20} proposed a mutual learning method providing soft labels.\nFor more reliable pseudo labels, SSG~\\cite{conf\/iccv\/FuWWZSUH19} clusters samples in three scales and validate each other.\nMEB-Net~\\cite{conf\/eccv\/ZhaiYLJJ020} respectively introduces multiple groups of prototypes or homogeneous networks to generate the pseudo labels.\nUNRN~\\cite{conf\/aaai\/ZhengLZZZ21} and GLT~\\cite{conf\/cvpr\/ZhengLH0LZ21} design a memory bank to save anchors for aligning the distribution and learning identities in a contrastive learning manner.\nLimited by the constraints in the feature level these methods rely on, the models that collaborate to generate pseudo labels are homogeneous.\nThese characteristics determine that the model can only learn similar knowledge from others.\nNevertheless, these approaches alleviate the domain gap only considering the single embedding space inevitably makes some mistakes.\n\n\\subsection{Knowledge Distillation}\nKnowledge distillation makes a student network learns from a strong teacher network to improve the student's ability.\nThe common approaches can be summarized as hard distillation and soft distillation.\nSoft distillation~\\cite{journals\/corr\/HintonVD15, conf\/eccv\/WeiXXZ0T20} minimizes the distribution difference between the prediction generated by teacher and student.\nThe soft label generated by the teacher model can alleviate overfitting just like labels smoothing~\\cite{conf\/cvpr\/YuanTLWF20}.\nUnlike the soft, hard-label distillation regards the prediction result of the teacher as a valid label.\nAnd positive pairs predicted by the teacher are used to transfer identity knowledge from the teacher to a student network in semi-supervised and unsupervised learning tasks.\nTemporal ensembling~\\cite{conf\/iclr\/LaineA17} put the former networks as the teacher and use memory saving average predictions for each sample as supervision for the unlabeled samples.\nTo avoid storing predictions for saving memory, Mean Teacher~\\cite{conf\/iclr\/TarvainenV17} averaged student model weights as the parameter of the teacher.\nDuring the training, the predictions made by the teacher are seen as supervision for unlabeled samples.\nThe models consider similar information in these methods because the teacher and the student have the same structure and similar initialization.\nIt makes the networks focus within a certain range and limit the knowledge student can learn.\nThe proposed DAML exchange knowledge utilizes both soft and hard distillation in the different training stages.\nThanks to the heterogeneous networks, the proposed DAML gives the student model a broader perspective and can generate pseudo labels from different views.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Fig2_Framework.pdf}\n\\vspace{-3mm}\n\\caption{ \\small\nOverview of our Dual-level Asymmetric Mutual Learning method (DAML).\nThe teacher network is trained under the supervision of pseudo labels and ground-truth labels for target-domain and source-domain samples.\nAnd the student only directly learns knowledge from target-domain samples with pseudo labels.\nAt the beginning of epochs, we first generate the pseudo labels for target dataset, and update the classifiers based on the predictions of cluster centers.\nTo distill the different subspace knowledge from the teacher to the student,  $\\mathcal{L}_{id}$ makes the student predictions of target-domain samples close to the teacher.\nMeanwhile, for student can better adopt the identity knowledge learned by the teacher, we minimize the distribution differences of the same source-domain samples with $\\mathcal{L}_{dom}$.\n}\n\\label{fig2:Framework}\n\\end{figure*}\n\n\\subsection{CNN and ViT}\nSince AlexNet~\\cite{conf\/nips\/KrizhevskySH12} achieve great success on ImageNet~\\cite{conf\/cvpr\/DengDSLL009}, a variety of convolutional neural networks (CNN)~\\cite{conf\/cvpr\/SzegedyVISW16, conf\/cvpr\/HeZRS16, conf\/cvpr\/HuangLMW17, conf\/eccv\/PanLST18} is proposed to solve different tasks.\nAs Transformers~\\cite{conf\/nips\/VaswaniSPUJGKP17} were proposed for machine translation and were seen with significant results in many NLP tasks, the application of self-attention to images is widely concerned. \nA new model without any convolution, Vision Transformers (ViT)~\\cite{conf\/iclr\/DosovitskiyB0WZ21}, has been proposed for computer vision tasks and shows its potential.\nDuring the calculation process, the CNN keeps the spatial information and can only focus on the surrounding area in one layer due to the nature of convolution.\nIn contrast, ViT emphasizes the correlation between two patches, and its receptive field involves the whole feature map.\nThese differences make the CNN and ViT learn different knowledge from the training set for the same task.\nAnd in our paper, we take advantage of this difference to achieve asymmetric distillation, making ViT a better performer with our DAML.\nThe ViT works as a student because the receptive field of a patch in the ViT covers the area that one convolution kernel can consider, not vice-versa. \n\n\\section{Methodology}\n\\label{Sec:Methodology}\n\\subsection{Overview}\n\nThe ultimate goal of the unsupervised domain adaptation (UDA) person Re-ID is to gain a model work on a target-domain dataset based on a labeled source-domain dataset and an unlabeled target-domain dataset.\nLet $\\mathcal{S}=\\{(\\mathbf{x}_s^i, \\mathbf{y}_s^i)\\}_{i=1}^{N_s}$ and $\\mathcal{T}=\\{\\mathbf{x}_t^i\\}_{i=1}^{N_t}$ respectively denote the source-domain images with ground-truth labels and the unlabeled target-domain images, where $N_s$ and $N_t$ are the numbers of samples from these two domains.\n\nAs shown in Fig.~\\ref{fig2:Framework}, the Dual-level Asymmetric Mutual Learning (DAML) method trains the student to extract discriminative representations from two different subspaces to perform the UDA person Re-ID task.\nFirstly, DAML adopts two heterogeneous networks: teacher CNN ${\\rm E}_T(\\cdot)$ and student ViT ${\\rm E}_S(\\cdot)$ which are pre-trained on the source-domain dataset in a supervised manner to extract features in different subspaces.\nAt each epoch, we first group target-domain samples into $K$ classes by the clustering algorithm.\nThe distance between two target-domain samples will be calculated according to the features ${\\rm E}_T(\\mathbf{x}_t^i)=\\mathbf{t}_i^T \\in \\mathbb{R}^{c_T}$ and ${\\rm E}_S(\\mathbf{x}_t^i)=\\mathbf{t}_i^S \\in \\mathbb{R}^{c_S}$ extracted by the teacher and student models with corresponding weights.\nThe $\\{\\hat{\\mathbf{y}}_i\\}_{i=1}^{N_t}$ are the pseudo labels for the target-domain samples.\nThen, for each class center $\\mathbf{c}_y$, we generate its prediction with the classifiers ${\\rm C}(\\cdot|\\mathbf{W}_t^S)$ and ${\\rm C}(\\cdot|\\mathbf{W}_t^T)$ for updating the parameter $\\mathbf{W}_t^S$ and $\\mathbf{W}_t^T$ in a smooth method.\n\nAfter that, we train the teacher and the student models with the pseudo labels in a supervised manner.\nFor the teacher model, classifier ${\\rm C}(\\cdot|[\\mathbf{W}_s^T, \\mathbf{W}_t^T])$ will learn both source-domain and target-domain knowledge.\nWhile the classifier ${\\rm C}(\\cdot|\\mathbf{W}_t^S)$ for the student model only directly learns the target-domain knowledge.\nThe constraints between two networks transfer the identity knowledge learned by the teacher to the target and help the student learn from diverse subspaces. \nFinally, we only adopt the features $\\mathbf{t}_i^S = {\\rm E}_S(\\mathbf{x}_t^i)$ extracted by the student model for testing.\n\n\\subsection{Smooth Classifier Update (SCU)}\n\nAt the beginning of epochs, we extract the target-domain features $\\mathbf{t}_i^T = E_T(\\mathbf{x}^i_t)$ and $\\mathbf{t}_i^S = E_S(\\mathbf{x}^i_t)$ with two heterogeneous networks.\nTo better utilize the knowledge from the two models, we first define the neighborhood of an instance according to its relations in two different subspaces:\n\\begin{equation}\n    N_i = \\Bigg\\{x_j \\Bigg| 1 - \\frac{\\langle t_i^M, t_j^M \\rangle}{\\Vert t_i^M\\Vert_2 \\Vert t_j^M\\Vert_2} < \\alpha, M \\in \\{T, S\\} \\Bigg\\},\n\\end{equation}\nthe $\\alpha$ here is a hyper-parameter.\nWith the limitation of neighbor selection considering both teacher features and student features simultaneously, the neighbors of an instance should be close to it in both subspaces.\nThe above constraint ensures that instances with apparent differences will not be clustered as the same identity because the patterns of the two models applied to recognize an instance are different.\n\nTo exploit the information from two different subspaces and make the pseudo labels more reliable, we combine features from heterogeneous networks and define the distance between two samples as:\n\\begin{equation}\n    d_{i,j} =\n\\begin{cases}\n\\begin{aligned}\n& 1 - \\frac{\\langle[\\mathbf{t}_i^T, \\mathbf{t}_i^S],[\\mathbf{t}_j^T, \\mathbf{t}_j^S]\\rangle}\n           {\\Vert[\\mathbf{t}_i^T, \\mathbf{t}_i^S]\\Vert_2 \\Vert[\\mathbf{t}_j^T, \\mathbf{t}_j^S]\\Vert_2}, & x_i \\in N_j \\ and \\ x_j \\in N_i \\\\\n& \\rm{Inf}, & Others\n\\end{aligned}\n\\end{cases}\n\\end{equation}\nwhere $[\\cdot,\\cdot]$ represents the concatenation of two features, and $\\langle\\cdot,\\cdot\\rangle$ denotes the dot product between two features.\nIn short, we define the similarity between two samples as the cos similarity between the features constructed by concatenating their teacher feature and student feature.\nThen the pseudo labels can be generated based on the relationship among instances with the clustering algorithm.\n\nWith the pseudo labels, some methods \\cite{conf\/iclr\/GeCL20,conf\/aaai\/ZhengLZZZ21} directly update the classifier by replacing the classifier parameters with the new class centers to adapt the count of classes change.\nThese methods will make the knowledge lost because the class centers may not represent the corresponding class well.\nTo protect the knowledge involved in the classifiers, we update the classifiers more smoothly as follows:\n\\begin{equation}\n\\begin{aligned}\n\\mathbf{W}^i_t=\\sum_{k=1}^{\\hat{K}} \\mathbf{\\hat{W}}^k_t \\cdot \\frac{e^{\\mathbf{p}_i^k}}{\\sum_{j=1}^{\\hat{K}} e^{\\mathbf{p}_i^j}},\n\\end{aligned}\n\\label{update_the_target_classifiers}\n\\end{equation}\nwhere $\\mathbf{W}_t^i$ is the parameters for the $i^{th}$ target-domain identity in the next epoch, $\\mathbf{p}_i ={\\rm C}(\\mathbf{c}_i|\\mathbf{\\hat{W}}_t)$ is the prediction of class center $\\mathbf{c}_i$ with the parameters $\\mathbf{\\hat{W}}_t$ from the last epoch which includes $\\hat{K}$ classes.\nNote that the momentum for SGD is updated following the parameters in the process.\n\n\\subsection{Identity Learning}\n\nThe core of person re-identification is identifying the persons.\nFor two heterogeneous networks learning to extract discriminative representation, there are two level objective functions are applied.\nFirstly, at the feature level, the triplet loss:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{tri}(\\mathbf{f})=\\frac{1}{n}\\sum_{i=1}^n \\max \\{ \\rho + d_p - d_n, 0 \\},\n\\end{aligned}\n\\label{loss_triplet}\n\\end{equation}\nis applied to guarantee the features can well represent their corresponding samples.\nWhere $\\mathbf{f}$ represents a batch of the features, $n=|\\mathbf{f}|$ is the size of the batch, \n$\\rho$ is the tiniest margin between the distance to the furthest positive instance $d_p$ and the distance to the nearest negative instance $d_n$.\nThe relationship between two instances from the source domain depends on the ground-truth labels and the pseudo labels for target-domain samples.\nDue to the different dimensions of the features extracted by heterogeneous networks, the triplet loss can only be applied in a certain subspace.\n\nThen, in the logits level, we apply the cross-entropy loss with classifiers:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{Ttid}=-\\frac{1}{n}\\sum_{i=1}^n\\log{P(\\hat{\\mathbf{y}}_i|{\\rm C}(\\mathbf{t}_i^T | [\\mathbf{W}_s^T,\\mathbf{W}_t^T]))},\n\\end{aligned}\n\\label{loss_teacher_learn_target_identity}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{Stid}=-\\frac{1}{n}\\sum_{i=1}^n\\log{P(\\hat{\\mathbf{y}}_i|{\\rm C}(\\mathbf{t}_i^S | \\mathbf{W}_t^S))},\n\\end{aligned}\n\\label{loss_student_learn_target_identity}\n\\end{equation}\nwhere $\\hat{\\mathbf{y}}_i$ is the pseudo label for target-domain example $\\mathbf{x}_t^i$.\nThe trainable parameters $\\mathbf{W}_s^T$, $\\mathbf{W}_t^T$ and $\\mathbf{W}_t^S$ respectively denote the classifier parameters for the teacher classifying source-domain samples, the teacher classifying target-domain samples, and the student classifying target-domain samples.\nMeanwhile, to take advantage of the ground-truth label, the teacher also learns the source-domain knowledge by:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{Tsid}=-\\frac{1}{n}\\sum_{i=1}^n\\log{P(\\mathbf{y}_i|{\\rm C}(\\mathbf{s}_i^T | [\\mathbf{W}_s^T,\\mathbf{W}_t^T]))},\n\\end{aligned}\n\\label{loss_teacher_learn_source_identity}\n\\end{equation}\nhere, $\\mathbf{y}_i$ is the ground-truth label for source-domain sample $\\mathbf{x}_s^i$.\nNote that, with Eq.(\\ref{loss_teacher_learn_target_identity}) and Eq.(\\ref{loss_teacher_learn_source_identity}), the classifier ${\\rm C}(\\mathbf{t}_i^T | [\\mathbf{W}_s^T,\\mathbf{W}_t^T])$ in the teacher has learned both two domain knowledge while classifier ${\\rm C}(\\mathbf{t}_i^T | \\mathbf{W}_t^S)$ for the student learns the target-domain knowledge only.\n\n\\subsection{Asymmetric Mutual Learning (AML)}\n\nCompare the structure of the Convolutional Neural Network (CNN) and Vision Transformer (ViT), the most evident difference is that the ViT can capture long-range information with its cascaded self-attention modules. \nHowever, CNN only focuses on the local limited by the size of the convolution kernel.\nIn addition, the CNN inductive bias, which includes assumptions of the data, can involve information that ViT may not consider and the convolution kernel with a deterministic shape guarantees spatial information.\nMore intuitively, the features extracted by the two networks have different dimensions.\nIt ensures the subspaces learned by the heterogeneous networks are different but makes feature-level constraints unusable.\nThe asymmetric distillation benefit from the difference in the patterns that two heterogeneous networks predict the identity.\nAnd focus on twofold: to allow students access to knowledge from the different subspaces and transfer the knowledge from the source to the target.\n\nTo make the student learn from different subspaces and take advantage of the reliable source-domain labels, the proposed DAML transfers the identity knowledge from the teacher by reducing the Kullback-Leibler divergence between the predictions of the target-domain features as:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{id}=\\frac{1}{n}\\sum_{i=1}^n&{\\rm C}(\\mathbf{t}^T_i|\\mathbf{W}_t^T)\\log{\\frac{{\\rm C}(\\mathbf{t}^S_i|\\mathbf{W}_t^S)}{{\\rm C}(\\mathbf{t}^T_i|\\mathbf{W}_t^T)}},\n\\end{aligned}\n\\label{equation_knwoledge_distillation}\n\\end{equation}\nwith the above objective function, the student can learn the knowledge from the teacher which adopts knowledge from both source and target domain with Eq.(\\ref{loss_teacher_learn_target_identity}) and Eq.(\\ref{loss_teacher_learn_source_identity}).\nHowever, domain knowledge is also transferred to students and may harm the performance in the target domain.\nThe ideal way to alleviate the distribution effect is to make the teacher predict identities under the target domain.\n\nLimited by the domain gap, the knowledge learned from the source domain can not be directly applied to the target domain.\nAnd the goal of the proposed asymmetric mutual learning is to gain a student network that adapts to the target domain while benefiting from the source-domain identity knowledge.\nMaking source-domain predictions from the teacher similar to the student will transfer the classifying knowledge learned from the source domain to the target domain:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{dom}=\\frac{1}{n}\\sum_{i=1}^n&{\\rm C}(\\mathbf{s}^S_i|\\mathbf{W}_t^S)\\log{\\frac{{\\rm C}(\\mathbf{s}^T_i|\\mathbf{W}_t^T)}{{\\rm C}(\\mathbf{s}^S_i|\\mathbf{W}_t^S)}},\n\\end{aligned}\n\\label{equation_eliminate_domain_gap}\n\\end{equation}\nhere, $\\mathbf{W}_t^T$ learns source-domain knowledge with Eq.(\\ref{loss_teacher_learn_source_identity}) while $\\mathbf{W}_t^S$ only learns the knowledge from target domain.\nEq.(\\ref{equation_eliminate_domain_gap}) focuses on making the teacher predict source-domain samples in the same way as the student.\nIn this way, the student can better adopt identity knowledge from the source domain without a domain gap as much as possible.\nCompared with Eq.(\\ref{equation_knwoledge_distillation}), the above equation distills the knowledge in a different direction and together make the student model can distinguish pedestrian in the target domain. \n\n\\subsection{Optimization}\nThe total loss $\\mathcal{L}$ of DAML can be summarized as:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}=&\\big(\\mathcal{L}_{Ttid} + \\mathcal{L}_{tri}(\\mathbf{t}^T)\\big) +\n             \\big(\\mathcal{L}_{Stid} + \\mathcal{L}_{tri}(\\mathbf{t}^S)\\big) \\\\\n            &\\lambda_1\\big(\\mathcal{L}_{Tsid} + \\mathcal{L}_{tri}(\\mathbf{s}^T)\\big)\n            +\\lambda_2\\mathcal{L}_{id} + \\lambda_3\\mathcal{L}_{dom}\n\\end{aligned}\n\\label{full_loss_functions}\n\\end{equation}\nwhere $\\lambda_1$, $\\lambda_2$, and $\\lambda_3$ are hype-parameters to balance the contributions of individual loss terms.\n\n\\section{Experiments}\n\n\\subsection{Datasets}\n\n\\textbf{Datasets}\nWe evaluated our method on three public datasets \\textbf{Market-1501}~\\cite{conf\/iccv\/ZhengSTWWT15}, \\textbf{CUHK-SYSU}~\\cite{journals\/corr\/XiaoLWLW16} and \\textbf{MSMT17}~\\cite{conf\/cvpr\/WeiZ0018}.\n\n\\begin{itemize}\n    \\item \\textbf{Market-1501} contains $32,668$ labeled images captured from $1,501$ identities by $6$ cameras.\n    %\n    The training set has $12,936$ images of $751$ identities. \n    %\n    In addition, $3,368$ query images and $19,732$ gallery images from the other $750$ identities are used as the testing set.\n    %\n    \\item \\textbf{CUHK-SYSU} includes $33,901$ labeled images of $8,432$ identities taken in diverse scenes.\n    %\n    The training set is constructed with $5,532$ identities having $15,088$ images, and the rest is used for testing.\n    %\n    There are $2,900$ images for the query and $6,978$ images for the gallery in the testing set.\n    %\n    \\item \\textbf{MSMT17} is a large-scale dataset consisting of $126,441$ bounding boxes of $4,101$ identities caught on $12$ outdoor and $3$ indoor cameras.\n    %\n    Among them, $32,621$ images of $1,041$ identities are used for training and $93,820$ of $3,060$ identities are used for testing.\n\\end{itemize}\n\n\\begin{table*}[t]\n\n\\caption{\nComparison of CMC (\\%) and \\emph{m}AP (\\%) performances with the SOTA methods on \\textbf{Market-1501}, \\textbf{CUHK-SYSU} and \\textbf{MSMT17}.\n}\n\n\\centering\n\\resizebox{1.0\\linewidth}{!}{\n\\setlength{\\tabcolsep}{2mm}\n\\begin{tabular}{l||c|ccc||c|ccc}\n\\hline\n\\multirow{2}{*}{Method}  & \\multicolumn{4}{c||}{Market-1501 $\\rightarrow$ CUHK-SYSU} & \\multicolumn{4}{c}{CUHK-SYSU $\\rightarrow$ Market-1501}  \\\\\n\\cline{2-9} & \\emph{m}AP & R1 & R5 & R10    & \\emph{m}AP & R1 & R5 & R10   \\\\\n\\hline\nDirectly Transfer (IBN-ResNet-50)                   & 74.1  & 77.2  & 85.7  & 88.7  & 38.8  & 63.7  & 79.4  & 85.2 \\\\\nDirectly Transfer (ViT-Base)                        & 86.0  & 87.2  & 94.1  & 95.0  & 36.2  & 60.3  & 76.5  & 82.9 \\\\\n\\hline\nUNRN~\\cite{conf\/aaai\/ZhengLZZZ21}(AAAI'21)          & 62.3  & 64.1  & 76.9  & 82.0  & 70.9  & 86.7  & 92.8  & 94.5  \\\\\nMMT(IBN-ResNet-50)~\\cite{conf\/iclr\/GeCL20}(ICLR'20) & 78.4  & 81.0  & 89.7  & 92.2  & 76.0  & 88.8  & 95.2  & 97.0  \\\\\nMEB-Net~\\cite{conf\/eccv\/ZhaiYLJJ020}(ECCV'20)       & 81.1  & 83.2  & 90.9  & 93.1  & 69.3  & 84.0  & 92.9  & 95.2  \\\\\n\\hline\nDAML (Ours)                                         & \\textbf{84.3}  & \\textbf{86.2}  & \\textbf{92.6}  & \\textbf{94.6}  & \\textbf{84.1}  & \\textbf{93.1}  & \\textbf{97.7}  & \\textbf{98.2} \\\\\n\\hline\nSupervised (IBN-ResNet-50)                          & 90.8  & 95.2  & 96.6  & 89.0  & 83.0  & 94.1  & 97.4  & 98.4 \\\\\nSupervised (ViT-Base)                               & 93.1  & 97.2  & 97.8  & 92.1  & 82.3  & 93.2  & 97.9  & 98.8 \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Method}  & \\multicolumn{4}{c||}{Market-1501 $\\rightarrow$ MSMT17} & \\multicolumn{4}{c}{CUHK-SYSU $\\rightarrow$ MSMT17}  \\\\\n\\cline{2-9} & \\emph{m}AP & R1 & R5 & R10    & \\emph{m}AP & R1 & R5 & R10   \\\\\n\\hline\nDirectly Transfer (IBN-ResNet-50)                   & 8.4   & 23.8  & 34.5  & 39.6  & 10.3  & 26.3  & 38.3  & 44.3 \\\\\nDirectly Transfer (ViT-Base)                        & 13.0  & 33.3  & 45.3  & 51.1  & 12.5  & 28.2  & 41.1  & 47.5 \\\\\n\\hline\nMEB-Net~\\cite{conf\/eccv\/ZhaiYLJJ020}(ECCV'20)       & 20.6  & 44.1  & 58.3  & 64.3  & 21.3  & 45.6  & 59.5  & 65.6  \\\\\nUNRN~\\cite{conf\/aaai\/ZhengLZZZ21}(AAAI'21)          & 25.3  & 52.4  & 64.7  & 69.7  & 12.6  & 31.1  & 43.8  & 49.7 \\\\\nMMT(IBN-ResNet-50)~\\cite{conf\/iclr\/GeCL20}(ICLR'20) & 26.6  & 54.4  & 67.6  & 72.9  & 24.0  & 49.0  & 63.0  & 68.6 \\\\\n\\hline\nDAML (Ours)                                         & \\textbf{41.4}  & \\textbf{65.4}  & \\textbf{76.0}  & \\textbf{80.2}  & \\textbf{44.0}  & \\textbf{67.0}  & \\textbf{78.0}  & \\textbf{81.9}  \\\\\n\\hline\nSupervised (ViT-Base)                               & 54.1  & 76.6  & 87.5  & 90.8  & 54.1  & 76.6  & 87.5  & 90.8 \\\\\nSupervised (IBN-ResNet-50)                          & 49.9  & 79.2  & 88.2  & 91.3  & 49.9  & 79.2  & 88.2  & 91.3 \\\\\n\\hline\n\\end{tabular}\n}\n\n\\label{table_comparison_SOTA}\n\\end{table*}\n\n\\begin{table}[!t]\n\\centering\n\\setlength{\\tabcolsep}{1.0mm}\n{\n\\caption{\\small\nAblation study in terms of \\emph{m}AP (\\%) and CMC (\\%) on \\textbf{CUHK-SYSU (CS) $\\rightarrow$ Market-1501 (M)}.\n}\n\\label{table_ablation}\n\\resizebox{1.0\\linewidth}{!}{\n\\renewcommand{\\multirowsetup}{\\centering}\n\\begin{tabular}{l||c|c}\n\\hline\n\\multirow{2}{*}{\\ \\ \\ \\ \\ \\ \\ \\ \\ Method}  & \\multicolumn{2}{c}{CS $\\rightarrow$ M} \\\\\n\\cline{2-3} & \\ \\ \\emph{m}AP \\ \\  & \\ \\ R1 \\ \\\\\n\\hline\nIBN-ResNet-50(Directly)                                         & 38.8 & 63.7 \\\\\nViT-Base(Directly)                                              & 36.2 & 60.3 \\\\\n\\hline\nDAML w\/o $\\mathcal{L}_{Tsid} + \\mathcal{L}_{tri}(\\mathbf{s}^T)$ & 83.6 & 92.8 \\\\\nDAML w\/o $\\mathcal{L}_{id}$                                     & 83.3 & 92.3 \\\\\nDAML w\/o $\\mathcal{L}_{dom}$                                    & 83.6 & 92.5 \\\\\nDAML w\/o SCU                                                    & 81.0 & 91.0 \\\\\n\\hline\nDAML                                                            & 84.1 & 93.1 \\\\\n\\hline\nIBN-ResNet-50(Supervised)                                       & 83.0 & 94.1 \\\\\nViT-Base(Supervised)                                            & 82.3 & 93.2 \\\\\n\\hline\n\\end{tabular}\n}\n}\n\\end{table}\n\n\n\\subsection{Experiment Setting}\n\n\\textbf{Performance Metric:}\nAs a UDA task, we select one dataset as the source-domain dataset and another as the target-domain dataset.\nThe model is trained with the labeled source-domain training set and adapts the target domain through the unlabeled target-domain training set.\nThen the performance is evaluated according to the student network which work on the target-domain testing set.\nIn our experiments, following the standard metrics, we employ the cumulative matching characteristic (CMC) curve and the mean average precision (\\emph{m}AP) score.\nOur experiments report rank-1, rank-5, and rank-10 accuracy and mAP scores.\n\n\\textbf{Implementation Details:} \nIn the most common setting, we select IBN-ResNet-50~\\cite{conf\/eccv\/PanLST18} as the teacher network and ViT-Base~\\cite{conf\/iclr\/DosovitskiyB0WZ21} as the student network.\nThe batch size is set to $64$ for both source-domain and target-domain datasets. \nIn one batch, the sampler will select $16$ identities and $4$ images for each identity according to the ground-truth label or pseudo label for two domains.\nThe input image has a fixed size of $256 \\times 128$.\n\nIn the pre-training stage, we first train models $120$ epochs on the source-domain dataset.\nThe teacher CNN model is optimized by SGD with an initial learning rate of $1\\times{10}^{-2}$ and weight decay of $5\\times{10}^{-4}$ with a learning rate decays at $40^{th}$ and $70^{th}$ epoch.\nThe SGD optimizer is employed with a momentum of $0.9$ and the weight decay of $1\\times{10}^{-4}$ for student ViT.\nThe learning rate is set to $8\\times{10}^{-3}$, and the cosine schedule is applied.\nThe input images are augmented with random flip and randomly erase with $50\\%$ probability.\n\nIn the fine-tuning stage, we adopted half the learning rate of the previous stage.\nSpecifically, the learning rate is set to $5\\times10^{-3}$ for teacher CNN and $4\\times10^{-3}$ for student ViT.\nAnd the total number of training epochs is set to $60$.\nThe input images for two heterogeneous networks are randomly flipped and erased with $50\\%$ probability.\nWhen calculating the neighbors of an instance, the maximum acceptable distance $\\alpha$ is $0.6$.\nWe generate the pseudo labels by DBSCAN~\\cite{conf\/kdd\/EsterKSX96}.\nFor DBSCAN, we select $0.6$ as the maximum distance between neighbors and set the minimal number of neighbors to $2$ for \\textbf{CUHK-SYSU} and $4$ for others.\nThe hype-parameters $\\alpha$, $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$ are set to $0.5$, $0.1$, $0.7$ and $1.2$, respectively.\nAt the feature level, the margin $\\rho$ for triplet loss is set to $1.2$.\n\n\\subsection{Comparison with State-of-the-art Methods}\nSince Duke University terminated the \\textbf{DukeMTMC}~\\cite{conf\/eccv\/RistaniSZCT16} dataset, which has been widely used for evaluation of unsupervised domain adaptation person Re-ID task, the comparison becomes difficult.\nTo meet the moral and ethical requirements and provide a new baseline for comparison, we evaluate the performance of some representative works which have official open-source codes based on the \\textbf{CUHK-SYSU} dataset.\nAnd the results in \\textbf{Market-1501 $\\rightarrow$ MSMT17} setting is from the authors' reports.\nWe compare our DAML with state-of-the-art (SOTA) unsupervised domain adaptation person Re-ID approaches.\nMMT~\\cite{conf\/iclr\/GeCL20} applies two networks that have the same structure for learning from each other with both feature-level and logit-level constraints.\nMEB-Net~\\cite{conf\/eccv\/ZhaiYLJJ020} introduces three homogeneous networks, and the output of each network is considered comprehensively in the pseudo label generation.\nMoreover, UNRN~\\cite{conf\/aaai\/ZhengLZZZ21} designs a memory bank storing class centers from both source and target domains to mitigate the influence of noise labels.\nAs shown in Tab.~\\ref{table_comparison_SOTA}, we evaluate the performance in four different manners, \\textit{i.e.}, \\textbf{Market-1501 $\\rightarrow$ CUHK-SYSU}, \\textbf{CUHK-SYSU $\\rightarrow$ Market-1501}, \\textbf{Market-1501 $\\rightarrow$ MSMT17}, and \\textbf{CUHK-SYSU $\\rightarrow$ MSMT17}.\n\n\\textbf{Comparisons on large-scale datasets:}\nThe comparison results on \\textbf{Market-1501 $\\rightarrow$ MSMT17} and \\textbf{CUHK-SYSU $\\rightarrow$ MSMT17} are shown in the bottom of Table~\\ref{table_comparison_SOTA}.\nThe proposed DAML outperforms existing SOTAs by large margins.\nSpecifically, DAML achieves the Rank-1 accuracy of $65.4\\%$ and \\emph{m}AP of $41.4\\%$ in the \\textbf{Market-1501 $\\rightarrow$ MSMT17} setting, significantly improving the Rank-1 accuracy by $11.0\\%$ and \\emph{m}AP by $14.8\\%$ over the SOTA MMT.\nWhen compared to the SOTAs in \\textbf{CUHK-SYSU $\\rightarrow$ MSMT17} setting, the performance margin between our DAML and MMT is also significantly, e.g., the Rank-1 boost is $18.0\\%$, and the \\emph{m}AP boost is $20.0\\%$.\n\n\\begin{table*}[!t]\n\\centering\n\\setlength{\\tabcolsep}{1.0mm}\n{\n\\caption{\\small\nInfluence of different backbones in terms of \\emph{m}AP (\\%) and Rank-1 (\\%) on \\textbf{CUHK-SYSU (CS)} $\\rightarrow$ \\textbf{Market-1501 (M)}.\n}\n\\resizebox{1.0\\linewidth}{!}{\n\\renewcommand{\\multirowsetup}{\\centering}\n\\begin{tabular}{l|l|c|c||c|ccc}\n\\hline\n\\multirow{2}{*}{Method}  & \\multirow{2}{*}{Backbone} & \\multirow{2}{*}{\\makecell{Training \\\\Parameter}} & \\multirow{2}{*}{\\makecell{Testing \\\\Parameter}} & \\multicolumn{4}{c}{CS $\\rightarrow$ M} \\\\\n\\cline{5-8} & & & & \\ \\emph{m}AP \\ & \\ R1 \\ & \\ R5 \\ & \\ R10 \\ \\\\\n\\hline\nMMT     & IBN-ResNet-50 + IBN-ResNet-50             & 99.8M     & 24.9M     & 76.0 & 88.8 & 95.2 & 97.0 \\\\\nMMT     & ViT-Base + ViT-Base                       & 345.2M    & 86.3M     & 75.2 & 86.7 & 94.2 & 96.4 \\\\\nUNRN     & ResNet-50-NL                             & 77.1M     & 38.5M     & 70.9 & 86.7 & 92.8 & 94.5 \\\\\nUNRN     & ViT-Base                                 & 185.4M    & 86.3M     & 73.2 & 86.8 & 93.2 & 95.2 \\\\\n\\hline\nDAML & IBN-ResNet-50 + ViT-Base                     & 111.2M    & 86.3M     & 84.1 & 93.1 & 97.7 & 98.2 \\\\\n\\hline\n\\end{tabular}\n\\label{table_parameter}\n}\n\n}\n\\end{table*}\n\n\\textbf{Comparisons on small-scale dataset:}\nWe also evaluate DAML on two small-scale target-domain datasets settings, \\textbf{Market-1501 $\\rightarrow$ CUHK-SYSU} and \\textbf{CUHK-SYSU $\\rightarrow$ Market-1501}, as shown in the top of Table~\\ref{table_comparison_SOTA}.\nSimilar to the results on large-scale datasets, DAML consistently outperforms current SOTAs.\nSpecifically, we achieve Rank-1 accuracy of $86.2\\%$ and \\emph{m}AP of $84.3\\%$ in \\textbf{Market-1501 $\\rightarrow$ CUHK-SYSU} setting.\nCompared with the SOTA MEB-Net, the Rank-1 and \\emph{m}AP respectively improved by $3.0\\%$ and $3.2\n\\%$.\nMeanwhile Rank-1 accuracy of $93.1\\%$ and \\emph{m}AP of $84.1\\%$ are gained in \\textbf{CUHK-SYSU $\\rightarrow$ Market-1501} setting.\nIt improves the Rank-1 accuracy and \\emph{m}AP by $4.3\\%$ and $8.1\\%$ compared with the SOTA MMT.\nNote that the performance on \\textbf{Market-1501 $\\rightarrow$ CUHK-SYSU} setting is even worse than direct transfer. \nBecause there are only two samples per class in \\textbf{CUHK-SYSU} on average and it harms the pseudo label generation.\nWe will discuss this problem in section \\textbf{Samples Augment}.\n\nThe above results demonstrate the outstanding performance of DAML thanks to its ability to learn knowledge from different subspaces and selectively transfer knowledge between two heterogeneous networks for unsupervised domain adaptation person Re-ID.\n\n\\subsection{Ablation Study}\n\nIn this section, we conduct ablation experiments on \\textbf{CUHK-SYSU $\\rightarrow$ Market-1501} setting to assess the contribution of each component by separately removing them from DAML for training and evaluation.\n\nAs shown in Table~\\ref{table_ablation}, when removing $\\mathcal{L}_{Tsid} + \\mathcal{L}_{tri}(\\mathbf{s}^T)$, the Rank-1 accuracy drops by $0.3\\%$ and \\emph{m}AP drops by $0.5\\%$, since the reliable identity information is underutilized.\nIt illustrates that the ability to use the information in the source domain effectively is an essential factor in determining the model's performance.\nIt illustrates the essential to effectively use the information in the source domain\nWhen removing $\\mathcal{L}_{id}$, which helps student network to learn the identity knowledge from the teacher, the performance drops of Rank-1 and \\emph{m}AP are $0.8\\%$ and $0.8\\%$, respectively, compared with the full DAML.\nThe performance drops due to the ignorance of the knowledge from the different subspaces in the logit level.\nAnd the knowledge can still transfer to each other through the pseudo label generation.\nSimilarly, to validate the effectiveness of $\\mathcal{L}_{dom}$, we remove it from DAML.\nThe result also shows the margin of the Rank-1 accuracy by $0.6\\%$ and \\emph{m}AP by $0.5\\%$ to the complete DAML, which demonstrates that $\\mathcal{L}_{dom}$ effectively helps to make the teacher network predict samples in \nThe smooth classifier update (SCU) saves the knowledge learned in the last epoch.\nWhen it is removed, the Rank-1 accuracy drops by $2.1\\%$, and \\emph{m}AP drops by $3.1\\%$.\nThe results prove that learning the knowledge from different subspaces and taking advantage of correct identity information from the source domain are the two keys to solving UDA person Re-ID.\n\n\\subsection{Discussions}\n\n\\subsubsection{Influence of Backbone}\nTo meet the requirement of heterogeneous networks in the proposed DAML, we introduce the ViT-Base, which contains more trainable parameters as the backbone.\nTo clarify the source of performance growth, we repeat the experiments of MMT~\\cite{conf\/iclr\/GeCL20} and UNRN~\\cite{conf\/aaai\/ZhengLZZZ21} while replacing the backbone with ViT-Base.\nAs shown in Tab.~\\ref{table_parameter}, \"Backbone\" represents the construction to extract features in the testing stage.\nWhen the ViT-base replaces the backbone, the optimization process follows the setting in TransReID \\cite{conf\/iccv\/He0WW0021}.\nAs shown in table.~\\ref{table_parameter}, after replacing the backbone with ViT-Base, there is no significant change in results.\nLimited by the symmetrical design in mutual learning manner and the high similarity in the classifying ways of ViTs, the Rank-1 and \\emph{m}AP of MMT dropped $2.1\\%$ and $0.8\\%$, respectively.\nFor the UNRN method, which focuses on making pseudo labels reliable through the memory mechanism, the ViT brings $0.1\\%$ and $2.3\\%$ in Rank-1 and \\emph{m}AP with much more parameters.\nBased on the above experiments, we can conclude that the heterogeneous networks and the asymmetric learning strategy play a major role in the growth of performance.\n\n\\subsubsection{Heterogeneous Networks Analysis}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=2.0\\columnwidth]{Fig3_CamGrad.pdf}\n\\caption{\\small\nVisualization results of the models on \\textbf{Market-1501}.\nFor each line, we show an input image, the area considered by the pre-trained ViT, the pre-trained CNN, and the different combinations of teacher and student in turn.\n}\n\\label{fig3:CamGrad}\n\\end{figure*}\n\\begin{table}[!t]\n\\centering\n\\setlength{\\tabcolsep}{1.0mm}\n{\n\\caption{\\small\nAsymmetric distillation analysis in terms of \\emph{m}AP (\\%) and Rank-1 (\\%) on \\textbf{CUHK-SYSU (CS)} $\\rightarrow$ \\textbf{Market-1501 (M)}.\n}\n\\resizebox{1.0\\linewidth}{!}{\n\\renewcommand{\\multirowsetup}{\\centering}\n\\begin{tabular}{l|l||c|c}\n\\hline\n\\multicolumn{2}{c||}{Method} & \\multicolumn{2}{c}{CS $\\rightarrow$ M} \\\\\n\\hline\n\\ \\ Teacher & \\ \\ Student & \\ \\emph{m}AP \\ & \\ \\ R1 \\   \\\\\n\\hline\nIBN-ResNet-50   & ViT-Base          & 84.1 & 93.1 \\\\\nViT-Base        & ViT-Base          & 82.0 & 91.7 \\\\\nViT-Base        & IBN-ResNet-50     & 80.1 & 91.3 \\\\\nIBN-ResNet-50   & IBN-ResNet-50     & 79.7 & 91.0 \\\\\n\\hline\n\\end{tabular}\n\\label{table_asymmetric}\n}\n}\n\\end{table}\nOne of the keys to improving UDA person Re-ID is learning knowledge from the different subspaces.\nTo illustrate the effect of heterogeneous networks, we train the proposed DAML with different combinations of teacher and student.\nFrom Table.~\\ref{table_asymmetric}, we can figure out that the student with a heterogeneous teacher will achieve better performance.\nSpecifically, the performance of student ViT improved by $0.7\\%$ and $1.1\\%$ in Rank-1 and \\emph{m}AP with the asymmetric teacher. \nThe results in the student CNN is similar, Rank-1 and \\emph{m}AP are enhanced by $2.5\\%$ and $5.3\\%$.\nThese experimental results strongly prove the necessity of using two heterogeneous networks to work as the teacher and student.\nAnd the knowledge from different subspaces has the capacity to help the student to learn broader knowledge.\n\nWhen ViT is seen as the student, the benefit from the heterogeneous teacher is more evident than the improvement that CNN works as the student.\nThe heterogeneous teacher for ViT brings in $1.4\\%$ and $2.1\\%$ on Rank-1 and \\emph{m}AP.\nWhile it only improves Rank-1 and \\emph{m}AP in $0.3\\%$ and $0.4\\%$ for the student CNN.\nThis phenomenon can be ascribed to the difference in the range of receptive field of these two networks that the former can consider the relationship between any two areas, but the size of convolution kernels limits the latter.\nIt gives ViT has the ability to learn the pattern that CNN applied to classify identities but not vice versa.\n\n\\subsubsection{Visualization}\nThe proposed DAML can make the student learn the knowledge from different subspaces.\nTo further illustrate the effectiveness of DAML, which can selectively transfer the knowledge between two networks, we apply Score-CAM \\cite{conf\/cvpr\/WangWDYZDMH20} to visualize the pixel-wise attention areas on \\textbf{CUHK-SYSU $\\rightarrow$ Market-1501} setting.\nFig.~\\ref{fig3:CamGrad} visualizes individual attention patterns for the three people from the target domain, where each column represents the attention area of the pre-trained CNN, ViT, and the different combinations of teacher-student.\nFrom the first two columns, we can observe that the classifying patterns of CNN and ViT are different, which states the difference between their embedding spaces.\nWith these discrepancies, the heterogeneous networks can be improved by learning knowledge from heterogeneous networks.\nIn the last four columns, we can find that the networks mutual learning with heterogeneous networks can better consider the individual by the whole pedestrian while also taking into account many details that identify the persons more efficiently.\nOn the contrary, the recognition patterns of the networks that mutual learning with the same network have no significant change.\nThe visualization demonstrates the function of DAML in learning the knowledge from different subspaces improving the performance of the student.\n\n\n\n\\subsubsection{Samples Augment}\n\\begin{table}[!t]\n\\centering\n\\setlength{\\tabcolsep}{1.0mm}\n{\n\\caption{\\small\nInfluence of Sample Augment for clustering algorithm in terms of \\emph{m}AP (\\%) and CMC (\\%) on \\textbf{Market-1501 (M) $\\rightarrow$ CUHK-SYSU (CS)}.\n}\n\\label{table_augment}\n\\resizebox{1.0\\linewidth}{!}{\n\\renewcommand{\\multirowsetup}{\\centering}\n\\begin{tabular}{l|c||c|ccc}\n\\hline\n\\multirow{2}{*}{\\ Method} & \\multirow{2}{*}{\\makecell{Repeat \\\\Times}} & \\multicolumn{4}{c}{M $\\rightarrow$ CS} \\\\\n\\cline{3-6} & & \\emph{m}AP  & \\ R1 \\ & R5 & R10 \\\\\n\\hline\nDirectly Transfer                                           & - & 86.0 & 87.2 & 94.1 & 95.0 \\\\\n\\hline\nDAML                                                        & 0 & 84.3 & 86.2 & 92.6 & 94.6 \\\\\n\\hline\n\\multirow{2}{*}{+ Random Crop}                              & 1 & 89.3 & 90.6 & 95.5 & 96.9 \\\\\n                                                            & 2 & 89.1 & 90.4 & 95.4 & 96.6 \\\\\n\\hline\n\\multirow{2}{*}{+ Random Erase}                             & 1 & 88.3 & 90.0 & 95.0 & 96.3 \\\\\n                                                            & 2 & 89.2 & 90.6 & 95.5 & 96.6 \\\\\n\\hline\n\\multirow{2}{*}{\\makecell{+ Random Crop \\\\+ Random Erase}}  & 1 & 88.5 & 89.8 & 95.4 & 96.8 \\\\\n                                                            & 2 & 87.6 & 89.0 & 95.2 & 96.6 \\\\\n\\hline\nSupervised                                                  & - & 90.8 & 95.2 & 96.6 & 89.0 \\\\\n\\hline\n\\end{tabular}\n}\n\n}\n\\end{table}\nDue to the small number of samples for each class in \\textbf{CUHK-SYSU}, the performance of clustering algorithm is severely limited\nThe simplest and most direct way to address this problem is by augmenting the samples with random erase~\\cite{conf\/aaai\/Zhong0KL020} and random crop, which can generate new samples while keeping the original identity when extracting features for the clustering algorithm.\nAs shown in Tab.~\\ref{table_augment}, the performance with augmented data is better than the original. \nSpecifically, the Rank-1 and \\emph{m}AP are enhanced by $4.4\\%$ and $5.0\\%$ when applying the random crop method.\nSimilarly, the random erase improves Rank-1 and \\emph{m}AP by $4.4\\%$ and $4.9\\%$.\nThe above experiment results state the necessity of enough samples for each class in the clustering algorithm.\nNevertheless, applying both random crop and random erase is not as effective as applying only one. \nThe Rank-1 and \\emph{m}AP are only increased by $3.6\\%$ and $4.2\\%$.\nIt suggests that the excessive augment method may harm identity knowledge and reduce the benefits from the augmented samples.\n\nCompared to datasets collected for research, the number of identities and the number of samples in each identity are unknown in a real-world system.\nAnd this information cannot be counted on the raw data unless annotated on them.\nHowever, one of the advantages of unsupervised domain adaptation Re-ID is avoiding the annotation on the target domain, and it means the class-dependent super parameters are not available.\nBecause clustering algorithms elapse a long time to run on large datasets and take up most of the total training time, the super parameter selection experiments may be unacceptable for real-world systems.\nThus, a method without any clustering algorithm that still can mine identity knowledge from the target domain may make more sense for applying to the real world.\nOn the other hand, an efficient data augment method can restore the UDA methods based on clustering algorithm.\n\n\\section{Conclusion}\nIn this paper, we proposed the Dual-level Asymmetric Mutual Learning, termed DAML, to learn knowledge from a broader scope via asymmetric mutual learning with heterogeneous networks for unsupervised domain adaptation person Re-ID.\nOur method aims to learn the knowledge from various subspaces and transfer the identity knowledge from the source to the target domain.\nThe former can improve feature expressiveness while also rectifying potential faults during training.\nThe latter takes full advantage of the identity knowledge from the source domain to improve the performance in the target domain.\nSpecifically, DAML first generates the pseudo labels according to the features extracted by heterogeneous networks, which are more reliable due to the consideration of various subspaces.\nAnd the knowledge from two subspaces can be exchanged in a hard distillation manner in this process.\nWith the smooth classifier update, the classifiers can maintain the knowledge from the last epoch.\nThen, the teacher will train on both source-domain and target-domain datasets to utilize the ground-truth label and transfer the knowledge to the target domain with the domain knowledge from the student.\nTo better adapt to the target domain, the student only trained on the target-domain dataset and benefited from the guidance from the teacher, which had learned the source-domain knowledge.\nExperiments on four different experiment settings prove essential to learn the knowledge from various subspaces and demonstrate the effectiveness of the proposed DAML for unsupervised domain adaptation person Re-ID.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Riemann-squared modified gravities}\n Let me not to list all papers which mention modified gravity\ntheories with Lagrangian composed of the three invariants\nquadratic in the Riemannian curvature.\n To my thinking, all Riemann-squared-gravities (as well\nas $f(R)-$gravities, $f(R)\\neq R$) are not appropriate.\n\n So,\n$(a+bR+R^2\/2)-$gravity ($R$ is the Ricci scalar) leads to\nincompatible PDE system (the trace part,\n$\\mathbf{E}_\\mu{}^\\mu=0$, can be added or subtracted freely):\n\\[\n\\mathbf{E}_{\\mu\\nu}=R_{;\\mu;\\nu} -\nR_{\\mu\\nu}(b+R)+g_{\\mu\\nu}(...R..R^2)=0;\n\\]\nsome people prefer to move the principal derivatives (4-th\norder) to RHS, perhaps trying to hide them in the\nenergy-momentum tensor; however it is not good way to deal with\nPDEs. The next combination of prolonged equations,\n\\[ \\mathbf{E}_{\\mu\\nu;\\lambda} - \\mathbf{E}_{\\mu\\lambda;\\nu } =0, \\]\nafter cancellation of principal derivatives (5-th order), gives\n  new 3-d order equations which are irregular in the second jets:\n the term\n \\[ R_{;\\ve}R^{\\ve}{}_{\\mu\\nu\\lambda} \\]\n can not be cancelled by other terms which contain only Ricci tensor\n and scalar.\nThe rank of the new subsystem depends on the second derivatives,\n $g_{\\mu\\nu,\\lambda\\rho}$\n(for the definition of PDE regularity  see \\cite{eima}).\n\n As a rule, researchers of modified gravities concentrate their\n efforts on most symmetrical problems including cosmological\n solutions with the spherical symmetry. In this case the irregularity\n of the above system is safely masked: the new subsystem becomes just\n identity due to skew-symmetry of its two indices.\n\nAlso irregular in second jets are equations of Gauss-Bonnet (or\nLovelock) gravity with extra dimension(s).\n\nThe most interesting case, $R_{\\mu\\nu} G^{\\mu\\nu}-$gravity\n(the Ricci tensor is contracted with the Einstein tensor), gives the\nfollowing compatible system:\n \\begin{equation}}            \\def\\ee{\\end{equation} \\label{rg}  -\\mathbf{D}_{\\mu\\nu}=\n G_{\\mu\\nu;\\lambda}{}^{;\\lambda}+\n G^{\\epsilon \\tau} (2R_{\\epsilon\n\\mu\\tau\\nu } - \\fr12g_{\\mu\\nu}R_{\\epsilon \\tau }) =0;\n \\ \\, \\mathbf{D}_{\\mu\\nu;\\lambda}g^{\\nu\\lambda}\\equiv 0\\,.\n\\ee\n In linear approximation, there are simple evolution\nequations for Ricci tensor and scalar (total $D(D-1)\/2\\,$\npolarizations---including one scalar polarization):\n\\[ \\square R=0, \\ \\square R_{\\mu\\nu}=0. \\]\nUsing the Bianchi identity,\n $R_{\\mu\\nu[\\lambda\\epsilon;\\tau]}\\equiv 0$,\n  its prolongation and contractions,\n\\[ R_{\\mu\\nu[\\lambda\\epsilon;\\tau];\\rho} g^{\\tau\\rho}\\equiv0, \\\n R_{\\mu\\nu[\\lambda\\epsilon;\\tau]}g^{\\mu\\tau}\\equiv0, \\]\none can write the evolution equation\n for the Riemann (or Weyl) tensor\n(in linear approximation again):\n \\begin{equation}}            \\def\\ee{\\end{equation} \\label{riem} \\square\n R_{\\mu\\epsilon\\nu\\tau}=R_{\\mu\\nu,\\epsilon\\tau}-\n R_{\\mu\\tau,\\nu\\epsilon}+R_{\\epsilon\\tau,\\mu\\nu}\n -R_{\\epsilon\\nu,\\mu\\tau}.\n \\ee\n This equation is more complex: it has the source term (in its\n RHS)\n composed from the Ricci tensor. As a result in general case, when\n the Ricci-polarizations do not vanish, the polarizations\n related to the Weyl tensor [and responsible for gravity, tidal\n forces; their number is $D(D-3)\/2$] should grow linearly with\n time,\n \\[a(t) = (c_0 + c_1 t)\\exp(- i \\omega t),\\]\n  while linear\n approximation is valid.\n\n\nThis means that the regime of weak gravity  is linearly\nunstable, as well as the trivial solution itself\n (i.e.,  \\emph{nothing is unreal} in this theory). Hence\nthe theory is physically irrelevant---we still live in very weak\ngravity. (Note that in General Relativity, when the Ricci tensor is\nexpressed\n through  the energy-momentum tensor\n [which does not expand into plane waves---with dispersion of light in\n vacuum],\n  the equation (\\ref{riem}) defines\n  radiation of gravitation waves.)\n\nThe linear instability here does not contradict the correctness\nof Cauchy problem; the modern compatibility theory (e.g., the\nPommaret's book \\cite{eima}) gives easy answers about the Cauchy\nproblem, number of polarizations, and so on (especially easy for\nanalytical PDE systems).\n\nAnd the last remark. Let ${\\cal L}=\\sqrt{-g}L$ is a homogeneous\nLagrangian density of order $p$ in metrics, that is\n\\[  {\\cal L}(\\k g_{\\mu\\nu})=\\k^p {\\cal L}(g_{\\mu\\nu})\\, . \\]\nThe result of variation, the symmetric tensor\n \\[ \\mathbf{D}^{\\mu\\nu} = \\frac1 {\\sqrt{-g}}\n \\frac{\\delta}          \\def\\eps{{\\epsilon}}     \\def\\ve{\\varepsilon {\\cal L}}{\\delta}          \\def\\eps{{\\epsilon}}     \\def\\ve{\\varepsilon g_{\\mu\\nu}}, \\\n \\mathbf{D}^{\\mu\\nu}{}_{;\\nu}\\equiv0, \\]\n has the next relation to its Lagrangian (the trace is\n proportional to the Lagrangian scalar up to a covariant divergence):\n \\[ \\mathbf{D}^{\\mu\\nu} g_{\\mu\\nu} = p\\, L + A^{\\mu}{}_{;\\mu}. \\]\n\nIf $ \\mathbf{D}_{\\mu\\nu}$  (with\n$\\mathbf{D}^{\\mu\\nu}{}_{;\\nu}\\equiv0$) is found just by using the\nBianchi identity, this relation reveals the corresponding\nLagrangian, see {eg.}\\\n equation (\\ref{rg}).\n\n\\section{Frame field theory}\nThe theory of frame field, $h^a{}_\\mu$, also known as Absolute\nParallelism (AP), has large symmetry group\nwhich includes both global symmetries of Special Relativity (this\n defines the signature) and local symmetries, the\npseudogroup {\\it Diff}$(D)$, of General Relativity.\n\n AP is more appropriate as a\nmodified gravity, or just a good theory with topological charges and\nquasi-charges (their phenomenology, at some\nconditions and to a certain extent, can look\n like a quantum field theory) \\cite{tc}. In this\ncase,  the Ricci tensor has very specific form (due to field\nequations; linear approximation): \n\\[ R_{\\mu\\nu}\\propto \\Phi_{(\\mu,\\nu)}, \\, \\\n \\\n \\Phi_\\mu=h_a{}^\\nu (h^a{}_{\\nu,\\mu}-h^a{}_{\\mu,\\nu})\n\\ \\mbox{-- trace of torsion};\n\\]\nthis form does not cause the Weyl polarizations growth, see\n(1); so the  weak gravity regime (but not the trivial\nsolution!$\\cdot\\!$) is stable.\n\n \\subsection{{Co- and contra-singularities\n and unique equation }}\n\n\nThere is one unique equation of AP (non-Lagrangian, with the\nunique $D$) which solutions are free of arising singularities.\n The formal integrability test\n \\cite{pommaret} can be extended to the cases of degeneration of\neither co-frame matrix, $h^a{}_\\mu$ (co-singularities),  or\ncontra-variant frame (or density of some weight), serving as a\nlocal and covariant test for singularities. This test singles out\nthe next equation (and $D$=5  \\cite{tc};\n$\\eta_{ab}=\\mbox{diag}(-1,1,\\ldots,1)$, then $h=\\det\nh^a{}_\\mu=\\sqrt{-g}$):\n \\begin{equation} \\label{ue}\n {\\bf E}_{a\\mu}=L_{a\\mu\\nu;\\nu}- \\fr13 (f_{a\\mu}\n +L_{a\\mu \\nu }\\Phi _{\\nu })=0\\, ;\n\\end{equation}\n here \\qquad \\qquad  \\qquad $ L_{a\\mu \\nu }=L_{a[\\mu \\nu]}=\n\\Lambda_{a\\mu \\nu }-S_{a\\mu \\nu }-\\fr23 h_{a[\\mu }\\Phi_{\\nu]},$\n \\[\n\\Lambda_{a\\mu \\nu }=2h_{a[\\mu,\\nu]}, \\ S_{\\mu \\nu \\l}=3\\L_{[\\mu\n\\nu \\l]}, \\ \\Phi_\\mu=\\L_{aa \\mu}, \\ f_{\\mu\\nu}=2\\Phi_{[\\mu,\\nu]}\n. \\]\n Coma \",\" and semicolon \";\" denote partial derivative and\ncovariant differentiation with symmetric Levi-Civita connection,\nrespectively. One should retain the identities:\n \\begin{equation}}            \\def\\ee{\\end{equation}\\label{iden}\n \\Lambda_{a[\\mu\\nu;\\lambda]} \\equiv 0\\,,\n  \\ \\  h_{a\\l}\\Lambda_{abc;\\l}\\equiv f_{cb}\\,\n  (= f_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}h_c^\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi h_b^\\n), \\ f_{[\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n;\\l]}\\equiv0\n  .\\ee\n Equation ${\\bf E}_{a\\mu;\\mu}=0$ gives\n `Maxwell-like equation' (I omit\n $\\eta_{ab}$ and $g^{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}=h_a^\\mu h_a^\\nu$\n in contractions):\n\\begin{equation}\\label{max}{\n(f_{a\\mu}\n +L_{a\\mu \\nu }\\Phi _{\\nu })_{;\\mu}=0, \\mbox{ or \\ }\n f_{\\mu\\nu;\\nu}=(S_{\\mu \\nu\\l }\\Phi _{\\l })_{;\\n} \\ \\\n(= -\\fr1 2 S_{\\mu \\nu\\l }f_{\\n\\l}, \\mbox{ see below}) \\, .}\n\\end{equation}\nReally (\\ref{max})\nfollows from the symmetric part, because\nskewsymmetric one gives the identity; the\ntrace part\n becomes irregular  (principal\n derivatives vanish) if $D=4$ (forbidden $D$):\n \\[{\n2{\\bf E}_{[\\nu\\mu]}=S_{\\mu\\nu\\l;\\l}=0, \\ {\\bf\nE}_{[\\nu\\mu];\\nu}\\equiv 0; \\ \\ {\\bf E}_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\m}={\\bf\nE}_{a\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi}h_b^\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\eta^{ab} =\\fr{4-D}3 \\Phi_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi;\\mu}+ (\\L^2)=0.\n } \\]\nSystem (\\ref{ue}) remains compatible under adding\n$f_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}=0$, see (\\ref{max});  this is not the case for other\ncovariants, $S, \\Phi$, or Riemannian curvature, which relates to\n$\\L$ as usually:\n \\[ R_{a\\mu\\nu\\lambda}= 2h_{a\\mu;[\\nu;\\lambda]}; \\\nh_{a\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi}h_{a\\nu;\\l}=\\fr12 S_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n\\l}-\\L_{\\l\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}.\\]\n\nGR is a special case of AP. Using 3-minors (ie., co-rank 3) of\nco-metric,\n  \\[  [\\mu \\nu ,\\varepsilon \\tau ,\\alpha \\beta ] \\equiv\n \\partial^3 (-g)\/(\\partial g_{\\mu \\nu }\n\\partial g_{\\varepsilon \\tau }\\partial g_{\\alpha \\beta }) ,\\,\\]\nand their skew-symmetry, one can write the vacuum GR equation\n   as\nfollows:  $\\, 2(-g)G^{\\mu \\nu }=$\n \\begin{equation}}            \\def\\ee{\\end{equation} \\label{gr}{\n [\\mu \\nu ,\\varepsilon \\tau  ]_{,\\ve\\tau}+ (g'^2)=\n [\\mu \\nu ,\\varepsilon \\tau , \\alpha\n\\beta ](g_{\\alpha \\beta ,\\varepsilon \\tau }+ g^{\\rho \\phi }\\Gamma\n_{\\rho ,\\varepsilon \\tau } \\Gamma _{\\phi ,\\alpha \\beta })=\n [\\mu \\nu ,\\varepsilon \\tau , \\alpha\n\\beta ]R_{\\alpha \\beta \\varepsilon \\tau } =0.}\\ee\n Similarly,  all AP equations  can be rewritten\n that 2-minors of co-frame,\n\\[{ \\pmatrix{\\mu\\; \\nu\n\\vspace{-2.mm}\\cr a\\;b} =\\frac{\\partial^2 h}{\\partial h^{a}{}_{\\mu}\n\\partial h^{b}{}_{\\nu}}\n= 2!\\, h h^{\\mu}_{[a} h^{\\nu}_{b]} \\  \\ \n\\mbox{i.e. } \\\n [\\mu_1 \\nu_1 ,\\ldots, \\mu_k \\nu_k  ]=\\fr1{k!}\n \\pmatrix{\\mu_1\\,\\cdots\\, \\mu_k\n\\vspace{-2.mm}\\cr a_1\\,\\cdots\\, \\alpha}          \\def\\b{\\beta}            \\def\\g{\\gamma_k}\\!\\! \\pmatrix{\\nu_1\\,\\cdots\\,\n\\nu_k\n\\vspace{-2.mm}\\cr a_1\\,\\cdots\\, \\alpha}          \\def\\b{\\beta}            \\def\\g{\\gamma_k}  \\\n),} \\] completely define the coefficients at the principal\nderivatives. The simplest compatible equation\n(see Einstein--Mayer classification of compatible\nequations in 4$D$ AP \\cite{eima}),\n \\begin{equation}}            \\def\\ee{\\end{equation} \\label{estar}  {\\bf\nE^*}_{a\\mu}=\\L_{a\\mu\\nu;\\nu}=0, \\ {\\bf\nE^*}_{a\\mu;\\mu}\\equiv0 \\ee\n gives\n \\[{ h^{2}{\\bf E^*}_{a}{}^{\\mu }=\n(h_{a\\alpha ,\\beta \\nu }-h_{a\\beta,\\alpha  \\nu } )(-g)g^{\\alpha\n\\mu}g^{\\beta \\nu}+(h^\\prime{}^{2})\n= h_{a\\alpha ,\\beta \\nu }[\\vspace{1mm}\\alpha \\mu ,\\beta \\nu\n\\vspace{-1mm}] +(h^\\prime{}^{2})\\ .}\\]\n Like determinant,\n$k$-minors ($k\\leq D$)\n are multi-linear expressions in\n elements of $h^a{}_\\mu$-matrix, and some minors do not\nvanish when rank$\\,h^a{}_\\mu=D-1$.\n\n For any AP equation [including\n Eqs.~(\\ref{gr}) and (\\ref{estar})],\n with the \\emph{unique exception}, Eq.~(\\ref{ue}),\n  (where only skew-symmetric\n part participates in identity and can be written with 2- and 3-minors,\n while symmetric part needs 1-minors vanishing too\n rapidly), the\n regularity of principal terms survives (and\n symbol $G_2$ keeps involutive)\n  if\n rank$\\,g_{\\mu\\nu}=D-1$.\n\nThis observation is important and relevant to the problem of\nsingularities; it means seemingly that the unique equation\n(\\ref{ue}) does not suffer of nascent co-singularities in\nsolutions of general position.\n\nThe other case is contra-singularities \\cite{tc} relating to\ndegeneration of contra-variant density of some weight:\n \\begin{equation}}            \\def\\ee{\\end{equation}\n\\label{dens} { H_a{}^\\mu= h^{1\/D_*} h_a{}^\\mu;\n H=\\det H^a{}_\\mu, \\\nh_a{}^\\mu= H^{1\/(D-D_*)} H_a{}^\\mu\\, .} \\ee\n Here $D_*$ depends on\nequation: $D_*=2$ for GR, $D_*=\\infty$ for Eq.~(\\ref{estar}), and\n$D_*=4$ for the unique equation (which can be written 3-linearly\nin $H_a{}^\\mu$ and its derivatives \\cite{tc}).  %\n\nIf integer, $D_*$ is the forbidden spacetime dimension. The nearest\npossible  $D=5$ is of special interest: in this case minor\n$H^{-1} H^a{}_\\mu$ simply coincides with $h^a{}_\\mu$; that is,\ncontra-singularity simultaneously implies co-singularity (of high\nco-rank), but that is impossible! The possible interpretation of\nthis observation is:\n for the unique equation,\n contra-singularities are impossible if $D=5$\n (perhaps due to some specifics of\n\\emph{Diff}-orbits on $H_a{}^\\mu$-space).\n\\ %\nThis leaves no room for changes in the theory (assuming\n Nature does not like singularities).\n\n \\subsection{{Tensor $T_{\\mu\\nu}$ and post-Newtonian effects\n (Pauli's questions to AP)}}\n One might rearrange ${\\bf E}_{(\\mu\\nu)}=0$ picking out\n (into LHS) the Einstein tensor,\n $G_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}=R_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}-\\fr12g_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}R$, but the rest terms are not\n proper energy-momentum tensor: they contain linear terms\n $\\Phi_{(\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi;\\nu)}$ (no positive energy (!); instead one more\n presentation of `Maxwell equation' (\\ref{max}) is\n  possible---as divergence of symmetrical tensor).\n\n However, the prolonged equation\n${\\bf E}_{(\\mu\\nu);\\l;\\l}=0$ can be written as $RG$-gravity:\n \\begin{equation}}            \\def\\ee{\\end{equation} \\label{tmunu}{\n G_{\\mu \\nu\n;\\lambda ;\\lambda }+ G_{\\epsilon \\tau} (2R_{\\epsilon \\mu \\tau \\nu\n} - \\fr12g_{\\mu \\nu }R_{\\epsilon \\tau }) =T_{\\mu\\nu} (\\Lambda\n'^{2},\\cdots), \\ T_{\\mu\\nu;\\nu}=0; }\\ee\n up to\nquadratic terms, \\ \\ ${ T_{\\mu\\nu}=\n\\fr29(\\fr14g_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}f^2-f_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\l}f_{\\n\\l})\n+A_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\eps\\n\\tau}(\\L^2){}_{,\\eps\\tau};\n } $\\\ntensor $A$ has symmetries of Riemann tensor, so the term $A''$\nadds nothing to momentum and angular momentum.\n\nIt is worth noting that:\n\\\\\n(a) the theory does not match GR, but reveals $RG$-gravity\n(sure, (\\ref{tmunu}) does not contain all);\n\\\\\n(b) only $f$-component (three transverse polarizations in $D=5$)\ncarries $D$-momentum and angular momentum (`powerful' waves); other\n12 polarizations are `powerless', or `weightless'. This is a very\nunusual feature---impossible\n in Lagrangian tradition; how to quantize ?\n\n\\\\\n(c) $f$-component feels only metric and $S$-field,\n see (\\ref{max}), but $S$\nhas effect only on polarization of $f$: $S_{[\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n\\l]}$ does not\nenter eikonal equation, and $f$ moves along usual Riemannian\ngeodesic;\ntrace $T_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\m}=\\fr1{18}f_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}f_{\\mu}             \\def\\n{\\nu}              \\def\\p{\\pi\\n}$ can be non-zero\nif  $f^2\\neq0$;\n\\\\\n(e) it should be stressed that  $f$-component is\nnot usual (quantum) EM-field---just important covariant\nresponsible for energy-momentum (there is\nno gradient invariance for $f$; phenomenological quantum fields\nshould  account somehow for topological (quasi)charges \\cite{tc}).\n\nAnother strange feature is the instability of trivial solution:\n some `powerless' polarizations grow linearly with time\n in presence of\n `powerful' $f$-polarizations. Really,  the linearized\n Eq.~(\\ref{ue}) and identity (\\ref{iden}) give\n (following equations should be understood as linearized):\n \\[ {\n \\Phi_{a,a}=0 \\ (D\\neq 4), \\\n 3\\Lambda_{abd,d}= \\Phi_{a,b}-2\\Phi_{b,a},\n \\ \\Lambda_{a[bc,d],d}\\equiv0\\, \\ \\Rightarrow \\\n 3\\Lambda_{abc,dd}=-2 f_{bc,a}\\, . }\\]\n The last D`Alembert equation has the \\emph{source} in its RHS.\n  Some components of $\\Lambda$\n (most symmetrical irreducible parts)\n do not grow (as well as curvature), because\n (linearized equations)\n \\[{ S_{abc,dd}=0, \\ \\Phi_{a,dd}=0, \\ \\,\n   f_{ab,dd}=0, \\ R_{abcd,ee}=0. }\\]\n  However the least symmetrical $\\L$-components\n  do go up with time (three growing but powerless\n   polarizations),\n  if  the ponderable waves (three $f$-polarizations)\n  do not vanish; this should be the case for\n  solutions of general position. Again, nothing is unreal!\n\n\n\\subsection{Expanding\n O$_4$-symmetrical solutions and cosmology}\nThe unique symmetry of AP equations gives scope for symmetrical\n solutions. In contrast to GR,  Eqn.~(\\ref{ue}) has\nnon-stationary spherically\n symmetric solutions.\nThe $O_4$-symmetric field  can be generally written \\cite{sn} as\n\\begin{equation}  \\label{spsy}{\n h^{a}{}_{\\mu }(t,x^i)=\n \\pmatrix{a&bn_{i}  \\cr\n cn_{i} & en_{i}n_{j}+d\\Delta _{ij} };\n\\ \\ i,j=(1,2,3,4), \\ n_i=\\frac{x^i}{r}. }\n\\end{equation}\n Here $a,\\ldots,e$ are functions of time, $t=x^0$, and radius\n $r$, $\\Delta_{ij}=\\delta_{ij}-n_i n_j, \\ r^2=x^i x^i$.\n As functions of radius, $b,c$ are odd,\n  while the others are even;\n  other boundary conditions: $e=d$ at $r=0$,\n and $h^a{}_\\mu\\to \\delta^{\\,a}_\\mu$ as $r\\to \\infty$.\nPlacing in (\\ref{spsy}) $b=0, e=d$ (the other interesting choice\nis $b{=}c{=}0$)\n and making integrations one can arrive to the next system\n (resembling dynamics of Chaplygin gas; dot and prime\n denote derivation on time and radius, resp.)\n\\begin{equation}\\label{gas}{\nA^{\\cdot}=AB^\\prime -BA^\\prime +\\frac{\\,3}r AB, \\  B^{\\,\\cdot\n}=AA^\\prime -BB^\\prime -\\frac{\\,2}r B^{2}, \\mbox{where }A=\n \\frac {\\,a} e=e^{1\/2},\\ B=-\\frac c {\\,e}\\, .}\n\\end{equation}\nThis system has non-stationary solutions, and a single-wave\nsolution (of proper `sign') might serve as a suitable (stable)\ncosmological expanding background.\n  The condition $f_{\\mu\\nu}{=}0$ is a must for solutions with such\n a high symmetry (as well as\n $S_{\\mu\\nu\\l}{=}0$); so, these $O_4$-solutions\n carry no energy, weight nothing---some lack of \\emph{gravity} !\n\n More realistic cosmological model might look like a single\n $O_4$-wave\n (or a sequence of waves) moving along the radius and being\n filled with chaos, or  stochastic waves, both\n powerful (\\emph{weak}, $\\Delta h\\ll1$) and\n  powerless ($\\Delta h<1$, but intense enough that\n  to give non-linear\n  fluctuations with $\\Delta h\\sim1$).\n  The development and examination of stability\n of this model is an interesting problem.\n The inhomogeneity of metric  in giant $O_4$-wave\n can serve as\n a time-dependent `shallow dielectric guide' for that weak\n $f-$waves. The ponderable  waves (which slow down\nthe large wave) should have wave-vectors almost tangent to the\n$S^3$-sphere of wave-front  to be trapped inside this\nshallow wave-guide; the imponderable waves can grow up, and\npartly escape from the wave-guide,\n and their wave-vectors\ncan be less tangent to the $S^3$-sphere.\nThe waveguide thickness can be small for an `observer' in the center\nof $O_4$-symmetry, but in co-moving coordinates it can be very\nlarge (still $\\ll R$).  This model can explain well the SNe1a redshift\n data \\cite{sn}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\Gamma=\\operatorname{PSL}(2,\\Z)$ be the modular group and $\\Hb$ the upper half-plane.\nWe consider the (completed) \nnonholomorphic Eisenstein series $E^\\ast(z,s)$ for the modular\ngroup which is given for $z\\in\\Hb$ and $\\Re(s)>1$ by\n\\begin{equation}\nE^\\ast(z,s):=\\pi^{-s}\\Gamma(s)\\left(\\frac{1}{2}\\sum_{(m,n)\\in\\Z^2\\setminus (0,0)}\n\\frac{y^s}{|cz+d|^{2s}}\\right).\n\\end{equation}\nIt is well known that for fixed $z$, $E^\\ast(z,s)$ admits a meromorphic \ncontinuation to the whole $s$-plane, and satisfies the functional equation\n\\[E^\\ast(z,s)=E^\\ast(z,1-s).\\]\nIts only singularities are simple poles at $s=1$ and $s=0$. As a function of\n$z$, it is invariant under $\\Gamma$\n\\[E^\\ast(\\gamma z,s)=E^\\ast(z,s), \\quad \\gamma\\in\\Gamma.\\]\nIn particular $E^\\ast(z,s)$ is invariant under $z\\mapsto z+1$ and so it has a\nFourier expansion\n\\[E^\\ast(x+iy,s)=\\sum_{n\\in\\Z}a_n(y,s)e^{2\\pi ix}.\\]\nThe zeroth Fourier coefficient $\\varphi_0(y,s)$ is given by\n\\[\\varphi_0(y,s)=\\Lambda(2s)y^s+\\Lambda(2s-1)y^{1-s},\\]\nwhere $\\Lambda(s)$ is the completed zeta function\n\\[\\Lambda(s)=\\pi^{-s\/2}\\Gamma\\left(\\frac{s}{2}\\right)\\zeta(s).\\]\nLet $a>0$. The zeros of  $\\varphi_0(a,s)$ \nhave been studied by various authors. Hejhal\n\\cite[Proposition 5.3(f)]{He} proved that for all $a\\ge 1$, the complex zeros\nof $\\varphi_0(a,s)$ are on the critical line $\\Re(s)=1\/2$. Lagarias and Suzuki \n\\cite{LS} reproved this result and also determined the occurrence of real\nzeros. Ki \\cite{Ki} proved that all complex zeros are simple.\nPutting these results together, we have following theorem.\n\\begin{theo}\\label{th0.1}\nFor each $a\\ge 1$ all complex zeros of $\\varphi_0(y,s)$ are simple and lie on\nthe \ncritical line $\\Re(s)=1\/2$. Moreover there is a critical value\n$a^\\ast=4\\pi e^{-\\gamma}=7.055...$\nsuch that the following holds:\n\\begin{itemize}\n\\item[1)] For $1\\le a\\le a^\\ast$ all zeros are on the critical line.\n\\item[2)] For $a>a^\\ast$ there are exactly two zeros off the critical\n  line. These \nare real simple zeros $\\rho_a, 1-\\rho_a$ with $\\rho_a\\in(1\/2,1)$. The zero\n$\\rho_a$ is a nondecreasing function of $a$ and $\\lim_{a\\to\\infty}\\rho_a=1$.\n\\end{itemize}\n\\end{theo}\nThe first aim of this paper is to point out that there is actually a\nspectral interpretation of the zeros of $\\varphi_0(a,s)$, which  \ngives another proof of this theorem. For $a> 0$ let $\\Delta_a$ be the\ncut-off Laplacian $\\Delta_a$ introduced by Lax and Phillips \\cite{LP}.\nIt acts in the subspace $\\H_a\\subset L^2(\\Gamma\\ba \\Hb)$  of all\n$f\\in L^2(\\Gamma\\ba \\Hb)$ satisfying\n$\\int_0^1f(x+iy)\\,dx=0 $ for almost all $y\\ge a$.\nThe cut-off Laplacian $\\Delta_a$ is a nonnegative self-adjoint operator with\npure point spectrum. The spectrum has been studied by Colin de Verdiere \n\\cite{CV}. Let \n\\begin{equation}\\label{0.2}\nc(s)=\\frac{\\Lambda(2s-1)}{\\Lambda(2s)},\\quad s\\in\\C.\n\\end{equation}\nThe following theorem is an immediate consequence of \\cite[Th\\'eor\\`eme 5]{CV}.\n\n\\begin{theo}\\label{th0.2}\nFor every $a>0$, the spectrum of $\\Delta_a$ is the union of the cuspidal\neigenvalues \n$0<\\lambda_1\\le\\lambda_2\\le\\cdots\\to\\infty$ of $\\Delta$ and a sequence of\neigenvalues\n$$0<\\mu_0(a)< \\mu_1(a)<\\cdots$$\nwith the following properties:\n\\begin{enumerate}\n\\item[1)] Each eigenvalue $\\mu_j(a)$ is a decreasing function of $a$.\n\\item[2)] If $a\\ge1$, each eigenvalue $\\mu_j(a)$ has multiplicity 1. Moreover\n $\\lim_{a\\to\\infty}\\mu_0(a)=0$ and if $j\\ge 1$, then $\\mu_j(a)\\ge 1\/4$ and\n $\\lim_{a\\to\\infty}\\lambda_j(a)= 1\/4$. \n\\item[3)] Let $a\\ge1$. Then the  map $s\\mapsto s(1-s)$ is a bijection between\n  the zeros $\\rho\\not=1\/2$ of $\\varphi_0(a,s)$ and the eigenvalues\n  $\\mu_j(a)\\not=1\/4$ of $\\Delta_a$.\n\\item[4)] $1\/2$ is a zero of $\\varphi_0(a,s)$ for all $a>0$.  \nThere is $j$ with $\\mu_j(a)=1\/4$ if and only if $c^\\prime(1\/2)=-2\\log a$.\n\\end{enumerate}\n\\end{theo}\nThis theorem implies that for $a\\ge 1$  there is at most one\n zero of $\\varphi_0(a,s)$ which is off the\nline $\\Re(s)=1\/2$. The simplicity of \nthe zeros follows from the consideration of the corresponding eigenfunctions\nof $\\Delta_a$. \n\nThe main purpose of this article is to extend these results to the constant\nterm of other Eisenstein series. In the  \npresent paper we will consider the Eisenstein series attached to\n$\\operatorname{PSL}(2,\\cO_K)$, \nwhere $\\cO_K$ is the ring of integers of\nan imaginary quadratic field $K=\\Q(\\sqrt{-D})$ of class number one, $D$ being\na square free positive integer. These are exactly the fields\n$\\Q(\\sqrt{-D})$ with $D=1,2,3,7,11,19,43,67,163$ \\cite{St}.  Let $d_K$ be the\ndiscriminant of $K$ and let $\\zeta_K(s)$ be the Dedekind zeta function of\n$K$. Let\n\\begin{equation}\n\\Lambda_K(s)=\\left(\\frac{\\sqrt{|d_K|}}{2\\pi}\\right)^s\\Gamma(s)\\zeta_K(s)\n\\end{equation}\nbe the completed zeta function. \nThen $\\Lambda_K(s)$ satisfies the functional equation \n$\\Lambda_K(s)=\\Lambda_K(1-s)$.\nFor $a>0$ let \n\\begin{equation}\n\\varphi_{K}(a,s):=a^s\\Lambda_K(s)+a^{2-s}\\Lambda_K(s-1).\n\\end{equation}\nNote that $\\varphi_{K}(a,s)$ is a Dirichlet series which satisfies the\nfunctional  equation $\\varphi_{K}(a,s)=\\varphi_{K}(a,2-s)$.\nLet $\\xi_K(s)=s(s-1)\\Lambda_K(s)$. Then $\\xi_K(s)$ is entire. Put\n\\[a^\\ast_K:=\\exp\\left(1+\\frac{\\xi_K^\\prime(1)}{\\xi_K(1)}\\right). \\]\nThen our main result is the following theorem.\n\\begin{theo}\\label{th0.3}\nLet $K=\\Q(\\sqrt{-D})$ be an imaginary quadratic field of class number one. \nThen for each $a\\ge 1$ all complex zeros of \n$\\varphi_K(a,s)$ are simple and lie on the line $\\Re(s)=1$. Moreover \n\\begin{enumerate}\n\\item[1)] For $a>\\max\\{a^\\ast_K,1\\}$ there are exactly two zeros off the \ncritical line.\nThese are simple zeros $\\rho_a,2-\\rho_a$ with $\\rho_a\\in(1,2)$. The zero\n$\\rho_a$ is a nondecreasing function of $a$, and $\\lim_{a\\to\\infty}\\rho_a=2$. \n\\item[2)] If  $1\\le a^\\ast_K$ and $1\\le a\\le a^\\ast_K$,\n all zeros of $\\varphi_K(a,s)$ are on the critical line.\n\\end{enumerate}\n\\end{theo}  \nTo prove Theorem \\ref{th0.3}, we extend Colin de Verdiere's Theorem \n\\cite[Th\\'eor\\`eme 5]{CV} to our setting. Let $\\cO_K$ be the ring of integers\nof $K$ and let $\\Gamma=\\operatorname{PSL}(2,\\cO_K)$. Then $\\Gamma$ is a discrete subgroup of \n$\\operatorname{PSL}(2,\\C)$ which acts properly discontinuously on the 3-dimensional\nhyperbolic space $\\Hb^3$. The quotient $\\Gamma\\ba\\Hb^3$ has finite volume\nand a single cups at $\\infty$. The function $\\varphi_K(y,s)$ appears to be the \nzeroth Fourier coefficient of the modified Eisenstein series attached to the\ncusp $\\kappa=\\infty$. Let $\\Delta_a$ be the corresponding cut-off Laplacian\non $\\Gamma\\ba\\Hb^3$. Then we generalize Theorem \\ref{th0.2} to this setting.\nAs above, this leads to a spectral interpretation of the zeros of\n$\\varphi_K(a,s)$, and we deduce Theorem \\ref{th0.3} from this spectral\ninterpretation. \n\nThe constant $a^\\ast_K$ can be computed using the Kronecker limit formula \n\\cite[p. 273]{La} \nand the Chowla-Selberg formula \\cite[(2), p.110]{SC}.\nFor example, we get\n\\[a^\\ast_{\\Q(i)}=\\frac{4\\pi^2 e^{2+\\gamma}}{\\Gamma(1\/4)^4}\\approx 3.00681,\n\\quad\na^\\ast_{\\Q(\\sqrt{-2})}=\\frac{8\\pi^2e^{2+\\gamma}}\n{\\left(\\Gamma\\left(\\frac{1}{8}\\right)\\Gamma\\left(\\frac{3}{8}\\right)\\right)^{2}}\n\\approx 3.2581 .\\] \nThus $a^\\ast_{\\Q(\\sqrt{-D})}>1$ for $D=1,2$ and therefore, in the range \n$1\\le a\\le a^\\ast_{\\Q(\\sqrt{-D})}$  all zeros of\n$\\varphi_{\\Q(\\sqrt{-D})}(a,s)$ are  \non the line $\\Re(s)=1$. \n\nThe method used to prove Theorem \\ref{th0.3} can be extended so that\nan arbitrary number field $K$ of class number one can be treated. \nThe underlying global Riemannian symmetric \nspace is $X=\\Hb^{r_1}\\times (\\Hb^3)^{r_2}$, where $r_1$ is the number of real\nplaces and $r_2$ the number of pairs of complex conjugate places of $K$. The\nstructure of $\\Gamma\\ba X$ is slightly more complicated, however\nthe proof of the corresponding statement is completely analogous. More\ngenerally, it seems to be possible to prove similar results for the\nconstant terms of rank one cuspidal Eisenstein series attached to\n$\\operatorname{PSL}(n,\\Z)$.  \n \n\\section{The cut-off Laplacian}\n\\setcounter{equation}{0}\n\nThe cut-off Laplacian can be defined for every rank one locally symmetric \nspace. In the present paper we are dealing only with hyperbolic\nmanifolds of dimension 2 and 3. So  we discuss only the case of a\nhyperbolic 3-manifold. The general case is similar.\n\nLet  \n\\[\\Hb^3=\\{(x_1,x_{2},y)\\in\\R^3\\colon y>0\\}\\]\nbe the hyperbolic 3-space with its hyperbolic metric\n\\[ds^2=\\frac{dx_1^2+dx_{2}^2+dy^2}{y^2}.\\]\nThe hyperbolic Laplacian $\\Delta$ is given by\n\\[\\Delta=-y^2\\left(\\frac{\\partial^2}{\\partial x_1^2}\n+\\frac{\\partial^2}{\\partial x_2^2}+ \n\\frac{\\partial^2}{\\partial y^2}\\right)+y\\frac{\\partial}{\\partial y}.\\]\nIf we regard $\\Hb^3$ as the set of all quaternions $z+yj$ with $z\\in\\C$\nand $y>0$, then $G=\\operatorname{PSL}(2,\\C)$ is the group of all orientation preserving\nisometries of $\\Hb^3$. It acts by linear fractional transformations, i.e., for\n$\\gamma=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\in G$,\n\\[\\gamma(w)=(aw+b)(cw+d)^{-1},\\quad w\\in\\Hb^3.\\]\nLet\n$$B(\\C):=\\left\\{\\begin{pmatrix}\\lambda&z\\\\0&\\lambda^{-1}\\end{pmatrix}\\colon\n  \\lambda\\in\\C^\\ast,\\; z\\in\\C\\right\\}\/\\{\\pm \\operatorname{Id}\\},\\quad\nN(\\C):=\\left\\{\\begin{pmatrix}1&z\\\\0&1\\end{pmatrix}\\colon z\\in\\C\\right\\}.$$ \nLet $\\Gamma\\subset G$ be a discrete subgroup of finite co-volume. \nLet $\\kappa_1=\\infty, \\kappa_2,...,\\kappa_m\\in \\Pf^1(\\C)$ be a complete set of\n$\\Gamma$-inequivalent cusps, and let $\\Gamma_i$ be the stabilizer of\n$\\kappa_i$ in $\\Gamma$. Choose $\\sigma_i\\in G$ such that\n$\\sigma_i(\\kappa_i)=\\infty$, $i=1,...,m$. Then\n\\begin{equation}\\label{1.1}\n\\sigma_i\\Gamma_i\\sigma_i^{-1}\\cap\nN(\\C)=\\left\\{\\begin{pmatrix}1&\\lambda\\\\0&1\\end{pmatrix}\\colon \n\\lambda\\in L_i\\right\\},\n\\end{equation}\nwhere $L_i$ is a lattice in $\\C$ \\cite[Theorem 2.1.8]{EGM}. Note that\nfor $D\\not=1,3$  the intersection actually coincides with  \n$\\sigma_i\\Gamma_i\\sigma_i^{-1}$. \nChoose closed fundamental sets $\\cP_i$  for\nthe action of $\\sigma_i\\Gamma_i \\sigma_i^{-1}$ \non $\\Pf^1(\\C)\\setminus\\{\\infty\\}=\\C$. For $T>0$ define\n\\[\\tilde \\cF_i(T):=\\left\\{ (z,y)\\in\\Hb^3\\colon z\\in\\cP_i,\\;\\; y\\ge T\\right\\}.\\] \nLet $\\cF_i(T)=\\sigma_i^{-1}(\\tilde \\cF_i(T))$. There exists $T_i>0$ such \nthat\nany two points in $\\cF_i(T_i)$ are $\\Gamma$-equivalent if and only if they are\n$\\Gamma_i$-equivalent. For such a choice of $T_1,...,T_m$  there exists a\ncompact subset $\\cF_0\\subset \\Hb^3$ such that\n\\begin{equation}\\label{1.2}\n\\cF:=\\cF_0\\cup \\cF_1(T_1)\\cup\\cdots\\cup \\cF_m(T_m)\n\\end{equation}\nis a fundamental domain for $\\Gamma$ \\cite[Proposition 2.3.9]{EGM}. Let\n\\begin{equation}\\label{1.3}\nb:=\\max\\{T_1,...,T_m\\}.\n\\end{equation}\nBy $C^\\infty_c(\\Gamma\\ba\\Hb^3)$ we denote\nthe space of $\\Gamma$-invariant $C^\\infty$-functions on $\\Hb^3$ with compact \nsupport in $\\cF$. For $f\\in C^\\infty_c(\\Gamma\\ba \\Hb^3)$ let\n$$\\parallel f\\parallel^2=\\int_{\\cF}|f(x_1,x_2,y)|^2\\;\\frac{dx_1dx_2dy}{y^3},$$\nand let $L^2(\\Gamma\\ba \\Hb^3)$ be the completion of $C^\\infty_c(\\Gamma\\ba\n\\Hb^3)$ with respect to this norm.\nSimilarly let\n$$\\parallel df\\parallel^2=\\int_{\\cF} df\\wedge\\ast \\overline{df}\n=\\int_{\\cF}\\parallel\ndf(x_1,x_2,y)\\parallel^2\\;\\frac{dx_1dx_2dy}{y}.$$ \nLet $H^1(\\Gamma\\ba \\Hb^3)$ denote the completion of $C^\\infty_c(\\Gamma\\ba\n\\Hb^3)$ with respect to the norm\n$$\\parallel f\\parallel^2_1:=\\parallel f\\parallel^2+\\parallel df\\parallel^2.$$\nDenote by $|\\cP_i|$ the Euclidean area of the fundamental domain \n$\\cP_i\\subset\\C$ of\n$\\sigma_i\\Gamma_i\\sigma_i^{-1}$.\nFor $f\\in L^2(\\Gamma\\ba \\Hb^3)$ let \n\\[f_{j,0}(y)=\\frac{1}{|\\cP_i|}\\int_{\\cP_i} f(\\sigma_i^{-1}(x_1,x_2,y))\\;dx_1dx_2\\]\nbe the zeroth coefficient of the  Fourier expansion of $f$ at the cusp\n$\\kappa_j$. Note that for $f\\in H^1(\\Gamma\\ba \\Hb^3)$, each $f_{j,0}$ belongs to\n$H^1(\\R^+)$ and therefore, each $f_{j,0}$ is a continuous function on $\\R^+$. \nFor $a>0$ let\n\\[H^1_a(\\Gamma\\ba \\Hb^3):=\\left\\{f\\in H^1(\\Gamma\\ba \\Hb^3)\\colon f_{j,0}(y)=0\\;\n\\;{\\mathrm for}\\;y\\ge a,\\;j=1,...,m\\right\\}.\\]\nThen $H^1_a(\\Gamma\\ba\\Hb^3)$ is a closed subspace of $H^1(\\Gamma\\ba\\Hb^3)$. \nHence the quadratic form \n\\[q_a(f)=\\parallel df\\parallel^2,\\quad f\\in H^1_a(\\Gamma\\ba\\Hb^3),\\]\nis closed. Let $\\Delta_a$ denote the self-adjoint operator which represents\nthe quadratic form $q_a$. It acts in the Hilbert space $\\H_a$ which is the\nclosure of $H^1_a(\\Gamma\\ba\\Hb^3)$ in $L^2(\\Gamma\\ba\\Hb^3)$. By definition,\n$\\Delta_a$ is nonnegative. Its domain can be described as follows.\nLet $\\psi_{j,a}\\in\\cD^\\prime(\\Gamma\\ba\\Hb^3)$ be defined by \n\\[\\psi_{j,a}(f):=f_{j,0}(a),\\quad f\\in C^\\infty_c(\\Gamma\\ba\\Hb^3).\\]\nLet $b>0$ be defined by (\\ref{1.3}). \n\\begin{lem} Let $a\\ge b$. Then the domain of $\\Delta_a$ consists of all \n$f\\in H^1_a(\\Gamma\\ba \\Hb^3)$ for which there exist $C_1,...,C_m\\in\\C$ \nsuch that\n\\begin{equation}\\label{1.4}\n\\Delta f-\\sum_{j=1}^mC_j\\psi_{j,a}\\in L^2(\\Gamma\\ba \\Hb^3).\n\\end{equation}\n\\end{lem}\nThe proof of the lemma is analogous to the proof of Th\\'eor\\`em 1 in\n\\cite{CV}. \nLet  $f\\in \\operatorname{dom}(\\Delta_a)$. By the lemma, there exist $C_1,...,C_m\\in\\C$ such \nthat (\\ref{1.4}) holds. Then $\\Delta_af$ is given by\n\\begin{equation}\\label{1.5}\n\\Delta_af=\\Delta f-\\sum_{j=1}^mC_j\\psi_{j,a}.\n\\end{equation}\n\n\nFurthermore, we have\n\\begin{lem}\\label{l1.2}\n$\\Delta_a$ has a compact resolvent.\n\\end{lem}\n\\begin{proof} The proof is a simple extension of the argument in \n\\cite[p.206]{LP}. \n\\end{proof}\nSo the spectrum of $\\Delta_a$  consists of a sequence of eigenvalues \n$0\\le \\lambda_1(a)\\le \\lambda_2(a)\\le\\cdots\\to\\infty$ with finite\nmultiplicities. To describe the eigenvalues and eigenfunctions of $\\Delta_a$\nmore explicitely, we need  to consider the Eisenstein series. Let $\\sigma_j\\in\nG$, $j=1,...,m$, be as above. \nFor each cusp $\\kappa_i$,  the Eisenstein series $E_i(w,s)$ attached to\n$\\kappa_i$ is defined to be\n\\[E_i(w,s):=\\sum_{\\gamma\\in\\Gamma_i\\ba \\Gamma}y(\\sigma_i\\gamma w)^s,\\quad\n\\Re(s)>2,\\]\nwhere $y(\\sigma_i\\gamma w)$ denotes the $y$-component of $\\sigma_i\\gamma w$. \nSelberg has shown that $E_i(w,s)$ can be meromorphically continued in $s$ to \nthe whole complex plane $\\C$, and it is an automorphic eigenfunction of\n$\\Delta$ with \n$$\\Delta E_i(w,s)=s(2-s)E_i(w,s).$$\nSince $E_i(w,s)$ is $\\Gamma$-invariant, it is invariant under the stabilizer\n$\\Gamma_j$ of the cusp $\\kappa_j$ and therefore, admits a Fourier expansion\nat  $\\kappa_j$. The constant Fourier coefficient is given by\n\\begin{equation}\\label{1.3a}\n\\delta_{ij}y^s+C_{ij}(s)y^{2-s}.\n\\end{equation}\nPut\n\\[C(s):=\\left(C_{ij}(s)\\right)_{i,j=1,...,m}.\\]\nThis is the so called ``scattering matrix''. The scattering matrix and the \nEisenstein series satisfy the following system of functional equations.\n\\begin{equation}\\label{1.7}\n\\begin{split}\n&C(s)C(2-s)=\\operatorname{Id},\\\\\n&E_i(w,s)=\\sum_{j=1}^m C_{ij}(s)E_j(w,2-s), \\quad i=1,...,m.\n\\end{split}\n\\end{equation}\n\nNow recall that a square integrable eigenfunction $f$ of $\\Delta$ is called \ncuspidal, if the zeroth Fourier coefficient $f_{j,0}$ of $f$\nat the cusp $\\kappa_j$ vanishes for all $j=1,...,m$. Denote by\n$S(\\lambda;\\Gamma)$ the space of cuspidal eigenfunctions of $\\Delta$ with \neigenvalue $\\lambda$. A function $f\\in C^0(\\Gamma\\ba \\Hb^3)$ is called of \nmoderate growth, if there exists $R\\in\\R^+$ such that\n\\[f(\\sigma^{-1}_i(x_1,x_2,y))=O(y^R),\\quad {\\mathrm\n  for}\\;\\;(x_1,x_2,y)\\in\\cF_i(b),\\; i=1,...,m.\\] \nIt follows from the Fourier expansion in the cusps \\cite[Section 3.3.3]{EGM}\nthat a  cuspidal eigenfunction $f$ of $\\Delta$ is rapidly  decreasing in each\ncusp. Therefore for $\\psi\\in\\E(\\lambda,\\Gamma)$ and $\\varphi\\in\nS(\\lambda,\\Gamma)$ , the inner product $\\langle \\psi, \\varphi\\rangle$ \nis well defined. For $\\lambda\\in\\C$ let\n\\begin{equation}\n\\E(\\lambda;\\Gamma):=\\left\\{\\psi\\in L^2_{\\loc}(\\Gamma\\ba\\Hb^3)\\colon\n  \\Delta\\psi=\\lambda\\psi,\\;\\psi\\;\\;{\\mathrm is \\; of\\;\n  moderate\\; growth},\\;\\psi\\perp S(\\lambda;\\Gamma)\\right\\}.\n\\end{equation}\n\\begin{lem}\\label{l1.3} Let $m$ be the number of cusps of $\\Gamma\\ba\\Hb^3$. \nThen \n\\begin{enumerate}\n\\item[1)] For every $\\lambda\\in\\C$ we have $\\dim\\E(\\lambda;\\Gamma)\\le m$. \n\\item[2)] Suppose that $\\lambda=s(2-s)$, where $\\Re(s)\\ge 1$, $s\\not=1$,\n  and $s$ is not a pole of any Eisenstein series. Then\n$E_1(w,s),...,E_m(w,s)$ is a basis of $\\E(\\lambda;\\Gamma)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe proof is analogous to the proof of Satz 11 in \\cite[p.171]{Ma}. Let \n$\\phi,\\psi\\in\\E(\\lambda;\\Gamma)$ and $\\lambda=s(2-s)$. The zeroth Fourier \ncoefficient of $\\phi$, $\\psi$ at the $j$-th cusp is given by\n\\[\\phi_{0,j}(y)=a_jy^s+b_jy^{2-s},\\quad\n\\psi_{0,j}(y)=c_jy^s+d_jy^{2-s},\\;s\\not=1,\\] \nand for $s=1$ by\n\\[\\phi_{0,j}(y)=a_jy+b_jy\\log y,\\quad \\psi_{0,j}(y)=c_jy+d_jy\\log y.\\]\nLet $\\chi_{j,[a,\\infty)}$ denote the characteristic function of $\\Gamma_i\\ba\n\\cF_i(a)$ in $\\Gamma\\ba\\Hb^3$. Let\n\\[\\phi_a=\\phi-\\sum_{j=1}^m\\chi_{j,[a,\\infty)}\\phi_{0,j},\\quad  \n\\psi_a=\\psi- \\sum_{j=1}^m\\chi_{j,[a,\\infty)}\\psi_{0,j}\\] \nbe the truncations of $\\phi$ and $\\psi$, respectively, at level $a\\ge b$. \nLet $s\\not=1$. Integrating by parts, we get\n\\begin{equation}\\label{1.9}\n\\begin{split}\n0=\\int_{\\Gamma\\ba \\Hb^3}(\\phi_a\\Delta\\psi_a-\\psi_a\\Delta\\phi_a)\\;d{\\mathrm \nvol}&=\n\\sum_{j=1}^m\\left(\\psi_{0,j}(y)\\frac{\\partial\\phi_{0,j}}{\\partial y}(y)-\n\\phi_{0,j}(y)\\frac{\\partial\\psi_{0,j}}{\\partial y}(y)\\right)\\bigg|_{y=a}\\\\\n&=(2-2s)\\sum_{j=1}^m(a_jd_j-b_jc_j).\n\\end{split}\n\\end{equation}\nFor $s=1$ the right hand side equals $\\sum_{j=1}^m(a_jd_j-b_jc_j)$.\n\nAn element of $\\E(\\lambda)$ is uniquely determined by its zeroth Fourier \ncoefficients. Define $i\\colon \\E(\\lambda)\\to \\C^{2m}$ by\n$i(\\phi):=(a_1,b_1,...,a_m,b_m),$\nwhere $a_j,b_j$ are the $0$-th Fourier coefficients of $\\phi$ at the $j$-th\ncusp. Then $i$ is an \nembedding. Let \n\\[I(x,y):=\\sum_{j=1}^m(x_{2j-1}y_{2j}-x_{2j}y_{2j-1})\\]\nbe the standard symplectic form on $\\C^{2m}$. Then by (\\ref{1.9}) and the\ncorresponding statement for $s=1$,\n$\\E(\\lambda)$ is an isotropic subspace for $I$. Hence $\\dim\\E(\\lambda)\\le m$. \n\nIf $s\\not=1$ and $s$ is not a pole of $E_j(w,s)$, $j=1,...,m$, the Eisenstein \nseries have linearly independent $0$-th Fourier coefficients given by \n(\\ref{1.3a}). This implies that $E_1(w,s),...,E_m(w,s)$ are linearly\nindependent and hence, form a basis of $\\E(\\lambda,\\Gamma)$, where\n$\\lambda=s(2-s)$.\n\\end{proof}\nWe are now ready to describe the spectrum of $\\Delta_a$. Let\n$0=\\Lambda_0<\\Lambda_1<\\cdots<\\Lambda_N<1$ be the eigenvalues of $\\Delta$ with\nnon-cuspidal eigenfunctions. Then $\\Lambda_j$ has the form \n$\\Lambda_j=\\sigma_j(2-\\sigma_j)$ with $\\sigma_j\\in (1,2]$ and $\\sigma_j$ is a \nsimple pole of some \nEisenstein series $E_k(w,s)$. The corresponding residue is an \neigenfunction of $\\Delta$ with eigenvalue $\\Lambda_j$ \n\\cite[Proposition 6.2.2]{EGM}.\nThe following theorem\nis an extension of Theorem 5 in \\cite{CV} to the 3-dimensional case.\n\\begin{theo}\\label{th1.4}\nThe spectrum of $\\Delta_a$ is the union of the cuspidal \neigenvalues $(\\lambda_i)_i $ of $\\Delta$ and a sequence\n$0<\\mu_0(a)\\le \\mu_1(a)\\le \\cdots $\nwith the following properties:\n\\begin{enumerate}\n\\item[1)] Each $\\mu_j(a)$ is a decreasing function of $a$,\n\\item[2)] Let $a\\ge b$. Then each $\\mu_j(a)$ has multiplicity $\\le m$, and\n the map $s\\mapsto s(2-s)$ is a bijection between the\neigenvalues $\\mu_j(a)\\not\\in\\{1,\\Lambda_0,...,\\Lambda_N\\}$ and the zeros of\n$\\rho\\in\\C\\setminus\\{1,\\sigma_0,...,\\sigma_N\\}$ of \n$\\varphi(s):=\\det(C(s)+a^{2s-2}\\operatorname{Id})$.\n\\end{enumerate}\n\\end{theo}\n\\begin{proof}\nLet $f\\in L^2(\\Gamma\\ba\\Hb^3)$ be a cuspidal eigenfunction of $\\Delta$.\nThen $f$ belongs to\n$H^1_a(\\Gamma\\ba \\Hb^3)$ and $\\Delta f$ is square integrable.  Hence\nby (\\ref{1.4}) and (\\ref{1.5}), $f\\in\\operatorname{dom}(\\Delta_a)$ and $\\Delta_af=\\Delta f$.\nThus all cuspidal eigenfunctions of $\\Delta$ are also eigenfunctions of\n$\\Delta_a$ with the same eigenvalues. \n\nFor 1)\nobserve that for $a\\le a^\\prime$ we have\n\\[H^1_a(\\Gamma \\ba\\Hb^3)\\subset H^1_{a^\\prime}(\\Gamma \\ba\\Hb^3).\\]\nSince $\\Delta_a$ is a non-negative self-adjoint operator with compact\nresolvent, 1) follows from the mini-max characterization of its eigenvalues.\n\n\n\n2) Let $\\Phi$ be an eigenfunction of $\\Delta_a$ with eigenvalue\n$\\mu_j(a)=s_j(2-s_j)$ and suppose that $\\Phi\\perp\nS(\\mu_j(a);\\Gamma)$.  Let $\\Phi_{i,0}$ be the zeroth Fourier coefficient of\n$\\Phi$ at the cusp $\\kappa_i$. There exists $1\\le j\\le m$ such that\n$\\Phi_{j,0}\\not\\equiv 0$. Let $\\hat\\Phi_{i,0}$ denote the analytic continuation\nof $\\Phi_{i,0}$ from $(0,a)$ to $\\R^+$. Put\n\\[\\hat\\Phi(w):=\\Phi(w)+\\sum_{i=1}^m\\chi_{i,[a,\\infty)}\\hat\\Phi_{i,0}.\\]\nThen $\\hat\\Phi\\in C^\\infty(\\Gamma\\ba \\Hb^3)$ and $\\Delta \\hat\\Phi\n=\\mu_j(a)\\hat\\Phi$. \nMoreover $\\hat\\Phi$\nis of moderate growth and $\\hat\\Phi\\perp S(\\mu_j(a),\\Gamma)$. Therefore $\\hat\\Phi\\in\\E(\\mu_j(a);\\Gamma)$. \nBy Lemma \\ref{l1.3},\n1), it follows that the multiplicity of $\\mu_j(a)$ is bounded by $m$.\n\nNext suppose that\n$\\mu_j(a)=s_j(2-s_j)\\not\\in\\{1,\\Lambda_0,...,\\Lambda_N\\}$. We can  \nassume that $\\Re(s_j)\\ge 1$. As explained above, $s_j$ is not a pole of any\nEisenstein series $E_k(w,s)$, $k=1,...,m$. Therefore, by Lemma \\ref{l1.3} \nthere exist \n$c_1,...,c_m\\in\\C$ such that \n\\[\\hat\\Phi(w)=\\sum_{i=1}^m c_i E_i(w,s_j).\\]\nBy construction, the zeroth Fourier coefficient $\\hat\\Phi_{l,0}(y)$ of \n$\\hat\\Phi(w)$ at\n$\\kappa_l$ vanishes at $y=a$ for all $l=1,...,m$. By (\\ref{1.3a}) this implies\n\\[\\sum_{i=1}^mc_i\\left(\\delta_{il}a^{s_j}+C_{il}(s_j)a^{2-s_j}\\right)=0,\\quad\nl=1,...,m.\\]\nHence $\\det(C(s_j)+a^{2s_j-2}\\operatorname{Id})=0$.\n\nNow assume that $\\det(C(s_j)+a^{2s_j-2}\\operatorname{Id})=0$, $\\Re(s_j)\\ge 1$,\n and $s_j\\not\\in\\{1,\\sigma_0,...,\\sigma_N\\}$. Then $s_j$ is not a pole of any\nEisenstein series. Let $0\\not=\\psi\\in\\C^m$ such\nthat\n\\begin{equation}\\label{1.10}\nC(s_j)\\psi=-a^{2s_j-2}\\psi.\n\\end{equation}\nLet $\\psi=(c_1,...,c_m)$ and set\n\\[\\hat\\Phi(w):=\\sum_{k=1}^m c_kE_k(w,s_j).\\] \nBy (\\ref{1.3a}) and (\\ref{1.10}), the $0$-th Fourier coefficient of \n$\\hat\\Phi(w)$ at the cusp $\\kappa_l$ is given by\n\\begin{equation}\n\\sum_{k=1}^m\nc_k\\left(\\delta_{kl}y^{s_j}+C_{kl}(s_j)y^{2-s_j}\\right)\n=c_l\\left(y^{s_j}-a^{2s_j-2}y^{2-s_j}\\right). \n\\end{equation}\nTherefore, the $0$-th Fourier coefficient $\\hat\\Phi_{l,0}(y)$ vanishes at\n$y=a$ for all $l=1,...,m$. Put\n\\[\\Phi(w)=\\hat\\Phi(w)-\\sum_{i=1}^m\\chi_{i,[a,\\infty)}\\hat\\Phi_{l,0}(y).\\]\nThen $\\Phi\\in H^1_a(\\Gamma\\ba\\Hb)$ and it follows from the description of\nthe domain of $\\Delta_a$ that $\\Phi\\in\\operatorname{dom}(\\Delta_a)$ and \n$\\Delta_a \\Phi=s_j(2-s_j)\\Phi$. \n\\end{proof}\n\nNow assume that $\\Gamma\\ba \\Hb^3$ has a single cusp $\\kappa=\\infty$. Then\nthere is a single  \nEisenstein series $E(w,s)$, which is given by\n\\[E(w,s)=\\sum_{\\gamma\\in\\Gamma_\\infty\\ba\\Gamma}y(\\gamma w)^s,\\quad \\Re(s)>2.\\]\nThe zeroth Fourier coefficient of $E(w,s)$ at $\\infty$ equals\n\\begin{equation}\n\\varphi_0(y,s):=y^s+c(s)y^{2-s},\n\\end{equation}\nwhere $c(s)$ is a meromorphic function of $s\\in\\C$. \nThe following facts are well known \n\\cite[pp.243-245]{EGM} .\nThe poles of $E(w,s)$ in the half-plane $\\Re(s)>1$ are \ncontained in the interval $(1,2]$ and are all simple. The residue of $E(w,s)$\nat a pole \n$\\sigma\\in (1,2]$ is a square integrable eigenfunction of $\\Delta$ with \neigenvalue $\\sigma(2-\\sigma)$, which  is non-cuspidal. Moreover all\nnon-cuspidal eigenfunctions of $\\Delta$ are obtained in this way. Thus\nthe non-cuspidal eigenvalues have multiplicity one. So in this case we get \nthe following refinement of Theorem \\ref{th1.4}.\n\\begin{theo}\\label{th1.5}\nAssume that $\\Gamma\\ba\\Hb^3$ has a single cusp.\nThen the spectrum of $\\Delta_a$ is the union of the cuspidal \neigenvalues $(\\lambda_i)_i $ of $\\Delta$ and a sequence\n$0<\\mu_0(a)< \\mu_1(a)< \\cdots $\nwith the following properties:\n\\begin{enumerate}\n\\item[1)] Each $\\mu_j(a)$ is a decreasing function of $a$;\n\\item[2)] Let $a\\ge b$. Then each $\\mu_j(a)$ has multiplicity 1 and the map\n$s\\mapsto s(2-s)$ is a bijection between the zeros $\\rho\\not=1$ of \n$\\varphi_0(a,s)$ and the eigenvalues $\\mu_j(a)\\not=1$ of $\\Delta_a$. \n\\item[3)] There is $j$ with $\\mu_j(a)=1$  if and only if $c(1)=-1$ and\n $c^\\prime(1)=-2\\log a$.  \n\\item[4)] Let $0=\\Lambda_0<\\Lambda_1<\\cdots<\\Lambda_N<1$ be the\n eigenvalues of $\\Delta$ \nwith  non-cuspidal eigenfunctions. If $a\\ge b$, then \n\\[0=\\Lambda_0<\\mu_0(a)<\\Lambda_1<\\mu_1(a)<\\cdots <\\Lambda_N< \n\\mu_N(a).\\]\nMoreover, \n\\begin{equation}\n\\lim_{a\\to\\infty}\\mu_j(a)=\\begin{cases}\\Lambda_j&,\\; 0\\le j\\le N;\\\\\n1&, \\;j>N.\n\\end{cases}\n\\end{equation}\n\\end{enumerate}\n\\end{theo}\n\\begin{proof}\n1) follows from Theorem \\ref{th1.4}. If $s\\not\\in\\{1,\\sigma_0,...,\\sigma_N\\}$,\nthen 2) follows also from 2) of Theorem \\ref{th1.4}. Now suppose that $s_0$\nis a pole of $E(w,s)$ in the half-plane $\\Re(s)\\ge 1$. Then the residue\n$\\psi$ of $E(w,s)$ at $s_0$ is an eigenfunction of $\\Delta$ with non-vanishing\n$0$-th Fourier coefficient. Moreover $\\psi\\perp S(\\lambda,\\Gamma)$.\nTherefore $\\psi\\in\\E(\\lambda,\\Gamma)$, where \n$\\lambda=s_0(2-s_0)$. By Lemma \\ref{l1.3} it follows that\n $\\dim\\E(\\lambda,\\Gamma)=1$.\nMoreover the constant term $\\psi_0(y)$ of $\\psi$ has the form \n$\\psi_0(y)=cy^{2-s_0}$. Especially, it never vanishes on $\\R^+$. On the other\nhand, if $\\Phi$ were an eigenfunction of $\\Delta_a$ with eigenvalue $\\lambda$,\nthen the corresponding eigenfunction $\\hat\\Phi\\in\\E(\\lambda,\\Gamma)$, \nconstructed in the proof of Theorem \\ref{th1.4} would have a constant term\nwhich  \nvanishes at $y=a$. Hence the eigenvalues $\\Lambda_j$ can not be eigenvalues of\n$\\Delta_a$. This proves 2)\n\nFor 3) we note that $c(1)^2=1$ by the\nfunctional equation (\\ref{1.7}). Thus $c(1)=\\pm 1$. If $c(1)=-1$,  then \n$E(w,1)\\equiv 0$. Put\n$\\psi(w):=\\frac{d}{ds}E(w,s)\\big|_{s=1}.$ Then $\\Delta \\psi=\\psi$ and the \nzeroth Fourier coefficient of $\\psi$ is given by\n\\[\\psi_0(y)=(2\\log y+c^\\prime(1))y.\\]\nSuppose that $c^\\prime(1)=-2\\log a$. Put \n$\\hat \\psi_a(w):=\\psi(w)-\\chi_{[a,\\infty)}\\psi_0(y(w))$. Then $\\hat \\psi_a$\nis in the domain of $\\Delta_a$ and $\\Delta_a\\hat\\psi_a=\\hat\\psi_a$. \n\nFor the other direction suppose that $\\hat\\psi$ is an eigenfunction of\n$\\Delta_a$ with eigenvalue \n$1$ and $\\hat\\psi\\perp S(1,\\Gamma)$. Let $\\hat\\psi_0(y)$ be the $0$-th Fourier\ncoefficient of $\\hat\\psi$.  \nExtend $\\hat\\psi_0(y)$ in the obvious way  from $(b,a)$  to a smooth function \n$\\psi_0$ on $(b,\\infty)$. Set\n\\[\\psi:=\\hat\\psi+\\chi_[a,\\infty)\\psi_{0}.\\]\nThen $\\psi$ is smooth, of moderate growth, satisfies $\\Delta\\psi=\\psi$ and\n$\\psi\\perp S(1,\\Gamma)$. Thus $\\psi\\in\\E(1,\\Gamma)$. Therefore by Lemma\n\\ref{l1.3} it follows that $\\dim\\E(1,\\Gamma)=1$. On the other hand, we have\n$0\\not=\\frac{d}{ds}E(s,w)\\big|_{s=1}\\in\\E(1,\\Gamma)$.  \nTherefore it follows that there exists\n$c\\not=0$ such that $\\psi(w)=c\\frac{d}{ds}E(w,s)\\big|_{s=1}$. \nComparing the constant terms, it follows that $c^\\prime(1)=-2\\log a$. \n\n4) follows from the mini-max principle and the fact that $\\cup_{a\\ge b}\nH^1_a(\\Gamma \\ba\\Hb^3)$ is dense in $H^1(\\Gamma \\ba\\Hb^3)$.\n\\end{proof}\n\n\\section{Hyperbolic surfaces of finite area}\n\\setcounter{equation}{0}\n\nIn this section we prove Theorem \\ref{th0.2} and deduce Theorem\n\\ref{th0.1} from it.\n\nLet $\\Gamma=\\operatorname{PSL}(2,\\Z)$ be the modular group. Then $\\Gamma\\ba \\Hb$ has a single\ncusp $\\kappa=\\infty$. As fundamental domain we take the standard domain\n\\[F:=\\left\\{z\\in\\Hb\\colon |\\Re(z)|<1\/2,\\,|z|>1\\right\\}.\\]\nSo we can take $b=1$. \nThe Eisenstein series attached to the cusp $\\infty$ is given by\n\\[E(z,s)=\\sum_{\\gamma\\in\\Gamma_\\infty\\ba\\Gamma}\\Im(\\gamma z)^s=\\sum_{(m,n)=1}\n\\frac{y^s}{|mz+n|^{2s}},\\quad \\Re(s)>1.\\]\nThe constant term of $E(z,s)$ equals\n\\begin{equation}\\label{2.1}\ny^s+c(s)y^{1-s},\n\\end{equation}\nwhere $c(s)$ is the meromorphic function defined by (\\ref{0.2}). \n\n\\noindent\n{\\bf Proof of Theorem 0.2:}\nSince the \nthe zeros of the completed zeta function $\\Lambda(s)$ are all contained in the\nstrip $0<\\Re(s)<1$, it follows that $\\Lambda(2s-1)$ and $\\Lambda(2s)$ have no\ncommon  zeros. This implies that the zeros of \n\\[\\varphi_0(a,s)=\\Lambda(2s)a^s+\\Lambda(2s-1)a^{1-s}\\]\ncoincide with the zeros of\n$a^s+c(s)a^{1-s}$. Now note that $\\Lambda(s)$ has poles of order 1 at $s=1,0$.\nThe residue at $s=1$ is 1 and the residue at $s=0$ is $-1$. Hence\n\\begin{equation}\\label{2.2}\nc(1\/2)=\\lim_{s\\to 1\/2}\\frac{\\Lambda(2s-1)}{\\Lambda(2s)}=-1.\n\\end{equation}\nNow the statements 1), 3) and 4) follow immediately from \\cite[Th\\'eor\\`em\n5]{CV}.  For 2) we use that by Roelcke \\cite{Ro}, the smallest positive\neigenvalue $\\lambda_1$ of $\\Delta$ on $\\Gamma\\ba \\Hb$ satisfies\n$\\lambda_1>1\/4$. Then 2) follows from Th\\'eor\\`em 5, (iii), of \\cite{CV}. \n\n\\noindent\n{\\bf Proof of Theorem 0.1:} Let $a\\ge 1$ and let $\\rho\\not=1\/2$ be a zero of\n$\\varphi_0(a,s)$. Then by Theorem 0.2, 3), $\\rho(1-\\rho)$ is an eigenvalue of\n$\\Delta_a$. Hence $\\rho(1-\\rho)$ is a non-negative real number. If \n$\\rho(1-\\rho)> 1\/4$, then $\\rho$ is a complex zero with $\\Re(\\rho)=1\/2$. \nBy Theorem 0.2, 2), there exist at most two zeros, $\\rho_a$ with \n$0<\\rho_a(1-\\rho_a)<1\/4$. Thus $\\rho_a$ and $1-\\rho_a$ are zeros and we\nmay assume that \n$\\rho_a\\in (1\/2,1)$. Let $a<a^\\prime$. Then by 1), we have\n\\[-(\\rho_a-1\/2)^2=\\rho_a(1-\\rho_a)-1\/4\\ge\\rho_{a^\\prime}(1-\\rho_{a^\\prime})\n-1\/4=-(\\rho_{a^\\prime}-1\/2)^2.\\]\nThus $\\rho_a\\le\\rho_{a^\\ast}$ and by Theorem \\ref{th0.2}, 2), it follows that \n$\\lim_{a\\to\\infty}\\rho_a=1$. This proves the second part of 2).\n \nNow consider the eigenvalue $1\/4$. Let\n\\[\\xi(s)=s(s-1)\\Lambda(s).\\]\nThen  we get\n\\[c^\\prime(1\/2)=-\\frac{c^\\prime(1\/2)}{c(1\/2)}=-4\\lim_{s\\to 0}\n\\left(\\frac{\\Lambda^\\prime(s)}{\\Lambda(s)}+\\frac{1}{s}\\right)=-4\\left(1+\n\\frac{\\xi^\\prime(0)}{\\xi(0)}\\right).\\]\nBy \\cite[pp.80-81]{Da} it follows that\n\\[2\\left(1+\\frac{\\xi^\\prime(0)}{\\xi(0)}\\right)=-\\gamma+\\log 4\\pi,\\]\nwhere $\\gamma$ denotes Euler's constant. Let $a^\\ast:=4\\pi e^{-\\gamma}$. \nThen we get\n\\begin{equation}\\label{2.3}\nc^\\prime(1\/2)=-2\\log a^\\ast.\n\\end{equation}\nBy Theorem 0.2, 4), $1\/4$ is an eigenvalue \nof $\\Delta_a$ if and only if $a=a^\\ast$.\nSince all  eigenvalues $\\mu_i(a)$ of $\\Delta_a$ are non-increasing\nfunctions of $a$ and have multiplicity one, it follows that\n$\\mu_0(a^\\ast)=1\/4$, $\\mu_0(a)<1\/4$, if $a>a^\\ast$, and $\\mu_0(a)>1\/4$, if\n$1\\le a<a^\\ast$. Since $\\mu_0(a)=\\rho_a(1-\\rho_a)$, this proves 1) and the\nfirst part of 2). \n\nIt remains to show that all zeros $\\rho\\not=1\/4$ are simple. Let $\\rho$ be\na zeros of $\\varphi_0(a,s)$ of multiplicity $\\ge 2$. Then $\\rho$ is also a \nzero of\n$1+c(s)a^{1-2s}$ of multiplicity $\\ge 2$. Thus\n\\[0=\\frac{d}{ds}\\left(1+c(s)a^{1-2s}\\right)\\big|_{s=\\rho}=a^{1-2s}\n\\left(c^\\prime(\\rho)-2c(\\rho)\\log a \\right).\\]\nand hence,  we get\n\\begin{equation}\\label{2.4}\n2\\log a-\\frac{c^\\prime}{c}(\\rho)=0.\n\\end{equation}\n  \nLet $\\chi_{[a,\\infty)}$ be the characteristic function of the interval\n$[a,\\infty)$ \nin $\\R^+$. For $a\\ge 1$ let\n\\[\\hat E(z,s):=E(z,s)-\\chi_{[a,\\infty)}(y^s+c(s)y^{1-s})\\]\nbe the truncated Eisenstein series. Then $\\hat E(z,s)$ is square integrable\nand its norm is given by the Maass-Selberg relation. \nFor $r\\in\\R$, $r\\not=0$, we have \\cite[(7.42')]{Se}\n\\begin{equation}\n\\parallel \\hat E(1\/2+ir)\\parallel^2=2\\log a-\\frac{c^\\prime}{c}(1\/2+ir)+\n\\frac{1}{r}\\Im\\left(c(1\/2+ir)a^{-2ir}\\right).\n\\end{equation} \nLet $\\rho=1\/2+ir$, $r\\not=0$, be a zero of $1+c(s)a^{1-2s}$ of multiplicity\n$\\ge 2$. Then \n$$c(1\/2+ir)=-a^{2ir}.$$\n Thus $\\Im\\left(c(1\/2+ir)a^{-2ir}\\right)=0$.\nThis together with (\\ref{2.4}) implies that $\\hat E(1\/2+ir)=0$. On the other\nhand, the zeroth Fourier coefficient is not identically zero. Hence we arrive\nat a contradiction.\n\nThe case of real zeros is treated similarly. Let $\\sigma\\in(1\/2,1)$ be a \nzero of $1+c(s)a^{1-2s}$ of multiplicity $\\ge 2$. In this case the \nMaass-Selberg relation is\n\\begin{equation}\n\\parallel \\hat E(\\sigma)\\parallel^2=\\frac{a^{2\\sigma-1}\n-|c(\\sigma)|^2a^{1-2\\sigma}}{2\\sigma-1}+2c(\\sigma)\\log a- c^\\prime(\\sigma)\n\\end{equation} \n\\cite[(7.42'')]{Se}.\nSince $\\sigma$ is a zero, we have $|c(\\sigma)|=a^{2\\sigma-1}$. \nThis together with (\\ref{2.4}) implies\n$\\hat E(\\sigma)=0$, which again leads to a contradiction. \n\nFinally note that by (\\ref{2.2}) and (\\ref{2.3}), $1\/2$ is a double zero \nof $\\varphi_0(a^\\ast,s)$.  This completes the proof of Theorem 0.1.\n\\hfill$\\Box$\n\nWe note that Th\\'eor\\`em 5 of \\cite{CV} has an extension to hyperbolic surfaces\nwith  several cusps similar to Theorem \\ref{th1.4}. In the case of $m$ cusps\nthe scattering matrix $C(s)$ is  an $m\\times m$ matrix. In this case we study\nthe zeros of the function\n\\begin{equation}\\label{2.5}\n\\phi(a,s):=\\det(a^s+a^{1-s}C(s)).\n\\end{equation}\nFor a congruence subgroup\n$\\Gamma\\subset \\operatorname{PSL}(2,\\Z)$ the entries of $C(s)$ consist of ratios of Dirichlet\n$L$-series \\cite{Hu}. \nIf we multiply (\\ref{2.5}) by the common denominator, we get a certain linear\n combination \nof products of Dirichlet $L$-functions.  As example, consider the case of the\n Hecke group \n$\\Gamma=\\Gamma_0(n)$, where $n=p_1\\cdots p_r$ is a product of pairwise distinct\nprime numbers $p_i$. Then the number of cusps of $\\Gamma_0(n)\\ba \\Hb$ is \n$m=2^r$. The scattering matrix can be described in the following way. \nFor a prime $p$ let\n\\[N_p(s):=\\frac{1}{p^{2s}-1}\\begin{pmatrix}p-1& p^s-p^{1-s}\\\\\np^s-p^{1-s}& p-1\n\\end{pmatrix}\n\\]\nThen \n\\[C(s)=\\frac{\\Lambda(2s-1)}{\\Lambda(2s)}\\left(N_{p_1}(s)\\otimes\\cdots \n\\otimes N_{p_r}(s)\\right)\\]\n\\cite{He}. Suppose that  $n=p$ is a prime number. Then we get\n\\begin{equation}\n\\begin{split}\n\\det&\\left(a^s\\operatorname{Id}+a^{1-s}C(s)\\right)\\\\\n&=a^{2s}+2a\\frac{p-1}{p^{2s}-1}\\cdot\n\\frac{\\Lambda(2s-1)}{\\Lambda(2s)}+a^{2-2s}\\frac{p^{2-2s}-1}{p^{2s}-1}\\cdot\n\\frac{\\Lambda(2s-1)^2}{\\Lambda(2s)^2}\\\\\n&=\\frac{\\prod_{k=1}^2\\left(a^s(p^s+(-1)^k)\\Lambda(2s)\n+a^{1-s}(p^{1-s}+(-1)^k)\\Lambda(2s-1)\\right)}{(p^{2s}-1)\\Lambda(2s)^2}. \n\\end{split}\n\\end{equation}\nNote that the numerator and the denominator have no common zeros. Therefore the\nzeros of the function $\\phi(a,s)$ coincide with the zeros\nof the numerator. Let $a\\ge 1$. Then it follows as above that the map\n$s\\mapsto s(1-s)$ induces a bijection between the zeros $\\rho\\not=1\/2$ of \n$\\phi(a,s)$ and the eigenvalues $\\mu_j(a)\\not=1\/4$ of $\\Delta_a$.\nThis implies that for $a\\ge 1$ the complex zeros of\n\\[a^s(p^s\\pm 1)\\Lambda(2s)+a^{1-s}(p^{1-s}\\pm 1)\\Lambda(2s-1)\\] \nlie all on the line $\\Re(s)=1\/2$. \nAs for the real zeros, we observe that $\\Gamma_0(p)$ has no residual\neigenvalues other than 0 \\cite[Theorem 11.3]{Iw}. By the mini-max principle it\nfollows that \n$\\mu_1(a)\\ge 1\/4$. Therefore, there exist at most two real zeros, $\\rho_a$\nand $1-\\rho_a$, with $\\rho_a\\in(1\/2,1)$. Moreover $\\lim_{a\\to\\infty}\\rho_a=1$.\n\n  \n\\section{Imaginary quadratic fields}\n\\setcounter{equation}{0} \n\nIn this section we derive Theorem \\ref{th0.3} from  Theorem \\ref{th1.5}. \n\nLet $K=\\Q(\\sqrt{-D})$ be an imaginary quadratic field, where $D$ is a square\nfree positive integer, \nand let $\\cO_K$ be the ring of integers in $K$. Let \n$\\Gamma_K=\\operatorname{PSL}(2,\\cO_K)$. \nThen $\\Gamma_K\\subset\\operatorname{PSL}(2,\\C)$ is a discrete subgroup of finite co-volume. The\nnumber of cusps of $\\Gamma_K\\ba\\Hb^3$ equals the class number $h_K$ of \n$K$. \nAssume that $h_K=1$. This is the case if and only if \n$D=1,2,3,7,11,19,43,67,163$ \\cite{St}. Then $\\Gamma_K\\ba\\Hb^3$ has a single \ncusps  which\nwe choose to be at $\\kappa=\\infty$. A fundamental domain $\\cF_K$ of \n$\\Gamma_K$ can be constructed as in \\cite[Section 7.3]{EGM}, \n\\cite[pp. 17-19]{Sw}. \nLet $F_K$ be a fundamental domain for the group\n\\[\\mathrm{PSL}(2,\\cO_K)_\\infty=\\left\\{\\begin{pmatrix}\\alpha&\\beta\\\\0&\\lambda\n\\end{pmatrix}\\in\\mathrm{PSL}(2,\\cO_K)\\right\\}.\\]\nIf\n$\\mu,\\lambda\\in\\cO_K$ \ngenerate $\\cO_K$ and $\\mu\\not=0$, let \n\\[H(\\mu,\\lambda):=\\{(z,r)\\in\\Hb^3\\colon |\\mu z+\\lambda|^2+|\\mu|^2r^2\\ge 1\\}.\\]\nThis is the set of points $w\\in\\Hb^3$ which lie above or on the hemisphere\n$S(\\mu,\\lambda)$ in $\\Hb^3$ which is  the set of all $(z,r)$ such that\n$|\\mu z+\\lambda|^2+|\\mu|^2 r^2=1$. This hemisphere has center $(\\lambda\/\\mu,0)$\nand radius $1\/|\\mu|$. Let \n\\[B_K:=\\bigcap H(\\mu,\\lambda),\\]\nwhere the intersection is taken over all $\\mu,\\lambda\\in\\cO_K$ as above. Then\na fundamental domain $\\cF_K$ of $\\Gamma_K$ is given by\n\\[\\cF_K=\\{(z,r)\\in B_K\\colon z\\in F_K\\}\\]\n\\cite[Theorem 7.3.4]{EGM}, \\cite[Section 3]{Sw}. Thus $\\cF_K$ is the complement\nin $F_K\\times \\R^+$ of Euclidean semi-balls with center at $(\\lambda\/\\mu,0)$ and\nradius $1\/|\\mu|$, $\\mu\\not=0$. Since $\\mu\\in\\cO_K$, it follows that \n$|\\mu|\\ge 1$ and therefore, we have\n\\begin{equation}\\label{3.2}\nF_K\\times [1,\\infty)\\subset \\cF_K.\n\\end{equation}\nFor $\\Q(i)$ and $\\Q(\\sqrt{-3})$,\nthe fundamental domain is a hyperbolic pyramid with one vertex at $\\infty$.\n\nLet $d_K$ be the discriminant of $K$ and\nlet $\\zeta_K(s)$ be the Dedekind zeta function of $K$. Let\n\\[\\Lambda_K(s)=\\left(\\frac{\\sqrt{|d_K|}}{2\\pi}\\right)^s\\Gamma(s)\\zeta_K(s)\\]\nbe the completed zeta function. \nThen the constant term of the Eisenstein\nseries $E(w,s)$ is given by\n\\begin{equation}\\label{3.3}\nE_0(y,s)=y^s+\\frac{\\Lambda_K(s-1)}{\\Lambda_K(s)}y^{2-s}\n\\end{equation}\n\\cite[Section 8.3]{EGM}, \\cite{ES}, and therefore, the scattering matrix\n $c_K(s)$ equals\n\\begin{equation}\\label{3.4}\nc_K(s)=\\frac{\\Lambda_K(s-1)}{\\Lambda_K(s)}.\n\\end{equation} \nThe completed zeta function satisfies\nthe functional equation $\\Lambda_K(s)=\\Lambda_K(1-s)$. All its zeros are\ncontained in the strip $0<\\Re(s)<1$. This implies that $\\Lambda_K(s-1)$\nand $\\Lambda_K(s)$ have no common zeros. Therefore for $a>0$, the zeros of \n\\[\\varphi_K(a,s)=a^s\\Lambda_K(s-1)+a^{2-s}\\Lambda_K(s)\\]\ncoincide with the zeros of $E_0(a,s)$. \n\nWe can now apply Theorem \\ref{th1.5}. By (\\ref{3.2}) we can take $b=1$.\nLet $a\\ge 1$ and let $\\rho\\not=1$ be a zero of $\\psi_K(a,s)$. Then by\nTheorem  \\ref{th1.5}, 2), $\\rho(2-\\rho)$ is an eigenvalue of $\\Delta_a$. This\nimplies \nthat if $\\rho(2-\\rho)\\ge 1$, then $\\rho=1+ir$ with $r\\in\\R$, $r\\not=0$. \nIf $\\rho(2-\\rho)< 1$, then $\\rho$ is real. Thus all complex zeros of \n$\\varphi_K(a,s)$ are on the line $\\Re(s)=1$.  \n\nFor the real zeros we need to consider the non-cuspidal eigenvalues \n of $\\Delta$. By the spectral resolution of the Laplacian, these eigenvalues \nare in \none-to-one correspondence with the poles of the Eisenstein series $E(w,s)$ in \nthe interval $(1,2]$ \\cite[Proposition 6.2.2]{EGM}.\n On the other hand, using the\nMaass-Selberg relations, it follows that\nthe poles of $E(w,s)$ in $(1,2]$ coincide with the poles of the scattering\nmatrix $c_K(s)$ in $(1,2]$. Since $\\Lambda_K(s)$ has no zeros in $\\Re(s)>1$\nand the only pole in $\\Re(s)>0$ is a simple pole at $s=1$, it follows from\n(\\ref{3.4}) that the only pole of $c_K(s)$ in $\\Re(s)>1$ is a simple pole at \n$s=2$ which corresponds to the eigenvalue 0. This shows that $\\Delta$ has no\nnon-cuspidal eigenvalues in the interval $(0,1)$.  \n By Theorem \\ref{th1.5}, 4), it follows that  $\\varphi_K(a,s)$ has  at most \ntwo real zeros $\\rho_a$ and $2-\\rho_a$ with\n$\\rho_a\\in (1,2)$. Let $a<a^\\prime$. Then by Theorem \\ref{th1.5}, 1), we \nhave\n\\[-(\\rho_a-1)^2=\\rho_a(2-\\rho_a)-1\\ge \\rho_{a^\\prime}(2-\\rho_{a^\\prime})-1=\n-(\\rho_{a^\\prime}-1)^2.\\]\nThis implies $\\rho_a\\le\\rho_{a^\\prime}$, and by 4) it follows that\n$\\lim_{a\\to\\infty}\\rho_a=1$. \n\nNext we consider the eigenvalue $1$. Recall that $\\Lambda_K(s)$ has simple\npoles at $s=0$ and $s=1$. \nUsing the functional equation, we get ${\\mathrm Res}_{s=1}\\Lambda_K(s)=-\n{\\mathrm Res}_{s=0}\\Lambda_K(s)$. This implies $c_K(1)=-1$.\nBy Theorem \\ref{th1.5}, 3), $1$ occurs as an eigenvalue $\\mu_j(a)$ of\n$\\Delta_a$ if and only if $c_K^\\prime(1)=-2\\log a$. Let\n\\[\\xi_K(s)=s(s-1)\\Lambda_K(s).\\]\nThen\n\\[c_K^\\prime(1)=-\\frac{c_K^\\prime(1)}{c_K(1)}=\n\\lim_{s\\to 1}\\left(\\frac{\\Lambda_K^\\prime(s)}{\\Lambda_K(s)}\n-\\frac{\\Lambda_K^\\prime(s-1)}{\\Lambda_K(s-1)}\\right)\n=-2\\left(1+\\frac{\\xi_K^\\prime(0)}{\\xi_K(0)}\\right).\\]\nLet \n\\[a^\\ast_K:=\\exp\\left(1+\\frac{\\xi_K^\\prime(0)}{\\xi_K(0)}\\right).\\]\nThen by the above, there is $j$ such that  $\\mu_j(a)=1$ if and only if \n$a=a^\\ast_K$. Assume that $a^\\ast_K\\ge 1$. Using that the eigenvalues $\\mu_j(a)$\nof $\\Delta_a$ are non-increasing and have multiplicity one, we get \n$\\mu_0(a^\\ast_K)=1$, $\\mu_0(a)<1$, if $a>a^\\ast_K$, and $\\mu_0(a)>1$, if\n$a<a^\\ast_K$. \nIf $a^\\ast_K<1$, then $\\mu_0(a)<1$ for all $1\\le a$.\nUsing that $\\mu_0(a)=\\rho_a(2-\\rho_a)$, we get 1) and 2) of Theorem \n\\ref{th0.3}. The simplicity of the zeros $\\rho\\not=1$ follows again from\nthe Maass-Selberg relations \\cite[p.270]{EGM} in the same way as in the case \nof $\\operatorname{PSL}(2,\\Z)$.\n\\hfill$\\Box$\n\nThe constant $a^\\ast_K$  in Theorem \\ref{th0.3} can be computed as follows. \nLet $w_K$ be the order of the group of units of $\\cO_K$. Let $1,\\omega$ be \na basis for $\\cO_K$ as a $\\Z$ module. \nSince $K$ has class number one, we have\n\\[\\zeta_K(s)=\\frac{1}{w_K}{\\sum}^\\prime_{m,n}\\frac{1}{|m+n\\omega|^{2s}},\\] \nthe sum being taken for all integers $(m,n)\\not=(0,0)$. \nUsing the Kronecker limit formula \\cite[p. 273]{La}, we get for \n$K=\\Q(\\sqrt{-D})$\n\\begin{equation}\na^*_{K}=\\frac{e^{2+\\gamma}}{2\\pi\\sqrt{D}}\\begin{cases}\n\\frac{1}{2|\\eta(i\\sqrt{D})|^4},&\\quad D\\equiv 2,3 \\mod 4\\\\ \n\\frac{1}{\\big|\\eta\\left(\\frac{1+i\\sqrt{D}}{2}\\right)\\big|^4},&\\quad D\\equiv 1\n\\mod 4, \\end{cases}\n\\end{equation}\nwhere $\\gamma$ denotes Euler's constant and $\\eta(z)$ is the Dedekind eta \nfunction. The values of the eta function  can be computed by the\nChowla-Selberg formula \\cite[(2), p.110]{SC}.  Let $\\Delta(z)=\\eta(z)^{24}.$ \nLet $D\\equiv 2,3 \\mod 4$ and assume that $h(D)=1$. Then the Chowla-Selberg\nformula gives \n\\begin{equation} \n\\Delta(\\sqrt{D}i)=\\frac{1}{(8\\pi D)^6}\\left\\{\\prod_{m=1}^{4D}\n\\Gamma\\left(\\frac{m}{4D}\\right)^{\\left(\\frac{-4D}{m}\\right)}\\right\\}^{3w_K}.\n\\end{equation}\nSo for $D=1$ we get\n\\[\\Delta(i)=\\frac{1}{(8\\pi)^6}\\left(\\frac{\\Gamma\\left(\\frac{1}{4}\\right)}\n{\\Gamma\\left(\\frac{3}{4}\\right)}\\right)^{12}=\n\\frac{\\Gamma\\left(\\frac{1}{4}\\right)^{24}}{2^6(2\\pi)^{18}},\\] \nwhere in the right equality we used the duplication formula of the Gamma\nfunction. This implies\n$$a^\\ast_{\\Q(i)}=\\frac{4\\pi^2 e^{2+\\gamma}}{\\Gamma(1\/4)^4}\\approx\n3.0068.$$\nSimilarly for $D=2$ we get \n\\[\\Delta(\\sqrt{2}i)=\\frac{1}{(2\\pi)^68^6}\n\\left(\\frac{\\Gamma\\left(\\frac{1}{8}\\right)\\Gamma\\left(\\frac{3}{8}\\right)}\n{\\Gamma\\left(\\frac{5}{8}\\right)\\Gamma\\left(\\frac{7}{8}\\right)}\\right)^6\n=\\frac{\\left(\\Gamma\\left(\\frac{1}{8}\\right)\\Gamma\\left(\\frac{3}{8}\\right)\\right)^{12}}{2^{15}(2\\pi)^{18}},\\]\nand therefore\n\\[a^\\ast_{\\Q(\\sqrt{-2})}=\\frac{8\\pi^2e^{2+\\gamma}}\n{\\left(\\Gamma\\left(\\frac{1}{8}\\right)\n\\Gamma\\left(\\frac{3}{8}\\right)\\right)^{2}}\\approx \n3.2581 .\\]\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis paper deals with  two problems related to {\\em algebraic\nshifting} that were raised by Gil Kalai in \\cite{K00}.\n\nAlgebraic shifting is an algebraic operation introduced\nby  Kalai \\cite{BK}, \\cite{K91} that transforms a simplicial\ncomplex $\\Gamma$ into a simpler ({\\em shifted}) complex \n$\\Delta(\\Gamma)$, while preserving\nimportant combinatorial, topological and algebraic invariants such as\nface numbers, reduced Betti numbers and extremal algebraic Betti\nnumbers.  \nThere are two versions of algebraic shifting ---\nexterior and symmetric: the first one amounts to computing \nthe (degree) reverse lexicographic generic initial ideal ($\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}$) of \nthe Stanley-Reisner ideal of $\\Gamma$ in the exterior\nalgebra, while the second one amounts to computing  $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}$\nin the symmetric algebra and then applying  a certain ``squarefree''\noperation $\\Phi$. In this paper we consider only the symmetric version\nof algebraic shifting. We refer to  this operation as {\\em revlex shifting}\nand denote it by $\\Delta_{\\mbox{\\upshape {\\tiny rl}}}$.\n\n\nClearly $\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)\\neq \\Gamma$ if $\\Gamma$ is not shifted.\nAmong the many beautiful properties of revlex shifting is the fact that\nthe converse statement holds as well, namely that\n\\begin{equation}           \\label{P5}\n\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)=\\Gamma \\quad \\mbox{if } \\Gamma \\mbox{ is shifted,}\n\\end{equation}\nand hence that $\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma))=\n\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)$ for an arbitrary complex $\\Gamma$. \nThis result was stated in \\cite{K91}\nand a somewhat hard proof was given in \\cite{AHH}.\nEq.~(\\ref{P5}) along with the two problems on algebraic shifting\n posed by Gil Kalai \\cite[Problems 16 \\& 5]{K00} is the starting\npoint of our paper.\n\nIn \\cite[Problem 16]{K00} Kalai asks if algebraic shifting\ncan be axiomatized. In that direction we prove the following result.\n(We denote by $[n]$  the set $\\{1, 2, \\ldots, n\\}$, and by $f(\\Gamma)$ and \n$\\beta_i(\\Gamma)$, $i\\geq 0$, the $f$-vector and \nthe reduced simplicial Betti numbers of $\\Gamma$ computed with coefficients\nin a field {\\bf k}, respectively.)\n\\begin{theorem}     \\label{stable}\nLet $\\Delta$ be an operation  that associates with every $n\\geq 0$ \nand every\nsimplicial complex $\\Gamma$ on the vertex set $V=[n]$ a \nshifted simplicial complex $\\Delta(\\Gamma)$ on the same vertex set.\n Assume further that $\\Delta$ satisfies the following properties:\n\\begin{enumerate}\n\\item $f(\\Delta(\\Gamma))=f(\\Gamma)$;\n\\item \n$\\Delta(\\Gamma\\ast\\{n+1\\})=\\Delta(\\Gamma)\\ast\\{n+1\\}$;\n\\item if $\\Gamma'\\subseteq\\Gamma$, then\n$\\Delta(\\Gamma')\\subseteq \\Delta(\\Gamma)$; \n\\item \n$\\sum_{i=0}^{\\dim\\Gamma}\\beta_i(\\Gamma)\\leq \n\\sum_{i=0}^{\\dim\\Delta(\\Gamma)}\\beta_{i}(\\Delta(\\Gamma))$.\n\\end{enumerate}\nThen for every shifted complex $\\Gamma$, $\\Delta(\\Gamma)=\\Gamma$.\n\\end{theorem}\nAs a corollary we obtain a new and much simpler proof of Eq.~(\\ref{P5}).\n(Here $\\Gamma\\ast\\{n+1\\}$ is {\\em the cone} over $\\Gamma$, \nthat is, a simplicial\ncomplex on the vertex set $[n+1]$ whose set of faces consists of faces of \n$\\Gamma$ together\nwith $\\{F\\cup\\{n+1\\} : F\\in \\Gamma\\}$.)\n\n\nProblem 5 in \\cite{K00} asks whether the\nproperty given by Eq.~(\\ref{P5}) holds if\none considers symmetric shiftings with respect to arbitrary\nterm orders. \nSince in the case of exterior shiftings the answer is positive (as was shown \nby Kalai \\cite[Prop.~4.2]{Ka90}),\none may expect to have the same result in the symmetric case as well. \nHere we consider (degree) lexicographic order, \nand denote the corresponding shifting operation by $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}$. \nTo our surprise we discover that only very few shifted complexes are fixed\nby lex shifting.\nOur results are summarized in Theorem \\ref{Thm2} below. \n\nDenote by $\\mathbb{N}$ the set of all positive integers. We say that an ideal \n$I\\subset S={\\bf k}[x_i: i\\in\\mathbb{N}]$ is a \n{\\em universal squarefree lexsegment ideal} (abbreviated USLI) if it is \n finitely generated in each degree and is a squarefree\nlexsegment ideal of $S$. \n(Equivalently, an ideal $I$ of $S$ that is finitely generated\nin each degree is a USLI if $I\\cap S_{[n]}$  is a \nsquarefree lexsegment ideal of $S_{[n]}:={\\bf k}[x_1, \\ldots, x_n]$\nfor every $n$.)\nThus, for example, the ideal \n$\\langle x_1x_2, x_1x_3, x_1x_4x_5x_6x_7\\rangle$ is a USLI, while the ideal\n$\\langle x_1x_2, x_1x_3, x_2x_3\\rangle$\nis a squarefree lexsegment of $S_{[3]}$ \nbut is not a squarefree lexsegment of $S$,\nand hence is not a USLI. \nA simplicial complex $\\Gamma$ is a {\\em USLI complex}\nif its Stanley-Reisner ideal, $I_\\Gamma$, is a USLI\n\nRecall that for a monomial ideal $J\\subset S_{[n]}$\nthe {\\em (bi-graded) Betti numbers} of $J$ are the invariants \n$\\beta_{i,j}(J)$ that appear in the minimal free resolution\nof $J$ as an $S_{[n]}$-module.\n\\begin{equation}    \\label{alg_betti}\n\\ldots \n \\bigoplus_j S_{[n]}(-j)^{\\beta_{i,j}(J)} \\rightarrow \\ldots\n\\rightarrow \\bigoplus_j S_{[n]}(-j)^{\\beta_{1,j}(J)} \\rightarrow\n\\bigoplus_j S_{[n]}(-j)^{\\beta_{0,j}(J)} \\rightarrow J \\rightarrow 0\n\\end{equation}\nHere $S_{[n]}(-j)$ denotes $S_{[n]}$ with grading shifted by $j$.\nFollowing \\cite{Eis2}, we define the {\\em $B$-sequence} of $J$,\n$B(J):=\\{B_j(J) : j\\geq 1\\}$, where $B_j(J):=\\sum_{i=0}^j (-1)^i\n\\beta_{i,j}(J)$. (The $B$-sequence of an ideal contains the same\ninformation as its Hilbert series --- see Section\n\\ref{infinite_section} for more details as well as for the definition\nof the $B$-sequence for a monomial ideal of $S$ that is finitely\ngenerated in each degree.)\n\n\\begin{theorem} \\qquad       \\label{Thm2}\n\\begin{enumerate} \n\\item A (finite) shifted simplicial complex $\\Gamma$ satisfies\n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(\\Gamma)=\\Gamma$ if and only if $\\Gamma$ is a USLI complex.\n Moreover, if $\\Gamma$\nis not a USLI complex, then  all the complexes\nin the sequence $\\{\\Delta_{\\mbox{\\upshape {\\tiny lex}}}^i(\\Gamma)\\}_{i=0}^{\\infty}$ are distinct.\n(Here $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}^i(\\Gamma)$ denotes the complex obtained from \n$\\Gamma$ by $i$\nconsecutive applications of $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}$.)\n\n\\item The {\\em limit} complex\n$\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}(\\Gamma):=\n\\lim_{i\\rightarrow\\infty}\\Delta_{\\mbox{\\upshape {\\tiny lex}}}^i(\\Gamma)$\n is well defined and is a (usually infinite) USLI complex.\n Moreover, $\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}(\\Gamma)$ is \nthe unique USLI complex whose Stanley-Reisner ideal has the same $B$-sequence \nas $I_\\Gamma$.\n\\end{enumerate}\n\\end{theorem}\n\nThe last part of the theorem implies that\nif two simplicial complexes $\\Gamma_1$ and $\\Gamma_2$\nthat have the same $h$-vector \n(up to possibly several zeros appended at the end), then\n$\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}(\\Gamma_1)=\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}(\\Gamma_2)$.\nThus, in contrast to revlex shifting,\n the operation $\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}$\nforgets all the information that $\\Gamma$ carries (including the dimension\nof $\\Gamma$) except its $h$-numbers.\n\nOur theorems establish for simplicial complexes, results similar in\nspirit to those in commutative algebra due to Bigatti-Conca-Robbiano\n\\cite{BCR} and Pardue \\cite{Pardue}.\nTheorem~4.3 in \\cite{BCR} asserts that if $I$ is a strongly stable\nideal in $S_{[n]}$ and $\\mathcal{L}$ is a distraction matrix, then\n$\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(D_{\\mathcal L}(I))=I$, while Proposition~30 in\n\\cite{Pardue} asserts that sufficiently (but finitely) many\napplications of the operation $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}} \\circ D_{\\mathcal L}$ to a\nmonomial ideal $I\\subset S_{[n]}$ results in the unique lexsegment\nideal of $S_{[n]}$ having the same Hilbert function as $I$.\n\nThe structure of the paper is as follows. Section \\ref{axiom_section}\nis devoted to the proof of Theorem \\ref{stable}. \nIn Section \\ref{revlex_section}\nafter recalling basic facts and definitions related to generic initial ideals\nand revlex shifting we provide a short new proof of Eq.~(\\ref{P5}). \nIn Section \\ref{USLI_section} we introduce and study the class of\nuniversal squarefree lexsegment ideals (USLIs) \nand the class of {\\em almost USLIs} ---\nthe notions that play a crucial role in the proof of Theorem \\ref{Thm2}.\n Finally in Section \\ref{infinite_section} we prove Theorem \\ref{Thm2}.\nWe close with a brief discussion of arbitrary term orders.\n\n\\section{Axiomatizing Algebraic Shifting}    \\label{axiom_section}\nThis section is devoted to the proof of Theorem \\ref{stable}.\n We start by reviewing several\nnotions pertaining to simplicial complexes.\n\nDenote the collection of all subsets of \n$[n]:=\\{1, 2, \\ldots, n\\}$ by $2^{[n]}$.\nRecall that a simplicial complex $\\Gamma$ on the vertex set $[n]$ \nis a collection $\\Gamma\\subseteq 2^{[n]}$ that  is\nclosed under inclusion. \n(We do not require that every singleton $\\{i\\}\\subseteq[n]$\nis  an element of $\\Gamma$.)\nThe elements of $\\Gamma$ are called faces\nand the maximal faces (under inclusion) are called {\\em facets}.\n $F\\in\\Gamma$ is an\n{\\em $i$-dimensional face} (or an $i$-face) if\n$|F|=i+1$. The {\\em dimension} of $\\Gamma$,\n$\\dim \\Gamma$, is the maximal dimension of its faces.\n The number of $i$-dimensional faces of $\\Gamma$\nis denoted by $f_i(\\Gamma)$, and the sequence \n$f(\\Gamma):=(f_{-1}(\\Gamma), f_0(\\Gamma), f_1(\\Gamma), \n\\ldots f_{\\dim\\Gamma}(\\Gamma))$ is called the\n$f$-vector of $\\Gamma$. \nAnother set of invariants associated\nwith $\\Gamma$ is the set of its  reduced\nBetti numbers \n$\\beta_i(\\Gamma):=\\dim_{\\bf k}\\widetilde{H}_i(\\Gamma; {\\bf k})$,\nwhere $\\widetilde{H}_i(\\Gamma; {\\bf k})$ is the $i$-th reduced\nsimplicial homology of $\\Gamma$ with coefficients in a field ${\\bf k}$.\n\nA simplicial complex $\\Gamma$ on the vertex set \n$[n]$ is called {\\em shifted} if for every\n$F\\in\\Gamma$, $i\\in F$, and $i<j\\leq n$, the set \n$(F\\setminus\\{i\\})\\cup\\{j\\}$ is a face of $\\Gamma$ as well.\nThe Betti numbers of a shifted complex $\\Gamma$ are combinatorial\ninvariants and can be  computed via the \nfollowing well-known formula \\cite[Thm.~4.3]{BK}:\n\\begin{lemma}   \\label{betti}\nIf $\\Gamma$ is a shifted complex on the vertex set $[n]$, then \n$$\\beta_i(\\Gamma)=|\\{F\\in\\max(\\Gamma) \\,:\\, |F|=i+1,\\, n\\notin \\Gamma\\}|,$$\nwhere $\\max(\\Gamma)$ denotes the set of \nfacets of $\\Gamma$.\n\\end{lemma}\n\nFor a simplicial complex $\\Gamma$ and a vertex $v$ of $\\Gamma$ define\nthe {\\em antistar of $v$ in $\\Gamma$} as\n$\\mbox{\\upshape ast}\\,_{\\Gamma} (v) = \\{F\\in\\Gamma: v \\notin F\\}$. Define also \nthe {\\em link of $v$ in $\\Gamma$} by $\\mbox{\\upshape lk}\\,_\\Gamma (v):=\n\\{F\\in\\mbox{\\upshape ast}\\,_{\\Gamma}(v) :  F\\cup\\{v\\}\\in\\Gamma\\}$.\nNote that if $\\Gamma$ is a shifted complex on the vertex set $[n]$,\nthen $\\mbox{\\upshape lk}\\,_\\Gamma (n)$ and $\\mbox{\\upshape ast}\\,_\\Gamma (n)$ are \nshifted complexes on $[n-1]$. \n\nIf $\\Gamma$ is a simplicial complex on $V$ and $u\\not\\in V$,\nthen the {\\em cone} over $\\Gamma$ with apex $u$ is a simplicial complex,\ndenoted  $\\Gamma\\ast \\{u\\}$, on the vertex set $V\\cup\\{u\\}$\nwhose faces are all sets of the form   \n$F\\cup A$, where $F\\in\\Gamma$ and $A\\subseteq \\{u\\}$.\nThus \nfor any vertex $v$ of $\\Gamma$, \n$\\Gamma=\\mbox{\\upshape lk}\\,_\\Gamma(v)\\ast\\{v\\} \\cup \\mbox{\\upshape ast}\\,_\\Gamma(v)$ and \n$\\mbox{\\upshape lk}\\,_\\Gamma(v)\\ast\\{v\\} \\subseteq \\Gamma \\subseteq \n             \\mbox{\\upshape ast}\\,_\\Gamma(v)\\ast\\{v\\}$.\n\nNow we are ready to verify Theorem \\ref{stable} asserting that\nif $\\Delta$ is an operation  that associates with every $n\\geq 0$ \nand every\nsimplicial complex $\\Gamma$ on the vertex set $V=[n]$ a \nshifted simplicial complex $\\Delta(\\Gamma)$ on the same vertex set,\nand if\n$\\Delta$ satisfies the following properties:\n\\begin{enumerate}\n\\item $f(\\Delta(\\Gamma))=f(\\Gamma)$;\n\\item \n$\\Delta(\\Gamma\\ast\\{n+1\\})=\\Delta(\\Gamma)\\ast\\{n+1\\}$;\n\\item if $\\Gamma'\\subseteq\\Gamma$, then\n$\\Delta(\\Gamma')\\subseteq \\Delta(\\Gamma)$; \n\\item \n$\\sum_{i=0}^{\\dim\\Gamma}\\beta_i(\\Gamma)\\leq \n\\sum_{i=0}^{\\dim\\Delta(\\Gamma)}\\beta_{i}(\\Delta(\\Gamma))$,\n\\end{enumerate}\nthen for every shifted complex $\\Gamma$, $\\Delta(\\Gamma)=\\Gamma$.\n\n\\smallskip\\noindent {\\it Proof of Theorem \\ref{stable}: \\ } \nFix a shifted complex $\\Gamma$ on $n$ vertices.\nIf $n=0$ or $n=1$ then $\\Delta(\\Gamma)=\\Gamma$ by property (1). \nWe proceed by induction on~$n$. Since\n the link and the antistar of the vertex $n$\nin $\\Gamma$, $\\Gamma'=\\mbox{\\upshape lk}\\,_\\Gamma (n)$ and \n$\\Gamma''=\\mbox{\\upshape ast}\\,_\\Gamma (n)$, \nrespectively,\nare shifted complexes on the vertex set $[n-1]$\nand since \n$\\Gamma'\\ast \\{n\\}\\subseteq \\Gamma \\subseteq \\Gamma''\\ast\\{n\\}$,\nthe induction hypothesis together with\nproperties (2) and (3) yield\n$$\\Gamma'\\ast \\{n\\}\\subseteq \\Delta(\\Gamma) \n\\subseteq \\Gamma''\\ast\\{n\\}.$$\nTherefore,\n\\begin{eqnarray*}\nA &:=& \\{F\\in\\max(\\Delta(\\Gamma)) \\,:\\, n\\notin F\\} \n=\\{F\\in\\max(\\Delta(\\Gamma)) \\,:\\,  F\\in \\Gamma''\\} \\\\\n&\\subseteq& \\Gamma''\\setminus\\Gamma' \\stackrel{(\\star)}{=}\n\\{F\\in\\max(\\Gamma) \\,:\\, n\\notin F\\}=:B,\n\\end{eqnarray*}\nwhere \n$(\\star)$ follows from the shiftedness of $\\Gamma$:\n$$F\\in\\max(\\Gamma)\\cap 2^{[n-1]}  \\Longleftrightarrow  \nF\\in\\Gamma \\mbox{ but } F\\cup\\{n\\} \\notin \\Gamma \\Longleftrightarrow \nF\\in \\mbox{\\upshape ast}\\,_\\Gamma(n)\\setminus \\mbox{\\upshape lk}\\,_\\Gamma(n)=\\Gamma''\\setminus \\Gamma'.$$ \nOn the other hand, Lemma \\ref{betti} and property (4) imply that\n$$|A|=\\sum_{i=0}^{\\dim\\Delta(\\Gamma)}\\beta_i(\\Delta(\\Gamma))\\geq \n\\sum_{i=0}^{\\dim\\Gamma} \\beta_i(\\Gamma)=|B|,$$\nand thus that $A=B$. Hence\n$\\Delta(\\Gamma)\\supseteq A=\\Gamma''\\setminus \\Gamma'$, and\nwe infer that \n$$\\Delta(\\Gamma)\\supseteq (\\Gamma'\\ast\\{n\\})\\cup( \\Gamma''\\setminus \\Gamma')=\n\\Gamma.$$\n Since $f(\\Gamma)=f(\\Delta(\\Gamma))$ by property (1), it follows\nthat $\\Delta(\\Gamma)=\\Gamma$. \\hfill$\\square$\\medskip\n\n\n\n\\section{Generic Initial Ideals and revlex shifting}          \n                        \\label{revlex_section}\nIn this section we review basic facts and definitions related to\ngeneric initial ideals and revlex shifting.  We also provide a new short proof\nof Eq.~(\\ref{P5}) asserting that\n $\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)=\\Gamma$ for a shifted \n$\\Gamma$.\nLet $S_{[n]}={\\bf k}[x_1, \\ldots, x_n]$ \nbe the ring of polynomials in $n$  variables\n over  a field ${\\bf k}$ of characteristic zero, and let $\\Gamma$ be a\n simplicial complex on the vertex set $[n]$. We recall that\nthe {\\em Stanley-Reisner ideal} of  $\\Gamma$ \n\\cite{St} is the squarefree monomial  ideal $I_\\Gamma \\subset S_{[n]}$ \nwhose generators correspond to nonfaces of $\\Gamma$:\n$$ I_{\\Gamma} := \\langle \\prod_{j=1}^k x_{i_j} \\in S_{[n]}\\,:\\,\n\\{i_1<i_2<\\ldots<i_k\\}\\notin\\Gamma\\rangle.\n$$\nThe Stanley-Reisner ideal of a shifted complex is called a\n{\\em squarefree strongly stable ideal}. \n(Equivalently, a squarefree monomial ideal  \n$I$ is squarefree strongly stable, if for every minimal generator $m$ of \n$I$ and for\nevery $1\\leq i<j$ such that $x_j|m$ but $x_i\\not|m$, the monomial $mx_i\/x_j$\nlies in $I$.)\n\nLet $\\succ$ be a term order on $S_{[n]}$ that refines the partial\norder by degree where lower degree monomials are more expensive than\nhigher degree monomials, and satisfies $x_1\\succ\\ldots\\succ x_n$. Let\n$I\\subset S_{[n]}$ be a homogeneous ideal such as the Stanley-Reisner\nideal of $\\Gamma$.  Consider a generic $n\\times n$ matrix $g$.  Then\n$g$ acts on the set of linear forms of $S_{[n]}$ by\n$gx_j=\\sum_{i=1}^{n}g_{ij}x_i$ and this action can be extended\nuniquely to a ring automorphism on $S_{[n]}$ that we also denote by\n$g$.  Following \\cite[Thm.~15.18]{Eis} define the {\\em generic initial\n  ideal of} $I$ with respect to $\\succ$ as\n$$\\mbox{\\upshape Gin}_\\succ(I):=\\mbox{\\upshape in}_\\succ(gI),$$\nwhere $\\mbox{\\upshape in}_\\succ(gI)$ is the\ninitial ideal of $gI$ with respect to $\\succ$ in the sense of\nGr\\\"obner basis theory. The same theorem in \\cite{Eis} asserts that we\ncan choose $g$ to be upper triangular and hence we assume from now on\nthat $gx_j=\\sum_{i=1}^jg_{ij}x_i$.\n\nWe briefly outline how to compute $\\mbox{\\upshape Gin}_\\succ(I)$ (for a detailed description\nthe reader is referred to \\cite[Thm.~15.18]{Eis}).\n\\begin{definition}\nAn exterior monomial in  $\\bigwedge^l(S_{[n]})_d$ is an element of the form \n$m_1\\wedge\\ldots\\wedge m_l$ where each $m_i$ is a monomial of $S_{[n]}$ \nof degree $d$ and $m_1\\succ \\ldots \\succ m_l$. The \nextension of $\\succ$ to an order on monomials\nof $\\bigwedge^l(S_{[n]})_d$ is the order in which \n$m_1\\wedge\\ldots\\wedge m_l \\succ n_1\\wedge\\ldots\\wedge n_l$ if for some $s$\nwe have that $m_s\\succ n_s$ and $m_i=n_i$ for $i<s$. For a non-zero element \n$f$ of $\\bigwedge^l(S_{[n]})_d$, define the {\\em initial term}  of $f$, \n$\\mbox{\\upshape in}_\\succ(f)$, to be\nthe $\\succ$-largest monomial appearing in $f$ with nonzero coefficient\nwhen $f$ is written as a linear combination of (distinct) monomials.\n\\end{definition}\n\nConsider a generic $n\\times n$ upper-triangular matrix $g$ \nand its action on $S_{[n]}$.\nLet $I_d$ be the $d$-th homogeneous component of a homogeneous ideal $I$, \nand let  $f_1, \\ldots,  f_t$ be a basis of $I_d$.\nThen $g(f_1)\\wedge \\ldots \\wedge g(f_t)\\in\\wedge^t(S_{[n]})_d$.\nDenote by  $M_d=m_1\\wedge \\ldots \\wedge m_t$ the monomial\n$\\mbox{\\upshape in}_\\succ (g(f_1)\\wedge \\ldots \\wedge g(f_t))$ \nand by $V_d$  the subspace of $(S_{[n]})_d$\nspanned by $m_1, \\ldots, m_t$.\n\n\\begin{proposition}  \\label{gin_constr}\n $\\mbox{\\upshape Gin}_\\succ(I)=\\bigoplus V_d$.\n\\end{proposition}\n\n\nSeveral basic properties of Gins are summarized in the following lemma.\n\\begin{lemma}  \\label{cone}\nLet $I\\subset S_{[n]}$  be a homogeneous ideal. Then \n\\begin{enumerate}\n\\item $\\mbox{\\upshape Gin}_\\succ(I)$ is a strongly stable monomial ideal (that is, if \n$m\\in \\mbox{\\upshape Gin}_\\succ(I)$, $x_j|m$ and $1\\leq i <j$, then \n$x_im\/x_j\\in \\mbox{\\upshape Gin}_\\succ(I)$ as well). \n\\item $\\mbox{\\upshape Gin}_\\succ(I)$ and $I$ have the same Hilbert function \n(that is, $\\dim_{\\bf k}(\\mbox{\\upshape Gin}_\\succ(I)_d)=\\dim_{\\bf k}(I_d)$ for all $d$).\n\\item If $J\\subseteq I$ is a homogeneous ideal of $S_{[n]}$, then \n$\\mbox{\\upshape Gin}_\\succ(J)\\subseteq \\mbox{\\upshape Gin}_\\succ(I)$.\n\\item Let\n$\\succ'$ be an extension of  $\\succ$ to a term order on $S_{[n+1]}$ satisfying\n$x_n\\succ' x_{n+1}$. Then \n$\\mbox{\\upshape Gin}_{\\succ'}(IS_{[n+1]})=(\\mbox{\\upshape Gin}_\\succ I) S_{[n+1]}.$\nIn particular, for a simplicial complex $\\Gamma$ on $[n]$,\n$\\mbox{\\upshape Gin}_{\\succ'}(I_{\\Gamma\\ast\\{n+1\\}})=(\\mbox{\\upshape Gin}_\\succ I_\\Gamma) S_{[n+1]}.$\n\\end{enumerate}\n\\end{lemma}\n\\smallskip\\noindent {\\it Proof: \\ }\nPart (1)  is \\cite[Thm.~15.18 and Thm.~15.23]{Eis}. \nPart (2) follows from \\cite[Thm 15.3]{Eis}.\nPart (3) is obvious from the definitions.\nTo  prove part (4),  consider a generic upper-triangular \n$(n+1)\\times(n+1)$ matrix  $\\widetilde{g}$ and \nits left-upper $n\\times n$ submatrix $g$. \nThen $g$ acts on $S_{[n]}$, $\\widetilde{g}$\nacts on $S_{[n+1]}$, and \n$\\widetilde{g}x_i=gx_i$ for all $1\\leq i\\leq n$. \nTherefore for every (homogeneous)  element \n$h$ of $I\\subset S_{[n]}\\subset S_{[n+1]}$,\n$\\widetilde{g}h=gh$. Thus for $h\\in I$,\n $\\mbox{\\upshape in}_{\\succ'}(\\widetilde{g}h)=\\mbox{\\upshape in}_{\\succ}(gh)$, implying that\n$\\mbox{\\upshape Gin}_\\succ I \\subseteq \\mbox{\\upshape Gin}_{\\succ'}(IS_{[n+1]})$, and hence that\n$(\\mbox{\\upshape Gin}_\\succ I)S_{[n+1]}\\subseteq \\mbox{\\upshape Gin}_{\\succ'}(IS_{[n+1]})$. \nThe lemma follows, since both  the ideals $(\\mbox{\\upshape Gin}_\\succ I)S_{[n+1]}$ and \n$\\mbox{\\upshape Gin}_{\\succ'}(IS_{[n+1]})$\nhave the same Hilbert function.  \\hfill$\\square$\\medskip\n\nIn the later sections we compare Gins of the same ideal $I$\ncomputed with respect to different term orders. \nFor that  we need the following definition:\n\n\\begin{definition}\n  Let $I_1\\neq I_2$ be two monomial ideals of $S_{[n]}$ and let $\\succ$ be\n  a term order.  We say that $I_1\\succ I_2$ if the largest monomial in\n  the symmetric difference of $I_1$ and $I_2$ is in $I_1$.\n  Equivalently, $I_1\\succ I_2$ if the largest monomial in the\n  symmetric difference of $G(I_1)$ and $G(I_2)$ is in $G(I_1)$, where\n  $G(I_1)$ and $G(I_2)$ are the sets of minimal generators of $I_1$\n  and $I_2$ respectively.\n\\end{definition}\n\nOne immediate observation is\n\\begin{lemma}  \\label{gin>gin}\nLet $\\sigma$ and $\\tau$ be two term orders on $S_{[n]}$.\nThen $\\mbox{\\upshape Gin}_{\\sigma}(I)\\geq_\\sigma \\mbox{\\upshape Gin}_{\\tau}(I)$\nfor any homogeneous ideal $I\\subset S_{[n]}$.\n\\end{lemma} \n\\smallskip\\noindent {\\it Proof: \\ }\nLet $f_1, \\ldots, f_t$ be a basis of $I_d$, and let \n$g$ be a generic $n \\times n$\nupper-triangular matrix. \nSince $M'_d:=\\mbox{\\upshape in}_{>_{\\tau}}(g(f_1)\\wedge\\ldots\\wedge g(f_t))$ appears\nin $g(f_1)\\wedge\\ldots\\wedge g(f_t)$ with a non-zero coefficient, it follows\nthat \n$M_d:=\\mbox{\\upshape in}_{\\sigma}(g(f_1)\\wedge\\ldots\\wedge g(f_t)) \\geq_{\\sigma} M'_d$\n(for every $d\\geq 0$).\nProposition \\ref{gin_constr} implies the lemma.\n\\hfill$\\square$\\medskip\n\nWe remark that a stronger version of Lemma \\ref{gin>gin} was proved in \n\\cite[Cor.~1.6]{Conca}.\n\nAnother ingredient needed for defining revlex shifting\nis the notion of the squarefree operation. This is a bijection $\\Phi$\nbetween the set of all monomials in $\\{x_i : i\\in \\mathbb{N}\\}$ and the set of all\nsquarefree monomials in $\\{x_i : i\\in \\mathbb{N}\\}$, defined by\n$$\\Phi(\\prod_{j=1}^k x_{i_j})=\\prod_{j=1}^k x_{i_j+j-1}, \\mbox{ where }\n   i_1\\leq i_2\\leq \\ldots\\leq i_k.\n$$\n Note that for a monomial  $m\\in S_{[n]}$, \n$\\Phi(m)$ may not belong to  $S_{[n]}$. However the graded \nreverse lexicographic order \n has the following remarkable property \n\\cite[Lemma 6.3(ii)]{K91}, \\cite[Lemma 1.1]{AHH}: if \n$m$ is a minimal generator of $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}} I_\\Gamma$ \n(where $\\Gamma$ is a simplicial \ncomplex on $[n]$), then $\\Phi(m)$ is an element of $S_{[n]}$.\nThis leads to the following definition (due to Kalai):\n\\begin{definition}   \\label{equiv}\nLet $\\Gamma$ be a simplicial complex on the vertex set $[n]$.\nThe reverse lexicographic shifting of $\\Gamma$,\n$\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)$, is a simplicial  complex on $[n]$ whose \nStanley-Reisner ideal is given by\n$$I_{\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)}=\\langle \\Phi(m) \\; : \\; m\\in\nG(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}} I_\\Gamma) \\rangle,$$\nwhere $G(I)$ denotes the set of\nthe minimal generators of a monomial ideal $I$.\n\\end{definition}\n\nWe now provide a new and simple proof of Eq.~(\\ref{P5})\n (due originally to Aramova, Herzog, and Hibi \\cite{AHH}).\n\\begin{theorem}   \\label{AHH}\nThe revlex shifting $\\Delta_{\\mbox{\\upshape {\\tiny rl}}}$ satisfies all the conditions of \nTheorem \\ref{stable}. Thus $\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma)=\\Gamma$ for every\n shifted complex $\\Gamma$.\n\\end{theorem} \n\\smallskip\\noindent {\\it Proof: \\ } It is well-known that (symmetric) revlex shifting satisfies all\nthe conditions of Theorem \\ref{stable}, except possibly for property\n(2) whose proof appears to be missing in the literature (for the\nexterior version of algebraic shifting it was recently verified by\nNevo \\cite{Nevo}): the fact that $\\Delta(\\Gamma)$ is a shifted\nsimplicial complex follows from Lemma~\\ref{cone}(1); property (1) is\n\\cite[Lemma 6.3(i)]{K91}; property (3) is a consequence of \nLemma~\\ref{cone}(3); property (4) follows from \\cite[Cor.~8.25]{H} asserting\nthat $\\beta_i(\\Gamma)=\\beta_i(\\Delta(\\Gamma))$ for all $i$. To prove\nproperty (2) it suffices to check that $\\Delta(\\Gamma)$ and\n$\\Delta(\\Gamma\\star \\{n+1\\})$ have the same set of minimal nonfaces\n(equivalently, $I_{\\Delta(\\Gamma)}\\subset S_{[n]}$ and\n$I_{\\Delta(\\Gamma\\star \\{n+1\\})}\\subset S_{[n+1]}$ have the same set\nof minimal generators).  This follows from Definition \\ref{equiv} and\nLemma \\ref{cone}(4). \\hfill$\\square$\\medskip\n\n\\paragraph{\\bf Remarks} \\quad\n\n(1)\n  We note that to verify the inequality $\\sum \\beta_i(\\Gamma)\\leq \\sum\n  \\beta_i(\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma))$ one does not need to use the fact\n  that $\\beta_i(\\Gamma)=\\beta_i(\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma))$ for all\n  $i$, which is a consequence of the deep result due to\n  Bayer--Charalambous--Popescu \\cite{BCP} and Aramova--Herzog~\\cite{AH} \nthat revlex shifting preserves extremal (algebraic) Betti\n  numbers.  Instead one can use the standard flatness argument (see\n  \\cite[Thm.~3.1]{H}) to show that $\\beta_{i,j}(I_\\Gamma) \\leq\n  \\beta_{i,j}(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I_\\Gamma)) = \\beta_{i,j}(I_{\\Delta(\\Gamma)})$ \nfor all $i$, $j$, where the equality comes from\n  the fact that $\\Phi$ applied to (minimal generators of)\n a strongly stable ideal\n  $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I_\\Gamma)$ preserves algebraic Betti numbers (see\n  \\cite[Lemma 2.2]{AHH}). The Hochster formula \\cite{Hoc} then asserts\n  that the reduced Betti numbers of a simplicial complex are equal to\n  certain algebraic graded Betti numbers of its Stanley-Reisner ideal.\n\n(2)\n In algebraic terms, the statement of\nTheorem \\ref{AHH} translates to the fact that if\n$I\\subset S_{[n]}$ is a squarefree strongly stable ideal, then\n$\\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I))=I$, where  \n$\\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I)):=\\langle \\Phi(m): m \\in G(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I)) \\rangle\n$.\nHence $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I)=\\langle\\Phi^{-1}(\\mu) : \\mu\\in G(I)\\rangle$, that is,\n computing the revlex Gin of a squarefree strongly stable ideal $I$\nsimply amounts to applying $\\Phi^{-1}$ to the  minimal generators\nof $I$.\n\n(3) Our proof (as well as the original proof  in \\cite{AHH})\nof the equation $\\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I))=I$ for a squarefree strongly\nstable ideal $I$ works only over a field ${\\bf k}$ of characteristic zero.\nWe however do not know of any counterexamples in the case of a field\nof positive characteristic.\n\n\n\n\\section{Combinatorics of USLIs, almost USLIs, and lex Gins}   \n                 \\label{USLI_section}\nIn this section we introduce and study \nthe class of universal squarefree lexsegment ideals (USLIs)\nand the class of almost USLIs. These notions turn out to be crucial \nin the proof of Theorem \\ref{Thm2}.\nTo allow for infinitely generated ideals \n(as we  need in the following section) we \nconsider the system of rings \n$S_{[n]}$, $n\\in\\mathbb{N}$,\nendowed with natural embeddings $S_{[n]}\\subseteq S_{[m]}$ for $m\\geq n$, and\nprovide definitions suitable\nfor the direct limit \nring $S=\\lim_{n\\rightarrow\\infty}S_{[n]}={\\bf k}[x_i : i\\in\\mathbb{N}]$.\n\n\nRecall that a squarefree monomial ideal \n$I\\subset S$ ($I\\subset S_{[n]}$, respectively) is a \n{\\em squarefree lexsegment ideal} of $S$ ($S_{[n]}$, respectively)\n if for every\nmonomial $m\\in I$ and every\nsquarefree monomial $m'\\in S$ ($m'\\in S_{[n]}$, respectively)\nsuch that $\\deg(m')=\\deg(m)$ and $m'>_{\\mbox{\\upshape {\\tiny lex}}} m$, \n$m'$ is an element of $I$ as well.\n\\begin{definition}   \\label{USLI}\nAn ideal $L$ of $S$ (or of $S_{[n]}$)\nis\na {\\em universal squarefree lexsegment ideal} (abbreviated USLI)\nif it is finitely generated in each degree and $LS$ is\na squarefree lexsegment ideal of $S$. \nEquivalently, an ideal \n$L=L(k_\\bullet)$ (here $k_\\bullet=\\{k_i\\}_{i\\in\\mathbb{N}}$ is a \nsequence of nonnegative integers) is a \nUSLI with $k_i$  minimal generators\nof degree $i$ (for $i\\in\\mathbb{N}$) if and only if\n the set of minimal generators of $L$, \n$G(L)$, is given by\n$$G(L)=\n\\bigcup_{r=1}^{\\infty}\\left\\{\n(\\prod_{j=1}^{r-1} x_{R_j})\\cdot x_l \\;:\\; R_{r-1}+1 \n\\leq l \\leq R_r-1\\right\\},  \\mbox{ where } R_j=j+\\sum_{i=1}^j k_i. \n \n$$\n\\end{definition}\n\nThe easiest way to verify the description of the set\n$G(L)=\\{m_1 >_{\\mbox{\\upshape {\\tiny lex}}} m_2  >_{\\mbox{\\upshape {\\tiny lex}}} \\cdots \\ >_{\\mbox{\\upshape {\\tiny lex}}} m_s  >_{\\mbox{\\upshape {\\tiny lex}}} \\cdots \\}$ \nof a USLI $L$ is by induction on $s$. Indeed, if $m_1, \\cdots, m_s$ satisfy\nthe above description and\n$m_s=(\\prod_{j=1}^{r-1} x_{R_j})\\cdot x_l$, \nthen there are two possibilities for\n$m_{s+1}$: either $\\deg(m_{s+1})=\\deg(m_s)=r$ (equivalently, $l<R_r-1$)\nor $\\deg(m_{s+1})=r'>r$ (equivalently, $l=R_r-1$ and $k_i=0$ for all $r<i<r'$).\nIn the former case, since $m_{s}>_{\\mbox{\\upshape {\\tiny lex}}} m_{s+1}$ and since $m_s$ is the \nimmediate lex-predecessor of\n$m':=(\\prod_{j=1}^{r-1} x_{R_j})\\cdot x_{l+1}$,\nit follows that\n$m'\\geq_{\\mbox{\\upshape {\\tiny lex}}} m_{s+1}\\in L$ \nwhich together\nwith  $L$ being a USLI implies that \n$m'\\in L$. Since $m'$ is not divisible by any of $m_1, \\cdots, m_s$,\nthis yields\n$m_{s+1}=m'$. The treatment\nof the latter case is similar: just observe that every squarefree monomial\nof degree $r'$ that is lex-smaller than\n $m':=(\\prod_{j=1}^{r-1} x_{R_j})\\cdot (\\prod_{j=1}^{r'-r+1} x_{l+j})=\n (\\prod_{j=1}^{r'-1} x_{R_j})\\cdot x_{R_{r'-1}+1}$ is divisible by at least\none of\n$m_1, \\ldots, m_s$ and hence is in $L-G(L)$, while $m'$ is not divisible\nby any of $m_1, \\cdots, m_s$.\n\n\\begin{example} \\quad\n\\begin{enumerate} \n\\item\n  The ideal $\\langle x_1x_2, x_1x_3,\n  x_2x_3\\rangle$ (the Stanley-Reisner ideal of three isolated points)\n  is a lexsegment in $S_{[3]}$, but is not a lexsegment in $S$, and\n  hence is not a USLI.  \n\n\\item\n  The ideal $I = \\langle x_1x_2, x_1x_3,\n  x_1x_4x_5x_6x_7\\rangle$ is the USLI with $k_1 = 0, k_2 = 2, k_3 =\n  k_4 = 0, k_5 = 1$ and $k_i = 0$ for all $i > 5$. In this example,\n  check that $R_1 = 1, R_2 = 4, R_3 = 5, R_4 = 6$ and $R_5 = 8$. \n\\end{enumerate}\n\\end{example}\n\nNote that every USLI is a squarefree strongly stable ideal, and hence\nis the Stanley-Reisner ideal of a shifted (possibly infinite)\nsimplicial complex (we refer to such complex as a {\\em USLI complex}).\nAll complexes considered in this section are assumed to be finite.\n\nThe following lemma describes  certain combinatorial properties of \nUSLI complexes. This lemma together with\n Lemmas \\ref{main_lemma} and  \\ref{Pardue_lemma}\nbelow provides a key step in the proof of \nTheorem \\ref{Thm2}.\n\\begin{lemma}   \\label{comb_USLI}\nLet $\\Gamma$ be a USLI complex on the vertex set $[n]$\nwith $I_\\Gamma=L(k_\\bullet)$. \n\\begin{enumerate}\n\\item If $I_\\Gamma\\neq 0$ and $k_d$ is the last nonzero entry\nin the sequence $k_\\bullet$, then $\\Gamma$ has exactly $d$ facets. \nThey are given by\n$$F_i=\\left\\{\\begin{array}{ll}\n    \\{R_j : 1\\leq j \\leq i-1\\}\\cup [R_i+1, n] &  \\mbox{ if  $1\\leq\n    i\\leq d-1$,}\\\\ \n    \\{R_1, \\ldots, R_{d-1}\\}\\cup [R_d, n] & \\mbox{ if $i=d$.}\n\\end{array}\\right.\n$$\n\\item If $\\Gamma'$ is a shifted complex on $[n]$ such that \n$f(\\Gamma)=f(\\Gamma')$, then $\\Gamma=\\Gamma'$. (In other words\nevery USLI complex is the only shifted complex in its $f$-class).\n\\end{enumerate}\n\\end{lemma} \n\\smallskip\\noindent {\\it Proof: \\ } We verify part (1) by induction on $n+d+\\sum k_i$. The assertion\nclearly holds if $d=1$ or if $\\sum k_i=1$. For instance, if $d=1$ and $k_1=n$\n(equivalently, $R_1=n+1$), then $F_1=[n+1, n]=\\emptyset$ is the only \nfacet of $\\Gamma$.\n\nNote that $R_d$\nis the index of the first variable that does\nnot divide any of the minimal generators of $I_\\Gamma$.\nThus if $R_d\\leq n$, then $\\Gamma=\\mbox{\\upshape lk}\\,_\\Gamma(n)\\star\\{n\\}$, and we are done\nby applying induction hypothesis to the USLI complex $\\mbox{\\upshape lk}\\,_\\Gamma(n)$.\nSo assume that $R_d=n+1$.\nThen $\\mbox{\\upshape lk}\\,_\\Gamma(n)$ and $\\mbox{\\upshape ast}\\,_\\Gamma(n)$\nare easily seen to be the USLI complexes on the vertex set $[n-1]$\nwhose Stanley-Reisner ideals are given\nby $L_1=L(k_1, \\ldots, k_{d-2}, k_{d-1}+1)$ and \n$L_2=L(k_1, \\ldots, k_{d-1}, k_d-1)$,\nrespectively. \nHence by induction hypothesis the complex\n$\\mbox{\\upshape lk}\\,_\\Gamma(n)\\star\\{n\\}$ has exactly $d-1$ facets, namely\nthe sets $F_1, \\ldots, F_{d-1}$ from the list above.\nNow if $k_d>1$, then  by induction hypothesis\nthe facets of $\\mbox{\\upshape ast}\\,_\\Gamma(n)$ are the sets $F_1-\\{n\\}, \\ldots,\nF_{d-1}-\\{n\\}, F_d$. Since\n$\\Gamma= (\\mbox{\\upshape lk}\\,_\\Gamma(n)\\star\\{n\\}) \\cup \\mbox{\\upshape ast}\\,_\\Gamma(n)$,\nit follows that  $\\max(\\Gamma)=\\{F_1, \\ldots, F_d\\}$.\nSimilarly, if $k_d=1$ and $k_j$ is the last nonzero\nentry in the sequence $(k_1, \\ldots, k_{d-1})$, then \n the facets of $\\mbox{\\upshape ast}\\,_\\Gamma(n)$ are the sets $F_1-\\{n\\}, \\ldots,\nF_{j-1}-\\{n\\}, F_d$, and the result follows in this case as well.\n\nTo prove part (2) we induct on $n$. The assertion is obvious for $n=1$.\nFor $n>1$ we consider two cases.\n\n{\\bf Case 1:} $R_d\\leq n$. In this case \n$\\Gamma=\\mbox{\\upshape lk}\\,_\\Gamma(n)\\star\\{n\\}$, so $\\beta_i(\\Gamma)=0$ for all $i$.\nSince among all squarefree strongly stable\nideals with the same Hilbert function\nthe squarefree lexsegment ideal has the largest \nalgebraic Betti numbers \\cite[Thm.~4.4]{AHHlex}, \nand since by Hochster's formula \\cite{Hoc}, \n$\\beta_{n-i-1}(\\Lambda)=\\beta_{i-1,n}(I_\\Lambda)$ \nfor any simplicial complex $\\Lambda$ on the vertex set $[n]$,\n it follows that $\\beta_i(\\Gamma')\\leq\\beta_i(\\Gamma)=0$,\nand so $\\beta_i(\\Gamma')=0$ for all $i$. Since $\\Gamma'$ is shifted,\nLemma \\ref{betti} implies that all facets of $\\Gamma'$ contain $n$.\nThus $\\Gamma'=\\mbox{\\upshape lk}\\,_{\\Gamma'}(n)\\star\\{n\\}$, and the assertion follows\nfrom induction hypothesis applied to \n$\\mbox{\\upshape lk}\\,_\\Gamma(n)$ and $\\mbox{\\upshape lk}\\,_{\\Gamma'}(n)$.\n\n{\\bf Case 2:} $R_d=n+1$. In this case all facets of $\\Gamma$ but $F_d$ \ncontain vertex $n$ (this follows from part (1) of the Lemma), and we infer\nfrom Lemma \\ref{betti}  that\n$$\\beta_i(\\Gamma)=\\left\\{\\begin{array}{ll} 0, \\mbox{ if $i\\neq d-2$} \\\\\n                                          1, \\mbox{ if $i= d-2$.} \\\\\n                        \\end{array}\n                 \\right.\n$$\nRecall the Euler-Poincar\\'e formula asserting that for any simplicial\ncomplex $\\Lambda$, \n$$\\sum_{j\\geq -1}(-1)^j f_j(\\Lambda) \n= \\sum_{j\\geq -1}(-1)^j \\beta_j(\\Lambda)\n=:\\widetilde{\\chi}(\\Lambda).$$\nTherefore, $\\widetilde{\\chi}(\\Gamma')=\\sum_{j\\geq -1}(-1)^j f_j(\\Gamma')=\n\\sum_{j\\geq -1}(-1)^j f_j(\\Gamma)=\\widetilde{\\chi}(\\Gamma)=(-1)^{d-2}$, and\nhence not all Betti numbers of $\\Gamma'$  vanish. The\nsame reasoning as in Case 1 then shows that  \n$\\beta_i(\\Gamma')=\\beta_i(\\Gamma)$ for all $i$. Applying\nLemma \\ref{betti} once again, we obtain that \n$\\Gamma'=(\\mbox{\\upshape lk}\\,_{\\Gamma'}(n)\\star\\{n\\})\\cup\\{ F'\\}$, where $|F'|=d-1$\nand $F'$\nis the only facet of $\\Gamma'$ that does not contain $n$. Thus \n$f(\\mbox{\\upshape lk}\\,_{\\Gamma}(n))= f(\\mbox{\\upshape lk}\\,_{\\Gamma'}(n))$ and\n $f(\\mbox{\\upshape ast}\\,_{\\Gamma}(n))= f(\\mbox{\\upshape ast}\\,_{\\Gamma'}(n))$, and so\n$\\mbox{\\upshape lk}\\,_{\\Gamma}(n)=\\mbox{\\upshape lk}\\,_{\\Gamma'}(n)$ and \n$\\mbox{\\upshape ast}\\,_{\\Gamma}(n)=\\mbox{\\upshape ast}\\,_{\\Gamma'}(n)$ (by induction hypothesis),\nyielding that $\\Gamma=\\Gamma'$.\n\\hfill$\\square$\\medskip\n\nWe now turn to the class of {\\em almost USLIs}. \n(Recall our convention that lower degree monomials are \nlex-larger than  higher degree monomials.)\n\n\n\\begin{definition}\nLet $I\\subset S$ (or $I\\subset S_{[n]}$)\nbe a squarefree strongly stable monomial  ideal with \n$G(I)=\\{m_1>_{\\mbox{\\upshape {\\tiny lex}}}  \\ldots >_{\\mbox{\\upshape {\\tiny lex}}} m_l>_{\\mbox{\\upshape {\\tiny lex}}} m_{l+1} \\}$. \nWe say that $I$ is {\\em an almost USLI}\nif  $I$ is not a USLI, but $L=\\langle m_1, \\ldots, m_l\\rangle$ is a USLI.\nWe say that a simplicial complex $\\Gamma$ is {\\em an almost USLI complex} \nif $I_\\Gamma$ is an almost USLI.\n\\end{definition}\n\nAs we will see in the next section (see also Lemma \\ref{Pardue_lemma} below),\n what makes almost USLI complexes noninvariant under\nlex shifting is the following combinatorial property. (We recall that the \n{\\em regularity} of a finitely generated stable monomial ideal $I$, $\\mbox{\\upshape reg}(I)$,\nis the maximal degree of its minimal generators.)\n\n\\begin{lemma}  \\label{main_lemma}\nLet $\\Gamma$ be an almost USLI complex.\nThen $|\\max(\\Gamma)|>\\mbox{\\upshape reg}(I_\\Gamma)$.\n\\end{lemma}\n\n\\smallskip\\noindent {\\it Proof: \\ } \nAssume $\\Gamma$ is a simplicial complex on $[n]$\nwith $G(I_\\Gamma)=\\{m_1>_{\\mbox{\\upshape {\\tiny lex}}}\\ldots>_{\\mbox{\\upshape {\\tiny lex}}}m_l>_{\\mbox{\\upshape {\\tiny lex}}}m_{l+1}\\}$.\nWe have to show that $|\\max(\\Gamma)|>\\deg(m_{l+1})=:d$.\nWe verify this by induction on $d$. To simplify the notation\n assume without loss of generality  that every singleton \n$\\{i\\}\\subset[n]$ is a vertex of $\\Gamma$\n(equivalently, $I_\\Gamma$ has no generators of degree 1).\nIf there are generators of degree 1 then the proof given below can\nbe modified by letting the index $R_1$ play the role of the index $1$. \nAs $I_\\Gamma$ is an almost USLI, and so\n$\\langle m_1, \\ldots, m_l\\rangle$ is a USLI,\nthis leaves two possible cases:\n\n{\\bf Case 1:}\n{\\em $m_1, \\ldots, m_l$ are divisible by $x_1$, but\n$m_{l+1}$ is not divisible by $x_1$.}\nSince $I_\\Gamma$ is squarefree strongly stable, it follows that \n$m_{l+1}=\\prod_{j=2}^{d+1}x_j$. In this case each set $F_i=[n]-\\{1, i\\}$,\n$i=2, \\ldots, d+1$, is a facet of $\\Gamma$. \n(Indeed the product $\\prod\\{x_j : j\\in F_i\\}$\n is not divisible by $m_{l+1}$,\nand it is also not divisible by $x_1$, and hence\nby $m_1, \\ldots, m_{l}$, implying that $F_i$ is a face. To show that $F_i$\nis a maximal face observe that \n $F_i\\cup \\{i\\}$ \ncontains the support of $m_{l+1}$, and hence is not a face,\nbut then shiftedness of $\\Gamma$ implies that\nneither is $F_i\\cup\\{1\\}$.)\nSince there also should be a facet containing $1$, we conclude\nthat $\\max(\\Gamma)\\geq d+1>\\deg(m_{l+1})$, \ncompleting the proof of this case.\n\n{\\bf Case 2:} \n{\\em All minimal generators of $I$ are divisible by $x_1$.}\nIn this case consider an almost USLI\n$I_\\Gamma':=\\langle x_1, m_1\/x_1, \\ldots, m_{l+1}\/x_1 \\rangle$.\n By induction hypothesis $\\Gamma'$\nhas $s>\\deg(m_{l+1})-1$ facets which we denote by $F_1, \\ldots, F_s$.\nOne  easily verifies that\n$\\max(\\Gamma)=\\left\\{\\{1\\}\\cup F_1, \\ldots, \\{1\\}\\cup F_s, [2,n]\\right\\},\n$\nand so $|\\max(\\Gamma)|=s+1>\\deg(m_{l+1})$.\n\\hfill$\\square$\\medskip\n\n\nWe close this section with an algebraic lemma that relates regularity of\n$\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I_\\Gamma)$ to the number of facets of $\\Gamma$ (for an arbitrary\ncomplex $\\Gamma$).\n\n\\begin{lemma}  \\label{Pardue_lemma}\nFor a (finite) simplicial complex $\\Gamma$, \n$\\mbox{\\upshape reg}(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I_{\\Gamma}))\\geq |\\max(\\Gamma)|$.\n\\end{lemma}\n\\smallskip\\noindent {\\it Proof: \\ }\nThis fact is a corollary of \\cite[Lemma 23]{Pardue}\napplied to squarefree (and hence radical)  ideal $I_\\Gamma\\in S_{[n]}$. \nFor  $\\sigma\\subseteq[n]$, we denote by $P_\\sigma$ the (prime)\nideal in $S_{[n]}$ generated by $\\{x_j : j\\notin\\sigma\\}$. It is well known\nthat $I_\\Gamma$ has the following prime decomposition:\n$\nI_\\Gamma=\\cap_{\\sigma\\in\\max(\\Gamma)} P_\\sigma.\n$\n Thus the variety of $I_\\Gamma$, $\\mathcal{V}(I_\\Gamma)$,\nis the union (over $\\sigma\\in\\max(\\Gamma)$)\nof the irreducible subvarieties $\\mathcal{V}(P_\\sigma)$.\nEach such subvariety is a \nlinear subspace of ${\\bf k}^n$ of codimension $n-|\\sigma|$.\n\\cite[Lemma 23]{Pardue} then implies that the monomial $m:=\\prod x_i^{r_i}$,\nwhere $r_i=|\\{\\sigma\\in\\max(\\Gamma): |\\sigma|=n-i\\}|$,\nis a minimal generator of $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I_\\Gamma)$.\nHence $\\mbox{\\upshape reg}(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I_\\Gamma))\\geq \\deg(m)=|\\max(\\Gamma)|$.\n\\hfill$\\square$\\medskip\n\n\n\n\n\n\n\\section{Lex shifting,  $B$-numbers and the limit complex}         \n                     \\label{infinite_section}\nIn this section after defining the notion of\nlexicographic shifting and the notion of $B$-numbers \n(a certain analog of the Hilbert function) we prove Theorem~\\ref{Thm2}. \nWe remark that extending the notion of algebraic shifting to an arbitrary term \norder\n$\\succ$ is not entirely automatic since the $\\Phi$-image of the set of\nminimal generators of $\\mbox{\\upshape Gin}_{\\succ}(I_\\Gamma)\\subset S_{[n]}$, \n$G(\\mbox{\\upshape Gin}_{\\succ}(I_\\Gamma))$, may not be a subset of $S_{[n]}$.\n This however can be easily corrected if one considers the system of rings \n$S_{[n]}$, $n\\in\\mathbb{N}$,\nendowed with natural embeddings $S_{[n]}\\subseteq S_{[m]}$ \nfor $m\\geq n$, and makes \nall the computations in the direct limit ring \n$S=\\lim_{n\\rightarrow\\infty}S_{[n]}={\\bf k}[x_i : i\\in\\mathbb{N}]$. This is the approach\nwe adopt here.\n We work with the class of monomial ideals $I\\subset S$\nfinitely generated in each degree.\nThroughout this section we use the graded \nlexicographic term order on $S$.\n\n\\begin{definition}   \\label{gin_def}\nLet $I$ be a monomial ideal of $S$ that is finitely generated\nin each degree.\nDefine \n$$\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I):=\\lim_{n\\rightarrow\\infty}\\,\n\\left(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I\\cap S_{[n]})\\right)S,\n$$ \nwhere we consider $I\\cap S_{[n]}$ as an ideal of $S_{[n]}$.\n\\end{definition}\nSince the $d$-th component of $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I\\cap S_{[n]})$ depends only on the\n$d$-th component of $I\\cap S_{[n]}$,\n or equivalently on the minimal generators of \n$I\\cap S_{[n]}$ of degree $\\leq d$,\nLemma \\ref{cone}(4) implies that \n$\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I)$ is well-defined and that for every\n$d$ there is $n(d)$ such that\n$(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}I)_d=((\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I\\cap S_{[n]}))S)_d$ \nfor all $n\\geq n(d)$.\nThus $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I)$ is a monomial ideal finitely generated\nin each degree. (It is finitely generated if $I$ is.)\n  Moreover, it follows from Lemma \\ref{cone}(1) that\n$\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I)$ is a strongly stable ideal.  \n\n\nRecall that the squarefree operation $\\Phi$ \ntakes monomials of $S$ to squarefree\nmonomials of $S$. \nIf $I\\subset S$ is a monomial ideal finitely generated in each degree,\nwe define $\\Phi(I):=\\langle \\Phi(m) : m\\in G(I)\\rangle$, where $G(I)$ is\nthe set of minimal generators of $I$.\n\\begin{definition}\nLet $I$ be a homogeneous ideal of $S$ that is finitely generated\nin each degree. The {\\em lexicographic shifting} \nof $I$ is the squarefree strongly stable ideal \n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)=\\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I))$. \nThe {\\em $i$-th lexicographic shifting} of $I$\nis the ideal $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}^i(I)$, where $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}^i$ stands \nfor $i$ successive applications of $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}$. \nWe also define the {\\em limit ideal}\n$\\overline{\\Delta}(I):=\\lim_{k\\rightarrow\\infty}\\Delta_{\\mbox{\\upshape {\\tiny lex}}}^k(I)$.\n\\end{definition}\n\nThe rest of the section is devoted to the proof of Theorem \\ref{Thm2}.\nFirst however we digress and review\nseveral facts on algebraic Betti numbers (defined by Eq.~(\\ref{alg_betti})).\n\n\n\\begin{lemma}      \\label{betti-prop}\nLet $I$ and $J$ be monomial ideals of $S_{[n]}$.\n\\begin{enumerate}\n\\item If $I_j=J_j$ for all $0\\leq j\\leq j_0$, then \n$\\beta_{i,j}(I)=\\beta_{i,j}(J)$ for all $i$ and all $j\\leq j_0$. \n\\item The Betti numbers of $I\\subset S_{[n]}$ coincide with those\nof $IS_{[n+1]}\\subset S_{[n+1]}$, that is,\n $\\beta_{i,j}(I)=\\beta_{i,j}(IS_{[n+1]})$ for all $i, j$.\n\\end{enumerate}\n\\end{lemma}\n\\smallskip\\noindent {\\it Proof: \\ }\nPart (1) follows from the standard facts that\n$$\\beta_{i,j}(I)=\n\\dim_{\\bf k} \\mbox{\\upshape Tor}_i^{S_{[n]}}({\\bf k}, I)_{j}=\n\\dim_{\\bf k} \\mbox{\\upshape Tor}_i^{S_{[n]}}(I, {\\bf k})_{j},$$ \nwhere we identify ${\\bf k}$ with the $S_{[n]}$-module \n$S_{[n]}\/\\langle x_1, \\ldots, x_n\\rangle$. \nFor part (2) note that if $\\mathbb{F}$ is\nthe free minimal \nresolution of $I$ over $S_{[n]}$, then  $\\mathbb{F}\\otimes_{S_{[n]}} S_{[n+1]}$ \nis the free minimal resolution of $IS_{[n+1]}$ over $S_{[n+1]}$,\nyielding the lemma.\n\\hfill$\\square$\\medskip\n\nThe above properties allow to extend the definition\nof the Betti numbers to the class of monomial ideals of $S$\nthat are finitely generated in each degree.\n\n\\begin{definition}   \\label{betti_def}\nLet $I\\subset S$ be a monomial ideal finitely generated in each\ndegree.\n Define \n$$\\beta_{i,j}(I):=\n\\lim_{n\\rightarrow\\infty}\\beta_{i,j}(I\\cap S_{[n]}) \\quad \\mbox{for all } \ni, j\\geq0,\n$$\nwhere we consider $I\\cap S_{[n]}$ as an ideal of $S_{[n]}$.\n\\end{definition}\nWe remark that since $I$ is finitely generated in each degree,\nfor a fixed $j_0$\nthere exists $n_0$ such that $(I\\cap S_{[n+1]})_j=((I\\cap S_{[n]})S_{[n+1]})_j$\nfor all $0\\leq j \\leq j_0$ and $n\\geq n_0$.\n  Hence it follows from Lemma \\ref{betti-prop}\nthat (for a fixed $i$)\nthe sequence $\\{\\beta_{i,j_0}(I\\cap S_{[n]})\\}_{n\\in\\mathbb{N}}$\nis a constant for indices starting with $n_0$, and thus\n $\\beta_{i,j_0}(I)$ is well-defined. \n\n\nThe Betti numbers of strongly stable ideals (of $S_{[n]}$) were computed by\nEliahou and Kervaire \\cite{ElKer}, and the analog of this\nformula for squarefree strongly stable ideals (of $S_{[n]}$) was established\nby Aramova, Herzog, and Hibi \\cite{AHHlex}. Definition \\ref{betti_def}\nallows to state these results as follows. (For a monomial $u$\ndefine $m(u):=\\max\\{i : x_i|u\\}$.)\n\\begin{lemma}   \\label{EK}\nLet $I\\subset S$ be a monomial ideal finitely generated in each degree, \nlet $G(I)$ denote its set of minimal generators, and let \n$G(I)_j=\\{u\\in G(I): \\deg u=j\\}$. \n\\begin{enumerate}\n\\item If $I$ is strongly stable, then\n$\\beta_{i, i+j}(I)=\\sum_{u\\in G(I)_j} {m(u)-1 \\choose i}$;\n\\item If $I$ is squarefree strongly stable, then\n$\\beta_{i, i+j}(I)=\\sum_{u\\in G(I)_j} {m(u)-j \\choose i}$.\nIn particular, if $I=L(k_\\bullet)$ is a USLI, then\n$\\beta_{i, i+j}(I)=\\sum_{l=1}^{k_j}{k_1+\\ldots+k_{j-1}+l-1 \\choose i}$.\n\\end{enumerate}\n\\end{lemma}\n\nUsing  the notion of the Betti numbers, one can define \na certain analog of the Hilbert function ---\n the $B$-numbers --- of a monomial ideal $I$ of $S$ that is finitely generated \nin each degree.\n\n\\begin{definition}   \\label{B-definition}\nLet $I\\subset S$ (or $I\\subset S_{[n]}$)\nbe a monomial ideal finitely generated in each degree, and \nlet $\\beta_{i,j}(I)$ be its graded Betti numbers. Define\n$$\nB_j(I):=\\sum_{i=0}^j (-1)^i\\beta_{i,j}(I) \\quad \\mbox{ for all }\nj\\geq 0 \\quad  (\\mbox{e.g., $B_0=0$ and $B_1(I)=|G(I)_1|$}).\n$$\n The sequence \n$B(I):=\\{B_j(I): j\\geq 1\\}$ is called the {\\bf $B$-sequence} of $I$. \n\\end{definition}\n\\begin{remark}\nIt is well known and is easy to prove (see \\cite[Section 1B.3]{Eis2})\n that for every $n\\in\\mathbb{N}$ the polynomial\n $\\sum_j B_j(I\\cap S_{[n]})x^j$\nequals $(1-x)^n \\text{Hilb}(I\\cap S_n, x)$, where \n$\\text{Hilb}(I\\cap S_n, x)$ is the Hilbert series of $I\\cap S_{[n]}$. \nIn particular, if $\\Gamma$ is a $(d-1)$-dimensional\nsimplicial complex on $[n]$ and \n$I_{\\Gamma}\\subset S_{[n]}$\nis its Stanley-Reisner ideal then \n\\[\n\\frac{1-\\sum_j B_j(I_\\Gamma)x^j}{(1-x)^n} = \\text{Hilb}(S_{[n]}\/I_\\Gamma, x)\n=\\sum_{i=0}^{d} \\frac{f_{i-1}(\\Gamma)x^i}{(1-x)^i}=\n\\frac{\\sum_{i=0}^d h_i(\\Gamma)x^i}{(1-x)^{d}}, \n\\]\nwhere $\\{h_i(\\Gamma)\\}_{i=0}^d$ is the $h$-vector of $\\Gamma$ \\cite{St}.\n(Recall that\n$h_j=\\sum_{i=0}^j (-1)^{j-i}{d-i \\choose j-i}f_{i-1}$ for\n$0\\leq j \\leq d$. In particular, $h_1=f_0-d$.)\nThus \n$\\sum_j B_j(I_\\Gamma)x^j=1-(1-x)^{h_1}\\sum_i h_ix^i$\n(if one assumes that $\\{i\\}\\in\\Gamma$ for every $i\\in[n]$), and so\nthe $h$-vector of $\\Gamma$ \ndefines the $B$-sequence of $I_\\Gamma$.\n\\end{remark}\n\nThe following lemma provides the\nanalog of the ``$f(\\Gamma)=f(\\Delta_{\\mbox{\\upshape {\\tiny rl}}}(\\Gamma))$-property\".\n\\begin{lemma} \\label{cone2}\nIf $I\\subset S$  is a monomial ideal that is\n finitely generated in each degree, then \nthe ideals $I$ and $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)$ have the same $B$-sequence.\nIn particular, if $I$ is finitely generated, then for a sufficiently large $n$,\nthe ideals $I\\cap S_{[n]}$ and $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)\\cap S_{[n]}$ \nhave the same Hilbert function\n(in $S_{[n]}$).\n\\end{lemma}\n\\smallskip\\noindent {\\it Proof: \\ }\nSince for every $n\\in\\mathbb{N}$ the ideals\n$I\\cap S_{[n]}$ and $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I\\cap S_{[n]})$ have the same Hilbert function\n(in $S_{[n]}$) (see Lemma \\ref{cone}), and since \n$B_i(I)=\\lim_{n\\rightarrow\\infty} B_i(I\\cap S_{[n]})$, \nthe above remark implies that $B(I)=B(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I))$.  Finally,\n since $\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I)$\nis a strongly stable ideal (Lemma \\ref{cone}),  we infer (by comparing\nthe two formulas of Lemma \\ref{EK})\nthat \n$\\beta_{i,j}(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I))=\\beta_{i,j}(\\Phi\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I))=\\beta_{i,j}\n(\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I))$ for all $i, j$,\nand so $B(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I))=B(\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I))$.\nThe result follows. \\hfill$\\square$\\medskip\n\n\n\n\nNow we are ready to verify the first part of Theorem \\ref{Thm2}. In fact\nwe prove the following slightly more general result.\n\n\\begin{theorem} \\label{main}\nLet $I$ be a squarefree strongly stable ideal of $S$ finitely \ngenerated in each degree. \nThen $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)>_{\\mbox{\\upshape {\\tiny lex}}} I$ unless $I$ is a USLI in which case\n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)= I$. \nMoreover if $I$ is finitely\ngenerated and is not a USLI, then \nall ideals in the sequence\n$\\{\\Delta^i_{\\mbox{\\upshape {\\tiny lex}}}(I)\\}_{i\\geq 0}$ are distinct. \n\\end{theorem}\n\n\n\\smallskip\\noindent {\\it Proof: \\ } There are several possible cases.\n\n{\\bf Case 1:} $I=L(k_\\bullet)$ is a USLI. To prove that \n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)=I$,\nit suffices to show that for every $d\\geq 1$, \n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(L(k^{(d)})= L(k^{(d)})$, where \n$k^{(d)}:=\\{k_1, \\ldots, k_d, 0,0,\\ldots\\}$ is the sequence $k_\\bullet$\ntruncated at $k_d$.\nBut this is immediate from Lemmas \\ref{comb_USLI}(2) and \\ref{cone2}. \nIndeed, for $n=n(d)$ sufficiently large\nthe simplicial complexes on the vertex set $[n]$ whose Stanley-Reisner\nideals are given by $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(L(k^{(d)})\\cap S_{[n]}$ and \n$L(k^{(d)})\\cap S_{[n]}$, respectively,\nare shifted and have the same $f$-numbers. \nSince the second complex is a USLI complex,\nit follows that those complexes, and hence their ideals, coincide. \n \n{\\bf Case 2:} $I=\\langle m_1, \\ldots, m_l, m_{l+1} \\rangle$\n is an almost USLI.\nLet $n$ be the largest index of a variable appearing in $\\prod_{i=1}^{l+1}m_i$,\nand let $\\Gamma$ be a simplicial complex on $[n]$ with \n$I_\\Gamma = I\\cap S_{[n]}$.\nThen\n$$\\mbox{\\upshape reg}(\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I))=\\mbox{\\upshape reg}(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I_\\Gamma)) \n\\stackrel{\\mbox{\\tiny {Lemma \\ref{Pardue_lemma}}}}{\\geq} |\\max(\\Gamma)|\n\\stackrel{\\mbox{\\tiny {Lemma \\ref{main_lemma}}}}{>}\\mbox{\\upshape reg}(I_\\Gamma)=\\mbox{\\upshape reg}(I),\n$$\nyielding that $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)\\neq I$ in this case.\nMoreover, since by Eq.~(\\ref{P5}),  \n$\\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I_\\Gamma))=I_\\Gamma$ and since $\\Phi$ is a \nlex-order preserving map,\nwe infer from Lemma \\ref{gin>gin} that \n$\\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny lex}}}(I_\\Gamma))\\geq_{\\mbox{\\upshape {\\tiny lex}}} \\Phi(\\mbox{\\upshape Gin}_{\\mbox{\\upshape {\\tiny rl}}}(I_\\Gamma))\n=I_\\Gamma$,\nand hence that $\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)>_{\\mbox{\\upshape {\\tiny lex}}} I$.\n\n{\\bf Case 3:} I is squarefree strongly stable, but is not a USLI. \nIn this case we sort $G(I)=\\{m_1, \\ldots, m_l, m_{l+1}, \\ldots\\}$ by graded lex-order\nand assume that $m_{l+1}$ is the first non-USLI generator of $I$.\nLet  \n$I_1=\\langle m_1, \\ldots, m_l \\rangle$  and let \n$I_2=\\langle m_1, \\ldots, m_{l+1} \\rangle$. \nThen $I_1$ is a USLI, $I_2$ is an almost USLI, and $I_1\\subset I_2\\subseteq I$. \nHence by the previous two cases \n$I_1=\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I_1)\\subset\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I_2)$ \nand \n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I_2)>_{\\mbox{\\upshape {\\tiny lex}}} I_2$, and so \n there exists a monomial $m$, $m_l>_{\\mbox{\\upshape {\\tiny lex}}} m>_{\\mbox{\\upshape {\\tiny lex}}} m_{l+1}$, such that\n$m \\in G(\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I_2)) \\subseteq G(\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I))$. \nThus\n$\\Delta_{\\mbox{\\upshape {\\tiny lex}}}(I)>_{\\mbox{\\upshape {\\tiny lex}}} I$. \n\nFinally to show that for a finitely generated ideal $I$,\nall ideals in the sequence $\\{\\Delta^i_{\\mbox{\\upshape {\\tiny lex}}}(I)\\}_{i\\geq 0}$ are distinct, \nit suffices to check that none of those ideals is a USLI. \nThis is an immediate corollary of Lemmas \\ref{comb_USLI}(2) and  \\ref{cone2}. \\hfill$\\square$\\medskip\n\n\n\nOur next goal is to prove the second part of Theorem \\ref{Thm2}. \nTo do that we  fix a sequence of  integers\n$B=\\{B_j : j\\geq 1\\}$ and study the class $\\mathcal{M}(B)$ of all monomial ideals $I\\subset S$ \nthat are finitely generated in each degree  and satisfy $B(I)=B$.\n\n\\begin{lemma}\nThere is at most one USLI in the class $\\mathcal{M}(B)$.\n\\end{lemma}\n\\smallskip\\noindent {\\it Proof: \\ } \nRecall that  a USLI $L=L(k_\\bullet)$ \nis uniquely defined by its $k$-sequence\n$k_\\bullet=\\{k_i : i\\geq 1\\}$, where $k_i=\\beta_{0,i}(L)=|G(L)_i|$. \nRecall also that  \n$B(L)$ is a function of \n$k_\\bullet$ (see Lemma \\ref{EK}(2)), and so to complete the proof it suffices\nto show that this function is\none-to-one, or more precisely that\n $k_j$ is determined by\n$k_1, \\ldots, k_{j-1}, B_j$ (for every $j\\geq 1$). \nAnd indeed, \n\\begin{eqnarray*}\nk_j&=&\\beta_{0,j}(L)=B_j-\\sum_{i=1}^j (-1)^i\\beta_{i,j}(L)  \n   \\quad (\\mbox{by definition of } B_j)\\\\\n&=& B_j-\\sum_{i=1}^j(-1)^i \\sum_{l=1}^{k_{j-i}}{k_1+\\ldots+k_{j-i-1}+l-1 \\choose i}\n\\quad (\\mbox{by Lemma } \\ref{EK}(2)).  \n\\end{eqnarray*}\n\\hfill$\\square$\\medskip\n\nNow we are ready to prove (the slightly more general \nversion of) the second part of Theorem \\ref{Thm2}.\n\\begin{theorem}\nFor every ideal $I\\in\\mathcal{M}(B)$, the limit ideal $\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}(I)$ \nis well defined and is the unique USLI of $\\mathcal{M}(B)$.\n\\end{theorem}\n\\smallskip\\noindent {\\it Proof: \\ }\nFix $I\\in \\mathcal{M}(B)$. To show that $\\overline{\\Delta}_{\\mbox{\\upshape {\\tiny lex}}}(I)$ \nis well defined, it suffices to check that for every $d\\geq 0$,\nthere exists $s=s(d)$ such that \n\\begin{equation}   \\label{stab}\nG(\\Delta^{s}_{\\mbox{\\upshape {\\tiny lex}}}(I))_{\\leq d}=G(\\Delta^{s+1}_{\\mbox{\\upshape {\\tiny lex}}}(I))_{\\leq d} \n\\end{equation}\n(where $G(J)_{\\leq d}:=\\cup_{j\\leq d} G(J)_j$),\nand hence that all ideals $\\Delta^{i}_{\\mbox{\\upshape {\\tiny lex}}}(I)$, $i\\geq s$,\nhave the same $d$-th homogeneous component.\nWe verify this fact by showing that the collection of all \npossible sets of minimal generators \n\\begin{equation}  \\label{finite}\n       \\mathcal{G}_{\\leq d}:=\\{ G(J)_{\\leq d} : \n       J\\in\\mathcal{M}(B), J \\mbox{ is squarefree strongly stable}\\} \n \\quad \\mbox{is finite}.\n\\end{equation}\n(This yields (\\ref{stab}), since all ideals $\\Delta^{i}_{\\mbox{\\upshape {\\tiny lex}}}(I)$, $i\\geq 1$,\nare squarefree strongly stable, and since \n$\\Delta^{i}_{\\mbox{\\upshape {\\tiny lex}}}(I)\\leq_{\\mbox{\\upshape {\\tiny lex}}} \\Delta^{i+1}_{\\mbox{\\upshape {\\tiny lex}}}(I)$ \nby Theorem \\ref{main}.)\nEq.~(\\ref{finite}) can be easily proved by induction.\n It clearly holds for \n$d=0$. Now if $J\\in\\mathcal{M}(B)$ is squarefree strongly stable, then\nby Lemma \\ref{EK}(2) and Definition \\ref{B-definition}, \n$$ |G(J)_d|=\\beta_{0,d}(J)=\nB_d-\\sum_{i=1}^{d}(-1)^i\\sum_{u\\in G(J)_{d-i}}{m(u)-(d-i) \\choose i},\n$$\nso assuming that the collection \n$\\mathcal{G}_{\\leq d-1}$ is finite, or equivalently that the set of integers\n$\\{m(u): u\\in G(J)_{\\leq d-1}\\in\\mathcal{G}_{\\leq d-1}\\}$  is bounded \n(say by $n(d)$), \nwe obtain that \nthere exists a constant $g(d)$ such that\n$|G(J)_d|\\leq g(d)$ for all squarefree strongly stable ideals $J\\in\\mathcal{M}(B)$.\nBut then the squarefree strongly stable property implies that\n$m(u)< n(d)+g(d)+d$ for every $u\\in G(J)_{\\leq d}\\in \\mathcal{G}_{\\leq d}$,\nand (\\ref{finite}) follows.\n\nThe second part of the statement is now immediate:\nindeed if $G(\\Delta^s(I))_{\\leq d} = G(\\Delta^{s+1}(I))_{\\leq d}$,\n then by Theorem \\ref{main}, \n$G(\\Delta^s(I))_{\\leq d}= G(\\overline{\\Delta}(I))_{\\leq d}$ \nis the set of minimal generators of a USLI.\n\\hfill$\\square$\\medskip\n\n\\section{Remarks on other term orders}\nWe close the paper by discussing several results and conjectures\nrelated to algebraic shifting with respect to arbitrary term orders. \n To this end, we say that an order $\\succ$ \non monomials of $S$ is a {\\em term order} if \n$x_i\\succ x_{i+1}$ for  $i\\geq 1$, \n$m\\succ m'$ as long as $\\deg(m)<\\deg(m')$,\nand  the restriction of $\\succ$\n to $S_{[n]}$\nis a term order on $S_{[n]}$ for all $n\\geq 1$. In addition,\nwe restrict our discussion\nonly to those term orders on $S$ that are compatible with the squarefree\noperation $\\Phi$, that is, $\\Phi(m)\\succ\\Phi(m')$ if $m\\succ m'$.\n\nSimilarly to Definition \\ref{gin_def}, for a term order $\\succ$ on $S$ and\na homogeneous ideal $I\\subset S$ that is finitely generated in each degree,\nwe define $\\Delta_\\succ(I):=\\Phi(\\mbox{\\upshape Gin}_\\succ(I))$. Thus $\\Delta_\\succ(I)$\nis a squarefree strongly stable ideal that has the same $B$-sequence as $I$.\n(Indeed, the proof of Lemma \\ref{cone2} carries over to this more \ngeneral case.)\n\nWe say that a squarefree monomial ideal $I\\subset S$ is a {\\em US$\\succ$I}\nif for every monomial $m\\in I$ and every squarefree monomial $m'$\nsuch that\n$\\deg(m)=\\deg(m')$ and  $m'\\succ m$, $m'$ is an element of $I$ as well. \nBeing US$\\succ$I implies being squarefree strongly stable.\n\nIn view of Theorems  \\ref{Thm2} and \\ref{AHH}\nit is natural to ask the following:\n\\begin{enumerate}\n\\item Does $\\Delta_\\succ(I)=I$ hold for every US$\\succ$I I?\n\\item Is there  a term order $\\succ$ other than the lexicographic order\nfor which the equality $\\Delta_\\succ(I)=I$ implies that $I$ is a \nUS$\\succ$I?\n\\item Is there  a term order $\\succ$ other than the \nreverse lexicographic order such that the equation $\\Delta_\\succ(I)=I$ \nholds for all squarefree strongly stable ideals $I$?\n\\end{enumerate}\n\nThe next proposition answers the first question in the affirmative.\n\\begin{proposition}\nIf $I$ is a  US$\\succ$I, then $\\Delta_\\succ(I)=I$\nfor every term order on $S$ that is compatible with $\\Phi$.\n\\end{proposition}\n\\smallskip\\noindent {\\it Proof: \\ } Exactly as in the proof of Theorem \\ref{main} (see the\nlast three lines of Case 2), \none can show that $\\Delta_\\succ(I)\\succeq I$. Hence either\n$\\Delta_\\succ(I)= I$, in which case we are done, or the \n$\\succ$-largest monomial, $m$, in the symmetric difference of \n$G(\\Delta_\\succ(I))$ and $G(I)$ is an element of $G(\\Delta_\\succ(I))$.\nSince $I$ is a US$\\succ$I, we obtain in the latter case that \n$G(\\Delta_\\succ(I))_i=G(I)_i$ for all $i<\\deg(m)$ and\n$$\nG(I)_{i_0}=\\{m'\\in G(\\Delta_\\succ(I))_{i_0} : m'\\succ m\\}\n\\quad \\mbox{ for } i_0=\\deg(m),\n$$\nthat is, $G(I)_{i_0}$ is a strict subset of $ G(\\Delta_\\succ(I))_{i_0}$.\nThis is however impossible, since it contradicts the fact that\nthe ideals $I$ and $\\Delta_\\succ(I)$ have the same $B$-sequence.\n\\hfill$\\square$\\medskip\n\nThe answer to the second question is negative as follows from \nthe following result.\n\n\\begin{proposition}\nIf $I$ is a USLI, then $\\Delta_\\succ(I)=I$ for all term orders $\\succ$.\n\\end{proposition}\nWe omit the proof as it is completely analogous to that of \nTheorem \\ref{main}, Case 1.\n\nWhile we do not know the answer to the third question, we believe that it\nis negative. In fact it is tempting to conjecture that the following holds.\nLet $\\succ$ be a term order on $S$ other than the (graded) \nreverse lexicographic order, and let $k\\geq 2$ be the smallest degree\non which $\\succ$ and revlex disagree. Write $m_i$ to denote the $i$th \nsquarefree monomial of $S$ of degree $k$ with respect to the revlex order.\n(It is a fundamental property of the revlex order that every squarefree \nmonomial of $S$ of degree $k$ is of the form $m_i$ for some finite $i$.)\n\n\\begin{conjecture}\nLet $i_0\\geq 1$ be the smallest index for which \n$I_{i_0}:=\\langle m_1, \\cdots, m_{i_0}\\rangle$ is not a US$\\succ$I. \nThen $\\Delta_{\\succ}(I_{i_0})\\neq I_{i_0}$.\n\\end{conjecture}\n\n\n\\section*{Acknowledgments} We are grateful to Aldo Conca for\nhelpful discussions and to the anonymous referees for insightful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nMonocular Visual Odometry (MVO) is a popular method for camera pose estimation, but due to the scale ambiguity \\cite{song2015high,zhou2016reliable, wu2020eao,2009Absolute,2010scale}, the MVO system cannot provide real odometry data. Therefore, an accurate and robust scale recovery algorithm is of great significance in the application of MVO \\cite{hawkeye-zhu}.\nThe key of scale recovery is to integrate absolute reference information, such as the gravity orientation from IMU \\cite{qin2018vins} or the depth measured by Lidar \\cite{zhang2014real, li2017hybrid}. The baseline of stereo cameras can also serve as such a reference \\cite{mur2017orb}. However, these sensors are not always available in real-world applications. Moreover, a complicated sensor calibration and fusion process is needed to align the MVO system with other sensors. \n\nAnother frequently used method for removing scale ambiguity is to take as reference the height of a mounted camera above the ground plane, which remains a stable signal during the navigation of vehicles. The idea is to estimate a ratio between the real camera height and the relative one calculated from image features. The ratio then can be used to recover the real scale. The advantages of this method are significant since it does not depend on other sensors and is with high feasibility. The method is also regarded as one of the most promising solutions in this research area.\n\nPrior work like \\cite{zhou2019ground,zhou2016reliable,song2015high,1211380} typically leverages the results of feature matching to calculate the homography matrix and then decompose the matrix to estimate the parameters of the ground plane, based on which the relative camera height can be obtained. The major deficiency of this method is that the decomposition is very sensitive to noises and multiple solutions exist, which requires additional operations to eliminate the ambiguity. Some other work like \\cite{grater2015robust,1211380} chooses to directly fit the ground plane using feature points that lie on the ground, e.g., the center regions of a road. By removing low-quality image features outside the target region, the robustness is improved. However, the target region may be occluded sometimes, which interferes with the detection of image features, thus degrading the accuracy of scale recovery.\n\n\nIn recent work \\cite{yin2017scale,andraghetti2019enhancing,xue2020toward,wagstaff2020self, wang2020tartanair}, deep learning based MVO algorithms are proposed, in which the camera pose with a real scale is directly predicted by the neural network in an end-to-end manner. Such methods have received much attention in recent years, but their generalization ability across different scenarios is very limited \\cite{wang2020tartanair}. Some other deep learning based methods take scale recovery as an independent problem. For instance, DNet \\cite{xue2020toward} is proposed to perform ground segmentation and depth regression simultaneously. Based on the predicted depth points within the ground region, a dense geometrical constraint is then formulated to help recover the scale. In \\cite{wagstaff2020self}, a scale-recovery loss is developed based on the idea of enforcing the consistency between the known camera height and the predicted one. Constrained by this loss, the neural network can predict more accurate ego poses. Nonetheless, these methods usually require a large-scale training process, and the computational cost is prohibitively expensive.\n\nIn this paper, we propose a light-weight method for accurate and robust scale recovery based on plane geometry. The method includes an efficient Ground Point Extraction (GPE) algorithm based on Delaunay triangulation \\cite{shewchuk1996triangle} and a Ground Points Aggregation (GPA) algorithm for aggregating ground points from consecutive frames. Based on these two algorithms, a large number of high-quality ground points are selected. For scale recovery, we first formulate a least-square problem to fit the ground plane and then estimate the relative camera height to calculate the real scale. By leveraging the high-quality points and a RANSAC-based optimizer, the scale can be estimated accurately and robustly. Benefiting from the light-weight design of the algorithms, our method can achieve a 20Hz running frequency on the benchmark dataset. \n\nThe main contributions of this work are as follows:\n\\begin{itemize}\n\t\\item We propose a GPE algorithm based on Delaunay triangulation, which can accurately extract ground points.  \n\t\\item We propose a GPA algorithm that can effectively aggregate local ground points and perform robust optimization of the ground plane parameters.  \n\t\\item Based on the proposed algorithms, we implement a real-time MVO system with accurate and robust scale recovery, aiming to reduce scale drift and provide accurate odometry in long-distance navigations without loop closure. \n\n\\end{itemize}\n\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[height=6.3cm]{sys2.pdf}\n\t\\caption{Overview of the proposed system. There are two parallel threads in the system: 1) The MVO thread takes image frames as input and estimates the current camera pose, 2) The GPE-GPA thread fetches image features from the MVO thread and selects high-quality points for ground plane estimation.}\n\n\t\\label{overview}\n\t\\vspace{-2mm}\n\\end{figure*}\n\n\n\n\\section{System Overview}\nThe notations used in this paper are as follows:\n\\begin{itemize}\n\t\\item $T_t\\in{R^{4\\times4}}$ - The camera pose of image frame $I_t$ in the global frame, which is composed of a camera orientation $R_t\\in{R^{3\\times3}}$ and a translation $\\boldsymbol{t}_t\\in{R^{3\\times1}}$.\n\t\\item $T_{t,t-1}$ - The relative pose of frame $I_{t-1}$ w.r.t. frame $I_t$.\n\t\\item $K$ - The intrinsic matrix of a pinhole camera model.\n\t\\item $\\mathbf{x}_i, \\mathbf{u}_i$ - The 3D map point in camera frame and its corresponded 2D point on image plane after projection.\n\n\t\\item $\\mathbf{n}_t, h_t$ - The plane parameters, i.e.,  $\\mathbf{n}_t \\cdot \\mathbf{x}_i-h_t=\\mathbf{0}$, where $\\mathbf{n}_t$ is the normal vector and $h_t$ is the distance to the plane.\n\t\\item $h, {h}^{\\dagger}, h^*$ -- The calculated camera height form image features, the estimated camera height after scale recovery, and the real camera height.\n\\end{itemize}\n\n\n\\subsection{Problem Definition}\nGiven consecutive image frames from a calibrated monocular camera, our goal is to estimate the absolute scale of camera poses and then recover the real camera trajectory by making use of the prior known camera height $h^*$. Under scale ambiguity, the camera height $h$ calculated from image features maintains a ratio with the real one, i.e., $s=h^*\/h$. Therefore, scale recovery is essentially to compute $s$, and the key lies in the accurate estimation of the ground plane.\n\n\n\n\n\n\n\\subsection{System Architecture}\nThe proposed system in this work is shown in Fig. \\ref{overview}. There are two parallel threads in the system: The first one is the MVO thread, which takes consecutive images as input and estimates the current camera pose, e.g., the ORB-SLAM2 framework. The second thread is used to run the GPE and GPA algorithms for scale recovery. The proposed system is based on such an assumption that the ground is locally flat and can be approximated by a plane with a surface normal. The workflow of the second thread is as follows.\n\nAs shown in the red block in Fig.\\ref{overview}, for each image frame from the MVO thread, the Delaunay triangulation is first applied to segment the matched feature points into a set of triangles. Each triangle is then back-projected into the camera frame, and the associated plane parameters are also estimated. After that, several geometrical constraints are leveraged to select and then refine ground points. \n\nNote that selected ground points are not enough for an accurate estimation of the plane parameters. We thus propose the GPA algorithm to aggregate ground points from multiple frames using a sliding windows method, as shown in the orange block of Fig.\\ref{overview}. Based on the aggregated local points, a robust parameter estimation procedure is then performed to fit the ground plane. Accordingly, the relative camera height of each frame can be estimated, and the absolute camera trajectory is recovered, shown in the blue block of Fig.\\ref{overview}.\n\n\n\n\\section{Ground Plane Estimation}\n\n\\subsection{Ground Point Extraction}\nFor a given set of matched feature points $\\mathbf{u}^t_i$, $i\\in\\{1,2,\\dots,N\\}$, in the current image frame $I_t$, the Delaunay triangulation uses each of the feature points as a triangle vertex. We back-project the triangles from the image plane into the current camera frame and denote them by $\\Delta_i^t, i\\in\\{1,2,\\dots M\\}$ associated with a set of vertices $\\mathbf{x}^t_{ij},j\\in\\{1,2,3\\}$. The normal vector $\\mathbf{n}_i^t = ({n}_{i,x}^t, {n}_{i,y}^t, {n}_{i,z}^t)$ of each triangle can be obtained by the cross product,\n\\begin{equation}\n\t\\mathbf{n}_i^t=\\frac{(\\mathbf{x}^t_{i1}-\\mathbf{x}^t_{i2})\\times(\\mathbf{x}^t_{i1}-\\mathbf{x}^t_{i3})}{||(\\mathbf{x}^t_{i1}-\\mathbf{x}^t_{i2})\\times(\\mathbf{x}^t_{i1}-\\mathbf{x}^t_{i3})||_2},\n\t\\label{est-n}\n\\end{equation}\nwhere $\\mathbf{n}_i^t$ has an unit length. For each vertex of the triangle, the following geometrical constraint then holds:\n\\begin{equation}\n\t\\mathbf{n}^t_i \\cdot \\mathbf{x}^t_{ij}-h^t_i=0.\n\t\\label{est-h}\n\\end{equation}\n\nTherefore, $h^t_i$ can also be estimated. Here, we also add two additional constraints, i.e., ${n}_{i,y}^t>0$ and $h^t_i>0$, based on the fact that the camera is mounted on the top of the vehicle and is above the ground plane, as shown in Fig.\\ref{overview}.\n\nNote that the triangles are scattered in the whole image plane, hence we need to identify the ones located on the ground plane, named \\textit{ground triangles}, for estimating the plane parameters. Based on the fact that the normal of a ground triangles is orthogonal to camera translation $\\boldsymbol{t}_{t,t-1}$, and that the pitch angle of the camera is zero, the ground triangles can be identified by testing with the following constraints,\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\arccos(\\mathbf{n}^t_i, \\boldsymbol{t}_{t,t-1}) &= 0, \\\\\n\t\t|\\arctan(-\\frac{R_{32}}{R_{33}})| = 0,\\, &R_{33}\\neq0. \\\\\n\t\\end{aligned}\n\\end{equation}\n\nIn practice, the equality condition cannot be strictly satisfied. We thus set a tolerance value of $5^\\circ$ in the actural implementation.\n\n\nFor ground triangles that satisfy the above constraints, their vertices are categorized into a new point set $\\tilde{\\mathbf{x}}^t_{ij}, i\\in\\{1,2,\\dots K\\}, j\\in\\{1,2,3\\}, K\\small{<}M$. Since the same vertex point may be shared by multiple triangles, we also need to remove the repetitive ones from the point set. This will ensure the same contribution of each point to the ground plane estimation. \n\nThe ground points are now initially segmented out, denoted by $\\tilde{\\mathbf{x}}^t_{k} \\in \\mathcal{G}$, but there may still exist some outliers introduced by moving objects and some remote points.\nTo further improve the quality of $\\mathcal{G}$, a RANSAC-based method is leveraged to optimize $\\tilde{\\mathbf{x}}^t_{k}$, which minimizes a plane-distance error as follows,\n\\begin{equation}\n\t\\min_{\\tilde{\\mathbf{x}}^t_{g} \\in \\mathcal{G}} \\;\\sum_{g=1}^{|\\mathcal{G}|}||\\mathbf{n}^t\\cdot\\tilde{\\mathbf{x}}^t_{g}-h^t||_2.\n\t\\label{ransac}\n\\end{equation}\n\nIn the implementation, we randomly sample three points to estimate a new plane with eq. \\eqref{est-n}-\\eqref{est-h}, and then we calculate the total distance of the remaining points to the estimated plane. Such a process repeats $Z$ times, and the plane that induces the minimum total distance error is reserved. The points with a distance larger than $0.01$m to the reserved plane are then removed.\nThis is actually a more strict criterion for ground point selection. After this process, only high-quality ground points are reserved. In Alg. \\ref{ag1}, we present a complete procedure of the GPE algorithm, which gives more details about the proposed implementation.\n\n\n\\begin{algorithm}[t]\n\t\\caption{ Ground Point Extraction (GPE) }\n\t\\label{ag1}\n\n\t\\KwIn{  $\\mathbf{u}^t_i$ , $R_{t,t-1}$, $\\boldsymbol{t}_{t,t-1}$}\n\t\\KwOut{$\\{\\tilde{\\mathbf{x}}^t_{g}\\}$}\n\t\n\n\t\n\t$\\{\\Delta_i^t\\}$$\\gets$ \\textsc{DelaunayTriangulation}($\\mathbf{u}^t_i$) \n\t\n\n\t\n\n\t\n\n\t\n\ttriangles points set $\\{\\tilde{\\mathbf{x}}^t_{ij}\\}\\gets\\emptyset$\n\t\n\tsegmented points set $\\{\\tilde{\\mathbf{x}}^t_{k}\\}\\gets\\emptyset$\n\t\n\tground points set $\\mathcal{G}_{best}\\gets\\emptyset$\n\t\n\ttemp ground points set $\\mathcal{G}_k\\gets\\emptyset$\n\t\n\t\n\t\n\t\\For{each  $\\mathbf{u}_{ij}^t\\in\\{\\Delta_i^t\\}, j=\\{1,2,3\\}$ }{\n\t\t\n\t\t\n\t\t\n\t\tback-project $\\mathbf{x}^t_{ij}$=$K^{-1}\\mathbf{u}_{ij}^t$\n\t\t\n\t\tcalculate $\\mathbf{n}_i^t$ by Eq.(1)\n\t\t\n\t\tcalculate $h_i^t$ by Eq.(2)\n\t\t\n\t\t\\If{$\t|\\arctan(-\\frac{R_{32}}{R_{33}})|<\\theta_{1}$ \\& $h^t_i>0$}\n\t\t{\n\t\t\t\\If{$\\arccos(\\mathbf{n}^t_i, \\boldsymbol{t}_{t,t-1})<\\theta_{2}$}\n\t\t\t{\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t$\\{\\tilde{\\mathbf{x}}^t_{ij}\\}\\cup\\mathbf{x}^t_{ij}$\t\n\t\t\t}\n\t\t}\t\n\t}\n\n\tdelete repeate vertices and get $ \\{\\tilde{\\mathbf{x}}^t_{k}\\}\\subset\\{\\tilde{\\mathbf{x}}^t_{ij}\\}$\n\t\n\n\t{\/* \\emph{ensure enough points} *\/} \\\\\n\t\\While{size ($ \\{\\tilde{\\mathbf{x}}^t_{k}\\}$) $>$ 5 }{ \n\t\t\n\t\t\\For{iterations $<Z$}{\n\t\t\trandomly select  $\\mathbf{x}^t_{1},\\mathbf{x}^t_{2},\\mathbf{x}^t_{3}\\in \\{\\tilde{\\mathbf{x}}^t_{k}\\}$\n\t\t\t\n\t\t\tcalculate $\\mathbf{n}^t$ by Eq.(1)\n\t\t\t\n\t\t\t\\For{$\\tilde{\\mathbf{x}}^t_{g}$ in $\\{\\tilde{\\mathbf{x}}^t_{k}\\}$}\n\t\t\t{\n\t\t\t\t$D\\gets|\\mathbf{n}^t\\cdot\\tilde{\\mathbf{x}}^t_{g}-h^t|$ by Eq.(4) \n\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\If{$D<d_{thresh}$}{\n\t\t\t\t\t$\\mathcal{G}_k\\cup \\{\\tilde{\\mathbf{x}}^t_{g}\\}$\n\t\t\t\t}\n\t\t\t}\n\t\t\t\\If{size $(\\mathcal{G}_k)$ $>$ size $(\\mathcal{G}_{best})$ }\n\t\t\t{\n\t\t\t\t$\\mathcal{G}_{best}\\gets\\mathcal{G}_k$ \\tcp{best model select}\n\t\t\t}\t\n\t\t\t\n\t\t}\n\t\t$\\{\\tilde{\\mathbf{x}}^t_{g}\\}\\gets\\mathcal{G}_{best}$\n\t\t\n\t\t\\Return {$\\{\\tilde{\\mathbf{x}}^t_{g}\\}$}\n\t}\n\\end{algorithm}\t\n\n\n\n\\subsection{Ground Point Aggregation}\n\nDue to the strict criteria by GPE, the inliers are not enough for accurate estimation of the ground plane. Therefore, we propose the GPA algorithm to aggregate ground points from consecutive image frames. As shown in Fig. \\ref{lpg}, we leverage the sliding window method to select image frames, and a frame buffer is maintained to store the camera poses and ground points in the current window. At each time step, with the arrival of a new image frame, we update the buffer and then estimate the ground plane by solving a least-squares problem.    \n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[height=5.5cm]{SW.pdf}\n\t\\caption{Illustration of the GPA algorithm. In the bottom-left figure, the red dots indicate ground points, and the green line segment is the normal of each triangle. In the bottom-right figure, the red quadrilateral is the estimated ground plane based on the aggregated ground points in the current window.}\n\t\\label{lpg}\n\\end{figure}\n\nFrom the MVO thread, we can get the pose $T_t$ and the inliers $\\mathbf{x}^t_g$ of each frame from the buffer. We then can transform each of the inliers, denoted by $\\mathbf{x}^t_{gi}$, into the global frame,\n\\begin{equation}\n\t\\mathbf{p}_i=T_t \\,[\\mathbf{x}^t_{gi}, 1]^T.\n\\end{equation}\n\nSuppose there are $N$ local ground points in the buffer, the least-squares problem that minimizes the plane-distance error of these points is formulated as follows:\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\min_{\\boldsymbol{\\mu}}\\sum_{i=1}^{N}\\|\\boldsymbol{\\mu}^T \\mathbf{p}_i\\|_2&=\\min_{\\boldsymbol{\\mu}}\\boldsymbol{\\mu}{P_t}{P_t}^T\\boldsymbol{\\mu}^T, \\\\\n\t\t\\boldsymbol{\\mu} = &[\\mathbf{n}_t, -h_t]^T,\n\t\\end{aligned}\n\t\\label{mat}\n\\end{equation}\nwhere ${P_t}=[\\mathbf{p}_1, \\mathbf{p}_2, \\cdots, \\mathbf{p}_N ]\\in R^{4\\times N}$. Equation (\\ref{mat}) can be rewritten as follows:\n\\begin{equation}\n\t\\begin{aligned}\n\t\t&\\min_{\\boldsymbol{\\mu}} \\boldsymbol{\\mu} Q_t\\boldsymbol{\\mu}^T,  \\\\ \n\t\tQ_t&={P}_t{P}_t^T\\in R^{4\\times4},\t\n\t\\end{aligned}\n\t\\label{opt}\n\\end{equation}\nwhich can then be efficiently solved by the SVD method.\n\nTo further improve the estimation accuracy of $\\boldsymbol{\\mu}$, we also introduce a weighting matrix $\\Sigma={\\sigma^{-2}_z}I$, where $\\sigma_z$ is the normalized distance of the point depth to their mean value. As a result, the matrix $Q$ in Eq. \\eqref{opt} becomes,\n\\begin{equation}\n\tQ_t={P}_t \\Sigma {P}_t^T\\in R^{4\\times4}.\n\\end{equation}\n\nAnother important refinement on the estimated plane parameter is to conduct a RANSAC-based optimization, which shares the same idea as Eq. \\eqref{ransac}. In each iteration of the optimization, we first estimate $\\boldsymbol{\\mu}$, and then calculate the distance between $\\mathbf{p}_i$ and the estimated plane. Points with a distance larger than $0.01$m are removed, and the remaining is then leveraged to estimate a new $\\boldsymbol{\\mu}$. Such a process continues until convergence. We denote the final plane normal by $\\mathbf{n}_t^*$ and the reserved ground points by $\\mathbf{p}_k^*, k\\in\\{1, 2, \\cdots, K\\}$. The relative camera height then can be calculated by projecting the camera center to the ground plane:\n\\begin{equation}\n\th^t_j=\\mathbf{n}_t^* \\cdot (\\mathbf{p}_c-\\mathbf{p}_k^*),\n\t\\label{multi-h}\n\\end{equation}\nwhere $\\mathbf{p}_c$ is the camera center of frame $I_t$ in the global frame. It is worth noting that there are $K$ estimated camera heights, which will be further processed to recover a smooth scale. Details of the GPA algorithm are presented in Alg. \\ref{alg2}.\n\n\\begin{algorithm}[t]\n\t\\label{alg2}\n\t\\caption{Ground Points Aggregation (GPA)}\n\t\n\t\\KwIn{ $I_t$, $\\{\\mathbf{x}^t_{g}\\}$, $T_t$}\n\t\n\t\\KwOut{$\\{h^t\\}$}\n\t\n\tbuffer $\\{queue\\}\\gets \\emptyset$\n\t\n\tinliers in global frame $\\{\\mathbf{p}_t\\} \\gets \\emptyset$\n\t\n\tinliers in current frame $\\{\\mathbf{x}_g^t\\} \\gets \\emptyset$\n\t\n\treserved ground points $\\{\\mathbf{p}_k^*\\}\\gets \\emptyset$\n\t\n\tcamera heights $\\{h_t\\}\\gets \\emptyset$\n\t\n\t\n\t\\While{$I_t$ is not empty}{\n\t\t$\\{queue\\} \\cup \\{I_t\\} $\t\n\t\t\n\t\t\\If{$size(\\{queue\\})>4$}{\n\t\t\t\n\t\t\t\\textsc{pop$\\_$front}($\\{queue\\}$) \n\t\t\t\n\t\t\t\\tcp{fixed number of frames}\n\t\t\t\\For{$each$ $I_t$ in $\\{queue\\}$}{\n\t\t\t\tcalculate $\\mathbf{p}_i$ by Eq.(5)\n\t\t\t\t\n\t\t\t\t$\\{\\mathbf{p}_t\\} \\cup \\{\\mathbf{p}_i\\} $\n\t\t\t}\n\t\t\t\n\t\t\tMatrix $P_t$\n\t\t\t\n\t\t\tcalculate $\\Sigma={\\sigma^{-2}_z}I$ \\tcp{Weighting Matrix}\n\t\t\t\n\t\t\tcalculate $Q_t$ by Eq.(8)\n\t\t\t\n\t\t\t$\\boldsymbol{\\mu} \\gets \\textsc{SVDDecomposition}(Q)$\n\t\t\t\n\t\t}\n\t\n\t\t\n\t\t\\For{iteration $<Z$}{\n\t\t\tRandomly select $\\mathbf{p}_s \\in \\{\\mathbf{p}_t\\}$\n\t\t\t\n\t\t\t$D \\gets \\boldsymbol{\\mu}^T \\cdot \\mathbf{p}_s $\n\t\t\t\n\t\t\t\\If{$D > d_{thresh}$}{\n\t\t\t\t\n\t\t\t\tRemove $\\mathbf{p}_s$ in $\\{\\mathbf{p}_t\\}$\n\t\t\t\t\n\t\t\t\t\n\t\t\t}\n\t\t\t$\\{\\mathbf{p}_k^*\\}\\gets \\{\\mathbf{p}_t\\}$\n\t\t\t\n\t\t\tcalsulate $Q_s$ by Eq.(8)\n\t\t\t\n\t\t\t$Q \\gets Q_s$\n\t\t\t\n\t\t\t$\\boldsymbol{\\mu_s} \\gets \\textsc{SVDDecomposition}(Q)$\n\t\t\t\n\t\t\t$\\boldsymbol{\\mu} \\gets \\boldsymbol{\\mu_s}$ \\tcp{update model}\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t}\n\t\t\n\t\t\\For{$\\mathbf{p}^*$ in $\\{\\mathbf{p}_k^*\\}$}{\n\t\t\tcalculate $h_j^t$ by Eq.(9)\n\t\t\t\n\t\t\t$\\{h^t\\}\\cup \\{h_j^t\\}$\n\t\t}\n\t\t\\Return{$\\{h^t\\}$}\n\t}\n\t\n\t\n\n\\end{algorithm}\n\n\\subsection{Filter-Based Scale Recovery}\nAfter we compute the relative camera height $h$ of each frame, the scale factor is then obtained by $s_t=h^*\/h$, the motion scale of each frame is recovered by\n\\begin{equation}\n\t\\boldsymbol{t}^{\\dagger}_{t,t-1}=s_t \\cdot \\boldsymbol{t}_{t,t-1}.\n\\end{equation}\n\nCorresponding to the multiple $h$ values in Eq. \\eqref{multi-h}, there are also multiple estimated scales.\n\n\nBy plotting the scaled camera heights of each frame in the figure, shown in Fig. \\ref{gaussian}, we find the data do not strictly follow a Gaussian distribution. Therefore, we choose the median point as the scale of the current frame. In the time domain, a moving average filter is applied, shown in Fig. \\ref{filter-b}, which can give a more smooth result.\n\n\\begin{figure}[!htbp]\n\t\\centering\t\n\t\\subfigure[The distribution of the scaled camera heights.]{\n\t\t\\includegraphics[width=0.90\\linewidth]{distribution.pdf}\n\t\t\\label{gaussian}\n\t}\n\t\\subfigure[The estimated camera height on sequence-02 and -05.]{\n\t\t\\includegraphics[width=0.50\\linewidth]{Figure01.png}\\hspace{-5mm}\n\t\t\\includegraphics[width=0.50\\linewidth]{Figure02.png}\n\t\t\\label{filter-b}\n\t}\t\n\t\\centering\n\t\\caption{Demonstration of the filter-based scale recovery. The green points are the scaled camera heights. The red curve is the smoothed one.}\n\t\\label{filter}\n\\end{figure}\n\n\n\\section{Experiments}\nWe conduct experiments to evaluate the performance of our proposed method. The MVO system used in the experiments is implemented based on ORB-SLAM2, and the proposed scale recovery method is integrated as an independent thread. The system architecture is demonstrated in Fig. \\ref{overview}. The KITTI dataset \\cite{geiger2012we} is adopted as the benchmark dataset, in which sequence-01 is not used since it fails most feature-based VO systems. All the experiments are conducted using a laptop with Intel(R) Core(TM) i5-6300HQ CPU @ 2.30 GHz. \n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[height=8.2cm]{trajectoryv2.pdf}\n\t\\caption{The re-scaled trajectories on KITTI dataset. The blue and green trajectories are generated by our system and ORB-SLAM2-noLC, respectively.}\n\t\\label{fig3}\n\\end{figure*}\n\n\\subsection{Qualitative Evaluation}\n\nThe qualitative evaluation results of the proposed method are visualized in Fig. \\ref{fig3}. The trajectories outputted by the system are recovered using the proposed method, which means similarity transformation is not necessary when compared with ground-truth trajectories\\cite{grupp2017evo}. The baseline trajectories, indicated by green color in Fig. \\ref{fig3}, are generated by ORB-SLAM2 with loop closure detection disabled, denoted by ORB-SLAM2-noLC. We can see that our re-scaled trajectories can eliminate scale drift and form correct loops, which demonstrates the effectiveness of the proposed method. \n\nThe comparison of trajectory length between the ground truth and the proposed method is shown in Table \\ref{tabel-1}, in which the Relative Length Error (RLE) is computed by $e=|l_{gt}-l_{ours}|\/|l_{gt}|$, where $l$ is the length of a trajectory.\nFor sequence-02, -04, -06, -07, and -10, the RLE is less than 1$\\%$. The high performance is due to the fact that most roads in these sequences are straight lines and contain rich features. Sequence-00 and -08 are more complicated cases, in which the scenario is composed of a lot of curves and turns. The path lengths are relatively longer than other sequences. The RLE is thus slightly higher, 2.17$\\%$ and 2.72$\\%$, respectively. Nevertheless, the results show that the proposed system can estimate an accurate scale of the entire trajectory.\n\n\\begin{table}[t]\n\t\\renewcommand\\tabcolsep{16.0pt} \n\t\\caption{Comparison of re-scaled trajectory \\protect\\\\length with ground truth}\n\t\\begin{center}\n\t\t\\label{table_time}\t\n\t\t\\begin{tabular}{lllllll}\n\t\t\t\\toprule\n\t\t\tSeq & GT (m)&Ours (m) & RLE (\\%)\\\\\n\t\t\t\\midrule\n\t\t\t00 & 3645.213 & 3724.419 & 2.173   \\\\\n\t\t\t02 &  5071.151 &5067.233 &0.757   \\\\\n\t\t\t03 &547.774 &560.888 &0.558 \\\\\n\t\t\t04 &391.861 &393.645 &0.455   \\\\\n\t\t\t05 &2175.159 &2205.576 &1.398   \\\\\n\t\t\t06 &1235.982 &1232.876 &0.251   \\\\\n\t\t\t07 &709.065 &694.697 &0.767   \\\\\n\t\t\t08 &3137.398 &3222.795 &2.722   \\\\\n\t\t\t09 &1738.601 &1705.051 &1.929   \\\\\n\t\t\t10 &911.399 &919.518 &0.890   \\\\\n\t\t\t\\bottomrule  \t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\label{tabel-1}\n\\end{table}\n\\renewcommand{\\arraystretch}{1} \n\\begin{table*}[t]\n\t\\centering\n\t\\fontsize{6}{8.4}\\selectfont\n\t\\begin{threeparttable}\n\t\t\\caption{comparison of average translation errors and rotation errors with the latest visual odometry methods on kitti dataset}\n\t\t\\label{table2}\n\t\t\\begin{tabular}{ccccccccccccccccccc}\n\t\t\t\\toprule\n\t\t\t\\multirow{3}{*}{Seq}& \t\n\t\t\t\\multicolumn{2}{c}{ORB-SLAM2-noLC \\cite{mur2017orb}}\n\t\t\t&\\multicolumn{2}{c}{ VISO2-M\\cite{2011StereoScan}}\n\t\t\t&\\multicolumn{2}{c}{VISO2-Stereo\\cite{2011StereoScan}}\n\t\t\t&\\multicolumn{2}{c}{Song et.al\\cite{song2015high}}\n\t\t\t&\\multicolumn{2}{c}{Zhou et.al\\cite{zhou2019ground}}\n\t\t\t&\\multicolumn{2}{c}{Brandon et.al\\cite{wagstaff2020self}}\n\t\t\t&\\multicolumn{2}{c}{DNet\\cite{xue2020toward}}\n\t\t\t\n\t\t\t&\\multicolumn{2}{c}{Ours}\\cr\n\t\t\t\\cmidrule(lr){2-3} \\cmidrule(lr){4-5}\n\t\t\t\\cmidrule(lr){5-6} \\cmidrule(lr){6-7}\n\t\t\t\\cmidrule(lr){8-9} \\cmidrule(lr){10-11}\n\t\t\t\\cmidrule(lr){12-13} \\cmidrule(lr){14-15}\n\t\t\t\\cmidrule(lr){16-17}\n\t\t\t&Trans&Rot&Trans&Rot&Trans&Rot&Trans&Rot\n\t\t\t&Trans&Rot&Trans&Rot&Trans&Rot&Trans&Rot\\cr\n\t\t\t&(\\%)&(deg\/m)&(\\%)&(deg\/m)&(\\%)&(deg\/m)&(\\%)&(deg\/m)\n\t\t\t&(\\%)&(deg\/m)&(\\%)&(deg\/m)&(\\%)&(deg\/m)&(\\%)&(deg\/m)\\cr\n\t\t\t\\midrule\n\t\t\t00&20.8&$-$&11.91&0.0209&2.32&0.0109&2.04&0.0048&2.17&0.0039&1.86& &1.94&$-$  &\\textbf{1.41}&0.0054\\cr\n\t\t\t\n\t\t\t01&$-$&$-$&$-$&$-$&$-$&$-$&$-$ &$-$ & $-$&$-$ &$-$ &$-$ &$-$&$-$&$-$&$-$\\cr\n\t\t\t\n\t\t\t02&9.52&$-$&3.33&0.0114&\\textbf{2.01}&0.0074&1.50&0.0035& $-$&$-$ &2.27&$-$ &3.07&$-$  &2.18&0.0046\\cr\n\t\t\t\n\t\t\t03&$-$&$-$&10.66&0.0197&2.32&0.0107&3.37&0.0021&2.70&0.0044&$-$& $-$&$-$&$-$  &\\textbf{1.79}&0.0041\\cr\n\t\t\t\n\t\t\t04&$-$&$-$&7.40&0.0093&0.99&0.0081&2.19&0.0028&$-$ &$-$ &$-$&$-$ &$-$&$-$  &\\textbf{1.91}&0.0021\\cr\n\t\t\t\n\t\t\t05&18.63&$-$&12.67&0.0328&1.78&0.0098&1.43&0.0038&$-$ &$-$ &\\textbf{1.50}&$-$ &3.32&$-$  &1.61&0.0064\\cr\n\t\t\t\n\t\t\t06&18.98&$-$&4.74&0.0157&\\textbf{1.17}&0.0072&2.09&0.0081& $-$& $-$&2.05&$-$ &2.74&$-$  &2.03&0.0044\\cr\n\t\t\t\n\t\t\t07&13.82&$-$&$-$&$-$&$-$&$-$&$-$&$-$&$-$ &$-$ &1.78& $-$&2.74& $-$   &\\textbf{1.77} &0.0230\\cr\n\t\t\t\n\t\t\t08&22.06&$-$&13.94&0.0203&2.35&0.0104&2.37&0.0044& $-$& $-$&2.05&$-$ &2.72&$-$  &\\textbf{1.51}&0.0076\\cr\n\t\t\t\n\t\t\t09&12.74&$-$&4.04&0.0143&2.36&0.0094&1.76&0.0047&$-$ &$-$ &\\textbf{1.50}&$-$ &3.70& $-$ &1.77&0.0118\\cr\n\t\t\t\n\t\t\t10&4.86&$-$&25.20&0.0388&1.37&0.0086&2.12&0.0085&2.09&0.0054&3.70& &5.09&  &\\textbf{1.25} &0.0031& \\cr\n\t\t\t\\midrule\n\t\t\tAvg&18.17& $-$&14.39&0.0245&2.32&0.0095&2.03&\\textbf{0.0045}&2.32&0.045&2.03 & $-$&3.17&$-$&\\textbf{1.72} & 0.0068&\\cr\n\t\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{threeparttable}\n\\end{table*}\n\n\n\n\n\\subsection{Quantitative Evaluation}\n\n{The quantitative comparison} between our method and the baseline methods, including \\cite{wagstaff2020self,xue2020toward,song2015high,zhou2019ground,2011StereoScan,mur2017orb}, is presented in Table \\ref{table2}. The average translation error and rotation error are adopted as evaluation metrics.\n\nWe can see that ORB-SLAM2-noLC and VISO2-M have the worst performance due to the lack of loop closure detection. The scale drift of the two methods induces a large translation error, 18.17$\\%$ and 14.39$\\%$ respectively, while the VO systems with scale recovery all maintain a low translation error, $<4\\%$. It can also be seen that a MVO system with scale recovery \\cite{song2015high, zhou2019ground, xue2020toward, wagstaff2020self} can exhibit competitive performance with a stereo VO system like VISO2-M \\cite{2011StereoScan}, which significantly demonstrates the importance of scale recovery for MVO. \n\nIn terms of monocular systems, we can see our proposed method achieves the minimum translation error while maintaining a competitive performance on rotation error. The methods proposed by Song \\textit{et al.} \\cite{song2015high} and Zhou \\textit{et al.} \\cite{zhou2019ground} can not work with sequence-07, because they both rely on a fixed region to extract ground points, whereas occlusions by moving vehicles occur frequently in this sequence. In contrast with \\cite{song2015high, zhou2019ground}, the proposed method works well with sequence-07 with a translation error of 1.77$\\%$, benefiting from the GPA algorithm.  \n\n\nIn \\cite{xue2020toward}, a deep neural network, named DNet, is proposed for monocular depth prediction and scale recovery. Compared with this method, our method shows a better accuracy in all the sequences. In \\cite{wagstaff2020self}, a real-time MVO system is implemented based on self-supervised learning techniques. This method can slightly outperform our proposed method in sequence-05 and -09, but has a much lower accuracy in sequence-00, -08, and -10. A similar phenomenon can be observed when comparing with \\cite{song2015high}. This indicates a significant variance on the performance of \\cite{wagstaff2020self}. Actually, this is the limitation of most deep learning based methods, which has been discussed in detail by \\cite{wang2020tartanair}.    \n\nThe comparative experiments in Table \\ref{table2} significantly verify the effectiveness of our method and demonstrates the advantages over the latest methods in the literature.\n\\subsection{Efficiency Evaluation}\nAnother significant advantage of our method lies in its high efficiency. We evaluate the run-time of our system on all the KITTI sequences mentioned above, and the experiment repeats five times. The median run-time is reported in Fig. \\ref{time}. In all the experiments, the MVO thread requires $50$-$55$ ms, while the GPE and GPA requires less than $10$ ms, which makes the system suitable for real-time applications.   \n\\section{CONCLUSIONS}\n\nIn this work, we present a light-weight MVO system with accurate and robust scale recovery, aiming to reduce scale drift and provide accurate odometry in long-distance navigations without loop closure. We solve the scale ambiguity for MVO by implementing our GPE-GPA algorithm for selecting high-quality points and optimizing them in a local sliding window. Sufficient data and robust optimizer provide accurate metric trajectory leveraging the ratio of the estimated camera height and the real one. Extensive experiments show that our proposed framework can achieve state-of-the-art accuracy and recover a metric trajectory without additional sensors. The system is designed to be a light-weight framework, which can achieve real-time performance with 20 Hz running frequency. Our proposed light-weight MVO system facilitates the localization and navigation of low-cost autonomous vehicles in long-distance scenarios.\n\nFurther study into integrating the uncertainty of plane estimation will be considered, which will further improve the accuracy of scale recovery. The light-weight neural network for ground segmentation will also be considered to help constrain the extraction of high-quality ground points.\n\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[scale=0.45]{time-consum2.pdf}\n\t\\caption{Run-time of our MVO system on KITTI dataset. The time costs of MVO thread, the GPE algorithm, and the GPA algorithm are reported.}\n\t\\label{time}\n\\end{figure}\n\\enlargethispage{-7.8cm}\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nChiral Perturbation Theory (ChPT) is the effective field theory of QCD at low energies \\cite{weinberg75,sielder}. \nIts  paradigmatic application is the purely mesonic sector in $SU(2)$.\\footnote{Which even presents one corner of \nconcern due to the enhanced role of the right-hand-cut in the isoscalar scalar pion-pion scattering \n\\cite{npa620,nd,zeros,gamma,alba12,bernasigma,truong1988}, with an important \nimpact as well in the pion-nucleon ($\\pi N$) sector \\cite{sainiosumrule,aco2013,sigma2012}.}\nIts extension to the one-baryon sector presents some complications due to the \nlarge nucleon mass that does not vanish in the chiral limit \\cite{sainio88,manohar91}, which posed interesting problems to \nthe theory.\\footnote{A faster stabilization of the chiral series in this case has been recently accomplished \\cite{sigma2012,aco2013,strangeness14} by combining the covariant formalism of the Extended on Mass Shell Regularization Scheme (EOMS) \\cite{eoms} with the explicit inclusion \nof the $\\Delta(1232)$ in the $\\delta$-counting \\cite{delta-c} .}\n  For reviews on ChPT on these topics see e.g. \\cite{bijnens09,ecker,bernard,pich,ulf}.\n\nThe extension of ChPT to systems with a larger  baryonic number was  \nconsidered in  Ref.~\\cite{weinn}, where the chiral counting is applied to the calculation \nof the multi-nucleon potential. In these cases one also has to face the problem associated with the \ninfrared enhancement associated with the small nucleon kinetic energies, which requires \nto resum the infinite string of diagrams due to the iteration of  intermediate multi-nucleon states. \nThe extension of the chiral power counting to finite density system, including the contributions of multi-nucleon \nreducible diagrams, is given in Ref.~\\cite{finiteden}. For related reviews see e.g. \n\\cite{Epelbaum:2008ga,Machleidt:2011zz,Epelbaum:2005pn,Bedaque:2002mn,vanKolck:1999mw}.\n\n The application of the set up of Ref.~\\cite{weinn} to nucleon-nucleon ($NN$) scattering \nhas been phenomenologically successful \\cite{ordo94,entem,thesis,epe042}. However, the sensitivity \nof the results on the values of the cutoff taken to solve the associated  Lippmann-Schwinger equation for the \niteration of two-nucleon intermediate states has given rise to a flurry of publications, whose fair \n and comprehensive consideration is beyond this introduction. For more detailed accounts on this \nrespect the reader is referred to \\cite{Machleidt:2011zz,pavon06,nogga,kswa,phillipssw,pavon11,longyang,zeoli,Epelbaum:2008ga}.\n\n\nWe continue here  the application of the $N\/D$ method \\cite{chew} to $NN$ scattering \n extending the previous work of Refs.~\\cite{paper1,paper2,gor2013}. For this method the dynamical \ninput is not the $NN$ potential but the discontinuity of a $NN$ partial-wave amplitude along the \nleft-hand-cut (LHC), which is denoted in the following by $2i\\Delta(A)$. Here $A$ is   \nthe center of mass (c.m.) three-momentum squared of a $NN$ state. In other words, \n$\\Delta(A)$ is the imaginary part of a $NN$ partial-wave amplitude along the LHC, that \n extends for real $A$ with $A<-M_\\pi^2\/4$, being $M_\\pi$ the pion ($\\pi$) mass.  \nThe function $\\Delta(A)$  is due to the multi-exchange of pions driving the finite-range nuclear \nforces, while in a low-energy effective field theory the short-range nuclear forces are accounted for by \nlocal interactions of zero range that do not contribute to $\\Delta(A)$ for finite $A$.\n The two-nucleon irreducible contributions to $\\Delta(A)$ are amenable to a straightforward ChPT expansion, \nin much the same way as discussed  in Ref.~\\cite{weinn} for the calculation of the chiral $NN$ potential.\nHowever, $\\Delta(A)$ has also contributions from two-nucleon reducible diagrams but, \nas explained in Ref.~\\cite{gor2013},  these contributions require to cut all \nthe pion lines simultaneously when iterating one-pion exchange (OPE). \nIn this way, when including an extra $NN$ intermediate state in the iteration of the unitarity two-nucleon \ndiagrams their contribution to $\\Delta(A)$ starts further away in the LHC. It then results that the $n$th iteration of \n two-nucleon intermediate states, which at least requires $n+1$ OPE ladders, \n gives contribution to $\\Delta(A)$ only for $A<-(n+1)^2 M_\\pi^2\/4$. \nThis makes that its relevance \nfor physical values of $A$ ($A\\geq 0$) in the low-energy region clearly dismisses \nwith increasing $n$. As a result, because of the chiral expansion together with this other effect that\n numerically suppresses the proliferation of two-nucleon reducible diagrams in the calculation of $\\Delta(A)$,  \none can determine this function reliably in ChPT.\\footnote{Notice that the suppression \nof the iteration of two-nucleon reducible diagrams only occurs for $\\Delta(A)$, and it does not occur to any \nother ``component'' of a $NN$ partial wave amplitude.}\n\n In Refs.~\\cite{paper1,paper2} the $N\/D$ method was \nsolved with $\\Delta(A)$  calculated at leading order (LO) from OPE, while in Ref.~\\cite{gor2013}\nthe NLO contributions to  $\\Delta(A)$ were also included. \nThese contributions comprise two-nucleon irreducible two-pion exchange and once-iterated OPE, whose sum gives \nthe  leading two-pion exchange (TPE). \nReference~\\cite{gor2013} obtained a clear improvement in the reproduction of the phase shifts and mixing angles  \n given by the Nijmegen partial-wave analysis (PWA) \\cite{Stoks:1994wp} as compared with the LO \nstudy, so that a  global and rather good agreement  is achieved at NLO. \n We want to give one step forward and consider here the next-to-next-to-leading (NNLO) contributions to \n$\\Delta(A)$, which are given by the imaginary part along the LHC of the \ntwo-nucleon irreducible TPE diagrams with a NLO $\\pi N$ vertex in Heavy-Baryon ChPT (HBChPT) \\cite{peripheral}. \nWe see that the chiral expansion within our approach is well behaved, so that there is a steady improvement \nin the reproduction of the Nijmegen PWA results when passing from LO to NLO and then to NNLO, where a quite good\n reproduction of the Nijmegen PWA is  finally obtained. \nThis is accomplished in a progressive and smooth way, without violent variations in the results obtained at \nevery order.\\footnote{This was not the case in previous studies, e.g. in the model calculation of $NN$ scattering by Ref.~\\cite{lutz} \nthat uses a modified version of the $N\/D$ method by truncating the integrals along the LHC with a sharp cutoff.}\n In addition, we deal with convergent integrals by taking  enough number of subtractions so that the above \nreferred regulator dependence that arises when solving the Lippmann-Schwinger equation with a chiral $NN$ potential \nis avoided by construction in our approach. An interesting outcome from our study is that we\n corroborate the long-range correlations between the effective range and scattering length for each of the $NN$ $S$ waves,  \n$^1S_0$ and $^3S_1$, when only the corresponding scattering length is taken as experimental input. \nThese correlations, first noticed in Ref.~\\cite{pavon06}, were also obtained in the NLO $N\/D$ study of Ref.~\\cite{gor2013}, and \nwithin our approach they are deduced solely from basic principles of $NN$ partial-wave amplitudes, namely, \n chiral symmetry, unitarity and analyticity.\nThey  are typically fulfilled at the level of around a $10\\%$ when comparing with \nthe experimental values for the effective ranges.  \nWe should say that we can proceed further and include more subtractions, so that we can implement within our formalism \nthe exact values of the effective ranges, something not possible in the tight scheme of Ref.~\\cite{pavon06}.\n \nRegarding the subtraction constants we elaborate below a chiral power counting for them, by \ntaking into account the change in their values due to variations in the subtraction point.\n We show that at NLO and NNLO in the calculation of $\\Delta(A)$ one properly takes twice-subtracted dispersion relations (DRs). \nNevertheless, on top of this criterion we impose that one should obtain the proper threshold behavior \nfor higher partial waves, as well as having meaningful solutions of the integral equations (IEs) \nthat result from the corresponding DRs.\\footnote{By a meaningful solution we mean here a mathematical solution to the IE \nthat does not depend on the the number of points employed and \nin the arbitrary large extension of the LHC on which they lie  \n when performing the numerical discretization to solve the IE.} \nThese two requirements often imply the necessity of taking  more than two subtractions \nin the corresponding DRs relations. \nRegarding the number of subtractions used to guarantee the threshold behavior \nfor higher partial waves we use here the formalism developed in Ref.~\\cite{gor2013}, so that partial waves with \norbital angular momentum $\\ell\\geq 1$ and mixing partial waves with total angular momentum $J\\geq 1$ vanish  \nat threshold as $ A^\\ell$ and $A^J$, respectively. \nThis requires to take at least $\\ell$ or $J$ subtractions, in order, \nwith $\\ell-1$ or $J-1$ free parameters, respectively. \nBut at the end, as emphasized in Ref.~\\cite{gor2013}, \nnone or only one  of the resulting subtraction constants for a given partial wave with $\\ell>1$ (or $J>1$ for a mixing wave) is necessary to reproduce data. \nThis interesting point, which allows to treat easily higher partial waves,  is called in Ref.~\\cite{gor2013} the \nprinciple of maximal smoothness.\n\n  In our study we have also paid special attention to the issue concerning the impact on the results of the rather large size \nof the NLO $\\pi N$ counterterms, typically denoted by $c_i$ \\cite{ulf},  which first appear \n in the calculation of $\\Delta(A)$ at NNLO. \nIt is discussed in Ref.~\\cite{epe04} that the $\\pi N$ monomials, proportional to the $c_i$ counterterms, \n produce a too large contribution to the  $NN$ potential at medium and short distances   when it is \ncalculated at NNLO in dimensional regularization, which worsens the properties of the chiral expansion. \nBecause of this Ref.~\\cite{epe04} argued to better use  a cutoff regularization to calculate the NNLO potential, or equivalently, \nto cut  the energy spectral representation of the NNLO $NN$ potential at around the chiral symmetry breaking scale. \nThis last point would be equivalent to truncate the full extent of the LHC in our dispersive integrals. \n However, it is interesting to remark that we do not need to do that in order to obtain a good reproduction of the Nijmegen \nPWA when employing $\\Delta(A)$ determined up to NNLO. \nIn fact,  we observe that the definitive improvement of our results compared \nwith the Born approximation does not arise by \nmodifying  the two-nucleon \nirreducible diagrams at NNLO,  but by performing the iteration of two-nucleon unitarity diagrams as required\n by analyticity and  unitarity  in a well-defined way.\n\n\nAfter this introduction we review the $N\/D$ method for coupled and uncoupled partial waves in \nSec.~\\ref{unformalism}.\n The function $\\Delta(A)$, calculated in ChPT up to NNLO, is discussed in Sec.~\\ref{delta}, \nwhere we also elaborate the chiral power counting for the subtraction constants.\n Sections \\ref{1s0} to \n\\ref{gi5w} are devoted to discuss the application of the $N\/D$ method to \n the different $NN$ partial waves up to $J=5$.\n There it is shown that a quite good reproduction of the Nijmegen PWA phase shifts and mixing angles results.\n In these sections we also compare \nwith the Born approximation for higher partial waves and discuss on the relative importance of the different contributions \nto $\\Delta(A)$.\n Our concluding remarks are given in Sec.~\\ref{conc}. \nFinally, we discuss in Appendix \\ref{appen:vs} a method to calculate higher order shape parameters \n of the $NN$ $S$ waves. \n\n\n\n\\section{The $N\/D$ method}\n\\label{unformalism}\n\nA detailed presentation of the formalism for the $N\/D$ method \\cite{chew}\n can be found in Ref.~\\cite{gor2013}.\n Here we only reproduce the main facets of the approach. \n\n\\subsection{Uncoupled partial waves}\n\\label{upw}\nAn uncoupled $NN$ partial wave is written as the quotient of two functions, where the\n numerator is the function $N(A)$ and the denominator is $D(A)$. \nThen, one writes\n\\begin{align}\nT(A)&=\\frac{N(A)}{D(A)}~,\n\\label{eq.ta}\n\\end{align}\nwith $T(A)$ the corresponding $NN$ partial wave in the c.m. frame. In the following we use the spectroscopic notation and denote by \n$^{2S+1}L_J$ the different $NN$ partial waves with $S$ the total spin, $L$ the orbital angular momentum and $J$ the \ntotal angular momentum. The point for the splitting of $T(A)$ in two functions is because $N(A)$ has only LHC while $D(A)$ \nhas only right-hand cut (RHC), also called unitarity cut.\n The following expressions for the discontinuities of the functions $N(A)$ and $D(A)$ \nalong their respective cuts then arise,\n\\begin{align}\n\\mathrm{Im} D(A)=-\\rho(A) N(A)~,~A>0~,\\nonumber \\\\\n\\mathrm{Im} N(A)=\\Delta(A) D(A)~,~A<L~.\n\\label{disconts}\n\\end{align}\nHere $L=-M_\\pi^2\/4$  and it represents the onset of the LHC for $A<L$ due to OPE, \nand $\\rho(A)$ is the phase space factor\n\\begin{align}\n\\rho(A)=\\frac{m \\sqrt{A}}{4\\pi}~~,\n\\label{rhodef}\n\\end{align}\nwhere $m$ is the nucleon mass. \n  The first of the relations in Eq.~\\eqref{disconts} is a consequence of elastic unitarity for a $NN$ partial wave, which reads \n\\begin{align}\n\\mathrm{Im}T(A)=\\rho |T(A)|^2~,~A>0~.\n\\end{align}\n In terms of $1\/T(A)$ this can be recast simply as \n\\begin{align}\n\\mathrm{Im}\\,\\frac{1}{T(A)}=-\\rho(A)~,~ A>0~.\n\\label{invTun}\n\\end{align} \nWith this normalization the relation between the $T$ and $S$ matrices is $S(A)=1+2i\\rho(A) T(A)$.  \nThe  discontinuity of a $NN$ partial wave \n$T(A)$ along the LHC is given by $2i \\Delta(A)$, which directly implies the second expression in Eq.~\\eqref{disconts}. \n\n\n Standard DRs for the functions $D(A)$ and $N(A)$ are derived in Ref.~\\cite{gor2013} under the assumption that \nthe function $D(A)$  does not diverge faster than a polynomial of degree $n_0$  for $A\\to \\infty$. Then for $n>n_0$ one \ncan write \\cite{gor2013}\n\\begin{align}\nD(A)&=\\sum_{i=1}^n \\delta_i (A-C)^{i-1}-\\frac{(A-C)^n}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)N(q^2)}{(q^2-A)(q^2-C)^n}~,\\nonumber\\\\\nN(A)&=\\sum_{i=1}^n \\nu_i (A-C)^{i-1}+\\frac{(A-C)^n}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2-C)^n}~,\n\\label{standardr}\n\\end{align}\nwhere $C$ is the subtraction point. Notice that the same number of subtractions is taken both in $D(A)$ and $N(A)$. \nThe argument given in Ref.~\\cite{gor2013} makes use of the fact that $N(A)=T(A)D(A)$ and $T(A)$,\n because of unitarity, vanish at least as $A^{-1\/2}$ for $A\\to +\\infty$. \n As a result if $D(A)$ diverges at most as $A^{n_0}$ then $N(A)$ does not diverge faster than $A^{n_0-1\/2}$. \nHere we take into account the\nSugawara and Kanazawa theorem \\cite{barton,suga}, as a consequence of which any function \nlike $D(A)$ or $N(A)$ with only one cut  of infinite extent along the\n real axis has the same limit for $A\\to \\infty$ in any direction of the $A$-complex plane. \n In addition, it is clear from Eq.~\\eqref{standardr} and the standard theory of DRs \\cite{spearman}, \nthat we can take different values for the corresponding subtraction points for each function \nseparately. Indeed, for many partial waves  we will take the subtractions for the function $D(A)$   in two \ndifferent subtraction points, one at $C=0$ and the other at $C=-M_\\pi^2$. This is motivated by the fact that \nwe impose the normalization \n\\begin{align}\nD(0)=1~,\n\\label{nor.da}\n\\end{align}\nwhich can always be done by dividing simultaneously $D(A)$ and $N(A)$ by a constant without altering their ratio \ncorresponding to $T(A)$, Eq.~\\eqref{eq.ta}. \n In this way,  one subtraction for $D(A)$ is always taken at $C=0$ in order \nto guarantee straightforwardly the normalization Eq.~\\eqref{nor.da}.\n\nTo solve $D(A)$ in terms of the input $\\Delta(A)$ and the subtraction constants  we substitute   in Eq.~\\eqref{standardr} the expression \nfor $N(A)$ into the DR of $D(A)$, so that we end with the following IE for $D(A)$ with $A<L$,\n\\begin{align}\nD(A)&=\\sum_{i=1}^n \\delta_i (A-C)^{n-i}-\\sum_{i=1}^n \\nu_i\\frac{(A-C)^n}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-C)^{n-i+1}}\\nonumber\\\\\n&+\\frac{(A-C)^n}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^n}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-k^2)}~.\n\\label{inteq1}\n\\end{align}\nThe key point of the method is to solve this IE numerically which provides the knowledge of $D(A)$ for $A<L$. \nOnce $D(A)$ is known  along the LHC  we can calculate all the \nfunctions $D(A)$, $N(A)$ and $T(A)$ in the whole $A$-complex plane. To obtain $D(A)$ one can use \nEq.~\\eqref{inteq1} and for $N(A)$ one has the second of the DRs in Eq.~\\eqref{standardr}. \nOnce $N(A)$ and $D(A)$ are known one can calculate $T(A)$ by applying Eq.~\\eqref{eq.ta}.\n\n Notice also that \nthe integrations along the RHC in Eq.~\\eqref{inteq1} can be done algebraically \nin terms of the function\n\\begin{align}\ng(A,k^2)&\\equiv \\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-k^2)}=\\frac{i m \/ 4\\pi}{\\sqrt{A+i0^+}+\\sqrt{k^2+i0^+}}~.\n\\label{gdef}\n\\end{align}\n The term $+i0$  is necessary  for negative $A$ or $k^2$, with the prescription \n $\\sqrt{-1\\pm i0}=\\pm i\\pi$. \nWe can calculate the other RHC integrals \n of Eq.~\\eqref{inteq1} with higher powers of the factor $(q^2-C)$ in the denominator by simple differentiation \nwith respect to $C$ of the function $g(A,k^2)$, \n\\begin{align}\n\\frac{\\partial^{p-1}\\,g(A,C)}{\\partial C^{p-1}}=\\frac{(p-1)!}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2-C)^p}~.\n\\label{derg}\n\\end{align}\n\n\\subsection{Coupled partial waves}\n\\label{cpw}\n\nFor the case of the triplet partial waves with total angular momentum $J$ we have the mixing between the partial waves \nwith $\\ell=J-1$ and $\\ell'=J+1$, except for the $^3P_0$. In this case we denote the different coupled partial waves \nby $t_{ij}(A)$ with $i,~j=1,~2$, where 1 labels the lower angular momentum $\\ell\\equiv \\ell_1$ and 2 the higher one \n$\\ell' \\equiv \\ell_2$. All of \nthem are gathered together in the $2\\times 2$ matrix $T(A)$, in terms of which the $S$-matrix reads\n\\begin{align}\nS(A)&=I+2i\\rho(A)T(A)\\nonumber\\\\\n& = \\left(\n \\begin{array}{cc}\n \\cos 2\\epsilon_J\\ e^{2i\\delta_1}            & i\\sin 2\\epsilon_J\\ e^{i(\\delta_1+\\delta_2)} \\\\ \ni\\sin 2\\epsilon_J\\ e^{i(\\delta_1+\\delta_2)} &   \\cos 2\\epsilon_J\\ e^{2i\\delta_2}\n \\end{array} \\right)~,\n\\label{relst}\n\\end{align}\n where $I$ is the $2\\times 2 $ unit matrix, $\\epsilon_J$ is the mixing angle, and $\\delta_{1}$ and $\\delta_2$ are the phase\n shifts for the channels with orbital angular momentum $\\ell$ and $\\ell'$, in this order. Equation \n\\eqref{relst} corresponds to the Stapp parameterization \\cite{stapp}.\n\nNow, the $N\/D$ method explained for the uncoupled waves in Sec.~\\eqref{unformalism} is extended to the coupled channel case \n\\cite{paper2,gor2013} by writing down three $N\/D$ equations, one for every $t_{ij}(A)$ [notice that because of time reversal \n$t_{12}(A)=t_{21}(A)$]. \n The main difference with respect to the uncoupled case is that now the discontinuity along the RHC \nof the inverse of $t_{ij}(A)$ does not \nsimply correspond to $-\\rho(A)$, but it also contains information on the other coupled partial waves. \n In the following let us employ the notation \n\\begin{align}\n\\mathrm{Im} \\frac{1}{t_{ij}(A)} \\equiv -\\nu_{ij}(A)~,A>0~.\n\\label{nuij.def}\n\\end{align}\n From Eq.~\\eqref{relst} it is straightforward to obtain  the following expressions for the $\\nu_{ij}(A)$ \\cite{paper2,gor2013},  \n\\begin{align}\n\\nu_{11}(A) & =   \\rho(A) \\left[ 1- \\frac{\\frac{1}{2}\\sin^2 2\\epsilon_J}{1-\\cos 2\\epsilon_J \\cos 2\\delta_1} \\right]^{-1} ~,\\nonumber\\\\\n\\nu_{22}(A) & =   \\rho(A) \\left[ 1- \\frac{\\frac{1}{2}\\sin^2 2\\epsilon_J}{1-\\cos 2\\epsilon_J \\cos 2\\delta_2} \\right]^{-1}~,\\nonumber \\\\\n\\nu_{12}(A) & = 2 \\rho(A) \\frac{\\sin(\\delta_1 + \\delta_2)}{\\sin 2\\epsilon_J} \\label{nuij}~.\n\\end{align}\nIn terms of them we have the analogous  DRs for $D(A)$ and $N(A)$ of Eq.~\\eqref{standardr}, but now distinguishing \nbetween the different $D_{ij}(A)$ and $N_{ij}(A)$ such that $t_{ij}(A)=N_{ij}(A)\/D_{ij}(A)$, and employing $ \\nu_{ij}(A)$ \ninstead of simply $\\rho(A)$. The following expressions are obtained \\cite{gor2013}:\n\\begin{align}\nD_{ij}(A)&=\\sum_{p=1}^n \\delta^{(ij)}_p (A-C)^{p-1}-\\sum_{p=1}^n \\nu^{(ij)}_p\\frac{(A-C)^n}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)}{(q^2-A)(q^2-C)^{n-p+1}}\\nonumber\\\\\n&+\\frac{(A-C)^n}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2-C)^n}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)}{(q^2-A)(q^2-k^2)}~,\\\\\nN_{ij}(A)&=\\sum_{p=1}^n \\nu^{(ij)}_p (A-C)^{p-1}+\\frac{(A-C)^n}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2-A)(k^2-C)^n}~.\n\\label{standardrcc}\n\\end{align}\n Here, we also impose the normalization condition at $A=0$,\n\\begin{align}\nD_{ij}(0)=1~.\n\\end{align}\nOf course, the same remark concerning the subtraction point as done in Sec.~\\ref{upw} is also in order here. Namely, \nwe can use different subtraction points for the functions $D_{ij}(A)$ and $N_{ij}(A)$, as well as to use even different subtraction \npoints in the same function, as we will do below for $D_{ij}(A)$. \n\n\\subsection{Higher partial waves}\n\\label{hpw}\n\nAn uncoupled $NN$ partial wave with $\\ell\\geq 1$ should vanish at threshold as $A^\\ell$.\n Similarly for a coupled partial wave we have the analogous results but in terms of $\\ell_{ij}\\equiv (\\ell_i+\\ell_j)\/2$, with   $i,~j=1,~2$. \n As discussed in Ref.~\\cite{gor2013} this threshold behavior is enforced by taken at least $\\ell$ \nor $\\ell_{ij}$ subtractions\n at $C=0$ in the DR for $N(A)$ in  Eq.~\\eqref{standardr} or Eq.~\\eqref{standardrcc}, respectively, \nand setting $\\nu_p=0$ ($\\nu_p^{(ij)}=0$) for $p=1,\\ldots,\\ell$ ($\\ell_{ij}$). In this way we end with the DRs:\n\\begin{align}\n&\\underline{\\mathrm{Uncoupled~ case}:}\\nonumber \\\\\n\\label{highd}\nD(A)&=1+\\sum_{p=2}^{\\ell}\\delta_p A^{p-1}+\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^\\ell}g(A,k^2)~,\\\\\n\\label{highn}\nN(A)&=\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^\\ell dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^\\ell (k^2-A)}~,\\\\\n\\label{tayloruc}\n\\delta_p&=\\frac{1}{(p-1)!}D^{(p-1)}(0)~,~p=2,3,\\ldots\\\\\n&\\underline{\\mathrm{Coupled~ case}:}\\nonumber \\\\\n\\label{highdcc}\nD_{ij}(A)&=1+\\sum_{p=2}^{\\ell_{ij}}\\delta^{(ij)}_p A(A-C)^{p-2}  +\\frac{A(A-C)^{\\ell_{ij}-1}}{\\pi} \n\\int_{-\\infty}^L\\!\\! dk^2 \\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^{\\ell_{ij}}} g_{ij}(A,k^2,C;\\ell_{{ij}-1})~,\\\\\n\\label{highncc}\nN_{ij}(A)&=\\frac{A^{\\ell_{ij}}}{\\pi}\\int_{-\\infty}^L \\!\\!dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^{2})^{\\ell_{ij}}(k^2-A)}~,\\\\\n\\delta_p^{(ij)}&=\\frac{(-1)^p}{C^{p-1}}\\left[\n\\sum_{n=0}^{p-2}\\frac{(-1)^n}{n!}C^n D^{(n)}_{ij}(C)-1\n\\right]~,~p=2,3,\\ldots\n\\label{taylor}\n\\end{align}\nwhere we have denoted the derivative of $D(A)$ of order $n$ by $D^{(n)}(A)$. In addition, we have introduced the function $g_{ij}(A,k^2,C;m)$ defined \nas\n\\begin{align}\ng_{ij}(A,k^2,C;m)&=\\frac{1}{\\pi}\n\\int_0^\\infty\\!\\! dq^2\n\\frac{ \\nu_{ij}(q^2) (q^2)^m }{ (q^2-A) (q^2-k^2) (q^2-C)^{m}}~,\n\\label{gij.a.k2.c} \n\\end{align}\nwhich can be expressed algebraically as a combination of $g(A,B)$'s, Eq.~\\eqref{gdef}, with \ndifferent arguments.\n\nAlthough in this way there is  a proliferation of subtraction constants \n(which are not constrained) in the function $D(A)$ as $\\ell$ ($\\ell_{ij}$)\n grows, most of them play a negligible role. \nThis is so because $NN$ partial waves with $\\ell$ or $\\ell_{ij}$ greater than 2 are \n quite perturbative \\cite{peripheral,gor2013}.  \nIn practical terms we have found in our NNLO study, as well as in the previous \n one at NLO \\cite{gor2013}, that for higher partial waves\n only $\\delta_{\\ell}$ (or $\\delta^{(ij)}_{\\ell_{ij}}$), if  any,  \n is needed to fit data, with the rest of them fixed to zero. Furthermore,\n  no significant improvement in the reproduction of data or in the fitted values is observed \nby releasing  $\\delta_i$ or $\\delta_i^{(ij)}$ with $i<\\ell$ or $\\ell_{ij}$, respectively, so that the fit is stable. This is called in Ref.~\\cite{gor2013} \nthe {\\it principle of  maximal smoothness} because it implies for the uncoupled case \nthat  the derivatives of $D(A)$ at $A=0$ with order $< \\ell-1$ are zero, as it follows from Eqs.~\\eqref{highd} and \n\\eqref{tayloruc}. Similarly, for the coupled case\n it implies that $D_{ij}(C)=1$ and $D_{ij}^{(n)}(C)=0$ for $1\\leq n \\leq \\ell_{ij}-3$, cf. Eqs.~\\eqref{highdcc} and \\eqref{taylor}. \n In some cases, it happens that  $\\delta_\\ell$ or $\\delta_{\\ell_{ij}}^{(ij)}$ is also zero and then we say that for this partial wave the subtraction constants have the {\\it pure perturbative values}. \n\nWe further illustrate in this work the perturbative character of $NN$ partial waves with $\\ell\\,(\\ell_{ij})\\geq 3$ by \ncomparing the full outcome from the $N\/D$ method with the perturbative result corresponding to the leading Born \napproximation, cf. Sec.~\\ref{born}. In this case there is no dependence on any of the \nsubtraction constants $\\delta_p$ or $\\delta_p^{(ij)}$ and, indeed, we show below  that the results are typically \nrather similar to the full ones, although the latter reproduce closer the Nijmegen PWA, as one should expect.\n\n\n\\section{The input function $\\Delta(A)$}\n\\label{delta}\nThe discontinuity along the LHC of a NN partial wave, $2i \\Delta(A)$, is taken from the  \ncalculation of Ref.~\\cite{peripheral} in Baryon ChPT (BChPT)  up to ${\\cal O}(p^3)$ or NNLO, which includes OPE plus leading and subleading TPE.\n At this order $\\Delta(A)$ \nfor a given partial wave diverges at most as $\\lambda (-A)^{3\/2}$ for $A\\to-\\infty$, with $\\lambda$ a constant. \nAs discussed in Ref.~\\cite{gor2013}, when $\\lambda<0$ one can  have solutions for the integral \nequation providing $D(A)$ for $A<L$ in the once-subtracted case even  with a divergent $\\Delta(A)$ for  \n$A\\to-\\infty$. \nHowever, as we will see below $\\lambda$ is not always negative and more subtractions are then required.\n\n\\subsection{NLO $\\pi N$ counterterms}\nAt NNLO the function $\\Delta(A)$ is sensitive to the NLO $\\pi N$ ChPT low-energy constants (LECs) \n $c_1$, $c_3$ and $c_4$. \n We take their values from different works in the literature, that are summarized in Table~\\ref{tab:cis}. \nWithin the same reference we distinguish, when appropriate, between those values obtained by \nfitting phase shifts from the  Karlsruhe-Helsinki group (KH) \\cite{kh} or the George Washington University group (GW) \\cite{gw}. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|r|l|l|l|}\n\\hline\n  Analysis      & $c_1$ [GeV$^{-1}$]   &   $c_3$ [GeV$^{-1}$]   &   $c_4$ [GeV$^{-1}$]  \\\\  \n\\hline   \nGW-HBChPT \\cite{epe12} & $-1.13$  & $-5.51$ & $3.71$\\\\\nKH-HBChPT \\cite{epe12} & $-0.75$ & $-4.77$ & $3.34$ \\\\\n\\hline\nKH \\cite{buttiker} & $-0.81\\pm 0.12$ & $8\\pm 57$ & $3.40\\pm 0.04$ \\\\\n\\hline\nGW-EOMS \\cite{aco2013} & $-1.50\\pm 0.007$ & $-6.63\\pm 0.31$ & $3.68\\pm 0.14$ \\\\\nKH-EOMS \\cite{aco2013} & $-1.26\\pm 0.14$ & $-6.74\\pm 0.38$ & $3.74\\pm 0.16$ \\\\\n\\hline\nGW-IR\\cite{ir2011}  & $-1.32\\pm 14$ & $-6.9\\pm 6 $ & $3.66\\pm 0.33$ \\\\\nKH-IR \\cite{ir2011}  & $-1.08\\pm 0.15$ & $-7.0\\pm 0.7$ & $3.72\\pm 0.32$ \\\\  \n\\hline\nNN data \\cite{rentmeester2003} & $-0.76\\pm 0.7$ & $-4.78\\pm 0.10$ & $3.96\\pm 0.22$ \\\\\n\\hline\nGW-UChPT \\cite{aco2013}& $-1.11\\pm 0.02$ & $-4.78\\pm 0.04$ & $3.04\\pm 0.02$ \\\\\nKH-UChPT \\cite{aco2013} & $-1.04 \\pm 0.02$ & $-4.48\\pm 0.05$ & $3.00\\pm 0.02$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Different sets of values for the ${\\cal O}(p^2)$ $ \\pi N$ LECs $c_1$, $c_3$ and $c_4$.\n\\label{tab:cis}}\n\\end{center}\n\\end{table}\n\nReference~\\cite{epe12} performs an ${\\cal O}(p^4)$ HBChPT study of $\\pi N$ scattering data. \nWe take its values instead of the ones from the older HBChPT studies at ${\\cal O}(p^3)$ and ${\\cal O}(p^4)$ \\cite{fettes98}. \n We include too the values from Lorentz covariant BChPT obtained in Ref.~\\cite{aco2013} \nby fitting $\\pi N$ phase shifts making use of EOMS at ${\\cal O}(p^3)$. \n Furthermore, we show in the table the $c_i$'s obtained in the covariant ${\\cal O}(p^3)$ BChPT study of Ref.~\\cite{ir2011} \nwithin Infrared Regularization (IR).\n However, due to the better convergence of the $\\pi N$ \nscattering amplitude in EOMS than in IR \\cite{sigma2012,aco2013} we give results only for\n the values obtained within EOMS \\cite{aco2013}. \nThe resulting uncertainty band is already wide enough to \ntake into account further uncertainties by considering explicitly the $c_i$'s  from the IR study of \nRef.~\\cite{ir2011}, which indeed are rather close to those obtained in EOMS \\cite{aco2013}. \n We also notice that the values from Ref.~\\cite{rentmeester2003}, obtained in a $NN$ scattering study, \nare very similar to those of KH \\cite{epe12}, so that in the following we  consider only the latter ones. \nAgain the uncertainty estimated  takes into account the variation in the results by \nemploying the $c_i$'s from Ref.~\\cite{rentmeester2003}.\n The work \\cite{buttiker} fixes $c_4$ accurately but its analysis is insensitive to $c_3$, \nprecisely the ${\\cal O}(p^2)$ $\\pi N$ LEC on which our results mostly depend. \nThis is  why we do not show results for this set of $c_i$'s either. \nFinally, we also give the resulting values from the fits to $\\pi N$ data within Unitarized  EOMS BChPT\n obtained in Ref.~\\cite{aco2013}.  \nThese are the fits  that provide more stable values under the change of the data between KH and GW. \nThese values are quite similar to those from the set KH \\cite{epe12}. \nIn summary, when discussing our results we will take into account the \n values for the LECs $c_i$  obtained in Refs.~\\cite{epe12} and \\cite{aco2013}, namely, the rows 2, 3, 5, 6, 10 and 11 \nin Table~\\ref{tab:cis}.  \n\n\\subsection{Number of subtractions in the chiral expansion of $\\Delta(A)$}\n\\label{nschpt}\n\nAn interesting point to discuss is   the appropriate number of subtractions for a given chiral \norder in the calculation of $\\Delta(A)$. In other terms, we want to establish a chiral power counting \nfor the subtraction constants involved in the calculation of the functions $D(A)$ and $N(A)$. \n\nIn the previous works in which we applied the $N\/D$ method to study $NN$ interactions \\cite{paper1,paper2,gor2013} our main \ncriterion for fixing the number of subtractions was to end with a well-defined IE for $D(A)$ with $A<L$. \n We could also add more subtractions and fit low-energy data with more precision by having more free parameters\n at our disposal, a point actually  used in these works too. \n However, by having a chiral power counting for the subtraction constants \none has a connection between the number of subtraction  and the chiral order for the \ncalculation of $\\Delta(A)$. \n\n \nA chiral power counting for the subtraction constants can be established by studying their variation when changing the \nsubtraction point in the low-energy region.\n Let us consider first the chiral order for the subtraction constants appearing in $N(A)$,  \ndenoted by $\\nu_i$ in Eq.~\\eqref{standardr}. \nFor definiteness  let us employ a twice-subtracted DR, which reads \n\\begin{align}\nN(A)&=\\nu_1+\\nu_2 A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2(k^2-A)}~.\n\\label{sc.co1}\n\\end{align}\nNow, let us move the subtraction point from zero to $C={\\cal O}(M_\\pi^2)$. It is then straightforward to show that the \nprevious DR can be rewritten as\\footnote{To show this, one can rewrite the factor $A^2\/k^2$ in the integral \nof Eq.~\\eqref{sc.co1} as  $([A-C]+C)^2\/(k^2-C)^2\\times (k^2-C)^2\/(k^2)^2$ and then isolate the term $(A-C)^2\/(k^2-C)^2$. \nThe rest of terms can be reabsorbed in the polynomial on the right-hand side (r.h.s.) of Eq.~\\eqref{sc.co1}.}\n\\begin{align}\nN(A)&=\\nu'_1+\\nu'_2 A+\\frac{(A-C)^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^2(k^2-A)}~,\\nonumber\\\\\n\\nu'_1&=\\nu_1-\\frac{C^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^2 k^2}~,\\nonumber\\\\\n\\nu'_2&=\\nu_2+\\frac{C}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta(k^2)D(k^2)}{(k^2-C)^2 k^2}\\frac{2k^2-C}{k^2}~.\n\\label{sc.co2}\n\\end{align}\nFor  $C={\\cal O}(p^2)$, $k^2={\\cal O}(p^2)$ because the result of the convergent integral  at low-energies \nis dominated by the low-energy region of the integrand and $D(k^2)=1+\\ldots={\\cal O}(p^0)$. Furthermore, since at \nleading order $\\Delta(k^2)={\\cal O}(p^0)$,  it  follows then from Eq.~\\eqref{sc.co2} that \n$\\nu_1={\\cal O}(p^0)$ and $\\nu_2= {\\cal O}(p^{-2})$. This\n procedure can be easily generalized so that $ \\nu_n={\\cal O}(p^{-2(n-1)})$. By increasing the chiral order \nin the calculation of $\\Delta(A)$ up to ${\\cal O}(p^m)$ the $\\nu_n$ will receive an extra contribution \nstarting at ${\\cal O}(p^{-2(n-1)+m})$, as it is also clear from Eq.~\\eqref{sc.co2}.\n Now, the point is to demand that for a given $m$ the maximum value of $n$, denoted by $n_0$,\n should not be so large that $-2(n_0-1)+m<0$. \nBy this condition we are  requiring that the chiral dimension for a given subtraction  constant\n with $n\\leq n_0$  be positive or zero, since \nshort-distance physics gives rise to contributions that do not vanish in the chiral limit.\\footnote{As a \nresult they are counted as ${\\cal O}(p^0)$} Then  the raising in the chiral dimension of $\\nu_n$ until the nominal one, \n$-2(n-1)+m\\geq 0$,  must come from powers of $M_\\pi$, $|C|^{\\frac{1}{2}}\\sim  M_\\pi$.\\footnote{One could ask about the fact that the chiral dimension for \nthe other contributions to $\\Delta(A)$ of order $m'<m$ could imply a negative $-2(n-1)+m'$ with $n\\leq n_0$.\n This already occurs e.g. in \nthe paradigmatic example of ChPT, namely, meson-meson scattering. The point is to realize that these extra long-range physics \ncontributions cancel explicitly with other contributions stemming from the rearrangement of the dispersive integral, \nwhich was done already with less subtractions when including only lower orders in $\\Delta(A)$.} This power \ncounting coincides with the standard Weinberg chiral power counting \\cite{weinn}, that is applied to the calculation\n of the $NN$ potential. It is also  worth noticing \nthat $\\nu_n$ is multiplied by $(A-C)^{n-1}$, so that the chiral order of the product is always $m$ for any $n$, which \ncorresponds to the chiral order of the dispersive integral with the ${\\cal O}(p^m)$ contribution of $\\Delta(A)$. \n\nOne can proceed analogously also for the function $D(A)$. \nWe also exemplify it by writing down a \ntwice-subtracted DR for $D(A)$, \n\\begin{align}\nD(A)&=1+\\delta_2 A-\\frac{A(A-C)}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)N(q^2)}{q^2(q^2-C)(q^2-A)}~.\n\\label{sc.co3}\n\\end{align}\nWe now move the subtraction point from $C$ to $E$. \nLet us recall that the normalization $D(0)=1$ is fixed \nand this is why we do not change the position of the first subtraction taken at $A=0$.\n As a result of this rewriting we obtain the evolution \n\\begin{align}\n\\delta_2&\\rightarrow \\delta_2+\\frac{C-E}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)N(q^2)}{q^2(q^2-E)(q^2-C)}\n\\label{sc.co4}\n\\end{align}\nFor ascribing the chiral order to $\\delta_n$, $n\\geq 2$, we have, as before, that\n $C\\sim E\\sim M_\\pi^2$, $q^2={\\cal O}(p^2)$. \nAdditionally, we count $\\rho(q^2)={\\cal O}(p^0)$ because it involves the product $m \\sqrt{q^2}$ with $m\\gg M_\\pi$, a large number.  Let us also recall at this point that along the RHC, the extent of the integral in Eq.~\\eqref{sc.co3},\n the strong effects due to the infrared enhancement of the $NN$ \nintermediate states \\cite{weinn}, which is directly related with \nthe large nucleon mass,   should be resummed.\n We then conclude from Eq.~\\eqref{sc.co4} that\n $\\delta_2={\\cal O}(p^{-2})$ for the LO contribution of $N(A)={\\cal O}(p^0)$. \nThis result can be generalized \neasily to more subtractions so that $\\delta_n={\\cal O}(p^{-2(n-1)})$. \nHowever, this chiral order increases when considering higher orders contributions to $N(A)$ stemming in turn \nfrom  higher orders in the calculation of $\\Delta(A)$.\n As just discussed above in connection with Eq.~\\eqref{sc.co2}, these ${\\cal O}(p^m)$ contributions to\n $\\Delta(A)$   give rise to contributions of the same order in $N(A)$. \nThus, once they are taken into account, one has the corresponding rise in the chiral order of $\\delta_n$ so that now \nit counts as $\\delta_n={\\cal O}(p^{-2(n-1)+m})$.\n In this way,  the chiral orders of $\\nu_n$ and $\\delta_n$ are the same for the same $n$ and $m$. \nIndeed, this is a necessary result because according with the general \nformalism of Sec.~\\ref{unformalism}  the same number of subtractions are taken \nboth in $D(A)$ and $N(A)$.  \nTo satisfy this requirement is also another reason for taking  $\\rho={\\cal O}(p^0)$ in the \nchiral counting.  \n We also stress that the chiral power counting that we have established for the subtraction \nconstants $\\delta_n$ corresponds to two-nucleon reducible diagrams,\\footnote{As it is apparent \nfrom the factor $q^2-A$ in the denominator of the RHC integral in Eq.~\\eqref{sc.co3}.} \nwhile the standard Weinberg chiral power counting for  nuclear interactions \\cite{weinn} only involves \ntwo-nucleon irreducible diagrams.\n\nAlthough we have offered here the arguments for the uncoupled case the same results follow \nfor the coupled-channel partial waves because the function $\\nu_{ij}(A)$, Eq.~\\eqref{nuij.def},\n share the same chiral counting \nas $\\rho(A)$, since the $T$-matrix is ${\\cal O}(p^0)$.\\footnote{With $\\rho={\\cal O}(p^0)$ it is also true that $\\text{Im}t_{ij}={\\cal O}(p^0)$ because \nof unitarity  for $A\\geq 0$, $\\text{Im}t_{ij}=\\rho\\sum_k t_{ik}t_{jk}^*$~.}  \nIn summary, for $\\Delta(A)$ calculated up to ${\\cal O}(p^m)$ we have \nthe following power counting for \nthe subtractions constants, \n\\begin{align}\n\\nu_n~,~\\delta_n&\\sim {\\cal O}(p^{-2(n-1)+m})~.\n\\label{summarypwc}\n\\end{align}\n\nNow, by applying the requirement that $-2(n-1)+m\\geq 0$  it results \nthat in our present study at NNLO  one should properly take two subtractions ($n=2$) since $m=3$. \nHowever, on top of this criterion we first require that the resulting IE  has  \nwell-defined solutions and for this to happen it is necessary to introduce more than two subtractions in \nsome $NN$ partial waves, as discussed below. \nIn addition, we have to satisfy the right threshold behavior for higher partial waves, which for $\\ell\\geq 3$ ($J\\geq 3$ for \nthe mixing partial waves)\nrequires to take $\\ell>2$ ($J>2$ subtractions), cf. Sec.~\\ref{hpw}.\n\n\n\\section{Uncoupled $^1 S_0$ wave}\n\\label{1s0} \n\nIn this section we study the $^1 S_0$ partial wave. We first take the once-subtracted DRs:\n\\begin{align}\n\\label{onceD}\nD(A)&=1-\\nu_1 A  g(A,0)+\\frac{A}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{k^2}g(A,k^2)~,\\\\\n\\label{onceN}\nN(A)&=\\nu_1+\\frac{A}{\\pi}\\int_{-\\infty}^Ldk^2\\frac{\\Delta(k^2)D(k^2)}{k^2(k^2-A)}~.\n\\end{align}\nWe have one free parameter $ \\nu_1$ that can be fixed in terms of the $^1S_0$ scattering length $a_s$ \n\\begin{align}\n\\nu_1=-\\frac{4\\pi a_s}{m}~,\n\\label{1s0nu1}\n\\end{align}\nwith the experimental value $a_s=-23.76\\pm 0.01$~fm \\cite{thesis}.  \n\nThe phase shifts obtained by solving the IE of Eq.~\\eqref{onceD} are shown in \n Fig.~\\ref{fig:1fp1s0} as a function of the c.m. three-momentum, denoted by $p$ ($p= \\sqrt{A}$)  \nin the axis of abscissas. \nThe (red) hatched area   corresponds to our results from Eqs.~\\eqref{onceD}-\\eqref{1s0nu1}\n with $\\Delta(A)$ calculated up-to-and-including ${\\cal O}(p^3)$ contributions and by taking into account the variation in the results from the different values employed for the NLO $\\pi N$ ChPT counterterms in Table~\\ref{tab:cis}.\nOur present results are compared with \n the neutron-proton ($np$) $^1S_0$ phase shifts of the Nijmegen PWA \\cite{Stoks:1994wp}  (black dashed line), \n the OPE results  of Ref.~\\cite{paper1}  (blue dotted line) and the NLO results of Ref.~\\cite{gor2013} (magenta solid line).\n As we see,  the Nijmegen PWA phase shifts are better reproduced at lower energies at NNLO than  at smaller orders, \nthough one also observes an excess of repulsion at this order.  \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.7\\textwidth]{.\/1s0.ps} \n\\end{center}\n\\caption[pilf]{\\protect {  (Color online.) Phase shifts of the $^1S_0$ $NN$ partial wave where the number \nof subtractions taken is indicated by the value of $n$ given in the legend of each type of line. \n The once-subtracted DR results are shown by the (red) hatched areas at NNLO, the \n(magenta) solid lines at NLO \\cite{gor2013} and the (blue) dotted lines at LO (OPE) \\cite{paper1}. \nThe twice-subtracted DR results correspond to the (cyan) band at NNLO and \nthe (green) dash-dotted  line at NLO  \\cite{gor2013}. \n The Nijmegen PWA phase shifts are shown by the (black) dashed lines.\n}\n\\label{fig:1fp1s0}\n}\n\\end{figure}\n\nNext, we work out the effective range expansion (ERE) parameters for the $^1S_0$ . \nBy taking into account the relation in our normalization\n\\begin{align}\n\\frac{4\\pi}{m}\\frac{D}{N}=-\\frac{1}{a_s}+\\frac{1}{2}r_s A+\n\\sum_{i=1}^{10}v_i A^i - i\\sqrt{A}+{\\cal O}(A^{11})~,\n\\label{efr1}\n\\end{align}\nwith $r_s$ the $^1S_0$ effective range and the shape parameters $v_i$, $i=2,\\ldots,10$. \n We designate by $I_m$, $m=1,2,\\ldots$, the  integral along the LHC,\n\\begin{align}\n I_{2n}&=\\int_{-\\infty}^Ldk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^n}~,\\nonumber\\\\\n I_{2n+1}&=\\int_{-\\infty}^Ldk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^n\\sqrt{-k^2}}~.\\nonumber\\\\\n\\end{align}\nFrom Eqs.~\\eqref{onceD}, \\eqref{onceN} and \\eqref{efr1} we derive the following expressions for \n$r_s$ and the shape parameters in the ERE up to $i=4$\n\\begin{align}\nr_s&=-\\frac{m (a_s I_3+I_4)}{2 \\pi ^2 a_s^2} ~,\\nonumber\\\\\nv_2&=-\\frac{m \\left(I_4 m (a_s I_3+I_4)+4 \\pi ^2 a_s (a_s\n   I_5+I_6)\\right)}{16 \\pi ^4 a_s^3}~,\\nonumber\\\\\nv_3&=-\\frac{m \\left[16 \\pi ^4 a_s^2 (a_s I_7+I_8)+I_4^2 m^2 (a_s I_3+I_4)+4\n   \\pi ^2 a_s m (a_s I_3 I_6+a_s I_4 I_5+2 I_4 I_6)\\right]}{64 \\pi\n   ^6 a_s^4}~\\,\\\\\nv_4&=-\\frac{m}{256 \\pi ^8 a_s^5}\\left[64 \\pi ^6 a_s^3 (a_s I_9+I_{10})+16 \\pi ^4 a_s^2 m \\left(a_s (I_3\n   I_8+I_4 I_7+I_5 I_6)+2 I_4 I_8+I_6^2\\right)+I_4^3 m^3\n   (a_s I_3+I_4)\\right.\\nonumber\\\\\n&\\left.+4 \\pi ^2 a_s I_4 m^2 (2 a_s I_3 I_6+a_s I_4\n   I_5+3 I_4 I_6)\\right]~.\n\\label{ere.1s0}\n\\end{align}\nFor higher order shape parameters is more efficient to use the numerical method developed in Appendix \\ref{appen:vs}, \nto which we refer.\n\nThe resulting values for $r_s$ and \n the shape parameters $v_i$, $i=1,\\ldots,6$, are given in \nTable~\\ref{table:vs1s0a} and for $v_i$, $i=7,\\ldots, 10$ are shown \nin Table~\\ref{table:vs1s0b} in the second and third rows for NLO and \nNNLO, respectively. The latter are indicated by NNLO-I.\n These results are compared \nwith the results from the calculation based on the NNLO $NN$ potential of Refs.~\\cite{epe04} and \\cite{thesis},\n and with the Nijmegen PWA values.\n Our results for $v_3$ and $v_4$ are very similar to those obtained in Ref.~\\cite{thesis}. The difference between \n\\cite{thesis} and \\cite{epe04} stems from the fact that in the latter reference a different method to regularize pion exchanges was introduced, the so-called spectral function regularization, instead of the dimensional regularization used in Ref.~\\cite{thesis}. We also observe a clear improvement in the reproduction of the ERE parameters from NLO to NNLO. \nAt NLO the errors in Tables~\\ref{table:vs1s0a} and \\ref{table:vs1s0b} reflect the numerical uncertainty in the \ncalculation of higher order derivatives. At NNLO in addition they take into account the spread in the results \nfrom the different sets of $c_i$'s used. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|}\n\\hline\n    &$r_s$   & $v_2$  & $v_3$   & $v_4$ & $v_5$ & $v_6$    \\\\\n\\hline\nNLO & $2.32$ & $-1.08$ & 6.3  & $-36.2$  & 225 & $-1463$  \\\\\n\\hline\nNNLO-I & $2.92(6)$ & $-0.32(8)$ & $4.9(1)$ & $-27.7(8)$ & $177(4)$  & $-1167(30)$  \\\\\n\\hline\nNNLO-II & $2.699(4)$ & $-0.657(3)$ & $5.20(2)$  & $-30.39(9)$  & $191.9(6)$ & $-1263(3)$  \\\\\n\\hline\nRef.~\\cite{thesis} & $2.68$ & $-0.61$ & $5.1$ & $-30.0$ & & \\\\\n\\hline\nRef.~\\cite{epe04} & $2.62\\sim 2.67$ & $-0.52\\sim -0.48$ & $4.0\\sim 4.2$ & $-20.5\\sim -19.9$ & & \\\\\n\\hline\nRef.~\\cite{Stoks:1994wp} & $2.68$ & $-0.48$ & $4.0$ & $-20.0$ & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Values for effective range $r_s$ [fm] and the shape parameters \n$v_i$, $i=2,\\ldots,6$ in units of fm$^{2i-1}$\n for our present results at NNLO with once-subtracted DRs [Eq.~\\eqref{onceD}] (NNLO-I in the third row) \nand with twice-subtracted DRs [Eq.~\\eqref{twiceD}] (NNLO-II in the fourth row). \nThe second row shows the results at  NLO with once-subtracted DRs [Eq.~\\eqref{onceD}].\n We also give the values obtained by using the NNLO $NN$ potential in \nRefs.\\cite{thesis} and \\cite{epe04} (fifth and\n sixth  rows, respectively). The values corresponding to the Nijmegen PWA \\cite{Stoks:1994wp}, \n as obtained in Refs.~\\cite{epe04,thesis}, are given in the last row.  \n\\label{table:vs1s0a}}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n    &  $v_7\\times 10^{-1}$ & $v_8\\times 10^{-2}$  & $v_{9}\\times 10^{-3}$  & $v_{10}\\times 10^{-4}$   \\\\\n\\hline\nNLO & $985$ & $-681$ & 480  & $-344(3)$  \\\\\n\\hline\nNNLO-I & $795(18)$ & $-554(12)$ & $393(8)$ & $-284(6)$  \\\\\n\\hline\nNNLO-II & $857.1(1.9)$ & $-595.7(1.3)$ & $421.7(9)$ & $-304(3)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Values for the shape parameter $v_i$, $i=7,\\ldots,10$ in units of fm$^{2i-1}$. \nFor the meanings of the rows see Table~\\ref{table:vs1s0a}.\n\\label{table:vs1s0b}}\n\\end{center}\n\\end{table}\n\n\n\nFrom Eq.~\\eqref{ere.1s0}  we can also derive a power series expansion of the ERE parameters as a function of $a_s$, as it was done previously for $r_s$ in Ref.~\\cite{gor2013} at NLO. We refer to that reference for further details. The important point is that $D(A)$ satisfies the linear IE of  Eq.~\\eqref{onceD} with an inhomogeneous term that is a polynomial of first degree in $a_s$. As a result,  $D(A)=D_0(A)+a_s D_1(A)$, with $D_0(A)$ and $D_1(A)$ independent of $a_s$. This also implies that the different $I_n$ can be expressed as $I_n^{(0)}+a_s I_n^{(1)}$ with $I_n^{(0)}$ and $I_n^{(1)}$ independent of $a_s$. \nIn this way, the ERE parameters satisfies the following expansions \n\\begin{align}\nr_s&=\\alpha_0+\\frac{\\alpha_{-1}}{a_s}+\\frac{\\alpha_{-2}}{a_s^2}~,\\nonumber\\\\\nv_n&=\\sum_{m=-n-1}^0 \\frac{v_n^{(m)}}{a_s^m}~,\n\\label{expvs.1s0}\n\\end{align}\nwith the coefficients $\\alpha_i$ and $v_n^{(i)}$ independent of $a_s$. \nThe relation between $r_s$ and $a_s$ was first realized in Ref.~\\cite{pavon06} in the context of $NN$ scattering.\\footnote{The correlation between the effective range and the scattering length\n in Eq.~\\eqref{expvs.1s0} was derived earlier in atomic physics for Van der Waals potentials\n \\cite{flambaum}, and throughly confronted    with data \\cite{calle2010}.} The explicit expressions \nof $\\alpha_i$ ($i=-2,-1,0)$ in terms of $D_0(A)$ and $D_1(A)$ were given in Ref.~\\cite{gor2013}.\n Its values at NNLO are\n\\begin{align}\n\\alpha_0&= 2.61\\sim 2.73~\\text{fm}~,\\nonumber\\\\\n\\alpha_{-1}&=-5.93\\sim -5.65~\\text{fm}^2~,\\nonumber\\\\\n\\alpha_{-2}&=5.92\\sim 6.12~ \\text{fm}^3~.\n\\label{exp.1s0}\n\\end{align}\n The expressions for the coefficients $v_n^{(m)}$ in Eq.~\\eqref{expvs.1s0} can also be worked straightforwardly \nin terms of $D_0(A)$ and $D_1(A)$ by the interested reader. For conciseness we do not reproduce them here. \nThe  results in Eq.~\\eqref{exp.1s0}  are perfectly compatible with those obtained in the first entry of \nRef.~\\cite{pavon06}, $\\alpha_0=2.59\\sim 2.67$~fm, $\\alpha_{-1}=-5.85\\sim -5.64$~fm$^2$ and $\\alpha_{-2}=5.95\\sim \n6.09$~fm$^3$. This reference employs the chiral $NN$ potential in a Lippmann-Schwinger equation that is \n renormalized  with boundary conditions and imposing \nthe hypothesis of orthogonality of the wave functions determined with\n different energy.\\footnote{Since the potentials involved are singular this orthogonality condition is imposed in \n the formalism of Ref.~\\cite{pavon06}.} \nIn our case, however, the expansions in Eq.~\\eqref{expvs.1s0} are consequences of basic principles of a $NN$ \npartial wave like unitarity, analyticity and chiral symmetry.  \n The resulting phase shifts in Fig.~\\ref{fig:1fp1s0} from Eq.~\\eqref{onceD}, and shown by the (red) hatched area,\n are also coincident with those obtained by Ref.~\\cite{pavon06}.  \nThey are also rather similar to those obtained when employing only one contact term in the third entry of Ref.~\\cite{phillipssw},\n which studies the independence of its results as a function of the cutoff used to solve the Lippmann-Schwinger equation.\n Nevertheless, in this case  the NNLO chiral potential is calculated by truncating its spectral representation \\cite{epe04},  \n while Ref.~\\cite{pavon06} uses the dimensional regularized result \n(which requires to take to infinity the cutoff(s) used in Ref.~\\cite{phillipssw}.)\n \n\n\nNext,  we consider the twice-subtracted DRs: \n\\begin{align}\n\\label{twiceD}\nD(A)&=1+\\delta_2 A-\\nu_1\\frac{A(A+M_\\pi^2)}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)}{(q^2-A)(q^2+M_\\pi^2)q^2}\n-\\nu_2 A(A+M_\\pi^2) g(A,-M_\\pi^2)\\nonumber\\\\\n&+\\frac{A(A+M_\\pi^2)}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2} \n\\int_0^\\infty dq^2\\frac{\\rho(q^2)q^2}{(q^2-A)(q^2+M_\\pi^2)(q^2-k^2)}\n~, \\\\\n\\label{twiceN}\nN(A)&=\\nu_1+\\nu_2 A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^2}~,\n\\end{align}\nwhere the two subtractions in the function $N(A)$ and one  for $D(A)$ are taken at $C=0$, while\n the other subtraction in $D(A)$ is placed at $C=-M_\\pi^2$. \nTaking into account Eq.~\\eqref{gdef} it is straightforward to rewrite\n\\begin{align}\n\\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\rho(q^2)q^2}{(q^2-A)(q^2-k^2)(q^2-C)}=\\frac{C g(A,C) - k^2 g(A,k^2)}{C-k^2}~.\n\\end{align} \n The subtraction constant $\\nu_1$ is given by Eq.~\\eqref{1s0nu1}, while $\\nu_2$ and $\\delta_2$ \nare directly fitted to \n the $np$ Nijmegen PWA phase shifts.\\footnote{Since Ref.~\\cite{Stoks:1994wp} does not provide \nerrors we always perform a least square fit,  without weighting.}  \n The best fit occurs for\n\\begin{align}\n\\nu_2&=-23(1)~M_\\pi^{-4}\\nonumber\\\\\n\\delta_2&=-8.0(3)~M_\\pi^{-2}~,\n\\label{nu2delta2}\n\\end{align}        \nwhere the intervals of values stem from the uncertainty due to the different values of $c_i$'s taken. \nThe reproduction of data is very good, as shown by the (cyan) filled area in Fig.~\\ref{fig:1fp1s0} which \nlies on top of the Nijmegen PWA $np$ phase shifts.  In the same figure \n  we show by the (green) dash-dotted line \nthe twice-subtracted DR result at NLO,  which reproduces  \nthe  Nijmegen data equally well as obtained at NNLO, with  \n the fitted values $\\nu_2=-11.9$~$M_\\pi^{-4}$ and $\\delta_2=-4.6~M_\\pi^{-2}$. \n   The resulting ERE shape parameters for the fit in Eq.~\\eqref{nu2delta2} \nare shown in the fourth rows of Tables~\\ref{table:vs1s0a} and \\ref{table:vs1s0b}, where \n we observe a remarkable good agreement with Ref.~\\cite{thesis}. \nWe predict $r_s=2.70$~fm which is compatible with its experimental value \n$r_s=2.75\\pm 0.05$~fm \\cite{thesis}. A similar good reproduction of the $^1S_0$ phase shifts is also achieved by Ref.~\\cite{phillipssw} in \nterms of two contact terms, although in this case there is a strong sensitivity   on the cutoff employed to \nregularize the Lippmann-Schwinger equation near those values that give rise to poles in the domain of validity \nof the effective field theory. \n\n The value of $\\nu_2$ in \nEq.~\\eqref{nu2delta2} is rather large, of similar size in absolute value to $\\nu_1\\simeq 31~M_\\pi^{-2}$, Eq.~\\eqref{1s0nu1}. A linear correlation between $\\nu_2$ and $\\delta_2$ can be observed in  a $\\chi^2$ contour plot,  along which \n there is an absolute minimum corresponding to the parameters given \nin Eq.~\\eqref{nu2delta2}.  \nThe subtraction constant $\\nu_2$ that results from the once-subtracted DR \n Eq.~\\eqref{onceN}, and that we denote by $\\nu_2^{\\mathrm{pred}}$, is given by the \nexpression\n\\begin{align}\n\\nu^{\\mathrm{pred}}_2&=\\frac{1}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}~,\n\\label{nu2predicted}\n\\end{align}\nwith the numerical value $\\nu_2^{\\mathrm{pred}}\\simeq -6.0$, $-6.4$ and $-7.5\\pm 0.2~M_\\pi^{-4}$ when \n $\\Delta(A)$ is calculated up to ${\\cal O}(p^0)$, ${\\cal O}(p^2)$ and ${\\cal O}(p^3)$, respectively.\n The difference between the predicted and fitted values for $\\nu_2$   at NLO  \n is denoted by $\\delta\\nu_2^{(0)}$.\n The superscript takes into account the chiral order for $\\nu_2$, ${\\cal O}(p^{-2+m})$ according to the new \ncontribution to $\\Delta(A)$ of ${\\cal O}(p^m)$, Eq.~\\eqref{summarypwc}. \n  The value obtained is $\\delta \\nu_2^{(0)}\\simeq -5.5~M_\\pi^{-4}$. At NNLO in order \nto calculate  $\\delta \\nu_2^{(1)}$ one has to subtract  $\\delta\\nu_2^{(0)}$ to the difference \nbetween the fitted value in Eq.~\\eqref{nu2delta2} and the predicted one from Eq.~\\eqref{nu2predicted}. \n Then, one has $\\delta\\nu_2^{(1)}\\simeq -15+5.5=-9.5~M_\\pi^{-4}$.\n This implies that in order to overcome the excess of repulsion at NNLO   one needs\n to incorporate a significant contribution from short-distance physics to give account \nof ``missing physics'', beyond the pure long-range physics\\footnote{We mean here the physics driven by the \n multi-pion exchanges giving rise to the LHC and  to $\\Delta(A)$.}\n that stems from the once-subtracted DR case and that is not able to provide an accurate \nreproduction of data as shown in Fig.~\\ref{fig:1fp1s0} by the (red) hatched areas. \n  The large value for $\\delta\\nu_2^{(1)}$ is mainly due to the ${\\cal O}(p^2)$ $\\pi N$ counterterms \n$c_i$'s, which in turn are dominated by the  $\\Delta(1232)$ resonance contribution \\cite{bernard93,aco2012}. \n  This can be easily seen by performing a fit to data in which we set  $c_i=0$ for all of them.\n A good reproduction \nof the Nijmegen PWA phase shifts results but now $\\delta\\nu_2^{(1)}\\simeq -1.5~M_\\pi^{-4}$, which is \nmuch smaller than $\\delta\\nu_2^{(0)}$, with a ratio $\\delta\\nu_2^{(1)}\/\\delta\\nu_2^{(0)}\\sim 30\\% \\sim \n{\\cal O}(p)$. This indicates that once the large  contributions \n that stem from the  $c_i$ coefficients are discounted a quite natural (baryon) chiral expansion emerges. \n \nRegarding the absolute value of $\\delta\\nu_2^{(0)}$ one should expect on dimensional grounds that\n\\begin{align}\n|\\delta\\nu_2^{(0)}|\\sim  \\frac{4\\pi\\,|a_s|}{m \\Lambda^2}~,\n\\label{abvalue_v2}\n\\end{align}\nwith $\\Lambda$ the expansion scale. The factor $4\\pi\/m$ is due to our normalization, cf. Eq.~\\eqref{efr1}. \nThere should be also another contribution to $\\delta\\nu_2^{(0)}$ not proportional to $a_s$, but since \nthe scattering length is so large the contribution shown in Eq.~\\eqref{abvalue_v2}\n is expected to be the most important. For $\\Lambda\\simeq 350$~MeV, one would have \n$|\\delta\\nu_2^{(0)}|\\sim 5~M_\\pi^{-4}$, which is very similar indeed to the reported value above. This \nvalue of $\\Lambda$ is also consistent with the ratio $\\delta\\nu_2^{(1)}\/\\delta\\nu_2^{(0)}\\sim 1\/3$ given above \nas $M_\\pi\/\\Lambda \\sim 1\/3$.\n\nLet us consider now the relevance  of the different contributions to $\\Delta(A)$ by evaluating the double\n integral in Eq.~\\eqref{twiceD}, namely,\n\\begin{align}\n\\frac{A(A+M_\\pi^2)}{\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}\\int_0^\\infty dq^2\\frac{\\rho(q^2)q^2}{(q^2-A)(q^2+M_\\pi^2)(q^2-k^2)}~,\n\\label{1s0.quanty}\n\\end{align}\n with the full result for $D(A)$ but with $\\Delta(A)$ in the integrand of Eq.~\\eqref{1s0.quanty} evaluated \npartially with some contributions or all of them. The result of this exercise is given\n in the left panel of Fig.~\\ref{fig:1s0quanty} for the $c_i$ coefficients of Ref.~\\cite{epe12}, \n collected in the first row of Table~\\ref{tab:cis}. \n In turn,  we show directly $\\Delta(A)$ along the LHC in the right panel of Fig.~\\ref{fig:1s0quanty}. \n The (black) dash-dotted lines correspond to OPE, \nthe  (blue) dotted lines take into account the full ${\\cal O}(p^2)$ TPE,\n including both two-nucleon reducible and irreducible TPE, and  \n the (cyan) double-dotted lines contain  the ${\\cal O}(p^3)$  two-nucleon irreducible TPE. \nIn the right panel we show by the (cyan) filled area the variation in the ${\\cal O}(p^3)$ irreducible TPE \ncontribution by varying between the different sets of $c_i$'s from Refs.~\\cite{epe04} and \\cite{aco2013}, as discussed above. \nThis band indicates a large source of uncertainty  in $\\Delta(A)$.\n  In the left panel the (red) solid line  results by keeping  all the contributions to $\\Delta(A)$, and \none can quantify from this panel the fact that   the ${\\cal O}(p^3)$ irreducible TPE is the largest subleading contribution.\n At $\\sqrt{A}=100$~MeV it is around 28\\% of the OPE contribution, and it raises with energy\n so that at $\\sqrt{A}=200$~MeV it is 44\\% and at 300~MeV it becomes 66\\%. The increase in energy \nof the relative size of the subleading TPE contribution should be expected because at low energies\n the suppression mechanism due to the earlier onset of the OPE source of $\\Delta(A)$ along the LHC at $L$ is more efficient.\n In addition, it is well-known that the $\\Delta(1232)$ plays a prominent role in $\\pi N$ scattering because\n its proximity to the $\\pi N$ threshold and its strong coupling to this channel.\n This manifests in the large size of the LECs $c_3$ and $c_4$ in Table~\\ref{tab:cis} due to the $\\Delta(1232)$ contribution \nto them, evaluated in Refs.~\\cite{bernard93,aco2012}. \nThe large impact of the $\\Delta(1232)$ is  the well-known reason for the large size of subleading TPE,\n but once its leading effects are taken into account at $ {\\cal O}(p^3)$ the chiral expansion\n stabilizes \\cite{entem,epe04}, as we have also concluded  in the discussion following Eq.~\\eqref{nu2predicted}. \n  In the following, we skip the discussion on the relative importance of the different contributions \nto $\\Delta(A)$ for those $NN$ partial waves with a similar situation to the one discussed \nconcerning the $^1S_0$. \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/lhcd1s0.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/dis.lhc1s0.ps}\n\\end{tabular}\n\\caption{ { (Color online.) Left panel: different contributions to the integral in Eq.~\\eqref{1s0.quanty} for the $^1S_0$.\n Right panel: contributions to $\\Delta(A)$. These contributions comprise OPE (black dash-dotted line), \nleading TPE (blue dotted line) and \nthe subleading TPE contribution, shown by the (cyan) double-dotted line in the left panel \nand by the (cyan) filled area in the right one. The total result, only shown for the left panel, \nis the (red) solid line.}\n\\label{fig:1s0quanty}}\n\\end{center}\n\\end{figure}\n\n\\section{Uncoupled $P$ waves}\n\\label{pw} \nIn this section we discuss the application of the method to the  uncoupled $P$ waves. \n At NNLO one has for these waves that \n\\begin{align}\n\\lambda_P=\\lim_{A\\to-\\infty}\\frac{\\Delta(A)}{(-A)^{(3\/2)}}>0~,\n\\label{unlambdap}\n\\end{align}\nso that, according to the results of Ref.~\\cite{gor2013}, its Proposition 4, a once-subtracted DR for $D(A)$, \nEq.~\\eqref{standardr}, does not converge and more subtractions should be taken. Then, we directly discuss \nthe twice- and three-time subtracted DRs. \n\nThe twice-subtracted DRs are given by:\n\\begin{align}\n\\label{twiceDNl}\nD(A)&=1+\\delta_2 A-\\nu_2 A^2 g(A,0)\n+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}g(A,k^2)~,\\nonumber \\\\\nN(A)&=\\nu_2 A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^2}~,\n\\end{align}\nwith all the subtractions in Eq.~\\eqref{standardr} taken at $C=0$. \n The three-time subtracted DRs are:\n\\begin{align}\nD(A)&=1+\\delta_2 A + \\delta_3 A(A+M_\\pi^2)+(\\nu_2-\\nu_3 M_\\pi^2)  A(A+M_\\pi^2)^2  \\frac{\\partial g(A,-M_\\pi^2)}{\\partial M_\\pi^2}\n-\\nu_3 \\, A(A+M_\\pi^2)^2 g(A,-M_\\pi^2)\\nonumber\\\\\n&+\\frac{A(A+M_\\pi^2)^2}{\\pi}\\int_{-\\infty}^Ldk^2 \\frac{\\Delta(k^2)D(k^2)}{(k^2)^3}g(A,k^2,-M_\\pi^2;2)~,\\nonumber\\\\\nN(A)&=\\nu_2 A+ \\nu_3 A^2+\\frac{A^3}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^3}~.\n\\label{pw.tdr}\n\\end{align}\nHere all the subtractions in $N(A)$ and one in $D(A)$ are taken at $C=0$, while the other two subtractions \nin $D(A)$ are taken at $C=-M_\\pi^2$. This is done in order to avoid handling an infrared diverging \nintegral along the RHC multiplying $\\nu_2$ that would result if all the subtractions were taken at $C=0$. \nThe function $g(A,k^2,C;m)$ appearing in Eq.~\\eqref{pw.tdr} is defined as\n\\begin{align}\ng(A,k^2,C;m)=\\int_0^\\infty\\!\\! dq^2\\frac{\\rho(q^2) (q^2)^m}{(q^2-A)(q^2-k^2)(q^2-C)^m}~.\n\\label{def.el.gm}\n\\end{align}\n\nIn all the cases the subtraction constant $\\nu_2$ is fixed in terms of the \nscattering volume, $a_V$, \n\\begin{align}\n\\nu_2=4\\pi a_V\/m~.\n\\label{pnu2fix}\n\\end{align}\nFor $a_V$ we take the values $0.890$, $-0.543$ and $-0.939~M_\\pi^{-3}$ for the partial waves $^3P_0$, $^3P_1$ and $^1P_1$, \nin order, as deduced from Ref.~\\cite{Stoks:1994wp}.\n\n\\subsection{$^3P_0$ wave}\n\\label{3p0}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{.\/3p0.ps} \n\\caption{ {\\small }\n\\label{fig:3p0}\n Phase shifts of the $^3P_0$ $NN$ partial wave. \n The three-time subtracted DR results at NNLO are shown by the (red) hatched area and the twice-subtracted DR results  at NLO \\cite{gor2013} are given by the (magenta) solid line. \nThe (blue) dotted line corresponds to the OPE results \\cite{paper1} and \nthe Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\end{center}\n\\end{figure}\n\n \n For the $^3P_0$ wave the twice-subtracted DRs at NNLO, Eq.~\\eqref{twiceDNl}, do not provide \nstable results under the increase in absolute value \nof the lower limit of integration along the LHC. \nHowever, the three-time subtracted DRs, Eq.~\\eqref{pw.tdr}, are convergent and provide meaningful results. \n Notice that, as stated in Sec.~\\ref{nschpt}, \non top of the number of subtractions required by the chiral counting, two at NNLO,\n we impose the requirement of having well-defined IEs providing stable \n solutions. \nRegarding the subtractions constants $\\nu_3$, $\\delta_2$ and $\\delta_3$ in Eq.~\\eqref{pw.tdr}, \nwe can fix $\\nu_3=0$ because it plays a negligible role in the fits and, if released,  the fit remains stable. \nThe fitted values for $\\delta_2$ and $\\delta_3$  are\n\\begin{align}\n\\delta_2&= 2.82(5)~M_\\pi^{-2}\\nonumber\\\\\n\\delta_3&=0.18(6) ~M_\\pi^{-4}~,\n\\end{align}\nwhere the intervals of values take into account the dispersion in the results \nthat stems from the different sets of $c_i$'s in Table~\\ref{tab:cis}.\n The phase shifts calculated, shown by the (red) hatched area \nin Fig.~\\ref{fig:3p0}, reproduce exactly the Nijmegen PWA phase shifts \\cite{Stoks:1994wp}, \ngiven by the (black) dashed line. Indeed, the two lines overlap each other. \nThe results with different sets of values for the $c_i$ counterterms cannot be distinguished  either between each other. \n The (magenta) solid line shows the results with twice-subtracted DRs at NLO \\cite{gor2013}, \nwhich are already almost on top of the data, and the OPE results \\cite{paper1} are shown by the (blue) dotted line. \n We have also checked that a tree-time-subtracted DR at LO and NLO provide already a prefect reproduction of data as well. \nThen, the wave $^3P_0$ studied at ${\\cal O}(p^3)$ is not a good partial wave to learn above chiral dynamics, because \n independently of order up to which $\\Delta(A)$ is calculated the reproduction of data is excellent when \nthree-subtractions are taken. \n\n\n\n\\subsection{$^3P_1$ wave}\n\\label{3p1}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{.\/3p1.ps} \n\\caption{ {\\small }\n\\label{fig:3p1}\n Phase shifts of the $^3P_1$ $NN$ partial wave. \n The three-time subtracted DR results at NNLO are shown by the (red) hatched area. \nThe (blue) dotted line corresponds to the OPE results \\cite{paper1} and\n the Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\end{center}\n\\end{figure}\n\nFor this partial wave the situation is similar to that discussed for  the $^3P_0$. \nThe twice-subtracted DRs, Eq.~\\eqref{twiceDNl}, do not provide stable results and we have \nto consider then the three-time subtracted DRs, Eq.~\\eqref{pw.tdr}. \n The free parameters are $\\delta_2$ and $\\delta_3$, with $\\nu_3$ fixed to 0 (the fit is stable if this \nsubtraction constant is released). The fitted values  are\n\\begin{align}\n\\delta_2& = 2.7(1)~M_\\pi^{-2},\\nonumber\\\\\n\\delta_3& = 0.47(3)~M_\\pi^{-4}~.\n\\end{align}\n\nThe resulting phase shifts are  shown in  Fig.~\\ref{fig:3p1} by the (red) hatched area and \n reproduce perfectly the Nijmegen PWA phase shifts (shown by \nthe black dashed line),  independently of the set of values\n for the $c_i$'s chosen from Refs.~\\cite{epe12,aco2013} in Table~\\ref{tab:cis}. \n At NLO \\cite{gor2013} it is also necessary to take three-subtracted DRs in order to obtain stable results and the reproduction \nof data is equally perfect. \nThis is why we have not included the NLO results in Fig.~\\ref{fig:3p1}. \n Similarly to the $^3P_0$ case, we cannot discern the impact of chiral dynamics at ${\\cal O}(p^3)$ once three-time subtracted DRs \nare considered.\n\n\\subsection{$^1P_1$ wave}\n\\label{1p1}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.5\\textwidth]{.\/1p1.ps} \n\\caption{ {\\small }\n\\label{fig:1p1}\n Phase shifts of the $^1P_1$ $NN$ partial wave. \n The twice subtracted DR results at NNLO are shown by the (red) hatched area, while at NLO \\cite{gor2013}\ncorrespond to  the (magenta) solid line. \nThe (blue) dotted line corresponds to the OPE results \\cite{paper1} and \nthe Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\end{center}\n\\end{figure}\n \nFor this partial wave the twice-subtracted DR results from Eq.~\\eqref{twiceDNl} are quite stable  at low energies. \nThe free parameters are now $\\nu_2$ and $\\delta_2$. \nThe resulting fitted value for $\\delta_2$ to  the Nijmegen PWA phase shifts  is  \n\\begin{align}\n\\delta_2 =0.4(1)~M_\\pi^{-2}~,\n\\end{align}\nwith the variation in the value due to the set of $c_i$'s taken [$\\nu_2$ is given by Eq.~\\eqref{pnu2fix}]. \nWe show by the (red) hatched area in Fig.~\\ref{fig:1p1} our results by employing the different $c_i$ sets of values. \nFor this case the curves obtained with the $c_i$ from \\cite{aco2013}, by reproducing the $\\pi N$ phase shifts with \nLorentz covariant EOMS BChPT, are the closest to data and determine the upper\n limit of the hatched area in Fig.~\\ref{fig:1p1}. \n The improvement in the reproduction of data for the $^1P_1$ partial wave by the twice-subtracted DRs at NNLO compared with the \nresults obtained at NLO with the same number of subtractions \n  (hatched area versus (magenta) solid line)  is a notorious effect from $\\pi N$ physics. One should \nnotice that for the $^1P_1$ wave the dispersive integral on the r.h.s. of Eq.~\\eqref{twiceDNl} for the function $D(A)$ is clearly \ndominated by the OPE contribution\n This is the reason why for the $^1P_1$ one does not need to take three subtractions but two are enough. \n Although, as much as for the other partial waves discussed until now, the ${\\cal O}(p^3)$ two-nucleon irreducible \nTPE is the dominant contribution between the subleading effects to $\\Delta(A)$. \n\n\n\n\n\\section{Uncoupled $D$ waves}\n\\label{dw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1d2.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3d2.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1D_2$ (left panel) and $^3D_2$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome \\cite{gor2013}. In both cases twice-subtracted DRs are used. The \nphase shifts in the Born approximation   are shown by the (cyan) filled bands, \nthe OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines and \n the Nijmegen PWA phase shifts are given by the (black) dashed lines.}\n\\label{fig:dw} }\n\\end{center}\n\\end{figure}\n\n\nHere, we discuss the $D$ waves. In order to preserve the right threshold behavior we \nemploy the twice-subtracted DRs of Eqs.~\\eqref{highd} and \\eqref{highn} with $\\ell=2$. \nFor the uncoupled $D$ waves one has that\n\\begin{align}\n\\lambda_D&=\\lim_{A\\to -\\infty}\\frac{\\Delta(A)}{(-A)^{3\/2}}<0\n\\end{align}\nand for this sign we do not have numerical problems \nin the solution of the resulting IE even for diverging $\\Delta(A)$  \\cite{gor2013}.\n\n\\begin{align}\n\\label{twiceDNl_Dw}\nD(A)&=1+\\delta_2 A\n+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2)^2}g(A,k^2)~,\\nonumber \\\\\nN(A)&=\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)D(k^2)}{(k^2-A)(k^2)^2}~.\n\\end{align}\n\nThe only free parameter per partial wave is $\\delta_2=D^{(1)}(0)$ which is  fitted to the \nNijmegen PWA phase shifts. Taking into account the \ndifferent sets of values for the $c_i$ counterterms we have the following \nresults,\n\\begin{align}\n^1D_2:& ~D^{(1)}(0)= 0.07(1) ~M_\\pi^{-2},\\nonumber\\\\\n^3D_2:& ~D^{(1)}(0)= -0.017(3)~M_\\pi^{-2}~.\n\\end{align} \nThe reproduction of data is excellent as shown by the (red) \nhatched areas in Fig.~\\ref{fig:dw}, where the phase shifts for the $^1D_2$ are given in the left panel \nand those of the $^3D_2$ in the right one. Our results indeed overlap the Nijmegen PWA phase shifts given by \nthe (black) dashed lines.  \nReference~\\cite{gor2013} obtained the (magenta) solid line making use also of twice-subtracted DRs at NLO. \n We see a remarkable   improvement from NLO to NNLO due to the inclusion of NLO  $\\pi N$ dynamics, \nparticularly for the $^1D_2$ partial wave.  \n\n\\subsection{Perturbative and Born approximation phase shifts}\n\\label{born}\n\nThe higher is the orbital angular momentum $\\ell$ the more perturbative is expected to be the \ncorresponding $NN$ partial wave. \nThis statement was studied in detail in the perturbative study of  \n Ref.~\\cite{peripheral} by making use \nof the one-loop approximation in BChPT.\n Indeed, we can easily obtain  from \nour formalism both the leading perturbative solution to the \nIEs of the $N\/D$ method in powers of $\\Delta(A)$,  as well as the leading term in the Born series approximation \nfor the chiral $NN$ amplitude calculated up to ${\\cal O}(p^3)$ in Ref.~\\cite{peripheral}. \nThe point is that for a \nweak  interaction  (small $\\Delta(A)$ at low three-momentum) one can expect that $D(A)\\simeq 1$ at low energies.\n It is then reasonable to consider that substituting $D(A)\\to 1$ in the integral on the r.h.s.  of \nEq.~\\eqref{highn} would be meaningful to calculate $N(A)$, \n because we have a rapid converging integral due to the factor $(k^2)^\\ell$ in the denominator for a sufficiently \nlarge value of $\\ell$.\\footnote{Of course, the precise meaning of this statement could vary from one case to \nthe other due to characteristic facets of the considered  partial wave.}  The perturbative \nresult for $N(A)$, denoted by $N_p(A)$, is then\n\\begin{align}\nN^{(p)}(A)=\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta(k^2)}{(k^2)^\\ell (k^2-A)}~.\n\\label{eq.hw.per}\n\\end{align}\nHad we included only the two-nucleon irreducible contributions to  $\\Delta(A)$, which is then \ndenoted as $\\Delta_B(A)$,   \nthe previous integral becomes  the DR representation of the $NN$ potential that \nwe denominate $N_B(A)$,\n\\begin{align}\nN_B(A)=\\frac{A^\\ell}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_B(k^2)}{(k^2)^\\ell (k^2-A)}~.\n\\label{eq.nborn}\n\\end{align}\nThis is due to the fact that the $NN$ potential projected in a given \npartial wave  is an analytical function that only has LHC and it can be \nwritten in terms of a  DR  along the latter cut. \n We have checked numerically that  the DR representation \nEq.~\\eqref{eq.nborn} for the $NN$ potential coincides\n with its explicit partial wave decomposition taking into account \nthe expressions given in Ref.~\\cite{peripheral}.  \n In our notation the relation between $N_B(A)$ and the phase shifts in the Born approximation, $\\delta_B(A)$, reads\n\\begin{align}\n\\delta_B(A)=\\rho(A) N_B(A)~.\n\\label{deltab}\n\\end{align}\nAn analogous  expression holds for the perturbative phase shifts $\\delta^{(p)}(A)$ calculated in terms of $N^{(p)}(A)$. \n The difference between the perturbative phase shifts and the Born approximation ones \nfor $\\ell\\geq 2$ is typically not very significant and quite small. \nIn the following we compare our full results  with $\\delta_B(A)$, \nsince these phase shifts can be also calculated straightforwardly in potential models.\nWe proceed in the same way for the coupled channel case as well by evaluating $N_{ij}(A)$ in the Born approximation \nby substituting  $D_{ij}(A)\\to 1$ in Eq.~\\eqref{highncc}, and keeping only the two-nucleon irreducible contributions \nto $ \\Delta_{ij}(A)$.\n\nTurning back to the uncoupled $D$ waves we also show in Fig.~\\ref{fig:dw} the leading Born approximation phase shifts \nobtained from the NNLO two-nucleon  irreducible contributions to $\\Delta(A)$ by the (cyan) filled areas.\n One observes that these curves are quite different from our full results \ngiven by the hatched areas. This clearly indicates that the perturbative \ntreatment of the $NN$ $D$ waves is not accurate.\n\n\\section{Uncoupled $F$ waves}\n\\label{fw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1f3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3f3.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1F_3$ (left panel) and $^3F_3$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome. In both cases three-time-subtracted DRs are used.\nThe (cyan) filled bands give $\\delta_B(A)$, \nthe OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines and \nthe Nijmegen PWA phase shifts correspond to the (black) dashed lines.}\n\\label{fig:fw} }\n\\end{center}\n\\end{figure}\n\nFor the $F$ waves we have three subtractions with two free parameters $\\delta_2$ and $\\delta_3$. We fix $\\delta_2=0$ in the following (according to the principle of maximal smoothness) and fit $\\delta_3$ to data. \n At NNLO the fitted values for $D^{(2)}(0)=2 \\delta_3$, Eq.~\\eqref{tayloruc}, are: \n\\begin{align}\n^1F_3:~& D^{(2)}(0)= 0.057(3)~M_\\pi^{-4}~,\\nonumber\\\\\n^3F_3:~& D^{(2)}(0)= 0.035(5)~M_\\pi^{-4}~,\n\\end{align}\nwhere the variation in the values is due to the different sets of $c_i$ counterterms employed. \nThe NNLO results are shown by the (red) hatched areas in Fig.~\\ref{fig:fw} which reproduce the \nNijmegen PWA phase shifts (black dashed line) better than the NLO results (magenta lines) and \nthe perturbative phase shifts (cyan filled areas).  This improvement \nis particularly noticeable for the $^3F_3$ partial wave.\n\nWe also observe that for the $F$ waves the phase shifts in the leading Born approximation, Eq.~\\eqref{deltab}, \nrun much closer to our full results than for the $D$ waves, which clearly indicates that \n$F$ waves are more perturbative. \n Nevertheless, the relative deviation of the perturbation results compared \nwith the full solution is still around a 50\\% at the end of the interval shown in Fig.~\\ref{fig:fw}. \n A similar conclusion on the more perturbative nature of the $F$ waves \nwas also reached in the pure perturbative study of Ref.~\\cite{peripheral} by comparing with experimental data.  \n However, here we can also compare with the full unambiguous solution of the corresponding IE. \n For example, we can learn from Fig.~\\ref{fig:fw}\n that the widths of the (cyan) filled bands for the Born approximation results \nreflect a much larger dependence on the $c_i$ \ncoefficients than the one corresponding to the full nonperturbative results given by the (red) hatched\n areas.\n Thus, within our approach the failure reported in Refs.~\\cite{epe04,thesis} to reproduce simultaneously \nthe $D$ and $F$ waves by using the NNLO chiral potential \ncalculated in dimensional regularization in Ref.~\\cite{peripheral}  \n because the large values of the $c_i$  counterterms does not happen. \nNamely, we are able to \ndescribe properly both the  uncoupled $D$ and $F$ waves, Figs.~\\ref{fig:dw} and \\ref{fig:fw}, respectively, \nand the dependence on the precise set of $c_i$'s taken is quite mild for the full results. \nIndeed our calculation at NNLO describe the Nijmegen PWA phase shifts better than  the NLO ones \\cite{gor2013},\n which is not the case for all of these waves in  Ref.~\\cite{thesis} \nbased on the (modified) Weinberg approach when comparing their NLO and NNLO results.\n\n \n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/lhcd1f3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/dis.lhc.1f3.ps} \\\\\n\\includegraphics[width=.4\\textwidth]{.\/lhcd3f3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/dis.lhc.3f3.ps} \\\\\n\\end{tabular}\n\\caption{ { (Color online.) Left panel: different contributions to the integral on the \nr.h.s. of Eq.~\\eqref{highd} for $\\ell=3$.\n The meanings of the lines are the same as in Fig.~\\ref{fig:1s0quanty}. \nFor definiteness we consider the $c_i$'s given in the last row of Table~\\ref{tab:cis}.  }\n\\label{fig:unFquanty}}\n\\end{center}\n\\end{figure}\n\nThe increase in the perturbative character of the $F$ waves can also be seen by considering the \nrelevance of the different contributions of $\\Delta(A)$ to the integral on the r.h.s. of\n Eq.~\\eqref{highd}, proceeding in a similar way to that already performed for the $^1S_0$ partial wave \nin Sec.~\\ref{1s0}.\n The result is shown in the left panels of Fig.~\\ref{fig:unFquanty}, where\n the first row corresponds to $^1F_3$ and the second to $^3F_3$.\n In the right panels we show directly the different contributions to $\\Delta(A)$. \nThe meanings of the lines \nin Fig.~\\ref{fig:unFquanty} are the same as in Fig.~\\ref{fig:1s0quanty}, though here \n the $c_i$'s are taken from Ref.~\\cite{aco2013}, given in the last row of Table~\\ref{tab:cis}, \nwhich is enough for the present purposes. \nNotice, that now a qualitative different situation is found with respect to what is shown in \nFig.~\\ref{fig:1s0quanty}, that also holds for the $P$ and $D$ waves discussed in Secs.~\\ref{pw} and \\ref{dw}. \n For the $F$ and higher waves the subleading two-nucleon irreducible TPE contribution is much less important and \nOPE is by far the dominant contribution, as it should correspond to a perturbative high-$\\ell$ wave.\n \n\n \n\n\n\\section{Uncoupled $G$ waves}\n\\label{gw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1g4.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3g4.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1G_4$ (left panel) and $^3G_4$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome. In both cases four-time--subtracted DRs are used. \n The  (cyan) filled areas represent\n the outcome from the leading Born approximation,  \n the OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines \nand the Nijmegen PWA analysis is the (black) dashed lines.}\n\\label{fig:gw} }\n\\end{center}\n\\end{figure}\n\nFor the $G$ waves we have four subtractions of which $\\delta_i$ $(i=2,3,4)$ are free but, according to the \nprinciple of maximal smoothness, all of them are fixed to 0 except \n$\\delta_4=D^{(3)}(0)\/3!$ that is fitted to data. \nAt  NNLO the fitted values for $D^{(3)}(0)$ are: \n\\begin{align}\n^1G_4:~& D^{(3)}(0)=-0.014(2)~M_\\pi^{-6}~,\\nonumber\\\\\n^3G_4:~& D^{(3)}(0)=-0.055(5)~M_\\pi^{-6}~,\n\\end{align}\nwhere the variation in the values is due to the different sets of $c_i$ counterterms employed. \nThe corresponding results are shown by the (red) hatched areas in Fig.~\\ref{fig:gw}. \nFor both partial waves the actual dependence on the $c_i$ coefficients for the resulting phase shifts \nis almost negligible and the hatched  areas degenerate to lines. \n  The low-energy results are very similar at NLO and NNLO and reproduce the Nijmegen \nPWA phase shifts quite well.\n These results are better than the perturbative ones in the Born approximation,  Eq.~\\eqref{deltab}, which  \nare shown by the (cyan) filled areas. \nAs indicated for the uncoupled $F$ waves here OPE overwhelmingly dominates \n the different contribution to the dispersive integral on the r.h.s. of Eq.~\\eqref{highd}. \nThis indicates that these waves are rather perturbative, though still we observe differences \naround 30\\% for $p\\lesssim 300$~MeV in Fig.~\\ref{fig:gw} between the full and perturbative results.\n\n\n\n\n\\section{Uncoupled $H$ waves}\n\\label{hw}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/1h5.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3h5.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) Phase shifts for $^1H_5$ (left panel) and $^3H_5$ (right panel). \n The (red) hatched areas  correspond to the NNLO results while the (magenta) solid lines are \nthe NLO outcome. \n The (cyan) filled bands correspond to $\\delta_B(A)$, \nthe OPE result from Ref.~\\cite{paper1} is the (blue) dotted lines and  \nthe Nijmegen PWA is the (black) dashed lines.}\n\\label{fig:hw} }\n\\end{center}\n\\end{figure}\n\nFor the case of the uncoupled $H$ waves, $^1H_5$ and $^3H_5$, we apply the five-time subtracted DRs of  \nEqs.~\\eqref{highd} and \\eqref{highn} with $\\ell=5$. \nWe fit $\\delta_5=D^{(4)}(0)\/4 !$ to the Nijmegen PWA phase shifts, which for $\\ell\\geq 5$ \ncorrespond to  those obtained from the $NN$ potential model of Ref.~\\cite{obe}, while $\\delta_{2,3,4}$ are fixed to 0 (principle of maximal smoothness). \nWe obtain the fitted values: \n\\begin{align}\n^1H_5:~D^{(4)}(0)&=0.156~ M_\\pi^{-8}~,\\nonumber\\\\\n^3H_5:~D^{(4)}(0)&=0.066~M_\\pi^{-8}.\n\\end{align} \nThe resulting fit is stable if  we release $\\delta_i$ $(i=2,3,4)$. \nThe  phase shifts obtained are shown by the (red) hatched areas in  Fig.~\\ref{fig:hw}  by taking into account the spread of the results \ndepending of the set of $c_i$'s chosen. \nIn this figure the left panel corresponds to  $^1H_5$ and  the right one to $^3H_5$.  \n For the former the resulting curve indeed  overlaps the Nijmegen PWA phase shifts \\cite{Stoks:1994wp}. \nWe also show by the (cyan) filled bands the phase shifts in the leading Born approximation\n which run rather close to \nthe full results, indeed for the $^3H_5$ case the (cyan) filled band is overlapped by the (red) hatched one.   \nThis clearly indicates the perturbative nature for the $H$ waves. \nFor them it is also true that OPE overwhelmingly dominates \nthe dispersive integral on the r.h.s. of Eq.~\\eqref{highd}, which is also the expected behavior \nfor a perturbative partial wave.\nNotice that for the $^1H_5$ wave the dependence on the actual values of the \n$c_i$ coefficients is so small that at the end the hatched and filled areas collapse to lines.\n For the $^3H_5$ case there is a visible, albeit small, dependence on the set of $c_i$'s employed. \nIn both cases the NNLO results reproduce the Nijmegen PWA phase shifts closer than the NLO and OPE results. \n\n \n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/N3h5.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/D3h5.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) The functions $N(A)$ and $N_p(A)$ are shown by the (red) solid and (blue) \ndash-dotted lines in the left panel, respectively. The real part of the function $D(A)$ is plotted in the right panel. }\n\\label{fig:hw.nd} }\n\\end{center}\n\\end{figure}\n\nIt is interesting to discuss in this case the behavior of the function $N(A)$ compared with $N_p(A)$, given in Eq.~\\eqref{eq.hw.per}. \nThe main point is that here both $N(A)$ and $D(A)$ have a zero at around 450~MeV.\n We consider only the $^3H_5$ wave because a similar discussion would follow for $^1H_5$ as well, \nthat we skip for brevity.\n In the left panel of Fig.~\\ref{fig:hw.nd} we show by the (red) solid line the full $N(A)$ \n and by the (blue) dashed line the perturbative result $N_p(A)$. \nWe see that they  are very similar, as expected for a partial wave with an $\\ell$ as high as 5.\n In addition, we display  in the right panel of the same figure the real part of $D(A)$ from Eq.~\\eqref{highd}, \nwhich is very close to 1, as expected for a situation with a weak interaction as well. \nAll these curves are obtained by employing the  $c_i$'s from Ref.\\cite{aco2013}.\n A bit higher in energy both  $N_p(A)$ and  $N(A)$ have a zero \n at around $\\sqrt{A}=450$~MeV.\n Since $T(A)=N(A)\/D(A)$ this would imply that $T(A)=0$ at that energy, which is at odds \nwith the values of the phase shifts given by the Nijmegen PWA \\cite{Stoks:1994wp} that do not vanish at this point. \nThe only remedy is that  $D(A)$ is also zero at the same point so that \n one had a limit 0\/0 that is finally finite. \nThis is indeed the case and it is is the reason why \n $D(A)$ starts to decrease for $\\sqrt{A}>200$~MeV in Fig.~\\ref{fig:hw.nd}.\n\n\nAnother question of interest to think about is  what have we  gained by solving exactly Eq.~\\eqref{highd} instead of \n using only the perturbative solution, Eq.~\\eqref{eq.hw.per}, or the Born approximation, Eq.~\\eqref{eq.nborn}, with \nthe related $ \\delta_B(A)$, Eq.~\\eqref{deltab}?.  \nThe main point that one should consider \n in connection with this question is that by solving the full and nonperturbative Eq.~\\eqref{highd} \n(furthermore, in good agreement with data) one can then state that Eq.~\\eqref{eq.hw.per} is a  perturbation of a \n well-defined and existing nonperturbative solution. \nBy solving exactly Eq.~\\eqref{highd} we have needed to consider explicitly $\\delta_5$ as a free parameter  for the uncoupled \n$H$ waves  and fit it to the Nijmegen PWA. \nIndeed, $\\delta_5$ is not only necessary for a good fit, \n but it is also required in order to keep $D(A)\\simeq 1$ at low \nthree-momentum. \nOtherwise, the contribution from the dispersive integral to $D(A)$ on the r.h.s. of Eq.~\\eqref{highn} would \nbe too large and negative and would render a too strong function $N(A)$  in plain disagreement with $N_p(A)$. \nNotice as well that in the case of the partial wave $^1H_5$ a better reproduction of data is achieved than with $\\delta_B(A)$. \n It is also worth recalling the previous finding in Sec.~\\ref{fw} for the $F$ waves, \nwhere the full results show a much smaller dependence on \nthe set of $c_i$ coefficients used than the perturbative or Born approximation phase shifts, cf. Fig.~\\ref{fig:fw}.\n\n\n\\section{Coupled $^3S_1-{^3D_1}$ waves}\n\\label{sd12w}\n\n\nWe start our study of the $^3S_1-{^3D_1}$ coupled-partial-wave system in terms of just one free parameter, \nthat we choose as the pole position of the deuteron in the $A$-complex plane, $k^2_d=-m E_d$, with $E_d= 2.225$~MeV \nthe deuteron binding energy. \n Thus we implement once-subtracted DRs for the $^3S_1$ and twice-subtracted ones for the $^3D_1$. \nIn the case of the mixing partial wave we have a mixed situation with a once-subtracted DR \nfor $N_{12}(A)$ and a twice-subtracted one for $D_{12}(A)$.\n In this way we guarantee both the right threshold behavior as well as the experimental \n deuteron-pole position in all the partial waves. \nWe write now explicitly  the DRs considered. For the $^3S_1$ one has,\n\\begin{align}\n\\label{3s1_a}\nD_{11}(A)&=1-\\frac{A}{k_d^2}\\frac{g_{11}(A,0)}{g_{11}(k_d^2,0)}\n+\\frac{A}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{k^2}\n\\Bigg[g_{11}(A,k^2)-g_{11}(A,0)\\frac{g_{11}(k_d^2,k^2)}{g_{11}(k_d^2,0)}\\Bigg]~,\\nonumber\\\\\nN_{11}(A)&=\\nu_1^{(11)}+\\frac{A}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{k^2(k^2-A)}~,\n\\end{align}\nwith all the subtractions taken at $A=0$ and the new function $g_{ij}(A)$ is defined as\n \\begin{align}\ng_{ij}(A,k^2)&=\\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)}{(q^2-A)(q^2-k^2)}~,\n\\label{gij}\n\\end{align}\n The subtraction constant $\\nu_1$ in $N_{11}(A)$ is fixed by imposing that $D_{11}(k_d^2)=0$,\n\\begin{align}\n\\nu_1^{(11)}&=\\frac{1}{k_d^2 \\,g_{11}(k_d^2,0)}\\Bigg[\n1+\\frac{k_d^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2) D_{11}(k^2)}{k^2}g_{11}(k^2,k_d^2)\n\\Bigg]~,\n\\label{once_nu0}\n\\end{align}\na result that is  already implemented in Eq.~\\eqref{3s1_a} for $D_{11}(A)$.\n \nThe corresponding DRs for the $^3D_1$ and the mixing wave can be grouped together in the same form,\n\\begin{align}\n\\label{3sd1_a}\nD_{ij}(A)&=1-\\frac{A}{k_d^2}+\\frac{A(A-k_d^2)}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^{\\ell_{ij}}}\ng_{ij}^{(d)}(A,k^2;\\ell_{ij})~,\\nonumber\\\\\nN_{ij}(A)&=\\frac{A^{\\ell_{ij}}}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^{\\ell_{ij}}(k^2-A)}~.\n\\end{align}\nwhere $\\ell_{12}=1$ and $\\ell_{22}=2$ and all the subtractions for the $N_{ij}(A)$ are taken at $A=0$,\n while in the function $D(A)$ one is taken at $A=0$ and \nthe other at $A=k_d^2$. The  function $g^{(d)}_{ij}(A,k^2;m)$ is defined as\n\\begin{align}\ng^{(d)}_{ij}(A,k^2;m)&=\\frac{1}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{ij}(q^2)(q^2)^{m-1}}{(q^2-A)(q^2-k^2)(q^2-k_d^2)}~.\n\\label{gijd}\n\\end{align}\n\nThe results obtained by solving the IEs for the functions $D_{ij}(A)$ along the LHC\n from Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a} are \nshown in Fig.~\\ref{fig:3sd1_a} by the (cyan) filled areas.\n These results are indicated as NNLO-I and \n all the subtraction constants are fixed in terms of $k_d^2$, without any other freedom. \n The spread in the results originates by taking  different  sets of $c_i$'s from Refs.~\\cite{epe12,aco2013} \nand varying the input in the iterative procedure.\n The present NNLO calculation from Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a}\n reproduces the  Nijmegen PWA  mixing angle $\\epsilon_1$ much better than the NLO \nresult from the same set of equations,  which is shown by the (magenta) dot-dashed lines.\n This improvement in the description of $\\epsilon_1$ \nwhen passing from NLO to NNLO is also seen in Ref.~\\cite{thesis} by \nemploying the Weinberg scheme. \nThe $^3S_1$ phase shifts are also reproduced better at NNLO than at NLO, while \nthe $^3D_1$ phase shifts are somewhat worse described by the former. \nOur results for the  $^3S_1$ and $^3D_1$ phase shifts are quite similar to those obtained \nin Ref.~\\cite{pavon06}, but not for $\\epsilon_1$ where our outcome is closer to the Nijmegen PWA. \nThe comparison is not so straightforward with the results of Ref.~\\cite{phillipssw},  which depend\n very much on the type of chiral $NN$ potential used.\nFor the $^3S_1-{^3D_1}$ coupled partial waves we do not show the Born approximation results  in Fig.~\\ref{fig:3sd1_a} because \nthey are specially poor, see e.g. Refs.\\cite{epe04,peripheral} for the $^3D_1$ phase shifts.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3s1_all.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3d1_all.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e1_all.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From left to right and top to bottom:\n Phase shifts for $^3S_1$, $^3D_1$ and the mixing angle  $\\epsilon_1$, respectively.\nThe (cyan) filled areas  correspond to the NNLO-I outcome obtained by solving \n Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a}. The  hatched areas with (red) crossed lines are the NNLO-II results \n that stem from Eqs.~\\eqref{2subs.3s1}, \\eqref{2subs.mix} and \\eqref{3sd1_a}.  \nIn addition, for the $^3D_1$ we show by the hatched areas with (gray) parallel lines the results obtained \nby employing three-time subtracted DRs for $^3D_1$, Eq.~\\eqref{3d1.extra}. \nAs usual, the (magenta) dot-dashed lines are the NLO phase shifts and mixing angle, \n the  LO ones are given by the  (blue) dotted lines and the \nNijmegen PWA results correspond to the (black) dashed lines.}\n\\label{fig:3sd1_a} }\n\\end{center}\n\\end{figure}\n\nWe can also predict from Eqs.\\eqref{3s1_a} and \\eqref{3sd1_a} the $^3S_1$ scattering length ($a_t$) and effective range ($r_t$). The former is given in terms of $\\nu_1^{(11)}$, Eq.~\\eqref{once_nu0}, as \n\\begin{align}\na_t&=-\\frac{m \\nu_1^{(11)}}{4\\pi}~.\n\\end{align} \nRegarding $r_t$ we can proceed similarly as discussed in detail in Ref.~\\cite{gor2013} \nwhere the following expression  is derived, \n\\begin{align}\nr_t=-\\frac{m}{2\\pi^2 a_t}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2}\\left\\{\\frac{1}{a_t}+\\frac{4\\pi k^2}{m}g_{11}(0,k^2)  \\right\\}-\\frac{8}{m}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)-\\rho(q^2)}{(q^2)^2}~,\n\\label{3s1.rt}\n\\end{align}\nThis equation also exhibits a correlation between $r_t$ and $a_t$, although in a more complicated manner than \nfor the $^1S_0$ partial wave, as shown in Eq.~\\eqref{expvs.1s0}, because $\\nu_{11}(A)$ depends nonlinearly on $D_{11}(A)$. \n \nAnother observable that we also consider is the slope at threshold of $\\epsilon_1$, \nindicated as $a_\\varepsilon$, and defined by  \n\\begin{align}\na_\\varepsilon=\\lim_{A\\to 0^+}\\frac{\\sin 2\\epsilon_1}{A^\\frac{3}{2}}=1.128~M_\\pi^{-3}~,\n\\label{avarepsilon}\n\\end{align}\nwhere the numerical value is deduced from the Nijmegen PWA phase shifts. From the DRs in Eq.~\\eqref{3sd1_a} we obtain \nthe following expression for $a_\\varepsilon$,\n\\begin{align}\na_\\varepsilon&=\\frac{m}{4\\pi^2}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^2}~.\n\\label{aep.predicted}\n\\end{align}\n\nIt is also interesting to diagonalize the $^3S_1-{^3D_1}$ $S$-matrix around the deuteron pole position. This can \nbe done by means of a real orthogonal matrix \\cite{swart},\n\\begin{align}\n{\\cal O}&=\\left(\n\\begin{array}{ll}\n \\cos \\varepsilon_1 & -\\sin\\varepsilon_1 \\\\\n\\sin\\varepsilon_1 & \\cos \\varepsilon_1\n\\end{array}\n\\right)~.\n\\label{mat.ort}\n\\end{align}\nSuch that\n\\begin{align}\nS&={\\cal O}\\left(\n\\begin{array}{ll}\nS_0 & 0 \\\\\n0 & S_2\n\\end{array}\n\\right) {\\cal O}^T~,\n\\label{eigen3s1}\n\\end{align}\nwith $S_0$ and $S_2$ the $S$-matrix eigenvalues.  The asymptotic $D\/S$ ratio of the deuteron, $\\eta$, \ncan be expressed in terms of $\\varepsilon_1$ as\n\\begin{align}\n\\eta=-\\tan \\varepsilon_1~.\n\\label{eta.ep}\n\\end{align}\nThe residue of $S_0$ at the deuteron pole position is denoted by $N_p^2$,\n\\begin{align}\nN_p^2&=\\lim_{A\\rightarrow k_d^2} \\left(\\sqrt{-k_d^2}+i\\sqrt{A}\\right) S_0~.\n\\end{align} \n\nAs discussed in  Ref.~\\cite{cohen}  the shape parameters are a good testing ground for the range of \napplicability of the underlying EFT. \nWe then study our results for the shape parameters of the lowest \neigenphase $\\delta_0$ (also called $^3S_1$ eigenphase), Eq.~\\eqref{eigen3s1}, with the diagonalization of the \n$S$-matrix performed in the physical region $A\\geq 0$,\\footnote{This can also be done in terms of an \northogonal matrix Eq.~\\eqref{mat.ort} because of two-body unitarity.}\n \\begin{align}\n\\sqrt{A}\\,\\text{cot}\\delta_0=-\\frac{1}{a_t}+\\frac{1}{2}r_t A +\\sum_{i=2}^{10}v_i A^i+{\\cal O}(A^{11})~.\n\\label{3s1.ere}\n\\end{align}\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n    & $a_t$ [fm]  & $r_t$ [fm]  & $\\eta$ & $ N_p^2$ [fm$^{-1}$] & $a_\\varepsilon$ [$M_\\pi^{-3}$] \\\\\n\\hline\nNLO     & 5.22 & 1.47 & 0.0295 & 0.714 & 1.372 \\\\\n\\hline\nNNLO-I & $5.52(3)$ & $1.89(3)$ & $0.0242(3)$ & $0.818(10)$& $1.270(9)$\\\\\n\\hline\nNNLO-II & $5.5424^\\star$ & $ 1.759^\\star$ & $0.02535(13)$ & $0.78173(2)$ & $1.293(8)$\\\\\n\\hline\nRef.~\\cite{swart}& $5.4194(20)$ & $1.7536(25)$ & $0.0253(2)$ &  $0.7830(15)$ & \\\\\n\\hline\nRef.~\\cite{thesis} & $5.424$ & $1.753$ & 0.0245 & & \\\\ \n\\hline\n\\end{tabular}\n\\caption{Values for $a_t$, $r_t$, $\\eta$, $N_p^2$ and $a_\\varepsilon$. The results \n predicted from Eqs.~\\eqref{3s1_a} \n and \\eqref{3sd1_a} are given in the second (NLO) and third row (NNLO-I). \nThe values given in the fourth row (NNLO-II) are obtained once $a_t$ and $r_t$ are fixed to the experimental \n figures, which is indicated by a star on top of the values. \nWe also show the results from Refs.~\\cite{swart} and \\cite{thesis} in the fifth and sixth rows, respectively. \n\\label{table:eta}}\n\\end{center}\n\\end{table}\n\nThe scattering length and effective range in the previous equation are the same as given above because \ncoupled-wave effects with the $^3D_1$ only affects the shape parameters $v_i$, $i\\geq 2$. The values obtained at NLO and NNLO from Eqs.~\\eqref{3s1_a} and \\eqref{3sd1_a} for $a_t$, $r_t$, $\\eta$, \n$N_p^2$ and $a_\\varepsilon$ are shown in Table~\\ref{table:eta} in the \nsecond and third rows, respectively. \nWe observe that the numbers at NNLO (indicated by NNLO-I) are already \n rather close to those of Ref.~\\cite{swart}, obtained from the Nijmegen PWA of $n p$ data,\n and Ref.~\\cite{thesis}. \nIt is interesting to remark that our value for $r_t$ is a prediction  in terms of only one  \nsubtraction constant (fixed by the deuteron pole position) and \n$NN$ forces stemming from $\\pi N$ physics. This value deviates from experiment \n $r_t=1.759\\pm 0.005$~fm \\cite{thesis} around a $10\\%$ at NNLO ($\\sim 20\\%$  at NLO), while the relative \n experimental error is  around $3\\%$. \n Other determinations for the parameter $\\eta$,  not shown in Table~\\ref{table:eta}, are $\\eta=0.0256(4)$ \\cite{rodning}, \n$\\eta=0.0271(4)$ \\cite{ericson:82}, \n$\\eta=0.0263(13)$ \\cite{conzett:79} and \n$\\eta=0.0268(7)$ \\cite{martorell}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n    & $v_2$  & $v_3$   & $v_4$ & $v_5$ & $v_6$    \\\\\n\\hline\nNLO & -0.10572(12)& 0.8818(11) & $-5.427(11)$  & 36.73(11) & $-259.9(1.1)$ \\\\\n\\hline\nNNLO-I & 0.157(22)& 0.645(9) & $-3.41(13)$  & 23.2(8) & $-161(6)$ \\\\\n\\hline\nNNLO-II & $0.0848(4)$ & $0.762(7)$ & $-4.33(2)$ & $29.0(2)$ & $-198(2)$\\\\\n\\hline\nRef.~\\cite{swart} & $0.040(7)$ & $0.673(2)$ & $-3.95(5)$ & $27.0(3)$ & \\\\\n\\hline\nRef.~\\cite{thesis} & $0.046$ & $0.67$ & $-3.9$ & & \\\\\n\\hline\n\\end{tabular}\n\\caption{Values for the shape parameters $v_i$, $i=2,\\ldots,6$ in units of fm$^{2i-1}$.\n The results \n predicted from Eqs.~\\eqref{3s1_a}\n and \\eqref{3sd1_a} are given in the second  (NLO) and third row (NNLO-I). \nThe errors for the NLO results correspond  entirely to the numerical accuracy in the calculation. \nThose values corresponding to NNLO-II are given in the fourth row. \nThe values from Refs.~\\cite{swart} and \\cite{thesis} appear in the fifth and sixth rows, in order.\n\\label{table:vs3s1a}}\n\\end{center}\n\\end{table}\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n    &  $v_7$ & $v_8\\times 10^{-1}$  & $v_{9}\\times 10^{-2}$  & $v_{10}\\times 10^{-3}$   \\\\\n\\hline\nNLO &  1867(11) & $-1375(11)$ & $1008(11)$ & $-760(12)$ \\\\\n\\hline\nNNLO-I & 1161(41) & $-840(30)$ & $625(22)$ & $-463(17)$ \\\\ \n\\hline\nNNLO-II& $1426(13)$ & $-1015(15)$ & $764(17)$ & $-545(20)$\\\\\n\\hline \n\\end{tabular}\n\\caption{Values for the shape parameters $v_i$, $i=7,\\ldots,10$ in units of fm$^{2i-1}$. \nFor the meanings of the rows see Table~\\ref{table:vs3s1a}.\n\\label{table:vs3s1b}}\n\\end{center}\n\\end{table}\n\nThe values for the shape parameters $v_i$, $i=2,\\ldots,6$, are given in Table~\\ref{table:vs3s1a} and \nfor $i=7,\\ldots,10$ in Table~\\ref{table:vs3s1b}. Up to \nour knowledge the values of the shape parameters with $i>5$ were not given before. We detailed in \nAppendix \\ref{appen:vs} the numerical method that allows us to perform the appropriate \nderivatives up to so high order.\\footnote{For example in Ref.~\\cite{swart} it is stated that their numerical \nset up is not precise enough to calculate  $v_6$ and that it already casts doubts about the numerical \n accuracy for $v_5$.}  \n We could  have also given shape parameters of even higher orders\n within a numerical precision of a few per cent, \nbut this is skipped because its apparent little relevance in practice.\n One can appreciate the numerical precision in the calculation of the shape parameters by considering the errors in \n Tables~\\ref{table:vs3s1a} and \\ref{table:vs3s1b} for the NLO results, which entirely  correspond to  the numerical accuracy.\n Notice that for the highest shape parameter shown,  $v_{10}$, its relative error is 1.5$\\%$, \njust slightly worse than for $v_9$ with a relative error of 1.1\\%. \nWe then see that by increasing the order of the shape parameter \nthe numerical accuracy only worsens little by little. \nMorever,  the errors at NNLO take into account additionally \nthe variation in the results from the different sets of $c_i$'s employed and the dependence in the \ninput for starting the iterative process.\n For the shape parameters with large order, $i\\geq 5$,  their absolute values increase \ntypically as ${\\cal O}(1\/M_\\pi)^{2i-1}$, which is the expected behavior for long-range interactions\n mediated by OPE. It is clear from Table~\\ref{table:vs3s1a} that the shape parameters $v_i$, $i=2,\\ldots,5$ \npredicted by the NNLO-I calculation (third row) are typically\n closer to the values of Refs.~\\cite{swart,thesis} than those at NLO (second row). \nThis is a positive feature indicating a well-behaved expansion of the results obtained by \napplying the $N\/D$ method with the discontinuity $\\Delta(A)$ expanded in BChPT.\n\n\nAccording to  the power counting for the subtraction constants, Eq.~\\eqref{summarypwc}, at NNLO it is appropriate\n to consider twice-subtracted DRs. \nFor the $^3S_1-{^3D_1}$ system this implies to take into account two more free parameters for the $^3S_1$ \nwave and one more for the mixing partial wave.\n The three parameters for the $^3S_1$ wave are fixed in terms \nof the experimental values of $k_d^2$, $r_t$ and $a_t$.\nThe DR for the $^3D_1$ wave is the same as in Eq.~\\eqref{3sd1_a}. \n The twice-subtracted DRs taken now regarding the $^3S_1$ partial wave are\n\\begin{align}\nD_{11}(A)&=1-\\frac{A}{k_d^2}-\\nu_1^{(11)} \\,A(A-k_d^2) g_{11}^{(d)}(A,0;1)\n-\\nu_2^{(11)} \\,A(A-k_d^2) g_{11}(A,k_d^2)\\nonumber\\\\\n&+\\frac{A(A-k_d^2)}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2} \ng_{11}^{(d)}(A,k^2;2)~,\\nonumber\\\\\nN_{11}(A)&=\\nu_1^{(11)}+\\nu_2^{(11)}\\,A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2(k^2-A)}~,\\nonumber\\\\\n\\nu_1^{(11)}&=-\\frac{4\\pi a_t}{m}~,\\nonumber\\\\\n\\nu_2^{(11)}&=\\frac{\\nu_1^{(11)}}{\\nu_1^{(11)} \\,k_d^2\\, g_{11}(0,k_d^2)-1}\\left\\{\n\\frac{1}{k_d^2}+a_t\\Bigg(\n\\frac{4 k_d^2}{m}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)-\\rho(q^2)}{(q^2)^2(q^2-k_d^2)}+\\frac{1}{\\sqrt{-k_d^2}}\n-\\frac{r_t}{2}\\Bigg)\\right. \\nonumber\\\\\n&\\left. +\\frac{k_d^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^2}g_{11}(k_d^2,k^2)\n\\right\\}~.\n\\label{2subs.3s1}\n\\end{align}\n\nFor the mixing partial wave the DRs are \n\\begin{align}\nD_{12}(A)&=1-\\frac{A}{k_d^2}-\\nu_2^{(12)} A(A-k_d^2)g_{12}(A,k_d^2)\n+\\frac{A(A-k_d^2)}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{12}(k^2)D_{12}(k^2)}{(k^2)^2}g_{12}^{(d)}(A,k^2;2)~,\\nonumber\\\\\nN_{12}(A)&=\\nu_2^{(12)} A+\\frac{A^2}{\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{12}(k^2)D_{12}(k^2)}{(k^2)^2(k^2-A)}~,\n\\label{2subs.mix}\n\\end{align}\n The results obtained by solving the IEs of Eqs.~\\eqref{2subs.3s1}, \\eqref{2subs.mix} and Eq.~\\eqref{3sd1_a} \nwith $\\ell_{22}=2$ are denoted in the following by NNLO-II and correspond to the (red) hatched areas with crossed lines \nin Fig.~\\ref{fig:3sd1_a}. \nIt turns out that we cannot obtain a solution of the resulting IE for $D_{12}(A)$ by \nimplementing any arbitrary value for $\\nu_2^{(12)}$.\n We have further checked this statement by employing the following expression for $\\nu_2^{(12)}$,\n\\begin{align}\n\\nu_{2}^{(12)}&=\\frac{\\Theta}{2\\pi}\\int_{-\\infty}^L dk^2\\frac{\\Delta_{ij}(k^2)D_{ij}(k^2)}{(k^2)^2}~.\n\\end{align}\nHere, the integral is the same as in Eq.~\\eqref{aep.predicted}, so that if we take $ \\Theta=1$ we would simply \nrewrite the IE of Eq.~\\eqref{3sd1_a} in terms of twice-subtracted DRs. Then, we  vary $\\Theta$ and whenever we find \na meaningful solution the obtained value for $a_\\varepsilon=m\\nu_2^{(12)}\/2\\pi$ is always basically the same,  \n$a_\\varepsilon\\simeq 1.30~M_\\pi^{-3}$.\n In our opinion this difficulty in our approach to reproduce the value for $a_\\varepsilon$ \n that follows from the Nijmegen PWA, Eq.~\\eqref{avarepsilon}, casts doubts on this number. \n  Notice that  the calculated values for $\\epsilon_1$ at low momentum, \ne.g. for $\\sqrt{A}\\lesssim 100$~MeV, lie on top of the curve for the Nijmegen PWA results\n as shown in the third panel of Fig.~\\ref{fig:3sd1_a} by the coincident hatched and filled areas that overlap the Nijmegen PWA line. \n  The phase shifts and $\\epsilon_1$  are quite similar to the NNLO-I results in terms of just one free parameter.\nNevertheless,  the $^3S_1$ phase shifts for  NNLO-II are closer to the Nijmegen PWA ones  \nat lower energies, but the change for this S-wave by going from once- to twice-subtracted DRs\n is much less notorious than in the case of the partial wave $^1S_0$, discussed in Sec.~\\ref{1s0}. \nWe can also see in the fourth row of\n Table~\\ref{table:eta} that the NNLO-II values for $\\eta$ and $N_p^2$ are compatible \nwith those of  Ref.~\\cite{swart}, which is quite remarkable.\n The value for $a_\\varepsilon$ mentioned above is shown in \nthe last column of the same table.\n The shape parameters are shown in the forth rows of Tables~\\ref{table:vs3s1a} \nand \\ref{table:vs3s1b}, where we observe \na better agreement with the numbers given in Ref.~\\cite{swart} for \n $v_4$ and $v_5$ than for $v_2$ and $v_3$. The variation of the values\n between NNLO-I and NNLO-II for the higher order shape parameters allows us to guess \nin a  conservative way the systematic uncertainty affecting their calculation. \n\nOn the other hand, we would like  to elaborate further on the fact that at NNLO the results for the $^3D_1$ phase \nshifts do not still offer a good reproduction of the Nijmegen PWA ones, being even worse than those obtained \nat NLO.\n In Ref.~\\cite{epe04} one can find a discussion on the difficulties arisen in their calculation \n because of the large values of the  NLO $\\pi N$ \ncounterterms, namely $c_3$ and $c_4$, in order to reproduce simultaneously the $D$ and $F$ waves \nwithin the Weinberg scheme using the NNLO  chiral potential  \ncalculated in dimensional regularization. \n Considering this observation \n we obtain that when all the $c_i=0$ our NNLO result \nfor $\\delta_2$ is then essentially the same as the NLO one in Fig.~\\ref{fig:3sd1_a}, \ncorresponding to the (magenta) dot-dashed line. \n In view of this, we study now the influence in the results by including one more subtraction in the DRs for $^3D_1$ \nwith the aim of determining whether this worsening \nis an effect that can be counterbalanced in a natural way at ${\\cal O}(p^4)$. \n In this way we use the same twice-subtracted DRs for $^3S_1$ and the mixing partial wave \ngiven in Eqs.~\\eqref{2subs.3s1} and \\eqref{2subs.mix}, respectively, while  \n the following three-time subtracted DRs are used for the $^3D_1$\n\\begin{align}\nD_{22}(A)&=1-\\frac{A}{k_d^2}+\\delta_3^{(22)} A(A-k_d^2)\n-\\nu_3^{(22)} A(A-k_d^2)^2\\frac{\\partial g_{22}^{(d)}(A,0;2)}{\\partial k_d^2}\\nonumber\\\\\n&+\\frac{A(A-k_d^2)^2}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2)D_{22}(k^2)}{(k^2)^3} \n\\frac{\\partial g_{22}^{(d)}(A,k^2;3)}{\\partial k_d^2}~,\\nonumber\\\\\nN_{22}(A)&=\\nu_3^{(22)} A^2+\\frac{A^3}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2)D_{22}(k^2)}{(k^2)^3(k^2-A)}~,\n\\label{3d1.extra}\n\\end{align}\nwith two additional subtraction constants  $\\delta_3^{(22)}$ and $\\nu_3^{(22)}$. \nConsidering the results  obtained from the twice-subtracted DRs for all the waves in the system\n $^3S_1-{^3D_1}$, and denoting by  $\\hat{D}_{22}(A)$ the function $D_{22}(A)$ obtained then, we have \nthe following predictions for the subtraction constants $\\delta_3^{(22)}$ and $\\nu_3^{(22)}$, \n\\begin{align}\n\\nu_3^{\\mathrm{pred}}&=\\frac{1}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2)\\hat{D}_{22}(k^2)}{(k^2)^3}~,\\nonumber\\\\\n\\delta_3^{\\mathrm{pred}}&=\\frac{1}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{22}(k^2) \\hat{D}_{22}(k^2)}{(k^2)^2}g_{22}^{(d)}(k^2,k_d^2;2)~.\n\\label{nudelta2pre}\n\\end{align}\nThe numerical values that stem from the previous expressions are \n$\\delta_3^{\\mathrm{pred}}\\simeq 1~m_\\pi^{-4}$ and $\\nu_3^{\\mathrm{pred}}\\simeq -2.5~m_\\pi^{-6}$~. A fit \nto the $^3D_1$ phase shifts only requires to vary $\\nu_3^{(22)}$ around that value with the final \nresult $\\nu^{(22)}_3= -2.05(5)$~$m_\\pi^{-6}$, while \n$\\delta^{(22)}_3$ stays put.\n Then, it is only necessary a relatively small change of around 20\\% in \n $\\nu_3^{(22)}$ \n from the one predicted by the twice-subtracted DRs in Eq.~\\eqref{nudelta2pre} in order to end with a much better\n reproduction of the \n$^3D_1$ phase shifts that is compatible with the Nijmegen PWA, as shown \n by the hatched areas with (gray) parallel lines in Fig.~\\ref{fig:3sd1_a} (denoted as \nNNLO-III results).\n Since the reproduction of \nthe $^3S_1$ phase shifts and mixing angle $\\epsilon_1$ is the same as the one obtained already in terms of the \ntwice-subtracted DRs, the so-called NNLO-II results, we do not show them \nnor the values for the other parameters given  in Tables~\\ref{table:eta}, \\ref{table:vs3s1a} and \\ref{table:vs3s1b}, \nthat would be also basically coincident with the NNLO-II ones in these tables. \n\n We now elaborate on the difference between the value of $\\nu_3^{(22)}$ fitted and the one predicted, $\\nu_3^{\\mathrm{pred}}$. \nAccording to the power counting of Sec.~\\ref{nschpt}, cf. Eq.~\\eqref{summarypwc}, $\\nu_3^{(22)}={\\cal O}(p^{-1})$\n in our present NNLO calculation. If we consider that this difference is an effect that stems from the ${\\cal O}(p^4)$ \ncontributions to $\\Delta(A)$, which are not considered here yet, one would have that  \n$\\delta\\nu_3 \\equiv \\nu_3^{(22)}-\\nu_3^{\\mathrm{pred}}\\simeq 0.6~M_\\pi^{-6} \n={\\cal O}(p^0)$. It also follows then that nominally\n $\\delta \\nu_3\/\\nu_3^{\\mathrm{pred}}={\\cal O}(p)$ and taking into account the \n numerical values\n\\begin{align}\n\\frac{\\delta \\nu_3}{\\nu_3^{\\mathrm{pred}}}=0.23\\sim \\frac{M_\\pi}{\\Lambda}~,\n\\end{align}\nwe can estimate that $\\Lambda\\sim 4 M_\\pi$, which is \n similar to the estimate of $\\Lambda$ obtained in Sec.~\\ref{1s0} \nfor  the  $^1S_0$ partial wave.\n As a result, $\\delta \\nu_3$ is consistent with a naturally sized ${\\cal O}(p^4)$ effect.\n\nThe fact that the matrix of limiting values\n\\begin{align}\nM_{ij}=\\lim_{A\\to -\\infty}\\frac{\\Delta_{ij}(A)}{(-A)^{3\/2}}\n\\label{mij.3s1}\n\\end{align}\nhas two negative eigenvalues  is certainly related with the possibility of obtaining \nmeaningful DRs with only one free parameter as first obtained in this section. \nWe base this statement on the necessity condition of Ref.~\\cite{gor2013} \nin order to obtain meaningful once-subtracted DRs for $\\lambda<0$,\n a condition also introduced in Sec.~\\ref{nschpt}. \nIndeed, since the mixing between different partial waves is very small \nthese eigenvalues are given in good approximation by $M_{11}$ and $M_{22}$; this rule applies \nindeed  not only to the $^3S_1-{^3D_1}$ coupled waves but to any other one.\n\n\\section{Coupled $^3P_2-{^3F_2}$ waves}\n\\label{pf2w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3p2.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3f2.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e2.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3P_2$, $^3F_2$ and the mixing angle  $\\epsilon_2$, respectively.\nThe (red) hatched areas  correspond to the NNLO results and the (cyan) filled bands are the leading Born approximation results.\n The NLO phase shifts and mixing angle are shown by the (magenta) dot-dashed lines and the LO ones are given by the \n (blue) dotted lines.\n The Nijmegen PWA phase shifts correspond to the (black) dashed lines.\n} \n\\label{fig:3pf2} }\n\\end{center}\n\\end{figure}\n\nWe dedicate this section to the study of the coupled wave system $^3P_2-{^3F_2}$. By direct computation \none has in this case that \n\\begin{align}\n\\lambda_{11}=\\lim_{A\\to -\\infty}\\frac{\\Delta_{11}(A)}{(-A)^{3\/2}}>0~,\n\\label{couplambdap}\n\\end{align}\nwhich requires one to consider DRs with more than one subtraction for the $^3P_2$ wave \\cite{gor2013}.\n Indeed, similarly to the $^3P_0$ and $^3P_1$ partial waves, studied in Secs.~\\ref{3p0} and \\ref{3p1}, respectively,  \nwe need to take at least three subtractions in the DRs for the $^3P_2$ wave in order to obtain \nstable and meaningful results.\n Thus,   we have the following  three-time subtracted DRs for the $^3P_2$ wave, \n\\begin{align}\n\\label{d.3p2}\nD_{11}(A)&=1+\\delta^{(11)}_2 A+\\delta^{(11)}_3 A(A-C)-\\nu^{(11)}_{2}\\frac{A(A-C)^2}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)}{(q^2-A)(q^2-C)^2}\\nonumber\\\\\n&-\\nu^{(11)}_3\\frac{A(A-C)^2}{\\pi}\\int_0^\\infty dq^2\\frac{\\nu_{11}(q^2)q^2}{(q^2-A)(q^2-C)^2}\\nonumber\\\\\n&+\\frac{A(A-C)^2}{\\pi}\\int_{-\\infty}^L dk^2\n\\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^3} g_{11}(A,k^2,C;2)~,\\\\\n\\label{n.3p2}\nN_{11}(A)&=\\nu^{(11)}_2 A+\\nu^{(11)}_3 A^2+\\frac{A^3}{\\pi}\\int_{-\\infty}^L dk^2 \\frac{\\Delta_{11}(k^2)D_{11}(k^2)}{(k^2)^3(k^2-A)}~.\n\\end{align}\nWith respect to the mixing and $^3F_2$  partial waves we use the standard formalism \n for the coupled waves  given in Eqs.~\\eqref{highdcc} and \\eqref{highncc} with $\\ell_{12}=2$ and $\\ell_{22}=3$, respectively. \nAs a result  2 and 3 subtractions are taken in order.\n\n As usual for the $P$ waves, we fix $\\nu_2^{(11)}=4\\pi a_V\/m$  by requiring the exact reproduction  of the $^3P_2$ scattering volume \nextracted from the Nijmegen PWA  \\cite{Stoks:1994wp},\n\\begin{align}\na_V=0.0964~M_\\pi^{-3}~,\n\\end{align}\nwhile $\\nu_3^{(11)}$ is fitted to the results of this PWA. \nRegarding the subtraction constants $\\delta_i^{(11)}$, $i=1,$~2, we  follow the principle of maximal smoothness in virtue of which we  fix $\\delta^{(11)}_{2}=0$\n and fit $D_{11}^{(1)}(-M_\\pi^2)$.\\footnote{In the following we use \n$D_{ij}^{p-2}(-M_\\pi^2)$ as free parameter in terms of which one can calculate $\\delta_{p}^{(ij)}$ from Eq.~\\eqref{taylor}.}  \n The resulting fitted values are: \n\\begin{align}\nD_{11}^{(11)}(-M_\\pi^2)&=0.025(5)~M_\\pi^{-2}~,\\nonumber\\\\\n\\nu_3^{(11)}&=0.155(5)~M_\\pi^{-6}~,\\\\\nD_{22}^{(11)}(-M_\\pi^2)&=0.011(4)~M_\\pi^{-2}~,\n\\end{align}\n with the interval of values reflecting the dependence on the $c_i$'s chosen.\n The free parameter associated with  the mixing wave is fixed to its pure\n perturbative value, cf. Sec.~\\ref{hpw}, $D_{12}(-M_\\pi^2)=1$.\n\nAll in all the resulting phase shifts are shown by the (red) hatched areas in Fig.~\\ref{fig:3pf2}. \nThere we see a clear improvement at NNLO in the reproduction \nof the $^3P_2$ phase shifts  compared with the results at NLO, given by the (magenta) dot-dashed lines, so that now the (red) hatched area overlaps the Nijmegen PWA phase shifts.\n The $^3F_2$ phase shifts and mixing angle $\\epsilon_2$ are reproduced with a similar quality to that already achieved at NLO.\n We also give by the (cyan) filled bands the results obtained by the leading Born approximation, Eq.~\\eqref{deltab}, \nwith $\\Delta(A)$ calculated at NNLO. Due to the fact that the latter diverges as $(-A)^{3\/2}$ for $A\\to-\\infty$ at least two \nsubtractions have to be taken in the DR for $N_B(A)$, Eq.~\\eqref{eq.nborn}. \nThis is immediately accomplished for the $D$ and \nhigher partial waves but for a $P$-wave with $\\ell=1$ one needs to include one \nextra subtraction.\n In particular, for our present case we use Eq.~\\eqref{n.3p2} with $D_{11}(A)\\to 1$ and with $\\Delta_{11}(k^2)$\n restricted to its two-nucleon irreducible contributions, with the subtraction constants $\\nu^{(11)}_{2}$ and $\\nu^{(11)}_{3}$ \ntaking the same values as discussed before. \nWe see that our full results provide a clear improvement in the reproduction of the Nijmegen PWA phase shifts and \nmixing angle  with respect to the Born approximation. \n One should mention that the Born approximation  phase shifts for \n$^3F_2$ and $^3F_3$ have a striking resemblance to the full NNLO results of Ref.~\\cite{thesis} obtained within the Weinberg \nscheme. \n We have obtained this improvement without dismissing the strength of the TPE at NNLO, as advocated in Ref.~\\cite{epe04}. \n This makes that our full results are not so much sensitive to the particular set of $c_i$'s taken as \npreviously thought in the literature from the results of Refs.~\\cite{thesis,epe04}.\n\n\\section{Coupled $^3D_3-{^3G_3}$ waves}\n\\label{dg3w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3d3.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3g3.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e3.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3D_3$, $^3G_3$ and the mixing angle  $\\epsilon_3$, in order.\nThe (red) hatched areas correspond to the NNLO results and the (cyan) filled bands are the leading order Born approximation.\n The NLO results are shown by the (magenta) dot-dashed lines and the LO ones are given by the \n (blue) dotted lines. The Nijmegen PWA phase shifts correspond to the (black) dashed lines.}\n\\label{fig:3dg3} }\n\\end{center}\n\\end{figure}\n\nFor the study of the $^3D_3-{^3G_3}$ coupled waves we follow the formalism for coupled waves, Eqs.~\\eqref{highdcc} and \n\\eqref{highncc}, with $\\ell_{11}=2$, $\\ell_{12}=3$ and $\\ell_{22}=4$, so that $\\ell_{ij}$ subtractions are taken in the DRs for the  \ncoupled wave $ij$. \n Regarding  the free parameters we follow the principle of maximal smoothness, although for the mixing wave \nthe subtraction constants take their pure perturbative values. So that  we fit to data \n$D_{11}(-M_\\pi^2)$ and $D_{22}^{(2)}(-M_\\pi^2)$, with the resulting values:\n\\begin{align}\nD_{11}(-M_\\pi^2)&=0.90(5)~,\\nonumber\\\\\nD_{22}^{(2)}(-M_\\pi^2)&=-0.09(1)~M_\\pi^{-4}~,\n\\label{fit.3dg3}\n\\end{align}\n The interval of values in Eq.~\\eqref{fit.3dg3} reflect the dependence on the set of  values considered for the $c_i$'s. \nThe resulting phase shifts are shown by the (red) hatched areas in Fig.~\\ref{fig:3dg3}.\n Importantly at NNLO the phase shifts for the $^3D_3$ wave   follow  closely the Nijmegen PWA phase shifts \nso that a remarkable improvement is obtained in comparison with both \nthe NLO and Born results.\n  Notice that this is accomplished without any need of dismissing the strength of\n TPE as directly obtained from the NLO $\\pi N$ amplitudes. \nWe have been able to improve the situation by taking into account the subtraction constant $ \\delta_2$ or $D_{11}(-M_\\pi^2)$, \nwhose presence is required by the nonperturbative unitarity implementation\\footnote{In more general terms, by generating the analytical properties \nassociated with the RHC while respecting unitarity in the full amplitudes.} at NNLO, cf. Eq.~\\eqref{summarypwc}.\n  We also observe a good reproduction of the Nijmegen PWA results for the waves $^3G_3$ and $\\epsilon_3$, \nwhich are already well reproduced at NLO \\cite{gor2013} as shown  by the (magenta) dot-dashed lines.\n \n\n\n\n\\section{Coupled $^3F_4-{^3H_4}$ waves}\n\\label{gh4w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3f4.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3h4.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e4.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3F_4$, $^3H_4$ and the mixing angle  $\\epsilon_4$, in order.\nThe (red) hatched areas correspond to the NNLO results and the (cyan) filled ones to the leading\nBorn approximation.\n The NLO results are shown by the (magenta) solid line and the LO ones are given by the \n (blue) dotted lines. \nThe Nijmegen PWA phase shifts are given by the (black) dashed lines.}\n\\label{fig:3fh4} }\n\\end{center}\n\\end{figure}\n\nThe discussion of the $^3F_4-{^3H_4}$ coupled-wave system follows the  standard formalism for coupled waves, Eq.~\\eqref{highdcc} and \\eqref{highncc},\n with $\\ell_{11}=3$, $\\ell_{12}=4$ and $\\ell_{22}=5$. The free parameters are then fitted to data according to the \n principle of maximal smoothness. However, for $^3H_4$ and the mixing partial wave there is no improvement in the reproduction of data\n with respect to the situation in which the pure perturbative values are taken, so that \nat the end we only have to fit $D_{11}^{(1)}(-M_\\pi^2)$  to the Nijmegen PWA results. \nThe fitted value is\n\\begin{align}\nD_{11}^{(1)}(-M_\\pi^2)&= -0.009(3)~M_\\pi^{-2}~.\n\\label{free.fh4}\n\\end{align}\n The resulting phase shifts and mixing angle are shown by the (red) hatched areas  in  Fig.\\ref{fig:3fh4},\nwith the width of the band reflecting the dependence on values for the $\\pi N$ NLO counterterms. \nOne can observe  a clear improvement in the description of the $^3F_4$ phase shifts compared with the results from OPE (blue dotted lines), NLO \n(magenta dot-dashed lines) and leading Born approximation (cyan filled areas). \n Similarly to the $^3D_3$ wave in the previous section, \n this improvement is related with the effect of the subtraction constant $\\delta_3^{(11)}$ which  \nis not directly related with an improvement in  the \ncalculation of $\\Delta_{11}(A)$, and hence of the $NN$ potential. \nLet us recall that the subtraction constants $\\delta_p^{(ij)}$ arise because of the rescattering process that the \n$N\/D$  method allows to treat in a clear and well-defined way, \novercoming the obscurities that still remain in the literature associated with the use of the cutoff regularized Lippmann-Schwinger  with \na higher-order $NN$ potential. \nFor the  mixing angle $\\epsilon_4$ the quality in the reproduction of data is  similar to that obtained \nby the other approximations just quoted. \nHowever, for the  $^3 H_4$ phase shifts the outcome at NNLO is a bit worse than at \nNLO and OPE, though one should also notice the tiny values for the $^3H_4$ phase shifts so that this discrepancy \nis certainly small  in absolute value. \nWe have also checked that it cannot be removed by releasing the other subtraction constants $\\delta_p^{(22)}$, with \n$p=2$, 3 and 4. \nLikely, the origin of this difference in the $^3H_4$ phase shifts  between our full results and the Nijmegen PWA \ncan be tracked back to the change in the leading Born approximation once the ${\\cal O}(p^3)$ two-nucleon irreducible \ncontributions are included in $\\Delta_{22}(A)$.\n\n\n\\section{Coupled $^3G_5-{^3I_5}$ waves}\n\\label{gi5w}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=.4\\textwidth]{.\/3g5.ps} & \n\\includegraphics[width=.4\\textwidth]{.\/3i5.ps}\\\\  \n\\includegraphics[width=.4\\textwidth]{.\/e5.ps}  \n\\end{tabular}\n\\caption[pilf]{\\protect { (Color online.) From top to bottom and left to right: Phase shifts for $^3G_5$, $^3I_5$ and the mixing angle  $\\epsilon_5$, in order.\nThe (red) hatched areas correspond to the NNLO results and the filled ones to the leading\nBorn approximation.\n The NLO results are shown by the (magenta) dot-dashed line and the LO ones are given by the \n (blue) dotted lines.\n The Nijmegen PWA phase shifts are shown by the (black) dashed lines.}\n\\label{fig:3gi5} }\n\\end{center}\n\\end{figure}\n\n\nThe standard formalism for coupled waves with high angular momentum, Eqs.~\\eqref{highdcc} and \\eqref{highncc}, is followed here \nwith  $\\ell_{11}=4$, $\\ell_{12}=5$ and $\\ell_{22}=6$.\n The application of the  principle of maximal smoothness \nto fit the free parameters  provides a good reproduction of the Nijmegen PWA phase shifts \\cite{Stoks:1994wp}.\\footnote{For $5\\leq J\\leq 8$ the Nijmegen PWA phase shifts \\cite{Stoks:1994wp} are those obtained from the $NN$ potential model of Ref.~\\cite{obe}.}\nThe range of values obtained for the free parameters $D_{11}^{(2)}(-M_\\pi^2)$ and $D_{22}^{4}(-M_\\pi^2)$ is\n\\begin{align}\nD_{11}^{(2)}(-M_\\pi^2)&=-0.0025(5)~M_\\pi^{-4}~,\\nonumber\\\\\nD_{22}^{(4)}(-M_\\pi^2)&=-0.0125(5)~M_\\pi^{-8}~,\n\\label{free.gi5}\n\\end{align}\nwhile basically the same results are obtained for any  $D_{12}^{(3)}(-M_\\pi^2)\\leq 0~M_\\pi^{-6}$. \n The results are shown in Fig.\\ref{fig:3gi5} by the (red) hatched areas whose widths take into account the \nuncertainty from the set of $c_i$'s   taken and some numerical noise from the iterative process. \n A clear improvement results in the description of the $^3G_5$ phase shifts compared with the OPE (blue dotted lines), NLO (magenta dot-dashed lines)\n and leading Born approximation results (cyan filled areas). \nIt is worth stressing that this partial wave cannot be well reproduced\neven at NNNLO in the Weinberg potential scheme  neither by \nkeeping a finite value for the three-momentum cutoff entering \nin the solution of the Lippmann-Schwinger equation \\cite{epe042}, \nnor by  sending it to $\\infty$ as in Ref.~\\cite{zeoli}.\n A similar situation occurs too  \nfor the leading Born approximation results at NNLO, as shown by the (cyan) filled area in the first panel, \n a result also obtained in Ref.~\\cite{epe04}. \nEven more, the modification of the TPE mechanism proposed in this reference by making use of the \nso-called spectral-function regularization  is inoperative here to provide \nan improvement in the Born approximation results.\n A similar problem was also observed in the perturbative calculation at NNNLO in \nRef.~\\cite{entem}.\n From ours results this is not surprising because the improvement in the reproduction of the  \nNijmegen PWA phase shifts for the $^3G_5$ wave is accomplished through the subtraction constant $\\delta_4^{(11)}$. \nThis constant is directly related to the $NN$ rescattering (from which the final function $D_{11}(A)$ stems nonperturbatively) \nand not to the $NN$ potential or $\\Delta_{22}(A)$.\n In the case of the mixing angle $\\epsilon_5$ and the \n$^3I_5$ phase shifts  there is a slight worsening in the reproduction of Nijmegen \nPWA compared with the NLO ones, but still our results run very close to the Nijmegen PWA ones.\n\n\n\\begin{table}\n\\begin{center}\n{\\small\n\\begin{tabular}{|l|l|l|}\n\\hline\nWave & Type of DRs & Parameters \\\\\n\\hline\n$^1S_0$ & 1DR & $\\nu_1=30.69$ \\\\\n       &  2DR & $\\nu_1=30.69$~,~$\\nu_2=-23(1)$, $\\delta_2=-8.0(3)$ \\\\\n\\hline\n$^3P_0$ & 3DR & $\\nu_2=1.644$~,~$\\delta_2=2.82(5)$~,~$\\delta_3=0.18(6)$ \\\\\n\\hline\n$^3P_1$ & 3DR & $\\nu_2=-1.003$~,~$\\delta_2=2.7(1)$~,~$\\delta_3=0.47(3)$ \\\\\n\\hline\n$^1 P_1$ & 2DR & $\\nu_2=-1.723$~,~$\\delta_2=0.4(1)$ \\\\\n\\hline\n$^1D_2$ &  LTS & $D^{(1)}(0)=0.07(1)$  \\\\\n\\hline\n$^3D_2$ & LTS &   $D^{(1)}(0)=-0.017(3)$  \\\\\n\\hline\n$^1F_3$ & LTS & $D^{(2)}(0)=0.057(3)$  \\\\\n\\hline\n$^3F_3$ & LTS & $D^{(2)}(0)= 0.035(5)$  \\\\\n\\hline\n$^1G_4$ & LTS & $D^{(3)}(0)=-0.014(2)$  \\\\\n\\hline\n$^3G_4$ & LTS & $D^{(3)}(0)=-0.055(5)$  \\\\\n\\hline\n$^1H_5$ & LTS & $D^{(4)}(0)=0.156$  \\\\\n\\hline\n$^3H_5$ & LTS & $D^{(4)}(0)=0.066$  \\\\\n\\hline\n$^3S_1-{^3D_1}$ & $1$DR $^3S_1$, 2DR $^3D_1$, mixing & $E_d$ \\\\\n               & 2DR all & $a_t$, $r_t$, $E_d$ \\\\\n               & 2DR $^3S_1$, mixing, 3DR $^3D_1$ &  $a_t$, $r_t$, $E_d$, $\\nu_3^{(22)}=-2.05(5)$ \\\\  \n\\hline\n$^3P_2-{^3F_2}$ & 3DR for $^3P_2$ and LTS for the others & $\\nu^{(11)}_2=0.178$~,~$D^{(1)}_{11}(-M_\\pi^2)=0.025(5)$~,~\n$\\nu_3^{(11)}=0.155(5)$\\\\\n               &  &  $D_{22}(-M_\\pi^2)=0.011(4)$\\\\\n\\hline\n$^3D_3-{^3G_3}$ & LTS & $D_{11}(-M_\\pi^2)=0.90(5)$~,~$D^{(2)}_{22}(-M_\\pi^2)=-0.09(1)$  \\\\\n\\hline\n$^3F_4-{^3H_4}$ & LTS & $D_{11}^{(1)}(-M_\\pi^2)=-0.009(3)$  \\\\\n\\hline\n$^3G_5-{^3I_5}$ & LTS & $D_{11}^{(2)}(-M_\\pi^2)=-0.0025(5)$~,~$D_{22}^{(4)}(-M_\\pi^2)=-0.0125(5)$ \\\\\n\\hline\n\\end{tabular} }\n\\caption[pilf]{\\protect {  We give in the columns from left to right, in order,  \nthe partial wave, the type of DRs employed to study it\n and the values for the free parameters involved.  }\n\\label{tab:allparam} }\n\\end{center}\n\\end{table}\n\nFinally, we give in Table~\\ref{tab:allparam} the values of the free parameters employed in the different partial waves according \nto the type of DRs employed, which is  indicated in the second column.\n This is done by following the notation, already introduced in Ref.~\\cite{gor2013}, \n $m$DR with $m=1,2,\\ldots$, and it should be read as $m$-time subtracted DR.\n For the higher $NN$ partial waves we use the abbreviation  LTS to indicate that \n$\\ell$ (or $J$ for the mixing partial waves) subtractions are taken  to satisfy the threshold behavior, following \nthe standard formalism explained in Sec.~\\ref{hpw}.\n According to the \nprinciple of maximal smoothness only the highest derivative $D^{(n)}(C)$ \nrelated to the subtraction constants in $D(A)$ is not fixed to its perturbative value \n(1 for $n=0$ and 0 for $n\\neq 0$) and released, \nif appropriate.  The units correspond to appropriate powers of $M_\\pi^2$, although they are not explicitly shown. \nThere is a proliferation of free parameters for the $P$ waves because for them $\\lambda>0$, Eqs.~\\eqref{unlambdap} \nand \\eqref{couplambdap}, so that, except for the $^1P_1$ wave,  three-time-subtracted DRs are needed.\n This could be a specific feature for the NNLO calculation of $\\Delta(A)$ that has to be investigated for\n higher-orders.\\footnote{If then $\\lambda<0$ \none would need to invoke  less free parameters for the $P$ waves than in Table~\\ref{tab:allparam}.}\n \n\n\\section{Conclusions}\n\\label{conc}\n\nWe have discussed in this paper the application of the $N\/D$ method when its dynamical input, namely, \nthe imaginary part of the $NN$ partial waves along the LHC, is calculated in ChPT up to NNLO.  \nIt then comprises OPE, leading and subleading two-nucleon irreducible TPE and once-iterated OPE ~\\cite{peripheral}. \nWe have obtained a quite good reproduction of the Nijmegen PWA phase shifts and mixing angles, in better agreement \nthan the one achieved in the previous lower order studies at LO \\cite{paper1,paper2} and NLO \\cite{gor2013}.   In particular, our NNLO results are able to reproduce \n the phase shifts for the triplet waves with $\\ell_{11}=J-1$, $^3P_2$, $^3D_3$, $^3F_4$ \nand $^3G_5$,  while at NLO they were not properly accounted for.\nWe do not need to modify the NNLO two-nucleon irreducible diagrams (or chiral $NN$ potential) in order to obtain such a good agreement \nwith the Nijmegen PWA, contrary to common wisdom.  \nThe point that stems from our study is that one should perform in a well-defined way\n the iteration of diagrams along the RHC, which are responsible \nfor unitarity and analyticity attached to this cut, rather than reshuffling  the $NN$ potential with contributions  \nfrom higher orders. \nIn this respect, the use of DRs allows one to perform the iteration of two-nucleon intermediate states independently of regulator. We have also compared our full results for the higher partial waves with the Born approximation. From this comparison, as well as from \nthe direct study \nof the importance of the different contributions of $\\Delta(A)$ to the dispersive integrals,  \nit follows that  the $NN$  $D$ waves cannot be treated perturbatively.\n\nIt is also worth remarking that up to the order studied here we reproduce the long-range correlation between the \neffective ranges and the scattering lengths for the $NN$ $S$ waves when only once-subtracted DRs are applied. \nIn this way one can predict values for the $S$-wave effective ranges in agreement with experiment up to around a $10 \\%$. \nWe have also elaborated a chiral power counting for the subtraction constants, so that twice-subtracted DRs \nare appropriate when $\\Delta(A)$ is calculated at NLO and NNLO.  \nFrom these considerations it turns out also that the chiral power expansion is made over a scale $\\Lambda\\sim 400$~MeV.  \nOne should consider further  the impact of higher orders in $\\Delta(A)$, which are partially calculated already in \nthe literature, as an interesting extension of the present work in order to settle the applicability of the $N\/D$\nmethod to $NN$ scattering in ChPT with a high degree of accurateness.\n\n\n\n\\section*{Acknowledgments}\n This work is partially funded by the grants MINECO (Spain) and ERDF (EU), grant FPA2010-17806 and the Fundaci\\'on S\\'eneca 11871\/PI\/09.\n We also thank the financial support from the EU-Research Infrastructure\nIntegrating Activity\n ``Study of Strongly Interacting Matter\" (HadronPhysics2, grant n. 227431)\nunder the Seventh Framework Program of EU and   \nthe Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we study  the following problem raised by Coman-Guedj-Zeriahi.\n\\begin{prob}[\\cite{CGZ13}]\\label{prob:1}\nLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, equipped with a K\\\"ahler metric $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $k>0$. Is the following holds\n\\begin{align*}\n\\mbox{Psh}(V,\\omega|_V)=\\mbox{Psh}(X,\\omega)|_V?\n\\end{align*}\n\\end{prob}\n\nRecently, there are many progress towards this problem. \n\\begin{itemize}\n\\item When $\\omega$ is a Hodge metric and $\\varphi$ is a smooth quasi-psh function on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>0$, then Problem \\ref{prob:1} has a positive answer by Schumacher \\cite{Sch98}.\n\\item When $\\omega$ is a Hodge metric, then Problem \\ref{prob:1} has a positive answer by Coman-Guedj-Zeriahi \\cite{CGZ13}, and when $\\omega$ is a K\\\"ahler metric and $\\varphi$ is a smooth quasi-psh function on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>0$, then Problem \\ref{prob:1} has a positive answer in the same paper \\cite{CGZ13}.\n\\item When  $\\omega$ is a K\\\"ahler metric and $\\varphi$ is a  quasi-psh function  on $V$, which has analytic singularities, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>\\epsilon\\omega|_V$ for some $\\epsilon>0$, there is a quasi-psh function $\\Phi$ on $X$, such that $\\Phi|_V=\\varphi$ and $\\omega+\\sqrt{-1}\\partial\\bar\\partial \\Phi>\\epsilon'\\omega$ on $X$ by Collins-Tosatti \\cite{CT14}.\n\\item When  $\\omega$ is a K\\\"ahler metric and $\\varphi$ is a  quasi-psh function with arbitrary singularity  on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>\\epsilon\\omega|_V$ for some $\\epsilon>0$. Suppose that $V$ has a holomorphic tubular neighborhood in $X$, then there is a quasi-psh function $\\Phi$ on $X$, such that $\\Phi|_V=\\varphi$ and $\\omega+\\sqrt{-1}\\partial\\bar\\partial \\Phi>\\epsilon'\\omega$ on $X$ for some $\\epsilon'>0$ by Wang-Zhou \\cite{WZ20}.\n\\end{itemize}\n\nThe main theorem of this paper is as follows.\n\\begin{thm}\\label{thm:main}\nLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, equipped with a K\\\"ahler metric $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $k>0$. Let $\\varphi$ be  a  quasi-psh function with arbitrary singularity  on $V$, such that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial \\varphi>\\epsilon\\omega|_V$ for some $\\epsilon>0$.  Suppose that there is a open neighborhood $U$ of $V$ in $X$, and  a holomorphic retraction $\\pi:U\\rightarrow V$. Then  there is a quasi-psh function $\\Phi$ on $X$, such that $\\Phi|_V=\\varphi$ and $\\omega+\\sqrt{-1}\\partial\\bar\\partial \\Phi>\\epsilon'\\omega$ on $X$  for some $\\epsilon'>0$.\n\\end{thm}\n\n\\begin{rem}\nThe main theorem is a slightly stronger than the result in \\cite{WZ20}, by weakening the assumption that $V$ has a holomorphic tubular neighborhood structure in $X$ to the assumption that $V$ has a holomorphic retraction structure in $X$. By a holomorphic retraction, we mean that there is an  open neighborhood $U$ of $V$ in $X$, and  a holomorphic map $\\pi:U\\rightarrow V$, such that $\\pi|_V:V\\rightarrow V$ is the identity map.  Without the holomorphic tubular neighborhood structure,  we need to compute the complex Hessian of the square of the distance function to $V$ on $X$. The main advantage of this generalization is that there many examples of compact  K\\\"ahler  manifolds  which are  not necessarily projective.\n\\end{rem}\nWe also consider the extension of K\\\"ahler currents in a big class.\n\\begin{thm}\nLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, and $V\\subset X$ be a complex submanifold of positive dimension. Suppose that $V$ has a holomorphic retraction structure  in $X$.\nLet $\\alpha\\in H^{1,1}(X,\\mathbb R)$ be a big class and $E_{nK}(\\alpha)\\subset V$. \nThen any  K\\\"ahler current in  $\\alpha|_V$ is the restriction of a K\\\"ahler current in $\\alpha$.\n\\end{thm}\n\n\\subsection*{Acknowledgement}\nThe second author would like to thank G. Hosono and T. Koike for helpful discussions, and especially T. Koike for  sharing the note \\cite{Ko21}.\n\n\n\n\n\n\n\n\\section{Complex Hessian of square of  distance to  a complex  submanifold}\\label{sect:dist}\nWe follow  Matsumoto's notations in \\cite{Ma}.\nLet $(M,g)$ be a $C^\\infty$ Rimannian manifold of dimension $n$.\nFor $x,y\\in M$, we denote by $\\delta(x,,y)$ the distance between $x$ and $y$ induced by the metric $g$.\n\nIt is known that for any $p\\in M$, there is an open coordinate neighborhood $U$ of $p$,\nchoose a coordinate $x_1,x_2,\\cdots,x_n$ on $U$, with $x(p)=0$ and\n$g_{ij}=g(\\frac{\\partial}{\\partial x_i},\\frac{\\partial}{\\partial x_j}),\\, i,j=1,2,\\cdots,n$.\nFor $v=(v_1,v_2,\\cdots,v_n)\\in\\mathbb{R}^n$,\nwe may view $v\\in T_xM$ as $\\sum_{j=1}^{n}v_j\\frac{\\partial}{\\partial x_j}|_x$.\nWe may shrink $U$ if necessary, there is an open neighborhood $B\\subset \\mathbb{R}^n$ of $0$,\nsuch that $\\Phi(x,v)=(x,\\exp_x(v))$\nis bijection from $U\\times B$ to $\\Phi(U\\times B)$, both $\\Phi$ and $\\Phi^{-1}$ are $C^\\infty$.\nAs $\\Phi(x,0)=(x,x)$, and from the property of exponential mapping $y=\\exp_x v$,\nwe can get\n\\begin{equation}\\label{eq1}\nJ\\Phi(0,0)\n=\\left[\n\\begin{array}{cc}\nI & 0\\\\\nI & I\n\\end{array}\n\\right].\n\\end{equation}\n\nAs $\\Phi(U\\times B)$ is an open neighborhood of $(p,p)$,\nwe may take an open set $V\\subset U$, such that $p\\in V$, and  $\\Phi(U\\times B)\\supset V\\times V$.\nWrite $(x,v(x,y))=\\Phi^{-1}(x,y)$, then\n$$y=\\exp_x(v(x,y)),\\, \\delta(x,y)^2=\\sum_{i,j=1}^n g_{ij}(x)v_i(x,y)v_j(x,y)$$\nand from (\\ref{eq1}), we have\n\\begin{equation}\\label{eq2}\nv(0,0)=0, \\,\\, \\frac{\\partial v_i}{\\partial y_j}(0,0)=-\\frac{\\partial v_i}{\\partial x_j}(0,0)=\\delta_{ij},\\, 1\\leq i,j\\leq n.\n\\end{equation}\nLet $S\\in M$ be a $C^\\infty$ submanifold of $M$ with $dim S=k$, $0\\leq r<n$, define\n$$d(x)=\\delta(x,S)=inf\\{\\delta(x,y):y\\in S\\}, \\; x\\in M.$$\nLet $h(x)=d(x)^2$, it is  proved in \\cite{Ma} that $h(x)$ is $C^\\infty$ in a tube neighborhood of $S$.\nAs we need do calculation on $h$, and for completeness, we introduce Matsumoto's proof of smoothness.\n\nGiven $p\\in S$, we can choose a coordinate $(U,x=(x_1,x_2,\\cdots,x_n))$ around $p$, such that\n$x(p)=0$, $S\\cap U=\\{x_{k+1}=\\cdots=x_n=0\\}$, and\n\\begin{equation}\\label{eq3}\ng_{ij}(0)=g(\\frac{\\partial}{\\partial x_i},\\frac{\\partial}{\\partial x_j})(0)=\\delta_{ij},\\, i,j=1,2,\\cdots,n.\n\\end{equation}\nTake a small neighborhood $V\\subset U$ of $p$, such that $d(x)=\\delta(x,S\\cap U)$ for any $x\\in V$.\nLet $$f(x,y)=\\delta(x,y)^2=\\sum_{i,j=1}^{n}g_{ij}(x)v_i(x,y)v_j(x,y)$$\nand $$F_\\mu(x,y)=\\frac{\\partial f}{\\partial y_\\mu}(x,y),\\; \\mu=1,2,\\cdots,k$$\nfor $x\\in V$ and $y=(y_1,\\cdots,y_k,0,\\cdots,0)\\in S\\cap U$. Notice that $F_\\mu(0,0)=0, \\mu=1,2,\\cdots,k$.\n\n\nFrom (\\ref{eq2}), (\\ref{eq3}), we can get\n\\begin{equation}\\label{eq4}\n\\begin{split}\n&\\frac{\\partial F_\\mu}{\\partial y_\\nu}(0,0)\\\\\n=&2\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial v_i}{\\partial y_\\mu}(0,0)\\frac{\\partial v_j}{\\partial y_\\nu}(0,0)\\\\\n=&2\\delta_{\\mu\\nu}.\n\\end{split}\n\\end{equation}\n\nTherefore, from the implicit function theorem, we can find a neighborhood $V_0\\subset V$ of $p$,\nso that each $x\\in V_0$ has a unique solution $y=y(x)\\in S\\cap U$ of equations $F_\\mu(x,y)=0, \\mu=1,2,\\cdots,n$,\n$y(0)=0$ and $y=y(x)$ is $C^\\infty$ on $V_0$. As for each $x\\in V_0$, there is at least one point $y\\in S\\cap U$, \nsuch that $d(x)=\\delta(x,S\\cap U)=\\delta(x,y)$. Therefore, the point $y$ is uniquely determined by $x$ and \nit must coincide to $y(x)$ because $f(x,y)=\\delta(x,y)^2$ is minimal at $y=y(x)$ for each $x$.\n\n\nHence $$h(x)=\\delta(x,S\\cap U)^2=\\delta(x,y(x))^2=f(x,y(x))$$\nfor $x\\in V_0$.\n\nFrom $F_\\mu(x,y(x))=0$, taking the partial derivatives of this equation, we can get \n$$\\frac{\\partial F_\\mu}{\\partial x_j}(0,0)+\\sum_{\\nu=1}^{k}\\frac{\\partial F_\\mu}{\\partial y_\\nu}(0,0)\n\\frac{\\partial y_\\nu}{\\partial x_j}(0)=0.$$ \nCombining with  equation (\\ref{eq4}),\nwe can get $\\frac{\\partial y_\\mu}{\\partial x_j}(0)=-\\frac{1}{2}\\frac{\\partial F_\\mu}{\\partial x_j}(0,0)$\nfor $\\mu=1,2,\\cdots,k$ and $j=1,2,\\cdots,n$.\n\nFrom (\\ref{eq2}), (\\ref{eq3}), we have\n\n\\begin{equation*\n\\begin{split}\n&\\frac{\\partial F_\\mu}{\\partial x_l}(0,0)\\\\\n=&\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial v_i}{\\partial y_\\mu}(0,0)\\frac{\\partial v_j}{\\partial x_l}(0,0)+\n\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial v_i}{\\partial x_l}(0,0)\\frac{\\partial v_j}{\\partial y_\\mu}(0,0)\\\\\n=&-2\\delta_{\\mu l}.\n\\end{split}\n\\end{equation*}\n\nHence, \n\\begin{equation}\\label{eq d}\n\\frac{\\partial y_\\mu}{\\partial x_j}(0)=\\delta_{\\mu j}\n\\; \\text{for}\\; \\mu=1,2,\\cdots,k  ; j=1,2,\\cdots,n.\n\\end{equation}\n\nLet $a_j(x)=v(x,y(x)), j=1,2,\\cdots,n$, then $a_j(0)=0$, and \n\\begin{equation}\\label{eq6}\n\\begin{split}\n\\frac{\\partial a_j}{\\partial x_l}(0)&=\\frac{\\partial v_j}{\\partial x_l}(0,0)+\n\\sum_{\\mu=1}^{k}\\frac{\\partial v_j}{\\partial y_\\mu}(0,0)\\frac{\\partial y_\\mu}{\\partial x_k}(0)\\\\\n&=\\sum_{\\mu=1}^{k}\\delta_{j\\mu}\\delta_{\\mu k}-\\delta_{jk}.\n\\end{split}\n\\end{equation}\nAs $h(x)=\\sum_{i,j=1}^{n}g_{ij}(x)v_i(x,y(x))v_j(x,y(x))=\\sum_{i,j=1}^{n}g_{ij}(x)a_i(x)a_j(x)$, from (\\ref{eq6}), then\n\\begin{equation}\\label{eq 7}\n\\begin{split}\n\\frac{\\partial^2h}{\\partial x_s\\partial x_t}(0)\n&=2\\sum_{i,j=1}^{n}g_{ij}(0)\\frac{\\partial a_i}{\\partial x_s}(0)\\frac{\\partial a_j}{\\partial x_t}(0)\\\\\n&=\n\\begin{cases}\n0 \\quad\\quad s \\; \\text{or} \\; t\\leq k \\\\\n2\\delta_{st} \\quad s,t>k\n\\end{cases}\n\\end{split}\n\\end{equation}\n\nNow we let $(X,\\omega)$ be a compact Hermitian manifold with  a Hermitian metric $\\omega$. Let $g$ be the Riemannian metric on $X$ induced by $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $r>0$.  Fix any  $p\\in V$. There is a holomorphic coordinate  $(U, z=(z_1,\\cdots, z_k, z_{k+1}, \\cdots,z_n))$ centered at $p$ in $X$, such that $U\\cap V=\\{z_{k+1}=\\cdots=z_n=0\\}$, and $g_{ij}(0)=\\delta_{ij}$ for $i,j=1,\\cdots, 2n$, here we write $z_i=x_{2i-1}+\\sqrt{-1}x_{2i}$. Since  $$\\frac{\\partial}{\\partial z_i}=\\frac{1}{2}(\\frac{\\partial}{\\partial x_{2i-1}}-\\sqrt{-1}\\frac{\\partial}{\\partial x_{2i}}), \\frac{\\partial}{\\partial\\bar z_i}=\\frac{1}{2}(\\frac{\\partial}{\\partial x_{2i-1}}+\\sqrt{-1}\\frac{\\partial}{\\partial x_{2i}}),$$\nwe get that \n\\begin{align}\\label{equ: chessian}\n\\frac{\\partial^2h}{\\partial z_i\\partial \\bar z_j}=\\frac{1}{4}\\left(\\frac {\\partial^2h}{\\partial x_{2i-1}\\partial x_{2j-1}}       +\\frac{\\partial^2h}{\\partial x_{2i}\\partial x_{2j}}-\\sqrt{-1}\\frac{\\partial^2h}{\\partial x_{2j}\\partial x_{2j-1}} +\\sqrt{-1}\\frac{\\partial^2h}{\\partial x_{2j-1}\\partial x_{2j}}    \\right).\n\\end{align}\n\nCombining (\\ref{eq 7}) and (\\ref{equ: chessian}), we obtain the following \n\n\\begin{prop}\\label{prop:hessian}Let $(X,\\omega)$ be a complex $n$-dimensional  Hermitian manifold with a Hermitian metric $\\omega$. Let $V\\subset X$ be a complex submanifold of complex dimension $k$. Let $p\\in V$ be an arbitrarily fixed point in $V$, then there is a holomorphic coordinate chart  $(U,z=(z_1,\\cdots,z_k,z_{k+1},\\cdots, z_n))$ centered at $p$ such that $U\\cap V=\\{z_{k+1}=\\cdots=z_n=0\\}$ and \n\t\\begin{align*}\n\t\\frac{\\partial^2h}{\\partial z_i\\partial \\bar z_j}(0)=\n\t\\begin{cases}\n\t0 \\quad\\quad i \\; \\text{or} \\; j\\leq k \\\\\n\t2\\delta_{ij} \\quad i,j>k.\n\t\\end{cases}\n\t\\end{align*}\n\t\\end{prop}\n\n\\section{Proof of the main theorem}\n\nIn this section, we give the proof of Theorem \\ref{thm:main}. The idea of the proof is similar with that in \\cite{WZ20}. The main difference lies in the construction of the local uniform extension. For the sake of completeness, we give the detailed proof.\n\n\\begin{lemma}[\\cite{BK07, WZ20}]\\label{key lemma}\n\tLet $\\varphi$ be a quasi-psh function on a compact Hermitian  manifold $(X,\\omega)$,  such that $\\omega+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi\\geq \\varepsilon\\omega$ and $\\varphi<C<0$.\n\tThen there is a  sequence of smooth functions $\\varphi_m$ and a decreasing sequence $\\varepsilon_m>0$ converging to $0$, satisfying the following\n\t\\begin{itemize}\n\t\t\\item [(a)] $\\varphi_m\\searrow \\varphi$;\n\t\t\\item [(b)]$\\omega+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi_m\\geq (\\varepsilon-\\varepsilon_m)\\omega$;\n\t\t\\item[(c)] $\\varphi_m\\leq -\\frac{C}{2}$.\n\t\\end{itemize}\n\\end{lemma}\n\n\\begin{lemma}[c.f. \\cite{DP04}]\\label{reference function}\n\tThere exists a function $F:X\\rightarrow [-\\infty, +\\infty)$ which is smooth on $X\\setminus V$, with logarithmic singularities along $V$, and such that $\\omega+\\sqrt{-1}\\partial\\bar{\\partial}F\\geq \\varepsilon \\omega$ is  a K\\\"{a}hler current on $X$.\n\tBy subtracting a large constant, we can make that $F<0$ on $X$.\n\\end{lemma}\n\n\nLet $T=\\omega|_V+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi\\geq \\varepsilon\\omega|_V$ be the given K\\\"{a}hler current  in the K\\\"{a}hler class $[\\omega|_V]$, where $\\varphi$ is a strictly $\\omega|_V$-psh function.\nBy subtracting a large constant, we may assume that $\\sup_V \\varphi<-C$ for some positive constant $C$.\n\nBy  Lemma \\ref{key lemma}, we have that there is a sequence of non-increasing  smooth strictly $\\omega|_V$-psh functions $\\varphi_{m}$ on $V$,\nand a decreasing sequence of positive numbers $\\varepsilon_m$ such that as $m\\rightarrow \\infty$\n\\begin{itemize}\n\n\t\\item $\\varphi_{m} \\searrow \\varphi$;\n\t\\item$\\omega|_V+\\sqrt{-1}\\partial\\bar{\\partial}\\varphi_m> \\frac{\\varepsilon}{2}\\omega|_V$;\n\t\\item $\\varphi_m\\leq -\\frac{C}{2}$.\n\\end{itemize}\n\nWe say a smooth strictly $\\omega|_V$-psh function $\\phi$ on $V$ satisfies \\textbf{assumption $\\bigstar_{\\varepsilon, C}$}, if $\\omega|_V+\\sqrt{-1}\\partial\\bar{\\partial}\\phi>\\frac{\\varepsilon}{2}\\omega|_V$ and $\\phi<-\\frac{C}{2}$.\n\nNote that for all $m\\in \\mathbb{N}^+$, $\\varphi_m$ satisfy  \\textbf{assumption $\\bigstar_{\\varepsilon, C}$}.  In the following, we will extend all the $\\varphi_m$ simultaneously to non-increasing  strictly $\\omega$-psh functions on   the ambient manifold $X$.\n\n\\noindent\\textbf{Step1: Local uniform extensions of $\\varphi_m$ for all $m$.} Let $U\\subset X$ be an open neighborhood of $V$ and let $r:U\\rightarrow V$ be a holomorphic retraction. Let $\\phi$ be a function satisfying  \\textbf{assumption $\\bigstar_{\\varepsilon, C}$}.  Let $h$ be the square of the distance function , which is a smooth function defined in \\S\\ref{sect:dist}.\n We define \t\n\\begin{align*}\n\\bar{\\phi}:=\\phi\\circ r+Ah\n\\end{align*}\nwhere $A$ is a positive constant to be determined later. \n\nFix arbitrary $p\\in V$, choose a holomorphic coordinate chart $(W_p,z=(z_1,\\cdots,z_n))$ centered at $p$ and $W_p\\cap V=\\{z_{k+1}=\\cdots=z_n=0\\}$, $g_{i\\bar j}(0)=\\delta_{ij}$  and $W\\subset U$, where $\\omega=\\sum_{i,j=1}^ng_{i\\bar j}dz_i\\wedge d\\bar z_j$. Then on $W_p$, we have that \n\\begin{align*}\n\\bar{\\phi}(z):=(\\phi\\circ r)(z)+Ah(z).\n\\end{align*}\nNote that on $W_p$, \n\\begin{align*}\n\\omega+\\sqrt{-1}\\partial\\bar\\partial \\bar\\phi(z)&=(\\omega-r^*(\\omega|_V))+r^*(\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial\\phi)+A\\sqrt{-1}\\partial\\bar\\partial h\\\\\n&\\geq (\\omega-r^*(\\omega|_V))+\\varepsilon r^*(\\omega|_V)+A\\sqrt{-1}\\partial\\bar\\partial h\n\\end{align*}\nThe second inequality follows from the fact that $\\omega|_V+\\sqrt{-1}\\partial\\bar\\partial\\phi\\geq \\varepsilon\\omega|_V$  on $V$ and $r$ is a holomorphic retraction map.  The key point is that the last term in above inequality  is independent of $\\phi$.\n\n\n\n\\begin{claim}\\label{claim: 1}There is  an open neighborhood  $W_p$ (independent of $\\phi$), of $p$ in $U$, and  positive constants $A>0$ and  $\\varepsilon'>0$ (independent of $\\phi$), such that on $W_p$,\n\t\n\\begin{align*}\n\\omega+\\sqrt{-1}\\partial\\bar\\partial \\bar\\phi\\geq \\frac{\\varepsilon'}{2}\\omega \\mbox{~~and~~} \\bar\\phi\\leq -\\frac{C}{4}.\n\\end{align*}\n\\end{claim}\n\\begin{proof}Under the local coordinate chosen as above, one can see that \n\t\\begin{align*}\n\t&r(z)=(r_1(z_1,\\cdots,z_n),\\cdots, r_k(z_1,\\cdots,z_n),0,\\cdots,0);\\\\\n&\tr(z_1,\\cdots, z_k,0,\\cdots, 0)=(z_1,\\cdots,z_k,0,\\cdots,0);\\\\\n&dr_i(z_1,\\cdots,z_k,0,\\cdots,0)=dz_i+\\sum_{k+1\\leq j\\leq n}\\frac{\\partial r_i}{\\partial z_j}(z_1,\\cdots, z_k,0\\cdots,0)dz_j.\n\t\\end{align*}\nSince $\\omega|_V=\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}(z_1,\\cdots,z_k,0,\\cdots,0)dz_i\\wedge d\\bar z_j$, it follows that at $(z_1,\\cdots, z_k,0,\\cdots,0)$, \n\\begin{align*}\nr^*(\\omega|_V)=&\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}(dz_i+\\sum_{k+1\\leq l\\leq n}\\frac{\\partial r_i}{\\partial z_l}dz_l)\\wedge(d\\bar z_j+\\sum_{k+1\\leq m\\leq n}\\frac{\\partial \\bar r_j}{\\partial \\bar z_m} d\\bar z_m)\\\\\n=&\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}dz_i\\wedge d\\bar z_j+\\sum_{1\\leq i\\leq k,k+1\\leq m\\leq n}\\sum_{1\\leq j\\leq k}g_{i\\bar j}\\frac{\\partial \\bar r_j}{\\partial \\bar z_m}dz_i\\wedge d\\bar z_m\\\\\n&+\\sum_{1\\leq j\\leq k,k+1\\leq l\\leq n}\\sum_{1\\leq i\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial \\bar z_l}dz_i\\wedge d\\bar z_l\n+\\sum_{k+1\\leq l,m\\leq n}\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial  z_l}\\frac{\\partial  \\bar r_j}{\\partial \\bar z_m}dz_l\\wedge d\\bar z_m.\n\\end{align*}\n\nThus,  at $(z_1,\\cdots,z_k,0,\\cdots,0)$, we get the following\n\\begin{align*}\n\\omega+\\sqrt{-1}\\partial\\bar\\partial \\bar\\phi(z)\\geq &\\sum_{1\\leq i,j\\leq k}(\\varepsilon g_{i\\bar j}+Ah_{i\\bar j})dz_i\\wedge d\\bar z_j+\\sum_{1\\leq i\\leq k,k+1\\leq m\\leq n}(g_{i\\bar m}+Ah_{i\\bar m}+(\\varepsilon-1)\\sum_{1\\leq j\\leq k}g_{i\\bar j}\\frac{\\partial \\bar r_j}{\\partial \\bar z_m})dz_i\\wedge d\\bar z_m\\\\\n&+\\sum_{1\\leq j\\leq k,k+1\\leq l\\leq n}(g_{j\\bar l}+Ah_{j\\bar l}+(\\varepsilon-1)\\sum_{1\\leq i\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial \\bar z_l})dz_i\\wedge d\\bar z_l\\\\\n&+\\sum_{k+1\\leq i,j\\leq n}(g_{i\\bar j}+Ah_{i\\bar j}+(\\varepsilon-1)\\sum_{1\\leq i,j\\leq k}g_{i\\bar j}\\frac{\\partial  r_i}{\\partial  z_l}\\frac{\\partial  \\bar r_j}{\\partial \\bar z_m})dz_i\\wedge d\\bar z_j.\n\\end{align*}\nSince $(g_{i\\bar j})_{1\\leq i,j\\leq k}$ is positive definite, from Proposition \\ref{prop:hessian}, we can see that when $A>0$ is sufficiently large (independent of $\\phi$), there is an open neighborhood $W_p$ (independent of $\\phi$), such that the conclusion of Claim \\ref{claim: 1} holds.\n\n\n\n\t\\end{proof}\n\nTo emphasis the uniformity, it is worth to point out again that the chosen of the open set $W_p$, and  the constant $\\varepsilon'$ is independent of $\\phi$, as long as $\\phi$ satisfies \\textbf{assumption $\\bigstar_{\\varepsilon,C}$}.\nWe call the above data $ (W_p,\\varepsilon',-\\frac{C}{4},\\bar\\phi)$ an \\textbf{admissible local extension} of $\\phi$.\n\n\nSince all the $\\varphi_m$ satisfy the same \\textbf{assumption $\\bigstar_{\\varepsilon,C}$}, thus near $p$, we can choose a \\textbf{uniform admissible local extension $ (W_p,A,\\varepsilon',-\\frac{C}{4},\\bar\\varphi_m)$} of $\\varphi_m$, for all $m\\in\\mathbb{N}^+$.\nSince $V$ is compact, one may choose an  open neighborhood $W$ of $V$ in $X$, and universal constants $A>0$ and $\\varepsilon'>0$, such that the functions $\\widetilde \\varphi_m:=\\varphi_m\\circ r+Ah$ are defined on $W$, such that $\\omega+i\\partial\\bar\\partial \\widetilde \\varphi_m\\geq \\varepsilon'\\omega$ on $W$ for all $m$. Since $\\{\\varphi_m\\}$ is a non-increasing sequence, one obtains that $\\{\\widetilde{\\varphi}_m\\}$ is a non-increasing sequence.\n\n\\noindent\\textbf{Step 2: Global extensions of $\\varphi_m$ for all $m$.} Up to shrinking, we may assume that $\\widetilde{\\varphi}_m$ are defined on the closure of $W$ for all $m\\in \\mathbb{N}^+$.\nLet $F$  be the quasi-psh function in Lemma \\ref{reference function}.\nNear $\\partial W$ (the boundary of $W$), the function $F$ is smooth, and $\\sup_{\\partial W}\\widetilde{\\varphi}_{1}=-C''$ for some positive constant $C''>0$.\nNow we choose a small positive $\\nu$, such that $\\inf_{\\partial W}(\\nu F)>-\\frac{C''}{2}$ and   $\\omega+i\\partial\\bar{\\partial}\\nu F\\geq\\varepsilon'\\omega$.\nThus $\\nu F >\\widetilde{\\varphi}_{1}\\geq \\widetilde{\\varphi}_m$ in a neighborhood of $\\partial W$ for all $m\\in \\mathbb{N}^+$, since $\\widetilde{\\varphi}_m$ is non-increasing.\nTherefore, we can finally define\n\\begin{align*}\n\\Phi_m=\\left\\{\n\\begin{array}{ll}\n\\max\\{\\widetilde{\\varphi}_m, \\nu F\\}, & \\hbox{on $W$;} \\\\\n\\nu F, & \\hbox{on $X\\setminus W$,}\n\\end{array}\n\\right.\n\\end{align*}\nwhich is defined on the whole of $X$. It  is easy to check that $\\Phi_m$ satisfies the following properties:\n\\begin{itemize}\n\t\\item $\\Phi_m$ is non-increasing in  $m$,\n\t\\item $\\Phi_m\\leq 0$ for all $m\\in \\mathbb{N}^+$,\n\t\\item $\\omega+i\\partial\\bar{\\partial}\\Phi_m\\geq \\varepsilon'\\omega$ for all $m\\in \\mathbb{N}^+$,\n\t\\item $\\Phi_m|_V=\\varphi_m$ for all $m\\in \\mathbb{N}^+$.\n\\end{itemize}\n\n\\noindent\\textbf{Step 3: Taking limit to complete the proof of Theorem \\ref{thm:main}.}\nFrom above steps, we get a sequence of non-increasing, non-positive strictly $\\omega$-psh functions $\\Phi_m$ on $X$. Then  either $\\Phi_m\\rightarrow -\\infty $ uniformly on $X$, or $\\Phi:=\\lim\\limits_m\\Phi_m\\in$ Psh$(X,\\omega)$.\nBut $\\Phi_m|_V=\\varphi_m\\searrow \\varphi\\not\\equiv -\\infty$, the first case will not appear.\nMoreover,   we can see that $\\Phi:=\\lim\\limits_m\\Phi_m$ is a strictly $\\omega$-psh function on $X$ from the property $\\omega+i\\partial\\bar{\\partial}\\Phi_m\\geq \\varepsilon'\\omega$ for all $m\\in \\mathbb{N}^+$, and $\\Phi|_V=\\lim\\limits_m\\Phi_m|_V=\\lim\\limits_m\\varphi_m=\\varphi$.\nIt follows that   $(\\omega+i\\partial\\bar{\\partial}\\Phi)|_V=\\omega|_V+i\\partial\\bar{\\partial}\\varphi$.\nThus we complete the proof of Theorem \\ref{thm:main}.\n\n\n\\begin{rem} By similar arguments as in \\cite{WZ20}, we can get the following extension results for K\\\"ahler currents in a big class.\n\t\\begin{thm}\n\t\tLet $(X,\\omega)$ be a compact K\\\"ahler manifold of complex dimension $n$, and $V\\subset X$ be a complex submanifold of positive dimension. Suppose that $V$ has a holomorphic retraction structure  in $X$.\n\t\tLet $\\alpha\\in H^{1,1}(X,\\mathbb R)$ be a big class and any of the irreducible components of $E_{nK}(\\alpha)$ either does not intersect with $V$, or is contained in $V$. \n\t\tThen any  K\\\"ahler current in  $\\alpha|_V$ is the restriction of a K\\\"ahler current in $\\alpha$.\n\t\\end{thm}\n\t\\end{rem}\n\n\\section{Examples}\n\nIn \\cite{HK20}, Hosono-Koike point out that in Nakayama's example and Zariski's example, the submanifold  have holomorphic tubular neighborhood structure in  the ambient manifold, thus have  holomorphic retraction structure. \n\n\\noindent{\\textbf{Product manifold.}} Let $Y_1$ and $Y_2$ be two compact K\\\"ahler manifold and set $X:=Y_1\\times Y_2$. Fix an arbitrary point $p\\in Y_2$, let $V=Y_1\\times p$, then the natural map $\\pi:Y_1\\times Y_2\\rightarrow Y_1\\times p$  serves as a holomorphic retraction map. \n\n\nAn interesting  example of non-product manifold, communicated to us by Koike \\cite{Ko21}, is  the following famous example of Serre.\n\n\\noindent{\\textbf{Serre's example.}} Let $X:=\\mathbb P_{[x;y]}\\times \\mathbb C_z\/\\sim$, where $\\tau\\in \\mathbb H$ with $\\mathbb H$ be the upper half plane, and $$([x;y],z)\\sim ([x;y+x],z+1)\\sim ([x;y+\\bar\\tau\\cdot x],z+\\tau).$$\nLet $V:=\\{x=0\\}\\subset X$ as a submanifold of $X$ which is obviously isomorphic to the elliptic curve $\\mathbb C\/\\langle 1,\\tau\\rangle$.  It is easy to check  that  the projection map $\\pi: X\\rightarrow \\mathbb C\/\\langle 1,\\tau\\rangle=:V$ is a holomorphic retraction. It can also be verified that $V$ does not have a holomorphic tubular neighborhood structure in $X$.\n\\begin{rem}\n\tIn \\cite{Ko21}, Koike gives a very interesting  proof of Theorem \\ref{thm:main} for Serre's example,  which however seems not applicable to general case treated in this paper.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{{\\bf Introduction}}\n\n\nThe loop space $LM$ of a manifold $M$ appears frequently in mathematics and\n mathematical physics. In this paper, using an infinite dimensional version of\n Chern-Simons theory associated to the Wodzicki residue for\n pseudodifferential operators ($\\Psi{\\rm DO}$s), we develop a  computable theory of secondary\n characteristic classes  on the tangent\n bundle to loop spaces.   We apply these secondary classes to distinguish circle actions\n on $S^2\\times S^3$, and we prove that $\\pi_1({\\rm Diff} (S^2\\times S^3))$ is infinite.  To our knowledge, these applications are the first examples of nonzero Wodzicki-type characteristic classes. \n\n\nSince Chern-Weil and Chern-Simons theory are geometric, it is necessary\nto understand connections and curvature on loop spaces.  A  Riemannian metric\n$g$ \non $M$ induces a family of metrics $g^s$ on $LM$ parametrized by a Sobolev\nspace parameter $s \\geq 0$, where $s=0$ gives the usual $L^2$ metric, and the\nsmooth case is a kind of limit as $s\\to\\infty.$  Thus we think of\n$s$ as a regularizing parameter, and pay attention to  the parts of the theory which\nare independent of $s$. \n\nIn Part I, we compute the connection and curvature  for the Levi-Civita\nconnection for $g^s$ for $s>\\frac{1}{2}$.  The \nclosed form expressions obtained for the Levi-Civita connection for general $LM$ \nextend Freed's results for loop groups \\cite{Freed}.  The connection and \ncurvature forms take values in\nzeroth order $\\Psi{\\rm DO}$s acting on a trivial bundle over $S^1$. \nFor Wodzicki-Chern-Simons classes, we only need\nthe principal and subprincipal symbols for these\nforms, which we calculate.\n\n\n\n\nIn Part II, we develop a theory of Chern-Simons classes on loop spaces.\nThe structure group for the Levi-Civita connection for\n$(LM, g^s)$ is the set of invertible zeroth order $\\Psi{\\rm DO}$s, so we need\n invariant polynomials on the corresponding Lie algebra.  The naive choice is\nthe standard polynomials $\\operatorname{Tr}(\\Omega^k)$ of the curvature $\\Omega = \\Omega^s$,\nwhere Tr is the operator trace.  However, $\\Omega^k$ is zeroth order\nand hence not trace class, and in any case the operator trace\nis impossible to compute in general.  Instead, as in \\cite{P-R2} we use the \nWodzicki residue, the only trace on the full\nalgebra of $\\Psi{\\rm DO}$s.  \nFollowing Chern-Simons\n \\cite{C-S} as much as possible, we build a theory of\nWodzicki-Chern-Simons (WCS) classes, which gives classes in $H^{2k-1}(LM^{2k-1})$ associated to partitions of $k$.\n\nThere are two main differences from the finite\ndimensional theory. The absence of a Narasimhan-Ramanan universal connection\ntheorem means that we do not have a theory of differential characters\n\\cite{Ch-Si}.  However, since we have a family of connections on $LM$, we can define real valued, not just ${\\mathbb R}\/{\\mathbb Z}$-valued, WCS classes. \n\n In contrast to the operator trace, the Wodzicki residue is locally\n computable, so we can write explicit expressions for the WCS classes.\nIn particular, we can see how the WCS classes depend on the Sobolev parameter $s$, and \nhence define a  ``regularized\" or $s$-independent WCS classes.  \nThe local expression also yields some vanishing results for WCS classes.  More importantly,\nwe produce a nonvanishing \nWCS class on $L(S^2\\times S^3).$  This leads to the topological results described in the first paragraph.\n\nFor related results on characteristic classes on infinite rank bundles with a group of $\\Psi{\\rm DO}$s as structure group, see \n \\cite{lrst, P-R2}.\n \n\n\\medskip\nThe paper is organized as follows.  Part I treates the family of metrics $g^s$ on $LM$\nassociated to $(M,g)$.  \\S2 discusses connections associated to $g^s.$  After some preliminary material,\nwe compute the Levi-Civita connection for $s=0$ (Lemma \\ref{lem:l2lc}), $s=1$\n(Theorem \\ref{old1.6}),  $s\\in {\\mathbb Z}^+$ (Theorem \\ref{thm:sinz}), and general \n$s>\\frac{1}{2}$ (Theorem \\ref{thm25}).   These connections allow us to track how the geometry\nof $LM$ depends on $s$.  \n\nBoth the Levi-Civita and $H^s$ connections have connection and curvature forms taking \nvalues in $\\Psi{\\rm DO}$s of order zero.  In \\S3, we compute the symbols of these forms needed in Part II.  In \\S4, we show that our results extend Freed's on loop groups \\cite{Freed}.\n\nPart II covers Wodzicki-Chern-Simons classes.  In \\S5, we review the finite dimensional\nconstruction of Chern and Chern-Simons classes, and use the Wodzicki residue to define Wodzicki-Chern (WC) and WCS classes (Definition \\ref{def:WCS}).  We prove the necessary vanishing\nof the WC classes for mapping spaces (and in particular for $LM$) in Proposition\n\\ref{prop:maps}.  In Theorem \\ref{thm:5.5}, we give the explicit local expression for the\nrelative WCS class $CS_{2k-1}^W(g)\\in H^{2k-1}(LM^{2k-1})$ associated to the trivial\npartition of $k$.  We then define the regularized or $s$-independent WCS class.  \nIn Theorem \\ref{WCSvan}, we give a vanishing result\nfor WCS classes.  \n\nIn particular, the WCS class which is the analogue of the classical dimension three Chern-Simons class vanishes on loop spaces of \n $3$-manifolds, so we look for nontrivial examples on $5$-manifolds.\n In \\S\\ref{dimfive}, we use a Sasaki-Einstein\nmetric constructed in \\cite{gdsw} to produce a nonzero WCS class  $CS_5^W\\in\nH^5(L(S^2\\times\nS^3)).$  We prove $CS_5^W\\neq 0$ by an exact computer calculation showing\n $\\int_{[a^L]} CS_5^W \\neq 0$, where\n$[a^L]\\in H_5(LM)$ is a cycle associated to a simple\ncircle action on $S^2\\times S^3.$  From this\nnonvanishing, we conclude both that  the circle action is not \nsmoothly homotopic to the trivial action and that $\\pi_1({\\rm Diff} (S^2\\times S^3))$ is infinite.\nWe expect other similar results in the future.  \n\n\n\nOur many discussions with Sylvie Paycha are gratefully\nacknowledged. We also thank Kaoru Ono and Dan Freed for pointing out errors in  previous versions of \nthe paper.\n\n\n\n\n\\bigskip\n\n\n\\large\n\\noindent {{\\bf Part I. The Levi-Civita Connection on the Loop Space $LM$}}\n\\normalsize\n\n\\bigskip\n\nIn this part, we compute the Levi-Civita connection on $LM$\nassociated to a Riemannian metric on $M$ and a Sobolev parameter $s=0$ \nor $s>\\frac{1}{2}.$  The standard $L^2$ metric on $LM$ is the case $s=0$, and otherwise we avoid technical issues by assuming that $s$ is greater than the critical exponent $\\frac{1}{2}$ for analysis on bundles over $S^1.$  \n In \\S\\ref{LCconnection}, the\nmain results are  Lemma \\ref{lem:l2lc}, Theorem \\ref{old1.6}, \nTheorem \\ref{thm:sinz}, and Theorem \\ref{thm25},\nwhich compute the Levi-Civita connection for $s =0$, $s=1$, \n$s\\in {\\mathbb Z}^+$,  and general $s >\\frac{1}{2},$ respectively.\n\n\nIn\n\\S3, we compute the relevant symbols of the connection one-forms and the\ncurvature two-forms.  In \\S4, we compare our results with work of Freed\n\\cite{Freed} on loop groups.    \n\n\n\n\\section{{\\bf The Levi-Civita Connection for Sobolev Parameter $s\\geq 0$}}\n\\label{LCconnection}\n\nThis section covers background material and computes the Levi-Civita\nconnection on $LM$ for Sobolev parameter $s=0$ and $s>\\frac{1}{2}$.  \nIn \\S2.1, we review\nmaterial on $LM$, and in \\S2.2 we review pseudodifferential operators and the\nWodzicki residue.  In \\S2.3, we give the crucial computations of the Levi-Civita connections\nfor $s=0,1$.\nThis computation is extended to $s\\in {\\mathbb Z}^+$ in \\S2.4, and to general $s>\\frac{1}{2}$ in\n\\S2.5.    In \\S2.6, we discuss how the geometry of $LM$ forces an extension of\nthe structure group of $LM$ from a gauge group to a group of bounded\ninvertible $\\Psi{\\rm DO}$s.\n     \n\\subsection{{\\bf Preliminaries on $LM$}}\n\n${}$\n\\medskip\n\nLet $(M, \\langle\\ ,\\ \\rangle)$ \nbe a closed, connected,  oriented Riemannian $n$-manifold with  loop space $LM\n= C^\\infty(S^1,M)$ of smooth loops. \n$LM$ is a smooth infinite dimensional Fr\\'echet manifold, but it is\n technically simpler \nto work \nwith the smooth Hilbert manifold $H^{s'}(S^1,M)$ of loops in some Sobolev class $s' \\gg 0,$\nas we now recall. For $\\gamma\\in LM$, the formal\ntangent space $T_\\gamma LM$ is \n$\\Gamma(\\gamma^*TM)$, the space\n of smooth sections of the pullback bundle $\\gamma^*TM\\to\nS^1$.   The actual tangent space of $H^{s'}(S^1, M)$ at $\\gamma$ is \n$H^{s'-1}(\\gamma^*TM),$     the sections of $\\gamma^*TM$ of Sobolev class $s'-1.$\nWe will fix $s'$ and use $LM, T_\\gamma LM$ for $H^{s'}(S^1, M), H^{s'-1}(\\gamma^*TM)$, respectively.\n\nFor each $s>1\/2,$ we can complete $\\Gamma(\\gamma^*TM\\otimes {\\mathbb C})$\n with respect to the Sobolev inner product\n \\begin{equation}\\label{eq:Sob1}\n\\langle X,Y\\rangle_{s}=\\frac{1}{2\\pi}\\int_0^{2\\pi} \\langle(1+\\Delta)^{s}\nX(\\alpha),Y(\\alpha)\n\\rangle_{\\gamma (\\alpha)}d\\alpha,\\  X,Y\\in \\Gamma(\\gamma^*TM).\n\\end{equation}\nHere $\\Delta=D^*D$, with $D=D\/d\\gamma$ the covariant derivative along\n$\\gamma$. (We use this notation instead of the classical $D\/dt$ to keep track\nof $\\gamma$.)\nWe need the complexified pullback bundle $\\gamma^*TM\\otimes {\\mathbb C}$, denoted from now on\njust as $\\gamma^*TM$, in order to apply the\npseudodifferential operator  $(1+\\Delta)^{s}.$\nThe construction of $(1+\\Delta)^{s}$ is reviewed in\n\\S\\ref{pdoreview}.  \nWe denote this completion by  $H^{s'}(\\gamma^*TM)$.   We can consider the\n$s$ metric on $TLM$ for any $s\\in {\\mathbb R}$, but we will only consider \n$s=0$ or $1\/2 < s\\leq s'-1.$\n\n\nA small real neighborhood $U_\\gamma$ \nof the zero section in $H^{s'}(\\gamma^*TM)$ is a\ncoordinate chart near $\\gamma\\in LM$ \nvia the pointwise exponential map\n\\begin{equation}\\label{pointwiseexp}\n\\exp_\\gamma:U_\\gamma\n\\to L M, \\ X \\mapsto \n\\left(\\alpha\\mapsto \\exp_{\\gamma(\\alpha)} X(\\alpha)\\right).  \n\\end{equation}\nNote that the domain of the exponential map is not contained in $T_\\gamma LM.$\nThe differentiability of the transition functions $\\exp_{\\gamma_1}^{-1}\\cdot\n\\exp_{\\gamma_2}$ is proved in\n\\cite{E} and \\cite[Appendix A]{Freed1}.\nHere $\\gamma_1, \\gamma_2$ are close loops in the sense that\na geodesically convex neighborhood of $\\gamma_1(\\theta)$ contains\n$\\gamma_2(\\theta)$ and vice versa for all $\\theta.$\nSince \n$\\gamma^*TM$\nis (noncanonically) isomorphic to the trivial bundle ${\\mathcal R} =\nS^1\\times {\\mathbb C}^n\\to S^1$, \nthe model space for $LM$ is the set of \n$H^{s'}$ sections of this trivial bundle.  \nThe $s$ metric is a weak Riemannian metric for $s<s'$ in the sense that the topology induced on $H^{s'}(S^1, M)$ by the exponential map applied to $H^{s}(\\gamma^*TM)$ is weaker than the $H^{s'}$ topology.\n\n\nThe complexified tangent bundle\n$TLM$ has\ntransition functions $d(\\exp_{\\gamma_1}^{-1} \n\\circ \\exp_{\\gamma_2})$.  Under\nthe isomorphisms $\\gamma_1^*TM \\simeq {\\mathcal R} \\simeq\n\\gamma_2^*TM$, the transition functions lie in the gauge group\n${{\\mathcal G}}({\\mathcal R})$, so this is the structure group of $TLM.$\n\n\n\n\n\n\n\\subsection{{\\bf Review of $\\Psi{\\rm DO}$ Calculus}}\\label{pdoreview}\n${}$\n\\medskip\n\nWe recall the construction of classical\npseudodifferential operators ($\\Psi{\\rm DO}$s)\n on a closed \nmanifold $M$ from \\cite{gilkey, See}, assuming knowledge of $\\Psi{\\rm DO}$s on\n ${\\mathbb R}^n$ (see e.g. \\cite{hor, shu}).   \n \n\n\n\n\n A linear operator $P:C^\\infty(M)\\to C^\\infty(M)$\nis a $\\Psi{\\rm DO}$ of  order $d$ if for\nevery open chart $U\\subset M$ and functions $\\phi,\\psi\\in C_c^\\infty(U)$,\n$\\phi P\\psi$ is a $\\Psi{\\rm DO}$ of order $d$ on ${\\mathbb R}^n$, where we do not\n distinguish between $U$ and its diffeomorphic image in ${\\mathbb R}^n$.\n Let\n  $\\{U_i\\}$ be a finite cover of $M$\nwith subordinate partition of unity\n  $\\{\\phi_i\\}.$  Let $\\psi_i\\in C^\\infty_c(U_i)$ have $\\psi_i \\equiv 1$ on\n  supp$(\\phi_i)$  and set $P_i = \\psi_iP\\phi_i.$\nThen\n  $ \\sum_i \\phi_iP_i\\psi_i$ is a $\\Psi{\\rm DO}$ on $M$, and\n$P$ differs from $ \\sum_i \\phi_iP_i\\psi_i$ by a smoothing operator, denoted\n$P\\sim \\sum_i \\phi_iP_i\\psi_i$. \n In particular, this sum is independent of the\n  choices up to smoothing operators.\nAll this carries over to $\\Psi{\\rm DO}$s acting on sections of a bundle over $M$.\n\nAn example is the $\\Psi{\\rm DO}$ $(1+\\Delta-\\lambda)^{-1}$ for $\\Delta$ a positive\norder nonnegative elliptic $\\Psi{\\rm DO}$ and $\\lambda$\noutside the spectrum of $1+\\Delta.$  In each $U_i$, we construct a parametrix\n$P_i$ for $A_i = \\psi_i\n(1+\\Delta-\\lambda)\\phi_i$ by formally inverting $\\sigma(A_i)$ and then\nconstructing a $\\Psi{\\rm DO}$ with the inverted symbol.  By \\cite[App.~A]{A-B2},\n$ B = \\sum_i \\phi_iP_i\\psi_i$ is a parametrix for $(1+\\Delta-\\lambda)^{-1}$.\n  Since\n  $B \\sim (1+\\Delta-\\lambda)^{-1}$,\n  $(1+\\Delta-\\lambda)^{-1}$ is itself a $\\Psi{\\rm DO}$.\nFor $x\\in U_i$, by definition\n$$\\sigma((1+\\Delta-\\lambda)^{-1})(x,\\xi) = \\sigma(P)(x,\\xi) = \\sigma(\\phi\nP\\phi)(x,\\xi),$$\nwhere $\\phi$ is a bump function with $\\phi(x) = 1$ \\cite[p.~29]{gilkey};  \nthe symbol depends on the choice of $(U_i, \\phi_i).$  \n\nThe operator $(1+\\Delta)^s$ for Re$(s) <0$,\nwhich exists as a bounded\noperator on $L^2(M)$ by the functional calculus, is also a $\\Psi{\\rm DO}$.\nTo see this,\n we construct the putative symbol $\\sigma_i$\nof $ \\psi_i(1+\\Delta)^s\\phi_i$ in each $U_i$\nby a contour integral $\\int_\\Gamma \\lambda^{s}\\sigma[(1+\\Delta-\\lambda)^{-1}]\nd\\lambda$\naround the spectrum of $1+\\Delta$.\nWe then\nconstruct a $\\Psi{\\rm DO}$ $Q_i$ on $U_i$ with $\\sigma(Q_i) = \\sigma_i$,\nand set $Q  = \\sum_i\\phi_i Q_i\\psi_i.$\nBy arguments in \\cite{See},\n$(1+\\Delta)^s \\sim Q$,\nso $(1+\\Delta)^s$ is a $\\Psi{\\rm DO}$.\n\n\n\n\n Recall that the {\\it Wodzicki residue} of a $\\Psi{\\rm DO}$ $P$ on sections\n  of a bundle $E\\to\nM^n$ is\n\\begin{equation}\\label{wresdef} {\\rm res}^{\\rm w}(P) = \n\\int_{S^*M} \\operatorname{tr}\\ \\sigma_{-n}(P)(x,\\xi) d\\xi dx,\\end{equation}\nwhere $S^*M$ is the unit cosphere bundle for some metric.  The Wodzicki\nresidue is independent of choice of local coordinates, and up to scaling is \nthe unique trace on the algebra of $\\Psi{\\rm DO}$s if dim$(M)>1$ (see\ne.g.~\\cite{fgl} in general and \\cite{ponge} for the case $M=S^1.$).  \n\n\nThe Wodzicki residue will be used in\nPart II to define characteristic classes on $LM$.  In our particular case, the operator $P$ \nwill be an $\\Psi{\\rm DO}$ of order $-1$ acting on sections of a bundle over $S^1$\n (see (\\ref{cswint})), so\n$\\sigma_{-1}(P)$ is globally defined. Of course, $\\int_{S^*S^1}\n\\operatorname{tr}\\sigma_{-1}(P) d\\xi d\\theta = 2\\int_{S^1}\\operatorname{tr}\\sigma_{-1}(P)d\\theta$.  It is\neasy to check that this integral,\nwhich strictly speaking involves a choice of cover of $S^1$ and a partition of unity,\nequals the usual $2\\int_0^{2\\pi} \\operatorname{tr}\\sigma_{-1}(P) d\\theta.$\n\n\n\n\n\n\n\n\n\n\n\\subsection{The Levi-Civita Connection for $s=0, 1$}\n${}$\n\\medskip\n\n\nThe smooth Riemannian manifold $LM = H^{s'}(S^1,M)$ has tangent bundle $TLM$ with \n$T_\\gamma LM = H^{s'-1}(\\gamma^*TM).$  For the $s'-1$ metric on $TLM$ (i.e., \n$s = s'-1$ in (\\ref{eq:Sob1})), \nthe \nLevi-Civita connection exists and is determined by the six term formula\n\\begin{eqnarray}\\label{5one}\n2\\ip{\\con{s}{X}Y,Z}_{s} &=& X\\ip{Y,Z}_{s}+Y\\ip{X,Z}_{s}-Z\\ip{X,Y}_{s}\\\\\n&&\\qquad +\\ip{[X,Y],Z}_{s}+\\ip{[Z,X],Y}_{s}-\\ip{[Y,Z],X}_s\\nonumber\n\\end{eqnarray}\n\\cite[Ch. VIII]{lang}.  The point is that each term on the RHS of (\\ref{5one}) \nis\na {\\it continuous} linear functional $T_i:H^{s=s'-1}(\\gamma^*TM) \\to {\\mathbb C}$ in $Z$.  Thus \n$T_i(Z) = \\ip{T_i'(X,Y),Z}_s$ for a unique $T'(X,Y)\\in H^{s'-1}(\\gamma^*TM)$, and $\\con{s}{Y}X\n= \\frac{1}{2}\\sum_i T'_i.$  \n\nIn general, the Sobolev parameter $s$ in (\\ref{eq:Sob1}) differs from the parameter $s'$ defining the loop space.  We discuss how this affects the existence of a Levi-Civita connection. \n\n\\begin{rem}\\label{lcrem}  For general $s >\\frac{1}{2}$, the Levi-Civita connection for the $H^s$ \nmetric is guaranteed to exist on the bundle $H^s(\\gamma^*TM)$, as above.  However, it is inconvenient to have the bundle depend on the Sobolev parameter, for several reasons:  \n(i) $H^s(\\gamma^*TM)$ is strictly speaking not the tangent bundle of $LM$, (ii) for the\n$L^2$ ($s=0$) metric, the Levi-Civita connection should be given by the Levi-Civita connection on $M$ applied pointwise along the loop (see Lemma \\ref{lem:l2lc}), and on $L^2(\\gamma^*TM)$  this would have to be interpreted in the distributional sense; (iii) to compute Chern-Simons classes on\n$LM$ in Part II, we need to compute with a pair of connections corresponding to $s=0, s=1$ on the\nsame bundle.  These problems are not fatal: (i) and (ii) are essentially  aesthetic issues,\nand for (iii), the connection one-forms will take values in zeroth order $\\Psi{\\rm DO}$s, which are bounded operators on any \n$H^{s'-1}(\\gamma^*TM)$, so $s' \\gg 0$ can be fixed.  \n\nThus it is more convenient\nto fix $s'$ and consider the family of $H^s$ metrics on $TLM$ for \n$\\frac{1}{2} < s < s'-1$. \n  However, the existence of the Levi-Civita connection for the $H^s$ metric is trickier.\n  For a sequence $Z\\in H^{s'-1} = H^{s'-1}(\\gamma^*TM)$ with $Z\\to 0$\n  in $H^{s'-1}$ or in \n  $H^s$, the RHS of (\\ref{5one}) goes to $0$ for fixed $X, Y\\in H^s.$  Since\n  $H^{s'-1}$ is dense in $H^{s}$, the RHS of (\\ref{5one}) extends to a continuous linear functional on $H^s$.  Thus the RHS of (\\ref{5one}) is given by\n  $\\langle L(X,Y), Z\\rangle_s$ for some $L(X,Y)\\in H^s.$  We set $\\nabla^{s}_YX = \n  \\frac{1}{2}L(X,Y)$.  Note that even if we naturally demand that\n  $X, Y\\in H^{s'-1}$, we only get $\\nabla^s_YX\\in H^s\\supset H^{s'-1}$ without additional work.  Part of the content of Theorem \\ref{thm25} is that the Levi-Civita connection exists in the {\\it strong sense}:  given a tangent vector $X\\in H^{s'-1}(\\gamma^*TM)$ and a smooth vector field\n$Y(\\eta)\\in H^{s'-1}(\\eta^*TM)$ for all $\\eta$,\n   $\\nabla^s_XY(\\gamma)\\in H^{s'-1}(\\gamma^*TM).$  See Remark 2.6.\n  \n \n\n\\end{rem}\n\n\n\n\nWe need to discuss local coordinates on $LM$.\nFor motivation, recall that\n\\begin{equation}\\label{lie}[X,Y]^a = X(Y^a)\\partial_a - Y(X^a)\\partial_a\n\\equiv \\delta_X(Y) -\\delta_Y(X)\n\\end{equation}\nin local coordinates on a finite dimensional manifold.  Note that\n$X^i\\partial_iY^a = X(Y^a) =\n(\\delta_XY)^a$ in this notation.\n\nLet $Y$ be a vector field on $LM$, and let $X$ be a tangent vector at\n$\\gamma\\in LM.$  The local variation $\\delta_XY$ of $Y$ in the direction of $X$ at $\\gamma$ is \ndefined as usual: let $\\gamma(\\varepsilon,\\theta)$ be a family of loops in $M$\nwith $\\gamma(0,\\theta) = \\gamma(\\theta), \\frac{d}{d\\varepsilon}|_{_{\\varepsilon=0}}\n\\gamma(\\varepsilon,\\theta) = X(\\theta).$  Fix $\\theta$, and let $(x^a)$ be\ncoordinates near $\\gamma(\\theta)$.  We call these coordinates \n{\\it manifold coordinates.} Then\n$$\\delta_XY^a(\\gamma)(\\theta) \\stackrel{{\\rm def}}{=}\n\\frac{d}{d\\varepsilon}\\biggl|_{_{_{\\varepsilon =0}}} Y^a(\\gamma(\\varepsilon,\\theta)).$$\nNote that $\\delta_XY^a = (\\delta_XY)^a$ by definition.\n\n\\begin{rem} Having $(x^a)$ defined only near a fixed $\\theta$ is inconvenient.\nWe can find coordinates that work for all $\\theta$ as follows. For\n  fixed $\\gamma$, there is an $\\varepsilon$ such that for all $\\theta$,\n  $\\exp_{\\gamma(\\theta)} X$ is inside the cut locus of $\\gamma(\\theta)$ if\n  $X\\in T_{\\gamma(\\theta)}M$ has $|X|<\\varepsilon.$  Fix such an $\\varepsilon.$ Call \n  $X\\in H^{s'-1}(\\gamma^*TM)$ {\\it\n  short} if $|X(\\theta)|<\\varepsilon$ for all $\\theta.$  Then\n$$U_\\gamma = \\{\\theta \\mapsto \\exp_{\\gamma(\\theta)}X(\\theta) | X\\ {\\rm is\\\n    short}\\}\\subset LM$$\nis a coordinate neighborhood of $\\gamma$ parametrized by $\\{ X: X\\  {\\rm is\\ \n    short}\\}.$  \n\n We know\n$H^{s'-1}(\\gamma^*TM) \\simeq H^{s'-1}(S^1\\times {\\mathbb R}^n)$ noncanonically, so\n$U_\\gamma$ is parametized by short sections of $H^{s'-1}(S^1\\times {\\mathbb R}^n)$ for\na different $\\varepsilon.$  In particular, we have a smooth diffeomorphism $\\beta$ from\n$U_\\gamma$ to short sections of $H^{s'-1}(S^1\\times {\\mathbb R}^n)$.\n\nPut coordinates $(x^a)$ on ${\\mathbb R}^n$, which we identify canonically with the fiber ${\\mathbb R}^n_\\theta$\nover $\\theta$ in $S^1\\times {\\mathbb R}^n$. For $\\eta\\in\nU_\\gamma$, we have $\\beta(\\eta) = (\\beta(\\eta)^1(\\theta),...,\\beta(\\eta)^n(\\theta)).$\nAs with finite dimensional coordinate systems, we will drop $\\beta$ and just\nwrite\n$\\eta = (\\eta(\\theta)^a).$ These coordinates work for all\n$\\eta$ near $\\gamma$ and for all $\\theta.$  The definition of $\\delta_XY$ above carries over to exponential coordinates.\n\nWe will call these coordinates {\\it exponential coordinates}.\n\\end{rem}\n\n(\\ref{lie}) continues to hold\n for vector fields on $LM$, in either \n manifold or exponential coordinates.\n   To see this, one checks that the coordinate-free proof that $L_XY(f) =\n [X,Y](f)$ for $f\\in C^\\infty(M)$ (e.g.~\\cite[p.~70]{warner}) carries over to\n functions on $LM$.  In brief, the usual proof involves a map $H(s,t)$ of a\n neighborhood of the origin in ${\\mathbb R}^2$ into $M$, where $s,t$ are parameters for\n the flows of $X, Y,$ resp.  For $LM$, we have a map $H(s,t,\\theta)$, where\n $\\theta$ is the loop parameter.  \nThe  usual proof uses\n only $s, t$ differentiations,\n so $\\theta$ is unaffected.    The point is that the $Y^i$ are local functions\n on the $(s,t,\\theta)$  parameter space, whereas\nthe $Y^i$ are not\n  local functions on $M$ at points where loops cross or self-intersect.\n\nWe first compute the $L^2$ ($s=0$) Levi-Civita connection invariantly and in \nmanifold coordinates.\n\n\n\n\\begin{lem} \\label{lem:l2lc} Let $\\nabla^{LC}$ be the Levi-Civita connection on $M$.  \n Let $\\operatorname{ev}_\\theta:LM\\to M$ be $\\operatorname{ev}_\\theta(\\gamma) = \\gamma(\\theta).$ \n Then $D_XY(\\gamma)(\\theta) \\stackrel{\\rm def}{=} \n (\\operatorname{ev}_\\theta^*\\nabla^{LC})_XY(\\gamma)(\\theta)$ is the $L^2$ Levi-Civita connection on $LM$.  In manifold coordinates,\n \\begin{equation}\\label{l2lc} (D_XY)^a(\\gamma)(\\theta) = \\delta_XY^a(\\gamma)(\\theta) +\n  \\cch{bc}{a}(\\gamma(\\theta))X^b(\\gamma)(\\theta) Y^c(\\gamma)(\\theta).\n  \\end{equation}\n\\end{lem}\n\\medskip\n\nAs in Remark \\ref{lcrem}, we may assume that\n$X, Y\\in H^{s'-1}(\\gamma^*TM)$ with $s' \\gg 0$, so (\\ref{l2lc}) makes sense.\n\n\\begin{proof} $\\operatorname{ev}_\\theta^*\\nabla^{LC}$ is a connection on\n$\\operatorname{ev}_\\theta^*TM\\to LM$. We have\n$\\operatorname{ev}_{\\theta,*}(X) = X(\\theta)$.  If $U$ is a coordinate\nneighborhood on $M$ near some $\\gamma(\\theta)$, then on $\\operatorname{ev}_\\theta^{-1}(U)$, \n\\begin{eqnarray*}(\\operatorname{ev}_\\theta^*\\nabla^{LC})_XY^a(\\gamma)(\\theta) &=& (\\delta_{X}Y)^a(\\gamma)(\\theta) +\n((\\operatorname{ev}_\\theta^*\\omega^{LC}_{X})Y)^a (\\theta)\\\\\n&=& (\\delta_{X}Y)^a(\\gamma)(\\theta) + \n  \\chw{b}{c}{a}(\\gamma(\\theta))X^b(\\gamma)(\\theta) Y^c(\\gamma)(\\theta)\n\\end{eqnarray*}\nSince $\\operatorname{ev}_\\theta^*\\nabla^{LC}$ is a connection, for each fixed $\\theta$, $\\gamma$ and $X\\in\n T_\\gamma LM$, \n $Y\\mapsto$\\\\\n $ (\\operatorname{ev}^*_\\theta\\nabla^{LC})_XY(\\gamma)$\n has Leibniz rule with respect to\nfunctions on $LM$.   Thus $D$ is a connection on $LM.$\n\n\n$D$ is torsion free, as from the local expression\n  $D_XY - D_YX = \\delta_XY - \\delta_YX = [X,Y].$\n\nTo show that  $D_XY$ is compatible with the $L^2$\n  metric, first recall that  for a function $f$ on $LM$, $D_Xf = \\delta_Xf =\n  \\frac{d}{d\\varepsilon}|_{_{\\varepsilon=0}}f(\\gamma(\\varepsilon,\\theta))$ for $X(\\theta)\n   = \\frac{d}{d\\varepsilon}|_{_{\\varepsilon=0}}\\gamma(\\varepsilon, \\theta).$\n  (Here $f$ depends only on\n  $\\gamma$.)  Thus (suppressing the partition of unity, which is independent of $\\varepsilon$)\n\\begin{eqnarray*} D_X\\langle Y,Z\\rangle_0 &=& \n  \\frac{d}{d\\varepsilon}\\biggl|_{_{_{\\varepsilon=0}}}\\int_{S^1} g_{ab}(\\gamma(\\varepsilon,\\theta))\nY^a(\\gamma(\\varepsilon,\\theta))Z^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&=& \\int_{S^1}\\partial_c\ng_{ab}(\\gamma(\\varepsilon,\\theta))\nX^cY^a(\\gamma(\\varepsilon,\\theta))Z^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&&\\qquad + \\int_{S^1} \ng_{ab}(\\gamma(\\varepsilon,\\theta))\n(\\delta_XY)^a(\\gamma(\\varepsilon,\\theta))Z^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&&\\qquad \n+ \\int_{S^1} \ng_{ab}(\\gamma(\\varepsilon,\\theta))\nY^a(\\gamma(\\varepsilon,\\theta))(\\delta_XZ)^b(\\gamma(\\varepsilon,\\theta))d\\theta\\\\\n&=& \\int_{S^1}\\Gamma{}_{c a}^{e}g_{eb} X^cY^aZ^b +\n\\Gamma{}_{c b}^{e}g_{ae}X^cY^aZ^b\\\\\n&&\\qquad +g_{ab}(\\delta_XY)^aZ^b + g_{ab}Y^a(\\delta_X Z)^bd\\theta\\\\\n&=& \\langle D_XY,Z\\rangle_0 + \\langle Y, D_XZ\\rangle_0.\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{rem} The local expression for $D_XY$ also holds in exponential coordinates. More precisely, let $(e_1(\\theta),...,e_n(\\theta))$\nbe a global frame of $\\gamma^*TM$ given by the trivialization of\n$\\gamma^*TM.$  Then $(e_i(\\theta))$ is also naturally a frame of\n$T_XT_{\\gamma(\\theta)}M$ for all $X\\in T_{\\gamma(\\theta)}M.$  We use\n$\\exp_{\\gamma(\\theta)}$ to pull back the metric on $M$ to a metric on\n$T_{\\gamma(\\theta)}M$: \n$$g_{ij}(X) = (\\exp^*_{\\gamma(\\theta)}g)(e_i, e_j) =\n  g(d(\\exp_{\\gamma(\\theta)})_X (e_i),   d(\\exp_{\\gamma(\\theta)})_X\n  (e_j))_{\\exp_{\\gamma(\\theta)}X}.$$\nThen the Christoffel symbols  \n$\\Gamma_{b c}^{a}(\\gamma(\\theta))$\nare computed with respect to\n  this metric.  For example, the term $\\partial_\\ell g_{bc}$ means $e_\\ell\n  g(e_a, e_b)$, etc.  The proof that $D_XY$ has the local expression (\\ref{l2lc}) \n  then carries over to exponential coordinates.\n\n\\end{rem}\n\nThe  $s=1$ Levi-Civita connection on $LM$ is given as follows.\n\n\\begin{thm} \\label{old1.6}\nThe $s=1$ Levi-Civita connection $\\nabla = \\nabla^1$ on $LM$ is given at the loop\n$\\gamma$ by\n\\begin{eqnarray*}  \\nabla_XY &=& D_XY + \\frac{1}{2}(1+\\Delta)^{-1}\\left[\n-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y\\right.\\\\\n&&\\qquad \\left. -\\nabla_{\\dot\\gamma}(R(Y,\\dot\\gamma)X) - R(Y,\\dot\\gamma)\\nabla_{\\dot\\gamma} X\\right.\\\\\n&&\\qquad \\left. +R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma\\right].\n\\end{eqnarray*}\n\\end{thm}\n\n\nWe prove this in a series of steps.  The assumption in the next Proposition will be dropped later.\n\n\n\\begin{prop} \\label{old1.3}\nThe Levi-Civita connection for the $s=1$ metric is given by\n$$\\nabla_X^1Y = D_XY + \\frac{1}{2}(1+\\Delta)^{-1}[D_X, 1+\\Delta]Y +\n  \\frac{1}{2}(1+\\Delta)^{-1}[D_Y, 1+\\Delta]X \n+ A_XY,$$\nwhere  we assume that for $X, Y\\in H^{s'-1}$, $A_XY$ is well-defined by\n\\begin{equation}\\label{insert2}-\\frac{1}{2}\\langle [D_Z,1+\\Delta]X,Y\\rangle_0 = \\langle A_XY,Z\\rangle_1.\n\\end{equation}\n\\end{prop}\n\n\n\n\n\\begin{proof} By Lemma \\ref{lem:l2lc},\n\\begin{eqnarray*} X\\langle Y,Z\\rangle_1 &=& X\\langle (1+\\Delta)Y,Z\\rangle_0 =\n  \\langle D_X((1+\\Delta)Y),Z\\rangle_0 + \\langle (1+\\Delta)Y, D_XZ\\rangle_0\\\\\nY\\langle X,Z\\rangle_1 &=& \\langle D_Y((1+\\Delta)X),Z\\rangle_0 + \\langle (1+\\Delta)X,\nD_YZ\\rangle_0\\\\\n-Z\\langle X,Y\\rangle_1 &=& -\\langle D_Z((1+\\Delta)X),Y\\rangle_0 - \\langle (1+\\Delta)X,\nD_ZY\\rangle_0\\\\\n\\langle [X,Y],Z\\rangle_1 &=& \\langle(1+\\Delta)(\\delta_XY - \\delta_YX), Z\\rangle_0\n= \\langle (1+\\Delta)(D_XY - D_YX),Z\\rangle_0\\\\\n\\langle[Z,X],Y\\rangle_1 &=& \\langle(1+\\Delta)(D_ZX-D_XZ),Y\\rangle_0\\\\\n-\\langle[Y,Z],X\\rangle_1 &=& -\\langle(1+\\Delta)(D_YZ-D_ZY),X\\rangle_0.\n\\end{eqnarray*}\nThe six terms on the left hand side must sum up to $2\\langle \\nabla^1_XY,Z\\rangle_1$ \nin the sense of Remark \\ref{lcrem}.\nAfter some cancellations, for $\\nabla =\\nabla^1$ we get\n\\begin{eqnarray*} 2\\langle\\nabla_XY,Z\\rangle_1 &=& \\langle D_X((1+\\Delta)Y),Z\\rangle_0 +\n  \\langle D_Y((1+\\Delta)X),Z\\rangle_0\\nonumber\\\\\n&&\\qquad + \\langle (1+\\Delta)(D_XY - D_YX),Z\\rangle_0 - \\langle\n  D_Z((1+\\Delta)X),Y\\rangle_0\\nonumber\\\\\n&&\\qquad +\\langle(1+\\Delta)D_ZX),Y\\rangle_0\\nonumber\\\\\n&=& \\langle (1+\\Delta)D_XY,Z\\rangle_0 + \\langle [D_X,1+\\Delta] Y, Z\\rangle_0\\\\\n&&\\qquad + \\langle (1+\\Delta)D_YX,Z\\rangle_0 + \\langle [D_Y,1+\\Delta] X, Z\\rangle_0\\nonumber\\\\\n&&\\qquad + \\langle (1+\\Delta)(D_XY - D_YX),Z\\rangle_0 -\\langle\n  [D_Z,1+\\Delta]X,Y\\rangle_0\\\\\n&=& 2\\langle D_XY,Z\\rangle_1 + \\langle (1+\\Delta)^{-1}[D_X,1+\\Delta]Y,Z\\rangle_1\\\\\n&&\\qquad + \\langle (1+\\Delta)^{-1}[D_Y,1+\\Delta]X,Z\\rangle_1 +2\n\\langle A_XY,Z\\rangle_1.\n\\end{eqnarray*}\n\n\\end{proof}\n\nNow we compute the bracket terms in the Proposition.  We have $[D_X,1+\\Delta] =\n[D_X,\\Delta]$. Also,\n$$0 = \\dot\\gamma\\langle X, Y\\rangle_0 = \\langle\\nabla_{\\dot\\gamma}X,Y\\rangle_0\n+ \\langle X,\\nabla_{\\dot\\gamma}Y\\rangle_0,$$\nso \n\\begin{equation}\\label{one}\\Delta = \\nabla_{\\dot\\gamma}^* \\nabla_{\\dot\\gamma}\n  = -\\nabla_{\\dot\\gamma}^2.\n\\end{equation}\n\n\\begin{lem} $[D_X,\\nabla_{\\dot\\gamma}]Y = R(X,\\dot\\gamma)Y.$\n\\end{lem}\n\n\\begin{proof} \nNote that $\\gamma^\\nu, \\dot\\gamma^\\nu$ are locally defined functions on \n$S^1\\times LM.$\nLet $\\tilde\\gamma:\n[0,2\\pi]\\times (-\\varepsilon,\\varepsilon)\\to M$ be a smooth map with $\\tilde\\gamma(\\theta,0) =\n\\gamma(\\theta)$, and\n$\\frac{d}{d\\tau}|_{\\tau = 0}\\tilde\\gamma(\\theta,\\tau) = Z(\\theta).$\nSince $(\\theta,\\tau)$ are coordinate functions on\n$S^1\\times (-\\varepsilon,\\varepsilon)$, we have\n\\begin{eqnarray}\\label{badterms} Z(\\dot\\gamma^\\nu) &=& \\delta_Z(\\dot\\gamma^\\nu) = \\ptau{Z}(\\dot\\gamma^\\nu) =\n\\dtau\\right.\\left(\\frac{\\partial}{\\partial\\theta}\n(\\tilde\\gamma(\\theta,\\tau)^\\nu\\right)\\\\\n&=& \\frac{\\partial}{\\partial\\theta}\n\\dtau\\right. \\tilde\\gamma(\\theta,\\tau)^\\nu = \\partial_\\theta Z^\\nu \\equiv\n\\dot Z^\\nu.\\nonumber\n\\end{eqnarray}\n\n We compute\n\\begin{eqnarray*} \n(D_X\\nabla_{\\dot\\gamma} Y)^a\n&=& \\delta_X(\\nabla_{\\dot\\gamma} Y)^a +\n  \\chw{b}{c}{a}X^b\\nabla_{\\dot\\gamma} Y^c\\\\\n&=& \\delta_X(\\dot\\gamma^j\\partial_jY^a + \\chw{b}{c}{a}\\dot\\gamma^bY^c\n+ \\chw{b}{c}{a}X^b(\\dot\\gamma^j\\partial_jY^c + \\chw{e}{f}{c}\\dot\\gamma^e Y^f)\\\\\n&=& \\dot X^j\\partial_jY^a + \\dot\\gamma^j\\partial_j\\delta_XY^a +\n  \\partial_m\\chw{b}{c}{a}X^m\\dot\\gamma^bY^\n+ \\chw{b}{c}{a}\\dot X^bY^c + \\chw{b}{c}{a}\\dot\\gamma^b\\delta_XY^c\\\\\n&&\\qquad \n+ \\chw{b}{c}{a}X^b\\dot\\gamma^j\\partial_jY^c +\n  \\chw{b}{c}{a}\\chw{e}{f}{c}X^b\\dot\\gamma^eY^f.\\\\\n(\\nabla_{\\dot\\gamma} D_XY)^a &=& \\dot\\gamma^j(\\partial_j(D_XY)^a +\n  \\chw{b}{c}{a}\\dot\\gamma^b (D_XY)^c)\\\\\n&=& \\dot\\gamma^j\\partial_j(\\delta_XY^a + \\chw{b}{c}{a}X^b Y^c) \n+ \\chw{b}{c}{a}\\dot\\gamma^b(\\delta_XY^c + \\chw{s}{f}{c}X^eY^f)\\\\\n&=& \\dot\\gamma^j\\partial_j\\delta_XY^a + \\dot\\gamma^j\\partial_j\\chw{b}{c}{a}X^bY^c +\n  \\chw{b}{c}{a}\\dot X^bY^c\n+ \\chw{b}{c}{a}X^b\\dot Y^c + \\chw{b}{c}{a}\\dot\\gamma^b\\delta_XY^c \\\\\n&&\\qquad + \\chw{b}{c}{a}\\chw{e}{f}{c}\\dot\\gamma^b X^eY^f.\n\\end{eqnarray*}\nTherefore\n\\begin{eqnarray*} (D_X\\nabla_{\\dot\\gamma}Y - \\nabla_{\\dot\\gamma}D_XY)^a &=& \\partial_m\n  \\chw{b}{c}{a}X^m\\dot\\gamma^bY^c - \\partial_j \\chw{b}{c}{a}\\dot\\gamma^j X^bY^c\n  + \\chw{b}{c}{a}\\chw{e}{f}{c}X^b\\dot\\gamma^e Y^f   \\\\\n&&\\qquad \n-\\chw{b}{c}{a}\\chw{e}{f}{c}\\dot\\gamma^b X^e Y^f \\\\\n&=& (\\partial_j \\Gamma_{bc}^{a} - \\partial_b \\chw{j}{c}{a} +\\chw{j}{e}{a}\\chw{b}{c}{e}-\n\\chw{b}{e}{a}\\chw{j}{c}{e})\\dot\\gamma^b X^j Y^c \\\\\n&=& R_{jbc}^{\\ \\ \\ a}X^j\\dot\\gamma^b Y^c,\n\\end{eqnarray*}\nso \n$$D_X\\nabla_{\\dot\\gamma}Y - \\nabla_{\\dot\\gamma}D_XY =  R(X,\\dot\\gamma)Y.$$\n\\end{proof}\n\n\\begin{cor}\\label{cor:zero}\n At the loop $\\gamma$, $[D_X,\\Delta]Y = -\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) -\n  R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}Y.$  In particular, $[D_X,\\Delta]$ is a zeroth order\n  operator.\n\\end{cor}\n\n\\begin{proof}  \n\\begin{eqnarray*} [D_X,\\Delta]Y &=& (-D_X\\nabla_{\\dot\\gamma}\\ndg + \\nabla_{\\dot\\gamma}\\ndg D_X)Y \\\\\n&=& -(\\nabla_{\\dot\\gamma} D_X\\nabla_{\\dot\\gamma} Y+ R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y) +\\nabla_{\\dot\\gamma}\\ndg D_XY\\\\\n&=& -(\\nabla_{\\dot\\gamma}\\ndg D_XY + \\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) + R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y) \n+\\nabla_{\\dot\\gamma}\\ndg D_XY\\\\\n&=& -\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y.\n\\end{eqnarray*}\n\\end{proof}\n\nNow we complete the proof of Theorem \\ref{old1.6}, showing in the process that $A_XY$ exists. \n\\medskip\n\n\\noindent {\\it Proof of Theorem \\ref{old1.6}.}\n By Proposition \\ref{old1.3} and Corollary \\ref{cor:zero}, we have\n\\begin{eqnarray*} \\nabla_XY &=& D_XY + \\frac{1}{2}(1+\\Delta)^{-1}[D_X,1+\\Delta]Y +\n  (X\\leftrightarrow Y) + A_XY\\\\\n&=& D_XY + \\frac{1}{2}(1+\\Delta)^{-1}(-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y) + \n  (X\\leftrightarrow Y) + A_XY,\n\\end{eqnarray*}\nwhere $(X\\leftrightarrow Y)$ denotes the previous term with $X$ and $Y$ switched.\n\nThe curvature tensor satisfies \n$$-\\langle Z, R(X,Y)W\\rangle = \\langle R(X,Y)Z,W\\rangle = \\langle R(Z,W)X,Y\n\\rangle$$\n pointwise, so\n\\begin{eqnarray*} \\langle A_XY,Z\\rangle_1 &=&\n  -\\frac{1}{2}\\langle[D_Z,1+\\Delta]X,Y\\rangle_0\\\\\n&=& -\\frac{1}{2}\\langle (-\\nabla_{\\dot\\gamma}(R(Z,\\dot\\gamma)X) - R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma} X,Y\\rangle_0\\\\\n&=& -\\frac{1}{2} \\langle R(Z,\\dot\\gamma)X,\\nabla_{\\dot\\gamma} Y\\rangle_0 + \\frac{1}{2}\\langle\n  R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma} X,Y\\rangle_0\\\\\n&=& -\\frac{1}{2} \\langle R(X,\\nabla_{\\dot\\gamma} Y)Z,\\dot\\gamma\\rangle_0 + \\frac{1}{2}\\langle R(\\nabla_{\\dot\\gamma} X,\n  Y)Z,\\dot\\gamma\\rangle_0\\\\\n&=& \\frac{1}{2}\\langle Z, R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma\\rangle_0 - \\frac {1}{2} \\langle Z,\n  R(\\nabla_{\\dot\\gamma} X,Y)\\dot\\gamma\\rangle_0\\\\\n&=&\\frac{1}{2}\\langle Z, (1+\\Delta)^{-1}(R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma)\\rangle_1.\n\\end{eqnarray*}\nThus $A_XY$ must equal $\\frac{1}{2} (1+\\Delta)^{-1}(R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma).$  \nThis makes sense: for $X, Y\\in H^{s'-1}$, \n $A_XY\\in H^{s'}\\subset H^1,$ since $R$ is zeroth order.  \n\n\\hfill$\\Box$\n\\medskip \n\n\n\n\\begin{rem} Locally on $LM$, we should have \n$D_XY = \\delta_X^{LM}Y +  \\omega_X^{LM}(Y)$.  \nNow\n$\\delta_X^{LM}Y$  can only mean $\\frac{d}{d\\tau}|_{\\tau =\n  0}\\frac{d}{d\\epsilon}|_{\\epsilon = 0}\\gamma(\\epsilon,\\tau,\\theta)$, where\n$\\gamma(0,0,\\theta) = \\gamma(\\theta)$, ${d\\epsilon}|_{\\epsilon =\n  0}\\gamma(\\epsilon,0,\\theta) = X(\\theta)$, \n${d\\tau}|_{\\tau = 0}\\gamma(\\epsilon,\\tau,\\theta) = Y_{\\gamma(\\epsilon, 0,\\cdot)}(\\theta).$\n In other words, $\\delta_X^{LM}Y$ equals $ \\delta_XY$.\nSince $D_XY^a = \\delta_XY^a + \\chw{b}{c}{a}(\\gamma(\\theta))$, the connection one-form for the $L^2$ Levi-Civita connection on $LM$ is given by\n$$\\omega^{LM}_X(Y)^a(\\gamma)(\\theta) = \\chw{b}{c}{a}(\\gamma(\\theta))X^bY^c\n= \\omega^M_X(Y)^a(\\gamma(\\theta)).$$\n\\end{rem}\n\nBy this remark, we get\n\\begin{cor}\\label{cor2} The connection one-form $\\omega = \\omega^1$ for $\\nabla^1$ in \nexponential coordinates is\n\\begin{eqnarray}\\label{two}\\omega_X(Y)(\\gamma)(\\theta) &=& \\omega^M_X(Y)(\\gamma(\\theta))\n  +  \n\\frac{1}{2}\\bigl\\{(1+\\Delta)^{-1}\\left[\n-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) - R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma} Y\\right.\\nonumber\\\\\n&&\\qquad \\left. -\\nabla_{\\dot\\gamma}(R(Y,\\dot\\gamma)X) - R(Y,\\dot\\gamma)\\nabla_{\\dot\\gamma} X\\right.\\\\\n&&\\qquad  \\left.+R(X,\\nabla_{\\dot\\gamma} Y)\\dot\\gamma - R(\\nabla_{\\dot\\gamma} X, Y)\\dot\\gamma\\right]\\bigr\\}(\\theta).\\nonumber\n\\end{eqnarray}\n\\end{cor}\n\n\n\\subsection{The Levi-Civita Connection for $s\\in{\\mathbb Z}^+$}\n${}$\n\\medskip\n\nFor $s>\\frac{1}{2}$, the proof of Prop.~\\ref{old1.3} extends directly to give\n\n\\begin{lem} \\label{lem: LCs}\nThe Levi-Civita connection for the $H^s$ metric is given by\n$$\\nabla_X^sY = D_XY + \\frac{1}{2}(1+\\Delta)^{-s}[D_X, (1+\\Delta)^s]Y +\n  \\frac{1}{2}(1+\\Delta)^{-s}[D_Y, (1+\\Delta)^s]X \n+ A_XY,$$\nwhere we assume that for $X, Y\\in H^{s'-1}$, $A_XY\\in H^s$ is characterized by\n\\begin{equation}\\label{axy}\n-\\frac{1}{2}\\langle [D_Z,(1+\\Delta)^s]X,Y\\rangle_0 = \\langle A_XY,Z\\rangle_s.\n\\end{equation}\n\\end{lem}\n\\bigskip\n\nWe now compute the bracket terms.\n\n\\begin{lem}\\label{bracketterms}\nFor $s\\in {\\mathbb Z}^+$, at the loop $\\gamma$,\n\\begin{equation}\\label{bracket}\n[D_X,(1+\\Delta)^s]Y = \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} \\nabla_{\\dot\\gamma}^j(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}Y).\n\\end{equation}\nIn particular, $[D_X,(1+\\Delta)^s]Y$ is a $\\Psi{\\rm DO}$ of order $2s-1$ in either $X$ or $Y$.\n\\end{lem}\n\n\\begin{proof}  The sum over $k$ comes from the binomial expansion of $(1+\\Delta)^s$, so\nwe just need an inductive formula for \n$[D_X,\\Delta^s].$   \nThe case $s=1$ is Proposition \\ref{old1.3}. For the induction step, we have\n\\begin{eqnarray*} [D_X,\\Delta^s] &=& D_X\\Delta^{s-1}\\Delta - \\Delta^sD_X\\\\\n&=& \\Delta^{s-1}D_X\\Delta + [D_X,\\Delta^{s-1}]\\Delta - \\Delta^sD_X\\\\\n&=& \\Delta^sD_X +\\Delta^{s-1}[D_X,\\Delta] + [D_X,\\Delta^{s-1}]\\Delta\n-\\Delta^sD_X\\\\\n&=& \\Delta^{s-1}(-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) -R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}Y)\\\\\n&&\\qquad  -  \n\\sum_{j=0}^{2s-3}(-1)^{s-1}\n\\nabla^j_{\\dot\\gamma}(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-j-1}(-\\nabla^2_{\\dot\\gamma}Y)\\\\\n&=& (-1)^{s-1}(-\\nabla_{\\dot\\gamma}^{2s-1}(R(X,\\dot\\gamma)Y) - (-1)^{s-1}\\nabla_{\\dot\\gamma}^{2s-2}(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}Y)\\\\\n&&\\qquad + \\sum_{j=0}^{2s-3}(-1)^{s}\n\\nabla^j_{\\dot\\gamma}(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-j-1}(-\\nabla^2_{\\dot\\gamma}Y)\\\\\n&=& \\sum_{j=0}^{2s-1}(-1)^s \\nabla_{\\dot\\gamma}^j(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}Y).\n\\end{eqnarray*} \n\\end{proof}\n\nWe check that $A_XY$ is a $\\Psi{\\rm DO}$ in $X$ and $Y$ for $s\\in {\\mathbb Z}^+.$\n\n\\begin{lem} \\label{insert3} For $s\\in{\\mathbb Z}^+$ and fixed $X, Y\\in H^{s'-1}$, $A_XY$ in (\\ref{axy})\n is an explicit $\\Psi{\\rm DO}$ in $X$ and $Y$ of order at most $-1.$\n\n\\end{lem}\n\n\\begin{proof}  By (\\ref{bracket}), for  $j, 2k-1-j \\in \\{0,1,...,2s-1\\}$, a typical term on\nthe left hand side of (\\ref{axy}) is\n\\begin{eqnarray*}   \\ipo{\\nabla^j_{\\dot\\gamma}(R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}X)}{Y} &=& \n(-1)^j\n \\ipo{R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}X}{\\nabla^j_{\\dot\\gamma} Y}\\\\\n&=& (-1)^j\\int_{S^1} g_{i\\ell} (R(Z,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}X)^i(\\nabla^j_{\\dot\\gamma} Y)^\\ell d\\theta\\\\\n&=& (-1)^j\\int_{S^1} g_{i\\ell} Z^k R_{krn}^{\\ \\ \\ i}\\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell d\\theta\\\\\n&=& (-1)^j\n \\int_{S^1} g_{tm}g^{kt}  g_{i\\ell} Z^m R_{krn}^{\\ \\ \\ i}\\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell d\\theta\\\\\n&=& (-1)^j \\ipo{Z}\n{g^{kt} g_{i\\ell} R_{krn}^{\\ \\ \\ i}\\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell\\partial_t}\\\\ \n&=& (-1)^j\\ipo{Z}{R^t_{\\ rn\\ell} \\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell\\partial_t}\\\\ \n&=&(-1)^{j+1} \\ipo{Z}{R^{\\ \\ \\ t}_{n\\ell r} \\dot\\gamma^r (\\nabla_{\\dot\\gamma}^{2k-1-j}X)^n (\\nabla_{\\dot\\gamma}^jY)^\\ell\\partial_t}\\\\\n&=& (-1)^{j+1}\\ipo{Z}{R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma}\\\\\n&=& (-1)^{j+1} \\ips{Z}{(1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma}.\n  \\end{eqnarray*}\n  (In the integrals and inner products, the local expressions are in fact globally defined one-forms on $S^1$, resp.~vector fields along $\\gamma$, so we do not need a partition of unity.)\n$(1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma$ is of order at most $-1$ in either $X$ or $Y$, so this term is in \n$H^{s'}\\subset H^s.$  Thus the last inner product is well defined.  \n\\end{proof}\n\nBy  (\\ref{axy}), (\\ref{bracket}) and the proof of Lemma \\ref{insert3}, we get\n$$A_XY = \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} (-1)^{j+1}   (1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma.$$\nThis gives:\n\n\\begin{thm} \\label{thm:sinz}\nFor $s\\in{\\mathbb Z}^+$, the Levi-Civita connection for the $H^s$ metric at the\nloop $\\gamma$ is given by\n\\begin{eqnarray*} \\nabla_X^sY(\\gamma) &=& D_XY(\\gamma) + \\frac{1}{2}(1+\\Delta)^{-s}\n \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} \\nabla_{\\dot\\gamma}^j(R(X,\\dot\\gamma)\\nabla_{\\dot\\gamma}^{2k-1-j}Y)\\\\\n&&\\qquad + (X\\leftrightarrow Y)\\\\\n  &&\\qquad \n +  \\sum_{k=1}^s(-1)^k\\left(\\begin{array}{c}s\\\\k\\end{array}\\right)\n\\sum_{j=0}^{2k-1} (-1)^{j+1}   (1+\\Delta)^{-s} R(\\nabla_{\\dot\\gamma}^{2k-1-j}X,\\nabla_{\\dot\\gamma}^jY)\\dot\\gamma.\n\\end{eqnarray*}\n\\end{thm}\n\n\\subsection{The Levi-Civita Connection for General $s>\\frac{1}{2}$}\n\n${}$\n\\medskip\n\nIn this subsection, we show that the $H^s$ Levi-Civita connection for general $s>\\frac{1}{2}$ exists in the strong sense of Remark \\ref{lcrem}.\nThe formula is less explicit than in the \n$s\\in {\\mathbb Z}^+$ case, but is good enough for symbol calculations.\n\n\n\n\nBy Lemma \\ref{lem: LCs}, we have to examine the term $A_XY$, which, if it exists, is\n characterized by (\\ref{axy}):\n$$-\\frac{1}{2}\\ipo{[D_Z,(1+\\Delta)^s]X}{Y} = \\ips{A_XY}{Z}$$\nfor $Z\\in H^s$.  As explained in Remark \\ref{lcrem}, we may take\n$X, Y\\in H^{s'-1}.$\nThroughout this section we assume that $s'\\gg s$.\n\nThe following lemma extends Lemma \\ref{bracketterms}.\n\\begin{lem}\\label{pdo}\n (i)   For fixed $Z\\in H^{s'-1}$,  $[D_Z,(1+\\Delta)^s] X$ is a $\\Psi$DO of\n  order $2s-1$ in $X$. For ${\\rm Re}(s)\\neq 0$, the principal symbol of $[D_Z,(1+\\Delta)^s]$ is \n  linear in $s$.\n  \n  (ii) For fixed $X\\in H^{s'-1}$, $[D_Z,(1+\\Delta)^s]X$ is a $\\Psi{\\rm DO}$ \n  of order $2s-1$   in $Z$.\n\\end{lem}\n\nAs usual, ``of order $2s-1$\" means ``of order at most $2s-1.$\"  \n\n\n\n\\begin{proof}\n(i) For $f:LM\\to {\\mathbb C}$, we get $[D_Z,(1+\\Delta)^s]fX = f[D_Z,(1+\\Delta)^s]X$, since $[f,(1+\\Delta)^s]=0.$  \nTherefore, $[D_Z,(1+\\Delta)^s]X$ depends only on $X|_\\gamma.$\n\nBy Lemma \\ref{lem:l2lc}, $D_Z = \\delta_Z + \\Gamma \\cdot Z$ in shorthand exponential \ncoordinates.  The Christoffel symbol term is zeroth order and $(1+\\Delta)^s$ has scalar leading order symbol, so $[\\Gamma\\cdot Z,(1+\\Delta)^s]$ has order $2s-1.$  \n\nFrom the integral expression for \n$(1+\\Delta)^s$, it is immediate that \n\\begin{eqnarray}\\label{immediate}\n[\\delta_Z,(1+\\Delta)^s]X &=& (\\delta_Z(1+\\Delta)^s) X + (1+\\Delta)^s\\delta_Z X - (1+\\Delta)^s\\delta_ZX\\\\\n&=& (\\delta_Z(1+\\Delta)^s) X.\\nonumber\n\\end{eqnarray}\n$\\delta_Z(1+\\Delta)^s$ is a limit of differences of $\\Psi{\\rm DO}$s on bundles isomorphic to $\\gamma^*TM$.\nSince the algebra of $\\Psi{\\rm DO}$s is closed in the Fr\\'echet topology\nof all $C^k$ seminorms\nof symbols and smoothing terms\non compact sets, $\\delta_Z(1+\\Delta)^s$ is a $\\Psi{\\rm DO}.$\n\nSince $(1+\\Delta)^s$ has order $2s$ and has scalar leading order symbol, \n$[D_Z,(1+\\Delta)^s]$ have order $2s-1$.  For later purposes (\\S3.2), we compute some explicit symbols.  \n\nAssume Re$(s)<0.$  As in the construction of $(1+\\Delta)^s$,\n we will compute what the symbol asymptotics\nof $\\delta_Z(1+\\Delta)^s$ should\nbe, and then construct an operator with these asymptotics.\nFrom the functional calculus for unbounded operators, we have\n\\begin{eqnarray}\\label{funcalc}\n\\delta_Z(1+\\Delta)^s &=& \\delta_Z\\left(\\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s(1+\\Delta-\\lambda)^{-1}d\\lambda\\right)\\nonumber\\\\\n&=& \\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s\\delta_Z (1+\\Delta-\\lambda)^{-1}d\\lambda\\\\\n&=& -\\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s (1+\\Delta-\\lambda)^{-1} (\\delta_Z\\Delta)\n(1+\\Delta-\\lambda)^{-1}d\\lambda,\\nonumber\n\\end{eqnarray}\nwhere $\\Gamma$ is a contour around the spectrum of $1+\\Delta$, and the\nhypothesis on $s$ justifies the exchange of $\\delta_Z$ and the integral.  The\noperator $A =\n(1+\\Delta-\\lambda)^{-1} \\delta_Z\\Delta  (1+\\Delta-\\lambda)^{-1}$ is a $\\Psi{\\rm DO}$\nof order $-3$\n with top order symbol\n\\begin{eqnarray*} \\sigma_{-3}(A)(\\theta,\\xi)^\\ell_j &=&\n(\\xi^2-\\lambda)^{-1}\\delta^\\ell_k (-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{\\mu}{k}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{\\mu}{k}\\dot Z^\\nu) \\xi (\\xi^2-\\lambda)^{-1}\\delta^\\mu_j\\\\\n&=&\n(-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\n\\xi (\\xi^2-\\lambda)^{-2}.\n\\end{eqnarray*}\nThus the top order symbol of $\\delta_Z(1+\\Delta)^s$ should be\n\\begin{eqnarray}\\label{tsmo}\n \\sigma_{2s-1}(\\delta_Z(1+\\Delta)^s)(\\theta,\\xi)^\\ell_j\n&=& -\\frac{i}{2\\pi}\\int_\\Gamma\n\\lambda^s (-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\n\\xi (\\xi^2-\\lambda)^{-2} d\\lambda  \\nonumber\\\\\n&=& \\frac{i}{2\\pi}\\int_\\Gamma s\\lambda^{s-1}\n(-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\n\\xi (\\xi^2-\\lambda)^{-1} d\\lambda \\nonumber\\\\\n&=& s(-2Z^i\\partial_i{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot\\gamma^\\nu\n   -2{\\operatorname{ch}}{\\nu}{j}{\\ell}\\dot Z^\\nu)\\xi (\\xi^2-\\lambda)^{s-1}.\n\\end{eqnarray}\nSimilarly, all the terms in the symbol asymptotics for $A$ are of the form\n$B^\\ell_j \\xi^n(\\xi^2-\\lambda)^m$ for some matrices $B^\\ell_j =\nB^\\ell_j(n,m).$   This produces a symbol sequence $\n\\sum_{k\\in {\\mathbb Z}^+}\\sigma_{2s-k}$,  and there exists a  $\\Psi{\\rm DO}$ $P$ with $\\sigma(P) =\n\\sum \\sigma_{2s-k}$.  (As in \\S\\ref{pdoreview}, we \nfirst produce operators $P_i$ on a coordinate cover $U_i$ of $S^1$, \nand then set\n$P = \\sum_i\\phi_iP_i\\psi_i$.) The construction\ndepends on\nthe choice of local coordinates\ncovering $\\gamma$, the partition of unity and cutoff\nfunctions as above, and a cutoff function in $\\xi$; as\nusual, different choices change the operator by a smoothing operator.\nStandard estimates \nshow that $P-\\delta_Z(1+\\Delta)^s$ is a smoothing\noperator, this verifies explicitly that  $\\delta_Z(1+\\Delta)^s$ is a $\\Psi{\\rm DO}$ of order $2s-1.$\n\n\nFor Re$(s) >0$,  motivated by differentiating $(1+\\Delta)^{-s}\\circ(1+\\Delta)^s = {\\rm\n  Id}$, we set\n\\begin{equation}\\label{abc}\n\\delta_Z(1+\\Delta)^s = -(1+\\Delta)^s\\circ\\delta_Z(1+\\Delta)^{-s}\\circ(1+\\Delta)^s.\n\\end{equation}\nThis is again a $\\Psi{\\rm DO}$ of order $2s-1$ with principal symbol linear in $s$. \n\n (ii) As a $\\Psi{\\rm DO}$ of order $2s$, $(1+\\Delta)^s$ has the expression\n $$(1+\\Delta)^s X(\\gamma)(\\theta) = \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi,$$\n where we omit the cover of $S^1$ and its partition of unity on the right hand side.\n Here \n p(\\theta,\\xi)$ is the symbol of $(1+\\Delta)^s$, which has the asymptotic expansion\n $\n p(\\theta,\\xi) \\sim \\sum_{k=0}^\\infty p_{2s-k }(\\theta,\\xi).$$\n The covariant derivative along $\\gamma$ on\n$Y\\in\\Gamma(\\gamma^*TM)$ is given by\n\\begin{eqnarray*}\\frac{DY}{d\\gamma} &=&\n(\\gamma^*\\nabla^{M})_{\\partial_\\theta}(Y) =\n\\partial_\\theta Y + (\\gamma^*\\omega^{M})(\\partial_\\theta)(Y)\\\\\n&=& \\partial_\\theta(Y^i)\\partial_i + \\dot\\gamma^t Y^r\n\\Gamma^j_{tr}\\partial_j,\n\\end{eqnarray*}\nwhere $\\nabla^{M}$ is the Levi-Civita connection on $M$ and $\\omega^{M}$ is the\nconnection one-form in exponential coordinates on $M$. \nFor $\\Delta =\n(\\frac{D}{d\\gamma})^* \\frac{D}{d\\gamma}$, an integration by parts using the\nformula\n$\\partial_tg_{ar} = \\Gamma_{\\ell t}^ng_{rn} + \\Gamma_{rt}^ng_{\\ell n}$  gives\n$$(\\Delta Y)^k = -\\partial^2_\\theta Y^k\n-2\\Gamma_{\\nu\\mu}^k\\dot\\gamma^\\nu\\partial_\\theta Y^\\mu -\\left\n( \\partial_\\theta\\Gamma_{\\nu\\delta}^k\\dot\\gamma^\\nu\n+\\Gamma_{\\nu\\delta}^k\\ddot\\gamma^\\nu +\n\\Gamma_{\\nu\\mu}^k\\Gamma_{\\varepsilon\\delta}^\\mu\\dot\\gamma^\\varepsilon\\dot\\gamma^\\nu\\right)\nY^\\delta.$$\nThus $p_{2s}(\\theta, \\xi) =  |\\xi|^2$ is independent of $\\gamma$, but the lower order symbols  depend on \n derivatives of both $\\gamma$ and the metric on $M$. \n \n We have\n \\begin{eqnarray} [D_Z,(1+\\Delta)^s]X(\\gamma)(\\theta) &=&\n D_Z \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\label{216}\\\\\n &&\\quad - \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) D_ZX(\\gamma)(\\theta')d\\theta' d\\xi. \\label{217}\n \\end{eqnarray}\n In local coordinates, (\\ref{216}) equals\n \\begin{eqnarray} \\label{218}\n \\lefteqn{\n\\left[ D_Z \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\nonumber\\\\\n &=& \\delta_Z\\left[\n \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a(\\theta)\\\\\n &&\\quad + \\Gamma^a_{bc} Z^b(\\gamma)(\\theta) \n \\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^c(\\theta).\\nonumber\n \\end{eqnarray}\n Here we have suppressed matrix indices in $p$ and $X$.\nWe can bring $\\delta_Z$ past the integral on the right hand side of (\\ref{218}).  If\n$\\gamma_\\epsilon$ is a family of curves with $\\gamma_0 = \\gamma, \\dot\\gamma_\\epsilon = Z$, then\n$$\\delta_Zp(\\theta, \\xi) = \\frac{d}{d\\epsilon}\\biggl|_{_{_{\\epsilon=0}}} p(\\gamma_\\epsilon,\n\\theta,\\xi) = \\frac{d\\gamma_\\epsilon^k}{d\\epsilon}\\biggl|_{_{_{\\epsilon=0}}}\n\\partial_k p(\\gamma,\\theta, \\xi) = Z^k(\\gamma(\\theta))\\cdot \\partial_k p(\\theta,\\xi).$$ Substituting this into (\\ref{218}) gives\n\\begin{eqnarray}\\label{219}\n\\lefteqn{\n\\left[ D_Z \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\nonumber\\\\\n&=& \n\\lefteqn{\n\\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n Z^k(\\gamma)(\\theta)\\cdot \\partial_k p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\\\\n &&\\quad \n+ \\Gamma^a_{bc} Z^b(\\gamma)(\\theta) \n \\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^c(\\theta).\\nonumber\\\\\n&&\\qquad + \n \\left[ \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) \\delta_Z X(\\gamma)(\\theta')d\\theta' d\\xi\\right]^c(\\theta).\\nonumber\n\\end{eqnarray}\nSimilarly, (\\ref{217}) equals\n\\begin{eqnarray}\\label{220}\n\\lefteqn{\\left[\\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) D_ZX(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a}\\nonumber\\\\\n &=& \\left[\\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi) \\delta_ZX(\\gamma)(\\theta')d\\theta' d\\xi\\right]^a\\\\\n &&\\quad + \n \\int_{T^*S^1}\n e^{i(\\theta-\\theta')\\cdot \\xi}\n p(\\theta,\\xi)^a_e\\Gamma^e_{bc}Z^b(\\gamma)(\\theta') X^c(\\gamma)(\\theta')d\\theta' d\\xi.\n \\nonumber\n \\end{eqnarray}\n Substituting (\\ref{219}), (\\ref{220}), into (\\ref{216}), (\\ref{217}), respectively, gives\n \\begin{eqnarray}\\label{221}\n\\lefteqn{( [D_Z,(1+\\Delta)^s]X(\\theta))^a}\\\\\n&=&\nZ^b(\\theta)\\cdot\\left[\\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}\\left(\\partial_bp^a_e(\\theta,\\xi) +\n\\Gamma_{bc}^a(\\gamma(\\theta)p_e^c(\\theta, \\xi)\\right)X^e(\\theta')d\\theta'd\\xi\\right]\\nonumber\\\\\n&&\\quad - \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}p(\\theta,\\xi)^a_e\\Gamma_{bc}^e\n(\\gamma(\\theta'))Z^b(\\theta') X^c(\\theta')d\\theta' d\\xi,\\nonumber\n  \\end{eqnarray}\n where $X(\\theta') = X(\\gamma)(\\theta)$ and similarly for Z.\n \n The first term on the right hand side of (\\ref{221}) is order zero in $Z$; note that\n $0<2s-1$, since $s>\\frac{1}{2}$.  For the last term in (\\ref{221}), we do a change of variables typically used in the proof that the composition of $\\Psi{\\rm DO}$s is a $\\Psi{\\rm DO}.$  Set\n \\begin{equation}\\label{221a}q(\\theta, \\theta', \\xi)^a_b = p(\\theta,\\xi)^a_e \\Gamma_{bc}^e(\\gamma(\\theta'))X^c\n (\\theta'),\n \\end{equation}\n so the last term equals\n \\begin{eqnarray*}\n(PZ)^a(\\theta) &\\stackrel{\\rm def}{=}&  \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}q(\\theta, \\theta', \\xi)^a_b Z^b(\\theta') d\\theta' d\\xi\\\\\n &=& \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\xi}q(\\theta, \\theta', \\xi)^a_b e^{i(\\theta'-\\theta'')\n \\cdot\\eta} Z^b(\\theta'') d\\theta'' d\\eta \\ d\\theta' d\\xi,\n \\end{eqnarray*}\n by applying  Fourier transform and its inverse to $Z$. A little algebra gives\n \\begin{equation}\\label{222}\n (PZ)^a(\\theta) = \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot\\eta}r(\\theta,\\eta)^a_b Z^b(\\theta')\n d\\theta' d\\eta,\n \\end{equation}\n with \n \\begin{eqnarray*}r(\\theta, \\eta) &=& \\int_{T^*S^1} e^{i(\\theta-\\theta')\\cdot(\\xi-\\eta)}\n q(\\theta,\\theta', \\xi) d\\theta' d\\xi\\\\\n &=&  \\int_{T^*S^1} e^{it\\cdot\\xi} q(\\theta,\\theta - t, \\eta + \\xi) dt\\  d\\xi.\n \\end{eqnarray*} \n In the last line we continue to abuse notation by treating the integral in local coordinates in \n $t = \\theta-\\theta'$ lying in an interval $I\\subset {\\mathbb R}$ and implicitly\n summing over a cover and partition of unity of $S^1;$ thus we can consider $q$ as a compactly supported function in $t\\in{\\mathbb R}.$\n Substituting in the Taylor expansion of $q(\\theta,\\theta - t, \\eta + \\xi)$ in $\\xi$ gives in local coordinates\n  \\begin{eqnarray}\\label{223a}\n  r(\\theta, \\eta) &=& \\int_{T^*{\\mathbb R}} e^{it\\cdot \\xi} \\left[ \\sum_{\\alpha, |\\alpha|=0}^N\n \\frac{1}{\\alpha!} \\partial_\\xi^\\alpha|_{\\xi=0} q(\\theta,\\theta-t, \\eta+\\xi)\\xi^\\alpha + {\\rm O}\n (|\\xi|^{N+1})\\right] dt \\  d\\xi\\nonumber\\\\\n &=& \\sum_{\\alpha, |\\alpha|=0}^N \\frac{i^{|\\alpha|}}{\\alpha!} \\partial^\\alpha_t\\partial^\\alpha\n _\\xi q(\\theta, \\theta, \\eta) + {\\rm O} (|\\xi|^{N+1}).\n \\end{eqnarray}\n Thus $P$ in (\\ref{222}) is a $\\Psi{\\rm DO}$ with apparent top order symbol \n $q(\\theta, \\theta, \\eta)$, which by (\\ref{221a}) has order $2s.$  The top order symbol can be computed in any local coordinates on $S^1$ and $\\gamma^*TM$.  If we choose\n manifold coordinates (see \\S2.3) which are \nRiemannian normal coordinates centered at $\\gamma(\\theta)$, the Christoffel symbols vanish at this point, and\n so\n $$q(\\theta, \\theta, \\eta)^a_b = p(\\theta,\\xi)^a_e\\Gamma_{bc}^e(\\gamma(\\theta)) X^c(\\theta)\n =0.$$\n Thus $P$ is in fact of order $2s-1$, and so both terms on the right hand side of (\\ref{221}) have order at most $2s-1$.\n\n\\end{proof}\n\n\n\\begin{rem} (i) For $s\\in{\\mathbb Z}^+$,  $\\delta_Z(1+\\Delta)^s$ differs\nfrom the usual definition by a smoothing operator.  \n\n(ii) For all $s$, the proof of Lemma \\ref{pdo}(i) shows that\n$\\sigma(\\delta_Z(1+\\Delta)^s) = \\delta_Z(\\sigma((1+\\Delta)^s)).$\n\n\\end{rem}\n\nWe can now complete the computation of the Levi-Civita connection for general $s.$\n\nLet $[D_\\cdot,(1+\\Delta)^s]X^*$ be the formal $L^2$ adjoint  of $[D_\\cdot,(1+\\Delta)^s]X$.\nWe abbreviate $[D_\\cdot,(1+\\Delta)^s]X^*(Y)$ by $[D_Y,(1+\\Delta)^s]X^*.$\n\n\n\n\\begin{thm}  \\label{thm25} (i) For $s>\\frac{1}{2}$, \nThe Levi-Civita connection for the $H^s$ metric is given by\n\\begin{eqnarray}\\label{quick}\\nabla_X^sY &=& D_XY + \\frac{1}{2}(1+\\Delta)^{-s}[D_X, (1+\\Delta)^s]Y +\n  \\frac{1}{2}(1+\\Delta)^{-s}[D_Y, (1+\\Delta)^s]X \\nonumber\\\\\n&&\\quad -\\frac{1}{2} (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*.\n\\end{eqnarray}\n\n(ii) The connection one-form $\\omega^s$ in exponential coordinates is given by\n\\begin{eqnarray}\\label{223}\\lefteqn{\\omega^s_X(Y)(\\gamma) (\\theta)}\\\\\n&=& \\omega^M(Y)(\\gamma(\\theta)) + \n\\left(\\frac{1}{2}(1+\\Delta)^{-s}[D_X, (1+\\Delta)^s]Y +\n  \\frac{1}{2}(1+\\Delta)^{-s}[D_Y, (1+\\Delta)^s]X \\right.\\nonumber\\\\\n  &&\\quad \\left.\n-\\frac{1}{2} (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*\\right)(\\gamma)(\\theta).\\nonumber\n\\end{eqnarray}\n\n(iii) The connection one-form takes values in zeroth order $\\Psi{\\rm DO}$s.\n\\end{thm}\n\n\\begin{proof}  Since $[D_Z,(1+\\Delta)^s]X$ is a $\\Psi{\\rm DO}$ in $Z$ of order $2s-1$, its formal adjoint is\na $\\Psi{\\rm DO}$ of the same order.  Thus\n$$\\langle [D_Z,(1+\\Delta)^s]X,Y\\rangle_0 = \\langle Z, [D_\\cdot, (1+\\Delta)^s]X^*(Y)\\rangle\n= \\langle Z, (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*\\rangle_s.$$\nThus $A_XY$ in (\\ref{axy}) satisfies\n$A_XY = (1+\\Delta)^{-s}[D_Y,(1+\\Delta)^s]X^*.$  Lemma \\ref{lem: LCs} applies to all $s>\\frac{1}{2}$, \nso (i) follows.  (ii) follows as \nin Corollary \\ref{cor2}.  Since $\\omega^M$ is zeroth order and all \nother terms have order $-1$, (iii) holds as well.\n\\end{proof}\n\n\\begin{rem}  This theorem implies that the Levi-Civita connection exists for the \n$H^s$ metric in the strong sense:  for $X\\in  T_\\gamma LM =H^{s'-1}(\\gamma^*TM)$\nand $Y\\in H^{s'-1}(\\cdot^*TM)$ a smooth vector field on $LM = H^{s'}(S^1,M)$,\n $\\nabla^s_XY(\\gamma)\\in H^{s'-1}(\\gamma^*TM).$  (See Remark\n2.1.)  For each term except $D_XY$ on the right hand side of (\\ref{quick}) is order\n$-1$ in $Y$, and so takes $H^{s'-1}$ to $H^{s'}\\subset H^{s'-1}.$  For $D_XY = \\delta_XY + \\Gamma\\cdot Y$, $\\Gamma$ is zeroth order and so bounded on $H^{s'-1}.$  Finally, \nthe definition of a smooth vector field on $LM$ implies that $\\delta_XY$ stays in $H^{s'-1}$\nfor all $X$.\n\\end{rem}\n\n\n\n\n\n\n\n\\subsection{{\\bf Extensions of the Frame Bundle of $LM$}}\\label{extframe}\n\nIn this subsection we discuss the choice of structure group for the\n$H^s$ and Levi-Civita connections on $LM.$\n\nLet ${\\mathcal H}$ be the Hilbert space\n $H^{s_0}(\\gamma^*TM)$ for \na fixed $s_0$ and $\\gamma.$ \n Let $GL({\\mathcal H})$ be the group of bounded invertible linear\noperators on ${\\mathcal H}$; inverses of elements are bounded by the closed graph\ntheorem.   $GL({\\mathcal H})$ has the subset\ntopology of the norm topology on ${\\mathcal B}({\\mathcal H})$, the bounded linear\noperators on ${\\mathcal H}$.\n$GL({\\mathcal H})$ is an infinite dimensional Banach Lie group, as a group which\nis an open subset of the infinite dimensional Hilbert manifold \n${\\mathcal B}({\\mathcal H})$\n\\cite[p.~59]{Omori}, and has Lie algebra \n${\\mathcal B}({\\mathcal H})$. Let $\\Psi{\\rm DO}_{\\leq 0}, \n\\Psi{\\rm DO}_0^*$ denote the algebra of classical\n$\\Psi{\\rm DO}$s of nonpositive order \nand the group of invertible zeroth order $\\Psi{\\rm DO}$s, respectively,\nwhere all $\\Psi{\\rm DO}$s act on ${\\mathcal H}.$   \nNote that $\\Psi{\\rm DO}_0^*\\subset GL({\\mathcal H}).$\n   \n\\begin{rem} \nThe inclusions of $\\Psi{\\rm DO}_0^*, \\Psi{\\rm DO}_{\\leq 0}$ into $GL({\\mathcal H}), {\\mathcal\n    B}({\\mathcal H})$ are trivially continuous in the subset topology.\nFor the Fr\\'echet topology on $\\Psi{\\rm DO}_{\\leq 0}$, \nthe  inclusion is \ncontinuous as in \\cite{lrst}.\n\\end{rem}\n\nWe recall\nthe relationship between\n the  connection one-form $\\theta$ on the frame bundle $FN$ of a\n manifold $N$\nand\nlocal expressions for the connection on $TN.$ For $U\\subset N$,\n let $\\chi:U\\to FN$ be a local section.\n A metric  connection $\\nabla$ on $TN$ with local\nconnection one-form $\\omega$ determines a connection $\\theta_{FN}\\in\n \\Lambda^1(FN, {\\mathfrak o}(n))$ on $FN$\nby {\\it (i)} $\\theta_{FN}$ is the Maurer-Cartan one-form on each fiber,\nand {\\it (ii) }\n$\\theta_{FN}(Y_u)=\\omega (X_p),$ for $ Y_u=\\chi_*X_p$\n\\cite[Ch.~8, Vol.~II]{Spi}, or equivalently\n$\\chi^*\\theta_{FN} = \\omega.$\n\n\nThis applies to $N=LM.$\nThe frame bundle $FLM\\to LM$ is constructed\nas in the finite dimensional case. The\nfiber over $\\gamma$ is isomorphic to the gauge group ${\\mathcal G}$ of ${\\mathcal R}$\nand fibers are glued by the transition functions for\n$TLM$. Thus the frame bundle is\ntopologically a\n${\\mathcal G}$-bundle.\n\nHowever, by Theorem \\ref{thm25},\nthe Levi-Civita connection one-form $\\omega^s_X$\ntakes\nvalues in $\\Psi{\\rm DO}_{\\leq 0}$. \nThe curvature two-form $\\Omega^{s} = d_{LM}\\omega^{s} + \\omega^{s}\\wedge\n\\omega^s$ also takes values in $\\Psi{\\rm DO}_{\\leq 0}.$  (Here $d_{LM}\\omega^{s}(X,Y)$\nis defined by the Cartan formula for the exterior derivative.)\nThese\nforms should take values in the Lie algebra of the structure\ngroup.  Thus we should extend the structure group to the Fr\\'echet Lie group\n $\\Psi{\\rm DO}_0^*$, since its Lie\nalgebra is $\\Psi{\\rm DO}_{\\leq 0}.$  \nThis leads to an extended\nframe bundles, also denoted $FLM$. The  transition\n functions are unchanged, since \n${\\mathcal G} \\subset \\Psi{\\rm DO}_0^*$.\n Thus $(FLM,\\theta^s)$ as a geometric\nbundle (i.e.~as a  bundle with connection $\\theta^s$ associated to\n$\\nabla^{1,s}$) is a $\\Psi{\\rm DO}_0^*$-bundle.\n\nIn summary, for the Levi-Civita connections we have\n$$ \\begin{array}{ccc}\n{\\mathcal G}&\\longrightarrow &FLM\\\\\n& & \\downarrow\\\\\n& & LM\n\\end{array}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\begin{array}{ccc}\n\\Psi{\\rm DO}_0^*&\\longrightarrow &(FLM,\\theta^s)\\\\\n& & \\downarrow\\\\\n& & LM\n\\end{array}\n$$\n\n\n\\begin{rem}\\label{rem:ext}  If\n we extend the structure group of the frame bundle with\n  connection from $\\Psi{\\rm DO}_0^*$ to\n  $GL({\\mathcal H})$, the frame bundle becomes trivial by Kuiper's theorem.  \n \nThus\nthere is a potential loss of information if \nwe pass to the larger frame\n  bundle. \n\n\nThe situation is similar to the following examples.  Let $E\\to S^1$ be\nthe $GL(1,{\\mathbb R})$ (real line)\n bundle with gluing functions (multiplication by) $1$ at $1\\in\nS^1$ and $2$ at $-1\\in S^1.$  $E$ is trivial as a $GL(1,{\\mathbb R})$-bundle, \nwith global section $f$ with $\\lim_{\\theta\\to -\\pi^+}f(e^{i\\theta}) = 1, \nf(1) = 1,\n\\lim_{\\theta\\to\\pi^-}f(e^{i\\theta}) = 1\/2.$  \nHowever, as a $GL(1,{\\mathbb Q})^+$-bundle, $E$ is nontrivial, as a\nglobal section is locally constant. As a second example,\n let $E\\to M$ be a nontrivial\n$GL(n,{\\mathbb C})$-bundle. Embed ${\\mathbb C}^n$ into a Hilbert space ${\\mathcal H}$, and extend $E$\nto an $GL({\\mathcal H})$-bundle ${\\mathcal E}$ \nwith fiber ${\\mathcal H}$ and with the  transition functions for $E$ (extended by the identity in\ndirections perpendicular to the image of $E$).  Then ${\\mathcal E}$ is\ntrivial.\n\n\\end{rem}\n\n\n\n\\section{{\\bf Local Symbol Calculations}}\\label{localsymbols}\n\nIn this section, we  compute the $0$ and $-1$ order symbols of the\nconnection one-form and the curvature two-form of\n the $s=1$ Levi-Civita connection. \n We also compute the $0$ and $-1$ order symbols of the\nconnection one-form for the general $s>\\frac{1}{2}$ connection, and the $0$ order symbol of the \ncurvature of the general $s$ connection.\n These results are used in the calculations of Wodzicki-Chern-Simons\nclasses in \\S6. The formulas  show that\nthe $s$-dependence of these symbols is\nlinear, which will be used to define regularized Wodzicki-Chern-Simons classes\n(see Definition \\ref{def:regularized}).\n\n\\subsection{{\\bf Connection and Curvature Symbols for $s=1$}}\n${}$\n\\medskip\n\nIn this subsection $\\omega = \\omega^1, \\Omega = \\Omega^1.$\n\nUsing Corollary \\ref{cor2}, we can compute these symbols easily. \n\n\\begin{lem} \\label{old2.1}\n(i) At $\\gamma(\\theta)$,\n$\\sigma_0(\\omega_X)^a_b =  (\\omega^M_X)^a_b = \\chw{c}{b}{a}X^c.$\n\n(ii)  \\begin{eqnarray*}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\omega_X) &=& \\frac{1}{2}(-2R(X,\\dot\\gamma)\n-R(\\cdot,\\dot\\gamma)X + R(X,\\cdot)\\dot\\gamma).\n\\end{eqnarray*}\nEquivalently,\n\\begin{eqnarray*}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\omega_X)^a_b &=& \n\\frac{1}{2}\n(-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\  a})X^c\\dot\\gamma^d.\n\\end{eqnarray*}\n\\end{lem}\n\n\\begin{proof} (i) For $\\sigma_0(\\omega_X)$, the only term in (\\ref{two}) of order\n  zero is the Christoffel term.  \n\n(ii) For $\\sigma_{-1}(\\omega_X)$, label the last six terms on the right hand side of\n(\\ref{two}) by (a), ..., (f).  By Leibniz rule for the tensors, the only\nterms of order $-1$ come from:\nin (a), $-\\nabla_{\\dot\\gamma}(R(X,\\dot\\gamma)Y) = -R(X, \\dot\\gamma) \\nabla_{\\dot\\gamma}Y +$ lower order in\n$Y$;\nin (b), the term  $-R(X, \\dot\\gamma) \\nabla_{\\dot\\gamma}Y$;\nin (c), the term $-R(\\nabla_{\\dot\\gamma}Y, \\dot\\gamma)X$;\nin (e), the term $R(X,\\nabla_{\\dot\\gamma}Y)\\dot\\gamma.$\n\n\nFor any vectors $Z, W$, the curvature endomorphism $R(Z, W): TM\\to TM$ has\n$$R(Z,W)^a_b = R_{c d b}^{\\ \\ \\ a}Z^cW^d.$$\n  Also, since $(\\nabla_{\\dot\\gamma}Y)^a =\n\\frac{d}{d\\theta}Y^a $ plus zeroth order terms,\n$\\sigma_{1}(\\nabla_{\\dot\\gamma}) =\ni\\xi\\cdot Id.$\nThus in (a) and (b), \n$\\sigma_1(-R(X, \\dot\\gamma) \\nabla_{\\dot\\gamma})^a_b = -R_{cdb}^{\\ \\ \\ a}X^c\\dot\\gamma^d\\xi.$\n\nFor (c), we have $-R(\\nabla_{\\dot\\gamma}Y, \\dot\\gamma)X = -R_{cdb}^{\\ \\ \\\n  a}(\\nabla_{\\dot\\gamma}Y)^c\\dot\\gamma^d X^b\\partial_a$, so the top order symbol is\n$-R_{cdb}^{\\ \\ \\ a}\\xi\\dot\\gamma^dX^b = -R_{bdc}^{\\ \\ \\ a}\\xi\\dot\\gamma^d X^c.$\n\nFor (e), we have $R(X,\\nabla_{\\dot\\gamma}Y)\\dot\\gamma = R_{cdb}^{\\ \\ \\ a}X^c(\\nabla_{\\dot\\gamma}Y)^d\n\\dot\\gamma^b\\partial_a$, so the top order symbol is \n$R_{cdb}^{\\ \\ \\ a}X^c\\xi \\dot\\gamma^b = R_{cbd}^{\\ \\ \\ a}X^c\\xi \\dot\\gamma^d.$\n\nSince the top order symbol of $(1+\\Delta)^{-1}$ is $|\\xi|^{-2}$, adding these four terms\nfinishes the proof. \n\\end{proof}\n \n We now compute the top symbols of the curvature tensor.  $\\sigma_{-1}(\\Omega)$ involves\n the covariant derivative of the curvature tensor on $M$, but fortunately this symbol\n will not be needed in Part II.\n \n\n\\begin{lem}\\label{old2.2}\n(i) \n$\\sigma_0(\\Omega(X,Y))^a_b =  R^M(X,Y)^a_b = R_{cdb}^{\\ \\ \\ a}X^cY^d.$\n\n(ii) \\begin{eqnarray*}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\Omega(X,Y)) &=&\n\\frac{1}{2}\\left(\\nabla_X[-2R(Y,\\dot\\gamma) - R(\\cdot,\\dot\\gamma)Y +\n  R(Y,\\cdot)\\dot\\gamma]\\right.\\\\\n&&\\qquad \\left. - (X\\leftrightarrow Y) \\right.\\\\\n&&\\qquad\\left. - [-2R([X,Y],\\dot\\gamma) -R(\\cdot,\\dot\\gamma)[X,Y] + R([X,Y],\\cdot)\\dot\\gamma] \\right).\n\\end{eqnarray*}\nEquivalently, in Riemannian normal coordinates on $M$ centered at $\\gamma(\\theta)$,\n\\begin{eqnarray}\\label{moc}\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\Omega(X,Y))^a_b &=& \\frac{1}{2}\nX[(-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a})\\dot\\gamma^d]Y^c - (X\\leftrightarrow Y)\\nonumber\\\\\n&=&\\frac{1}{2}\nX[-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}]\\dot\\gamma^dY^c  -(X\\leftrightarrow Y)\\\\\n&&\\qquad +\n\\frac{1}{2}[-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}]\\dot X^dY^c - (X\\leftrightarrow Y)\\nonumber\n\\end{eqnarray}\n\\end{lem}\n\n\n\n\n\n\\begin{proof} \n(i)\n\\begin{eqnarray*} \\sigma_0(\\Omega(X,Y))^a_b &=& \\sigma_0((d\\omega +\n  \\omega\\wedge\\omega)(X,Y))^a_b\\\\\n&=& [(d\\sigma_0(\\omega) + \\sigma_0(\\omega)\\wedge\\sigma_0(\\omega))(X,Y)]^a_b\\\\\n&=& [(d\\omega^M + \\omega^M\\wedge\\omega^M)(X,Y)]^a_b\\\\\n&=& R^M(X,Y)^a_b = R_{cdb}^{\\ \\ \\ a}X^cY^d.\n\\end{eqnarray*}\n\n(ii) Since $\\sigma_0(\\omega_X)$ is independent of $\\xi$, after dividing by\n$i|\\xi|^{-2}\\xi$ we have\n\\begin{eqnarray*}\\sigma_{-1}(\\Omega(X,Y))^a_b &=& (d\\sigma_{-1}(\\omega)\n  (X,Y))^a_b + \\sigma_0(\\omega_X)^a_c\\sigma_{-1}(\\omega_Y)^c_b\n+ \\sigma_{-1}(\\omega_X)^a_c\\sigma_{0}(\\omega_Y)^c_b\\\\\n&&\\qquad\n-\\sigma_0(\\omega_Y)^a_c\\sigma_{-1}(\\omega_X)^c_b\n+ \\sigma_{-1}(\\omega_Y)^a_c\\sigma_{0}(\\omega_X)^c_b.\n\\end{eqnarray*}\nAs an operator on sections of $\\gamma^*TM$, \n$\\Omega^{LM} - \\Omega^M$ has order $-1$  so $\\sigma_{-1}(\\Omega^{LM})\n= \\sigma_{-1}(\\Omega^{LM} -\\Omega^M)$ is independent of coordinates.\nIn Riemannian normal coordinates at $\\gamma(\\theta)$, $\\sigma_0(\\omega_X) = \\sigma_0(\\omega_Y) = 0$, so\n\\begin{eqnarray*}\\sigma_{-1}(\\Omega(X,Y))^a_b &=&\n  (d\\sigma_{-1}(\\omega)(X,Y))^a_b\\\\\n&=& X(\\sigma_{-1}(\\omega_Y))^a_b - Y(\\sigma_{-1}(\\omega_X))^a_b\n  -\\sigma_{-1}(\\omega_{[X.Y]})^a_b\\\\\n&=& \\frac{1}{2} X[(-2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}]Y^c\\dot\\gamma^d] - (X\\leftrightarrow Y)\\\\\n&&\\qquad -\\frac{1}{2}( -2R_{cdb}^{\\ \\ \\ a} -R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\\n  a}][X,Y]^c\\dot\\gamma^d.\n\\end{eqnarray*}\nThe terms involving $X(Y^c) - Y(X^c) - [X,Y]^c$ cancel (as they must, since the symbol two-form \ncannot involve derivatives of $X$ or $Y$).  Thus\n$$\\sigma_{-1}(\\Omega(X,Y))^a_b = \\frac{1}{2} X[(-2R_{cdb}^{\\ \\ \\ a} \n-R_{bdc}^{\\ \\ \\ a} + R_{cbd}^{\\ \\ \\  a})Y^c\\dot\\gamma^d] - (X\\leftrightarrow Y).$$\n\nThis gives the first coordinate expression in (\\ref{moc}). The second expression follows from\n $X(\\dot\\gamma^d) = \\dot X^d $ (see (\\ref{badterms})).\n\nTo convert from the coordinate expression to the covariant expression, we follow the \nusual procedure of changing ordinary derivatives to covariant derivatives and adding bracket terms.  For example,\n\\begin{eqnarray*}\\nabla_X(R(Y,\\dot\\gamma)) &=& (\\nabla_XR)(Y,\\dot\\gamma) + R(\\nabla_XY,\\dot\\gamma)\n  + R(Y,\\nabla_X\\dot\\gamma) \\\\\n&=& X^iR_{cdb\\ ;i}^{\\ \\ \\ a}Y^c\\dot\\gamma^d + R(\\nabla_XY,\\dot\\gamma) + R_{cdb}^{\\ \\ \\\n    a}Y^c(\\nabla_X\\dot\\gamma)^d.\n\\end{eqnarray*}\nIn Riemannian normal coordinates at\n$\\gamma(\\theta)$, we have $X^iR_{cdb\\ ;i}^{\\ \\ \\ a} = X^i\\partial_i R_{cdb}^{\\\n  \\ \\ a} = X(R_{cdb}^{\\ \\ \\ a})$ and $(\\nabla_X\\dot\\gamma)^d =  X(\\dot\\gamma^d).$ \n \n  Thus \n  $$\\nabla_X(R(Y,\\dot\\gamma)) -(X\\leftrightarrow Y) - R([X,Y],\\dot\\gamma) = X(R_{cdb}^{\\ \\ \\ a}\\dot\\gamma^d)Y^c - (X\\leftrightarrow Y).$$\n  The other terms are handled similarly.\n  \\end{proof} \n\n\\subsection{{\\bf Connection and Curvature Symbols for General $s$}}\n${}$\n\\medskip\n\nThe noteworthy feature of these computations is the linear dependence of $\\sigma_{-1}(\\omega^{s})$ on $s$.  \n\nLet $g$ be the Riemannian metric on $M$.\n\n\\begin{lem}\\label{lem:33}\n(i) At $\\gamma(\\theta)$,\n$\\sigma_0(\\omega^s_X)^a_b =  (\\omega^M_X)^a_b = \\chw{c}{b}{a}X^c.$\n\n(ii) $\\sigma_0(\\Omega^s(X,Y))^a_b =  R^M(X,Y)^a_b = R_{cdb}^{\\ \\ \\ a}X^cY^d.$\n\n\n(iii)  $\n\\frac{1}{i|\\xi|^{-2}\\xi}\\sigma_{-1}(\\omega^s_X)^a_b = s T(X,\\dot\\gamma, g)$,\nwhere $T(X, \\dot\\gamma, g)$  is tensorial and independent of $s$. \n\\end{lem}\n\n\n\\begin{proof} (i)  By Lemma \\ref{pdo}, the only term of order zero in (\\ref{223}) \n is $\\omega^M_X.$\n\n(ii)  The proof of Lemma \\ref{old2.2}(ii) carries over.  \n\n(iii) By Theorem \\ref{thm25}, we have to compute $\\sigma_{2s-1}$ for $[D_X,(1+\\Delta)^s]$, \n$[D_\\cdot,(1+\\Delta)^s]X$, and $[D_\\cdot,(1+\\Delta)^s]X^*$, as \n$\\sigma_{-1} ((1+\\Delta)^{-s}[D_X,(1+\\Delta)^s]) =  |\\xi|^{-2s}\\sigma_{-1}([D_X,(1+\\Delta)^s])$, etc.\n\nWrite $D_X = \\delta_X + \\Gamma\\cdot X$ in shorthand.  Since $(1+\\Delta)^s$ has scalar leading order symbol, $[\\Gamma\\cdot X,(1+\\Delta)^s]$ has order $2s-1.$  Thus\nwe can compute $\\sigma_{2s-1}([\\Gamma\\cdot X,(1+\\Delta)^s])$ in any coordinate system.  \nIn  Riemannian normal coordinates centered at $\\gamma(\\theta)$, as in the proof of Lemma \\ref{pdo}(ii),  the Christoffel symbols vanish.\nThus $\\sigma_{2s-1}([\\Gamma\\cdot X,(1+\\Delta)^s]) =0.$\n\nBy (\\ref{tsmo}), $\\sigma_{2s-1}([\\delta_X,(1+\\Delta)^s])$ is $s$ times a tensorial expression in $X, \\dot\\gamma, g$,\nsince $\\partial_i\\Gamma_{\\nu j}^{\\ell} = \\frac{1}{3}(R_{i\\nu j}^{\\ \\ \\ \\ell} +\nR_{ij\\nu }^{\\ \\ \\ \\ell})$ in normal coordinates.  The term with $\\Gamma$ vanishes, so \n$\\sigma_{2s-1}( [D_X,(1+\\Delta)^s]) $ is $s$ times this tensorial expression.\n\nThe argument for $\\sigma_{2s-1}([D_\\cdot,(1+\\Delta)^s]X$ is similar.  The term\nwith \n$\\Gamma  $ vanishes. \nBy (\\ref{222}), (\\ref{223a}), \n$$\\sigma_{2s-1}([\\delta_\\cdot,(1+\\Delta)^s]X)^a_b = \ni\\sum_j\\partial_t^j\\partial_\\xi^j|_{t=0, \\xi=0} (p(\\theta, \\xi)^a_e\\Gamma_{bc}^e(\\gamma-t, \\eta +\\xi) X^c(\\theta-t)).$$\nBy (\\ref{tsmo}), the right hand side is linear in $s$ for Re$(s) <0$.  By (\\ref{abc}), this implies \nthe linearity in $s$ for Re$(s)>0.$  \n\nSince $\\sigma_{2s-1}([D_\\cdot,(1+\\Delta)^s]X^*) = (\\sigma_{2s-1}([D_\\cdot, (1+\\Delta)^s]X))^*$, this \nsymbol is also linear in $s$.\n\\end{proof}  \n\n\n\\section{{\\bf The Loop Group Case}}\n\nIn this section, we relate our work to Freed's work on based loop groups\n$\\Omega G$\n\\cite{Freed}.  We find a particular representation of the loop algebra that\ncontrols \nthe order of the curvature of the $H^1$ metric on $\\Omega G.$\n\n\n$\\Omega G\\subset LG$ has tangent space $T_\\gamma\\Omega G\n= \\{X\\in T_\\gamma LG: X(0) = X(2\\pi) = 0\\}$ in some Sobolev topology.  \nInstead of using \n$D^2\/d\\gamma^2$ to define the Sobolev spaces, the usual choice is\n$\\Delta_{S^1} = -d^2\/d\\theta^2$ coupled to the\nidentity operator on the Lie algebra ${\\mathfrak g}$.  Since this operator has\nno kernel on $T_\\gamma\\Omega M$, \n$1 + \\Delta$ is replaced by\n $\\Delta$.  These changes in the $H^s$ inner product\ndo not alter the spaces of Sobolev sections, but the $H^s$ metrics on $\\Omega G$ are \nno longer induced from a metric on $G$ as in the previous sections.\n\n\nThis simplifies the calculations of the Levi-Civita connections.\nIn particular,\\\\\n $[D_Z,\\Delta^s] = 0$, so there is no term $A_XY$ as in (\\ref{axy}).  \nAs a result, one can work directly with the six term formula (\\ref{5one}).\nFor $X, Y, Z$ left\ninvariant vector fields, the first three terms on the right hand side of \n(\\ref{5one}) vanish. Under the standing assumption that $G$ has a left\ninvariant, \nAd-invariant inner product,\none obtains\n$$2\\nabla^{(s)}_XY = [X,Y] + \\Delta^{-s}[X,\\Delta^sY] +\n\\Delta^{-s}[Y,\\Delta^sX]$$\n\\cite{Freed}.  \n\n\nIt is an interesting question to compute the order of the curvature operator\nas a function of $s$.  For based loops, Freed proved that this order is at\nmost $-1$.  In \\cite{andres}, it is\nshown that the order of $\\Omega^s$ is at most $-2$ for all $s\\neq 1\/2, 1$ on\nboth $\\Omega G$ and $LG$, and is exactly $-2$ for $G$ nonabelian.  \n For the case $s=1$, we have a much stronger result.  \n\n\n\\begin{prop} The curvature of the\nLevi-Civita connection for the $H^1$ inner product on $\\Omega\nG$ associated to $-\\frac{d^2}{d\\theta^2}\\otimes {\\rm Id}$ is a $\\Psi{\\rm DO}$ of order $-\\infty.$\n\\end{prop}\n\n\\noindent {\\sc Proof:}\nWe give two quite different proofs.  \n\nBy \\cite{Freed}, the $s=1$ curvature operator $\\Omega = \\Omega^{1}$\nsatisfies\n$$\\left\\langle \\Omega(X,Y)Z,W\\right\\rangle_1 = \\left(\\int_{S^1}[Y,\\dot\nZ],\\int_{S^1}[X,\\dot W]\\right)_{\\mathfrak g} - (X\\leftrightarrow Y),$$\nwhere  the inner product is the Ad-invariant form on the Lie algebra ${\\mathfrak g}$.  We\nwant to write the right hand side of \nthis equation as an $H^1$ inner product with $W$, in\norder to recognize $\\Omega(X,Y)$ as a $\\Psi{\\rm DO}.$\n\n\nLet $\\{e_i\\}$ be an orthonormal basis of ${\\mathfrak g}$, considered\nas a left-invariant frame of $TG$ and as global sections of $\\gamma^*TG.$\n Let $\\cc{i}{j}{k} = ([e_i,e_j],\ne_k)_{\\mathfrak g}$ be the structure constants of ${\\mathfrak g}.$\n(The Levi-Civita connection on left invariant vector fields\nfor the left invariant metric is\ngiven by $\\nabla_XY = \\frac{1}{2}[X,Y]$, so the structure constants\nare twice the Christoffel symbols.)  For $X = X^ie_i =\nX^i(\\theta)e_i, Y = Y^je_j,$ etc., integration by parts \ngives\n$$\\left\\langle\\Omega(X,Y)Z,W\\right\\rangle_1 = \\left(\\int_{S^1} \\dot\nY^iZ^jd\\theta\\right)\\left( \\int_{S^1}\\dot X^\\ell W^m d\\theta\\right)\n\\cc{i}{j}{k}\\cc{\\ell}{m}{n}\\delta_{kn} - (X\\leftrightarrow Y).$$\nSince\n$$\\int_{S^1}\\cc{\\ell}{m}{n}\\dot X^\\ell W^m =\n\\int_{S^1}\\left(\\delta^{mc}\\cc{\\ell}{c}{n}\\dot X^\\ell\ne_m,W^be_b\\right)_{\\mathfrak g} = \\left\n\\langle \\Delta^{-1}(\\delta^{mc}\\cc{\\ell}{c}{n} \\dot X^\\ell e_m),\nW\\right\\rangle_1,$$\nwe get\n\\begin{eqnarray*}\n\\langle\\Omega(X,Y)Z,W\\rangle_1 &=& \\left\\langle\n \\left[\\int_{S^1} \\dot Y^i Z^j\\right]\n\\cc{i}{j}{k}\\delta_{kn}\\delta^{ms}\\cc{\\ell}{s}{n} \\Delta^{-1}(\\dot\nX^\\ell e_m),W\\right\\rangle_1- (X\\leftrightarrow Y)\\\\\n&=&\\left\\langle \\left[ \\int_{S^1}\na_j^k(\\theta,\\theta')Z^j(\\theta')d\\theta'\\right] e_k,W\\right\\rangle_1,\n\\end{eqnarray*}\nwith\n\\begin{equation}\\label{a}a_j^k(\\theta,\\theta') = \\dot Y^i(\\theta')\n\\cc{i}{j}{r}\\delta_{rn}\\delta^{ms}\\cc{\\ell}{s}{n} \n\\left( \\Delta\n^{-1}( \\dot X^\\ell\ne_m)\\right)^k(\\theta) - (X\\leftrightarrow Y).\n\\end{equation}\n\n\n\n\nWe now show that $Z\\mapsto \\left(\\int_{S^1}\na_j^k(\\theta,\\theta')Z^j(\\theta')d\\theta'\\right)e_k$ is a smoothing\noperator.  Applying Fourier transform and Fourier inversion to $Z^j$\nyields\n\\begin{eqnarray*} \\int_{S^1} a_j^k(\\theta,\\theta')Z^j(\\theta')d\\theta'\n&=& \\int_{S^1\\times{\\mathbb R}\\times S^1}\na_j^k(\\theta,\\theta')e^{i(\\theta'\n-\\theta'')\\cdot\\xi}Z^j(\\theta'')d\\theta''d\\xi d\\theta'\\\\\n&=&\n\\int_{S^1\\times{\\mathbb R}\\times S^1} \\left[ a_j^k(\\theta,\\theta')e^{-i(\\theta\n-\\theta')\\cdot\\xi}\\right]e^{i(\\theta\n-\\theta'')\\cdot\\xi}Z^j(\\theta'')d\\theta''d\\xi d\\theta',\n\\end{eqnarray*}\nso $\\Omega(X,Y)$ is a $\\Psi{\\rm DO} $ with symbol \n\\begin{equation}\\label{b} b_j^k(\\theta,\\xi) =\n\\int_{S^1} a_j^k(\\theta,\\theta') e^{i(\\theta-\\theta')\\cdot\\xi} d\\theta',\n\\end{equation}\nwith the usual mixing of local and global notation.\n\nFor fixed $\\theta$,\n(\\ref{b})  contains the Fourier transform of $\\dot Y^i(\\theta')$  and $\\dot X^i(\\theta')$, as\nthese are the only $\\theta'$-dependent terms in (\\ref{a}).\nSince the\nFourier transform is taken in a local chart with respect to a\npartition of unity, and since in each chart $\\dot Y^i$ and $\\dot X^i$ times the\npartition of unity function is compactly supported, the Fourier\ntransform of $a_j^k$ in each chart is rapidly decreasing.  Thus\n$b_j^k(\\theta,\\xi)$ is the product of a rapidly decreasing function\nwith $e^{i\\theta\\cdot\\xi}$, and hence is of order $-\\infty.$\n\n\nWe now give a second proof.  For all $s$,\n$$\\nabla_X Y = \\frac{1}{2}[X,Y] -\\frac{1}{2} \\Delta^{-s}[\\Delta^sX,Y]\n+\\frac{1}{2}\\Delta^{-s}[X,\\Delta^sY].$$\nLabel the terms on the right hand side (1) -- (3).\n As an operator on $Y$ for fixed $X$, the symbol of (1) is\n$\\sigma((1))^a_\\mu = \\frac{1}{2}X^ec_{\\varepsilon\\mu}^a.$\n Abbreviating $\\xii{-s}$ by $\\xi^{-2s}$, we have\n\\begin{eqnarray*} \\sigma((2))^a_\\mu &\\sim & -\\frac{1}{2}c_{\\varepsilon\\mu}^a\n\\left[ \\xi^{-2s}\\Delta^sX^\\varepsilon -\\frac{2s}{i}\\xi^{-2s-1}\n\\partial_\\theta\\Delta^s X^\\varepsilon  \\right.\\\\\n&&\\ \\ \\  \\left. +\\sum_{\\ell=2}^\\infty\\frac{(-2s)(-2s-1)\n\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!}\\xi^{-2s-\\ell}\n\\partial_\\theta^\\ell\\Delta^s X^\\varepsilon \\right]\\\\\n\\sigma((3))^a_\\mu &\\sim &  \\frac{1}{2}c_{\\varepsilon\\mu}^a\n\\left[ X^\\varepsilon+ \\sum_{\\ell=1}^\\infty \\frac{(-2s)(-2s-1)\n\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!} \\xi^{-\\ell}\\partial_\\theta^\\ell X^\\varepsilon\\right].\n\\end{eqnarray*}\nThus\n\\begin{eqnarray}\\label{fourone}\n\\sigma(\\nabla_X)^a_\\mu  &\\sim& \\frac{1}{2}c_{\\varepsilon\\mu}^a\\left[ 2X^\\varepsilon\n   -\\xi^{-2s}\\Delta^sX^\\varepsilon\n+\\frac{2s}{i}\n\\xi^{-2s-1}\\partial_\\theta\\Delta^sX^\\varepsilon\\right. \\nonumber\\\\\n&&\\ \\ \\\n   -\\sum_{ \\ell=2}^\\infty\\frac{(-2s)(-2s-1)\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!}\n\\xi^{-2s-\\ell}\\partial_\\theta^\\ell\\Delta^s X^\\varepsilon \\\\\n&&\\ \\ \\  \\left. + \\sum_{\\ell=1}^\\infty \\frac{(-2s)(-2s-1)\n\\ldots(-2s-\\ell+1)}{i^\\ell \\ell!} \\xi^{-\\ell}\\partial_\\theta^\\ell\n X^\\varepsilon. \\right].\\nonumber\n\\end{eqnarray}\n\nSet $s=1$ in (\\ref{fourone}), and replace $\\ell$\nby\n$\\ell-2$ in the first infinite sum.  Since $\\Delta = -\\partial_\\theta^2$, a\nlittle algebra gives\n\\begin{equation}\\label{fourtwo}\n\\sigma(\\nabla_X)^a_\\mu \\sim c_{\\varepsilon\\mu}^a\\sum_{\\ell=0}^\\infty\n\\frac{(-1)^\\ell}{i^\\ell}\n\\partial_\\theta^\\ell X^\\varepsilon\\xi^{-\\ell}\n=  {\\operatorname{ad\\,}}\\left( \\sum_{\\ell=0}^\\infty\n\\frac{(-1)^\\ell}{i^\\ell}\\partial_\\theta^\\ell\nX\\xi^{-\\ell}\n\\right).\n\\end{equation}\n\nDenote the infinite sum in the last term of (\\ref{fourtwo})\nby $W(X,\\theta,\\xi)$. The map\n$X\\mapsto W(X,\\theta,\\xi)$ takes the  Lie algebra of left invariant vector\nfields on $LG$ to the Lie algebra\n$L{\\mathfrak g}[[\\xi^{-1}]], $\nthe space of formal $\\Psi{\\rm DO}$s of nonpositive integer order on the trivial bundle\n$S^1\\times{\\mathfrak g} \\to S^1$, where the Lie bracket on the\ntarget involves multiplication of power series and bracketing in\n${\\mathfrak g}.$  We claim that this map is a Lie algebra homomorphism.\nAssuming this, we see that\n\\begin{eqnarray*} \\sigma\\left(\\Omega(X,Y)\\right) &=&\n  \\sigma\\left([\\nabla_X,\\nabla_Y] -\\nabla_{[X,Y]}\\right)\n\\sim \\sigma\\left( [{\\operatorname{ad\\,}} W(X), {\\operatorname{ad\\,}} W(Y)] - {\\operatorname{ad\\,}} W([X,Y]) \\right)\\\\\n&=& \\sigma\\left( {\\operatorname{ad\\,}} ( [W(X), W(Y)]) - {\\operatorname{ad\\,}} W([X,Y]) \\right) = 0,\n\\end{eqnarray*}\nwhich proves that $\\Omega(X,Y)$ is a smoothing operator.\n\nTo prove the claim,\nset $X = x^a_n\\eff{n}e_a, Y =y^b_m\\eff{m}e_b$.\nThen\n\\begin{eqnarray*} W([X,Y]) &=&\n W( x^ny^m\\eff{(n+m)}c_{ab}^k e_k) =\\sum_{\\ell=0}^\\infty \\frac{(-1)^\\ell}\n{i^\\ell } c_{ab}^k \\partial_\\theta^\\ell\n \\left(x^a_ny^b_m\\eff{(n+m)}\\right) \\xi^{-\\ell}e_k\\\\\n  {[} W(X)  ,   W(Y)]\n&=& \\sum_{\\ell=0}^\\infty \\sum_{p+q = \\ell}\n\\frac{(-1)^{p+q}}{i^{p+q}} \\partial_\\theta^p \\left(\nx^a_n\\eff{n}\\right) \\partial_\\theta^q\n\\left( y^b_m\\eff{m}\\right)\\xi^{-(p+q)}c_{ab}^k e_k,\n\\end{eqnarray*}\nand these two sums are clearly equal.\n\\hfill $\\Box$\n\n\\bigskip\n\nIt would be interesting to understand how the map $W$ fits into the\nrepresentation theory of the loop algebra $L{{\\mathfrak g}}.$ \n\\bigskip\n\n\\large\n\\noindent {{\\bf Part II. Characteristic Classes on $LM$}}\n\\normalsize\n\\bigskip\n\n\nIn this part, we construct a general theory of Chern-Simons\nclasses on certain infinite rank bundles including the frame\/tangent bundle of \nloop spaces,\nfollowing the construction of primary characteristic classes\nin \\cite{P-R2}. The primary  classes vanish on the tangent bundles of\nloop spaces, which forces the\nconsideration of secondary classes. \nThe key ingredient is to replace the ordinary matrix trace in the Chern-Weil\ntheory of\n invariant polynomials\non  finite dimensional Lie groups with the Wodzicki residue on invertible bounded\n$\\Psi{\\rm DO}$s.  \n\nAs discussed in the Introduction, there are absolute and relative versions of Chern-Simons theory.  We use the relative version, which assigns an odd degree form to a pair \nof connections.\nIn particular, for $TLM$, we can use the $L^2$ (i.e. $s=0$) and\n $s=1$ Levi-Civita connections to form Wodzicki-Chern-Simons (WCS) classes associated to a metric on $M$. \n \n In \\S\\ref{CSCLS}, we develop the general theory of Wodzicki-Chern and WCS classes for \n bundles with structure group $\\Psi{\\rm DO}_0^*$, the group of invertible classical zeroth order pseudodifferential operators.   We show the vanishing of \n  the Wodzicki-Chern classes of $LM$ and more general mapping spaces. \nAs in finite dimensions, we show the existence of  WCS classes in \n  $H^n(LM,{\\mathbb C})$ if dim$(M) = n$ is odd (Definition \\ref{def:WCS})\n  and give the local expression for the WCS classes associated to the Chern character\n  (Theorem \\ref{thm:5.5}).\n   In Theorem \\ref{WCSvan}, we prove that the\n    Chern character WCS class vanishes if dim$(M) \\equiv 3\n   \\ ({\\rm mod}\\ 4)$.\nIn \\S\\ref{dimfive}, we associate to every circle action $a:S^1\\times M^n\\to M^n$\n an $n$-cycle $[a]$\n in $LM$.  For a specific metric on $S^2\\times S^3$ and a specific circle action $a,$\n we prove via exact computer calculations that the WCS class is nonzero by integrating it over $[a].$\n Since the corresponding integral for the cycle associated to the trivial action \n is zero, $a$ cannot be homotoped to the trivial action. \nWe use this result to prove that $\\pi_1({\\rm Diff}\n (S^2\\times S^3))$ is infinite.\n\n\nThroughout this part, $H^*$ always refers to de Rham cohomology for complex valued forms.  By \\cite{beggs}, $H^*(LM)\\simeq H^*_{\\rm sing}(LM,{\\mathbb C}).$\n\n\n\\section{{\\bf Chern-Simons Classes on Loop Spaces}}\\label{CSCLS}\n\nWe begin in \\S5.1 with a review of Chern-Weil and Chern-Simons theory in\nfinite dimensions, following  \\cite{C-S}.  \n\nIn\n\\S5.2, we discuss Chern-Weil and Chern-Simons theory on a class of infinite rank bundles\nincluding the frame bundles of loop spaces.  As in \\S2.7,  the geometric\nstructure group of these bundles\n is $\\Psi{\\rm DO}_0^*$, so we need a trace on the Lie algebra\n$\\Psi{\\rm DO}_{\\leq 0}$ to define invariant polynomials.  There are two\ntypes of  traces, one given by taking the zeroth order symbol and one given by\nthe Wodzicki residue \\cite{paycha-lescure}, \\cite{ponge}.  Here we only consider the \nWodzicki residue trace.  \n\n\nUsing this trace, we generalize the usual definitions of Chern and Chern-Simons classes in\nde Rham cohomology.  In particular,\ngiven a $U(n)$-invariant polynomial $P$ of degree $k$, we define a corresponding \nWCS class $CS^W_P\\in H^{2k-1}(LM)$ if dim$(M) = 2k-1.$    We are forced to consider these secondary classes, because the Wodzicki-Chern classes of mapping spaces\n${\\rm Maps} (N,M)$ vanish.  In Theorem \\ref{thm:5.5}, we give an exact expression for the WCS\nclasses associated to the Chern character.\nIn Theorem \\ref{WCSvan}, we show that these WCS classes in $H^{4k+3}(LM^{4k+3})$\nvanish; in contrast, in finite dimensions, the Chern-Simons classes associated to the Chern character vanish in $H^{4k+1}(M^{4k+1}).$\n\n\n\n\n\\subsection{{\\bf Chern-Weil  and Chern-Simons Theory for Finite Dimensional\n  Bundles} }\n\nWe first review the Chern-Weil construction. \nLet $G$ be a finite dimensional Lie group with Lie algebra ${\\mathfrak  g}$, and let\n $G\\to F\\to M$ be a principal $G$-bundle over a manifold $M$. \nSet $\n {\\mathfrak  g}^k={\\mathfrak  g}^{\\otimes k}$ and let\n\\begin{equation*}I^k(G)\n= \\{P:{\\mathfrak  g}^k\\to {\\mathbb C}\\ | P\\ \\text{symmetric,\n  multilinear, Ad-invariant}\\}\n\\end{equation*}\nbe the degree $k$ Ad-invariant polynomials on ${\\mathfrak  g}.$\n\n\n\\begin{rem}\nFor classical Lie groups $G$, $I^k(G)$ is generated by the polarization of\nthe Newton polynomials $\\operatorname{Tr}(A^\\ell)$, where $\\operatorname{Tr}$ is the usual trace on finite\ndimensional matrices.\n\\end{rem}\n\n\nFor $\\phi\\in\\Lambda^\\ell(F,{\\mathfrak  g}^k)$, $P\\in I^k(G)$, set\n$P(\\phi)=P\\circ \\phi\\in\\Lambda^\\ell(F)$. \n\n\n\\begin{thm}[The Chern-Weil Homomorphism \\cite{K-N}] \\label{previous}\nLet $F\\to M$ have a connection $\\theta$ with curvature $\\Omega_F\\in\n\\Lambda^2(F,{\\mathfrak  g})$. For $P\\in I^k(G)$, $P(\\Omega_F)$ is a closed\n invariant real form on $F$, and so\ndetermines a closed form\n$P(\\Omega_M)\\in \\Lambda^{2k}(M)$.\nThe Chern-Weil map\n\\begin{equation*}\n\\oplus_{k}I^k(G)\\to H^{*}(M), \\ P\\mapsto [P(\\Omega_M)]\n\\end{equation*}\nis a well-defined algebra homomorphism, and in particular is independent of the choice of\nconnection on $F$.\n\\end{thm} \n\nThe proof depends on:\n\\begin{itemize}\n\\item (The {\\it commutativity property}) \nFor $\\phi\\in\\Lambda^{\\ell}(F,{\\mathfrak  g}^k)$, \n\\begin{equation}\\label{eq:deri}\nd(P(\\phi))=P(d\\phi).\n\\end{equation}\n\\item (The {\\it infinitesimal invariance property})\nFor $\\psi_i\\in\\Lambda^{\\ell_i}(F,{\\mathfrak  g})$, $\\phi\\in\\Lambda^{1}(F,{\\mathfrak  g})$ and $P\\in\n  I^k(G)$, \n\\begin{equation}\\label{eq:inva}\n\\sum^k_{i=1}\n(-1)^{\\ell_1+\\dots+\\ell_i}P(\\psi_1\\wedge\\dots\\wedge[\\psi_i,\\phi]\\wedge\\dots\n\\psi_l)=0. \n\\end{equation}\n\\end{itemize}\n$[P(\\Omega_M)]$ is\ncalled the {\\it characteristic class} of $P$. For example, the characteristic class\n associated to  $\\operatorname{Tr}(A^k)$ is the k${}^{\\rm th}$ component of the Chern character of $F$.\n\n\nPart of the theorem's content\nis that for any two connections on $F$,\n$P(\\Omega_1) - P(\\Omega_0) = \ndCS_P(\\theta_1,\\theta_0)$  \nfor some odd form $CS_P(\\nabla_1, \\nabla_0)$.  Explicitly, \n\\begin{equation}\\label{5.1}\nCS_P(\\theta_1,\\theta_0) = \\int_0^1 P(\\theta_1-\\theta_0,\\overbrace{\\Omega_t,...,\\Omega_t}^{k-1})\n\\ dt\n\\end{equation}\nwhere \n$$\\theta_t = t\\theta_0+(1-t)\\theta_1,\\ \\ \\Omega_t = d\\theta_t+\\theta_t\\wedge\\theta_t$$ \\cite[Appendix]{chern}.  \n\n\\begin{rem}\nFor $F\\stackrel{\\pi}{\\to} M$,  \n$\\pi^*F\\to F$ is trivial.\nTake $\\theta_1$ to be the flat connection on $\\pi^*F$\nwith respect to a fixed trivialization.\nLet $\\theta_1$ also\ndenote the connection $\\chi^*\\theta_1$ on $F$, \nwhere $\\chi$ is the global section of $\\pi^*F.$  For any other connection $\\theta_0$ on $F$,  $\\theta_t = t\\theta_0, \\Omega_t = t\\Omega_0 + (t^2-t)\\theta_0\\wedge \\theta_0$. \n Assume an invariant  polynomial $P$ takes values in ${\\mathbb R}.$  Then  we\nobtain the formulas for the transgression form $TP(\\Omega_1)$\non $F$: for \n\\begin{equation}\\label{eq:ChernSimons}\n\\phi_t =t\\Omega_1+\\frac{1}{2}(t^2-t)[\\theta,\\theta],\\ \\ \nTP(\\theta)=l\\int_0^1 P(\\theta\\wedge \\phi^{k-1}_t)dt,\n\\end{equation}\n$dTP(\\theta)=P(\\Omega_1)\\in \\Lambda^{2l}(F)$\n\\cite{C-S}.  $TP(\\Omega_1)$ pushes down to an ${\\mathbb R}\/{\\mathbb Z}$-class on $M$,\nthe absolute Chern-Simons class.\n\\end{rem}\n\nAs usual, these formulas carry over to connections $\\nabla = d+\\omega$\non vector bundles $E\\to M$ in the form\n\\begin{equation}\\label{5.11}\nCS_{P}(\\nabla_1,\\nabla_0) = \\int_0^1 P(\\omega_1-\\omega_0,\\Omega_t,..., \n\\Omega_t)\\ dt,\n\\end{equation}\nsince $\\omega_1-\\omega_0$ and\n$\\Omega_t$ are globally defined forms.  \n\n \n \n \\subsection{{\\bf Chern-Weil and Chern-Simons Theory for $\\Psi{\\rm DO}_0^*$-Bundles}}\n\n \n Let $\\mathcal E\\to\\mathcal M$ be an infinite rank bundle\n over a paracompact Banach manifold\n $\\mathcal M$, with the fiber of $\\mathcal E$ \n modeled on a fixed Sobolev class of sections of \n a finite rank hermitian vector  bundle $E\\to N$, and with structure group $\\pdo_0^*(E)$.  \n For such $\\pdo_0^*$-bundles,\n we can produce\n primary and secondary characteristic classes \n once we choose a trace on $\\Psi{\\rm DO}_{\\leq 0}(E)$.\n Since the adjoint action of $\\pdo_0^*$ on $\\Psi{\\rm DO}_{\\leq 0}$ is by conjugation,  a trace on $\\Psi{\\rm DO}_{\\leq 0}$ will extend to a polynomial on forms\nsatisfying (\\ref{eq:deri}), (\\ref{eq:inva}), so the finite dimensional proofs extend.  \n \nThese traces were classified  in \\cite{lesch-neira, paycha-lescure}, although there are slight variants\nin our special case $N= S^1$ \\cite{ponge}.  Roughly speaking, the traces fall into two classes, the leading order symbol trace \\cite{P-R2} and the Wodzicki residue.  In this paper,\nwe consider only the Wodzicki residue, and refer to \\cite{lrst} for the leading order symbol\ntrace.\n\nFor simplicity, we mainly restrict to the generating invariant polynomials $P_k(A) = A^k$, and \nonly consider $\\mathcal E = TLM$, which we recall is the complexified tangent bundle.  We will work with vector bundles rather than principal bundles.  \n\n\n\n\\begin{defn}  \\label{def:WCS}\n(i) The k${}^{\\rm th}$ {\\it Wodzicki-Chern (WC) form} of a $\\Psi{\\rm DO}_0^*$-connection\n$\\nabla$ on $TLM$ with curvature $\\Omega$ is\n\\begin{equation}\\label{5.1a}\nc_k^W(\\Omega)(\\gamma) =\\frac{1}{k!}\n \\int_{S^*S^1}\\operatorname{tr}\\sigma_{-1}(\\Omega^{k}) \\ d\\xi  dx.\n\\end{equation}\nHere we recall that for each $\\gamma\\in LM$,\n$\\sigma_{-1}(\\Omega^k)$ is a $2k$-form with values in  endomorphisms\n of a trivial bundle\nover $S^*S^1$. \n\n\n\n(ii) The k${}^{\\rm th}$ {\\it Wodzicki-Chern-Simons (WCS) form} of two $\\Psi{\\rm DO}_0^*$-connections \n$\\nabla_0,\\nabla_1$ on $TLM$ is\n\\begin{eqnarray}\\label{5.22}\nCS^W_{2k-1}(\\nabla_1,\\nabla_0) &=&\\frac{1}{k!}\n \\int_0^1 \\int_{S^*S^1}\\operatorname{tr}\\sigma_{-1}((\\omega_1-\\omega_0)\\wedge \n(\\Omega_t)^{k-1})\\ dt\\\\ \n&=&\\frac{1}{k!} \\int_0^1 {\\rm res}^{\\rm w} \n[(\\omega_1-\\omega_0)\\wedge \n(\\Omega_t)^{k-1}]\\ dt.\\nonumber\n\\end{eqnarray}\n\n(iii) The  k${}^{\\rm th}$ {\\it Wodzicki-Chern-Simons form} associated to a Riemannian metric \n$g$ \non $M$, denoted $CS^W_{2k-1}(g)$,  is $CS^W_{2k-1}(\\nabla_1,\\nabla_0)$, where $\\nabla_0, \\nabla_1$ refer to the \n$L^2$ and $s=1$ Levi-Civita connections on $LM$, respectively.\n\n\n(iv) Let $\\Sigma = \\{\\sigma\\}$ be the group of permutations of $\\{1,...,k\\}$. Let $I:\n1\\leq i_1< ...< i_\\ell = k$ be a partition of $k$ (i.e. with $i_0=0$, $\\sum_{j=1}^k\n (i_j-i_{j-1}) = k$) .  For  the symmetric, $U(n)$-invariant, \nmultilinear form on ${\\mathfrak u}(n)$\n\\begin{eqnarray*} P_I(A_1,A_2,...,A_k) &=& \\frac{1}{k!}\n\\sum_\\sigma \\operatorname{tr}(A_{\\sigma(1)}\\cdot...\\cdot A_{\\sigma(i_1)})\n\\operatorname{tr}(A_{\\sigma(i_1+1)}\\cdot...\\cdot A_{\\sigma(i_2)})\\\\\n&&\\qquad \\cdot ...\\cdot \\operatorname{tr}(A_{\\sigma(i_{\\ell-1})}\n\\cdot ...\\cdot A_{\\sigma(k)}),\n\\end{eqnarray*}\ndefine the symmetric, $\\pdo_0^*$-invariant, multilinear form on $\\Psi{\\rm DO}_{\\leq 0}$ by\n\\begin{eqnarray*} P_I^W(B_1,...,B_k) &=&  \\frac{1}{k!}\n \\sum_\\sigma\\left( \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(B_{\\sigma(1)}\\cdot...\\cdot B_{\\sigma(i_1)}) \\right.\\\\\n&&\\qquad \\left. \\cdot\n \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}\n(B_{\\sigma(i_1+1)}\\cdot...\\cdot B_{\\sigma(i_2)})\\right. \\\\\n&&\\qquad \\left. \\cdot ...\\cdot \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(B_{\\sigma(i_{\\ell-1})} \n\\cdot ...\\cdot B_{\\sigma(k)})\\right).\n\\end{eqnarray*}\n  The {\\it Wodzicki-Chern form associated\nto $P_I$} for a $\\Psi{\\rm DO}_0^*$-connection on $TLM$ with curvature $\\Omega$ is \n\\begin{eqnarray}\\label{wcpi} c_{P_I}^W(\\Omega) &=& \nP_I^W(\\Omega,\\Omega,...,\\Omega)\\\\\n &=& \\frac{1}{k!}\n\\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(\\Omega^{k_1}) \\cdot \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(\\Omega^{k_2})\n\\cdot\n...\\cdot \\int_{S^*S^1} \\operatorname{tr}\\sigma_{-1}(\\Omega^{k_\\ell}\\nonumber )\\\\\n&=& \\frac{k_1!k_2!\\cdot...\\cdot k_\\ell !}{k!} \nc_{k_1}^W(\\Omega)c_{k_2}^W(\\Omega)\\cdot ...\\cdot c_{k_\\ell}^W(\\Omega),\\nonumber\n\\end{eqnarray}\nwhere $k_1=i_1-i_0, k_2 = i_2-i_1,...,\nk_\\ell = i_\\ell - i_{\\ell-1}$.\n\nSetting $K = (k_1,...,k_\\ell)$, we also denote $c_{P_I}^W(\\Omega)$ by $c_K^W(\\Omega).$\n\n\n\n(v) \nLet $\\nabla_0,\\nabla_1$ be $\\Psi{\\rm DO}_0^*$-connections on $TLM$ with connection forms\n$\\omega_0, \\omega_1,$ respectively.  The \n {\\it Wodzicki-Chern-Simon form associated to $P_I$ and $\\nabla_0, \\nabla_1$}\nis\n$$\nCS^W_{P_I}(\\nabla_1,\\nabla_0) = \\int_0^1 P_I^W(\\omega_1-\\omega_0,\\Omega_t,...,\n\\Omega_t)dt. $$ \n\\end{defn}\n\nIn (iv) and (v), we do not bother with a normalizing constant, since we do not claim that \nthere is a normalization which gives classes with integral periods.  \nNote that the k${}^{\\rm th}$ WCS class is associated to $P_k(A_1,...,A_k) = \\operatorname{tr}(A_1\\cdot\n...\\cdot A_k)$, i.e. the partition $K = (k)$, or in other words to the polynomial giving the \nk${}^{\\rm th}$ component of the Chern character.\n\nAs in finite dimensions, $c_k^W(\\nabla)$ is a closed $2k$-form, with de Rham cohomology\nclass $c_k(LM)$\n independent of $\\nabla$, as $c_k^W(\\Omega_1)  - c_k^W(\\Omega_0) =\ndCS^W_{2k-1}(\\nabla_1,\\nabla_0).$  \n \n\\begin{rem}  It is an interesting question to determine all the $\\Psi{\\rm DO}_0^*$-invariant polynomials on \n$\\Psi{\\rm DO}_{\\leq 0}.$  As above, $U(n)$-invariant polynomials combine with the Wodzicki residue \n(or the other traces on $\\Psi{\\rm DO}_{\\leq 0}$) to give $\\Psi{\\rm DO}_0^*$-polynomials,\nbut there may be others.  \n\\end{rem}  \n\n\n\nThe tangent space $TLM$, and more generally mapping spaces\nMaps$(N,M)$  with $N$ closed\nhave vanishing Wodzicki-Chern classes.  Here we take a Sobolev topology on Maps$(N,M)$ for some\nlarge Sobolev parameter, so that Maps$(N,M)$ is a paracompact Banach manifold.\nWe denote the de Rham class of $c_{P_I}^W(\\Omega)$ for a connection on $\\mathcal E$ by\n$c_{P_I}(\\mathcal E).$ \n\n\n\n\\begin{prop} \\label{prop:maps} Let $N, M$ be closed manifolds, and let {\\rm Maps}${}_f(N,M)$ denote\nthe component of a fixed $f:N\\to M$.  Then the  cohomology classes\n$c_{P_I}^W({\\rm Maps}_f(N,M)) $ of $T{\\rm Maps}(M,N)$ vanish.\n\\end{prop}\n\n\\begin{proof}\nFor $TLM$, the $L^2$ connection in Lemma \\ref{lem:l2lc}\nhas curvature $\\Omega$ which is a multiplication operator.  Thus $\\sigma_{-1}(\\Omega)$ and hence $\\sigma_{-1}(\\Omega^{i})$ are zero, \nso the WC forms $c_{P_I}(\\Omega)$ also vanish.\n\nFor $n\\in N$ and $h:N\\to M$,\nlet $\\operatorname{ev}_n: {\\rm Maps}_f(N,M)$ be $\\operatorname{ev}_n(h) = h(n).$ \n Then $D_XY(h)(n) \\stackrel{\\rm def}{=} \n (\\operatorname{ev}_h^*\\nabla^{LC,M})_XY(h)(n)$ is the $L^2$ Levi-Civita connection on \\\\\nMaps$(N,M).$\nAs in Lemma \\ref{lem:l2lc},\nthe curvature of $D$ is a \na multiplication operator.  Details are left to the reader.\n\\end{proof}\n\n\n\\begin{rem}  (i) These mapping spaces fit into the framework of the Families Index Theorem in \nthe case of a trivial fibration\n$Z\\to M\\stackrel{\\pi}{\\to} B$ of closed manifolds.  Given a\nfinite rank bundle $E\\to M$, we get an associated infinite rank bundle ${\\mathcal E} = \\pi_*E\n\\to B$.  For the fibration $N\\to N\\times {\\rm Maps}(N,M)\\to {\\rm Maps}(N,M)$ and $E = {\\rm\n  ev}^*TM$, ${\\mathcal E}$ is $T{\\rm Maps}(N,M).$  A connection $\\nabla$ on $E$ induces a connection $\\nabla^{{\\mathcal E}}$ on ${\\mathcal E}$ defined by\n\\begin{equation*}\n(\\nabla^{{\\mathcal E}}_Z s)(b)\n(z)=\\left( (\\operatorname{ev}^*\\theta^u)_{(Z,0)} u_s\\right)(b,z).\n\\end{equation*}\nHere $u_s(b,z)=s(b)(z)$.\nThe curvature $\\Omega^{{\\mathcal E}}$ satisfies\n\\begin{equation*}\\label{eq:pullback}\n\\Omega^{{\\mathcal E}}(Z,W)s(b)(z)=(\\operatorname{ev}^*\\Omega ) ((Z,0),(W,0)) u_s(b,z).\n\\end{equation*}\nThis follows from\n\\begin{equation*}\n\\Omega^{{\\mathcal E}}(Z,W)s(b)(z)= [\\nabla^{{\\mathcal E}}_Z \\nabla^{{\\mathcal E}}_W\n-\\nabla^{{\\mathcal E}}_W \\nabla^{{\\mathcal E}}_Z -\\nabla^{{\\mathcal E}}_{[Z,W]}]\ns(b)(z).\n\\end{equation*}\nThus the connection and curvature forms take values in multiplication operators, and \nso $c_k^W({\\mathcal E}) = 0.$\n\n\nIf the fibration is nontrivial, the connection on ${\\mathcal E}$ depends on the choice of a horizontal complement to $TZ$ in $TM$, and the corresponding connection and curvature forms take\nvalues in first order differential operators.  \n\n(ii) In finite dimensions, odd Chern forms of complexified real bundles like\\\\\n $T{\\rm Maps}(N,M)$ vanish, because the form involves a composition of an odd number of skew-symmetric matrices.  In contrast, odd WC forms involve terms like\n$\\sigma_{-1}(\\Omega^1)\\wedge\\Omega^M\\wedge...\\wedge\\Omega^M,$ where $\\Omega^1$ is the curvature of the $s=1$ Levi-Civita connection.  By Lemma \n\\ref{old2.2}(ii), $\\sigma_{-1}(\\Omega^1)$ is not skew-symmetric as an endomorphism.  Thus\nit is not obvious that the odd WC forms vanish.\n\nSimilarly, in finite dimensions the Chern-Simons form for the odd Chern classes of complexified real bundles vanish, but this need not be the case for WCS forms.  In fact, we will produce nonvanishing \nWCS classes associated to $c_3^W(TLM^5)$ in \\S\\ref{dimfive}.\n\n\\end{rem}\n\n\nIn finite dimensions, Chern classes are topological obstructions to the\nreduction of the structure group and geometric obstructions to the existence\nof a flat connection.  \nWodzicki-Chern classes for $\\Psi{\\rm DO}_0^*$-bundles \nare also topological and geometric obstructions, but\nthe geometric information is a little more refined due to the grading on the\nLie algebra \n $\\Psi{\\rm DO}_{\\leq 0}$.\n\n\\begin{prop}\n Let ${\\mathcal E}\\to{\\mathcal B}$ be an infinite rank $\\pdo_0^*$-bundle, for\n  $\\pdo_0^*$ acting on\n$E\\to N^n$.  \nIf ${\\mathcal E}$ admits a reduction to the gauge group ${\\mathcal G}(E)$, then\n  $c_k^W({\\mathcal E})  =   0$ for all $k$, and hence $c_{P_I}^W({\\mathcal E}) =0$ for all $P_I$.\nIf ${\\mathcal E}$ admits a \n   $\\pdo_0^*$-connection whose\n  curvature has order $-k$, then $\n  c_{\\ell}({\\mathcal E}) =0$ for $\\ell \\geq [n\/k].$\n \\end{prop}\n\n\\begin{proof}  If the structure group of  ${\\mathcal E}$ reduces to the gauge\n  group, there exists a connection one-form\n  with values in Lie$({\\mathcal G}) = {\\rm End}(E)$, the Lie algebra of multiplication\n  operators.  Thus the Wodzicki residue of powers of the curvature vanishes,\n  so the Wodzicki-Chern classes vanish.\nFor the second statement, the order of the curvature is less than\n$-n$ for $\\ell \\geq [n\/k]$,  so the Wodzicki residue\n  vanishes in this range. \n  \\end{proof}\n  \n  However, we do not have examples of nontrivial WC classes; cf.~\\cite{lrst}, where it is \n  conjectured that these classes always vanish.  \n  \\bigskip\n\n\n\n\nThe relative WCS form is not difficult to compute.  \n\n\\begin{prop}  Let $\\sigma$ be in the group of permutations of $\\{1,\\ldots,2k-1\\}.$ Then\n\\begin{eqnarray}\\label{5.4}\n\\lefteqn{CS^W_{2k-1}(g)(X_1,...,X_{2k-1}) }\\\\\n&=&\n\\frac{2}{(2k-1)!} \\sum_{\\sigma} {\\rm sgn}(\\sigma) \\int_{S^1}\\operatorname{tr} [\n(-2R(X_{\\sigma(1)},\\dot\\gamma)\n-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma) \\nonumber\\\\\n&&\\qquad \n\\cdot (\\Omega^M)^k(X_{\\sigma(2)},..X_{\\sigma(2k-1)} )].\\nonumber\n\\end{eqnarray}\n\\end{prop}\n\n\n\n\n\\begin{proof}\n$$\\sigma_0((\\omega_1-\\omega_0)_X)^a_b = \\Gamma_{cb}^aX^c -\\Gamma_{cb}^aX^c = 0.$$\nThus \n\\begin{equation}\\label{cswint}\nCS^W_{2k-1}(g) = \\int_0^1 \\int_{S^*S^1}\\operatorname{tr}\\sigma_{-1}(\\omega_1-\\omega_0)\\wedge (\\sigma_0(\\Omega_t))^k\\ dt.\n\\end{equation}\nMoreover,\n\\begin{eqnarray*}\\sigma_0(\\Omega_t) &=& td(\\sigma_0(\\omega_0)) + (1-t)d(\\sigma_0(\\omega_1)) \\\\\n&&\\qquad + \n(t\\sigma_0(\\omega_0) + (1-t)\\sigma_0(\\omega_1))\\wedge (t\\sigma_0(\\omega_0) + (1-t)\\sigma_0(\\omega_1))\\\\\n&=& d\\omega^M + \\omega^M\\wedge \\omega^M\\\\\n&=& \\Omega^M.\n\\end{eqnarray*}\nTherefore\n\\begin{equation}\\label{5.3}\nCS^W_{2k-1}(g) = \\int_0^1 \\int_{S^*S^1}\\operatorname{tr} [\\sigma_{-1}(\\omega_1)\n\\wedge (\\Omega^M)^k]\\ dt,\n\\end{equation}\nsince $\\sigma_{-1}(\\omega_0) = 0.$  We can drop the integral over $t$. \nThe integral over the $\\xi$ variable contributes a factor of $2$: the integrand has\na factor of $|\\xi|^{-2}\\xi$, which equals $\\pm 1$ on the two components of $S^*S^1$.\nSince the fiber of $S^*S^1$ at a fixed $\\theta$ consists of two points \nwith opposite orientation, the ``integral\" over each fiber is $1-(-1) = 2.$ \nThus\n\\begin{eqnarray}\\label{5.4a}  \\lefteqn{\nCS^W_{2k-1}(g)(X_1,...X_{2k-1})    }  \\\\\n&=& =  \\frac{2}{(2k-1)!}   \\sum_\\sigma {\\rm sgn}(\\sigma)  \\int_{S^1}\\operatorname{tr}[\n(-2R(X_{\\sigma(1)},\\dot\\gamma)\n-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma)\\nonumber\\\\\n&&\\qquad\n\\cdot (\\Omega^M)^k(X_{\\sigma(2)},..X_{\\sigma(2k-1)} )]\\nonumber\n\\end{eqnarray}\nby Lemma \\ref{old2.1}.\n\\end{proof} \n\n\nThis produces odd classes in the de Rham cohomology of the loop space of an odd\ndimensional manifold.\n\n\\begin{thm}\\label{thm:5.5}\n (i)  Let dim$(M) = 2k-1$ and let $P$ be a $U(n)$-invariant polynomial of degree \n$k.$  Then $c^W_P(\\Omega) \\equiv 0$ for any $\\Psi{\\rm DO}_0^*$-connection $\\nabla$ on\n $TLM.$  Thus $CS^W_P(\\nabla_1,\\nabla_0)$ is closed and defines a \n class $[CS^W_P(\\nabla_1,\\nabla_0)]\\in H^{2k-1}(LM).$  In particular, we can\n define $[CS^W_P(g)]\\in H^{2k-1}(LM)$ for a Riemannian metric $g$ on $M$. \n \n \n \n(ii)  For dim$(M) = 2k-1$, the k${}^{\\it th}$ Wodzicki-Chern-Simons form $CS^W_{2k-1}(g)$\nsimplifies to \n \\begin{eqnarray}\\label{csg}\n\\lefteqn{CS^W_{2k-1}(g)(X_1,...,X_{2k-1}) }\\nonumber \\\\\n&=&\n\\frac{2}{(2k-1)!} \\sum_{\\sigma} {\\rm sgn}(\\sigma) \\int_{S^1}\\operatorname{tr}[\n(-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma)\\\\\n&&\\qquad \n\\cdot (\\Omega^M)^{k-1}(X_{\\sigma(2)},..X_{\\sigma(2k-1)} )].\\nonumber\n\\end{eqnarray}\n\n\n \n\n \\end{thm}\n \n \\begin{proof} (i) Let $\\Omega$ be the curvature of $\\nabla.$\n $c^W_P(\\Omega)(X_1,\\dots, X_{2k})(\\gamma)$ is a sum of monomials of the form\n(\\ref{wcpi}).  This is\na $2k$-form on $M$, and hence vanishes.  \n\n \n (ii) Since\n $$R(X_{1},\\dot\\gamma)\n\\cdot (\\Omega^M)^k(X_{2},..X_{2k-1}) = \n[i_{\\dot\\gamma}\\operatorname{tr}(\\Omega^{k})](X_1,...X_{2k-1}) = \\operatorname{tr}(\\Omega^k)(\\dot\\gamma, X_1,\\ldots,X_{2k-1}),$$\nthe first term on the right hand side of (\\ref{5.4a}) vanishes on a $(2k-1)$-manifold.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{rem} There are several variants to the construction of relative WCS classes.\n\n(i) If we define the transgression form $Tc_k(\\nabla)$ with the Wodzicki residue\nreplacing the trace in (\\ref{eq:ChernSimons}), it is easy to check that $Tc_k(\\nabla)$\ninvolves $\\sigma_{-1}(\\Omega).$  For $\\nabla$ the $L^2$ connection, this WCS class vanishes.  For $\\nabla$ the $H^s$ connection, $s>0$, $\\sigma_{-1}(\\Omega)$ involves \nthe covariant derivative of the curvature of $M$ (cf.~Lemma \\ref{old2.2} for $s=1.$)  Thus the\nrelative WCS class is easier for computations than the absolute class $[Tc_k(\\nabla)].$\n\n\n(ii) If we define $CS_k^W(g)$ using the Levi-Civita connection for the $H^s$ \nmetric instead of\nthe $H^1$ metric, the WCS class is simply multiplied by the artificial parameter $s$ by \nLemma \\ref{lem:33}.  Therefore setting $s=1$ is not only computationally convenient, it \nregularizes the WCS, in that it extracts the $s$-independent information.\nThis justifies the following definition:\n\n\n\\begin{defn}  \\label{def:regularized}\nThe {\\it regularized  $k^{th}$ WCS class} associated to a Riemannian metric \n$g$ on $M$ is $CS_k^{W, {\\rm reg}}(g) \\stackrel{\\rm def}{=} \nCS_k^W(\\nabla^{1},\\nabla^0)$, where $\\nabla^{1}$ is the $H^1$ connection\nand $\\nabla^0$ is the $L^2$ Levi-Civita connection.  \n\\end{defn}  \n\n\\end{rem}\n\n\\bigskip\n\nWe conclude this section with a vanishing result that does not have a finite dimensional \nanalogue.\n  \\begin{thm} \\label{WCSvan}\n  The {\\it  k}${}^{\\it  th}$ WCS class $CS_k^W(g)$\n   vanishes if ${\\rm dim}(M) \\equiv 3 \\ ({\\rm mod}\\  4).$\n  \\end{thm}\n\n\\begin{proof}  Let dim$(M) =2 k-1$. Since $\\Omega^M$ takes values in skew-symmetric endomorphisms, \nso does\n$(\\Omega^M)^{k-1}$ if $k$ is even, i.e. if ${\\rm dim}(M) \\equiv 3 \\ ({\\rm mod}\\  4).$\n  The term \n  $-R(\\cdot,\\dot\\gamma)X_{\\sigma(1)} + R(X_{\\sigma(1)},\\cdot)\\dot\\gamma$ in (\\ref{csg}) is a symmetric\n  endomorphism.  For in Riemannian normal coordinates, this term is\n  $(-R_{bdca} +R_{cbda})X^c\\dot\\gamma^d \\equiv A_{ab}$, say, so the curvature terms in \n  $A_{ab} - A_{ba}$ are\n  \\begin{eqnarray*}\n  -R_{bdca} +R_{cbda} + R_{adcb} - R_{cadb} &=& -R_{bdca} +R_{cbda} \n  + R_{cbad} - R_{dbca}\\\\\n  &=& -R_{bdca} +R_{cbda} -R_{cbda} + R_{bdca}=0.\n  \\end{eqnarray*}\nThus the integrand in  (\\ref{csg}) is the trace of a symmetric endomorphism composed with a skew-symmetric endormorphism, and so \nvanishes.\n\\end{proof}\n\n\\begin{exm}  We contrast Theorem \\ref{WCSvan} with the situation in finite dimensions. \nLet dim$(M)=3.$ \nThe only invariant monomials of degree two are $\\operatorname{tr}(A_1A_2)$ and \n$\\operatorname{tr}(A_1)\n\\operatorname{tr}(A_2)$ (corresponding to $c_2$ and $c_1^2$, respectively).  \n\nFor $M$, $\\operatorname{tr}(A_1A_2)$ gives rise \nto the classical Chern-Simons invariant for $M$.  However, the Chern-Simons class associated to \n$\\operatorname{tr}(A_1)\\operatorname{tr}(A_2)$ involves $\\operatorname{tr}(\\omega_1-\\omega_0)\\operatorname{tr}(\\Omega_t)$, \nwhich vanishes since both forms take values in skew-symmetric endomorphisms.\n\nIn contrast, on $LM$ we know that the WCS class $CS^W_3$ associated to \n$\\operatorname{tr}(A_1A_2)$ vanishes.  The WCS associated to $\\operatorname{tr}(A_1)\\operatorname{tr}(A_2)$ involves \n$\\operatorname{tr}\\sigma_{-1}(\\omega_1-\\omega_0) = \\operatorname{tr}\\sigma_{-1}(\\omega_1)$ and $\\operatorname{tr}\\sigma_{-1}(\\Omega_t).$  \nBoth $\\omega_1$ and $ \\Omega_t$ take values in skew-symmetric $\\Psi{\\rm DO}$s, but\nthis does not imply that the terms in their symbol expansions are skew-symmetric.  In fact, a calculation using Lemma \\ref{old2.1} shows that $\\sigma_{-1}(\\omega_1)$ is not skew-symmetric. \nThus the WCS class associated to $\\operatorname{tr}(A_1)\\operatorname{tr}(A_2)$ may be nonzero.\n\n\n\\end{exm}\n\n\n\n\n\n\n\\section{{\\bf An Application of Wodzicki-Chern-Simons Classes to Circle\n    Actions}}\\label{dimfive}\n\n\nIn this section we use WCS classes to distinguish different $S^1$ actions on\n$M=S^2\\times S^3$. We use this to conclude that $\\pi_1({\\rm Diff}(M) , id)$ is infinite.  \n\nRecall that  $H^*(LM)$ denotes de Rham cohomology of complex valued \nforms.   In particular, integration of closed forms over homology cycles gives a pairing of\n$H^*(LM)$ and $H_*(LM,{\\mathbb C})$. \n\n For any closed oriented manifold $M$, let $a_0,a_1:S^1\\times M\\to M$ be two smooth actions.  Thus\n$$a_i(0,m) = m, \\ a_i(\\theta,a(\\psi,m)) = a_i(\\theta + \\psi, m).$$\n\n\\begin{defn} (i)  $a_0$ and $a_1$\n are {\\it smoothly  homotopic} if there exists a smooth map \n$$F:[0,1]\\times S^1\\times M\\to M,\\ F(0,\\theta,m) = a_0(\\theta,m),\\\n F(1,\\theta,m) = a_1(\\theta,m).$$\n\n(ii)  $a_0$ and $a_1$ are {\\it smoothly  homotopic through actions} if\n $F(t,\\cdot,\\cdot):S^1\\times M\\to M$ is an action for all $t$.\n\n\\end{defn}\n\nWe can rewrite an action in two equivalent ways.\n\n\n\\begin{itemize}\n\\item $a$ determines (and is determined by)\n$a^D:S^1\\to {\\rm Diff} (M)$ given by\n$a^D(\\theta)(m) = a(\\theta,m).$  $a^D(\\theta)$ is a diffeomorphism because \n$$a^D(-\\theta)(a^D(\\theta,m)) = a(-\\theta, a(\\theta,m)) = m.$$\nSince $a^D(0) = id,$  we get a class $[a^D]\\in \\pi_1({\\rm Diff} (M), id)$, the\nfundamental group of ${\\rm Diff} (M)$ based at $id.$  Here Diff$(M)$ is a Banach manifold\nas an open subset of the Banach manifold of ${\\rm Maps} (M) = {\\rm Maps} (M,M)$ of some fixed Sobolev class.\n\n\n\\item $a$ determines (and is determined by)\n$a^L:M\\to LM$ given by $ a^L(m)(\\theta) = a(\\theta,m)$. This determines a class\n$[a^L]\\in H_n(LM,{\\mathbb Z})$ with $n = {\\rm dim}(M)$ by setting $[a^L] = a^L_*[M].$\n  In concrete terms, if we triangulate $M$ as the $n$-cycle $\\sum_i n_i\\sigma_i$,\nwith $\\sigma_i:\\Delta^n\\to M$, \nthen $[a^L]$ is the homology class of\nthe cycle $\\sum_i n_i (a^L\\circ \\sigma_i).$  \n\n\\end{itemize}\n\nWe give a series of elementary lemmas comparing these maps.\n\n\n\\begin{lem}\\label{lem:one} $a_0$ is smoothly homotopic to $a_1$ through actions iff $[a^D_0] =\n\t  [a^D_1]\\in \\pi_1({\\rm Diff} (M), id).$\n\\end{lem}\n\n\\begin{proof} ($\\Rightarrow$) Given $F$ as above, set $G:[0,1]\\times S^1\\to\n\t  {\\rm Diff} (M)$ by $G(t,\\theta)(m) = F(t,\\theta,m).$  We have $G(0,\\theta)(m)\n\t  = a_0(\\theta,m) = a^D(\\theta)(m)$, $G(1,\\theta)(m) = a_1(\\theta, m)\n\t  = a^D_1(\\theta)(m)$.\n$G(t,\\theta)\\in{\\rm Diff} (M)$, because \n$$G(t,-\\theta)(G(t,\\theta)(m)) = F(t,-\\theta,F(t,\\theta,m)) = F(t,0,m) = m.$$\n(This uses that $F(t,\\cdot,\\cdot)$ is an action.)\nSince $F$ is\nsmooth, $G$ is a continuous (in fact, smooth) map of ${\\rm Diff} (M)$.\nThus $a^D_0, a^D_1$ \nare homotopic as elements of\\\\\n ${\\rm Maps} (S^1,{\\rm Diff} (M))$, so $[a^D_0] = [a^D_1].$\n\\bigskip\n\n\\noindent ($\\Leftarrow$)  Let $G:[0,1]\\times S^1\\to {\\rm Diff} (M)$ be a continuous\nhomotopy from\n$a^D_0(\\theta) = G(0,\\theta) $ to $a^D_1(\\theta) = G(1,\\theta)$ with $G(t,0) = id$\nfor all $t$. \nIt is possible to approximate\n$G$ arbitrarily well by a smooth map, since $[0,1]\\times S^1$ is compact.  Set\n$F:[0,1]\\times S^1\\times M\\to M$ by\n$F(t,\\theta,m) = G(t,\\theta)(m).$  \n $F$ is smooth.  Note that\n$F(0,\\theta,m) = G(t,\\theta)(m) = a^D_0(\\theta)(m) = a_0(\\theta,m)$, and\n$F(1,\\theta,m) = a_1(\\theta,m).$  Thus $a_0$ and $a_1$ are smoothly homotopic.\n\\end{proof}\n\n\nThere are similar results for $a^L.$\n\n\\begin{lem}\\label{lem:three}  $a_0$ is smoothly homotopic to $a_1$ iff $a^L_0,\na^L_1:M\\to LM$ are smoothly homotopic.\n\\end{lem}\n\n\n\\begin{proof}  Let $F$ be the homotopy from $a_0$ to $a_1$.  Set\n\t  $H:[0,1]\\times M \\to LM $ by $H(t,m)(\\theta) = F(t,\\theta,m).$  Then\n$H(0,m)(\\theta) = F(0,\\theta,m) = a_0(\\theta,m) =  a^L_0(m)(\\theta)$,\n  $H(1,m)(\\theta) =  a^L_1(m)(\\theta),$  so $H$ is a homotopy from $\n  a^L_0$ to $ a^L_1.$ \nIt is  easy to check that $H$ is smooth.\n\n\nConversely, if $H:[0,1]\\times M \\to LM $ is a smooth homotopy from $a^L_0$ to\n$a^L_1$, set $F(t,\\theta, m) = H(t,m)(\\theta).$  \n\\end{proof}\n\n\\begin{cor}\\label{cor:one} If $a_0$ is smoothly homotopic to $a_1$, then\n$[a^L_0] =  [a^L_1]\\in H_n(LM,{\\mathbb Z}).$  \n\\end{cor}\n\n\\begin{proof} By the last Lemma, $a^L_0$ and $a^L_1$ are homotopic. Thus \n$[a^L_0] = a^L_{0,*}[M] = a^L_{1,*}[M] = [a^L_1].$\n\\end{proof}\n\nThis yields a technique to use WCS classes to distinguish actions and to investigate \n$\\pi_1({\\rm Diff}(M) ,id).$  From now on, ``homotopic\" means ``smoothly homotopic.\"\n\n\n\\begin{prop} \\label{prop:two} Let dim$(M)=2k-1.$ Let $a_0, a_1:S^1\\times M\\to M$ be actions.\n\n\n(i)  If $\\int_{[a^L_0]} CS^W_{2k-1 } \\neq \\int_{[a^L_1]} CS^W_{2k-1 }$, then $a_0$ and $a_1$\n  are not homotopic through actions, and $[a^D_0]\\neq [a^D_1]\\in \\pi_1({\\rm Diff} (M),id).$\n\n(ii)  If $\\int_{[a_1^L]} CS^W_{2k-1 } \\neq 0,$ then\n  $\\pi_1({\\rm Diff} (M), id)$  is infinite.\n\n\\end{prop}\n\n\\begin{proof} \n\n(i) By Stokes' Theorem, $[a^L_0]\\neq [a^L_1]\\in H_n(LM,{\\mathbb C})$.\n  By Corollary \\ref{cor:one}, $a_0$ and $a_1$ are not homotopic,\n   and hence not homotopic\n  through actions.  By\nLemma \\ref{lem:one}, $[a^D_0]\\neq [a^D_1]\\in \\pi_1({\\rm Diff} (M),id).$\n\n(ii) Let $a_n$ be the $n^{\\rm th}$ iterate of \n $a_1$, i.e. $a_n(\\theta,m) =\na_1(n\\theta,m).$  \n\nWe claim that \n $\\int_{[a^L_n]}CS^W_{2k-1 } =\nn\\int_{[a^L_1]}CS^W_{2k-1 }$.  By (\\ref{5.4}), every term in $CS^W_{2k-1 }$ is of the\nform $\\int_{S^1}\\dot\\gamma(\\theta) f(\\theta)$, where $f$ is a periodic function on the\ncircle.  Each loop $\\gamma\\in\na^L_1(M)$ corresponds to the loop $\\gamma(n\\cdot)\\in a^L_n(M).$  Therefore the term\n$\\int_{S^1}\\dot\\gamma(\\theta) f(\\theta)$ is replaced by \n$$\\int_{S^1} \\frac{d}{d\\theta}\\gamma(n\\theta) f(n\\theta)d\\theta \n = n\\int_0^{2\\pi} \\dot\\gamma(\\theta)f(\\theta)d\\theta.$$\nThus $\\int_{[a^L_n]}CS^W_{2k-1 } = n\\int_{[a^L_1]}CS^W_{2k-1 }.$\n By (i), the $[a^L_n]\\in \n\\pi_1({\\rm Diff} (M), id)$\nare all distinct.  \n\n\\end{proof}\n\n\n\n\n\\begin{rem}\nIf two actions\nare homotopic through actions,\nthe $S^1$ index of an equivariant operator of the two actions is the same. (Here equivariance\nmeans for each action $a_t, t\\in [0,1].$)\nIn contrast to Proposition \\ref{prop:two}(ii), the $S^1$ index of an equivariant operator\ncannot distinguish actions on odd dimensional manifolds, as the\n$S^1$ index vanishes. This can be seen from the\nlocal version of the\n$S^1$ index theorem \\cite[Thm. 6.16]{BGV}. For the normal bundle to the\nfixed point set is always even dimensional, so the fixed point set consists of\nodd dimensional submanifolds.  The integrand in the fixed point submanifold\ncontribution to the $S^1$-index is the constant term in the short time\nasymptotics of the appropriate heat kernel.  In odd dimensions, this constant\nterm is zero.\n\n  In \\cite{MRT2}, we  interpret the $S^1$ index theorem as\nthe integral of an equivariant characteristic class over $[a^L]$.\n\\end{rem}\n\n\nWe now apply these methods to a Sasaki-Einstein metric on $S^2\\times S^3$\nconstructed in \\cite{gdsw}\n to prove the following:\n\n\\begin{thm} (i) There is an $S^1$ action on $S^2\\times S^3$ that is not smoothly homotopic\nto the trivial action.\n\n(ii) $\\pi_1({\\rm Diff} (S^2\\times S^3), id)$ is infinite.\n\\end{thm}\n\nThe content of  (i) is that although the $S^1$-orbit\n $\\gamma_x$ through $x\\in S^2\\times S^3$\nis contractible to $x$, the contraction cannot be constructed to be \nsmooth in $x$.  \n\n\\begin{proof}\nAccording to  \\cite{gdsw}, the locally defined metric \n\\begin{eqnarray}\\label{metric} g &=&\\frac{1-cy}{6}(d\\theta^2 + \\sin^2\\theta d\\phi^2) + \n\\frac{1}{w(y)q(y)} dy^2 + \\frac{q(y)}{9}[d\\psi^2 -\\cos\\theta d\\phi^2]\\nonumber\\\\\n&&\\qquad + w(y)\\left[d\\alpha + \\frac{ac-2y+y^2c}{6(a-y^2)}[d\\psi -\\cos\\theta d\\phi]\\right]^2,\n\\end{eqnarray}\nwith \n$$w(y) = \\frac{2(a-y^2)}{1-cy}, q(y) = \\frac{a-3y^2+2cy^3}{a-y^2},$$\nis a family of Sasaki-Einstein metrics on a coordinate ball in the variables\n$(\\phi, \\theta, \\psi, y, \\alpha).$  Here $a$ and $c$ are constants, and we can take $a\\in (0,1], c=1$.  \nFor $p,q$ relatively prime, $q<p$, and satisfying $4p^2-3q^2 = n^2$ for some integer $n$,  \nand for  $a = a(p,q)< 1$, the metric extends to a $5$-manifold $Y^{p,q}$ which has\nthe coordinate ball as a dense subset.  \nIn this case, $( \\phi, \\theta, \\psi, y)$\nare  spherical coordinates on $S^2\\times S^2$ with a nonstandard metric, and $\\alpha$ is the fiber coordinate of an \n$S^1$-fibration $S^1\\to Y^{p,q}\\to S^2\\times S^2.$  \n$Y^{p,q}$ is diffeomorphic to $S^2\\times S^3,$ and\n has first Chern class which integrates over the two $S^2$ factors to $p+q$ and $p$ \\cite[\\S2]{gdsw}.\nThe coordinate ranges are $\\phi\\in (0,2\\pi), \\theta \\in (0,\\pi), \\psi\\in (0,2\\pi)$,\n$\\alpha\\in (0,2\\pi\\ell)$,\nwhere $\\ell = \\ell(p,q)$, and \n$y\\in (y_1, y_2)$, with the $y_i$ the two smaller roots of $a-3y^2+2y^3=0$.  $p$ and\n$q$ determine $a, \\ell, y_1, y_2$ explicitly \\cite[(3.1), (3.4), (3.5), (3.6)]{gdsw}.  \n\n\n\n\nFor these choices of $p, q$, we get an $S^1$-action $a_1$ on $Y^{p,q}$ by rotation in the $\\alpha$-fiber.\nWe claim that for e.g. $(p,q) = (7,3)$\n\\begin{equation}\\label{neqo} \\int_{[a_1^L]} CS_5^W(g) \\neq 0.\n\\end{equation}\nBy Proposition \\ref{prop:two}(iii), this implies $\\pi_1({\\rm Diff} (S^2\\times S^3), id)$ is infinite.\nSince the trivial action $a_0$ has $\\int_{[a_0^L]} CS_5^W(g) = 0$ (by the proof of \nProposition \\ref{prop:two}(ii) with $n=0$), $a_0$ and $a_1$ are not smoothly homotopic by\nProposition \\ref{prop:two}(i).   Thus showing (\\ref{neqo}) will prove the theorem.\n\nSet $M = S^2\\times S^3$. Since $a_1^L:M\\to LM$ has degree one on its image,\n\\begin{equation}\\label{cswf} \n\\int_{[a_1^L]} CS_5^W(g) = \\int_M a_1^{L,*} CS_5^W(g).\n\\end{equation}\nFor $m\\in M$, \n$$a_1^{L,*}CS_5^W(g)_m = f(m)d\\phi\\wedge d\\theta\\wedge dy\\wedge d\\psi\\wedge \nd\\alpha$$ \nfor some $f\\in C^\\infty(M)$.  We determine\n$f(m)$ by explicitly computing $a_{1,*}^L(\\partial_\\phi),..., a_{1,*}^L(\\partial_\\alpha),$ (e.g. \n$a_{1,*}^L(\\partial_\\phi)(a^L(m))(t) = \\partial_\\phi|_{a(m,t)}$ ),\nand noting\n\\begin{eqnarray}\\label{f} f(m) &=& f(m)d\\phi\\wedge d\\theta\\wedge dy\\wedge d\\psi\\wedge \nd\\alpha(\\partial_\\phi,\\partial_\\theta,\\partial_y,\\partial_\\psi,\\partial_\\alpha)\\nonumber\\\\\n&=& a_1^{L,*}CS_5^W(g)_m(\\partial_\\phi,...,\\partial_\\alpha)\\\\\n&=& CS_5^W(g)_{a_1^L(m)}(a_{1,*}^L(\\partial_\\phi),...,a_{1,*}^L(\\partial_\\alpha)).\\nonumber\n\\end{eqnarray}\nSince $CS^W_5(g)$ is explicitly computable from the formulas in \\S\\ref{localsymbols}, we can\ncompute $f(m)$ from (\\ref{f}).  Then $\\int_{[a_1^L]} CS_5^W(g) = \\int_{M} f(m)\nd\\phi\\wedge d\\theta\\wedge dy\\wedge d\\psi\\wedge \nd\\alpha$ can be computed as an ordinary integral in the dense coordinate space. \n\n\nVia this method, in the Mathematica file {\\tt ComputationsChernSimonsS2xS3.pdf} at {\\tt http:\/\/math.bu.edu\/people\/sr\/},  $\\int_{[a_1^L]} CS_5^W(g)$ is computed as\n a function of $(p,q).$  For example, \n$(p,q) = (7,3)$,\n$$\\int_{[a_1^L]} CS_5^W(g)  = -\\frac{1849\\pi^4}{22050}.$$\nThis formula is exact; the rationality up to $\\pi^4$ follows from $4p^2-3q^2$ being a perfect\nsquare, as then the various integrals computed in (\\ref{cswf}) with respect to our coordinates\nare rational functions evaluated at rational endpoints. \n In particular, (\\ref{neqo})\nholds.\n  \\end{proof}\n\n\n\\begin{rem}  \nFor $a=1$, the metric extends to the closure of the coordinate chart, but the total space is $S^5$ with the standard metric.  \n$\\pi_1({\\rm Diff} (S^5))$ is torsion \\cite{F-H}.  By Proposition \\ref{prop:two}(ii), $\\int_{[a^L]} CS^W_5 = 0$\nfor any circle action on $S^5.$  In the formulas in the Mathematica file, $\\int_{[a^L]}CS^W_5$ is proportional to\n$(-1+a)^2$, which vanishes at $a=1$.  \nThis gives a check of the validity of the computation.\n\\end{rem}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSchwinger long ago predicted that a strong, constant electric field will create electron-positron pairs \\cite{schwinger1951gauge}. This has, however, never been experimentally observed as the electric field strengths required are several orders of magnitude larger than has ever been achieved in the laboratory \\cite{danson2015petawatt,Turcu:2016dxm}. The rate of pair production becomes large as the electric field strength approaches an appreciable fraction of the Schwinger critical field, $E_c= m_e^2c^3\/ e \\hbar$, corresponding to an electric field intensity $I_c=\\tfrac{1}{2}\\ensuremath{\\epsilon_0} c E_c^2 \\approx 2\\times 10^{29}~\\mr{Wcm}^{-2}$. It has also long ago been predicted that one can create electron-positron pairs from a thermal bath of photons \\cite{weaver1976reaction}, via the two photon Breit-Wheeler process, $\\gamma\\gamma\\to e^+e^-$ \\cite{breit1934collision}. This too has never been observed due to the unattainably high temperatures required (although an experiment has been proposed that uses a quasi-thermal radiation field for one of the two photons \\cite{pike2014photon}). In this case, the rate of pair production becomes large when $k_B T$ becomes an appreciable fraction of the rest mass energy of an electron and positron, $2m_e c^2\\sim 1~\\mr{MeV}$.\n\nOne can, however, combine these two ingredients. Starting from an initial state containing a thermal bath of photons at a high temperature and adding a strong, constant electric field, one finds the rate of pair production is significantly faster than either the Schwinger process or the purely thermal process alone. The nature of the process depends on the relative magnitudes of the electric field strength and the temperature. At low temperatures, when $E\/E_c\\gg k_B T\/2m_ec^2$, the process is essentially Schwinger pair production, in which virtual electron-positron pairs tunnel quantum mechanically through their energy barrier to existence. In the opposite limit, $E\/E_c\\ll k_B T\/2m_ec^2$, at high temperatures, virtual electron-positron pairs are given sufficient energy from the thermal bath to transition over the barrier classically. At intermediate temperatures the process can be described as thermally enhanced quantum tunnelling, whereby the virtual electron-positron pairs tunnel from an excited state at nonzero separation.\n\nHere we consider the viability of observing the thermal Schwinger process. If this were realised, it would be the first observation of semiclassical pair production in quantum electrodynamics (QED), a class of nonperturbative phenomena with applications in many branches of physics, including astrophysics, cosmology, heavy ion collisions, and plasma physics (see for example \\cite{Ruffini:2009hg}). It would also open up the controlled study of semiclassical pair production in general, a very basic process in quantum field theory and one that has proved elusive experimentally.\n\nTo put our work in context, we note that several other mechanisms have been proposed to lower the intensities required for Schwinger pair production, by including high frequency fields \\cite{brezin1970pair,Ringwald:2001ib,Popov:2001ak,Piazza:2004sv,dunne2009catalysis,Otto:2015gla,Panferov:2015yda,Torgrimsson:2016ant,Kohlfurst:2012rb,Jansen:2013dea}, or the Coulomb fields of highly charged nuclei \\cite{Muller:2003zzd,PhysRevA.73.053409,DiPiazza:2009py,FillionGourdeau:2012zs}. Further, it has been pointed out that once an initial seed pair has been produced, a cascade of pair production will follow for sufficiently strong field intensities \\cite{Bell:2008zzb,PhysRevLett.105.080402,Nerush:2010fe}. This may dramatically amplify any signal of Schwinger pair production. On the experimental side of strong-field QED, there has been much progress on a variety of fronts \\cite{burke1997positron,mackenroth2011nonlinear,thomas2012strong,blackburn2014quantum,cole2018experimental,poder2018experimental}, and the next generation of high intensity lasers offer exciting possibilities to discover and investigate qualitatively new phenomena in QED \\cite{DiPiazza:2011tq,Turcu:2016dxm}.\n\nIn the regime we consider here, there is an additional significance, in that the formula for the rate of pair production is a rare example of an all-orders and all-loop result in QED \\cite{gould2017thermal,Gould:2018ovk}. Perturbative estimates of the rate of pair production are orders of magnitude slower, so could be distinguished experimentally. Hence, one can probe QED beyond the perturbative loop expansion, and could decide experimentally on the validity of conjectured all-order behaviours of QED \\cite{cvitanovic1977asymptotic,affleck1981pair,lebedev1984virial,huet2017asymptotic,gould2017thermal,Gould:2018ovk}. Further, an experimental verification of the formulae would directly carry over to strongly-coupled physics, and hence provide an important validation of principle in the search for magnetic monopoles \\cite{acharya2016search,gould2017magnetic}. As such, an experimental search for the thermal Schwinger process is well motivated even independently of its connection to the pure Schwinger process.\n\n\\section{Theoretical results}\nAt zero temperature, and for electric field strengths somewhat below $E_c$, the rate of electron-positron pair production per unit volume in a constant electric field is given by Schwinger's result \\cite{schwinger1951gauge},\n\\begin{equation}\n \\Gamma_{\\mr{Schwinger}}(E)\\approx \\frac{(eE)^2}{4\\pi^3 c \\hbar^2}\\mr{e}^{-\\frac{\\pi m_e^2 c^3}{e  E \\hbar}}. \\label{eq:rate_schwinger}\n\\end{equation}\nLoop corrections to this formula have been computed at two loops \\cite{ritus1975lagrange,ritus1977connection,lebedev1984virial,ritus1998effective}, (see also \\cite{Gies:2016yaa}) and have been resummed to all loops (within the quenched approximation) \\cite{affleck1981pair,lebedev1984virial}. However, the loop corrections are small, and only give an $O(1\\%)$ enhancement over Schwinger's one-loop result. At leading loop order, there are also corrections which are exponentially subdominant for small $E\/E_c$ \\cite{schwinger1951gauge}.\n\nIf one adds a thermal bath of photons, the rate is enhanced. Note that in this analysis it is crucial that there are very few charged particles in the initial thermal state as their presence would Debye screen the electric field. The Debye screening length should be longer than the scales relevant for pair production (which we give in Section \\ref{sec:schematic}). Hence, we assume $k_B T\\ll m_e c^2$.\n\nDepending on the relative magnitudes of $E$ and $T$ one finds three different regimes. In the lowest temperature regime, the energy of the thermal bath is less than the energy that an electron-positron pair would gain when accelerated by the electric field over their Compton wavelength. That is, for temperatures lower than\n\\begin{equation}\n    T_{CW}:=e E\\hbar\/(2m_e c k_B). \\label{eq:lowT}\n\\end{equation}\nIn this regime, the thermal bath is negligible and electron-positron pairs are produced by quantum tunnelling from virtuality in vacuum, to reality. Above $T_{CW}$ the thermal bath excites virtual electron-positron pairs significantly above their ground state. Pair production then takes the form of quantum tunnelling from an excited state. At higher temperatures still, virtual electron-positron pairs acquire sufficient energy from thermal fluctuations to go over the energy barrier classically. This process dominates over quantum tunnelling for temperatures greater than\n\\begin{equation}\n    T_{WS}:=\\left(4 e E^3\\ensuremath{\\epsilon_0}\\hbar^4\/\\pi^3 m_e^2 k_B^4\\right)^{1\/4}\n\\end{equation}\nThis temperature can be understood as that when the lowest thermal (Matsubara) frequency $2\\pi k_BT\/\\hbar$ is equal to the exponential decay rate of the unstable electron-positron transition state, $2\\sqrt{2}(eE^3\\ensuremath{\\epsilon_0}\/m_e^2)^{1\/4}$.\\footnote{The labels $C$, $W$ and $S$ follow the notation of Ref.~\\cite{gould2017thermal}. They stand for \\emph{circular}, \\emph{wavy} and \\emph{straight} (or \\emph{sphaleron}) respectively, and refer to the shape of the instanton describing the processes at low, intermediate and high temperatures.}\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{allApproxQEDRatesExpPlusWriting5.pdf}\n  \\caption[Thermal Schwinger Rate]{Approximate rates of pair production for low, medium and high temperatures. The coloured region is where the approximations leading to Eqs. \\eqref{eq:rate_schwinger}, \\eqref{eq:rate_w} and \\eqref{eq:rate_sphaleron} are well satisfied. All rates are normalised by $\\Gamma_{\\mr{Ref}}=0.1~\\muup\\mr{m}^{-3}\\muup\\mr{s}^{-1}$. For intermediate temperatures we use only the leading two contributions in the fine structure constant whereas in the high temperature regime all orders are included. This leads to the apparent discontinuity which will be smoothed out by contributions at higher orders.}\n \n  \\label{fig:allRatesApprox}\n\\end{figure}\n\n\\emph{Low temperatures,} $T<T_{CW}$:\nFor very low temperatures, $T\\ll T_{CW}$, corrections to the Schwinger rate can be considered perturbatively. The dominant contribution to the rate in this regime is thus given by Fig.~\\ref{fig:feyn}~(a), just as at $T=0$. The leading thermal corrections arise at two loops and are \\cite{gies2000qed}\n\\begin{equation}\n    \\Gamma_C=\\Gamma_{\\mr{Schwinger}}\\left[1+\\frac{\\pi^4 e^2}{1440\\ensuremath{\\epsilon_0}\\hbar c} \\left(\\frac{T}{T_{CW}}\\right)^4+\\dots\\right].\n   \n\\end{equation}\nAs $T$ increases towards $T_{CW}$, higher order perturbative corrections become important. The exponential enhancement due to these corrections has been computed at leading \\cite{monin2010photon} and next-to-leading \\cite{gould2017thermal} order. The corrections to the exponent of the rate are small if $E\\lesssim E_c$, except if one is very close to $T_{CW}$ in which case even the exponent is unknown. Where one can trust the calculations, the thermal enhancement at low temperatures is less that 1\\%.\n\n\\emph{Intermediate temperatures,} $T_{CW}<T<T_{WS}$:\nAt $T_{CW}$, the rate goes through a sharp transition. In this intermediate temperature range, the rate of pair production is significantly (exponentially) enhanced by the presence of the thermal bath. For intermediate temperatures, the exponent of the rate is given by \\cite{selivanov1986tunneling,ivlev1987tunneling,brown2015schwinger,gould2017thermal,Draper:2018lyw}\n\\begin{align}\n  &\\Gamma_{W}(E,T)\\sim \\nonumber \\\\\n  &\\exp\\left[-2\\frac{m_e^2c^3}{eE\\hbar}\\left( \\arcsin\\left(\\frac{T_{CW}}{T}\\right) +  \\frac{T_{CW}}{T} \\sqrt{1 - \\frac{T_{CW}^2}{T^2}}\\right)\\right], \\label{eq:rate_w}\n\\end{align}\nthough there has been some dispute on this \\cite{medina2015schwinger,Korwar:2018euc}. The precise formula, including the prefactor of the exponential has recently been worked out in Ref.~\\cite{Torgrimsson:2019sjn}. There it was also shown that the dominant contribution to the rate in this regime is given by Fig.~\\ref{fig:feyn}~(b).\n\n\\emph{High temperatures,} $T_{WS}<T$:\nAt $T_{WS}$ there is again a sharp transition in the rate. In the high temperature regime, the pair production process can be seen as a thermal process enhanced by the presence of the electric field. Unlike the lower temperature regimes, the rate is not dominated by the diagrams in Fig.~\\ref{fig:feyn}; infinitely more such diagrams contribute to the leading approximation to the rate, leading to a significant nonperturbative enhancement. Some of the present authors have recently calculated the rate \\cite{gould2017thermal,Gould:2018ovk}. It is given by\n\\begin{equation}\n \\Gamma_{S}(E,T)\\approx \\frac{4 k_B T_{WS}\\left(m_e k_B T\\right)^{3\/2}\\mr{e}^{-\\frac{2m_e c^2}{k_B T}+\\frac{\\sqrt{e^3E\/\\pi\\ensuremath{\\epsilon_0}}}{k_B T}}}{(4\\pi)^{3\/2}\\hbar^4\\sin \\left(\\frac{\\pi T_{WS} }{T }\\right)\\sinh^2\\left(\\frac{\\pi T_{WS}}{\\sqrt{2} T }\\right)} .\\label{eq:rate_sphaleron}\n\\end{equation}\nIn deriving this expression we assumed the following three strong inequalities: $e E\\hbar\/m_e^2c^3\\ll 1$, $k_BT\/m_e c^2\\ll 1$ and $e (k_B T)^2 \/E c^2 \\ensuremath{\\epsilon_0} \\hbar^2\\ll 1$. Note that these imply the calculation is only valid when the rate of pair production is not too fast. For temperatures much larger than $T_{WS}$, this equation simplifies to \n\\begin{align}\n\\Gamma_S(E,T) & \\approx \\frac{m_e^{3\/2} T^2 \\left(k_B T\\right)^{5\/2}}{\\pi^{9\/2}\\hbar^4 T_{WS}^2}  \\mr{e}^{-\\frac{2m_e c^2}{k_B T}+\\frac{\\sqrt{e^3E\/\\pi\\ensuremath{\\epsilon_0}}}{k_B T}} \\nonumber \\\\\n&\\approx \\frac{ m_e^{5\/2} (k_B T)^{9\/2}}{ 2 \\pi ^3 \\hbar^{6} \\sqrt{ \\ensuremath{\\epsilon_0} e E^{3}} } e^{-\\frac{2 m_e c^2}{k_BT}+\\frac{\\sqrt{e^3E\/\\pi\\ensuremath{\\epsilon_0}}}{k_B T}}.\\label{eq:rate_sphaleron_high}\n\\end{align}\nAs one can see from this expression, surprisingly the rate is higher for weaker fields. This behaviour cannot be extrapolated to arbitrarily weak fields as the validity of our approximations breaks down for $E\\lesssim e (k_B T)^2 \/c^2 \\ensuremath{\\epsilon_0} \\hbar^2$.\n\nFor a thermal bath of photons in zero electric field, electron-positron pairs can be produced by two photon fusion (the Breit-Wheeler process). Integrating the cross section for this process over the photon thermal distribution one finds \\cite{weaver1976reaction}\\footnote{Note that there are several typos in Ref.~\\cite{weaver1976reaction}. We have repeated the calculation for low temperatures, both fully numerically and in the nonrelativistic approximation, finding agreement with Eq. \\eqref{eq:breit_wheeler}.}\n\\begin{align}\n \\Gamma_{BW}(T)&\\approx 2 \\int \\frac{d^3 p}{(2\\pi )^3}\\frac{d^3 q}{(2\\pi )^3} \\frac{p^\\mu q_\\mu}{p^0 q^0}\\theta\\left((p^\\mu+q^\\mu)^2-4m^2\\right)\\nonumber \\\\\n & \\qquad\\qquad\\qquad\\qquad f(p^0)f(q^0)\\sigma_{BW}\\left((p^\\mu+q^\\mu)^2\\right)\\nonumber \\\\\n &\\approx \\frac{e^4 m_e (k_B T)^3}{2(2\\pi)^4 c^3 \\ensuremath{\\epsilon_0}^2 \\hbar^6}\\mr{e}^{-\\frac{2m_e c^2}{k_B T}},\\label{eq:breit_wheeler}\n\\end{align}\nwhere $f(E)=1\/(\\mr{e}^{E\/k_BT}-1)$ and $\\sigma_{BW}(s)$ is the Breit-Wheeler cross section \\cite{breit1934collision}. This expression is valid for low temperatures, $k_B T\/m_e c^2 \\ll 1$, up to about $O(100~\\mr{keV}\/k_B)$ with an accuracy of a few percent. In the absence of an intense electric field, higher order perturbative effects involving more photons are expected to be subdominant if $e k_B T\/\\sqrt{c \\hbar \\ensuremath{\\epsilon_0}} \\ll m_ec^2$, or $k_BT\\ll 3 m_e c^2$ \\cite{king2012pair}.\n\nWe note that the addition of a constant electric field does not enhance the perturbative Breit-Wheeler process as a constant field consists of zero energy photons. Beyond this idealised limit, corrections to the Breit-Wheeler rate due to the presence of an additional source of photons with wavelength $\\lambda$ are suppressed by $\\exp(-\\lambda m^2 c^3\/(2\\pi \\hbar k_B T))$. For a typical laser source with $\\lambda = 0.8~\\muup \\mr{m}$ and a thermal bath of temperature $T=20~\\mr{keV}\/k_B$, say, this correction is completely negligible $\\sim 10^{-10^6}$.\n\nThe Breit-Wheeler process should not be thought of as a competing process to thermal Schwinger pair production. Fig.~\\ref{fig:feyn}~(b) shows the dominant Feynman diagram for thermal Schwinger pair production in the intermediate temperature regime. Applying the Optical Theorem, one can see that a unitarity cut of diagram (b) gives the Breit-Wheeler process~\\cite{Torgrimsson:2019sjn}, except that the effect of the electric field has been accounted for to all orders, giving a nonperturbative enhancement over the pure Breit-Wheeler process. In fact, Eq.~\\eqref{eq:rate_w} reduces to Eq.~\\eqref{eq:breit_wheeler} for temperatures $T\\gg T_{CW}$.\n\nHowever, in the high temperature regime, $T>T_{WS}$, the diagrams of Fig.~\\ref{fig:feyn} cease to dominate the rate of pair production and there is a further nonperturbative enhancement that cannot easily be understood diagramatically. As can be seen from Eqs.~\\eqref{eq:rate_sphaleron} and \\eqref{eq:rate_sphaleron_high}, the dependence of the rate, $\\Gamma_S$, on the fine-structure constant is non-analytic even after one absorbs one power of $e$ into the electric field.\n\\begin{figure}\n  \\includegraphics[width=0.7\\columnwidth]{feyn.pdf}\n  \\caption{Feynman diagrams which dominate the rate of thermal Schwinger pair production in the (a) low and (b) intermediate temperature regimes. Double lines denote the electron propagator including the effect of the external electric field to all orders and the wiggly lines denote photons from the thermal bath. In the high temperature regime, the rate is not dominated by a single such Feynman diagram but infinitely many diagrams contribute to the leading approximation to the rate.}\n  \\label{fig:feyn}\n\\end{figure}\n\nFor completeness, we note that the addition of a single electromagnetic plane wave to a thermal bath of photons also leads to nonperturbatively enhanced electron-positron pair production. This is true even in the long-wavelength limit, $\\lambda m c\/\\hbar \\to \\infty$, showing the collective, nonperturbative nature of the phenomenon. In this case, the Breit-Wheeler rate is additively enhanced by \\cite{king2012pair},\n\\begin{equation}\n\\Gamma_{\\mr{Plane}} \\approx \\frac{3^{3\/4} e^2 (k_B T)^2 m^2 }{16 \\pi ^{5\/2} \\ensuremath{\\epsilon_0} \\hbar^5} \\left(\\frac{e E \\hbar k_B T}{\n   m^3 c^5}\\right)^{1\/4} \n   \\mr{e}^{-\\sqrt{\\frac{16c^5 m^3}{3 e E \\hbar k_B T }}}. \\label{eqn:rate_king} \n\\end{equation}\nThis result is valid for $\\sqrt{k_B T E\/(m c^2 E_c)}\\ll 1$. The crucial difference from that of a constant electric field is that the electromagnetic invariant $E^2-c^2B^2$ of a plane wave vanishes. As we will note later, Eq. \\eqref{eqn:rate_king} is orders of magnitude smaller than the thermal Schwinger rate, showing that the absence of the magnetic field is crucial for pair production.\n\n\\section{Observability}\n\nWe would like to understand exactly how high the temperatures and how strong the electric fields need to be to get a measurable rate of pair production. To answer that, we will make a simple comparison with the experiment of Ref.~\\cite{burke1997positron}, which was the first experiment to observe the (multi-photon) Breit-Wheeler process. They observed $106\\pm14$ positrons produced in this way, from a total spacetime interaction volume of order $10^{-21}~\\mr{m}^3 \\mr{s}$ (when integrated over all laser shots). Hence we take as our observable reference rate $\\Gamma_{\\mr{Ref}}=10^{23} ~\\mr{m}^{-3}\\mr{s}^{-1}=0.1~\\muup\\mr{m}^{-3}\\muup\\mr{s}^{-1}$, which is approximately Avogadro's number of positrons per metre cubed per second. One can therefore reasonably expect that a normalised rate greater than 1 will be required for the rate of pair production to be measurable. In Fig.~\\ref{fig:allRatesApprox} we show the thermal Schwinger rate in all three regimes, normalised by this reference rate.\n\nThe almost perfectly vertical lines of constant rate in the low temperature regime reflect that, in this regime, the thermal enhancements are small. As such, this regime offers no advantages over pure Schwinger pair production for experimentally observing pair production. On the other hand, in the intermediate and high temperature regimes, the thermal enhancements are very significant. Of these two regimes, observing pair production in the high temperature regime is easier, because the electric field intensities required are orders of magnitude smaller, while the temperatures required are very similar.\n\nFrom Figure \\ref{fig:allRatesApprox} one can see that temperatures around $O(20~\\mr{keV}\/k_B)$ or above are needed in order to produce an observable number of positrons. Perhaps the leading method of producing high temperature thermal photons is with a laser and cavity, or holhraum. The aim of achieving inertial confinement fusion (ICF) has been a powerful incentive in developing these technologies. Thermal distributions of $0.3~\\mr{keV}\/k_B$ have been achieved since 1990, though about $0.4~\\mr{keV}\/k_B$ is likely the upper limit of this approach \\cite{lindl2004physics}. Unfortunately at these temperatures, the thermal enhancement of the Schwinger rate is negligible.\n\nWhen ICF is achieved, the burning thermonuclear plasma leads to significantly higher energy densities. Charged particles in the plasma are expected to reach temperatures from $O(20~\\mr{keV}\/k_B)$ to $O(200~\\mr{keV}\/k_B)$, depending on the composition and size of the burning plasma \\cite{tabak1996role,rose2013electron}. Burning deuterium (D) plasmas are expected to be hotter than burning deuterium-tritium (DT) plasmas, as the peak nuclear reaction rate is at higher energies for D-D nuclear reactions. For a fixed composition, larger plasmas reach higher temperatures.\n\nAs the plasma is not optically thick, the effective temperature of the photons is lower than that of the charged particles. For representative examples, of burning deuterium plasma with radii $r=120~\\muup\\mr{m}$ and $r=150~\\muup\\mr{m}$, one finds that the photon energy density is equal to that of a Planck distribution with two degrees of freedom at $T=22~\\mr{keV}\/k_B$ and $T=26~\\mr{keV}\/k_B$ respectively. The photon distribution can be calculated using the approach of Ref.~\\cite{rose2013electron}. However, the result is further from equilibrium than that of the charged particles. For now, we will assume a thermal distribution of photons, though we will return to this point in Section \\ref{sec:distributions}.\n\n\\section{An experimental scheme} \\label{sec:schematic}\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{experimentSchematic3.pdf}\n  \\caption{Schematic of the experimental set-up. Two counter propagating high energy beams are focused into an X-ray radiation field produced by a burning fusion.}\n  \\label{fig:schematic}\n\\end{figure}\n\nSo, in order to observe the thermal Schwinger process, we would propose combining two lasers with combined intensity $O(10^{23}~\\mr{Wcm}^{-2})$ with a source of thermal photons with temperature $O(20~\\mr{keV}\/k_B)$. A possible schematic for such an experiment is shown in Figure \\ref{fig:schematic}. The region of interest for our purposes is on the left-hand side, outside the ignited thermonuclear plasma. A window is needed to hold up the material expansion long enough to allow the high-intensity lasers to interact with the radiation from the ICF capsule. The wall of the hohlraum would in principle be able to act in this way whilst transmitting the majority of the radiation, though this would require specific design. As long as the distances from the nuclear plasma are small compared with its radius, the geometric reduction of the intensity will not be significant.\n\nThe electric field is provided by a high intensity laser, split into two counter-propagating beams. These are focused so that the magnetic fields cancel in the vicinity of a given point, whereas the electric fields reinforce. Assuming standard parameters for the laser, with wavelength $\\lambda \\sim 0.8~\\muup m$, the field maxima of the two beams are expected to be approximately of size $O(\\lambda^3)$ and of time extent $O(\\lambda\/c)$, with approximately 10-20 field maxima per shot, amounting to a possible pair production region of size $5\\times 10^{-32}~\\mr{m}^3\\mr{s}$. The integrals of the rate over the interaction region can be carried out in the locally constant field approximation (see for example Refs.~\\cite{galtsov1983macroscopic,Gavrilov:2016tuq}), within which the region around the field maxima will dominate the pair production. However, in what follows we simply multiply the rates by the approximate spacetime volume of the field maxima, which is sufficient to get the order of magnitude correct.\n\nTo achieve $\\gtrsim 1$ electron-positron pair produced per shot, requires a rate $5\\times 10^6$ times faster than $\\Gamma_{\\mr{Ref}}$ (see Fig.~\\ref{fig:sphaleron_rate}). Assuming a thermal distribution of photons from the burning nuclear plasma, this could be achieved at\n\\begin{align}\n T_\\star &\\approx 20~\\mr{keV}\/k_B, \\nonumber \\\\\n I_{E\\star} &\\approx 1.3\\times 10^{23}~ \\mr{W cm}^{-2}, \\label{eq:parameters}\n\\end{align}\nwhere $I_E$ refers to the combined intensity of the two beams. This parameter point is shown as a star in Fig.~\\ref{fig:sphaleron_rate}. For comparison, we also consider a second point with a significantly higher production rate, at \n\\begin{align}\n T_\\blacktriangle &\\approx 26~\\mr{keV}\/k_B, \\nonumber \\\\\n I_{E\\blacktriangle} &\\approx 3.7\\times 10^{23}~\\mr{W cm}^{-2}. \\label{eq:parameters_triangle}\n\\end{align}\nFor these two sets of parameters, the numbers of positrons produced per shot via the thermal Schwinger, Breit-Wheeler and pure Schwinger (without thermal enhancement) processes are given in Table \\ref{table:positrons_per_shot}. We also include the nonperturbative enhancement to the number of positrons produced due to only one of the two laser beams, given by Eq. \\eqref{eqn:rate_king}. \n\nNote that the pure Breit-Wheeler process can take place in a larger region than that of the Schwinger pair production, which is not accounted for in Table~\\ref{table:positrons_per_shot}. In order to ensure that the thermal Schwinger process dominates, and taking into account its $O(10^6)$ times higher rate, the volume of the interaction region should be significantly less than about $10^7 \\lambda^3\\approx 5\\times 10^{-3}\\mr{mm}^3$. This can be achieved by modifying the diameter of the hohlraum window and by focusing the laser fairly close to the window. Further, the directionality of emitted electrons and positrons can help distinguish between production mechanisms, with the Breit-Wheeler process producing pairs more or less isotropically and the Schwinger process producing pairs along the electric field of the counterpropagating lasers.\n\nIt is encouraging that a relatively small increase in both radiation temperature and laser intensity\nproduces such a significant increase in the production rate. One can also see that the thermal Schwinger process has a huge nonperturbative enhancement. A simple perturbative estimate of the number of positrons produced by the Breit-Wheeler process underestimates the actual number by a factor of $10^6$. Such large enhancements are a generic feature of the thermal Schwinger process in the high temperature regime \\cite{Gould:2018ovk}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ c c c c c } \n  \\toprule\n   & Thermal Schwinger & Breit-Wheeler & Eq. \\eqref{eqn:rate_king} & Schwinger\\\\\n  \\colrule\n  $\\star$ & 3 & $3\\times 10^{-6}$ & $10^{-167}$ & $10^{-1817}$\\\\\n  $\\blacktriangle$ & $1\\times 10^6$ & 1 & $10^{-105}$ & $10^{-1062}$ \\\\\n\\end{tabular}\n\\caption{Numbers of positrons produced per shot via different mechanisms for the two sets of parameters given by Eqs. \\eqref{eq:parameters} and \\eqref{eq:parameters_triangle}. The column labelled Eq. \\eqref{eqn:rate_king} is that for a thermal bath plus a single laser beam (rather than counter-propagating beams).}\n\\label{table:positrons_per_shot}\n\\end{center}\n\\end{table}\n\nOne can also compare the thermal Schwinger rate to that obtained in a thermal bath plus only a single high intensity laser beam (in which case the magnetic field does not cancel). In this case, the enhancement of the rate of pair production due to the high intensity laser is given by Eq.~\\eqref{eqn:rate_king}. For the parameters of either Eqs. \\eqref{eq:parameters} or \\eqref{eq:parameters_triangle}, one finds that the enhancement is negligible and the rate of this process is smaller than the thermal Schwinger rate by a factor of $\\sim 10^{\u2212100}$ or more, as can be seen in Table \\ref{table:positrons_per_shot}.\n\nFor the validity of the locally constant field approximation, it is important that the electric field, as well as the photon distribution from the plasma, are slowly varying on the time and length scales of the pair creation process, described by an instanton. The time scale of the instanton is $t_{\\mr{inst}}\\sim \\hbar\/k_B T$ and the length scale is $\\sqrt{e\/(4\\pi \\ensuremath{\\epsilon_0} E)}$ \\cite{gould2017thermal,Gould:2018ovk}. Using temperature and electric field strengths determined by Eq. \\eqref{eq:parameters}, this amounts to $3\\times 10^{-20}~s\\approx 10^{-11}~\\mr{m}\/c$ and $5\\times 10^{-12}~\\mr{m}$ respectively. The smallest length scale on which the electric field varies is the wavelength of the laser. Assuming a laser with wavelength of $\\lambda \\sim 0.8~\\muup m$, one can safely treat the electric field as constant. Further, one would expect the photon distribution from the plasma to vary on a length scale of order the size of the hohlraum window. This will likewise be much larger than the length scale of the instanton, $\\sim 5\\times 10^{-12}~\\mr{m}$, and hence the locally constant field approximation is applicable.\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{sphaleronQEDRateExp2.pdf}\n  \\caption[Thermal Schwinger rate at high temperature]{The number of electron-positron pairs produced per shot in the experiment proposed here. The coloured region is the high temperature region, and where the approximations leading to Eq. \\eqref{eq:rate_sphaleron} are valid. The solid black line is the boundary between the intermediate and high temperature regions, defined by $T=T_{WS}$. The dashed black line is defined by $E=0.2e(k_BT)^2\/\\ensuremath{\\epsilon_0} c^2 \\hbar^2$ and the dotted black line by $T=0.2m_ec^2\/k_B$. The star and diamond are the points referred to in Eqs. \\eqref{eq:parameters} and \\eqref{eq:parameters_triangle}.}\n  \\label{fig:sphaleron_rate}\n\\end{figure}\n\nIn the region where the electric field and thermal photons collide, electrons and positrons will be produced with an approximately thermal spectrum of velocities and then accelerated in opposite directions antiparallel and parallel respectively to the electric field. Their thermal velocities are isotropic in the lab frame and are expected to be rather large, $\\tfrac{1}{3}\\bar{v^2}\\sim k_B T\/m_e \\sim (0.2 c)^2$. The field then accelerates the particles over a distance $\\lesssim \\lambda$, giving them a highly relativistic velocity, $v\\approx c$, parallel to the electric field and up to energies of order $eE\\lambda \\sim 1~\\mr{GeV}$. Once produced, the electrons and positrons may be deflected in opposite directions with a magnet, after which their momenta can be measured by a calorimeter, as in Ref. \\cite{burke1997positron}. If the combined intensity of the lasers is greater than around $10^{24}~\\mr{Wcm}^{-1}$, a seed electron-positron pair produced by the thermal Schwinger process will induce a cascade of pair production, so amplifying any positive signal~\\cite{Bell:2008zzb,PhysRevLett.105.080402,Nerush:2010fe}.\n\nIn the absence of charged particles, the thermal Schwinger process is the dominant mechanism of electron-positron pair production. However, if charged particles are not adequately shielded, other pair production processes are possible, such as the trident mechanism ($e^-Z\\to e^- e^+e^- Z$) and the Bethe-Heitler process ($\\gamma Z\\to e^+e^- Z$). Another possibility is for non-linear Compton scattering of charged particles in the laser field, producing high energy photons which then take part in the Breit-Wheeler process. Debye screening by charged particles will also inhibit the thermal Schwinger process if the Debye length is not much longer than the length scale of the pair creation process, $\\sqrt{e\/(4\\pi \\ensuremath{\\epsilon_0} E)}$. For the parameters of Eq. \\eqref{eq:parameters}, one requires the density of charged particles to be much less than one per $\\mr{pm}^3$. In the regime we have considered here, the purely thermal and the purely Schwinger pair production rates are orders of magnitude lower than the combination. Thus by performing null shots, with either only the burning plasma or only the high intensity laser, one can measure the presence of any backgrounds.\n\n\\section{Photon distributions}\\label{sec:distributions}\n\nLet us return to consider the distribution of photons in the burning plasma. This must be close to equilibrium for our approach to be valid. To investigate this we have solved the Boltzmann equation for the distribution of photons for a range of different plasma sizes and compositions. We have followed the method of Ref.~\\cite{rose2013electron}, including the effect of Compton scattering. For our representative example, of a burning deuterium plasma of radius $r=150~\\muup\\mr{m}$, the photon intensity at the surface of this plasma is shown as the full black line in Fig.~\\ref{fig:intensity}.\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\columnwidth]{planckLogIntensities2.pdf}\n  \\caption[Photon Intensity]{Photon intensity in a D-burning target of radius $150~\\muup\\mr{m}$, along with various approximations to it. Note that a purely thermal distribution at $T=148~\\mr{keV}\/k_B$ would lie at much higher intensities.}\n  \\label{fig:intensity}\n\\end{figure}\n\nEquating the photon energy density to that of a thermal distribution, one finds that the effective temperature of the distribution is $26~\\mr{keV}\/k_B$. Doing the same for the photon number density, one instead finds a somewhat lower effective temperature of $16~\\mr{keV}\/k_B$, showing that the distribution is shifted to higher energies with respect to a thermal distribution. Plotting the photon intensity of a Planck distribution at $T=26~\\mr{keV}\/k_B$, the blue dotted line in Fig.~\\ref{fig:intensity}, one can see the shift to higher energies.\n\nThe high energy tail of the distribution, above about $700~\\mr{keV}$, is an exponential fall off and hence fits well a Boltzmann tail with an effective temperature of $T=148~\\mr{keV}\/k_B$, though scaled down by a normalisation, or equivalently a negative photon chemical potential\\footnote{The photon chemical potential must be zero in equilibrium, but not necessarily out of equilibrium. In this context its presence is natural as photon number conserving processes dominate.}, $\\mu=-1097~\\mr{keV}$, plotted as the dot-dashed green line in Fig.~\\ref{fig:intensity}. At the lowest energies, the distribution rises above this, and can be better described by a purely thermal distribution at a much lower temperature, $T=7.9~\\mr{keV}\/k_B$, plotted as the dashed red line in Fig.~\\ref{fig:intensity}. At intermediate energies, the distribution is not well described by a Bose-Einstein distribution. Nevertheless, the overall shape of the distribution is qualitatively similar to a thermal distribution, being smooth and highly occupied with a power-like rise at low energies and an exponential decrease at high energies, though we have used four different effective temperatures to describe different aspects of it, ranging from $7.9~\\mr{keV}\/k_B$ to $148~\\mr{keV}\/k_B$.\n\nIn two counterpropagating laser beams with intensity given by Eq.~\\eqref{eq:parameters} or \\eqref{eq:parameters_triangle}, one finds that the intermediate temperature regime of thermal Schwinger pair production would be reached at temperatures above\n\\begin{align}\n    T_{CW,\\star} &=0.20~\\mr{keV}\/k_B, \\nonumber \\\\\n    T_{CW,\\blacktriangle} &= 0.32~\\mr{keV}\/k_B, \\label{eq:tcw} \n\\end{align}\nand the high temperature regime would be reached at temperatures above\n\\begin{align}\n    T_{WS,\\star} &=2.5~\\mr{keV}\/k_B, \\nonumber \\\\ \n    T_{WS,\\blacktriangle} &= 3.7~\\mr{keV}\/k_B. \\label{eq:tws} \n\\end{align}\nAll four effective temperatures we have used to describe the distribution of photons in the burning plasma are well above these temperatures. We thus expect the high temperature regime to provide a better description of pair production in this setup than either the low or intermediate temperature regimes, which would imply that the diagrams of Fig.~\\ref{fig:feyn} do not dominate pair production and there is a nonperturbative enhancement over both pure Schwinger and pure thermal pair production.\n\nPhysically, it is clear that the process of pair production should not depend on the photon gas being precisely in equilibrium: if we use the picture of tunnelling from an excited state, one would expect that it is the energy and density of the photon distribution, rather than the nearness to equilibrium, that matters.\n\nOn the other hand, the condition of equilibrium is necessary for the calculation because it leads to important simplifications in the calculation of the production rate. The nonperturbative calculation of Eq.~\\eqref{eq:rate_sphaleron} \\cite{gould2017thermal,Gould:2018ovk} relied heavily on the Matsubara formalism~\\cite{Bloch1932,Matsubara:1955ws}, which is only valid in equilibrium. In the high temperature regime a resummation of all-orders of the perturbative loop expansion was required. Generalising the result to any out-of-equilibrium distribution is beyond the scope of this paper.\n\nWe note however that the diagrams of Fig.~\\ref{fig:feyn} can be calculated in an arbitrary photon distribution, following the approach of Ref.~\\cite{Torgrimsson:2019sjn}, though in the high temperature regime these diagrams are not dominant. Considering the calculation in this distribution, it can be seen that these diagrams reproduce the perturbative Breit-Wheeler rate up to very small corrections, essentially because the photon gas is highly occupied at energies much greater than $k_B T_{CW}$ (see Eqs.~\\eqref{eq:lowT} and \\eqref{eq:tcw}). Further, perturbative corrections in this distribution due to the high intensity laser require one photon from the high energy tail of the distribution and hence are suppressed by $\\exp(-\\lambda m^2 c^3\/(2\\pi \\hbar k_B T) + \\mu\/k_B T)\\sim 10^{-10^{5}}$, where $T$ and $\\mu$ here refer to the green dot-dashed line in Fig.~\\ref{fig:intensity}. Thus any enhancement due to the high intensity laser must be a nonperturbative phenomenon which goes beyond the diagrams of Fig.~\\ref{fig:feyn}.\n\nBecause the full nonperturbative calculation of the rate of pair production is beyond the scope of this paper, the possibility of nonperturbative enhancements in our proposed setup is conjectural. However, as all the effective temperatures we have used to describe the photon distribution are larger than $T_{WS}$, Eq.~\\eqref{eq:tws}, we expect the high temperature regime to best describe the photon distribution in question. We thus expect a nonperturbative enhancement over the perturbative prediction, as is the case in equilibrium where the enhancement to the positron yield was $O(10^6)$. The experiment we have proposed here would be able to test this plausible conjecture, by performing null shots without the counterpropagating laser beams, for which only the perturbative process is possible. This would be able to determine which features of a photon distribution are important for the nonperturbative enhancements to pair production which feature in the thermal Schwinger effect, and how generic such enhancements are.\n\n\\section*{Acknowledgements}\nO.G. would like to thank Holger Gies and Greger Torgrimsson for discussions related to this work. O.G. was supported from the Research Funds of the University of Helsinki. S.M. was supported by Engineering and Physical Sciences Research Council grant No. EP\/M018555\/1 and by Horizon 2020 under European Research Council Grant Agreement No. 682399. A.R. was supported by the U.K. Science and Technology Facilities Council grant ST\/P000762\/1. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Background}\n\nIn this paper, let $\\Sigma$ be a fixed, compact, smooth, oriented surface of genus \n$g > 1$. We also denote $\\sigma |dz|^2$ as a {hyperbolic metric} on $\\Sigma$, for conformal \ncoordinates $z$. On $(\\Sigma, \\sigma |dz|^2)$, we denote the Laplacian as\n\\begin{center}\n$\\Delta = {\\frac{4}{\\sigma}}{\\frac{\\partial^2}{\\partial z \\partial \\bar{z}}}$,\n\\end{center}\nwith nonpositive eigenvalues.\n\n\n{Teichm\\\"{u}ller space} is a complex manifold with the {mapping class group} acting by biholomorphisms, of {complex dimension} \n$3g-3$, the number of independent closed curves in any pair-of-pants decomposition \nof the surface. \n\n\nThe {Weil-Petersson} metric is invariant under the action of the {mapping class group}, hence it decends to a \nmetric on the {moduli space}. It is shown that the {Weil-Petersson} metric is K\\\"{a}hlerian (\\cite {Ah1}), \nwith negative {sectional curvature} (\\cite {Tr}, \\cite {Wp3}). The {Weil-Petersson} Riemannian curvature \ntensor is given as the Tromba-Wolpert formula\n(\\cite {Tr}, \\cite {Wp3}):\n\\begin{center} \n$R_{\\alpha\\bar{\\beta}\\gamma\\bar{\\delta}} = \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\beta})\\mu_{\\gamma}\\bar\\mu_{\\delta}dA + \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\delta})\\mu_{\\gamma}\\bar\\mu_{\\beta}dA$,\n\\end{center}\nwhere $\\Delta$ is the Laplacian for the {hyperbolic metric} $\\sigma$ on the surface, and \n$D = -2(\\Delta -2)^{-1}$ is a self-adjoint, compact operator.\nThose $\\mu$'s in the formula are tangent vectors, i.e., harmonic {Beltrami differential}s. There is a natural \npairing between \n$QD(\\Sigma) =\\{\\phi(z) dz^2\\}$ and $HB(\\Sigma) =\\{\\mu(z) {\\frac{d\\bar{z}}{dz}}\\}$:\n\\begin{center}\n$<\\phi dz^2,\\mu {\\frac{d\\bar{z}}{dz}}> = Re \\int_{\\Sigma}\\phi \\mu dzd\\bar{z}$.\n\\end{center}\n\n\nA remarkable property of the {Weil-Petersson} metric is that it is incomplete (\\cite {Ma}, \n\\cite {Wp1}). This is caused by pinching off at least one short closed geodesic on the \nsurface. The {Weil-Petersson} completion modulo the {mapping class group} is topologically the Deligne-Mumford \ncompactification of the {moduli space}. The compactification divisor, thus \nconsists of a union of lower dimensional {Teichm\\\"{u}ller space}s, each such space consists of noded \n{Riemann surface}s, obtained by pinching nontrivial short closed geodesics on the surface \n(\\cite {B}, \\cite {Ma}). Therefore the compactification divisor can be described \nvia the systole $l_{0}(\\sigma)$ as the set $\\{l_{0}(\\sigma) = 0\\}$.\n\n\nSince we will analyze {harmonic map}s between compact hyperbolic surfaces, we recall some \nfundamental facts here. For a Lipschitz map $w:(\\Sigma, \\sigma |dz|^2) \\rightarrow \n(\\Sigma, \\rho |dw|^2)$, where $\\sigma |dz|^2$ and $\\rho |dw|^2$ are {hyperbolic metric}s on \n$\\Sigma$, and $z$ and $w$ are conformal coordinates on $\\Sigma$, \none follows Sampson (\\cite {S}) to define \n\\begin{center}\n${\\mathcal{H}}(z) = {\\frac{\\rho(w(z))}{\\sigma(z)}}|w_z|^2, {\\mathcal{L}}(z) = \n{\\frac{\\rho(w(z))}{\\sigma(z)}}|w_{\\bar{z}}|^2$.\n\\end{center}\nWe call ${\\mathcal{H}}(z)$ the {\\it holomorphic energy density}, and ${\\mathcal{L}}(z)$ \nthe {\\it anti-holomorphic energy density}. Then the energy density function of $w$ is \nsimply $e(w)= {\\mathcal{H}} + {\\mathcal{L}}$, and the total energy is then given by\n\\begin{center}\n$E(w,\\sigma,\\rho) = \\int_{\\Sigma}e\\sigma|dz|^2$.\n\\end{center}\n\n\nWe also note that the {Jacobian determinant} relative to the $\\sigma$ metric is therefore given by \n$J(z) = {\\mathcal{H}}(z) - {\\mathcal{L}}(z)$.\n\n\nThe map $w$ is called {\\it harmonic} if it is a critical point of this energy \nfunctional, i.e., it satisfies Euler-Lagrange equation:\n\\begin{center}\n$w_{z\\bar{z}}+ {\\frac{\\rho_w}{\\rho}}w_z w_{z\\bar{z}} = 0$.\n\\end{center}\n\n\nThe $(2,0)$ part of the pullback $w^{*}\\rho$ is the so-called {\\it {Hopf differential}}:\n\\begin{center}\n$\\phi(z)dz^2 = (w^{*}\\rho)^{(2,0)} = \\rho(w(z)) w_z {\\bar{w}}_zdz^2$.\n\\end{center}\nIt is routine to check that $w$ is harmonic if and only if $\\phi dz^2 \\in QD(\\Sigma)$, \nand $w$ is conformal if and only if $\\phi=0$.\n\n\nIn our situation, there is a unique \n{harmonic map} $w:(\\Sigma, \\sigma) \\rightarrow (\\Sigma, \\rho)$ in the homotopy class of the \nidentity, moreover, this map $w$ is a diffeoemorphism with positive Jacobian $J$, and \n${\\mathcal{H}}>0$ (\\cite {ES}, \\cite {Hr}, \\cite {S}, \\cite {SY}). \n\n\nA key observation to link the harmonic maps to Teichm\\\"{u}ller theory is that one \nobtains a map from {Teichm\\\"{u}ller space} to $QD(\\Sigma)$, for some fixed {hyperbolic metric} $\\sigma$. More \nspecifically, this map sends any {hyperbolic metric} on $\\Sigma$ to a holomorphic quadratic \ndifferential associated to the unique {harmonic map} in the homotopy \nclass of the identity. This map is a diffeomorphism (\\cite {S}, \\cite{Wf1}).\n\n\\section{Proof of Main Theorems} \n\\noindent \nLet $\\mu = \\mu (z){\\frac{d\\bar{z}}{dz}}$ be a unit {Weil-Petersson} normed harmonic {Beltrami differential}, thus \n$\\int_{\\Sigma}|\\mu|^2 dA = 1$. The {Weil-Petersson} {holomorphic sectional curvature} in the direction of $\\mu$ is given by\n\\begin{center}\n$K_h = -2\\int_{\\Sigma}D(|\\mu|^2)|\\mu|^2 dA$.\n\\end{center} \nIts upper bound, in terms of the genus, is known, proved by Wolpert:\n\\begin{lem}(\\cite {Wp4})\n$K_h < -{\\frac{1}{2\\pi(g-1)}}$.\n\\end{lem}\n\n\n\\begin{rem}\nThis upper bound holds without restriction on the systole. Since all \n{sectional curvature}s are negative, so the {Ricci curvature} and scalar curvature are bounded from \nabove by $-{\\frac{1}{2\\pi(g-1)}}$, and $-{\\frac{3(3g-2)}{4\\pi}}$, respectively (\\cite {Wp4}).  \n\\end{rem}\n\n\nWe now show the following pointwise estimate on $|\\mu(z)|$, where the tangent \nvector $\\mu(z){\\frac{d\\bar{z}}{dz}}$ is normalized to have unit {Weil-Petersson} \nnorm, and we shall apply this estimate to prove our main theorems.\n\\begin{theorem}\nFor $\\mu(z){\\frac{d\\bar{z}}{dz}} \\in HB(\\Sigma)$ with $||\\mu||_{WP} =1$, there \nexists a positive constant $h_0$, independent of $g$, such that \n$|\\mu (z)| \\le h_0$, for all $z \\in \\Sigma$, where the surface $\\Sigma$ is in \nthe thick part of the {moduli space}.\n\\end{theorem}\n\\begin{proof}\nRecall that $\\mu(z){\\frac{d\\bar{z}}{dz}}$ is a harmonic {Beltrami differential}, hence is a symmetric \ntensor given as $\\bar{\\phi}(ds^2)^{-1}$ for $\\phi$ a {holomorphic quadratic differential} with at most simpole poles \nat the cusps and $ds^2$ the {hyperbolic metric} tensor (\\cite{Wp2}). Since the surface $\\Sigma$ lies in the \nthick part of {moduli space}, hence no cusps and this {holomorphic quadratic differential} $\\phi = \\phi(z)dz^{2}$ has no poles.\n\n\nNote that $\\phi \\in QD(\\Sigma)$, as stated in the previous section, by a theorem of \nWolf (\\cite{Wf1}), there exists a {hyperbolic metric} $\\rho$ on surface $\\Sigma$, and a unique \n{harmonic map} $w: (\\Sigma,\\sigma) \\rightarrow (\\Sigma,\\rho)$, such that $\\phi$ is the \nHopf differential associated to this {harmonic map} $w$, i.e., \n$\\phi(z)dz^{2} = \\rho(w(z)) w_z {\\bar{w}}_zdz^{2}$.\n\n\nMuch of our study will be analyzing this {harmonic map} $w$. Note that even though \n$inj_{\\sigma}(\\Sigma) > r_{0} >0$, the metric $\\rho$ might not lie in the thick part \nof the {moduli space}. We recall that the holomorphic and anti-holomorphic energy density \nfunctions of $w$ are defined as \n${\\mathcal{H}}(z) = {\\frac{\\rho(w(z))}{\\sigma(z)}}|w_z|^2$, and ${\\mathcal{L}}(z) = \n{\\frac{\\rho(w(z))}{\\sigma(z)}}|w_{\\bar{z}}|^2$, respectively. \n\n\nThe energy density function of $w$ is $e(w)= {\\mathcal{H}} + {\\mathcal{L}}$, while the {Jacobian determinant} \nbetween {hyperbolic metric}s $\\sigma$ and $\\rho$ is therefore \n$J(z) = {\\mathcal{H}}(z) - {\\mathcal{L}}(z)$. Since the map $w$ is a diffeomorphism \nwith positive {Jacobian determinant}, we have ${\\mathcal{H}}(z) > {\\mathcal{L}}(z) \\ge 0$.\n\n\nWe also find that \n${\\mathcal{H}}{\\mathcal{L}} = {\\frac{|\\phi|^{2}}{\\sigma^{2}}} = |\\mu|^{2}$, so \nthe zeros of ${\\mathcal{L}}$ are the zeros of $|\\mu|$, or equivalently, the zeros of $\\phi$.\n\n\nLet $\\nu$ be the {Beltrami differential} of the map $w$, defined by \n$\\nu = {\\frac{w_{\\bar z}d\\bar z}{w_{z}dz}}$. It measures the failure of $w$ to be \nconformal, and since $J > 0$, we have $|\\nu|<1$. \n\n\nOne easily finds that\n $|\\nu|^{2} = {\\frac{\\mathcal{L}} {\\mathcal{H}}}$, therefore,\n \\begin{center}\n $|\\mu|= {\\sqrt{{\\mathcal{H}}{\\mathcal{L}}}} = {\\mathcal{H}}|\\nu| < {\\mathcal{H}}$.\n \\end{center}\nThus it suffices to estimate ${\\mathcal{H}}(z)$ to bound $|\\mu|$ pointwisely.\n\n\nLet $z_{0} \\in \\Sigma$ such that ${\\mathcal{H}}(z_{0}) = max_{z \\in \\Sigma}{\\mathcal{H}}(z)$. \nWe follow a calculation of Schoen-Yau to define a local one-form $\\theta = \\sqrt{\\sigma(z)}dz$, \nand find (\\cite {SY}):\n \\begin{center}\n $|w_{\\theta}|^{2}= {\\frac{\\rho}{\\sigma}}|w_{z}|^{2} = {\\mathcal{H}}(z)$,\n \\end{center}\n and \n \\begin{eqnarray}\n \\Delta |w_{\\theta}|^{2}= 4|w_{\\theta \\theta}|^{2}+2J|w_{\\theta}|^{2}-2|w_{\\theta}|^{2}.\n \\end{eqnarray}\n We rewrite this as \n \\begin{eqnarray}\n \\Delta{\\mathcal{H}}= 4|w_{\\theta \\theta}|^{2}+2J{\\mathcal{H}}-2{\\mathcal{H}} > -2{\\mathcal{H}}.\n \\end{eqnarray}\n \n \n Therefore ${\\mathcal{H}}$ is a subsolution to an elliptic equation $(\\Delta+2)f=0$.\n \n \n Recalling that $inj_{\\sigma}(\\Sigma)>r_{0}>0$, we embed a hyperbolic ball $B_{z_{0}}({\\frac{r_{0}}{2}})$ \n into $\\Sigma$, centered at $z_{0}$ with radius ${\\frac{r_{0}}{2}}$. Morrey's theorem (\\cite{Mo}, theorem 5.3.1) \n on subsolutions of elliptic differential equations guarantees that there is a constant $C(r_{0})$, such that, \n \\begin{center}\n ${\\mathcal{H}}(z_{0}) = sup_{B_{z_{0}}({\\frac{r_{0}}{4}})}{\\mathcal{H}}(z) \n \\le C(r_{0})\\int_{B_{z_{0}}({\\frac{r_{0}}{2}})}{\\mathcal{H}}(z)\\sigma dzd\\bar{z}$.\n \\end{center}\n \n \n Another consequence of formula $(3)$ is the Bochner identity (see \\cite {SY}), as now $log{\\mathcal{H}}$ \n is well defined:\n \\begin{eqnarray}\n \\Delta log{\\mathcal{H}}= 2{\\mathcal{H}}-2{\\mathcal{L}} -2.\n \\end{eqnarray} \n The minimal principle implies ${\\mathcal{H}}(z) \\ge 1$ for all $z \\in \\Sigma$, and we find\n \\begin{center}\n $\\int_{\\Sigma}{\\mathcal{L}}dA \\le \\int_{\\Sigma}{\\mathcal{H}}{\\mathcal{L}}dA = \n \\int_{\\Sigma}|\\mu|^{2}dA = ||\\mu||_{WP}= 1$. \n \\end{center}\n \n \nIt is not hard to see that we can actually bound the total energy of this {harmonic map} $w$ from above, \nin terms of the genus. More precisely, we recall that $w$ is a diffeomorphism, so\n\\begin{center}\n$ \\int_{\\Sigma}({\\mathcal{H}}-{\\mathcal{L}})dA =  \\int_{\\Sigma}JdA = Area(w(\\Sigma)) = 4\\pi (g-1)$,\n\\end{center}\ntherefore the total energy satisfies\n\\begin{eqnarray*}\nE(w) & = & \\int_{\\Sigma}e(w)dA =  \\int_{\\Sigma}({\\mathcal{H}}+{\\mathcal{L}})dA \\\\\n& = & \\int_{\\Sigma}({\\mathcal{H}}-{\\mathcal{L}})dA + 2\\int_{\\Sigma}{\\mathcal{L}}dA \\\\\n&\\le& Area(\\Sigma,\\sigma) + 2 \\\\\n&= & 4\\pi(g-1) +2.\n\\end{eqnarray*}\nTherefore\n \\begin{eqnarray*}\n\\int_{B_{z_{0}}({\\frac{r_{0}}{2}})}{\\mathcal{H}}(z)\\sigma dzd\\bar{z} \n& < & \\int_{B_{z_{0}}({\\frac{r_{0}}{2}})}({\\mathcal{H}}(z) + {\\mathcal{L}}(z))\\sigma dzd\\bar{z} \\\\\n& = & E(w) - \\int_{\\Sigma \\backslash B_{z_{0}}({\\frac{r_{0}}{2}})}e(z) \\sigma dzd\\bar{z} \\\\\n& \\le & 4\\pi(g-1) +2 - (4\\pi(g-1)-A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}}))) \\\\\n&=& A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}})) + 2.\n\\end{eqnarray*}\nwhere $A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}}))$ is the hyperbolic area of the ball \n$B_{z_{0}}({\\frac{r_{0}}{2}})$.  \n\n\nWe set $h_0 = C(r_0)(A_{1}({B_{z_{0}}}({\\frac{r_{0}}{2}})) + 2)$, then $h_0$ \nis independent of the genus $g$, as it is obtained from a local estimate in a \ngeodesic ball. Therefore, \n \\begin{center}\n $|\\mu(z)| < {\\mathcal{H}}(z) \\le {\\mathcal{H}}(z_0) < h_0$,\n \\end{center}\nfor all $z \\in \\Sigma$.\n\\end{proof}\n\\begin{rem}\nThe assumption of the surface lying in the thick part of the {moduli space} is essential to this \nargument, since we used an estimate in an embedded geodesic ball.\n\\end{rem}\n\\noindent\nAs an application of this estimate, we can prove theorem 1.1 easily. \n\n\\begin{proof}(of theorem 1.1) Recall that the operator $D = -2(\\Delta -2)^{-1}$ is self-adjoint, and $||\\mu||_{WP} = 1$:\n\\begin{center}\n$|K_h| = 2\\int_{\\Sigma}D(|\\mu|^2)|\\mu|^2 dA < 2h_0^2 \\int_{\\Sigma}D(|\\mu|^2)dA = 2h_0^2$.\n\\end{center}\n\\end{proof}.\n\n\nWe now shift our attention to general {Weil-Petersson} {sectional curvature}s and to prove theorem 1.2.\n\\begin{proof}(of theorem 1.2)\nWe recall from the previous section the Riemannian curvature tensor of the {Weil-Petersson} metric \nis given by (\\cite {Tr}, \\cite{Wp3}):\n\\begin{center} \n$R_{\\alpha\\bar{\\beta}\\gamma\\bar{\\delta}} = \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\beta})\\mu_{\\gamma}\\bar\\mu_{\\delta}dA + \n\\int_{\\Sigma}D(\\mu_{\\alpha}\\bar\\mu_{\\delta})\\mu_{\\gamma}\\bar\\mu_{\\beta}dA$,\n\\end{center}\nwhere $\\mu$'s in the formula are harmonic {Beltrami differential}s, and $D$ again is the operator \n$-2(\\Delta -2)^{-1}$.\n\n\nTo calculate the {sectional curvature}, we choose two arbitrary orthonormal harmonic {Beltrami differential}s \n$\\mu_0$ and $\\mu_1$. In other words, we have \n\\begin{center}\n$\\int_{\\Sigma}|\\mu_0|^2 dA = \\int_{\\Sigma}|\\mu_1|^2 dA = 1$ and \n$\\int_{\\Sigma}\\mu_0 {\\bar{\\mu}}_1 dA = 0$.\n\\end{center}\nThen the Gaussian curvature of the plane spanned by $\\mu_0$ and $\\mu_1$ is \n(\\cite {Wp3})\n\\begin{eqnarray*}\nK(\\mu_0,\\mu_1) &=& {\\frac{1}{4}}(R_{0\\bar{1}0\\bar{1}} - R_{0\\bar{1}1\\bar{0}} - \nR_{1\\bar{0}0\\bar{1}}+R_{1\\bar{0}1\\bar{0}}) \\nonumber \\\\\n&=& Re(\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{0}\\bar\\mu_{1}dA) - \n{\\frac{1}{2}}Re(\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{1}\\bar\\mu_{0}dA) \\nonumber \\\\\n&-& {\\frac{1}{2}}\\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA. \n\\end{eqnarray*}\nFrom (\\cite {Wp3}, lemma 4.3) and H\\\"{o}lder inequality, we have\n\\begin{eqnarray*}\n|Re(\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{0}\\bar\\mu_{1}dA)| &\\le& \n\\int_{\\Sigma}|D(\\mu_{0}\\bar\\mu_{1})||\\mu_{0}\\bar\\mu_{1}|dA \\nonumber \\\\\n&\\le& \\int_{\\Sigma}\\sqrt{D(|\\mu_{0}|^{2})}\\sqrt{D(|\\mu_{0}|^{2})}|\\mu_{0}\\bar\\mu_{1}|dA \\nonumber \\\\\n&\\le&  \\int_{\\Sigma}D(|\\mu_{0}|^2)|\\mu_{1}|^2dA = \\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA, \n\\end{eqnarray*}\nand similarly\n\\begin{center}\n$|\\int_{\\Sigma}D(\\mu_{0}\\bar\\mu_{1})\\mu_{1}\\bar\\mu_{0}dA| \\le \\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA$. \n\\end{center}\nTherefore \n\\begin{center}\n$|K(\\mu_0,\\mu_1)| \\le 2\\int_{\\Sigma}D(|\\mu_{1}|^2)|\\mu_{0}|^2dA$.  \n\\end{center}\nWe apply theorem 3.3 to find that there is a $h_0 > 0$, independent of $g$, such that $|\\mu_{0}| < h_0$. \nSo $|K| < 2h_0^2 \\int_{\\Sigma}D(|\\mu_{1}|^2)dA = 2h_0^2$.\n\\end{proof}\n\n\nIt is now straightforward to see that theorem 1.3 holds since all {sectional curvature}s are negative. \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn this note we consider non-spurious solutions by using a critical point\ntheory to the following Dirichlet problem \n\\begin{equation}\n\\begin{array}{l}\n\\ddot{x}\\left( t\\right) =f\\left( t,x\\left( t\\right) \\right) \\\\ \nx\\left( 0\\right) =x\\left( 1\\right) =\n\\end{array}\n\\label{par}\n\\end{equation}\nwhere $f:\\left[ 0,1\\right] \\times \\mathbb{R} \\rightarrow \\mathbb{R}$ is a\njointly continuous function. Further we will make precise what is meant by\nthe solutions to (\\ref{par}).\n\nThe existence of non-spurious solutions is very important for the\napplications since in such a case one can approximate solutions to (\\ref{par\n) with a sequence of solutions to a suitably chosen family of discrete\nproblems and one is sure that this approximation converges to the solution\nof the original problem, see \\cite{kelly}. There are many ways in which a\nboundary value problem can be discretized and the existence and multiplicity\ntheory on difference equations is very vast, see for example \\cite{agrawal}, \n\\cite{candito1}, \\cite{guojde}, \\cite{MRT}. However, as underlined by\nAgarwal, \\cite{agarvalpaper}, there are no clear relations between\ncontinuous problems and their discretization which means that both problems\ncan be solvable, but the approximation approaches nothing but the solution\nto the continuous problem or else, the discrete problem is solvable and the\ncontinuous one is not or the other way round. Let us recall his examples:\n\n\\begin{example}\nThe continuous problem $\\ddot{x}(t) + \\frac{\\pi^2}{n^2}x(t)=0$, $x(0)=x(n)=0$\nhas an infinite number of solutions $x(t)= c \\sin \\frac{\\pi t}{n}$ ($c$ is\narbitrary) whereas its discrete analogue $\\Delta^2x(k)+\\frac{\\pi^2}{n^2\nx(k)=0$, $x(0)=x(n)=0$ has only one solution $x(k)\\equiv 0$. The problem \n\\ddot{x}(t)+\\frac{\\pi^2}{4n^2}x(t)=0$, $x(0)=0$, $x(n)=1$ has only one\nsolution $x(t)=\\sin\\frac{\\pi t}{2n}$, and its discrete analogue $\\Delta^2\nx(k)+ \\frac{\\pi^2}{4n^2}x(k)=0$, $x(0)=0$, $x(n)=1$ also has one solution.\nThe continuous problem $\\ddot{x}(t)+4\\sin^2\\frac{\\pi}{2n}x(t)=0$, $x(0)=0$, \nx(n)=\\varepsilon \\neq 0$ has only one solution $x(t)= \\varepsilon \\frac{\\sin\n(2 \\sin\\frac{\\pi}{2n})t]}{\\sin[(2 \\sin\\frac{\\pi}{2n})n]}$, whereas its\ndiscrete analogue $\\Delta^2x(k)+4\\sin^2\\frac{\\pi}{2n}x(k)=0$, $x(0)=0$, \nx(n)=\\varepsilon \\neq 0$ has no solution.\n\\end{example}\n\nThus, the nature of the solution changes when a continuous boundary value\nproblem is being discretized. Moreover, two-point boundary value problems\ninvolving derivatives lead to multipoint problems in the discrete case.\n\nThe above remarks and examples show that steal it is important to consider\nboth continuous and discrete problems simultaneously and investigate\nrelation between solutions which is the key factor especially when the\nexistence part follows by standard techniques.\n\nThere have been some research in this case addressing mainly problems whose\nsolutions where obtained by the fixed point theorems and the method of lower\nand upper solutions, \\cite{rech1}, \\cite{rachunkowa2}, \\cite{thomsontisdell\n. In this submission we are aiming at using critical point theory method,\nnamely the direct method of the calculus of variations (see for example \\cit\n{Ma} for a nice introduction to this topic) in order to show that in this\nsetting one can also obtain suitable convergence results. The advance over\nworks mentioned is that we can have better growth conditions imposed on $f$\nat the expense of not putting derivative of $x$ in $f$. As expected we will\nhave to get the uniqueness of solutions for the associated discrete problem,\nwhich is not always easy to be obtained, see \\cite{uni2}.\n\nIn \\cite{kelly} following \\cite{gaines}, it is suggested which family of\ndifference equations for $n\\in \\mathbb{N}$ is to be chosen when\napproximating problem (\\ref{par}). For $a$, $b$ such that $a<b<\\infty $, \na\\in \\mathbb{N}\\cup \\{0\\}$, $b\\in \\mathbb{N}$ we denote $\\mathbb{N\n(a,b)=\\{a,a+1,...,b-1,b\\}$. For a fixed $n\\in \\mathbb{N}$ the nonlinear\ndifference equation with Dirichlet boundary conditions is given as follows\nfor $k\\in \\mathbb{N}(0,n-1)$ \n\\begin{equation}\n\\Delta ^{2}x\\left( k-1\\right) =\\frac{1}{n^{2}}f\\left( \\frac{k}{n},x\\left(\nk\\right) \\right) ,\\,\\,\\,x\\left( 0\\right) =x\\left( n\\right) =0.\n\\label{diffequ}\n\\end{equation\nHere $\\Delta $ is the forward difference operator, i.e. $\\Delta x\\left(\nk-1\\right) =x\\left( k\\right) -x\\left( k-1\\right) $ and we see that $\\Delta\n^{2}x\\left( k-1\\right) =x\\left( k+1\\right) -2x\\left( k\\right) +x\\left(\nk-1\\right) $. Assume that both continuous boundary value problem (\\ref{par})\nand for each fixed $n\\in \\mathbb{N}$ discrete boundary value problem (\\re\n{diffequ}) are uniquely solvable by, respectively $x$ and $x^{n}=\\left(\nx^{n}(k)\\right) $. Moreover, let there exist two constants $Q,N>0$\nindependent of $n$ and such that \n\\begin{equation}\nn|\\Delta x^{n}(k-1)|\\leq Q\\text{ and }|x^{n}(k)|\\leq N  \\label{ewa1}\n\\end{equation\nfor all $k\\in \\mathbb{N}(0,n)$ and all $n\\geq n_{0}$, where $n_{0}$ is fixed\n(and arbitrarily large). Lemma 9.2. from \\cite{kelly} says that for some\nsubsequence $x^{n_{m}}=\\left( x^{n_{m}}(k)\\right) $ of $x^{n}$ it holds \n\\begin{equation}\n\\lim_{m\\rightarrow \\infty }\\max_{0\\leq k\\leq n_{m}}\\left\\vert\nx^{n_{m}}\\left( k\\right) -x\\left( \\frac{k}{n_{m}}\\right) \\right\\vert =0.\n\\label{ewa2}\n\\end{equation\nIn other words, this means that the suitable chosen discretization\napproaches the given continuos boundary value problem. Such solutions to\ndiscrete BVPs are called non-spurious in contrast to spurious ones which\neither diverge or else converge to anything else but the solution to a given\ncontinuous Dirichlet problem.\n\n\\section{Non spurious solutions for (\\protect\\ref{par})}\n\n\\subsection{The continuous problem}\n\nIn the existence part we apply variational methods. This means that with\nproblem under consideration we must associate the Euler action functional,\nprove that this functional is weakly lower semicontinuous in a suitable\nfunction space, coercive and at least G\\^{a}teaux differentiable. Given this\nthree conditions one knows that at least a weak solution to problem under\nconsideration exists whose regularity can further be improved with known\ntools. Such scheme, commonly used within the critical point theory is well\ndescribed in the first chapters of \\cite{Ma}.\n\nThe solutions to (\\ref{par}) will be investigated in the space \nH_{0}^{1}\\left( 0,1\\right) $ consisting of absolutely continuous functions\nsatisfying the boundary conditions and with a.e. derivative being integrable\nwith square. Such a solution is called a weak one, i.e. a function $x\\in\nH_{0}^{1}\\left( 0,1\\right) $ is a weak $H_{0}^{1}\\left( 0,1\\right) $\nsolution to (\\ref{par}), if \n\\begin{equation*}\n\\int_{0}^{1}\\dot{x}\\left( t\\right) \\dot{v}\\left( t\\right)\ndt+\\int_{0}^{1}f\\left( t,x\\left( t\\right) \\right) v\\left( t\\right) dt=0\n\\end{equation*\nfor all $v\\in H_{0}^{1}\\left( 0,1\\right) $. The classical solution to (\\re\n{par}) is then defined as a function $x:$ $\\left[ 0,1\\right] \\rightarrow \n\\mathbb{R}$ belonging to $H_{0}^{1}\\left( 0,1\\right) $ such that $\\ddot{x}$\nexists a.e. and $\\ddot{x}\\in L^{1}\\left( 0,\\pi \\right) $. Since $f$ is\njointly continuous, then it is known from the Fundamental Theorem of the\nCalculus of Variations, see \\cite{Ma}, that $x$ is in fact twice\ndifferentiable with classical continuous second derivative. Thus $x\\in\nH_{0}^{1}\\left( 0,1\\right) \\cap C^{2}\\left( 0,1\\right) $.\n\nLet $F\\left( t,x\\right) =\\int_{0}^{x}f\\left( t,s\\right) ds$ for $\\left(\nt,x\\right) \\in \\left[ 0,1\\right] \\times \\mathbb{R}$. We link solutions to \n\\ref{par}) with critical points to a $C^{1}$ functional $J:H_{0}^{1}\\left(\n0,1\\right) \\rightarrow \\mathbb{R}$ given by \n\\begin{equation*}\nJ\\left( x\\right) =\\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right)\ndt+\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt.\n\\end{equation*\nLet us examine $J$ for a while. Due to the continuity of $f$ functional $J$\nis well defined. Recall that the norm in $H_{0}^{1}\\left( 0,1\\right) $ reads \n\\begin{equation*}\n\\left\\Vert x\\right\\Vert =\\sqrt{\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt}.\n\\end{equation*\nThen we see $\\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt=\\frac{1}{2\n\\left\\Vert x\\right\\Vert ^{2}$ is a $C^{1}$ functional by standard facts. Its\nderivative is a functional on $H_{0}^{1}\\left( 0,1\\right) $ which reads \n\\begin{equation*}\nv\\rightarrow \\int_{0}^{1}\\dot{x}\\left( t\\right) \\dot{v}\\left( t\\right) dt.\n\\end{equation*\nConcerning the nonlinear part we see that for any fixed $v\\in\nH_{0}^{1}\\left( 0,1\\right) $ (which is continuous of course) function \n\\varepsilon \\rightarrow \\int_{0}^{1}F\\left( t,x\\left( t\\right) +\\varepsilon\nv\\left( t\\right) \\right) dt$ (where the integral we can treat as the Riemann\none) due to the Leibnitz differentiation formula under integral sign is \nC^{1}$ and the derivative of $\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right)\ndt$ is a functional on $H_{0}^{1}\\left( 0,1\\right) $ which reads \n\\begin{equation*}\nv\\rightarrow \\int_{0}^{1}f\\left( t,x\\left( t\\right) \\right) v\\left( t\\right)\ndt\n\\end{equation*\nif we recall that $F\\left( t,x\\right) =\\int_{0}^{x}f\\left( t,s\\right) ds$.\nSince the above is obviously continuous in $x$ uniformly in $v$ form unit\nsphere, we see that $J$ is in fact $C^{1}.$\n\nRecall also Poincar\\'{e} inequality $\\int_{0}^{1}x^{2}\\left( t\\right) dt\\leq \n\\frac{1}{\\pi ^{2}}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt$ and Sobolev's\none $\\max_{t\\in \\left[ 0,1\\right] }\\left\\vert x\\left( t\\right) \\right\\vert\n\\leq \\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt$.\n\nWe sum up the assumptions on the nonlinear term in (\\ref{par}) since in\norder to get the above mentioned observations continuity of $f$ is\nsufficient. We assume that\\newline\n\\textit{\\textbf{H1}} $f:\\left[ 0,1\\right] \\times \\mathbb{R} \\rightarrow \n\\mathbb{R}$ is a continuous function such that $f\\left( t,0\\right) \\neq 0$\nfor $t\\in \\left[ 0,1\\right]$;\\newline\n\\textit{\\textbf{H2}} $f$ is nondecreasing in $x$ for all $t\\in \\left[ 0,\n\\right] $\n\n\\begin{proposition}\nAssume that \\textbf{H1} and \\textbf{H2} are satisfied. Then problem (\\re\n{par}) has exactly one nontrivial solution.\n\\end{proposition}\n\n\\begin{proof}\nFirstly, we consider the existence part. Note that by Weierstrass Theorem\nthere exists $c>0$ such that \n\\begin{equation*}\n\\left\\vert f\\left( t,0\\right) \\right\\vert \\leq c\\text{ for all }t\\in \\left[\n0,1\\right] .\n\\end{equation*\nSince $f$ is nondecreasing in $x$ \\textit{\\textbf{H2}} it follows that $F$\nis convex. Since $F\\left( t,0\\right) =0$ for all $t\\in \\left[ 0,1\\right] $\nwe obtain from the well known inequality \n\\begin{equation}\nF(t,x)=F(t,x)-F(t,0)\\geq f\\left( t,0\\right) x\\geq -\\left\\vert f\\left(\nt,0\\right) x\\right\\vert   \\label{aaa}\n\\end{equation\nvalid for any $x$ and all for all $t\\in \\left[ 0,1\\right] $. We observe that\nfrom (\\ref{aaa}) we get \n\\begin{equation}\nF\\left( t,x\\right) \\geq -c\\left\\vert x\\right\\vert \\text{ for all }t\\in \\left[\n0,1\\right] \\text{ and all }x\\in \\mathbb{R}.  \\label{estimF}\n\\end{equation\nHence for any $x\\in H_{0}^{1}\\left( 0,1\\right) $ we see by Schwartz and\nPoincar\\'{e} inequality \n\\begin{equation*}\n\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt\\geq\n-c\\int_{0}^{1}\\left\\vert x\\left( t\\right) \\right\\vert dt\\geq -\\frac{c}{\\pi \n\\left\\Vert x\\right\\Vert .\n\\end{equation*\nTherefor\n\\begin{equation}\n\\begin{array}{l}\nJ\\left( x\\right) \\geq \\frac{1}{2}\\left\\Vert x\\right\\Vert ^{2}-\\left\\vert\nc\\right\\vert \\left\\Vert x\\right\\Vert \n\\end{array}\n\\label{Ju_}\n\\end{equation\nHence from (\\ref{Ju_}) we obtain that $J$ is coercive. Note that $\\frac{1}{2\n\\left\\Vert x\\right\\Vert ^{2}$ is obviously w.l.s.c. on $H_{0}^{1}\\left(\n0,1\\right) $. Next, by the Arzela-Ascoli Theorem and Lebesgue Dominated\nConvergence, see these arguments in full detail in \\cite{Ma} in the proof of\nTheorem 1.1 we see that $x\\rightarrow \\int_{0}^{1}F\\left( t,x\\left( t\\right)\n\\right) dt$ is weakly continuous. Thus $J$ is weakly l.s.c. as a sum of a\nw.l.s.c. and weakly continuous functionals. Since $J$ is $C^{1}$ and convex\nfunctional it has exactly one argument of a minimum which is necessarily a\ncritical point and thus a solution to (\\ref{par}). Putting $x=0$ in (\\re\n{par}) one see that we have a contradiction, so any solution is nontrivial.\n\\end{proof}\n\nIn order to get the existence of nontrivial solution to (\\ref{par}) it would\nsuffice to assume that $f\\left( t_{0},0\\right) \\neq 0$ for some $t_{0}\\in\n\\left[ 0,1\\right] $ but since we need to impose same conditions on discrete\nproblem it is apparent that our assumption is more reasonable. Moreover,\nthere is another way to prove the weak lower semincontinuity of $J$, namely\nshow that $J$ is continuous. Then it is weakly l.s.c. since it is convex.\nHowever, in proving continuity of $J$ on $H_{0}^{1}\\left( 0,1\\right) $ one\nuses the same arguments.\n\n\\subsection{The discrete problem}\n\nNow we turn the discretization of (\\ref{par}), i.e. to problem (\\ref{diffequ\n).\nconsidered in the $n$-dimensional Hilbert space $E$ consisting of functions \nx:\\mathbb{N}(0,n)\\rightarrow \\mathbb{R}$ such that $x(0)=x(n)=0$. Space $E$\nis considered with the following nor\n\\begin{equation}\n\\left\\Vert x\\right\\Vert =\\left( \\sum\\limits_{k=1}^{n}|{\\Delta \nx(k-1)|^{2}\\right) ^{\\frac{1}{2}}.  \\label{norm_operator}\n\\end{equation\nWe can also consider $E$ with the following norm \n\\begin{equation*}\n\\left\\Vert u\\right\\Vert _{0}=\\left( \\sum\\limits_{k=1}^{n}|u(k)|^{2}\\right) ^\n\\frac{1}{2}}.\n\\end{equation*\nSince $E$ is finite dimensional there exist constants $c_{b}=\\frac{1}{2}$ \\\nand $c_{a}=\\left( \\left( n-1\\right) n\\right) ^{1\/2}$ such that \n\\begin{equation}\nc_{b}\\left\\Vert u\\right\\Vert \\leq \\left\\Vert u\\right\\Vert _{0}\\leq\nc_{a}\\left\\Vert u\\right\\Vert \\text{ for all }u\\in E.  \\label{c_a_c_b}\n\\end{equation\nSolutions to (\\ref{diffequ}) correspond in a $1-1$ manner to the critical\npoints to the following $C^{1}$ functional $\\mathcal{I}:E\\rightarrow \\mathbb\nR}$ \n\\begin{equation*}\n\\mathcal{I}(x)=\\sum\\limits_{k=1}^{n}\\tfrac{1}{2}|\\Delta x(k-1)|^{2}+\\frac{1}\nn^{2}}\\sum\\limits_{k=1}^{n-1}F(\\frac{k}{n},x(k))\n\\end{equation*\nwith $F$ defined as before. This means that \n\\begin{equation*}\n\\frac{d}{dx}\\mathcal{I}(x)=0\\text{ if and only if }x\\text{ satisfies (\\re\n{diffequ}).}\n\\end{equation*\nNow we do not need to introduce the notion of the weak solution that is why\nwe have only one type of variational solution. We know that by the discrete\nSchwartz Inequality by (\\ref{estimF}) and by (\\ref{c_a_c_b}) \n\\begin{equation}\n\\begin{array}{l}\n\\mathcal{I}(x)\\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\frac{1}{n^{2}}\\left\\vert\nc\\right\\vert \\sqrt{n}\\left( \\sum\\limits_{k=1}^{n-1}\\left\\vert x\\left(\nk\\right) \\right\\vert ^{2}\\right) ^{1\/2} \\\\ \n\\\\ \n\\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\left\\vert c\\right\\vert \\frac{\\sqrt{n-1}}{\n}\\left\\Vert x\\right\\Vert \\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\left\\vert\nc\\right\\vert \\left\\Vert x\\right\\Vert \n\\end{array}\n\\label{relcoer}\n\\end{equation\nHence $\\mathcal{I}(x)\\rightarrow +\\infty $ as $\\Vert x\\Vert \\rightarrow\n+\\infty $ and we are in position to formulate the following\n\n\\begin{proposition}\n\\label{solvability_diff_equ}Assume that \\textbf{H1}, \\textbf{H2} hold. Then\nproblem (\\ref{diffequ}) has exactly one nontrivial solution.\n\\end{proposition}\n\n\\subsection{Main result}\n\n\\begin{theorem}\n\\label{first convergence theorem}Assume that conditions \\textbf{H1}, \\textbf\nH2} are satisfied. Then there exists $x\\in H_{0}^{1}\\left( 0,1\\right) \\cap\nC^{2}\\left( 0,1\\right) $ which solves uniquely (\\ref{par}) and for each \nn\\in \\mathbb{N}$ there exists $x^{n}$ which solves uniquely (\\ref{diffequ}).\nMoreover, there exists a subsequence $x^{n_{m}}$ of $x^{n}$ such that\ninequalities (\\ref{ewa2}) are satisfied.\n\\end{theorem}\n\n\\begin{proof}\nWe need to show that there exist two constants independent of $n$ such that\ninequalities (\\ref{ewa1}) hold.\nwhere $n_{0}$ is fixed. Then Lemma 9.2. from \\cite{kelly} provides the\nassertion of the theorem. In our argument we use some observations used in\nthe investigation of continuous dependence on parameters for ODE, see \\cit\n{LedzewiczWalczak}. Fix $n$. By Proposition \\ref{solvability_diff_equ},\nthere exists $x^{n}$ solving uniquely (\\ref{diffequ}) and which is an\nargument of a minimum to $\\mathcal{I}$ such that it holds that $\\mathcal{I\n(x^{n})\\leq \\mathcal{I}(0)=0$. Thus relation (\\ref{relcoer}) leads to the\ninequality \n\\begin{equation*}\n\\frac{1}{2}\\Vert x^{n}\\Vert\\leq \\left\\vert c\\right\\vert \\frac{\\sqrt{n-1}}{n}.\n\\end{equation*}\nSince $\\max_{k\\in \\mathbb{N}(0,n)}\\left\\vert x^{n}\\left( k\\right)\n\\right\\vert \\leq \\frac{\\sqrt{n+1}}{2}\\left\\Vert x^{n}\\right\\Vert$ we get\nthat for all $k\\in \\mathbb{N}(0,n)$ \n\\begin{equation*}\n\\left\\vert x^{n}\\left( k\\right) \\right\\vert \\leq 2\\left\\vert c\\right\\vert \n\\frac{\\sqrt{n-1}}{n}\\frac{\\sqrt{n+1}}{2}\\leq \\left\\vert c\\right\\vert =N.\n\\end{equation*}\nBy Lemma 9.3 in \\cite{kelly} we now obtain that there is a constant $Q$ such\nthat condition \n\\begin{equation*}\nn\\vert \\Delta x^{n}(k-1) \\vert \\leq Q \\text{ and }\\vert x^{n}(k) \\vert \\leq N\n\\end{equation*}\nfor all $k\\in \\mathbb{N}(0,n)$ and all $n\\in \\mathbb{N}$ is satisfied. This\nmeans that the application of Lemma 9.2 from \\cite{kelly} finishes the proof.\n\\end{proof}\n\n\\section{Final comments and examples}\n\nIn this section we provide the examples of nonlinear terms satisfying our\nassumptions and we will investigate the possibility of replacing the\nconvexity assumption imposed on $F$ with some weaker requirement as well as\nwe comment on exisiting results in the literature.\n\nConcerning the examples of nonlinear terms any nondecreasing $f$ is of order\nbounded or unbounded, see \n\n\\begin{enumerate}\n\\item[a)] $f\\left( t,x\\right) =g\\left( t\\right) \\exp \\left( x-t^{2}\\right)$;\n\n\\item[b)] $f\\left( t,x\\right) =g\\left( t\\right) \\arctan \\left( x\\right)$;\n\n\\item[c)] $f\\left( t,x\\right) =g\\left( t\\right) x^{3}+\\exp \\left(\nx-t^{2}\\right)$,\n\\end{enumerate}\n\nwhere $g$ is any lower bounded continuous function with positive values. \n\nIn view of remarks contained in \\cite{Ma} functional $J$ can be written  \n\\begin{equation*}\nJ\\left( x\\right) =\\left( \\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right)\ndt-\\frac{a}{2\\pi }\\int_{0}^{1}x^{2}\\left( t\\right) dt\\right) \n\\end{equation*\n\\begin{equation*}\n+\\left( \\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt+\\frac{a}{2\\pi \n\\int_{0}^{1}x^{2}\\left( t\\right) dt\\right) .\n\\end{equation*\nThen functional \n\\begin{equation*}\nx\\rightarrow \\left( \\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt\n\\frac{a}{2\\pi }\\int_{0}^{1}x^{2}\\left( t\\right) dt\\right) \n\\end{equation*\nis strictly convex as long as $a\\in \\left( 0,1\\right) $. Note that the first\neigenvalue of the differential operator $-\\frac{d^{2}}{dt^{2}}$ with\nDirichlet boundary conditions on $\\left[ 0,1\\right] $ is $\\frac{1}{\\pi }$\n(note this is the best constant in Poincar\\'{e} inequality). Hence we can\nrelax convexity assumption $F$ by assuming that \n\\begin{equation*}\nx\\rightarrow F\\left( t,x\\right) +\\frac{a}{2\\pi }x^{2}\n\\end{equation*\nis convex for any $t\\in \\left[ 0,1\\right] $. Then $F_{1}\\left( t,x\\right)\n=F\\left( t,x\\right) +\\frac{a}{2\\pi }x^{2}$ satisfies (\\ref{estimF}). \n\nThe natural question arises if similar procedure is possible as far as the\ndiscrete problem (\\ref{diffequ}) is concerned. However there is one big\nproblem here since the first eigenvalue for $-\\Delta ^{2}$ reads $\\lambda\n_{1}=2-2\\cos \\left( \\frac{\\pi }{n+1}\\right) $ and of course $\\lambda\n_{1}\\rightarrow 0$ as $n\\rightarrow \\infty $. This means that the above idea\nwould not work, since we cannot find $a$ for all $n$ idependent of $n$ (for\neach $n$ such $a=a\\left( n\\right) $ exists ). \n\nA comparison with existing results is also in order. The only papers\nconcerning the existence of non-spurious solutions are \\cite{rech1}, \\cit\n{rachunkowa2}, \\cite{thomsontisdell} which follow ideas developed in \\cit\n{gaines} and which were mentioned already in the Introduction. We not only\nuse different methods, namely critical point theory, but also we are not\nlimited as far as the growth is concerned since in sources mentioned $f$ is\nsublinear. However, we could not incorporate the derivative of $x$ into the\nnonlinear term. This is not possible by variational approach but could be\nmade possible by connecting variational methods with Banach contraction\nprinciple and it shows that the research concerning the existence of\nnon-spurious solutions with critical point approach can be further developed.\n\nWe cannot use sublinear growth as in sources mentioned since it does not\nprovide the inequality \n\\begin{equation}\nF\\left( t,x\\right) -F\\left( t,0\\right) \\geq f\\left( t,0\\right) x\\text{ for\nall }t\\in \\left[ 0,1\\right] \\text{ and all }x\\in \\mathbb{R}.  \\label{ccccc}\n\\end{equation\nWith our approach inequality (\\ref{ccccc}) is essential in proving the\nrequired estimations which lead to the existence of non-spurious solutions.\nThis is shown by the below remarks where direct calculations are performed.\n\nThe relevant growth condition reads\\newline\n\\textit{\\textbf{H2a} There exist constants} $a,b>0$ and $\\gamma \\in \\left[\n0,1\\right) $ such that \n\\begin{equation}\nf\\left( t,x\\right) \\leq a+b\\left\\vert x\\right\\vert ^{\\gamma }\\text{ for all \nt\\in \\left[ 0,1\\right] \\text{ and all }x\\in \\mathbb{R}.\n\\label{cond_unboud_below}\n\\end{equation\nBy (\\ref{cond_unboud_below}) for all $t\\in \\left[ 0,1\\right] $ and all $x\\in \n\\mathbb{R}$ it holds \n\\begin{equation*}\nF\\left( t,x\\right) \\leq a\\left\\vert x\\right\\vert +\\frac{b}{\\gamma +1\n\\left\\vert x\\right\\vert ^{\\gamma +1}.\n\\end{equation*\nSince $F\\left( t,x\\right) \\geq -\\left\\vert F\\left( t,x\\right) \\right\\vert $\nwe see by Schwartz, Holder and Poincar\\'{e} inequality for any $x\\in\nH_{0}^{1}\\left( 0,1\\right) $ \n\\begin{equation*}\n\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt\\geq -c_{1}\\left\\Vert\nx\\right\\Vert -c_{2}\\left\\Vert x\\right\\Vert ^{\\gamma +1},\n\\end{equation*\nwhere $c_{1}=a$ and $c_{2}>0$ (the exact value of $c_{2}$ is not important\nsince $\\gamma +1<2$ and functional $J$ is coercive disregarding of the value\nof $c_{2}$. Then problem (\\ref{par}) has at least one solution by the direct\nmethod of the calculus of variations.\n\nIn order to consider problem (\\ref{diffequ}) we need to perform exact\ncalculations since in this case, in view of the convergence Theorem \\re\n{first convergence theorem}, the precise values of constants are of utmost\nimportance. In case of \\textit{\\textbf{H2a}} from H\\\"{o}lder's inequality\nand (\\ref{c_a_c_b}) we get \n\\begin{equation*}\n\\begin{array}{ll}\n\\sum\\limits_{k=1}^{n-1}\\vert u(k)\\vert^{\\gamma +1} & =\\su\n\\limits_{k=1}^{n-1}\\vert u(k)\\vert^{\\gamma +1} \\cdot 1 \\\\ \n&  \\\\ \n& \\leq \\left( \\sum\\limits_{k=1}^{n-1}\\vert u(k)\\vert^{\\gamma +1}\\vert^{\\frac\n2}{\\gamma +1} }\\right) ^{\\frac{\\gamma +1}{2}}\\left(\n\\sum\\limits_{k=1}^{n-1}\\vert1\\vert^{\\frac{1}{1- \\frac{\\gamma +1}{2}}}\\right)\n^{1-\\frac{2}{\\gamma +1}} \\\\ \n&  \\\\ \n& =\\left( n-1\\right) ^{\\frac{1-\\gamma }{2}}\\left\\Vert u\\right\\Vert\n_{0}^{\\gamma +1}\\leq \\left( \\left( n-1\\right) n\\right) ^{\\frac{\\gamma +1}{2\n}\\left( n-1\\right) ^{\\frac{1-\\gamma }{2}}\\left\\Vert u\\right\\Vert ^{\\gamma +1}\n\\\\ \n&  \\\\ \n& =\\left( n-1\\right) n^{\\frac{\\gamma +1}{2}}\\left\\Vert u\\right\\Vert ^{\\gamma\n+1}\\leq \\left( n-1\\right) n\\left\\Vert u\\right\\Vert ^{\\gamma +1}\n\\end{array\n\\end{equation*}\nThus \n\\begin{equation*}\n\\frac{1}{n^{2}}\\frac{b}{\\gamma +1}\\sum\\limits_{k=1}^{n-1}\\vert\nu(k)\\vert^{\\gamma +1}\\leq \\frac{b}{\\gamma +1}n^{\\frac{\\gamma -1}{2\n}\\left\\Vert u\\right\\Vert ^{\\gamma +1}\n\\end{equation*}\n\nHence by the above calculations and (\\ref{relcoer}) we get for any $x\\in E$ \n\\begin{equation}\n\\mathcal{I}(x)\\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\left\\vert a\\right\\vert \n\\frac{\\sqrt{n-1}}{n}\\left\\Vert x\\right\\Vert -\\frac{b}{\\gamma +1}n^{\\frac\n\\gamma -1}{2}}\\left\\Vert x\\right\\Vert ^{\\gamma +1}.  \\label{rel_add_coer}\n\\end{equation\nThus $\\mathcal{I}(x)\\rightarrow +\\infty $ as $\\Vert x\\Vert \\rightarrow\n+\\infty .$ By Lemma 9.2. from \\cite{kelly}, we need to show that (\\ref{ewa1\n) holds. Fix $n$. Since $\\mathcal{I}(x^{n})\\leq \\mathcal{I}(0)=0$, the\nrelation (\\ref{rel_add_coer}) leads to the inequality \n\\begin{equation}\n\\frac{1}{2}\\Vert x^{n}\\Vert \\leq \\left\\vert a\\right\\vert \\frac{\\sqrt{n-1}}{n\n+\\frac{b}{\\gamma +1}n^{\\frac{\\gamma -1}{2}}\\left\\Vert x^{n}\\right\\Vert\n^{\\gamma }.  \\label{ineq}\n\\end{equation\nSince $\\gamma <1$ we see $n^{\\frac{\\gamma -1}{2}}\\rightarrow 0$. Thus there\nis some $n_{0}$ that for all $n\\geq n_{0}$ it holds $\\frac{b}{\\gamma +1}n^\n\\frac{\\gamma -1}{2}}<\\frac{1}{4}$. Take $n\\geq n_{0}$. Let us consider two\ncases, namely $\\left\\Vert x^{n}\\right\\Vert \\leq 1$ and $\\left\\Vert\nx^{n}\\right\\Vert >1$. In case $\\left\\Vert x^{n}\\right\\Vert >1$ we get from \n\\ref{ineq}) that \n\\begin{equation*}\n\\frac{1}{2}\\Vert x^{n}\\Vert \\leq \\left\\vert a\\right\\vert \\frac{\\sqrt{n-1}}{n\n+\\frac{1}{4}\\left\\Vert x^{n}\\right\\Vert \n\\end{equation*\nRecall $\\max_{k\\in \\mathbb{N}(0,n)}\\left\\vert x^{n}\\left( k\\right)\n\\right\\vert \\leq \\frac{\\sqrt{n+1}}{2}\\left\\Vert x^{n}\\right\\Vert $ we get\nthat for all $k\\in \\mathbb{N}(0,n)$ \n\\begin{equation*}\n\\left\\vert x\\left( k\\right) \\right\\vert \\leq 4\\left\\vert a\\right\\vert \\frac\n\\sqrt{n-1}}{n}\\frac{\\sqrt{n+1}}{2}\\leq 2\\left\\vert a\\right\\vert =N.\n\\end{equation*\nFor the case $\\left\\Vert x^{n}\\right\\Vert \\leq 1$ we however we cannot\nproceed without (\\ref{ccccc}). The reason is what while on space $E$\ndisregarding of $n$ the sequence is norm bounded by $1$ (uniformely in $n$)\nin norm given by (\\ref{norm_operator}), this is not the case with the\nmax-norm where it is unbounded as $n\\rightarrow \\infty $.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThis paper considers estimation of parameters of distributions whose domain is a particular non-Euclidean geometry: a topological space divided into $M$ equivalence classes by actions of a finite spherical symmetry group. A well known example of a finite spherical symmetry group is the point group in 3 dimensions describing the soccer ball, or football, with truncated icosahedral symmetry that also corresponds to the symmetry of the Carbon-60 molecule. This paper formulates a general approach to parameter estimation in distributions defined over such domains. We use a restricted finite mixture representation introduced in~\\cite{chen_parameter_2015} for probability distributions that are invariant to actions of any topological group. This representation has the property that the number of mixture components is equal to the order of the group, the distributions in the mixture are all parameterized by the same parameters,  and the mixture coefficients are all equal. This is practically significant since many reliable algorithms have been developed for parameter estimation when samples come from finite mixture distributions~\\cite{dempster_maximum_1977,sohn_efficient_2011}.\n\nWe apply the representation to an important problem in materials science: analysis of mean orientation in polycrystals. \nCrystal orientation characterizes properties of materials including electrical conductivity and thermal conductivity. Polycrystalline materials are composed of grains of varying size and orientation, where each grain contains crystal forms with similar orientations. The quality of the material is mainly determined by the grain structure i.e. the arrangement of the grains, their orientations, as well as the distribution of the precipitates. Thus accurate estimation of crystal orientation of the grains is useful for predicting how materials fail and what modes of failure are more likely to occur \\cite{de_graef_structure_2007}.\n\nThe mean orientation of the grain, characterized for example by its Euler angles, can only be specified modulo a set of angular rotations determined by the symmetry group associated with the specific type of crystal, e.g. hexagonal, cubic. This multiplicity of equivalent Euler angles complicates the development of reliable mean orientation estimators. The problem becomes even harder when the orientations are sampled from a region encompassing more than one grain such that the orientations cluster over different mean directions. In such a case, we would like to identify whether the orientations are multi-modally distributed and also estimate the mean direction for each cluster. \n\nIn our previous work~\\cite{chen_parameter_2015}, we introduced the finite mixture of Von Mises-Fisher (VMF) distribution for observations that are invariant to actions of a spherical symmetry group. We applied the expectation maximization (EM) maximum likelihood (ML) algorithm, called EM-VMF, to estimate the group-invariant parameters of this distribution. In this paper, we develop a hyperbolic representation simplification of the EM-VMF algorithm that reduces the computation time by a factor of $2$. We also introduce a new group invariant distribution for spherical symmetry groups, called the $\\mathcal{G}$-invariant Watson distribution, which like VMF is a density parameterized by location (angle mean) and scale (angle concentration) over the $p$-dimensional sphere. An EM algorithm is presented for estimation of the parameters, called the EM-Watson algorithm. Furthermore, mixture-of-$\\mathcal{G}$-invariant Watson (mGIW) and von Mises-Fisher (mGIV) distributions are introduced to perform clustering on the $\\mathcal{G}$-invariant sphere. An EM algorithm is presented for estimation of the parameters of the mGIW and mGIV distributions. We illustrate how the Generalized Likelihood Ratio Test (GLRT) can be used to detect the presence of multiple modes in a sample and how it can be combined with the EM algorithm for mGIW and mGIV distributions to cluster multiple orientations on the sphere. \n\nThe performance of the proposed EM orientation estimators is evaluated by simulation and compared to other estimators. The EM orientation estimators are then illustrated on Electron Backscatter Diffraction EBSD data collected from a Nickel alloy whose crystal form induces the $m\\overline{3}m$~\\cite{newnham_properties_2004} cubic point symmetry group. We establish that the EM orientation estimators result in significantly improved estimates of the mean direction in addition to providing an accurate estimate of concentration about the mean. Furthermore, with the extended mixture models, we are able to identify and cluster multi-modally distributed samples more accurately than the K-means algorithm.\n\nThe paper is organized as follows. Section~\\ref{sec:group-invariant} describes group invariant random variables and gives the mixture representation for their densities. Section \\ref{sec:spherical_symmetry_group} specializes to random variables invariant relative to actions of the spherical symmetry group and develops the $\\mathcal G$-invariant VMF and Watson distributions along with  EM-ML parameter estimator. The clustering methods based on the $\\mathcal{G}$-invariant distributions along with the GLRT are elaborated in Section \\ref{sec:clustering_spherical_symmetry_group}. The crystallography application and data simulation are presented in Section~\\ref{sec:app_crystal_orientation_estimation} and the experiment results are shown in Section \\ref{sec:experiment}. Section~\\ref{sec:conclusion} has concluding remarks.\n\n\\section{Group-invariant random variables}\n\\label{sec:group-invariant}\n\\def\\bfx{{\\mathbf x}}\nConsider a finite topological group $\\mathcal G=\\{G_1, \\ldots, G_M\\}$ of $M$ distinct actions on a topological space $\\mathcal X$, $G_i: \\mathcal X \\rightarrow \\mathcal X$ and a binary operation \"*\" defining the action composition $G_i * G_j$, denoted $G_i G_j$. $\\mathcal G$ has the properties that composition of multiple actions is associative, for every action there exists an inverse action, and there exists an identity action \\cite{birkhoff_brief_1963}. A real valued function $f(\\bx)$ on $\\mathcal X$ is said to be invariant under $\\mathcal G$ if: $f(G\\bx)=f(\\bx)$ for $G\\in \\mathcal G$.   Let $\\bX$ be a random variable defined on $\\mathcal X$. We have the following theorem for the probability density $f(\\bx)$ of $\\bX$.\n\\begin{theorem}\n\\label{thm:1}\nThe density function $f: \\mathcal X\\rightarrow \\Reals$ is invariant under $\\mathcal G$ if and only if\n\\begin{eqnarray}\n\\label{eq:thm1_representation}\n\\!\\begin{aligned}\n\\exists\\ &h: \\mathcal X\\rightarrow \\Reals\\ s.t. \\\\\n&f(\\bx)= \\frac{1}{M} \\sum_{i=1}^M h(G_i\\bx).\n\\end{aligned}\n\\end{eqnarray}\n\\end{theorem}\nThis theorem is a slight generalization of \\cite[Thm. 2.1]{chen_parameter_2015} in that the density $h(.)$ is not necessarily the same as $f(.)$.  The proof is analogous to that of \\cite[Thm. 2.1]{chen_parameter_2015}.\n\nTheorem \\ref{thm:1} says that any density $f(\\bx)$ that is invariant under group $\\mathcal G$ can be represented as a finite mixture of a function and its translates $h(G_i\\bx)$ under the group's actions $G_i \\in \\mathcal G$. As pointed out in~\\cite{chen_parameter_2015}, Thm.~\\ref{thm:1} has important implications on $\\mathcal G$-invariant density estimation and parameter estimation. In particular it can be used to construct maximum likelihood estimators for parametric densities. Let $h(\\bx;\\btheta)$ be a density on $\\mathcal X$ that is parameterized by a parameter $\\btheta$ in a parameter space $\\Theta$. We extend  $h(\\bx;\\btheta)$ to a $\\mathcal G$-invariant density $f$ by using Thm. \\ref{thm:1}, obtaining:\n\\begin{eqnarray}\nf(\\bx;\\btheta)=\\frac{1}{M} \\sum_{i=1}^M h_i(\\bx;\\btheta),\n\\label{eq:SSG}\n\\end{eqnarray}\nwhere $h_i(\\bx;\\btheta)=h(G_i\\bx;\\btheta)$. This density is of the form of a finite mixture of densities $h_i(\\bx;\\btheta)$ of known parametric form where the mixture coefficients are all identical and equal to $1\/M$. Maximum likelihood (ML) estimation of the parameter $\\btheta$ from an i.i.d. sample $\\{\\bx_i\\}_{i=1}^n$ from any $\\mathcal G$-invariant density $f$ can now be performed using finite mixture model methods \\cite{mclachlan_finite_2004} such as the  Expectation-Maximization (EM) algorithm~\\cite{dempster_maximum_1977} or the restricted Boltzman machine (RBM) \\cite{sohn_efficient_2011}.\n\n\\section{ML within a Spherical Symmetry Group}\n\\label{sec:spherical_symmetry_group}\nAs in~\\cite{chen_parameter_2015} we specialize Thm.~\\ref{thm:1} to estimation of parameters for the case that the probability density is on a sphere and is invariant to actions in a spherical symmetry group. In Section~\\ref{sec:app_crystal_orientation_estimation} this will be applied to a crystallography example under spherical distribution likelihood models for the mean crystal orientation. In general, the measured and mean orientations can be represented by Euler angles~\\cite{eberly_euler_2008}, Rodrigues Vectors~\\cite{rodrigues_lois_1840}, or Quaternions~\\cite{altmann_rotations_2005}. As in~\\cite{chen_parameter_2015}, we use the quaternion representation to enable orientations to be modeled by spherical distributions since the quaternion representation is a $4$D vector on the $3$-sphere $S^3$, i.e. $\\bq = (q_1, q_2, q_3, q_4)$ such that $\\|\\bq\\| = 1$.\n\nAny of the aforementioned orientation representations have inherent ambiguity due to crystal symmetries. For example, if the crystal has cubic symmetry, its orientation is only uniquely defined up to a 24-fold set of proper rotations of the cube about its symmetry axes.\nThese actions form a point symmetry group, called $432$, a sub-group of $m\\overline{3}m$. In quaternion space, since each orientation corresponds to two quaternions with different sign $\\{\\bq,-\\bq\\}$, these rotations reflections, and inversions can be represented as a spherical symmetry group $\\mathcal{G}$ of quaternionic matrices $\\{\\bP_1,\\ldots,\\bP_{M}\\}$, with sign symmetry such that $\\bP_i=-\\bP_{i-M\/2}\\ \\forall M\/2<i\\le M$, where $M=48$ for cubic symmetry. \n\nBased on the symmetry group $\\mathcal{G}$, we can define the distance between two quaternions under $\\mathcal{G}$ as:\n\\begin{equation}\n\\label{eq:sym_dist}\nd_{\\mathcal{G}}(\\bq_1,\\bq_2) = \\min_{\\bP\\in\\mathcal{G}} \\arccos{(\\bq_1^T\\bP\\bq_2)}\n\\end{equation}\nTwo quaternions $\\bq_1,\\bq_2$ are called symmetry-equivalent to each other if they are mapped to an equivalent orientation under $\\mathcal{G}$, i.e. $d_{\\mathcal{G}}(\\bq_1,\\bq_2)=0$. A fundamental zone (FZ), also called the fundamental domain, is a conic solid subset of the sphere that can be specified to disambiguate any particular orientation $\\bx$. However, as will be seen in Sec. \\ref{sec:app_crystal_orientation_estimation}, reduction of the entire data sample $\\{\\bx_i\\}_{i=1}^n$ to a FZ destroys information necessary for maximum likelihood estimation: the entire $\\mathcal{G}$-invariant density (\\ref{eq:SSG}) must be used. In the following two subsections, we introduce two $\\mathcal{G}$-invariant spherical distributions: von Mises-Fisher and Watson distributions~\\cite{mardia_directional_1999} to model orientations in quaternion space.\n\n\\subsection{Hyperbolic $\\mathcal G$-invariant von Mises-Fisher Distribution}\n\\label{sec:ginv_VMF_dist}\nFirst we review the $\\mathcal G$-invariant von Mises-Fisher distribution presented in~\\cite{chen_parameter_2015}. The von Mises-Fisher (VMF) distribution arises in directional statistics~\\cite{mardia_directional_1999} as an analogue of the multivariate Gaussian distribution on the $(p-1)$-dimensional sphere $S^{(p-1)}\\subset \\Reals^p$, where $p\\geq 2$. The VMF distribution is parameterized by the mean direction $\\bmu\\in S^{(p-1)}$ and the concentration parameter $\\kappa\\ge 0$:\n\\begin{equation}\n\\phi(\\mathbf{x};\\mathbf{\\bmu},\\kappa) = c_p(\\kappa)\\exp{(\\kappa\\bmu^T\\mathbf{x})},\n\\label{eq:VMF_pdf}\n\\end{equation}\nwhere $c_p(\\kappa) = \\frac{\\kappa^{p\/2-1}}{(2\\pi)^{p\/2}I_{p\/2-1}(\\kappa)}$ and $I_p(\\cdot)$ is the modified Bessel function of the first kind of order $p$. Given an i.i.d. sample $\\{\\bx_i\\}_{i=1}^n$ from the VMF distribution, the ML estimator has the closed-form expressions \\cite{mardia_directional_1999}\n\\begin{align}\n\\label{eq:ML_estimator}\n\\hat{\\bmu}=\\frac{\\mathbf{\\bgamma}}{\\|\\mathbf{\\bgamma}\\|},\\ \\hat{\\kappa}=A_p^{-1}\\left(\\frac{\\|\\mathbf{\\bgamma}\\|}{n}\\right),\\\n\\end{align}\nwhere $\\mathbf{\\bgamma}=\\sum_{i=1}^{n}\\mathbf{\\bx}_i$ and $A_p(u)=\\frac{I_{p\/2}(u)}{I_{p\/2-1}(u)}$.\n\nLet $\\mathcal{G}$ be a group of symmetric actions $\\{\\bP_1, \\ldots, \\bP_M\\}$ acting on the quaternionic representation of orientation on the $3$-dimensional sphere $S^{3}$. We extend the VMF distribution (\\ref{eq:VMF_pdf}) using the mixture representation in Thm~\\ref{thm:1}:\n\\begin{eqnarray}\n\\label{eq:mixture_densityp}\nf_v(\\mathbf{x};\\bmu,\\kappa)\n&=&\\frac{1}{M}\\sum_{m=1}^M\\phi(\\bP_m\\mathbf{x};\\bmu, \\kappa)\\\\\n\\label{eq:mixture_density}\n&=&\\frac{1}{M}\\sum_{m=1}^M\\phi(\\mathbf{x};\\bP_m\\bmu, \\kappa)\n\\end{eqnarray}\nwhere in going from (\\ref{eq:mixture_densityp}) to (\\ref{eq:mixture_density}) we used the inner product form $\\bmu^T \\bx$ in (\\ref{eq:VMF_pdf}) and the symmetry of $\\bP_m$. The expression (\\ref{eq:mixture_density}) for the extended VMF distribution is in the form of\na finite mixture of standard VMF distributions on the same random variable $\\bx$ having different mean parameters $\\bmu_m =\\bP_m\\bmu$ but having the same concentration parameter $\\kappa$.\n\nThe finite mixture (\\ref{eq:mixture_density}) for the $\\mathcal G$-invariant density $f_v(\\mathbf{x};\\bmu,\\kappa)$ is in a form for which an EM algorithm~\\cite{dempster_maximum_1977} can be implemented to compute the ML estimates of $\\bmu$ and $\\kappa$. Denoting the parameter pair as $\\bomega=\\{\\bmu,\\kappa\\}$, the EM algorithm generates a sequence $\\{\\bomega^{(k)}\\}$ of estimates that monotonically increase the likelihood. These estimates are given by $\\bomega^{(k+1)}= \\arg\\max_{\\bomega}  E_{S|\\bX,\\bomega^{(k)}}[\\log{L(\\bomega;\\{\\bx_i,s_i\\})}]$, where $s_i$ is a latent variable assigning $\\bx_i$ to a particular mixture component in (\\ref{eq:mixture_density}) and $L(\\bomega;\\{\\bx_i, s_i\\})$ is the likelihood function of $\\bomega$ given the complete data $\\{\\bx_i, s_i\\}_{i=1}^n$. Specifically,\n\\begin{eqnarray}\n\\label{eq:qfunction}\n&&E_{S|\\bX,\\bomega}[\\log{L(\\bomega;\\{\\bx_i,s_i\\})}] \\\\\n&=& \\sum_{i=1}^{n}\\sum_{m=1}^Mr_{i,m}(\\log{c_p(\\kappa)}+\\kappa(\\bP_m\\bmu)^T\\mathbf{x}_i),\n\\nonumber\n\\end{eqnarray}\nwhere $r_{i,m}=P(s_i=m|\\bx_i,\\bomega)$. The EM algorithm takes the form:\n\nE-step:\n\\begin{equation}\n\\label{eq:VMF_Estep}\n\\begin{split}\nr_{i,m} = \\frac{\\phi(\\bx_i; \\bP_m\\bmu, \\kappa)}{\\sum_{l=1}^M\\phi(\\bx_i; \\bP_l\\bmu, \\kappa)}, m\\in\\left\\{1,2,\\ldots,M\\right\\}.\n\\end{split}\n\\end{equation}\n\nM-step:\n\\begin{align}\n\\label{eq:EM_Mstep}\n\\hat{\\bmu}&=\\frac{\\bgamma}{\\|\\bgamma\\|},\\ \\hat{\\kappa}=A_p^{-1}\\left(\\frac{\\|\\bgamma\\|}{n}\\right),  \\\\\n\\bgamma&=\\sum_{i=1}^{n}\\sum_{m=1}^Mr_{i,m}\\bP_m^T\\bx_i. \\label{eq:VMF_Mstep_gamma}\n\\end{align}\n\nThe hyberbolic $\\mathcal G$-invariant von Mises-Fisher distribution is obtained by exploiting the sign symmetry in $\\mathcal G$. In particular, (\\ref{eq:VMF_Mstep_gamma}) in the M-step can be re-written as:\n\\begin{equation}\n\\label{eq:EM_Mstep_simplified}\n\\begin{split}\n\\bgamma&=\\sum_{i=1}^{n}\\sum_{m=1}^Mr_{i,m}\\bP_m^T\\mathbf{x}_i \\\\\n&=\\sum_{i=1}^{n}\\left(\\sum_{m=1}^{M\/2}r_{i,m}\\bP_m^T\\bx_i - r_{i,\\frac{M}{2}+m}\\bP_m^T\\bx_i\\right) \\\\\n&=\\sum_{i=1}^{n}\\sum_{m=1}^{M\/2} \\frac{\\sinh{(\\kappa(\\bP_m\\bmu)^T\\bx_i)}}{\\sum_{l=1}^{M\/2}\\cosh{(\\kappa(\\bP_l\\mu)^T\\bx_i)} } \\bP_m^T\\bx_i,\n\\end{split}\n\\end{equation}\nwhere $\\sinh,\\cosh$ are the hyperbolic sinusoidal functions. Equation (\\ref{eq:VMF_Estep}) in E-step is simplified as:\n\\begin{equation}\n\\label{eq:EM_Estep_simplified}\nr'_{i,m} = \\frac{\\sinh{(\\kappa(\\bP_m\\bmu)^T\\bx_i)}}{\\sum_{l=1}^{M\/2}\\cosh{(\\kappa(\\bP_l\\mu)^T\\bx_i)}}, m\\in\\left\\{1,2,\\ldots,\\frac{M}{2}\\right\\}.\n\\end{equation}\n\nIn Section~\\ref{sec:experiment} we demonstrate the computational improvement of the hyperbolic form of the EM algorithm obtained by substituting (\\ref{eq:EM_Mstep_simplified}), (\\ref{eq:EM_Estep_simplified}) into (\\ref{eq:EM_Mstep}), (\\ref{eq:VMF_Estep}) respectively.\n\n\\subsection{$\\mathcal G$-invariant Watson Distribution}\n\\label{sec:ginv_Watson_dist}\nAs described at the beginning of this section, each orientation corresponds to two quaternions with different sign, which is equivalent to an axis of the sphere. For axial data it is more natural to use the Watson distribution~\\cite{watson_equatorial_1965}, which models the probability distribution of axially symmetric vectors on the $(p-1)$-dimensional unit sphere, i.e. $\\pm\\bx\\in S^{p-1}$ are equivalent. Similar to VMF, the distribution is parametrized by a mean direction $\\pm\\bmu\\in S^{p-1}$, and a concentration parameter $\\kappa\\in\\Reals$ that is no longer necessarily non-negative. Its probability density function is\n\\begin{equation}\n\\label{eq:Watson_pdf}\nW_p(\\bx; \\bmu,\\kappa) = \\frac{1}{\\bbM(\\frac{1}{2},\\frac{p}{2},\\kappa)}\\exp{\\left(\\kappa(\\bmu^T\\bx)^2\\right)},\n\\end{equation}\nwhere $\\bbM$ is the Kummer confluent hypergeometric function defined in~\\cite{bateman_higher_1955}. According to (\\ref{eq:Watson_pdf}), the positive-negative pair of group actions $\\{\\bP_m,-\\bP_m\\}$ contribute the same value in the density function. The set of the group action pairs $\\mathcal{G}'=\\{\\{\\bP_m,-\\bP_m\\}\\}_{m=1}^{M\/2}$ is the quotient group $\\mathcal{G}\/\\mathcal{I}$, where $\\mathcal{I}=\\{\\bI_p,-\\bI_p\\}\\subset\\mathcal{G}$ and $\\bI_p$ is the identity matrix of dimension $p$. Therefore, $\\mathcal{G}'$ is also a group and we can use Thm~\\ref{thm:1} to extend the Watson distribution to the mixture representation under $\\mathcal{G}'$:\n\\begin{equation}\n\\label{eq:Watson_mixture_density}\nf_w(\\bx;\\bmu,\\kappa) = \\frac{1}{M'} \\sum_{m=1}^{M'} W_p(\\bx; \\bP_m\\bmu,\\kappa),\n\\end{equation}\nwhere $M'=M\/2$. The ML estimates of $\\bmu$ and $\\kappa$ can also be calculated by the EM algorithm. The E-step for the Watson mixture distribution is\n\\begin{equation}\n\\label{eq:Watson_EM_Estep}\n\\begin{split}\nr_{i,m} = \\frac{\\exp{\\{\\kappa((\\bP_m\\bmu)^T\\mathbf{x}_i)^2\\}}}{\\sum_{l=1}^{M'}\\exp{\\{\\kappa((\\bP_l\\bmu)^T\\mathbf{x}_i)^2\\}}},m\\in\\left\\{1,2,...,M'\\right\\}.\n\\end{split}\n\\end{equation}\n\nFor the M-step, we take a similar approach as~\\cite{mardia_directional_1999} as follows:\n\\begin{equation}\n\\label{eq:Watson_Mstep_Dev}\n\\begin{split}\n&E_{S|\\bX,\\bomega}[\\log{L(\\bomega;\\{\\bx_i,s_i\\})}] \\\\\n=& n\\left( \\kappa\\bmu^T\\tilde{T}\\bmu - \\log{\\left(M'\\bbM\\left(\\frac{1}{2},\\frac{p}{2},\\kappa\\right)\\right)}\\right),\n\\end{split}\n\\end{equation}\nwhere $\\tilde{T}=\\frac{1}{n}\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,m}(\\bP_m^T\\bx_i\\bx_i^T\\bP_m)$ is the scatter matrix of $\\bx_1,...,\\bx_n$. Let $\\tilde{t}_1,...,\\tilde{t}_p$ be the eigenvalues of $\\tilde{T}$ with \n\\begin{equation}\n\\tilde{t}_1\\ge...\\ge\\tilde{t}_p,\n\\end{equation}\nand let $\\pm\\bt_1,...,\\pm\\bt_p$ be the corresponding unit eigenvectors.\nSince we want to find $\\bmu$ which maximizes (\\ref{eq:Watson_Mstep_Dev}) such that $\\bmu^T\\bmu=1$, the estimator of $\\bmu$ for fixed $\\kappa$ has the following form:\n\\begin{equation}\n\\begin{split}\n\\label{eq:Watson_mu_est}\n\\hat\\bmu &= \\bt_1, \\hat{\\kappa} > 0, \\\\\n\\hat\\bmu &= \\bt_p, \\hat{\\kappa} < 0.\n\\end{split}\n\\end{equation}\n\nSimilarly by fixing $\\bmu$ and setting to zero the derivative of (\\ref{eq:Watson_Mstep_Dev}) with respect to $\\kappa$, we have:\n\\begin{equation}\n\\label{eq:Watson_Tp_func}\n\\begin{split}\n&Y_p(\\kappa) = \\frac{\\bbM'(\\frac{1}{2},\\frac{p}{2},\\kappa)}{\\bbM(\\frac{1}{2},\\frac{p}{2},\\kappa)} = \\frac{\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,m}(\\bmu^T\\bP_m^T\\bx_i)^2}{n} \\\\\n\\Rightarrow& \\hat{\\kappa} = Y_p^{-1}\\left( \\frac{\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,m}(\\bmu^T\\bP_m^T\\bx_i)^2}{n}\\right),\n\\end{split}\n\\end{equation}\nThe final estimates of $\\bmu$ and $\\kappa$ are obtained by checking both cases ($\\hat{\\kappa}>0$, $\\hat{\\kappa}<0$) and choosing the one which is consistent for (\\ref{eq:Watson_mu_est})(\\ref{eq:Watson_Tp_func}).\n\n\\section{Clustering with a Spherical Symmetry Group}\n\\label{sec:clustering_spherical_symmetry_group}\nIn this section we extend the parameter estimation problem to the situation where there are multiple group-invariant distributions with different parameters that govern the samples. This problem arises, for example, in poly-crystaline materials when estimating the mean crystal orientation over a region containing more than one grain (perhaps undetected). This problem can be solved by first applying some standard clustering methods, e.g. K-means\\cite{hartigan_algorithm_1979}, and then estimating the parameters for each cluster. However, clustering methods based on the distance relation between the samples are complicated by the presence of spherical symmetry because it is necessary to distinguish modes that are due only to symmetry from those that distinguish different clusters. Therefore, we propose a model-based clustering algorithm which accommodates symmetry to handle this problem.\n\nConsider the situation where the samples $\\{\\bx_i\\}_{i=1}^n$ follow a mixture of $\\mathcal{G}$-invariant density functions. For the VMF distribution, the mixture density has the following form:\n\\begin{equation}\n\\label{eq:mixture_of_Ginv_VMF}\ng_v(\\bx;\\{\\bmu_c,\\kappa_c,\\alpha_c\\}) = \\sum_{c=1}^C\\alpha_c\\left(\\sum_{m=1}^M \\frac{1}{M} \\phi(\\bx;\\bP_m\\bmu_c,\\kappa_c)\\right),\n\\end{equation}\nwhere $C$ is the number of clusters assumed to be fixed a priori, $\\bmu_c,\\kappa_c$ are the parameters for the $c$-th cluster and $\\alpha_c$ are the mixing coefficients where $\\sum_{c=1}^C\\alpha_c=1$ and $\\alpha_c>0$ for all $c$. The parameters of (\\ref{eq:mixture_of_Ginv_VMF}) can be estimated by the EM algorithm:\n\nE-step:\n\\begin{equation}\n\\label{eq:mVMF_mClusters_Estep}\nr_{i,c,m}=\\frac{\\alpha_c\\phi(\\bx_i; \\bP_m\\bmu_c,\\kappa_c)}{\\sum_{h=1}^C\\alpha_{h}\\sum_{l=1}^M\\phi(\\bx_i;\\bP_{l}\\mu_{h}, \\kappa_{h})}\n\\end{equation} \n\nM-step:\n\\begin{align}\n\\alpha_c&=\\sum_{i=1}^n\\sum_{m=1}^Mr_{i,c,m}, \\hat{\\bmu}_c=\\frac{\\bgamma_c}{\\|\\bgamma_c\\|}, \\hat{\\kappa}_c=A_p^{-1}\\left(\\frac{\\|\\bgamma_c\\|}{n\\alpha_c}\\right),\\\\\n\\label{eq:mVMF_mClusters_Mstep}\n\\bgamma_c&=\\sum_{i=1}^n\\sum_{m=1}^M r_{i,c,m}\\bP_m^T\\bx_i,\n\\end{align}\nwhere $r_{i,c,m}$ is the probability of sample $\\bx_i$ belonging to the $c$-th cluster and the $m$-th symmetric component.\n\nFor the Watson distribution, the mixture of $\\mathcal{G}$-invariant Watson density is\n\\begin{equation}\n\\label{eq:mixture_of_Ginv_Watson}\ng_w(\\bx;\\{\\bmu_c,\\kappa_c, \\alpha_c\\}) = \\sum_{c=1}^C\\alpha_c\\left(\\sum_{m=1}^{M'} \\frac{1}{M'}W_p(\\bx; \\bP_m\\bmu_c,\\kappa_c)\\right)\n\\end{equation}\n\nThe E-step is similar to (\\ref{eq:mVMF_mClusters_Estep}) with $\\phi$ replaced by $W_p$ function. The M-step can be computed with a similar approach as in Section~\\ref{sec:ginv_Watson_dist} with the following modifications:\n\\begin{align}\n\\label{eq:Watson_mClusters_Mstep_hatT}\n\\tilde{T_c}&=\\frac{1}{n\\alpha_c}\\sum_{i=1}^n\\sum_{m=1}^{M'} r_{i,c,m}(\\bP_m^T\\bx_i\\bx_i^T\\bP_m), \\\\\n\\label{eq:Watson_mClusters_Mstep_kappa}\n\\hat{\\kappa}_c&=Y_p^{-1}\\left(\\frac{\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,c,m}(\\bmu_c^T\\bP_m^T\\bx_i)^2}{n\\alpha_c}\\right),\n\\end{align}\nwhere $\\alpha_c=\\sum_{i=1}^n\\sum_{m=1}^Mr_{i,c,m}$.\n\\subsection{Multi-modality Tests on $\\mathcal{G}$-invariant Spherical Distributions}\nGiven sample set $\\{\\bx_i\\}_{i=1}^n$ on $S^{p-1}$, the objective is to determine whether the $n$ samples are drawn from one single distribution or a mixture of $C$ distributions. For polycrystalline materials, the result of this determination can be used to discover undetected grains within a region. We propose to use a multi-modal hypothesis test based on the $\\mathcal{G}$-invariant distributions to solve this problem. The two hypotheses are $H_0$: The samples are from a single $\\mathcal{G}$-invariant distribution $f(\\bx;\\{\\bmu,\\kappa\\})$; and $H_1$: The samples are from a mixture of $C$ distributions $g(\\bx;\\{\\bmu_c,\\kappa_c,\\alpha_c\\}_{c=1}^C)$. The Generalized Likelihood Ratio Test (GLRT)~\\cite{hero_statistical_2000} has the following form:\n\\begin{equation}\n\\label{eq:multi_sample_GLRT}\n\\begin{aligned}\n\\Lambda_{GLR} &= \\frac{\\max_{\\{\\bmu_c,\\kappa_c,\\alpha_c\\}_{c=1}^C\\in\\Theta_1}g(\\{\\bx_i\\}_{i=1}^n;\\{\\bmu_c,\\kappa_c,\\alpha_c\\}_{c=1}^C)}{\\max_{\\{\\bmu,\\kappa\\}\\in\\Theta_0}f(\\{\\bx_i\\}_{i=1}^n;\\{\\bmu,\\kappa\\})} \\\\\n&\\gtrless^{H_1}_{H_0} \\eta\n\\end{aligned}\n\\end{equation}\nwhere $\\Theta_0,\\Theta_1$ are the parameter spaces for the two hypotheses. The $f$ and $g$ functions for VMF and Watson distributions are defined in (\\ref{eq:mixture_density}), (\\ref{eq:Watson_mixture_density}) and (\\ref{eq:mixture_of_Ginv_VMF}), (\\ref{eq:mixture_of_Ginv_Watson}) respectively and the test statistic $\\Lambda_{GLR}$ can be calculated by the proposed EM algorithm. According to Wilks's theorem~\\cite{wilks_large-sample_1938} as $n$ approaches $\\infty$, the test statistic $2\\log{\\Lambda_{GLR}}$ will be asymptotically $\\chi^2$-distributed with degrees of freedom equal to $(p+1)(C-1)$, which is the difference in dimensionality of $\\Theta_0$ and $\\Theta_1$. Therefore, the threshold $\\eta$ in (\\ref{eq:multi_sample_GLRT}) can be determined by a given significance level $\\alpha$.\n\n\\section{Application to Crystallographic Orientation}\n\\label{sec:app_crystal_orientation_estimation}\nCrystal orientation and the grain distribution in polycrystalline materials determine the mechanical properties of the material, such as, stiffness, elasticity, and deformability. Locating the grain regions and estimating their orientation and dispersion play an essential role in detecting anomalies and vulnerable parts of materials.\n\nElectron backscatter diffraction (EBSD) microscopy acquires crystal orientation at multiple locations within a grain by capturing the Kikuchi diffraction patterns of the backscatter electrons ~\\cite{saruwatari_crystal_2007}. A Kikuchi pattern can be translated to crystal orientation through Hough Transformation analysis~\\cite{lassen_automated_1994} or Dictionary-Based indexing~\\cite{park_ebsd_2013}. The process of assigning mean orientation values to each grain is known as indexing. Crystal forms  possess  point symmetries, e.g. triclinic, tetragonal, or cubic, leading to  a probability density of measured orientations that is invariant over an associated spherical symmetry group $\\mathcal{G}$. Therefore, when the type of material has known symmetries, e.g., cubic-type symmetry for nickel or gold, the $\\mathcal{G}$-invariant VMF and Watson models introduced in Section~\\ref{sec:spherical_symmetry_group} can be applied to estimate the mean orientation $\\bmu_g$ and the concentration $\\kappa_g$ associated with each grain. Furthermore, the clustering method along with the multi-sample hypothesis test in Section~\\ref{sec:clustering_spherical_symmetry_group} can be used to detect the underlying grains within a region.\n\n\\subsection{Simulation of Crystallographic Orientation}\n\\label{sec:simulation_orientations}\nTo simulate the crystallographic orientations, we first draw random samples from VMF and Watson distributions with $p=4$. The random variable $\\bx$ in a spherical distribution can be decomposed~\\cite{mardia_directional_1999}:\n\\begin{equation}\n\\label{eq:normal_tangent_decompose}\n\\bx=t\\bmu+\\sqrt{1-t^2}S_\\bmu(\\bx),\n\\end{equation}\nwhere $t=\\bmu^T\\bx$ and $S_\\bmu(\\bx)=(I_p-\\bmu\\bmu^T)\\bx\/\\|(I_p-\\bmu\\bmu^T)\\bx\\|$. Let $f(\\bx;\\bmu)$ be the p.d.f. of the distribution where $\\bmu$ is the mean direction. According to the normal-tangent decomposition property, for any rotationally symmetric distribution, $S_\\bmu(\\bx)$ is uniformly distributed on $S_{\\bmu^\\bot}^{p-2}$, the $(p-2)$-dimensional sphere normal to $\\bmu$, and the density of $t=\\bx^T\\bmu$ is given by:\n\\begin{equation}\n\\label{eq:tangent_density}\nt\\mapsto cf(t)(1-t^2)^{(p-3)\/2}.\n\\end{equation}\n\nFor VMF distribution, substituting (\\ref{eq:normal_tangent_decompose}) into (\\ref{eq:VMF_pdf}) and combining with (\\ref{eq:tangent_density}), we have the density of the tangent component $t$ as:\n\\begin{equation}\n\\label{eq:VMF_tangent_density}\n\\begin{split}\nf_v(t)&= C_v\\exp{\\{\\kappa t\\}}(1-t^2)^{(p-3)\/2} \\\\\nC_v&=\\left(\\frac{\\kappa}{2}\\right)^{(p\/2-1)}\\left(I_{p\/2-1}(\\kappa)\\Gamma\\left(\\frac{p-1}{2}\\right)\\Gamma\\left(\\frac{1}{2}\\right)\\right)^{-1}.\n\\end{split}\n\\end{equation}\n\nSimilarly, the density of the tangent component of Watson distribution is:\n\\begin{equation}\n\\label{eq:Watson_tangetn_density}\n\\begin{split}\nf_w(t)&= C_w\\exp{\\{\\kappa t^2\\}}(1-t^2)^{(p-3)\/2} \\\\\nC_w&=\\frac{\\Gamma(\\frac{p}{2})}{\\Gamma(\\frac{p-1}{2})\\Gamma(\\frac{1}{2})}\\frac{1}{\\bbM(\\frac{1}{2},\\frac{p}{2},\\kappa)}.\n\\end{split}\n\\end{equation}\nRandom samples from the density functions (\\ref{eq:VMF_tangent_density}) and (\\ref{eq:Watson_tangetn_density}) can be easily generated by rejection sampling. \n\nThe generated quaternions from VMF and Watson distributions are then mapped into the Fundamental Zone (FZ) with the symmetric group actions to simulate the wrap-around problem we observe in real data, i.e. observations are restricted to a single FZ. For cubic symmetry, the FZ in quaternion space is defined in the following set of equations: \\\\\n\\begin{minipage}{0.48\\linewidth}\n\\begin{equation}\n\\begin{cases}\n|q_2\/q_1|\\le\\sqrt{2}-1 \\\\\n|q_3\/q_1|\\le\\sqrt{2}-1 \\\\\n|q_4\/q_1|\\le\\sqrt{2}-1 \\\\\n|q_2\/q_1 + q_3\/q_1 + q_4\/q_1|\\le 1 \\\\\n|q_2\/q_1 - q_3\/q_1 + q_4\/q_1|\\le 1 \\\\\n|q_2\/q_1 + q_3\/q_1 - q_4\/q_1|\\le 1 \\\\\n|q_2\/q_1 - q_3\/q_1 - q_4\/q_1|\\le 1 \\nonumber\n\\end{cases}\n\\end{equation}\n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n\\begin{equation}\n\\label{eq:FZ_equations}\n\\begin{cases}\n|q_2\/q_1 - q_3\/q_1|\\le\\sqrt{2} \\\\\n|q_2\/q_1 + q_3\/q_1|\\le\\sqrt{2} \\\\\n|q_2\/q_1 - q_4\/q_1|\\le\\sqrt{2} \\\\\n|q_2\/q_1 + q_4\/q_1|\\le\\sqrt{2} \\\\\n|q_3\/q_1 - q_4\/q_1|\\le\\sqrt{2} \\\\\n|q_3\/q_1 + q_4\/q_1|\\le\\sqrt{2} \\\\\n\\end{cases}\n\\end{equation}\n\\end{minipage}\nwhere $q_i$ is the $i$-th component of quaternion $\\bq$.\n\n\\section{Experimental Results}\n\\label{sec:experiment}\n\\subsection{$\\mathcal G$-invariant EM-ML Parameter Estimation on Simulated Data}\nSets of $n$ i.i.d. samples were simulated from the VMF or Watson distributions using the method described in Sec.\\ref{sec:simulation_orientations} with given $\\bmu=\\bmu_o,\\kappa=\\kappa_o$ for the $m\\overline{3}m$ point symmetry group associated with the symmetries of cubic crystal lattice planes. The number of samples for each simulation was set to $n=1000$ and $\\kappa_o$ was swept from  $1$ to $100$ while, for each simulation run,  $\\bmu_o$ was selected uniformly at random. The experiment was repeated $100$ times and the average values of $\\hat{\\kappa}$ and the inner product $\\hat{\\bmu}^T \\bmu_o$ are shown in Fig.~\\ref{fig:mu_est} and \\ref{fig:kappa_est}. In the figures we compare performance for the following methods: (1) the naive ML estimator for the standard VMF or Watson model that does not account for the point group structure (labeled \"ML Estimator\"). (2) Mapping each of the samples $\\bx_i$ toward a reference direction $\\bx_{r}$ (randomly selected from $\\{\\bx_i\\}_{i=1}^n$), i.e. $\\bx_i\\mapsto\\bP_m\\bx_i$, where $\\bP_m=\\arg\\min_{\\bP\\in\\mathcal{G}} \\arccos{(\\bx_{r}^T\\bP\\bx)}$, to remove possible ambiguity. Then performing ML for the standard VMF or Watson distribution (labeled \"Modified ML\"). (3) Applying our proposed EM algorithm directly to the $n$ samples using the mixture of VMF distribution (\\ref{eq:VMF_Estep})-(\\ref{eq:VMF_Mstep_gamma}) (labeled \"EM-VMF\") (4) Applying our proposed EM algorithm to the mixture of Watson distribution (\\ref{eq:Watson_EM_Estep})-(\\ref{eq:Watson_Tp_func}) (labeled \"EM-Watson\").\n\n\nFigure \\ref{fig:mu_est} shows the inner product values $\\bmu_o^T\\hat{\\bmu}$. The proposed EM-VMF and EM-Watson estimators have similar performance in that they achieve perfect recovery of the mean orientation ($\\bmu_o^T\\hat{\\bmu}=1$) much faster than the other methods as the concentration parameter $\\kappa_o$ increases (lower dispersion of the samples about the mean) no matter whether the data is generated from VMF (Fig.~\\ref{fig:mu_est_VMFdata}) or Watson distribution (Fig.~\\ref{fig:mu_est_Watsondata}), indicating the robustness of the proposed approaches under model mismatch. Notice that when $\\kappa_o$ is small ($\\kappa_o<20$ for VMF data and $\\kappa_o<10$ for Watson data), none of the methods can accurately estimate the mean orientation. The reason is that when $\\kappa_o$ is small the samples become nearly uniformly distributed over the sphere. The threshold $\\kappa_o$ value at which performance starts to degrade depends on the choice of point symmetry group and the distribution used to simulate the data. In Fig.~\\ref{fig:kappa_est} it is seen that the biases of the proposed EM-VMF~\\cite{chen_parameter_2015} and EM-Watson $\\kappa$ estimators are significantly lower than that of the other methods compared. While the modified ML performs better than the naive ML estimator, its bias is significantly worse than the proposed EM-VMF and EM-Watson approaches. \n\n\\begin{figure}\n  \\centering\n  \\subfigure[VMF Simulated Data]{\n    \\label{fig:mu_est_VMFdata}\n    \\includegraphics[width=4.25cm]{figures\/Mu_est_VMF}}\n  \\subfigure[Watson Simulated Data]{\n    \\label{fig:mu_est_Watsondata}\n    \\includegraphics[width=4.25cm]{figures\/Mu_est_Watson}}\n  \\caption{Mean orientation estimator comparisons for $\\mathcal G$-invariant densities. Shown is the average inner product $\\bmu_o^T\\hat{\\bmu}$ of four estimators $\\hat{\\bmu}$ when $\\bmu_o$ is the true mean orientation as a function of the true concentration parameter $\\kappa_o$ for the data simulated from VMF (Fig.~\\ref{fig:mu_est_VMFdata}) and from Watson (Fig.~\\ref{fig:mu_est_Watsondata}) distribution. The naive estimator (\"ML Estimator\" in blue line) does not attain perfect estimation (inner product $=1$) for any $\\kappa_o$ since it does not account for the spherical symmetry group structure. The modified ML (green dashed line) achieves perfect estimation as $\\kappa_o$ becomes large.  The proposed EM-ML methods (\"EM-VMF\", \"EM-Watson\") achieve perfect estimation much faster than the other methods even under model mismatch (EM-VMF for Watson simulated data and vice versa).}\n  \\label{fig:mu_est}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\subfigure[VMF Simulated Data]{\n    \\label{fig:kappa_est_VMFdata}\n    \\includegraphics[width=4.25cm]{figures\/Kappa_est_VMF}}\n  \\subfigure[Watson Simulated Data]{\n    \\label{fig:kappa_est_Watsondata}\n    \\includegraphics[width=4.25cm]{figures\/Kappa_est_Watson}}\n  \\caption{Concentration parameter estimator bias as a function of the true concentration $\\kappa_o$ for data simulated from VMF (Fig.~\\ref{fig:kappa_est_VMFdata})\\cite{chen_parameter_2015} and from Watson (Fig.~\\ref{fig:kappa_est_Watsondata}) distributions. The bias of the naive ML (blue solid line) is large over the full range of $\\kappa_o$. The modified ML (green dashed line) estimates $\\kappa$ more accurately when $\\kappa_o$ is small. Our proposed EM-VMF and EM-Watson estimators (black dotted line and magenta dashed line) have lower bias than the other estimators.}\n  \\label{fig:kappa_est}\n\\end{figure}\n\nFigure \\ref{fig:computation_time} shows the computation time of the estimation algorithms presented in Fig.~\\ref{fig:mu_est} and Fig.~\\ref{fig:kappa_est}. The computation time for all methods decreases as $\\kappa_o$ becomes larger. When $\\kappa_o$ is small ($\\kappa_o<20$ for VMF data and $\\kappa_o<10$ for Watson data), because the samples are almost uniformly distributed around the sphere, it is difficult for the EM algorithms to converge to the optimal solution and they therefore require maximum number of iterations to stop, forming the plateaus in Fig.~\\ref{fig:computation_time}. Notice that EM-Watson requires less time than EM-VMF even though it has more complicated E and M-steps. The reason is that EM-Watson uses only half of the symmetry operators, which corresponds to the size of the quotient group $\\mathcal{G}\/\\mathcal{I}$ as described in Section~\\ref{sec:ginv_Watson_dist}. By applying the hyperbolic sinusoidal simplification in Section~\\ref{sec:ginv_VMF_dist} (labeled \"EM-VMF-Hyper\"), we can further reduce the computation time by more than a factor of $2$ compared to the original EM-VMF.  \n\n\\begin{figure}\n  \\centering\n  \\subfigure[VMF Simulated Data]{\n    \\label{fig:computation_time_VMFdata}\n    \\includegraphics[width=4cm]{figures\/ComputationTime_VMF}}\n  \\subfigure[Watson Simulated Data]{\n    \\label{fig:computation_time_Watsondata}\n    \\includegraphics[width=4cm]{figures\/ComputationTime_Watson}}\n  \\caption{Computation time for calculating the result in Fig.~\\ref{fig:mu_est} and Fig.~\\ref{fig:kappa_est}. EM-Watson (magenta dashed line) has less computation time than EM-VMF (black dotted line) because it uses only half of the symmetry operators. EM-VMF-Hyper (cyan circle line) which uses the hyperbolic sinusoidal simplification of EM-VMF reduces the computation time by more than a factor of $2$.}\n  \\label{fig:computation_time} \n\\end{figure}\n\n\\subsection{$\\mathcal G$-invariant Clustering on Simulated Data}\nIn this section, we demonstrate the performance of our proposed EM approaches for clustering. Sets of $n$ i.i.d. samples were simulated from the VMF or Watson distributions with $\\kappa=\\kappa_o$ and one of two mean directions ($\\bmu_1,\\bmu_2$) to generate two clusters of samples. The spherical symmetry group is $m\\overline{3}m$ as before. The number of samples for each set was set to $n=1000$ and $\\kappa_o$ was swept from $1$ to $100$ while, for each set, $\\bmu_1,\\bmu_2$ was selected uniformly at random. The experiment was repeated $100$ times and the average values of the inner product $(\\hat{\\bmu}_1^T\\bmu_1+\\hat{\\bmu}_2^T\\bmu_2)\/2$ are shown in Fig.~\\ref{fig:mu_est_2clusters}. In the figure we compare performances of the following methods: (1) Cluster the samples by standard K-means algorithm with the distance defined by the arc-cosine of the inner product and then use the naive ML within each cluster to estimate the mean directions (labeled \"K-means\"). (2) \nCluster the samples by K-means with the distance defined as (\\ref{eq:sym_dist}) and then use the aforementioned modified ML estimator (labeled \"Modified K-means\"). (3) Apply our proposed multi-cluster EM-VMF algorithm to the $n$ samples directly (\\ref{eq:mVMF_mClusters_Estep})-(\\ref{eq:mVMF_mClusters_Mstep}) (labeled \"EM-VMF\") (4) Apply our multi-cluster EM-Watson algorithm to the $n$ samples directly (\\ref{eq:Watson_mClusters_Mstep_hatT})-(\\ref{eq:Watson_mClusters_Mstep_kappa}) (labeled \"EM-Watson\").\n\nFigure \\ref{fig:mu_est_2clusters} shows the average inner product values $(\\hat{\\bmu}_1^T\\bmu_1+\\hat{\\bmu}_2^T\\bmu_2)\/2$ from the mean direction estimation. The proposed EM-VMF and EM-Watson are able to correctly cluster the samples and achieve perfect recovery of the two mean orientations much faster than the other K-means approaches. Notice that the region where all the methods fail is larger than the single cluster case since multiple clusters increase the difficulty of parameter estimation. Again, no matter whether the samples are simulated from VMF or Watson distribution, our proposed approaches perform equally well under both cases.\n\nTo further test the ability to detect multiple clusters given a set of samples, we generate $1000$ sets of samples. Each set has $1000$ samples and is assigned randomly to label $0$ or $1$. If the set is labeled $0$, the samples are generated from a single distribution; If the set is labeled $1$, then the samples in the set are randomly generated from two distributions with different means. The GLRT is used with the four aforementioned clustering methods to test whether the samples in each set are uni-modal or multi-modal. The Receiver Operating Characteristic (ROC) curves of the test results are shown in Fig.~\\ref{fig:ROC}. The naive K-means with ML estimator which does not consider the symmetry group actions fails to distinguish whether the multiple modes are from actual multiple distributions or due to the wrap-around effect from the fundamental zone mapping. Therefore, this approach tends to over-estimate the goodness of fit of the $H_1$ model for true negative cases and under-estimate it for true positive cases, resulting in a result that is even worse than random guessing. The modified K-means performs better than K-means but worse than our proposed EM-VMF and EM-Watson algorithms. \n\n\\begin{figure}\n  \\centering\n  \\subfigure[VMF Simulated Data]{\n    \\label{fig:mu_est_2clusters_VMFdata}\n    \\includegraphics[width=4.25cm]{figures\/Mu_est_2clusters_VMF}}\n  \\subfigure[Watson Simulated Data]{\n    \\label{fig:mu_est_2clusters_Watsondata}\n    \\includegraphics[width=4.25cm]{figures\/Mu_est_2clusters_Watson}}\n  \\caption{Mean orientation estimator comparisons for samples generated from two different means. Shown is the average inner product $(\\hat{\\bmu}_1^T\\bmu_1+\\hat{\\bmu}_2^T\\bmu_2)\/2$ of four methods when $\\bmu_1,\\bmu_2$ are the true mean orientations as a function of the true concentration parameter $\\kappa_o$ for the data simulated from VMF (Fig.~\\ref{fig:mu_est_2clusters_VMFdata}) and from Watson (Fig.~\\ref{fig:mu_est_2clusters_Watsondata}) distributions. The K-means with naive estimator (\"K-means\" in blue line) does not attain perfect estimation for any $\\kappa_o$. A modified K-means with ML estimator (\"modified K-means\" in green dashed line) achieve perfect estimation as $\\kappa_o$ becomes large.  The proposed EM-VMF and EM-Watson methods (\"EM-VMF\", \"EM-Watson\") achieves perfect estimation much faster than the other methods.}\n  \\label{fig:mu_est_2clusters} \n\\end{figure}\n\n\n\\begin{figure}\n  \\centering\n  \\subfigure[VMF Simulated Data]{\n    \\label{fig:ROC_VMFdata}\n    \\includegraphics[width=4cm]{figures\/Hypotest_VMF_kappa50}}\n  \\subfigure[Watson Simulated Data]{\n    \\label{fig:ROC_Watsondata}\n    \\includegraphics[width=4cm]{figures\/Hypotest_Watson_kappa50}}\n  \\caption{ROC curve for detecting bi-modally distributed samples. The samples are uni-modal or bi-modal distributed from VMF (Fig.~\\ref{fig:ROC_VMFdata}) or Watson (\\ref{fig:ROC_Watsondata}) distributions with $\\kappa_o=50$. The naive K-means with ML estimator cannot cluster the samples well and estimate the mean directions accurately, resulting in poor detection which is even worse than random guessing. The modified K-means (green dashed line) performs better than K-means but is still unsatisfactory. Our proposed EM-VMF (black dots) and EM-Watson (magenta dashed line) methods have very good performance in this detection task.}\n  \\label{fig:ROC} \n\\end{figure}\n\n\\subsection{EM-ML orientation estimation for IN100 Nickel Sample}\nWe next illustrate the proposed EM-VMF and EM-Watson orientation estimators on a real IN100 sample acquired from US Air Force Research Laboratory (AFRL) \\cite{park_ebsd_2013}. The IN100 sample is a polycrystalline Ni superalloy which has cubic symmetry in the $m\\overline{3}m$ point symmetry group.  EBSD orientation measurements were acquired on a $512\\times 384$ pixel grid, corresponding to spatial resolution of $297.7$ nm. The Kikuchi diffraction patterns were recorded on a $80\\times 60$ photosensitive detector for each of the pixels. \n\nFigure \\ref{fig:IN100} (a) shows a $200\\times 200$ sub-region of the full EBSD sample where the orientations are shown in the inverse pole figure (IPF) coloring obtained from the OEM EBSD imaging software and (b) is the back-scattered electron (BSE) image. Note that the OEM-estimated orientations in some grain regions of the IPF image are very inhomogeneous, having a mottled appearance, which is likely due to a fundamental zone wrap-around problem. As an alternative, we apply a combination of the proposed EM estimators (EM-VMF or EM-Watson) and the GLRT~(\\ref{eq:multi_sample_GLRT}) with $C=2$ and significance level $\\alpha=0.05$ to detect multi-modal distributions within each OEM-segmented region. Figure \\ref{fig:IN100} (c)(e) show the estimates of the mean orientations of the regions\/sub-regions, where the sub-regions surrounded by white boundaries indicate those that have been detected as deviating from the distribution of the majority of samples from the same region. The multi-modally distributed regions may be due to undetected grains, inaccurate segmentation, or noisy orientation observations. To distinguish the latter situations from the first in which the region really consists of two grains, the misalignment\/noise test introduced in~\\cite{chen_coercive_2015} can be used. Figures \\ref{fig:IN100} (d)(f) show the estimated concentration parameter $\\kappa$ for the regions\/sub-regions. Note that the estimated $\\kappa$ are large for most of the regions\/sub-regions because those regions which have multi-modally distributed samples are detected and their concentration parameters are estimated separately for each sub-region. \n\n\n\\begin{figure}[htb]\n  \\centering\n  \\subfigure[IPF from OEM]{\n  \t\\includegraphics[width=3.2cm]{figures\/EA}}\n  \\subfigure[BSE from OEM]{\n  \t\\includegraphics[width=3.8cm]{figures\/BSE}}\n \n \n  \\subfigure[EM-VMF $\\hat{\\bmu}$]{\n  \t\\includegraphics[width=3.2cm]{figures\/Detected_Grains_VMF}}\t\n  \\subfigure[EM-VMF $\\hat{\\kappa}$]{\n  \t\\includegraphics[width=3.8cm]{figures\/Detected_Kappa_VMF}}\n \n \n  \\subfigure[EM-Watson $\\hat{\\bmu}$]{\n  \t\\includegraphics[width=3.2cm]{figures\/Detected_Grains_Watson}}\n  \\subfigure[EM-Watson $\\hat{\\kappa}$]{\n  \t\\includegraphics[width=3.8cm]{figures\/Detected_Kappa_Watson}}\n  \\caption{A $200\\times 200$ sub-region of the IN100 sample. (a) is the IPF image for the Euler angles extracted from EBSD by OEM imaging software. IPF coloring in some grains is not homogeneous, likely due to symmetry ambiguity. (b) is the BSE image of the sample. (c)(e) show the estimates of the mean orientations of the regions\/sub-regions using a combination of the proposed EM estimators, EM-VMF and EM-Watson respectively, and the GLRT~(\\ref{eq:multi_sample_GLRT}) to detect multi-modal distributions within each OEM-segmented region. The sub-regions surrounded by white boundaries indicate those that have been detected as deviating from the distribution of the majority of samples from the same region. (d)(f) show the estimated concentration parameter $\\kappa$ for the regions\/sub-regions. Note that the estimated $\\kappa$ are large for most of the regions\/sub-regions because those regions which have multi-modally distributed samples are detected and their concentration parameters are estimated separately for each sub-region.}\n\\label{fig:IN100}\n\\end{figure}\n\\section{Conclusion}\n\\label{sec:conclusion}\nA hyperbolic $\\mathcal G$-invariant von Mises-Fisher distribution was shown to be equivalent to the distribution proposed in~\\cite{chen_parameter_2015}. The advantage of the hyperbolic form is parameter estimation can be performed with substantially fewer computations. A different group invariant orientation distribution was introduced, called the $\\mathcal{G}$-invariant Watson distribution, and an EM algorithm was presented that iteratively estimates its orientation and concentration parameters. We introduced multi-modal generalizations of these $\\mathcal G$-invariant distributions using mixture models and showed that these can be used to effectively cluster populations of orientations that have spherical symmetry group invariances. The mixture of VMF and Watson models were applied to the problem of estimation of mean grain orientation parameters in polycrystalline materials whose orientations lie in the $m\\overline{3}m$ point symmetry group. Application of the finite mixture representation to other types of groups would be worthwhile future work.\n\n\n\\section*{Acknowledgment}\nThe authors are grateful for inputs from Megna Shah, Mike Jackson and Mike Groeber. AOH would like to acknowledge financial support from USAF\/AFMC grant FA8650-9-D-5037\/04 and AFOSR grant FA9550-13-1-0043. MDG would like to acknowledge financial support from AFOSR MURI grant FA9550-12-1-0458.\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n$B$-meson decays are an excellent source of information about the CKM mechanism and allow us to test our\n understanding of the CP violation. In nonleptonic $B$\ndecays we must deal with final states interactions (FSI) as well, since they may modify the values of the extracted\n parameters. It is hard to take FSI into consideration properly since\n there are a lot of possible decay channels.\n\\\\\n\nDuring the recent years several authors have investigated various possible corrections\n due to FSI. Most of the analyses take into account the elastic and inelastic effects\n arising from intermediate states containing light quarks ($u$, $d$, $s$) [1-6] and apply\n  symmetries of strong interactions (isospin, SU(3)) to reduce the number of parameters.\n  Some authors argued that intermediate states containing charmed quarks ($c$) may also play an important role [7-13].\n\\\\\n\nIn the present paper we analyse $B$ decays  into two light noncharmed pseudoscalar mesons.\n We consider FSI originating only from the intermediate states containing c quarks.\nIn sections 2 and 3 we introduce our parametrisation and relations between the amplitudes.\n Section 4 contains the description of the fit procedure and our results together with the CP\n asymmetry predictions. Finally, a short summary is given in section 5.\n\n\\section{Short-distance amplitudes}\n\n The decays of $B$ meson into two noncharmed pseudoscalar mesons are characterised by 10 $SU(3)_{f}$ invariant\n amplitudes corresponding to the specific quark-line diagrams. As in \\cite{ZL,ZL2} we use four\n dominant amplitudes: tree $T(T^{'})$, colour-suppressed $C(C^{'})$, penguin\n $P(P^{'})$ and singlet penguin $S^{'}$. Unprimed (primed) amplitudes denote\n strangeness conserving (violating) processes and are related to each other.\n Topological decompositions of decay amplitudes can be found in \\cite{ZL}.\n\\\\\n\n We use the Wolfenstein parameters: $\\lambda=0.222$,\n $A=0.832$, $\\bar{\\rho}=0.224$ and $\\bar{\\eta}=0.317$ \\cite{fl}.\n All relations bellow are calculated up to $O(\\lambda^4)$ unless explicitly written\n otherwise. Terms proportional to $\\lambda^4$ are kept on account of complex factor in $P^{'}$, which may interfere with FSI correction. We assume that all short-distance (SD) strong phases\n are negligible. For the tree amplitude we have\n\\begin{equation}\nT^{'} = \\frac{V_{us}}{V_{ud}} \\frac{f_{K}}{f_{\\pi}} T =0.278 T,\n\\end{equation}\nwhere $\\frac{f_{K}}{f_{\\pi}}$ is the SU(3) breaking factor. Both $T$ and $T^{'}$\namplitudes have a weak phase equal $\\gamma$.\nWe assume that SD penguin amplitudes are dominated by $t$ quark contribution.\nWhen terms of order $\\lambda^{4}$ are included, the strangeness violating penguin amplitude $P^{'}$ acquires a small weak\nphase. Thus, $P^{'}$ can be represented as a sum of two terms, the second one due to the $O(\\lambda^4)$ correction:\n\\begin{equation}\nP^{'} = \\frac{V_{ts}}{V_{td}} P = -(5.241+0.105 e^{i\\gamma} )|P|\n\\end{equation}\nPenguin amplitude $P$ has weak phase -$\\beta$. We used in our fits the value of $\\beta=24^{o}$ consistent with the world average.\nThe singlet penguin has the same phase as penguin $P'$:\n\\begin{equation}\nS^{'}=e^{i arg(P')}|S'|\n\\end{equation}\n\\\\\n\n\nFinally, we accept relations between the tree and colour-suppressed amplitudes:\n\\begin{equation}\nC=\\xi T,\n\\end{equation}\n\n\\begin{equation}\nC^{'}=(\\xi -(1+\\xi )\\delta_{EW} e^{-i\\gamma})T^{'}\n\\end{equation}\nwhere $\\xi=0.17$  and $\\delta_{EW}=0.65$. The last equation includes electroweak penguin\n$P_{EW}^{'}$. The EW penguin contribution $\\sim \\delta_{EW} e^{-i\\gamma}$ was calculated (see e.g. \\cite{Neu2,Neu:98})\nwithout $\\lambda^{4}$ corrections. This fact should not affect the fits much  since $P_{EW}^{'} \\sim S^{'}$ \\cite{h} and the small correction\n in $S^{'}$ is\npractically invisible in the fits (the only changes we\nobserved were in the asymmetry for the $B^{+} \\to \\eta^{'} K^{+}$ decay channel).\n\n\n\n\n\\section{Long-distance charming penguins }\nIt was argued [7-13] that the intermediate states composed of charmed mesons ($D\\bar{D}$, etc.), generated from the\n$b \\rightarrow c \\bar{c} d(s) $ tree amplitudes $T_{c}^{(')}$, may lead via rescattering to amplitudes of  \npenguin topology with an internal $c$ quark (the \"charming penguin\"). Our calculations are similar as in\nthe case of long-distance $u$-type penguins \\cite{ZL2}. Assuming SU(3) symmetry, we can redefine\npenguins:\n\\begin{equation}\nP^{(')} \\rightarrow P^{(')}+id_{c}T_{c}^{(')}\n\\end{equation}\nwhere $d_{c}$ is related to the size of the LD charming penguin and is a complex number in general.\nBecause we do not have information about $d_{c}$ (or $T_{c}^{(')}$), it is convenient to\nintroduce the following parametrisation:\n\\begin{equation}\nid_{c}T_{c}^{(')}=P^{(')}_{cLD}e^{i \\delta_{c}}\n\\end{equation}\nStrong phase $\\delta_{c}$ and size $P^{(')}_{cLD}$ of the charming penguin are additional free\nparameters\nin our fits. The weak phases are determined by the tree amplitudes $T^{(')}_{c}$ and are either $\\pi$ or 0.\nWe can eliminate $P^{'}_{cLD}$ using the relation\n\\begin{equation}\n\\frac{P^{'}_{cLD}}{P_{cLD}}=\\frac{T^{'}_{c}}{T_{c}}=\\frac{V_{cs}}{V_{cd}}=-4.388\n\\end{equation}\n\\\\\n\nShort-distance charming penguin $P_{c}^{'}$ has the same weak phase as $P_{cLD}^{'}$.\nIt can be included in a new redefined charming penguin\n\\begin{equation}\nP^{(')}_{cef}e^{i \\delta} = P^{(')}_{c}+P^{(')}_{cLD}e^{i \\delta_{c}}\n\\end{equation}\nwith new effective size and strong phase.\n\n\n\n\n\\section{Results of fits }\nWe minimise function f defined as:\n\\begin{equation}\nf =\\sum_i{\\frac{(B_i^{\\rm theor}-B_i^{\\rm exp})^2}{(\\Delta B_i^{\\rm\nexp})^2}}\n\\end{equation}\nwhere $B_i^{\\rm theor(exp)}$ denote theoretical (experimental) CP-averaged\n branching fractions\nand $\\Delta B_{i}^{\\rm exp}$ is an experimental error for i-th decay channel.\nThe sum is over all 16 decay channels as in \\cite{ZL2,chpZ}.\nExperimental branching ratios and their errors are listed in Tables 1 and 2.\nThe connection between the amplitudes and branching ratios was corrected in our calculations for the lifetime\ndifference between $B^{+}$ and $B^{0}$:\n\n\\begin{equation}\n\\frac{\\tau_{B^{+}}}{\\tau_{B^{0}}}=1.068\n\\end{equation}\n\\\\\nWe considered two sets of data. The first one was the same as in \\cite{ZL2}. The second one was used\nin \\cite{chpZ}. Data in Table 2 are more recent and differ from the previous ones in a couple of entries.\nWe performed fits in three general cases:\n\\begin{enumerate}\n\\item {without long-distance charming penguin contributions and with $|T|$, $|P|$, $|S'|$, $\\gamma$\ntreated as free parameters}\n\\item {with long-distance charming penguins described by real $P_{cef}$ as an additional parameter and\n$\\delta=0$, which is consistent with\n calculations done in \\cite{chpZ} but without any assumed connection between $P_{c}$ and $P_{t}$}\n\\item {with long-distance charming penguins described by two additional free parameters: $\\delta$, $P_{cef}$.}\n\\end{enumerate}\n\n\n\\begin{table}[h]\n\\caption{Fits to the first set of data (in units of $10^{-6}$)} \\label{tab:fit1}\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nDecay channel                 & Exp          &SD amplitudes  &\\multicolumn{3}{|c|}{Charming penguin}       \\\\\n\n\n                             &              &only   &(case 2)    &   \\multicolumn{2}{|c|}{(case3)}\\\\\n\n                            &              & (case 1)  &$\\delta=0^{o}$ &$\\gamma$ free  &$\\gamma=64.5^o$  \\\\\n\n\n\n\\hline \\hline\n$(B^+ \\to \\pi ^+ \\pi ^0)$    &$5.8\\pm 1.0$  &$5.01$  &5.65\t      &$5.73$    &$5.85$\t\t \\\\\n$(B^+ \\to K ^+ \\bar{K}^0)$   &$0.0\\pm 2.0$  &$0.68$  &0.71\t      &$2.10$    &$1.81$\t\t \\\\\n$(B^+ \\to\\pi ^+ \\eta)$       &$2.9\\pm 1.1$  &$2.15$  &1.76\t      &$2.47$    &$2.24$\t\t \\\\\n$(B^+ \\to\\pi ^+ \\eta ')$     &$0.0\\pm 7.0$  &$1.07$  &0.88\t      &$1.24$    &$1.12$\t\t \\\\\n\\hline\n$(B^0_d \\to \\pi ^+ \\pi ^- )$ &$4.7\\pm 0.5$  &$4.90$  &4.78\t      &$4.76$    &$4.75$\t\t \\\\\n$(B^0_d \\to\\pi ^0 \\pi ^0)$   &$1.9\\pm 0.7$  &$0.62$  &0.73\t      &$1.50$    &$1.36$\t\t \\\\\n$(B^0_d \\to K^+ K^-)$        &$0.0\\pm 0.6$  &$0.00$  &0 \t      &$0.00$    &$0.00$\t\t \\\\ \t\t\t    \n$(B^0_d \\to K^0 \\bar{K}^0)$  &$0.0\\pm 4.1$  &$0.62$  &0.66\t      &$1.94$    &$1.67$\t\t \\\\ \t   \t    \n\n\\hline\n$(B^+ \\to \\pi ^+ K ^0 )$     &$18.1\\pm 1.7$ &$18.40$ &19.21\t      &$18.67$   &$20.41$\t\t \\\\ \t   \n$(B^+ \\to \\pi ^0 K ^+ )$     &$12.7\\pm 1.2$ &$13.11$ &13.10\t      &$11.61$   &$10.63$\t\t \\\\ \t   \n$(B^+ \\to\\eta K^+)$          &$4.1\\pm 1.1$  &$2.46 $ &2.30\t      &$4.30 $   &$3.96$\t\t \\\\ \t   \n$(B^+ \\to\\eta ' K^+)$        &$75\\pm 7.0$   &$73.00$ &73.37\t      &$68.91$   &$69.69$\t\t \\\\ \t   \n\\hline\n$(B^0_d \\to\\pi ^- K^+)$      &$18.5\\pm 1.0$ &$18.76$ &18.60\t      &$18.38$   &$18.60$\t\t \\\\ \t   \n$(B^0_d \\to\\pi ^0 K^0)$      &$10.2\\pm 1.2$ &$6.20$  &6.57\t      &$7.76$    &$9.12$\t\t \\\\ \t   \n$(B^0_d \\to\\eta K^0)$        &$0.0\\pm 9.3$  &$1.81$  &1.79\t      &$3.19$    &$4.22$\t\t \\\\ \t   \n$(B^0_d \\to\\eta ' K^0)$      &$56\\pm 9.0$   &$66.28$ &67.36\t      &$62.35$   &$66.12$\t\t \\\\ \t   \n\\hline\n\n$|T|$&                                      &$2.60$  &2.76\t      &$2.78$    &$2.81$                 \\\\\n\n$|P|$&                                      &$0.79$  &1.45\t      &$2.59$    &$1.92$                 \\\\\n\n$|S'|$&                                     &$1.75$  &1.72\t      &$2.46$    &$3.02$                 \\\\\n\n$P_{cef}$&                                  &        &-0.77\t      &$-2.81$   &$-2.32$                \\\\\n\n$\\gamma$&                                   &$103^o$ &$94^o$\t      &$110^o$   &$64.5^o$               \\\\\n\n$\\delta$&                                   &        &$0^o$\t      &$\\pm18^o$ &$\\pm26^o$              \\\\\n\n$f_{m}$&                                    &$15.36$ &14.79\t      &$6.37$    &$9.39$   \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\\begin{table}[h]\n\\caption{Fits to the second set of data (in units of $10^{-6}$)} \\label{tab:fit2}\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nDecay channel                & Exp          &SD amplitudes  &\\multicolumn{3}{|c|}{Charming penguin}       \\\\\n\n\n                             &              &only   &(case 2)    &   \\multicolumn{2}{|c|}{(case3)}\\\\\n\n                            &              & (case 1)  &$\\delta=0^{o}$ &$\\gamma$ free  &$\\gamma=64.5^o$  \\\\\n\n\n\\hline \\hline\n$(B^+ \\to \\pi ^+ \\pi ^0)$    &$5.3\\pm 0.8$  &$4.27$  &5.32\t   &$5.05$    &$5.40$\t\t    \\\\\n$(B^+ \\to K ^+ \\bar{K}^0)$   &$0.0\\pm 2.4$  &$0.69$  &0.96\t   &$2.55$    &$1.58$\t\t    \\\\\n$(B^+ \\to\\pi ^+ \\eta)$       &$4.2\\pm 0.9$  &$2.66$  &2.04\t   &$3.04$    &$2.29$\t\t    \\\\\n$(B^+ \\to\\pi ^+ \\eta ')$     &$0.0\\pm 4.5$  &$1.33$  &1.02\t   &$1.52$    &$1.14$\t\t    \\\\\n\\hline\t\t\t\t\t\t     \t\t   \t     \n$(B^0_d \\to \\pi ^+ \\pi ^- )$ &$4.6\\pm 0.4$  &$5.09$  &4.76\t   &$4.75$    &$4.70$\t\t    \\\\\n$(B^0_d \\to\\pi ^0 \\pi ^0)$   &$1.9\\pm 0.5$  &$0.51$  &0.83\t   &$1.65$    &$1.18$\t\t    \\\\\n$(B^0_d \\to K^+ K^-)$        &$0.0\\pm 0.6$  &$0.00$  &0 \t   &$0.00$    &$0.00$\t\t    \\\\\n$(B^0_d \\to K^0 \\bar{K}^0)$  &$0.0\\pm 1.8$  &$0.64$  &0.89\t   &$2.35$    &$1.46$\t\t    \\\\\n\t\t\t\t\t\t     \t\t   \t     \n\\hline\n$(B^+ \\to \\pi ^+ K ^0 )$     &$21.8\\pm 1.4$ &$19.10$ &22.11\t   &$22.44$   &$21.57$  \t    \\\\\n$(B^+ \\to \\pi ^0 K ^+ )$     &$12.8\\pm 1.1$ &$11.97$ &12.45\t   &$10.92$   &$11.39$  \t    \\\\\n$(B^+ \\to\\eta K^+)$          &$3.2\\pm 0.7$  &$2.03 $ &1.57\t   &$2.71$    &$3.04$\t\t    \\\\\n$(B^+ \\to\\eta ' K^+)$        &$77.6\\pm 4.6$ &$74.02$ &76.18\t   &$75.27$   &$74.64$  \t    \\\\\n\\hline\t\t\t\t\t\t     \t\t   \t     \n$(B^0_d \\to\\pi ^- K^+)$      &$18.2\\pm 0.8$ &$17.57$ &18.20\t   &$19.33$   &$19.01$  \t    \\\\\n$(B^0_d \\to\\pi ^0 K^0)$      &$11.9\\pm 1.5$ &$6.86$  &8.03\t   &$9.86$    &$9.14$\t\t    \\\\\n$(B^0_d \\to\\eta K^0)$        &$0.0\\pm 4.6$  &$1.76$  &1.63\t   &$3.85$    &$3.26$\t\t    \\\\\n$(B^0_d \\to\\eta ' K^0)$      &$65.2\\pm 6.0$ &$68.66$ &72.32\t   &$73.14$   &$70.76$  \t    \\\\\n\\hline\n\n$|T|$&                                      &$2.36$  &2.68\t   &$2.61$    &$2.7$\t\t    \\\\\n\t\t\t\t\t\t     \t\t   \t     \n$|P|$&                                      &$0.83$  &2.06\t   &$2.63$    &$1.9$\t\t    \\\\\n\t\t\t\t\t\t     \t\t   \t     \n$|S'|$&                                     &$1.77$  &1.69\t   &$2.96$    &$2.61$\t\t    \\\\\n\t\t\t\t\t\t     \t\t   \t     \n$P_{cef}$&                                  &\t     &-1.45\t   &$-3.05$   &$-2.07$  \t    \\\\\n\t\t\t\t\t\t     \t\t   \t     \n$\\gamma$&                                   &$85^o$  &$68^o$\t   &$22^o$    &$64.5^o$ \t    \\\\\n$\\delta$&                                   &\t     &$0^o$\t   &$\\pm19^o$ &$\\pm26^o$\t    \\\\\n\n$f_{m}$&                                    &$27.98$ &24.73\t   &$14.97$   &$15.71$  \t   \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure}[h]\n\n\\includegraphics[angle=-90,width=0.5\\textwidth]{cv.eps}\n\\includegraphics[angle=-90,width=0.5\\textwidth]{cv2.eps}\n\n\\caption{\\small Dependence of $f_{m}(\\gamma)$ on $\\gamma$ for the first (left) and second (right) set of data.\nSolid lines denote case without charming penguin, dashed lines - case with charming  penguin and $\\delta =0$, \ndotted lines - case with charming penguin and $\\delta$ let free.}\n\n\\end{figure}\n\n\n\n\nResults of the fits are contained in Tables 1, 2. The branching fractions were calculated for the best\nfits and for the fit with fixed $\\gamma=64.5^{o}$. The minimums $f_{m}$ obtained by minimising $f$ of (10) are showed in the last rows of the tables.\nIn general, the fitted values of  $\\gamma$ are far from the standard model prediction.\nTo find out what happens one should study the dependence of the fitted function on $\\gamma$.\nLet us denote by $f_{m}(\\gamma)$ the minimum values of $f$ obtained when keeping $\\gamma$ fixed.\nThe function $f_{m}(\\gamma)$ is obtained either by setting $P_{cef}$=0, or by assuming $\\delta=0$ while letting\n $P_{cef}$ free, or by letting both  $P_{cef}$ and $\\delta$ free.\nFigure 1 shows $f_{m}(\\gamma)$ for the first (left) and second (right) set of data. The worst fits are those\nwithout charming penguins (solid lines). The minimal values were achieved for $\\gamma=103^o$ and\n$\\gamma=85^o$ respectively. For both fits with charming penguin and the strong phase $\\delta_{c}=0$ (dashed lines), the\nbest fit corresponds to $\\gamma$\nshifted down by  $9^{o}$($17^{o}$)  and a slightly lower value of $f_{m}$. In the third  case shown (dotted\n lines) $\\delta$ was let free. For the first set of data, the $f_{m}(\\gamma)$ is fairly small over the whole\n region shown $(\\gamma \\in (0,120^{o}))$.\n  For the second, more recent set of data this region is restricted to about $10^{o}-80^{o}$. Since the values of\n $f_{m}$ differ a little in the above-mentioned region we should rather think of an allowed range of $\\gamma$.\n\\\\\n\nThe values of fitted parameters $|P|$,$|S^{'}|$,$|P_{cef}|$ vary for different values of $\\gamma$.\nThe most stable are the ratio $\\frac{P_{cef}}{P}$ and the strong phase $\\delta$. $|\\frac{P_{cef}}{P}|$\nchanges from 1.1 to 1.3(1.2) only. The function $f_{m}(\\delta ,\\gamma)$ has a deep minimum around\n$ \\delta \\approx \\pm 20^{o}$ (Fig.2) for a wide range of fits with fixed $\\gamma$. Both positive and negative signs of $\\delta$ are allowed\nas the fitted function is\nsymmetric under $\\delta \\leftrightarrow -\\delta$. The fact that the ratio $|\\frac{P_{cef}}{P}|$ is close to unity  is in agreement with the calculation in \\cite{buras}. On the other hand, for the\n best fits with $\\delta=0$ the ratio $|\\frac{P_{cef}}{P}|$ is about  0.53(0.7). This value for the second set of data is higher than that assumed in \\cite{chpZ}.\n\\begin{figure}[h]\n\\begin{center}\n\n\\includegraphics[width=0.9\\textwidth]{delta_23.eps}\n\n\\caption{\\small Dependence of $f_{m}(\\gamma ,\\delta)$ on $\\delta$ for selected\nvalues of $\\gamma$, dashed (solid) lines denote the first (second) set of data.}\n\\end{center}\n\n\\end{figure}\n\n\\begin{table}[h]\n\\caption{Asymmetries generated by charming penguin for $\\delta>0$ (for $\\delta<0$ asymmetries are of opposite\nsign)}\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{|l|c|c|c|c|c|}\n\\hline\nDecay channel &  \\multicolumn{2}{|c|}{First set of data}&\\multicolumn{2}{|c|}{Second set of data} &Experiment\\\\\n              & $\\gamma$ fitted&$\\gamma=64.5^o$ &$\\gamma$ fitted&$\\gamma=64.5^o$                  &        \\\\\n\\hline \\hline\n$(B^+ \\to \\pi ^+ \\pi ^0)$ &$0$ &0              &$0$ &0&$-0.07 \\pm 0.14$\\\\\t     \t\t\t\t\t\t\t\t   \n$(B^+ \\to K ^+ \\bar{K}^0)$ &-0.93&-0.90        &-0.90&-0.98&-\\\\ \t\t     \t\t\t\t\t\t\t\t   \n$(B^+ \\to\\pi ^+ \\eta)$   &0.48&0.87            &0.47&0.76&$-0.44 \\pm 0.18 \\pm 0.01$\\\\\t\t\t\t\t\t\t\t   \n$(B^+ \\to\\pi ^+ \\eta ')$&0.48&0.87             &0.47&0.76&-\\\\\t\t\t     \t\t\t\t\t\t\t\t   \n\\hline\n$(B^+ \\to \\pi ^+ K ^0 )$&0.11&0.08             &0.04&0.07&$0.02 \\pm 0.06$\\\\\t     \t\t\t\t\t\t\t\t   \n$(B^+ \\to \\pi ^0 K ^+ )$ &-0.21&-0.28          &-0.09&-0.23&$0.00 \\pm 0.12$\\\\\t     \t\t\t\t\t\t\t\t    \n$(B^+ \\to\\eta K^+)$   &0&0                     &0&0&$-0.52 \\pm 0.24 \\pm 0.01$\\\\      \t\t\t\t\t\t\t\t   \n$(B^+ \\to\\eta ' K^+)$ &0.006&-0.004            &0.005&-0.004&$0.02 \\pm 0.042$\\\\      \t\t\t\t\t\t\t\t   \n\\hline\n$(B^0_d \\to\\pi ^- K^+)$ & -0.19&-0.24          & -0.075&-0.21&$-0.09 \\pm0.03$\\\\      \t\t\t\t\t\t\t\t   \n\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\nCharming penguins with a nonvanishing strong phase may be a source of direct CP asymmetries.\nThe predicted values were calculated for the same points as in Tables 1,2. The results are\ngiven in Table 3 together with the averages from Belle, BABAR and CLEO experiments \\cite{Rosner}.\nThe main features are large asymmetries in the $\\Delta S=0$ sector with relatively small asymmetries\nfor the $\\Delta S=1$ decays channels. We are not able to predict the absolute signs of the asymmetries\nsince we have two allowed signs of $\\delta$. The asymmetry for $(B^+ \\to \\pi ^+ K ^0 )$\nis a pure $\\lambda^{4}$ effect and shows a potential influence of this correction.\n\n\n\n\\section{Conclusions}\nOur results permit to draw the following conclusions:\n\\begin{enumerate}\n\\item{Even without the charming penguins the value of angle $\\gamma$  extracted from the fit depends on the details of data. More recent data prefer the value of $\\gamma$ \nmore in accordance with the expectations of the standard model.  }\n\\item{If we admit the non-zero value of the charming penguin (with strong phase equal zero), the fitted values of $\\gamma$ may move toward the SM value by $10^{o}-15^{o}$.}\n\\item{Admitting strong phase of the charming penguin as a free parameter leads to a relatively flat function $f_{m}(\\gamma)$ i.e. it allows a wide range of $\\gamma$.\n This means that there is probably  too much freedom in the fits. However, the fitted strong phase $\\delta$ is\n relatively stable and close to $\\pm 20^{o}$.}\n\\end{enumerate}\n\n\n{\\it Acknowledgements}. I would like to thank P.  \\.Zenczykowski for helpful discussions and comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum states can be well described in terms of Wigner functions.\nThe states with Gaussian Wigner function have been of particular\ninterests in context of quantum information processing. Entanglement\nin such states in terms of quadratures are also well studied. On the\nother hand, the quantum states with non-Gaussian Wigner function are\nalso quite important. For example, a single-photon state, which\nfinds many applications in quantum information processing,  shows\nnon-Gaussian behavior in phase-space. In a recent experiment, a\nnon-Gaussian state has been produced by homodyne detection technique\nfrom a single-mode squeezed state of light \\cite{wenger,kim}.\nFor certain non-Gaussian states, Wigner functions can take negative\nvalues. Such negativity refers to nonclassicality of these states.\nThese states are useful in entanglement distillation\n\\cite{eisert,cirac}, loophole-free tests of Bell's inequality\n\\cite{bell}, and quantum computing \\cite{sanders}. A specific class\nof such nonclassical states has been shown to be similar to the\nSchrodinger kitten state, in the sense that their Wigner functions\nshow negativity at the origin of phase space \\cite{science,polzik}.\nIt is well known that the Schrodinger cat states \\cite{knight},\nwhich are quantum superpositions of coherent states, are\nnon-classical in nature and are very important to study the interface\nof quantum and classical worlds. Superposition of coherent states\nwith low amplitudes creates Schrodinger kitten states. Most of the\nexperiments to prepare the Schrodinger cat states have been\nperformed in cavities or bound systems. Thus they are not much\nuseful in quantum information networks though they have the\nnon-Gaussian nature which is required in certain quantum\ncommunication protocols. In \\cite{science,polzik}, it has been shown\nhow to prepare an Schrodinger kitten state in an optical system, by\nsubtracting a single photon from a squeezed vacuum state. This\noptical kitten state would overcome the limitations of bound\nsystems. Repeated photon-subtractions can lead to conditional\ngeneration of arbitrary single-mode state \\cite{cerf}. We note that\nsimilar non-Gaussian states could be prepared by adding a single\nphoton to a squeezed vacuum state (see \\cite{tara,bellini} for\ndetails of photon-added coherent states, which are also non-Gaussian\nstates). These states are equivalent to single-photon subtracted\nsqueezed vacuum state and exhibit similar behavior in phase space.\nIt is worth to mention that non-Gaussian two-mode entangled states\ncan be prepared by subtracting a photon from a two-mode squeezed\nstate \\cite{sasaki,paris}.\n\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{sq_figs}Plots of Wigner functions of single-photon\nsubtracted squeezed states for (a) $r=0.31$ and (b) $r=0.8$ with\n$\\theta=0$.}\n\\end{center}\n\\end{figure*}\n\\begin{figure}\n\\caption{\\label{cond}Variation of $C$ in phase space for $r=0.31$\nand $\\theta=0$.}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\caption{\\label{ellipse}Contour plot for $C=$ constant in phase space for\n(a) $r=0.31$ and (b) $r=0.8$.}\n\\end{center}\n\\end{figure}\n\nIn this paper, we focus our study on the nonclassical properties and\ndecoherence of single-photon subtracted squeezed vacuum states which\nare optically produced single-mode non-Gaussian states. The\nstructure of the paper is as follows. In Sec. II, we introduce the\nphoton-subtracted squeezed states and discuss its nonclassical\nproperties in terms of the sub-Poissonian statistics and the\nnegativity of its Wigner function. We derive a compact expression of\nthe Wigner function and find the region in phase space where it\nbecomes negative. We show that there is an upper bound of the\nsqueezing parameter for this state to exhibit sub-Poissonian\nstatistics. In Sec. III, we study the effects of two different model\nof decoherence: photon-number decay and phase damping. In both\ncases, we derive analytical expressions for the time-evolution of\nthe state and its Wigner function. We discuss the loss of\nnonclassicality due to decoherence. We show through the study of\nevolution of the Wigner function how the state decays to vacuum as a\nresult of photon-number decay. We further show that phase damping\nleads to much slower decoherence than the photon-number decay.\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics{fig5.eps}}\n\\end{center}\n\\caption{\\label{Qdecay}Variation of the $Q$-parameter\nwith time in presence of decoherence due to decay of\nphoton for squeezing parameters $r=0.31$ (red line) and $r=0.8$\n(green line).}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\\label{negative_time}Variation of $P\/(a^2-4|c|^2)^{5\/2}$ in\nphase space for $r=0.31$, $\\theta=0$, and $\\kappa t=0.1$.}\n\\end{figure}\n\n\\section{Photon-subtracted squeezed states}\nAn unnormalized single-mode squeezed vacuum state is given by\n\\begin{equation}\n|\\psi\\rangle\\equiv\nS(\\zeta)|0\\rangle\\;,\\;S(\\zeta)=\\exp\\left[\\frac{\\zeta}{2} a^{\\dag\n2}-\\frac{\\zeta^*}{2} a^2\\right]\\;,\n\\end{equation}\nwhere $S(\\zeta)$ is the squeezing operator, $\\zeta=re^{i\\theta}$ is\nthe complex squeezing parameter, and $a$ is the annihilation\noperator. The Wigner function of this state is Gaussian and positive\nin phase space. When $p~(>0)$ number of photons are subtracted from\nsuch states, the state can be written as\n\\begin{equation}\n|\\psi\\rangle_p \\equiv a^p S(\\zeta)|0\\rangle \\equiv\na^p\\exp\\left[\\frac{\\xi}{2}a^{\\dag 2}\\right]|0\\rangle\\;,\n\\end{equation}\nwhere $\\xi=\\tanh(\\zeta)$. For odd $p=2m+1$, the normalized form of\nthis state can be written as\n\\begin{equation}\n\\label{odd}|\\psi\\rangle_p =\\frac{1}{N_o}\\sum_{s=0}^\\infty\n\\frac{(\\xi\/2)^{s+m+1}}{(s+m+1)!}\\frac{(2s+2m+2)!}{\\sqrt{2s+1}}|2s+1\\rangle\\;,\n\\end{equation}\nwhile for even $p=2m$, the state becomes\n\\begin{equation}\n\\label{even}|\\psi\\rangle_p=\\frac{1}{N_e}\\sum_{s=0}^\\infty\n\\frac{(\\xi\/2)^{s+m}}{(s+m)!}\\frac{(2s+2m)!}{\\sqrt{2s}}|2s\\rangle\\;,\n\\end{equation}\nwhere $N_o$ and $N_e$ are the normalization constants. In this\npaper, we focus on the case when a single photon is subtracted from\nthe squeezed vacuum, i.e., for $m=0$ in (\\ref{odd}).\n\\begin{figure}\n\\begin{center}\n\\caption{\\label{ellipse1}Plot of $16C'\/(1-e^{-2\\kappa t})$ in phase\nspace for (a) $r=0.31$ and (b) $r=0.8$ at times (i) $\\kappa t=0.05$,\n(ii) $\\kappa t=0.1$, (iii) $\\kappa t=0.3$, and (iv) $\\kappa t=0.5$.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{small_sq}Wigner function of single-photon\nsubtracted squeezed states for $\\theta=0$ and $r=0.31$ at (a)\n$\\kappa t=0.05$, (b) $\\kappa t=0.1$, (c) $\\kappa t=0.3$, and (d)\n$\\kappa t=0.5$.}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{large_sq}Wigner function of single-photon\nsqueezed states for $\\theta=0$ and $r=0.8$ at (a) $\\kappa t=0.1$,\n(b) $\\kappa t=0.3$, (c) $\\kappa t=0.5$, and (d) $\\kappa t=0.7$.}\n\\end{center}\n\\end{figure*}\n\\subsection{Negativity of the Wigner function}\nThe Wigner function of the squeezed vacuum state $|\\psi\\rangle$ is\ngiven by\n\\begin{equation}\n\\label{w_sq}W_{\\rm\nsq}(\\alpha,\\alpha^*)=\\frac{2}{\\pi}\\exp(-2|\\tilde{\\alpha}|^2)\\;,\n\\end{equation}\nwhere $\\tilde{\\alpha}=\\alpha\\cosh(r)-\\alpha^*e^{i\\theta}\\sinh(r)$.\nThe function is Gaussian in phase space, as shown in Fig.\n\\ref{vacuum}. We now calculate the Wigner function of the state\n$|\\psi\\rangle_p$. This state can be rewritten as\n\\begin{equation}\n\\label{rel}|\\psi\\rangle_p =a^p S(\\zeta)|0\\rangle=S(\\zeta)S^\\dag\n(\\zeta)a^pS(\\zeta)|0\\rangle\\;.\n\\end{equation}\nUsing the relation\n\\begin{equation}\nS^\\dag(\\zeta)a^pS(\\zeta)=[\\cosh(r)a+e^{i\\theta}\\sinh(r)a^{\\dag}]^p\\;,\n\\end{equation}\nwe get the following from (\\ref{rel})\n\\begin{equation}\n\\label{state}|\\psi\\rangle_1\\equiv S(\\zeta)|1\\rangle\\;,\n\\end{equation}\nfor $p=1$. For a density matrix $\\tilde{\\rho}(a,a^\\dag)$, we can\nwrite the following:\n\\begin{equation}\n\\label{relation}S(\\zeta)\\tilde{\\rho}(a,a^\\dag)S^\\dag\n(\\zeta)=\\tilde{\\rho}[S(\\zeta)aS^\\dag (\\zeta),S(\\zeta)a^\\dag\nS^\\dag(\\zeta)]\\;.\n\\end{equation}\nWe write the Wigner function of $\\tilde{\\rho}(a,a^\\dag)$ as\n$W_{\\tilde{\\rho}}(\\alpha,\\alpha^*)$. Using the identities\n\\begin{eqnarray}\nS(\\zeta)aS^\\dag (\\zeta) &=&a\\cosh(r)-a^\\dag e^{i\\theta}\\sinh(r)\\;,\\\\\nS(\\zeta)a^\\dag S^\\dag(\\zeta) &=& a^\\dag\n\\cosh(r)-ae^{-i\\theta}\\sinh(r)\\;,\n\\end{eqnarray}\nin (\\ref{relation}), we can thus write the Wigner function of the\ndensity matrix $\\rho(a,a^\\dag)=S(\\zeta)\\tilde{\\rho}(a,a^\\dag)S^\\dag\n(\\zeta)$ as\n\\begin{equation}\n\\label{formula}W_{\\rho}(\\alpha,\\alpha^*)=W_{\\tilde{\\rho}}(\\tilde{\\alpha},\\tilde{\\alpha}^*)\\;,\n\\end{equation}\nwhere we have used the linearity property of the Wigner function.\nFor the state (\\ref{state}), $\\tilde{\\rho}=|1\\rangle\\langle 1|$ and\nits Wigner function is given by\n\\begin{equation}\n\\label{wsq}W_{\\tilde{\\rho}}(\\alpha,\\alpha^*)=\\frac{2}{\\pi}(4|\\alpha|^2-1)e^{-2|\\alpha|^2}\\;.\n\\end{equation}\nThus, the Wigner function of the single-photon subtracted squeezed\nvacuum state becomes\n\\begin{equation}\n\\label{wig1}W_\\rho(\\alpha,\\alpha^*)=\\frac{2}{\\pi}(4|\\tilde{\\alpha}|^2-1)e^{-2|\\tilde{\\alpha}|^2}\\;,\n\\end{equation}\nwhere we have used (\\ref{formula}) and (\\ref{wsq}). Clearly, the\nWigner function (\\ref{wig1}) is non-Gaussian in phase-space. We show\nthe plot of this Wigner function in the phase-space in Figs.\n\\ref{sq_figs} for different squeezing parameters. As an evidence of\nnon-classicality of the state, squeezing in one of the quadratures\nis clear in the plots. Also there is some negative region of the\nWigner function in the phase-space which is another evidence of the\nnon-classicality of the state. The function becomes negative in\nphase space, when\n\\begin{equation}\n\\label{neg_cond}|\\tilde{\\alpha}|^2<\\frac{1}{4}\\;.\n\\end{equation}\nWe show in Fig. \\ref{cond} the variation of\n$C=|\\tilde{\\alpha}|^2-\\frac{1}{4}$ in phase space. The negative\nregion of $C$ corresponds to the negativity of the Wigner function.\nNote that $C=$ constant corresponds to ellipse in phase space, as\nshown in Fig. \\ref{ellipse}.\n\nNote that the photon-subtracted squeezed states are similar to\nSchrodinger kitten states \\cite{science,polzik} because Wigner functions\nexhibit the same characteristics in phase-space especially for the\nlarge values of the squeezing parameters. Moreover, in both cases,\nWigner function becomes negative in the center of phase space.\n\n\\subsection{Sub-Poissonian nature of the photon-subtracted state}\nThe nonclassicality of the state $|\\psi\\rangle_1$ can also be\nanalyzed by studying its sub-Poissonian character in terms of the\nMandel's $Q$-parameter \\cite{mandel} which is defined by\n\\begin{equation}\n\\label{Qpara}Q=\\frac{\\langle a^{\\dag 2}a^2\\rangle-\\langle a^\\dag\na\\rangle^2}{\\langle a^\\dag a\\rangle}\\;.\n\\end{equation}\nThe negativity of the $Q$-parameter refers to sub-Poissonian\nstatistics of the state. However in \\cite{tara_Q}, it has been shown\nthat a state can be nonclassical even if $Q$ is positive. A similar\nsituation occurs in the present case. For the state (\\ref{odd}), we\nfind\n\\begin{eqnarray}\n\\langle a^{\\dag\n2}a^2\\rangle &=&\\sum_{n=0}^\\infty n(n-1)\\rho_{n,n}=\\frac{3|\\xi|^4(3+2|\\xi|^2)}{N_o^2(1-|\\xi|^2)^{7\/2}}\\;,\\nonumber\\\\\n\\label{avg1}\\langle a^\\dag a\\rangle &=&\\sum_{n=0}^\\infty\nn\\rho_{n,n}=\\frac{|\\xi|^2(1+2|\\xi|^2)}{N_o^2(1-|\\xi|^2)^{5\/2}}\\;,\n\\end{eqnarray}\nwhere the normalization constant is given by\n\\begin{equation}\nN_o^2=\\frac{|\\xi|^2}{(1-|\\xi|^2)^{3\/2}}\\;.\n\\end{equation}\nFrom (\\ref{avg1}), we find that $Q$ becomes negative for\n$|\\xi|\\lesssim 0.43$, which is satisfied for $r\\lesssim 0.46$. We\nemphasize that the Wigner function has negative region for all\nvalues of $r$, and thus the photon-subtracted squeezed state is\nnonclassical for all $r$, though it does not exhibit sub-Poissonian\nphoton statistics above certain squeezing threshold.\n\n\\section{Models of decoherence}\nWe next consider how this state evolves under decoherence. The\ndecoherence of the single-mode state (\\ref{odd}) can be due to decay\nof photons to the reservoir or due to phase damping.\n\n\\subsection{Amplitude decay model}\nWhen the photons decay to reservoir, the corresponding Markovian\ndynamics of the state is well described by the following equation:\n\\begin{equation}\n\\frac{d}{dt}\\rho=-\\kappa(a^\\dag a\\rho-2a\\rho a^\\dag+\\rho a^\\dag a)\\;,\n\\end{equation}\nwhere $\\kappa$ is the rate of decay. The solution of this equation\ncan be written as\n\\begin{equation}\n\\label{sol}\\rho(t)=\\sum_{n,n'}\\rho_{n,n'}(t)|n\\rangle\\langle n'|\\;,\n\\end{equation}\nwhere the density matrix element $\\rho_{n,n'}(t)$ can be found by\nusing the Laplace transformation and the iteration methods\n\\cite{gsa_po}. To see this, let us start with the time-dependent\nequation for $\\rho_{n,n'}$:\n\\begin{equation}\n\\dot{\\rho}_{n,n'}=-\\kappa(n+n')\\rho_{n,n'}+2\\kappa\\sqrt{(n+1)(n'+1)}\\rho_{n+1,n'+1}\\;.\n\\end{equation}\nUsing the new subscripts $q=n-n'$ and $p=(n+n')\/2$, the above\nequation transforms into\n\\begin{equation}\n\\label{pq}\\dot{\\rho}_{p,q}=-2\\kappa p\\rho_{p,q}+2\\kappa\n\\sqrt{(p+1)^2-(q\/2)^2}\\rho_{p+1,q}\\;.\n\\end{equation}\nTaking Laplace transformation of (\\ref{pq}) and using the original\nsubscript $n$ and $n'$, we can write the time-dependent solution for\nthe density matrix elements as\n\\begin{eqnarray}\n\\rho_{n,n'}(t)&=&e^{-\\kappa\nt(n+n')}\\sum_{r=0}^\\infty\\sqrt{\\left(^{n+r}C_r\\right)\\left(^{n'+r}C_r\\right)}\\nonumber\\\\\n\\label{rhonn}&&\\times (1-e^{-2\\kappa t})^r\\rho_{n+r,n'+r}(t=0)\\;,\n\\end{eqnarray}\nwhere for the single-photon subtracted squeezed vacuum\n\\begin{eqnarray}\n\\rho_{n+r,n'+r}(t=0)&=&\\frac{1}{N_o^2}\\frac{(\\xi\/2)^{(n+r+1)\/2}(\\xi^*\/2)^{(n'+r+1)\/2}}{(\\frac{n+r+1}{2})!(\\frac{n'+r+1}{2})!}\\nonumber\\\\\n&&\\label{init}\\times\\frac{(n+r+1)!(n'+r+1)!}{\\sqrt{(n+r)!(n'+r)!}}\\;.\n\\end{eqnarray}\n\nWe next calculate the parameter $Q$ [Eq. (\\ref{Qpara})] for the\nstate (\\ref{sol}). We have found that\n\\begin{eqnarray}\n\\langle a^{\\dag 2}a^2\\rangle &=&\\sum_{n=0}^\\infty\nn(n-1)\\rho_{n,n}(t)\\;,\\nonumber\\\\\n\\label{avg2}\\langle a^\\dag a\\rangle &=&\\sum_{n=0}^\\infty n\\rho_{n,n}(t)\\;,\n\\end{eqnarray}\nwhere $\\rho_{n,n}(t)$ is given by (\\ref{rhonn}) and (\\ref{init}) for\n$n=n'$ in case of state $|\\psi\\rangle_1$. Using Eqs. (\\ref{avg2}), we\nplot $Q$ with time in Fig. \\ref{Qdecay}. It is easy to see that\nat long times ($\\kappa t\\rightarrow \\infty$), $Q$ vanishes. This is\nbecause at this limit, $\\rho_{n,n}(t)$ vanishes for all non-zero $n$\nand $\\rho_{0,0}(t\\rightarrow \\infty)=1$, i.e., the state decays to\nvacuum. Thus the averages (\\ref{avg2}) vanish and $Q$ also vanishes.\n\n\\subsubsection{Evolution of Wigner function}\nThe evolution of the Wigner function is governed by the following\nequation:\n\\begin{equation}\n\\label{wignereq}\\frac{\\partial W}{\\partial\nt}=\\kappa\\left[\\frac{\\partial}{\\partial\n\\alpha}\\alpha+\\frac{\\partial}{\\partial\n\\alpha^*}\\alpha^*+\\frac{\\partial^2}{\\partial\n\\alpha\\partial\\alpha^*}\\right]W(\\alpha,\\alpha^*)\\;.\n\\end{equation}\nThe solution can be written as\n\\begin{eqnarray}\n\\label{wignersol}W(\\alpha,\\alpha^*,t)&=&\\frac{2}{\\pi(1-e^{-2\\kappa\nt})}\\int\nd^2\\alpha_0W(\\alpha_0,\\alpha_0^*,0)\\nonumber\\\\\n&&\\exp\\left\\{-2\\frac{|\\alpha-\\alpha_0e^{-\\kappa\nt}|^2}{(1-e^{-2\\kappa t})}\\right\\}\\;,\n\\end{eqnarray}\nwhere $W(\\alpha_0,\\alpha_0^*,0)$ is the Wigner function of the\ninitial state. It is easy to verify this solution putting\n(\\ref{wignersol}) in (\\ref{wignereq}). The time-evolution of the\nWigner function of the squeezed vacuum state $|\\psi\\rangle$ can be\neasily calculated analytically  using the following integral\nidentity \\cite{gsa_prdeq,puri}:\n\\begin{eqnarray}\n\\label{ident}&&\\int\nd^2\\alpha\\exp[-|\\alpha|^2]\\exp\\left(-\\frac{\\mu}{\\tau}\\alpha^2-\\frac{\\nu}{\\tau}\\alpha^{*2}-\\frac{z^*\\alpha}{\\sqrt{\\tau}}+\\frac{z\\alpha^*}{\\sqrt{\\tau}}\\right)\\nonumber\\\\\n&&=\\frac{\\pi\\tau}{\\sqrt{\\tau^2-4\\mu\\nu}}\\exp\\left(-\\frac{\\mu\nz^2+\\nu z^{*2}+\\tau|z|^2}{\\tau^2-4\\mu\\nu}\\right)\\;.\n\\end{eqnarray}\nUsing Eq. (\\ref{wsq}) and the above identity in Eq.\n(\\ref{wignersol}), we get the following:\n\\begin{eqnarray}\nW(\\alpha,\\alpha^*,t)&=&\\frac{4}{\\pi(1-e^{-2\\kappa t})}\n\\frac{\\exp\\left[-\\frac{2|\\alpha|^2}{1-e^{-2\\kappa t}}\\right]}{\\sqrt{a^2-4|c|^2}}\\nonumber\\\\\n&&\\times\\exp\\left[\\frac{b^2c^*+b^{*2}c+a|b|^2}{a^2-4|c|^2}\\right]\\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na&=&2\\cosh(r)+2\\frac{e^{-2\\kappa t}}{1-e^{-2\\kappa t}}\\;,\\nonumber\\\\\nb&=&\\frac{2\\alpha^*e^{-\\kappa t}}{1-e^{-2\\kappa\nt}}\\;,\\;c=e^{-i\\theta}\\sinh(r)\\;.\n\\end{eqnarray}\nClearly the Wigner function of the squeezed vacuum state is Gaussian\nat all times.\n\nWe now calculate the time-dependence of the Wigner function of the\nstate $|\\psi\\rangle_1$. The initial Wigner function, as given by\n(\\ref{wig1}), can be rewritten as\n\\begin{equation}\n\\label{alter_W}W(\\alpha,\\alpha^*,0)=\\frac{2}{\\pi}D\\left.\\left[e^{-\\lambda|\\tilde{\\alpha}|^2}\\right]\\right|_{\\lambda=2}\\;,\\;D=-4\\frac{d}{d\\lambda}-1\\;.\n\\end{equation}\nUsing (\\ref{alter_W}) and (\\ref{wignersol}), we can find the\nfollowing expression for the Wigner function:\n\\begin{eqnarray}\nW(\\alpha,\\alpha^*,t)&=&\\left(\\frac{2}{\\pi}\\right)^2\\frac{1}{1-e^{-2\\kappa\nt}}D\\int\nd^2\\alpha_0\\exp[-\\lambda|\\tilde{\\alpha}_0|^2]\\nonumber\\\\\n&&\\times \\left.\\exp\\left[-2\\frac{|\\alpha-\\alpha_0e^{-\\kappa\nt}|^2}{1-e^{-2\\kappa t}}\\right]\\right|_{\\lambda=2}\\;,\n\\end{eqnarray}\nwhere\n$\\tilde{\\alpha}_0=\\alpha_0\\cosh(r)-\\alpha_0^*e^{i\\theta}\\sinh(r)$.\nSimplifying the above expression using Eq. (\\ref{ident}), we get\n\\begin{eqnarray}\nW(\\alpha,\\alpha^*,t)&=&\\frac{32P}{\\pi(1-e^{-2\\kappa\nt})}\\frac{\\exp\\left[-\\frac{2|\\alpha|^2}{1-e^{-2\\kappa\nt}}\\right]}{(a^2-4|c|^2)^{5\/2}}\\nonumber\\\\\n&&\\label{timeW}\\times\\exp\\left[\\frac{b^2c^*+b^{*2}c+a|b|^2}{a^2-4|c|^2}\\right]\\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nP&=&(1-x^2)\\{\\sinh(2r)(e^{i\\theta}b^2+e^{-i\\theta}b^{*2})\\nonumber\\\\\n&&+2[(x+1)^2+4x\\sinh^2(r)]\\}\\nonumber\\\\\n\\label{P}&&+2\\{(1+x^2)\\cosh(2r)+2x\\}|b|^2\n\\end{eqnarray}\nand\n\\begin{equation}\nx=\\frac{e^{-2\\kappa t}}{1-e^{-2\\kappa t}}\\;.\n\\end{equation}\nClearly, the Wigner function is non-Gaussian due to the presence of\nthe polynomial $P$. This becomes negative when the polynomial $P$\nbecomes negative. In Fig. \\ref{negative_time}, we have plotted\n$C'(\\alpha,\\alpha^*,t)=P\/(a^2-4|c|^2)^{5\/2}$ in phase space to show\nthe negative region for Wigner function.\n\nNote that at the center of the phase space ($\\alpha= \\alpha^*=0$),\nthe Wigner function is maximally negative. At the center,\n\\begin{equation}\nC'(0,0,t)=\\frac{2(1-x^2)\\{(x+1)^2+4x\\sinh^2(r)\\}}{(a^2-4|c|^2)^{5\/2}}\\;,\n\\end{equation}\nwhich becomes negative when $(1-x^2)$ becomes negative. This leads\nto the following condition:\n\\begin{equation}\n\\kappa t<\\kappa t_0=\\frac{1}{2}\\ln(2)\\;,\n\\end{equation}\nwhich is independent of the squeezing parameter $r$. Thus the Wigner\nfunction has certain negative region for the time\n$t<t_0=\\ln(2)\/2\\kappa$. However the negative value does depend on\nthe squeezing parameter.\n\nWe find from Eq. (\\ref{P}) that at $t=t_0$ (i.e, when $x^2=1$), $P$\nbecomes a circle in phase space. Thus beyond $t>t_0$, the ellipse\n$16C'\/(1-e^{-2\\kappa t})=$ constant interchanges its minor and major\naxes. We show this behavior in Figs. \\ref{ellipse1} for different\nvalues of $r$. Note that at times much larger than decoherence\ntime-scale $1\/\\kappa$ (i.e., for $\\kappa t\\rightarrow \\infty$),\n$P\\rightarrow 2$ and thus becomes constant throughout the phase\nspace.\n\nUsing Eq. (\\ref{timeW}), we show the variation of Wigner function at\ndifferent time-scales in Figs. \\ref{small_sq}. It is easy to see how\nthe negative region of the Wigner function gradually diminishes. At\nlong times $\\kappa t\\rightarrow \\infty$, the Wigner function becomes\n\\begin{equation}\nW(\\alpha,\\alpha^*,\\infty)= \\frac{2}{\\pi}e^{-2|\\alpha|^2}\\;,\n\\end{equation}\nwhich corresponds to vacuum state. We have shown this in Fig.\n\\ref{vacuum}(b). This can also be understood from Eq. (\\ref{rhonn}).\nFor $\\kappa t\\rightarrow \\infty$, $\\rho_{0,0}$ approaches unity,\nwhereas all other density matrix elements vanish. This means that at\nlong times, the state decays to vacuum, as we have discussed\nearlier.\n\nWe next study the time-evolution of the Wigner function for the case\nof large squeezing, i.e., large values of $\\zeta$. In this case the\nsingle photon subtracted squeezed state becomes similar to a\nSchrodinger cat state. For large times, such an optical cat state\ndecays to vacuum. Thus the Wigner function becomes Gaussian, as\ndiscussed above. We show this evolution for large squeezing in Figs.\n\\ref{large_sq}.\n\n\\subsection{Effect of phase damping}\nWe now study the effect of phase-damping on the state\n$|\\psi\\rangle_1$. Such damping can be described by the following\nmaster equation:\n\\begin{equation}\n\\dot{\\rho}=-\\kappa_p(A^{\\dag}A\\rho-2A\\rho A^\\dag+\\rho A^\\dag A)\\;,\n\\end{equation}\nwhere $A=a^\\dag a$ is the number operator and $\\kappa_p$ is the\ncorresponding rate of decoherence. The solution of this equation can\nbe easily found as (\\ref{sol}) where\n\\begin{equation}\n\\label{sol_phase}\\rho_{n,n'}(t)=\\exp[-(n-n')^2\\kappa_p\nt]\\rho_{n,n'}(0)\\;.\n\\end{equation}\nIt is easy to see that only the diagonal elements $\\rho_{n,n}$ do\nnot decay due to dephasing. Thus at long times, we can write\n\\begin{equation}\n\\rho(t\\rightarrow \\infty)=\\sum_{n=0}^\\infty\n\\rho_{n,n}(0)|n\\rangle\\langle n|\\;,\n\\end{equation}\nwhich refers to a mixed state.\n\nUsing Eqs. (\\ref{avg2}) and (\\ref{sol_phase}), we next calculate the\nparameter $Q$. We find that the averages\n$\\langle a^{\\dag 2}a^2\\rangle$ and $\\langle a^\\dag a\\rangle$ do not\ndepend upon time, because in case of phase damping\n$\\rho_{n,n}(t)=\\rho_{n,n}(0)$. Thus $Q$ remains the same for all\ntimes.\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{wig_phase_fig}Wigner function in phase\nspace at long times in presence of phase damping for (a) $r=0.31$\nand (b) $r=0.8$.}\n\\end{center}\n\\end{figure*}\nHowever, the corresponding Wigner function has certain\ntime-dependence. We find that at long times, the Wigner function\nbecomes\n\\begin{equation}\n\\label{wigner_phase}W(\\alpha,\\alpha^*,\\infty)=\\sum_{n=0}^\\infty\n\\rho_{n,n}(0)W_{|n\\rangle\\langle n|}(\\alpha,\\alpha^*)\\;,\n\\end{equation}\nwhere $W_{|n\\rangle\\langle n|}(\\alpha,\\alpha^*)$ is the Wigner\nfunction of a Fock state $|n\\rangle$ as given by\n\\begin{equation}\nW_{|n\\rangle\\langle\nn|}(\\alpha,\\alpha^*)=(-1)^n\\frac{2}{\\pi}e^{-2|\\alpha|^2}L_n(4|\\alpha|^2)\\;.\n\\end{equation}\nThe function (\\ref{wigner_phase}) refers to a highly nonclassical\nstate. It is interesting to note that all the Fock states have\nindependent contributions to the Wigner function at long times,\nweighted by their initial population $\\rho_{n,n}(0)$. On the other\nhand, in case of decoherence due to photon-number decay, only the\nvacuum state survives. In Fig. \\ref{wig_phase_fig}, we plot the\nWigner function (\\ref{wigner_phase}) in phase space for different\nsqueezing. Note that the Wigner function has negative region at long\ntimes representing nonclassicality for all $r$, even if the state\ndoes not exhibit sub-Poissonian statistics for $r\\gtrsim 0.46$\n(because $Q$ is positive). In fact, if $Q$ is positive, it does not\nmean that the the state is classical. In such cases, we have to use\nother parameters to test the nonclassicality. Several parameters\nhave been introduced in this context \\cite{tara_Q,nonclass}. We can use\nhierarchy of these parameters which have been shown to be especially\nuseful in context of cat states. Here we illustrate the utility of one such\nparameter, e.g., the $A_3$ parameter as defined by \\cite{tara_Q}\n\\begin{equation}\n\\label{a3}A_3=\\frac{{\\rm det}[m^{(3)}]}{{\\rm det}[\\mu^{(3)}]-{\\rm\ndet}[m^{(3)}]}\\;,\n\\end{equation}\nwhere\n\\begin{equation}\nm^{(3)}=\\left(\\begin{array}{ccc} 1&m_1&m_2\\\\\nm_1&m_2&m_3\\\\\nm_2&m_3&m_4\n\\end{array}\\right)\\;,\\;\\mu^{(3)}=\\left(\\begin{array}{ccc} 1&\\mu_1&\\mu_2\\\\\n\\mu_1&\\mu_2&\\mu_3\\\\\n\\mu_2&\\mu_3&\\mu_4\n\\end{array}\\right)\\;,\n\\end{equation}\n$m_s=\\langle a^{\\dag s}a^s\\rangle$, $\\mu_s=\\langle (a^\\dag\na)^s\\rangle$ and det indicates determinant of the matrix. The state exhibits phase-insensitive nonclassical\nproperties if $A_3$ lies between 0 and -1 \\cite{tara_Q}. For the\nstate $|\\psi\\rangle_1$ we have found that $A_3$ remains negative for\n$|\\xi|\\lesssim 0.6$ which corresponds to $r\\lesssim 0.7$. Clearly\n$A_3$ is a stronger measure of nonclassicality than $Q$ because it\nleads to a larger upper bound of $r$ to exhibit nonclassicality.\nFurther, comparing the Wigner functions in Figs. \\ref{wig_phase_fig} with those at $t=0$ [see\nFigs. \\ref{sq_figs}], we find that the Wigner function varies very\nslowly with time for small squeezing. But for large squeezing, the\nvariation is faster. Although we can conclude that phase damping\nleads to much slower decoherence than amplitude damping.\n\n\\section{Conclusions}\nIn conclusion, we have studied how a class of non-Gaussian states\nevolves in presence of decoherence. We have considered a\nsingle-photon subtracted squeezed vacuum state, the Wigner function\nof which is similar to that of a Schrodinger kitten state. We have\nfound an upper bound for squeezing parameter for which this state\nexhibits sub-Poissonian photon statistics. However, the state\nremains nonclassical for all values of the squeezing parameter\nbecause the Wigner function becomes negative around central region\nin phase space. Next, we have studied how the state evolves in\npresence of two different kinds of decoherence, viz., amplitude\ndecay and phase damping. We have found analytical expressions for\nthe time-evolution of the state and the Wigner function in both\ncases. In case of amplitude decay, the Wigner function loses its\nnon-Gaussian nature and becomes Gaussian at long times,\ncorresponding to vacuum. On the other hand, phase damping leads to\nmuch slower decoherence than amplitude damping. The state remains\nnonclassical at long times.\n\n\\begin{acknowledgments}\nA.B. gratefully acknowledges the partial support from the Women in\nScience and Engineering program in University of Southern\nCalifornia, Los Angeles, USA. G.S.A. kindly acknowledges support\nfrom NSF grant no. CCF0524673.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThis  paper continues  the  series on  spectroscopic  orbits of  stars\nbelonging to hierarchical  systems \\citep{paper1,paper2,paper3}. It is\nmotivated by  the need  to improve statistics  of orbital  elements in\nstellar  hierarchies. Statistics will  inform us  on the  processes of\ntheir formation  and dynamical evolution, as outlined  in the previous\npapers  of  this  series.    This  work  augments  the  collection  of\nobservational data  on stellar  hierarchies assembled in  the multiple\nstar catalog \\citep[MSC;][]{MSC}.\n\n\n\\begin{deluxetable*}{c c rr   l cc rr r c }\n\\tabletypesize{\\scriptsize}     \n\\tablecaption{Basic parameters of observed multiple systems\n\\label{tab:objects} }  \n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{WDS} & \n\\colhead{Comp.} &\n\\colhead{HIP} & \n\\colhead{HD} & \n\\colhead{Spectral} & \n\\colhead{$V$} & \n\\colhead{$V-K$} & \n\\colhead{$\\mu^*_\\alpha$} & \n\\colhead{$\\mu_\\delta$} & \n\\colhead{RV} & \n\\colhead{$\\overline{\\omega}$\\tablenotemark{a}} \\\\\n\\colhead{(J2000)} & \n & &   &  \n\\colhead{Type} & \n\\colhead{(mag)} &\n\\colhead{(mag)} &\n\\multicolumn{2}{c}{ (mas yr$^{-1}$)} &\n\\colhead{(km s$^{-1}$)} &\n\\colhead{(mas)} \n}\n\\startdata\n04125$-$3609 &A & 19639 & 26758 & F3V     & 7.12 & 1.12 & 61   &   24  & 35.40    & 7.94 \\\\\n             &B & 19646 & 26773 & F2IV    & 7.91 & 0.92 & 61   &   24  & 35.98    & 8.03 \\\\\n12059$-$4951 &A & \\ldots & 105080 & G3V   & 9.13 & 1.43 & 25   &$-$15  & 50.19    & 9.99 \\\\\n             &B & \\ldots & 105081 & G0V   & 9.18 & 1.42 & 31   &$-$20  & 50.09    & 7.20 \\\\\n12283$-$6146 &A & 60845  & 108500 & G3V   & 6.82 & 1.64 & 71   &$-$160 & 40.02    & 19.93 \\\\\n             &D & \\ldots & \\ldots & \\ldots& 13.70 & 4.54& 73   &$-$169 & \\ldots    & 19.91 \\\\\n12404$-$4924 &A & 61840  & 110143 & G0V & 7.60 & 2.00 & $-$28 & $-$112 & 6.76     & 18.59 \\\\  \n15275$-$1058 &A & 75663  & 137631 & G0    & 8.14  & 1.35&$-$65 &$-$35  & $-$56.0 v & 9.29 \\\\\n             &B & \\ldots & \\ldots & G0    & 9.21  & 1.50&$-$61 &$-$35  & $-$56.82 & 7.69 \\\\\n15410$-$1449 &A & 76816  & 139864 & F8V   & 9.47  & 1.62&$-$26 &$-$1   & $-$38.94 & 3.23 \\\\\n             &B & \\ldots & \\ldots & \\ldots& 9.74 & 2.51&$-$25  &$-$2   & $-$50.9 v & 3.15 \\\\  \n15577$-$3915 &A & 78163  & 142728 & G3V   & 9.04 & 1.54& 17    & 6     & 9.41     & 10.42 \\\\\n             &B & \\ldots & \\ldots & \\ldots&10.30 & \\ldots& 31  & 4     & 6.78 v     & 13.57 \\\\\n16005$-$3605 &A & 78416  & 143215 & G1V   &8.65 & 1.32 &$-$26  &$-$41  & 1.60     & 9.33 \\\\\n             &B & \\ldots & \\ldots & \\ldots&9.32 & 1.31 &$-$28  &$-$41  & 1.43     & 9.31 \\\\\n16253$-$4909 &AB& 80448  & 147633 & G2V   & 7.5? &2.3? &$-$95  &$-$94  & $-$2.08  & 19.66 \\\\\n17199$-$1121 &A & 84789  & 156769 & F2    & 9.11 & 1.37&   6   & 13    & 5.62    & 5.33 \\\\\n             &B & \\ldots & \\ldots & \\ldots& 9.89 & 1.37&  5    & 12    & 5.97    & 5.34    \n\\enddata\n\\tablenotetext{a}{Proper motions and parallaxes are \n  from the {\\it Gaia} DR2 \\citep{Gaia,Gaia1}.}\n\\end{deluxetable*}\n\nThe  systems studied  here are  presented  in Table~\\ref{tab:objects}.\nOnly one of them (HIP 61840) is a simple binary belonging to the 67 pc\nsample of solar-type stars; others  contain from three to five components and\nare also relatively  close to the Sun. Their  principal components are\nmain sequence stars with spectral types  from F2V to G3V.  The data in\nTable~\\ref{tab:objects} are collected from  Simbad and {\\it Gaia} DR2 \\citep{Gaia},\nthe  radial velocities  (RVs) are  determined here  (variable  RVs are\nmarked by 'v').\n\nThe structure of this paper is similar to the previous ones.  The data\nand methods  are briefly recalled in  Section~\\ref{sec:obs}, where the\nnew  orbital elements  are also  given. Then  in Section~\\ref{sec:obj}\neach system is  discussed individually. The paper closes  with a short\nsummary in Section~\\ref{sec:sum}.\n\n\n\\section{Observations and data analysis}\n\\label{sec:obs}\n\n\n\\subsection{Spectroscopic observations}\n\nThe spectra used here were taken with the 1.5 m telescope sited at the\nCerro Tololo  Inter-American Observatory (CTIO) in  Chile and operated\nby             the             SMARTS            Consortium.\\footnote{\n  \\url{http:\/\/www.astro.yale.edu\/smarts\/}}  The   observing  time  was\nallocated  through  NOAO.   Observations  were made  with  the  CHIRON\noptical echelle spectrograph \\citep{CHIRON} by the telescope operators\nin service mode.  In two runs,  2017 August and 2018 March, the author\nalso observed in classical mode.  All spectra are taken in the slicer\nmode with a resolution of $R=80,000$ and a signal to noise ratio of at\nleast 20.  Thorium-Argon calibrations were recorded for each target.\n\nRadial  velocities are  determined  from the  cross-correlation\nfunction (CCF) of echelle orders with the binary mask based on the solar\nspectrum,  as detailed  in  \\citep{paper1}. The  RVs  derived by  this\nmethod should be  on the absolute scale if  the wavelength calibration\nis  accurate.  The  CHIRON  RVs were  checked  against standards  from\n\\citep{Udry1998}, and a small  offset of $+0.15$ km~s$^{-1}$ was found\nin  \\cite{paper3}.   \n\nThe CCF contains two dips  in the case of double-lined systems studied\nhere.  The  dip width  is related to  the projected rotation  speed $V\n\\sin i$, while its area depends on the spectral type, metallicity, and\nrelative  flux.  Table~\\ref{tab:dip} lists  average parameters  of the\nGaussian  curves fitted  to  the CCF  dips.   It gives  the number  of\naveraged  measurements  $N$ (blended  CCFs  were  not  used), the  dip\namplitude  $a$,  its  dispersion  $\\sigma$,  the  product  $a  \\sigma$\nproportional to  the dip  area (hence to  the relative flux),  and the\nprojected rotation velocity $V \\sin i$, estimated from $\\sigma$ by the\napproximate  formula   given  in  \\citep{paper1}.    The  last  column\nindicates  the presence  or absence  of the  lithium 6708\\AA  ~line in\nindividual components.\n\n\\begin{deluxetable*}{l l c cccc c}    \n\\tabletypesize{\\scriptsize}     \n\\tablecaption{CCF parameters\n\\label{tab:dip}          }\n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{HIP\/HD} & \n\\colhead{Comp.} & \n\\colhead{$N$} & \n\\colhead{$a$} & \n\\colhead{$\\sigma$} & \n\\colhead{$a \\sigma$} & \n\\colhead{$V \\sin i$ } & \n\\colhead{Li}\n\\\\\n &  &  & &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{  6708\\AA}\n}\n\\startdata\nHIP 19639  & Aa &  7 & 0.061   &  12.53  &    0.765 &   21.7 &N\\\\\nHIP 19639  & Ab &  7 & 0.033   &   5.90  &    0.193 &    8.7 &N\\\\\nHIP 19646  & B  &  2 & 0.042    &  18.87  &    0.787 &   33: &N\\\\ \nHD 105080  & A  &  3 & 0.389   &   3.68  &    1.429 &    2.5 &N\\\\ \nHD 105081  & Ba & 11 & 0.179   &   3.82  &    0.684 &    3.1 &N\\\\\nHD 105081  & Bb & 11 & 0.167   &   3.78  &    0.630 &    3.0 &N\\\\\nHIP 60845  & Aa &  2 & 0.183   &   3.83  &    0.702 &    3.2 &N\\\\\nHIP 60845  & Ab &  2 & 0.124   &   4.01  &    0.497 &    3.8 &N\\\\\nHIP 60845  & BC &  2 & 0.430   &   3.57  &    1.535 &    2.0 &N\\\\ \nHIP 61840  & Aa &  9 & 0.189   &   4.51  &    0.853 &    5.3 &Y\\\\\nHIP 61840  & Ab &  9 & 0.124   &   4.13  &    0.511 &    4.2 &Y\\\\\nHIP 75663  & A  &  5 & 0.223   &   6.85  &    1.528 &   10.7 &Y\\\\\nHIP 75663  & Ba & 12 & 0.161   &   4.88  &    0.789 &    6.3 &Y\\\\ \nHIP 75663  & Bb & 12 & 0.155   &   4.87  &    0.754 &    6.3 &Y\\\\\nHIP 76816  & Aa &  6 & 0.110   &   8.14  &    0.899 &   13.3 &Y\\\\ \nHIP 76816  & Ab &  6 & 0.030   &   4.49  &    0.137 &    5.3 &Y\\\\\nHIP 76816  & B  &  5 & 0.506   &   3.78  &    1.913 &    3.0 &Y\\\\\nHIP 78163  & Aa &  9 & 0.197   &   4.08  &    0.803 &    4.0 &Y\\\\\nHIP 78163  & Ab &  9 & 0.192   &   4.05  &    0.778 &    4.0 &Y\\\\\nHIP 78163  & B  &  3 & 0.371   &   4.67  &    1.732 &    5.8 &N\\\\ \nHIP 78416  & Aa &  9 & 0.047   &  15.43  &    0.725 &   27: &Y\\\\\nHIP 78416  & Ab &  9 & 0.041   &  13.18  &    0.536 &   23: &Y\\\\\nHIP 78416  & B  &  4 & 0.058   &  22.68  &    1.324 &   40: &Y\\\\ \nHIP 80448  & Aa &  2 & 0.101   &  12.30  &    1.247 &   21.3 &Y\\\\ \nHIP 80448  & Ab &  2 & 0.023   &   9.78  &    0.221 &   16.5 &Y\\\\\nHIP 80448  & B  &  3 & 0.171   &   8.62  &    1.469 &   14.3 &Y\\\\    \nHIP 84789  & Aa &  6 & 0.043   &  12.41  &    0.536 &   21.5 &N\\\\\nHIP 84789  & Ab &  6 & 0.048   &  10.88  &    0.522 &   18.6 &N\\\\\nHIP 84789  & B  &  2 & 0.036   &  27.49  &    0.996 &   49: & N\n\\enddata \n\\end{deluxetable*}\n\n\n\n\\subsection{Speckle interferometry}\n\nInformation  on   the  resolved  subsystems  is   retrieved  from  the\nWashington Double Star Catalog \\citep[WDS;][]{WDS}. It is complemented\nby  recent  speckle   interferometry  at  the  Southern  Astrophysical\nResearch  (SOAR)  telescope.   The  latest  publication  \\citep{SAM18}\ncontains references to previous papers.\n\n\\subsection{Orbit calculation}\n\nAs in Paper  3 \\citep{paper3}, orbital elements and  their errors are\ndetermined   by  the   least-squares  fits   with   weights  inversely\nproportional   to   the   adopted   errors.    The   IDL   code   {\\tt\n  orbit} \\citep{ORBIT}\\footnote{Codebase: \\url{http:\/\/www.ctio.noao.edu\/~atokovin\/orbit\/} and \n\\url{http:\/\/dx.doi.org\/10.5281\/zenodo.61119} }\nis   used.   It   can   fit  spectroscopic,   visual,   or   combined\nvisual\/spectroscopic  orbits. Formal  errors of  orbital  elements are\ndetermined from these fits.   The elements of spectroscopic orbits are\ngiven in Table~\\ref{tab:sborb} in common notation.\n\n\n\n\\begin{deluxetable*}{l l cccc ccc c c}    \n\\tabletypesize{\\scriptsize}     \n\\tablecaption{Spectroscopic orbits\n\\label{tab:sborb}          }\n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{$P$} & \n\\colhead{$T$} & \n\\colhead{$e$} & \n\\colhead{$\\omega_{\\rm A}$ } & \n\\colhead{$K_1$} & \n\\colhead{$K_2$} & \n\\colhead{$\\gamma$} & \n\\colhead{rms$_{1,2}$} &\n\\colhead{$M_{1,2} \\sin^3 i$} \n\\\\\n& & \\colhead{(days)} &\n\\colhead{(+24,00000)} & &\n\\colhead{(degree)} & \n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{ (${\\cal M}_\\odot$) } \n}\n\\startdata\nHIP 19639 & Aa,Ab & 2.35254 & 58002.5732 & 0.0 & 0.0          & 39.474 & 49.194 & 35.400 & 0.36 & 0.094 \\\\\n         && $\\pm$0.00005 & $\\pm$0.0019 & fixed & fixed & $\\pm$0.160 & $\\pm$0.298 & $\\pm$0.100 & 0.87 & 0.076 \\\\\nHD 105081 & Ba,Bb & 30.427 & 58214.038 & 0.418 & 82.1         & 46.856 & 47.493 & 50.099 & 0.07 & 1.00 \\\\\n         && $\\pm$0.006 & $\\pm$0.023 & $\\pm$0.001 & $\\pm$0.3 & $\\pm$0.117 & $\\pm$0.118 & $\\pm$0.041 & 0.05 & 0.98 \\\\\nHIP 60845 & Aa,Ab & 6.3035 & 58195.6279 & 0.0 & 0.0           & 31.361 & 32.128 & 40.018 & 0.10 & 0.084 \\\\\n         && $\\pm$0.0001 & $\\pm$0.0017 & fixed & fixed & $\\pm$0.063 & $\\pm$0.114 & $\\pm$0.036 & 0.27 & 0.082\\\\\nHIP 61840 & Aa,Ab& 9.6717 & 58194.383 & 0.007 & 357.0         & 56.665 & 60.744 & 6.745 & 0.05 & 0.84 \\\\\n         && $\\pm$0.0008 & $\\pm$0.345 & $\\pm$0.001 & $\\pm$13.0 & $\\pm$0.525 & $\\pm$0.562 & $\\pm$0.048 & 0.05 & 0.78\\\\\nHIP 75663 & Ba,Bb &  22.8704 & 58204.4963 & 0.613 & 260.1     & 49.001 & 49.622 & $-$56.847 & 0.29 & 0.56 \\\\\n         &&  $\\pm$0.0047 & $\\pm$0.020 & $\\pm$0.001 & $\\pm$0.3 & $\\pm$0.126 & $\\pm$0.134 & $\\pm$0.045 & 0.27 & 0.56\\\\\nHIP 76816 & Aa,Ab & 6.95176 & 58197.3528 & 0.0 & 0.0          & 45.790 & 62.448 & $-$39.145 & 0.29 & 0.53 \\\\\n         && $\\pm$0.00002 & $\\pm$0.0063 & fixed & fixed & $\\pm$0.306 & $\\pm$0.570 & $\\pm$0.186 & 1.13 & 0.39 \\\\\nHIP 78163 & Aa,Ab &  21.8186 & 58202.091 & 0.577 & 301.0      & 45.429 & 45.671 & 9.411 & 0.04 & 0.47\\\\\n         &&  $\\pm$0.0015 & $\\pm$0.014 & $\\pm$0.002 & $\\pm$0.3 & $\\pm$0.161 & $\\pm$0.186 & $\\pm$0.045 & 0.04 & 0.46\\\\\nHIP 78416 & Aa,Ab & 21.0802 & 58197.2344 & 0.708 & 99.1       & 64.636 & 71.737 & 1.636 &  0.16 & 1.03 \\\\\n         &&  $\\pm$0.0030 & $\\pm$0.0072 & $\\pm$0.003 & $\\pm$0.3 & $\\pm$0.372 & $\\pm$0.509 & $\\pm$0.112 & 0.61 & 0.93\\\\\nHIP 80448 &Aa,Ab  & 2.2699 & 58195.5948 & 0.0 & 0.0           & 73.124 & 108.452 & $-$1.848 & 0.64 & 0.84 \\\\\n         && $\\pm$0.0002 & $\\pm$0.0033 & fixed & fixed & $\\pm$0.592 & $\\pm$0.698 & $\\pm$0.270 & 0.51 & 0.57\\\\\nHIP 84789 &Aa,Ab & 2.2758 & 58196.7968 & 0.0 & 0.0            & 78.044 & 79.004 & 5.624 & 0.40 & 0.46 \\\\\n         && $\\pm$0.0001 & $\\pm$0.0020 & fixed & fixed & $\\pm$0.181 & $\\pm$0.182 & $\\pm$0.073 & 0.21 & 0.45\n\\enddata \n\\end{deluxetable*}\n\n\\begin{deluxetable*}{l l cccc ccc}    \n\\tabletypesize{\\scriptsize}     \n\\tablecaption{Visual orbits\n\\label{tab:vborb}          }\n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{$P$} & \n\\colhead{$T$} & \n\\colhead{$e$} & \n\\colhead{$a$} & \n\\colhead{$\\Omega_{\\rm A}$ } & \n\\colhead{$\\omega_{\\rm A}$ } & \n\\colhead{$i$ }  \\\\\n& & \\colhead{(years)} &\n\\colhead{(years)} & &\n\\colhead{(arcsec)} & \n\\colhead{(degree)} & \n\\colhead{(degree)} & \n\\colhead{(degree)} \n}\n\\startdata\nHD 105080 & Aa,Ab  & 91.6       & 1999.15   & 0.40      & 0.176             &  5.4      & 224.1    & 40.4  \\\\\nHIP 60845 &   A,BC  & 690        & 1826.7    & 0.20      & 2.485             & 104.4    & 156.6      &  141.4  \\\\\nHIP 60845 &   B,C   & 28.2      & 1990.2     & 0.166     & 0.221             & 162.8    & 77.3       & 156.1  \\\\\n      &         & $\\pm$0.2  & $\\pm$0.3 & $\\pm$0.009  & $\\pm$0.003        & $\\pm$6.5  & $\\pm$5.9  & $\\pm$2.5  \\\\\nHIP 80448 &   A,B   & 1950      & 1926.58    & 0.63      & 4.644             & 75.5     & 267.9      & 152.1 \\\\\nHIP 80448 &   Ba,Bb & 20.0      & 2018.08    & 0.42      & 0.176             & 6.1      & 150.9      & 109.1  \\\\\n      &         & $\\pm$0.3  & $\\pm$0.24 & $\\pm$0.02  & $\\pm$0.003        & $\\pm$1.2  & $\\pm$4.9  & $\\pm$0.8  \n\\enddata \n\\end{deluxetable*}\n\n\n\nFor two multiple systems, the resolved measurements of inner and outer\npairs are represented by  visual orbits. Simultaneous fitting of inner\nand outer orbits is done  using the code {\\tt orbit3.pro} described by\n\\citet{TL2017}; the code is available in \\citep{ORBIT3}.\\footnote{Codebase: \\url{http:\/\/dx.doi.org\/10.5281\/zenodo.321854}}\nIt accounts for the wobble in  the trajectory of the outer pair caused\nby the subsystem.  The wobble amplitude is $f$  times smaller than the\ninner   semimajor  axis,  where   the  wobble   factor  $f   =  q_{\\rm\n  in}\/(1+q_{\\rm in})$  depends on the  inner mass ratio  $q_{\\rm in}$.\nThe elements  of visual orbits are given  in Table~\\ref{tab:vborb}. As\nouter orbits are  poorly constrained, I do not  list their errors. The\nouter  orbits  serve primarily  to  model  the  observed part  of  the\ntrajectory for the determination  of $f$.  In the figures illustrating\nthese orbits,  the observed trajectories  are plotted relative  to the\nprimary component of each system, on the same scale.\n\nIndividual RVs  of spectroscopic binaries  and their residuals  to the\norbits are presented  in Table~\\ref{tab:rv}. The HIP or  HD number and\nthe system  identifier (components joined  by comma) in the  first two\ncolumns define the binary. Then follow  the Julian date, the RV of the\nprimary  component $V_1$,  its adopted  error $\\sigma_1$  (blended CCF\ndips  are assigned large  errors), and  its residual  (O$-$C)$_1$. The\nlast  three columns  give  velocities, errors,  and  residuals of  the\nsecondary  component. Table~\\ref{tab:rvconst}  contains  RVs of  other\ncomponents,     both      constant     and     variable.      Finally,\nTable~\\ref{tab:speckle}  lists  position  measurements  used  for  the\ncalculation of visual orbits. It contains the HIP or HD number, system\nidentification,   date  of   observation,  position   angle  $\\theta$,\nseparation  $\\rho$,  adopted  error  $\\sigma_\\rho$ (errors  in  radial  and\ntangential directions  are considered to be equal),  and the residuals\nto the  orbits in $\\theta$ and  $\\rho$. The last  column indicates the\nmeasurement technique.   Measurements of the outer systems  are of two\nkinds: when the inner pair  is unresolved (e.g.  HIP 60845 A,BC), they\nrefer  to   the  photo-center  of  the  inner   pair,  while  resolved\nmeasurements  refer  to the  individual  components  (e.g.  HIP  60845\nA,B). The orbit-fitting code  accounts for reduced wobble amplitude of\nunresolved (photo-center) measurements compared to resolved ones.\n\n\n\n\\begin{deluxetable*}{r l c rrr rrr}    \n\\tabletypesize{\\scriptsize}     \n\\tablecaption{Radial velocities and residuals (fragment)\n\\label{tab:rv}          }\n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{Date} & \n\\colhead{$V_1$} & \n\\colhead{$\\sigma_1$} & \n\\colhead{(O$-$C)$_1$ } &\n\\colhead{$V_2$} & \n\\colhead{$\\sigma_2$} & \n\\colhead{(O$-$C)$_2$ } \\\\\n & & \n\\colhead{(JD $+$24,00000)} &\n\\multicolumn{3}{c}{(km s$^{-1}$)}  &\n\\multicolumn{3}{c}{(km s$^{-1}$)}  \n}\n\\startdata\n 19639 & Aa,Ab &  57985.8910 &  68.74 &   0.30 &   0.16  &    $-$6.37 &   0.60 &  $-$0.42   \\\\\n 19639 & Aa,Ab &  57986.8980 &  15.11 &   0.30 &   0.18  &    59.76 &   0.60 &  $-$1.15   \\\\\n 19639 & Aa,Ab &  58052.6260 &  31.22 &  10.00 &   2.23  & \\ldots   &\\ldots  & \\ldots   \\\\\n 19639 & Aa,Ab &  58053.6080 &  22.97 &   0.50 &   1.26  &    54.35 &   1.00 &   1.89   \n\\enddata \n\\end{deluxetable*}\n\n\\begin{deluxetable}{r l r r }    \n\\tabletypesize{\\scriptsize}     \n\\tablecaption{Radial velocities of other components\n\\label{tab:rvconst}          }\n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{HIP\/HD} & \n\\colhead{Comp.} & \n\\colhead{Date} & \n\\colhead{RV}   \\\\ \n & & \n\\colhead{(JD $+$24,00000)} &\n\\colhead {(km s$^{-1}$)}  \n}\n\\startdata\n19646  &  B   & 57985.8932&  36.005 \\\\\n19646  &  B   & 58193.5386&  35.944 \\\\ \n105080 &  A   & 58193.7546&  50.182 \\\\  \n105080 &  A   & 58194.6308&  50.198 \\\\\n105080 &  A   & 58195.6489&  50.179 \\\\\n60845  &  BC  & 57985.4627&  42.462  \\\\\n60845  &  BC  & 58193.7521&  42.524 \\\\ \n60845  &  BC  & 58194.6447&  42.514 \\\\\n60845  &  BC  & 58195.6586&  42.523 \\\\\n60845  &  BC  & 58177.7617&  42.513 \\\\\n60845  &  BC  & 58232.5972&  42.540 \\\\\n60845  &  BC  & 58242.5361&  42.513 \\\\\n75663  &  A   & 57986.4885& $-$50.737 \\\\ \n75663  &  A   & 58177.8100& $-$56.446 \\\\\n75663  &  A   & 58193.8257& $-$57.181 \\\\\n75663  &  A   & 58195.8323& $-$57.305 \\\\\n76816  &  B   & 57986.4996& $-$50.912 \\\\\n76816  &  B   & 58193.8413& $-$50.916 \\\\ \n76816  &  B   & 58194.8267& $-$50.921  \\\\\n76816  &  B   & 58195.8509& $-$50.912 \\\\  \n78163  &  B   & 57986.5130&   6.116 \\\\  \n78163  &  B   & 58194.8533&   7.152 \\\\ \n78163  &  B   & 58195.7872&   7.082 \\\\ \n78416  &  B   & 57986.5221&   0.752 \\\\\n78416  &  B   & 58193.8595&   1.568 \\\\ \n78416  &  B   & 58194.8446&   1.816 \\\\ \n78416  &  B   & 58195.7774&   1.581 \\\\ \n80448  &  B   & 58193.8678&   7.509  \\\\\n80448  &  B   & 58195.8008&   7.752  \\\\\n80448  &  B   & 58194.8621&   7.679  \\\\\n80448  &  B   & 58228.8113&   7.515 \\\\ \n80448  &  B   & 58233.8463&   7.353  \\\\\n80448  &  B   & 58246.6693&   8.395  \\\\\n80448  &  B   & 58248.8529&   8.176  \\\\\n80448  &  B   & 58256.7395&   7.532  \\\\\n80448  &  B   & 58257.8020&   7.916  \\\\\n84789  &  B   & 57986.5346&   6.669 \\\\\n84789  &  B   & 58195.8671&   5.260 \n\\enddata \n\\end{deluxetable}\n\n\\begin{deluxetable*}{r l l rrr rr l}    \n\\tabletypesize{\\scriptsize}     \n\\tablecaption{Position measurements and residuals (fragment)\n\\label{tab:speckle}          }\n\\tablewidth{0pt}                                   \n\\tablehead{                                                                     \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{Date} & \n\\colhead{$\\theta$} & \n\\colhead{$\\rho$} & \n\\colhead{$\\sigma_\\rho$} & \n\\colhead{(O$-$C)$_\\theta$ } & \n\\colhead{(O$-$C)$_\\rho$ } &\n\\colhead{Ref.\\tablenotemark{a}} \\\\\n & & \n\\colhead{(yr)} &\n\\colhead{(\\degr)} &\n\\colhead{(\\arcsec)} &\n\\colhead{(\\arcsec)} &\n\\colhead{(\\degr)} &\n\\colhead{(\\arcsec)} &\n}\n\\startdata\n 60845 & B,C &  1939.4600 &    357.8 &   0.3500 &   0.1500 &      4.2 &   0.1351 & M  \\\\\n 60845 & B,C &  1956.3800 &    184.1 &   0.3100 &   0.0500 &      8.8 &   0.0956 & M \\\\\n 60845 & B,C &  2018.1639 &     92.6 &   0.1769 &   0.0050 &      1.9 &   0.0070 & S \\\\\n 60845 & A,BC &  1880.3800 &    270.5 &   2.4300 &   0.2500 &      2.2 &   0.3463 & M \\\\\n 60845 & A,BC &  1991.2500 &    201.1 &   2.0510 &   0.0100 &     $-$1.2 &  $-$0.0042 & H \\\\\n 60845 & A,B &  2018.1639 &    187.1 &   2.0965 &   0.0050 &      0.2 &  $-$0.0018 &  S\n\\enddata \n\\tablenotetext{a}{\nH: Hipparcos;\nS: speckle interferometry at SOAR;\ns: speckle interferometry at other telescopes;\nM: visual micrometer measures;\nG: Gaia DR2.\n}\n\\end{deluxetable*}\n\n\n\\section{Individual objects}\n\\label{sec:obj}\n\nFor  each observed system,  the corresponding  Figure shows  a typical\ndouble-lined CCF (the Julian date and components' designatios are marked\non the plot) together with the RV curve representing the orbit. In the\nRV curves, squares denote  the primary component, triangles denote the\nsecondary component, while the full and dashed lines plot the orbit. \n\n\\subsection{HIP 19639 and 19646 (triple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig1.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 19639 Aa,Ab.\n\\label{fig:19639}\n}\n\\end{figure}\n\nThe 50\\arcsec ~pair of bright stars HIP~19639 and 19646 was identified\nas  a visual binary  by J.F.W.~Hershel  in 1838.   The {\\it  Gaia} DR2\nastrometry leaves no doubt that  this pair is physical: the components\nhave common  proper motion (PM),  distance, and RV. The  outer orbital\nperiod  is  of the  order  of 240  kyr.   \\citet{N04}  found that  the\ncomponent A is  a double-lined binary; its 2.35-day circular orbit is\ndetermined  here  for the  first  time (Figure~\\ref{fig:19639}).   Two\nspectra (JD 2458052 and 2458053) were taken with the NRES spectrograph,\nas described in \\citep{paper3}.\n\nThe CCF of the stronger component Aa is wide and asymmetric, while the\nCCF  of  Ab  is  narrower;  their  widths  correspond  to  approximate\nprojected rotation velocities $ V \\sin i$ of 21.7 and 8.7 km~s$^{-1}$,\nrespectively, while the ratio of  the CCF areas implies $\\Delta V_{\\rm\n  Aa,Ab} =  1.50$ mag. Wide and  shallow dips lead to  large RV errors\nand large residuals to the orbit.  The mass ratio in the inner pair is\n$q_{\\rm Aa,Ab}  = 0.82$.  The RV of  the component B its  close to the\ncenter-of-mass velocity  of A.   The component B  also has a  wide CCF\ncorresponding to $ V \\sin i$ of $\\sim$33 km~s$^{-1}$. \n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig2.eps}\n\\caption{Location of  three components of the HIP~19639  system on the\n  color-magnitude  diagram  (squares).  The  full  line  is  a 2  Gyr\n  isochrone for solar  metallicity \\citep{Dotter2008}, where asterisks\n  and numbers mark masses.\n\\label{fig:iso}\n}\n\\end{figure}\n\nComponents  of   the  triple  system  HIP~19639  are   placed  on  the\ncolor-magnitude  diagram  (CMD)  in  Figure~\\ref{fig:iso},  using  the\ndistance modulus  of 5.47  mag. The  $V$ magnitudes of  Aa and  Ab are\nestimated  from the  combined  magnitude of  A  and the  spectroscopic\nmagnitude difference of  1.5 mag. The $V-K$ color  of Ab, not measured\ndirectly, is  assumed to place it  on the main sequence.   It is clear\nthat  both Aa  and B  are located  above the  main sequence,  near the\nturn-off. Their  positions match reasonably  well the 2  Gyr isochrone\nand correspond to the masses of 1.6 and 1.5 ${\\cal M}_\\odot$. The mass\nof Ab  deduced from the  isochrone is 1.28 ${\\cal  M}_\\odot$, matching\nthe spectroscopic mass ratio, while the radii of Aa and Ab are 2.7 and\n1.1 $R_\\odot$.  The orbital axis $a_1 + a_2 = 10.6 R_\\odot$ means that\nthe  binary  is detached.   However,  contact  and  mass transfer  are\nimminent when Aa expands further.\n\nThe spectroscopic  mass sum  of the inner  binary is only  0.18 ${\\cal\n  M}_\\odot$.   The mass  sum estimated  above, 2.88  ${\\cal M}_\\odot$,\nimplies  an inclination  $i_{\\rm Aa,Ab}  = 23\\fdg4$,  or  $\\sin i_{\\rm\n  Aa,Ab} = 0.40$, hence the  synchronous rotation velocities of Aa and\nAb are 23.8  and 9.7 km~s$^{-1}$, respectively, in  agreement with the\nmeasured CCF width.  Summarizing, this is an interesting triple system\nwhere the inner close binary is caught at evolutionary phase preceding\nthe mass transfer.\n\n\n\\subsection{HD 105080 and 105081 (2+2 quadruple) }\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig3.eps}\n\\caption{CCF (left)  and RV curve (right) of HD 105081 Ba,Bb.\n\\label{fig:105080}\n}\n\\end{figure}\n\n\nTwo nearly equal stars HD~105080  and 105081 form a 12\\farcs9 physical\nbinary, first measured by J.~Hershel  in 1835.  Each of these stars is\na close pair, making the whole  system quadruple.  The pair Aa,Ab is a\nknown visual binary RST~4958  with a small magnitude difference. Since\nits discovery in 1942 by R. A. Rossiter, it was observed only episodically.\nBy adding  three speckle  measures made  at SOAR in  2017 and  2018, a\npreliminary  orbit  with  $P  =  91.6$  years  can  be  fitted  to  the\nobservations (Table~\\ref{tab:vborb}).  Double  lines were reported for\nthis star in the literature,  although there could be a confusion with\nthe double-lined component  B.  The RV of A  is practically coincident\nwith the center-of-mass velocity of B.  The {\\it Gaia} DR2 parallax of\nA has  a large error of 1\\,mas,  being affected by the  Aa,Ab pair.  I\nadopt  the  parallax  of  B  as  the distance  to  the  system.   Both\ncomponents are then located on the CMD very close to each other, above\nthe  main  sequence. This  distance  and  the  visual orbit  of  Aa,Ab\ncorrespond to the mass sum of 1.7 ${\\cal M}_\\odot$; however, the orbit\nis poorly constrained.\n\nThe component B (HD~105081), which  is only slightly fainter than A in\nthe $V$ and $G$ bands, is revealed here to be a twin double-lined pair\nwith      $P=30.4$       days      and      eccentricity      $e=0.42$\n(Figure~\\ref{fig:105080}).  The spectroscopic mass sum of\nBa and Bb, 1.98 ${\\cal M}_\\odot$, and the mass sum inferred from the\nabsolute magnitudes, 2.26 ${\\cal M}_\\odot$, imply inclination\n$i_{\\rm Ba,Bb}=  73\\degr$. The CCF  dips of Ba  and Bb are  narrow and\ndeep,   hence   the  residuals   to   the   orbit   are  small,   only\n0.07~km~s$^{-1}$ .\n\n\n\n\\subsection{HIP 60845 (quintuple)}\n\n\n\\begin{figure}\n\\plotone{fig4.eps}\n\\caption{Quintuple system HIP 60845 (WDS J12283$-$6146). The positions\n  of three  components A, BC, and D  on the sky and  their motions are\n  illustrated in  the upper panel.  Periods and  masses are indicated.\n  The lower panel shows the observed motion of the subsystems A,BC and\n  B,C and their  orbits. In this plot, the  coordinate origin coincides\n  with  the main  star  A, the  wavy  line shows  the  motion of  the\n  component B  around A  according to the  orbit. Small  crosses depict\n  measurements  of A,BC where  the pair  BC was  unresolved, asterisks\n  depict the resolved  measurents of A,B. The orbit  of B,C is plotted\n  on the  same scale around the   coordinate origin  by the dashed\n  line and triangles. \n\\label{fig:HIP60845}\n}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig5.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 60845 Aa,Ab. \n\\label{fig:60845}\n}\n\\end{figure}\n\n\nThis  multiple  system is  located  within  50\\,pc  from the  Sun  and\ncontains  at least  five stars  arranged  in a  rare 3-tier  hierarchy\nillustrated  in Figure~\\ref{fig:HIP60845}.   The widest  7\\farcs9 pair\nRST~4499 AB,D is  physical, based on its stable  relative position and\nthe {\\it  Gaia} DR2  parallaxes and PMs  of the components.   The fast\n(175 mas~yr$^{-1}$) PM facilitates discrimination between physical and\noptical companions, despite the high stellar density in this field and\nthe faintness  of D ($V  = 13.4$ mag).   The period of  AB,D estimated\nfrom its projected separation $\\rho_{\\rm AB,D} = 7\\farcs88$ is about 4\nkyr and corresponds to the  chracteristic orbital velocity  $\\mu^* =\n2 \\pi \\rho_{\\rm AB,D}\/P_{\\rm  AB,D} = 13$ mas~yr$^{-1}$.  The relative\nPM  between A  and D,  measured by  {\\it Gaia}  and corrected  for the\norbital  motion of  A,  is  11 mas~yr$^{-1}$;  it  is directed  almost\nexactly toward A (position angle  240\\degr), as indicated by the arrow\nin  Figure~\\ref{fig:HIP60845}. If D  moves on  an eccentric  orbit, it\nwill come  close to  A,BC in $\\sim$700  years, disrupting  the system.\nAlternatively,  the  observed  motion  might correspond  to  a  highly\ninclined and not very eccentric  outer orbit, in which case the system\ncould  be dynamically stable.   If the  pair AB,D  is bound,  the true\nseparation between A  and D cannot exceed its  projected separation by\nmore   than  $\\sim$2  times,   given  their   relative  speed   of  11\nmas~yr$^{-1}$ and the total mass sum of 4.5 ${\\cal M}_\\odot$.\n\n\nThe  visual  binary  A,BC  (CPO~12),  for which  a  crude  orbit  with\n$P=2520$\\,years  and semimajor  axis of  5\\farcs4 has  been  published by\n\\citet{USN2002}, occupies  the intermediate hierarchical  level.  This\norbit  is poorly  constrained  by the  century-long  observed arc.   I\ncomputed an  alternative orbit  with $P_{\\rm A,BC}  = 690$ years,  with \nsmaller eccentricity and semimajor axis (Table~\\ref{tab:vborb}).  This\norbit makes more sense, given  the threat to dynamical stability posed\nby the  outer component D.  Even  then, the period  ratio $P_{\\rm AB,D}\/\nP_{\\rm  A,BC}  \\sim  5$  is  comparable  to  the  dynamical  stability\nlimit. On the other hand, the  nearly circular orbit of the inner pair\nB,C (RST~4499) with $P=28.2$\\,years  is definitive.  The visual orbits and\nthe estimated mass  sums match the {\\it Gaia}  DR2 parallax reasonably\nwell.  Both orbits are retrograde and have  small inclinations.\n\nThe visual primary star A  was identified as a spectroscopic binary by\n\\citet{N04}.   Now  its  6.3-day  double-lined  orbit  is  determined\n(Figure~\\ref{fig:60845}).  The eccentricity  does not differ from zero\nsignificantly, hence the circular orbit  is imposed.  The masses of Aa\nand  Ab are  almost equal,  as are  their CCF  dips.  Given  the small\nangular distance  between A and BC,  2\\farcs06, the light  is mixed in\nthe  fiber,  so   the  CCF  often  has  3  dips;   the  CCF  shown  in\nFigure~\\ref{fig:60845} is  an exception recorded with  good seeing and\ncareful guiding.  The  magnitude difference between Ab and  Aa is 0.37\nmag, hence their individual $V$  magnitudes are 7.79 and 8.16 mag.  By\ncomparing  the mass sum  of Aa  and Ab  estimated from  their absolute\nmagnitudes, 2.2 ${\\cal M}_\\odot$,  with the spectroscopic mass sum of\n0.167 ${\\cal  M}_\\odot$, I find that  the orbit of Aa,Ab  has a small\ninclination  of  $i_{\\rm  Aa,Ab}  \\approx 25\\degr$.   The  synchronous\nrotation  of  the component  Aa,  of  one  solar radius,  implies  the\nprojected  rotation of  $V \\sin  i =  3.3$ km~s$^{-1}$,  close  to the\nmeasured value.  The three inner orbits could be close to coplanarity,\ngiven their small inclinations. \n\nThe CCF  dip corresponding to the  combined light of BC  is narrow and\nhas a  constant RV  of 42.5 km~s$^{-1}$,  close to  the center-of-mass\nvelocity of  A.  Slow  axial rotation, location  of components  on the\nmain  sequence in  the  CMD,  and non-detection  of  the lithium  line\nsuggest that this quintuple system is relatively old.\n\n\n\\subsection{HIP 61840 (binary)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig6.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 61840 Aa,Ab.\n\\label{fig:61840}\n}\n\\end{figure}\n\nUnlike  the rest  of the objects  in  this paper,  this star  is a  simple\nspectroscopic binary without additional  components. It belongs to the\n67 pc  sample  of solar-type  stars  \\citep{FG67a}  and  is young,  as\ninferred from the chromoshperic activity, X-ray flux, and the presence\nof lithium  in its atmosphere.  The double-lined  nature was announced\nby \\citet{Wichman2003} and \\citet{N04}, but the orbital period was, so\nfar, unknown.  The object  has been observed by speckle interferometry\nat SOAR in 2011 and 2016 and found unresolved.\n\nThe orbit with  $P=9.67$ days has a small,  but significantly non-zero\neccentricity   $e=  0.007 \\pm  0.001$  (Figure~\\ref{fig:61840}). The\nresiduals to the circular orbit are 0.3 km~s$^{-1}$,  6$\\times$\nlarger than to the eccentric orbit.  The masses of Aa and Ab estimated\nfrom  absolute magnitudes  are  1.24 and  1.13  ${\\cal M}_\\odot$,  the\nspectroscopic mass sum is 1.62 ${\\cal M}_\\odot$, hence the inclination\nis $i =  62\\degr$.  The measured projected rotation  speed matches the\nsynchronous speed.\n\n\\subsection{HIP 75663 (2+2 quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig7.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 75663 Ba,Bb.\n\\label{fig:75663}\n}\n\\end{figure}\n\n\nThis system is  quadruple.  The outer 9\\farcs4 pair  was discovered in\n1825 by W.~Struve.  Its estimated period is 17\\,kyr. \\citet{N04} noted\ndouble lines  in the visual secondary  B.  Its orbital  period is 22.9\ndays (Figure~\\ref{fig:75663}),  with a large  eccentricity of $e=0.61$\nand the mass ratio $q_{\\rm Ba,Bb}  = 0.997$ (a twin). The areas of the\nCCF dips  of Ba and Bb are  equal to within 2\\%.  The masses estimated\nfrom absolute  magnitudes are 1.02  ${\\cal M}_\\odot$ each,  leading to\nthe  orbital  inclination   of  $i_{\\rm  Ba,Bb}=55\\degr$.   The  axial\nrotation of Ba and Bb is faster than synchronous, as expected for such\neccentric orbit.\n\n\nThe RV  of the main  component A is  variable according to  the CHIRON\ndata (range  from $-$50.7 to $-$57.3 km~s$^{-1}$)  and the literature.\n\\citet{N04} made two measurements averaging at $-58.2$ km~s$^{-1}$ and\nsuspected   RV  variability,   while  {\\it   Gaia}   measured  $-55.4\n\\pm 2.$ km~s$^{-1}$.   The photo-center motion  of A  caused by  the subsystem\nAa,Ab  could  explain  the  discrepancy  between the  {\\it  Gaia}  DR2\nparallaxes of A and B {\\bf (9.29$\\pm$0.16 and 7.69$\\pm$0.07 mas,\n  respectively). A similar discrepancy of parallaxes exists in the HD\n  105080\/81 system, where A is a visual binary. } The period of Aa,Ab is not known; presumably it\nis longer  than a year.  \\citet{Isaacson2010} classified  this star as\nchromospherically    active    and    found    the   RV    jiter    of\n4.2\\,m~s$^{-1}$. The location of the  component A on the CMD indicates\nthat  it  is slightly  evolved  and  matches  approximately the  4-Gyr\nisochrone. Lithium is detectable in the spectra of A and B.\n\n\n\\subsection{HIP 76816 (2+2 quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig8.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 76816 Aa,Ab.\n\\label{fig:76816}\n}\n\\end{figure}\n\n\nThis  is a  quadruple system  located at  309\\,pc from  the  Sun.  The\n5\\farcs4  visual binary  HWE~37  is known  since  1876; its  estimated\nperiod is 33\\,kyr.  Double lines in the component A were discovered by\n\\citet{Desidera2006}. I used their measurement to refine the period of\nthe  circular  orbit  of  Aa,Ab  with $P=6.95$  days,  determined  here\n(Figure~\\ref{fig:76816}).  The eccentric  orbit has similar residuals,\nhence the circular solution is retained.  The components Aa and Ab are\nunequal in the amplitudes of their  CCF dips (area ratio 0.152, or 2.0\nmag difference) and of the RV  variation (mass ratio 0.735). The RV of\nthe visual  component B  is also variable  with a long,  still unknown\nperiod.  I  measured its RV  at $-$50.9 km~s$^{-1}$  (constant), while\n{\\it  Gaia}  measured  $-41.4$  km~s$^{-1}$  and  \\citet{Desidera2006}\nmeasured $-37.5$ km~s$^{-1}$; these RVs differ from the center-of-mass\nvelocity of A, $-39.94$ km~s$^{-1}$.\n\nThe matching  {\\it Gaia} DR2 parallaxes  place both A and  B above the\nmain sequence. The component B is  more evolved: it is brighter than A\nin the  $K$ band (unless  its $K$ magnitude  measured by 2MASS  is \ndistorted  by  the  proximity  of  A,  as  happens  with  other  close\npairs). The mass sum of Aa and Ab, estimated crudely from the absolute\nmagnitudes,  is almost 3  ${\\cal M}_\\odot$,  leading to  the\norbital inclination  of 42\\fdg5.    The stars\nAa and Ab apparently rotate synchronously with the orbit.\n\n\n\\subsection{HIP 78163 (2+2 quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig9.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 78163 Aa,Ab.\n\\label{fig:78163}\n}\n\\end{figure}\n\n\nThis multiple  system is  composed by the  outer 7\\farcs9  pair WG~185\n(estimated period 8\\,kyr) and  the inner subsystem Aa,Ab discovered by\n\\citet{N04}. For  the latter,  I determined here  the orbit  with $P=\n21.8$ days (Figure~\\ref{fig:78163}),  $e=0.58$, and mass ratio $q_{\\rm\n  Aa,Ab} = 0.996$  (a twin).  The RV of the  component B measured with\nCHIRON  ranges  from 6.1  to  7.1  km~s$^{-1}$  and differs  from  the\ncenter-of-mass RV of the  component A, 9.4 km~s$^{-1}$.  Considering\nalso  the {\\it  Gaia} RV(B)=16.3  km~s$^{-1}$, I  infer that  B  is a\nsingle-lined  binary, possibly  with  a  long period  and  a small  RV\namplitude.   Its  photo-center motion  could  explain  the slight  \n discrepancy between  {\\it Gaia} parallaxes and PMs of A\nand B. Therefore, the parallax of A, 10.42\\,mas, is likely the correct\none.  \n\nThe  twin  components  Aa  and  Ab  have masses  of  one  solar  each.\nComparting  them  to $M  \\sin^3  i$,  the  inclination of  50\\degr  ~is\nderived.   The   stars   A   and   B   are   located   on   the   main\nsequence. Interestingly,  lithium is detectable  in the spectra  of Aa\nand Ab, but not in B, which is a similar solar-mass star.\n\n\n\n\\subsection{HIP 78416 (triple or quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig10.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 78416 Aa,Ab.\n\\label{fig:78416}\n}\n\\end{figure}\n\n\nThe outer  6\\farcs5 pair  HWE~81, known since  1876, has  an estimated\nperiod  of  10\\,kyr.   \\citet{N04}  detected  RV  variability  of  the\ncomponent   A,  later   found   to  be   a   double-lined  binary   by\n\\citet{Desidera2006}.   The   orbital  period   is  21  days   and  the\neccentricity  $e=0.766$ is  unusually  high for  such  a short  period\n(Figure~\\ref{fig:78416}).  The  wide and  shallow CCF dips  imply fast\naxial  rotation.  For  this  reason,  the RVs  are  not measured  very\naccurately  and the  residuals to  the orbit  are large,  0.2  and 0.6\nkm~s$^{-1}$.  By comparing the estimated masses of Aa and Ab, 1.15 and\n1.03 ${\\cal M}_\\odot$ respectively, with  $M \\sin^3 i$, I estimate the\norbital inclination of 74\\degr.\n\nThe visual component B  also has a fast axial rotation of  $V \\sin i \\sim\n40$ km~s$^{-1}$, degrading the accuracy of its RV measurement.  The\nRVs of B measured with CHIRON, by {\\it Gaia}, and by \\citet{Desidera2006}\n(1.4, $-$0.7,  and 0.5 km~s$^{-1}$ respectively)  are reasonably close\nto  the center-of-mass  RV of  A, 1.7  km~s$^{-1}$. Therefore,  B is\nlikely a  single star. All three  stars Aa, Ab, and  B have comparable\nmasses and  similar colors.  The component A,  being a close  pair, is\nlocated on the CMD just above B, as expected.\n\nThe RVs of Aa and Ab measured by \\citet{Desidera2006}, 58.9 and $-1.8$\nkm~s$^{-1}$,  correspond  to   the  center-of-mass  velocity  of  30.2\nkm~s$^{-1}$  and do  not fit  the present  orbit with  $\\gamma  = 1.7$\nkm~s$^{-1}$.   This  discrepancy suggests  that  Aa,Ab  is orbited  by\nanother close  companion.  Further  monitoring is needed,  however, to\nprove this hypothesis. \n\nAccording to  \\citet{Rizzuto2011}, this system belongs to  the Sco OB2\nassociation with  a probability of 74\\%. Fast  axial rotation and the\npresence of lithium indicate a young age.\n\n\n\n\\subsection{HIP 80448 (2+2 quadruple)}\n\n\nThis young multiple system is  located within 50\\,pc from the Sun.  It\ncontains  four components  in  a  small volume.   The  outer pair  A,B\n(COO~197) has  an uncertain visual orbit with  a millenium-long period\n\\citep{Ary2015b}.  Its secondary component was resolved in 2004 into a\n0\\farcs13  pair CVN~27  Ba,Bb by  \\citet{Chauvin2010},  using adaptive\noptics.   Independently,   a  subsystem  TOK~50   Aa,Ab  with  similar\nseparation  was  discovered in  2009  by  \\citet{TMH10} using  speckle\ninterferometry.  In  fact,  the   same  subsystem  Ba,Bb  was  wrongly\nattributed to  the primary component;  its published measures  at SOAR\nwith  angle inverted  by 180\\degr  ~match the  preliminary  orbit with\n$P_{\\rm Ba,Bb}=20$ years.  The pair  TOK~50 Aa,Ab does not exist.  The\ncomponent Bb is fainter than Ba by  3.5 mag in the $V$ band and by 1.1\nmag  in the  $K$ band;  its  lines are  not detected  in the  combined\nspectrum of all stars.\n\n\n\n\\begin{figure}\n\\plotone{fig11.eps}\n\\caption{Visual orbits of HIP~80448  A,B and Ba,Bb (WDS J16253$-$4909,\n  COO~197 and CVN~27).\n\\label{fig:COO197}\n}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig12.eps}\n\\caption{CCF (left)  and RV curve (right) of HIP 80448 Aa,Ab. \n\\label{fig:80448}\n}\n\\end{figure}\n\n\n\nFigure~\\ref{fig:COO197} shows the  positions of resolved components on\nthe sky and the fitted orbits. Considering that the outer orbit is not\nconstrained by  the data, I fixed  its period to $P_{\\rm  A,B} = 1950$\nyears  and  its  axis to  $a_{\\rm  A,B}  =  4\\farcs64$ to  adjust  the\ndynamical mass sum to its  estimated value, 3.5 ${\\cal M}_\\odot$.  The\norbit of Ba,Bb with $P_{\\rm Ba,Bb}  = 20$ years yields the mass sum of\n1.8 ${\\cal  M}_\\odot$, close to  the estimated one. The  ``wobble'' in\nthe positions  of A,Ba  caused by the  subsystem is clearly  seen. Its\nrelative amplitude  $f = 0.40$  corresponds to the mass  ratio $q_{\\rm\n  Ba,Bb} = f\/(1-f) = 0.67$ that argees with the magnitude difference.\n\nThe spectrum  of the main component  A (in fact, blended  light of Aa,\nAb, and Ba) shows stationary lines of Ba and double lines of Aa and Ab\nin  rapid  motion; the  subsystem  Aa,Ab  was  discovered with  CHIRON\n\\citep{survey}.  It is found here  that the orbital period  is\n$P_{\\rm   Aa,Ab}   =   2.3$    days   and   the   orbit   is   circular\n(Figure~\\ref{fig:80448}).  The  mass ratio  $q_{\\rm Aa,Ab} =  0.67$ is\nsimilar  to the mass  ratio $q_{\\rm  Ba,Bb}$, while  the ratio  of dip\nareas corresponds to $\\Delta V_{\\rm  Aa,Ab} = 1.9$ mag.  Comparison of\nestimated and spectroscopic mass sums leads to the orbital inclination\n$i_{\\rm Aa,Ab}  = 66\\degr$  or $i_{\\rm Aa,Ab}  = 114\\degr$. It  is not\ndissimilar to the inclination of   other inner pair, $i_{\\rm Ba,Bb}\n= 109\\degr$, but it is  difficult to believe that these two subsystems\nhave coplanar orbits, given the huge difference of their periods.\n\nThe  components Aa  and Ab  rotate synchronously  with the  orbit. The\nprojected rotation of Ba, $V \\sin i = 14.3$ km~s$^{-1}$, is relatively\nfast, supporting the thesis that this system is young. The presence of\nlithium also suggests  youth.  The four components are  located in the\nCMD at about 0.7 mag above the main sequence.\n\n\nThe  pair  Ba,Bb is  presently  near  the  periastron of  its  20 year\norbit. I  measured the  RV(Ba) from 7.51  to 8.18  km~s$^{-1}$, quite\ndifferent from the center-of-mass  velocity of A, $-2.08$ km~s$^{-1}$.\nThis positive  difference and the positive trend actually  match the orbit\nof  Ba,Bb; I predict  that RV(Ba)  will soon  start to  decrease. The\norbits  of  A,B  and  Ba,Bb   are  not  coplanar,  although  both  are\nretrograde. \n\n\n\n\\subsection{HIP 84789 (triple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig13.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 84789 Aa,Ab. Note that\n  the secondary component Ab has a dip with larger amplitude.\n\\label{fig:84789}\n}\n\\end{figure}\n\n\nThis 5\\farcs6 visual binary STF~2148, discovered in 1832 by W.~Struve,\nhas  an estimated  period of  17\\,kyr.   Double lines  in its  primary\ncomponent  A were  noted by  \\citet{N04}.  The  orbital period  of the\nsubsystem Aa,Ab determined here is $P_{\\rm  Aa,Ab} = 2.3$ days; it is a\ntwin pair  with $q_{\\rm  Aa,Ab} =  0.988$ (Figure~\\ref{fig:84789}). The  deeper CCF\ndip belongs to the less massive component Ab; the more massive star Aa\nrotates a  little faster and has the  dip area 3\\% larger  than Ab, as\nwell  as the  smaller RV  amplitude. The  RVs of  both  components are\nmeasured  with large  errors owing  to the  wide and  low-contrast CCF\ndips; the residuals  to the orbits are also large.   An attempt to fit\nthe  orbit  with non-zero  eccentricity  does  not  result in  smaller\nresiduals, hence the orbit is circular.\n\nThe  estimated masses of  Aa and  Ab are  1.30 ${\\cal  M}_\\odot$ each,\nleading   to   the   orbital   inclination   of   $i_{\\rm   Aa,Ab}   =\n45\\degr$.  Assuming  the  radii  of  1.3  $R_\\odot$,  the  synchronous\nrotation velocity  is  $V  \\sin  i  =  20.2$  km~s$^{-1}$.  The  width  of  the\ncorrelation dips  matches this estimate and suggests  that Aa rotates\nslightly faster and Ab slightly slower than synchronous.\n \nThe two  components A and  B are located  on the CMD above  each other\n(they have  the same  color) because A  contains two equal  stars; the\nmass of B is  very similar to the masses of Aa  and Ab, 1.3 solar. The\ncomponent B is single, as inferred  from the equality of its RV to the\ncenter-of-mass velocity of  A. However, it rotates much  faster, at $V\n\\sin i  \\sim 49$ km~s$^{-1}$. Very likely,  the rotation of Aa  and Ab has\nbeen slowed down by tidal synchronization with the orbit.\n\n\n\n\\section{Summary}\n\\label{sec:sum}\n\n\n\\begin{figure}\n\\plotone{fig14.eps}\n\\caption{Eccentricity vs.  period for members  of hierarchical systems\n  studied here  (large triangles)  and for 467  spectroscopic binaries\n  from  the   MSC  with   primary  masses  from   0.5  to   1.5  solar\n  (crosses). The  dashed line shows  the locus of HIP~78416  Aa,Ab for\n  evolution with  constant angular momentum,  $P(1 - e^2)^{3\/2}  = {\\rm\n    const}$.\n\\label{fig:pe}\n}\n\\end{figure}\n\nProbably by  accident, the periods  of 9 spectroscopic  systems within\nhierarchical  multiples  are equally  divided  between three  distinct\ngroups:  (i)   circular  orbits  with   $P  \\approx  2.3$   days,  (ii)\nintermediate  periods  between  6  and  9  days,  circular  or  nearly\ncircular, and (iii) eccentric orbits  with periods from 21 to 30 days.\nFigure~\\ref{fig:pe}  places these  orbits  on the  period-eccentricity\nplot.   The plus  signs are  467 spectroscopic  binaries  with primary\nmasses from 0.5 to 1.5 ${\\cal M}_\\odot$ from the MSC \\citep{MSC}. When\nthe  eccentric orbits  of the  group (iii)  are  tidally circularized,\ntheir periods  will match those  of group (ii), suggesting  that these\nsubsystems could be formed by  a common mechanism, such as Kozai-Lidov\ncycles with dynamical tides \\citep{Moe2018}.  The periods of group (i)\nare  substantially  shorter,  so  their  formation  history  could  be\ndifferent.\n\nSix out  of the 10 double-lined  binaries studied here  are twins with\nmass  ratio  $q >  0.95$,  while the  lowest  measured  mass ratio  is\n0.67. If  the mass ratios  were uniformly distributed in  the interval\n(0.7, 1.0), where double lines  are detectable, the fraction of twins\nwould be only 0.15, whereas in  fact it is 0.6.  It is established that\ntwins correspond to a well-defined peak in the mass ratio distribution\nof solar-type spectroscopic binaries \\citep{twins}.  They are believed\nto be formed when a low-mass binary accretes a major part of its mass.\nThe mass  influx also creates  conditions for formation  of additional\ncomponents, building  stellar hierarchies ``from  inside out''.  Thus,\ntwins are  naturally produced as inner components  of multiple systems\nin the process of mass assembly.\n\nThe  goal  of   this  study  was  to  determine   unknown  periods  of\nspectroscopic subsystems in several multiple stars. Although this goal\nis reached,  I discovered RV  variability of other  visual components\n(HIP 75663A,  76816B, and 78163B),  converting these triples  into 2+2\nquadruples.  The periods  of new  subsystems, presumably  long, remain\nunknown so far.\n\n\n\n\n\n\\acknowledgements\n\nI thank the operator of  the 1.5-m telescope R.~Hinohosa for executing\nobservations  of  this  program  and  L.~Paredes  for  scheduling  and\npipeline processing.  Re-opening of CHIRON  in 2017 was largely due to\nthe enthusiasm and energy of T.~Henry.\n\nThis work  used the  SIMBAD service operated  by Centre  des Donn\\'ees\nStellaires  (Strasbourg, France),  bibliographic  references from  the\nAstrophysics Data  System maintained  by SAO\/NASA, and  the Washington\nDouble Star Catalog maintained at USNO.\n\nThis work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia}\nData Processing and Analysis Consortium (DPAC,\n\\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC\nhas been provided by national institutions, in particular the institutions\nparticipating in the {\\it Gaia} Multilateral Agreement.\n\n\\facilities{CTIO:1.5m, SOAR}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nLet $M$ be an $n$-dimensional  Riemannian manifold, and let\n$\\Sigma$ be a compact surface with boundary which\ncarries a conformal structure; we do {\\bf not} assume\n$\\Sigma$ is orientable.  Many existence theorems for minimal\nsurfaces in the literature \\cite{D31}, \\cite{R}, \\cite{C} find\nsolutions by minimizing the {\\bf  energy} of a mapping \n$f:\\Sigma \\to M$, where both $f$ and the conformal structure of\n$\\Sigma$ are allowed to vary. The energy may be written as\n\\begin{equation} \\label{energy}\n E(f) := \\frac12 \\int_\\Sigma (|f_x|^2 + |f_y|^2)\\, dx\\, dy \n\\end{equation}\nwhere $(x,y)$ are local conformal coordinates for $\\Sigma$, and\nsubscripts are used to denote partial derivatives. Write\n$\\frac{D}{\\partial x}$, etc., for\ncovariant partial derivatives in the Riemannian manifold $M$.\nIf the mapping $f$ and the conformal structure on $\\Sigma$ are\nstationary for $E$, then the  resulting mapping is {\\bf harmonic}: \n\\begin{equation} \\label{harm}\n\\Delta f :=\n\\frac{D}{\\partial x}\\frac{\\partial f}{\\partial x} +\n\\frac{D}{\\partial y}\\frac{\\partial f}{\\partial y} =0\n\\end{equation}\nand {\\bf conformal}:\n\\begin{equation} \\label{conf}\n |f_x| \\equiv |f_y|, \\langle f_x, f_y \\rangle \\equiv 0. \n\\end{equation}\n\nWe shall refer to a conformally parameterized harmonic mapping as\na {\\em conformally parameterized minimal surface} (CMS).  Observe that\nfor any $W^{1,2}$ mapping, $E(f)$ is bounded below by the area \n\\begin{equation} \\label{area}\n A(f) := \\int_\\Sigma |f_x \\wedge f_y |\\, dx\\, dy,\n\\end{equation}\nwith equality if and only if $f$ is a conformal mapping almost\neverywhere.  In particular, a mapping which minimizes $E(f)$ in a\ngeometrically defined class of mappings has minimum area among\nmappings of the admissible class (see \\cite{D39}, p. 232 or Remark\n\\ref{serrin} below).  Moreover, a conformal mapping which is\nharmonic, that is, stationary for $E$, is minimal, that is,\nstationary for area.\n\nA CMS is an immersion except at a discrete set of {\\bf branch\npoints.} Let a point of $\\Sigma$ be given by $(0,0)$ in some local\nconformal coordinates $(x,y)$ for $\\Sigma$. Write $z=x+iy$. Then\n$(0,0)$ is a {\\em branch point} of $f$ of {\\em order} $m-1$ if for some\nsystem of coordinates $u^1, \\dots, u^n$ for $M$ and for some\ncomplex vector $c$, $f(x,y)$ satisfies the asymptotic description \n$$f^1(x,y) + i f^2(x,y) = c z^m + O(z^{m+1})$$ and \n$$ f^k(x,y) = O(z^{m+1}),$$ \n$k=3,\\dots, n$, as $(x,y)\\to (0,0)$.  Here we have written \n$f^k(x,y)$ for the value of the $k^{th}$ coordinate $u^k$ at \n$f(x,y)$, $k=1, \\dots , n$, and $O(z^{m+1})$ denotes any\n``remainder\"\nfunction bounded by a constant times $|z^{m+1}|$. We shall\nrefer to a mapping which is an immersion except at a discrete set\nof branch points as a {\\bf branched immersion} (see \\cite{GOR}).\n\nOur main theorem is \n\\begin{thm} \\label{main}\nSuppose $\\Sigma$ is of the topological type of the real projective\nplane $\\R P^2$. Let a CMS $f:\\Sigma^2 \\to M^3$ have minimum area\namong all $h:\\Sigma \\to M$ which are not homotopic to a constant\nmapping.  Then $f$ is an immersion.\n\\end{thm}\nIn order to prove this theorem, we will distinguish between two\ntypes of branch points: see \\cite{O}, \\cite{GOR}, \\cite{Alt72},\n\\cite{Alt73}, and \\cite{G73}. A {\\bf false branch point} of a\nbranched immersion $f:\\Sigma \\to M$ is a branch point $z_0$\nsuch that the image set $f(U)$ is an embedded surface, under\nanother parameterization, for some neighborhood $U$ of $z_0$ in\n$\\Sigma$.  Otherwise, we call it a {\\bf true branch point}. A\nbranched immersion $f:\\Sigma \\to M$ is said to be {\\bf ramified}\nif there are two disjoint open sets $V,W \\subset \\Sigma$ with\n$f(V) = f(W)$. If $f$ is ramified in every neighborhood of a point\n$z_0$, we say that $z_0$ is a {\\bf ramified branch point}.  Note\nthat any false branch point of a branched immersion must be\nramified.  Osserman showed that in codimension one, a branched\nimmersion $f:\\Sigma^2 \\to M^3$ with a true branch point cannot\nminimize area, see \\cite{O} and Theorem \\ref{oss} below,  in\ncontradiction to assertions of Douglas (p. 239 of \\cite{D32}) and\nof Courant (footnote p. 46 of \\cite{C41}). \nOn the other hand, regarding false branch points, we\nshall extend to nonorientable surfaces the fundamental theorem of\nbranched immersions in \\cite{G75}, and show that if a branched CMS\n$f:\\R P^2 \\to M^n$ is ramified, with any codimension, then there\nis another CMS $\\widetilde{f}:\\R P^2 \\to M$ with at most half the\narea of $f$. \n\nWe would like to acknowledge the interest of Simon Brendle in this\nproblem, whose questions, not used in \\cite{BBEN}, stimulated us\nto investigate this research topic. We are also indebted to the\nlate Jim Serrin for pointing us toward Remark \\ref{serrin}.\n\n\\section{Analysis of branch points}\\label{anal}\n\nThis section reports on material that has appeared in the\nliterature, see especially \\cite{G73}. In this paper, we shall\ndiscuss certain steps in the interest of clarity and completeness. \n\nLet $\\Sigma^2$ be a compact surface with a conformal structure, $M^n$ a\nRiemannian manifold, and let $f: \\Sigma \\to M$ be a CMS. Consider a\nbranch point $z_0 \\in \\Sigma$ for $f$. Write $D$ for the\nRiemannian connection on $M$. Let local conformal\ncoordinates $(x,y)$ for $\\Sigma$ and local coordinates \n$(q_1, \\dots, q_n)$ for $M$ be introduced with $z_0 = (0,0)$ and\n$f(z_0) = (0,\\dots,0)$ in these coordinates. Then equation\n\\eqref{harm} may be rewritten\n$$\\frac{D}{\\partial\\overline{z}}\\frac{\\partial f}{\\partial z}=0,$$\nwhere we write the complex coordinate $z=x+iy$, \n$\\frac{\\partial f}{\\partial z}= \n\\frac12[\\frac{\\partial f}{\\partial x}-i\\frac{\\partial f}{\\partial\ny}]$, and \n$\\frac{D}{\\partial \\overline z}= \n\\frac12[\\frac{D}{\\partial x} +i \\frac{D}{\\partial y}].$\nIn this form,\nwe see that harmonicity implies that the complex tangent vector\n$\\frac{\\partial f}{\\partial z}$ is holomorphic to first order. It\nis readily shown that for some positive integer $m$ and for some\ncomplex tangent vector $c=a+ib$ to $M$ at $f(z_0)$, \n\\begin{equation}\nf(z)=\\mathcal{R}\\{c z^m\\} +O_2(|z|^{m+1}).\n\\end{equation}\nHere we write $\\mathcal{R}\\{v\\}$ for the real part of a complex\nvector $v$, and we have used the big-O notation with the subscript\n$2$, meaning that as $z \\to 0$, the remainder term is bounded by a\nconstant times $|z|^{m+1}$, its first partial derivatives are\nbounded by a constant times $|z|^m$ and its second partial\nderivatives are bounded by a constant times $|z|^{m-1}$. It\nfollows from the conformality condition \\eqref{conf} that the\ncomplex-bilinear inner product \n$\\langle c,c \\rangle = |a|^2 - |b|^2 +2i \\langle a,b \\rangle =0.$ \nChoose a new system of coordinates $p_1, \\dots, p_n$ for $M$ near \n$f(z_0)$ with $\\frac{\\partial}{\\partial p_1}=a$ and \n$\\frac{\\partial}{\\partial p_2}=b$; and a new system of coordinates\n$(\\widetilde{x},\\widetilde{y})$ for $\\Sigma$ with\n$\\widetilde{z}=\\widetilde{x}+i\\widetilde{y}=|a|^{1\/m} z$.\nThen along the mapping $f$,\n$$p_1+ip_2=\\widetilde{z}^m +\\sigma(\\widetilde{z})$$\n and\n$$p_\\ell=\\psi_\\ell(\\widetilde{z}),$$\n$\\ell = 3,\\dots,n$, where \n$\\sigma(\\widetilde{z}), \\psi_\\ell(\\widetilde{z}) = O_2(\\widetilde{z}^{m+1})$. \nWe now define a non-conformal complex parameter $w=u_1+iu_2$ on a\nneighborhood of the branch point in $\\Sigma:$\n\\begin{equation}\\label{defw}\nw:=\\widetilde{z}\\,\n\\Big[1+\\widetilde{z}^{-m}\\sigma(\\widetilde{z})\\Big]^{1\/m}.\n\\end{equation}\nThen $w$ is a $C^{1,\\alpha}$ coordinate on $\\Sigma,$ for some\nH\\\"{o}lder exponent $\\alpha >0$, in terms of\nwhich the coordinate representation of $f$ is simplified:\n\\begin{equation}\\label{anonpar}\np_1+ip_2=w^m, \n\\end{equation} \nand\n$$p_\\ell=\\phi_\\ell(w)=O_2(w^{m+1}),$$\n$\\ell=3,\\dots,n.$\n\nWe now turn our attention to the case n=3 of {\\bf codimension\none}. The self-intersection of the surface is determined by the\nsingle real-valued function $\\phi(w)=\\phi_3(w).$ Define \n$\\overline\\phi(w)= \\phi(\\zeta_m w)$, where \n$\\zeta_m = e^{2\\pi i\/m}$ is a primitive $m^{\\it th}$ root of unity,\nand let $\\Phi(w)=\\phi(w)-\\overline\\phi(w).$ Then the zeroes of $\\Phi$\ncorrespond to curves of intersection of the surface with itself.\nBut both $\\phi$ and $\\overline\\phi$ satisfy the {\\it same}\nquasilinear minimal surface equation in $M$, with the {\\it same}\ncoefficients. Therefore, their difference \n$\\Phi:= \\phi-\\overline\\phi$ satisfies a linear homogeneous PDE:\n\\begin{equation}\\label{Phieq}\n\\sum_{i,j=1}^2 a_{ij}\\Phi_{u_i u_j}+\\sum_{i=1}^2 a_i \\Phi_{u_i} +\na \\Phi =0, \n\\end{equation}\nwhose coefficients, as functions of $w$, are obtained by \nintegrating from the PDE\nsatisfied by $\\phi$ to the PDE satisfied by $\\overline\\phi$ along\nconvex combinations. We have $a_{ij}(0,0) =\\delta_{ij}$. \n\nIt follows that $\\Phi$ satisfies an asymptotic formula \n\\begin{equation}\\label{asymp}\n\\Phi(w) = {\\mathcal R}\\{A w^N\\} + O_2(w^{N+1}),\n\\end{equation}\nfor some integer $N >m$ and some complex constant $A\\neq 0$ \n(see \\cite{HW}). We shall call\n$N-1$ the {\\bf proper index} of the branch point.\nWe may sketch the proof of Hartman and Wintner in \\cite{HW}. We \nrewrite the PDE \\eqref{Phieq} in terms of the complex \ngradient $\\Phi_w= \\frac12(\\Phi_{u_1}- i \\Phi_{u_2})$. \nIf $\\nabla\\Phi(w)=o(w^{k-1})$, then  we test against the function\n$g(w) = w^{-k} (w-\\zeta)^{-1}$, where $\\zeta \\neq 0$ is small, to\nshow that $\\Phi_\\zeta(\\zeta) = a \\zeta^k+o(\\zeta^{k+1})$ for some\n$a\\in \\C$. Proceeding by induction on $k$, one finds formula \n\\eqref{asymp} for some integer $N$ and some $A\\neq 0$, unless \n$\\nabla\\Phi(w) \\equiv 0$. Details are as in \\cite{HW}, pp. 455-458.\n\nThus, there are two alternatives: $\\Phi$ is either identically\nzero or satisfies the asymptotic formula \\eqref{asymp} where \nthe integer $N$ is $\\geq m+1$ and the complex number\n$A\\neq 0$. If $\\Phi \\equiv 0$, then $z_0$ is a false branch point:\nsee section \\ref{false} below.\n\nIf $\\Phi$ is not $\\equiv 0$, then we have a {\\bf true branch point}. \n\n\\begin{thm} \\label{oss}\n(\\cite{O}, \\cite{G73}.)  Suppose $\\Sigma$ is a\nsurface with a conformal structure, $M^3$ a Riemannian manifold \nand $f:\\Sigma \\to M$ a\nmapping which has smallest area in a $C^0$ neighborhood of\n$f$. Then $f$ has no true branch points.\n\\end{thm}\n\\pf\nAs we have just seen, a true branch point $z_0$ has an order \n$m-1 \\geq 1$ and a coordinate neighborhood in $\\Sigma$ with a\n$C^{1,\\alpha}$ complex coordinate $w$, $w=0$ at $z_0$, such that\n$f$ has the representation \\eqref{anonpar} near $z_0$. Adjacent\nsheets $p_3=\\phi(w)$ and $p_3=\\overline\\phi(w)=\\phi(\\zeta_m w)$ \nintersect when $\\Phi(w)=0$, which, according to formula\n\\eqref{asymp}, occurs along $2N\\geq 6$ arcs in $\\Sigma$ forming\nequal angles $\\pi\/N$ when they leave $w=0$. Let one of these\narcs be parameterized as $\\gamma_1:[0,\\varepsilon]\\to \\Sigma$, and\nlet the corresponding arc be $\\gamma_2:[0,\\varepsilon]\\to \\Sigma$, \ndefined by $w(\\gamma_2(t)) =\\zeta_m w(\\gamma_1(t))$.\nThen for all $0\\leq t \\leq \\varepsilon$, \n$\\phi(\\gamma_1(t))=\\phi(\\gamma_2(t))$. Note from formula\n\\eqref{anonpar} that all three coordinates coincide: the mapping\n$f(\\gamma_1(t))= f(\\gamma_2(t)), 0\\leq t \\leq \\varepsilon$.\n\nWe may now construct a Lipschitz-continuous and piecewise smooth\nsurface $\\widetilde{f}$ which has the same area as $f$, but has\ndiscontinuous tangent planes, following Osserman \\cite{O}. The\nidea of the following construction is that the parameter domain\n$D$ may be cut along the arcs $\\gamma_1((0,\\varepsilon))$ and\n$\\gamma_2((0,\\varepsilon))$, opened up to form a lozenge, with two\npairs of adjacent sides originally identified, and then closed up\nalong the remaing two pairs of adjacent sides. \n\nIn detail: choose an open topological disk $D\\subset \\C$ on which the\ncoordinate $w$ is defined, $(0,0)\\in D$, and which is invariant \nunder the rotation taking $w$ to $\\zeta_m w$.  Assume\n$\\gamma_1(\\varepsilon)$ and $\\gamma_2(\\varepsilon)$ are the first\npoints along $\\gamma_1$ resp. $\\gamma_2$ which lie on the boundary of\n$D$. We shall construct a discontinuous,\npiecewise $C^1$ mapping $Q:B_1 \\to D$, such that\n$\\widetilde{f}(\\zeta):=f(Q(\\zeta))$ is nonetheless continuous, and\n$Q$ is one-to-one and onto except for sets of measure $0$. Here,\n$B_1$ is the disk $\\{z\\in \\C: |z|<1\\}$. Choose\npoints $A_i=\\gamma_i(\\varepsilon\/2)$, $i=1,2$. Then $D$ is\nbroken along $\\gamma_1$ and $\\gamma_2$ into two curvilinear pentagons \nwith vertices $\\gamma_1(\\varepsilon), A_1, (0,0), A_2$ and\n$\\gamma_2(\\varepsilon).$ The edges of these pentagons are \n$\\gamma_1([\\varepsilon\/2,\\varepsilon])$,  $\\gamma_1([0,\\varepsilon\/2])$,\n$\\gamma_2([0,\\varepsilon\/2])$, $\\gamma_2([\\varepsilon\/2,\\varepsilon])$ \nand one of the two arcs of $\\partial D$ with endpoints \n$\\gamma_1(\\varepsilon)$ and $\\gamma_2(\\varepsilon)$.\nSimilarly, break the unit disc $B_1$ \nalong the interval $(-1,1)$ of the $x$-axis and the interval\n$[-\\frac12, \\frac12]$ of the $y$-axis into two pentagons.\nEach pentagon in $B_1$ will be bounded by four line segments, an\ninterval along the $y$-axis being used twice, plus the upper or\nlower half-circle of $\\partial B_1$.  Denote the points\n$a=(0,\\frac12),$ $e=(0,-\\frac12)$, $c_1=(1,0)$, $c_2=(-1,0)$ and\ngive the origin $(0,0)$ four different names: $b_1$ when approached\nfrom the first quadrant $\\{x>0,y>0\\}$, \n$b_2$ when approached from the second quadrant $\\{x<0, y>0\\}$,\n$d_2$ when approached from the third quadrant $\\{x<0, y<0\\}$, and\n$d_1$ when approached from the fourth quadrant $\\{x>0, y<0\\}$. $Q$\nwill map the pentagon in $B_1$ in the upper half-plane $y>0$ to the pentagon\nin $D$ lying counterclockwise from $\\gamma_1$ and clockwise from\n$\\gamma_2$, with $Q(c_1)=\\gamma_1(\\varepsilon)$, $Q(b_1)=A_1$,\n$Q(a)=(0,0)$, $Q(b_2)=A_2$, and $Q(c_2)=\\gamma_2(\\varepsilon)$.\nThis describes $Q$ on the boundary of one of the two pentagons;\nthe other pentagon is similar. The\ninterior of each pentagon may be made to correspond by a $C^1$\ndiffeomorphism. We require $Q$ to be continuous along the\n$x$-axis. Of course, $Q$ is discontinuous along the intervals\n$0<y<\\frac12$ and $-\\frac12<y<0$\nof the $y$-axis. However, we can regain the continuity of\n$\\widetilde{f}$ by requiring that for $0\\leq y \\leq \\frac12$, along\nthe interval from $b_2$ to $a$, $Q(0,y)=\\gamma_2(\\varepsilon y)$ and\nalong the interval from $b_1$ to $a$, \n$Q(0,y)=\\gamma_1(\\varepsilon y)$. Similarly, $Q$ will map the\npentagon in the lower half-plane to the pentagon lying\ncounterclockwise from $\\gamma_2$ and clockwise from \n$\\gamma_1$, taking care that for $-\\frac12 \\leq y \\leq 0$, \n$Q(0,y)=\\gamma_2(-\\varepsilon y)$ as approached from the third\nquadrant, and $Q(0,y)=\\gamma_1(-\\varepsilon y)$ as approached from\nthe fourth quadrant.\n\nThen the continuous, piecewise $C^1$ image surfaces\n$\\widetilde{f}(B_1)$ and $f(D)$ consist of the same pieces of\nsurface in $M$, and therefore have the same area. But the tangent\nplanes of $\\widetilde{f}$ are discontinuous along the $y$-axis at\neach $y$ in the open intervals $-\\frac12 < y < 0$ and \n$0 < y < \\frac12$, which implies that $\\widetilde{f}$, and\ntherefore $f$, does not have minimum area: smoothing\n$\\widetilde{f}$ near these arcs reduces its area to first order.\n\nNote that by choosing $\\varepsilon$ small, we may make $\\widetilde{f}$\narbitrarily close to $f$ in the uniform topology.\n\\qed\n\n\\begin{obs}\\label{courant} \nR. Courant (see p. 123 of \\cite{C}) asked whether the curves of \nself-intersection near any true branch point of a minimal surface in\nEuclidean $\\R^3$ meet at equal angles at the branch point. We may\nobserve that this conjecture is partially correct, as seen above:\nthe curves of intersection given by $\\Phi(w)=0$ make equal angles\nat the branch point.  However, the totality of the curves of\nintersection may \nmake a variety of angles at a branch point of order \n$m-1\\geq 3$. Specifically, if $m=4$ and the proper index $N-1=5$,\nthen the zeroes of $\\Phi(w)=\\phi(w)-\\phi(\\zeta_4 w)$ will occur\nalong $2N=12$ curves forming equal angles in the parameter plane,\nand therefore (since angles in the $w$-plane at $0$ are multiplied\nby $m=4$ in $\\R^3$) equal angles in the\ntangent plane to $f(\\Sigma)$ at the branch point.  But their images are\nonly those curves of intersection coming from {\\bf successive} pairs of\nthe four ``sheets\" of the surface. We also have the intersection\nof {\\bf non-successive} sheets, which are the images of zeroes of\n$\\Phi_2(w):=\\phi(w)-\\phi(-w)$, since $\\zeta_4^2=i^2=-1$. The leading\nterm $A w^6$ of the asymptotic formula \\eqref{asymp} cancels when\nwe compute $\\Phi_2(w)=\\phi(w)-\\phi(-w)=\\Phi(w)+\\Phi(iw)$. \n\nIt follows from the PDE satisfied by $\\phi(w)$ and by $\\phi(-w)$ \nthat $\\Phi_2$ satisfies an\nelliptic PDE analogous to \\eqref{Phieq}, and therefore an\nasymptotic relation analogous to \\eqref{asymp} with leading term\n$A_2w^{N_2}$ for some integer $N_2>N$ and some \n$A_2\\in \\C\\backslash \\{0\\}$. That is, the curves of intersection\nof non-successive sheets form a family of equally spaced\ndirections, which are presumably independent of the directions of\nthe curves of intersection of successive sheets.  This philosophy\nis justified by the following explicit example with $N=6$ and\n$N_2=7$.\n\nChoose $a,b\\in\\C\\backslash \\{0\\}$. Using the Weierstra\\ss\\ representation\n(see \\cite{O69}, p. 63) for a minimal surface $f:\\C \\to \\R^3$ in\nEuclidean 3-space, based on the polynomials $4z^3$ and\n$2az^2+2bz^3$ (the latter representing the Gau\\ss\\ map in\nstereographic projection), we have the specific CMS with \n\\begin{eqnarray}\nf^1_z(z) = \\Big[1-(az^2+bz^3)^2\\Big]2z^3\\\\\nf^2_z(z) = -i\\Big[1+(az^2+bz^3)^2\\Big]2z^3\\\\\nf^3_z(z) = 4z^3(az^2+bz^3),\n\\end{eqnarray}\nwhich leads to \n$$w^4:=f^1+if^2 = z^4-\\frac{\\ol{a}^2\\ol{z}^8}{2}-\n\\frac{8}{9}\\ol{a}\\ol{b}\\ol{z}^9- \\frac{2}{5}\\ol{b}^2\\ol{z}^{10}$$\nand to\n$$z=w\\Big(1+\\frac{\\ol{a}^2\\ol{w}^8}{8 w^4}+\n\\frac{2\\ol{a}\\ol{b}\\ol{w}^9}{9 w^4}+O_2(|w|^6)\\Big),$$\nvia an extensive, but straightforward, computation. \nRecall that each component $f^k$ of\n$f$ is real and harmonic as a function of $z$. Rewriting \n$$f^3(z)=8\\,\\mathcal{R} \\{ \\frac{a}{6} z^6 + \\frac{b}{7} z^7\\}$$\nas a (non-harmonic) function of $w$, we find \n$$ \\phi(w) = 8\\,\\mathcal{R}\\{\\frac{a}{6}w^6 + \\frac{b}{7}w^7+\n\\frac{\\ol{a}|a|^2}{8} \\ol{w}^8 w^2 +O_2(|w|^{11})\\}:$$\nthe difference of $\\phi$ on successive sheets is \n\\begin{equation}\\label{succ}\n\\Phi(w):=\\phi(w)-\\phi(iw)=\n8\\,\\mathcal{R}\\{\\frac{a}{3}w^6+\\frac{b}{7}(1+i)w^7\\}+ O_2(|w|^{10})\n\\end{equation}\nand on non-successive sheets is\n\\begin{equation}\\label{nonsucc}\n\\Phi_2(w):=\\phi(w)-\\phi(-w)=\n8\\,\\mathcal{R}\\{bw^7\\}+ O_2(|w|^{10}).\n\\end{equation}\n\nFrom the formula \\eqref{succ}, we see that $N=6$, $A=\\frac{8a}{3}$ \nand the curves of intersection of\nsuccessive sheets are curves in $\\R^3$ leaving the branch point\nalong the $(x_1,x_2)$-plane, which is the tangent plane to\n$\\Sigma$ at the branch point, in the $12$ directions\n$(\\cos(4\\theta),\\sin(4\\theta),0)$, where $6\\theta+\\arg(a)$ is an\ninteger multiple of $\\pi$. The $12$ directions are\npaired off to form $6$ curves in $\\R^3$ leaving the branch point \nat equal angles $\\frac{2\\pi}{3}$.\n\nSimilarly, from the formula \\eqref{nonsucc}, we see that $N_2=7$,\n$A_2=8b$ and the curves \nof intersection of nonsuccessive sheets are seven curves leaving\nthe branch point and making equal angles (images of 14 curves in\nthe $w$-plane, paired). The arguments of \n$a,b \\in \\C\\backslash\\{0\\}$ may be given arbitrary values, so that the\nangle between a representative of the family of six curves of\nself-intersection and a representative of the family of seven\ncurves of self-intersection may be chosen arbitrarily. For most\nchoices, these 13 curves in $\\R^3$ will {\\bf not} form equal\nangles at the branch point.\n\n\\end{obs}\n\n\\section{False branch points}\\label{false}\n\nThe elimination of false branch points from an area-minimizing CMS\n$f:\\Sigma \\to M$ is in general only possible by comparison with\nsurfaces $\\Sigma_0$ of {\\it reduced topological type} (see\n\\cite{D39}, \\cite{G77}): for orientable surfaces, $\\Sigma_0$ has \nsmaller genus or the same total genus and more connected components. \nAs an oriented example, we may\nchoose $\\Sigma$ to be a surface of genus $2$ and $\\Sigma_0$ to be\na torus. Then there is a branched covering $\\pi: \\Sigma \\to\n\\Sigma_0$ with two branch points of order one. (Think of $\\Sigma$\nas embedded in $\\R^3$ so that it is invariant under a rotation by\n$\\pi$ about the $z$-axis, and meets the $z$-axis only at two\npoints: the quotient under this rotation is a torus.) Now choose a\nminimizing CMS $f_0: \\Sigma_0 \\to M^n$, and let $f:\\Sigma \\to M^n$\nbe $f = f_0 \\circ \\pi$. Then $f$ has two false branch points.  In\norder to be sure that $f$ minimizes area in its homotopy class, we\nmay choose $M^3$ to be a flat $3$-torus with two small periods and\none large period.  As one sees from this example, in order to show \nthat false branch points do not\noccur, we must assume that $f$ minimizes area among mappings from\nsurfaces of the topological  type of $\\Sigma$ {\\bf and} of lower\ntopological type. This hypothesis was used by J. Douglas (see\n\\cite{D39}), in a strict form, to find the existence of minimal\nsurfaces $f:\\Sigma \\to \\R^3$ with prescribed boundary. For \n$\\R P^2$, however, there are no nonorientable surfaces of lower type.\n\nResults in the literature for false branch points have until now\nassumed that $\\Sigma$ is {\\it oriented}, see \\cite{G73},\n\\cite{Alt73}, \\cite{GOR}, \\cite{G75}, \\cite{G77} and \\cite{T}.\nIn order to treat false branch points for nonorientable\nsurfaces, we will need to extend certain known results. In\nparticular, the following theorem appears in \\cite{G75} for \n{\\em orientable} surfaces, possibly with boundary, including\nsurfaces of prescribed mean curvature vector not necessarily zero.\n\n\\begin{thm}\\label{fund}\n(Fundamental theorem of branched immersions)\nLet $\\Sigma^2$ be a compact surface with boundary endowed with a \nconformal structure, $\\partial\\Sigma$ possibly empty \nand $\\Sigma$ not necessarily orientable.  Let $M^n$ be a Riemannian\nmanifold and $f:\\Sigma \\to M$ a CMS. Assume that the restriction\nof $f$ to $\\partial\\Sigma$ is injective. Then there exists a compact\nRiemann surface with boundary $\\widetilde\\Sigma$, a branched covering \n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ and a CMS \n$\\widetilde{f}: \\widetilde\\Sigma \\to M$ such that \n$f = \\widetilde{f} \\circ \\pi$.  Moreover, the restriction of\n$\\widetilde{f}$ to $\\partial\\widetilde\\Sigma$ is injective.\nFurther, $\\widetilde\\Sigma$ is orientable if and only if $\\Sigma$\nis orientable.\n\\end{thm}\n\\pf\nIf $\\Sigma$ is orientable, then \nTheorem 4.5 of \\cite{G75} provides an orientable quotient surface\n$\\widetilde\\Sigma$, a branched covering \n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ and an unramified CMS \n$\\widetilde f: \\widetilde\\Sigma \\to M$ such that \n$f=\\widetilde f \\circ \\pi$. \n\nThere remains the case where $\\Sigma$ is {\\bf not orientable}.\nAssume, without loss of generality, that $\\Sigma$ is connected. \n\nLet $p:\\widehat\\Sigma \\to \\Sigma$ be the oriented double cover of\n$\\Sigma$, with the induced conformal structure. Then \n$\\widehat\\Sigma$ is connected and orientable, and $p$ is two-to-one. \nThe composition \n$\\widehat{f}=f \\circ p: \\widehat\\Sigma \\to M$ is a CMS, defined on\nan orientable surface, and we may apply Theorem 4.5 of \\cite{G75}\nto find a compact orientable surface with boundary \n$\\widehat{\\widetilde\\Sigma}$, an unramified\nCMS $\\widehat{\\widetilde{f}}:\\widehat{\\widetilde\\Sigma}\\to M$ and\nan orientation-preserving branched covering \n$\\widehat{\\pi}:\\widehat\\Sigma \\to \\widehat{\\widetilde\\Sigma}$ so that\n$\\widehat{f}$ factors as $\\widehat{\\widetilde{f}}\\circ\\widehat{\\pi}$.\n\nNow let $\\widetilde\\Sigma$ be the quotient surface of\n$\\widehat{\\widetilde\\Sigma}$ under the identification of \n$\\widehat{\\pi}(x^+) \\in \\widehat{\\widetilde\\Sigma}$ with \n$\\widehat{\\pi}(x^-) \\in \\widehat{\\widetilde\\Sigma}$ whenever \n$x^\\pm \\in \\widehat{\\Sigma}$ and $p(x^+)= p(x^-)$ in $\\Sigma$.\nThen for each $x \\in \\Sigma$, $p^{-1}(x)$ consists of two points\n$x^+,\\ x^-\\in\\widehat{\\Sigma}$ and there are diffeomeorphic\nneighborhoods of $\\widehat{\\pi}(x^+)$ and of $\\widehat{\\pi}(x^-)$\nwhich are thereby identified in $\\widetilde\\Sigma$, with reversal\nof orientation. This implies \nthat $\\widetilde\\Sigma$ is a differentiable $2$-manifold.  Write\n$\\widetilde{p}:\\widehat{\\widetilde\\Sigma}\\to\\widetilde\\Sigma$ for\nthe quotient mapping. Then $\\widetilde{f}:\\widetilde\\Sigma \\to M$\nis well defined such that \n$\\widehat{\\widetilde{f}} = \\widetilde{f}\\circ\\widetilde{p}$.\nAlso, for $x\\in \\Sigma$, the two pre-images $x^+, x^-\\in\n\\widehat{\\Sigma}$ have\n$\\widetilde{p}\\circ\\widehat{\\pi}(x^+)=\\widetilde{p}\\circ\\widehat{\\pi}(x^-),$ \nso that we may define\n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ by \n$\\pi(x):= \\widetilde{p}\\circ \\widehat{\\pi}(x^\\pm)$.\n\nNote that the mappings $p, \\widehat{\\pi}$ and $\\widetilde p$ are\nsurjective, and therefore also $\\pi:\\Sigma\\to\\widetilde{\\Sigma}$.\n\nIn the event that $\\partial{\\Sigma}$ is nonempty, since\n$f=\\widetilde f \\circ \\pi$ restricted to $\\partial\\Sigma$  is\ninjective, it follows readily that the restriction of\n$\\widetilde{f}$ to $\\partial{\\widetilde{\\Sigma}}$ is injective. \n\nThen in the above construction, for each $x\\in \\Sigma$, \n$f$ defines the\nsame piece of surface, with opposite orientations, on\nneighborhoods of $x^+$ and of $x^-$. The branched covering\n$\\widehat\\pi: \\widehat\\Sigma \\to \\widehat{\\widetilde\\Sigma}$\npreserves orientation, implying that\n$\\widehat\\pi(x^+) \\neq \\widehat\\pi(x^-)$.\nSince $\\widehat{\\widetilde\\Sigma}$ is connected, there is a path\nfrom $\\widehat\\pi(x^+)$  to $\\widehat\\pi(x^-)$ whose image in\n$\\widetilde\\Sigma$ reverses orientation. Therefore\n$\\widetilde\\Sigma$ is not orientable. \n\\qed\n\n\\section{An immersion of $\\R P^2$}\n\nWe are now ready to give the proof of the main Theorem \\ref{main}.\nLet $f: \\R P^2 \\to M^3$ be a CMS into a three-dimensional\nRiemannian manifold, which has minimum area among all mappings\n$\\R P^2 \\to M^3$ not homotopic to a constant. Write\n$\\Sigma = \\R P^2$. From Theorem \\ref{oss}, we see that $f$ has no\ntrue branch points. (For this conclusion, it would suffice that $f$\nminimizes area in a $C^0$ neighborhood of each branch point.)\n\nThere remains the possibility of false branch points.\n\nWe first recall the computation of the Euler characteristic of a\nsurface. For a compact, connected surface which is either\norientable or nonorientable, the Euler characteristic\n$$ \\chi(\\Sigma) = 2 - r(\\Sigma),$$\nwhere $r(\\Sigma)$ is the topological characteristic of $\\Sigma$\n\\cite{D39}, also known as the nonorientable genus;\nwe\nshall adopt the term {\\bf demigenus}. If $\\Sigma$ is orientable,\nthen it has even demigenus and genus $\\frac12 r(\\Sigma)$. If \nit is non-orientable, then $\\Sigma$ may be constructed\nby adding $r(\\Sigma)$ cross-caps to the sphere.\nThe demigenus of the sphere equals zero, of $\\R P^2$ equals one, of\nthe torus and the Klein bottle equals two. For other compact\nsurfaces without boundary, the demigenus is $\\geq 3$. \n\nNow according to Theorem \\ref{fund}, there is a compact Riemann\nsurface $\\widetilde\\Sigma$, a branched covering\n$\\pi:\\Sigma \\to \\widetilde\\Sigma$ and an unramified CMS\n$\\widetilde f:\\widetilde\\Sigma \\to M$ such that\n$f = \\widetilde f \\circ \\pi$. We will apply the Riemann-Hurwitz\nformula to the branched covering $\\pi$:\n\\begin{equation}\\label{RH}\n \\chi(\\Sigma) = d \\, \\chi(\\widetilde\\Sigma) - {\\mathcal O}(\\pi),\n\\end{equation}\nwhere $d$ is the degree of $\\pi$, \n${\\mathcal O}(\\pi)$ is the total order of branching of $\\pi$, and\n$\\chi$ is the Euler number.  Suppose that $f$ has a false\nbranch point, or more generally, a ramified branch point.  Then\n${\\mathcal O}(\\pi)\\geq 1,$ and the branched covering $\\pi$ has\ndegree $d \\geq 2$.\n\nUsing the formula \\eqref{RH}, we can determine the\ntopological type of $\\widetilde\\Sigma$. Since $\\Sigma$ is\nhomeomorphic to $\\R P^2$, it has $\\chi(\\Sigma)=1$.  We also know\nthat $d>0$\nand ${\\mathcal O}(\\pi) \\geq 1$. It follows that the demigenus\n$r(\\widetilde\\Sigma) \\leq 1$. Otherwise, the integer\n$r(\\widetilde\\Sigma)$ would be $\\geq 2$, which implies\n$\\chi(\\widetilde\\Sigma) \\leq 0$ and by the formula \\eqref{RH}, \n$1 \\leq -{\\mathcal O}(\\pi)\\leq -1$,  a contradiction.  That is,\n$\\widetilde\\Sigma$ is either the sphere or the projective plane.\nBut according to Theorem \\ref{fund}, since $\\Sigma$ is not\norientable, $\\widetilde\\Sigma$ is not orientable; therefore,\n$\\widetilde\\Sigma$ is homeomorphic to $\\R P^2$.\n\nNote that if $\\widetilde f:\\widetilde\\Sigma \\to M$ were homotopic\nto a constant mapping, then so would be $f = \\widetilde f \\circ\n\\pi$.\n\nOn the other hand,\nthe area of $f$ equals the area of $\\widetilde f$ times the degree\n$d$ of $\\pi$. But $d\\geq 2$, so the area of $\\widetilde f$ is at\nmost one-half the area of $f$.  But this would mean that $f$ does\nnot have minimum area among maps $:\\R P^2 \\to M$ not homotopic to\na constant mapping, contradicting our hypothesis.  This implies\nthat $f$ has no branch points, and is therefore an immersion.  \n\\qed\n\n\\begin{rem}\nWe have treated conformally parameterized {\\em minimal} surfaces\nin this paper. However, the proofs go through with only minor\nchanges for projective planes of nonzero {\\em prescribed} mean\ncurvature $H:M^3 \\to \\R$, provided that $f(\\Sigma)$ has a\ntransverse orientation.  This can occur only when $M$ is\nnon-orientable.\n\nIt also appears plausible that a version of Theorem \\ref{fund} can be\nextended to the more general case of mappings satisfying the {\\em\nunique continuation property}, see \\cite{GOR}.  \n\\end{rem}\n\n\\begin{rem}\\label{serrin}\nAn alternative approach to branch points of minimal surfaces\nof the type of the disk appears in the recent book \\cite{T} by \nTromba. The second and higher variations of energy $E$ \n(see \\eqref{energy}) of a CMS $f$ are computed in a\nneighborhood of a branch point $z_0$, and the lowest nonvanishing\nvariation is shown to be negative if the branch point is\nnonexceptional. This is defined in terms of the {\\bf index} $i>m$ \nof $z_0$, where $i+1$ is the order of contact of the mapping\nwith the tangent plane at the branch point. Note that the proper\nindex $N$ is $\\geq i$ (recall the definition \\eqref{asymp}).\nIf $i+1$ is an integer multiple of $m$, where $m-1$ is the order\nof the branch point, then the branch point is called {\\em\nexceptional}. Tromba also shows that exceptional interior branch\npoints will not occur, provided that the mapping has minimum area\n$A$ among surfaces with the same boundary curve. \n\nIn fact, minimizing area and minimizing energy, under such Plateau\nboundary conditions, are equivalent properties of a Lipschitz\ncontinuous mapping $f:B \\to \\R^n$, where $B$ is the unit disk in\n$\\R^2$.  Namely, if $E(f) \\leq E(g)$ for all Lipschitz-continuous\nmappings $g:B \\to \\R^n$ defining the same boundary curve, then\n$A(f) \\leq A(g)$ for all such $g$, as we now show. \n\nOtherwise, for some $g:B \\to \\R^n$ with the same Plateau boundary \nconditions as $f$, $A(f) > A(g)$. Write $\\eta = A(f)-A(g)>0$.\nApproximate $g$ with $g^\\delta(w):=(g(w),\\delta w) \\in \\R^{n+2}$.\nThen $g^\\delta$ is a Lipschitz immersion, so there are conformal\ncoordinates $\\widetilde w=:F^{-1}(w)$ for some bi-Lipschitz \nhomeomorphism $F:B \\to B$ which preserves $\\partial B$ (see\n\\cite{M}). Write\n$\\widetilde g^\\delta(\\widetilde w):= g^\\delta(F(\\widetilde w))$ for\nthe conformal mapping with the same image as $g^\\delta$. Define\n$\\widetilde g:B \\to \\R^n$ by composing $\\widetilde g^\\delta$ with\nthe projection from $\\R^{n+2} \\to \\R^n$.  Then the energy \n$E(\\widetilde g^\\delta) = E(\\widetilde g) + \\delta^2 E(F)$.\nAlso, the area $A(g^\\delta) \\leq A(g) + C\\delta$ for some constant\n$C$. It follows that\n$$E(\\widetilde g)\\leq E(\\widetilde g^\\delta)=A(\\widetilde\ng^\\delta)=A(g^\\delta)\\leq A(g) + C\\delta <A(f) \\leq E(f)$$\nif $\\delta$ is chosen small enough that $C\\delta <\\eta$. This\nimplies that $f$ does not minimize energy, a contradiction.\n\nThe converse implication, that if $f$ minimizes area then it\nminimizes energy, follows similarly.\n\nWe may observe that with this remark, Tromba's book \\cite{T} gives\nan independent proof that a mapping from the disk into $\\R^3$\nwhich minimizes energy for prescribed Plateau boundary conditions\nis an immersion in the interior. \n\nIn addition, the book contains new, partial results on\nboundary branch points.\n\n\\end{rem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec:intro}}\n\nDark energy is a premier mystery of cosmology and high energy physics. \nTo address this, NASA and the Department of Energy (DOE) are funding the \nJoint Dark Energy Mission (JDEM).  Because the physical nature of \ndark energy is so unknown, it is challenging to quantify simply and \naccurately the science requirements for learning about the physical \norigin.  To assess approaches to a figure of merit for the dark \nenergy science reach of different JDEM architectures, NASA and DOE \nformed the Figure of Merit Science Working Group (FOMSWG). \n\nFOMSWG found in a preliminary report \\cite{fomswg} that \nthe figure of merit used by the Dark Energy Task Force \\cite{detf}, \ngiven by the inverse area of the likelihood contour in the dark energy \nequation of state plane $w_0$-$w_a$ was reasonable for the task.  Here the \nequation of state (EOS) as a function of scale factor is given by \n$w(a)=w_0+w_a(1-a)$.  \nThis form has been tested for physical accuracy and against bias, and has \nbeen shown to faithfully reproduce relations for observables such as \ndistances and Hubble parameters to an accuracy of $10^{-3}$ \\cite{calib}, \nbetter than needed for JDEM. \n\nThese results, and especially the calibration relations of \\cite{calib}, \nshould substantially allay concerns about using a particular functional \nform.  Recall that the current issue is how to {\\it project\\\/} simulated \nconstraints of various JDEM scenarios; once the data are in hand one will \ncarry out the analysis through a diversity of methods, and test \nspecific models directly against the data without an intermediate form. \nNevertheless, one can reasonably consider an {\\it additional\\\/} \n(not replacement) approach to build confidence that the conclusions \non science design were robust.  An example is principal component \nanalysis (see, e.g., \\cite{hutstar,albern,deplpca,mort} for \napplication to dark \nenergy), which has the capability of covering a wide variety of \nfunctional forms.  \n\nThe assessment of whether principal components (PCs) add new insights \nto JDEM design projections depends on appropriate scientific criteria. \nHere we present brief cautions about possible oversimplifications of \ninterpretation, titling the following section headings with some of the \npossible misunderstandings. \n\n\n\\section{Uncertainties are Small, So We Know the Answer?} \n\nThe principal components compose the EOS through \n\\begin{equation} \nw(a)-w_{\\rm b}(a)=\\sum_i \\alpha_i\\,e_i(a), \\label{eq:pc} \n\\end{equation} \nwhere $w_{\\rm b}$ is the baseline EOS to compare to (e.g.\\ $w=-1$), \n$e_i$ are the eigenmodes \nof the Fisher matrix for the particular experiment, and $\\alpha_i$ \nare the mode coefficients.  \n\nGiven the information from some \n(simulated) experiment, one forms the Fisher information matrix and \ndiagonalizes it.  The rows of the diagonalization matrix are the \neigenvectors, or modes $e_i(a)$.  If \nthe Fisher matrix was formed with respect to the baseline model, then \nthe expectation value of the coefficients $\\langle\\alpha_i\\rangle=0$.  \nThe rms about the mean is $\\sigma_i\\equiv\\sigma(\\alpha_i)$, i.e.\\ the \ninverse square root of the eigenvalues. \n\nOne of the key misapprehensions of PCA is the physical interpretation \nof the uncertainties on the amplitudes $\\alpha_i$ of each mode.  \nAlthough PCs are often ordered by the uncertainties, these values \n$\\sigma_i$ have no physical meaning by themselves.  \nOne has to interpret their magnitude in terms of some distance between \nmodels.  One possibility is adopting the range of $w\\in [-1,0]$, valid \nfor many models (with the upper limit to avoid disrupting early \nmatter domination), and considering values of $\\sigma_i$ as useful \nif they are smaller than one.  For example, FOMSWG imposes priors on \nthe EOS such that $\\sigma_i$ cannot exceed one.  This prescription is \nequivalent to a redefinition \n\\begin{equation} \n\\sigma_{\\rm true} \\rightarrow \\sigma_{\\rm FOM}= \n\\frac{\\sigma_{\\rm true}}{\\sqrt{1+\\sigma_{\\rm true}^2}}\\,. \\label{eq:sigscale}\n\\end{equation} \n\nOne might choose to form a figure of merit given as the reciprocal of \nthe product of all the $\\sigma_i$'s.  This appears similar to a  \nmulti-dimensional extension (cf.\\ \\cite{albern}) of the DETF figure \nof merit written in terms of \n$1\/[\\sigma(w_p)\\,\\sigma(w_a)]$ where $w_p$ is the pivot EOS value. \nHowever, none of this addresses the meaning.  Since every PC by \nconstruction has an uncertainty smaller than or equal to one, does \nthis mean that all give physical insight?  As argued by \n\\cite{linhut,leach,deplpca}, the $\\sigma_i$'s alone capture no physics. \nThe imposition of a prior does define a goal for ``smallness'' of \nthe uncertainty, if somewhat artificially.  However, just as in \nmost astronomy, the true information is held by the \nsignal-to-noise ratio, \n\\begin{equation} \nS\/N=\\left[\\sum \\frac{\\alpha_i^2}{\\sigma_i^2}\\right]^{1\/2}, \\label{eq:snr}\n\\end{equation} \nwhere each mode contributes $\\alpha_i\/\\sigma_i$ worth of $S\/N$ \n\\cite{kadota,leach}.  This is equivalent to the $\\chi^2$ comparison of \nmode coefficients between a model and the baseline.  So the important \nquantity is not $\\sigma_i$ but $\\sigma_i\/\\alpha_i$.  \n\nConsider if one misestimated the value of some PC coefficient $\\alpha_i$ \nby $\\sigma_i$ (i.e.\\ $1\\sigma$) -- would it change the physics conclusion? \nIn particular, the cosmological constant \ncorresponds to $\\alpha_i=0$ for all PCs.  If some $\\alpha_i$ is \nwithin $1\\sigma$ (or some other confidence level) of zero, then \nthat PC is not effectively contributing to distinction of the \nphysics from a cosmological constant, since there is a high statistical \nprobability that $\\alpha_i$ is consistent with zero.  (This can of course \nbe generalized to differences between any two models.)  \n\nTherefore, how many PCs truly contribute to distinguishing some \nmodel from the cosmological constant $\\Lambda$?  We test this \nfor a continuum of models, over various model classes, assuming \nan imaginary experiment giving 0.3\\% \ndistance measurements between $z=0-3$, plus Planck CMB \ninformation.  Every one of 36 PCs between $a=0.1-1$ \nhave $\\sigma_i$'s less than one, by the construction of \nEq.~(\\ref{eq:sigscale}).  \nHowever, when we carry out the test for various model parameters, \nand model classes, below, we find generically that only $\\sim$2-3 \nPCs have $\\sigma_i\/\\alpha_i<1\/2$, i.e.\\ a $2\\sigma$ deviation from \nthe cosmological \nconstant, even with the extraordinary assumptions of experimental \naccuracy.  This is in agreement with the multiple studies of this \nissue by \\cite{linhut}. \n\nFigures~\\ref{fig:sigalf}--\\ref{fig:sigalf2} illustrate the physical \ndiscriminating power of this idealized experiment as a function of \ncosmological model.  For most models viable under current data, \n2-3 PCs have sufficient $S\/N$ to be useful in distinguishing between \nthe true cosmology and the cosmological constant.  Of course as the \nmodel approaches the cosmological constant, distinction becomes more \ndifficult.  \n\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\psfig{file=sigalfp.ps,width=3.4in}\n\\psfig{file=sigalfm.ps,width=3.4in}\n\\caption{Noise-to-signal ratios $\\sigma_i\/\\alpha_i$ vs.\\ deviation\nfrom the cosmological constant for models with $w(a)=-1+w_a(1-a)$ in\nthe freezing class (top), and $w(a)=(-1-w_a)+w_a(1-a)$ in the thawing \nclass (bottom). \nApproaching the cosmological constant, $w_a\\to0$, the models become \ndifficult to distinguish from $\\alpha=0$, i.e.\\ $\\Lambda$.  \nOnly 3-5 PCs have $S\/N>1$, for any of the models. \nThe intersection of the horizontal line \nwith a PC shows where that mode gives a $2\\sigma$ indication of difference \nfrom $\\Lambda$.  For $|w_a|<0.4$, only 2 PCs meet this criterion. \n(Note breaks in the PC5 curve come from $\\alpha_5$ switching sign.)  \n}\n\\label{fig:sigalf}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\psfig{file=sigalfo.ps,width=3.4in}\n\\caption{As Fig.~\\ref{fig:sigalf}, but for $w(a)=-1+A\\,[1-\\cos(\\ln a)]$ \nwith non-monotonic behavior.  Note $w$ would exceed 0 at some redshifts \nfor $A>0.5$. \n}\n\\label{fig:sigalf2}\n\\end{center}\n\\end{figure}\n\n\nTo exhibit robustness of these results against the \nspecific assumed model, we scan over model parameters for classes of \nqualitatively different dark energy physics, representing the thawing \nclass, freezing class, and a non-monotonic EOS.  In the last case, we \nsee that the ability of PCA not to be locked into a particular \nfunctional form, e.g.\\ a monotonic parametrization, does not make \nmore degrees of freedom significant.  \n\nAgain, what we really care about is the $S\/N$.  In Table~\\ref{tab:sn2} \nwe list the fraction of the total $S\/N$ (Eq.~\\ref{eq:snr}) \ncontributed by the two best modes -- for the case \nfor each dark energy class where higher modes contribute \nthe {\\it most\\\/}.  For the thawing class the higher modes add less \nthan 0.3\\% to the total, and for the freezing class less than 2.8\\% in \nthe most sensitive case, dropping to less than 0.5\\% for modes above \nthe third.  And recall this was for a highly idealized experiment.  In \nthe oscillating model, an ad hoc case designed to be especially \nPCA-friendly, the most extreme case with oscillations reaching $w=0$ \nallows higher modes to contribute up to 14\\% (8\\% for above the third \nmode).  These are actually {\\it overestimates\\\/} of the importance of high \nmodes because as discussed in the next section the $S\/N$ of the higher \nmodes degrades when $w(z>9)$ is marginalized over rather than fixed. \n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular*}{0.95\\columnwidth} \n{@{\\extracolsep{\\fill}} l c c }\n\\hline\nModel & $(S\/N)_2\/(S\/N)_{\\rm all}$ & \\ $(S\/N)_3\/(S\/N)_{\\rm all}$ \\\\ \n\\hline\nFreezing ($w_a=0.7$)& 0.972& \\ 0.995 \\\\ \nThawing ($w_a=-0.5$) & 0.997& \\ 0.9998 \\\\ \nOscillating ($A=0.5$) & 0.862& \\ 0.922 \\\\ \n\\hline \n\\end{tabular*}\n\\caption{Fraction of total signal-to-noise contributed by the first \ntwo, or three, principal components for the case in each dark energy \nclass {\\it most\\\/} favoring PCA high modes.} \n\\label{tab:sn2}\n\\end{center}\n\\end{table}\n\n\nInterestingly, fitting $w_0$-$w_a$ \nrather than using PCA provides distinction from the cosmological \nconstant at the $1\\sigma$ ($2\\sigma$) level for $|w_a|=0.13$ (0.24) -- \nvery comparable to the PCA approach.  That is, the second parameter \nbecomes useful at almost the same values as in the top panel of \nFig.~\\ref{fig:sigalf}, for the PCA freezing case, and is more \nsensitive than in the bottom panel for the PCA thawing case. \nBoth methods demonstrate that significant physical constraints on \nthe dark energy EOS are described by of order two quantities. \n\nThe main point though is that the important information is not in \n$\\sigma_i$, but $\\sigma_i\/\\alpha_i$.  Just because PCA may say \nuncertainties $\\sigma_i$ are small, \nthis does not mean that we know the physics answer. \n\n\n\\section{What Happens at High Redshifts, Stays at High Redshifts?} \n\nWhile the previous demonstration of PC uncertainties and \nsignal-to-noise is \nthe most important of this paper, we also note that assumptions about \nthe high redshift EOS behavior have significant effects.  \nIn practice, the PCs are often computed assuming a cut off at some \nmaximum redshift to avoid complications in calculating \nthe cosmic microwave background (CMB) primordial power spectra and the \ninitial conditions for growth of matter perturbations.  At higher \nredshifts one must therefore choose a particular form or value for \nthe EOS.  FOMSWG fixes $w(z>9)=-1$.  One justification \nis that for the cosmological constant the dark energy density \nfades away quickly into the past, so the exact value of $w$ there is \nunimportant.  However, we have no guarantee that the cosmological \nconstant is the correct model, nor even essentially any current \ninformation on the behavior of the EOS at $z>1$ \\cite{kowalski}. \nAssumptions about the high redshift behavior can lead to significant \nbiases and improper conclusions about the nature \nof dark energy (see, e.g., \\cite{deplpca}).  \n\nBias should not be as severe a problem for a high transition redshift, \n$z=9$, and with many redshift bins at $z\\gtrsim3$ the extra degrees of \nfreedom should ameliorate bias from \nthe prior at $z>9$.  However, fixing $w(z>9)=-1$ does demonstrably \ninfluence the PCs: for example some of the uncertainties \n$\\sigma_i\/\\alpha_i$ that are apparently tightly determined \ncan degrade by a factor three when $w(z>9)$ is not fixed. \nThe modes themselves also change shape, as we discuss next. \n\nThus, what happens at high redshift \ndoes {\\it not\\\/} stay at high redshift, but can affect some important \naspects of the principal component analysis. \n\n\n\\section{Highest is Best?} \n\nDoes the location in redshift of the maximum of a PC, say the first one, \nsay something fundamental about the science reach of the survey or probe \nemployed?  No -- as is clear from Eq.~(\\ref{eq:pc}) the EOS constraints \nfollow from the sum -- with both positive and negative contributions -- \nover all the PCs, not any single one.  \n\nMoreover, Figure~\\ref{fig:pchiz} demonstrates that artificially fixing \nthe high redshift EOS behavior changes the PC shape and peak location.  \nThis shift has nothing to do with the experimental design and so the \npeak location is not a signpost to experiment optimization.  \nAssumptions on $w(z>9)$ can affect probes differently: e.g.\\ supernova \ndistances do not involve $w(z>9)$ while baryon acoustic oscillations, \nbeing tied to high redshift, do. \n\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\psfig{file=PCA_w01wa7.ps,width=3.4in}\n\\caption{The location and shape of peaks in principal component modes \ndepend on the high redshift treatment of the EOS.  The peak location \ntherefore does not in itself translate into the science impact of a given \nexperiment.  Note the effects are more severe for a more realistic \nexperiment. \n}\n\\label{fig:pchiz}\n\\end{center}\n\\end{figure}\n\n\nFinally, even if a PC peak did mean that the experiment is most sensitive \nto the dark energy EOS at a higher redshift, say, that would not imply \nthat the experiment is most sensitive to the nature of dark energy.  \nFor example, dark energy is most influential today, so perhaps one wants \nan experiment most sensitive to the low redshift behavior.   \n(We emphasize that understanding dark energy at low redshift still \nrequires measuring expansion and growth to high redshift, to break \ndegeneracies.)  At best, one could say that probes that weight the dark \nenergy differently in redshift have some complementarity.  \n\nBut there is \nno justification for claiming that the probe with the highest peak, or \nwith the peak at the highest redshift, is the best probe. \n\n\n\n\n\\section{Conclusions} \n\nPrincipal component analysis is a valid technique, \nused appropriately.  Oversimplifying PCA interpretation or \ninadequately appreciating the effect of assumptions \nemployed can lead to misunderstandings and false beliefs.  We \npresent cautionary examples of three apparently plausible but \nunjustified extrapolations.  While data \nshould be analyzed in every reasonable manner, for understanding the \ngeneric cosmology reach the more complicated PCA approach \ndemonstrates no extraordinary advantage over the well-tested and highly \ncalibrated phase space dynamics approach of $w_0$-$w_a$. \n\n\n\\acknowledgments \n\nWe thank Andy Albrecht and Dragan Huterer for detailed discussions of \nPCA issues, and Bob Cahn for useful suggestions.  \nThis work has been supported in part by the Director, Office of Science, \nOffice of High Energy Physics, of the U.S.\\ Department of Energy under \nContract No.\\ DE-AC02-05CH11231. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA well known item of the string\/gauge theory holographic dictionary states that closed strings are the duals of glueballs in the corresponding gauge theories. On the other hand, using the gravity\/gauge theory duality, glueball operators of the boundary field theory correspond to fields in the gravity bulk theory, in particular modes of the dilaton, the graviton, and the RR field. Using this latter correspondence a spectrum of holographic glueballs has been determined \\cite{Csaki:1998qr,Brower:2000rp,Elander:2013jqa}. However, from the same reasoning as in \\cite{Sonnenschein:2014jwa} and \\cite{Sonnenschein:2014bia}, it is probably the spectrum of the strings and not of the bulk fields that would really correspond to the experimental data when moving from large $N_c$ and large $\\lambda$ to the to the realistic values of $N_c=3$ and $\\lambda\\sim 1$. For  mesons \\cite{Sonnenschein:2014jwa} and baryons \\cite{Sonnenschein:2014bia} it was argued that the (open) string configurations admit a modified Regge behavior that matches that of the observed hadrons whereas bulk modes do not admit this property. In this paper we argue that a similar correspondence exists for glueballs, and that the glueballs will probably have a better description in terms of closed strings rather than modes of bulk fields. The idea of glueballs as closed strings has previously been discussed in various terms in works such as \\cite{Bhanot:1980fx,Niemi:2003hb,Sharov:2007ag,Solovev:2000nb,Talalov:2001cp,LlanesEstrada:2000jw,Szczepaniak:2003mr,Pons:2004dk,Abreu:2005uw, Brau:2004xw,Mathieu:2005wc,Simonov:2006re,Mathieu:2006bp,BoschiFilho:2002vd}.\n\nIt is common lore that glueballs and flavorless mesons cannot be distinguished since they carry the same quantum numbers and that the corresponding resonances encountered in experiments are in fact generically linear combinations of the two kinds of states. If, however, we refer to the stringy description of hadrons then, since mesons and glueballs correspond to open and closed strings respectively, there are certain characterizing features with which one can distinguish between them.\n\nThe most important difference between the open string mesons and the closed string glueballs is the slope, or equivalently the (effective) tension. It is a basic property of strings (see sections \\ref{sec:lightcone} and \\ref{sec:flat_string}) that the effective tension of a closed string is twice that of an open string and hence there is a major difference between the two types of strings, as\n\\begin{equation} \\ensuremath{\\alpha^\\prime}_{closed}= \\frac12 \\ensuremath{\\alpha^\\prime}_{open}\\qquad  \\rightarrow \\qquad \\ensuremath{\\alpha^\\prime}_{gb}= \\frac12 \\ensuremath{\\alpha^\\prime}_{meson} \\,. \\end{equation}\nThus the basic idea of this paper is that one should be able to distinguish between glueballs and flavorless mesons by assigning some of them to certain trajectories with a mesonic slope $\\ensuremath{\\alpha^\\prime}_{meson}\\sim 0.9$ GeV$^{-2}$ and others to trajectories with a glueball slope of $\\ensuremath{\\alpha^\\prime}_{gb}\\sim 0.45$ GeV$^{-2}$.\n\nThe slope is not the only thing that is different between open and closed strings. It follows trivially from the spectrum of closed strings (see section \\ref{sec:lightcone}) that in the critical dimension it has an intercept that is twice that of the open string. However, we are interested in strings in four dimensions rather than the critical dimension, and there, as will be discussed in section \\ref{sec:non_critical_string}, the determination of the intercept is still not fully understood. Thus the intercept cannot currently serve as a tool for identifying glueballs.\n\nAnother important difference between open and closed string hadrons is in their decay mechanisms. Based on the holographic description of a meson as a string that connects to flavor branes at its two endpoints, it was determined in \\cite{Peeters:2005fq} that the width of decay of a meson of mass \\(M\\) behaves like\n\\begin{equation} \\Gamma \\sim \\left(\\frac{ 2 M}{\\pi T} - \\frac{m^1_{sep}+m^2_{sep}}{2T}\\right)  e^{\\frac{-{m^q_{sep}}^2}{T}} \\,, \\label{eq:holo_width}\\end{equation}\nwhere $M$ is the mass of the meson, $m^1_{sep}$ and $m^2_{sep}$ are the masses of the string endpoint quarks in the initial state (assumed here to be small), $m^q_{sep}$ is the mass of the quark and antiquark pair generated by the split of the string and $T$ is the string tension. The factor preceding the exponent is in fact the string length.\n\nAs we discuss in section \\ref{sec:decays}, for the case of a closed string decaying into two open strings the width will be proportional to the string length squared, and the single exponential suppressing factor will be replaced by\n\\begin{equation} e^{\\frac{-m^q_{sep}{}^2}{T}}e^{\\frac{-m^{q^\\prime}_{sep}{}^2}{T}} \\,,\\end{equation}\nwhere $m^{q}_{sep}$ and $ m^{q^\\prime}_{sep}$ are the masses of each of the two quark-antiquark pairs that will have to be created in the process.\n\nThus it is clear that the width of a glueball should be narrower than that of the corresponding meson open string, particularly for decay channels involving heavier quarks like $s$, $c$ and $b$. This can serve as an additional tool of disentangling between mesons and glueballs. We list one distinguishing feature of a glueball decaying into two mesons in section \\ref{sec:decays}.\n\nThe main motivation of reviving the description of mesons and baryons in terms of open strings in \\cite{Sonnenschein:2014jwa} and \\cite{Sonnenschein:2014bia} has been the holographic string\/gauge duality. The same applies also to the closed string picture of glueballs.   The spectra of closed strings in a class of holographic confining models was analyzed in \\cite{PandoZayas:2003yb}. The result was that the relation between the mass and angular momentum takes the following form:\n\\begin{equation} J = \\ensuremath{\\alpha^\\prime}_{gb} (E^2 - 2 m_0 E) + a \\,, \\end{equation}\nwhere $\\ensuremath{\\alpha^\\prime}_{gb}$ is the corresponding slope, $E$ is the mass of the glueball, $a$ is the intercept, and $m_0$ is a parameter that can be either positive or negative and is determined by the particular holographic model used. Note that this relation modifies the well known linear relation between $J$ and $E^2$. In section \\ref{sec:holo_fits} we discuss the phenomenological implications of this relation and analyze the possibility of grouping flavorless hadrons along such holographic trajectories.\n\nThe main goal of this paper is to perform an explicit comparison between observational data of flavorless hadrons and the resonance states predicted by the models of rotating open string with massive endpoints for the mesons and rotating folded closed strings for glueballs.\n\nUnfortunately there exists no unambiguous way to assign the known flavorless hadrons (the focus in this paper is on the \\(f_0\\) and \\(f_2\\) resonances) into trajectories of mesons and glueballs, but it is clear that \\textbf{one cannot consistently sort all the known resonances into meson trajectories alone}. One of the main problems in identifying glueball trajectories is simply the lack of experimental data, particularly in the mass region between \\(2.4\\) GeV and the \\(c\\bar{c}\\) threshold, the region where we expect the first excited states of the glueball to be found. It is because of this that we cannot find a glueball trajectory in the angular momentum plane.\n\nWe mostly focused then on the radial trajectories of the \\(f_0\\) (\\(J^{PC} = 0^{++}\\)) and \\(f_2\\) (\\(2^{++}\\)) resonances. For the \\(f_0\\) we examined the possibility of identifying one of the states \\(f_0(980)\\), \\(f_0(1370)\\), \\(f_0(1500)\\), or \\(f_0(1710)\\) as the glueball ground state and building the trajectories beginning from those states. This procedure did not show any significant preference for any one of the glueball candidates over the other. For the \\(f_2\\) there is less ambiguity, but still no positive identification. Between the different \\(2^{++}\\) state we find that the two very narrow resonances \\(f_2(1430)\\) and \\(f_J(2220)\\) (the latter being a popular candidate for the tensor glueball) do not belong on meson trajectories.\n\nThe paper is organized as follows. Section \\ref{sec:theory} is devoted to the theory of rotating closed strings. In section \\ref{sec:lightcone} we review the light-cone quantization of the basic bosonic string and describe its spectrum. Next we address the rotating folded string. We present the classical solution and the corresponding Regge trajectory, starting by discussing the case of flat spacetime. We introduce the Polchinski-Strominger term needed to assure two dimensional conformal invariance in non-critical dimension and discuss the problematic result for the intercept for a folded closed string in four dimensions. In section \\ref{sec:holo_string} we review the results of \\cite{PandoZayas:2003yb} for the rotating folded string in holographic backgrounds, and the semiclassical correction obtained there. Section \\ref{sec:decays} is devoted to the decay process of string decaying into two strings. We summarize the result for the decay of an open string into two open strings \\cite{Peeters:2005fq} and generalize it also to the case of a closed string decaying into two open strings. Section \\ref{sec:phenomenology} deals with the phenomenology of the rotating folded string models and the comparison between them and the observational data. We begin by spelling out the basic assumptions of the phenomenological models in section \\ref{sec:fitting_models}. We then present the key experimental players: the $f_0$ and $f_2$ resonances. In \\ref{sec:f0_fits} we propose several assignments of the $f_0$ resonances into radial $(n,M^2)$ trajectories, first into only various mesonic trajectories and then into various possible combinations when singling out some states as glueballs. In \\ref{sec:f2_fits} we describe possible assignments of the $f_2$, first into orbital $(J,M^2)$ trajectories, then into \\((n,M^2)\\) trajectories. Section \\ref{sec:holo_fits} expands on previous sections by using the non-linear trajectory that characterizes the glueballs of holographic models. In section \\ref{sec:lattice} we discuss the spectrum of glueballs that follows from lattice gauge theory models. We review the trajectories determined in lattice simulations and their corresponding slopes. Both types of trajectories, $(J,M^2)$ and $(n,M^2)$, are discussed. Section \\ref{sec:summary} is a summary and discussion of the results and states some open questions. In the appendix \\ref{sec:predictions} we list the predictions of our models for the yet unobserved excited partners of the glueball candidates, based on their Regge trajectories.\n\n\\section{The rotating closed string} \\label{sec:theory}\n\\subsection{Quantized closed string in light cone gauge} \\label{sec:lightcone}\nWe review here the derivation of the spectrum of the bosonic closed string in the light cone gauge. We simply present the derivation in chapter 1 of \\cite{Polchinski:Vol1}, omitting some of the details for brevity's sake. The following treatment is essentially true only for the critical dimension \\(D = 26\\), but we keep a general \\(D\\) in the formulae. We return to this point in section \\ref{sec:non_critical_string}.\n\nWe start from the Polyakov action\n\\begin{equation} S = -\\frac{1}{4\\pi\\ensuremath{\\alpha^\\prime}}\\int d\\tau d\\sigma \\sqrt{-\\gamma}\\gamma^{\\alpha\\beta}\\eta_{\\mu\\nu}\n\\partial_\\alpha X^\\mu \\partial_\\beta X^\\nu \\,.\\end{equation}\nWe define the light cone coordinates \\(x^\\pm = \\frac{1}{\\sqrt{2}}(x^0\\pm ix^1)\\), and set the gauge by making the three requirements\n\\begin{equation} X^+ = \\tau\\,, \\qquad \\partial_\\sigma \\gamma_{\\sigma\\sigma} = 0\\,, \\qquad \\sqrt{-\\gamma} = 1  \\label{eq:lightcone} \\,.\\end{equation}\nThe equations of motion for the transverse coordinates are then simple wave equations and they are generally solved (with closed string boundary conditions, for \\(\\sigma \\in (-\\ell,\\ell)\\)) by\n\\begin{equation} X^i(\\sigma,\\tau) = x^i + \\frac{p^i}{p^+}\\tau + i\\left(\\frac{\\ensuremath{\\alpha^\\prime}}{2}\\right)^{1\/2}\n\\sum_{n\\neq0}\\left[\\frac{\\alpha^i_n}{n}\\exp\\left(-i\\frac{2\\pi n (\\sigma+c\\tau)}{2\\ell}\\right)\n + \\frac{\\beta^i_n}{n}\\exp\\left(i\\frac{2\\pi n (\\sigma-c\\tau)}{2\\ell}\\right)\\right] \\,.\\end{equation}\nThe constant \\(c\\) is related to the coordinate length \\(\\ell\\) and the conserved quantity \\(p^+\\) via \\(c = \\ell\/(\\pi\\ensuremath{\\alpha^\\prime} p^+)\\). Aside from \\(\\ell\\), which is proportional to the physical string length, these constants do not have any significance on their own except in keeping track of units.\n\nThe left and right moving modes, \\(\\alpha^i_n\\) and \\(\\beta^i_n\\), are independent of each other (and hence, commute) and are normalized in such a way that\n\\begin{equation} [\\alpha^i_m,\\alpha^j_n] = [\\beta^i_m,\\beta^j_n] = m\\delta^{ij}\\delta_{m,-n} \\,.\\end{equation}\nThe Hamiltonian has the mode expansion\n\\begin{equation} H = \\frac{p^i p^i}{2p^+} + \\frac{1}{\\pi\\ensuremath{\\alpha^\\prime}}\\left[\\sum_{n>0}\\left(\\alpha^i_{-n}\\alpha^i_n+\\beta^i_{-n}\\beta^i_n\\right)+ A + \\tilde{A}\\right] \\,, \\end{equation}\nnoting that \\((\\alpha^i_n)^\\dagger = \\alpha^i_{-n}\\). \\(A\\) and \\(\\tilde{A}\\) are the c-numbers one gets when normal-ordering the sums. After regularizing the appropriate infinite sums, identical for the left and the right moving modes, and taking contributions from \\(D-2\\) transverse modes, we get the result\n\\begin{equation} A = \\tilde{A} = \\frac{2-D}{24} \\,.\\end{equation}\n\nFrom here we get the spectrum using the mass shell condition \\(M^2 = -p^2 = 2p^+ H - p^i p^i\\), which translates to\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}\\left(\\sum_{n>0}\\left(\\alpha^i_{-n}\\alpha^i_n+\\beta^i_{-n}\\beta^i_n\\right)+ A + \\tilde{A}\\right) \\,,\\end{equation}\nor,\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}\\left(N+\\tilde{N}+A+\\tilde{A}\\right) \\,,\\end{equation}\nwhere \\(N\\) and \\(\\tilde{N}\\) are the total population numbers of the left and right moving modes.\n\nFor comparison, the same treatment of the open string leads to the result\n\\begin{equation} M^2_{open} = \\frac{1}{\\ensuremath{\\alpha^\\prime}}\\left(N+A\\right) \\,.\\end{equation}\nHere we have neither the constant prefactor of two which halves the slope of the closed string, nor do we have two different kinds of modes on the string and the resulting doubling of the intercept.\n\n\\subsubsection{Quantized closed string: The spectrum}\nWhile the left and right moving modes on the closed string are independent, there is one constraint that relates them, affecting the spectrum. After making the gauge choice by imposing the three conditions of eq. \\ref{eq:lightcone} we still have a residual symmetry of \\(\\tau\\)-independent translations of \\(\\sigma\\). This results in the additional constraint\n\\begin{equation} N = \\tilde{N} \\,.\\end{equation}\nThe total number of excitations has to be equal for the left and right moving modes.\n\nThe vacuum state of the closed string is defined as the state annihilated by all \\(\\alpha^i_n\\) and \\(\\beta^i_n\\), for positive \\(n\\). It has \\(N = \\tilde{N} = 0\\), we denote it simply \\(|0\\rangle\\),\\footnote{The vacuum state may also have some center of mass momentum \\(p\\), but we suppress it in this notation.} and its mass is determined by the intercepts:\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}(A+\\tilde{A}) = \\frac{2-D}{6\\ensuremath{\\alpha^\\prime}} \\,.\\end{equation}\nFor \\(D = 26\\) this state is a tachyon, with \\(M^2 = -4\/\\ensuremath{\\alpha^\\prime}\\). The first excited state has \\(N = \\tilde{N} = 1\\), and so is of the form\n\\begin{equation} \\alpha^{i}_{-1}\\beta^j_{-1}|0\\rangle \\end{equation}\nand its mass is\n\\begin{equation} M^2 = \\frac{2}{\\ensuremath{\\alpha^\\prime}}(2+A+\\tilde{A}) = \\frac{26-D}{6\\ensuremath{\\alpha^\\prime}} \\,.\\end{equation}\nIn the critical dimension we have here a massless tensor and a massless scalar.\n\nThe most important feature of the spectrum for our uses is that it forms an infinite tower of states, with the difference between each pair of consecutive states being\n\\begin{equation} \\Delta M^2 = \\frac4\\ensuremath{\\alpha^\\prime} \\,, \\end{equation}\nwith one factor of two coming from the halving of the slope, and the other from the fact that \\(N+\\tilde{N}\\) takes only even values: \\(0,2,4,6,\\ldots\\).\n\n\\subsection{The rotating closed string solution} \\label{sec:rotating_string}\n\\subsubsection{Classical rotating folded string} \\label{sec:flat_string}\n\nHere we use the Nambu-Goto action for the string\n\\begin{equation} S = -\\frac{1}{2\\pi\\ensuremath{\\alpha^\\prime}}\\int d\\tau d\\sigma \\sqrt{-h} \\,,\\end{equation}\nwith\n\\begin{equation} h = \\det h_{\\alpha\\beta}\\,, \\qquad h_{\\alpha\\beta} = \\eta_{\\mu\\nu}\\partial_\\alpha X^\\mu \\partial_\\beta X^\\nu \\,,\\end{equation}\nand \\begin{equation} \\ensuremath{\\alpha^\\prime} = \\frac{1}{2\\pi T} \\,. \\end{equation}\n\nThe rotating folded string is the solution\n\\begin{equation} X^0 = \\tau \\qquad X^1 = \\frac{1}{\\omega}\\sin(\\omega\\sigma)\\cos(\\omega\\tau) \\qquad X^2 = \\frac{1}{\\omega}\\sin(\\omega\\sigma)\\sin(\\omega\\tau) \\,.\\label{eq:rotsol}\\end{equation}\nWe take \\(\\sigma \\in (-\\ell,\\ell)\\) and correspondingly \\(\\omega\\) takes the value \\(\\omega = \\pi\/\\ell\\). The energy of this configuration is\n\\begin{equation} E = T \\int_{-\\ell}^\\ell d\\sigma \\partial_\\tau X^0 = 2T\\ell \\,.\\end{equation}\nThe angular momentum we can get by going to polar coordinates (\\(X^1 = \\rho\\cos\\theta, X^2 = \\rho\\sin\\theta\\)), then\n\\begin{equation} J = T \\int_{-\\ell}^\\ell d\\sigma \\rho^2 \\partial_\\tau \\theta =\n\\frac{T}{\\omega} \\int_{-\\ell}^\\ell d\\sigma \\sin^2(\\omega\\sigma) = \\frac{\\pi T}{\\omega^2} = \\frac{T\\ell^2}{\\pi}\\,.\\end{equation}\nFrom the last two equations we can easily see that for the classical rotating folded string\n\\begin{equation} J = \\frac{1}{4\\pi T}E^2 = \\frac{1}{2}\\ensuremath{\\alpha^\\prime} E^2 \\,.\\end{equation}\n\n\\subsubsection{Quantization of the rotating folded string} \\label{sec:non_critical_string}\nIn a previous section we reviewed the quantization of the bosonic closed string in the critical dimension, \\(D = 26\\). There we have the result\n\\begin{equation} \\frac{1}{2}\\ensuremath{\\alpha^\\prime} M^2 = N + \\tilde{N} - \\frac{D-2}{12} \\,. \\end{equation}\nWe would like to obtain a correction to the classical trajectory of a similar form when quantizing the rotating folded string in \\(D = 4\\) dimensions. In \\cite{Hellerman:2013kba} the intercept was computed in the context of effective string theory where the Polchinski-Strominger (PS) term \\cite{Polchinski:1991ax},\n\\begin{equation} \\mathcal{L}_{PS} = \\frac{26-D}{24\\pi}\\frac{(\\partial_+^2X\\cdot\\partial_-X)(\\partial_-^2X\\cdot\\partial_+X)}{(\\partial_+X\\cdot\\partial_-X)^2} \\label{eq:psterm} \\,, \\end{equation}\ncompensates for the conformal anomaly when working outside the critical dimension. The derivatives are with respect to the variables \\(\\sigma^\\pm \\equiv \\tau\\pm\\sigma\\).\n\nAs was described in the introduction and will be further discussed in section \\ref{sec:holo_string}, a major candidate for describing the glueball is a rotating closed string in a holographic background which lives, by definition, in the critical dimension. One may conclude that in this case the PS term is not needed. However, as was argued in \\cite{Aharony:2009gg}, upon integrating out the massive degrees of freedom of the closed string that resides in the  critical holographic dimension one gets the PS action as part of the effective string action in the non-critical $D$ dimensions.  \n\nThe calculation in \\cite{Hellerman:2013kba} is for a general dimension \\(D\\), with, as already mentioned, the PS term included. In dimensions larger than four the string will rotate in two planes and the angular momentum is characterized by two quantum numbers \\(J_1\\) and \\(J_2\\). The result obtained there for the Regge trajectory of the closed string is\n\\begin{equation} \\frac{\\ensuremath{\\alpha^\\prime}}{2}M^2 = (J_1+J_2) - \\frac{D-2}{12} + \\frac{26-D}{24}\n\\left((\\frac{J_1}{J_2})^\\frac{1}{4}-(\\frac{J_2}{J_1})^\\frac{1}{4}\\right)^2 \\,. \\end{equation}\nThis expression is singular when \\(J_2 = 0\\), which is necessarily the case when \\(D = 4\\), since in four dimensions the rotation is in a single plane. Therefore the expression is not usable precisely in the context in which we would like to use it.\n\nWe can see where this originates by inserting the 4D rotating solution from eq. \\ref{eq:rotsol} into the expression for the PS term, eq. \\ref{eq:psterm}. The expression obtained,\n\\begin{equation} \\mathcal{L}_{PS} = -\\frac{D-26}{24\\pi}\\omega^2\\tan^2(\\omega\\sigma) \\,,\\end{equation}\nis singular when \\(\\omega\\sigma = \\pm\\frac{\\pi}{2}\\), i.e. at the two points \\(\\sigma = \\pm\\frac{\\ell}{2}\\), which are the ``endpoints'', or folding-points, of the rotating folded string, and the integral on \\(\\mathcal{L}_{PS}\\) giving the correction diverges:\n\\begin{equation} \\int_{-\\ell}^\\ell d\\sigma \\mathcal{L}_{PS} = -\\frac{D-26}{12\\pi}\\omega^2\\int_{-\\ell\/2}^{\\ell\/2}d\\sigma\\tan^2(\\omega\\sigma) = -\\frac{D-26}{12\\pi}\\omega\\left(\\tan x-x\\right)|_{x=-\\pi\/2}^{\\pi\/2} \\,.\\end{equation}\nWe see that beneath the divergent \\(\\tan x\\) there is also the finite part\n\\begin{equation} \\frac{D-26}{12}\\frac{\\pi}{\\ell} \\,.\\end{equation}\n\nThe denominator in the PS term is simply \\((\\dot{X}^2)^2\\), so the problem emerges because the endpoints move at the speed of light. The same problem is encountered in the treatment of the open string, but as was shown in \\cite{Hellerman:2013kba} in that case one can introduce a counterterm at the string boundaries that renders the action and correspondingly the intercept finite. In fact it was found out that summing up the contributions to the latter from the PS and from the Casimir term, the \\(D\\) dependence is canceled out between the two terms, and the intercept is given simply by $a= 1$, for all \\(D\\). Another possible approach for regularizing the rotating open string is to add masses to its endpoints. However, the quantization of the system of a rotating string with massive particles on its ends is still not fully understood \\cite{ASY}.\n\nFor the closed string it is not clear how to regularize the system. One potential way to do it might be to add two masses at the two endpoints of the folded string. The resulting system looks like two open strings connected at their boundaries by these masses, but not interacting in any other way. In the rotating solution the two strings lie on top of one another. The boundary condition, which is the equation of motion of the massive endpoint is modified: it is the same as for the open string, but with an effective double tension \\(T \\rightarrow 2T\\), in accordance with the ratio of the slopes of the open and closed strings discussed above. In fact everything else is doubled too. If this process of adding masses on the closed string and taking then the limit of zero mass is a legitimate way to regularize, then it is probable that the result is also simply double that of the open string, as it is for the critical dimension. Obviously, though, even in that case we cannot perform the quantization of the folded closed string since, as mentioned above, we do not fully control the quantization of an open string with massive endpoints.   \n\n\\subsubsection{The closed string in a curved background} \\label{sec:holo_string}\nThe full analysis of rotating closed string in holographic curved backgrounds was performed in \\cite{PandoZayas:2003yb}. We present here the key points in short form.\n\nIf we look at a curved background metric of the form\n\\begin{equation} ds^2 = h(r)^{-1\/2}(-dX^0dX^0+dX^idX^i) + h(r)^{1\/2}dr^2 + \\ldots \\,,\\end{equation}\nwith \\(i = 1,2,3\\) and the ellipsis denoting additional transverse coordinates, the rotating folded string, namely the configuration,\n\\begin{equation} X^0 = l\\tau \\qquad X^1 = l\\sin\\sigma\\cos\\tau \\qquad X^2 = l\\sin\\sigma\\sin\\tau \\,,\\end{equation}\nis still\\footnote{We follow a somewhat different normalization here, taking \\(\\omega = \\pi\/\\ell\\) from the previous section to be \\(1\\), and introducing a common prefactor \\(l\\), but the solution is essentially the same as the flat space solution of section \\ref{sec:flat_string}.} a solution to the string equations of motion provided we take\n\\begin{equation} r(\\sigma,\\tau) = r_0 = Const. \\end{equation}\nwhere \\(r_0\\) is a point where the metric satisfies the condition\n\\begin{equation} \\partial_r g_{00}(r)|_{r=r_0} = 0, \\qquad g_{00}(r)|_{r=r_0} \\neq 0 \\,.\\end{equation}\nThe existence of such a point is also one of the sufficient conditions for the dual gauge theory to be confining \\cite{Kinar:1998vq}. Compared to the folded string in flat spacetime, the energy and angular momentum take each an additional factor in the form of \\(g_{00}(r_0)\\):\n\\begin{equation} E = \\frac{1}{2\\pi\\ensuremath{\\alpha^\\prime}}\\int_{-\\pi}^\\pi g_{00}(r_0)d\\sigma = g_{00}(r_0) \\frac{l}{\\ensuremath{\\alpha^\\prime}} \\,,\\end{equation}\n\\begin{equation} J = T\\int_{-\\pi}^\\pi g_{00}(r_0)\\sin^2\\sigma d\\sigma = g_{00}(r_0) \\frac{l^2}{2\\ensuremath{\\alpha^\\prime}} \\,.\\end{equation}\nDefining an effective string tension \\(T_{eff} = g_{00}(r_0)T\\) and slope \\(\\ensuremath{\\alpha^\\prime}_{eff} = (2\\pi T_{eff})^{-1}\\), we can write the relation\n\\begin{equation} J = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}_{eff} E^2 \\,.\\end{equation}\nThe same factor of \\(g_{00}(r_0)\\) multiplies the effective tension in the open string case, and therefore the closed and open string slopes are still related by the factor of one half, although the open string trajectories have the additional modification which can be ascribed to the presence of endpoint masses \\cite{Kruczenski:2004me,Sonnenschein:2014jwa}. We draw the two types of strings in figure \\ref{fig:closed_open_map}.\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.95\\textwidth]{closed_and_open.png} \\\\\n\t\\caption{\\label{fig:closed_open_map} Closed and open strings in holographic backgrounds (top), and their mappings into flat spacetime (bottom). For the open string, the mapping from curved to flat background adds endpoint masses to the strings \\cite{Kruczenski:2004me,Sonnenschein:2014jwa}, with the vertical segments mapped to the point-like masses in flat space. For the closed string, we look at the simple folded string in both cases. Note that classically the rotating folded string has zero width, and as such would look like an open string with no endpoint masses, and not like in the drawing.}\n\t\t\t\\end{figure}\n\n\nCalculations of the quantum corrected trajectory of the folded closed string in a curved background in different holographic backgrounds were performed in \\cite{PandoZayas:2003yb} and \\cite{Bigazzi:2004ze} using semiclassical methods. This was done by computing the spectrum of quadratic fluctuations, bosonic and fermionic, around the classical configuration of the folded string.\nIt was shown in \\cite{PandoZayas:2003yb} that the Noether charges of the energy  $E$ and angular momentum  $J$ that incorporate the quantum fluctuations, are related to the expectation value of the world-sheet Hamiltonian in the following manner:\n\\begin{equation}\nlE -J = \\int d\\sigma <{\\cal H}_{ws}> \\,.\n\\end{equation}\nThe contributions to the expectation value of the world-sheet  Hamiltonian are from several massless bosonic modes, ``massive\" bosonic modes and massive fermionic modes.  \n For the ``massive\" bosonic fluctuations around the rotating solution one gets a \\(\\sigma\\)-dependent mass term, with equations of motion of the form\n\\begin{equation} (\\partial_\\tau^2-\\partial_\\sigma^2+2m_0^2l^2\\cos^2\\sigma)\\delta X^i = 0 \\end{equation}\nappearing in both analyses, \\(m_0\\) being model dependent. A similar mass term, also with \\(\\cos\\sigma\\), appears in the equations of motion for some fermionic fluctuations as well, the factor of \\(\\cos^2\\sigma\\) in the mass squared coming in both cases from the induced metric calculated for the rotating string, which is \\(h_{\\alpha\\beta} \\sim \\eta_{\\alpha\\beta}\\cos^2\\sigma\\).\n\nThe result in both papers is that the Regge trajectories are of the form\n\\begin{equation} J = \\ensuremath{\\alpha^\\prime}_{closed}(E^2- 2m_0 E) +a  \\,.\\end{equation}\nwhere $m_0$ is a mass parameter that characterizes the holographic model and $a$ is the intercept which generically takes the form $a= \\frac{\\pi}{24}(\\#\\text{bosonic massless  modes} - \\#\\text{fermionic massless modes})$. The two papers \\cite{PandoZayas:2003yb} and \\cite{Bigazzi:2004ze} use different holographic models (Klebanov-Strassler and Maldacena-N\\'{u}\\~{n}ez backgrounds in the former and Witten background in the latter) and predict different signs for \\(m_0\\), which is given as a combination of the parameters specific to the background. In \\cite{PandoZayas:2003yb} \\(m_0\\) is positive, while in \\cite{Bigazzi:2004ze} it is negative. According to \\cite{PandoZayas:2003yb} the slope of the closed string trajectory is left unchanged from the classical case\n\\begin{equation} \\ensuremath{\\alpha^\\prime}\\!_{closed} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}\\!_{open} \\,,\\end{equation}\nwhile the model used in \\cite{Bigazzi:2004ze} predicts an additional renormalization of the slope,\n\\begin{equation} \\ensuremath{\\alpha^\\prime}\\!_{closed} = \\frac{1}{2}\\left(1-\\frac{c}{\\lambda}\\right)\\ensuremath{\\alpha^\\prime}\\!_{open} \\,,\\end{equation}\nfor some small constant \\(c\\), which makes this a smaller effect than that caused by the addition of the \\(m_0\\) mass term.\n\n\\subsection{Other string models of the glueball and the Regge slope}\nIn previous sections we have shown that the expected Regge slope for the closed string is\n\n\\begin{equation} \\ensuremath{\\alpha^\\prime}_{closed} = \\frac12\\ensuremath{\\alpha^\\prime}_{open} \\,, \\end{equation}\nbut other string models of the glueball predict different values for the effective slope of the glueballs, \\(\\ensuremath{\\alpha^\\prime}_{gb}\\).\n\nOne such prediction is based on the potential between two static adjoint SU(N) charges, that, according to lattice calculations, is expected to be proportional to the quadratic Casimir operator. For small distances this added group theory factor can be obtained easily from perturbation theory, and calculations in \\cite{Bali:2000un} show that what is referred to as the ``Casimir scaling hypothesis'' holds in lattice QCD for large distances as well, and this means that the effective string tension also scales like the Casimir operator (as the potential at large distances is simply \\(V(\\ell) \\approx T_{eff}\\ell\\)). Therefore, a model of the glueball as two adjoint charges (or constituent gluons) joined by a flux tube predicts the ratio between the glueball and meson (two fundamental charges) slopes to be\n\\begin{equation} \\frac{\\ensuremath{\\alpha^\\prime}\\!_{gb}}{\\ensuremath{\\alpha^\\prime}_{meson}} = \\frac{C_2(\\text{Fundamental})}{C_2(\\text{Adjoint})} = \\frac{N^2-1}{2N^2} = \\frac{4}{9}\\,, \\end{equation}\nwhere for the last equation we take \\(N = 3\\). For \\(N \\rightarrow \\infty\\) we recover the ratio of \\(1\/2\\), as can be easily seen. An argument from field theory for the double tension of the adjoint string at large \\(N\\) is in \\cite{Armoni:2006ri}.\n\nOther models attempt to tie the closed string to the phenomenological pomeron. The pomeron slope is measured to be \\cite{Donnachie:1984xq}\n\\begin{equation} \\ensuremath{\\alpha^\\prime}\\!_{pom} = 0.25\\:\\text{GeV}^{-2} \\approx 0.28\\times\\ensuremath{\\alpha^\\prime}\\!_{meson}\\,, \\end{equation}\nand the pomeron trajectory is commonly associated with both glueballs and closed strings. One string model that predicts a pomeron-like slope was proposed in \\cite{Isgur:1984bm} and is presented in \\cite{Meyer:2004jc} or in more detail in \\cite{Meyer:2004gx}. It is simply the model of a rotating closed string, with a fixed circular shape. This string has two types of trajectories, a phononic trajectory (excitations propagating along the string) which has \\(\\ensuremath{\\alpha^\\prime}_{phonon} = \\frac{1}{4}\\ensuremath{\\alpha^\\prime}_{open}\\), and an orbital trajectory (the circular string rotating around an axis in the circle's plane), for which \\(\\ensuremath{\\alpha^\\prime}\\!_{orbital} = \\frac{3\\sqrt{3}}{16}\\ensuremath{\\alpha^\\prime}_{open} \\approx 0.32\\times\\ensuremath{\\alpha^\\prime}_{open}\\). If the rotating circular loop were allowed to deform, it would have necessarily flowed towards the flattened folded string configuration that we have been discussing, which always maximizes the angular momentum at a given energy.\n\nThere are also other possibilities of rigidly rotating closed string of other shapes, as in \\cite{Burden:1982zb}, which may give yet another prediction of the ratio between open and closed string Regge slopes. Another related object is the ``\\(\\Delta\\)-shaped'' string, which we mentioned in \\cite{Sonnenschein:2014bia} as one of the stringy models of the baryon. The model is that of three masses with each pair of them connected by a string. This results in what is essentially a closed string with three quarks placed on it, which has lead 't Hooft to remark that such a configuration could be related to a quark-gluon hybrid \\cite{'tHooft:2004he}, rather than a pure glueball.\n\n\\subsection{The decays of the holographic closed string} \\label{sec:decays}\n\\subsubsection{Open string decays}\nThe open string hadron decays when it tears at a point along the string and the two loose ends connect via quantum fluctuations to a flavor brane, creating a quark-antiquark pair. Another way to think of this process is that the string fluctuates, before tearing,\tand when it reaches a flavor brane it connects to it, tears, and the pair is created. When thinking of the decay in this second way, with the fluctuation preceding the tear, it is clear that the quark and antiquark are of the same flavor, a result not a priori guaranteed when the strings tears and then reconnects to the branes. This is illustrated in figure \\ref{fig:decay_open}.\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.95\\textwidth]{decay_open.png}\n\t\\caption{\\label{fig:decay_open} A schematic look at the decay of a holographic open string, in this case a strange meson decaying into a strange meson and a light meson. \\textbf{Top}: the picture where the string tears first, then reconnects to the flavor branes. \\textbf{Bottom}: the string fluctuates up to the brane before tearing, the splits. We prefer the second picture since it assures that flavor is conserved, which is not a priori the case when the string tears at the bottom.}\n\t\t\t\\end{figure}\n\nThe probability that a fluctuation reaches the flavor brane of a quark of flavor \\(q\\) is \\cite{Peeters:2005fq}\n\\begin{equation} e^{-(m_{sep}^q)^2\/T} \\,, \\end{equation}\nwhere the quark mass \\(m_{sep}^q\\) in this context is equal to the string tension times the distance of the brane from the holographic wall.\\footnote{The fact that in this model the mass \\(m_{sep}^q\\) is proportional to \\(T\\) is especially important when considering the opposing limits \\(T \\rightarrow 0\\) and \\(T\\rightarrow\\infty\\).}\n\nSince the tear can occur at any point along the string, we expect the total probability (and hence the total decay width) to be proportional to the string length \\(L\\).\\footnote{In the holographic picture, it is the length of the horizontal segment of the string that is considered. When moving into flat space, it is the length between the two endpoint masses, and the relation \\(M \\propto TL\\) receives corrections from the endpoint masses, as already written in eq. \\ref{eq:holo_width} in the introduction.} We then expect that the total decay width behave like\n\\begin{equation} \\Gamma \\propto Le^{-(m_{sep}^q)^2\/T}\\,, \\end{equation}\nwhere \\(m_{sep}^q\\) is the quark produced in the decay. In \\cite{Sonnenschein:2014jwa} we extracted some values of the quark masses as obtained from the Regge trajectories of mesons. For the light \\(u\/d\\) quarks the masses were small enough so the exponent is close to one, while the \\(s\\) quark showed a mass for which \\(m_s^2\/T \\sim 1\\). We would then say that decays where an \\(\\ensuremath{s\\bar{s}}\\) pair is created are suppressed by a factor of \\(e^{-1}\\) (before taking into account the smaller phase space).\\footnote{In an alternative description \\cite{Cotrone:2005fr,Bigazzi:2006jt}, the decay rate is power-like (rather than exponentially) suppressed with the mass of the quark-antiquark pair.}\n\n\\subsubsection{Rotating closed string}\nThe decay process of a closed string is less simple as the string has to tear twice.\\footnote{Another holographic approach to describe the decay of a glueball into two mesons, based on fields in the bulk and not closed strings was discussed in \\cite{Hashimoto:2007ze,Brunner:2015oqa,Brunner:2015yha}} A single tear in the closed would produce an open string, and it in turn will have to tear again, so at the end of the process we have two open strings. If the closed string is the glueball, then this is the process of a glueball decaying to two mesons. In the total decay width we will have then the string length squared, one factor of \\(L\\) for each time the string tears, as well as two exponents for the two pair creation events:\n\\begin{equation} \\Gamma \\propto L^2\\exp(-\\frac{m_q^2}{T})\\exp(-\\frac{m_{q^\\prime}^2}{T}) \\,. \\label{eq:decay_closed} \\end{equation}\nThis process is illustrated in figure \\ref{fig:decay_closed}.\n\nIf we want to identify a glueball from this basic prediction we have to look at the branching ratios of processes where the presence of the second exponent is significant, namely at processes where pairs of \\(s\\) and \\(\\bar{s}\\) are produced.\n\nThe glueball unlike the meson will have the possibility of decaying into either of the three options: decay into two light mesons with two pairs of light quarks created, into \\(K\\bar{K}\\) with one pair of \\(\\ensuremath{s\\bar{s}}\\) and the other light, or into \\(\\phi\\phi\\) when two pairs of \\(\\ensuremath{s\\bar{s}}\\) are created. The exponents predict the following hierarchy between the three modes:\n\\begin{equation} \\Gamma(Gb\\rightarrow \\text{2 light}) : \\Gamma(Gb\\rightarrow K\\bar{K}) : \\Gamma(Gb\\rightarrow \\phi\\phi) = 1\\,:e^{-1}\\,:e^{-2} \\,. \\end{equation}\nThis ratio will still need to be modified by phase space factors, which in any realistic scenario will be significant and will suppress the \\(\\ensuremath{s\\bar{s}}\\) modes even further. This is because the states we would measure are not too far from the \\(\\phi\\phi\\) threshold of approximately 2 GeV.\n\n\\begin{figure}[tp!] \\centering\n\t\\includegraphics[width=0.95\\textwidth]{decay_closed.png} \\\\\n\t\\includegraphics[width=0.67\\textwidth]{decay_closed_worldsheet.png}\n\t\\caption{\\label{fig:decay_closed} A schematic look at the decay of a holographic closed string to two mesons. (I) The string tears for the first time. (II) An \\(\\ensuremath{s\\bar{s}}\\) pair is produced and the string tears for the second time. (III) A second pair is created, this time of light quarks (i.e. \\(u\\bar{u}\\) or \\(d\\bar{d}\\)), and two open strings are formed. (IV) A different perspective showing more clearly the final product of this decay: a \\(K\\) meson and a \\(\\bar{K}\\). Note that the distances in this schematic between the flavor branes and the wall are not in scale. The bottom figure is the corresponding world sheet, of a closed string opening up at two points and forming two open strings.}\n\t\t\t\\end{figure}\n\n\\clearpage\n\\section{Phenomenology} \\label{sec:phenomenology}\n\\subsection{Basic assumptions and fitting models} \\label{sec:fitting_models}\nWe will be looking at unflavored isoscalar resonances below the \\(c\\bar{c}\\) threshold. These states will be either mesons with the quark contents \\(\\frac{1}{\\sqrt{2}}(u\\bar{u}-d\\bar{d})\\) or \\(s\\bar{s}\\), or glueballs.\\footnote{Some states, such as the \\(f_0(500)\/\\sigma\\), may also be exotic multiquark states.} Correspondingly, we have several types of trajectories. For the light mesons we have the usual linear form,\n\\begin{equation} J + n = \\ensuremath{\\alpha^\\prime} M^2 + a \\,, \\label{eq:traj_lin}\\end{equation}\nwith \\(\\ensuremath{\\alpha^\\prime} = (2\\pi T)^{-1}\\). Note that whenever we use \\(\\ensuremath{\\alpha^\\prime}\\) without a subscript in this paper, it refers to this slope of the linear meson trajectories.\n\nFor \\(s\\bar{s}\\) states, we use the formula for the mass corrected trajectory (as was used in \\cite{Sonnenschein:2014jwa}) defined by\n\t\\begin{equation} E = 2m_s\\left(\\frac{\\beta\\arcsin \\beta+\\sqrt{1-\\beta^2}}{1-\\beta^2}\\right) \\label{eq:massFitE} \\,,\\end{equation}\n\t\\begin{equation} J + n = a + 2\\pi\\ensuremath{\\alpha^\\prime} m_s^2\\frac{\\beta^2}{(1-\\beta^2)^2}\\left(\\arcsin \\beta+\\beta\\sqrt{1-\\beta^2}\\right) \\,.\\label{eq:traj_mass} \\end{equation}\nThese are the trajectories of a rotating string with two masses \\(m_s\\) at its endpoints, and with an added intercept and extrapolated \\(n\\) dependence. \\(\\beta\\) is the velocity of the endpoint mass. The limit \\(m_s \\rightarrow 0\\) (with \\(\\beta \\rightarrow 1\\)) takes us back to the linear trajectory of eq. \\ref{eq:traj_lin}, with the first correction in the expression for \\(J\\) being proportional to \\(\\ensuremath{\\alpha^\\prime} m_s^{3\/2} E^{1\/2}\\).\n\nFor the glueballs we assume linear trajectories of the form\n\\begin{equation} J + n = \\ensuremath{\\alpha^\\prime}\\!_{gb} M^2 + a \\,,\\end{equation}\nand we take \\(\\ensuremath{\\alpha^\\prime}\\!_{gb}\\) to be \\(\\frac{1}{2}\\ensuremath{\\alpha^\\prime}\\), where \\(\\ensuremath{\\alpha^\\prime}\\) is the slope of the mesons as obtained in our fits of the various meson trajectories. A typical value would be between 0.80 and 0.90 GeV\\(^{-2}\\).\n\nIn a later section we examine the possible application of the formula based on the holographic prediction,\n\\begin{equation} J + n = \\ensuremath{\\alpha^\\prime}_{gb}E^2-2\\ensuremath{\\alpha^\\prime}_{gb}m_0E + a\\,.\\end{equation}\nWhen using this formula we will also take \\(\\ensuremath{\\alpha^\\prime}\\!_{gb} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}\\), ignoring the possible correction to the slope, which we assume to be small.\n\nOne assumption which we must state explicitly before continuing to the fits is that there is no mixing of light mesons, \\(\\ensuremath{s\\bar{s}}\\) mesons, and glueballs. It is an open question how strongly glueballs and mesons are mixed, with results varying greatly between different models, from almost maximal mixing to very weak (different results based on different models are collected in \\cite{Crede:2008vw}). In a stringy model, where glueballs are represented by closed strings and mesons by open strings, it seems more natural that they will not mix at all. We also assume that the mixing between the light quark states and the \\(\\ensuremath{s\\bar{s}}\\) is weak, in placing states either on the linear trajectories of the light mesons or on the mass corrected trajectories of the \\(\\ensuremath{s\\bar{s}}\\), the same assumption that was used in \\cite{Sonnenschein:2014jwa} in fitting the \\(\\omega\\) and \\(\\phi\\) mesons. It is not obvious how the possible mixing between the two types of mesons affects the trajectories.\n\n\\subsubsection{The two types of trajectories}\nAlong \\emph{radial trajectories}, or trajectories in the \\((n,M^2)\\) plane, the states differ only by the radial\\footnote{The term should not be confused as having something to do with the radial coordinate of holography.} excitation number \\(n\\), all other quantum numbers constant. Since \\(n\\) is not actually measured we have to assign a value ourselves to the different states, and from there emerges a great ambiguity that we have to solve.\n\nMesons belong on trajectories in the \\((n,M^2)\\) plane with a slope that seems to be slightly smaller than in the \\((J,M^2)\\) plane. The typical values are 0.80--0.85 GeV\\(^{-2}\\) for the former and \\(0.90\\) GeV\\(^{-2}\\) for the latter type of trajectories, as our fits in \\cite{Sonnenschein:2014jwa} have shown. We implicitly assume in the following sections that for the glueballs there will be a similar difference between the slopes in the different planes. When we write that \\(\\ensuremath{\\alpha^\\prime}_{gb} = \\frac12\\ensuremath{\\alpha^\\prime}_{meson}\\) we refer to \\(\\ensuremath{\\alpha^\\prime}_{meson}\\) as it is obtained for the meson fits in the same plane, rather than taking fixed values of \\(\\ensuremath{\\alpha^\\prime}\\). This also serves to restrict the number of parameters in a given fit: we always try to describe all the trajectories using a single value of \\(\\ensuremath{\\alpha^\\prime}\\).\n\nWe should also note that while for the mesons, \\(n\\) naturally takes the values \\(n = 0,1,2,\\ldots\\) along the radial trajectories, the case is not so for glueballs. For the closed strings we noted in section \\ref{sec:lightcone} that the number of left and right moving modes has to be equal, and so \\(n\\), which is really \\(N+\\tilde{N}\\) in this case, should be even: \\(n = 0,2,4,\\ldots\\).\n\nFor the \\emph{orbital trajectories}, or trajectories in the \\((J,M^2)\\) plane, we expect to find, along the leading trajectory of the glueball, the ground state with \\(J^{PC} = 0^{++}\\) followed by the tensor glueball (\\(2^{++}\\)) as its first excited state, and continue to higher states with even \\(J\\) and \\(PC = ++\\).\n\nThe orbital trajectories of the mesons will be constructed as usual, and using the known quark model relations \\(P = (-1)^{L+1}\\) and \\(C = (-1)^{L+S}\\). The relevant trajectories are then expected to have states with \\(J^{PC} = 1^{--}, 2^{++}, 3^{--}, 4^{++}, \\ldots\\). It is worth noting then that for mesons, a \\(0^{++}\\) state is an excited state with \\(L = 1\\) and \\(S = 1\\), and not a part of what we usually take for the trajectory when we use states of increasing \\(J\\).\n\n\\subsection{The glueball candidates: The \\texorpdfstring{$f_0$}{f0} and \\texorpdfstring{$f_2$}{f2} resonances}\nThere is an abundance of isoscalar states with the quantum numbers \\(J^{PC} = 0^{++}\\) (the \\(f_0\\) resonances) or \\(J^{PC} = 2^{++}\\) (\\(f_2\\)). The Particle Data Group's (PDG) latest Review of Particle Physics \\cite{PDG:2014}, which we we use as the source of experimental data throughout this paper, lists 9 \\(f_0\\) states and 12 \\(f_2\\) states, with an additional 3 \\(f_0\\)'s and 5 \\(f_2\\)'s listed as unconfirmed ``further states''. These are listed in tables \\ref{tab:allf0} and \\ref{tab:allf2}. In the following we make a naive attempt to organize the known \\(f_0\\) and \\(f_2\\) states into trajectories, first in the plane of orbital excitations \\((J,M^2)\\), then in the radial excitations plane \\((n,M^2)\\).\n\nThe states classified as ``further states'' are generally not used unless the prove to be necessary to complete the trajectories formed by the other states. The ``further states'' will be marked with an asterisk below.\\footnote{Note that the asterisk is not standard notation nor a part of the PDG given name of a state, we only use it to make clear the status of given states throughout the text.}\n\nIt is not the purpose of this paper to review all the information available on the \\(f_0\\) and \\(f_2\\) resonances, nor to present the different theories and speculations regarding their meson or glueball nature. We usually attempt to form Regge trajectories first, using just the masses and basic quantum numbers, and then verify if the implications regarding the contents of a given state make sense in the light of additional experimental data, namely the different states' decay modes.\n\nFor a more complete picture regarding the spectrum and specifically the interpretation of the different resonances as glueballs, the reader is referred to reviews on glueball physics and their experimental status such as \\cite{Klempt:2007cp,Mathieu:2008me,Crede:2008vw,Ochs:2013gi}, citations therein, and subsequent works citing these reviews.\n\n\n\\begin{table}[t!] \\centering\n\t\t\\begin{tabular}{|l|l|l|l|l|} \\hline\n\t\t\\textbf{State} & \\textbf{Mass} [MeV] & \\textbf{Width} [MeV] & \\textbf{Width\/mass} & \\textbf{Decay modes} \\\\ \\hline\\hline\n\t\t\\(f_0(500)\/\\sigma\\) & 400--550 & 400--700 & 1.16\\plm0.36 & \\(\\pi\\pi\\) dominant\\\\ \\hline\n\t\t\\(f_0(980)\\) & \\(990\\pm20\\) & 40--100 & 0.07\\plm0.03 & \\(\\pi\\pi\\) dominant, \\(K\\overline{K}\\) seen \\\\ \\hline\n\t\t\\(f_0(1370)\\) & 1200--1500 & 200--500 & 0.26\\plm0.11 & \\(\\pi\\pi\\), \\(4\\pi\\), \\(\\eta\\eta\\), \\(K\\overline{K}\\) \\\\ \\hline\n\t\t\\(f_0(1500)\\) & \\(1505\\pm6\\) & \\(109\\pm7\\) & 0.072\\plm0.005 & \\(\\pi\\pi\\) \\([35\\%]\\), \\(4\\pi\\) \\([50\\%]\\), \\\\\n\t\t\t\t\t\t\t\t\t&  & & & \\(\\eta\\eta\\)\/\\(\\eta\\eta\\prime\\) \\([7\\%]\\), \\(K\\overline{K}\\) \\([9\\%]\\) \\\\ \\hline\t\t\n\t\t\\(f_0(1710)\\) & \\(1720\\pm6\\) & \\(135\\pm8\\) & 0.078\\plm0.005 & \\(K\\overline{K}\\), \\(\\eta\\eta\\), \\(\\pi\\pi\\) \\\\ \\hline\n\t\t\\(f_0(2020)\\) & \\(1992\\pm16\\) & \\(442\\pm60\\) & 0.22\\plm0.03 & \\(\\rho\\pi\\pi\\), \\(\\pi\\pi\\), \\(\\rho\\rho\\), \\(\\omega\\omega\\), \\(\\eta\\eta\\) \\\\ \\hline\n\t\t\\(f_0(2100)\\) & \\(2103\\pm8\\) & \\(209\\pm19\\) & 0.10\\plm0.01 & \\\\ \\hline\n\t\t\\(f_0(2200)\\) & \\(2189\\pm13\\) & \\(238\\pm50\\) & 0.11\\plm0.02 & \\\\ \\hline\n\t\t\\(f_0(2330)\\) & \\(2325\\pm35\\) & \\(180\\pm70\\) & 0.08\\plm0.03 &  \\\\ \\hline\n\t\t*\\(f_0\\)(1200--1600) & 1200--1600 & 200--1000 & 0.43\\plm0.29 & \\\\ \\hline\n\t\t*\\(f_0\\)(1800) & \\(1795\\pm25\\) & \\(95\\pm80\\) & 0.05\\plm0.04 & \\\\ \\hline\n\t\t*\\(f_0\\)(2060) & \\(\\sim2050\\) & \\(\\sim120\\) & \\(\\sim0.04\\)--\\(0.10\\) & \\\\ \\hline\n\t\t\\end{tabular} \\caption{\\label{tab:allf0} All the \\(f_0\\) states as listed by the PDG. The last few states, marked here by asterisk, are classified as ``further states''.}\n\t\\end{table}\n\n\t\t\\begin{table}[t!] \\centering\n\t\t\\begin{tabular}{|l|l|l|l|l|} \\hline\n\t\t\\textbf{State} & \\textbf{Mass} [MeV] & \\textbf{Width} [MeV] & Width\/mass & \\textbf{Decay modes} \\\\ \\hline\\hline\n\t\t\\(f_2(1270)\\) & 1275.1\\plm1.2 & 185.1\\plm2.9 & 0.15\\plm0.00 & \\(\\pi\\pi\\) \\([85\\%]\\), \\(4\\pi\\) \\([10\\%]\\), \\(KK\\), \\(\\eta\\eta\\), \\(\\gamma\\gamma\\), ... \\\\ \\hline\n\t\t\\(f_2(1430)\\) & 1453\\plm4 & 13\\plm5 & 0.009\\plm0.006 & \\(KK\\), \\(\\pi\\pi\\) \\\\ \\hline\n\t\t\\(f^\\prime_2(1525)\\) & 1525\\plm5 & 73\\plm6 & 0.048\\plm0.004 & \\(KK\\) \\([89\\%]\\), \\(\\eta\\eta\\) \\([10\\%]\\), \\(\\gamma\\gamma\\) [seen], ... \\\\ \\hline\n\t\t\\(f_2(1565)\\) & 1562\\plm13 & 134\\plm8 & 0.09\\plm0.01 & \\(\\pi\\pi\\), \\(\\rho\\rho\\), \\(4\\pi\\), \\(\\eta\\eta\\), ... \\\\ \\hline\n\t\t\\(f_2(1640)\\) & 1639\\plm6 & 99\\plm60 & 0.06\\plm0.04 & \\(\\omega\\omega\\), \\(4\\pi\\), \\(KK\\) \\\\ \\hline\n\t\t\\(f_2(1810)\\) & 1815\\plm12 & 197\\plm22 & 0.11\\plm0.01 & \\(\\pi\\pi\\), \\(\\eta\\eta\\), \\(4\\pi\\), \\(KK\\), \\(\\gamma\\gamma\\) [seen] \\\\ \\hline\n\t\t\\(f_2(1910)\\) & 1903\\plm9 & 196\\plm31 & 0.10\\plm0.02 & \\(\\pi\\pi\\), \\(KK\\), \\(\\eta\\eta\\), \\(\\omega\\omega\\), ... \\\\ \\hline\n\t\t\\(f_2(1950)\\) & 1944\\plm12 & 472\\plm18 & 0.24\\plm0.01 & \\(K^*K^*\\), \\(\\pi\\pi\\), \\(4\\pi\\), \\(\\eta\\eta\\), \\(KK\\), \\(\\gamma\\gamma\\), \\(pp\\) \\\\ \\hline\n\t\t\\(f_2(2010)\\) & 2011\\plm76 & 202\\plm67 & 0.10\\plm0.03 & \\(KK\\), \\(\\phi\\phi\\) \\\\ \\hline\n\t\t\\(f_2(2150)\\) & 2157\\plm12 & 152\\plm30 & 0.07\\plm0.01 & \\(\\pi\\pi\\), \\(\\eta\\eta\\), \\(KK\\), \\(f_2(1270)\\eta\\), \\(a_2\\pi\\), \\(pp\\) \\\\ \\hline\n\t\t\\(f_J(2220)\\) & 2231.1\\plm3.5 & 23\\plm8 & 0.010\\plm0.004 & \\(\\pi\\pi\\), \\(KK\\), \\(pp\\), \\(\\eta\\eta^\\prime\\) \\\\ \\hline\n\t\t\\(f_2(2300)\\) & 2297\\plm28 & 149\\plm41 & 0.07\\plm0.02 & \\(\\phi\\phi\\), \\(KK\\), \\(\\gamma\\gamma\\) [seen] \\\\ \\hline\n\t\t\\(f_2(2340)\\) & 2339\\plm55 & 319\\plm81 & 0.14\\plm0.04 & \\(\\phi\\phi\\), \\(\\eta\\eta\\) \\\\ \\hline\n\t\t*\\(f_2(1750)\\) & 1755\\plm10 & 67\\plm12 & 0.04\\plm0.01 & \\(KK\\), \\(\\gamma\\gamma\\), \\(\\pi\\pi\\), \\(\\eta\\eta\\) \\\\ \\hline\n\t\t*\\(f_2(2000)\\) & 2001\\plm10 & 312\\plm32 & 0.16\\plm0.02 & \\\\ \\hline\n\t\t*\\(f_2(2140)\\) & 2141\\plm12 & 49\\plm28 & 0.02\\plm0.01 & \\\\ \\hline\n\t\t*\\(f_2(2240)\\) & 2240\\plm15 & 241\\plm30 & 0.11\\plm0.01 & \\\\ \\hline\n\t\t*\\(f_2(2295)\\) & 2293\\plm13 & 216\\plm37 & 0.10\\plm0.02 & \\\\ \\hline\n\t\t\\end{tabular} \\caption{\\label{tab:allf2} All the \\(f_2\\) states as listed by the PDG. The last few states, marked here by asterisk, are classified as ``further states''.}\n\t\\end{table}\n\n\t\\subsection{Assignment of the \\texorpdfstring{$f_0$}{f0} into trajectories} \\label{sec:f0_fits}\n\tIn a given assignment, we generally attempt to include all the \\(f_0\\) states listed in table \\ref{tab:allf0}, sorting them into meson and, if possible, glueball trajectories.\n\t\n\tWe make an exception of the \\(f_0(500)\/\\sigma\\) resonance, which we do not use in any of the following sections. Its low mass and very large width are enough to make it stand out among the other \\(f_0\\) states listed in the table. There is no common consensus regarding the composition of the \\(\\sigma\\). We find that it does not belong on a meson Regge trajectory. If we assume it is a glueball then our model predicts the next state to be at around 2.2 GeV, and, since we assume its width to be proportional to its mass squared (as implied by eq. \\ref{eq:decay_closed}), it would have a width of at least 8 GeV. We hope, in that case, that there is no reason to make such an assumption.\\footnote{The authors of \\cite{Nebreda:2011cp} state that the interpretation of the \\(f_0(500)\/\\sigma\\) as a glueball is ``strongly disfavored'', from what they consider a model independent viewpoint. We found no references that suggest the opposite.} Therefore, we simply ``ignore'' the \\(f_0(500)\\) in the following sections.\n\n\t\\subsubsection{Assignment of all states as mesons}\n\t\tSorting the \\(f_0\\) states into trajectories with a meson-like slope leads to an assignment of the \\(f_0\\)'s into two groups of four:\n\t\\[\\mathrm{Light}:\\qquad980, 1500, 2020, 2200, \\]\n\t\\[\\ensuremath{s\\bar{s}}:\\qquad1370, 1710, 2100, 2330. \\]\nWhile this simple assignment includes all the confirmed \\(f_0\\) states (except the \\(f_0(500)\\)) on two parallel trajectories, it remains unsatisfactory. If there are no glueballs we expect the states in the lower trajectory to be (predominantly) composed of light quarks, while the higher states should be \\(\\ensuremath{s\\bar{s}}\\). This does not match what we know about the decay modes of the different states. For example, the \\(f_0(1370)\\) does not decay nearly as often to \\(K\\bar{K}\\) as one would expect from an \\(\\ensuremath{s\\bar{s}}\\) state. In fact, this assignment of the \\(f_0\\)'s into meson trajectories was proposed in some other works \\cite{Anisovich:2000kxa,Anisovich:2002us,Masjuan:2012gc}, and the mismatch with the decay modes was already addressed in greater detail in \\cite{Bugg:2012yt}.\n\n\t\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(980) as glueball}\n\t\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_980_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0 &\t1350\\plm150 & 1317 & 1505\\plm6 & 1505 & 990\\plm20 & 990 \\\\\n\t\t\t1 &\t1720\\plm6 & 1738 & 1992\\plm16 & 1984 & - & - \\\\\n\t\t\t2 &\t2103\\plm8 & 2075 & ? & 2340 & ? & 2470 \\\\\n\t\t\t3 &\t2325\\plm35 & 2365 & & & & \\\\\n\t\t\t4 & ? & 2620 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit980} The results of the fit to the assignment with \\(f_0(980)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.788\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 3.78\\). The intercepts obtained are (-1.35) for light mesons, (-0.52) for \\(\\ensuremath{s\\bar{s}}\\), and (-0.38) for glueballs. We also list the predicted mass of the next state in each trajectory.}\n\t\t\t\\end{table}\n\n\tIn this and the following sections we pick and single out a state as the glueball ground state and try to build the meson trajectories without it.\n\n\tFirst is the the \\(f_0(980)\\). Assuming it is the glueball then the \\(f_0(2330)\\) is at the right mass to be its first excited (\\(n = 2\\)) partner. However, we find that the two meson trajectories given this assignment,\n\\[\\mathrm{Light:}\\qquad 1370, 1710, 2100,\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1500, 2020,\\]\nalso predict a state very near the mass of the \\(f_0(2330)\\), and according to this assignment, there should be two more \\(f_0\\) states near the \\(f_0(2330)\\), for a total of three. The \\(f_0(2200)\\) has to be excluded.\n\nWe again have to put some states on trajectories that are not quite right for them: the \\(f_0(1710)\\) has a significant branching ratio for its decay into \\(K\\overline{K}\\), while the \\(f_0(1500)\\), which is taken as the head of the \\(\\ensuremath{s\\bar{s}}\\) trajectory, decays to \\(K\\overline{K}\\) less than \\(10\\%\\) of the time.\n\nNote that the assignment above is the same as the one we would make if we excluded the \\(f_0(980)\\) on the grounds of it being an exotic (but non-glueball) state and assumed all the other states are mesons. The \\(f_0(980)\\) is commonly believed to be a multiquark state or a \\(K\\bar{K}\\) ground state,\\footnote{See the PDG's ``Note on scalar mesons below 2 GeV'' and references therein.} and in fact, we will find in following sections that even it is not a glueball, it is better to exclude it from the meson trajectories. The trajectories and masses obtained are in table \\ref{tab:fit980}.\n\t\n\t\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(1370) as glueball}\n\t\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_1370_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0 &\t990\\plm20 & 1031 & 1720\\plm6 & 1733 & 1350\\plm150 & 1350 \\\\\n\t\t\t1 &\t1505\\plm6 & 1488 & 2189\\plm13 & 2103 & - & - \\\\\n\t\t\t2 &\t1795\\plm25 & 1835 & ? & 2400 & ? & 2530 \\\\\n\t\t\t3 &\t2103\\plm8 & 2123 & & & & \\\\\n\t\t\t4 & 2325\\plm35 & 2377 & & & & \\\\\n\t\t\t5 & ? & 2610 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit1370} The results of the fit to the assignment with \\(f_0(1370)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.873\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 10.01\\). The intercepts obtained are (-0.93) for light mesons, (-1.06) for \\(\\ensuremath{s\\bar{s}}\\), and (-0.80) for glueballs. We also list the predicted mass of the next state in each trajectory.}\n\t\t\t\\end{table}\n\nFrom here onwards the states singled out as glueballs are too high in mass for their excited states to be in the range of the \\(f_0\\) states listed in table \\ref{tab:allf0}, that is beneath 2.4 GeV.\n\nExcluding the \\(f_0(1370)\\), we have:\n\\[\\mathrm{Light:}\\qquad [980], 1500, *1800, 2100, 2330\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1710, 2200.\\]\nThe \\(f_0(980)\\) is put here in brackets to emphasize that it is optional. Including or excluding it can affect some of the fitting parameters but the trajectory is certainly not incomplete if we treat \\(f_0(980)\\) as a non-meson resonance and take \\(f_0(1500)\\) as the head of the trajectory.\n\nThe main issue here is that we have to use the state \\(*f_0(1800)\\) to fill in a hole in the meson trajectory, a state that is still considered unconfirmed by the PDG and whose nature is not entirely known. It was observed so far only as an enhancement in the radiative decay \\(J\/\\psi \\rightarrow \\gamma\\omega\\phi\\) and its observers at BESIII \\cite{Ablikim:2012ft} suggest it is an exotic state - a tetraquark, a hybrid, or itself a glueball. More experimental data is needed here.\n\nOther than that we have \\(f_0(2100)\\) as a light meson and \\(f_0(2200)\\) as \\(\\ensuremath{s\\bar{s}}\\). This is the option that is more consistent with the decays, as \\(f_0(2200)\\) is the one state of the two which is known to decay into \\(K\\overline{K}\\) (we again refer to the comments in \\cite{Bugg:2012yt} and references therein). However, in terms of the fit, we might do better to exchange them. It is possible that the proximity of these two resonances to each other affects their masses in such a way that our model can not predict, and this affects badly the goodness of our fit, as can be seen in table \\ref{tab:fit1370}.\n\n\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(1500) as glueball}\n\n\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_1500_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0 &\t1350\\plm150 & 1031 & 1720\\plm6 & 1723 & 1505\\plm6 & 1505 \\\\\n\t\t\t1 &\t1795\\plm25 & 1723 & 2103\\plm8 & 2097 & - & - \\\\\n\t\t\t2 &\t1992\\plm16 & 2029 & ? & 2400 & ? & 2620 \\\\\n\t\t\t3 &\t2325\\plm35 & 2295 & & & & \\\\\n\t\t\t4 & ? & 2530 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit1500} The results of the fit to the assignment with \\(f_0(1500)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.870\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 2.51\\). The intercepts obtained are (-1.58) for light mesons, (-1.03) for \\(\\ensuremath{s\\bar{s}}\\), and (-0.99) for glueballs. We also list the predicted mass of the next state in each trajectory.}\n\t\t\t\\end{table}\n\t\t\t\nTaking the \\(f_0(1500)\\) to be the glueball, then the light meson trajectory will start with \\(f_0(1370)\\), giving:\n\t\\[\\mathrm{Light:}\\qquad 1370, *1800, 2020, 2330,\\]\n\t\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1710, 2100.\\]\nWith \\(f_0(1500)\\) identified as the glueball, this assignment includes all the states except \\(f_0(2200)\\). Incidentally though, the \\(f_0(2200)\\) would have belonged on the glueball trajectory if we had allowed odd values of \\(n\\) for the glueball. In other words, it matches the prediction for the \\(n = 1\\) state of the half slope trajectory beginning with \\(f_0(1500)\\). We could also use \\(f_0(2200)\\) as the \\(\\ensuremath{s\\bar{s}}\\) state and leave out \\(f_0(2100)\\) instead.\n\nThere is no glaring inconsistency in this assignment with the decay modes, but we are again confronted with the state \\(*f_0(1800)\\), which we need to complete the light meson trajectory. We can see from table \\ref{tab:allf0} that the \\(f_0(2020)\\) is wider than other states in its trajectory, whereas we maintain that the ratio between width and mass \\(\\Gamma\/M\\) should be roughly constant along a trajectory. In particular, the last state in the trajectory, \\(f_0(2330)\\), is much narrower than \\(f_0(2020)\\). We can assign the \\(f_0(2330)\\) to the \\(\\ensuremath{s\\bar{s}}\\) trajectory instead, but there is no other argument for that state being \\(\\ensuremath{s\\bar{s}}\\), considering it was observed only in its decays to \\(\\pi\\pi\\) and \\(\\eta\\eta\\). Perhaps the fact that \\(f_0(1370)\\) and \\(f_0(2020)\\) are both quite wide means that there should be two additional states, with masses comparable to those of \\(*f_0(1800)\\) and \\(f_0(2330)\\), that are also wide themselves, and those states will better complete this assignment. The results of the assignment are presented in table \\ref{tab:fit1500}.\n\t\t\t\n\\subsubsection{Assignment with \\texorpdfstring{$f_0$}{f0}(1710) as glueball}\n\n\\begin{table}[tp!] \\centering\n\t\\includegraphics[width=0.60\\textwidth]{glue_fit_1710_f1n.png} \\qquad\n\t\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t\t\t\t\t\\(n\\) & \\multicolumn{2}{|c|}{Light} & \\multicolumn{2}{|c|}{\\(s\\bar{s}\\)} & \\multicolumn{2}{|c|}{Glueball} \\\\ \\hline\n\t\t\t\t\t& Exp. & Thry. & Exp. & Thry. & Exp. & Thry. \\\\ \\hline\n\t\t\t0&\t1350\\plm150 & 1378 & 1505\\plm6 & 1506 & 1720\\plm6 & 1720 \\\\\n\t\t\t1&\t1795\\plm25 & 1777 & 1992\\plm16 & 1977 & - & - \\\\\n\t\t\t2&\t2103\\plm8 & 2102 & ? & 2330 & ? & 2830 \\\\\n\t\t\t3&\t2325\\plm35 & 2383 & & & & \\\\\n\t\t\t4&\t? & 2640 & & & & \\\\ \\hline\n\t\t\t\t\\end{tabular} \\caption{\\label{tab:fit1710} The results of the fit to the assignment with \\(f_0(1710)\\) as the glueball ground state. The slope is \\(\\ensuremath{\\alpha^\\prime} = 0.793\\) GeV\\(^{-2}\\) and the mass of the \\(s\\) quark \\(m_s = 500\\) MeV. This fit has \\(\\chi^2 = 0.71\\). The intercepts obtained are (-1.51) for light mesons, (-0.53) for \\(\\ensuremath{s\\bar{s}}\\), and (-1.17) for glueballs.}\n\t\t\t\\end{table}\n\t\t\t\nExcluding the \\(f_0(1710)\\) from the meson trajectories we can make an assignment that includes all states except the \\(f_0(500)\\) and \\(f_0(980)\\):\n\\[\\mathrm{Light:}\\qquad 1370, *1800, 2100, 2330\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1500, 2020, 2200\\]\n\\[\\mathrm{Glue:}\\qquad 1710\\]\nThe disadvantage here is that we again have to use \\(f_0(1500)\\) as the head of the \\(\\ensuremath{s\\bar{s}}\\) trajectory despite knowing that its main decay modes are to \\(4\\pi\\) and \\(\\pi\\pi\\), as well as the fact the we - once again - need the \\(*f_0(1800)\\) resonance to fill in a hole for \\(n = 1\\) in the resulting light meson trajectory. This trajectory can be seen in table \\ref{tab:fit1710}.\n\n\\subsubsection{Conclusions from the \\texorpdfstring{$f_0$}{f0} fits}\nIt is not hard to see that the \\(f_0\\) resonances listed in the PDG's Review of Particle Physics all fit in quite neatly on two parallel trajectories with a slope similar to that of other mesons. However, upon closer inspection, these trajectories - one for light quark mesons and one for \\(\\ensuremath{s\\bar{s}}\\) - are not consistent with experimental data, as detailed above. For us the naive assignment is also inconsistent with what we have observed for the other \\(\\ensuremath{s\\bar{s}}\\) trajectories in \\cite{Sonnenschein:2014jwa}, namely that the \\(\\ensuremath{s\\bar{s}}\\) trajectories are not purely linear, and have to be corrected by adding a non-zero string endpoint mass for the \\(s\\) quark, usually of at least 200 MeV.\n\nThe other novelty that we hoped to introduce, the half slope trajectories of the glueball, proved to be impractical - given the current experimental data which only goes up to less than \\(2.4\\) GeV for the relevant resonances.\n\n\tOne conclusion that can be drawn is that the state \\(f_0(980)\\) can be comfortably excluded from any of the meson trajectories, which is consistent with its being the \\(K\\overline{K}\\) ground state.\n\n\tThe unconfirmed state \\(*f_0(1800)\\) turns up in the assignments with glueballs in them, usually to fill in a hole in the light meson trajectory. If the \\(*f_0(1800)\\) is not in itself a meson as mentioned before, then we would hope that there is another yet unobserved \\(f_0\\) state with a very similar mass, say 1800--1850 MeV.\n\t\n\tThere is no one assignment that seems the correct one, although the two assignments singling out either \\(f_0(1370)\\) or \\(f_0(1500)\\) as the glueball ground states seem more consistent than the other possibilities. The best way to determine which is better is, as always, by finding more experimental data. We list our predictions for higher resonances based on these assignments in section \\ref{sec:predictions} of the appendix.\n\n\\subsection{Assignment of the \\texorpdfstring{$f_2$}{f2} into trajectories} \\label{sec:f2_fits}\nWe now turn to the \\(f_2\\) tensor resonances, that were listed in the beginning of the section in table \\ref{tab:allf2}. We will first examine trajectories in the \\((J,M^2)\\) plane, then move on to the attempt to assign all the \\(f_2\\) states to trajectories in the \\((n,M^2)\\) plane.\n\\subsubsection{Trajectories in the \\texorpdfstring{$(J,M^2)$}{(J,M2)} plane} \\label{sec:f2_orbital}\nThe only way to get a linear trajectory connecting a \\(0^{++}\\) and a \\(2^{++}\\) state with the slope \\(\\ensuremath{\\alpha^\\prime}\\!_{gb} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}_{meson}\\) is to take the lightest \\(f_0\\) glueball candidate and the heaviest known \\(f_2\\). Then we have the pair \\(f_0(980)\\) and \\(f_2(2340)\\), and the straight line between them has a slope of 0.45 GeV\\(^{-2}\\). There is no \\(J = 1\\) resonance near the line stretched between them. However, this example mostly serves to demonstrate once again the difficulty of forming the glueball trajectories in practice. The glueball states are predicted to be fewer and farther apart then the mesons in their respective Regge trajectories.\n\nTherefore, it is a more sound strategy to look again for the meson trajectories, see what states are excepted from them, and check for overall consistency of the results. In forming the meson trajectories, we know that we can expect the \\(\\omega\\) mesons with \\(J^{PC} = 1^{--}\\) to be part of the trajectories, in addition to some states at higher spin, which will allow us to form trajectories with more points.\n\nMoving on from \\(J = 0^{++}\\) and \\(2^{++}\\) to higher spin states, we see two \\(J^{PC} = 4^{++}\\) states that could belong to a trajectory: \\(f_4(2050)\\) and \\(f_4(2300)\\). The first of those, \\(f_4(2050)\\), belongs to a well known meson trajectory in the \\((J,M^2)\\) plane, following \\(\\omega(782)\\), \\(f_2(1270)\\), and \\(\\omega_3(1670)\\). The slope of the fit to that trajectory is \\(\\ensuremath{\\alpha^\\prime} = 0.91\\) GeV\\(^{-2}\\), and we can even include in it states of spin 5 and 6: \\(*\\omega_5(2250)\\) and \\(f_6(2510)\\).\n\nThe mass of the \\(f_4(2300)\\) is too low for it to belong to a linear trajectory with a glueball slope. Taking it to be a meson one can put it on a linear trajectory following \\(\\omega(1420)\\) and \\(f_2(1810)\\). To complete this trajectory we need a \\(J^{PC} = 3^{--}\\) state with a mass near 2070 MeV. The PDG lists one unconfirmed state, \\(X(2080)\\), with the quantum numbers \\(I(J^{PC}) = ?(3^{-?})\\), which might be a match.\n\nWe also find another meson trajectory involving the second excited \\(\\omega\\) meson - \\(\\omega(1650)\\). This trajectory would be comprised of \\(\\omega(1650)\\), \\(*\\omega_3(2255)\\), and with one of \\(f_2(1950)\\) or \\(f_2(2010)\\) between them.\n\nWe also have the meson trajectories of the \\(\\ensuremath{s\\bar{s}}\\). The first joins the ground state \\(\\phi(1020)\\) with \\(f_2^\\prime(1525)\\) and \\(\\phi_3(1850)\\). We can form a daughter trajectory starting with the \\(\\phi(1680)\\), and going on to include \\(f_2(1950)\\) or \\(f_2(2010)\\), as well as the unconfirmed \\(*\\omega_3(2285)\\). This trajectory is nearly identical to that of the \\(\\omega(1650)\\) of the last paragraph.\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.76\\textwidth]{traj_j_mes.png}\n\t\t\t\t\\caption{\\label{fig:traj_j_mes} The trajectory of the \\(\\omega\\) (blue) and \\(\\phi\\) (red) mesons in the \\((J,M^2)\\) plane and their daughter trajectories. The fits have the common slope \\(\\ensuremath{\\alpha^\\prime} = 0.903\\) GeV\\(^{-2}\\), and the \\(\\ensuremath{s\\bar{s}}\\) trajectories are fitted using a mass of \\(m_s = 250\\) MeV for the \\(s\\) quark. The states forming the trajectories are as follows: With \\(J^{PC} = 1^{--}\\), \\(\\omega(782)\\), \\(\\phi(1020)\\), \\(\\omega(1420)\\), \\(\\omega(1650)\\), \\(\\phi(1680)\\). With \\(J^{PC} = 2^{++}\\), \\(f_2(1270)\\), \\(f_2^\\prime(1525)\\), \\(f_2(1810)\\), \\(f_2(1950)\\), and \\(f_2(2010)\\). With \\(J^{PC} = 3^{--}\\), \\(\\omega_3(1670)\\), \\(\\phi_3(1850)\\), \\(*\\omega_3(2255)\\), and \\(\\omega_3(2285)\\). And with \\(J^{PC} = 4^{++}\\), \\(f_4(2050)\\) and \\(f_4(2300)\\). We also plot at \\(J^{PC} = 0^{++}\\) the \\(f_0(980)\\) and \\(f_0(1370)\\) which are found to lie near the trajectories fitted, but were not included themselves in the fits, as they are not theoretically expected to belong to them.}\n\t\t\t\\end{figure}\n\nThe meson trajectories described above are plotted in figure \\ref{fig:traj_j_mes}.\n\nTo summarize, we have found several meson trajectories in the \\((J,M^2)\\) plane of at least three states. As shown in the figure, these trajectories pass quite closely to the states \\(f_0(980)\\) and \\(f_0(1370)\\), but as meson trajectories these should begin with a \\(J^{PC} = 1^{--}\\) state (with orbital angular momentum \\(L = 0\\) and spin \\(S = 1\\)). A \\(0^{++}\\) meson state could only be included as an excited state with \\(L = 1\\) and \\(S = 1\\), but we found that for each trajectory we can use an existing \\(f_2\\) state in that place. The \\(f_2\\) states classified in this assignment as mesons are \\(f_2(1270)\\), \\(f_2^\\prime(1525)\\), \\(f_2(1810)\\), \\(f_2(1950)\\), and \\(f_2(2010)\\). These can perhaps be partnered to existing \\(f_0\\) states as members of triplets of states with \\(J = 0, 1, 2\\) and \\(PC = ++\\) split by spin-orbit interactions. We do not know the exact magnitude of the splitting. There are some \\(f_0\\) states close (within 20--100 MeV) to the \\(f_2\\) states mentioned above, and the PDG lists some \\(f_1\\) (\\(1^{++}\\)) resonances that may be useful, but we do not find any such trio of states with similar properties and masses that could be said to belong to such a spin-orbit triplet. Therefore, we limit our conclusions from these Regge trajectories to the \\(f_2\\) which we found we could directly place on them.\n\n\\subsubsection{\\texorpdfstring{Trajectories in the $(n,M^2)$}{(n,M2)} plane} \\label{sec:f2_radial}\nSorting the \\(f_2\\) resonances into trajectories, the situation is somewhat simpler than with the \\(f_0\\) scalars, as here we have two states that belong on meson trajectories in the \\((J,M^2)\\) plane, as we found in previous sections. In particular, the \\(f_2(1270)\\) belongs to the trajectory of the \\(\\omega\\) meson, and the \\(f^\\prime_2(1525)\\) is an \\(s\\bar{s}\\) and sits on the \\(\\phi\\) trajectory. Their decay modes and other properties are also well known and there is no real doubt about their nature.\n\nThe linear trajectory beginning with the \\(f_2(1270)\\) meson includes the states \\(f_2(1640)\\)and \\(f_2(1950)\\). We can include one of the further states \\(*f_2(2240)\\) as the fourth point in the trajectory. We can also use the \\(f_J(2220)\\) in place of the \\(*f_2(2240)\\), but it seems an unnatural choice because of the widths of the states involved (the \\(f_J(2220)\\) is much narrower than the others).\n\nThe projected trajectory of the \\(f^\\prime_2(1525)\\), using the same slope as the \\(f_2(1270)\\) trajectory and adding mass corrections for the \\(s\\) quark, includes the \\(f_2(2010)\\) and the \\(f_2(2300)\\).\n\nThis leaves out the states \\(f_2(1430)\\), \\(f_2(1565)\\), \\(f_2(1810)\\), \\(f_2(1910)\\), \\(f_J(2220)\\), and \\(f_2(2340)\\), as well as the five resonances classified as further states.\n\nThe next state we look at is \\(f_2(1810)\\), classified as a light meson in the \\((J,M^2)\\) fits of the previous section. Its mass is not right for it to belong to the trajectory of the \\(f_2(1270)\\), so we try to use it as the head of another light meson trajectory. If it belongs to a parallel trajectory to that of the \\(f_2(1270)\\) then the state that follows it is \\(f_2(2150)\\). The next state could be \\(f_2(2340)\\), except that it has been observed to decay to \\(\\phi\\phi\\), making it very unlikely to be a light quark meson.\n\nThe state \\(f_2(1430)\\) is intriguing. In part because of the very small width reported by most (but not all) experiments cited in the PDG, and in part because it is located in mass between the two lightest mesons of \\(J^{PC} = 2^{++}\\), that is between \\(f_2(1270)\\) (light) and \\(f_2^\\prime(1525)\\) (\\(\\ensuremath{s\\bar{s}}\\)). If we had to assign the \\(f_2(1430)\\) to a Regge trajectory, then it is best placed preceding the \\(f_2(1810)\\) and \\(f_2(2150)\\) in the linear meson trajectory discussed in the last paragraph.\n\nThe \\(f_J(2200)\\), previously known as \\(\\xi(2230)\\), is also a narrow state. It is currently listed by the PDG as having either \\(J^{PC} = 2^{++}\\) or \\(4^{++}\\), but some of the experiments cited by the PDG tend towards \\(J = 2\\). It has been considered a candidate for the tensor glueball \\cite{Bai:1996wm,Crede:2008vw}. It can be assigned to a linear meson trajectory, as already discussed, but it is clear already from its narrow width that it is not the best choice, even before addressing other experimental finds regarding it (for example, the fact that it was not observed in \\(\\gamma\\gamma\\) scattering \\cite{Benslama:2002pa} and the resulting bounds on its decay into photons).\n\nThe \\(f_2(1565)\\) is also left out, but it could be paired with \\(f_2(1910)\\) to form another linear meson trajectory. To continue we need another state with a mass of around 2200 MeV.\n\nTo summarize, we may organize the \\(f_2\\) resonances by picking first the resonances for the trajectories of the two known mesons,\n\\[\\mathrm{Light:}\\qquad 1270, 1640, 1950\\]\n\\[\\ensuremath{s\\bar{s}}\\mathrm{:}\\qquad 1525, 2010, 2300 \\]\nthen find the trajectories starting with the lightest states not yet included. This gives us another meson trajectory using the states\n\\[\\mathrm{Light:}\\qquad 1810, 2150\\]\n\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=0.76\\textwidth]{traj_n_f2.png}\n\t\t\t\t\\caption{\\label{fig:traj_n_f2} Some radial trajectories of the \\(f_2\\), with blue lines for light mesons and red for \\(\\ensuremath{s\\bar{s}}\\). The fits have the common slope \\(\\ensuremath{\\alpha^\\prime} = 0.846\\) GeV\\(^{-2}\\), and the \\(\\ensuremath{s\\bar{s}}\\) trajectories are fitted using a mass of \\(m_s = 400\\) MeV for the \\(s\\) quark. The states forming the trajectories are as follows: The first light meson trajectory with \\(f_2(1270)\\), \\(f_2(1640)\\), and \\(f_2(1950)\\), and followed by the unconfirmed state \\(*f_2(2240)\\) which was not used in the fit. The \\(\\ensuremath{s\\bar{s}}\\) trajectory with \\(f_2^\\prime(1525)\\), \\(f_2(2010)\\), and \\(f_2(2300)\\). And the second light meson trajectory with \\(f_2(1810)\\) and \\(f_2(2150)\\).}\n\t\t\t\\end{figure}\n\nThe trajectories formed by these eight states are drawn in figure \\ref{fig:traj_n_f2}.\n\n\\subsubsection{Conclusions from the \\texorpdfstring{$f_2$}{f2} fits}\nThere are some simplifications in assigning the \\(f_2\\) to radial trajectories compared to assigning the \\(f_0\\) resonances, as we can look at both orbital and radial trajectories and it is easier to classify some states as mesons. The radial trajectories described in section \\ref{sec:f2_radial} are consistent with the orbital trajectories of section \\ref{sec:f2_orbital}: states classified as mesons in the latter are also classified as mesons in the former, and with the same quark contents.\n\nIn the previous sections we for the most part avoided using the five \\(f_2\\) states classified in the PDG as ``further states'', although some of them could have played a role in the radial trajectory assignments. Counting confirmed and unconfirmed states alike, the PDG lists a total of 11 states with masses between 1900 and 2340 MeV. Since the different states have been observed in different processes, and hence have different decay modes, it would be useful to clarify experimentally the status of all these states and then reattempt the assignments of the then confirmed states into trajectories. We also note that the fact that there are many resonances with identical quantum numbers near to each other can interfere with the naive mass predictions of the Regge trajectories. In any case, and like for the \\(f_0\\), further experimental data on resonances between 2.3--3.0 GeV will likely prove useful.\n\nWe have not addressed yet the issue of the decay modes of the different states and how consistent they are with the assignments of the previous sections. The \\(f_2(1270)\\) and \\(f_2^\\prime(1525)\\) are well established as a light quark meson and an \\(\\ensuremath{s\\bar{s}}\\) respectively, and they were the basis from which we built the different trajectories. As for their excited states, the data on their branching ratios cited by the PDG is very partial for higher states. However, we find an interesting case when looking at the trio of states \\(f_2(1910)\\), \\(f_2(1950)\\), and \\(f_2(2010)\\). We have classified \\(f_2(1950)\\) as a light meson and \\(f_2(2010)\\) as \\(\\ensuremath{s\\bar{s}}\\), which is what fits best with the Regge trajectories. Another option would be to use \\(f_2(1910)\\) as a light meson and \\(f_2(1950)\\) as \\(\\ensuremath{s\\bar{s}}\\), which is still consistent. Then the \\(f_2(2010)\\), which was observed to decay to \\(\\phi\\phi\\) (despite the very small phase space), could perhaps be classified as a \\(\\phi\\phi\\) bound state, in an analogous fashion to the \\(f_0(980)\\).\n\nThe most interesting states after that remain the \\(f_2(1430)\\) and \\(f_J(2220)\\). While the latter has been considered a candidate for the glueball and has been the object of some research (see papers citing \\cite{Bai:1996wm}), the former is rarely addressed, despite its curious placement in the spectrum between the lightest \\(2^{++}\\) light and \\(\\ensuremath{s\\bar{s}}\\) mesons. It seems a worthwhile experimental question to clarify its status - and its quantum numbers, as the most recent observation \\cite{Vladimirsky:2001ek} can not confirm whether it is a \\(0^{++}\\) or \\(2^{++}\\) state, a fact which led to at least one suggestion \\cite{Vijande:2004he} that the \\(f_2(1430)\\) could be itself the scalar glueball.\n\n\\subsection{Assignments with non-linear trajectories for the glueball} \\label{sec:holo_fits}\nIn this section we check the applicability of a glueball trajectory of the form\n\\begin{equation} J = \\ensuremath{\\alpha^\\prime}_{gb}E^2-2\\ensuremath{\\alpha^\\prime}_{gb}m_0E + a \\,, \\label{eq:holotraj} \\end{equation}\nwhich is the general form we expect from a semi-classical calculation of the corrections to the trajectory in a curved background, and as put forward in section \\ref{sec:holo_string}. The novelty here is a term linear in the mass \\(E\\), which makes the Regge trajectory \\(\\alpha(t)\\) non-linear in \\(t = E^2\\). The constant \\(m_0\\) can be either negative or positive, depending on the specific holographic background, and a priori we have to examine both possibilities. It was also noted in section \\ref{sec:holo_string} that there may be a correction to the slope, but we assume it is small compared to the uncertainty in the phenomenological value of the Regge slope, and we use\n\\begin{equation} \\ensuremath{\\alpha^\\prime}_{gb} = \\frac{1}{2}\\ensuremath{\\alpha^\\prime} \\end{equation}\nthroughout this section. We also substitute \\(J \\rightarrow J + n\\) as usual to apply the formula to radial trajectories.\n\nWith the \\(m_0\\) term we can write\n\\begin{equation} \\frac{\\partial J}{\\partial E^2} = \\frac{\\ensuremath{\\alpha^\\prime}}{2}\\left(1-\\frac{m_0}{E}\\right) \\,. \\label{eq:eff_holo_slope} \\end{equation}\nWe can look at this as an effective slope, and it is the easiest way to see that when \\(m_0\\) is negative, the effective slope is higher than that of the linear trajectory, and vice versa.\n\n\\subsubsection{Fits using the holographic formula}\nUsing the simple linear formula we could not, in most cases, find glueball trajectories among the observed \\(f_0\\) and \\(f_2\\) states. This is because the first excited state is expected to be too high in mass and outside the range of the states measured in experiment.\n\nAdding an appropriate \\(m_0\\) term can modify this behavior enough for us to find some pairs of states on what we would then call glueball trajectories, and by appropriate we mean a negative value that will make the effective slope of eq. \\ref{eq:eff_holo_slope} higher. The problem is then that we have only pairs of states, with two fitting parameters: \\(m_0\\) and \\(a\\) (and \\(\\ensuremath{\\alpha^\\prime}\\) which is fixed by the meson trajectory fits). We form these pairs by picking a state left out from the meson trajectories proposed in sections \\ref{sec:f0_fits} and \\ref{sec:f2_fits} and assigning it as the excited partner of the appropriate glueball candidate.\n\nThere is a solution for \\(m_0\\) and \\(a\\) for any pair of states which we can take, and the question then becomes whether there is a reason to prefer some values of the two parameters over others. We list some other values obtained for \\(m_0\\) and \\(a\\) in table \\ref{tab:holo_fits}.\n\n\\begin{table}[t!] \\centering\n\t\\begin{tabular}{|c|c|c|c|c|} \\hline\n\tGround state & Excited state & \\(\\ensuremath{\\alpha^\\prime}\\) [GeV\\(^{-2}\\)] & $m_0$ [GeV] & \\(a\\) \\\\ \\hline\n\t\n\t\\(f_0(980)\\) & \\(f_0(2200)\\) & 0.79 & -0.52 & -0.78 \\\\\n\t\n\t\\(f_0(1370)\\) & \\(f_0(2020)\\) & 0.87 & -2.00 & -3.21 \\\\\n\t\n\t\\(f_0(1500)\\) & \\(f_0(2200)\\) & 0.87 & -1.51 & -2.97 \\\\\n\t\n\t\\(f_2(1430)\\) & \\(f_J(2220)\\) & 0.81 & -1.33 & -2.42 \\\\ \\hline\n\t\n\t\\end{tabular} \\caption{\\label{tab:holo_fits} Values obtained for the parameters \\(m_0\\) and \\(a\\) for some of the possible pairs of states on glueball trajectories. The states selected as the excited state of the glueball are those not included in the meson trajectories of the assignments of sections \\ref{sec:f0_fits} and \\ref{sec:f2_fits}, and the slopes are selected based on the results of the meson fits presented in the same sections.}\n\\end{table}\n\n\\subsubsection{Using the holographic formula with a constrained intercept}\n\\cite{Bigazzi:2004ze} implies that a universal form of the first semi-classical correction of the Regge trajectory of the rotating folded string is\n\\begin{equation} J + n = \\frac{1}{2}\\ensuremath{\\alpha^\\prime}(E-m_0)^2 \\,, \\end{equation}\nup to further (model dependent) modifications of the slope, which in the cases calculated are small. In other words, the intercept obtained then from the semi-classical calculation is\n\\begin{equation} a = \\frac{1}{2}\\ensuremath{\\alpha^\\prime} m_0^2 \\,. \\end{equation}\n\nThe intercept is always positive in this scenario. If we want to include the ground state with \\(J = n = 0\\) the only way to do it is to take a positive \\(m_0\\), specifically we should take \\(m_0 = M_{gs}\\), where \\(M_{gs}\\) is the mass of the ground state. There is no problem with the resulting expression theoretically, but it is not very useful in analyzing the observed spectrum. The trouble is that when using this expression the energy rises much too fast with \\(J\\) and we end up very quickly with masses outside the range of the glueball candidates. If we take, for instance, \\(f_0(980)\\) as the ground state then the first excited state is expected to have a mass of around 2500 MeV, and the heavier candidates naturally predict even heavier masses for the excited states.\n\nAnother way to use eq. \\ref{eq:holotraj} is to begin the trajectory with a \\(J = 2\\) state. Then \\(m_0\\) can be either positive or negative.  We can then proceed as usual: we pick the head of a trajectory and see if there are any matches for its predicted excited states. We can see, for example, that we can again pair \\(f_2(1430)\\) with \\(f_J(2220)\\). Constraining \\(\\ensuremath{\\alpha^\\prime}\\) to be \\(0.90\\) GeV\\(^{-2}\\), the best fit has \\(m_0 = -0.72\\) GeV, and the masses calculated are 1390 and 2260 MeV for the experimental values of \\(1453\\pm4\\) and \\(2231\\pm4\\) MeV.\n\n\\subsection{Glueball Regge trajectories in lattice QCD} \\label{sec:lattice}\nThe glueball spectrum has been studied extensively in lattice QCD. Some works have compared results with different stringy models, e.g. \\cite{Athenodorou:2010cs,Bochicchio:2013aha,Bochicchio:2013sra,Caselle:2015tza}. However, the question whether or not the glueballs form linear Regge trajectories is not often addressed, due to the difficulty involved in computing highly excited states. When linear Regge trajectories are discussed, it is often when trying to identify the glueball with the pomeron and searching for states along the given pomeron trajectory,\n\\begin{equation} \\alpha(t) = \\ensuremath{\\alpha^\\prime}_p t + 1 + \\epsilon \\, \\end{equation}\n where the slope and the intercept are known from experiment to be \\(\\ensuremath{\\alpha^\\prime}_p = 0.25\\) GeV\\(^{-2}\\) and \\(1 + \\epsilon \\approx 1.08\\) \\cite{Donnachie:1984xq}.\n\nThe most extensive study of glueball Regge trajectories is that of Meyer and Teper \\cite{Meyer:2004jc,Meyer:2004gx}, where a relatively large number of higher mass states is computed, including both high spin states and some highly excited states at low spin.\n\nWe quote in table \\ref{tab:lat_masses} some lattice results for glueball masses from different calculations. The results are for \\(SU(3)\\) and \\(D = 4\\), and more results are collected in \\cite{Gregory:2012hu}. Most of these give only the masses of the lowest glueball states for different quantum numbers. These are low spin states with different combinations of parity and charge parity. While a spectrum is obtained, most states are isolated, in the sense that they cannot be grouped with other states to form Regge trajectories.\n\nIn the table \\ref{tab:lat_masses} we list the lattice results for the \\(0^{++}\\) ground state, the lowest \\(2^{++}\\) state, and the first excited \\(0^{++}\\) glueball, as well as for the \\(0^{-+}\\) and \\(2^{-+}\\). We may straight lines between the first spin-0 state and its excited partner to calculate the slope.\n\nOne thing we see at this first glance at the spectrum is that the spin-2 state is, in most studies, lower than we would expect it based on the Regge slope assumption.\\footnote{The fact that the tensor glueball is close to the scalar seems to have been long known in lattice QCD, see e.g. \\cite{Albanese:1987ds}.} The second spin-0 state, on the other hand, is about where we want it to be, assuming a closed string model, where the slope is half that of meson trajectories, and the first excited state has the excitation number \\(n = 2\\) (for one left moving and one right moving mode excited). In the next section we do some fits to some trajectories with more than two states, based on the results in \\cite{Meyer:2004gx}.\n\n\\begin{table}[t!] \\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline\n\t& Meyer \\cite{Meyer:2004gx}  & M\\&P \\cite{Morningstar:1999rf} & Chen \\cite{Chen:2005mg} & Bali \\cite{Bali:1993fb} & Gregory \\cite{Gregory:2012hu} \\\\ \\hline\\hline\n\n\t\\(0^{++}\\) & 1475\\plm30\\plm65\t\t\t& 1730\\plm50\\plm80 & 1710\\plm50\\plm80 & 1550\\plm50\\plm80 & 1795\\plm60 \\\\ \\hline\n\n\t\\(2^{++}\\) & 2150\\plm30\\plm100\t\t& 2400\\plm25\\plm120 & 2390\\plm30\\plm120 & 2270\\plm100\\plm110 & 2620\\plm50\\\\ \\hline\n\n\t\\(0^{++}\\) & 2755\\plm30\\plm120\t\t& 2670\\plm180\\plm130 & - & - & 3760\\plm240 \\\\ \\hline\\hline\n\n\t\\(0^{-+}\\) & 2250\\plm60\\plm100\t\t\t& 2590\\plm40\\plm130 & 2560\\plm35\\plm120 & 2330\\plm260\\plm120 & - \\\\ \\hline\n\n\t\\(2^{-+}\\) & 2780\\plm50\\plm130\t\t& 3100\\plm30\\plm150 & 3040\\plm40\\plm150 & 3010\\plm130\\plm150 & 3460\\plm320\\\\ \\hline\n\n\t\\(0^{-+}\\) & 3370\\plm150\\plm150\t\t& 3640\\plm60\\plm180 & - & - & 4490\\plm590 \\\\ \\hline\\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{++}\\) (in $J$) & 0.82\\plm0.17 & 0.72\\plm0.18 & 0.72\\plm0.17 & 0.73\\plm0.19 & 0.55\\plm0.05 \\\\ \\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{++}\\) (in $n$) & 0.37\\plm0.05 & 0.48\\plm0.14 & - & - & 0.18\\plm0.03 \\\\ \\hline\\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{-+}\\) (in $J$) & 0.75\\plm0.26 & 0.69\\plm0.28 & 0.74\\plm0.32 & 0.55\\plm0.27 & - \\\\ \\hline\n\n\t\\(\\ensuremath{\\alpha^\\prime}\\!_{-+}\\) (in $n$) & 0.32\\plm0.08 & 0.31\\plm0.07 & - & - & - \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{\\label{tab:lat_masses} Lattice predictions from different studies for glueball masses [MeV] and resulting Regge slopes [GeV\\(^{-2}\\)], in the \\((J,M^2)\\) plane or in the \\((n,M^2)\\) plane. The slope is calculated assuming the first excited state has \\(n = 2\\).}\n\t\\end{table}\n\n\\subsubsection{Regge trajectory fits to results from the lattice}\nResults in lattice computations are for the dimensionless ratio between the mass of a state and the square root of the string tension: \\(M\/\\sqrt{T}\\). To get the masses \\(M\\) in MeV one has to fix the scale by setting the value of \\(T\\). This introduces an additional uncertainty in the obtained values. In table \\ref{tab:lat_masses} we listed the masses in MeV and calculated the dimensionful slope, but for the purpose of identifying Regge trajectories we can work directly with dimensionless quantities, avoiding this extra error. Thus, for the following, our fitting model will be\n\\begin{equation} \\frac{M^2}{T} = \\frac{2\\pi}{q} (N + a) \\end{equation}\nIn this notation the ratio \\(q\\), which is the primary fitting parameter (in addition to the intercept \\(a\\)), is expected to be 1 for open strings and \\(1\/2\\) for closed strings. It is referred to below as the ``relative slope''. \\(N\\) will be either the spin \\(J\\) or the radial excitation number \\(n\\).\n\n\\paragraph{Trajectories in the \\texorpdfstring{$(J,M^2)$}{(J,M2)} plane:} As mentioned above, \\cite{Meyer:2004gx} has the most high spin states. The analysis there observes that the first \\(2^{++}\\)and \\(4^{++}\\) states can be connected by a line with the relative slope\n\\begin{equation} q = 0.28\\pm0.02, \\end{equation}\nwhich, when taking a typical value of the string tension \\(\\sqrt{T} = 430\\) MeV (\\(\\ensuremath{\\alpha^\\prime} = 0.84\\) GeV\\(^{-2}\\)), gives a slope virtually identical to that expected for the pomeron, \\(0.25\\) GeV\\(^{-2}\\). This trajectory can be continued with the calculated \\(6^{++}\\) state. A fit to the three state trajectory gives the result\n\\begin{equation} q = 0.29\\pm0.15.\\end{equation}\nThis trajectory leaves out the \\(0^{++}\\) ground state. In \\cite{Meyer:2004gx} the lowest \\(0^{++}\\) is paired with the second, excited, \\(2^{++}\\) state, giving a trajectory with\n\\begin{equation} q = 0.40\\pm0.04.\\end{equation}\nA possibility not explored in \\cite{Meyer:2004gx} is that of continuing this trajectory, of the first \\(0^{++}\\) and the excited \\(2^{++}\\), and with the \\(4^{++}\\) and \\(6^{++}\\) states following. Then we have the result\n\\begin{equation} q = 0.43\\pm0.03\\end{equation}\nThis second option not only includes more points, it is also a better fit in terms of \\(\\chi^2\\) per degrees of freedom (0.37 instead of 1.24). This is a nice result from the closed string perspective, but the lowest \\(2^{++}\\) state is then left out. There is also a \\(J = 3\\) state in the \\(PC = ++\\) sector that lies very close to the trajectory of the \\(0^{++}\\) ground state. In our model it is not expected to belong to the trajectory, so that state is also left out of the fit. The trajectories of the \\(PC = ++\\) states are in the left side of figure \\ref{fig:lat_Meyer}.\n\n\\begin{figure}[tp!] \\centering\n\t\\includegraphics[width=0.49\\textwidth]{lattice_Meyer_pp.png}\n\t\\includegraphics[width=0.49\\textwidth]{lattice_Meyer_0pp.png}\n\t\\caption{\\label{fig:lat_Meyer} The trajectories of the \\(PC = ++\\) glueball states found in lattice calculations in \\cite{Meyer:2004gx}. \\textbf{Left:} Trajectories in the \\((J,M^2)\\) plane. The full line is the fit to a proposed trajectory using four states with \\(J = 0, 2, 4, 6\\), where the relative slope is \\(0.43\\) and the lightest tensor is excluded (\\(\\chi^2 = 0.37\\)). The dotted line is the leading trajectory proposed in the analysis in \\cite{Meyer:2004gx}, with a pomeron-like slope. It includes the \\(J = 2, 4,\\) and \\(6\\) states. (\\(\\chi^2 = 1.24\\)). In this second option the scalar is excluded. Also plotted is the \\(3^{++}\\) state, which was not used in the fit. \\textbf{Right:} trajectory of four states with \\(J^{PC} = 0^{++}\\). The relative slope is exactly \\(0.50\\) (\\(\\chi^2 = 1.48\\)).}\n\\end{figure}\n\n\\paragraph{Trajectories in the \\texorpdfstring{$(n,M^2)$}{(n,M2)} plane:}\nIn trajectories in the \\((n,M^2)\\) plane we assume \\(n\\) takes only even values, i.e. \\(n = 0,2,4,\\ldots\\), as it does for the closed string. The results when taking \\(n = 0,1,2,\\ldots\\) will be half those listed.\n\nIn this section, we again have to rely mostly on \\cite{Meyer:2004gx}, as it offers calculations of several excited states with the same \\(J^{PC}\\). Most notably we see there four states listed with \\(J^{PC} = 0^{++}\\). We observe that those points are well fitted by a trajectory with the slope\n\\begin{equation} q = 0.50\\plm0.07, \\end{equation}\nwhere \\(\\chi^2 = 1.48\\) for the fit. It is interesting to compare this with the trajectory that can be drawn from the \\(0^{++}\\) ground state in the \\((J,M^2)\\) plane. The \\((n,M^2)\\) trajectory with \\(n = 0,2,4,6\\) is very similar to the trajectory beginning with the same state and continuing to \\(J = 2, 4,\\) and \\(6\\). This is what we see also for mesons and baryons in experiment: two analogous trajectories with similar slopes in the different planes.\n\nOther than the trajectory of the four \\(0^{++}\\) states (plotted in figure \\ref{fig:lat_Meyer}), we list the slopes calculated for pairs of states who share other quantum numbers. This is in table \\ref{tab:lat_n}.\n\\begin{table} \\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|} \\hline\n\t\t\\(J^{PC}\\) & \\(0^{++}\\) & \\(2^{++}\\) & \\(4^{++}\\) & \\(0^{-+}\\) & \\(2^{-+}\\) \\\\ \\hline\\hline\n\t\tMeyer \\cite{Meyer:2004gx} &\n\t\t\t\t0.50\\plm0.07 & 0.67\\plm0.10 & 0.30\\plm0.06 & 0.39\\plm0.07 & 0.56\\plm0.13 \\\\ \\hline\n\n\t\tM\\&P \\cite{Morningstar:1999rf} &\n\t\t\t\t0.51\\plm0.12 & -\t\t\t\t\t\t& -\t\t\t\t\t\t & 0.32\\plm0.02 & 0.38\\plm0.03 \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{\\label{tab:lat_n} Relative slopes \\(q\\) of trajectories in the \\((n,M^2)\\) plane. The first result (Meyer\/\\(0^{++}\\)) is that of a fit to the four point trajectory drawn in \\ref{fig:lat_Meyer}. The other results are obtained when calculating the slopes between pairs of states, where the lowest state is assumed to have \\(n = 0\\), and the first excited state is taken to have \\(n = 2\\).}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\\texorpdfstring{$SU(N)$ vs. $SU(3)$}{SU(N) vs. SU(3)} and the quenched approximation}\nMost of the studies of glueballs on the lattice utilize the ``quenched'' approximation, which in this case amounts to calculating the spectrum of the pure SU(3) Yang-Mills gauge theory without matter. The degree to which the quenched results are modified when fermions are added to the theory is still unknown. However, if our purpose is to see whether or not glueballs form Regge trajectories, the spectrum of the pure gluon theory should be as useful as that of real QCD.\n\nThere have also been some calculations of the ``glueball'' spectrum of \\(SU(N)\\) Yang-Mills for other values of \\(N\\) \\cite{Lucini:2014paa}. These results taken from \\cite{Lucini:2004my}, are fitted in \\cite{Lucini:2014paa} to the formulae (the numbers in brackets are the errors in the last significant digits):\n\\begin{equation} \\frac{M_{0^{++}}}{\\sqrt{T}} = 3.28(8)+\\frac{2.1(1.1)}{N^2} \\,,\\end{equation}\n\\[ \\frac{M_{2^{++}}}{\\sqrt{T}} = 4.78(14)+\\frac{0.3(1.7)}{N^2} \\,,\\]\n\\[ \\frac{M_{0^{++*}}}{\\sqrt{T}} = 5.93(17)-\\frac{2.7(2.0)}{N^2} \\,.\\]\nUsing these values, we get for \\(SU(3)\\), the relative slopes (the prefactor of 2 in these formulae is \\(J\\) or \\(n\\)):\n\\begin{equation} 2\\frac{2\\pi T}{M^2_{2^{++}}-M^2_{0^{++}}} = 1.16\\plm0.27, \\qquad\n 2\\frac{2\\pi T}{M^2_{0^{++*}}-M^2_{0^{++}}} = 0.65\\plm0.11 \\,, \\end{equation}\nwhile for the \\(N\\rightarrow\\infty\\) limit,\n\\begin{equation} 2\\frac{2\\pi T}{M^2_{2^{++}}-M^2_{0^{++}}} = 1.04\\plm0.13, \\qquad\n 2\\frac{2\\pi T}{M^2_{0^{++*}}-M^2_{0^{++}}} = 0.52\\pm0.05 \\,.\\end{equation}\n\nWhile this is too little data to be significant, we observe that the value approaches 1 as \\(N\\) grows for the excitation in \\(J\\), and it approaches \\(\\frac12\\) (the closed string value) in \\(n\\). However, as was already seen from the results of \\cite{Meyer:2004gx}, the first \\(2^{++}\\) does not seem to lie on the trajectory of the \\(0^{++}\\) ground state, and these results seem to confirm this further. The radial trajectory on the other hand is again perfectly consistent with the closed string picture, and more so when going to the limit of this large \\(N\\) computation.\n\n\\section{Summary} \\label{sec:summary}\n\nFor many years the identification of glueballs, a basic prediction of QCD, in the experimental spectrum of flavorless isoscalar hadrons has been an open question.  Moreover, the common lore is that there is no way to disentangle glueballs from flavorless mesons since there is no quantum number that distinguish between them. \n\nHere in this paper we have attempted to identify glueballs by turning to a well known feature of the hadron spectrum, its Regge trajectories. Stating it differently we use a stringy picture of rotating folded closed strings to describe the glueball in a similar way to the description of mesons and baryons in terms of open string with massive endpoints of \\cite{Sonnenschein:2014jwa} and \\cite{Sonnenschein:2014bia}.\n\nThe great disadvantage in using trajectories is that they are a property not of single states, but of a spectrum of states. Thus, for positive identification, we need to have in our spectrum, to begin with, several glueballs which we would then assign to a trajectory. The fact that the ratio between the open and closed string slopes is exactly half adds some ambiguity to the \\((n,M^2)\\) trajectories where the value of \\(n\\) cannot be determined by experiment: two states whose mass difference is, for instance, \\(\\Delta M^2 = 4\/\\ensuremath{\\alpha^\\prime}\\) can be either open strings with \\(\\Delta n = 4\\) between them, or closed strings with \\(\\Delta n = 2\\). The difference between the open and closed string trajectories would be in the number of states between those two: there would be more open strings for \\(\\Delta n = 1\\), 2, and 3. Thus we have to rely on experiment to observe all the relevant states in the given mass range, so that the absence of a state from a Regge trajectory could reasonably be used as evidence.\n\nDue to this situation it is clearly advisable to use additional predictions pertaining to the properties of single states to identify them as open or closed string hadrons. We have presented, qualitatively, the decay mechanism of the closed string to two open strings, which would be the decay of a glueball into two mesons. We included one prediction of the branching ratios of glueballs when decaying into light mesons, kaons, or \\(\\phi\\) (\\(s\\bar{s}\\)) mesons. If there were measurements of a state which has those three decay modes with the hierarchy we predict between them, we could have declared it a glueball, based on our model of holographic strings. One has to look more closely to find more ways in which open and closed strings vary.\n\nThere are obviously additional tasks and questions to further explore the closed string picture of glueballs. Here we list some of them:\n\\begin{itemize}\n\\item\nAs was emphasized in this note the most urgent issue is to gain additional data about flavorless hadrons. This calls for a further investigation of experiments that yield this kind of resonances and for proposing future experiments of potential glueball production, in particular in the range above 2.4 GeV. This can follow the predictions of the masses and width of the resonances as were listed in appendix \\ref{sec:predictions}.\n\\item\nRelated to the exploration of experimental data is the investigation of efficient mechanisms of creating glueballs. This issue was not addressed in this paper. Among possible glueball formation one finds radiative $J\/\\psi$ decays, pomeron pomeron collisions in hadron-hadron central production and in $p$-$\\bar p$ annihilation. Naturally, we would like to understand possible glueball formation in LHC experiments. It is known that we can find in the latter processes of gluon-gluon scattering and hence it may serve as a device for glueball creation. \n\\item\nAs was mentioned in section \\ref{sec:non_critical_string}, the quantization of folded closed strings in D non-critical dimensions has not yet been deciphered. In \\cite{Hellerman:2013kba} the expression derived for the intercept is singular in the case where is only one rotation plane - as it naturally is in $D=4$. We mentioned a potential avenue to resolve this issue by introducing massive particles on the folds, quantize the system as that of a string with massive endpoints \\cite{ASY}, and then take the limit of zero mass. \n\\item\nWe have mentioned that the rotating closed strings are in fact rotating folded closed strings. However, we did not make any attempt in this note to explore the role of the folds. In fact it seems that very few research has been devoted to the understanding of folded strings \\cite{Ganor:1994rm}. It would be interesting to use the rotating closed string as a venue to the more general exploration of strings with folds which may be related to certain systems in nature.\n\\item\nA mystery related to the closed string description of glueballs is the relation between the pomeron and the glueball. Supposedly  both the glueball and the pomeron are described by a closed string. As we have emphasized in this note the slope of the closed string is half that of the open string and hence we advocated the search of trajectories with that slope. However, it was found from fitting the differential cross section of $p$-$p$ collisions that the slope of the pomeron is $\\ensuremath{\\alpha^\\prime}_{pomeron}\\approx 0.25$ GeV$^{-2}$. That is, a slope which is closer to a quarter of that of the meson open string rather than half. Thus the stringy structure of the pomeron and its exact relation to the glueball is still an open question.\n\\item\nThe closed string description of the glueball faces a very obvious question. In QCD one can form a glueball as a bound state of two, three, or in fact any number of gluons. The stringy picture seems to describe the composite of two gluons, and it is not clear how to realize those glueballs constructed in QCD from more than two gluons.\n\\end{itemize} \n\n\\acknowledgments{\nWe would like to thank Ofer Aharony and Abner Soffer for their comments on the manuscript and for insightful conversations, and Shmuel Nussinov and Shimon Yankielowicz for useful discussions.  This work was supported in part by a centre of excellence supported by the Israel Science Foundation (grant  number 1989\/14), and by the US-Israel bi-national fund (BSF) grant number 2012383 and the Germany Israel bi-national fund GIF grant number I-244-303.7-2013. \n}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\n\nIn Ref.~\\cite{norma93} the representations in Grassmann  and in Clifford space were discussed.\nIn Ref.~(\\cite{nh2018} and the references therein) the second quantization procedure in both\nspaces --- in Clifford space and in Grassmann space --- were discussed in order to try to understand  \n\"why nature made a choice of Clifford rather than Grassmann space\" during the expansion \nof our universe, although in both spaces the creation operators $ \\hat{b}^{\\dagger}_{j} $  \nand the annihilation operators $ \\hat{b}_{j} $ exist fulfilling the anticommutation relations \nrequired for fermions~\\cite{nh2018}\n\\begin{eqnarray}\n\\{ \\hat{b}_i, \\hat{b}^{\\dagger}_{j} \\}_{+}  |\\phi_{o}> &=&\\delta_{i j}\\; |\\psi_{o}>\\,,\n\\nonumber\\\\\n\\{\\hat{b}_i, \\hat{b}_{j} \\}_{+}  |\\psi_{o}>&=& 0\\; |\\psi_{o}> \\,,\\nonumber\\\\\n\\{\\hat{b}^{\\dagger}_i,\\hat{b}^{\\dagger}_{j}\\}_{+} |\\psi_{o}>\n&=&0\\; |\\psi_{o}> \\,,\\nonumber\\\\\n\\hat{b}^{\\dagger}_{j} |\\psi_{o}>& =& \\, |\\psi_{j}>\\, \\nonumber\\\\\n\\hat{b}_{j} |\\psi_{o}>& =&0\\,  |\\psi_{o}>\\,.\n\\label{ijthetaprod} \n\\end{eqnarray} \n$|\\psi_{o}>$ is  the vacuum state. We use $|\\psi_o>= |1>$.\n\n\n Some observations to be included into introduction:\n However, even in the Grassmann case and with gravity only the scalar fields would appear in  \n$d=(3+1)$.\n\nThe creation operators can be expressed in both spaces as products of eigenstates \nof the Cartan subalgebra, Eq.~(\\ref{choicecartan}), of the Lorentz algebra, \nEqs.~(\\ref{cartaneigengrass}, \\ref{signature}).\nStarting with one state (Ref.~\\cite{nh2018})\nall the other states\nof the same representation are reachable  by the  generators of the Lorentz transformations\n(which do not belong to the Cartan subalgebra), with ${\\cal {\\bf S^{ab}}}$ presented in \nEq.~(\\ref{Lorentztheta}) in  Grassmann space and with either $S^{ab}$ or $\\tilde{S}^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}), in Clifford space.\n\n But while there are in Clifford case two kinds of the generators of the Lorentz \ntransformations --- $S^{ab}$ and $\\tilde{S}^{ab}$, the first transforming members of one \nfamily among themselves, and the second transforming one member of a particular family into\nthe same member of other families --- there is in Grassmann space only one kind \nof the Lorentz generators --- ${\\cal {\\bf S^{ab}}}$. Correspondingly are all the states in \nClifford space, which can be second quantized as products of nilpotents and projectors~%\n\\cite{nh02,nh03,nh2018},\nreachable with one of the two kinds of the operators $S^{ab}$ and $\\tilde{S}^{ab}$, \nwhile different representations are in Grassmann space disconnected. \n\nOn the other hand the vacuum state is in Grassmann case simple --- $|\\psi_o>= |1>$ ---\nwhile in Clifford case is the sum of products of projectors, Eq.~(\\ref{vac1}). \n\nIn Grassmann space states are in the adjoint representations with respect to the Lorentz \ngroup, while states in Clifford space belong to the fundamental representations with respect \nto both generators,  $S^{ab}$ and $\\tilde{S}^{ab}$, or they are singlets. Correspondingly \nare properties  of fermions, described  with the {\\it spin-charge-family} theory~%\n\\cite{IARD2016,n2014matterantimatter,JMP2013,normaJMP2015,nh2017,nd2017}, which \nuses the Clifford space to describe fermion degrees of freedom, in agreement with the \nobservations, offering explanation for all the assumptions of the {\\it standard model} \n(with families included) and also other observed phenomena. \n\nIn Grassmann case the spins \nmanifest, for example, in the case of $SO(6)$ or $SO(5,1)$ decuplets or singlets --- triplets and \nsinglets in Clifford case, \nTable~\\ref{Table grassdecupletso51.} --- while with respect to the subgroups $SU(3)$ and \n$U(1)$ of $SO(6)$ the states belong to either singlets, or triplets or sextets, \nTables~\\ref{Table grassdecuplet.},~\\ref{Table grasssextet.}\n --- triplets and singlets in the Clifford case.\n\n\nIn what follows we discuss representations, manifesting as charges and spins of fermions, of \nsubgroups of $SO(13,1)$, when internal degrees of freedom of fermions are described in \nGrassmann  space and compare properties of these representations with the properties of \nthe corresponding representations appearing in Clifford space. We assume, as in the\n{\\it spin-charge-family} theory, that both spaces, the internal and the ordinary space, \nhave $d=2(2n+1)$-dimensions, $n$ is positive integer, $d \\ge 14$ and that all the degrees \nof freedom of fermions and bosons originate in $d=2(2n+1)$, in which fermions interact with \ngravity only. \n\nAfter the break of the starting symmetry $SO(13,1)$ into $SO(7,1) \\times SU(3) \\times U(1)$, and\nfurther to $SO(3,1) \\times SU(2) \\times SU(2) \\times SU(3) \\times U(1)$, fermions manifest \nin $d=(3+1)$ the spin and the corresponding charges and interact with the gauge fields, which \nare indeed the spin connections with the space index $m=(0,1,2,3)$, originating in  \n$d=(13,1)$~\\cite{nd2017}. Also scalar fields originate in gravity: Those spin connections with \nthe space index $a =(5,6,7,8)$ determine masses of fermions, those with the space index \n$a=(9,10,\\dots,14)$ contribute to particle\/antiparticle asymmetry in our \nuniverse~\\cite{n2014matterantimatter}. \n\nWe pay attention in this paper mainly on fermion fields with spin $1$, the creation and \nannihilation operators of which fulfill the anticommutation relations of Eq.~(\\ref{ijthetaprod})\nin Grassmann space. \n\n\n\n\n\\subsection{Creation and annihilation operators in Grassmann space}\n\\label{grassmann}\n\nIn Grassmann $d=2(2n+1)$-dimensional space the creation and annihilation operators follow from\nthe starting two creation and annihilation operators, both with an odd Grassmann character,  \nsince those with an even Grassmann character do not obey the anticommutation relations of \nEq.~(\\ref{ijthetaprod})~\\cite{nh2018} \n\\begin{eqnarray}\n\\hat{b}^{\\theta 1 \\dagger}_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}} \\,\n(\\theta^0 - \\theta^3) (\\theta^1 + i \\theta^2) (\\theta^5 + i \\theta^6) \\cdots (\\theta^{d-1} +\n i \\theta^{d}) \\,,\\nonumber\\\\\n\\hat{b}^{\\theta 1}_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}}\\,\n (\\frac{\\partial}{\\;\\partial \\theta^{d-1}} - i \\frac{\\partial}{\\;\\partial \\theta^{d}})\n\\cdots (\\frac{\\partial}{\\;\\partial \\theta^{0}}\n-\\frac{\\partial}{\\;\\partial \\theta^3})\\,,\\nonumber\\\\\n\\hat{b}^{\\theta 2 \\dagger }_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}} \\,\n(\\theta^0 + \\theta^3) (\\theta^1 + i \\theta^2) (\\theta^5 + i \\theta^6) \\cdots (\\theta^{d-1} +\n i \\theta^{d}) \\,,\\nonumber\\\\\n\\hat{b}^{\\theta 2}_{1} &=& (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}}\\,\n (\\frac{\\partial}{\\;\\partial \\theta^{d-1}} - i \\frac{\\partial}{\\;\\partial \\theta^{d}})\n\\cdots (\\frac{\\partial}{\\;\\partial \\theta^{0}}\n+\\frac{\\partial}{\\;\\partial \\theta^3})\\,.\n\\label{start(2n+1)2theta}\n\\end{eqnarray}\nAll the creation operators are products of the eigenstates of the Cartan subalgebra operators,%\nEq.~(\\ref{choicecartan})\n\\begin{eqnarray}\n\\label{cartaneigengrass}\n{\\cal {\\bf S}}^{ab} (\\theta^a \\pm \\epsilon \\theta^b) &=& \\mp i  \n\\frac{\\eta^{aa}}{\\epsilon} (\\theta^a \\pm \\epsilon \\theta^b)\\,, \\nonumber\\\\\n\\epsilon = 1\\,, \\;\\, {\\rm for}\\;\\, \\eta^{aa}=1\\,,&&\n\\epsilon = i \\,,\\;\\, {\\rm for}\\;\\, \\eta^{aa}= -1\\,, \\nonumber\\\\\n{\\cal {\\bf S}}^{ab}\\, (\\theta^a \\theta^b \\pm \\epsilon \\theta^c \\theta^d)=0\\,,&& \n {\\cal {\\bf S}}^{cd}\\, (\\theta^a \\theta^b \\pm \\epsilon \\theta^c \\theta^d)=0\\,.\n\\end{eqnarray}\n\n\nThe two creation operators, $\\hat{b}^{\\theta 1 \\dagger}_{1}$ and \n$\\hat{b}^{\\theta 2 \\dagger}_{1}$, if applied on the vacuum state, form the starting two states \n$\\phi^{1}_{1}$ and $\\phi^{2}_{1}$ of the two representations, respectively. The vacuum state \nis chosen to be the simplest one~\\cite{nh2018} --- $|\\phi_{0}> = |1>$. The rest of creation operators\nof each of the two groups, $\\hat{b}^{\\theta 1 \\dagger}_{i}$ and $\\hat{b}^{\\theta 2 \\dagger}_{i}$, \nfollow from the starting one by the application of the generators of the Lorentz transformations in \nGrassmann space ${\\cal {\\bf S}}^{ab}$, Eq.~(\\ref{Lorentztheta}), which do not belong to the \nCartan subalgebra, Eq.~(\\ref{choicecartan}), of the Lorentz algebra. They generate either \n$|\\phi^{1}_{j}>$ of the first group or $|\\phi^{2}_{j}>$ of the second group.\n\nAnnihilation operators $\\hat{b}^{\\theta 1}_{i}$ and $\\hat{b}^{\\theta 2}_{i}$ follow from the \ncreation ones by the Hermitian conjugation~\\cite{nh2018}, when taking into account the \nassumption\n\\begin{eqnarray}\n\\label{grassher}\n(\\theta^a)^{\\dagger} &=&  \\frac{\\partial}{\\partial \\theta_{a}} \\eta^{aa}=\n-i \\,p^{\\theta a} \\eta^{aa}\\,, \n\\end{eqnarray}\nfrom where it follows\n\\begin{eqnarray}\n\\label{grassp}\n(\\frac{\\partial}{\\partial \\theta_{a}})^{\\dagger} &=& \\eta^{aa}\\, \\theta^a\\,,\\quad\n(p^{\\theta a})^{\\dagger} = -i \\eta^{aa} \\theta^a\\,.\n\\end{eqnarray}\n\n\nThe annihilation operators $\\hat{b}^{\\theta 1}_{i}$ and $\\hat{b}^{\\theta 2}_{i}$ annihilate \nstates $|\\phi^{1}_{i}>$ and $|\\phi^{2}_{i}>$, respectively.  \n\nThe application of ${\\cal {\\bf S}}^{01}$ on $\\hat{b}^{\\theta 1 \\dagger}_{1}$, for example,\n transforms this creation operator into $\\hat{b}^{\\theta 1 \\dagger}_{2} = $ \n$(\\frac{1}{\\sqrt{2}})^{\\frac{d}{2} -1} \\,(\\theta^0  \\theta^3 +i \\theta^1 \\theta^2)$\n$ (\\theta^5 + i \\theta^6) \\cdots (\\theta^{d-1} - i \\theta^{d})$. Correspondingly its Hermitian \nconjugate annihilation operator is equal to\n$\\hat{b}^{\\theta 1}_{2} = (\\frac{1}{\\sqrt{2}})^{\\frac{d}{2}-1}\\,\n (\\frac{\\partial}{\\;\\partial \\theta^{d-1}} - i \\frac{\\partial}{\\;\\partial \\theta^{d}})\n\\cdots (\\frac{\\partial}{\\;\\partial \\theta^{3}} \\,\\frac{\\partial}{\\;\\partial \\theta^0} - i \n\\frac{\\partial}{\\;\\partial \\theta^{2}} \\,\\frac{\\partial}{\\;\\partial \\theta^1})$.\n\nAll the states are normalized with respect to the integral over the Grassmann coordinate  \nspace~\\cite{norma93}\n\n\\begin{eqnarray}\n\\label{grassnorm}\n<\\phi^{a}_{i}|\\phi^{b}_{j} > &=&  \\int d^{d-1} x  d^d \\theta^a\\, \\,\\omega \n <\\phi^{a}_{i}|\\theta> <\\theta|\\phi^{b}_{j} > = \\delta^{ab}\\,\\delta_{ij} \\,, \\nonumber\\\\\n\\omega&=& \\Pi^{d}_{k=0}(\\frac{\\partial}{\\;\\,\\partial \\theta_k} + \\theta^{k})\\,,\n\\end{eqnarray}\nwhere $\\omega$ is a weight function, defining the scalar product $<\\phi^a_{i}|\\phi^b_{j} >$,\n and we require that~\\cite{norma93}\n\\begin{eqnarray}\n\\label{grassintegral}\n\\{ d\\theta^a, \\theta^b \\}_{+} &=&0, \\,\\;\\;  \\int d\\theta^a  =0\\,,\\,\\;\\; \n\\int d\\theta^a \\theta^a =1\\,,\\nonumber\\\\\n\\int d^d \\theta \\,\\,\\theta^0 \\theta^1 \\cdots \\theta^d &=&1\\,,\n\\nonumber\\\\\nd^d \\theta &=&d \\theta^d \\dots d\\theta^0\\,,\n\\end{eqnarray}\nwith $ \\frac{\\partial}{\\;\\,\\partial \\theta_a} \\theta^c = \\eta^{ac}$. \n\n\nThere are $\\frac{1}{2}\\, \\frac{d!}{\\frac{d!}{2} \\frac{d!}{2}}$ in each of these two groups \nof creation operators of an odd Grassmann character in $d=2(2n+1)$-dimensional space.\n\nThe rest of creation operators (and the corresponding annihilation operators) would have  rather \nopposite  Grassmann character than the ones studied so far: like {\\bf a.} $\\theta^0 \\theta^1$ \nfor the creation operator and \n[$\\frac{\\partial}{\\partial \\theta^1}\\frac{\\partial}{\\partial \\theta^0}$]  for the \ncorresponding annihilation operator in $d=(1+1)$\n (since $\\{\\theta^0 \\theta^1, \\frac{\\partial}{\\partial \\theta^{1}}$ \n $\\frac{\\partial}{\\partial \\theta^{0}} \\}_{+}$ gives $(1+ (1+1) \\theta^0 \\theta^1 $\n $\\frac{\\partial}{\\partial \\theta^{1}} \\frac{\\partial}{\\partial \\theta^{0}}) $), and like {\\bf b.}\n $(\\theta^0 \\mp \\theta^3) (\\theta^1\\pm i \\theta^2)$ for creation operator and \n[$ (\\frac{\\partial}{\\partial \\theta^1}\\mp i \\frac{\\partial}{\\partial \\theta^2}) \n(\\frac{\\partial}{\\partial \\theta^0} \\mp \\frac{\\partial}{\\partial \\theta^3})$] for the\nannihilation operator,   or $\\theta^0 \\theta^3\n \\theta^1  \\theta^2$ for the creation operator and [$\\frac{\\partial}{\\partial \\theta^2} \n\\,\\frac{\\partial}{\\partial \\theta^1}\\,\\frac{\\partial}{\\partial \\theta^3}\\,\n\\frac{\\partial}{\\partial \\theta^0}$] for the annihilation operator in $d=(3+1)$ \n (since, let say, $\\{\\frac{1}{2} (\\theta^0 - \\theta^3) (\\theta^1 + i \\theta^2),$ \n $\\frac{1}{2} (\\frac{\\partial}{\\partial \\theta^1} - i \\frac{\\partial}{\\partial \\theta^2}) \n (\\frac{\\partial}{\\partial \\theta^0} - \\frac{\\partial}{\\partial \\theta^3})\\}_{+}$ gives\n $(1 + \\frac{1}{4} (1+1)  (\\theta^0 - \\theta^3) (\\theta^1 + i \\theta^2)  \n (\\frac{\\partial}{\\partial \\theta^1} - i \\frac{\\partial}{\\partial \\theta^2}) \n (\\frac{\\partial}{\\partial \\theta^0} - \\frac{\\partial}{\\partial \\theta^3})$ and equivalently for \nother  cases), but applied on a vacuum states some of them still fulfill some of the relations\n of Eq.~(\\ref{ijthetaprod}), but not all (like $\\{\\frac{1}{2}(\\theta^0 - \\theta^3) \n(\\theta^1 + i \\theta^2),$  $\\frac{1}{2} (\\theta^0 + \\theta^3) (\\theta^1 - i \\theta^2)\\}_{+}\n=$  $i \\theta^0 \\theta^1\\theta^2 \\theta^3 $, while it should be zero). \n\nLet us add that, like in Clifford case, one can simplify the scalar product in Grassmann case \nby recognizing that the scalar product  is equal to $\\delta^{ab}\\,\\delta_{ij}$\n\\begin{eqnarray}\n\\label{grassscalar}\n<\\phi^{a}_{i}|\\theta> <\\theta|\\phi^{b}_{j} > &=& \\delta^{ab}\\,\\delta_{ij}\\,,\n\\end{eqnarray}\nwithout  integration over the Grassmann coordinates. Let us manifest this in the case\nof $d=(1+1)$:$<1|\\frac{1}{\\sqrt{2}}(\\frac{\\partial}{\\partial \\theta^0} - \n\\frac{\\partial}{\\partial \\theta^1})\n\\frac{1}{\\sqrt{2}}(\\theta^0 - \\theta^1)|1>=1 $, $|1>$ is the normalized vacuum state,\n$<1|1>=1$. It is true \nin all dimensions, what can easily be understood for all the states, which are defined by \nthe creation operators $\\hat{b}_{i}^{\\dagger}$ on the vacuum state $|1>$, \n$|\\phi^{b}_{i} >= \\hat{b}_{i}^{\\dagger}|1>$, fulfilling the anticommutation relations\nof Eq.~(\\ref{ijthetaprod}).\n\n \n\\subsection{Creation and annihilation operators in Clifford space}\n\\label{clifford}\n\nThere are two kinds of Clifford objects~\\cite{norma93}, (\\cite{IARD2016} and Refs. therein),\n$\\gamma^a$ and $\\tilde{\\gamma}^a$, both fulfilling the anticommutation relations\n\\begin{eqnarray}\n\\label{tildecliffcomrel}\n \\{\\gamma^a, \\gamma^b \\}_{+} &=& 2 \\eta^{a b} = \n\\{\\tilde{\\gamma}^a, \\tilde{\\gamma}^b \\}_{+}\\,, \\nonumber\\\\ \n \\{\\gamma^a, \\tilde{\\gamma}^b \\}_{+}&=&0\\,. \n\\end{eqnarray}\nBoth Clifford algebra objects are expressible with $\\theta^a $ and \n$\\frac{\\partial}{\\,\\;\\partial \\theta^a}$~\\cite{norma93,nh2018}, \n (\\cite{IARD2016} and Refs. therein)\n\\begin{eqnarray}\n\\label{cliffthetarel}\n\\gamma^a &=& (\\theta^a + \\frac{\\partial}{\\;\\partial \\theta_a})\\,,\\nonumber\\\\\n\\tilde{\\gamma}^a &=& i \\, (\\theta^a - \\frac{\\partial}{\\;\\partial \\theta a})\\,, \\nonumber\\\\\n\\theta^a &=&\\frac{1}{2} (\\gamma^a - i \\tilde{\\gamma}^a)\\,,\\nonumber\\\\\n\\frac{\\,\\partial}{\\partial \\theta_a} &=&\\frac{1}{2} \\, (\\gamma^a + i \\tilde{\\gamma}^a)\\,,\n\\end{eqnarray}\nfrom where it follows: $(\\gamma^a)^{\\dagger} = \\gamma^a \\eta^{aa}$, \n$(\\tilde{\\gamma}^a)^{\\dagger} = \\tilde{\\gamma}^a \\eta^{aa}$,\n$\\gamma^a \\gamma^a = \\eta^{aa}$, $\\gamma^a (\\gamma^a)^{\\dagger} =1$,\n$ \\tilde{\\gamma}^a  \\tilde{\\gamma}^a = \\eta^{aa}$, \n$ \\tilde{\\gamma}^a  (\\tilde{\\gamma}^a)^{\\dagger} =1$.\n\n\nCorrespondingly we can use either $\\gamma^a$  or  $\\tilde{\\gamma}^a$ instead of \n$\\theta^a$ to span the internal space of fermions. Since both, $\\gamma^a$ and \n$\\tilde{\\gamma}^a$, are expressible with $\\theta^a$ and the derivatives with respect to \n$\\theta^a$, the norm of vectors in Clifford space can be defined by the same integral as in \nGrassmann space, Eq.(\\ref{grassnorm}), or we can simplify the scalar product (as in the \nGrassmann case, Eq.~(\\ref{grassscalar}) by introducing\nthe Clifford vacuum state $|\\psi_{oc}>$, Eq.~(\\ref{vac1}), instead of $|1>$ in Grassmann\ncase. \n\nWe make use of $\\gamma^a$ to span the vector space. As in the case of Grassmann\nspace we require that the basic states are eigenstates of the Cartan subalgebra operators of \n$S^{ab}$ and $\\tilde{S}^{ab}$, Eq.~(\\ref{choicecartan}). \n\\begin{eqnarray}\n\\stackrel{ab}{(k)}:&=& \n\\frac{1}{2}(\\gamma^a + \\frac{\\eta^{aa}}{ik} \\gamma^b)\\,,\\quad \n\\stackrel{ab}{(k)}^{\\dagger} = \\eta^{aa}\\stackrel{ab}{(-k)}\\,,\\nonumber\\\\\n\\stackrel{ab}{[k]}:&=&\n\\frac{1}{2}(1+ \\frac{i}{k} \\gamma^a \\gamma^b)\\,,\\quad \\;\\,\n\\stackrel{ab}{[k]}^{\\dagger} = \\,\\stackrel{ab}{[k]}\\,,\\nonumber\\\\\nS^{ab}\\,\\stackrel{ab}{(k)} &=& \\frac{1}{2} k\\, \\stackrel{ab}{(k)}\\,,\\quad \\quad \\quad\nS^{ab}\\,\\stackrel{ab}{[k]}  = \\frac{1}{2} k\\, \\stackrel{ab}{[k]}\\,, \\nonumber\\\\\n\\tilde{S}^{ab}\\,\\stackrel{ab}{(k)} &=& \\frac{1}{2} k\\, \\stackrel{ab}{(k)}\\,,\\quad \\quad \\quad\n\\tilde{S}^{ab}\\,\\stackrel{ab}{[k]}  = -\\frac{1}{2} k\\, \\stackrel{ab}{[k]}\\,,\n\\label{signature}\n\\end{eqnarray}\nwith $k^2 = \\eta^{aa} \\eta^{bb}$. To calculate $\\tilde{S}^{ab}\\,\\stackrel{ab}{(k)}$ and \n$\\tilde{S}^{ab}\\,\\stackrel{ab}{[k]}$ we define~\\cite{nh03,nh02} the application of \n$\\tilde{\\gamma}^a$  on any Clifford algebra object A  as follows\n\\begin{eqnarray}\n(\\tilde{\\gamma^a} A = i (-)^{(A)} A \\gamma^a)|\\psi_{oc}>\\,,\n\\label{gammatildeA}\n\\end{eqnarray}\nwhere $A$ is any Clifford algebra object and  $(-)^{(A)} = -1$,  if $A$ is an odd Clifford algebra\n object and $(-)^{(A)} = 1$,  if $A$  is an even Clifford algebra\n object, $|\\psi_{oc}>$ is the vacuum state, replacing  the vacuum state $|\\psi_o>= |1>$,\n used in Grassmann case, with the \none  of Eq.~(\\ref{vac1}), in accordance with the relation of Eqs.~({\\ref{cliffthetarel},\n\\ref{grassnorm}, \\ref{grassintegral}}), Ref.~\\cite{nh2018}.\n\n\nWe can define now the creation and annihilation operators in Clifford space so that they fulfill the\nrequirements of Eq.~(\\ref{ijthetaprod}).\nWe write the starting creation operator and its Hermitian conjugate one (in \naccordance with Eq.~(\\ref{signature}) and Eq.(\\ref{choicecartan})) in $2(2n+1)$-dimensional \nspace as follows~\\cite{nh2018}\n\\begin{eqnarray}\n\\hat{b}^{1 \\dagger}_1&=& \\stackrel{03}{(+i)} \\stackrel{12}{(+)} \\stackrel{56}{(+)}\\cdots\n\\stackrel{d-1\\;d}{(+)}\\,,\\nonumber\\\\\n\\hat{b}^{1}_1&=& \\stackrel{d-1\\;d}{(-)} \\cdots \\stackrel{56}{(-)} \\stackrel{12}{(-)}\n\\stackrel{03}{(-i)}\\,.\n\\label{bstart}\n\\end{eqnarray}\nThe starting creation operator $\\hat{b}^{1  \\dagger}_1$, when applied on the vacuum state\n$|\\psi_{oc}>$, defines the starting family member of the starting ''family\". The corresponding\nstarting annihilation operator is its Hermitian conjugated one, Eq.~(\\ref{signature}).\n\nAll the other creation operators of the same family can be obtained by the application of the \ngenerators of the Lorentz transformations $S^{ab}$, Eq.~(\\ref{Lorentzgammatilde}), which do \nnot belong to the Cartan subalgebra of $SO(2(2n+1) -1,1)$, Eq.~(\\ref{choicecartan}). \n\\begin{eqnarray}\n\\hat{b}^{1\\dagger}_i &\\propto & S^{ab} ..S^{ef} \\hat{b}^{1\\dagger}_1\\,,\\nonumber\\\\\n\\hat{b}^{1}_i&\\propto & \\hat{b}^{1}_1 S^{ef}..S^{ab}\\,,\n\\label{b1i}\n\\end{eqnarray}\nwith $S^{ab\\dagger} = \\eta^{aa} \\eta^{bb} S^{ab}$. The proportionality factors are chosen \nso, that the corresponding states $|\\psi^{1}_{1}>= \\hat{b}^{1\\dagger}_i |\\psi_{oc}>$ are \nnormalized, where $|\\psi_{oc}>$ is the normalized vacuum state, $<\\psi_{oc}|\\psi_{oc}> =1$. \n\nThe creation operators creating different \"families\" with respect to the starting \"family\",\n Eq.~(\\ref{bstart}), \ncan be obtained from the starting one by the application of $\\tilde{S}^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}), which do not belong to the Cartan subalgebra of \n$\\widetilde{SO}(2(2n+1) -1,1)$, Eq.~(\\ref{choicecartan}). They all keep the \"family member\" \nquantum number unchanged.\n\\begin{eqnarray}\n\\hat{b}^{\\alpha \\dagger}_i &\\propto & \\tilde{S}^{ab} \\cdots \\tilde{S}^{ef}\\,\n\\hat{b}^{1\\dagger}_{i}\\,.\n\\label{balpha1}\n\\end{eqnarray}\nCorrespondingly we can define (up to the proportionality factor) any creation operator for any\n\"family\" and any \"family member\" with the application of $S^{ab}$ and $\\tilde{S}^{ab}$%\n~\\cite{nh2018}\n\\begin{eqnarray}\n\\hat{b}^{\\alpha \\dagger}_i&\\propto & \\tilde{S}^{ab} \\cdots \\tilde{S}^{ef} \n{S}^{mn}\\cdots {S}^{pr}\n\\hat{b}^{1\\dagger}_{1}\\nonumber\\\\\n&\\propto & {S}^{mn}\\cdots {S}^{pr} \\hat{b}^{1\\dagger}_{1} {S}^{ab} \\cdots {S}^{ef}\\,.\n\\label{anycreation}\n\\end{eqnarray}\nAll the corresponding annihilation operators follow from the creation ones by the Hermitian \nconjugation.\n\nThere are $2^{\\frac{d}{2}-1}$ $\\times \\;\\, 2^{\\frac{d}{2}-1}$ creation operators of an odd \nClifford character and the same number of annihilation operators, which fulfill the anticommutation\nrelations of Eq.~(\\ref{ijthetaprod}) on the vacuum state $|\\psi_{oc}>$ with\n$2^{\\frac{d}{2}-1}$ summands\n\\begin{eqnarray}\n|\\psi_{oc}>&=& \\alpha\\,( \\stackrel{03}{[-i]} \\stackrel{12}{[-]} \n\\stackrel{56}{[-]}\\cdots\n\\stackrel{d-1\\;d}{[-]} + \\stackrel{03}{[+i]} \\stackrel{12}{[+]} \\stackrel{56}{[-]} \\cdots\n\\stackrel{d-1\\;d}{[-]} + \\stackrel{03}{[+i]} \\stackrel{12}{[-]} \\stackrel{56}{[+]}\\cdots\n\\stackrel{d-1\\;d}{[-]} + \\cdots ) |0>\\,, \\quad \\nonumber\\\\\n&&\\alpha =\\frac{1}{\\sqrt{2^{\\frac{d}{2}-1}}}\\,, \\nonumber\\\\\n&&{\\rm for}\\; d=2(2n+1)\\,,\n\\label{vac1}\n\\end{eqnarray}\n$n$ is a positive integer. For a chosen $\\alpha =\\frac{1}{\\sqrt{2^{\\frac{d}{2}-1}}}$\nthe vacuum is normalized: $<\\psi_{oc}|\\psi_{oc}>=1$. \n\n\nIt is proven in Ref.~\\cite{nh2018} that the creation and annihilation operators fulfill the\nanticommutation relations required for fermions, Eq.~(\\ref{ijthetaprod}). \n\n\n\n\n\\section{Properties of representations of the Lorentz group $SO(2(2n+1))$ and of subgroups\nin Grassmann and in Clifford space}\n\\label{grassmanncliffordrepresentations}\n\nThe purpose of this contribution is to compare properties of the representations of the Lorentz\ngroup $SO(2(2n+1))$,  $n \\ge 3$, when for the description of the internal degrees of freedom \nof fermions either {\\bf i.}  Grassmann space or {\\bf ii.} Clifford space is used. The \n{\\it spin-charge-family} theory~(\\cite{JMP2013,normaJMP2015,IARD2016,%\nn2014matterantimatter,nh2017,nd2017,n2012scalars} and the references therein) namely\npredicts that all the properties of the observed either quarks and leptons or vector gauge fields \nor scalar gauge fields originate in $d \\ge (13+1)$, in which massless fermions interact with the \ngravitational field only --- with its spin connections and \nvielbeins. \n\nHowever, both --- Clifford space and Grassmann space --- allow second quantized states, the \ncreation and annihilation operators of which fulfill the anticommutation relations for fermions of \nEq.~(\\ref{ijthetaprod}). \n\nBut while Clifford space offers the description of spins, charges and families of fermions in \n$d=(3+1)$, all in the fundamental representations of the Lorentz group $SO(13,1)$  and the \nsubgroups of the Lorentz group, in agreement with the observations, the representations of the \nLorentz group are in Grassmann space the adjoint ones, in disagreement with what we  \nobserve.\n\nWe compare properties of the representations in Grassmann case with those in Clifford case \nto be able  to better understand \"the choice of nature in the expanding universe, making use of the \nClifford degrees of freedom\", rather than Grassmann degrees of freedom.\n\nIn introduction we briefly reviewed properties of creation and annihilation operators in both spaces,\npresented in Ref.~\\cite{nh2018} (and the references therein). \nWe pay attention on spaces with $d=2(2n+1)$ of ordinary coordinates and  $d=2(2n+1)$ internal \ncoordinates, either of Clifford or of Grassmann character. \n\n{\\bf i.} $\\;\\;$ In Clifford case there are $2^{\\frac{d}{2} - 1}$ creation operators of an odd \nClifford character, creating \"family members\" when applied on the vacuum state. We choose \nthem to be eigenstates of the Cartan subalgebra operators, Eq.(\\ref{choicecartan}), of the Lorentz \nalgebra. All the members\ncan be reached from any of the creation operators  by the application of $S^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}).\nEach \"family member\" appears in $2^{\\frac{d}{2} - 1}$ \"families\", again of an odd Clifford \ncharacter, since the corresponding creation operators are  reachable by $\\tilde{S}^{ab}$, \nEq.~(\\ref{Lorentzgammatilde}), which are Clifford even objects. \n\nThere are correspondingly $2^{\\frac{d}{2} - 1} \\cdot$ $2^{\\frac{d}{2} - 1}$ creation and the \nsame number ($2^{\\frac{d}{2} - 1} \\cdot$ $2^{\\frac{d}{2} - 1}$) of annihilation operators. Also \nthe annihilation operators, annihilating states of $2^{\\frac{d}{2} - 1}$ \"family members\" in\n $2^{\\frac{d}{2} - 1}$ \"families\", have an odd Clifford character, since they are Hermitian conjugate\n to the creation ones. \n\nThe rest of $2 \\cdot$ $2^{\\frac{d}{2} - 1}\\cdot$ $2^{\\frac{d}{2} - 1}$ members\nof the Lorentz representations  have an even Clifford character, what means that the corresponding \ncreation and annihilation operators can not fulfill the anticommutation relations required for \nfermions, Eq.~(\\ref{ijthetaprod}). \nAmong these $2^{\\frac{d}{2} - 1}$ products of projectors determine the vacuum state, \nEq.~(\\ref{vac1}).\n\n\n{\\bf ii.} $\\;\\;$ In Grassmann case there are $\\frac{d!}{\\frac{d}{2}!\\,\\frac{d}{2}! }$ \noperators of an odd\nGrassmann character, which form the creation operators, fulfilling with the corresponding \nannihilation operators the requirements of  Eq.~(\\ref{ijthetaprod}). All the creation operators \nare chosen to be products of the eigenstates of the Cartan subalgebra ${\\cal {\\bf S}}^{ab}$, \nEq.~(\\ref{choicecartan}).  The corresponding annihilation operators are the Hermitian conjugated \nvalues of the creation operators, Eqs.~(\\ref{grassher}, \\ref{grassp}, \\ref{start(2n+1)2theta}). \nThe creation operators form, when applied on the simple vacuum state $|\\phi_{o}> = |1>$,\ntwo independent groups of states. The members of each of the two groups are \nreachable from any member of a group by the application of ${\\cal {\\bf S}}^{ab}$, \nEq.~(\\ref{Lorentztheta}). All the states of any of the two decuplets are orthonormalized.\n\nWe comment in what follows the representations in $d=(13+1)$ in Clifford and in Grassmann case.\nIn {\\it spin-charge family} theory there are breaks of the starting symmetry from $SO(13,1)$ to\n$SO(3,1)\\times SU(2) \\times SU(3) \\times U(1)$ in steps, which lead to the so far observed \nquarks and leptons, gauge and scalar fields and gravity. One \nof the authors (N.S.M.B.), together with H.B. Nielsen, defined the discrete symmetry operators for \nKaluza-Klein theories for spinors in Clifford space~\\cite{nhds}. In Ref.~\\cite{nh2018} the same \nauthors define the discrete \nsymmetry operators in the case that for the description of fermion degrees of \nfreedom Grassmann space is used. Here we comment symmetries in both spaces for some of\nsubgroups of the $SO(13,1)$ group, as well as the appearance of the Dirac sea.  \n\n\n\\subsection{Currents in Grassmann space} \n\\label{currents}\n\n\n\n\\begin{eqnarray}\n\\{ \\theta^a p_a, \\frac{\\partial }{\\partial \\theta_b} p_b \\}_{+}&=&\n \\theta^a p_a  \\frac{\\partial }{\\partial \\theta_b} p_b + \n \\frac{\\partial }{\\partial \\theta_b} p_b \\theta^a p_a = \\eta^{ab}p_a p_b \\,, \n\\label{KGgrass}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\phi^{\\dagger}(\\theta^0 + \\frac{\\partial }{\\partial \\theta_0})\n(\\theta^a +  \\frac{\\partial }{\\partial \\theta_a}) \\phi &=& j^{a} \\,, \n\\label{currentgrass}\n\\end{eqnarray}\n\n\\subsection{Equations of motion in Grassmann and Clifford space}\n\\label{equationinCandG}\n\nWe define~\\cite{nh2018} the action in Grassmann  space, for which we require --- similarly as in \n Clifford case --- that the action for a free massless object \n\\begin{eqnarray}\n{\\cal A}\\,  &=& \\frac{1}{2}  \\int \\; d^dx \\;d^d\\theta\\; \\omega \\, \\{\\phi^{\\dagger} \n(1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\,\\frac{1}{2}\\,\n (\\theta^a p_{a}+ \\eta^{aa} \\theta^{a \\dagger}\n p_{a}) \\phi \\} \\,, \n\\label{actionWeylGrass}\n\\end{eqnarray}\nis Lorentz invariant. The corresponding equation of motion is \n\\begin{eqnarray}\n\\label{Weylgrass}\n\\frac{1}{2}[ (1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\,\\theta^a + \n((1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\,\\theta^a)^{\\dagger}]\\,\n\\, p_{a} \\,|\\phi^{\\theta}_{i}>\\,&= & 0\\,,\n\\end{eqnarray}\n$p_{a} = i\\, \\frac{\\partial}{\\partial x^a}$, leading to the Klein-Gordon equation\n\\begin{eqnarray}\n\\label{LtoKGgrass}\n\\{(1-2\\theta^0 \\frac{\\partial}{\\partial \\theta^0}) \\theta^a p_{a}\\}^{\\dagger}\\,\\theta^b p_{b}\n|\\phi^{\\theta}_{i}>&= &  \np^a p_a |\\phi^{\\theta}_{i}>=0\\,.\n\\end{eqnarray}\nIn the Clifford case the action for massless fermions is well known\n\\begin{eqnarray}\n{\\cal A}\\,  &=& \\int \\; d^dx \\; \\frac{1}{2}\\, (\\psi^{\\dagger}\\gamma^0 \\, \\gamma^a p_{a} \\psi) +\n h.c.\\,, \n\\label{actionWeyl}\n\\end{eqnarray}\n leading to the equations of motion \n\\begin{eqnarray}\n\\label{Weyl}\n\\gamma^a p_{a}  |\\psi^{\\alpha}>&= & 0\\,, \n\\end{eqnarray}\nwhich fulfill also the Klein-Gordon equation\n\\begin{eqnarray}\n\\label{LtoKG}\n\\gamma^a p_{a} \\gamma^b p_b |\\psi^{\\alpha}_{i}>&= &   \np^a p_a |\\psi^{\\alpha}_{i}>=0\\,.\n\\end{eqnarray}\n\n\n\n\n\\subsection{Discrete symmetries in Grassmann and Clifford space}\n\\label{CPT}\n\n\nWe follow also here Ref.~\\cite{nh2018} and the references therein.\nWe distinguish in $d$-dimensional space two kinds of dicsrete operators ${\\cal C}, {\\cal P}$ \nand ${\\cal T}$\n operators with respect to the internal space which we use.\n\nIn the Clifford case~\\cite{nhds}, when the whole $d$-space is treated equivalently, we have \n\\begin{eqnarray}\n\\label{calCPTH}\n{\\cal C}_{{\\cal H}}&=& \\prod_{\\gamma^a \\in \\Im} \\gamma^a \\,\\, K\\,,\\quad\n{\\cal T}_{{\\cal H}}= \\gamma^0 \\prod_{\\gamma^a \\in \\Re} \\gamma^a \\,\\, K\\, I_{x^0}\\,\\,\\,,\n \\quad {\\cal P}^{(d-1)}_{{\\cal H}} = \\gamma^0\\,I_{\\vec{x}}\\,,\\nonumber\\\\\nI_{x} x^a &=&- x^a\\,, \\quad I_{x^0} x^a = (-x^0,\\vec{x})\\,, \\quad I_{\\vec{x}} \\vec{x} =\n -\\vec{x}\\,, \\nonumber\\\\\nI_{\\vec{x}_{3}} x^a &=& (x^0, -x^1,-x^2,-x^3,x^5, x^6,\\dots, x^d)\\,.\n\\end{eqnarray}\nThe product $\\prod \\, \\gamma^a$ is meant in the ascending order in $\\gamma^a$.\n\nIn the Grassmann case we correspondingly define\n\\begin{eqnarray}\n\\label{calCPTG}\n{\\cal C}_{G}&=& \\prod_{\\gamma^a_{ G} \\in \\Im \\gamma^a} \\, \\gamma^a_{ G}\\, K\\,,\\quad\n{\\cal T}_{G} = \\gamma^0_{G} \\prod_{\\gamma^a_{G} \\in \\Re\n \\gamma^a} \\,\\gamma^a_{ G}\\, K \\, I_{x^0}\\,,\\quad\n{\\cal P}^{(d-1)}_{G} = \\gamma^0_{G} \\,I_{\\vec{x}}\\,\n\\end{eqnarray}\nwith $\\gamma^a_{G}$  defined as \n\\begin{eqnarray}\n\\label{gammaG}\n\\gamma^{a}_{G} &=& \n(1- 2 \\theta^a \\eta^{aa} \\frac{\\partial}{\\partial \\theta_{a}})\\,, \n\\end{eqnarray}\nwhile $I_{x}$,\n$I_{\\vec{x}_{3}}$ \nis defined in Eq.~(\\ref{calCPTH}).\nLet be noticed, that since $\\gamma^a_{G}$ ($= - i \\eta^{aa}\\, \\gamma^a \\tilde{\\gamma}^a$) is \nalways real as there is $ \\gamma^a i \\tilde{\\gamma}^a$, while $ \\gamma^a$ is either real or \nimaginary,\nwe use in Eq.~(\\ref{calCPTG}) $\\gamma^a$  to make a choice of appropriate $\\gamma^a_{G}$. \nIn what follows we shall use the notation as in Eq.~(\\ref{calCPTG}).\n\n\nWe define, according to Ref.~\\cite{nh2018} (and the references therein)  in both cases --- Clifford \nGrassmann case --- the operator\n \"emptying\"~\\cite{JMP2013,normaJMP2015} (arxiv:1312.1541) the Dirac sea, so that operation of \n\"emptying$_{N}$\" after the charge conjugation ${\\cal C }_{{\\cal H}}$ in the Clifford case and \n\"emptying$_{G}$\" after the charge conjugation ${\\cal C }_{G}$\nin the Grassmann case (both transform the state put on the top of either the Clifford or the Grassmann\nDirac sea into the corresponding negative energy state) creates the anti-particle state to the starting \nparticle state, both  put on the top of the Dirac sea and both solving the Weyl equation, either in the\nClifford case, Eq.~(\\ref{Weyl}),  or in the Grassmann case, Eq.~(\\ref{Weylgrass}), for free massless \nfermions\n\\begin{eqnarray}\n\\label{empt}\n\"{\\rm emptying}_{N}\"&=& \\prod_{\\Re \\gamma^a}\\, \\gamma^a \\,K\\, \\quad {\\rm in} \\, \\;\n{\\rm Clifford}\\, {\\rm space}\\,, \n\\nonumber\\\\\n\"{\\rm emptying}_{G}\"&=& \\prod_{\\Re \\gamma^a}\\, \\gamma^a_{G} \\,K\\, \\quad {\\rm in}\\;\\, \n{\\rm Grassmann} \\,  {\\rm space}\\,, \n\\end{eqnarray}\nalthough we must keep in mind that indeed the anti-particle state is a hole in the Dirac sea from the \nFock space point of view. The operator \"emptying\" is bringing the single particle operator \n${\\cal C }_{{\\cal H}}$ in the Clifford case and ${\\cal C }_{G}$ in the Grassmann case into the operator \non the Fock space in each of the two cases.\nThen the anti-particle  state creation operator --- \n${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{a}[\\Psi_{p}]$ --- to the corresponding  particle state \ncreation operator --- can be obtained also as follows\n\\begin{eqnarray}\n\\label{makingantip}\n{\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{a}[\\Psi_{p}]\\, |vac>  &=& \n{\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal H}}}\\, \n{\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{p}[\\Psi_{p}]\\, |vac>  =  \n\\int \\, {\\mathbf{\\Psi}}^{\\dagger}_{a}(\\vec{x})\\, \n({\\bf \\mathbb{C}}_{\\cal H}\\,\\Psi_{p} (\\vec{x})) \\,d^{(d-1)} x  \\, \\,|vac> \\,,\\nonumber\\\\\n{\\bf \\mathbb{C}}_{\\cal H} &=& \"{\\rm emptying}_{N}\"\\,\\cdot\\, {\\cal C}_{{\\cal H}}  \\,\n\\end{eqnarray}\nin both cases.\n\nThe operators ${\\bf \\mathbb{C}}_{\\cal H}$ and ${\\bf \\mathbb{C}}_{G}$ \n\\begin{eqnarray}\n\\label{emptCHG}\n{\\bf \\mathbb{C}}_{\\cal H} &=& \"{\\rm emptying}_{N}\" \\,\\cdot\\, {\\cal C}_{{\\cal H}} \\,,\\nonumber\\\\\n{\\bf \\mathbb{C}}_{G} &=& \"{\\rm emptying}_{NG}\" \\,\\cdot\\, {\\cal C}_{G}\\,,\n\\end{eqnarray}\n operating on \n$\\Psi_{p} (\\vec{x})$ transforms the positive energy spinor state (which solves the corresponding \nWeyl equation for a massless free fermion) put on the top of the Dirac sea into the positive energy \nanti-fermion state, which again solves the corresponding Weyl equation for a massless free \nanti-fermion put on the top of the Dirac sea. Let us point out that either the operator \n$\"{\\rm emptying}_{N}\" $ \nor the operator $\"{\\rm emptying}_{NG}\"$ transforms the single particle operator either\n$ {\\cal C}_{\\cal H}$ or ${\\cal C}_{G}$ into the operator operating in the Fock space. \n\n\nWe use the Grassmann even, Hermitian and real operators $\\gamma^{a}_{G}$, \nEq.~(\\ref{gammaG}), to define discrete symmetry in Grassmann space, first we did in \n$((d+1)-1)$ space, Eq.~(\\ref{calCPTG}), now we do in $(3+1)$  space, Eq.~(\\ref{calCPTNG}),\n as it is done \nin~\\cite{nhds} in the Clifford case. \nIn the Grassmann case we do this in analogy with the operators in the Clifford case~\\cite{nhds}\n\\begin{eqnarray}\n\\label{calCPTNG}\n{\\cal C}_{NG}&=& \\prod_{\\gamma^m_{ G} \\in \\Re  \\gamma^m} \\, \\gamma^m_{ G}\\, K \\,\n I_{x^6 x^8...x^d}\\,,\\nonumber\\\\\n{\\cal T}_{NG}&=& \\gamma^0_{G} \\prod_{\\gamma^m_{G} \\in \\Im\n \\gamma^m} \\, K \\, I_{x^0} I_{x^5 x^7...x^{d-1}}\\,,\\nonumber\\\\\n{\\cal P}^{(d-1)}_{NG} &=& \\gamma^0_{G} \\, \\prod_{s=5}^{d}\\, \\gamma^s_{ G} I_{\\vec{x}}\\,,\n\\nonumber\\\\\n{\\bf \\mathbb{C}}_{NG} &=&\\prod_{\\gamma^s_{ G} \\in \\Re  \\gamma^s} \\,\\gamma^s_{ G}\\,, \n I_{x^6 x^8...x^d}\\,,\\quad\n{\\bf \\mathbb{C}}_{NG} {\\cal P}^{(d-1)}_{NG} = \\gamma^0_{G} \\, \\prod_{\n\\gamma^s_{ G} \\in \\Im  \\gamma^s, s=5}^{d}\\, \\gamma^s_{ G}\\, I_{\\vec{x}_{3}}\\,\n I_{x^6 x^8...x^d}\\,,\n\\nonumber\\\\\n{\\bf \\mathbb{C}}_{NG} {\\cal T}_{NG} {\\cal P}^{(d-1)}_{NG} &=& \\prod_{\\gamma^s_{ G} \\in \n\\Im  \\gamma^a} \\,\\gamma^a_{ G}\\,I_x K\\,.\n\\end{eqnarray}\n\n\n\\subsection{Representations in Grassmann and in Clifford space in $d=(13+1)$}\n\\label{so13+1}\nIn the {\\it spin-charge-family} theory the starting dimension of space must be $\\ge(13+1)$, in \norder that the theory manifests in $d=(3+1)$ all the observed properties of quarks and leptons,\n gauge and scalar fields (explaining the appearance of  higgs and \nthe Yukawa couplings), offering as well the explanations for the observations in \ncosmology.\n\nLet us therefore comment properties of representations in both spaces when $d=(13+1)$,\n if we analyze one group of \"family members\" of one of families in Clifford space, and \none of the two representations of  $\\frac{1}{2}\\, \\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$.\n\n\n{\\bf a.} $\\;\\;$  Let us start \nwith Clifford space~\\cite{IARD2016,normaJMP2015,n2014matterantimatter,JMP2013,pikanorma,%\nportoroz03,norma93}. Each \"family\" representation has $2^{\\frac{d}{2} - 1}= 64$\n \"family members\". If we analyze this representation with respect to the subgroups $SO(3,1)$, \n$(SU(2)\\times SU(2))$ of $SO(4)$ and ($SU(3)\\times$ $U(1)$) of $SO(6)$ of  the \nLorentz group $SO(13,1)$, we find that the representations have quantum numbers of all the so \nfar observed quarks and leptons and antiquarks and antileptons, all with spin up and spin down, \nas well as of the left and right handedness, with the right handed neutrino included as the\nmember of this representation.\n\nLet us make a choice of the \"family\", which follows by the application of $\\tilde{S}^{15}$ on the \n\"family\", for which the creation operator of the right-handed neutrino with spin $\\frac{1}{2}$ \nwould be $ \\stackrel{03}{(+i)}\\,\\stackrel{12}{(+)}|\\stackrel{56}{(+)}\\,\n\\stackrel{78}{(+)}||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$.\n(The corresponding annihilation operator of this creation operator is  $\\stackrel{13 \\;14}{(-)}\\;\\;\n\\stackrel{11\\;12}{(-)}\\;\\;\\stackrel{9\\;10}{(-)}||\\stackrel{78}{(-)}\\,\\stackrel{56}{(-)}| \n\\stackrel{12}{(-)}\\,\\stackrel{03}{(-i)}$). In Table~\\ref{Table so13+1.} presented creation\noperators for all the \"family members\" of this family follow \nby the application of $S^{ab}$ on \n$\\tilde{S}^{15}$ $ \\stackrel{03}{(+i)}\\,\\stackrel{12}{(+)}|\\stackrel{56}{(+)}\\,\n\\stackrel{78}{(+)}||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$.\n(The annihilation operator of $\\tilde{S}^{15}$ $ \\stackrel{03}{(+i)}\\,\n\\stackrel{12}{(+)}|\\stackrel{56}{(+)}\\,\\stackrel{78}{(+)}||\\stackrel{9\\;10}{(+)}\\;\\;\n\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $ is $\\stackrel{13 \\;14}{[-]}\\;\\;\n\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{9\\;10}{(-)}||\\stackrel{78}{(-)}\\,\\stackrel{56}{[+]}| \n\\stackrel{12}{[+]}\\,\\stackrel{03}{(-i)}$.) \n\nThis is the representation of \nTable~\\ref{Table so13+1.}, in which all the 'family members'' of one \"family\"  are classified with \nrespect to the subgroups $SO(3,1)\\times SU(2) \\times SU(2)\\times SU(3) \\times U(1)$.\nThe vacuum state on which the creation operators, represented in the third column, apply \nis defined in Eq.~(\\ref{vac1}). All the creation operators of all the states are of an odd Clifford \ncharacter, fulfilling together with the annihilation operators (which have as well the equivalent\nodd Clifford character, since the Hermitian conjugation do not change the \nClifford character) the requirements of Eq.~(\\ref{ijthetaprod}). Since the Clifford even operators \n$S^{ab}$ and $\\tilde{S}^{ab}$ do not change the Clifford character, all the creation and \nannihilation operators, obtained by products of $S^{ab}$ or $\\tilde{S}^{ab}$ or both,\nfulfill the requirements of Eq.~(\\ref{ijthetaprod}).\n\nWe recognize in Table~\\ref{Table so13+1.} that quarks distinguish from leptons only in the \n$SO(6)$ part of the creation operators. Quarks belong to the colour ($SU(3)$) triplet carrying\nthe \"fermion\" $(U(1))$ quantum number $\\tau^{4} =\\frac{1}{6}$, antiquarks belong to the colour \nantitriplet, carrying the \"fermion\" quantum number $\\tau^{4} = -\\frac{1}{6}$. Leptons belong \nto the colour ($SU(3)$) singlet, carrying the \"fermion\" $(U(1))$ quantum number $\\tau^{4} =\n -\\frac{1}{2}$, while antileptons belong to the colour antisinglet, carrying the \"fermion\" quantum \nnumber $\\tau^{4} = \\frac{1}{2}$. \n \nLet us also comment that the oddness and evenness of part of states in the subgroups of the \n$SO(13,1)$ group change: While quarks and leptons have in the part of $SO(6)$ an odd Clifford \ncharacter, have antiquarks and antileptons in this part an even odd Clifford character. \nCorrespondingly the Clifford character changes in the rest of subgroups \n\n\\bottomcaption{\\label{Table so13+1.}%\n\\tiny{\nThe left handed ($\\Gamma^{(13,1)} = -1$~\\cite{IARD2016})\nmultiplet of spinors --- the members of the fundamental representation of the $SO(13,1)$ group,\nmanifesting the subgroup $SO(7,1)$\n of the colour charged quarks and antiquarks and the colourless\nleptons and antileptons --- is presented in the massless basis using the technique presented in\nRefs.~\\cite{nh02,nh03,IARD2016,normaJMP2015}.\nIt contains the left handed  ($\\Gamma^{(3,1)}=-1$)\n weak ($SU(2)_{I}$) charged  ($\\tau^{13}=\\pm \\frac{1}{2}$, Eq.~(\\ref{so42})),\nand $SU(2)_{II}$ chargeless ($\\tau^{23}=0$, Eq.~(\\ref{so42}))\nquarks and leptons and the right handed  ($\\Gamma^{(3,1)}=1$)\n weak  ($SU(2)_{I}$) chargeless and $SU(2)_{II}$\ncharged ($\\tau^{23}=\\pm \\frac{1}{2}$) quarks and leptons, both with the spin $ S^{12}$  up and\ndown ($\\pm \\frac{1}{2}$, respectively). \nQuarks distinguish from leptons only in the $SU(3) \\times U(1)$ part: Quarks are triplets\nof three colours  ($c^i$ $= (\\tau^{33}, \\tau^{38})$ $ = [(\\frac{1}{2},\\frac{1}{2\\sqrt{3}}),\n(-\\frac{1}{2},\\frac{1}{2\\sqrt{3}}), (0,-\\frac{1}{\\sqrt{3}}) $], Eq.~(\\ref{so64}))\ncarrying  the \"fermion charge\" ($\\tau^{4}=\\frac{1}{6}$, Eq.~(\\ref{so64})).\nThe colourless leptons carry the \"fermion charge\" ($\\tau^{4}=-\\frac{1}{2}$).\nThe same multiplet contains also the left handed weak ($SU(2)_{I}$) chargeless and $SU(2)_{II}$\ncharged antiquarks and antileptons and the right handed weak ($SU(2)_{I}$) charged and\n$SU(2)_{II}$ chargeless antiquarks and antileptons.\nAntiquarks distinguish from antileptons again only in the $SU(3) \\times U(1)$ part: Antiquarks are\nantitriplets,\n carrying  the \"fermion charge\" ($\\tau^{4}=-\\frac{1}{6}$).\nThe anticolourless antileptons carry the \"fermion charge\" ($\\tau^{4}=\\frac{1}{2}$).\n $Y=(\\tau^{23} + \\tau^{4})$ is the hyper charge, the electromagnetic charge\nis $Q=(\\tau^{13} + Y$).\nThe vacuum state,\non which the nilpotents and projectors operate, is presented in Eq.~(\\ref{vac1}).\nThe reader can find this  Weyl representation also in\nRefs.~\\cite{n2014matterantimatter,pikanorma,portoroz03,normaJMP2015} and the references\ntherein. }\n}\n\\tablehead{\\hline\ni&$$&$|^a\\psi_i>$&$\\Gamma^{(3,1)}$&$ S^{12}$&\n$\\tau^{13}$&$\\tau^{23}$&$\\tau^{33}$&$\\tau^{38}$&$\\tau^{4}$&$Y$&$Q$\\\\\n\\hline\n&& ${\\rm (Anti)octet},\\,\\Gamma^{(7,1)} = (-1)\\,1\\,, \\,\\Gamma^{(6)} = (1)\\,-1$&&&&&&&&& \\\\\n&& ${\\rm of \\;(anti) quarks \\;and \\;(anti)leptons}$&&&&&&&&&\\\\\n\\hline\\hline}\n\\tabletail{\\hline \\multicolumn{12}{r}{\\emph{Continued on next page}}\\\\}\n\\tablelasttail{\\hline}\n\\begin{center}\n\\tiny{\n\\begin{supertabular}{|r|c||c||c|c||c|c||c|c|c||r|r|}\n1&$ u_{R}^{c1}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n2&$u_{R}^{c1}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n3&$d_{R}^{c1}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n4&$ d_{R}^{c1} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n5&$d_{L}^{c1}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n6&$d_{L}^{c1} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n7&$ u_{L}^{c1}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\n8&$u_{L}^{c1}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\\hline\n\\shrinkheight{0.3\\textheight}\n9&$ u_{R}^{c2}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n10&$u_{R}^{c2}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n11&$d_{R}^{c2}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$\n&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$ - \\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n12&$ d_{R}^{c2} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $\n&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n13&$d_{L}^{c2}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$\n&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n14&$d_{L}^{c2} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n15&$ u_{L}^{c2}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\n16&$u_{L}^{c2}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$-\\frac{1}{2}$&$\\frac{1}{2\\,\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\\hline\n17&$ u_{R}^{c3}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n18&$u_{R}^{c3}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{2}{3}$&$\\frac{2}{3}$\\\\\n\\hline\n19&$d_{R}^{c3}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n20&$ d_{R}^{c3} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$-\\frac{1}{3}$&$-\\frac{1}{3}$\\\\\n\\hline\n21&$d_{L}^{c3}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n22&$d_{L}^{c3} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$-\\frac{1}{3}$\\\\\n\\hline\n23&$ u_{L}^{c3}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\n24&$u_{L}^{c3}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$-\\frac{1}{\\sqrt{3}}$&$\\frac{1}{6}$&$\\frac{1}{6}$&$\\frac{2}{3}$\\\\\n\\hline\\hline\n25&$ \\nu_{R}$&$ \\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $ &1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$0$&$-\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n26&$\\nu_{R}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$ &$0$&$0$&$-\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n27&$e_{R}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$-\\frac{1}{2}$&$-1$&$-1$\\\\\n\\hline\n28&$ e_{R} $&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$-\\frac{1}{2}$&$-1$&$-1$\\\\\n\\hline\n29&$e_{L}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-1$\\\\\n\\hline\n30&$e_{L} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&-1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-1$\\\\\n\\hline\n31&$ \\nu_{L}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$ &-1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$0$\\\\\n\\hline\n32&$\\nu_{L}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$0$&$-\\frac{1}{2}$&$-\\frac{1}{2}$&$0$\\\\\n\\hline\\hline\n33&$ \\bar{d}_{L}^{\\bar{c1}}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $ &-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n34&$\\bar{d}_{L}^{\\bar{c1}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n35&$\\bar{u}_{L}^{\\bar{c1}}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n36&$ \\bar{u}_{L}^{\\bar{c1}} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n37&$\\bar{d}_{R}^{\\bar{c1}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$&1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n38&$\\bar{d}_{R}^{\\bar{c1}} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n39&$ \\bar{u}_{R}^{\\bar{c1}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$ &1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0 &$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\n40&$\\bar{u}_{R}^{\\bar{c1}}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$\n&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$-\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\\hline\n41&$ \\bar{d}_{L}^{\\bar{c2}}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $\n&-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n42&$\\bar{d}_{L}^{\\bar{c2}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n43&$\\bar{u}_{L}^{\\bar{c2}}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n44&$ \\bar{u}_{L}^{\\bar{c2}} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $\n&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n45&$\\bar{d}_{R}^{\\bar{c2}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n46&$\\bar{d}_{R}^{\\bar{c2}} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $\n&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n47&$ \\bar{u}_{R}^{\\bar{c2}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n &1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0 &$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\n48&$\\bar{u}_{R}^{\\bar{c2}}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$\n&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$\\frac{1}{2}$&$-\\frac{1}{2\\,\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\\hline\n49&$ \\bar{d}_{L}^{\\bar{c3}}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $ &-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n50&$\\bar{d}_{L}^{\\bar{c3}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$\\frac{1}{3}$&$\\frac{1}{3}$\\\\\n\\hline\n51&$\\bar{u}_{L}^{\\bar{c3}}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n52&$ \\bar{u}_{L}^{\\bar{c3}} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}  $&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$&$-\\frac{2}{3}$\\\\\n\\hline\n53&$\\bar{d}_{R}^{\\bar{c3}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n54&$\\bar{d}_{R}^{\\bar{c3}} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$\\frac{1}{3}$\\\\\n\\hline\n55&$ \\bar{u}_{R}^{\\bar{c3}}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $ &1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0 &$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\n56&$\\bar{u}_{R}^{\\bar{c3}}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$\\frac{1}{\\sqrt{3}}$&$-\\frac{1}{6}$&$-\\frac{1}{6}$&$-\\frac{2}{3}$\\\\\n\\hline\\hline\n57&$ \\bar{e}_{L}$&$ \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\n\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $ &-1&$\\frac{1}{2}$&0&\n$\\frac{1}{2}$&$0$&$0$&$\\frac{1}{2}$&$1$&$1$\\\\\n\\hline\n58&$\\bar{e}_{L}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$-\\frac{1}{2}$&0&\n$\\frac{1}{2}$ &$0$&$0$&$\\frac{1}{2}$&$1$&$1$\\\\\n\\hline\n59&$\\bar{\\nu}_{L}$&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&-1&$\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n60&$ \\bar{\\nu}_{L} $&$ - \\stackrel{03}{(+i)}\\,\\stackrel{12}{(-)}|\n\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&-1&$-\\frac{1}{2}$&0&\n$-\\frac{1}{2}$&$0$&$0$&$\\frac{1}{2}$&$0$&$0$\\\\\n\\hline\n61&$\\bar{\\nu}_{R}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$0$\\\\\n\\hline\n62&$\\bar{\\nu}_{R} $&$ - \\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $&1&$-\\frac{1}{2}$&\n$-\\frac{1}{2}$&0&$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$0$\\\\\n\\hline\n63&$ \\bar{e}_{R}$&$\\stackrel{03}{(+i)}\\,\\stackrel{12}{[+]}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$ &1&$\\frac{1}{2}$&\n$\\frac{1}{2}$&0 &$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$1$\\\\\n\\hline\n64&$\\bar{e}_{R}$&$\\stackrel{03}{[-i]}\\,\\stackrel{12}{(-)}|\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}\n||\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$&1&$-\\frac{1}{2}$&\n$\\frac{1}{2}$&0&$0$&$0$&$\\frac{1}{2}$&$\\frac{1}{2}$&$1$\\\\\n\\hline\n\\end{supertabular}\n}\n\\end{center}\n\n\n\n\nFamilies are generated by $\\tilde{S}^{ab}$ applying on any one of the  \"family members\".  \nAgain all the \"family members\" of this \"family\" follow by the application of all $S^{ab}$ (not \nbelonging to Cartan subalgebra). \n\nThe spontaneous break of symmetry from \n$SO(13,1)$ to $SO(7,1) \\times SU(3) \\times U(1)$, Refs.~\\cite{IARD2016,n2014matterantimatter,%\nnormaJMP2015}, makes in the {\\it spin-charge-family} theory  all the families, generated by  \n$\\tilde{S}^{mt}$ and $\\tilde{S}^{st}$,\n[$m=(0,1,2,3)$, $s=(5,6,7,8), t=(9,10,11,12,13,14)$], massive of the scale \nof $\\ge 10^{16}$ GeV~\\cite{DHN,DN012,%\nfamiliesDNproc}. Correspondingly there are only eight families of\nquarks and leptons, \nwhich split into two groups of four families, both manifesting  the symmetry $\\widetilde{SU}(2)\n\\times \\widetilde{SU}(2)$ $\\times U(1)$. (The fourth of the lower four families is predicted to be \nobserved at the LHC, the stable  of the upper four families\ncontributes to the dark  \nmatter~\\cite{gn2009}.)\n\nIn the {\\it spin-charge-family} theory fermions interact with only gravity, which manifests\nafter the break of the starting symmetry in $d=(3+1)$ as all the known vector gauge fields, \nordinary gravity and the higgs and the Yukawa couplings~\\cite{nd2017,IARD2016,%\nn2014matterantimatter,normaJMP2015,n2012scalars}. There are scalar fields which bring masses \nto family members. The theory explains not only all the assumptions of the {\\it standard model}\nwith the appearance of families, the vector gauge fields and the scalar fields, it also explains \nappearance of the dark matter~\\cite{gn2009}, \nmatter\/antimatter asymmetry~\\cite{n2014matterantimatter} and other phenomena, like the\nmiraculous cancellation of the triangle anomalies  in the {\\it standard model}~\\cite{nh2017}. \n\n\n{\\bf b.} $\\;\\;$ We compare representations of $SO(13,1)$ in Clifford space with those in \nGrassmann space. We have {\\bf no \"family\" quantum numbers in Grassmann space}. \nWe only have two groups of creation operators, defining --- when applied on the vacuum state\n $|1>$ --- $\\frac{1}{2}$ $\\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$ equal in $d=(13+1)$ to $1716$\nmembers in each of the two groups in comparison  in Clifford case with $64$ \"family \nmembers\" in one \"family\" and $64$ \"families\", which  the breaks of symmetry reduce to \n$8$ \"families\", making all the $(64 - 8)$ \"families\" massive and correspondingly not observable at low \nenergies~(\\cite{normaJMP2015,DHN} and the references therein).\n\n \nSince the  $1716$ members are hard to be mastered, let us look therefore at each subgroup ---\n$SU(3) \\times U(1)$, $SO(3,1)$ and $SU(2)\\times SU(2)$  of  $SO(13,1)$ --- separately. \n\n\nLet us correspondingly analyze  the subgroups: $SO(6)$ from the point of view of\n the two subgroups $SU(3)\\times U(1)$, and  $SO(7,1)$ from the point of view of\n the two subgroups $SO(3,1) \\times SO(4)$, and let us also analyze $SO(4)$ as \n$SU(2)\\times$ $SU(2)$.\n\n\n\n\n\n\\subsection{Examples of second quantizable states in Grassmann and in Clifford space} \n\\label{examples}\n\nWe compare properties of representations in Grassmann and in Clifford space for several\nchoices of subgroups of  $SO(13,1)$ in the case \nthat in both spaces creation and annihilation operators fulfill requirements of\n Eq.~(\\ref{ijthetaprod}), that is that both kinds of states can be second quantized.\nLet us again point out that in Grassmann case fermions carry integer spins, while in Clifford\ncase they carry half integer spin.\n\n\\subsubsection{ States in Grassmann and in Clifford space for $d=(5+1)$}\n\\label{examples51}\n\nWe study properties of representations of the subgroup $SO(5,1)$ (of the group $SO(13,1)$),\nin Clifford and in Grassmann space, requiring that states can be in both spaces second quantized, \nfulfilling therefore Eq.~(\\ref{ijthetaprod}).\n\n{\\bf a. }\nIn Clifford space there are $2^{\\frac{d}{2}-1}$, each with  $2^{\\frac{d}{2}-1}$ family \nmembers, that is $4$ families, each with $4$ members. All these sixteen states are of \nan odd Clifford character, since all can be obtained by products of $S^{ab}$, \n$\\tilde{S}^{ab}$\nor both from a Clifford odd starting state and are correspondingly second quantizable\nas required in Eq.~(\\ref{ijthetaprod}). All the states are  the eigenstates of the Cartan \nsubalgebra of the Lorentz algebra in Clifford space, Eq.~(\\ref{choicecartan}), solving the Weyl\nequation for free massless spinors in Clifford space, Eq.~(\\ref{Weyl}). \n The four familes, with four members each, are presented in\nTable~\\ref{Table clifffourxfour51.}. All of these $16$ states are reachable from the first one\nin each of the four families by $S^{ab}$, or by $\\tilde{S}^{ab}$ if applied on any family member.\n\nEach of these four families have positive and negative energy solutions, as presented in~%\n\\cite{nhds},  in Table I.. We present in Table~\\ref{Table clifffourxfour51.} only states of a\npositive energy, that is states  above the Dirac sea. The antiparticle states are reachable from\nthe particle states by the application of the operator $\\mathbb{C}_{{\\cal N}}\\,\n $$ {\\cal P}^{(d-1)}_{{\\cal N}} = \\gamma^0 \\gamma^5 I_{\\vec{x}_3} I_{x^6}$, keeping \nthe spin $\\frac{1}{2}$, while changing the charge from $\\frac{1}{2}$ to $- \\frac{1}{2}$. \nAll the states above the Dirac sea are indeed the hole in the Dirac sea, as explained in\nRef.~\\cite{nhds}.\n\\begin{table}\n\\begin{tiny}\n  \\centering\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n    \\hline\\hline\n    $$ & $\\psi$ & $S^{03}$ & $S^{12}$ & $S^{56 }$ \n                         & $\\Tilde{S}^{03}$ & $\\Tilde{S}^{12}$ & $\\Tilde{S}^{56 }$\\\\\n    \\hline\\hline\n        $\\psi^{ I}_{1}$  & $\\stackrel{03}{(+i)} \\stackrel{12}{(+)} \\stackrel{56}{(+)} $\n                               & $\\frac{i}{2}$ & $\\frac{1}{2}$ &  $\\frac{1}{2}$\n    & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$\\\\\n        $\\psi^{ I}_{2}$ &$\\stackrel{03}{[-i]}\\stackrel{12}{[-]} \\stackrel{56}{(+)}$\n        & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$  & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$                    \\\\\n        $\\psi^{ I}_{3}$ &$\\stackrel{03}{[-i]} \\stackrel{12}{(+)} \\stackrel{56}{[-]} $\n                               & $- \\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$\n      & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$                              \\\\\n        $\\psi^{ I}_{4}$ &$\\stackrel{03}{(+i)} \\stackrel{12}{[-]} \\stackrel{56}{[-]}$\n      & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$    & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$                    \\\\\n    \\hline \n         $\\psi^{II}_{1}$  & $\\stackrel{03}{[+i]} \\stackrel{12}{[+]} \\stackrel{56}{(+)}$ \n                               & $\\frac{i}{2}$ & $\\frac{1}{2}$ &  $\\frac{1}{2}$\n      & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$\\\\\n        $\\psi^{II}_{2}$  &$\\stackrel{03}{(-i)} \\stackrel{12}{(-)} \\stackrel{56}{(+)}  $\n       & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$                        \\\\\n        $\\psi^{II}_{3}$  & $\\stackrel{03}{(-i)} \\stackrel{12}{[+]} \\stackrel{56}{[-]}$\n         & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$       & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$               \\\\\n        $\\psi^{II}_{4}$  & $\\stackrel{03}{[+i]}\\stackrel{12}{(-)} \\stackrel{56}{[-]}$\n       & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$         & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$                       \\\\\n       \\hline\n       $\\psi^{III}_{1}$ & $\\stackrel{03}{[+i]} \\stackrel{12}{(+)} \\stackrel{56}{[+]}$\n                                 & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$\n       & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$\\\\\n        $\\psi^{III}_{2}$ & $\\stackrel{03}{(-i)} \\stackrel{12}{[-]} \\stackrel{56}{[+]}$\n         & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$  & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$                              \\\\\n       $\\psi^{III}_{3}$ & $\\stackrel{03}{(-i)} \\stackrel{12}{(+)} \\stackrel{56}{(-)} $\n          & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$   & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$                   \\\\\n       $\\psi^{III}_{4}$ & $\\stackrel{03}{[+i]} \\stackrel{12}{[-]} \\stackrel{56}{(-)} $\n         & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$ & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$\\\\\n        \\hline\n        $\\psi^{IV}_{1}$ &  $\\stackrel{03}{(+i)}\\stackrel{12}{[+]} \\stackrel{56}{[+]} $\n                               & $\\frac{i}{2}$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ \n                              & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$\\\\\n        $\\psi^{IV}_{2}$ & $\\stackrel{03}{[-i]} \\stackrel{12}{(-)}\\stackrel{56}{[+]}  $ \n         & $-\\frac{i}{2}$ & $-\\frac{1}{2}$ & $\\frac{1}{2}$ & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$                     \\\\\n       $\\psi^{IV}_{3}$ & $\\stackrel{03}{[-i]} \\stackrel{12}{[+]}\\stackrel{56}{(-)}  $\n        & $-\\frac{i}{2}$ & $\\frac{1}{2}$ & $-\\frac{1}{2}$  & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$                              \\\\\n       $\\psi^{IV}_{4}$ & $\\stackrel{03}{(+i)} \\stackrel{12}{(-)} \\stackrel{56}{(-)}  $ \n        & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$  & $\\frac{i}{2}$ & $-\\frac{1}{2}$ & $-\\frac{1}{2}$                       \\\\\n    \\hline\\hline\n  \\end{tabular}\n  \\caption{\\label{Table clifffourxfour51.} The four families, each with four members. For the \nchoice $p^a=(p^0,0,0,p^3,0,0)$  have the first and the second member  the space part \nequal to $e^{- i |p^0| x^0+i |p^3|x^3}$ and $e^{- i |p^0| x^0-i |p^3|x^3}$,  \nrepresenting the particles with spin up and down, respectively. The third and the fourth member\nrepresent the antiparticle states, with the space part equal to  $e^{- i |p^0| x^0-i |p^3|x^3}$\nand $e^{- i |p^0| x^0+i |p^3|x^3}$, with the spin up and down respectively.\nThe antiparticle states follow from the particle state by the application of $\\mathbb{C}_{{\\cal N}}\\,\n $$ {\\cal P}^{(d-1)}_{{\\cal N}} = \\gamma^0 \\gamma^5 I_{\\vec{x}_3} I_{x^6}$.\nThe charge of the particle states is $\\frac{1}{2}$, for antiparticle states $-\\frac{1}{2}$.}\n\\end{tiny}\n\\end{table}\n\n\n{\\bf b.0}\nIn Grassmann space there are $ \\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$ second quantizable\nstates as required in Eq.~(\\ref{ijthetaprod}), forming in $d=(5+1)$ two decuplets --- \neach with  $ \\frac{1}{2} \\,\\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}$ states --- all are  the \neigenstates of the Cartan subalgebra of the Lorentz algebra in (internal) Grassmann space. \nAll the states of one (anyone of the two) decuplets are reachable by  the application of the \noperators ${\\cal {\\bf S}}^{ab}$ on a starting state. The two decouplets are presented in\nTable~\\ref{Table grassdecupletso51.}\n\nLet us first find the solution of the equations of motion for free massless fermions,\n Eq.~(\\ref{Weylgrass}),  with the momentum $p^a =(p^0, p^1,p^2,p^3,0,0)$. One obtains\nfor $\\psi_I = \\alpha (\\theta^0-\\theta^3) (\\theta^1 +i \\theta^2) (\\theta^5 +i \\theta^6)$\n$+ \\beta (\\theta^0 \\theta^3 + i\\theta^1\\theta^2) (\\theta^5 +i \\theta^6) +  $ \n$\\gamma (\\theta^0+\\theta^3) (\\theta^1 - i \\theta^2) (\\theta^5 +i \\theta^6)$ the solution\n\\begin{eqnarray}\n\\label{5+1I}\n\\beta&=& \\frac{2\\gamma (p^1-i p^2)}{(p^0 -p^3)}=\n\\frac{2\\gamma (p^0+ p^3)}{(p^1 +i p^2)}=- \\frac{2\\alpha (p^0- p^3)}{(p^1 -i p^2)}\n= - \\frac{2\\alpha (p^1+i p^2)}{(p^0 +p^3)}\\,,\\nonumber\\\\\n(p^0)^2 &=& (p^1)^2 + (p^2)^2 +(p^3)^2\\,, \\nonumber\\\\\n\\frac{\\beta}{- \\alpha } &=& \\frac{2 (p^0- p^3)}{(p^1 -i p^2)}\\,, \\quad\n\\frac{\\gamma}{- \\alpha } =\\frac{(p^0- p^3)^2}{(p^1 -i p^2)^2}\\,.\n\\end{eqnarray}\n\nOne has for $p^0= |p^0|$ the positive energy solution, describing a fermion above the \"Dirac sea\", \nand for   $p^0=- |p^0|$ the negative energy solution, describing a fermion in  the \"Dirac sea\".\nThe \"charge\" of the \"fermion\" is $1$. \nSimilarly one finds the solution for the other three states with the negative \"charge\" $-1$, again\nwith the positive and negative energy.\nThe space part of the \"fermion\" state is for \"spin up\" equal to \n$e^{- i |p^0| x^0+i \\vec{p}\\vec{x}}$, for his antiparticle for the same internal spin \n$e^{- i |p^0| x^0 - i \\vec{p}\\vec{x}}$.  \n\nThe discrete symmetry operator ${\\bf \\mathbb{C}}_{NG}$ ${\\cal P}^{(d-1)}_{NG}$, which is \nin our case equal to $\\gamma^0_{G} \\gamma^5_{G}  I_{\\vec{x}_{3}} I_{x^6}$, transforms\nthe  first state in Table~\\ref{Table grassdecupletso51.} into the sixth, the  second state\ninto the fifth, the third state into the fourth, keeping the same spin while changing the \"charge\" of\nthe superposition of the three states $\\psi_{Ip}$.\nBoth superposition of states, Eq.~(\\ref{5+1I}) represent the positive energy states put on \nthe top of the \"Dirac\" sea, the first \ndescribing a particle with \"charge\" $1$ and the second superposition of the second three states \n$\\psi_{Ia}$, describing the antiparticle with the\"charge\" $-1$. \nWe namely apply\n ${\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal NG}}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$\n on ${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{p}[\\Psi^{pos}_{I}]$ by applying \n $\\mathbb{C}_{{\\cal NG}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$ on $ \\Psi^{pos}_{I}$ as follows: \n ${\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal NG}}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$ \n ${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{p}\n \\Psi^{pos}_{I}]$ $({\\underline {\\bf \\mathbb{C}}}_{{{\\bf \\cal NG}}}\\, $\n$ {\\cal P}^{(d-1)}_{{\\cal NG}})^{-1} =$ \n ${\\underline {\\bf {\\Huge \\Psi}}}^{\\dagger}_{aNG}$$[\\mathbb{C}_{{\\cal NG}}$\n ${\\cal P}^{(d-1)}_{{\\cal NG}}\\Psi^{pos}_{1}]$.\n\n One recognizes that it is $\\mathbb{C}_{{\\cal NG}}\\, $$ {\\cal P}^{(d-1)}_{{\\cal NG}}$ \n$ \\Psi^{pos}_{I}=$\n $\\Psi^{pos}_{II}$ (Table~\\ref{Table grassdecupletso51.}), \n which must be put on the top of the \"Dirac\" sea, representing the hole in the particular state \n in the \"Dirac\" sea, which solves the corresponding  equation of motion for the negative energy.\n\n\n\n\n \\begin{table}\n \\begin{center}\n\\includegraphics{TabDecupletsGrasspdf-fig.pdf}\n \\end{center}\n \\caption{\\label{Table grassdecupletso51.}  The creation operators of the decuplet and the \nantidecuplet  of the orthogonal group $SO(5,1)$ in Grassmann space are presented. \n Applying on the vacuum state $|\\phi_{0}> = |1>$ the creation operators form eigenstates of\nthe Cartan subalgebra, Eq.~(\\ref{choicecartan}), (${\\cal {\\bf S}}^{0 3}, {\\cal {\\bf S}}^{1 2}$, \n${\\cal {\\bf S}}^{5 6}$). The states within each decuplet are reachable from any \nmember by ${\\cal {\\bf S}}^{ab}$. The product of the discrete operators\n${\\bf \\mathbb{C}}_{NG}$ ($=\\prod_{\\Re \\gamma^s} \\, \\gamma^s_{G}\\, I_{x^6 x^8...x^d}$, \ndenoted as ${\\bf \\mathbb{C}}$ in the last column) \n${\\cal P}^{(d-1)}_{NG}$  ($ = \\gamma^0_{G} \\, \\prod_{s=5}^{d}\\, \\gamma^s_{ G}\n I_{\\vec{x}_{3}}$)\ntransforms, for example, $\\psi^{I}_{1}$ into  $\\psi^{I}_{6}$,  $\\psi^{I}_{2}$  into  $\\psi^{I}_{5}$ \nand $\\psi^{I}_{3}$ into  $\\psi^{I}_{4}$. Solutions of the Weyl equation, Eq.~(\\ref{Weylgrass}), \nwith the negative energies belong to the \"Grassmann sea\", with the positive energy to the particles \nand antiparticles.\nAlso the application of the discrete operators ${\\cal C}_{GN}$, Eq.~(\\ref{calCPTNG}) and \n${\\cal C}_{NG}$ ${\\cal P}^{(d-1)}_{NG}$, Eq.~(\\ref{calCPTNG}) is demonstrated.\n}\n \\end{table}\n\n\n\\subsubsection{Properties of  $SO(6)$ in Grassmann and in Clifford space when $SO(6)$ is embedded\ninto $SO(13,1)$}\n\\label{so6}\n{\\bf a.}\nLet us first repeat properties of the $SO(6)$ part of the $SO(13,1)$ representation of $64$ \n\"family members\" in Clifford space, presented in Table~\\ref{Table so13+1.}. As seen in\nTable~\\ref{Table so13+1.} there are one quadruplet ($2^{\\frac{d}{2}-1}=4$) --- \n($\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]} $,  \n$\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]}$,\n$\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)} $,\n$\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)}$), representing\nquarks and leptons --- and one antiquadruplet --- \n($\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{(+)} $,  \n$\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{(+)}$,\n$\\stackrel{9 \\;10}{(+)}\\;\\;\\stackrel{11\\;12}{(+)}\\;\\;\\stackrel{13\\;14}{[-]} $,\n$\\stackrel{9 \\;10}{[-]}\\;\\;\\stackrel{11\\;12}{[-]}\\;\\;\\stackrel{13\\;14}{[-]}$), representing\nantiquarks and antileptons, which both belong to the $64^{th}$-plet, if $SO(6)$ is embedded into \n$SO(13,1)$. The creation operators (and correspondingly their annihilation operators) have \nfor $32$ members (representing quarks and leptons) the $SO(6)$ part of an odd Clifford \ncharacter (and can be correspondingly second quantized (by itselves~\\cite{nh2018} or) together\nwith the rest of space, manifesting $SO(7,1)$ (since it has an even Clifford character).  The rest \nof $32$ creation operators \n(representing antiquarks and antileptons) has in the $SO(6)$ part an even Clifford character and \ncorrespondingly in the rest of the Clifford space in $SO(7,1)$ an odd Clifford character.\n\n\nLet us discuss the case with the quadruplet of $SO(6)$ with an odd Clifford character. From the \npoint of view \nof the subgroups $SU(3)$ (the colour subgroup) and $U(1)$ (the $U(1)$ subgroup carrying the\n \"fermion\"  quantum number),  the quadruplet consists of one $SU(3)$ singlet with the \n\"fermion\" quantum number $-\\frac{1}{2}$ and one triplet  with the \"fermion\" quantum number \n$\\frac{1}{6}$. The Clifford even $SO(7,1)$ part of $SO(13,1)$ define together with the Clifford \nodd  $SO(6)$ part the quantum numbers of the right handed \nquarks and  leptons and of the left handed quarks and leptons of the {\\it standard model}, the\nleft handed weak charged and the right handed weak chargeless. \n\n In the same representation of $SO(13,1)$ there is also one antiquadruplet, which has the even \nClifford character of $SO(6)$ part and the odd Clifford character in the $SO(7,1)$ part of the \n$SO(13,1)$. The antiquadruplet of the $SO(6)$ part consists of one \n$SU(3)$ antisinglet with the \"fermion\" quantum number $\\frac{1}{2}$ and one antitriplet \n with the \"fermion\" quantum number $-\\frac{1}{6}$. The $SO(7,1) \\times SO(6)$ antiquadruplet\n of $SO(13,1)$ carries quantum numbers of left handed weak chargeless antiquarks and \nantileptons and of the right handed weak charged antiquarks and antileptons of the \n{\\it standard model}. \n\nBoth, quarks and leptons and antiquarks and antileptons, belong to the same representation of \n$SO(13,1)$,\nexplaining the miraculous cancellation of the triangle anomalies in the {\\it standard model}\nwithout connecting by hand the handedness and the charges of quarks and leptons~\\cite{nh2017},\nas it must be done in the $SO(10)$ models. \n\n{\\bf b.}\nIn Grassmann space there are one ($\\frac{1}{2}\\,\\frac{d!}{\\frac{d}{2}! \\frac{d}{2}!}= 10$)\ndecuplet representation of $SO(6)$ and one antidecuplet, both presented in \nTable~\\ref{Table grassdecuplet.}. To be able to second quantize the theory,\nthe whole representation must be Grassmann odd. Both decuplets in \nTable~\\ref{Table grassdecuplet.} have an odd Grassmann character, what means that products\nof eigenstates of the Cartan subalgebra in the rest of Grassmann space must be of an \nGrassmann even character to be second quantizable. Both decuplets would, however, appear in \nthe same representation of $SO(13,1)$, and one can expect also decuplets of an even\nGrassmann character, if $SO(6)$ is embedded into $SO(13,1)$%\n~\\footnote{This can easily be understood, if we look at the subgroups of the group $SO(6)$. \n{\\bf i.} $\\,$ Let us look at the subgroup $SO(2)$. There are two creation operators of an odd \nGrassmann character, in this case $(\\theta^{9} - i \\theta^{10})$ and $(\\theta^{9} +\n i \\theta^{10})$. Both appear in either decuplet or in antidecuplet --- together with \n$\\theta^{9} \\theta^{10}$ with an even Grassmann character ---\nmultiplied by the part appearing from the rest of space $d=(11,12,13,14)$. But if $SO(2)$ is \nnot embedded in $SO(6)$, then the two states, corresponding to the creation operators,  \n$(\\theta^{9} \\mp i \\theta^{10})$, belong to different representations, and so is  \n$\\theta^{9} \\theta^{10}$.\n{\\bf ii.} $\\,$ Similarly we see, if we consider the subgroup $SO(4)$ of the group \n$SO(6)$. All six states, $(\\theta^{9} + i \\theta^{10})\n\\cdot (\\theta^{11} + i \\theta^{12})$, $(\\theta^{9} - i \\theta^{10})\n\\cdot (\\theta^{11} - i \\theta^{12})$, $(\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})$,\n $(\\theta^{9} + i \\theta^{10})\\cdot (\\theta^{11} - i \\theta^{12})$, \n$(\\theta^{9} - i \\theta^{10})\\cdot (\\theta^{11} + i \\theta^{12})$, $(\\theta^{9} \\theta^{10} - \n\\theta^{11} \\theta^{12})$, appear in the \ndecuplet and in the antidecuplet, multiplied with the part appearing from the rest of space, in this \ncase in $d=(13,14)$, if $SO(4)$ is embedded in $SO(6)$. But, in $d=4$ \nspace there are two decoupled groups of three states~\\cite{norma93}: \n[$(\\theta^{9} + i  \\theta^{10})\\cdot (\\theta^{11} + i  \\theta^{12})$, $(\\theta^{9} \\theta^{10}\n + \\theta^{11} \\theta^{12})$, $(\\theta^{9} - i  \\theta^{10}) \\cdot\n (\\theta^{11} - i \\theta^{12})$] and [$(\\theta^{9} - i  \\theta^{10}) \\cdot (\\theta^{11} + \ni  \\theta^{12})$, $(\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})$, $(\\theta^{9} + i  \n\\theta^{10}) \\cdot (\\theta^{11} - i \\theta^{12})$]. Neither of these six members could be \nsecond quantized in $d=4$ alone.}.\n\nWith respect to $SU(3)\\times U(1)$ subgroups of the group $SO(6)$  the decuplet manifests as one \nsinglet, one triplet and one sextet, while the antidecuplet manifests as one antisinglet, one \nantitriplet and one antisextet. All the corresponding quantum numbers of either the Cartan \nsubalgebra operators or of the corresponding diagonal operators  of the $SU(3)$ or $U(1)$ subgroups\nare presented in Table~\\ref{Table grassdecuplet.}. \n \\begin{table}\n \\begin{center}\n\\begin{tiny}\n \\begin{tabular}{|c|r|r|r|r|r|r|r|r|}\n \\hline\n$I$& &$\\rm{decuplet}$&${\\cal {\\bf S}}^{9\\,10}$&${\\cal {\\bf S}}^{11\\,12}$&\n${\\cal {\\bf S}}^{13\\,14}$&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n \\hline \n& $1$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$1$&$1$&$1$&$-1$&$0$&$0$\\\\\n\\hline\n&$2$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14})$ &$1$&$0$&$0$&$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\n\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$3$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$1$&$-1$&$-1$&$ +\\frac{1}{3}$&$+ 1$&$+\n\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $4$  & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$0$&$0$&$1$&$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$-1$&$-1$&$-1$&$ +\\frac{1}{3}$&$0$&$-\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$6$  & $ (\\theta^{11} + i \\theta^{12}) (\\theta^{9}\\theta^{10}\n+ \\theta^{13} \\theta^{14})$ &$0$&$1$&$0$&$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$7$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$-1$&$1$&$-1$&$ +\\frac{1}{3}$&$- 1$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$8$  & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$0$&$0$&$-1$&$+\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $9$  & $ (\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} - i \\theta^{12})$ &$0$&$-1$&$0$&$+\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$10$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14})$ &$-1$&$0$&$0$&$+\\frac{1}{3}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline \\hline \n $II$& &$\\rm{decuplet}$&${\\cal {\\bf S}}^{9\\,10}$&${\\cal {\\bf S}}^{11\\,12}$&\n${\\cal {\\bf S}}^{13\\,14}$&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n\\hline\n& $1$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$-1$&$-1$&$-1$&$+1$&$0$&$0$\\\\\n\\hline\n&$2$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14})$ &$-1$&$0$&$0$&$+\\frac{1}{3}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$3$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$-1$&$1$&$1$&$ -\\frac{1}{3}$&$- 1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $4$  & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$0$&$0$&$-1$&$+\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$1$&$1$&$-1$&$ -\\frac{1}{3}$&$0$&$+\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$6$  & $ (\\theta^{11} - i \\theta^{12}) (\\theta^{9}\\theta^{10} +\n \\theta^{13} \\theta^{14})$ &$0$&$-1$&$0$&$+\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$7$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$1$&$-1$&$1$&$ -\\frac{1}{3}$&$+ 1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$8$  & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$0$&$0$&$1$&$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n& $9$  & $ (\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} + i \\theta^{12})$ &$0$&$1$&$0$&$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$10$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14})$ &$1$&$0$&$0$&$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n \\hline\n \\end{tabular}\n\\end{tiny}\n \\end{center}\n \\caption{\\label{Table grassdecuplet.}  The creation operators of the decuplet and the \nantidecuplet  of the orthogonal group $SO(6)$ in Grassmann space are presented.  Applying on the vacuum state $|\\phi_{0}> = |1>$ the creation operators form eigenstates of the Cartan subalgebra, \nEq.~(\\ref{choicecartan}), (${\\cal {\\bf S}}^{9\\,10}, {\\cal {\\bf S}}^{11\\,12}$, \n${\\cal {\\bf S}}^{13\\,14}$). The states within each decouplet are reachable from any \nmember by ${\\cal {\\bf S}}^{ab}$. The quantum numbers (${\\bf \\tau^{33}}, {\\bf \\tau^{38}}$) and \n${\\bf \\tau^{4}}$ of \nthe subgroups $SU(3)$ and  $U(1)$ of the group $SO(6)$ are also presented, Eq.~(\\ref{so64}).\n}\n \\end{table}\n\nWhile in Clifford case the representations of $SO(6)$, if  the group $SO(6)$ is embedded into  \n$SO(13,1)$,  are defining a Clifford odd quadruplet and a Clifford even antiquadruplet, the \nrepresentations in Grassmann case \ndefine one decuplet and one antidecuplet, both of the same Grassmann character, the odd one in \nour case. The two quadruplets in Clifford case manifest \nwith respect to the subgroups $SU(3)$ and $U(1)$ as a triplet and a singlet, and as an antitriplet\nand an antisinglet, respectively. In Grassmann case the two decuplets manifest with \nrespect to the subgroups $SU(3)$ and $U(1)$ as a (triplet, singlet, sextet) and  as an (antitriplet,\nantisinglet, antisextet), respectively. The corresponding multiplets are presented in \nTable~\\ref{Table grasssextet.}. \n \\begin{table}\n \\begin{center}\n\\begin{tiny}\n \\begin{tabular}{|c|r|r|r|r|r|}\n \\hline \\hline\n$I$& &&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n \\hline \n$\\rm{singlet}$&$ $  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$-1$&$0$&$0$\\\\\n\\hline\\hline\n$\\rm{triplet}$&$1$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14}) $ &$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$2$  & $ (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$3$  & $ (\\theta^{11} + i \\theta^{12}) (\\theta^{9}\\theta^{10}\n+ \\theta^{13} \\theta^{14}) $ &$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline \\hline\n$\\rm{sextet}$&$1$  & $(\\theta^{9}+ i \\theta^{10}) (\\theta^{11} - i \\theta^{12})  \n(\\theta^{13} - i \\theta^{14})\n$ &$\\frac{1}{3}$&$+1$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$2$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$\\frac{1}{3}$&$0$&$-\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$3$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14}) $ &$\\frac{1}{3}$&$-1$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$4$  & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14}) $ &$\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$  & $(\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} - i \\theta^{12}) $ &$\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$6$  & $(\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14}) $ &$\\frac{1}{3}$&$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\\hline\n$II$&&&${\\bf \\tau^{4}}$&${\\bf \\tau^{33}}$& ${\\bf \\tau^{38}}$\\\\\n \\hline \n$\\rm{antisinglet}$&$ $  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$+1$&$0$&$0$\\\\\n\\hline\n\\hline\n$\\rm{antitriplet}$&$1$  & $ (\\theta^{9} - i \\theta^{10}) (\\theta^{11}\\theta^{12} +\n \\theta^{13} \\theta^{14})$ &$+\\frac{1}{3} $ &$-\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$2$  & $  (\\theta^{9} \\theta^{10} + \\theta^{11} \\theta^{12})\n (\\theta^{13} - i \\theta^{14})$ &$+\\frac{1}{3}$&$0$&$+\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$3$  & $(\\theta^{11} - i \\theta^{12}) (\\theta^{9}\\theta^{10} +\n \\theta^{13} \\theta^{14}) $ &$+\\frac{1}{3}$&$+\\frac{1}{2}$&$-\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline \\hline\n$\\rm{antisextet}$&$1$  & $(\\theta^{9} - i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} + i \\theta^{14})$ &$ -\\frac{1}{3} $ &$-1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$2$  & $(\\theta^{9} + i \\theta^{10}) (\\theta^{11} + i \\theta^{12})\n (\\theta^{13} - i \\theta^{14}) $ &$-\\frac{1}{3}$&$0$&$+\\frac{2}{\\sqrt{3}}$\\\\\n\\hline\n&$3$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11} - i \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$-\\frac{1}{3}$&$+1$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$4$  & $ (\\theta^{9} \\theta^{10} - \\theta^{11} \\theta^{12})\n (\\theta^{13} + i \\theta^{14}) $ &$-\\frac{1}{3}$&$0$&$-\\frac{1}{\\sqrt{3}}$\\\\\n\\hline\n&$5$  & $ (\\theta^{9} \\theta^{10} - \\theta^{13} \\theta^{14})\n (\\theta^{11} + i \\theta^{12}) $ &$-\\frac{1}{3}$&$-\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n&$6$  & $ (\\theta^{9} + i \\theta^{10}) (\\theta^{11}\\theta^{12} -\n \\theta^{13} \\theta^{14}) $ &$-\\frac{1}{3}$&$+\\frac{1}{2}$&$+\\frac{1}{2\\sqrt{3}}$\\\\\n\\hline\n \\end{tabular}\n\\end{tiny}\n \\end{center}\n \\caption{\\label{Table grasssextet.} The creation operators in Grassmann space of the decuplet \nof Table~\\ref{Table grassdecuplet.} are arranged with respect to the $SU(3)$ and $U(1)$ \nsubgroups of the group $SO(6)$ into a singlet, a triplet and a sextet. \nThe corresponding  antidecuplet  manifests as an antisinglet, an antitriplet and an antisextet.  \n${\\bf \\tau^{33}}= \\frac{1}{2} ({\\cal {\\bf S}}^{9\\,10} - {\\cal {\\bf S}}^{11\\,12})$, \n${\\bf \\tau^{38}}=$ $\\frac{1}{2 \\sqrt{3}} ({\\cal {\\bf S}}^{9\\,10} + {\\cal {\\bf S}}^{11\\,12} - \n2 {\\cal {\\bf S}}^{13\\,14})$,  ${\\bf \\tau^{4}}=$ - \n$\\frac{1}{3} ({\\cal {\\bf S}}^{9\\,10} + {\\cal {\\bf S}}^{11\\,12} + {\\cal {\\bf S}}^{13\\,14})$;\n${\\cal {\\bf S}}^{a b}$ $= i (\\theta^a \\frac{\\partial}{\\partial \\theta_b} - \\theta^b \n\\frac{\\partial}{\\partial \\theta_a})$. \n}\n \\end{table}\nThe \"fermion\" quantum number ${\\bf \\tau^{4}}$ has for either singlets or triplets in Grassmann \nspace, Table~\\ref{Table grasssextet.}, twice the value of the corresponding singlets and triplets\n in Clifford space, Table~\\ref{Table so13+1.}: $(-1, +1)$ in \nGrassmann case to be compared with $(- \\frac{1}{2}, +\\frac{1}{2})$ in Clifford case and \n$(+ \\frac{1}{3}, -\\frac{1}{3})$ in  Grassmann \ncase to be compared with $(+ \\frac{1}{6}, -\\frac{1}{6})$ in Clifford case. \n\n\\begin{figure}\n  \\centering\n   \\includegraphics[width=0.45\\textwidth]{sl1a-sing-8May.pdf} \n   \\hfill\n   \\includegraphics[width=0.45\\textwidth]{sl1sing-8May.pdf}\n  \\caption{\\label{FigSO6} Representations of the subgroups $SU(3)$ and $U(1)$ of the group \n$SO(6)$ in Grassmann space for two Grassmann odd representations of \nTable~\\ref{Table grasssextet.} are presented. \nOn the abscissa axis and on the ordinate axis the values of the two diagonal operators, \n${\\bf \\tau^{33}}$ and ${\\bf \\tau^{38}}$ of the coulour ($SU(3)$)\nsubgroup are presented, respectively, with full circles. On the third axis the values of the \nsubgroup of the \"fermion number\" $U(1)$ is presented with the open circles, the same for all \nthe representations of each  multiplet. There are one singlet, one triplet and one sextet on the \nleft hand side and one antisinglet, one antitriplet and one antisextet on the right hand side.} \n\\end{figure}\n\nWhen  $SO(6)$ is embedded into $SO(13,1)$, the $SO(6)$ representations of either even or odd \nGrassmann character contribute to both  of the decoupled, \n$1716$ states of $SO(13,1)$ representations\ncontribute, provided that the $SO(8)$ content has the opposite Grassmann character than the $SO(6)$\ncontent. The product of both representations must be Grassmann odd in order that the corresponding \ncreation and annihilation operators fulfill the required anticommutation relations for fermions, \nEq.~(\\ref{ijthetaprod}).\n\n\\subsubsection{Properties of the subgroups $SO(3,1)$ and $SO(4)$ of the group $SO(8)$ in \nGrassmann and in Clifford space, when  $SO(8)$ is embedded into  $SO(13,1)$}\n\\label{so8}\n\n{\\bf a.}\nLet us again repeat first properties of the $SO(3,1)$ and $SO(4)$ parts of the $SO(13,1)$\nrepresentation of  $64$ \"family members\" in Clifford space, presented in Table~\\ref{Table so13+1.}. \nAs seen in Table~\\ref{Table so13+1.} there are four octets and four antioctets of $SO(8)$. \nAll four octets, having an even Clifford character and forming $32$ states when embedded into \n$SO(13,1)$, are the same for either\nquarks or for leptons, they distinguish only in the $SO(6)$ part (of a Clifford odd character) of the \n$SO(13,1)$ group, that is in the colour ($SU(3)$) part and  the \"fermion quantum number\" \n($U(1)$) part. \nAlso the four antioctets, having an odd Clifford character, are all the same for the $32$ \nfamily members of antiquarks and antileptons, they again distinguish only in the Clifford \neven $SO(6)$ part of $SO(13,1)$, that is in the anticolour ($SU(3)$) part and the \"fermion \nquantum number\" ($U(1)$) part.\n\nThe $64^{th}$-plet of creation operators has an odd Clifford character either for quarks and \nleptons  or for antiquarks and antileptons --- correspondingly have an odd \nClifford character also their annihilation operators --- and can be  second quantized~\\cite{nh2018}.\n\n\nLet us analyze first the octet ($2^{\\frac{8}{2}-1}=8$), which  is the same for all $32$ members of\nquarks and leptons. The octet has an even Clifford character. All the right handed $u_{R}$-quarks \nand  $\\nu_{R}$-leptons have the $SO(4)$ part of $SO(8)$ equal to\n$\\stackrel{56}{[+]}\\,\\stackrel{78}{(+)}$, while their left handed partners have  the $SO(4)$ part of \n$SO(8)$ equal to $\\stackrel{56}{[+]}\\,\\stackrel{78}{[-]}$.\nAll the right handed $d_{R}$-quarks  and  $e_{R}$-leptons have the $SO(4)$ part of $SO(8)$ \nequal to $\\stackrel{56}{(-)}\\,\\stackrel{78}{[-]}$, while their left handed partners have the $SO(4)$ \npart of $SO(8)$ equal to $\\stackrel{56}{(-)}\\,\\stackrel{78}{(+)]}$. The left handed quarks and \nleptons are doublets with respect to $\\vec{\\tau}^{1}$ and singlets with $\\vec{\\tau}^{2}$, while \nthe right handed quarks and leptons are  singlets with respect to $\\vec{\\tau}^{1}$ and doublets\n with $\\vec{\\tau}^{2}$. \nThe left and right handed quarks and lepton belong with respect to the $SO(3,1)$ group to either\nleft handed or the right handed spinor representations, respectively.\n\n{\\bf b.}\nIn Grassmann space the $SO(8)$ group of an odd Grassmann character has $\\frac{1}{2}$ \n$\\frac{8!}{4! 4!} = 35$ creation operators in each of the two groups and the same number of \nannihilation operators, obtained from the creation operators by Hermitian conjugation, \nEq.~(\\ref{grassher}). The corresponding states, created by the creation operators on the vacuum \nstate $|\\phi_{o}>$, can be therefore second quantized. But if embedded the group $SO(8)$ into\nthe group $SO(13,1)$ the subgroup $SO(6)$ must have an even Grassmann character in oder \nthat the states in $SO(13,1)$ can be second quantized according to Eq.~(\\ref{ijthetaprod}). \n\nAccording to what we learned in the case of the group $SO(6)$, each of the two independent \nrepresentations of the group $SO(13,1)$ of an odd Grassmann character must   \ninclude either the even $SO(7,1)$ part and the odd $SO(6)$ part or  the odd \n$SO(7,1)$ part and the even $SO(6)$ part. To the even $SO(7,1)$ representation \neither the odd $SO(3,1)$  and the odd $SO(4)$  parts  contribute or both must be of the \nGrassmann even character. In the case that the $SO(7,1)$ part has an odd Grassmann character \n(in this case the $SO(6)$ has an even Grassmann character) then one of the\ntwo parts $SO(3,1)$  and $SO(4)$ must be odd and the other even.\n\n\n\\section{Concluding remarks}\n\\label{conclusions}\n\nWe learned in this contribution that although  either Grassmann or Clifford space offer the second\nquantizable description of the internal degrees of freedom of fermions (Eq.~(\\ref{ijthetaprod})),\nthe Clifford space offers more: It offers  not only the description of all the \"family members\", \nexplaining all the degrees of freedom of the observed quarks and leptons and antiquark and antileptons, \nbut also the explanation for the appearance of families. \n\nThe interaction of fermions with the gravity fields --- the vielbeins and the spin connections\n --- in the $2(2n+1)$-dimensional space can be achieved, as suggested by the {\\it spin-charge-family}\ntheory~(\\cite{normaJMP2015,n2014matterantimatter} and references therein), by replacing the \nmomentum $p_{a}$ in the Lagrange density function for a free particle by the covariant momentum,  \nequally appropriate for both representations. In Grassmann space we have: $p_{0a}= f^{\\alpha}{}_a$\n $p_{0\\alpha}$, with $p_{0\\alpha} = p_{\\alpha}  - \\frac{1}{2}\\, {\\cal {\\bf S}}^{ab} \n\\Omega_{ab \\alpha}$, where $ f^{\\alpha}{}_a$ is the vielbein in $d=2(2n+1)$-dimensional space and\n$\\Omega_{ab \\alpha}$ is the spin connection field of the Lorentz generators ${\\cal {\\bf S}}^{ab}$.\nIn Clifford space we have equivalently: $p_{0a}= f^{\\alpha}{}_a$  $p_{0\\alpha}$,  $p_{0\\alpha}= \n p_{\\alpha}  -  \\frac{1}{2}  S^{ab} \\omega_{ab \\alpha} - \\frac{1}{2}  \\tilde{S}^{ab}  \n \\tilde{\\omega}_{ab \\alpha}$. Since ${\\cal {\\bf S}}^{ab} = S^{ab}  + \\tilde{S}^{ab}$ we find that\nwhen no fermions are present either $\\Omega_{ab \\alpha}$ or $\\omega_{ab \\alpha}$ or \n$\\tilde{\\omega}_{ab \\alpha}$ are uniquely expressible by vielbeins $f^{\\alpha}{}_a$\n (\\cite{normaJMP2015,n2014matterantimatter} and references therein). It might be that\n \"our universe made a choice between the Clifford and the Grassmann algebra\" when breaking\n the starting symmetry by making condensates of fermions, since that for breaking symmetries  \nClifford space offers better opportunity\".\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{intro}\n\nThe study of topological superconducting wires, which host Majorana zero\nmodes (MZMs) at their ends, is a field of intense research in condensed\nmatter physics, not only because of the interesting basic physics involved \n\\cite{sato}, but also because of possible applications in decoherence-free\nquantum computing.\\cite{kitaev,nayak,alicea,lobos}\n\nIn 2010, Lutchyn \\textit{et al.} \\cite{lutchyn2010} and Oreg \\textit{et al.} \n\\cite{oreg2010} proposed a model for topological superconducting wires\ndescribing a system formed by a semiconducting wire with spin-orbit coupling\n(SOC) and proximity-induced s-wave superconductivity under an applied\nmagnetic field perpendicular to the direction of the SOC. This yields a\ntopological superconducting phase with MZMs localized at its ends. The\nobservation of these MZMs in these types of wires was reported in different\nexperimental studies.\\cite{wires-exp1,wires-exp2,wires-exp3,wires-exp4}\n\nThe search for different models and mechanisms leading to topological\nsuperconducting phases continues being a very active avenue of research\ntheoretically and experimentally.\n\nMore recently, there has been experimental research as well as theoretical\nstudies in similar models, including those for time-reversal invariant\ntopological superconductors,\\cite{review-tritops,volpez} of the effects of\nMZMs in Josephson junctions, in particular because the dependence on the\napplied magnetic flux introduces an additional control knob.\\cite{volpez,zazu,pientka,hell,ren,fornie,cata,tomo}\n\nIn particular, it has been recently proposed that the current-phase relation\nmeasured in Josephson junctions may be used to find the parameters that\ndefine the MZMs.\\cite{tomo} A possible difficulty in these experiments is\nthe slow thermalization to the ground state\nin the presence of a gap.\\cite{bondy}\nA way to circumvent this problem is to rotate the magnetic field\nslowly from a direction not perpendicular to the SOC in which the system is\nin a gapless superconducting phase, in which thermalization is easier.\\cite{tomo}\nTherefore, it is convenient to know the phase diagram of the system\nand the extension of this gapless phase.\n\nIn this work we calculate the phase diagram of the lattice version of the\nmodel and discuss in particular the gapless phase. The paper is organized as\nfollows. In Sec. \\ref{model} we describe the model. The topological\ninvariants use to define the phase diagram are presented in Sec. \\ref{inv}. In Sec. \\ref{res} we show the numerical results, analytical expressions for\nthe boundaries of the topological phase and discuss briefly the Majorana zero modes.\nWe summarize the results in Sec. \\ref{sum}.\n\n\\section{Model}\n\n\\label{model}\n\nThe model for topological superconducting wires studied in this work is the\nlattice version of that introduced by Lutchyn \\textit{et al.} \n\\cite{lutchyn2010} and Oreg \\textit{et al.} \\cite{oreg2010}. The Hamiltonian can\nbe written as \\cite{tomo} \n\\begin{eqnarray}\nH &=&\\sum_{\\ell }[\\mathbf{c}_{\\ell }^{\\dagger }\\left( -t\\;\\sigma _{0}-i\\vec{\n\\lambda}\\cdot \\vec{\\sigma}\\right) \\mathbf{c}_{\\ell +1}+\\Delta c_{\\ell\n\\uparrow }^{\\dagger }c_{\\ell \\downarrow }^{\\dagger }+\\text{H.c.}  \\notag \\\\\n&-&\\mathbf{c}_{\\ell }^{\\dagger }\\left( \\vec{B}\\cdot \\vec{\\sigma}+\\mu \\sigma\n_{0}\\right) \\mathbf{c}_{\\ell }],\\;\\;  \\label{ham}\n\\end{eqnarray}\nwhere $\\ell $ labels the sites of a chain, $\\mathbf{c}_{\\ell }=(c_{\\ell\n\\uparrow },c_{\\ell \\downarrow })^{T}$, $t$ is the nearest-neighbor hopping, $\n\\vec{\\lambda}$ is the SOC, $\\Delta $ represents the magnitude of the\nproximity-induced superconductivity, $\\vec{B}$ is the applied magnetic field\nand $\\mu $ is the chemical potential. As usual, the components of the vector $\n\\vec{\\sigma}=\\left( \\sigma _{x},\\sigma _{y},\\sigma _{z}\\right) $ are the\nPauli matrices and $\\sigma _{0}$ is the 2$\\times $2 unitary matrix. The\npairing amplitude $\\Delta $ can be assumed real. Otherwise, the phase can be\neliminated by a gauge transformation in the operators $c_{\\ell \\sigma\n}^{\\dagger }$ that absorbs the phase.\n\nWithout loss of generality, we choose the $z$ direction as that of the\nmagnetic field ($\\vec{B}=B\\mathbf{\\hat{z}}$) and $x$ perpendicular to the\nplane defined by $\\vec{\\lambda}$ and $\\vec{B}$ ($\\vec{\\lambda}=\\lambda _{y} \n\\mathbf{\\hat{y}}+\\lambda_{z} \\mathbf{\\hat{z}}$). After Fourier\ntransformation, the Hamiltonian takes the form $H=\\sum_{k}H_{k}$, with\n\n\\begin{eqnarray}\nH_{k} &=&-(\\mu +2t\\cos (k))(c_{k\\uparrow }^{\\dagger }c_{k\\uparrow\n}+c_{k\\downarrow }^{\\dagger }c_{k\\downarrow }) \\\\\n&& -B(c_{k\\uparrow }^{\\dagger }c_{k\\uparrow }-c_{k\\downarrow }^{\\dagger\n}c_{k\\downarrow }) -2\\sin (k)\\left[ i\\lambda _{y}(c_{k\\uparrow }^{\\dagger\n}c_{k\\downarrow }-c_{k\\downarrow }^{\\dagger }c_{k\\uparrow }) \\right.  \\notag\n\\\\\n&& \\left. +\\lambda _{z}(c_{k\\uparrow }^{\\dagger }c_{k\\uparrow\n}-c_{k\\downarrow }^{\\dagger }c_{k\\downarrow })\\right] +\\Delta (c_{k\\uparrow\n}^{\\dagger }c_{-k\\downarrow }^{\\dagger }+c_{-k\\downarrow }c_{k\\uparrow }). \n\\notag  \\label{hk}\n\\end{eqnarray}\nUsing the four-component spinor $(c_{k\\uparrow }^{\\dagger },c_{k\\downarrow\n}^{\\dagger },c_{-k\\uparrow },c_{-k\\downarrow })$,\\cite{tewari} the\ncontribution to the Hamiltonian for wave vector $k$ can be written in the\nform\n\n\\begin{eqnarray}\nH_{k} &=&-(\\mu +2t\\cos (k))\\tau _{z}\\otimes \\sigma _{0}-B\\tau _{z}\\otimes\n\\sigma _{z}-\\Delta \\tau _{y}\\otimes \\sigma _{y}  \\notag \\\\\n&&+2\\lambda _{y}\\sin (k)\\tau _{z}\\otimes \\sigma _{y}-2\\lambda _{z}\\sin\n(k)\\tau _{0}\\otimes \\sigma _{z},  \\label{hk2}\n\\end{eqnarray\nwhere the Pauli matrices $\\sigma _{\\alpha }$ act on the spin space, while\nthe $\\tau _{\\alpha }$ act on the particle-hole space. Writing the matrix\nexplicitly, $H_{k}$ takes the form\n\n\\begin{equation}\nH_{k}\n\\begin{pmatrix}\n-a-B-z & -iy & 0 & \\Delta \\\\ \niy & -a+B+z & -\\Delta & 0 \\\\ \n0 & -\\Delta & a+B-z & iy \\\\ \n\\Delta & 0 & -iy & a-B+z\n\\end{pmatrix}\n\\label{hk3}\n\\end{equation\nwhere $a=\\mu +2t\\cos (k)$, $B=|B|=B_{z}$, $y=2\\lambda _{y}\\sin (k)$ and $\nz=2\\lambda _{z}\\sin (k)$.\n\n\\section{Topological invariants}\n\n\\label{inv} In this section we define the topological invariants we use to\ncharacterize the topological phases. In general, the Hamiltonian belongs to\ntopological class D with a \n$\\mathbb{Z}_{2}$ topological invariant.\\cite{Schn,ryu} However, for perpendicular $\\vec{\\lambda}$ and $\\vec{B}$ ($z=0$),\nthe system has a chiral symmetry and belongs to the topological class BDI\nwith a $\\mathbb{Z}$ (integer) topological invariant corresponding to a\nwinding number.\\cite{tewari}  In this case, the calculation of the\ntopological invariant is simpler, as shown by Tewari and Sau.\\cite{tewari}  \n\nFollowing this work, we perform a rotation in $\\pi \/2$ around the $\\hat{y}$\naxis in particle-hole space, which transforms $\\tau _{z}$ to $\\tau _{x}$: $\nH_{k}^{\\prime }=UH_{k}U^{\\dagger }$ with $U=$exp$(-i\\pi \/4)\\tau _{y}$. With\nthis transformation $H_{k}^{\\prime }$ becomes\n\n\\begin{equation}\nH_{k}^{\\prime }=\n\\begin{pmatrix}\n-z & 0 & -a-B & \\Delta -iy \\\\ \n0 & z & -\\Delta +iy & -a+B \\\\ \n-a-B & -\\Delta -iy & -z & 0 \\\\ \n\\Delta +iy & -a+B & 0 & z\n\\end{pmatrix}\n\\label{hkp}\n\\end{equation}\nTaking  $z=0$, this rotation yields an off-diagonal (chiral symmetric)\nHamiltonian. This allows us to define a winding number $W$ (a topological $\n\\mathbb{Z}$ invariant) from the phase of the determinant of the $2\\times 2$\nmatrix $A(k)$, which is the upper right corner of Eq. (\\ref{hkp}).\\cite{tewari}\nSpecifically $\\mathrm{Det}(A(k))$$=|\\mathrm{Det}(A(k))|$$e^{i\\theta (k)}=$$a^{2}-B^{2}-(\\Delta -iy)^{2}$, and\n\n\\begin{equation}\nW=\\frac{-i}{\\pi }\\int\\limits_{0}^{\\pi }\\frac{d(e^{i\\theta (k)})}{e^{i\\theta\n(k)}}.  \\label{win}\n\\end{equation}\nIn addition, a $\\mathbb{Z}_{2}$ invariant $I$ can be defined from the\nrelative sign of $\\mathrm{Det}(A)$ (which is real for $k=0$ and $k=\\pi$)\nbetween the points $k=0$ and $k=\\pi$:\n\n\\begin{equation}\nI=(-1)^{W}=\\text{sign}\\frac{\\mathrm{Det}(A(\\pi ))}{\\mathrm{Det}(A(0))}\n\\label{z2}\n\\end{equation}\nLooking for the condition that $I\\equiv -1$ (mod 2), we obtain that the\nconditions for the system to be in the topological phase are that $\\lambda\n_{y}=|\\vec{\\lambda}|\\neq 0\\neq \\Delta $ and the remaining parameters should\nsatisfy \n\\begin{equation}\n|2|t|-r|<|\\mu |<|2|t|+r|,\\text{ \\ \\ \\ with }r=\\sqrt{B^{2}-\\Delta ^{2}}>0.\n\\label{bound}\n\\end{equation}\nWe note that changing the sign of any of the parameters does not change the\nboundary of the topological phase. This is due to the symmetry properties of\nthe Hamiltonian.\\cite{tomo}\n\nIn the more general case, when $\\vec{\\lambda}$ and $\\vec{B}$ are not\nperpendicular, it is not possible to follow the approach outlined above. In\nthis case, we use the Zak Berry phase to construct the topological\ninvariant. \\cite{zak,king,resta,ortiz,bf1,bf2,hatsu,bf3,ryu,deng,budich}\nSpecifically, the Hamiltonian $H_{k}$ has four different eigenvectors and\nfor each of them, following Zak,\\cite{zak} one can calculate a Berry phase\nfrom the Bloch functions as the wave vector $k$ varies in the loop $0\\leq\nk\\leq 2\\pi $ (with $k=2\\pi $ equivalent to $k=0$). For each eigenstate $\n|u(k)\\rangle $ of $H_{k}$, the Berry phase is \n\n\\begin{equation}\n\\gamma =-\\text{Im }\\int\\limits_{0}^{2\\pi }dk\\langle u(k)|\\frac{\\partial }{\n\\partial k}|u(k)\\rangle .  \\label{gam}\n\\end{equation}\n\n\\ In addition (as noted before \\cite{tomo}) choosing a suitable coordinate\nframe ( $\\vec{\\lambda}\\cdot \\mathbf{\\hat{y}}=\\vec{B}\\cdot \\mathbf{\\hat{y}}=0$\n), the Hamiltonian Eq. (\\ref{ham}) is invariant under an antiunitary\noperator defined as the product of inversion (defined by the transformation $\n\\ell \\leftrightarrow N+1-\\ell $, for a chain with $N$ sites) and complex\nconjugation, implying that the Berry phase $\\gamma $ is quantized with only\ntwo possible values $0$ and $\\pi $ (mod $2\\pi $).\\cite{hatsu} Naturally the\nvalue of the Berry phase does not depend on the choice of the reference\nframe. Therefore, as for an insulator, if the system has a gap, the sum of\nthe Berry phases of all one-particle states of energies below the gap mod $2\\pi $, defines a \n$\\mathbb{Z}_{2}$ topological number, indicating that the system is trivial\n(topological) if this sum is equivalent to 0 ($\\pi $) mod $2\\pi .\n$\\cite{ryu,budich} Moreover, from Eq. (\\ref{hk}) it is easy to realize that the\ncharge conjugation $c_{\\ell \\sigma }^{\\dagger }\\leftrightarrow c_{\\ell\n\\sigma }$, which in Fourier space means $c_{k\\sigma }^{\\dagger }=(1\/\\sqrt{N\n)\\sum_{l}e^{-ik\\ell }c_{\\ell \\sigma }^{\\dagger }\\leftrightarrow c_{-k\\sigma }\n$, transforms $H_{k}\\leftrightarrow -H_{-k}$. Therefore the sum of the Berry\nphases of all positive eigenvalues gives the same topological number as the\nsum of all negative eigenvalues.\n\nIn our model, $H_{k}$ has four eigenvalues $E(k)$. The lowest one $E_{1}(k)$\nis always negative and the corresponding eigenvector has always a Berry\nphase 0. From the above mentioned charge-transfer symmetry, the fourth\neigenvalue (the highest one) has energy $E_{4}(k)=-E_{1}(-k)>0$. Therefore,\nthe Berry phase of the second eigenvalue (which is equal to that of the\nthird one) determines the $\\mathbb{Z}_{2}$ invariant. We have calculated the\nBerry phase $\\gamma $ of each of the four bands (and particularly the second\none) from the normalized eigenvectors $|u_{j}\\rangle =|u(k_{j})\\rangle $ of\nthe $4\\times 4$ matrix obtained numerically at $M$ wave vectors $k_{j}=2\\pi\n(j-1)\/M$, using a numerically invariant expression \\cite{ortiz,deng}. This\nexpression is derived in the following way. Discretizing Eq. (\\ref{gam}) and\napproximating $\\partial |u(k)\/\\partial k=(M\/2\\pi )(|u(k_{j+1})\\rangle\n-|u(k_{j})\\rangle )$, one obtains\n\n\\begin{equation}\n\\gamma =-\\text{Im}\\sum_{j=1}^{M}\\left[ \\langle u_{j}|\\left( |u_{j+1}\\rangle\n-|u_{j}\\rangle \\right) \\right] .  \\label{gam2}\n\\end{equation}\nIf $M$ is large enough so that $k_{j}$ and $k_{j+1}$ are very close, then \n$x=\\langle u_{j}|u_{j+1}\\rangle -1$ is very small and one can retain only the\nfirst term in the Taylor series expansion ln$(1+x)=x-x^{2}\/2+...$ Replacing in Eq. (\n\\ref{gam2}) one obtains\n\n\\begin{eqnarray}\n\\gamma  &=&-\\mathrm{{Im}\\left[ \\ln (P)\\right] }\\text{, where }  \\notag \\\\\nP &=&\\langle u_{1}|u_{2}\\rangle \\langle u_{2}|u_{3}\\rangle ...\\langle\nu_{M-1}|u_{1}\\rangle   \\label{gam4}\n\\end{eqnarray}\nIt is easy to see that Eq. (\\ref{gam}) is gauge invariant. This means that\nthe result does not change if $|u(k)\\rangle $ is replaced by $e^{i\\varphi\n(k)}|u(k)\\rangle $, where $\\varphi (k)$ is a smooth function with $\\varphi\n(2\\pi )=\\varphi (0).$ Similarly, the product $P$ is independent of the base\nchosen by the numerical algorithm to find the eigenstates $|u_{j}\\rangle .$\nTherefore Eq. (\\ref{gam4}) is numerically gauge invariant. Analyzing the\nchange in the results with increasing $M$, we find that  $M\\sim 250$ is\nenough to obtain accurately all phase boundaries shown below. A further\nincrease in  $M$ leads to changes that are not visible in the scale of the\nfigures.\n\nThis $\\mathbb{Z}_{2}$ topological invariant defined by the Berry phase \nof the second (or third) state can be trivially extended to the gapless case \nif the energies of the second and third state do not cross as a function of $k$.\nEven if the energies cross the Berry phases can be calculated switching \nthe states at the crossing. However, this case is not of interest here.\n\n\\section{Results}\n\n\\label{res}\n\n\\subsection{Phase diagram}\n\n\\label{phdi}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{4p_D-fase_varioB_TeX.eps}\n\\end{center}\n\\caption{Phase diagram in the $\\mu,\\Delta$ plane for perpendicular \n$\\vec{\\lambda}$ and $\\vec{B}$, $t=1$, \n$\\lambda=|\\vec{\\lambda}|=2$, and several values of $B$. \nGray region I denotes the topological sector and white region II the \nnon topological one.}\n\\label{variob}\n\\end{figure}\n\nWe start by discussing the simplest case of perpendicular $\\vec{\\lambda}$\nand $\\vec{B}$. In Fig. \\ref{variob} we display the resulting phase diagram\nfor some parameters, showing the possible different shapes. There are two\ngapped phases, the trivial (white region II) and the topological one (light\ngray I), separated in general by two circular arcs defined by Eqs. \n(\\ref{bound}). For simplicity we discuss the case $t, B > 0$. The topological character\nis independent of the sign of the different parameters. If $B<2t$, the\nregion of possible values of $|\\mu|$ inside the topological sector extends\nfrom $2t-B$ to $2t+B$ for $\\Delta \\rightarrow 0$ and shrinks for increasing \n\\Delta$ until it reduces to the point $|\\mu|=2t$ for $\\Delta \\rightarrow B$.\nIf $B=2t$, the semicircle touches the point $\\mu=0$. For larger $B$, the\nregion $|\\mu|< \\sqrt{B^{2}-\\Delta^{2}}-2t$ for $\\Delta^2 < B^{2}-4t^2$ is\nexcluded from the topological region.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{4p_fase_phiB0_beta_TeX.eps}\n\\end{center}\n\\caption{(Color online) Phase diagram in the $\\mu,\\Delta$ plane for $t=1$, \n$\\lambda=2$, $B=4$ and several values of the angle \n$\\beta_{\\lambda B}$ between $\\vec{\\lambda}$ and $\\vec{B}$. \nRegions I and II as in Fig. \\ref{variob}. Region III (IV)\nin dark gray (black) corresponds to the gapless phase\nwith Berry phase $\\pi$ (0).\nThe red points at the top left corresponds to numerical calculations\nwhich detected localized states at the ends.}\n\n\\label{variobeta}\n\\end{figure}\n\nWhile for perpendicular $\\vec{\\lambda}$ and $\\vec{B}$, the gap vanishes only\nat particular lines in the phase diagram (black lines in Fig. \\ref{variob})\nfor which the topological transition takes place, for general angles $\\beta\n_{\\lambda B}$ between both vectors, there is a finite region in the $\\mu\n,\\Delta $ plane for which the gap vanishes, in particular for $|\\Delta\n|<\\Delta _{c}$, where $\\Delta _{c}$ is a critical value, independent of $\\mu \n$, determined analytically below. Before presenting the analytical\ncalculation, we describe the general features of each phase in the phase\ndiagram, as shown in Fig. \\ref{variobeta}. The gapped regions in the figure\nare denoted by I and II. The remaining two regions are gapless. We separate them by the trivial (topological) character of the Berry phases of the second and third eigenstate, indicating the\ncorresponding regions with black (dark gray) color and roman number IV (III). \nIn spite of the topological Berry phases of the latter gapless phase, \nMZMs in a finite chain are not expected to be protected against small\nperturbations because of the absence of a gap. Therefore we describe this\nphase as non topological. Furthermore we do not find numerically signatures\nof localized end states in this phase.\n\nWe have also checked the boundaries of the\ntopological phase solving numerically finite chains and searching for\nlocalized states at their ends and the presence of the finite gap. The\nlocalized states are described in Sec. \\ref{majo}. The presence of the\ngap is defined by the condition that the determinant $D(k)$ of $H_{k}$ is\npositive for each $k$. As it can be seen in Fig. \\ref{variobeta} top left,\nthe results of both approaches agree.\n\n\\subsection{Analytical expressions for the boundaries of the topological\nphase}\n\n\\label{anal}\n\nFor perpendicular $\\vec{\\lambda}$ and $\\vec{B}$, the boundaries of the\ntopological phase are defined by Eqs. (\\ref{bound}) and the conditions \n$|\\vec{\\lambda}|\\neq 0$ and $\\Delta \\neq 0$. As the angle is changed \nfrom  90\\textdegree, the gap reduces and a non-zero $|\\Delta |$ is necessary to keep the gap\nopen (see Fig. \\ref{variobeta}). For convenience, we discuss first the case $\n\\lambda _{z}=0$ (perpendicular $\\vec{\\lambda}$ and $\\vec{B}$) and later\nconsider the general case $\\vec{\\lambda}=\\lambda _{y}\\mathbf{\\hat{y}}\n+\\lambda _{z}\\mathbf{\\hat{z}}$ with $\\lambda _{z}\\neq 0$. For $\\lambda _{z}=0\n$, the determinant $D_{0}(k)$ of $H_{k}$ [see Eqs. (\\ref{hk3}) or (\\ref{hkp}\n)]\n\n\\begin{eqnarray}\nD_{0}(k) &=&C^{2}+4\\Delta ^{2}y^{2},  \\notag \\\\\nC &=&a^{2}+\\Delta ^{2}-B^{2}-y^{2},  \\label{d0k}\n\\end{eqnarray\nis positive semidefinite. It can vanish only for $y=0$ implying either $k=0$\nor $k=\\pi $. For $k=0$ ($k=\\pi )$, $C=0$ implies $|\\mu +2t|=r$ ($|\\mu -2t|=r$\n). Comparing with Eqs. (\\ref{bound}), one realizes that the gap vanishes in\ngeneral only at one wave vector and only at the transition between\ntopological and non-topological gapped phases, as expected. The exception is\nthe case $|2t|=r$ and $\\mu =0$, for which the gap vanishes at both wave\nvectors.\n\nIn the general case with $z=2\\lambda _{z}\\sin (k)$ non zero, the determinant\nof $H_{k}$ is [see Eq. (\\ref{hkp})]\n\n\\begin{equation}\nD(k)=D_{0}+2z^{2}(\\Delta ^{2}+y^{2}-a^{2}-B^{2})+z^{4}  \\label{dk}\n\\end{equation}\nWe can consider $D(k)$ as a function of $x=\\cos (k)$. For large enough $\n|\\lambda _{z}|$, it turns out that, at the wave vector $k=0$, and parameters\nfor which $C=y=z=0$ [implying $D(0)=0$], $dD(x)\/dx>0$ and as a consequence\nfor small positive $k$ ($x<1$) the determinant becomes negative signaling\nthe instability of the gapped phase. For $\\lambda _{z}=0$, as in the previous\ncase the derivative is negative, but $x$ cannot be increased beyond 1, so\nthat $D(k)\\geq 0$. A similar reasoning with the corresponding changes in the\nsign can be followed for $k=\\pi $. An explicit calculation of the derivative\nusing the conditions $C=\\sin (k)=0$ gives \n\\begin{equation}\n\\frac{dD}{dx}=32[B^{2}\\lambda _{z}^{2}-\\Delta ^{2}(\\lambda _{z}^{2}+\\lambda\n_{y}^{2})]x.  \\label{dddx}\n\\end{equation}\nThis implies that to have a gap one needs that $|\\Delta |>\\Delta _{c}$ where \n\\begin{equation}\n\\Delta _{c}^{2}=B^{2}\\frac{\\lambda _{z}^{2}}{\\lambda _{z}^{2}+\\lambda\n_{y}^{2}}=B^{2}\\cos ^{2}(\\beta _{\\lambda B}).  \\label{deltac}\n\\end{equation}\nThis condition has been found before for a model similar to ours in the\ncontinuum with quadratic dispersion.\\cite{rex}\n\nAfter some algebra, the determinant in the general case can be written in\nthe form \n\\begin{equation}\nD=(C-z^{2})^{2}+16(\\lambda _{z}^{2}+\\lambda _{y}^{2})(\\Delta ^{2}-\\Delta\n_{c}^{2})(1-x^{2}),  \\label{d2}\n\\end{equation}\nwhich is again positive semidefinite for $|\\Delta |>\\Delta _{c}$ and\npositive definite for $0\\neq k\\neq \\pi $, indicating a gapped phase. Since $\nx=1$ implies $y=z=1$, the remaining boundaries of the topological phase\nremain the same as for perpendicular $\\vec{\\lambda}$ and $\\vec{B}.$ For $\n|\\Delta|=\\Delta _{c}$ (as in Fig. \\ref{eigenmu}), the values of $k$ for\nwhich the determinant vanishes are given by the solutions with $|x|\\leq 1$\nof the following quadratic equation\n\n\\begin{eqnarray}\n0 &=&4(t^{2}+\\lambda ^{2})x^{2}+4t\\mu x  \\notag \\\\\n&&+\\mu ^{2}+\\Delta _{c}^{2}-B^{2}-4\\lambda ^{2},  \\label{xc}\n\\end{eqnarray}\nwhere $\\lambda =|\\vec{\\lambda}|$.\n\n\\subsection{Transition from the topological phase to the gapless phases}\n\n\\label{topogap}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{3p_autovalores_beta_TeX_pi_sc_rojo.eps}\n\\end{center}\n\\caption{(Color online) Second (black thin lines) and third (red thick lines) eigenvalues of $H_k$ as a function of wave vector\nfor $t=1$, $\\lambda=\\Delta=2$, $B=\\mu=5$, and several values\nof the angle $\\beta_{\\lambda B}$ between \n$\\vec{\\lambda}$ and $\\vec{B}$.}\n\\label{eigenbeta}\n\\end{figure}\n\nTo gain insight into the transition from the topological phase to the\ngapless phases, we represent in Fig. \\ref{eigenbeta} the second and third\neigenvalues of $H_{k}$ [$E_{2}(k)$ and $E_{3}(k)$, respectively] for\ndifferent values $\\beta _{\\lambda B}$ of the angle between $\\vec{\\lambda}$\nand $\\vec{B}$. The parameters are such that, for $\\vec{\\lambda}\\cdot \\vec{B}=0\n$, the system is in the topological phase with a finite gap. As the angle is\nchanged (in either direction) the gap between the second and third\neigenvalue decreases until at a certain critical angle [given by the\nsolution of Eq. (\\ref{xc})] $E_{2}(k_{c})=E_{3}(-k_{c})=0$ at one particular\nwave vector $k_{c}$ ($0.3613\\pi$ in the figure), denoting the onset of the\ngapless phase. Further turning $\\vec{\\lambda}$ and $\\vec{B}$ to the parallel\n(or antiparallel) direction, both eigenvalues vanish at two different wave\nvectors.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{mus_betacritico_TeX_sc_rojo.eps}\n\\end{center}\n\\caption{(Color online) Same as Fig. \\ref{eigenbeta} \nfor $t=1$, $\\lambda=\\Delta=2$, $B=5$, \n$\\beta_{\\lambda B}=66.42$\\textdegree \\ and several values of $\\mu$.}\n\\label{eigenmu}\n\\end{figure}\n\n\nIf keeping the other parameters fixed, the chemical potential $\\mu $ is\nchanged towards one border $\\mu _{c}$ of the topological phase for $\\vec\n\\lambda}\\cdot \\vec{B}=0$ [given by Eq. (\\ref{bound})]; the critical wave\nvector $k_{c}$ is displaced either to $k_{c}=0$ or to $k_{c}=\\pi $ depending\non the border. This is illustrated in Fig. \\ref{eigenmu}. At the\ncorresponding border $\\mu =\\mu _{c}$,  one has $E_{2}(k_{c})=E_{3}(k_{c})=0$, indicating a crossing of the levels which is also accompanied by a change\nin the Berry phases of the corresponding eigenvectors. Further displacing \n\\mu $ the system enters the non topological gapped phase. Therefore, the\npoint $\\mu =\\mu _{c}$,  $\\Delta =\\Delta _{c}$ is at the border of the\ntopological phase, the nontrivial gapless phase with Berry phase $\\pi$, and the non-topological gapped phase. In fact also the trivial gapless phase reaches\nthis tetracritical point in the phase diagram (see Fig. \\ref{variobeta}).\n\n\n\\subsection{Majorana modes}\n\n\\label{majo}\n\nThe topological phase is characterized \nby the presence of Majorana modes zero modes at the ends of an infinite chain.\nFor a finite chain, the modes at both ends mix, giving rise to a \nfermion $\\Gamma$ and its Hermitian conjugate with energies $\\pm E$ which \ndecay exponentially with the length $L$ of the chain. We have obtained $\\Gamma$ numerically in chains of $L \\sim 200$ sites. The probability $p(i)$ of finding\na fermion at site $i$ (adding both spins and creation and annihilation) is shown\nin Fig. \\ref{pis}. The main feature of the top figure is  a decay of $p(i)$\nas the distance from any of the ends increases. We have chosen a case with a rather slow decay to facilitate visualization. In addition to this decay, some oscillations are visible with a short period.\n\nIn order to quantify the decay length of the localization of the end modes, we have fit the probability with an exponentially decaying function \n$p(i) \\sim A$exp$(-i\/\\xi)$ at the left end. At the bottom of \nFig. \\ref{pis} we show the dependence\nof $\\xi$ inside the topological phase \\textrm{I} as one of the parameters is varied.\nAs expected, $\\xi$ diverges at the boundary with the non topological gapped phase \\textrm{II}, which has a different $\\mathbb{Z}_{2}$ topological invariant \n(at $\\Delta_{c_2}=3.872983346$ in the figure). We also find that  \n$\\xi$ diverges at the boundary with the gapless phase \\textrm{III} \n(at $\\Delta_{c_1}=0.694592711$ in the figure), a phase with the same topological invariant but gapless. These facts allow us to obtain numerically\nthe transitions from the localization of the end states \n(see top left panel of Fig. \\ref{variobeta}). \n\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{2p_c2_xi_TeX.eps}\n\\end{center}\n\t\\caption{Top: probability of finding a fermion at each site of a chain  for the eigenstate of lowest positive energy for $t=1$, $\\lambda=2$, $B=4$, $\\beta_{\\lambda B}=80$\\textdegree, $\\mu=3$, and $\\Delta=0.75$. Bottom: inverse of the localization length as a function of $\\Delta$. The transition between phases I and III is at $\\Delta_{c_1}=0.694592711$ and the transition between phases I and II is at $\\Delta_{c_2}=3.872983346$.}\n\\label{pis}\n\\end{figure}\n\n\n\\section{Summary and discussion}\n\n\\label{sum}\n\nUsing numerical and analytical methods, \nwe calculate the phase diagram of a widely used model for topological superconducting\nwires, the essential ingredients of which are local s-wave pairing $\\Delta$, spin-orbit coupling $\\vec{\\lambda}$ and magnetic field $\\vec{B}$. \nWe determine the boundary of the gapped topological phase analytically. \nThis phase contains robust Majorana zero modes at both ends that are of great interest.\nWe expect that\nthis result will be relevant for future studies in the field.\n\nThe optimal situation for topological superconductivity is when $\\vec{B}$ is perpendicular\nto $\\vec{\\lambda}$. In this case, both the topological and non-topological phases are gapped. If instead $\\vec{B}$ has a component in the direction of $\\vec{\\lambda}$, a gapless\nsuperconducting phase appears for certain parameters. This phase can also be separated in two \nphases differing in a $\\mathbb{Z}_{2}$ topological invariant. However, due to the absence of a gap,\nwe do not find Majorana zero-modes at the ends of the phase with \nnontrivial $\\mathbb{Z}_{2}$, in contrast to those present in the gapped topological phase.\n\nTilting the magnetic field to enter the gapless phase might be used as a trick to relax\nthe system to the ground state in some measurements, like Josephson current.\nIn the gapped topological phase, in the absence of \nlow-frequency phonons or other excitations, the physics is dominated by a few \nbound states inside the gap, completely isolated from the continuum, and the current would oscillate, without reaching a steady state.\\cite{chung} One way to avoid this problem would be to use a magnetic field so that the system is in the gapless phase, with low-energy\nexcitations available for thermalization,\nand then rotate adiabatically the field to the desired value so that the system remains in the ground state.\n\n\\section*{Acknowledgments}\n\nWe thank L. Arrachea for helpful discussions. We are sponsored by PIP\n112-201501-00506 of CONICET, PICT-2017-2726, PICT-2018-04536 and\nPICT-Raices-2018.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Ongoing work - scratch}\n\n\n\\section{Introduction}\n\nAutonomous Underwater Vehicles (AUVs) have found applications in a variety of underwater exploration and monitoring tasks including high-resolution, geo-referenced optical\/acoustic ocean floor mapping and measurements of water column properties such as currents, temperature and salinity \\cite{yoerger1991autonomous}. An advantage of AUVs over other methods of ocean observation is the autonomy and decoupling from a surface vessel that a self-contained robot provides.\n\nThe ability to geo-reference, or to compute the absolute position in a global reference frame, is essential for AUVs for the purposes of path planning for mission requirements, registration with independently navigated information, or revisiting a previous mission. Geo-referenced navigation is often achieved by initializing the navigation solution to the GPS while on the surface and, once submerged, relaying on velocity measurements from a  Doppler Velocity Log (DVL).  When the water depth is less than the range of the DVL (a 300kHz DVL has a range of $\\sim$200m), the DVL has continuous bottom lock throughout the mission.  The DVL sensor provides measurements of the seafloor relative velocity of the AUV.  By combining this information with an appropriate heading reference, the observations are placed in the global reference frame and integrated to facilitate underwater dead reckoning. The result is accuracies of 22m per hour (2\\(\\sigma\\)) in position error growth attainable during diving and 8m per hour error growth (2\\(\\sigma\\)) is possible if coupled with a navigation-grade (\\(>\\)\\$100K) Inertial Measurement Unit (IMU) \\cite{napolitano2004phins}.\n\nIn cases where the seafloor depth is greater than the DVL bottom lock range, transitioning from the surface to the seafloor presents a localization problem \\cite{kinseynonlinear2014}, since both GPS and DVL are unavailable in the mid-water column. Traditional solutions include range-limited Long Base Line (LBL) acoustic networks requiring deployment, Ultra Short Base Line (USBL) navigation requiring a dedicated ship, or single range navigation from an acoustic beacon attached to a ship \\cite{webster2012advances} or an autonomous surface vehicle (ASV) \\cite{kinsey2013-auv-asv}. In addition to requiring dedicated infrastructure, acoustic positioning also suffers from multipath returns and the need to accurately measure the sound speed profile through the water column. Acoustic methods typically give \\(\\mathcal{O}({10m})\\) accuracy at 1km ranges \\cite{kinsey2006survey} \\cite{mandt2001integrating}.\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth] {pictures\/20151203-ff_ocean.jpg}\n      \\caption{The \\textit{FlatFish} AUV \\cite{albiez2015flatfish} during sea trails. Image: Jan Albiez, SENAI CIMATEC}\n      \\label{fig:flatfish}\n\\end{figure}\n\nIMUs provide a strap down navigation capability through providing body accelerations and rotation rates without external aiding such as GPS, acoustic ranging, or DVL velocities. However, IMUs quickly accumulate position errors, with an unaided tactical grade IMU (\\(>\\)\\$10K) drifting at $\\sim$100km per hour, and a navigation grade IMU drifting at $\\sim$1km per hour \\cite{titterton2004strapdown}. There also exists cases where DVL bottom-lock is not possible, when the altitude is very low, such as in inspection or docking scenarios.\n\nIn \\cite{hegrenaes2011model}, a model-aiding Inertial Navigation System (INS) is applied with water-track from the DVL. Comparatively, the novel contributions of the work presented in this paper are as follows: \n\\begin{enumerate}\n\t\\item Utilizing and validating through experiment a manifold based Unscented Kalman Filter (UKF) which can observe and utilize the Earth rotation for heading estimation,\n\t\\item Incorporating and validating a novel drag and thrust model-based aiding, which accounts for the systematic uncertainty in vehicle parameters by incorporating them as states in the UKF and\n\t\\item Incorporating and validating the use of ADCP measurements in a novel form to further aid the estimation in cases of DVL bottom-lock loss.\n\\end{enumerate}\n\nIMUs with low gyro bias uncertainty allow gyrocompassing by measuring the Earth rotation to estimate heading. The price range for navigation grade IMUs (as used in \\cite{hegrenaes2011model}) with a low bias uncertainty are typically in the \\(>\\)\\$100K USD price range. In this paper, the KVH1750 IMU, in the \\(>\\)\\$10K USD price range, is utilized. In order for this price range IMU to be utilized, the biases are estimated in a fully coupled approach in the navigation filter. Real-world experiments with the \\textit{FlatFish} AUV (Fig. \\ref{fig:flatfish}) show that less than 1$^{\\circ}$ (2$\\sigma$) heading uncertainty is possible in the filter following an initialization within 15$^{\\circ}$ of the true heading (possible from a magnetic sensor). Further experiments also show that the filter is capable of consistent positioning, and data denial validates the method for DVL dropouts due to very low or high altitude scenarios. Additionally this work was implemented using the MTK \\cite{hertzberg2013integrating} and ROCK \\cite{rock} framework in C++\\footnote{The implementation is under open source license available on \\url{https:\/\/github.com\/rock-slam\/slam-uwv_kalman_filters}}, and is capable of running in real-time on computing available on the \\textit{FlatFish} AUV.\n\nThe work in this paper utilizes vehicle model-based aiding and the ADCP sensor for further ocean water current and vehicle velocity constraints. Model-aiding allows physics based constraints on the positioning, and the uncertainty in each parameter can be set to account for the systematic error associated with a system identification. Thus even a low accuracy system identification can still be used with this filter without resulting in filter overconfidence. Additionally, by modeling the vehicle parameters as time-varying, the model itself has become uncertain, as any small deviations in dynamics from the modeling equations can be absorbed by the time-varying parameters. ADCP-aiding in cases where DVL dropout would occur, due to being higher altitude than the bottom-lock range, also can aid the model by providing independent vehicle velocity constraints. ADCP also gives information regarding the surrounding water currents when there is a DVL dropout and the vehicle state estimation relies more on model-aiding.\nGenerally, inertial navigation is achieved using error-state filtering \\cite{hegrenaes2011model}, but this is not necessary as is shown in this paper. This paper gives a more conceptually simplified approach, while also utilizing manifold methods \\cite{forster2015manifold} to represent attitude, which is more general than other methods.\n\n\n\n\n\n\\section{Model-aided Inertial filter design}\n\n\n\n\n\nOur filter design is conceptually simple, since we model all modalities in one filter and model the attitude of the vehicle as a manifold.\nWe utilize an Unscented Kalman Filter (UKF) since it doesn't require the Jacobians of the process or measurement models and can handle non-linearities better than an Extended Kalman Filter \\cite{wan2000unscented}.\n\nThe attitude of the vehicle is an element of $SO(3)$, the group of orientations in $\\mathbb{R}^3$. To directly estimate the attitude in the filter it can be either modeled by a minimal parametrization (Euler angles) or by a over-parametrization (quaternion or rotation matrix). A minimal parametrization has singularities, i.e. small changes in the state space might require large changes in the parameters. An over-parametrization has redundant parameters and needs to be re-normalized as required. In both cases it requires special treatment in the filter, which leads to a conceptually more complex filter design.\nRepresenting the attitude as a manifold is a more general solution in which the filter operates on a locally mapped neighborhood of $SO(3)$ in $\\mathbb{R}^3$ \\cite{hertzberg2013integrating}.\n\n\\begin{table}[h]\n\\centering\n\\caption{Filter state}\n\\begin{tabular}{l|l}\n\\hline\n\\thead{Elements of \\\\ the state vector} & Description \\\\\n\\hline\n$\\mathbf{p}^n \\in \\mathbb{R}^3$\t\t& Position of the IMU in the navigation frame \\\\\n$\\boldsymbol{\\phi}^n \\in \\mathbb{R}^3$ \t& Attitude of the IMU in the navigation frame \\\\\n$\\mathbf{v}^n \\in \\mathbb{R}^3$ \t& Velocity of the IMU in the navigation frame \\\\\n$\\mathbf{a}^n \\in \\mathbb{R}^3$ \t& Acceleration of the IMU in the navigation frame \\\\\n$\\mathbf{M_{\\text{sub}}} \\in \\mathbb{R}^{2\\times3}$  & Inertia sub-matrix of the motion model \\\\\n$\\mathbf{D}_{l,\\text{sub}} \\in \\mathbb{R}^{2\\times3}$  & Linear damping sub-matrix of the motion model \\\\\n$\\mathbf{D}_{q,\\text{sub}} \\in \\mathbb{R}^{2\\times3}$  & Quadratic damping sub-matrix of the motion model \\\\\n$\\mathbf{v}_{c,v}^{n} \\in \\mathbb{R}^{2}$\t& \\begin{tabular}{@{}l@{}}Water current velocity surrounding \\\\ the vehicle in navigation frame\\end{tabular} \\\\\n$\\mathbf{v}_{c,b}^{n} \\in \\mathbb{R}^{2}$\t& \\begin{tabular}{@{}l@{}}Water current velocity below the \\\\ vehicle in navigation frame\\end{tabular} \\\\\n$g^n \\in \\mathbb{R}$\t\t& Gravity in the navigation frame \\\\\n$\\mathbf{b}_{\\omega} \\in \\mathbb{R}^3$ \t& Gyroscope bias \\\\\n$\\mathbf{b}_{a} \\in \\mathbb{R}^3$ \t& Accelerometer bias \\\\\n$\\mathbf{b}_{c} \\in \\mathbb{R}^{2}$ \t& Bias in the ADCP measurements \\\\\n\\hline\n\\end{tabular}\n\\label{state_table}\n\\end{table}\n\nTable \\ref{state_table} shows the state vector of the filter as element of $\\mathbb{R}^{43}$ and  gives a detailed description of the higher dimensional elements of the state vector. The navigation frame is North-East-Down (NED).\nThe body and IMU frames are x-axis pointing forward, y-axis pointing left and z-axis pointing up. \nIn the filter design we consider the IMU frame not to be rotated with respect to the body frame.\n\n\n\n\\subsection{Inertial prediction equations}\n\nThe following equations describe the prediction models for position, velocity, acceleration and attitude, applying a constant acceleration model for translation and a constant angular velocity model for rotation:\n\\begin{equation}\n\\mathbf{p}^n_{t} = \\mathbf{p}^n_{t-1} + \\mathbf{v}_{t-1}^n \\delta t\n\\label{eq:pred1}\n\\end{equation}\n\\begin{equation}\n\\label{vn}\n\\mathbf{v}^n_{t} = \\mathbf{v}^n_{t-1} + \\mathbf{a}_{t-1}^n \\delta t\n\\end{equation}\n\\begin{equation}\n\\label{vn}\n\\mathbf{a}^n_{t} = \\mathbf{a}^n_{t-1}\n\\end{equation}\n\\begin{equation}\n\\boldsymbol{\\phi}^n_{t} = \\boldsymbol{\\phi}^n_{t-1} \\boxplus [C^n_{b,t-1}(\\boldsymbol{\\omega}_{t-1}^{b} - \\mathbf{b}_{\\omega,t-1}) - \\boldsymbol{\\Omega}_{e}^{n} \\delta t]\n\\label{eq:pred2}\n\\end{equation}\n\nwhere $\\mathbf{p}^n_{t}$ is the position of the IMU in the navigation frame at time $t$, $\\mathbf{v}^n_{t}$ is the velocity of the IMU in the navigation frame, $\\mathbf{a}^n_{t}$ is the acceleration of the IMU in the navigation frame, $\\mathbf{C}^n_{b,t}$ is the coordinate transformation from body to navigation frame, $\\boldsymbol{\\phi}^n_{t}$ is the attitude of the IMU in the navigation frame, $\\boldsymbol{\\omega}^b_t$ is the rotation rates in the body frame, $\\mathbf{b}_{\\omega,t}$ is the gyroscope bias and $\\boldsymbol{\\Omega}_{e}^{n}$ is the Earth rotation in the navigation frame. The $\\boxplus$ operator in \\eqref{eq:pred2} is a manifold based addition, as defined in \\cite{hertzberg2013integrating}.\nEquations \\eqref{eq:pred1} to \\eqref{eq:pred2} each have corresponding prediction noise added.\n\nThe accelerometer measurements are handled with an update equation on the acceleration state as follows:\n\n\\begin{align}\n\\textbf{z}_{a}(t) = \\mathbf{f}^b_t+ \\mathbf{b}_{a,t} + \\mathbf{C}^{n}_{b,t}\\mathbf{g}^{n}_{t} + \\nu_{a}\n\\end{align}\n\nwhere $\\mathbf{f}^b_t$ is the specific force acting on the vehicle at time $t$, $\\mathbf{b}_{a,t}$ is the accelerometer bias and $\\mathbf{g}^{n}_{t}$ is the gravity vector $\\begin{bmatrix} 0, 0, g_t^n \\end{bmatrix}^T$ in the navigation frame.\nThe gravity state is modeled applying a constant gravity model in order to refine the theoretical gravity according to the WGS-84 ellipsoid earth model starting with a small initial uncertainty.\nThe acceleration state in the filter allows both the accelerometer and model-aiding to act on the filter in a consistent fashion, without resorting to virtual correlation terms when an acceleration state does not exist, such as in \\cite{hegrenaes2011model}.\nAccelerometer and gyro bias terms are modeled as a first order Markov process as follows:\n\n\\begin{equation}\n\\dot{b} = -\\frac{1}{\\tau_{b}}(b-b_{0}) + \\nu_{b}\n\\label{bias_equation}\n\\end{equation}\nwhere $\\tau_{b}$ is the expected rate change of the bias, $b_{0}$ is the mean bias value, and $\\nu_{b}$ is a zero-mean normally distributed random variable with\n\\begin{equation}\n\\sigma_{b} = \\sqrt{\\frac{2f\\sigma_{b\\,drift}^2}{\\tau_{b}}}\n\\end{equation}\nwhere $\\sigma_{b\\,drift}$ is a bound to the bias drift and $f$ is the measurement frequency. The accelerometer and gyro bias are assumed to be zero mean.\n\n\n\\subsection{Model-aiding update equations}\n\nIn this section we show a model-aiding measurement function using a simplified vehicle motion model for which a subset of the parameter space is part of the filter state.\nThis allows the filter to refine the parameters at runtime and to account for the systematic uncertainty in the vehicle parameters.\n\nThe nonlinear equations for motion \\cite{fossen2002marine} of a rigid body with 6 DOF can be written as\n\\begin{equation}\n\\label{eq:motion_fossen}\n\\boldsymbol{\\tau} = \\mathbf{M} \\dot{\\boldsymbol{\\nu}} + \\mathcal{C}(\\boldsymbol{\\nu}) \\boldsymbol{\\nu} + \\mathbf{D}(\\boldsymbol{\\nu}) \\boldsymbol{\\nu} + \\mathbf{g}(\\mathbf{R}_{b}^n)\n\\end{equation}\nwhere $\\boldsymbol{\\tau}$ is the vector of forces and torques, $\\boldsymbol{\\nu}$ is the vector of linear and angular velocities, $\\mathbf{M}$ is the inertia matrix including added mass, $\\mathcal{C}(\\boldsymbol{\\nu})$ is the Coriolis and centripetal matrix, $\\mathbf{D}(\\boldsymbol{\\nu})$ is the hydrodynamic damping matrix and $\\mathbf{g}(\\mathbf{R}_{b}^n)$ is the vector of gravitational forces and moments given the rotation matrix from the body to the navigation frame $\\mathbf{R}_{b}^n$.\n\n\\begin{equation}\n\\mathbf{g}(\\mathbf{R}) = \\begin{bmatrix} \\mathbf{R}^{-1} \\hat{k} (W-B) \\\\ \\mathbf{r}_{G} \\times \\mathbf{R}^{-1}\\hat{k}W - \\mathbf{r}_{B} \\times \\mathbf{R}^{-1}\\hat{k}B \\end{bmatrix}\n\\label{eq:gravity}\n\\end{equation}\nEquation~\\eqref{eq:gravity} shows how the gravitational forces and moments are calculated given the weight $W$, buoyancy $B$, center of gravity $\\mathbf{r}_{G}$ and center of buoyancy $\\mathbf{r}_{B}$ of the vehicle,\nwhere $\\hat{k}$ is the unit vector $\\begin{bmatrix} 0, 0, 1 \\end{bmatrix}^T$.\n\nWe assume the Coriolis and centripetal forces as well as damping terms higher than second order are negligible for vehicles operating within lower speeds (typically below $1.5$ m\/s).\nThis allows us the define the measurement function for the forces and torques in the body frame from \\eqref{eq:motion_fossen} as\n\\begin{equation}\n\\textbf{z}_{\\boldsymbol{\\tau}}(t) = \\mathbf{M}_{t} \\begin{bmatrix}\\mathbf{a}_{t}^{b} \\\\ \\boldsymbol{\\alpha}_{t}^{b}\\end{bmatrix} + \\mathbf{D}(\\begin{bmatrix}\\mathbf{v}_{t}^{b} \\\\ \\boldsymbol{\\omega}^b_t\\end{bmatrix},t) + \\mathbf{g}(\\mathbf{R}_{b,t}^n) + \\nu_{\\boldsymbol{\\tau}}\n\\label{eq:meas_tau}\n\\end{equation}\nwhere $\\mathbf{a}_{t}^{b}$ is the linear acceleration, $\\boldsymbol{\\alpha}_{t}^{b}$ is the angular acceleration, $\\mathbf{v}_{t}^{b}$ is the linear velocity and $\\boldsymbol{\\omega}^b_t$ is the angular velocity, all expressed in the body-fixed frame at time $t$. $\\nu_{\\boldsymbol{\\tau}}$ is the random noise of the force and torque measurement, with a standard deviation given by the thruster manufacturer.\n\n$\\mathbf{a}_{t}^{b}$ can be computed given the acceleration in the navigation frame $\\mathbf{a}_{t}^{n}$ as\n\\begin{equation}\n\\mathbf{a}_{t}^{b} = \\mathbf{C}_{n,t}^{b}\\mathbf{a}_{t}^{n} - \\boldsymbol{\\omega}^b_t \\times (\\boldsymbol{\\omega}^b_t \\times \\mathbf{p}^b)\n\\label{eq:acc_body}\n\\end{equation}\nwhere $\\mathbf{C}_{n,t}^{b}$ is the coordinate transformation matrix from navigation to body frame at time $t$ and $\\mathbf{p}^b$ is the position of the IMU in the body frame.\n\n$\\mathbf{v}_{t}^{b}$ can be computed given the velocity in the navigation frame $\\mathbf{v}_{t}^{n}$ as\n\\begin{equation}\n\\mathbf{v}_{t}^{b} = \\mathbf{C}_{n,t}^{b} (\\mathbf{v}_{t}^{n} - \\mathbf{v}_{c,v,t}^{n}) - \\boldsymbol{\\omega}^b_t \\times \\mathbf{p}^b\n\\label{eq:velocity_body}\n\\end{equation}\nwhere $\\mathbf{v}_{c,v,t}^{n}$ is the water current velocity surrounding the vehicle at time $t$.\n\nEquation~\\eqref{eq:damping} shows how the damping is defined given the linear and angular velocities at time $t$.\n\\begin{equation}\n\\mathbf{D}(\\mathbf{\\nu}_{t}, t) = \\mathbf{D}_{l,t} \\cdot \\mathbf{\\nu}_{t} + |\\mathbf{\\nu}_{t}|^{T} \\cdot \\mathbf{D}_{q,t} \\cdot \\mathbf{\\nu}_{t}\n\\label{eq:damping}\n\\end{equation}\nThe linear damping matrix $\\mathbf{D}_{l,t}$, the quadratic damping matrix $\\mathbf{D}_{q,t}$ and the inertia matrix $\\mathbf{M}_{t}$ are time dependent, since for all of them a sub matrix is part of the filter state.\nThe filter states $\\mathbf{D}_{l,\\text{sub},t}$, $\\mathbf{D}_{q,\\text{sub},t}$, $\\mathbf{M}_{\\text{sub},t}$ $\\in \\mathbb{R}^{2\\times3}$ are defined by removing the rows 3 to 6 and columns 3 to 5 from the full damping and inertia matrices $\\mathbf{D}_{l}$, $\\mathbf{D}_{q}$, $\\mathbf{M}$ $\\in \\mathbb{R}^{6\\times6}$. In other words we model the $x,xy,x\\psi,yx,y,y\\psi$ terms of the matrices in the filter, where $\\psi$ is the yaw. Because we expect them to have the major impact with respect to the horizontal accelerations and velocities, in case of an AUV keeping roll and pitch stable. It would be easy to extended the filter states and add more model terms, however it is a trade-off between the additional benefit, the computational complexity and potential filtering instability.\n\nThe damping and inertia state prediction models have a base time varying component, with a timescale of around one hour, modeled as in \\eqref{bias_equation}.\nThe vehicle parameters are initialized using a prior system identification, with the means of the states set at these values in the first order Markov process equation. Since the vehicle parameters are states in the filter, the systematic uncertainty in their error can be accounted for, which acts like a bias rather than a noise. This allows even a low accuracy system identification, or very crude estimates of the parameter values, to still allow estimation without resulting in overconfidence, due to the bias error having a stronger and different effect to simply increasing the uncertainty in the vehicle model noise. \nThis also allows the vehicle modeling to adapt to different scenarios, such as surfacing or changes to the vehicle following the system identification, while constraining their value range by utilizing the first order Markov process model. \nAlthough these parameters could have been modeled as a constant, by allowing the parameters to have a time-varying component it acts as a way to implement ``model uncertainty''.\nIn this way, the robustness of the filter improves as we no longer fully trust our model to be a perfect representation of the true dynamics, which is most definitely the case with applying a simplified and computationally tractable model for real-time usage to the real-world.\n\n\\subsection{ADCP-aiding update equations}\n\nGiven the 3D velocities output from the ADCP, the observation function for each ADCP measurement is\n\\begin{equation}\n\\textbf{z}_{c,i}(t) = \\textbf{C}_{n,t}^{b}(- \\textbf{v}_{t}^{n} + \\frac{d_{max}-d_{i}}{d_{max}}\\textbf{v}_{c,v,t}^{n} + \\frac{d_{i}}{d_{max}}\\textbf{v}_{c,b,t}^{n}) + \\textbf{b}_{a,t} + \\nu_{a}\n\\label{z_adcp}\n\\end{equation}\n\n\nwhere $\\textbf{z}_{c,i}$ is the ADCP measured current vector in the i$^{th}$ measurement cell, $\\textbf{C}_{n,t}^{b}$ is the coordinate transform from navigation\/world frame to ADCP\/body frame at time $t$,\n$\\textbf{v}_{t}^{n}$ is the vehicle velocity in the world\/navigation frame, $\\textbf{v}_{c,v,t}^{n}$ is the water current velocity surrounding the vehicle, $\\textbf{v}_{c,b,t}^{n}$ is the water current velocity at the maximum ADCP range, $\\textbf{b}_{a,t}$ is the bias in the ADCP measurement and $\\nu_{a}$ is the random noise in the ADCP measurement, with a standard deviation given by the sensor manufacturer.\n\nTo reduce the state number of the filter, the vertical velocity of the water currents are not estimated. The ADCP measurement model is a depth dependent function with two water current states, which linearly interpolates between them. The states are located at the vehicle position, and at a water volume at end of the ADCP measurement range. The water velocity and the ADCP bias state prediction models have a base time varying component, with a timescale of around one hour for the water current and half an hour for the bias, modeled as in \\eqref{bias_equation}. In addition to this, the water velocity state will vary more given spatial motion through a water current vector field. This component scales the process model uncertainty of the water velocity state according to the vehicle velocity. In this way, if the vehicle is slowly traveling through the water current vector field, it can account for the spatial scale of the water currents, which can depend on the environment. For example, water currents near complex bathymetry or strong wind and tides can contribute to smaller spatial scale water current velocity changes, compared to the case of the mid-water ocean \\cite{medagoda2015autonomous}.\n\n\\section{Results}\n\n\n\n\n\nAll the experiments have been made using the \\textit{FlatFish} AUV \\cite{albiez2015flatfish} shown in Fig. \\ref{fig:flatfish}.\nAs relevant sensors for our experiments, the vehicle is equipped with a KVH 1750 IMU, a Rowe SeaProfiler DualFrequency 300\/1200 kHz ADCP\/DVL, a Paroscientific 8CDP700-I pressure sensor, a u-blox PAM-7Q GPS receiver and six 60N Enitech ring thrusters. For heading evaluation purposes we also use a Tritech Gemini 720i Multibeam Imaging Sonar attached to the AUV.\nThe data sets have been collected during the sea trails of the second phase of the \\textit{FlatFish} project close to the shore of Salvador (Brazil) during April 2017.\n\nSince the experiments took place in the open ocean in all data sets, a fiber optic tether was attached to the vehicle for safety reasons. As a result, even though the vehicle model parameters were estimated with a prior system identification, there would be a large error associated with the model given the tether, so there is $\\sim$20\\% uncertainty in the parameter values. Nonetheless, the filter is robustly capable of accounting for this increase in the uncertainty of the vehicle model parameters. This allows the filter to adaptively change the parameters while keeping them in a constrained range through the use of the first order Markov process model.\nThe filter is capable of running in 14$\\times$ real-time with an integration frequency of 100 Hz on computing available on the \\textit{FlatFish} AUV.\n\n\\subsection{Heading estimation experiment}\n\\label{sec:heading_experiment}\n\n  \n  \n\nIn this data set we show that the filter is able to find its true heading without a global positioning reference, given an initial guess.\nThe mission consists of a initialization phase on the surface followed by a submerged phase before resurfacing. During the initialization phase the vehicle moves for around 8 minutes on a straight line in order to estimate its true heading and position by incorporating GPS measurements. In the submerged phase the vehicle changes its heading to face the target coordinate and follows a straight line for about 112 meters to reach it.\n\n\\begin{figure}[!h]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth,trim=0 0 0 5,clip] {pictures\/random_init_min_max_heading_with_and_without_gps.pdf}\n      \\caption{The plots show the estimated heading during the mission given different filter configurations and initial headings distributed over 30$^{\\circ}$.\n\t       The green crosses show independent land mark based heading measurements. 200 seconds in the mission the heading offset was corrected resulting in the short change of attitude.}\n      \\label{fig:heading_comp}\n\\end{figure}\n\nFig. \\ref{fig:heading_comp} shows six runs of the same data set in different filter configurations. Three GPS-aided runs with a different initial heading distributed over 30$^{\\circ}$, one with a close initial guess (black line), one with a 15$^{\\circ}$ positive offset (cyan line) and one with a 15$^{\\circ}$ negative offset (blue line). Due to the help of the GPS measurements the estimated headings converge in the first 5 minutes.\nThe three runs not integrating a global position reference starting with the same heading distribution show that the filter is able to find its true heading by observing the rotation of the earth (gyro compassing), relying only on Inertial and velocities.\nAfter 15 minutes the GPS-aided and the non-GPS-aided estimated headings have converged with an uncertainty below 0.5$^{\\circ}$ (1$\\sigma$).\nInitial errors $>$15$^{\\circ}$ will converge as well given more time. Critical however are initial errors close to 180$^{\\circ}$.\nThe green crosses show multiple independent measurements of the expected vehicle heading based on landmarks (poles) visible in the multibeam imaging sonar on the vehicle.\nThe average difference between the landmark based headings to the filter estimates is below 1$^{\\circ}$.\nWe expect the uncertainties of these measurements to be within 5$^{\\circ}$ due to the uncertainties associated with the pole positions in surveyed maps and in the sonar images.\n\n\\begin{figure}[!h]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth,trim=0 0 0 5,clip] {pictures\/random_init_min_max_heading_convergence_with_and_without_gps.pdf}\n      \\caption{The blue solid line is the error in heading with integrated GPS measurements.\n\t       The red solid line is the error in heading without the integration of GPS measurements.\n\t       The dashed lines are the corresponding uncertainties (1$\\sigma$).}\n      \\label{fig:heading_diff}\n\\end{figure}\n\nUsing the GPS-aided heading with a close initial guess shown in Fig. \\ref{fig:heading_comp} (black solid line) as ground truth, we can have a closer look in Fig. \\ref{fig:heading_diff} on the uncertainties and how the estimates improve.\nIn Fig. \\ref{fig:heading_diff} both filter configurations start with an offset of -15$^{\\circ}$ to the ground truth and an initial uncertainty of 30$^{\\circ}$ (1$\\sigma$).\nThe GPS-aided heading estimate converges, as expected, quickly to the ground truth while staying in the 1$\\sigma$ bound.\nFor the heading estimate without global positioning reference we can see that the strong offset and high uncertainty in the beginning leads to a fast compensation in the correct direction with an overshot slightly exceeding the 1$\\sigma$ bound. As the experiment progresses we can see that observing different orientations helps to estimate the gyroscope bias and therefore helps to detect the error between the expected rotation of the earth given the current orientation. We have shown that our filter is able to estimate its true heading by observing the rotation of the earth and that observations from different attitudes help to improve the process.\n\n\\subsection{Repeated square path experiment}\n\n  \n  \n  \n\nIn this experiment we show how the filter performs when the vehicle travels a longer distance of 1 km without horizontal position aiding measurements, such as GPS.\nThe vehicle was following a 5 times repeated square trajectory with an edge length of 50 meter for $\\sim$1 hour. After resurfacing, the position difference to the GPS ground truth is within 0.5\\% of the traveled distance.\n\nStarting with an initialization phase (same as in \\ref{sec:heading_experiment}) on the surface, to estimate its heading and position using GPS measurements, the vehicle submerges to 10 m depth, performs the mission and surfaces at the end.\nThe blue line in Fig. \\ref{fig:square_trajectory} shows the trajectory of the vehicle from minute 20 to minute 80 in the mission, i.e. 1 minute before submerging and 2 minutes after surfacing.\nThe red dots are the GPS measurements including outliers.\n\n\\begin{figure}[!ht]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth,trim=0 0 0 5,clip] {pictures\/5x50m_square_trajectory.pdf}\n      \\caption{The blue solid line shows the trajectory of the vehicle performing 5 times a 50 meter square trajectory in a depth of 10 meter.\n\t       After traveling the distance of 1 km the horizontal (North\/East plane) position difference is withing 5 meter (0.5\\% of distance traveled).}\n      \\label{fig:square_trajectory}\n\\end{figure}\n\nThe pose filter used on the vehicle at the time the data set was created was not aware of the drift and the initial error in heading.\nOur filter can correct the heading by observing the rotation of the earth and compensate for DVL dropouts utilizing the motion model.\nHowever during the mission a fiber optic tether was attached to the vehicle which represents an unmodeled source of error.\n\n\\begin{figure}[!ht]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth,trim=0 0 0 -2,clip] {pictures\/5x50m_square_horizontal_pos_error.pdf}\n      \\caption{The blue solid line shows the horizontal (North\/East plane) position difference with respect to the GPS measurements (including outliers).\n\t       The red and magenta dashed lines represent the corresponding uncertainty (1$\\sigma$ and 2$\\sigma$).}\n      \\label{fig:square_pos_diff}\n\\end{figure}\n\nThe blue line in Fig. \\ref{fig:square_pos_diff} shows the position difference on the North\/East plane with respect to the GPS measurements (including outliers).\nDuring the first 20 minutes of the mission the GPS measurements are integrated in the filter allowing initialization.\nAfter resurfacing (minute 78 and onward) the GPS measurements are not integrated allowing us to observe the difference to the ground truth.\nAfter traveling a distance of 1 km the position difference is withing 5 meter (0.5\\% of distance traveled) and in the 2$\\sigma$ bound of the position uncertainty.\n\n\\begin{figure}[!ht]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth,trim=0 0 0 5,clip] {pictures\/5x50m_square_water_current.pdf}\n      \\caption{Estimated water current in north (red) and east (blue) direction. The dashed lines represent the corresponding uncertainties (2$\\sigma$).}\n      \\label{fig:5x50m_square_water_current}\n\\end{figure}\n\nIn the case that ADCP measurements are not available the filter will estimate the water currents only by the difference between the motion model based velocity and the DVL based velocity, as modeled in \\eqref{eq:velocity_body}.\nFig. \\ref{fig:5x50m_square_water_current} shows the estimated water current velocities in North and East direction during this experiment without the aiding of ADCP measurements.\nDuring the first 20 minutes the uncertainties of the water current velocities stay constant, since we apply the model-aiding measurements with an increased uncertainty in case the vehicle is surfaced.\nWhen the mission starts and the vehicle submerges (starting around minute 21) to a depth of 10 meters we can see that the estimated water flow changes to the one on the surface and that its velocity continuously increases during the 1 hour mission. The uncertainties reduce during this phase since we trust the model more when submerged.\nThe impact of the tether attached to the vehicle is seen as an unmodeled but estimated drag, which changes depending on the direction the vehicle travels.\n\n\\begin{figure}[t]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth] {pictures\/5x50m_square_damping_terms.pdf}\n      \\caption{Linear damping in x (red) and y (blue) direction in the body frame. The dashed lines represent the corresponding uncertainties (2$\\sigma$).}\n      \\label{fig:5x50m_damping}\n\\end{figure}\n\nFig. \\ref{fig:5x50m_damping} shows the linear damping terms on the x and y-axis in the body frame of the vehicle and how they are refined during the mission.\nBecause the vehicle travels during the mission mainly in the forward direction, the damping term on the x-axis is refined and the corresponding uncertainty reduces more compared to the y-axis damping term. The uncertainty reduction reaches a limit however due to observability, and the first order Markov process model ensures that the parameters become neither overconfident nor unconstrained. In this way, the model parameters can adapt with time to new conditions and implicitly represents some uncertainty in the model equations themselves.\n\n\n\\subsection{Square path with ADCP}\n\n\\begin{figure}[!ht]\n      \\centering\n      \\includegraphics [width=0.45\\textwidth,trim=5 10 5 25,clip] {pictures\/adcp_traj.pdf}\n      \\caption{The solid blue line shows the trajectory of the vehicle performing a square path in a depth of 2 meter while surfacing in each corner.}\n      \\label{fig:adcp_traj}\n\\end{figure}\n\n\\begin{figure*}[!ht]\n      \\centering\n      \\includegraphics [width=0.75\\textwidth,trim=0 5 0 0,clip] {pictures\/adcp_compare.pdf}\n      \\caption{Square path with ADCP - The position uncertainties and differences from the ground truth are compared for different data denials.}\n      \\label{fig:adcp_compare}\n\\end{figure*}\n\n\\begin{table}[]\n\\caption{Filter position difference from ground truth and estimated uncertainty}\n\\label{adcp_table}\n\\centering\n\\begin{tabular}{|l|l|l|}\n\\hline\nFilter measurements used & \\thead{Estimated uncertainty\\\\after 1000 seconds} & \\thead{Position difference \\\\from ground truth\\\\ after 1000 seconds} \\\\ \\hline\nInertial + ADCP  & 50.9 m (2$\\sigma$) & 22.0 m \\\\ \\hline\nInertial + model & 45.7 m (2$\\sigma$) & 21.5 m \\\\ \\hline\nInertial + model + ADCP & 32.3 m (2$\\sigma$) & 16.1 m \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nThis mission undergoes a 600 second initialization phase on the surface (as in \\ref{sec:heading_experiment}), then 1000 seconds of data denial to show the performance of the filter in different scenarios. During the data denial phase, the vehicle completes a square trajectory, and surfaces at the corners. The ground truth trajectory is shown in Fig. \\ref{fig:adcp_traj}. The ground truth is determined using Inertial, DVL, GPS, ADCP and model-aiding.\n\nSince this mission also includes ADCP measurements interleaved with DVL, the ADCP-aiding update is applied. During this mission, there are cases where the downward facing DVL drops out due to very low altitude (between 0-2m during the mission), and there is collision with the sandy bottom. Despite this challenging data set, the filter is capable of estimating the position of the vehicle, validated by the smooth trajectory without sudden corrections at the GPS measurements during the corner surfacing shown in Fig. \\ref{fig:adcp_traj}.\n\nWith the full measurement filter (without data denial), the filter is able to handle DVL drop outs, which could be the case in low-altitude scenarios such as inspection or docking, by letting the model-aiding fill in during these time periods. Data denial further validates the filter performance in DVL loss scenarios, as shown in Fig. \\ref{fig:adcp_compare}. In cases of DVL bottom-lock loss due to altitude being too high (simulated through data-denial), the ADCP and model-aiding combined gives the best solution, compared to either ADCP or model-aiding alone.\n\nThe position estimate differences compared to the ground truth for these data denials are consistent with the 2$\\sigma$ uncertainty bounds, while remaining stable. At approximately 400 seconds following data denial, the filter with only ADCP and inertial measurements appears to slightly exceed the 2$\\sigma$ bounds, due to a low altitude section with very little valid ADCP measurements, and some ADCP outliers are incorporated into the filter since the innovation gate increases due to inertial-only dead-reckoning. The ground truth also increases in uncertainty at this stage due to the lack of DVL measurements, relying more on the model-aiding. Following further measurements, the filter recovers and is able to reduce the difference between the filter estimate and the ground truth. This is possible since the water current estimate will not vary significantly in this timescale, so that the vehicle can use this state when there are ADCP measurements available again to estimate the velocity and thus position of the vehicle.\n\nThe ADCP-aiding typically performed worse in this case than the model-aiding, but this can be attributed to low altitude where there are very few valid ADCP measurements available. Nonetheless, incorporating these ADCP measurements into the model-aiding improved on the performance of either option. In addition to another source of velocity-aiding information from the ADCP, it also allows an independent source of information regarding the water currents surrounding the vehicle, which is required to transform the water relative velocity of the vehicle model to the navigation frame position used in the filter.\n\nThe results are further quantitatively compared in Table \\ref{adcp_table}. The combination of the ADCP-aiding and model-aiding results in a significant improvement compared to model-aiding alone, reducing position uncertainty from 45.7m (2$\\sigma$) to 32.3m (2$\\sigma$) during 1000 seconds of data denial.\n\n\n\n\n\\section{Conclusions}\n\n\nThe filter designed and implemented in this paper would be appropriate for general AUV navigation, despite not using a navigation grade IMU. In comparison to \\cite{hegrenaes2011model}, the primary insight to the design of this filter is the incorporation of the acceleration state, and adding many parameters as states to account for their correlated error, while modeling with a first order Markov process to constrain the change the filter can apply. The engineering design trade-off is that adding too many states will unnecessarily add computational complexity and potential filtering instability. \n\nThis furthers the state-of-the-art for robust filter design for INS, model-aiding and ADCP measurements, capable of real-time performance, consistency and stability as outlined in the experiments, while remaining conceptually simple. \nThis paper has shown a manifold based UKF that applies a novel strategy for inertial, model-aiding and ADCP measurement incorporation.\nThe filter is capable of observing and utilizing the Earth rotation for heading estimation to within 1$^{\\circ}$ (2$\\sigma$) by estimating the KVH 1750 IMU biases. \nThe drag and thrust model-aiding accounts for the correlated nature of vehicle model parameter error by applying them as states in the filter. The usage of the model-aiding is validated through observing that the filter remains consistent and does not become overconfident or unstable in the real-world experiments, despite uncertain vehicle model parameters. \n\nIt is hypothesized that the usage of time varying first order Markov processes to model these parameters act as a way to implement ``model uncertainty'', improving the robustness of the filter as we no longer fully trust our model to be a perfect representation of the true dynamics, which is most definitely the case with applying simplified and computationally tractable model for real-time usage to the real-world.\nADCP-aiding provides further information for the model-aiding in the case of DVL bottom-lock loss. The importance of water current estimation is highlighted in underwater navigation in the absence of external aiding, justifying the use of the model-aiding and ADCP sensor. Through data denial, scenarios with no DVL bottom lock are shown to be consistently estimated. Additionally this work was implemented using the MTK and ROCK framework in C++, and is capable of running in 14$\\times$ real-time on computing available on the \\textit{FlatFish} AUV.\n\n\nFuture work would include full spatiotemporal real-time ADCP based methods to more accurately model and observe the water current state around vehicle. This requires implementing a mapping approach, such as the work from \\cite{medagoda2016mid} \\cite{medagoda2015autonomous}.\nThe primary source of bias uncertainty for the KVH 1750 IMU is due to temperature change. If the temperature of the IMU can be controlled, or this bias can be calibrated with further experiments, then the performance can be further improved. Further heading evaluation will be possible with better ground truth, such as a visual confirmation or by utilizing an independent heading estimator such as an iXblue PHINS, so that a more accurate heading comparison can be undertaken. The error in alignments of sensors could also be further compensated, perhaps by adding states to the filter similar to the strategy for other systematic biases. Finally, further experiments and implementations in a variety of scenarios are planned to further test and refine the proposed filtering strategy.\n\n\n\n\n\n\\addtolength{\\textheight}{-12cm}  \n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\n\nWe like to thank Shell and SENAI CIMATEC for the opportunity to test the presented work on \\textit{FlatFish}.\n\nWe also like to thank all colleagues of the \\textit{FlatFish} team for their support and Javier Hidalgo-Carri\\'o for his review.\n\nThis work was supported in part by the EurEx-SiLaNa project\n(grant No. 50NA1704) which is funded by the German Federal\nMinistry of Economics and Technology (BMWi).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nDislocations can alter different stages of the precipitation process\nin crystalline solids, which consists of nucleation, growth and\ncoarsening \\cite{Larche_1979,Wagner_Kampmann_1991}. Distortion of the\nlattice in proximity of a dislocation can enhance nucleation in\nseveral ways \\cite{Porter_Easterling_1981,Christian_2002}. The main\neffect is the reduction in the volume strain energy associated with\nthe phase transformation. Nucleation on dislocations can also be\nhelped by solute segregation which raises the local concentration of\nthe solute in the vicinity of a dislocation, caused by migration of\nsolutes toward the dislocation, the Cottrell atmosphere effect. When\nthe Cottrell atmosphere becomes supersaturated, nucleation of a new\nphase may occur followed by growth of nucleus. Moreover, dislocation\ncan aid the growth of an embryo beyond its critical size by providing\na diffusion passage with a lower activation energy.\n\n\nPrecipitation of second-phase along dislocation lines has been\nobserved in a number of alloys\n\\cite{Aaronson_et_al_1971,Aaron_Aaronson_1971}. For example, in\nAl-Zn-Mg alloys, dislocations not only induce and enhance nucleation\nand growth of the coherent second-phase MgZn$_2$ precipitates, but\nalso produce a spatial precipitate size gradient around them\n\\cite{Allen_Vandesande_1978,Deschamps_et_al_1999,Deschamps_Brechet_1999}.\nCahn \\cite{Cahn_1957} provided the first quantitative model for\nnucleation of second-phase on dislocations in solids.  In Cahn's\nmodel, it is assumed that a cross-section of the nucleus is circular,\nwhich is strictly valid for a screw dislocation\n\\cite{Larche_1979}. Also, it is posited that the nucleus is incoherent\nwith the matrix so that a constant interfacial energy can be allotted\nto the boundary between the new phase and the matrix. An incoherent\nparticle interface with the matrix has a different\natomic configuration than that of the phases. The matrix is\nan isotropic elastic material and the formation of the precipitate\nreleases the elastic energy initially stored in its volume. Moreover,\nthe matrix energy is assumed to remain constant by precipitation. In\nthis model, besides the usual volume and surface energy terms in the\nexpression for the total free energy of formation of a nucleus of a\ngiven size, there is a term representing the strain energy of the\ndislocation in the region currently occupied by the new phase. Cahn's\nmodel predicts that both a larger Burgers vector and\na more negative chemical free energy change between the precipitate\nand the matrix induce higher nucleation rates, in agreement with\nexperiment \\cite{Aaronson_et_al_1971,Aaron_Aaronson_1971}.\n\n\nSegregation phenomenon around dislocations, i.e. the Cottrell\natmosphere effect, has been observed among others in Fe-Al alloys\ndoped with boron atoms \\cite{Blavette_et_al_1999} and in silicon\ncontaining arsenic impurities \\cite{Thompson_et_al_2007}, in\nqualitative agreement with Cottrell and Bilby's predictions\n\\cite{Cottrell_Bilby_1949}. Cottrell and Bilby considered segregation\nof impurities to straight-edge dislocations with the Coulomb-like\ninteraction potential of the form $\\phi=A\\sin \\theta\/r$, where $A$\ncontains the elasticity constants and the Burgers vector, and\n$(r,\\theta)$ are the polar coordinates. Cottrell and Bilby ignored the\nflow due concentration gradients and solved the simplified diffusion\nequation in the presence of the aforementioned potential field. The\nmodel predicts that the total number of impurity atoms removed from\nsolution to the dislocation increases with time $t$ according to $N(t)\n\\sim t^{2\/3}$, which is good agreement with the early stages of\nsegregation of impurities to dislocations, e.g. in iron containing\ncarbon and nitrogen \\cite{harper_1951}. A critical review of the\nBilby-Cottrell model, its shortcomings and its improvements are given\nin \\cite{Bullough_Newman_1970}.\n\n\nThe object of our present study is the diffusion-controlled growth of\na new phase, i.e., a post nucleation process in the presence of\ndislocation field rather than the segregation effect. As in Cahn's\nnucleation model \\cite{Cahn_1957}, we consider an incoherent\nsecond-phase precipitate growing under the action of a screw\ndislocation field.  This entails that the stress field due to\ndislocation is pure shear. The equations used for diffusion-controlled\ngrowth are radially symmetric. These equations for second-phase in a solid or from a\nsupercooled liquid have been, in the absence of an external field, solved\nby Frank \\cite{Frank_1950} and discussed by Carslaw and Jaeger\n\\cite{Carslaw_Jaeger_1959}. The exact analytical solutions of the\nequations and their various approximations thereof have been\nsystematized and evaluated by Aaron et al. \\cite{Aaron_et_al_1970},\nwhich included the relations for growth of planar\nprecipitates. Applications of these solutions to materials can\nbe found in many publications, e.g. more recent papers on growth of\nquasi-crystalline phase in Zr-base metallic glasses\n\\cite{Koster_et_al_1996} and growth of Laves phase in Zircaloy\n\\cite{Massih_et_al_2003}.  We should also mention another\ntheoretical approach to the problem of nucleation and growth of an\nincoherent second-phase particle in the presence of dislocation\nfield \\cite{Sundar_Hoyt_1991}. Sundar and Hoyt \n\\cite{Sundar_Hoyt_1991} introduced the dislocation field, as in Cahn\n\\cite{Cahn_1957}, in the nucleation part of the model,\nwhile for the growth part the steady-state solution of the concentration field\n(Laplace equation) for elliptical particles was utilized.\n\n\nThe organization of this paper as follows. The formulation of the\nproblem, the governing equations and the formal solutions are given in\nsection \\ref{sec:formul}. Solutions of specific cases are presented in\nsection \\ref{sec:comp}, where the supersaturation as a function of the\ngrowth coefficient is evaluated as well as the spatial variation of\nthe concentration field in the presence of dislocation. In section\n\\ref{sec:disc}, besides a brief discourse on the issue of interaction\nbetween point defects and dislocations, we calculate the\nsize-dependence of the concentration at the curved precipitate\/matrix\nfor the problem under consideration. We have carried out our\ncalculations in space dimensions $d=2$ and $d=3$.  Some mathematical\nanalyses for $d=3$ are relegated to appendix \\ref{sec:appa}.\n\n\n\n\\section{Formulation and general solutions}\n\\label{sec:formul}\n\n  We consider the problem of growth of the new phase, with radial\nsymmetry (radius $r$), governed by the diffusion of a single entity,\n$u\\equiv u(r,t)$, which is a function of space and time $(r,t)$. $u$\ncan be either matter (solvent or solute) or heat (the latent heat of\nformation of new phase). The diffusion in the presence of an external\nfield obeys the Smoluchowski equation \\cite{Chandra_1943} of the form\n\\begin{eqnarray}\n  \\label{eqn:smolu}\n \\frac{\\partial u}{\\partial\nt} & = & \\nabla \\cdot \\mathbf{J}, \\\\\n\\label{eqn:smolu-flux}\n \\mathbf{J} & = & D(\\nabla u-\\beta\\mathbf{F}u),\n\\end{eqnarray}\n\\noindent\nwhere $D$ is the diffusivity, $\\beta=1\/k_BT$, $k_B$ the Boltzmann\nconstant, $T$ the temperature, and $\\mathbf{F}$ is an external field\nof force. The force can be local (e.g., stresses due to dislocation\ncores in crystalline solids) or caused externally by an applied field (e.g., electric field\nacting on charged particles). If the acting force is conservative, it\ncan be obtained from a potential $\\phi$ through $\\mathbf{F}=-\\nabla\n\\phi$. The considered geometric condition applies\nto the case of second-phase particles growing in a\nsolid solution under phase transformation \\cite{Massih_et_al_2003} or\ndroplets growing either from vapour or from a second liquid\n\\cite{Frank_1950}. A steady state is reached when\n$\\mathbf{J}=\\mathrm{const.}=0$, resulting in\n$u=u_0\\exp(-\\beta\\phi)$.\n\nHere, we suppose that the diffusion field is along\nthe core of dislocation line and that a cross-section of the\nprecipitate (nucleus), perpendicular to the dislocation, is circular,\ni.e., the precipitate surrounds the dislocation. Furthermore, we treat\nthe matrix and solution as linear elastic isotropic media. The elastic\npotential energy of a stationary dislocation of length $l$ is given by\n\\cite{Kittel_1996,Friedel_1967}\n\\begin{equation}\n  \\label{eqn:screw}\n  \\phi = A\\ln\\frac{r}{r_0}, \\qquad \\qquad \\qquad \\textrm{for}\\quad r\\ge r_0\n\\end{equation}\n\\noindent\nwhere $A=Gb^2l\/4\\pi$ for screw dislocation, $G$ is the elastic shear\nmodulus of the crystal, $b$ the magnitude of the Burgers vector, $\\nu$\nPoisson's ratio, and $r_0$ is the usual effective core radius. Also,\nwe assume that the dislocation's elastic energy is relaxed within the\nvolume occupied by the precipitate and that the precipitate is\nincoherent with the matrix. Hence the interaction energy between the\nelastic field of the screw dislocation and the elastic field of the solute\nis zero. In the case of an edge dislocation and coherent\nprecipitate\/matrix interface, this interaction is non-negligible.\n\n\nWe study the effect of the potential field (\\ref{eqn:screw}) on\ndiffusing atoms in solid solution using the Smoluchowski\nequation (\\ref{eqn:smolu}). The governing\nequation in spherical symmetry, in $d$ spatial dimension, with $B \\equiv \\beta A$, is \n\\begin{equation}\n  \\label{eqn:pde-1}\n  \\frac{1}{D}\\frac{\\partial u}{\\partial t}=\n  \\frac{\\partial^2 u}{\\partial r^2}+(d-1+B)\\frac{1}{r}\\frac{\\partial\nu}{\\partial r}+(d-2)B\\frac{u}{r^2}.\n\\end{equation}\n\\noindent\n Making a usual change of variable to the\ndimensionless reduced radius $s=r\/\\sqrt{Dt}$, the partial differential\nequation (\\ref{eqn:pde-1}) is reduced to an ordinary differential\nequation of the form\n\\begin{equation}\n  \\label{eqn:ode-1}\n  \\frac{{\\rm d^2} u}{{\\rm d}s^2}+\\Big(\\frac{s}{2}+\\frac{d-1+B}{s}\\Big)\\frac{{\\rm\nd}u}{{\\rm d}s}\n+(d-2)B\\frac{u}{s^2}=0,\n\\end{equation}\n\\noindent\nwith the boundary conditions, $u(\\infty) = u_m$, and $u(2\\lambda) =\nu_s$, where $u_m$ is the mean (far-field) solute concentration in the\nmatrix and $u_s$ is the concentration in the matrix at the\nnew-phase\/matrix interface determined from thermodynamics of new\nphase, i.e., phase equilibrium and the capillary effect.  Moreover,\nthe conservation of flux at the interface radius $R=2\\lambda\\sqrt{Dt}$ gives\n\\begin{equation}\n  \\label{eqn:flux-1}\n  K_d R^{d-1}\\vert \\mathbf{J}\\vert_{r=R}  = q \\frac{{\\rm d}V_d}{{\\rm d}t},\n\\end{equation}\n\\noindent\nwhere $K_d=2\\pi^{d\/2}\/\\Gamma(d\/2)$, $\\Gamma(x)$ the usual\n$\\Gamma$-function, $V_d=2\\pi^{d\/2}R^d\/d\\Gamma(d\/2)$, and  $q$  the\namount of the diffusing entity ejected at the boundary of the growing\nphase per unit volume of the latter (new phase) formed. In $s$-space,\nequation (\\ref{eqn:flux-1}) is written as  \n\\begin{equation}\n  \\label{eqn:conserve-flux}\n  \\Big(\\frac{{\\rm d}u}{{\\rm d}s}\\Big)_{s=2\\lambda} = -\\Big(\\frac{Bu_s}{2\\lambda}+q\\lambda\\Big).\n\\end{equation}\n\\noindent\nThe boundary condition $u(2\\lambda) = u_s$ and equation (\\ref{eqn:conserve-flux})\nwill provide a relationship between $u_s$ and $u_m$ through $\\lambda$.\n\nFor $d=2$, equation (\\ref{eqn:ode-1}) is very much simplified, and  we find \n\\begin{equation}\n  \\label{eqn:sol-2d}\n  u(s)=u_m+\\frac{(Bu_m+2q\\lambda^2)\\lambda^B e^{\\lambda^2}\\Gamma(-B\/2,s^2\/4)}\n  {2-B\\lambda^Be^{\\lambda^2}\\Gamma(-B\/2,\\lambda^2)},\n\\end{equation}\n\\noindent\nwhere we utilized $u(\\infty) = u_m$ and equation (\\ref{eqn:conserve-flux}). Here\n $\\Gamma(a,z)$ is the incomplete\ngamma function defined by the integral $\\Gamma(a,z)=\\int_z^\\infty\nt^{a-1}e^{-t}dt$ \\cite{Abramowitz_Stegun_1964}. The yet unknown parameter\n$\\lambda$ is found from relation (\\ref{eqn:sol-2d}) at $u(2\\lambda) =\nu_s$ for a set of input parameters $u_s$, $u_m$ $q$, and $B$, through which the concentration\nfield, equation (\\ref{eqn:sol-2d}), and the growth of second-phase\n($R=2\\lambda\\sqrt{Dt}$) are determined.\n\n\nLet us consider the case of $d=3$, that is assume that the\npotential in equation (\\ref{eqn:screw}) is meaningful for a\nspherically symmetric system. In this case, for $B \\ne 0$, the point $z=0$ is\na regular singularity of  equation\n(\\ref{eqn:ode-1}), while $z=\\infty$ is an irregular singularity for\nthis equation, see appendix \\ref{sec:appa} for further\nconsideration. Nevertheless, for\n$d=3$, the general solution of equation (\\ref{eqn:ode-1}) is expressed in the form\n\\begin{equation}\n  \\label{eqn:sol-3d}\n u(s)=\n2C_1\\,{_1\\!F}_1\\Big(-\\frac{1}{2};\\frac{1+B}{2};-\\frac{s^2}{4}\\Big)s^{-1}\n+ 2^{B}C_2\\,{_1\\!F}_1\\Big(-\\frac{B}{2};\\frac{3-B}{2};-\\frac{s^2}{4}\\Big) s^{-B},\n\\end{equation}\n\\noindent\nwhere  ${_1\\!F}_1(a;b;z)$ is Kummer's confluent hypergeomtric function,\nsometimes denoted by $M(a,b,z)$ \\cite{Abramowitz_Stegun_1964}. The\nintegration constants $C_1$ and $C_2$ in equation (\\ref{eqn:sol-3d}) can be determined\nby invoking  equation (\\ref{eqn:conserve-flux}) and also the condition\n$u(\\infty)=u_m$, cf. appendix \\ref{sec:appa}.\n\n\\section{Computations}\n\\label{sec:comp}\n\nTo study the growth behavior of a second-phase in a solid solution\nunder the action of screw dislocation field, we attempt to compute the\ngrowth rate constant as a function of the supersaturation parameter $k$,\ndefined as $k \\equiv (u_s-u_m)\/q_u$ with $q_u=u_p-u_s$, where $u_p$\n is the composition of the nucleus \\cite{Aaron_et_al_1970}. For $d=2$,\ni.e., a cylindrical second-phase platelet, equation (\\ref{eqn:sol-2d}) with\n$u(2\\lambda) = u_s$ yields\n\n\\begin{equation}\n  k  = \\Bigg[\\frac{2\\lambda^2+Bu_m(u_p-u_s)^{-1}}{2-B\\lambda^B e^{\\lambda^2}\\Gamma\\big(-B\/2,\\lambda^2\\big)}\\Bigg]\n  \\lambda^B e^{\\lambda^2}\\Gamma\\big(-B\/2,\\lambda^2\\big).\n\\label{eqn:supsat-2d-exact}\n\\end{equation}\n\\noindent\nFor $B=0$, the relations obtained by Frank \\cite{Frank_1950} are recovered,\nnamely\n\\begin{eqnarray}\n u(z)  & =  & u_m + q_u \\lambda^2e^{\\lambda^2}E_1(z^2\/4), \\label{eqn:sol-0d2} \\\\\n k & = & \\lambda^2 e^{\\lambda^2}E_1(\\lambda^2), \\label{eqn:flux-0d2}\n\\end{eqnarray}\n\\noindent\nwhere $E_1(x)$ is the exponential integral of order one, related to the\nincomplete gamma function through the identity\n$E_n(x)=x^{n-1}\\Gamma(1-n,x)$ \\cite{Abramowitz_Stegun_1964}.\n\nFrom equation (\\ref{eqn:supsat-2d-exact}), it is seen that a complete separation of the supersaturation parameter\n$k \\equiv (u_s-u_m)(u_p-u_s)^{-1}$ is not possible for $B \\ne\n0$. However, for $u_s << u_p$ (a reasonable proviso) we write\n\\begin{equation}\n  k  = \\Big(\\lambda^2 + \\frac{B}{2}\\,\\epsilon\\Big)\\, \n  \\lambda^B e^{\\lambda^2}\\Gamma\\big(-B\/2,\\lambda^2\\big)+\\mathcal{O}(\\epsilon^2),\n\\label{eqn:supsat-2d-approx}\n\\end{equation}\n\\noindent\nwith $\\epsilon \\equiv u_s\/u_p$. For $B=1$, equations\n(\\ref{eqn:sol-2d}) and (\\ref{eqn:supsat-2d-approx}) yield, respectively\n\\begin{eqnarray}\n u(z)  & =  & u_m + \\frac{2\\lambda\\,e^{\\lambda^2}\\,(u_m + 2q_u\\lambda^2)\\,E_{3\/2}(z^2\/4)}\n{[2-e^{\\lambda^2} E_{3\/2}(\\lambda^2)]z},\n \\label{eqn:sol-1d2} \\\\\n k & = & \\Big(\\lambda^2 + \\frac{\\epsilon}{2}\\Big)\\,e^{\\lambda^2}E_{3\/2}(\\lambda^2)+\\mathcal{O}(\\epsilon^2).\n\\label{eqn:flux-1d2}\n\\end{eqnarray}\n\\noindent\nSimilarly for $B=2$, we have \n\\begin{eqnarray}\n u(z)  & =  & u_m + \\frac{4\\lambda^2e^{\\lambda^2}(u_m + q_u \\lambda^2)E_2(z^2\/4)\n}{[1-e^{\\lambda^2} E_{2}(\\lambda^2)]z^2},\n \\label{eqn:sol-2d2} \\\\\n k & = & (\\lambda^2 + \\epsilon)E_2(\\lambda^2)+\\mathcal{O}(\\epsilon^2).\n\\label{eqn:flux-2d2}\n\\end{eqnarray}\n\\noindent\n\nWe have plotted the growth coefficient $\\lambda=R\/2\\sqrt{Dt}$ as a\nfunction of the supersaturation parameter $k$ in figure\n\\ref{fig:k-2d} and the spatial variation of the concentration field\nin figure \\ref{fig:u-2d} for $d=2$ and several values of $B$. The\ncomputations are performed to $\\mathcal{O}(\\epsilon^2)$ with\n$\\epsilon=0.01$. Figure \\ref{fig:k-2d} shows that $\\lambda$ is an\nincreasing function of $k$; and also, as $B$ is raised $\\lambda$ is\nelevated. This means that an increase in the amplitude of dislocation\nforce (e.g., the magnitude of the Burgers vector) enhances second-phase\ngrowth in an alloy.\n\nFigure \\ref{fig:u-2d} displays the reduced concentration versus the\nreduced radius $z=r\/\\sqrt{Dt}$ for $\\lambda=1$. The reduced\nconcentration is calculated via equation (\\ref{eqn:sol-2d}). It is seen that for\n$z \\lesssim 1.6$ the concentration is enriched with increase in $B$,\nwhereas for $z \\gtrsim 1.6$, it is vice versa. So, for\n$\\lambda=1$, the crossover $z$-value is $z_c \\approx 1.6$. Also, as\n$\\lambda$ is reduced, $z_c$ is decreased.\n\n\\begin{figure}[htbp]\n\\begin{center}\n    \\includegraphics[width=0.80\\textwidth]{fig\/lambda_k2d}\n \\caption{ Growth coefficient $\\lambda$ as a function of supersaturation $k$ \n   at various levels of dislocation force\namplitude $B$ for a circular plate ($d=2$) and $u_s=0.01u_p$.}\n\\label{fig:k-2d}\n  \\end{center}\n\\end{figure}\n\\begin{figure}[htbp]\n \\begin{center}\n    \\includegraphics[width=0.80\\textwidth]{fig\/u2d}\n\\caption{Reduced concentration field as a function of reduced distance\n from the surface of the circular plate ($d=2$) at various levels of dislocation force\namplitude $B$ and at $\\lambda=1$.}\n\\label{fig:u-2d}\n\\end{center}\n\\end{figure}\n\nFor $d=3$, i.e., a spherical second-phase particle in the absence of\ndislocation field ($B=0$), we find\n\\begin{eqnarray}\n u(z)  & =  & u_m + 2 q_u\n\\lambda^3 e^{\\lambda^2}\\Big[\\frac{2e^{-z^2\/4}}{z}\n-\\sqrt{\\pi}\\,\\mathrm{erfc}(z\/2)\\Big],\n \\label{eqn:sol-3d-b0} \\\\\n k & = & 2\\lambda^2\\Big[1-\\sqrt{\\pi}\\,\\lambda\\, e^{\\lambda^2}\\mathrm{erfc}(\\lambda)\\Big].\n\\label{eqn:flux-3d-b0}\n\\end{eqnarray}\n\\noindent\nThis corresponds to the results obtained by Frank\n\\cite{Frank_1950}.\n\n For $d=3$ and $B=2$, equation (\\ref{eqn:ode-1}) is\nsimplified and an analytical solution can be found, resulting in\n\\begin{eqnarray}\n u(z) & = & \\Bigg(\\frac{e^{z^2\/4}(z^2+2)\\Big[\\sqrt{\\pi}\\lambda\ne^{\\lambda^2}\\Big(\\mathrm{erf}(\\frac{z}{2})-\\mathrm{erf}(\\lambda)\\Big)-1\\Big]\n+2\\lambda e^{\\lambda^2}z}{\\sqrt{\\pi}\\lambda e^{\\lambda^2}\\mathrm{erfc}(\\lambda)-1}\\Bigg)\n\\frac{e^{-z^2\/4}}{z^2}u_m +\n\\nonumber\\\\\n & & + \\;\n \\Bigg(\\frac{\\lambda^3 e^{\\lambda^2}\\Big[2z-\\sqrt{\\pi}e^{z^2\/4}(z^2+2)\\mathrm{erfc}(\\frac{z}{2})\\Big]}\n{\\sqrt{\\pi}\\lambda e^{\\lambda^2}\\mathrm{erfc}(\\lambda)-1}\\Bigg)\\frac{e^{-z^2\/4}}{z^2}q_u.\n \\label{eqn:sol-3d2}\n\\end{eqnarray}\n\\noindent\nPutting $u(2\\lambda)=u_s$, we obtain\n\\begin{equation}\n k   =  \\frac{1+2\\lambda^2\\Big(1-\\sqrt{\\pi}\\lambda\\,e^{\\lambda^2}\\mathrm{erfc}(\\lambda)\\Big)}\n{2\\lambda^2\\Big(\\sqrt{\\pi}\\lambda\\,e^{\\lambda^2}\\mathrm{erfc}(\\lambda)-1\\Big)}\\;\\frac{u_m}{q_u}+\n\\frac{2\\lambda^2-(1+2\\lambda^2)\\sqrt{\\pi}\\lambda\\,e^{\\lambda^2}\\mathrm{erfc}(\\lambda)}\n{2\\Big(\\sqrt{\\pi}\\lambda\\,e^{\\lambda^2}\\mathrm{erfc}(\\lambda)-1\\Big)}.\n\\label{eqn:k-3d-2e}\n\\end{equation}\nFor $u_s << u_p$,  we write\n\\begin{equation}\n k  =\n-2\\lambda^4+\\sqrt{\\pi}\\lambda^3(1+2\\lambda^2)\\,e^{\\lambda^2}\\mathrm{erfc}(\\lambda)\n + \\Big(1-2\\lambda^2 + 2\\sqrt{\\pi}\\lambda^3\\,e^{\\lambda^2}\\mathrm{erfc}(\\lambda)\\Big)\\epsilon\n+\\mathcal{O}(\\epsilon^2).\n\\label{eqn:k-3d-2a}\n\\end{equation}\n\\noindent\n\n\nGeneral analytical expressions of $u(z)$ and $k$, in terms of confluent\nhypergeometric functions, can also be found for even values of $B$ as\ndetailed in appendix \\ref{sec:appa}. Furthermore, asymptotic forms of\n$u(z)$ for large and small $z$ can be calculated, see appendix\n\\ref{sec:appa} for analysis of $z>>1$. Figure \\ref{fig:k-23d} compares $k$\nversus $\\lambda$ for $d=2$ and $d=3$ in the absence of dislocation\nfield ($B=0$).\n\n\\begin{figure}[htbp]\n\\begin{center}\n    \\includegraphics[width=0.80\\textwidth]{fig\/lambda_k2d3d}\n \\caption{Growth coefficient $\\lambda$ as a function of supersaturation parameter $k$ \n  at $B=0$ for a circular plate ($d=2$) versus a sphere ($d=3$).}\n\\label{fig:k-23d}\n  \\end{center}\n\\end{figure}\n\n\\section{Discussion}\n \\label{sec:disc}\n\nThe potential energy in equation (\\ref{eqn:screw}) describes the\nelastic energy of the dislocation relaxed within the volume\noccupied by the second-phase precipitate \\cite{Cahn_1957}. It was treated\nhere as an external field affecting the diffusion-limited growth of\nsecond-phase precipitate. The interaction energy of impurities in a\ncrystalline with dislocations depends on the specific model or\nconfiguration of a solute atom and a matrix which is used. Commonly,\nit is assumed that the solute acts as an elastic center of\ndilatation. It is a fictitious sphere of radius $R^\\prime$ embedded\nconcentrically in a spherical hole of radius $R$ cut in the matrix. If\nthe elastic constants of the solute and matrix are the same, the work\ndone in inserting the atom in the presence of dislocation is\n$w=p\\Delta v$, where $p$ is the hydrostatic pressure and $\\Delta v$ is\nthe difference between the volume of the hole in the matrix and the\nsphere of the fictitious impurity. For a screw dislocation $p=0$,\nwhile near an edge dislocation\n$p=\\frac{(1+\\nu)bG\\sin\\theta}{3\\pi(1-\\nu)r}$ for an impurity with\npolar coordinates $(r,\\theta)$ with respect to the dislocation $0z$,\nhence $w \\propto \\Delta v\\sin\\theta\/r$\n\\cite{Cottrell_Bilby_1949}. Using a nonlinear elastic theory\n\\cite{Nabarro_1987}, a screw dislocation may also interact with the\nspherical impurity with the interaction energy $w \\propto \\Delta\nv\/r^2$. Moreover, accounting for the differences in the elastic\nconstants of a solute and a matrix, the solute will relieve shear\nstrain energy as well as dilatation energy, which will also interact\nwith a screw dislocation with a potential $w \\propto \\Delta v\/r^2$\n\\cite{Friedel_1967}. Indeed, Friedel \\cite{Friedel_1967} has\nformulated that by introducing a dislocation into a solid solution of\nuniform concentration $c_0$, the interaction energy between the\ndislocation and solute atoms can be written as $w \\backsimeq\nw_0(b\/\\delta)^n f(\\theta)$, where $\\delta$ is the distance between the two\ndefects, $w_0$ the binding energy when $\\delta=b$, and $f(\\theta)$\naccounts for the angular dependence of the interaction along the\ndislocation. Also, $n=1$ for size effects and $n=2$ for effects due to\ndifferences in elastic constants.  The discussed model for the\ninteraction energy between solute atoms and dislocations has been used\nto study the precipitation process on dislocations by number of\nworkers in the past \\cite{Ham_1959,Bullough_Newman_1962} and thoroughly\nreviewed in \\cite{Bullough_Newman_1970}. These studies concern\nprimarily the overall phase transformation (precipitation of a new\nphase) rather than the growth of a new phase considered in our\nnote. That is, they used different boundary conditions as compared to the\nones used here.\n\n\nLet us now link the supersaturation parameter $k$ to an experimental\nsituation. For this purpose, the values of $u_s$, i.e. the\nconcentration at the interface between the second-phase and matrix\nshould be known. The capillary effect leads to a relationship between\n$u_s$ and the equilibrium composition $u_{eq}$ (solubility line in a\nphase diagram). To obtain this relationship, we consider an incoherent\nnucleation of second-phase on a dislocation \\`a la Cahn\n\\cite{Cahn_1957}. A Burgers loop around the dislocation in the matrix\nmaterial around the incoherent second-phase (circular plate) will have\na closure mismatch equal to $b$. Following Cahn, on forming the\nincoherent plate of radius $R$, the total free energy change per unit\nlength is\n\\begin{equation}\n\\label{eqn:cahn-fe}\n\\mathcal{G}=-\\pi R^2 \\Delta g_v +2\\pi\\gamma\nR-A^\\prime\\ln(R\/r_0),\n\\end{equation}\n\\noindent\nwhere $\\Delta g_v$ is the volume free energy of\nformation, $\\gamma$ the interfacial energy and the last term is the\ndislocation energy, $A^\\prime=Gb^2\/4\\pi$ for screw dislocations,\ncf. equation (\\ref{eqn:screw}).  Setting ${\\rm d}\\mathcal{G}\/{\\rm d}R=0$, yields\n\\begin{equation}\n  \\label{eqn:crit-rad}\n  R = \\frac{\\gamma}{2\\Delta g_v}\\Big(1 \\pm \\sqrt{1-\\alpha}\\Big),\n\\end{equation}\n\\noindent\nwhere $\\alpha=2A^\\prime \\Delta g_v\/\\pi\\gamma^2$. So, if $\\alpha>1$,\nthe nucleation is barrierless, i.e., the phase transition kinetics is\nonly governed by growth kinetics, which is the subject of our\ninvestigation here. If, however, $\\alpha < 1$, there is an energy\nbarrier and the local minimum of $\\mathcal{G}$ at $R=R_0$, which\ncorresponds to the negative sign in equation (\\ref{eqn:crit-rad}),\nensued by a maximum at $R=R^\\ast$ corresponding to the positive sign\nin this equation. The local minimum corresponds to a subcritcal\nmetastable particle of the second-phase surrounding the dislocation\nline, and it is similar to the Cottrell atmosphere of solute atoms in\na segregation problem. When $\\alpha = 0$, corresponding to $B=0$, the\ntwo phases are in equilibrium and the maximum in $\\mathcal{G}$ is\ninfinite, as for homogeneous nucleation.\n\n\nFor a dilute regular solution, $\\Delta\ng_v=(k_BT\/V_p)\\ln(u_s\/u_{eq})$, where $V_p$ is the atomic volume of\nthe precipitate compound, $u_s$ is the concentration of the matrix at\na curved particle\/matrix interface and $u_{eq}$ that of a flat\ninterface, which is in equilibrium with the solute concentration in\nthe matrix. Equation (\\ref{eqn:crit-rad}) gives\n$\\Delta g_v=\\gamma\/R-A^\\prime\/2\\pi R^2$. Hence, for a dilute regular\nsolution, we write\n\\begin{equation}\n  \\label{eqn:gibbs-thom}\n u_s=u_{eq}\\exp{\\Big[\\frac{\\zeta}{R}\\Big(1-\\frac{\\eta}{R}\\Big)\\Big]},\n\\end{equation}\n\\noindent\nwhere $\\zeta = \\beta V_p\\gamma$, $\\beta=1\/k_BT$ and $\\eta = A^\\prime\/2\\pi\\gamma$. Subsequently, the\nsupersaturation parameter is expressed by\n\\begin{equation}\n  \\label{eqn:supsat}\n  k = \\frac{u_{eq}\\exp[{\\frac{\\zeta}{R}(1-\\frac{\\eta}{R})}]-u_m}{u_p-u_{eq}\\exp[\\frac{\\zeta}{R}(1-\\frac{\\eta}{R})]}.\n\\end{equation}\n\\noindent\nTaking the following typical values: $\\gamma=0.2$ Jm$^{-2}$, $G=40$\nGPa, and $b=0.25$ nm, then $A^\\prime \\approx 2.0\\times10^{-10}$ N and\n$\\eta=0.16$ nm. Figure \\ref{fig:thoms-2d} depicts $u_s\/u_{eq}$, from\nequation (\\ref{eqn:gibbs-thom}), as a function of scaled radius\n$R\/\\zeta$ for $V_p=1.66 \\times 10^{-29}$ m$^3$, $\\eta=0$ and\n$\\eta=0.16$ nm at $T=600$ K. Equation\n(\\ref{eqn:gibbs-thom}) is analogous to the Gibbs-Thomson-Freundlich\nrelationship \\cite{Christian_2002} comprising a dislocation defect.\n\n\nRecalling now the values used for the interaction parameter $B$ in the\ncomputations presented in the foregoing section, we note that for\n$B=2$ and the above numerical values for $G$ and $b$ at  $T=1000$ K, we\nfind $l\\approx 0.14$ nm, which is close to the calculated value of\n$\\eta$.\n\nIn Cahn's model, the assumption that all the strain energy of the\ndislocation within the volume occupied by the nucleus can be relaxed\nto zero demands that the nucleus is incoherent. For a coherent nucleus\nforming on or in proximity of dislocations, this supposition is not\ntrue. Instead, it is necessary to calculate the elastic interaction\nenergy between the nucleus and the matrix, which for an edge\ndislocation is in the form $Gb^2\/[4\\pi(1-\\nu)r]$ for the energy\ndensity per unit length \\cite{Barnett_1971}. In the same manner, to\nextend our calculations for growth of coherent precipitate, we must\nemploy this kind of potential energy, i.e. the potential energy of the\nform $\\phi(r)=-A\\ln(r\/r_0)+C \\sin\\theta\/r$, in the governing kinetic\nequation rather than relation (\\ref{eqn:screw}).\n\\begin{figure}[htbp]    \n\\begin{center}\n\\includegraphics[width=0.80\\textwidth]{fig\/thoms_2d}{} \\\\\n\\caption{The size dependence of the concentration at the curved\nprecipitate\/matrix interface $u_s$ relative to that of the flat\ninterface $u_{eq}$ for a set of parameter values given in the\ntext, cf. eq. (\\ref{eqn:gibbs-thom}).}\n\\label{fig:thoms-2d}\n\\end{center}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\n \n This paper presents a novel unsupervised representation learning framework for complex multivariate time series, called Temporal Neighborhood Coding (TNC). This framework is designed to learn the underlying dynamics of non-stationary signals and to model the progression over time by defining a temporal neighborhood. The problem is motivated by the medical field, where patients transition between distinct clinical states over time, and obtaining labels to define these underlying states is challenging. We evaluate the performance of TNC on multiple datasets and show that our representations are generalizable and can easily be used for diverse tasks such as classification and clustering. \n We finally note that TNC is flexible to be used with arbitrary encoder architectures; therefore, the framework is applicable to many time series data domains. Moreover, in addition to tasks presented in this paper, general representations can be used for several other downstream tasks, such as anomaly detection, which is challenging in supervised learning settings for time series data in sparsely labeled contexts.\n\n\n\n\n\\section{Results}\nIn this section we present the results for clusterability of the latent representations and downstream classification performance for all datasets and across all baselines. Clusterability indicates how well each method recovers appropriate states, and classification assesses how  informative our representations are for downstream tasks. \n\n\\subsection{Evaluation: Clusterability}\nMany real-world time series data have underlying multi-category structure, naturally leading to representations with clustering properties. Encoding such general priors is a property of a good representation \\citep{bengio2013representation}.\nIn this section, we assess the distribution of the representations in the encoding space. If information of the latent state is properly learned and encoded by the framework, the representations of signals from the same underlying state should cluster together. Figures \\ref{fig:tcl_dist}, \\ref{fig:trip_dist}, and \\ref{fig:cpc_dist} show an example of this distribution for simulated data across compared approaches. Each plot is a 2-dimensional t-SNE visualization of the representations where each data point in the scatter plot is an encoding $Z \\in R^{10}$ that represents a window of size $\\delta=50$ of a simulated time series. \nWe can see that without any information about the hidden states, representations learned using TNC cluster windows from the same hidden state better than the alternative approaches. The results show that CPC and Triplet Loss have difficulty separating time series that are generated from non-linear auto-regressive moving average (NARMA) models with variable regression parameters.\n\n\\begin{figure}[!h]\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[scale=0.32]{TCL\/figures\/encoding_distribution.pdf}\n\\captionof{figure}{TNC representations}\n\\label{fig:tcl_dist}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[scale=0.32]{TCL\/figures\/encoding_distribution_trip.pdf}\n\\captionof{figure}{T-loss representations}\n\\label{fig:trip_dist}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n    \\centering\n\\includegraphics[scale=.32]{TCL\/figures\/encoding_distribution_cpc.pdf}\n\\caption{CPC representations}\n\\label{fig:cpc_dist}\n\\end{subfigure}\n\\label{fig:distribution}\n\\caption{T-SNE visualization of signal representations for the simulated dataset across all baselines. Each data point in the plot presents a 10-dimensional representation of a window of time series of size $\\delta=50$, and the color indicates the latent state of the signal window. See Appendix \\ref{app:plots} for similar plots from different datasets.}\n\\end{figure}\n\n\nTo compare the representation clusters' consistency for each baseline, we use two very common cluster validity indices, namely, the Silhouette score and the Davies-Bouldin index. We use K-means clustering in the representation space to measure these clusterability scores. The Silhouette score measures the similarity of each sample to its own cluster, compared to other clusters. The values can range from $-1$ to $+1$, and a greater score implies a better cohesion. The Davies-Bouldin Index measures intra-cluster similarity and inter-cluster differences. This is a positive index score, where smaller values indicate low within-cluster scatter and large separation between clusters. Therefore, a lower score represents better clusterability (more details on the cluster validity scores and how they are calculated can be found in Appendix \\ref{app:clustering_metrics}). \n\n\\begin{table}[!h]\n\\begin{tabular}{lcccccc}\n   &\\multicolumn{2}{c}{Simulation} & \\multicolumn{2}{c}{ECG Waveform} & \\multicolumn{2}{c}{HAR}\\\\\n    \\toprule\n    Method & Silhouette $\\uparrow$ & DBI $\\downarrow$ & Silhouette $\\uparrow$ & DBI $\\downarrow$ & Silhouette $\\uparrow$ & DBI $\\downarrow$\\\\\n    \\midrule\n     \\bf{TNC}   & {\\bf{0.71$\\pm$0.01}} & {\\bf{0.36$\\pm$0.01}} & {\\bf{0.44$\\pm$0.02}} & {\\bf{0.74$\\pm$0.04}} & {\\bf{0.61$\\pm$0.02}} & {\\bf{0.52$\\pm$0.04}} \\\\\n     CPC            & 0.51$\\pm$0.03 & 0.84$\\pm$0.06 & 0.26$\\pm$0.02 & 1.44$\\pm$0.04 & 0.58$\\pm$0.02 & 0.57$\\pm$0.05\\\\\n     T-Loss   & 0.61$\\pm$0.08 & 0.64$\\pm$0.12 & 0.25$\\pm$0.01 &  1.30$\\pm$0.03 & 0.17$\\pm$0.01 &  1.76$\\pm$0.20\\\\\n     \\midrule\n     K-means & 0.01$\\pm$0.019 & 7.23$\\pm$0.14 & 0.19$\\pm$0.11 & 3.65$\\pm$0.48 & 0.12$\\pm$0.40 & 2.66$\\pm$0.05\\\\\n     \\bottomrule\n    \\end{tabular}\n     \\caption{Clustering quality of representations in the encoding space for multiple datasets. }\n    \\label{tab:sil_score}\n\\end{table}\n\n\nTable \\ref{tab:sil_score} summarizes the scores for all baselines and across all datasets, demonstrating that TNC is superior in learning representations that can distinguish the latent dynamics of time series. CPC performs closely to Triplet loss on waveform data but performs poorly on the simulated dataset, where signals are highly non-stationary, and transitions are less predictable. However, for the HAR dataset, CPC clusters the states very well because most activities are recorded in a specific order, empowering predictive coding. Triplet loss performs reasonably well in the simulated setting; however, it fails to distinguish states $0$ and $2$, where signals come from autoregressive models with different parameters and have a relatively similar generative process. Performing K-means on the original time series generally does not generate coherent clusters, as demonstrated by the scores. However, the performance is slightly better in time series like the ECG waveforms, where the signals are formed by consistent shapelets, and therefore the DTW measures similarity more accurately.\n\n\n\n\\subsection{Evaluation: Classification}\nWe further evaluate the quality of the encodings using a classification task. We train a linear classifier to evaluate how well the representations can be used to classify hidden states. The performance of all baselines is compared to a supervised classifier, composed of an encoder and a classifier with identical architectures to that of the unsupervised models, and a K-nearest neighbor classifier that uses DTW metric. The performance is reported as the prediction accuracy and the area under the precision-recall curve (AUPRC) score since AUPRC is a more accurate reflection of model performance for imbalance classification settings like the waveform dataset.\n\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{lcccccc}\n         &\\multicolumn{2}{c}{Simulation}& \\multicolumn{2}{c}{ECG Waveform}& \\multicolumn{2}{c}{HAR}\\\\\n         \\toprule\n         Method & AUPRC & Accuracy & AUPRC & Accuracy & AUPRC & Accuracy\\\\\n         \\midrule\n        \\bf{TNC}& {\\bf{0.99$\\pm$0.00}} & {\\bf{97.52$\\pm$0.13}} & {\\bf{0.55$\\pm$0.01}} & {\\bf{77.79$\\pm$0.84}} & {\\bf{0.94$\\pm$0.007}} & {\\bf{88.32$\\pm$0.12}}\\\\\n         CPC & 0.69$\\pm$0.06 & 70.26$\\pm$6.48 & 0.42$\\pm$0.01 & 68.64$\\pm$0.49 & 0.93$\\pm$0.006 & 86.43$\\pm$1.41\\\\\n         T-Loss & 0.78$\\pm$0.01 & 76.66$\\pm$1.40 & 0.47$\\pm$0.00 & 75.51$\\pm$1.26 & 0.71$\\pm$0.007 & 63.60$\\pm$3.37   \\\\\n         \\midrule\n         KNN & 0.42$\\pm$0.00 & 55.53$\\pm$0.65 & 0.38$\\pm$0.06 & 54.76$\\pm$5.46 & 0.75$\\pm$0.01& 84.85$\\pm$0.84\\\\\n         \\midrule\n         Supervised&  {\\bf{0.99$\\pm$0.00}} & {\\bf{98.56$\\pm$0.13}} & {\\bf{0.67$\\pm$0.01}} & {\\bf{94.81$\\pm$0.28}} & {\\bf{0.98$\\pm$0.00}} & {\\bf{92.03$\\pm$2.48}}\\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Performance of all baselines in classifying the underlying hidden states of the time series, measured as the accuracy and AUPRC score. }\n    \\label{tab:classification_result}\n\\end{table}{}\n\n\n\nTable \\ref{tab:classification_result} demonstrates the classification performance for all datasets. The performance of the classifiers that use TNC representations are closer to the end-to-end supervised model in comparison to CPC and Triplet Loss. This provides further evidence that our encodings capture informative parts of the time series and are generalizable to be used for downstream tasks. In datasets like the HAR, where an inherent ordering usually exists in the time series, CPC performs reasonably. However, in datasets with increased non-stationarity, the performance drops. Triplet Loss is also a powerful framework, but since it samples positive examples from overlapping windows of time series, it is vulnerable to map the overlaps into the encoding and, therefore, fail to learn more general representations. TNC, on the other hand, samples similar windows from a wider distribution, defined by the temporal neighborhood, where many of the neighboring signals do not necessarily overlap. \nThe lower performance of the CPC and Triplet Loss methods can also be partly because none of these methods explicitly account for the potential sampling bias that happens when randomly selected negative examples are similar to the reference $W_t$.\n\n\n\n\n\\subsection{Evaluation: Trajectory}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[scale=0.2]{TCL\/figures\/embedding_trajectory_hm.pdf}\n    \\caption{Trajectory of a signal encoding from the simulated dataset. The top plot shows the original time series with shaded regions indicating the underlying state. The bottom plot shows the 10 dimensional encoding of the sliding windows $W_t$ where $\\delta=50$. }\n    \\label{fig:trajectory}\n\\end{figure}\n\n\nThis section investigates the trajectories of our learned encodings over time to understand how the state transitions are captured and modeled in the representation space. This is an important property for non-stationary time series where underlying states change over time, and capturing those changes is critical in many application domains such as healthcare. Figure \\ref{fig:trajectory} shows a sample from the simulated dataset. The top panel shows the signal measurements over time, and the shaded regions indicate the underlying latent states. The bottom panel illustrates the $10$-dimensional representation of a sliding window $W_t$ estimated over time. From the bottom panel of Figure \\ref{fig:trajectory}, we can see that the encoding pattern changes at state transitions and settle into a different pattern, corresponding to the new state. This change happens at every transition, and we can see the distinct patterns for all $4$ underlying states in the representations. This analysis of the trajectory of change could be very informative for the users' post-analysis; for instance, in clinical applications, it could help clinicians visualize the evolution of the patient state over time and plan treatment based on the state progression\n\n\n\\section{Related work}\nWhile integral for many applications, unsupervised representation learning has been far less studied for time series \\citep{langkvist2014review}, compared to other domains such as vision or natural language processing \\citep{denton2017unsupervised, radford2015unsupervised, gutmann2012noise, wang2015unsupervised}. One of the earliest approaches to unsupervised end-to-end representation learning in time series is the use of auto-encoders \\citep{choi2016multi, amiriparian2017sequence, malhotra2017timenet} and seq-to-seq models \\citep{lyu2018improving}, with the objective to train an encoder jointly with a decoder that reconstructs the input signal from its learned representation. Using fully generative models like variational auto-encoders is also useful for imposing properties like disentanglement, which help with the interpretability of the representations \\citep{dezfouli2019disentangled}. However, in many cases, like for high-frequency physiological signals, the reconstruction of complex time series can be challenging; therefore, more novel approaches are designed to avoid this step. Contrastive Predictive Coding \\citep{oord2018representation, lowe2019putting} learns representations by predicting the future in latent space, eliminating the need to reconstruct the full input. The representations are such that the mutual information between the original signal and the concept vector is maximally preserve using a lower bound approximation and a contrastive loss. Very similarly, in Time Contrastive Learning \\citep{hyvarinen2016unsupervised}, a contrastive loss is used to predict the segment-ID of multivariate time-series as a way to extract representation.  \\cite{franceschi2019unsupervised} employs time-based negative sampling and a triplet loss to learn scalable representations for multivariate time series. Some other approaches use inherent similarities in temporal data to learn representations without supervision. For instance, in similarity-preserving representation learning \\citep{lei2017similarity}, learned encodings are constrained to preserve the pairwise similarities that exist in the time domain, measured by DTW distance. Another group of approaches combines reconstruction loss with clustering objectives to cluster similar temporal patterns in the encoding space \\citep{ma2019learning}.\n\nIn healthcare, learning representation of rich temporal medical data is extremely important for understanding patients' underlying health conditions. However, most of the existing approaches for learning representations are designed for specific downstream tasks and\nrequire labeling by experts \\citep{choi2016medical, choi2016learning, tonekaboni2020went}. Examples of similar works to representation learning in the field of clinical ML include computational phenotyping for discovering subgroups of patients with similar underlying disease mechanisms from temporal clinical data \\citep{lasko2013computational, suresh2018learning, schulam2015clustering}, and disease progression modeling, for learning the hidden vector of comorbidities representing a disease over time \\citep{wang2014unsupervised, alaa2018forecasting}.\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nReal-world time-series data is high dimensional, complex, and has unique properties that bring about many challenges for data modeling \\citep{yang200610}. In addition, these signals are often sparsely labeled, making it even more challenging for supervised learning tasks. Unsupervised representation learning can extract informative low-dimensional representations from raw time series by leveraging the data's inherent structure, without the need for explicit supervision. These representations are more generalizable and robust, as they are less specialized for solving a single supervised task. Unsupervised representation learning is well studied in domains such as vision \\citep{donahue2019large, denton2017unsupervised, radford2015unsupervised} and natural language processing \\citep{radford2017learning, young2018recent, mikolov2013efficient}, but has been underexplored in the literature for time series settings.\nFrameworks designed for time series need to be efficient and scalable because signals encountered in practice can be long, high dimensional, and high frequency. Moreover, it should account for and be able to model dynamic changes that occur within samples, i.e., non-stationarity of signals. \n\n\nThe ability to model the dynamic nature of time series data is especially valuable in medicine. Health care data is often organized as a time series, with multiple data types, collected from various sources at different sampling frequencies, and riddled with artifacts and missing values. Throughout their stay at the hospital or within the disease progression period, patients transition gradually between distinct clinical states, with periods of relative stability, improvement, or unexpected deterioration, requiring escalation of care that alters the patient's trajectory. A particular challenge in medical time-series data is the lack of well-defined or available labels that are needed for identifying the underlying clinical state of an individual or for training models aimed at extracting low-dimensional representations of these states.\nFor instance, in the context of critical-care, a patient's stay in the critical care unit (CCU) is captured continuously via streaming physiological signals by the bedside monitor. Obtaining labels for the patient's state for extended periods of these signals is practically impossible as the underlying physiological state can be unknown even to the clinicians. This further motivates the use of unsupervised representation learning in these contexts. Learning rich representations can be crucial in facilitating the tracking of disease progression, predicting the future trajectories of the patients, and tailoring treatments to these underlying states. \n\nIn this paper, we propose a self-supervised framework for learning representations for complex multivariate non-stationary time series. This approach, called Temporal Neighborhood Coding (TNC), is designed for temporal settings where the latent distribution of the signals changes over time, and it aims to capture the progression of the underlying temporal dynamics. TNC is efficient, easily scalable to high dimensions, and can be used in different time series settings. We assess the quality of the learned representations on multiple datasets and show that the representations are general and transferable to many downstream tasks such as classification and clustering. We further demonstrate that our method outperforms existing approaches for unsupervised representation learning, and it even performs closely to supervised techniques in classification tasks. The contributions of this work are three-fold:\n\\begin{enumerate}\n    \\item We present a novel neighborhood-based unsupervised learning framework for \\emph{non-stationary} multivariate time series data.\n    \\item We introduce the concept of a temporal neighborhood with stationary properties as the distribution of similar windows in time. The neighborhood boundaries are determined automatically using the properties of the signal and statistical testing.\n    \\item We incorporate concepts from Positive Unlabeled Learning, specifically, sample weight adjustment, to account for potential bias introduced in sampling negative examples for the contrastive loss.\n\\end{enumerate}\n\n\\section{Method}\n\nWe introduce a framework for learning representations that encode the underlying state of a multivariate, non-stationary time series. Our self-supervised approach, TNC, takes advantage of the local smoothness of the generative process of signals to learn generalizable representations for windows of time series. This is done by ensuring that in the representation space, the distribution of signals proximal in time is distinguishable from the distribution of signals far away, i.e., proximity in time is identifiable in the encoding space. We represent our multivariate time series signals as ${X}\\in R^{D\\times T}$, where $D$ is the number of features and $T$ is the number of measurements over time.\n$X_{[t-\\frac{\\delta}{2}, t+\\frac{\\delta}{2}]}$ represents a window of time series of length $\\delta$, centered around time $t$, that includes measurements of all features taken in the interval $[t-\\frac{\\delta}{2}, t+\\frac{\\delta}{2}]$. Throughout the paper, we refer to this window as $W_t$ for notational simplicity. Our goal is to learn the underlying representation of $W_t$, and by sliding this window over time, we can obtain the trajectory of the underlying states of the signal.\n\nWe define the temporal neighborhood ($N_t$) of a window $W_t$ as the set of all windows with centroids $t^*$, sampled from a normal distribution $t^*\\sim\\mathcal{N}(t,\\eta\\cdot\\delta)$. Where $\\mathcal{N}$ is a Gaussian centered at $t$, $\\delta$ is the size of the window, and $\\eta$ is the parameter that defines the range of the neighborhood. Relying on the local smoothness of a signal's generative process, the neighborhood distribution is characterized as a Gaussian to model the gradual transition in temporal data, and intuitively, it approximates the distribution of samples that are similar to $W_t$. The $\\eta$ parameter determines the neighborhood range and depends on the signal characteristics and how gradual the time series's statistical properties change over time. This can be set by domain experts based on prior knowledge of the signal behavior, or for more robust estimation, it can be determined by analyzing the stationarity properties of the signal for every $W_t$. Since the neighborhood represents similar samples, the range should identify the approximate time span within which the signal remains stationary, and the generative process does not change. For this purpose, we use the Augmented Dickey-Fuller (ADF) statistical test to determine this region for every window. Proper estimation of the neighborhood range is an integral part of the TNC framework. If $\\eta$ is too small, many samples from within a neighborhood will overlap, and therefore the encoder would only learn to encode the overlapping information. On the other hand, if $\\eta$ is too big, the neighborhood would span over multiple underlying states, and therefore the encoder would fail to distinguish the variation among these states. Using the ADF test, we can automatically adjust the neighborhood for every window based on the signal behavior. More details on this test and how it is used to estimate $\\eta$ is described in section \\ref{sec:neighbourhood}.  \n\n\nNow, assuming windows within a neighborhood possess similar properties, signals outside of this neighborhood, denoted as $\\bar{N_t}$, are considered non-neighboring windows. Samples from $\\bar{N_t}$ are likely to be different from $W_t$, and can be considered as negative samples in a context of a contrastive learning framework. However, this assumption can suffer from the problem of \\emph{sampling bias}, common in most contrastive learning approaches \\citep{chuang2020debiased, saunshi2019theoretical}. This bias occurs because randomly drawing negative examples from the data distribution may result in negative samples that are actually similar to the reference. This can significantly impact the learning framework's performance, but little work has been done on addressing this issue \\citep{chuang2020debiased}. In our context, this can happen when there are windows from $\\bar{N_t}$ that are far away from $W_t$, but have the same underlying state. To alleviate this bias in the TNC framework, we consider samples from $\\bar{N_t}$ as unlabeled samples, as opposed to negative, and use ideas from Positive-Unlabeled (PU) learning to accurately measure the loss function. In reality, even though samples within a neighborhood are all similar, we cannot make the assumption that samples outside this region are necessarily different. For instance, in the presence of long term seasonalities, signals can exhibit similar properties at distant times. In a healthcare context, this can be like a stable patient that undergoes a critical condition, but returns back to a stable state afterwards. \n\nIn PU learning, a classifier is learned using labeled data drawn from the positive class ($P$) and unlabeled data ($U$) that is a mixture of positive and negative samples with a positive class prior $\\pi$ \\citep{du2014analysis, kiryo2017positive, du2014class}. Existing PU learning methods fall under two categories based on how they handle the unlabeled data:\n1) methods that identify negative samples from the unlabeled cohort \\citep{li2003learning};\n2) methods that treat the unlabeled data as negative samples with smaller weights \\citep{lee2003learning, elkan2008learning}.\nIn the second category, unlabeled samples should be properly weighted in the loss term in order to train an unbiased classifier. \\cite{elkan2008learning} introduces a simple and efficient way of approximating the expectation of a loss function by assigning individual weights $w$ to training examples from the unlabeled cohort. This means each sample from the neighborhood is treated as a positive example with unit weight, while each sample from $\\bar{N}$ is treated as a combination of a positive example with weight $w$ and a negative example with complementary weight $1-w$. In the original paper \\citep{elkan2008learning}, the weight is defined as the probability for a sample from the unlabeled set to be a positive sample, i.e. $w = p(y = 1|x)$ for $x \\in U$. In the TNC framework, this weight represents the probability of having samples similar to $W_t$ in $\\bar{N}$. By incorporating weight adjustment into the TNC loss (Equation \\ref{eq:obj}), we account for possible positive samples that occur in the non-neighboring distribution. $w$ can be approximated using prior knowledge of the underlying state distribution or tuned as a hyperparameter. Appendix \\ref{app:w} explains how the weight parameter is selected for our different experiment setups and also demonstrates the impact of weight adjustment on performance for downstream tasks.\n\nAfter defining the neighborhood distribution, we train an objective function that encourages a distinction between the representation of samples of the same neighborhood from the outside samples. An ideal encoder preserves the neighborhood properties in the encoding space. Therefore representations $Z_l=Enc(W_l)$ of samples from a neighborhood $W_l \\in N_t$, can be distinguished from representation $Z_k=Enc(W_k)$ of samples from outside the neighborhood $W_k \\in \\bar{N}_t$.\nTNC is composed of two main components:\n\n\\begin{enumerate}\n    \\item An Encoder $Enc(W_t)$ that maps $W_t\\in \\mathbb{R}^{D\\times\\delta}$ to a representation $Z_t\\in\\mathbb{R}^M$, in a lower dimensional space ($M \\ll D\\times\\delta$), where $ D\\times\\delta$ is the total number of measurements in $W_t$.\n    \\item A Discriminator $\\mathcal{D}(Z_t,Z)$ that approximates the probability of $Z$ being the representation of a window in $N_t$. More specifically, it receives two samples from the encoding space and predicts the probability of those samples belonging to the same temporal neighborhood. \n\\end{enumerate}\n\nTNC is a general framework; therefore, it is agnostic to the nature of the time series and the architecture of the encoder. The encoder can be any parametric model that is well-suited to the signal properties \\citep{oord2016wavenet, bai2018empirical, fawaz2019deep}. For the Discriminator $\\mathcal{D}(Z_t,Z)$ we use a simple multi-headed binary classifier that outputs $1$ if $Z$ and $Z_t$ are representations of neighbors in time, and $0$ otherwise. In the experiment section, we describe the architectural details of the models used for our experiments in more depth. \n\n\\begin{figure}\n    \\begin{subfigure}{.5\\textwidth}\n    \\centering\n    \\includegraphics[scale=0.4]{TCL\/figures\/tnc_explain_a.pdf} \n    \\caption{Neighborhood samples}\n  \\label{fig:explanation_a}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n    \\centering\n    \\includegraphics[scale=0.4]{TCL\/figures\/tnc_explain_b.pdf} \n    \\caption{Non-neighboring samples}\n  \\label{fig:explanation_b}\n\\end{subfigure}\n\\caption{Overview of the TNC framework components. For each sample window $W_t$ (indicated with the dashed black box), we first define the neighborhood distribution. The encoder learns the distribution of windows sampled from $N_t$ and $\\bar{N_t}$, in the representation space. Then samples from these distributions are fed into the discriminator alongside $Z_t$, to predict the probability of the windows being in the same neighborhood.}\n\\label{fig:explanation}\n\\end{figure}\n\nFigure \\ref{fig:explanation} provides a summary overview of the TNC framework. We formalize the objective function of our unsupervised learning framework in Equation \\ref{eq:obj}. In essence, we would like the probability likelihood estimation of the Discriminator to be accurate, i.e., close to $1$ for the representation of neighboring samples and close to $0$ for windows far apart. Samples from the non-neighboring region ($\\bar{N}$) are weight-adjusted using the $w$ parameters to account for positive samples in this distribution.\n\n\\begin{equation}\\label{eq:obj}\n    \\mathcal{L} = -\\mathbb{E}_{W_t \\sim X}[ \n        \\mathbb{E}_{W_l \\sim N_t}[\\log \\underbrace{\\mystrut{1.5ex}\\mathcal{D} (Z_t, Z_l)}_{\\makebox[0pt]{\\text{$\\mathcal{D}(Enc(W_t), Enc(W_l))$}}}] + \n        \\mathbb{E}_{W_k \\sim {\\bar{N}_t}} [(1-w_t) \\times \\log{\\underbrace{\\mystrut{1.5ex}(1-\\mathcal{D}(Z_t, Z_k))}_{\\makebox[0pt]{$\\mathcal{D}(Enc(W_t), Enc(W_k))$}}} + \n        w_t \\times \\log{\\mathcal{D}(Z_t, Z_k)}]\n        ]\n\\end{equation}{}\n\\vspace{-3mm}\n\n\nWe train the encoder and the discriminator hand in hand by optimizing for this objective. Note that the Discriminator is only part of training and will not be used during inference. Similar to the encoder, it can be approximated using any parametric model. However, the more complex the Discriminator, the harder it becomes to interpret the latent space's decision boundaries since it allows similarities to be mapped on complex nonlinear relationships. \n\n\\paragraph{Defining the neighborhood parameter using the ADF test:}\\label{sec:neighbourhood} \nAs mentioned earlier, the neighborhood range can be specified using the characteristics of the data. In non-stationary time series, the generative process of the signals changes over time. We define the temporal neighborhood around every window as the region where the signal is relatively stationary. Since a signal may remain in an underlying state for an unknown amount of time, each window's neighborhood range may vary in size and must be adjusted to signal behavior. \nTo that end, we use the Augmented Dickey-Fuller (ADF) statistical test to derive the neighborhood range $\\eta$. The ADF test belongs to a category of tests called \"Unit Root Test\", and is a method for testing the stationarity of a time series. For every $W_t$, we want to find the neighborhood range around that window that indicates a stationary region. To determine this, we start from $\\eta=1$ and gradually increase the neighborhood size $\\eta$, measuring the $p$-value from the test at every step. Once $p$-value is above a threshold (in our setting $0.01$), it means that it fails to reject the null hypothesis and suggests that within this neighborhood region, the signal is no longer stationary. This way, we find the widest neighborhood within which the signal remains relatively stationary. Note that the window size $\\delta$ is constant throughout the experiment, and during ADF testing, we only adjust the neighborhood's width.\n\n\n\\section{Experiments}\nWe evaluate our framework's usability on multiple time series datasets with dynamic latent states that change over time.\nWe compare classification performance and clusterability against two state-of-the-art approaches for unsupervised representation learning for time series: \n\\begin{inparaenum}\n    \\item Contrastive Predictive Coding (CPC) \\citep{oord2018representation} that uses predictive coding principles to train the encoder on a probabilistic contrastive loss.\n    \\item Triplet-Loss (T-Loss), introduced in \\citep{franceschi2019unsupervised}, which employs time-based negative sampling and a triplet loss to learn representations for time series windows. The triplet loss objective ensures similar time series have similar representations by minimizing the pairwise distance between positive samples (subseries) while maximizing it for negative ones. \n\\end{inparaenum}\n(See Appendix \\ref{app:baseline_imp} for more details on each baseline.)\n\nFor a fair comparison and to ensure that the difference in performance is not due to the differences in the models' architecture, the same encoder network is used across all compared baselines. Our objective is to compare the performance of the learning frameworks, agnostic to the encoder's choice. Therefore, we selected simple architectures to evaluate how each framework can use a simple encoder's limited capacity to learn meaningful representations.\nWe assess the generalizability of the representations by 1) evaluating clusterability in the encoding space and 2) using the representations for a downstream classification task. \nIn addition to the baselines mentioned above, we also compare clusterability performance with unsupervised K-means and classification with a K-Nearest Neighbor classifier, using Dynamic Time Warping (DTW) to measure time series distance. All models are implemented using Pytorch $1.3.1$ and trained on a machine with Quadro 400 GPU \\footnote{Code implementation can be found at \\url{https:\/\/github.com\/sanatonek\/TNC_representation_learning}}. Below we describe the datasets for our experiments in more detail.\n\n\n\\subsection{Simulated data}\nThe simulated dataset is designed to replicate very long, non-stationary, and high-frequency time series for which the underlying dynamics change over time. Our generated time series consists of $2000$ measurements for $3$ features, generated from $4$ different underlying states. We use a Hidden Markov Model (HMM) to generate the random latent states over time, and in each state, the time series is generated from a different generative process, including Gaussian Processes (GPs) with different kernel functions and Nonlinear Auto-regressive Moving Average models with different sets of parameters ($\\alpha$ and $\\beta$). Besides, for it to further resemble realistic (e.g., clinical) time series, two features are always correlated. More details about this dataset are provided in the Appendix \\ref{app:simulated_data}. \nFor this experimental setup, we use a two-directional, single-layer recurrent neural network encoder. We have selected this simple architecture because it handles time series with variable lengths, and it easily extends to higher-dimensional inputs. The encoder model encodes multi-dimensional signal windows of $\\delta=50$ into $10$ dimensional representation vectors. The window size is selected such that it is long enough to contain information about the underlying state but not too long to span over multiple underlying states. A more detailed discussion on the window size choice is presented in Appendix \\ref{app:window_size}. \n\n\n\\subsection{Clinical waveform data}\nFor a real-world clinical experiment, we use the MIT-BIH Atrial Fibrillation dataset \\citep{moody1983new}.\nThis dataset includes $25$ long-term Electrocardiogram (ECG) recordings ($10$ hours in duration) of human subjects with atrial fibrillation. It consists of two ECG signals; each sampled at 250 Hz. The signals are annotated over time for the following different rhythm types: 1) Atrial fibrillation, 2) Atrial flutter, 3) AV junctional rhythm, and 4) all other rhythms. Our goal in this experiment is to identify the underlying type of arrhythmia for each sample without any information about the labels. This dataset is particularly  interesting and makes this experiment challenging due to the following special properties:\n\n\n\\begin{itemize}\n    \\item The underlying heart rhythm changes over time in each sample. This is an opportunity to evaluate how different representation learning frameworks can handle alternating classes in non-stationary settings; \n    \\item The dataset is highly imbalanced, with atrial flutter and AV junctional rhythm being present in fewer than $0.1\\%$ of the measurements. Data imbalance poses many challenges for downstream classification, further motivating the use of unsupervised representation learning;\n    \\item The dataset has samples from a small number of individuals, but over an extended period (around 5 million data points). This realistic scenario, common in healthcare data, shows that our framework is still powerful in settings with a limited number of samples.\n\\end{itemize}\n\nThe simple RNN encoder architecture used for other experiment setups cannot model the high-frequency ECG measurements. Therefore, inspired by state-of-the-art architectures for ECG classification problems, the encoder $Enc$ used in this experiment is a 2-channel, 1-dimensional strided convolutional neural network that runs directly on the ECG waveforms. We use six convolutional layers with a total down-sampling factor of 16. The window size is $2500$ samples, meaning that each convolutional filter covers at least half a second of ECG recording, and the representations are summarized in a 64-dimensional vector.\n\n\\subsection{Human Activity Recognition (HAR) data}\nHuman Activity Recognition (HAR) is the problem of predicting the type of activity using temporal data from accelerometer and gyroscope measurements. We use the HAR dataset from the UCI Machine Learning Repository \\footnote{\\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/human+activity+recognition+using+smartphones}} that includes data collected from 30 individuals using a smartwatch. Each person performs six activities: 1) walking, 2) walking upstairs, 3) walking downstairs, 4) sitting, 5) standing, and 6) laying. The time-series measurements are pre-processed to extract 561 features. For our purpose, we concatenate the activity samples from every individual over time using the subject identifier to build the full-time series for each subject, which includes continuous activity change. Similar to the simulated data setting, we use a single-layer RNN encoder. The selected window size is $4$, representing about 15 seconds of recording, and the representations are encoded in a $10$-dimensional vector space. \n\\section{Submission of conference papers to ICLR 2021}\n\nICLR requires electronic submissions, processed by\n\\url{https:\/\/openreview.net\/}. See ICLR's website for more instructions.\n\nIf your paper is ultimately accepted, the statement {\\tt\n  {\\textbackslash}iclrfinalcopy} should be inserted to adjust the\nformat to the camera ready requirements.\n\nThe format for the submissions is a variant of the NeurIPS format.\nPlease read carefully the instructions below, and follow them\nfaithfully.\n\n\\subsection{Style}\n\nPapers to be submitted to ICLR 2021 must be prepared according to the\ninstructions presented here.\n\n\nAuthors are required to use the ICLR \\LaTeX{} style files obtainable at the\nICLR website. Please make sure you use the current files and\nnot previous versions. 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Lines should be dark\nenough for purposes of reproduction; art work should not be\nhand-drawn. The figure number and caption always appear after the\nfigure. Place one line space before the figure caption, and one line\nspace after the figure. The figure caption is lower case (except for\nfirst word and proper nouns); figures are numbered consecutively.\n\nMake sure the figure caption does not get separated from the figure.\nLeave sufficient space to avoid splitting the figure and figure caption.\n\nYou may use color figures.\nHowever, it is best for the\nfigure captions and the paper body to make sense if the paper is printed\neither in black\/white or in color.\n\\begin{figure}[h]\n\\begin{center}\n\\fbox{\\rule[-.5cm]{0cm}{4cm} \\rule[-.5cm]{4cm}{0cm}}\n\\end{center}\n\\caption{Sample figure caption.}\n\\end{figure}\n\n\\subsection{Tables}\n\nAll tables must be centered, neat, clean and legible. Do not use hand-drawn\ntables. The table number and title always appear before the table. See\nTable~\\ref{sample-table}.\n\nPlace one line space before the table title, one line space after the table\ntitle, and one line space after the table. The table title must be lower case\n(except for first word and proper nouns); tables are numbered consecutively.\n\n\\begin{table}[t]\n\\caption{Sample table title}\n\\label{sample-table}\n\\begin{center}\n\\begin{tabular}{ll}\n\\multicolumn{1}{c}{\\bf PART}  &\\multicolumn{1}{c}{\\bf DESCRIPTION}\n\\\\ \\hline \\\\\nDendrite         &Input terminal \\\\\nAxon             &Output terminal \\\\\nSoma             &Cell body (contains cell nucleus) \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Default Notation}\n\nIn an attempt to encourage standardized notation, we have included the\nnotation file from the textbook, \\textit{Deep Learning}\n\\cite{goodfellow2016deep} available at\n\\url{https:\/\/github.com\/goodfeli\/dlbook_notation\/}.  Use of this style\nis not required and can be disabled by commenting out\n\\texttt{math\\_commands.tex}.\n\n\n\\centerline{\\bf Numbers and Arrays}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1in}p{3.25in}}\n$\\displaystyle a$ & A scalar (integer or real)\\\\\n$\\displaystyle {\\bm{a}}$ & A vector\\\\\n$\\displaystyle {\\bm{A}}$ & A matrix\\\\\n$\\displaystyle {\\tens{A}}$ & A tensor\\\\\n$\\displaystyle {\\bm{I}}_n$ & Identity matrix with $n$ rows and $n$ columns\\\\\n$\\displaystyle {\\bm{I}}$ & Identity matrix with dimensionality implied by context\\\\\n$\\displaystyle {\\bm{e}}^{(i)}$ & Standard basis vector $[0,\\dots,0,1,0,\\dots,0]$ with a 1 at position $i$\\\\\n$\\displaystyle \\text{diag}({\\bm{a}})$ & A square, diagonal matrix with diagonal entries given by ${\\bm{a}}$\\\\\n$\\displaystyle {\\textnormal{a}}$ & A scalar random variable\\\\\n$\\displaystyle {\\mathbf{a}}$ & A vector-valued random variable\\\\\n$\\displaystyle {\\mathbf{A}}$ & A matrix-valued random variable\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Sets and Graphs}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {\\mathbb{A}}$ & A set\\\\\n$\\displaystyle \\mathbb{R}$ & The set of real numbers \\\\\n$\\displaystyle \\{0, 1\\}$ & The set containing 0 and 1 \\\\\n$\\displaystyle \\{0, 1, \\dots, n \\}$ & The set of all integers between $0$ and $n$\\\\\n$\\displaystyle [a, b]$ & The real interval including $a$ and $b$\\\\\n$\\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$\\\\\n$\\displaystyle {\\mathbb{A}} \\backslash {\\mathbb{B}}$ & Set subtraction, i.e., the set containing the elements of ${\\mathbb{A}}$ that are not in ${\\mathbb{B}}$\\\\\n$\\displaystyle {\\mathcal{G}}$ & A graph\\\\\n$\\displaystyle \\parents_{\\mathcal{G}}({\\textnormal{x}}_i)$ & The parents of ${\\textnormal{x}}_i$ in ${\\mathcal{G}}$\n\\end{tabular}\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Indexing}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {a}_i$ & Element $i$ of vector ${\\bm{a}}$, with indexing starting at 1 \\\\\n$\\displaystyle {a}_{-i}$ & All elements of vector ${\\bm{a}}$ except for element $i$ \\\\\n$\\displaystyle {A}_{i,j}$ & Element $i, j$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{i, :}$ & Row $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{:, i}$ & Column $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\etens{A}}_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor ${\\tens{A}}$\\\\\n$\\displaystyle {\\tens{A}}_{:, :, i}$ & 2-D slice of a 3-D tensor\\\\\n$\\displaystyle {\\textnormal{a}}_i$ & Element $i$ of the random vector ${\\mathbf{a}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Calculus}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle\\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$\\\\ [2ex]\n$\\displaystyle \\frac{\\partial y} {\\partial x} $ & Partial derivative of $y$ with respect to $x$ \\\\\n$\\displaystyle \\nabla_{\\bm{x}} y $ & Gradient of $y$ with respect to ${\\bm{x}}$ \\\\\n$\\displaystyle \\nabla_{\\bm{X}} y $ & Matrix derivatives of $y$ with respect to ${\\bm{X}}$ \\\\\n$\\displaystyle \\nabla_{\\tens{X}} y $ & Tensor containing derivatives of $y$ with respect to ${\\tens{X}}$ \\\\\n$\\displaystyle \\frac{\\partial f}{\\partial {\\bm{x}}} $ & Jacobian matrix ${\\bm{J}} \\in \\mathbb{R}^{m\\times n}$ of $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$\\\\\n$\\displaystyle \\nabla_{\\bm{x}}^2 f({\\bm{x}})\\text{ or }{\\bm{H}}( f)({\\bm{x}})$ & The Hessian matrix of $f$ at input point ${\\bm{x}}$\\\\\n$\\displaystyle \\int f({\\bm{x}}) d{\\bm{x}} $ & Definite integral over the entire domain of ${\\bm{x}}$ \\\\\n$\\displaystyle \\int_{\\mathbb{S}} f({\\bm{x}}) d{\\bm{x}}$ & Definite integral with respect to ${\\bm{x}}$ over the set ${\\mathbb{S}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Probability and Information Theory}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle P({\\textnormal{a}})$ & A probability distribution over a discrete variable\\\\\n$\\displaystyle p({\\textnormal{a}})$ & A probability distribution over a continuous variable, or over\na variable whose type has not been specified\\\\\n$\\displaystyle {\\textnormal{a}} \\sim P$ & Random variable ${\\textnormal{a}}$ has distribution $P$\\\\% so thing on left of \\sim should always be a random variable, with name beginning with \\r\n$\\displaystyle  \\mathbb{E}_{{\\textnormal{x}}\\sim P} [ f(x) ]\\text{ or } \\mathbb{E} f(x)$ & Expectation of $f(x)$ with respect to $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Var}(f(x)) $ &  Variance of $f(x)$ under $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Cov}(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P({\\textnormal{x}})$\\\\\n$\\displaystyle H({\\textnormal{x}}) $ & Shannon entropy of the random variable ${\\textnormal{x}}$\\\\\n$\\displaystyle D_{\\mathrm{KL}} ( P \\Vert Q ) $ & Kullback-Leibler divergence of P and Q \\\\\n$\\displaystyle \\mathcal{N} ( {\\bm{x}} ; {\\bm{\\mu}} , {\\bm{\\Sigma}})$ & Gaussian distribution %\nover ${\\bm{x}}$ with mean ${\\bm{\\mu}}$ and covariance ${\\bm{\\Sigma}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Functions}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle f: {\\mathbb{A}} \\rightarrow {\\mathbb{B}}$ & The function $f$ with domain ${\\mathbb{A}}$ and range ${\\mathbb{B}}$\\\\\n$\\displaystyle f \\circ g $ & Composition of the functions $f$ and $g$ \\\\\n  $\\displaystyle f({\\bm{x}} ; {\\bm{\\theta}}) $ & A function of ${\\bm{x}}$ parametrized by ${\\bm{\\theta}}$.\n  (Sometimes we write $f({\\bm{x}})$ and omit the argument ${\\bm{\\theta}}$ to lighten notation) \\\\\n$\\displaystyle \\log x$ & Natural logarithm of $x$ \\\\\n$\\displaystyle \\sigma(x)$ & Logistic sigmoid, $\\displaystyle \\frac{1} {1 + \\exp(-x)}$ \\\\\n$\\displaystyle \\zeta(x)$ & Softplus, $\\log(1 + \\exp(x))$ \\\\\n$\\displaystyle || {\\bm{x}} ||_p $ & $L^p$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle || {\\bm{x}} || $ & $L^2$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle x^+$ & Positive part of $x$, i.e., $\\max(0,x)$\\\\\n$\\displaystyle \\bm{1}_\\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\n\\section{Final instructions}\nDo not change any aspects of the formatting parameters in the style files.\nIn particular, do not modify the width or length of the rectangle the text\nshould fit into, and do not change font sizes (except perhaps in the\n\\textsc{References} section; see below). Please note that pages should be\nnumbered.\n\n\\section{Preparing PostScript or PDF files}\n\nPlease prepare PostScript or PDF files with paper size ``US Letter'', and\nnot, for example, ``A4''. The -t\nletter option on dvips will produce US Letter files.\n\nConsider directly generating PDF files using \\verb+pdflatex+\n(especially if you are a MiKTeX user).\nPDF figures must be substituted for EPS figures, however.\n\nOtherwise, please generate your PostScript and PDF files with the following commands:\n\\begin{verbatim}\ndvips mypaper.dvi -t letter -Ppdf -G0 -o mypaper.ps\nps2pdf mypaper.ps mypaper.pdf\n\\end{verbatim}\n\n\\subsection{Margins in LaTeX}\n\nMost of the margin problems come from figures positioned by hand using\n\\verb+\\special+ or other commands. We suggest using the command\n\\verb+\\includegraphics+\nfrom the graphicx package. Always specify the figure width as a multiple of\nthe line width as in the example below using .eps graphics\n\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.eps}\n\\end{verbatim}\nor\n\\begin{verbatim}\n   \\usepackage[pdftex]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.pdf}\n\\end{verbatim}\nfor .pdf graphics.\nSee section~4.4 in the graphics bundle documentation (\\url{http:\/\/www.ctan.org\/tex-archive\/macros\/latex\/required\/graphics\/grfguide.ps})\n\nA number of width problems arise when LaTeX cannot properly hyphenate a\nline. Please give LaTeX hyphenation hints using the \\verb+\\-+ command.\n\n\\subsubsection*{Author Contributions}\nIf you'd like to, you may include  a section for author contributions as is done\nin many journals. This is optional and at the discretion of the authors.\n\n\\subsubsection*{Acknowledgments}\nUse unnumbered third level headings for the acknowledgments. All\nacknowledgments, including those to funding agencies, go at the end of the paper.\n\n\n\n\n\\section{Appendix}\n\n\n\\subsection{Simulated Dataset}\\label{app:simulated_data}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[scale=0.42]{TCL\/figures\/simulation_sample.pdf}\n    \\caption{A normalized time series sample from the simulated dataset. Each row represents a single feature, and the shaded regions indicate one of the $4$ underllying simulated states.}\n    \\label{fig:sim_data_example}\n\\end{figure}{}\n\nThe simulated time series consists of $3$ features generated from different underlying hidden states. Figure \\ref{fig:sim_data_example} shows a sample from this dataset. Each panel in the figure shows one of the features, and the shaded regions indicate the underlying state of the signal in that period. We use a Hidden Markov Model (HMM) to generate these random latent states over time. The transition probability is set equal to \\%5 for switching to an alternating state, and \\%85 for not changing state. In each state, the time series is generated from a different signal distribution. Table \\ref{tab:state_signal_distribution} describes the generative process of each signal feature in each state. Note that feature $1$ and $2$ are always correlated, mainly to mimic realistic clinical time series. As an example, physiological measurements like pulse rate and heart rate are always correlated. \n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{lllll}\n         \\toprule\n          & State 1 &  State 2 &  State 3 &  State 4\\\\\n          \\midrule\n         Feature 1& GP (periodic) & NARMA$_\\alpha$ &  GP (Squared Exp.) & NARMA$_\\beta$ \\\\\n         \n         Feature 2 & GP (periodic) &  NARMA$_\\alpha$ &  GP (Squared Exp.) & NARMA$_\\beta$ \\\\\n         \n         Feature 3& GP (Squared Exp.) & NARMA$_\\beta$ & GP (periodic) & NARMA$_\\alpha$ \\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Signal distributions for each time series feature of the simulated dataset}\n    \\label{tab:state_signal_distribution}\n\\end{table}{}\n\nIn the state $1$, the correlated features are generated by a Gaussian Process (GP) with a periodic kernel. Feature 3, which is uncorrelated with the other two features, comes from another GP with a squared exponential kernel. In addition to GPs, we also have multiple Non-Linear Auto-Regressive Moving Average (NARMA) time series models. The linear function of NARMA$_\\alpha$ and NARMA$_\\beta$ are shown in Equation \\ref{eq:narma_alpha} and \\ref{eq:narma_beta}.\n\n\\begin{equation}\n\\text{NARMA}_\\alpha: y(k+1) = 0.3 y(k) + 0.05 y(k) \\sum_{i=0}^{n-1} y(k-i) + 1.5 u(k-(n-1)) u(k) + 0.1\n    \\label{eq:narma_alpha}\n\\end{equation}\n\n\\begin{equation}\n\\text{NARMA}_\\beta: y(k+1) = 0.1 y(k) + 0.25  y(k)  \\sum_{i=0}^{n-1} y(k-i) + 2.5  u(k-(n-1))  u(k) -0.005\n    \\label{eq:narma_beta}\n\\end{equation}\n\nA white Gaussian noise with $\\sigma=0.3$ is added to all signals, and overall, the dataset consists of 500 instances of $T=2000$ measurements.\n\n\n\n\\subsection{Baseline implementation details}\\label{app:baseline_imp}\nImplementation of all baselines are included in the code base for reproducibility purposes, and hyper-parameters for all baselines are tuned using cross-validation.\n\n\n\\paragraph{Contrastive Predictive Coding (CPC):} The CPC baseline first processes the sequential signal windows using an encoder $Z_t = Enc(X_t)$, with a similar architecture to the encoders of other baselines. Next, an autoregressive model $g_{ar}$ aggregates all the information in $Z_{\\leq t}$ and summarizes it into a context latent representation $c_t = g_{ar}(Z_{\\leq t})$. In our implementation, we have used a single layer, a one-directional recurrent neural network with GRU cell and hidden size equal to the encoding size as the auto-regressor. Like the original paper, the density ratio is estimated using a linear transformation, and the model is trained for $1$ step ahead prediction.\n\\vspace{-2mm}\n\\paragraph{Triplet-Loss (T-Loss):} The triplet loss baseline is implemented using the original code made available by the authors on Github\\footnote{\\url{https:\/\/github.com\/White-Link\/UnsupervisedScalableRepresentationLearningTimeSeries}}. \n\\vspace{-2mm}\n\\paragraph{KNN and K-means:} These two baselines for classification and clustering are implemented using the tslearn library \\footnote{\\url{https:\/\/tslearn.readthedocs.io\/en\/stable\/index.html}}, that integrates distance metrics such as DTW. Note that evaluating DTW is computationally expensive, and the tslearn implementation is not optimized. Therefore, for the waveform data with windows of size 2500, we had to down-sampled the signal frequency by a factor of two. \n\n\n\\subsection{TNC implementation extra details}\\label{app:TNC_imp}\nTo define the neighborhood range in the TNC framework, as mentioned earlier, we use the Augmented-Dickey Fuller (ADF) statistical test to determine this range ($\\eta$) as the region for which the signals remain stationary. More precisely, we gradually increase the range, from a single window size up to 3 times the window size (the upper limit we set), and repeatedly perform the ADF test. We use the $p$-value from this statistical test to determine whether the Null hypothesis can be rejected, meaning that the signal is stationary. At the point where the $p$-value is above our defined threshold ($0.01$), we can no longer assume that the signal is stationary, and this is where we set the $\\eta$ parameter. Now, once the neighborhood is defined, we make sure the non-neighboring samples are taken from the distribution of windows with at least $4\\times \\eta$ away from $W_t$, ensuring a low likelihood of belonging to the neighborhood. Note that for implementation of ADF, we use the stats model library \\footnote{\\url{https:\/\/www.statsmodels.org\/dev\/generated\/statsmodels.tsa.stattools.adfuller.html}}. Unfortunately, this implementation is not optimized and does not support GPU computation, therefore evaluating the neighborhood range using ADF slows down TNC framework training. As a future direction, we are working on an optimized implementation of the ADF score for our framework.\n\n\n\n\n\\subsection{Selecting the window size}\\label{app:window_size}\n\nThe window size $\\delta$ is an important factor in the performance of a representation learning framework, not only for TNC but also for similar baselines such as CPC and triplet loss. Overall, the window size should be selected such that it is long enough to contain information about the underlying state and not too long to span over multiple underlying states. In our settings, we have selected the window sizes based on our prior knowledge of the signals. For instance, in the case of an ECG signal, the selected window size is equivalent to 7 seconds of recording, which is small enough such that the ECG remains in a stable state and yet has enough information to determine that underlying state. Our understanding of the time series data can help us select an appropriate window size, but we can also experiment with different $\\delta$ to learn this parameter. Table \\ref{tab:window_size} shows classification performance results for the simulation setups, under different window sizes. We can clearly see the drop in performance for all baseline methods when the window size is too small or too large.\n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{lcccccc}\n          &\\multicolumn{2}{c}{$\\delta=10$}& \\multicolumn{2}{c}{$\\delta=50$}& \\multicolumn{2}{c}{$\\delta=100$} \\\\\n          \\toprule\n           & AUPRC & Accuracy & AUPRC & Accuracy & AUPRC & Accuracy \\\\\n           \\midrule\n          TNC & 0.74 $\\pm$ 0.01 & 71.60 $\\pm$ 0.59 & 0.99 $\\pm$ 0.00 & 97.52 $\\pm$ 0.13 & 0.84 $\\pm$ 0.11 & 84.25 $\\pm$ 9.08\\\\\n          CPC & 0.49 $\\pm$ 0.02 & 51.85 $\\pm$ 1.81 & 0.69 $\\pm$ 0.06 & 70.26 $\\pm$ 6.48 & 0.49 $\\pm$ 0.05 & 56.65 $\\pm$ 0.81\\\\\n          T-Loss & 0.48 $\\pm$ 0.06 & 56.70 $\\pm$ 1.07 & 0.78 $\\pm$ 0.01 & 76.66 $\\pm$ 1.14 & 0.73 $\\pm$ 0.008 & 73.29 $\\pm$ 1.58 \\\\\n          \\bottomrule\n         \n    \\end{tabular}\n    \\caption{Downstream classification performance for different window size $\\delta$ on the simulated dataset}\n    \\label{tab:window_size}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Clustering metrics}\\label{app:clustering_metrics}\nMost cluster validity measures assess certain structural properties of a clustering result. In our evaluation, we have used two measures, namely the Silhouette score and Davies-Bouldin index, to evaluate the representations' clustering quality. Davies-Bouldin measures intra-cluster similarity (coherence) and inter-cluster differences (separation).\nLet $\\mathcal{C} = \\{\\mathcal{C}_1,..., \\mathcal{C}_k\\}$ be a clustering of a set $D$ of objects. The Davies-Bouldin score is evaluated as follows:  \n\n\\begin{equation}\n    DB = \\frac{1}{k}\\sum_{i=1}^{k} max_{j} \\frac{s(\\mathcal{C})+s(\\mathcal{C})}{\\delta \\mathcal{C}_i\\mathcal{C}_j}\n    \\label{eq:db_score}\n\\end{equation}\n\nWhere $s(\\mathcal{C})$ measures the scatter within a cluster, and $\\delta$ is a cluster to cluster distance measure. On the other hand, the silhouette score measures how similar an object is to its cluster \\emph{compared} to other clusters. Both measures are commonly used for the evaluation of clustering algorithms. A comparison of 2 metrics has shown that the Silhouette index produces slightly more accurate results in some cases. However, the Davies-Bouldin index is generally much less complex to compute \\cite{petrovic2006comparison}. \n\n\n\n\n\n\n\\subsection{Setting the weights for PU learning}\\label{app:w}\n\nAs mentioned in the Experiment section, the weight parameter in the loss is the probability of sampling a positive window from the non-neighboring region. One way to set this parameter is using prior knowledge of the number and the distribution of underlying states. Another way is to learn it as a hyperparameter. Table \\ref{tab:weight} shows the TNC loss for different weight parameters. The loss column reports the value measured in Equation \\ref{eq:obj}, and the accuracy shows how well the discriminator identifies the neighboring samples from non-neighboring ones for settings with different weight parameters. To also assess the impact of re-weighting the loss on downstream classification performance, we compared these performance measures for weighted and non-weighted settings. Table \\ref{tab:weight_TF} demonstrates these results and confirms that weight adjusting the loss for non-neighboring samples improves the quality of learned representations.\n\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{lcccccc}\n         &\\multicolumn{2}{c}{Simulation}& \\multicolumn{2}{c}{ECG Waveform}& \\multicolumn{2}{c}{HAR}\\\\\n         \\toprule\n         Weight & Loss & Accuracy & Loss & Accuracy & Loss & Accuracy\\\\\n         \\midrule\n        0.2 & 0.582$\\pm$0.002  & 74.29$\\pm$0.61 & 0.631$\\pm$0.011 & 60.44$\\pm$2.56 & 0.475$\\pm$0.004 & 85.75$\\pm$0.5 \\\\\n        0.1 & 0.571$\\pm$0.011 & 75.41$\\pm$0.37 & 0.637$\\pm$0.011 & 63.67$\\pm$1.29 & 0.413$\\pm$0.003 & 88.21$\\pm$1.29\\\\\n        0.05 & 0.576$\\pm$0.002 & 75.73$\\pm$0.24 & 0.622$\\pm$0.023 & 66.04$\\pm$3.46 & 0.383$\\pm$0.001 & 87.33$\\pm$0.17\\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Training the TNC framework using different weight parameters. The loss is the measured value determined in Equation \\ref{eq:obj}, and the Accuracy is the accuracy of the discriminator.}\n    \\label{tab:weight}\n\\end{table}{}\n\n\n\n\n\\begin{table}[!h]\n    \\centering\n    \\begin{tabular}{lccc}\n         Weighting? &\\multicolumn{1}{c}{Simulation}& \\multicolumn{1}{c}{ECG Waveform}& \\multicolumn{1}{c}{HAR}\\\\\n         \\toprule\n         True & {\\bf{97.52$\\pm$0.13}} & {\\bf{77.79$\\pm$0.84}} & {\\bf{88.32$\\pm$0.12}}\\\\\n         False & 97.17$\\pm$0.44 & 75.26$\\pm$1.48 & 75.25$\\pm$13.6\\\\\n         \\bottomrule\n    \\end{tabular}\n    \\caption{Downstream classification accuracy on simulated data with the TNC frameworks, using 2 different weighting strategies: 1)Trained with weight adjustment, 2)Trained with $w=0$.}\n    \\label{tab:weight_TF}\n\\end{table}{}\n\n\n\n\\subsection{Supplementary Figure}\\label{app:plots}\n\n\n\\subsubsection{Clinical waveform data}\nIn order to understand what TNC framework encodes from the high dimensional ECG signals, we visualize the trajectory of the representations of an individual sample over time. Figure \\ref{fig:trajectory_wf} demonstrates this example, where the top 2 rows are ECG signals from two recording leads and the bottom row demonstrates the representation vectors. We see that around second $40$ the pattern in the representations change as a result of an artifact happening in one of the signals. With the help from our clinical expert, we also tried to interpret different patterns in the encoding space. For instance, between time $80$ and $130$, where features 0-10 become more activated, the heart rate (HR) has increased. Increase in HR can be seen as increased frequency in the ECG signals and is one of the indicators of arrhythmia that we believe TNC has captured. Figure \\ref{fig:distribution_wf} shows the distribution of the latent encoding of ECG signals for different baselines, with colors indicating the arrhythmia class. \n\n\\begin{figure}[!h]\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[scale=0.32]{TCL\/figures\/encoding_distribution_wf.pdf}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[scale=0.32]{TCL\/figures\/encoding_distribution_wf_trip.pdf}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n    \\centering\n\\includegraphics[scale=.32]{TCL\/figures\/encoding_distribution_wf_cpc.pdf}\n\\end{subfigure}\n\\caption{T-SNE visualization of {\\bf{waveform}} signal representations for unsupervised representation learning baselines. Each point in the plot is a 64 dimensional representation of a window of time series, with the color indicating the latent state.}\n\\label{fig:distribution_wf}\n\\end{figure}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[scale=0.2]{TCL\/figures\/embedding_trajectory_hm_wf.pdf}\n    \\caption{Trajectory of a {\\bf{waveform}} signal encoding. The top two plots show the ECG recordings from 2 ECG leads. The bottom plot shows the 64 dimensional encoding of the sliding windows $W_t$ where $\\delta=2500$. }\n    \\label{fig:trajectory_wf}\n\\end{figure}\n\n\n\n\n\\subsubsection{HAR data}\nFigure \\ref{fig:distribution_har} and \\ref{fig:trajectory_har} are similar plots to the ones demonstrated in the previous section, but for the HAR dataset. As shown in Figure \\ref{fig:trajectory_har}, the underlying states of the signal are clearly captured by the TNC framework as different patterns in the latent representations.\n\n\\begin{figure}[!h]\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[scale=0.32]{TCL\/figures\/encoding_distribution_har.pdf}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[scale=0.32]{TCL\/figures\/encoding_distribution_har_trip.pdf}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n    \\centering\n\\includegraphics[scale=.32]{TCL\/figures\/encoding_distribution_har_cpc.pdf}\n\\end{subfigure}\n\\caption{T-SNE visualization of {\\bf{HAR}} signal representations for all baselines. Each point in the plot is a 10 dimensional representation of a window of $\\delta=4$, with colors indicating latent states.}\n\\label{fig:distribution_har}\n\\end{figure}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[scale=0.2]{TCL\/figures\/embedding_trajectory_hm_har.pdf}\n    \\caption{Trajectory of a {\\bf{HAR}} signal encoding. The top plot shows the original time series with shaded regions indicating the underlying state. The bottom plot shows the 10 dimensional encoding of the sliding windows $W_t$ where $\\delta=4$. }\n    \\label{fig:trajectory_har}\n\\end{figure}\n\n\n\n\n\\subsubsection{Simulation data}\nIn addition to the initial experiment, we also show the trajectory of the encoding for a smaller encoding size ($3$). In this setting, we have 4 underlying states in the signal, and only 3 dimensions for the encoding.\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[scale=0.2]{TCL\/figures\/embedding_trajectory_3.pdf}\n    \\caption{Trajectory of a {\\bf{simulation}} signal encoding. The top plots shows the signals and the bottom plot shows the 3 dimensional encoding of the sliding windows $W_t$ where $\\delta=50$. }\n    \\label{fig:trajectory_sim_3}\n\\end{figure}\n\\subsubsection*{Acknowledgments}\nResources used in preparing this research were provided, in part, by the Government of Canada through CIFAR, and companies sponsoring the Vector Institute. This research was undertaken, in part, thanks to funding from the Canadian Institute of Health Research (CIHR) and the Natural Sciences and Engineering Research Council of Canada (NSERC).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nNovae occur in binary systems in which a Roche lobe filling secondary is\nlosing hydrogen-rich material through the inner Lagrangian point onto a\nwhite dwarf (WD) primary.  Mass transfer can also occur in long period \nsystems if the secondary has a significant wind, {\\it e.g.} the giant \nsecondary in RS Oph or V407 Cyg.  Core material is mixed into the accreted \nmaterial and is violently ejected into space when the pressure at the \nWD-accretion interface becomes great enough to initiate a thermonuclear \nrunaway (TNR).  Novae eject, into the interstellar medium (ISM), a \nmixture of material accreted from the companion star, highly processed \nmaterial from the underlying WD, and products of nucleosynthesis occurring \nduring the TNR.  As a result of the TNR, up to 10$^{-4}$ M$_{\\odot}$ of \nmaterial can be ejected from the WD enriched in C, N, O, Ne, Mg, Al \nand other species \n\\btxt{\\citep{2006NuPhA.777..550J}}\nat $v \\sim 10^2 - 10^4$ km s$^{-1}$.  Any remaining \nhydrogen still bound to the WD continues to burn in hydrostatic \nequilibrium until it is consumed or ejected via a wind.\n\nInitially, the radiative output of a nova occurs in the optical but as the\nphotosphere of the WD recedes, the spectral energy distribution\nshifts to higher energies \\citep{1978ARA&A..16..171G}.  The rate of the \noptical decline defines a nova's primary characteristics\n\\citep[{\\it e.g.},][and references therein]{Warner2008}, namely the time \nto decline 2 magnitudes from visual maximum, t$_2$.  The decline \nrate depends on the amount of mass ejected, its velocity, composition,\nand if it runs into circumbinary material.  The bolometric luminosity \nduring the outburst is high, near or exceeding the Eddington limit\n(for the fastest novae), and thus additional \nmaterial is ejected via a strong stellar wind\n\\citep{1998MNRAS.300..931S,2001MNRAS.320..103S}.  In some novae the \ncollision between this fast wind and the initial exploded mass or any\npre-existing circumbinary material can produce X-ray emission from shocks.  \nThe emission from this early X-ray phase is hard, has a low luminosity, \nof order 10$^{33-35}$ erg s$^{-1}$, and declines relatively rapidly \n\\citep{1998ApJ...499..395B,2001MNRAS.326L..13O}.  As fuel continues to burn, \nmass loss causes the photosphere of the WD to shrink \n\\citep{1985ApJ...294..263M}.  \n\\btxt{The effective temperature increases, \npeaking in the soft X-rays, at (2-8)$\\times$10$^5$ K \n\\citep{1996ApJ...456..788K,1996ApJ...463L..21S,2010ApJ...717..363R}}.  \nOnce the ejecta have cleared sufficiently, and if the line of sight extinction \nis not severe, some novae exhibit characteristics similar to the\nSuper Soft X-ray binary sources \n\\citep[SSSs:][]{1997ARA&A..35...69K} with strong and soft, \nE$_{peak}$ $<$ 1 keV, X-ray emission.  This point in novae evolution is \ncalled the SSS phase.  At low spectral resolution, the UV\/X-ray\nspectral energy distributions (SED) resembles\nblackbodies, but higher resolution {\\it Chandra}\\ or {\\it XMM}\\ grating observations\nreveal a significantly more complex picture.  The spectra frequently have\nP-Cygni profiles or emission lines superimposed on a line blanketed\natmosphere.  Models sophisticated enough to interpret the high resolution\ndata are only now becoming available \\citep{2010AN....331..175V}.  Once \nnuclear burning ends, the X-ray light curve rapidly declines as the WD cools \nmarking the end of the SSS phase and the outburst.  At some point mass\ntransfer resumes and eventually another eruption occurs.  These are\ncalled classical novae (CNe) until a second outburst is observed then they\nbecome recurrent novae (RNe).  \n\\btxt{Detailed reviews of nova evolution are \npresented by \\citet{Starrfield08} and \\citet{2010AN....331..160B}.\n\\citet{2010AN....331..169H} discuss the theoretical implications of X-ray \nobservations of novae while \\citet{2010ApJS..187..275S} discusses the \ncurrent understanding of the RN class.}\n\nAn important, but not the sole\ndriver of the nova phenomenon is the mass of the WD. \nExplosions on larger mass WDs expel less mass but at higher velocities. \nThey have larger luminosities, are in outburst for less time, and (should)\nhave shorter recurrence times than novae on lower mass WDs.  \nHigh mass ($>$ 1.25 M$_{\\odot}$) WDs reach TNR ignition more rapidly\nthan low mass WDs and thus do not have the chance to accrete as much\nmaterial.  They also reach higher peak temperatures during the TNR leading\nto a more energetic explosion.  However, other factors are believed to \nplay important roles leading to a nova event.  These include\nthe composition of the WD, either CO or ONe, \nthe initial temperature of the WD during accretion, the mass accretion \nrate \\citep{2005ApJ...623..398Y}, the composition of the accreted \nmaterial \\citep{2000AIPC..522..379S}, and the mixing history\nof the core\/envelope. All impact  \nhow much mass can be transfered to the WD before \nan outburst begins.  Models show that different combinations of these \ncharacteristics can reproduce a wide range of nova outbursts \n\\citep{2005ApJ...623..398Y,WoodStar11}. \nUnfortunately very few of these parameters\nhave been observationally verified in any nova.\n\nThe X-ray regime is a crucial component for the study of novae providing \ninsight into TNR burning processes, WD mass and composition, accretion and \nmixing mechanisms, dust grain formation and destruction, and mass loss \nprocesses.  In addition, high mass novae such as RNe are potential\nSN Ia progenitors via the single degenerate scenario \n\\citep[e.g.][]{2008A&A...484L...9W,2010ASPC..429..173W,2010Ap&SS.329..287M}.\nTo make progress understanding the physics of these important astrophysical\nphenomena, observations of a large number of novae are required \nto sample all the contributing \nfactors. Prior to the launch of {\\it Swift}\\ the general X-ray temporal\nevolution of novae was far from complete \nas only a few novae had been observed at more than one epoch in X-rays.  \n\n{\\it Swift}\\ is an excellent facility for studying novae as it has a superb\nsoft X-ray response with its XRT instrument \\citep{2005SSRv..120..165B}.\n\\citet{2007ApJ...663..505N} show how the XRT favorably compares with \ncurrently available X-ray instruments.  \n{\\it Swift}\\ also has a co-aligned UV\/optical instrument, UVOT\n\\citep[see][for details]{2005SSRv..120...95R}, which provides either 6 \nfilter photometry or low resolution grism spectroscopy.  The other \n{\\it Swift}\\ instrument is a $\\gamma$-ray detector, BAT. \n\\btxt{However, novae are generally not strong $\\gamma$-ray \nsources \\citep{2005NuPhA.758..721H}.}\nThe decay of $^{22}$Na\n(half-life 2.6 yrs) generates a 1275 keV emission line but only $>$ \n1.25 M$_{\\odot}$ WDs are predicted to produce sufficient \n$^{22}$Na during the TNR.  This line \nhas not yet been definitively detected by satellites \\citep{Hernanz08}\nbut there is a recent claim by \\citet{2010ApJ...723L..84S}\nthat their models with Compton decay of $^{22}$Na can account for\nthe hard X-ray flux in V2491 Cyg provided an exceptionally large amount\nof $^{22}$Na, 3$\\times$10$^{-5}$M$_{\\odot}$, was synthesized.\nAnother $\\gamma$-ray emission mechanism is electron-positron \nannihilation very early in the outburst but this is expected to be\ndetectable only in nearby novae.  \nThe symbiotic RN RS Oph, at 1.6 kpc, was clearly detected \nin the lowest energy channels of the {\\it Swift}\/BAT \\citep{2008A&A...485..223S},\nbut that emission is consistent with that from high temperature\nshocks as the outburst ejecta plow into the pre-existing red giant\nwind.  Recently the symbiotic RN V407 Cyg was detected in the GeV band \nby {\\it Fermi-LAT} \\citep{2010Sci...329..817A} only a few days after\nvisual maximum.  \\citet{2010Sci...329..817A} show that the $\\gamma$-ray\nemission can be explained by either Compton scattering of infrared photons \nin the red giant wind or $\\pi^0$ decay from proton-proton collisions.\n\\citet{2011arXiv1101.6013L} predict that $\\pi^0$ $\\gamma$-rays\nwill be created in the high circumbinary densities of very long orbital \nperiods systems such as V407 Cyg, with a period of $\\sim$ 43 years \n\\citep{1990MNRAS.242..653M,2011arXiv1109.5397S}.\n\n{\\it Swift}\\ has a rapid response ToO procedure and flexible scheduling \nwhich is critical in obtaining well sampled X-ray light curves of \ntransient events.  Initial {\\it Swift}\\ results of 11 novae were \npresented in \\citet{2007ApJ...663..505N}.\nSince that time significantly more data have been obtained by the \n{\\it Swift}\\ Nova-CV group\\footnote{The current members of the group and \nobservation strategy are provided at \\url{http:\/\/www.swift.ac.uk\/nova-cv\/}.}\nwhich has devised an observing strategy to efficiently utilize the\nsatellite's unique capabilities and maximize the science return by \nobserving interesting and bright novae with low extinction\nrecently discovered in the Milky Way and Magellanic Clouds.  In five \nyears {\\it Swift}\\ has performed multiple visits for \\totalswift\\ classical and \nrecurrent Galactic\/Magellanic Cloud novae totaling well over 2 Ms of \nexposure time.  \n\nHere we present a summary of all the Galactic\/Magellanic Cloud {\\it Swift}\\\nnova observations from launch (2004 November 20) to 2010 July 31 using\nthe XRT (0.3-10 keV) X-ray instrument (count rates and hardness ratios) \nand the available UVOT (1700-8000\\AA) filter photometry. {\\it Swift}\\ \nobservations of novae in the M31 group are reported in \n\\citet{2010A&A...523A..89H}, \\citet{2010AN....331..187P} and \nreferences within.  We combine the {\\it Swift}\\ Galactic\/Magellanic Cloud \ndata with archival pointed observations of CNe and RNe from {\\it ROSAT}, \n{\\it XMM}, {\\it Chandra}, {\\it BeppoSax}, {\\it RXTE}, and {\\it ASCA} to produce\nthe most comprehensive X-ray sample of local nova.  \nThe sample includes \\totalSSS\\ systems that were observed during the SSS phase.\n\nIn Section 2, we summarize the properties of the \\total\\ novae in the\nX-ray sample.  The averaged {\\it Swift}\\ XRT count rates and UVOT magnitudes\nfor each observational session are also provided.  Studies of high \nfrequency phenomena in individual objects are either left for future\nwork or have previously been published \\citep[V458 Vul, V2941 Cyg, V598 Pup, \nRS Oph, V407 Cyg, and V723 Cas in][ respectively]{2009AJ....137.4160N,\n2010MNRAS.401..121P,2009A&A...507..923P,2011ApJ...727..124O,\n2011A&A...527A..98S,2008AJ....135.1328N}.  \nSections 3 and 4 detail the observations and results \nduring the hard and SSS phases, respectively.  A discussion \nfollows in Section 5 articulating trends between\nthe SSS duration and t$_2$, expansion velocity of the ejecta, and\norbital period plus the role of SSS emission in \ndust-forming novae.  Also included is a discussion on the origin of the \ndifferent variability observed in the X-ray and UV light curves of the \n{\\it Swift}\\ sources.  Optical characteristics indicative of SSS emission\nin CN and RN are also presented.  The last section, Section 6, \nprovides a summary of this work.\n\n\\section{THE X-RAY DATA SET}\n\n\\subsection{Characteristics}\n\nTable \\ref{chartable} presents the primary characteristics of all the \nGalactic\/Magellanic Cloud novae with pointed X-ray observations prior\nto July 31st, 2010.  In addition to the {\\it Swift}\\ data, the sample includes \nall the publicly available pointed observations from the {\\it ROSAT}, {\\it XMM}, \n{\\it Chandra}, {\\it BeppoSax}, {\\it RXTE}, and {\\it ASCA} archives.\nThe columns give the nova name, visual magnitude at maximum, Julian date of\nvisual maximum, time to decline two magnitudes from visual maximum,\nthe Full-Width at Half-Maximum, FWHM, of H$\\alpha$ or H$\\beta$ taken near\nvisual maximum, E(B-V) and averaged Galactic hydrogen column density, N$_H$, \nalong the line of sight, proposed orbital period, estimated distance, \nwhether the nova was observed to form dust, and if the nova is a \nknown RN.  The numbers in the parentheses are the literature \nreferences given in the table notes.  The names of novae with {\\it Swift}\\ \nobservations are shown in {\\it bold}.\n\n\\begin{deluxetable}{lllrrrrrrll}\n\\tablecaption{Observable Characteristics of \nGalactic\/Magellanic Cloud novae with X-ray observations\\label{chartable}}\n\\rotate\n\\tablewidth{700pt}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Name\\tablenotemark{a}} & \\colhead{V$_{max}$\\tablenotemark{b}} & \n\\colhead{Date\\tablenotemark{c}} & \\colhead{t$_2$\\tablenotemark{d}} & \n\\colhead{FWHM\\tablenotemark{e}} & \\colhead{E(B-V)} & \n\\colhead{N$_H$\\tablenotemark{f}} & \\colhead{Period} & \n\\colhead{D} & \\colhead{Dust?\\tablenotemark{g}} & \\colhead{RN?} \\\\ \n\\colhead{} & \\colhead{(mag)} & \\colhead{(JD)} & \\colhead{(d)} & \n\\colhead{(km s$^{-1}$)} & \\colhead{(mag)} & \\colhead{(cm$^{-2}$)} &\n\\colhead{(d)} & \\colhead{(kpc)} & \\colhead{} & \\colhead{}\n} \n\\startdata\nCI Aql & 8.83 (1) & 2451665.5 (1) & 32 (2) & 2300 (3) & 0.8$\\pm0.2$ (4) & 1.2e+22 & 0.62 (4) & 6.25$\\pm5$ (4) & N & Y \\\\\n{\\bf CSS081007}\\tablenotemark{h} & \\nodata & 2454596.5\\tablenotemark{i} & \\nodata & \\nodata & 0.146 & 1.1e+21 & 1.77 (5) & 4.45$\\pm1.95$ (6) & \\nodata & \\nodata \\\\\nGQ Mus & 7.2 (7) & 2445352.5 (7) & 18 (7) & 1000 (8) & 0.45 (9) & 3.8e+21  & 0.059375 (10) & 4.8$\\pm1$ (9) & N (7) & \\nodata \\\\\nIM Nor & 7.84 (11) & 2452289 (2) & 50 (2) & 1150 (12) & 0.8$\\pm0.2$ (4) & 8e+21 & 0.102 (13) & 4.25$\\pm3.4$ (4) & N & Y \\\\\n{\\bf KT Eri} & 5.42 (14) & 2455150.17 (14) & 6.6 (14) & 3000 (15) & 0.08 (15) & 5.5e+20 & \\nodata & 6.5 (15) & N & M \\\\\n{\\bf LMC 1995} & 10.7 (16) & 2449778.5 (16) & 15$\\pm2$ (17) & \\nodata & 0.15 (203) & 7.8e+20  & \\nodata & 50 & \\nodata & \\nodata \\\\\nLMC 2000 & 11.45 (18) & 2451737.5 (18) & 9$\\pm2$ (19) & 1700 (20) & 0.15 (203) & 7.8e+20  & \\nodata & 50 & \\nodata & \\nodata \\\\\n{\\bf LMC 2005} & 11.5 (21) & 2453700.5 (21) & 63 (22) & 900 (23) & 0.15 (203) & 1e+21 & \\nodata & 50  & M (24) & \\nodata \\\\\n{\\bf LMC 2009a} & 10.6 (25) & 2454867.5 (25) & 4$\\pm1$  & 3900 (25) & 0.15 (203)  & 5.7e+20 & 1.19 (26) & 50 & N & Y \\\\\n{\\bf SMC 2005} & 10.4 (27) & 2453588.5 (27) & \\nodata & 3200 (28) & \\nodata & 5e+20  & \\nodata & 61 & \\nodata & \\nodata \\\\\n{\\bf QY Mus} & 8.1 (29) & 2454739.90 (29) & 60:  & \\nodata & 0.71 (30) & 4.2e+21  & \\nodata & \\nodata & M & \\nodata \\\\\n{\\bf RS Oph} & 4.5 (31) & 2453779.44 (14) & 7.9 (14) & 3930 (31) & 0.73 (32) & 2.25e+21 & 456 (33) & 1.6$\\pm0.3$ (33) & N (34) & Y \\\\\n{\\bf U Sco} & 8.05 (35) & 2455224.94 (35) & 1.2 (36) & 7600 (37) & 0.2$\\pm0.1$ (4) & 1.2e+21 & 1.23056 (36) & 12$\\pm2$ (4) & N & Y \\\\\n{\\bf V1047 Cen} & 8.5 (38) & 2453614.5 (39) & 6 (40) & 840 (38) & \\nodata & 1.4e+22  & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n{\\bf V1065 Cen} & 8.2 (41) & 2454123.5 (41) & 11 (42) & 2700 (43) & 0.5$\\pm0.1$ (42) & 3.75e+21 & \\nodata & 9.05$\\pm2.8$ (42) & Y (42) & \\nodata \\\\\nV1187 Sco & 7.4 (44) & 2453220.5 (44) & 7: (45) & 3000 (44) & 1.56 (44) & 8.0e+21 & \\nodata & 4.9$\\pm0.5$ (44) & N & \\nodata \\\\\n{\\bf V1188 Sco} & 8.7 (46) & 2453577.5 (46) & 7 (40) & 1730 (47) & \\nodata & 5.0e+21  & \\nodata & 7.5 (39) & \\nodata & \\nodata \\\\\n{\\bf V1213 Cen} & 8.53 (48) & 2454959.5 (48) & 11$\\pm2$ (49) & 2300 (50) & 2.07 (30) & 1.0e+22 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n{\\bf V1280 Sco} & 3.79 (51) & 2454147.65 (14) & 21 (52) & 640 (53) & 0.36 (54) & 1.6e+21  & \\nodata & 1.6$\\pm0.4$ (54) & Y (54) & \\nodata \\\\\n{\\bf V1281 Sco} & 8.8 (55) & 2454152.21 (55) & 15:& 1800 (56) & 0.7 (57) & 3.2e+21 & \\nodata & \\nodata & N & \\nodata \\\\\n{\\bf V1309 Sco} & 7.1 (58) & 2454714.5 (58) & 23$\\pm2$ (59) & 670 (60) & 1.2 (30) & 4.0e+21 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n{\\bf V1494 Aql} & 3.8 (61) & 2451515.5 (61) & 6.6$\\pm0.5$ (61) & 1200 (62) & 0.6 (63) & 3.6e+21  & 0.13467 (64) & 1.6$\\pm0.1$ (63) & N & \\nodata \\\\\n{\\bf V1663 Aql} & 10.5 (65) & 2453531.5 (65) & 17 (66) & 1900 (67) & 2: (68) & 1.6e+22  & \\nodata & 8.9$\\pm3.6$ (69) & N & \\nodata \\\\\nV1974 Cyg & 4.3 (70) & 2448654.5 (70) & 17 (71) & 2000 (19) & 0.36$\\pm0.04$ (71) & 2.7e+21  & 0.081263 (70) & 1.8$\\pm0.1$ (72) & N & \\nodata \\\\\n{\\bf V2361 Cyg} & 9.3 (73) & 2453412.5 (73) & 6 (40) & 3200 (74) & 1.2: (75) & 7.0e+21 & \\nodata & \\nodata & Y (40) & \\nodata \\\\\n{\\bf V2362 Cyg} & 7.8 (76) & 2453831.5 (76) & 9 (77) & 1850 (78) & 0.575$\\pm0.015$ (79) & 4.4e+21  & 0.06577 (80) & 7.75$\\pm3$ (77) & Y (81) & \\nodata \\\\\n{\\bf V2467 Cyg} & 6.7 (82) & 2454176.27 (82) & 7 (83) & 950 (82) & 1.5 (84) & 1.4e+22  & 0.159 (85) & 3.1$\\pm0.5$ (86) & M (87) & \\nodata \\\\\n{\\bf V2468 Cyg} & 7.4 (88) & 2454534.2 (88) & 10: & 1000 (88) & 0.77 (89) & 1.0e+22  & 0.242 (90) & \\nodata & N & \\nodata \\\\\n{\\bf V2491 Cyg} & 7.54 (91) & 2454567.86 (91) & 4.6 (92) & 4860 (93) & 0.43 (94) & 4.7e+21  & 0.09580: (95) & 10.5 (96) & N & M \\\\\nV2487 Oph & 9.5 (97) & 2450979.5 (97) & 6.3 (98) & 10000 (98) & 0.38$\\pm0.08$ (98) & 2.0e+21 & \\nodata & 27.5$\\pm3$ (99) & N (100) & Y (101) \\\\\n{\\bf V2540 Oph} & 8.5 (102) & 2452295.5 (102) & \\nodata & \\nodata & \\nodata & 2.3e+21 & 0.284781 (103) & 5.2$\\pm0.8$ (103) & N & \\nodata \\\\\nV2575 Oph & 11.1 (104) & 2453778.8 (104) & 20: & 560 (104) & 1.4 (105) & 3.3e+21 & \\nodata & \\nodata & N (105) & \\nodata \\\\\n{\\bf V2576 Oph} & 9.2 (106) & 2453832.5 (106) & 8: & 1470 (106) & 0.25 (107) & 2.6e+21  & \\nodata & \\nodata & N & \\nodata \\\\\n{\\bf V2615 Oph} & 8.52 (108) & 2454187.5 (108) & 26.5 (108) & 800 (109) & 0.9 (108) & 3.1e+21  & \\nodata & 3.7$\\pm0.2$ (108) & Y (110) & \\nodata \\\\\n{\\bf V2670 Oph} & 9.9 (111) & 2454613.11 (111) & 15: & 600 (112) & 1.3: (113) & 2.9e+21  & \\nodata & \\nodata & N (114) & \\nodata \\\\\n{\\bf V2671 Oph} & 11.1 (115) & 2454617.5 (115) & 8: & 1210 (116) & 2.0 (117) & 3.3e+21  & \\nodata & \\nodata & M (117) & \\nodata \\\\\n{\\bf V2672 Oph} & 10.0 (118) & 2455060.02 (118) & 2.3 (119) & 8000 (118) & 1.6$\\pm0.1$ (119) & 4.0e+21  & \\nodata & 19$\\pm2$ (119) & \\nodata & M \\\\\nV351 Pup & 6.5 (120) & 2448617.5 (120) & 16 (121) & \\nodata & 0.72$\\pm0.1$ (122) & 6.2e+21 & 0.1182 (123) & 2.7$\\pm0.7$ (122) & N & \\nodata \\\\\n{\\bf V382 Nor} & 8.9 (124) & 2453447.5 (124) & 12 (40) & 1850 (23) & \\nodata & 1.7e+22 & \\nodata & \\nodata & \\nodata & \\nodata \\\\\nV382 Vel & 2.85 (125) & 2451320.5 (125) & 4.5 (126) & 2400 (126) & 0.05: (126) & 3.4e+21  & 0.146126 (127) & 1.68$\\pm0.3$ (126) & N & \\nodata \\\\\n{\\bf V407 Cyg} & 6.8 (128) & 2455266.314 (128) & 5.9 (129) & 2760 (129) & 0.5$\\pm0.05$ (130) & 8.8e+21 & 15595 (131) & 2.7 (131) & \\nodata & Y \\\\\n{\\bf V458 Vul} & 8.24 (132) & 2454322.39 (132) & 7 (133) & 1750 (134) & 0.6 (135) & 3.6e+21 & 0.06812255 (136) & 8.5$\\pm1.8$ (133) & N (135) & \\nodata \\\\\n{\\bf V459 Vul} & 7.57 (137) & 2454461.5 (137) & 18 (138) & 910 (139) & 1.0 (140) & 5.5e+21  & \\nodata & 3.65$\\pm1.35$ (138) & Y (140) & \\nodata \\\\\nV4633 Sgr & 7.8 (141) & 2450895.5 (141) & 19$\\pm3$ (142) & 1700 (143) & 0.21 (142) & 1.4e+21  & 0.125576 (144) & 8.9$\\pm2.5$ (142) & N & \\nodata \\\\\n{\\bf V4643 Sgr} & 8.07 (145) & 2451965.867 (145) & 4.8 (146) & 4700 (147) & 1.67 (148) & 1.4e+22 & \\nodata & 3 (148) & N & \\nodata \\\\\n{\\bf V4743 Sgr} & 5.0 (149) & 2452537.5 (149) & 9 (150) & 2400 (149) & 0.25 (151) & 1.2e+21 & 0.281 (152) & 3.9$\\pm0.3$ (151) & N & \\nodata \\\\\n{\\bf V4745 Sgr} & 7.41 (153) & 2452747.5 (153) & 8.6 (154) & 1600 (155) & 0.1 (154) & 9.0e+20  & 0.20782 (156) & 14$\\pm5$ (154) & \\nodata & \\nodata \\\\\n{\\bf V476 Sct} & 10.3 (157) & 2453643.5 (157) & 15 (158) & \\nodata & 1.9 (158) & 1.2e+22  & \\nodata & 4$\\pm1$ (158) & M (159) & \\nodata \\\\\n{\\bf V477 Sct} & 9.8 (160) & 2453655.5 (160) & 3 (160) & 2900 (161) & 1.2: (162) & 4e+21  & \\nodata & \\nodata & M (163) & \\nodata \\\\\n{\\bf V5114 Sgr} & 8.38 (164) & 2453081.5 (164) & 11 (165) & 2000 (23) & \\nodata & 1.5e+21  & \\nodata & 7.7$\\pm0.7$ (165) & N (166) & \\nodata \\\\\n{\\bf V5115 Sgr} & 7.7 (167) & 2453459.5 (167) & 7 (40) & 1300 (168) & 0.53 (169) & 2.3e+21  & \\nodata & \\nodata & N (169) & \\nodata \\\\\n{\\bf V5116 Sgr} & 8.15 (170) & 2453556.91 (170) & 6.5 (171) & 970 (172) & 0.25 (173) & 1.5e+21 & 0.1238 (171) & 11$\\pm3$ (173) & N (174) & \\nodata \\\\\n{\\bf V5558 Sgr} & 6.53 (175) & 2454291.5 (175) & 125 (176) & 1000 (177) & 0.80 (178) & 1.6e+22  & \\nodata & 1.3$\\pm0.3$ (176) & N (179) & \\nodata \\\\\n{\\bf V5579 Sgr} & 5.56 (180) & 2454579.62 (180) & 7: & 1500 (23) & 1.2 (181) & 3.3e+21 & \\nodata & \\nodata & Y (181) & \\nodata \\\\\n{\\bf V5583 Sgr} & 7.43 (182) & 2455051.07 (182) & 5: & 2300 (182) & 0.39 (30) & 2.0e+21 & \\nodata & 10.5 & \\nodata & \\nodata \\\\\n{\\bf V574 Pup} & 6.93 (183) & 2453332.22 (183) & 13 (184) & 2800 (184) & 0.5$\\pm0.1$  & 6.2e+21 & \\nodata & 6.5$\\pm1$  & M (185) & \\nodata \\\\\n{\\bf V597 Pup} & 7.0 (186) & 2454418.75 (186) & 3: & 1800 (187) & 0.3 (188) & 5.0e+21  & 0.11119 (189) & \\nodata & N (188) & \\nodata \\\\\n{\\bf V598 Pup} & 3.46 (14) & 2454257.79 (14) & 9$\\pm1$ (190) & \\nodata & 0.16 (190) & 1.4e+21 & \\nodata & 2.95$\\pm0.8$ (190) & \\nodata & \\nodata \\\\\n{\\bf V679 Car} & 7.55 (191) & 2454797.77 (191) & 20: & \\nodata & \\nodata & 1.3e+22  & \\nodata & \\nodata & \\nodata & \\nodata \\\\\n{\\bf V723 Cas} & 7.1 (192) & 2450069.0 (192) & 263 (2) & 600 (193) & 0.5 (194) & 2.35e+21  & 0.69 (195) & 3.86$\\pm0.23$ (196) & N & \\nodata \\\\\nV838 Her & 5 (197) & 2448340.5 (197) & 2 (198) & \\nodata & 0.5$\\pm0.1$ (198) & 2.6e+21  & 0.2975 (199) & 3$\\pm1$ (198) & Y (200) & \\nodata \\\\\n{\\bf XMMSL1 J06}\\tablenotemark{j} & 12 (201) & 2453643.5 (202) & 8$\\pm2$ (202) & \\nodata & 0.15 (203) & 8.7e+20 & \\nodata & 50 & \\nodata & \\nodata \\\\\n\\enddata\n\\tablenotetext{a}{Novae with {\\it Swift}\\ observations are presented in {\\it bold}.}\n\\tablenotetext{b}{Visual maximum.}\n\\tablenotetext{c}{Date of visual maximum.}\n\\tablenotetext{d}{As measured from the visual light curve. A \":\" indicates\nan uncertain value due to an estimate from the AAVSO light curve.}\n\\tablenotetext{e}{Of Balmer lines measured at or near visual maximum.}\n\\tablenotetext{f}{Average Galactic N$_H$ within 0.5$\\arcdeg$ of the nova \nposition as given in the HEASARC N$_H$ tool.}\n\\tablenotetext{g}{Dust forming nova? (Y)es, (N)o, or (M)abye.\nNovae with \"N\" but no dust reference were sufficiently observed\nbut no dust was specifically reported in any of the references.\n}\n\\tablenotetext{h}{Full nova name is CSS081007030559+054715.}\n\\tablenotetext{i}{An averaged date based on available photometry.}\n\\tablenotetext{j}{Full nova name is XMMSL1 J060636.2-694933.} \\\\\n\\tablecomments{Numbers in parenthesis are the reference codes.} \n\\tablerefs{\n1 = \\citet{2000IAUC.7411....3H};\n2 = \\citet{2010AJ....140...34S};\n3 = \\citet{2000IAUC.7409....1T};\n4 = \\citet{2010ApJS..187..275S};\n5 = \\citet{2010AN....331..156B};\n6 = \\citet{2008ATel.1847....1S};\n7 = \\citet{1984MNRAS.211..421W};\n8 = \\citet{1983IAUC.3766....1C};\n9 = \\citet{1984A&A...137..307K};\n10 = \\citet{1989ApJ...339L..41D};\n11 = \\citet{2002IAUC.7791....2K};\n12 = \\citet{2002IAUC.7799....3D};\n13 = \\citet{2003MNRAS.343..313W};\n14 = \\citet{2010ApJ...724..480H};\n15 = \\citet{2009ATel.2327....1R};\n16 = \\citet{1995IAUC.6143....2L};\n17 = \\citet{2004IBVS.5582....1L};\n18 = \\citet{2000IAUC.7453....1L};\n19 = \\citet{2003A&A...405..703G};\n20 = \\citet{2000IAUC.7457....1D};\n21 = \\citet{2005IAUC.8635....1L};\n22 = \\citet{2007JAVSO..35..359L};\n23 = Average of our SMARTS spectra;\n24 = Evidence from our SMARTS IR lightcurve;\n25 = \\citet{2009IAUC.9019....1L};\n26 = \\citet{2009ATel.2001....1B};\n27 = \\citet{2005IAUC.8582....2L};\n28 = \\citet{2005IAUC.8582....3M};\n29 = \\citet{2008IAUC.8990....2L};\n30 = \\citet{1998ApJ...500..525S};\n31 = \\citet{2007ApJ...665L..63B};\n32 = \\citet{1985IAUC.4067....2S};\n33 = \\citet{2008ApJ...673.1067N};\n34 = \\citet{2007ApJ...671L.157E};\n35 = \\citet{2010ATel.2419....1S};\n36 = \\citet{2001A&A...378..132E};\n37 = \\citet{2010ATel.2411....1A};\n38 = \\citet{2005IAUC.8596....1L};\n39 = \\citet{2007ApJ...663..505N};\n40 = \\citet{2007ApJ...662..552H};\n41 = \\citet{2007IAUC.8800....1L};\n42 = \\citet{2010AJ....140.1347H};\n43 = \\citet{2007IAUC.8800....2W};\n44 = \\citet{2006ApJ...638..987L};\n45 = \\citet{2004AAS...20515003S};\n46 = \\citet{2005IAUC.8574....1P};\n47 = \\citet{2005IAUC.8576....2N};\n48 = \\citet{2009IAUC.9043....1P};\n49 = From AAVSO lightcurve;\n50 = \\citet{2009IAUC.9043....2P};\n51 = \\citet{2007IAUC.8807....1Y};\n52 = \\citet{2008MNRAS.391.1874D};\n53 = \\citet{2007CBET..852....1M};\n54 = \\citet{2008A&A...487..223C};\n55 = \\citet{2007IAUC.8810....1Y};\n56 = \\citet{2007IAUC.8812....2N};\n57 = \\citet{2007IAUC.8846....2R};\n58 = \\citet{2008IAUC.8972....1N};\n59 = From AAVSO light curve;\n60 = \\citet{2008IAUC.8972....2N};\n61 = \\citet{2000A&A...355L...9K};\n62 = \\citet{1999IAUC.7324....1F};\n63 = \\citet{2003A&A...404..997I};\n64 = \\citet{2001IAUC.7665....2B};\n65 = \\citet{2005IAUC.8540....1P};\n66 = \\citet{2006JBAA..116..320B};\n67 = Average of our SMARTS spectra;\n68 = \\citet{2005IAUC.8640....2P};\n69 = \\citet{2007ApJ...669.1150L};\n70 = \\citet{1994ApJ...431L..47D};\n71 = \\citet{1996AJ....111..869A};\n72 = \\citet{1997A&A...318..908C};\n73 = \\citet{2005IAUC.8483....1N};\n74 = \\citet{2005IAUC.8484....1N};\n75 = \\citet{2005IAUC.8524....2R};\n76 = \\citet{2006CBET..466....2W};\n77 = \\citet{2008A&A...479L..51K};\n78 = \\citet{2006ATel..792....1C};\n79 = \\citet{2006IAUC.8702....2S};\n80 = \\citet{2009ATel.2137....1B};\n81 = \\citet{2008AJ....136.1815L};\n82 = \\citet{2007IAUC.8821....1N};\n83 = \\citet{2009AAS...21349125L};\n84 = \\citet{2007IAUC.8848....1M};\n85 = \\citet{2008ATel.1723....1S};\n86 = \\citet{2009AN....330...77P};\n87 = \\citet{WoodStar11};\n88 = \\citet{2008IAUC.8927....2N};\n89 = \\citet{2008IAUC.8936....2R};\n90 = \\citet{2009ATel.2157....1S};\n91 = \\citet{2008IAUC.8934....1N};\n92 = \\citet{2008ATel.1485....1T};\n93 = \\citet{2008ATel.1475....1T};\n94 = \\citet{2008IAUC.8938....2R};\n95 = \\citet{2008ATel.1514....1B} but Darnley et al. (2011, submitted) do not find convincing evidence of this period in their data.;\n96 = \\citet{2008CBET.1379....1H};\n97 = \\citet{1998IAUC.6941....1N};\n98 = \\citet{2000ApJ...541..791L};\n99 = \\citet{2000ApJ...541..791L};\n100 = \\citet{1998IAUC.7049....1R};\n101 = \\citet{2009AJ....138.1230P};\n102 = \\citet{2002IAUC.7809....1R};\n103 = \\citet{2005PASA...22..298A};\n104 = \\citet{2006IAUC.8671....1P};\n105 = \\citet{2006IAUC.8710....2R};\n106 = \\citet{2006IAUC.8700....1W};\n107 = \\citet{2006IAUC.8730....5L};\n108 = \\citet{2008MNRAS.387..344M};\n109 = \\citet{2007IAUC.8824....1N};\n110 = \\citet{2007IAUC.8846....2R};\n111 = \\citet{2008IAUC.8948....2A};\n112 = \\citet{2008IAUC.8948....3N};\n113 = \\citet{2008IAUC.8956....1R};\n114 = \\citet{2008IAUC.8998....3S};\n115 = \\citet{2008IAUC.8950....1N};\n116 = \\citet{2008CBET.1448....1H};\n117 = \\citet{2008IAUC.8957....1R};\n118 = \\citet{2009IAUC.9064....1N};\n119 = \\citet{2010MNRAS.tmp.1484M};\n120 = \\citet{1992IAUC.5422....1C};\n121 = \\citet{1996ApJ...466..410O};\n122 = \\citet{1996MNRAS.279..280S};\n123 = \\citet{2001MNRAS.328..159W};\n124 = \\citet{2005IAUC.8497....1L};\n125 = \\citet{1999IAUC.7176....1L};\n126 = \\citet{2002A&A...390..155D};\n127 = \\citet{2006AJ....131.2628B};\n128 = \\citet{2010CBET.2199....1H};\n129 = \\citet{2011MNRAS.410L..52M};\n130 = \\citet{2011A&A...527A..98S};\n131 = \\citet{1990MNRAS.242..653M};\n132 = \\citet{2007CBET.1029....1M};\n133 = \\citet{2008Ap&SS.315...79P};\n134 = \\citet{2007IAUC.8862....2B};\n135 = \\citet{2007IAUC.8883....1L};\n136 = \\citet{2010MNRAS.407L..21R};\n137 = \\citet{2007CBET.1183....1M};\n138 = \\citet{2010NewA...15..170P};\n139 = \\citet{2007CBET.1181....1N};\n140 = \\citet{2008IAUC.8936....3R};\n141 = \\citet{1998IAUC.6847....2N};\n142 = \\citet{2001MNRAS.328.1169L};\n143 = \\citet{1998IAUC.6848....1D};\n144 = \\citet{2008MNRAS.387..289L};\n145 = \\citet{2001IAUC.7590....2K};\n146 = \\citet{2001IBVS.5138....1B};\n147 = \\citet{2001IAUC.7589....2A};\n148 = \\citet{2008AstL...34..249B};\n149 = \\citet{2002IAUC.7975....2K};\n150 = \\citet{2003MNRAS.344..521M};\n151 = \\citet{2007AAS...210.0402V};\n152 = \\citet{2003IAUC.8176....3W};\n153 = \\citet{2003IAUC.8126....2L};\n154 = \\citet{2005A&A...429..599C};\n155 = \\citet{2003IAUC.8132....2K};\n156 = \\citet{2006MNRAS.371..459D};\n157 = \\citet{2005IAUC.8607....1S};\n158 = \\citet{2006MNRAS.369.1755M};\n159 = \\citet{2005IAUC.8638....1P};\n160 = \\citet{2006A&A...452..567M};\n161 = \\citet{2005IAUC.8617....1P};\n162 = \\citet{2005IAUC.8644....1M};\n163 = \\citet{2005IAUC.8644....1M};\n164 = \\citet{2004IAUC.8306....1N};\n165 = \\citet{2006A&A...459..875E};\n166 = \\citet{2004IAUC.8368....3L};\n167 = \\citet{2005IAUC.8502....1S};\n168 = \\citet{2005IAUC.8501....2K};\n169 = \\citet{2005IAUC.8523....4R};\n170 = \\citet{2005IAUC.8559....2G};\n171 = \\citet{2008A&A...478..815D};\n172 = \\citet{2005IAUC.8559....1L};\n173 = \\citet{2008ApJ...675L..93S};\n174 = \\citet{2005IAUC.8579....3R};\n175 = \\citet{2007CBET.1010....1M};\n176 = \\citet{2008NewA...13..557P};\n177 = \\citet{2007CBET.1006....1I};\n178 = \\citet{2007IAUC.8884....2R};\n179 = \\citet{2009AAS...21442806R};\n180 = \\citet{2008CBET.1352....1M};\n181 = \\citet{2008IAUC.8948....1R};\n182 = \\citet{2009IAUC.9061....1N};\n183 = \\citet{2004IAUC.8445....3S};\n184 = \\citet{2005IBVS.5638....1S};\n185 = \\citet{Heltonthesis};\n186 = \\citet{2007IAUC.8895....1P};\n187 = \\citet{2007IAUC.8896....2N};\n188 = \\citet{2008IAUC.8911....2N};\n189 = \\citet{2009MNRAS.397..979W};\n190 = \\citet{2008A&A...482L...1R};\n191 = \\citet{2008IAUC.8999....1W};\n192 = \\citet{1996A&A...315..166M};\n193 = \\citet{1996IAUC.6365....2I};\n194 = \\citet{2008AJ....135.1328N};\n195 = \\citet{2007AstBu..62..125G};\n196 = \\citet{2009AJ....138.1090L};\n197 = \\citet{1991IAUC.5222....1S};\n198 = \\citet{1996MNRAS.282..563V};\n199 = \\citet{1994ApJ...420..830S};\n200 = \\citet{1992ApJ...384L..41W};\n201 = \\citet{2009A&A...506.1309R};\n202 = \\citet{2009A&A...506.1309R};\n203 = Standard LMC value.\n}\n\\end{deluxetable}\n\nAlthough P-Cygni absorption profiles provide the best values for\nthe early velocities of the ejecta, they are not nearly as well reported \nin the literature as FWHMs of Balmer lines near maximum.  Since nearly \nevery nova has a FWHM citation as part of the spectroscopic confirmation of \nthe initial visual detection, they are used as the expansion velocity\nproxy.  Expansion velocities provide another way to classify a nova since \nmore massive WDs eject less mass and at a greater velocity than low mass WDs.  \nThis characteristic can be preferable to t$_2$ since the rate of decline \ncan be difficult to determine for novae with secondary maxima, dust \nformation, or that have poorly sampled early light curves.  Both FWHM \nand t$_2$ are used as simple proxies for the WD mass.\n\\footnote{While many other parameters also affect these observables, such as \nthe accretion rate, these parameters are generally not known for specific\nnovae and thus their contributions to the secular evolution can not be\ndetermined.}\nThe N$_H$ values were \nobtained from the HEASARC N$_H$ \ntool\\footnote{http:\/\/heasarc.gsfc.nasa.gov\/cgi-bin\/Tools\/w3nh\/w3nh.pl}\nusing the averaged LAB \\citep{2005A&A...440..775K} and DL \n\\citep{1990ARA&A..28..215D} maps within a 0.5$\\arcdeg$ area around each nova.\n\n\\begin{figure*}[htbp]\n\\plotone{newt2vsfwhm.ps}\n\\caption{The t$_2$ vs. FWHM near maximum for the novae in the sample.  Filled\ncircles are known RNe.  Half filled circles are suspected\nRNe based on their characteristics.  Circles with asterisks inside\nindicate dusty novae.  The distribution histograms for t$_2$ and the FWHM \nare also shown in the secondary graphs.  The dotted lines in the t$_2$ \nhistogram show the boundaries between the \"very fast\", \"fast\", \"moderately \nfast\", and \"slow\" light curve classifications \\citep{Warner2008}.  The \nmajority of our novae belong to the \"fast\" or \"very fast\" classifications.\n\\label{t2vsfwhm}}\n\\end{figure*}\n\nFigure \\ref{t2vsfwhm} shows the t$_2$ and FWHM distribution for all the novae \nwith both values.  The 7 filled circles are known RNe while the 3\nhalf filled circles are suspected RNe.  Dusty novae have an asterisk\ninside their circle symbols.  As expected, the RNe tend \ntoward large FWHM and fast t$_2$ times.  In this sample the dusty novae \nare scattered throughout the FWHM-t$_2$ phase space showing no particular \npreference for any type of nova.  Figure \\ref{t2vsfwhm} also shows\nthat there is a wide dispersion between FWHM and t$_2$, e.g. novae\nwith t$_2$ of 10 days have FWHM values between 1000 and 3000 km s$^{-1}$.\n\nThe top panel of Figure \\ref{t2vsfwhm} shows the distribution of t$_2$ in \none day bins.  Using the light curve classifications of \\citet{Warner2008},\nthe sample is heavily weighted toward very fast (t$_2$ $<$ 10 days) and \nfast (11 $>$ t$_2$ $<$ 25 days) novae.  These are intrinsically more luminous,\nwith a larger rise from quiescence to maximum light.  The peak is \nat 8 days and with a median t$_2$ of 9 days.  There are only 5 novae \nin the entire sample with t$_2$ times greater than 50 days, \nIM Nor, LMC 2005, QY Mus, V723 Cas, and V5558 Sgr. The far\nright panel in Figure \\ref{t2vsfwhm} gives the distribution for \nFWHM in 500 km s$^{-1}$ bins.  The majority of the novae in the sample\nhave low expansion velocities with the peak in the 1500-2000 km s$^{-1}$ bin. \nThe median FWHM is 1800 km s$^{-1}$.  There are only 5 novae with \nFWHM $\\geq$ 4000 km s$^{-1}$ in the sample and all but V4643 Sgr \nare RNe or suspected RNe.\n\nThe X-ray sample is biased toward fast novae for multiple reasons. \nThe bulk of the observations are from {\\it Swift}, and {\\it Swift}\\ has only been \noperational for 5 years.  Fast systems, like the RN RS Oph will rise \nand fall on time-scales of months (see Section \\ref{var}) while slow \nnovae, such as V1280 Sco, have not yet had sufficient time to evolve \ninto soft X-ray sources (and may not) thus are therefore underrepresented. \nSlow novae also require more observing time \nto be monitored over their lifetime, particularly if the same coverage of\nthe X-ray evolution is desired.  Allocations of {\\it Swift}\\ observing time\nover multiple cycles are difficult to justify and execute unless a compelling\nscientific rationale is forthcoming, such as unusual or significant \nspectral variations (see Section \\ref{fex}), count rate oscillations, \nabundance pattern changes, etc.\nSlow and old novae (many tens of months post-outburst)\nare generally sampled once a year in part due to their slow evolution.\nHowever, the main reason the sample depicted in Fig. \\ref{t2vsfwhm}\nfavors fast novae is due to the strong \nselection effect toward outbursts on high mass WDs.  While high mass WDs, \n{\\it e.g.} $\\geq$ 1.2 M$_{\\odot}$, are relatively rare in the field,\nthe time-scale between outbursts is significantly shorter than for low-mass \nWDs, meaning they dominate any observational sample\n\\citep{1994ApJ...425..797L}.  Finally, high mass WDs give rise to more\nluminous outbursts and the {\\it Swift}\\ Nova-CV group has a V $<$ 8 magnitude\nselection criterion which leads to preferentially selecting brighter sources.\n\n\\begin{figure*}[htbp]\n\\plotone{dusthistogram.ps}\n\\caption{The distribution of dusty novae in the sample.  The cross-hashed\nregion is for novae that showed strong dust characteristics; however, the\npresence of dust in these systems has not been spectrophotometrically \ncorroborated at IR wavelengths. The majority of novae in our X-ray\nselected sample (Table \\ref{chartable}) did not form dust.\n\\label{dusthistogram}}\n\\end{figure*}\n\nFigure \\ref{dusthistogram} shows the distribution of our sample \n(Table \\ref{chartable}) with respect to dust formation frequency.  \nOnly $\\sim$ 16\\% of the novae in the sample had clear indications\nin the literature of dust formation in the ejecta.  The dust formation \nfrequency increases to 31\\% when including the 7 novae where dust \nlikely formed based on characteristics of the visual light curve but not \nyet confirmed by a measured SED excess in the thermal- and mid-infrared, \n{\\it e.g.} the \"maybe\"s in column 10 of Table \\ref{chartable}.\nThis is consistent with the expectations of the general population\nof dusty novae which ranges from 18\\% to $\\gtrsim 40$\\%.  The lower limit\nis set by \\citet{2010AJ....140...34S} who find that 93 well sampled \nAmerican Association of Variable Star Observers (AAVSO) novae have\nthe large dip in their visual light curves indicative of strong \ndust formation \\citep[see][]{1998PASP..110....3G}.  The upper limit\nis from a recent \\textit{Spitzer} survey of IR bright novae that finds\nmany novae have weak dust emission signatures with little or no dip in the \nvisual light curve especially at late epochs (many 100s of days post-outburst) \nwhen emission from the dust envelope is a few $\\mu$Jy\n\\citep{WoodStar11,Heltonthesis}\n\nIn order to obtain the best X-ray and UV data, it is desirable to target\nnovae with low extinction along the line of sight.  However, determining\nthe extinction early in the outburst is challenging.  N$_H$ maps are crude\nsince they sample large regions of the sky.  The region size used to \nderive the N$_H$ values in Table \\ref{chartable} was 0.5$^{\\arcdeg}$.\nTypically just a handful of sight lines are available in regions of this\nrelatively small size.  The problem is exacerbated in inhomogeneous areas \nlike the Galactic plane where most novae are found.  The \nextinction maps of \\citet{1998ApJ...500..525S} can be used to obtain E(B-V) \nsince their spatial resolution is significantly higher. However, the\n\\citet{1998ApJ...500..525S} maps suffer from large errors in the Galactic\nplane, $|b| <$ 5$^{\\arcdeg}$.  Maps also give the total Galactic line of\nsight with no information versus distance and thus provide only an upper \nlimit.  E(B-V) can also be determined from\nindirect methods but these require either high resolution spectroscopy \nto measure ISM absorption lines \n\\citep[{\\it e.g.} \\ion{Na}{1}~D $\\lambda$5890\\AA;][]{munari97}, \nthe line strengths of optical and near-IR spectroscopy of \\ion{O}{1} lines\n\\citep{rudy89}, or extensive B and V photometry during\nthe early outburst \\citep[{\\it e.g.} intrinsic (B-V) at V$_{max}$ or\nt$_2$;][]{1987A&AS...70..125V}.  Finally, \nE(B-V) estimates can be affected by other factors occurring during \nthe outburst such as dust formation or intrinsic absorption from the ejecta\nwhile the expanding material is still dense.\n\nIt is therefore desirable to check \nthat the general relationship between N$_H$ and E(B-V) holds for novae.  \nFigure \\ref{ebmvnh} shows N$_H$ versus E(B-V) for the novae in this \npaper with the dotted line showing the average Milky Way extinction law, \nE(B-V) = N$_H$\/4.8$\\times$10$^{21}$ \\citep{1978ApJ...224..132B}, as the \ndotted line.  The circles represent novae with Galactic latitudes, \n$|b| \\geq$ 5$^{\\arcdeg}$ while pluses are novae found within the disk,\n$|b| <$ 5$^{\\arcdeg}$.  Filled circles are Magellanic novae.  Errors are \npresent on all sources when available in the literature.  There is good \nagreement with the relationship for novae with E(B-V) $\\leq$0.6 and \nN$_H$ $\\leq$ 2.9$\\times$10$^{21}$ with a correlation coefficient of 0.85. \nThese are primarily novae found outside of the galactic disk and thus\nfit the relationship well.  Novae with these low extinction values and\ncolumn densities are ideal for soft X-ray detection.  The relationship \nbreaks down at larger values with a lower correlation coefficient of 0.64 \nfor the entire sample as it is dominated by novae embedded within\nthe Galactic disk.  Novae with E(B-V) values greater than 1.5 generally \nmake poor SSS candidates due to the large extinction.\n\nThe maximum magnitude vs. rate of decline relationship of\n\\citet{1995ApJ...452..704D} provides an estimate of the distances\nfor the Galactic novae in Table \\ref{chartable}.  The distance estimate\nrange extends from the relatively nearby V1280 Sco ($\\sim$ 1 kpc) \nto the other side of the Galaxy for V2576 Oph ($\\sim$ 28 kpc).\nThe median Galactic distance from this relationship is 5.5 kpc\nfor this sample.\n\n\\begin{figure*}[htbp]\n\\plotone{ebmvnh.ps}\n\\caption{Local N$_H$ value versus the estimated E($B-V$). The values\nare from Table~\\ref{chartable}. The dotted line shows the E(B-V) vs. N$_H$ \nrelationship of \\citet{1978ApJ...224..132B}. Circles are $|b| \\geq$ \n5$^{\\arcdeg}$ novae, the $|b| <$ 5$^{\\arcdeg}$ novae are shown as pluses,\nand filled circles are Magellanic novae.  Errors bars are given when\navailable.\n\\label{ebmvnh}}\n\\end{figure*}\n\n\\subsection{X-ray evolution\\label{xrayevol}}\n\nAll the available {\\it Swift}\\ XRT and UVOT data of novae in the public archive\nup to 2010 July 31 are presented in Table \\ref{fullswift}.  The data were\nprimarily obtained from pointed observations but a few serendipitous \nobservations are also included.  The full data \nset is available in the electronic edition with only V1281 Sco shown as an\nexample here.  The columns provide the {\\it Swift}\\ observation identification,\nexposure time, day of the observation from visual maximum (see Table \n\\ref{chartable}), XRT total (0.3-10 keV) count rate, the Hard (1-10 keV)\nto Soft (0.3-1 keV) hardness ratio, HR1, the Soft and Hard band count rates,\nthe (Hard-Soft)\/(Hard+Soft) hardness ratio, HR2, and the uvw2 \n($\\lambda_c$ = 1928\\AA), uvm2 (2246\\AA), uvw1 (2600\\AA), $u$ (3465\\AA), \n$b$ (4392\\AA), and $v$ (5468\\AA) UVOT filter magnitudes if available.  \nThe UVOT magnitudes do not include the systematic photometric calibration \nerrors from \\citet[][Table 6]{2008MNRAS.383..627P}.\n\nThere is one row in the table per observation ID, however this is not a \nfixed unit of time; most observation IDs are less than 0.13 days \nduration and the median exposure time is 1.76 ks. For this exposure time, \nour 3 sigma detection limit is 0.0037 counts s$^{-1}$ (0.3-10 keV, corrected \nfor typical PSF coverage). This corresponds to an unabsorbed flux limit in \nthe same band, assuming absorption by N$_H$ = 3$\\times$ 10$^{20}$ of \n1.5$\\times$ 10$^{-13}$ and 2.0$\\times$ 10$^{-13}$ erg cm$^{-2}$ s$^{-1}$ for\na 5 keV optically thin thermal spectrum and a 50 eV blackbody spectrum, \nrespectively.\n\nTo create a self consistent dataset for Table~\\ref{fullswift} we used the \nsoftware described by \\citet{2009MNRAS.397.1177E,2007A&A...469..379E}.\nThis extracts source and background event lists from the data (using an \nannular source region where necessary to eliminate pile up), and then bins \nthese data to form the light curve, applying corrections for pile up, bad \npixels and the finite size of the source region as necessary. \nSince novae tend to be soft, we chose the energy bands for the\nhardness ratio to be 0.3--1 keV and 1--10 keV. There is also evidence\nthat, for very soft sources, pile up occurs at lower count rates than for\nhard sources; we thus set the threshold at which pile up is considered a\nrisk to be 0.3 (80) count s$^{-1}$ in PC (WT) mode \\citep[the defaults from]\n[are 0.6 and 150 respectively]{2007A&A...469..379E}.\n\nWe chose to group the data in one bin per {\\it Swift}\\ observation. In the\ncurrent version of the online software (for this binning mode), background\nsubtraction is only carried out using Gaussian statistics, and does not\nproduce upper limits if this results in a non-detection. We thus took\nthe `detailed' light curves produced by the web tools, which include the\nnumber of measured counts in each bin, the exposure time, and the\ncorrection factor (accounting for pile up etc.). Following the approach\nof \\citet{2007A&A...469..379E} for other binning methods, where any bins had\nfewer than 15 detected source counts, we used the Bayesian method of\n\\citet{1991ApJ...374..344K} to determine whether the source was\ndetected at the 3-sigma level. If this was not the case, a 3-$\\sigma$\nupper limit was produced using this Bayesian method, otherwise a data\npoint with standard 1-$\\sigma$ uncertainty was produced using the \n\\citet{1991ApJ...374..344K} approach.\n\nThe hardness ratios were always calculated using Gaussian statistics, \nunless one band had zero detected source photons: in this case no ratio\ncould be produced.  The hardness ratios were defined as HR1 = H\/S and \nHR2 = (H-S)\/(H+S) where H = 1.0-10 keV and S = 0.3-1.0 keV.\n\n\\begin{deluxetable}{cccccccccccccc}\n\\tablecaption{{\\it Swift}\\ XRT\/UVOT data for novae in the archive\\label{fullswift}}\n\\tablewidth{0pt}\n\\tabletypesize{\\scriptsize}\n\\rotate\n\\tablecolumns{14}\n\\tablehead{\n\\colhead{ObsID} & \\colhead{Exp} & \\colhead{Day\\tablenotemark{a}} & \n\\colhead{CR\\tablenotemark{b}} & \n\\colhead{HR1\\tablenotemark{c}} & \\colhead{Soft} &\n\\colhead{Hard} & \\colhead{HR2\\tablenotemark{c}} &\n\\colhead{uvw2} & \\colhead{uvm2} & \\colhead{uvw1} & \\colhead{u} & \n\\colhead{b} & \\colhead{v} \\\\ \n\\colhead{} & \\colhead{(ksec)} & \\colhead{(d)} & \\colhead{(ct s$^{-1}$)} &\n\\colhead{} & \\colhead{(ct s$^{-1}$)} & \\colhead{(ct s$^{-1}$)} &\n\\colhead{} & \\colhead{(mag)} & \\colhead{(mag)} & \\colhead{(mag)} & \n\\colhead{(mag)} & \\colhead{(mag)} & \\colhead{(mag)}\n}\n\\startdata\n\\cutinhead{V1281\\,Sco}\n00030891001 & 3.87&   2.95&$<$ 0.0030                       & \\nodata           & \\nodata&\\nodata&\\nodata&\\nodata&\\nodata &\\nodata&\\nodata&\\nodata&\\nodata \\\\\n00037164001 & 5.24& 338.66&    0.1634$^{+0.0079}_{-0.0079}$ & 0.0090$\\pm$0.0074 & 0.1619& 0.0015 &-0.98  &\\nodata&\\nodata &\\nodata&\\nodata&\\nodata&\\nodata \\\\\n00037164002 & 3.45& 344.07&    0.2429$^{+0.0120}_{-0.0120}$ & 0.0062$\\pm$0.0057 & 0.2414& 0.0015 &-0.99  &19.50  &19.64   &18.20  &\\nodata&\\nodata&\\nodata \\\\\n00037164003 & 4.24& 351.05&    0.6376$^{+0.0282}_{-0.0282}$ & 0.0047$\\pm$0.0053 & 0.6346& 0.0030 &-0.99  &20.32  &\\nodata &\\nodata&\\nodata&\\nodata&\\nodata \\\\\n00037164005 & 1.66& 361.10&    0.2727$^{+0.0185}_{-0.0185}$ & 0.0012$\\pm$0.0081 & 0.2723& 0.0003 &-1.00  &20.43  &\\nodata &\\nodata&\\nodata&\\nodata&\\nodata \\\\\n00037164006 & 2.89& 366.41&    0.2284$^{+0.0129}_{-0.0129}$ & 0.0097$\\pm$0.0089 & 0.2262& 0.0022 &-0.98  &20.42  &\\nodata &\\nodata&\\nodata&\\nodata&\\nodata \\\\\n00037164007 & 2.02& 432.69&    0.0853$^{+0.0079}_{-0.0079}$ & 0.0002$\\pm$0.0063 & 0.0853& 0.0000 &-1.00  &20.22  &\\nodata &19.11  &\\nodata&\\nodata&\\nodata \\\\\n00090248001 & 4.68& 819.99&$<$ 0.0013                       & \\nodata           & \\nodata&\\nodata&\\nodata&20.32  &$>$20.56&20.07  &\\nodata&\\nodata&19.30 \\\\\n\\enddata\n\\tablenotetext{a}{Days after visual maximum, see Table \\ref{chartable}.}\n\\tablenotetext{b}{corrected for PSF losses and bad columns.\nThe 3 sigma upper limits are given when there is no detection \nbetter than 3 sigma.}\n\\tablenotetext{c}{Hardness ratios defined as HR1=H\/S and HR2=(H-S)\/(H+S) with \nHard(H)=1-10\\,keV and Soft(S)=0.3-1\\,keV}\n\\tablecomments{Table \\ref{fullswift} is published in its entirety in the \nelectronic edition of the {\\it Astrophysical Journal}.  A portion is shown \nhere for guidance regarding its form and content.}\n\\end{deluxetable}\n\n\\begin{figure*}[htbp]\n\\includegraphics[angle=90,scale=0.60]{sssgood.ps}\n\\caption{The X-ray epochs for {\\it Swift}\\ sources with the best SSS\nphase coverage.\nThe novae are arranged by increasing optical emission line FWHM with \nthe FWHM values shown either left or right of the source.  \"U\" is used \nfor novae with unknown FWHM velocities.  Refer to Table \\ref{colordescriptions}\nfor a summary of the color coding.  \\label{sssgood}}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\includegraphics[angle=90,scale=0.60]{sssbad.ps}\n\\caption{Same as Figure \\ref{sssgood} except for the {\\it Swift}\\ sources without\nsignificant SSS detections.  \\label{sssbad}}\n\\end{figure*}\n\nFigures \\ref{sssgood} and \\ref{sssbad} show the X-ray observations of the \n{\\it Swift}\\ novae as a function of time since visual maximum.  The novae are \norganized from bottom to top in increasing FWHM values (Table \n\\ref{chartable}), with the FWHM alternating on the left and right sides\nof the figures.  Novae with unknown FWHM are labeled \"U\" and placed\nat the bottom.  Figure \\ref{sssgood} shows the novae with confirmed SSS \nemission while Figure \\ref{sssbad} shows the\nnovae with no current SSS detections.  Note that some novae in \nFigure \\ref{sssbad}, particularly the slowly evolving ones V5558 Sgr \nand V2468 Cyg,\nmay eventually evolve to the SSS phase.  Figure \\ref{sssother} is the same \nbut for well observed SSS novae observed prior to the launch of {\\it Swift};\nthese novae typically have much poorer observational coverage.\nThe black stars are the individual {\\it Swift}\\ observations.  \nThe figures also contain supplemental observations obtained with other\nX-ray facilities, {\\it Chandra}, {\\it XMM}, {\\it ASCA}, {\\it RXTE}, \n{\\it BeppoSax}, and {\\it ROSAT}\\ which are shown as circles, downward pointing \ntriangles, upward pointing triangles, yellow squares, diamonds, and red \nsquares, respectively.  The colors associated with each bar give the \ntype of emission observed based on the hardness ratio.  Red bars indicate\ntime intervals when the HR2 of an individual source was $\\lesssim -0.3$ \nand the uncertainty in the relative error was $<$ 5\\%; the photons in this\ncase are primarily soft and these regions are associated with the SSS phase.\nOrange bars designate observations with the same hardness ratio but \nlarger errors.  Yellow\nshows regions between observations where hard\/soft change occurred.\nThe orange and yellow regions represent the maximum limits of the soft \nphase since the transition occurred at some point during these times.  \nSection \\ref{sssphase} describes the SSS phase in greater detail.  \nGreen regions show times when the overall detected spectrum was \nhard, HR2 $>$ -0.3 and section \\ref{xrayevol} discusses this phase.  \nFinally, blue represents \ntimes of non-detections.  Table \\ref{colordescriptions} also gives the \ncolor descriptions for Figures \\ref{sssgood} - \\ref{sssother}.\n\n\\begin{deluxetable}{lll}\n\\tablecaption{Figures \\ref{sssgood} - \\ref{sssother} detection\ndefinitions and descriptions and symbol legend\\label{colordescriptions}}\n\\tablewidth{0pt}\n\\tabletypesize{\\scriptsize}\n\\tablehead{\n\\colhead{Color} & \\colhead{HR2\\tablenotemark{a} and error} & \n\\colhead{X-ray emission} \n}\n\\startdata\nBlue   & \\nodata & Undetected \\\\\nGreen  & $>$-0.3 & Hard \\\\\nYellow & \\nodata & Transition between Green and Orange\/Red classification. \\\\\nOrange & $\\lesssim$-0.3 and $>$ 5\\% error & Soft but with large uncertainty, highly variable during initial rise. \\\\\nRed    & $\\lesssim$-0.3 and $<$ 5\\% error & Soft X-rays \\\\\n\\hline\n\\multicolumn{3}{c}{Symbol legend} \\\\\n\\hline\n{\\it Swift} & stars & \\\\\n{\\it Chandra} & circles & \\\\\n{\\it XMM} & downward pointing triangles & \\\\\n{\\it ASCA} & downward pointing triangles & \\\\\n{\\it RXTE} & yellow squares & \\\\\n{\\it BeppoSax} & diamonds & \\\\\n{\\it ROSAT} & red squares & \\\\\n\\enddata\n\\tablenotetext{a}{HR2=(H-S)\/(H+S) with\nHard(H)=1-10\\,keV and Soft(S)=0.3-1\\,keV}\n\\end{deluxetable}\n\nSeveral trends are evident in Figures \\ref{sssgood} - \\ref{sssother}.\nAs the FWHM decreases, the novae in the sample become SSS later and remain \nin the SSS phase longer.  This behavior is consistent with \nlarger expansion velocity novae originating on higher mass WDs\n\\citep{2009cfdd.confE.199S}.  In addition the early, hard detections \nare generally only observed in the high FWHM novae.  The trends evident in \nFigures \\ref{sssgood} and \\ref{sssother} allow for a straightforward \ninterpretation of Figure \\ref{sssbad} - fast novae (loci at the top \nof the panel) are infrequently observed in the SSS phase as \nearly X-ray observations of these systems is often absent.  The slower \nnovae at the bottom of the figure have not been followed with \nsufficient temporal coverage late in their evolution or they \nhave not yet reached the SSS phase or have ceased nuclear burning before\ntheir ejecta clears sufficiently to observe SSS emission.\n\nA note of caution about using Figures \\ref{sssgood} and \\ref{sssbad} to\ndetermine nuclear burning time scales is appropriate.  These figures \nare based only on the strength and error of HR2 as provided in \nTable \\ref{fullswift} which is based on a fixed hardness threshold for\nall novae in the table.\nNovae that have significant intrinsic hard emission such as V407 Cyg\nmay not be classified as SSSs by this definition even though they \nhave soft X-ray light curves typical of nuclear burning and cessation\non the WD (see Section \\ref{v407cygSSS}). \nHigh extinction will have a similar effect.  The red regions\ngenerally also overestimate the duration of the SSS since that phase is\nalso defined by a tremendous rise in the soft X-ray count rate.\nSections \\ref{S:ton} and \\ref{S:toff} provide the determination of nuclear \nburning time scales for the X-ray nova sample.\n\n\\begin{figure*}[htbp]\n\\includegraphics[angle=90,scale=0.60]{non_swift.ps}\n\\caption{Same as Figure \\ref{sssgood} but for pre-{\\it Swift}\\ \nSSS novae.\\label{sssother}}\n\\end{figure*}\n\nFor completeness, Table \\ref{othertable} gives a summary of all the \npublicly available, pointed {\\it XMM}\\ and {\\it Chandra}\\ nova observations.\nThe columns are the nova name, the observational \nidentifier, the exposure time, Julian date and day after visual \nmaximum of the observation, and a short comment on the result of \nthe observation.  The instrument set up is also given in the 2nd \ncolumn for the {\\it Chandra}\\ observations.  In some cases this data set \nprovides important information on the SSS status of some sources\ndue to a lack of or weak {\\it Swift}\\ detections. \nAn example would be the {\\it XMM}\\ observations\nof V574 Pup which confirms that there was a strong SSS during the \ninterval between 2005 and 2007 when there were no {\\it Swift}\\ data \n\\citep{Heltonthesis}.\n\n\\begin{deluxetable}{llrcrl}\n\\tablecaption{Pointed Chandra and XMM observations of recent novae \n\\label{othertable}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Name} & \\colhead{Obs ID\\tablenotemark{a}} & \n\\colhead{Exp (ks)} & \\colhead{JD} & \\colhead{Days\\tablenotemark{b}} & \n\\colhead{Result\\tablenotemark{c}}\n}\n\\startdata\nCI Aql    & 2465 (ACIS-S)     &  2.2 & 2452062 &  396 & Faint (1) \\\\\n          & 2492 (ACIS-S)     & 19.9 & 2452123 &  457 & Faint (1) \\\\\n          & 0652760201        & 26.9 & 2455577 & 3912 & NPA \\\\\nCSS081007\\tablenotemark{d} & 9970 (HRC-S\/LETG) & 35.2 & 2454818 &  222:& SSS (2) \\\\ \nIM Nor    & 3434 (ACIS-S)     &  5.6 & 2452317 &   28 & Not detected (3) \\\\ \n          & 2672 (ACIS-S)     &  4.9 & 2452425 &  136 & Faint and hard (3) \\\\\nKT Eri    & 12097 (HRC-S\/LETG)& 15.2 & 2455219 &   69 & SSS (4) \\\\\n          & 12100 (HRC-S\/LETG)&  5.1 & 2455227 &   77 & SSS \\\\\n          & 12101 (HRC-S\/LETG)&  5.1 & 2455233 &   83 & SSS \\\\\n\t  & 12203 (HRC-S\/LETG)&  5.1 & 2455307 &  157 & SSS \\\\\nNova LMC 2000 & 0127720201    & 16.3 & 2451751 &   14 & Faint and hard (5) \\\\\n              & 0127720301    & 10.0 & 2451785 &   48 & Hard (5) \\\\\n\t      & 0127720401    & 10.5 & 2451998 &  291 & Not detected (5) \\\\\nNova LMC 2009a& 0610000301    & 37.7 & 2454957 &   90 & SSS \\\\\n              & 0610000501    & 58.1 & 2455032 &  165 & SSS \\\\\n              & 0604590301    & 31.9 & 2455063 &  196 & SSS \\\\ \n              & 0604590401     & 51.2 & 2455097 &  230 & SSS \\\\ \nNova SMC 2005 & 0311590601    & 11.6 & 2453807 &  219 & Not detected \\\\\nU Sco     & 12102 (HRC-S\/LETG)& 23.2 & 2455241 &   17 & SSS (6) \\\\\n          & 0650300201        & 63.8 & 2455247 &   23 & SSS (7) \\\\\n          & 0561580301        & 62.8 & 2455259 &   35 & SSS (7) \\\\\nV1065 Cen & 0555690301        &  9.4 & 2454837 &  714 & Not detected \\\\\nV1187 Sco & 4532 (ACIS-S)     &  5.2 & 2453305 &   96 & $<$ 2keV + NVII line \\\\\n          & 4533 (HRC-S\/LETG) & 26.1 & 2453401 &  181 & Not detected \\\\\n          & 0404431101        &  4.7 & 2454161 &  941 & Not detected \\\\\n          & 0404430301        &  9.4 & 2454161 &  941 & Not detected \\\\\n          & 0555691001        &  7.1 & 2454904 & 1684 & Not detected \\\\ \nV1280 Sco & 0555690601        &  4.5 & 2454903 &  756 & 1-keV emission \\\\\nV2361 Cyg & 0405600101        & 11.1 & 2453868 &  456 & Not detected \\\\\n          & 0405600401        & 14.9 & 2454028 &  616 & Not detected \\\\\nV2362 Cyg & 0506050101        &  9.9 & 2454225 &  394 & thermal plasma (8) \\\\\n          & 0550190501        & 27.9 & 2454821 &  990 & Very weak \\\\\nV2467 Cyg & 0555690501        &  7.0 & 2454780 &  605 & SSS \\\\\nV2487 Oph & 0085580401        &  8.3 & 2451965 &  986 & thermal plasma (9)\\\\\n          & 0085581401        &  8.1 & 2452157 & 1178 & thermal plasma (9)\\\\\n          & 0085581701        &  7.6 & 2452331 & 1352 & thermal plasma (9)\\\\\n          & 0085582001        &  8.5 & 2452541 & 1562 & thermal plasma + Fe K$\\alpha$ line (9)\\\\\nV2491 Cyg & 0552270501        & 39.3 & 2454606 &   39 & SSS (10) \\\\\n          & 0552270601        & 30.0 & 2454616 &   49 & SSS (11) \\\\\nV2575 Oph & 0506050201        & 14.9 & 2454347 &  569 & Not detected \\\\ \nV2576 Oph & 0506050301        & 11.5 & 2454376 &  544 & Not detected \\\\\nV2615 Oph & 0555690401        &  9.7 & 2454922 &  735 & Not detected \\\\\nV351 Pup  & 0304010101        & 51.8 & 2453525 & 4908 & Faint \\\\\nV458 Vul  & 0555691401        & 11.7 & 2454780 &  459 & weak 1-keV emission \\\\\nV4633 Sgr & 0085580301        & 10.2 & 2451828 &  933 & weak (12) \\\\\n          & 0085581201        &  7.3 & 2451977 & 1082 & weak (12)\\\\\n\t  & 0085581301        & 11.6 & 2452159 & 1264 & weak (12)\\\\\nV4643 Sgr & 0148090101        & 11.9 & 2452716 &  750 & Not detected \\\\\n          & 0148090501        & 11.0 & 2452894 &  928 & Not detected \\\\\nV5114 Sgr & 0404430401        &  7.9 & 2454167 & 1086 & Not detected \\\\\n          & 0404431201        &  3.6 & 2454167 & 1086 & Not detected \\\\\nV5115 Sgr & 0405600301        &  9.2 & 2454005 &  566 & weak SSS \\\\\n          & 0550190201        & 14.9 & 2454925 & 1486 & weak detection \\\\ \nV5116 Sgr & 0405600201        & 12.9 & 2454164 &  608 & SSS (13) \\\\\n          & 7462 (HRC-S\/LETG) & 35.2 & 2454336 &  780 & SSS (14) \\\\\n          & 0550190101        & 26.6 & 2454893 & 1337 & Not detected \\\\ \nV574 Pup  & 0404430201        & 16.6 & 2454203 &  872 & SSS  \\\\\nV598 Pup  & 0510010901        &  5.5 & 2454402 &  146 & SSS (15) \\\\\nXMMSL1 J060636\\tablenotemark{d} & 0510010501   &  8.9 & 2454270 &  627 & SSS (16) \\\\\n\\enddata\n\\tablenotetext{a}{{\\it Chandra}\\ observations have a four digit IDs and are followed\nby the instrument configuration. {\\it XMM}\\ observations have 10 digit IDs.}\n\\tablenotetext{b}{Days since visual maximum, see Table \\ref{chartable}.}\n\\tablenotetext{c}{The number in parenthesis is the code to the published data.\nNPA stands for \"Not Publicly Available\" and indicates proprietary observations\nat the time of this publication.}\n\\tablenotetext{d}{Full novae names are CSS081007030559+054715 and\nXMMSL1 J060636.2-694933.} \\\\\n\\tablecomments{\n(1) \\citet{2002A&A...387..944G};\n(2) \\citet{2009ATel.1910....1N};\n(3) \\citet{2005ApJ...620..938O};\n(4) \\citet{2010ATel.2418....1N};\n(5) \\citet{2003A&A...405..703G};\n(6) \\citet{2010ATel.2451....1O};\n(7) \\citet{2010ATel.2469....1N};\n(8) \\citet{2007ATel.1226....1H};\n(9) \\citet{2007ASPC..372..519F}; \n(10) \\citet{2008ATel.1561....1N};\n(11) \\citet{2008ATel.1573....1N};\n(12) \\citet{2007ApJ...664..467H};\n(13) \\citet{2008ApJ...675L..93S};\n(14) \\citet{2007ATel.1202....1N};\n(15) \\citet{2008A&A...482L...1R};\n(16) \\citet{2009A&A...506.1309R}.\n}\n\\end{deluxetable}\n\n\\section{THE EARLY HARD X-RAY PHASE}\n\nSome novae have hard X-ray emission, e.g. $>$ 1 keV, early \nin the outburst.  These novae tend to be fast or recurrent novae.  This \ninitial hard emission is thought to arise from shock heated gas \ninside the ejecta or from collisions with external material, {\\it e.g} the \nwind of the red giant secondary in RS Oph \\citep{2006ApJ...652..629B,\n2006Natur.442..276S,2007ApJ...665..654V, 2009ApJ...691..418D}.  Early hard \nX-ray emission observed in the very fast nova V838 Her has been attributed \nto intra-ejecta shocks from a secularly increasing ejection velocity \n\\citep{1992Natur.356..222L,1994MNRAS.271..155O}.  Much later in the outburst\nwhen nuclear burning has ceased hard X-rays can again dominate.  These hard\nX-rays come from line emission from the ejected shell and\/or emission \nfrom the accretion disk \\citep{2002AIPC..637..345K}, or in the case of\nRS Oph, the re-emergence of the declining shocked wind emission once the \nSSS emission has faded \\citep{2008ASPC..401..269B}.\n\nEvery nova with a FWHM $\\ge$ 3000 km s$^{-1}$ and observations within 100 \ndays after visual maximum in the {\\it Swift}\\ sample exhibited hard X-rays.  \nThis detection rate is partially due to the fact that many of these novae \nwere high interest targets, {\\it e.g.} very bright at visual maximum (KT Eri),\nextreme ejection velocity (V2672 Oph), RN (RS Oph, V407 Cyg and\nU Sco), detected prior to outbursts as an X-ray source (V2491 Cyg), etc.; \nthus their early X-ray evolution was well documented.  In addition, \na higher cadence of observations during the early phases greatly \nincreased the probability of discovery.  \n\nThe evidence of initial hard X-ray emission for slow novae is sparse\nas few were well sampled early in their outbursts.  Only V458 Vul\n\\citep{2009AJ....137.4160N,2009PASJ...61S..69T} had early \nobservations which showed a hard component with a duration of hundreds\nof days from its first observation $\\sim$ 70 days after visual maximum.\nThe lack of significant evidence of hard emission in the early outburst \nof slow novae is consistent with\nshocks, either within the ejecta or with a pre-outburst ambient medium, \nbeing the primary source of early hard X-ray emission in the\nfaster novae.  Slower novae have lower ejection speeds\nand thus should either have less or delayed shock emission\n\\citep[see equ. 3 in][]{2006ApJ...652..629B}.  Hard X-rays\nwere also detected late in the outburst of novae with extreme and multiple \nejection events.\nThe best example of this is V2362 Cyg \nwhich at the time of its unusually bright secondary maximum had\nalready doubled the width of its emission lines and was detected \nas a hard X-ray source \\citep{2008AJ....136.1815L}.  \nSimilarly the slow nova V5558 Sgr was also \na late hard source.  Its early light curve was marked by numerous \nsecondary maxima similar to V723 Cas \\citep{2008NewA...13..557P}.  \n\nAnother interesting case is the slow nova V1280 Sco which was detected \nmultiple times between days 834 - 939 after outburst as an X-ray source.  \n\\citet{2009ATel.2063....1N} found that the X-ray count rate was \nrelatively low and the SED was best fit with multiple thermal plasma \nmodels consistent with line emission.  They attributed the lines \nfrom shock heating of the ejecta but this is difficult to reconcile \nwith how rapidly shock emission declines.  \\citet{2010ApJ...724..480H} showed\nthat V1280 Sco had two bright secondary peaks after maximum. Thus, it is\npossible that this nova experienced additional ejection events later in\nthe outburst that contributed the necessary energy to power shocks.\nContemporary optical spectra from our Small and Moderate Aperture\nResearch Telescope System (SMARTS) nova monitoring program\nshow that the photosphere of V1280 Sco remains optically thick with \nP-Cygni profiles still present more than 4 years after outburst. \nAlternatively, the line emission may be from circumstellar gas \nphotoionized by the initial X-ray pulse of the explosion.  Given the \nrelative proximity of V1280 Sco, ranging from 0.63$\\pm$0.10 kpc\n\\citep{2010ApJ...724..480H} to 1.6 kpc \\citep{2008A&A...487..223C}, any X-ray \nemission lines would be much brighter than most novae in our sample which has\na larger median distance of 5.5 kpc.  Unfortunately, V1280 Sco was X-ray \nfaint making it impossible to determine the source of its X-ray emission.\n\n\n\\section{THE SSS PHASE\\label{sssphase}}\n\n\\subsection{Rise to X-ray Maximum and the ``Turn-on'' Time\\label{S:ton}}\n\nThe unprecedented temporal coverage of the early outburst in X-rays with\n{\\it Swift}\\ has fully revealed a new phenomenon during the rise to X-ray maximum.\nPrior to {\\it Swift}, V1974 Cyg had the best sampled X-ray light curve\n\\citep[see Fig. 1 in ][]{1996ApJ...456..788K}.  The 18 {\\it ROSAT}\\ \nobservations showed a slow and monotonic rise to maximum. This light \ncurve evolution was expected as the obscuration from the ejecta clears \nand the effective temperature of the WD photosphere increases\n\\citep{1985ApJ...294..263M}.  However {\\it Chandra}\\ observations of\nV1494 Aql \\citep{2003ApJ...584..448D} and V4743 Sgr\n\\citep{2003ApJ...594L.127N} hinted that this transition was not as smooth\nas previously observed, with short term \"bursts\", periodic oscillations,\nand sudden declines.  \n\nWith daily and sometimes hourly {\\it Swift}\\ coverage,\nthe rise to X-ray maximum is unequivocally highly chaotic with large \nchanges in the count rate evident in all well observed {\\it Swift}\\ \nnovae to date.  Figure \\ref{kteriearlylc} illustrates this \nphenomenon in KT Eri from the data available in Table \\ref{fullswift}.\nDuring the initial rise to X-ray maximum, it exhibited large\noscillations.  The numerous large declines are even more dramatic when the\nobservational data sets are not grouped by observation\nID number as in Table \\ref{fullswift} \nbut broken into small increments (Walter et al. in prep).\nAt 76 days after visual maximum the variability became much smaller \nand the count rate stabilized around $\\sim$150 ct s$^{-1}$. \nIn addition to KT Eri \\citep{2010ATel.2392....1B},\nRS Oph \\citep{2011ApJ...727..124O}, U Sco \\citep{2010ATel.2430....1S},\nnova LMC 2009a \\citep{2009ATel.2025....1B,Bode2011},\nV2672 Oph \\citep{2009ATel.2173....1S}, \nV2491 Cyg \\citep{2010MNRAS.401..121P,2011arXiv1103.4543N}, \nand V458 Vul \\citep{2009AJ....137.4160N} \nall showed this large amplitude variability.  The first three novae \nare known RNe while the next two and KT Eri are suspected to be\nRNe based on their observational characteristics.  \nThe fact that the less energetic V458 Vul \nalso exhibited this phenomenon indicates that it is not just associated\nwith very fast or recurrent novae. \nSee Section \\ref{var} for further discussion of nova variability.\n\n\\begin{figure*}[htbp]\n\\plotone{KTEriearlylc.ps}\n\\caption{The early X-ray light curve of KT Eri in days since visual maximum.\nThe top panel shows the count rate and the lower panel gives the hardness\nratio, HR1.  Dotted lines are added to the top panel to emphasize the\nvariability.  Prior to day 65 KT Eri was faint and hard.\nBetween days 65 and 75 the source transitioned to the bright SSS phase\nwith large amplitude oscillations in the count rate and some corresponding\nchanges in HR1.  After day 76 the both the count rate and\nhardness ratio significantly stabilized but still showed variability\n(see Section \\ref{var}).\n\\label{kteriearlylc}}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\plotone{turnonvsvel.ps}\n\\caption{SSS turn-on time of novae (Table \\ref{timescales})\nas a function of the ejection velocity (estimated from the FWHMs\nin Table \\ref{chartable}).  \nFilled circles are known RNe.  Half filled circles\nare suspected RNe based on their characteristics.\nFrom the top to the bottom the lines\nshow the relationship from Shore (2008; Eqn. 9.2) for ejected masses\nof 1$\\times$10$^{-3}$, 1$\\times$10$^{-4}$, 1$\\times$10$^{-5}$,\n1$\\times$10$^{-6}$, and 1$\\times$10$^{-7}$ M$_{\\odot}$, respectively.\nThe downward and upward arrows are estimated upper and lower limits.\n\\label{velturnon}}\n\\end{figure*}\n\nThe emergence of the SSS, referred to as ``turn-on'' time or t$_{on}$ \nhereafter, provides information on the mass of the ejected shell.  \nThe turn-on times for the novae in this sample are given in \nTable \\ref{timescales}.  The t$_{on}$ time is defined as the time \nafter visual maximum when the HR2 $<$ -0.8 and there is significant\nincrease in the soft count rate.  Similarly the ``turn-off'' \ntime (t$_{off}$) is defined as the time after t$_{on}$ when the\nhardness ratio becomes harder, HR2 $>$ -0.8, and the soft count rate\ndeclines rapidly as nuclear burning ends.  Note that these\ndefinitions should not be confused with the SSS phases as shown in \nFigures \\ref{sssgood} - \\ref{sssother} as t$_{on}$ and t$_{off}$ also\ninclude the change in the soft count rate.  SSS emission can only be \nobserved when the ejecta column density declines \nto the point where the source can be observed. With the expansion\nvelocity and turn-on time, upper limits on the ejected mass can be\nestablished.  \\citet{Shore08} gives the relationship (see Equation 9.2),\n\\begin{equation}\nM_{eject} \\sim 6 \\times 10^{-7} \\phi N_{H}(22) v_{exp}(1000)^2 t_{on}^2 M_{\\odot} \\\\\n\\end{equation}\nwhere $\\phi$ is the filling factor, N$_H$(22) is the column density \nin units of 10$^{22}$ cm$^{-2}$, v$_{exp}$(1000) is the expansion velocity in \nunits of 1000\\ km\\ s$^{-1}$, t$_{on}$ is the soft X-ray turn-on time\nin days and assumes spherical geometry. In this study, $\\phi$ = 0.1 and a\ncolumn density of 10$^{22}$ cm$^{-2}$ is used as the minimum N$_H$ for\nthe ejected shell to become transparent to soft X-rays.\nThe expansion velocity is determined from v$_{exp}$ = FWHM\/2.355\n\\citep{2010PASP..122..898M} where FWHM is the width of the Balmer\nlines near visual maximum as given in Table \\ref{chartable}.\nUsing the t$_{on}$ times from Table \\ref{timescales}, \nFigure \\ref{velturnon} shows the estimated ejected masses as a function \nof ejection velocity.  \n\\btxt{Note that the velocities derived from these FWHMs are lower limits\nas the X-ray opacity in the ejecta depends on faster material.  This has\nthe effect of shifting all the points in Figure \\ref{velturnon} to the \nright.}\nAccordingly the fastest novae, at the bottom \nright, U Sco and V2672 Oph, must have ejected much less than 10$^{-5}$ \nM$_{\\odot}$ otherwise they would not have been observed as SSS sources \nso early after outburst.  This inference is consistent with independent \nejected mass estimates \n\\citep[{\\it e.g.} U Sco,][]{2000AJ....119.1359A,2010AJ....140.1860D,\n2010ApJ...720L.195D}.\nConversely, novae in the upper left corner must eject a significant \namount of material.  Large mass ejection events are also inferred from \nthe optical spectra of novae like V1280 Sco which still showed P-Cygni \nlines 3 years after outburst \\citep{2010PASJ...62L...5S} and a year \nlater in our recent SMARTS spectroscopy.  \n\nNote that external extinction from the ISM is not taken into account \nin Figure \\ref{velturnon} nor is the evolution of the \neffective temperature of the WD photosphere. Novae with large extinction\nmay never be observed in the SSS phase while a slow increase in the \nWD temperature after the ejecta has sufficiently cleared will delay\nthe onset of t$_{on}$ resulting in an overestimate of the ejected mass\nderived from Equ. 1.   Both factors along with deviations\nfrom the underlying assumptions such as different filling factors\nand non-spherical symmetry, can lead to different mass values given\nin Figure \\ref{velturnon}.  These limitations explain why two novae\nwith the same ejection velocities, V2468 Cyg and V5558 Sgr at \n425 km s$^{-1}$, can have divergent mass estimates due to different\nturn-on times.\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{SSS X-ray time scales\\label{timescales}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Name} & \\colhead{turn-on} & \\colhead{turn-off} \\\\\n\\colhead{} & \\colhead{(d)} & \\colhead{(d)}\n}\n\\startdata\nCI Aql & \\nodata & $<$396 \\\\\nCSS 081007 & 185$\\pm$68 & 314$\\pm$68 \\\\\nGQ Mus & \\nodata & 3484.5$\\pm$159.5 \\\\\nIM Nor & $>$28 & $<$136 \\\\\nKT Eri & 71$\\pm$1 & 280$\\pm$10 \\\\\nLMC 1995 & $<$1087 & 2545$\\pm$426 \\\\\nLMC 2000 & $>$48 & $<$291 \\\\\nLMC 2009a & 95$\\pm$5 & 270$\\pm$10 \\\\\nRS Oph & 35$\\pm$5 & 70$\\pm$2 \\\\\nU Sco & 23$\\pm$1 & 34$\\pm$1 \\\\\nV1047 Cen & $>$144 & $<$972 \\\\\nV1065 Cen & \\nodata & $<$744\\tablenotemark{a} \\\\\nV1187 Sco & \\nodata & $<$181\\tablenotemark{a} \\\\\nV1213 Cen & $<$322 & $>$494 \\\\\nV1280 Sco & $>$928 & \\nodata \\\\\nV1281 Sco & $<$339 & 627$\\pm$194 \\\\\nV1494 Aql & 217.5$\\pm$30.5 & 515.5$\\pm$211.5 \\\\\nV1974 Cyg & 201$\\pm$54 & 561.5$\\pm$50.5 \\\\\nV2361 Cyg & \\nodata & $<$456 \\\\\nV2362 Cyg & \\nodata & $<$990 \\\\\nV2467 Cyg & $<$456 & 702$\\pm$97 \\\\\nV2468 Cyg & $<$586  & \\nodata \\\\\nV2487 Oph & \\nodata & $<$986 \\\\\nV2491 Cyg & 40$\\pm$2 & 44$\\pm$1 \\\\\nV2672 Oph & 22$\\pm$2 & 28$\\pm$2 \\\\\nV351 Pup & \\nodata & $<$490 \\\\\nV382 Vel & $<$185 & 245.5$\\pm$22.5 \\\\\nV407 Cyg & 15$\\pm$5 & 30$\\pm$5 \\\\\nV458 Vul & 406$\\pm$4 & $>$1051 \\\\\nV4633 Sgr & \\nodata & $<$934 \\\\\nV4743 Sgr & 115$\\pm$65 & 634$\\pm$108 \\\\\nV5114 Sgr & $<$1086 & \\nodata \\\\\nV5115 Sgr & $<$546 & 882$\\pm$336 \\\\\nV5116 Sgr & 332.75$\\pm$275.25 & 938$\\pm$126 \\\\\nV5558 Sgr & $>$850\\tablenotemark{b} & \\nodata \\\\\nV5583 Sgr & $<$81 & 149$\\pm$68 \\\\\nV574 Pup & 571$\\pm$302 & 1192.5$\\pm$82.5 \\\\\nV597 Pup & 143$\\pm$23 & 455$\\pm$15 \\\\\nV598 Pup & \\nodata & $<$127 \\\\\nV723 Cas & $<$3698 & $>$5308 \\\\\nV838 Her & \\nodata & $<$365 \\\\\nXMMSL1 J060636 & \\nodata & $<$291  \\\\\n\\enddata\n\\tablecomments{t$_{on}$ and t$_{off}$ bracket the time after visual maximum\nwhen the hardness ratio HR2 is softer than -0.8.}\n\\tablenotetext{a}{Evolution of $[$\\ion{Fe}{7}$]$ (6087\\AA) and lack of \n$[$\\ion{Fe}{10}$]$ (6375\\AA) in our SMARTS optical\nspectra are consistent with this upper limit from the X-ray non-detection.\nSee Section \\ref{fex}.}\n\\tablenotetext{b}{Optical spectra are slowly becoming more ionized which\nis consistent with slowly increasing SSS emission observed with {\\it Swift}.}\n\\end{deluxetable}\n\n\\subsection{Turn-off time\\label{S:toff}}\n\nTable \\ref{timescales} shows t$_{off}$ times or upper\/lower limits\nfor the novae in our sample.  If optical light curve decline times,\n{\\it e.g.} t$_2$, are used as \nsimple proxies for WD masses then \nthere should be a relationship between \nt$_2$ and duration of the SSS phase.  In Figure \\ref{t2turnoff} the\nturn-off time, t$_{off}$, is shown versus t$_2$.\nOverplotted as the solid line is the turn-off versus decline relationship\nof \\citet[][Equ. 31]{2010ApJ...709..680H} where t$_3$ was converted \nto t$_2$ using Equ. 7 in \\citet{2007ApJ...662..552H}.\nThe combined uncertainties of both equations is represented by \nthe two dotted lines.  \\citet{2010ApJ...709..680H} find that the time when\nnuclear burning ends is $\\propto$ t$_{break}^{1.5}$ (Equ. 26), where\nt$_{break}$ is the time of the steepening of their model free-free \noptical-IR light curves.  This relationship is derived using a series of \nsteady state models with a decreasing envelope mass to fit the observed \nmultiwavelength light curves.  The X-ray and UV light curves are fit with \nblackbodies while the optical and IR curves use optically thin, free-free \nemission.  The parameters of the model are the WD mass, composition of\nthe WD envelope, and its mass prior to outburst.  While the general trend\nis similar, the observed data do not fit the \\citet{2010ApJ...709..680H} \nrelationship, especially when the sample is expanded to include the novae \nwith only upper or lower limits.\n\nThe relationship derived by \\citet{2010ApJ...709..680H} utilizes the\nt$_2$ derived from the $y$ band light curve instead of the $V$ band as in \nthis paper.  The $y$ band is used by \\citet{2010ApJ...709..680H} since it \ngenerally samples the continuum where as the $V$ band can have a contribution\nin the red wing from strong H$\\alpha$ line emission. However, the difference\nin filters can not explain the poor agreement between the data and the\nrelationship in Fig. \\ref{t2turnoff} since there are similar numbers of\nnovae that fall above the line as below.  If a contribution from H$\\alpha$\nin $V$ was significant then the disagreement would not be symmetric.\n\nSimilarly, Figure \\ref{velturnoff} shows the relationship between the \nFWHM and turn-off time with the dotted line depicting the \n\\citet{2003A&A...405..703G} turn-off vs. velocity relation.  This relationship\nwas derived from all the SSS nova data available at the time which was\nonly 4 well constrained SSS novae and 4 novae with turnoff limits.  With \nthe significantly larger sample currently available it is clear that \nthere is not a tight fit to the relationship.\nThis discrepancy is particularly acute for the slower novae in our sample \nwhich have turned off much sooner than expected.  These Figures illustrate \nthat the gross behavior of novae is still poorly understood\nand confirm that the observational characteristics of an individual\nnova is governed by more than just the WD mass.\n\n\\begin{figure*}[htbp]\n\\plotone{newturnoffvst2.ps}\n\\caption{SSS turn-off time as a function of t$_2$ time with the\n\\citet{2010ApJ...709..680H} relationship (solid line) and its\nassociated uncertainty (dotted lines) overplotted.  Upper and lower\nlimits are also shown. \nFilled circles are known RNe.  Half filled circles \nare suspected RNe based on their characteristics.\n\\label{t2turnoff}}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\plotone{turnoffvsvel.ps}\n\\caption{SSS turn-off time as a function of the FWHM of H$\\alpha$ or\nH$\\beta$ near visual maximum.  The relationship of \\citet{2003A&A...405..703G}\nis shown as the dotted line.  Upper and lower limits are also shown.\nFilled circles are known RNe.  Half filled circles\nare suspected RNe based on their characteristics.\n\\label{velturnoff}}\n\\end{figure*}\n\nAccurate determinations of the duration of nuclear burning\ncan also provide an independent ejected mass estimate.  Recently,\n\\citet{2010ApJ...712L.143S} found that the ejected mass is only\ndependent on the total radiated energy, E$_{rad}$, and does not\nrequire knowledge about the geometry and structure of the shell\nas with other methods. E$_{rad}$ is not a trivial value to determine\nas it depends on the bolometric luminosity of the source and the\nduration of the outburst. {\\it Swift}\\ observations can potentially\ndetermine the bolometric flux both when the bulk of the emission is\nin a narrow wavelength region such as the early, optically thick\nphases in the UV and optical or later in the soft X-ray during the\nSSS phase.  Estimates of the luminosity during both phases requires\nan accurate determination of the extinction and the distance.\nPerhaps the best example to use the \\citet{2010ApJ...712L.143S}\ntechnique on is RS Oph.  With a bolometric luminosity of\n3$\\times$10$^{4}$ L$_{\\odot}$ from TMAP atmosphere models\n\\citep{2011ApJ...727..124O} and a t$_{off}$ of 70 days the estimated\nejected mass $\\sim$ 2$\\times$10$^{-6}$ M$_{\\odot}$.  This is consistent\nwith the low mass estimates from the radio\n\\citep[(4$\\pm$2)$\\times$10$^{-7}$ M$_{\\odot}$;][]{2009MNRAS.395.1533E}\nand hydro-dynamical models of the X-ray behavior\n\\citep[1.1$\\times$10$^{-6}$ M$_{\\odot}$;][]{1992MNRAS.255..683O} and\n\\citep[$\\sim$5$\\times$10$^{-6}$ M$_{\\odot}$;][]{2009ApJ...691..418D}\n\n\\subsubsection{SSS phase durations}\n\nFigure \\ref{histogram}a shows the distribution of the duration of the SSS \nphase for this sample of novae.  Since there are still relatively few \nnovae with well established turn-off times a coarse histogram with only three \nbins is used.  The bins have durations of less than one year, between \none and three years, and greater than three years.  Due to large \nuncertainties in their exact turn-off times, ten of the sample novae \ncannot be placed within a single bin and thus are shown as the smaller \ncross-hatched columns between the bins in which they might belong.\nOf the \\totallimitSSS\\ novae with detected SSS emission or with strong \nlimits on the duration of the SSS phase, 89\\%, have turned off in under \n3 years.  There are only four novae, GQ Mus, LMC 1995, V574 Pup, and \nV723 Cas, with detected SSS emission beyond 3 years.  V458 Vul and \nV1213 Cen were still SSSs at their last observations and could also \nexceed 3 years.  A similar rapid turn-off was inferred from a search of\nthe {\\it ROSAT}\\ archive of novae with SSS detections.  \\citet{2001A&A...373..542O} \nfound only 3 SSS novae among the 39 Galactic and Magellanic cloud novae \nin the {\\it ROSAT}\\ archive observed at least once within 10 years after \nvisual maximum.  The median age of the 19 novae with documented turn-off \ntimes from this sample is 1.4 years.\n\n\n\\begin{figure*}[htbp]\n\\plottwo{histogram.ps}{M31histogram.ps}\n\\caption{Distribution of the durations of well established SSS novae \nin the Galaxy\/Magellanic Clouds (left figure) and M31 (right figure)\nfrom \\citet{2010A&A...523A..89H}.\nThe three duration bins are less than one year, between 1 and 3 years, \nand greater than three years.  The hashed areas include the novae with \nonly limits on their turnoff time that precludes placing them in a \nspecific bin.\n\\label{histogram}}\n\\end{figure*}\n\n\nThe situation is different for nova surveys of M31 \n\\citep{2007A&A...465..375P,2010A&A...523A..89H} where SSSs identified as \nclassical novae 5-10 years after outburst are fairly common, {\\it e.g.}\n1995-05b, 1995-11c, and 1999-10a.  Figure \\ref{histogram}b shows the same\nbins as before but with the 18 M31 novae detected as SSSs given in Table 9 \nof \\citet{2010A&A...523A..89H}.  The difference can be explained by the\npredominance of slower novae in the M31 sample.  The mean t$_2$ time \nof the nine M31 novae with reported decline times in the \n\\citet{2010A&A...523A..89H} SSS sample is 31 days whereas the peak \nfor our sample is significantly faster at 8 days (Figure \\ref{t2vsfwhm}). \nThe discrepancy in speed class between the two samples is due to selection \neffects.  By design the Galactic\/Magellanic sample consists primarily \nof bright and hence faster novae.  M31 surveys sample the entire \ngalaxy but with fewer {\\it Chandra}, {\\it XMM}\\ and {\\it Swift}\\ observations that are \nrandomly scattered in time.  The M31 strategy finds many novae since\nthe observed M31 nova rate is greater, $\\sim$ 30 novae yr$^{-1}$ \n\\citep{1989AJ.....97.1622C} than that of the Milky Way\n\\citep[$\\sim$ 5 novae yr$^{-1}$][]{1997ApJ...487..226S}, however, with\nlimited time sampling slower novae with longer SSS phases are easier to\ndetect than fast novae with rapid turn-offs.\n\n{\\it ROSAT}\\ detected 2 SSS novae out of 21 Galactic novae for a Milky Way \ndetection frequency of 9.5\\% \\citep{2001A&A...373..542O}. \nIf the 4 RNe in the {\\it ROSAT}\\ list\nare discarded because their observations were taken $\\gtrsim$ 1 year\nafter outburst, the detection frequency increases to 11.8\\%.  The M31\nsurvey has a similar low SSS detection frequency of 6.5\\% \n\\citep{2010AN....331..187P}.  These two results show that it is difficult\nto catch novae during their SSS phase via random time sampling.  \nHowever, a more systematic approach that 1) targets only \nbright and low extinction novae and 2) obtains multiple\nobservations early in the outburst may have a greater detection\nfrequency.  Indeed, {\\it Swift}\\ has a significantly greater SSS detection rate of \n$\\sim$ 45\\% during its five years of operation with this more systematic\napproach.  \n\n\\subsection{SSS emission in the hard X-ray spectrum of \nV407 Cyg\\label{v407cygSSS}}\n\nIn the initial analysis of the {\\it Swift}\\ data in \\citet{2011A&A...527A..98S} \na second soft component was required to fit some of the {\\it Swift}\\ X-ray spectra.\nHowever, there were insufficient counts to distinguish between a blackbody \nand an optically-thin plasma model. Assuming a distance of 2.7 kpc\n\\citep{1990MNRAS.242..653M}, the unabsorbed flux of the soft component\nin the day $<$30 model of Table 3 in \\citet{2011A&A...527A..98S} gives a \nblackbody luminosity of 2$\\times$10$^{37}$ erg s$^{-1}$ which is reasonable\nfor nuclear burning on a WD.  To investigate whether the soft emission\nin V407 Cyg can be attributed to nuclear burning, we reanalyze the {\\it Swift}\\\nX-ray data with twice as many time bins as previously used.  Figure\n\\ref{v407cygmodel} shows the results.  As in \\citet{2011A&A...527A..98S}\nthe model abundances are allowed to vary but the temperatures are not \nsignificantly different if the abundances are constrained to be solar.\nThe data prior to day 10 and after day 50 can be fit with a single optically \nthin plasma model.  The remaining 4 time bins all require a soft component\nwhich in this analysis is assumed to be a blackbody.  Both the derived\nN$_H$ and the optically thin component temperature decline with time\nin the models.  The blackbody effective temperature increases until the\nday 36 bin and declines in the day 45 bin, although the error bars are\nlarge enough that it could be constant over the last two dates.  The \nderived luminosities (over the 0.3--10~keV X-ray band) \nfor the four dates with blackbody components are \n2.3$\\times$10$^{42}$ erg s$^{-1}$, 9.3$\\times$10$^{37}$ erg s$^{-1}$,\n1.9$\\times$10$^{35}$ erg s$^{-1}$, and 3.1$\\times$10$^{35}$ erg s$^{-1}$,\nrespectively, assuming a distance of 2.7 kpc.  The extreme luminosity \nfor the day 16 bin cannot be considered reliable, given the very low \nfitted temperature of $\\sim$ 25~eV below the XRT 0.3~keV low-energy cut off.\nNevertheless, the results of fitting blackbodies to the {\\it Swift}\\ V407 Cyg \ndata are consistent with a scenario where the nuclear burning proceeded \non the WD surface near Eddington limits until about 30 days after \nvisual maximum.  The fuel was consumed after that point leading to a \nrapid drop in the luminosity.  Thus, although V407 Cyg was not a \ntrue SSS, its soft photon light curve was consistent \nwith expected evolution as seen in other novae.\n\n\\begin{figure*}[htbp]\n\\plotone{v407cygmodel.ps}\n\\caption{Results of model fits to the {\\it Swift}\\ V407 Cyg data set.  The top\nleft panel shows the total 0.3-10 keV (squares) and soft 0.3-1 keV (circles)\nlight curves.  The derived N$_H$ column for the 6 date bins is shown in \nthe top right panel.  The Mekal temperature of the hotter, optical thin\nplasma model is shown in the middle left panel while the right middle panel\nshows the temperature of the blackbody fit to the softer component. A \nsecond, soft component is not needed in the first and last date bins.\nThe bottom panels show the observed (left) and unabsorbed (right) fluxes.\nSquares give the total from all components while the circles show just\nthe blackbody contributions.  The right axis of the last panel also shows\nthe corresponding 0.3-10 keV luminosity assuming a 2.7 kpc distance.\n\\label{v407cygmodel}}\n\\end{figure*}\n\n\n\\section{DISCUSSION} \n\n\\subsection{Orbital period and turn-off time}\n\n\\citet{2003A&A...405..703G} found a correlation between the orbital \nperiod and X-ray turn-off time.  However, at that time only four novae \nhad both well determined periods and  X-ray turn-off times, GQ Mus, \nV1974 Cyg, V1494 Aql, and V382 Vel, and limits on CI Aql and U Sco.  \nThe observed trend implied \nthat novae with short orbital periods had the longest duration SSS phases.  \n\\citet{2003A&A...405..703G} attributed this relationship to \na feedback loop between the WD and its secondary.  The luminous X-rays \nproduced during the SSS phase excessively heat the facing side of the\nsecondary in short period systems.  The energy added to the outer\nlayers of the secondary causes it to expand, producing \nhigher mass loss leading to enhanced accretion of material onto \nthe WD.  \n\nSince 2003, the turn-off times of \\newturnoffperiod\\ additional \nnovae with known periods have been determined.  There are also strong \nlimits on the turn-off times of \\newturnoffperiodlimit\\\nother novae with known orbital periods. \nInclusion of this expanded sample, shown in Figure \\ref{periodvstoff}, \ncauses the trend between orbital period and duration of the \nSSS phase noted by \\citet{2003A&A...405..703G} to disappear. \nThe new distribution, with an increased sample size, shows \nno discernible correlation.  Orbital separation apparently\nhas no effect on the duration of nuclear burning. \n\n\\begin{figure*}[htbp]\n\\plotone{newperiod.ps}\n\\caption{SSS turn-off time as a function of orbital period for novae with\nwell established turn-off times and novae with good upper (i.e. still in\nthe SSS phase) and lower limits. \nFilled circles are known RNe.  Half filled circles\nare suspected RNe based on their characteristics.\nThe top plot shows the distribution\nhistogram of our sample (solid line) and of all the known novae (dotted\nline) from Table 2.5 in \\citet{Warner2008}.\n\\label{periodvstoff}}\n\\end{figure*}\n\nTo see if the lack of a trend could be explained by having a \nnon-representative sample of novae, the top panel of Figure \n\\ref{periodvstoff} shows the distribution in 1 hour orbital period \nbins of the updated \\citet{Warner2008} sample as the solid line.  \nThe distribution of all novae with orbital periods is shown as the \ndotted line and shows that the SSS sample is a consistent sub-sample\nof the known nova period distribution.\n\n\\citet{2010AJ....139.1831S} claim a similar relationship between turn-off \ntime and orbital period, albeit in highly magnetized systems.  They find that\nof the eight novae with quiescent luminosities $>$10$\\times$ brighter \nthan pre-eruption, all have long SSS phases, short orbital periods, highly \nmagnetized WDs, and very slow declines during quiescence.  \nSimilar to \\citet{2003A&A...405..703G}, \\citet{2010AJ....139.1831S}\npropose that nuclear burning on the WD is prolonged by increased \naccretion from the close secondary but in this case efficiently funneled \non the WD by the strong magnetic fields.  The 8 novae \n\\citet{2010AJ....139.1831S} cite are CP Pup, RW UMi, T Pyx, V1500 Cyg, \nGQ Mus, V1974 Cyg, V723 Cas, and V4633 Sgr.  \n\nThe hypothesis that these specific characteristics enhance the SSS duration \ncan be directly evaluated using V4633 Sgr, GQ Mus, V1974 Cyg, and V723 Cas, \nsince they all have X-ray observations within the first 3 years of \noutburst.  For CP Pup, RW UMi, T Pyx, and V1500 Cyg the assertion of a long \nlasting SSS emission phase depends on secondary evidence as none had any \ndirect X-ray observations during outburst.  Lacking direct X-ray observations\nwe will ignore these 4 sources for the test.\n\nThe first X-ray observation of V4633 Sgr was obtained 934 \ndays after visual maximum but it and subsequent observations were of \na hard source implying that any SSS emission was missed.\nWith an upper limit of 2.5 years for its SSS emission, V4633 Sgr can not\nbe considered a long-lived SSS nova based on the distribution shown \nin Figure \\ref{histogram}a.  The SSS duration in V1974 Cyg was\neven shorter and much better constrained at 1.53$\\pm$0.14 years.  In\naddition, V1974 Cyg was not \"excessively\" luminous in outburst as alleged in \n\\citet{2010AJ....139.1831S}.  Its early UV plus optical fluxes were\nconsistent with the Eddington luminosity of a WD with a mass range of \n0.9-1.4 M$_{\\sun}$ \\citep{1994ApJ...421..344S}.  The later \"excessive\" X-ray \nluminosities of \\citet{1998ApJ...499..395B} were derived from blackbody fits \nwhich are known to predict higher luminosities than model atmospheres \nfit to the same data.\nWhile V723 Cas has the longest SSS duration known among novae ($\\gtrsim 15$ \nyrs), its orbital period is very long at 16.62 hrs and significantly longer\nthan that of GQ Mus, 1.43 hrs.  The claim of magnetic activity in V723 Cas \nis based on the different periodicities observed in the early light curve \nindicating an intermediate polar (IP).  However, the multiple periodicities \nused as evidence by \\citet{2010AJ....139.1831S} were from data \nobtained early in the outburst while the nova ejecta were still clearing \n\\citep{1998CoSka..28..121C}.  Photometry obtained at this early stage of\ndevelopment frequently results in noisy periodograms.  Data obtained\nlater in the outburst by \\citet{2007AstBu..62..125G} and over the\nthe last 4 years from our own photometric monitoring \\citep[Hamilton C.,\nprivate communication,][]{2007AAS...210.0404S} reveal a well defined\n16.7 hr period with a large $\\sim$ 1.5 magnitude amplitude in the UV, \noptical and NIR bands.  There is no other evidence in the literature \nto support that V723 Cas is magnetic. \nOf the 4 novae with supporting X-ray observations, only GQ Mus fully\nmatches the criteria of a long lasting SSS on a magnetic WD in \na short period system.  \n\nWith our expanded X-ray sample there are 3 additional novae with well \nconstrained SSS durations that can potentially be used to test the hypothesis.\nV4743 Sgr \\citep{2006AJ....132..608K}, V597 Pup \\citep{2009MNRAS.397..979W}, \nand V2467 Cyg \\citep{2008ATel.1723....1S} are IP candidates and thus \nbelieved to have strong magnetic fields.  The orbital periods for \nV597 Pup and V2467 Cyg are relatively short at 2.66 and 3.8 hrs,\nrespectively, but the period in V4743 Sgr is much longer at 6.74\nhrs, see Table \\ref{chartable}.  While the turn-off times for\nthese novae are all longer than one year they are not exceptionally\nlong, with durations of 1.74$\\pm$0.29, 1.25$\\pm$0.04, and 1.85$\\pm$0.33\nyears for V4743 Sgr, V597 Pup, and V2467 Cyg, respectively.  Thus\nthe data available do not imply that short orbital period or\nstrong magnetic fields produce significantly longer SSS \nthan the average novae from our sample.\n\nAn interesting question is why there is no trend between orbital\nperiod and SSS duration as the underlying assumption of enhanced\naccretion due to heating of the secondary is sound.  One reason \nwould be that there is no significant enhancement in the \nmass transfer rate from the illuminated secondary, perhaps from shielding\ndue to a thick disk.  Another possibility is that there is an\neffect but it is subtle and affected by other variables such as the\nstrength of the magnetic field, composition of the accreted material,\nWD mass, etc.  Another possibility is that an accretion disk can not\nform under the harsh conditions during the SSS phase which inhibits \nadditional mass transfer.  More observations of novae with different \ncharacteristics are required in order to understand the underlying\nphysics.\n\n\\subsection{Dusty novae}\n\nThe creation, evolution, and eventual destruction of dust occurs on \nrelatively rapid time-scales in novae making them excellent objects \nfor understanding dust grain formation.  One curious aspect of dust\nin novae is how grains can grow within the harsh photoionizing \nenvironment.  A correlation of the recent \\textit{Spitzer}\nspectroscopic observations of dusty novae \n\\citep[see][for examples]{WoodStar11,Heltonthesis}\nwith this large X-ray sample can bring insights to why most novae do \nnot form dust and the reasons for the large differences in composition \nand amounts in the novae that do form dust.\n\nIn general, it is believed that grain growth occurs within dense clumps \nin the ejecta where they are shielded from hard radiation.  Spectroscopic \nand direct imaging show that nova shells are inherently clumpy \n\\citep{1997AJ....114..258S,OBB08}.  Grain formation inside dense clumps \nalso explains the higher frequency of dust in slow novae \n\\citep[see Table 13.1 in][]{ER08} as they eject more material at lower \nvelocities and suffer greater remnant shaping than fast novae and thus \nprovide more protection for grain formation. However, even fast novae \nwith small ejected masses have shown some dust formation, such as V838 Her\n\\citep{2007ApJ...657..453S}.  A contrary view was proposed that ionization \nactually promotes dust formation via the accretion of grain clusters through\ninduced dipole interactions \\citep{2004A&A...417..695S}.\n\nKnown and likely dusty novae represent 31\\% of the X-ray sample but only \ntwo, V2467 Cyg and V574 Pup, were also SSSs.  While there were no \ncharacteristic dips in either visual light curve indicating significant \ndust formation \\citep{2009AAS...21349125L,2005IBVS.5638....1S}, both novae \nshowed evidence of some dust formation from the presence of weak silicate \nemission features in the late \\textit{Spitzer} mid-IR spectra \n\\citep{WoodStar11}.  In V2467 Cyg the first {\\it Swift}\\ X-ray detection was 458 \ndays after maximum. It was weak but dominated by soft photons.  The following \n{\\it Swift}\\ observation on day 558 revealed the nova was still soft but\nwas also almost 3 times brighter.  The {\\it Spitzer} spectra showing \nweak dust features were taken between these {\\it Swift}\\ observations, \naround day 480.  V574 Pup was detected as a SSS by \n{\\it XMM}\\ and {\\it Swift}\\ 872 and 1116 days after visual maximum, respectively.\n{\\it Spitzer} observations taken around the same time as the {\\it Swift}\\ data \nshowed the same weak silicate emission features seen in V2467 Cyg.  \nThe X-ray observations confirm that dust, albeit weak, can exist in the \nejecta when the amount of photoionizing radiation is at its peak.  Detailed\nphotoionization modeling of these novae is required to determine if clumps\nexisted in the ejecta during this time and if the conditions were sufficient\nto protect the dust grains.\n\nThere are also two strong dust formers in the sample with hard X-ray emission.\nV2362 Cyg was detected numerous times by {\\it Swift}\\ \\citep{2008AJ....136.1815L} \nand twice with {\\it XMM}\\ \\citep{2007ATel.1226....1H} but none of the observations\nwere consistent with a SSS.  However, V2362 Cyg had significant dust \nemission at the times of the {\\it Swift}\\ and first {\\it XMM}\\ observations.  \nThe dust likely formed in the later extraordinary mass ejection event\nthat produced the large secondary peak in the light curve and increased\nejection velocities.  The additional material would have absorbed the soft\nX-ray emission and delayed the onset of any SSS phase.\nIn the last {\\it XMM}\\ observation it was extremely faint indicating that \nif there was a SSS phase it was over by 990 days after maximum.  \nV1280 Sco was detected as an X-ray source late in its outburst but the X-rays \nwere relatively hard and faint \\citep{2009ATel.2063....1N}.  V1280 Sco \nhas yet to be observed as an SSS and its internal extinction is still large.\nIn both V2362 Cyg and V1280 Sco, grain growth was likely enhanced to\nproduce the large dust events due to the effective shielding of the \nlarge mass ejections.\n\n\\subsection{Variability during SSS phase \\label{var}}\n\nAt the maximum effective temperature, (2-8)$\\times$10$^5$ K, the bulk of\nthe emission in a nova outburst comes from X-rays which are primarily soft. \nAssuming the external column is low enough and the effective temperature\nis suitably high enough, this X-ray emission can be detected.  The theory\nof constant bolometric luminosity predicts that at X-ray maximum the \nlight curve should be relatively constant since one is observing the \nmajority of the emitted flux.  Constant bolometric luminosity has been \nobservationally verified in the early phase of the outburst from the\ncombined UV, optical and near-IR light data, {\\it e.g.} FH Ser\n\\citep{1974ApJ...189..303G}, V1668 Cyg \\citep{1981MNRAS.197..107S} and \nLMC 1988\\#1 \\citep{1998MNRAS.300..931S}.  However, the expected X-ray \nplateau in all well studied {\\it Swift}\\ novae has been far from constant.  \nIn addition, the rise to X-ray maximum also shows large amplitude \noscillations.  What is the source of the variability during both phases?\n\nOne important caveat when discussing the {\\it Swift}\\ data is that the XRT \ncount rate is not a direct measure of the bolometric flux, only the \nportion that is emitted between 0.3 and 10 keV.  During the SSS phase the \nvast majority of photons are emitted within this range but if the effective \ntemperature varies due to photospheric expansion or contraction, the XRT \ncount rate will change even if the source has a constant bolometric \nluminosity \\citep[see also][]{2011ApJ...727..124O}.\nFigure \\ref{xrtevol} illustrates how the estimated XRT count \nrate varies as a function of effective temperature for simple blackbody \nmodels (WebPIMMS\\footnote{http:\/\/heasarc.nasa.gov\/Tools\/w3pimms.html})\nassuming a constant luminosity and column density\nsee Section \\ref{Xrayvar}.  A decline from 500,000 K\nto 400,000 K drops the total {\\it Swift}\\ XRT count rate by a factor of 6.\nThe change in HR1 is almost a factor of 10 while there is\nessentially no change in the HR2 hardness ratio.  Thus changes\nin temperature might in principle account for the observed X-ray \noscillations, see Section \\ref{teffvar}.  Why the temperature or radius \nof the WD photosphere would change on the observed \ntime scales remains an open question however.  The next sections outline \npossible explanations for the variations seen during the SSS phase.\n\n\\subsubsection{Variable visibility of the WD\\label{Xrayvar}}\n\nFigures 2 and 5 in \\citet{2011ApJ...727..124O} show in exquisite detail\nthe rapid and extreme variability in the X-ray light curve and hardness\nratio evolution in RS Oph.  In general the trend was for RS Oph to be softer\nat high X-ray flux but counter examples were also observed.\n\\citet{2011ApJ...727..124O} cite variable visibility of the hot WD as a \npossible explanation of the observed phenomena.  Changes in the \nextinction can come from either variable ionization of the ejecta leading \nto changing extinction at higher ionization states or neutral absorption \nfrom high density clumps passing through the line of sight.  Changes in \nthe ionization structure of the ejecta are unlikely given the rapid \nhour to day time-scales but are consistent with the crossing times of \nsmall, dense clumps traveling across the line of sight assuming transverse \nvelocities of a few percent of the radial velocity.  There is evidence \nfor this at other wavelengths.  For example, a sudden absorption component \nthat appeared in the Balmer lines of V2214 Oph in July 1988 was interpreted \nby \\citet{1991ApJ...376..721W} as the passage of an absorbing clump in \nfront of the the emitting region.  Both types of absorption changes \nshould be manifest as a hardening of the X-ray spectrum or increase in \nthe hardness ratio with increasing soft flux emission consistent with \nthe counter examples of \\citet{2011ApJ...727..124O}.\n\nAs a test of the neutral absorption theory we use the model results \nfrom a recent photoionization analyses in WebPIMMS to determine the\ncount rates and hardness ratios for different column densities and \nsimulate the effect of clumps.  The photoionization models require \ntwo components, high density clumps embedded within a larger diffuse medium\n\\citep[see][for details]{2007ApJ...657..453S,2010AJ....140.1347H}\nto fit the emission lines of the ejected shell.  For convenience we\nuse the May 24th, 1991\nmodel parameters for V838 Her in Table 2 of \\citet{2007ApJ...657..453S}.\nThe model uses a blackbody with an effective temperature of 200,000 K to\nphotoionize a  two component spherical shell.  The model shell has a\nclump-to-diffuse density ratio of 3 with a radius equal to the expansion\nvelocity multiplied by the time since outburst.  To facilitate comparisons \nwith the results in Figure \\ref{xrtevol}, the same unabsorbed bolometric \nflux is assumed.  WebPIMMS predicts a {\\it Swift}\\ soft band count rate \nof 5.3$\\times$10$^{-3}$ ct s$^{-1}$ through the \nlower density diffuse gas (N$_H$ = 1.2$\\times$10$^{21}$ cm$^{-2}$)\nand 8.5$\\times$10$^{-6}$ ct s$^{-1}$ from the higher density clumps\n(N$_H$ = 3.7$\\times$10$^{21}$ cm$^{-2}$).  While the total count rate\ndeclines by over 100$\\times$, the HR2 hardness ratio does not change with\nthis particular model. The HR2 can vary significantly when using different \nmodel parameters such as lower initial densities or higher clump to diffuse\ndensity ratios.  Care is required when using hardness ratios of \nlow resolution data.  In a SSS source with a hard X-ray component, such as \nRS Oph, the hardness ratio will increase if the soft component decreases \nfor any reason, not just due to absorption.  \n\nAnother problem with variable visibility in RNe and very fast CNe is that\nthe amount of mass ejected is very low thus minimizing any effect the ejecta\nhave on the obscuration of the WD.  The effect should be greater\nin slower novae with more ejected mass such as V458 Vul.  In addition,\n\\citet{2011ApJ...727..124O} find that in RS Oph the ratio of high flux\nstates to low flux states as a function of energy is not consistent with\neither type of variable visibility of the WD.  Rather the best fit comes from\nan increase in the effective temperature and declining radius,\nsee Section \\ref{teffvar}.\n\n\n\\begin{figure*}[htbp]\n\\plotone{webpimms.ps}\n\\caption{The logarithmic X-ray count rates, hardness ratios and logarithmic\n$uvw2$ count rates\nas a function of blackbody temperature as calculated by webPIMMS.\nAn unabsorbed, bolometric flux of 3.3$\\times$10$^{-8}$ erg\/s\/cm$^{-2}$ \n(1$\\times$10$^{38}$ erg s$^{-1}$ at 5 kpc) and N$_H$ of \n3$\\times$10$^{21}$ cm$^{-2}$ was used in all models.\nThe top panel shows the soft (0.3-1 keV, solid line and filled circles) \nand hard (0.1-10 keV, dashed line and triangles) count rates.  The soft\ncontribution dominates at all effective temperatures.  The middle panels \nshow the hardness ratios HR1(=H\/S) and HR2(=(H-S)\/(H+S)).  The bottom panel \nshows how the $uvw2$ count rate increases as the blackbody temperature \ndeclines.  \\label{xrtevol}}\n\\end{figure*}\n\n\\subsubsection{Periodic oscillations\\label{sssperiods}}\n\nThere are several proposed explanations of the periodic X-ray variations.\nIn the X-ray light curve of V1494 Aql, \\citet{2003ApJ...584..448D} found \nperiodicities that they attributed to non-radial g$^+$-mode pulsations.  \nSimilar oscillations have been observed in V4743 Sgr \n\\citep{2003ApJ...594L.127N,2010MNRAS.405.2668D}.\n\nThe factor of almost ten decline in the {\\it XMM}\\ X-ray light curve of \nV5116 Sgr was interpreted by \\citet{2008ApJ...675L..93S} as a partial \neclipse of the WD since its duration was consistent with the orbital period.  \nFiner binning of the day 762, 764, and 810 {\\it Swift}\\ observations of \nV5116 Sgr reveals the presence of a 500-800 second oscillation.  This\nX-ray periodicity is significantly shorter than the 2.97 h orbital\nperiod found by \\citet{2008A&A...478..815D}.  In addition,\nthe day 810 data show a strong flare that increases the count rate by\na factor of three with no significant change in the hardness ratio.\nThis was similar to the flare seen in V1494 Aql \\citep{2003ApJ...584..448D}.\nNo other flares were seen in the V5116 Sgr data set.\n\nOther orbital periods have been detected with {\\it Swift}.\nU Sco is a high inclination system with deep eclipses and an orbital\nperiod of 1.23 days \\citep{2001A&A...378..132E}.  Deep eclipses were\nobserved in the 2010 outburst in the {\\it Swift}\\ UVOT light curves while the\nXRT light curves showed generally lower flux levels during the UV eclipses,\nbut did not otherwise exhibit clear eclipse signatures.\n\\citep{2010ATel.2442....1O}.  A 1.19 day orbital period was deduced from \nthe {\\it Swift}\\ UVOT light curves in the RN nova LMC 2009a \n\\citep{2009ATel.2001....1B}.  This orbital period was also observed in the \nXRT light curve during the SSS phase, but with a lag with respect to the \nUV\/optical of 0.24 days \\citep{Bode2011}.\n\nThe X-ray behavior in CSS 081007:030559+054715 was extremely unusual.  This\nodd source was discovered well after optical maximum by the Catalina Real-time\nTransient Survey \\citep{2008ATel.1835....1P}.  Its X-ray spectra were\nextremely soft, consistent with the low extinction along its position\nhigh above the Galactic plane ($b$ = -43.7$\\arcdeg$) \nwhich places it well outside the plane of the Galaxy where novae\nare generally not located.  Figure \\ref{csslc} shows the {\\it Swift}\\ XRT\/UVOT\nlight curves compiled from the data in Table 2.  \nTo first order both light curves are in phase with \nsignificant variability superimposed over three major maxima.  \n\\citet{2010AN....331..156B} report that the \n{\\it Swift}\\ light curves are unique with a 1.77 day periodicity.  They speculate \nthat the period is due to obscuration of the X-ray source in a high \ninclination system with a 1.77 day orbital period.\n\nOscillations significantly shorter than the hours-to-days of typical \nnovae orbital periods have also been detected with {\\it Swift}.  Oscillations \nof order 35 s have been observed in RS Oph \n\\citep{2006ATel..770....1O,2011ApJ...727..124O}\nand KT Eri \\citep{2010ATel.2423....1B}.  Some WDs have rotation \nperiods in this range \\citep[{\\it e.g.} 33 s in AE Aqr;][]{2008PASJ...60..387T}.\nIt seems unlikely that RS Oph and KT Eri should both have nearly identical \nrotation periods unless the pulsations are tied to the mass of the WD\nwhich for both novae are predicted near the Chandrashkar limit.  \nAnother reason the observed\nvariability might not be associated with the rotating WD is that \nthe $\\sim$ 35 second periodicity is not always detected in the {\\it Swift}\\\nand {\\it Chandra}\\ X-ray light curves.  The 35 second pulsations could be due to \na nuclear burning instability on the WD surface \n\\citep[see ][]{2011ApJ...727..124O}.  If so, then the period then is a \nfunction of WD mass, and perhaps indicates that the WDs in RS Oph and KT Eri\nare near the Chandrasekhar mass.\n\n\\subsubsection{Temperature variations\\label{teffvar}}\n\nLong lived SSS, such as Cal 83, have non-periodic X-ray on\/off states.  \n\\citet{2000A&A...354L..37R} speculate that the \ndecline in X-ray flux is due to accretion disk interactions such\nas an increase in the mass accretion rate causing the WD photosphere \nto expand and shifting the SED into the EUV.  These sources then become \noptically brighter from the irradiation of the accretion disk and \nsecondary by the larger WD photosphere.  The source remains X-ray faint \nuntil the WD photosphere shrinks back to its\noriginal size.  Figure \\ref{v458vullc} shows similar behavior in the \n{\\it Swift}\\ X-ray and UV light curves of V458 Vul\ncompiled from the data in Table 2.  The 100$\\times$ decline\nin the X-ray light curve is matched by a 1.5 magnitude increase\nin the UV light curve.  Figure \\ref{xrtevol} shows that similar X-ray\nand UV variations can be achieved by large declines in the effective \ntemperature. For example, a decline from 700,000 K to 500,000 K \nproduces a factor of 85 decline in the X-ray count rate and a 1.1 \nmagnitude $uvw1$ band increase.  \nIf the underlying phenomenon in V458 Vul \nis the same as proposed for RX J0513.9-6951 \\citep{2000A&A...354L..37R}\nand Cal 83 \\citep{2002A&A...387..944G} and the accretion disk has been\nreestablished, V458 Vul should have an orbital period of order one\nday to produce an accretion rate high enough to drive stable nuclear burning.\nHowever, \\citet{2010MNRAS.407L..21R} find a short orbital period of $\\sim$ \n98 minutes implying that V458 Vul will not have a long term SSS phase.\n\n\\begin{figure*}[htbp]\n\\plotone{csslc.ps}\n\\caption{The X-ray and uvw2 light curves of the particular nova\nCSS 081007:030559+054715.  The X-ray and UV evolution are in phase.\n\\label{csslc}}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\plotone{v458vullc.ps}\n\\caption{{\\it Swift}\\ X-ray (top panel) and uvw1 (bottom panel) light curve \nfor V458 Vul.  The X axis is the number of days after visual maximum.  \nPrior to day 400, V458 Vul was in transition to X-ray maximum.  \nAfter day 400 the majority of the X-ray observations had a count rate \nof $\\sim$ 1 ct s$^{-1}$.  However, during the later phase there \nare three periods where the X-ray counts declined by about a factor \nof 100.  During these times the uvw1 ($\\lambda_c$ = 2600\\AA)\nphotometric brightness increased by a magnitude.\n\\label{v458vullc}}\n\\end{figure*}\n\n\\subsection{Estimating time scales in a variable environment}\n\nThe variability of novae also raises questions about how confident \none can be in the determination of turn-off times.  A prime example can be \nseen in the X-ray light curve of V458 Vul in Figure \\ref{v458vullc}.  If \nmonitoring had stopped following the four observations between days 450 \nand 480 the subsequent recovery would never have been found, and it\nwould have been noted that V458 Vul had a turn-off time of 1.2 years \ninstead of $\\gtrsim$ 2.9 years.  While this could be a significant \nproblem with the determination of a turn-off time in most novae, it is \nlikely that the phenomenon observed in V458 Vul is rare.  The X-ray \nbehavior of V458 Vul, a 100$\\times$ decline in flux and a subsequent \nrecovery, is the \\textit{only} case observed in the \\totallimitswiftSSS\\ \nnovae studied by \\textit{Swift} with SSS emission.\n\\citet{2001A&A...373..542O} found no similar \"reborn\" SSSs in their review\nof the {\\it ROSAT}\\ all sky survey although some of the novae in M31 previously\nthought to be RNe with very rapid outburst time scales may actually be\nnormal novae but with on\/off states similar to those in V458 Vul.  Since the \nsudden X-ray declines in V458 Vul also had corresponding UV rises, if these\nsource exist in M31, they should be easily found with X-ray and UV\ncapable facilities such as {\\it Swift}\\ and {\\it XMM}.\n\n\\subsection{SSS in RNe and the light curve plateau\\label{RNplateau}}\n\nA plateau in visible light of RNe is speculated to arise from the reradiation \nof the SSS emission from an accretion disk dominating the emission after \nthe free-free emission has faded \\citep{2008ASPC..401..206H}.  Once nuclear \nburning ends and the accretion disk is no longer irradiated, the light curve \ncontinues its decline to quiescence.  Figure 17.1 and \n17.2 show that the optical plateaus are nearly coincident with \nthe SSS emission in RS Oph and U Sco.\n\nThe other well observed RNe in the {\\it Swift}\\ archive are novae LMC 2009a \nand V407 Cyg.  LMC 2009a was previously seen in outburst in 1971 \n\\citep{2009IAUC.9019....1L}.  It had a much longer SSS phase than RS Oph \nand U Sco and it ended 270 days after maximum.  Unfortunately, the V band light \ncurve compiled from the AAVSO archives and our own SMARTS photometry \ndoes not extend beyond 110 days after visual maximum so we can not \ndetermine whether an optical plateau was observed later in this outburst,\nsee Figure 17.3.  However, the {\\it Swift}\\ uvw2 and SMARTS B\nband light curves are relatively flat during the SSS phase (see Bode et al. \nsubmitted) indicating LMC 2009a did go through an optical plateau phase.\nThe data\nare not as extensive for V407 Cyg but the rise in the soft X-ray emission\nconsistent with nuclear burning on the WD (see Section \\ref{v407cygSSS})\nis coincident with a short plateau in the optical light curve as shown\nin Figure 17.4.\n\nThere are three other novae with well observed SSS phases in the {\\it Swift}\\\narchive that are suspected to be RNe based on their outburst characteristics.\nThe novae are V2491 Cyg, KT Eri, and V2672 Oph.  Figure 17.5\nshows that there is no indication of a plateau in V2491 Cyg while it was\na SSS.  However, the SSS phase in V2491 Cyg was extremely short, $<$10 days,\nwhich may not be sufficient time to produce a noticeable optical plateau\nor the system did not have its accretion disk reform this early in the\noutburst. \n\nThe early outburst spectra of KT Eri were indicative of the He\/N class with \nhigh expansion velocities typical of RNe \\citep{2009ATel.2327....1R}.  \nKT Eri also had short X-ray light curve modulation similar \nto RS Oph, see Section \\ref{sssperiods} and\n\\citet{2010ATel.2392....1B}, while Bode et al. (2011, in prep) draw \nattention to KT Eri's similarities with the X-ray behavior of LMC 2009a.\nThe X-ray and V band observations are shown in Figure 17.6.\nThe AAVSO V band light curve shows a flattening at 80 days after visual maximum\nor about 10 days after KT Eri became a SSS implying there was an\noptical plateau.\n\nThe case for V2672 Oph as a RNe is based on its extreme expansion velocities\nat maximum \\citep{2009IAUC.9064....2A} and early radio synchrotron emission\nsimilar to that observed in RS Oph \\citep{2009ATel.2195....1K}.  \n\\citet{2010MNRAS.tmp.1484M} also find many similarities between V2672 Oph\nand U Sco.  Unfortunately, the X-ray and optical observations were \nhampered due to the faintness at visual maximum and the relatively \nlarge column density.\nBased on the hardness ratio, V2672 Oph was in its SSS phase between days 15 \nand 30 after visual maximum (Figure 17.7).  The AAVSO \nV band light curve is supplemented with SMARTS V band photometry which\nshows a plateau between day 10 through 50 after visual maximum.\n\nOf the 4 known RNe and 3 suspected RNe, there are sufficient optical data \nto reveal the presence of a plateau in six.  Of those six all but V2491\nCyg have evidence of an optical plateau correlated with the X-ray SSS\nemission.  However, \\citet{2010ApJS..187..275S} finds that not all Galactic \nRNe have optical plateaus.  It is interesting to note that if the plateau\nphase is caused by reradiation off an accretion disk as suggested by\n\\citet{2008ASPC..401..206H} then there is no effect on the presence \nor strength of the plateau due to the inclination of the system.  One\nwould expect the effect in more face-on systems like RS Oph,\n$i$ = 39$^{+1}_{-10}$$^{\\circ}$ \\citep{2009ApJ...703.1955R} than in\nedge-on systems such as U Sco, $i = 82.7\\pm2.9^{\\circ}$ \n\\citep{2001MNRAS.327.1323T}.  Regardless of the root cause of optical \nplateaus, their presence can clearly be used as a proxy signature of \nSSS emission.  \nHowever, it should be stressed that while optical plateaus likely \nindicate soft X-ray emission, the start and ending of this phase in the \noptical light curve does not necessarily correspond to the turn-on and\nturn-off times in the SSS phase.  Relationships between the optical\ntimescales and the X-ray are only weakly correlated, e.g. \nFig. \\ref{t2turnoff}, and the two phases do not always align in the \nRN and suspected RN in this sample (Fig. 17.1-17.7).\n\nOptical\/NIR plateaus should only be observed in RNe and other fast \nnovae that eject very little mass.  In slower novae the later spectra \n({\\it i.e.} several tens of weeks after maximum light) are dominated by \nhydrogen recombination and nebular line emission effectively hiding any \nirradiation effects.  The continuum from the WD or a hot accretion disk \ncan only be observed after the ejecta have sufficiently cleared.  \n\n{\\bf Fig. Set} \n\\figsetnum{17}\n\\figsettitle{X-ray and optical evolution}\n\n\n\\figsetgrpnum{17.1}\n\\figsetgrptitle{RS Oph}\n\\figsetplot{f17_1.eps}\n\\figsetgrpnote{X-ray and optical evolution of RS Oph.  The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve.  \\label{rsophplat}}\n\n\n\n\\figsetgrpnum{17.2}\n\\figsetgrptitle{U Sco}\n\\figsetplot{f17_2.eps}\n\\figsetgrpnote{X-ray and optical evolution of U Sco. The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve. \\label{uscoplat}}\n\n\n\n\\figsetgrpnum{17.3}\n\\figsetgrptitle{Nova LMC 2009 A}\n\\figsetplot{f17_3.eps}\n\\figsetgrpnote{X-ray and optical evolution of Nova LMC 2009a.  The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve and includes our own SMARTS photometry.  \\label{nlmc09plat}}\n\n\n\n\\figsetgrpnum{17.4}\n\\figsetgrptitle{V407 Cyg}\n\\figsetplot{f17_4.eps}\n\\figsetgrpnote{X-ray and optical evolution of V407 Cyg.  The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve.  To accentuate the soft contribution to the total in the top panel, the squares show the soft, 0.3-1 keV, light curve.  The V band light curve includes the AAVSO data (filled circles) and the photometry of \\citet{2011MNRAS.410L..52M} (diamonds).  \\label{v407cygplat}}\n\n\n\n\\figsetgrpnum{17.5}\n\\figsetgrptitle{V2491 Cyg}\n\\figsetplot{f17_5.eps}\n\\figsetgrpnote{X-ray and optical evolution of V2491 Cyg. The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve. \\label{v2491cygplat}}\n\n\n\n\\figsetgrpnum{17.6}\n\\figsetgrptitle{KT Eri}\n\\figsetplot{f17_6.eps}\n\\figsetgrpnote{X-ray and optical evolution of KT Eri. The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve.  The gaps in the light curves are due to KT Eri being behind the Sun.  \\label{kteriplat}}\n\n\n\n\\figsetgrpnum{17.7}\n\\figsetgrptitle{V2672 Oph}\n\\figsetplot{f17_7.eps}\n\\figsetgrpnote{X-ray and optical evolution of V2672 Oph.  The top panel is the {\\it Swift}\\ XRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, (H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel shows the AAVSO V band light curve and includes our own SMARTS photometry.  \\label{v2672ophplat}}\n\n\n\n\n\\begin{figure*}[htbp]\n\\plotone{f17_1.eps}\n\\caption{X-ray and optical evolution of RS Oph.  The top panel is the {\\it Swift}\\\nXRT (0.3-10 keV) count rate and the middle panel is the hardness ratio, \n(H-S)\/(H+S) where H = 1-10 keV and S = 0.3-1 keV.  The bottom panel\nshows the AAVSO V band light curve.  Similar figures for U Sco, Nova LMC 2009\nA, V407 Cyg, V2491 Cyg, KT Eri, and V2672 Oph are available in the electronic\nedition.  \\label{rsophplat}}\n\\end{figure*}\n\n\\subsection{SSS proxies at other wavelengths: The $[$\\ion{Fe}{10}$]$\\ line\\label{fex}}\n\n\\btxt{\\citet{2001AJ....121.1126V} used the evolution of UV emission\nline light curves developed in \\citet{1996ApJ...463L..21S} for V1974 Cyg\nto estimate turn-off times.  This allowed \\citet{2001AJ....121.1126V} to\ndetermine the nuclear burning timescales of five novae with no pointed\nX-ray observations but significant amounts of {\\it IUE} data.  Unfortunately,\nit is currently difficult to obtain sufficient UV emission line data to \nutilize this technique while the optical plateau (\\S \\ref{RNplateau})\nonly applies to fast and recurrent novae.  Another X-ray proxy is \nneeded for slower novae.}\n\nThe emergence of the coronal $[$\\ion{Fe}{10}$]$\\ 6375\\AA\\ line in the nebular spectra\nof novae has been long recognized as a strong indication of photoionization \nof the ejecta from a hot source \\citep[e.g.,][]{1989ApJ...341..968K}.  \nWith an ionization potential of 235 eV, an ejected shell must be highly \nionized by a hot WD to produce $[$\\ion{Fe}{10}$]$.  While shocks can produce \nhigh temperatures, they only contribute very early in the outburst\nand are insignificant during the later nebular phase when $[$\\ion{Fe}{10}$]$\\ is typically\nobserved, in all but the RS Oph-type RNe. For example, strong $[$\\ion{Fe}{10}$]$\\ \nand [\\ion{Fe}{14}] 5303\\AA\\ emission has been observed in RS Oph in all \noutbursts with adequate spectroscopic coverage \\citep{2009ApJ...703.1955R}. \nHowever, these lines appear well before the SSS phase begins.  A \nrelationship between $[$\\ion{Fe}{10}$]$\\ and soft X-ray emission has not been previously \ndemonstrated but can be strengthened with our larger nova sample.\n\nSeven novae with confirmed SSS emission, GQ Mus \\citep{1989ApJ...341..968K},\nV1974 Cyg \\citep{1995A&A...294..488R}, V1494 Aql \\citep{2003A&A...404..997I},\nV723 Cas \\citep{2008AJ....135.1328N}, V574 Pup \\citep{Heltonthesis}, V597 Pup, \nand V1213 Cen \\citep{2010ATel.2904....1S} all had strong $[$\\ion{Fe}{10}$]$\\ lines in \ntheir late nebular spectra.  Example spectra of V597 Pup and V1213 Cen from \nour SMARTS archive are shown in Figure \\ref{fexplots}.  In addition, \nextensive optical spectra from our Steward Observatory northern \nhemisphere nova monitoring \ncampaign shows that V2467 Cyg may also have had weak $[$\\ion{Fe}{10}$]$\\ emission at the \nsame time it was a SSS but this can not be confirmed due to nearby \n\\ion{O}{1} lines.  These novae clearly show that the presences of strong\n$[$\\ion{Fe}{10}$]$\\ in the optical spectrum is indicitive of underlying SSS emission.\nTo our knowledge there has never been a nova with strong $[$\\ion{Fe}{10}$]$\\ emission \nthat was not also a SSS during contemporaneous X-ray observations.  \nWhile additional optically and X-ray observations are needed to fully test\nthis hypothesis, ground based spectroscopic monitoring is a powerful tool \nfor detecting SSS novae from $[$\\ion{Fe}{10}$]$\\ emission in novae with significant\nejected mass.  The RNe and very fast CNe \nwith rapid turn-on\/off times are not strong photoionization sources \nlong enough to produce $[$\\ion{Fe}{10}$]$\\ in their meager ejected shells.\n\n\\begin{figure*}[htbp]\n\\plottwo{v1213cen_100627.ps}{v597pup080326.ps}\n\\caption{$[$\\ion{Fe}{10}$]$\\ 6375\\AA\\ emission in V1213 Cen (left) and V597 Pup (right)\nobtained on June 27th, 2010 (415 days from visual maximum) \nand March 26th, 2008 (133 days from visual maximum), respectively.\nThe $[$\\ion{Fe}{7}$]$ 6087\\AA\\ line is also visible in both spectra. Strong\n$[$\\ion{Fe}{10}$]$\\ emission relative to $[$\\ion{Fe}{7}$]$ is a hallmark of novae in\ntheir SSS phase.  \n\\label{fexplots}}\n\\end{figure*}\n\n\n\\section{SUMMARY}\n\nOver the last decade our knowledge of the X-ray behavior of \nnovae has increased dramatically with the launch of \nthe latest generation\nX-ray facilities.  Observations of novae when they are radiating the majority \nof their flux in the soft X-ray band provide critical insight into \nthe behavior of the WD and TNR processes.  Currently \\totalSSS\\ \nGalactic\/Magellanic novae have been observed as SSSs of which \\totalswiftSSS\\ \nsuch classifications have come from \nover 2 Ms of {\\it Swift}\\ observations during the last five years.  \n\nThis large sample shows that individual novae can differ significantly \nfrom fits to smaller ensemble data sets such as the t$_2$ relationship of \n\\citet{2010ApJ...709..680H} and the expansion velocity relationship of \n\\citet{2003A&A...405..703G}.  Surprisingly, there is also no relationship \nbetween orbital period and the duration of nuclear burning.  This large\ndata set confirms that many factors are in play in the evolution of the \nSSS phase.\n\nThe duration of nuclear burning on the WD is short, with 89\\% of the \nnovae have turned off within 3 years in this expanded sample.  The median\nduration of the sample is 1.4 years. This contrasts with the same distribution\nin M31 which is peaked at longer burning novae.  The difference is likely\na selection effect between the two surveys.\n\nThe new {\\it Swift}\\ data are also challenging our understanding of novae \nwith highly variable X-ray light curves both during the rise to and at\nX-ray maximum.  Various mechanisms are likely at work to produce the \nvariability.  Additional observations are warranted not only to help\ndecipher the current peculiar observations but also to be sure that we\nhave captured the full range of variability behaviors both periodic and\nnon-periodic that novae may yet produce.  \nLong {\\it XMM}\\ and {\\it Chandra}\\ grating observations can explore the short \nterm oscillations more effectively than {\\it Swift}\\ whereas {\\it Swift}\\ can\neasily track the long term behavior such as turn-on and turn-off times.\nIn addition, simultaneous X-ray\/UV \nobservations only available through {\\it XMM}\\ and {\\it Swift}\\ will continue to\nbe a powerful tool to test the evolution of the emission from the WD \nduring the outburst.\n\nTo date no strong dust-forming novae have been detected as a SSS.  \nV2362 Cyg did have detectable soft X-ray photons but it was not similar \nto any of the other SSS novae.  While V574 Pup and V2467 Cyg were in\nthe SSS phase they had IR features indicating weak silicate dust emission.\nV1280 Sco had a large DQ Her-like dust event but also ejected so much \nmaterial and at a low velocity that it is still optically thick several\nyears after visual maximum.  Any SSS phase will not be detected until\nthis material clears.\n\nThere are optical behaviors that track SSS emission in novae.  \nFor the RNe with well defined plateaus \nin their optical light curves, RS Oph and U Sco, the X-ray light curves\nreach maximum around the same time.  However, not all RNe and suspected\nRNe in the sample had optical plateaus even though they had well \ndocumented observations during X-ray maximum.  An optical spectroscopic \nsignature indicative of an SSS phase is the presence\nof strong $[$\\ion{Fe}{10}$]$\\ 6375\\AA\\ emission.  In the sample, all novae with $[$\\ion{Fe}{10}$]$\\ that\nwere subsequently observed in the X-ray were SSSs.  These were slower\nnovae that ejected significantly more material than the RNe.  The inverse \nof the $[$\\ion{Fe}{10}$]$\\ relationship does not hold since the source may turn off before \n$[$\\ion{Fe}{10}$]$\\ can be created in the ejecta.  While the presences of neither optical\nplateaus or $[$\\ion{Fe}{10}$]$\\ has yet been shown to be simultaneous with SSS emission,\nthese relationships offer excellent oppertunities to use ground-based \nmonitoring to coordinate X-ray observations during the important \nSSS phase.  \n\nAdditional X-ray data need to be collected since the sample is statistically \nmeager with only \\totalSSS\\ known SSS novae and is smaller still for novae\nwith early, hard X-ray detections.  Trends can be difficult to confirm given\nthe wide range of behavior observed during the different X-ray phases.\nWith the sample heavily biased toward fast and recurrent novae, efforts \nshould be expended on novae that are not \ncurrently well represented in the X-ray sample such as slow and \ndust forming novae. The monitoring of the two slow novae that \nhave been detected as X-ray sources, V5558 Sgr and V1280 Sco, but \nhave not yet evolved to a SSS state, will help in \nunderstanding slow systems.  Likewise, {\\it Swift}\\ monitoring of the \ntwo long lasting SSSs, V723 Cas and V458 Vul, are also of interest \nsince they are rare, and thus, important to our understanding\nof why they persist.  \n\nFinally, it is important to continue to collect X-ray observations of \nnovae and build on this sample.  This analysis shows that each nova is in\nsome ways unique and that attempts to predict their behavior based on a \nrelationship to a single observational value, {\\it e.g.} t$_2$ versus the \nnuclear burning timescale, is fraught with difficulties.  Some of these\nproblems can be addressed by expanding the sample to include regions of\nthe parameter space that are not well represented.  This X-ray sample\nincludes few slow novae which likely explains the differences between\nthe nuclear burning timescale of the Milky Way and M31 surveys.\nIt is also equally important to obtain numerous, high \nquality data for all bright novae through their evolution and at different\nwavelengths from X-ray to radio.  Multiwavelength observations are critical\nto properly interpret nova phenomena such as the apparent early turn-off in\nV458 Vul and to verify periodicities seen in the X-ray, particularly\npotential orbital periods. With the understanding that comes from a few \nwell observed novae \nlike RS Oph and U Sco, the entire nova data set can be anchored to nova \ntheory.  These large data sets also reveal new phenomena such as the \nstrong X-ray variability that is not appreciated in novae with \nsparser observations or detected at other wavelengths.\n\n\\acknowledgments\n\nThis research has made use of data obtained from NASA's {\\it Swift}\\ satellite.  \nWe thank Neil Gehrels and the {\\it Swift}\\ team for generous allotments of ToO \nand fill in time.  Funding support from NASA NNH08ZDA001N1. \nStony Brook University's initial participation in the SMARTS consortium was \nmade possible by generous contributions from the Dean of Arts and Sciences, \nthe Provost, and the Vice President for Research of Stony Brook University.  \nWe acknowledge with thanks the variable star observations from the AAVSO \nInternational Database contributed by observers worldwide and used in this \nresearch.  \nJPO, KP, PE \\& AB acknowledge the support of the STFC.  \nSS acknowledges partial support from NASA and NSF grants to ASU.  \nJJD was supported by NASA contract NAS8-39073 to the {\\it Chandra}\\ X-ray Center.\n\n{\\it Facilities:} \\facility{Swift(UVOT\/XRT)}, \\facility{AAVSO}, \n\\facility{CTIO:1.3m}, \\facility{CTIO:1.5m}, \\facility{Bok(B\\&C spectrograph)},\n\\facility{Spitzer(IRS)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Acknowledgments}\nWe thank M. Calandra for useful discussions. We acknowledge funding from EU Graphene Flagship, ERC Grant Hetero2D, EPSRC Grant Nos. EP\/509K01711X\/1, EP\/K017144\/1, EP\/N010345\/1, EP\/M507799\/ 5101, and EP\/L016087\/1 and the Joint Project for the Internationalization of Research 2015 by Politecnico di Torino and Compagnia di San Paolo.\n\nThe authors declare no competing financial interests.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nSecondary cosmic ray (CR) particles of an extensive air shower (EAS) approaching towards the ground as a thin disk through the atmosphere from the direction of their parent primary CR particle at the speed of light. After the first interaction point, the disk begins to form, and continues to grow, and then starts attenuating after the depth of shower maximum. The transverse and longitudinal momenta imparted on the shower particles emerging from their parent particles via the hadronic interactions would cause the lateral and longitudinal spreads for these particles in an EAS. consequently the periphery of successive iso-density contours get shortened about the EAS axis, which is analogous to an inclined inverted truncated cone.\n\\section{Cone model: An elliptic lateral density function}\nThe evolution of a conical shower profile of an inclined EAS is shown in the Fig. \\ref{Cone_Prof}. The geometric correction is done through the projection of the horizontal elliptic surface to the shower plane. Corresponding equi-density contours are shown in Fig. \\ref{2D_IsoDen}(a), (b). The projected electron density in the shower plane ($\\rho_s$) is\n\\begin{equation}\n\\label{eq:Geom_Corr}\n\\rho_s(r_s)  =  {\\rho_g(r_g)}\/{ \\cos\\Theta}\n\\end{equation}\nAn exponential fall of the density of the shower electrons results from the EAS attenuation with a factor $ e ^ {-\\eta \\cdot AB}$, where $\\eta$ is the attenuation length [1]. Electron density in the ground plane $(\\rho_g)$ would be \n\\begin{equation}\n\\label{eq:Attn_Dens}\n\\rho_g(r_g)= \\cos\\Theta \\cdot \\rho_s(r_s) \\cdot e^{-\\eta \\cdot AB}\n\\end{equation}\n\\begin{figure}[htbp]\n\t\\begin{minipage}{18pc}\n\t\t\\includegraphics[trim=0.8cm 0.8cm 0.8cm 1.5cm, clip=true, totalheight=0.22\\textheight, angle=0]{Conical_Geometry.pdf}\n\t\t\\caption{\\label{Cone_Prof}Conical shower profile.}\n\t\\end{minipage}\\hspace{1pc}%\n\t\\begin{minipage}{18pc}\n\t\n\t\t\\includegraphics[trim=0.6cm -0.2cm 0.6cm -2.2cm, clip=true, width=0.16\\textheight]{Density_IsoLines_Ground.eps}\n\t\t\\includegraphics[trim=0.6cm -0.2cm 0.6cm -2.2cm, clip=true, width=0.16\\textheight]{Density_IsoLines_Shower.eps}\n\t\t\\caption{\\label{2D_IsoDen} 2-dimensional iso-density contours in the ground and shower plane.}\n\t\\end{minipage}\n\\end{figure}\n\nA characteristic function (CF) to describe the exponential behaviour of LDD with $r_s$ is proposed as follows,\n\\begin{equation}\n\\label{eq:CF}\n\\rho(r_s) \\simeq c \\cdot e^{-\\alpha (\\frac{r_s}{r_0})^\\kappa}\n\\end{equation}\nFinally the gap length parameter is given by,\n\\begin{equation}\n\\label{eq:Xc_1}\nx_C = 6813 y_R^{2-\\kappa} r_0^\\kappa \\eta (\\alpha \\kappa)^{-1}\n\t\\frac{\\tan \\Theta}{\\cos(\\Theta+\\sigma)} \n\t\\cdot\n\t\\frac{\\cos\\sigma}{H-r_s \\sin \\Theta}\n\\end{equation}\nSince $x_C>0$, the shower attenuation does shift the center of the ellipse towards the early part of the shower. Let us write the above equation as $x_C = 2 A_f y_R \\tan\\Theta$, where $A_f$ stands for,\n\\begin{equation}\n\\label{eq:Af}\nA_f = 6813 r_0^\\kappa \\eta (\\alpha \\kappa)^{-1}\n\\cdot\n\\frac{\\cos\\sigma}{2\\cos(\\Theta+\\sigma)(H-r_s \\sin \\Theta)}\n\\end{equation}\nThe modified length of semi-minor axis of an equi-density ellipse is,\n\\begin{equation}\n\\label{eq:yR}\n\ty_R = -2 A_f r_g \\cos\\beta_g \\tan\\Theta \\cdot \\frac{\\cos^2(\\Theta + \\sigma)}{\\cos^2\\sigma}\n\t+ r_g\\sqrt{1-\\cos^2\\beta_g \\sin^2(\\Theta+\\sigma)}\n\\end{equation}\nWe are familiar with most commonly used LDF as NKG function in CR air shower physics.\nThe polar angle dependent elliptic-LDF (ELDF) can be found from the NKG type Symmetric-LDF (SLDF) by making a substitution for the variable $r_s$ by $y_R$, and the equation for the ELDF finally takes the following structure. \n\\begin{equation\n\\label{eq:ELDF}\n\t\\rho(r_g,\\beta_g)=\\cos\\Theta\\cdot C(s_\\perp)N_e \\cdot (y_R\/r_0)^{s_\\perp-2} (1+y_R\/r_0)^{s_\\perp-4.5}\n\\end{equation}\nWhere, $C(s_\\perp)=\\frac{\\Gamma(4.5-s_\\perp)}{2\\pi r_0^2\\Gamma(s_\\perp)\\Gamma(4.5-2s_\\perp)}$ is the normalization factor, while $s_{\\perp}$, $r_0$ and $N_e$ respectively are called the age parameter, the moli\\`{e}re radius and shower size.\n\\section{Results and discussions}\nThe MC simulation code \\textit{CORSIKA} of version 7.69 with the hadronic interaction models QGSJet-01c and UrQMD is used. The reconstructed polar electron densities obtained using SLDF, ELDF including the projection and ELDF including both the attenuation and projection [2], at a core distance 50 m for an average 100 PeV proton shower with $\\Theta =50^o$. These are shown in Fig. \\ref{Polar_Dens_ELDF}, and the result reconfirms that the ELDF with GL is more appropriate for reconstruction of non-vertical EASs.\nIn the Fig. \\ref{LDD_PI}, the polar averaged LDD for P and Fe initiated showers are approximated by the CF (Eq. \\ref{eq:CF}). From the best possible fitting of the simulated data the parameter $\\alpha$ picks values 4.3 and 3.7 while $\\kappa$ takes 0.36 and 0.43 respectively for P and Fe.\nIn the Fig. \\ref{Iso_dens}, the center of the equi-density ellipse experiences a translation from \\textit{O} to \\textit{C} $(\\overline{OC} \\sim 9.75~m)$ solely due to attenuation of EAS electrons. On the other hand, the model predicted GL is about $6.35$~m, evaluated using the Eq. \\ref{eq:Xc_1}. \nThe GL parameter exhibits sensitivity to P and Fe initiated showers significantly for low-end values of $\\rho_e$ (Fig. \\ref{XcYr_PI}). GL is found to increase  with energy of CRs (Fig. \\ref{XcYr_E}). The elongation of the iso-density curve with increasing $\\Theta$ is evident from the values of GL for different zenith angles (Fig. \\ref{XcYr_Z}). The model predicted values for GL which are shown by the dotted and short dashed lines (Fig. \\ref{XcYr_PI}-\\ref{XcYr_Z}) are in good agreement with the simulated data. We have studied the dependence of the GL on $\\Theta$ for a fixed electron density (Fig. \\ref{GL_Z_IsoD}) as well as at fixed $y_R$ value (Fig. \\ref{GL_Z_yR}), which shows a mass sensitivity of primary CR. A correlation between the mean GL with primary energy ($E$) corresponding to $\\Theta = 50^o$ and $\\rho_e = 1.5~m^{-2}$ for P and Fe induced EASs, is depicted in Fig. \\ref{GL_E}.\n\\begin{figure}[h]\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 1.cm 1.05cm, clip=true, totalheight=0.17\\textheight]{PDD_R50m_ELDF.eps}\n\t\t\\caption{\\label{Polar_Dens_ELDF}Ground plane polar density distribution.}\n\t\\end{minipage}\\hspace{1pc}%\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.9cm 0.8cm, clip=true, totalheight=0.17\\textheight]{CF_PI.eps}\n\t\t\\caption{\\label{LDD_PI}Electron LDD fitted by CF.}\n\t\\end{minipage}\\hspace{1pc}%\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{IsoDen_Curve.eps}\n\t\t\\caption{\\label{Iso_dens}Formation of GL from equi-density ellipse.}\n\t\\end{minipage}\n\\end{figure}\n\\begin{figure}[h]\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{XcYr_PI.eps}\n\t\t\\caption{\\label{XcYr_PI}$x_C$ vs $y_R$ for P and Fe initiated showers.}\n\t\\end{minipage}\\hspace{1pc}%\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{XcYr_E.eps}\n\t\t\\caption{\\label{XcYr_E}$x_C$ vs $y_R$ for two energies.}\n\t\\end{minipage}\\hspace{1pc}%\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{XcYr_Z.eps}\n\t\t\\caption{\\label{XcYr_Z}$x_C$ vs $y_R$ at two zenith angles.}\n\t\\end{minipage}\n\\end{figure} \n\\begin{figure}[h]\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{GL_Z_IsoD.eps}\n\t\t\\caption{\\label{GL_Z_IsoD}Mass sensitivity of GL from its variation with $\\Theta$ at fixed $\\rho_e$.}\n\t\\end{minipage}\\hspace{0.8pc}%\n\t\\begin{minipage}[b]{2in}\n\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{GL_Z_yR.eps}\n\t\t\\caption{\\label{GL_Z_yR}Mass sensitivity of GL from its variation with $\\Theta$ at fixed $y_R$.}\n\t\\end{minipage}\\hspace{0.8pc}%\n\t\\begin{minipage}[b]{2in}\n\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{GL_E.eps}\n\t\\caption{\\label{GL_E} Mass sensitivity of GL from its variation with $E$.}\n\\end{minipage}\n\n\\end{figure}\n\nApplication of ELDF has been incarnated in terms of the local age parameter (LAP) in Fig. \\ref{LAP_PI}.\nThe analytical expression for the LAP [3] between two adjacent radial distances $[r_i, r_j]$ is:\n\\begin{equation\n\\label{eq:LAP}\ns_{ij}=\\frac{ln(F_{ij}X_{ij}^2Y_{ij}^{4.5})}{X_{ij}Y_{ij}} \n\\end{equation}\nHere, $F_{ij}=\\rho(y_R(i))\/\\rho(y_R(j))$,\n$X_{ij} = y_R(i)\/y_R(j)$, \n$Y_{ij} = (\\frac{y_R(i)}{r_0}+1)\/(\\frac{y_R(j)}{r_0}+1)$\nand $r_0$ is the moli\\`{e}re radius obtained from the best fit value of LDD by CF. A correlation between the mean LAP with primary energy for P and Fe induced EASs are given in Fig. \\ref{MPAL_PI}.\n\\begin{figure}[htbp]\n\t\\begin{center}\n\t\t\\begin{minipage}[b]{2.5in}\n\t\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{LAP_PI.eps}\n\t\t\t\\caption{\\label{LAP_PI}Variation of LAP with $r_g$.}\n\t\t\\end{minipage}\\hspace{1pc}%\n\t\t\\begin{minipage}[b]{2.5in}\n\t\t\t\\includegraphics[trim=0.6cm 0.6cm 0.6cm 1.05cm, clip=true, totalheight=0.17\\textheight]{MLAP_PI.eps}\n\t\t\t\\caption{\\label{MPAL_PI} Mean LAP versus $E$.}\n\t\t\\end{minipage}\n\t\\end{center}\n\\end{figure}\n\\section{Conclusions \\& future outlook}\nIn this work a modeling of the atmospheric attenuation effect on the LDD of electrons is made considering the conical shower profile. \nThe magnitude of the GL that determines the attenuation power of shower particles for a non-vertical EAS, possesses a clear primary CR mass dependence. The ELDF has been used to the simulated electron densities to estimate the LAP, which manifests different radial variation. The variation of mean LAP with primary energy\/shower size clearly shows sensitivity to CR mass composition. \n\nThere is a scope to judge the high energy hadronic interaction model dependence of our results in future. An analysis of simulation data considering the LDD of muons and also the combined LDD of electrons and muons are in progress.\n\n\n\\section*{Acknowledgments}\n\\noindent    \nRKD acknowledges the financial support under grant. no. 1513\/R-2020. from the University of North Bengal.\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\n\n\n\n\nIn the so-called unitary limit of a quantum Bose or Fermi gas,  the\nscattering length $a$ diverges.   This occurs at a fixed point of the \nrenormalization group,  thus these systems provide interesting examples of \ninteracting,  scale-invariant theories with dynamical exponent $z=2$, \ni.e. non-relativistic.  \nThey can be realized experimentally by tuning the scattering\nlength to $\\pm \\infty$ using a Feshbach resonance.\n(See for instance \\cite{Experiment1,Experiment2} and references\ntherein.)   They are also thought to occur at the surface of\nneutron stars.  These systems  have also attracted much theoretical \ninterest\\cite{Leggett,Nozieres,Ohashi,Ho,HoMueller,Astrakharchik,Nussinov,Perali,LeeShafer,Wingate,Bulgac,Drummond,Burovski,Nishida,Nikolic}.   \nThere have even been some proposals to use the AdS\/CFT correspondence \nto learn about these models\\cite{Son,Maldacena,Herzog,Adams}.  \n\n\nBecause of the scale-invariance,  the only length scales in the problem\nare based on the density $n^{1\/d}$ where $d$ is the spatial dimension, \nand the thermal wavelength $\\lambda_T = \\sqrt{2\\pi\/mT}$.   Equivalently,\nthe only energy scales are the chemical potential $\\mu$ and the temperature $T$.  \nThe problem is challenging since there is no small paramater to expand in\nsuch as $n a^3$.    Any possible critical point must occur at a \nspecific value of $x=\\mu\/T$.   This can be translated into \nuniversal values for $n_c \\lambda_T^3$, or for fermions \nuniversal values for $T_c\/T_F$ where $\\epsilon_F = k_B T_F$ is the Fermi energy.  \nFor instance the critical point of an ideal Bose gas is the simplest example,\nwhere $n_c \\lambda_T^3 = \\zeta (3\/2) = 2.61$.  \n\n\n\nThe present work is the sequel to \\cite{PyeTon2}, where we used the S-matrix \nbased formulation\nof the quantum statistical mechanics developed in\\cite{LeClairS,PyeTon}.   \nThis approach is very well-suited to the problem because in the unitary limit\nthe S-matrix $S=-1$, and kernels in the integral equations simplify.   \nIn fact, this approach can be used to develop an expansion in $1\/a$.  \nThe main formulas for the 2 and 3 dimensional cases of both bosons and fermions\nwere presented,  however only the 2-dimensional case was analyzed in detail\nin \\cite{PyeTon2}.   \nHere we analyze the 3-dimensional case.   \n\n\n\nThe models considered  are the simplest \nmodels of non-relativistic bosons or  fermions with quartic\ninteractions.    The bosonic model is defined by the action\nfor a complex scalar field $\\phi$. \n\\begin{equation}\n\\label{bosonaction}\nS =  \\int d^3 {\\bf x} dt \\(  i \\phi^\\dagger  \\partial_t \\phi - \n\\frac{ |\\vec{\\nabla} \\phi |^2}{2m}  - \\frac{g}{4} (\\phi^\\dagger  \\phi)^2 \\)\n\\end{equation}\nFor fermions, due to the fermionic statistics, \none needs at least a 2-component field \n$\\psi_{\\uparrow , \\downarrow} $:\n\\begin{equation}\n\\label{fermionaction}\nS = \\int d^3 {\\bf x} dt \\(  \\sum_{\\alpha=\\uparrow, \\downarrow}  \ni \\psi^\\dagger_\\alpha \\partial_t  \\psi_\\alpha  - \n\\frac{|\\vec{\\nabla}  \\psi_\\alpha|^2}{2m}   - \\frac{g}{2} \n\\psi^\\dagger_\\uparrow \\psi_\\uparrow \\psi^\\dagger_\\downarrow \\psi_\\downarrow \\) \n\\end{equation}\nIn both cases,  positive $g$ corresponds to repulsive interactions.   \nThe bosonic theory only has a $U(1)$ symmetry.   The fermionic theory\non the other hand has the much larger SO(5) symmetry.   \nThis is evident from the work\\cite{Kapit} which considered a relativistic\nversion, since  the same arguments apply to a non-relativistic kinetic term. \nThis is also clear from the work\\cite{Nikolic} which considered \nan $N$-component version with Sp(2N) symmetry,  and noting that\nSp(4) = SO(5).   \n\n\n\nThe interplay between the scattering length, the bound state,  \nand the renormalization group fixed point was discussed \nin detail from the point of view of the\nS-matrix in \\cite{PyeTon2}.   In 3 spatial dimensions the fixed point occurs\nat negative coupling $g_*= - 4 \\pi^2 \/m\\Lambda$,  where $\\Lambda$ is \nan ultra-violet cut-off.  For $g$  less than $g_*$, there is a bound\nstate that can Bose-Einstein condense (BEC), and for the fermionic case, \nthis is referred to as the BEC side.  As $g_*$ is approached from this side,\nthe scattering length goes to $+\\infty$.   The bound state disappears at\n$g_*$.  When $g$ approaches $g_*$ from above,  the scattering length \ngoes to $-\\infty$, and this is referred to as the BCS side.   In this paper we\nwork on the BCS side of the cross-over since the bound state does not have\nto be incorporated into the thermodynamics.  On this side the interactions\nare effectively attractive.       \n\n\nTheoretical studies have mainly focussed on the fermionic case,   \nand for the most part at zero temperature,   which is appropriate for a large Fermi energy.\n  The bosonic case has been less studied,  since \na homogeneous \nbosonic gas with attractive interactions is thought to be unstable against\nmechanical collapse, and the collapse occurs before any kind of BEC.  \nThe situation is actually different for harmonically  trapped gases,  where BEC can occur\\cite{trapped}.  \nHowever studies of the homogeneous bosonic case were  based on a small, \nnegative scattering length\\cite{Stoof,MuellerBaym,Thouless,YuLiLee},\nand it is not clear that the conclusions reached  there can be extrapolated \nto the unitary limit.    Since the density of\ncollapse is proportional to $1\/a$\\cite{MuellerBaym},   extrapolation to infinite scattering length \nsuggests that the gas  collapses at zero density,  which seems unphysical,  since the gas could in\nprinciple be stabilized at finite temperature by thermal pressure.   \n One can also point out that  in the van der Waals gas,  the\ncollapse is stabilized by a finite size of the atoms,  which renders\nthe compressibility finite.  In the unitary limit, there is nothing to play\nsuch a role.  \n  In the sequel we will present evidence that the unitary \nBose gas undergoes BEC when $n \\lambda_T^3 \\approx 1.3$. \n This lower value\nis consistent with the attractive interactions.    We also estimate \nthe critical exponent describing how the compressibility diverges at the critical point. \n\n\n\n\nIn the next section we define the  scaling functions  that determine the free energy\nand density,  and derive expressions for the energy and entropy per particle,  specific heat per\nparticle, and compressibility.     In section III the formulation of the unitary limit in\n\\cite{PyeTon2}  is summarized.     The two-component fermion case is analyzed in section IV.\nEvidence is presented for a critical point with $T_c\/T_F \\approx 0.1$,  which is consistent with\nlattice  Monte Carlo simulations.    Bosons are analyzed in section V,  where we present evidence\nfor BEC in this strongly interacting gas.     Motivated by  the conjectured lower bound\\cite{Kovtun} \nfor\nthe ratio of the viscosity to entropy density  $\\eta\/s > \\hbar\/4\\pi k_B$ for relativistic systems,\nwe study this ratio for both the fermionic and bosonic cases in section VI.\n   Our results for fermions are\nconsistent with experiments,  with  $\\eta\/s >   4.72$  times the conjectured lower bound.\nFor bosons,  this ratio is minimized at the critical point  where \n$\\eta\/s > 1.26$ times the bound.    \n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Scaling functions in the unitary limit.}  \n\n\n\nThe scale invariance in the unitary limit implies some\nuniversal scaling forms\\cite{Ho}.   In this section we \ndefine various scaling functions with a meaningful normalization\nrelative to free particles.    \n\nFirst consider a single species of bosonic or fermionic particle with\nmass $m$ at chemical potential $\\mu$ and temperature $T$. \n  The free energy density\nhas the form\n\\begin{equation}\n\\label{freeenergy}\n\\CF =  - \\zeta (5\/2) \\( \\frac{mT}{2\\pi} \\)^{3\/2} T  \\, c(\\mu\/T) \n\\end{equation}\nwhere the scaling function $c$ is only a function of $x\\equiv \\mu\/T$.\n($\\zeta$ is Riemann's zeta function.)   \nThe combination $\\sqrt{mT\/2\\pi} = 1\/\\lambda_T$, where \n$\\lambda_T$ is the thermal wavelength.   For a single free boson  or fermion:\n\\begin{equation}\n\\label{cfreelim}\n\\lim_{\\mu\/T \\to 0} ~~  c_{\\rm boson} = 1, ~~~~~~~\n c_{\\rm fermion} = 1- \\inv{2 \\sqrt{2}}\n~~~~~~({\\rm free ~ particles}).  \n\\end{equation}\nIt is also convenient to define the  scaling function $q$, which is a measure\nof the quantum degeneracy, in terms of the density   as follows: \n\\begin{equation}\n\\label{nhat}\nn \\lambda_T^3 = q\n\\end{equation}\nThe two scaling functions $c$ and $q$ are of course related since\n$n= - \\partial \\CF \/ \\partial \\mu$,  which leads to \n\\begin{equation}\n\\label{qcprime}\nq = \\zeta (5\/2) c'\n\\end{equation}   \nwhere $c'$ is the derivative of $c$ with respect to $x$.   \nHenceforth $g'$ will always denote the derivative of $g$ with respect to $x$.\nThe expressions  for $c$ and $q$ for free theories will be implicit\nin the next section.    \n\n\nAlso of interest are several energy per particle scaling functions.   \nAt a renormalization group fixed point, the energy density\nis related to the free energy in the same way as for free particles:\n\\begin{equation}\n\\label{energyden}\n\\frac{E}{V} = -\\frac{3}{2} \\CF \n\\end{equation}\nwhere $V$ is the volume.    For a free fermion,   in the zero \ntemperature limit,  the energy per particle $E\/N \\to \\frac{3}{5} \\mu$.\nThe Fermi energy is \n\\begin{equation}\n\\label{EF}\n\\epsilon_F =\\inv{m}  \\( 3 \\pi^2 n\/\\sqrt{2} \\)^{2\/3} \n\\end{equation}\nThe above definition can  also  be used for bosons.  \nSince $\\epsilon_F = \\mu$ in the zero temperature free  fermionic gas,  this leads\nus to define the scaling function $\\xi$:\n\\begin{equation}\n\\label{xi}\n\\xi (x)  =  \\frac{5}{3} \\frac{E}{N \\epsilon_F} =  \\frac{5 \\zeta(5\/2)}{3}\n\\( \\frac{6}{\\pi} \\)^{1\/3}      \\,       \\frac{c}{q^{5\/3}}\n\\end{equation}\nIn the limit $T\\to 0$,  i.e. $x\\to \\infty$,   $\\xi \\to 1$ for a free\nfermion.   \n\nA different energy per particle scaling function, $\\tilde{\\xi}$,  is meaningful\nas $x\\to 0$:\n\\begin{equation}\n\\label{xitilde}\n\\frac{E}{N T}  =  \\frac{3 (2 \\sqrt{2} -1 ) \\zeta(5\/2) }{2(2 \\sqrt{2} -2) \n\\zeta(3\/2)} \n   \\,  \\tilde{\\xi} (x) \n\\end{equation}\nWith the above normalization $\\tilde{\\xi} = 1$ for a free fermion in the limit\n$x\\to 0$.   In terms of the above scaling functions:\n\\begin{equation}\n\\label{xitildesc}\n\\tilde{\\xi} (x) =   \\frac{ \\zeta (3\/2)   \n( 2 \\sqrt{2} -2)}{(2 \\sqrt{2} -1)}  \\,  \\frac{c}{q} \n\\end{equation}\n\nThe entropy density is $s = - \\partial \\CF \/ \\partial T$,  and the entropy per particle\ntakes the form\n\\begin{equation}\n\\label{sn}\n\\frac{s}{n} =   \\zeta (5\/2) \n\\( \\frac{ 5 c\/2 - x c'}{q} \\) \n\\end{equation}\n\n   \n\n\nNext consider the \n specific heat per particle at constant volume and particle number,\ni.e. constant density.   One needs $\\partial x\/ \\partial T$ at constant density.\nUsing the fact that $n \\propto T^{3\/2} q$,  at constant density\n$q \\propto T^{-3\/2}$.   This gives\n\\begin{equation}\n\\label{CV.1}\nT \\(  \\frac{\\partial x} {\\partial T} \\)_n =   -\\frac{3}{2}  \\frac{ q}{q'} \n\\end{equation}\nThe specific heat per particle is then:\n\\begin{equation}\n\\label{CV.2}\n\\frac{C_V}{N} = \\inv{N}  \\( \\frac{\\partial E}{\\partial T} \\)_{N,V} =\n\\frac{\\zeta(5\/2)}{4} \\( 15 \\frac{c}{q} - 9 \\frac{c'}{q'} \\) \n\\end{equation}\n\nThe isothermal compressibility is defined as \n\\begin{equation}\n\\label{comp.1}\n\\kappa =  - \\inv{V} \\( \\frac{ \\partial V}{\\partial p} \\)_T \n\\end{equation}\nwhere the pressure $p= - \\CF$.   Since $n=N\/V$ and $N$ is kept fixed, \n\\begin{equation}\n\\label{comp.2}\n\\kappa =  - n  \\( \\frac{ \\partial n^{-1} }{\\partial p} \\)_T =  \\inv{n T}  \\frac{ q'}{q}  =\n\\inv{T} \\( \\frac{mT}{2\\pi} \\)^{3\/2}   \\frac{q'}{q^2}   \n\\end{equation}\n\n\nFinally the equation of state can be expressed parametrically \nas follows.  Given $n$ and $T$, one uses eq. (\\ref{nhat}) to find\n$x$ as a function of $n,T$.   The pressure can then be written as\n\\begin{equation}\n\\label{eqnstate}  \np =  \\(  \\frac{\\zeta(5\/2) c(x(n,T))}{q(x(n,T))} \\)  n T\n\\end{equation}\n\n\n\nIn order to compare with numerical simulations and experiments,\nit will be useful to plot various quantities as a function of  $q$ or \n$T\/T_F$:\n\\begin{equation}\n\\label{TTF}\n\\frac{T}{T_F}  =  \\(  \\frac{4}{3 \\sqrt\\pi  q} \\)^{2\/3} \n\\end{equation}\n\n\n\n\n\\section{Two-body scattering approximation}\n\n\n\nThe main features of the two-body scattering approximation developed in\n\\cite{PyeTon} are the following.   Consider again first a single component\ngas.   The filling fractions,  or \noccupation numbers,  are parameterized in terms of a pseudo-energy\n$\\varepsilon ({\\bf k} )$:\n\\begin{equation}\n\\label{fill}\nf({\\bf k} )  =  \\inv{ e^{\\beta \\varepsilon ({\\bf k} ) } -s }\n\\end{equation}\nwhich determine the density:\n\\begin{equation}\n\\label{dens}\nn = \\int \\frac{d^3 {\\bf k}}{(2\\pi)^3} ~ \\inv{ e^{\\beta \\varepsilon  ({\\bf k} ) } -s }\n\\end{equation}\nwhere $s=1,-1$ corresponds to bosons, fermions respectively\nand $\\beta = 1\/T$.    \nThe consistent summation of 2-body scattering leads to \nan integral equation for the \n pseudo-energy $\\varepsilon ({\\bf k})$.   It is convenient to define the\nquantity:\n\\begin{equation}\n\\label{ydef}\ny ({\\bf k} ) = e^{-\\beta (\\varepsilon({\\bf k}) - \\omega_{\\bf k} + \\mu )}\n\\end{equation}\nwhere $\\omega_{\\bf k} = {\\bf k}^2 \/ 2m$.   Then $y$ satisfies the integral\nequation \n\\begin{equation}\n\\label{yinteq}\ny ({\\bf k} ) =  1 + \\beta   \\int  \\frac{d^3 {\\bf k}'}{(2\\pi)^3} \\, \nG({\\bf k} - {\\bf k}' )  \\frac{y({\\bf k}' )^{-1}}{e^{\\beta \\varepsilon ({\\bf k}')} -s}\n\\end{equation}\nThe free energy density is then\n\\begin{equation}\n\\label{freefoam2}\n\\CF = -T  \\int \\frac{d^3 {\\bf k}}{ (2\\pi)^3} \\[ \n- s  \\log ( 1- s e^{-\\beta \\varepsilon  } ) \n-\\inv{2}  \\frac{ (1-y^{-1} )}{e^{\\beta \\varepsilon} -s}   \\]  \n\\end{equation}\n\nThe kernel has the following structure:\n\\begin{equation}\n\\label{Gstructure}\nG  =  - \\frac{i}{\\mathcal{I}} \\log ( 1 + i \\mathcal{I} {\\cal M}}\t\\def\\CN{{\\cal N}}\t\\def\\CO{{\\cal O})\n\\end{equation}\nwhere ${\\cal M}}\t\\def\\CN{{\\cal N}}\t\\def\\CO{{\\cal O}$ is the scattering amplitude and $\\mathcal{I}$ represents\nthe available phase space for two-body scattering.   The argument\nof the $\\log$ can be identified as the S-matrix function. \nIn the unitary limit,  \n\\begin{equation}\n\\label{CMunit}\n{\\cal M}}\t\\def\\CN{{\\cal N}}\t\\def\\CO{{\\cal O} =   \\frac{2i}{\\mathcal{I}} =  \\frac{16 \\pi i}{m |{\\bf k} - {\\bf k}'|},\n\\end{equation}\nand the S-matrix equals $-1$.    The kernel becomes\n\\begin{equation}\n\\label{kernel}\nG({\\bf k} - {\\bf k}' ) =  \\mp  \\frac{8 \\pi^2}{m |{\\bf k} - {\\bf k}'|} , \n\\end{equation} \nwhere the $-$ sign corresponds to $g$ being just below the fixed point\n$g_*$, where the scattering length $a\\to +\\infty$ on the BEC side,\nwhereas the $+$ sign corresponds to $a\\to - \\infty$ on the BCS side.  \nAs explained in the Introduction,  we work on the BCS side.\n\nThe angular integrals in eq. (\\ref{yinteq}) are easily performed.  \nDefining the dimensionless variable $\\kappa = {\\bf k}^2 \/ 2mT$,  \nthe integral equation becomes \n\\begin{equation}\n\\label{ykappa}\ny (\\kappa) =  1 +  4  \\int_0^\\infty  d\\kappa'  \\[  \n\\Theta (\\kappa - \\kappa') \\sqrt{\\kappa'\/\\kappa}  + \n\\Theta (\\kappa' - \\kappa) \\]   \\frac{z}{e^{\\kappa'}  - s z y(\\kappa')} \n\\end{equation}\nwhere $z = e^{\\mu\/T}$ is the fugacity and $\\Theta(\\kappa)$ is the standard\nstep function equal to 1 for $\\kappa > 0$, zero otherwise.  \n\n\nFinally comparing with the definitions in the last section \nthe scaling function for the density  and free energy are\n\\begin{equation}\n\\label{nhatsc}\nq (x)   =  \\frac{2}{\\sqrt\\pi} \\int_0^\\infty  d\\kappa  \\sqrt{\\kappa} \n  \\frac{  y(\\kappa) z }{e^\\kappa - s  y(\\kappa) z}  \n\\end{equation}\nand \n\\begin{equation}\n\\label{cscale}\nc =  \\frac{2}{\\sqrt{\\pi} \\zeta (5\/2) }   \n\\int_0^\\infty  d\\kappa \\sqrt{\\kappa}  \\(  -s  \\log \\( 1- s z y(\\kappa)\n e^{-\\kappa} \\) \n- \\inv{2}  \\frac{ z ( y(\\kappa) - 1 ) }{e^\\kappa - s zy(\\kappa)} \\)\n\\end{equation}\nThe ideal, free  gas limit corresponds to $y=1$ \nwhere $q= s {\\rm Li}_{3\/2} (s z)$ and  $c= s {\\rm Li}_{5\/2} (sz)\/ \\zeta(5\/2)$,\nwhere ${\\rm Li}$ is the polylogarithm.    The BEC critical point of the\nideal gas occurs at $\\mu=0$, i.e. $q=\\zeta(3\/2)$.  \n\n\n\n\nConsider now two-component fermions with the action (\\ref{fermionaction}).  \nHere the phase space factor $\\mathcal{I}$ is doubled and since $G\\propto 1\/\\mathcal{I}$, \nthe kernels have an extra $1\/2$:\n\\begin{equation}\n\\label{Gbosferm}\nG_{\\rm fermi} = \\inv{2} G_{\\rm bose} \n\\end{equation}\nDue to the SU(2) symmetry,  the two-component fermion reduces to\ntwo identical copies of the above 1-component expressions, with the\nmodification (\\ref{Gbosferm}).   \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Analysis of fermions}\n\n\n\n Recall that for 2 component fermions,\nthe $4$ is replaced by $2$ in eq. (\\ref{ykappa}), and by the SU(2) \nsymmetry, the system reduces to two identical copies of the one-component \nequations of the last sections;  below we present results for \na single  component, it being implicit  that the free energy and density\nare doubled.    \n\nThe integral equation for $y(\\kappa)$, eq. (\\ref{ykappa}),  can be\nsolved numerically by iteration.\n One first substitutes \n$y_0 = 1$ on the right hand side and this gives the  approximation\n$y_1$ for $y$.  One then substitutes $y_1$ on the right hand side\nto generate $y_2$,  etc.   For regions of $z$ where there are no\ncritical points,  this procedure converges rapidly, and as little\nas 5 iterations are needed.    For fermions,  as one approaches \nzero temperature, i.e. $x$ large and positive,   more iterations are needed\nfor convergence.  The following results are based on 50 iterations.  \n\nWhen $z\\ll 1$,   $y \\approx 1$, and the properties of the free ideal \ngas are recovered, since the gas is very dilute.   There are solutions\nto eq. (\\ref{ykappa}) for all   $-\\infty < x < \\infty$.  ($x=\\mu\/T$).  \nThe scaling function $c$, and it's comparision with a free theory, \nare shown in Figure \\ref{cF} as a function of $x$.\n   The corrections to the free \ntheory become appreciable when $x>-2$.   At $x=0$:\n\\begin{equation}\n\\label{czero}\nc(0) = 0.880, ~~~~~~\\tilde{\\xi} = 0.884, \n\\end{equation}\ncompared to the free gas values of $c(0) = 0.646$ and $\\tilde{\\xi} =1$.  \n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{x}{$x=\\mu\/T$}\n\\psfrag{y}{$c$}\n\\psfrag{a}{$\\rm free$}\n\\includegraphics[width=10cm]{cF.eps} \n\\end{center}\n\\caption{$c(x)$ and its equivalent for a free theory as a function of\n$x=\\mu\/T$.}  \n\\vspace{-2mm}\n\\label{cF} \n\\end{figure} \n\n\nThe scaling function $q$ for the density is shown as function of \n$x$ in Figure \\ref{qF}. Note that the density in the interacting case\nis always higher than for a free gas,  due to the attractive interactions.\nAt $x=0$,  $q(0) = 1.18$, whereas for a free gas $q=0.765$.   \nAt low temperatures and high densities, $\\mu\/T \\gg 1$, the\noccupation numbers resemble that of a degenerate Fermi gas,\nas shown in Figure \\ref{fF}.\n\n \n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{x}{$x=\\mu\/T$}\n\\psfrag{y}{$q$}\n\\psfrag{a}{$\\rm free$}\n\\includegraphics[width=10cm]{qF.eps} \n\\end{center}\n\\caption{$q(x)$ and its equivalent for a free theory as a function of\n$x=\\mu\/T$.}  \n\\vspace{-2mm}\n\\label{qF} \n\\end{figure} \n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{x}{$\\kappa =  \\beta {\\bf k}^2 \/2m$}\n\\psfrag{y}{$f$}\n\\psfrag{a}{$x=15$}\n\\includegraphics[width=10cm]{fF.eps} \n\\end{center}\n\\caption{The occupation numbers as a function of $\\kappa$ for $x=5,10,15$.}  \n\\vspace{-2mm}\n\\label{fF} \n\\end{figure} \n\n\n\n\nWhereas $c$ and $q$ are nearly featureless,  other quantities \nseem to indicate a critical point, or phase transition, at \nlarge density.    For instance,  the entropy per particle \ndecreases with decreasing temperature up to $x < x_c \\approx 11.2$,\nas shown in Figure \\ref{snF}.  Beyond this point the entropy per particle\nhas the unphysical behavior of increasing with temperature.  \nA further indication that the region $x>x_c$ is unphysical is\nthat the specific heat per particle becomes negative, as shown in\nFigure \\ref{CVNF}.  When $x\\ll 0$,  $C_V\/N$ approaches the \nclassical value $3\/2$.   This leads us to suggest a phase transition,\nat $x=x_c$, corresponding to the critical temperature  \n$ T_c \/T_F  \\approx 0.1$.  As we will show, our analysis of\nthe viscosity to entropy-density ratio suggests a higher $T_c\/T_F$.  \nThere have been numerous estimates of $T_c\/T_F$ based on various\napproximation schemes, mainly using Monte Carlo methods on the lattice\n\\cite{Perali, LeeShafer,Wingate,Bulgac,Drummond,Burovski}, \nquoting results for $T_c\/T_F$ between $0.05$ and $0.23$.  The work\n\\cite{LeeShafer} puts an upper bound $T_c \/ T_F < 0.14$,\nand the  most recent results of Burovski et. al. quote $T_c\/T_F =0.152(7)$.\nOur result is thus consistent with previous work.    \nThe equation of state at this point follows from eq. (\\ref{eqnstate}):\n\\begin{equation}\n\\label{eqstF}\np = 4.95 n T\n\\end{equation}\n\n\n  \n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{x}{$x=\\mu\/T$}\n\\psfrag{y}{$s\/n$}\n\\includegraphics[width=10cm]{snF.eps} \n\\end{center}\n\\caption{Entropy per fermionic particle as a function of $x$.}  \n\\vspace{-2mm}\n\\label{snF} \n\\end{figure} \n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{x}{$x=\\mu\/T$}\n\\psfrag{y}{$C_V\/N$}\n\\includegraphics[width=10cm]{CVNF.eps} \n\\end{center}\n\\caption{Specific heat per particle as a function of $x$ for fermions.}  \n\\vspace{-2mm}\n\\label{CVNF} \n\\end{figure} \n\n\n\nThe energy per particle, normalized to the Fermi energy $\\epsilon_F$,\ni.e. $E\/N \\epsilon_F = 3 \\xi \/5$,  and the entropy per particle,  \nare  shown in Figures \\ref{xiF},\\ref{snFTTF} as a function of $T\/T_F$, where\n$k_B T_F = \\epsilon_F$.   At high temperatures it matches that of a free\nFermi gas, in agreement with the Monte Carlo simulations in\n\\cite{Bulgac,Burovski}.  Note that there is no sign of \npair-breaking at $T^*\/T_F = 0.5$ predicted in \\cite{Perali},\nand this also agrees with the Monte Carlo simulations.\n However at low temperatures in the vicinity\nof $T_c$, the agreement is not as good.   This suggests our approximation\nis breaking down for very large $z$, i.e. the limit of zero temperature. \nThe same conclusion is reached by examining $\\mu\/\\epsilon_F$,\ndisplayed in Figure \\ref{muepF}, since the zero temperature \nepsilon expansion and Monte Carlo give\n $\\mu\/\\epsilon_F \\approx 0.4 - 0.5$\\cite{Son,Burovski}. \n\n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{X}{$T\/T_F$}\n\\psfrag{Y}{$\\frac{E}{N \\epsilon_F}$}\n\\includegraphics[width=10cm]{xiF.eps} \n\\end{center}\n\\caption{Energy per particle normalized to $\\epsilon_F$  as a function of\n$T\/T_F$.}  \n\\vspace{-2mm}\n\\label{xiF} \n\\end{figure} \n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{X}{$T\/T_F$}\n\\psfrag{Y}{$s\/n$}\n\\includegraphics[width=10cm]{snFTTF.eps} \n\\end{center}\n\\caption{Entopy  per particle   as a function of\n$T\/T_F$.}  \n\\vspace{-2mm}\n\\label{snFTTF} \n\\end{figure} \n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{X}{$T\/T_F$}\n\\psfrag{Y}{$\\mu \/ \\epsilon_F$}\n\\includegraphics[width=10cm]{muepF.eps} \n\\end{center}\n\\caption{Chemical potential normalized to $\\epsilon_F$ as a function of\n$T\/T_F$.}  \n\\vspace{-2mm}\n\\label{muepF} \n\\end{figure} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Analysis of bosons}  \n\n\n\n\n\n\nFor bosons we again solved the integral equation (\\ref{ykappa}) \nby iteration, starting from $y=1$.   Since the occupation numbers decay\nquickly as a function of $\\kappa$,  we introduced a cut-off $\\kappa < 10$.  \nFor $x$ less than approximately $-2$,  the gas behaves nearly classically.  \n\nThe main feature of the solution to the integral equation is that for\n$x>x_c \\equiv -1.2741$,  there is no solution that is smoothly connected to\nthe classical limit $x\\to -\\infty$.  Numerically,  when there is no solution\nthe iterative procedure fails to converge.  \nThe free energy scaling function is plotted in Figure \\ref{cB}. \nNote that $c<1$, where $c=1$ is the free field value.   \n  We thus take the physical region to \nbe $x< x_c$.    We find strong evidence that the gas undergoes BEC \nat $x=x_c$.   In Figure \\ref{epofx}, we plot $\\varepsilon ({\\bf k}=0 )$ as a function of\n$x$,  and ones sees that it goes to zero at $x_c$.   This implies the occupation\nnumber $f$ diverges at ${\\bf k} =0$ at this critical point.   \nOne clearly sees this behavior in Figure \\ref{fB}.  \n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{x}{$x=\\mu\/T $}\n\\psfrag{y}{$c$}\n\\psfrag{a}{$\\rm ideal$}\n\\includegraphics[width=10cm]{cB.eps} \n\\end{center}\n\\caption{The free-energy scaling function $c$ as a function of $\\mu\/T$ compared \nto the ideal gas case.}  \n\\vspace{-2mm}\n\\label{cB} \n\\end{figure} \n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{x}{$x=\\mu\/T$}\n\\psfrag{y}{$\\varepsilon ({\\bf k}=0)\/T$}\n\\psfrag{a}{$x_c$}\n\\includegraphics[width=10cm]{epofx.eps} \n\\end{center}\n\\caption{The pseudo-energy $\\varepsilon$ at ${\\bf k} =0$  as a function of $x=\\mu\/T$.}  \n\\vspace{-2mm}\n\\label{epofx} \n\\end{figure} \n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{x}{$\\kappa =  \\beta {\\bf k}^2 \/2m $}\n\\psfrag{y}{$f$}\n\\psfrag{a}{$x_c$}\n\\includegraphics[width=10cm]{fB.eps} \n\\end{center}\n\\caption{The occupation number $f(\\kappa)$  for  $x=-1.275$ and $x_c = -1.2741$.}  \n\\vspace{-2mm}\n\\label{fB} \n\\end{figure} \n\n\nThe compressibility is shown in Figure \\ref{compressB},  and diverges at\n$x_c$, again consistent with BEC.      \nWe thus conclude that there is a critical point at $x_c$ which a \nstrongly interacting, scale invariant version of the ideal BEC.   \nIn terms of the density, the critical point is:\n\\begin{equation}\n\\label{xcb}\nn_c \\lambda_T^3  =  1.325, ~~~~~~~~~( \\mu\/T = x_c = -1.2741 )\n\\end{equation}\nThe negative value of the chemical potential is consistent with the\neffectively attractive interactions.   The above should be compared with \nthe ideal BEC of the free theory,  where $x_c = 0$ and \n$n_c \\lambda_T^3 = \\zeta (3\/2) = 2.61$,  which is higher by a factor of 2.  \nAt the critical point the equation of state is \n\\begin{equation}\n\\label{eqstB}\np = 0.318 n T\n\\end{equation}\ncompared to $p = 0.514 nT$ for the free case.   ($0.514 = \\zeta(5\/2)\/ \\zeta (3\/2)$). \n\nA  critical exponent $\\nu$ characterizing the diverging  compressibility can be defined as \n\\begin{equation}\n\\label{expnu}\n\\kappa  \\sim  (T-T_c)^{-\\nu}\n\\end{equation}\nA log-log plot of the compressibility verses $T-T_c$ shows an approximately\nstraight line,   and we obtain   $\\nu \\approx 0.69$.    This should be compared with\nBEC in an ideal gas,   where $\\nu \\approx 1.0$.     Clearly the unitary gas version \nof BEC  is  in a different universality class.   \n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{x}{$x=\\mu\/T $}\n\\psfrag{y}{$\\kappa T (mT)^{3\/2}$}\n\\includegraphics[width=10cm]{compressB.eps} \n\\end{center}\n\\caption{The compressibility  $\\kappa $  as a function of $\\mu\/T$.}  \n\\vspace{-2mm}\n\\label{compressB} \n\\end{figure} \n\n\n\n\n\n\n\n\n\n\n\nThe energy per particle scaling function $\\tilde{\\xi}$ at the critical point is\n$\\tilde{\\xi} (x_c) = 0.281$ compared to $0.453 = (2 \\sqrt{2} -2)\/(2 \\sqrt{2} -1)$ \nfor the free case.    The entropy per particle and specific heat per particle\nare plotted in Figures \\ref{snB}, \\ref{CVNB} as a function of $T\/T_c$.  \nAt large temperatures,  as expected $C_V\/N = 3\/2$, i.e. the classical value. \nIt increases as $T$ is lowered,  however in contrast to the ideal gas case,\nit then begins to decrease as $T$ approaches $T_c$.      \n\n\n\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{x}{$T\/T_c$}\n\\psfrag{y}{$s\/n$}\n\\includegraphics[width=10cm]{snB.eps} \n\\end{center}\n\\caption{The entropy per particle as a function of  $T\/T_c$.}  \n\\vspace{-2mm}\n\\label{snB} \n\\end{figure} \n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm}\n\\psfrag{x}{$T\/T_c$}\n\\psfrag{y}{$C_V\/N $}\n\\includegraphics[width=10cm]{CVNB.eps} \n\\end{center}\n\\caption{The specific heat per particle as a function of  $T\/T_c$.}  \n\\vspace{-2mm}\n\\label{CVNB} \n\\end{figure} \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Entropy to viscosity ratio}\n\n\nConsider first a single component gas.  \nIn kinetic theory, the shear viscosity can be expressed as \n\\begin{equation}\n\\label{shear.1}\n\\eta = \\inv{3}  n  \\bar{v} m \\ell_{\\rm free} \n\\end{equation}\nwhere $\\bar{v}$ is the average speed and $\\ell_{\\rm free} $ is the mean\nfree path.   The mean free path is \n$\\ell_{\\rm free} = 1\/(\\sqrt{2} n \\sigma)$ where $\\sigma$ is the total\ncross-section.   (The $\\sqrt{2}$  comes from the ratio of the mean speed\nto the mean relative speed\\cite{Reif}.)     In the unitary limit\nthe S-matrix $S=-1$,  which implies the scattering amplitude in\neq. (\\ref{CMunit}).   This leads to \n\\begin{equation}\n\\label{shear.2}\n\\sigma =  \\frac{ m^2 |{\\cal M}}\t\\def\\CN{{\\cal N}}\t\\def\\CO{{\\cal O}|^2}{4\\pi}  \n=  \\frac{16 \\pi}{|{\\bf k}|^2}\n\\end{equation}\nwhere $|k|$ is the momentum of one of the particles in the center of mass\nframe,  i.e. $|{\\bf k}_1 - {\\bf k}_2| = 2 |{\\bf k}|$.    This gives\n\\begin{equation}\n\\label{shear.3}\n\\eta =  \\frac{m^3 \\bar{v}^3}{48 \\sqrt{2} \\pi} \n\\end{equation}\n\nSince the equation (\\ref{energyden}) is  the same relation \nbetween the pressure and energy of a free gas,  and the pressure\nis due to the kinetic energy, this implies\n\\begin{equation}\n\\label{shear.4}\n\\inv{2} m \\bar{v}^2  =  E\/N  =  \\frac{3}{2} \\frac{c}{c'} T   \n\\end{equation}\nSince the  entropy density $s= - \\partial \\CF \/ \\partial T$, one finally has\n\\begin{equation}\n\\label{etas}\n\\frac{\\eta}{s} =  \\frac{\\sqrt{3\\pi}}{8 \\zeta (5\/2)} \n \\(  \\frac{c}{c'} \\)^{3\/2}  \\inv{ 5 c \/2 - x c' } \n\\end{equation}\n\n\nFor two-component fermions,  the available phase space $\\mathcal{I}$ is doubled.\nAlso,  spin up particles only scatter with spin down.  This implies\n$\\eta$ is $8$ times the above expression.   Since the entropy density is doubled,\nthis implies that $\\eta\/s$ is $4$ times the expression eq. (\\ref{etas}).  \n\n\nThe ratio $\\eta\/s$ for fermions as a function of $T\/T_F$ is shown in\nFigure \\ref{etasF},  and is in good agreement both quantitatively and\nqualitatively  with the experimental data \nsummarized in \\cite{Schafer}.   The lowest value occurs at $x=2.33$, \nwhich corresponds to $T\/T_F = 0.28$, and \n\\begin{equation}\n\\label{etasFlim}\n\\frac{\\eta}{s} > 4.72 \\frac{\\hbar}{4 \\pi k_B}\n\\end{equation}\nThe experimental data\nhas a minimum that is about $6$ times this bound.    \nIn the free fermion theory the minimum occurs at $\\mu\/T \\approx 2.3$,\nwhich gives $\\eta\/s >  7.2  \\hbar \/ 4 \\pi k_B$.   \n\n\n\n \n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{X}{$T\/T_F$}\n\\psfrag{Y}{$\\eta \/ s$}\n\\includegraphics[width=10cm]{etasF.eps} \n\\end{center}\n\\caption{The viscosity to entropy-density ratio  as a function of\n$T\/T_F$ for fermions.  The horizontal line is $1\/4\\pi$.}  \n\\vspace{-2mm}\n\\label{etasF} \n\\end{figure} \n\n\n\n\n\n\n\nFor bosons, the ration $\\eta\/s$ is plotted in Figure \\ref{etasB} as\na function of $T\/T_c$.   One sees that it has a minimum at the critical point,\nwhere \n\\begin{equation}\n\\label{etasBlim}\n\\frac{\\eta}{s} > 1.26  \\frac{\\hbar}{4 \\pi k_B}\n\\end{equation}\n   Thus the bosonic gas \nat the unitary critical point is a more perfect fluid than that of fermions. \nOn the other hand,  the ideal Bose gas at the critical point has a lower value:\n\\begin{equation}\n\\label{etasBfree}\n\\frac{\\eta}{s} \\Bigg\\vert_{\\rm ideal}  =  \\frac{ \\sqrt{ 3 \\pi \\zeta (5\/2)}}\n{20 \\zeta (3\/2)^{3\/2}} =  0.53 \\frac{\\hbar}{4 \\pi k_B}\n\\end{equation}\n\n\n\n\\begin{figure}[htb] \n\\begin{center}\n\\hspace{-15mm} \n\\psfrag{x}{$T\/T_c$}\n\\psfrag{y}{$\\eta \/ s$}\n\\includegraphics[width=10cm]{etasB.eps} \n\\end{center}\n\\caption{The viscosity to entropy-density ratio  as a function of\n$T\/T_c$ for bosons.  The horizontal line is $1\/4\\pi$.}  \n\\vspace{-2mm}\n\\label{etasB} \n\\end{figure} \n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\n\n\nWe presented a novel analytic treatment of unitary Bose and Fermi gases\nat finite temperature and chemical potential \nusing  a new formulation of statistical mechanics\nbased on the exact, zero temperature,  2-body scattering.    \nOur results appear to be consistent with lattice Monte Carlo methods. \nAll of the thermodynamic functions, such as entropy per particle, energy\nper particle,  specific heat, compressibility, and viscosity \nare readily calculated once one numerically solves the integral equation\nfor the pseudo-energy.    \n\nFor fermions, our 2-body approximation is good if the temperatures are not\ntoo low.   We estimated  $T_c\/T_F \\approx 0.1$,  where the critical \npoint occurs at $\\mu\/T \\approx 11.2$.        For bosons we presented\nevidence for a strongly interacting version of BEC at the critical \npoint $n \\lambda_T^3 \\approx 1.3$,  corresponding to \n$\\mu \/ T = -1.27$. \n\n\n\n\n\n\n\\section{Acknowledgments}\n\n\nWe wish to thank Erich Mueller for helpful discussions.    \n   This work is supported by the National Science Foundation\nunder grant number  NSF-PHY-0757868. \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\\IEEEPARstart{T}o perform diagnosis and prognosis of cardiovascular disease (CVD) medical experts depend on the reliable quantification of cardiac function \\cite{white1987left}. Cardiac magnetic resonance imaging (CMRI) is currently considered the reference standard for quantification of ventricular volumes, mass and function \\cite{grothues2002comparison}. Short-axis CMR imaging, covering the entire left and right ventricle (LV resp. RV) is routinely used to determine quantitative parameters of both ventricle's function. This requires manual or semi-automatic segmentation of corresponding cardiac tissue structures for end-diastole (ED) and end-systole (ES). \n\nExisting semi-automated or automated segmentation methods for CMRIs regularly require (substantial) manual intervention caused by lack of robustness. Manual or semi-automatic segmentation across a complete cardiac cycle, comprising \\num{20} to \\num{40} phases per patient, enables computation of parameters quantifying cardiac motion with potential diagnostic implications but due to the required workload, this is practically infeasible. Consequently, segmentation is often performed at end-diastole and end-systole precluding comprehensive analysis over complete cardiac cycle. \n\nRecently\\cite{litjens2017survey, leiner2019machine}, deep learning segmentation methods have shown to outperform traditional approaches such as those exploiting level set, graph-cuts, deformable models, cardiac atlases and statistical models \\cite{petitjean2011review, peng2016review}. However, recent comparison of a number of automatic methods showed that even the best performing methods generated anatomically implausible segmentations in more than 80\\% of the CMRIs \\cite{bernard2018deep}. Such errors do not occur when experts  perform segmentation. To achieve acceptance in clinical practice these shortcomings of the automatic approaches need to be alleviated by further development. This can be achieved by generating more accurate segmentation result or by development of approaches that automatically detect segmentation failures. \n\nIn manual and automatic segmentation of short-axis CMRI, largest segmentation inaccuracies are typically located in the most basal and apical slices due to low tissue contrast ratios \\cite{suinesiaputra2015quantification}. To increase segmentation performance, several methods have been proposed \\cite{tan2018fully, zheng20183, savioli2018automated, bai2018automated}. Tan et al. \\cite{tan2018fully} used a convolutional neural network (CNN) to regress anatomical landmarks from long-axis views (orthogonal to short-axis). They exploited the landmarks to determine most basal and apical slices in short-axis views and thereby constraining the automatic segmentation of CMRIs. This resulted in increased robustness and performance. Other approaches leverage spatial \\cite{zheng20183} or temporal \\cite{savioli2018automated, bai2018automated} information to increase segmentation consistency and performance in particular in the difficult basal and apical slices.\n\nAn alternative approach to preventing implausible segmentation results is by incorporating knowledge about the highly constrained shape of the heart. Oktay et al. \\cite{oktay2017anatomically} developed an anatomically constrained neural network (NN) that infers shape constraints using an auto-encoder during segmentation training. Duan et al. \\cite{duan2019automatic} developed a deep learning segmentation approach for CMRIs that used atlas propagation to explicitly impose a shape refinement. This was especially beneficial in the presence of image acquisition artifacts. Recently, Painchaud et al. \\cite{painchaud2019cardiac} developed a post-processing approach to detect and transform anatomically implausible cardiac segmentations into valid ones by defining cardiac anatomical metrics. Applying their approach to various state-of-the-art segmentation methods the authors showed that the proposed method provides strong anatomical guarantees without hampering segmentation accuracy. \n\n\\begin{figure*}[t]\n\t\\center\n\t\\includegraphics[width=6in, height=3.5in]{figures\/overview_approach.pdf}%\n\t\n\t\\caption{Overview of proposed two step approach. Step 1 (left): Automatic CNN segmentation of CMR images combined with assessment of segmentation uncertainties. Step 2 (right): Differentiate tolerated errors from segmentation failures (to be detected) using distance transform maps based on reference segmentations. Detection of image regions containing segmentation failures using CNN which takes CMR images and segmentation uncertainties as input. Manual corrected segmentation failures (green) based on detected image regions.}\n\t\\label{fig_overview_method}\n\\end{figure*}\n\n\nA different research trend focuses on detecting segmentation failures, i.e. on automated quality control for image segmentation. These methods can be divided in those that predict segmentation quality using image at hand or corresponding automatic segmentation result, and those that assess and exploit predictive uncertainties to detect segmentation failure. \n\nRecently, two methods were proposed to detect segmentation failures in large-scale cardiac MR imaging studies to remove these from subsequent analysis\\cite{alba2018automatic, robinson2019automated}. Robinson et al. \\cite{robinson2019automated} using the approach of Reverse Classification Accuracy (RCA) \\cite{valindria2017reverse} predicted CMRI segmentation metrics to detect failed segmentations. They achieved good agreement between predicted metrics and  visual quality control scores. Alba et al. \\cite{alba2018automatic} used statistical, pattern and fractal descriptors in a random forest classifier to directly detect segmentation contour failures without intermediate regression of segmentation accuracy metrics. \n\nMethods for automatic quality control were also developed for other applications in medical image analysis. Frounchi et al. \\cite{frounchi2011automating} extracted features from the segmentation results of the left ventricle in CT scans. Using the obtained features the authors trained a classifier that is able to discriminate between consistent and inconsistent segmentations. To distinguish between acceptable and non-acceptable segmentations Kohlberger el al. \\cite{kohlberger2012evaluating} proposed to directly predict multi-organ segmentation accuracy in CT scans using a set of features extracted from the image and corresponding segmentation. \n\nA number of methods aggregate voxel-wise uncertainties into an overall score to identify insufficiently accurate segmentations. For example, Nair et al. \\cite{nair2018exploring} computed an overall score for target segmentation structure from voxel-wise predictive uncertainties. The method was tested for detection of Multiple Sclerosis in brain MRI. The authors showed that rejecting segmentations with high uncertainty scores led to increased detection accuracy indicating that correct segmentations contain lower uncertainties than incorrect ones. Similarly, to assess segmentation quality of brain MRIs Jungo et al. \\cite{jungo2018uncertainty} aggregated voxel-wise uncertainties into a score per target structure\nand showed that the computed uncertainty score enabled identification of erroneous segmentations.\n\nUnlike approaches evaluating segmentation directly, several methods use predictive uncertainties to predict segmentation metrics and thereby evaluate segmentation performance \\cite{roy2019bayesian, devries2018leveraging}. For example, Roy et al. \\cite{roy2019bayesian} aggregated voxel-wise uncertainties into four scores per segmented structure in brain MRI. The authors showed that computed scores can be used to predict the Intersection over Union and hence, to determine segmentation accuracy. \nSimilar idea was presented by DeVries et al. \\cite{devries2018leveraging} that predicted segmentation accuracy per patient using an auxiliary neural network that leverages the dermoscopic image, automatic segmentation result and obtained uncertainties. The researchers showed that a predicted segmentation accuracy is useful for quality control.\n\nWe build on our preliminary work where automatic segmentation of CMR images using a dilated CNN was combined with assessment of two measures of segmentation uncertainties \\cite{sander2019towards}. For the first measure the multi-class entropy per voxel (entropy maps) was computed using the output distribution. For the second measure Bayesian uncertainty maps were acquired using Monte Carlo dropout (MC-dropout) \\cite{gal2016dropout}. In \\cite{sander2019towards} we showed that the obtained uncertainties almost entirely cover the regions of incorrect segmentation i.e. that uncertainties are calibrated. In the current work we extend our preliminary research in two ways. First, we assess impact of CNN architecture on the segmentation performance and calibration of uncertainty maps by evaluating three existing state-of-the-art CNNs. Second, we employ an auxiliary CNN (detection network) that processes a cardiac MRI and corresponding spatial uncertainty map (Entropy or Bayesian) to automatically detect segmentation failures. We differentiate errors that may be within the range of inter-observer variability and hence do not necessarily require correction (tolerated errors) from the errors that an expert would not make and hence require correction (segmentation failures). Given that overlap measures do not capture fine details of the segmentation results and preclude us to differentiate two types of segmentation errors, in this work, we define segmentation failure using a metric of boundary distance. In \\cite{sander2019towards} we found that degree of calibration of uncertainty maps is dependent on the loss function used to train the CNN. Nevertheless, in the current work we show that uncalibrated uncertainty maps are useful to detect local segmentation failures. \t\nIn contrast to previous methods that detect segmentation failure per-patient or per-structure\\cite{roy2019bayesian, devries2018leveraging}, we propose to detect segmentation failures per image region. We expect that inspection and correction of segmentation failures using image regions rather than individual voxels or images would simplify correction process. To show the potential of our approach and demonstrate that combining automatic segmentation with manual correction of the detected segmentation failures per region results in higher segmentation performance we performed two additional experiments. In the first experiment, correction of detected segmentation failures was simulated in the complete data set. In the second experiment, correction was performed by an expert in a subset of images. Using publicly available set of CMR scans from MICCAI 2017 ACDC challenge \\cite{bernard2018deep}, the performance was evaluated before and after simulating the correction of detected segmentation failures as well as after manual expert correction.\n\n\\section{Data}\n\nIn this study data from the MICCAI \\num{2017} Automated Cardiac Diagnosis Challenge (ACDC) \\cite{bernard2018deep} was used. The dataset consists of cardiac cine MR images (CMRIs) from 100 patients uniformly distributed over normal cardiac function and four disease groups: dilated cardiomyopathy, hypertrophic cardiomyopathy, heart failure with infarction, and right ventricular abnormality. Detailed acquisition protocol is described by Bernard et al.~\\cite{bernard2018deep}. Briefly, short-axis CMRIs were acquired with two MRI scanners of different magnetic strengths (\\num{1.5} and \\num{3.0} T). Images were made during breath hold using a conventional steady-state free precession (SSFP) sequence. CMRIs have an in-plane resolution ranging from \\num{1.37} to \\SI{1.68}{\\milli\\meter} (average reconstruction matrix \\num{243} $\\times$ \\num{217} voxels) with slice spacing varying from \\num{5} to \\SI{10}{\\milli\\meter}. Per patient 28 to 40 volumes  are provided covering partially or completely one cardiac cycle. Each volume consists of on average ten slices covering the heart. Expert manual reference segmentations are provided for the LV cavity, RV endocardium and LV myocardium (LVM) for all CMRI slices at ED and ES time frames. To correct for intensity differences among scans, voxel intensities of each volume were scaled to the [\\num{0.0}, \\num{1.0}] range using the minimum and maximum of the volume. Furthermore, to correct for differences in-plane voxel sizes, image slices were resampled to \\num{1.4}$\\times\\SI{1.4}{\\milli\\meter}^2$. \n\n\\begin{figure*}[!t]\n\t\\captionsetup[subfigure]{justification=centering}\n\t\\centering\n\n\t\\subfloat[]{\\includegraphics[width=5in]{figures\/qualitative\/auto\/patient099_slice02_ES_emap_auto.pdf}%\n\t\t\\label{fig_seg_qual_example1}}\n\t\n\t\\subfloat[]{\\includegraphics[width=5in]{figures\/qualitative\/auto\/patient097_slice00_ES_emap_auto.pdf}%\n\t\t\\label{fig_seg_qual_example2}}\n\t\n\t\\caption{Examples of automatic segmentations generated by different segmentation models for two cardiac MRI scans (rows) at ES at the base of the heart.}\n\t\\label{fig_seg_qualitative_results}\n\\end{figure*}\n\n\\section{Methods}\n\nTo investigate uncertainty of the segmentation, anatomical structures in CMR images are segmented using a CNN. To investigate whether the approach generalizes to different segmentation networks, three state-of-the-art CNNs were evaluated. For each segmentation model two measures of predictive uncertainty were obtained per voxel. Thereafter, to detect and correct local segmentation failures an auxiliary CNN (detection network) that analyzes a cardiac MRI was used. Finally, this leads to the uncertainty map allowing detection of image regions that contain segmentation failures. Figure~\\ref{fig_overview_method} visualizes this approach.\n\n\n\\subsection{Automatic segmentation of cardiac MRI}\n\nTo perform segmentation of LV, RV, and LVM in cardiac MR images i.e. \\num{2}D CMR scans, three state-of-the-art CNNs are trained. Each of the three networks takes a CMR image as input and has four output channels providing probabilities for the three cardiac structures (LV, RV, LVM) and background. Softmax probabilities are calculated over the four tissue classes. Patient volumes at ED and ES are processed separately. During inference the \\num{2}D automatic segmentation masks are stacked into a \\num{3}D volume per patient and cardiac phase. After segmentation, the largest \\num{3}D connected component for each class is retained and volumes are resampled to their original voxel resolution. Segmentation networks differ substantially regarding architecture, number of parameters and receptive field size. To assess predictive uncertainties from the segmentation models \\textit{Monte Carlo dropout} (MC-dropout) introduced by Gal \\& Ghahramani \\cite{gal2016dropout} is implemented in every network. The following three segmentation networks were evaluated: Bayesian Dilated CNN, Bayesian Dilated Residual Network, Bayesian U-net.\n\n\\vspace{1ex}\n\\noindent \\textbf{Bayesian Dilated CNN (DN)}: The Bayesian DN architecture comprises a sequence of ten convolutional layers. Layers \\num{1} to \\num{8} serve as feature extraction layers with small convolution kernels of size \\num{3}$\\times$\\num{3} voxels. No padding is applied after convolutions. The number of kernels increases from \\num{32} in the first eight layers, to \\num{128} in the final two fully connected classification layers, implemented as \\num{1}$\\times$\\num{1} convolutions. The dilation level is successively increased between layers \\num{2} and \\num{7} from \\num{2} to \\num{32} which results in a receptive field for each voxel of \\num{131}$\\times$\\num{131} voxels, or \\num{18.3}$\\times$ $\\SI{18.3}{\\centi\\meter}^2$. All trainable layers except the final layer use rectified linear activation functions (ReLU). To enhance generalization performance, the model uses batch normalization in layers \\num{2} to \\num{9}. In order to convert the original DN~\\cite{wolterink2017automatic} into a Bayesian DN dropout is added as the last operation in all but the final layer and \\num{10} percent of a layer's hidden units are randomly switched off. \n\n\\vspace{1ex}\n\\noindent \\textbf{Bayesian Dilated Residual Network (DRN)}: The Bayesian DRN is based on the original DRN from Yu et al. \\cite{yu2017dilated} for image segmentation. More specifically, the DRN-D-22\\cite{yu2017dilated} is used which consists of a feature extraction module with output stride eight followed by a classifier implemented as fully convolutional layer with \\num{1}$\\times$\\num{1} convolutions. Output of the classifier is upsampled to full resolution using bilinear interpolation. The convolutional feature extraction module comprises eight levels where the number of kernels increases from \\num{16} in the first level, to \\num{512} in the two final levels. The first convolutional layer in level \\num{1} uses \\num{16} kernels of size \\num{7}$\\times$\\num{7} voxels and zero-padding of size \\num{3}. The remaining trainable layers use small \\num{3}$\\times$\\num{3} voxel kernels and zero-padding of size \\num{1}. Level \\num{2} to \\num{4} use a strided convolution of size \\num{2}. To further increase the receptive field convolutional layers in level \\num{5}, \\num{6} and \\num{7} use a dilation factor of \\num{2}, \\num{4} and \\num{2}, respectively. Furthermore, levels \\num{3} to \\num{6} consist of two residual blocks. All convolutional layers of the feature extraction module are followed by batch normalization, ReLU function and dropout. Adding dropout and switching off \\num{10} percent of a layer's hidden units converts the original DRN~\\cite{yu2017dilated} into a Bayesian DRN.\n\n\\vspace{1ex}\n\\noindent \\textbf{Bayesian U-net (U-net)}: The standard architecture of the U-net~\\cite{ronneberger2015u} is used. The network is fully convolutional and consists of a contracting, bottleneck and expanding path. The contracting and expanding path each consist of four blocks i.e. resolution levels which are connected by skip connections. The first block of the contracting path contains two convolutional layers using a kernel size of \\num{3}$\\times$\\num{3} voxels and zero-padding of size \\num{1}. Downsampling of the input is accomplished by employing a max pooling operation in block \\num{2} to \\num{4} of the contracting path and the bottleneck using a convolutional kernel of size \\num{2}$\\times$\\num{2} voxels and stride \\num{2}. Upsampling is performed by a transposed convolutional layer in block \\num{1} to \\num{4} of the expanding path using the same kernel size and stride as the max pooling layers. Each downsampling and upsampling layer is followed by two convolutional layers using \\num{3}$\\times$\\num{3} voxel kernels with zero-padding size \\num{1}. The final convolutional layer of the network acts as a classifier and uses \\num{1}$\\times$\\num{1} convolutions to reduce the number of output channels to the number of segmentation classes. The number of kernels increases from \\num{64} in the first block of the contracting path to \\num{1024} in the bottleneck. In contrast, the number of kernels in the expanding path successively decreases from \\num{1024} to \\num{64}. In deviation to the standard U-net instance normalization is added to all convolutional layers in the contracting path and ReLU non-linearities are replaced by LeakyReLU functions because this was found to slightly improve segmentation performance. In addition, to convert the deterministic model into a Bayesian neural network dropout is added as the last operation in each block of the contracting and expanding path and \\num{10} percent of a layer's hidden units are randomly switched off.\n\n\\subsection{Assessment of predictive uncertainties} \\label{uncertainty_maps}\nTo detect failures in segmentation masks generated by CNNs in testing, spatial uncertainty maps of the obtained segmentations are generated. For each voxel in the image two measures of uncertainty are calculated. First, a computationally cheap and straightforward measure of uncertainty is the entropy of softmax probabilities over the four tissue classes which are generated by the segmentation networks. Using these, normalized entropy maps $\\bf E \\in [0, 1]^{H\\times W}$ (e-map) are computed where $H$ and $W$ denote the height and width of the original CMRI, respectively.\n\nSecond, by applying MC-dropout in testing, softmax probabilities with a number of samples $T$ per voxel are obtained. As an overall measure of uncertainty the mean standard deviation of softmax probabilities per voxel over all tissue classes $C$\\label{ref:maximum_variance} is computed\n\n\n\\begingroup\n\\small\n\\begin{align}\n\\textbf{B} (I)^{(x, y)} &= \\frac{1}{C} \\sum_{c=1}^{C} \\sqrt{\\frac{1}{T-1}  \\sum_{t=1}^{T} \\big(p_t(I)^{(x, y, c)} - \\hat{\\mu}^{(x, y, c)} \\big)^2 } \\; ,\n\\end{align}\n\\endgroup\n\nwhere $\\textbf{B}(I)^{(x, y)} \\in [0, 1]$ denotes the normalized value of the Bayesian uncertainty map (b-map) at position $(x, y)$ in \\num{2}D slice $I$, $C$ is equal to the number of classes, $T$ is the number of samples and $p_t(I)^{(x, y, c)}$ denotes the softmax probability at position $(x, y)$  in image $I$ for class $c$. The predictive mean per class $\\hat{\\mu}^{(x, y, c)}$ of the samples is computed as follows:\n\n\\begingroup\n\\small\n\\begin{align}\n\\hat{\\mu}^{(x, y, c)} &= \\frac{1}{T} \\sum_{t=1}^{T} p_t(I)^{(x, y, c)} \\; .\n\\end{align}\n\\endgroup\n\nIn addition, the predictive mean per class is used to determine the tissue class per voxel.\n\n\\subsection{Calibration of uncertainty maps} \nIdeally, incorrectly segmented voxels as defined by the reference labels should be covered by higher uncertainties than correctly segmented voxels. In such a case the spatial uncertainty maps are perfectly calibrated. \\textit{Risk-coverage curves} introduced by Geifman et al.\\cite{geifman2017selective} visualize whether incorrectly segmented voxels are covered by higher uncertainties than those that are correctly segmented. Risk-coverage curves convey the effect of avoiding segmentation of voxels above a specific uncertainty value on the reduction of segmentation errors (i.e. risk reduction) while at the same time quantifying the voxels that were omitted from the classification task (i.e. coverage). \n\nTo generate risk-coverage curves first, each patient volume is cropped based on a minimal enclosing  parallelepiped bounding box that is placed around the reference segmentations to reduce the number of background voxels. Note that this is only performed to simplify the analysis of the risk-coverage curves. Second, voxels of the cropped patient volume are ranked based on their uncertainty value in descending order. Third, to obtain uncertainty threshold values per patient volume the ranked voxels are partitioned into \\num{100} percentiles based on their uncertainty value. Finally, per patient volume each uncertainty threshold is evaluated by computing a coverage and a risk measure. Coverage is the percentage of voxels in a patient volume at ED or ES that is automatically segmented. Voxels in a patient volume above the threshold are discarded from automatic segmentation and would be referred to an expert. The number of incorrectly segmented voxels per patient volume is used as a measure of risk. Using bilinear interpolation risk measures are computed per patient volume between $[0, 100]$ percent.\n\n\n\\subsection{Detection of segmentation failures}\n\nTo detect segmentation failures uncertainty maps are used but direct application of uncertainties is infeasible because many correctly segmented voxels, such as those close to anatomical structure boundaries, have high uncertainty. Hence, an additional patch-based CNN (detection network) is used that takes a cardiac MR image together with the corresponding spatial uncertainty map as input. For each patch of \\num{8}$\\times$\\num{8} voxels the network generates a probability indicating whether it contains segmentation failure. In the following, the terms patch and region are used interchangeably.\n\nThe detection network is a shallow Residual Network (S-ResNet) \\cite{he2016deep} consisting of a feature extraction module with output stride eight followed by a classifier indicating the presence of segmentation failure. The first level of the feature extraction module consists of two convolutional layers. The first layer uses \\num{16} kernels of \\num{7}$\\times$\\num{7} voxels and zero-padding of size \\num{3} and second layer \\num{32} kernels of \\num{3}$\\times$\\num{3} voxels and zero-padding of \\num{1} voxel. Level \\num{2} to \\num{4} each consist of one residual block that contains two convolutional layers with \\num{3}$\\times$\\num{3} voxels kernels with zero-padding of size \\num{1}. The first convolutional layer of each residual block uses a strided convolution of \\num{2} voxels to downsample the input. All convolutional layers of the feature extraction module are followed by batch normalization and ReLU function. The number of kernels in the feature extraction module increases from \\num{16} in level \\num{1} to \\num{128} in level \\num{4}. The network is a \\num{2}D patch-level classifier and requires that the size of the two input slices is a multiple of the patch-size. \\label{patch_size} The final classifier consists of three fully convolutional layers, implemented as \\num{1}$\\times$\\num{1} convolutions, with \\num{128} feature maps in the first two layers. The final layer has two channels followed by a softmax function which indicates whether the patch contains segmentation failure. Furthermore, to regularize the model dropout layers ($p=0.5$) were added between the residual blocks and the fully convolutional layers of the classifier.\n\n\\begin{table*}\n\t\\caption{Segmentation performance of different combination of model architectures, loss functions and evaluation modes (without or with MC dropout enabled during testing) in terms of Dice coefficient (top) and Hausdorff distance (bottom) (mean $\\pm$ standard deviation). Each combination comprises a block of two rows. A row in which column \\textit{Uncertainty map for detection} indicates e- or b-map shows results for the combined segmentation and detection approach. Numbers accentuated in black\/bold are ranked first in the segmentation only task whereas numbers accentuated in red\/bold are ranked first in the combined segmentation \\& detection task. The last row states the performance of the winning model in the ACDC challenge (on \\num{100} patient images) \\cite{isensee2017automatic}. Number with asterisk indicates statistical significant at \\num{5}\\% level w.r.t. the segmentation-only approach. Best viewed in color.}\n\t\\label{table_overall_segmentation_performance}\n\t\\centering\n\t\\tiny\n\t\\subfloat[Dice coefficient]{\n\t\t\\begin{tabular}{| C{1.6cm} | C{0.8cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\\hline\n\t\t\t& \\textbf{Uncertainty} & \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\\textbf{Model} & \\textbf{map for detection} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\\hline\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDN-Brier &  & \\phantom{x}0.962$\\pm$0.02 & \\phantom{x}0.928$\\pm$0.04) & \\phantom{x}0.875$\\pm$0.03)  & \\phantom{x}0.901$\\pm$0.11 & \\phantom{x}0.832$\\pm$0.10) & \\phantom{x}0.884$\\pm$0.04 \\\\\n\t\t\t& \\textbf{e-map} & *0.965$\\pm$0.01 & *0.949$\\pm$0.02 & *0.885$\\pm$0.03  & *0.937$\\pm$0.06 & *0.905$\\pm$0.05 & *0.909$\\pm$0.03 \\\\ \n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDN-Brier+MC  & & \\phantom{x}0.961$\\pm$0.02 & \\phantom{x}0.922$\\pm$0.04 & \\phantom{x}0.875$\\pm$0.04 & \\phantom{x}0.912$\\pm$0.08  & \\phantom{x}0.839$\\pm$0.11 & \\phantom{x}0.882$\\pm$0.04  \\\\ \n\t\t\t& \\textbf{b-map} & *0.966$\\pm$0.01 & *0.950$\\pm$0.01 & *0.886$\\pm$0.03  & \\textbf{\\textcolor{red}{*0.942}}$\\pm$0.03 & *0.916$\\pm$0.04 & *0.912$\\pm$0.03 \\\\ \n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDN-soft-Dice  & & \\phantom{x}0.960$\\pm$0.02 & \\phantom{x}0.921$\\pm$0.04 & \\phantom{x}0.870$\\pm$0.04 &  \\phantom{x}0.909$\\pm$0.08 & \\phantom{x}0.812$\\pm$0.12 & \\phantom{x}0.879$\\pm$0.04 \\\\\n\t\t\t& \\textbf{e-map} & *0.965$\\pm$0.01 & *0.945$\\pm$0.02 & *0.879$\\pm$0.04  & *0.938$\\pm$0.03 & *0.891$\\pm$0.06 & *0.905$\\pm$0.03 \\\\ \n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDN-soft-Dice+MC & & \\phantom{x}0.958$\\pm$0.02 & \\phantom{x}0.913$\\pm$0.05  & \\phantom{x}0.868$\\pm$0.04 & \\phantom{x}0.907$\\pm$0.07  & \\phantom{x}0.818$\\pm$0.12 & \\phantom{x}0.875$\\pm$0.04  \\\\\n\t\t\t& \\textbf{b-map} & *0.964$\\pm$0.01 & *0.944$\\pm$0.02 & *0.877$\\pm$0.04  & *0.939$\\pm$0.03 & *0.900$\\pm$0.05 & *0.904$\\pm$0.03 \\\\  \t\t\t\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDRN-CE &  & \\phantom{x}0.961$\\pm$0.02 & \\phantom{x}0.929$\\pm$0.03 & \\phantom{x}0.878$\\pm$0.03  & \\phantom{x}0.912$\\pm$0.06 & \\phantom{x}0.850$\\pm$0.09 & \\phantom{x}0.891$\\pm$0.03 \\\\  \n\t\t\t& \\textbf{e-map} & \\phantom{x}0.964$\\pm$0.01 & *0.943$\\pm$0.02 & *0.886$\\pm$0.03  & *0.937$\\pm$0.03 & *0.899$\\pm$0.04 & *0.908$\\pm$0.03 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDRN-CE+MC &  & \\phantom{x}0.961$\\pm$0.02 & \\phantom{x}0.926$\\pm$0.03 & \\phantom{x}0.877$\\pm$0.03  & \\phantom{x}0.913$\\pm$0.06 & \\phantom{x}0.847$\\pm$0.10 & \\phantom{x}0.890$\\pm$0.03 \\\\  \n\t\t\t& \\textbf{b-map} & *0.965$\\pm$0.01 & *0.948$\\pm$0.01 & *0.887$\\pm$0.03  & *0.939$\\pm$0.03 & *0.911$\\pm$0.04 & *0.909$\\pm$0.03 \\\\ \n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDRN-soft-Dice &  & \\phantom{x}0.964$\\pm$0.01 & \\phantom{x}\\textbf{0.937}$\\pm$0.02 & \\phantom{x}0.888$\\pm$0.03  & \\phantom{x}\\textbf{0.919}$\\pm$0.06 & \\phantom{x}0.856$\\pm$0.09 & \\phantom{x}\\textbf{0.900}$\\pm$0.03 \\\\ \n\t\t\t& \\textbf{e-map} & \\phantom{x}0.967$\\pm$0.01 & *0.945$\\pm$0.02 & \\phantom{x}0.893$\\pm$0.03  & \\phantom{x}0.934$\\pm$0.04 & *0.892$\\pm$0.06 & *0.911$\\pm$0.03 \\\\ \n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDRN-soft-Dice+MC &  & \\phantom{x}0.963$\\pm$0.02 & \\phantom{x}0.935$\\pm$0.03 & \\phantom{x}0.886$\\pm$0.03  & \\phantom{x}0.921$\\pm$0.06 & \\phantom{x}\\textbf{0.857}$\\pm$0.09 & \\phantom{x}0.899$\\pm$0.03 \\\\ \n\t\t\t& \\textbf{b-map} & \\phantom{x}0.967$\\pm$0.01 & *0.947$\\pm$0.02 & \\phantom{x}0.893$\\pm$0.03  & *0.938$\\pm$0.03 & *0.907$\\pm$0.04 & *0.912$\\pm$0.03 \\\\\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tU-net-CE &  & \\phantom{x}0.962$\\pm$0.02 & \\phantom{x}0.923$\\pm$0.05 & \\phantom{x}0.878$\\pm$0.03  & \\phantom{x}0.907$\\pm$0.07 & \\phantom{x}0.840$\\pm$0.08 & \\phantom{x}0.885$\\pm$0.03 \\\\ \n\t\t\t& \\textbf{e-map} & \\phantom{x}0.966$\\pm$0.01 & *0.946$\\pm$0.02 & *0.890$\\pm$0.03  & *0.935$\\pm$0.04 & *0.901$\\pm$0.06 & *0.909$\\pm$0.03 \\\\ \n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tU-net-CE+MC &  & \\phantom{x}0.962$\\pm$0.02 & \\phantom{x}0.926$\\pm$0.04 & \\phantom{x}0.879$\\pm$0.03  & \\phantom{x}0.909$\\pm$0.07 & \\phantom{x}0.849$\\pm$0.07 & \\phantom{x}0.887$\\pm$0.03 \\\\ \n\t\t\t& \\textbf{b-map} & \\phantom{x}0.967$\\pm$0.01 & \\textbf{\\textcolor{red}{*0.954}}$\\pm$0.02 & *0.893$\\pm$0.03  & *0.940$\\pm$0.04 & \\textbf{\\textcolor{red}{*0.920}}$\\pm$0.04 & \\textbf{\\textcolor{red}{*0.914}}$\\pm$0.03 \\\\\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tU-net-soft-Dice &  & \\phantom{x}\\textbf{0.965}$\\pm$0.02 & \\phantom{x}0.928$\\pm$0.04 & \\phantom{x}0.888$\\pm$0.03  & \\phantom{x}0.914$\\pm$0.08 & \\phantom{x}0.844$\\pm$0.09 & \\phantom{x}0.896$\\pm$0.03 \\\\ \n\t\t\t& \\textbf{e-map} & \\phantom{x}0.968$\\pm$0.01 & *0.943$\\pm$0.03 & *0.898$\\pm$0.03  & \\phantom{x}0.930$\\pm$0.05 & *0.886$\\pm$0.07 & *0.911$\\pm$0.03 \\\\ \n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tU-net-soft-Dice+MC &  & \\phantom{x}\\textbf{0.965}$\\pm$0.02 & \\phantom{x}0.929$\\pm$0.04 & \\phantom{x}\\textbf{0.889}$\\pm$0.03  & \\phantom{x}0.911$\\pm$0.10 & \\phantom{x}0.845$\\pm$0.09 & \\phantom{x}0.897$\\pm$0.03 \\\\ \n\t\t\t& \\textbf{b-map} & \\phantom{x}\\textbf{\\textcolor{red}{0.968}}$\\pm$0.01 & *0.948$\\pm$0.03 & \\textbf{\\textcolor{red}{*0.900}}$\\pm$0.03  & \\phantom{x}0.928$\\pm$0.09 & *0.895$\\pm$0.06 & \\textbf{\\textcolor{red}{*0.914}}$\\pm$0.03 \\\\\n\t\t\t\n\t\t\t\\hdashline\n\t\t\tIsensee et al. & & \\phantom{x}0.966& \\phantom{x}0.941 & \\phantom{x}0.899 & \\phantom{x}0.924 & \\phantom{x}0.875\t& \\phantom{x}0.908  \\\\\n\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{table_seg_perf_dsc}\n\t} \n\t\\vspace{13ex}\n\t\\centering\n\t\\tiny\n\t\\subfloat[Hausdorff Distance]{\n\t\t\\begin{tabular}{| C{1.6cm} | C{0.8cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\\hline\n\t\t\t& \\textbf{Uncertainty}& \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\\textbf{Model} & \\textbf{map for detection} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\\hline\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDN-Brier &  &  \\phantom{x}6.7$\\pm$3.1 & \\phantom{x}13.5$\\pm$5.9 & \\phantom{x}10.2$\\pm$6.9 & \\phantom{x}10.7$\\pm$7.7 & \\phantom{x}16.7$\\pm$6.8 & \\phantom{x}12.3$\\pm$5.8 \\\\\n\t\t\t& \\textbf{e-map} &  *5.7$\\pm$2.7 & *11.7$\\pm$5.2 & *\\phantom{x}8.3$\\pm$5.9 & *\\phantom{x}8.0$\\pm$6.5 & *14.2$\\pm$5.6 & *\\phantom{x}9.7$\\pm$5.0 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDN-Brier+MC &  &  \\phantom{x}6.9$\\pm$3.3 & \\phantom{x}13.1$\\pm$5.2 & \\phantom{xx}9.9$\\pm$5.9 & \\phantom{xx}9.9$\\pm$5.7 & \\phantom{x}15.0$\\pm$6.1 & \\phantom{x}12.0$\\pm$5.2 \\\\\n\t\t\t& \\textbf{b-map} &  *5.5$\\pm$2.6 & *10.6$\\pm$5.1 & *\\phantom{x}7.4$\\pm$4.2 & *\\phantom{x}7.5$\\pm$6.0 & *12.6$\\pm$5.6 & *\\phantom{x}8.8$\\pm$4.0 \\\\\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDN-soft-Dice &  &  \\phantom{x}7.1$\\pm$3.5 & \\phantom{x}14.8$\\pm$6.8 & \\phantom{x}11.0$\\pm$6.6 & \\phantom{x}10.2$\\pm$5.6 & \\phantom{x}17.7$\\pm$7.8 & \\phantom{x}12.9$\\pm$6.2 \\\\\n\t\t\t& \\textbf{e-map} &  *5.6$\\pm$2.8 & *12.6$\\pm$5.5 & *\\phantom{x}8.6$\\pm$4.6 & *\\phantom{x}8.0$\\pm$5.0 & *14.6$\\pm$5.9 & *\\phantom{x}9.6$\\pm$4.5 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDN-soft-Dice+MC &  &  \\phantom{x}7.7$\\pm$3.9 & \\phantom{x}14.4$\\pm$6.0 & \\phantom{x}10.5$\\pm$4.9 & \\phantom{x}10.1$\\pm$5.3 & \\phantom{x}17.2$\\pm$8.0 & \\phantom{x}12.5$\\pm$5.3 \\\\\n\t\t\t& \\textbf{b-map} &  *6.3$\\pm$3.4 & *11.5$\\pm$4.0 & *\\phantom{x}8.6$\\pm$4.8 & *\\phantom{x}7.8$\\pm$4.6 & *13.6$\\pm$4.9 & *\\phantom{x}9.6$\\pm$4.7 \\\\\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDRN-CE &  &  \\phantom{x}5.5$\\pm$2.6 & \\phantom{x}11.7$\\pm$5.4 & \\phantom{xx}8.2$\\pm$6.2 & \\phantom{xx}9.1$\\pm$6.4 & \\phantom{x}13.7$\\pm$5.6 & \\phantom{xx}8.9$\\pm$5.3 \\\\\n\t\t\t& \\textbf{e-map} &  *4.5$\\pm$1.9 & *\\phantom{x}9.0$\\pm$4.5 & *\\phantom{x}6.3$\\pm$4.1 & *\\phantom{x}6.2$\\pm$4.4 & *11.1$\\pm$5.3 & \\textbf{\\textcolor{red}{*\\phantom{x}6.7}}$\\pm$4.2 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDRN-CE+MC &  &  \\phantom{x}5.6$\\pm$2.6 & \\phantom{x}11.9$\\pm$5.5 & \\phantom{xx}8.0$\\pm$5.9 & \\phantom{xx}8.7$\\pm$5.5 & \\phantom{x}13.5$\\pm$5.9 & \\phantom{xx}\\textbf{8.5}$\\pm$4.5 \\\\\t\t \n\t\t\t& \\textbf{b-map} &  \\textbf{\\textcolor{red}{*4.2}}$\\pm$1.6 & \\textbf{\\textcolor{red}{*\\phantom{x}8.1}}$\\pm$3.7 & \\textbf{\\textcolor{red}{*\\phantom{x}6.1}}$\\pm$4.2 & \\textbf{\\textcolor{red}{*\\phantom{x}5.4}}$\\pm$3.6 & \\textbf{\\textcolor{red}{*10.1}}$\\pm$5.5 & *\\phantom{x}6.8$\\pm$3.8 \\\\\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDRN-soft-Dice &  &  \\phantom{x}\\textbf{5.5}$\\pm$2.8 & \\phantom{x}11.9$\\pm$6.1 & \\phantom{xx}\\textbf{7.7}$\\pm$5.9 & \\phantom{xx}8.5$\\pm$5.0 & \\phantom{x}13.5$\\pm$5.5 & \\phantom{xx}8.9$\\pm$5.1 \\\\\n\t\t\t& \\textbf{e-map} &  *4.6$\\pm$2.2 & *\\phantom{x}9.4$\\pm$4.5 & \\phantom{xx}6.7$\\pm$4.7 & *\\phantom{x}6.7$\\pm$4.4 & *11.6$\\pm$5.4 & *\\phantom{x}7.0$\\pm$3.3 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDRN-soft-Dice+MC &  &  \\phantom{x}5.7$\\pm$3.2 & \\phantom{x}\\textbf{11.5}$\\pm$5.1 & \\phantom{xx}8.0$\\pm$5.5 & \\phantom{xx}\\textbf{8.3}$\\pm$4.5 & \\phantom{x}\\textbf{13.3}$\\pm$5.1 & \\phantom{xx}8.9$\\pm$5.1 \\\\\n\t\t\t& \\textbf{b-map} &  *4.5$\\pm$2.2 & *\\phantom{x}9.3$\\pm$4.5 & *\\phantom{x}6.3$\\pm$4.0 & *\\phantom{x}6.2$\\pm$4.1 & *10.4$\\pm$5.0 & *\\phantom{x}7.0$\\pm$3.4 \\\\\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tU-net-CE &  &  \\phantom{x}6.4$\\pm$4.3 & \\phantom{x}15.7$\\pm$8.6 & \\phantom{xx}9.0$\\pm$6.0 & \\phantom{xx}9.7$\\pm$5.3 & \\phantom{x}17.0$\\pm$7.7 & \\phantom{x}12.7$\\pm$8.2 \\\\\n\t\t\t& \\textbf{e-map} &  *4.9$\\pm$3.9 & *12.2$\\pm$8.1 & *\\phantom{x}7.1$\\pm$5.6 & *\\phantom{x}6.1$\\pm$3.2 & *12.6$\\pm$6.5 & *\\phantom{x}8.4$\\pm$6.3 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tU-net-CE+MC &  &  \\phantom{x}6.2$\\pm$4.2 & \\phantom{x}15.3$\\pm$8.4 & \\phantom{xx}8.8$\\pm$5.8 & \\phantom{xx}9.2$\\pm$5.0 & \\phantom{x}16.5$\\pm$7.6 & \\phantom{x}12.0$\\pm$8.0 \\\\\n\t\t\t& \\textbf{b-map} &  *4.3$\\pm$1.6 & *\\phantom{x}9.9$\\pm$6.6 & *\\phantom{x}6.7$\\pm$4.8 & *\\phantom{x}5.4$\\pm$2.8 & *10.3$\\pm$4.7 & *\\phantom{x}7.6$\\pm$6.2 \\\\\n\t\t\t\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tU-net-soft-Dice &  &  \\phantom{x}6.1$\\pm$3.9 & \\phantom{x}14.1$\\pm$7.6 & \\phantom{x}10.6$\\pm$8.4 & \\phantom{xx}9.2$\\pm$7.1 & \\phantom{x}16.3$\\pm$7.5 & \\phantom{x}12.6$\\pm$9.6 \\\\\n\t\t\t& \\textbf{e-map} &  *4.6$\\pm$2.3 & *11.3$\\pm$7.2 & *\\phantom{x}7.5$\\pm$5.5 & *\\phantom{x}7.3$\\pm$6.5 & *13.7$\\pm$7.6 & *\\phantom{x}9.8$\\pm$8.0 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tU-net-soft-Dice+MC &  &  \\phantom{x}6.2$\\pm$3.9 & \\phantom{x}14.1$\\pm$7.7 & \\phantom{x}10.5$\\pm$8.7 & \\phantom{xx}9.0$\\pm$7.0 & \\phantom{x}15.8$\\pm$7.5 & \\phantom{x}12.1$\\pm$9.2 \\\\\n\t\t\t& \\textbf{b-map} &  *4.5$\\pm$2.1 & *10.4$\\pm$7.2 & *\\phantom{x}7.6$\\pm$7.0 & *\\phantom{x}7.3$\\pm$6.9 & *12.9$\\pm$6.6 & *\\phantom{x}9.8$\\pm$8.4 \\\\\n\t\t\t\\hdashline\n\t\t\tIsensee et al. &  & \\phantom{x}7.1 &  \\phantom{x}14.3 & \\phantom{xx}8.9 &  \\phantom{xx}9.8 &  \\phantom{x}16.3 & \\phantom{x}10.4 \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{table_seg_perf_hd}\n\t}  \n\t\n\\end{table*} \n\n\n\\section{Evaluation}\\label{evaluation}\n\nAutomatic segmentation performance, as well as performance after simulating the correction of detected segmentation failures and after manual expert correction was evaluated. For this, the \\num{3}D Dice-coefficient (DC) and \\num{3}D Hausdorff distance (HD) between manual and (corrected) automatic segmentation were computed. Furthermore, the following clinical metrics were computed for manual and (corrected) automatic segmentation: left ventricle end-diastolic volume (EDV); left ventricle ejection fraction (EF); right ventricle EDV; right ventricle ejection fraction; and left ventricle myocardial mass. Following Bernard et al.\\cite{bernard2018deep} for each of the clinical metrics three performance indices were computed using the measurements based on manual and (corrected) automatic segmentation: Pearson correlation coefficient; mean difference (bias and standard deviation); and mean absolute error (MAE).\n\nTo evaluate detection performance of the automatic method precision-recall curves of identification of slices that require correction were computed. A slice is considered positive in case it consists of at least one image region with a segmentation failure. To achieve accurate segmentation in clinic, identification of slices that contain segmentation failures might ease manual correction of automatic segmentations in daily practice. To further evaluate detection performance detection rate of segmentation failures was assessed on a voxel level. More specific, sensitivity against the number of false positive regions was evaluated because manual correction is presumed to be performed at this level.\n\nFinally, after simulation and manual correction of the automatically detected segmentation failures, segmentation was re-evaluated and significance of the difference between the DCs, HDs and clinical metrics was tested with a Mann\u2013Whitney U test.\n\n\n\\section{Experiments}\n\nTo use stratified four-fold cross-validation the dataset was split into training (75\\%) and test (25\\%) set. The splitting was done on a patient level, so there was no overlap in patient data between training and test sets. Furthermore, patients were randomly chosen from each of the five patient groups w.r.t. disease. Each patient has one volume for ED and ES time points, respectively.  \n\n\\subsection{Training segmentation networks} \\label{training_segmentation}\n\nDRN and U-net were trained with a patch size of \\num{128}$\\times$\\num{128} voxels which is a multiple of their output stride of the contracting path. In the training of the dilated CNN (DN) images with \\num{151}$\\times$\\num{151} voxel samples were used. Zero-padding to \\num{281}$\\times$\\num{281} was performed to accommodate the \\num{131}$\\times$\\num{131} voxel receptive field that is induced by the dilation factors. Training samples were randomly chosen from training set and augmented by \\num{90} degree rotations of the images. All models were initially trained with three loss functions: soft-Dice\\cite{milletari2016v} (SD); cross-entropy (CE); and Brier loss\\cite{brier1950verification}. However, for the evaluation of the combined segmentation and detection approach for each model architecture the two best performing loss functions were chosen: soft-Dice for all models; cross-entropy for DRN and U-net and Brier loss for DN. For completeness, we provide the equations for all three used loss functions.\n\n\n\\begingroup\n\\small\n\\begin{align}\n\\text{soft-Dice}_{c} = \\frac{\\sum_{i=1}^{N} R_{c}(i) \\; A_{c}(i) }{\\sum_{i=1}^{N} R_{c}(i) + \\sum_{i=1}^{N} A_{c}(i)}  \\; ,\n\\end{align}\n\\endgroup\nwhere $N$ denotes the number of voxels in an image, $R_{c}$ is the binary reference image for class $c$ and $A_{c}$ is the probability map for class $c$.\n\n\\begingroup\n\\small\n\\begin{align}\n\\begin{split}\n\\text{Cross-Entropy}_{c} &= - \\; \\sum_{i=1}^{N} t_{ic} \\; \\log \\; p(y_i=c|x_i) \\; , \\\\& \\text{ where } t_{ic} = 1 \\text{ if } y_{i}=c, \\text{ and \\num{0} otherwise.}\n\\end{split}\n\\end{align}\n\\endgroup\n\n\\begingroup\n\\small\n\\begin{align}\n\\begin{split}\n\\text{Brier}_{c} &= \\sum_{i=1}^{N}  \\big(t_{ic} - p(y_i=c|x_{i}) \\big)^2 \\; , \\\\ &\\text{ where } t_{ic} = 1 \\text{ if } y_{i}=c, \\text{ and \\num{0} otherwise.}\n\\end{split}\n\\end{align}\n\\endgroup\n\nwhere $N$ denotes the number of voxels in an image and $p$ denotes the probability for a specific voxel $x_i$ with corresponding reference label $y_i$ for class $c$.\n\nChoosing Brier loss to train the DN model instead of CE was motivated by our preliminary work which showed that segmentation performance of DN model was best when trained with Brier loss\\cite{sander2019towards}.\n\nAll models were trained for 100,000 iterations. DRN and U-net were trained with a learning rate of \\num{0.001} which decayed with a factor of \\num{0.1} after every 25,000 steps. Training DN used the snapshot ensemble technique~\\cite{huang2017snapshot}, where after every 10,000 iterations the learning rate was reset to its original value of \\num{0.02}.\n\nAll three segmentation networks were trained using mini-batch stochastic gradient descent using a batch size of \\num{16}. Network parameters were optimized using the Adam optimizer \\cite{kingmadp}. Furthermore, models were regularized with weight decay to increase generalization performance. \n\n\\subsection{Training detection network}\\label{label_training_detection}\n\nTo train the detection model a subset of the errors performed by the segmentation model is used. Segmentation errors that presumably are within the range of inter-observer variability and therefore do not inevitably require correction (tolerated errors) are excluded from the set of errors that need to be detected and corrected (segmentation failures). To distinguish between tolerated errors and the set of segmentation failures $\\mathcal{S}_I$ the Euclidean distance of an incorrectly segmented voxel to the boundary of the reference target structure is used. For each anatomical structure a \\num{2}D distance transform map is computed that provides for each voxel the distance to the anatomical structure boundary. To differentiate between tolerated errors and the set of segmentation failures $\\mathcal{S}_I$ an acceptable tolerance threshold is applied. A more rigorous threshold is used for errors located inside compared to outside of the anatomical structure because automatic segmentation methods have a tendency to undersegment cardiac structures in CMRI. Hence, in all experiments the acceptable tolerance threshold was set to three voxels (equivalent to on average \\SI{4.65}{\\milli\\meter}) and two voxels (equivalent to on average \\SI{3.1}{\\milli\\meter}) for segmentation errors located outside and inside the target structure. Furthermore, a segmentation error only belongs to $\\mathcal{S}_I$ if it is part of a \\num{2}D \\num{4}-connected cluster of minimum size \\num{10} voxels. This value was found in preliminary experiments by evaluating values $\\{1, 5, 10, 15, 20\\}$. However, for apical slices all segmentation errors are included in $\\mathcal{S}_I$ regardless of fulfilling the minimum size requirement because in these slices anatomical structures are relatively small and manual segmentation is prone to large inter-observer variability~\\cite{bernard2018deep}. Finally, segmentation errors located in slices above the base or below the apex are always included in the set of segmentation failures.\n\nUsing the set $\\mathcal{S}_I$ a binary label $t_j$ is assigned to each patch $P_j^{(I)}$ indicating whether $P_j^{(I)}$ contains at least one voxel belonging to set $\\mathcal{S}_I$ where $j \\in \\{1 \\dots M \\}$ and $M$ denotes the number of patches in a slice $I$.  \n\nThe detection network is trained by minimizing a weighted binary cross-entropy loss:\n\n\\begingroup\n\\small\n\\begin{equation} \\label{eq_detection_loss}\n\\mathcal{L}_{DT} = - \\sum_{j \\in P^{(I)}} w_{pos} \\; t_j \\log p_j + (1 - t_j) \\log (1 - p_j)  \\; ,\n\\end{equation}\n\\endgroup\n\nwhere $w_{pos}$ represents a scalar weight, $t_j$ denotes the binary reference label and $p_j$ is the softmax probability indicating whether a particular image region $P_j^{(I)}$ contains at least one segmentation failure. The average percentage of regions in a patient volume containing segmentation failures ranges from \\num{1.5} to \\num{3} percent depending on the segmentation architecture and loss function used to train the segmentation model. To train a detection network $w_{pos}$ was set to the ratio between the average percentage of negative samples divided by the average percentage of positive samples.\n\nEach fold was trained using spatial uncertainty maps and automatic segmentation masks generated while training the segmentation networks. Hence, there was no overlap in patient data between training and test set across segmentation and detection tasks. In total \\num{12} detection models were trained and evaluated resulting from the different combination of \\num{3} model architectures (DRN, DN and U-net), \\num{2} loss functions (DRN and U-net with CE and soft-Dice, DN with Brier and soft-Dice) and \\num{2} uncertainty maps (e-maps, b-maps).\n\n\n\\begin{table*}\n\t\\caption{Segmentation performance of different combination of model architectures, loss functions and evaluation modes (without or with MC dropout (MC) enabled during testing) in terms of clinical metrics: left ventricle (LV) end-diastolic volume (EDV); LV ejection fraction (EF); right ventricle (RV) EDV; RV ejection fraction; and LV myocardial mass. Quantitative results compare clinical metrics based on reference segmentations with 1) automatic segmentations and 2) simulated manual correction of automatic segmentations using spatial uncertainty maps. $\\rho$ denotes the Pearson correlation coefficient, \\textit{bias} denotes the mean difference between the two measurements (mean $\\pm$ standard deviation) and \\textit{MAE} denotes the mean absolute error between the two measurements. Each combination comprises a block of two rows. A row in which column \\textit{Uncertainty map for detection} indicates e- or b-map shows results for the combined segmentation and detection approach. Numbers accentuated in black\/bold are ranked first in the segmentation only task. Numbers in red indicate statistical significant at \\num{5}\\% level w.r.t. the segmentation-only approach for the specific clinical metric. Best viewed in color.}\n\t\\label{table_cardiac_function_indices}\n\t\\tiny\n\t\\centering\n\t\\begin{tabular}{| C{1.6cm} | C{1.cm} | C{0.3cm} c C{0.3cm} | C{0.2cm} C{0.8cm} C{0.3cm} | C{0.3cm} C{0.8cm} C{0.3cm} | C{0.3cm} C{0.8cm} C{0.3cm} | C{0.3cm} C{0.8cm} C{0.3cm} |}\n\t\t\\hline\n\t\t& \\multicolumn{1}{c|}{\\thead{\\textbf{Uncertainty} \\\\ \\textbf{map for} \\\\ \\textbf{detection}}}  & \\multicolumn{3}{c}{\\textbf{LV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{LV$_{EF}$}} & \\multicolumn{3}{c}{\\textbf{RV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{RV$_{EF}$}} & \\multicolumn{3}{c|}{\\textbf{LVM$_{Mass}$}}  \\\\\n\t\n\t\n\t\t\\textbf{Method} &  & \\textbf{$\\rho$} & \\multicolumn{1}{l}{\\textbf{bias$\\pm\\sigma$}} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} & \\multicolumn{1}{c}{\\textbf{$\\rho$}} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} \\\\\n\t\t\\hline\n\t\t\\rowcolor{LightGreen}\n\t\tDN-Brier &  & 0.997 & \\phantom{x}\\textbf{0.0$\\pm$6.1} & 4.5 & 0.892 & \\phantom{x}2.2$\\pm$\\phantom{x}9.2 & 4.2 & 0.977 & \\textbf{-0.2$\\pm$11.8} & \\phantom{x}8.5 & 0.834 & 5.3$\\pm$10.3 & \\phantom{x}8.5 & 0.984 & -2.7$\\pm$\\phantom{x}9.0 & \\phantom{x}\\textbf{7.0} \\\\\n\t\t& e-map  & 0.997 & \\phantom{x}0.0$\\pm$5.5 & 4.0 & 0.982 & \\phantom{x}0.1$\\pm$\\phantom{x}3.8 & 2.2 & 0.992 & \\phantom{x}0.0$\\pm$\\phantom{x}6.9 & \\phantom{x}5.2 & 0.955 & 1.9$\\pm$\\phantom{x}5.5 & \\phantom{x}4.1 & 0.986 & -2.1$\\pm$\\phantom{x}8.4 & \\phantom{x}6.6 \\\\\n\t\t\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightCyan}\n\t\tDN-Brier+MC &  & 0.997 & \\phantom{x}1.6$\\pm$6.0 & 4.4 & 0.921 & \\phantom{x}1.1$\\pm$\\phantom{x}7.9 & 3.9 & 0.975 & \\phantom{x}6.7$\\pm$12.4 & \\phantom{x}9.6 & 0.854 & 3.5$\\pm$\\phantom{x}9.9 & \\phantom{x}7.7 & 0.984 & \\phantom{x}0.7$\\pm$\\phantom{x}9.2 & \\phantom{x}7.1 \\\\\n\t\t& b-map  & 0.998 & \\phantom{x}1.0$\\pm$5.3 & 3.9 & 0.991 & \\phantom{x}0.0$\\pm$\\phantom{x}2.7 & 1.9 & 0.993 & \\phantom{x}3.2$\\pm$\\phantom{x}6.7 & \\phantom{x}5.7 & 0.975 & 0.8$\\pm$\\phantom{x}4.0 & \\phantom{x}3.0 & 0.987 & \\phantom{x}0.1$\\pm$\\phantom{x}8.3 & \\phantom{x}6.5 \\\\\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightGreen}\n\t\tDN-soft-Dice &  & 0.996 & \\phantom{x}1.2$\\pm$6.5 & 4.9 & 0.918 & \\phantom{x}1.5$\\pm$\\phantom{x}8.0 & 3.9 & 0.972 & \\phantom{x}\\textbf{0.2$\\pm$13.0} & \\phantom{x}9.6 & 0.802 & 7.2$\\pm$11.3 & 10.2 & 0.982 & -4.5$\\pm$\\phantom{x}9.6 & \\phantom{x}8.5 \\\\\n\t\t& e-map  & 0.997 & \\phantom{x}1.0$\\pm$5.5 & 4.2 & 0.989 & \\phantom{x}0.2$\\pm$\\phantom{x}3.0 & 2.2 & 0.990 & \\phantom{x}0.2$\\pm$\\phantom{x}7.6 & \\phantom{x}5.9 & \\textcolor{red}{0.940} & \\textcolor{red}{3.3$\\pm$\\phantom{x}6.2} & \\textcolor{red}{\\phantom{x}5.2} & 0.983 & -4.3$\\pm$\\phantom{x}9.3 & \\phantom{x}8.2 \\\\\n\t\t\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightCyan}\n\t\tDN-soft-Dice+MC &  & 0.996 & \\phantom{x}3.2$\\pm$7.1 & 5.6 & 0.958 & \\phantom{x}0.4$\\pm$\\phantom{x}5.7 & 3.6 & 0.964 & \\phantom{x}8.1$\\pm$14.9 & 12.3 & 0.827 & 4.8$\\pm$11.0 & \\phantom{x}8.9 & 0.978 & -0.7$\\pm$10.7 & \\phantom{x}8.3 \\\\\n\t\t& b-map  & 0.997 & \\phantom{x}2.2$\\pm$5.6 & 4.4 & 0.988 & -0.2$\\pm$\\phantom{x}3.1 & 2.2 & 0.990 & \\phantom{x}4.0$\\pm$\\phantom{x}7.7 & \\phantom{x}7.0 & 0.959 & 1.8$\\pm$\\phantom{x}5.1 & \\phantom{x}4.1 & 0.982 & -1.4$\\pm$\\phantom{x}9.5 & \\phantom{x}7.6 \\\\\n\t\t\n\t\t\\hdashline[5pt\/5pt]\n\t\t\\rowcolor{LightGreen}\n\t\tDRN-CE &  & 0.997 & -0.2$\\pm$5.5 & 4.1 & 0.968 & \\phantom{x}1.2$\\pm$\\phantom{x}5.0 & 3.5 & 0.976 & \\phantom{x}1.5$\\pm$12.1 & \\phantom{x}8.5 & 0.870 & 1.3$\\pm$\\phantom{x}9.2 & \\phantom{x}6.9 & 0.980 & \\phantom{x}\\textbf{0.6$\\pm$10.2} & \\phantom{x}7.8 \\\\\n\t\t& e-map  & 0.998 & \\phantom{x}0.2$\\pm$4.5 & 3.5 & 0.992 & \\phantom{x}0.2$\\pm$\\phantom{x}2.5 & 1.9 & 0.988 & \\phantom{x}1.4$\\pm$\\phantom{x}8.5 & \\phantom{x}6.2 & 0.952 & 0.8$\\pm$\\phantom{x}5.6 & \\phantom{x}4.2 & 0.985 & \\phantom{x}0.4$\\pm$\\phantom{x}8.7 & \\phantom{x}6.8 \\\\\n\t\t\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightCyan}\n\t\tDRN-CE+MC &  & \\textbf{0.998} & \\phantom{x}1.0$\\pm$4.9 & \\textbf{3.9} & 0.972 & \\phantom{x}0.8$\\pm$\\phantom{x}4.6 & 3.1 & 0.973 & \\phantom{x}4.8$\\pm$12.8 & \\phantom{x}9.4 & 0.876 & \\textbf{0.4$\\pm$\\phantom{x}9.1} & \\phantom{x}\\textbf{6.6} & 0.981 & \\phantom{x}1.9$\\pm$\\phantom{x}9.9 & \\phantom{x}7.6 \\\\\n\t\t& b-map  & 0.998 & \\phantom{x}0.7$\\pm$4.6 & 3.6 & 0.992 & -0.1$\\pm$\\phantom{x}2.5 & 1.8 & 0.992 & \\phantom{x}2.9$\\pm$\\phantom{x}6.9 & \\phantom{x}5.7 & 0.967 & 0.6$\\pm$\\phantom{x}4.6 & \\phantom{x}3.4 & 0.987 & \\phantom{x}1.2$\\pm$\\phantom{x}8.3 & \\phantom{x}6.6 \\\\\n\t\t\\hdashline[1pt\/2pt]\n\t\t\n\t\t\\rowcolor{LightGreen}\n\t\tDRN-soft-Dice &  & \\textbf{0.998} & \\phantom{x}0.8$\\pm$5.1 & 4.0 & 0.976 & \\phantom{x}0.2$\\pm$\\phantom{x}4.4 & 3.0 & 0.980 & \\phantom{x}\\textbf{0.2$\\pm$11.0} & \\phantom{x}\\textbf{7.5} & \\textbf{0.882} & 3.1$\\pm$\\phantom{x}8.7 & \\phantom{x}6.8 & 0.984 & -3.5$\\pm$\\phantom{x}9.1 & \\phantom{x}7.5 \\\\\n\t\t& e-map  & 0.998 & \\phantom{x}0.7$\\pm$4.4 & 3.5 & 0.987 & -0.1$\\pm$\\phantom{x}3.1 & 2.2 & 0.987 & \\phantom{x}0.1$\\pm$\\phantom{x}9.1 & \\phantom{x}6.4 & 0.938 & 1.9$\\pm$\\phantom{x}6.3 & \\phantom{x}4.9 & 0.986 & -3.5$\\pm$\\phantom{x}8.7 & \\phantom{x}7.1 \\\\\n\t\t\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightCyan}\n\t\tDRN-soft-Dice+MC &   & \\textbf{0.998} & \\phantom{x}1.8$\\pm$5.1 & \\textbf{3.9} & \\textbf{0.979} & -0.3$\\pm$\\phantom{x}4.1 & 2.9 & 0.977 & \\phantom{x}3.5$\\pm$11.7 & \\phantom{x}8.1 & 0.868 & 1.7$\\pm$\\phantom{x}9.5 & \\phantom{x}6.8 & 0.983 & -1.4$\\pm$\\phantom{x}9.5 & \\phantom{x}7.4 \\\\\n\t\t&  b-map  & 0.998 & \\phantom{x}1.7$\\pm$4.7 & 3.7 & 0.990 & -0.2$\\pm$\\phantom{x}2.9 & 2.1 & 0.989 & \\phantom{x}2.3$\\pm$\\phantom{x}8.1 & \\phantom{x}5.8 & 0.959 & 0.8$\\pm$\\phantom{x}5.2 &\\phantom{x}3.8 & 0.986 & -1.3$\\pm$\\phantom{x}8.5 & \\phantom{x}6.8 \\\\\n\t\t\n\t\t\\hdashline[5pt\/5pt]\n\t\t\\rowcolor{LightGreen}\n\t\tU-net-CE &  & 0.995 & -4.7$\\pm$7.2 & 6.1 & 0.954 & \\phantom{x}4.1$\\pm$\\phantom{x}6.0 & 5.1 & 0.963 & -7.6$\\pm$15.2 & 12.1 & 0.870 & 5.6$\\pm$\\phantom{x}9.0 & \\phantom{x}8.1 & 0.971 & -8.5$\\pm$12.2 & 11.5 \\\\\n\t\t& e-map  & 0.998 & -3.2$\\pm$4.8 & 4.4 & 0.992 & \\phantom{x}1.7$\\pm$\\phantom{x}2.6 & 2.4 & 0.987 & -4.1$\\pm$\\phantom{x}9.1 & \\phantom{x}6.7 & 0.957 & 2.6$\\pm$\\phantom{x}5.2 & \\phantom{x}4.1 & 0.983 & -5.7$\\pm$\\phantom{x}9.3 & \\phantom{x}8.2 \\\\\n\t\t\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightCyan}\n\t\tU-net-CE+MC &   & 0.995 & -4.3$\\pm$7.2 & 5.9 & 0.958 & \\phantom{x}3.8$\\pm$\\phantom{x}5.8 & 4.9 & 0.968 & -4.8$\\pm$14.1 & 10.7 & 0.867 & 5.0$\\pm$\\phantom{x}9.1 & \\phantom{x}7.9 & 0.972 & -8.1$\\pm$12.0 & 11.1 \\\\\n\t\t& b-map  & 0.997 & -3.5$\\pm$5.5 & 4.9 & 0.990 & \\phantom{x}1.6$\\pm$\\phantom{x}2.9 & 2.6 & 0.992 & -1.8$\\pm$\\phantom{x}7.0 & \\phantom{x}4.9 & 0.974 & 1.6$\\pm$\\phantom{x}4.1 & \\phantom{x}3.3 & 0.981 & -6.8$\\pm$10.0 & \\phantom{x}9.4 \\\\\n\t\t\\hdashline[1pt\/2pt]\n\t\t\n\t\t\\rowcolor{LightGreen}\n\t\tU-net-soft-Dice &  & 0.997 & -2.0$\\pm$6.0 & 4.5 & 0.853 & \\phantom{x}3.6$\\pm$10.9 & 5.0 & 0.968 & -1.0$\\pm$14.1 & 10.0 & 0.782 & 4.8$\\pm$11.6 & \\phantom{x}9.0 & \\textbf{0.985} & -7.7$\\pm$\\phantom{x}8.8 & \\phantom{x}9.2 \\\\\n\t\t& e-map  & 0.997 & -1.7$\\pm$5.3 & 4.1 & 0.969 & \\phantom{x}1.9$\\pm$\\phantom{x}4.9 & 3.3 & 0.981 & -0.1$\\pm$10.9 & \\phantom{x}7.5 & 0.919 & 3.3$\\pm$\\phantom{x}7.0 & \\phantom{x}5.9 & 0.984 & -6.6$\\pm$\\phantom{x}9.0 & \\phantom{x}8.7 \\\\\n\t\t\n\t\t\\hdashline[1pt\/2pt]\n\t\t\\rowcolor{LightCyan}\n\t\tU-net-soft-Dice+MC &  & 0.997 & -1.8$\\pm$5.9 & 4.4 & 0.941 & \\phantom{x}3.0$\\pm$\\phantom{x}6.7 & 4.4 & 0.969 & \\phantom{x}0.6$\\pm$13.9 & \\phantom{x}9.8 & 0.792 & 4.4$\\pm$11.3 & \\phantom{x}8.7 & \\textbf{0.985} & -7.2$\\pm$\\phantom{x}8.9 & \\phantom{x}8.9 \\\\\n\t\t& b-map  & 0.997 & -1.5$\\pm$5.3 & 4.1 & 0.979 & \\phantom{x}1.1$\\pm$\\phantom{x}4.1 & 2.9 & 0.985 & \\phantom{x}1.2$\\pm$\\phantom{x}9.4 & \\phantom{x}6.5 & 0.939 & 2.9$\\pm$\\phantom{x}6.2 & \\phantom{x}4.9 & 0.984 & -5.9$\\pm$\\phantom{x}9.0 & \\phantom{x}8.5 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\nThe patches used to train the network were selected randomly (\\nicefrac{2}{3}), or were forced (\\nicefrac{1}{3}) to contain at least one segmentation failure by randomly selecting a scan containing segmentation failure, followed by random sampling of a patch containing at least one segmentation failure. During training the patch size was fixed to \\num{80}$\\times$\\num{80} voxels. To reduce the number of background voxels during testing, inputs were cropped based on a minimal enclosing, rectangular bounding box that was placed around the automatic segmentation mask. Inputs always had a minimum size of \\num{80}$\\times$\\num{80} voxels or were forced to a multiple of the output grid spacing of eight voxels in both direction required by the patch-based detection network. The patches of size \\num{8}$\\times$\\num{8} voxels did not overlap. In cases where the automatic segmentation mask only contains background voxels (scans above the base or below apex of the heart) input scans were center-cropped to a size of \\num{80}$\\times$\\num{80} voxels. \n\nModels were trained for 20,000 iterations using mini-batch stochastic gradient descent with batch-size \\num{32} and Adam as optimizer\\cite{kingmadp}. Learning rate was set to \\num{0.0001} and decayed with a factor of \\num{0.1} after \\num{10,000} steps. Furthermore, dropout percentage was set to \\num{0.5} and weight decay was applied to increase generalization performance.\n\n\\begin{figure*}[t]\n\t\\center\n\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/all_models_froc_voxel_detection_rate.pdf}%\n\t\t\\label{fig_froc_voxel_detection}}\n\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/all_models_slices_prec_recall.pdf}%\n\t\t\\label{fig_prec_rec_slice_detection}}\n\t\n\t\\caption{Detection performance of segmentation failures generated by different combination of segmentation architectures and loss functions. (a) Sensitivity for detection of segmentation failures on voxel level (y-axis) as a function of number of false positive image regions (x-axis). (b) Precision-recall curve for detection of slices containing segmentation failures (where AP denotes average precision). Results are split between entropy and Bayesian uncertainty maps. Each figure contains a curve for the six possible combination of models (three) and loss functions (two). SD denotes soft-Dice and CE cross-entropy, respectively.}\n\t\\label{fig_dt_perf_all_models}\n\\end{figure*}\n\n\\subsection{Segmentation using correction of the detected segmentation failures}\n\nTo investigate whether correction of detected segmentation failures increases segmentation performance two scenarios were performed. In the first scenario manual correction of the detected failures by an expert was simulated for all images at ED and ES time points of the ACDC dataset. For this purpose, in image regions that were detected to contain segmentation failure predicted labels were replaced with reference labels. In the second scenario manual correction of the detected failures was performed by an expert in a random subset of \\num{50} patients of the ACDC dataset. The expert was shown CMRI slices for ED and ES time points together with corresponding automatic segmentation masks for the RV, LV and LV myocardium. Image regions detected to contain segmentation failures were indicated in slices and the expert was only allowed to change the automatic segmentations in these indicated regions. Annotation was performed following the protocol described in\\cite{bernard2018deep}. Furthermore, expert was able to navigate through all CMRI slices of the corresponding ED and ES volumes.\n\n\\section{Results}\n\nIn this section we first present results for the segmentation-only task followed by description of the combined segmentation and detection results. \n\n\\subsection{Segmentation-only approach} \\label{results_seg_only}\n\nTable~\\ref{table_overall_segmentation_performance} lists quantitative results for segmentation-only and combined segmentation and detection approach in terms of Dice coefficient and Hausdorff distance. These results show that DRN and U-net achieve similar Dice coefficients and outperformed the DN network for all anatomical structures at end-systole. Differences in the achieved Hausdorff distances among the methods are present for all anatomical structures and for both time points. The DRN model achieved the highest and the DN network the lowest Hausdorff distance.\n\nTable~\\ref{table_cardiac_function_indices} lists results of the evaluation in terms of clinical metrics. These results reveal noticeable differences between models for ejection fraction (EF) of left and right ventricle, respectively. We can observe that U-net trained with the soft-Dice and the Dilated Network (DN) trained with Brier or soft-Dice loss achieved considerable lower accuracy for LV and RV ejection fraction compared to DRN. Overall, the DRN model achieved highest performance for all clinical metrics.\n\n\\noindent \\textbf{Effect of model architecture on segmentation}: Although quantitative differences between models are small, qualitative evaluation discloses that automatic segmentations differ substantially between the models. Figure~\\ref{fig_seg_qualitative_results} shows that especially in regions where the models perform poorly (apical and basal slices) the DN model more often produced anatomically implausible segmentations compared to the DRN and U-net. This seems to be correlated with the performance differences in Hausdorff distance.\n\n\\noindent \\textbf{Effect of loss function on segmentation}: The results indicate that the choice of loss function only slightly affects the segmentation performance. DRN and U-net perform marginally better when trained with soft-Dice compared to cross-entropy whereas DN performs better when trained with Brier loss than with soft-Dice. For DN this is most pronounced for the RV at ES.\n\nA considerable effect of the loss function on the accuracy of the LV and RV ejection fraction can be observed for the U-net model. On both metrics U-net achieved the lowest accuracy of all models when trained with the soft-Dice loss.\n\n\\noindent \\textbf{Effect of MC dropout on segmentation}: The results show that enabling MC-dropout during testing seems to result in slightly improved HD while it does not affect DC. \n\\begin{table}\n\t\\caption{Average precision and percentage of slices with segmentation failures generated by Dilated Network (DN), Dilated Residual Network (DRN) and U-net when trained with soft-Dice (SD), CE or Brier loss. Per patient, average precision of detected slices with failure using e- or b-maps (\\num{2}$^{nd}$ and \\num{3}$^{rd}$ columns). Per patient, average percentage of slices containing segmentation failures (reference for detection task) (\\num{4}$^{th}$ and \\num{5}$^{th}$ columns).}\n\t\\label{table_evaluation_slice_detection}\n\t\\begin{tabular}{l  C{1.4cm}  C{1.4cm} C{1.4cm} C{1.4cm}  }\n\t\t\\textbf{Model} & \\multicolumn{2}{c}{\\textbf{Average precision}} & \\multicolumn{2}{c}{ \\thead{\\textbf{\\% of slices} \\\\ \\textbf{with segmentation failures} }} \\\\\n\t\t& e-map & b-map & e-map & b-map \\\\\n\t\t\\hline\n\t\tDN-Brier &  84.0 & 83.0 & 53.7 & 52.4   \\\\\n\t\tDN-SD  &  87.0 & 85.0 & 58.3 & 58.1   \\\\\n\t\t\\hdashline\n\t\tDRN-CE  & 75.0 & 69.0  &  39.5 & 39.4 \\\\\n\t\tDRN-SD  & 67.0 & 67.0 &  34.9 & 33.7\\\\\n\t\t\\hdashline\n\t\tU-net-CE   & 81.0 & 75.0 & 54.8 & 52.5 \\\\\n\t\tU-net-SD   & 76.0 & 76.0 & 46.7 & 45.5 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Detection of segmentation failures}\n\n\\noindent \\textbf{Detection of segmentation failures on voxel level}: To evaluate detection performance of segmentation failures on voxel level Figure~\\ref{fig_froc_voxel_detection} shows average voxel detection rate as a function of false positively detected regions. This was done for each combination of model architecture and loss function exploiting e- (Figure~\\ref{fig_froc_voxel_detection}, left)  or b-maps (Figure~\\ref{fig_froc_voxel_detection}, right). These results show that detection performance of segmentation failures depends on segmentation model architecture, loss function and uncertainty map. \n\nThe influence of (segmentation) model architecture and loss function on detection performance is slightly stronger when e-maps were used as input for the detection task compared to b-maps. Detection rates are consistently lower when segmentation failures originate from segmentation models trained with soft-Dice loss compared to models trained with CE or Brier loss. Overall, detection rates are higher when b-maps were exploited for the detection task compared to e-maps.\n\n\\begin{table*}\n\t\\caption{Comparing performance of segmentation-only approach (auto-only) with combined segmentation and detection approach for two scenarios: simulated correction of detected segmentation failures (auto$+$simulation); and manual correction of detected segmentation failures by an expert (auto$+$expert). Automatic segmentations were obtained from a U-net trained with cross-entropy. Evaluation was performed on a subset of \\num{50} patients from the ACDC dataset. Scenarios are compared against segmentation-only approach (auto-only) in terms of (a) Dice Coefficient (b) Hausdorff Distance and (c) Clinical metrics. Results obtained from simulated manual correction represent an upper bound on the maximum achievable performance. Detection network was trained with e-maps. Number with asterisk indicates statistical significant at \\num{5}\\% level w.r.t. the segmentation-only approach. Best viewed in color.}\n\t\\label{table_manual_corr_performance}\n\t\\centering\n\t\\small\n\t\\subfloat[\\textbf{Dice coefficient:} Mean $\\pm$ standard deviation for left ventricle (LV), right ventricle (RV) and left ventricle myocardium (LVM).]{\n\t\t\\begin{tabular}{| C{2.cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\\hline\n\t\t\t& \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\\textbf{Scenario} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\\hline\n\t\t\tauto-only & \\phantom{x}0.964$\\pm$0.02 & \\phantom{x}0.927$\\pm$0.04 & \\phantom{x}0.883$\\pm$0.03  & \\phantom{x}0.916$\\pm$0.05 & \\phantom{x}0.854$\\pm$0.08 & \\phantom{x}0.886$\\pm$0.04 \\\\ \n\t\t\tauto$+$simulation & \\phantom{x}0.967$\\pm$0.01 & *0.948$\\pm$0.03 & *0.894$\\pm$0.03  & *0.939$\\pm$0.03 & *0.915$\\pm$0.04 & *0.910$\\pm$0.03 \\\\ \n\t\t\n\t\t\tauto$+$expert & \\phantom{x}0.965$\\pm$0.02 & \\phantom{x}0.940$\\pm$0.03 & \\phantom{x}0.885$\\pm$0.03  & \\phantom{x}0.927$\\pm$0.04 & \\phantom{x}0.868$\\pm$0.07 & \\phantom{x}0.894$\\pm$0.03 \\\\ \n\t\t\t\n\t\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{table_manual_seg_perf_dsc}\n\t} \n\t\n\t\\centering\n\t\n\t\\subfloat[\\textbf{Hausdorff Distance:} Mean $\\pm$ standard deviation for left ventricle (LV), right ventricle (RV) and left ventricle myocardium (LVM).]{\n\t\t\\begin{tabular}{| C{2.cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\\hline\n\t\t\t& \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\\textbf{Scenario} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\\hline\n\t\t\tauto-only  &  \\phantom{x}5.6$\\pm$3.3 & \\phantom{x}15.7$\\pm$9.7 & \\phantom{x}8.5$\\pm$6.4 & \\phantom{x}9.2$\\pm$5.8 & \\phantom{x}16.5$\\pm$8.8 & \\phantom{x}13.4$\\pm$10.5 \\\\\n\t\t\tauto$+$simulation &  \\phantom{x}4.5$\\pm$2.1 & *\\phantom{x}9.0$\\pm$4.6 & *5.9$\\pm$3.4 & *5.2$\\pm$2.5 & *10.3$\\pm$3.7 & *\\phantom{x}6.6$\\pm$2.9 \\\\\n\t\t\n\t\t\tauto$+$expert &  \\phantom{x}4.9$\\pm$2.8 & *\\phantom{x}9.8$\\pm$4.3 & \\phantom{x}7.3$\\pm$4.3 & \\phantom{x}7.2$\\pm$3.3 & *12.5$\\pm$4.7 & *\\phantom{x}8.3$\\pm$3.5 \\\\\n\t\t\t\n\t\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\label{table_manual_seg_perf_hd}\n\t}  \n\t\n\t\\subfloat[\\textbf{Clinical metrics:} a) Left ventricle (LV) end-diastolic volume (EDV) b) LV ejection fraction (EF) c) Right ventricle (RV) EDV d) RV ejection fraction e) LV myocardial mass. Quantitative results compare clinical metrics based on reference segmentations with 1) automatic segmentations; 2) simulated manual correction and 3) manual expert correction of automatic segmentations using spatial uncertainty maps. $\\rho$ denotes the Pearson correlation coefficient, \\textit{bias} denotes the mean difference between the two measurements (mean $\\pm$ standard deviation) and \\textit{MAE} denotes the mean absolute error between the two measurements.]{\n\t\t\\label{table_manual_cardiac_function_indices}\n\t\t\\small\n\t\t\\begin{tabular}{| C{2.cm} | C{0.5cm}  C{0.8cm} C{0.55cm} | C{0.5cm}  C{0.8cm}C{0.55cm} | C{0.5cm}  C{0.8cm} C{0.55cm} | C{0.5cm}  C{0.8cm} C{0.55cm} | C{0.5cm} C{0.8cm} C{0.55cm} |}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{3}{c}{\\textbf{LV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{LV$_{EF}$}} & \\multicolumn{3}{c}{\\textbf{RV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{RV$_{EF}$}} & \\multicolumn{3}{c|}{\\textbf{LVM$_{Mass}$}}  \\\\\n\t\t\t\n\t\t\t\\textbf{Scenario} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\multicolumn{1}{c}{\\textbf{$\\rho$}} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} \\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\tauto-only  & 0.995 & -4.4 $\\pm$7.0 & 5.7 & 0.927 & 5.0 $\\pm$7.1 & 5.8 & 0.962 & -6.4 $\\pm$16.2 & 11.9 & 0.878 & 5.8 $\\pm$8.7 & 8.0 & 0.979 & -6.4 $\\pm$10.6 & 9.5 \\\\\n\t\t\t\n\t\t\tauto$+$simulation & 0.998 & -3.9 $\\pm$5.2 & 4.8 & 0.989 & 2.3 $\\pm$2.9 & 2.9 & 0.984 & -3.7 $\\pm$10.4 & 6.8 & 0.954 & 2.7 $\\pm$5.5 & 4.5 & 0.983 & -5.5 $\\pm$9.6 & 8.1 \\\\\n\t\t\n\t\t\t\n\t\t\tauto$+$expert & 0.996 & -4.3 $\\pm$6.5 & 5.5 & 0.968 & 2.7 $\\pm$4.8 & 4.3 & 0.976 & -3.2 $\\pm$12.9 & 8.3 & 0.883 & 5.1 $\\pm$8.6 & 7.7 & 0.980 & -6.2 $\\pm$10.2 & 9.1 \\\\\n\t\t\t\t\\hline\n\t\t\\end{tabular}\n\t}\n\t\n\\end{table*}\n\n\\vspace{1ex}\n\\noindent \\textbf{Detection of slices with segmentation failures}: To evaluate detection performance w.r.t. slices containing segmentation failures precision-recall curves for each combination of model architecture and loss function using e-maps (Figure~\\ref{fig_prec_rec_slice_detection}, left) or b-maps (Figure~\\ref{fig_prec_rec_slice_detection}, right) are shown. The results show that detection performance of slices containing segmentation failures is slightly better for all models when using e-maps. Furthermore, the detection network achieves highest performance using uncertainty maps obtained from the DN model and the lowest when exploiting e- or b-maps obtained from the DRN model. Table~\\ref{table_evaluation_slice_detection} shows the average precision of detected slices with segmentation failures per patient, as well as the average percentage of slices that do contain segmentation failures (reference for detection task). The results illustrate that these measures are positively correlated i.e. that precision of detected slices in a patient volume is higher if the volume contains more slices that need correction. On average the DN model generates cardiac segmentations that contain more slices with at least one segmentation failure compared to U-net (ranks second) and DRN (ranks third). A higher number of detected slices containing segmentation failures implies an increased workload for manual correction.\n\n\\subsection{Calibration of uncertainty maps} \\label{result_eval_quality_umaps}\n\nFigure~\\ref{fig_risk_cov_comparison} shows risk-coverage curves for each combination of model architectures, uncertainty maps and loss functions (Figure~\\ref{fig_risk_cov_comparison} left: CE or Brier loss, Figure~\\ref{fig_risk_cov_comparison} right: soft-Dice). The results show an effect of the loss function on slope and convergence of the curves. Segmentation errors of models trained with the soft-Dice loss are less frequently covered by higher uncertainties than models trained with CE or Brier loss (steeper slope and lower minimum are better). This difference is more pronounced for e-maps. Models trained with the CE or Brier loss only slightly differ concerning convergence and their slopes are approximately identical. In contrast, the curves of the models trained with the soft-Dice differ regarding their slope and achieved minimum. Comparing e- and b-map of the DN-SD and U-net-SD models the results reveal that the curve for b-map has a steeper slope and achieves a lower minimum compared to the e-map. For the DRN-SD model these differences are less striking. In general for a specific combination of model and loss function the risk-coverage curves using b-maps achieve a lower minimum compared to e-maps.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=4.7in, height=2.8in]{figures\/risk_cov\/cov_risk_curve_both_seg_errors.pdf}\n\t\n\t\\caption{Comparison of risk-coverage curves for different combination of model architectures, loss functions and uncertainty maps. Results are separated for loss functions (left cross-entropy and Brier, right soft-Dice loss). \\num{100}\\% coverage means that none of the voxels is discarded based on its uncertainty whereas a coverage of \\num{0}\\% denotes the scenario in which all predictions are replaced by their reference labels. Note, all models were trained with two different loss functions (1) soft-Dice (SD) for all models (2) cross-entropy (CE) for DRN and U-net and Brier loss for DN.}\n\t\\label{fig_risk_cov_comparison}\n\\end{figure*}\n\n\\begin{table*}\n\t\\caption{Effect of number of Monte Carlo (MC) samples on segmentation performance in terms of (a) Dice coefficient (DC) and (b) Hausdorff Distance (HD) (mean $\\pm$ standard deviation). Higher DC and lower HD is better. Abbreviations: Cross-Entropy (CE), Dilated Residual Network (DRN) and Dilated Network (DN).} \n\t\\label{table_seg_perf_per_samples}\n\t\\small\n\t\\centering\n\t\\subfloat[Dice coefficient]{\n\t\t\\begin{tabular}{|c  C{1.5cm}  C{1.5cm} c|}\n\t\t\t\\hline\n\t\t\t\\textbf{\\thead{Number of \\\\ MC samples}} & DRN-CE & U-net-CE & DN-soft-Dice\\\\\n\t\t\t\\hline\n\t\t\t1 & 0.894$\\pm$0.07 & 0.896$\\pm$0.07 & 0.871$\\pm$0.09 \\\\\n\t\t\t3 & 0.900$\\pm$0.07 & 0.901$\\pm$0.07 & 0.883$\\pm$0.08 \\\\\n\t\t\t5 & 0.902$\\pm$0.07 & 0.901$\\pm$0.07 & 0.887$\\pm$0.08 \\\\\n\t\t\t7 & 0.903$\\pm$0.07 & 0.901$\\pm$0.07 & 0.888$\\pm$0.08 \\\\\n\t\t\t10 & 0.904$\\pm$0.06 & 0.902$\\pm$0.07 &0.890$\\pm$0.08  \\\\\n\t\t\t20 & 0.904$\\pm$0.07 & 0.902$\\pm$0.07 & 0.890$\\pm$0.08 \\\\\n\t\t\t30 & 0.904$\\pm$0.07 & 0.902$\\pm$0.07 & 0.891$\\pm$0.08 \\\\\n\t\t\t60 & 0.904$\\pm$0.07 & 0.902$\\pm$0.07 & 0.891$\\pm$0.08 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t}\n\t\\subfloat[Hausdorff Distance]{\n\t\t\\begin{tabular}{|c  C{1.5cm}  C{1.5cm} c|}\n\t\t\t\\hline\n\t\t\t\\textbf{\\thead{Number of \\\\ MC samples}}& DRN-CE & U-net-CE & DN-soft-Dice \\\\\n\t\t\t\\hline\n\t\t\t1 & 9.88$\\pm$5.76 & 11.79$\\pm$8.23 & 13.54$\\pm$7.14 \\\\\n\t\t\t3 & 9.70$\\pm$6.13 & 11.40$\\pm$7.78 & 12.71$\\pm$6.79 \\\\\n\t\t\t5 & 9.54$\\pm$6.07 & 11.37$\\pm$7.81 & 12.06$\\pm$6.29 \\\\\n\t\t\t7 & 9.38$\\pm$5.86 & 11.29$\\pm$7.86 & 12.08$\\pm$6.38 \\\\\n\t\t\t10 & 9.38$\\pm$5.91 & 11.24$\\pm$7.71 & 11.85$\\pm$6.34 \\\\\n\t\t\t20 & 9.37$\\pm$5.83 & 11.27$\\pm$7.79 & 11.90$\\pm$6.52 \\\\\n\t\t\t30 & 9.39$\\pm$5.91 &  11.32$\\pm$7.93 & 11.90$\\pm$6.48 \\\\\n\t\t\t60 & 9.39$\\pm$5.93 & 11.22$\\pm$7.83 & 11.89$\\pm$6.56 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\t\n\t}\n\\end{table*}\n\n\\subsection{Correction of automatically identified segmentation failures} \\label{results_combined_approach}\n\n\\textbf{Simulated correction:} The results listed in Table~\\ref{table_overall_segmentation_performance} and \\ref{table_cardiac_function_indices} show that the proposed method consisting of segmentation followed by simulated manual correction of detected segmentation failures delivers accurate segmentation for all tissues over ED and ES points. Correction of detected segmentation failures improved the performance in terms of DC, HD and clinical metrics for all combinations of model architectures, loss functions and uncertainty measures. Focusing on the DC after correction of detected segmentation failures the results reveal that performance differences between evaluated models decreased compared to the segmentation-only task. This effect is less pronounced for HD where the DRN network clearly achieved superior results in the segmentation-only and combined approach. The DN performs the least of all models but achieves the highest absolute DC performance improvements in the combined approach for RV at ES. Overall, the results in Table~\\ref{table_overall_segmentation_performance} disclose that improvements attained by the combined approach are almost all statistically significant ($p \\leq 0.05$) at ES and frequently at ED (\\num{96}\\% resp. \\num{83}\\% of the cases). Moreover, improvements are in \\num{99}\\% of the cases statistically significant for HD compared to \\num{81}\\% of the cases for DC.\n\nResults in terms of clinical metrics shown in Table~\\ref{table_cardiac_function_indices} are inline with these findings. We observe that segmentation followed by simulated manual correction of detected segmentation failures resulted in considerably higher accuracy for LV and RV ejection fraction. Achieved improvements for clinical metrics are only statistically significant ($p \\leq 0.05$) in one case for RV ejection fraction.\t\n\nIn general, the effect of correction of detected segmentation failures is more pronounced in cases where the segmentation-only approach achieved relatively low accuracy (e.g. DN-SD for RV at ES). Furthermore, performance gains are largest for RV and LV at ES and for ejection fraction of both ventricles.\n\n\n\\begin{figure*}[!t]\n\t\\captionsetup[subfigure]{justification=centering}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=6in]{figures\/qualitative\/sim_corr\/patient018_slice07_ES_bmap_acorr.pdf}%\n\t\t\\label{fig_qual_result_sim_corr_example1}}\n\t\n\t\\subfloat[]{\\includegraphics[width=6in]{figures\/qualitative\/sim_corr\/patient070_slice05_ED_emap_acorr.pdf}%\n\t\t\\label{fig_qual_result_sim_corr_example2}}\n\t\n\t\\subfloat[]{\\includegraphics[width=6in]{figures\/qualitative\/sim_corr\/patient081_slice01_ES_emap_acorr.pdf}%\n\t\t\\label{fig_qual_result_sim_corr_example3}}\n\t\\caption{Three patients showing results of combined segmentation and detection approach consisting of segmentation followed by simulated manual correction of detected segmentation failures. First column shows MRI (top) and reference segmentation (bottom). Results for automatic segmentation and simulated manual correction respectively achieved by: Dilated Network (DN-Brier, \\num{2}$^{nd}$ and \\num{5}$^{th}$ columns); Dilated Residual Network (DRN-soft-Dice, \\num{3}$^{rd}$ and \\num{6}$^{th}$ columns); and U-net (soft-Dice, \\num{4}$^{th}$ and \\num{7}$^{th}$ columns).}\n\t\\label{fig_seg_detection_qualitative_results}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\t\\captionsetup[subfigure]{justification=centering}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient048_slice02_mcorr.pdf}%\n\t\t\\label{fig_qual_result_man_corr_example1} \\hspace{3ex}}\n\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient091_slice01_mcorr.pdf}%\n\t\t\\label{fig_qual_result_man_corr_example2}}\n\t\n\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient075_slice06_mcorr.pdf}%\n\t\t\\label{fig_qual_result_man_corr_example3} \\hspace{3ex}}\n\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient006_slice01_mcorr.pdf}%\n\t\t\\label{fig_qual_result_man_corr_example4}}  \n\t\n\t\\caption{Four patients showing results of combined segmentation and detection approach consisting of segmentation followed by manual expert correction of detected segmentation failures. Expert was only allowed to adjust the automatic segmentations in regions where the detection network predicted segmentation failures (orange contour shown in 2$^{nd}$ column). Automatic segmentations were generated by a U-net trained with the cross-entropy loss. Segmentation failure detection was performed using entropy maps.}\n\t\\label{fig_qualitative_results_man_corr}\n\\end{figure*}\n\nThe best overall performance is achieved by the DRN model trained with cross-entropy loss while exploiting entropy maps in the detection task. Moreover, the proposed two step approach attained slightly better results using Bayesian maps compared to entropy maps. \n\n\\vspace{1ex}\n\\textbf{Manual correction}: Table~\\ref{table_manual_corr_performance} lists results for the combined automatic segmentation and detection approach followed by \\textit{manual} correction of detected segmentation failures by an expert. The results show that this correction led to improved segmentation performance in terms of DC, HD and clinical metrics. Improvements in terms of HD are in \\num{50} percent of the cases statistically significant ($p \\leq 0.05$) and most pronounced for RV and LV at end-systole. \n\nQualitative examples of the proposed approach are visualized in Figures~\\ref{fig_seg_detection_qualitative_results} and \\ref{fig_qualitative_results_man_corr} for simulated correction and manual correction of the automatically detected segmentation failures, respectively. For the illustrated cases (simulated) manual correction of detected segmentation failures leads to increased segmentation performance.\nOn average manual correction of automatic segmentations took less than \\num{2} minutes for ED and ES volumes of one patient compared to \\num{20} minutes that is typically needed by an expert for the same task.\n\n\n\\section{Ablation Study}\n\nTo demonstrate the effect of different hyper-parameters in the method, a number of experiments were performed. In the following text these are detailed.\n\n\\subsection{Impact of number of Monte Carlo samples on segmentation performance}\n\nTo investigate the impact of the number of Monte Carlo (MC) samples on the segmentation performance validation experiments were performed for all three segmentation architectures (Dilated Network, Dilated Residual Network and U-net) using $T$ $\\in \\{1, 3, 5, 7, 10, 20, 30, 60\\}$ samples. Results of these experiments are listed in Table~\\ref{table_seg_perf_per_samples}. We observe that segmentation performance started to converge using \\num{7} samples only. Performance improvements using an increased number of MC samples were largest for the Dilated Network. Overall, using more than \\num{10} samples did not increase segmentation performance. Hence, in the presented work $T$ was set to \\num{10}.\t\n\n\\subsection{Effect of patch-size on detection performance}\n\nThe combined segmentation and detection approach detects segmentation failures on region level. To investigate the effect of patch-size on detection performance three different patch-sizes were evaluated: \\num{4}$\\times$\\num{4}, \\num{8}$\\times$\\num{8}, and \\num{16}$\\times$\\num{16} voxels. The results are shown in Figure~\\ref{fig_grid_compare}. We can observe in Figure~\\ref{fig_fn_grid_froc_voxel_detection} that larger patch-sizes result in a lower number of false positive regions. The result is potentially caused by the decreasing number of regions in an image when using larger patch-sizes compared to smaller patch-sizes. Furthermore, Figure~\\ref{fig_fn_grid_prec_rec_slice_detection} reveals that slice detection performance is only slightly influenced by patch-size. To ease manual inspection and correction by an expert, it is desirable to keep region-size i.e. patch-size small. Therefore, in the experiments a patch-size of \\num{8}$\\times$\\num{8} voxels was used.\n\n\\begin{figure*}[t]\n\t\\center\n\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/compare_grid_froc_voxel_detection_rate.pdf}%\n\t\t\\label{fig_fn_grid_froc_voxel_detection}}\n\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/compare_grid_slices_prec_recall.pdf}%\n\t\t\\label{fig_fn_grid_prec_rec_slice_detection}} \n\t\n\t\\caption{Detection performance for three different patch-sizes specified in voxels. (a) Sensitivity for detection of segmentation failures on voxel level (y-axis) versus number of false positive image regions (x-axis). (b) Precision-recall curve for detection of slices containing segmentation failures (where AP denotes average precision). Results are split between entropy and Bayesian uncertainty maps. In the experiments patch-size was set to \\num{8}$\\times$\\num{8} voxels.}\n\t\\label{fig_grid_compare}\n\\end{figure*}\n\n\\subsection{Impact of tolerance threshold on number of segmentation failures}\n\nTo investigate the impact of the tolerance threshold separating segmentation failures and tolerable segmentation errors, we calculated the ratio of the number of segmentation failures and all errors i.e. the sum of tolerable errors and segmentation failures. Figure~\\ref{fig_threshold_compare} shows the results. We observe that at least half of the segmentation failures are located within a tolerance threshold i.e. distance of two to three voxels of the target structure boundary as defined by the reference annotation. Furthermore, the mean percentage of failures per volume is considerably lower for the Dilated Residual Network (DRN) and highest for the Dilated Network. This result is inline with our earlier finding (see Table~\\ref{table_evaluation_slice_detection}) that average percentage of slices that do contain segmentation failures is lowest for the DRN model.\n\n\\section{Discussion}\n\nWe have described a method that combines automatic segmentation and assessment of uncertainty in cardiac MRI with detection of image regions containing segmentation failures. The results show that combining automatic segmentation with manual correction of detected segmentation failures results in higher segmentation performance.\nIn contrast to previous methods that detected segmentation failures per patient or per structure, we showed that it is feasible to detect segmentation failures per image region. In most of the experimental settings, simulated manual correction of detected segmentation failures for LV, RV and LVM at ED and ES led to statistically significant improvements. These results represent the upper bound on the maximum achievable performance\tfor the manual expert correction task. Furthermore, results show that manual expert correction of detected segmentation failures led to consistently improved segmentations. However, these results are not on par with the simulated expert correction scenario. This is not surprising because inter-observer variability is high for the presented task and annotation protocols may differ between clinical environments. Moreover, qualitative results of the manual expert correction reveal that manual correction of the detected segmentation failures can prevent anatomically implausible segmentations (see Figure~\\ref{fig_qualitative_results_man_corr}).\nTherefore, the presented approach can potentially simplify and accelerate correction process and has the capacity to increase the trustworthiness of existing automatic segmentation methods in daily clinical practice.\n\n\nThe proposed combined segmentation and detection approach was evaluated using three state-of-the-art deep learning segmentation architectures. The results suggest that our approach is generic and applicable to different model architectures. Nevertheless, we observe noticeable differences between the different combination of model architectures, loss functions and uncertainty measures. In the segmentation-only task the DRN clearly outperforms the other two models in the evaluation of the boundary of the segmented structure. Moreover, qualitative analysis of the automatic segmentation masks suggests that DRN generates less often anatomically implausible and fragmented segmentations than the other models. We assume that clinical experts would prefer such segmentations although they are not always perfect. Furthermore, even though DRN and U-net achieve similar performance in regard to DC we assume that less fragmented segmentation masks would increase trustworthiness of the methods. \n\n\\begin{figure*}[t]\n\t\\center\n\t\\subfloat[]{\n\t\t\\includegraphics[width=3.in]{figures\/dt\/tp_per_threshold_out_struc.pdf}%\n\t\t\\label{fig_threshold_struc_out_compare}\n\t}\n\t\\subfloat[]{\n\t\t\\includegraphics[width=3.in]{figures\/dt\/tp_per_threshold_in_struc.pdf}%\n\t\t\\label{fig_threshold_struc_in_compare}\n\t}\n\t\\caption{Mean percentage of the segmentation failures per volume (y-axis) in the set of all segmentation errors (tolerable errors$+$segmentation failures) depending on the tolerance threshold (x-axis). Red, dashed vertical line indicates threshold value that was used throughout the experiments. Results are split between segmentation errors located (a) outside and (b) inside the target structure. Each figure contains a curve for U-net, Dilated Network (DN) and Dilated Residual Network (DRN) trained with the soft-Dice (SD) loss. Segmentation errors located in slices above the base or below the apex are always included in the set of segmentation failures and therefore, they are independent of the applied tolerance threshold.}\n\t\\label{fig_threshold_compare}\n\\end{figure*}\n\nIn agreement with our preliminary work we found that uncertainty maps obtained from a segmentation model trained with soft-Dice loss have a lower degree of uncertainty calibration compared to when trained with one of the other two loss functions (cross-entropy and Brier)\\cite{sander2019towards}. Nevertheless, the results of the combined segmentation and detection approach showed that a lower degree of uncertainty calibration only slightly deteriorated the detection performance of segmentation failures for the larger segmentation models (DRN and U-net) when exploiting uncertainty information from e-maps. Hendrycks and Gimpel \\cite{hendrycks2016baseline} showed that softmax probabilities generated by deep learning networks have poor direct correspondence to confidence. However, in agreement with Geifman et al. \\cite{geifman2017selective} we presume that probabilities and hence corresponding entropies obtained from softmax function are ranked consistently i.e. entropy can potentially be used as a relative uncertainty measure in deep learning. In addition, we detect segmentation failures per image region and therefore, our approach does not require perfectly calibrated uncertainty maps. Furthermore, results of the combined segmentation and detection approach revealed that detection performance of segmentation failures using b-maps is almost independent of the loss function used to train the segmentation model. In line with Jungo et al. \\cite{jungo2019assessing} we assume that by enabling MC-dropout in testing and computing the mean softmax probabilities per class leads to better calibrated probabilities and b-maps. This assumption is in agreement with Srivastava et al.~\\cite{srivastava2014dropout} where a CNN with dropout used at testing is interpreted as an ensemble of models.\n\nQuantitative evaluation in terms of Dice coefficient and Hausdorff distance reveals that proposed combined segmentation and detection approach leads to significant performance increase. However, the results also demonstrate that the correction of the detected failures allowed by the combined approach does not lead to statistically significant improvement in clinical metrics. This is not surprising because state-of-the-art automatic segmentation methods are not expected to lead to large volumetric errors \\cite{bernard2018deep} and standard clinical measures are not sensitive to small segmentation errors. Nevertheless, errors of the current state-of-the-art automatic segmentation methods may lead to anatomically implausible segmentations \\cite{bernard2018deep} that may cause distrust in clinical application. Besides increase in trustworthiness of current state-of-the-art segmentation methods for cardiac MRIs, improved segmentations are a prerequisite for advanced functional analysis of the heart e.g. motion analysis\\cite{bello2019deep} and very detailed morphology analysis such as myocardial trabeculae in adults\\cite{meyer2020genetic}.\n\nFor the ACDC dataset used in this manuscript, Bernard et al.\\cite{bernard2018deep} reported inter-observer variability ranging from \\num{4} to \\SI{14.1}{\\milli\\meter} (equivalent to on average \\num{2.6} to \\num{9} voxels). To define the set of segmentation failures, we employed a strict tolerance threshold on distance metric to distinguish between tolerated segmentation errors and segmentation failures (see Ablation study). Stricter tolerance threshold was used because the thresholding is performed in \\num{2}D, while evaluation of segmentation is done in \\num{3}D. Large slice thickness in cardiac MR  could lead to a discrepancy between the two. As a consequence of this strict threshold results listed in Table~\\ref{table_evaluation_slice_detection} show that almost all patient volumes contain at least one slice with a segmentation failure. This might render the approach less feasible in clinical practice. Increasing the threshold decreases the number of segmentation failures and slices containing segmentation failures (see Figure~\\ref{fig_threshold_compare}) but also lowers the upper bound on the maximum achievable performance. Therefore, to show the potential of our proposed approach we chose to apply a strict tolerance threshold. Nevertheless, we realize that although manual correction of detected segmentation failures leads to increased segmentation accuracy the performance of precision-recall is limited (see Figure~\\ref{fig_dt_perf_all_models}) and hence, should be a focus of future work. \n\nThe presented patch-based detection approach combined with (simulated) manual correction can in principle lead to stitching artefacts in the resulting segmentation masks. A voxel-based detection approach could potentially solve this. However, voxel-based detection methods are more challenging to train due to the very small number of voxels in an image belonging to the set of segmentation failures.\n\nEvaluation of the proposed approach for \\num{12} possible combinations of segmentation models (three), loss functions (two) and uncertainty maps (two) resulted in an extensive number of experiments. Nevertheless, future work could extend evaluation to other segmentation models, loss functions or combination of losses. Furthermore, our approach could be evaluated using additional uncertainty estimation techniques e.g. by means of ensembling of networks \\cite{lakshminarayanan2017simple} or variational dropout \\cite{kingma2015variational}. In addition, previous work by Kendall and Gal~\\cite{kendall2017uncertainties}, Tanno et al. \\cite{tanno2019uncertainty} has shown that the quality of uncertainty estimates can be improved if model (epistemic) and data (aleatoric) uncertainty are assessed simultaneously with separate measures. The current study focused on the assessment of model uncertainty by means of MC-dropout and entropy which is a combination of epistemic and aleatoric uncertainty. Hence, future work could investigate whether additional estimation of aleatoric uncertainty improves the detection of segmentation failures. \n\nFurthermore, to develop an end-to-end approach future work could incorporate the detection of segmentation failures into the segmentation network. Besides, adding the automatic segmentations to the input of the detection network could increase the detection performance. \n\nFinally, the proposed approach is not specific to cardiac MRI segmentation. Although data and task specific training would be needed the approach could potentially be applied to other image modalities and segmentation tasks.\n\n\\section{Conclusion}\n\nA method combining automatic segmentation and assessment of segmentation uncertainty in cardiac MR with detection of image regions containing local segmentation failures has been presented. The combined approach, together with simulated and manual correction of detected segmentation failures, increases performance compared to segmentation-only. The proposed method has the potential to increase trustworthiness of current state-of-the-art segmentation methods for cardiac MRIs. \n\n\\section*{Data and code availability}\nAll models were implemented using the PyTorch\\cite{paszke2017automatic} framework and trained on one Nvidia GTX Titan X GPU with \\num{12} GB memory. The code to replicate the study is publicly available at  \\href{https:\/\/github.com\/toologicbv\/cardiacSegUncertainty}{https:\/\/github.com\/toologicbv\/cardiacSegUncertainty}.\n\n\n\\section*{Acknowledgements}\n\nThis study was performed within the DLMedIA program (P15-26) funded by Dutch Technology Foundation with participation of PIE Medical Imaging.\n\n\\section*{Author contributions statement}\n\nJ.S., B.D.V. and I.I. designed the concept of the study. J.S. conducted the experiments. J.S., B.D.V. and I.I. wrote the manuscript. All authors reviewed the manuscript. \n\n\\section*{Additional information}\n\n\\textbf{Competing interests}: The authors declare that they have no competing interests. \n\n\\ifCLASSOPTIONcaptionsoff\n\\newpage\n\\fi\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section*{Introduction}\n\t\n\t\\IEEEPARstart{T}o perform diagnosis and prognosis of cardiovascular disease (CVD) medical experts depend on the reliable quantification of cardiac function \\cite{white1987left}. Cardiac magnetic resonance imaging (CMRI) is currently considered the reference standard for quantification of ventricular volumes, mass and function \\cite{grothues2002comparison}. Short-axis CMR imaging, covering the entire left and right ventricle (LV resp. RV) is routinely used to determine quantitative parameters of both ventricle's function. This requires manual or semi-automatic segmentation of corresponding cardiac tissue structures for end-diastole (ED) and end-systole (ES). \n\t\n\tExisting semi-automated or automated segmentation methods for CMRIs regularly require (substantial) manual intervention caused by lack of robustness. Manual or semi-automatic segmentation across a complete cardiac cycle, comprising \\num{20} to \\num{40} phases per patient, enables computation of parameters quantifying cardiac motion with potential diagnostic implications but due to the required workload, this is practically infeasible. Consequently, segmentation is often performed at end-diastole and end-systole precluding comprehensive analysis over complete cardiac cycle. \n\t\n\tRecently\\cite{litjens2017survey, leiner2019machine}, deep learning segmentation methods have shown to outperform traditional approaches such as those exploiting level set, graph-cuts, deformable models, cardiac atlases and statistical models \\cite{petitjean2011review, peng2016review}. However, recent comparison of a number of automatic methods showed that even the best performing methods generated anatomically implausible segmentations in more than 80\\% of the CMRIs \\cite{bernard2018deep}. Such errors do not occur when experts  perform segmentation. To achieve acceptance in clinical practice these shortcomings of the automatic approaches need to be alleviated by further development. This can be achieved by generating more accurate segmentation result or by development of approaches that automatically detect segmentation failures. \n\t\n\tIn manual and automatic segmentation of short-axis CMRI, largest segmentation inaccuracies are typically located in the most basal and apical slices due to low tissue contrast ratios \\cite{suinesiaputra2015quantification}. To increase segmentation performance, several methods have been proposed \\cite{tan2018fully, zheng20183, savioli2018automated, bai2018automated}. Tan et al. \\cite{tan2018fully} used a convolutional neural network (CNN) to regress anatomical landmarks from long-axis views (orthogonal to short-axis). They exploited the landmarks to determine most basal and apical slices in short-axis views and thereby constraining the automatic segmentation of CMRIs. This resulted in increased robustness and performance. Other approaches leverage spatial \\cite{zheng20183} or temporal \\cite{savioli2018automated, bai2018automated} information to increase segmentation consistency and performance in particular in the difficult basal and apical slices.\n\t\n\tAn alternative approach to preventing implausible segmentation results is by incorporating knowledge about the highly constrained shape of the heart. Oktay et al. \\cite{oktay2017anatomically} developed an anatomically constrained neural network (NN) that infers shape constraints using an auto-encoder during segmentation training. Duan et al. \\cite{duan2019automatic} developed a deep learning segmentation approach for CMRIs that used atlas propagation to explicitly impose a shape refinement. This was especially beneficial in the presence of image acquisition artifacts. Recently, Painchaud et al. \\cite{painchaud2019cardiac} developed a post-processing approach to detect and transform anatomically implausible cardiac segmentations into valid ones by defining cardiac anatomical metrics. Applying their approach to various state-of-the-art segmentation methods the authors showed that the proposed method provides strong anatomical guarantees without hampering segmentation accuracy. \n\t\n\t\\begin{figure}[t]\n\t\t\\center\n\t\t\\includegraphics[width=6in, height=3.5in]{figures\/overview_approach.pdf}%\n\t\t\n\t\t\\caption{Overview of proposed two step approach. Step 1 (left): Automatic CNN segmentation of CMR images combined with assessment of segmentation uncertainties. Step 2 (right): Differentiate tolerated errors from segmentation failures (to be detected) using distance transform maps based on reference segmentations. Detection of image regions containing segmentation failures using CNN which takes CMR images and segmentation uncertainties as input. Manual corrected segmentation failures (green) based on detected image regions.}\n\t\t\\label{fig_overview_method}\n\t\\end{figure}\n\t\n\t\n\tA different research trend focuses on detecting segmentation failures, i.e. on automated quality control for image segmentation. These methods can be divided in those that predict segmentation quality using image at hand or corresponding automatic segmentation result, and those that assess and exploit predictive uncertainties to detect segmentation failure. \n\t\n\tRecently, two methods were proposed to detect segmentation failures in large-scale cardiac MR imaging studies to remove these from subsequent analysis\\cite{alba2018automatic, robinson2019automated}. Robinson et al. \\cite{robinson2019automated} using the approach of Reverse Classification Accuracy (RCA) \\cite{valindria2017reverse} predicted CMRI segmentation metrics to detect failed segmentations. They achieved good agreement between predicted metrics and  visual quality control scores. Alba et al. \\cite{alba2018automatic} used statistical, pattern and fractal descriptors in a random forest classifier to directly detect segmentation contour failures without intermediate regression of segmentation accuracy metrics. \n\t\n\tMethods for automatic quality control were also developed for other applications in medical image analysis. Frounchi et al. \\cite{frounchi2011automating} extracted features from the segmentation results of the left ventricle in CT scans. Using the obtained features the authors trained a classifier that is able to discriminate between consistent and inconsistent segmentations. To distinguish between acceptable and non-acceptable segmentations Kohlberger el al. \\cite{kohlberger2012evaluating} proposed to directly predict multi-organ segmentation accuracy in CT scans using a set of features extracted from the image and corresponding segmentation. \n\t\n\tA number of methods aggregate voxel-wise uncertainties into an overall score to identify insufficiently accurate segmentations. For example, Nair et al. \\cite{nair2018exploring} computed an overall score for target segmentation structure from voxel-wise predictive uncertainties. The method was tested for detection of Multiple Sclerosis in brain MRI. The authors showed that rejecting segmentations with high uncertainty scores led to increased detection accuracy indicating that correct segmentations contain lower uncertainties than incorrect ones. Similarly, to assess segmentation quality of brain MRIs Jungo et al. \\cite{jungo2018uncertainty} aggregated voxel-wise uncertainties into a score per target structure\n\tand showed that the computed uncertainty score enabled identification of erroneous segmentations.\n\t\n\tUnlike approaches evaluating segmentation directly, several methods use predictive uncertainties to predict segmentation metrics and thereby evaluate segmentation performance \\cite{roy2019bayesian, devries2018leveraging}. For example, Roy et al. \\cite{roy2019bayesian} aggregated voxel-wise uncertainties into four scores per segmented structure in brain MRI. The authors showed that computed scores can be used to predict the Intersection over Union and hence, to determine segmentation accuracy. \n\tSimilar idea was presented by DeVries et al. \\cite{devries2018leveraging} that predicted segmentation accuracy per patient using an auxiliary neural network that leverages the dermoscopic image, automatic segmentation result and obtained uncertainties. The researchers showed that a predicted segmentation accuracy is useful for quality control.\n\t\n\tWe build on our preliminary work where automatic segmentation of CMR images using a dilated CNN was combined with assessment of two measures of segmentation uncertainties \\cite{sander2019towards}. For the first measure the multi-class entropy per voxel (entropy maps) was computed using the output distribution. For the second measure Bayesian uncertainty maps were acquired using Monte Carlo dropout (MC-dropout) \\cite{gal2016dropout}. In \\cite{sander2019towards} we showed that the obtained uncertainties almost entirely cover the regions of incorrect segmentation i.e. that uncertainties are calibrated. In the current work we extend our preliminary research in two ways. First, we assess impact of CNN architecture on the segmentation performance and calibration of uncertainty maps by evaluating three existing state-of-the-art CNNs. Second, we employ an auxiliary CNN (detection network) that processes a cardiac MRI and corresponding spatial uncertainty map (Entropy or Bayesian) to automatically detect segmentation failures. We differentiate errors that may be within the range of inter-observer variability and hence do not necessarily require correction (tolerated errors) from the errors that an expert would not make and hence require correction (segmentation failures). Given that overlap measures do not capture fine details of the segmentation results and preclude us to differentiate two types of segmentation errors, in this work, we define segmentation failure using a metric of boundary distance. In \\cite{sander2019towards} we found that degree of calibration of uncertainty maps is dependent on the loss function used to train the CNN. Nevertheless, in the current work we show that uncalibrated uncertainty maps are useful to detect local segmentation failures. \t\n\tIn contrast to previous methods that detect segmentation failure per-patient or per-structure\\cite{roy2019bayesian, devries2018leveraging}, we propose to detect segmentation failures per image region. We expect that inspection and correction of segmentation failures using image regions rather than individual voxels or images would simplify correction process. To show the potential of our approach and demonstrate that combining automatic segmentation with manual correction of the detected segmentation failures per region results in higher segmentation performance we performed two additional experiments. In the first experiment, correction of detected segmentation failures was simulated in the complete data set. In the second experiment, correction was performed by an expert in a subset of images. Using publicly available set of CMR scans from MICCAI 2017 ACDC challenge \\cite{bernard2018deep}, the performance was evaluated before and after simulating the correction of detected segmentation failures as well as after manual expert correction.\n\t\n\t\\section*{Data}\n\t\n\tIn this study data from the MICCAI \\num{2017} Automated Cardiac Diagnosis Challenge (ACDC) \\cite{bernard2018deep} was used. The dataset consists of cardiac cine MR images (CMRIs) from 100 patients uniformly distributed over normal cardiac function and four disease groups: dilated cardiomyopathy, hypertrophic cardiomyopathy, heart failure with infarction, and right ventricular abnormality. Detailed acquisition protocol is described by Bernard et al.~\\cite{bernard2018deep}. Briefly, short-axis CMRIs were acquired with two MRI scanners of different magnetic strengths (\\num{1.5} and \\num{3.0} T). Images were made during breath hold using a conventional steady-state free precession (SSFP) sequence. CMRIs have an in-plane resolution ranging from \\num{1.37} to \\SI{1.68}{\\milli\\meter} (average reconstruction matrix \\num{243} $\\times$ \\num{217} voxels) with slice spacing varying from \\num{5} to \\SI{10}{\\milli\\meter}. Per patient 28 to 40 volumes  are provided covering partially or completely one cardiac cycle. Each volume consists of on average ten slices covering the heart. Expert manual reference segmentations are provided for the LV cavity, RV endocardium and LV myocardium (LVM) for all CMRI slices at ED and ES time frames. To correct for intensity differences among scans, voxel intensities of each volume were scaled to the [\\num{0.0}, \\num{1.0}] range using the minimum and maximum of the volume. Furthermore, to correct for differences in-plane voxel sizes, image slices were resampled to \\num{1.4}$\\times\\SI{1.4}{\\milli\\meter}^2$. \n\t\n\t\\begin{figure}[!t]\n\t\t\\captionsetup[subfigure]{justification=centering}\n\t\t\\centering\n\t\n\t\t\\subfloat[]{\\includegraphics[width=5in]{figures\/qualitative\/auto\/patient099_slice02_ES_emap_auto.pdf}%\n\t\t\\label{fig_seg_qual_example1}}\n\t\n\t\t\\subfloat[]{\\includegraphics[width=5in]{figures\/qualitative\/auto\/patient097_slice00_ES_emap_auto.pdf}%\n\t\t\\label{fig_seg_qual_example2}}\n\t\t\n\t\t\\caption{Examples of automatic segmentations generated by different segmentation models for two cardiac MRI scans (rows) at ES at the base of the heart.}\n\t\t\\label{fig_seg_qualitative_results}\n\t\\end{figure}\n\t\n\t\\section*{Methods}\n\t\n\tTo investigate uncertainty of the segmentation, anatomical structures in CMR images are segmented using a CNN. To investigate whether the approach generalizes to different segmentation networks, three state-of-the-art CNNs were evaluated. For each segmentation model two measures of predictive uncertainty were obtained per voxel. Thereafter, to detect and correct local segmentation failures an auxiliary CNN (detection network) that analyzes a cardiac MRI was used. Finally, this leads to the uncertainty map allowing detection of image regions that contain segmentation failures. Figure~\\ref{fig_overview_method} visualizes this approach.\n\t\n\t\n\t\\subsection*{Automatic segmentation of cardiac MRI}\n\t\n\tTo perform segmentation of LV, RV, and LVM in cardiac MR images i.e. \\num{2}D CMR scans, three state-of-the-art CNNs are trained. Each of the three networks takes a CMR image as input and has four output channels providing probabilities for the three cardiac structures (LV, RV, LVM) and background. Softmax probabilities are calculated over the four tissue classes. Patient volumes at ED and ES are processed separately. During inference the \\num{2}D automatic segmentation masks are stacked into a \\num{3}D volume per patient and cardiac phase. After segmentation, the largest \\num{3}D connected component for each class is retained and volumes are resampled to their original voxel resolution. Segmentation networks differ substantially regarding architecture, number of parameters and receptive field size. To assess predictive uncertainties from the segmentation models \\textit{Monte Carlo dropout} (MC-dropout) introduced by Gal \\& Ghahramani \\cite{gal2016dropout} is implemented in every network. The following three segmentation networks were evaluated: Bayesian Dilated CNN, Bayesian Dilated Residual Network, Bayesian U-net.\n\t\n\t\\vspace{1ex}\n\t\\noindent \\textbf{Bayesian Dilated CNN (DN)}: The Bayesian DN architecture comprises a sequence of ten convolutional layers. Layers \\num{1} to \\num{8} serve as feature extraction layers with small convolution kernels of size \\num{3}$\\times$\\num{3} voxels. No padding is applied after convolutions. The number of kernels increases from \\num{32} in the first eight layers, to \\num{128} in the final two fully connected classification layers, implemented as \\num{1}$\\times$\\num{1} convolutions. The dilation level is successively increased between layers \\num{2} and \\num{7} from \\num{2} to \\num{32} which results in a receptive field for each voxel of \\num{131}$\\times$\\num{131} voxels, or \\num{18.3}$\\times$ $\\SI{18.3}{\\centi\\meter}^2$. All trainable layers except the final layer use rectified linear activation functions (ReLU). To enhance generalization performance, the model uses batch normalization in layers \\num{2} to \\num{9}. In order to convert the original DN~\\cite{wolterink2017automatic} into a Bayesian DN dropout is added as the last operation in all but the final layer and \\num{10} percent of a layer's hidden units are randomly switched off. \n\t\n\t\\vspace{1ex}\n\t\\noindent \\textbf{Bayesian Dilated Residual Network (DRN)}: The Bayesian DRN is based on the original DRN from Yu et al. \\cite{yu2017dilated} for image segmentation. More specifically, the DRN-D-22\\cite{yu2017dilated} is used which consists of a feature extraction module with output stride eight followed by a classifier implemented as fully convolutional layer with \\num{1}$\\times$\\num{1} convolutions. Output of the classifier is upsampled to full resolution using bilinear interpolation. The convolutional feature extraction module comprises eight levels where the number of kernels increases from \\num{16} in the first level, to \\num{512} in the two final levels. The first convolutional layer in level \\num{1} uses \\num{16} kernels of size \\num{7}$\\times$\\num{7} voxels and zero-padding of size \\num{3}. The remaining trainable layers use small \\num{3}$\\times$\\num{3} voxel kernels and zero-padding of size \\num{1}. Level \\num{2} to \\num{4} use a strided convolution of size \\num{2}. To further increase the receptive field convolutional layers in level \\num{5}, \\num{6} and \\num{7} use a dilation factor of \\num{2}, \\num{4} and \\num{2}, respectively. Furthermore, levels \\num{3} to \\num{6} consist of two residual blocks. All convolutional layers of the feature extraction module are followed by batch normalization, ReLU function and dropout. Adding dropout and switching off \\num{10} percent of a layer's hidden units converts the original DRN~\\cite{yu2017dilated} into a Bayesian DRN.\n\t\n\t\\vspace{1ex}\n\t\\noindent \\textbf{Bayesian U-net (U-net)}: The standard architecture of the U-net~\\cite{ronneberger2015u} is used. The network is fully convolutional and consists of a contracting, bottleneck and expanding path. The contracting and expanding path each consist of four blocks i.e. resolution levels which are connected by skip connections. The first block of the contracting path contains two convolutional layers using a kernel size of \\num{3}$\\times$\\num{3} voxels and zero-padding of size \\num{1}. Downsampling of the input is accomplished by employing a max pooling operation in block \\num{2} to \\num{4} of the contracting path and the bottleneck using a convolutional kernel of size \\num{2}$\\times$\\num{2} voxels and stride \\num{2}. Upsampling is performed by a transposed convolutional layer in block \\num{1} to \\num{4} of the expanding path using the same kernel size and stride as the max pooling layers. Each downsampling and upsampling layer is followed by two convolutional layers using \\num{3}$\\times$\\num{3} voxel kernels with zero-padding size \\num{1}. The final convolutional layer of the network acts as a classifier and uses \\num{1}$\\times$\\num{1} convolutions to reduce the number of output channels to the number of segmentation classes. The number of kernels increases from \\num{64} in the first block of the contracting path to \\num{1024} in the bottleneck. In contrast, the number of kernels in the expanding path successively decreases from \\num{1024} to \\num{64}. In deviation to the standard U-net instance normalization is added to all convolutional layers in the contracting path and ReLU non-linearities are replaced by LeakyReLU functions because this was found to slightly improve segmentation performance. In addition, to convert the deterministic model into a Bayesian neural network dropout is added as the last operation in each block of the contracting and expanding path and \\num{10} percent of a layer's hidden units are randomly switched off.\n\t\n\t\\subsection*{Assessment of predictive uncertainties} \\label{uncertainty_maps}\n\tTo detect failures in segmentation masks generated by CNNs in testing, spatial uncertainty maps of the obtained segmentations are generated. For each voxel in the image two measures of uncertainty are calculated. First, a computationally cheap and straightforward measure of uncertainty is the entropy of softmax probabilities over the four tissue classes which are generated by the segmentation networks. Using these, normalized entropy maps $\\bf E \\in [0, 1]^{H\\times W}$ (e-map) are computed where $H$ and $W$ denote the height and width of the original CMRI, respectively.\n\t\n\tSecond, by applying MC-dropout in testing, softmax probabilities with a number of samples $T$ per voxel are obtained. As an overall measure of uncertainty the mean standard deviation of softmax probabilities per voxel over all tissue classes $C$\\label{ref:maximum_variance} is computed\n\t\n\t\n\t\\begingroup\n\t\\small\n\t\\begin{align}\n\t\\textbf{B} (I)^{(x, y)} &= \\frac{1}{C} \\sum_{c=1}^{C} \\sqrt{\\frac{1}{T-1}  \\sum_{t=1}^{T} \\big(p_t(I)^{(x, y, c)} - \\hat{\\mu}^{(x, y, c)} \\big)^2 } \\; ,\n\t\\end{align}\n\t\\endgroup\n\t\n\twhere $\\textbf{B}(I)^{(x, y)} \\in [0, 1]$ denotes the normalized value of the Bayesian uncertainty map (b-map) at position $(x, y)$ in \\num{2}D slice $I$, $C$ is equal to the number of classes, $T$ is the number of samples and $p_t(I)^{(x, y, c)}$ denotes the softmax probability at position $(x, y)$  in image $I$ for class $c$. The predictive mean per class $\\hat{\\mu}^{(x, y, c)}$ of the samples is computed as follows:\n\t\n\t\\begingroup\n\t\\small\n\t\\begin{align}\n\t\\hat{\\mu}^{(x, y, c)} &= \\frac{1}{T} \\sum_{t=1}^{T} p_t(I)^{(x, y, c)} \\; .\n\t\\end{align}\n\t\\endgroup\n\t\n\tIn addition, the predictive mean per class is used to determine the tissue class per voxel.\n\t\n\t\\subsection*{Calibration of uncertainty maps} \n\tIdeally, incorrectly segmented voxels as defined by the reference labels should be covered by higher uncertainties than correctly segmented voxels. In such a case the spatial uncertainty maps are perfectly calibrated. \\textit{Risk-coverage curves} introduced by Geifman et al.\\cite{geifman2017selective} visualize whether incorrectly segmented voxels are covered by higher uncertainties than those that are correctly segmented. Risk-coverage curves convey the effect of avoiding segmentation of voxels above a specific uncertainty value on the reduction of segmentation errors (i.e. risk reduction) while at the same time quantifying the voxels that were omitted from the classification task (i.e. coverage). \n\t\n\tTo generate risk-coverage curves first, each patient volume is cropped based on a minimal enclosing  parallelepiped bounding box that is placed around the reference segmentations to reduce the number of background voxels. Note that this is only performed to simplify the analysis of the risk-coverage curves. Second, voxels of the cropped patient volume are ranked based on their uncertainty value in descending order. Third, to obtain uncertainty threshold values per patient volume the ranked voxels are partitioned into \\num{100} percentiles based on their uncertainty value. Finally, per patient volume each uncertainty threshold is evaluated by computing a coverage and a risk measure. Coverage is the percentage of voxels in a patient volume at ED or ES that is automatically segmented. Voxels in a patient volume above the threshold are discarded from automatic segmentation and would be referred to an expert. The number of incorrectly segmented voxels per patient volume is used as a measure of risk. Using bilinear interpolation risk measures are computed per patient volume between $[0, 100]$ percent.\n\t\n\t\n\t\\subsection*{Detection of segmentation failures}\n\t\n\tTo detect segmentation failures uncertainty maps are used but direct application of uncertainties is infeasible because many correctly segmented voxels, such as those close to anatomical structure boundaries, have high uncertainty. Hence, an additional patch-based CNN (detection network) is used that takes a cardiac MR image together with the corresponding spatial uncertainty map as input. For each patch of \\num{8}$\\times$\\num{8} voxels the network generates a probability indicating whether it contains segmentation failure. In the following, the terms patch and region are used interchangeably.\n\t\n\tThe detection network is a shallow Residual Network (S-ResNet) \\cite{he2016deep} consisting of a feature extraction module with output stride eight followed by a classifier indicating the presence of segmentation failure. The first level of the feature extraction module consists of two convolutional layers. The first layer uses \\num{16} kernels of \\num{7}$\\times$\\num{7} voxels and zero-padding of size \\num{3} and second layer \\num{32} kernels of \\num{3}$\\times$\\num{3} voxels and zero-padding of \\num{1} voxel. Level \\num{2} to \\num{4} each consist of one residual block that contains two convolutional layers with \\num{3}$\\times$\\num{3} voxels kernels with zero-padding of size \\num{1}. The first convolutional layer of each residual block uses a strided convolution of \\num{2} voxels to downsample the input. All convolutional layers of the feature extraction module are followed by batch normalization and ReLU function. The number of kernels in the feature extraction module increases from \\num{16} in level \\num{1} to \\num{128} in level \\num{4}. The network is a \\num{2}D patch-level classifier and requires that the size of the two input slices is a multiple of the patch-size. \\label{patch_size} The final classifier consists of three fully convolutional layers, implemented as \\num{1}$\\times$\\num{1} convolutions, with \\num{128} feature maps in the first two layers. The final layer has two channels followed by a softmax function which indicates whether the patch contains segmentation failure. Furthermore, to regularize the model dropout layers ($p=0.5$) were added between the residual blocks and the fully convolutional layers of the classifier.\n\t\t\n\t\\begin{table*}\n\t\t\\caption{Segmentation performance of different combination of model architectures, loss functions and evaluation modes (without or with MC dropout enabled during testing) in terms of Dice coefficient (top) and Hausdorff distance (bottom) (mean $\\pm$ standard deviation). Each combination comprises a block of two rows. A row in which column \\textit{Uncertainty map for detection} indicates e- or b-map shows results for the combined segmentation and detection approach. Numbers accentuated in black\/bold are ranked first in the segmentation only task whereas numbers accentuated in red\/bold are ranked first in the combined segmentation \\& detection task. The last row states the performance of the winning model in the ACDC challenge (on \\num{100} patient images) \\cite{isensee2017automatic}. Number with asterisk indicates statistical significant at \\num{5}\\% level w.r.t. the segmentation-only approach. Best viewed in color.}\n\t\t\\label{table_overall_segmentation_performance}\n\t\t\\centering\n\t\t\\tiny\n\t\t\\subfloat[Dice coefficient]{\n\t\t\t\\begin{tabular}{| C{1.6cm} | C{0.8cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\t\\hline\n\t\t\t\t& \\textbf{Uncertainty} & \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\t\\textbf{Model} & \\textbf{map for detection} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\t\\hline\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDN-Brier &  & \\phantom{x}0.962$\\pm$0.02 & \\phantom{x}0.928$\\pm$0.04) & \\phantom{x}0.875$\\pm$0.03)  & \\phantom{x}0.901$\\pm$0.11 & \\phantom{x}0.832$\\pm$0.10) & \\phantom{x}0.884$\\pm$0.04 \\\\\n\t\t\t\t& \\textbf{e-map} & *0.965$\\pm$0.01 & *0.949$\\pm$0.02 & *0.885$\\pm$0.03  & *0.937$\\pm$0.06 & *0.905$\\pm$0.05 & *0.909$\\pm$0.03 \\\\ \n\t\t\t\t\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDN-Brier+MC  & & \\phantom{x}0.961$\\pm$0.02 & \\phantom{x}0.922$\\pm$0.04 & \\phantom{x}0.875$\\pm$0.04 & \\phantom{x}0.912$\\pm$0.08  & \\phantom{x}0.839$\\pm$0.11 & \\phantom{x}0.882$\\pm$0.04  \\\\ \n\t\t\t\t& \\textbf{b-map} & *0.966$\\pm$0.01 & *0.950$\\pm$0.01 & *0.886$\\pm$0.03  & \\textbf{\\textcolor{red}{*0.942}}$\\pm$0.03 & *0.916$\\pm$0.04 & *0.912$\\pm$0.03 \\\\ \n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDN-soft-Dice  & & \\phantom{x}0.960$\\pm$0.02 & \\phantom{x}0.921$\\pm$0.04 & \\phantom{x}0.870$\\pm$0.04 &  \\phantom{x}0.909$\\pm$0.08 & \\phantom{x}0.812$\\pm$0.12 & \\phantom{x}0.879$\\pm$0.04 \\\\\n\t\t\t\t& \\textbf{e-map} & *0.965$\\pm$0.01 & *0.945$\\pm$0.02 & *0.879$\\pm$0.04  & *0.938$\\pm$0.03 & *0.891$\\pm$0.06 & *0.905$\\pm$0.03 \\\\ \n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDN-soft-Dice+MC & & \\phantom{x}0.958$\\pm$0.02 & \\phantom{x}0.913$\\pm$0.05  & \\phantom{x}0.868$\\pm$0.04 & \\phantom{x}0.907$\\pm$0.07  & \\phantom{x}0.818$\\pm$0.12 & \\phantom{x}0.875$\\pm$0.04  \\\\\n\t\t\t\t& \\textbf{b-map} & *0.964$\\pm$0.01 & *0.944$\\pm$0.02 & *0.877$\\pm$0.04  & *0.939$\\pm$0.03 & *0.900$\\pm$0.05 & *0.904$\\pm$0.03 \\\\  \t\t\t\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDRN-CE &  & \\phantom{x}0.961$\\pm$0.02 & \\phantom{x}0.929$\\pm$0.03 & \\phantom{x}0.878$\\pm$0.03  & \\phantom{x}0.912$\\pm$0.06 & \\phantom{x}0.850$\\pm$0.09 & \\phantom{x}0.891$\\pm$0.03 \\\\  \n\t\t\t\t& \\textbf{e-map} & \\phantom{x}0.964$\\pm$0.01 & *0.943$\\pm$0.02 & *0.886$\\pm$0.03  & *0.937$\\pm$0.03 & *0.899$\\pm$0.04 & *0.908$\\pm$0.03 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDRN-CE+MC &  & \\phantom{x}0.961$\\pm$0.02 & \\phantom{x}0.926$\\pm$0.03 & \\phantom{x}0.877$\\pm$0.03  & \\phantom{x}0.913$\\pm$0.06 & \\phantom{x}0.847$\\pm$0.10 & \\phantom{x}0.890$\\pm$0.03 \\\\  \n\t\t\t\t& \\textbf{b-map} & *0.965$\\pm$0.01 & *0.948$\\pm$0.01 & *0.887$\\pm$0.03  & *0.939$\\pm$0.03 & *0.911$\\pm$0.04 & *0.909$\\pm$0.03 \\\\ \n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDRN-soft-Dice &  & \\phantom{x}0.964$\\pm$0.01 & \\phantom{x}\\textbf{0.937}$\\pm$0.02 & \\phantom{x}0.888$\\pm$0.03  & \\phantom{x}\\textbf{0.919}$\\pm$0.06 & \\phantom{x}0.856$\\pm$0.09 & \\phantom{x}\\textbf{0.900}$\\pm$0.03 \\\\ \n\t\t\t\t& \\textbf{e-map} & \\phantom{x}0.967$\\pm$0.01 & *0.945$\\pm$0.02 & \\phantom{x}0.893$\\pm$0.03  & \\phantom{x}0.934$\\pm$0.04 & *0.892$\\pm$0.06 & *0.911$\\pm$0.03 \\\\ \n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDRN-soft-Dice+MC &  & \\phantom{x}0.963$\\pm$0.02 & \\phantom{x}0.935$\\pm$0.03 & \\phantom{x}0.886$\\pm$0.03  & \\phantom{x}0.921$\\pm$0.06 & \\phantom{x}\\textbf{0.857}$\\pm$0.09 & \\phantom{x}0.899$\\pm$0.03 \\\\ \n\t\t\t\t& \\textbf{b-map} & \\phantom{x}0.967$\\pm$0.01 & *0.947$\\pm$0.02 & \\phantom{x}0.893$\\pm$0.03  & *0.938$\\pm$0.03 & *0.907$\\pm$0.04 & *0.912$\\pm$0.03 \\\\\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tU-net-CE &  & \\phantom{x}0.962$\\pm$0.02 & \\phantom{x}0.923$\\pm$0.05 & \\phantom{x}0.878$\\pm$0.03  & \\phantom{x}0.907$\\pm$0.07 & \\phantom{x}0.840$\\pm$0.08 & \\phantom{x}0.885$\\pm$0.03 \\\\ \n\t\t\t\t& \\textbf{e-map} & \\phantom{x}0.966$\\pm$0.01 & *0.946$\\pm$0.02 & *0.890$\\pm$0.03  & *0.935$\\pm$0.04 & *0.901$\\pm$0.06 & *0.909$\\pm$0.03 \\\\ \n\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tU-net-CE+MC &  & \\phantom{x}0.962$\\pm$0.02 & \\phantom{x}0.926$\\pm$0.04 & \\phantom{x}0.879$\\pm$0.03  & \\phantom{x}0.909$\\pm$0.07 & \\phantom{x}0.849$\\pm$0.07 & \\phantom{x}0.887$\\pm$0.03 \\\\ \n\t\t\t\t& \\textbf{b-map} & \\phantom{x}0.967$\\pm$0.01 & \\textbf{\\textcolor{red}{*0.954}}$\\pm$0.02 & *0.893$\\pm$0.03  & *0.940$\\pm$0.04 & \\textbf{\\textcolor{red}{*0.920}}$\\pm$0.04 & \\textbf{\\textcolor{red}{*0.914}}$\\pm$0.03 \\\\\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tU-net-soft-Dice &  & \\phantom{x}\\textbf{0.965}$\\pm$0.02 & \\phantom{x}0.928$\\pm$0.04 & \\phantom{x}0.888$\\pm$0.03  & \\phantom{x}0.914$\\pm$0.08 & \\phantom{x}0.844$\\pm$0.09 & \\phantom{x}0.896$\\pm$0.03 \\\\ \n\t\t\t\t& \\textbf{e-map} & \\phantom{x}0.968$\\pm$0.01 & *0.943$\\pm$0.03 & *0.898$\\pm$0.03  & \\phantom{x}0.930$\\pm$0.05 & *0.886$\\pm$0.07 & *0.911$\\pm$0.03 \\\\ \n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tU-net-soft-Dice+MC &  & \\phantom{x}\\textbf{0.965}$\\pm$0.02 & \\phantom{x}0.929$\\pm$0.04 & \\phantom{x}\\textbf{0.889}$\\pm$0.03  & \\phantom{x}0.911$\\pm$0.10 & \\phantom{x}0.845$\\pm$0.09 & \\phantom{x}0.897$\\pm$0.03 \\\\ \n\t\t\t\t& \\textbf{b-map} & \\phantom{x}\\textbf{\\textcolor{red}{0.968}}$\\pm$0.01 & *0.948$\\pm$0.03 & \\textbf{\\textcolor{red}{*0.900}}$\\pm$0.03  & \\phantom{x}0.928$\\pm$0.09 & *0.895$\\pm$0.06 & \\textbf{\\textcolor{red}{*0.914}}$\\pm$0.03 \\\\\n\t\t\t\t\n\t\t\t\t\\hdashline\n\t\t\t\tIsensee et al. & & \\phantom{x}0.966& \\phantom{x}0.941 & \\phantom{x}0.899 & \\phantom{x}0.924 & \\phantom{x}0.875\t& \\phantom{x}0.908  \\\\\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\t\\label{table_seg_perf_dsc}\n\t\t} \n\t\t\\vspace{13ex}\n\t\t\\centering\n\t\t\\tiny\n\t\t\\subfloat[Hausdorff Distance]{\n\t\t\t\\begin{tabular}{| C{1.6cm} | C{0.8cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\t\\hline\n\t\t\t\t& \\textbf{Uncertainty}& \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\t\\textbf{Model} & \\textbf{map for detection} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\t\\hline\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDN-Brier &  &  \\phantom{x}6.7$\\pm$3.1 & \\phantom{x}13.5$\\pm$5.9 & \\phantom{x}10.2$\\pm$6.9 & \\phantom{x}10.7$\\pm$7.7 & \\phantom{x}16.7$\\pm$6.8 & \\phantom{x}12.3$\\pm$5.8 \\\\\n\t\t\t\t& \\textbf{e-map} &  *5.7$\\pm$2.7 & *11.7$\\pm$5.2 & *\\phantom{x}8.3$\\pm$5.9 & *\\phantom{x}8.0$\\pm$6.5 & *14.2$\\pm$5.6 & *\\phantom{x}9.7$\\pm$5.0 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDN-Brier+MC &  &  \\phantom{x}6.9$\\pm$3.3 & \\phantom{x}13.1$\\pm$5.2 & \\phantom{xx}9.9$\\pm$5.9 & \\phantom{xx}9.9$\\pm$5.7 & \\phantom{x}15.0$\\pm$6.1 & \\phantom{x}12.0$\\pm$5.2 \\\\\n\t\t\t\t& \\textbf{b-map} &  *5.5$\\pm$2.6 & *10.6$\\pm$5.1 & *\\phantom{x}7.4$\\pm$4.2 & *\\phantom{x}7.5$\\pm$6.0 & *12.6$\\pm$5.6 & *\\phantom{x}8.8$\\pm$4.0 \\\\\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDN-soft-Dice &  &  \\phantom{x}7.1$\\pm$3.5 & \\phantom{x}14.8$\\pm$6.8 & \\phantom{x}11.0$\\pm$6.6 & \\phantom{x}10.2$\\pm$5.6 & \\phantom{x}17.7$\\pm$7.8 & \\phantom{x}12.9$\\pm$6.2 \\\\\n\t\t\t\t& \\textbf{e-map} &  *5.6$\\pm$2.8 & *12.6$\\pm$5.5 & *\\phantom{x}8.6$\\pm$4.6 & *\\phantom{x}8.0$\\pm$5.0 & *14.6$\\pm$5.9 & *\\phantom{x}9.6$\\pm$4.5 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDN-soft-Dice+MC &  &  \\phantom{x}7.7$\\pm$3.9 & \\phantom{x}14.4$\\pm$6.0 & \\phantom{x}10.5$\\pm$4.9 & \\phantom{x}10.1$\\pm$5.3 & \\phantom{x}17.2$\\pm$8.0 & \\phantom{x}12.5$\\pm$5.3 \\\\\n\t\t\t\t& \\textbf{b-map} &  *6.3$\\pm$3.4 & *11.5$\\pm$4.0 & *\\phantom{x}8.6$\\pm$4.8 & *\\phantom{x}7.8$\\pm$4.6 & *13.6$\\pm$4.9 & *\\phantom{x}9.6$\\pm$4.7 \\\\\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDRN-CE &  &  \\phantom{x}5.5$\\pm$2.6 & \\phantom{x}11.7$\\pm$5.4 & \\phantom{xx}8.2$\\pm$6.2 & \\phantom{xx}9.1$\\pm$6.4 & \\phantom{x}13.7$\\pm$5.6 & \\phantom{xx}8.9$\\pm$5.3 \\\\\n\t\t\t\t& \\textbf{e-map} &  *4.5$\\pm$1.9 & *\\phantom{x}9.0$\\pm$4.5 & *\\phantom{x}6.3$\\pm$4.1 & *\\phantom{x}6.2$\\pm$4.4 & *11.1$\\pm$5.3 & \\textbf{\\textcolor{red}{*\\phantom{x}6.7}}$\\pm$4.2 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDRN-CE+MC &  &  \\phantom{x}5.6$\\pm$2.6 & \\phantom{x}11.9$\\pm$5.5 & \\phantom{xx}8.0$\\pm$5.9 & \\phantom{xx}8.7$\\pm$5.5 & \\phantom{x}13.5$\\pm$5.9 & \\phantom{xx}\\textbf{8.5}$\\pm$4.5 \\\\\t\t \n\t\t\t\t& \\textbf{b-map} &  \\textbf{\\textcolor{red}{*4.2}}$\\pm$1.6 & \\textbf{\\textcolor{red}{*\\phantom{x}8.1}}$\\pm$3.7 & \\textbf{\\textcolor{red}{*\\phantom{x}6.1}}$\\pm$4.2 & \\textbf{\\textcolor{red}{*\\phantom{x}5.4}}$\\pm$3.6 & \\textbf{\\textcolor{red}{*10.1}}$\\pm$5.5 & *\\phantom{x}6.8$\\pm$3.8 \\\\\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tDRN-soft-Dice &  &  \\phantom{x}\\textbf{5.5}$\\pm$2.8 & \\phantom{x}11.9$\\pm$6.1 & \\phantom{xx}\\textbf{7.7}$\\pm$5.9 & \\phantom{xx}8.5$\\pm$5.0 & \\phantom{x}13.5$\\pm$5.5 & \\phantom{xx}8.9$\\pm$5.1 \\\\\n\t\t\t\t& \\textbf{e-map} &  *4.6$\\pm$2.2 & *\\phantom{x}9.4$\\pm$4.5 & \\phantom{xx}6.7$\\pm$4.7 & *\\phantom{x}6.7$\\pm$4.4 & *11.6$\\pm$5.4 & *\\phantom{x}7.0$\\pm$3.3 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tDRN-soft-Dice+MC &  &  \\phantom{x}5.7$\\pm$3.2 & \\phantom{x}\\textbf{11.5}$\\pm$5.1 & \\phantom{xx}8.0$\\pm$5.5 & \\phantom{xx}\\textbf{8.3}$\\pm$4.5 & \\phantom{x}\\textbf{13.3}$\\pm$5.1 & \\phantom{xx}8.9$\\pm$5.1 \\\\\n\t\t\t\t& \\textbf{b-map} &  *4.5$\\pm$2.2 & *\\phantom{x}9.3$\\pm$4.5 & *\\phantom{x}6.3$\\pm$4.0 & *\\phantom{x}6.2$\\pm$4.1 & *10.4$\\pm$5.0 & *\\phantom{x}7.0$\\pm$3.4 \\\\\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tU-net-CE &  &  \\phantom{x}6.4$\\pm$4.3 & \\phantom{x}15.7$\\pm$8.6 & \\phantom{xx}9.0$\\pm$6.0 & \\phantom{xx}9.7$\\pm$5.3 & \\phantom{x}17.0$\\pm$7.7 & \\phantom{x}12.7$\\pm$8.2 \\\\\n\t\t\t\t& \\textbf{e-map} &  *4.9$\\pm$3.9 & *12.2$\\pm$8.1 & *\\phantom{x}7.1$\\pm$5.6 & *\\phantom{x}6.1$\\pm$3.2 & *12.6$\\pm$6.5 & *\\phantom{x}8.4$\\pm$6.3 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tU-net-CE+MC &  &  \\phantom{x}6.2$\\pm$4.2 & \\phantom{x}15.3$\\pm$8.4 & \\phantom{xx}8.8$\\pm$5.8 & \\phantom{xx}9.2$\\pm$5.0 & \\phantom{x}16.5$\\pm$7.6 & \\phantom{x}12.0$\\pm$8.0 \\\\\n\t\t\t\t& \\textbf{b-map} &  *4.3$\\pm$1.6 & *\\phantom{x}9.9$\\pm$6.6 & *\\phantom{x}6.7$\\pm$4.8 & *\\phantom{x}5.4$\\pm$2.8 & *10.3$\\pm$4.7 & *\\phantom{x}7.6$\\pm$6.2 \\\\\n\t\t\t\t\n\t\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightGreen}\n\t\t\t\tU-net-soft-Dice &  &  \\phantom{x}6.1$\\pm$3.9 & \\phantom{x}14.1$\\pm$7.6 & \\phantom{x}10.6$\\pm$8.4 & \\phantom{xx}9.2$\\pm$7.1 & \\phantom{x}16.3$\\pm$7.5 & \\phantom{x}12.6$\\pm$9.6 \\\\\n\t\t\t\t& \\textbf{e-map} &  *4.6$\\pm$2.3 & *11.3$\\pm$7.2 & *\\phantom{x}7.5$\\pm$5.5 & *\\phantom{x}7.3$\\pm$6.5 & *13.7$\\pm$7.6 & *\\phantom{x}9.8$\\pm$8.0 \\\\\n\t\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\t\n\t\t\t\t\\rowcolor{LightCyan}\n\t\t\t\tU-net-soft-Dice+MC &  &  \\phantom{x}6.2$\\pm$3.9 & \\phantom{x}14.1$\\pm$7.7 & \\phantom{x}10.5$\\pm$8.7 & \\phantom{xx}9.0$\\pm$7.0 & \\phantom{x}15.8$\\pm$7.5 & \\phantom{x}12.1$\\pm$9.2 \\\\\n\t\t\t\t& \\textbf{b-map} &  *4.5$\\pm$2.1 & *10.4$\\pm$7.2 & *\\phantom{x}7.6$\\pm$7.0 & *\\phantom{x}7.3$\\pm$6.9 & *12.9$\\pm$6.6 & *\\phantom{x}9.8$\\pm$8.4 \\\\\n\t\t\t\t\\hdashline\n\t\t\t\tIsensee et al. &  & \\phantom{x}7.1 &  \\phantom{x}14.3 & \\phantom{xx}8.9 &  \\phantom{xx}9.8 &  \\phantom{x}16.3 & \\phantom{x}10.4 \\\\\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\t\\label{table_seg_perf_hd}\n\t\t}  \n\t\t\n\t\\end{table*} \n\t\n\t\n\t\\section*{Evaluation}\\label{evaluation}\n\t\t\n\tAutomatic segmentation performance, as well as performance after simulating the correction of detected segmentation failures and after manual expert correction was evaluated. For this, the \\num{3}D Dice-coefficient (DC) and \\num{3}D Hausdorff distance (HD) between manual and (corrected) automatic segmentation were computed. Furthermore, the following clinical metrics were computed for manual and (corrected) automatic segmentation: left ventricle end-diastolic volume (EDV); left ventricle ejection fraction (EF); right ventricle EDV; right ventricle ejection fraction; and left ventricle myocardial mass. Following Bernard et al.\\cite{bernard2018deep} for each of the clinical metrics three performance indices were computed using the measurements based on manual and (corrected) automatic segmentation: Pearson correlation coefficient; mean difference (bias and standard deviation); and mean absolute error (MAE).\n\n\tTo evaluate detection performance of the automatic method precision-recall curves of identification of slices that require correction were computed. A slice is considered positive in case it consists of at least one image region with a segmentation failure. To achieve accurate segmentation in clinic, identification of slices that contain segmentation failures might ease manual correction of automatic segmentations in daily practice. To further evaluate detection performance detection rate of segmentation failures was assessed on a voxel level. More specific, sensitivity against the number of false positive regions was evaluated because manual correction is presumed to be performed at this level.\n\t\n\tFinally, after simulation and manual correction of the automatically detected segmentation failures, segmentation was re-evaluated and significance of the difference between the DCs, HDs and clinical metrics was tested with a Mann\u2013Whitney U test.\n\t\n\n\t\n\t\\section*{Experiments}\n\t\n\tTo use stratified four-fold cross-validation the dataset was split into training (75\\%) and test (25\\%) set. The splitting was done on a patient level, so there was no overlap in patient data between training and test sets. Furthermore, patients were randomly chosen from each of the five patient groups w.r.t. disease. Each patient has one volume for ED and ES time points, respectively.  \n\t\n\t\\subsection*{Training segmentation networks} \\label{training_segmentation}\n\t\n\tDRN and U-net were trained with a patch size of \\num{128}$\\times$\\num{128} voxels which is a multiple of their output stride of the contracting path. In the training of the dilated CNN (DN) images with \\num{151}$\\times$\\num{151} voxel samples were used. Zero-padding to \\num{281}$\\times$\\num{281} was performed to accommodate the \\num{131}$\\times$\\num{131} voxel receptive field that is induced by the dilation factors. Training samples were randomly chosen from training set and augmented by \\num{90} degree rotations of the images. All models were initially trained with three loss functions: soft-Dice\\cite{milletari2016v} (SD); cross-entropy (CE); and Brier loss\\cite{brier1950verification}. However, for the evaluation of the combined segmentation and detection approach for each model architecture the two best performing loss functions were chosen: soft-Dice for all models; cross-entropy for DRN and U-net and Brier loss for DN. For completeness, we provide the equations for all three used loss functions.\n\t\n\t\n\t\\begingroup\n\t\\small\n\t\\begin{align}\n\t\t\\text{soft-Dice}_{c} = \\frac{\\sum_{i=1}^{N} R_{c}(i) \\; A_{c}(i) }{\\sum_{i=1}^{N} R_{c}(i) + \\sum_{i=1}^{N} A_{c}(i)}  \\; ,\n\t\\end{align}\n\t\\endgroup\n\twhere $N$ denotes the number of voxels in an image, $R_{c}$ is the binary reference image for class $c$ and $A_{c}$ is the probability map for class $c$.\n\t\n\t\\begingroup\n\t\\small\n\t\\begin{align}\n\t\t\\text{Cross-Entropy}_{c} = - \\; \\sum_{i=1}^{N} t_{ic} \\; \\log \\; p(y_i=c|x_i) \\; , \\text{ where } t_{ic} = 1 \\text{ if } y_{i}=c, \\text{ and \\num{0} otherwise.}\n\t\\end{align}\n\t\\endgroup\n\t\n\t\\begingroup\n\t\\small\n\t\\begin{align}\n\t\t\\text{Brier}_{c} = \\sum_{i=1}^{N}  \\big(t_{ic} - p(y_i=c|x_{i}) \\big)^2 \\; , \\text{ where } t_{ic} = 1 \\text{ if } y_{i}=c, \\text{ and \\num{0} otherwise.}\n\t\\end{align}\n\t\\endgroup\n\t\n\twhere $N$ denotes the number of voxels in an image and $p$ denotes the probability for a specific voxel $x_i$ with corresponding reference label $y_i$ for class $c$.\n\t\n\tChoosing Brier loss to train the DN model instead of CE was motivated by our preliminary work which showed that segmentation performance of DN model was best when trained with Brier loss\\cite{sander2019towards}.\n\t\n\tAll models were trained for 100,000 iterations. DRN and U-net were trained with a learning rate of \\num{0.001} which decayed with a factor of \\num{0.1} after every 25,000 steps. Training DN used the snapshot ensemble technique~\\cite{huang2017snapshot}, where after every 10,000 iterations the learning rate was reset to its original value of \\num{0.02}.\n\t\n\tAll three segmentation networks were trained using mini-batch stochastic gradient descent using a batch size of \\num{16}. Network parameters were optimized using the Adam optimizer \\cite{kingmadp}. Furthermore, models were regularized with weight decay to increase generalization performance. \n\t\n\t\\subsection*{Training detection network}\\label{label_training_detection}\n\t\n\tTo train the detection model a subset of the errors performed by the segmentation model is used. Segmentation errors that presumably are within the range of inter-observer variability and therefore do not inevitably require correction (tolerated errors) are excluded from the set of errors that need to be detected and corrected (segmentation failures). To distinguish between tolerated errors and the set of segmentation failures $\\mathcal{S}_I$ the Euclidean distance of an incorrectly segmented voxel to the boundary of the reference target structure is used. For each anatomical structure a \\num{2}D distance transform map is computed that provides for each voxel the distance to the anatomical structure boundary. To differentiate between tolerated errors and the set of segmentation failures $\\mathcal{S}_I$ an acceptable tolerance threshold is applied. A more rigorous threshold is used for errors located inside compared to outside of the anatomical structure because automatic segmentation methods have a tendency to undersegment cardiac structures in CMRI. Hence, in all experiments the acceptable tolerance threshold was set to three voxels (equivalent to on average \\SI{4.65}{\\milli\\meter}) and two voxels (equivalent to on average \\SI{3.1}{\\milli\\meter}) for segmentation errors located outside and inside the target structure. Furthermore, a segmentation error only belongs to $\\mathcal{S}_I$ if it is part of a \\num{2}D \\num{4}-connected cluster of minimum size \\num{10} voxels. This value was found in preliminary experiments by evaluating values $\\{1, 5, 10, 15, 20\\}$. However, for apical slices all segmentation errors are included in $\\mathcal{S}_I$ regardless of fulfilling the minimum size requirement because in these slices anatomical structures are relatively small and manual segmentation is prone to large inter-observer variability~\\cite{bernard2018deep}. Finally, segmentation errors located in slices above the base or below the apex are always included in the set of segmentation failures.\n\t\n\tUsing the set $\\mathcal{S}_I$ a binary label $t_j$ is assigned to each patch $P_j^{(I)}$ indicating whether $P_j^{(I)}$ contains at least one voxel belonging to set $\\mathcal{S}_I$ where $j \\in \\{1 \\dots M \\}$ and $M$ denotes the number of patches in a slice $I$.  \n\t\n\tThe detection network is trained by minimizing a weighted binary cross-entropy loss:\n\t\n\t\\begingroup\n\t\\small\n\t\\begin{equation} \\label{eq_detection_loss}\n\t\t\\mathcal{L}_{DT} = - \\sum_{j \\in P^{(I)}} w_{pos} \\; t_j \\log p_j + (1 - t_j) \\log (1 - p_j)  \\; ,\n\t\\end{equation}\n\t\\endgroup\n\t\n\twhere $w_{pos}$ represents a scalar weight, $t_j$ denotes the binary reference label and $p_j$ is the softmax probability indicating whether a particular image region $P_j^{(I)}$ contains at least one segmentation failure. The average percentage of regions in a patient volume containing segmentation failures ranges from \\num{1.5} to \\num{3} percent depending on the segmentation architecture and loss function used to train the segmentation model. To train a detection network $w_{pos}$ was set to the ratio between the average percentage of negative samples divided by the average percentage of positive samples.\n\t\n\tEach fold was trained using spatial uncertainty maps and automatic segmentation masks generated while training the segmentation networks. Hence, there was no overlap in patient data between training and test set across segmentation and detection tasks. In total \\num{12} detection models were trained and evaluated resulting from the different combination of \\num{3} model architectures (DRN, DN and U-net), \\num{2} loss functions (DRN and U-net with CE and soft-Dice, DN with Brier and soft-Dice) and \\num{2} uncertainty maps (e-maps, b-maps).\n\t\n\t\n\t\\begin{table}\n\t\t\\caption{Segmentation performance of different combination of model architectures, loss functions and evaluation modes (without or with MC dropout (MC) enabled during testing) in terms of clinical metrics: left ventricle (LV) end-diastolic volume (EDV); LV ejection fraction (EF); right ventricle (RV) EDV; RV ejection fraction; and LV myocardial mass. Quantitative results compare clinical metrics based on reference segmentations with 1) automatic segmentations and 2) simulated manual correction of automatic segmentations using spatial uncertainty maps. $\\rho$ denotes the Pearson correlation coefficient, \\textit{bias} denotes the mean difference between the two measurements (mean $\\pm$ standard deviation) and \\textit{MAE} denotes the mean absolute error between the two measurements. Each combination comprises a block of two rows. A row in which column \\textit{Uncertainty map for detection} indicates e- or b-map shows results for the combined segmentation and detection approach. Numbers accentuated in black\/bold are ranked first in the segmentation only task. Numbers in red indicate statistical significant at \\num{5}\\% level w.r.t. the segmentation-only approach for the specific clinical metric. Best viewed in color.}\n\t\t\\label{table_cardiac_function_indices}\n\t\t\\tiny\n\t\t\\begin{tabular}{| C{1.6cm} | C{1.cm} | C{0.3cm} c C{0.3cm} | C{0.2cm} C{0.8cm} C{0.3cm} | C{0.3cm} C{0.8cm} C{0.3cm} | C{0.3cm} C{0.8cm} C{0.3cm} | C{0.3cm} C{0.8cm} C{0.3cm} |}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{1}{c|}{\\thead{\\textbf{Uncertainty} \\\\ \\textbf{map for} \\\\ \\textbf{detection}}}  & \\multicolumn{3}{c}{\\textbf{LV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{LV$_{EF}$}} & \\multicolumn{3}{c}{\\textbf{RV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{RV$_{EF}$}} & \\multicolumn{3}{c|}{\\textbf{LVM$_{Mass}$}}  \\\\\n\t\t\n\t\t\n\t\t\t\\textbf{Method} &  & \\textbf{$\\rho$} & \\multicolumn{1}{l}{\\textbf{bias$\\pm\\sigma$}} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} & \\multicolumn{1}{c}{\\textbf{$\\rho$}} & \\textbf{bias$\\pm \\sigma$} & \\textbf{MAE} \\\\\n\t\t\t\\hline\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDN-Brier &  & 0.997 & \\phantom{x}\\textbf{0.0$\\pm$6.1} & 4.5 & 0.892 & \\phantom{x}2.2$\\pm$\\phantom{x}9.2 & 4.2 & 0.977 & \\textbf{-0.2$\\pm$11.8} & \\phantom{x}8.5 & 0.834 & 5.3$\\pm$10.3 & \\phantom{x}8.5 & 0.984 & -2.7$\\pm$\\phantom{x}9.0 & \\phantom{x}\\textbf{7.0} \\\\\n\t\t\t& e-map  & 0.997 & \\phantom{x}0.0$\\pm$5.5 & 4.0 & 0.982 & \\phantom{x}0.1$\\pm$\\phantom{x}3.8 & 2.2 & 0.992 & \\phantom{x}0.0$\\pm$\\phantom{x}6.9 & \\phantom{x}5.2 & 0.955 & 1.9$\\pm$\\phantom{x}5.5 & \\phantom{x}4.1 & 0.986 & -2.1$\\pm$\\phantom{x}8.4 & \\phantom{x}6.6 \\\\\n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDN-Brier+MC &  & 0.997 & \\phantom{x}1.6$\\pm$6.0 & 4.4 & 0.921 & \\phantom{x}1.1$\\pm$\\phantom{x}7.9 & 3.9 & 0.975 & \\phantom{x}6.7$\\pm$12.4 & \\phantom{x}9.6 & 0.854 & 3.5$\\pm$\\phantom{x}9.9 & \\phantom{x}7.7 & 0.984 & \\phantom{x}0.7$\\pm$\\phantom{x}9.2 & \\phantom{x}7.1 \\\\\n\t\t\t& b-map  & 0.998 & \\phantom{x}1.0$\\pm$5.3 & 3.9 & 0.991 & \\phantom{x}0.0$\\pm$\\phantom{x}2.7 & 1.9 & 0.993 & \\phantom{x}3.2$\\pm$\\phantom{x}6.7 & \\phantom{x}5.7 & 0.975 & 0.8$\\pm$\\phantom{x}4.0 & \\phantom{x}3.0 & 0.987 & \\phantom{x}0.1$\\pm$\\phantom{x}8.3 & \\phantom{x}6.5 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDN-soft-Dice &  & 0.996 & \\phantom{x}1.2$\\pm$6.5 & 4.9 & 0.918 & \\phantom{x}1.5$\\pm$\\phantom{x}8.0 & 3.9 & 0.972 & \\phantom{x}\\textbf{0.2$\\pm$13.0} & \\phantom{x}9.6 & 0.802 & 7.2$\\pm$11.3 & 10.2 & 0.982 & -4.5$\\pm$\\phantom{x}9.6 & \\phantom{x}8.5 \\\\\n\t\t\t& e-map  & 0.997 & \\phantom{x}1.0$\\pm$5.5 & 4.2 & 0.989 & \\phantom{x}0.2$\\pm$\\phantom{x}3.0 & 2.2 & 0.990 & \\phantom{x}0.2$\\pm$\\phantom{x}7.6 & \\phantom{x}5.9 & \\textcolor{red}{0.940} & \\textcolor{red}{3.3$\\pm$\\phantom{x}6.2} & \\textcolor{red}{\\phantom{x}5.2} & 0.983 & -4.3$\\pm$\\phantom{x}9.3 & \\phantom{x}8.2 \\\\\n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDN-soft-Dice+MC &  & 0.996 & \\phantom{x}3.2$\\pm$7.1 & 5.6 & 0.958 & \\phantom{x}0.4$\\pm$\\phantom{x}5.7 & 3.6 & 0.964 & \\phantom{x}8.1$\\pm$14.9 & 12.3 & 0.827 & 4.8$\\pm$11.0 & \\phantom{x}8.9 & 0.978 & -0.7$\\pm$10.7 & \\phantom{x}8.3 \\\\\n\t\t\t& b-map  & 0.997 & \\phantom{x}2.2$\\pm$5.6 & 4.4 & 0.988 & -0.2$\\pm$\\phantom{x}3.1 & 2.2 & 0.990 & \\phantom{x}4.0$\\pm$\\phantom{x}7.7 & \\phantom{x}7.0 & 0.959 & 1.8$\\pm$\\phantom{x}5.1 & \\phantom{x}4.1 & 0.982 & -1.4$\\pm$\\phantom{x}9.5 & \\phantom{x}7.6 \\\\\n\t\t\t\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDRN-CE &  & 0.997 & -0.2$\\pm$5.5 & 4.1 & 0.968 & \\phantom{x}1.2$\\pm$\\phantom{x}5.0 & 3.5 & 0.976 & \\phantom{x}1.5$\\pm$12.1 & \\phantom{x}8.5 & 0.870 & 1.3$\\pm$\\phantom{x}9.2 & \\phantom{x}6.9 & 0.980 & \\phantom{x}\\textbf{0.6$\\pm$10.2} & \\phantom{x}7.8 \\\\\n\t\t\t& e-map  & 0.998 & \\phantom{x}0.2$\\pm$4.5 & 3.5 & 0.992 & \\phantom{x}0.2$\\pm$\\phantom{x}2.5 & 1.9 & 0.988 & \\phantom{x}1.4$\\pm$\\phantom{x}8.5 & \\phantom{x}6.2 & 0.952 & 0.8$\\pm$\\phantom{x}5.6 & \\phantom{x}4.2 & 0.985 & \\phantom{x}0.4$\\pm$\\phantom{x}8.7 & \\phantom{x}6.8 \\\\\n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDRN-CE+MC &  & \\textbf{0.998} & \\phantom{x}1.0$\\pm$4.9 & \\textbf{3.9} & 0.972 & \\phantom{x}0.8$\\pm$\\phantom{x}4.6 & 3.1 & 0.973 & \\phantom{x}4.8$\\pm$12.8 & \\phantom{x}9.4 & 0.876 & \\textbf{0.4$\\pm$\\phantom{x}9.1} & \\phantom{x}\\textbf{6.6} & 0.981 & \\phantom{x}1.9$\\pm$\\phantom{x}9.9 & \\phantom{x}7.6 \\\\\n\t\t\t& b-map  & 0.998 & \\phantom{x}0.7$\\pm$4.6 & 3.6 & 0.992 & -0.1$\\pm$\\phantom{x}2.5 & 1.8 & 0.992 & \\phantom{x}2.9$\\pm$\\phantom{x}6.9 & \\phantom{x}5.7 & 0.967 & 0.6$\\pm$\\phantom{x}4.6 & \\phantom{x}3.4 & 0.987 & \\phantom{x}1.2$\\pm$\\phantom{x}8.3 & \\phantom{x}6.6 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tDRN-soft-Dice &  & \\textbf{0.998} & \\phantom{x}0.8$\\pm$5.1 & 4.0 & 0.976 & \\phantom{x}0.2$\\pm$\\phantom{x}4.4 & 3.0 & 0.980 & \\phantom{x}\\textbf{0.2$\\pm$11.0} & \\phantom{x}\\textbf{7.5} & \\textbf{0.882} & 3.1$\\pm$\\phantom{x}8.7 & \\phantom{x}6.8 & 0.984 & -3.5$\\pm$\\phantom{x}9.1 & \\phantom{x}7.5 \\\\\n\t\t\t& e-map  & 0.998 & \\phantom{x}0.7$\\pm$4.4 & 3.5 & 0.987 & -0.1$\\pm$\\phantom{x}3.1 & 2.2 & 0.987 & \\phantom{x}0.1$\\pm$\\phantom{x}9.1 & \\phantom{x}6.4 & 0.938 & 1.9$\\pm$\\phantom{x}6.3 & \\phantom{x}4.9 & 0.986 & -3.5$\\pm$\\phantom{x}8.7 & \\phantom{x}7.1 \\\\\n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tDRN-soft-Dice+MC &   & \\textbf{0.998} & \\phantom{x}1.8$\\pm$5.1 & \\textbf{3.9} & \\textbf{0.979} & -0.3$\\pm$\\phantom{x}4.1 & 2.9 & 0.977 & \\phantom{x}3.5$\\pm$11.7 & \\phantom{x}8.1 & 0.868 & 1.7$\\pm$\\phantom{x}9.5 & \\phantom{x}6.8 & 0.983 & -1.4$\\pm$\\phantom{x}9.5 & \\phantom{x}7.4 \\\\\n\t\t\t&  b-map  & 0.998 & \\phantom{x}1.7$\\pm$4.7 & 3.7 & 0.990 & -0.2$\\pm$\\phantom{x}2.9 & 2.1 & 0.989 & \\phantom{x}2.3$\\pm$\\phantom{x}8.1 & \\phantom{x}5.8 & 0.959 & 0.8$\\pm$\\phantom{x}5.2 &\\phantom{x}3.8 & 0.986 & -1.3$\\pm$\\phantom{x}8.5 & \\phantom{x}6.8 \\\\\n\t\t\t\n\t\t\t\\hdashline[5pt\/5pt]\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tU-net-CE &  & 0.995 & -4.7$\\pm$7.2 & 6.1 & 0.954 & \\phantom{x}4.1$\\pm$\\phantom{x}6.0 & 5.1 & 0.963 & -7.6$\\pm$15.2 & 12.1 & 0.870 & 5.6$\\pm$\\phantom{x}9.0 & \\phantom{x}8.1 & 0.971 & -8.5$\\pm$12.2 & 11.5 \\\\\n\t\t\t& e-map  & 0.998 & -3.2$\\pm$4.8 & 4.4 & 0.992 & \\phantom{x}1.7$\\pm$\\phantom{x}2.6 & 2.4 & 0.987 & -4.1$\\pm$\\phantom{x}9.1 & \\phantom{x}6.7 & 0.957 & 2.6$\\pm$\\phantom{x}5.2 & \\phantom{x}4.1 & 0.983 & -5.7$\\pm$\\phantom{x}9.3 & \\phantom{x}8.2 \\\\\n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tU-net-CE+MC &   & 0.995 & -4.3$\\pm$7.2 & 5.9 & 0.958 & \\phantom{x}3.8$\\pm$\\phantom{x}5.8 & 4.9 & 0.968 & -4.8$\\pm$14.1 & 10.7 & 0.867 & 5.0$\\pm$\\phantom{x}9.1 & \\phantom{x}7.9 & 0.972 & -8.1$\\pm$12.0 & 11.1 \\\\\n\t\t\t& b-map  & 0.997 & -3.5$\\pm$5.5 & 4.9 & 0.990 & \\phantom{x}1.6$\\pm$\\phantom{x}2.9 & 2.6 & 0.992 & -1.8$\\pm$\\phantom{x}7.0 & \\phantom{x}4.9 & 0.974 & 1.6$\\pm$\\phantom{x}4.1 & \\phantom{x}3.3 & 0.981 & -6.8$\\pm$10.0 & \\phantom{x}9.4 \\\\\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\n\t\t\t\\rowcolor{LightGreen}\n\t\t\tU-net-soft-Dice &  & 0.997 & -2.0$\\pm$6.0 & 4.5 & 0.853 & \\phantom{x}3.6$\\pm$10.9 & 5.0 & 0.968 & -1.0$\\pm$14.1 & 10.0 & 0.782 & 4.8$\\pm$11.6 & \\phantom{x}9.0 & \\textbf{0.985} & -7.7$\\pm$\\phantom{x}8.8 & \\phantom{x}9.2 \\\\\n\t\t\t& e-map  & 0.997 & -1.7$\\pm$5.3 & 4.1 & 0.969 & \\phantom{x}1.9$\\pm$\\phantom{x}4.9 & 3.3 & 0.981 & -0.1$\\pm$10.9 & \\phantom{x}7.5 & 0.919 & 3.3$\\pm$\\phantom{x}7.0 & \\phantom{x}5.9 & 0.984 & -6.6$\\pm$\\phantom{x}9.0 & \\phantom{x}8.7 \\\\\n\t\t\t\n\t\t\t\\hdashline[1pt\/2pt]\n\t\t\t\\rowcolor{LightCyan}\n\t\t\tU-net-soft-Dice+MC &  & 0.997 & -1.8$\\pm$5.9 & 4.4 & 0.941 & \\phantom{x}3.0$\\pm$\\phantom{x}6.7 & 4.4 & 0.969 & \\phantom{x}0.6$\\pm$13.9 & \\phantom{x}9.8 & 0.792 & 4.4$\\pm$11.3 & \\phantom{x}8.7 & \\textbf{0.985} & -7.2$\\pm$\\phantom{x}8.9 & \\phantom{x}8.9 \\\\\n\t\t\t& b-map  & 0.997 & -1.5$\\pm$5.3 & 4.1 & 0.979 & \\phantom{x}1.1$\\pm$\\phantom{x}4.1 & 2.9 & 0.985 & \\phantom{x}1.2$\\pm$\\phantom{x}9.4 & \\phantom{x}6.5 & 0.939 & 2.9$\\pm$\\phantom{x}6.2 & \\phantom{x}4.9 & 0.984 & -5.9$\\pm$\\phantom{x}9.0 & \\phantom{x}8.5 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\n\tThe patches used to train the network were selected randomly (\\nicefrac{2}{3}), or were forced (\\nicefrac{1}{3}) to contain at least one segmentation failure by randomly selecting a scan containing segmentation failure, followed by random sampling of a patch containing at least one segmentation failure. During training the patch size was fixed to \\num{80}$\\times$\\num{80} voxels. To reduce the number of background voxels during testing, inputs were cropped based on a minimal enclosing, rectangular bounding box that was placed around the automatic segmentation mask. Inputs always had a minimum size of \\num{80}$\\times$\\num{80} voxels or were forced to a multiple of the output grid spacing of eight voxels in both direction required by the patch-based detection network. The patches of size \\num{8}$\\times$\\num{8} voxels did not overlap. In cases where the automatic segmentation mask only contains background voxels (scans above the base or below apex of the heart) input scans were center-cropped to a size of \\num{80}$\\times$\\num{80} voxels. \n\t\n\tModels were trained for 20,000 iterations using mini-batch stochastic gradient descent with batch-size \\num{32} and Adam as optimizer\\cite{kingmadp}. Learning rate was set to \\num{0.0001} and decayed with a factor of \\num{0.1} after \\num{10,000} steps. Furthermore, dropout percentage was set to \\num{0.5} and weight decay was applied to increase generalization performance.\n\t\n\t\\begin{figure}[t]\n\t\t\\center\n\t\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/all_models_froc_voxel_detection_rate.pdf}%\n\t\t\t\\label{fig_froc_voxel_detection}}\n\t\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/all_models_slices_prec_recall.pdf}%\n\t\t\t\\label{fig_prec_rec_slice_detection}}\n\t\t\n\t\t\\caption{Detection performance of segmentation failures generated by different combination of segmentation architectures and loss functions. (a) Sensitivity for detection of segmentation failures on voxel level (y-axis) as a function of number of false positive image regions (x-axis). (b) Precision-recall curve for detection of slices containing segmentation failures (where AP denotes average precision). Results are split between entropy and Bayesian uncertainty maps. Each figure contains a curve for the six possible combination of models (three) and loss functions (two). SD denotes soft-Dice and CE cross-entropy, respectively.}\n\t\t\\label{fig_dt_perf_all_models}\n\t\\end{figure}\n\n\t\\subsection*{Segmentation using correction of the detected segmentation failures}\n\t\n\tTo investigate whether correction of detected segmentation failures increases segmentation performance two scenarios were performed. In the first scenario manual correction of the detected failures by an expert was simulated for all images at ED and ES time points of the ACDC dataset. For this purpose, in image regions that were detected to contain segmentation failure predicted labels were replaced with reference labels. In the second scenario manual correction of the detected failures was performed by an expert in a random subset of \\num{50} patients of the ACDC dataset. The expert was shown CMRI slices for ED and ES time points together with corresponding automatic segmentation masks for the RV, LV and LV myocardium. Image regions detected to contain segmentation failures were indicated in slices and the expert was only allowed to change the automatic segmentations in these indicated regions. Annotation was performed following the protocol described in\\cite{bernard2018deep}. Furthermore, expert was able to navigate through all CMRI slices of the corresponding ED and ES volumes.\n\n\t\n\t\\section*{Results}\n\t\n\tIn this section we first present results for the segmentation-only task followed by description of the combined segmentation and detection results. \n\t\n\t\\subsection*{Segmentation-only approach} \\label{results_seg_only}\n\t\n\tTable~\\ref{table_overall_segmentation_performance} lists quantitative results for segmentation-only and combined segmentation and detection approach in terms of Dice coefficient and Hausdorff distance. These results show that DRN and U-net achieve similar Dice coefficients and outperformed the DN network for all anatomical structures at end-systole. Differences in the achieved Hausdorff distances among the methods are present for all anatomical structures and for both time points. The DRN model achieved the highest and the DN network the lowest Hausdorff distance.\n\t\n\tTable~\\ref{table_cardiac_function_indices} lists results of the evaluation in terms of clinical metrics. These results reveal noticeable differences between models for ejection fraction (EF) of left and right ventricle, respectively. We can observe that U-net trained with the soft-Dice and the Dilated Network (DN) trained with Brier or soft-Dice loss achieved considerable lower accuracy for LV and RV ejection fraction compared to DRN. Overall, the DRN model achieved highest performance for all clinical metrics.\n\n\t\\noindent \\textbf{Effect of model architecture on segmentation}: Although quantitative differences between models are small, qualitative evaluation discloses that automatic segmentations differ substantially between the models. Figure~\\ref{fig_seg_qualitative_results} shows that especially in regions where the models perform poorly (apical and basal slices) the DN model more often produced anatomically implausible segmentations compared to the DRN and U-net. This seems to be correlated with the performance differences in Hausdorff distance.\n\t\n\t\\noindent \\textbf{Effect of loss function on segmentation}: The results indicate that the choice of loss function only slightly affects the segmentation performance. DRN and U-net perform marginally better when trained with soft-Dice compared to cross-entropy whereas DN performs better when trained with Brier loss than with soft-Dice. For DN this is most pronounced for the RV at ES.\n\t\n\tA considerable effect of the loss function on the accuracy of the LV and RV ejection fraction can be observed for the U-net model. On both metrics U-net achieved the lowest accuracy of all models when trained with the soft-Dice loss.\n\t\n\t\\noindent \\textbf{Effect of MC dropout on segmentation}: The results show that enabling MC-dropout during testing seems to result in slightly improved HD while it does not affect DC. \n\n\t\\begin{table}\n\t\t\\caption{Average precision and percentage of slices with segmentation failures generated by Dilated Network (DN), Dilated Residual Network (DRN) and U-net when trained with soft-Dice (SD), CE or Brier loss. Per patient, average precision of detected slices with failure using e- or b-maps (\\num{2}$^{nd}$ and \\num{3}$^{rd}$ columns). Per patient, average percentage of slices containing segmentation failures (reference for detection task) (\\num{4}$^{th}$ and \\num{5}$^{th}$ columns).}\n\t\t\\label{table_evaluation_slice_detection}\n\t\t\\begin{tabular}{l  C{1.4cm}  C{1.4cm} C{1.4cm} C{1.4cm}  }\n\t\t\t\\textbf{Model} & \\multicolumn{2}{c}{\\textbf{Average precision}} & \\multicolumn{2}{c}{ \\thead{\\textbf{\\% of slices} \\\\ \\textbf{with segmentation failures} }} \\\\\n\t\t\t& e-map & b-map & e-map & b-map \\\\\n\t\t\t\\hline\n\t\t\tDN-Brier &  84.0 & 83.0 & 53.7 & 52.4   \\\\\n\t\t\tDN-SD  &  87.0 & 85.0 & 58.3 & 58.1   \\\\\n\t\t\t\\hdashline\n\t\t\tDRN-CE  & 75.0 & 69.0  &  39.5 & 39.4 \\\\\n\t\t\tDRN-SD  & 67.0 & 67.0 &  34.9 & 33.7\\\\\n\t\t\t\\hdashline\n\t\t\tU-net-CE   & 81.0 & 75.0 & 54.8 & 52.5 \\\\\n\t\t\tU-net-SD   & 76.0 & 76.0 & 46.7 & 45.5 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\subsection*{Detection of segmentation failures}\n\t\n\t\\noindent \\textbf{Detection of segmentation failures on voxel level}: To evaluate detection performance of segmentation failures on voxel level Figure~\\ref{fig_froc_voxel_detection} shows average voxel detection rate as a function of false positively detected regions. This was done for each combination of model architecture and loss function exploiting e- (Figure~\\ref{fig_froc_voxel_detection}, left)  or b-maps (Figure~\\ref{fig_froc_voxel_detection}, right). These results show that detection performance of segmentation failures depends on segmentation model architecture, loss function and uncertainty map. \n\t\n\tThe influence of (segmentation) model architecture and loss function on detection performance is slightly stronger when e-maps were used as input for the detection task compared to b-maps. Detection rates are consistently lower when segmentation failures originate from segmentation models trained with soft-Dice loss compared to models trained with CE or Brier loss. Overall, detection rates are higher when b-maps were exploited for the detection task compared to e-maps.\n\t\t\n\\begin{table*}\n\t\\caption{Comparing performance of segmentation-only approach (auto-only) with combined segmentation and detection approach for two scenarios: simulated correction of detected segmentation failures (auto$+$simulation); and manual correction of detected segmentation failures by an expert (auto$+$expert). Automatic segmentations were obtained from a U-net trained with cross-entropy. Evaluation was performed on a subset of \\num{50} patients from the ACDC dataset. Scenarios are compared against segmentation-only approach (auto-only) in terms of (a) Dice Coefficient (b) Hausdorff Distance and (c) Clinical metrics. Results obtained from simulated manual correction represent an upper bound on the maximum achievable performance. Detection network was trained with e-maps. Number with asterisk indicates statistical significant at \\num{5}\\% level w.r.t. the segmentation-only approach. Best viewed in color.}\n\t\\label{table_manual_corr_performance}\n\t\\centering\n\t\\small\n\t\\subfloat[\\textbf{Dice coefficient:} Mean $\\pm$ standard deviation for left ventricle (LV), right ventricle (RV) and left ventricle myocardium (LVM).]{\n\t\t\\begin{tabular}{| C{2.cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\\hline\n\t\t\t& \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\\textbf{Scenario} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\\hline\n\t\t\tauto-only & \\phantom{x}0.964$\\pm$0.02 & \\phantom{x}0.927$\\pm$0.04 & \\phantom{x}0.883$\\pm$0.03  & \\phantom{x}0.916$\\pm$0.05 & \\phantom{x}0.854$\\pm$0.08 & \\phantom{x}0.886$\\pm$0.04 \\\\ \n\t\t\tauto$+$simulation & \\phantom{x}0.967$\\pm$0.01 & *0.948$\\pm$0.03 & *0.894$\\pm$0.03  & *0.939$\\pm$0.03 & *0.915$\\pm$0.04 & *0.910$\\pm$0.03 \\\\ \n\t\t\n\t\t\tauto$+$expert & \\phantom{x}0.965$\\pm$0.02 & \\phantom{x}0.940$\\pm$0.03 & \\phantom{x}0.885$\\pm$0.03  & \\phantom{x}0.927$\\pm$0.04 & \\phantom{x}0.868$\\pm$0.07 & \\phantom{x}0.894$\\pm$0.03 \\\\ \n\t\t\t\n\t\t\t\\bottomrule\n\t\t\t\n\t\t\\end{tabular}\n\t\t\\label{table_manual_seg_perf_dsc}\n\t} \n\n\t\\centering\n\n\t\\subfloat[\\textbf{Hausdorff Distance:} Mean $\\pm$ standard deviation for left ventricle (LV), right ventricle (RV) and left ventricle myocardium (LVM).]{\n\t\t\\begin{tabular}{| C{2.cm} | R{1.7cm} R{1.7cm} R{1.7cm} | R{1.7cm} R{1.7cm} R{1.7cm} | }\n\t\t\t\\hline\n\t\t\t& \\multicolumn{3}{c|}{\\textbf{End-diastole}} & \\multicolumn{3}{c|}{\\textbf{End-systole}} \\\\\n\t\t\t\\textbf{Scenario} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} & \\multicolumn{1}{l}{\\textbf{LV}} & \\multicolumn{1}{l}{\\textbf{RV}} & \\multicolumn{1}{l|}{\\textbf{LVM}} \\\\ \n\t\t\t\\hline\n\t\t\tauto-only  &  \\phantom{x}5.6$\\pm$3.3 & \\phantom{x}15.7$\\pm$9.7 & \\phantom{x}8.5$\\pm$6.4 & \\phantom{x}9.2$\\pm$5.8 & \\phantom{x}16.5$\\pm$8.8 & \\phantom{x}13.4$\\pm$10.5 \\\\\n\t\t\tauto$+$simulation &  \\phantom{x}4.5$\\pm$2.1 & *\\phantom{x}9.0$\\pm$4.6 & *5.9$\\pm$3.4 & *5.2$\\pm$2.5 & *10.3$\\pm$3.7 & *\\phantom{x}6.6$\\pm$2.9 \\\\\n\t\t\n\t\t\tauto$+$expert &  \\phantom{x}4.9$\\pm$2.8 & *\\phantom{x}9.8$\\pm$4.3 & \\phantom{x}7.3$\\pm$4.3 & \\phantom{x}7.2$\\pm$3.3 & *12.5$\\pm$4.7 & *\\phantom{x}8.3$\\pm$3.5 \\\\\n\t\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\label{table_manual_seg_perf_hd}\n\t}  \n\n\t\\subfloat[\\textbf{Clinical metrics:} a) Left ventricle (LV) end-diastolic volume (EDV) b) LV ejection fraction (EF) c) Right ventricle (RV) EDV d) RV ejection fraction e) LV myocardial mass. Quantitative results compare clinical metrics based on reference segmentations with 1) automatic segmentations; 2) simulated manual correction and 3) manual expert correction of automatic segmentations using spatial uncertainty maps. $\\rho$ denotes the Pearson correlation coefficient, \\textit{bias} denotes the mean difference between the two measurements (mean $\\pm$ standard deviation) and \\textit{MAE} denotes the mean absolute error between the two measurements.]{\n\t\t\\label{table_manual_cardiac_function_indices}\n\t\t\\small\n\t\t\\begin{tabular}{| C{2.cm} | C{0.5cm}  C{0.8cm} C{0.55cm} | C{0.5cm}  C{0.8cm}C{0.55cm} | C{0.5cm}  C{0.8cm} C{0.55cm} | C{0.5cm}  C{0.8cm} C{0.55cm} | C{0.5cm} C{0.8cm} C{0.55cm} |}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{3}{c}{\\textbf{LV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{LV$_{EF}$}} & \\multicolumn{3}{c}{\\textbf{RV$_{EDV}$}} &  \\multicolumn{3}{c}{\\textbf{RV$_{EF}$}} & \\multicolumn{3}{c|}{\\textbf{LVM$_{Mass}$}}  \\\\\n\n\t\t\t \\textbf{Scenario} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\textbf{$\\rho$} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} & \\multicolumn{1}{c}{\\textbf{$\\rho$}} & \\textbf{bias $\\pm\\sigma$} & \\textbf{MAE} \\\\\n\t\t\t\\hline\n\t\t\n\t\t     auto-only  & 0.995 & -4.4 $\\pm$7.0 & 5.7 & 0.927 & 5.0 $\\pm$7.1 & 5.8 & 0.962 & -6.4 $\\pm$16.2 & 11.9 & 0.878 & 5.8 $\\pm$8.7 & 8.0 & 0.979 & -6.4 $\\pm$10.6 & 9.5 \\\\\n\t\t\t \n\t\t\t auto$+$simulation & 0.998 & -3.9 $\\pm$5.2 & 4.8 & 0.989 & 2.3 $\\pm$2.9 & 2.9 & 0.984 & -3.7 $\\pm$10.4 & 6.8 & 0.954 & 2.7 $\\pm$5.5 & 4.5 & 0.983 & -5.5 $\\pm$9.6 & 8.1 \\\\\n\t\t\t\n\t\t\t \n\t\t\t auto$+$expert & 0.996 & -4.3 $\\pm$6.5 & 5.5 & 0.968 & 2.7 $\\pm$4.8 & 4.3 & 0.976 & -3.2 $\\pm$12.9 & 8.3 & 0.883 & 5.1 $\\pm$8.6 & 7.7 & 0.980 & -6.2 $\\pm$10.2 & 9.1 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\n\\end{table*}\n\t\n\t\\vspace{1ex}\n\t\\noindent \\textbf{Detection of slices with segmentation failures}: To evaluate detection performance w.r.t. slices containing segmentation failures precision-recall curves for each combination of model architecture and loss function using e-maps (Figure~\\ref{fig_prec_rec_slice_detection}, left) or b-maps (Figure~\\ref{fig_prec_rec_slice_detection}, right) are shown. The results show that detection performance of slices containing segmentation failures is slightly better for all models when using e-maps. Furthermore, the detection network achieves highest performance using uncertainty maps obtained from the DN model and the lowest when exploiting e- or b-maps obtained from the DRN model. Table~\\ref{table_evaluation_slice_detection} shows the average precision of detected slices with segmentation failures per patient, as well as the average percentage of slices that do contain segmentation failures (reference for detection task). The results illustrate that these measures are positively correlated i.e. that precision of detected slices in a patient volume is higher if the volume contains more slices that need correction. On average the DN model generates cardiac segmentations that contain more slices with at least one segmentation failure compared to U-net (ranks second) and DRN (ranks third). A higher number of detected slices containing segmentation failures implies an increased workload for manual correction.\n\t\n\t\\subsection*{Calibration of uncertainty maps} \\label{result_eval_quality_umaps}\n\t\n\tFigure~\\ref{fig_risk_cov_comparison} shows risk-coverage curves for each combination of model architectures, uncertainty maps and loss functions (Figure~\\ref{fig_risk_cov_comparison} left: CE or Brier loss, Figure~\\ref{fig_risk_cov_comparison} right: soft-Dice). The results show an effect of the loss function on slope and convergence of the curves. Segmentation errors of models trained with the soft-Dice loss are less frequently covered by higher uncertainties than models trained with CE or Brier loss (steeper slope and lower minimum are better). This difference is more pronounced for e-maps. Models trained with the CE or Brier loss only slightly differ concerning convergence and their slopes are approximately identical. In contrast, the curves of the models trained with the soft-Dice differ regarding their slope and achieved minimum. Comparing e- and b-map of the DN-SD and U-net-SD models the results reveal that the curve for b-map has a steeper slope and achieves a lower minimum compared to the e-map. For the DRN-SD model these differences are less striking. In general for a specific combination of model and loss function the risk-coverage curves using b-maps achieve a lower minimum compared to e-maps.\n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[width=4.7in, height=2.8in]{figures\/risk_cov\/cov_risk_curve_both_seg_errors.pdf}\n\t\t\n\t\t\\caption{Comparison of risk-coverage curves for different combination of model architectures, loss functions and uncertainty maps. Results are separated for loss functions (left cross-entropy and Brier, right soft-Dice loss). \\num{100}\\% coverage means that none of the voxels is discarded based on its uncertainty whereas a coverage of \\num{0}\\% denotes the scenario in which all predictions are replaced by their reference labels. Note, all models were trained with two different loss functions (1) soft-Dice (SD) for all models (2) cross-entropy (CE) for DRN and U-net and Brier loss for DN.}\n\t\t\\label{fig_risk_cov_comparison}\n\t\\end{figure}\n\t\n\t\\begin{table}\n\t\t\\caption{Effect of number of Monte Carlo (MC) samples on segmentation performance in terms of (a) Dice coefficient (DC) and (b) Hausdorff Distance (HD) (mean $\\pm$ standard deviation). Higher DC and lower HD is better. Abbreviations: Cross-Entropy (CE), Dilated Residual Network (DRN) and Dilated Network (DN).} \n\t\t\\label{table_seg_perf_per_samples}\n\t\t\\small\n\t\t\\subfloat[Dice coefficient]{\n\t\t\t\\begin{tabular}{|c  C{1.5cm}  C{1.5cm} c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\thead{Number of \\\\ MC samples}} & DRN-CE & U-net-CE & DN-soft-Dice\\\\\n\t\t\t\t\\hline\n\t\t\t\t1 & 0.894$\\pm$0.07 & 0.896$\\pm$0.07 & 0.871$\\pm$0.09 \\\\\n\t\t\t\t3 & 0.900$\\pm$0.07 & 0.901$\\pm$0.07 & 0.883$\\pm$0.08 \\\\\n\t\t\t\t5 & 0.902$\\pm$0.07 & 0.901$\\pm$0.07 & 0.887$\\pm$0.08 \\\\\n\t\t\t\t7 & 0.903$\\pm$0.07 & 0.901$\\pm$0.07 & 0.888$\\pm$0.08 \\\\\n\t\t\t\t10 & 0.904$\\pm$0.06 & 0.902$\\pm$0.07 &0.890$\\pm$0.08  \\\\\n\t\t\t\t20 & 0.904$\\pm$0.07 & 0.902$\\pm$0.07 & 0.890$\\pm$0.08 \\\\\n\t\t\t\t30 & 0.904$\\pm$0.07 & 0.902$\\pm$0.07 & 0.891$\\pm$0.08 \\\\\n\t\t\t\t60 & 0.904$\\pm$0.07 & 0.902$\\pm$0.07 & 0.891$\\pm$0.08 \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\subfloat[Hausdorff Distance]{\n\t\t\t\\begin{tabular}{|c  C{1.5cm}  C{1.5cm} c|}\n\t\t\t\t\\hline\n\t\t\t\t\\textbf{\\thead{Number of \\\\ MC samples}}& DRN-CE & U-net-CE & DN-soft-Dice \\\\\n\t\t\t\t\\hline\n\t\t\t\t1 & 9.88$\\pm$5.76 & 11.79$\\pm$8.23 & 13.54$\\pm$7.14 \\\\\n\t\t\t\t3 & 9.70$\\pm$6.13 & 11.40$\\pm$7.78 & 12.71$\\pm$6.79 \\\\\n\t\t\t\t5 & 9.54$\\pm$6.07 & 11.37$\\pm$7.81 & 12.06$\\pm$6.29 \\\\\n\t\t\t\t7 & 9.38$\\pm$5.86 & 11.29$\\pm$7.86 & 12.08$\\pm$6.38 \\\\\n\t\t\t\t10 & 9.38$\\pm$5.91 & 11.24$\\pm$7.71 & 11.85$\\pm$6.34 \\\\\n\t\t\t\t20 & 9.37$\\pm$5.83 & 11.27$\\pm$7.79 & 11.90$\\pm$6.52 \\\\\n\t\t\t\t30 & 9.39$\\pm$5.91 &  11.32$\\pm$7.93 & 11.90$\\pm$6.48 \\\\\n\t\t\t\t60 & 9.39$\\pm$5.93 & 11.22$\\pm$7.83 & 11.89$\\pm$6.56 \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\t\n\t\t}\n\t\\end{table}\n\t\n\t\\subsection*{Correction of automatically identified segmentation failures} \\label{results_combined_approach}\n\t\n\t\\textbf{Simulated correction:} The results listed in Table~\\ref{table_overall_segmentation_performance} and \\ref{table_cardiac_function_indices} show that the proposed method consisting of segmentation followed by simulated manual correction of detected segmentation failures delivers accurate segmentation for all tissues over ED and ES points. Correction of detected segmentation failures improved the performance in terms of DC, HD and clinical metrics for all combinations of model architectures, loss functions and uncertainty measures. Focusing on the DC after correction of detected segmentation failures the results reveal that performance differences between evaluated models decreased compared to the segmentation-only task. This effect is less pronounced for HD where the DRN network clearly achieved superior results in the segmentation-only and combined approach. The DN performs the least of all models but achieves the highest absolute DC performance improvements in the combined approach for RV at ES. Overall, the results in Table~\\ref{table_overall_segmentation_performance} disclose that improvements attained by the combined approach are almost all statistically significant ($p \\leq 0.05$) at ES and frequently at ED (\\num{96}\\% resp. \\num{83}\\% of the cases). Moreover, improvements are in \\num{99}\\% of the cases statistically significant for HD compared to \\num{81}\\% of the cases for DC.\n\t\n\tResults in terms of clinical metrics shown in Table~\\ref{table_cardiac_function_indices} are inline with these findings. We observe that segmentation followed by simulated manual correction of detected segmentation failures resulted in considerably higher accuracy for LV and RV ejection fraction. Achieved improvements for clinical metrics are only statistically significant ($p \\leq 0.05$) in one case for RV ejection fraction.\t\n\t\n\tIn general, the effect of correction of detected segmentation failures is more pronounced in cases where the segmentation-only approach achieved relatively low accuracy (e.g. DN-SD for RV at ES). Furthermore, performance gains are largest for RV and LV at ES and for ejection fraction of both ventricles.\n\t\n\n\t\\begin{figure}[!t]\n\t\t\\captionsetup[subfigure]{justification=centering}\n\t\t\\centering\n\t\t\\subfloat[]{\\includegraphics[width=6in]{figures\/qualitative\/sim_corr\/patient018_slice07_ES_bmap_acorr.pdf}%\n\t\t\t\\label{fig_qual_result_sim_corr_example1}}\n\t\t\n\t\t\\subfloat[]{\\includegraphics[width=6in]{figures\/qualitative\/sim_corr\/patient070_slice05_ED_emap_acorr.pdf}%\n\t\t\t\\label{fig_qual_result_sim_corr_example2}}\n\t\t\n\t\t\\subfloat[]{\\includegraphics[width=6in]{figures\/qualitative\/sim_corr\/patient081_slice01_ES_emap_acorr.pdf}%\n\t\t\t\\label{fig_qual_result_sim_corr_example3}}\n\t\t\\caption{Three patients showing results of combined segmentation and detection approach consisting of segmentation followed by simulated manual correction of detected segmentation failures. First column shows MRI (top) and reference segmentation (bottom). Results for automatic segmentation and simulated manual correction respectively achieved by: Dilated Network (DN-Brier, \\num{2}$^{nd}$ and \\num{5}$^{th}$ columns); Dilated Residual Network (DRN-soft-Dice, \\num{3}$^{rd}$ and \\num{6}$^{th}$ columns); and U-net (soft-Dice, \\num{4}$^{th}$ and \\num{7}$^{th}$ columns).}\n\t\t\\label{fig_seg_detection_qualitative_results}\n\t\\end{figure}\n\t\n\t\\begin{figure}[!t]\n\t\t\\captionsetup[subfigure]{justification=centering}\n\t\t\\centering\n\t\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient048_slice02_mcorr.pdf}%\n\t\t\t\\label{fig_qual_result_man_corr_example1} \\hspace{3ex}}\n\t\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient091_slice01_mcorr.pdf}%\n\t\t\t\\label{fig_qual_result_man_corr_example2}}\n\t\t\n\t\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient075_slice06_mcorr.pdf}%\n\t\t\t\\label{fig_qual_result_man_corr_example3} \\hspace{3ex}}\n\t\t\\subfloat[]{\\includegraphics[width=3.3in]{figures\/qualitative\/man_corr\/with_region_contour\/patient006_slice01_mcorr.pdf}%\n\t\t\t\\label{fig_qual_result_man_corr_example4}}  \n\t\t\n\t\t\\caption{Four patients showing results of combined segmentation and detection approach consisting of segmentation followed by manual expert correction of detected segmentation failures. Expert was only allowed to adjust the automatic segmentations in regions where the detection network predicted segmentation failures (orange contour shown in 2$^{nd}$ column). Automatic segmentations were generated by a U-net trained with the cross-entropy loss. Segmentation failure detection was performed using entropy maps.}\n\t\t\\label{fig_qualitative_results_man_corr}\n\t\\end{figure}\n\t\n\n\tThe best overall performance is achieved by the DRN model trained with cross-entropy loss while exploiting entropy maps in the detection task. Moreover, the proposed two step approach attained slightly better results using Bayesian maps compared to entropy maps. \n\t\n\t\\vspace{1ex}\n\t\\textbf{Manual correction}: Table~\\ref{table_manual_corr_performance} lists results for the combined automatic segmentation and detection approach followed by \\textit{manual} correction of detected segmentation failures by an expert. The results show that this correction led to improved segmentation performance in terms of DC, HD and clinical metrics. Improvements in terms of HD are in \\num{50} percent of the cases statistically significant ($p \\leq 0.05$) and most pronounced for RV and LV at end-systole. \n\t\n\tQualitative examples of the proposed approach are visualized in Figures~\\ref{fig_seg_detection_qualitative_results} and \\ref{fig_qualitative_results_man_corr} for simulated correction and manual correction of the automatically detected segmentation failures, respectively. For the illustrated cases (simulated) manual correction of detected segmentation failures leads to increased segmentation performance.\n\tOn average manual correction of automatic segmentations took less than \\num{2} minutes for ED and ES volumes of one patient compared to \\num{20} minutes that is typically needed by an expert for the same task.\n\t\n\t\n\t\\section*{Ablation Study}\n\t\n\tTo demonstrate the effect of different hyper-parameters in the method, a number of experiments were performed. In the following text these are detailed.\n\t\n\t\\subsection*{Impact of number of Monte Carlo samples on segmentation performance}\n\n\tTo investigate the impact of the number of Monte Carlo (MC) samples on the segmentation performance validation experiments were performed for all three segmentation architectures (Dilated Network, Dilated Residual Network and U-net) using $T$ $\\in \\{1, 3, 5, 7, 10, 20, 30, 60\\}$ samples. Results of these experiments are listed in Table~\\ref{table_seg_perf_per_samples}. We observe that segmentation performance started to converge using \\num{7} samples only. Performance improvements using an increased number of MC samples were largest for the Dilated Network. Overall, using more than \\num{10} samples did not increase segmentation performance. Hence, in the presented work $T$ was set to \\num{10}.\t\n\t\t\n\t\\subsection*{Effect of patch-size on detection performance}\n\t\n\tThe combined segmentation and detection approach detects segmentation failures on region level. To investigate the effect of patch-size on detection performance three different patch-sizes were evaluated: \\num{4}$\\times$\\num{4}, \\num{8}$\\times$\\num{8}, and \\num{16}$\\times$\\num{16} voxels. The results are shown in Figure~\\ref{fig_grid_compare}. We can observe in Figure~\\ref{fig_fn_grid_froc_voxel_detection} that larger patch-sizes result in a lower number of false positive regions. The result is potentially caused by the decreasing number of regions in an image when using larger patch-sizes compared to smaller patch-sizes. Furthermore, Figure~\\ref{fig_fn_grid_prec_rec_slice_detection} reveals that slice detection performance is only slightly influenced by patch-size. To ease manual inspection and correction by an expert, it is desirable to keep region-size i.e. patch-size small. Therefore, in the experiments a patch-size of \\num{8}$\\times$\\num{8} voxels was used.\n\t\n\t\\begin{figure}[t]\n\t\t\\center\n\t\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/compare_grid_froc_voxel_detection_rate.pdf}%\n\t\t\t\\label{fig_fn_grid_froc_voxel_detection}}\n\t\t\\subfloat[]{\\includegraphics[width=3.4in, height=1.7in]{figures\/dt\/compare_grid_slices_prec_recall.pdf}%\n\t\t\t\\label{fig_fn_grid_prec_rec_slice_detection}} \n\t\t\n\t\t\\caption{Detection performance for three different patch-sizes specified in voxels. (a) Sensitivity for detection of segmentation failures on voxel level (y-axis) versus number of false positive image regions (x-axis). (b) Precision-recall curve for detection of slices containing segmentation failures (where AP denotes average precision). Results are split between entropy and Bayesian uncertainty maps. In the experiments patch-size was set to \\num{8}$\\times$\\num{8} voxels.}\n\t\t\\label{fig_grid_compare}\n\t\\end{figure}\n\t\n\t\\subsection*{Impact of tolerance threshold on number of segmentation failures}\n\t\n\tTo investigate the impact of the tolerance threshold separating segmentation failures and tolerable segmentation errors, we calculated the ratio of the number of segmentation failures and all errors i.e. the sum of tolerable errors and segmentation failures. Figure~\\ref{fig_threshold_compare} shows the results. We observe that at least half of the segmentation failures are located within a tolerance threshold i.e. distance of two to three voxels of the target structure boundary as defined by the reference annotation. Furthermore, the mean percentage of failures per volume is considerably lower for the Dilated Residual Network (DRN) and highest for the Dilated Network. This result is inline with our earlier finding (see Table~\\ref{table_evaluation_slice_detection}) that average percentage of slices that do contain segmentation failures is lowest for the DRN model.\n\t\n\t\\section*{Discussion}\n\t\n\tWe have described a method that combines automatic segmentation and assessment of uncertainty in cardiac MRI with detection of image regions containing segmentation failures. The results show that combining automatic segmentation with manual correction of detected segmentation failures results in higher segmentation performance.\n\tIn contrast to previous methods that detected segmentation failures per patient or per structure, we showed that it is feasible to detect segmentation failures per image region. In most of the experimental settings, simulated manual correction of detected segmentation failures for LV, RV and LVM at ED and ES led to statistically significant improvements. These results represent the upper bound on the maximum achievable performance\tfor the manual expert correction task. Furthermore, results show that manual expert correction of detected segmentation failures led to consistently improved segmentations. However, these results are not on par with the simulated expert correction scenario. This is not surprising because inter-observer variability is high for the presented task and annotation protocols may differ between clinical environments. Moreover, qualitative results of the manual expert correction reveal that manual correction of the detected segmentation failures can prevent anatomically implausible segmentations (see Figure~\\ref{fig_qualitative_results_man_corr}).\n\tTherefore, the presented approach can potentially simplify and accelerate correction process and has the capacity to increase the trustworthiness of existing automatic segmentation methods in daily clinical practice.\n\n\t\n\t\n\tThe proposed combined segmentation and detection approach was evaluated using three state-of-the-art deep learning segmentation architectures. The results suggest that our approach is generic and applicable to different model architectures. Nevertheless, we observe noticeable differences between the different combination of model architectures, loss functions and uncertainty measures. In the segmentation-only task the DRN clearly outperforms the other two models in the evaluation of the boundary of the segmented structure. Moreover, qualitative analysis of the automatic segmentation masks suggests that DRN generates less often anatomically implausible and fragmented segmentations than the other models. We assume that clinical experts would prefer such segmentations although they are not always perfect. Furthermore, even though DRN and U-net achieve similar performance in regard to DC we assume that less fragmented segmentation masks would increase trustworthiness of the methods. \n\t\n\t\\begin{figure}[t]\n\t\t\\center\n\t\t\\subfloat[]{\n\t\t\t\\includegraphics[width=3.in]{figures\/dt\/tp_per_threshold_out_struc.pdf}%\n\t\t\t\\label{fig_threshold_struc_out_compare}\n\t\t}\n\t\t\\subfloat[]{\n\t\t\t\\includegraphics[width=3.in]{figures\/dt\/tp_per_threshold_in_struc.pdf}%\n\t\t\t\\label{fig_threshold_struc_in_compare}\n\t\t}\n\t\t\\caption{Mean percentage of the segmentation failures per volume (y-axis) in the set of all segmentation errors (tolerable errors$+$segmentation failures) depending on the tolerance threshold (x-axis). Red, dashed vertical line indicates threshold value that was used throughout the experiments. Results are split between segmentation errors located (a) outside and (b) inside the target structure. Each figure contains a curve for U-net, Dilated Network (DN) and Dilated Residual Network (DRN) trained with the soft-Dice (SD) loss. Segmentation errors located in slices above the base or below the apex are always included in the set of segmentation failures and therefore, they are independent of the applied tolerance threshold.}\n\t\t\\label{fig_threshold_compare}\n\t\\end{figure}\n\t\n\tIn agreement with our preliminary work we found that uncertainty maps obtained from a segmentation model trained with soft-Dice loss have a lower degree of uncertainty calibration compared to when trained with one of the other two loss functions (cross-entropy and Brier)\\cite{sander2019towards}. Nevertheless, the results of the combined segmentation and detection approach showed that a lower degree of uncertainty calibration only slightly deteriorated the detection performance of segmentation failures for the larger segmentation models (DRN and U-net) when exploiting uncertainty information from e-maps. Hendrycks and Gimpel \\cite{hendrycks2016baseline} showed that softmax probabilities generated by deep learning networks have poor direct correspondence to confidence. However, in agreement with Geifman et al. \\cite{geifman2017selective} we presume that probabilities and hence corresponding entropies obtained from softmax function are ranked consistently i.e. entropy can potentially be used as a relative uncertainty measure in deep learning. In addition, we detect segmentation failures per image region and therefore, our approach does not require perfectly calibrated uncertainty maps. Furthermore, results of the combined segmentation and detection approach revealed that detection performance of segmentation failures using b-maps is almost independent of the loss function used to train the segmentation model. In line with Jungo et al. \\cite{jungo2019assessing} we assume that by enabling MC-dropout in testing and computing the mean softmax probabilities per class leads to better calibrated probabilities and b-maps. This assumption is in agreement with Srivastava et al.~\\cite{srivastava2014dropout} where a CNN with dropout used at testing is interpreted as an ensemble of models.\n\t\n\tQuantitative evaluation in terms of Dice coefficient and Hausdorff distance reveals that proposed combined segmentation and detection approach leads to significant performance increase. However, the results also demonstrate that the correction of the detected failures allowed by the combined approach does not lead to statistically significant improvement in clinical metrics. This is not surprising because state-of-the-art automatic segmentation methods are not expected to lead to large volumetric errors \\cite{bernard2018deep} and standard clinical measures are not sensitive to small segmentation errors. Nevertheless, errors of the current state-of-the-art automatic segmentation methods may lead to anatomically implausible segmentations \\cite{bernard2018deep} that may cause distrust in clinical application. Besides increase in trustworthiness of current state-of-the-art segmentation methods for cardiac MRIs, improved segmentations are a prerequisite for advanced functional analysis of the heart e.g. motion analysis\\cite{bello2019deep} and very detailed morphology analysis such as myocardial trabeculae in adults\\cite{meyer2020genetic}.\n\t\n\tFor the ACDC dataset used in this manuscript, Bernard et al.\\cite{bernard2018deep} reported inter-observer variability ranging from \\num{4} to \\SI{14.1}{\\milli\\meter} (equivalent to on average \\num{2.6} to \\num{9} voxels). To define the set of segmentation failures, we employed a strict tolerance threshold on distance metric to distinguish between tolerated segmentation errors and segmentation failures (see Ablation study). Stricter tolerance threshold was used because the thresholding is performed in \\num{2}D, while evaluation of segmentation is done in \\num{3}D. Large slice thickness in cardiac MR  could lead to a discrepancy between the two. As a consequence of this strict threshold results listed in Table~\\ref{table_evaluation_slice_detection} show that almost all patient volumes contain at least one slice with a segmentation failure. This might render the approach less feasible in clinical practice. Increasing the threshold decreases the number of segmentation failures and slices containing segmentation failures (see Figure~\\ref{fig_threshold_compare}) but also lowers the upper bound on the maximum achievable performance. Therefore, to show the potential of our proposed approach we chose to apply a strict tolerance threshold. Nevertheless, we realize that although manual correction of detected segmentation failures leads to increased segmentation accuracy the performance of precision-recall is limited (see Figure~\\ref{fig_dt_perf_all_models}) and hence, should be a focus of future work. \n\t\n\tThe presented patch-based detection approach combined with (simulated) manual correction can in principle lead to stitching artefacts in the resulting segmentation masks. A voxel-based detection approach could potentially solve this. However, voxel-based detection methods are more challenging to train due to the very small number of voxels in an image belonging to the set of segmentation failures.\n\t\n\tEvaluation of the proposed approach for \\num{12} possible combinations of segmentation models (three), loss functions (two) and uncertainty maps (two) resulted in an extensive number of experiments. Nevertheless, future work could extend evaluation to other segmentation models, loss functions or combination of losses. Furthermore, our approach could be evaluated using additional uncertainty estimation techniques e.g. by means of ensembling of networks \\cite{lakshminarayanan2017simple} or variational dropout \\cite{kingma2015variational}. In addition, previous work by Kendall and Gal~\\cite{kendall2017uncertainties}, Tanno et al. \\cite{tanno2019uncertainty} has shown that the quality of uncertainty estimates can be improved if model (epistemic) and data (aleatoric) uncertainty are assessed simultaneously with separate measures. The current study focused on the assessment of model uncertainty by means of MC-dropout and entropy which is a combination of epistemic and aleatoric uncertainty. Hence, future work could investigate whether additional estimation of aleatoric uncertainty improves the detection of segmentation failures. \n\t\n\tFurthermore, to develop an end-to-end approach future work could incorporate the detection of segmentation failures into the segmentation network. Besides, adding the automatic segmentations to the input of the detection network could increase the detection performance. \n\t\n\tFinally, the proposed approach is not specific to cardiac MRI segmentation. Although data and task specific training would be needed the approach could potentially be applied to other image modalities and segmentation tasks.\n\t\n\t\\section*{Conclusion}\n\t\n\tA method combining automatic segmentation and assessment of segmentation uncertainty in cardiac MR with detection of image regions containing local segmentation failures has been presented. The combined approach, together with simulated and manual correction of detected segmentation failures, increases performance compared to segmentation-only. The proposed method has the potential to increase trustworthiness of current state-of-the-art segmentation methods for cardiac MRIs. \n\t\n\t\\section*{Data and code availability}\n\tAll models were implemented using the PyTorch\\cite{paszke2017automatic} framework and trained on one Nvidia GTX Titan X GPU with \\num{12} GB memory. The code to replicate the study is publicly available at  \\href{https:\/\/github.com\/toologicbv\/cardiacSegUncertainty}{https:\/\/github.com\/toologicbv\/cardiacSegUncertainty}.\n\t\n\n\t\\input{main_article.bbl}\n\t\n\t\\section*{Acknowledgements}\n\t\n\tThis study was performed within the DLMedIA program (P15-26) funded by Dutch Technology Foundation with participation of PIE Medical Imaging.\n\t\n\t\\section*{Author contributions statement}\n\t\n\tJ.S., B.D.V. and I.I. designed the concept of the study. J.S. conducted the experiments. J.S., B.D.V. and I.I. wrote the manuscript. All authors reviewed the manuscript. \n\t\n\t\\section*{Additional information}\n\t\n\t\\textbf{Competing interests}: The authors declare that they have no competing interests. \n\t\n\t\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEntropy production plays a fundamental role in both classical and quantum thermodynamics: by being related to the second law  at a fundamental level, it enables to identify and quantify the irreversibility of physical phenomena \\cite{DeGrootMazur}. This intimate connection has raised a great deal of interest in relation to the theory of open quantum systems, where one is concerned about the dynamics of a system interacting with the infinitely many environmental degrees of freedom \\cite{Breuer-Petruccione}. In this scenario, a plethora of genuine quantum effects is brought about and a general and exhausting theory of entropy production is hitherto missing \\cite{Mauro_thermo2018}.\n\nThe second law of thermodynamics can be expressed in the form of a lower bound to the entropy change $\\Delta S$ undergone by the state of a given system that exchanges an amount of heat $Q$ when interacting with a bath at temperature $T$, that is\n\\begin{align}\n\\Delta S \\ge \\int \\frac{\\delta Q}{T} \\ .\n\\end{align}\nThe strict inequality holds if the process that the system is undergoing is irreversible. One can thus define the entropy production $\\Sigma$ as\n\\begin{align}\n\\label{eq:entropy_prod_def}\n\\Sigma \\equiv \\Delta S - \\int \\frac{\\delta Q}{T} \\ge 0 \\ .\n\\end{align}\nFrom \\Cref{eq:entropy_prod_def}, one can obtain the following expression involving the rates~\\cite{Landi2013, PhysRevLett.118.220601}:\n\\begin{align}\n\\frac{{\\rm{d}} S}{{\\rm{d}} t} = \\Pi(t) - \\Phi(t) \\ ,\n\\end{align}\nwhere $\\Pi(t)$ is the entropy production rate and $\\Phi(t)$ is the entropy flux from the system to the environment: at any time $t$, in addition to the entropy that is flowing from the system to the environment, there might thus be a certain amount of entropy intrinsically produced by the process and quantified by $\\Pi(t)$. \n\nEntropy production is an interesting quantity to monitor in the study of open quantum systems, since this is the context where irreversibility is unavoidably implied. The issue has been addressed in order to obtain an interesting characterisation and measure of the irreversibility of the system dynamics \\cite{Mauro_thermo2018}. In particular, it has been recently shown that the entropy production of an open quantum system can be split into different contributions: one is classically related to population unbalances, while the other is a genuine quantum contribution due to coherences \\cite{Landi:19, POLKOVNIKOV:2011, PhysRevE.99.042105}. This fundamental result holds whenever the system dynamics is either described by a map microscopically derived through the Davies approach or in the case of a finite map encompassing thermal operations~\\cite{Landi:19}. \nMost of these works, though,  are solely focused on the Markovian case, when the information is monotonically flowing from the system to the environment. Under this hypothesis, the open dynamics is formally described by a quantum dynamical semigroup; this is essential to mathematically prove that the entropy production is a non-negative quantity \\cite{Spohn1978, Alicki_1979, PhysRevA.68.032105}. \nMoreover, whenever the quantum system undergoing evolution is composite (i.e., multipartite) beside system-environment correlations, also inter-system correlations will contribute to the overall entropy production.  A full account of the role of such correlations (entanglement above all) on the entropy balance is not known.\n\n\nHowever, a strictly Markovian description of the dynamics does not encompass all possible evolutions. There might be circumstances in which there is no clear separation of time-scales between system and environment: this hampers the application of the Born-Markov approximation \\cite{Breuer-Petruccione}. In some cases, a backflow of information going from the environment to the system is observable, usually interpreted as a signature of a quantum non-Markovian process \\cite{Rivas:14, Breuer:16}. From a thermodynamical perspective the non-negativity of the entropy production rate is not always guaranteed, as there might be intervals of time in which it attains negative values. It has been argued that this should not be interpreted as a violation of the second law of thermodynamics \\cite{Benatti2017}, but it should call for a careful use of the theory, in the sense that -- in the entropy production balance -- the role of the environment cannot be totally neglected. This idea can be justified in terms of the backflow of information that quantum non-Markovianity entails: the system retrieves some of the information that has been previously lost because of its interaction with the surroundings.\n\nIn this paper,  we investigate the way initial correlations affect the entropy production rate in an open quantum system by considering the case of non-Markovian Brownian motion. \nWe focus on the case of an uncoupled bipartite system connected to two independent baths. The rationale behind this choice is related to the fact that any interaction between the two oscillators would likely generate, during the evolution, quantum correlations between the two parties. In general, the entanglement dynamically generated through the interaction would be detrimental to the transparency of the picture we would like to deliver, as it would be difficult to isolate the contribution to $\\Pi(t)$ coming from the initial inter-system correlations. To circumvent this issue, in our study we choose a configuration where the inter-system dynamics is trivial (two independent relaxation processes) but the bipartite state is initially correlated. \\textit{De facto}, the entanglement initially present in the state of our ``medium'' acts as an extra knob which can be tuned to change the rate of entropy production, thus steering the thermodynamics of the open system that we consider.\n\nThe paper is organised as follows. In \\Cref{sec:Gauss} we introduce a closed expression of the entropy production rate for a system whose dynamics is described in terms of a differential equation in the Lyapunov equation. In \\Cref{sec:QBM} we introduce the model we would like to study: a system of two uncoupled harmonic oscillators, described by a non-Markovian time-local master equation. We also discuss the spectral properties of the two local reservoirs. This minimal, yet insightful, setting allows us to investigate -- both numerically and analytically -- how different initial states can affect the entropy production rate. We investigate this relation in depth in \\Cref{sec:main}, where, by resorting to  a useful parametrisation for two-mode entangled states, we focus on the role of the purity of the total two-mode state and on the link between the entanglement we input in the initial state and the resulting entropy production. In \\Cref{sec:Markovian_Limit} we assess whether our results survive when we take the Markovian limit. Finally, in \\Cref{sec:conclusions}, we summarise the evidence we get and we eventually draw our conclusions.\n\n\\section{Entropy production rate for Gaussian systems}\n\\label{sec:Gauss}\n\nWe restrict our investigation to the relevant case of Gaussian systems.\n\\cite{Ferraro:05, Serafini:17, Carmichael}. This choice dramatically simplifies the study of our system dynamics, since the evolution equations only involve the finite-dimensional covariance matrix (CM) of the canonically conjugated quadrature operators. \nAccording to our notation, the CM $\\boldsymbol{\\sigma}$, defined as\n\\begin{align}\n\\label{eq:cov_def}\n\\sigma_{ij} =\\braket{\\{X_i, X_j \\}} - 2\\braket{X_i}\\braket{X_j}  ,\n\\end{align}\nsatisfies the Lyapunov equation\n\\begin{align}\n\\label{eq:Lyapunov}\n\\dot{\\boldsymbol{\\sigma}} = \\mathbf{A} \\boldsymbol{\\sigma} + \\boldsymbol{\\sigma} {\\mathbf{A}^{\\rm{T}}} + \\mathbf{D} ,\n\\end{align}\nwhere $\\mathbf{A}$ and $\\mathbf{D}$ are the drift and the diffusion matrices, respectively, and $\\mathbf{X} = \\{q_1,p_1,\\ldots, q_N,p_N\\}^{\\rm T}$ is the vector of quadratures for $N$ bosonic modes. \nIn particular, the CM representing a two-mode Gaussian state can always be brought in the standard form \\cite{Ferraro:05, Serafini:17}:\n\\begin{align}\n\\label{eq:sigma_sf}\n\\boldsymbol{\\sigma} = \\begin{pmatrix}\na & 0 & c_{+} & 0 \\\\\n0 & a & 0 & c_{-} \\\\\nc_{+} & 0 & b & 0 \\\\\n0 & c_{-} & 0 & b \n \\end{pmatrix},\n\\end{align}\nwhere the entries $a, \\ b$, and $c_{\\pm}$ are real numbers. Furthermore, a necessary and sufficient condition for separability of a two-mode Gaussian state is given by the Simon criterion \\cite{PhysRevA.72.032334}:\n\\begin{align}\n\\label{eq:uncertainty}\n\\tilde{\\nu}_{-} \\ge 1, \n\\end{align}\nwhere $\\tilde{\\nu}_{-}$ is the smallest symplectic eigenvalue of the partially transposed CM $\\tilde{\\boldsymbol{\\sigma}} = \\boldsymbol{P\\sigma P}$, being $\\mathbf{P} = {\\rm diag} (1,1,1,-1)$. This bound expresses  in the phase-space language the Peres-Horodecki PPT (Positive Partial Transpose) criterion for separability \\cite{Peres:96, Horodecki:97, Simon:14, PhysRevA.72.032334}.\n\nTherefore, the smallest symplectic eigenvalue encodes all the information needed to quantify the entanglement for arbitrary two-modes Gaussian states. For example, one can measure the entanglement through the violation of the PPT  criterion \\cite{PhysRevLett.90.027901}. Quantitatively, this is given by the logarithmic negativity of a quantum state $\\varrho$, which -- in the continuous variables formalism -- can be computed considering the following formula \\cite{PhysRevA.65.032314, PhysRevA.72.032334}:\n\\begin{align}\n\\label{eq:log_neg}\nE_{\\mathcal{N}} ( \\varrho) = {\\rm max} \\left [ 0, - \\ln{\\tilde{\\nu}_{-}}\\right ] .\n\\end{align}\nGiven the global state $\\varrho$ and the two single-mode states $\\varrho_{i} = {\\rm Tr}_{j \\ne i} \\varrho$,  global $\\mu \\equiv {\\rm Tr} \\varrho^2$ and the local  $\\mu_{1,2} \\equiv {\\rm Tr} \\varrho_{1,2}^2$ purities can be used to characterise  entanglement in Gaussian systems. It has been shown that two different classes of extremal states can be identified: states of maximum negativity for fixed global and local purities (GMEMS) and states of minimum negativity for fixed global and local purities (GLEMS) \\cite{PhysRevLett.92.087901}.\n\nMoreover, the continuous variables approach provides a remarkable advantage:  the open quantum system dynamics can be remapped into a Fokker-Plank equation for the Wigner function of the system. This formal result enables us to carry out our study of the entropy production using a different approach based on phase-space methods, instead of resorting to the usual approach based on von Neumann entropy. The harmonic nature of the system we would like to consider makes our choice perfectly appropriate to our study and, as we will show in \\Cref{sec:main,sec:Markovian_Limit}, well suited to systematically scrutinise inter-system correlations. Our analysis is thus based on the Wigner entropy production rate \\cite{PhysRevLett.118.220601}, defined as\n\\begin{align}\n\\Pi(t) \\equiv - \\partial_t K(W(t) || W_{\\rm eq}),\n\\end{align}\nwhere $K(W|| W_{\\rm eq})$ is the Wigner relative entropy between the Wigner function $W$ of the system and its expression for the equilibrium state $W_{\\rm eq}$.\n\nFurthermore, we are in the position of using the closed expressions for $\\Phi(t)$ and $\\Pi(t)$ coming from the theory of classical stochastic processes \\cite{Brunelli, Landi2013}. In particular, it has been shown that the entropy production rate $\\Pi(t)$ can be expressed in terms of the matrices $\\mathbf{A},\\mathbf{D},\\boldsymbol{\\sigma}$ as~\\cite{Brunelli}\n\\begin{equation}\n\\label{eq:entropy_prod_rate}\n\\begin{aligned}\n\\Pi(t) &= \\frac{1}{2} {\\rm{Tr}} [ \\boldsymbol{\\sigma}^{-1} \\mathbf{D}] + 2 {\\rm{Tr}} [ \\mathbf{A}^{{\\rm irr}}]   \\\\ \n&+ 2 {\\rm{Tr}} [ \\left (\\mathbf{A}^{{\\rm irr}}\\right )^{{\\rm T}} \\mathbf{D}^{-1} \\mathbf{A}^{{\\rm irr}} \\boldsymbol{\\sigma}] \\ ,\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbf{A}^{{\\rm irr}}$ is the irreversible part of matrix $\\mathbf{A}$, given by\n$\\mathbf{A}^{{\\rm irr}} =  \\left ( \\mathbf{A} + \\mathbf{E} \\mathbf{A} \\mathbf{E}^{{\\rm T}}\\right )\/2$,\nwhere $\\mathbf{E} = {\\rm diag}(1,-1,1-1)$ is the symplectic representation of time reversal operator.\n\n\\section{Quantum Brownian motion}\n\\label{sec:QBM}\nWe study the relation between the preparation of the initial state and the entropy production rate considering a rather general example: the quantum Brownian motion \\cite{Breuer-Petruccione, Weiss1999}, also known as Caldeira-Leggett model \\cite{CaldeiraLeggett1983a} . More specifically, we consider the case of a harmonic oscillator interacting with a bosonic reservoir made of independent harmonic oscillators. The study of such a paradigmatic system has been widely explored in both the Markovian \\cite{Breuer-Petruccione, CaldeiraLeggett1983a} and non-Markovian \\cite{PhysRevD.45.2843} regimes using the influence functional method: in this case, one can trace out the environmental degrees of freedom exactly. \nOne can also solve the dynamics of this model using the open quantum systems formalism \\cite{Breuer-Petruccione, Rivas2012}, where the Brownian particle represents the system, while we identify the bosonic reservoir with the environment. \nThe usual approach relies on the following set of assumptions, which are collectively known as Born-Markov approximation~\\cite{Breuer-Petruccione}:\n\\begin{enumerate}\n\\item The system is weakly coupled to the environment.\n\\item The initial system-environment state is factorised.\n\\item \\label{Born-Markov3} It is possible to introduce a separation of the timescales governing the system dynamics and the decay of the environmental correlations.\n\\end{enumerate}\nHowever, we aim to solve the dynamics in a more general scenario, without resorting to assumption \\ref{Born-Markov3}. We are thus considering the case in which, although the system-environment coupling is weak, non-Markovian effects may still be relevant. Under such conditions, one can derive a time-local master equation for the reduced dynamics of the system \\cite{PhysRevA.67.042108, Int1}.\n\nMore specifically, we consider a system consisting of two quantum harmonic oscillators, each of them interacting with its own local reservoir (see \\Cref{fig:system}). Each of the two reservoirs is modelled as a system of system of $N$ non-interacting bosonic modes. \nIn order to understand the dependence of the entropy production upon the initial correlations, we choose the simplest case in which the two oscillators are identical, i.e., characterised by the same bare frequency $\\omega_0$ and the same temperature $T$, and they are uncoupled, so that only the initial preparation of the global state may entangle them. The Hamiltonian of the global system thus reads as (we consider units such that $\\hbar = 1$ throughout the paper)\n\\begin{align}\n\\label{eq:Hamiltonian}\nH & = \\sum_{j=1,2} \\omega_0 \\, a_{j}^{\\dagger} a_j + \\sum_{j=1,2} \\sum_k \\omega_{jk} \\, b_{jk}^{\\dagger} b_{jk} \\nonumber \\\\\n& + \\alpha \\sum_{j=1,2} \\sum_k \\left ( \\frac{a_j + a_j^{\\dagger}}{\\sqrt{2}}\\right ) \\left ( g_{jk}^{*} b_{jk} + g_{jk} b_{jk}^{\\dagger} \\right ),\n\\end{align}\nwhere $a_j^{\\dagger}$ ($a_j$) and $b_{jk}^{\\dagger}$ ($b_{jk}$) are the system and reservoirs creation (annihilation) operators, respectively, while $\\omega_{1k}$ and $\\omega_{2k}$ are the frequencies of the reservoirs modes. The dimensionless constant $\\alpha$ represents the coupling strength between each of the two subsystems and the their local bath, while the constants $g_{jk}$ quantify the coupling between the $j-$th oscillator ($j=1,2$) and the $k-$th mode of its respective reservoir. These quantities therefore appear in the definition of the spectral density (SD)\n\\begin{align}\n\\label{eq:SD-def}\nJ_{j} (\\omega) = \\sum_{k} |g_{jk}|^2 \\, \\delta (\\omega - \\omega_{jk}) \\ .\n\\end{align}\nIn what follows, we will the consider the case of symmetric reservoirs, i.e., $J_1(\\omega) = J_2(\\omega) \\equiv J(\\omega)$. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{system.pdf}\n\\caption{\\small{System of two uncoupled quantum harmonic oscillators interacting with their local reservoirs. The latter are characterised by the same temperature $T$ and the same spectral properties. The two parties of the systems are initially correlated and we study their dynamics under the secular approximation so that non-Markovian effects are present.}}\n\\label{fig:system}\n\\end{figure}\n\nWe would also like to work in the secular approximation by averaging over the fast oscillating terms after tracing out the environment: unlike the rotating-wave approximation, in this limit not all non-Markovian effects are washed out \\cite{PhysRevA.67.042108}.\n\nUnder these assumptions, the dynamics of this system is governed by a time-local master equation, that in the interaction picture reads as\n\\begin{equation}\n\\label{eq:ME_sec}\n\\begin{aligned}\n\\dot{\\rho} (t) = &-  \\frac{\\Delta(t) + \\gamma(t)}{2} \\sum_{j=1,2} \\left ( \\{\\adag_j a_j, \\rho\\} - 2 a_j \\rho \\adag_j \\right ) \\\\\n& - \\frac{\\Delta(t) - \\gamma(t)}{2} \\sum_{j=1,2}  \\left (\\{a_j \\adag_j, \\rho \\}- 2 \\adag_j \\rho a_j \\right ),\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ is the reduced density matrix of the global system, while the time dependent coefficients $\\Delta(t)$ and $\\gamma(t)$ account for diffusion and dissipation, respectively. \nThe coefficients in \\Cref{eq:ME_sec} have a well-defined physical meaning: $(\\Delta(t) + \\gamma(t))\/2$ is the rate associated with the incoherent loss of excitations from the system, while $(\\Delta(t) - \\gamma(t))\/2$ is the rate of incoherent pumping.\n\n\nThe coefficients $\\Delta(t)$ and $\\gamma(t)$ are ultimately related to the spectral density $J(\\omega)$ as\n\\begin{align}\n\\label{eq:QBM_delta}\n\\Delta(t) \\equiv \\frac{1}{2} \\int_{0}^{t}  \\kappa (\\tau) \\cos{(\\omega_0 \\tau}) \\, d \\tau,\n\\end{align}\n\\begin{align}\n\\label{eq:QBM_gamma}\n\\gamma (t) \\equiv \\frac{1}{2} \\int_{0}^{t}  \\mu (\\tau)  \\sin{(\\omega_0 \\tau})  \\, d \\tau,\n\\end{align}\nwhere $\\kappa(\\tau)$ and $\\mu(\\tau)$ are the noise and dissipation kernels, respectively, which -- assuming reservoirs in thermal equilibrium -- are given by  \n\n\n\\begin{equation}\n\\label{eq:kappamu}\n\\left[\n\\begin{matrix}\n\\kappa(\\tau)\\\\\n\\mu(\\tau)\n\\end{matrix}\\right]=\n2 \\alpha^2 \\int\\limits_{0}^{\\mathcal{1}} J (\\omega) \\left[\\begin{matrix}\n\\cos{(\\omega \\tau)} \\coth{\\left(\\frac{\\beta}{2} \\omega \\right)}\\\\\n\\sin(\\omega\\tau)\\end{matrix}\\right] d \\, \\omega,\n\\end{equation}\nwhere $\\beta = (k_B T)^{-1}$ is the inverse temperature and $k_B$ the Boltzmann constant.\n\nMoreover, it can be shown that the dynamics of a harmonic system that is linearly coupled to an environment can be described in terms of a differential equation in the Lyapunov form given by \\Cref{eq:Lyapunov} \\cite{Serafini:17}. We can indeed notice that in \\Cref{eq:Hamiltonian} the interaction between each harmonic oscillator and the local reservoir is expressed by a Hamiltonian that is bilinear (i.e., quadratic) in the system and reservoir creation and annihilation operators. Hamiltonians of this form lead to a master equation as in \\Cref{eq:ME_sec}, where the dissipators  are quadratic in the system creation and annihilation operators $\\adag_j, a_j$. Under these conditions, one can recast the dynamical equations in the Lyapunov form in \\Cref{eq:Lyapunov}  \\cite{Ferraro:05}, where the matrices $\\mathbf{A}$ and $\\mathbf{D}$ are time-dependent, due to non-Markovianity. Indeed, we get $\\mathbf{A} = -\\gamma(t) \\mathbbm{1}_4$ and $\\mathbf{D}= 2 \\Delta(t)  \\mathbbm{1}_4$ (here $ \\mathbbm{1}_4$ is the $4 \\times 4$ identity matrix).\n\nThe resulting Lyapunov equation can be analytically solved, giving the following closed expression for the CM at a time $t$:\n\\begin{align}\n\\label{eq:sigma_t}\n\\boldsymbol{\\sigma}(t) = \\boldsymbol{\\sigma}(0) e^{-\\Gamma(t)} + 2 \\Delta_{\\Gamma}(t) \\mathbbm{1}_4,\n\\end{align}\nwith\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:del_gamma}\n\\Gamma(t) \\equiv 2 \\int_{0}^{t}  d \\tau \\,  \\gamma(\\tau)\\,\\,\n\\text{and}\\,\\,\n\\Delta_{\\Gamma} (t) \\equiv e^{-\\Gamma(t)} \\int_{0}^{t} d \\tau \\, \\Delta(\\tau) e^{\\Gamma(\\tau)}.\n\\end{aligned}\n\\end{equation}\nMoreover, a straightforward calculation allows us to determine the steady state of our two-mode system. By imposing $\\dot{\\boldsymbol{\\sigma}} \\equiv 0$ in  \\Cref{eq:Lyapunov}, one obtains that the system relaxes towards a diagonal state with associated CM ${\\boldsymbol{\\sigma}}_{\\mathcal{1}} \\equiv \\Delta(\\infty)\/\\gamma(\\infty) \\mathbbm{1}_4$. By plugging ${\\boldsymbol{\\sigma}}_{\\mathcal{1}}$ in \\Cref{eq:entropy_prod_rate}, we find $\\Pi_\\infty\\equiv\\lim_{t\\rightarrow \\infty}\\Pi(t)=0$, showing a vanishing entropy production at the steady state. This instance can also be justified by noticing that, as $t \\to \\mathcal{1}$, we approach the Markovian limit. Therefore, the Brownian particles, exclusively driven by the interaction with their local thermal baths, will be relaxing toward the canonical Gibbs state with a vanishing associated entropy production rate \\cite{Breuer-Petruccione, Spohn1978, Landi:19}.\n\n\n\\subsection{Choice of the spectral density} \nIn order to obtain a closed expression for the time-dependent rates $\\Delta(t)$ and $\\gamma(t)$, one has to assume a specific form for the spectral density $J(\\omega)$, which -- to generate an irreversible dynamics -- is assumed to be a continuous function of the frequency $\\omega$. In quite a general way, we can express the SD as\n\\begin{align}\nJ(\\omega) = \\eta \\ \\omega_c^{1-\\epsilon} \\ \\omega^\\epsilon \\; f(\\omega, \\omega_c),\n\\end{align}\nwhere $\\epsilon>0$ is known as the Ohmicity parameter and $\\eta > 0$. Depending on the value of $\\epsilon$, the SD is said to be Ohmic ($\\epsilon=1$), super-Ohmic ($\\epsilon>1)$, or sub-Ohmic ($\\epsilon<1$). The function $f(\\omega, \\omega_c)$ represents the SD cut-off and $\\omega_c$ is the cut-off frequency. Such  function is introduced so that  $J(\\omega)$ vanishes for $\\omega \\to 0$ and $\\omega \\to \\mathcal{1}$. We focus on two different functional forms for $f(\\omega, \\omega_c)$, namely, the Lorentz-Drude cut-off $f(\\omega, \\omega_c) \\equiv \\omega_c^2 \/ (\\omega_c^2 + \\omega^2)$ and the exponential cut-off $f(\\omega, \\omega_c) \\equiv e^{-\\omega\/\\omega_c}$.\nIn particular, we choose an Ohmic SD with a Lorentz-Drude cut-off\n\\begin{align}\n\\label{eq:SD_Ohm_LD}\nJ(\\omega) = \\frac{2 \\omega}{\\pi} \\frac{\\omega_c^2}{\\omega_c^2 + \\omega^2},\n\\end{align}\nwhere $\\eta \\equiv 2 \/ \\pi$. Note that this choice is mathematically convenient, but is inconsistent from a physical point of view, as it implies instantaneous dissipation, as acknowledged in Refs.~\\cite{PhysRevD.45.2843, PazZurek2001}.\nWe also consider the following SDs\n\\begin{align}\n\\label{eq:SD_exp}\nJ(\\omega) =  \\omega_c^{1-\\epsilon} \\ \\omega^\\epsilon \\; e^{-\\omega\/\\omega_c},\n\\end{align}\nwith $\\epsilon=1, \\, 3, \\,1\/2$ and $\\eta \\equiv 1$, as the coupling strength is already contained in the constant $\\alpha$. In all these cases, the time-dependent coefficients $\\Delta(t)$ and $\\gamma(t)$ can be evaluated analytically~\\cite{PhysRevA.80.062324}. \n\n\n\\section{Initial correlations and entropy production}\n\\label{sec:main}\nWe can now use our system to claim that initial correlations shared by the non-interacting oscillators do play a role in the entropy production rate. We do this by employing a parametrisation that covers different initial preparations~\\cite{PhysRevA.72.032334}.  The entries of the matrix given by \\Cref{eq:sigma_sf} can be expressed as follows\n\\begin{equation}\n\\label{eq:sigma_a_b}\na = s +d, \\qquad b = s-d\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:sigma_c}\nc_{\\pm} = \\frac{ \\sqrt{\\left (4d^2 + f \\right )^2{-}4g^2} \\pm  \\sqrt{\\left (4s^2 + f  \\right )^2{-}4g^2} }{4 \\sqrt{s^2 -d^2}},\n\\end{aligned}\n\\end{equation}\nwith $f = (g^2 +1)(\\lambda -1) \/2- (2d^2+g)(\\lambda +1)$. This allows us to parametrise the CM using four parameters: $s, d, g,  \\lambda$. The local purities are controlled by the parameters $s$ and $d$ as $\\mu_1 = (s+d)^{-1}$ and $\\mu_2 = (s-d)^{-1}$, while the global purity is $\\mu=1\/g$. Furthermore, in order to ensure  legitimacy of a CM, the following constraints should be fulfilled\n\\begin{align}\n\\label{eq:constraints}\ns \\ge 1, \\quad | d | \\le s -1, \\quad g \\ge 2| d | + 1.\n\\end{align}\nOnce the three aforementioned purities are given, the remaining degree of freedom required to determine the negativities is controlled by the parameter $\\lambda$, which encompasses all the possible entangled two-modes Gaussian states. The two classes of extremal states are obtained upon suitable choice of  $\\lambda$. For $\\lambda=-1$ ($\\lambda = +1$) we recover the GLEMS (GMEMS) mentioned in~\\Cref{sec:Gauss}. \n\nTo show a preview of our results, we start with a concrete case shown in \\Cref{fig1}. We prepare the system in a pure ($g=1$) symmetric ($d=0$) state, and investigate the effects of initial correlations on $\\Pi(t)$ by comparing the value taken by this quantity for such an initial preparation with what is obtained by considering the covariance matrix associated with the tensor product of the local states of the oscillators, i.e., by forcefully removing the correlations among them. Non-Markovian effects are clearly visible in the oscillations of the entropy production and lead to negative values of $\\Pi(t)$ in the first part of the evolution. This is in stark contrast with the Markovian case, which entails non-negativity of the entropy production rate. Crucially, we see that, for a fixed initial value of the local energies, the presence of initial correlations enhances the amount of entropy produced at later times, increasing the amplitude of its oscillations. We also stress that both curves eventually settle to zero (on a longer timescale than shown in \\Cref{fig1}) as argued in Sec.~\\ref{sec:QBM}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{correlations.pdf}\n\\caption{\\small{Entropy production rate in a system of two non-interacting oscillators undergoing the non-Markovian dynamics described in~\\Cref{sec:QBM}. We compare the behaviour of the entropy production rate resulting from a process where the system is initialised in a state with no initial correlations (solid line) to what is obtained starting from a correlated state (dashed-dotted line). The latter case refers to the preparation of a system in a pure ($g=1$), symmetric ($d=0$) squeezed state ($\\lambda = 1$). The former situation, instead, corresponds to taking the tensor product of the local states. In this plot we have taken $s=2$ and an Ohmic SD with Lorentz-Drude cut-off. The system parameters are  $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.1 \\omega_0^{-1}$.}}\n\\label{fig1}\n\\end{figure}\n\nWe now move to a more systematic investigation of $\\Pi(t)$ and its dependence on the specific choice of  $s, d, g,  \\lambda$.\nIn order to separate the contributions, we first study the behaviour of $\\Pi(t)$ when we vary one of those parameters, while all the others are fixed. We can first rule out the contribution of thermal noise by considering the case in which the reservoirs are in their vacuum state. Such zero-temperature limit can be problematic, as some approaches to the quantification of entropy production fail to apply in this limit~\\cite{PhysRevLett.118.220601}. In contrast, phase-space methods based on the R\\'{e}nyi-$2$-Wigner entropy allow to treat such a limit without pathological behaviours associated with such {\\it zero-temperature catastrophe}~\\cite{PhysRevLett.118.220601}. This formal consistency is preserved also in the case of a system whose dynamics is described by \\Cref{eq:ME_sec}, as shown in \\Cref{fig_vacuum}.\nWe take $T=0$ and choose an Ohmic SD with an exponential cut-off -- given by \\Cref{eq:SD_exp} -- with $\\epsilon=1$. \nThe map describing the dynamics converges to a stationary state characterised by a vanishing $\\Pi(t)$,  although the oscillations are damped to zero more slowly, as non-Markovian effects are more persistent in the presence of zero-temperature reservoirs. Furthermore, we notice that the differences between different initial states are most pronounced in correspondence of the first peak: this suggests that the maximum value for the entropy production can be reasonably chosen as an apt figure of merit to distinguish the differences due to state preparation. \nSupported by this evidence, we adopt the value of the first maximum of $\\Pi(t)$ as an indicator of the irreversibility generated in the relaxation dynamics by different initial preparations.\n\nIn the inset of \\Cref{fig_vacuum} we show the logarithmic negativity given by \\Cref{eq:log_neg}. The interaction with zero-temperature reservoirs does not cause detrimental effects to entanglement, as the latter is preserved over time \\cite{PhysRevLett.100.220401,PhysRevA.75.062119, PhysRevA.80.062324}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Ohm_vacuum_g.pdf}\n\\caption{\\small{Entropy production rates corresponding to independent zero-temperature reservoirs. We consider preparations of the initial global state corresponding to different values of parameter $g$ (related to the global purity of the state), while fixing $s=2$, $d=0$, and $\\lambda = 1$. In the inset we plot the logarithmic negativity for the same choice of parameters: entanglement persists over time up to the reach of a steady state of the dynamics. We have taken an Ohmic SD with an exponential cut-off with  $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$.}}\n\\label{fig_vacuum}\n\\end{figure}\n\n\nWe also address the case of finite-temperature reservoirs and an Ohmic SD with Lorentz-Drude cut-off given by \\Cref{eq:SD_Ohm_LD}~\\footnote{the analysis can easily be extended to the exponential cut-off in \\Cref{eq:SD_exp} for the Ohmic ($\\epsilon=1$), super-Ohmic ($\\epsilon=3$) and sub-Ohmic ($\\epsilon=1\/2$) case}. \nWe thus fix $s, d, \\lambda$ and  let $g$ vary to explore the role played by the global purity. \\Cref{fig2} shows that, by increasing $g$ -- i.e., by reducing the purity of the global state -- $\\Pi(t)$ decreases: an initial state with larger purity lies far from an equilibrium state at the given temperature of the environment and is associated with a larger degree of initial entanglement [cf. inset of \\Cref{fig2} and the analysis reported in \\Cref{sec:entanglement}], which translates in a larger entropy production rate. Furthermore, our particular choice of the physical parameters leads to the observation of ``entanglement sudden death''~\\cite{PhysRevLett.100.220401, PhysRevA.80.062324}: an initial state with non-null logarithmic negativity completely disentangles in a finite time due to interaction with environment, the disentangling time being shortened by a growing $g$ [cf. inset of \\Cref{fig2}].\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Ohm_LD_g.pdf}\n\\caption{\\small{Entropy production rates corresponding to different preparations of the initial global state. We consider different values of parameter $g$, while taking $s=2$, $d=0$, and $\\lambda = 1$. In the inset we plot the logarithmic negativity for the same choice of the parameters. The dynamics of the system has been simulated using an Ohmic SD with a Lorentz-Drude cut-off. The system parameters are  $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.1 \\omega_0^{-1}$.}}\n\\label{fig2}\n\\end{figure}\n\nSimilarly, we can bias the local properties of the oscillators by varying $d$ and, in turn, $g= 2d +1$, while keeping $s, \\lambda$ fixed: in \\Cref{fig3} we can observe that, when the global energy is fixed, the asymmetry in the local energies -- and purities $\\mu_1$ and $\\mu_2$ -- reduces the entropy production rate. In the inset we show that, by increasing the asymmetry between the two modes, the entanglement takes less time to die out. These results are consistent with the trends observed in \\Cref{fig2}.\nIndeed, a bias in the local energies would make the reduced state of one of the two oscillators more mixed, and thus less prone to preserve the entanglement that is initially set in the joint harmonic state. Such imbalance would give different weights to the two local dissipation processes, thus establishing an effective preferred local channel for dissipation. In turn, this would result in a lesser weight to the contribution given by correlations.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Ohm_LD_d.pdf}\n\\caption{\\small{Entropy production rates corresponding to different values of $d$ and $g=2d+1$ in the parametrisation of the initial state (we have taken $s=4$ and $\\lambda = 1$). In the inset, the behaviour of the logarithmic negativity is shown. In this figure, we take an Ohmic SD with a Lorentz-Drude cut-off and $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.1 \\omega_0^{-1}$.}}\n\\label{fig3}\n\\end{figure}\n\nWe conclude our analysis in this Section by exploring the parameter space in a more systematic way by fixing the global energy $s$ and randomly choosing the three parameters left, provided that the constraints in \\Cref{eq:constraints} are fulfilled. We see from \\Cref{fig4} that the curve for $\\Pi(t)$ comprising all the others is the one corresponding to unit global purity, i.e., $g=1$, and $d=0$, $\\lambda=1$ (dashed line). The globally pure state is indeed the furthest possible from a diagonal one: the rate at which  entropy production varies is increased in order to reach the final diagonal state ${\\boldsymbol{\\sigma}}_{\\mathcal{1}}$. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Ohm_LD_pure.pdf}\n\\caption{\\small{Entropy production rates $\\Pi(t)$ (absolute value)  as a function of time. The initial CM is parametrised by fixing $s$ ($s=10$ in the figure) and randomly choosing $d, g, \\lambda$ such that they are uniformly distributed in the intervals $[0,s-1]$, $[2d+1,d+10]$ and $[-1,1]$ respectively. The figure reports $N_{R} = 1000$ different realisations of the initial state. The dashed line corresponds to the globally pure state ($g=1$) with $d=0$, $\\lambda=1$. All the plots are obtained considering an Ohmic SD with a Lorentz-Drude cut-off and $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.1 \\omega_0^{-1}$.}}\n\\label{fig4}\n\\end{figure}\n\n\\subsection{Dependence on the initial entanglement}\n\\label{sec:entanglement}\n\nWe now compare the trends corresponding to different choices of the parameters characterising the initial state. As non-Markovian effects are reflected in oscillating behaviour of the entropy production, we can contrast cases corresponding to different initial preparations by looking at the maximum and the minimum values $\\Pi_{\\rm max}$ and $\\Pi_{\\rm min}$ that the entropy production rate assumes for each choice of the parameters. Taking into account the evidence previously gathered, in the simulations reported in this Section we fix the minimum value for $g$, i.e., $g=2d+1$, and $\\lambda = +1$ as significant for the points that we want to put forward. In fact, with such choices we are able to parametrise the initial state with a minimum number of variables, while retaining the significant features that we aim at stressing. We can further assume, without loss of generality, $d \\ge 0$: this is simply equivalent to assuming that the first oscillator is initially prepared in a state with a larger degree of mixedness than the second one, i.e., $\\mu_1 \\le \\mu_2$. In this case, we can express $d$ in terms of the smallest symplectic eigenvalue of the partially transposed CM $\\tilde{\\nu}_{-}$. Therefore, taking into account the constraints given by \\Cref{eq:constraints}, one has that $d = -\\frac{1}{2} ( \\tilde{\\nu}_{-}^2 - 2s \\tilde{\\nu}_{-} +1)$.\nWe already mentioned in \\Cref{sec:QBM} that we are able to derive a closed expression for the CM at any time $t$, given by \\Cref{eq:sigma_t}. We can further notice that the positive and negative peaks in the entropy production rate are attained at short times. We can thus perform a Taylor expansion of $\\Delta(t)$ in \\Cref{eq:del_gamma} to obtain\n\\begin{equation}\n\\begin{aligned}\n\\Delta_{\\Gamma} (t) =  \n[1- \\Gamma(t)]  \\int_{0}^{t} d \\tau \\Delta(\\tau) + \\int_{0}^{t} d \\tau \\; \\Gamma(\\tau) \\Delta(\\tau)  + \\mathcal{O} (\\alpha^4) \\ .\n\\end{aligned}\n\\end{equation}\nAs $\\Delta(t) \\propto \\alpha^2$ and $\\Gamma(t) \\propto \\alpha^2$, we can retain only the first term consistently with the weak coupling approximation we are resorting to. Therefore, we can recast \\Cref{eq:sigma_t} in a form that is more suitable for numerical evaluations, namely\n\\begin{align}\n\\label{eq:sigma_t_wc}\n\\boldsymbol{\\sigma}(t) = \\left [ 1 - \\Gamma(t) \\right ]  \\boldsymbol{\\sigma}(0) + \\left [ 2 \\int_{0}^{t} d \\tau \\Delta(\\tau) \\right ] \\mathbbm{1}_4 \\ .\n\\end{align}\nBy substituting \\Cref{eq:sigma_t_wc} into \\Cref{eq:entropy_prod_rate}, we get the analytic expression for the entropy production rate, given in \\Cref{app:a} for the sake of completeness but whose explicit form is not crucial for our analysis here. \n\nIn this way, all the information about the initial state is encoded in the value of $\\tilde{\\nu}_{-}$ while $s$ is fixed. Note that this expression holds for any SD: once we choose the latter, we can determine the time-dependent coefficients $\\Delta(t)$ and $\\gamma(t)$ and thus the entropy production rate $\\Pi(t)$. We can then compute the maximum of the entropy production rate and study the behaviour of $\\Pi_{\\rm max}$ and $\\Pi_{\\rm min}$ as functions of the entanglement negativity $E_{\\mathcal{N}}$  at $t=0$. In \\Cref{fig5} we compare numerical results to the curve obtained by considering the analytical solution discussed above and reported in \\Cref{eq:entropy_prod_rate_general}. Remarkably, we observe a monotonic behaviour of our chosen figure of merit with the initial entanglement negativity: the more entanglement we input at $t=0$ the higher the maximum of the entropy production rate is. We can get to the same conclusion (in absolute value) when we consider the negative peak $\\Pi_{\\rm min}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Ohm_LD_in_neg.pdf}\n\\caption{\\small{Maximum and minimum of the entropy production rate $\\Pi_{\\rm max}$ and $\\Pi_{\\rm min}$  as functions of the  entanglement negativity at $t=0$. We take $s=4$, $g=2d +1$, $\\lambda = 1$, while $0 \\le d \\le 3$. We compare the numerical results (triangles and circles) to the analytical solution in \\Cref{eq:entropy_prod_rate_general} (solid line). We have used an Ohmic SD with a Lorentz-Drude cut-off and $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.01 \\omega_0^{-1}$.}}\n\\label{fig5}\n\\end{figure}\n\nThe monotonic behavior highlighted above holds regardless of the specific form of the spectral density.  In \\Cref{fig6} we study $\\Pi_{\\rm max}$ against the smallest symplectic eigenvalue $\\tilde{\\nu}_{-}$ of the partially transposed CM for the various spectral densities we have considered, finding evidence of a power law of the form $\\Pi_{\\rm max} \\propto \\tilde{\\nu}_{-}^{\\delta}$, where the exponent $\\delta$ depends on the reservoir's spectral properties.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Exponents.pdf}\n\\caption{\\small{Plot of $\\Pi_{\\rm max}$ against the smallest symplectic eigenvalue of the partially transposed CM at $t=0$ (logarithmic scale) for different SDs (as stated in the legend). The initial state is prepared using the same parametrisation chosen in \\Cref{fig5}, while we have taken $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.1 \\omega_0^{-1}$.}}\n\\label{fig6}\n\\end{figure}\n\n\n\n\\section{Markovian limit}\n\\label{sec:Markovian_Limit}\nWe are now interested in assessing whether the analytical and numerical results gathered so far bear dependence on the non-Markovian character of the dynamics. With this in mind, we explore the Markovian limit, in which the problem is fully amenable to an analytical solution, that can also be used to validate our numerical results. Such limit is obtained by simply choosing an Ohmic SD with a Lorentz-Drude regularisation -- \\Cref{eq:SD_Ohm_LD} --  and taking the long time and high temperature limits, i.e. $\\omega_0 t \\gg 1$ and $\\beta^{-1} \\gg \\omega_0$. This yields the time-independent coefficients\n\\begin{align}\n\\frac{\\Delta(t) - \\gamma(t)}{2}& \\longrightarrow \\gamma_M \\left ( 2 \\bar{n}(\\omega_0) + 1\\right ) , \\\\\n\\frac{\\Delta(t) + \\gamma(t)}{2}& \\longrightarrow \\gamma_M \\, \\bar{n}(\\omega_0) \\ ,\n\\end{align}\nwhere $\\bar{n}(\\omega_0) = (e^{\\beta \\omega_0} -1)^{-1}$ is the average number of excitations at a given frequency $\\omega_0$, whereas $\\gamma_M \\equiv 2 \\alpha^2 \\omega_c^2  \\omega_0 \/ (\\omega_c^2 + \\omega_0^2)$. \nTherefore, \\Cref{eq:ME_sec} reduces to a master equation describing the dynamics of two uncoupled harmonic oscillators undergoing Markovian dynamics, for which we take $\\mathbf{A} = -\\gamma_M \\mathbbm{1}_4$ and $\\mathbf{D}= 2 \\gamma_M (2 \\bar{n}(\\omega_0) +1)  \\mathbbm{1}_4$ in \\Cref{eq:Lyapunov}.\n\nWorking along the same lines  as in the non-Markovian case, we study the behavior of $\\Pi(t)$ by suitably choosing the parameters encoding the preparation of the initial state. For example, in \\Cref{fig7} we plot the entropy production rate as a function of time for different values of $g$. The limiting procedure gives back a coarse-grained dynamics monotonically decreasing toward the thermal state, to which it corresponds a non-negative entropy production rate, asymptotically vanishing in the limit $t \\to \\mathcal{1}$ . Moreover, the memoryless dynamics leads to a monotonic  decrease of the entanglement negativity, as shown in the inset of \\Cref{fig7}.\nIn this case, the globally pure state ($g=1$, dashed line in \\Cref{fig8})  still plays a special role: all the curves corresponding to value of $g$ smaller that the unity remains below it.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Markov_g.pdf}\n\\caption{\\small{Entropy production rates corresponding to different preparations of the initial global state in the Markovian limit. We have taken different values of $g$ (thus varying the global purity of the state of the system) with $s=2$, $d=0$, $\\lambda = 1$, $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.01 \\omega_0^{-1}$.}}\n\\label{fig7}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Markov_pure.pdf}\n\\caption{\\small{Entropy production rate $\\Pi(t)$  plotted against the dimensionless time $\\omega_0 t$ in the Markovian regime. The initial CM is parametrised by setting $s=10$ and randomly sampling (in a uniform manner) $d, g, \\lambda$ from the intervals $[0,s-1]$, $[2d+1,d+10]$ and $[-1,1]$, respectively. We present $N_{R} = 100$ different realisations of the initial state. The dashed line represent the state with unit global purity ($g=1$) and $d=0, \\; \\lambda=1$. For the dynamics, we have taken $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.01 \\omega_0^{-1}$.}}\n\\label{fig8}\n\\end{figure}\n\nThe Markovian limit provides a useful comparison in terms of integrated quantities. In this respect, we can study what happens to the entropy production $\\Sigma = \\int_{0}^{+\\mathcal{1}} \\Pi(t) dt$. Although the non-Markovian dynamics entails the negativity of the entropy production rate in certain intervals of time, the overall entropy production is larger than the quantity we would get in the corresponding Markovian case, as can be noticed in \\Cref{fig9}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{sigma.pdf}\n\\caption{\\small{Entropy production in the non-Markovian case (solid line) as a function $E_{\\cal N}(t=0)$, compared with its counterpart achieved in the corresponding Markovian limit (dashed line).  We have taken $s=2$, $d=0$, $\\lambda = 1$, $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.01 \\omega_0^{-1}$.}}\n\\label{fig9}\n\\end{figure}\n\nFinally, we can study the dependence of the Markovian entropy production rate on the initial entanglement. Note that, in this limit, \\Cref{eq:entropy_prod_rate} yields an analytic expression for the entropy production rate at a generic time $t$, which we write explicitly in \\Cref{eq:entropy_prod_rate_Markov}. From our numerical inspection, we have seen that the entropy production rate is maximum at $t=0$, so that\n\\begin{align}\n\\label{eq:entropy_prod_rate_Markov_max}\n\\Pi_{\\rm max} \\equiv \\Pi(0) & = - 8 \\gamma_M + 4 s \\, \\gamma_M \\tanh\\left (\\frac{\\beta \\omega_0}{2} \\right ) \\nonumber \\\\ \n& +\\frac{4 s \\, \\gamma_M \\coth \\left (\\frac{\\beta \\omega_0}{2} \\right )}{ (2 s -\\tilde{\\nu}_{-}) \\tilde{\\nu}_{-}} \\ .\n\\end{align} \n\nIf we fix the parameter $s$ and plot $\\Pi_{\\rm max}$ against $\\tilde{\\nu}_{-}$, we can contrast analytical and numerical results [cf.~\\Cref{fig10}]. We can draw the same conclusion as in the non-Markovian case: the more entanglement we input, the higher the entropy production rate.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.83\\linewidth]{Markov_nu.pdf}\n\\caption{\\small{Markovian limit: maximum entropy production rate $\\Pi_{\\rm max}$  as a function of the minimum symplectic eigenvalue of the partially transposed CM at $t=0$. We have taken $s=4$, $g=2d +1$, $\\lambda = 1$, $0 \\le d \\le 3$, $\\alpha = 0.1 \\omega_0$, $\\omega_c = 0.1 \\omega_0$, $\\beta = 0.1 \\omega_0^{-1}$. We compare the curve obtained numerically (triangles) to the analytical trend (solid line) found through \\Cref{eq:entropy_prod_rate_Markov_max}.}}\n\\label{fig10}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe have studied -- both numerically and analytically -- the dependence of the entropy production rate on the initial correlations between the components of a composite system. We have considered two non-interacting oscillators exposed to the effects of local thermal reservoirs. By using a general parametrisation of the initial state of the system, we have systematically explored different physical scenarios. We have established that correlations play an important role in the rate at which entropy is intrinsically produced during the process. Indeed, we have shown that, when the system is prepared in a globally pure state,  we should expect a higher entropy production rate. This is the case -- regardless of the spectral density chosen -- for initial entangled states of the oscillators: larger initial entanglement is associated with higher rates of entropy production, which turns out to be a monotonic function of the initial degree of entanglement. \nRemarkably, our analysis takes into full consideration signatures of non-Markovianity in the open system dynamics.  \n\nIt would be interesting, and indeed very important, to study how such conclusions are affected by the possible interaction between the constituents of our system, a situation that is currently at the focus of our investigations, as well as non-Gaussian scenarios involving either non-quadratic Hamiltonians or spin-like systems.\n\n\\acknowledgements\nWe thank R. Puebla for insightful discussions and valuable feedback about the work presented in this paper.\nWe acknowledge support from the H2020 Marie Sk{\\l}odowska-Curie COFUND project SPaRK (Grant nr.~754507), the H2020-FETPROACT-2016 HOT (Grant nr.~732894), the H2020-FETOPEN-2018-2020 project TEQ (Grant nr.~766900), the DfE-SFI Investigator Programme (Grant 15\/IA\/2864), COST Action CA15220, the Royal Society Wolfson Research Fellowship (RSWF\\textbackslash R3\\textbackslash183013) and International Exchanges Programme (IEC\\textbackslash R2\\textbackslash192220), the Leverhulme Trust Research Project Grant (Grant nr.~RGP-2018-266). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{H\\,{\\scshape ii}~regions}, that are an outcome of the photoionization of newly forming high-mass stars ($\\textup{M} \\gtrsim 8~M_{\\odot}$), not only play a crucial role in understanding processes involved in high-mass star formation but also reveal the various feedback mechanisms at play on the surrounding ambient interstellar medium (ISM) and the natal molecular cloud. Numerous observational and theoretical studies of {H\\,{\\scshape ii}~regions} have been carried out in the last two decades. However, dedicated multiwavelength studies of star-forming complexes add to the valuable observational database that provide a detailed and often crucial insight into the intricacies involved.\n \nIn this work, we study the massive star-forming region IRAS 17149$-$3916. This region is named RCW 121 in the catalog of $\\rm H{\\alpha}$ emission in the Southern Milky Way \\citep{1960MNRAS.121..103R}. The mid-infrared (MIR) dust bubble, S6, from the catalog of \\citet{2006ApJ...649..759C} is seen to be associated with this complex. IRAS 17149$-$3916 has a bolometric luminosity of $\\sim~9.4 \\times 10^4~L_{\\odot}$ \\citep{2006A&A...447..221B}. In literature, several kinematic distance estimates are found for this complex. The near and far kinematic distance estimates range between 1.6 -- 2.2 and 14.5 -- 17.7~kpc, respectively \\citep{{1997MNRAS.291..261W},{2004ApJS..154..553S},{2006A&A...447..221B},{2010ApJ...716.1478W}}. In a recent paper, \\citet{2014MNRAS.437..606T} use the spectral classification of the candidate ionizing star along with near-infrared (NIR) photometry to place this complex at 2~kpc. This is in agreement with the near kinematic distance estimates and conforms to the argument of  \\citet{2006A&A...447..221B} for assuming the near kinematic distance of 2.1~kpc based on the height above the Galactic plane. Based on the above discussion, we assume a distance of 2.0~kpc in this work. \n\nThis star-forming region has been observed as part of several radio continuum surveys at 2.65~GHz \\citep{1969AuJPA..11...27B}, 4.85~GHz \\citep{1994ApJS...91..111W}, and more recently at 18 and 22.8~GHz by \\citet{2013A&A...550A..21S}. Using NIR data, \\citet{2006AJ....131..951R} detect a cluster of young massive stars associated with this IRAS source. These authors also suggest IRS-1, the brightest source in the cluster, to be the likely ionizing star of the {H\\,{\\scshape ii}~region} detected in radio wavelengths. \n\\citet{2008A&A...486..807A} probed the $^{12}\\rm CO$ molecular gas in the region. Based on this observation, they conclude that RCW 121 and RCW 122 are possibly unrelated star-forming regions belonging to a giant molecular complex while negating the speculation of these being triggered by Wolf-Rayet stars located in the HM~1 cluster. In the most recent work on this source, \\citet{2014MNRAS.437..606T} re-visit the cluster detected by \\citet{2006AJ....131..951R}. \nThese authors also detect three bright {\\it Herschel} clumps, the positions of which coincide with three of the 1.2~mm clumps of \\citet{2006A&A...447..221B}.    \n\nIntroducing the IRAS 17149$-$3916 complex, in Fig.~\\ref{fig_intro}, we show the colour composite image of the associated region. The 5.8\\,$\\rm \\mu m$ IRAC band, which is mostly a dust tracer \\citep{2010ApJ...716.1478W}, displays an almost closed, elliptical ring emission morphology. The extent of the bubble, S6, as estimated by \\citet{2006ApJ...649..759C} traces this. The cold dust component, revealed by the \\textit{Herschel} 350\\,$\\rm \\mu m$ emission, is distributed along the bubble periphery with easily discernible cold dust clumps. Ionized gas sampled in the SuperCosmos $\\rm H{\\alpha}$ survey \\citep{2005MNRAS.362..689P} fills the south-west part of the bubble and extends beyond towards south. MIR emission at 21\\,$\\rm \\mu m$ is localized towards the south-west rim of the bubble. This emission is seen to spatially correlate with the central, bright region of ionized gas (see Fig.~\\ref{radioim})\nand is generally believed to be due to the Ly$\\alpha$ heating of dust grains to temperatures of around 100~K \\citep{1991MNRAS.251..584H}.\n\nIn this paper, we present an in-depth multiwavelength study of this star-forming region. In discussing the investigation carried out, we present the radio observations and the related data reduction procedure followed in Section \\ref{obs-data}. This section also briefly discusses the salient features of the archival data used in the study. Section \\ref{results} presents the results obtained for the associated ionized gas and dust environment. Section \\ref{discussion} delves into the detailed discussion and interpretation related to the observed morphology of the ionized gas, investigation of the pillar structures, dust clumps in the realm of triggered star formation, and the nature of the detected dust clumps and cores. Section \\ref{conclusion} highlights the main results obtained in this study. \n \n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.5]{Fig1.eps}\n\\caption{Colour-composite image towards IRAS 17149$-$3916 with the MSX 21\\,$\\rm \\mu m$ (red), SuperCosmos $\\rm H_{\\alpha}$ (blue) and IRAC 5.8 \\,$\\rm \\mu m$ (green) are shown overlaid with the contours of the  \\textit{Herschel} 350\\,$\\rm \\mu m$ map. The contour levels are 600, 700, 1000, 1500, 2000, 2500, 4000, and 6000~MJy\/sr. The ellipse shows the position and the extent of the bubble, S6, as identified by \\citet{2006ApJ...649..759C}.}\n\\label{fig_intro}\n\\end{figure*}\n\n\\section{Observation, data reduction and archival data} \\label{obs-data}\n\\subsection{Radio Continuum Observation } \\label{radio_obs}\nTo probe the ionized component of IRAS 17149$-$3916, we have carried out low-frequency radio continuum observations of the region at 610 and 1280~MHz with the Giant Meterwave Radio Telescope (GMRT), Pune, India. GMRT offers a hybrid configuration of 30 antennas (each of diameter 45~m) arranged in a Y-shaped layout. The three arms contain 18 evenly placed antennas and provide the largest baseline of $\\sim 25$~km. The central $\\rm 1\\,km^2$ region houses 12 antennas that are randomly oriented with shortest possible baseline of $\\sim 100$\\,m.\nA comprehensive overview of GMRT systems can be found in \\citet{1991ASPC...19..376S}.\nThe target was observed with the full array for nearly full-synthesis to maximize the {\\it uv} coverage which is required to detect the extended, diffuse emission. Observations were carried out during August 2014 at 610 and 1280\\,MHz with a bandwidth of 32 MHz over 256 channels. In the full-array mode, the resolution is $\\sim$5 and 2~arcsec and the largest detectable spatial scale is $\\sim$17 and 7~arcmin at 610 and 1280~MHz, respectively. Radio sources 3C 286 and 3C 48 were selected as the primary flux calibrators. The phase calibrator, 1714-252, was observed after each 40-min scan of the target to calibrate the phase and amplitude variations over the full observing run. The details of the GMRT radio observations and constructed radio continuum maps are listed in Table \\ref{radio_tab}.\n\nAstronomical Image Processing System (AIPS) was used to reduce the radio continuum data where we follow the procedure detailed in \\citet{2017MNRAS.472.4750D} and \\citet{2019MNRAS.485.1775I}. The data sets are carefully examined to identify bad data (non-working antennas, bad baselines, RFI, etc.), employing the tasks, {\\tt TVFLG} and {\\tt UVPLT}. Following standard procedure, the gain and bandpass calibration is carried out after flagging bad data. Subsequent to bandpass calibration, channel averaging is done while keeping bandwidth smearing negligible. Continuum maps at both frequencies are generated using the task {\\tt IMAGR}, adopting wide-field imaging technique to account for w-term effects. Several iterations of self-calibration (phase-only) are performed to minimize phase errors and improve the image quality. Subsequently, primary beam correction was carried out for all the generated maps. \n\nIn order to obtain a reliable estimate of the flux density, contribution from the Galactic diffuse emission needs to be accounted for. This emission follows a power-law spectrum with a steep negative spectral index of $-2.55$ \\citep{1999A&AS..137....7R} and hence has a significant contribution at the low GMRT frequencies (especially at 610~MHz). This results in the increase in system temperature, which becomes particularly crucial when observing close to the Galactic plane as is the case with our target, IRAS 17149$-$3916. The flux calibrators lie away from the Galactic plane and for such sources at high Galactic latitudes, the Galactic diffuse emission would be negligible. This makes it essential to quantify the system temperature correction to be applied in order to get an accurate estimate of the flux density. Since measurement of the variation in the system temperature of the antennas at GMRT are not automatically implemented during observations, we adopt the commonly used Haslam approximation discussed in \\citet{2015MNRAS.451...59M} and implemented in \\citet{2019MNRAS.485.1775I}.\n\nThe sky temperature, ${T_{\\rm sky}}$, at frequency $\\nu$ for the location of our source is determined using the equation\n\\begin{equation}\nT_{\\rm sky,\\nu} = T_{\\rm sky}^{408}\\bigg(\\frac{\\nu}{408~\\textrm{MHz}}\\bigg)^\\gamma\n\\end{equation}\n\\noindent\nwhere $\\gamma = -2.55$ is the spectral index of the Galactic diffuse emission and $\\it{T_{\\rm sky}^{\\rm 408}}$ \nis the sky temperature at 408~MHz obtained from the all-sky 408~MHz survey of \\citet{1982A&AS...47....1H}. Using this method, we estimate the scaling factors of 2.2 and 1.24 at 610 and 1280~MHz, respectively, which are used to rescale and obtain the final maps.\n\\begin{table}\n\\caption{Details of the GMRT radio continuum observations.} \n\\begin{center}\n\\label{radio_tab}\n\\begin{tabular}{lll}\n\\\\ \n\\hline \\hline\n & 610 MHz & 1280 MHz \\\\\n\\hline\nDate of Obs. & 8 August 2014 & 14 August 2014 \\\\\nFlux Calibrators & 3C286, 3C48 & 3C286, 3C48 \\\\\nPhase Calibrators & 1714-252 & 1714-252 \\\\\nOn-source integration time  & $\\sim$4~h & $\\sim$4~h \\\\\nSynth. beam & 10.69\\arcsec$\\times$6.16\\arcsec & 4.41\\arcsec$\\times$2.24\\arcsec \\\\\nPosition angle. (deg) & 7.04 & 6.37 \\\\\n{\\it rms} noise (mJy\/beam) & 0.41 & 0.07 \\\\\nInt. Flux Density (Jy) & $14.1 \\pm 1.4$  & $12.6 \\pm 1.3$   \\\\\n{\\small (integrated within $3\\sigma$ level)} & &  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Other available data}\\label{data_archive}\n\\subsubsection{Near-infrared data from 2MASS} \\label{data_2mass}\nNIR ($\\rm JHK_s$) data for point sources around our region of interest has been obtained from the Two Micron All Sky Survey (2MASS) Point Source Catalog (PSC). Resolution of 2MASS images is $\\sim$5.0 arcsec. We select those sources for which the ``read-flag'' values are 1 - 3 to ensure a sample with reliable photometry. This data is used to study the stellar population associated with IRAS 17149$-$3916.\n\n\\subsubsection{Mid-infrared data from Spitzer} \\label{data_spitzer}\n\nThe MIR images towards IRAS 17149$-$3916 are obtained from the archives of the Galactic Legacy Infrared Midplane Survey Extraordinaire (GLIMPSE) survey of the {\\it Spitzer} Space Telescope. We retrieve images towards the region in the four Infrared Array Camera (IRAC; \\citealt{2004ApJS..154...10F}) bands (3.6, 4.5, 5.8, 8.0\\,$\\rm \\mu m$). These images have an angular resolution of $\\lesssim 2$ arcsec with a pixel size of $\\sim 0.6$ arcsec. We utilize these images in our study to present the morphology of the MIR emission associated with the region.\n\n\\subsubsection{Far-infrared data from Herschel} \\label{data_herschel}\nThe far-infrared (FIR) data used in this paper have been obtained from the {\\it Herschel} Space Observatory archives. Level 2.5 processed 70 - 500\\,$\\rm \\mu m$ images from Spectral and Photometric Imaging Receiver (SPIRE; \\citealt{2010A&A...518L...3G}) and JScanam images from the Photodetector Array Camera and Spectrometer (PACS; \\citealt{2010A&A...518L...2P}), that were observed as part of the {\\it Herschel} infrared Galactic plane Survey (Hi-GAL; \\citealt{2008A&A...481..345M}), were retrieved. We use the FIR data to examine cold dust emission and investigate the cold dust clumps in the regions.\n\n\\subsubsection{Atacama Large Millimeter Array archival data} \\label{data_alma}\nWe make use of the 1.4\\,mm (Band~6) continuum maps obtained from the archives of {\\it Atacama Large Millimeter Array (ALMA)} to identify the compact dust cores associated with IRAS 17149$-$3916. These observations were made in 2017 (PI: A.Sanchez-Monge \\#2016.1.00191.S) using the extended 12m-Array configuration towards four pointings, S61, S62, S63, and S64. Each of these pointings sample different regions of the IRAS 17149$-$3916 complex. The retrieved maps have an angular resolution of 1.4\\,arcsec $\\times$ 0.9\\,arcsec and a pixel scale of 0.16\\,arcsec.\n\n\\subsubsection{Molecular line data from MALT90 survey}\n\nThe Millimeter Astronomy Legacy Team 90 GHz survey (MALT90; \\citealt{{2011ApJS..197...25F},{2013PASA...30...57J}}) was carried out using the Australia Telescope National Facility (ATNF) Mopra 22-m telescope with an aim to characterize molecular clumps associated with {H\\,{\\scshape ii}~regions}. The survey dataset contains molecular line maps of more than 2000 dense cores lying in the plane of the Galaxy, the corresponding sources of which were taken from the ATLASGAL 870\\,$\\rm \\mu m$ continuum survey. The MALT90 survey covers 16 molecular line transitions lying near 90 GHz with a spectral resolution of $0.11$ km s$^{-1}$ and an angular resolution of 38\\,arcsec. In this study we use the optically thin $\\rm N_2 H^+$ line spectra to carry out the virial analysis of the detected dust clumps associated with this complex.\n\n\\section{Results}\n\\label{results}\n\\subsection{Ionized gas} \\label{ionized}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.3]{Fig2a.eps}\n\\includegraphics[scale=0.3]{Fig2b.eps}\n\\caption{{\\it Top}: 610\\,MHz GMRT map of the region associated with IRAS 17149$-$3916 overlaid with the 3$\\sigma$ ($\\sigma = 0.6\\,\\rm mJy\\,beam^{-1}$) level contour in light blue. White contours correspond to the 18\\,GHz emission mapped with ATCA \\citep{2013A&A...550A..21S}, with contour levels starting from 3$\\sigma$ and increasing in steps of 8$\\sigma$ ($\\sigma = 1.2\\times 10^{-2}\\,\\rm Jy\\,beam^{-1}$). {\\it Bottom:} Same as the top panel but for 1280\\,MHz GMRT ($\\sigma = 0.1\\,\\rm mJy\\,beam^{-1}$) and 22.8\\,GHz ATCA map with the contour levels starting from 3$\\sigma$ and increasing in steps of 5$\\sigma$ ($\\sigma = 1.2\\times 10^{-2}\\,\\rm Jy\\,beam^{-1}$). The GMRT maps presented here have been convolved to a resolution of 12\\,arcsec for 610~MHz and 5\\,arcsec for 1280~MHz. The beams sizes of the GMRT and ATCA maps are shown in the lower left and right hand corner, respectively.}\n\\label{radioim}\n\\end{figure}\nWe present the first low-frequency radio maps of the region associated with IRAS 17149$-$3916 obtained using the GMRT. The continuum emission mapped at 610 and 1280~MHz are shown in Fig.~\\ref{radioim}. The ionized gas reveals an interesting, large-extent cometary morphology where the head lies in the west direction with a fan-like diffuse tail opening in the east. The tail has a north-south extension of $\\sim$ 6 arcmin. The bright radio emission near the head displays a `horse shoe' shaped structure that opens towards the north-east and mostly traces the south-western portion of the dust ring structure presented in Fig.~\\ref{fig_intro}. This is enveloped within the extended and faint, diffuse emission. In addition, there are several discrete emission peaks seen at both frequencies. The {\\it rms} noise and the integrated flux density values estimated are listed in Table \\ref{radio_tab}. For the latter, the flux density is integrated within the respective 3$\\sigma$ contours. The quoted errors are estimated following the method discussed in \\citet{2018A&A...612A..36D}.\n\nAlso included in the figure are contours showing the high-frequency ATCA observations at 18 and 22.8~GHz, from \\citet{2013A&A...550A..21S}. These snapshot ($\\sim$ 10~mins) ATCA maps sample only the brightest region towards the head and the emission at 18~GHz is seen to be more extended. The GMRT and ATCA maps reveal the presence of several distinct peaks which are likely to be externally ionized density enhanced regions thus suggesting a clumpy medium. \nThe fact that some of these peaks could also be internally ionized by newly formed massive stars cannot be ruled out. Hence, further detailed study is required to understand the nature of these radio peaks. \nA careful search of the SIMBAD\/NED database rules out any possible association with background\/foreground radio sources in the line of sight. Comparing with Fig.~\\ref{fig_intro}, the ionized emission traced in the $\\rm H{\\alpha}$ image agrees well with the GMRT maps.  \\citet{2006AJ....131..951R} and \\citet{2014MNRAS.437..606T} present the ionized emission mapped in the Br$\\gamma$ line which is localized to the central part and mostly correlates with the bright emission seen in the GMRT maps. \n\nAssuming the radio emission at 1280~MHz to be optically thin and emanating\nfrom a homogeneous, isothermal medium, we derive several physical parameters of the detected {H\\,{\\scshape ii}~region} using the following expressions from \\citet{2016A&A...588A.143S},\\\\\n\n{\\it Lyman continuum photon flux ($N_{\\rm Ly}$):}\n\n\\begin{equation}\n\\begin{split}\n\\left( \\frac{N_{\\rm Ly}}{\\rm s^{-1}}\\right) = 4.771 \\times 10^{42} \\left(\\frac{F_\\nu}{\\rm Jy}\\right) \\left( \\frac{T_{\\rm e}}{\\rm K}\\right)^{-0.45} \\\\\n\\times \\left( \\frac{\\nu}{\\rm GHz}\\right)^{0.1} \\left( \\frac{D}{\\rm pc}\\right) ^{2}\n\\end{split}\n\\label{Lyman_flux}\n\\end{equation}\n\n{\\it Electron number density ($n_{\\rm e}$):}\n\\begin{equation}\n\\begin{split}\n\\left ( \\frac{n_{\\rm e}}{\\rm cm^{-3}}\\right)  = 2.576 \\times 10^6 \\left(\\frac{F_\\nu}{\\rm Jy}\\right)^{0.5} \\left( \\frac{T_{\\rm e}}{\\rm K}\\right)^{0.175} \\left( \\frac{\\nu}{\\rm GHz}\\right)^{0.05} \\\\\n\\times   \\left( \\frac{\\theta_{\\rm source}}{\\rm arcsec}\\right)^{-1.5} \\left( \\frac{D}{\\rm pc}\\right) ^{-0.5}\n\\end{split}\n\\label{e_no_density}\n\\end{equation}\n\n{\\it Emission measure (EM):}\n\\begin{equation}\n\\begin{split}\n\\left( \\frac{\\rm EM}{\\rm pc\\ cm^{-6}}\\right) = 3.217 \\times 10^7  \\left(\\frac{F_\\nu}{\\rm Jy}\\right) \\left( \\frac{T_{\\rm e}}{\\rm K}\\right)^{0.35} \\\\\n\\times \\left( \\frac{\\nu}{\\rm GHz}\\right)^{0.1} \\left( \\frac{\\theta_{\\rm source}}{\\rm arcsec}\\right)^{-2}\n\\end{split}\n\\label{emission_measure}\n\\end{equation}\nwhere, $F_{\\nu}$ is the integrated flux density of the ionized region, $T_{\\rm e}$ is the electron temperature, $\\nu$ is the frequency, $\\theta_{\\rm source}$ is the angular diameter of the {H\\,{\\scshape ii}~region} and D is the distance to this region. $T_{\\rm e}$ is taken to be 5000~K from the radio recombination line estimates by \\citet{1987A&A...171..261C}. Approximating the emission region to an ellipse, the angular source size ($\\theta_{\\rm source}$) is taken to be the geometric mean of the axes of the ellipse and is estimated to be 6.25~arcmin (3.6~pc).  The derived physical parameters are listed in Table \\ref{radio-physical-param}. \n\n\\begin{table}\n\\caption{Derived physical parameters of the {H\\,{\\scshape ii}~region} associated with IRAS 17149$-$3916.}\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\nSize & log $N_{\\rm Ly}$ & EM            & $n_{\\rm e}$    & Spectral Type \\\\\n(pc) &              & (cm$^{-6}$pc) & (cm$^{-3}$) &               \\\\\n\\hline      \n\n3.6  & 48.73        & 5.8$\\times$10$^{5}$ & 1.3$\\times$10$^{2}$ &  O6.5V -- O7V \\\\ \\hline\n\\end{tabular}\n\\label{radio-physical-param}\n\\end{center}\n\\end{table}\n \nIf we assume a single star to be ionizing the H\\,{\\scshape ii}~region\\ and compare the Lyman-continuum photon flux obtained from the 1280~MHz map with the parameters of O-type stars presented in \\citet[Table 1;][]{2005A&A...436.1049M}, we estimate its spectral type to be O6.5V -- O7V. \nThis can be considered as a lower limit as the emission at 1280~MHz could be optically thick as well. In addition, one needs to account for dust absorption of Lyman continuum photons, which can be significant as shown by many studies \\citep[e.g.][]{2011A&A...525A.132P}. \nThe estimated spectral type suggests a mass range of $\\sim 20 - 40~ M_{\\odot}$ for the ionizing star \\citep{2005A&A...436.1049M}. \n\nTo decipher the nature of the ionized emission, we determine the spectral index, $\\alpha$ which is defined as $F_{\\nu} \\propto \\nu^{\\alpha}$. The flux density, $F_{\\nu}$, is calculated from the GMRT radio maps. For this, we generate two new radio maps at 610 and 1280~MHz by setting the $\\it uv$ range to a common value of $0.14 - 39.7 $~k$\\lambda$. This ensures similar spatial scales being probed at both frequencies. Further, the beam size for both the maps is set to $\\rm 12~arcsec \\times 12~arcsec$. $F_{\\nu}$ is obtained by integrating within the area defined by the 3$\\sigma$ contour of the new 610~MHz map. \nThe integrated flux density values are estimated to be $\\rm 13.7 \\pm 1.3 \\,Jy\\,$, $\\rm 12.1 \\pm 1.2 \\,Jy\\,$  at 610 and 1280~MHz, respectively. \nThese yield a spectral index of $-0.17 \\pm 0.19$. Similar values are obtained for the central, bright radio emission as well. Within the quoted uncertainties, the average spectral index is fairly consistent with optically thin, free-free emission as expected from {H\\,{\\scshape ii}~regions} which are usually dominated by thermal emission. Spectral index estimate of $-0.1$, consistent with optically thin thermal emission, is also obtained by combining the GMRT flux density values with the available single dish measurements at 2.65~GHz \\citep{1969AuJPA..11...27B} and 4.85~GHz \\citep{1994ApJS...91..111W}.\n\n\\subsection{The dust environment}\n\\label{mir-dust}\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.4]{Fig3.eps}\n\\caption{Dust emission associated with IRAS 17149$-$3916 from the {\\it Spitzer}-GLIMPSE and the {\\it Herschel}-Hi-GAL surveys.} \n\\label{iremission}\n\\end{figure*}\nThe warm and the cold dust emission associated with IRAS 17149$-$3916 unravels interesting morphological features like a bubble, pillars, filaments, arcs, and clumps, that strongly suggest this to be a very active star forming complex where the profound radiative and mechanical feedback of massive stars on the surrounding ISM is clearly observed. Fig.~\\ref{iremission} compiles the MIR and FIR emission towards IRAS 17149$-$3916 from the GLIMPSE and Hi-GAL surveys. Apart from the stellar population probed at the shorter wavelengths, the diffuse emission seen in the IRAC-GLIMPSE images would be dominated by the emissions from polycyclic aromatic hydrocarbons (PAHs) excited by the UV photons in the photodissociation regions \\citep{2012A&A...542A..10A,2012ApJ...760..149P}. Close to the hot stars, there would also be significant contribution from thermally emitting warm dust that is heated by stellar radiation \\citep{2008ApJ...681.1341W}.  In {H\\,{\\scshape ii}~regions}, emission from dust heated by trapped Ly$\\alpha$ photons \\citep{1991MNRAS.251..584H}, would also be present in these IRAC bands. In the wavelength regime of the 21~$\\rm \\mu m$ MSX, the emission is either associated with stochastically heated Very Small Grains (VSGs) or thermal emission from hot big grains (BGs). As we go to the FIR Hi-GAL maps, cold dust emission dominates and shows up as distinct clumps and filamentary structures where, emission in the 70~$\\rm \\mu m$ band is dominated by the VSGs and the longer wavelength bands like 250~$\\rm \\mu m$ band trace emissions from the BGs \\citep{2012ApJ...760..149P}.\n\n\\subsubsection*{Dust temperature and column density maps}\nTo understand the nature of cold dust emission, we generate the dust temperature and the molecular hydrogen column density maps following the procedure detailed in \\citet{2018A&A...612A..36D} and \\citet{2019MNRAS.485.1775I} and briefly stated here. A pixel-by-pixel modified blackbody modelling to the observed spectral energy distribution is carried out. As discussed in these two papers, the 70\\,$\\rm \\mu m$ data is not used because there would be appreciable contribution from the warm dust component hence rendering a single modified blackbody fit inaccurate. Thus, we have the FIR emission at 160, 250, 350, and 500\\,$\\rm \\mu m$ mostly on the Rayleigh-Jeans part to constrain the model given by\n\\begin{equation}\nF_{\\nu}-I_{\\rm bg} = B_{\\nu}(\\nu,T_{\\rm d})~\\Omega~(1-{\\rm e}^{-\\tau_{\\nu}}) \n\\label{MBB-Eqn}\n\\end{equation}\nwhere,  $F_{\\nu}$ is the observed flux density, $I_{\\rm bg}$ is the background flux density,  $B_{\\nu}(\\nu,T_{\\rm d})$ is the Planck function at the dust temperature $\\rm T_{\\rm d}$, $\\Omega$ is the solid angle subtended by a pixel (all maps are convolved to a common resolution of 35.7~arcsec and regridded to a common pixel size of  $\\rm 14~arcsec\\times14~arcsec$). The background flux is estimated from a nearby region relatively free of clumpy and bright emission.\nThe optical depth $\\tau_{\\nu}$ in Eqn. \\ref{MBB-Eqn} can be expressed as\n\\begin{equation}\n\\tau_{\\nu} = \\mu_{\\rm H_2}~ N({\\rm H_2})~ m_{\\rm H}~ \\kappa_{\\nu}\n\\end{equation}\nwhere, $\\mu_{\\rm H_2}$ is the mean molecular weight which is taken as 2.8 \\citep{2008A&A...487..993K}, $N({\\rm H_2})$ is the column density, $m_{\\rm H}$ is the mass of hydrogen atom and $\\kappa_{\\nu}$ ($\\rm cm^2\\,g^{-1}$) is the dust opacity which is given as \\citep{1983QJRAS..24..267H}\n\\begin{equation}\n\\kappa_{\\nu} = 0.1\\left ( \\frac{\\nu}{1200~{\\rm GHz}} \\right )^{\\beta} \n\\label{kappa}\n\\end{equation}\nHere, $\\beta$ denotes the dust emissivity spectral index and a typical value of 2 estimated in several star regions is assumed. \nIn fitting the modified blackbody to the observed flux densities, $N({\\rm H_2})$ and $T_{\\rm d}$ are kept as free parameters. \n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.32]{Fig4a.eps}\n\\includegraphics[scale=0.32]{Fig4b.eps}\n\\includegraphics[scale=0.32]{Fig4c.eps}\n\\caption{Column density (a), dust temperature (b), and chi-square ($\\chi^{2}$) (c) maps of the region associated with the H\\,{\\scshape ii}\\ region. The 610~MHz ionized emission overlaid as magenta and gray contours on column density and dust temperature maps, respectively. The contour levels are 0.01, 0.03, 0.06, 0.12, and 0.2~Jy\/beam. The retrieved clump apertures are shown on the column density (in blue) and dust temperature (in black) maps following the nomenclature as discussed in the text. The black contour in (a) shows the area integrated for estimating $n_0$ (refer Section \\ref{clumps-CC}.)}\n\\label{cdtchi}\n\\end{figure*}\nThe column density and dust temperature maps generated are shown in Fig.~\\ref{cdtchi}. The goodness of the fits for each pixel can be seen in the $\\chi^{2}$ map where the maximum $\\chi^{2}$ value is seen to be $\\sim 8$.\nThe column density map presents a triangular morphology with three distinct, bright and dense regions.\nA network of broad filaments are also seen in the map. The dust temperature map is relatively patchy with regions of higher temperature within the radio nebula. A region with warm temperature is seen to be located towards the south-east of IRAS 17149$-$3916, the signature of which can be seen in the {\\it Herschel} maps shown in Fig.~\\ref{iremission}. The western side of the {H\\,{\\scshape ii}~region} shows comparatively cold temperatures. Furthermore, the filamentary features seen in the column density map are mostly revealed as distinct low temperature lanes. \n\n\\subsubsection*{Dust clumps and cores}\\label{dust_clump}\nThe FIR and the column density maps show the presence of dust clumps. These clumps are identified using the \\textit{Herschel} 350\\,$\\rm \\mu m$ map and the {\\it Dendrogram}\\footnote{\\url{https:\/\/dendrograms.readthedocs.io\/en\/stable\/}} algorithm. Using this algorithm, we identify the smallest structures, called the `leaves', in the 350\\,$\\rm \\mu m$ map, which in this case are the cold dust clumps. The key input parameters for the identification of the clumps are (1) {\\it min\\_value} = $3\\sigma$ and (2) {\\it min\\_delta} = $\\sigma$, where $\\rm \\sigma (= 191.2\\,MJy\\,sr^{-1})$ is the {\\it rms} level of the 350\\,$\\rm \\mu m$ map. An additional parameter, {\\it min\\_pix = N}, is also used, which is the minimum number of pixels required for a `leaf' to be considered an independent entity. To ensure that the clumps are resolved, the value of {\\it N} is chosen to be 7\\,pixels, the beam area of the 350\\,$\\rm \\mu m$ map. Setting these parameters, we extract three cold dust clumps. The central panel of Fig.~\\ref{dustclumps} shows the 350\\,$\\rm \\mu m$ map overlaid with the retrieved apertures of the three detected clumps labelled 1, 2, and 3. The physical parameters of the detected clumps are listed in Table \\ref{clump-param}. These are derived from the 350\\,$\\rm \\mu m$, column density and dust temperature maps. The peak positions are determined from the 350\\,$\\rm \\mu m$ map. The clump radii, $r=(A\/\\pi)^{0.5}$, where $A$ is the enclosed area within the retrieved clump apertures. \n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.5]{Fig5.eps}\n\\caption{ The central panel is the \\textit{Herschel} 350\\,$\\rm \\mu m$ map overlaid with the {\\it Dendrogram} retrieved clump apertures in black. White circles represent four pointings of the {\\it ALMA} observations, S61, S62, S63, and S64. The 1.4\\,mm {\\it ALMA} map towards each pointing is placed at the corners of the 350\\,$\\rm \\mu m$ image. The apertures of the dense cores extracted using the {\\it Dendrogram} algorithm are overlaid and labelled on these maps. The beam sizes of the 1.4\\,mm maps are given towards the lower left hand corner of each image.}\n\\label{dustclumps}\n\\end{figure*}\n\\begin{table*}\n\\caption{Physical parameters of the detected dust clumps associated with IRAS 17149$-$3916}\n\\begin{center}\n\\centering \n\\scalebox{0.85}{\n\\begin{tabular}{ccccccccccc}\n\\hline\nClump & \\multicolumn{2}{c}{Peak position} & Mean $T_{\\rm d}$ & $\\Sigma N({\\rm H_2})$       & Radius & Mass          & Mean $N({\\rm H_2})$         & No. density $(n_{\\textup{H}_{2}})$ & \\multicolumn{1}{l}{$M_{\\rm vir}$} & \\multicolumn{1}{l}{$\\alpha_{\\rm vir}$} \\\\\n      & RA (J2000)      & DEC (J2000)     & (K)              & $(\\times 10^{23}$ cm$^{-2})$ & (pc)   & ($M_{\\odot}$) & $(\\times 10^{22}$ cm$^{-2}$) & $(\\times 10^{4}$ cm$^{-3}$)        & ($M_{\\odot}$)                     & \\multicolumn{1}{l}{}                   \\\\ \\hline\n1     & 17 18 24.32     & -39 19 28.04    & 27.8             & 6.1                          & 0.3    & 250           & 5.5                          & 5.0                                & 452                               & 1.75                                   \\\\\n2     & 17 18 18.77     & -39 19 04.54    & 25.0             & 7.1                          & 0.3    & 292           & 5.4                          & 5.1                                & 600                               & 2.15                                   \\\\\n3     & 17 18 22.57     & -39 18 36.44    & 27.3             & 2.8                          & 0.2    & 117           & 3.6                          & 4.6                                & 450                               & 4.08                                   \\\\ \\hline\n\\end{tabular}}\n\\label{clump-param}\n\\end{center}\n\\end{table*}\nTo estimate the masses of the detected clumps, we utilize the column density map and use the following expression\n\\begin{equation}\nM_{\\rm clump} =\\mu_{\\rm H_2}~ \\Sigma N({\\rm H_2})~A_{\\rm pixel}~m_{\\rm H}\n\\label{mass_eqn}\n\\end{equation}\nwhere, $\\mu_{\\rm H_2}$ is the mean molecular weight taken as 2.8, $\\Sigma N({\\rm H_2})$ is the integrated column density over the clump area, $A_{\\rm pixel}$ is the pixel area in $\\rm cm^2$ and $m_{\\rm H}$ is the mass of hydrogen atom. The number density is determined using the expression $n_{\\rm H_2} = 3\\,N({\\rm H_2})\/4r$. The peak positions of the Clumps 1, 2, and 3 agree fairly well with Clumps III, I, and II, respectively, detected by \\citet{2014MNRAS.437..606T} using {\\it Herschel} maps. Comparing the masses presented in Table \\ref{clump-param}, the estimates given in \\citet{2014MNRAS.438.2716T} are higher by a factor of 8, 1.5, and 3 for the Clumps 1, 2, and 3, respectively. \n\n{\\it ALMA} continuum data enables investigation of the detected clumps at high resolution.  In Fig.~\\ref{dustclumps}, we show {\\it ALMA} dust continuum maps at 1.4\\,mm towards the four pointings marked and labelled as S61, S62, S63, and S64 where the first three pointings lie mostly within three clumps and S64 lies outside towards the east. Using the same {\\it Dendrogram} algorithm, several cores are identified. The key input parameters to the {\\it Dendrogram} algorithm are, {\\it min\\_value} = $3\\sigma$, {\\it min\\_delta} = $\\sigma$, and {\\it min\\_pix = N}, where $\\sigma$ is the {\\it rms} level and $N(=60)$ is the beam area of the 1.4\\,mm maps. In order to avoid detection of spurious cores, we only retain those with peak flux density greater than $5\\sigma$. Applying these constraints, seven cores are identified towards S61 and S62 each, one towards S63 and two towards S64. \n\nTo further study these dense cores, we estimate their physical parameters. Adopting the formalism described by \\citet{2018ApJ...853..160C} and assuming the emission at 1.4\\,mm to be optically thin, the masses are estimated using the following equation\n\n\\begin{eqnarray}\n  M & = &\n  \\displaystyle 0.0417 \\, M_{\\odot}\n  \\left( {\\textrm e}^{0.533 (\\lambda \/ {1.3\\, \\textrm {mm}})^{-1}\n      (T \/ {20\\, \\textrm {K}})^{-1}} - 1 \\right) \\left( \\frac{F_{\\nu}}{\\textrm {mJy}} \\right) \\nonumber \\\\\n  & & \\displaystyle\n \\times \\left( \\frac{\\kappa_{\\nu}}{0.00638\\,\\textrm{cm}^2\\,\\textrm{g}^{-1}} \\right)^{-1}\n  \\left( \\frac{d}{\\textrm {kpc}} \\right)^2\n  \\left( \\frac{\\lambda}{1.3\\, \\textrm {mm}} \\right)^{3} \n  \\label{core_mass}\n\\end{eqnarray}\nHere, $F_\\nu$ is the integrated flux density of each core, $d$ is the distance to the source and $\\lambda$ is the wavelength. Opacity, $\\kappa_\\nu$ is estimated using equation~\\ref{kappa} with the dust emissivity spectral index $\\beta$ fixed at 2.0. For cores detected in S61, S62, and S63, mean dust temperatures of the respective clumps are taken. For S64, the mean dust temperature for the region covering the S64 pointing is used. The effective radius, $r=(A\/\\pi)^{0.5}$, of each core is also estimated where $A$ is area enclosed within each core aperture.\nThe identified cores with the retrieved apertures are shown in Fig.~\\ref{dustclumps} and the estimated physical parameters are list in Table \\ref{alma-cores}. The uncertainties related to the missing flux effect are not taken into account in deriving these parameters since it may not be significant given that the largest recoverable scales is quoted to be $\\sim$10~arcsec for this ALMA dataset which is appreciably larger than the typical sizes of the detected cores. Barring the largest detected core which has an angular size of $\\sim$7~arcsec, the average size of the cores is 3~arcsec.\n\\begin{table*}\n\\caption{Parameters of the detected dust cores extracted associated with IRAS 17149$-$3916}\n\\begin{center}\n\\centering \n\\begin{tabular}{ccccccc}\n\\hline\n    & Core & \\multicolumn{2}{c}{Peak position}    &  Flux density  & Radius     & Mass \\\\\n    &   & RA (J2000)      & DEC (J2000)       &  (mJy)    & (pc)      & ($M_\\odot$)       \\\\\n\\hline\nS61 & 1 & 17 18 23.03   & -39 19 12.95      & 9.7   & 0.01  & 1.7   \\\\\n    & 2 & 17 18 23.75   & -39 19 18.47      & 62.2  & 0.02  & 10.7  \\\\\n    & 3 & 17 18 23.45   & -39 19 19.53      & 32.8  & 0.02  & 5.7   \\\\\n    & 4 & 17 18 24.17   & -39 19 22.35      & 11.2  & 0.01  & 1.9   \\\\\n    & 5 & 17 18 24.34   & -39 19 25.09      & 106.7 & 0.02  & 18.4  \\\\\n    & 6 & 17 18 24.76   & -39 19 19.50      & 4.5   & 0.01  & 0.8  \\\\\n    & 7 & 17 18 24.93   & -39 19 28.12      & 31.8  & 0.02  & 5.5   \\\\\n\\hline \nS62 & 1 & 17 18 20.38   & -39 18 54.32      & 46.0  & 0.02  & 9.4  \\\\\n    & 2 & 17 18 20.09   & -39 18 54.43      & 9.8   & 0.01  & 2.0  \\\\\n    & 3 & 17 18 19.64   & -39 18 56.07      & 7.9   & 0.02  & 1.6  \\\\\n    & 4 & 17 18 19.63   & -39 19 04.70      & 2.1   & 0.01  & 0.4  \\\\\n    & 5 & 17 18 18.82   & -39 19 04.94      & 3.1   & 0.01  & 0.6   \\\\\n    & 6 & 17 18 18.35   & -39 19 01.65      & 19.2  & 0.02  & 3.9  \\\\\n    & 7 & 17 18 18.42   & -39 19 05.37      & 8.2   & 0.01  & 1.7   \\\\\n\\hline \nS63 & 1 & 17 18 23.48   & -39 18 40.70      & 357.7 & 0.03  & 62.3  \\\\\n\\hline \nS64 & 1 & 17 18 29.98   & -39 19 11.22      & 7.0   & 0.02  & 1.1  \\\\\n    & 2 & 17 18 29.10   & -39 18 54.11      & 1.9   & 0.01  & 0.3  \\\\\n\n\\hline\n\\end{tabular}\n\\label{alma-cores}\n\\end{center}\n\\end{table*}\n\n\\subsection{Molecular line observation of identified clumps}\nWe use the optically thin $\\rm N_2 H^+$ line emission to determine the $V_{\\textup{LSR}}$ and the line width, $=\\Delta V$ of the clumps. The line spectra are extracted by integrating over the retrieved apertures of the clumps as shown in Fig.~\\ref{n2hplus_spectra}. The $\\rm N_2 H^+$ spectra have seven hyperfine structures and the {\\tt hfs} method of {\\tt CLASS90} is used to fit the observed spectra. The line parameters retrieved from spectra are listed in Table \\ref{n2hplus_fit_parameters}. The $V_{\\textup{LSR}}$ determined agrees well with the value of $\\rm 13.7\\, km s^{-1}$ obtained from $\\rm CS(2-1)$ observations of the region by \\citet{1996A&AS..115...81B} \n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.2]{Fig6a.eps}\n\\includegraphics[scale=0.2]{Fig6b.eps}\n\\includegraphics[scale=0.2]{Fig6c.eps}\n\\caption{Spectra of the optically thin N$_{2}$H$^{+}$ line emission extracted over the three identified clumps associated with IRAS 17149$-$3916. The red curves are the {\\tt hfs} fit to the spectra. The estimated $V_{\\textup{LSR}}$ is denoted by the dashed blue line and the location of the hyperfine components by magenta lines.}\n\\label{n2hplus_spectra}\n\\end{figure*}\n\\begin{table}\n\\centering\n\\caption{The retrieved $\\rm N_2 H^+$ line parameters, $V_{\\textup{LSR}}$, $\\Delta V$, $T_{\\rm mb}$ and $\\int T_{\\textup{mb}} {\\rm dV}$ for the identified clumps associated with IRAS 17149$-$3916.}\n\\begin{tabular}{ccccc}\n\\hline\nClump & $V_{\\textup{LSR}}$ & $\\Delta V$    & $T_{\\rm mb}$ & $\\int T_{\\textup{mb}} {\\rm dV}$ \\\\\n      & (km s$^{-1}$)      & (km s$^{-1}$) & (K)          & (K km s$^{-1}$)             \\\\ \\hline\n1     & -13.0              & 3.2           & 1.5          & 5.4                         \\\\\n2     & -13.8              & 3.7           & 1.6          & 6.8                         \\\\\n3     & -13.8              & 3.9           & 0.8          & 3.1                         \\\\ \\hline\n\\end{tabular}\n\\label{n2hplus_fit_parameters}\n\\end{table}\n\n\\section{Discussion}\n\\label{discussion}\n\\subsection{Understanding the morphology of the ionized gas}\nAs seen, the GMRT maps reveal the large extent and prominent cometary morphology of the {H\\,{\\scshape ii}~region} associated with IRAS 17149$-$3916 which was earlier discussed as a roundish {H\\,{\\scshape ii}~region} by \\citet{2014MNRAS.437..606T}. In this section, we attempt to investigate the likely mechanism for this observed morphology. \nThe widely accepted models to explain the formation of cometary {H\\,{\\scshape ii}~regions} are (1) the bow-shock model (e.g. \\citealt{1992ApJ...394..534V}), (2) the champagne-flow model (e.g. \\citealt{1979A&A....71...59T}), and (3) the mass loading model (e.g. \\citealt{1997ApJ...476..166W}). However, subsequent studies, like those conducted by \\citet{2003ApJ...596..344C} and \\citet{2006ApJS..165..283A}, find \nthe `hybrid' models, that are a combination of these, to better represent the observed morphologies. \n\nThe bow-shock model assumes a wind-blowing, massive star moving supersonically through a dense cloud. Whereas, the champagne-flow model invokes a steep density gradient encountered by the expanding {H\\,{\\scshape ii}~region} around a newly formed stationary, massive star possibly located at the edge of a clump. Here, the ionized gas flows out towards regions of minimum density. In comparison, the model proposed by \\citet{1997ApJ...476..166W} invokes the idea of strong stellar winds mass loading from the clumpy molecular cloud and the cometary structure unfolds when a gradient in the geometrical distribution of mass loading centres are introduced. In this model the massive, young star is considered to be stationary as in the case of the champagne-flow model. \n\nWhile observation of ionized gas kinematics is required to understand the origin of the observed morphology, in the discussion that follows, we discuss a few aspects based on the\nradio, column density, and FIR maps of the region associated with IRAS 17149$-$3916 along with the identification of E4 as the likely ionizing star (refer Section \\ref{ionizing_star}). Following the simple analytic expressions discussed in \\citet{2018A&A...612A..36D}, we derive a few shock parameters to probe the bow-shock model. Taking the spectral type of E4 to be O6.5V -- O7V as estimated from the radio flux density, and assuming it to move at a typical speed of $\\rm 10~km\/s$ through the molecular cloud, we calculate  the `stand-off' distance to range between $\\rm 2.6~arcsec (0.02~pc) - 3.1~arcsec (0.03~pc)$. This is defined as the distance from the star at which the shock occurs and where the momentum flux of the stellar wind equals the ram pressure of the surrounding cloud. The theoretically estimated value is significantly less than the observed distance of $\\sim \\rm 84~arcsec (0.8~pc)$ between E4 and the cometary head. Taking viewing angle into consideration would decrease the theoretical estimate thus widening the disparity further. Based on the above estimations, it is unlikely that the bow-shock model would explain the cometary morphology. To confirm further, we determine the trapping parameter which is the inverse of the ionization fraction. As the ionizing star moves supersonically through the cloud, the swept off dense shells trap the {H\\,{\\scshape ii}~region} within it and its expansion is eventually inhibited by the ram pressure. Trapping becomes more significant when recombinations far exceed the ionizing photons. Studies of a large number of cometary {H\\,{\\scshape ii}~regions} show the trapping parameter to be much greater than unity \\citep{1991ApJ...369..395M}. For IRAS 17149$-$3916, we estimate the value to lie in the range $3.2 - 3.5$ which indicates either weak or no bow shock. Similar interpretations are presented in \\citet{2018A&A...612A..36D} and \\citet{2016MNRAS.456.2425V}. The trapping parameters obtained by these authors lie in the range 1.2 -- 4.3.\n\nTo investigate the other models, namely the champagne-flow and clumpy\/mass loading wind models, we compare the observed spatial distribution of the dust component and the ionized gas. The FIR and column density maps presented in Section \\ref{mir-dust} show a complex morphology of pillars, arcs, filaments in the region with detected massive clumps towards the cometary head. The steep density gradient towards the cometary head is evident. Without the ionized and molecular gas kinematics information, it is difficult to invoke the champagne-flow model. However, the maps do show the presence of clumps towards the cometary head which could act as potential mass loading centres and thus support the clumpy cloud model. Further observations and modelling are essential before one can completely understand the mechanisms at work.\n\n\\subsection{Ionizing massive star(s)}\\label{ionizing_star}\n\\citet{2006AJ....131..951R} have studied the associated stellar population towards IRAS 17149$-$3916 in the NIR. Using the colour-magnitude diagram, these authors show the presence of a cluster of massive stars within the infrared nebula and suggest IRS-1 to be the likely ionizing source. In a later study, \\citet{2014MNRAS.437..606T} have supported this view citing the spectroscopic classification of IRS-1 as O5 -- O6 by \\citet{2005A&A...440..121B} and consistency with the Lyman continuum photon flux estimated from the radio observations by  \\citet{2013A&A...550A..21S}. \n\\begin{figure*}\n\\centering\n\\includegraphics[width=9cm,height=6.6cm]{Fig7a.eps}\n\\includegraphics[width=8cm,height=6cm]{Fig7b.eps}\n\\includegraphics[width=9cm,height=7cm]{Fig7c.eps}\n\\caption{(a) J vs J-H colour magnitude diagram of the sources (cyan dots) associated with IRAS 17149$-$3916 and located within the 3$\\sigma$ radio contour. The nearly vertical solid lines represent the ZAMS loci with 0, 15, and 30 magnitudes of visual extinction corrected for the distance. The slanting lines show the reddening vectors for spectral types B0 and O5. \n(b) J-H vs H-K colour-colour diagram for sources (magenta dots) in the same region as (a). The cyan and black curves show the loci of main sequence and giants, respectively, and are taken from \\citet{1983A&A...128...84K} and \\citet{1988PASP..100.1134B}. The locus of classical T Tauri adopted from \\citet{1997AJ....114..288M} is shown as long dashed line. The locus of Herbig AeBe stars shown as short dashed line is adopted from \\citet{1992ApJ...393..278L}. The parallel lines are the reddening vectors where cross marks indicate intervals of 5 mag of visual extinction. The colour-colour plot is divided into three regions, namely, `F', `T', and `P' (see text for more discussion). The interstellar reddening law assumed is from \\citet{1985ApJ...288..618R}. The magnitudes, colours and various loci plotted in both the diagrams are in the \\citet{1988PASP..100.1134B} system. \nThe identified early type (earlier than B0) and YSOs candidates are highlighted as blue and red stars, respectively.  Of these, the ones located towards the central, bright radio emission are labelled as `E' and `Y', respectively. (c) An enlarged view of the bottom left portion of (b) showing the position of spectral types on the main sequence locus. The location of the source E4 and the errors on the colour are also shown.}\n\\label{NIR-CC-CM}\n\\end{figure*}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.2]{Fig8.eps}\n\\caption{Panel a: {\\it Spitzer} 5.8\\,$\\rm \\mu m$ image in grey scale overlaid by 610~MHz radio contours. The contour levels are 0.002, 0.01, 0.03, 0.06, 0.12, and 0.2~Jy\/beam. The YSOs and massive stars detected towards the central ionized region are shown in red and blue coloured stars, respectively. Panel b: This shows the zoom-in view of central part of the H\\,{\\scshape ii}\\ region.}\n\\label{EY-Spa-dis}\n\\end{figure}\nWhile the spectral type estimated from the GMRT radio emission at 1280~MHz is consistent with that of IRS-1, we investigate the stellar population within the radio emission for a better understanding. \nIn Figs.~\\ref{NIR-CC-CM}(a) and (b), we plot the NIR colour-magnitude and colour-colour diagrams, respectively, of 2MASS sources located within the 3$\\sigma$ radio contour. \nFig.~\\ref{NIR-CC-CM}(c) shows an enlarged view of the bottom left portion of Fig.~\\ref{NIR-CC-CM}(b) to highlight the location of the star E4 with respect to the main sequence locus.\nFollowing the discussion given in \\citet[Fig. 7;][]{2006A&A...452..203T} the colour-colour plot is classified into `F', `T', and `P' regions. The `F' region is occupied by mostly field stars or Class III sources, the `T' region is for T-Tauri stars (Class II YSOs) and protostars (Class I YSOs) populate the `P' region.\nAs seen from the figures, there are sixteen sources earlier than spectral type B0 and eighteen identified YSOs. The sample of identified YSOs fall in the `T' region, the sources of which are believed to be Class II objects with NIR excess. Sources that lie towards the central bright, radio emitting region are labelled in the figures with prefixes of `E' for the sources earlier than B0 and `Y' for the YSOs. As indicated in Fig.~\\ref{NIR-CC-CM}(b), early type sources E1 and E2 are also the identified YSOs, Y1 and Y3, respectively. The coordinates and NIR magnitudes of these selected sources are listed in Table~\\ref{EY-sources}. Fig.~\\ref{EY-Spa-dis} shows the spatial distribution of the above sources with respect to the radio and 5.8\\,$\\rm \\mu m$ emission.\n\nAs seen from the above analysis, in addition to the presence of possible discrete radio sources that could be internally ionized, several massive stars are also identified from the NIR colour-magnitude and colour-colour diagrams. Hence, it is likely that ionization in this {H\\,{\\scshape ii}~region} is the result of this cluster of massive stars.  However, the observed symmetrical, cometary morphology of the ionized emission strongly suggests that the ionization is mostly dominated by a single star. \nAs seen in Fig.~\\ref{NIR-CC-CM}(a), out of the early type sources that lie towards the central, bright radio emission, the colour and magnitude of the source E4 is consistent with a spectral type of $\\sim$O6.\nA careful scrutiny of the Fig.~\\ref{NIR-CC-CM}(b) show that early type stars E1 and E2 are embedded Class II sources and hence unlikely to be the main driving source. Sources E3, E5, E6 are possibly reddened giants or field stars. The location of early type star, E4 (which is the source IRS-1) in the colour-colour diagram (see enlarged view shown in Fig.~\\ref{NIR-CC-CM}c) agrees fairly well with the spectral type estimate of $\\sim$O6 obtained from the colour-magnitude diagram and strongly advocates it as the dominating exciting source. This is consistent with the identification of IRS-1 as the ionizing star in previous studies. Spatially also, the location of E4 clearly suggests its role in the formation of the network of pillar like structures observed (see Section \\ref{pillars}). As mentioned earlier, the spectral type of E4, estimated from NIR spectroscopy, is in good agreement with the radio flux. Location wise, however, it is 30~arcsec away from the radio peak. This offset could be attributed to density inhomogeneity or clumpy structure of the surrounding, ambient ISM. Supporting this scenario of E4 being the dominant player, are the interesting pillar like structures revealed in the MIR images discussed in the next section. \n\\begin{table*}\n\\caption{Early type and YSOs detected within the central, bright, radio emission of IRAS 17149$-$3916}\n\\begin{center}\n\\centering\n\\scalebox{0.85}{\n\\begin{tabular}{cccccc}\n\\hline\nSource             & \\multicolumn{2}{c}{Coordinates} & J      & H      & K      \\\\\n                   & RA (J2000)     & DEC (J2000)    &        &        &        \\\\ \\hline\n &               & {\\it Early-type sources}  &        &        &   \\\\[1mm]\nE1 (Y1) & 17 18 22.21   & -39 18 42.24   & 11.364 & 10.039 & 8.957  \\\\\nE2 (Y3) & 17 18 22.85   & -39 18 22.45   & 15.356 & 12.086 & 10.086 \\\\\nE3      & 17 18 25.11   & -39 18 46.41   & 15.398 & 12.486 & 11.361 \\\\\nE4      & 17 18 25.45   & -39 19 08.61   & 8.654  & 8.208  & 7.927  \\\\\nE5      & 17 18 25.68   & -39 18 26.65   & 11.790  & 9.617  & 8.606  \\\\\nE6      & 17 18 25.94   & -39 18 00.89   & 7.715  & 7.191  & 7.021  \\\\ \\hline\n  &               &   {\\it YSOs}             &        &        &        \\\\[1mm]\nY1 (E1) & 17 18 22.21   & -39 18 42.24   & 11.364 & 10.039 & 8.957  \\\\\nY2      & 17 18 22.28   & -39 18 12.69   & 15.064 & 14.161 & 13.578 \\\\\nY3 (E2) & 17 18 22.85   & -39 18 22.45   & 15.356 & 12.086 & 10.086 \\\\\nY4      & 17 18 22.87   & -39 18 58.45   & 14.413 & 13.346 & 12.393 \\\\\nY5      & 17 18 23.28   & -39 19 07.77   & 12.993 & 11.906 & 11.135 \\\\\nY6      & 17 18 24.44   & -39 19 11.04   & 14.324 & 12.958 & 12.166 \\\\\nY7      & 17 18 25.14   & -39 19 25.95   & 14.741 & 13.657 & 12.664 \\\\\nY8      & 17 18 25.28   & -39 18 24.76   & 12.818 & 11.372 & 10.420  \\\\\nY9      & 17 18 28.30   & -39 19 49.71   & 14.677 & 13.449 & 12.680 \\\\\n\\hline\n\\end{tabular}}\n\\label{EY-sources}\n\\end{center}\n\\end{table*}\n\n\\subsection{Triggered star formation}\\label{pillars}\n\n\\subsubsection*{Pillar Structures}\nIn Fig.~\\ref{pillar-structure}, we illustrate the identification of pillar structures in the IRAC 8\\,$\\rm \\mu m$ image. The MIR emission presents a region witnessing a complex interplay of the neutral, ambient ISM with the ionizing radiation of newly formed massive star(s). The boxes labelled `A' and `B' show prominent pillar structures, the orientation of these are clearly pointed towards E4. This strongly suggests E4 as the main sculptor of the detected pillars. Furthermore, it also supports the identification of E4 as the main ionizing source of the H\\,{\\scshape ii}~region\\ . \n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.15]{Fig9.eps}\n\\caption{(a) 8.0\\,$\\rm \\mu m$ map of IRAS 17149$-$3916 from the {\\it Spitzer}-GLIMPSE survey. Blue arrows highlight the pillar structures identified within Boxes `A' and `B'. The white `+' mark shows the position of E4 (IRS-1). (b) A zoom-in on `A' at IRAC 5.8\\,$\\rm \\mu m$. (c) 1280\\,MHz map covering the pillar `A'.}\n\\label{pillar-structure}\n\\end{figure}\nOne of the mechanisms widely accepted to explain the formation of these pillars is the radiative driven implosion (RDI) \\citep{1994A&A...289..559L}. Here, pre-existing clouds exposed to newly forming massive star(s) are sculpted into pillars by slow photoevaporation caused due to strong impingement of ionizing radiation. The other being the classical collect and collapse model of triggered star formation proposed by \\citet{1977ApJ...214..725E}. Under this framework, the expanding {H\\,{\\scshape ii}~region} sweeps up the surrounding material, creating dense structures that could eventually form pillars in their shadows. \n\nFigs.~\\ref{pillar-structure}(b) and (c) show the zoomed in IR and radio view of pillar `A'. Clearly seen is a slightly elongated and bright radio source at the pillar head.  To ascertain the nature of the bright radio source, we estimate few physical parameters using the 1280~MHz GMRT map. Using the 2D fitting tool of {\\small CASA} viewer we fit a 2D Gaussian and determine the deconvolved size and flux density of this source to be $\\rm 19.05~arcsec \\times 8.07~arcsec$ ($\\theta_{\\rm source} = 12.4 \\rm arcsec; 0.12 \\rm pc$) and $\\rm 445 \\,mJy\\,$, respectively. Inserting these values in equations \\ref{Lyman_flux}, \\ref{e_no_density}, and \\ref{emission_measure}, we get $ log N_{\\rm Ly} = 47.30$ , $n_{\\rm e} = 3.9\\times10^3~\\rm cm^{-3}$ and EM= $\\rm 1.8\\times10^6~pc~cm^{-6}$. The ionized mass ($\\rm M_{ion} = \\frac{4}{3}\\pi r^{3} \\mathit{n_{\\rm e}} m_{p}$, where $\\rm r$ is the radius of the source and $\\rm m_{p}$ is the mass of proton) is also calculated to be $\\rm 0.09 ~ M_{\\odot}$. \nThe estimated values of these physical parameters lie in between the typical values of compact and UCHII region \\citep{2002ASPC..267...81K,2005A&A...433..205M,2021A&A...645A.110Y}.\nHence this radio source could well represent an intermediate evolutionary stage between a compact and an UC{H\\,{\\scshape ii}~region} thus indicating a direct signature of triggered star formation at the tip of the pillar. An alternate picture for the bright radio emission at the head of the pillar could also be external ionization by the ionizing front emanating from E4. Such externally ionized, tadpole-shaped structures have been studied in the Cygnus region, where the ionized front heads point towards the central, massive Cygnus OB2 cluster  \\citep{2019A&A...627A..58I}. In support of the former scenario of the UC{H\\,{\\scshape ii}~region}, a bright and compact 5.8\\,$\\rm \\mu m$ emission region is seen that is co-spatial with radio emission. This compact IR emission is seen in all IRAC bands and {\\it Herschel} images. While it is not listed as an IRAC point source in the GLIMPSE catalog, it is included in the PACS 70~$\\rm \\mu m$ point source catalog \\citep{2017arXiv170505693M}. There also exists a 2MASS counterpart (within $\\sim$3~arcsec) but has been excluded from the YSO identification procedure owing to poor quality photometry in one or more 2MASS bands. It is thus likely that the compact IR emission sampled in the {\\it Spitzer}-GLIMPSE and {\\it Herschel} images is the massive YSO powering the UC{H\\,{\\scshape ii}~region}.\n\nSeveral studies (e.g. \\citealt{2010ApJ...712..797B,2017MNRAS.470.4662P}) have shown evidence of star formation in the pillar tips in the form jets, outflows, YSO population, etc. \nThe driving mechanism for this triggered star formation, RDI, is initiated when the propagating ionizing front traverses the pillar head creating a shell (known as the ionized boundary layer, IBL) of ionized gas. If the pressure of the IBL exceeds the internal pressure of the neutral gas within the pillar head then shocks are driven into it. This leads to compression and subsequent collapse of the clump leading to star formation. However, to comment further on the detected UC{H\\,{\\scshape ii}~region} on the tip of pillar `A' and link its formation to RDI, one needs to conduct pressure balance analysis using molecular line data as discussed in \\citet{2017MNRAS.470.4662P} and \\citet{2013A&A...556A.105O}. These authors have used $\\rm ^{13}CO$ transitions for their analysis which is not possible in our case as CO molecular line data with adequate spatial resolution is not available for this region of interest. Furthermore, attempting any study with the detected MALT90 transitions is difficult given the limited spatial resolution. In a recent study, \\citet{2020MNRAS.493.4643M} have carried out involved hydrodynamical simulations to study pillar formation in turbulent clouds. As discussed by these authors, star formation triggered in pillar heads can be explained without invoking the RDI mechanism. Gravitational collapse of pre-existing clumps can lead to star formation without the need for ionizing radiation to play any significant role. From their simulations, they conclude that compressive turbulence driven in {H\\,{\\scshape ii}~regions}, which competes with the reverse process of photoevaporation of the neutral gas, ultimately dictates the triggering of star formation in these pillars. Further high-resolution studies are required to understand the nature of the compact radio and IR emission at the head of the pillar `A'.\n\n\\subsubsection*{Dust clumps and the collect and collapse mechanism}\n\\label{clumps-CC}\nThe detection of clumps and the signature of fragmentation to cores is evident from the FIR and sub-mm dust continuum maps presented in Fig.~\\ref{dustclumps}. Investigating the collect and collapse hypothesis is necessary to explain whether the dust clumps are a result of swept-up material that is accumulated or these clumps are pre-existing entities. Towards this, we carry out a simple analysis and evaluate few parameters such as the dynamical age ($t_{\\rm dyn}$) of the {H\\,{\\scshape ii}~region} and the fragmentation time ($t_{\\rm frag}$) of the cloud. \n\nAssuming that the {H\\,{\\scshape ii}~region} associated with IRAS 17149$-$3916 expands in a homogeneous cloud, the dynamical timescale can be estimated using the classical expressions from \\citet{1978ppim.book.....S} and \\citet{1980pim..book.....D},\n\\begin{equation}\nt_{\\rm dyn} = \\frac{4}{7}\\frac{R_{\\rm St}}{a_{\\rm Hii}}\\left [ \\left ( \\frac{R_{\\rm if}}{R_{\\rm St}} \\right )^{7\/4} - 1  \\right ] \n\\end{equation}\nwhere, $a_{\\rm Hii}$ is the isothermal sound speed and is assumed to be 10~km s$^{-1}$, $R_{\\rm if} = 1.8$~pc is the radius of the {H\\,{\\scshape ii}~region} determined from the geometric mean of an ellipse visually fit to encompass the ionized emission in the 610~MHz GMRT map. $ R_{\\rm St}$ is the Str\\\"{o}mgren radius, given by the following equation\n\\begin{equation}\nR_{\\rm St} = \\left ( \\frac{3 N_{\\rm Ly}}{4 \\pi n^{2}_{0} \\alpha_{\\rm B}} \\right )^{1\/3} \n\\end{equation}\nwhere, $N_{\\rm Ly}$ is the Lyman continuum flux, $n_{0}$ is the initial particle density of the ambient gas. To derive $n_{0}$, we assume that the dense, bright region seen in the column density map is swept-up material due to the expansion of the H\\,{\\scshape ii}~region\\ and it was initially homogeneously distributed within the radius of the {H\\,{\\scshape ii}~region}. To estimate the mass of this swept-up material, we use equation \\ref{mass_eqn} and the column density map (refer Section \\ref{dust_clump}). Integrating within the black contour shown in Fig.~\\ref{cdtchi}(a), the mass is estimated to be $1645~ {M_{\\odot}}$. Taking the observed estimate of the radius, we calculate $n_{0}$ to be $9.5 \\times 10^{2}$ cm$^{-3}$.  $\\alpha_{B}$ is the coefficient of radiative recombination and is determined using the expression \\citep{1980pim..book.....D}:\n\\begin{equation}\n\\alpha_{\\rm B} = 2.6\\times10^{-13} \\left ( \\frac{10^{4}\\textup{K}}{T_{\\rm e}} \\right )^{0.7} \\textup{cm}^{3} \\: \\textup{s}^{-1}\n\\end{equation}\nwhere, $T_{\\rm e} = 5000$~K, is the electron temperature. Using the above parameters and from the spectral type of the ionizing source (O6.5V - O7V), we estimate the $t_{\\rm dyn}$ to be $\\sim$ 0.2~Myr.\n\nUsing the formalism discussed in \\citet{1994MNRAS.268..291W}, we proceed next to estimate the fragmentation time scale of a cloud and that can be written as\n\\begin{equation}\nt_{\\rm frag} = 1.56 \\: a^{7\/11}_{{\\rm s},0.2}\\:  \\mathit{N^{-1\/11}_{{\\rm Ly},\\textup{49}}}\\:  n^{-5\/11}_{0,3} \\: \\textup{Myr}\n\\end{equation}\nwhere, $a_{\\rm s} = a_{\\rm s,0.2} \\times 0.2$ km s$^{-1}$ is the speed of sound in the shocked layer and is taken as 0.3 km s$^{-1}$ \\citep{2017MNRAS.472.4750D}. $N_{\\rm Ly} = N_{\\rm Ly,\\textup{49}} \\times 10^{49}$ s$^{-1}$ is the ionizing photon flux and $n_{0} = n_{0,3}\\times 10^{3}$ cm$^{-3}$ is the initial particle density of the ambient gas. Plugging in the values in the above expression, we estimate $t_{\\rm frag}$ to be $\\sim$ 2.2~Myr.\nComparing the estimates of the two time scales involved, it is seen that the fragmentation time scale is more than a factor of 10 larger than the dynamical time scale of the {H\\,{\\scshape ii}~region}. This essentially indicates that if the clumps detected are the result of swept-up material due to expansion of the {H\\,{\\scshape ii}~region}, then the shell has not got enough time to fragment thus making the collect and collapse process highly unlikely here. Such a scenario has been invoked by \\citet{2012A&A...544A..39J} for the dust bubble N22. In contrast, \\citet{2017MNRAS.472.4750D}, in their investigation of bubble CS51 found support for the collect and collapse hypothesis. Thus, further studies, as indicated earlier, are required to probe the RDI process not only with regards to the pillar structures but also the detected clumps. \n\n\\subsection{Nature of the detected dust clumps and cores}\n\\subsubsection*{Virial analysis of the dust clumps}\nHere, we investigate the gravitational stability of the identified dust clumps associated with IRAS 17149$-$3916. This would enable us to determine whether these clumps are gravitationally bound or not. The virial mass, $M_{\\rm vir}$, of a dust clump is the amount of mass that can be supported against self-gravity purely by thermal and non-thermal gas motion. This is given by \\citet{2016MNRAS.456.2041C}\n\\begin{equation}\nM_{\\rm vir} = \\frac{5\\ r\\ \\Delta V^2}{8\\ {\\rm ln}(2)\\ a_1\\ a_2\\ G} \\sim 209\\ \\frac{1}{a_1\\ a_2} \\left(\\frac{\\Delta V}{\\rm km\\ s^{-1}} \\right)^2\\ \\left(\\frac{r}{\\rm pc}\\right) M_{\\odot}\n\\end{equation}\nIn the above equation, $\\Delta V$ is the line width of the optically thin $\\rm N_2 H^+$ line, $r$ is the radius of clumps taken from Table \\ref{clump-param}, the constant $a_1$ accounts for the correction for power-law density distribution, and is given by $a_1 = (1-p\/3)\/(1-2p\/5)$, for $p< 2.5$ \\citep{1992ApJ...395..140B} where we adopt $p=1.8$ \\citep{2016MNRAS.456.2041C}. The constant $a_2$ accounts for the shape of the clump which we assume to be spherical and take $a_2$ as 1.\nWe also calculate the virial parameter, $\\alpha_{\\rm vir} = M_{\\rm vir}\/M_{\\rm clump}$. The estimated values of $M_{\\rm vir}$ and $\\alpha_{\\rm vir}$ are listed in Table \\ref{clump-param}. \nAs discussed in \\citet{2013ApJ...779..185K} and \\citet{2019ApJ...878...10T}, $\\alpha_{\\rm vir} = 2$ sets a lower threshold for gas motions to prevent collapse in the absence of magnetic field and\/or external pressure. The virial parameter estimate for Clump 1 is $ < 2$ indicating that it is gravitationally bound and hence likely to collapse. However, Clump 2 is marginally above this threshold and Clump 3, which shows signature of star formation in the form of an UC{H\\,{\\scshape ii}~region} has a higher virial parameter value of 4.1. Similar values of $\\alpha > 2$ has been observed for protostellar and prestellar dense cores by \\citet{2019ApJ...878...10T}. These authors have used the $\\rm C^{18}O$ line and discuss the contribution from turbulence as a primary factor that would significantly affect the line width and hence overestimate the virial mass. While turbulence gets dissipated in the densest region of molecular clouds and the $\\rm N_2 H^+$ line used here is a dense gas tracer, it is likely that the resolution of the MALT90 survey does not probe the inner dense cores and the observed velocity dispersion is influenced by the outer and more turbulent region. High-resolution molecular line observations are thus essential to probe the nature of the clumps.  \n\n\\subsubsection*{Clump fragmentation and the detected cores}\n{\\it ALMA} 1.4~mm continuum map (Fig.~\\ref{dustclumps}) is seen to resolve the identified dust clumps into a string of cores, with masses ranging between $0.3 - 62.3~M_{\\odot}$ and radii $\\sim 0.01$~pc, thus indicating a scenario of hierarchical fragmentation. If we assume that fragmentation of the clumps is governed by thermal Jeans instability, then the initially homogeneous gas clump has a Jeans length and mass given by \\citet{2019ApJ...886..102S}\n\\begin{equation}\n    \\lambda_J = \\sigma_{\\rm th} \\left ( \\frac{\\pi}{G\\rho} \\right )^{1\/2}\n\\end{equation}\nand \n\\begin{equation}\n    M_J = \\frac{4\\pi\\rho}{3}\\left ( \\frac{\\lambda_J}{2} \\right )^3 = \\frac{\\pi^{5\/2}}{6}\\frac{\\sigma_{\\rm th}^3}{\\sqrt{G^3\\rho}}\n\\end{equation}\nwhere $\\rho$ is the mass density, $G$ the gravitational constant and $\\sigma_{\\rm th}$ the thermal velocity dispersion (the isothermal sound speed) and is given by\n\\begin{equation}\n    \\sigma_{\\rm th} = \\left ( \\frac{k_B T}{\\mu m_{\\rm H}} \\right )^{1\/2}\n\\end{equation}\nwhere $ k_B$ is the Boltzmann constant and $\\mu$ the mean molecular weight. \nAs the thermal velocity dispersion will be dominated by $\\rm H_2$ and He, we consider $\\mu = 2.37$ \\citep{{2008A&A...487..993K},{2014MNRAS.439.3275W},{2019ApJ...886..102S}}. Using the clump parameters tabulated in Table \\ref{clump-param}, we estimate $\\lambda_J$ and $M_J$ of the clumps which are listed in Table \\ref{clump-Jeans}. If turbulence drives the fragmentation instead, then the turbulent Jeans length and mass for each clump is derived by replacing the thermal velocity dispersion with the clump velocity dispersion estimated from the observed line width of the dense gas tracer $\\rm N_2 H^+$ which is a good approximation for the turbulent line width. The calculated $\\lambda_{\\rm turb}$ and $M_{\\rm turb}$ values are given in Table \\ref{clump-Jeans}. \n\\begin{table*}\n\\caption{Hierarchical fragmentation of clumps associated with IRAS$-$3916}\n\\begin{center}\n\\centering\n\\begin{tabular}{ccccccccc}\n\\hline\nClump & $\\sigma_{\\rm th}$ & $\\sigma^a$ & $\\lambda_J$ & $M_J$ &$\\lambda_{\\rm turb}$ & $M_{\\rm turb}$  \\\\\n      & ($\\rm km\\,s^{-1}$) & ( $\\rm km\\,s^{-1}$) & (pc) & ($M_{\\odot}$)  & (pc)  & ($M_{\\odot}$) \\\\  \n\\hline \n\n1   & 0.3   & 1.4   & 0.2   & 6.8   & 0.8   & 543 \\\\\n2   & 0.3   & 1.6   & 0.2   & 5.3   & 0.9   & 807 \\\\\n3   & 0.3   & 1.7   & 0.1   & 5.6   & 0.8   & 819\\\\\n\n\\hline\n\\end{tabular}\n\\label{clump-Jeans}\n\\end{center}\n$^a$ $\\sigma = \\Delta V\/\\sqrt{8\\,{\\rm ln}2}; \\Delta V$ being the $\\rm N_2 H^+$ line width.\n\\end{table*}\nThe turbulent Jeans masses are $\\sim 80 - 150$ times larger than the thermal Jeans mass. \nComparing with the derived core masses, it is seen that 11 out of the 15 detected cores ($\\sim 73\\%$) in the three clumps have masses less than the Jeans mass. This suggests that the observed cores are consistent with the prediction of Jeans fragmentation without invoking turbulence indicating that it does not play a significant role in the fragmentation process. Similar results are obtained by \\citet{2019ApJ...886..102S} who studied the 70\\,$\\rm \\mu m$ dark massive clumps in early stages using {\\it ALMA} data. As discussed by these authors, the majority of detected cores having masses less than the thermal Jeans mass supports competitive accretion and hierarchical fragmentation frameworks. The four cores whose mass exceeds the Jeans mass (the `super-Jeans' cores) are suitable candidates for forming high-mass cores. However, further high-resolution observations are essential to completely understand the fragmentation process, if any, at the core level. \n\n\\begin{figure*}\n\\centering \n\\includegraphics[scale=0.6]{Fig10.eps}\n\\caption{Masses of the dense cores identified from the 1.4\\,mm {\\it ALMA} maps and the cold dust clumps identified using the 350\\,$\\rm \\mu m$ \\textit{Herschel} map of IRAS 17149$-$3916 are plotted as a function of their effective radii, depicted by circles and `$\\star$'s, respectively. The shaded area corresponds to the low-mass star-forming region that do not satisfy the condition, $ M > 870\\,M_\\odot(r\/\\rm pc)^{1.33}$ \\citep{2010ApJ...723L...7K}. Black-dashed lines indicate the surface density thresholds of 0.05 and 1\\,$\\rm g\\,cm^{-2}$ defined by \\citet{2014MNRAS.443.1555U}  and \\citet{2008Natur.451.1082K}, respectively. The red lines represent the surface density thresholds of 116\\,$M_\\odot\\,\\rm pc^{-2}$ ($\\sim 0.024\\,\\rm g\\,cm^{-2}$) and 129\\,$M_\\odot\\,\\rm pc^{-2}$ ($\\sim 0.027\\,\\rm g\\,cm^{-2}$) for active star formation proposed by \\citet{2010ApJ...724..687L} and \\citet{2010ApJ...723.1019H}, respectively.}\n\\label{alma_cores_mass_radius}\n\\end{figure*}\nIn Fig.~\\ref{alma_cores_mass_radius}, we plot the estimated mass and radius of the identified clumps and cores. The plot also compiles several surface density thresholds proposed by various studies to identify clumps\/cores with efficient and active star formation  \\citep{2010ApJ...724..687L,2010ApJ...723.1019H,2014MNRAS.443.1555U}. In addition, criteria for these to qualify as high-mass star-forming ones are also included. All the detected clumps and cores associated with IRAS 17149$-$3916 are seen to be active star-forming regions. The three clumps satisfy the empirical mass-radius criteria, $ M > 870\\,M_\\odot(r\/\\rm pc)^{1.33}$ defined by \\citet{2010ApJ...723L...7K}, and hence are likely to harbour massive star formation. At the core scale ($\\rm < 0.1~pc$), \\citet{2008Natur.451.1082K} have posed a theoretical surface density threshold of $\\rm 1\\,g\\,cm^{-2}$, below which cores would be devoid of high-mass star formation. From the figure, we see that there are four cores (2 in Clump 1, 1 each in Clumps 2 and 3) which have masses $\\gtrsim 10~M_{\\odot}$ and above this surface density limit. These are the `super-Jeans' cores discussed above. \nHigh-resolution molecular line observations are essential to shed better light on the nature of the cores, the gas kinematics involved and for accurate determination of physical parameters like temperature, mass, etc.\n\n\\subsection{Conclusion}\n\\label{conclusion}\nUsing multiwavelength data, we have carried out a detailed analysis of the region associated with IRAS 17149$-$3916. The important results of this study are summarized below.\n\n\\begin{enumerate}\n\\item Using the GMRT, we present the first low-frequency radio continuum maps of the region mapped at 610 and 1280~MHz. The {H\\,{\\scshape ii}~region}, previously believed to be nearly spherical, displays a large-extent cometary morphology. The origin of this morphology is not explained by the bow shock model. The presence of dense clumps towards the cometary head indicates either the champagne flow or the clumpy cloud model but further observations of the ionized gas kinematics are essential to understand the observed morphology. \n\n\\item The integrated flux densities yields an average spectral index value of $-0.17\\pm0.19$ consistent with thermal {\\it free-free} emission. If powered by a single massive star, the estimated Lyman continuum photon flux suggests an exciting star of spectral type  O6.5V -- O7V star. \n\n\\item NIR colour-magnitude and colour-colour diagrams show the presence of a cluster of massive stars (earlier than spectral type B0) located within the bright, central radio emitting region. MIR and FIR images show complex and interesting features like a bubble, pillars, clumps, filaments,and arcs revealing the profound radiative and mechanical feedback of massive stars on the surrounding ISM. \n\n\\item The spatial location of source, E4 (IRS-1), and the orientation of observed pillar structures with respect to it, strongly suggest it as the dominant driving source for the cometary {H\\,{\\scshape ii}~region}. This view finds support from the position of E4 in the colour-magnitude and colour-colour diagrams. Further, its spectral type estimation from literature agrees well with that estimated for the exciting source of the {H\\,{\\scshape ii}~region} from GMRT data. \n\n\\item The column density map reveals the presence of dust clumps towards the cometary head while the dust temperature map appears to be relatively patchy with regions of higher temperature within the radio nebula. The dust clumps identified using the  \\textit{Herschel} 350\\,$ \\rm \\mu m$ map have masses ranging between $\\sim$100 - 300~$\\rm M_\\odot$ and radius $\\sim$0.2 - 0.3~pc. Virial analysis using the $\\rm N_2 H^+$ shows that the south-east clump (\\#1) is gravitationally bound. For the other two clumps (\\# 2 and 3), the line widths would possibly have contribution from turbulence thus rendering larger values of the virial parameter.\n\n\\item A likely compact\/UC{H\\,{\\scshape ii}~region} is seen at the tip of a pillar structure oriented towards the source E4 thus suggesting evidence of triggered star formation under the RDI framework. In addition, the detected dust clumps are investigated to probe the collect and collapse model of triggered star formation. The estimated dynamical time scales are seen to be smaller by a factor of $\\sim$10 compared to the fragmentation timescale of the clumps thus clearly negating the collect and collapse mechanism at work. \n\n\\item The {\\it ALMA} 1.4~mm dust continuum map probes the dust clumps at higher resolution and reveal the presence of 17 compact dust cores with masses and radii in the range of $0.3 - 62.3~M_\\odot$ and 0.01 -- 0.03~pc, respectively. The largest and the most massive core is located within Clump 3. The estimated core masses are consistent with thermal Jeans fragmentation and support the competitive accretion and hierarchical fragmentation scenario. \n\n\\item Four `super-Jeans' fragments are detected and are suitable candidates for forming high-mass stars and their mass and radius estimates satisfy the various threshold defined in literature for the potential high-mass star-forming cores.\n\n\\end{enumerate}\n\n\\section*{Acknowledgements}\nWe would like to thank the referee for comments and suggestions which helped in improving the quality of the manuscript. We thank the staff of the GMRT that made the radio observations possible. GMRT is run by the National Centre for Radio Astrophysics\nof the Tata Institute of Fundamental Research. The authors would like to thank Dr. Alvaro S\\'{a}nchez-Monge for providing the FITS image of the radio maps presented in \\citet{2013A&A...550A..21S}. CHIC acknowledges the support of the Department of Atomic Energy, Government of India, under the project  12-R\\&D-TFR-5.02-0700. This work is based\n[in part] on observations made with the {\\it Spitzer} Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This publication also made use of data products from {\\it Herschel} (ESA space observatory). This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center\/California Institute of Technology, funded by the NASA and the NSF. This work makes use of the ATLASGAL data, which is a collaboration between the Max-Planck-Gesellschaft, the European Southern Observatory (ESO) and the Universidad de Chile. This paper makes use of the following ALMA data: ADS\/JAO.ALMA\\#2016.1.00191.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. \n\n\\addcontentsline{toc}{section}{Acknowledgements}\n\\section*{Data Availability}\nThe original data underlying this article will be shared on reasonable request to the corresponding author.\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec: intro}\nIn the physical sciences, simple mathematical models are often used as a means of developing intuition and capturing phenomenology for complex systems. Whilst more complex, faithful models might require computational methods or advanced analytical techniques to study them, simple models are commonly amenable to succinct pen-and-paper analysis, and can sometimes yield dynamical predictions that agree with those of more intricate representations, at least qualitatively. This minimalistic approach is readily exemplified in the study of microswimmers, where minimal models of complex, shape deforming swimmers on the microscale are used in a range of settings, from back-of-the-envelope calculation and undergraduate teaching through to state-of-the-art, application-driven research contributions \\cite{Zottl2012,Thery2020,Gutierrez-Ramos2018,Qi2022,Spagnolie2015,Elgeti2016,Malgaretti2017,Mathijssen2016,Lauga2020,Omori2022,Lauga2009}.\n\nA common, if not ubiquitous, hydrodynamic representation of a microswimmer is the \\emph{force dipole} model, wherein the flow field generated by a swimmer is taken to correspond simply to that of a force dipole. This approximate representation is valid in the far field of a microswimmer, with force-free swimming conditions being appropriate in the inertia-free limit of low-Reynolds-number swimming that applies to many microswimmers, including the well-studied spermatozoa and the breaststroke-swimming algae \\textit{Chlamydomonas reinhardtii}. Invariably, the force dipole is assumed to be aligned along a swimmer-fixed axis and taken to be of constant signed strength. These parameters can be estimated from experimental measurements and hydrodynamic simulations \\cite{Klindt2015,Ishimoto2020}, and typically involve averaging out rapid temporal variations that can be present in biological swimmers. This leads to a minimal model of microswimming: instead of studying a complex, shape-deforming swimmer and the associated time-varying flow field, we can instead consider the motion of a constant-strength force dipole in the same environment. With this approximation, one can often derive surrogate equations of motion for the swimmer \\cite{Kim2005,Lauga2009}, which can then be analysed with ease, at least when compared to models that capture the intricate, time-dependent details of the microswimmer and the flow that it generates.\n\nThough many of the assumptions associated with this modelling approach are well understood, such as the limitations of the far-field approximation when studying near-field interactions, the impact of assuming a constant dipole strength is less clear. More generally, the impact of adopting constant, \\textit{a priori}-averaged parameters in simple models of temporally evolving microswimmers has not been thoroughly investigated, to the best of our knowledge. However, it is clear that employing rapidly varying parameters can have significant consequences for the predictions of simple models. For example, the work of \\citet{Omori2022} recently explored a simple model of shape-changing swimmers, explicitly including rapid variation in the parameters that describe the swimmer shape and its speed of self propulsion. The subsequent analysis of \\citet{Walker2022a} highlighted, amongst other observations, that the fast variation in the parameters was key to the behavioural predictions of the model, which were found to align with the experimental observations of \\citet{Omori2022}. Hence, the study of the effects of employing time-dependent parameters in even simple models, in comparison to adopting constant, averaged parameters, is warranted.\n\nThus, the primary aim of this study will be to explore minimal models of microswimming, incorporating fast variation in model parameters. To do this, motivated by a number of recent works in a similar vein \\cite{Walker2022,Walker2022a,Gaffney2022}, we will employ a multiple scales asymptotic analysis \\cite{Bender1999} to systematically derive effective governing equations from non-autonomous models. In particular, we will incorporate the effects of rapid variation by exploiting the separation of timescales often associated with microswimming, yielding leading-order autonomous dynamical systems. In what follows, we will focus on two example scenarios: the interaction of a dipole swimmer with a boundary, and the angular dynamics of two hydrodynamically interacting dipoles. Through our analysis, which will be simple, if not elementary, we will compare the dynamics of multi-timescale models with the predictions of the simplest, constant-parameter models, seeking to ascertain both if qualitative differences arise and if they can be systematically corrected for by informed parameter choices.\n\n\\section{A dipole near a no-slip boundary}\\label{sec: no slip}\n\\subsection{Model equations}\nConsider a swimmer moving in a half space that is bounded by an infinite plane, with the swimmer moving in a plane perpendicular to the boundary. We parameterise the orientation of the swimmer by the angle $\\theta$ between a swimmer-fixed director $\\d$ and the boundary normal, and parameterise its position by the distance $h$ from its centre to the boundary, as illustrated in \\cref{fig: no slip: setup}. With all quantities dimensionless, a minimal model for the swimmer dynamics is presented in part by \\citet{Lauga2020} as\n\\begin{subequations}\\label{eq: no slip: original system}\n\\begin{align}\n    \\diff{h}{t} &= \\frac{3p}{16h^2}\\left(1 + 3\\cos{2\\theta}\\right) + u \\cos{\\theta}\\,,\\\\\n    \\diff{\\theta}{t} &= \\frac{3p\\sin{2\\theta}}{64h^3}\\left[4 + B(3 + \\cos{2\\theta})\\right]\\,,\\label{eq: no slip: original system: angular}\n\\end{align}\n\\end{subequations}\nwhere $u$ is speed of self propulsion and we have shifted \\citeauthor{Lauga2020}'s definition of $\\theta$ by $\\pi\/2$. In this minimal model, the flow generated by the swimmer in the absence of the boundary is assumed to be purely that of a force dipole with vector strength $\\vec{p}=p\\d$, aligned along the body fixed director that defines $\\theta$, and the swimmer shape is captured only through the Bretherton parameter $B$ \\cite{Bretherton1962}. This modelling approach can be justified by considering a far-field limit of a swimmer, though here we focus on analysing the model of \\cref{eq: no slip: original system} rather than on its origin and motivation. In particular, we focus on the angular dynamics contained within \\cref{eq: no slip: original system: angular}.\n\nThe standard approach to modelling this system would be to assume that $p$, $u$, and $B$ are constant in time, as is the case in the textbook of \\citet{Lauga2020}. This can be interpreted as averaging away any time dependence of the three parameters, which one would generically expect to be present for a multitude of shape-changing microswimmers, for instance. Here, we do not perform this \\emph{\\textit{a priori}} averaging of the parameters, and will instead suppose that $p$, $u$, and $B$ are indeed functions of time. In particular, we suppose that $p=p(\\omega t)$, $u=u(\\omega t)$, and $B=B(\\omega t)$ are periodic functions of $\\omega t$, where $\\omega\\gg1$ is a large dimensionless frequency of oscillation and we assume that $p$, $u$, and $B$ share a period, in line with the rapid shape changes undergone by many microswimmers. For later convenience, we make the additional assumption that the average of $p$ over a period is non-zero, and will impose the minimal restriction that $B\\in(-1,1)$, which holds for all but the most elongated of objects \\cite{Bretherton1962}. Hence, we study the non-autonomous system\n\\begin{subequations}\\label{eq: no slip: full system}\n\\begin{align}\n    \\diff{h}{t} &= \\frac{3p(\\omega t)}{16h^2}\\left(1 + 3\\cos{2\\theta}\\right) + u(\\omega t) \\cos{\\theta}\\,,\\\\\n    \\diff{\\theta}{t} &= \\frac{3p(\\omega t)\\sin{2\\theta}}{64h^3}\\left[4 + B(\\omega t)(3 + \\cos{2\\theta})\\right]\\,,\n\\end{align}\n\\end{subequations}\nwith $\\omega\\gg1$ and all other quantities being $\\bigO{1}$ as $\\omega\\to\\infty$. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{figs\/no_slip_setup\/no_slip_setup.png}\n    \\caption{Geometry and parameterisation of a minimal model of a swimmer above a no-slip boundary. The swimmer is parameterised by the angle $\\theta$ between the vector dipole strength $\\vec{p}$ and the separation $h$ of its centroid from the boundary. The velocity due to self-propulsion is assumed to be along the direction of $\\vec{p}$, shared with swimmer-fixed director $\\d$.}\n    \\label{fig: no slip: setup}\n\\end{figure}\n\n\\subsection{Multi-scale analysis}\nWe will attempt to exploit the large frequency $\\omega \\gg 1$ in order to make progress in analysing the non-autonomous system of \\cref{eq: no slip: full system}, employing the method of multiple scales \\cite{Bender1999}. Following this approach, we introduce the fast timescale $T \\coloneqq \\omega t$, so that $p = p(T)$ etc., and formally treat $t$ and $T$ as independent. The proper time derivative $\\mathrm{d}\/\\mathrm{d}t$ accordingly transforms as\n\\begin{equation}\\label{eq: no slip: time transform}\n    \\diff{}{t} \\mapsto \\pdiff{}{t} + \\omega \\pdiff{}{T}\\,,\n\\end{equation}\ntransforming our non-autonomous system of ordinary differential equations (ODEs) into a system of partial differential equations (PDEs). We now seek asymptotic expansions of $h$ and $\\theta$ in inverse powers of $\\omega$, which we write as\n\\begin{equation}\n    h \\sim h_0(t,T) + \\frac{1}{\\omega}h_1(t,T) + \\cdots\\,, \\quad \\theta \\sim \\theta_0(t,T) + \\frac{1}{\\omega}\\theta_1(t,T) + \\cdots\\,.\n\\end{equation}\nTransforming \\cref{eq: no slip: full system} via \\cref{eq: no slip: time transform} and inserting these asymptotic expansions gives the $\\bigO{\\omega}$ balance simply as\n\\begin{equation}\n    \\pdiff{h_0}{T} = 0\\,, \\quad \\pdiff{\\theta_0}{T} = 0 \\quad \\implies \\quad h_0 = h_0(t)\\,, \\quad \\theta_0 = \\theta_0(t)\\,,\n\\end{equation}\nso that the leading order solutions are independent of the fast timescale $T$. This should be expected, as the forcing of the system is strictly $\\bigO{1}$, so that the dominant contribution to the evolution occurs on the long timescale $t$.\n\nAt the next asymptotic order we pick up the $\\bigO{1}$ forcing, and we have\n\\begin{subequations}\\label{eq: no slip: order unity system}\n\\begin{align}\n    \\diff{h_0}{t} + \\pdiff{h_1}{T} &= \\frac{3p(T)}{16h_0^2}(1 + 3\\cos{2\\theta_0}) + u(T)\\cos{\\theta_0}\\,,\\\\\n    \\diff{\\theta_0}{t} + \\pdiff{\\theta_1}{T} &= \\frac{3p(T)\\sin{2\\theta_0}}{64h_0^3}\\left[4 + B(T)(3 + \\cos{2\\theta_0})\\right]\\,,\n\\end{align}\n\\end{subequations}\nwriting $t$-derivatives of $h_0$ and $\\theta_0$ as proper due to their established independence from $T$. The appropriate solvability conditions for this first-order system are obtained by averaging the equations over a period in $T$ and imposing periodicity in $T$, equivalent to the Fredholm Alternative Theorem for this system \\cite{Bender1999}. To do so, we assume, without loss of generality, that the period of the fast oscillations is $2\\pi$, defining the averaging operator $\\avg{\\cdot}$ via\n\\begin{equation}\\label{eq: no slip: averaging operator}\n    \\avg{a} = \\frac{1}{2\\pi}\\int_0^{2\\pi}a(T)\\mathop{}\\!\\mathrm{d}{T}\\,.\n\\end{equation}\nComputing the average of \\cref{eq: no slip: order unity system}, we arrive at\n\\begin{subequations}\\label{eq: no slip: systematic model}\n\\begin{align}\n    \\diff{h_0}{t} &= \\frac{3\\avg{p}}{16h_0^2}(1 + 3\\cos{2\\theta_0}) + \\avg{u}\\cos{\\theta_0}\\,,\\\\\n    \\diff{\\theta_0}{t} &= \\frac{3\\avg{p}\\sin{2\\theta_0}}{64h_0^3}\\left[4 + \\frac{\\avg{pB}}{\\avg{p}}(3 + \\cos{2\\theta_0})\\right]\\,,\n\\end{align}\n\\end{subequations}\nwith the imposed periodicity eliminating the fast-time derivatives. Comparing these leading order differential equations with those of \\cref{eq: no slip: full system}, we see that we have essentially replaced the parameters $p$, $u$, and $B$ with the effective parameters $\\avg{p}$, $\\avg{u}$, and $\\avg{pB}\/\\avg{p}$, the precise forms of which have arisen through our brief, systematic analysis. Whilst the modifications to $p$ and $u$ are as might be naively expected, the effective shape constant $\\avg{pB}\/\\avg{p}$ is perhaps less obvious at first glance, with one perhaps expecting the average parameter $\\avg{B}$. Indeed, with these authors having previously been guilty of employing such parameters in back-of-the-envelope calculations, we refer to this model as the \\emph{\\textit{a priori}-averaged model}, given explicitly by \n\\begin{subequations}\\label{eq: no slip: a priori model}\n\\begin{align}\n    \\diff{h^a}{t} &= \\frac{3\\avg{p}}{16h^2}\\left(1 + 3\\cos{2\\theta}\\right) + \\avg{u} \\cos{\\theta}\\,,\\\\\n    \\diff{\\theta^a}{t} &= \\frac{3\\avg{p}\\sin{2\\theta}}{64h^3}\\left[4 + \\avg{B}(3 + \\cos{2\\theta})\\right]\\,,\n\\end{align}\n\\end{subequations}\nusing a superscript of $a$ to denote the solutions of the \\textit{a priori}-averaged system. This model makes use of the averaged parameters $\\avg{p}$, $\\avg{u}$, and $\\avg{B}$ in place of the rapidly oscillating quantities.\n\nThough an elementary observation, it is worth highlighting that, without any additional assumptions on $p(T)$ and $B(T)$, it is in general not the case that $\\avg{pB}\/\\avg{p} = \\avg{B}$, so that we should expect to observe differences between the systematically determined, leading-order dynamics of \\cref{eq: no slip: systematic model} and those of the \\textit{a priori}-averaged model of \\cref{eq: no slip: a priori model}. In what follows, through a brief consideration of the angular evolution equations, we will highlight how these differences can be more than simply quantitative.\n\nFocussing on the angular dynamics, we specifically consider the abstracted scalar autonomous ODE\n\\begin{equation}\\label{eq: no slip: scalar ode}\n    \\diff{x}{t} = f(x;\\alpha,\\beta)\\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq: no slip: f}\n    f(x;\\alpha,\\beta) \\coloneqq \\frac{3\\alpha\\sin{2x}}{64h^3}\\left[4 + \\beta(3 + \\cos{2x})\\right]\\,,\n\\end{equation}\nso that $(\\alpha,\\beta) = (\\avg{p},\\avg{B})$ corresponds to the \\textit{a priori}-averaged model, whilst $(\\alpha,\\beta) = (\\avg{p},\\avg{pB}\/\\avg{p})$ gives the leading order, systematically averaged dynamics. Note that, for the purposes of a stability analysis of the angular dynamics, we can treat the swimmer separation from the boundary as a positive parameter, abusing notation and generically writing $h$ in the denominator of \\cref{eq: no slip: f}, without materially modifying a steady state analysis of the angular dynamics.\n\n\\subsection{Exploring the autonomous dynamics}\nThe fixed points of \\cref{eq: no slip: scalar ode} are readily seen to be $x = n\\pi$, $x=\\pi\/2 + n\\pi$, and solutions of $4 + \\beta(3 + \\cos{2x})=0$ (if they exist), for $n\\in\\mathbb{Z}$. Notably, if $\\beta\\in(-1,1)$, as is the case in the \\textit{a priori}-averaged model, then the only steady states are at integer multiples of $\\pi\/2$. Focussing on these steady states, their linear stability is given by\n\\begin{equation}\\label{eq: no slip: simple steady states}\n    x = \\left\\{\\begin{array}{lr}\n        n\\pi & \\text{ is stable } \\iff\\alpha(1+\\beta) < 0\\,,\\\\\n        \\pi\/2 + n\\pi & \\text{ is stable } \\iff\\alpha(2+\\beta) > 0\\,.\\\\\n        \\end{array}\\right.\n\\end{equation}\nHence, for $\\beta\\in(-1,1)$, the stability of the steady states is determined solely by the sign of $\\alpha$, with $\\alpha>0$ giving rise to unstable states at $x=n\\pi$ and stable states at $x=\\pi\/2+n\\pi$. Identifying $\\alpha$ with the signed dipole strength, this is precisely in line with the classical analysis of pusher and puller swimmers via the \\textit{a priori}-averaged model, as summarised by \\citet{Lauga2009}, with pushers and pullers corresponding to $\\alpha > 0$ and $\\alpha < 0$, respectively.\n\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[permil,width=0.7\\textwidth]{figs\/no_slip_dynamics\/no_slip_dynamics.png}\n    \\put(100,140){$\\beta<-2$}\n    \\put(425,140){$-2<\\beta<-1$}\n    \\put(810,140){$\\beta>-1$}\n    \\put(0,300){(a)}\n    \\put(350,300){(b)}\n    \\put(700,300){(c)}\n    \\put(115,300){$\\theta=0$}\n    \\put(-30,140){$\\frac{\\pi}{2}$}\n    \\put(136,-27){$\\pi$}\n    \\put(293,140){$\\frac{3\\pi}{2}$}\n    \\end{overpic}\n    \\caption{Steady states and stability of angular dynamics for the autonomous system of \\cref{eq: rollers: abstract autonomous system} are shown as dynamics on a circle, for $\\alpha>0$ fixed and for various values of $\\beta$. Swimming parallel to the boundary corresponds to states with $\\theta=\\pi\/2$, $3\\pi\/2$, whilst $\\theta=0$, $\\pi$ corresponds to swimming aligned with the normal to the boundary. (a) With $\\beta < -2$, the system evolves to a steady state with $x=n\\pi$, $n\\in\\mathbb{Z}$. (b) For $-2<\\beta<-1$, $x=n\\pi\/2$ are stable for all $n\\in\\mathbb{Z}$, with unstable configurations present between these attractors. (c) For $\\beta > -1$, the system evolves to a steady state with $x=\\pi\/2 + n\\pi$, for $n\\in\\mathbb{Z}$. Stable states are shown as solid points, whilst unstable points are shown hollow. Stabilities for $\\alpha<0$ are obtained by reversing the illustrated dynamics.}\n    \\label{fig: no slip: dynamics}\n\\end{figure}\n\nHowever, if $\\beta<-1$, the profile of stability can change significantly. Bifurcations at $\\beta=-1$ and $\\beta=-2$ see the creation and destruction of additional steady states (the solutions of $4 + \\beta(3 + \\cos{2x})=0$) accompanied by changes in stability of the steady states at $x=n\\pi$ and $x=\\pi\/2 + n\\pi$, respectively. When they exist, the additional states have the opposite stability to the other steady states, so that they are stable for $\\alpha<0$. For $\\beta<-2$, the equation defining the additional steady states admits no real solutions, so that these steady states cease to exist and the angular equilibria are the same as for $\\beta>-1$, though with opposite linear stabilities. Each of these dynamical regimes is illustrated in \\cref{fig: no slip: dynamics} for $\\alpha>0$, highlighting a strong dependence of the dynamics on $\\beta$. The linear stability of each state is flipped upon taking $\\alpha<0$.\n\n\n\\subsection{Comparing the emergent dynamics}\nThe elementary analysis of the previous section highlights how qualitative changes in the globally attractive behaviour of the model depend strongly on the parameter $\\beta$. However, the predictions of the \\textit{a priori}-averaged model are simple: if the swimmer is a pusher, with $\\alpha=\\avg{p}>0$, then the states $\\theta^a=n\\pi$ are unstable, and the swimmer instead evolves to a state where $\\theta^a = \\pi\/2 + n\\pi$ and swims parallel to the boundary. If the swimmer is a puller, with $\\alpha=\\avg{p}<0$, then the swimmer instead evolves to $\\theta^a=n\\pi$, thereafter moving perpendicular to the boundary.\n\nHowever, the predictions of the systematically averaged model are more complex. Whilst switching the sign of $\\avg{p}$ still switches the stability of each steady state, the value of $\\beta = \\avg{pB}\/\\avg{p}$, which need not be smaller than unity in magnitude, can materially alter both the steady states in existence and the stability of the states that correspond to parallel and perpendicular swimming. For instance, if $\\alpha=\\avg{p}>0$ and $\\beta\\in(-2,-1)$, both the $\\theta_0=n\\pi$ and the $\\theta_0=\\pi\/2+n\\pi$ states are linearly stable, accompanied by four steady states in the range $\\theta_0\\in(0,2\\pi)$ that are unstable; for $\\alpha=\\avg{p}<0$, the stability of each state is swapped. Hence, swimmers with $\\avg{p}<0$ and $\\avg{pB}\/\\avg{p}\\in(-2,-1)$ will evolve to a steady state that is not a multiple of $\\pi\/2$; in other words, they will neither align parallel nor perpendicular to the boundary, a behaviour that is never predicted by the \\textit{a priori}-averaged model in any admissible parameter regime.\n\nAs an explicit illustration of how the two models can qualitatively differ, we take $p(T) = 4A\\sin{T} + 1$ and $B(T) = \\sin{(T)}\/2$, so that $\\avg{p} = 1$, $\\avg{B}=0$, and $\\avg{pB}\/\\avg{p} = A$. The \\textit{a priori}-averaged model predicts the dynamics shown in \\cref{fig: no slip: dynamics}c for all values of $A$, whilst the systematically averaged dynamics follow \\cref{fig: no slip: dynamics}b for $A\\in(-2,-1)$ and \\cref{fig: no slip: dynamics}a for $A<-2$. Fixing $A=-3\/2$, the temporal evolution of both of these models is shown in \\cref{fig: no slip: example}, along with a numerical solution to angular dynamics of the full system of \\cref{eq: no slip: full system}. Here, we have fixed $h>0$ as a parameter, recalling that the swimmer separation serves only to modify the rate at which the angular dynamics approach a steady state. In agreement with our analysis, the \\textit{a priori}-averaged model incorrectly predicts the qualitative evolution of the model swimmer, whilst the leading-order, systematically averaged model is in agreement with the full numerical solution.\n\nDespite these general qualitative differences, it should be noted that there are parameter regimes in which the dynamics are qualitatively indistinct between the models. For instance, suppose that we are in a regime where $\\beta=\\avg{pB}\/\\avg{p}\\not\\in(-2,-1)$, so that the only steady states are those given in \\cref{eq: no slip: simple steady states}. Then, if $p(T)$ is of fixed sign for all $T$, so that the swimmer is unambiguously a pusher or a puller, it is simple to note that $\\avg{p} + \\avg{pB} =\\int_0^{2\\pi}p(T)[1+B(T)]\\mathop{}\\!\\mathrm{d}{T}\/2\\pi$ has the same sign as $\\avg{p}$, recalling that $B\\in(-1,1)$. Similarly, $2\\avg{p} + \\avg{pB}$ has the same sign as $\\avg{p}$, so that the long-time behaviour is completely determined by the sign of $\\avg{p}$, following \\cref{eq: no slip: simple steady states}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{figs\/no_slip_example\/no_slip_example.png}\n    \\caption{Angular evolution of a swimmer at fixed separation from a no-slip boundary. The prediction of the \\textit{a priori}-averaged model can be seen to not align with the prediction of the systematically averaged model of \\cref{eq: no slip: systematic model} or the dynamics of the full model of \\cref{eq: no slip: original system}, with qualitatively distinct steady configurations from the same initial condition. Rapid oscillations in the solution full dynamics are not visible at the resolution of this plot. Here, we have fixed $h=1$, $\\omega = 100$, and taken $p(T) = -6\\sin{(T)} + 1$ and $B(T) = \\sin{(T)}\/2$.}\n    \\label{fig: no slip: example}\n\\end{figure}\n\n\\section{Interacting dipole rollers}\\label{sec: rollers}\n\\subsection{Model equations}\\label{sec: rollers: model eqs}\nConsider a pair of particles in the plane that are pinned in place in the laboratory frame, so that they are free to rotate in the plane but unable to translate. The particles, which we term \\emph{rollers}, are assumed to interact through dipolar flow fields, with vector dipole strengths $\\vec{p}_1$ and $\\vec{p}_2$, respectively, which are both assumed to lie in the plane containing the particles. Taking $\\{\\e_x,\\e_y\\}$ to be an orthogonal basis for the laboratory frame that spans this plane, we define the orientation of the $i$\\textsuperscript{th} particle, $i\\in\\{1,2\\}$, via the angle $\\theta_i$ between a body-fixed axis and $\\e_x$, so that the direction of the roller can be captured as $\\d_i = \\cos{\\theta_i}\\e_x + \\sin{\\theta_i}\\e_y$. Analogously to the previous section, we assume that the vector dipole strength of each particle is aligned with $\\d_i$, so that we can write $\\vec{p}_i = p_i\\d_i$, where $p_i$ is the scalar dipolar strength and may take any sign. We further assume that $p_1=p_2$, so that the dipoles are of equal strength, though we remark that retaining generality is straightforward but notationally cumbersome.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{figs\/roller_setup\/roller_setup.png}\n    \\caption{Geometry of fixed-position dipole rollers. Each roller is pinned in place at its centre, but is free to rotate, driven by the dipolar flow generated by the other. Their orientation is captured by their directors $\\d_1$ and $\\d_2$, parameterised by $\\theta_1$ and $\\theta_2$, respectively. The axis of each roller's dipolar flow field is assumed to be directed along their orientation vector.}\n    \\label{fig: roller setup}\n\\end{figure}\nWithout loss of generality, we assume that the displacement of particle $1$ from particle $2$ is $r\\e_x$, where $r>0$ is the distance between them. In this setting, illustrated in \\cref{fig: roller setup} and following \\citet[p. 295]{Lauga2020}, the effects of each particle's dipolar flow on the orientation of the other particle are captured by the following dimensionless coupled system of ODEs:\n\\begin{subequations}\\label{eq: rollers: full system}\n\\begin{align}\n    \\diff{\\theta_1}{t} &= f(\\theta_1,\\theta_2;p,B)\\,,\\\\\n    \\diff{\\theta_2}{t} &= f(\\theta_2,\\theta_1;p,B)\\,,\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{equation}\\label{eq: rollers: def f}\n    f( x,y;\\alpha,\\beta) \\coloneqq -\\frac{3\\alpha}{r^3}\\left(\\sin{ y}\\cos{ y} + \\beta\\left[\\sin{ x}\\cos{ x}\\left(1 - 5\\cos^2{ y}\\right) + \\cos{ y}\\sin{(2 x- y)}\\right]\\right)\\,,\n\\end{equation}\ndistinct from the notation of the previous section. Here, $B$ is once again the shape-capturing parameter of \\citet{Bretherton1962}, and we assume that both of the particles are associated with the same shape parameter. This assumption can be relaxed at the expense of notational convenience.\n\nAs with the previous model, the standard approach would be to assume that $p$ and $B$ are constant, so that the above system is autonomous. Here, we take $p=p(T)$ and $B=B(T)$, interpreting the constant-parameter model as the \\textit{a priori}-averaged model, as above. Hence, we study the non-autonomous system\n\\begin{subequations}\\label{eq: rollers: non-autonomous system}\n\\begin{align}\n    \\diff{\\theta_1}{t} &= f(\\theta_1,\\theta_2;p(\\omega t),B(\\omega t)) \\,,\\\\\n    \\diff{\\theta_2}{t} &= f(\\theta_2,\\theta_1;p(\\omega t), B(\\omega t))\\,,\n\\end{align}\n\\end{subequations}\nwith $\\omega\\gg1$ and all other quantities being $\\bigO{1}$ as $\\omega\\to\\infty$. It should be noted that, whilst we study this system as a simple extension of an established model, these equations can be rigorously derived by considering the dynamics of shape-changing spheroids, subject to the far-field approximation and the assumption that their instantaneous deformation is both force- and torque-free. This derivation is somewhat elementary and follows the approach given in \\citet{Gaffney2022b}, from which this model can also be extended to accommodate more general deformations through the addition of uncomplicated terms. Here, seeking simplicity and clarity, we pursue the model given in \\cref{eq: rollers: non-autonomous system}.\n\n\\subsection{Multi-scale analysis}\nThe multi-scale analysis of this problem proceeds entirely analogously to that of the previous section, but is reproduced here in detail in the interest of clarity. As in \\cref{sec: no slip}, we formally introduce the fast timescale $T=\\omega t$, transforming the proper time derivative as\n\\begin{equation}\\label{eq: rollers: time transform}\n    \\diff{}{t} \\mapsto \\pdiff{}{t} + \\omega \\pdiff{}{T}\\,.\n\\end{equation}\nWe seek an asymptotic expansion of $\\theta_1$ and $\\theta_2$ in inverse powers of $\\omega$, which we write as\n\\begin{equation}\\label{eq: rollers: expansions}\n    \\theta_1 \\sim \\theta_{1,0}(t,T) + \\frac{1}{\\omega}\\theta_{1,1}(t,T) + \\cdots\\,, \\quad \\theta_2 \\sim \\theta_{2,0}(t,T) + \\frac{1}{\\omega}\\theta_{2,1}(t,T) + \\cdots\\,.\n\\end{equation}\nTransforming \\cref{eq: rollers: non-autonomous system} via \\cref{eq: rollers: time transform} and inserting the expansions of \\cref{eq: rollers: expansions}, at $\\bigO{\\omega}$ we have\n\\begin{equation}\n    \\pdiff{\\theta_{1,0}}{T} = 0\\,, \\quad \\pdiff{\\theta_{2,0}}{T} = 0 \\quad \\implies \\quad \\theta_{1,0} = \\theta_{1,0}(t)\\,, \\quad \\theta_{2,0} = \\theta_{2,0}(t)\\,,\n\\end{equation}\nso that the leading order solutions for $\\theta_1$ and $\\theta_2$ are independent of the fast timescale and, thus, are functions only of $t$. As in \\cref{sec: no slip}, this arises due to the forcing of the ODEs being $\\bigO{1}$, so that the forcing doesn't contribute at $\\bigO{\\omega}$. At $\\bigO{1}$, we have\n\\begin{subequations}\\label{eq: rollers: first-order system}\n\\begin{align}\n    \\diff{\\theta_{1,0}}{t} + \\pdiff{\\theta_{1,1}}{T} &= f(\\theta_{1,0},\\theta_{2,0};p(T), B(T)) \\,,\\label{eq: rollers: first-order system: theta1}\\\\\n    \\diff{\\theta_{2,0}}{t} + \\pdiff{\\theta_{2,1}}{T} &= f(\\theta_{2,0},\\theta_{1,0};p(T), B(T))\\,,\\label{eq: rollers: first-order system: theta2}\n\\end{align}\n\\end{subequations}\nusing the fact that $\\theta_{1,0}$ and $\\theta_{2,0}$ are independent of $T$ to write their time derivatives as total derivatives. As in \\cref{sec: no slip}, imposing periodicity and averaging over a period in $T$ closes the system of PDEs at this order. Without loss of generality, we assume that the period of oscillations of $p$ and $B$ is $2\\pi$, and we recall the averaging operator $\\avg{\\cdot}$ from \\cref{eq: no slip: averaging operator} as\n\\begin{equation}\\label{eq: rollers: avg}\n    \\avg{a} = \\frac{1}{2\\pi}\\int_0^{2\\pi} a(T)\\mathop{}\\!\\mathrm{d}{T}\\,.\n\\end{equation}\nTo compute the average of \\cref{eq: rollers: first-order system} in $T$, it is instructive to consider the dependence of $f$ on its arguments explicitly. To that end, we explicitly compute the average of \\cref{eq: rollers: first-order system: theta1} as\n\\begin{equation}\\label{eq: rollers: explicit average}\n    \\diff{\\theta_{1,0}}{t} = -\\frac{3}{r^3}\\left(\\avg{p}\\sin{\\theta_{2,0}}\\cos{\\theta_{2,0}} + \\avg{pB}\\left[\\sin{\\theta_{1,0}}\\cos{\\theta_{1,0}}\\left(1 - 5\\cos^2{\\theta_{2,0}}\\right) + \\cos{\\theta_{2,0}}\\sin{(2\\theta_{1,0}-\\theta_{2,0})}\\right] \\right)\n\\end{equation}\nwith the average of \\cref{eq: rollers: first-order system: theta2} following similarly. Comparing the right-hand side of \\cref{eq: rollers: explicit average} with the definition of $f$ in \\cref{eq: rollers: def f}, we identify the systematically averaged governing equations with those of the original system:\n\\begin{subequations}\\label{eq: rollers: effective eqs}\n\\begin{align}\n    \\diff{\\theta_{1,0}}{t} & = f(\\theta_{1,0}, \\theta_{2,0}; \\avg{p}, \\avg{pB} \/ \\avg{p})\\,,\\\\\n    \\diff{\\theta_{2,0}}{t} & = f(\\theta_{2,0},\\theta_{1,0}; \\avg{p}, \\avg{pB} \/ \\avg{p})\\,.\n\\end{align}\n\\end{subequations}\nHence, in order to understand the leading order behaviour of the non-autonomous system of \\cref{eq: rollers: non-autonomous system}, it is sufficient to explore the autonomous system\n\\begin{subequations}\\label{eq: rollers: abstract autonomous system}\n\\begin{align}\n    \\diff{x}{t} &= f(x,y;\\alpha,\\beta)\\,,\\\\\n    \\diff{y}{t} &= f(y,x;\\alpha,\\beta)\\,,\n\\end{align}\n\\end{subequations}\nidentifying $x$ and $y$ with the leading order solutions for $\\theta_1$ and $\\theta_2$, respectively. \n\nAs noted in \\cref{sec: rollers: model eqs}, the commonplace model presented by \\citet{Lauga2020} makes use of constant parameters in place of $p$ and $B$ in \\cref{eq: rollers: full system}, which we interpret as the \\textit{a priori} averages of $p(T)$ and $B(T)$ for flow-generating particles. In symbols, when interpreted in this way, the model of \\citet{Lauga2020} is equivalent to taking $(\\alpha,\\beta) = (\\avg{p},\\avg{B})$ in \\cref{eq: rollers: abstract autonomous system}. In line with having taken these model-agnostic averages of the parameters, we refer to this model as the \\emph{\\textit{a priori}-averaged model}, which we state explicitly as\n\\begin{subequations}\\label{eq: rollers: naive system}\n\\begin{align}\n    \\diff{\\theta_1^a}{t} &= f(\\theta_1^a,\\theta_2^a;\\avg{p},\\avg{B})\\,,\\\\\n    \\diff{\\theta_2^a}{t} &= f(\\theta_2^a,\\theta_1^a;\\avg{p},\\avg{B})\\,,\n\\end{align}\n\\end{subequations}\nwith the superscript of $a$ distinguishing the solution from that of the full system of \\cref{eq: rollers: full system}. In contrast, we have seen that the true leading order behaviour actually corresponds to taking $(\\alpha,\\beta) = (\\avg{p}, \\avg{pB}\/\\avg{p})$, with $\\avg{pB}\\neq\\avg{p}\\avg{B}$ in general. As we have seen throughout our analysis, this difference in parameters results directly from the employed processes of averaging: one is performed independent of the dynamical systems, whilst the other systematically determines the appropriate averaged parameters for this particular dynamical system. In the next two sections, we seek to determine if these differences in employed parameters between the systematically averaged equations and the \\textit{a priori}-averaged model can result in differences in behaviour, which we establish through an elementary exploration of the autonomous dynamical system of \\cref{eq: rollers: abstract autonomous system}.\n\n\\subsection{Exploring the autonomous dynamics}\nFirst, we identify and classify the fixed points of \\cref{eq: rollers: abstract autonomous system} in terms of $\\alpha$ and $\\beta$, before returning to consider the particular parameter combinations of the previous section. The dynamical system of \\cref{eq: rollers: abstract autonomous system} can be explored via standard methods with relative ease, so we refrain from providing a full and detailed account of the analysis. It is worth noting that, due to the periodicity of the trigonometric functions in $f$, the forcing of the dynamics is periodic with period $\\pi$ in both $x$ and $y$, so that we only need to characterise the dynamics up to multiples of $\\pi$. This periodicity, along with a visual summary of the analysis that follows, is illustrated in \\cref{fig: rollers: portraits,fig: rollers: fixed points}.\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[width=0.8\\textwidth]{figs\/roller_configurations\/roller_configurations.png}\n    \\put(0,50){(a)}\n    \\put(56,50){(b)}\n    \\put(0,25){(c)}\n    \\put(56,25){(d)}\n    \\end{overpic}\n    \\caption{Fixed points of the two-roller system for $\\alpha>0$. Up to symmetry and $\\pi$-periodicity, the identified fixed points correspond to four distinct configurations whose stability depend on $\\alpha$ and $\\beta$ nontrivially. Here, we report the stability for $\\alpha>0$, corresponding to a swimmer with $\\avg{p}>0$ in \\cref{eq: rollers: effective eqs}, which we might refer to as pusher-type particle. The dynamics of a puller-type particle, with $\\alpha<0$, correspond to reversing the direction of motion from $\\alpha>0$, swapping the stability of non-saddle fixed points. (a) Parallel alignment along the direction of separation is stable if $\\beta<-1$ and unstable for $\\beta>-1$. (b) Orthogonal alignment, with one particle pointing directly towards the other, is stable only if $\\beta>0$. (c) Parallel alignment that is perpendicular to the relative displacement is only stable if $\\beta<-1\/2$. (d) Steady parallel alignment that is neither parallel nor perpendicular to the relative displacement is unstable when it exists, which is for $\\beta < -1\/2$ and $\\beta>1\/3$.}\n    \\label{fig: rollers: fixed points}\n\\end{figure}\n\nIt is simple to identify fixed points along the manifolds $x=y$ and $x=-y$, seeking solutions of\n\\begin{equation}\n    \\frac{3\\alpha}{2r^3}\\sin{2x}\\left(1 + \\beta\\left[2-5\\cos^2{x}\\right]\\right) = 0 \\quad \\text{ and } \\quad \\frac{3\\alpha}{2r^3}\\sin{2x}\\left(1 + \\beta\\cos^2{x}\\right) = 0\\,,\n\\end{equation}\nrespectively. For non-zero $\\alpha$, these both admit solutions $x=n\\pi\/2$, $n\\in\\mathbb{Z}$, whilst the former admits the additional solutions of $1+\\beta\\left[2-5\\cos^2{x}\\right]=0$ whenever $\\beta \\leq -1\/2$ or $\\beta \\geq 1\/3$. There are additional steady states on the $x=-y$ manifold that satisfy $1+\\beta\\cos^2{x}=0$, which exist whenever $\\beta\\leq-1$. Further, we note the existence of additional fixed points on the manifold $x + y = \\pi\/2$, which again correspond to $x=n\\pi\/2$ for $n\\in\\mathbb{Z}$. Hence, the fixed points of the system are given by $(x,y)\\in\\{(0,0),(\\pi\/2,0),(0,\\pi\/2),(\\pi\/2,\\pi\/2)\\}$, up to periodicity, in addition to solutions of $1+\\beta\\left[2-5\\cos^2{x}\\right]=0$ on the $x=y$ manifold and solutions of $1+\\beta\\cos^2{x}=0$ on the $x=-y$ manifold. The fixed points and their readily computed linear stabilities are summarised in \\cref{tab: rollers: stab} in \\cref{app: rollers: stab}, with the steady configurations interpreted in terms of the particles in \\cref{fig: rollers: fixed points}. \n\nWe illustrate the overall dynamics in various parameter regimes through phase portraits in \\cref{fig: rollers: portraits}, capturing the full range of qualitatively distinct behaviours that emerge from \\cref{eq: rollers: abstract autonomous system}. Noting that $\\alpha$ plays only a simple role in the dynamics, with changing the sign of $\\alpha$ simply reversing the direction of evolution, we fix $\\alpha>0$ in \\cref{fig: rollers: portraits}, focussing instead on the impact of varying $\\beta$. From these portraits, it is clear that changing the value of $\\beta$ can have a drastic effect on the dynamical system. For instance, $\\beta$ crossing the thresholds of $-1\/2$ and $1\/3$ modifies the character of the phase plane through the emergence or destruction of saddle points and nodes, accompanied by qualitative changes in phase-plane trajectories. Though there are multiple further bifurcations, a notable switch in stability occurs when crossing $\\beta=0$, with $\\beta=0$ corresponding to an integrable system with truly closed orbits \\footnote{Excluding heteroclinic trajectories.} that bifurcate into stable and unstable spirals either side of the bifurcation point.\n\nThese local bifurcations, though effecting changes in linear stability, also give rise to changes in the globally attracting dynamics of the system. This drastic alteration to the overall behaviour is illustrated via the sample trajectories highlighted in blue in \\cref{fig: rollers: portraits}, which either approach closed, heteroclinic connections in \\cref{fig: rollers: portraits}d or the fixed point at the centre of stable spirals in \\cref{fig: rollers: portraits}f and \\cref{fig: rollers: portraits}g, for instance.\n\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[permil,width=\\textwidth]{figs\/all_phase_portraits\/all_phase_portraits.png}\n    \\put(10,557){(a)}\n    \\put(273,557){(b)}\n    \\put(520,557){(c)}\n    \\put(760,557){(d)}\n    \\put(10,282){(e)}\n    \\put(273,282){(f)}\n    \\put(520,282){(g)}\n    \\put(760,282){(h)}\n    \\end{overpic}\n    \\caption{Diverse phase portraits of the autonomous system of \\cref{eq: rollers: abstract autonomous system}. Varying $\\beta$, we illustrate the qualitatively distinct portraits of the autonomous dynamics, with stable and unstable fixed points shown as solid and empty circles, respectively. Moving from (a) to (b), we identify the transition of $(0,0)$ from a stable node to a saddle point via the coalescence of two saddles on the $x=-y$ manifold; following this transition, the parallel alignment of \\cref{fig: rollers: fixed points}c is the globally attracting state. Moving from (b) to (c) and from (h) to (g) transitions stable\/unstable nodes at $(0,\\pi\/2)$ and $(\\pi\/2,0)$ to spirals of the same stability. Significant qualitative changes occur through bifurcations of the saddles at $(0,0)$ and $(\\pi\/2,\\pi\/2)$, each into a node and two saddles, visible between panels (c) and (d) and panels (f) and (g). The blue trajectories in each panel, which have a common initial condition, highlight the qualitative change in behaviour that can result from changing $\\beta$, potentially altering the dynamics from the almost-heteroclinic orbits of panel (d) to the distinct stable attractors of panels (c) and (f), for instance. Here, we have fixed $\\alpha>0$, noting that $\\alpha<0$ corresponds purely to a reversal of the dynamics.}\n    \\label{fig: rollers: portraits}\n\\end{figure}\n\n\\subsection{Comparing the emergent dynamics}\nFrom the above exploration of the autonomous system of \\cref{eq: rollers: abstract autonomous system}, it is clear that changes to the parameter $\\beta$ can have significant qualitative impacts on the emergent dynamics. In this section, we showcase how adopting $(\\alpha,\\beta) = (\\avg{p},\\avg{B})$ in the \\textit{a priori}-averaged model can give rise to predictions that are wholly different  to those of the systematically motivated parameters $(\\alpha,\\beta)=(\\avg{p},\\avg{pB}\/\\avg{p})$.\n\nBefore we exemplify such cases, it is worth highlighting that certain classes of $p(T)$ and $B(T)$ trivially result in $\\avg{pB}\/\\avg{p}=\\avg{B}$, so that the dynamics associated with the \\textit{a priori} and systematically averaged models are identical to leading order. This equality trivially holds if at least one of $p(T)$ or $B(T)$ is constant, with $\\avg{pB}=\\avg{p}\\avg{B}$ in this case. Hence, if the rollers can be associated with a constant dipole strength, or if their Bretherton parameters do not not change in time, then we can naively average any remaining fast-time dependencies before inserting them into the model and achieve the correct leading order behaviour. Further, if we are concerned only with the eventual configuration of the rollers, and not the details of any transient dynamics, we can identify additional $p(T)$ and $B(T)$ that can be \\textit{a priori}-averaged without consequence. For instance, if $p(T)>0$ and $B(T)>0$ for all $T$, then we have $\\avg{pB}\/\\avg{p}>0$ and $\\avg{B}>0$, so that $\\beta > 0$ in both cases and the configuration shown in \\cref{fig: rollers: fixed points}b is globally attracting, up to symmetry, as can be deduced from \\cref{fig: rollers: portraits}f-h. These particular constraints are compatible, for example, with requiring that the particle be spheroidal, always prolate, and consistently generating a dipolar flow field that can be associated with a hydrodynamic pusher.\n\nFor more general $p(T)$ and $B(T)$, however, it is clear that we cannot guarantee that the dynamics predicted by the \\textit{a priori}-averaged model of \\cref{eq: rollers: naive system} will be at all reminiscent of the leading order systematically averaged dynamics of \\cref{eq: rollers: effective eqs}. Seeking a minimal example in order to highlight this general observation, we take $p(T)=2A\\sin{T} + 1$ and $B(T) = \\sin{T} + D$. Clearly, $\\avg{p}=1$ and $\\avg{B}=D$, so that the choice of $D$ uniquely determines the panel of \\cref{fig: rollers: portraits} that corresponds to the dynamics of the \\textit{a priori}-averaged model. However, computing $\\avg{pB}\/\\avg{p} = A + D$ highlights that we can choose $A$ so that the systematically averaged dynamics occupy any given panel of \\cref{fig: rollers: portraits}. \n\nTo illustrate this concretely, we take $A = -5\/4$ and $D = 1\/2$ and numerically solve \\cref{eq: rollers: naive system,eq: rollers: effective eqs,eq: rollers: full system} from the same initial condition, taking $\\omega=100$,  with the solutions shown in \\cref{fig: rollers: comparison}. The \\textit{a priori}-averaged system shown in \\cref{fig: rollers: comparison}a corresponds to $\\beta=1\/2$, so follows the dynamics of \\cref{fig: rollers: portraits}g, with the numerical solution approaching the $(0,\\pi\/2)$ steady state, in which the rollers are perpendicular to one another. In contrast, the systematically averaged dynamics of \\cref{fig: rollers: comparison}b correspond to $\\beta = -3\/4$ and the portrait of \\cref{fig: rollers: portraits}c, evolving to the parallel steady state $(\\pi\/2,\\pi\/2)$. The numerical solution to the full problem of \\cref{eq: rollers: full system} is also shown in \\cref{fig: rollers: comparison}b, highlighting good agreement with the asymptotic solution and evidencing the potential for disparity between the predictions of the \\textit{a priori}-averaged model and the dynamics of temporally evolving bodies.\n\n\\begin{figure}\n    \\centering\n    \\begin{overpic}[width=0.8\\textwidth]{figs\/roller_comparison\/roller_comparison.png}\n    \\put(3,36){(a)}\n    \\put(53,36){(b)}\n    \\end{overpic}\n    \\caption{Comparing the results of \\textit{a priori} and systematic averaging. With fixed initial conditions of $(\\pi\/8,0)$, numerical solutions of (a) the \\textit{a priori}-averaged system of \\cref{eq: rollers: naive system}, (b) the systematically averaged dynamics of \\cref{eq: rollers: effective eqs}, and the full system of \\cref{eq: rollers: non-autonomous system} are shown as solid curves. (a) With $p(T) = -5\\sin{T}\/2 + 1$ and $B(T) = \\sin{T} + 1\/2$, the \\textit{a priori}-averaged dynamics approach the $(0,\\pi\/2)$ steady state, following the dynamics illustrated in \\cref{fig: rollers: portraits}g, as $\\beta=\\avg{B}=1\/2$. (b) The leading order, systematically derived dynamics evolve to a distinct steady state, with $\\theta_{1,0}=\\theta_{2,0} = \\pi\/2$, following the dynamics shown in \\cref{fig: rollers: portraits}c, as $\\beta = \\avg{pB}\/\\avg{p} = -3\/4$. The corresponding steady state configurations of the rollers are illustrated in the lower right corner of each panel, highlighting the qualitatively distinct behaviours predicted by the models. Here, we have taken $\\omega=100$.}\n    \\label{fig: rollers: comparison}\n\\end{figure}\n\n\n\n\\section{Discussion and conclusions}\nThe use of \\textit{a priori}-averaged parameters in minimal modelling approaches can be appealing, seemingly commensurate with seeking the simplest possible model of a given setting. However, through the simple examples presented in this study, we have seen how such model-agnostic averaging can result in behavioural predictions that differ qualitatively from those of models that incorporate fast-timescale parameter oscillations, which are common to many microswimmers. Hence, the key conclusion of our simple analysis is that such a modelling approach can be unreliable even at the level of back-of-the-envelope calculations, with the intuition gained from such explorations potentially rendered invalid. This observation is expected to hold somewhat generically across such simple models, with our approach extending to a range of employed minimal swimmer representations.\n\nThough one might interpret the conclusion of this study as an argument against the use of minimal models of microswimming, in fact, asymptotic analysis of these models revealed that they \\emph{can} capture the leading order dynamics, but only with appropriate parameterisation. In particular, our analysis highlights that it is the use of \\textit{a priori}-averaged parameters, rather than the use of constant parameters more generally, that gives rise to inaccurate predictions. Hence, this study supports the use of minimal models in developing intuition and understanding of microswimmer systems, though only when used with systematically derived, effective parameters. \n\nFurther, we have seen how an asymptotic analysis can show that, in certain parameter regimes, one can reliably employ \\textit{a priori}-averaged parameters without qualitatively affecting the predictions of the model. However, the explorations of \\cref{sec: no slip,sec: rollers} revealed that such robust parameter regimes are far from being universal; on the contrary, we have seen that they depend strongly on the model in question. Hence, in general, a bespoke analysis is required for any given model in order to determine these regimes of serendipitous agreement.\n\nThe analysis presented in this study is simple, even elementary, relying on the commonplace method of multiple scales and the basic observation that the average of a product need not be the product of individual averages. Despite this simplicity, we have identified potential missteps in the use of the simplest models of microswimming, of which these authors have previously been guilty. Further, we have demonstrated how a straightforward, multi-timescale analysis can inform reliable, systematic parameterisation of minimal models, such that they recover their marked utility in the generation of intuition, basic understanding, and back-of-the-envelope predictions.\n\n\\section*{Acknowledgments}\nB.J.W. is supported by the Royal Commission for the Exhibition of 1851. K.I. acknowledges JSPS-KAKENHI for Young Researchers (Grant No. 18K13456), JSPS-KAKENHI for Transformative Research Areas (Grant No. 21H05309) and JST, PRESTO, Japan (Grant No. JPMJPR1921).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec:intro}}\n\nThe study of $\\delta$ Scuti stars is expected to provide an\nimportant diagnostic tool for the structure and evolution of\nintermediate-mass stars at the main sequence evolution\nand thick-shell hydrogen burning stages. Having masses between 1.5 and\n2.5 $M_{\\odot}$, these pulsating variables develop convective\ncores during their central hydrogen-burning phases that make them\nsuitable for investigating the hydrodynamic processes occurring in\nthe core.\n\nAlthough most of the general properties of these stars are well\nunderstood, the application of seismological methods to derive their\nproperties has proved to be a difficult task. The problems\nconcerning the modelling of $\\delta$ Scuti stars have been\ndiscussed in details in recent reviews (e.g.\\ Christensen-Dalsgaard\n\\cite{christensen}; Kjeldsen \\cite{kjeldsen}). The most acute seem\nto be the mode selection mechanism and the influence of rotation.\nConcerning the former, although there is fairly good agreement\nbetween the observed frequency range and that derived from\ninstability calculations (e.g.\\ Michel et al.\\ \\cite{michel1}), far\nmore modes than observed are predicted to be unstable. \nWhile it is expected that the forthcoming\nasteroseismology space mission COROT (Baglin et al.\\ \\cite{baglin})\nwill be able to  disentangle the whole spectra of $\\delta$ Scuti\nstars, for the moment the\nmechanism that could cause an amplitude limitation in some\nmodes is still unknown. As a consequence, mode identification\nbased on the comparison between theoretical and observed\nfrequencies is difficult to attain in general. On the other hand,\nalthough several observational techniques for mode identification\nhave been developed in recent years (Watson \\cite{watson};\nGarrido et al.\\ \\cite{garrido}; Aerts \\cite{aerts}; Kenelly \\&\nWalker \\cite{kenelly}; Viskum et al.\\ \\cite{viskum}), in only a few\ncases have these techniques  been successfully applied.\n\n\nIt is well known that most $\\delta$ Scuti stars\nare rapid rotators. This has important effects on the\nmode frequencies, both directly as a consequence of the changes that must be\nintroduced in the oscillation equations and indirectly through the changes in the\nequilibrium models.\nTherefore, successful analysis of  rotating $\\delta$ Scuti stars\nrequires that rotation should be taken  into account consistently  at\nall levels in the analysis.\n\nDespite these problems, in recent years several \nattempts have been made to interpret the observed spectra\nof $\\delta$ Scuti stars (Goupil et al.\\ \\cite{goupil}; P\\'erez\nHern\\'andez et al.\\ \\cite{perez1}; Guzik \\& Bradley \\cite{guzik};\nPamyatnykh et al.\\ \\cite{pamyatnykh}; Bradley \\& Guzik\n\\cite{bradley}; Hern\\'andez et al.\\ \\cite{hernandez}; Su\\'arez et\nal.\\ \\cite{suarez1}). Here we address the problem of mode\nidentification for stars in an open cluster. These stars are\nexpected to have a common age and chemical composition.\nFurthermore, the distance to the cluster is usually known with\nhigh accuracy. The constraints imposed by the cluster parameters\nhave proved to be very useful when modelling an ensemble of\n$\\delta$ Scuti stars. The best examples of such studies are found\nfor the variables in the Praesepe cluster (Michel et al.\\\n\\cite{michel1}; Hern\\'andez et al.\\ \\cite{hernandez}). In a similar\nway, we consider here several $\\delta$ Scuti stars of the Pleiades\ncluster and search for a best fit solution in the sense of a set\nof stellar parameters that allows the simultaneous modelling of\nall the stars considered, and that satisfies all the observables,\nincluding the frequencies.\n\nThe paper is organized as follows. In Sect.~\\ref{sec:stars}, we\ndescribe the main characteristics of the Pleiades cluster and the\nsix $\\delta$ Scuti stars under study. The modelling, the range of\nparameters and the calculations of the eigenfrequencies are\ndiscussed in Sect.~\\ref{sec:models}. The details of the comparison\nbetween observed and theoretical frequencies are discussed in\nSect.~\\ref{sec:comp}. The results and their\ndiscussion are given in Sect.~\\ref{sec:results}. Finally, we\npresent our conclusions in Sect.~\\ref{sec:conclusions}.\n\n\n\\section{The observational material \\label{sec:stars}}\n\nThe Pleiades (M45) is a young ($\\sim 75$--100\\, Myr), Population I\ncluster. Because of its proximity, the observational parameters of the\nPleiades have been intensively studied. In particular, the\nmetallicity of the cluster is estimated to be between $-0.14 \\leq$\n[Fe\/H] $\\leq +0.13$ (e.g. $-$0.034 $\\pm$ 0.024 Boesgaard \\& Friel\n\\cite{boesgaard}, 0.0260 $\\pm$ 0.103 Cayrel de Strobel\n\\cite{cayrel}, $-$0.11 $\\pm$ 0.03 Grenon \\cite{grenon}),\n\nUntil recently, there was a dispute regarding the distance of the\nPleiades cluster from the Earth. While the determination of the\ncluster distance  from direct parallax measurements of\nHIPPARCOS gives 116.3$\\pm^{3.3}_{3.2}$~pc\n($m_{V}-M_{V}= 5.33\\,\\pm\\, 0.06$ mag, Mermilliod et al.\\\n\\cite{mermilliod}), the previously accepted distance, which is\nbased on comparing the cluster's main sequence with that of nearby\nstars, was  $\\approx$130 pc corresponding to a distance modulus of\nabout 5.60 mag (e.g.\\ 5.65 $\\pm$ 0.38 mag [Eggen \\cite{eggen}],\n5.60 $\\pm$ 0.05 mag [Pinsonneault et al.\\ \\cite{pinsonneault}],\n5.61 $\\pm$ 0.03 mag [Stello \\& Nissen \\cite{stello}]).\nAs  will be discussed in Sect.~\\ref{sec:results}, the distance derived\nin the  research presented here agrees with the latter value.\n\n\\begin{table*}[ht]\n  \\caption{Observational properties of the target stars}\n  \\begin{tabular}{lccccccc}\n  \\hline\n  Star        & HD &  ST  & $m_{V}$&$B-V$ & $E(B-V)$ & $V \\sin\\, i$ & $\\beta$ \\\\\n              &    &      &        &      &       & $(\\mathrm{km\\, s}^{-1})$&  \\\\\n  \\hline\n  -       & 23628 & A7V   &  $7.66\\pm0.03 $& $+0.211\\pm0.005$ &$0.063 \\pm0.008$ & $ 150 \\pm15$& 2.884  \\\\\n  V650 Tau& 23643 & A3V   &  $7.77\\pm0.02 $& $+0.157\\pm0.006$& $0.027\\pm0.008$ &  $175\\pm18$ & 2.862  \\\\\n  -       & 23194 & A5V   &  $8.06\\pm0.02 $& $+0.202\\pm0.010$& $0.076\\pm0.008$ & $20 \\pm2$& 2.881  \\\\\n  V624 Tau& 23156 & A7V   &  $8.23\\pm0.01$ & $+0.250\\pm0.005$& $0.046\\pm0.008$ & $ 20 \\pm2$& 2.823  \\\\\n  V647 Tau& 23607 & A7V   &  $8.26\\pm0.01$ & $+0.255 \\pm0.001$&$0.057 \\pm0.008$ & $ 20 \\pm2$& 2.841  \\\\\n  V534 Tau& 23567 & A9V   &  $8.28\\pm0.02$ & $+0.360\\pm0.001$ &$ 0.084\\pm0.008$ & $ 90\\pm 9$& 2.788  \\\\\n\n  \\hline\n  \\end{tabular}\n  \\label{tab:stars}\n\\end{table*}\n\nTo date, six $\\delta$ Scuti stars are known in\nthe Pleiades. In a survey of variability in the\ncluster, Breger (\\cite{breger}) found   $\\delta$ Scuti type oscillations\nin four stars. The remaining two were recently reported to be\n$\\delta$ Scuti stars  by Koen (\\cite{koen1}) and Li et al.\\ (\\cite{li}).\nSome observational properties of these stars are given in\nTable~\\ref{tab:stars}.  The projected rotational\nvelocities, $v\\sin  i$, were obtained from Morse et al.\\\n(\\cite{morse}) and Uesugi \\& Fukuda (\\cite{uesugi}). A 10\\%\nuncertainty in these quantities has been assumed. The spectral\ntypes (ST) and $\\beta$ parameters were obtained from the SIMBAD database in\nStrasbourg (France). The errors in the photometric parameters $(B-V)$\nand $m_{V}$ are estimated from the dispersion between different\nmeasurements of these quantities. The errors in reddening are\ntaken from Breger (\\cite{breger1}). We shall use these\nuncertainties in Section 3.2.\n\nIn recent years, the $\\delta$ Scuti stars of the Pleiades cluster have been\nobserved in several campaigns of the STEPHI multi-site network\n(Michel et al.\\ \\cite{michel2}). The information on\nthe oscillation frequencies of the stars used here has\nbeen obtained from those campaigns and are published elsewhere\n(Belmonte \\& Michel \\cite{belmonte1}; Michel et al.\\\n\\cite{michel4}; Liu et al.\\ \\cite{liu}; Li et al.\\ \\cite{li};\nLi et al.\\ \\cite{li1}; Fox-Machado et al.\\ \\cite{fox}).\nThe frequency peaks detected\nwith a confidence level above 99\\% are summarized\nin Table~\\ref{tab:freq}. We note that the frequency resolution in a typical\nSTEPHI campaign (three weeks) is $\\Delta \\nu \\sim 0.5\\, \\mu$Hz.\n\n\\begin{table}[!t]\n\\begin{center}\n\\caption{Observed frequencies used in this work. The data were obtained from\ndifferent STEPHI campaigns as detailed in the text.}\n\\begin{tabular}{lc|lc}\n\\hline \\hline\nstar   & $\\nu$   & star& $\\nu $     \\\\\n& $(\\mu {\\rm Hz})$&  &   $(\\mu {\\rm Hz})$ \\\\\n\\hline\nHD 23628& 191.8 & V624 Tau &242.9    \\\\\n    & 201.7  &&409.0    \\\\\n    & 376.6  &&413.5   \\\\\\cline{1-2}\nV650 Tau&197.2&&416.4    \\\\\n    &292.7&& 451.7   \\\\\n    &333.1&& 489.4   \\\\\n    &377.8 &&529.1   \\\\\\hline\nHD 23194&533.6&V647 Tau&236.6 \\\\\n&574.9&&304.7    \\\\\\cline{1-2}\nV534 Tau &234.2 &&315.6    \\\\\n    &252.9&&374.4    \\\\\n    &307.6&&405.8     \\\\\n    &377.9 &&444.1     \\\\\n    &379.0 &&\\\\\n    &448.1 &&  \\\\\n    &525.0&&  \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:freq}\n\\end{table}\n\nFigure~\\ref{fig:dscutis} illustrates a colour--magnitude diagram of the Pleiades\ncluster in the region where the target stars are located.\nThe filled circles corresponds to the $\\delta$ Scuti stars.\nThe apparent magnitudes, $V$, colours, $(B-V)$ and\nmembership used are contained in the WEBDA database (Mermilliod\n\\cite{mermilliod1}). The Pleiades shows\ndifferential reddening with significant excess in the southwest with an\naverage value of $E(B-V) \\approx 0.04$ (e.g.\\\nvan Leeuwen \\cite{vanleeuwen}; Breger \\cite{breger1}; Hansen-Ruiz\n\\& van Leeuwen \\cite{hansen}; Pinsonneault et al.\\\n\\cite{pinsonneault}). In particular, we adopted the reddening for\nindividual stars as given by Breger (\\cite{breger1}) if present, otherwise\nthe average of $E(B-V) = 0.04$ was used. Stars known to be spectroscopic binaries\nwere rejected.\n\nGiven the young age of the Pleiades cluster, we expect  all the\n$\\delta$-Scuti stars to be at an early evolutionary stage on the main sequence.\nFrom the figure it follows that\nwhile V647 Tau and V624 Tau have similar masses,\nV650 Tau, HD 23194 and HD 23628 are slightly more massive.\nV534 Tau, located near the red edge of the instability\nstrip, is the coolest star in the ensemble.\n\n\n\\begin{figure}[!t]\n\\resizebox{\\hsize}{!} {\\includegraphics{3791fig1.eps}}\n\\caption{HR diagram of the Pleiades cluster. Only the region around the instability\nstrip is shown. The target stars are represented with filled circles.\nThe blue and red edges of the instability strip are indicated by continuous lines\nand were taken from Rodr\\'{\\i}guez et al.\\\n(\\cite{rodriguez}). An isochrone of 70 Myr is also shown.}\n\\label{fig:dscutis}\n\\end{figure}\n\n\\section{The modelling \\label{sec:models}}\n\nIn this section we shall explain how we have computed the stellar models,\nthe corresponding eigenfrequencies and how we have constrained the stellar\nparameters used in the forward comparison with the observed frequencies.\n\n\\subsection{The stellar models}\n\nWe have considered stellar models\nwith input physics appropriate to the mass range covered by\n$\\delta$ Scuti stars. In particular, the stellar models were computed\nusing the CESAM evolutionary code (Morel \\cite{morel}).\nThe nuclear reaction rates are from Caughlan \\& Fowler\n(\\cite{caughlan}), the equation of state is from Eggleton et al.\\\n(\\cite{eggleton}), the opacities from the OPAL project (Iglesias\n\\& Roger \\cite{iglesias}) complemented at low temperatures by\nKurucz data (\\cite{kurucz}), and  the\natmosphere is computed using the Hopf's $T(\\tau)$ law (Mihalas\n1978). The convection is described according\nto the classical mixing-length theory (MLT hereafter). In particular,\nwe have considered an MLT parameter of $\\alpha_{\\rm MLT} =\nl\/H_{p}=1.52$, where $H_{p}$ is the pressure scale-height.\nWe have considered a fixed value for  $\\alpha_{\\rm  MLT}$ because this parameter is not expected\nto have any significant influence on the global structure of the evolution models\nconsidered for our target stars.\n\nWe have considered different sets of initial chemical  compositions\nin order to cover the Pleiades [Fe\/H] observed range (see\nSection~\\ref{sec:stars}) and different initial He abundances. The\nvalues used are given in Table~\\ref{tab:metal}. For each initial\nchemical composition we have computed models with and without core\novershooting,  $\\alpha_{{\\rm ov}}=0.20$ in the former case \n(Schaller et al.\\ \\cite{schaller}).\n\n\\begin{table}[!t]\n\\begin{center}\n\\caption{Metallicity of the models. [Fe\/H] = $\\log (Z\/X)_{\\ast}- \\log (Z\/X)_{\\odot}$,\nwith $(Z\/X)_{\\odot}=0.0245$ from Grevesse \\& Noels (\\cite{grevesse}).\n\\label{tab:metal}}\n\\begin{tabular}[h]{ccccc}\n\\noalign{\\smallskip} \\hline \\hline \\noalign{\\smallskip}\n$X_{0}$&$Y_{0}$&$Z_{0}$&[Fe\/H]&$\\alpha_{\\rm ov}$\\\\\n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip}\n\n0.735&0.250&0.015&$-$0.0794\\,\\,\\,\\,&0.0--0.2\\\\\n\n0.685&0.300&0.015&$-$0.0488\\,\\,\\,\\,&0.0--0.2\\\\\n\n0.700&0.280&0.020&0.0668&0.0--0.2\\\\\n\n0.725&0.250&0.025&0.1484&0.0--0.2\\\\\n\n0.675&0.300&0.025&0.1794&0.0--0.2\\\\\n\n\\noalign{\\smallskip} \\hline  \\noalign{\\smallskip}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nAlthough the final comparison with observations is done\nconsidering rotating models, our procedure requires the\ncomputation of non-rotating models and the corresponding\nisochrones. Hence, sequences of non-rotating models were\ncalculated for masses between 0.8 $M_{\\odot}$ and 5.0 $M_{\\odot}$,\nfrom the ZAMS to the sub-giant branch and the corresponding\nisochrones computed for each sets of parameters in\nTable~\\ref{tab:metal}. In the following analysis we shall consider\nthree ages, $A$, of 70 Myr, 100 Myr and 130 Myr. Finally, for\ncomparison with the observations three distance moduli, $d$, of\n5.50, 5.60 and 5.70 mag were considered\n except for models with [Fe\/H]=$-$0.0488, in which case \nan additional value of $d=5.39$ mag was used.\n\n\nFigure~\\ref{fig:dscutis} shows an example of such isochrones\ncomputed with the following parameters: [Fe\/H]=0.0668,\n$\\alpha_{\\rm ov} = 0.2$, $A = 70$ Myr and $d= 5.70$ mag. The\nisochrones were calibrated from [$T_{\\rm eff}, \\log\n(L\/L_{\\odot})$] to $(B-V, M_{V})$ by using the Schmidt--Kaler\n(\\cite{schmidt}) calibration for magnitudes and the relationship\nbetween $T_{\\rm eff}$ and $B-V$ of Sekiguchi \\& Fukugita\n(\\cite{sekiguchi}) for the colours. As can be seen in\nFig.~\\ref{fig:dscutis}, in the case illustrated here the isochrone\nmatches the observed colour--magnitude diagram.  For some combinations \nof high metallicity and low\ndistance modulus the isochrone fit is not satisfactory but we have\nnot rejected those combinations to allow for an independent\ndetermination of the cluster parameters from the oscillation\nfrequencies.   \nHence we have considered \nthe entire combination of cluster parameters given in\nTable~\\ref{tab:metal} and the three ages given above. \n\n\n\\subsection{Photometric effect of rotation on the HR diagram}\n\nIt is known that fast rotation affects the position of a star in the\ncolour--magnitude diagram (e.g.\\ Tassoul \\cite{tassoul}).\nIn particular a ZAMS rotating model has a smaller luminosity and mean\n$T_{\\mathrm{eff}}$ than a non-rotating model with the same mass and chemical\ncomposition.\nAlso, the magnitude and the colour of a rotating star depend on\nthe aspect angle, $i$, between the line of sight and the rotation\naxis. In particular, a rotating star seen pole-on\nappears brighter and hotter than the same star seen equator-on.\n\nWe now search for the range of masses and angular rotations suitable for\neach star. To do this we shall use the\nresults of P\\'erez Hern\\'andez et al.\\ (\\cite{perez2}), in which the\ncorrection to the photometric magnitudes of non-rotating models needed to obtain\nthose of rotating models were computed for main sequence stars\nof spectral type A0 to F5.\nIn this calculation, we shall take into account the observed $v \\sin i$ for each star.\nAs an example,\nFig.~\\ref{fig:cor} illustrates the corrections due to rotation in\nthe particular case of the $\\delta$ Scuti type star V650 Tau with $v \\sin\ni=175$ km s$^{-1}$. The actual position of the star is shown with\na thick dot upon which there is a cross that gives the errors in\nthe estimation of $(M_{V})_{0}$ and $(B-V)_{0}$ calculated\naccording the following expressions:\n\n\\begin{figure}\n\\resizebox{\\hsize}{!} {\\includegraphics[]{3791fig2.eps}}\n\\caption{Colour--magnitude diagram illustrating the photometric\ncorrections due to rotation in the case of V650 Tau\n($v\\sin i = 175\\,$km s$^{-1}$). The  thick continuous line is the\nsame isochrone represented in Fig.~\\ref{fig:dscutis} ([Fe\/H] =\n0.0668, $\\alpha_{\\rm ov} = 0.2$, $A = 70$ Myr, $d=5.7$ mag).\nSome models are shown with filled circles upon the isochrone. The masses of the \nfirst and last models are indicated.\nThe small tracks represent the photometric corrections due to rotation for each  \n the models indicated on the isochrone when $i$ runs from\n$i=90^{\\circ}$ to $i=i_{\\rm min}$. The corrections at fixed $i$ are represented by dotted lines.}\n\\label{fig:cor}\n\\end{figure}\n\n\\begin{equation}\n\\sigma[(B-V)_{0}] = \\sqrt{\\sigma^{2}(B-V) +\\sigma^{2}[E(B-V)]},\n\\end{equation}\n\n\n\\begin{equation}\n\\sigma[(M_{V})_{0}] = \\sqrt{\\sigma^{2}(d)+\\sigma^{2}(m_{V}) +\n(3.2\\sigma[E(B-V)])^{2}}\n\\, ,\n\\end{equation}\n\n\\noindent\nwhere $\\sigma(d)=0.06$ is the error in the distance modulus. The\nerrors in the magnitude $M_{V}$, colour index $(B-V)$ and reddening\n$E(B-V)$ are given in Table~\\ref{tab:stars}.\n\nThe range of masses and rotational velocities of the star depends\non the stellar parameters considered and hence needs to be computed\nfor each combination in Table~\\ref{tab:metal}. In particular, in\nthe example illustrated in Fig.~\\ref{fig:cor}, the same cluster\nparameters as in Fig.~\\ref{fig:dscutis} were considered\n([Fe\/H]=0.0668, $\\alpha_{\\rm ov} = 0.2$, $A = 70$ Myr and $d=\n5.70$ mag). The isochrone is represented by a thick continuous\nline and some of its models are shown by dots. For each model on\nthe isochrone there is a small track corresponding to the\nphotometric corrections due to rotation computed by using the\nprojected velocity  $v \\sin i$ of the star and changing the angle\nof inclination from $i=90^{\\circ}$ to the minimum angle $i=i_{\\rm\nmin}$ corresponding to $\\sim 0.90 \\Omega_{\\rm br}$, where\n$\\Omega_{\\rm br}$ is the break-up angular rotational velocity.\nThis angular velocity is given by the balance between the\ngravitational attraction and the centrifugal force at the equator.\nThe dotted lines give the corrections at fixed angles:\n$i=90^{\\circ}$ (the closest line to the isochrone), $i=50^{\\circ}$\n(intermediate line) and $i=39^{\\circ}$ (the reddest line). It is\nevident that the actual position of the star, including the error\nbox, is associated to a domain of non-rotating models. In this\ncase we have obtained for the  star considered a range\nof masses of $[1.86,1.91]M_{\\odot}$ and a range of angles of\ninclination  of $[39^{\\circ},50^{\\circ}]$.  Nonetheless, in order\nto take into account possible errors coming from the isochrone\ncalibration we shall consider a wider range of aspect angles\n$i=[i_{\\rm min}, 90^{\\circ}]$. The same procedure is carried out\nfor the other stars.  Also one needs to consider all the\nmetallicities, ages, overshooting parameters and distance moduli for\nthe whole set of stars.\n\nHaving obtained ranges of $M$ and values of $i_{\\rm min}$, we proceed to find the\ncorresponding range of angular rotational velocities $\\Omega$, for\nwhich an estimate of the equatorial radius, $R_{\\rm eq}$, of the\nrotating model is needed. Under the assumption of uniform rotation\nand approximating the surface of the star by a Roche surface (see\ne.g. P\\'erez Hern\\'andez et al. \\cite{perez2}) one has:\n\n\\begin{equation}\nR_{\\rm eq} \\simeq R_0 \\frac{3}{\\omega} \\cos \\left\\{ \\frac{1}{3}\n\\left[ \\pi + {\\rm arcos}\\, \\omega \\right] \\right\\}\n\\; ,\n\\end{equation}\n\n\\noindent where $\\omega \\simeq \\Omega\/\\Omega_{\\rm br}$,\n$\\Omega^2_{\\rm bq} \\simeq 8GM\/(27R^3_0)$ and $R_0$ is the polar\nradius. As noted in P\\'erez Hern\\'andez et al.~(\\cite{perez2}), the\npolar radius can be approximated by the radius of a spherical\nnon-rotating model with the same mass and evolutionary state\n(since our stars are close to the ZAMS, a non-rotating model of\nthe same mass and age is suitable for this error-box estimation).\n\nSince on the other hand the angular rotation is related to the equatorial radius by\n\n\\begin{equation}\n\\Omega \\simeq \\frac{(v\\sin i)_{\\rm obs}}{R_{\\rm eq}\\sin i}\n\\; ,\n\\end{equation}\n\n\\noindent it is possible to carry out a simple iterative process\nto obtain a range of possible rotations, $[\\Omega_{\\rm min},\\Omega_{\\rm max}]$, \nfrom the previously obtained range of\nangle of inclinations $[i_{\\rm min},90^{\\circ}]$ and masses.\nSince for given data parameters, the mass of the star is obtained with an \nuncertainty of $\\sim 0.02$ $M_{\\odot}$ this\niterative process was carried out for just one mass.\n\nAs an example, Figure~\\ref{fig:om} shows the ranges of $\\Omega$\nobtained for the six stars as a function of the mass of the models\nconsidering the same cluster parameters as in\nFigure~\\ref{fig:cor}. Lines of different types give the estimates\nfor the different stars as indicated in the figure. A similar\nprocedure needs to be considered for all sets of parameters in\nTable~\\ref{tab:metal}.\n\n\\begin{figure}[!t]\n\\resizebox{\\hsize}{!} {\\includegraphics[width=22cm]{3791fig3.eps}}\n\\caption{Range of rotation rates against masses estimated for the\nsix stars from the photometric corrections due to rotation applied\nto the same isochrone depicted in Figure~\\ref{fig:cor}. The vertical error\nbars give the estimated range of mass for each star. The crosses\ngive the values of the break-up angular rotational velocity,\n$\\Omega_{\\rm br}$. The ranges of $\\Omega$ and $M$ are associated\nwith stars as follows: dot-dashed line for V650 Tau $(v\\sin i=175$\nkm s$^{-1})$; long-dashed line for HD 23628 $(v\\sin i=150\\,$km\ns$^{-1})$; three-dots dashed line for HD 23194 $(v\\sin i=20$\nkm s$^{-1})$; dashed-line for V647 Tau $(v\\sin i=20$ km s$^{-1})$;\ncontinuous line for V624 Tau $(v\\sin i=20\\,$km s$^{-1})$ and\ndotted line for V534Tau $(v\\sin i=90\\,$km s$^{-1})$. }\n\\label{fig:om}\n\\end{figure}\n\nWith the information obtained above we compute evolutionary\nsequences of rotating models with the same input physics as the\nnon-rotating models but with appropriate initial angular\nrotational velocities in order to match as closely as possible the\nextreme values in the interval $[\\Omega_{\\rm min},\\Omega_{\\rm\nmax}]$ at the final age. Solid-body rotation and conservation of\nglobal angular momentum during the evolution were assumed during\nthe calculus. In order to have a better overview of the rotation rates of\nthe stars we have also computed models with an intermediate value\nof $\\Omega_{\\rm mid} \\approx 0.5 \\Omega_{\\rm br}$. A total amount\nof 1620 sequences of rotating models were finally obtained for the\nwhole ensemble of stars.\n\n\\subsection{Theoretical oscillation frequencies}\n\nThe eigenfrequencies of the rotating models previously described\nhave been calculated using the oscillation code FILOU (Tran Minh\net al.\\ \\cite{tran}; Su\\'arez  \\cite{suarez2}). We have considered\nfrequency perturbations up to second order in the rotation rate.\nFrequencies of the modes are labelled in the usual way: $n, l, m$\nfor the radial order, degree and azimuthal order. The radial\norders of a given mode are assigned according to the Scuflaire\ncriterion (\\cite{scuflaire}) with $n>0$ for the $p$ modes, $n<0$\nfor the $g$ modes, $n=1$ for the fundamental radial mode of degree\n0 and 1, and $n=0$ for the fundamental mode with the $l=2$.\nEigenfrequencies were computed up to $l=2$. For geometrical\nreasons higher degree modes are expected to have considerably\nsmaller amplitudes. The theoretical frequencies cover the\nfrequency range of the observed pulsation peaks (see\nTable~\\ref{tab:freq}). Coupling between modes is not considered in\nthe present work.\n\nThe estimated interval of $\\Omega_{\\rm\nrot}$ for all the stars may be as large as \n$\\Delta \\Omega_{\\rm rot} \\sim 1.6 \\times 10^{-4}$ rad\/s \nwhich in terms of cyclic rotational frequency corresponds\nto $\\Delta \\nu_{\\rm rot} \\sim  25\\,\\mu$Hz (see for instance\nFig.~\\ref{fig:om}). After some tests, we found that a\nsatisfactory comparison between the observed and theoretical\nfrequencies cannot be achieved by using only the frequencies\ncomputed for models with three $\\Omega_{\\rm rot}$ within this\ninterval. To overcome this difficulty once we had the mode\nfrequencies for each series of models (three $\\Omega_{\\rm rot}$\nfor fixed $M$, [Fe\/H], $\\alpha_{\\rm MLT}$, $\\alpha_{\\rm ov}$, $d$\nand $A$), we proceeded to interpolate the results on $\\Omega_{\\rm\nrot}$ in 21 equally spaced points covering the interval\n[$\\Omega_{\\rm min}, \\Omega_{\\rm max}$]. A quadratic spline\ninterpolation was applied. In order to\nevaluate the goodness of the interpolation we also computed the\neigenfrequencies for several randomly selected models. We have\nfound that there is a good agreement between the interpolated  and\ntheoretical frequencies up to $\\nu \\sim 600\\, \\mu$Hz ($n \\sim 6$),\n the average of absolute separation being $|\\nu_{\\rm int} -\\nu_{\\rm\ncal} |\\sim 0.3\\,\\mu$Hz. The disagreement found beyond this value can be\nexplained by the fact that the second order effect of rotation is\nenhanced at higher frequencies. In any case, as can be seen in\nTable~\\ref{tab:freq}, the highest frequency we are dealing with\nis $\\simeq 574\\,\\mu$Hz.\n\n\\section{Method of comparison \\label{sec:comp}}\n\nOnce we have the mode frequencies for all sets of parameters we\ncompare the observational frequencies ($\\nu_{\\rm obs}$)\nwith the theoretical ones ($\\nu_{\\rm cal}$) at each interpolated\n$\\Omega_{\\rm rot}$.\n\nLet us consider a rotating model with given parameters for just\none star. Then, for all the possible combinations between the\nobserved and computed frequencies, we compute the quantity $\\chi^{2}$ given by\n\n\\begin{equation}\n\\chi^{2}= \\frac{1}{N}\\sum^{N}_{j=1}(\\nu_{{\\rm obs},j} - \n\\nu_{{\\rm cal},j})^{2}\n\\, ,\n\\end{equation}\n\n\\noindent\nwhere $N$ is the number of observational frequencies.\nIn this computation it is understood that each theoretical frequency is\nassigned to one observed frequency at most.\n\nAs a first step, in order fully to  exploit the collective behaviour\nof stars within open cluster we consider the solutions of the six\nstars with given cluster parameters of [Fe\/H], $\\alpha_{\\rm ov}$,\n$d$ and $A$. The models corresponding to each star differ in the\nmass and the rotation rate. As an example,  Fig.\\ref{fig:ang}\nshows $\\chi^{2}$ against  angular rotational velocity,\n$\\Omega$, for the six models corresponding to the same parameters\nas in Fig.~\\ref{fig:cor} but without overshooting ([Fe\/H]=0.0668,\n$\\alpha_{\\rm ov}=0.0$, $d$=5.70 mag and $A$=70 Myr). Each panel\ncorresponds to the star indicated in the figure. For clarity,\nonly solutions with $\\chi^{2}< 20 \\, (\\mu\\rm{Hz}^{2})$ are shown. Similar\nfeatures are found for other metallicity, distance moduli and ages,\nbut the lowest $\\chi^{2}$ may appear with rather high values.\n\n\\begin{figure}[!ht]\n\\resizebox{\\hsize}{!} {\\includegraphics[width=22cm]{3791fig4.eps}}\n\\caption{$\\chi^{2}$ against $\\Omega$ for six models with common parameters\n [Fe\/H], $\\alpha_{\\rm ov}$, $d$ and $A$.}\n\\label{fig:ang}\n\\end{figure}\n\nSince we expect the same [Fe\/H], $d$ and $A$ for all the stars in\nthe cluster we shall assume that the best fits should happen\nsimultaneously in the six stars despite\n differences in rotation\nrate and masses of the models. Hence we proceed to calculate the\nmean square root of the lowest $\\chi^{2}$ found in each model for\ngiven common parameters by means of the following expression:\n\n\\begin{equation}\n\\epsilon= \\sqrt{\\frac{1}{6}\\sum^{6}_{i=1}(\\chi_{{\\rm min},i})^{2}}.\n\\end{equation}\n\n\n\\noindent Figure~\\ref{fig:sig} shows $\\epsilon$ against\n[Fe\/H] for models without overshooting.  For the sake of clarity\nonly the points with $\\epsilon \\leq 3.0\\,\\mu$Hz are shown. It\ncan be seen that the solution with the smallest $\\epsilon $\ncorresponds to [Fe\/H]=0.0668. A plot of similar characteristics is\nobtained for models with overshooting.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!} {\\includegraphics[width=22cm]{3791fig5.eps}}\n\\caption{Minimum values of the mean square root of the difference\nbetween observed and theoretical frequencies for models with the\nsame values of [Fe\/H], $\\alpha_{\\rm ov}$, $d$ and $A$  against\n[Fe\/H]. For clarity, only models without overshooting are shown.\nThe symbols are related to different values of $A$ and $d$ as\nfollows:  asterisks ($d=5.39$ mag), squares ($d=5.50$ mag), diamonds ($d=5.60$ mag), circles\n($d=5.70$ mag), small (70 Myr), middle (100 Myr) and big (130\nMyr). } \\label{fig:sig}\n\\end{figure}\n\nSince only those identifications obtained from eq.~(5) with low\n$\\chi^{2}$ values are of interest, we require that for each star\nthe solution must have $\\chi \\leq 3.5\\,\\mu$Hz. \nIn Fig.~\\ref{fig:sig} we reject those solutions that have at least one\nmodel with $\\chi> 3.5\\,\\mu$Hz for all the parameters and\ncombinations between observed and theoretical frequencies. With\nthis restriction we found that only those solutions associated with\nthe models represented by the four lowest points in\nFig.~\\ref{fig:sig} should be considered.  Similar\nsolutions were found for models with overshooting. All these models are listed\nin Table~\\ref{tab:dom} and  will be analysed in detail later.\n\n\\begin{table}[]\n\\begin{center}\n\\caption{Ensembles of models with and without overshooting, which,\nafter applying a threshold of $\\chi \\leq 3.5\\,\\mu$Hz, have potential valid\nsolution (see text for details).}\n\\begin{tabular}{cccccc}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multispan6 \\hfill [Fe\/H] = 0.0668 \\hfill \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multispan3 \\hfill $\\alpha_{\\rm ov} = 0.00$ \\hfill &\\multispan3 \\hfill\n$\\alpha_{\\rm ov} = 0.20$ \\hfill \\\\\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n& $d$ & $A$ &   &  $d$ &   $A$ \\\\\n\n& (mag)&(Myr)& & (mag)&(Myr)          \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nC1&  5.70 & \\,\\,70&D1& 5.70 & \\,\\,70 \\\\\nC2& 5.70 & 130& D2&5.60 & 100\\\\\nC3&     5.60 & \\,\\,70 &D3&5.60 &130 \\\\\nC4&     5.60 & 100&D4&5.70 &100 \\\\\n\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\end{tabular}\n\\normalsize\n\\end{center}\n\\label{tab:dom}\n\\end{table}\n\nWe shall use a geometrical argument  to  reduce further the\nnumber of solutions so far available for each star. To this end,\nwe take into account that the visibility of each mode\ndepends on the angle of inclination. \nFollowing Pesnell~(\\cite{pesnell}), we illustrate in\nFig.~\\ref{fig:vis} the spatial response function\n$S_{lm}$ against $i$, for mode degree $l=0,1,2$. For\nsimplicity, limb-darkening has been neglected.  It can be seen that for $i \\approx\n90^{\\circ}$ only even $l+m$ values can be detected, while for $i\n\\approx 0^{\\circ}$ only modes with $m = 0$ will be visible.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!} {\\includegraphics[width=22cm]{3791fig6.eps}}\n\\caption{Variation of visibility amplitude with the inclination angle for radial\n$l=0$ (continuous line) and non-radial $l=1$ (dotted lines) and $l=2$ (dashed lines)\nmodes.}\\label{fig:vis}\n\\end{figure}\n\nWe introduce an hypothesis based on the visibility of the modes:\nwe  reject any solution with one or more modes\nwith $S_{lm}^{\\prime} < 0.1$. After applying this constraint we\nwere able to limit the analysis to four sets in \nTable~\\ref{tab:dom}, two with $\\alpha_{\\rm ov}=0.0$: C1\n$(d=5.70,\\, A=70)$, C3 $(d=5.60,\\, A=70)$ and two with\n$\\alpha_{\\rm ov}=0.2$: D1 $(d=5.70,\\, A=70)$, D2 $(d=5.60,\\, A=100\n)$. We note that while for the stars HD 23628 and HD 23194 only\na few identifications ($<$ 8) remain, the number of identifications\nfor V624 Tau, V534 Tau, V647 Tau and V650 Tau remained larger than\n$100$ in most  cases.\n\n\n\\section{Results and discussion \\label{sec:results}}\n\nIn order to discuss the results we shall introduce a parameter\n $\\Delta$ associated with each star in each identification and given\nby\n\n\\begin{equation}\n\\Delta = {\\rm max} ( |\\nu_{\\rm obs}- \\nu_{\\rm cal}| ).\n\\end{equation}\n\nTable~\\ref{tab:delta} lists the number of solutions for the six\nstars in each ensemble. Different maximum values of $\\Delta$ have\nbeen considered. Certain features can be derived from these\ngeneral results:\n\n\n\\begin{table*}[]\n\\begin{center}\n\\caption{Number of possible solutions for models in each ensemble\nfor different values of $\\Delta$ (in $\\mu$Hz). \\label{tab:delta}}\n\\begin{tabular}{lcccccccccc}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n&\\multispan{10} \\hfill [Fe\/H] = 0.0668 \\hfill \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n&\\multispan5 \\hfill $\\alpha_{\\rm ov} = 0.00$ \\hfill &\\multispan5 \\hfill $\\alpha_{\\rm ov} = 0.20$ \\hfill \\\\\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n&$\\Delta\\,(\\mu{\\rm Hz})\\, \\leq$  &1.0 & 2.0&3.0 &3.5 &  $\\Delta\\,(\\mu{\\rm Hz})\\, \\leq$    &1.0   &2.0 & 3.0 & 3.5   \\\\\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n&$\\frac{M}{M_{\\odot}}$&  & & &&$\\frac{M}{M_{\\odot}}$&  & & & \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n&C1  & & & & &D1&  & & & \\\\\nV624 Tau&1.70 &0 &0 &12 &18&1.69 &0 &0 &0 &12 \\\\\nV534 Tau&1.67 &0 &0 &0 &16&1.67 & 0& 0& 0&0 \\\\\nV647 Tau&1.70 &0 &0 &0 &2 &1.70 &0 & 0&0 &2 \\\\\nV650 Tau&1.86 &0 &0 &7 &13&1.86 &0 &2 &4 &4 \\\\\nHD 23194&1.80 &0 &1 &2 & 3&1.81 &0 &0 &3 &3 \\\\\nHD 23668&1.84 &0 &1 &2 & 4&1.84 &0 &0 &2 &3 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n&C3  & & & & &D2 & & & & \\\\\nV624 Tau&1.68 &0 &0 &6 &12&1.68 &0 &0 &0 &0\\\\\nV534 Tau&1.66 &0 &0 &0 &24&1.66 & 0& 0& 0&0 \\\\\nV647 Tau&1.69 &0 &0 &0 &2 &1.69 &0 & 0&0 &3 \\\\\nV650 Tau&1.86 &0 &0 &0 &0&1.86 &0 &0 &0 &3\\\\\nHD 23194&1.79 &1 &1 &2 &2&1.79 &1 &1 &1 &2 \\\\\nHD 23668&1.82 &0 &0 &3 &4&1.84 &0 &2 &3 &6 \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\end{tabular}\n\\normalsize\n\\end{center}\n\\end{table*}\n\n\n\\begin{itemize}\n\\item The resulting masses for each star are similar\nfor all the solutions. \n\n\\item The number of solutions for a given star at fixed $\\Delta$\nassociated with models with overshooting is smaller than without it.\n\n\\item There are no solutions with $\\Delta \\leq 3.0\\,\\mu$Hz for the stars V534 Tau\nand V647 Tau, while the star HD 23194 shows solutions even with\n$\\Delta \\leq 1.0\\mu$Hz.\n\n\\item In order to find a set of models that has at least one solution simultaneously\nfor all the stars a value of $\\Delta \\simeq 3.5\\,\\mu$Hz is needed.\nThis solution is found for C1 that corresponds to the cluster\nparameters [Fe\/H]=0.0668, $\\alpha_{\\rm ov}=0.0$, $d=5.70$ and\n$A=70$, and it coincides with the minimum in Figure~\\ref{fig:sig}.\n\n\\item\nThe remaining ensembles in Table~\\ref{tab:delta} have at least\none solution simultaneously when\nthe value of $\\Delta$ is increased slightly. In particular, for\nC3 a value of $\\Delta \\simeq 4.5\\,\\mu$Hz is needed to obtain a\nsolution while for D1 and D2 a value of $\\Delta \\simeq 3.8\\,\\mu$Hz\nis required. \n\n\\end{itemize}\n\nWe have found that the\nidentifications and the range of aspect angles derived for each star are \nsimilar for all the solutions. \nThis agreement may be due to the fact that the estimated range of mass and radius,\nand hence of mean densities, is similar for all the solutions. \nAt the  ages considered the overshooting has a negligible\neffect on the models.\n\nTable~\\ref{tab:results} summarizes the parameters estimated for\nthe cluster as well as for each star associated with the possible\nidentifications. The radial orders of these\nmodes are consistent with growth rate predictions\n(Su\\'arez et al.\\ \\cite{suarez}). \nWhile these identifications cannot be considered as\ndefinitive but rather as ``best fit solutions'', some\ninformation on the stars could be derived.\nOn the one hand, those stars with  smaller masses\n(V624 Tau, V534 Tau and V647 Tau) have more than six frequencies and could have \nsimultaneously excited radial and non-radial oscillations although\nat most two radial modes. \nFollowing Hern\\'andez et al.\\ (\\cite{hernandez}), \nat most two radial modes were found to be excited for the $\\delta$\nScuti stars in the Praesepe cluster.\nOn the other hand, the most massive stars, V650 Tau, HD\n23628 and HD 23194, with a mean mass of 1.83 $M_{\\odot}$\npresent on average few observational frequencies ($N \\le 4$) that\nmight be  associated only with non-radial modes. With the\npresent observational data it remains unclear, however, whether this\nis a general trend for $\\delta$ Scuti stars. \n\n\\begin{figure}\n\\resizebox{\\hsize}{!} {\\includegraphics[width=22cm]{3791fig7.eps}}\n\\caption{Comparison of observed (solid lines) and theoretical (dot lines)\nfrequencies of the best fit solutions in C1-models (Table~\\ref{tab:delta}).}\n\\label{fig:comp}\n\\end{figure}\n\n\n\\begin{center}\n\\begin{table*}[]\n\\caption{Summary of parameters derived for the Pleiades cluster as well as for\n$\\delta$ Scuti stars with the possible identification of the frequency modes.\n\\label{tab:results}}\n\\begin{tabular}{lcclcc}\n\n\\noalign{\\smallskip}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n \\multispan{6}\\hfill [Fe\/H] = 0.0668, $\\alpha_{\\rm ov} = $[0.00-0.20], $m_{V}-M_{V}=$[5.60-5.70], $A=$[70-100]${\\times}10^{6}$ years \\hfill \\\\\n\\noalign{\\smallskip}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n Star&$\\nu $ &Identification &Star&$\\nu $ &identification  \\\\\n         &($\\mu$Hz)& ($n,l,m)$&        &($\\mu$Hz)& ($n,l,m)$  \\\\\n\\noalign{\\smallskip}\n\\hline\n\\hline\n\\noalign{\\smallskip}\nV624 Tau&242.9& (1,0,0) &V650 Tau& 197.2&(1,1,1) \\\\\n$M = [1.68$-1.72]\\,$M_{\\odot}$         &409.0& (3,1,1)    &$M =$ [1.84-1.88]\\,$M_{\\odot}$       & 292.7&(0,2,$-$2),(2,2,2)   \\\\\n  $\\nu_{\\rm rot}= [3$-5]\\,$\\mu$Hz &413.5& (3,1,0) & $\\nu_{\\rm rot}= [$25-28]\\, $\\mu$Hz  & 333.1&(3,1,1),(3,2,2)  \\\\\n $i=[37^{\\circ}$-$67^{\\circ}]$&416.4&(3,1,$-$1) & $i=[60^{\\circ}$-$70^{\\circ}]$ & 377.8&(2,2,$-$2),(3,1,0)  \\\\\n& 451.7& (3,2,$-$2),(4,0,0) & &&(3,1,$-$1),(3,2,1)\\\\\n                &489.4 & (4,1,0),(4,1,1) &&&\\\\\n              &529.1 & (4,2,$-$1),(4,2,$-$2)&&&\\\\\n            & &(5,0,0) &&&\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nV534 Tau&234.2& (1,1,1)&HD 23628 &191.8 &(0,2,2)   \\\\\n$M = [1.65$-1.69]\\,$M_{\\odot}$        &252.9& (1,1,0)&$M = [1.82$-1.86]\\,$M_{\\odot}$ &201.7 & (1,1,1)    \\\\\n$\\nu_{\\rm rot}= [14$-16]\\,$\\mu$Hz &307.6&(2,0,0),(2,1,1)& $\\nu_{\\rm rot}= [24$-26]\\,$\\mu$Hz  &376.6&(2,2,$-$2)\\\\\n$i=[59^{\\circ}$-$79^{\\circ}]$  &377.9& (2,2$-$1),(3,0,0)   &$i=[53^{\\circ}$-$59^{\\circ}]$&&\\\\\n         &379.0 & (2,2$-$1),(3,0,0)  &  &&\\\\\n         & & (3,1,1)  &&\\\\\n        &448.1& (3,2,$-$1),(4,0,0)      &&&\\\\\n               &525.0& (4,2,$-$1)  &&&\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\nV647 Tau &236.6&(1,1,1)  &HD 23194&533.6&(5,1,1),(5,1,$-$1) \\\\\n $M = [1.68$-1.72]\\,$M_{\\odot}$  &304.7&(1,2,1) &$M = [1.78$-1.82]\\,$M_{\\odot}$&574.9&(5,2,0),(5,2,1)  \\\\\n $\\nu_{\\rm rot}= 10-11\\,\\mu$Hz&315.6&(2,0,0),(2,1,1)& $\\nu_{\\rm rot}= [14$-23]\\,$\\mu$Hz & & \\\\\n $i=17^{\\circ}$-18$^{\\circ}$ &374.4&(2,2,$-$1)&$i=7^{\\circ}$-$12^{\\circ}$ &&\\\\\n     &405.8&(3,1,0)  &&&\\\\\n         &444.1&(3,2,0)    &&&\\\\\n\\noalign{\\smallskip}\n\\hline\n\\hline\n\\noalign{\\smallskip}\n\\end{tabular}\n\\end{table*}\n\\end{center}\n\n\n\n\nA comparison of observed and computed frequencies is shown in\nFigure~\\ref{fig:comp}. In each panel the solid lines represent the \nobservational frequencies with their corresponding amplitudes.\nThe ``best fit'' theoretical frequencies\nare represented by dotted lines. The stars are arranged from top to bottom\naccording to their magnitude. \n\nFrom this figure it follows that\nthere is an asymmetric triplet in the observational spectrum of\nV624 Tau. We have found that for all the solutions the\nidentification is always a $(l=1,\\,n=3)$ mode split by rotation.\nIn turn, this implies that an independent estimate of the mean rotation\nrate can be obtained from the rotational splitting, up to second\norder, given by\n\n\\begin{equation}\n\\frac{\\nu_{-m} - \\nu_{m}}{2 m} \\sim \\frac{\\Omega_{\\rm rot}}{2 \\pi}\n(1-C_{nl}),\\;\\;\\;\\; m=1,2,\\ldots,l\n\\end{equation}\n\n\\noindent\nWe have found complete agreement between the cyclic\nrotational frequency computed from the above expression and those\nderived from the mode identification,  the differences being \n$\\sim 0.03-0.04\\,\\mu$Hz. We have also found a fairly\ngood agreement between the observations and the models in the differences\n$\\nu_{-m}-\\nu_{0}$ and $\\nu_{+m}-\\nu_{0}$.\n\n\\medskip\nIf the estimated inclination angles for\nthe stars are correct, all of them could be rapid rotators ($\\nu_{\\rm rot}>10\\mu$Hz)\nexcept for V624 Tau with $\\nu_{\\rm rot}\\leq 5\\mu$Hz.\nHD 23194 and V647 Tau, whose projected rotational velocities are low,\n$v\\,\\sin\\,i = 20$ kms$^{-1}$, would have equatorial velocities as high as\n90 and 70 kms$^{-1}$ respectively.\n\n\\medskip\nIn the present work the distance is found to be\n$m_{V}-M_{V} = 5.60-5.70$ mag $(132 - 138$ pc) and agrees very well\nwith that derived recently\nby independent determinations using independent techniques and data\n(Pan et al.\\ \\cite{pan}; Munari et al.\\ \\cite{munari}; Soderblom et al.\\\n\\cite{soderblom}; Percival et al.\\ \\cite{percival}).\n However, although\nthe isochrones computed with sub-solar metallicity\n  ($Z=0.015$ and $Y=0.30$) match well the observational data\n with a distance modulus of $d=5.39$ mag in agreement with\nthat of HIPPARCOS within 1-$\\sigma$ error,\nthe resulting solutions (shown by asterisks in\nFig.~\\ref{fig:sig}) do not lead to\n good fits.  Moreover, the plot of $\\epsilon$ against $d$ which we do\nnot reproduce here, shows better fits for further distances.\nFor solar metallicity, in order to reproduce the HIPPARCOS MS of the Pleiades,\nunrealistically high helium abundances are required (e.g. $Y=0.34$ Belikov et al.\n\\cite{belikov}, $Y=0.37$ Pinsonneault et\n al. \\cite{pinsonneault}).\n\n\\section{Conclusions \\label{sec:conclusions}}\n\nIn this study we have performed a seismological\nanalysis of six $\\delta$ Scuti stars belonging to the Pleiades\ncluster to identify their frequency modes.\nTo the best of our knowledge this group of\nvariables constitutes the most statistically significant sample of\n$\\delta$ Scuti stars analysed in an open cluster to date.\n\nRotational effects  were considered in different stages of\nthe analysis: first when determining the star positions in the HR\ndiagram and second when computing stellar models that approximately\nreproduce the evolutionary stage of the stars, and finally when\ncomputing theoretical oscillation frequencies in order to\nconstruct seismic models for target stars. The comparison between\nobservational and computed frequencies was carried out by\na least-squares fit.\nThere is a large number of possible solutions partly because\n neither the equatorial velocity nor the inclination\nangle are known a priori. In order to limit the number of possible\nsolutions we used the relationship between the\namplitude visibility, $S_{lm}$ and the aspect angle, $i$.\nAs a result we found  few solutions for each star,\nsuggesting the existence of only $p$ modes of low degree, low radial order\nin all the stars. For the less massive stars,\nsolutions with at most two radial modes were also possible. These solutions have\nassociated ranges of stellar parameters for each star. Most of the\nstars could be rapid rotators according to the estimated angle of inclination,\n$i$.\n\nThe best fits between observational and theoretical frequencies\nare achieved for global cluster parameters of [Fe\/H] = 0.0668\n$(Z_{0}=0.02,\\, X_{0}=0.70)$, $A=70 - 100\\,$Myr and $d=5.60 -\n5.70$ mag. The derived distance modulus for the Pleiades cluster\nagrees with that of the main sequence fitting method, in spite of the fact that\nthe isochrones with sub-solar metallicity\nclosely matches the Pleiades HR diagram with the HIPPARCOS distance.\n\n\n\n\\begin{acknowledgements}\nThis work has been partially funded by grants AYA2001-1571, ESP2001-4529-PE and \nESP2004-03855-C03-03 of the Spanish national research plan. L.F.M acknowledges the partial \nfinancial support by the Spanish AECI. J.C.S. acknowledges the financial support by the Spanish Plan of Astronomy and Astrophysics, under project AYA2003-04651, by the Spanish ``Consejer\\'{\\i}a de Innovaci\\'on, Ciencia y Empresa'' from the ``Junta de Andaluc\\'{\\i}a local government, and by the Spanish Plan Nacional del Espacio under project ESP2004-03855-C03-01.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe problem of predicting network structure can be both of great practical importance as well as a case-study in understanding the usefulness of deep learning in network settings. An accurate model can be used to suggest friend recommendations, product recommendations, and even predict individual user actions. A system which solves this problem is generally referred to in the literature as a recommender system, and such systems are quite common at large Internet companies such as Amazon \\cite{Linden:2003:ARI:642462.642471}, Netflix \\cite{Zhou:2008:LPC:1424237.1424269}, and Google.\n\nThe main approaches typically taken fall into two categories - \\textit{content based} and \\textit{collaborative filtering} approaches. The first makes use of text, meta-data, and other features in order to identify potentially related items, while the latter leans more towards making use of aggregated behavior and of a large number of training samples (ie, users and businesses). Collaborative filtering approaches have proven useful in recommender systems in industry, and are typically the preferred method due to how expensive it typically is (in both computational resources and engineering effort) to extract useful features from large amounts of meta-data. However, with advances in deep learning (extracting features from videos and text that are useful for many tasks), it seems feasible that revisiting content-based approaches with additional network-level data will prove fruitful.\n\nIn this paper, we seek to explore a novel method combining both deep learning feature extraction (a \\textit{content-based} approach) with network prediction models (a quasi-\\textit{collaborative filtering} approach). We focus on a real-world, practical network - the Yelp Review Network. The network consists of 4.7M review (edges), 156K businesses, 200K pictures, covering over 12 metropolitan areas in the united state.\n\nSpecifically, we seek to model the problem of predicting a user's star rating of a previously unrated business by using features about the business, the user, as well as existing interactions between the user and other businesses.\n\nFrom a general view point, we hypothesize that the final star rating given by a users is a mixture of all of the above interactions. In particular, we would expect that rating at time $t$ between user $i$ and business $j$ could be modeled as:\n$$\nr_t = f(i_t, j_t, \\delta_{i,j,t}) + \\mathcal{N}(0,\\epsilon_{i,j,t})\n$$\n\nHere, we have $i_t$ is the overall user-based bias at time $t$. For example, some users simply tend to give higher or lower ratings based on previous experience -- one could argue this is inherent to the user directly. We also have $j_t$, the overall business bias at time $t$. For example, some business are objectively better across the board, by having better food, websites, or being at better locations. Finally, the term $\\delta_{i,j,t}$ which is an interaction term reflecting the interaction between this user and the business as time $t$. One might imagine for example that a user who really enjoys Mexican food will tend to give those restaurants a higher rating.\n\nIn the end, these three terms should be combined in some way (with normalization, etc.) to arrive at a final rating. As such, we essentially have four models which can be combined to give better predictive power:\n\n\\begin{itemize}\n\\item a user model, trained only on user properties\n\\item a business model, trained on business properties\n\\item interaction model trained on a mixture of both properties with additional features known only to the network (such as previous business interactions, etc).\n\\end{itemize}\n\n\\section{Related Work}\nIn general, there are three areas of interest in the literature. We have (1) work which focuses and provides techniques for predicting results based on network structures, (2) work which has applied some ML techniques to the features extracted from networks (and sometimes elements themselves), and (3) work which throws away a lot of the network structure and focuses exclusively on using the data to make predictions. All of these are supervised learning methods which varying degrees of complexity. We provide a brief overview of them, followed by a section discussing the mathematical underpinnings of the models.\n\n\\subsection{Graph-Based Approaches}\n\nLiben-Nowell and Kleinberg \\cite{TheLinkPredictionProblemForSocialNetworks} formalize the \\textit{link prediction problem} and develop a proximity-based approach to predict the formation of links in a large co-authorship network. The model focuses on the network topology alone, ignoring any additional meta-data associated with each node since its basic hypothesis is that the known network connections offer sufficient insight to accurately predict network growth over time. They formally tackle the problem of given a social graph $G = (V,E)$ where each edge represents an interaction between $u,v$ and a particular timestamps $t$, can we use a subset of the graph across time (ie, with edges only in the interval $[t,t']$ to predict a future subset of the graph $G'$). The methods presented ignore the creation of new nodes, focusing only on edge prediction.\n\nMultiple predictors $p$ are presented, each focusing on only network structure. For example, some intuitive predictors (there are many others studied, though not necessarily as intuitive) for the edge creation between $x$ and $y$:\n\n\\begin{enumerate}\n\\item graph distance -- (negated) length of the shortest path between $x$ and $y$\n\\item preferential attachments -- $|\\Gamma(x)| \\cdot |\\Gamma(y)|$ where $\\Gamma: V \\to 2^V$ is a map from nodes to neighbors of nodes.\n\\end{enumerate}\n\nEach of the above predictors $p$ can output a ranked list of most likely edges. The paper evaluates effectiveness by comparing calculating the percentage of edges which are correctly predicted to exists in the test data. The baseline for the paper appears to be a random predictor based on the training graph and the graph distance predictor. The predictors are evaluated over five difference co-authoring networks. =\n\nThe predictors can be classified into essentially three categories:\n\n\\begin{itemize}\n\\item Predictors based on local network structure\n\\item Predictors based on global network structure\n\\item Meta predictors based on a mixture of the above two \n\\end{itemize}\n\nAll predictors performed above the random baseline, on average. The hitting time predictors performed below the graph distance baseline, with a much narrower positive gap for the remaining predictors. Most predictors performed on-par with just a common neighbors predictors.\n\n\\subsection{Introducing ML}\n\nFurther work by Leskovec et al. \\cite{Leskovec:2010:PPN:1772690.1772756} seeks to introduce the nuance of both ``positive'' and ``negative'' relationships to the link prediction problem, addressing limitations of previous work. In concrete, it seeks to predict the sign of each edge in a graph based on the local structure of the surrounding edges. Such predictions can be helpful in determining future interactions between users, as well as determining polarization of groups and communities. \n\nLeskovec et al. introduce the ``edge sign prediction problem'' and study it in three social networks where explicit trust\/distrust is recorded as part of the graph structure, work which is later expanded by Chiang et al. \\cite{Chiang:2011:ELC:2063576.2063742}. The explicit sign of the edges is given by a vote for or a vote against, for example, in the Wikipedia election network. They find that their prediction performance degrades only slightly across these three networks, even when the model is trained on one network and evaluated against another.\n\nThey also introduces social-psychological theories of balance and status and demonstrates that these seems to agree, in some predictions, with the models explored.\n\nFurthermore, they introduces the novel idea of using a machine learning approach built on top of the network features to improve the performance of the model. Rather than rely directly on any one network features, it instead extracts these features from the network and uses them in a machine learning model, achieving great performance. The features selected are, roughly speaking:\n\n\\begin{itemize}\n\\item Degree features for pair $(u,v)$ - there are seven such features, which are (1) the number of incoming positive edges to $v$, (2) the number of incoming negative edges to $v$, (3) the number of outgoing positive edges from $u$, (4) the number of outgoing negative edges from $u$, (5) the total number of common neighbors between $u$ and $v$, (6) the out-degree of $u$ and the (7) in-degree of $v$.\n\\item Triad features - We consider 16 distinct triads produced by $u,v,w$ and count how many of each type of triad.\n\\end{itemize}\n\nThe above features are fed into a logistic regression model and are used to relatively successfully predict the sign of unknown edges.\n\nOverall, while previous network predictions problems have attempted to make use of machine learning, most still rely on relatively simple models and have not yet made the jump to deeper architectures.\n\n\\subsection{Content-Based Deep Learning}\nHasan et. al in \\cite{Hasan06linkprediction} introduce the very important idea of using features of the node to assist in link prediction. The paper also significantly expands on the set of possible models to use for ML, demonstrating that for their data, SVMs work the best when it comes to predicting the edge. They formulate their problem as a supervised machine learning problem. Formally, we take two snapshots of a network at different times $t$ and $t'$ where $t' > t$. The training set of generated by choosing pairs of nodes $(u,v)$ which are not connected by an edge in $G_t$, and labeling as positive if they are connected in $G_{t'}$ and negative if they are not connected in $G_{t'}$. The task then becomes a classification problem to predict whether the edges $(u,v)$ is positive or negative. \n\nIn particular,they make use of the following features:\n\n\\begin{itemize}\n\\item Proximity features - computed from the similarity between nodes.\n\\item Aggregated features - how \"prolific\" a scientists is, or other features that belong to each node.\n\\item Network topology features - (1) shortest distance among pairs of nodes, (2) number of common neighbors, (3) Jaccard's coefficient, etc.\n\\end{itemize}\n\nThe authors rigorously describes the sets of features it found the most predictive, and takes into account node-level information extractable from the network as well as some amount of ``meta''-level information (for example, how similar two nodes are to each other). The results demonstrate great success (with accuracies up to 90\\% compared to a baseline of 50\\% or so). Overall, The authors presents a novel approach of using machine learning to assist in the link prediction problem by rephrasing the problem as a supervised learning task.\n\n\\section{Methodology and Data}\nIn this section, we describe the architecture of our feature extraction networks as well as lay the ground work for our predictive models. We define our loss function and presents some additional details used for training, such as learning rate and other hyper-parameters.\n\nWe convert the original data from JSON format to CSV. The data set contains 156,639 businesses (with 101 distinct attributes), 196,278 photos (associated with businesses), 1,028,802 tips (these are between users and businesses), 135,148 check-ins (again, associated with each business), and 1,183,362 users.\n\n\\subsection{Dataset}\nOur dataset is the set released for the Yelp Data Set Challenge Round 10 \\cite{YelpDataSet} in 2017. The entirety of the dataset consists of the following entities:\n\\begin{itemize}\n\\item \\textbf{Businesses}: Consists of exactly 156,639 businesses. It contains data about businesses on Yelp including geographical location, attributes, and categories.\n\\item \\textbf{Reviews}: 4,736,897 reviews. It contains full review text (for NLP processing) as well as the user id that wrote the review and the business id the review is written for. It also contains the number of stars given, as well as the number of useful, funny, and cool up-votes (finally, it also contains the date).\n\\item \\textbf{Users}: 1,183,362 Yelp users. It includes the user's friend mapping and all the meta-data associated with the user. Just this single dataset consists contains 538,440,966 edges.\n\\item \\textbf{Tips}: 1,028,802 tips. Tips are associated with each business and are written by users. Tips are similar to reviews, but without rating and usually much shorter.\n\\item \\textbf{Photos:} 196,278 Photos, each associated with businesses. The photos are also associated with captions.\n\\item \\textbf{Check-ins:} 135,148 check-ins on a business (this a business only attribute).\n\\end{itemize}\n\nAs we can see from above, the dataset is relatively rich, with many possible graph structures to study on top of it. In general, given that we are trying to predict review ratings, we focus on the following bipartite graph with users and businesses:\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=0.2]\n\\tikzstyle{every node}+=[inner sep=0pt]\n\\draw [black] (21.9,-11.9) circle (3);\n\\draw (21.9,-11.9) node {$1$};\n\\draw [black] (21.9,-20.2) circle (3);\n\\draw (21.9,-20.2) node {$2$};\n\\draw [black] (21.9,-28.1) circle (3);\n\\draw (21.9,-28.1) node {$3$};\n\\draw [black] (21.9,-36.9) circle (3);\n\\draw (21.9,-36.9) node {$4$};\n\\draw [black] (44.1,-8.1) circle (3);\n\\draw (44.1,-8.1) node {$1$};\n\\draw [black] (44.1,-8.1) circle (2.4);\n\\draw [black] (44.1,-18.7) circle (3);\n\\draw (44.1,-18.7) node {$2$};\n\\draw [black] (44.1,-18.7) circle (2.4);\n\\draw [black] (44.1,-27.4) circle (3);\n\\draw (44.1,-27.4) node {$3$};\n\\draw [black] (44.1,-27.4) circle (2.4);\n\\draw [black] (44.1,-35.9) circle (3);\n\\draw (44.1,-35.9) node {$4$};\n\\draw [black] (44.1,-35.9) circle (2.4);\n\\draw [black] (44.1,-43.8) circle (3);\n\\draw (44.1,-43.8) node {$5$};\n\\draw [black] (44.1,-43.8) circle (2.4);\n\\draw [black] (24.86,-11.39) -- (41.14,-8.61);\n\\fill [black] (41.14,-8.61) -- (40.27,-8.25) -- (40.44,-9.23);\n\\draw (34.64,-10.84) node [below] {$reviews$};\n\\draw [black] (24.36,-13.62) -- (41.64,-25.68);\n\\fill [black] (41.64,-25.68) -- (41.27,-24.81) -- (40.7,-25.63);\n\\draw [black] (23.94,-14.1) -- (42.06,-33.7);\n\\fill [black] (42.06,-33.7) -- (41.89,-32.77) -- (41.15,-33.45);\n\\draw [black] (24.89,-20) -- (41.11,-18.9);\n\\fill [black] (41.11,-18.9) -- (40.27,-18.46) -- (40.34,-19.46);\n\\draw [black] (24.75,-21.13) -- (41.25,-26.47);\n\\fill [black] (41.25,-26.47) -- (40.64,-25.75) -- (40.33,-26.7);\n\\draw [black] (24.66,-35.72) -- (41.34,-28.58);\n\\fill [black] (41.34,-28.58) -- (40.41,-28.44) -- (40.8,-29.35);\n\\draw [black] (24.35,-29.83) -- (41.65,-42.07);\n\\fill [black] (41.65,-42.07) -- (41.29,-41.2) -- (40.71,-42.01);\n\\draw [black] (24.73,-29.09) -- (41.27,-34.91);\n\\fill [black] (41.27,-34.91) -- (40.68,-34.17) -- (40.35,-35.11);\n\\draw [black] (23.73,-34.52) -- (42.27,-10.48);\n\\fill [black] (42.27,-10.48) -- (41.38,-10.8) -- (42.18,-11.41);\n\\draw [black] (24.13,-26.09) -- (41.87,-10.11);\n\\fill [black] (41.87,-10.11) -- (40.94,-10.27) -- (41.61,-11.01);\n\\end{tikzpicture}\n\\caption{Simplified graph model of user reviews of businesses. The graph is bipartite, with users and businesses connected by directed \"review\" edges.}\n\\label{fig:graph_structure}\n\\end{figure}\n\nand further propose making use of the friend-friend explicit graph (it is possible that it might be meaningful to see if we can find any relationship between friend reviews and user reviews) and the tip edges (without ratings, but possible meaningful information about a business). With this additional information, the structure of the graph itself becomes increasingly complex, as shown in Diagram \\ref{fig:graph_complex_structure}.\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=0.2]\n\\tikzstyle{every node}+=[inner sep=0pt]\n\\draw [black] (28.4,-27.4) circle (3);\n\\draw (28.4,-27.4) node {$1$};\n\\draw [black] (30.9,-16.9) circle (3);\n\\draw (30.9,-16.9) node {$2$};\n\\draw [black] (20.3,-19.9) circle (3);\n\\draw (20.3,-19.9) node {$3$};\n\\draw [black] (21.9,-36.9) circle (3);\n\\draw (21.9,-36.9) node {$4$};\n\\draw [black] (44.1,-8.1) circle (3);\n\\draw (44.1,-8.1) node {$1$};\n\\draw [black] (44.1,-8.1) circle (2.4);\n\\draw [black] (44.1,-18.7) circle (3);\n\\draw (44.1,-18.7) node {$2$};\n\\draw [black] (44.1,-18.7) circle (2.4);\n\\draw [black] (44.1,-27.4) circle (3);\n\\draw (44.1,-27.4) node {$3$};\n\\draw [black] (44.1,-27.4) circle (2.4);\n\\draw [black] (44.1,-35.9) circle (3);\n\\draw (44.1,-35.9) node {$4$};\n\\draw [black] (44.1,-35.9) circle (2.4);\n\\draw [black] (44.1,-43.8) circle (3);\n\\draw (44.1,-43.8) node {$5$};\n\\draw [black] (44.1,-43.8) circle (2.4);\n\\draw [black] (33.4,-15.24) -- (41.6,-9.76);\n\\fill [black] (41.6,-9.76) -- (40.66,-9.79) -- (41.22,-10.62);\n\\draw (41.47,-13) node [below] {$5\\mbox{ }review$};\n\\draw [black] (41.115,-27.585) arc (-94.24841:-162.7529:11.037);\n\\fill [black] (41.11,-27.58) -- (40.35,-27.03) -- (40.28,-28.02);\n\\draw (33.5,-25.71) node [below] {$tip$};\n\\draw [black] (30.21,-19.82) -- (29.09,-24.48);\n\\fill [black] (29.09,-24.48) -- (29.77,-23.82) -- (28.79,-23.59);\n\\draw (28.89,-21.73) node [left] {$friend$};\n\\draw [black] (29.09,-24.48) -- (30.21,-19.82);\n\\fill [black] (30.21,-19.82) -- (29.53,-20.48) -- (30.51,-20.71);\n\\draw [black] (22.5,-21.94) -- (26.2,-25.36);\n\\fill [black] (26.2,-25.36) -- (25.95,-24.45) -- (25.27,-25.19);\n\\draw (21.45,-24.14) node [below] {$friend$};\n\\draw [black] (26.2,-25.36) -- (22.5,-21.94);\n\\fill [black] (22.5,-21.94) -- (22.75,-22.85) -- (23.43,-22.11);\n\\draw [black] (26.71,-29.88) -- (23.59,-34.42);\n\\fill [black] (23.59,-34.42) -- (24.46,-34.05) -- (23.63,-33.48);\n\\draw (24.55,-30.8) node [left] {$friend$};\n\\draw [black] (23.59,-34.42) -- (26.71,-29.88);\n\\fill [black] (26.71,-29.88) -- (25.84,-30.25) -- (26.67,-30.82);\n\\draw [black] (21.62,-33.91) -- (20.58,-22.89);\n\\fill [black] (20.58,-22.89) -- (20.16,-23.73) -- (21.15,-23.64);\n\\draw (21.73,-28.32) node [right] {$friend$};\n\\draw [black] (20.58,-22.89) -- (21.62,-33.91);\n\\fill [black] (21.62,-33.91) -- (22.04,-33.07) -- (21.05,-33.16);\n\\draw [black] (18.908,-36.812) arc (-98.51807:-309.93742:12.583);\n\\fill [black] (18.91,-36.81) -- (18.19,-36.2) -- (18.04,-37.19);\n\\draw (8.57,-18.2) node [left] {$friend$};\n\\draw [black] (18.908,-36.811) arc (-98.5328:-309.92269:12.582);\n\\fill [black] (28.85,-14.72) -- (28.56,-13.82) -- (27.92,-14.59);\n\\draw [black] (31.02,-25.95) -- (41.48,-20.15);\n\\fill [black] (41.48,-20.15) -- (40.53,-20.1) -- (41.02,-20.98);\n\\draw (37.8,-23.55) node [below] {$tip$};\n\\draw [black] (20.708,-16.933) arc (166.48658:66.25769:15.116);\n\\fill [black] (41.49,-6.63) -- (40.96,-5.85) -- (40.56,-6.76);\n\\draw (27.15,-6.41) node [above] {$tip$};\n\\draw [black] (24.9,-36.77) -- (41.1,-36.03);\n\\fill [black] (41.1,-36.03) -- (40.28,-35.57) -- (40.33,-36.57);\n\\draw [black] (24.76,-37.79) -- (41.24,-42.91);\n\\fill [black] (41.24,-42.91) -- (40.62,-42.19) -- (40.32,-43.15);\n\\draw (34.33,-39.78) node [above] {$tip$};\n\\draw [black] (24.66,-35.72) -- (41.34,-28.58);\n\\fill [black] (41.34,-28.58) -- (40.41,-28.44) -- (40.8,-29.35);\n\\draw (36.85,-32.7) node [below] {$4\\mbox{ }review$};\n\\end{tikzpicture}\n\\caption{Proposed Complex Graph Models Based on Users, Reviews, Businesses, User-User Interactions, and Tips}\n\\label{fig:graph_complex_structure}\n\\end{figure}\n\n\\subsection{Predictive Models}\nThe rich meta-data about the network makes it quite interested to analyze, and opens up a lot of venues for possible improvements in terms of link prediction. We have multiple networks available for explorations, including \\textit{user-user} network (based on friendships, comments, etc.), \\textit{user-business} network, based on reviews given by a specific business to a user.\n\nFurthermore, we also have the raw text of the Yelp Review as well as geographical information about the business and photos for some businesses, which opens the possibility of using moderns visual image recognition and natural language processing techniques to further extract node-level meta-data to incorporate into our model.\n\nConcretely, we focus our work on predicting the rating that a user will assign a particular business. This problem has immediate and obvious utility: it would be useful to help users discover new businesses to visit (if the predicted rating is high) and also help business determine positive and negative trends. The dataset can be broken into three sets so we can train, evaluate, and test our models. One set will have edges, reviews, and information for businesses for a certain time $[t_0, t_1)$, the second set will have the edges created from $[t_1, t_2)$ and will be used to cross-validate our models and tune hyper-parameters, and the third set will he a hold out containing edges from $[t_2, t_3)$ and will be used for testing only.\n\n\\subsection{Network-Only Predictor}\nWe first present a predictive model which focus ``only'' on the structure of the graph, and uses this information to predict the ratings. For this purposes, we focus on the smaller user\/business graph as shown in Figure \\ref{fig:graph_structure}. We therefore have an undirected, weighed graph. In later sections, we explore alternative representations as well as additional data which can be input to our learning models.\n\n\n\\subsubsection{Data Preprocessing}\nGiven this representation $G$, we define three sets - training, validation, and test. We split the graph naturally -- edges and nodes are added to the graph as time progresses. However, we make special care to only use the nodes which remained and were available in the graph for the extent of our study. We can see the distribution of the reviews (edges) in our graph over time in Figure \\ref{fig:reviews_over_time}. Given the skewed nature of the graph, we subset it to include only the latest reviews. Let us consider $G, G_{train}, G_{val}$ and $G_{test}$ where $G = G_{train} \\cup G_{val} \\cup G_{test}$. We first perform the following to obtain $G$:\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{distribution_of_reviews_over_time}\n\\caption{Number of reviews in original dataset as a measure of time. We can see readily that the number of reviews increases drastically in the later years.}\n\\label{fig:reviews_over_time}\n\\end{figure}\n\n\\begin{itemize}\n\\item Remove all reviews before ``2016-08-24''. This is primarily to (1) remove bias from early users and reviewers and instead focus on later reviews (see Figure \\ref{fig:reviews_subset_over_time} for the distribution over time, which is far more uniform) and (2) reduce the size of our graphs to a manageable data set. We then have a graph with 428,795 users, 107,138 businesses, and 1,000,277 edges. We therefore have an extremely sparse graph, as only 0.0103109977941166\\% of all possible edges even exist.\n\\end{itemize}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{distribution_of_reviews_subset_over_time}\n\\caption{Number of reviews by date for $G$}\n\\label{fig:reviews_subset_over_time}\n\\end{figure}\n\nWe now proceed to split the graph into $G_{train}, G_{test}$ and $G_{val}$. We split time-wise using three split-points, $t_0, t_1,$ and $t_2$. Then we have $G_{train}$ as the subset of $G$ from $[t_0, t_1)$ and $G_{val}$ as the subset in $[t_1, t_2)$ with $G_{test}$ containing the subset to the latest date $[t_2, \\infty)$. Furthermore, we set all nodes in $G_{val}$ and $G_{test}$ to be the same set as those in $G_{train}$ to avoid running into issues with unseen nodes in the network.\n\nAfter the above, we end up with the following networks:\n\\begin{itemize}\n\\item $G_{train} = (V_{train}, E_{train})$ with $|V_{train}| = 375,149$ where we have $283,085$ users and $92,064$. We also have $|E_{train}| = 599,133$, which is an incredibly sparse graph given (only $0.00143945169412\\%$ of edges exists, even taking into account the bipartite structure of the graph).\n\\item $G_{val} = (V_{val}, E_{val})$ with $|V_{val}| = 75,466$ and $|E_{val}| = 88,079$.\n\\item $G_{test} = (V_{test}, E_{test})$ with $|V_{test}| = 67.125$ and $|E_{val}| = 73,730$.\n\\end{itemize}\n\nNote that we've split the data essentially into an $80\\%, 10\\%, 10\\%$ split. For more details on the graph structures, see Appendix \\ref{sec:graph_distributions}. Given the sparsity of the graph, we focus on predicting the star rating given that an edge is created between user $u$ and business $b$. As such, our dataset does not contain any negative examples. We leave this predictive problem for open investigation. Furthermore, given the extreme size of our data, we process and train our models using Google Compute Engine with 8 CPUs and 30GB of memory.\n\n\\subsubsection{Graph Features}\nNow that we have partitioned our data into training, validation, and testing, we move forward with calculating some rating prediction scores. We first focus on calculating multiple properties from our generated graph. In fact, we calculate the following:\n\n\\begin{itemize}\n\\item \\textbf{Number of Common Raters:} For each pair $(u,b)$ of user and business, we calculate the number of common raters. A common rater is an extension of neighbors, in the sense that this is someone who has also rated $b$.\n\\item \\textbf{Number of Common Business:} For each pair $(b,u)$ of user and business, we calculate the number of common businesses. A common business is an extension of neighbors, in the sense that this is a business someone who has also rated $b$.\n\\item \\textbf{Average Rating of Common Raters:} For each pair $(u,b)$ of user and business, we calculate the number of common raters. A common rater is an extension of neighbors, in the sense that this is someone who has also rated $b$.\n\\item \\textbf{Average Rating of Common Business:} For each pair $(u,b)$ of user and business, we calculate the number of common raters. A common rater is an extension of neighbors, in the sense that this is someone who has also rated $b$.\n\\item \\textbf{Preferential Attachment}: We take the product of the average star rating of businesses rated by $u$ and the average star rating of raters of business $b$. We expect this value to indicate the relative popularity.\n\\item \\textbf{Page Rank:} We treat the graph as an unweighed undirected graph and calculate the page rank value for all nodes and assign their sum as a feature.\n\\item \\textbf{Eigenvector Centrality}: We calculate the global centrality of a node (compared to its neighbors) and use the sum as a feature.\n\\item \\textbf{Adamic-Adar measure}: We look at common neighbors (as defined previously) and sum the inverse of the sum of their degrees (considering the graph to be weighed). Intuitively, this creates a measure for similarity where nodes with the same degreed neighbors are more similar.\n\\end{itemize}\n\n\nOnce calculate for our training, validation, and test data sets, all of the features are normalized to have unit mean and unit variance as is standard practice in machine learning problems.\n\n\\subsubsection{Models for Prediction}\nWe now present and describe the machine learning models used from the extracted features. Let $X$ be our training matrix, which is of shape $(n,d)$ where $n$ is the number of training examples and $d$ is the number of features extracted (in our case, $d = 9$) and $n = 599,133$. The most straight forward approach is simply to integrate our extracted features individually and directly train our models to predict the ratings, in a scale from $0$ to $5$. We now present the models we attempted.\n\n\\begin{enumerate}\n\\item \\textbf{Linear Regression}: We attempt to fit a standard linear regressors to our input feature set. That is to say, our model takes the form of $r_i = \\sum_{d=1}^{D} w_d x_{id}$ where $x_i$ is a single feature vector in our training set and $r_i$ is the corresponding rating. We train the model directly using the generated data from above and directly on the raw ratings for each edge. Linear regression is a simple model which attempts to minimize the mean square error, and can be thought of as a data generating process where we assume the ratings $r$ are generated by $r = W^TX + b + \\epsilon$ where $\\epsilon \\sim N(0,\\sigma)$ is some noise introduced into the system. The models is then able to recover the best plane of fit $W$ such that the error is minimized. In terms of loss functions, we can consider this as minimizing the loss function $L:$\n\\begin{align*}\nL(W,b; X,y) &= \\sum_{i=1}^{|X|} (\\hat{y}_i - y_i)^2 \\\\\n&=\\sum_{i=1}^{|X|} (w_i^Tx_i + b_i - y_i)^2 \\\\\n&=\\sum_{i=1}^{|X|} \\left(\\sum_{d = 1}^{|x_i|} w_dx_{id} + b_i - y_i\\right)^2\n\\end{align*}\nwhere $x_i$ is a the $i$-th row in our feature matrix $X$. We minimize over the parameters $W$.\n\\item \\textbf{Ridge Regression}: This is an improvement of linear regression. A possible issue with normal linear regression is that the possibility of over-training on the training set. It is possible to generate an extremely ``peaky'' set of weights such that the training error is reduced significantly yet the test error increases. The issue here is that we lack any term enforcing generalization in our loss function. The most typical method to enforce this generalization is to add a regularizer to the weights $W$. The loss function then becomes:\n\\begin{align*}\nL(W, b; X,y) &= \\sum_{i=1}^{|X|} (\\hat{y}_i - y_i)^2 + \\alpha|W|\\\\\n&=\\sum_{i=1}^{|X|} (w_i^Tx_i + b_i - y_i)^2 + \\alpha \\left|\\sum_{i,j} W_{ij}^2\\right|\\\\\n&=\\sum_{i=1}^{|X|} \\left(\\sum_{d = 1}^{|x_i|} w_dx_{id} + b_i - y_i\\right)^2 + \\alpha \\left|\\sum_{i,j} W_{ij}^2\\right|\n\\end{align*}\n\nThe above encourages the model to minimize the squared loss for the training data while still maintaining a relatively sparse matrix $W$. This further prevent values in $W$ from becoming too large. In our case, we find $\\alpha = 0.0001$ to be the optimal hyper-parameter (tuned on the validation set).\n\\item \\textbf{Bayesian Regression}: Bayesian regression is essentially equivalent to ridge regression, but it is self-regularizing -- this means we do not need to choose an optimal parameter $\\alpha.$ The theory behind Bayesian regression is to consider finding the parameters $W$ in our mode $y = WX$ which maximize the model probability. Given Bayes' rule, we have:\n\n\\begin{align*}\nP(W,b \\mid X) &= \\frac{P(X \\mid W,b)P(W,b)}{P(X)} \\\\\n&\\propto P(X \\mid W,b)P(W,b) \\\\\n\\end{align*}\nIf we consider the case where $P(W,b) \\sim N(\\mu, \\Sigma)$, then we arrive at ridge regression. We use this Bayesian model to also directly predict our ratings $r$. We optimize the above using the ADAM gradient descent optimizer where we use $\\alpha_1 = \\alpha_2 = \\lambda = \\lambda_2 = 0.000001$. The parameters are not tuned using the validation set due to lack of computational resources.\n\n\\item \\textbf{Deep Neural Networks}: The latest research has had great success using ``deep learning'' to extract more details from the data and to learn the values and result more directly. We make use of this approach by constructing a relatively shallow network consists of a fully connected layer with 200 neurons, followed by a second fully-connected layer with 40 neurons, followed by a fully connected layer of 8 neurons, and a final fully connected layer of 2 neurons.\n\nGiven the recent effectiveness in a large range of tasks of this model, we expect that it will likewise be useful for rating prediction.\n\nThis gives us a total of $200x(9 + 1) + 40x200 + 8x40 + 2x8$ with a relu nonlinearity:\n$$\nrelu(x) = \\max(0,x)\n$$\nWe use a final softmax at the end to generate the distribution of ratings.\n\nWe use Adam to perform gradient descent (with parameters $\\beta_1 = 0.9$ and $\\beta_2 = 0.999$ and $\\epsilon = 1\\times10^{-8}$) on the loss function with a regularization factor or $\\alpha = 0.0001$ and batch size of $200$. We maintain a constant learning rate of $0.001$, randomly shuffle the input data. The parameters are selected based on past experience with neural network training and are not optimized using cross-validation or the validation set. \n\n\\item \\textbf{Random Forest}: We make use also of a random forest estimator. The random forest is a meta estimator that fits a number of classifying decision trees on various sub-samples of the dataset and uses averaging to improve the predictive accuracy and control over-fitting. We generate the sub-samples by sampling from the original data set with replacement. We use a total of 100 estimators, where we look at all features in the data set when considering the best split. \n\\end{enumerate}\n\n\\subsection{Method Evaluation}\n\nFor all of the above approaches, we evaluate our effectiveness on the validation set and use this to tune our hyper-parameters (ie, network size, learning rate, etc.). In the end, we evaluate the results on the test set (previously unseen and untouched by our models) and make predictions for ratings in the seen edges. \n\nWe evaluate our models across three metrics:\n\n\\begin{itemize}\n\\item The root mean squared error. This evaluates how close our predictions achieve our desired ratings:\n$$\nRMSE = \\sqrt{\\frac{1}{|N|}\\sum_{i = 1}^{|X|} (\\hat{y}_i - y_i)^2}\n$$\n\\item The relative error. This is a metric that evaluates, on average, how wrong our star rating is compared to the true star rating. We take the method to be more indicate of improvements in our algorithms and with our data extraction. Formally, we define the relative error as:\n$$\nRELERROR = 100*\\frac{1}{|X|}\\sum_{i=1}^{|X|} \\frac{|\\hat{y_i} - y_i|}{\\max_{i=1}^{|X|} \\max\\{\\hat{y_i}, \\hat{y}\\}}\n$$\n\\item The last metric we use for evaluating our regression models is the $R^2$ score. This gives us a way to evaluate our models against other models in literature, as is standard across regression problems. The best possible score is 1.0. The score ranges from $(-\\infty, 1.0]$, where the worse a model is the more negative the value. Note that in the case where we have a model which simply predicts the constant expected value of the final output (disregarding any input features):\n$$\nE[Y] = \\frac{1}{|X|}\\sum_{i=1}^{|X|} y_{i}\n$$\nwe will have a score of $0.0$. The formula for computing this score is:\n$$\nR^2 = 1 - \\frac{\\sum_{i=1}^{|X|}(y_i - \\hat{y}_i)^2}{\\sum_{i=1}^{|X|} \\left(y_i - \\frac{1}{|X|}\\sum_{i=1}^{|X|}y_i\\right)^2}\n$$\n\n\\end{itemize}\n\n\\subsection{Extracting Item Features}\n\nGiven our results (see Results section) from the above models, we continue forward with our deep neural network. We begin by augmenting the data available for the business nodes. \n\n\\begin{itemize}\n\\item We make use of the pre-trained SqueezeNet network included in the PyTorch model zoo for visual image processing. We first down-sample the images to the expected 256x256x3 input (we do this simply by cropping and averaging pixel values over regions mapping into the 256x256xspace).\n\\item We can then feed these smaller images directly into the pre-trained squeezenet (see Figure \\ref{fig:squeezenet_architecture} for architecture) which has been modified to remove the final soft-max layer (and instead we produce a vector in $\\mathbb{R}^1000$.\n\\item For a business $b$, we take the $p_i^b \\in \\mathbb{R}^1000$ and compute their mean. We use this embedding as a representation of the business.\n\n\\item Furthermore, we make use of the pre-trained word-embedding and take the business description and generate a small 256-dimensional vector.\n\\item We concatenate the above vectors into a 1000 + 256 + 9 vector, which we take as input into a modified neural net.\n\\end{itemize}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{squeezenet.jpg}\n\\caption{Original SqueezeNet Architecture. We modify it to remove the final soft-max layer and instead output a $1000$ embedding for our images.}\n\\label{fig:squeezenet_architecture}\n\\end{figure}\n\nThe meat of the model consists of a neural net which takes a input as 1265-length vector for each $(u,b)$ pair and runs through through a single layer with 200 hidden units (so number of parameters is 200x1265). We then take this and feed it into the successful network described in the previous section and evaluate it in the same way as described before.\n\n\\subsubsection{Training and Date}\n\nDue to the large size of the above networks, we subset the data significantly into a much smaller amount of only ~15k reviews. We select the businesses with the most photos as the candidates to subset by, and make sure we take the reviews which include these businesses. With the reduced data size, we are able to successfully train our specified model end-to-end and achieve a marginal improvement over our previous models.\n\n\n\\section{Results and Discussion}\nIn this section, we presents the results on our test set and provide some discussion as to the extent to which our models successfully predicted Yelp review ratings.\n\n{\\renewcommand{\\arraystretch}{2}%\n\\begin{table*}[]\n\\centering\n\\caption{Supervised Training Results on Training Set}\n\\label{table:trainint_set_results}\n\\begin{tabular}{|l|lll}\n\\hline\n\\textit{\\textbf{Model}}      & \\multicolumn{1}{l|}{\\textbf{RMSE}} & \\multicolumn{1}{l|}{\\textbf{RELERROR}} & \\multicolumn{1}{l|}{\\textbf{R\\textasciicircum 2}} \\\\ \\hline\n\\textit{Baseline}            & 1.50142049076                      & 25.7312431633                          & 0.0                                               \\\\ \\cline{1-1}\n\\textit{Linear Regression}   & 1.29409210615                      & 20.397681968                           & 0.257107970487                                    \\\\ \\cline{1-1}\n\\textit{Ridge Regression}    & 1.29409210617                      & 20.3976788744                          & 0.257107970462                                    \\\\ \\cline{1-1}\n\\textit{Bayesian Regression} & 1.29409213097                      & 20.3975770983                          & 0.257107941987                                    \\\\ \\cline{1-1}\n\\textit{Neural Network}      & 1.26509191767                      & 18.7831282852                          & 0.290030838364                                    \\\\ \\cline{1-1}\n\\textit{Random Forest}       & \\textbf{0.749654164334}            & \\textbf{10.0184313324}                 & \\textbf{0.750702893173}                           \\\\ \\cline{1-1}\n\\textit{\\textbf{Business Features}}       & 1.24943163247            & 16.2852635532                 & 0.32123445344                           \\\\ \\cline{1-1}\n\\end{tabular}\n\\end{table*}\n}\n\n\n{\\renewcommand{\\arraystretch}{2}%\n\\begin{table*}[t]\n\\centering\n\\caption{Supervised Training Results on Validation Set}\n{}\\label{table:validation_set_results}\n\\begin{tabular}{|l|lll}\n\\hline\n\\textit{\\textbf{Model}}      & \\multicolumn{1}{l|}{\\textbf{RMSE}} & \\multicolumn{1}{l|}{\\textbf{RELERROR}} & \\multicolumn{1}{l|}{\\textbf{R\\textasciicircum 2}} \\\\ \\hline\n\\textit{Baseline}            & 1.42776997765                      & 24.1191750901                          & 0.0                                               \\\\ \\cline{1-1}\n\\textit{Linear Regression}   & 1.18380441761                      & 18.2853412809                          & 0.327892972837                                    \\\\ \\cline{1-1}\n\\textit{Ridge Regression}    & 1.18380391943                      & 18.2853539128                          & 0.312515072251                                    \\\\ \\cline{1-1}\n\\textit{Bayesian Regression} & 1.18378759889                      & 18.2857790551                          & 0.312534028173                                    \\\\ \\cline{1-1}\n\\textit{Neural Network}      & \\textbf{1.16442192777}             & \\textbf{16.0869919882}                 & \\textbf{0.334842664919}                           \\\\ \\cline{1-1}\n\\textit{Random Forest}       & 1.18801209881                      & 18.6598898159                          & 0.307618649842                                    \\\\ \\cline{1-1}\n\\textit{\\textbf{Business Features}}       & 1.14952444234            & 14.8854451849                 & 0.35986245424                           \\\\ \\cline{1-1}\n\\end{tabular}\n\\end{table*}\n}\n{\\renewcommand{\\arraystretch}{2}%\n\\begin{table*}[t]\n\\centering\n\\caption{Supervised Training Results on Test Set}\n\\label{table:test_set_results}\n\\begin{tabular}{|l|lll}\n\\hline\n\\textit{\\textbf{Model}}      & \\multicolumn{1}{l|}{\\textbf{RMSE}} & \\multicolumn{1}{l|}{\\textbf{RELERROR}} & \\multicolumn{1}{l|}{\\textbf{R\\textasciicircum 2}} \\\\ \\hline\n\\textit{Baseline}            & 1.4634860104                       & 24.7465614817                          & -0.000855120695457                                \\\\ \\cline{1-1}\n\\textit{Linear Regression}   & 1.19928440313                      & 18.5669587686                          & 0.327892972837                                    \\\\ \\cline{1-1}\n\\textit{Ridge Regression}    & 1.19928405085                      & 18.5669755633                          & 0.327893367682                                    \\\\ \\cline{1-1}\n\\textit{Bayesian Regression} & 1.19927256299                      & 18.567542783                           & 0.327906243749                                    \\\\ \\cline{1-1}\n\\textit{Neural Network}      & \\textbf{1.1838237529}              & \\textbf{16.3219078547}                 & \\textbf{0.34511029369}                            \\\\ \\cline{1-1}\n\\textit{Random Forest}       & 1.19281377927                      & 18.7137709175                          & 0.335125985417                                    \\\\ \\cline{1-1}\n\\textit{\\textbf{Business Features}}       & 1.1694425252            & 15.4556500245                 & 0.35111454552                           \\\\ \\cline{1-1}\n\\end{tabular}\n\\end{table*}\n}\n\nWe now present the final results from our models, each evaluated on the test set. We compare the different methods used, and discuss their differences and possible improvements. The main results are presented in Table \\ref{table:test_set_results}, with the validation data set results in Table \\ref{table:validation_set_results} and the training data set results in Table \\ref{table:trainint_set_results}. \n\nWe have implemented a complete end-to-end pipeline which begins with the (1) raw Yelp JSON data, (2) construct training, validation, and testing graphs over the data by our pre-determined timescales, (3) extracts training, validation, and testing sets from the graphs generated and computes a variety of network properties to be used by our machine learning models and (4) trains a variety of machine learning models on the extracted data and tunes their hyper-parameters when possible, (5) culminating in the evaluation of the models on the known results from the test set. We implements this work, wrote optimized code for feature extraction, and built our networks. Everything was implemented ourselves with the use of SNAP, Python, scikit-learn, and PyTorch -- the code base is publicly available at GitHub \\footnote{https:\/\/github.com\/kandluis\/cs224w-project}.\n\nThe major challenge faced when experimenting was the sheer size of the dataset -- even after sub-setting the data to a more manageable size in the millions and using the extremely powerful Google Compute Engine to add additional memory and processing power, more complex models such as random forest and the convolution neural networks could take in the order of days to fully train. Even extracting the word embeddings and pre-trained image feature vectors with SqueezeNet and ResNet would alone take a significant amount of time -- so much so that it proved unfeasible to do for a large portion of the dataset.\n\nAs such, as described in our Methods section, we were able to sample only approximately one years worth of data from the Yelp review network. However, despite this, we were nonetheless able to train the network predictors on over 500k training samples (reviews) which contained over 280k users and over 92k businesses (for more than 375k nodes), and validate and test our network models on over 88K and 73K examples respectively.\n\nFinally, to evaluate the performance of all of our models we make use of RMSE, relative error, and the score function defined in our methods. Our results can be see in Table \\ref{table:trainint_set_results}, Table \\ref{table:test_set_results}, and Table \\ref{table:validation_set_results}.\n\n\\section{Discussion}\nWe begin the discussion by analyzing some network properties.\n\n\\subsection{Summary Statistics}\n\\subsubsection{Users}\nWe present some overview of the user meta-data. In Figure \\ref{fig:user_characteristics}, we can see that multiple characteristics of the users follow a power-law degree distributions -- and not just the node degrees. This is to say that the distribution can be modeled as $P(k) \\propto k^{-\\gamma}$. The power-law distribution is immediately evident in:\n\n\\begin{itemize}\n\\item Number of review -- we have a few users that write many reviews and many users that write few reviews.\n\\item Number of friends -- this means the network follows a true-power law distribution.\n\\item useful\/funny\/cool\/fans -- this appears to demonstrate that social ranking\/status also follows a power-law distribution in the Yelp social network. This trend is further demonstrated by Figure \\ref{fig:user_compliment_distribution}.\n\\end{itemize} \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{distribution_user_characteristics}\n\\caption{Frequency of countable user characteristics -- the majority exhibit a power-law distributions}\n\\label{fig:user_characteristics}\n\\end{figure}\n\nFurthermore, we can look at the average rating given by users, across the network. The results are shown in a log plot in Figure \\ref{fig:user_rating_distribution}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{average_user_rating_distribution}\n\\caption{Distribution of Average user Rating}\n\\label{fig:user_rating_distribution}\n\\end{figure}\n\nWe notice that the ratings tend to be inflated (3-5) stars being quite frequent, while 1-2 stars being very infrequent. Presumable this might be due to the fact that people do not frequent poor restaurants. The other aspect that is immediately apparent is the spikes at even numbered ratings -- this is likely due to users who have rated only one once, of which we have many.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{compliement_type_distribution}\n\\caption{Distribution of Received user Compliments}\n\\label{fig:user_compliment_distribution}\n\\end{figure}\n\n\\subsubsection{Businesses}\nWe present some overview of the user meta-data. In Figure \\ref{fig:business_review_distribution}, we can see that the power-law distribution is also respected in the business side. Furthermore, we can also see that businesses tend to be rated quite highly, on average, with most businesses either having in the range from 3-5 (see Figure \\ref{fig:business_star_distribution}).\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{business_review_count_distribution}\n\\caption{Business Review Distribution}\n\\label{fig:business_review_distribution}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{business_star_distribution}\n\\caption{Business Average Star Distribution}\n\\label{fig:business_star_distribution}\n\\end{figure}\n\n\\subsection{Review Prediction}\nWe continue our discussing by now focusing on the results for our model predictions.\n\nWe can see that extracting network-only data and using machine learning models to fit the ratings seems to perform relatively well, even with simple, un-regularized linear regression. It appears that the features we selected, for example the nearest neighbors and the average ratings, are quite effective at both capturing network properties as well as capturing the user ratings. We did not need to extend the network to include further metadata information, and the results were nonetheless quite good, especially when compared to our non-trivial baseline.\n\nFurthermore, we note that our feature extraction proved extremely effective at generalizing across models. We see that in particular, the deep neural network and the random forest models both performed extremely well. It's interesting to note that the random forest model appears to have over-fit the data by a significant margin. This appeared promising on the training set but did not pan-out when we took the model to unseen data. However, we note that the neural net performed the best -- this appears to lead credence to the idea that the function learned is inherently non-linear, at least in the feature space we selected. This is somewhat counter to what we originally hypothesized, since all of the original features are approximately in the same scale as the ratings and would, intuitively, appear to predict the ratings rather directly. This idea is supported by the t-SNE embedding in Figure \\ref{fig:tsne_embedding}, where we embed our test set in a lower dimensional space (given the features extracted) and we color each based on the rating given (from $1,2,3,4,5$). \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TSE.png}\n\\caption{t-SNE embedding of of $G_{test}$}\n\\label{fig:tsne_embedding}\n\\end{figure}\n\n\nThis intuitively gives us a good foundation for why the features we choose appear to be so well correlated. Furthermore, we note that the embedding shows a clear non-linear distribution, which appears to corroborate the results where our neural network performed the best. This seems to imply that a neural network would be the best approach to disambiguate between the possible ratings a user might assign to a business. We found the issue of predicting the lack of edges to be somewhat more nuanced and subtle, though initial experiments with this approach proved promising.\n\nAnother promising aspect involves using the photographic and textual descriptions of businesses as input features to our predictive models. Despite making use of pre-trained word-embeddings and pre-trained image models, the computational cost for the models proved extreme for our dataset. Sub-setting into smaller set showed some initially results which appeared positive, however the smaller dataset makes it difficult to determine whether the additional information was accurately fed into the models and used effectively. However, it does appear that the networks performed well overall, and at least learned to make use of the features from the images.\n\n\\section{Conclusion}\nIn this paper, we have presented a novel approach towards review rating prediction in a subset of the Yelp review network. We have investigated the effectiveness of network-only machine learning models, determining 9 key structural features of the network that proved effective in predicting the weight of unseen edges after using supervised algorithms to generate the models. We demonstrated that a deep neural net even with the limited feature set was the most effective and most general approach. Furthermore, we performed early experiments in making use of none-network features to improve the predictions of the neural network. We did this by creating a pipeline building on previous work were for each $(u,b)$ pair, the business descriptions were converted into their respective word-embeddings using the popular word2vect network followed by an RNN which output a fixed sized 256 feature vector for each business. Furthermore, we selected key images from the business and the photo dataset provided by yelp and ran them through pre-trained SqueezeNet network with the final classification layer removed to generate multiple 4096-dimensional feature vectors per image. These feature vectors were then averaged and fed as additional input into a final fully-connected neural network. These preliminary results showed marginal improvement in the accuracy of the results. This shows not only that our original models are able to understand higher order relationships within the Yelp review network, but are also able to understand features (and build on them) specific to each node.\n\n\n\\section{Future Work}\nThe project could be continued in several directions. In particular, we could continue to follow the example set by \\cite{PintrestProject} and consider some of the temporal features of the graph structure. They proposed using a sliding window approach to achieve improved accuracy in link-prediction, which could easily be modified to support review prediction in the Yelp network.\n\nFurthermore, our preliminary work incorporating deep convolution neural nets and recurrent neural nets to extract feature embeddings for the businesses have demonstrated marginal capabilities of improving the predictive power of our models. Further work could be done in this area by, rather than extracting static embeddings, incorporating the visual and textual networks into an end-to-end model which could tweak the learned weights for visual and textual processing in order to have better understanding of how these features related to the ratings given to businesses by users. Furthermore, would like to see further work placed into whether user features can similar be used to improve performance -- for example, finding embeddings of users based on their features and using these embeddings as inputs to our model.\n\nLastly, there's additional work to be done to incorporate even more graph features into the predictive model. Given the effectiveness of the network structure itself at predicting the values of unseen ratings alone, we would like to explore further network features and models and see how this additional information can improve our models. This can include incorporating the information about tips -- we would expect someone that has given a tip to be more likely to rate the business positively (or negatively).\n\nIn any case, we believe that there is yet much work to be done in this field and many potential interesting developments in the area of combining non-network features with network features.\n\n\n\\section{Acknowledgments}\nThe author would like to thank Jure Leskovec and the rest of the CS224W staff for their continued guidance and advice during the course of the project. In particular, for emphasizing a focus on network properties.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nModeling the regression relationship between a response $Y$ and a multivariate Euclidean predictor vector $\\tbf{X}$ corresponds to specifying the form of the conditional means  $h(\\tbf{x})=\\mbb{E}{(Y|\\tbf{X} = \\tbf{x})}$ and higher dimensionality of $\\tbf{X}$ can be problematic when one is interested to go beyond the standard multiple linear model. This provides strong motivation to consider regression models  that provide dimension reduction, and single index models are one of the most popular approaches to achieve this, under the assumption that the influence of the predictors can be collapsed to an index, i.e., a projection on one direction complemented by a nonparametric link function. This reduces the predictors to  a univariate index while still capturing relevant features of the  high-dimensional data and is thus not subject to the curse of dimensionality.  This model generalizes linear regression (where the link function is the identity)  and is a special case of   nonparametric regression  \\citep{heck:86, rice:86, rupp:03}, which in its general form is subject to the curse of dimensionality.  For a real-valued response $Y$ and a $p$-dimensional predictor $\\tbf{X}$, the semiparametric single index regression model is \n\\begin{equation}\\begin{gathered} \\label{model:real}\\mbb{E}{(Y|\\tbf{X}= \\tbf{x})} = m(\\tbf{x} ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}). \\end{gathered}\\end{equation} \nIn model~\\eqref{model:real},\nthe dependence between $Y$ and $\\tbf{X}$ is  characterized by the conditional mean, i.e.,  \nthe conditional mean is a function of ${\\boldsymbol{\\bar{\\theta}_0}},$ and this reduces the dimensionality of the predictor from $p$ to essentially 1. \n\nThe function $m(\\cdot)$ is nonparametric and thus includes  location and level changes, and therefore the vector $\\tbf{X}_i$ cannot include a constant that would serve as  intercept. For identifiability reasons, ${\\boldsymbol{\\bar{\\theta}_0}}$ is often assumed to be a unit vector with non-negative first coordinate. A second approach that has been used is to require  one component  to equal one. This presupposes  that the component that is set to equal 1  indeed  has a non-zero coefficient \\citep{lin:07,cui:11, ichi:93}.\nModel (\\ref{model:real})   is only meaningful if  the Euclidean predictor vector $\\tbf{X}_i$ is  of dimension $2$ or larger. If $\\tbf{X}_i$ is one-dimensional, the\ncorresponding special case of the  model is the one-dimensional nonparametric regression $\\mbb{E}{(Y|X =x)} = m(x)$, which does not feature any parametric component.\n\nDue to its flexibility, the interpretability of the (linear) coefficients  and the nonparametric link function $m(\\cdot)$, as well as due to its wide applicability in many scientific fields, the classical single index regression model with Euclidean responses has attracted attention from the scientific community for a long time.  The coefficient ${\\boldsymbol{\\bar{\\theta}_0}}$ that defines the single index $\\tbf{x} ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}$ along with the shape of the nonparametric component $m(\\cdot)$  characterizes the  relationship between the response and the predictor. \nThe parametric component ${\\boldsymbol{\\bar{\\theta}_0}}$ is \n of primary interest\nin this model.  \nThe problem of recovering the true direction ${\\boldsymbol{\\bar{\\theta}_0}}$ can be viewed as a  subclass of sufficient dimension reduction (SDR) techniques, where identifying the  central subspace of $\\tbf{X}$ that explains most of the variation in $Y$  has been a prime target \\citep{chen:98, li:89,cook:94, li:07}. \n\nIn addition to sufficient dimension reduction techniques, a multitude of related  approaches to estimate ${\\boldsymbol{\\bar{\\theta}_0}}$ in \\eqref{model:real} have been studied.  These include projection pursuit regression (PPR) \\citep{frie:81, hall:89},  the average derivative approach \\citep{hard:89a, stok:86, fan:95, powe:86}, sliced inverse regression (SIR) \\citep{li:91, cook:91}, conditional minimum average variance estimation (MAVE) \\citep{xia:09} and various other methods \\citep{wang:10, xia:99, xia:06, xia:07, lian:10,yu:02}.\nVarious approaches focused on nonparametric estimation of the link function to recover the index parameter in \\eqref{model:real}   \\citep{hard:93, huh:02, hris:01} along with  partially linear versions \\citep{carr:97,cui:11,ichi:93}, various noise models \\citep{chan:10a,wang:10}\n\n\nVarious extensions of single index regression have  been considered more recently  \\citep{zhao:20,kere:20}, including models with  multiple indices or \nhigh- dimensional predictors \\citep{zhu:09b,zhou:08, kuch:17, kuch:20} and longitudinal and functional data as predictors \\citep{jian:11, chen:11, ferr:11, novo:19}. However, none of these extensions considers the case where responses are not in a Euclidean vector space, even though this case is increasingly important for data application. An exception is    \\cite{ying:20}, who  considered extending sufficient dimension reduction approaches for the case of random objects.\nThis lack of available methodology for single index models with random object responses motivates our approach.\nNon-Euclidean complex data structures arising in areas such as biological or social sciences are becoming increasingly common, due to  technological advances have made it possible to record and efficiently store time courses of images \\citep{peyr:09, gonz:18}, shapes \\citep{smal:12} or networks \\citep{tsoc:04},\nin addition to sensor data and  other complex data. For example, one might be interested in the functional connectivity quantified as correlation matrices obtained from neuroimaging  studies to study the effect of predictors on brain connectivity, an application that we explore further below.\n\n\nOther examples of general metric space objects include probability distributions  \\citep{deli:17}, \nsuch as age-at-death distributions as observed in demography or network objects, such as internet traffic networks. Such ``object-oriented data'' \\citep{marr:alon:14}  or ``random objects'' \\citep{mull:16} can be viewed as random variables taking  values in a separable metric space that is devoid of a vector space structure and where only pairwise distances between the observed data are available. Existing methodology for single index models as briefly reviewed above assumes that one has Euclidean responses, and these methods rely in a fundamental way on the vector space structure of the responses. When there is no linear structure, new  methodology is needed and this paper contributes to this development. \n\nThe natural notion of a mean for random elements of a metric space is the  Fr\\'echet mean \\citep{frec:48}, which  is a direct generalization of the standard mean, and is defined as the element of the metric space for which the expected squared distance to all other elements, the so-called Fr\\'echet function, is minimized. Depending on the space and metric, Fr\\'echet means may or may not exist as unique minimizers of the Fr\\'echet function. Fr\\'echet regression is an extension of Fr\\'echet means  to the notion of  conditional Fr\\'echet means, and has been  recently studied in several papers \\citep{pete:19, dube:19, pete:19b,chen:18}, including  both  global and local versions. \n\nIn this paper, we introduce a novel method for the single Index Fr\\'echet Regression (IFR) when the response variable is a random object lying in a general metric space and the predictor is a $p$-dimensional Euclidean vector. \nOur goal is to develop a simple and straightforward extension of the conventional estimation\nparadigm for single index models for this challenging case.\nSince there is no notion of direction or sign in a  general metric space, we interpret the index parameter in the proposed index Fr\\'echet regression model (IFR)  as the direction in the predictor space along which the variability of the response is maximized.\nIt turns out to be useful to view the direction as an M-estimator of an appropriate objective function, and to use empirical process theory to show consistency of the proposed estimate.\nWe also develop a   \nbootstrap method to obtain inference in finite sample situations. \n\nThe paper is organized as follows: The basic set up is defined in Section~\\ref{sec:model:methods} and theory on the asymptotic behavior of the index parameter is provided in Section~\\ref{sec:theory}\nThe index vector is assumed to lie on a hypersphere, with non-negative first element to facilitate identifiability. Then it is natural to quantify the  performance of the proposed estimators  by the geodesic distances between the estimated and true directions.  Empirical studies with different types of random object responses are conducted in Section~\\ref{sec:simul} to validate the estimation method. In Section~\\ref{sec:data:ADNI} we apply the methods to infer and analyze the effect of age, sex, total ADAS score, and the stage of propagation of  Alzheimer's Disease (AD) on the brain connectivity networks of patients with dementia. The networks are derived from fMRI signals of certain brain regions \\citep{thom:11} and for our analysis we represent them as correlation matrices. We conclude with a brief discussion in Section~\\ref{sec:concl}. \n\n\\section{Model and estimation methods}\n\\label{sec:model:methods}\nMore formally, in all of the following $(\\Omega,d,P)$ is a totally bounded metric space with metric $d$ and probability measure $P.$ The random objects $Y$ take values in $\\Omega$. \nThis is coupled with a $p$-dimensional real-valued predictor $\\tbf{X}.$  The conditional Fr\\'echet mean of $Y|\\tbf{X}$ is a generalization of $\\mbb{E}{(Y|\\tbf{X} = \\tbf{x})}$ to metric spaces, as the minimizer of $\\mbb{E}{(d^2(Y,\\omega)|\\tbf{X} = \\tbf{x}))},$ $\\omega \\in \\Omega.$ The latter is the corresponding generalized measure of dispersion around the conditional Fr\\'echet mean and can be viewed as a conditional Fr\\'echet function. \n\nAdopting the framework of  Fr\\'echet regression for random objects with Euclidean predictors, we define the Index Fr\\'echet Regression (IFR) model as\n\\begin{equation}\\begin{gathered} \\label{model:sim}\n\\mop{\\tbf{x} \\t {\\boldsymbol{\\bar{\\theta}_0}}} := \\mbb{E}_\\oplus{(Y|\\tbf{X} = \\tbf{x})} := \\underset{\\omega \\in \\Omega}{\\argmin} \\ \\mbb{E}{(d^2(Y,\\omega)|\\tbf{X} = \\tbf{x})},\n\\end{gathered}\\end{equation}\nwhere ${\\boldsymbol{\\bar{\\theta}_0}}$ is the true direction parameter of interest. The conditional Fr\\'echet mean is assumed to be a function of ${\\boldsymbol{\\bar{\\theta}_0}}$ in such a way that the distribution of $Y$ only depends on $\\tbf{X}$ through the index $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}},$ that is, $$Y \\perp \\mbb{E}{(Y|\\tbf{X})}|(\\tbf{X} ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}).$$\nModel \\eqref{model:real} emerges as a  special case of model \\eqref{model:sim} for a Euclidean response, as the conditional Fr\\'echet mean coincides with the conditional expectation $\\mbb{E}{(Y|\\tbf{X})}$ for the choice of the squared distance metric for the case $\\Omega = \\mbb{R}.$\n\n\nThe identifiability condition for is rephrased following the state-of-the-art literature \\citep{carr:97, lin:07, cui:11, zhu:06}. We assume the parameter space to be $\\Theta$ rather than the entire $\\mbb{R}^p$ in order to ensure that $\\boldsymbol{\\theta}$ in the representation \\eqref{sim:obj} can be uniquely defined, where $\\Theta :=  \\{\\boldsymbol{\\bar{\\theta}} = (\\theta_1, \\dots, \\theta_p)^{\\top} : \\ltwoNorm{\\boldsymbol{\\bar{\\theta}}} = 1,\\ \\theta_1 \\geq 0, \\ \\boldsymbol{\\bar{\\theta}} \\in \\mbb{R}^p\\}.$ \nWe first choose an identifiable parametrization which transforms the boundary of a unit ball in $\\mbb{R}^p$ to the interior of a unit ball in in $\\mbb{R}^{(p-1)}.$ By eliminating $\\theta_1,$ the parameter space $\\Theta$ can be rearranged to a form $\\{( (1- \\sum_{r=2}^p \\theta_r^2)^{1\/2}, \\theta_2,\\dots, \\theta_p)^{\\top} : \\sum_{r=2}^p \\theta_r^2 \\leq 1\\}.$ This re-parametrization is the key to analyzing the asymptotic properties of the estimates for $\\boldsymbol{\\theta}$ and to facilitating an efficient computation algorithm.  \n\nThe true parameter is then partitioned into $\\boldsymbol{\\bar{\\theta}} = (\\theta_1, \\boldsymbol{\\theta})^{\\top},$ where  $\\boldsymbol{\\theta} = (\\theta_2,\\dots, \\theta_p)^{\\top}.$ We need to estimate the $(p-1)-$ dimensional vector $\\boldsymbol{\\theta}$ in the single-index model, and then use the fact that $\\theta_1 = (1- \\sum_{r=2}^p \\theta_r^2)^{1\/2}$ to obtain $\\hat{\\theta}_1.$\n\\begin{prop}[Identifiability of model \\eqref{model:sim}]\n\\label{lem:iden:sim:obj}\nSuppose $h_\\oplus({\\tbf{x}}) = \\mbb{E}{(Y|\\tbf{X} = \\tbf{x})}.$ The support $S$ of $h_\\oplus({\\cdot})$ is a convex bounded set with at least one interior point and $h_\\oplus({\\cdot})$ is a non-constant continuous function on $S.$ If\n$$h_\\oplus(\\tbf{x}) = g_{1\\oplus}(\\boldsymbol{\\alpha}^{\\top} \\tbf{x}) = g_{2\\oplus}(\\boldsymbol{\\beta} ^{\\top} \\tbf{x}), \\text{ for all } \\tbf{x} \\in S, $$  for some continuous link function objects $g_{1\\oplus}$ and $g_{2\\oplus}$, and some $\\boldsymbol{\\alpha}, \\boldsymbol{\\beta}$ with positive first element such that $\\ltwoNorm{\\boldsymbol{\\alpha}} = \\ltwoNorm{\\boldsymbol{\\beta}} =1$ then $\\boldsymbol{\\alpha} = \\boldsymbol{\\beta}$ and $g_{1\\oplus} \\equiv g_{2\\oplus}$ on $\\{\\boldsymbol{\\alpha} ^{\\top} \\tbf{x}| \\tbf{x} \\in S\\}.$\n\\end{prop}\n\nThe above result can be proven using a similar argument as given in Theorem $1$ of \\cite{lin:07}. \n\nStudying the special case of the Euclidean response $Y$ in detail one may observe that the variation in $Y$ results from the variation in $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}$ and also from the variation in the error, $\\varepsilon$ \\citep{ichi:93}. On the contour line $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}} = c$, the variability in $Y$ only results from the variability in $\\varepsilon$. Along the contour line $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}= c,\\ \\boldsymbol{\\bar{\\theta}}\\neq {\\boldsymbol{\\bar{\\theta}_0}}$, the value of $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}$ changes. Therefore the variability in $Y$ on the contour line $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}= c,\\ \\boldsymbol{\\bar{\\theta}}\\neq {\\boldsymbol{\\bar{\\theta}_0}}$ comes from both the variation in $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}$ and in  $\\varepsilon$.  Since ${\\rm Var}\\left(Y|\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}= c\\right)$ measures the variability in $Y$ on a contour line $\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}= c,\\ \\boldsymbol{\\bar{\\theta}}\\neq {\\boldsymbol{\\bar{\\theta}_0}}$, a sensible way to alternatively interpret ${\\boldsymbol{\\bar{\\theta}_0}}$ would be finding the minimizer of the objective function $H(\\boldsymbol{\\bar{\\theta}})$, where\n$H(\\boldsymbol{\\bar{\\theta}}) : = \\mbb{E}{\\left( {\\rm Var}\\left(Y|\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}\\right)\\right)} \\text{ and } {\\boldsymbol{\\bar{\\theta}_0}} = \\underset{\\boldsymbol{\\bar{\\theta}}:\\ \\boldsymbol{\\bar{\\theta}} ^{\\top} \\boldsymbol{\\bar{\\theta}} =1}{\\argmin}\\ H(\\boldsymbol{\\bar{\\theta}}).$\nIt is indeed important to impose the constraint $\\boldsymbol{\\bar{\\theta}} ^{\\top} \\boldsymbol{\\bar{\\theta}} = 1,$ with the first element of the index $\\theta_{01}>0$ to ensure the identifiability of the objective function. Under such constraint we note that $H({\\boldsymbol{\\bar{\\theta}_0}}) \\leq H(\\boldsymbol{\\bar{\\theta}}).$\n\n\n\nThe method for recovering the true direction of the single index from model~\\eqref{model:sim} can be generalized in a similar way. The conditional variance of $Y$ given $\\tbf{X} = \\tbf{x}$ for a real-valued response can be directly generalized to the conditional Fr\\'echet variance  $d^2(Y,\\mop{\\tbf{x} ^{\\top} \\boldsymbol{\\bar{\\theta}}})$ for any given unit orientation vector $\\boldsymbol{\\bar{\\theta}}.$ Thus, for a general object response $Y\\in (\\Omega,d),$ ${\\boldsymbol{\\bar{\\theta}_0}}$ can alternatively be expressed as\n\\begin{equation}\\aligned\n\\label{sim:obj}\n{\\boldsymbol{\\bar{\\theta}_0}} = &\\underset{\\boldsymbol{\\bar{\\theta}} \\in\\ \\Theta}{\\argmin}\\ H(\\boldsymbol{\\bar{\\theta}}),  \\ \\text{where } H(\\boldsymbol{\\bar{\\theta}})= \\mbb{E}{\\left(d^2(Y,\\mop{\\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}}})\\right)},\\\\\n\\mop{t} &= \\underset{\\omega\\in \\Omega}{\\argmin} \\ M(\\omega, t),  \\ \\text{with } M(\\omega,t):= \\mbb{E}{\\left(d^2(Y,\\omega) |\\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}= t\\right)}.\n\\endaligned\\end{equation}\n\nTo recover ${\\boldsymbol{\\bar{\\theta}_0}}$ from the representation~\\eqref{sim:obj}, one needs to also estimate the conditional Fr\\'echet mean, as  in the IFR model \\eqref{model:sim}. We employ the local Fr\\'echet regression estimate \\citep{pete:19} for this. The conditional Fr\\'echet mean in \\eqref{sim:obj} can be approximated by a locally weighted Fr\\'echet mean, with weight function $S(\\cdot,\\cdot,\\cdot)$ that depends on a chosen kernel function $K(\\cdot)$ and a bandwidth parameter $b.$ For any given unit direction index $\\boldsymbol{\\bar{\\theta}},$ this intermediate localized weighted Fr\\'echet mean is defined as\n\\begin{equation}\\aligned\n\\label{intermed:local:fr}\n\\tmop{t} &= \\underset{\\omega\\in \\Omega}{\\argmin} \\ \\tilde{L}_b(\\omega, t),  \\ \\text{with } \\tilde{L}_b(\\omega,t):= \\mbb{E}{\\left(S(\\tbf{X}^{\\top} \\boldsymbol{\\bar{\\theta}},\\ t,b)d^2(Y,\\omega)\\right)},\n\\endaligned\\end{equation}\nwhere\n\\begin{equation}\\aligned \n\\label{local:Fr:weight}  \n&S(\\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}},\\ t,b) = \\frac{1}{\\sigma_0^2}K_b(\\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}}- t ) [\\mu_2 - \\mu_1(\\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}}- t)],\\\\\n&\\mu_k = \\mbb{E}{(K_b( \\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}} - t ) \\ ( \\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}}- t )^k)},\\ k = 0,1,2,\\quad \n\\sigma_0^2 = \\mu_2 \\mu_0- \\mu_1^2, \n\\endaligned\\end{equation}\nwhere $M(\\cdot,t) = \\tilde{L}_b(\\cdot,t) + O(b)$ for all $t$ \\citep{pete:19}. Suppose we observe a random sample of paired observations  $(\\tbf{X}_i,Y_i),\\ l=1,\\dots,n$, where $\\tbf{X}_i$ is a $p-$ dimensional Euclidean predictor and $Y_i \\in (\\Omega,d)$ is an object response situated in a general metric space $(\\Omega,d).$ Using the form of the intermediate target in \\eqref{intermed:local:fr} and replacing the auxiliary parameters by their corresponding empirical estimates, the local Fr\\'echet regression estimator at a given value $t$ of the single index  is defined as\n\n\\begin{equation}\\aligned\n\\label{est:local:fr}\n\\hmop{t} &= \\underset{\\omega \\in \\Omega}{\\argmin} \\ \\hat{L}_n(\\omega, t),  \\ \\text{with } \\hat{L}_n(\\omega,t):= \\frac{1}{n}\\sum_{i=1}^n \\wh{S}(\\tbf{X}_i ^{\\top} \\boldsymbol{\\bar{\\theta}},\\ t,b)d^2(Y_i,\\omega),\n\\endaligned\\end{equation}\nwhere\n\\begin{equation}\\aligned \n\\label{est:local:Fr:weight}  \n&\\wh{S}(\\tbf{X}_i ^{\\top} \\boldsymbol{\\bar{\\theta}},\\ t,b) = \\frac{1}{\\hat{\\sigma}_{0}^2}K_b( \\tbf{X}_i ^{\\top} \\boldsymbol{\\bar{\\theta}} - t) [\\hat{\\mu}_{2} - \\hat{\\mu}_{1}(\\tbf{X}_i ^{\\top} \\boldsymbol{\\bar{\\theta}} -  t)],\\\\\n&\\hat{\\mu}_{p} = \\frac{1}{n} \\sum_{j=1}^n K_b( \\tbf{X}_i ^{\\top} \\boldsymbol{\\bar{\\theta}} - t ) \\ (\\tbf{X}_i ^{\\top} \\boldsymbol{\\bar{\\theta}} - t)^p, \\ p = 0,1,2,\\quad\n\\hat{\\sigma}_{0}^2 = \\hat{\\mu}_{2} \\hat{\\mu}_{0}- \\hat{\\mu}_{1}^2.\n\\endaligned\\end{equation}\nAssuming  that the support of $T:= \\tbf{X}^{\\top} \\boldsymbol{\\bar{\\theta}}$, for any given unit direction $\\boldsymbol{\\bar{\\theta}}$ is compact and is denoted by $\\mathcal{T} = [0,1],$ we partition the interval $\\mathcal{T}$ into $M$ equal-width non-overlapping bins $\\{B_1,B_2,\\dots,B_M\\},$ such that data belonging to different bins are independent and identically distributed.\nWe define the mean observations $\\tilde{\\tbfX}_l$ and $\\tilde{Y}_l$ for the data points belonging to the $l-$th bin, where the latter are defined as the appropriate Fr\\'echet barycenters,\n\\begin{equation}\\aligned\n\\label{binned_dat}\n\\tilde{\\tbfX}_l &= \\sum_{i=1}^{n} W_{il} \\tbf{X}_i, \\ \\tilde{Y}_l = \\underset{\\omega \\in \\Omega} {\\text{argmin}}\\ \\sum_{i=1}^{n}  W_{il} d^2(Y_i,\\omega), \\text{ where } W_{il} &=  \\frac{\\indicator{\\tbf{X}_i^{\\top}\\boldsymbol{\\bar{\\theta}} \\in B_l} } {\\sum_{i=1}^{n} \\indicator{\\tbf{X}_i^{\\top}\\boldsymbol{\\bar{\\theta}} \\in B_l}}.\n\\endaligned\\end{equation}\n\nHere  the number of bins $M$ depends on the sample size $n$. The appropriate choice of $M = M(n)$ will be discussed later.\nThe proposed estimator for the true direction ${\\boldsymbol{\\bar{\\theta}_0}}$ in \\eqref{sim:obj} is then \n\\begin{equation}\\aligned \n\\label{est:sim:obj}\n\\boldsymbol{\\wh{\\bar{\\theta}}} &= \\underset{\\boldsymbol{\\bar{\\theta}} \\in\\ \\Theta}{\\argmin}\\ V_n(\\boldsymbol{\\bar{\\theta}}), \\text{ where }\nV_n(\\boldsymbol{\\bar{\\theta}}) = \\frac{1}{M} \\sum_{l=1}^{M} d^2(\\tilde{Y}_l, \\hmop{\\tilde{\\tbfX}_l^{\\top}\\boldsymbol{\\bar{\\theta}}}).\n\\endaligned\\end{equation}\nHere $\\hmop{\\tilde{\\tbfX}_l^{\\top}\\boldsymbol{\\bar{\\theta}}}, \\ l =1,\\dots,M,\\ $ is the local Fr\\'echet regression estimator, constructed based on the sample $(\\tbf{X}_i,Y_i), \\ i=1,\\dots,n$, and evaluated at each sample point of the binned sample $(\\tilde{\\tbfX}_l,\\tilde{Y}_l), \\ l = 1,\\dots, M,$ as described in \\eqref{est:local:fr} and \\eqref{est:local:Fr:weight}.\nDefine an intermediate quantity that corresponds to the empirical version of $H(\\cdot)$ in \\eqref{sim:obj} as \n\\begin{equation}\\aligned \n\\label{intermed:sim:obj}\n\\boldsymbol{\\tilde{\\bar{\\theta}}} &= \\underset{\\boldsymbol{\\bar{\\theta}} \\in\\ \\Theta}{\\argmin}\\ \\tilde{V}_n(\\boldsymbol{\\bar{\\theta}}), \\text{ where }\n\\tilde{V}_n(\\boldsymbol{\\bar{\\theta}}) = \\frac{1}{M} \\sum_{l=1}^{M} d^2(\\tilde{Y}_l, \\mop{\\tilde{\\tbfX}_l^{\\top}\\boldsymbol{\\bar{\\theta}}}).\n\\endaligned\\end{equation}\nThis auxilary quantity is used to prove the asymptotic results for $\\boldsymbol{\\wh{\\bar{\\theta}}}$ in the next section.\n\nThe bandwidth $b = b(n)$ is a tuning parameter involved in the estimation and the rate of convergence for $\\hmop{\\cdot}$ to $\\mop{\\cdot}$ is contingent on $b.$ It is important to note here that, another possible estimator for $\\mop{\\cdot}$ could be given by the global Fr\\'echet regression estimator introduced by \\cite{pete:19}. This is developed by generalizing multiple linear regression to the case of a metric-valued response by viewing the regression function as a sequence of weighted Fr\\'echet means, with weights derived from those of the corresponding standard linear regression. Using this alternative estimate for the unknown link function in the IFR model~\\eqref{model:sim} avoids the tuning parameter $b$ that is needed for local Fr\\'echet regression.\n\n\n\\section{Theory}\n\\label{sec:theory}\nThe unknown quantities that constitute the Index Fr\\'echet Regression (IFR) model consist of the nonparametric link function and the index parameter, and thus the asymptotic properties of the estimate of the true unit direction rely on those of the estimates  of the link function (with local Fr\\'echet regression) and the index parameter (through an M-estimator of the criterion function $H(\\cdot)$ in \\eqref{sim:obj}). The separable metric space $(\\Omega,d)$ is assumed to be totally bounded with diameter $D$, hence separable.  In addition, with regard to the quantities in \\eqref{sim:obj}, \\eqref{est:local:fr}, and \\eqref{est:sim:obj} we require the following assumptions.\n\\begin{enumerate}[label = (A\\arabic*), series = fregStrg, start = 1]\n\\item \\label{ass:fr:exist} The conditional and weighted Fr\\'echet means in \\eqref{sim:obj}, \\eqref{intermed:local:fr}, \\eqref{est:local:fr} and \\eqref{binned_dat} are well defined, i.e., they exist and are unique.\n\\item \\label{ass:reg:cont} The link function $\\mop$ is Lipschitz continuous, that is,  that is, there exists a real constant $K > 0$, such that, for all $\\boldsymbol{\\bar{\\theta}}_1,  \\boldsymbol{\\bar{\\theta}}_2 \\in \\bar{\\Theta},$\n\\[\nd(\\mop{\\tbf{x} ^{\\top}\\boldsymbol{\\bar{\\theta}}_1}, \\mop{\\tbf{x} ^{\\top}\\boldsymbol{\\bar{\\theta}}_2}) \\le\nK \\ltwoNorm{\\tbf{x} ^{\\top} (\\boldsymbol{\\bar{\\theta}}_1-\\boldsymbol{\\bar{\\theta}}_2)}.\n\\]\n\\item For any $\\varepsilon>0$ and $\\beta_1,\\beta_2>1,$ define\n\\begin{equation}\\aligned\n\\label{rate:an}\na_n =  \\max\\{ b^{2\/(\\beta_1 -1)}, (nb^2)^{-1\/(2(\\beta_2 -1)+\\varepsilon)}, (nb^2(-\\log b)^{-1})^{1\/2(\\beta_2 -1)}\\}.\n\\endaligned\\end{equation}\n\\label{ass:tuning:M} \nThe number of non-overlapping bins defined in Section~\\ref{sec:model:methods}, $M$ is a function of the sample size $n,$ that is $M = M(n)$ such that $Ma_n \\rightarrow 0.$\n\\end{enumerate}\nWe note that for $\\beta_1 = \\beta_2 =2,$ $a_n$ reduces to\n\\[\na_n =  \\max\\{ b^{2}, (nb^2)^{-1\/(2 +\\varepsilon)}, (nb^2(-\\log b)^{-1})^{1\/2}\\}.\n\\]\t\n\n\nThe above assumptions are commonly imposed when one studies M-estimators.  Whether Fr\\'echet means are well defined depends on the nature of the space, as well as the metric considered. For example, in case of Euclidean responses Fr\\'echet means coincide with the usual means for random vectors with finite second moments. For finite-dimensional Riemannian manifolds additional regularity conditions are required \\citep{afsa:11,penn:18}. For Hadamard spaces,   unique Fr\\'echet means are known  to exist \\citep{bhat:03,bhat:05,patr:15,kloe:10}.\nAssumption~\\ref{ass:reg:cont} limits how fast the object $\\mop{\\cdot}$ can change, introducing a concept of smoothness in the link function for the IFR model \\eqref{model:sim}.\nAssumption~\\ref{ass:fr:exist} is satisfied for the space $(\\Omega,d_W)$ of univariate probability distributions with the 2-Wasserstein metric and also for the space $(\\Omega,d_F)$ of covariance matrices\nwith the Frobenius metric $d_F$ \\citep{dube:19,pete:19}.\nAssumption~\\ref{ass:tuning:M} serves to  connect the uniform rate of convergence $a_n$ for the local Fr\\'echet regression estimator as given in ~\\eqref{rate:an} with the number of bins $M$. A basic additional assumption is that the predictors needed for the nonparametric Fr\\'echet regression are randomly distributed over the domain where the function is to be estimated, and that on average they  become denser as more data are collected. \nThis requires that there is at least one continuous predictor since if all the predictors are binary then the predictor locations cannot become denser with larger sample size. For any given direction $\\boldsymbol{\\bar{\\theta}},$ the univariate index variable $T := \\tbf{X} ^{\\top} \\boldsymbol{\\bar{\\theta}}$ is assumed to have a density $f_T(\\cdot)$ with a compact support $\\mathcal{T}$ and that the multivariate random variable $\\tbf{X}$ is bounded.\n\nAdditional assumptions  ~\\ref{ass:minUnif}-\\ref{ass:curvatureUnif}, and ~\\ref{ass:ker}-\\ref{ass:jointdtn} have been used previously in \\cite{pete:19} and are stated in the Appendix. They concern metric entropy  and curvature  for M estimators  and are commonly used  in their asymptotic analysis utilizing  empirical process theory \\citep{vand:00}. They\nare specifically  required to establish  consistency and uniform rate of convergence for the local Fr\\'echet regression estimator in \\eqref{est:sim:obj} \\citep{chen:20}.  Assumptions ~\\ref{ass:ker}-\\ref{ass:jointdtn} are commonly used  in the local regression literature \\citep{silv:78,fan:96}.\n\\begin{prop}\n\\label{lem:H}\nUnder assumption~\\ref{ass:fr:exist}-\\ref{ass:reg:cont}, $H(\\cdot)$ in model \\eqref{sim:obj} is a continuous function of $\\boldsymbol{\\bar{\\theta}}$ and for any $\\boldsymbol{\\bar{\\theta}} \\in \\bar{\\Theta},$ $H({\\boldsymbol{\\bar{\\theta}_0}}) \\leq H(\\boldsymbol{\\bar{\\theta}}).$ \n\\end{prop}\n\nMost types of random objects, such as those in the Wasserstein space (the space of probability distributions equipped with the  2-Wasserstein distance) or  the space of symmetric, positive semidefinite matrices endowed with the Frobenius or power metric satisfy assumptions ~\\ref{ass:minUnif}-\\ref{ass:curvatureUnif} (see Appendix) with $\\beta_1= \\beta_2 =2$. \tIf one chooses the bandwidth sequence $b$ for  the local Fr\\'echet regression such that, for a given $\\varepsilon>0,$ $b\\sim n^{-(\\beta_1 -1)\/(2\\beta_1 + 4\\beta_2 - 6 +2\\varepsilon)},$ $a_n$ is of the order $n^{-\\frac{1}{(\\beta_1 +2\\beta_2 -3 +\\varepsilon)}}$ \\citep{chen:20}. For  $\\beta_1 = \\beta_2 =2,$ this becomes \n$a_n \\sim n^{-\\frac{1}{3+\\varepsilon}},$  leading to a uniform convergence rate that is arbitrarily close to $O_P(n^{-1\/3}).$ Any choice $M=M(n)=n^{\\gamma}$ with \n$0 <\\gamma<\\frac{1}{3}$ will then satisfy Assumption~\\ref{ass:tuning:M}.  \t\n\nWe will make use of the following known  result to deal with the link function part when investigating  the  asymptotic convergence rates of the proposed IFR estimator.\n\\begin{lem}[\\cite{chen:20} Theorem $1$]\n\\label{lem:unif:local:fr:rate}\nUnder assumptions~\\ref{ass:minUnif}-\\ref{ass:curvatureUnif},~\\ref{ass:ker}-\\ref{ass:jointdtn}, and if $b\\to 0,$ such that $nb^2(-\\log b)^{-1} \\to \\infty$ as $n\\to \\infty,$  for any $\\varepsilon>0,$ and $\\beta_1,\\beta_2 >1$ as per  assumption~\\ref{ass:curvatureUnif},\n\\begin{equation}\\begin{gathered}\n\\underset{t \\in \\mathcal{T}}{\\sup} \\ d(\\hmop{t},\\mop{t}) = O_P(a_n),\n\\end{gathered}\\end{equation}\nwhere $a_n$ is as given in equation \\eqref{rate:an} in Assumption~\\ref{ass:tuning:M}.\n\\end{lem}\n\nThe following result demonstrates the consistency of the proposed estimator for the true index direction. All proofs can be found in the Supplementary Material\n\\begin{thm} \\label{thm:probConv}\nUnder assumptions~\\ref{ass:fr:exist}-\\ref{ass:reg:cont}, ~\\ref{ass:minUnif}-\\ref{ass:curvatureUnif}, and ~\\ref{ass:ker}-\\ref{ass:jointdtn}\n$$\\boldsymbol{\\wh{\\bar{\\theta}}} -{\\boldsymbol{\\bar{\\theta}_0}} \\overset{P}{\\longrightarrow} 0 \\text{ on } \\bar{\\Theta}.$$\n\\end{thm}\nAny $\\boldsymbol{\\bar{\\theta}} \\in \\bar{\\Theta}$ is decomposed into $(\\theta_1,\\boldsymbol{\\theta})^{\\top},$ where $\\theta_1>0,$ and for purposes of modeling the single index, keeping identifiability in mind,  can be expressed solely as a function of $\\boldsymbol{\\theta}.$\nTo this end we rewrite  the criteria function and the corresponding minimizers in terms of the sub-vector $\\boldsymbol{\\theta}$ only, \n\\begin{equation}\\aligned\n\\boldsymbol{\\theta_0} = \\underset{\\boldsymbol{\\theta} : \\boldsymbol{\\theta} \\in \\Theta}{\\argmin} \\, H(\\boldsymbol{\\theta}),\\quad \n\\tilde{\\boldsymbol{\\theta}} = \\underset{\\boldsymbol{\\theta} : \\boldsymbol{\\theta} \\in \\Theta}{\\argmin} \\, \\tilde{V}_n(\\boldsymbol{\\theta}),\\quad\t\n\\hat{\\boldsymbol{\\theta}}   = \\underset{\\boldsymbol{\\theta} : \\boldsymbol{\\theta} \\in \\Theta}{\\argmin} \\, V_n(\\boldsymbol{\\theta}).\n\\endaligned\\end{equation}\nWe note that $\\boldsymbol{\\theta_0},$ $\\tilde{\\boldsymbol{\\theta}},$ and $\\hat{\\boldsymbol{\\theta}}$ are the unconstrained minimizers for the criteria functions $H(\\cdot),$ $\\tilde{V}_n(\\cdot),$ and $V_n(\\cdot)$ respectively, which \nare continuous functions of $\\boldsymbol{\\theta},$ the latter two almost surely. \n\nCombining the consistency result for the direction vector from Theorem~\\ref{thm:probConv} with the uniform convergence of the local Fr\\'echet regression estimator in Lemma~\\ref{lem:unif:local:fr:rate}, the asymptotic consistency of the estimated single index regression (IFR) model follows.\n\\begin{cor}\n\\label{cor:probConv:ifr}\nUnder the conditions required for Theorem~\\ref{thm:probConv}, for any $\\tbf{x} \\in \\mbb{R}^p,$\n$$d(\\hmop{\\tbf{x} ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}}}, \\mop{\\tbf{x} ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}}) = o_P(1).$$\n\\end{cor}\nThe above corollary justifies the effectiveness of the use of local Fr\\'echet regression in the context of IFR.\n\n\\section{Simulation studies}\n\\label{sec:simul}\nThere are two tuning parameters involved in the implementation of the single index Fr\\'echet regression (IFR) model in~\\eqref{sim:obj}, namely the bandwidth $b = b(n)$ involved in the local Fr\\'echet regression as per \\eqref{model:sim} and the number of i.i.d data points, $M = M(n),$ where the local Fr\\'echet regression is fitted as per Assumption~\\ref{ass:tuning:M}.\n\nThe pair $(b,M)$ can be chosen by leave-one-out cross validation, where the objective function to be minimized is the mean discrepancy between the local Fr\\'echet regression estimates and the observed distributions for the binned data; specifically, \n\\[b_{opt} = \\argmin_{(b,M)} \\frac{1}{Mn} \\sum_{l=1}^M \\sum_{i=1}^n d^2(\\tilde{Y}_l, \\loomop{\\tilde{\\tbfX}_l ^{\\top}\\boldsymbol{\\bar{\\theta}}}) \\]\nwhere $\\loomop{\\tilde{\\tbfX}_l ^{\\top}\\boldsymbol{\\bar{\\theta}}}$ is the local Fr\\'echet regression estimate at $\\tilde{\\tbfX}_l ^{\\top}\\boldsymbol{\\bar{\\theta}}$ obtained with bandwidth $b$ based on the sample excluding the $i-$th pair $(\\tbf{X}_i,Y_i)$, i.e., \n\\[\n\\loomop{\\tilde{\\tbfX}_l ^{\\top}\\boldsymbol{\\bar{\\theta}}} = \\underset{\\omega \\in \\Omega}{\\argmin} \\frac{1}{(n-1)} \\sum_{j\\neq i}  \\wh{S}(\\tbf{X}_j ^{\\top} \\boldsymbol{\\bar{\\theta}},\\tilde{\\tbfX}_l ^{\\top} \\boldsymbol{\\bar{\\theta}} ,b)d^2(Y_j,\\omega),\n\\]\nIn practice, we replace leave-one-out cross validation by $5-$fold cross validation when $n > 30$.\n\nThe performance of the estimation is measured through simulations under various settings. The random objects we consider include samples of univariate distributions equipped with the Wasserstein$-2$ metric, samples of adjacency matrices from networks with the Frobenius metric and samples of multivariate data with the usual Euclidean metric. It is important to recall that true direction ${\\boldsymbol{\\bar{\\theta}_0}}$ is chosen to lie on the unit sphere in $\\mbb{R}^p$ with $\\theta_{01}>0.$ In each case, the optimal direction is estimated as the minimizer of $V_n(\\boldsymbol{\\theta}) : \\boldsymbol{\\theta} \\in \\mbb{R}^{p-1} \\text{ and } \\boldsymbol{\\theta} ^{\\top} \\boldsymbol{\\theta} \\leq 1,$ in \\eqref{est:sim:obj}. \n\nTo evaluate the accuracy of the estimate, we repeat the data generating mechanism $500$ times in each simulation setting, and for each such replication, obtain the optimal direction as $\\boldsymbol{\\wh{\\bar{\\theta}}}^{(i)} \\ i = 1,\\dots, 500.$ The intrinsic Fr\\'echet mean of these $500$ estimates on the unit sphere is computed as $\\widehat{\\bar{\\boldsymbol{\\theta}}}.$ Since each $\\boldsymbol{\\wh{\\bar{\\theta}}}^{(i)}$ lies on the manifold (the unit sphere in $\\mbb{R}^p$), the bias and deviance of the estimator is estimated as\n\\begin{align}\n\\text{bias}(\\boldsymbol{\\wh{\\bar{\\theta}}}) &= \\arccos \\langle \\widehat{\\bar{\\boldsymbol{\\theta}}},{\\boldsymbol{\\bar{\\theta}_0}} \\rangle, \\nonumber \\\\\n\\text{dev}(\\boldsymbol{\\wh{\\bar{\\theta}}}) &= {\\rm Var}\\left(  \\arccos \\langle \\boldsymbol{\\wh{\\bar{\\theta}}}^{(i)}, \\widehat{\\bar{\\theta}} \\rangle\\right) \n\\label{simul:bias:var}\n\\end{align}\n\n\\noindent Essentially, we estimate the $(p-1)-$ dimensional parameter $\\boldsymbol{\\theta_0} = (\\theta_{20},\\dots,\\theta_{p0})$ freely and then estimate $\\theta_{10}$ from the relation  $\\theta_{10} = \\sqrt{1- \\ltwoNorm{\\boldsymbol{\\theta_0}}^2}.$ \n\n\\subsection{Distributional responses}\n\nThe  space of distributions with the Wasserstein$-2$ metric provides an ideal setting for illustrating the efficacy of the proposed methods through simulation experiments. We consider distributions on a bounded domain $\\mathcal{T}$ as the response, $Y(\\cdot)$, and they are represented by the respective quantile functions $Q(Y)(\\cdot)$.  A $p-$ dimensional Euclidean predictor $\\tbf{X}$ is considered.\nThe random response is generated conditional on $\\tbf{X}$, by adding noise to the true regression quantile\n\\begin{align}\nQ(m_\\oplus(\\tbf{x}))(\\cdot) &= \\mbb{E}{\\left(Q(Y)(\\cdot)|\\tbf{X} = \\tbf{x}\\right)}\n\\label{simul:dens1}\n\\end{align}\n\n\\noindent As emphasized, the conditional distribution of $Y$ depends on $\\tbf{X}$ only through the true index parameter $\\boldsymbol{\\theta_0}.$\nTwo different simulation scenarios are examined as we generate the distribution objects from location-scale shift families (see Table~\\ref{table:sim}). In the first setting, the response is generated, on average, as a normal distribution with parameters that depend on $\\tbf{X}.$ For  $\\tbf{X} =\\tbf{x}$, the distribution parameters $\\mu \\sim N(\\zeta(\\tbfx \\t \\true), \\nu_1)$ is independently sampled, and for a fixed parameter $\\sigma=0.1$ the corresponding distribution is given by $Q(Y)(\\cdot) = \\mu + \\sigma \\Phi^{-1}(\\cdot)$. Here, the relevant sub-parameter is chosen as $\\nu_1 = 0.1$ and three different link functions are considered, namely $\\zeta(y) = y$ $\\ \\zeta(y) = y^2,$ and $\\zeta(y) = \\exp(y),$ and $\\Phi(\\cdot)$ is the standard normal distribution function. The second setting is slightly more complicated. The distributional parameter $\\mu|\\tbf{X} = \\tbf{x}$ is sampled as before and $\\sigma = 0.1$ is assumed to be a fixed parameter. The resulting distribution is then ``transported'' in Wasserstein space via a random transport map $T$, that is uniformly sampled from the collection of maps $T_k(a) = a - \\sin (ka)\/|k|$ for $k \\in \\{\\pm1, \\pm 2, \\pm 3\\}$. The distributions thus generated are not Gaussian anymore due to the transportation. Nevertheless, one can show that the Fr\\'echet mean is exactly $ \\mu + \\sigma \\Phi^{-1}(\\cdot)$ as before.\n\n\\begin{table}[!htb]\n\t\\centering\n\t\\begin{tabular}{|l|l|}\n\t\t\\hline\n\t\tSetting I &\n\t\tSetting II \\\\ \\hline\n\t\t\\begin{tabular}[c]{@{}l@{}}$Q(Y)(\\cdot) = \\mu + \\sigma \\Phi^{-1}(\\cdot) $, \\\\ where \\\\ $\\mu \\sim N(\\zeta(\\tbfx \\t \\true), \\nu_1), $\\ $\\sigma = 0.1.$\n\t\t\\end{tabular} &\n\t\t\\begin{tabular}[c]{@{}l@{}}$Q(Y)(\\cdot) = T \\#(\\mu +\\sigma \\Phi^{-1}(\\cdot)) $, \\\\ where\\\\ $\\mu \\sim N(\\zeta(\\tbfx \\t \\true), \\nu_1), $\\ $\\sigma = 0.1,$ \\\\ $T_k(a) = a - \\sin(ka)\/|a|,  k \\in \\{\\pm1,\\pm 2, \\pm 3\\}.$\\end{tabular} \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Table showing two different simulation scenarios.}\n\t\\label{table:sim}\n\\end{table}\n\nTo this end, we generate a random sample of size $n$ of density objects and multivariate Euclidean predictors from the true models, incorporating measurement error as described in the two situations above. We sample the predictors $\\tbf{X}_i$ to be of dimension $p=4,$ with the components of the vectors to be distributed independently as $Beta(1,1).$ Three different link functions are used to simulate the densities from the ``true'' model, namely, the identity link, the square link, and the exponential link. The bias and deviance of the estimated direction vectors for varying sample sizes are displayed in Tables~\\ref{tab:dens:set1} and~\\ref{tab:dens:set2}. We note that, the bias encountered due to the local Fr\\'echet estimation is generally low and the variance of the estimates also diminish with a higher sample size.\n\\begin{table}[!htb]\n\t\\begin{center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multicolumn{7}{|c|}{Setting I}                                                                 \\\\ \\hline\n\t\t\t& \\multicolumn{2}{c|}{link1 ($x \\mapsto x$)} & \\multicolumn{2}{c|}{link2 ($x \\mapsto x^2$)} & \\multicolumn{2}{c|}{link3 ($x \\mapsto e^x$)} \\\\ \\hline\n\t\t\t& bias       & dev     & bias         & dev         & bias         & dev         \\\\ \\hline\n\t\t\t$n = 100$  & 0.033      & 0.027         & 0.031        & 0.029       & 0.039        & 0.041       \\\\ \\hline\n\t\t\t$n = 1000$ & 0.011      & 0.013        & 0.017        & 0.012       & 0.020        & 0.013       \\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Table showing bias and variance of $\\boldsymbol{\\wh{\\bar{\\theta}}}$ (measured in radians) based on $500$ replications when the predictor dimension is $p=4.$ We took the four components of the vectors to be distributed independently as $Beta(1,1).$  The tuning parameters $(b,M)$ are  chosen by a $5-$fold cross validation method.}\n\t\\label{tab:dens:set1}\n\\end{table}\nThe performance of the fits is evaluated by computing the Mean Square Error (MSE) between the observed and the fitted distribution. Denoting the simulated true and estimated distribution objects at $(\\tilde{\\tbfX}_l,\\tilde{Y}_l),$ by $\\mop{\\tilde{\\tbfX}_l ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}}$ and $\\hmop{\\tilde{\\tbfX}_l ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}}}$ respectively, for $l=1,\\dots,M,$ the utility of the estimation was measured quantitatively by \n\\begin{align}\n\\label{simu:dens:mse:ifr}\nMSE = \\frac{1}{M} \\sum_{l=1}^M d^2_W(\\tilde{Y}_l, \\hmop{\\tilde{\\tbfX}_l ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}}}),\n\\end{align}\nwhere $d_W(\\cdot,\\cdot)$ is the Wasserstein-2 distance between two distributions.\nWe also compared the estimation performance of the proposed single index Fr\\'echet regression (IFR) method to a baseline Global Fr\\'echet regression (GFR) method, which can handle multivariate predictors being a generalization of global least squares regression \\citep{pete:19}. Denoting the GFR estimate of the distribution at $(\\tilde{\\tbfX}_l,\\tilde{Y}_l),$ by $\\hat{g}_\\oplus(\\tilde{\\tbfX}_l)$  for each $l=1,\\dots,M,$ we express the MSE of the fits as\n\\begin{align}\n\\label{simu:dens:mse:gfr}\nMSE = \\frac{1}{M} \\sum_{l=1}^M d^2_W(\\tilde{Y}_l, \\hat{g}_\\oplus(\\tilde{\\tbfX}_l)),\n\\end{align}\nwhere $d_W(\\cdot,\\cdot)$ is the Wasserstein-2 distance between two distributions. Figure~\\ref{fig:dens:mspe} shows the boxplots for different link functions used to generate the distribution data in the two simulation settings I and II for a sample size of $n=1000.$ We observe that, the IFR method outperforms the baseline GFR method in all the cases. Perhaps, the closest comparison between the two methods would be when an identity link function is used in the data generation mechanism. This is indeed expected since, in this case, the true model essentially reduces to a linear model.\nWe also display the fitted and the true distributions represented as densities (Figure~\\ref{fig:dens:fits}). The estimates from the IFR method matches the true observed densities quite well, thus the proposed IFR method can be used to validate the estimation method.\n\\begin{figure}[!htb]\n\t\\centering\n\t\\begin{subfigure}[b]{0.49\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{plots\/boxplot_dens_n100}\n\t\t\\caption{Simulation Setting I.}\n\t\t\\label{fig:dens_set1:mspe}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.49\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{plots\/boxplot_dens_set2_n100}\n\t\t\\caption{Simulation Setting II.}\n\t\t\\label{fig:dens_set2:mspe}\n\t\\end{subfigure}\n\t\\caption{Boxplot of MSPE of the fits using the single index Fr\\'echet regression model (IFR) and the Global Fr\\'echet regression (GFR) model for a sample size $n=1000.$ The left and the right panel correspond to the simulation settings I and II, respectively. The left, middle, and right columns in each of the panels correspond to the three different link functions used in the data generation mechanism, namely, identity, square, and exponential link functions, respectively, while the link functions in all cases are estimated from the data.}\n\t\\label{fig:dens:mspe}\n\\end{figure}\n\n\n\\begin{table}[!htb]\n\t\\begin{center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t\\multicolumn{7}{|c|}{Setting II}                                                                \\\\ \\hline\n\t\t\t& \\multicolumn{2}{c|}{link1 ($x \\mapsto x$)} & \\multicolumn{2}{c|}{link2 ($x \\mapsto x^2$)} & \\multicolumn{2}{c|}{link3 ($x \\mapsto e^x$)} \\\\ \\hline\n\t\t\t& bias       & dev      & bias         & dev        & bias         & dev        \\\\ \\hline\n\t\t\t$n = 100$  & 0.029      & 0.027         & 0.022        & 0.037       & 0.028        & 0.044       \\\\ \\hline\n\t\t\t$n = 1000$ & 0.010      & 0.012         & 0.011        & 0.014       & 0.017        & 0.021       \\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Table showing bias and variance of $\\boldsymbol{\\wh{\\bar{\\theta}}}$ (measured in radians) based on $500$ replications when the predictor dimension is $p=4.$ \tWe took the five components of the vectors to be distributed independently as $Beta(1,1).$ The tuning parameters $(b,M)$ are chosen by $5-$fold cross validation method.}\n\t\\label{tab:dens:set2}\n\\end{table}\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\t\\includegraphics[width=\\textwidth, height= .4\\textheight]{plots\/fits_dens_set2_n100}\n\t\\caption{Figure showing the density estimates of the distribution objects generated in simulation setting I. The blue and red curves are the observed and estimated densities, respectively. The left, middle, and right panels correspond to the three different link functions used in the data generation mechanism, namely, identity, square, and exponential link functions, respectively.}\n\t\\label{fig:dens:fits}\n\\end{figure}\n\n\n\\subsection{Adjacency matrix responses}\nHere the response objects are assumed to reside in the space of adjacency matrices arising out of a weighted graph equipped with the Frobenius norm. The $(p,q)-$th entry of the adjacency matrix $Y$ is given by,\n\\begin{align}\n\\label{simu:adj:mat}\nY_{pq} = m(\\tbf{x}^{\\top} \\boldsymbol{\\theta_0}) + \\epsilon_{pq},\n\\end{align}\nwhere $\\epsilon_{pq}$ are independently sampled errors and the link function $m(\\cdot)$ is such that $Y_{pq} \\in (0,1)$ for all $p, q.$ To ensure this, an appropriate link function in this case is taken as the expit link function, that is, $m(\\tbf{x}^{\\top} \\boldsymbol{\\theta_0}) = 1\/(1 + \\exp(- m(\\tbf{x}^{\\top} \\boldsymbol{\\theta_0}))).$ For a given index $\\tbf{x}^{\\top} \\boldsymbol{\\theta_0},$ $\\epsilon_{pq}$ was sampled from a Uniform distribution on $[\\max\\{0,-m(\\tbf{x}^{\\top} \\boldsymbol{\\theta_0})\\}, \\min\\{1,1-m(\\tbf{x}^{\\top} \\boldsymbol{\\theta_0})\\}]$. \n\\begin{table}[!htb]\n\t\\begin{center}\n\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{2}{c|}{link ($x \\mapsto 1\/(1+\\exp(-x))$} \\\\ \\hline\n\t\t\t& bias       & dev \\\\ \\hline\n\t\t\t$n = 100$  & 0.044      & 0.052 \\\\ \\hline\n\t\t\t$n = 1000$ & 0.021      & 0.019 \\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Table showing bias and variance of $\\hat{\\theta}$ (measured in radians) based on $500$ replications for weighted adjacency matrix responses. The tuning parameters for the model-fitting are chosen by $5-$ fold cross validation method.}\n\t\\label{tab:adjacency:set1}\n\\end{table}\nWe generated samples of networks with $10$ nodes, as one might encounter in brain networks, with the weighted adjacency computed as per \\eqref{simu:adj:mat}. The predictors are sampled from a $4-$dimensional multivariate normal distributions, where each of the components was truncated to lie between $[-5,5].$ While the mean vector for the multivariate normal distribution in the data generation scheme is assumed to be the zero vector, we assume the associated covariance matrix to be non-identity with ${\\rm cor}(X_1,X_2) = {\\rm cor}(X_1,X_3) =  {\\rm cor}(X_2,X_3) = 0.3,$ and  ${\\rm cor}(X_1,X_4) = {\\rm cor}(X_2,X_4) = -0.4.$ The variances for each of the four components are assumed equal to $0.25.$ We note here that the non-zero correlation among the components of the predictor vector does not influence the performance of the nonparametric regression fit negatively. Table~\\ref{tab:adjacency:set1} presents the bias and variance of the estimator computed based on $500$ replication of the data generating process. \n\\subsection{Euclidean responses}\nHere the object response of interest is assumed to lie in the Euclidean space. For generating the predictor vectors we consider a $5-$dimensional vector distributed as truncated multivariate normal distributions, where each of the components is truncated to lie between $[-10,10].$ The components are assumed to be correlated such that $X_1$ correlates with $X_2$ and $X_3$ with $r = 0.5$, and $X_2$ and $X_3$ correlate with $r = 0.25.$ The variances for each of the five components are $0.1.$ \nThe consistency of the estimates is illustrated in Table~\\ref{tab:euclidean:set1} based on $500$ replications of the simulation scenario.\n\\begin{table}[!htb]\n\t\\begin{center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{2}{c|}{link1 ($x \\mapsto x$)} & \\multicolumn{2}{c|}{link2 ($x \\mapsto x^2$)} & \\multicolumn{2}{c|}{link3 ($x \\mapsto e^x$)} \\\\ \\hline\n\t\t\t& bias       & dev     & bias         & dev         & bias         & dev         \\\\ \\hline\n\t\t\t$n = 100$  & 0.013      & 0.061         & 0.025        & 0.048       & 0.037        & 0.029       \\\\ \\hline\n\t\t\t$n = 1000$ & 0.006      & 0.021         & 0.014        & 0.019       & 0.013        & 0.009      \\\\ \\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Table showing bias and variance of $\\boldsymbol{\\wh{\\bar{\\theta}}}$ (measured in radians) based on $500$ replications for a Euclidean vector response. The predictors $X_1,\\dots, X_5$ are generated from a truncated multivariate normal distribution.}\n\t\\label{tab:euclidean:set1}\n\\end{table}\n\\section{Data analysis}\n\\label{sec:data:ADNI}\nModern functional Magnetic Resonance Imaging (fMRI) methodology has made it possible to study structural elements of the brain and identify brain regions or cortical hubs that  exhibit similar behavior, especially when subjects are in the resting state \\citep{alle:14, ferr:13}\nIn resting state fMRI, a time series of Blood Oxygen Level Dependent (BOLD) signal is observed for the seed voxels in selected functional hubs. For each hub, a seed voxel is identified as the voxel whose signal has the highest correlation with the signals of nearby voxels. Alzheimer's Disease has been found to have associations with anomalies in functional integration of brain regions and target regions or hubs of high connectivity in the brain \\citep{damo:12,zhan:10c}. \n\n\nData used in the preparation of this article were obtained from the Alzheimer's Disease Neuro-imaging Initiative (ADNI) database (\\url{adni.loni.usc.edu}).\nBOLD signals for $V= 11$ brain seed voxels for each subject were extracted. These Regions of Interest are aMPFC (Anterior medial prefrontal cortex), PCC (Posterior cingulate cortex), dMFPC (Dorsal medial prefrontal cortex), TPJ (Temporal parietal junction), LTC (Lateral temporal cortex), TempP (Temporal pole), vMFPC (Ventral medial prefrontal cortex), pIPL (Posterior inferior parietal lobule), Rsp (Retrosplenial cortex), PHC (Parahippocampal cortex), and HF$^+$ (Hippocampal formation)  \\citep{andr:10}. The pre-processing of the BOLD signals was implemented by adopting the standard procedures of slice-timing correction, head motion correction and normalization and other standard steps. The signals for each subject were recorded over the interval $[0, 270]$ (in seconds), with $K=136$ measurements available at $2$ second intervals. From this the temporal correlations were computed to construct the connectivity correlation matrix, also referred to as the Pearson correlation matrix in the area of fMRI studies. \n\nThe data set in our analysis consists of $n=830$ subjects at the four stages of the disease: $372$ CN (cognitively normal), $113$ EMCI (early mild cognitive impairment), $200$ LMCI ( late mild cognitive impairment), and $145$ AD subjects were considered. The inter-hub connectivity Pearson correlation matrix for the $i-th$ subject is denoted as $Y_{i}$,and has the  $(q,r)-$th element \n\\begin{align}\n(Y_{i})_{qr} = \\frac{\\sum_{p=1}^{K}(s_{ipq} - \\bar{s}_{iq}) (s_{ipr} - \\bar{s}_{ir})}{\\left[\\left(\\sum_{p=1}^{K}(s_{ipq} - \\bar{s}_{iq})^2\\right) \\left(\\sum_{p=1}^{K}(s_{ipq} - \\bar{s}_{iq})^2 \\right) \\right]^{1\/2}},\n\\label{data:ADNI:pearson:corr}\n\\end{align}\nwhere $s_{ipq}$ is the $(p,q)^{\\text{th}}$ element of the signal matrix for the $i^{\\text{th}}$ subject and $\\bar{s}_{iq} := \\frac{1}{K}\\sum_{p=1}^{K} s_{ipq}$ is the mean signal strength for the $q^{\\text{th}}$ voxel. For Alzheimer's disease trials, ADAS-Cog-13 is a widely-used measure of cognitive performance. It measures impairments across several cognitive domains that are considered to be affected early and characteristically in Alzheimer's disease \\citep{rock:07, kuep:18}. It is important to note that higher scores are associated with more serious cognitive deficiency.  \n\nTo demonstrate the validity of the model, we consider the out-of-sample prediction performance of the proposed IFR. For this, we first randomly split the dataset into a training set with sample size $n_{\\text{train}}$ and a test set with the remaining $n_{\\text{test}}$ subjects. The IFR method was implemented as follows - for any given unit direction $\\boldsymbol{\\bar{\\theta}} \\in \\bar{\\Theta},$ we partition the domain of the projections into $M$ equal-width non-overlapping bins and compute the mean observations $\\tilde{\\tbfX}_l$ and $\\tilde{Y}_l$ for the data points belonging to the $l-$th bin, where the latter are defined as the appropriate Fr\\'echet barycenters. Observe that $M$ is dependent on the sample size. The ``true'' index is estimated as $\\boldsymbol{\\wh{\\bar{\\theta}}}$ as per~\\eqref{est:sim:obj}. We then take the fitted objects obtained  from the training set, and predict the responses in the test set using the covariates present in the test set. As a measure of the efficacy of the fitted model, we  compute root mean squared prediction error (RMPE) as\n\\begin{align}\n\\text{RMPE} = \\left[\\frac{1}{M_{n_{\\text{test}}}}\\sum_{i=1}^{M_{n_{\\text{test}}}}  d_F^2\\left(\\tilde{Y}_l^{\\text{test}},  \\hmop{\\tilde{\\tbfX}_l ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}}} \\right)  \\right]^{-1\/2},\n\\end{align}\nwhere $\\tilde{Y}_l^{\\text{test}}$ and $\\hmop{\\tilde{\\tbfX}_l ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}}}$ denote, respectively, the $l^{\\text{th}}$ observed and predicted responses in the test set, evaluated at the binned average $\\tilde{\\tbfX}_l.$  We repeat this process $500$ times, and compute RMPE for each split for the subjects separately (See Table~\\ref{tab:ADNI:rmpe}).\nThe tuning parameters $(b,M)$ are chosen by a $5-$fold cross validation method for each replication of the process.\n\\begin{table}[h!]\n\t\\centering\n\t\\begin{tabular}{ ccc } \n\t\t\\hline\n\t\t$n_{\\text{train}}$ &  $n_{\\text{test}}$ & RMPE for the IFR method\\\\\n\t\t\\hline\n\t\t$500$ & $330$ &  $0.206$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Average Root Mean Prediction Error (RMPE) over $1000$ repetitions for the subjects, as obtained from the local fits of the single Index Fr\\'echet Regression (IFR) model.  Here, $n_{\\text{train}}$ and $n_{\\text{test}}$ denote the sample sizes for the  split training and testing datasets respectively.}\n\t\\label{tab:ADNI:rmpe}\n\\end{table}\nWe observe that the out-of-sample predictions error is quite low for after fitting the IFR model. In fact it is very close to the in-sample-prediction error $(0.251)$, calculated as the average distance between the observed training sample and the predicted objects based on the covariates in the training sets, which supports the validity of the proposed IFR models. \n\n\nAnother interest of our study was to understand the effect of various relevant predictors, such as age, gender, total score, and stage of the disease on the inter-hub functional connectivity. In particular, the hypothesis to test is given by\n$H_0: \\mop{\\cdot} = \\mop{\\theta_1X_1} \\text{ vs. } H_1: \\mop{\\cdot} = \\mop{\\sum_{j=1}^p \\theta_jX_j},$ or \nequivalently \n$$H_0: \\boldsymbol{\\theta} = \\mathbf{0}_{(p-1) \\times 1} \\ vs. \\ H_1: \\text{ not all } \\theta_j \\text{ are }0,  \\ j =2,\\dots,p $$ \nwhere $\\boldsymbol{\\bar{\\theta}} = (\\theta_1,\\boldsymbol{\\theta})^{\\top}$ and $\\boldsymbol{\\theta} = (\\theta_2,\\dots,\\theta_p).$\nHere we consider $p =10,$ predictors, namely, $X_{1}=$ stages for the disease, $X_2 =$ age, $X_3 =$  sex, $X_4 =$ total ADAS score, and the pairwise interaction terms between them as\n$X_5 = X_1X_2$, $X_6 = X_1X_3$, $X_7 = X_1X_4$, $X_8 = X_2X_3$, $X_9 = X_2X_4$, and $X_{10} = X_3X_4$. \n\nWe employ a bootstrap procedure to test for $H_0$.\nBased on a standard bootstrap approach, we resample $(\\tbf{X}_i^\\ast, Y_i^\\ast)$ a large number of times, $B$. For each of these $B$ bootstrap samples, the estimated direction as $\\hat{\\boldsymbol{\\theta}}^{\\ast}$ ($(p-1)-$ dimensional vector) is computed. \tWe estimate the full $p-$ dimensional vector $\\boldsymbol{\\wh{\\bar{\\theta}}}^\\ast$ for each of the bootstrap sample, where  $\\boldsymbol{\\wh{\\bar{\\theta}}}^\\ast = (\\widehat{\\theta}_1^\\ast, \\hat{\\boldsymbol{\\theta}}^\\ast),$ with $\\widehat{\\theta}_1^\\ast = \\sqrt{1 - \\ltwoNorm{\\hat{\\boldsymbol{\\theta}}^\\ast}^2}.$ Denote the $p-$ dimensional unit vector estimated from the original sample  as  $\\boldsymbol{\\wh{\\bar{\\theta}}}.$  $\\boldsymbol{\\wh{\\bar{\\theta}}}$  and $\\boldsymbol{\\wh{\\bar{\\theta}}}^\\ast$ for all $B$ bootstrap samples, lie on a sphere is $\\mbb{R}^p.$ We can compute the  the geodesic distance between two points $\\boldsymbol{\\bar{\\gamma}_1} $ and $\\boldsymbol{\\bar{\\gamma}_2} $ situated on the boundary of a unit sphere by $d_g(\\boldsymbol{\\bar{\\gamma}_1} , \\boldsymbol{\\bar{\\gamma}_2} ) = \\arccos \\langle \\boldsymbol{\\bar{\\gamma}_1} , \\boldsymbol{\\bar{\\gamma}_2} \\rangle.$ To this end, we proceed to conduct a bootstrap test as follows. \n\\begin{itemize}\n\t\\item[1.] Compute $d_g(\\boldsymbol{\\wh{\\bar{\\theta}}}^{\\ast (b)}, {\\boldsymbol{\\bar{\\theta}_0}}),$where $\\boldsymbol{\\wh{\\bar{\\theta}}}^{\\ast (b)}$ is the $p-$ dimensional unit vector estimated based on the $b-$th bootstrap sample, $b = 1,\\dots, B$ and ${\\boldsymbol{\\bar{\\theta}_0}} = (1,0,\\dots)_{p \\times 1}.$\n\t\\item[2.] The Achieved Significance Level (ASL) of the bootstrap test is given by\n\t$$ASL = \\frac{1}{B}\\sum_{b=1}^B d_g(\\boldsymbol{\\wh{\\bar{\\theta}}}^{\\ast (b)},{\\boldsymbol{\\bar{\\theta}_0}}) > d_g(\\boldsymbol{\\wh{\\bar{\\theta}}}, \\boldsymbol{\\wh{\\bar{\\theta}}}^{\\ast (b)}),$$\n\t\\item[3.] If $ASL <\\alpha$ reject the null hypothesis at level $\\alpha.$\n\\end{itemize}\nWe carried out the procedure for testing the significance of the other predictors when ``stages of the disease'' is assumed to be included in the model. The $ASL$ for the bootstrap test  came out to be $0.012,$ thus giving evidence for rejecting $H_0$ at level $\\alpha = 0.05.$ That is, not all of the $(p-1)$ predictors are insignificant. \n\\begin{table}[H]\n\t\\begin{center}\n\t\t\\begin{tabular}{|l|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{2}{c|}{Step 1} & \\multicolumn{2}{c|}{Step 2} & \\multicolumn{2}{c|}{Step 3} \\\\ \\hline\n\t\t\t& Coeff.       & ASL      & Coeff.       &ASL      & Coeff        & ASL      \\\\ \\hline\n\t\t\tAge         & -0.364       & 0.005        & -0.394       & -            & -0.401       & -            \\\\ \\hline\n\t\t\tGender      & 0.371        & 0.122        & 0.558        & 0.161        & 0.279        & 0.113        \\\\ \\hline\n\t\t\tTotal Score & 0.198        & 0.094        & 0.207        & 0.010        & 0.173        & -            \\\\ \\hline\n\t\t\\end{tabular}\n\t\t\\caption{Table shows the coefficients and p-values for the step-wise addition of predictors to quantify their relative significance.}\n\t\t\\label{tab:data:step_reg}\n\t\\end{center}\n\\end{table}\nWe further performed a sequential addition of predictors to quantify their relative significance. For this, we specify an ``alpha-to-enter'' significance level at $\\alpha = 0.05$ and fit each of the one-predictor model. To this end, we consider $X_1$ to be in the model and fit model~\\eqref{sim:obj} to estimate the direction index. In particular, we include each of $X_2,$ $X_3,$ and $X_4$ in the model along with $X_1$ and test whether the corresponding effect is significant, i.e., we test $\\theta_j=0, \\ j=2,3,4$ separately. The first predictor to be included in the step-wise model is the predictor that has the smallest $ASL$. We stop if no predictor has a test $ASL$ less than $\\alpha.$ Table~\\ref{tab:data:step_reg} illustrates the step-wise addition of predictors for all one-predictor models. The first predictor to be added at level $0.05,$ when $X_1$ (``stage of the disease'') is already in the model, is $X_2$ (age). Interestingly, a negative value of $\\hat{\\theta}_2$ $(-0.364)$ signifies a possible negative effect of the predictor on the response. This is quite expected since Alzheimer's is a disease that is known to progress with age.\nIn step 2, $X_4$ (total score) is added as a significant predictor at $0.05$ level. Here the estimate $\\hat{\\theta}_4 = 0.207$ can be interpreted as the possible association of a higher value of the total score with greater cognitive impairment. The effect of $X_3$ (gender) is deemed not significant.\n\nFinally, with $X_1,$ $X_2,$ and $X_4$ in the model, we test for significance of the pairwise interactions terms. The hypothesis of interest is $H_0 : \\theta_5 =\\theta_6 = \\dots = \\theta_{10} =0.$ The $ASL$ for the test comes out to be $0.106,$ giving evidence for no significant pairwise interaction to be included in the model. \nThus we estimate the relevant ``true'' model in this case as $E_\\oplus(Y|\\tbfX \\t \\para) = \\mop{X_1\\theta_1 + X_2\\theta_2 + X_4\\theta_4}.$\nThe estimated average Fr\\'echet error $\\frac{1}{n} \\sum_{i=1}^n d^2(Y_i,\\hmop{X_{1i}\\hat{\\theta}_1 + X_{2i}\\hat{\\theta}_2 + X_{4i}\\hat{\\theta}_4})$ is \nquite small $(0.239).$ \n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=.9\\textwidth]{.\/plots\/obs_vs_fits_ADNI_corrplots}\n\n\n\t\\caption{The observed and fitted functional connectivity matrices for an increasing value of the single index are plotted. The estimated direction $\\boldsymbol{\\wh{\\bar{\\theta}}}$ is computed for the final model, and the estimated index $\\tbf{X}_i ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}}$ are calculated. The panels in the top row, from left to right, depict the observed functional connectivity correlation matrices for the subjects, for whom the estimated index values are the closest to the $25\\%, 50\\%,$ and $75\\%$ of the estimated index respectively. The bottom row shows the fitted functional connectivity correlation matrices for the same subjects, the link functions estimated using the nonparametric Fr\\'echet regression at the given quantiles (from left to right). Positive (negative) values are drawn in red (blue) and larger circles correspond to larger absolute values. The figure illustrates the dependence of functional connectivity on the overall index effect.}\n\t\\label{fig:ADNI:obs_vs_fits}\n\\end{figure}\n\n\nTo demonstrate the validity of the IFR method, we compute the estimated indices for the final model as $X_{1}\\hat{\\theta}_1 + X_{2}\\hat{\\theta}_2 + X_{4}\\hat{\\theta}_4,$ for   each subject and calculate the $25\\%, 50\\%,$ and $75\\%$ quantiles of the index. These come out to be $q_1 = 15.048,$ $q_2 = 16.430,$ and $q_3 = 18.250,$ respectively. We find the subjects who have their estimated index values closest to $q_1,$ $q_2,$ and $q_3$ respectively. Table~\\ref{tab:ADNI:fits} shows the details on the three subjects selected. We compare the observed and fitted functional connectivity correlation matrices for these three subjects, where the object link function is fitted by the local Fr\\'echet regression method at the estimated index values corresponding to each subject. This gives an intuitive idea of how the estimated link function at the estimated direction vector given  by $\\hmop{\\tbf{x} ^{\\top}\\boldsymbol{\\wh{\\bar{\\theta}}}}$ changes with an increasing value of the index $\\tbf{x} ^{\\top} \\boldsymbol{\\wh{\\bar{\\theta}}},$ and thus brings about the effectiveness of the IFR model. In Figure~\\ref{fig:ADNI:obs_vs_fits} we display the observed (top row) and fitted (bottom row) correlation matrices for an increasing  value of the single index at $q_1,$ $q_2,$ and $q_3$ respectively, in the columns from left to right. We observe that the fits match the general pattern of the observed matrices quite well. Indeed the Frobenius distance between the observed and the estimated matrices at $q_1,$ $q_2,$ and $q_3$ are calculated as $1.68,$ $1.10,$ and $0.79,$ respectively. A seeming tendency to have more negative correlation values in the connectivity matrices with increasing index values is an interesting finding. An overall effect of stages of the disease, age, and total score tends to influence the connectivity pattern in the brain of the Alzheimer's patients. A higher index value would imply more cognitive deficiency in this case, which matches the widely held beliefs. \n\\begin{table}[]\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|}\n\t\t\\hline\n\t\t\\begin{tabular}[c]{@{}c@{}}Subject\\\\ number\\end{tabular} &\n\t\t\\begin{tabular}[c]{@{}c@{}}Estd.\\\\ index value\\end{tabular} &\n\t\t\\begin{tabular}[c]{@{}c@{}}Stage of the\\\\ disease\\end{tabular} &\n\t\tAge &\n\t\tGender &\n\t\tTotal score \\\\ \\hline\n\t\t726 & 15.045 & 2 & 66.10 y & M & 20.33 \\\\ \\hline\n\t\t695 & 16.430 & 2 & 78.12 y & M & 14    \\\\ \\hline\n\t\t556 & 18.252 & 1 & 72.55 y & M & 51.67 \\\\ \\hline\n\t\\end{tabular}\n\t\\caption{Table showing the details on the subjects who have their estimated index values closest to the first three quantiles of the estimated index, $q_1 (15.048),$ $q_2 (16.430),$ and $q_3 (18.250),$ respectively. The subject number $726$ has his estimated index value closest to $q_1$ and so on.}\n\t\\label{tab:ADNI:fits}\n\\end{table}\n\\section{Discussion}\n\\label{sec:concl}\nThe proposed single Index Fr\\'echet Regression (IFR) model provides a new tool for the regression analysis of random object data, which are increasingly encountered in modern data analysis. Instead of a two step procedure to estimate the link function and the index parameter separately, we discuss a direct M-estimation approach. In fact, the proposed method is a combination of multivariate M-estimation (for the index vector) and Fr\\'echet regression (for the link function) that extends the regression regime to object data responses. The index parameter is recovered as the unit direction minimizing the ``residual sum of squares'' after fitting the Fr\\'echet model. In fact, any other convex loss function can be used for this purpose.  For an efficient computation, we use the \\emph{Julia} language and parallel programming to estimate the minimizer direction $\\hat{\\boldsymbol{\\theta}}$ by searching over $1000$ directions such that the Fr\\'echet variance is minimized. In a $25$ core computing system, for a sample size $n=1000,$ determining the optimal direction takes about $2$ hours, while employing a $5-$fold cross validation method to select the tuning parameters $(b,M).$\n\nIn this project, we provide the asymptotic results involving the estimation of the index parameter which could be extended for inference. In particular, single index models being a generalization of linear regression, the interpretability of the index parameter is important for testing the effect of a subset of predictors in modulating the response. The same is true in our IFR model, despite the responses being situated in a general metric space, as illustrated by the FMRI brain imaging example.\n\n\\section*{Appendix}\n\\subsection*{A.1. Technical assumptions regarding local Fr\\'echet regression}\n\\label{appen:assump}\nRecall, for any given direction $\\boldsymbol{\\theta},$ such that $\\tbfx \\t \\para = t,$ the conditional Fr\\'echet mean is given by \n\\begin{equation}\\aligned\n\\label{loc:fr:target}\n\\mop{t} = \\underset{\\omega \\in \\Omega}{\\argmin} \\ M(\\omega,t); \\quad M(\\omega,t) := \\mbb{E}{(d^2(Y,\\omega)|\\tbf{X}^{\\top} \\boldsymbol{\\theta} = t)}.\n\\endaligned\\end{equation}\nThe local Fr\\'echet regression estimate is given by \n\\begin{equation}\\aligned\n\\label{loc:fr::inter:target}\n\\hmop{t} = \\underset{\\omega \\in \\Omega}{\\argmin} \\ \\hat{L}_n(\\omega,t);\\quad  \\hat{L}_n(\\omega,t); :=\\frac{1}{n}\\sum_{i=1}^n\\wh{S}(\\tbf{X}_i ^{\\top} \\boldsymbol{\\theta}_1, t, h)d^2(Y_i,\\omega)).\n\\endaligned\\end{equation}\nLet us define the intermediate localized weighted Fr\\'echet mean as \n\\begin{equation}\\aligned\n\\label{loc:fr:estd}\n\\tmop{t} = \\underset{\\omega \\in \\Omega}{\\argmin} \\ \\tilde{L}_b(\\omega,t); \\quad \\tilde{L}_b(\\omega,t):=\\mbb{E}{(S(\\tbfX \\t \\para_1, t, h)d^2(Y,\\omega))}.\n\\endaligned\\end{equation}\nThe marginal density $f_T$ of $T := \\tbf{X} ^{\\top} \\boldsymbol{\\theta},$ for any $\\boldsymbol{\\theta} \\in \\mbb{R}^{p-1} \\text{ and } \\boldsymbol{\\theta} ^{\\top} \\boldsymbol{\\theta} \\leq 1,$ is bounded away from zero on its support $\\mathcal{T},$ i.e., $\\inf_{t\\in\\mathcal{T}}\\ f_T(t) >0.$ We require the following assumptions for as described in \\cite{chen:20}.\n\\begin{enumerate}[label=(U\\arabic*)]\n\\item \\label{ass:minUnif} \nFor all $t \\in \\mathcal{T},$ \nthe minimizers $\\mop{t}$, $\\hmop{t}$, and $\\tmop{t}$ exist and are unique, the latter two almost surely. In addition, for any $\\varepsilon>0$, \n\\begin{equation}\\begin{gathered}\n\\inf_{t \\in \\mathcal{T}}\n\\inf_{d(\\mop{t},\\omega)>\\varepsilon} [M(\\omega,t)-M(\\mop{t},t)]>0,\\\\\n\\liminf_{b \\to 0} \\inf_{t \\in \\mathcal{T}} \n\\inf_{d(\\omega, \\tmop{t}) > \\varepsilon} [\\tilde{L}_b(\\omega,t) - \\tilde{L}_b(\\tmop{t},t)]>0,\n\\end{gathered}\\end{equation}\nand there exists $c = c(\\varepsilon)>0$ such that\n\\begin{equation}\\begin{gathered}\nP\\left(\\inf_{t \\in \\mathcal{T}} \\inf_{d(\\hmop{t},\\omega) >\\varepsilon} [\\hat{L}_n(\\omega,t) - \\hat{L}_n(\\hmop{t},t)]\\geq c \\right) \\to 1.\n\\end{gathered}\\end{equation}\n\\item \\label{ass:entropyUnif} \nLet $\\mathcal{B}_r(\\mop{t}) \\subset \\Omega$ be a ball of radius $r$ centered at $\\mop{t}$ and $\\mathcal{N}(\\varepsilon,\\mathcal{B}_r(\\mop{t}),d)$ be its covering number using balls of radius $\\epsilon.$ Then\n\\begin{equation}\\begin{gathered}\n\\underset{r\\to 0+}{\\lim} \\int_0^1 \\underset{t \\in \\mathcal{T}}{\\sup}  \\sqrt{1 + \\log \\mathcal{N}(r\\varepsilon,\\mathcal{B}_r(\\mop{t}),d)} d \\epsilon = O(1).\n\\end{gathered}\\end{equation}\n\\item \\label{ass:curvatureUnif} There exists $r_1,r_2>0,$ $c_1,c_2>0,$ and $\\beta_1,\\beta_2>1$ such that\n\\begin{equation}\\begin{gathered}\n\\inf_{t \\in \\mathcal{T}} \\ \\inf_{d(\\mop{t},\\omega)<r_1}  [M(\\omega, t) - M(\\mop{t},t) - c_1d^2(\\omega,\\mop{t})^{\\beta_{1}]} \\geq 0,\\\\\n\\underset{b \\to 0}{\\liminf} \\ \\inf_{t \\in \\mathcal{T}} \t\\inf_{d(\\omega, \\tmop{t}) <r_2}\\ [\\tilde{L}_b(\\omega, t) - \\tilde{L}_b(d(\\tmop{t}, t) - c_2d^2(\\omega, \\tmop{t})^{\\beta_2}] \\geq 0.\n\\end{gathered}\\end{equation}\n\\end{enumerate}\nFurthermore, we require the following assumptions.\n\\begin{enumerate}[label = (R\\arabic*)]\n\\item \\label{ass:ker} The kernel $K$ is a probability density function, symmetric around zero, uniformly continuous on $\\mbb{R}$ such that $\\int_\\mbb{R} K(x)^j x^k < \\infty,$ for $r,s=1,\\dots 6.$  The derivative $K'$ exists and is bounded on the support of $K$, i.e., $\\sup_{x : K(x)>0}|K'(x)| < \\infty.$ Additionally, $\\int_\\mbb{R} x^2 |K'(x)|\\sqrt{|x\\log|x||} dx <\\infty.$\n\\item \\label{ass:jointdtn} \nThe marginal density $f_T$ of $T = \\tbfX \\t \\para$ for any given unit direction $\\boldsymbol{\\theta}$ and the conditional densities $f_{T|Y}(\\cdot,y)$ of $T = \\tbfX \\t \\para$ given $Y= y$ exist and are twice continuously differentiable on the interior of $\\mathcal{T}$, the latter for all $y \\in\\Omega$. The marginal density $f_T$ is bounded away from zero on its support $\\mathcal{T},$ i.e., $\\inf_{t\\in\\mathcal{T}} f_T(t) >0.$\nThe second-order derivative $f_T''$ is uniformly bounded, $\\sup_{^{\\top}}|f_T''(t)|<\\infty$. \nThe second-order partial derivatives $(\\partial^2 f_{T|Y}\/\\partial t^2)(\\cdot,y)$ are uniformly bounded, $\\sup_{y,t} |(\\partial^2 f_{T|Y}\/\\partial t^2)(\\cdot,y)| < \\infty$. \nAdditionally, for any open set $U\\subset \\Omega$, $P(Y\\in U | T = t)$ is continuous as a function of $t$. \n\\end{enumerate}\n\\bibliographystyle{apalike}\n\n\\section*{\\bf \\sf 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{\\underset{\\varepsilon \\to 0}{\\lim}\\ }\n\\def \\underset{h \\to 0}{\\lim}\\ {\\underset{h \\to 0}{\\lim}\\ }\n\\def \\frac{1}{\\eps}{\\frac{1}{\\varepsilon}}\n\\def \\frac{1}{\\eps^2}{\\frac{1}{\\varepsilon^2}}\n\\def \\boldsymbol{\\theta}_\\mathbf{\\ast}{\\boldsymbol{\\theta}_\\mathbf{\\ast}}\n\\def \\boldsymbol{\\theta}^\\mathbf{\\ast}{\\boldsymbol{\\theta}^\\mathbf{\\ast}}\n\\def \\triangledown{\\triangledown}\n\\def \\tilde{\\Dtri}{\\tilde{\\triangledown}}\n\\def \\wh{\\Dtri}{\\widehat{\\triangledown}}\n\n\\def \\boldsymbol{\\alpha}{\\boldsymbol{\\alpha}}\n\\def \\boldsymbol{\\beta}{\\boldsymbol{\\beta}}\n\\def g_{1\\oplus} {g_{1\\oplus}}\n\\def g_{2\\oplus} {g_{2\\oplus}}\n\\def h_\\oplus {h_\\oplus}\n\\def \\wh{\\sigma}_{rs}(\\boldsymbol{\\theta_0}){\\widehat{\\sigma}_{rs}(\\boldsymbol{\\theta_0})}\n\\def \\wh{S}{\\widehat{S}}\n\\def \\wh{\\Sigma}(\\boldsymbol{\\theta_0}){\\widehat{\\Sigma}(\\boldsymbol{\\theta_0})}\n\\def \\wh{\\Lambda}(\\boldsymbol{\\theta_0}){\\widehat{\\Lambda}(\\boldsymbol{\\theta_0})}\n\n\\def \\bm{\\mathit{\\Delta}}H {\\bm{\\mathit{\\Delta}}H}\n\\def \\bm{\\mathit{\\Delta}}\\tilde{V}_n {\\bm{\\mathit{\\Delta}}\\tilde{V}_n}\n\\def V_n{V_n}\n\\def \\bm{\\mathit{\\Delta}}\\Vn {\\bm{\\mathit{\\Delta}}V_n}\n\n\\def \\bm{\\mathit{\\Delta^2}}H {\\bm{\\mathit{\\Delta^2}}H}\n\\def \\bm{\\mathit{\\Delta^2}}\\tilde{V}_n {\\bm{\\mathit{\\Delta^2}}\\tilde{V}_n}\n\\def V_n{V_n}\n\\def \\bm{\\mathit{\\Delta^2}}\\Vn {\\bm{\\mathit{\\Delta^2}}V_n}\n\\def \\overset{P}{\\rightarrow}{\\overset{P}{\\rightarrow}}\n\\def \\overset{D}{\\rightarrow}{\\overset{D}{\\rightarrow}}\n\\def   \\underset{\\boldsymbol{\\theta} \\in \\Theta}{\\sup}{  \\underset{\\boldsymbol{\\theta} \\in \\Theta}{\\sup}}\n\\def \\underset{\\omega \\in \\Omega}{\\argmin}{\\underset{\\omega \\in \\Omega}{\\argmin}}\n\\def \\underset{\\boldsymbol{\\bar{\\theta}} \\in\\ \\Theta}{\\argmin}{\\underset{\\boldsymbol{\\bar{\\theta}} \\in\\ \\Theta}{\\argmin}}\n\\def \\underset{\\boldsymbol{\\theta} : \\boldsymbol{\\theta} \\in \\Theta}{\\argmin}{\\underset{\\boldsymbol{\\theta} : \\boldsymbol{\\theta} \\in \\Theta}{\\argmin}}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\argmin}{argmin}\n\n\\def\\msr{B}{\\mathscr{B}}\n\\def\\mbb{E}{\\mathbb{E}}\n\\defP{P}\n\\def\\mbb{R}{\\mathbb{R}}\n\\def\\mbb{N}{\\mathbb{N}}\n\\def\\ntnum_+{\\mbb{N}_+}\n\\def\\mbb{Z}{\\mathbb{Z}}\n\\defO{O}\n\\defo{o}\n\\defO_P{O_P}\n\\defo_P{o_P}\n\\def\\rightarrow\\infty{\\rightarrow\\infty}\n\\def\\indicator#1{\\mathbb I\\left(#1\\right)}\n\\def\\ball#1#2{B_{#2}(#1)}\n\n\\def^{\\top}{^{\\top}}\n\\def {\\boldsymbol{\\bar{\\theta}_0}}{{\\boldsymbol{\\bar{\\theta}_0}}}\n\\def \\tbf{x} \\t {\\boldsymbol{\\bar{\\theta}_0}}{\\tbf{x} ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}}\n\\def \\tbf{X} \\t {\\boldsymbol{\\bar{\\theta}_0}}{\\tbf{X} ^{\\top} {\\boldsymbol{\\bar{\\theta}_0}}}\n\\def \\boldsymbol{\\tilde{\\bar{\\theta}}}{\\boldsymbol{\\tilde{\\bar{\\theta}}}}\n\\def \\boldsymbol{\\wh{\\bar{\\theta}}}{\\boldsymbol{\\widehat{\\bar{\\theta}}}}\n\\def \\boldsymbol{\\bar{\\theta}}{\\boldsymbol{\\bar{\\theta}}}\n\n\\def\\boldsymbol{\\theta_0}{\\boldsymbol{\\theta_0}}\n\\def\\tilde{\\boldsymbol{\\theta}}{\\tilde{\\boldsymbol{\\theta}}}\n\\def\\boldsymbol{\\theta}{\\boldsymbol{\\theta}}\n\\def\\boldsymbol{\\theta}^{\\mathbf{(1)}}{\\boldsymbol{\\theta}^{\\mathbf{(1)}}}\n\\def\\tbf{X}{\\textbf{X}}\n\\def\\tbf{x}{\\textbf{x}}\n\\def m(\\cdot){m(\\cdot)}\n\\def \\hat{m}(\\cdot){\\hat{m}(\\cdot)}\n\\defK{K}\n\\def \\hat{\\boldsymbol{\\theta}}{\\hat{\\boldsymbol{\\theta}}}\n\\def \\tbfX \\t \\true{\\tbf{X} ^{\\top} \\boldsymbol{\\theta_0}}\n\\def \\tbfx \\t \\true{\\tbf{x} ^{\\top} \\boldsymbol{\\theta_0}}\n\\def \\tbfX \\t \\para{\\tbf{X} ^{\\top} \\boldsymbol{\\theta}}\n\\def \\tbfx \\t \\para{\\tbf{x} ^{\\top} \\boldsymbol{\\theta}}\n\\def \\tilde{Y}_l{\\tilde{Y}_l}\n\\def \\tilde{\\tbfX}_l{\\tilde{\\tbf{X}}_l}\n\\def \\tilde{L}_b{\\tilde{L}_b}\n\\def \\hat{L}_n{\\hat{L}_n}\n\\def \\tilde{V}_n{\\tilde{V}_n}\n\n\n\\newcommand \\mop[1]{m_\\oplus(#1)}\n\\newcommand \\hmop[1]{\\hat{m}_\\oplus(#1)}\n\\newcommand \\tmop[1]{\\tilde{m}_\\oplus(#1)}\n\\newcommand \\loomop[1] {\\hat{m}_{\\oplus(-i)}(#1)}\n\n\n\\newcommand{\\ltwoNorm}[2][]{%\n\t\\ifthenelse{ \\equal{#1}{} }\n\t{\\ensuremath{\\|#2\\|}}\n\t{\\ensuremath{\\|#2\\|_{#1}}}\n} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe study of heavy quarkonia, i.e. mesons build from heavy (with $m_Q\\gg\\Lambda_\\mathrm{QCD}$) quark-antiquark pair is a very interesting task from both theoretical and experimental points of view. The theoretical model that is usually used for theoretical description of these particles is the Nonrelativistic Quantum Chromodynamic (NRQCD) \\cite{Bodwin:1994jh}, that allows one to describe with pretty good accuracy the processes of charmonia (e.g. ${J\/\\psi}$ or $\\chi_{cJ}$ or bottomonia (e.g. $\\Upsilon$, $\\chi_{bJ}$) production on hadronic colliders as well as various decays of these particles \\cite{Aad:2011sp,Chatrchyan:2011kc,TheATLAScollaboration:2013bja,Butenschoen:2012qr,Butenschoen:2012px, Likhoded:2016zmk, Aaij:2016bqq}. It should be mentioned, however, that NRQCD predictions depend on a number of parameters (so called NRQCD matrix elements), whose numerical values are determined phenomenologically from analysis of available experimental data \\cite{Likhoded:2014kfa,Abe:1997yz,Abulencia:2007bra,Chatrchyan:2012ub,LHCb:2012ac,Aaij:2013dja}.\n\nIn addition to mentioned above processes there are also reactions that can be considered in almost model independent way. Among such processes one can name, for example, lepton pair production in $\\chi_{cJ}\\to{J\/\\psi}\\ell\\ell$ and $\\chi_{bJ}\\to\\Upsilon\\ell\\ell$ decays. In \\cite{Faessler:1999de} it was shown that the branching fractions of these decays and distributions over the invariant mass of $(\\ell\\ell)$ pair can be calculated using very general assumptions on the basis of experimentally known branching fractions of the corresponding radiative decays $\\chi_{cJ}\\to{J\/\\psi}\\gamma$ and $\\chi_{bJ}\\to\\Upsilon\\gamma$ (see also \\cite{Eichten:1979ms,Brambilla:2004wf, Barnes:2005pb,Cao:2016xqo}). Currently only $\\chi_{c1,2}\\to{J\/\\psi} ee$ process was studied experimentally \\cite{Ablikim:2017kia}. It could be interesting also to consider muon pair production in $\\chi_{cJ}\\to{J\/\\psi}\\mu\\mu$ and $\\chi_{bJ}\\to\\Upsilon\\mu\\mu$ decays. In the current paper we consider theoretically these reactions.\n\n\n\n\nIn the recent experimental paper \\cite{Ablikim:2017kia} BESIII Collaboration analysed electron-positron pair production in $\\chi_{c1,2}$-meson decays\n\\begin{align}\n  \\label{eq:decay}\n  \\chi_{cJ} &\\to{J\/\\psi} e^+ e^-.\n\\end{align}\n In this article branching fractions of the named reactions and distributions over invariant mass of the lepton pair were presented. It is interesting to note, that these results can be obtained from the branching fractions of the radiative decays $\\chi_{cJ}\\to{J\/\\psi}\\gamma$ using a very simple relation. In paper \\cite{Faessler:1999de} for example, it was shown, that $q^2=m_{ee}^2$ distribution of the (\\ref{eq:decay}) decay is equal to\n \\begin{align}\n   \\label{eq:II}\n   \\frac{d\\Br{J}{{\\ell\\ell}}}{dq^2} &= \n                               \\frac{\\alpha}{3\\pi q^2}\\frac{\\lambda(M_\\chi,M_\\psi,\\sqrt{q^2})}{\\lambda(M_\\chi,M_\\psi,0)} \n                               \\left(1+\\frac{2m_\\ell^2}{q^2}\\right)\\sqrt{1-\\frac{4m_\\ell^2}{q^2}}\n                               \\Br{J}{\\gamma},\n \\end{align}\nwhere $\\Br{J}{{\\ell\\ell}}$ and $\\Br{J}{\\gamma}$ are the branching fractions of $\\chi_{cJ}\\to{J\/\\psi}{\\ell\\ell}$ and $\\chi_{cJ}\\to{J\/\\psi}\\gamma$ decays respectively and\n\\begin{align}\n  \\lambda(M,m_1,m_2) &= \\sqrt{1-\\left(\\frac{m_1+m_2}{M}\\right)^2} \\sqrt{1-\\left(\\frac{m_1-m_2}{M}\\right)^2}\n\\end{align}\nis the velocity of the final particle in $M\\to m_1m_2$ decay. It should be noted that this result is almost model independent since it is based on the gauge invariance of $\\chi_c\\to{J\/\\psi}\\gamma^*$ vertex. The only assumption that was made is that one can neglect the form factors' dependence on photon virtuality. This assumption seems pretty reasonable since according to energy conservation $q^2<(M_\\chi-M_\\psi)^2\\sim 0.025\\,\\mathrm{GeV}^2$, that is much smaller than the typical hard scale $\\sim M_\\psi^2\\sim 10\\,\\mathrm{GeV}^2$. It is clear from presented in Fig.~\\ref{fig:hQEE} figures, that obtained by BESIII Collaboration experimental data are in good agreement with theoretical predictions (\\ref{eq:II}). The same is true also for the integrated values of the branching fractions: theoretical predictions in comparison with BESIII results are\n\\begin{align}\n \\fBr{0}{ee} &= 8.1\\times 10^{-3},\\qquad   \\left(\\fBr{0}{ee}\\right)_\\mathrm{exp} = (9.5\\pm1.9\\pm0.7)\\times 10^{-3},\\\\\n \\fBr{1}{ee} &= 8.6\\times 10^{-3},\\qquad   \\left(\\fBr{1}{ee}\\right)_\\mathrm{exp} = (10.1\\pm0.3\\pm0.5)\\times 10^{-3},\\\\\n \\fBr{2}{ee} &= 8.7\\times 10^{-3},\\qquad   \\left(\\fBr{2}{ee}\\right)_\\mathrm{exp} = (11.3\\pm0.4\\pm0.5)\\times 10^{-3}.\n\\end{align}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{hQ_EE.pdf}\n  \\caption{$Q$ distribution in comparison with BESSIII data \\cite{Ablikim:2017kia, Zhang:PC}}\n  \\label{fig:hQEE}\n\\end{figure}\n\n\n\nIt could be interesting to use described above formalism for muon pair production in $\\chi_{cJ}\\to{J\/\\psi}\\mu\\mu$ decays. It is clear that in this case we can also use (\\ref{eq:decay}) relation to describe $q^2$ distributions, the corresponding results are shown in figure \\ref{fig:hQ2MM}(a). One can easily see that the form of the distributions depends strongly on the spin of the initial particle and differs significantly from the presented in figure \\ref{fig:hQEE} curves, but all these distributions are given by all universal relation (\\ref{eq:II}), only the difference in participating particles' masses is important. As for the integrated branching fractions, we have\n\\begin{align}\n\\fBr{0}{\\mu\\mu} &= 2.2\\times 10^{-4},\n\\quad \\fBr{1}{\\mu\\mu}= 5.1 \\times 10^{-4},\n\\quad\\fBr{2}{\\mu\\mu}= 6.4\\times 10^{-4}.\n\\end{align}\nUsing the same approach we have calculated also branching fractions of $\\chi_{bJ}\\to\\Upsilon\\mu\\mu$ decays:\n\\\n\\begin{align}\n  \\fBrb{0}{\\mu\\mu} &= 4.7\\times 10^{-4},\n\\quad \\fBrb{1}{\\mu\\mu}= 5.7 \\times 10^{-4},\n\\quad\\fBrb{2}{\\mu\\mu}= 6.2\\times 10^{-4}.\n\\end{align}\nDistributions over leptons' invariant mass are shown in Figure \\ref{fig:hQ2MM}(b).\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{Q2_MM}\n  \\caption{Normalized $Q^2$ distribution for $\\chi_{cJ}\\to J\/\\psi\\mu\\mu$ (left figure) and $\\chi_{bJ}\\to\\Upsilon\\mu\\mu$ (right figure) decays}\n  \\label{fig:hQ2MM}\n\\end{figure}\n\n\n\n\nIf one wants to obtain other types of distributions more detailed information on the physics of the underlying process is required. In our work we use the following expressions for $P$-wave quarkonia  decay verticies \\cite{Baranov:2011ib}:\n\\begin{align}\n  \\A{0} &= g_0 M_V \\left( g_{\\mu\\nu} - \\frac{p_\\mu q_\\nu}{(pq)}\\right)\\epsilon_V^\\nu \\epsilon^{(\\gamma)}_\\mu, \\\\\n  \\A{1} &= g_1 e_{\\mu\\nu\\alpha\\beta} q^\\nu\\epsilon_V^\\alpha\\epsilon_\\chi^\\beta \\epsilon^{(\\gamma)}_\\mu,\\\\\n  \\A{2} &= \\frac{g_2}{M_V} p^\\mu \\epsilon_V^\\alpha \\epsilon_{\\chi_{c2}}^{\\alpha\\beta}\\left[q_\\mu\\epsilon^{(\\gamma)}_\\beta-q_\\beta\\epsilon^{(\\gamma)}_\\mu\\right],\n\\end{align}\nwhere $\\epsilon^{(\\gamma)}$, $\\epsilon_V$, and $\\epsilon_\\chi$ are polarization vectors of final photon, vector quarkonia and initial $\\chi_Q$ meson respectively, while  dimensionless coupling constants $g_{0,1,2}$ can be determined from experimental values of the corresponding radiative decays.\nUsing these expressions it is easy to obtain presented in Fig.~\\ref{fig:hM2PsiM} distributions over squared invariant of $\\psi\\mu$ pair.\n\nExperimentally vector quakonium  is detected via its leptonic decay $V\\to\\mu^+\\mu^-$. In order to obtain the information of the polarization picture of\n\\begin{align}\n  \\label{eq:dec2}\n  \\chi_{QJ} &\\to V\\mu^+\\mu^- \\to (\\mu^+\\mu^-)\\mu^+\\mu^-\n\\end{align}\nit could be interesting to study also distributions over squared invariant mass of the same-signed final leptons. These distributions are shown in Fig.~\\ref{fig:hm2MM}.\n\n\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{PsiK.pdf}\n  \\caption{Normalized $m^2_{V\\mu}$ distribution for $\\chi_c$- and $\\chi_b$ meson decays (left and right panels respectively)}\n  \\label{fig:hM2PsiM}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{K1KK1.pdf}\n  \\caption{Normalized \n$m_{\\mu^+\\mu^+}^2$ distribution for $\\chi_c$- and $\\chi_b$ meson decays (left and right panels respectively)}\n  \\label{fig:hm2MM}\n\\end{figure}\n\nThe author would like to thank I. Belyaev for useful discussions and J. Zhang for providing BESIII experimental data.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\nLet $K$ be a number field, $\\mathcal{O}_K$ be its ring of integers and $\\mathfrak{A}\\subseteq\\mathcal{O}_K$ be a principal ideal.\nOur aim is to investigate the following property of $\\mathfrak{A}$:\n\n\\begin{equation}\\label{eq:property}\\hbox{ $\\mathfrak{A}$ admits a generator $x$ such that $|\\sigma(x)|\\geq 1$ for every embedding $\\sigma: K\\to\\mathbb{C}$}.\n\\end{equation}\n\nWe shall call such a generator a \\emph{large} element of $\\mathcal{O}_K$ (\\emph{strictly large} when the strict inequality holds), and we shall call \\emph{large} every ideal satisfying property \\eqref{eq:property}.\n\nWe shall see that all but finitely many principal ideals $\\mathfrak{A}$ are strictly large; in particular this happens  when the \\emph{logarithmic norm} $n(\\mathfrak{A})=\\frac {N(\\mathfrak{A})}{[K:\\mathbb{Q}]}$ exceeds the \\emph{covering radius} of the lattice of units with respect to the $L_\\infty$ norm. Moreover, every non-trivial ideal becomes large in a suitable finite extension of $K$.\n\nIt is possible to relate the notion of largeness of a principal ideal to the Weil height of its generators. Therefore, lower bounds of the Weil height on $K$, as given by the Bogomolov property, may help to prove that some ideals are not large. \nTo this aim, we shall state some inequalities concerning \nthe covering radius, the regulator  and  the Weil height of systems of multiplicatively independent units of $K$. We shall apply the technique described above in some concrete example.\n\nAs soon as the group of units of $K$ is known, it is relatively easy to decide if a principal ideal is large, and we give the corresponding algorithm. Then, we shall present\nthe results of applying it to some particular ideal in cyclotomic fields.\n\nAs a last application, we shall define the notion of floor function for $K$ relatively to $\\mathfrak{A}$ and show that condition \\eqref{eq:property} allows to explicitly construct a bounded floor function.\nThis turns out to be a good property for the the resulting continued fractions (\\cite{CapuanoMurruTerracini2022}).\n\n\\section{The largeness property and general results}\\label{sect:theproblem}\nLet $K$ be a number field of degree $d$.\nWe denote by $\\calo_K$ the ring of integers of $K$ and $\\calo_K^\\times$ the group of units.\nThe number field $K$ has $r_1$ real embeddings $\\sigma_1,\\dots,\\sigma_{r_1}$\nand $2r_2$ complex embeddings $\\sigma_{r_1+1},\\dots,\\sigma_{r_1+2r_2}$,\nwhere $r_1+2r_2 = d$ and $\\sigma_{r_1+i}$ and $\\sigma_{r_1+r_2+i}$\nare conjugates for $1\\leq i \\leq r_2$. We denote by $\\Sigma$ the whole set of Archimedean embeddings. We shall denote by $|\\cdot |$ the standard complex absolute value.\nAn element $x \\in K$ therefore has\n$r_1+r_2$ Archimedean absolute values, namely $|\\sigma_1(x)|, \\dots, |\\sigma_{r_1+r_2}(x)|$,\nand we have $|\\sigma_{r_1+i}(x)| = |\\sigma_{r_1+r_2+i}(x)|$\nfor all $1\\leq i \\leq r_2$.\nWe put $s=r_1+r_2$, $r=s-1$.\n\nLet\n\\begin{align*} \\iota: K& \\longrightarrow  \\mathbb{R}^{r_1}\\times \\mathbb{C}^{r_2}\\\\\n\\lambda &\\longmapsto (\\sigma_1(\\lambda),\\ldots,\\sigma_{r_{1}}(\\lambda),\\sigma_{r_1+1}(\\lambda),\\ldots,\\sigma_{r_1+r_2}(\\lambda))\n\\end{align*} be the canonical embedding of $K$, and\n$$\\ell: K^\\times \\to \\mathbb{R}^{r_1+r_2}$$ be the logarithmic embedding, i.e., the composition $\\mathcal{L}\\circ\\iota$ where \n\\begin{align*} \\mathcal{L}: (\\mathbb{R}^\\times)^{r_1}\\times (\\mathbb{C}^\\times)^{r_2}& \\longrightarrow  \\mathbb{R}^{r_1}\\times \\mathbb{R}^{r_2}\\\\\n(x_1,\\ldots, x_{r_1}, y_1,\\ldots, y_{r_2})  &\\longmapsto (\\log|x_1|,\\ldots, \\log|x_{r_1}|, 2\\log|y_1|,\\ldots, 2\\log|y_{r_2}|).\n\\end{align*} \\\\\nFor $\\mathbf{x}=(x_1,\\ldots, x_{r_1},y_1,\\ldots, y_{r_2})\\in \\mathbb{R}^{r_1}\\times \\mathbb{C}^{r_2}$, let us define\n$$N(\\mathbf{x})=\\prod_{i=1}^{r_1}|x_i \n|\\cdot \\prod_{j=1}^{r_2}|y_j|^2;$$ \nthen, $N(\\iota(a))=|N_{K\/\\mathbb{Q}}(a)|$ for every $a\\in K.$\nThe {\\em absolute norm} of $x\\in K$ is equal to the absolute value of the norm of $x$.\\\\\nWe shall denote $\\Lambda_K=\\ell(\\mathcal{O}_K^\\times)$; it is a lattice of rank $r$ in $\\mathbb{R}^s$ by Dirichlet's Unit Theorem. We recall also that  the \\emph{regulator} $R_K$ of $K$ is the determinant of any submatrix of order $r$ of the $r\\times (r+1)$ matrix whose rows are $\\ell(u_1),\\ldots, \\ell(u_r)$ \nfor a system $u_1,\\ldots, u_r$ of fundamental units for $K$. If $V_K$ is  the volume  of a fundamental domain for $\\Lambda_K$ then the relation $V_K=\\sqrt{s}R_K$ holds.\n\n\\begin{definition}\\ \\begin{itemize}\n    \\item[a)] We say that $x\\in \\mathcal{O}_K$ is \\emph{large} (resp. \\emph{strictly large}) if\n$|\\sigma(x)|\\geq  1$ (resp. $|\\sigma(x)|> 1$) for every $\\sigma\\in\\Sigma$ . \n\\item[b)] An ideal $\\mathfrak{A}\\subseteq\\mathcal{O}_K$ is \\emph{large} (resp. \\emph{strictly large}) if it principal and has a large (resp. strictly large) generator.\n\\end{itemize}  \n\\end{definition}\n\nFor $x \\in \\mathcal{O}_K$\nas in the above definition,\nwe observe that\n$x$ is large in $\\mathcal{O}_K$ if and only if $x$ is large in $\\mathcal{O}_L$ for any finite extension $L$ of $K$.\nFor the ideal $\\mathfrak{A}$\nit is true that if it is large in $K$, then\nthe ideal $\\mathfrak{A} \\mathcal{O}_L$ is large in $\\mathcal{O}_L$, but the converse is not true.\n\nThen the following definition makes sense and extends the notion of largeness to possibly infinite extensions:\n\\begin{definition}\nLet $L$ be an infinite extension of $K$.\nFor an element $x \\in \\mathcal{O}_K$\nand an ideal $\\mathfrak{A} \\subseteq \\mathcal{O}_K$, we say that:\n\\begin{itemize}\n\\item[a)]\n$x$ is \\emph{large} (resp. \\emph{strictly large}) in $L$ if there is a number field $K'$ with $K \\subseteq K'\\subseteq L$ such that $x$ is large (resp. strictly large) in $K'$.\n\\item[b)]\n$\\mathfrak{A}$ is \\emph{large} (resp. \\emph{strictly large}) in $L$ if there is a number field $K'$ with $K \\subseteq K'\\subseteq L$ such that $\\mathfrak{A} \\mathcal{O}_{K'}$ is large (resp. strictly large) in $K'$.\n\\end{itemize}\n\\end{definition}\n\nSince every unit in $\\mathcal{O}_K$ has norm $\\pm 1$, a unit $x$ is large if and only if $|\\sigma(x)|=1$ for every $\\sigma\\in \\Sigma$; by a theorem of Kronecker \\cite{Kronecker1857}, this happens if and only if $x$ is a root of unity. Therefore no unit can be strictly large. So every strictly large element $x$ in $\\mathcal{O}_K$ must satisfy $|N_{K\/\\mathbb{Q}}(x)|\\geq 2$.\\\\\nWe shall see in Proposition \\ref{prop:normalarge} that almost all principal ideals in $\\mathcal{O}_K$ are (strictly) large. In order to prove this fact we need to recall some terminology from lattice theory. \n\nLet $\\Lambda$ be a lattice in $\\mathbb{R}^n$ of rank $r$ and for a real number $p\\in [1,\\infty)\\cup\\{\\infty\\}$ let $|| \\cdot||_p$ be the norm $L_p$ in $\\mathbb{R}^n$. The \\emph{distance function} relatively to $p$ is by definition\n    $$\\rho_p(\\mathbf{v},\\Lambda)=\\min_{\\mathbf{w}\\in\\Lambda} ||\\mathbf{v}-\\mathbf{w}||_p.$$\n    The \\emph{covering radius} of $\\Lambda$ with respect to $|| \\cdot||_p$ is\n    $$\\rho_p(\\Lambda)=\\sup_{\\mathbf{v}\\in \\mathrm{span}(\\Lambda)} \\rho_p(\\mathbf{v},\\Lambda).$$\n     Balls of radius $\\rho_p(\\Lambda)$ centered around all lattice points cover the whole space $\\mathrm{span}(\\Lambda)$.\\\\\n    By the well known inequality\n     \\begin{equation}\\label{eq:disuglp} ||\\mathbf{v}||_p\\leq ||\\mathbf{v}||_r\\leq n^{\\frac 1 r-\\frac 1 p}||\\mathbf{v}||_p\\quad \\hbox{for } \\infty\\geq p\\geq r,\\end{equation}\n     we get \n    \\begin{equation}\\label{eq:disugcovrad} \\rho_p(\\Lambda)\\leq \\rho_r(\\Lambda) \\leq n^{\\frac 1 r-\\frac 1 p}\\rho_p(\\Lambda)\\quad \\hbox{ for } \\infty\\geq p\\geq r.\\end{equation} \n    If $K$ is a number field we shall write $\\rho_p(K)$ instead of $\\rho_p(\\Lambda_K)$.\n    \n    For every algebraic number $x\\in\\overline{\\mathbb{Q}}^\\times$ we define the \\emph{logarithmic norm}\n    $$n(x)=\\frac{\\log|N_{\\mathbb{Q}(x)\/\\mathbb{Q}}(x)|}{[\\mathbb{Q}(x):\\mathbb{Q}]}.$$\n    Analogously, if $\\mathfrak{A}\\subseteq \\mathcal{O}_K$ is any non-zero ideal, we write\n    $$n(\\mathfrak{A})=\\frac{\\log|N_{K\/\\mathbb{Q}}(\\mathfrak{A})|}{[K:\\mathbb{Q}]}.$$\n    Notice that $$n(x)=\\frac{\\log|N_{K\/\\mathbb{Q}}(x)|}{[K:\\mathbb{Q}]},$$\n    for every finite extension $K$ of $\\mathbb{Q}(x)$; moreover\n    $$n(x)=n(ux),$$\n    for every  algebraic unit $u\\in\\overline{\\mathbb{Q}}$. \n    Then $n(a)=n(a\\mathcal{O}_K)$\n     depends only on the principal ideal generated by $a$ in the ring of integers of every number field containing $a$.\\\\\n     We also observe that $n:\\overline{\\QQ}^\\times\\to \\mathbb{R}$ is a morphism; in particular $n(x^k)=kn(x)$  for every $k\\in\\mathbb{N}$.\n     We also have $n(x) \\ge 0$ when $x$ is an algebraic integer.\n\\begin{proposition}\\label{prop:normalarge}  \n Every principal ideal $\\mathfrak{A}$ of $\\mathcal{O}_K$ such that $n(\\mathfrak{A})>\\rho_\\infty(K)$ is strictly large.\\\\\n Therefore all but finitely many integral principal ideals of $\\mathcal{O}_K$ are strictly large.\n\\end{proposition}\n\\begin{proof}\nLet $x\\in \\mathcal{O}_K$ be a generator of $\\mathfrak{A}$ and put $N=|N_{K\/\\mathbb{Q}}(x)|$. The image of units $\\ell(\\mathcal{O}_K^\\times)$ is a lattice in  $\\mathbb{R}^s$ of rank $r=s-1$; it spans the hyperplane $\\mathcal{H}$ of $\\mathbb{R}^s$ with equation $\\sum_{i=1}^{r_1} x_i+\\sum_{i=1}^{r_2}y_i=0$. The vector $$\\mathbf{y}=\\ell(x)-\\frac 1 d \\log(N)(1,\\ldots,1,2,\\ldots,2)$$ lies on $\\mathcal{H}$ . Let $\\rho=\\rho_\\infty(K)$ and assume $n(x)>\\rho$; by definition of covering radius, there exists $u\\in\\mathcal{O}_K^\\times$ such that $||\\mathbf{y}+\\ell(u)||_\\infty \\leq \\rho$. This means that $|\\log|\\sigma(ux)|-n(x)|\\leq \\rho$ for every Archimedean embedding $\\sigma$ of $K$, so that\n$$|\\log|\\sigma(ux)||\\geq n(x)-\\rho> 0.$$\nThe second assertion follows from the fact that the ideals of norm $\\le\\rho$ are finitely many.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:esisteL} Every non-trivial integral ideal $\\mathfrak{A} \\subsetneq \\mathcal{O}_K$ is strictly large in $\\overline{\\mathbb{Q}}$.\n\\end{proposition}\n\\begin{proof} The statement is a consequence of \\cite[Th\u00e9or\u00e8me 5.1]{BergeMartinet1989}, here we present a more direct proof.\nFirst of all, by class field theory, it is well known that $\\mathfrak{A}$ becomes principal in a suitable finite extension $K'$ of $K$.\nWe have $\\mathfrak{A} \\mathcal{O}_{K'} = x \\mathcal{O}_{K'}$ for some $x \\in \\mathcal{O}_{K'}$.\nBy Proposition \\ref{prop:normalarge} there exists a positive integer $j$ such that $x^j$ is strictly large in $K'$. Let $u\\in \\mathcal{O}_{K'}^\\times$ be such that $|\\sigma(u x^j)|>1$ for every embedding $\\sigma:K'\\to\\mathbb{C}$. Let $L=K'(\\omega)$ where $\\omega^j=u$. Let $\\tau:L\\to\\mathbb{C}$ be any embedding and $\\sigma$ be the restriction of $\\tau$ to $K'$. Then\n$$|\\tau(\\omega x)|^j=|\\sigma(ux^j)|>1$$\nso that $|\\tau(\\omega x)|>1$.\n\\end{proof}\n\nBy looking at the proof of Proposition \\ref{prop:esisteL}, we see that a uniform and stronger version holds true.\nFor every number field $K$ and every positive $j\\in\\mathbb{N}$, we denote by $K_j$ the field obtained from $K$ by adding the $j$-th roots of every unit of $K$; it is a finite extension of $K$ by Dirichlet's Unit Theorem.\n\\begin{proposition}\\label{prop:esisteLuniforme}\nLet $K$ be a number field, and let $j> \\frac{\\rho_\\infty(K)[K:\\mathbb{Q}]}{\\log2}$. Every non-trivial principal ideal $\\mathfrak{A} \\subsetneq \\mathcal{O}_K$ is strictly large in $K_j$.\n  \\end{proposition}\n  \\begin{proof}\n  Let $x \\in \\mathcal{O}_K$ be a generator of $\\mathfrak{A}$.\n  We have $n(x)\\geq \\frac{\\log 2}{[K:\\mathbb{Q}]}$, so that $n(x^j)=jn(x) >\\rho_\\infty(K)$. Then one can choose $L=K_j$ in the proof of Proposition \\ref{prop:esisteL}.\n  \\end{proof}\n\\section{Largeness and Weil height}\\label{sect:height}\nLet $h$ denote the logarithmic Weil height of an algebraic number (see for example \\cite[\\S 1.5.7]{BombieriGubler2006}). For $x\\in K$\n$$h(x)=\\frac 1 d\\sum_{\\sigma\\in \\Sigma} \\max\\{0,\\log|\\sigma(x)|\\}+\\log|a|$$\nwhere $a$ is the leading coefficient of a primitive equation for $x$ over $\\mathbb{Z}$; in particular\nfor an algebraic integer $x$ in $\\mathcal{O}_K$\n$$h(x)=\\frac 1 d\\sum_{\\sigma\\in \\Sigma} \\max\\{0,\\log|\\sigma(x)|\\}.$$\nIt follows that \n\\begin{equation}\\label{eq:h>n} h(x)\\geq \\frac 1 d\\log|N_{K\/\\mathbb{Q}}(x)|=n(x) \\quad \\hbox{ for every non-zero algebraic integer $x$},\\end{equation}\nand equality holds exactly when $x\\mathcal{O}_K$ is large.\\\\\nThen we can draw necessary conditions for largeness of ideals when some explicit minoration for the height of elements in $\\mathcal{O}_K$ is known. Namely, if there is a constant $c>0$ such that \n\\begin{equation}\\label{eq:maggiorazioneperinteri} h(x)>c\\hbox{ for every } x\\in\\mathcal{O}_K\\setminus\\mathcal{O}_K^\\times\\end{equation}\nand  $\\mathfrak{A}$ is  a principal ideal such that $n(\\mathfrak{A})\\leq c$, then $\\mathfrak{A}$ cannot be large.\\\\\nWe are thus lead to make use of the well known \\emph{Bogomolov property} $\\PB$  and an additional property $\\PS$ defined below.\\\\\nLet $\\mathcal{A}$ be a set of algebraic numbers.\n  We put\n    \\begin{align*}\n    b(\\mathcal{A})&=\\inf\\{h(x)\\ |\\ x\\in \\mathcal{A}, x\\not=0, x\\hbox{ not a root of unity }\\};\\\\\n    s(\\mathcal{A})&=\\inf\\{n(x)\\ |\\ x\\in\\mathcal{A}, x\\not=0, N_{\\mathbb{Q}(x)\/\\mathbb{Q}}(x)\\not=\\pm 1\\}.\n    \\end{align*}\n\\begin{definition}\\label{def:propBS} We say that a set $\\mathcal{A}$ of algebraic numbers satisfies\n \\begin{itemize} \n    \\item[a)]   \\emph{property $\\PB$} if $b(\\mathcal{A})>0$;\n\\item[b)]  \\emph{property $\\PS$}  if $s(\\mathcal{A})>0$.\n\\end{itemize}\n\\end{definition}\n\nIn particular, if $x\\in\\mathcal{O}_L\\setminus \\mathcal{O}_L^\\times $ for  an (infinite) extension $L$  with the property $\\PB$, then $h(x)\\geq c_L$ for some $c_L>0$ depending only on $L$ and thus, by \\eqref{eq:h>n}, \n$$x\\mathcal{O}_{\\mathbb{Q}(x)} \\hbox{ large} \\Longrightarrow \nn(x)\\geq c_L.\n$$\nProperty $\\PB$ is known for some special algebraic extensions, as the compositum ${\\mathbb Q}^{\\rm tr}$ of all totally real fields; it is also known for extensions having bounded local degrees at some finite place, and for Abelian extensions of number fields (see~\\cite[Remark 5.2, p.1902]{AmorosoDavidZannier2014}). \\\\\nNote however that property $\\PB$ for the whole field  $L$, and even for the ring of integers of $L$, is much stronger that condition \\eqref{eq:maggiorazioneperinteri}, which assumes a lower bound only for the height of algebraic \\emph{integers} which are not units. \n\n\\begin{examples}\\label{exs:esempi}~\n\n\\begin{itemize}\n\\item[a)] Of course $\\PS\\Rightarrow \\PB$ if $\\mathcal{A}$ is a set of algebraic integers containing only a finite number of units. \n \n\\item[b)] On the other hand, there exist sets of algebraic integers satisfying \n$\\PB$ but not $\\PS$: for example   the ring $\\mathcal{O}^{\\rm ab}$ of integers of $\\mathbb{Q}^{\\rm ab}$ satisfies property $\\PB$ with $b(\\mathcal{O}^{\\rm ab})\\geq \\frac{\\log 5}{12}$, (see the main theorem in \\cite{AmorosoDvornicich2000}), but  \n$$n(1-\\zeta_p)=\\frac{\\log(p)}{p-1}$$ \nfor a prime $p$ and $\\zeta_p$ a primitive $p$-th root of unity; therefore $s(\\mathcal{O}^{\\rm ab})=0$.\n\\item[c)] It is  proven in \\cite[Corollary 1]{AmorosoDvornicich2000} that  property $\\PS$ holds for the set $\\mathcal{A}$ of algebraic integers $x$ lying in an Abelian extension of $\\mathbb{Q}$ and such that $  x\/{\\overline{x}}$ is not a root of unity. More precisely \n$$n(x)\\geq \\frac{\\log 5 } {12},\\hbox{ for every $x\\in\\mathcal{A}$ }.$$ \n\\item[d)] Recall that ${\\mathbb Q}^{\\rm tr}(i)$ is the compositum of all CM fields, see \\cite[page 1902]{AmorosoDavidZannier2014}; therefore $x\\in {\\mathbb Q}^{\\rm tr}(i)$ if and only if  $\\mathbb{Q}(x)$ is either a totally real or a CM field. Since the complex conjugation commutes with all the embeddings of ${\\mathbb Q}^{\\rm tr}(i)$ in $\\mathbb{C}$, \nwe have $|\\sigma(x)| = 1$ for some $\\sigma \\in \\Sigma$ if and only if \n$|\\sigma(x)| = 1$ for all $\\sigma \\in \\Sigma$.\nIn this case, we just write $|x| = 1$.\\\\\nBy a result of Schinzel (apply~\\cite[Corollary 1', p. 386]{Schinzel1973}, to the linear polynomial $P(z)=z-x$), if\n$\\mathcal{A}=\\{x\\in {\\mathbb Q}^{\\rm tr}(i)\\ |\\ |x|\\not=1\\}$ then \n\\begin{equation}\\label{eq:Schinzel} b(\\mathcal{A})\\geq \\frac 1 2\\log \\frac{1+\\sqrt{5}} 2.\\end{equation}\n\\end{itemize}\n\\end{examples}\n\nBy Example \\ref{exs:esempi}.d) we obtain the following\n\\begin{proposition}\\label{prop:notlargeperSchinzel}\nLet  $L={\\mathbb Q}^{\\rm tr}(i)$, and let $x\\in\\mathcal{O}_L $ be a non-zero element.\nIf $n(x)< \\frac 1 2 \\log\\frac{1+\\sqrt{5}}2$ then $x \\mathcal{O}_{\\mathbb{Q}(x)}$ is not large in $L$ except if $x$ is a unit.\n\\end{proposition}\n\\begin{proof}\nIf $x$ is a unit, then $x\\mathcal{O}_{\\mathbb{Q}(x)} = \\mathcal{O}_{\\mathbb{Q}(x)}$ is trivially large. If not, we have $|x|\\not=1$, so that Schinzel result \\eqref{eq:Schinzel} implies that $ h(x)\\geq \\frac 1 2\\log \\frac{1+\\sqrt{5}} 2>n(x)$. Then the result follows from \\eqref{eq:h>n}.\n\\end{proof}\n\n\n\n\n\\subsection{Example}  Let $p$ be one of the primes for which ${\\QQ}(\\zeta_{p-1})$ has class number one. Note that $p$ splits completely in ${\\QQ}(\\zeta_{p-1})$. Recall (\\cite{MasleyMontgomery1976}) that the cyclotomic field ${\\QQ}(\\zeta_m)$ has class number one if and only if $m$ is one of the following forty-four numbers:\n\\begin{multline*}\n3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28,\\\\ \n30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.\n\\end{multline*}\n(which corresponds to twenty-nine distinct cyclotomic fields). Thus the relevant primes are\n\\begin{equation}\n\\label{list}\n5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 61, 67, 71.\n\\end{equation} \n\\begin{question}\n\\label{qu:continued}\nLet $p$ be one of the fifteen primes~\\eqref{list} and let $\\mathfrak{P}$ be a prime ideal over $p$ in the ring of integers of $\\mathbb{Q}(\\zeta_{p-1})$. Is $\\mathfrak{P}$ large?\n\\end{question}\nSince ${\\mathbb Q}^{\\rm ab}\\subseteq{\\mathbb Q}^{\\rm tr}(i)$ and\n$$\np\\leq \\Big(\\frac{1+\\sqrt{5}}2\\Big)^{\\varphi(p-1)\/2}\\hbox{ for }p=41,67\\hbox{ and }71\n$$\nthe answer to Question~\\ref{qu:continued} is negative for these primes, by Proposition \\ref{prop:notlargeperSchinzel}. \n\nNote that we could have tried the same strategy as in Proposition \\ref{prop:notlargeperSchinzel}\nbut using the inequality of Example \\ref{exs:esempi}b. This would work only for the primes $p$ satisfying\n\\begin{equation}\\label{eq:valetutti}\np < 5^{\\varphi(p-1)\/12}.\n\\end{equation}\nHowever, this inequality is satisfied by none of the primes in the list~\\eqref{list}.\n\nIn the subsequent Theorem \\ref{teo:risposta},  Question \\eqref{qu:continued} will receive a complete answer.\n \\subsection{Number theoretic minorations of the covering radius}\n In the light of the propositions \\ref{prop:normalarge} and \\ref{prop:esisteLuniforme}, it is useful to have some quantitative information on the covering radius $\\rho_\\infty(K)$ of a number field $K$ of degree $d$. \\\\\n Let $\\lambda_1,\\ldots,\\lambda_r$ be the successive minima (w.r.t. the Euclidean norm $||\\cdot ||_2$) of the lattice $\\Lambda_K$.  \n It is well known (see for example \\cite[Theorem 7.9]{MicciancioGoldwasser2002} that \n\\begin{equation}\\label{eq:disugminsucccovrad} \\lambda_1\\leq\\ldots\\leq \\lambda_r \\leq 2\\rho_2\\leq \\sqrt{s}\\lambda_r.\\end{equation}\nMoreover by \\eqref{eq:disugcovrad} we have\n \\begin{equation}\\label{eq:disugcovrad2}\n \\rho_\\infty(K)\\leq \\rho_2(K)\\leq  \\sqrt{s}\\rho_\\infty(K).\\end{equation}\n Therefore \n \\begin{align} \\rho_\\infty(K)&\\geq \\frac 1 {\\sqrt{s}}\\rho_2(K)\\quad\\hbox{from \\eqref{eq:disugcovrad2}}\\nonumber \\\\\\\n& \\geq \\frac 1 {2\\sqrt{s}}\\lambda_r\\quad\\hbox{from \\eqref{eq:disugminsucccovrad}}\\label{eq:keyminoration}\n \\end{align} \n \n \\begin{theorem}\\label{teo:regolatore}~\n  \\begin{itemize}\n     \\item [a)] Let $K$ be a number field such that $r\\geq 1$. Then  $\\rho_\\infty(K)\\geq \\frac 1 2 R_K^{\\frac 1r}$. \n     \\item[b)] \n  There exists a constant $c>0$ such that $\\rho_\\infty(K)\\geq c$ for every number field $K$ such that $r\\geq1$.\n  \\end{itemize}\n \\end{theorem}\n \n \\begin{proof}\n Recall that $V_K$ is the volume of a fundamental domain for $\\Lambda_K$.\n By Minkowski's Second Theorem \\cite[Theorem 1.5]{MicciancioGoldwasser2002}\n $$(\\lambda_1\\cdot\\ldots\\cdot \\lambda_r)^{\\frac 1 r}\\geq \\sqrt{r}\\cdot  V_K^{\\frac 1r}= \\sqrt{r}\\cdot (\\sqrt{s}R_K)^{\\frac 1r}.$$\n Then from \\eqref{eq:keyminoration}\n \\begin{align}\\label{eq:covradreg} \\rho_\\infty(K)\\geq \\frac {\\sqrt{r}}{2\\sqrt{s}}(\\sqrt{s}R_K)^{\\frac 1r}\\geq c' R_K^{\\frac 1r},\\end{align}\n for a suitable constant $c'$.  Studying the function $\\frac {\\sqrt{r}}{2\\sqrt{s}}(\\sqrt{s})^{\\frac 1 r}$ with $s=r+1$ we see that $c'=\\frac 1 2$. This proves a). Then b)  follows from the well known fact that there exist constants $c_0>0$ and $c_1>1$ such that $R_K>c_0\\cdot c_1^d$ (\\cite[\\S 3]{Zimmert1981}, see also \\cite{FriedmanSkoruppa1999}. \\end{proof}\n \n The next results provide some lower bounds for the covering radius involving the Weil height on $K$.\\\\\nFor $n=1,...,r$ we put, as in \\cite[Page 9]{AmorosoDavid2021}\n$$\\mu_K(n)=\\inf_{\\atop{v_1,\\ldots, v_n\\in\\mathcal{O}_K^\\times}{{\\rm multipl. indep.}}}(h(v_1)\\cdot\\ldots\\cdot h(v_n)),$$\n \n \\begin{theorem}\n \\label{teo:eccolo}\n For $n=1,\\ldots,r$, we have\n $$\\rho_\\infty(K)\\geq \\frac d s\\mu_K(n)^{\\frac 1 n}.$$\n In particular $$\\rho_\\infty(K)\\geq \\frac d s b(\\mathcal{O}_K^\\times).$$\n \\end{theorem}\n \\begin{proof}\nFor every $u\\in\\mathcal{O}_K^\\times$, by \\eqref{eq:disuglp},\n\\begin{equation}\\label{eq:due} 2dh(u)= ||\\ell(u)||_1 \\leq  \\sqrt{s}||\\ell(u)||_2.\\end{equation} \n \nThere exist multiplicatively independent $u_1,\\ldots, u_r \\in\\mathcal{O}_K^\\times$ such that  $\\lambda_i=||\\ell(u_i)||_2$ (\\cite[Theorem 1.2]{MicciancioGoldwasser2002}). \nWe notice that for $n=1,\\ldots, r$\n\\begin{align*} \\lambda_n & =\\inf_{\\atop{v_1,\\ldots, v_n\\in\\mathcal{O}_K^\\times} {\\rm multipl. indep.}}  \\max\\{ ||\\ell(v_1)||_2,\\ldots,  ||\\ell(v_n)||_2\\}\\\\\n&\\geq \\frac {2d} {\\sqrt{s}} \\inf_{\\atop{v_1,\\ldots, v_n\\in\\mathcal{O}_K^\\times} {\\rm multipl. indep.}} \\max \\{ h(v_1),\\ldots,  h(v_n)\\}\\quad\\hbox{ by \\eqref{eq:due}.}\n\\end{align*}\nIt follows from \\eqref{eq:keyminoration} that\n\\begin{align*} \\rho_\\infty(K) &\\geq \\frac 1 {2\\sqrt{s}} \\lambda_n\\geq \\frac d s\\inf_{\\atop{v_1,\\ldots, v_n\\in\\mathcal{O}_K^\\times} {\\rm multipl. indep.}}  \\max \\{ h(v_1),\\ldots,  h(v_n)\\}\\geq \\frac d s\\mu_K(n)^{\\frac 1 n}.\\end{align*}\n\\end{proof}\n\nBy inequality \\eqref{eq:covradreg}, it is possible to use known lower bounds for the regulator $R_K$ in order to deduce lower bounds for the covering radius $\\rho_\\infty(K)$. See \\cite[\\S 3.5, 15]{Narkiewicz2004} for an overview on evaluations of the regulator and \\cite[\\S 8]{Narkiewicz2004} for the special case of Abelian extensions. Moreover \\cite[Proposition 3.3]{AmorosoDavid2021} provides a tool allowing to improve the bound from below of extensions $K$ for which $\\mathcal{O}_K^\\times$ has property $\\PB$. In particular \\cite[Corollaire 3.5]{AmorosoDavid2021} deals with the case of totally real and CM field.\n\nSome remarkable results are collected below:\n \n\\begin{itemize}\n\\item[a)] Let $L$ be an infinite extension. Assume that $\\mathcal{O}_L^\\times$ satisfies property $\\PB$ and let $c_L= b(\\mathcal{O}_L^\\times)=\\inf_{u\\in\\mathcal{O}_L^\\times\\setminus\\mathcal{O}_L^{\\times,\\rm{tors}}} h(u) >0$. Then by Theorem \\ref{teo:eccolo}$$ \\rho_\\infty(K)\\geq c_L,$$ for every number field $K \\subseteq L$ such that $r(K) \\geq 1$.\n\\item[b)] In particular, if a number field $K$ is contained in $\\mathbb{Q}^{\\rm{tr}}(i)$, then by Example \\ref{exs:esempi} d),\n$$\\rho_\\infty(K)\\geq \\frac 1 2 \\log \\frac{1+\\sqrt{5}} 2.$$\n\\item[c)] Silverman's theorem \\cite{Silverman1984} allows to construct fields with  fixed degree and covering radius arbitrarily large. It suffices to choose a non-CM field of discriminant large enough.  \n\\end{itemize}\n\n\n\n\n\n\\section{An algorithm for largeness}\\label{sect:algorithm}\n\n  We describe an algorithm that solves the following problem:\n  \\begin{problem}\\label{problem1}\n  Given a non-zero algebraic number $x\\in K$ and a bound $B>0$,\n  find all units $u\\in \\calo_K^\\times$ such that\n  $|\\sigma(xu)| \\geq  B$ for all $\\sigma \\in \\Sigma$.\n\\end{problem}\nWhen $B=1$ this algorithm detects when a principal ideal is large.\n\\subsection{Basic results}\n\n\\begin{lemma}\\label{upperbound}\n  If $u \\in \\calo_K^\\times$ is a solution of Problem \\ref{problem1},\n  then for all $S \\subset \\Sigma$, we have\n  $$B^{\\# S} \\leq \\prod_{\\sigma \\in S} |\\sigma(xu)| \\leq B^{\\# S - d} N(x)$$\n  where $\\# S$ denotes the cardinality of $S$.\n\\end{lemma}\n\\begin{proof}\nLet $u \\in \\mathcal{O}_K^\\times$ be a solution of Problem \\ref{problem1}.\nFor $\\sigma \\in S$, we have the trivial inequality $B \\leq |\\sigma(xu)|$.\nMultiplying these inequalities gives the announced left inequality.\n\n\nFor $\\sigma \\in S$, we have the inequality $|\\sigma(xu)| \\leq |\\sigma(xu)|$, and\nfor $\\sigma \\not\\in S$, we have $B \\leq |\\sigma(xu)|$.\nMultiplying these inequalities gives $B^{d-\\# S} \\prod_{\\sigma \\in S} |\\sigma(xu)| \\leq N(xu)$.\nBut $u$ is a unit, hence $N(xu) = N(x)$, whence the result.\n\\end{proof}\n\n\\begin{proposition}\\label{smallnorm}\n  If $N(x) < B^d$, then Problem \\ref{problem1} has no solution.\n\\end{proposition}\n\\begin{proof}\nApply Lemma \\ref{upperbound} with $S = \\Sigma$.\n\\end{proof}\n\n\\begin{proposition}\\label{finitesol}\n  Given $x\\in K$ and $B>0$, Problem \\ref{problem1} has only finitely many solutions.\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{smallnorm}, Problem \\ref{problem1} has no solution for $x=0$.\nWe assume now that $x\\neq 0$.\n\nDividing the right inequality of Lemma \\ref{upperbound} by $\\prod_{\\sigma \\in S}|\\sigma(x)| \\neq 0$ gives\n$$ \\prod_{\\sigma\\in S} |\\sigma(u)| \\leqslant  B^{\\# S - d} \\prod_{\\sigma \\not\\in S}|\\sigma(x)|\n\\leqslant B^{\\#S-d} \\left( \\frac {||x||_1}{d-\\# S} \\right)^{d-\\# S} \\; .$$\nLet us consider the characteristic polynomial of $u$ for the extension $K\/{\\QQ}$ and\ndenote it by $P_u$.\nSince $u$ is a unit, $P_u \\in \\mathbb{Z}[X]$ and $P_u$ is monic.\nThe roots of $P_u$ in $\\mathbb{C}$ are the real or complex numbers $\\sigma(u)$.\nUsing the above inequality and expressing the coefficient $a_k$ of $X^k$ in $P_u$ in terms of the roots of $P_u$,\nwe deduce that $|a_k|$ is bounded independently of $u$.\nFor example we have\n$|a_d| = |a_0|=1$ since $u$ is a unit\nand \n$$|a_k| \\leq \\binom{d}{k} \\left( \\frac {||x||_1}{kB} \\right)^k$$\nfor the other values of $k$.\nSince this bound does not depend on $u$, there are only finitely many\npossibilities for $P_u$, hence for $u$.\n\\end{proof}\n\n\\noindent{\\bf Remark.}\nWe could turn the proof of Proposition \\ref{finitesol} into an algorithm\nthat tests all polynomials with coefficients within some bounds depending on $B$ and $x$.\nExplicitly, using the bounds given during the proof,\nwe see that, for a given number field of fixed degree $d$,\nthe number of polynomials that need to be tested, is proportional to\n$\\displaystyle \\left( \\frac {||x||_1}{B} \\right)^\\alpha$\nwith $\\displaystyle \\alpha = \\sum_{k=1}^{d-1} k = \\frac {d(d-1)}2$.\nThe number of polynomials that need to be tested in the algorithm\nis therefore exponential in the input $x$, hence very large,\nand the resulting algorithm is very slow.\n\nWe will give another algorithm in the next section.\n\n\\subsection{An algorithm to solve Problem \\ref{problem1}}\nIf $A=(a_{i,j})$ is a matrix (or a vector) with real entries,\nwe write $A \\geq 0$ to indicate that $a_{i,j} \\geq 0$ for all $i$ and $j$.\nWe also write $A\\geq B$ if $A-B \\geq 0$.\nWe will use the fact that, if  $A\\geq 0$ and $B\\geq 0$, then $AB \\geq 0$.\n\nWe recall that, for a number field $K$ of degree $d$, with $r_1$ real embeddings and $r_2$ complex embeddings, we have set $s = r_1+r_2$ and $r = s-1$.\nIf $u_1,\\dots,u_r$ are generators of $\\calo_K^\\times$ modulo torsion,\nwe define the matrix $L$ of size $r\\times s$ by\n$$ L_{i,j} = \\log | \\sigma_j(u_i)|$$\nif $\\sigma_j$ is real and \n$$ L_{i,j} = 2 \\log | \\sigma_j(u_i)|$$\nif $\\sigma_j$ is complex.\nThe $i$-th row of $L$ is equal to $\\ell(u_i)$.\nAt last, we define the column vector $V = (1,\\dots,1)^t \\in \\mathbb{Z}^s$.\n\n\\medskip\nUsing logarithms, we can reformulate our Problem \\ref{problem1} as:\n\n\\begin{problem}\\label{problem2}\n  Given a non-zero algebraic number $x\\in K^\\times$ and a bound $B>0$,\n  find all rows $U \\in \\mathbb{Z}^r$ such that\n  $$ UL + X \\geq 0 $$\n  where $X = \\ell(x)-\\mathcal{L}(B)$.\n\\end{problem}\n\nFormulated in this way, we see that Problem \\ref{problem2} can be solved by integer linear programming. However, the situation is not generic here, and a simpler algorithm is given below.\n\n\\begin{algo}\\label{algo}~\n  \n  Input : $x\\in K$, $x\\neq 0$, and $B>0$.\n\n  Output : all solutions $U \\in \\mathbb{Z}^r$ of Problem \\ref{problem2}.\n  \n  {\\tt\n    \\begin{enumerate}\n    \\item Compute the matrix $L$ of size $r\\times s$\n      and the column vector $V$ of size $s$ as in the above definition.\n    \\item Remove from $L$ its last column and call $M$ the inverse of this matrix.\n      Concatenate $M$ with the row vector of size $r$ whose all entries are $0$\n      and obtain a new matrix $M$ of size $s\\times r$.\n      \n      For the next steps, we use the notation $N_{,j}$ for the $j$-th column of a matrix $N$.\n    \\item Define the matrix $N^+$ of size $s\\times r$, such that, for $1\\leq j \\leq r$, \n      $N^+_{,j} = M_{,j} - \\min_i\\{M_{i,j}\\} V$.\n    \\item Define the matrix $N^-$ of size $s\\times r$, such that, for $1\\leq j \\leq r$, \n      $N^-_{,j} = M_{,j} - \\max_i\\{M_{i,j}\\} V$.\n            \n\n    \\item Compute the row vector $X = \\ell(x) - \\mathcal{L}(B)$.\n    \\item For all row vector $U \\in \\mathbb{Z}^r$ in the range\n      $ -XN^+ \\leq U \\leq -XN^-$, test if $UL+X \\geq 0$.\n      If this is the case, output $U$.\n    \\end{enumerate}\n    }\n\\end{algo}\n\n\\begin{proposition}\\label{proofalgo}\n  Algorithm \\ref{algo} is correct.\n\n  Furthermore, when the number field $K$ is fixed,\n  the number of $U$ that need to be tested during step 6\n  is at most proportional to $( \\log N(x)-  d\\log B +1)^r$.\n\n\\end{proposition}\n\\begin{proof}\nWe follow the algorithm step by step.\n\\begin{enumerate}\n\\item By Dirichlet's Unit Theorem, the matrix $L$ constructed in step 1 has rank $r$.\n  Since the absolute norm of a unit is equal to $1$, we have $LV = 0$,\n  hence $V$ is in the right kernel of $L$.\n\\item By Dirichlet's Unit Theorem,\n  when we remove any column of $L$, the determinant of the remaining square matrix\n  is always the same and equals the regulator of $K$, which is not $0$.\n  This matrix of size $r \\times r$ is invertible.\n  By construction, we have $LM = I_r$.\n\\item For all columns of $N^+$, we have $N^+_{,j} = M_{,j} - \\min_i\\{M_{i,j}\\} V$.\n  Let $i(j)$ be the index such that $\\min_i\\{M_{i,j}\\} = M_{i(j),j}$.\n  We have $N^+_{i,j} = M_{i,j} - M_{i(j),j} \\geq 0$ by minimality.\n  Hence $N^+ \\geq 0$.\n  Because $V$ is in the right kernel of $L$, we deduce that\n  $LN^+ = LM = I_r$.\n\\item Using a similar argument, we can prove that\n  $N^- \\leq 0$ and $LN^- = I_r$.\n\\item There is nothing to say here.\n\\item If $U$ is a solution of Problem \\ref{problem2},\n  then $UL+X \\geq 0$.\n  But $N^+ \\geq 0$, hence $ULN^+ + XN^+ \\geq 0$.\n  By the relation $LN^+ = I_r$, we deduce $U \\geq -XN^+$.\n  By $N^- \\leq 0$, we deduce $ULN^- + XN^- \\leq 0$, and $U \\leq -XN^-$.\n\\end{enumerate}\n\nIn order to bound the number of $U$ tested in step $6$,\nwe observe that $-XN^+ \\leq U \\leq -XN^-$.\nFor the $j$-th entry, this is explicitly\n$-XN^+_{,j} \\leq U_j \\leq -XN^-_{,j}$\nhence the number of $U_j$ that need to be tested is at most equal to\n$-XN^-_{,j} + XN^+_{,j}+1$. But we have\n\\begin{align*} -XN^-_{,j} + XN^+_{,j} & = X(M_{,j}-\\min_i\\{M_{i,j}\\}V - M_{,j}+\\max_i\\{M_{i,j}\\}V)\\\\\n& = (\\max\\{M_{,j}\\} - \\min\\{M_{,j}\\}) XV\n\\end{align*}\nWe also have $XV = \\log N(x)- d\\log B$.\nIf $\\log N(x) -d\\log B < 0$, we have seen in Proposition \\ref{smallnorm} that the problem has no solution. When\n$\\log N(x) -d\\log B \\geq 0$, we have\n$$-XN^-_{,j} + XN^+_{,j}+1\n\\leq \n(\\max\\{M_{,j}\\} - \\min\\{M_{,j}\\}+1) (\\log N(x) - d \\log B + 1)$$\nwhence a bound for the number of $U$ by\n$$ (\\log N(x) - d\\log B+1)^r \\times \\prod_j (\\max\\{M_{,j}\\} - \\min\\{M_{,j}\\}+1)$$\n\n\n\\end{proof}\n\n\\section{A complete example}\\label{sect:cyclotomic}\n\nIn this section, we shall give a detailed execution of Algorithm \\ref{algo}, which answers Question \\ref{qu:continued} for $p=17$.\nAll computations were done using PARI\/gp \\cite{PARI2}.\n\n\\medskip\nLet us consider the $16$-th cyclotomic field $K$ equal to ${\\QQ}(\\zeta) = {\\QQ}[X]\/\\Phi_{16}(X)$,\nwhere $\\Phi_{16}(X) = X^8+1$.\n\nIn this field, we consider $x = -\\zeta^7 - \\zeta^3 + \\zeta^2$.\nWe have $N(x) = 17$, hence $x$ is a generator of a principal prime ideal above $17$.\nWe are looking for another generator $x'$ of this principal ideal\nsuch that $|\\sigma(x')| \\geq 1$ for all $\\sigma\\in\\Sigma$. We need to solve Problem \\ref{problem2} with $B = 1$.\n\nWe follow here the steps of Algorithm \\ref{algo}.\n\n\\begin{enumerate}\n\\item\n  For this field, we have $d=8$, $r_1 = 0$ and $r_2 = 4$.\n  In this case, we have $s = r_1+r_2 = 4$ and $r = 3$.\n  \n  The units of $K$ are generated by\n  $u_0 = \\zeta$, $u_1 = -\\zeta^6 + \\zeta^2 - 1$, $u_2 = \\zeta^2+\\zeta+1$ and $u_3=-\\zeta^6+\\zeta^3-\\zeta$,\n  where $u_0$ generates the torsion part and $u_1,u_2,u_3$ generate the free part.\n  The matrix $L$ is equal to\n  $$L =\n  \\left(\n  \\begin{array}{rrrr}\n  -1.76274&-1.76274& 1.76274& 1.76274\\\\\n  -0.33031& 2.09306&-2.89946& 1.13671\\\\\n   1.13671&-2.89946&-0.33031& 2.09306\n  \\end{array} \\right)\n  $$\n  We easily check that $L\\begin{pmatrix}1\\\\1\\\\1\\\\1\\end{pmatrix} = 0$.\n\\item\n  We have\n  $$M = \n  \\left( \\begin{array}{rrrr}\n  -0.46575&-0.29144& 0.07276\\\\\n  -0.17430&-0.07276&-0.29144\\\\\n  -0.07276&-0.36421&-0.21868\\\\\n  0 & 0 & 0\n  \\end{array} \\right)\n  $$\n  We can check that $LM = I_3$, the identity matrix of order $3$.\n\\item We have\n  $\\min(M_{,1}) = -0.46575$,\n  $\\min(M_{,2}) = -0.36421$,\n  $\\min(M_{,3}) = -0.29144$.\n  This gives\n $$N^+ = \\left( \\begin{array}{rrr}\n 0       & 0.07276 & 0.36421 \\\\\n 0.29144 & 0.29144 & 0       \\\\\n 0.39298 & 0       & 0.07276 \\\\\n 0.46575 & 0.36421 & 0.29144 \\\\\n \\end{array}\\right)$$\n  We can check that $N^+ \\geq 0$ and $LN^+ = I_3$.\n\\item We have\n  $\\max(M_{,1}) = 0$,\n  $\\max(M_{,2}) = 0$,\n  $\\max(M_{,3}) = 0.07276$.  This gives\n $$N^- = \\left( \\begin{array}{rrr}\n -0.46575 & -0.29144 & 0       \\\\\n -0.17430 & -0.07276 & -0.36421\\\\\n -0.07276 & -0.36421 & -0.29144\\\\\n 0        & 0        & -0.07276\\\\\n \\end{array}\\right)$$\n  We can check that $N^- \\leq 0$ and $LN^- = I_3$.\n\\item Since $B=1$, we have $\\mathcal{L}(B) = 0$.\n  For $x = -\\zeta^7 - \\zeta^3 + \\zeta^2$, we have\n  $$X = \\ell(x) = (1.40668, 0.65107, 1.72510, -0.94965)$$\n\\item\n  We compute\n  $$-XN^+ = (-0.42539, 0.05376, -0.36109)$$\n  $$-XN^- = ( 0.89419, 1.08567,  0.67081)$$\n  In this example, the only $U \\in \\mathbb{Z}^3$ within the bounds is\n  $U = (0,1,0)$.\n  However, for this $U$, we have\n  $$UL+X = (1.07636, 2.74414, -1.17435, 0.18706)$$\n  hence this is not a solution.\n\n\n\\end{enumerate}\n  This computation shows that, in this example, Problem \\ref{problem2} has no solution.\\\\\nAnalogous computations applied to all prime $p$ in the list \\eqref{list} allow to give a complete answer to Question \\ref{qu:continued}:\n\\begin{theorem}\\label{teo:risposta}\nLet $p$ be one of the primes for which $\\mathbb{Q}(\\zeta_{p-1})$ has class number 1, as listed in \\eqref{list}, and let $\\mathfrak{P}$ be a prime ideal over $p$ in the ring of integers of $\\mathbb{Q}(\\zeta_{p-1})$. Then $\\mathfrak{P}$ is large if and only if\n\\begin{equation}\\label{eq:listalarge} p\\in \\{5,7,11,13,19,31\\}.\\end{equation}\n\\end{theorem}\nMore precisely, for each primes in the list \\eqref{eq:listalarge} the following table gives (up to Galois conjugation and multiplication by a root of unity) the elements $\\pi$ of absolute norm $p$ having all the components $\\geq 1$ in the canonical embedding ($\\zeta=\\zeta_{p-1}$ in each case): \n\n\\begin{table}[h]\n\\def1.5{1.5}\n\\begin{tabular}{l|l}    $p$ & $\\pi$\\\\      \\hline $5$ & $2\\zeta+1$\\\\\n    $7$ & $\\zeta-3$\\\\\n    $11$ & $2\\zeta^3-1$, \\quad $2\\zeta^2-\\zeta+1$\\\\\n    $13$ & $-2\\zeta^3-\\zeta^2$, \\quad $\\zeta^3-\\zeta^2+2$\\\\\n    $19$  & $-\\zeta^4-\\zeta^3+\\zeta^2+\\zeta+1$\\\\\n    $31$ & $-\\zeta^7-\\zeta^3-\\zeta$,\\quad $ -\\zeta^6-\\zeta^5+\\zeta^3+\\zeta^2+\\zeta-1$\\\\\n    &\n    \\end{tabular}\n   \n    \\caption{Strictly large integers of absolute norm $p$ in $\\mathbb{Q}(\\zeta_{p-1})$} \\label{tavola} \n\\end{table}\n\n\n\n\n\\section{Another application: floor functions and types}\\label{sect:floorfunctions}\n\nLet $K$ be a number field of degree $d$ over $\\mathbb{Q}$, and let $\\mathcal{O}_K$ be its ring of integers. We fix an ideal $\\mathfrak{A}$ of $\\mathcal{O}_K$.\nThe aim of this section is to apply largeness (when possible) in order to define  complete sets of representatives of $K\/\\mathfrak{A}$  (which will be called \\emph{types}) satisfying some integrality properties and having all the Archimedean embeddings bounded in a controlled way. \\\\ Types associated to a prime ideal $\\mathfrak{P}$ of a number field were introduced in \\cite{CapuanoMurruTerracini2022} with the aim of  constructing a general notion of $\\mathfrak{P}$-adic continued fractions and studying their finiteness and periodicity properties.\n\nLet $\\mathcal{M}_K^0$ be a set of representatives for the non-Archimedean places of $K$. For every rational prime $p$ and every  $v\\in\\mathcal{M}_K^0$ above $p$ let $K_{v}\\subseteq \\overline{\\mathbb{Q}}_p$ be the completion of $K$ w.r.t. the $v$-adic valuation and $\\mathcal{O}_v$ be its valuation ring; we put $d_v=[K_v:\\mathbb{Q}_p]$. Let $|\\cdot |_v=|N_{K_v\/\\mathbb{Q}_p}(\\cdot)|_p^{\\frac 1{d_v}}$ be the unique extension  of $|\\cdot |_p$ to $K_v$.  \nLet $\\widetilde{K}=\\prod_{v \\mid \\mathfrak{A}} K_v$ be the $\\mathfrak{A}$-adic completion of $K$, with $K$ diagonally embedded, and $\\widetilde{\\mathcal{O}}=\\prod_{v \\mid \\mathfrak{A}}\\mathcal{O}_v$.\\\\\nLet $S_0=\\{v\\in \\mathcal{M}_K^0 \\mid  v \\mid \\mathfrak{A}\\}$.\n\\begin{definition}\nAn \\emph{$\\mathfrak{A}$-adic floor function} for $K$ is a function $s:\\widetilde K\\to K$ such that\n\\begin{itemize} \n\\item[a)] $\\alpha-s(\\alpha)\\in \\mathfrak{A}\\widetilde\\mathcal{O}$ for every $\\alpha\\in \\widetilde K$;\n\\item[b)] $|s(\\alpha)|_{v}\\leq 1$ for every  $v\\in \\mathcal{M}_K^0\\setminus S_0$;\n\\item[c)] $s(0)=0$;\n\\item[d)] $s(\\alpha)=s(\\beta)$ if $\\alpha-\\beta\\in\\mathfrak{A}\\widetilde\\mathcal{O}$.\n\\end{itemize}\n\\end{definition}\n\n\n    \nBy the Strong Approximation Theorem in number fields (see for example \\cite[Theorem 4.1]{Cassels}),  $\\mathfrak{A}$-adic floor functions always exist, and there are infinitely many. \\\\\nWe define the ring of $S_0$-integers\n\\[\\mathcal{O}_{K,S_0}=\\{\\alpha\\in K\\ |\\ |\\alpha|_v\\leq 1\\hbox{ for every } v \\in \\mathcal{M}_K^0 \\setminus S_0\\}.\n\\]\nThen, we can regard an $\\mathfrak{A}$-adic floor function as a map $s: \\widetilde K\/\\mathfrak{A}\\widetilde\\mathcal{O}\\to \\mathcal{O}_{K,S_0}$ such that $s(\\mathfrak{A}\\widetilde\\mathcal{O})=0$ and which is a section of the projection map $\\widetilde K\\to \\widetilde K\/\\mathfrak{A}\\widetilde\\mathcal{O}$.\nTherefore the choice of an $\\mathfrak{A}$-adic floor function amounts to choose  a set $\\mathcal{Y}$ of representatives of the cosets of $\\mathfrak{A}\\widetilde\\mathcal{O}$ in $\\widetilde K$ containing $0$ and contained in\n$\\mathcal{O}_{K,S_0}$.\\\\\nWe shall call the data $\\tau=(K,\\mathfrak{A},s)$ (or $(K,\\mathfrak{A},\\mathcal{Y})$) a \\emph{type}. \n\\begin{remark} The absolute Galois group $\\mathrm{Gal}(\\overline{\\QQ}\/\\mathbb{Q})$ acts on the set of types; indeed, if $\\tau=(K,\\mathfrak{A}, s)$ is a type, then $\\sigma\\in \\mathrm{Gal}(\\overline{\\QQ}\/\\mathbb{Q})$ induces a continuous map $\\widetilde K \\to \\widetilde{K^\\sigma}$, where $\\widetilde{K^\\sigma}$  is the completion of $K^\\sigma$ with respect to the ideal $\\mathfrak{A}^\\sigma$.  Then  $\\tau^\\sigma=(K^\\sigma,\\mathfrak{A}^\\sigma, s^\\sigma)$ is also a type, where $s^\\sigma=\\sigma\\circ s\\circ \\sigma^{-1}$. In particular, if $K\/\\mathbb{Q}$ is a Galois extension and $\\sigma$ belongs to the decomposition group \n$$D_\\mathfrak{A}=\\{\\sigma\\in\\mathrm{Gal}(\\overline{\\QQ}\/\\mathbb{Q})\\ |\\  \\mathfrak{A}^\\sigma=\\mathfrak{A}\\},$$\n   then $\\tau^\\sigma=(K,\\mathfrak{A}, s^\\sigma)$ is again an $\\mathfrak{A}$-adic type.\n\\end{remark}\n\\subsection{Types arising from generators of $\\mathfrak{A}$} \\label{sec:special_type}\nIn the case $\\mathfrak{A}$ is principal, there is a natural way of defining an $\\mathfrak{A}$-adic floor function.\nIndeed, let $\\pi\\in\\mathfrak{A}$ be  generator and let $\\mathcal{R}$ be a  complete set of representatives of $\\mathcal{O}_K\/\\mathfrak{A}$ containing $0$. Then, every $\\alpha\\in \\widetilde K$ can be expressed uniquely as a Laurent series  $\\alpha=\\sum_{j=-n}^\\infty c_j\\pi^j$, where $c_j\\in\\mathcal{R}$ for every $j$. It is possible to define an  $\\mathfrak{A}$-adic floor function by\n$$s(\\alpha)=\\sum_{j=-n}^0c_j\\pi^j\\in K. $$\nWe shall denote the types $\\tau=(K,\\mathfrak{A},s)$ obtained in this way by $\\tau=(K,\\pi,\\mathcal{R})$, and we will usually call them \\emph{special types}.\n\n\\begin{example}[Browkin and Ruban types over $\\mathbb{Q}$]\nWhen $K=\\mathbb{Q}$ and $\\pi=p$ odd prime, two main special types have been studied in the literature:\n\\begin{itemize}\n\\item the \\emph{Browkin type} $\\tau_B=(\\mathbb{Q},p,\\mathcal{R}_B)$ where $\\mathcal{R}_B=\\{- \\frac {p-1} 2,\\ldots, \\frac {p-1} 2\\}$ (see \\cite{Browkin1978, Bedocchi1988, Bedocchi1989, Bedocchi1990, Browkin2000, CapuanoMurruTerracini2020});\n\\item the \\emph{Ruban type} $\\tau_R=(\\mathbb{Q},p,\\mathcal{R}_R)$ where $\\mathcal{R}_R=\\{0,\\ldots, p-1\\}$ (see \\cite{Ruban1970, Laohakosol1985, Wang1985, CapuanoVenezianoZannier2019}).\n\\end{itemize}\n\\end{example}\n\n\\subsection{Bounded types} We say that a type $\\tau=(K,\\mathfrak{A},s)$ is \\emph{bounded} if there exists a real number $C>0$ such that  $|\\sigma(s(\\alpha))| <C$ for every $\\alpha\\in K$ and every Archimedean embedding $\\sigma$ of $K$.\n\\begin{proposition}\\label{prop:existboundedtype} For every number field $K$ and prime ideal $\\mathfrak{A}$ there exist a bounded type $(K,\\mathfrak{A},s)$. \n\\end{proposition}\n\\begin{proof}\nLet $\\iota: K\\to \\mathbb{R}^{r_1}\\times\\mathbb{C}^{r_2}\\simeq \\mathbb{R}^d$ be the canonical embedding. Then $\\iota(\\mathfrak{A})$ is a lattice in $\\mathbb{R}^d$. Let $\\mathcal{D}_\\mathfrak{A}$ be a bounded fundamental domain containing $0$. We define a floor function $s$ in the following way: firstly choose any $\\mathfrak{A}$-adic floor function $s'$ for $K$. Let $\\alpha\\in \\widetilde K$ and put $\\alpha'=s'(\\alpha)$; then $\\beta= \\alpha'+\\gamma\\in D_\\mathfrak{A}$ for a suitable $\\gamma\\in \\mathfrak{A}$; define $s(\\alpha)=\\beta$. Since $D_\\mathfrak{A}$ is a bounded subset of $\\mathbb{R}^d$, the claim is proven.\n\\end{proof}\n\\begin{remark}  Let $\\rho$ be the covering radius of the lattice $\\iota(\\mathfrak{A})$ with respect to the sup norm. Then the closed ball centered in $0$ of radius $\\rho$ with respect to this norm contains a fundamental domain $D_\\mathfrak{A}$ as in the proof of Proposition \\ref{prop:existboundedtype}. In particular we see that there exists a type $(K,\\mathfrak{A},s)$ such that $||\\iota(s(x))||_\\infty\\leq \\rho$, for every $x\\in\\widetilde K$.\\end{remark} \n\n\\begin{proposition}\\label{prop:specialtypesbounded} Assume that $\\mathfrak{A}$ is a non-zero principal ideal of $\\mathcal{O}_K$ having a strictly large generator $\\pi$. Let $\\mathcal{R}$ be any complete set of representatives of $\\mathcal{O}_K\/\\mathfrak{A}$ containing $0$. Then the special type $(K,\\pi,\\mathcal{R})$ is bounded.\n\\end{proposition}\n \n\\begin{proof}\nFor every Archimedean embedding $\\sigma:K\\to\\mathbb{C}$ let $\\lambda_\\sigma=|\\sigma(\\pi)|$ and $L_\\sigma=\\max\\{|\\sigma(c)|\\ |\\ c\\in \\mathcal{R}\\}$. \nThen for every $\\sigma$\n$$\\left|\\sum_{j=-n}^0 \\sigma(c_j)\\sigma(\\pi^j)\\right | \\leq  \\frac {L_\\sigma\\lambda_\\sigma}{\\lambda_\\sigma-1}.$$\n\\end{proof}\n\n\\begin{remark}\nFor each $p$ in the list \\eqref{list}, and every prime ideal $\\mathfrak{P}$ over $p$ in $\\mathbb{Z}(\\zeta_{p-1})$, the set $\\mathcal{R}_p=\\{\\zeta^i\\ |\\ i=0,\\ldots, p-2\\}\\cup\\{0\\}$ is a complete set of representatives of $\\mathbb{Z}[\\zeta]\/\\mathfrak{P}$. Then by Proposition \\ref{prop:specialtypesbounded} the special types $(\\mathbb{Q}(\\zeta_{p-1}),\\pi,\\mathcal{R}_p)$ are bounded for every $p$  and $\\pi$ as in Table \\ref{tavola}.\n\\end{remark}\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLanguage is an indispensable and important part of human daily life. Natural language is everywhere as a most direct and simple tool of expression. Natural language processing is to transform the language used for human communication into a machine language that can be understood by machines. It is a model and algorithm framework for studying language capabilities. In recent years, NLP research has increasingly used new deep learning methods. As an important branch of artificial intelligence, language models are models that can estimate the probability distribution of a group of language units (usually word sequences). These models can be built at a lower cost and have significantly improved several NLP tasks, such as machine translation, speech recognition and parsing. The processing flow of natural language can be roughly divided into five steps: obtaining anticipation, preprocessing the corpus, characterizing, model training, and evaluating the effect of modeling.\n\nWith the rapid development of the Internet, the frequency of online communication on social software such as Weibo, Twitter, and forums is getting higher and higher, and the Internet itself has also changed from \"reading Internet\" to \"interactive Internet\". The Internet has not only become an important source for people to obtain information, but also an important platform for people to express their opinions and share their own experiences and directly express their emotions. The achievements of NLP research laid a good foundation for text sentiment analysis. Text sentiment analysis is an important research branch in the field of natural language understanding, involving theories and methods in the fields of linguistics, psychology, artificial intelligence, etc. It mainly includes the processing of text sources, the subjective and objective classification of network text, and the subjective text Analysis and other steps.\n\nDue to the huge inclusiveness and openness of the Internet itself, it attracts users of different races, different languages, different cultural backgrounds and different religious beliefs to communicate with each other here. Therefore, mixed language sentiment classification will be an important research for NLP direction.\n\n\\fn{This work is licensed under a Creative Commons Attribution 4.0 International Licence. Licence details: http:\/\/creativecommons.org\/licenses\/by\/4.0\/.}\n\n\\section{Related Work}\nSentiment analysis is a research with a long history that helps us understand the connections and relationships between objects. In recent years, many scholars have made great progess on sentiment analysis. A basic task in sentiment analysis is classifying the polarity of a given text at the document, sentence, or feature\/aspect level\u2014whether the expressed opinion in a document, a sentence or an entity feature\/aspect is positive, negative, or neutral. Subsequently, the method described in a patent by Volcani and Fogel~\\cite{Volcanietal2001}, looked specifically at sentiment and identified individual words and phrases in text with respect to different emotional scales. Many other subsequent efforts were less sophisticated, using a mere polar view of sentiment, from positive to negative, such as work by Turney~\\cite{PeterD.Turneyetal2002}, and Pang~\\cite{BoPangetal2002} who applied different methods for detecting the polarity of product reviews and movie reviews respectively. One can also classify a document's polarity on a multi-way scale, which was attempted by Pang~\\cite{BoPangetal2005} and Snyder~\\cite{Snyderetal2007}.\n\nBut according to our findings, this research becomes particularly difficult in multilingual societies, especially in many code-mixed texts. Though some researchers have explored in the field, there is still a long way to go. Sharma and Srinivas explore various methods to normalize the text and judged the polarity of the statement as positive or negative using various sentiment resources~\\cite{S.Sharmaetal2015}. Bhargava and Sharma develop a flexible and robust system for mining sentiments from code mixed sentences for English with combination of four other Indian languages (Tamil, Telugu, Hindi and Bengali)~\\cite{R.Bhargava2016}. Ghosh and Das extract sentiment (positive or negative) from Facebook posts in the form of code-mixed social media data using a machine learning approach~\\cite{SouvickGhoshetal2017}.\n\n\\section{Data and Methodology}\n\n\\subsection{Data Description}\nThis task is to predict the sentiment of the mixed tweets of a given code. The sentiment tags are divided into three categories: positive, negative and pertinent. Words are also given unique language tags, such as en (English), spa (Spanish), hi (Hindi), mixed and univ (for example, symbols, @mentioned, hashtags). The given data set is divided into training data set and validation data set, which contains emoticons, symbols, Spanish and English, Hindi and English mixed tweets. Since expressions and symbols cannot be directly put into classification and recognition, preprocessing is required to convert them into recognizable English phrases; for other languages mixed with English, we need to use English phrases to recognize them, and we need to label emotions. So use Multi-task to simultaneously recognize English words and perform emotion prediction.\n\n\n\n\\subsection{Preprocessing}\nWe first process the data before feeding the data set to any model. In this section we will introduce the core methods and strategies of processing the raw data.\n\n\\textbf{Emoji Substitution} - We design a function to map emoji unicode to replacement phrases. We treat these phrases as regular English phrases so their semantics can be preserved, especially when the dataset size is limited.\n\n\\textbf{Character Filtering} - We also convert all the text into lower case. Since 'URL' does not have embedding representation in some pre-trained embedding and models, 'URL' is substituted by 'http'. Besides, we delete some uncommon characters without emotions such as '@' and 'https'.\n\n\\subsection{Methodology}\n\n\\textbf {Bert} - The input part of BERT is a sequence where two sentences are connected. The two sentences are separated by a separator, and an identification symbol is added to the front and the end of each sentence to indicate the beginning and end of the sentence. For each word, BERT performs three different embedding operations, namely encoding the word position information, Word2vec encoding the word, and encoding the entire sentence. Vector stitching these three embedding results can get BERT input. In order to train the bidirectional feature and obtain the connection between the two sentences, BERT uses the Masked Language Model pre-training method, which randomly covers part of the words in the sentence, and uses the training model to predict this part of the word and the next sentence.This article uses the word vector pre-trained by the BERT method as the vector of the input short text. Because the mixed short texts of tweets are mostly replaced by English words in Spanish and Hindi words or Spanish and Hindi words in English, that is, the context is similar, so Spanish and English, Hindi and English The co-trained word vector is used as the vector for inputting short text.\n\n\\textbf{Fine-tune} - The method NFT-TM refers to adding a complex network structure to the upper layer of the BERT model. During training, part of the convolutional layer of the pre-trained model is frozen, the parameters of the BERT are fixed, and only the upper task model network is trained separately. Matthew Peter and others from the Allen Institute for Artificial Intelligence in the United States compared the effects of the FT-TM and NFT-TM methods on the two pre-training models of ELMo and BERT. For BERT, the fine-tune effect is slightly better. Therefore, this experiment uses the NFT-TM method, followed by a simple specific task layer after the Bert model. During training, the BERT is fine-tuned according to the training sample set of the task. The practical results of this task show Excellent results.\n\n\\textbf{Multi-task} - Multitask Learning is a derivation transfer learning method. The main tasks use domain-specific information possessed by the training signals of related tasks as a constant task. A machine learning method for inductive bias to improve the generalization performance of main tasks. Multi-task learning involves the simultaneous parallel learning of multiple related tasks and the simultaneous back propagation of gradients. Multiple tasks help each other learn through the underlying shared representation and improve the generalization effect. In this task, due to the mixed sentences of Spanish and English, Hindi and English, and the need to predict emotions.Therefore, the Multi-task model is used to identify English and predict emotion at the same time, reduce network overfitting and improve generalization effect.\n\n\\textbf{Adam} - The Adam algorithm is different from the traditional stochastic gradient descent. Stochastic gradient descent keeps a single learning rate to update all the weights. The learning rate does not change during the training process, and Adam calculates the first and second moment estimates of the gradient design independent adaptive learning rates for different parameters. The Adam algorithm records the first moment of the gradient, that is, the average of all gradients in the past and the current gradient, so that each time the update is performed, the gradient of the last update and the current update will not differ much, that is, the gradient is smooth and stable can adapt to unstable objective functions. Adam recorded the second moment of the gradient, that is, the average of the square of the past gradient and the square of the current gradient, which reflects the environmental perception ability, and generates an adaptive learning rate for different parameters. This task uses the Adam algorithm to improve the overall efficiency of solving the problem, and performs more quickly and excellently when the gradient becomes sparse.\n\n\n\n\\textbf {Tfidf} - We use Tf-Idf to digitize the text in the data set as a string, and then evaluate the importance of a word for a file set or for a corpus where it is located. The importance of a word increases in proportion to the number of times it appears in the document, but at the same time it decreases in inverse proportion to its frequency in the corpus. Both high-frequency words and low file frequencies in the file collection will produce a higher weight TF-IDF, so we use it to filter common words and retain important words, thereby improving the efficiency and accuracy of the overall task.\n\n\n\n\n\\section{Experiment Results}\n\\begin{table}[h]\n    \\centering\n        \\begin{tabular}{ccc}\n        \\hline\n            System& F1(averaged)&\\\\\n            \\hline\n            Fine-tuned Multi-task BERT& 0.730\\\\\n            Fine-tuned BERT& 0.687\\\\\n            Original BERT& 0.589\\\\\n            \\hline\n        \\end{tabular}\n    \\caption{The results of experiments}\n    \\label{table:data}\n\\end{table}\nThe results on the official test sets for this task are presented in Table 1. Our multi-task BERT model, fine-tuned on the provided dataset.After exhausting top3 submissions, we achieved an averaged F1 score of 0.730. \n\n\n\n\\section{Conclusion}\n\nIn this article, we introduce the results of SemEval-2020 Sentimix Task 9: Sentiment Analysis of Code-Mixed Tweets recognition and classification. The goal of this task is to determine and classify mixed languages, and label mixed sentences of English-Hindi and English-Spanish with positive, negative or neutral emotions. In the task, we first preprocess the data with Emoji Substitution and Character Filtering, convert the emoji recognition into recognizable English words, and then use the Fine-tune method on the processed data to access a specific network after the bert model Layer, through Multi-task and Adam to reduce the network overfitting and improve the generalization effect, improve the efficiency and accuracy of the overall task, and  After exhausting top3 submissions, we achieved a score of 0.730 in this task.\n\nIn the future, we will conduct more in-depth research on the entiment Analysis of Code-Mixed Tweets, and use other models and methods to practice and try to continuously improve the stability and accuracy of classification recognition.\n\n\n\\printbibliography\n\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn interferometric fiber-optic gyroscope (FOG) is a device that measures the rotation rate by making use of the Sagnac effect, which induces a phase difference between two counterpropagating light waves traveling in the same fiber coil~\\cite{Lefevre2014,Lefevre2020,Korkishko2017}.\nBecause of their high precision and stability, FOGs have been widely used in numerous applications, such as satellite stabilization, inertial navigation, and rotational seismometers~\\cite{Lefevre2016,Sanders2016,Napoli2016}.\nIn evaluating the performance of FOGs, two aspects are often considered: long-term stability and short-term sensitivity.\nLong-term stability is crucial, for example, in terms of reducing drift and errors that can accumulate when FOGs are operated in an actual inertial navigation system (INS)~\\cite{Paturel2014}.\nBetter short-term sensitivity not only improves the performance of INSs but also expands the application of FOGs to other areas, such as seismology~\\cite{Schmelzbach2018,Kurzych2018}.\nOne way to improve the short-term sensitivity is to increase the scaling factor by using a larger fiber coil~\\cite{Toldi2017,Li2019}.\nHowever, optical power is attenuated in such a long fiber, and thereby the effect of shot noise could degrade the sensitivity~\\cite{Guattari2016}.\nIn addition, thermal phase noise becomes nonnegligible for long fibers~\\cite{Li2019,Knudsen1995,Moeller1996}.\n\nAnother method is to suppress the noises that impose a limitation on the short-term sensitivity~\\cite{Morris2022}.\nFour types of noise limit the sensitivity of a km-scale FOG: thermal phase noise (TPN), relative intensity noise (RIN), shot noise, and detection noise.\nSince the TPN originates from the thermodynamic fluctuation of the optical fiber, it decreases as the modulation frequency increases~\\cite{Li2019,Knudsen1995,Moeller1996}.\nThe TPN was successfully reduced in an experiment using a FOG with a 30\\,km-long single-mode (SM) fiber coil, where modulation frequency corresponding to the 33rd-order harmonic of the eigenfrequency was employed~\\cite{Li2019}.\nHowever, the RIN eventually limited the angular random walk in this experiment.\nIn modern FOGs, broadband light sources with low temporal coherence are the preferred choice since they reduce coherence-induced noise.\nHowever, the random beating between frequency components within the broad spectrum causes significant intensity noise called RIN~\\cite{Burns1990,Burns1996,Shin2010,Blake1994}.\nIn comparison to the shot noise level of the light source, this RIN is often referred to as excess RIN~\\cite{Burns1990}, and many methods of reducing or compensating for the excess RIN have been proposed and tested~\\cite{Shin2010,Blake1994,Yurek1986,Moeller1991,Guattari2014,Polynkin2000,Rabelo2000,Honthaas2014,He2020,Suo2021}.\nOne example is a technique called noise subtraction, which can be realized electronically~\\cite{Yurek1986,Moeller1991} or optically~\\cite{Guattari2014,Polynkin2000,Rabelo2000}.\nIn both cases, some portion of the light source is picked up as a reference, and its associated noise is considered to be correlated to that of the output power from the FOG.\nBy carefully adjusting the delay between the reference and the output, the noise in the output is compensated for with that in the reference.\nMany other efforts for RIN suppression have been made, such as optical spectrum filtering~\\cite{Honthaas2014}, a dual-polarization scheme~\\cite{He2020}, and a technique employing a semiconductor optical amplifier~\\cite{Shin2010,Suo2021}.\nWhen the TPN and excess RIN are sufficiently suppressed, the shot noise of the light source and detection noise finally limit the sensitivity.\n\nIn this paper, we propose and experimentally demonstrate simultaneous suppression of TPN and RIN.\nWe reduce TPN by employing a relatively high phase modulation frequency and suppress RIN by direct feedback to the drive current of a superluminescent diode (SLD) having a center wavelength of 1.5\\,$\\mu$m.\n\nSome portion of the SLD is picked up and detected by a photodetector, whose output voltage is used as a feedback signal to compensate for the intensity noise, i.e., excess RIN.\nWe examine the property of the RIN in a sinusoidal-wave modulation-demodulation scheme and use a simple method to suppress only certain frequency components of the RIN, namely, around the modulation frequency and its third-order harmonic.\nThe short-term sensitivity is evaluated by the angular random walk~(ARW) coefficient.\nOur simultaneous suppression of TPN and RIN yields an ARW of $\\sim 15\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$, which is a 6-fold improvement compared to $\\sim 93\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ obtained by the conventional eigenfrequency modulation-demodulation technique.\nOur FOG also shows good long-term stability, having a bias instability of $\\sim 33\\,\\mu\\mathrm{deg}\/\\mathrm{h}$ for a measurement time of 40 hours.\n\n\\section{Dominant noise factors and their suppression}\n\nTo improve the short-term sensitivity of a FOG, it is essential to examine the ratio of the FOG signal to the total noise of the system.\nWhen a phase shift $\\Delta \\phi_\\mathrm{R}$ is induced because of the Sagnac effect in the FOG, the detected signal photocurrent is described by\n\\begin{equation}\nI = \\frac{I_0}{2} \\Big\\{1+ \\cos \\left[ \\Delta \\phi_\\mathrm{R} + \\phi_\\mathrm{m} \\sin \\omega_\\mathrm{m} t \\right] \\Big\\},\n\\label{eq:fog}\n\\end{equation}\nwhere $\\omega_\\mathrm{m}$, $\\phi_\\mathrm{m}$, and $I_0$ represent the modulation frequency, modulation index, and signal current without rotation and modulation, respectively.\nThe corresponding rotation rate $\\Omega_\\mathrm{R}$ is calculated by $\\Omega_\\mathrm{R} = (c \\lambda \/4\\pi RL) \\Delta \\phi_\\mathrm{R} \\equiv \\Delta \\phi_\\mathrm{R}\/S_\\mathrm{F}$, where $c$, $\\lambda$, $R$, $L$, and $S_\\mathrm{F}$ are the speed of light, central wavelength,  radius of the fiber coil, fiber length, and scaling factor, respectively.\nThe output $I$ can be expanded using the harmonics of the modulation frequency, namely, $n\\omega_\\mathrm{m} \\,(n=1,2,\\cdots)$, and the corresponding Bessel functions $J_n (\\phi_\\mathrm{m})$ of integer order $n$ as follows:\n\\begin{eqnarray}\nI & \\approx & \\frac{I_0}{2} \\Big[1+ J_0 (\\phi_\\mathrm{m}) + 2 \\sum_{k=1}^{\\infty} J_{2k} (\\phi_\\mathrm{m}) \\cos 2k\\omega_\\mathrm{m} t\\Big] \\nonumber \\\\\n& &-I_0 \\Delta \\phi_\\mathrm{R} \\sum_{k=1}^{\\infty} J_{2k-1} (\\phi_\\mathrm{m}) \\sin (2k-1)\\omega_\\mathrm{m} t,\n\\label{eq:bess}\n\\end{eqnarray}\nwhere we assume $\\Delta \\phi_\\mathrm{R} \\ll 1$.\n\nThe component of $I$ oscillating at the modulation frequency $\\omega_\\mathrm{m}$ is $I_\\omega \\approx I_0 \\Delta \\phi_\\mathrm{R} J_1  (\\phi_\\mathrm{m})$.\nWe denote the demodulated current noises for TPN, RIN, shot noise, and detection noise by $\\sigma_\\mathrm{TPN}$, $\\sigma_\\mathrm{RIN}$, $\\sigma_\\mathrm{SN}$, and $\\sigma_\\mathrm{DN}$, respectively.\nEventually, an ARW is given by\n\\begin{equation}\n\\mathrm{ARW} = \\frac{1}{S_\\mathrm{F} J_1 (\\phi_\\mathrm{m}) I_0 \\sqrt{B}} \\sqrt{ \\sigma^2_\\mathrm{TPN} + \\sigma^2_\\mathrm{RIN} +\\sigma^2_\\mathrm{SN} +\\sigma^2_\\mathrm{DN}},\n\\label{eq:arw}\n\\end{equation}\nwhere $B$ represents the detection bandwidth.\n\nThe TPN originates from the random thermal motion of silica particles in the optical fiber~\\cite{Li2019,Knudsen1995,Moeller1996}.\nThis noise becomes more detrimental in a longer fiber.\nThe corresponding noise $\\sigma_\\mathrm{TPN}$ has been extensively studied~\\cite{Li2019,Knudsen1995} and expressed by\n\\begin{eqnarray}\n\\sigma_\\mathrm{TPN} & = & I_0 \\bigg\\{ \\pi B \\sum_{n=1}^{\\infty} \\Big[ J_{2n-1} (\\phi_\\mathrm{m}) - J_{2n+1} (\\phi_\\mathrm{m}) \\Big]^2 \\nonumber \\\\\n& & \\quad \\times \\langle \\Delta \\phi_{N,\\mathrm{rms}}^2 (2n \\omega_\\mathrm{m}) \\rangle \\bigg\\}^{1\/2},\n\\label{eq:tpn}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n& & \\langle \\Delta \\phi_{N,\\mathrm{rms}}^2 (\\omega) \\rangle \\nonumber \\\\\n& & = \\frac{k_B T^2 L}{\\kappa \\lambda^2} \\left( \\frac{\\mathrm{d}n_\\mathrm{eff}}{\\mathrm{d}T}+n_\\mathrm{eff} \\alpha_L\\right)^2 \\times \\ln \\left[ \\frac{\\left(\\frac{2}{W_0}\\right)^4+\\left(\\frac{\\omega}{D}\\right)^2}{\\left(\\frac{4.81}{d}\\right)^4+\\left(\\frac{\\omega}{D}\\right)^2}\\right] \\nonumber \\\\\n& & \\quad \\times \\left[ 1-\\mathrm{sinc} \\left(\\frac{\\omega L n_\\mathrm{eff}}{c} \\right) \\right].\n\\end{eqnarray}\nHere, $\\langle \\Delta \\phi_{N,\\mathrm{rms}}^2 (\\omega) \\rangle$ is the spectral density of phase noise introduced by the TPN.\n$k_\\mathrm{B}$ is Boltzmann's constant, and $\\mathrm{d}n_\\mathrm{eff}\/\\mathrm{d}T$ is the temperature coefficient of the effective refractive index $n_\\mathrm{eff}$ of the fiber.\nThe fiber is characterized by mode field diameter $2W_0$, cladding diameter $d$, fiber length $L$, linear thermal expansion coefficient $\\alpha_L$, thermal conductivity $\\kappa$, and thermal diffusivity $D$.\nNote that the expression in (\\ref{eq:tpn}) differs from the equation in \\cite{Li2019,Knudsen1995}, which considers that the spectral densities $\\langle \\Delta \\phi_{N,\\mathrm{rms}}^2 (\\omega) \\rangle$ and $\\langle \\Delta \\phi_{N,\\mathrm{rms}}^2 (-\\omega) \\rangle$ are statistically independent. However, they are correlated, and our equation accounts for this fact~\\cite{MM2022}.\nBased on this expression, the effect of TPN can be reduced by increasing the modulation frequency, which corresponds to the odd-order harmonic of the eigenfrequency of the coil.\nWe apply this approach to our coil made of a polarization-maintaining~(PM) fiber with $L \\sim 4920$\\,m below.\n\nExcess RIN is another dominant noise originating from the random beating between all the frequency components within the broad spectrum of a light source~\\cite{Burns1990,Burns1996,Shin2010,Blake1994}.\nThe amount of noise is simply given by the inverse of the frequency spectrum linewidth $\\Delta \\nu$ of the light source. \nIn the lock-in measurement, the RIN is distributed between the signal and quadrature phases, and the signal phase component $\\sigma_\\mathrm{RIN}$ can be calculated as follows:\n\\begin{eqnarray}\n\\sigma_\\mathrm{RIN} &=& \\frac{I_0 \\sqrt{B}}{2 \\sqrt{\\Delta \\nu} } \\bigg\\{ \\Bigl[ 1+J_0 (\\phi_\\mathrm{m})-J_2 (\\phi_\\mathrm{m}) \\Bigr]^2 \\nonumber \\\\\n& & \\quad +\\sum_{n=1}^{\\infty} \\Bigl[J_{2n} (\\phi_\\mathrm{m})-J_{2(n+1)} (\\phi_\\mathrm{m}) \\Bigr]^2 \\bigg\\}^{1\/2},\n\\label{eq:rin}\n\\end{eqnarray}\nunder the assumption that the RIN is white and each of its spectral components is statistically independent~\\cite{Blake1994}.\nThe spectrum bandwidth $\\Delta \\nu$ of the light source can be calculated using the expression\n\\begin{equation}\n\\Delta \\nu =\\frac{\\left[ \\int P(\\nu) d\\nu\\right]^2}{\\int P^2(\\nu) d\\nu}, \n\\label{eq:dnu}\n\\end{equation}\nwhere $P(\\nu)$ is the power spectral density of the light source with respect to frequency $\\nu$~\\cite{Burns1990}.\nWhen RIN is the dominant noise source, the ARW becomes a minimum at $\\phi_\\mathrm{m} \\approx 2.7$~\\cite{Blake1994}.\nThe essential point is that the first term in the square root in (\\ref{eq:rin}) comes from the RIN at frequency $\\omega_\\mathrm{m}$, whereas the second comes from the RIN at the odd-order harmonics.\nAs seen in (\\ref{eq:bess}), mixing between the $2k\\omega_\\mathrm{m}$ frequency component generated by phase modulation and the RIN at the odd-order harmonics leads to the $\\omega_\\mathrm{m}$ component and contributes to the demodulated current noise. \nAround $\\phi_\\mathrm{m}=2.7$, the contribution from the odd-order harmonics is not negligible, and the RIN suppression for only the first-order harmonics is insufficient to suppress $\\sigma_\\mathrm{RIN}$.\n\nNotably, the contributions from the first- and third-order components to $\\sigma_\\mathrm{RIN}$ are comparable, while that from the higher order is significantly small around $\\phi_\\mathrm{m}=2.7$.\nTherefore, RIN suppression at first- and third-order harmonic components effectively reduces $\\sigma_\\mathrm{RIN}$.\nWhen the dominant noises such as TPN and RIN are sufficiently suppressed, shot noise, which is given by\n\\begin{eqnarray}\n\\sigma_\\mathrm{SN} & = & \\bigg\\{e B I_0 \\sum_{s=0}^{\\infty} \\left[ J_{2s}\\left(\\frac{\\phi_\\mathrm{m}}{2}\\right) - J_{2(s+1)}\\left(\\frac{\\phi_\\mathrm{m}}{2}  \\right) \\right]^2 \\bigg\\}^{1\/2}   \\nonumber \\\\\n& = & \\bigg\\{\\frac{e B I_0}{2} \\left[ 1+J_0 (\\phi_\\mathrm{m})-J_2 (\\phi_\\mathrm{m}) \\right] \\bigg\\}^{1\/2},\n\\label{eq:pnsn}\n\\end{eqnarray}\nand detection noise will finally limit the sensitivity.\nNote that the expression in (\\ref{eq:pnsn}) differs from the conventional equation, which only includes the effects of the DC component in the detected signal~\\cite{Burns1990,Moeller1989}. In contrast, our equation also accounts for the AC contribution, which modifies the amount of noise in the case of a lock-in measurement~\\cite{MM2022}.\n\nDetection noise is generated mainly from a transimpedance amplifier with a photodiode~\\cite{Scott2001} and a lock-in amplifier, which is given by\n\\begin{equation}\n\\sigma_\\mathrm{DN} = \\sqrt{\\frac{B}{2}} \\bigg\\{2 e i_\\mathrm{D} + i_\\mathrm{NT}^2 + \\left( \\frac{v_\\mathrm{NT}}{R_\\mathrm{F}}\\right)^2 + \\frac{4k_\\mathrm{B}T}{R_\\mathrm{F}}+\\left( \\frac{v_\\mathrm{NL}}{R_\\mathrm{F}} \\right)^2 \\bigg\\}^{1\/2},\n\\label{eq:pndn}\n\\end{equation}\nwhere $i_\\mathrm{D}$ is the dark current of the photodiode, $i_\\mathrm{NT}$ and $v_\\mathrm{NT}$ are the input current noise spectral density (NSD) and input voltage NSD of a transimpedance amplifier, $R_\\mathrm{F}$ is the feedback resistance, and $v_\\mathrm{NL}$ is the input voltage NSD of a lock-in amplifier.\n\n\\begin{figure}[!ht]\n\\centering\\includegraphics[width=8.8cm]{Figure1.pdf}\n\\caption{Schematic setup of the FOG apparatus. SLD, superluminescent diode; iso, isolator; att, variable attenuator; MIOC, multifunctional integrated optical chip; PD, photodiode; PC, personal computer; FB, feedback circuit.} \n\\label{fig:setup}\n\\end{figure}\n\n\\section{Experimental setup}\nOur all-fiber-based experimental apparatus is schematically shown in Fig.~\\ref{fig:setup}. The light source is an SLD with a central wavelength $\\lambda$ of $\\sim 1544$\\,nm and a bandwidth of 52\\,nm (full width at half maximum) (see Fig.~\\ref{fig:spectrum}).\nThe central wavelength is calculated as an expectation value of the spectrum distribution in Fig.~\\ref{fig:spectrum}.\nThe SLD output goes through an isolator to avoid back reflection.\nNext, the primary output from the PM fiber coupler goes to a standard setup of a FOG, which consists of a circulator, a multifunctional integrated optical chip (MIOC), and a fiber coil. \nThe fiber coil is produced by winding a PM fiber with a length $L$ of $\\sim 4920$\\,m and an average radius $R$ of 115\\,mm using a quadruple winding pattern.\nIts eigenfrequency $f_\\mathrm{e}$ is estimated to be $20.3$\\,kHz.\nSinusoidal modulation is applied to the MIOC.\nThe output from the circulator is detected by an InGaAs photodiode (PD1) with a sensitivity of 0.95\\,A\/W, \nfollowed by a transimpedance amplifier with a gain of 12\\,kV\/A. The optical attenuation in the coil, including the MIOC, is 16.3\\,dB, whereas the attenuation of the bare fiber used for making the coil is less than 1\\,dB\/km. This difference comes from bending the fiber and using an epoxy glue in the fabrication process.\nThe typical power without phase modulation is 450\\,$\\mu$W at PD1.\nThe detected signal is measured by a lock-in amplifier.\nThe minor output from the coupler is used for RIN suppression. The signal detected by a photodiode (PD2) is sent to a homemade feedback circuit, whose output voltage is fed back to the current controller of the SLD. The details are mentioned later.  \n\nAs shown in (\\ref{eq:bess}), in the case of sinusoidal phase modulation, the output current $I$ can be expressed using the harmonics of the modulation frequency and the Bessel functions.\nThe magnitudes of the $i$-th harmonics $h_i$ can be measured by a lock-in amplifier with a multifrequency demodulation function.\nUsing the ratio of the first and second harmonic components $h_1$ and $h_2$, the Sagnac phase shift $\\Delta \\phi_\\mathrm{R}$ can be obtained in the following form~\\cite{Bohm1983}:\n\\begin{equation}\n\\Delta \\phi_\\mathrm{R} =\\tan^{-1} \\left( \\frac{J_2 (\\phi_\\mathrm{m} )}{J_1 (\\phi_\\mathrm{m} )} \\frac{h_1}{h_2} \\right).\n\\end{equation}\nHere the information of $\\phi_\\mathrm{m}$ is extracted from the following relation:\n\\begin{equation}\n\\frac{h_2}{h_4} = \\frac{J_2 (\\phi_\\mathrm{m} )}{J_4 (\\phi_\\mathrm{m} )},\n\\end{equation}\nwhere $h_4$ is the magnitude of the fourth-order harmonic component.\nNote that the fluctuation of the SLD power and that of the modulation index can be compensated for by this method.\n\n\\begin{figure}[!t]\n\\centering\\includegraphics[width=7.5cm]{Figure2.pdf}\n\\caption{Measured spectrum of the SLD at the photodetector PD1.} \n\\label{fig:spectrum}\n\\end{figure}\n\nThe fiber coil and MIOC are placed in a vacuum chamber with pressure below $5\\times10^{-2}$\\,Pa, which is surrounded by a magnetic shield and temperature stabilized at 24.0 $^\\circ \\mathrm{C}$ within 1\\,mK.\n\n\n\\section{Results}\n\nFig.~\\ref{fig:arw} shows the measured ARW (red circle data points) as a function of modulation frequency $f_\\mathrm{m}=\\omega_\\mathrm{m}\/2\\pi$.\nHere, we measure the Allan deviation of the rotation angular rate $\\Omega_\\mathrm{R}$ as a function of integration time $\\tau$ and fit it with a function $\\mathrm{ARW}\/\\sqrt{\\tau}$. \nNote that excess RIN is not yet suppressed at this stage, and we use the modulation index $\\phi_\\mathrm{m} = 2.7$ where the optimal ARW can be expected.\nIt is clearly shown that the measured values of ARW decrease monotonically with the modulation frequency, similar to the case of the SM fiber in \\cite{Li2019}.\nThe measured values reach a lower limit of $\\sim 31\\,\\mathrm{\\mu deg}\/\\sqrt{\\mathrm{h}}$ in a frequency range larger than $386$\\, kHz, which corresponds to the 19-th order harmonic of the eigenfrequency $f_\\mathrm{e}$ in our setting.\n\nThis figure also shows a fitting curve to the measured data points with (\\ref{eq:arw}), which considers TPN, RIN, shot noise, and detection noise.\nTo calculate $\\sigma_\\mathrm{TPN}$, we use the values written in \\cite{Li2019} regarding the fiber characteristics such as $n_\\mathrm{eff}$, $\\mathrm{d}n_\\mathrm{eff}\/\\mathrm{d}T$, $\\alpha_L$, and $\\kappa$.\nOther parameters are $2W_0= 6.7\\,\\mu$m and $d=80\\,\\mu$m from our manufacturer's data sheet.\nThe thermal diffusivity of an optical fiber is used as a fitting parameter, since that of fused silica glasses is different between bulk material and optical fiber as discussed in \\cite{Chardon1983}.\nIn fact, the measured value for the fiber in \\cite{Chardon1983} was $1.21\\times10^{-7} \\mathrm{m}^2 \\mathrm{s}^{-1}$ and was smaller than the bulk diffusivity of $8.3\\times10^{-7} \\mathrm{m}^2 \\mathrm{s}^{-1}$, which is typically used in the calculations on TPN~\\cite{Li2019,Knudsen1995,Moeller1996}.\nWe obtain a thermal diffusivity of $(5.5\\pm0.8)\\times 10^{-7} \\mathrm{m}^2 \\mathrm{s}^{-1}$, which has no discrepancy with the above values. \nThe excess RIN is calculated by (\\ref{eq:rin}), and the spectrum bandwidth $\\Delta \\nu$ of the light source in (\\ref{eq:dnu}) is obtained as $\\Delta \\nu = 9.71$\\,THz from the measured spectrum in Fig.~\\ref{fig:spectrum}.\nThis model assumes pure spontaneous emission as a light source, but due to the amplification process in the SLD, the experimental value of the RIN can be smaller than the calculated one~\\cite{Suo2021}.\nTherefore, the previous work introduced the noise suppression factor $\\eta$ and divided the noise $\\sigma_\\mathrm{RIN}$ by $\\sqrt{\\eta}$, where the $\\eta$ values were 4.17, 5.28, and 4.65~\\cite{Shin2010}.\nSimilarly, we use this factor $\\eta$ as a fitting parameter and obtain $\\eta = 1.91\\pm 0.17$. \nThe magnitudes of shot noise in (\\ref{eq:pnsn}) and detection noise in (\\ref{eq:pndn}) are calculated with our experimental parameters.\nThe contributions of RIN, TPN, shot noise, and detection noise are plotted in Fig.~\\ref{fig:arw} as dashed lines.\n\n\\begin{figure}[!t]\n\\centering\\includegraphics[width=8.8cm]{Figure3.pdf}\n\\caption{Modulation frequency dependence of an angular random walk (ARW). Measured values of an ARW are plotted as a function of modulation frequency $f_\\mathrm{m} = \\omega_\\mathrm{m}\/2\\pi$ for the case without RIN suppression (red circle points).\nThe red solid line shows the fitted result of the total noise. Calculation results for RIN and TPN using the fitted values are presented by orange and grey dashed lines, respectively.\nThe purple and green dashed lines show the calculations for shot noise and detection noise, respectively.\nThe modulation depth is set to $\\phi_\\mathrm{m}= 2.7$.\nThe error bars of the data points, which are the standard deviation of the fitting of the ARW, are smaller than the marker size.\n} \n\\label{fig:arw}\n\\end{figure}\n\nClearly, in the lower modulation-frequency range below $100$\\,kHz, the TPN is dominant, whereas its contribution decreases in the higher frequency range, and the RIN becomes dominant.\nThe short-term sensitivity of the FOG can be improved by suppressing the RIN in the higher frequency range.\nHereafter, we set our modulation frequency $f_\\mathrm{m}$ to $630.6$\\,kHz ($f_\\mathrm{m}=31f_\\mathrm{e}$), at which the TPN contributes less than the shot noise.\n\n\nAs discussed in the theoretical section, we suppress excess RIN at particular frequencies, namely, $f_\\mathrm{m}$ and\/or $3f_\\mathrm{m}$, using the feedback to the current driver of the SLD.\nAs shown in Fig.~1, some portion of the SLD output is picked off by the fiber coupler and detected by PD2.\nThe output voltage of the transimpedance amplifier after PD2 is split into two.\nEach of the outputs goes through a bandpass filter whose resonance frequency is set to $f_\\mathrm{m}$ or $3f_\\mathrm{m}$, and its output is used in the feedback.\nFig.~\\ref{fig:nd} shows the noise spectral density detected by PD1 when phase modulation to the MIOC is not applied.\nThe RIN is successfully suppressed around the frequencies $f_\\mathrm{m}$ and $3f_\\mathrm{m}$ by a factor of $\\sim$2.7 and $\\sim$2.1, respectively.\n\n\n\\begin{figure}[!t]\n\\centering\\includegraphics[width=8.8cm]{Figure4.pdf}\n\\caption{\nNoise spectral densities of the light source measured at photodiode PD1.\nThe red trace shows the result without RIN suppression.\nThe green and blue traces show the results with RIN suppression at $f_\\mathrm{m}$ only and at $f_\\mathrm{m}$ and $3f_\\mathrm{m}$, respectively.\nThe vertical dashed lines indicate $f_\\mathrm{m}$ and $3f_\\mathrm{m}$.\n} \n\\label{fig:nd}\n\\end{figure}\n\nUnder these RIN-suppression conditions, we measure the dependence of the ARW on the modulation index $\\phi_\\mathrm{m}$, as shown in Fig.~\\ref{fig:arw31}. \nFirst, we examine the case without RIN suppression.\nThe red circle points in Fig.~\\ref{fig:arw31} are the measured results, showing a minimum value of $\\sim 31\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ at $\\phi_\\mathrm{m} \\approx 2.7$, as expected.\nThe red solid line shows the theoretical calculation using the fitted values obtained in Fig.~\\ref{fig:arw}.\nThese two results are in good agreement.\nWe also measure the case with RIN suppression at $f_\\mathrm{m}$ only.\nThe green triangle points are the measured results, clearly showing that the ARW values are significantly improved compared to the case without RIN suppression.\nThe minimum value in this case is $\\sim 22\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$, and its position is shifted to $\\phi_\\mathrm{m} \\approx 2.0$.\nThese results are in good agreement with a theoretical calculation based on (\\ref{eq:rin}) by considering the amount of RIN suppression using a factor of $\\sim 2.7$ around $f_\\mathrm{m}$.\nSimilarly, the measured and theoretical results for the case of RIN suppression at $f_\\mathrm{m}$ and $3f_\\mathrm{m}$ are presented by the blue square points and blue solid line, respectively.\nThe calculation considers the RIN suppression using a factor of $\\sim 2.7$ at $f_\\mathrm{m}$ and $\\sim 2.1$ at $3f_\\mathrm{m}$, showing good agreement with the experiment.\nAll the excellent agreements between the measured results and the calculations suggest that our understanding of the RIN based on (\\ref{eq:rin}) is correct.\nThis understanding shows that even if all the frequency components of the RIN are suppressed by the same amount as the $f_\\mathrm{m}$ component ($\\sim$2.7), the ARW value does not improve significantly (the blue dotted line).\nOn the other hand, completely suppressing only the $f_\\mathrm{m}$ component improves the ARW, showing a value of $\\sim 13\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ (the black dotted line), and completely suppessing $f_\\mathrm{m}$ and $3f_\\mathrm{m}$ leads to further improvement, reaching a value of $\\sim 8\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ (the black solid line).\n\n\\begin{figure}[!t]\n\\centering\\includegraphics[width=8.8cm]{Figure5.pdf}\n\\caption{Modulation index dependence of an ARW at $f_\\mathrm{m}=630.6$\\,kHz.\nMeasured values of an ARW are plotted for the cases without RIN suppression (red circle points) and with RIN suppression at $f_\\mathrm{m}$ only (green triangle points) and at $f_\\mathrm{m}$ and $3f_\\mathrm{m}$ (blue square points).\nThe red, green, and blue solid lines are the corresponding theoretical calculations using the fitted values obtained in Fig.~\\ref{fig:arw}.\nThe blue dotted line is the theoretical calculation assuming that all the frequency components of the RIN are suppressed by the same amount as the $f_\\mathrm{m}$ component.\nThe black dotted and solid lines correspond to ARW values when the RIN is completely suppressed at $f_\\mathrm{m}$ only and at $f_\\mathrm{m}$ and $3f_\\mathrm{m}$, respectively.\nThe orange, green, purple, and grey dashed lines show the calculated results for the RIN, detection noise, shot noise, and TPN, respectively.\nThe error bars of the data points, which are the standard deviation of the fitting of the ARW, are smaller than the marker size.} \n\\label{fig:arw31}\n\\end{figure}\nWe obtain the minimum ARW value $\\sim 15\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ at $\\phi_\\mathrm{m} = 2.24$, and we also examine the long-term stability under this experimental condition.\nFig.~\\ref{fig:allan} shows the measured Allan deviation of the rotation rate $\\Omega_\\mathrm{R}$ for a measurement time of 40 hours.\nDespite such a long measurement time, we achieve a bias stability of $\\sim 33\\,\\mu\\mathrm{deg\/h}$, showing that our FOG has excellent long-term stability.\nFor a time scale longer than several hours, a ramp rate with a slope of $+1$ is observed, indicating a very slow monotonic change in the FOG signal over a long period of time~\\cite{IEEE1998}.\nThis change could be due to a drift of the central wavelength and the corresponding scale factor.\nAn investigation of the ramp rate is necessary but beyond the scope of the present study.\nThe ARW of $15\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ corresponds to an interferometer phase noise of $7\\times 10^{-8}\\,\\mathrm{rad}\/\\sqrt{\\mathrm{Hz}}$ and provides a root mean square phase noise of $7\\times 10^{-9}\\,\\mathrm{rad}$ after $100$ seconds of average time. The total phase accumulated after the propagation in the fiber coil is $3\\times 10^{10}\\,\\mathrm{rad}$. Therefore, reciprocity is estimated to be $\\sim 2\\times10^{-19}$, which is the same order as the current high-performance FOG~\\cite{Lefevre2020}.\n\n\\begin{figure}[!t]\n\\centering\\includegraphics[width=7.5cm]{Figure6.pdf}\n\\caption{Allan deviation of the measured rotation rate $\\Omega_\\mathrm{R}$. The experimental condition is the same as that corresponding to the blue square point at $\\phi_\\mathrm{m} \\approx 2.24$ in Fig.~\\ref{fig:arw31}.\nThe red dashed line shows the fitted result with a function of $\\mathrm{ARW}\/\\sqrt{\\tau}$.\n}\n\\label{fig:allan}\n\\end{figure}\n\n\\begin{table}[!ht]\n\\caption{Parameters of high-performance interferometric FOGs}\n\\label{table:ARW}\n\\centering\n\\begin{tabular}{c c c c}\n\\hline\nARW & fiber length & fiber-coil diameter & Ref.\\\\\n\\hline\n$69\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ & $5000$\\,m & $250$\\,mm & \\cite{Korkishko2017}\\\\\n$32\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ & $5000$\\,m & $180$\\,mm & \\cite{Lefevre2020}\\\\\n$12\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ & $30630$\\,m & $380$\\,mm & \\cite{Li2019}\\\\\n$15\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$ & $4920$\\,m & $230$\\,mm & present work\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\n\nWe have demonstrated the simultaneous suppression of TPN and RIN. \nWe have revisited the derivation of the RIN distribution in the lock-in detection scheme and shown that it is sufficient to suppress the RIN components at only the modulation frequency and its third-order harmonic. \nWe observed significant improvement of the ARW coefficient by simply suppressing the first- and third-order components, where we employed the modulation frequency of the 31st harmonics of the eigenfrequency of the fiber coil.\nAlthough suppression of RIN at all frequency components is technically difficult, especially for coils with large eigenfrequencies, suppression of only two low-frequency components is relatively simple.\nOur simultaneous suppression of TPN and RIN yielded an ARW coefficient of $\\sim 15\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$, as well as a bias instability of $\\sim 33\\,\\mu\\mathrm{deg}\/\\mathrm{h}$ for a measurement time of 40 hours.\nThis ARW is, to our knowledge, the best one in the world for FOGs using a kilometer-length fiber coil as represented in Table~\\ref{table:ARW}.\nAchieving such a small ARW is very important in practical INS applications, reducing the time required for initial INS alignment.\nAs discussed in~\\cite{Paturel2014}, bias instability below $20\\,\\mu\\mathrm{deg}\/\\mathrm{h}$ is necessary to maintain a navigational accuracy of a one nautical mile per month.\nIn order to obtain this value within one-hour alignment, the ARW must be less than $20\\,\\mu\\mathrm{deg}\/\\sqrt{\\mathrm{h}}$~\\cite{Paturel2014}.\nOur performance has reached this level, and therefore paves the way for practical application to ultra-precise inertial navigation in long-term operation.\n\n\n\\section*{Acknowledgments}\nWe thank K. Aikawa, Y. Kamino, H. Inaba, Y. Yatsu, T. Mukaiyama, D. Akamatsu, T. Ido, M. Yasuda, and K. Tsuchiya for fruitful discussions. We are grateful to H. Ishijima for his technical assistance with the experiments.\n\n\n\n\\bibliographystyle{apsrev4-2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFlux-dominated solar dynamo models, which use a solar like velocity\nfield, together with an estimated diffusivity \\linebreak profile and\nan $\\alpha$ effect resembling the Babcock-Leighton mechanism for\nproducing poloidal magnetic fields, have \\linebreak  demonstrated to provide\nsolutions that resemble quite well the observations of the large scale\nsolar magnetic cycle \\linebreak (Dikpati et al. 2004, Chatterjee, Nandy \\&\nChoudhuri 2004, Guerrero \\& de Gouveia Dal Pino 2007, hereafter GDP).\n\nIn this kind of models, the usually accepted scenario for the dynamo\noperation can be described as follows: starting with a dipolar\nfield, the $\\Omega$ effect takes place both at the convection zone\nand the tachocline, and a toroidal magnetic field is formed and\namplified by the radial and latitudinal components of the\ndifferential rotation. Once this field reaches values between $10^4$\n- $10^5$ G, it becomes buoyant unstable and arises through the\nconvection zone being twisted by the Coriolis force until it emerges\nat the surface and forms bipolar magnetic regions. This part of the\nprocess is known as Babcock - Leighton $\\alpha$ effect. At the\nsurface, the leading part of a bipolar region points to the equator,\nwhile the rear part is drifted polarward by the meridional\ncirculation. This migration contributes to the formation of big\nloops in each hemisphere, which later will reconnect in order to\nform a new poloidal field of opposite polarity to the initial one.\n\nIn view of the present status of the\nobservations, the model proposed above presents several problems, as\n\\linebreak remarked by Brandenburg (2005), the most important of which\nis related to the location, the sign and the amplitude of the alpha\neffect. In the scenario discussed above, the $\\alpha$ term\nshould  reproduce the emergence of magnetic flux tubes of strong\ntoroidal magnetic fields and then the migration of these fields\ntoward the poles. In this way, the most probable location for the\ndominance of this\nterm would be the upper layers of the convection zone, with a\nlatitudinal  distribution peaking around $30 ^{\\circ}$, which is the\nlatitude at which the strongest magnetic activity is observed. If\nthe solar dynamo is operating at the tachocline, where the shear has\npositive values, the natural sign of the alpha effect should be\nnegative (positive) in the north (south) hemisphere, in order to get\nequatorward branches in agreement with the mean field dynamo theory.\nHowever, the presence of an equartorward meridional flow at deeper\nlayers could bring the possibility of a positive (negative) $\\alpha$\neffect at the north (south) hemisphere and still give the correct\nmigration. Recent works have suggested that the amplitude of the\n$\\alpha$ term could be important in the determination of the parity of\nthe dynamo (Dikpati \\& Gilman 2001, Bonnano et al. 2002), this effect is\nnot considered in this work since our domain corresponds to one\nhemisphere only, but it will be addressed in a forthcoming paper\n(Guerrero \\& de Gouveia Dal Pino 2007, in preparation).\n\nAnother important question regards the real importance of the\ntachocline in the dynamo process. As a layer of strong radial shear,\nthe tachocline could be the ideal site for the amplification of the\nmagnetic field, though the maximum amplitude of  the shear is\nlocated at high latitudes, which suggests an intense activity  close\nto the poles. In this case, meridional circulation could be again an\nappropriate solution to this problem, since a velocity of about $3$\nm s$^{-1}$ could advect down and equatorward the magnetic field\nacross a stable layer until reaching the latitudes where the strong\nactivity is observed (Nandy \\& Choudhuri 2002, Chatterjee et al.\n2004). However, the use of a deep meridional  flow in a flux\ndominated dynamo has been found to be very sensitive to the adopted\n$\\alpha$ effect and  the diffusion and velocity profiles (Guerrero\n\\& Mu\\~{n}oz 2004) and can also generate undesirable abundance\nvariations in the convection zone. In addition, several numerical\nsimulations (Gilman \\& Miesh 2004, R\\\"{u}diger, Kitchatinov \\& Arlt\n2005) have demonstrated the impossibility of penetration of the\nmeridional flow below the tachocline more than a few kilometers. In\norder to allow some shallower penetration, other possibilities, such\nas overshooting or turbulent magnetic pumping have been invoked in the \nliterature (Rogers, Glatzmaier \\& Jones 2006, Ziegler \\& R\\\"{u}diger\n2003, respectively), where magnetic flux tubes are drifted\nto the overshoot layer and can be amplified to the desired\nmagnitudes.\n\nIndeed, if a flow penetrates inside the tachocline, there should be\na mechanism that could prevent either the formation of strong\ntoroidal magnetic fields at higher latitudes or the participation of\nthese fields in the cycle which is responsible for the observed\nphenomena. In recent work (GDP), we have shown that a possible\nsolution to the problem above could be the choice of an appropriate\nset of physical parameters, since the model is sensitive to the\nlocation of the transition between the sub-adiabatic and the\nsuper-adiabatic layers, to the diffusivity value at the bulk of the\nconvection zone and to the thickness of the solar tachocline (see\nFigs. 8a and 8b of GDP). We have also found that the latitudinal\nshear term is dominant over the radial shear term throughout the\ncycle, and that the radial shear peaks only in the early years of\nthe cycle and decays afterward  to values about two orders of\nmagnitude smaller than the latitudinal component.\n\nIn this paper, we present a detailed description on how the\nsuppression of the magnetic fields at high latitudes is obtained. We\nalso present the results when two different prescriptions for the\nmeridional circulation profile are employed. They seem to be\ninsensitive to these variations.\n\n\\section{Model}\n\nOur model is quite similar to the one explained in detail in\nGuerrero \\& Mu\\~{n}oz 2004 and GDP. In essence, we solve the mean\nfield induction equation in the $r$ and $\\theta$\ncoordinates by using a second order finite difference scheme for the\nspatial derivatives (Lax-Wendroff for the first derivatives and FTCS\nfor the second ones, Press et al. 2002) and the second order ADI\nmethod for the temporal advance \\footnote{For sake of\n    simplicity we have not included in our induction equation a\n    diamagnetic diffusion term which could be important when gradients \n    of turbulent diffusivity are present (see e.g., Kitchatinov \\&\n    R\\\"{u}diger, 1992)}. For the present \nwork, we have written a new parallel version of the code using the MPI\nlibraries, obtaining a speedup factor around $3$ in a four-processor\ncomputer. This has allowed us to use a resolution of 200 x 200 grid\npoints, with an adjustable time step to fulfill the Courant condition\nand perform long and fast numerical simulations.\n\n\\begin{table}\n\\centerin\n\\caption{Parameters employed in the present model compared to those of\n  GDP. $U_0$ is the maximum amplitude of the meridional circulation at\n  the surface, $R_p$ is the penetration depth of the flow, $\\beta_1$\n  and $\\beta_2$ fit the latitudinal profile of the meridional\n  circulation to the helioseismology results, $\\eta_c$ and $\\eta_s$\n  are the values of the magnetic diffusivity at the convection zone\n  and at the surface, respectively.}\n\\label{tlab}\n\\begin{tabular}{ccc}\\hline\nParameter & Value in GDP & Present value \\\\ \\hline\n$U_0$ & $25$ m s$^{-1}$ & $10$ m s$^{-1}$ \\\\\n$R_p$ & $0.69 R_{\\odot}$ & $0.70 R_{\\odot}$ \\\\\n$\\beta_1$ & $6.06 \\times 10^9$ cm$^{-1}$ & $3.47 \\times 10^{10}$\ncm$^{-1}$\\\\\n$\\beta_2$ & $4.6 \\times 10^9$ cm$^{-1}$ & $1.39 \\times 10^{10}$\ncm$^{-1}$\\\\\n\\hline\n$\\eta_c$  & $5\\times 10^9$cm$^2$s$^{-1}$ & $3\\times 10^{10}\n$cm$^2$s$^{-1}$ \\\\\n$\\eta_s$  & $1\\times 10^{12}$cm$^2$s$^{-1}$ & $3\\times 10^{12}\n$cm$^2$s$^{-1}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Results}\n\n\\begin{figure*}\n\\includegraphics[scale=0.72]{fig1Low.eps}\\\\\n\\caption{Snapshots of the positive (negative) toroidal field contours\n  are shown in blue (red) scale together with the field lines of the\n  positive (negative) poloidal field in continuous (dashed) lines,\n  for four different times ($T\/8$, $T\/4$, $3 T \/8$ and $T\/2$) along a\n  half 11-year cycle. The upper (a), middle\n  (b) and bottom (c) panels present the same snapshots for models\n  with a thin ($0.02 R_{\\odot}$), intermediate ($0.06 R_{\\odot}$) and thick ($0.1\n  R_{\\odot}$) tachocline.}\n\\label{label1}\n\\end{figure*}\n\nFor comparison, we have considered the same velocity and diffusion\nprofiles of GDP, but have altered the value of some of the\nparameters, as indicated in Table 1. We note that, as\nin GDP (see Fig. 3 of that work) we have considered two different\nprofiles for the magnetic diffusivity in the convective zone. In a\nthin layer near the top, we have assumed  a supergranular diffusivity,\nwhile in the inner region of the  convective zone  we have adopted a\nlower value. Indeed, there is no physical reason why this value should\nbe smaller than that near the top. However, if we take the same large\nvalue ($10^{12}$ cm$^2$ s$^{-1}$) for the entire convection zone, the\ndynamo will enter in the diffusion dominated regime.\nAnother important alteration is the location of the\neruption of the magnetic flux tubes. In GDP, the toroidal magnetic\nfields are transported from the top of the tachocline, i.e. $\\alpha$\n$\\propto$ $B_{\\phi}(R_c + \\frac{d_1}{2},\\theta)$ (where $B_{\\phi}$ is\nthe azimuthal component of the magnetic field, and $R_c$ is the\nlocation of the center of the \\linebreak tachocline of thickness\n$d_1$), while in the present \nwork the eruption point is placed in the middle of the overshoot layer\n($r_c$ $=$ $0.715 R_{\\odot}$, see the dashed line in the snapshots of\nFig. 1). Besides being more realistic, this assumption allows us to\nmake direct comparisons with the stability analysis of magnetic flux \ntubes in the overshoot layer (see Fig. 1 of Ferriz-Mas \\&\nSch\\\"{u}ssler 1994). \\footnote{We note that this scheme is slightly\ndifferent from the one\n  assumed by Dikpati et. al. (2004), who take the average value for the\n  toroidal magnetic field along the overshoot layer.}\nFig. 1 displays four snapshots of the evolution of the toroidal\nmagnetic field (in color-scale)  and poloidal field (lines) for four\ndifferent stages of a half (11-year) cycle. The top, middle and\nbottom panels correspond to a thin, intermediate and a thick\ntachocline, respectively. As it can be seen, at the beginning of each\nnew cycle, toroidal magnetic fields start to be formed at two\ndifferent places: one part inside of the convective layer,\naround $70$ degrees, where the poloidal shear term has its maximum\nvalue; and another one at the tachocline, very close to the poles\nwhere the radial shear reaches its maximum amplitude.\n\nWhen a thin tachocline is considered (top of Fig. 1), the toroidal\nfield generated at the poles remains confined below the center of\nthe overshoot layer.  At the place where we consider the eruption of\nthe magnetic flux tubes, this high latitude toroidal field does not\nreach the necessary values to become buoyant. This effect can be\nseen also in the upper panel of  Fig. 2, where a time-latitude\nbutterfly diagram for the contours of both, the toroidal magnetic\nfield at $r_c$ (lines) and the radial magnetic field (grey scale\nbackground) at the surface are shown. The contours for the toroidal\nfield are equally log-spaced for amplitudes above $5 \\times 10^4$ G,\nwhich is the value at which the buoyancy begins to develop. Along\nwith the confinement, we can note that the high latitude toroidal\nfield diffuses very efficiently since its generation stops, as\nquickly as, the radial shear component decreases. On the other hand,\nthe field which is developed inside the convection zone is advected\ndown and equartorward by the meridional flow, subtly penetrating the\ntachocline and reaching values above $5  \\times 10^4$ G .  The\nsnapshots show that when the toroidal fields are going downward, the\ntoroidal magnetic field for the new cycle begins to be formed. The\nappearance of this new field seems to dislocate the older one and\ncause an additional push downward.\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.78]{fig2Low.eps}\n\\caption{Time-latitude butterfly diagram for the model of Table 1 and\n  top of Fig. 1 with a thin tachocline (top panel), and for the\n  meridional circulation profile used by Dikpati \\&\n  Charbonneau 1999 (bottom panel). The continuous (dashed) lines\n  represent the positive (negative) strength of the toroidal field at\n  the center of the overshoot layer ($r_c$$=$$0.715 R_{\\odot}$). The lines\n  are log-spaced and cover the interval between $5 \\times 10^4$ -\n  $10^5$  G. The background gray scale represents the positive (dark)\n  and negative (clear) radial field at the surface. }\n\\label{label1}\n\\end{figure}\n\nThe middle (b) and bottom (c) panels of Fig. 1 show the same stages\nof the cycle for models with an intermediate and a thicker\ntachocline. These diagrams show that the radial shear, in spite of\nhaving a smaller amplitude than in the previous case, is distributed\nalong a larger area, so that a larger portion of magnetic field is\namplified leading to a strong solar activity at high latitudes which\nis incompatible  with the observations. Note that this result\ndiffers from the one previously obtained in GDP due to the distinct\nlocation of the domain of influence of the $\\alpha$ effect. When the\nbuoyancy point is placed in the center of the overshoot layer, as in\nthe present case, only a thin tachocline is able to lead to solar\nlike results (top panel of Fig. 2).\n\nWe have  also tested  the validity of this mechanism to avoid the\nformation of high latitude toroidal fields with another meridional\ncirculation profile. The bottom panel of Fig. 2 shows the butterfly\ndiagram obtained with the analytical profile used by Dikpati \\&\nCharbonneau (1999). We note that it reproduces quite well the\nobservations, with the butterfly wings of the toroidal field only\nbelow $\\sim 45$ degrees and with a half cycle period of $15$ years.\n\nFinally, we have also performed simulations similar to those of Fig.\n2, but (artificially) turning off the radial shear term (i.e.\n$\\partial \\Omega \/ \\partial r$$=$$0$) in the induction equation\n(see, e.g. equations 1 and 2 of GDP) and found that the latitudinal\ndistribution of the toroidal magnetic fields remains unaltered in\nthe butterfly diagrams. This result indicates that the fields\ngenerated in the tachocline do not significantly contribute to the\nobserved activity, suggesting  that in a flux dominated solar dynamo\nprocess with a Babcock-Leighton $\\alpha$ effect the real role of the\ntachocline is to be, essentially, a storage region, provided that it\ncoincides with the overshoot layer. This is in agreement with the\nconclusions of Dikpati, Gilman \\& MacGregor (2005).\n\n\\section{Conclusions}\n\nWe have described a physical scenario in which a simple\nflux-dominated solar dynamo model with an appropriate set of\nparameters produces results which are in good agreement with the\nobservations. The main assumptions and results are summarized below:\n\n\\begin{enumerate}\n\\item\nSince an analysis of the evolution of the maximum of the shear terms\nalong the cycle shows that the latitudinal shear is always dominant\nover the radial one (which attains appreciable  values only during\nthe first years and quickly decays afterward), a possible mechanism\nto prevent the formation of toroidal magnetic fields  by the radial\nshear at high latitudes could be the  consideration of a thin\ntachocline. In this case, we find that  the dynamo will form\na thin layer of toroidal field below the overshoot layer  with\ninsufficient amplitude to produce buoyant flux tubes. Several\nnumerical tests have demonstrated the robustness of this assumption\nin order to avoid the appearance of sunspots near the poles.\n\\item\nThe bulk of the toroidal magnetic field is formed at the interior of\nthe convection zone thanks to the latitudinal shear and is drifted\nby convection and\/or magnetic pumping towards the sub-adiabatic\novershoot layer \\linebreak where we assume that it\nundergoes magnetic buoyancy and rises to the surface at values above\nof $5$$\\times$$10^4$ G.\nNote that the turbulent diamagnetism (which\nwas mentioned in the section 2) may also play an important role in\norder to pile-up the toroidal magnetic field in the overshoot layer,\nsince it behaves like a downward velocity term in the induction\nequation, as demonstrated by \\linebreak Kitchatinov \\& R\\\"{u}diger\n(1992). The importance of this term has not yet been studied in\ndetail, but will be considered in forthcoming studies.\n\\item\nThe above mentioned results suggest  that the dynamo which is\nresponsible for the observed activity does not operate at the\ntachocline, as usually assumed, but inside the convection zone and\nis due to the strong latitudinal shear. In this sense, the real role\nof the tachocline, if it coincides with the overshoot\nlayer, is to store the magnetic flux tubes until they reach the\nnecessary amplification to become buoyantly unstable.\n\\end{enumerate}\n\n\\acknowledgements We would like to thank to Dr. G. R\\\"{u}diger for\nthe invitation to participate of the very interesting Thinkshop in\nPostdam and to an anonymous referee for helpful comments on the\nmanuscript. We also acknowledge partial support from the Brazilian\nAgencies CNPq and FAPESP. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe study of open quantum systems is important both for fundamental and practical purposes. When an open system interacts with its environment, this typically leads to decoherence and loss of quantum properties~\\cite{Breuer2007,Rivas2012}, which -- in turn -- makes it often difficult to implement quantum protocols in an ideal manner in experiments~\\cite{Suter2016}.\nIn general, during the last ten years, there have been significant developments in understanding and characterizing the abundant and diverse features of\n open system dynamics~\\cite{RevRivas,Breuer2016,devega,LiRev,p1,p2}.\nThese developments have influenced, and have been influenced by, the increasing ability to realize experimentally reservoir engineering~\\cite{wineland}, various fundamental open system models in non-Markovian regime~\\cite{exp2},  and the control of open system dynamics~\\cite{zdliu}. Indeed, by now a number of various physical platforms have been used for this purpose\nincluding, e.g.,  optical systems~\\cite{exp2,zdliu,achiu,bhliu13,fff,Ber2015,sdc,scia1,syu,acu,sau}, NV-centers~\\cite{haa,nori2020}, trapped ions~\\cite{mwit}, and NMR-systems~\\cite{Ber2016,dku}. In addition to fundamental studies and tests, recent experimental work also includes some of the first exploitations of non-Markovian memory effects in basic quantum information protocols including, e.g., single qubit Deutsch-Josza algorithm~\\cite{DJ}.\n\nConsidering entanglement, as a matter of fact, it plays a dual role when considering implementation of  quantum protocols with systems interacting with their environments. To start with, we need entanglement -- as a quantum resource -- when implementing quantum protocols and to go beyond what can be achieved by classical resources. However, when an open system interacts with its environment, the entanglement within the open system decreases due to decoherence,  and the efficiency of the quantum protocol typically diminishes. At the same time, the open system often gets entangled with its environment. This means that the total system-environment state still contains useful  resource while it does not anymore reside in the degrees of freedom which are explicitly used for the implementation of the quantum protocol.\n\nTherefore, we arrive to the following question: Is it possible to make efficient experimental realization of a quantum information protocol via open quantum system?\nWe answer this question affirmatively and demonstrate a proof-of-principle experiment by using teleportation~\\cite{Bennett} as an example. \nThis also means that we combine the central concepts of quantum information and open quantum systems in a new fundamental manner in an experiment.\nNote that for technological purposes, there has been recently impressive experiments, e.g., demonstrating ground-to-satellite teleportation~\\cite{g2sat}\nand superdense teleportation for information transfer using a pair of hyper-entangled photons~\\cite{Kwiat2015}. \nWe, instead, rather see our current contribution dealing with fundamental questions on open quantum systems and quantum information. Our experiment uses the concept of nonlocal memory effects~\\cite{nlnm,bhliu13,sdc} and a recent theoretical proposal how to exploit them in teleportation~\\cite{telep}. For completeness, we next recall the basic steps of the scheme, including some changes compared to the original theoretical proposal, and then continue to the experimental part and results.\n\n\\section{Theoretical description}\n\nFirst, we prepare a polarization entangled state $\\ket{\\phi^+}_{ab}=\\frac{1}{\\sqrt{2}}( \\ket{HH}+\\ket{VV})$, where $H(V)$ denotes the horizontal (vertical) polarization of the photon.\nThe total initial polarization-frequency two-photon state is\n\\begin{equation}\\label{eq:1}\n\\ket{\\psi(0)}=\\ket{\\phi^+}_{ab}\\otimes \\int{d\\omega_a d\\omega_b g(\\omega_a,\\omega_b)\\ket{\\omega_a}\\ket{\\omega_b}},\n\\end{equation}\nwhere $g(\\omega_a,\\omega_b)$ is the joint frequency amplitude distribution of the photons $a$ and $b$ with $\\int d\\omega_a d\\omega_b|g(\\omega_a,\\omega_b)|^2=1$. \nWe describe further down below in more detail what role the properties of $|g(\\omega_a,\\omega_b)|^2$ play in the protocol.\nBefore Alice receives her photon, its polarization and frequency are coupled in a quartz plate. This local interaction is given by the time-evolution operator\n\\begin{equation}\nU(t) |\\lambda \\rangle |\\omega\\rangle =  \\exp(i n_{\\lambda} \\omega t) |\\lambda\\rangle |\\omega\\rangle,\n \\end{equation}\n where $\\lambda$ denotes the given polarization direction \nand $n_{\\lambda}$ its index of refraction in the quartz plate. Even though the polarization-frequency state remains pure, this leads to dephasing of the polarization degree of freedom~\\cite{exp2,bhliu13}. Indeed, after the local interaction in the side of Alice, the two-photon polarization-frequency state is both pure and entangled\n\\begin{equation}\n\\ket{\\psi(t_a)}=\\frac{1}{\\sqrt{2}}(\\ket{HH}\\ket{\\xi_{HH}(t_a)}+\\ket{VV}\\ket{\\xi_{VV}(t_a)}), \n\\end{equation}\nwhere $\\ket{\\xi_{\\lambda \\lambda}(t_a)}=\\int d\\omega_a d \\omega_b g(\\omega_a,\\omega_b)e^{i n_{\\lambda}^a\\omega_a t_a}\\ket{\\omega_a}\\ket{\\omega_b}$.\nThe joint polarization state, in turn, is not anymore fully entangled and has become a mixed state \n\\begin{eqnarray}\\label{eq:3}\n\\rho_{ab}(t_a)&=&\\frac{1}{2}(\\ket{HH}\\bra{HH}+\\kappa_1(t_a)\\ket{HH}\\bra{VV} \\nonumber\\\\\n&&+\\kappa_1^*(t_a)\\ket{VV}\\bra{HH}+\\ket{VV}\\bra{VV}),\n\\end{eqnarray}\nwhere the local decoherence function is\n\\begin{equation}\n\\label{eq6}\n\\kappa_a(t_a)=\\int d\\omega_ad\\omega_b |g(\\omega_a,\\omega_b)|^2 e^{-i \\Delta n_a \\omega_a t_a}\n\\end{equation}\nwith $\\Delta n_a=n_V^a-n_H^a$. \n\nAlice prepares now a third qubit in a state, that she wants to teleport to Bob. Unlike the original proposal where a third photon is used for this purpose~\\cite{telep},\nshe introduces binary path degree of freedom of the photon she possesses.\nIn general this path-qubit state is \n$\\ket{\\phi}_s=\\alpha\\ket{0}+\\beta\\ket{1}$,\nwhere $|0\\rangle$ and $|1\\rangle$ denote the two spatial paths she uses. \nTherefore, Alice's and Bob's over-all state is now \n$\\ket{\\Psi(t_a)}=\\frac{1}{\\sqrt{2}}\\ket{\\phi}_s\\left[(\\ket{HH}\\ket{\\xi_{HH}(t_a)}+\\ket{VV}\\ket{\\xi_{VV}(t_a)}\\right].$ \nWe can now write this state by using the Bell-state basis of the two qubits of Alice and obtain\n\\begin{eqnarray} \\label{eq:6}\n\\ket{\\Psi(t_a)}&=&\\frac{1}{2}\\ket{\\Phi^+}_{sa} [\\alpha\\ket{H}_b\\ket{\\xi_{HH}(t_a)}+\\beta\\ket{V}_b\\ket{\\xi_{VV}(t_a)}]\\nonumber\\\\\n&+&\\frac{1}{2}\\ket{\\Phi^-}_{sa}  [\\alpha\\ket{H}_b\\ket{\\xi_{HH}(t_a)}-\\beta\\ket{V}_b\\ket{\\xi_{VV}(t_a)}]\\nonumber\\\\\n&+&\\frac{1}{2}\\ket{\\Psi^+}_{sa} [\\beta\\ket{H}_b\\ket{\\xi_{HH}(t_a)}+\\alpha\\ket{V}_b\\ket{\\xi_{VV}(t_a)}]\\nonumber\\\\\n&+&\\frac{1}{2}\\ket{\\Psi^-}_{sa} [\\alpha\\ket{V}_b\\ket{\\xi_{VV}(t_a)}-\\beta\\ket{H}_b\\ket{\\xi_{HH}(t_a)}],\\nonumber \\\\\n\\end{eqnarray}\nwhere $\\ket{\\Phi^{\\pm}}_{sa}=\\frac{1}{\\sqrt{2}}( \\ket{0}\\ket{H}\\pm\\ket{1}\\ket{V})$ and $\\ket{\\Psi^{\\pm}}_{sa}=\\frac{1}{\\sqrt{2}}( \\ket{0}\\ket{V}\\pm\\ket{1}\\ket{H})$. \nIt is worth noting that in each line -- corresponding to four outcomes of Alice's measurement -- we have a pure and entangled state between Bob's polarization and the frequencies of the two-photons, in addition to the amplitudes $\\alpha$ and $\\beta$ being transferred.\n\nAlice communicates her measurement result to Bob. What should he do now? Bob first applies one of the four unitary transformations on his qubit -- according to the standard teleportation scheme -- and after this applies local polarization-frequency dephasing interaction to his photon.\nThe cases corresponding to four outcomes of Alice are\n(i) $\\ket{\\Phi^+}_{sa} \\Rightarrow \\mathbb{I}, \\quad \\Delta n_b=\\Delta n_a$;\n(ii) $\\ket{\\Phi^-}_{sa} \\Rightarrow  \\sigma_z, \\quad \\Delta n_b=\\Delta n_a$;\n(iii) $\\ket{\\Psi^+}_{sa} \\Rightarrow \\sigma_x, \\quad \\Delta n_b=-\\Delta n_a$;\n(iv) $\\ket{\\Psi^-}_{sa} \\Rightarrow i \\sigma_y, \\quad \\Delta n_b=-\\Delta n_a$.\nHere, $\\sigma_x, \\sigma_y$, and $\\sigma_z$ are the unitary Pauli rotations, and $\\Delta n_b$ indicates the conditional choice for Bob's  birefringence. Let us as an example to check one of the four cases in detail. \n\nSuppose that Alice's measurement outcome was  $\\ket{\\Phi^+}_{sa}$ corresponding to the first line of Eq.~(\\ref{eq:6}).\nFor this case, Bob's unitary qubit transformation is the indentity operator and he only need to apply dephasing noise with $\\Delta n_b=\\Delta n_a$.\nOnce Bob applies the dephasing interaction for the duration $t_b$, the state of his polarization qubit is\n\\begin{eqnarray} \\label{eq:10}\n\\rho_b &=&|\\alpha|^2 \\ket{H}\\bra{H}+\\alpha\\beta^*\\kappa(t_a,t_b) \\ket{H}\\bra{V}\\\\\n&&+\\alpha^*\\beta \\kappa^*(t_a,t_b) \\ket{V}\\bra{H}+|\\beta|^2 \\ket{V}\\bra{V}, \\nonumber\n\\end{eqnarray}\nwhere the decoherence function $\\kappa(t_a,t_b)$ is given by \n\\begin{equation}\n\\kappa(t_a,t_b)=\\int d \\omega_a d \\omega_b |g(\\omega_a,\\omega_b)|^2 e^{-i \\Delta n_b (\\omega_a t_a+\\omega_b t_b)}.\n\\end{equation}\nNote that the photon on the side of Alice is already destroyed. However, the influence of Bob's local noise on his polarization state does depend on the initial joint two-photon frequency distribution $|g(\\omega_a,\\omega_b)|^2$. \n\nThe question now becomes whether it is possible to have $|\\kappa (t_a,t_b)|=1$ so that Bob eventually has pure polarization state after his noise -- whilst in all of the previous points of the protocol the state has been mixed. For this purpose, let us study the properties of the initial two-photon frequency distribution\n $|g(\\omega_a,\\omega_b)|^2$.\n \nWe consider the joint two-photon frequency distribution  $|g(\\omega_a,\\omega_b)|^2$ in a downconversion process as a bivariate Gaussian distribution~\\cite{nlnm,bhliu13} with covariance matrix elements  $C_{ij}=\\langle\\omega_i\\omega_j\n\\rangle-\\langle\\omega_i\\rangle\\langle\\omega_j\\rangle$.\nThe means and variances for the two photons are \n$\\langle \\omega_a \\rangle = \\langle \\omega_b \\rangle =\\omega_0\/2$, where $\\omega_0$ is the frequency of the downconversion pump, and $C_{11} = \nC_{22} = \\langle\\omega^2_i\\rangle-\\langle\\omega_i\\rangle^2$. The frequency-frequency correlation is quantified by the coefficient $K=C_{11}\/\\sqrt{C_{11}\nC_{22}}=C_{12}\/C_{11}$, such that $|K|\\leq1$. \nTaking initially a maximally anti-correlated frequency distribution with $K=-1$ having $\\omega_a+\\omega_b=\\omega_0$,\nand Bob using interaction time $t_b=t_a$, the magnitude of  his decoherence function becomes $|\\kappa (t_a,t_b)|=1$ and he obtains -- after applying the local noise --  the pure polarization state   \n\\begin{equation} \\label{eq:12}\n|\\psi_F\\rangle=\\alpha |H\\rangle+\\beta e^{i \\omega_0 \\Delta n_b t_b} |V\\rangle.\n\\end{equation}\nBob can in straightforward manner cancel the extra relative phase with $\\omega_0$, and therefore succeeds in the teleportation with fidelity equal to one.\nNote that in general for dephasing, the width of the single-peak Gaussian frequency distribution defines how strongly the exponential damping of the  magnitude of the decoherence function occurs -- when having a single photon or noise only on one side of the two-photon system destroying the polarization entanglement  (see, e.g., Ref.~\\cite{exp2}). In the ideal case described above in Eq.~(\\ref{eq:12}), both photons still have finite local frequency distribution widths ($C_{11}$ and $C_{22}$) and subsequent source of dephasing even though the final teleported state can be pure state due to having $K=-1$.  \n\n\\section{Experimental set-up}\n\nWe display the experimental set-up in Fig.~\\ref{fig:set-up}.\nTo prepare the required initial state, we exploit SPDC process  where a $404$nm CW laser pumps a BBO crystal. This crystal is made of two orthogonal glued type-I phase matched BBOs and can be used to produce polarization entangled photons by using the $H+V$ direction polarized laser pump.\nThe frequency correlations, in turn, arise due to the narrow linewidth of the CW pump laser.\nIn our case, the $404$nm pump has measured linewidth below $0.06$nm.\nAfter downconversion, the photons have a bandwidth on the order of $135$nm. Then, $3$nm full width at half maximum (FWHM) bandpass filters (centered at $808$ nm) are used to choose the most indistinguishable SPDC photon pairs. Even though the filtering changes the frequency distribution, there is still high amount of frequency-frequency correlations left due the very narrow pump linewidth.\nNote also that in the used $3$nm frequency window, the difference between the indices of refraction for ordinary and extraordinary rays can be considered constant with value $0.00889$.\n\n\\begin{figure}[t!]\n\\begin{center}\n  \\includegraphics[keepaspectratio,width=0.45\\textwidth]{tele-v6.jpg}\n  \\caption{\\label{fig:set-up}  Experimental set-up. The two-photon state is prepared by the spontaneous parametric down conversion (SPDC) pumping the beta-barium-borate (BBO) crystal with a continuous wave (CW) laser. The BFs  are used to choose the photons.\nQPs implement the noise in Alice's side. To prepare Alice's path qubit state and to prepare for path-qubit Bell measurement, a specifically designed Sagnac interference ring\nwith very high stability is used.\nHere, we use a combined beam splitter, abbreviated with CO-BS, which consists of beam splitter and polarizing beam splitter parts.\n The path qubit state itself is prepared with LAAs along the paths inside the interferometer.\n Alice's HWPs, PBSs and SPDs complete the Bell-measurement. In Bob's side, HWPs induce unitary transformation, QPs dephasing noise, and state tomography \nis performed with MRP, PBS and SPDs.\nAbbreviations: HWP: half wave plate, BF: bandpass filter, SMF: single mode fiber, MRP: motor rotating plate, QP: quartz plate, PBS: polarizing beam splitter, LAA: linear adjustable attenuator, SPD: single photon detector.\n}\n  \\end{center}\n\\end{figure}\n\n\nOn Alice's side, we first insert quartz plates, with increasing thickness corresponding to increasing interaction times,  to induce dephasing noise for Alice's polarization qubit. \nThen Alice prepares her path qubit to be teleported, and makes the polarization-path Bell-measurement. Earlier proposals for path-polarization measurement have used, e.g., Mach-Zehnder  interferometer~\\cite{zender}. However, to improve significantly the stability of the scheme required for teleportation, we have designed a specific Sagnac interferometer for this purpose.\nHere, the crucial component is a specific beam splitter (CO-BS) which consists half of BS and half of PBS.\nWhen the photon enters the Sagnac interferometer, it goes through the BS part of the CO-BS. Along the paths of the interferometer, there are LAAs\n that can produce arbitrary ratio of the two paths and prepares the path qubit state to be teleported. Note that the path degree of freedom is fully independent of the polarization and frequency degrees of freedom of the photon. \nAlice can then make the Bell measurement by interfering her two paths, when the photon exits the interferometer in the PBS part of CO-BS, and by using other PBS at $45^{o}$ at each output path before the photon hits the SPDs at the outputs.\n\nOn Bob's side, he first implements with HWP the unitary operation on his polarization qubit based on Alice's Bell-measurement outcome. After this, he induces dephasing noise by using quartz plates to couple the polarization (open system) and frequency (environment). Finally, using MRPs and PBS, he performs the state tomography of his qubit and completes the protocol.\n\n\n\\section{Results}\n\nWe present now two sets of experimental results.\nIn both cases, the success of the teleportation is quantified in usual manner by  fidelity \n$F(\\rho_{\\text{out}}, \\rho_{\\text{in}}) = \\left(\\tr \\sqrt{\\sqrt{\\rho_{\\text{out}}} \\rho_{\\text{in}}\n\\sqrt{\\rho_{\\text{out}}}} \\right)^2$ between the state Alice prepared $\\rho_{\\text{in}}$ and  $\\rho_{\\text{out}}$ that Bob measured after completing the protocol.\n\nAlice's qubit resides in path degree of freedom of the photon while Bob's qubit corresponds to polarization state of another photon. \nNote that the maximum fidelity with fully classical mixed states and using LOCC is $2\/3$~\\cite{verst}.\nThe results are presented for all possible four Bell-state measurement outcomes of Alice $\\{|\\Phi^+\\rangle,~|\\Phi^-\\rangle,~|\\Psi^+\\rangle,~|\\Psi^-\\rangle\\}$.\n\n \\begin{figure}[t!]\n\\begin{center}\n  \\includegraphics[keepaspectratio,width=0.45\\textwidth]{different_noise_square2}\n  \\caption{\\label{fig:res1} The measured teleportation fidelity with error bars as a function of amount of noise first on Alice's side and followed by increasing amount of noise on Bob's side. The vertical lines indicate the sides of Alice and Bob for the noise. The dashed line is the theoretical fit for the experimental results including the possibly non-ideal value of $K$~\\cite{telep}.\n The fits give an estimate $-0.997 \\leqslant K \\leqslant -0.963$.\n  Teleported input state is $\\rho_{\\text{in}}=\\rho_{+}$ [Eq.~\\eqref{eq:in1}]. The results correspond to the first set of experimental results (see the main text). The optical path difference is expressed with the unit of $808$nm. When Alice increases noise on her side, the fidelity decreases. With increasing amount of noise on Bob's side, the fidelity recovers. The black horizontal line indicates the classical limit of the fidelity with value of $2\/3$.\nThe error bars are standard deviations calculated by the Monte-Carlo method and mainly due to the counting statistics.\nIn  most cases, the error bars are smaller than the symbols. }\n  \\end{center}\n\\end{figure}\n \n \\begin{figure}[t!]\n\\begin{center}\n  \\includegraphics[keepaspectratio,width=0.45\\textwidth]{same_noise_square3}\n  \\caption{\\label{fig:res2} The measured teleportation fidelity with error bars  as a function of increasing and equal duration of noise on both Alice's and Bob's side.\n Teleported input states $\\rho_{\\text{in}}$  are $\\rho_{+}$ [Eq.~\\eqref{eq:in1}] (\"+\" symbols),  $\\rho_{1}$ [Eq.~\\eqref{eq:in11}] (squares),\n  $\\rho_{2}$ [Eq.~\\eqref{eq:in2}] (circles), and  $\\rho_{i}$ [Eq.~\\eqref{eq:ini}] (\"x\" symbols).\nSince the earlier fits for Fig.~\\ref{fig:res1} show that we are very close to the ideal case, the solid and dashed lines above are for simplicity linear fits for the experimental results. \n The results correspond to the second set of experimental results (see the main text). The optical path difference corresponds to interaction time $t_a+t_b$ and is expressed with the unit of $808$nm. The fidelity remains essentially constant despite of having more and more noise. The black horizontal line indicates the classical limit of the fidelity with value of $2\/3$.The error bars are standard deviations calculated by the Monte-Carlo method and mainly due to the counting statistics. In  most cases, the error bars are smaller than the symbols.}\n  \\end{center}\n\\end{figure}\n\nIn the first part of the first experiment,  the duration of the noise $t_a$ in Alice's side is increased in stepwise manner before her Bell measurement.\nShe has prepared the path qubit state to be teleported\nwhich is given by, based on experimental state tomography as,\n\\begin{equation}\n\\label{eq:in1}\n\\rho_+=\\left(\\begin{array}{cc} 0.5507 &0.4871+i0.1007 \\\\0.4871-i0.1007& 0.4492\\end{array}\\right).\n\\end{equation}\nThis is expressed in the path qubit basis $|0\\rangle$ and $|1\\rangle$ and has the purity $0.999955$.\nBob, after receiving the Bell measurement outcome from Alice, implements the unitary transformation on his qubit in usual way but does not implement any noise.\nThe left sides of the four panels in Fig.~\\ref{fig:res1} show that in this case the teleportation fidelity decreases with increasing amount of noise, as expected and going below the classical limit.\nNow, in the second part of the experiment, in Alice's side there is always maximum duration of noise corresponding to optical path difference $237.6 \\lambda_0$ with \n$\\lambda_0=808$nm -- \n and Bob begins to increase the duration of noise in his side in stepwise manner. \nThe results are displayed on the right sides of the four panels in Fig.~\\ref{fig:res1}. With the increasing amount of noise in Bob's side, the teleportation fidelity increases. Note that in all of the cases, as well as for the results below, the experimental results for fidelity are based making state tomography also for the output state.\nWhen Bob has, in the last step, maximum duration of noise corresponding to the one on the side of Alice, the high-fidelity values correspond to the case as if there had essentially not been any noise in any part of the protocol, i.e., the ideal teleportation efficiency is recovered. For example, for Bell measurement outcome $|\\Psi^+\\rangle$,\nthe fidelity value without any noise [the first experimental point in the lower-left panel of Fig.~\\ref{fig:res1}] is $0.978\\pm0.003$ while fidelity value with maximum duration of noise in both sides [the last experimental point in the lower-left panel of Fig.~\\ref{fig:res1})] is $0.978\\pm0.006$. \n\n\nIn the second experiment, the duration of the local noise in both sides of Alice and Bob is increased in stepwise manner, in equal steps in both sides.\nFigure~\\ref{fig:res2} shows the results for four Bell measurement outcomes and using four different Alice's states to be teleported.\nOne of the input states is given by Eq.~\\eqref{eq:in1} and the three others, based on input state tomography, are\n\\begin{eqnarray}\n\\rho_1=\\left(\\begin{array}{cc} 0.9794 & 0.0151-i0.0321 \\\\0.0151+i0.0321 & 0.0206\\end{array}\\right),  \\label{eq:in11}\\\\\n\\rho_2=\\left(\\begin{array}{cc} 0.0255 & 0.01304+i0.0932  \\\\0.01304-i0.0932 & 0.9745\\end{array}\\right), \\label{eq:in2} \\\\\n\\rho_i=\\left(\\begin{array}{cc} 0.6060 &0.02178-i0.4827  \\\\0.02178+i0.4827 &0.3940\\end{array}\\right). \\label{eq:ini}\n\\end{eqnarray}\nThe purities these states are $0.962166,~0.968013$ and $0.989419$, respectively.\nWhen there is no noise at all, the left-most points in the panels, the high-fidelity teleportation is achieved, as expected. \nHowever, when there is increasing duration of local noise on both sides of Alice and Bob, the fidelity does not reduce and remains essentially constant.\nFor example with input state $\\rho_{\\text{in}}= \\rho_{i}$ [Eq.~\\eqref{eq:ini}]and Bell-measurement result $|\\Psi^+\\rangle$, the fidelity without any noise is $0.987\\pm0.002$ and with maximum duration of noise \n$0.985\\pm0.002$.\nThis gives clear experimental demonstration, that even though there is hardly any entanglement left in the joint two-photon polarization state,\none can in any case achieve high-fidelity teleportation when exploiting other useful resources available when considering also other degrees of freedom and the environment of an open system.\n\n\n\n\\section{Discussion and conclusions}\n\nWe have realized experimentally a scheme for high-fidelity teleportation with dephasing noise. \nHere, a pair of entangled qubits -- which  initially contain the quantum resource for the protocol -- is actually an open quantum system where each of the qubits interact with their local environments. Without the steps of the teleportation protocol, the dynamics of this bipartite open system displays non-Markovian features when the local environments of the qubits are initially correlated~\\cite{nlnm,bhliu13}. Note that there also exists a quantitave connection between the amount of non-Markovianity and the teleportation fidelity~\\cite{telep}.\nTherefore, we have demonstrated experimentally, that it is possible to implement high-fidelity teleportation via non-Markovian open quantum system.\n\nOur results also show that it is not necessary, that the original quantum resource resides anymore in the degrees of freedom or in the open system, which are explicitly used in the original protocol --  as long as useful resources still exist within or in the combination with the environment of the open system.\nIt is also worth noting here that in the described teleportation scheme, Alice's photon is destroyed in her Bell measurement while at this point Bob has not yet done anything with his qubit. Despite of this fact, Bob's subsequent open system qubit dynamics is influenced by the initially existing correlations between the two photons, even though Alices photon does not exist anymore. In a sense, in addition to teleporting a qubit state, the protocol allows to engineer in nonlocal manner -- both in time and in location --  the open system dynamics of Bob's qubit. \n\nIn general, we have given fundamental experimental results combining concepts from open quantum systems and quantum information.\nSo far, there exists a number of sophisticated experiments which, e.g.,  implement reservoir engineering, quantum simulate Markovian open system dynamical maps, control Markovian to non-Markovian transition and decoherence (see e.g.~\\cite{wineland,exp2,zdliu,blatt}).\nHowever, very little is known or fundamentally tested yet when going beyond the traditional open system - environment setting, and combining the open system dynamics with sophisticated quantum information protocols or other well-known quantum physical or optical schemes including, e.g., interferometry.\nWe hope that our current results stimulate further work for this direction and helps to explore new areas of quantum physics where open quantum systems, and their study,\nis being used outside their traditional framework.\n\n\\acknowledgments\nThis work was supported by the National Key Research and Development Program of China (No.~2017YFA0304100), \nthe National Natural Science Foundation of China (Nos.~62005263, 11774335, 11821404, 11874345)\nKey Research Program of Frontier Sciences, CAS (No.~QYZDY-SSW-SLH003), the Fundamental Research Funds for the Central Universities (No.~WK2470000026), Science Foundation of the CAS (No.~ZDRW-XH-2019-1), Anhui Initiative in Quantum Information Technologies (AHY020100), and Science and Technological Fund of Anhui Province for Outstanding\nYouth (No.~2008085J02). \nZ. D. L acknowledges financial support from China Postdoctoral Science Foundation (Grant No. 2020M671862).\nS. H. R. acknowledges financial support from Finnish Cultural Foundation and Turku University Foundation.\nJ. P. acknowledges financial support from Magnus Ehrnrooth Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n \n\n Throughout the paper, $\\mathcal{H}$ denotes a real Hilbert space with\n\tinner product $\\langle \\cdot ,\\cdot \\rangle $ and norm $\\Vert \\cdot \\Vert $,\n\trespectively. Consider $\\operatorname{T}: \\mathcal{H} \\to 2^{\\mathcal{H}}$ is a set-valued monotone operator. The monotone inclusion problem is to find $x \\in \\mathcal{H}$ such that\n\\begin{equation}\\label{P4e1.1}\n\t0 \\in T(x).\n\\end{equation}\nThe monotone inclusion problem (\\ref{P4e1.1}) plays important role in nonlinear analysis. Many problems arising in engineering, economics and physics can be framed as monotone inclusion problem  (see \\cite{Barty2007,Bennar2007,Briceno2013,Combettes2005,Daubechies2004,Nemirovski2009}). Martinet \\cite{Martinet1972} has proposed proximal point algorithm, which is very popular to solve monotone inclusion problem. The proximal point algorithm is given by\n\\begin{equation}\\label{P4e1.2}\n\tx_{n+1}=J_{\\lambda_n {T} }(x_n)\\quad\\forall n \\in \\mathbb{N},\n\\end{equation}\nwhere $\\lambda_n > 0$ is a regularization parameter and $x_1 \\in \\mathcal{H}.$ Rockafellar \\cite{Rockafella1976,Rockafellar1976} has proved that proximal point algorithm converges weakly to solution set of inclusion problems in real Hilbert space framework. Further, he has introduced the inexact proximal point algorithm  as follows:\n\\begin{equation}\\label{P4e1.3}\n\tx_{n+1}=J_{\\lambda_n T} (x_n+v_n),\\quad\\forall n \\in \\mathbb{N},\n\\end{equation}\nwhere $v_n$ is the error term in $\\mathcal{H}$. The sequence $\\{x_n\\}$ also converges weakly to solution set of inclusion problem provided $\\sum_{n=1}^{\\infty} v_n < \\infty.$ Guler \\cite{Guler1991} has shown by an example that sequence generated by proximal point algorithm (\\ref{P4e1.2}) converges weakly, but not strongly. It becomes a matter of interest for the research community to modify the proximal point algorithm to obtain the strong convergence. In such consequences, Tikhonov method has been proposed which generates as follows:\n\\begin{equation}\n\tx_{n+1}=J_{\\lambda_n T} (x),\n\\end{equation}\nwhere $x \\in \\mathcal{H}$ and $\\lambda_n >0$ such that $\\lambda_n \\to \\infty.$ Detailed study of Tikhonov regularization method can be found in \\cite{Butnariu2008,Tikhonov1965,Tikhonov1977,Tikhonov1963,Xu2002}. Lehdili and Moudafi \\cite{Lehdili1996} have combined the idea of proximal algorithm and Tikhonov regularization to find an algorithm converges strongly to solution of inclusion problem (\\ref{P4e1.1}). They have solved the inclusion problem (\\ref{P4e1.1}) by solving inclusion problem of fixed approximation of $T$, which is $T_n=T+\\mu_n  Id$, i.e.,\n\\begin{equation*}\n\t\\text{find } x\\in \\mathcal{H} \\text{ such that } 0 \\in T_n(x),\n\\end{equation*}\nwhere $\\mu_n$ is regularization parameter of $T$.\nThe proximal-Tikhonov algorithm is given by\n$$x_{n+1}=J_{\\lambda_{n}T_n}(x_{n}).$$\nThe Tikhonov regularization term $\\mu_n Id$ has impelled the strong convergence to the algorithm. In absence of Tikhonov regularization term, proximal-Tikhonov algorithm becomes proximal algorithm which shows only weak convergence in most of the cases. The strong convergence of the algorithm can be obtained by using some other techniques also, some of them can be found in \\cite{Bauschke2001,Haugazeau1968}.\n\nEvaluation of resolvent is sometimes as hard as the original problem. This problem has been tried to resolve by splitting the operator in two operators, i.e., $T=A+B,$ whose resolvents are easy to compute. For $T=A+B,$ the monotone inclusion problem (\\ref{P4e1.1}) becomes\n\\begin{equation}\\label{P4e1.4}\n\t\\text{find } x \\in \\mathcal{H} \\text{ such that }0 \\in (A+B)x,\n\\end{equation}\nwhere $A : \\mathcal{H} \\to 2^\\mathcal{H}$ is  maximally monotone operator  and $B$ is an operator.\nForward-backward splitting algorithm and Douglas-Rachford algorithm have been proposed to solve Problem (\\ref{P4e1.4}). Forward-backward splitting method has been proposed by Lions and Mercier \\cite{Lions1979}, Passty \\cite{Passty1979}, which is given by\n\\begin{equation}\n\tx_{n+1}=(Id+\\lambda_n A)^{-1}(Id-\\lambda_n B)x_n,\n\\end{equation}\nwhere $\\lambda_n > 0$ and $B:\\mathcal{H} \\to \\mathcal{H}$ is cocoercive operator. Mercier \\cite{Mercier1980} and Gabay \\cite{Gabay1983} have studied the convergence behavior of forward-backward method when $A^{-1}$ is $\\gamma$-strongly monotone with $\\gamma >0.$ They have proved that forward-backward algorithm converges weakly to the point in the solution set provided $\\lambda_n < 2 \\gamma$, is constant. In addition, if  $A$ is strongly monotone, then $\\{x_n\\}$ shows strong convergence to the unique solution  of Problem (\\ref{P4e1.4}). Chen and Rockafellar \\cite{Chen1997} have also assumed the strong monotonicity of $A$ to prove the strong convergence of forward-backward method which depends on Lipschitz constant and modulus of strong monotonicity. Further, forward-backward method has been extensively studied, few of them can be found in (\\cite{Chen1994,Chen1997,Mouallif1991,Moudafi1997}) and references therein. \n\nDouglas-Rachford method has been proposed to solve problem (\\ref{P4e1.4}) when both $A$ and $B$ are set-valued. It has been originally proposed by Douglas and Rachford \\cite{Douglas1956} to solve linear equations arising in heat-conduction problems. Lions and Mercier \\cite{Lions1979} have extended the Douglas-Rachford algorithm to monotone operators. Douglas-Rachford algorithm is given as follows$\\colon$\n\\begin{equation}\n\tx_{n+1}=R_B R_A x_n, \\quad\\forall n \\in \\mathbb{N},\n\\end{equation}\nwhere $R_B$ and $R_A$ are reflected resolvent of operators $B$ and $A$, respectively. Lions and Mercier \\cite{Lions1979} have proved that Douglas-Rachford algorithm converges weakly to a fixed point of operator $T$ which helps to obtain the solution of the Problem (\\ref{P4e1.4}). Svaiter \\cite{Svaiter2011} has supported the results of Lions and Mercier by proving the weak convergence of the shadow sequence to a solution. Further the analysis of the Douglas-Rachford algorithm can be found in (\\cite{Artacho2013,Davis2015,Luke2020,Phan2016}).\n\nLet $\\mathcal{C}$ be a nonempty closed convex subset of $\\mathcal{H}$ and $\\mathcal{S}: \\mathcal{C} \\to \\mathcal{C}$ be a nonexpansive operator. There are a number of iterative methods for finding fixed points of nonexpansive operators.  We recall some well known fixed point methods, which are given below$\\colon$\n\\begin{enumerate}\n\n\n\n\n\t\\item[$\\bullet$] Mann iteration method \\cite{Mann1953}:\n\t\\begin{equation*}\n\t\tx_{n+1}=(1-\\beta_n) x_n+ \\beta_n \\mathcal{S}(x_n),\\quad\\forall n \\in \\mathbb{N};\n\t\\end{equation*}\n\t\\item[$\\bullet$] S-iteration method \\cite{Agarwal2007}:\n\t\\begin{equation*}\n\t\tx_{n+1}=(1-\\alpha_n)\\mathcal{S}(x_n) +\\alpha_n \\mathcal{S}[(1-\\beta_n) x_n + \\beta_n \\mathcal{S} x_n],\\quad\\forall n \\in \\mathbb{N};\n\t\\end{equation*}\n\t\\item[$\\bullet$] Normal S-iteration method \\cite{Sahu2011}:\n\t\\begin{equation*}\n\t\tx_{n+1}=\\mathcal{S}[(1-\\beta_n)x_n+\\beta_n \\mathcal{S} x_n],\\quad\\forall n \\in \\mathbb{N};\n\t\\end{equation*}\n\\end{enumerate}\nwhere $\\alpha_n, \\beta_n \\in (0,1).$ The importance of these algorithms are not limited to solve fixed point problems, but these algorithms are also useful for solving inclusion problems of sum of a set-valued maximally monotone operator and a single-valued cocoercive operator, and inclusion problems of sum of two set-valued maximally monotone operators. The S-iteration methodology has been applied for solving\n\tvarious nonlinear problems, inclusion problems, optimization problems and\n\timage recovery problems. Recently, it has been demonstrated by Avinash et al. \\cite{Dixit2019}  that the inertial normal S-iteration method has better performance compared to the inertial Mann iteration method. The S-iteration method and normal S-iteration method  are also useful  for finding common fixed points of nonexpansive operators. Since last few years, these properties of normal S-iteration make it popular among research community to find fixed point. Several research articles related to S-iteration and normal S-iteration can be found in \\cite{Chang2013,Chang2014,Cholamjiak2015,Sahu2020,Sahuajit2020}. The weak convergence of the fixed point algorithms has reduced its applicability in infinite dimensional spaces. To achieve the strong convergence of algorithms one assumes stronger assumptions like strong monotonicity and strong convexity, which is difficult to achieve in many applications. This situation leaves a question to research community: can we find the strongly convergent algorithms without assuming these strong assumptions? The answer to this question is replied positively by Bot et al. \\cite{Bot2019}. They have modified the Mann algorithm  as follows:\n\\begin{equation}\\label{P4e1.5}\n\tx_{n+1}=e_n x_n + \\theta_n(\\mathcal{S}(e_n x_n)-e_n x_n),\n\\end{equation}\nwhere $e_n, \\theta_n$ are positive real numbers. The strong convergence of algorithm (\\ref{P4e1.5}) for nonexpansive operator $\\mathcal{S}$  has been studied by Bot et al. \\cite{Bot2019}  when set of fixed points of $\\mathcal{S}$ is nonempty and parameters $\\theta_n$ and $e_n$ satisfy the following:\n\\begin{enumerate}\n\t\\item [(i)] $0< e_{n} < 1$  for all $n \\in \\mathbb{N},$ $\\lim\\limits_{n \\to \\infty} e_{n} = 1$, $\\sum_{n=1}^{\\infty}(1-e_{n})= \\infty$ and $\\sum_{n=1}^{\\infty}|e_{n}-e_{n-1}|< \\infty;$\n\t\\item  [(ii)] $0<\\theta_{n}\\leq 1$  for all $n \\in \\mathbb{N},$ $0<\\liminf_{n\\to \\infty}\\theta_{n},$  $\\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$\n\\end{enumerate}\n\n  We consider the following more general problem:\n\\begin{Problem}\\label{Prob1}\n\tConsider $\\mathcal{T},\\mathcal{S}:\\mathcal{H} \\to \\mathcal{H}$ are nonexpansive operators. Find an element $x \\in  \\mathcal{H}$ such that $x \\in \\operatorname{Fix} (\\mathcal{T})\\cap \\operatorname{Fix} (\\mathcal{S})$.\n\\end{Problem}\n\n\t\\begin{Remark} The algorithm (\\ref{P4e1.5}) proposed by Bot et al. \\cite{Bot2019} can not apply to solve inclusion problem (\\ref{P4e1.1}).\n\\end{Remark}\n\n\nIn this paper,  we introduce the normal S-iteration  method based fixed point algorithm   to find common fixed point of nonexpansive operators  $\\mathcal{T}, \\mathcal{S}:\\mathcal{H} \\to \\mathcal{H}$, which converges strongly to solutions of common fixed point problem of operators $\\mathcal{S}$ and $\\mathcal{T}$. Based on the proposed fixed point algorithm, we develop a forward-backward algorithm  and a Douglas-Rachford algorithm containing Tikhonov regularization term to solve  the monotone inclusion problems.\nIn many cases, monotone inclusion problems are very complex, they contain mixture of linear and parallel sum monotone operators. Recently, many researchers have proposed primal-dual algorithms to precisely solve the considered complex monotone inclusion system \\cite{Attouch1996,Bot2013,Brice2011,Combettes2012,BC2013}.  We have proposed a forward-backward type primal-dual algorithm and a Doughlas-Rachford type primal-dual algorithm having Tikhonov regularization term to find the common solution of the complexly structured monotone inclusion problems. The proposed algorithms have a special property that resolvents of all the operators are evaluated separately.\\\\\n\nThe paper is organized as follows: Next section recalls some important definitions and results in nonlinear analysis. In Section \\ref{sec3}, we propose a normal S-iteration based Tikhonov regularized fixed point algorithm and study its convergence behavior. In Section \\ref{sec4}, we propose a forward-backward-type algorithm and a forward-backward-type primal-dual algorithm to solve inclusion problem and complexly structured monotone inclusion problem, respectively. In Section \\ref{sec5}, we propose Douglas-Rachford-type algorithms to solve monotone inclusion problems and complexly structured monotone inclusion problems of set-valued operators. In the last, we perform a numerical experiment to show the importance of proposed algorithms in solving image deblurring problems.\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\nThis section devotes some important definitions and results from nonlinear analysis and operator theory.\nLet $\\mathbb{N}$ and $\\mathbb{R}$ denote set of natural numbers and set of real numbers, respectively and  `$Id$' denotes identity operator. Consider the operator $T:\\mathcal{H}\\to 2^\\mathcal{H}.$ Let $Gr(T)$ be denote  the graph of $T$, $\\operatorname{Zer}(T)$ denote  set of zeros of operator $T$ and $\\operatorname{Fix} (T)$ denote  set of fixed points of $T$. The symbol $m$ is used to denote a strictly positive integer throughout the paper. The set of proper convex lower semicontinuous functions from $\\mathcal{H}$ to $[-\\infty,+\\infty]$ is denoted by $\\Gamma(\\mathcal{H}).$\nLet $A:\\mathcal{H} \\to 2^\\mathcal{H}$ be an operator. Domain of $A$ is $\\operatorname{dom}$ $(A)=\\{x\\in \\mathcal{H}: Ax\\neq \\emptyset\\}$. Range of $A$ is denoted by ran $(A)$ = $\\cup_{x\\in \\mathcal{H}}$ $Ax$. $A$ is said to be monotone if\n$$\\langle x-y, u-v \\rangle \\geq 0, \\ \\forall (x,u), (y,v) \\in Gr (A).$$\n$A$ is said to be maximally monotone if there exists no monotone operator $B:\\mathcal{H} \\to 2^\\mathcal{H}$ such that $Gr (B)$ properly contains $Gr (A).$ $A$ is strongly monotone with constant $\\beta \\in (0,\\infty)$ if\n\\begin{equation*}\n\t\\langle x-y,u-v \\rangle \\leq \\beta \\|x-y\\|^2 \\  \\forall (x,u), (y,v) \\in Gr(A).\n\\end{equation*}\n The resolvent of $A$ is defined by $J_A= (Id+A)^{-1} $ and the reflected resolvent of $A$ is $R_A= 2J_A -Id.$\nConsider $f: \\mathcal{H} \\to [-\\infty,\\infty]$. The conjugate of $f$ is defined by $f^*:\\mathcal{H} \\to [-\\infty,\\infty],$ $$u \\mapsto \\sup_{x \\in \\mathcal{H}} \\left( \\langle x,u \\rangle -f(x)\\right). $$\nLet $f:\\mathcal{H} \\to [-\\infty,\\infty]$ be a proper function. The subdifferential of $f$ is $\\partial f:\\mathcal{H} \\to 2^\\mathcal{H}$ is defined by\n$$x \\mapsto \\{u \\in \\mathcal{H}| f(y) \\geq f(x)+\\langle y-x,u \\rangle \\  \\forall y \\in \\mathcal{H}\\}.$$\nIf $f \\in \\Gamma(\\mathcal{H}),$ then $\\partial f$ is maximally monotone. The resolvent of subdifferential of $f$ is $prox_f$,  where $prox_f:\\mathcal{H}\\to \\mathcal{H}$ defined by\n\\begin{equation*}\n\tprox_f(x)=argmin_{y\\in \\mathcal{H}}\\left\\lbrace f(y)+\\frac{1}{2}\\|y-x\\|^2\\right\\rbrace .\n\\end{equation*}\n\n\\begin{Definition}\n\tLet $\\mathcal{C}$ be a nonempty subset of $\\mathcal{H}$. Then:\n\t\\item [(i)] interior of $\\mathcal{C}$ is\n\t\\begin{equation*}\n\t\t\\text{ int } \\mathcal{C}=\\{x\\in \\mathcal{C}:(\\exists \\rho > 0) B(0;\\rho) \\subset \\mathcal{C}-x\\};\n\t\\end{equation*}\n\t\\item  [(ii)]strong relative interior of $\\mathcal{C}$ is\n\t\\begin{equation*}\n\t\t\\text{sri } \\mathcal{C} =\\{x\\in \\mathcal{C}: cone(\\mathcal{C}-x)=\\overline{span}(\\mathcal{C}-x) \\};\n\t\\end{equation*}\n\t\\item  [(iii)]strong quasi-relative interior of $\\mathcal{C}$ is\n\t\\begin{equation*}\n\t\t\\text{sqri } \\mathcal{C} = \\{x \\in \\mathcal{C}: \\bigcup_{\\rho > 0}\\rho(\\mathcal{C}-x) \\text{is a closed linear subspace of space } \\mathcal{H}\\}.\n\t\\end{equation*}\n\\end{Definition}\nIn the case $\\mathcal{H}$ is finite dimensional, sqri and sri are equivalent.\n\n\\begin{Definition}\n\tLet $C$ be a nonempty subset of $\\mathcal{H},$ and $T:C\\to \\mathcal{H}$ be a nonexpansive operator. $T$ is said to be\n\t\\begin{enumerate}\n\t\t\\item[(i)] nonexpansive if\n\t\t\\begin{equation*}\n\t\t\t\\|Tx-Ty\\|\\leq \\|x-y\\| \\ \\forall x,y \\in C;\n\t\t\\end{equation*}\n\t\t\\item[(ii)] firmly nonexpansive if\n\t\t\\begin{equation*}\n\t\t\t\\|Tx-Ty\\|^2 +\\|(Id-T)x-(Id-T)y\\|^2 \\leq \\|x-y\\|^2 \\ \\forall x,y \\in C;\n\t\t\\end{equation*}\n\t\t\\item[(iii)] $\\beta$-cocoercive ($\\beta>0$) if\n\t\t\\begin{equation*}\n\t\t\t\\langle x-y,Tx-Ty \\rangle \\leq \\beta \\|Tx-Ty\\|^2 \\ \\forall x, y \\in C\n\t\t\\end{equation*}\n\t\t\\item[(iv)] $\\alpha$-averaged for $\\alpha \\in (0,1)$ if there exists a nonexpansive operator $R:C\\to \\mathcal{H}$ such that $T=(1-\\alpha)Id +\\alpha R$.\n\t\\end{enumerate}\n\\end{Definition}\nAn operator $T:\\mathcal{H}\\to 2^\\mathcal{H}$ is strongly monotone with $\\beta \\in (0, \\infty)$ implies $T^{-1}:\\mathcal{H} \\to \\mathcal{H}$ is $\\beta$-cocoerceive.\n\\begin{Definition}\\cite{Bauschke2011}\n\tLet $\\mathcal{C}$ be a nonempty subset of $\\mathcal{H}.$ Then:\n\t\n\t\\item [(i)] The indicator function $i_\\mathcal{C}:\\mathcal{H} \\to[-\\infty,+\\infty]$ is defined by\n\t\\begin{eqnarray}{\n\t\t\ti_\\mathcal{C}(x)\t=\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t0, \\ \\ if \\ x \\in \\mathcal{C}\\\\\n\t\t\t\t\\infty \\ \\ otherwise\n\t\t\t\\end{array}\\right.}.\n\t\\end{eqnarray}\n\t\\item [(ii)] The projection of a point $x \\in \\mathcal{H}$ on $\\mathcal{C}$ is defined by $\\operatorname{proj}_C(x) =\\left\\lbrace u \\in \\mathcal{C}:u=argmin_{z \\in \\mathcal{C} }\\|x-z\\|\\right\\rbrace.$\n\t\n\t\\item [(iii)]  Suppose $\\mathcal{C}$ is convex, then normal cone to $\\mathcal{C}$ at $x$ is defined by\n\t\\begin{eqnarray}{\n\t\t\tN_\\mathcal{C}(x)\t=\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\tu \\in \\mathcal{H}:\\sup \\langle y-x,u \\rangle \\leq 0 ~\\forall y \\in \\mathcal{C}, \\ if \\ x \\in \\mathcal{C}\\\\\n\t\t\t\t\\emptyset, \\ \\ \\ \\ otherwise\n\t\t\t\\end{array}\\right.}.\n\t\\end{eqnarray}\n\\end{Definition}\n\n\\begin{Definition}\\cite[Proposition 4.32]{Bauschke2011}\n\tThe parallel sum of two operators $T_1, T_2:\\mathcal{H}\\to 2^\\mathcal{H}$ is $T_1 \\Box T_2 :\\mathcal{H} \\to 2^\\mathcal{H}$ defined by $T_1 \\Box T_2  = (T_1 ^{-1}+T_2 ^{-1})^{-1}$.\n\\end{Definition}\nThe subdifferential of parallel sum of operators $T_1$ and $T_2$ is $\\partial (T_1 \\square T_2)=\\partial T_1 \\square \\partial T_2$.\n\\begin{Remark}\\label{P3R2.1}\n\tIf $T_1 $ and $T_2 $ are monotone then the set of zeros of their sum $\\operatorname{Zer}(T_1+T_2 )=J_{\\gamma T_2} (\\operatorname{Fix}(R_{\\gamma T_1 } R_{\\gamma T_2 }))$ $\\forall \\gamma >0 $.\n\\end{Remark}\n\n\\begin{Proposition}\\cite{Bauschke2011}\n\tConsider $T_1,T_2 : \\mathcal{H} \\to \\mathcal{H}$ be $\\alpha_1,\\alpha_2$-averaged operates, respectively. Then the averaged operator $T_1 \\circ T_2$ is $\\alpha= \\frac{\\alpha_1+\\alpha_2-2\\alpha_1\\alpha_2}{1-\\alpha_1\\alpha_2}$-averaged.\n\\end{Proposition}\n\n\\begin{Lemma}\\emph{\\cite[Corollary 4.18]{Bauschke2011}}\\label{P4L2.1}\n\tLet  $T:\\mathcal{H} \\to \\mathcal{H}$ be a nonexpansive mapping. Let $\\{u_n\\}$ be a sequence in $\\mathcal{H}$ and $u\\in \\mathcal{H}$ such that $u_n \\rightharpoonup  u$ and $u_n-Tu_n \\to 0$ as $n \\to \\infty$.\n\tThen $u \\in \\operatorname{Fix}(T)$.\n\\end{Lemma}\n\\begin{Lemma}\\cite[Lemma 2.5]{Xu2002}\\label{P4L2.2}\n\tLet $\\{a_n\\}$ be a sequence of nonnegative real numbers satisfying the inequality:\n\t$$a_{n+1}\\leq (1-\\theta_{n})a_n+\\theta_n b_n+\\epsilon_n \\ \\ \\forall n \\geq 0,$$\n\t where\n\t\\begin{enumerate}\n\t\t\\item[(i)] $ 0\\leq \\theta_n \\leq 1 $ for all $n \\geq 0$ and $\\sum_{n\\geq 0} \\theta_n = \\infty;$\n\t\t\\item[(ii)] $\\limsup_{n\\to \\infty} b_n \\leq 0$;\n\t\t\\item[(iii)] $\\epsilon_n \\geq 0 \\text{ for all } n \\geq 0 \\text{ and } \\sum_{n\\geq 0} \\epsilon_n < \\infty.$\n\t\\end{enumerate}\n\tThen the sequence $\\{a_n\\}$ converges to $0$.\n\\end{Lemma}\n\n\n\n\n\n\n\n\n\\section{Tikhonov Regularized Strongly Convergent Fixed Point Algorithm}\\label{sec3}\nThis section devotes  to investigate a computational theory for finding common fixed points of nonexpansive operators. We introduce a common fixed point algorithm such that sequence generated by the algorithm strongly converges to the set of common fixed points of mappings.\n\\begin{Algorithm}\\label{P4A3.1}\n\tLet $\\mathcal{S},\\mathcal{T} :\\mathcal{H} \\to \\mathcal{H}$ be nonexpansive mappings. Select $\\{e_{n}\\}$, $\\{\\theta_{n}\\} \\subset (0,1)$ and compute the $(n+1)^{th}$ iteration as follows:\n\t\\begin{equation}\\label{P4e3.1}\n\t\ty_{n+1}= \\mathcal{S}[(1-\\theta_n)e_{n}y_n+\\theta_n \\mathcal{T}(e_{n}y_n)] \\ \\ \\ \\  \\textrm{for all} \\ n\\in\\mathbb{N}.\n\t\\end{equation}\n\\end{Algorithm}\n\n\nWe now study the convergence behavior of Algorithm \\ref{P4A3.1} for finding the common fixed point of $\\mathcal{S}$ and $\\mathcal{T}$.\n\\begin{Theorem}\\label{P4T3.1}\n\tLet $\\mathcal{S},\\mathcal{T} :\\mathcal{H} \\to \\mathcal{H}$ be nonexpansive mappings such that $\\Omega:=\\operatorname{Fix} (\\mathcal{T})\\cap \\operatorname{Fix} (\\mathcal{S})\\neq \\emptyset$. Let $\\{y_n\\}$ be sequence in $\\mathcal{H}$ defined by Algorithm \\ref{P4A3.1}, where  $\\{\\theta_{n}\\}$ and $\\{e_{n}\\}$ are real sequences satisfy the following conditions:\n\t\\begin{enumerate}\n\t\t\\item [\\textbf{(i)}]\\label{P4C1}\n\t\t$0< e_{n} < 1$  for all $n \\in \\mathbb{N},$ $\\lim\\limits_{n \\to \\infty} e_{n} = 1$, $\\sum_{n=1}^{\\infty}(1-e_{n})= \\infty$ and $\\sum_{n=1}^{\\infty}|e_{n}-e_{n-1}|< \\infty;$\n\t\t\\item [\\textbf{(ii)}]\\label{P4C2}\n\t\t$0<\\underline{\\theta}\\le \\theta_{n}\\leq\\overline{\\theta}< 1$   for all $n \\in \\mathbb{N},$  and  $\\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$\n\t\\end{enumerate}\n\tThen the sequence $\\{y_n\\}$  converges strongly to $\\operatorname{proj}_{\\Omega}(0).$\n\\end{Theorem}\n\\begin{proof}\n\tIn order to prove the convergence of the sequence $\\{y_n\\}$, we proceed with following steps:\n\t\\begin{enumerate}\n\t\t\\item[Step 1.]\\label{C3.1} Sequence $\\{y_n\\}$ is bounded.\\\\\n\t\t\n\t\tLet  $y\\in \\Omega$. Since $\\mathcal{S}$ and $\\mathcal{T}$ are nonexpansive, we have following\n\t\t\\begin{eqnarray}\\label{P4e1}\n\t\t\t\\|y_{n+1}-y\\|&=& \\|\\mathcal{S}[(1-\\theta_{n})e_{n}y_n + \\theta_{n} \\mathcal{T}(e_{n}y_n)]-y\\| \\nonumber\\\\ [6pt]\n\t\t\t&\\leq &\\|(1-\\theta_{n})e_{n}y_n + \\theta_{n} \\mathcal{T}(e_{n}y_n)-y\\| \\nonumber\\\\[6pt]\n\t\t\t&\\leq &(1-\\theta_{n})\\|e_{n}y_n -y\\|+ \\theta_{n}\\| \\mathcal{T}(e_{n}y_n)-y\\| \\nonumber\\\\[6pt]\n\t\t\t&\\leq & \\|e_{n}y_n-y\\| \\\\[6pt]\n\t\t\t&=&\\|e_{n}(y_n-y)-(1-e_{n})y\\|\\nonumber\\\\[6pt]\n\t\t\t&\\leq & e_{n}\\|(y_n-y)\\|+(1-e_{n})\\|y\\|\\nonumber\\\\[6pt]\n\t\t\t&\\leq & \\text{max}\\{\\|y_0-y\\|,\\|y\\|\\}\\nonumber.\n\t\t\\end{eqnarray}\n\t\tThus, $\\{y_n\\}$ is bounded.\\\\\n\t\t\n\t\t\\item[Step 2.]\\label{C3.2} $\\|y_{n+1}-y_n\\| \\to 0$ as $n \\to \\infty.$\\\\\n\t\t\n\t\tUsing nonexpansivity of $\\mathcal{S} \\text{ and } \\mathcal{T}$, we have\n\t\t\\begin{eqnarray}\n\t\t\t\\|y_{n+1}-y_n\\|&=&\\|\\mathcal{S}[(1-\\theta_{n})e_{n}y_n + \\theta_{n} \\mathcal{T}(e_{n}y_{n})]-\\mathcal{S}[(1-\\theta_{n-1})e_{n-1}y_{n-1} + \\theta_{n-1} \\mathcal{T}(e_{n-1}y_{n-1})]\\| \\nonumber\\\\[6pt]\n\t\t\t&\\leq&\\|(1-\\theta_{n})e_{n}y_n + \\theta_{n} \\mathcal{T}(e_{n}y_{n})-(1-\\theta_{n-1})e_{n-1}y_{n-1} - \\theta_{n-1} \\mathcal{T}(e_{n-1}y_{n-1})\\| \\nonumber\\\\[6pt]\n\t\t\t&=& \\|(1-\\theta_{n})e_{n}y_n-(1-\\theta_{n-1})e_{n-1}y_{n-1}+\\theta_{n} \\mathcal{T}(e_{n}y_{n}) - \\theta_{n-1} \\mathcal{T}(e_{n-1}y_{n-1})\\| \\nonumber\\\\[6pt]\n\t\t\t&\\leq&\\|(1-\\theta_{n})(e_{n}y_n-e_{n-1}y_{n-1})+(\\theta_{n-1}-\\theta_{n})e_{n-1}y_{n-1})\\|\\nonumber\\\\\n\t\t\t&+&\\|\\theta_{n} (\\mathcal{T}(e_{n}y_{n})- \\mathcal{T}(e_{n-1}y_{n-1}))+(\\theta_{n}-\\theta_{n-1})\\mathcal{T}(e_{n-1}y_{n-1})\\| \\nonumber\\\\[6pt]\n\t\t\t&\\leq&\\|e_{n}y_n-e_{n-1}y_{n-1}\\|+|\\theta_{n}-\\theta_{n-1}|\\mathcal{C}_1 \\nonumber\\\\[6pt]\n\t\t\t&=& \\|e_{n}(y_n-y_{n-1})+(e_{n}-e_{n-1})y_{n-1}\\|+|\\theta_{n}-\\theta_{n-1}|\\mathcal{C}_1\\nonumber\\\\[6pt]\n\t\t\t&\\leq& e_{n}\\|y_n-y_{n-1}\\|+|e_{n}-e_{n-1}| \\mathcal{C}_2 +|\\theta_{n}-\\theta_{n-1}|\\mathcal{C}_1, \\nonumber\n\t\t\\end{eqnarray}\n\t\tfor some $\\mathcal{C}_1,$ $\\mathcal{C}_2 >0$. By applying Lemma \\ref{P4L2.2} with $a_n=\\|y_n-y_{n-1}\\|, b_n=0$, $ \\epsilon_n=|e_{n}-e_{n-1}| \\mathcal{C}_2 +|\\theta_{n}-\\theta_{n-1}|\\mathcal{C}_1 \\text{ and } \\theta_{n}=1-e_{n}, \\forall n \\in \\mathbb{N}$, we obtain that $\\|y_{n+1}-y_n\\| \\to 0$.\\\\\n\t\t\n\t\t\\item[Step 3.]\\label{C3.3} $\\|y_n-\\mathcal{T}y_n\\|$ and $\\|y_n-\\mathcal{S}y_n\\| \\to 0$ as $n \\to \\infty$.\\\\\n\t\t\n\t\tLet  $y\\in \\Omega$. Note\\begin{eqnarray}\n\t\t\t\\|y_{n+1}-y\\|^2&=& \\|\\mathcal{S}[(1-\\theta_{n})e_{n}y_n + \\theta_{n} \\mathcal{T}(e_{n}y_n)]-y\\|^2 \\nonumber\\\\ [6pt]\n\t\t\t&\\leq &\\|(1-\\theta_{n})e_{n}y_n + \\theta_{n} \\mathcal{T}(e_{n}y_n)-y\\|^2 \\nonumber\\\\[6pt]\n\t\t\t&=&(1-\\theta_{n})\\|e_{n}y_n-y\\|^2 + \\theta_{n} \\|\\mathcal{T}(e_{n}y_n)-y\\|^2 -\\theta_{n}(1-\\theta_{n}) \\|e_{n}y_n-\\mathcal{T}(e_{n}y_n)\\|^2\\nonumber\\\\ [6pt]\n\t\t\t&\\leq&(1-\\theta_{n})\\|e_{n}y_n-y\\|^2 + \\theta_{n} \\|e_{n}y_n-y\\|^2 -\\theta_{n}(1-\\theta_{n}) \\|e_{n}y_n-\\mathcal{T}(e_{n}y_n)\\|^2\\nonumber\\\\[6pt]\n\t\t\t&=& \\|e_{n}y_n-y\\|^2 - \\theta_{n}(1-\\theta_{n})\\|e_{n}y_n - \\mathcal{T}(e_{n}y_n)\\|^2,\n\t\t\\end{eqnarray}\n\t\twhich implies that\n\t\t\\begin{eqnarray}\n\t\t\t\\theta_{n}(1-\\theta_{n})\\|e_{n}y_n - \\mathcal{T}(e_{n}y_n)\\|^2 &\\leq& \\|e_{n}y_n-y\\|^2 -\\|y_{n+1}-y\\|^2.\\nonumber\\\\\n\t\t\t&\\leq& (\\|e_{n}y_n-y\\|+\\|y_{n+1}-y\\|)\\|e_{n}y_n-y_{n+1}\\| \\nonumber\\\\\n\t\t\t&\\leq& (\\|e_{n}y_n-y\\|+\\|y_{n+1}-y\\|)\\|e_{n}y_n-e_{n}y_{n+1}+e_{n}y_{n+1}-y_{n+1}\\| \\nonumber\\\\\n\t\t\t&\\leq& (\\|e_{n}y_n-y\\|+\\|y_{n+1}-y\\|)(e_{n}\\|y_n-y_{n+1}\\| +(1-e_{n})\\|y_{n+1}\\|).\\nonumber\n\t\t\\end{eqnarray}\n\t\tNote \n\t\t\\begin{eqnarray*}\n\t\t\t\\|\\beta_{n}y_n-y\\|^2 &=& \\|\\beta_{n}y_n+\\beta_{n}y-\\beta_{n}y-y\\|^2\\\\\n\t\t\t&=&\\|\\beta_{n}(y_n-y)-(1-\\beta_{n})y\\|^2\\\\\n\t\t\t&=&(1-\\beta_{n})\\|y\\|^2 +\\beta_{n}(1-\\beta_{n})\\|y_n\\|^2-\\beta_{n}\\|y_n-y\\|^2\\\\\n\t\t\t&\\leq& (1-\\beta_{n})\\|y\\|^2 +\\beta_{n}(1-\\beta_{n})\\|y_n\\|^2.\n\t\t\\end{eqnarray*}\n\t\tThus  $\\|\\beta_{n}y_n-y\\|^2  \\to 0 \\text{ as }n \\to \\infty.$\n\t\tSince $\\lim\\limits_{n \\to \\infty} e_{n} = 1$ by the condition (\\textbf{i}), $0<\\underline{\\theta}\\le \\theta_{n}\\leq\\overline{\\theta}< 1$   for all $n \\in \\mathbb{N}$ by the condition (\\textbf{ii}) and  $\\|y_{n+1}-y\\|\\to 0$ by step \\ref{C3.2}, we have\n\t\t$\t\\|e_{n}y_n - \\mathcal{T}(e_{n}y_n)\\|  \\to 0$ as $n \\to \\infty.$\n\t\t\n\t\t\n\t\t\n\t\tNow \t\\begin{eqnarray}\n\t\t\t\\|y_n-\\mathcal{T}y_n\\| &=&\\|y_n-e_{n}y_n+e_{n}y_n-\\mathcal{T}(e_{n}y_n)+\\mathcal{T}(e_{n}y_n)-\\mathcal{T}y_n\\| \\nonumber\\\\\n\t\t\t&\\leq& \\|y_n-e_{n}y_n\\|+\\|e_{n}y_n-\\mathcal{T}(e_{n}y_n)\\|+\\|\\mathcal{T}(e_{n}y_n)-\\mathcal{T}y_n\\|  \\nonumber\\\\\n\t\t\t&\\leq& 2(1-e_{n}) \\|y_n\\|+\\|e_{n}y_n-\\mathcal{T}(e_{n}y_n)\\| \\to 0 \\text{ as } n \\to \\infty. \\nonumber\n\t\t\\end{eqnarray}\n\t\tObserve that\n\t\t\\begin{eqnarray}\n\t\t\t\\|y_n-\\mathcal{S}y_n\\| &\\leq& \\|y_n-y_{n+1}\\|+\\|y_{n+1}-\\mathcal{S}y_n\\| \\nonumber\\\\\n\t\t\t&=& \\|y_n-y_{n+1}\\|+\\|\\mathcal{S}[(1-\\theta_n)e_{n}y_n + \\theta_n \\mathcal{T}(e_{n}y_n)]-\\mathcal{S}y_n\\| \\nonumber\\\\\n\t\t\t&\\leq&\\|y_n-y_{n+1}\\|+\\|(1-\\theta_n)e_{n}y_n + \\theta_n \\mathcal{T}(e_{n}y_n)-y_n\\| \\nonumber\\\\\n\t\t\t&\\leq&\\|y_n-y_{n+1}\\|+(1-\\theta_n)\\|e_{n}y_n-y_n\\| + \\theta_n \\|\\mathcal{T}(e_{n}y_n)-y_n\\| \\nonumber\\\\\n\t\t\t&\\leq&\\|y_n-y_{n+1}\\|+(1-\\theta_n)(1-e_{n})\\|y_n\\|+\\theta_n \\|\\mathcal{T}(e_{n}y_n)-\\mathcal{T}y_n+\\mathcal{T}y_n-y_n\\| \\nonumber\\\\\n\t\t\t&\\leq&\\|y_n-y_{n+1}\\|+(1-\\theta_n)(1-e_{n})\\|y_n\\|+\\theta_n\\|e_{n}y_n-y_n\\|+\\theta_{n}\\|\\mathcal{T}y_n-y_n\\| \\nonumber\\\\\n\t\t\t&=&\\|y_n-y_{n+1}\\|+(1-e_{n})\\|y_n\\|+\\theta_n \\|\\mathcal{T}y_n-y_n\\| \\to 0 \\text{ as } n \\to \\infty. \\nonumber\n\t\t\\end{eqnarray}\n\t\tSince $e_{n} \\to 1$ and  $\\|y_n-y_{n-1}\\|$ and $\\|y_n-\\mathcal{T}y_n\\| \\to 0 \\text{ as } n \\to \\infty,$\n\t\twe have $\\|y_n-\\mathcal{S}y_n\\| \\to 0.$\n\t\t\n\t\t\\item[Step 4.] \\label{C3.4}$\\{y_n\\}$ converges strongly to $\\bar{y}=\\operatorname{proj}_{\\Omega}(0).$\\\\\n\t\t\n\t\tFrom (\\ref{P4e1}), we set\n\t\t\\begin{eqnarray}\n\t\t\t\\|y_{n+1}-\\bar{y}\\| &\\leq& \\|\\mathcal{S}[(1-\\theta_n)e_{n}y_n + \\theta_n \\mathcal{T}(e_{n}y_n)]-\\bar{y}\\|\n\t\t\t\\nonumber\\\\\n\t\t\t&=& \\|(1-\\theta_n)e_{n}y_n + \\theta_n \\mathcal{T}(e_{n}y_n)-\\bar{y}\\| \\nonumber\\\\\n\t\t\t&=& \\|(1-\\theta_n)(e_{n}y_n-\\bar{y}) + \\theta_n \\mathcal{T}((e_{n}y_n)-\\bar{y})\\| \\nonumber\\\\\n\t\t\t&\\leq& \\|e_{n}y_n -\\bar{y}\\|.\n\t\t\\end{eqnarray}\n\t\tHence\n\t\t\\begin{eqnarray}\\label{P4e3.4}\n\t\t\t\\|y_{n+1}-\\bar{y}\\|^2 &\\leq& \\|e_{n}y_n -\\bar{y}\\|^2\\nonumber\\\\\n\t\t\t&\\leq& \\|e_{n}(y_n -\\bar{y})-(1-e_{n})\\bar{y}\\|^2 \\nonumber\\\\\n\t\t\t&\\leq& e_{n}^2 \\|y_n -\\bar{y}\\|^2+2 e_n(1-e_{n}) \\langle -\\bar{y}, y_n-\\bar{y} \\rangle+(1-e_{n})^2\\|\\bar{y}\\|^2 \\nonumber\\\\\n\t\t\t&\\leq& e_{n} \\|y_n -\\bar{y}\\|^2+2e(1-e_{n})  \\langle -\\bar{y}, y_n-\\bar{y} \\rangle+(1-e_{n})^2\\|\\bar{y}\\|^2.\n\t\t\\end{eqnarray}\t\n\t\tNext we show that\n\t\t\\begin{equation}\\label{P4e3.5}\n\t\t\t\\limsup_{n\\to \\infty} \\langle -\\bar{y}, y_n-\\bar{y} \\rangle\\leq 0.\n\t\t\\end{equation}\n\t\tContrarily assume a real number $l$ and a subsequence $\\{y_{n_j}\\}$ of $\\{y_n\\}$ satisfying\n\t\t\\begin{equation}\n\t\t\t\\langle -\\bar{y}, y_{n_j}-\\bar{y}\\rangle\\geq l > 0 ~ \\forall j\\in \\mathbb{N}.\n\t\t\\end{equation}\n\t\tSince $\\{y_n\\}$ is bounded, there exists a subsequence $\\{y_{n_j}\\}$ which converges weakly to an element $y\\in \\mathcal{H}.$ Lemma \\ref{P4L2.1} alongwith Step \\ref{C3.2} implies that $y\\in \\Omega.$ By using variational characterazation of projection, we can easily derive\n\t\t\\begin{equation}\n\t\t\t\\lim_{j\\to\\infty}\\langle -\\bar{y}, y_{n_j}-\\bar{y}\\rangle = \\langle -\\bar{y}, y-\\bar{y}  \\rangle \\leq 0,\n\t\t\\end{equation}\n\t\twhich is a contradiction. Thus, (\\ref{P4e3.5}) holds and\n\t\t\\begin{eqnarray}\n\t\t\t\\limsup_{n\\to \\infty} \\left( 2 e_n \\langle -\\bar{y}, y_n-\\bar{y} \\rangle+(1-e_{n})\\|\\bar{y}\\|^2 \\right)  \\leq 0.\n\t\t\\end{eqnarray}\n\t\tConsider $a_n=\\|y_n-\\bar{y}\\|, b_n= 2 e_n \\langle -\\bar{y}, y_n-\\bar{y} \\rangle+(1-e_{n})\\|\\bar{y}\\|^2, \\epsilon_n= 0$ and $\\theta_{n}= 1-e_{n}$ in (\\ref{P4e3.4}) and apply Lemma \\ref{P4L2.2}, we get the desired conclusion.\n\t\\end{enumerate}\n\\end{proof}\n\n\n\\begin{Corollary}\n\t\\label{c3.1}  Let $R_1, R_2:\\mathcal{H}\\to \\mathcal{H}$ be $\\alpha_1,\\alpha_2\n\t$-averaged operators respectively, such that $\\operatorname{Fix}{(R_1)}\\cap \\operatorname{Fix}(R_2) \\neq\n\t\\emptyset$. For $y_1\\in \\mathcal{H}$, let $\\{y_n\\}$ be sequence in $\\mathcal{H}$ defined by\n\t\\begin{equation}\\label{key}\n\t\ty_{n+1}=R_2\\{e_n y_n +\\theta_{n}(R_1 (e_{n}y_n)-e_{n}y_n)\\}\\ \\ \\forall n\\in\n\t\t\\mathbb{N},\n\t\\end{equation}\n\twhere $\\{\\theta_{n}\\}$ and $\\{e_{n}\\}$ are real sequences satisfy the\n\tcondition (\\textbf{i}) given in Theorem \\ref{P4T3.1} and the conditions:\n\n\t$$0<\\underline{\\Theta }\\leq \\alpha _{1}\\theta _{n}\\leq\n\t\\overline{\\Theta }<1 \\textrm{ for all } n\\in \\mathbb{N}\\textrm{ and  }\\sum_{n=1}^{\\infty} |\\theta\n\t_{n}-\\theta _{n-1}|<\\infty. $$\n\n\tThen the sequence $\\{y_n\\}$ converges strongly to $\\operatorname{proj}_{\\operatorname{Fix}{(R_1)}\\cap\n\t\t\\operatorname{Fix}(R_2)}(0).$\n\\end{Corollary}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Tikhonov Regularized Forward-Backward-type Algorithms}\\label{sec4}\nIn this section, we propose a forward-backward algorithm based on Algorithm \\ref{P4A3.1} to simultaneously solve the monotone inclusion problems of the sum of two maximally monotone operators in which one is single-valued. Further, we also propose an forward-backward-type primal-dual algorithm based on Algorithm \\ref{P4A3.1}  to solve a complexly structured monotone inclusion problem containing composition with linear operators and parallel-sum operators.\n\n\\subsection{Tikhonov Regularized Forward-Backward Algorithm}\nLet $A_1,A_2: \\mathcal{H} \\to 2^\\mathcal{H}$ be maximally monotone operators and $B_1,B_2: \\mathcal{H} \\to \\mathcal{H}$ be $\\alpha_1, \\alpha_2$-cocoercive operators. We consider the monotone inclusion problem\n\\begin{equation}\\label{P4Al3.1}\n\t\\text{ find }x \\in \\mathcal{H} \\text{ such that } 0 \\in (A_1 +B_1)x \\cap (A_2+B_2)x.\n\\end{equation}\nWe propose a forward-backward algorithm to solve the monotone inclusion problem (\\ref{P4Al3.1}) such that generated sequence converges strongly to the solution set of the Problem (\\ref{P4Al3.1}).\n\\begin{Theorem}\\label{T4.1}\n\tSuppose $\\operatorname{Zer}(A_1+B_1)\\cap \\operatorname{Zer}(A_2+B_2) \\neq \\emptyset$ and  $\\gamma_1 \\in (0,2\\alpha_1)\\text{ and }\\gamma_2 \\in (0,2\\alpha_2).$ For $y_1 \\in \\mathcal{H}$, consider the forward-backward algorithm defined as follows:\n\t\\begin{equation}\\label{e4.1}\n\t\ty_{n+1}=J_{\\gamma_2A_2}(Id-\\gamma_2B_2)\\left\\lbrace (1-\\theta_n)e_{n}y_n + \\theta_n J_{\\gamma_1 A_1}(e_n y_n-\\gamma_1B_1(e_{n}y_n))\\right\\rbrace  \\forall n \\in \\mathbb{N}.\n\t\\end{equation}\n\twhere $\\{\\theta_{n}\\}$ and $\\{e_{n}\\}$ are real sequences satisfy the\n\tcondition (\\textbf{i}) given in Theorem \\ref{P4T3.1} and the conditions:\n\n\t$$0<\\underline{\\Theta }\\leq  \\frac{2\\alpha_1}{4\\alpha_1-\\gamma_1} \\theta _{n}\\leq\n\t\\overline{\\Theta }<1 \\textrm{ for all } n\\in \\mathbb{N} \\textrm{ and } \\sum_{n=1}^{\\infty}|\\theta\n\t_{n}-\\theta _{n-1}|<\\infty .$$\n\n\tThen $\\{y_n\\}$ converges strongly to $\\operatorname{proj}_{\\operatorname{Zer}(A_1+B_1)\\cap \\operatorname{Zer}(A_2+B_2)}(0).$\n\\end{Theorem}\n\\begin{proof}\n\tSet $T_1=J_{\\gamma_1A_1}(Id-\\gamma_1B_1)$ and $T_2=J_{\\gamma_2A_2}(Id-\\gamma_2B_2),$ then algorithm (\\ref{e4.1}) can be rewritten as:\n\t\\begin{equation}\n\t\ty_{n+1}=T_2\\{(1-\\theta_{n})e_n y_n +\\alpha_1\\theta_{n}(T_1 (e_{n}y_n)-e_{n}y_n)\\} \\forall n \\in \\mathbb{N}.\n\t\\end{equation}\n\tSince $J_{\\gamma_1A_1}$ is $\\frac{1}{2}$-cocoercive \\cite[Corollary 23.8]{Bauschke2011} and $Id-\\gamma_1B_1$ is $\\frac{\\gamma_1}{2\\alpha_1}$-averaged \\cite[proposition 4.33]{Bauschke2011}, $T_1$ is $\\frac{2\\alpha_1}{4\\alpha_1 -\\gamma_1}$-averaged. Using the fact that $\\operatorname{Zer}(A_i+B_i)= \\operatorname{Fix}(T_i), i=1,2$ and assumption  $\\operatorname{Zer}(A_1+B_1)\\cap \\operatorname{Zer}(A_2+B_2) \\neq \\emptyset$ \\cite[Proposition 25.1]{Bauschke2011}, $\\operatorname{Fix}(T_1) \\cap \\operatorname{Fix}(T_2) \\neq \\emptyset.$ Therefore, Theorem \\ref{T4.1} follows from Corollary \\ref{c3.1}.\n\\end{proof}\n\nFurther, we consider the following minimization problem and propose a proximal-point-type algorithm to solve it.\n\\begin{Problem}\\label{P4P4.1}\n\tConsider strictly positive real numbers $\\beta_1,\\beta_2.$ Let $f_1, f_2 :\\mathcal{H} \\to \\mathbb{R} \\cup \\{\\infty\\}$ be proper convex lower semicontinuous functions and $g_1,g_2: \\mathcal{H} \\to \\mathbb{R}$ be convex and Frechet-differentiable functions with $\\frac{1}{\\beta_1},\\frac{1}{\\beta_2}$-Lipschitz continuous gradients, respectively. The problem is to find a point $y\\in \\mathcal{H}$ satisfying\n\t\\begin{equation}\\label{P4.1}\n\t\tmin\\left\\lbrace (f_1+g_1) \\cap (f_2+g_2)\\right\\rbrace .\n\t\\end{equation}\n\\end{Problem}\n\\begin{Theorem}\\label{c4.1}\n\tConsider the functions $f_1, f_2, g_1 \\text{ and }g_2$ are as in Problem \\ref{P4P4.1}. Let $argmin(f_1+g_1) \\cap argmin(f_2+g_2) \\neq \\emptyset.$ For $\\gamma_1 \\in (0,2\\beta_1] \\text{ and } \\gamma_2 \\in (0,2\\beta_2],$ consider an algorithm with initial point $y_1 \\in \\mathcal{H},$\n\t\\begin{equation}\\label{e4.2}\n\t\ty_{n+1}=prox_{\\gamma_2f_2}(Id-\\gamma_2\\nabla g_2)\\{(1-\\theta_n)e_{n}y_n + \\theta_n prox_{\\gamma_1f_1}(e_n y_n-\\gamma_1\\nabla g(e_{n}y_n))\\}  \\  \\ \\forall n \\in \\mathbb{N},\n\t\\end{equation}\n\twhere $\\theta_n \\in (0,1] \\text{ and }e_{n} \\in (0,\\frac{4\\beta_1-\\gamma_1}{2\\beta_1})$ satisfy the condition (\\textbf{i}) in Theorem \\ref{P4T3.1} and the conditions:\n\n\t$$0< \\underline{\\Theta} \\leq \\frac{2\\beta_1}{4\\beta_1-\\gamma_1}\\theta_{n} \\leq \\overline{\\Theta } <1 \\textrm{ and } \\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$$\n\n\tThen $\\{y_n\\}$ converges strongly to minimal norm solution of Problem \\ref{P4P4.1}.\n\\end{Theorem}\n\\begin{proof}\n\tConsider $A_1=\\partial f_1, A_2=\\partial f_2, B_1= \\nabla g_1, B_1=\\nabla g_2$. Since $$\\operatorname{Zer}(\\partial f_i +\\nabla g_i)= argmin (f_i +g_i), \\ i=1,2$$ and $\\nabla g_1,\\nabla g_2$ are $\\beta_1,\\beta_2$-cocoercive, respectively (Ballion-Hadded Theorem \\cite[Corollary 16.18]{Bauschke2011}). Thus, by Theorem \\ref{T4.1}, $\\{y_n\\}$ converges strongly to a point in $argmin(f_1+g_1) \\cap argmin(f_2+g_2)$.\n\\end{proof}\n\n\n\\subsection{Tikhonov Regularized Forward-Backward-type Primal-Dual Algorithm}\n\n\\begin{Problem}\\label{P6.1}\n\tSuppose $\\Omega_1, \\dots,\\Omega_m$ are real Hilbert spaces. Consider the following operators:\n\t\\begin{enumerate}\n\t\t$\\bullet  A,B:\\mathcal{H} \\to 2^\\mathcal{H}$ are maximally monotone operators,\\\\\n\t\t$\\bullet C,D:\\mathcal{H} \\to \\mathcal{H}$ are $\\mu_1,\\mu_2$-cocoercive operators, respectively, $\\mu_1,\\mu_2> 0$\\\\\n\t\t$\\bullet P_i,Q_i,R_i,  S_i:\\Omega_i \\to 2^{\\Omega_i}$ are maximally monotone operators such that $Q_i, S_i$ are $\\nu_i,\\delta_i$-cocoercive, respectively, $\\nu_i,\\delta_i>0,$ $i=1, \\dots, m$,\\\\\n\t\t$\\bullet$ nonzero continuous linear operators $L_i: \\mathcal{H} \\to \\Omega_i, i=1, \\dots, m.$\n\t\\end{enumerate}\n\tThe primal inclusion problem is to find $\\bar{y} \\in \\mathcal{H}$ satisfying\n\t\\begin{eqnarray}\n\t\t& 0\\in A \\bar{y} +\\sum_{i=1}^{m} L_i^{*}(P_i \\square Q_i)(L_i \\bar{y})+C\\bar{y}\\nonumber\\\\\n\t\t&\\text{and}\\nonumber\\\\\n\t\t&0\\in B\\bar{y} +\\sum_{i=1}^{k} L_i ^*(R_i \\square S_i)(L_i \\bar{y})+D\\bar{y} \\nonumber\n\t\\end{eqnarray}\n\ttogether with dual inclusion problem\n\t\\begin{eqnarray}\\label{e6.11}{find \\ \\ \\bar{v}_1\\in \\varOmega_1, \\dots, \\bar{v}_m \\in \\varOmega_m \\text{ such that }\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t-\\sum_{i=1}^{m}L_i^* \\bar{v}_i \\in Ay+Cy\\\\\n\t\t\t\t\\bar{v}_i \\in (P_i \\square Q_i)(L_i x)\\\\\n\t\t\t\t\\text{ and }\\\\\n\t\t\t\t-\\sum_{i=1}^{m}L_i^* \\bar{v}_i \\in By+Dy\\\\\n\t\t\t\t\\bar{v}_i \\in (R_i \\square S_i)(L_i y), \\ \\ \\ i =1, \\dots m.\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\\end{Problem}\nA point $(\\bar{y}, \\bar{v}_1,\\dots, \\bar{v}_m) \\in \\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m$ be a primal-dual solution of Problem \\ref{P6.1} if it satisfies the following:\\\\\n\\begin{eqnarray}{\n\t\t\\left\\{\n\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t-\\sum_{i=1}^{m}L_i^* \\bar{v}_i \\in A\\bar{y}+C\\bar{y},\\\\\n\t\t\t-\\sum_{i=1}^{m}L_i^* \\bar{v}_i \\in B\\bar{y}+D\\bar{y}, \\\\\n\t\t\t\\bar{v}_i \\in (P_i \\square Q_i)(L_i \\bar{y}),\\\\\n\t\t\t\\bar{v}_i \\in (R_i \\square S_i)(L_i \\bar{y}) \\ \\ \\ i =1,\\dots, m.\n\t\t\\end{array}\\right.}\n\\end{eqnarray}\n\n\n\\begin{Theorem}\\label{P4T6.1}\n\tConsider the operators as in Problem \\ref{P6.1}. Assume\n\t\\begin{equation}\\label{e6.4}\n\t\t0\\in ran \\left( A+\\sum_{i=1}^{m}L_i ^* \\circ (P_i \\square Q_i)\\circ L_i +C\\right)  \\bigcap ran \\left( B+\\sum_{i=1}^{m}L_i ^* \\circ (R_i \\square S_i)\\circ L_i +D\\right).\n\t\\end{equation}\n\tLet $\\tau,\\sigma_1,\\dots, \\sigma_m >0 $ such that\n\t\\begin{equation*}\n\t\t2\\rho\\min\\{\\beta_1, \\beta_2\\} \\geq 1,\n\t\\end{equation*}\n\twhere\n\t\\begin{eqnarray*}\n\t\t\\rho=\\min\\left\\lbrace \\frac{1}{\\tau},\\frac{1}{\\sigma_1},\\dots, \\frac{1}{\\sigma_m}\\right\\rbrace \\left(1-\\sqrt{\\tau\\sum_{i=1}^{m}\\sigma_i\\|L_i\\|^2}\\right)\\\\\n\t\t\\beta_1=\\min\\{\\mu_1,\\nu_1,\\dots,\\nu_m\\} \\textrm{ and } \\beta_2= \\min\\{\\mu_2,\\delta_1,\\dots,\\delta_m\\}.\n\t\\end{eqnarray*}\n\t\n\tConsider the algorithm with intial point $(y_1,v_{1,1},\\dots, v_{m,1}) \\in \\mathcal{H} \\times \\varOmega_1 \\times \\cdots \\times\\varOmega_m$ and defined by\n\t\n\t\n\t\\begin{algorithm}[H]\n\t\t\\SetAlgoLined\n\t\t\n\t\t\n\t\t\\KwIn{\\begin{enumerate}\n\t\t\t\t\\item initial points $(y_1,v_{1,1},\\dots, v_{m,1}) \\in \\mathcal{H} \\times \\varOmega_1 \\times \\cdots \\times\\varOmega_m$\n\t\t\t\t\\item real numbers $\\tau, \\sigma_i > 0$, $i=1,2,...,m$ be such that $\\tau\\sum_{i=1}^{m} \\sigma_i \\|L_i\\|^2 < 4 $.\n\t\t\t\t\\item $\\theta_n \\in (0,\\frac{4\\beta_1 \\rho-1}{2\\beta_1 \\rho}], e_{n} \\in (0,1)$ .\n\t\t\t\\end{enumerate}\t\n\t\t}\n\t\t\\SetKwBlock{For}{For}{}\n\t\t\\SetKwProg{}{}{}{}\n\t\t\\For($n\\in \\mathbb{N}$; ){\n\t\t\t$p_n = J_{\\tau A}\\left[e_{n}y_n-\\tau \\left(e_{n}\\sum_{i=1}^{m}L_i^* v_{i,n}+C(e_{n}y_n)\\right)\\right]$\\\\\n\t\t\t$u_n= e_{n}y_n+\\theta_{n}(p_n-e_{n}y_n)$\\\\\n\t\t\t\n\t\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t\t\n\t\t\t\t$q_{i,n}=J_{\\sigma_iP_i^{-1}}\\left[e_{n}v_{i,n}+\\sigma_i(L_i(2p_n-e_{n}y_n)-Q_i^{-1}(e_{n}v_{i,n}))\\right]$\\\\\n\t\t\t\t$u_{i,n}=e_{n}v_{i,n}+\\theta_{n}(q_{i,n}-e_{n}v_{i,n})$\\\\\n\t\t\t}\n\t\t\t\n\t\t\t$y_{n+1}=J_{\\tau B}\\left[u_n-\\tau \\left(\\sum_{i=1}^{m}L_i^* u_{i,n}+D(u_n)\\right)\\right]$\\\\\n\t\t\t$v_{i,n+1}=J_{\\sigma_iR_i^{-1}}\\left[u_{i,n}+\\sigma_i(L_i(2y_{n+1}-r_n)-S_i^{-1}(u_{i,n}))\\right]$\t\n\t\t\t\n\t\t\t\n\t\t}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\KwOut{$(y_{n+1},v_{1,n+1},\\dots,v_{m,n+1})$}\t\t\n\t\t\\caption{To optimize the complexly structured Problem \\ref{P6.1} }\\label{P4Al4.11}\n\t\\end{algorithm}\n\t\n\twhere, sequencences $\\{\\theta_{n}\\}$ and $\\{e_{n}\\}$ satisfy the condition (\\textbf{i}) and the conditions:\n\n\t$$0< \\underline{\\Theta} \\leq \\frac{2\\beta_1 \\rho}{4\\beta_1 \\rho-1}\\theta_{n} \\leq \\overline{\\Theta } <1   \\textrm{ and } \\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$$\n\n\tThen there exists $(\\bar{y},\\bar{v}_1,\\dots,\\bar{v}_m)\\in  \\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m $ such that sequence $\\{(y_n,v_{1,n},\\dots,v_{m,n})\\}$ converges strongly to  $(\\bar{y},\\bar{v}_1,\\dots,\\bar{v}_m)$ and satisfies the Problem \\ref{P6.1}.\n\\end{Theorem}\n\n\\begin{proof}\n\tConsider the real Hilbert space $\\mathcal{K }\\equiv$ $\\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m$ endowed with innner product\n\t$$\\langle (x,u_1,\\dots, u_m),(y,v_1,\\dots,v_m)\\rangle_{\\mathcal{K}} =\\langle x,y\\rangle_\\mathcal{H} +\\sum_{i=1}^{m}\\langle u_i,v_i \\rangle_{\\varOmega_i}$$\n\tand corresponding norm\n\t$$\\|(x,u_1, \\dots,u_m)\\|_\\mathcal{K}=\\sqrt{\\|x\\|_\\mathcal{H} ^2+\\sum_{i=1}^{m}\\|u_i\\|^2 _{\\varOmega_i} },\\ \\ \\  \\forall (x,u_1,\\dots u_m),(y,v_1,\\dots,v_m) \\in \\mathcal{K}.$$\n\tFurther we consider following operators on real Hilbert space $\\mathcal{K}$:\n\t\n\t\\begin{enumerate}\n\t\t\\item $\\phi_1: \\mathcal{K} \\to 2^\\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto (Ax,P_1^{-1}u_1,\\dots,P_m^{-1}u_m),$\n\t\t\\item $\\phi_2: \\mathcal{K} \\to 2^\\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto (Bx,R_1^{-1}u_1,\\dots,R_m^{-1}u_m),$\n\t\t\\item $\\xi:\\mathcal{K} \\to \\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto \\left(\\sum_{i=1}^{m}L_i ^*u_i,-L_1 x,\\dots, -L_m x\\right),$\n\t\t\\item $\\psi_1:\\mathcal{K} \\to \\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto\\left(Cx,Q_1^{-1}u_1,\\dots, Q_m^{-1}u_m\\right),$\n\t\t\\item $\\psi_2:\\mathcal{K} \\to \\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto\\left(Dx,S_1^{-1}u_1,\\dots, S_m^{-1}u_m\\right).$\n\t\\end{enumerate}\n\tThese operators are maximally monotone as $A,B,P_i,R_i,Q_i, S_i, \\ i=1,\\dots,m$ are maximally monotone and $\\xi$ is skew-symmetric, i.e.,\n\t$\\xi^*=-\\xi$ (\\cite[Proposition 20.22, 20.23 and Example 20.30]{Bauschke2011}).\n\tNow, define the continuous linear operator $\\textbf{V}:\\mathcal{K} \\to \\mathcal{K}$ by,\n\t$$(x,u_1,\\dots,u_m) \\to \\left(\\frac{x}{\\tau}-\\sum_{i=1}^{m}L_i^*u_i, \\frac{u_1}{\\sigma_1}-L_1x,\\dots,\\frac{u_m}{\\sigma_m}-L_mx\\right),$$\n\twhich is self adjoint and $\\rho$-strongly positive, i.e., $\\langle \\textbf{x},\\textbf{Vx}\\rangle_{\\mathcal{K}} \\geq \\rho \\|\\textbf{x}\\|_{\\mathcal{K}}^2 \\ \\ \\forall \\textbf{x}\\in \\mathcal{K}$ \\cite{BC2013}. Therefore inverse of operator $\\textbf{V}$ exists and satisfy $\\|\\textbf{V}^{-1}\\|\\leq \\frac{1}{\\rho}.$\\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tNow, consider the sequences\n\t\\begin{eqnarray}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t\\textbf{y}_n$=$(y_n,v_{1,n},\\dots,v_{m,n}),\\\\ \\textbf{u}_n$=$(u_n,u_{1,n},\\dots,u_{m,n}),\\\\\n\t\t\t\t\\textbf{x}_n =(p_n,q_{1,n},\\dots,q_{m,n}).\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\t\n\tBy taking into account the sequences $\\{\\textbf{y}_n \\}$,$\\{\\textbf{x}_n\\}$ and $\\{\\textbf{u}_n\\}$ and operator \\textbf{V}, Algorithm \\ref{P4Al4.11} can be rewritten as\n\n\n\n\t\\begin{eqnarray}\\label{e6.22}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\te_{n}\\text{\\textbf{V}}(\\text{\\textbf{y}}_n)-\\text{\\textbf{V}}(\\text{\\textbf{x}}_n)-\\text{\\textbf{$\\psi_1$}}(e_{n} \\text{\\textbf{y}}_n) \\in (\\phi_1+\\xi)(\\text{\\textbf{x}}_n)\\\\\n\t\t\t\t\\text{\\textbf{u}}_n=e_{n}\\text{\\textbf{y}}_n+\\theta_n(\\text{\\textbf{x}}_n-e_{n}\\text{\\textbf{y}}_n)\\\\\n\t\t\t\t\\text{\\textbf{V}\\textbf{$\\textbf{u}_n$}-\\textbf{V}\\textbf{y}}_{n+1}-\\psi_2 \\text{\\textbf{$\\textbf{u}_n$}} \\in (\\phi_2+\\xi)\\textbf{y}_{n+1}.\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\tOn further analysing Algorithm $\\ref{P4Al4.11}$, we get\n\t\\begin{eqnarray}\\label{e6.22}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t\\textbf{x}_n=J_{\\textbf{A}_1}(e_{n}\\textbf{y}_n-\\textbf{B}_1(e_{n}\\textbf{y}_n)) \\\\\n\t\t\t\t\\textbf{u}_n=e_{n}\\textbf{y}_n+\\theta_n(\\textbf{x}_n-e_{n}\\textbf{y}_n)\\\\\n\t\t\t\t\\textbf{y}_{n+1}=J_{\\textbf{A}_2}(\\textbf{u}_n-\\textbf{B}_2 \\textbf{u}_n),\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\twhere $\\textbf{A}_1=\\textbf{V}^{-1}(\\phi_1 +\\xi)$ ,\\ $\\textbf{B}_1=\\textbf{V}^{-1}\\psi_1$,\\ $\\textbf{A}_2=\\textbf{V}^{-1}(\\phi_2 +\\xi)$ and $\\textbf{B}_2=\\textbf{V}^{-1}\\psi_2.$ The Algorithm (\\ref{e6.22}) is in the form of Algorithm (\\ref{P4Al3.1}) for $\\gamma=1$ and $A_i=\\textbf{A}_i$ and $B_i=\\textbf{B}_i, \\ i=1,2$.\n\tNow, we define the real Hilbert space $\\mathcal{K}_{\\textbf{V}} \\equiv \\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m$ endowed with inner product $\\langle \\textbf{x},\\textbf{y}\\rangle_{\\mathcal{K}_{\\textbf{V}}}=\\langle \\textbf{x},{\\textbf{V}}\\textbf{y}\\rangle_{\\mathcal{K}}$ and corresponding norm is given by, $\\|\\textbf{x}\\|_{\\mathcal{K}_{\\textbf{V}}}=\\sqrt{\\langle \\textbf{x},{\\textbf{V}}\\textbf{x}\\rangle_{\\mathcal{K}}} \\ \\ \\forall \\textbf{x},\\textbf{y} \\in \\mathcal{K}_{\\textbf{V}}.$ \\\\\n\tIn view of real Hilbert space $\\mathcal{K}_{\\textbf{V}}$ and Algorithm \\ref{P4Al4.11}, we observe the following:\n\t\\begin{enumerate}\n\t\t\\item[(i)] since dom$(\\xi)=\\mathcal{K}$, $\\phi_i +\\xi$ are maximally monotone on $\\mathcal{K}$ and thus maximal monotonocity of $\\textbf{A}_i$ and $\\textbf{B}_i$ are  followed, for $i=1,2$ \\cite{BC2013}.\n\t\t\\item[(ii)] $\\textbf{B}_i$  are $\\beta_i \\rho$-cocoercive on $\\mathcal{K}_{\\textbf{V}}$ as $\\psi_i$ are $\\beta_i$-cocoercive in $\\mathcal{K}$ \\cite[Page 672]{BC2013}, for $i=1,2$.\n\t\t\\item[(iii)] $\\operatorname{Zer}(\\textbf{A}_i+\\textbf{B}_i$)=$\\operatorname{Zer}({\\textbf{V}^{-1}}(\\phi_i+\\xi+\\psi_i))$=$\\operatorname{Zer}(\\phi_i+\\xi+\\psi_i),\\ i=1,2$ and from condition $(\\ref{e6.4})$, we can easily obtain that $\\operatorname{Zer}(\\textbf{A}_1+\\textbf{B}_1)\\cap \\operatorname{Zer}(\\textbf{A}_2+\\textbf{B}_2) \\neq \\emptyset.$\n\t\\end{enumerate}\n\t\\par \tSince \\textbf{V} is self-adjoint and $\\rho$-strongly positive, weak convergence and strong convergence of sequences are same in both Hilbert spaces $\\mathcal{K}$ and $\\mathcal{K}_{\\textbf{V}}$. Operators $\\textbf{A}_i, \\textbf{B}_i, \\ i=1,2$ and sequences $\\{\\theta_{n}\\},\\{e_{n}\\}$ satisfy the assumptions in Theorem $\\ref{T4.1}$, therefore, according to Theorem \\ref{T4.1}, $\\{\\textbf{x}_n\\}$  converges strongly to $(\\bar{y},\\bar{v}_1,\\dots,\\bar{v}_m)\\in$ $\\operatorname{proj}_{\\operatorname{Zer}(\\textbf{A}_1+\\textbf{B}_1)\\cap \\operatorname{Zer}(\\textbf{A}_2+\\textbf{B}_2)}(0,\\dots,0)$ in the space ${\\mathcal{K}}_\\textbf{V}$ as $n \\to \\infty.$ Thus, we obtain the conclusion as $(\\bar{y},\\bar{v}_1,\\dots,\\bar{v}_m)\\in \\operatorname{Zer}(\\phi_1+\\xi+\\psi_1)\\cap \\operatorname{Zer}(\\phi_2+\\xi+\\psi_2),$  also satisfy primal-dual Problem \\ref{P6.1}.\n\t\n\\end{proof}\nNext, we define a complexly structured convex optimization problem and their Fenchel duals. Further, we propose an algorithm to solve the considered problem and study the convergence property of algorithm to find simultaneously the common solutions of optimaztion problems and common solutions of their Fenchel duals. Let $m$ is a positive integer. We have considered the following problem:\n\\begin{Problem}\\label{P6.2}\n\tLet $f_1,f_2\\in \\varGamma(\\mathcal{H})$ and $h_1,h_2$ be convex differentiable function with $\\mu_1^{-1},\\mu_2^{-1}$- Lipschitz continuous gradients, for some $\\mu_1,\\mu_2 >0.$ Let $\\varOmega_i$ be real Hilbert spaces and $g_i, l_i,s_i,t_i \\in \\varGamma(\\varOmega_i)$ such that $l_i,t_i$ are $\\nu_i,\\delta_i (>0)$-strongly convex, respectively, and $L_i:\\mathcal{H} \\to \\varOmega_i$ be non-zero linear continuous operator $\\forall i=1,2,\\dots,m$. The opmization problem under consideration is\n\t\\begin{equation}\n\t\t\\inf_{x\\in \\mathcal{H}}\\left\\lbrace f_1(x)+\\sum_{i=1}^{m}(g_i\\square l_i)(L_i x)+h_1(x)\\right\\rbrace  \\bigcap \\inf_{x\\in \\mathcal{H}}\\left\\lbrace f_2(x)+\\sum_{i=1}^{m}(s_i\\square t_i)(L_i x)+h_2(x)\\right\\rbrace\n\t\\end{equation}\n\twith its Fenchel-dual problem\n\t\\begin{eqnarray}\n\t\t\\sup_{v_i \\in \\varOmega, i\\in 1,\\dots,m}\\left\\lbrace -(f_1^{*} \\square h_1^{*})\\left( -\\sum_{i=1}^{m}L_i^* v_i\\right) -\\sum_{i=1}^{m}\\left( g_i^*(v_i)+l_i^*(v_i)\\right) \\right\\rbrace \\nonumber\\\\\n\t\t\\bigcap \\sup_{v_i \\in \\varOmega, i\\in 1,\\dots,m}\\left\\lbrace -(f_2^{*} \\square h_2^{*})\\left( -\\sum_{i=1}^{m}L_i^* v_i\\right) -\\sum_{i=1}^{m}\\left( s_i^*(v_i)+t_i^*(v_i)\\right) \\right\\rbrace.\n\t\\end{eqnarray}\n\\end{Problem}\nIn following corollary, we propose an algorithm and study its convergence behaviour. The point of convergence will be the solution of Problem \\ref{P6.2}.\n\n\n\\begin{Corollary}\n\tAssume  in Problem \\ref{P6.2}\n\t\\begin{equation}\\label{e6.5}\n\t\t0\\in ran \\left( \\partial f_1+\\sum_{i=1}^{m}L_i ^* \\circ (\\partial g_i \\square \\partial l_i)\\circ L_i +\\nabla h_1 \\right)  \\bigcap ran \\left( \\partial f_2+\\sum_{i=1}^{m}L_i ^* \\circ (\\partial s_i \\square \\partial t_i)\\circ L_i +\\nabla h_2\\right).\n\t\\end{equation}\n\tConsider $\\tau, \\sigma_i >0,  \\ i=1,2,\\dots,m$ such that\n\t\\begin{equation*}\n\t\t2\\rho\\min\\{\\beta_1,\\beta_2\\} \\geq 1.\n\t\\end{equation*}\n\twhere\n\t\\begin{eqnarray*}\n\t\t\\rho= \\min\\{\\tau^{-1}, \\sigma_1^{-1}, \\dots, \\sigma_m^{-1}\\}\\left(1-\\sqrt{\\tau\\sum_{i=1}^{m}\\sigma_i\\|L_i\\|^2}\\right)\\\\\n\t\t\\beta_1=\\min\\{\\mu_1,\\nu_1,\\dots,\\nu_m\\}  \\textrm{ and }   \\beta_2= \\min\\{\\mu_2,\\delta_1,\\dots,\\delta_m\\}.\n\t\\end{eqnarray*}\n\t\n\tConsider the iterative scheme with intial point $(x_1,v_{1,1},\\dots, v_{m,1}) \\in \\mathcal{H} \\times \\varOmega_1 \\times \\cdots \\times\\varOmega_m$ and defined by\n\t\\begin{eqnarray}\\label{e6.6}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\tp_n = prox_{\\tau f_1}\\left[e_{n}x_n-\\tau \\left(e_{n}\\sum_{i=1}^{m}L_i^* v_{i,n}+\\nabla h_1(e_{n}x_n)\\right)\\right]\\\\\n\t\t\t\tu_n= e_{n}x_n+\\theta_{n}(p_n-e_{n}x_n)\\\\\n\t\t\t\tFor \\ i=1,2,\\dots, m\\\\\n\t\t\t\t\n\t\t\t\tq_{i,n}=prox_{\\sigma_ig_i^{*}}\\left[e_{n}v_{i,n}+\\sigma_i(L_i(2p_n-e_{n}x_n)-\\nabla l_i ^{*}(e_{n}v_{i,n}))\\right]\\\\\n\t\t\t\tu_{i,n}=e_{n}v_{i,n}+\\theta_{n}(q_{i,n}-e_{n}v_{i,n})\\\\\n\t\t\t\tx_{n+1}=prox_{\\tau f_2}\\left[u_n-\\tau \\left(\\sum_{i=1}^{m}L_i^* u_{i,n}+\\nabla h_2(u_n)\\right)\\right]\\\\\n\t\t\t\tv_{i,n+1}=prox_{\\sigma_i s_i^{*}}\\left[u_{i,n} +\\sigma_i(L_i(2x_{n+1}-u_n)-\\nabla t_i^{*}(u_{i,n}))\\right],\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\twhere sequences $\\{\\theta_{n}\\}$ and $\\{e_n\\}$ satisfy the condition (\\textbf{i}) in Theorem \\ref{P4T3.1} and the conditions:\n\n\t$$0< \\underline{\\Theta} \\leq \\frac{2\\beta_1 \\rho}{4\\beta_1 \\rho-1} \\theta_{n} \\leq \\overline{\\Theta } <1 \\textrm{ and } \\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$$\n\n\t\n\tThen, there exists $(\\bar{x},\\bar{v}_1,\\dots,\\bar{v}_m)\\in  \\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m $ such that sequence $(x_n,v_{1,n},\\dots,v_{m,n})$ converges strongly to  $(\\bar{x},\\bar{v}_1,\\dots,\\bar{v}_m)$ as $n \\to \\infty$ and $(\\bar{x},\\bar{v}_1,\\dots,\\bar{v}_m)$ satisfies Problem \\ref{P6.2}.\n\\end{Corollary}\t\n\n\n\n\n\n\n\n\n\n\\section{Tikhonov Regularized Douglas-Rachford-type Algorithms}\\label{sec5}\nIn this section, using Algorithm \\ref{P4A3.1} we propose a new Douglas-Rachford algorithm to solve monotone inclusion problem of sum of two maximally monotone operators. Further using proposed Douglas-Rachford algorithm, we propose a Douglas-Rachford-type primal-dual algorithm to solve complexly structured monotone inclusion problem containing linearly composite and parallel-sum  operators.\n\n\n\\subsection{Tikhonov Regularized Douglas-Rachford Algorithm}\nLet $A, B:\\mathcal{H} \\to 2^\\mathcal{H}$ be maximally monotone operators. In this section, we consider the following monotone inclusion problem:\n\\begin{equation}\n\t\\text{ find } x\\in \\mathcal{H} \\text{ such that } 0 \\in (A+B)x .\n\\end{equation}\nWe propose a new  Douglas-Rachford algorithm based on Algorithm \\ref{P4A3.1} such that generated sequence converges strongly to a point in the solution set.\n\n\\begin{Theorem}\\label{T5.1}\n\tConsider $x_1 \\in \\mathcal{H}$ and $\\gamma >0,$ then algorithm is given by:\n\t\\begin{eqnarray}\\label{e5.1}{\n\t\t\t\\forall n\\in \\mathbb{N}\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\ty_n=J_{\\gamma B}(e_{n}x_n)\\\\\n\t\t\t\tz_n=J_{\\gamma A}(2y_n-e_{n}x_n)\\\\\n\t\t\t\tu_n=e_{n}y_n +\\theta_{n}(z_n-y_n)\\\\\n\t\t\t\tx_{n+1}=(2J_{\\gamma A}-Id)(2J_{\\gamma B}-Id)u_n. \\ \\\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\tLet $\\operatorname{Zer}(A +B)  \\neq \\emptyset$ and sequences $ e_n \\in (0,1)$ and $\\theta_{n} \\in (0,2]$ satisfy the condition (\\textbf{i}) in Theorem \\ref{P4T3.1} and the conditions:\n\n\t\n\t$$0< \\underline{\\Theta} \\leq \\frac{\\theta_{n}}{2} \\leq \\overline{\\Theta } < 1 \\textrm{ and } \\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$$\n\t\n\n\tThen the following statements are true:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\{x_n\\}$ converges strongly to $\\bar{x}=\\operatorname{proj}_{\\operatorname{Fix}R_{\\gamma A} R_{\\gamma B}}(0)$.\n\t\t\\item[(ii)] \\label{P4i2} $\\{y_n\\}$ and $\\{z_n\\}$ converges strongly to $J_{\\gamma B}(\\bar{x})\\in \\operatorname{Zer}(A+B)$.\n\t\n\t\\end{enumerate}\n\\end{Theorem}\n\\begin{proof}\n\tConsider the operator $T\\equiv R_{\\gamma A} R_{\\gamma B}:\\mathcal{H} \\to 2^\\mathcal{H}$. From the definition of reflected resolvent, and definition of operator $T$, algorithm (\\ref{e5.1}) can be rewritten as\n\t\\begin{eqnarray}\n\t\tx_{n+1}&=&R_{\\gamma A}R_{\\gamma B}\\{e_{n}x_n+\\frac{\\theta_{n}}{2}(R_{\\gamma A}R_{\\gamma B})(e_{n}x_n)-e_{n}x_n\\}\\nonumber\\\\\n\t\t&=& T\\{e_{n}x_n +\\frac{\\theta_{n}}{2}(T(e_{n}x_n)-e_{n}x_n)\\}.\n\t\\end{eqnarray}\n\tSince resolvent operator is nonexpansive \\cite[Corollary 23.10(ii)]{Bauschke2011}, $T$ is nonexpansive. Suppose $x^* \\in \\operatorname{Zer}(A+B)$ and results from \\cite[Proposition 25.1(ii)]{Bauschke2011}, we have $\\operatorname{Zer}(A+B)=J_{\\gamma B}(\\operatorname{Fix}(T)),$ which collectively implies that $\\operatorname{Fix} {(T )} \\neq \\emptyset.$ Applying Theorem \\ref{P4T3.1} with $A_1=A_2=A,B_1=B_2=B$, we conclued that $\\{x_n\\}$ conveges strongly to $\\bar{x}=\\operatorname{proj}_{\\operatorname{Fix} (T)}(0)$ as $n \\to \\infty.$ \\\\\n\tThe continuity of resolvent operator forces the sequence $\\{y_n\\} $ to converge strongly to $J_{\\gamma B}\\bar{x} \\in \\operatorname{Zer}(A+B).$ Finally, since $z_n-y_n=\\frac{1}{2}(T(e_{n}x_n)-e_{n}x_n),$ which converges strongly to 0, concludes \\textbf{(ii)}. \\\\\n\\end{proof}\n\n\\begin{Problem}\\label{P4P4.11}\n\tLet $f,g:\\mathcal{H} \\to \\mathbb{R} \\cup \\{\\infty\\}$ are convex proper and lower semicontinuous functions. Consider the minimization problem\n\t\\begin{equation}\\label{P4e4.11}\n\t\t\\min_{x\\in \\mathcal{H}} f(x) +g(x).\n\t\\end{equation}\n\\end{Problem}\nUsing Karush-Kuhn-Tucker condition, (\\ref{P4e4.11}) is equivalent to solve the inclusion problem\n\\begin{equation}\n\t\\text{find} {\\ x \\in \\mathcal{H}} \\  0 \\in \\partial f(x) +\\partial g(x).\n\\end{equation}\n\n\n\nIn order to solve such type of problem, we propose an iterative scheme and study its convergence behaviour which can be summerized in following corollary.\n\\begin{Corollary}\n\tLet $f,g$ be as in Problem \\ref{P4P4.11} with $argmin_{x \\in \\mathcal{H}}\\{f (x)+g (x)\\} \\neq \\emptyset$ and $0 \\in sqri (dom \\ f-dom \\ g).$ Consider the following iterative scheme with $x_1 \\in \\mathcal{H}$:\n\t\\begin{eqnarray}\\label{e5.11}{\n\t\t\t\\forall n\\in \\mathbb{N}\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\ty_n=prox_{\\gamma g}(e_{n}x_n)\\\\\n\t\t\t\tz_n=J_{\\gamma f}(2y_n-e_{n}x_n)\\\\\n\t\t\t\tu_n=e_{n}x_n +\\theta_{n}(z_n-y_n)\\\\\n\t\t\t\tx_{n+1}=(2prox_{\\gamma f}-Id)(2prox_{\\gamma g}-Id)u_n, \\ \\\n\t\t\t\t\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\twhere $\\gamma >0$\n\tand sequences $\\{\\theta_{n}\\} \\text{ and } \\{e_{n}\\}$ satisfy the condition (\\textbf{i}) in Theorem \\ref{P4T3.1} and the conditions:\n\n\t\n\t$$0< \\underline{\\Theta} \\leq \\frac{\\theta_{n}}{2} \\leq \\overline{\\Theta } < 1 \\textrm{ and } \\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$$\n\tThen we have the following:\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\{x_n\\}$ converges strongly to $\\bar{x}= \\operatorname{proj}_{\\operatorname{Fix}(T)}(0)$ where $T=(2prox_{\\gamma f}-Id)(2prox_{\\gamma g}-Id)$.\n\t\t\\item[(ii)] $\\{y_n\\}$ and $\\{z_n\\}$ converge strongly to $prox_{\\gamma g}(\\bar{x})\\in argmin_{x \\in \\mathcal{H}}\\{f (x)+g (x)\\} $.\n\t\\end{enumerate}\n\t\\begin{proof}\n\t\tSince $argmin_{x \\in \\mathcal{H}}\\{f (x)+g (x)\\} \\neq \\emptyset$ and $0 \\in sqri (dom \\ f-dom \\ g)$ (\\cite[Proposition 7.2]{Bauschke2011}) ensures that $\\operatorname{Zer}(A+B)=argmin_{x \\in \\mathcal{H}}\\{f (x)+g (x)\\}.$ The results can be obtained by choosing $A =\\partial f, B= \\partial g $ in Theorem \\ref{T5.1}.\n\t\\end{proof}\n\\end{Corollary}\n\n\n\n\n\\subsection{Tikhonov Regularized Douglas-Rachford-type Primal-Dual Algorithm}\nIn this section, we propose Douglas-Rachford-type primal-dual algorithms to solve the complex-structured monotone inclusion problem having mixtures of composition of linear operators and parallel-sum operators. We consider the monotone inclusion problem is as follows:\n\\begin{Problem}\\label{P4P6.2}\n\tLet $A:\\mathcal{H} \\to 2^\\mathcal{H}$ be a maximally monotone operator. Consider for each $i=1, \\dots,m$, $\\varOmega_i$ is a real Hilbert space, $P_i, Q_i: \\varOmega_i \\to 2^{\\varOmega_i}$ are maximally monotone operators and $L_i:\\mathcal{H} \\to \\varOmega_i$ are nonzero linear continuous operator.\n\tThe primal inclusion problem is to find $\\bar{x} \\in \\mathcal{H}$ satisfying\n\t\\begin{eqnarray}\n\t\t0\\in A \\bar{x} +\\sum_{i=1}^{m} L_i^{*}(P_i \\square Q_i)(L_i \\bar{x})\\nonumber\n\t\\end{eqnarray}\n\ttogether with dual inclusion problem\n\t\\begin{eqnarray}\\label{e6.1}{find \\ \\ \\bar{v}_1\\in \\varOmega_1, \\dots, \\bar{v}_m \\in \\varOmega_m \\text{ such that } (\\exists x \\in \\mathcal{H})\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t-\\sum_{i=1}^{m}L_i^* \\bar{v}_i \\in Ax\\\\\n\t\t\t\t\\bar{v}_i \\in (P_i \\square Q_i)(L_i x), \\ i=1,\\dots,m.\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\\end{Problem}\n\nHere, operators $P_i ,Q_i, i=1,\\dots,m $ are not cocoercive, thus to solve the Problem \\ref{P4P6.2}, we have to evaluate resolvent of each operator, which makes the Douglas-Rachford algorithm based primal-dual algorithm is more appropriate to solve the problem.\n\n\\begin{Theorem}\\label{P4T5.11}\n\tIn addition to assumption in Problem \\ref{P4P6.2}, we assume that\n\t\n\t\\begin{equation}\\label{P4P6.10}\n\t\t0 \\in ran \\left( A+\\sum_{i=1}^{m}L_i ^* \\circ(P_i \\square Q_i) \\circ L_i \\right).\n\t\\end{equation}\n\tConsider the strictly positive integers $\\tau, \\sigma_i, i=1,\\dots,m$ satisfying\n\t\\begin{equation}\n\t\t\\tau \\sum_{i=1}^{m} \\sigma_i\\|L_i\\|^2 < 4.\n\t\\end{equation}\n\t\n\tConsider the initial point $(x_1,v_{1,1},\\dots,v_{m,1})\\in \\mathcal{H}\\times \\varOmega_i \\times \\cdots \\times \\varOmega_m.$ The primal- dual algorithm to solve Problem \\ref{P4P6.2} is given by\n\t\n\t\n\t\\begin{algorithm}[H]\n\t\t\\SetAlgoLined\n\t\t\n\t\t\\KwIn{\\begin{enumerate}\n\t\t\t\t\\item initial points $(x_1,v_{1,1},\\dots,v_{m,1})\\in \\mathcal{H}\\times \\varOmega_i \\times \\cdots \\times \\varOmega_m.$\n\t\t\t\t\\item strictly positive real numbers $\\tau, \\sigma_i$, $i=1,2,...,m$ be such that $\\tau\\sum_{i=1}^{m} \\sigma_i \\|L_i\\|^2 < 4 $.\n\t\t\t\t\\item sequences $e_{n} \\in (0,1),\\theta_n \\in (0,2]$\n\t\t\t\\end{enumerate}\t\n\t\t}\n\t\t\\SetKwBlock{For}{For}{}\n\t\t\\SetKwProg{}{}{}{}\n\t\t\\For($n\\in \\mathbb{N}$; ){\n\t\t\t$p_{1,n}=J_{\\tau A} (e_n x_n-\\frac{\\tau}{2}e_n \\sum_{i=1}^{m}L_i^{*}v_{i,n} ) $\\\\\n\t\t\t$w_{1,n}=2p_{1,n}-e_n x_n$\\\\\n\t\t\t\n\t\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t\t\n\t\t\t\t$\tp_{2,i,n}=J_{\\sigma_i P_i ^{-1}}(e_n v_{i,n} +\\frac{\\sigma_i}{2}L_i w_{1,n})$\\\\\n\t\t\t\t$\tw_{2,i,n}=2p_{2,i,n}-e_n v_{i,n} $  \\\\\n\t\t\t}\n\t\t\t\n\t\t\t$\tz_{1,n}=w_{1,n}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i^{*}w_{2,i,n}$\\\\\n\t\t\t$\tu_{1,n}=e_n x_n+\\theta_{n}(z_{1,n}-p_{1,n})$\\\\\n\t\t\t\n\t\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t\t$z_{2,i,n}=J_{\\sigma_i Q_i ^{-1}}(w_{2,i,n} + \\frac{\\sigma_i}{2}L_i (2z_{1,n}-w_{1,n}))$\\\\\n\t\t\t\t$u_{2,i,n}=e_n v_{i,n} +\\theta_{n}(z_{2,i,n}-p_{2,i,n})$\\\\\n\t\t\t\t\n\t\t\t}\n\t\t\t\n\t\t\t$\tq_{1,n}=J_{\\tau A}(u_{1,n}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i^{*}(u_{2,i,n}))$\\\\\n\t\t\t$\ts_{1,n}=2q_{1,n}-u_{1,n}$\\\\\n\t\t\t\n\t\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t\t$\tq_{2,i,n}=J_{\\sigma_i P_i ^{-1}}(u_{2,i,n}+\\frac{\\sigma_i}{2}L_i s_{1,n})$\\\\\n\t\t\t\t$\ts_{2,i,n}=2q_{2,i,n}-u_{2,i,n}$\\\\\n\t\t\t}\n\t\t\t\n\t\t\t$d_{1,n}=s_{1,n}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i^{*}(s_{2,i,n})$\\\\\n\t\t\t$x_{n+1}=2d_{1,n}-s_{1,n}$\\\\\n\t\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t\t$d_{2,i,n}=J_{\\sigma_i Q_i ^{-1}}(s_{2,i,n}+\\frac{\\sigma_i}{2}L_i(x_{n+1}))$\\\\\n\t\t\t\t$v_{2,i,n}=2d_{2,i,n}-s_{2,i,n}$\n\t\t\t\t\n\t\t\t}\n\t\t\t\n\t\t}\n\t\t\n\t\t\n\t\t\\KwOut{$(x_{n+1},v_{1,n+1},\\dots,v_{m,n+1})$}\t\t\n\t\t\\caption{To optimize the complexly structured monotone inclusion Problem \\ref{P4P6.2} }\n\t\t\\label{P3Al4.1}\n\t\\end{algorithm}\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\twhere sequences $\\{\\theta_{n}\\}$ and $\\{e_{n}\\}$ satisfy the condition (\\textbf{i}) in Theorem \\ref{P4T3.1} and the conditions:\n\n\t\n\t$$0< \\underline{\\Theta} \\leq \\frac{\\theta_{n}}{2} \\leq \\overline{\\Theta } < 1 \\textrm{ and } \\sum_{n=1}^{\\infty} |\\theta_{n}-\\theta_{n-1}| < \\infty.$$\n\t\n\tThen there exist an element $(\\bar{x}, \\bar{v}_1,\\dots,\\bar{v}_m)\\in \\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m$ such that following statements are true:\n\t\\begin{enumerate}\n\t\t\\item[(a)] Denote\\\\\n\t\t$\\bar{p}_1=J_{\\tau A}\\left( \\bar{x}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i ^* \\bar{v}_i\\right) $\\\\\n\t\t$\\bar{p}_{2,i}=J_{\\sigma_i P_i ^{-1}}\\left( \\bar{v}_i +\\frac{\\sigma_i}{2}L_i(2\\bar{p}_1-\\bar{x})\\right) , i=1,\\dots, m.$\n\t\tThen the point $(\\bar{p}_1, \\bar{p}_{2,1},\\dots, \\bar{p}_{2,m})\\in \\mathcal{H} \\times \\varOmega_1 \\times\\cdots \\times \\varOmega_m $ is a primal-dual solution of Problem \\ref{P4P6.2}.\n\t\t\\item[(b)] $(x_n,v_{1,n}, \\dots, v_{m,n})$ converges strongly to $(\\bar{x},\\bar{v}_1,\\dots, \\bar{v}_m)$.\n\t\t\\item[(c)] $(p_{1,n}, p_{2,1,n},\\dots, p_{2,m,n})$ and $(z_{1,n}, z_{2,1,n},\\dots, z_{2,m,n})$ converge strongly to $(\\bar{p}_1, \\bar{p}_{2,1},\\dots, \\bar{p}_{2,m})$.\n\t\\end{enumerate}\n\\end{Theorem}\n\n\\begin{proof}\n\tConsider the real Hilbert space $\\mathcal{K}$ and operators\n\t\\begin{enumerate}\n\t\t\\item[(i)] $\\phi: \\mathcal{K} \\to 2^\\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto (Ax,P_1^{-1}u_1,\\dots,P_m^{-1}u_m),$\n\t\t\n\t\t\\item[(ii)] $\\xi:\\mathcal{K} \\to \\mathcal{K}$, defined by $(x,u_1,\\dots,u_m)\\mapsto \\left(\\sum_{i=1}^{m}L_i ^*u_i,-L_1 x,\\dots, -L_m x\\right),$\n\t\t\\item[(iii)] $\\psi: \\mathcal{K} \\to \\mathcal{K}$, defined by\n\t\t$\\psi (x,u_1,\\dots,u_m)= \\left(0,Q_1^{-1}u_1,\\dots, Q_m^{-1}u_m\\right)$\n\t\\end{enumerate}\n\t\n\tWe can observe the following:\n\t\\begin{enumerate}\n\t\t\\item[(i)] operator $\\frac{1}{2} \\xi + \\psi \\text{ and } \\frac{1}{2}\\xi+\\phi $ are maximally monotone as dom $\\xi=\\mathcal{K}$ (\\cite[Corollary 24.4(i)]{Bauschke2011}),\n\t\t\\item[(ii)]  condition (\\ref{P4P6.10}) implies $\\operatorname{Zer}(\\phi+\\xi+\\psi) \\neq \\emptyset$,\n\t\t\\item[(iii)] \\label{P4R6.2}every point in $\\operatorname{Zer}(\\phi+\\xi+\\psi)$ solves Problem \\ref{P4P6.2}.\n\t\\end{enumerate}\n\tDefine the linear continuous operator $\\textbf{W}:\\mathcal{K}\\to \\mathcal{K}$, defined by\n\t$$\\textbf{W}(x,u_1,\\dots,u_m )= \\left( \\frac{x}{\\tau}-\\frac{1}{2}\\sum_{i=1}^{m}L_i ^*u_i,\\frac{u_1}{\\sigma_1}-\\frac{1}{2}L_1x, \\dots, \\frac{u_m}{\\sigma_m}-\\frac{1}{2}L_m x \\right) $$  which is self-adjoint. Consider $$\\rho=\\left( 1-\\frac{1}{2}\\sqrt{\\tau \\sum_{i=1}^{m}\\sigma_i\\|L_i\\|^2} \\right) \\min\\left\\lbrace \\frac{1}{\\tau}, \\frac{1}{\\sigma_1},\\dots, \\frac{1}{\\sigma_m} \\right\\rbrace >0.$$ The operator $\\textbf{W}$ is $\\rho$- strongly positive in $\\mathcal{K}_\\textbf{W}$ (\\cite{BC2013}) and satisfies the following inequality\n\t$$\\langle x, \\textbf{W}x \\rangle _\\mathcal{K} \\geq \\rho \\|x\\|_\\mathcal{K} ^2  \\ \\ \\forall x \\in \\mathcal{K}.$$ Thus the inverse of $\\textbf{W}$ exists and satisfies $\\|\\textbf{W}^{-1}\\| \\leq \\frac{1}{\\rho}.$ Consider the sequences\n\t\\begin{eqnarray}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t\\text{\\textbf{x}}_n = (x_n, v_{1,n}, \\dots,v_{m,n})\\\\\n\t\t\t\t\\text{\\textbf{y}}_n = (p_{1,n},p_{2,1,n},\\dots,p_{2,m,n})  \\\\\n\t\t\t\t\\text{\\textbf{z}}_n = (z_{1,n},z_{2,1,n},\\dots,z_{2,m,n})\\\\\n\t\t\t\t\\text{\\textbf{u}}_n = (u_{1,n},u_{2,1,n},\\dots,u_{2,m,n})\\\\\n\t\t\t\t\\text{\\textbf{q}}_n = (q_{1,n},q_{2,1,n},\\dots,q_{2,m,n}) \\\\\n\t\t\t\t\\text{\\textbf{s}}_n = (s_{1,n},s_{2,1,n},\\dots,s_{2,m,n}) \\\\\n\t\t\t\t\\text{\\textbf{d}}_n = (d_{1,n},d_{2,1,n},\\dots,d_{2,m,n}).\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\tUsing the definition of operators $\\phi, \\xi,\\psi$ and $\\textbf{W}$, Algorithm \\ref{P3Al4.1} can be written equivalently as\n\t\n\t\\begin{eqnarray}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t\\textbf{W}(e_n \\textbf{x}_n-\\textbf{y}_n ) \\in (\\frac{1}{2}\\xi+\\phi)\\textbf{y}_n\\\\\n\t\t\t\t\\textbf{W}(2\\textbf{y}_n-\\textbf{x}_n-\\textbf{z}_n ) \\in (\\frac{1}{2}\\xi+\\psi)\\textbf{z}_n\\\\\n\t\t\t\t\\textbf{u}_n= e_n \\textbf{x}_n+\\theta_{n}(\\textbf{z}_n -\\textbf{y}_n)\\\\\n\t\t\t\t\\textbf{W}(\\textbf{u}_n -\\textbf{q}_n)\\in (\\frac{1}{2}\\xi+\\phi)\\textbf{q}_n\\\\\n\t\t\t\t\\textbf{W}(\\textbf{s}_n -\\textbf{d}_n)\\in (\\frac{1}{2}\\xi+\\psi)(\\textbf{d}_n)\\\\\n\t\t\t\t\\textbf{x}_{n+1}=2\\textbf{d}_n-\\textbf{s}_n,\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\twhich is further equivalent to\n\t\\begin{eqnarray}{\n\t\t\t\\forall n\\in \\mathbb{N}\t\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t\\textbf{y}_n =(Id+\\textbf{W}^{-1}(\\frac{1}{2}\\xi+\\phi))^{-1}(\\textbf{x}_n)  \\\\\n\t\t\t\t{\\textbf{z}_n=(Id+\\textbf{W}^{-1}(\\frac{1}{2}\\xi+\\psi))^{-1}(2\\textbf{y}_n-\\textbf{x}_n) }\\\\\n\t\t\t\t{\\textbf{u}_n= \\textbf{x}_n+\\theta_{n}(\\textbf{z}_n -\\textbf{y}_n)}\\\\\n\t\t\t\t\\textbf{q}_n= (Id+\\textbf{W}^{-1}(\\frac{1}{2}\\xi+\\phi))^{-1}(\\textbf{u}_n)\\\\\n\t\t\t\t\\textbf{d}_n=(Id+\\textbf{W}^{-1}(\\frac{1}{2}\\xi+\\psi))^{-1}(\\textbf{s}_n)\\\\\n\t\t\t\t\\textbf{x}_{n+1}=2\\textbf{d}_n-\\textbf{s}_n.\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\tNow, consider the real Hilbert space $\\mathcal{K}_{\\text{\\textbf{W}}}=\\mathcal{H}\\times \\varOmega_1\\times \\cdots \\times \\varOmega_m$ with inner product and norm defined as \\\\\n\t$\\langle \\textbf{x}, \\textbf{y} \\rangle_{\\mathcal{K}_\\textbf{W}} = \\langle \\textbf{x}, \\textbf{W}\\textbf{y} \\rangle $ and $\\|\\textbf{x}\\|_{\\mathcal{K}_\\textbf{W}}= \\sqrt{\\langle \\textbf{x}, \\textbf{W}\\textbf{x} \\rangle_{\\mathcal{K}}}$ respectively.\\\\\n\tNow, define the operators $\\textbf{A} \\equiv \\textbf{W}^{-1}(\\frac{1}{2}\\xi+\\psi)$ and $\\textbf{B} \\equiv \\textbf{W}^{-1}(\\frac{1}{2}\\xi+\\phi),$ which are maximally monotone on $\\mathcal{K}_{\\textbf{W}}$ as $\\frac{1}{2}\\xi+\\phi$ and $\\frac{1}{2}\\xi+\\psi$ are maximally monotone on $\\mathcal{K}.$ The Algorithm \\ref{P3Al4.1} can be written in the form of Doughlas-Rachford algorithm as\n\t\\begin{eqnarray}{\n\t\t\t\\forall n\\in \\mathbb{N}\\left\\{\n\t\t\t\\begin{array}{lc@{}c@{}r}\n\t\t\t\t\\textbf{y}_n = \\textbf{J}_{\\textbf{B}}(e_n \\textbf{x}_n)\\\\\n\t\t\t\t\\textbf{u}_n = e_n\\textbf{x}_n-\\theta_{n}\\left( \\textbf{J}_{\\textbf{A}}(2\\textbf{y}_n - e_n \\textbf{x}_n)-\\textbf{y}_n\\right) \\\\\n\t\t\t\t\\textbf{x}_{n+1}=(2\\textbf{J}_{\\textbf{A}}- Id) (2\\textbf{J}_{\\textbf{B}}- Id)\\textbf{z}_n,\n\t\t\t\\end{array}\\right.}\n\t\\end{eqnarray}\n\twhich is of the form Algorithm (\\ref{e5.1}) for $\\gamma =1.$ From assumption (\\ref{P4P6.10}), we have\n\t$$\\operatorname{Zer}(\\textbf{A} +\\textbf{B})=\\operatorname{Zer}(\\textbf{W}^{-1}(\\textbf{M}+\\textbf{S}+\\textbf{Q}))=\\operatorname{Zer}(\\textbf{M}+\\textbf{S}+\\textbf{Q}).$$\n\tApplying Theorem \\ref{T5.1}, we can find $\\bar{\\textbf{x}} \\in \\operatorname{Fix}(R_{\\textbf{A}}R_{\\textbf{B}})$ such that $\\textbf{J}_{\\text{\\textbf{B}}}{\\bar{\\textbf{x}}} \\in \\operatorname{Zer}(\\textbf{A}+\\textbf{B}).$\n\\end{proof}\n\nAt the end of this section, we study iterative technique to solve  following convex optimization problem:\n\\begin{Problem}\\label{P4P5.33}\n\tLet $f \\in \\Gamma(\\mathcal{H})$. Consider $\\varOmega_i$ are real Hilbert spaces, $g_i,l_i \\in \\Gamma(\\varOmega_i)$ and $L_i : \\mathcal{H} \\to \\varOmega_i$ are linear continuous operators, $i=1,\\dots,m$. The optimization problem is given by\n\t\\begin{equation}\n\t\t\\inf_{x\\in H}\\left[f(x)+\\sum_{i=1}^{m}(g_i\\square l_i)(L_i x)\\right]\n\t\\end{equation}\n\twith conjugate-dual problem is given by\n\t\\begin{equation}\n\t\t\\sup_{v_i \\in \\varOmega, i=1,2,\\dots,m}\\left\\lbrace -f^{*}\\left( -\\sum_{i=1}^{m}L_i^* v_i\\right) -\\sum_{i=1}^{m}(g_i^*(v_i)+l_i^*(v_i))\\right\\rbrace.\n\t\\end{equation}\n\\end{Problem}\n\n\nConsider stricly positive integers $\\tau, \\sigma_i, i=1,\\dots,m$ and\ninitial point $(x_1,v_{1,1},\\dots,v_{m,1})\\in \\mathcal{H}\\times \\varOmega_i \\times \\cdots \\times \\varOmega_m.$ The primal-dual algorithm to solve Problem \\ref{P4P5.33} is given by\n\n\\begin{algorithm}[H]\n\t\\SetAlgoLined\n\t\n\t\\KwIn{\\begin{enumerate}\n\t\t\t\\item initial points $(x_1,v_{1,1},\\dots,v_{m,1})\\in \\mathcal{H}\\times \\varOmega_i \\times \\cdots \\times \\varOmega_m.$\n\t\t\t\\item \\label{P4As5.32}Positive real numbers $\\tau, \\sigma_i$, $i=1,2,...,m$ be such that $\\tau\\sum_{i=1}^{m} \\sigma_i \\|L_i\\|^2 < 4 $.\n\t\t\t\\item The sequences $\\{\\theta_n\\}, \\{e_n\\}$ satisfying the assumptions in  Theorem \\ref{P4T5.11}.\n\t\t\\end{enumerate}\t\n\t}\n\t\\SetKwBlock{For}{For}{}\n\t\\SetKwProg{}{}{}{}\n\t\\For($n\\in \\mathbb{N}$; ){\n\t\t$p_{1,n}=prox_{\\tau f} (e_n x_n-\\frac{\\tau}{2}e_n \\sum_{i=1}^{m}L_i^{*}v_{i,n} )$\\\\\n\t\t$w_{1,n}=2p_{1,n}-e_n x_n$\\\\\n\t\t\n\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t\n\t\t\t$\tp_{2,i,n}=prox_{\\sigma_i g_i ^*}(e_n v_{i,n} +\\frac{\\sigma_i}{2}L_i w_{1,n})$\\\\\n\t\t\t$\tw_{2,i,n}=2p_{2,i,n}-e_n v_{i,n}  $ \\\\\n\t\t}\n\t\t\n\t\t$z_{1,n}=w_{1,n}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i^{*}w_{2,i,n}$\\\\\n\t\t$u_{1,n}=e_n x_n+\\theta_{n}(z_{1,n}-p_{1,n})$\\\\\n\t\t\n\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t$z_{2,i,n}=prox_{\\sigma_i l_i ^ *}(w_{2,i,n} + \\frac{\\sigma_i}{2}L_i (2z_{1,n}-w_{1,n}))$\\\\\n\t\t\t$\tu_{2,i,n}=e_n v_{i,n} +\\theta_{n}(z_{2,i,n}-p_{2,i,n})$\\\\\n\t\t\t\n\t\t}\n\t\t\n\t\t$q_{1,n}=prox_{\\tau f}(u_{1,n}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i^{*}(u_{2,i,n}))$\\\\\n\t\t$s_{1,n}=2q_{1,n}-u_{1,n}$\\\\\n\t\t\n\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t$\tq_{2,i,n}=prox_{\\sigma_i g_i ^*}(u_{2,i,n}+\\frac{\\sigma_i}{2}L_i s_{1,n})$\\\\\n\t\t\t$\ts_{2,i,n}=2q_{2,i,n}-u_{2,i,n}$\\\\\n\t\t}\n\t\t\n\t\t$\tt_{1,n}=s_{1,n}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i^{*}(s_{2,i,n})$\\\\\n\t\t$\tx_{n+1}=2t_{1,n}-s_{1,n}$\\\\\n\t\t\\For($i=1,\\dots, m$; ){\n\t\t\t$t_{2,i,n}=prox_{\\sigma_i l_i ^*}(s_{2,i,n}+\\frac{\\sigma_i}{2}L_i(x_{n+1}))$\\\\\n\t\t\t$\tv_{2,i,n}=2t_{2,i,n}-s_{2,i,n}$\n\t\t\t\n\t\t}\n\t\t\n\t}\n\t\n\t\n\t\\KwOut{$(x_{n+1},v_{1,n+1},\\dots,v_{m,n+1})$}\t\t\n\t\\caption{To optimize the complexly structured monotone inclusion Problem \\ref{P4P5.33} }\\label{P3Al4.3}\n\\end{algorithm}\t\nwhere $\\{\\theta_{n}\\} \\text{ and } \\{e_n\\}$  are real sequences.\n\n\\begin{Corollary}\n\tIn addition to assumptions in Problem \\ref{P4P5.33}, consider\n\t\\begin{equation}\\label{e6.}\n\t\t0\\in ran\\left( \\partial f +\\sum_{i=1}^{m}L_i ^* \\circ (\\partial g_i \\square \\partial l_i)\\circ L_i \\right) .\n\t\\end{equation}\t\n\tThen, there exists an element $(\\bar{x}, \\bar{v}_1,\\dots,\\bar{v}_m)\\in \\mathcal{H}\\times \\varOmega_1 \\times \\cdots \\times \\varOmega_m$ such that following statements are true:\n\t\\begin{enumerate}\n\t\t\\item[(a)] Denote\\\\\n\t\t$\\bar{p}_1=prox_{\\tau f}\\left( \\bar{x}-\\frac{\\tau}{2}\\sum_{i=1}^{m}L_i ^* \\bar{v}_i\\right) $\\\\\n\t\t$\\bar{p}_{2,i}=prox_{\\sigma_i g^*_i}\\left( \\bar{v}_i +\\frac{\\sigma_i}{2}L_i(2\\bar{p}_i-\\bar{x})\\right) , i=1,\\dots, m.$\n\t\tThen the point $(\\bar{p}_1, \\bar{p}_{2,1},\\dots, \\bar{p}_{2,m})\\in \\mathcal{H} \\times \\varOmega_1 \\times\\cdots \\times \\varOmega_m $ is a primal-dual solution of Problem \\ref{P4P5.33}.\n\t\t\\item[(b)] $(x_n,v_{1,n}, \\dots, v_{m,n})$ converges strongly to $(\\bar{x},\\bar{v}_1,\\dots, \\bar{v}_m)$.\n\t\t\\item[(c)] $(p_{1,n}, p_{2,1,n},\\dots, p_{2,m,n})$ and $(z_{1,n}, z_{2,1,n},\\dots, z_{2,m,n})$ converge strongly to $(\\bar{p}_1, \\bar{p}_{2,1},\\dots, \\bar{p}_{2,m})$.\n\t\\end{enumerate}\n\\end{Corollary}\n\n\n\n\n\n\n\\section{Numerical Experiment}\nIn this section, we make an experimental setup to solve the wavelet based image deblurring problem. In image deblurring, we develop mathematical methods to recover the original, sharp image from blurred image. The mathematical formulation of the blurring process can be written as linear inverse problem,\n\\begin{equation} \\label{p4n1}\n\t\\text{ find } x \\in \\mathbb{R}^d \\text{ such that }\tAx=b+w\n\\end{equation}\nwhere $A\\in \\mathbb{R}^{m\\times d}$ is blurring operator, $b\\in \\mathbb{R}^m$ is blurred image and $w$ is an unknown noise.\nA classical approach to solve Problem (\\ref{p4n1}) is to minimize the least-square term\n$\\|Ax-b\\|^2$. In the deblurring case, the problem is ill-conditioned as the norm solution has a huge norm. To remove the difficulty, the ill-conditioned problem is replaced by a nearly well-conditioned problem. In wavelet domain, most images are sparse in nature, that's why we choose $l_1$ regularization. For $l_1$ regularization, the image processing problem becomes\n\\begin{equation}\\label{p4n2}\n\t\\min_{x\\in\\mathbb{R}^2}F(x)=\t\\|Ax-b\\|^2 +\\lambda \\|x\\|_1,\n\\end{equation}\nwhere $\\lambda$ is a sparsity controlling parameter and  provides a tradeoff between fidelity to the measurements and noise sensitivity. The $l_1$ regularization produces sparse images having sharp edges since it is less sensitive to outliers.\nUsing subdifferential characterization of the minimum of a convex function, a point $x^*$ minimizes $F(x)$ if and only if\n\\begin{equation}\\label{p4n3}\n\t0\\in A^T(Ax^*-b)+\\partial \\lambda \\|x^*\\|_1.\\nonumber\n\\end{equation}\nThus we can apply forward-backward Algorithm (\\ref{e4.1}) to solve the deblurring problem (\\ref{p4n2}).\n\nFor numerical experiment purposes, we have chosen images from publicly available domain and assumed reflexive (Neumann) boundary conditions. We blurred the images using gaussian blur of size $9 \\times 9$ and standard deviation 4. We have compared the algorithm (\\ref{e4.1}) with \\cite[Algorithm 8]{Bot2019}. The operator $A=RW$, where $W$ is the three stage haar wavelet transform and $R$ is the blur operator. The original and corresponding blurred images were shown in Figure \\ref{Fig6.5}.\n\\begin{figure}[htb!]\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{face}\n\t\t\\caption{Original.}\n\t\t\\label{Fig6.1}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{blurredface}\n\t\t\\caption{Blurred}\n\t\t\\label{Fig6.2}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{crowd}\n\t\t\\caption{Original}\n\t\t\\label{Fig6.3}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{blurredcrowd}\n\t\t\\caption{Blurred}\n\t\t\\label{Fig6.4}\n\t\\end{subfigure}\n\t\\caption{The original and blurred images of Lenna and crowd.}\n\t\\label{Fig6.5}\n\\end{figure}\nThe regularization parameter was chosen to be $\\lambda=2\\times10^{-5}$, and the initial image was the blurred image. The objective function value is denoted by $F(x^*)$ and function value at $n^{\\text{th}}$ iteration is denoted by $F(x_n)$. Sequences $\\{\\lambda_n\\}$ and $\\{\\beta_n\\}$ are chosen as $\\{1-\\frac{1}{n+1}\\}$ and $\\{0.9\\}$ respectively. The images recovered by the algorithms for 1000 iterations are shown in the figure.\nThe graphical representation of convergence of $F(x_n) - F(x^*)$ is depicted in Figure \\ref{Fig6.8}.\n\\begin{figure}[htb!]\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{face_plot_1000_9}\n\t\t\\caption{Lenna.}\n\t\t\\label{Fig6.6}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{crowd_plot_1000_9}\n\t\t\\caption{Crowd}\n\t\t\\label{Fig6.7}\n\t\\end{subfigure}\n\t\\caption{The variation of $F(x_n)-F(x^*)$ with respect to number of iteration for different images.}\n\t\\label{Fig6.8}\n\\end{figure}\nFor deblurring methods, lower the value of $F(x_n)-F(x^*)$ higher the quality of recovered images.\n\\begin{figure}[htb!]\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{face_mann_1000_9}\n\t\t\\caption{ Algorithm (\\ref{e4.1}).}\n\t\t\\label{Fig6.9}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{face_normalS_1000_9}\n\t\t\\caption{\\cite[Algorithm 8]{Bot2019}}\n\t\t\\label{Fig6.10}\n\t\\end{subfigure}\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{crowd_mann_1000_9}\n\t\t\\caption{ Algorithm (\\ref{e4.1}).}\n\t\t\\label{Fig6.11}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.48\\textwidth}\n\t\n\t\t\\includegraphics[width=\\linewidth]{crowd_normalS_1000_9}\n\t\t\\caption{\\cite[Algorithm 8]{Bot2019}.}\n\t\t\\label{Fig6.12}\n\t\\end{subfigure}\n\t\\caption{The recovered images using different algorithms for 1000 iterations.}\n\t\\label{Fig6.13}\n\\end{figure}\n\nIt can be observed from Figures \\ref{Fig6.8} and \\ref{Fig6.13}  that proposed Algorithm (\\ref{e4.1}) outperforms over \\cite[Algorithm 8]{Bot2019}.\n\n\n\\section*{Acknowledgements}\nThe first author acknowledges the Indian Institute of Technology (BHU), Varanasi for Senior Research Fellowship in terms of teaching assistantship. The third author is thankful to University Grant Commission for the Senior Research Fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSynchronization phenomena occur recurrently in \nphysical, chemical and biological systems.\nFew noteworthy examples include superconducting currents\nin Josephson junction arrays\n\\cite{strogatz_prl_1996,strogatz_pre_1998},\nemerging coherence in populations of chemical oscillators\n\\cite{kiss},\nor the accuracy of central circadian pacemakers in insects and vertebrates\n\\cite{reppert_97,winfree,sync_book}.\nThe latter serve as biomolecular time-keeping\ndevices, which most organisms have evolved to coordinate their\nphysiology and metabolic\nactivities with the geophysical light-dark and temperature cycles\n\\cite{schibler_naef}.\n\nThe present work was motivated by recent experiments in mammalian\ncell cultures in which the levels of\nproteins implicated in the circadian (with $\\sim 24$hr periods)\nclockwork were monitored using fluorescent reporters\n\\cite{nagoshi, welsh, Carr}.\nIt was demonstrated that\nindividual cellular oscillators generate self-sustained rhythms\nin protein abundance\nand that populations can be synchronized by treatment with a short\nserum shock or light pulse.\nImportantly frequencies of individual oscillators are not strictly\nconstant but drift in time\n(see for example Fig.~S2 in \\cite{Carr} showing the results in the zebrafish).\nSeveral chemical kinetics models thought to\ncapture the biochemistry responsible for generating\noscillations in living cells\nwere shown to exhibit oscillatory instabilities and limit-cycles\n\\cite{goldbeter, leibler}.\nExperimental and theoretical evidence therefore supports a\ndescription of oscillator populations in terms of phase variables.\n\nOur understanding of the onset of collective synchronization in coupled\nnonlinear oscillator models has greatly benefited from a large body of\nwork on the Kuramoto model\n\\cite{kuramoto_84,Sakaguchi_kuramoto_86,kuramoto_nishikawa_87,strogatz_prl_1992,stogatz_physicaD_2000}\n\\begin{eqnarray}\n\\label{kuramoto}\n\\dot\\phi_i=f_i+\\frac K N \\sum_{j=1}^N\\sin(\\phi_j-\\phi_i)+\\xi_i(t)~,\n\\end{eqnarray}\ndescribing the phase dynamics in a set of weakly coupled identical\nnon-linear oscillators.\nHere, $\\phi_i(t)$ represent the phase of the\n$i$-th oscillator at time $t$,\nand $\\xi_i(t)$ are white noise sources with\nexpectation and covariance\n\\begin{eqnarray*}\n{\\rm E}[\\xi_i(t)]&=&0~,\\\\\n{\\rm Cov}[\\xi_i(s),\\xi_j(t)]&=&2D\\delta _{ij}\\delta(s-t)~.\n\\end{eqnarray*}\nThe frequencies $f_i$ are static and taken from a distribution\n$g(f)$ symmetric around $\\mu_f$. Eq.~\\ref{kuramoto} is an effective\nmodel for the phase degrees of freedom in a population of\nlimit-cycle oscillators and assumes a regime where phase and amplitude\ndynamics decouple.\nThe parameter $K$ measures the strength of the all-to-all coupling.\nMost exact results are given for the coupling function\n$U(\\phi_j-\\phi_i) = \\sin(\\phi_j-\\phi_i)$. More general interactions\nlead to much greater\nanalytical complexity and were investigated in\n\\cite{crawford_prl95,crawford_physicaD_1999}.\nCritical properties of the model are conveniently studied using the complex\norder parameter $R(t)e^{i\\psi(t)}=\\frac{1}{N}\\sum_{j=1}^Ne^{i\\phi_j(t)}$\nso that collective synchronization occurs\nwhen $R_\\infty=\\lim_{T\\rightarrow\\infty}\\frac 1 T\\int_0^T\\! R(t)\\,dt$ remains positive\nin the infinite population limit.\nFor the sine coupling model a bifurcation occurs at\n$K_c = 2\/\\int\\frac D {D^2+f^2}g(f+\\mu_f)\\,\\!df$\nat which the incoherent desynchronized state $R_\\infty=0$ \nbecomes unstable and a macroscopic number of oscillators phase lock\nto the average phase $\\psi(t)=\\mu_f\\,t$ \\cite{crawford_jsp_1994,Sakaguchi,strogatz_jsp_1991}.\nFor $D\\rightarrow 0$ the classical Kuramoto result\n$K_c=\\frac 2{\\pi g(\\mu_f)}$ \\cite{kuramoto_book} is recovered.\nBelow the critical coupling, the incoherent state is linearly stable when $D>0$ \\cite{strogatz_jsp_1991}\nbut only neutrally stable when $D=0$ with $R(t)$ still decaying to zero \\cite{strogatz_prl_1992}.\n\n\\section{The Model}\n\nTo study the effects of the reported drifts on collective synchronization,\nwe generalize the Kuramoto model by introducing\na second time scale $1\/\\gamma$ (besides $1\/\\sigma_f$)\ncharacterizing the frequency drifts.\nThe frequency dynamics is formulated as an Ornstein-Uhlenbeck (O-U) process\nwhile the phases are coupled following the canonical\nall-to-all sine interaction. The model for $N$ oscillators reads\n\\begin{eqnarray}\n\\label{model}\n\\dot f_i(t)&=& -\\gamma(f_i(t)-\\mu_f)+\\eta_i(t)~,\\\\\n\\dot \\phi_i(t)&=& f_i(t)+\\frac K N \\sum_{j=1}^N\\sin\\bigl(\\phi_j(t)-\\phi\n_i(t)\\bigr)~,\\nonumber\n\\end{eqnarray}\nwhere $\\mu_f$ is the average frequency chosen identical\nfor each oscillator\nWe assume that the $\\eta_i$ are independent and identically\ndistributed white noise sources with\n\\begin{eqnarray*}\n{\\rm E}[\\eta_i(t)]&=&0~,\\\\\n{\\rm Cov}[\\eta_i(s),\\eta_j(t)]&=&\\eta^2\\delta _{ij}\\delta(s-t)~.\n\\end{eqnarray*}\nThe solution for $f_i$ \nis a Gaussian process with mean and covariance \n\\begin{eqnarray}\n{\\rm E}[f_i(t)]&=&\\mu_f+e^{-\\gamma t}(f_i(0)-\\mu_f)~,\\nonumber\\\\\n{\\rm Cov}[f_i(s),f_j(t)]&=&\\frac{\\eta^2}{2\\gamma}\\delta_{ij}\n\\left(e^{-\\gamma|t-s|}-e^{-\\gamma(t+s)}\\right)\\nonumber\\\\\n&\\stackrel{\\gamma t\\gg1}{\\longrightarrow}&\\frac{\\eta^2}{\\gamma^2}\\delta_{ij}\\delta(t-s)~.\n\\end{eqnarray}\n\nIn the following we use as independent parameters the\nasymptotic frequency dispersion $\\sigma_f^2=\\frac{\\eta^2}{2\\gamma}$\nand the damping $\\gamma$, which are in principle both accessible\nexperimentally.\nThen,\n$\n{\\rm Cov}[f_i(s),f_i(t)]\\rightarrow2\\sigma_f^2\/\\gamma\\,\\delta(t-s)\n$ when $\\gamma t\\gg1$~. \nTo remind the significance of this regime we note\nfor $K=0$ the phases $\\phi_i(t)$ also follow Gaussian processes with\n$\n{\\rm Cov}[\\phi_i(t),\\phi_i(t)]\n\\rightarrow\\frac{2\\sigma_f^2}{\\gamma}t\n$\nasymptotically for $\\gamma t\\gg 1$.\nBecause of the linear time dependence, this regime (Schmolukowski) describes\nphase diffusion with constant $D=\\frac{\\sigma_f^2}\\gamma$.\n\nAlthough it is {\\it a priori} unclear whether this model exhibits\na bifurcation, we expect that the large $\\gamma$ behavior\nreminiscent of phase diffusion will converge\nto the Kuramoto model (Eq.~\\ref{kuramoto}) with a\nfrequency distribution given by\n$g(f)=\\delta(f-\\mu_f)$ and white noise\nstrength $D=\\sigma_f^2\/\\gamma$, and thus exhibit a bifurcation at $K_c=2D$.\nHowever the small $\\gamma$ behavior is less obvious\nsince we are simultaneously concerned with long time properties.\nFor fixed $\\sigma_f$, we anticipate that synchronization should be hardest\nfor strictly static oscillators ($\\gamma=0$).\nThis case corresponds to the noiseless $D=0$ Kuramoto\nmodel with Gaussian frequency dispersion\n$g(f)=\\mathcal{N}_{\\mu_f,\\sigma_f}(f)\\equiv(\\sqrt{2\\pi}\\sigma_f)^{-1}e^{-(f-\\mu_f)^2\/2\\sigma_f^2}$, \nso that $K_c=2\\sqrt{2\/\\pi}\\,\\sigma_f$.\nAs the frequency dynamics loses stiffness (when $\\gamma$ increases),\nwe expect the synchronization threshold to be facilitated\nby the frequency drifts.\n\nWe study the infinite population model $N\\rightarrow\\infty$ by formulating\na Fokker-Planck equation for the time\ndependent joint density $p(\\phi, f, t)$.\nThe all-to-all interaction term\n\\begin{equation}\n\\label{roft}\n\\frac K N \\sum_{j=1}^N\\sin\\bigl(\\phi_j(t)-\\phi _i(t)\\bigr)\n\\,=\\,KR(t)\\sin(\\psi(t)-\\phi_i(t))\n\\end{equation}\nhas well known mean-field character and can be replaced\nfor $N\\rightarrow\\infty$\nby\n\\begin{equation}\n\\label{ergodic}\nK\\int_0^{2\\pi}\\!d\\theta\\int\\!dg\\,p(\\theta,g,t)\\sin(\\theta-\\phi_i(t))~.\n\\end{equation}\nWe obtain\n\\begin{eqnarray}\n\\label{fokker}\n\\frac{\\partial p}{\\partial t}&=&\n\\gamma\\sigma_f^2\\frac{\\partial^2p}{\\partial f^2}\n-f\\frac{\\partial p}{\\partial \\phi}\n+\\gamma(f-\\mu_f)\\frac{\\partial p}{\\partial f}+\\gamma p\\\\\n&&-K\\frac{\\partial \\bigl(c(p, \\phi)\\,p\\bigr)}{\\partial \\phi}\\nonumber~.\n\\end{eqnarray}\nThis is the known expression for an O-U process augmented by a phase\ncoupling involving $c(p,\n\\phi)=\\int_0^{2\\pi}\\!\\!d\\theta\\int\\!dg\\,p(\\theta,g,t)\\sin(\\theta-\\phi)$,\nwhich makes Eq.~\\ref{fokker} non-linear as a consequence of Eq.~\\ref{ergodic}.\n\n\\section{Stability Analysis}\n\nWe next discuss the linear stability of the incoherent\nstationary solution $p_0(\\phi, f)=\\mathcal{N}_{\\mu_f,\\sigma_f} (f)\/(2\\pi)$ in first\norder. For reasons that will become clear, we factorize a term \n$\\mathcal{N}^{1\/2}_{\\mu_f,\\sigma_f}(f)$\noff the perturbation and write\n$$\np(\\phi, f, t)=p_0(f)+\\mathcal{N}^{1\/2}_{\\mu_f,\\sigma_f}(f)\n\\epsilon\\bigl(\\phi-\\mu_ft,\\sigma_f^{-1}(f-\\mu_f),\\gamma t\\bigr)~,\n$$\nwhere $\\epsilon(\\tilde\\phi, \\tilde f, \\tilde t)$ is a small perturbation\nexpressed in a rotating frame using rescaled frequency and time variables.\nBy plugging this ansatz into Eq.~\\ref{fokker} we obtain the linearized problem\n$\\frac{\\partial\\epsilon}{\\partial t}=\\mathcal{L}\\epsilon+\\mathcal{O}(\\epsilon^2)$\nwhere $\\mathcal{L}$ is the linear operator\n\\begin{eqnarray*}\n\\mathcal{L}\\epsilon&=&\\frac{\\partial^2\\epsilon}{\\partial f^2}\n-\\frac{\\sigma_f}{\\gamma}f\\frac{\\partial\\epsilon}{\\partial \\phi}\n+\\left(\\frac{1}{2}-\\frac{1}{4} f^2\\right)\\epsilon\\hskip 2.5cm\\\\\n\\lefteqn{+\\frac{K}{2\\pi\\gamma}\\mathcal{N}^{1\/2}_{0,1}(f)\n\\int_0^{2\\pi}\\!\\!\\!\\!d\\theta\\!\n\\int\\!dg\\,\\mathcal{N}^{1\/2}_{0,1}(g)\\epsilon(\\theta,g,t)\\cos(\\theta-\\phi)~.}\n\\end{eqnarray*}\nDecomposing $\\epsilon$ as a Fourier series in $\\phi$, \n$\\epsilon (\\phi,f,t)\\,=\\,\\sum_{n=-\\infty}^\\infty\\epsilon_n(f,t)e^{-in\\phi}$,\nwe obtain for the coefficients $\\epsilon_n$\n\\begin{eqnarray}\n\\label{fourier}\n\\frac{\\partial\\epsilon _n}{\\partial t}\n&=&\\frac{\\partial^2\\epsilon _n}{\\partial f^2}\n+\\left(\\frac{1}{2}-\\frac{1}{4} f^2+\\frac{in\\sigma_f}{\\gamma}f\\right)\\epsilon_n\\nonumber\\\\\n&&+\\delta_{1|n|}\\frac{K}{2\\gamma}\n\\mathcal{N}^{1\/2}_{0,1}(f)\\int\\!dg\\,\\mathcal{N}^{1\/2}_{0,1}(g)\\epsilon_n(g,t)\\nonumber\\\\\n&\\equiv&\\mathcal{L}_n\\epsilon_n\n+\\delta_{1|n|}\\frac{K}{2\\gamma}\\langle\\epsilon _n,\\mathcal{N}^{1\/2}_{0,1}\\rangle\\mathcal{N}^{1\/2}_{0,1}~.\n\\end{eqnarray}\nWe notice that the first term representing the frequency dynamics resembles\nthe harmonic oscillator plus a complex part, which\ncan be removed by applying\nthe translation operator $U_\\theta$ defined by\n$\\bigl(U_\\theta f\\bigr)(x)=f(x-\\theta)$. We note that \n$\\mathcal{L}_n=U_{2in\\sqrt{a}}\\hat\\mathcal{L}_nU_{-2in\\sqrt{a}}$, \nwhere we have set $a=(\\sigma_f\/\\gamma)^2$ and the operator \n$\\hat\\mathcal{L}_n=\\partial_f^2+\\frac{1}{2}-n^2a-\\frac{1}{4} f^2$ is self-adjoint on\n$L^2(\\mathbb{R})$ and\nhas pure point spectrum\n$\\Sigma(\\hat\\mathcal{L}_n)=\\{\\lambda_{n\\ell}=-\\ell-n^2a\\,:\\,\\ell=0,1,2,\\dots\\}$.\nIts eigenfunctions are given in terms of the Hermite\nfunctions \\cite{reed-simon}\n\\begin{eqnarray*}\nH_0(x)&=&\\pi^{-\\frac{1}{4}}e^{-\\frac{1}{2} x^2}~\\mbox{and}~\\\\\nH_\\ell(x)&=&(2^\\ell\\ell!\\sqrt{\\pi})^{-\\frac{1}{2}}(-1)^\\ell\ne^{\\frac{1}{2} x^2}\\partial_x^\\ell e^{-x^2}~,~\\ell\\,=\\,1,2,\\dots\n\\end{eqnarray*}\nas follows:\n$$\n\\hat\\mathcal{L}_n\\Phi_\\ell\\,=\\,\\lambda_{n\\ell}\\Phi_\\ell~,\\quad\n\\Phi_\\ell(f)\\,=\\,2^{-1\/4}H_\\ell(2^{-1\/2}f)~.\\\\\n$\nTherefore $\\{U_{2in\\sqrt{a}}\\Phi_\\ell:\\ell=0,1,2,\\dots\\}$ forms an\northonormal family which diagonalizes $\\mathcal{L}_n$. \nNotice that the largest eigenvalue for each $n$ is\n$\\lambda_{n0}=-n^2a$.\n\nLinear stability follows directly except for $|n|\\neq 1$ and\n$K>0$. Indeed, \nfor $n=0$, we find $\\lambda_{00}=0$ with corresponding\neigenfunction is $\\Phi_0=\\mathcal{N}^{1\/2}_{0,1}$.\nHowever, this function lies outside the space of relevant perturbations\nbecause the normalization of $p$, \n$\\int_0^{2\\pi}d\\phi\\,\\int df\\,p(\\phi,f,t)=1$,\nrequires orthogonality of $\\mathcal{N}^{1\/2}_{0,1}$ and $\\epsilon_0(f,t)$\nthrough $\\int\\mathcal{N}^{1\/2}_{0,1}\\epsilon_0(f,t)df=0$.\nSubsequent eigenvectors have negative eigenvalues.\nFor all other $|n|\\neq 1$\nthe coupling term in Eq.~\\ref{fourier} vanishes\nand the incoherent state $p(\\phi, f, t)=p_0(f)$ \nis linearly stable as a consequence of the\nstrictly negative spectrum of $\\mathcal{L}_n$.\nThe same holds for all $n$ in the absence of coupling $K=0$.\n\\setlength{\\unitlength}{1mm}\n\\begin{figure}\n\\includegraphics[scale=0.8]{fig1}\n\\put(1,2){$\\displaystyle\\frac{\\gamma}{\\sigma_f}$}\n\\put(-75,45){$\\displaystyle\\frac{K_c}{\\sigma_f}$}\n\\caption{Behavior of $K_c$ as a function of $\\gamma$ as given by Eq.~\\ref{kcritical} with\n$a=(\\sigma_f\/\\gamma)^2$ (continuous line). \nDashed line represents the Kuramoto model with\nidentical frequencies and $K_c=2D=2\\sigma_f^2\/\\gamma$. The\n$\\gamma\\rightarrow 0$ limit reproduces the $\\gamma=0$ model with Gaussian\nfrequency dispersion and gives $K_c\/\\sigma_f=2\\sqrt{2\/\\pi}=1.5957$ (dotted \nline).}\n\\label{fig1}\n\\end{figure}\n\nFor the remaining case $n=\\pm 1$ and $K>0$,\nwe notice that the coupling term\nin Eq.~\\ref{fourier} also vanishes for all directions\northogonal to $\\mathcal{N}^{1\/2}_{0,1}$,\nleaving a one-dimensional space that could develop an instability.\nWe write the eigenvalue problem for Eq.~\\ref{fourier} implicitly as\n\\begin{eqnarray*}\n\\mathcal{L}_n\\epsilon_n\n+\\delta_{1|n|}\\frac{K}{2\\gamma}\\langle\\epsilon_n,\\mathcal{N}^{1\/2}_{0,1}\\rangle\n\\mathcal{N}^{1\/2}_{0,1}=\\lambda\\,\\epsilon_n~.\n\\end{eqnarray*}\nUsing the resolvent equation \n\\begin{eqnarray*}\n(\\lambda-\\mathcal{L}_n)^{-1}&=&(\\lambda-U_{2in\\sqrt{a}}\\hat\\mathcal{L}_nU_{-2in\\sqrt{a}})^{-1}\\\\\n&=&U_{2in\\sqrt{a}}(\\lambda-\\hat\\mathcal{L}_n)^{-1}U_{-2in\\sqrt{a}}\n\\end{eqnarray*}\nwe obtain\n\\begin{eqnarray*}\n\\epsilon_n&=&\\frac{K}{2\\gamma}\\langle\\epsilon _n,\\mathcal{N}^{1\/2}_{0,1}\\rangle\nU_{2in\\sqrt{a}}(\\lambda-\\hat\\mathcal{L}_n)^{-1}U_{-2in\\sqrt{a}}\\mathcal{N}^{1\/2}_{0,1}\\\\\n&=&\\frac{K}{2\\gamma}\\langle\\epsilon _n,\\mathcal{N}^{1\/2}_{0,1}\\rangle\n\\sum_{j=0}^\\infty\n\\frac{\\langle\\Phi_j,U_{-2in\\sqrt{a}}\\mathcal{N}^{1\/2}_{0,1}\\rangle}{\\lambda-\\lambda_{nj}}\nU_{2in\\sqrt{a}}\\Phi_j~,\n\\end{eqnarray*}\nwhere we have used the spectral decomposition\n$\\hat\\mathcal{L}_nf=\\sum_j\\lambda_j\\langle\\Phi_j,f\\rangle\\Phi_j$.\n\nTo find the critical coupling $K_c$ above which the incoherent state becomes\nlinearly unstable, we need to monitor when the\nlargest eigenvalue crosses the imaginary axis.\nAfter projecting onto $\\mathcal{N}^{1\/2}_{0,1}$, simplifying the factors \n$\\langle\\epsilon _n,\\mathcal{N}^{1\/2}_{0,1}\\rangle$ on both sides of the\nequation, and setting $\\lambda=0$ we\nfind an equation for $K_c$:\n\\begin{eqnarray}\n\\frac{2\\gamma}{K_c}&=&\n\\sum_{j=0}^\\infty\n\\frac{\\langle\\Phi_j,U_{-2i\\sqrt{a}}\\mathcal{N}^{1\/2}_{0,1}\\rangle^2}{-\\lambda_{1j}}\n\\,=\\,e^a\\sum_{j=0}^\\infty\\frac{(-a)^j}{j!(j+a)}\\nonumber\\\\\n&=&e^aa^{-a}\\!\\int_0^a\\!\\!t^{a-1}e^{-t}dt\n\\,=\\,e^aa^{-a}\\gamma(a,a)~,\n\\label{kcritical}\n\\end{eqnarray}\nwhere $\\gamma(a,x)$ is the lower incomplete $\\Gamma$-function.\n\nThe behavior of $K_c$ together with the Kuramoto model asymptotes for\n$\\gamma\\rightarrow\\infty$ and $\\gamma\\to0^+$ limit are shown in Fig.~\\ref{fig1}.\nIt is noticeable that we find a bifurcation for all values of $\\gamma$.\n$K_c$ strictly decreases from a finite\n$\\gamma=0$ as $\\gamma$ increases, asymptotically behaving as\n$K_c=2\\sigma_f^2\/\\gamma$. \nThe analytical result thus supports the following picture:\nfor small $\\gamma$, the dominant source of fluctuations against which\nthe coupling must work to achieve synchronization is the\n(Gaussian) frequency dispersion. As $\\gamma$ increases while $\\sigma_f$\nis kept fixed, faster frequency drifts help synchrony by preventing\nindividual oscillators with detuned frequency to stay out of tune for\ntoo long. Indeed with drifting frequencies every individual oscillators\nfluctuates around the mean frequency $\\mu_f$ with a time scale $\\gamma^{-1}$.\nIn the large $\\gamma$ regime, the effective frequency dispersion vanishes\nand the coupling force needs to synchronize noisy but otherwise identical frequency\noscillators. As predicted by the phase diffusion limit, the effective white\nnoise strength $D$ and hence $K_c$ decrease as $\\gamma^{-1}$.\n\nWe now discuss the asymptotic regimes in detail: \nthe small $\\gamma$ limit follows from\nreverting to the original variables and using the asymptotic expansion of\n$\\gamma(a,a)$ \n(using Stirling's formula and \\cite{abramowitz}:\n6.5.3, 6.5.22, and 6.5.35). We obtain in the\nlimit $\\gamma\\to0^+$\n$$\n2\\left(\\frac {K_c} \\sigma_f\\right)^{-1}\\,=\\,\n\\sqrt{\\frac{\\pi}{2}}+\n\\frac 1 3 \\frac{\\gamma}{\\sigma_f}+\\frac{\\sqrt{2\\pi}}{24}\\frac{\\gamma^2}{\\sigma_f^2}+\\mathcal{O}(\\gamma^3)~.\n$$\nThis proves that the model continuously interpolates to the noiseless ($D=0$)\nmodel and that the $\\gamma\\rightarrow 0$\nrecovers the $\\gamma=0$ transition predicted in the original\nKuramoto model at $K_c\/\\sigma_f=2\\sqrt{2\/\\pi}$.\nIn the opposite regime $\\gamma\\to\\infty$ (thus $a\\to 0$) we find\n(\\cite{abramowitz} 6.5.12, 13.1.2)\n$$\na^{-a}e^a\\gamma(a,a)\\,=\\,a^{-1}M(1,1+a,a)\\,\\sim\\,a^{-1}\\bigl(1+\\mathcal{O}(a)\\bigr)~,\n$$\nwhere $M(\\cdot,\\cdot,\\cdot)$ is the confluent hypergeometric\nfunction. This leads $K_c\\sim2\\sigma_f^2\/\\gamma+\\mathcal{O}(\\gamma^{-2})$\nand hence proves the convergence to the white noise model\n(Eq.~\\ref{kuramoto}) with $D=\\sigma_f^2\/\\gamma$.\n\nFinally, we mention a generalization that\nincludes a white noise source in the phase equation (as in Eq.~\\ref{kuramoto})\nin addition to the correlated frequency fluctuations.\nThis leads an additional diffusion term\n$\n-D\\frac{\\partial^2p}{\\partial \\phi^2}\n$\nin Eq.~\\ref{fokker}.\nFollowing the steps above readily extends Eq.~\\ref{kcritical} to\n$$\n\\frac{2\\gamma}{K_c}\\,=\\,\\,e^aa^{-(a+b)}\\gamma(a+b,a)~,\n$$\nwhere $b=D\/\\gamma$, with similar qualitative behavior.\nIn particular, $K_c$ asymptotes to $2(\\sigma_f^2\/\\gamma+D)$ for large\n$\\gamma$ and has finite $\\gamma\\rightarrow 0$ limit.\n\n\\section{Numerical Simulations}\n\nWe have performed numerical simulations of Eq.~\\ref{model} to explore\nthe behavior of $R(t)$ (see Eq.~\\ref{roft}) and in particular\n$R_\\infty$ in function of the reduced\ncoupling $K_r=(K-K_c)\/K_c$.\nTo verify the analytical results and study the scaling $R_\\infty=\\kappa K_r^\\beta$\nabove the bifurcation, we simulated a finite number of oscillators\nusing the exact solution for the frequency part, leading to the updates\n$f_i(t+\\!dt)=f_i(t)\\,e^{-\\gamma\\,\\!dt} + \\mu_f(1-e^{-\\gamma\\,\\!dt})+\n\\eta\\,\\sigma_f\\sqrt{1-e^{-2\\gamma\\,\\!dt}}$\nand $\\phi_i(t+\\!dt)=\\phi_i(t) + (f_i(t)+\\frac K N \\sum_j\\sin(\\phi_j-\\phi_i))\\,\\!dt$\nwhere $\\eta$ is a Gaussian random number.\nWe used Eq.~\\ref{roft} to compute $R(t)$ and transients were removed by waiting\nuntil the solutions from two different initial conditions\n$\\phi_i(t=0)=0$ and $\\phi_i(t=0)$ taken randomly converged to the same trajectory.\nThe steady state value $R_\\infty$ was subsequently estimated by averaging $R(t)$ over time.\n\nFig.~\\ref{kcfig} fully supports the analytical solution and also indicates that\nthe behavior of $K_c$ above the bifurcation depends only weakly on $\\gamma$ over the simulated range.\nTo inspect more closely whether $R_\\infty\\sim\\sqrt{K_r}$ as in the Kuramoto model,\nwe used refined spacing and larger sizes in the vicinity of $K_r=0^+$.\nAs shown in Fig.~\\ref{finer}, the simulations are compatible with an\nexponent $\\beta=0.5$, the slightly higher exponents probably reflect a\nfinite size effect.\nOn the other hand $\\kappa$ correlates negatively with $\\gamma$ which\nis visible in both Figs \\ref{kcfig} and \\ref{finer}.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.4,angle=270]{fig2}\n\\put(-58,-1){\\includegraphics[scale=0.17,angle=270]{fig3}}\n\\put(-70,-31){$R_\\infty$}\n\\put(-39,-68){$(K-K_c)\/K_c$}\n\\put(-60,-12){\\tiny$R_\\infty$}\n\\put(-45,-31){\\tiny$1\/N$}\n\\caption{Numerical simulation of Eq.~\\ref{model}. \nEstimation of $R_{\\infty}$ was obtained using finite size\nscaling for systems of sizes $N=5000$, $10000$ and\n$20000$. Eq.~\\ref{kcritical} was used for $K_c$ to set the reduced coupling\n$K_{r}=(K-K_c)\/K_c$. Values for $\\gamma$ were $4(\\circ)$,\n$3(\\bigtriangleup)$, $2.5(+)$,\n$2(\\times)$, $1.8(\\diamond)$ and $1.6(\\bigtriangledown)$\nand $\\sigma_f=1$. In each simulation, $10^5$ time steps of size\n$dt=0.01$ were performed. We verified that the dependence in the step size was\nweak. Inset: $1\/N$ finite size\nscaling for $\\gamma=1.8$. $K_r=0$ is the solid line, smaller (resp. larger)\n$K_r$ are below (resp. above) $K_r=0$.\nThe extrapolated value for $1\/N=0$ is used in the main panel.}\n\\label{kcfig}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.32,angle=270,origin=br]{fig4}\n\\includegraphics[scale=0.21]{fig5}\n\\put(-83,29){$R_\\infty$}\n\\put(-61,-3){$(K-K_c)\/K_c$}\n\\put(-26,40){$\\kappa$}\n\\put(-11,28){$\\gamma$}\n\\put(-26,11){$\\beta$}\n\\put(-11,-2){$\\gamma$}\n\\caption{Critical behavior above the bifurcation using step $dK_r=0.01$.\nHere, $\\gamma$ is $4(\\circ)$, $3(\\bigtriangleup)$, $2(+)$,\n$1.6(\\times)$, $1.2(\\diamond)$ and $0.8(\\bigtriangledown)$.\nNotice the log-log scale to emphasize the power law. Lines are fits to\n$R_\\infty=\\kappa\\,K_r^\\beta$ Systems of sizes $N=10000$, $20000$ and\n$50000$ were simulated with the same parameters and same scaling\nprocedure as in Fig.~\\ref{kcfig}. Right panels show parameter\nestimates from the left panel.}\n\\label{finer}\n\\end{figure}\n\n\\section{Discussion}\n\nWe have extended the Kuramoto model to frequencies which can drift\nin time following Ornstein-Uhlenbeck dynamics.\nThe net effect is that the white noise source in Eq.~\\ref{kuramoto} is\nreplaced by colored noise\n(with a Cauchy distributed power spectrum), hereby\nadding a new time scale describing memory or frequency stiffness to\nthe problem.\nApart from mean field coupling among the phases which introduces a\nnon-linearity, the stochastic phase and frequency dynamics lead to a linear\nFokker-Planck operator which can be solved.\nConsequently the linear stability of the\nincoherent state can be addressed analytically using\nspectral calculus. The expression for the critical coupling above which\nmacroscopic phase coherence emerges can be resummed and expressed in\nterms of incomplete $\\Gamma$-functions. Asymptotic expansion for small\nand large $\\gamma$ shows that the full model continuously interpolates\nbetween two limits of the original Kuramoto model: one dominated by\nnoise (large $\\gamma$) and the other by the frequency dispersion\n(small $\\gamma$).\nTherefore, the coupling force must counteract different sources of\nfluctuations to induce collective synchrony in drifting frequency\noscillators, depending on the regime set by $\\gamma$.\nSpecifically, for slowly drifting (small $\\gamma$) frequencies\nthe model approaches the noiseless model (Eq.~\\ref{kuramoto} with\n$D=0$ and $g(f)=\\mathcal{N}_{\\mu_f,\\sigma_f}(f)$)\nwhere the coupling splits the population into distinct locked and\nincoherent sub-populations, depending\non the proximity of individual frequencies to the population mode.\nAs $\\gamma$ increases, (while $\\sigma_f$ is held fixed) the frequencies\nlose their stiffness which results in a reduction in the critical\ncoupling $K_c$ needed for synchrony. Finally for very rapidly drifting\noscillators (large $\\gamma$) cancel out the frequency distribution and\ngenerate an effective white noise source acting on the phases of\notherwise equal frequency oscillators.\nAt the same time the locked and incoherent subgroups become indistinguishable.\nFor intermediate $\\gamma$, our numerical simulations indicate\nthat the model belongs to the same $\\beta=0.5$ universality class as the\nKuramoto model.\n\nBecause of analytical tractability and few parameters we expect this solution\nto be relevant for oscillatory systems in the presence of complex noise sources.\nSuch cases include populations of neural oscillators or biochemical oscillators\nwhere bioluminescence recordings have shown how intracellular noise sources\ngenerate frequency dispersion through drifts.\n\n\\subsection*{Acknowledgments}\nWe thank Benoit Kornmann and Ueli Schibler for initiating our interest in\nthe drifting frequency model and Olivier Hernandez for pointing out useful\nreferences. The simulations were performed on an Itanium2\ncluster from HP\/Intel at the Vital-IT facilities.\nFN acknowledges funding from the Swiss National Science Foundation\nNCCR Molecular Oncology program and NIH administrative supplement to parent\ngrant GM54339.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\\subsection{The $(\\mu,\\phi)$-$L^2$-Euler characteristic}\n\\label{subsec:The_(mu,phi)_L2-Euler_characteristic_cw-intro}\nLet $X$ be a finite connected CW-com\\-plex and let $\\mu\\colon \\pi_1(X) \\to G$ and $\\phi \\colon G \\to \\IZ$ be group\nhomomorphisms. We say that $(\\mu,\\phi)$ is an \\emph{$L^2$-acyclic Atiyah pair}\nif the $n$th $L^2$-Betti number\n$b_n^{(2)}(\\overline{X};\\caln(G))$ of the $G$-covering $\\overline{X} \\to X$ associated to\n$\\mu$ vanishes for all $n \\ge 0$, and $G$ is torsion-free and satisfies\nthe Atiyah Conjecture. (We will discuss the Atiyah Conjecture in \nSection~\\ref{sec:About_the_Atiyah_Conjecture}). \nThen one can define\nby twisting the cellular $\\IZ G$-chain complex with the infinite dimensional\n$G$-representation $\\phi^*\\IR \\IZ$ the \\emph{$(\\mu,\\phi)$-$L^2$-Euler characteristic}\n$\\chi^{(2)}(X;\\mu,\\phi)$ which is an  integer. We will not give the precise definition of $(\\mu,\\phi)$-$L^2$-Euler \ncharacteristic in the introduction but we refer to\nSection~\\ref{sec:Twisting_the_L2-Euler_characteristic_with_a_cocycle_in_the_first_cohomology}\nfor details and a summary of the key properties.\n\nThe $(\\mu,\\phi)$-$L^2$-Euler characteristic can be employed in many different contexts.\nFor example it is used by Funke--Kielak~\\cite{Funke-Kielak(2016)} to study descending HNN-extensions of \nfree groups and it is at least implicitly used by the authors and Tillmann~\\cite{Friedl-Lueck-Tillmann(2016)} to study one-relator groups.\n\n\n\n\\subsection{The $\\phi$-$L^2$-Euler characteristic of 3-manifolds}\n\\label{subsec:The_phi_L2-Euler_characteristic_for_universal_coverings_intro}\nIn this paper our main application of the $(\\mu,\\phi)$-$L^2$-Euler characteristic  lies in the study of 3-manifolds \nto which we restrict ourselves in the remainder of the introduction.\nMore precisely, our main focus  will be on the following class of $3$-manifolds.\n\n\\begin{definition}[Admissible $3$-manifold] \\label{def:admissible_3-manifold} \n A $3$-manifolds is called \\emph{admissible} if it is connected, orientable, and irreducible,\n its boundary is empty or a disjoint union of tori, and its fundamental group\n is infinite.\n\\end{definition}\n\nLet $M$ be an admissible $3$-manifold and let \n$\\phi \\colon \\pi_1(M) \\to \\IZ$ be a group homomorphism. Then \nall the conditions listed in Section~\\ref{subsec:The_(mu,phi)_L2-Euler_characteristic_cw-intro} are satisfied \nfor the triple $(X,\\id_{\\pi_1(M)},\\phi)$ and the  corresponding $L^2$-Euler characteristic \n$\\chi^{(2)}(M;\\id_{\\pi_1(M)},\\phi)$ is defined. We denote by $x_M(\\phi)$ the Thurston norm of $\\phi$ \nwhich is loosely speaking defined as the minimal complexity of a surface dual to $\\phi$.\n (We recall the precise definition of the Thurston norm in Section~\\ref{subsec:Brief_review_of_the_Thurston_norm}.) \nThe following is one of our main theorems.\n\n\n\\begin{theorem}[Equality of $(\\mu,\\phi)$-$L^2$-Euler characteristic and the Thurston norm\n for universal coverings]\n \\label{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_universal_covering_for_universal_coverings}\n Let $M\\ne S^1\\times D^2$ be an admissible $3$-manifold.\n\n Then we get for any $\\phi\\in H^1(M;\\IZ)$\n \\[\n -\\chi^{(2)}(M;\\id_{\\pi_1(M)},\\phi) \\,\\,=\\,\\, x_M(\\phi).\n \\]\n\\end{theorem}\n\nIf $M$ is not a closed graph manifold, then\nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_universal_covering_for_universal_coverings}\nis a direct consequence of the subsequent\nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}\ntogether with the fact that in this case the fundamental groups of $M$ satisfies the\nAtiyah Conjecture, see\nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}. If $M$ is a (closed) graph manifold\nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_universal_covering_for_universal_coverings}\nfollows from\nTheorem~\\ref{the:The_phi-L2-Euler_characteristic_and_the_Thurston_norm_for_graph_manifolds}.\n\\medskip\n\nIt is also interesting to consider group homomorphisms $\\pi_1(M)\\to G$ that are not the identity. \nFor example in Section~\\ref{sec:The_(mu,phi)-L2-Euler_characteristic_and_the_degree_of_higher_order_Alexander_polynomials}\nwe will see that if $G$ is a torsion-free elementary amenable group, then the\n$(\\mu,\\phi)$-$L^2$-Euler characteristic $\\chi^{(2)}(M;\\mu,\\phi)$ is basically the same as\nthe degrees of the non-commutative Alexander polynomials studied by\nCochran~\\cite{Cochran(2004)}, Harvey~\\cite{Harvey(2005)} and the first\nauthor~\\cite{Friedl(2007)}. In these three papers it was shown that the degrees of\nnon-commutative Alexander polynomials give lower bounds on the Thurston norm. The following\ntheorem can be viewed as a generalization of these results.\n\n\\begin{theorem}[The negative of the $(\\mu,\\phi)$-$L^2$-Euler characteristic is a lower bound for the Thurston norm]\n\\label{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic-intro}\nLet $M\\ne S^1\\times D^2$ be an admissible $3$-manifold and let $(\\mu,\\phi)$ be an $L^2$-acyclic Atiyah-pair.\n\nThen we get\n\\[\n-\\chi^{(2)}(M;\\mu,\\phi)~\\le~x_M(\\phi \\circ \\mu).\n\\]\n\\end{theorem}\n\nIn general the inequality of the above theorem is not an equality. But the following theorem shows \nthat for any `sufficiently large epimorphism', the inequality does become an equality.\n\n\\begin{theorem}[Equality of $(\\mu,\\phi)$-$L^2$-Euler characteristic and the Thurston norm]\n\\label{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}\nLet $M\\ne S^1\\times D^2$ be an admissible $3$-manifold which is not a closed graph manifold.\n\nThen there exists a virtually abelian group $\\Gamma$ and a factorization \n\\[\n\\pr_M \\colon \\pi_1(M) \\xrightarrow{\\alpha} \\Gamma \\xrightarrow{\\beta} H_1(M)_f := H_1(M)\/\\tors(H_1(M))\n\\]\nof the canonical projection $\\pr_M\\colon \\pi_1(M)\\to H_1(M)_f$ into epimorphisms such that the following holds:\n\n\nGiven a group $G$ satisfying the Atiyah Conjecture, a factorization of $\\alpha \\colon\n\\pi \\to \\Gamma$ into group homomorphisms $\\pi \\xrightarrow{\\mu} G \\xrightarrow{\\nu}\n\\Gamma$, and a group homomorphism $\\phi \\colon H_1(M)_f \\to \\IZ$, the pair $(\\mu, \\phi\n\\circ \\beta \\circ \\nu)$ is an $L^2$-acyclic Atiyah-pair, and we get\n\\[\n -\\chi^{(2)}(M;\\mu,\\phi \\circ \\beta \\circ \\nu)~=x_M(\\phi).\n\\]\n\\end{theorem}\n\nWith a little bit of extra effort one can use\nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm} to\nshow that one can use epimorphisms onto torsion-free elementary amenable groups to detect\nthe Thurston norm. Put differently, one can show that the aforementioned non-commutative\nAlexander polynomials detect the Thurston norm. We refer to\nCorollary~\\ref{cor:existence_of_torsion-free_elementary_coverings_with_equality} for the\nprecise statement.\n\n\n\n\\subsection{Inequality of the Thurston norm}\n\\label{subsec:Inequality_of_the_THurston_norm}\n\nOne of the key motivations for developing the theory of $L^2$-Euler characteristics is the following question by \nSimon~\\cite[Problem~1.12]{Kirby(1997)}.\n\n\\begin{question}\\label{que:Simons_question}\nLet $K$ and $K'$ be two knots. If there is an epimorphism from the knot group of $K$ to\nthe knot group of $K'$, does this imply that the genus of $K$ is greater than or equal to the\ngenus of $K'$?\n\\end{question}\n\nWe propose the following conjecture. \n\n\\begin{conjecture}[Inequality of the Thurston norm] \\label{con:Inequality_of_the_Thurston_norm} \n Let $f \\colon M \\to N$ be a map of\n admissible $3$-manifolds which is surjective on $\\pi_1(N)$ and induces an isomorphism\n $f_* \\colon H_n(M;\\IQ) \\to H_n(N;\\IQ)$ for $n \\ge 0$.\n \n Then we get for any $\\phi \\in H^1(N;\\IR)$ that\n \\[\n x_M(f^*\\phi) \\,\\,\\ge\\,\\, x_N(\\phi).\n \\]\n\\end{conjecture}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item In Section~\\ref{subsec:Brief_review_of_the_Thurston_norm} we will recall that the\n Thurston norm can be viewed as a generalization of the knot genus. In particular a proof\n of  Conjecture~\\ref{con:Inequality_of_the_Thurston_norm} \n would give an affirmative answer to Simon's question.\n\\item If $M$ and $N$ are closed 3-manifolds, then the conclusion of \n Conjecture~\\ref{con:Inequality_of_the_Thurston_norm} follows from~\\cite[Corollary~6.18]{Gabai(1983)}.\n\\item The condition on the induced map on rational homology cannot be dropped. For\n example, suppose that $M=S^1\\times \\Sigma$ with $\\Sigma$ a surface of genus $g\\geq 2$\n with boundary. Let $N$ be the exterior of a torus knot. Then $\\pi_1(N)$ is generated by\n two elements, therefore there exists an epimorphism $f_*\\colon \\pi_1(S^1\\times\n \\Sigma)\\to \\pi_1(N)$ which factors through the projection $\\pi_1(S^1\\times \\Sigma)\\to\n \\pi_1(\\Sigma)$. If $\\phi$ is a generator of $H^1(N;\\IZ)$, then $x_N(\\phi)\\ne 0$ but it\n is straightforward to see that $x_M(f^*\\phi)=0$. We are grateful to Yi Liu for pointing\n out this example.\n\\end{enumerate}\n\\end{remark}\n\nBefore we state our main contribution to\nConjecture~\\ref{con:Inequality_of_the_Thurston_norm} we need to recall one more\ndefinition. A group $G$ is called \\emph{locally indicable} if any finitely generated\nnon-trivial subgroup of $G$ admits an epimorphism onto $\\IZ$. For example\nHowie~\\cite{Howie(1982)} showed that the fundamental group of any admissible 3-manifold\nwith non-trivial boundary is locally indicable.\n\nOur main result is Theorem~\\ref{the:Inequality_of_the_Thurston_norm-intro}. \n\n\n\n\\begin{theorem}[Inequality of the Thurston norm] \\label{the:Inequality_of_the_Thurston_norm-intro}\nLet $f \\colon M \\to N$ be a map of admissible $3$-manifolds which is surjective on $\\pi_1(N)$ \nand induces an isomorphism $f_* \\colon H_n(M;\\IQ) \\to H_n(N;\\IQ)$ for $n \\ge 0$. Suppose that \n$\\pi_1(N)$ is residually locally indicable elementary amenable. \n\nThen we get for any $\\phi\\in H^1(N;\\IR)$ that\n \\[\n x_M(f^*\\phi ) \\,\\,\\ge\\,\\, x_N(\\phi).\n \\]\n\\end{theorem}\n\nBy Lemma~\\ref{lem:fibered-manifolds-acapl} the fundamental group of any fibered 3-manifold\nis residually locally indicable elementary amenable.  Thus we have proved\nConjecture~\\ref{con:Inequality_of_the_Thurston_norm} in particular in the case that $N$ is\nfibered. The conclusion of Conjecture~\\ref{con:Inequality_of_the_Thurston_norm} can be\nproved relatively easily for \\emph{fibered classes in $H^1(N;\\IR)$}, but it seems to us\nthat if $N$ is fibered, then there is no immediate reason why the inequality should hold\nfor \\emph{non-fibered} classes in $H^1(N;\\IR)$.\n\nWe propose the following\n\n\\begin{conjecture} \\label{con:res_loc_ind_elem_ama_always}\nThe fundamental group of any admissible $3$-manifold $M$ with $b_1(M) \\ge 1$\nis residually locally indicable elementary amenable. \n\\end{conjecture}\n\nA proof of Conjecture~\\ref{con:res_loc_ind_elem_ama_always} together with \nTheorem~\\ref{the:Inequality_of_the_Thurston_norm-intro} implies \nConjecture~\\ref {con:Inequality_of_the_Thurston_norm} and in particular an affirmative answer to Simon's \nQuestion~\\ref{que:Simons_question}.\n\n\n\n\\subsection{The $(\\mu,\\phi)$-$L^2$-Euler characteristic and the degree of the $L^2$-torsion \nfunction}\n\\label{subsec:The_(mu,phi)-L2-Euler_characteristic_and_the_degree_of_the_L2-torsion_function_intro}\n\nWe briefly discuss a relation of the $(\\mu,\\phi)$-$L^2$-Euler characteristic to the degree of the $L^2$-torsion \nfunction in Section~\\ref{sec:The_degree_of_the_L2-torsion_function}.\n\n\n\n\n\\subsection{Methods of proof}\n\\label{subsec:Methods_of_proof}\n\nOne key ingredient in this paper is to replace the Ore localization of a group ring $\\IZ G$,\nwhich is known to exist for torsion-free elementary amenable groups and is definitely not available,\nif $G$ contains a non-abelian free subgroup, by the division closure $\\cald(G)$ of\n$\\IZ G$ in the algebra $\\calu(G)$ of operators affiliated to the group von Neumann algebra $\\caln(G)$.\nThis is a well-defined skew field containing $\\IZ G$ if and only if $G$ is torsion-free and satisfies\nthe Atiyah Conjecture with rational coefficients. This is known to be true in many interesting cases. \n\n\nWe will also take advantage of the recent proof by Agol and others of the Virtual Fibering Conjecture.\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgments.}\nThe first author gratefully acknowledges the support provided by the SFB 1085 ``Higher\nInvariants'' at the University of Regensburg, funded by the Deutsche\nForschungsgemeinschaft {DFG}. The paper is financially supported by the Leibniz-Preis of the second author granted by the {DFG}\nand the ERC Advanced Grant ``KL2MG-interactions'' (no. 662400) of the second author granted by the European Research Council.\n\n\n\n\\tableofcontents\n\n\n\n\n\\typeout{------------ Section 1: Brief review of the Thurston norm -------------}\n\n\n\n\\section{Brief review of the Thurston norm}\n\\label{subsec:Brief_review_of_the_Thurston_norm}\n\nRecall the definition in~\\cite{Thurston(1986norm)} of the \\emph{Thurston norm} $x_M(\\phi)$\nof a compact connected orientable $3$-manifold $M$ with empty or non-empty boundary \nand an element $\\phi \\in H^1(M;\\IZ)=\\Hom(\\pi_1(M),\\IZ)$\n\\[\nx_M(\\phi)\\,\\,:=\\,\\,\\min \\{ \\chi_-(F)\\, | \\, F \\subset M \\mbox{ properly embedded surface dual to\n}\\phi\\},\n\\]\nwhere, given a surface $F$ with connected components $F_1, F_2, \\ldots , F_k$, we define\n\\[\\chi_-(F)\\,\\,:=\\,\\,\\smsum{i=1}{k} \\max\\{-\\chi(F_i),0\\}.\\]\n\n\nThurston~\\cite{Thurston(1986norm)} showed that this defines a seminorm on $H^1(M;\\IZ)$\nwhich can be extended to a seminorm on $H^1(M;\\IR)$ which we denote by $x_M$ again.\nIn particular we get for $r \\in \\IR$ and $\\phi \\in H^1(M;\\IR)$\n\\begin{eqnarray}\n x_M(r \\cdot \\phi) \n & = & \n |r| \\cdot x_M(\\phi).\n \\label{scaling_Thurston_norm}\n\\end{eqnarray}\n\n\nIf $K\\subset S^3$ is a knot, then we denote by $\\nu K$ an open tubular neighborhood of $K$\nand we refer to $X_K=S^3\\setminus \\nu K$ as the exterior of $K$. We refer to the minimal\ngenus of a Seifert surface of $K$ as the \\emph{genus $g(K)$ of $K$}. We have\n$H^1(X_K;\\IZ)\\cong \\IZ$ and an elementary exercise shows that for any generator $\\phi$ of\n$H^1(X_K;\\IZ)\\cong \\IZ$ we have\n\\begin{equation} x_{X_K}(\\phi)\\,\\,=\\,\\,\\operatorname{max}\\{\n 2g(K)-1,0\\}.\\label{equ:genus-thurston-norm} \\end{equation}\n\n\n\nIf $p \\colon M' \\to M$ is a finite covering with $n$ sheets, then Gabai~\\cite[Corollary~6.13]{Gabai(1983)} showed\n\\begin{eqnarray}\n x_{M'}(p^*\\phi) \n & = & \n n \\cdot x_M(\\phi).\n \\label{finite_coverings_Thurston_norm}\n\\end{eqnarray}\nIf $F \\to M \\xrightarrow{p} S^1$ is a fiber bundle for a compact connected orientable\n$3$-manifold $M$ and compact surface $F$ and $\\phi \\in H^1(M;\\IZ)$ is given by $H_1(p)\n\\colon H_1(M) \\to H_1(S^1)$, then\n\\begin{eqnarray}\n x_M(\\phi) & = & \n \\begin{cases}\n - \\chi(F) & \\text{if} \\;\\chi(F) \\le 0;\n \\\\\n 0 & \\text{if} \\;\\chi(F) \\ge 0.\n \\end{cases}\n \\label{fiber_bundles_Thurston_norm}\n\\end{eqnarray}\n\n\n\n\n\n\\typeout{-- Section 2: Twisting the $L^2$-Euler characteristic with a cocycle in the first cohomology -----------}\n\n\n\\section{Twisting the $L^2$-Euler characteristic with a cocycle in the first cohomology}\n\\label{sec:Twisting_the_L2-Euler_characteristic_with_a_cocycle_in_the_first_cohomology}\n\nIn this section we introduce our main invariant on the $L^2$-side, namely certain\nvariations of the $L^2$-Euler characteristic which are obtained in the special case of the\nuniversal covering $\\widetilde{X} \\to X$ with by twisting with an element $\\phi \\in H^1(X;\\IZ)$. More generally,\nwe will consider $G$-$CW$-complexes and twist with a group homomorphism $\\phi \\colon G \\to \\IZ$.\n\n\n\n\n\n \n\n\n\\subsection{Review of the $L^2$-Euler characteristic}\n\\label{subsec:Review_of_the_L2-Euler_characteristic}\n\nLet $G$ be a group. Denote by $\\caln(G)$ the group von Neumann algebra which can be\nidentified with the algebra $B(L^2(G),L^2(G)^G$ of bounded left $G$-equivariant operators\n$L^2(G) \\to L^2(G)$. Let $C_*$ be a finitely generated based free left $\\IZ G$-chain\ncomplex. Then we can consider the chain complex of finitely generated Hilbert\n$\\caln(G)$-chain complexes $L^2(G) \\otimes_{\\IZ G} C_*$. Its $L^2$-Betti numbers\n$b_n^{(2)}(L^2(G) \\otimes_{\\IZ G} C_*)$ are defined as the von Neumann dimension\nof its $L^2$-homology, see~\\cite[Section~1.1]{Lueck(2002)}.\n\nOne can also work entirely algebraically by applying $\\caln(G) \\otimes_{\\IZ G} -$ which\nyields a chain complex $\\caln(G) \\otimes_{\\IZ G} C_*$ of $\\caln(G)$-modules, where we\nconsider $\\caln(G)$ as a ring and forget the topology. There is a dimension function\ndefined for all $\\caln(G)$-modules, see~\\cite[Section~6.1]{Lueck(2002)}. So one gets\nanother definition of $L^2$-Betti numbers by taking this dimension of the\n$\\caln(G)$-module $H_n(\\caln(G) \\otimes_{\\IZ G} C_*)$. These two definitions agree\nby~\\cite[Section~6.2]{Lueck(2002)}.\n\nThe advantage of the algebraic approach is that it works and often still gives finite\n$L^2$-Betti numbers also in the case where we drop the condition that $C_*$ is a finitely\ngenerated free $\\IZ G$-chain complex and consider any $\\IZ G$-chain complex $C_*$. This is\nexplained in detail in~\\cite[Chapter~6]{Lueck(2002)}. We recall that for any\nchain complex of free left $\\IZ G$-chain modules $C_*$ we can define its $n$th $L^2$-Betti number a\n\\begin{eqnarray}\nb_n^{(2)}(\\caln(G) \\otimes_{\\IZ G} C_*)\n& := &\n \\dim_{\\caln(G)}\\bigl(H_n(\\caln(G) \\otimes_{\\IZ G} C_*)\\bigr)\n \\quad \\in [0,\\infty].\n\\label{b_n(2)(caln(G)_otimes_ZGC_ast)}\n\\end{eqnarray}\nIn particular given a $G$-$CW$-complex $X$ we can define its $n$th $L^2$-Betti number as \n\\begin{eqnarray}\n b_n^{(2)}(X;\\caln(G)) & := & \n \\dim_{\\caln(G)}\\bigl(H_n(\\caln(G) \\otimes_{\\IZ G} C_*(X))\\bigr)\n \\quad \\in [0,\\infty].\n \\label{b_n(2)(X;caln(G))}\n\\end{eqnarray}\n\nWe leave the superscript in the notation $b_n^{(2)}(\\caln(G) \\otimes_{\\IZ G} C_*)$ and $b_n^{(2)}(X;\\caln(G))$, although the definition is\npurely algebraic, in order to remind the reader that it is related to the classical notion\nof $L^2$-Betti numbers.\n\n\\begin{definition}[$L^2$-Euler characteristic]\n \\label{def:L2-Euler_characteristic}\n Let $X$ be a $G$-$CW$-complex. Define\n \\begin{eqnarray*}\n h^{(2)}(X;\\caln(G))\n & := & \\lmsum{p \\ge 0}{} b_p^{(2)}(X;\\caln(G)) \\in [0,\\infty];\n \\\\\n \\chi^{(2)}(X;\\caln(G))\n & := & \n \\lmsum{p \\ge 0}{} (-1)^p \\cdot b_p^{(2)}(X;\\caln(G))\n \\in \\IR, \\quad \\mbox{if}\\; h^{(2)}(X;\\caln(G)) < \\infty.\n \\\\\n \\end{eqnarray*}\n We call $\\chi^{(2)}(X;\\caln(G))$ the \\emph{$L^2$-Euler characteristic} of $X$.\n\\end{definition}\n\nThe condition $h^{(2)}(X;\\caln(G)) < \\infty$ ensures that the sum which appears in the\ndefinition of $\\chi^{(2)}(X;\\caln(G))$ converges absolutely. In the sequel we assume that\nthe reader is familiar with the notion of the $L^2$-Euler characteristic and its basic\nproperties, as presented in~\\cite[Section~6.6.1]{Lueck(2002)}. Another approach to\n$L^2$-Betti numbers for not necessarily finite $G$-$CW$-complexes is given by\nCheeger-Gromov~\\cite{Cheeger-Gromov(1986)}.\n\n\n\n\n\n\n\\subsection{The $\\phi$-twisted $L^2$-Euler characteristic}\n\\label{subsec:The_phi-twisted-L2-Euler_characteristic}\n\nWe will be interested in the following version of an $L^2$-Euler characteristic.\n\n\n\\begin{definition}[$\\phi$-twisted $L^2$-Betti number and $L^2$-Euler characteristic]\n \\label{def:phi-twisted_L2_betti_number_and_L2-Euler_characteristic}\n Let $X$ be a $G$-$CW$-complex. Let $\\phi \\colon G \\to \\IZ$ be a group homomorphism. Let\n $\\phi^*\\IZ[\\IZ]$ be the $\\IZ G$-module obtained from $\\IZ[\\IZ]$ regarded as module over\n itself by restriction with $\\phi$. If $C_*(X)$ is the cellular $\\IZ G$-chain complex,\n denote by $C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ]$ the $\\IZ G$-chain complex obtained by\n the diagonal $G$-action. Define\n \\begin{eqnarray*}\n b_n^{(2)}(X;\\caln(G),\\phi) & := &\n \\dim_{\\caln(G)}\\bigl(H_n(\\caln(G) \\otimes_{\\IZ G}(C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ]))\\bigr)\n \\in [0,\\infty];\n \\\\\n h^{(2)}(X;\\caln(G),\\phi)\n & := & \\lmsum{p \\ge 0}{} b_p^{(2)}(X;\\caln(G),\\phi) \\in [0,\\infty];\n \\\\\n \\chi^{(2)}(X;\\caln(G);\\phi)\n & := & \n \\lmsum{p \\ge 0}{} (-1)^p \\cdot b_p^{(2)}(X;\\caln(G),\\phi)\n \\in \\IR, \\hspace{2mm} \\mbox{if} \\; h^{(2)}(X;\\caln(G),\\phi) < \\infty.\n \\\\\n \\end{eqnarray*}\n We say that $X$ is \\emph{$\\phi$-$L^2$-finite} if $h^{(2)}(X;\\caln(G),\\phi) < \\infty$ holds. If this\n the case, we call the real number $\\chi^{(2)}(X;\\caln(G),\\phi)$ the \n \\emph{$\\phi$-twisted $L^2$-Euler characteristic} of $X$.\n \\end{definition}\n\nNotice that so far we are not requiring that the $G$-$CW$-complex $X$ is free or finite. \n\n\n\n\n\nWe collect the basic properties of this invariant.\n\n\\begin{theorem}[Basic properties of the $\\phi$-twisted $L^2$-Euler characteristic]\n \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic}\n Let $X$ be a $G$-$CW$-complex. Let $\\phi \\colon G \\to \\IZ$ be a group homomorphism.\n \n \\begin{enumerate}[font=\\normalfont]\n\n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic:homotopy_invariance}\n \\emph{$G$-homotopy invariance}\\\\\n Let $X$ and $Y$ be $G$-$CW$-complexes which are $G$-homotopy equivalent.\nThen $X$ is $\\phi$-$L^2$-finite if and only if $Y$ is $\\phi$-$L^2$-finite, and in this case\n we get\n \\[\n \\chi^{(2)}(X;\\caln(G),\\phi) = \\chi^{(2)}(Y;\\caln(G),\\phi);\n \\]\n\n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic:sum_formula}\n \\emph{Sum formula}\\\\\n Consider a $G$-pushout of $G$-$CW$-complexes\n \\[\n \\xymatrix@C1.2cm@R0.5cm{ X_0 \\ar[r] \\ar[d] & X_1 \\ar[d]\n \\\\\n X_2 \\ar[r]& X }\n \\]\n where the upper horizontal arrow is cellular, the left vertical arrow is an inclusion\n of $G$-$CW$-complexes and $X$ has the obvious $G$-$CW$-structure coming from the ones\n on $X_0$, $X_1$ and $X_2$. Suppose that $X_0$, $X_1$ and $X_2$ are $\\phi$-$L^2$-finite.\n Then $X$ is $\\phi$-$L^2$-finite and we get\n \\[\n \\chi^{(2)}(X;\\caln(G),\\phi) = \\chi^{(2)}(X_1;\\caln(G),\\phi) + \\chi^{(2)}(X_2;\\caln(G),\\phi) -\n \\chi^{(2)}(X_0;\\caln(G),\\phi);\n \\]\n\n \n\n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic:induction}\n \\emph{Induction}\\\\\n Let $i \\colon H \\to G$ be the inclusion of a subgroup of $G$. \n Then $X$ is $(\\phi \\circ i)$-$L^2$-finite if and only if $G \\times_HX$ is\n $\\phi$-$L^2$-finite. If this is the case, we get\n \\[\n \\chi^{(2)}(G \\times_H X;\\caln(G),\\phi) = \\chi^{(2)}(X;\\caln(H),\\phi \\circ i);\n \\]\n \n \n \n \n\n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic:restriction}\n \\emph{Restriction}\\\\\n Let $i\\colon H \\to G$ be the inclusion of a subgroup $H$ of $G$ with $[G:H] < \\infty$.\n Let $X$ be a $G$-$CW$-complex. Denote by $i^*X$ the $H$-$CW$-complex obtained from\n $X$ by restriction with $i$. \n Then $i^* X$ is $\\phi \\circ i$-$L^2$-finite if and only if $X$ is $\\phi$-$L^2$-finite, and in\n this case we get\n \\[\n \\chi^{(2)}(i^*X;\\caln(H),\\phi \\circ i) = [G:H]\\cdot \\chi^{(2)}(X;\\caln(G),\\phi);\n \\]\n\n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic:scaling_phi} \\emph{Scaling $\\phi$}\\\\\n We get for every integer $k \\ge 1$ that $X$ is $\\phi$-$L^2$-finite if and only if $X$\n is $(k \\cdot \\phi)$-$L^2$-finite, and in this case we get\n \\[\n \\chi^{(2)}(X;\\caln(G),k \\cdot \\phi) = k\\cdot \\chi^{(2)}(X;\\caln(G),\\phi);\n \\]\n\n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic:trivial_phi} \\emph{Trivial $\\phi$}\\\\\n Suppose that $\\phi$ is trivial. Then $X$ is $\\phi$-$L^2$-finite if and only if we have \n $b_n^{(2)}(X;\\caln(G)) = 0$ for all $n \\ge 0$. If this is the case, then\n \\[\n \\chi^{(2)}(X;\\caln(G),\\phi) = 0.\n \\]\n\n \n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:homotopy_invariance}\n This follows from the homotopy invariance of $L^2$-Betti numbers.\n \\\\[1mm]~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:sum_formula} The\n proof is analogous to the one of~\\cite[Theorem~6.80~(2) on page~277]{Lueck(2002)}, just\n replace the short split exact sequence of $\\IZ G$-chain complexes $0 \\to C_*(X_0) \\to\n C_*(X_1) \\oplus C_*(X_2) \\to C_*(X) \\to 0$ by the induced short split exact\n sequence of $\\IZ G$-chain complexes\n \\begin{multline*}\n 0 \\to C_*(X_0) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ] \n \\to C_*(X_1) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ] \\oplus C_*(X_2) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ] \n \\\\\n \\to C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ] \\to 0.\n \\end{multline*}\n \\\\[1mm]~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:induction} The\n proof is analogous to the one of~\\cite[Theorem~6.80~(8) on page~279]{Lueck(2002)} using\n the isomorphism of $\\IZ G$-chain complexes\n \\begin{multline*}\n C_*(G \\times_H X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ] \\cong \\bigl(\\IZ G \\otimes_{\\IZ H} C_*(X)\\bigr) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ])\n \\\\\n \\cong \\IZ G \\otimes_{\\IZ H}\\bigl(C_*(X) \\otimes_{\\IZ} (\\phi \\circ i)^*\\IZ[\\IZ]\\bigr),\n \\end{multline*}\n where the second isomorphism is given by $(g \\otimes u) \\otimes v \\mapsto g \\otimes u \\otimes g^{-1}v$.\n \\\\[1mm]~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:restriction} The\n proof is analogous to the one of~\\cite[Theorem~6.80~(7) on page~279]{Lueck(2002)} using\n the obvious identification of $\\IZ H$-chain complexes $i^*\\bigl(C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ]\\bigr)\n = C_*(i^*X) \\otimes_{\\IZ} i^*\\phi^*\\IZ[\\IZ]$.\n \\\\[1mm]~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:scaling_phi}\n Since there is an obvious isomorphism of $\\IZ G$-modules\n $(k \\cdot \\phi)^* \\IZ[\\IZ] \\cong \\bigoplus_{i = 1}^k \\phi^* \\IZ[\\IZ]$,\n we get\n\\begin{multline*}\nb_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} (C_*(X) \\otimes_{\\IZ} (k \\cdot \\phi)^*\\IZ[\\IZ]\\bigr)\n\\\\\n= k \\cdot b_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} (C_*(X) \\otimes_{\\IZ} \\phi^* \\IZ[\\IZ])\\bigr).\n\\end{multline*}\n\\\\[1mm]~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:trivial_phi} \nSince the triviality of $\\phi$ implies\nthat $C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ]$ is $\\IZ G$-isomorphic to $\\bigoplus_{\\IZ} C_*(X)$, we get\n\\begin{eqnarray*}\nb_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} (C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ]\\bigr)\n = \n\\begin{cases}\n0 & \\text{if} \\; b_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} C_*(X)\\bigr) = 0;\n\\\\\n\\infty & \\text{otherwise.}\n\\end{cases}\n\\end{eqnarray*}\nThis finishes the proof of Theorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic}.\n \\end{proof}\n\n\n \nWe can interpret the $\\phi$-twisted $L^2$-Euler characteristic also as an $L^2$-Euler characteristic \nfor surjective $\\phi$ as follows.\n\n\\begin{lemma} \\label{lem:phi-twisted_as_ordinary_L2_Euler_characteristic}\n Let $X$ be a $G$-$CW$-complex. Let $\\phi \\colon G \\to \\IZ$ be a surjective group homomorphism.\n Denote by $K$ the kernel of $\\phi$ and by $i \\colon K \\to G$ the inclusion. \n \n Then $X$ is $\\phi$-$L^2$-finite if and only if $h_n^{(2)}(i^*X;\\caln(K)) < \\infty$\n hold. If this is the case, then\n \\[\n \\chi^{(2)}(X;\\caln(G),\\phi)\\,\\, =\\,\\, \\chi^{(2)}(i^*X;\\caln(K)).\n \\]\n\n \n\\end{lemma}\n\\begin{proof} \n\nWe have the isomorphism of $\\IZ G$-chain complexes\n\\[\n\\IZ G \\otimes_{\\IZ K} i^*C_*(X) \\xrightarrow{\\cong} C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ],\n\\quad g \\otimes x \\mapsto gx \\otimes \\phi(g).\n\\]\nThe inverse sends $y\\otimes q$ to $g \\otimes g^{-1} y$ for any choice of $g \\in\n\\phi^{-1}(q)$. Since $\\caln(G)$ is flat as an $\\caln(K)$-module\nby~\\cite[Theorem~6.29~(1) on page~253]{Lueck(2002)}, we obtain a sequence of\nobvious isomorphisms of $\\caln(G)$-modules\n\\[\n\\begin{array}{ll}\n\\phantom{\\cong }\\caln(G) \\otimes_{\\caln(K)} H_n(\\caln(K) \\otimes_{\\IZ K} C_*(i^*X))\n\\hspace{-0.3cm}& \\cong\n\\caln(G) \\otimes_{\\caln(K)} H_n(\\caln(K) \\otimes_{\\IZ K} i^*C_*(X))\n\\\\\n \\cong \nH_n\\bigl(\\caln(G) \\otimes_{\\caln(K)} \\caln(K) \\otimes_{\\IZ K} i^*C_*(X)\\bigr)\n\n\\hspace{-0.3cm}& \\cong\nH_n\\bigl(\\caln(G) \\otimes_{\\IZ K} i^*C_*(X)\\bigr)\n\\\\\n \\cong \nH_n\\bigl(\\caln(G) \\otimes_{\\IZ G} \\IZ G \\otimes_{\\IZ K} i^*C_*(X))\\bigr)\n\n\\hspace{-0.3cm}& \\cong \nH_n\\bigl(\\caln(G) \\otimes_{\\IZ G} (C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ])\\bigr).\n\\end{array}\\]\nSince $\\dim_{\\caln(G)}(\\caln(G) \\otimes_{\\caln(K)} M) = \\dim_{\\caln(K)}(M)$ holds for\nevery $\\caln(K)$-module $M$ by~\\cite[Theorem~6.29~(2) on page~253]{Lueck(2002)}, we\nconclude for every $n \\ge 0$\n\\[\nb_n^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} C_*(i^*X);\\caln(K)\\bigr)\n=\nb_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} (C_*(X) \\otimes_{\\IZ} \\phi^*\\IZ[\\IZ]);\\caln(G)\\bigr).\n\\]\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{The $(\\mu,\\phi)$-$L^2$-Euler characteristic}\n\\label{subsec:The_(mu,phi)-L2-Euler_characteristic}\n\n We will be interested in this paper mainly in the case, where the $G$-$CW$-complex $X$ is\nfree. If we put $Y = X\/G$, then $X$ is the disjoint union of the preimages of the\ncomponents of $Y$. Therefore it suffices to study a connected $CW$-complex $Y$ and\n$G$-coverings $\\overline{Y} \\to Y$. Any such $G$-covering is obtained from the universal\ncovering $\\widetilde{Y} \\to Y$ and a group homomorphism $\\mu \\colon \\pi = \\pi_1(Y) \\to G$\nas the projection $G \\times_{\\mu} \\widetilde{Y} \\to Y$. Therefore we introduce the\nfollowing notation:\n\n \\begin{definition}[$(\\mu,\\phi)$-$L^2$-Euler characteristic]\n \\label{def:mu-phi-L2-Euler_characteristic}\n Let $X$ be a connected $CW$-complex. Let $\\mu \\colon \\pi_1(X) \\to G$ and $\\phi \\colon G \\to \\IZ$ be a group homomorphisms.\n Let $\\overline{X} \\to X$ be the $G$-covering associated to $\\mu$. We call $X$ \\emph{$(\\mu,\\phi)$-$L^2$-finite}\n if $\\overline{X}$ is $\\phi$-$L^2$-finite, and in this case we define the \\emph{$(\\mu,\\phi)$-$L^2$-Euler characteristic}\n $\\chi^{(2)}(X;\\mu,\\phi)$ to be $\\chi^{(2)}(\\overline{X};\\caln(G);\\phi)$, see \n Definition~\\ref{def:phi-twisted_L2_betti_number_and_L2-Euler_characteristic}.\n \\end{definition}\n\n\n The next lemma essentially reduces the general case $(\\mu,\\phi)$ to the special case,\n where $\\mu$ and $\\phi$ are surjective or $\\phi \\circ \\mu$ is trivial.\n\n\n \\begin{lemma} \\label{lem:reduction_to_surjective_mu} Let $X$ be a connected\n $CW$-complex. Let $\\mu \\colon \\pi_1(X) \\to G$ and $\\phi \\colon G \\to \\IZ$ be group\n homomorphisms. Let $G'$ be the image of $\\mu$. Let $\\mu' \\colon \\pi_1(X) \\to G'$ be the\n epimorphism induced by $\\mu$ and let $\\phi' \\colon G' \\to \\IZ$ be obtained by\n restricting $\\phi$ to $G'$.\n\n \\begin{enumerate}[font=\\normalfont]\n\n \\item \\label{lem:reduction_to_surjective_mu:mu'} \n Then $X$ is $(\\mu,\\phi)$-$L^2$-finite if and only if $X$ is $(\\mu',\\phi')$-$L^2$-finite.\n If this is the case, we get \n \\[\\chi^{(2)}(X;\\mu,\\phi)\\,\\, =\\,\\, \\chi^{(2)}(X;\\mu',\\phi');\\]\n\n\n \\item \\label{lem:reduction_to_surjective_mu:mu_circ_phi_is_not_trivial} \n\n Suppose that $\\mu \\circ \\phi \\not= 0$. Let\n $k\\ge 1 $ be the natural number such that the image of $\\phi'$ is $k \\cdot \\IZ$ and let $\\phi''\n \\colon G' \\to \\IZ$ be the epimorphism uniquely determined by $k \\cdot \\phi'' = \\phi'$. \n Then $X$ is $(\\mu,\\phi)$-$L^2$-finite, if and only if $X$ is\n $(\\mu',\\phi'')$-$L^2$-finite. If this is the case, we get\n \\[\n \\chi^{(2)}(X;\\mu,\\phi) \\,\\,=\\,\\, \\smfrac{1}{k} \\cdot \\chi^{(2)}(X;\\mu',\\phi'');\n \\]\n \\item \\label{lem:reduction_to_surjective_mu:mu_circ_phi_is_trivial} \n Suppose that $\\phi \\circ \\mu = 0$. Then\n $X$ is $(\\mu,\\phi)$-$L^2$-finite if and only if $b_n^{(2)}(\\overline{X};\\caln(G))$ vanishes \n for the $G$-covering $\\overline{X}$ associated to\n $\\mu$ and every $n \\ge 0$. If this is the case, then\n \\[\\chi^{(2)}(X;\\mu,\\phi)\\,\\, =\\,\\, 0.\n \\]\n \\end{enumerate}\n \\end{lemma}\n \n \\begin{proof} The first statement follows from Theorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic}.\n The second and third statement follow from (1) and Theorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic} (5) and (6).\n\\end{proof}\n\n\n\n\\begin{example}[Mapping torus] \\label{exa_mapping_torus} Let $Y$ be a connected finite\n $CW$-complex and $f \\colon Y \\to Y$ be a self-map. Let $T_f$ be its\n mapping torus. Consider any factorization $\\pi_1(T_f) \\xrightarrow{\\mu} G\n \\xrightarrow{\\phi} \\IZ$ of the epimorphism $\\pi_1(T_f) \\to \\pi_1(S^1) = \\IZ$ induced by\n the obvious projection $T_f \\to S^1$. Then $T_f$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n \\[\n \\chi^{(2)}(T_f;\\mu,\\phi) \\,\\,= \\,\\,\\chi(Y)\n \\]\n by the following argument. Let $\\overline{T_f}$ be the $G$-covering\n associated to $\\mu \\colon \\pi_1(T_f) \\to G$. Let $K$ be the kernel of \n $\\phi$ and $i \\colon K \\to G$ be the inclusion. The\n image of the composite $\\pi_1(Y) \\to \\pi_1(T_f) \\xrightarrow{\\mu} G$ is contained in\n $K$ and we can consider the $K$-covering $\\widehat{Y} \\to Y$ associated to it. The\n $K$-$CW$-complex $i^*\\overline{T_f}$ is $K$-homotopy equivalent to the $K$-$CW$-complex\n $\\widehat{Y}$, see~\\cite[Section~2]{Lueck(1994b)}. Hence we conclude \n $\\chi^{(2)}(T_f;\\mu,\\phi) = \\chi^{(2)}(\\widehat{Y};\\caln(K))$ \n from Lemma~\\ref{lem:phi-twisted_as_ordinary_L2_Euler_characteristic}\n and the $K$-homotopy invariance of $L^2$-Betti numbers. Since $Y$ is a finite \n $CW$-complex, we have $\\chi^{(2)}(\\widehat{Y};\\caln(K)) = \\chi(Y)$. \n\n Notice that the $L^2$-Betti numbers $b_n^{(2)}(\\overline{T_f};\\caln(G))$ are all trivial\n by~\\cite[Theorem~2.1]{Lueck(1994b)} and hence the $L^2$-Euler characteristic\n $\\chi^{(2)}(\\overline{T_f};\\caln(G))$ is trivial. So the passage to the subgroup of\n infinite index $K$ or, equivalently, the twisting with the $\\IZ G$-module $\\phi^*\\IZ[\\IZ]$ \n which is not finitely generated as abelian group,\n ensures that we get an interesting invariant by the\n $(\\mu,\\phi)$-$L^2$-Euler characteristic.\n \\end{example}\n\n \n\n\n\\begin{lemma}\\label{lem:tori} \n Let $T^n$ be the $n$-dimensional torus for $n \\ge 1$. Consider homomorphisms $\\mu \\colon\n \\pi_1(T^n) \\to G$ and $\\phi \\colon G \\to \\IZ$ such that the image of $\\mu$ is infinite.\n \nThen $T^n$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n\\[\n\\chi^{(2)}(T^n;\\mu,\\phi) = \n\\begin{cases} [\\IZ : \\im(\\phi \\circ \\mu)] \n& \n\\text{if}\\; n = 1, \\phi \\circ \\mu \\not = 0;\n\\\\\n0 \n& \n\\text{otherwise.}\n\\end{cases}\n\\]\n\\end{lemma}\n\\begin{proof}\n Because of\n Lemma~\\ref{lem:reduction_to_surjective_mu}~\\eqref{lem:reduction_to_surjective_mu:mu'} we\n can assume without loss of generality that $\\mu$ is surjective. Suppose that $\\phi \\circ \\mu$ is\n non-trivial. Because of\n Lemma~\\ref{lem:reduction_to_surjective_mu}~\\eqref{lem:reduction_to_surjective_mu:mu_circ_phi_is_not_trivial}\n it suffices to consider the case, where $\\mu$ and $\\phi$ are surjective.\n Then the claim follows from Example~\\ref{exa_mapping_torus} since there is a\n homeomorphism $h \\colon T^n \\xrightarrow{\\cong} T^{n-1} \\times S^1$ that that the composite of the map\n of $\\pi_1(T^{n-1} \\times S^1) \\to \\pi_1(S^1)$ induced by the projection onto $S^1$\n composed with $\\pi_1(h)$ is $\\phi$. Suppose that $\\phi$ is trivial. Since $\\mu$ has infinite image,\n one can show using~\\cite[Theorem~2.1]{Lueck(1994b)} that \n $b_m^{(2)}(\\overline{T^n};\\caln(G))$ vanishes for the $G$-covering\n $\\overline{T^n} \\to T^n$ associated to $\\mu$ for all $m \\ge 0$. Hence the claim follows from\n Lemma~\\ref{lem:reduction_to_surjective_mu}~\\eqref{lem:reduction_to_surjective_mu:mu_circ_phi_is_trivial}.\n\\end{proof}\n\n\n\n\n\\begin{theorem}[The $(\\mu,\\phi)$-$L^2$-Euler characteristic for $S^1$-$CW$-complexes] \n\\label{the:The_(mu,phi)-L2-Euler_characteristic_for_S1-CW-complexes}\nLet $X$ be a connected finite $S^1$-$CW$-complex. Let $\\mu \\colon \\pi_1(X) \\to G$ \nand $\\phi \\colon G \\to \\IZ$ be group homomorphisms. Suppose that for one and hence all $x \\in X$ the composite\n\\[\n\\eta \\colon \\pi_1(S^1,1) \\xrightarrow{\\pi_1(\\ev_x,1)} \\pi_1(X,x) \\xrightarrow{\\mu} G \\xrightarrow{\\phi} \\IZ\n\\]\nis injective, where $\\ev_x \\colon S^1 \\to X$ sends $z$ to $z \\cdot x$. \nDefine the $S^1$-orbifold Euler characteristic of $X$ by\n\\[\n\\chi^{S^1}_{\\orb}(X)\\,\\,=\\,\\, \\smsum{n \\ge 0}{} (-1)^n \\cdot \\smsum{e \\in I_n}{} \\smfrac{1}{|S^1_e|},\n\\]\nwhere $I_n$ is the set of open $n$-dimensional $S^1$-cells of $X$ and for \n$e \\in I_n $ we denote by $S^1_e$ the isotropy group of any point in $e$. \nThen $X$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n\\[\n\\chi^{(2)}(X;\\mu,\\phi)\\,\\, =\\,\\, \n\\chi^{S^1}_{\\orb}(X) \\cdot [\\IZ : \\im(\\eta)].\n\\]\n\\end{theorem}\n\\begin{proof}\n The strategy of proof is the same as the the one of~\\cite[Theorem~1.40 on page~43]{Lueck(2002)}, \n where one considers more general finite $S^1$-$CW$-complex $Y$ together with a\n $S^1$-map $f \\colon Y \\to X$ and does induction over the number of $S^1$-equivariant cells.\n A basic ingredient is the additivity of the two terms appearing in the desired equation in \n Theorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_S1-CW-complexes}.\n\\end{proof}\n\n \n\nWe mention that the condition about the injectivity of the map $\\eta$ appearing in\n Theorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_S1-CW-complexes} is necessary.\n\n\n\n\n\nFor the reader's convenience we record the next result \nwhich we will not need in this paper and whose proof is a variation of the one of\nTheorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_S1-CW-complexes}.\n\n\\begin{theorem}[The $(\\mu,\\phi)$-$L^2$-Euler characteristic for fibrations]\n \\label{the:(mu,phi)-Euler_characteristic_for_fibrations}\n Let $F \\xrightarrow{i} E \\xrightarrow{p} B$ be a fibration of connected $CW$-complexes. Suppose\n that $B$ is a finite $CW$-complex. Consider group homomorphisms\n $\\mu \\colon \\pi_1(E) \\to G$ and $\\phi \\colon G \\to \\IZ$. Suppose that\n $\\overline{F}$ is $(\\mu \\circ \\pi_1(i),\\phi)$-$L^2$-finite.\n\n Then $\\overline{E}$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n\\[\n\\chi^{(2)}(E;\\mu,\\phi) =\n\\chi(B) \\cdot \\chi^{(2)}(F,\\mu \\circ \\pi_1(i), \\phi). \n\\]\n\\end{theorem}\n\n\nIf $M$ is a compact connected orientable $3$-manifold with proper $S^1$-action, then $M$\nis a compact connected orientable Seifert manifold. The converse is not true\ngeneral. Associated to a compact connected orientable Seifert manifold is an orbifold $X$\nand $X$ has a orbifold Euler characteristic $\\chi_{\\orb}(X)$. For a basic introduction to\nthese notions we refer for instance to~\\cite{Scott(1983)}. If $M$ is a compact\n$3$-manifold with proper $S^1$-action, then $X$ is given by $M\/S^1$ and $\\chi_{\\orb}(X)$\nis the $S^1$-orbifold Euler characteristic $\\chi^{S^1}_{\\orb}(M)$. We omit the proof of\nthe next result since it is essentially a variation of the one of\nTheorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_S1-CW-complexes}, the role of\nthe cells $S^1\/H \\times D^n$ in\nTheorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_S1-CW-complexes} is now played\nby the typical neighborhoods of the Seifert fibers given by solid tori.\n\n\n\\begin{theorem}[The $(\\mu,\\phi)$-$L^2$-Euler characteristic for Seifert manifolds] \n\\label{the:The_(mu,phi)-L2-Euler_characteristic_for_Seifert_manifolds}\nLet $M$ be a compact connected orientable Seifert manifold. Let $\\mu \\colon \\pi_1(M) \\to G$ \nand $\\phi \\colon G \\to \\IZ$ be group homomorphisms. Suppose that for one (and hence all) $x \\in M$\n\\[\n\\eta \\colon \\pi_1(S^1,1) \\xrightarrow{\\pi_1(\\ev,1)} \\pi_1(M,x) \\xrightarrow{\\mu} G \\xrightarrow{\\phi} \\IZ\n\\]\nis injective, where $\\ev \\colon S^1 \\to M$ is the inclusion of a regular fiber.\nLet $X$ be the associated orbifold of $M$ and denote by $\\chi_{\\orb}(X)$ its orbifold Euler characteristic.\nThen $M$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n\\[\n\\chi^{(2)}(M;\\mu,\\phi)\\,\\, = \\,\\, \n\\chi_{\\orb}(X) \\cdot [\\IZ : \\im(\\eta)].\n\\]\n\\end{theorem}\n\n\n\n\\begin{lemma} \\label{lem:JSJ_decom_Euler} Let $M$ be a $3$-manifold,\nwhich is admissible, see Definition~\\ref{def:admissible_3-manifold}. Let\n $M_1$, $M_2$, \\ldots, $M_r$ be its pieces in the Jaco-Shalen-Johannson decomposition. Consider\n group homomorphisms $\\mu \\colon \\pi_1(M) \\to G$ and $\\phi \\colon G \\to \\IZ$. Suppose that\n the composite of $\\mu$ with $\\pi_1(j) \\colon \\pi_1(T^2) \\to \\pi_1(M)$ has infinite image for the inclusion\n $j \\colon T^2 \\to M$ of any splitting torus appearing in the Jaco-Shalen-Johannson decomposition.\n Let $\\mu_i \\colon \\pi_1(M_i) \\to G$ be the composite of $\\mu$ with the map $\\pi_1(M_i) \\to \\pi_1(M)$\n induced by the inclusion $M_i \\to M$. Suppose that $M_i$ is $(\\mu_i,\\phi)$-$L^2$-finite for $i = 1,2, \\ldots , r$. \n\n Then $M$ is $(\\mu,\\phi)$-$L^2$-finite and we have\n\\[\n\\chi^{(2)}(M;\\mu,\\phi)\\,\\,=\\,\\, \\lmsum{i = 1}{r} \\chi^{(2)}\\big(\\widetilde{M_i};\\mu_i,\\phi\\big).\n\\]\n\\end{lemma}\n\\begin{proof} \nThis follows from \nTheorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic}\n\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:sum_formula}\nand Lemma~\\ref{lem:tori}.\n\\end{proof}\n\n\n\\begin{theorem}[The $(\\phi,\\mu)$-$L^2$-Euler characteristic and the Thurston norm for graph manifolds]\n\\label{the:The_(mu,phi-L2-Euler_characteristic_and_the_Thurston_norm_for_graph_manifolds}\nLet $M$ be an admissible $3$-manifold, which is a graph manifold and not homeomorphic to $S^1 \\times D^2$.\nConsider group homomorphisms $\\mu \\colon \\pi_1(M) \\to G$ and $\\phi \\colon G \\to \\IZ$. \nSuppose that for each piece $M_i$ in the\nJaco-Shalen-Johannson decomposition the map $\\pi_1(S^1 ) \\xrightarrow{\\pi_1(\\ev_i)}\n\\pi_1(M_i) \\xrightarrow{j_i} \\pi_1(M) \\xrightarrow{\\mu} G \\xrightarrow{\\phi} \\IZ$ is\ninjective, where $\\ev_i \\colon S^1 \\to M_i$ is the inclusion of the regular fiber \nand $j_i \\colon M_i \\to M$ is the inclusion.\n\nThen $M$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n\\[\n- \\chi^{(2)}(M;\\mu,\\phi) \\,\\,=\\,\\, x_M(\\phi\\circ \\mu).\n\\]\n\\end{theorem}\n\\begin{proof} In the situation and notation of Lemma~\\ref{lem:JSJ_decom_Euler} \nwe conclude from~\\cite[Proposition~3.5 on page~33]{Eisenbud-Neumann(1985)}\n\\[\nx_M(\\phi)\\,\\, =\\,\\, \\lmsum{i = 1}{r} x_{M_i}(\\phi_i).\n\\]\nif $\\phi_i \\in H^1(M_i;\\IZ)$ is the restriction of $\\phi$ to $M_i$.\nMoreover, we get from Lemma~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_Seifert_manifolds}\nand from~\\cite[Lemma~A]{Herrmann(2016)}\nfor $i = 1,2 \\ldots, r$ \n\\[\n\\chi^{(2)}(M_i;\\mu_i,\\phi) \\,\\,=\\,\\, -x_M(\\phi_i),\n\\]\nif $\\mu_i$ is the composite of $\\mu$ with the homomorphism $\\pi_1(M_i) \\to \\pi_1(M)$ induced\nby the inclusion. \nNow the claim follows from Lemma~\\ref{lem:JSJ_decom_Euler}\nsince any splitting torus appearing in the Jaco-Shalen-Johannson decomposition contains a regular fiber\nof one of the pieces $M_i$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{The $\\phi$-$L^2$-Euler characteristic for universal coverings}\n\\label{subsec:The_phi-L2-Euler_characteristic_for_universal_coverings}\n\n\n\n\nIn this section we consider the special case of the universal covering and of a group\nhomomorphism $\\phi \\colon \\pi_1(X) \\to \\IZ$. This is in some sense the most canonical and\nimportant covering and in this case the formulations of the main results simplify in a\nconvenient way.\n\n\\begin{definition}[The $\\phi$-$L^2$-Euler characteristic for $\\phi \\in H^1(X;\\IZ)$]\n\\label{def:The_phi-L2-Euler_characteristic_for_phi_in_H1(X;Z)}\n \n Let $X$ be a connected $CW$-complex with fundamental group $\\pi$.\n Let $\\phi$ be an element in $H^1(X;\\IZ)$, or, equivalently, \n let $\\phi \\colon \\pi \\to \\IZ$ be a group homomorphism. We say that the universal covering\n $\\widetilde{X}$ of $X$ is \\emph{$\\phi$-$L^2$-finite}, if $X$ is $(\\id_{\\pi},\\phi)$-$L^2$-finite \n in the sense of Definition~\\ref{def:mu-phi-L2-Euler_characteristic}.\n If this is the case, we define its \\emph{$\\phi$-$L^2$-Euler characteristic}\n \\[\n \\chi(\\widetilde{X};\\phi) \n \\,\\, := \\,\\,\n \\chi^{(2)}(X;\\id_{\\pi},\\phi)\n \\]\n where $\\chi^{(2)}(X;\\id_{\\pi},\\phi)$ has been introduced in Definition~\\ref{def:mu-phi-L2-Euler_characteristic}.\n \n If $X$ is a (not necessarily connected) finite $CW$-complex and $\\phi \\in H^1(X;\\IZ)$, we say that\n $\\widetilde{X}$ is $\\phi$-$L^2$-finite if for each component $C \\in X$ the universal\n covering $\\widetilde{C} \\to C$ is $\\phi|_C$-$L^2$-finite and we put\n \\[\n \\chi^{(2)}(\\widetilde{X};\\phi) \\,\\,= \\,\\,\\lmsum{C \\in \\pi_0(X)}{} \\chi^{(2)}(\\widetilde{C},\\phi|_C).\n \\]\n\\end{definition}\n\n\n\n\nFor the reader's convenience we record the basic properties of the $\\phi$-$L^2$-Euler characteristic.\n\n\n\n\\begin{theorem}[Basic properties of the $\\phi$-$L^2$-Euler characteristic for universal coverings]\n\\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings}\n\\\n\n\\begin{enumerate}[font=\\normalfont]\n\n\n\\item \\emph{Homotopy invariance}\\\\[1mm]\n\\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:homotopy_invariance}\nLet $f \\colon X \\to Y $ be a homotopy equivalence of $CW$-complexes. Consider $\\phi \\in H^1(Y;\\IZ)$.\nLet $f^*\\phi \\in H^1(X;\\IZ)$ be its pullback with $f$.\n\nThen $\\widetilde{X}$ is $f^*\\phi$-$L^2$-finite if and only if $\\widetilde{Y}$ is $\\phi$-$L^2$-finite, and in this\n case we get\n \\[\n \\chi^{(2)}(\\widetilde{X};f^*\\phi)\\,\\, =\\,\\, \\chi^{(2)}(\\widetilde{Y};\\phi);\n \\]\n\n\\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:sum_formula}\n \\emph{Sum formula}\\\\\n Consider a pushout of $CW$-complexes\n \\[\n \\xymatrix@C1.2cm@R0.5cm{ X_0 \\ar[r] \\ar[d] & X_1 \\ar[d]\n \\\\\n X_2 \\ar[r]& X }\n \\]\n where the upper horizontal arrow is cellular, the left vertical arrow is an inclusion of\n $CW$-complexes and $X$ has the obvious $CW$-structure coming from the ones on $X_0$,\n $X_1$ and $X_2$. Consider $\\phi \\in H^1(X;\\IZ)$. For every $i \\in \\{0,1,2\\}$ suppose\n that for each base point $x_i \\in X_i$ the map $\\pi_1(j_i,x_i) \\colon \\pi_1(X_i,x_i) \\to \\pi_1(X,j_i(x_i))$ \n induced by the inclusion $j_i \\colon X_i \\to X$ is injective and that\n $\\widetilde{X_i}$ is $j_i^*\\phi$-$L^2$-finite.\n\n Then $\\widetilde{X}$ is $\\phi$-$L^2$-finite and we get\n \\[\n \\hspace{1cm} \\chi^{(2)}(\\widetilde{X};\\phi) \\,\\,=\\,\\, \\chi^{(2)}(\\widetilde{X_1};j_1^*\\phi) + \\chi^{(2)}(\\widetilde{X_2};j_2^*\\phi) -\n \\chi^{(2)}(\\widetilde{X_0};j_0^*\\phi);\n \\]\n \n \\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:finite_coverings}\n \\emph{Finite coverings}\\\\\n Let $p \\colon X \\to Y$ be a finite $d$-sheeted covering of connected $CW$-complexes, $\\phi$\nbe an element in $H^1(Y;\\IZ)$ and $p^*\\phi \\in H^1(X;\\IZ)$ be its pullback with $p$.\n\nThen $Y$ is $\\phi$-$L^2$-finite if and only if $X$ is $p^*\\phi$-$L^2$-finite, and in this case\n\\[\n\\chi^{(2)}(\\widetilde{X};p^*\\phi) \\,\\,:=\\,\\, d \\cdot \\chi^{(2)}(\\widetilde{Y};\\phi);\n\\]\n\n\n\\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:scaling_phi}\n \\emph{Scaling $\\phi$}\\\\\nLet $X$ be $CW$-complex and $\\phi$ be an element in $H^1(X;\\IZ)$\nConsider any integer $k \\not= 0$. \nThen $\\widetilde{X}$ is $\\phi$-$L^2$-finite if and only if $\\widetilde{X}$ is $(k \\cdot \\phi)$-$L^2$-finite, \nand in this case we get\n\\[\n\\chi^{(2)}(\\widetilde{X};k \\cdot \\phi)\\,\\, =\\,\\, |k| \\cdot \\chi^{(2)}(\\widetilde{X};\\phi);\n\\]\n\n\n\\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:trivial_phi}\n \\emph{Trivial $\\phi$}\\\\\nLet $X$ be a $CW$-complex. Let $\\phi$ be trivial.\nThen $\\widetilde{X}$ is $\\phi$-$L^2$-finite if and only if $b_n^{(2)}(\\widetilde{X};\\caln(\\pi_1(X)) = 0$\nholds for all $n \\ge 0$. If this is the case, we get\n\\[\n\\chi^{(2)}(\\widetilde{X};\\phi)\\,\\, =\\,\\, 0;\n\\]\n \n\n\n\n\\item \\label{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:tori}\n \\emph{Tori}\\\\\nLet $T^n$ be the $n$-dimensional torus for $n \\ge 1$. Consider any $\\phi \\in H^1(T^n;\\IZ)$. Then\n$\\widetilde{T^n}$ is $\\phi$-$L^2$-finite and we get\n\\[\n\\chi^{(2)}(\\widetilde{T^n};\\phi) \\,\\,=\\,\\, \n\\begin{cases} [\\IZ : \\im(\\phi)] \n& \n\\text{if}\\; n = 1, \\phi \\not = 0;\n\\\\\n0 \n& \n\\text{otherwise.}\n\\end{cases}\n\\]\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n The second statement follows from\n Theorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic}\n \\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:sum_formula}\n and~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic:induction} using the\n fact that for every $i \\in \\{0,1,2\\}$ and every base point $x_i \\in X_i$ the total space\n of the pullback of the universal covering of the component $D$ of $X$ containing\n $j_i(x_i)$ with $j_i|_C \\colon C \\to D$ for $C$ the component of $X_i$ containing $x_i$\n is $\\pi_1(D)$-homeomorphic to $\\pi_1(D) \\times_{\\pi_1(j_i)} \\widetilde{C}$ for the\n universal covering $\\pi_1(C,x_i) \\to \\widetilde{C} \\to C$ of $C$.\n\n The last statement is a special case of Lemma~\\ref{lem:tori}. Finally all other\n statements follow from the corresponding statements\n of~Theorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\begin{lemma} \\label{lem:JSJ_decom_Euler_universal} \n Let $M$ be an admissible $3$-manifold. Let\n $M_1, M_2, \\ldots, M_r$ be its pieces in the Jaco-Shalen-Johannson decomposition. Consider\n $\\phi \\in H^1(M;\\IZ)$. Let $\\phi_i \\in H^1(M_i;\\IZ)$ be the pullback of $\\phi$ with the\n inclusion $M_i \\to M$ for $i = 1,2 , \\ldots,r$.\n\n Then $M_i$ is $\\phi_i$-$L^2$-finite for $i = 1,2, \\ldots ,r$ and $M$ is\n $\\phi$-$L^2$-finite and we have\n\\[\n\\chi^{(2)}(\\widetilde{M};\\phi)\\,\\, = \\,\\,\\lmsum{i = 1}{r} \\chi^{(2)}(\\widetilde{M_i};\\phi_i).\n\\]\n\\end{lemma}\n\\begin{proof} \nIf $M_i$ is Seifert, then it is $\\phi_i$-$L^2$-finite by \nTheorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_for_Seifert_manifolds}.\nIf $M_i$ is hyperbolic, $b_n^{(2)}(\\widetilde{M_i})$ vanishes for all $n \\ge 0$\nby~\\cite[Theorem~0,1]{Lott-Lueck(1995)}, and hence $M_i$ is $\\phi_i$-$L^2$-finite by \nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}\n~\\eqref{the:Status_of_the_Atiyah_Conjecture:3-manifold_not_graph}\nand Theorem~\\ref{the:Atiyah_and_(mu,phi)-L2-Euler_characteristic}. Now the claim follows from\nTheorem~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings}\n\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:sum_formula}\nand~\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:tori}\nusing the fact that the splitting tori in the Jaco-Shalen-Johannson decomposition are incompressible.\n\\end{proof}\n\n\\begin{theorem}[The $\\phi$-$L^2$-Euler characteristic and the Thurston norm for graph manifolds]\n\\label{the:The_phi-L2-Euler_characteristic_and_the_Thurston_norm_for_graph_manifolds}\nLet $M$ be an admissible $3$-manifold, which is a graph manifold and not homeomorphic to $S^1 \\times D^2$. \nConsider $\\phi \\in H^1(M;\\IZ)$. Then $\\widetilde{M}$ is $\\phi$-$L^2$-finite and we get\n\\[\n- \\chi^{(2)}(\\widetilde{M};\\phi) \\,\\,=\\,\\, x_M(\\phi).\n\\]\n\\end{theorem}\n\\begin{proof} This follows from \nTheorem~\\ref{the:The_(mu,phi-L2-Euler_characteristic_and_the_Thurston_norm_for_graph_manifolds}.\n\\end{proof}\n\n\n\\begin{example}[$S^1\\times D^2$ and $S^1 \\times S^2$]\n\\label{exa:S1_timesD2_and_S1_times_S2}\nConsider a homomorphism $\\phi \\colon H_1(S^1 \\times D^2) \\xrightarrow{\\cong} \\IZ$.\nLet $k$ be the index $[\\IZ : \\im(\\phi)]$ if $\\phi$ is non-trivial, and let $k = 0$ if $\\phi$ is trivial.\nThen we conclude from~\\eqref{scaling_Thurston_norm},~\\eqref{fiber_bundles_Thurston_norm},\n Lemma~\\ref{lem:reduction_to_surjective_mu} and Example~\\ref{exa_mapping_torus}\n\\[\nx_{S^1 \\times D^2}(\\phi) \\,\\, = \\,\\, 0\\quad \\mbox{ and }\\quad \n-\\chi^{(2)}(\\widetilde{S^1 \\times D^2};\\phi) \n\\,\\, = \\,\\, k.\\]\nHence we have to exclude $S^1 \\times D^2$ in \nTheorem~\\ref{the:The_phi-L2-Euler_characteristic_and_the_Thurston_norm_for_graph_manifolds} and thus also in Theorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_universal_covering_for_universal_coverings}.\nAnalogously we get \n\\[\nx_{S^1 \\times S^2}(\\phi) \\,\\, = \\,\\, 0\\quad \\mbox{ and } \\quad\n-\\chi^{(2)}(\\widetilde{S^1 \\times S^2};\\phi) \n\\,\\, = \\,\\, 2 \\cdot k,\\]\nso that we cannot replace ``irreducible'' by ``prime'' in \nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_universal_covering_for_universal_coverings}.\n\\end{example}\n\n\n\n\n\n \\typeout{-------------------------- Section 3: About the Atiyah Conjecture -----------------------}\n\n\n\n\n\n\\section{About the Atiyah Conjecture}\n\\label{sec:About_the_Atiyah_Conjecture}\n\nSo far the definition and the analysis of the $\\phi$-twisted $L^2$-Euler characteristic\nhas been performed on an abstract level. In order to ensure that the condition\n$(\\mu,\\phi)$-$L^2$-finite is satisfied and that the $(\\mu,\\phi)$-$L^2$-Euler\ncharacteristic contains interesting information, we will need further input, namely, the\nfollowing Atiyah Conjecture.\n\n\n\n\\subsection{The Atiyah Conjecture}\n\\label{subsec:The_Atiyah_Conjecture}\n\n\n\\begin{definition}[Atiyah Conjecture]\n\\label{def:Atiyah_Conjecture}\nWe say that a torsion-free group $G$ satisfies the \\emph{Atiyah Conjecture} if for any\nmatrix $A \\in M_{m,n}(\\IQ G)$ the von Neumann dimension $\\dim_{\\caln(G)}(\\ker(r_A))$\nof the kernel of the $\\caln(G)$-homomorphism\n$r_A\\colon \\caln(G)^m \\to \\caln(G)^n$ given by right multiplication with $A$ is an integer.\n\\end{definition}\n\nThe Atiyah Conjecture can also be formulated for any field $F$ with $\\IQ \\subseteq F \\subseteq \\IC$\nand matrices $A \\in M_{m,n}(FG)$ and for any group with a bound on the order of its finite subgroups. \nHowever, we only need and therefore consider in this paper the case, where $F = \\IQ$ and $G$ is torsion-free.\n\n\\begin{theorem}[Status of the Atiyah Conjecture]\n\\label{the:Status_of_the_Atiyah_Conjecture}\n\\begin{enumerate}[font=\\normalfont]\n\n\\item \\label{the:Status_of_the_Atiyah_Conjecture:subgroups}\nIf the torsion-free group $G$ satisfies the Atiyah Conjecture,\nthen also each of its subgroups satisfies the Atiyah Conjecture;\n\n\n\n\n\\item \\label{the:Status_of_the_Atiyah_Conjecture:Linnell} Let $\\calc$ be the smallest\n class of groups which contains all free groups and is closed under directed unions and\n extensions with elementary amenable quotients. Suppose that $G$ is a torsion-free group\n which belongs to $\\calc$.\n\nThen $G$ satisfies the Atiyah Conjecture;\n\n\\item \\label{the:Status_of_the_Atiyah_Conjecture:3-manifold_not_graph} Let $G$ be an infinite group which is the\n fundamental group of an admissible $3$-manifold $M$. Suppose that one of the following conditions is satisfied:\n\n \\begin{itemize}\n \\item $M$ is not a closed graph manifold;\n \\item $M$ is a closed graph manifold which admits a Riemannian metric of non-positive sectional curvature.\n \\end{itemize}\n \n Then $G$ is torsion-free and belongs to $\\calc$. In particular $G$ satisfies the Atiyah Conjecture.\n\n\n\n\\item \\label{the:Status_of_the_Atiyah_Conjecture:approx}\nLet $\\cald$ be the smallest class of groups such that\n\\begin{itemize}\n\n\\item The trivial group belongs to $\\cald$;\n\n\\item If $p\\colon G \\to A$ is an epimorphism of a torsion-free group $G$ onto an\nelementary amenable group $A$ and if $p^{-1}(B) \\in \\cald$ for every finite group\n$B \\subset A$, then $G \\in \\cald$;\n\n\\item $\\cald$ is closed under taking subgroups;\n\n\\item $\\cald$ is closed under colimits and inverse limits over directed systems.\n\n\\end{itemize}\n\nIf the group $G$ belongs to $\\cald$,\nthen $G$ is torsion-free and the Atiyah Conjecture holds for $G$.\n\nThe class $\\cald$ is closed under direct sums, direct products and free\nproducts. Every residually torsion-free elementary amenable group\nbelongs to $\\cald$;\n\n\n\n\\end{enumerate}\n\n\\end{theorem}\n\\begin{proof}~\\eqref{the:Status_of_the_Atiyah_Conjecture:subgroups} \n This follows from~\\cite[Theorem~6.29~(2) on page~253]{Lueck(2002)}.\n \\\\[1mm]~\\eqref{the:Status_of_the_Atiyah_Conjecture:Linnell} This is due to Linnell,\n see for instance~\\cite{Linnell(1993)} or~\\cite[Theorem~10.19 on page~378]{Lueck(2002)}.\n \\\\[1mm]~\\eqref{the:Status_of_the_Atiyah_Conjecture:3-manifold_not_graph} It suffices to\n show that $G = \\pi_1(M)$ belongs to the class $\\calc$ appearing in\n assertion~\\eqref{the:Status_of_the_Atiyah_Conjecture:Linnell}. By the proof of the Virtual Fibering Theorem due to Agol, Liu, Przytycki-Wise, and \n Wise~\\cite{Agol(2008),Agol(2013),Liu(2013),Przytycki-Wise(2012), Przytycki-Wise(2014),Wise(2012raggs),Wise(2012hierachy)}\n there exists a finite normal covering $p \\colon \\overline{M} \\to M$ and a fiber\n bundle $F \\to \\overline{M} \\to S^1$ for some compact connected orientable surface $F$. Hence it\n suffices to show that $\\pi_1(F)$ belongs to $\\calc$. If $F$ has non-empty boundary,\n this follows from the fact that $\\pi_1(F)$ is free. If $M$ is closed, the commutator\n subgroup of $\\pi_1(F)$ is free and hence $\\pi_1(F)$ belongs to $\\calc$. Now\n assertion~\\eqref{the:Status_of_the_Atiyah_Conjecture:3-manifold_not_graph} follows from\n assertion~\\eqref{the:Status_of_the_Atiyah_Conjecture:Linnell}.\n \\\\[1mm]~\\eqref{the:Status_of_the_Atiyah_Conjecture:approx} This result is due to Schick,\n see for instance~\\cite{Schick(2001b)} or~\\cite[Theorem~10.22 on page~379]{Lueck(2002)}.\n\\end{proof}\n\n\n\n\\subsection{$L^2$-acyclic Atiyah pair}\n\\label{subsec:L2-acyclic_Atiyah_pair}\n\n\n\\begin{definition}[$L^2$-acyclic Atiyah-pair] \n \\label{def:L2-acyclic_Atiyah-pair} \n An \\emph{$L^2$-acyclic Atiyah-pair} $(\\mu,\\phi)$ for a finite connected $CW$-complex $X$\n consists of group homomorphisms $\\mu \\colon \\pi_1(X) \\to G$ and\n $\\phi \\colon G \\to \\IZ$ such that the $G$-covering $\\overline{X} \\to X$ associated to\n $\\mu$ is $L^2$-acyclic, i.e., the $n$th $L^2$-Betti number\n $b_n^{(2)}(\\overline{X};\\caln(G))$ vanish for every $n \\ge 0$, and $G$ is torsion-free and satisfies the\n Atiyah Conjecture.\n\\end{definition}\n\n\n \nNotice that the conditions appearing in Definition~\\ref{def:L2-acyclic_Atiyah-pair} only\nconcern $G$ and $\\mu$ but not $\\phi$. The Atiyah Conjecture enters in this paper because\nof the following theorem.\n\n\\begin{theorem}[The Atiyah Conjecture and the $(\\mu,\\phi)$-$L^2$-Euler characteristic]\n\\label{the:Atiyah_and_(mu,phi)-L2-Euler_characteristic}\nLet $X$ be a connected finite $CW$-complex. \nSuppose that $(\\mu,\\phi)$ is an $L^2$-acyclic Atiyah-pair.\nThen $X$ is $(\\mu,\\phi)$-$L^2$-finite, and the $(\\mu,\\phi)$-$L^2$-Euler characteristic\n$\\chi^{(2)}(X;\\mu,\\phi)$ is an integer.\n\\end{theorem}\n\nTheorem~\\ref{the:Atiyah_and_(mu,phi)-L2-Euler_characteristic} will be a direct consequence\nof Lemma~\\ref{lem:phi-twisted_as_ordinary_L2_Euler_characteristic}, \nLemma~\\ref{lem:reduction_to_surjective_mu} and the following \nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):chain},\nwhose formulation requires some preparation.\n\n\n\\subsection{The division closure $\\cald(G)$ of $\\IQ G$ in $\\calu(G)$}\n\\label{subsec:The_division_closure_cald(G)_of_FG_in_calu(G)}\n\n\n\n\nLet $S$ be a ring with subring $R \\subset S$. The \\emph{division closure} $\\cald(R \\subset S)$ \nis the smallest subring of $S$ which contains $R$ and is division closed, i.e., every\nelement in $\\cald(R \\subset S)$ which is a unit in $S$ is already a unit in $\\cald(R \\subset S)$.\n\n\n\\begin{notation}[$\\cald(G)$] \\label{not:T(Sigma_subset_R)} Let $G$ be a group.\n Denote by $\\calu(G)$ the algebra of operators\n affiliated to the (complex) group von Neumann algebra $\\caln(G)$,\n see~\\cite[Section~8.2]{Lueck(2002)}. (This is the Ore localization of $\\caln(G)$ with\n respect to the set of non-zero-divisors of $\\caln(G)$, see~\\cite[Theorem~8.22 on\n page~327]{Lueck(2002)}.) Denote by $\\cald(G)$ the division closure of $\\IQ G$ considered\n as a subring of $\\calu(G)$. \n\\end{notation}\n\nThe proof of Theorem~\\ref{the:Main_properties_of_cald(G)} will be based on ideas of Peter\nLinnell from~\\cite{Linnell(1993)} which have been explained in detail and a little bit extended\nin~\\cite[Chapter~10]{Lueck(2002)} and~\\cite{Reich(2006)}. \n\n\\begin{theorem}[Main properties of $\\cald(G)$]\\label{the:Main_properties_of_cald(G)}\nLet $G$ be a torsion-free group.\n\n\\begin{enumerate}[font=\\normalfont]\n\n\\item \\label{the:Main_properties_of_cald(G):skew_field}\nThe group $G$ satisfies the Atiyah Conjecture if and only if\n$\\cald(G)$ is a skew field;\n\n\n\\item \\label{the:Main_properties_of_cald(G):dim}\nSuppose that $G$ satisfies the Atiyah Conjecture.\nLet $C_*$ be a projective $\\IQ G$-chain complex.\nThen we get for all $n \\ge 0$\n\\[\nb_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IQ G} C_*\\bigr) = \\dim_{\\cald(G)}\\bigl(H_n(\\cald(G) \\otimes_{\\IQ G} C_*)\\bigr).\n\\]\nIn particular $b_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IQ G} C_*\\bigr)$ is either infinite or an integer;\n\n\\item \\label{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)} \n Suppose that $G$ satisfies the Atiyah Conjecture.\n Let $\\phi \\colon G \\to \\IZ$ be a surjective group\n homomorphism. Let $K \\subseteq G$ be the kernel of $\\phi$. Fix an element $\\gamma \\in G$\n with $\\phi(\\gamma) = 1$. If we define the semi-direct product $K \\rtimes \\IZ$ with\n respect to the conjugation automorphism $c_{\\gamma} \\colon K \\to K$ of $\\gamma$ on $K$,\n we can identify $G$ with $K \\rtimes \\IZ$ and $\\phi$ becomes the\n canonical projection $G = K \\rtimes \\IZ \\to \\IZ$. Let $\\cald(K)_{t}[u^{\\pm 1}]$\n be the ring of twisted Laurent polynomials with respect to the automorphism\n $t \\colon \\cald(K) \\xrightarrow{\\cong} \\cald(K)$ coming\n from $c_{\\gamma} \\colon K \\to K$. \n\n Then $\\cald(K_{t}[u^{\\pm 1}]$ is a non-commutative principal ideal domain, i.e.,\n it has no non-trivial zero-divisor and every left ideal is a principal left ideal and\n every right ideal is a principal right ideal. Furthermore the set $T$ of non-zero elements in\n $\\cald(K)_{t}[u^{\\pm 1}]$ satisfies the Ore condition and there is a canonical\n isomorphism of skew fields\n \\[\n T^{-1}\\cald(K)_{t}[u^{\\pm 1}] \\xrightarrow{\\cong} \\cald(G);\n \\]\n\n \n\n\\item \\label{the:Main_properties_of_cald(G):chain} \n Let $G$ be a torsion-free group which satisfies the\n Atiyah Conjecture. Let $\\phi \\colon G \\to \\IZ$ be a surjective group\n homomorphism. Denote by $i \\colon K \\to G$ the inclusion of the kernel $K$ of $\\phi$.\n Let $C_*$ be a finitely generated projective $\\IQ G$-chain complex such that \n $b_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*)$ vanishes for all $n \\ge 0$. Denote by $i^*C_*$ \n the restriction of the $\\IQ G$-chain complex $C_*$ to a $\\IQ K$-chain complex. \n\n \n\n Then $H_n(\\cald(K) \\otimes_{\\IQ K} i^*C_*)$ and $(H_n(\\cald(K) _{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*)$\n are finitely generated free as $\\cald(K)$-modules, $b_n^{(2)}\\bigl(\\caln(K) \\otimes_{\\IQ K} i^*C_*\\bigr)$ is finite,\n and we have\n\\begin{eqnarray*}\nb_n^{(2)}\\bigl(\\caln(K) \\otimes_{\\IQ K} i^*C_*\\bigr) \n& = &\n\\dim_{\\cald(K)}\\bigl(H_n(\\cald(K) _{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*)\\bigr).\n\\\\\n& = & \n\\dim_{\\cald(K)}\\bigl(H_n(\\cald(K) \\otimes_{\\IQ K} i^*C_*)\\bigr)\n\\end{eqnarray*}\nfor all $n \\ge 0$.\n\n \n\n\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}~\\eqref{the:Main_properties_of_cald(G):skew_field} This is proved in the\n case $F = \\IC$ in~\\cite[Lemma~10.39 on page~388]{Lueck(2002)}. The proof goes through\n for an arbitrary field $F$ with $\\IQ \\subseteq F \\subseteq \\IC$ without\n modifications. \n \\\\[1mm]~\\eqref{the:Main_properties_of_cald(G):dim} \n We have the following commutative diagram of inclusion of rings\n\\[\n\\xymatrix@C1.2cm@R0.5cm{\\IQ G \\ar[r] \\ar[d]\n& \\caln(G) \\ar[d]\n\\\\\n\\cald(G) \\ar[r] \n&\n\\calu(G).\n}\n\\]\nThere is a dimension function $\\dim_{\\calu{G}}$ for arbitrary (algebraic) $\\calu(G)$-modules\nsuch that for any $\\caln(G)$-module $M$ we have \n$\\dim_{\\calu{G}}(\\calu(G) \\otimes_{\\caln(G)} M) = \\dim_{\\caln(G)}(M)$ \nand basic features like additivity and continuity and cofinality are still satisfied, \nsee~\\cite[Theorem~8.29 on page~330]{Lueck(2002)}. Moreover, $\\calu(G)$ is flat over $\\caln(G)$,\nsee~\\cite[Theorem~8.22~(2) on page~327]{Lueck(2002)}. \nSince $\\cald(G)$ is a skew field by assertion~\\eqref{the:Main_properties_of_cald(G):skew_field}, \n$\\calu(G)$ is also flat as a $\\cald(G)$-module\nand we have for any $\\cald(G)$-module $M$ the equality\n$\\dim_{\\calu(G)}(\\calu(G) \\otimes_{\\cald(G)} M) = \\dim_{\\cald(G)}(M)$. We conclude\n\\begin{eqnarray*}\nb_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*) \n& = & \n\\dim_{\\caln(G)} \\bigl(H_n(\\caln(G) \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = & \n\\dim_{\\calu(G)} \\bigl(\\calu(G) \\otimes_{\\caln(G)} H_n(\\caln(G) \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = & \n\\dim_{\\calu(G)} \\bigl(H_n(\\calu(G) \\otimes_{\\caln(G)} \\caln(G) \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = & \n\\dim_{\\calu(G)} \\bigl(H_n(\\calu(G) \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = & \n\\dim_{\\calu(G)} \\bigl(H_n(\\calu(G) \\otimes_{\\cald(G)} \\cald(G) \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = & \n\\dim_{\\calu(G)} \\bigl(\\calu(G) \\otimes_{\\cald(G)} H_n(\\cald(G) \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = & \n\\dim_{\\cald(G)} \\bigl(H_n(\\cald(G) \\otimes_{\\IQ G} C_*)\\bigr).\n\\end{eqnarray*}\nThis finishes the proof of assertion~\\eqref{the:Main_properties_of_cald(G):dim}.\n\\\\[1mm]~\\eqref{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)} Since $G\n= K \\rtimes \\IZ$ satisfies the Atiyah Conjecture by assumption, the\nsame is true for $K$ by\nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:subgroups}.\nWe know already from assertion~\\eqref{the:Main_properties_of_cald(G):skew_field} that\n$\\cald(K)$ and $\\cald(G)$ are skew fields. The ring $\\cald(K)_{t}[u^{\\pm 1}]$ is\na non-commutative principal ideal domain, see~\\cite[2.1.1 on page~49]{Cohn(1995)}\nor~\\cite[Proposition~4.5]{Cochran(2004)}. The claim that the Ore localization\n$T^{-1}\\cald(K)_{t}[u^{\\pm 1}]$ exists and is isomorphic to $\\cald(G)$ is proved in\nthe case $F = \\IC$ in~\\cite[Lemma~10.60 on page~399]{Lueck(2002)}. The proof goes through\nfor an arbitrary field $F$ with $\\IQ \\subseteq F \\subseteq \\IC$ without\nmodifications. \n\\\\[1mm]~\\eqref{the:Main_properties_of_cald(G):chain} \nWe write the group ring $\\IQ G$ as the ring $\\IQ K_{t}[u^{\\pm 1}]$ of twisted Laurent\npolynomials with coefficients in $\\IQ K$. We get a commutative diagram of inclusions of rings,\nwhere $\\cald(K)_{t}[u^{\\pm 1}]$ is a (non-commutative) principal ideal domain and\n$\\cald(K)$ and $\\cald(G)$ are skew fields\n\n\\[\\xymatrix@C1.2cm@R0.5cm{\\IQ K \\ar[r] \\ar[d]\n&\n\\IQ G = \\IQ K_{t}[u^{\\pm 1}] \\ar[dd] \\ar[ldd]\n\\\\\n\\cald(K) \\ar[d]\n&\n\\\\\n\\cald(K)_{t}[u^{\\pm 1}] \\ar[r] \\ar[d]\n&\n\\cald(G) \\ar[d]^{\\id}\n\\\\\nT^{-1}\\cald(K)_{t}[u^{\\pm 1}] \\ar[r]^-{\\cong}\n&\n\\cald(G).\n}\n\\]\nSince $C_*$ is a finitely generated projective $\\IQ G$-chain complex by assumption,\nthe $\\cald(K)_{t}[u^{\\pm 1}]$-chain complex $\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{ \\IQ G}C_*$ \nis finitely generated projective. Since $\\cald(K \\rtimes \\IZ)$ is a\n(non-commutative) principal ideal domain, it follows from~\\cite[p.~494]{Cohn(1985)},\nthat there exist integers $r,s \\ge 0$ and non-zero elements $p_1, p_2, \\ldots , p_s \\in \\cald(K)_{t}[u^{\\pm 1}]$ \nsuch that we get an isomorphism of $\\cald(K)_{t}[u^{\\pm 1}]$-modules\n\\[\nH_n\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*\\bigr) \\cong \\cald(K)_{t}[u^{\\pm 1}]^r \\oplus \\bigoplus_{i = 1}^s \n\\cald(K)_{t}[u^{\\pm 1}]\/(p_i).\n\\]\nSince $\\cald(G) = T^{-1}\\cald(K)_{t}[u^{\\pm 1}]$ is flat over\n$\\cald(K)_{t}[u^{\\pm 1}]$, we conclude using\nassertion~\\eqref{the:Main_properties_of_cald(G):dim}\n\\begin{eqnarray*}\nr \n& = &\n\\dim_{\\cald(G)}\\bigl(\\cald(G)\\otimes_{\\cald(K)_{t}[u^{\\pm 1}]} \nH_n(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*)\\bigr)\n\\\\\n& = &\n\\dim_{\\cald(G)}\\bigl(\nH_n\\bigl(\\cald(G) \\otimes_{\\cald(K)_{t}[u^{\\pm 1}]} \\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*\\bigr)\\bigr)\n\\\\\n& = &\n\\dim_{\\cald(G)}\\bigl(\nH_n\\bigl(\\cald(G) \\otimes_{\\IQ G} C_*\\bigr)\\bigr)\n\\\\\n& = & \nb_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*).\n\\end{eqnarray*}\nSince by assumption $b_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*) = 0$ holds, we conclude\n\\[\nH_n(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*)\\,\\, \\cong \\,\\, \\bigoplus_{i = 1}^s \n\\cald(K)_{t}[u^{\\pm 1}]\/(p_i).\n\\]\nLemma~\\ref{lem:degree_and_rank} implies that $\\cald(K)_{t}[u^{\\pm 1}]\/(p_i)$ considered as\n$\\cald(K)$-module is finitely generated free. \nThis implies that $H_n\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*\\bigr)$\nconsidered as $\\cald(K)$-module is finitely generated free.\nAssertion~\\eqref{the:Main_properties_of_cald(G):dim} \napplied to $K$ instead of $G$ implies\n\\[\nb_n^{(2)}\\bigl(\\caln(K) \\otimes_{FK} i^*C_*\\bigr) \n=\n\\dim_{\\cald(K)}\\bigl(H_n\\bigl(\\cald(K) \\otimes_{\\IZ K} i^* C_*\\bigr)\\bigr). \n\\]\nThere is an obvious isomorphism of $\\cald(K)$-chain complexes\n\\[\n\\cald(K) \\otimes_{\\IZ K} i^*C_* \\xrightarrow{\\cong} \\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*,\n\\quad \nx \\otimes_{\\IZ K} y \\mapsto x \\otimes_{\\IZ G} y\n\\]\nwhich induces an isomorphism of $\\cald(K)$-modules\n\\[\nH_n\\bigl(\\cald(K) \\otimes_{\\IQ K} i^*C_*\\bigr) \\xrightarrow{\\cong} H_n\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*\\bigr).\n\\]\nHence we get\n\\[\n\\dim_{\\cald(K)}\\bigl(H_n\\bigl(\\cald(K) \\otimes_{FK} i^*C_*\\bigr)\\bigr) \n\\,\\,=\\,\\,\n\\dim_{\\cald(K)}\\bigl(H_n\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IQ G} C_*\\bigr)\\bigr). \n\\]\nThis finishes the proof of Theorem~\\ref{the:Main_properties_of_cald(G)}\nand hence also of Theorem~\\ref{the:Atiyah_and_(mu,phi)-L2-Euler_characteristic}.\n\\end{proof}\n\n\n\n\n \\typeout{-- Section 4: The $\\phi$-$L^2$-Euler characteristic is a lower bound for the Thurston norm ---------}\n\n\n\\section{The negative of the $(\\mu,\\phi)$-$L^2$-Euler characteristic is a lower bound for the Thurston norm}\n\\label{subsec:The_(mu,phi)-L2-Euler_characteristic_is_a_lower_bound_for_the_Thurston_norm}\n\n\n\n\\begin{theorem}[The negative of the $(\\mu,\\phi)$-$L^2$-Euler characteristic is a lower bound for the Thurston norm]\n\\label{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic}\nLet $M\\ne S^1\\times D^2$ be an admissible $3$-manifold and let $(\\mu,\\phi)$ be an $L^2$-acyclic Atiyah-pair.\nThen $M$ is $(\\mu,\\phi)$-$L^2$-finite and we get\n\\[\n-\\chi^{(2)}(M;\\mu,\\phi) \\,\\,\\le \\,\\,x_M(\\phi \\circ \\mu).\n\\]\n\\end{theorem}\n\nIts proof needs some preparation.\n\n\n\\subsection{Some preliminaries about twisted Laurent polynomials over skew fields}\n\\label{subsec:Some_preliminaries_about_twisted_Laurent_polynomials_over_skew_fields}\n\nIn this subsection we consider a skew field $\\cald$ together with an automorphism \n$t\\colon \\cald \\to \\cald$ of skew fields. Let $\\cald_{t}[u^{\\pm 1}]$ be the ring of twisted Laurent\npolynomials over $\\cald$. For a non-trivial element $x = \\sum_{i \\in \\IZ} d_i \\cdot u^i$\nin $\\cald_{t}[u^{\\pm 1}]$ we define its degree to be the natural number\n \\begin{eqnarray}\n \\deg(x) & := & n_+ - n_-\n \\label{degree_of_a_Laurent_polynomials}\n \\end{eqnarray}\n where $n_-$ the smallest integer such that $d_{n_-}$ does not vanish, and $n_+$ is largest\n integer such that $d_{n_+}$ does not vanish.\n\n \\begin{lemma} \\label{lem:degree_and_rank} \n Consider a non-trivial element $x$ in\n $\\cald_{t}[u^{\\pm 1}]$. Then the $\\cald_{t}[u^{\\pm 1}]$-homomorphism $r_x\n \\colon \\cald_{t}[u^{\\pm 1}] \\to \\cald_{t}[u^{\\pm 1}]$ given by right\n multiplication with $x$ is injective and its cokernel has finite dimension over\n $\\cald$, namely,\n \\[\\dim_{\\cald}(\\coker(r_x)) = \\deg(x).\n \\]\n \\end{lemma}\n \\begin{proof} Notice that for two non-trivial elements $x$ and $y$ we have $n_-(xy) = n_-(x) + n_-(y)$,\n $n_+(xy) = n_+(x) + n_+(y)$, and $\\deg(xy) = \\deg(x) + \\deg(y)$. Now one easily checks that\n $r_x$ is injective and that a $\\cald$-basis for the cokernel of $r_x$ is given by the image of the subset\n $\\{u^0, u^1, \\ldots , u^{\\deg(x)-1}\\}$ of $\\cald_{t}[u^{\\pm 1}]$ under the canonical projection\n $\\cald_{t}[u^{\\pm 1}] \\to \\coker(r_x)$.\n\n\\end{proof}\n\n\n\\begin{lemma} \\label{rank_estimate_for_a_plus_t_I_k} \nConsider integers $k,n$ with $0 \\le k$, $1 \\le n$ and $k \\le n$. Denote by\n $I_k^n$ the $(n,n)$-matrix whose first $k$ entries on the diagonal are $1$ and all of\n whose other entries are zero. Let $A$ be an $(n,n)$-matrix over $\\cald$. Suppose that\n the $\\cald_{t}[u^{\\pm 1}]$-map \n \\[\n r_{A + u \\cdot I_k^n} \\colon\n \\cald_{t}[u^{\\pm 1}]^n \\to \\cald_{t}[u^{\\pm 1}]^n\n \\] \n given by right multiplication with $A + u \\cdot I_k^n$ is injective. \n\n Then the dimension over $\\cald$ of its cokernel is finite and satisfies\n\\[\n\\dim_{\\cald}\\bigl(\\coker\\bigl(r_{A + u \\cdot I_k^n} \\colon \n\\cald_{t}[u^{\\pm 1}]^n \\to \\cald_{t}[u^{\\pm 1}]^n\\bigr)\\bigr) \\,\\,\\le\\,\\, k.\n\\]\n\\end{lemma}\n\\begin{proof}\n We use induction over the size $n$. If $ k = n$, the claim has already been proved\n in~\\cite[Proposition~9.1]{Harvey(2005)}. So we can assume in the sequel $k < n$. We\n perform certain row and column operations on matrices $B \\in\n M_{n,n}(\\cald_{t}[u^{\\pm 1}])$ and it will be obvious that they will respect the\n property that $r_B \\colon \\cald_{t}[u^{\\pm 1}]^n \\to \\cald_{t}[u^{\\pm 1}]^n$ is\n injective and will not change $\\dim_{\\cald}\\bigl(\\coker\\bigl(r_{B} \\colon\n \\cald_{t}[u^{\\pm 1}]^n \\to \\cald_{t}[u^{\\pm 1}]^n\\bigr)\\bigr)$. With the help\n of these operations we will reduce the size of $A$ by $1$ and this will finish the\n induction step. To keep the exposition easy, we explain the induction step from $(n-1)$\n to $n$ in the special case $k = 3$ and $n = 5$. The general induction step is\n completely analogous.\n\n\nSo we start with \n\\[\nA + u \\cdot I_3^5 =\\footnotesize{\\begin{pmatrix}\nu + * & * & * & * & * \n\\\\\n* & u+ * &* & * & *\n\\\\\n* & * & u+* & * & *\n\\\\\n* & * & * & * & *\n\\\\\n* & * & * & * & *\n\\end{pmatrix}}\n\\]\nwhere here and in the sequel $*$ denotes some element in $\\cald$.\nWe first treat the case, where the $(2,2)$-submatrix sitting in the right lower corner is non-trivial.\nBy interchanging rows and columns and right multiplying a row with a non-trivial element in $\\cald$, we can achieve \n\n\\[\n\\footnotesize{\\begin{pmatrix}\nu+* & * & * & * & * \n\\\\\n* & u+* &* & * & *\n\\\\\n* & * & u+* & * & *\n\\\\\n* & * & * & * & *\n\\\\\n* & * & * & * & 1\n\\end{pmatrix}}\n\\]\nBy subtracting appropriate right $\\cald$-multiplies of the lowermost row from the other rows and \nsubtracting appropriate left $\\cald$-multiplies of the rightmost column from the other columns,\nwe achieve\n\\[\n\\footnotesize{\n\\begin{pmatrix}\nu+* & * & * & * & 0\n\\\\\n* & u+* &* & * & 0\n\\\\\n* & * & u+* & * & 0\n\\\\\n* & * & * & * & 0\n\\\\\n0 & 0 & 0 & 0 & 1\n\\end{pmatrix}}\n\\]\nFor this matrix the claim follows from the induction hypothesis applied to the $(4,4)$-matrix\nobtained by deleting the lowermost row and the rightmost column. \n\nIt remains to treat the case, where the matrix looks like\n\\[\n\\footnotesize{\n\\begin{pmatrix}\nu+* & * & * & * & * \n\\\\\n* & u+* &* & * & *\n\\\\\n* & * & u+* & * & *\n\\\\\n* & * & * & 0 & 0\n\\\\\n* & * & * & 0 & 0\n\\end{pmatrix}}\n\\]\nAt least one of the entries in the lowermost row must be non-trivial since the map induced\nby right multiplication with it is assumed to be injective. By interchanging rows and\ncolumns and right multiplying a row with a non-trivial element in $\\cald$, we can achieve\n\\[\n\\footnotesize{\n\\begin{pmatrix}\nu+* & * & * & * & * \n\\\\\n* & u+* &* & * & *\n\\\\\n* & * & u+* & * & *\n\\\\\n* & * & * & 0 & 0\n\\\\\n1 & * & * & 0 & 0\n\\end{pmatrix}}\n\\]\nBy subtracting appropriate $\\cald$-multiples of first column from the second and third column, we\ncan arrange\n\\[\n\\footnotesize{\n\\begin{pmatrix}\nu+* & * \\cdot u+* & *\\cdot u+* & * & * \n\\\\\n* & u+* &* & * & *\n\\\\\n* & * & u+* & * & *\n\\\\\n* & * & * & 0 & 0\n\\\\\n1 & 0 & 0 & 0 & 0\n\\end{pmatrix}}\n\\]\nBy subtracting appropriate right $\\cald$-multiples of last row from the other rows we can achieve\n\\[\n\\footnotesize{\n\\begin{pmatrix}\n0 & * \\cdot u+* & * \\cdot u+* & * & * \n\\\\\n0 & u+* &* & * & *\n\\\\\n0 & * & u+* & * & *\n\\\\\n0 & * & * & 0 & 0\n\\\\\n1 & 0 & 0 & 0 & 0\n\\end{pmatrix}}\n\\]\nBy subtracting right $\\cald$-multiples of the second and the third row from the first row and then interchanging rows\nwe can arrange\n\\[\n\\footnotesize{\n\\begin{pmatrix}\n0 & u+* & * & * & * \n\\\\\n0 & * & u+* & * & *\n\\\\\n0 & * & * & * & *\n\\\\\n0 & * & * & 0 & 0\n\\\\\n1 & 0 & 0 & 0 & 0\n\\end{pmatrix}}\n\\]\nSince the induction hypothesis applies to the matrix obtained by deleting the first column and the lowermost row,\nwe have finished the induction step, and hence the proof of Lemma~\\ref{rank_estimate_for_a_plus_t_I_k}.\n\\end{proof}\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic}}\n\\label{subsec:Proof_of_Theorem_ref_the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic)}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic}]\nWe have already shown in Theorem~\\ref{the:Atiyah_and_(mu,phi)-L2-Euler_characteristic}\nthat $M$ is $(\\mu,\\phi)$-$L^2$-finite.\n\nBecause of~\\eqref{scaling_Thurston_norm} and Lemma~\\ref{lem:reduction_to_surjective_mu} \nwe can assume without loss of generality that $\\phi$ is surjective. \nLet $K$ be the kernel of $\\phi$ and $\\gamma$ be an element in $G$ with $\\phi(\\gamma) = 1$.\n It follows easily\n from~\\cite[Section~1]{Turaev(2002)} that we can find an oriented surface $\\Sigma\\subset M$ with\n components $\\Sigma_1,\\dots,\\Sigma_l$ and non-zero $r_1,\\dots,r_l\\in \\IN$ with the following\n properties: \n \\begin{itemize}\n\\item $r_1[\\Sigma_1]+\\dots+r_l[\\Sigma_l]$ is dual to $\\phi$,\n\\item $\\sum_{i=1}^l -r_i\\chi(\\Sigma_i)\\leq x_M(\\phi)$,\n\\item $M\\setminus \\Sigma$ is connected.\n\\end{itemize}\n(Here we use that $M\\ne S^1\\times D^2$ and that $N$ is irreducible.)\nFor $i=1,\\dots,l$ we pick disjoint oriented tubular neighborhoods $\\Sigma_i\\times [0,1]$\nand we identify $\\Sigma_i$ with $\\Sigma_i\\times \\{0\\}$. We write $M':=M\\setminus\n\\cup_{i=1}^l \\Sigma_i\\times [0,1]$. We pick once and for all a base point $p$ in $M'$ and\nwe denote by $\\widetilde{M}$ the universal cover of $M$. We write $\\pi=\\pi_1(M,p)$. For\n$i=1,\\dots,l$ we also pick a curve $\\nu_i'$ based at $p$ which intersects $\\Sigma_i$\nprecisely once in a positive direction and does not intersect any other component of\n$\\Sigma$. Put $\\nu_i = \\mu(\\nu_i')$. Note that $\\phi(\\nu_i)=r_i$. Finally for\n$i=1,\\dots,l$ we put\n\\[\nn_i :=- \\chi(\\Sigma_i)+2.\n\\]\n\nFollowing~\\cite[Section~4]{Friedl(2014twisted)} we now pick an appropriate $CW$-structure\nfor $M$ and we pick appropriate orientations and lifts of the cells to the universal\ncover. The resulting boundary maps are described in\ndetail~\\cite[Section~4]{Friedl(2014twisted)}. In order to keep the notation manageable we\nnow restrict to the case $l=2$, the general case is completely analogous.\n\nLet $\\overline{M} \\to M$ be the $G$-covering associated to $\\mu$. It follows from the\ndiscussion in~\\cite[Section~4]{Friedl(2014twisted)} and the definitions that the cellular\n$\\IZ G$-chain complex $C_*(\\overline{M})$ of $\\overline{M} $ looks like\n\\[\n\\xymatrix{0 \\ar[r] \n& \\IZ[G]^{4}\\ar[r]^-{B_3} \n& \\IZ[G]^{4+2n_1+2n_2+s}\\ar[r]^-{B_2} \n& \\IZ[G]^{4+2n_1+2n_2+s}\\ar[r]^-{B_1} \n& \\IZ[G]^4\\ar[r] \n& 0}\n\\]\nwhere $s$ is a natural number and the matrices $B_3,B_2,B_1$ are matrices of the form\n\\[\nB_3\n\\,\\, = \\,\\, \n\\footnotesize{\\begin{array}{ccccccc|c}\nn_1 & n_2 & 1 & 1 & 1 & 1 & s+n_1+n_2 &\n\\\\\n\\hline \n * & 0 & 1 & -\\nu_1& 0 & 0 & 0 & 1\n\\\\ \n 0 & 0 & 1 & -z_1 & 0 & 0 & * & 1\n\\\\\n 0 & * & 0 & 0 & 1 & -\\nu_2 & 0 & 1\n\\\\ \n 0 & 0 & 0 & 0 & 1 & -z_2 & * &1\n\\end{array} }\n\\]\n\\[\nB_2\n\\,\\,=\\,\\, \n\\footnotesize{\n\\begin{array}{ccccccccc|c}\n1 &1 & n_1 & n_1 & n_2 & n_2 & 1 & 1 & s\n\\\\\n\\hline\n * & 0 & \\id_{n_1} & -\\nu_1\\id_{n_1} & 0 & 0 & 0 & 0 & 0 & n_1\n\\\\ \n 0 & * & 0 & 0& \\id_{n_2} &-\\nu_2\\id_{n_2} & 0 & 0 & 0 & n_2\n\\\\ \n 0 & 0 & * & 0 & 0 & 0 & 0 & 0 & 0 & 1\n\\\\ \n 0 & 0 & 0 & * & 0 & 0 & 0 & 0 & 0 & 1\n\\\\ \n 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 & 1\n\\\\ \n 0 & 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 1\n\\\\ \n 0 & 0 & * & * & * & * & * & * & * & s+n_1+n_2\n\\end{array} }\n\\]\n\\[\nB_1\\,\\,=\\,\\,\n\\footnotesize{\n\\begin{array}{cccc|c}\n1 & 1 & 1 & 1\n\\\\\n\\hline\n 1& -\\nu_1 & 0 & 0 & 1\n\\\\ \n 0 & 0 & 1 &-\\nu_2 & 1\n\\\\ \n * & 0 & 0 & 0 & n_1\n\\\\\n 0 & * &0 & 0 & n_1\n\\\\\n0 & 0 & * & 0 & n_2\n\\\\\n0 & 0 & 0 & * & n_2\n\\\\\n1 & -x_1 & 0 & 0 & 1 \n\\\\\n0 & 0 & 1 & -x_2 & 1 \n\\\\ \n * & * & * & * & s\n\\end{array} }\n\\]\nwhere $x_1,x_2,z_1,z_2\\in K$, all the entries of the matrices marked by $*$ lie in $\\IZ K$\nand the entries above the horizontal line and right to the vertical arrow indicate the\nsize of the blocks. (Note that in~\\cite{Friedl(2014twisted)} we view the elements of\n$\\IZ[G]^n$ as column vectors whereas we now view them as row vectors.)\n\n\nDefine submatrices $B_i'$ of $B_i$ for $i = 1,2,3$ by \n\\[\n\\begin{array}{rcl} \nB_3'\n& = &\n\\footnotesize{\n\\begin{pmatrix} \n1 & -\\nu_1 & 0 & 0 \n\\\\ \n1 & -z_1 & 0 & 0 \n\\\\ \n0 & 0 & 1 & -\\nu_2 \n\\\\ \n0 & 0 & 1 &-z_2\n\\end{pmatrix}}\n\\\\\n&\n\\\\[-0.3cm]\nB_2'\n& = &\n\\footnotesize{\n\\begin{array}{ccccc|c}\n n_1 & n_1 & n_2 & n_2 & s & \n\\\\\n\\hline\n\\id_{n_1} & -\\nu_1\\id_{n_1} & 0 & 0 & 0 & n_1\n\\\\ \n0 & 0& \\id_{n_2} &-\\nu_2\\id_{n_2} & 0 & n_2\n\\\\ \n* & * & * & * & * & s+n_1+n_2\n\\end{array} }\n\\\\\n&\n\\\\[-0.3cm]\nB_1'\n& = &\n\\footnotesize{\n\\begin{pmatrix} \n1 & -\\nu_1 & 0 & 0 \n\\\\\n0 & 0 & 1 & -\\nu_2 \n\\\\\n1 & -x_1 & 0 & 0 \n\\\\\n0 & 0 & 1 & -x_2 \n\\end{pmatrix}}\n\\end{array}\n\\]\nwhere $B_1'$ and $B_3'$ are $(4,4)$-matrices and $B_2'$ is a $(2n_1+ 2n_2 + s,2n_1+2n_2+ s)$-matrix. \nFor $i = 1,2,3$ let $C_*[i]$ be the $\\IZ G$-chain complex\nconcentrated in dimensions $i$ and $i - 1$ whose $i$-th differential is given by the\nmatrix $B_i'$. There is an appropriate finitely generated free $\\IZ G$-chain complex\n$C_*'$ concentrated in dimensions $2$, $1$ and $0$ of the shape\n\\[\n\\ldots \\to 0 \\to 0 \\to \\IZ G^{2n_1 +2n_2+s} \\to \\IZ G^{4 + 2n_1 +2n_2+s} \\to \\IZ G^4\n\\]\nand obvious short based exact sequences of finitely generated based free $\\IZ G$-chain\ncomplexes\n\\[\n0 \\to C_*' \\to C_*(\\overline{M}) \\to C_*[3] \\to 0\n\\]\nand\n\\[\n0 \\to C_*[1] \\to C_*' \\to C_*[2] \\to 0.\n\\]\nConsider $\\nu' \\in G$ with $\\phi(\\nu) \\not= 0$ and $x' \\in H$. The matrix\n$\\begin{pmatrix}1& - \\nu' \\\\ 1 & x'\\end{pmatrix}$ can be transformed by subtracting the\nfirst row from the second row to the matrix \n$\\begin{pmatrix}1& -\\nu' \\\\ 0 & \\nu' + x'\\end{pmatrix}$. \nThe map $r_{\\nu' + x'} \\colon \\cald(K)_{t}[u^{\\pm 1}] \\to \\cald(K)_{t}[u^{\\pm 1}]$ \nis injective and its cokernel has dimension $|\\phi(\\nu')|$\nover $\\cald(K)$ by Lemma~\\ref{lem:degree_and_rank}.\nHence the map $\\cald(K)_{t}[u^{\\pm 1}]^2 \\to \\cald(K)_{t}[u^{\\pm 1}]^2$\ngiven by right multiplication with $\\begin{pmatrix}1& -\\nu' \\\\ 0 & \\nu' + x'\\end{pmatrix}$\nis injective and its cokernel has dimension $|\\phi(\\nu')|$\nover $\\cald(K)$. We conclude from Theorem~\\ref{the:Main_properties_of_cald(G)} for $l = 1,3$\nusing notation~\\eqref{b_n(2)(caln(G)_otimes_ZGC_ast)} and defining for a skewfield $\\cald$ and \na $\\cald$-chain complex $D_*$  its Betti number $b_n(D_*)$\nto be $\\dim_{\\cald}(H_n(D_*))$.\n\n\\begin{eqnarray*}\nb_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} C_*[l]\\bigr) \n& = & \n0 \\quad \\text{for} \\; n \\ge 0;\n\\\\\nb_n\\bigl(\\cald(K) \\otimes_{\\IZ K} i^*C_*[l]\\bigr) \n& = & \n\\begin{cases}\n0 & \\text{if}\\; n \\not= l-1;\\\\\nr_1 + r_2 & \\text{if}\\; n = l-1;\n\\end{cases}\n\\\\\n\\chi^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} i^*C_*[l]\\bigr) \n& = & \nr_1 + r_2.\n\\end{eqnarray*}\nSince $b_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr) = 0$ holds\nfor all $n \\ge 0$ by assumption, we get\n\\[\nb_n^{(2)}\\bigl(\\caln(G) \\otimes_{\\IZ G} C_*[2]\\bigr)\n = \n0 \\quad \\text{for} \\; n \\ge 0.\n\\]\nWe conclude from from Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):dim}\n\\[\n\\dim_{\\cald(G)}\\left(\\ker\\bigl(r_{B_2'} \\colon \\cald(G)^{1 +2n_1 +2n_2+s} \\to \\cald(G)^{1 +2n_1 +2n_2+s}\\bigr)\\right) = 0.\n\\]\nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)} \nimplies that \n\\[\nr_{B_2'} \\colon \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s} \\to \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s}\n\\]\nis injective. We get using Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):dim}\n\\begin{eqnarray}\n\\label{the:The_Thurston_norm_and_the_(mu,phi)_L2-Euler_characteristic:(3)}\n&&\n\\\\\n&&\\chi^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} i^*C_*(\\overline{M})\\bigr) \\,\\,=\\,\\,\\lmsum{l = 1}{3} \\chi^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} i^*C_*[l]\\bigr)\n\\nonumber\n\\\\\n& = & \n2r_1 + 2r_2 - \\chi^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} i^*C_*[2]\\bigr)\n\\nonumber\n\\\\\n& = & \n2r_1 + 2r_2 - \\chi\\bigl(\\cald(K) \\otimes_{\\IZ K} i^*C_*[2]\\bigr)\n\\nonumber\\\\\n& = &\n2r_1 + 2r_2 \n\\nonumber\\\\\n& & - \\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_{B_2'} \\colon \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s}\\bigr)\\bigr).\n\\nonumber\n\\end{eqnarray}\n\nSince $\\dim_{\\caln(G)}\\bigl(\\ker\\bigl(\\id_{\\caln(G)} \\otimes_{\\IZ G} r_{B_2'}) \\colon \\caln(G)^{1 +2n_1 +2n_2+s}\n\\to \\caln(G)^{1 +2n_1 +2n_2+s}\\bigr)\\bigr)$ coincides with $b_n^{(2)}(\\caln(G) \\otimes_{\\IZ  G} C_*')$ and $b_n^{(2)}(\\caln(G) \\otimes_{\\IZ G} C_*')$ is zero, one\nof the entries in the rightmost column of $B_2'$ is\nnon-trivial. By switching rows and multiplying a row with a non-trivial element in $\\cald(K)$, \nwe can arrange that the entry in the lower right corner is $1$. By elementary row and\ncolumn operations we can arrange that the lowermost row and the rightmost column have all\nentries zero except the element in the lower right corner which is still equal to $1$. By\niterating this process we can transform $B_2'$ by such row and column operations into a\nmatrix of the shape\n\\[\nB_2''\n\\,\\,=\\,\\,\n\\footnotesize{ \n\\begin{array}{ccccc|c}\n n_1 & n_1 & n_2 & n_2 & s & \n\\\\\n\\hline\n\\id_{n_1} & -\\nu_1\\id_{n_1} & 0 & 0 & 0 & n_1\n\\\\ \n0 & 0& \\id_{n_2} &-\\nu_2\\id_{n_2} & 0 & n_2\n\\\\ \n* & * & * & * & * & n_1+n_2\n\\\\ \n0 & 0 & 0 & 0 & \\id_s & s\n\\end{array} }\n\\]\nLet $B_2'''$ be the $(2n_1+ 2n_2)$-submatrix of $B_s''$ given by\n\\[\nB_2'''\n\\,\\, =\\,\\,\n\\footnotesize{ \n\\begin{array}{cccc|c}\n n_1 & n_1 & n_2 & n_2 & \n\\\\\n\\hline\n\\id_{n_1} & -\\nu_1\\id_{n_1} & 0 & 0 & n_1\n\\\\ \n0 & 0& \\id_{n_2} &-\\nu_2\\id_{n_2} & n_2\n\\\\ \n* & * & * & * & n_1+n_2\n\\end{array}}\n\\]\nAn inspection of the proof of~\\cite[Lemma~9.3]{Dubois-Friedl-Lueck(2014Alexander)} shows\nthat by elementary row and column operations and taking block sum with triangular matrices\nhaving $\\id$ on each diagonal entry,\nwe can transform $B_2'''$ into a matrix of the shape\n\\[\nB_2'''' = A''' + u \\cdot I_{r_1\\cdot n_1 + r_2\\cdot n_2}^{2r_r\\cdot n_1 +2r_2 \\cdot n_2}\n\\]\nfor some matrix $A'''' \\in M_{2n_1 +2n_2,2n_1 +2n_2}(\\cald(K))$ and\n$I_{r_1\\cdot n_1 + r_2\\cdot n_2}^{2r_r\\cdot n_1 +2r_2 \\cdot n_2}$ as introduced in Lemma~\\ref{rank_estimate_for_a_plus_t_I_k}. \nObviously we have\n\\begin{multline*}\n\\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_{B_2'} \\colon \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s}\\bigr)\\bigr)\n\\\\\n= \\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_{B_2''''} \\colon \\cald(K)_{t}[u^{\\pm 1}]^{2n_1 +2n_2} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{2n_1 +2n_2}\\bigr)\\bigr).\n\\end{multline*}\nWe conclude from Lemma~\\ref{rank_estimate_for_a_plus_t_I_k} applied to $B_2''''$\n\\begin{multline*}\n\\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_{B_2''''} \\colon \\cald(K)_{t}[u^{\\pm 1}]^{2n_1 +2n_2} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{2n_1 +2n_2}\\bigr)\\bigr)\n\\\\\n\\le r_1 \\cdot n_1 + r_2 \\cdot n_2.\n\\end{multline*}\nHence we get \n\\begin{multline}\n\\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_{B_2'} \\colon \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{1 +2n_1 +2n_2+s}\\bigr)\\bigr)\n\\label{the:The_Thurston_norm_and_the_(mu,phi)_L2-Euler_characteristic:(4)}\n\\\\\n\\le r_1 \\cdot n_1 + r_2 \\cdot n_2.\n\\end{multline}\nWe conclude from~\\eqref{the:The_Thurston_norm_and_the_(mu,phi)_L2-Euler_characteristic:(3)}\nand~\\eqref{the:The_Thurston_norm_and_the_(mu,phi)_L2-Euler_characteristic:(4)}\n\\[\n\\chi^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} i^*C_*(\\overline{M})\\bigr) \\ge 2r_1 + 2r_2 - r_1 \\cdot n_1 - r_2 \\cdot n_2.\n\\]\nThis together with Lemma~\\ref{lem:phi-twisted_as_ordinary_L2_Euler_characteristic} implies\n\\[ \\begin{array}{rcl}\n- \\chi^{(2)}(M;\\mu,\\phi) \n& = &\n- \\chi^{(2)}\\big(\\caln(K) \\otimes_{\\IZ K} i^*C_*(\\overline{M})\\big)\\\\\n& \\le &\n r_1 n_1 + r_2 n_2 - 2r_1 - 2r_2\n\\\\\n& = & \n r_1 (n_1 -2) + r_2 (n_2 -2)\n\\,\\, = \\,\\,\n- r_1 \\chi(\\Sigma_1) - r_2 \\chi(\\Sigma_2)\n\\,\\,\\le \\,\\,\nx_M(\\phi).\n\\end{array}\n\\]\nThis finishes the proof of Theorem~\\ref{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic}.\n\\end{proof}\n\n\n\n \\typeout{---- Section 5: Fox calculus and and the $(\\mu,\\phi)$-$L^2$-Euler characteristic ------}\n\n\n\\section{Fox calculus and the $(\\mu,\\phi)$-$L^2$-Euler characteristic}\n\\label{sec:Fox_calculus_and_the_(mu,phi)-L2-Euler_characteristic}\n\nThe following calculations of the $(\\mu,\\phi)$-$L^2$-Euler characteristic from a\npresentation of the fundamental group is adapted to the corresponding calculation for the\n$L^2$-torsion function appearing in~\\cite[Theorem~2.1]{Friedl-Lueck(2015l2+Thurston)} and\nwill be used when we will compare these two invariants and the higher order Alexander\npolynomials. For information about the Fox matrix and the Fox calculus we refer for instance \nto~\\cite[9B on page 123]{Burde-Zieschang(1985)}, and~\\cite{Fox(1953)}. \n\n\n\\begin{theorem}[Calculation of the $(\\mu,\\phi)$-$L^2$-Euler characteristic \nfrom a presentation of the fundamental group]\n \\label{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation} \nLet $M$ be an admissible $3$-manifold with fundamental group $\\pi$. Let\n\\[\n\\pi = \\langle x_1,x_2, \\ldots, x_a \\mid R_1,R_2, \\ldots, R_b \\rangle\n\\]\nbe a presentation of $\\pi$. Let the $(a,b)$-matrix over $\\IZ \\pi$\n\\[\nF = \n\\left( \\begin{array}{ccc} \\frac{\\partial R_1}{\\partial x_1}\n & \\ldots & \\frac{\\partial R_1}{\\partial x_a} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial R_b}{\\partial x_1} & \\ldots & \\frac{\\partial R_b}{\\partial x_a}\n\\end{array}\\right) \n\\]\nbe the Fox matrix of the presentation. \n\nLet $G$ be a torsion-free group which satisfies the Atiyah Conjecture.\nConsider two group homomorphisms \n$\\mu \\colon \\pi \\to G$ and $\\phi \\colon G \\to \\IZ$.\nSuppose that $\\mu$ is non-trivial and that $\\phi$ is surjective.\n\nLet $i \\colon K \\to G$ be the inclusion of the kernel $K = \\ker(\\phi)$ of \n$\\phi$ and let $t \\colon \\cald(K) \\xrightarrow{\\cong} \\cald(K)$ be the automorphism\nintroduced in \nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)}. \nLet $\\overline{M} \\to M$ be the $G$-covering associated to $\\mu$. \n\n\n\n\\begin{enumerate}\n\n\\item \\label{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation:non-empty} \nSuppose that $\\partial M$ is non-empty and and that $a = b + 1$. \nChoose $i \\in \\{1,2, \\ldots, r\\}$ with $|\\mu(x_i)| = \\infty$.\nLet $A$ be the $(a-1,a-1)$-matrix with entries in $\\IZ G$\nobtained from the Fox matrix $F$ by deleting the $i$-th column\nand then applying the homomorphism $M_{a-1,a-1}(\\IZ \\pi) \\to M_{a-1,a-1}(\\IZ G)$ \ninduced by $\\mu \\colon \\pi \\to G$.\n\nThen $b_n^{(2)}(\\overline{M};\\caln(G))$ vanishes for all $n \\ge 0$ if and only if we have\n$\\dim_{\\caln(G)}\\bigl(\\ker(r_A \\colon \\caln(G)^{a-1} \\to \\caln(G)^{a-1})\\bigr) = 0$.\nIf this is the case, then $(\\mu,\\phi)$ is an $L^2$-acyclic Atiyah-pair in the sense of \nDefinition~\\ref{def:L2-acyclic_Atiyah-pair} and we get \n\\begin{multline*}\n\\chi^{(2)}(M;\\mu,\\phi)\n\\\\\n= \n- \\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}\\bigr)\\bigr)\n\\\\\n+ |\\phi \\circ \\mu(x_i)|;\n\\end{multline*}\n\n\n\\item \\label{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation:empty} \nSuppose $\\partial M$ is empty. We make the assumption\nthat the given presentation comes from a Heegaard decomposition\nas described in~\\cite[Proof of Theorem~5.1]{McMullen(2002)}.\nThen $a = b$ and there is another set of dual generators \n$\\{x_1', x_2', \\ldots, x_a'\\}$ coming from the Heegaard decomposition as described \nin~\\cite[Proof of Theorem~5.1]{McMullen(2002)}. Choose $i,j \\in \\{1,2, \\ldots, a\\}$ \nwith $|\\mu(x_i)| = \\infty$ and $|\\mu(x_j')| = \\infty$.\nLet $A$ be the $(a-1,a-1)$-matrix with entries in $\\IZ G$\nobtained from the Fox matrix $F$ by deleting the $i$th column and the $j$th row\nand then applying the homomorphism $M_{a-1,a-1}(\\IZ \\pi) \\to M_{a-1,a-1}(\\IZ G)$\ninduced by $\\mu \\colon \\pi \\to G$.\n\n\nThen $b_n^{(2)}(\\overline{M};\\caln(G))$ vanishes for all $n \\ge 0$ if and only if we have\n$\\dim_{\\caln(G)}\\bigl(\\ker(r_A \\colon \\caln(G)^{a-1} \\to \\caln(G)^{a-1})\\bigr) = 0$.\nIf this is the case, then $(\\mu,\\phi)$ is an $L^2$-acyclic Atiyah-pair and we get \n\\begin{multline*}\n\\chi^{(2)}(M;\\mu,\\phi) \n\\\\\n= \n- \\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \n\\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}\\bigr)\\bigr)\n\\\\\n+ |\\phi \\circ \\mu(x_i)| + |\\phi \\circ \\mu(x_j')|.\n\\end{multline*}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} \n We only treat the case, where $\\partial M$ is empty, and leave it to the reader to\n figure out the details for the case of a non-empty boundary using the proof\n of~\\cite[Theorem~2.4]{Lueck(1994a)}. \n\n \n From~\\cite[Proof of Theorem~5.1]{McMullen(2002)} we\n obtain a compact $3$-dimensional $CW$-complex $X$ together with a homotopy equivalence\n $f \\colon X \\to M$ and a set of generators $\\{x_1', \\ldots ,x_a'\\}$ such that\n the $\\IZ G$-chain complex $C_*(\\overline{X})$ of $\\overline{X}$ looks like \n\\[\n\\IZ G \n\\xrightarrow{\\prod_{i=1}^a r_{\\mu(x_i') - 1}} \n\\bigoplus_{i=1}^a \\IZ G \n\\xrightarrow{r_{\\mu(F)}} \\bigoplus_{i=1}^a \\IZ G \n\\xrightarrow{\\bigoplus_{i=1}^a r_{\\mu(x_i) - 1}} \\IZ G,\n\\]\nwhere $\\mu(F)$ is the image of $F$ under the map $M_{a,a}(\\IZ \\pi) \\to M_{a,a}(\\IZ G)$ induced \nby $\\mu$ and $\\overline{X} \\to X$ is the pullback of $\\overline{M} \\to M$ with $f$.\n\nFor $g \\in G$ with $g \\not= 1$ let $D(g)_*$ be the $1$-dimensional\n$\\IZ G$-chain complexes which has as first differential\n$r_{g -1} \\colon \\IZ G\\to \\IZ G$. Since $g$ generates an infinite cyclic subgroup of $G$,\nwe conclude $b_n^{(2)}(\\caln(G) \\otimes_{|\\IZ G} D(g)_*) = 0$\nfor $n \\ge 0$ from~\\cite[Lemma~1.24~(4) on page~30 and Lemma~1.34~(1) on page~35]{Lueck(2002)}.\nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):chain} \nand Lemma~\\ref{lem:degree_and_rank} imply\n\\begin{eqnarray}\n\\chi^{(2)}\\bigl(\\caln(K) \\otimes_{\\IZ K} i^*D_*(g)) & = & |\\phi(g)|.\n\\label{chi(2)(D(g)}\n\\end{eqnarray}\nThere is a surjective $\\IZ G$-chain map \n$p_* \\colon C_*(\\overline{X}) \\to \\Sigma^2 D(\\mu(x_j'))_*$ \nwhich is the identity in degree $3$ and the\nprojection onto the summand corresponding to $j$ in degree $2$, and an injective \n$\\IZ G$-chain map $i_* \\colon D(\\mu(x_i))_* \\to C_*(\\overline{X})$\nwhich is the identity in degree $0$ and the inclusion to the summand corresponding to\n$i$ in degree $1$. Let $P_*$ be the kernel of $p_*$ and let $Q_*$ be the cokernel of\nthe injective map $j_* \\colon D(\\mu(x_1))_* \\to P_*$ induced by \n$i_* \\colon D(\\mu(x_1))_* \\to C_*(\\overline{X})$. \nThen $Q_*$ is concentrated in dimensions $1$ and $2$ and its second differential is\n$r_A \\colon \\IZ G^{a-1} \\to \\IZ G^{a-1}$. We conclude from the long exact\nhomology sequence of a short exact sequence of Hilbert $\\caln(G)$-chain complexes,\nthe homotopy invariance of $L^2$-Betti numbers and the additivity of the von Neumann dimension\n\\[\nb_n^{(2)}(\\overline{X};\\caln(G)) = \n\\begin{cases}\n\\dim_{\\caln(G)}\\bigl(\\ker(r_A \\colon \\caln(G)^{a-1} \\to \\caln(G)^{a-1}))\\bigr)& \\text{if} \\; n = 1,2;\n\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\]\nThis shows that $b_n^{(2)}(\\overline{M};\\caln(G))$ vanishes for all $n \\ge 0$ if and only if we have\n$\\dim_{\\caln(G)}\\bigl(\\ker(r_A \\colon \\caln(G)^{a-1} \\to \\caln(G)^{a-1}))\\bigr) = 0$.\n\nSuppose that this is the case. Then $(\\mu,\\phi)$ is an $L^2$-acyclic Atiyah-pair.\nWe conclude from Theorem~\\ref{the:Main_properties_of_cald(G)}\nthat\n$r_A \\colon \\cald(G)^{a-1} \\to \\cald(G)^{a-1}$ is injective and hence\n$r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}$ is injective\nand \n\\begin{multline}\n\\chi^{(2)}(\\caln(K) \\otimes_{\\IZ K} i^*Q_*)\n\\\\\n = \n\\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}\\bigr)\\bigr).\n\\label{chi(2)(caln(K)_otimes_iD(g)}\n\\end{multline}\nWe get from Lemma~\\ref{lem:phi-twisted_as_ordinary_L2_Euler_characteristic}\nand the chain complex version of~\\cite[Theorem~6.80~(2) on page~277]{Lueck(2002)}\napplied to the short exact sequences of $\\caln(K)$-chain complexes obtained by applying\n$\\caln(K) \\otimes_{\\IZ K} i^*-$ to the short exact sequences of $\\IZ G$-chain complexes\n$0 \\to P_* \\to C_*(\\overline{X}) \\to \\Sigma^2 D(\\mu(x_j'))_*\\to 0$\nand $0 \\to D(\\mu(x_1))_* \\to P_* \\to Q_*\\to 0$\n\\begin{eqnarray*}\n\\lefteqn{\\chi^{(2)}(\\overline{M};\\mu,\\phi)}\n& & \n\\\\\n& = & \n\\chi^{(2)}(\\caln(K) \\otimes_{\\IZ K} i^*C_*(\\overline{X}))\n\\\\\n& = & \n\\chi^{(2)}(\\caln(K) \\otimes_{\\IZ K} i^*Q_*) + \\chi^{(2)}(\\caln(K) \\otimes_{\\IZ K} i^*D(\\mu(x_j'))_*) \n\\\\ \n& & \\quad \\quad \\quad \\quad + \n\\chi^{(2)}(\\caln(K) \\otimes_{\\IZ K} i^*D(\\mu(x_i))_*) \n\\\\\n& \\stackrel{\\eqref{chi(2)(D(g)} \\,\\text{and} \\,\\eqref{chi(2)(caln(K)_otimes_iD(g)}}{=}& \n- \\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}\\bigr)\\bigr)\n\\\\\n& & \n\\quad \\quad \\quad \\quad + |\\phi \\circ \\mu(x_i) + |\\phi \\circ \\mu(x_j')|.\n\\end{eqnarray*}\nThis finishes the proof of Theorem~\\ref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation}.\n\\end{proof}\n\n\n\n\\begin{lemma} \\label{lem:b_1(2)_is_zero_implies_L2-acyclic}\n\\begin{enumerate}[font=\\normalfont]\n\\item \\label{lem:b_1(2)_is_zero_implies_L2-acyclic:b_1}\nLet $M$ be an admissible $3$-manifold. Consider an infinite group $G$ \nand a $G$-covering $\\overline{M} \\to M$. Then we get $b_n^{(2)}(\\overline{M}; \\caln(G)) = 0$ for\nall $n \\ge 0$ if $b_1(\\overline{M};\\caln(G)) = 1$;\n\n\n\\item \\label{lem:b_1(2)_is_zero_implies_L2-acyclic:universal}\nIf $M$ is an admissible $3$-manifold,\nthen we get $b_n^{(2)}(\\widetilde{M};\\caln(\\pi)) = 0$ for all $n \\ge 0$.\n\\end{enumerate}\n\n\\end{lemma}\n\\begin{proof}~\\eqref{lem:b_1(2)_is_zero_implies_L2-acyclic:b_1}\nSince $G$ is infinite, we have $b_0^{(2)}(\\overline{M};\\caln(G)) = 0$\nby~\\cite[Theorem~1.35~(8) on page~38]{Lueck(2002)}. Using Poincar\\'e duality in the closed case,\nsee~\\cite[Theorem~1.35~(3) on page~37]{Lueck(2002)} we conclude\n$b_m^{(2)}(\\overline{M};\\caln(G)) = 0$ for $m \\ge 3$. Since $\\chi(M) = 0$, we get\n$b_1^{(2)}(\\overline{M};\\caln(G)) = b_2^{(2)}(\\overline{M};\\caln(G))$. Hence the\nassumption $b_1^{(2)}(\\overline{M};\\caln(G)) = 0$ implies that we have\n$b_m^{(2)}(\\overline{M};\\caln(G)) = 0$ for all $m \\ge 0$. \n\\\\[1mm]~\\eqref{lem:b_1(2)_is_zero_implies_L2-acyclic:universal}\nThis follows from~\\cite[Theorem~0.1]{Lott-Lueck(1995)}.\n\\end{proof}\n\n\n\n\\begin{theorem}[The $(\\mu,\\phi)$-$L^2$-Euler characteristic in terms of the first homology]\n \\label{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology} \n Let $M$ be an admissible $3$-manifold. Let $G$ be a torsion-free group which satisfies the Atiyah\n Conjecture. Consider group homomorphisms $\\mu\n \\colon \\pi \\to G$ and $\\phi \\colon G \\to \\IZ$. Let $K$ be the kernel of $\\phi$ and $i\n \\colon K \\to G$ be the inclusion. Let $t \\colon \\cald(K) \\xrightarrow{\\cong} \\cald(K)$ \n be the automorphism introduced in \nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)}. \n Assume that $\\phi$ is surjective, $\\phi \\circ \\mu$ is not trivial, and the intersection $\\im(\\mu) \\cap \\ker(\\phi)$ is not trivial.\n Suppose that $b_1^{(2)}(\\overline{M};\\caln(G)) = 0$ holds for the $G$-covering \n $\\overline{M} \\to M$ associated to $\\mu$. Then:\n\n\n\\begin{enumerate} [font=\\normalfont]\n\n\\item \\label{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology:acyclic}\n\nThe pair $(\\mu,\\phi)$ is an $L^2$-acyclic Atiyah-pair;\n\n\\item \\label{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology:homology} \nWe have\n\\begin{multline*}\n\\hspace{12mm} \\chi^{(2)}(M;\\mu,\\phi) = - b_1^{(2)}(i^*\\overline{M};\\caln(K))\n\\\\\n= - \\dim_{\\cald(K)}\\bigl(H_1\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr).\n\\end{multline*}\nIn particular $\\chi^{(2)}(M;\\mu,\\phi)$ is an integer $\\ge 0$.\nMoreover, we have for $m \\not= 1$\n\\[\nH_m\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr) \\,\\, =\\,\\, 0,\n\\]\nand\n\\[\n\\hspace{1cm}\nb_m^{(2)}(i^*\\overline{M};\\caln(K))\n\\,\\,=\\,\\, \\dim_{\\cald(K)}\\bigl(H_m\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr) \\,\\,=\\,\\, 0.\n\\]\n\n\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\\begin{proof}~\\eqref{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology:acyclic}\nThis follows from \nLemma~\\ref{lem:b_1(2)_is_zero_implies_L2-acyclic}~\\eqref{lem:b_1(2)_is_zero_implies_L2-acyclic:b_1}.\n\\\\[1mm]~\\eqref{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology:homology} \nConsider any presentation\n\\[\n\\pi = \\langle x_1,x_2, \\ldots, x_a \\mid R_1,R_2, \\ldots, R_b \\rangle\n\\]\nof $\\pi$. We want to modify it to another presentation \n\\[\n\\pi = \\langle \\widehat{x}_1, \\widehat{x}_2, \\ldots, \\widehat{x}_{a}\\mid \\widehat{R}_1, \\widehat{R}_2, \\ldots, \\widehat{R}_{b}' \\rangle\n\\]\nby a sequence of Nielsen transformations,\ni.e., we replace the ordered set of generators \n$x_1, x_2, \\ldots, x_a$ by $x_{\\sigma(1)}, x_{\\sigma(2)}, \\ldots, x_{\\sigma(a)}$\nfor a permutation $\\sigma \\in S_a$ or we replace\n $x_1, x_2, \\ldots, x_a$ by $x_1, x_2, x_{i-1}, x_j^kx_i, x_{i+1}, \\ldots , x_{j-1}, x_j,x_{j+1}, \\ldots, x_a$ \nfor some integers $i,j,k$ with $1 \\le i < j \\le a$, and change the relations accordingly,\nsuch that $\\mu(\\widehat{x}_1) \\not= 0$ and $\\phi(\\widehat{x}_1) = 0$\nholds, provided that $\\phi$ is surjective and $\\im(\\phi) \\cap \\ker(\\phi)$ is non-trivial. \nWe use induction over $n = |\\{i \\in \\{1,2, \\ldots, a\\} \\mid \\mu(x_i) \\not= 0\\}|$. The induction beginning \n$n = 0,1$ is obvious since the case $n = 0$ cannot occur, since $\\phi \\circ \\mu \\not= 0$ and\n$\\phi$ is surjective, and in the case $n = 1$ we must have $\\phi \\circ \\mu(x_i) = 0$ for the only index\n$i$ with $\\mu(x_i) \\not= 0$, since $\\im(\\mu) \\cap \\ker(\\phi) \\not= \\{0\\}$.\nThe induction step from $(n-1)$ to $n \\ge 2$ is done as follows.\nWe use subinduction over $m = \\min\\{|\\phi(x_i)| \\mid i = 1,2, \\ldots, a, \\mu(x_i)\\not= 0\\}$. \nIf $m = 0$, then $\\mu(x_i) \\not = 0$ and $\\phi(x_i) = 0$ for some $i \\in\n\\{1,2, \\ldots, a\\}$, and we can achieve our goal by enumeration. The induction step from\n$(m-1)$ to $ m \\ge 1$ is done as follows. Choose $j \\in \\{1,2, \\ldots, a\\}$ such that\n$\\mu(x_j) \\not= 0$ and $\\phi(x_j) = m$. Since $n \\ge 2$, there must be an index \n$i \\in \\{1,2, \\ldots , a\\}$ with $i \\not = j$ and $\\mu(x_i) \\not= 0$. By changing the enumeration\nwe can arrange $i < j$. Choose integers $k,l$ with\n$0 \\le l < m$ and $\\phi(x_i) = k \\cdot \\phi(x_j) + l$. If we\nreplace $x_i$ by $x_i \\cdot x_j^{-k}$ and leave the other elements in $\\{x_1, x_2, \\ldots,x_a\\}$ \nfixed, we get a new set of generators which is part of a finite presentation with\n$a$ generators and $b$ relations of $\\pi$ for which the induction hypothesis applies since\neither $\\mu(x_i \\cdot x_j^{-k}) = 0$ holds or we have $\\mu(x_i \\cdot x_j^{-k}) \\not= 0$\nand $|\\phi(x_i \\cdot x_j^{-k})| = l \\le m-1$. \n\n\nWe have the following equations in $\\IZ \\pi$ for $u\\in \\IZ \\pi$\nand integers $i,j,k$ with $1 \\le i < j \\le a$ and $k \\ge 1$:\n\\begin{multline*}\nr_{x_j^{k}x_i - 1}(u) \n= \nu(x_j^kx_i - 1) \n= \nux_j^k(x_i - 1) + u(x_j^k - 1)\n= \nux_j^k(x_i - 1) + u\\left(\\sum_{i = 0}^{k-1} x_j^i\\right) \\cdot (x_j - 1)\n\\\\\n= \nux_j^k(x_i - 1) + \\left(\\sum_{i = 0}^{k-1} u x_j^i\\right) \\cdot (x_j - 1)\n= \nr_{x_i - 1}(ux_j^{k}) - r_{x_j -1} \\left(\\sum_{i = 0}^{k-1} u x_j^{i}\\right),\n\\end{multline*}\nand \n\\begin{multline*}\nr_{x_j^{-k}x_i - 1}(u) \n= \nu(x_j^{-k}x_i - 1) \n= \nux_j^{-k}(x_i - 1) - ux_j^{-k} (x_j^k- 1)\n\\\\\n= \nux_j^{-k}(x_i - 1) - ux_j^{-k}\\left(\\sum_{i = 0}^{k-1} x_j^i\\right) \\cdot (x_j - 1)\n\\\\\n= \nux_j^{-k}(x_i - 1) - \\left(\\sum_{i = 0}^{k-1} u x_j^{i-k}\\right) \\cdot (x_j - 1)\n= \nr_{x_i - 1}(ux_j^{-k}) - r_{x_j -1} \\left(\\sum_{i = 0}^{k-1} u x_j^{i-k}\\right).\n\\end{multline*}\nThis implies that we can find $\\IZ \\pi$-isomorphisms \n$f_1 \\colon \\IZ \\pi^a\\to \\IZ \\pi^a$ and $f_2 \\colon \\IZ \\pi^a\\to \\IZ \\pi^a$ \nsuch that the following diagrams commute\n\\begin{center}\n$\\begin{array}{rcl}\n\\hspace{7mm} \\xymatrix@R0.45cm@!C=8em{\n\\IZ \\pi^a \\ar[dd]_{f_1} ^{\\cong} \\ar[rd]^{\\quad \\bigoplus_{i = 1}^a r_{x_i-1}} &\n\\\\\n& \\IZ \\pi\n\\\\\n\\IZ \\pi^a \\ar[ru]_{\\quad \\bigoplus_{i = 1}^a r_{\\widehat{x}_i-1}} &\n}\n& \\hspace{20mm} &\n\\xymatrix@R0.45cm@!C=8em{\n& \n\\IZ \\pi^a \\ar[dd]_{f_2}^{\\cong} &\n\\\\\n\\IZ \\pi \\ar[ru]^{\\prod_{i = 1}^a r_{x_i'-1}} \\ar[rd]_{\\prod_{i = 1}^a r_{\\widehat{x}_i'-1}} & \n\\\\\n& \n\\IZ \\pi^a &\n}\n\\end{array}$\n\\end{center}\n\nNow we proceed as in the proof of \nTheorem~\\ref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation}\n\\eqref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation:empty},\nexplaining only the case where the boundary is empty. There we have\nwe constructed a compact $3$-dimensional $CW$-complex $X$ together with a homotopy equivalence\n $f \\colon X \\to M$ and set of generators $\\{x_1, \\ldots ,x_a\\}$ and $\\{x_1', \\ldots ,x_a'\\}$ such that\n the $\\IZ G$-chain complex $C_*(\\overline{X})$ of $\\overline{X}$ looks like \n\\[\n\\IZ G \n\\xrightarrow{\\prod_{j=1}^a r_{\\mu(x_i') - 1}} \n\\bigoplus_{i=1}^a \\IZ G \n\\xrightarrow{r_{\\mu(F)}} \\bigoplus_{i=1}^a \\IZ G \n\\xrightarrow{\\bigoplus_{i=1}^a r_{\\mu(x_i) - 1}} \\IZ G.\n\\]\nApplying the construction above yields new set of generators \n$\\{\\widehat{x}_1, \\ldots ,\\widehat{x}_a\\}$ and $\\{\\widehat{x}_1', \\ldots ,\\widehat{x}_a'\\}$\nsuch that $|\\mu(\\widehat{x}_1)| = |\\mu(\\widehat{x}_1')| = \\infty$\nand $\\phi \\circ \\mu(\\widehat{x_1}) = \\phi \\circ \\mu(\\widehat{x}_1') = 0$ \nand we can find a matrix $B \\in M_{a,a}(\\IZ G)$ and isomorphisms $f_2$ and $f_1$ making the following diagram\nof $\\IZ G$-modules commute.\n\\[\n\\xymatrix@!C=8em{\n\\IZ G \\ar[r]^{\\prod_{j=1}^a r_{\\mu(x_i') - 1}} \\ar[d]^{\\id}\n&\n\\bigoplus_{i=1}^a \\IZ G \\ar[r]^{r_{\\mu(F)}} \\ar[d]^{f_2}_{\\cong}\n&\n\\bigoplus_{i=1}^a \\IZ G \n\\ar[r]^{\\bigoplus_{i=1}^a r_{\\mu(x_i) - 1}} \\ar[d]^{f_1}_{\\cong}\n&\\IZ G \\ar[d]^{\\id}\n\\\\\n\\IZ G \\ar[r]_{\\prod_{j=1}^a r_{\\mu(\\widehat{x}_i') - 1}} \n&\n\\bigoplus_{i=1}^a \\IZ G \\ar[r]_{r_{B}} \n&\n\\bigoplus_{i=1}^a \\IZ G \n\\ar[r]_{\\bigoplus_{i=1}^a r_{\\mu(\\widehat{x}_i) - 1}} \n&\\IZ G\n}\n\\]\nNow proceed as in the proof of\nTheorem~\\ref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation}\n\\eqref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation:empty}\nusing the chain complex given by the lower row instead of the chain complex \ngiven by the upper row in the diagram above. If $A$ is the square matrix over $\\IZ G$\nobtained from $B$ by deleting the first row and the first\ncolumn and then applying the ring homomorphism $\\IZ \\pi\\to \\IZ G$ induced by $\\mu$\nto each entry, then $A$ is invertible over $\\cald(G)$ and we get because of \n$\\phi \\circ \\mu(\\widehat{x_1}) = \\phi \\circ \\mu(\\widehat{x}_1') = 0$ and \nLemma~\\ref{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic}\n\\begin{multline*}\n\\chi^{(2)}(M;\\mu,\\phi) \n= \n - \\dim_{\\cald(K)}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}\\bigr)\\bigr)\n\\\\\n= \\dim_{\\cald(K)}\\bigl(H_1\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr),\n\\end{multline*}\nand \n\\begin{multline*}\n0 = \n \\dim_{\\cald(K)}\\bigl(\\ker\\bigl(r_A \\colon \\cald(K)_{t}[u^{\\pm 1}]^{a-1} \\to \\cald(K)_{t}[u^{\\pm 1}]^{a-1}\\bigr)\\bigr).\n\\\\\n= \\dim_{\\cald(K)}\\bigl(H_2\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr),\n\\end{multline*}\nand\n\\[\n0 = \\dim_{\\cald(K)}\\bigl(H_m\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr)\n\\quad\\text{for}\\; m \\not = 1,2.\n\\]\nWe conclude from Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):chain} \nfor all $m \\ge 0$\n\\[\nb_m^{(2)}(i^*\\overline{M};\\caln(K))\n= \\dim_{\\cald(K)}\\bigl(H_m\\bigl(\\cald(K)_{t}[u^{\\pm 1}] \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr).\n\\]\nThis finishes the proof of Theorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology}.\n\\end{proof} \n\nIn view of Lemma~\\ref{lem:reduction_to_surjective_mu}~\\eqref{lem:reduction_to_surjective_mu:mu_circ_phi_is_not_trivial},\nthe next example   covers the case $\\im(\\mu) \\cap \\ker(\\phi) = \\{1\\}$\nwhich is not treated in Theorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology}.\n\n\n\\begin{example}[Infinite cyclic covering] \\label{exa:infinite_cyclic_covering}\nLet $M$ be an admissible $3$-manifold. Consider an epimorphism $\\mu \\colon \\pi \\to \\IZ$ and \na non-trivial homomorphism $\\phi \\colon \\IZ\\to \\IZ$. Let $k$ be the index of $\\im(\\phi)$ in $\\IZ$.\nLet $\\overline{M} \\to M$ be the infinite cyclic covering associated to $\\mu$.\n\nThen $(\\mu,\\phi)$ is an $L^2$-acyclic pair if and only if $b_1^{(2)}(\\overline{M};\\caln(\\IZ)) = 0$.\nThis follows from Lemma~\\ref{lem:b_1(2)_is_zero_implies_L2-acyclic} and the fact that $\\IZ$ satisfies\nthe Atiyah Conjecture by \nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:Linnell}.\n\nSuppose $b_1^{(2)}(\\overline{M};\\caln(\\IZ)) = 0$. Lemma~\\ref{lem:phi-twisted_as_ordinary_L2_Euler_characteristic},\nLemma~\\ref{lem:reduction_to_surjective_mu}~\\eqref{lem:reduction_to_surjective_mu:mu_circ_phi_is_not_trivial} and \nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):chain} imply that\n$\\dim_{\\IQ}(H_n(M;\\IQ)) < \\infty$ holds for $n \\ge 0$ and we get \n\\begin{equation}\n\\chi^{(2)}(M;\\mu,\\phi) \n= \nk \\cdot \\chi(\\overline{M}).\n\\label{exa:infinite_cyclic_covering:(1)}\n\\end{equation}\nNext we show\n\\begin{equation}\n\\chi(\\overline{M}) \n\\,\\,= \\,\\,\n\\begin{cases}\n1 - \\dim_{Q}(H_1(\\overline{M};\\IQ)) & \\text{if}\\; \\partial M \\not= \\emptyset;\n\\\\\n2 - \\dim_{Q}(H_1(\\overline{M};\\IQ)) & \\text{if}\\; \\partial M = \\emptyset.\n\\end{cases}\n\\label{exa:infinite_cyclic_covering:(2)}\n\\end{equation}\nSince $H_n(\\overline{M};\\IQ) =0$ for $n \\ge 3$ holds because $\\overline{M}$ is a non-compact $3$-manifold,\nthis will follow if we can show\n\\[\n\\dim_{Q}(H_2(\\overline{M};\\IQ)) \n\\,\\,=\\,\\, \n\\begin{cases} 0\n& \\text{if}\\; \\partial M \\not= \\emptyset;\n\\\\\n\\IQ & \\text{if}\\; \\partial M = \\emptyset.\n\\end{cases}\n\\]\nWe begin with the case that $M$ has\nnon-empty boundary. Then $M$ is homotopy equivalent to a $2$-dimensional complex $X$.\nSince $H_2(\\overline{M};\\IQ) \\cong_{\\IQ[\\IZ]} H_2(\\overline{X};\\IQ)$ is a $\\IQ[\\IZ]$-submodule of the free $\\IQ[\\IZ]$-module\n$C_2(\\overline{X}) \\otimes_{\\IZ} \\IQ$ for the pull back $\\overline{X} \\to X$ of $\\overline{M} \\to M$ with a homotopy\nequivalence $X \\to M$, the $\\IQ[\\IZ]$-module $H_2(\\overline{M};\\IQ)$ is free.\nSince $\\dim_{\\IQ}(H_2(\\overline{M};\\IQ)) < \\infty$ holds, this implies $H_2(\\overline{M};\\IQ) = 0$.\n\nNow suppose that $\\partial M$ is empty. Then we get a Poincar\\'e $\\IZ \\pi$-chain homotopy equivalence of\n$\\IZ \\pi$-chain complexes $\\Hom_{\\IZ \\pi}(C_{3-*}(\\widetilde{M}),\\IZ \\pi) \\to C_*(\\widetilde{M})$.\nIt induces a $\\IQ[\\IZ]$-chain homotopy equivalence $\\Hom_{\\IQ[\\IZ]}(\\IQ \\otimes_{\\IZ} C_{3-*}(\\overline{M}), \\IQ[\\IZ]) \n\\to \\IQ \\otimes_{\\IZ} C_{*}(\\overline{M})$ and hence a $\\IQ[\\IZ]$-isomorphism\n\\[H^1\\bigl(\\Hom_{\\IQ[\\IZ]}(\\IQ \\otimes_{\\IZ} C_{3-*}(\\overline{M}), \\IQ[\\IZ])\\bigr) \\xrightarrow{\\cong} H_2(\\overline{M}).\n\\]\nSince $\\IQ[\\IZ[$ is a principal ideal domain, we get from the Universal Coefficient Theorem for cohomology an exact sequence \nof $\\IQ[\\IZ]$-modules\n\\begin{multline*}\n0 \\to \\Ext^1_{\\IQ[\\IZ|}\\bigl(H_0(C_*(\\overline{M})),\\IQ[\\IZ]\\bigr)) \\to H^1\\bigl(\\Hom_{\\IQ[\\IZ]}(\\IQ \\otimes_{\\IZ} C_{3-*}(\\overline{M}), \\IQ[\\IZ])\\bigr) \n\\\\\n\\to \\Hom_{\\IQ[\\IZ]}\\bigl(H_1(\\overline{M};\\IQ),\\IQ[\\IZ]\\bigr) \\to 0.\n\\end{multline*}\nFrom $\\dim_{\\IQ}(H_1(\\overline{M};\\IQ)) < \\infty$, we conclude $\\Hom_{\\IQ[\\IZ]}\\bigl(H_1(\\overline{M};\\IQ),\\IQ[\\IZ]\\bigr) = 0$.\nSince $H_0(C_*(\\overline{M}))$ is the trivial $\\IQ[\\IZ]$-module $\\IQ$, we conclude that \n$\\Ext^1_{\\IQ[\\IZ|}\\bigl(H_0(C_*(\\overline{M})),\\IQ[\\IZ]\\bigr)$ is the trivial $\\IQ[\\IZ]$-module $\\IQ$\nand~\\eqref{exa:infinite_cyclic_covering:(2)} follows.\n\\end{example}\n\n\n\n\\typeout{------ Section 6: Equality of $(\\mu,\\phi)$-$L^2$-Euler characteristic and the Thurston norm ------}\n\n\\section{Equality of $(\\mu,\\phi)$-$L^2$-Euler characteristic and the Thurston norm}\n\\label{sec:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}\n\n\nThe section is devoted to the proof of\nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}\nwhich needs some preparation.\n\nFor the remainder of this section $G$ is a torsion-free group\nsatisfying the Atiyah Conjecture, $H$ is a finitely generated abelian\ngroup and $\\nu \\colon G \\to H$ a surjective group homomorphism.\n\n\n\\subsection{The non-commutative Newton polytope}\n\\label{subsec:The_non-commutative_Newton_polytope}\n\n\nChoose a set-theoretic section $s$ of $\\nu$, i.e., a map of sets $s \\colon H \\to G$ with $\\nu \\circ s = \\id_H$.\nNotice that we do \\emph{not} require that $s$ is a group homomorphism. Let $K_{\\nu}$ be\nthe kernel of $\\nu$. Then $K_{\\nu}$ also satisfies the Atiyah Conjecture\nby Theorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:subgroups},\nand $\\cald(K_{\\nu})$ and $\\cald(G)$ are skew fields by\nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):skew_field}.\nThere are obvious inclusions $\\IZ K_{\\nu} \\subseteq \\IZ G$ and $\\cald(K_{\\nu}) \\subseteq \\cald(G)$ \ncoming from the inclusion $j \\colon K_{\\nu} \\to G$. Let $c \\colon H \\to \\aut(\\cald(K_{\\nu}))$ be \nthe map sending $h$ to the automorphisms $c(h) \\colon \\cald(K_{\\nu}) \\xrightarrow{\\cong} \\cald(K_{\\nu})$\ninduced by the conjugation automorphism $K_{\\nu} \\to K_{\\nu}, \\; k \\mapsto s(h) \\cdot k \\cdot s(h)^{-1}$. Define\n$c \\colon H \\to \\aut(\\IZ K_{\\nu})$ analogously. Define $\\tau \\colon H \\times H \\to (\\IZ K_{\\nu})^{\\times}$ \nand $\\tau \\colon H \\times H \\to \\cald(K_{\\nu})^{\\times}$ by sending $(h,h')$\nto $s(h)\\cdot s(h')\\cdot s(hh')^{-1}$. Let $\\IZ K_{\\nu} \\ast_s H$ and $\\cald(K_{\\nu}) \\ast_s H$ be\nthe crossed product rings associated to the pair $(c,\\tau)$. Elements in $\\IZ K_{\\nu} \\ast_s H$\nand $\\cald(K_{\\nu}) \\ast_s H$ respectively are finite formal sums $\\sum_{h \\in H} x_h \\cdot h$ \nfor $x_h$ in $\\IZ K_{\\nu}$ and $\\cald(K_{\\nu})$ respectively. Addition is given by adding the\ncoefficients. Multiplication is given by the formula\n\\[\n\\Big(\\,\\smsum{h \\in H}{} x_{h} \\cdot h\\Big) \\cdot \\Big(\\,\\smsum{h \\in H}{} y_{h} \\cdot h\\Big)\\,\\, =\\,\\,\\smsum{h\\in H}{} \\Big(\\,\\,\\smsum{\\substack{h',h'' \\in H,\\\\h'h'' = h}}{} x_{h'} c_{h'}(y_{h''})\n \\tau(h',h'') \\Big) \\cdot h.\n\\]\nThis multiplication is uniquely determined by the properties $h\\cdot x = c(h)(x)\\cdot h$\nfor $x$ in $\\IZ K_{\\nu}$ and $\\cald(K_{\\nu})$ respectively, and $h \\cdot h' = \\tau(h,h') \\cdot\n(hh')$ for $h,h' \\in H$. We obtain an isomorphism of rings\n\n\\[\nj'_s \\colon \\IZ K_{\\nu} \\ast_s H \\,\\xrightarrow{\\cong}\\, \\IZ G, \\quad \\sum_{h \\in H} x_h \\cdot h\n\\,\\mapsto\\, \\lmsum{h \\in H}{} x_h \\cdot s(h),\n\\]\nsee~\\cite[Example~10.53 on page~396]{Lueck(2002)}, and an injective ring homomorphism\n\\[\nj_s \\colon \\cald(K_{\\nu}) \\ast_s H \\,\\to\\, \\cald(G), \\quad \\sum_{h \\in H} x_h \\cdot h\n\\,\\mapsto\\, \\lmsum{h \\in H}{} x_h \\cdot s(h).\n\\]\nMoreover, the set $T$ of non-trivial elements in $\\cald(K_{\\nu}) \\ast_s H$ satisfies the Ore\ncondition and $j_s$ induces an isomorphism of skew fields\n\\begin{equation}\n\\widehat{j_s} \\colon T^{-1} (\\cald(K_{\\nu}) \\ast_s H) \\xrightarrow{\\cong} \\cald(G).\n\\label{widehat(j_s)}\n\\end{equation}\nThis is proved for $F = \\IC$ in~\\cite[Lemma~10.68 on page~403]{Lueck(2002)}, the proof\ncarries over for any field $F$ with $\\IQ \\subseteq F \\subseteq \\IC$.\n\nFix an element $u = \\sum_{h \\in H} x_h \\cdot h$ in $\\cald(K_{\\nu}) \\ast_s H$ with $u \\not= 0$.\nNow we introduce a non-commutative analogue of the Newton polytope. \nA \\emph{polytope} in a finite dimensional real vector space is a subset which is the convex hull of a finite subset.\nAn element $p$ in a polytope is called \\emph{extreme} if the implication $p= \\frac{q_1}{2} + \\frac{q_2}{2}\n\\Longrightarrow q_1 = q_2 = p$ holds for all elements $q_1$ and $q_2$ in the polytope.\nDenote by $\\Ext(P)$ the set of extreme points of $P$. If $P$ is the convex hull of the finite set $S$,\nthen $\\Ext(P) \\subseteq S$ and $P$ is the convex hull of $\\Ext(P)$. \nThe \\emph{Minkowski sum} of two polytopes $P_1$ and $P_2$ is defined to be the polytope\n\\[\nP_1 + P_2 := \\{p_1 +p_2 \\mid p_1 \\in P_1, p \\in P_2\\}.\n\\] \nIt is the convex hull of the set $\\{p_1 + p_2 \\mid p_1 \\in \\Ext(P_1), p_2 \\in \\Ext(P_2)\\}$.\nDefine the \\emph{non-commutative Newton polytope} of $u$ \n\\begin{equation}\nP(u) \\subseteq \\IR \\otimes_{\\IZ} H\n\\label{P(u)}\n\\end{equation}\nto be the polytope given by the convex hull of the finite set $\\{1 \\otimes h \\mid x_h \\not= 0\\}$. \nThe definition of $P(u)$ is independent of the choice of the section $s$ by the following argument.\nConsider two set-theoretic sections $s$ and $s'$ of $\\nu \\colon G \\to H$. Then we get for\n$u \\in \\cald(K_{\\nu}) \\ast_s H$ and $u' \\in \\cald(K_{\\nu}) \\ast_{s'} H$ \n\\begin{eqnarray}\n\\widehat{j_s}(u) = \\widehat{j_{s'}}(u')\n& \\implies &\nP(u;s) = P(u';s')\n\\label{P(u;s)_independent_of_s}\n\\end{eqnarray} \nby the following argument. If we write $u =\\sum_{h \\in H} x_h \\cdot h$ and $u' = \\sum_{h \\in H} y_h \\cdot h$,\nthen $\\widehat{j_s}(u) = \\widehat{j_{s'}}(u')$ implies\n\\[\nu\\,\\, =\\,\\, \\lmsum{h \\in H}{} x_h \\cdot h \\,\\,= \\,\\, \\lmsum{h \\in H}{} \\bigl(y_h\\cdot s'(h) s(h)^{-1} \\bigr)\\cdot h,\n\\]\nand hence $x_h \\not = 0 \\Leftrightarrow y_h \\not= 0$. \n\n\nThe following lemma is well-known in the commutative setting and we explain its proof \nsince we could not find a reference for it in the literature.\n\n\\begin{lemma} \\label{lem:P(uv)_is_P(u)_plus_P(v)} \nFor $u,v \\in \\cald(K_{\\nu}) \\ast_s H$ with $u,v \\not= 0$, we have \n\\[\nP(uv) = P(u) + P(v).\n\\]\n\\end{lemma}\n\\begin{proof}\nConsider an extreme point $p \\in P(u;s) + P(v;s)$. Then we can find points \n$q_1 \\in P(u;s)$ and $q_2 \\in P(v;s)$ with $p = q_1 + q_2$. We want to show that $q_1$ and $q_2$ are extreme. \nConsider $q_1', q_1'' \\in P(u;s)$ and $q_2',q_2'' \\in P(v;s)$ with $q_1 = (q_1' +q_1'')\/2$ and $q_2 = (q_2' + q_1'')\/2$.\nThen $(q_1' + q_2')$, $(q_1'' + q_2')$, $(q_1' + q_2'')$, and $(q_1'' + q_2'')$ belong to $P(u;s) + P(v;s)$ and satisfy\n\\[\np\\,\\, =\\,\\, \\smfrac{q_1' + q_2'}{2} + \\smfrac{q_1''+ q_2''}{2}\\,\\, =\\,\\, \\smfrac{q_1' + q_2''}{2} + \\smfrac{q_1''+ q_2'}{2}.\n\\]\nSince $p \\in P(u;s) + P(v;s)$ is extreme, we conclude $q_1' + q_2' = q_1''+ q_2''$ and\n$q_1' + q_2'' = q_1''+ q_2'$. If we subtract the second equation from the first, we obtain\n$q_2' - q_2'' = q_2'' - q_2'$ and hence $q_2' = q_2''$. This implies also $q_1' = q_1''$.\nThis shows that $q_1 \\in P(u;s)$ and $q_2 \\in P(v;s)$ are extreme.\n\nSuppose that we have other points $q_1' \\in P(u;s)$ and $q_2' \\in P(v;s)$ with $p = q_1' + q_2'$.\nThen $q_1 + q_2'$ and $q_1' + q_2$ belong to $P(u;s) + P(v;s)$ and satisfy\n$p = \\frac{q_1 + q_2'}{2} + \\frac{q_1'+ q_2}{2}$. Since $p$ is extreme, this implies $p = q_1 + q_2' = q_1'+ q_2$.\nSince we also have $p = q_1 + q_2 = q_1' + q_2'$, we conclude $q_1 = q_1'$ and $q_2 = q_2'$.\n\nNow write $u = \\sum_{h \\in H} x_{h} \\cdot h$, $v = \\sum_{h \\in H} y_{h} \\cdot h$, and \n$uv = \\sum_{h \\in H} z_{h} \\cdot h$. Since $p \\in P(u;s) + P(v;s)$ is extreme, there is $h \\in H$ \nwith $p = 1 \\otimes h$. If we write $p = q_1 + q_2$ for $q_1 \\in P(u;s)$ \nand $q_2 \\in P(v;s)$, then we have already seen that $q_1$ and $q_2$ are extreme\nand hence there are $h_1,h_2 \\in H$ with $q_1 = 1 \\otimes h_1$, $q_2 = 1 \\otimes h_2$,\n$x_{h_1} \\not= 0$ and $y_{h_2} \\not= 0$. The equation $p = q_1 + q_2$ implies\n$h = h_1 + h_2$. Now consider elements $h_1', h_2' \\in H$ with $h = h_1' + h_2'$, $x_{h_1'} \\not= 0$ \nand $y_{h_2'} \\not= 0$. Put $q_1' = 1 \\otimes h_1'$ and $q_2' = 1 \\otimes h_2'$. Then $q_1' \\in P(u;s)$ \nand $q_2' \\in P(v;s)$ and we have $p = q_1' + q_2'$. We have already explained that this implies \n$q_1 = q_1'$ and $q_2 = q_2'$ and hence $h_1 = h_1'$ and $h_2 = h_2'$. Therefore we get\n$z_h = x_{h_1} c(h_1)(y_{h_2}) \\tau(h_1,h_2)$. Since $x_{h_1}$ and $y_{h_2}$ are non-trivial, we\nconclude $z_h \\not= 0$ and hence $p \\in P(uv;s)$. Hence every extreme point in \n$P(u;s) + P(v;s)$ belongs to $P(uv;s)$ which implies $P(u;s) + P(v;s) \\subseteq P(uv;s)$.\n\nOne easily checks that any point of the shape $1 \\otimes h$ for $z_h \\not= 0$ belongs\nto $P(u;s) + P(v;s)$ since $z_h \\not= 0$ implies the existence of $h_1$ \nand $h_2$ with $x_{h_1}, y_{h_2} \\not= 0$ and $h = h_1 + h_2$. We conclude $P(uv;s) \\subseteq P(u;s) + P(v;s)$.\nThis finishes the proof of Lemma~\\ref{lem:P(uv)_is_P(u)_plus_P(v)}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{The polytope homomorphism}\n\\label{subsec:The_polytope_homomorphism}\n\n\nWe obtain a finite dimensional real vector space $\\IR \\otimes_{\\IZ} H$. An integral polytope in \n$\\IR \\otimes_{\\IZ} H$ is a polytope such that $\\Ext(P)$ is contained in $H$, where we consider $H$ \nas a lattice in $ \\IR \\otimes_{\\IZ} H$ by the standard embedding \n$H \\to \\IR \\otimes_{\\IZ} H, \\; h \\mapsto 1 \\otimes h$. \nThe Minkowski sum of two integral polytopes is again an integral\npolytope. Hence the integral polytopes form an abelian monoid under the Minkowski sum with\nthe integral polytope $\\{0\\}$ as neutral element. \n\n\n\n\\begin{definition}[Grothendieck group of integral polytopes]\n\\label{def:The_Grothendieck_group_of_integral_polytope}\nLet $\\calp_{\\IZ}(H)$ be the abelian group given by the Grothendieck construction applied\nto the abelian monoid of integral polytopes in $\\IR \\otimes_{\\IZ} H$ under the Minkowski sum.\n\\end{definition}\n\n\n\nNotice that for polytopes $P_0$, $P_1$ and $Q$ in a finite dimensional real vector space\nwe have the implication $P_0 + Q = P_1 + Q \\Longrightarrow P_0 = P_1$, see~\\cite[Lemma~2]{Radstroem(1952)}.\nHence elements in $\\calp_{\\IZ}(H)$ are given by formal\ndifferences $[P] - [Q]$ for integral polytopes $P$ and $Q$ in $\\IR \\otimes_{\\IZ} H$ and we\nhave $[P_0] - [Q_0] = [P_1] - [Q_1] \\Longleftrightarrow P_0 + Q_1 = P_1 + Q_0$.\n\nThe polytope group $\\calp_{\\IZ}(H)$ is a free abelian group by Funke~\\cite{Funke(2016)}.\n\nIn the sequel we denote by $G_{\\abel}$ the abelianization\n$G\/[G,G]$ of a group $G$. Define the \\emph{polytope homomorphism} for a surjective homomorphism\n$\\nu \\colon G \\to H$ onto a finitely generated free abelian group $H$ \n\\begin{eqnarray}\n\\IP'_{\\nu} \\colon \\cald(G)^{\\times}_{\\abel} \\to \\calp_{\\IZ}(H).\n\\label{P_prime_colon_cald_to_p_Z}\n\\end{eqnarray}\nas follows. Choose a set-theoretic section $s$ of $\\nu$. Consider an element $z \\in \\cald(G)$ with $z \\not= 0$.\nChoose $u,v \\in \\cald(K_{\\nu}) \\ast_s H)$ such that $\\widehat{j_s}(uv^{-1}) = z$,\nwhere the isomorphism $\\widehat{j_s}$ has been introduced in~\\eqref{widehat(j_s)}.\nThen we define the image of the class $[z]$\nin $\\cald(G)^{\\times}_{\\abel}$ represented by $z$ under $\\IP'_{\\nu}$ to be $[P(u)]- [P(v)]$.\n\nWe have to show that this is independent of the choices of $s$,$u$ and $v$. \nSuppose that we have another set-theoretic section $s' \\colon H \\to G$ of $\\nu$ and \n$u',v' \\in \\cald(K_{\\nu}) \\ast_{s'} H$ with $u',v' \\not= 0$ and $z = \\widehat{j_{s'}}(u'v'^{-1})$.\nChoose $u'',v'' \\in \\cald(K_{\\nu}) \\ast_{s'}\u00a0H$ with $j_s(u) = j_{s'}(u'')$ and $j_s(v) = j_{s'}(v'')$.\nFrom $\\widehat{j_{s}}(uv^{-1}) = z = \\widehat{j_{s'}}(u'v'^{-1})$\nwe conclude $u'v'^{-1} = u''v''^{-1}$ in $T^{-1} \\cald(K_{\\nu}) \\ast_{s'} H$. Hence there exist\n$w',w'' \\in \\cald(K_{\\nu}) \\ast_{s'} H$ with $w',w'' \\not= 0$, $u' w' = u'' w''$ and $v'w' = v'' w''$.\nWe conclude \n\\begin{eqnarray*}\nP(u) - P(v)\n& \\stackrel{\\eqref{P(u;s)_independent_of_s}}{=}& \nP(u'') - P(v'')\n\\\\\n& = & \nP(u'') + P(w'') - P(w') - P(v'') - P(w'') + P(w') \n\\\\\n& \\stackrel{\\textup{Lemma}~\\ref{lem:P(uv)_is_P(u)_plus_P(v)}}{=} & \nP(u''w'') - P(w') - P(v''w'') + P(w') \n\\\\\n& = & \nP(u'w') - P(w') - P(v'w') + P(w) \n\\\\\n& \\stackrel{\\textup{Lemma}~\\ref{lem:P(uv)_is_P(u)_plus_P(v)}}{=} & \nP(u') + P(w') - P(w') - P(v') - P(w') + P(w') \n\\\\\n& = & \nP(u') - P(v'). \n\\end{eqnarray*}\nHence $\\IP'_{\\nu} \\colon \\cald(G)^{\\times} \\to \\calp_{\\IZ}(H)$ is well-defined. \nWe conclude from Lemma~\\ref{lem:P(uv)_is_P(u)_plus_P(v)} that $\\IP'_{\\nu}$\n is a homomorphism of abelian groups.\n\n\n\n\n\n\\subsection{Semi-norms and matrices over $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}$]}\n\\label{subsec:Semi-norms_and_matrices_over_cald(K_phi_circ_nu)_t(u,u(-1))}\n\n\n\n\n\nLet $P \\subseteq \\IR \\otimes_{\\IZ} H$ be a polytope. It defines a seminorm on \n$\\Hom_{\\IZ}(H,\\IR) = \\Hom_{\\IR}(\\IR \\otimes_{\\IZ} H,\\IR)$ by\n\\begin{eqnarray}\n \\|\\phi \\|_P \n&:= & \\tmfrac{1}{2}\n\\sup\\{\\phi(p_0) - \\phi(p_1) \\mid p_0, p_1 \\in P\\}.\n\\label{seminorm_of_a_polytope}\n\\end{eqnarray}\nIt is compatible with the Minkowski sum, namely, for two\nintegral polytopes $P,Q \\subseteq \\IR \\otimes_{\\IZ} H$ we have\n\\begin{eqnarray}\n \\|\\phi \\|_{P+ Q} \n&= & \n \\|\\phi \\|_{P} + \\|\\phi \\|_{Q}.\n\\label{seminorm_of_a_polytope_is_additive_in_polytope}\n\\end{eqnarray}\nPut \n\\begin{multline}\n\\calsn(H) \n := \n\\{f \\colon \\Hom_{\\IZ}(H;\\IR) \\to \\IR \\mid \\text{there exists integral polytopes}\n\\\\\n\\; P \\; \\text{and} \\; Q\\; \\text{in}\\; \\IR \\otimes_{\\IZ} H \\;\\text{with}\\; f = \\|\\; \\|_P - \\|\\; \\|_Q\\}.\n\\label{calsn(H)}\n\\end{multline}\nThis becomes an abelian group by $(f-g)(\\phi) = f(\\phi) - g(\\phi)$ because \nof~\\eqref{seminorm_of_a_polytope_is_additive_in_polytope}. Again because \nof~\\eqref{seminorm_of_a_polytope_is_additive_in_polytope}\nwe obtain an epimorphism of abelian groups\n\\begin{equation}\n\\sn \\colon \\calp_{\\IZ,\\Wh}(H) \\to \\calsn(H) \n\\label{norm_homomorphism}\n\\end{equation}\nby sending $[P] - [Q]$ for\ntwo polytopes $P,Q \\subseteq \\IR \\otimes_{\\IZ} H$ to the function\n\\[\n\\Hom_{\\IZ}(H,\\IR) \\to \\IR, \\quad \\phi \\mapsto \\|\\phi \\|_P - \\|\\phi \\|_Q.\n\\]\n\n\n\n\n\nConsider any finitely generated abelian group $H$ and group homomorphisms $\\nu \\colon G \\to H$ and\n$\\phi \\colon H \\to \\IZ$ such that $\\phi$ is surjective. Define a homomorphism\n\\begin{equation}\nD_{\\nu,\\phi} \\colon \\cald(G)^{\\times}_{\\abel} \\xrightarrow{\\IP'_{\\nu} }\\calp_{\\IZ}(H)\n\\xrightarrow{\\sn} \\calsn(H) \\xrightarrow{\\ev_{\\phi}} \\IR\n\\label{D_(nu,phi)}\n\\end{equation}\nto be the composite of the homomorphism defined in~\\eqref{P_prime_colon_cald_to_p_Z}\nand~\\eqref{norm_homomorphism} and the evaluation homomorphism $\\ev_{\\phi}$.\n\n\\begin{lemma} \\label{lem:D_(mu,psi_circ_omega_is_D_(omega_circ_nu,psi)}\nConsider finitely generated free abelian groups $H$ and $H'$ and surjective group homomorphisms\n$\\nu \\colon G \\to H$, $\\omega \\colon H \\to H'$ and a group homomorphism $\\psi \\colon H' \\to \\IZ$. \nThen we get the following equality of homomorphisms $\\cald(G)^{\\times}_{\\abel} \\to \\IR$\n\\[\nD_{\\mu,\\psi \\circ \\omega} \n= \nD_{\\omega \\circ \\nu,\\psi}.\n\\]\n\\end{lemma}\n\\begin{proof}\nChoose a set-theoretic section $s \\colon H \\to G$ of $\\nu$ and\na set-theoretic section $t \\colon H' \\to H$ of $\\omega$. Then $s \\circ t \\colon H' \\to G$\nis a set-theoretic section of $\\omega \\circ \\nu \\colon G \\to H'$. Let $K_{\\nu} \\subseteq G$ \nbe the kernel of $\\nu$, $K_{\\omega \\circ \\nu} \\subseteq G$ be the kernel of \n$\\omega \\circ \\nu$ and $K_{\\omega} \\subseteq H$ be the kernel of $\\omega$. Let \n$k \\colon K_{\\nu} \\to K_{\\omega \\circ \\nu}$ be the inclusion. We obtain an exact sequence \n$0 \\to K_{\\nu} \\xrightarrow{k} K_{\\omega \\circ \\nu} \\xrightarrow{\\nu|_{K_{\\omega \\circ \\nu}}}\nK_{\\omega}\\to 0$ of groups such that $K_{\\omega}$ is finitely generated free abelian. The\nsection $s$ induces a section $s|_{K_{\\omega \\circ \\nu}} \\colon K_{\\omega} \\to K_{\\omega \\circ \\nu}$ \nof $\\omega|_{K_{\\omega \\circ \\nu}} \\colon K_{\\omega \\circ \\nu} \\to K_{\\omega}$. We also have the exact sequence \n$0 \\to K_{\\omega \\circ \\nu} \\xrightarrow{l} G \\xrightarrow{\\omega \\circ \\nu} H' \\to 0$ \nfor $l$ the inclusion and the set-theoretic section $s \\circ t$ of $\\omega \\circ \\nu$. \nThus we get isomorphisms of skew fields\n\\begin{eqnarray*}\n\\widehat{j_s} \\colon T^{-1} \\cald(K_{\\nu}) \\ast_s H\n& \\xrightarrow{\\cong} & \n\\cald(G);\n\\\\\n\\widehat{k_{s|_{K_{\\omega}}}} \\colon T^{-1} \\cald(K_{\\nu}) \\ast_{s|_{K_{\\omega}}} K_{\\omega}\n& \\xrightarrow{\\cong} & \n\\cald(K_{\\omega \\circ \\nu});\n\\\\\n\\widehat{l_t} \\colon T^{-1} \\cald(K_{\\omega \\circ \\nu}) \\ast_{s \\circ t} H'\n& \\xrightarrow{\\cong} & \n\\cald(G),\n\\end{eqnarray*}\nwhere $T^{-1}$ always indicates the Ore localization with respect to the non-trivial elements.\nConsider $u = \\sum_{h \\in H} x_h \\cdot h$ in $\\cald(K_{\\nu}) \\ast_s H$. \nFor $h' \\in H'$ define an element in $\\cald(K_{\\nu}) \\ast_{s|_{K_{\\omega}}} K_{\\omega}$ by\n\\[\nu_{h'} = \\sum_{h \\in K_{\\omega}} \\bigl(x_{h \\cdot t(h')} \\cdot s(h \\cdot t(h') ) \\cdot s \\circ t(h')^{-1} \\cdot s(h)^{-1}\\bigr) \\cdot h.\n\\]\nIt is well-defined since $s(h \\cdot t(h') ) \\cdot s \\circ t(h')^{-1} \\cdot s(h)^{-1} \\in K_{\\nu}$ holds.\nDefine an element in $\\cald(K_{\\omega \\circ \\nu}) \\ast_{s \\circ t} H'$ by\n\\[\nv = \\sum_{h' \\in H'} \\widehat{k_{s|_{\\ker_{\\omega}}}} (u_{h'}) \\cdot h'.\n\\]\nThen we compute in $\\cald(G)$\n\\begin{eqnarray*}\n\\widehat{j_s}(u) \n& = & \n\\lmsum{h \\in H}{} x_h \\cdot s(h)\n\\,\\, = \\,\\, \n\\lmsum{h' \\in H'}{} \\lmsum{h \\in \\omega^{-1}(h')}{} x_h \\cdot s(h)\n\\\\\n& = & \n\\lmsum{h' \\in H'}{} \\bigg(\\,\\,\\lmsum{h \\in \\omega^{-1}(h')}{} x_h \\cdot s(h) \\cdot s \\circ t(h')^{-1} \\bigg) \\cdot s \\circ t(h')\n\\\\\n& = & \n\\lmsum{h' \\in H'}{} \\bigg(\\,\\lmsum{h \\in K_{\\omega}}{} x_{h \\cdot t(h')} \\cdot s(h \\cdot t(h') ) \\cdot s \\circ t(h')^{-1} \\bigg) \\cdot s \\circ t(h')\n\\\\\n& = & \n\\lmsum{h' \\in H'}{} \\bigg(\\,\\lmsum{h \\in K_{\\omega}}{} \\bigl(x_{h \\cdot t(h')} \n\\cdot s(h \\cdot t(h') ) \\cdot s \\circ t(h')^{-1} \\cdot s(h)^{-1}\\bigr) \\cdot s(h) \\bigg) \\cdot s \\circ t(h')\n\\\\\n& = & \n\\lmsum{h' \\in H'}{} \\widehat{k_{s|_{\\ker(\\omega)}}}(u_{h'}) \\cdot s \\circ t(h')\n\\,\\, = \\,\\,\n\\widehat{l_t}\\bigg(\\,\\lmsum{h' \\in H'}{} \\widehat{k_{s|_{\\ker(\\omega)}}}(u_{h'}) \\cdot h'\\bigg)\n\\,\\, = \\,\\,\n\\widehat{l_t}(v).\n\\end{eqnarray*}\nObviously we get for $h' \\in H'$\n\\[u_{h'} \\not= 0 \\Leftrightarrow \\exists h \\in \\omega^{-1}(h') \\; \\text{with}\\; x_h \\not= 0.\n\\]\nThis implies\n\\begin{multline*}\n\\sup\\{\\psi(h') - \\psi(k') \\mid h',k' \\in H', u_{h'} \\not= 0, u_{k'} \\not = 0\\}\n\\\\\n=\n\\sup\\{\\psi \\circ \\omega(h) - \\psi \\circ \\omega (k) \\mid h,k \\in H, x_{h} \\not= 0, x_{k} \\not = 0\\}.\n\\end{multline*}\nHence we get for the element $z \\in \\cald(G)$ given by $z = \\widehat{j_s}(u) = \\widehat{l_t}(v)$\n\\[\nD_{\\omega \\circ \\nu,\\psi}(z) = D_{\\nu,\\psi \\circ \\omega}(z).\n\\]\nSince $D_{\\omega \\circ \\nu,\\psi}$ and $D_{\\nu,\\psi \\circ \\omega}$ are homomorphisms, \nLemma~\\ref{lem:D_(mu,psi_circ_omega_is_D_(omega_circ_nu,psi)}\nfollows.\n\\end{proof}\n\n\nRecall from Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)} \nthat $\\cald(G)$ is the Ore localization of with respect to the set of non-zero\nelements of the ring $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ of twisted Laurent polynomials in the variable $u$\nwith coefficients in the skew-field $\\cald(K_{\\phi \\circ \\nu})$. Hence $\\cald(K_{\\phi \\circ\n  \\nu})_{t}[u^{\\pm 1}]$ is contained in $\\cald(G)$ and we can consider for any $x \\in\n\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ with $x \\not= 0$ its image $D_{\\nu,\\phi}([x])\n\\in \\IR$ under the homomorphism $D_{\\nu,\\phi}$ defined in~\\eqref{D_(nu,phi)}.\n\n\n\\begin{lemma} \\label{lem:image_of_an_element_x_under_D_nu,phi}\nConsider an element $x \\in \\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ with $x \\not = 0$.\n\nThen \n\\[\nD_{\\nu,\\phi}([x])\\, =\\,\\tmfrac{1}{2} \\deg(x),\n\\]\nwhere $\\deg(x)$ has been defined to be $k_+ -k_-$ if we write\n$x = \\sum_{k = k_-}^{k_+} z_n \\cdot u^k$ with $z_{k_+},z_{k_-} \\not= 0$.\n\\end{lemma}\n\\begin{proof}\nRecall that $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ does depend on a choice of a preimage of $1$ under $\\phi \\circ \\nu \\colon G \\to \\IZ$\nwhich is the same as a choice of a group homomorphism $t \\colon \\IZ \\to G$ with $\\phi \\circ \\nu \\circ t = \\id_{\\IZ}$.\nChoose a set theoretic map $s \\colon H \\to G$ with $\\nu \\circ s = \\id_H$.\nOne easily checks that $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ agrees with\n$\\cald(K_{\\phi \\circ \\nu}) \\ast_{t} \\IZ$. \nWe conclude $D_{\\nu,\\phi}(x) = D_{\\nu \\circ \\phi,\\id_{\\IZ}}(x)$ from Lemma~\\ref{lem:D_(mu,psi_circ_omega_is_D_(omega_circ_nu,psi)}. \nNow one easily checks $D_{\\phi \\circ \\nu,\\id_{\\IZ}}(x) = \\frac{1}{2} \\cdot \\deg(x)$ by inspecting the definitions, since for a polynomial\n$\\sum_{k = k_-}^{k_+} z_k u^k$ in one variable $u$ with $z_{k_-} ,z_{k_+} \\not= 0$ its Newton polytop is the interval\n$[k_-,k_+] \\subseteq \\IR$. (The factor $1\/2$ comes from the factor $1\/2$ in~\\eqref{seminorm_of_a_polytope}.)\n\\end{proof}\n\n\n\nThere is a Dieudonn\\'e determinant for invertible matrices over a skew field $D$ which takes\nvalues in the abelianization of the group of units $D^{\\times}_{\\abel}$ \nand induces an isomorphism, see~\\cite[Corollary~4.3 in page~133]{Silvester(1981)} \n\\begin{eqnarray}\n{\\det}_D \\colon K_1(D) \n& \\xrightarrow{\\cong} &\nD^{\\times}_{\\abel}\n\\label{K_1(K)_Dieudonne}\n\\end{eqnarray}\nThe inverse \n\\begin{eqnarray}\nJ_D \\colon D^{\\times}_{\\abel} & \\xrightarrow{\\cong} & K_1(D) \n\\label{K_1(K)_Dieudonne_inverse}\n\\end{eqnarray}\nsends the class of a unit in $D$ to the class of the corresponding $(1,1)$-matrix. \n\n\n\\begin{lemma} \\label{matrices_overcald(K_phi_circ_nu)_t(u,u(-1))}\nLet $A \\in \\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ be a matrix \nwhich becomes invertible over $\\cald(G)$.\nThen the composite of the homomorphisms defined in~\\eqref{D_(nu,phi)} and~\\eqref{K_1(K)_Dieudonne} \n\\[K_1(\\cald(G)) \\xrightarrow{\\det_{\\cald{G)}}} \\cald(G)^{\\times}_{\\abel} \n\\xrightarrow{D_{\\nu,\\phi}} \\IR\n\\]\nsends the class $[A] \\in K_1(\\cald(G))$ of $A$ to\n\\[\n\\tmfrac{1}{2}\\dim_{\\cald(K_{\\phi \\circ \\nu})}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]^n \\to \\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]^n\\bigr)\\bigr).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nThe twisted polynomial ring $\\cald(K_{\\phi \\circ \\nu})_{t}[u]$ has a Euclidean function given by\nthe degree and hence there is a Euclidean algorithm with respect to it. This algorithm\nallows to transform $A$ to a diagonal matrix over $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$ \nby the following operations\n\\begin{enumerate}\n\\item \nPermute rows or columns;\n\n\\item \nMultiply a row on the right or a column on the left\nwith an element of the shape $y u^m$ for some $y \\in \\cald(K_{\\phi \\circ \\nu})$ with $y \\not = 0$ and $m \\in \\IZ$;\n\n\\item \nAdd a right $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$-multiple of a row to another row and analogously for columns;\n\\end{enumerate}\nThese operations change the class $[A]$ of $A$ in $K_1(\\cald(G))$ by adding an element of the shape $J_{\\cald(G)}([yu^m])$ for \n$y \\in \\cald(K_{\\phi \\circ \\nu})$ with $y \\not = 0$ and $m \\in \\IZ$ for the homomorphism \n$J_{\\cald(G)}$ of~\\eqref{K_1(K)_Dieudonne_inverse}. Moreover, they do not change the \nkernel of the cokernel of $r_A$ since $yu^m$ is unit in $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$.\nSince $D_{\\nu,\\phi} [yu^m]) = 0$ follows from Lemma~\\ref{lem:image_of_an_element_x_under_D_nu,phi}, \nit suffices to treat the special case, \n where $A$ is a diagonal matrix over $\\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]$\nwith non-trivial entries on the diagonal.\n\nLet $x$ be the product of the diagonal entries of the diagonal matrix $A$. \nWe get in $\\cald(G)^{\\times}_{\\abel}$\n\\[\n{\\det}_{\\cald(G)^{\\times}}([A]) = [x].\n\\]\nUsing the obvious exact sequence for $\\cald(G)$-maps \n$f_1 \\colon M_0 \\to M_1 $ and $f_2 \\colon M_1 \\to M_2$\n\\[\n0 \\to \\ker(f_1) \\to \\ker(f_2 \\circ f_1) \\to \\ker(f_2) \\to \\coker(f_1) \\to \\coker(f_2 \\circ f_1) \\to \\coker(f_2) \\to 0,\n\\]\nwe conclude \n\\[\n\\dim_{\\cald(K_{\\phi \\circ \\nu})}(\\coker(r_A)) = \\dim_{\\cald(K_{\\phi \\circ \\nu})}\\bigl(\\coker(r_x \\colon \\cald(G) \\to \\cald(G))\\bigr).\n\\]\nWe conclude from Lemma~\\ref{lem:degree_and_rank} and Lemma~\\ref{lem:image_of_an_element_x_under_D_nu,phi}. \n\\[\n\\tmfrac{1}{2}\\dim_{\\cald(K_{\\phi \\circ \\nu})}(\\coker(r_x)) \\,\\,=\\,\\,\\tmfrac{1}{2} \\deg(x)\\,\\, =\\,\\, D_{\\nu,\\phi}([x]) \\,\\,=\\,\\, D_{\\nu,\\phi} \\circ {\\det}_{\\cald(G)}([A]).\n\\]\nThis finishes the proof of Lemma~\\ref{matrices_overcald(K_phi_circ_nu)_t(u,u(-1))}.\n\\end{proof}\n\n\n\n\\begin{lemma} \\label{continuity_of(mu,phi)-L2-Euler_in_phi}\n Let $M$ be an admissible $3$-manifold. Let $G$ be a torsion-free group\n which satisfies the Atiyah Conjecture. Consider any\n factorization $\\pi \\xrightarrow{\\mu} G \\xrightarrow{\\nu} H_1(M)_f$ of the canonical\n projection $\\pi \\to H_1(M)_f$. Assume that $b_1^{(2)}(\\overline{M};\\caln(G)) = 0$ holds for the $G$-covering \n $\\overline{M} \\to M$ associated to $\\mu$.\n\n Then there exist two seminorms $s_1$ and $s_2$ on $H^1(M;\\IR)$\n such that we get for every $\\phi \\in H^1(M;\\IZ) = \\Hom_{\\IZ}(H_1(M)_f;\\IZ)$ \n \\[\n \\chi^{(2)}(M;\\mu,\\phi \\circ \\nu) \\,\\, =\\,\\, s_1(\\phi) - s_2(\\phi).\n \\]\n\\end{lemma}\n\\begin{proof}\n We treat only the case, where $\\partial M$ is non-empty, the case of empty $\\partial M$ is completely analogous.\n Let $x_1, x_2, \\ldots , x_a$ be the element in $G$ and $A$ be the $(a-1,a-1)$-matrix over $\\IZ G$ appearing\n Theorem~\\ref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation}\n\\eqref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation:non-empty}. \n (Notice that they are independent of $\\phi$.) We conclude from\n Theorem~\\ref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation}\n\\eqref{the:calculation_of_(mu,phi)_L2-Euler_characteristic_from_a_presentation:non-empty}\nthat for any surjective group homomorphism $\\phi \\colon G \\to \\IZ$ we have\n \\begin{multline*}\n \\chi^{(2)}(M;\\mu,\\phi) = |\\phi \\circ \\nu \\circ \\mu(x_i)|\n \\\\\n - \\dim_{\\cald(K_{\\phi \\circ \\nu})}\\bigl(\\coker\\bigl(r_A \\colon \\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]^n \n \\to \\cald(K_{\\phi \\circ \\nu})_{t}[u^{\\pm 1}]^n\\bigr)\\bigr).\n\\end{multline*}\n Choose two seminorms $s_1$ and $s_2$ such that the image of the class $2\\cdot [A]$ in $K_1(\\caln(G))$  under the composite\n \\[\n K_1(\\cald(G)) \\xrightarrow{\\det_{\\cald{G)}}} \\cald(G)^{\\times}_{\\abel} \n \\xrightarrow{\\IP'_{\\nu}} \\calp_{\\IZ}(H_1(M)_f) \\xrightarrow{\\sn} \\calsn(H_1(M)_f)\n \\]\n is $s_1 - s_2$. We get from the definitions that\n for any surjective group homomorphism $\\phi \\colon G \\to \\IZ$ \n  the image of $2\\cdot [A]$ under the composite\n $K_1(\\cald(G)) \\xrightarrow{\\det_{\\cald{G)}}} \\cald(G)^{\\times}_{\\abel} \\xrightarrow{D_{\\nu,\\phi}} \\IR$ equals $s_1(\\phi) - s_2(\\phi)$.\n We conclude from Lemma~\\ref{matrices_overcald(K_phi_circ_nu)_t(u,u(-1))}\n \\[\n \\chi^{(2)}(M;\\mu,\\phi) = s_1(\\phi) + |\\phi \\circ \\nu \\circ \\mu(s_i)| - s_2(\\phi),\n \\]\n provided that $\\phi$ is surjective. \n We conclude from Lemma~\\ref{lem:reduction_to_surjective_mu} that the last equation holds for\n every group homomorphism $\\phi \\colon G \\to \\IZ$.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{The quasi-fibered case}\n\\label{subsec:The_quasi-fibered_case}\n\n\\begin{definition}[Fibered and quasi-fibered]\n \\label{def:fibered_and_quasi_fibered}\n Let $M$ be a $3$-manifold and consider an element $\\phi\\in H^1(M;\\IQ)$. We say that\n $\\phi$ is \\emph{fibered} if there exists a locally trivial fiber bundle $p\\colon M\\to S^1$ \n and $k \\in \\IQ$, $k > 0$ such that the induced map\n $p_*\\colon \\pi_1(M)\\to \\pi_1(S^1)=\\IZ $ coincides with $k \\cdot \\phi$. We call an\n element $\\phi \\in H^1(M;\\IR)$ \\emph{quasi-fibered}, if there exists a sequence of\n fibered elements $\\phi_n \\in H^1(M;\\IQ)$ converging to $\\phi$ in $H^1(M;\\IR)$.\n\\end{definition}\n\n\n\n\n\\begin{theorem}[Equality of $(\\mu,\\phi)$-$L^2$-Euler characteristic and the\n Thurston norm in the quasi-fibered case]\n \\label{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_in_the_quasi-fibered_case}\n Let $M$ be an admissible $3$-manifold, which is not homeomorphic to \n $S^1 \\times D^2$. Let $G$ be a torsion-free group which satisfies the\n Atiyah Conjecture. Consider any factorization \n $\\pr_M \\colon \\pi_1(M) \\xrightarrow{\\mu} G \\xrightarrow{\\nu} H_1(M)_f$ of the canonical projection \n $\\pr_M$. Let $\\phi \\colon H_1(M)_f \\to \\IZ$ be a quasi-fibered homomorphism. \n\n Then $(\\mu,\\phi \\circ \\nu)$ is an $L^2$-acyclic Atiyah pair and we get\n \\[\n -\\chi^{(2)}(M;\\mu,\\phi \\circ \\nu)\\,\\, =\\,\\, x_M(\\phi).\n \\]\n\\end{theorem}\n\\begin{proof} \nChoose a sequence of fibered elements $\\phi_n \\in H^1(M;\\IQ)$ converging to $\\phi$ in $H^1(M;\\IR)$.\nFor each $n$ choose a locally trivial fiber bundle \n$F_n \\to M \\xrightarrow{p_n} S^1$ and an $k_n \\in \\IQ$, $k_n > 0$ such that the induced map\n$(p_n)_*\\colon \\pi_1(M)\\to \\pi_1(S^1)=\\IZ $ coincides with $k_n \\cdot \\phi_n$.\nSince there is a finite covering $F_n \\to F_n'$ for a connected surface $F_n'$,\na locally trivially fiber bundle $F_n' \\to M \\to S^1$, and \n$M$ is not homeomorphic to $S^1 \\times S^2$ or $S^1 \\times D^2$,\nwe have $\\chi(F_n') \\le 0$ and hence $\\chi(F_n) \\le 0$.\nWe conclude from~\\eqref{fiber_bundles_Thurston_norm}, Example~\\ref{exa_mapping_torus},\nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:3-manifold_not_graph},\nand~\\cite[Theorem~2.1]{Lueck(1994b)} that \n$(\\mu,k_n \\cdot \\phi_n)$ is an $L^2$-acyclic Atiyah-pair for $M$ and\n\\begin{equation}\n-\\chi^{(2)}(M;\\mu,(k_n \\cdot \\phi_n) \\circ \\nu) = -\\chi(F_n) = x_{M}(k_n \\cdot \\phi_n).\n\\label{Coman}\n\\end{equation}\nLet $s_1$ and $s_2$ be the two seminorms appearing in Lemma~\\ref{continuity_of(mu,phi)-L2-Euler_in_phi}.\nRecall that we have for every $\\psi \\in H^1(M;\\IZ)$\n\\begin{equation}\n\\chi^{(2)}(M;\\mu,\\psi \\circ \\nu) \\,\\, =\\,\\, s_1(\\psi) - s_2(\\psi).\n\\label{Costa}\n\\end{equation}\nSince any seminorm on $H^1(M;\\IR)$ is continuous, we get\n\\begin{multline*}\nx_M(\\phi) \n= \n\\lim_{n \\to \\infty} x_M(\\phi_n)\n= \n\\lim_{n \\to \\infty} \\smfrac{x_M(k_n \\cdot \\phi_n)}{k_n}\n\\\\\n\\stackrel{\\eqref{Coman}}{=}\n\\lim_{n \\to \\infty} \\smfrac{-\\chi^{(2)}(M;\\mu,(k_n \\cdot \\phi_n) \\circ \\nu)}{k_n}\n\\stackrel{\\eqref{Costa}}{=}\n\\lim_{n \\to \\infty} \\smfrac{-s_1(k_n \\cdot \\phi_n) +s_2(k_n \\cdot \\phi_n)}{k_n}\n\\\\\n= \n\\lim_{n \\to \\infty} -s_1(\\phi_n) +s_2(\\phi_n)\n = \n-s_1(\\phi) +s_2(\\phi)\n\\stackrel{\\eqref{Costa}}{=} \n\\chi^{(2)}(M;\\mu,\\phi).\\qedhere\n\\end{multline*}\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}}\n\\label{subsec:Proof_of_Theorem_ref(the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm)}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}]\nThis is a variation of the proof of~\\cite[Theorem~5.1]{Friedl-Lueck(2015l2+Thurston)}. For the reader's convenience\nwe give some details here as well.\n\nAs explained in~\\cite[Section~10]{Dubois-Friedl-Lueck(2014Alexander)}, we conclude from\n combining~\\cite{Agol(2008),Agol(2013),Liu(2013),Przytycki-Wise(2012),Przytycki-Wise(2014),Wise(2012raggs),Wise(2012hierachy)}\n that there exists a finite regular\n covering $p \\colon N \\to M$ such that for any $\\phi \\in H^1(M;\\IR)$ its pullback\n $p^*\\phi \\in H^1(N;\\IR)$ is quasi-fibered. Let $k$ be the number of sheets of $p$.\n Let $\\pr_N \\colon \\pi_1(N) \\to H_1(N)_f$ be\n the canonical projection. Its kernel is a characteristic subgroup of $\\pi_1(N)$. The\n regular finite covering $p$ induces an injection $\\pi_1(p) \\colon \\pi_1(N) \\to \\pi_1(M)$\n such that the image of $\\pi_1(p)$ is a normal subgroup of $\\pi_1(M)$ of finite index.\n Let $\\Gamma$ be the quotient of $\\pi_1(M)$ by the normal subgroup\n $\\pi_1(p)(\\ker(\\pr_N))$. Let $\\alpha \\colon \\pi_1(M) \\to \\Gamma$ be the\n projection. Since $\\pi_1(p)(\\ker(\\pr_N))$ is contained in the kernel of the canonical\n projection $\\pr_M \\colon \\pi_1(M)\\to H_1(M)_f$ because of \n $H_1(p;\\IZ)_f \\circ \\pr_N = \\pr_M \\circ \\, \\pi_1(p)$, there exists precisely one epimorphism \n $\\beta \\colon \\Gamma \\to H_1(M)_f$ satisfying $\\pr_M = \\beta \\circ \\alpha$.\n Moreover, $\\alpha \\circ \\pi_1(p)$ factorizes over $\\pr_M$ into an injective homomorphism\n $j \\colon H_1(M)_f \\to \\Gamma$ with finite cokernel. \n Hence $\\Gamma$ is virtually finitely generated free abelian.\n\n\n\n Consider a factorization of $\\alpha \\colon \\pi_1(M) \\to \\Gamma$\ninto group homomorphisms $\\pi_1(M) \\xrightarrow{\\mu} G \\xrightarrow{\\nu} \\Gamma$ \nfor a group $G$ which satisfies the Atiyah Conjecture. Let $G' $ be the quotient of\n$\\pi_1(N)$ by the normal subgroup $\\pi_1(p)^{-1}(\\ker(\\mu))$ and\n$\\mu' \\colon \\pi_1(N) \\to G'$ be the projection. Since the kernels of\n$\\mu'$ and of $\\mu \\circ \\pi_1(p)$ agree, there is precisely\none injective group homomorphism $i \\colon G' \\to G$ satisfying \n$\\mu \\circ \\pi_1(p) = i \\circ \\mu'$. The kernel of $\\mu'$ is contained in the kernel of \n$\\pr_N \\colon \\pi_1(N) \\to H_1(N)_f$ since $j$ is injective and we have\n$j \\circ \\pr_N = \\nu \\circ i \\circ \\mu'$. Hence there is precisely one group homomorphism\n$\\nu' \\colon G' \\to H_1(N)_f$ satisfying $\\nu' \\circ \\mu' = \\pr_N$. One easily checks that \nthe following diagram commutes, and all vertical arrows are injective, the indices \n$[\\pi_1(N): \\im(\\pi_1(p)]$ and $[\\Gamma: H_1(MN_f]$ are finite,\nand $\\mu'$, $\\nu'$ and $\\beta$ are surjective.\n\\[\n\\xymatrix{\\pi_1(N) \\ar[r]^-{\\mu'} \\ar[d]^{\\pi_1(p)} \\ar@\/^{5mm}\/[rr]^{\\pr_N}\n& \nG' \\ar[r]^-{\\nu'} \\ar[d]^i\n& \nH_1(N)_f \\ar[rd]^{H_1(p)_f} \\ar[d]^j\n\\\\\n\\pi_1(M) \\ar[r]^-{\\mu} \\ar@\/_{5mm}\/[rr]_{t} \\ar@\/_{10mm}\/[rrr]_{\\pr_M}\n& \nG \\ar[r]^{\\nu}\n&\n\\Gamma \\ar[r]^-{\\beta}\n& \nH_1(M)_f \n}\n\\]\n\nSince $G$ satisfies the Atiyah Conjecture, the group $G'$ \nsatisfies the Atiyah Conjecture by \nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:subgroups}.\n\nSince $\\ker(\\mu) \\subseteq \\ker(\\alpha) \\subseteq \\im(\\pi_1(p))$ holds, we get $[G:G'] = k$ conclude \nfrom~\\eqref{finite_coverings_Thurston_norm} and from\nLemma~\\ref{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings}\n\\eqref{the:Basic_properties_of_the_phi_L2-Euler_characteristic_for_universal_coverings:finite_coverings}\n\\begin{eqnarray*}\n -\\chi^{(2)}(M;\\mu,\\phi \\circ \\beta \\circ \\nu) \n& = & \n -\\lmfrac{\\chi^{(2)}(N;\\mu',p^*\\phi \\circ \\nu')}{k};\n\\\\\nx_M(\\phi) \n& = & \n\\lmfrac{x_N(p^*\\phi)}{k}.\n\\end{eqnarray*}\nWe get from\nTheorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_in_the_quasi-fibered_case}\napplied to $N$, $\\mu'$, $\\nu'$ and $p^*\\phi$\n\\[\n -\\chi^{(2)}(N;\\mu',p^*\\phi \\circ \\nu')\\,\\, =\\,\\, x_N(p^*\\phi).\n\\]\nHence we get \n\\[\n -\\chi^{(2)}(M;\\mu,\\phi \\circ \\beta \\circ \\nu)\\,\\,=\\,\\, x_M(\\phi).\n\\]\nThis finishes the proof of Theorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm}.\n\\end{proof}\n\n\n\n\n\n\n \\typeout{---- Section 7: Epimorphism of fundamental groups and the Thurston norm}\n\n\\section{Epimorphism of fundamental groups and the Thurston norm}\n\\label{sec:Epimorphism_of_fundamental_groups_and_the_Thurston_norm}\n\n\n\n\nOn several occasions we will use the following lemma.\n\n \\begin{lemma} \\label{lem:elementary_amenable_and_ore} \n Let $G$ be a torsion-free elementary amenable group.\n Then the Ore localization \n $T^{-1} \\IQ G$ for $T$ the set of non-trivial elements in $\\IQ G$ exists\n and agrees with the skew field $\\cald(G)$. In particular $\\cald(G)$ is flat over $\\IQ G$.\n Moreover, $G$ satisfies the Atiyah Conjecture and we get for every finitely generated free $\\IQ G$-chain complex\n \\[\n b_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*) \\,\\, =\\,\\, \\dim_{\\cald(G)}(H_n(\\cald(G) \\otimes_{\\IQ G} C_*)).\n \\] \n\\end{lemma}\n\n \\begin{proof} This follows from \n Theorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:Linnell},\n Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):dim}\n and~\\cite[ Example~8.16 on page~324 and Lemma~10.16 on page~376]{Lueck(2002)}.\n The proofs there deal only with $\\IC$, but carry over without changes to any field $F$ with $\\IQ \\subseteq F \\subseteq \\IC$.\n\\end{proof}\n\n \n\n\n\\begin{theorem} \\label{the:localization_down_to_Z_versus_up_to_U(G)_new} Let $G$ be a group\n which is residually a locally indicable elementary amenable group. Let $f_* \\colon C_* \\to\n D_*$ be a $\\IQ G$-chain map of finitely generated free $\\IQ G$-chain complexes.\n Suppose that $\\id_{\\IQ} \\otimes_{\\IQ G} f_* \\colon \\IQ\\otimes_{\\IQ G} C_* \\to\n \\IQ\\otimes_{\\IQ G} D_*$ induces an isomorphism on homology. Then we get for $n \\ge 0$\n\\[\nb_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*)\\,\\, =\\,\\, b_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} D_*).\n\\]\n\\end{theorem}\n\\begin{proof}\n By considering the mapping cone, it suffices to show for a finitely generated free\n $\\IQ G$-chain complex $C_*$ that $b_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*)$ vanishes \nfor all $n \\ge 0$ if $H_n(\\IQ \\otimes_{\\IQ G} C_*)$ vanishes for all $n \\ge 0$. \n \nSince $G$ is residually a torsion-free locally indicable elementary amenable group,\nthere exists a sequence of epimorphisms $G\\to G_i$, $i\\in \\IN$ onto \ntorsion-free locally indicable elementary amenable groups such that the intersections of the kernels is trivial. We\nconclude from~\\cite{Clair(1999)},~\\cite[Theorem~1.14]{Schick(2001b)} \nor~\\cite[Theorem~13.3 on page~454]{Lueck(2002)}\n\\begin{eqnarray*}\nb_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*) & = & \n\\lim_{i \\in \\IN} \\;b_n^{(2)}(\\caln(G_i) \\otimes_{\\IQ G_i} \\IQ G_i \\otimes_{\\IQ G} C_*);\n\\\\\n\\IQ \\otimes_{\\IQ G} C_* & \\cong & \\IQ \\otimes_{\\IQ G_i} \\IQ G_i \\otimes_{\\IQ G} C.\n\\end{eqnarray*}\nIt follows that without loss\nof generality we can assume that $G$ itself is torsion-free locally indicable elementary amenable.\n\n\nThere is an involution of rings $\\IQ G \\to \\IQ G$ sending $\\sum_{g \\in G} \\lambda_g \\cdot g$ \nto $\\sum_{g \\in G} \\lambda_g \\cdot g^{-1}$. In the sequel we equip each\n$C_n$ with a $\\IQ G$-basis. With respect to this involution and $\\IQ G$-basis one can define the\ncombinatorial Laplace operator $\\Delta_n \\colon C_n \\to C_n$ which is the $\\IQ G$-linear\nmap given by $c_{n+1} \\circ c_n^* + c_{n-1}^* \\circ c_{n-1}$. Since the augmentation\nhomomorphism $\\IQ G \\to \\IQ$ sending $\\sum_{g \\in G} \\lambda_g \\cdot g $ to \n$\\sum_{g \\in G} \\lambda_g$ is compatible with the involution, \n$\\id_{\\IQ} \\otimes_{\\IQ G} \\Delta_n \\colon \\id_{\\IQ} \\otimes_{\\IQ G} C_n \\to \\id_{\\IQ} \\otimes_{\\IQ G} C_n$ \nis the combinatorial Laplace operator for $\\IQ \\otimes_{\\IQ G} C_*$. We conclude\nfrom~\\cite[Lemma~1.18 on page~24 and Theorem~6.25 on page~249]{Lueck(2002)}\n\\begin{eqnarray*}\nb_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*) & = & \n\\dim_{\\caln(G)}(\\ker(\\id_{\\caln(G)} \\otimes_{\\IQ G} \\Delta_n));\n\\\\\n\\dim_{\\IQ}(H_n(\\IQ \\otimes_{\\IQ G} C_*) & = & \\dim_{\\IQ}(\\ker(\\id_{\\IQ} \\otimes_{\\IQ G} \\Delta_n)).\n\\end{eqnarray*}\nSince  $G$ is amenable, we conclude \nfrom~\\cite[(6.74) on page~275,]{Lueck(2002)}\n\\begin{eqnarray*}\n\\dim_{\\caln(G)}(\\ker(\\id_{\\caln(G)} \\otimes_{\\IQ G} \\Delta_n))\n& = & \n\\dim_{\\caln(G)}(\\caln(G) \\otimes_{\\IQ G} \\ker(\\Delta_n)).\n\\end{eqnarray*}\nHence $b_n^{(2)}(\\caln(G) \\otimes_{\\IQ G} C_*) $ vanishes if $\\Delta_n$ is injective. The injectivity of $\\Delta_n$ follows from the\ninjectivity of $\\id_{\\IQ} \\otimes_{\\IQ G} \\Delta_n$ by~\\cite[Theorem~1]{Howie-Schneebeli(1983)} \nor~\\cite[Theorem~4.1]{Gersten(1983)}, since $G$ is locally indicable.\n\\end{proof}\n\n\\begin{theorem} \\label{the:Inequality} Let $f \\colon M \\to N$ be a map of admissible\n $3$-manifolds. Suppose that $\\pi_1(f)$ is surjective and $f$ induces an isomorphism \n $H_n(f;\\IQ) \\colon H_n(M;\\IQ) \\to H_n(N;\\IQ)$ for $n \\ge 0$. Suppose that $G$ is residually\n locally indicable elementary amenable. Let $\\mu \\colon \\pi_1(N) \\to G$, \n $\\nu \\colon G \\to H_1(N)_f$ and $\\phi \\colon H_1(N)_f \\to \\IZ$ be group \n homomorphisms. Let $\\overline{N} \\to N$ be the $G$-covering associated to $\\mu$ and\n $\\overline{M} \\to M$ be the $G$-covering associated to $\\mu \\circ \\pi_1(f)$. Suppose that\n $b_n^{(2)}(\\overline{N};\\caln(G))$ vanishes for $n \\ge 0$. \n\n Then $b_n^{(2)}(\\overline{M};\\caln(G))$ vanishes \n for $n \\ge 0$, $M$ is $(\\mu \\circ \\pi_1(f),\\phi \\circ \\nu)$-$L^2$-finite, \n $N$ is $(\\mu,\\phi \\circ \\nu)$-$L^2$-finite and we get\n \\[\n - \\chi^{(2)}(M;\\mu \\circ \\pi_1(f);\\phi \\circ \\nu) \\,\\,\\ge\\,\\, - \\chi^{(2)}(N;\\mu,\\phi \\circ \\nu).\n \\]\n\\end{theorem}\n\\begin{proof}\n Since a locally indicable group is torsion-free,\n $G$ is a residually torsion-free elementary amenable group and hence satisfies the Atiyah Conjecture\n by Theorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:approx}.\n Because of Lemma~\\ref{lem:reduction_to_surjective_mu} and \n Theorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:subgroups}\n we can assume without loss of generality\n that $\\mu$ and $\\phi \\circ \\nu$ are epimorphisms.\n Theorem~\\ref{the:localization_down_to_Z_versus_up_to_U(G)_new} implies that\n $b_n^{(2)}(\\overline{M};\\caln(G))$ vanishes for $n \\ge 0$. We conclude from \n Theorem~\\ref{the:Atiyah_and_(mu,phi)-L2-Euler_characteristic}\n that $M$ is $(\\mu \\circ \\pi_1(f),\\phi \\circ \\nu)$-$L^2$-finite and $N$ is $(\\mu,\\phi\\circ \\nu)$-$L^2$-finite. \n \n Since $\\pi_1(f)$ is surjective and hence the $G$-map $\\overline{f} \\colon \\overline{M} \\to\n \\overline{N}$ induced by $f$ is $1$-connected, we get \n $b_1^{(2)}(i^*\\overline{M};\\caln(K)) \\ge b_1^{(2)}(i^* \\overline{N};\\caln(K))$ for the inclusion \n $i \\colon K = \\ker(\\phi \\circ \\nu) \\to G$. If $\\phi \\circ \\nu \\circ \\mu = 0$, we conclude\n $\\chi^{(2)}(M;\\mu \\circ \\pi_1(f);\\phi \\circ \\nu) = \\chi^{(2)}(N;\\mu,\\phi \\circ \\nu) = 0$ from \n Lemma~\\ref{lem:reduction_to_surjective_mu}\n \\eqref{lem:reduction_to_surjective_mu:mu_circ_phi_is_trivial} and the claim follows.\n Hence we can assume without loss of generality that $\\phi \\circ \\nu \\circ \\mu$ is not trivial.\n \n We begin with the case $\\im(\\mu) \\cap \\ker(\\phi \\circ \\nu) \\not= \\{1\\}$. Then also \n $\\im(\\mu \\circ \\pi_1(f)) \\cap \\ker(\\phi \\circ \\nu) \\not= \\{1\\}$. We conclude from \n Theorem~\\ref{the:The_(mu,phi)-L2-Euler_characteristic_in_terms_of_the_first_homology} \n \\begin{eqnarray*}\n -\\chi^{(2)}(N;\\mu,\\phi \\circ \\nu) & = & b_1^{(2)}(i^*\\overline{N};\\caln(K));\n \\\\\n -\\chi^{(2)}(M;\\mu \\circ \\pi_1(f);\\phi \\circ \\nu) & = & b_1^{(2)}(i^*\\overline{M};\\caln(K)).\n \\end{eqnarray*}\n and Theorem~\\ref{the:Inequality} follows. \n\n It remains to treat the case, where $\\im(\\mu) \\cap \\ker(\\phi \\circ \\nu) = \\{1\\}$.\n Then $\\phi \\circ \\nu \\colon G \\to \\IZ$\n is an injection and hence a bijection, $K = \\{0\\}$, and we get from Example~\\ref{exa:infinite_cyclic_covering}\n \\begin{eqnarray*}\n - \\chi^{(2)}(N;\\mu,\\phi \\circ \\nu) \\,\\,=\\,\\, \n \\begin{cases} \n \\dim_{Q}\\bigl(H_1(\\overline{M};\\IQ)\\bigr) - 1 & \\text{if}\\; \\partial M \\not= \\emptyset;\\\\\n \\dim_{Q}\\bigl(H_1(\\overline{M};\\IQ)\\bigr) - 2 & \\text{if}\\; \\partial M = \\emptyset;\n \\end{cases}\n \\\\\n - \\chi^{(2)}(N;\\mu \\circ \\pi_1(f),\\phi \\circ \\nu)\\,\\, =\\,\\, \n \\begin{cases} \n \\dim_{Q}\\bigl(H_1(\\overline{N};\\IQ)\\bigr) - 1 & \\text{if}\\; \\partial N \\not= \\emptyset;\\\\\n \\dim_{Q}\\bigl(H_1(\\overline{N};\\IQ)\\bigr) - 2 & \\text{if}\\; \\partial N = \\emptyset;\n \\end{cases}\n\\end{eqnarray*}\n We already have shown $b_1^{(2)}(i^*\\overline{N};\\caln(K)) \\ge b_1^{(2)}(i^* \\overline{M};\\caln(K))$ which boils down\n in this special case to $\\dim_{Q}(H_1(\\overline{M};\\IQ))\\bigr) \\ge \\dim_{Q}(H_1(\\overline{N};\\IQ))\\bigr)$. We conclude\n from $H_3(M;\\IQ) \\cong H_3(N;\\IQ)$ that $\\partial M$ is empty if and only if $\\partial N$ is empty. This finishes the proof\n of Theorem~\\ref{the:Inequality}.\n \\end{proof}\n\n\n\n\n\n\n\n\\begin{theorem}[Inequality of the Thurston norm] \\label{the:Inequality_of_the_Thurston_norm}\nLet $f \\colon M \\to N$ be a map of admissible $3$-manifolds which is surjective \non $\\pi_1(N)$ and induces an isomorphism \n$f_* \\colon H_n(M;\\IQ) \\to H_n(N;\\IQ)$ for $n \\ge 0$. Suppose that $\\pi_1(N)$ is residually \nlocally indicable elementary amenable. \n Then we get for any $\\phi\\in H^1(N;\\IR)$ that\n \\[\n x_M(f^*\\phi ) \\,\\,\\ge\\,\\, x_N(\\phi).\n \\]\n\\end{theorem}\n\n\\begin{proof}\nSince seminorms are continuous and homogeneous it suffices to prove the \nstatement for all primitive classes $\\phi\\in H^1(N;\\IZ)=\\Hom(\\pi_1(N),\\IZ)$. \nThe case $N = S^1 \\times D^2$ is trivial. Hence we can assume that $N\\ne S^1\\times D^2$. \n We conclude from Theorem~\\ref{the:Inequality} applied in the case $G = \\pi_1(N)$ and $\\mu= \\id_{\\pi_1(N)}$\n \\[\n -\\chi^{(2)}(M; \\pi_1(f),\\phi) \\,\\,\\ge\\,\\, -\\chi^{(2)}(\\widetilde{N};\\phi)\n \\]\n Theorem~\\ref{the:The_Thurston_norm_ge_the_(mu,phi)-L2-Euler_characteristic} implies \n \\[\n x_M(\\phi \\circ \\pi_1(f)) \\,\\,\\ge\\,\\,-\\chi^{(2)}(M; \\pi_1(f),\\phi)\n \\]\nand Theorem~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm_universal_covering_for_universal_coverings} implies that \n  \\[\n  -\\chi^{(2)}(\\widetilde{N};\\phi) \\,\\,=\\,\\, x_N(\\phi).\n \\]\n Now Theorem~\\ref{the:Inequality_of_the_Thurston_norm} follows.\n\\end{proof}\n\n\nThe following lemma shows that Theorem~\\ref{the:Inequality_of_the_Thurston_norm} applies in particular \nif the manifold $N$ is fibered. Since it is well-known, thus we only provide a sketch of the proof.\n\n\n\\begin{lemma}\\label{lem:fibered-manifolds-acapl}\nThe fundamental group of any fibered 3-manifold is residually locally indicable elementary amenable.\n\\end{lemma}\n\n\n\\begin{proof}[Sketch of proof]\n Let $N$ be a fibered 3-manifold. Then $\\pi_1(N)\\cong \\IZ\\ltimes_\\varphi \\Gamma$ where\n $\\Gamma$ is a free group or a surface group and where $\\varphi\\colon \\Gamma\\to \\Gamma$\n is an automorphism. The derived series of $\\Gamma$ is defined by $\\Gamma^{(0)}=\\Gamma$\n and inductively by $\\Gamma^{(n+1)}=[\\Gamma^{(n)},\\Gamma^{(n)}]$. Since $\\Gamma$ is a\n free group or a surface group, each quotient $\\Gamma^{(n)}\/\\Gamma^{(n+1)}$ is free\n abelian and $\\bigcap_{n \\ge 1} \\Gamma^{(n)}=\\{1\\}$.\n\n The subgroups $\\Gamma^{(n)}$ are characteristic subgroups of $\\Gamma$, in particular\n they are preserved by $\\varphi$. Thus $\\varphi$ descends to an automorphism on\n $\\Gamma\/\\Gamma^{(n)}$. It is now straightforward to see that the epimorphisms\n $\\pi_1(N)=\\IZ\\ltimes \\Gamma\\to \\IZ\\ltimes \\Gamma\/\\Gamma^{(n)}$, $n\\in \\IN$ form a\n cofinal nested sequence of epimorphisms onto locally indicable elementary amenable\n groups.\n\\end{proof}\n\n\n\n\n\n \\typeout{---- Section 8: The $\\phi$-$L^2$-Euler characteristic and the degree of higher order Alexander polynomials}\n\n\\section{The $(\\mu,\\phi)$-$L^2$-Euler characteristic and the degree of non-commutative Alexander polynomials}\n\\label{sec:The_(mu,phi)-L2-Euler_characteristic_and_the_degree_of_higher_order_Alexander_polynomials}\n\n\n\n\n\n\nLet $M$ be an admissible $3$-manifold. Regard group homomorphisms $\\mu \\colon \\pi_1(M)\\to G$, \n$\\nu \\colon G \\to H_1(M)_f$ and $\\phi \\colon H_1(M)_f \\to \\IZ$ such that $\\nu\n\\circ \\mu$ is the projection $\\pi_1(M) \\to H_1(M)_f$ and $G$ is torsion-free elementary amenable.\nFor simplicity we discuss only the\ncase, where $\\phi$ is surjective. Let $\\overline{M} \\to M$ be the $G$-covering associated \nto $\\mu \\colon \\pi_1(M) \\to G$. Recall the following definition from\nHarvey~\\cite{Harvey(2005)} which extends ideas of Cochran~\\cite{Cochran(2004)}.\n(Actually they consider only certain solvable quotients $G$ of $\\pi_1(M)$ coming from the\nrational derived series, but their constructions apply directly to torsion-free elementary\namenable groups.) Let $T$ be the set of non-trivial elements in $\\IZ G$. As recorded\nalready in Lemma~\\ref{lem:elementary_amenable_and_ore}, the Ore localization $T^{-1}\\IZ G$\nis defined and is a skewfield. Define a natural number\n\\begin{eqnarray}\nr_n(M;\\mu) \n:= \n\\dim_{T^{-1}\\IZ G}\\bigl(H_n\\bigl(T^{-1}\\IZ G\\otimes_{\\IZ G} C_*(\\overline{M})\\bigr)\\bigr).\n\\label{r_n(M,mu)}\n \\end{eqnarray}\nLet $i \\colon K \\to G$ be the inclusion of the kernel of the composite $\\phi \\circ \\nu \\colon G \\to \\IZ$. \nIf $T$ is the set of non-zero elements in $\\IZ K$, we can\nagain consider its Ore localization $T^{-1}\\IZ K$ which is a skew field. We obtain an isomorphism\nfor an appropriate automorphism $t$ of $T^{-1}\\IZ K$, which comes from the conjugation automorphism\nof $K$ associated to a lift of a generator of $\\IZ$ to $G$, an isomorphism of skew-fields\n\\begin{equation}\n (T^{-1}\\IZ K)_{t}[u^{\\pm 1}] \\xrightarrow{\\cong} T^{-1}\\IZ G.\n\\label{Sanchez}\n\\end{equation}\nIf $r_1(M;\\mu,\\nu,\\phi)$ vanishes for all $n \\ge 0$, then we can define natural numbers\n\\begin{equation}\n\\delta_n(M;\\mu, \\nu, \\phi) \n:= \n\\dim_{T^{-1}\\IZ K}\\bigl(H_1(T^{-1}\\IZ G \\otimes_{\\IZ G} C_*(\\overline{M})\\bigr).\n\\label{delta_n(phi)_surjective_tors}\n\\end{equation}\nThis construction and the invariants above turn out to be special cases of the constructions defined in this paper.\nNamely, $K$ and $G$ satisfy the Atiyah Conjecture by \nTheorem~\\ref{the:Status_of_the_Atiyah_Conjecture}~\\eqref{the:Status_of_the_Atiyah_Conjecture:Linnell},\nand Lemma~\\ref{lem:elementary_amenable_and_ore} shows that we get identifications\n$T^{-1}\\IZ K = \\cald(K)$ and $T^{-1}\\IZ G = \\cald(G)$ under which the isomorphism~\\eqref{Sanchez}\ncorresponds to the isomorphism appearing in \nTheorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):cald(K)_and_(cald(G)}.\nMoreover $r_n(M;\\nu)$ agrees with $b_n^{(2)}(\\overline{M};\\caln(G))$\nby Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):dim}.\nHence Theorem~\\ref{the:Main_properties_of_cald(G)}~\\eqref{the:Main_properties_of_cald(G):chain} \nand Lemma~\\ref{lem:b_1(2)_is_zero_implies_L2-acyclic}~\\eqref{lem:b_1(2)_is_zero_implies_L2-acyclic:b_1}\nimply\n\n\n\\begin{theorem}[The $(\\mu,\\phi)$-$L^2$-Euler characteristic and $\\delta_n(M,\\mu,\\nu,\\phi)$]\n \\label{the:The_mu,phi_L2-Euler_characteristic_and_delta_n(phi)}\n Let $M$ be an admissible $3$-manifold. Consider group homomorphisms\n $\\mu \\colon \\pi_1(M) \\to G$, $\\nu \\colon G \\to H_1(M)_f$ and $\\phi \\colon H_1(M)_f \\to \\IZ$\nsuch that $\\nu \\circ \\mu$ is the projection $\\pi_1(M) \\to H_1(M)_f$ and $\\phi$ is surjective.\n\n Then $r_1(M;\\mu)$ vanishes if and only if $(\\mu,\\phi \\circ \\nu)$ is an $L^2$-Atiyah pair, and in this case we get\n \\[ \\chi^{(2)}(M;\\mu,\\phi \\circ \\nu)~=~-\\delta(M;\\mu,\\nu,\\phi).\\]\n\\end{theorem}\n\n\n\\begin{remark} \\label{rem:extending_Harveys_invariant} Another way of interpreting\n Theorem~\\ref{the:The_mu,phi_L2-Euler_characteristic_and_delta_n(phi)} is to say that our\n $L^2$-Euler characteristic invariant extends the original invariant due to\n Cochran, Harvey and the third author~\\cite{Cochran(2004),Harvey(2005),Friedl(2007)} to other\n coverings, in particular to the universal covering or to a $G$-covering for residually\n torsion-free elementary amenable group $G$ of an admissible $3$-manifold.\n\\end{remark}\n\n\n\n\nThe following lemma might also be of independent interest.\n\n\\begin{lemma} \\label{lem:G_can_be_arranged_to_be_torsion-free_elementary_amenable} Let\n $\\alpha \\colon \\pi_1(M) \\to \\Gamma$ be an epimorphism onto a group that is virtually torsion-free abelian. Then\n there exists a factorization of $\\alpha$ into group homomorphisms \n $\\pi \\xrightarrow{\\mu} G \\xrightarrow{\\nu} \\Gamma$ such that $G$ is torsion-free elementary amenable.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\alpha \\colon \\pi_1(M) \\to \\Gamma$ be an epimorphism onto a group $\\Gamma$ that admits a finite index subgroup $F$ that is free abelian. After possibly going to the core of $F$ we can assume that $F$ is normal.\n\nSince $\\alpha^{-1}(F)$ is a finite-index subgroup of $\\pi_1(M)$ we conclude from~\\cite[Theorem~4.3]{Friedl-Schreve-Tillmann(2015)} that there is a\n torsion-free elementary amenable group $G'$ together with an epimorphism \n $\\mu' \\colon \\pi_1(M) \\to G'$ such that $\\ker(\\mu')\\subset \\alpha^{-1}(F)$. Define the epimorphism $\\mu \\colon \\pi_1(M) \\to G$ to be the\n projection onto the quotient $G = \\pi_1(M)\/\\bigl(\\ker(\\mu') \\cap \\ker(\\alpha)\\bigr)$. Obviously there is \n an epimorphism $\\nu \\colon G \\to \\Gamma$ such that $\\nu \\circ \\mu = \\alpha$ since $\\alpha$ is by \n construction the projection from $\\pi_1(M)$ to the quotient $\\Gamma = \\pi_1(M)\/\\ker(\\alpha)$.\n It remains to show that $G$ is torsion-free elementary amenable. We have the obvious exact sequence\n \\[\n 1 \\to \\ker(\\mu')\/\\bigl(\\ker(\\mu') \\cap \\ker(\\alpha)\\bigr) \\to G \\to G' \\to 1.\n \\]\n and obviously $\\alpha$ defines an injection\n \\[\\ker(\\mu')\/\\bigl(\\ker(\\mu') \\cap \\ker(\\alpha)\\bigr) \n\\hookrightarrow F.\n\\]\nSince $F$ and $G'$ are torsion-free elementary amenable, the same is true for $G$.\n\\end{proof}\n\n\nTheorems~\\ref{the:Equality_of_(mu,phi)-L2-Euler_characteristic_and_the_Thurston_norm} and~\\ref{the:The_mu,phi_L2-Euler_characteristic_and_delta_n(phi)}\nand Lemma~\\ref{lem:G_can_be_arranged_to_be_torsion-free_elementary_amenable} imply\nthat the non-commutative Reidemeister torsions of~\\cite{Friedl(2007)} detect the Thurston norm of most 3-manifolds.\n\n\\begin{corollary} \\label{cor:existence_of_torsion-free_elementary_coverings_with_equality} \nLet $M$ be a $3$-manifold, which is admissible, see Definition~\\ref{def:admissible_3-manifold},\nis not a closed graph manifold and is not homeomorphic to $S^1 \\times D^2$. \nThen there is a torsion-free elementary amenable group $G$\nand a factorization $\\pr_M \\colon \\pi_1(M) \\xrightarrow{\\mu} G \\xrightarrow{\\nu} H_1(M)_f$\nof the canonical projection $\\pr_M$ into epimorphisms\nsuch that for any group homomorphism $\\phi \\colon H_1(M)_f \\to \\IZ$, \nthe pair $(\\mu, \\phi \\circ \\nu)$ is an $L^2$-acyclic Atiyah-pair and we get\n\\[\n \\delta(M;\\mu,\\nu,\\phi)~=~-\\chi^{(2)}(M;\\mu,\\phi \\circ \\nu)~=~x_M(\\phi).\n\\]\n\\end{corollary}\n\n\\begin{remark}\n The invariant $\\delta(M;\\mu,\\nu), \\phi$ of~\\cite{Friedl(2007)} is\n essentially the same as the Cochran-Harvey invariant~\\cite{Cochran(2004),Harvey(2005)},\n except that Cochran--Harvey only study solvable quotients of $\\pi_1(M)$. But as pointed\n out in~\\cite[Example~2.3]{Cochran(2004)}, in general invariants coming from solvable\n quotients do not suffice to detect the knot genus respectively the Thurston norm. It is\n necessary to work with the extra flexibility given by torsion-free elementary amenable\n groups.\n\\end{remark}\n\n\n\n\n\n\n \n\n\n\n \\typeout{---- Section 9: The degree of the $L^2$-torsion function}\n\n\n\\section{The degree of the $L^2$-torsion function}\n\\label{sec:The_degree_of_the_L2-torsion_function}\n\n\nIn~\\cite{Lueck(2015twisting)} the $\\phi$-twisted $L^2$-torsion function has been\nintroduced and analyzed for $G$-coverings of compact connected manifolds in all\ndimensions. In the sequel we will consider only $G$-coverings $\\overline{M} \\to M$ of\nadmissible $3$-manifolds $M$ for countable residually finite $G$. Then all the necessary\nconditions such as $\\det$-$L^2$-acyclicity for $\\overline{M}$ and the $K$-theoretic\nFarrell-Jones Conjecture for $\\pi_1(M)$ are automatically satisfied and do not have to be\ndiscussed anymore. One can assign to the $L^2$-torsion function by considering its asymptotic behavior at\ninfinity a real number called its degree and denoted by $\\deg\\bigl(\\rho^{(2)}(M;\\mu,\\phi)\\bigr)$.\nIf $G = \\pi_1(M)$ and $\\mu \\,\\,=\\,\\, \\id_{\\pi_1(M)}$, i.e., for the universal covering\n$\\widetilde{M}$, the equality\n\\[\n\\deg\\bigl(\\rho^{(2)}(M;\\mu,\\phi)\\bigr) = x_M(\\phi \\circ \\mu)\n\\]\nwas proved by the authors in~\\cite[Theorem~0.1]{Friedl-Lueck(2015l2+Thurston)} and\nindependently by Liu~\\cite{Liu(2015)}. Actually many more instances of $G$-coverings are\nconsidered in~\\cite[Theorem~5.1]{Friedl-Lueck(2015l2+Thurston)}, where this equality\nholds.\n\nWe just mention without proof the following theorem.\n\n\\begin{theorem}[The $(\\mu,\\phi)$-$L^2$-Euler characteristic is a lower bound for the\n degree of the $L^2$-torsion function]\n \\label{the:The_(mu,phi)-L2-Euler_characteristic_is_an_lower_bound_for_the_degree_of_the_L2-torsion_function}\n Let $M$ be an admissible $3$-manifold. Let $\\mu \\colon \\pi \\to G$ be a\n homomorphism to a torsion-free, elementary amenable, residually finite, countable group $G$ and\n $\\phi \\colon G \\to \\IZ$ be a group homomorphism. Let $\\overline{M} \\to M$ be the $G$-covering\n associated to $\\mu$. Suppose $b_1^{(2)}(\\overline{M};\\caln(G)) = 0$.\n Then $(\\mu,\\phi)$ is an $L^2$-acyclic Atiyah pair and we get\n \\[\n \\chi^{(2)}(M;\\mu,\\phi) \\le \\deg\\bigl(\\overline{\\rho}^{(2)}(M;\\mu,\\phi)\\bigr).\n \\]\n \\end{theorem}\n\n\n\n\n\n\\typeout{-------------------------------------- References ---------------------r------------------}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPhysicists like to think of `true' distributions of physics\nquantities, i.e those distributions \n(graphically represented by histograms and here also referred as `spectra') \none would observe under idealized conditions\nthat seldom -- never, strictly speaking --  happen in real live\n(ideal detector, no physical or instrumental background). \nThe `observed' distribution is then considered as a \n'noisy distortion' of the true one. \nAn important task of the experimental method is \ntherefore to infer the true distribution from the observed one, i.e.\nto correct the observed spectrum for distortion and noise. \nThis can be done by  different methods that follow\ndifferent approaches.\nSince this is not a review paper, \nI just outline the two different classes of strategies and then \nfocus on the specific issues of this work. \n\nIn a first kind of approach a mathematical function for \nthe true distribution\nis assumed (together with other functions to model the noise)\nand the task becomes that of \nestimating the free parameters of the function(s). Indeed, we\nfall in the so called domain of {\\it parametric inference} (`fits'), \nusually associated to names like `least-squares' or \n'maximum likelihood'.\\footnote{However one can show that it is preferable\nto base the data analysis on more \nsolid probabilistic ground,\nfrom which the mentioned methods might be derived\n(see e.g. Ref.~\\cite{lfits}) under special  conditions \nwhich fortunately hold in the large majority of \npractical cases (but it is better one  knows \nwhen the conditions of validity hold \nand what to do when they do not!).}  \n\nIn parametric inference all information contained in the observed \nspectrum is, to say, `distilled' into the model parameters.\nParametric inference, especially when the \nconditions to use least-squares methods\nhold, is usually the best and fastest way to proceed, \nif we have good reasons\nto believe the hypothesized family of functions.\n\nHowever, sometimes we wish\nto interpret the data as little as possible and just public \n`something equivalent' to an experimental distribution, \nwith the bin contents fluctuating according to an underlying \nmultinomial distribution, but having possibly got rid of physical\nand instrumental distortions, as well as of background.\nIn particle physics this second approach goes \nunder the name of {\\it unfolding}\n(or deconvolution, or restoration). \n\nSeveral years ago a simple algorithm \nbased on   Bayes' theorem was presented ~\\cite{BU}\nwith which it was possible\nto achieve rather `good' results (`good'\ncompared with the difficulty of the task). \nThe main improvements presented here concern the handling \nof small numbers and the evaluation of the uncertainty on the \nunfolded distribution, while the guiding ideas \nand the basic assumptions are substantially unchanged. \nTo be more clear, \nthe algorithm of Ref.~\\cite{BU} was relying on \nthe underlying hypotheses of normality and `small relative errors':\n`best estimates' were provided, with uncertainty calculated\nfrom standard `error propagation' \nformulas.\\footnote{It is worth remembering that these formulas\nrely on an hypothesis of linearity between {\\it input} and \n{\\it output quantities}, an hypothesis that holds approximately, \nin non-strictly linear problems, if the relative uncertainties of \nthe input quantities are small enough, such that the \ndependence input$\\rightarrow$output can be locally\nlinearized~\\cite{asymmetric}.} \nThe new algorithm handles better small numbers, \nin the way it will be described in Sec.~\\ref{ss:improvements},\nand performs the propagation of uncertainty by sampling,\ni.e. by Monte Carlo (MC) integration.\nThe algorithm has been implemented in a R language~\\cite{R} code available \non the author's web page~\\cite{R-code}.\n\nThe paper is organized in the following way. \nSection \\ref{sec:BayesIntro} gives a short introduction to\nBayesian inference and to the specific application subject of this paper. \nThen the algorithm of  Ref.~\\cite{BU} is reminded \nand the improvements are presented.\nThe issue of iterative use of the algorithm\nis also discussed, although the program now gives  also the possibility\nto use priors, provided by the user, over all possible true\nspectra. \nThis option should avoid the\nneed of iterations (but I anticipate that it might be  not that easy to \nmodel such priors and then the iterative strategy\nremains a pragmatic solution).\nFinally, some results on toy models are presented. \n\nSome technical issues concerning binomial and Dirichlet\ndistribution, including the use of the latter as prior\nconjugate of the former, are reminded in Appendix A. \nA second appendix is dedicated to the handling of the \nzeros occurring in the evaluation of the smearing matrix\n(a cause that produces no event in some effect bins, as it \nusually happens, depending also on the statistics of the\nMonte Carlo simulation). As it happens with intermediate \nsmoothing (or other kinds of regularization), this is\na suggestion of how the problem can be approached, whose\nsolution is delegated to the user, who is supposed to\nknow the physics case under investigation.\n\n\\section{Bayesian inference and Bayesian unfolding: \\\\\nfrom first principles to real life}\n\\label{sec:BayesIntro}\nThe so called Bayesian inference is a way to learn \nabout physical quantities and their relationships \n(all things we cannot directly see) \nfrom experimental data (what we can actually observe \nwith our senses, usually mediated by more or\nless sophisticated detectors)\nusing probability theory. This game\nis {\\it conceptually} rather\nsimple, proviso we accept that the intuitive meaning \nof probability -- a scale to rank our beliefs that \nseveral events might happen, or that several hypotheses \nmight be true -- is suitable in scientific reasoning\n(see Refs.~\\cite{BR} and \\cite{rpp} and references therein\nfor an introduction). \n\n\\begin{figure}[!t]\n\\begin{center}\n\\begin{tabular}{cc}\n\\epsfig{file=bn1.eps,clip=,width=0.37\\linewidth} &\n\\epsfig{file=bn3.eps,clip=,width=0.55\\linewidth} \\\\\n{\\large {\\it a)}} & {\\large {\\it b)}}\n\\end{tabular}\n\\end{center}\n\\caption{\\sl Bayesian network describing a fit model~\\cite{lfits}. \n$x_i$ and $y_i$ \nare the experimental observations, related to  $\\mu_{x_i}$ and $\\mu_{y_i}$\nby experimental errors. Instead, a deterministic `law' connects \nthe `true' values \n$\\mu_{y_i}$ to  $\\mu_{x_i}$ via the model parameters  $\\mvec\\theta$\n(solid arrows stand for probabilistic links, dashed for deterministic).\nNetwork {\\it a)} describes a simple model with errors only on \nthe ordinate. Network {\\it b)} takes into account errors on both\naxes, extra variability of the data points around the believed\nphysical law and systematic effects.}\n\\label{fig:bn1}\n\\end{figure}\nThe starting point of a Bayesian inference is to build up a model \nfor the deterministic \n(``{\\it B follows from A}'')\nand  the probabilistic \n(``{\\it B might follow from A}'')\nconnections\nthat relate the several entities that enter the problem.\nThis model has the interesting graphical representation\nof a network, usually called {\\it Bayesian network} or \n{\\it belief network} \n(to get an idea of their meaning and their application \n{\\it Google} these keywords, or browse the \n\\href{http:\/\/en.wikipedia.org\/wiki\/Bayesian_network}{\\it Wikipedia}). \nFor example, Fig.~\\ref{fig:bn1}, taken from \nRef.~\\cite{lfits}, shows a Bayesian network\nto model two physical quantities, $\\mu_x$ and $\\mu_y$,\nconnected to each other with a `law', whose parameters \nare denoted by  $\\mvec\\theta$. In the \nvery elementary case depicted in the \n left hand network of Fig.~\\ref{fig:bn1}\n the solution to the problem can be rather simple, \nunder some assumptions that, fortunately, hold\nvery often  in routine cases.\nBut, in general, the solution is not \nthat simple. Nevertheless, as stressed in Ref.~\\cite{lfits},\nafter one has built the graphical model, one is \noften more than half the way to the solution, \nthanks to the great progresses  recently made in \nBayesian network computing.\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=unfnet_t.eps,clip=,width=0.75\\linewidth}\n\\end{center}\n\\caption{{\\sl Probabilistic links from causes to effects. \nThe node indicated by $T$ (`trash') stand for the \ninefficiency bin and corresponds to $E_{n_E+1}$}}\n\\label{fig:unfnet}\n\\end{figure}\nThe essence of the Bayesian unfolding of Ref.~\\cite{BU},\nthat is the starting point also of this new version,\nis to make the problem discrete and to treat the `cause'\nbins as independent degrees of freedom, i.e. without \nconstraints among each other.\\footnote{An important \ndrawback of this feature will be discussed in section \\ref{sec:iter}.}\nFor this reason the algorithm\ncan virtually handle any kind of `smearing' and it is\neasily extensible, \nat least in principle and with the only limit due to computer power, \n to multidimensional\nproblems. In fact, the core of the algorithm only knows about \n`cause-cells' and `effect-cells',\\footnote{`Bin' \nand `cell' are used here as synonyms, although `cell' refers\nto a region on the configuration space and \n`bin' to histograms representing them.} \nbut it does not know the \nlocation of the cells in the configuration space of the problem.  \nFor the same reason, the treatment of background, \nand even of several independent sources of background, \ncan be easily embodied in the algorithm\nby just adding extra cause-cells, one cell per source of background.   \nAs a by-product, the algorithm also provides the number of events\nto be attributed to each source of background.\n(It is worth remembering that \nbackground might have an interesting physical meaning,\nand thus the estimation of the level `noise' \nmight provides indeed a physics measurement, \nas in the analysis of Ref.~\\cite{Raso2}.)\n\nGiven the discretization of the problem,\nthe Bayesian network relating \ncauses and effects is that shown in Fig.~\\ref{fig:unfnet},\nwhere we use the same notation of Ref.~\\cite{BU}, with the addition \nof the effect bin $T$ (`trash'), equivalent  to  $E_{n_E+1}$, \nto describe inefficiency (the reason to introduce this \nextra bin will become clear later).\n\nRephrasing the problem in probabilistic terms, the purpose of the \nunfolding is to find the `true' number of events  \nin each cause bin [$\\#(C_i)$ in Fig.~\\ref{fig:bn0},\nindicated by $x(C_i)$ in the text],\n\\begin{figure}[!t]\n\\begin{center}\n\\vspace{0.7cm}\n\\begin{tabular}{c}\n\\epsfig{file=unfnet_canc.eps,clip=,width=0.9\\linewidth}\n\\end{tabular}\n\\end{center}\n\\caption{{\\small From the `true' distribution (numbers of events in the \ncause-bins) \nto the observed distribution (numbers of events in the \neffect-bins). The number of events $\\#(\\,)$ is indicated with \n$x(\\,)$ in the text.}}\n\\label{fig:bn0}\n\\end{figure}\ngiven the observed spectrum and assuming \nsome knowledge about the smearing.\n  \nSince the links cause$\\rightarrow$effects have a probabilistic nature,\nit follows that also\nthe links effect$\\rightarrow$causes will be probabilistic, \nand therefore it will be\nuncertain  the number of events\nto be attributed to the cause-cells. We can only attempt to rank in \nprobability all possible spectra that might have caused the observed\none. \nIn other words, the realistic goal of our analysis\nis not to determine the {\\it the} true spectrum, but rather to assess\n\\begin{eqnarray}\nP(\\mvec x_C \\,|\\,\\mvec x_E,\\,\\Lambda,\\,I)\\,,\n\\label{eq:goal}\n\\end{eqnarray}\nwhere:\\footnote{In Ref.~\\cite{BU} \n$\\mvec x_C$, $\\mvec x_E$ and $\\Lambda$ were respectively indicated\nby  $\\underline{n}_C$, $\\underline{n}_E$ and $\\mvec S$.\nAs in Fig.~\\ref{fig:bn0} and in Ref.~\\cite{BU}, the symbols $i$\nand $j$ are used to index causes and effects, respectively, \nno matter if this choice leads to the unusual convention of \nindexing the $\\Lambda$ rows by $j$ and the columns by $i$, as in \nEq.~(\\ref{eq:Lambda}). \nNote that, when we refers to MC simulations\nto infer the smearing matrix, $\\mvec x_E$, we\n also need include the trash bin, as it will be reminded\nat the proper place.\nLater on [see Eq.~(\\ref{eq:lambda_i})] it will be convenient\nto name $\\mvec{\\lambda}_i$ the columns of the matrix $\\Lambda$. \nIn summary\n\\begin{eqnarray}\n\\Lambda &=& \\left(\\!\\!\n            \\begin{array}{cccc}\n             P(E_1\\,|\\,C_1,\\,I) &  P(E_1\\,|\\,C_2,\\,I) & \\ldots &  P(E_1\\,|\\,C_{n_c},\\,I) \\\\\n             P(E_2\\,|\\,C_1,\\,I) &  P(E_2\\,|\\,C_2,\\,I) & \\ldots &  P(E_2\\,|\\,C_{n_c},\\,I) \\\\ \n             \\ldots &  \\ldots &  \\ldots &  \\ldots  \\\\\n             P(E_{n_E+1}\\,|\\,C_1,\\,I) &  P(E_{n_E+1}\\,|\\,C_2,\\,I) \n             & \\ldots &  P(E_{n_E+1}\\,|\\,C_{n_c},\\,I)\n            \\end{array}\n            \\!\\!\\right) \\nonumber \\\\\n        &=& (\\mvec{\\lambda}_1,\\,\\mvec{\\lambda}_2,\n                          \\ldots,\\mvec{\\lambda}_{n_C})\\,. \\nonumber\n\\end{eqnarray}\n}\n\\begin{itemize}\n\\item\n$\\mvec x_C = \\{x(C_1),\\, x(C_2),\\, \\ldots,\\, x(C_{n_C}) \\}$ \nis the number of events in each bin of the true distribution,\ni.e. {\\it a} true spectrum.\\footnote{For \nthe use of the indefinite article in conjunction with \n`true values', see the ISO \n{\\it ``Guide to the expression of uncertainty\nin measurement''}\\,\\cite{ISO}, \naccording to which {\\it a} true value is\n{\\it ``a value compatible with the definition\nof  a given particular quantity.''} }\n\\item\n$\\mvec x_E = \\{x(E_1),\\, x(E_2),\\, \\ldots,\\, x(E_{n_E}) \\}$\nis the observed spectrum. \n\\item\n$\\Lambda$ stands for the smearing matrix, whose elements \n$\\lambda_{ji}$ (see remarks in footnote 4 concerning notation) are \ndefined in probabilistic terms as\n\\begin{eqnarray}\n\\lambda_{ji} &\\equiv& P(E_j\\,|\\,C_i,\\,I)\\,. \n\\label{eq:Lambda}\n\\end{eqnarray}\nThe knowledge of $\\Lambda$ comes usually from MC simulation and it is\ntherefore affected by an uncertainty, described,\n in general terms, by a pdf $f(\\Lambda\\,|\\,I)$.\n\\item\n$I$ stands for the state of information \nunder which the analysis is performed\n(this underlying condition  is often implicit in the pdf's).\n\\end{itemize}\nOnce we have stated clearly and in probabilistic terms our question\n({\\it  ``what is $P(\\mvec x_C \\,|\\,\\mvec x_E,\\,I)$?''}),\nprobability theory provides us the answer,\\footnote{Instead, \nthe idea of inverting the smearing matrix (assuming it square\nand not singular) is wrong in principle \n(i.e. besides the ascertainment \n that such a method  yields unacceptable results). \nIn fact, unfolding\nis a probabilistic problem and not a \ndeterministic linear one (or, equivalently, a geometric problem \nof rotating vectors). Therefore it needs to be solved by probabilistic\ntools and not by linear algebra methods (`rotations').\n It is easy to show that\n`rotation' works only when we \nhave an `infinite' number of events, such that stochastic\neffects are negligible and the observations coincides with the \nexpectations. In fact, each product $\\lambda_{ji}\\,x(C_i)$ is nothing but \nthe expected number of events in the effect-cell $E_j$ due to cause \n$C_i$ alone: \n\\begin{eqnarray*}\n\\mbox{E}[\\left.x(E_j)\\right|_{x(C_i)}] &=& P(E_j\\,|\\,C_i,\\,I)\\,x(C_i) =\n           \\lambda_{ji}\\,x(C_i) \n\\end{eqnarray*}\nSumming up the contributions from all cause-cells, we get the\nexpected value in the effect-cell $E_j$\n\\begin{eqnarray*}\n\\mbox{E}[\\left.x(E_j)\\right|_{\\mvec x_C}] &=& \n \\sum_i \\lambda_{ji}\\,x(C_i)\\,,\n\\end{eqnarray*}\nthat can be written in matrix form as\n\\begin{eqnarray*}\n\\mvec \\mu_E \\equiv \\mbox{E}[\\mvec x_E] &=& \\Lambda \\, \\mvec x_C\\,.\n\\end{eqnarray*}\nThen, if $\\Lambda$ is square and not singular, we get\n\\begin{eqnarray*}\n\\mvec x_C &=& \\Lambda^{-1}\\,\\mvec \\mu_E \\,.\n\\end{eqnarray*}\nBut this might be, at most, the solution of a text book exercise in \nmathematical probability theory, and does not help\nto solve real problems. \nThis is the reason why, besides\nthe fact that the matrix inversion gives notoriously \nbad results,\n{\\it the very idea \nof inverting the smearing matrix is logically\nflawed}: we can certainly apply $\\Lambda^{-1}$ to \na vector of numbers {\\it already known} to be sums \nof expected values of binomials, but we cannot apply\nit to a vector of numbers that {\\it might be} \n(we are not even sure of this, because some counts could be\ndue to background we do not take into account!) \nsums of binomial random variables. If we do it, there is no\nguarantee that  $\\Lambda^{-1}\\mvec x_E$ yields a vector\nof `valid numbers' of the $n$-parameters of binomials \n(the question that they might have the meaning of \na physical spectrum for the problem under study is \nsecondary at this level) and, in fact, even negative numbers\ncan be obtained!\nIt follows that unfolding methods which\nuse the matrix inversion as starting point and try to\ncure its bad features with some kitchen are not appealing from \na theoretical point view, although they might even provide \nacceptable results because of `mysterious' reasons\nI do not want to enter into (cooks might be extremely clever!).\n}\nat least in principle. \nIn fact, \n\\begin{enumerate}\n\\item\nBayes' theorem allows to calculate \n$P(\\mvec x_C \\,|\\,\\mvec x_E,\\,\\Lambda,\\,I)$, given the observation \n$\\mvec x_E$ and the smearing matrix $\\Lambda$, as\n\\begin{eqnarray}\nP(\\mvec x_C \\,|\\,\\mvec x_E,\\,\\Lambda,\\,I) &=& \n\\frac{P(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I) \\cdot P(\\mvec x_C\\,|\\,I)}\n     {\\sum_{\\mvec x_C}P(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I) \n     \\cdot P(\\mvec x_C\\,|\\,I)}\n\\label{eq:bayes1}\n\\end{eqnarray}\n(the formula will be explained in a while).\n\\item\nWe can take into account of the uncertainty about $\\Lambda$\nusing another theorem of probability theory, namely\n\\begin{eqnarray}\nP(\\mvec x_C \\,|\\,\\mvec x_E,I) &=& \\int \nP(\\mvec x_C \\,|\\,\\mvec x_E,\\Lambda,\\,I)\\,\nf(\\Lambda\\,|\\,I)\\,\\mbox{d}\\Lambda\\,.\n\\label{eq:marg_lambda}\n\\end{eqnarray}\n\\end{enumerate}\nLet us now go through the several ingredients needed to get \n$P(\\mvec x_C \\,|\\,\\mvec x_E,I)$\nand try to understand \nwere the problems arise. \n\\begin{itemize}\n\\item\nFirst of all, it is easy to realize that the denominator of \nEq.~(\\ref{eq:bayes1}) is just a normalization factor, and then \nwe can rewrite the Bayes' formula in a way that \nhighlights the main ingredients:\n\\begin{eqnarray}\nP(\\mvec x_C \\,|\\,\\mvec x_E,\\,\\Lambda,\\,I) & \\propto & \nP(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I) \\cdot P(\\mvec x_C\\,|\\,I)\\,,\n\\label{eq:bayes2}\n\\end{eqnarray}\nwhere $P(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I)$ is the \nso called {\\it likelihood}\n and $P(\\mvec x_C\\,|\\,I)$ the \n {\\it prior}. The left hand side of the Bayes' formula takes\nthe name of {\\it posterior}. As we see, \nin probabilistic inference the\nlikelihoods have the role of updating probabilities. \n\\item \nAlthough the presence of priors in the formula might\ncause anxiety in those who approach this kind\nof reasoning for the first time,\n it is a matter of fact that: 1) priors are logically necessary \nto get $P(\\mvec x_C \\,|\\,\\mvec x_E,\\,\\Lambda,\\,I)$\nstarting from the likelihood, i.e. to \nperform the so-called `probability inversion';\n2) they allow to plug into the model all relevant prior information,\nthat might come from previous data or from theoretical\nprejudices; \n3) priors are often so vague (or there are so many data -- that \nis the same for the relative balance of prior and likelihood \nin the Bayes' formula) that they influence negligibly the posterior,\nand the inference is often dominated by the likelihood. \n\\item \nLet us assume this last case of prior vagueness applies. \nThis is equivalent to have \n\\begin{eqnarray}\nP(\\mvec x_C\\,|\\,I) & = & \\mbox{\\it constant}\n\\label{eq:flat_prior}\n\\end{eqnarray}\nand the  inference is then performed according to the rule\n\\begin{eqnarray}\nP(\\mvec x_C \\,|\\,\\mvec x_E,\\,\\Lambda,\\,I) & \\propto & \nP(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I)\\,.\n\\label{eq:bayes3}\n\\end{eqnarray}\nIt is then not a surprise that the most probable spectrum  \n$\\mvec x_C$ is the one which maximizes the likelihood, \n{\\it if} we have no other good reason to believe \nthat this is not the case.\n(This is the meaning of the expression ``recovering \nmaximum likelihood estimators'' from the Bayesian approach.)   \n\\item\nIf we were able to write down a closed expression for the \nlikelihood, or at least to provide a simple \nalgorithmic expression of it,\nwe could somehow scan all possible spectra $\\mvec x_C$,\nfind the most probable one (or an `average' spectrum that\nsummarizes in some sense the variety of true spectra \ncompatible with the data)\n and assess somehow the uncertainty \nabout the result. \nBut, unfortunately, this is not the case\nwith $P(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I)$ \nof our interest. \nLet us see why.\nGiven a certain number of events in a cause-bin $x(C_i)$, \nthe number of events in the effect-bins,\nincluded the `trash' one, is described by a \n{\\it multinomial distribution} (see Appendix A.1):\n\\begin{eqnarray}\nP(\\mvec x_E\\,|\\,x(C_i), \\Lambda, I) &=& \n\\frac{x(C_i)!}{\\prod_j^{n_E+1} x(E_j)!}\\, \n\\prod_j^{n_E+1}  \\lambda_{ji}^{x(E_j)}\\,.\n\\end{eqnarray}\nIt follows that $P(\\mvec x_E \\,|\\,\\mvec x_C,\\,\\Lambda,\\,I)$ \nis the sum of  independent  multinomial distributions, \nfor which, unfortunately, a closed formula does not \nexist.\\footnote{It is well know that the sum of two Poisson variables\nis still a Poissonian. Similarly, the sum of two binomial variables \nhaving {\\it the same} parameter `$p$' is still a binomial. \n(This `reproductive property' applies to few other distributions). \nBut the sum of two binomial variables with different $p$ \ndoes not have a closed expression \n(this can be understood either intuitively, just thinking to the\nmeaning of a binomial, or, more formally, analyzing\nthe product of the characteristic functions of each binomial). \nThe same happens with a multinomial,\nthat is just an extension of the binomial to more than two \npossible outcomes.} This is the real serious \ntechnical problem that prevents\na straight application of the Bayes' formula.\n\\item\nThe elements of the smearing matrix $\\Lambda$ are obtained\nby MC: we generate a large number of events in each cell $C_i$ \nand count `where they end' after a realistic simulation. \nIntuitively, we expect $\\lambda_{ji}\\approx x(E_j)^{MC}\/x(C_i)^{MC}$, \nbut we also know  that this is just an estimate, with some \nuncertainty. Fortunately, in this case\nwe can make direct use of the Bayes' formula\napplied to MC events, if we model the prior using \na convenient pdf. \nIn fact, if we indicate by $\\mvec \\lambda_i$ the \n$i$-th column of $\\Lambda$ (see footnote 4), i.e. \n\\begin{eqnarray}\n\\mvec \\lambda_i = \\{\\lambda_{1,i},\\,\\lambda_{2,i},\\ldots,\\, \n\\lambda_{n_E+1,i}\\}\\,, \\label{eq:lambda_i}\n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\nf[\\mvec \\lambda_i\\,|\\, \\mvec x_E^{MC},\\,x(C_i)^{MC},\\, I] \n&\\propto& P[\\mvec x_E^{MC}\\,|\\,x(C_i)^{MC}, \\, \\mvec \\lambda_i,\\, I]\\cdot \nf(\\mvec \\lambda_i\\,|\\,I)\\,. \\nonumber \\\\\n&& \n\\end{eqnarray}\nSince $ P[\\mvec x_E^{MC}\\,|\\,x(C_i)^{MC}, \\, \\mvec \\lambda_i,\\, I]$\nis a multinomial,\nchoosing a Dirichlet prior\n(an extension of the Beta distribution to many dimensions \n-- see Appendix A.2),\nwe get a  Dirichlet posterior (Appendix A.3), i.e.\n\\begin{eqnarray} \n\\mvec \\lambda_i & \\sim & \\mbox{Dir}(\\mvec\\alpha_{posterior_i})\\,,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\mvec\\alpha_{posterior_i} &=& \\mvec\\alpha_{prior_i} + \n\\left.\\mvec x_E^{MC}\\right|_{x(C_i)}\\,.\\label{eq:update_alpha}\n\\end{eqnarray}\n(Hereafter the symbols `$\\sim$' stands for `follows the probability\ndistribution`.)\n\\item\nFinally there is the integral (\\ref{eq:marg_lambda}), which\nis in principle not a problem \n(for example, thanks to modern technologies, it can easily performed \nby MC methods). \n\\end{itemize}\nIn conclusion, the main serious problem is \n$P(\\mvec x_C \\,|\\,\\mvec x_E,\\Lambda,\\,I)$. Thus\nbeing unable to tackle the problem from the main door, \nwe need some `tricks' to circumvent the obstacle, but still under\nthe guidance of probability theory.\n\n\\section{Practical algorithm to perform an \nindependent-bin Bayesian unfolding}\nThe basic trick to avoid the mentioned difficulty is to apply\nBayes' theorem to causes and effects, instead than to \ntrue and observed spectrum. In practice, instead of using\nof Eq.~(\\ref{eq:bayes1}) we start from\n\\begin{eqnarray}\nP(C_i\\,|E_j,\\,I) &=& \\frac{P(E_j\\,|\\,C_i,\\,I)\\cdot P(C_i\\,|\\,I)}\n                         {\\sum_i P(E_j\\,|\\,C_i,\\,I)\\cdot P(C_i\\,|\\,I)}\\,,\n\\label{eq:bayes_CE} \n\\end{eqnarray}\nor\n\\begin{eqnarray}\n\\theta_{ij}&=& \\frac{\\lambda_{ji}\\cdot P(C_i\\,|\\,I)}\n         {\\sum_i \\lambda_{ji}\\cdot P(C_i\\,|\\,I)}\\,, \n\\label{eq:bayes_CE_theta}\n\\end{eqnarray}\nhaving defined $\\theta_{ij} \\equiv P(C_i\\,|E_j,\\,I)$\nin analogy to $\\lambda_{ji} \\equiv P(E_j\\,|\\,C_i,\\,I)$\\,.\n\nAt this point a very important remark is in order. \nThe prior  in Eqs.~(\\ref{eq:bayes_CE})-(\\ref{eq:bayes_CE_theta}) \nhas a different meaning from that of Eq.~(\\ref{eq:bayes1}). \nIn Eq.~(\\ref{eq:bayes1}) $P(\\mvec x_C\\,|\\,I)$ assigns different probabilities\nto all possible spectra.  Instead,\nin  Eqs.~(\\ref{eq:bayes_CE})-(\\ref{eq:bayes_CE_theta}) \n  $P(C_i\\,|\\,I)$ stands for \n{\\it a single spectrum} (more precisely, all spectra that \ndiffer from each other just by normalization). This can be better\nunderstood analyzing the meaning of `uniform' (or `flat') \nreferred to  $P(\\mvec x_C\\,|\\,I)$ and to $P(C_i\\,|\\,I)$.\n\\begin{itemize}\n\\item \n $P(\\mvec x_C\\,|\\,I) = \\mbox{\\it constant}$ means \n{\\it all spectra} are equally likely.\n\\item\n $P(C_i\\,|\\,I) = \\mbox{\\it constant}$ means we consider \nthe cause bins equally\nlikely, i.e. the prior assess an initial belief in {\\it flat spectra}. \n\\end{itemize}\nIn other words, while a flat $P(\\mvec x_C\\,|\\,I)$\nmeans indifference about all possible spectra in order \nto `let the data speak by themselves', \na flat $P(C_i\\,|\\,I)$ is a strong assumption that usually does \nnot correspond to our priors concerning the physics case.\nThis implies that we have to tune somehow the algorithm in order\nto take into account of this gross approximation. \nWe will come back to this issue in Sec.~\\ref{sec:iter}.\n\nAt this point, having evaluated  $P(C_i\\,|\\,E_j,\\,I)$, \nwe can use it to share the counts observed\nin each effect-bin among  all cause-bins\n(see Fig.~\\ref{fig:bn2}).\n\\begin{figure}[t]\n\\begin{center}\n\\begin{tabular}{c}\n\\epsfig{file=unfnet_xi.eps,clip=,width=0.9\\linewidth}\n\\end{tabular}\n\\end{center}\n\\caption{{\\sl Sharing counts observed in effect-cells\namong cause-cells according to $\\theta_{ij} = P(C_i\\,|\\,E_j,\\,I)$.}}\n\\label{fig:bn2}\n\\end{figure}\n The estimate of the true spectrum \nis obtained repeating this sharing for all observed bins\nand taking into account inefficiency. An uncertainty on the\nunfolded spectrum is also evaluated. Let us see how all this\nwas done in the old algorithm and how it has been improved.\n\n\\subsection{Old algorithm (Ref.~\\cite{BU})}\nIn the old algorithm the share of observed events among the causes\nwas done only considering\nexpectations and applying an inefficiency correction,\ngoing through the following steps:\n\\begin{itemize}\n\\item\nnumber of counts in $C_i$ due to the observation in $E_j$: \n\\begin{eqnarray}\n\\left.x(C_i)\\right|_{x(E_j)} & \\approx & P(C_i\\,|E_j,\\,I) \\cdot x(E_j) = \n                    \\theta_{ij}\\cdot x(E_j)\\,;  \\hspace{1.0cm}\n\\end{eqnarray}\n\\item\nnumber of counts in $C_i$ due to all observations: \n\\begin{eqnarray}\n\\left.x(C_i)\\right|_{\\mvec x_E} & \\approx & \n               \\sum_{j=1}^{n_E} P(C_i\\,|E_j,\\,I) \\cdot x(E_j) = \n               \\sum_{j=1}^{n_E}\\theta_{ij}\\cdot x(E_j)\n\\,; \\hspace{0.5cm}\n\\end{eqnarray}\n\\item\nnumber of counts in $C_i$ that also takes into account efficiency\n($\\epsilon_i$):\n\\begin{eqnarray}\nx(C_i) & \\approx &\\frac{1}{\\epsilon_i}\\, \\left.x(C_i)\\right|_{\\mvec x_E} \n =  \\frac{1}{\\epsilon_i}\\, \n               \\sum_{j=1}^{n_E} P(C_i\\,|E_j,\\,I) \\cdot x(E_j) \n\\label{eq:old_estimate}\n\\\\\n&& \\hspace{2.2cm}= \n       \\frac{1}{\\epsilon_i} \\sum_{j=1}^{n_E}\\theta_{ij}\\cdot x(E_j)\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\epsilon_i &=& \\sum_{j=1}^{n_E} P(E_j\\,|\\,C_i,\\,I) = \n              \\sum_{j=1}^{n_E} \\lambda_{ji} =\n              1 - \\lambda_{n_E+1,\\,i} \\,. \\label{eq:def_eps}\n\\end{eqnarray}\n\\item\nFinally, we have to remember that  the several $\\lambda_{ij}$ were\nestimated by MC simulation as \n\\begin{eqnarray}\n\\lambda_{ji}\\approx x(E_j)^{MC}\/x(C_i)^{MC}\\,,\n\\end{eqnarray}\nfrom which $\\epsilon_i $ and $\\theta_{ij}$ are calculated according\nto Eqs. (\\ref{eq:def_eps}) and (\\ref{eq:bayes_CE_theta}).\n\\end{itemize}\nEssentially, $x(C_i)$ of Eq.~(\\ref{eq:old_estimate}) was considered \nan `estimator', whose `error' was calculated by standard\n`error propagation' (this was a pragmatic compromise between `standard\nmethods' I was accustomed at that time \nand the Bayesian approach I was in the process of learning\n -- anyhow, not bad to begin, and not \nworst than other methods.)\n\nAt this point, there is still open the issue of the inappropriate priors,\nthat we have encountered  \nat the beginning of this section. \nSince this question remains in the improved algorithm, \nit will be discussed later in Sec.~\\ref{sec:iter}. \n\n\n\\subsection{Improvements}\\label{ss:improvements}\nAs it is easy to understand from the previous\ndescription, weak points of the \nold algorithm are the treatment of small numbers and \nthe calculation of uncertainty by `error propagation\nformulas'. \nLet us see how we can improve them. \n\\begin{itemize}\n\\item \nInstead of just `estimate' $\\lambda_{ij}$ as  \n$x(E_j)^{MC}\/x(C_i)^{MC}$, we can model their \npdf by a Dirichlet (see Appendix A): \n\\begin{eqnarray}  \n\\mvec \\lambda_i & \\sim & \\mbox{Dir}[\\mvec \\alpha_{prior}\\, + \\, \n                      \\left.\\mvec x_E^{MC}\\right|_{x(C_i)^{MC}}]\\,,\n\\label{eq:lambda_i_dir}\n\\end{eqnarray}\nwhere \n\\begin{itemize}\n\\item \n$\\mvec \\alpha_{prior}$ is the initial set of Dirichlet \nparameters -- typically unitary for uniform priors \n (see Appendix B for details); \n\\item\n$\\left.\\mvec x_E^{MC}\\right|_{x(C_i)^{MC}}$\nstands for the numbers of MC counts generated in $C_i$ \nand which end in each of \n{\\it all} possible $n_E+1$ `effects' \n(i.e. we also have to consider the inefficiency bin --\nsee Fig.~\\ref{fig:unfnet} -- in order to satisfy the condition \n$\\sum_i\\lambda_i = 1$\n of the Dirichlet variables).\n\\end{itemize}\n\\item\nThe uncertainty about $\\Lambda$ is taken into account by {\\it sampling}\nits values, that is equivalent to perform the integral \n(\\ref{eq:marg_lambda}).\\footnote{Note that if we are in doubt\nabout several smearing matrices, because e.g. they are obtained\nfrom different parameters of the simulation, the \n(`systematic') effect of this uncertainty can be taken\ninto account sampling from the different $\\Lambda$'s, with \nweights depending on our confidence on each of them.} \nTherefore, for  each sampling, \n   \\begin{itemize}\n   \\item the smearing matrix elements $\\lambda_{ji}$ are extracted  \n         according to a Dirichlet distribution;\n   \\item the efficiencies $\\epsilon_i = \\sum_{j=1}^{n_E}\\lambda_{ji}$ \n         are calculated;\n   \\item the inverse probabilities  $\\theta_{ij}$ are obtained \n         from the Bayes formula. \n   \\end{itemize}\n\\item \nThe sharing of the number of counts $x(E_j)$ observed in an effect-cell\namong all cause-bins is performed using a \nmultinomial distribution (see Appendix A.1):\n\\begin{eqnarray}  \n\\left.\\mvec x_C\\right|_{x(E_j)} & \\sim & \n\\mbox{Mult}[x(E_j),\\,\\mvec \\theta_j]\\,,\n\\label{eq:extract_xC}\n\\end{eqnarray}\nwith  $\\mvec \\theta_j = \\{\\theta_{1,j},\\,\n\\theta_{2,j},\\,\\ldots,\\,\\theta_{n_C,j},\\,\\}$.\n\\item\nThe  contributions of all {\\it observed} \neffect-bins are summed up, i.e. \n \\begin{eqnarray}  \n\\left.\\mvec x_C\\right|_{\\mvec x_E} \n& = & \\sum_{j=1}^{n_E} \\left.\\mvec x_C\\right|_{x(E_j)}\\,.\n\\label{eq:x_C_obs}\n\\end{eqnarray}\nInefficiency are taken into account as in the old algorithm:\n   \\begin{eqnarray}  \n   \\mvec x_C\n   & = & \\frac{\\left.\\mvec x_C\\right|_{\\mvec x_E}}\n              {\\mvec \\epsilon }\\,,\n   \\label{eq:x_C_mu_eps}\n   \\end{eqnarray}\n   where the ratio in  the r.h.s is done component by component.  \n\\end{itemize}\nHowever, we need to remember that the observed number of events\nin each effect bin, $x(E_j)$,\n comes from a Poisson distribution with unknown \nparameter $\\mu_j$, whose inference can be conveniently performed\nstarting from a  conjugate gamma distribution (see e.g. Ref.~\\cite{BR}).\nIt follows that \n\\begin{eqnarray}\n\\mu_j & \\sim & \\mbox{Gamma}[c_j+x(E_j),\\, r_j+1]\\,,\n\\label{eq:mu_j_Gamma}\n\\end{eqnarray}\nwhere $c_j$ and $r_j$ are the initial parameters of the gamma. \n[The case of a flat prior corresponds to \n$c_j=1$ and $r_j\\rightarrow 0$, i.e. an exponential with \nvanishing rate.] \n\nTherefore, what should be shared among \nthe cause-bins is not $x(E_j)$ but  \n$\\mu_j$,  extracted in each sampling \naccording to Eq.~(\\ref{eq:mu_j_Gamma}). \nHowever, $\\mu_j$ is a continuous variable and therefore\ncannot be used  \nas `trials parameter' of a multinomial distribution.\nNevertheless,\nfractional events can indeed be shared making use of a little trick:\n   \\begin{itemize}\n   \\item  \n   $\\mu_j$ is extracted according to Eq.~(\\ref{eq:mu_j_Gamma}) and \n   it is rounded to the closest {\\it positive} integer, \n   that we indicate here by $m_j$: \n   $m_j$ will be the number of trials used in the multinomial random\n   generator.\n   \\item \n   $\\left.\\mvec x_C\\right|_{m_j}$ is extracted and then \n   rescaled by the factor $\\mu_j\/m_j$, i.e.\n    \\begin{eqnarray}\n     \\left.\\mvec x_C\\right|_{m_j}  & \\sim & \n     \\mbox{Mult}(m_j,\\,\\mvec \\theta_j)\\,, \\\\\n    \\left.\\mvec x_C\\right|_{\\mu_j} &=& \\frac{\\mu_j}{m_j}\\,\n      \\left.\\mvec x_C\\right|_{m_j}\\,.\n    \\end{eqnarray}\n\\end{itemize}\nThis means that  Eq.~(\\ref{eq:x_C_obs}) has to be replaced by\n\\begin{eqnarray}  \n\\left.\\mvec x_C\\right|_{\\mvec x_E} \n& = & \\sum_{j=1}^{n_E} \\left.\\mvec x_C\\right|_{\\mu_j}\\,;\n\\label{eq:x_C_mu}\n\\end{eqnarray}\nthen the inefficiency corrections are applied as usual. \n\nEach sampling provides a spectrum $\\mvec x_C^{(t)}$, where \n$t$ is the `time' index of the sampling. After $N$ samplings\nwe can calculate an average spectrum, variance \nand covariance for each cause-bin, as well as covariances\namong bins, and any other statistical summary; or we can inspect \ngraphically each $x(C_i)$.\n\nThis algorithm has been implemented in the R language\\,\\cite{R}\nand it is freely available on the web~\\cite{R-code}. \nThe R language has been chosen\nbecause it is a powerful \nscripting language,\\footnote{I take the opportunity to make a point\nabout the teaching of scripting languages, and in particular of\n$R$, in the physics courses. Surely physics students might need to learn \nC (and perhaps C++) at some point, but it is a matter of fact that writing \nsome C code to solve little\/medium problems that occur everyday,\nand that might also need some graphics,\nis a pain, not only for students. The consequence is that\nstudents usually do not write little programs in C to solve\nthe problems they meet in the general physics  or \nlaboratory courses, continuing to use, instead,  spread-sheets,\nlearned in  high school (and forget real programming until they\nneed it for their theses, if they ever will need it).\nIn my opinion, teaching a scripting language\nlike R, perhaps in parallel to C, would be a good solution.\n(but if I really had\nto chose, I would opt for R, as the language to begin).\n(Personally, since I have discovered scripting languages,\nI use C  only for though professional tasks. \nIndeed, I have even almost abandoned {\\it Mathematica},\nunless I really need symbolic calculation.)\n}\nopen source, multi-platform, oriented towards statistics\napplications, well \nmaintained and with tons of contributed extension packages.\nIn practice a kind of programming {\\it lingua franca}. \n\nIn order to test the \nalgorithm, also a simple event generator has been included, \nthat reproduces the toy models of Ref.~\\cite{BU}. \nSome results will be presented in Sec.~\\ref{sec:results}\n\n\n\\section{Iterations and smoothing}\\label{sec:iter}\nLet us now go back to the issue of the priors, \nthat we have left open from the beginning of the \nprevious section. The crucial thing to understand \n is that, as we stated above, \n{\\it instead of using a prior flat over the possible spectra, \nwe are using a particular, flat spectrum as prior}. Therefore, \nthe posterior [i.e. the ensemble of $\\mvec x_C^{(t)}$\nobtained by sampling] is affected by this quite \nstrong assumption, that seldom holds in real cases.\n\nObviously, the first idea a practical physicist has,\nin order to fix the problem, is to do some {\\it fine tuning}.\nIn fact, simulations on toy models show that\nthe unfolded distribution reproduces rather well \nthe true one, even with a flat spectrum as prior.\nHowever, it still `remembers the flat prior'. \nThis effect can be cured iterating the procedure, using the posterior\nas prior in a subsequent unfolding. \nEmpirically we learn that, in `normal' cases, just \ntwo or three steps are sufficient to recover quite accurately\nthe true spectrum.\n\nAs it has been discussed in length in Ref.~\\cite{BU},\nwe cannot repeat the iterations for a very long time,\notherwise there will be a kind of positive feedback, driven \nby fluctuations, that makes the asymptotically\nunfolded spectrum `crazy' (it simply means that it is\nfar away from all reasonable expectations). Again, \nas we understand it, it is a question of \njudging the outcome by our physics priors,\\footnote{Good \nphysicists do have priors and always use them! \n(Only the perfect idiot has no priors.)}\nthat, although rather vague, tell us that, given  \nthe kind of measurement, a spectrum with wild oscillations  \nis far from our beliefs. This kind of problem happens also with\nother kind of unfolding algorithms\nand it is usually cured by {\\it regularization}.\n\nRegularization is based on the subjective scientific prejudice \nthat `wild' distributions are not physical\n(and, as the reader knows or might imagine, this very reasonable \nsubjective ingredient is unavoidably also\nembedded in all non-Bayesian methods).\nIn practice, regularization can be implemented constraining\nadjacent bins not to be `too discontinuous', for example limiting\nthe absolute value of the derivatives calculated numerically\non the unfolded spectrum.\nOr one might fit the spectrum with some orthonormal functions, \nbut dumping `high frequencies'. And so on. \n\nI do not have a strong opinion on the matter. My \nrecommendation is that one should know what one is doing.\nMy preferred regularization method \nfor one-dimensional problems consists \nin smoothing the posterior before injecting it \nas prior of the next iteration \n(but not after the last step!).\nThis smoothing can be performed by a low order polynomial fit,\nas it is implemented in the demo scripts accompanying the program.  \nI find this technique quite simple, well consistent with \nthe spirit of this unfolding method and with the reasons \nto go through the iterations. \nMoreover, it is easy to understand that the \nprocedure unfolding-smoothing-unfolding converges rapidly. \n\nIn conclusion, \nalthough the idea of iteration does not seem very\nBayesian (we cannot squeeze the same data twice!),\nit is in fact just an adaptive way to recover what \ncould have been obtained by a more reasonable uniform \nprior over the possible spectra. (Stated with other words,\nit would be non-Bayesian to stick to the \nfirst iteration, obtained from a prior that rarely\ncorresponds to physics spectra!) For this reason the \nidea of iteration has been kept, although some effort has \nbeen done to get rid of it. \nIn fact, in the new code the user can,\ninstead of providing a (typically flat) spectrum as prior, \nset up a function [\\,{\\tt priorf()}\\,] \nthat returns a normalized spectrum \nchosen at random according to a probability distribution\ndescribing prior knowledge.\nIn this way, in each sampling also the prior is sampled. \nHowever, I anticipate that implementing the function  {\\tt priorf()}\nmight not be that trivial, and iterative procedure still\nremains a pragmatic compromise \nto slalom among the difficulties. \n\n\\section{Results on toy models}\\label{sec:results}\nThe program has been checked with the same toy smearing \nmatrices used in Ref.~\\cite{BU}, reproduced here in \nTab.~\\ref{tab:smearing}. \n\\begin{table}\n{\\small \n\\begin{center}\n\\begin{tabular}{|c|cccccccccc|}\n\\multicolumn{11}{c}{Smearing 1} \\\\\n\\multicolumn{11}{c}{} \\\\ \n\\hline\n& $C_1$& $C_2$&  $C_3$& $C_4$&  $C_5$&  $C_6$& $C_7$&  $C_8$& $C_9$&  $C_{10}$  \\\\  \n\\hline\n$E_1$& 0.010& 0& 0& 0& 0& 0& 0& 0& 0& 0 \\\\\n$E_2$& 0.025& 0& 0& 0& 0& 0& 0& 0& 0& 0 \\\\\n$E_3$& 0.040& 0& 0& 0& 0& 0& 0& 0& 0&  0 \\\\\n$E_4$& 0.025& 0.050& 0& 0& 0& 0& 0&  0& 0&  0 \\\\\n$E_5$& 0& 0.100& 0& 0& 0& 0& 0&  0& 0&  0 \\\\\n$E_6$& 0& 0.050& 0.120& 0& 0& 0& 0& 0& 0& 0 \\\\\n$E_7$& 0& 0& 0.120& 0& 0& 0& 0& 0& 0 & 0 \\\\\n$E_8$&  0& 0& 0.060& 0.160& 0.025& 0& 0& 0& 0&  0 \\\\\n$E_9$&  0& 0& 0& 0.160& 0.100& 0.120& 0.140& 0& 0&  0\\\\\n$E_{10}$ &  0& 0& 0& 0.080& 0.250& 0.360& 0.280&  0.200& 0& 0 \\\\\n$E_{11}$ &  0& 0& 0& 0& 0.125& 0.120& 0.280&  0.400& 0.225&  0 \\\\\n$E_{12}$ &  0& 0& 0& 0& 0& 0& 0&  0.200& 0.450& 0.250 \\\\\n$E_{13}$ &  0& 0& 0& 0& 0& 0& 0&  0& 0.225&  0.500 \\\\\n$E_{14}$ &  0& 0& 0& 0& 0& 0& 0&  0& 0&  0.250 \\\\\n\\hline\n$\\epsilon$ & {\\it 0.1} & {\\it 0.2} & {\\it 0.3}& {\\it 0.4} & {\\it 0.5} &\n                  {\\it 0.6} & {\\it 0.7} & {\\it 0.8}& {\\it 0.9} & {\\it 1}\\\\ \n\\hline\n\\multicolumn{11}{c}{} \\\\ \n\\multicolumn{11}{c}{} \\\\ \n\\multicolumn{11}{c}{} \\\\ \n\\multicolumn{11}{c}{Smearing 2} \\\\\n\\multicolumn{11}{c}{} \\\\ \n\\hline\n& $C_1$& $C_2$&  $C_3$& $C_4$&  $C_5$&  $C_6$& $C_7$&  $C_8$& $C_9$&  $C_{10}$  \\\\  \n\\hline\n$E_1$& 0.045& 0 & 0 & 0 & 0 & 0 & 0 & 0 &  0 & 0.225 \\\\\n$E_2$& 0.090& 0.054 & 0 & 0 & 0 & 0 & 0 & 0 & 0.225 & 0.405  \\\\\n$E_3$& 0.045& 0.072 &  0.045 & 0 & 0 & 0 & 0 &  0.225& 0.360& 0.225 \\\\\n$E_4$& 0 & 0.054& 0.045& 0.045& 0.225& 0.180& 0.360& 0.405 &0.225 & 0 \\\\\n$E_5$&  0& 0& 0& 0.180& 0.360& 0.450& 0.315& 0.225& 0& 0 \\\\\n$E_6$& 0& 0& 0& 0.315& 0.180& 0.180 &0.180 &0 &0.045 &0 \\\\\n$E_7$&  0& 0& 0.135& 0.270& 0.045& 0& 0& 0.045& 0& 0 \\\\\n$E_8$&  0& 0.045& 0.270& 0.045& 0& 0& 0.045& 0& 0.045& 0 \\\\\n$E_9$& 0& 0.135& 0.270& 0& 0.045& 0.045& 0& 0& 0 &0.045 \\\\\n$E_{10}$& 0& 0.360& 0.045& 0& 0.045& 0.045& 0&0& 0& 0 \\\\\n$E_{11}$ & 0.180& 0.180& 0& 0& 0& 0& 0& 0& 0& 0 \\\\\n$E_{12}$ & 0.270& 0& 0& 0.045& 0& 0& 0& 0& 0& 0 \\\\\n$E_{13}$ & 0.270& 0& 0.045& 0& 0& 0& 0& 0& 0& 0 \\\\\n$E_{14}$ & 0.090& 0& 0.045& 0& 0& 0& 0& 0& 0& 0 \\\\\n\\hline\n$\\epsilon$ & {\\it 0.9} & {\\it 0.9} & {\\it 0.9}& {\\it 0.9} & {\\it 0.9} & \n                  {\\it 0.9} & {\\it 0.9} & {\\it 0.9}& {\\it 0.9} & {\\it 0.9}\\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\n\\caption{{\\sl Smearing matrices $P(E_j\\,|\\,C_i)$ of the toy models}}\n\\label{tab:smearing}\n\\end{table}\nThe results of the simulations are given in Fig.~\\ref{fig:toy1}. \n\\begin{figure}\n\\begin{center}\n\\epsfig{file=demo3_last_loop_sm1_df1_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df1_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df2_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df2_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df3_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df3_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df4_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df4_r1.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\end{center}\n\\caption{\\sl Result of unfolding applied to four toy models \n({\\rm data.func} parameter from 1 to 4 in the demo program) and two different \nsmearing matrices (left and right figures correspond, respectively,\nto `Smearing 1' and `Smearing 2' of Tab.~\\ref{tab:smearing}.).\n}\n\\label{fig:toy1}\n\\end{figure}\nThe generated \ndistributions are shown in black.\nThe `measured' distributions (in red) \nlook completely different from the `true', \ndue to the very severe smearing matrices used. \nThe figures also show \nthe results after the first iteration\n(light blue) and the intermediate ones \n(yellow). Note that, although in all simulations\ntwenty iterations have been performed, only a few intermediate\niterations are visible, because the others overlap with the\nfinal step. This allows you to get a feeling \nabout the speed of convergence of the algorithm, depending \non the difficulty of the problem\n -- let us\nremind that, thanks to the intermediate smoothing the algorithm\n{\\it does} converge.\n\nOther independent simulations are reported in Figs.~\\ref{fig:toy2} and \n\\ref{fig:toy3}.\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=demo3_last_loop_sm1_df1_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df1_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df2_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df2_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df3_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df3_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df4_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df4_r2.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\end{center}\n\\caption{{\\sl Same as Fig.~\\ref{fig:toy1}. Independent complete simulation.}}\n\\label{fig:toy2}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\epsfig{file=demo3_last_loop_sm1_df1_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df1_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df2_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df2_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df3_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth}\n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df3_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \\\\\n\\epsfig{file=demo3_last_loop_sm1_df4_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth}\n\\hspace{0.5cm}\n\\epsfig{file=demo3_last_loop_sm2_df4_r3.eps,bb=40 240 555 590,clip=,width=0.47\\linewidth} \n\\end{center}\n\\caption{{\\sl  Same as Fig.~\\ref{fig:toy1}. Independent complete simulation.}}\n\\label{fig:toy3}\n\\end{figure}\n\n\n\\section{Conclusions}\nThe evaluation of uncertainties of \nthe Bayesian unfolding of Ref.~\\cite{BU} has been improved\ntaking the probability density functions of the quantities\nof interest. Simplifications are achieved using prior conjugates\nand performing the relevant integrations, needed to propagate \nuncertainties, by sampling. \n\nThe paper also discusses the issue of the iterations and the\nrole of the {\\it intermediate} smoothing \nto reach fast convergence. It is important to note that, \ncontrary to other algorithms,\nthe intermediate smoothing acts as a {\\it regularization on the \npriors} and not on the unfolded spectrum. For this reason\nphysical peaks should still appear in the unfolded spectrum.\n\nThe R code of the algorithm is available, together with \na little simulation script to run it on toy models. \nThis script also implements \nof simple intermediate prior regularization, but \nthe reader should have clear in mind that\nthis task is left to the user, who is supposed to\nunderstand well his\/her physics case. \n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany of today's proposals for quantum information applications\\cite{bouwmeester:00} rely on the controlled \nand fast manipulation of the discrete states of the corresponding devices' underlying\nstructures. Semiconductor quantum dots (QDs) are frequently discussed as building blocks\nfor such materials because they hold out the prospect of tailor-made energy spectra\nand a high integrability in a solid-state environment. The excitonic QD states\nare promising candidates to be used as qubits for quantum computing\\cite{biolatti:00, chen:01, li:03, piermarocchi:02, boyle:08, nielsen:00},\nwhile the radiative decay from the biexciton cascade offers the possibility of an on-demand creation of indistinguishable entangled photon \npairs\\cite{moreau:01, stevenson:06, mueller:14}. \n\nIt has been shown that both exciton and biexciton states of a QD can be prepared by\nusing ultra-fast laser pulses under a variety of excitation conditions\\cite{reiter:14,ramsay:10c}.\nThe most commonly known schemes for this purpose are resonant excitation leading to Rabi rotations\\cite{zrenner:02,machnikowski:04, vagov:07, ramsay:10},\ndifferent protocols using chirped laser pulses exploiting the adiabatic rapid passage effect \\cite{schmidgall:10,simon:11,wei:14,lueker:12, glaessl:13, gawarecki:12, debnath:12, debnath:13, mathew:14},\nand phonon-assisted off-resonant driving \\cite{glaessl:11b, reiter:12, hughes:13, glaessl:13b}.\nRecently, the latter method has also been experimentally demonstrated\\cite{ardelt:14, quilter:15, bounouar:15}. \nIndeed, there is an increased interest in this approach,\nbecause it is not only stable against\nfluctuations of the applied field intensity, but also leaves the quantum dot transition laser-free, which can be\nimportant when the emitted photons need to be spectrally separated from the laser pulse.\nFurthermore, in contrast to the other two protocols the phonon-assisted scheme makes active use of the phonon coupling\nand even works the better the stronger this coupling is. \n\nIn this paper, we examine the influence of the pulse shape on the phonon-assisted state preparation.\nWe identify three distinct processes that take place during the laser-driven evolution of the QD states.\nWhen the pulse starts the coupling to the laser-field leads to a \\emph{dressing} of the bare QD states.\nThis enables a \\emph{phonon-induced relaxation} between the dressed states\\cite{leggett:87, dekker:87, glaessl:11b}\nand finally when the pulse is switched off an \\emph{undressing} takes place.\nWhile previous studies mostly concentrated on the phonon induced relaxation and the resulting exciton and biexciton\noccupation obtained when applying an off-resonant driving pulse\\cite{glaessl:13b}, here we will explain the impact of all three\ndifferent processes and examine in detail the role of the switch-on and switch-off phase of the excitation.\nWe will show that within the relaxation process there is a trade-off situation between a sufficiently fast \npreparation and an optimal preparation fidelity and that for a high fidelity \npreparation of the bare QD states an adiabatic undressing, that can be realized by a long enough switch-off\ntime, is indispensable in short pulse protocols.\nOur analysis also shows that even though Gaussian pulses, as used in very recent experiments\\cite{ardelt:14, quilter:15, bounouar:15},\nare not the ideal choice for the phonon-assisted protocol they fulfill the requirements for fast state preparation\nsurprisingly well for a wide intensity range provided that \nthe regime of non-adiabatic dynamics is not yet reached. The latter condition sets a lower bound to the pulse\nduration.\n\nThe pulse characteristics not only allow to control the fidelity of the achieved inversion,\nbut in addition can be used to select the QD state that is targeted by the phonon-assisted process.\nFor the two-level case the sign of the pulse detuning determines whether\nthe QD is driven towards the ground-state or the exciton state. For the exciton-biexciton system\nalso the biexciton binding energy and the pulse length play a critical role in determining the targeted QD state.\nUnderstanding that the preparation is a three-step process gives us an intuitive answer to the important question which state is selected\nby the phonon-assisted preparation scheme.\n\n\n\\section{Model}\\label{model}  \n\nWe consider a strongly confined GaAs QD driven by an external laser field and \ncoupled to a continuum of acoustic bulk phonons. Our model is defined by the Hamiltonian\n$H = H_{\\rm{dot,\\,light}} + H_{\\rm{dot-ph}}$, i.e., the electronic system coupled\nto the laser field and an additional phonon part.\nLet us first concentrate on \n\\begin{align}\n&H_{\\rm{dot,\\,light}} \\!= \\sum_\\nu \\hbar\\omega_\\nu |\\nu\\rangle\\langle\\nu|\n                      \\!+ \\sum_{\\nu\\nu'} \\hbar M_{\\nu\\nu'}|\\nu\\rangle\\langle\\nu'|,\n\\label{eq_H_dotlight}\n\\end{align}\nwhere $|\\nu\\rangle$ are the electronic basis states with corresponding energies $\\hbar \\omega_{\\nu}$. \nThe matrix element $M_{\\nu\\nu'}$ describes the coupling between the QD and the classical laser field using\nthe common dipole and rotating wave approximations.\nIn the first part of the paper we will restrict ourselves to a two-level system consisting of the ground-state $|0\\rangle$,\nfor which we set $\\hbar\\omega_0=0$, and a single exciton state $|X\\rangle$ with energy $\\hbar\\omega_X$ as illustrated in Fig.~\\ref{fig1}(a).\nThe two-level approximation is valid when considering circularly polarized light with a single polarization orientation\nand the exchange interaction is negligibly small. The exchange interaction strongly depends on the QD geometry\\cite{langbein:04} and \ncan be close to zero, as it is favorable for, e.g., entangled photon creation\\cite{stevenson:06}.\nIn this case the non-zero matrix elements of the light-matter coupling are given by\n\\begin{align}\n&M_{0X} = \\frac{1}{2}f(t) e^{i\\omega_L t}\\,, \\qquad M_{X0}=M_{0X}^*\\, ,\n\\label{eq_M2LS}\n\\end{align}\nwhere $f(t)$ is a real pulse envelope function, which in the following is referred to as field strength.\nThe field strength $f(t)$ is related to the electric field ${\\bf E}(t)$ with frequency $\\omega_L$ and the QD dipole matrix element\n${\\bf d}$ by $-{\\bf d}\\cdot{\\bf E}(t)=\\frac{\\hbar}{2}f(t)e^{-i\\omega_L t}$. \n\nA very useful picture for strongly driven few-level systems, which we will employ to analyze our results, is the dressed state picture. \nThe dressed states are the eigenstates of the coupled light-matter Hamiltonian\nin a frame rotating with the laser frequency \\cite{tannor:07}.\nFor the two-level system driven by a laser-field with a fixed field strength $f$ and a detuning $\\Delta$ they are given by the expressions\n\\begin{subequations} \\label{dressed_states}\n\\begin{eqnarray}\n |\\psi_{\\rm{up}}\\rangle &=& +\\cos(\\Theta)|0\\rangle + \\sin(\\Theta)|X\\rangle\\\\\n |\\psi_{\\rm{low}}\\rangle &=& -\\sin(\\Theta)|0\\rangle + \\cos(\\Theta)|X\\rangle\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\Theta$ is the mixing angle defined by $\\tan(\\Theta)=\\frac{\\hbar f}{\\Delta+ \\hbar\\Omega}$ and $\\Omega$ is the Rabi frequency given by \n\\begin{align}\n\\hbar \\Omega = \\sqrt{(\\hbar f)^2+\\Delta^2}.\n \\label{Omega}\n\\end{align}\nThe corresponding dressed state energies read\n\\begin{align}\n E_{\\rm{up\/low}}=\\frac{1}{2}(-\\Delta\\pm\\hbar\\Omega).\n \\label{dressed_state_energies}\n\\end{align}\nIt is worth noting that the contributions of the ground and exciton states to the dressed states vary depending on the\ndetuning and the field strength.\n\nThe final part of this paper will be devoted to excitations with linearly polarized light. In this case one also has to take into account\nthe biexciton state and the system described so far needs to be extended to a three-level model consisting\nof the ground state $|0\\rangle$, the single exciton state $|X\\rangle$ and the biexciton state $|B\\rangle$. \nNote that the single exciton state $|X\\rangle$ in the three-level system has different polarization than in the two-level system described further above.\nAs is illustrated in Fig.~\\ref{fig1}(b), the biexciton state has the energy $\\hbar\\omega_B=2 \\hbar\\omega_X- \\Delta_B$,\nwhere $\\Delta_B$ is the biexciton binding energy. \nIn the exciton-biexciton system, the non-vanishing dipole matrix elements are given by\n\\begin{align}\n&M_{0X} = M_{XB}= \\frac{1}{2}f(t)e^{i\\omega_L t}, M_{X0}=M_{BX}=M_{0X}^*.\n\\label{eq_M3LS}\n\\end{align}\n\nLet us now focus on the coupling of the QD to the phonon environment. We model the electron-phonon interaction \nby a pure-dephasing Hamiltonian which, together with the free phonon Hamiltonian, has the form\n\\begin{align}\n&H_{\\rm{dot-ph}} \\!= \\! \\sum_{\\bf q} \\hbar\\omega_{\\bf q}\\,b^\\dag_{\\bf q} b_{\\bf q} \n                       + \\sum_{{\\bf q} \\nu} \\hbar n_\\nu \\big(\n                      \\gamma_{\\bf q} b_{\\bf q} + \\gamma^{*}_{\\bf q} b^\\dag_{\\bf q}\n\\big)\n|\\nu\\rangle\\langle\\nu|.\n\\label{eq_H_phonon}\n\\end{align}\nThe operators $b^{\\dagger}_{\\bf q}$ ($b_{\\bf q}$) create (annihilate) a phonon with wave vector ${\\bf q}$ \nand linear dispersion $\\hbar \\omega_{{\\bf q}}=\\hbar c_s |{\\bf q}|$, where $c_s$ denotes the longitudinal sound velocity.\n$n_\\nu$ counts the number of excitons present in the state $|\\nu \\rangle$. The coupling constants $\\gamma_{\\bf q}$ are \nchosen specific for the deformation potential coupling to longitudinal acoustic phonons, which has been shown to be dominant \nfor typical self-assembled GaAs QDs\\cite{ramsay:10, krummheuer:02}. As described in more detail in Ref.~\\onlinecite{krummheuer:02},\nthe coupling constants $\\gamma_{\\bf q}$ depend on the electronic wave functions $\\Psi_{e(h)}$ as well as on the deformation potential\nconstants $D_{e(h)}$ for electrons (e) and holes (h), respectively. For simplicity, we assume the wave functions to be the ground-state\nsolutions of a spherically symmetric harmonic potential, i.e., $\\Psi_{e(h)}({\\bf r}) \\propto \\exp\\big(-r^2\/(2a_{e(h)}^2)\\big)$,\nand refer to  $a_e$ as the QD radius. The characteristics of the exciton-phonon coupling can be expressed by the spectral density\n$J(\\omega)=\\sum_{\\bf q}|\\gamma_{\\bf q}|^2\\delta(\\omega-\\omega_{\\bf q})$, which under the above assumptions reads\\cite{krummheuer:02}\n\\begin{align}\n&J(\\omega ) =\\frac{\\omega^{3}}{4\\pi^{2}\\rho \\hbar c_{s}^{5}} \n  \\big[D_{e}e^{-\\omega^{2}a_{e}^{2}\/(4c_{s}^{2})}   \n   -   D_{h}e^{-\\omega^{2}a_{h}^{2}\/(4c_{s}^{2})} \\big]^{2}\n\\label{eq_J}\n\\end{align}\nwith $\\rho$ being the mass density. For our present calculations we use material parameters specific for GaAs taken from the literature\\cite{krummheuer:05}\n$\\rho\\!=\\!5370$~kg\/m$^3$, $c_s\\!=\\!5110$~m\/s, $D_e=7.0$~eV, and $D_h=-3.5$~eV. For the QD size we choose $a_e=3$~nm and set $a_e\/a_h=1.15$ assuming equal potentials \nbut taking into account the different effective masses of electrons and holes for GaAs. \n\\begin{figure}[t]\n \\includegraphics[width=\\columnwidth]{fig1.eps}\n    \\caption{(Color online) Energetic level diagram of (a) the exciton system with circular polarized excitation and (b) the exciton-biexciton system with linear polarized excitation.\n    (c) Spectral density of the phonon coupling as a function of energy for the parameters used in our simulations (see text).}\n \\label{fig1}\n\\end{figure}\nThe spectral density for these parameters is shown in Fig.~\\ref{fig1}(c) where \nit can be seen that $J(\\omega)$ exposes a clear maximum at $\\hbar\\omega = E_J^{\\rm max} \\approx 2$~meV. \nThe electron-phonon interaction leads to a polaron-shift $\\delta_{ph}$ of the exciton and biexciton energy, such that when driving the exciton-to-ground state transition\nthe resonant excitation is on the shifted exciton energy $\\hbar \\tilde{\\omega}_X = \\hbar \\omega_X - \\delta_{ph}$. In this paper we are mostly interested\nin detuned excitations and define the detuning as the difference between the laser energy and the polaron-shifted exciton energy, i.e., \n\\[ \\Delta = \\hbar \\omega_L - \\hbar \\widetilde{\\omega}_X. \\]\n\nTo study the time evolution of the electronic QD occupations under excitation with the laser field we employ an implementation\nof a real-time path-integral approach\\cite{vagov:11}. This method allows a numerically exact treatment of the above model\ndespite the infinite number of LA phonon modes and yields the dynamics of the reduced electronic density matrix of the \nQD including arbitrary multi-phonon processes and taking into account all non-Markovian effects. \nWe assume the QD to be initially in a product state of the electronic ground-state and a thermal equilibrium of\nthe phonon modes at temperature $T=1$~K.\n\n\n\\section{Results}\\label{results}  \n\nWe start by analyzing the phonon-induced relaxation using continuous excitation that is switched on \ninstantaneously. Then we will apply short pulses with different pulse shapes to\nanalyze the influence of the adiabaticity of the dressing and undressing process \nin view of high-fidelity state preparation.\nFor this analysis we will also visualize the system trajectory on the Bloch sphere.\nFinally, we will demonstrate the selective addressing of all three states in the exciton-biexciton\nsystem using the phonon-assisted state preparation protocol.\n\n\\subsection{Phonon-induced relaxation}\\label{phonon_induced_relaxation}\n\n\\begin{figure}[ttt]\n \\includegraphics[width=8.7cm]{fig2.eps}\n    \\caption{(Color online)\n            (a) Exciton occupation $C_X$ as a function of time for different\n\t    field strengths $f=0.5$~ps$^{-1}$, $1.0$~ps$^{-1}$ and $1.5$~ps$^{-1}$. (b) Time $\\tau$ after which\n\t    the time average of the oscillations of $C_X$ has\n\t    reached 99\\% of $C_{X}^{\\infty}$ (see text) as a function of the field strength $f$.\n\t    Inset of (b): $C_{X}^{\\infty}$ as a function of the field strength $f$.\n\t    The detuning is $\\Delta=1.0$~meV.\n            }\n \\label{fig2}\n\\end{figure}\n\nThe phonon-induced relaxation can be best analyzed by considering a continuous excitation of the QD with a \nconstant field strength. Further, the laser field shall be circularly polarized such that our model\ncan be restricted to two electronic levels, as we explained in the previous section.\nGenerally, for a weak coupling between the electronic states and the phonon environment the standard expectation\nfor the driven QD dynamics is that the Markovian approximation is \nwell justified and that in the long-time limit the relaxation leads to a thermal occupation\nof the dot-photon dressed states\\cite{leggett:87, weiss:99} [c.f. Eq.~(\\ref{dressed_states})].\nIn a very good approximation, this has been shown to hold also true for the two-level model of the QD with standard GaAs-type parameters\nconsidered here\\cite{glaessl:11b}.\nFor very low temperatures, the system ends up mainly in the lower dressed state $|\\psi_{\\rm{low}}\\rangle$,\nwhich corresponds to an exciton occupation of\n\\begin{align}\n C_{X}^{\\infty} \\approx \\cos^2(\\Theta)= \\frac{1}{2} \\left(1 + \\frac{\\Delta}{\\sqrt{\\hbar^2 f^2 + \\Delta^2}}  \\right).\n \\label{CX_infty}\n\\end{align}\nFigure~\\ref{fig2}(a) shows the simulated temporal evolution of the exciton state under constant excitation for a detuning of $\\Delta = 1.0$~meV\nand three different field strengths $f$. Here, the laser field is switched on instantaneously at $t = 0$~ps.\nThe occupations perform damped Rabi oscillations around a mean value that approaches a constant value.\nFor a decreasing field strength the stationary exciton occupation rises as can also be seen in the inset of Fig.~\\ref{fig2}(b)\nwhere $C_{X}^{\\infty}$ given by Eq.~(\\ref{CX_infty}) is shown as a function of $f$ for $\\Delta = 1.0$~meV.\nMost importantly for phonon-assisted exciton preparation, the stationary exciton occupation is very close to one and only limited\nby the finite temperature for sufficiently small field strengths. Larger values of $f$, however, strongly reduce $C_{X}^{\\infty}$.\n\nOn the other hand, when we look at the time required to reach the final state, we find that for small field strengths $f$, \nthe time to reach the final state becomes longer. This is quantified in Fig.~\\ref{fig2}(b), where we have plotted the time $\\tau$ it\ntakes for the mean value of the oscillations to reach 99\\% of the exciton occupation expected for a thermal distribution of the\ndot-photon dressed states $C_{X}^{\\infty}$ as a function of the field strength.\nFor example at $f = 0.5$~ps$^{-1}$ a relaxation time of several hundred ps is required, which might exceed the time until other\nrelaxation processes occurring on longer time scales than the phonon-induced relaxation such as the radiative decay\nwhich is not considered here, come into play. Therefore such a low driving strength does not yield a sufficiently fast relaxation\nfor phonon-assisted state preparation. Indeed, we find that for $f\\rightarrow 0 $ the time needed for the state preparation diverges.\n\nTo understand the varying relaxation times for different driving strengths one has to keep in mind that the pure dephasing type\nphonon coupling does not induce direct transitions between the electronic states, but the phonon-induced relaxation is only\nenabled by the laser field. Transitions between the QD states only take place due to a non-vanishing overlap of both of the photon\ndressed states with the exciton state, which in turn is coupled to the phonon-environment. \nFor an efficient relaxation also the Rabi splitting $\\Omega$, i.e. the difference between the two dressed state energies, needs to be \nclose to typical phonon energies and ideally matches the maximum of the spectral density of the phonon coupling $J(\\omega)$ \n[c.f. Eq.~(\\ref{eq_J})]. Both of these properties of the relaxation rate are also captured by a simple application of Fermi's Golden Rule,\nwhich yields a relaxation time between the upper dressed state without phonons and the lower dressed state with one phonon of\n\\begin{align}\n\\tau_{\\rm relax} = \\left(\\frac{\\pi}{2}\\sin^2(2\\Theta)\\,J(\\Omega)\\right)^{-1}.\n\\label{golden_rule}\n\\end{align}\nHere it can be seen that the mixing angle $\\Theta$ as well as the spectral density evaluated at the Rabi splitting $J(\\Omega)$ enter\nthe relaxation rate. According to the simulations the maximal relaxation for the detuning $\\Delta = 1.0$~meV used here occurs for a field strength around $2.7$~ps$^{-1}$,\nwhich is reflected by the position of the minimum of the time needed for an almost complete relaxation plotted in Fig.~\\ref{fig2}(b).\nIn a good approximation this is expected from the rough estimation in Eq.~(\\ref{golden_rule}).\nDespite the short relaxation time at the optimal field strength, which is below $20$~ps, such a strong driving field is not applicable for high-fidelity\nstate preparation either, because the achieved final occupation of below $0.8$ is too far from the desired fidelity of one.\n\nLooking at our results, it becomes clear that the maximum exciton occupation for a given detuning can only be realized for almost\nvanishing field strengths where on the other hand it takes arbitrarily long times to complete the relaxation process.\nTherefore, we conclude, that when only the phonon relaxation process of the preparation takes place, \nthere is a trade-off between reaching the targeted state with high fidelity and realizing fast preparation times.\n\n\\subsection{Dressing and Undressing}\\label{dressing_undressing}\n\n\\begin{figure}[t]\n \\includegraphics[width=8.7cm]{fig3.eps}\n    \\caption{(Color online)\n         Dynamics for an off-resonant excitation of the QD with rectangular pulses (left) having a length of $40$~ps and Gaussian pulses (right) \n         with a FWHM of $20$~ps. The pulse areas are $\\alpha=10\\pi$ (green), $20\\pi$ (blue) and $30\\pi$ (red). \n    \t(a),(b) Exciton occupation $C_X$,\n       \t(c),(d) occupation of the energetically lower dressed state $C_{\\rm{low}}$, \n       \t(e),(f) instantaneous energy of the upper and lower dressed state $E_{\\rm{up\/low}}$, \n       \t(g),(h) pulse envelopes.\n       \tThe detuning is $\\Delta=1$~meV.\n\t    }\n \\label{fig3}\n\\end{figure}\n\nIn the previous section the switch-on of the laser field has been taken to be instantaneous. This implies that a sudden \ntransformation of the photon dressed states occurs at the beginning of the pulse.\nSimilarly, when switching off the laser field the photon dressed states are transformed back to the pure ground and exciton states.\nTo understand the importance of these \n\\emph{dressing} and \\emph{undressing} processes in terms of fast and high fidelity phonon-assisted state preparation,\nwe now look at excitations with different pulse shapes. First we compare an excitation of the two-level system with rectangular and Gaussian pulses.\nLater, we approximate the Gaussian pulse using a rectangular pulse with softened edges. \n\nFig.~\\ref{fig3} shows the temporal evolution under excitation for the rectangular (left) and Gaussian (right) pulses including (a),(b) the exciton occupation,\n(c),(d) the lower dressed state occupation, (e),(f) the instantaneous dressed state energies and (g),(h) the pulse envelopes of the electric field.\nThe length of the rectangular pulses is chosen as $40$~ps, which is twice the full width of half maximum (FWHM) of the Gaussian pulses, which is set to $20$~ps.\nAll calculations are performed for a detuning of $\\Delta = 1.0$~meV and for three different pulse areas $\\alpha=\\int_{-\\infty}^\\infty f(t)\\rm{d}t$ with $\\alpha=10\\pi$ (green curves),\n$20\\pi$ (blue curves), and $30\\pi$ (red curves). \n\nLet us start with the rectangular pulse (left panels in Fig.~\\ref{fig3}). The laser pulse sets in instantaneously at $t=-20$~ps and besides damped Rabi oscillations\nthere is an overall increase of the exciton occupation that depends on the strength of the pulse [Fig.~\\ref{fig3}(a) and see also Fig.~\\ref{fig2}(a)].\nIn the dressed state picture this behavior can be understood as follows:\nWhen there is no laser pulse, the dressed states are equal to the bare states where for a positive detuning the ground state corresponds to the upper dressed state $E_{\\rm{up}}$ and\nthe exciton state corresponds to the lower dressed state $E_{\\rm{low}}$. Note that in this picture the energies of the photons needed for the excitation\nare counted as part of the dressed state energies.\nAs soon as the laser pulse sets in the dressed states become a mixture of ground and exciton state [cf. Eq.~(\\ref{dressed_states})] with shifted energies shown in Fig.~\\ref{fig3}(e),\ni.e. the bare QD states become dressed by the laser field.\nDue to their overlap with the ground-state both dressed states instantly become occupied, which is reflected by Rabi oscillations in the bare states\\cite{hughes:13}.\nPhonons induce transitions between the dressed states \\cite{lueker:12, reiter:14} and\nat low temperatures transitions to the lower dressed state, that correspond to phonon emission, outweigh.\nTherefore, during the pulse the lower dressed state becomes more and more occupied as it can be seen in Fig.~\\ref{fig3}(c). This also means that the exciton occupation\nsuccessively approaches its stationary value $C_X^{\\infty}$, which depends on the exciton contribution to the lower dressed state but is above $0.5$ for all field strengths \nand positive detunings [c.f. Eq.~(\\ref{CX_infty})].\nWhen the laser pulse is switched off at $t=40$~ps the exciton occupation [Fig.~\\ref{fig3}(a)] keeps the value it has right before the switch-off.  \nAt the same time the dressed states are abruptly transformed back to the bare QD states due to the sudden stop of the rectangular pulse, which means that \nthe undressing takes place instantaneously. This is reflected by a step-like drop of the occupation of the lower dressed state that can be seen in Fig.~\\ref{fig3}(c).\nFor the weakest pulse ($\\alpha=10\\pi$) the final state is not reached within the time window of $40$~ps,\nwhile for the higher pulse areas the relaxation process is mostly completed. Either way, the final exciton occupation stays below $0.9$ for all of the three rectangular pulses and\nin fact no matter which rectangular shape is assumed for pulse lengths of a few tens of ps one never achieves a final value close to one due to the trade-off\nsituation described in the previous section.\n\nThe situation is quite different for the Gaussian pulses shown in the right column of Fig.~\\ref{fig3}.\nIn this case, no oscillations of the exciton occupation are visible and instead $C_X$ smoothly rises to its final value that for pulse areas\n$\\alpha=20\\pi$ and $30\\pi$ is considerably higher than for the rectangular pulses and practically reaches $1.0$ as it can be seen in Fig.~\\ref{fig3}(b).\nLike for the rectangular pulses, a phonon relaxation process takes place from the upper to the lower dressed state yielding a steady increase of the\noccupation of the lower dressed state in Fig.~\\ref{fig3}(d). However, because the field strength for Gaussian pulses is time-dependent,\nthe bare QD state contributions of the dressed states change during the pulse. This is also reflected by the time dependence of the dressed state energies shown in\nFig.~\\ref{fig3}(f), which can be used to extract the Rabi splitting at a given time. \nFor the weakest Gaussian pulse shown here ($\\alpha=10\\pi$), and similar to the case of the weakest rectangular pulse, the phonon-induced relaxation is \ntoo weak to complete the relaxation process within the duration of the pulse. This is because the Rabi splitting is well below the maximum of the phonon density\nat energy $E_J^{\\rm max}\\approx2$~meV [c.f. Fig.~\\ref{fig1}(c)] even at the maximum of the pulse where the Rabi splitting reaches its maximum value of approx.\n$1.4$~meV [c.f. Fig.~\\ref{fig3}(f)].\nIn contrast, for the higher pulse areas $\\alpha=20\\pi$ and $\\alpha=30\\pi$ the relaxation is more effective and leads to a full occupation of the lower dressed state.\nAt the pulse maximum the Rabi splitting for the $30\\pi$ pulse already becomes larger ($3.0$~meV) than $E_J^{\\rm max}$ temporarily leading to \na weakening of the relaxation which is visible by a reduced gain of the lower dressed state occupation [red curve in Fig.~\\ref{fig3}(d)] around $t=0$~ps.\nMost interestingly, the phonon-induced relaxation is practically complete around $t=10$~ps while the exciton occupation obtained by the stronger pulses is still\nfar from its final value and therefore the remaining increase of $C_X$ cannot be attributed to the relaxation.\nThe final value of $C_X$ is only reached within a second phase of increase that takes place while the pulse is switched-off. \nThis is only possible because during the switch-off the dressed states are \\emph{adiabatically undressed}, i.e. adiabatically transformed back to the bare QD states.\nIn this process the ground state component of the then almost fully occupied lower dressed state is reduced to zero which yields\na drastic increase of the exciton occupation. Importantly, a necessary precondition for this increase \nis that the undressing takes place slow enough such that the system can follow the transformation of the dressed states adiabatically and\nthe occupation stays in the lower dressed state. For example this is not fulfilled for the instantaneous undressing in case of the rectangular\npulses discussed earlier where the lower dressed state occupation experiences a step-like drop at the end of the pulse [Fig.~\\ref{fig3}(c)]\nthat does not occur for Gaussian pulses [Fig.~\\ref{fig3}(d)].\n\nAs it turns out the adiabatic undressing towards the end of the pulse is in fact essential for a successful {\\em fast} exciton preparation. \nThe absence of the adiabatic ending prevents the phonon-assisted state preparation protocol from working with high fidelity for short \nrectangular pulses and any other pulse shapes with too fast switch-off times.\n\n\\begin{figure}[t]\n \\includegraphics[width=8.7cm]{fig4.eps}\n    \\caption{(Color online)\n            Time dependent exciton occupation $C_X$ under pulsed excitation detuned by $\\Delta=1.0$~meV for different (a) \n\t    switch-on (b) and switch-off times (see key). (c),(d) corresponding\n\t    occupations of the energetically lower dressed state $C_{\\rm{low}}$ and  (e), (f) corresponding pulse envelopes. \n\t    \t}\n \\label{fig4}\n\\end{figure}\n\nHaving seen that the temporal evolution of the dressing and undressing plays a crucial role, we investigate in more detail the role of the switch-on and switch-off phase.\nTo this end we choose a special pulse shape that is designed to highlight the important role of these two processes on the preparation process. \nThe thermalization of the dressed states is obviously achieved best by a constant part of the pulse that is chosen such that the\ncoupling to the phonon environment is maximal ($f = 2.7$~ps$^{-1}$ for the parameters chosen here) and that is sufficiently long\nto complete the thermalization. In our case, the duration of this part of constant driving is chosen to be $40$~ps.\nIn order to systematically study the influences of the switch-on (switch-off) characteristics we add the left (right)\nhalf of a Gaussian pulse with FWHM $\\tau_{\\rm{on}}$ ($\\tau_{\\rm{off}}$) before (after) the constant part of the pulse envelope as illustrated in Fig.~\\ref{fig4}(e)[(f)]. \nHere, we compare a strictly rectangular pulse with $\\tau_{\\rm{on\/off}} = 0$~ps (green) with pulses of different switch-on\/off times \n$\\tau_{\\rm{on\/off}} =1$~ps (black), $3$~ps (blue) and $5$~ps (red).\nThe first row [panels (a) and (b)] shows the corresponding evolution of $C_X$, while the lower dressed state occupation is plotted in the second row [panels (c) and (d)] \nand the pulse envelopes are shown in the third row [panels (e) and (f)].\n\nLooking at the left panels of Fig.~\\ref{fig4}, we find that a longer switch-on time does not alter the resulting exciton occupation after the pulse,\nwhich is about $0.75$ for all three cases. However, we find  a significant reduction of the Rabi oscillations, which almost disappear for $\\tau_{\\rm{on}} = 5$~ps. \nThe insensitivity of the final occupation with respect to the switch-on dynamics is related to the irreversible nature of the relaxation process which,\nin case of a complete relaxation, leads to a complete loss of memory of the initial state. We thus find that the first phase, i.e., the dressing,\nis necessary to start the relaxation process, but for a sufficiently long relaxation phase the details of the switch-on are irrelevant for the final fidelity. \n\nIn contrast, as demonstrated in the right panels of Fig.~\\ref{fig4}, a slower switch-off gives rise to a further gain of exciton occupation\nwhen the field strength is reduced to zero. This effect is considerably less pronounced if the switch-off time becomes too short and does not occur at all\nif the pulse stops instantaneously. For example for $\\tau_{\\rm{off}} = 5$~ps (red curve) $C_X$ increases by more than $0.2$ during the switch-off, \nwhile for $\\tau_{\\rm{off}} = 1$~ps (black) the increase is below $0.05$.\nThus, we conclude, that to yield a high fidelity state preparation the transformation of the dressed states into the bare QD states needs to happen \nslow enough in order to allow for an adiabatic evolution, i.e., the undressing needs to be adiabatic.\n\nTo sum up, for an efficient and fast phonon-assisted state preparation there are basically two features of the pulse envelope that are significant:\nFirst, there has to be a phase of the pulse where the QD thermalizes in the dressed state basis, which must be long enough to complete the relaxation.\nThis phase can be chosen the shorter the stronger the phonon-induced relaxation is and therefore can be minimized when the field strength is such that\nthe strength of the phonon-coupling is maximal. Therefore in this phase a constant field strength corresponding to the maximal relaxation rate is optimal\nto achieve a complete thermalization with a minimal pulse length. Second, the pulse must be switched-off slowly enough to allow the QD system for \nan adiabatic undressing of the states.\n\nAlthough a Gaussian pulse shape does not meet these requirements in an optimal fashion it still works surprisingly well,\nmostly because the optimum for the ideal field strength at which\nthe dressed state relaxation is maximal is not a sharp maximum, but rather a broad peak as it can be seen from Fig.~\\ref{fig1}.\nTherefore, even though a Gaussian\npulse does not have a constant plateau, a sufficient relaxation takes place during most of the time of the pulse\nprovided a reasonable value of the pulse area is chosen. Most importantly, deviations from the Gaussian pulse shape\nthat go towards faster switch-off times, as they might be induced by a pulse-shaping setup, can be harmful to \nthe efficiency of the phonon-assisted preparation protocol.\n\n\\subsection{Interpretation on the Bloch sphere}\\label{bloch_sphere}\n\n\\begin{figure}[t]\n \\includegraphics[width=\\columnwidth]{fig5.eps}\n    \\caption{(Color online)\n            (a),(b) Illustration of the system trajectory on the Bloch sphere and\n            (c),(d) length of the Bloch vector as a function of time for \n            (a),(c) a Gaussian excitation with pulse areas $\\alpha=10\\pi$ (green), $20\\pi$ (blue) and $30\\pi$ (red)\n            and (b),(d) a rectangular excitation with different switch-on\/off times\n            $\\tau_{\\rm{on}}=5$~ps; $\\tau_{\\rm{off}}=0$~ps (blue) and  $\\tau_{\\rm{on}}=0$~ps;  $\\tau_{\\rm{off}}=5$~ps (red). \n            The detuning is $\\Delta=1$~meV.\n\t    }\n \\label{fig5}\n\\end{figure}\n\nAnother interesting aspect not highlighted so far is that the incoherent phonon scattering can result in a pure state, which even \ncan be transformed to a bare QD state.\nThis is best illustrated using the Bloch vector picture\\cite{allen:75}. In this picture the projection of the Bloch vector on the $z$-axis represents the inversion,\ni.e., the difference between the occupations of the upper and the lower level of the two-level system, while the in-plane component reflects the polarization.\nResonant lossless driving of a two-level system is reflected by the Bloch vector moving at the surface of the Bloch sphere taking the shortest path from one pole to the\nother corresponding to the well known coherent Rabi oscillations.\nPreparing the exciton state using a $\\pi$ pulse then means going from the lower pole corresponding to the ground level to the upper pole \ncorresponding to the excited level.\nA detuned excitation leads to a tilted oscillation, which starting from the lower pole does not reach the upper pole.\nIf the system is fully coherent, the Bloch vector stays on the surface, i.e., its length is constantly one. If decoherence takes place,\nthe length of the Bloch vector is decreased. \nClearly, the phonon-assisted preparation involves an incoherent phonon-induced relaxation, but as we have seen it can still \neventually lead to an almost perfect exciton state. \n\n\nTo analyze this in more detail we have calculated the trajectory of the Bloch vector of the driven QD system, which is shown in Fig.~\\ref{fig5}(a),(b)\nalongside the time evolution of the length of the Bloch vector presented in Fig.~\\ref{fig5} (c),(d) for an excitation with Gaussian pulses (left panel)\nand for rectangular pulses with softened switch-on or switch-off (right panel). The corresponding pulse shapes can be seen in Fig.~\\ref{fig3} (h) and Fig.~\\ref{fig4} (e),(f).\n\nLet us first look at the Gaussian pulses in the left column.\nWhen the pulse is switched on the Bloch vector leaves the surface of the sphere for all pulse areas in Fig.~\\ref{fig5}(a) \n[$\\alpha=10\\pi$ (green), $\\alpha=20\\pi$ (blue) and $\\alpha=30\\pi$ (red)]. The corresponding vector length\nshown in Fig.~\\ref{fig5}(c) decreases \nas it is expected as the result of the pure-dephasing phonon coupling. \nIndeed, the loss of coherence turns out to be as high as $95\\%$. \nThe time to reach the minimal Bloch vector length depends on the pulse area and is shorter for the stronger pulses where the relaxation is more\nefficient. \nThe vector length would vanish at the minimum in the ideal case of a completely adiabatic dressing process\nand provided that the phonons realize a fully incoherent occupation transfer between the dressed states. In that case the electronic density matrix would\nstay diagonal in the dressed state basis all the time and coherences between the dressed states would neither build up due to the laser driving nor \ndue to the phonons. The continuous occupation transfer from the upper to the lower dressed state would then necessarily lead to a zero of the Bloch vector length\nat some point in time. In our case, however, the minimal Bloch vector length has a small but finite value because the Gaussian pulses used in the simulations already\ninduce some small coherences between the dressed states (not shown).\nSubsequently, the coherence between $|0\\rangle$ and $|X\\rangle$ is restored to a large degree and the Bloch vector length approaches $1$ for the two stronger\npulses as the phonon-assisted transitions result in an almost complete occupation of the lower dressed state which is again a pure state lying \non the surface of the Bloch sphere. For the weaker pulse, $\\alpha=10\\pi$ (green), the trajectory ends inside the Bloch sphere and the vector\nlength stays well below $0.5$, because due to the insufficient pulse strength the relaxation does not complete.\nFor the stronger pulses, the dressed states transform from a superposition of ground and exciton state into\nthe pure exciton state during the adiabatic undressing, which corresponds to a motion of the Bloch vector on the surface towards the upper pole. \nIt thus becomes obvious that the laser-driven QD evolution describes a way {\\em through} the Bloch sphere to the upper pole, which is very\ndifferent to the motion along the surface in the case of an inversion yielded by applying a resonant $\\pi$ pulse.\n\nA similar analysis can be done for the Bloch vector trajectory for rectangular pulses where the switch-on or the switch-off\nedge is softened. The red curve in Fig.~\\ref{fig5}(b),(d) has a sharp switch-on and a smooth switch-off with $\\tau_{\\rm{off}}=5$~ps,\nwhile the blue curve corresponds  to a smooth switch-on with $\\tau_{\\rm{on}}=5$~ps and a sharp switch-off. \nWhen the switch-on is instantaneous (red curve), we see that the Bloch vector trajectory exhibits a spiral movement within the Bloch sphere \nreflecting the damped Rabi oscillations. In agreement with the essentially complete relaxation, the spiral ends up on a point close to the \nsurface of the Bloch sphere. For $\\tau_{\\rm{on}}=5$~ps (blue curve), the oscillations are less pronounced and the motion goes roughly \nalong the axis of the spiral movement. The spiraling will eventually vanish for even larger values of $\\tau_{\\rm{on}}$, \nbut the end point after the relaxation will be the same. If the laser pulse is switched off rapidly (blue curve), \nthe z-component of the Bloch vector stays constant after the end of the pulse and performs a circular motion.\nIn contrast, for $\\tau_{\\rm{off}}=5$~ps an adiabatic undressing takes place during the switch-off and the Bloch vector approaches the upper pole.\n\n\\subsection{Selective state preparation in the exciton-biexciton system}\\label{exciton_biexciton_system}\n\n\\begin{figure}[ttt]\n \\includegraphics[width=8.7cm]{fig6.eps}\n    \\caption{(Color online)\n            Occupations of the electronic levels $|0\\rangle$ (green), $|X\\rangle$ (blue) and $|B\\rangle$ (red) after optically exciting the\n            exciton-biexciton system with a Gaussian pulse of pulse area $\\alpha=20\\pi$ and FWHM$=20$~ps as a function of the detuning $\\Delta$\n            for a (a) positive biexciton binding energy $\\Delta_B=2.0$~meV and (b) negative biexciton binding energy $\\Delta_B=-2.0$~meV. (c) and (d) :\n            corresponding energies of the QD states in the rotating frame.\n\t    }\n \\label{fig6}\n\\end{figure}\n\nThe clear understanding of the different phases of the phonon-assisted relaxation mechanism worked out so far turns out to be highly valuable \nto predict in an easy way the QD state that can be obtained by pulsed off-resonant excitation for arbitrary initial states and also for the \nexciton-biexciton system illustrated in Fig.~\\ref{fig1}(b).\nTo this end it is required as before that during the switch-off phase of the pulse the system evolves adiabatically.\nWhen the pulse duration is sufficiently long such that the relaxation all the way to the lowest\ndressed state is fully completed the prepared state at the end of the pulse is determined exclusively by\nthe energetic order of the dressed states in the limit of vanishing pulse strength. \nRecalling that the dressed states are defined with respect to the rotating frame, the prepared state can technically be determined by subtracting \nfrom the energies of the QD states without light coupling the energy of the corresponding number of photons needed \nfor reaching that state and looking at the resulting order of the states. More specifically, zero photons have to be\nsubtracted for the ground state, one photon for an exciton state and two photons for the biexciton state.\nThe resulting state lowest in energy will be occupied predominantly at the end of the preparation process.\n\nFor example in the two-level system subtracting the laser energy from the laser-free QD state energies yields \ndressed state energies that are in the limit of vanishing field strength separated by the detuning between the laser\nand the polaron-shifted QD transition. For a positive detuning the energy of the exciton-like dressed state will be below\nthe corresponding ground state-like dressed state energy in the rotating frame and consequently the phonon-assisted preparation protocol with adiabatic undressing prepares \nthe exciton state.  On the other hand, a negative detuning reverses the order of the dressed state energies and at low temperatures where there is mostly phonon\nemission the protocol prepares the QD ground state independent of the initial state\\cite{glaessl:13b}.\n\nIn the case of an exciton-biexciton system subtracting the energy of the corresponding photons gives the energies of the QD states in the frame rotating\nwith the laser frequency as $E'_0=0$, $E'_X = -\\Delta$, $E'_B = -\\Delta_B - 2\\Delta$. Thus, the energetic ordering depends also on the \nbiexciton binding energy and we will consider both cases with positive and negative biexciton binding energy, which can both be realized \ndepending on the QD geometry\\cite{michler:09}. In the following we will refer to the dressed state that in the limit of vanishing field\nstrength transforms into the $|0\\rangle$ [$|X\\rangle$, $|B\\rangle$] state as $|0'\\rangle$ [$|X'\\rangle$, $|B'\\rangle$]. \n\nLet us first discuss the most commonly encountered situation of a QD with a positive biexciton binding energy $\\Delta_B = 2$~meV.\nFigure~\\ref{fig6} (a) shows the final occupation after excitation with a detuned Gaussian\npulse with a FWHM of $20$~ps and a pulse area of $\\alpha=20\\pi$ as a function of the detuning, while Fig.~\\ref{fig6} (c) shows the corresponding energies\nin the rotating frame. It can be seen that for detunings below the two-photon resonance, i.e., $\\Delta < -\\Delta_B\/2 = -1$~meV, \nthe occupation remains in the ground state (green curve). This is consistent with the energetic order of the states, since $E'_0$ is the lowest\nenergy in this parameter region. \nAt $\\Delta = -1$~meV the energetic order of the dressed states changes, because for detunings above the two-photon resonance $E'_B$ is \nthe lowest energy. This leads to a significant drop of the final ground state occupation in favor of the biexciton occupation (red curve), \nwhich approaches its maximum close to one at $\\Delta\\approx-0.5$~meV. At the one-photon resonance at $\\Delta = 0$, \nthe energetic order changes once again and for all positive detunings $|0'\\rangle$ is the highest energy dressed state, while $E'_B$ remains \nbeing the lowest energy. For sufficiently long pulses all the occupation would, of course, end up in the lowest branch resulting in\nthe preparation of the biexciton. However, for $\\Delta > 1$~meV the energetic splitting $E'_X - E'_B > 3$~meV exceeds the maximum of\nthe phonon spectral density $E_J^{\\rm max}$ lying around $2$~meV by far, which results in a very inefficient relaxation to the $|B'\\rangle$ state,\nwhich is not completed in the time window set by the pulse length.\nTherefore, instead, we observe a gain of the exciton occupation (blue curve) as soon as $E'_X$ crosses  $E'_0$.\nThe maximal exciton occupation of about $0.8$ is reached around $\\Delta = 2$~meV, where the energy splitting $E'_0 - E'_X = 2$~meV \nagrees with $E_J^{\\rm max}$.\nTherefore, it turns out that for a system with more than two states like the one considered here an incomplete relaxation\ncan also be advantageous for preparation purposes if a preparation of a QD state that in the rotating frame\nis not the lowest lying state, like in our case the exciton, is intended.\nAn even higher exciton occupation is possible for a larger biexciton binding energy which favors transitions to \n$|X'\\rangle$, because the coupling to the lowest dressed state $|B'\\rangle$ in this case gets even more\nout of resonance. \nIt also follows that whether the prepared state is \nthe exciton or the biexciton can in principle be selected by suitably adjusting the pulse length for all\npositive detunings, because the final state depends on whether the relaxation completes during the pulse \nor whether the intermediate level corresponding to the exciton \nis still  predominantly occupied when the pulse is switched off. \nDetunings higher than $2.0$~meV lead to a decrease of both the final \nexciton and the biexciton occupations, because the\nphonon-induced relaxation becomes weaker and weaker as the phonon environment becomes more and more out of resonance.\nFinally, for very large detunings above $\\Delta=4.0$~meV the energetic splittings between the dressed states are so large that for the given pulse length practically no relaxation\ntakes place and the QD remains in the ground-state.\n\nThe case of a negative biexciton binding energy of $\\Delta_B=-2$~meV is shown in the right column. Fig.~\\ref{fig6}(b) shows \nthe final occupations and the corresponding energies are plotted in Fig.~\\ref{fig6}(d). Similar to the case of \npositive biexciton binding energies the system remains in the ground state up to a detuning of $0$, since $E'_0$ is the lowest energy. \nBetween $\\Delta = 0$ and $2$~meV  $|X'\\rangle$ is the dressed state with the lowest energy. This energetic order appears\nexclusively for negative biexciton binding energies, resulting in a broad region where a complete preparation of the exciton occurs.\nThis finding is somewhat surprising, because the two-photon resonance at $\\Delta = 1$~meV also lies within this interval,\nwhere an efficient preparation of the exciton state might be unexpected, but can easily be understood in the context described here.\nAt $\\Delta = 2$~meV the energetically lowest state changes a second time and a sharp transition of the prepared state from the exciton to\nthe biexciton occurs.\nHowever, for $\\Delta > 2.5$~meV the energy splittings between the dressed states that are connected to the excitonic states and $|0'\\rangle$ already become too large for \na complete thermalization to take place during the length of the pulse.\nThis leads to a significant reduction of the biexciton occupation for higher detunings and above $\\Delta = 5$~meV the QD does not get affected by the pulse\nanymore staying in the ground-state.\nImportantly, also in the case of a negative biexciton binding energy the targeted state can be derived from the energetic order shown in Fig.~\\ref{fig6}(d),\ndemonstrating the correctness of the description found for the dynamics of the phonon-induced preparation process also in the case of the \nexciton-biexciton system.\n\nAs we have seen, to selectively address a state in an exciton-biexciton system by phonon-assisted state preparation the biexciton binding energy\nneeds to be considered carefully, while the detuning $\\Delta$ and the pulse length can act as control parameters to choose\nthe targeted state.\n\n\n\\section{Conclusions}\\label{conclusions}  \n\nWe have analyzed the dynamics of an optically driven semiconductor QD coupled to longitudinal acoustic phonons for different off-resonant\nexcitation conditions focusing on two main questions: a) what are the requirements for a fast state preparation using off-resonant\ndriving and b) how is the prepared state selected. We demonstrated that a fast high fidelity preparation process not only relies on\nan efficient phonon-induced relaxation between the dot-photon dressed states, but also on a successful adiabatic undressing of the states.\nTo understand the influence of the adiabatic undressing we compared the exciton occupations produced by Gaussian and rectangular pulse \nshapes and systematically studied the influence of the switch-on and switch-off times. This analysis revealed that while the switch-on \ntime plays a subordinate role it is crucial that the pulse is switched off slowly enough. Furthermore, we analyzed the coherence properties\nduring the state preparation process and revealed that the incoherent phonon scattering can also restore coherence and the system trajectory takes a way through\nthe Bloch sphere to the opposite pole.\nFinally, we showed that the concept of phonon-assisted state preparation by adiabatic undressing \nalso applies to the exciton-biexciton system, where an easy prediction of the prepared state is possible.\nWhen the pulse is long enough to support a full relaxation the prepared state is the one that is lowest in energy in the rotating frame\nin the limit of vanishing field strength.\nWe demonstrated that even pulses too short for a full relaxation can be used for preparing intermediate states that are not the lowest \nin energy in the rotating frame. \nAltogether, it is demonstrated that in an exciton-biexciton system the decisive parameters for selecting the prepared state \nare  the detuning, the biexciton binding energy and the pulse length.\n\n\\section{ACKNOWLEDGMENTS}\n\nA.M.B. and V.M.A. gratefully acknowledge the financial support from Deutsche Forschungsgemeinschaft\nvia the Project No. AX 17\/7-1. D.E.R. is thankful for financial support from the German Academic Exchange Service (DAAD) within the P.R.I.M.E. programme. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{intro}\n\nStudies to exorcise Maxwell's demon \\cite{leff2014maxwell} shed light on the relationship between information theory and statistical physics \\cite{szilard1929entropieverminderung,bennett1982thermodynamics,lloyd1989use,lloyd1997quantum,kim2011quantum,schaller2011probing}.\nResearchers reported models for the Maxwell demon, including the demon as a feedback controller \\cite{touchette2000information,touchette2004information,sagawa2008second,allahverdyan2009thermodynamic,cao2009thermodynamics,fujitani2010jarzynski,ponmurugan2010generalized,horowitz2010nonequilibrium,sagawa2010generalized,abreu2012thermodynamics,sagawa2012fluctuation,sagawa2012nonequilibrium,esposito2012stochastic,hartich2014stochastic,horowitz2014thermodynamics,horowitz2014second,shiraishi2015role,miyahara2018work,ribezzi2019large,ito2013information,sandberg2014maximum}, information reservoir such as a bit sequence \\cite{mandal2012work,mandal2013maxwell,deffner2013information,strasberg2014second,barato2014stochastic,merhav2015sequence,strasberg2015thermodynamics,chapman2015autonomous,boyd2016identifying,mcgrath2017biochemical,strasberg2017quantum,stopnitzky2019physical,boyd2017leveraging}, and the unifying approaches \\cite{horowitz2013imitating,barato2014unifying,shiraishi2016measurement}.\nIn those studies, researchers have shown second-law-like inequalities ({\\SLLI}s), establishing that the second law of thermodynamics (SLT)\n\\begin{IEEEeqnarray}{lCr}\n\\Delta\\shaent{\\Xtot}+\\Qtot\\geq 0,\n\\label{eq:SLT}\n\\end{IEEEeqnarray}\nis not violated by the demon in principle, where we denote by $\\shaent{\\Xtot}$ and $\\Qtot$ the entropy and the heat transfer of the whole system in contact with a heat bath.\nWe assume that all processes are isothermal processes with a heat bath of temperature $T$ in this Letter. We set $k_B T = 1$.\n\nThe {\\SLLI}s appeared in those previous studies can be tentatively classified into two types: the first one, which was reported in the studies of the feedback control model, asserting that the entropy of the system can be reduced by the amount of information gathered by the controller (demon):\n\\begin{IEEEeqnarray}{lCr}\n\\Delta \\shaent{X} + Q \\geq \\genI,\n\\IEEEeqnarraynumspace\n\\label{eq:HItype1}\n\\end{IEEEeqnarray}\nand the second one, which was reported in the studies of the information reservoir, asserting that work can be extracted in exchange for the entropy production of the information reservoir:\n\\begin{IEEEeqnarray}{lCr}\n\\Wavg \\leq \\Delta \\shaent{X},\n\\IEEEeqnarraynumspace\n\\label{eq:WHtype2}\n\\end{IEEEeqnarray}\nwhere $X$ denotes the state of the controlled system such as gases enclosed in a piston or a bit sequence;\n$Q$ denotes the heat transfer from the controlled system;\n$\\Wavg$ denotes the work extracted from the controlled system;\nFor $\\genI$ in the inequality~(\\ref{eq:HItype1}), several quantities such as transfer entropy have been employed.\nIn the following,\nwe specify it as the mutual information production: $\\genI = \\muI{\\Xsys\\prm}{\\memX}-\\muI{\\Xsys}{\\memX}$, where $\\memX$ is the memory's state.\nThen, the \\SLLI~(\\ref{eq:HItype1}) coincides with the one shown in Ref.~\\cite{sagawa2012fluctuation}, \n\n\n\nBoth types of the previous {\\SLLI}s do not imply that feedback control increases the \\textit{net maximum work}.\nHere, we call the upper limit of extracted work that accounts for the memory cost determined by the Landauer's principle \\cite{landauer1961irreversibility,sagawa2009minimal} the net maximum work. \nIn other words, the net maximum work is the upper limit of $\\Wavg$ minus the lower limit of the memory cost.\nAccording to the Landauer's principle, the work required to exploit the memory is greater than $\\genI$.\nThus, the \\SLLI~(\\ref{eq:HItype1}) does not imply that the net maximum work increases by measurements because $\\genI$ is canceled out by the memory cost.\nLikewise, the \\SLLI~(\\ref{eq:WHtype2}) does not provide the difference of the net maximum work between feedback and open-loop control because all of the terms in the \\SLLI~(\\ref{eq:WHtype2}) do not vary depending on whether the control is feedback or open-loop.\n\nOur first main result\nis that the sum of the entropy bounds for all subsystems leads to\nthe first \\SLLI\n\\begin{IEEEeqnarray}{lCr}\n\\Delta\\Htot + \\Qtot \\geq \\DelI\n\\IEEEeqnarraynumspace\n\\label{eq:HHQD5}\n\\end{IEEEeqnarray}\nimplying that feedback control increases the net maximum work.\nHere, $\\DelI$ is the positive quantity defined later.\nDue to the positivity of $\\DelI$, this \\SLLI is a stronger bound than the SLT~(\\ref{eq:SLT}).\nIn terms of extracting work, $\\DelI$ coincides with how much the system's entropy production is useless for a controller to extract the work from the system.\nAlthough $\\DelI$ can grow in proportional to the system size, the memory cost can be constant with the system size when there are correlations within the system.\nHence, the feedback control can vanish $\\DelI$ with significantly less memory cost than $\\DelI$.\nThus, the feedback control can increase the net maximum work.\n\n\nIn addition, the sum of the entropy bounds of the controlled subsystems leads to\nthe stronger inequality than {\\SLLI}~(\\ref{eq:HItype1}):\n\\begin{IEEEeqnarray}{lCr}\n\\Delta \\Hsys + \\Qsys \\geq \\genI + \\DelIsys,\n\\label{eq:HQID23}\n\\end{IEEEeqnarray}\nwhere $\\DelIsys$ denotes the positive quantity similar to $\\DelI$. \nThis inequality is our second main result and derived at the end of the main result part.\nLikewise, the stronger bound corresponding to \\SLLI~(\\ref{eq:WHtype2}) is immediately derived from \\SLLI~(\\ref{eq:HHQD5}) by\nregarding all subsystems as a controlled system such as a bit sequence and ignoring the change in the internal energy:\n\\begin{IEEEeqnarray}{lCr}\n\\Wavg \\leq \\Delta \\Htot - \\DelI,\n\\label{eq:WQHD4}\n\\end{IEEEeqnarray}\n\n\n\n\n\n\\textit{Main result --}\nWe study a classical system in contact with a heat bath of temperature $T$.\nWe assume that the system and the heat bath constitute a closed system.\nThe system consists of non-overlapping subsystems $1, 2,\\dots, \\Nx$.\nThe evolution is Markovian with discretized time intervals, and we focus on a single step of this discrete Markov process.\nThe initial and final states of the $k$-th subsystem are denoted by $\\Xs{k}$ and $\\Xs{k}\\prm$, respectively.\nLet $\\DepX{k}$ be the subset of $\\Xtot=\\{\\Xs{1},\\dots,\\Xs{\\Nx}\\}$ that influences the evolution of $\\Xk$ other than $\\Xk$ itself.\nIn other words, the state $\\Xs{k}$ evolves depending only on $\\{\\Xs{k}\\}\\cup\\DepX{k}$\n\\footnote{For the formal definition of the independence, see Definition~\\ref{def:dependence} in Section~\\ref{sect:Positivity of Co-dissipation}.\n}\n.\n\nWe premise the local detailed balance for each subsystem, which result in the entropy bound for a subsystem\n\\footnote{We demonstrate the inequality~(\\ref{eq:yuragi_k}) in\nSection~\\ref{ap:Fluctuation relation for a subsystem}.}\n:\n\\begin{IEEEeqnarray}{rCl}\n\\YuragiK := \\shaent{\\Xs{k}\\prm\\mid \\DepX{k}} - \\shaent{\\Xs{k}\\mid \\DepX{k}} + \\Qs{k} \\geq 0,\n\\label{eq:yuragi_k}\n\\end{IEEEeqnarray}\nwhere we denote by $\\YuragiK$ the total entropy production accompanied with the evolution of $\\Xs{k}$ and by $\\Qs{k}$ the heat transfer from the $k$-th subsystem to the heat bath.\n\nThe dependency among subsystems induces the graph $\\GcaC{0}=\\{\\VcaC{0},\\EcaC{0}\\}$ as follows:\n\\begin{IEEEeqnarray}{lCr}\n\\VcaC{0} := \\{\\{\\Xs{1}\\},\\{\\Xs{2}\\},\\dots,\\{\\Xs{\\Nx}\\}\\},\n\\label{eq:Vdef}\n\\\\\n\\EcaC{0} :=\\{\\EdgeX{j}{k}\\mid \\Xj \\in \\DepXX{\\Xk}, \\Xk\\in\\Xtot\\},\n\\label{eq:Edef}\n\\end{IEEEeqnarray}\nwhere we denote by $\\EdgeX{j}{k}$ the directed edge from the vertex $\\{\\Xs{j}\\}$ to the vertex $\\{\\Xs{k}\\}$.\nThe graph $\\GcaC{0}$ may contain cycles.\nTo eliminate dependencies causing these cycles, we consider the coarse-grained subsystems where original subsystems are merged if they are members of the same cycle in $\\Gorg$.\nHereinafter, $\\Xj$ denotes the states of the $j$-th subsystem in the coarse-grained subsystems, and $\\Nx$ denotes the number of the coarse-grained subsystems.\nWe denote by $\\Gca=(\\Vca,\\Eca)$ the graph induced from the coarse-grained subsystems by the same way shown in Eqs.~(\\ref{eq:Vdef}) and (\\ref{eq:Edef}).\nBy its definition, $\\Gca$ is a directed acyclic graph (DAG).\n\nLet $\\Gcomp=(\\Vcomp,\\Ecomp)$ be a connected component of $\\Gca$.\nThe graph sequence $\\GcompS{1}, \\GcompS{2}, \\cdots, \\GcompS{\\Ncomp}$ is constructed by the following procedure applying the edge contraction \\cite{gross2003handbook} repeatedly to $\\Gcomp$:\n\\begin{easylist}[enumerate]\n@ Initialization: let $j=1$, $\\GcompS{1}=\\Gcomp$.\n@ If the graph $\\GcompS{j}$ has exactly one vertex, terminate. Otherwise, go to step \\ref{list:step3}\n\\label{list:step2}\n@ Select a vertex couple $\\pcPair$ from $\\GcompS{j}$ such that every child of $\\p$ is a sink, and $\\c$ is a child of $\\p$.\n\\label{list:step3}\n@ Execute the edge contraction for $\\p$ and $\\c$, and let the result of it be $\\GcompS{j+1}$.\n\\label{list:step4}\n@ Increment $j$, and go back to step \\ref{list:step2}\n\\end{easylist}\n\\vspace{0.5em}\n\\noindent This procedure is feasible because a DAG with more than two vertices always has a vertex such that has only sink as its children.\nThe essence of this procedure is that any $\\GcompS{j}$ is a DAG, and the adjacency of $\\Gca$ preserves \\footnote{See Lemma~\\ref{lem:ReduceGraphProperties} in Section~\\ref{sect:Properties of the graph sequence}.}.\n\n\\newcommand{\\hat{V}}{\\hat{V}}\n\nFor simplicity, we introduce a convention regarding  collections of subsystems $\\hat{V}$.\nSuppose a function $f$ takes an arbitrary number of subsystems as arguments.\nThen, we just write $f(\\hat{V})$ in the sense of $f(\\Xs{V_1},\\Xs{V_2}\\cdots)$, where $\\Xs{V_j}$ is a $j$-th member of $\\hat{V}$.\nLikewise, we denote by $\\hat{V}\\prm$ the collection of the subsystem's final states in $\\hat{V}$.\nFurthermore, we denote by $\\Flatten(\\hat{V})$ the set of the subsystems created by flattening the nested structure of $\\hat{V}$.\nFor example, if $\\hat{V}=\\{\\{\\Xs{1},\\Xs{2}\\},\\{\\Xs{3},\\Xs{4}\\}\\}$, then\n\\begin{IEEEeqnarray}{cCc}\n\\Prb{}{\\hat{V}}=P(\\Xs{1},\\Xs{2},\\Xs{3},\\Xs{4}),\n\\label{eq:PVPXXXX8}\n\\\\\n\\hat{V}\\prm=\\{\\{\\Xs{1}\\prm,\\Xs{2}\\prm\\},\\{\\Xs{3}\\prm,\\Xs{4}\\prm\\}\\},\n\\label{eq:VXXXX9}\n\\\\\n\\Flatten(\\hat{V})=\\{\\Xs{1},\\Xs{2},\\Xs{3},\\Xs{4}\\}.\n\\end{IEEEeqnarray}\nWe denote by $\\Qset{\\hat{V}} := \\Sum{X\\in\\Flatten(\\hat{V})}{}\\Qs{}$ the heat transfer from all the subsystems composing $\\hat{V}$.\nWe denote by $\\DepXX{\\hat{V}}:=\\bigcup_{\\Xx\\in \\Flatten(\\hat{V})}{\\DepXX{\\Xx}}$ the sets of subsystems influencing the evolution of the members of $\\hat{V}$.\n\nWe use another notation to ignore the empty set as the conditioning event:\n\\begin{IEEEeqnarray}{cCr}\nP(V\\mid\\emptyset) := P(V),\n\\quad\n\\shaent{V\\mid\\emptyset} := \\shaent{V},\n\\\\ \n\\MutualInfo{V}{W\\mid\\emptyset}{}:=\\MutualInfo{V}{W}{}.\n\\label{eq:convention_empty_condition}\n\\end{IEEEeqnarray}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{F3_PC_graph.eps}\n\\caption{(Color online). The schematic diagram of the sets defined as Eqs.~(\\ref{eq:dpdpdp14}) and $\\DelITwoPC$, $\\DelIThreePC$, and $\\DelIFourPC$.\n}\n\\label{fig:pc_depend}\n\\end{center}\n\\end{figure}\n\nIn addition, \nwe prepare notations for sets of subsystems regarding the dependency of $\\p$ and $\\c$:\n\\begin{IEEEeqnarray}{rCl}\n\\aaa := \\DepXX{c} \\cap \\p&,&\n\\quad\n\\CE := \\DepXX{\\c} \\setminus \\p,\n\\nonumber\n\\\\\n\\D := \\DepXX{\\p} \\setminus \\DepXX{c}&,&\n\\label{eq:dpdpdp14}\n\\quad\n\\DCE := \\DepXX{\\p\\cup\\c} \\setminus \\p,\n\\\\\n\\C := \\DepXX{\\c} \\setminus \\DepXX{\\p} \\setminus \\vMergeP&,&\n\\quad\n\\E := \\DepXX{\\p} \\cap \\DepXX{\\c} \\setminus \\p\\,.\n\\nonumber\n\\end{IEEEeqnarray}\nAs illustrated in Fig.~\\ref{fig:pc_depend}, these sets have the following meaning: \n$\\aaa$ are the subsystems that are members of $\\p$ and influence to $\\c$; $\\D$ are the subsystems that influence $\\p$ but not $\\c$; $\\E$ are the subsystems that influence both $\\p$ and $\\c$; $\\C$ are the subsystems that influences $\\c$ but not $\\p$. We also note that $\\CE=\\C\\cup\\E$ and $\\DCE = \\C\\cup\\E\\cup\\D$. The minus symbol ``-'' put at the superscript of $\\Dep$ represents the elimination of $\\p$.\n\nThe key quantity $\\DelI$ is the sum of two modules, each caused by the independency of the evolution within or among connected components in $\\Gca$.\nThe first module $\\DelIinner{\\Gcomp}$, which is caused by the independency within a connected component, is provided by\n\\begin{IEEEeqnarray}{cCc}\n\\DelIinner{\\Gcomp}\n:= \n- \\Sum{(p,c)\\in \\Vpccc}{}\n\\left(\\DelIOne{p,c} + \\DelITwo{p,c}+\\DelIThree{p,c} +\\DelIFour{p,c}\\right),\n\\IEEEeqnarraynumspace\n\\label{eq:def_DELI_intra}\n\\end{IEEEeqnarray}\nwhere $\\Vpccc$ is the set of all $\\pcPair$ selected through constructing the graph sequence $\\GcompS{1},\\cdots,\\GcompS{\\Ncomp}$, and each term of $\\DelIinner{\\Gcomp}$ is mutual information production as follows:\n\\begin{IEEEeqnarray}{cCc}\n\\DelIOne{p,c} := \\muI{\\p\\prm}{\\c\\prm\\mid\\DCE}-\\MutualInfo{p}{c\\prm\\mid \\DCE}{},\n\\label{eq:def_delIOne}\n\\\\\n\\DelITwo{p,c} := \\MutualInfo{p\\prm}{\\C\\mid \\DE}{}- \\MutualInfo{p}{ \\C\\mid \\DE}{},\n\\label{eq:def_delITwo}\n\\\\\n\\DelIThree{p,c} := \\MutualInfo{c\\prm}{\\D\\mid p,\\CE}{}- \\MutualInfo{c}{\\D\\mid p,\\CE}{} ,\n\\label{eq:def_delIThree}\n\\\\\n\\DelIFour{p,c} :=\n\\MutualInfo{\\c\\prm}{\\p\\mid \\DepC}{}\n-\n\\MutualInfo{\\c}{\\p\\mid \\DepC}{},\n\\label{eq:def_delIFour}\n\\end{IEEEeqnarray}\nwhere we used the convention introduced in Eqs.~(\\ref{eq:PVPXXXX8}) and (\\ref{eq:VXXXX9}).\nThe independencies of the evolutions induced from the definitions~(\\ref{eq:dpdpdp14})\nlead to the negativity of $\\DelIOnePC$, $\\DelITwoPC, \\DelIThreePC$, and $\\DelIFourPC$ \n\\footnote{For the formal definition of the independent evolution, see Definition~\\ref{def:independent Markov process} in Section~\\ref{sect:Positivity of Co-dissipation}.\nIn addition, Lemma~\\ref{lem:ind_mutual_prod_nonincrease1} in this section establishes the negativity of the mutual information productions such as $\\DelIOnePC$, $\\DelITwoPC, \\DelIThreePC$ and $\\DelIFourPC$.}.\nFor example, $\\DelIOnePC$ is the mutual information production through the independent process of $\\p$ from $\\c\\prm$ conditioned on $\\DCE$.\nThis fact leads to the negativity of $\\DelIOnePC$.\nConsequently, $\\DelIinner{\\Gcomp}$ is positive.\nLikewise, the following quantity is caused by the independency among the connected components in $\\Gca$:\n\\begin{IEEEeqnarray}{rCl}\n\\DelIinter{\\Gca}\n :=\\Sum{j=2}{\\Ncc}\\left[\\MutualInfo{\\VcompI{j}}{\\VcompI{1:j-1}}{}\n - \\MutualInfo{{\\VcompIprm{j}}}{{\\VcompIprm{1:j-1}}}{}\n  \\right],\n\\IEEEeqnarraynumspace\n\\label{eq:defDelIinter}\n\\end{IEEEeqnarray}\nwhere we denote by $\\Ncc$ the number of connected components in $\\Gca$ and by $\\VcompI{j}$ all vertices in the $j$-th connected component of $\\Gca$.\nWe use the colons to represent the union of indexed sets such as $\\VcompI{1:j-1}:=\\bigcup_{k=1}^{j-1}\\VcompI{k}$. \nSince subsystems contained in different connected components evolve mutually independently, the mutual information production $\\MutualInfo{\\VcompIprm{j}}{\\VcompIprm{1:j-1}}{}\n - \\MutualInfo{{\\VcompI{j}}}{{\\VcompI{1:j-1}}}{}$ is negative, which leads to the positivity of $\\DelIinter{\\Gca}$.\nWe define $\\DelI$ as the sum of the two contributions:\n\\begin{IEEEeqnarray}{rCl}\n\\DelI := \\Sum{\\Gcomp\\in\\Gca}{}\\DelIinner{\\Gcomp}+\n\\DelIinter{\\Gca}.\n\\IEEEeqnarraynumspace\n\\label{eq:defDelI39}\n\\end{IEEEeqnarray}\nThe first main result of this Letter is the SLLI~(\\ref{eq:HHQD5}) with $\\DelI$ provided by Eq.~(\\ref{eq:defDelI39}).\nFor a rigorous proof of this result, see Theorem~\\ref{lem:sum_all_is_bound} in Section~\\ref{subsec:Calculation of the sum of YuragiRel using Gca}, where we demonstrate that the sum $\\Sum{k=1}{\\Nx}\\YuragiK$ results in the SLLI~(\\ref{eq:HHQD5}).\nMoreover, since all of its modules are positive, $\\DelI$ takes a positive value. \nThe positivity is formally proved in Theorem~\\ref{lem:JKLMI} in Section~\\ref{sect:Positivity of Co-dissipation}.\nThus, by the positivity of $\\DelI$, the \\SLLI~(\\ref{eq:HHQD5}) is a stronger bound than the SLT~(\\ref{eq:SLT}).\nIn particular, if all subsystems influence each other, then $\\DelI$ vanishes, and \\SLLI coincides with the SLT~(\\ref{eq:SLT}).\n\nTo obtain the \\SLLI~(\\ref{eq:HQID23}), we take the sum of the entropy bounds within the controlled system.\nWe regard $\\Xs{1}$ as the state $Y$ of the memory and $\\Xs{2:\\Nx}$ as the state $\\Xsys$ of the controlled system.\nLet $\\DelIsysS$ be $\\DelI$ determined by the definition~(\\ref{eq:defDelI39}) under the setting where the subsystems $2,3,\\dots,\\Nx$ constitute the whole system.\nIn addition, let $\\DelIsys$ be the quantity provided by adding $Y$ as the conditioning event in $\\DelIOnePC, \\DelITwoPC, \\DelIThreePC$, and $\\DelIFourPC$ when calculating $\\DelIsysS$.\nThen, we can show\n\\footnote{See Corollary~\\ref{th:strong_type1} in Section~\\ref{subsec:Calculation of the sum of YuragiRel using Gca}.}\nthat the sum $\\Sum{k=2}{\\Nx}\\YuragiK$ leads to the \\SLLI~(\\ref{eq:HQID23}).\nThe positivity of $\\DelIsys$ is established by a similar way of proving the positivity of $\\DelI$.\n\n\n\n\\textit{Example 1 --}\nWe observe that \\SLLI~(\\ref{eq:HHQD5}) implies that the feedback control increases the net maximum work contrary to the SLT~(\\ref{eq:SLT}) and the {\\SLLI}s~(\\ref{eq:HItype1})(\\ref{eq:WHtype2}) by the model inspired by the Szilard engine. \nWe regard subsystem 1 as a controller with a 1-bit memory and subsystems 2 through $N$ as a controlled system.\nAs illustrated in Fig.~\\ref{fig:f1}, the controlled system consists of $N-1$ ideal gas particles in a rigid box.\nIn the initial state, the box is separated by a wall, and all particles are on the left or right side of the wall with a probability of 0.5. Then, subsystems of the controlled system correlate with each other: $\\muI{\\Xj}{\\Xk}=\\ln2$ for $j,k\\geq2$.\nWe assume that feasible controls consist of three actions: the controller moves the wall to the left or right edge of the box or pulls the wall out from the box.\nWe investigate only when the final state is the maximum entropy state, where the particles move around the entire box.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.43\\textwidth]{PRL_F1.eps}\n    \\caption{The initial and final state in Example 1.}\n    \\label{fig:f1}\n\\end{figure}\n\nWe suppose that the feedback control consists of two steps: 1) the controller performs a binary measurement of whether a single particle stays on the left or right side without error; 2) the controller moves the wall to the opposite side of the particles.\nIn the second step, the entropy productions of the whole system and the controlled system are $\\Delta\\Htot=(N-1)\\ln2$ and $\\Delta\\Hsys=(N-2)\\ln2$. In addition, the increase in mutual information between the memory and the controlled system is $-\\ln2$, and the memory cost determined by the Landauer's principle is $\\ln2$.\n\nBesides the feedback control, we can consider three cases in terms of the measurement and the dependency.\nWith the measurement and without the influence from the memory's state on the controlled system's evolution, which we call the ``open-loop with measurement (\\OpenLoopWithMeasure),''\n$\\Delta\\Htot=(N-1)\\ln2$, $\\Delta\\Hsys=(N-2)\\ln2$ and the memory cost is $\\ln2$.\nWithout the measurement, regardless of the influence of the memory state on the controlled evolution, $\\Delta\\Htot=\\Delta\\Hsys=(N-2)\\ln2$ with the zero memory cost. \nWe call these cases the ``open-loop control without the measurement (\\OpenLoopWithoutMeasure).''\n\n\nThen, although the SLT~(\\ref{eq:SLT}) and the  {\\SLLI}s~(\\ref{eq:HItype1})(\\ref{eq:WHtype2}) claim the same net maximum work $(N-2)\\ln2$ for all four cases,\nour \\SLLI~(\\ref{eq:HHQD5}) asserts a strict limit such that no work can be extracted without the feedback control:\n\\begin{IEEEeqnarray}{lCr}\n\\Wavg \\leq\n\\begin{cases}\n(N-1)\\ln2 & (\\text{feedback control})\\\\\n0 & (\\text{\\OpenLoopWithMeasure or \\OpenLoopWithoutMeasure})\n\\end{cases}\n\\label{eq:NLN205}\n\\end{IEEEeqnarray}\nbecause\nwe have \\footnote{\nSee Section~\\ref{ap:Calculation of $\\DelI$ in Example 1}.\n}\n\\begin{IEEEeqnarray}{lCr}\n\\DelI=\n\\begin{cases}\n0 & (\\text{feedback control}) \\\\\n(N-1)\\ln2 & (\\text{\\OpenLoopWithMeasure})\\\\\n(N-2)\\ln2. & (\\text{\\OpenLoopWithoutMeasure})\n\\end{cases}\n\\end{IEEEeqnarray}\nAccordingly, only the feedback control enjoys the positive net maximum work $(N-2)\\ln2$ despite $-\\ln2$ or zero for the \\OpenLoopWithMeasure or the \\OpenLoopWithoutMeasure case.\nIn addition, this advantage of the feedback control \nincreases with the number of particles, which means the feedback control can be significantly beneficial more at a macroscopic level.\n\n\n\n\n\\textit{Example 2 --}\nTo observe how $\\DelIOnePC$ appears,\nwe consider a two-body system with the following dependency:\n\\begin{IEEEeqnarray}{lCr}\n\\DepX{1} = \\emptyset, \\quad \\DepX{2} = \\{\\Xs{1}\\}\n\\label{eq:x1tox2depen}\n\\end{IEEEeqnarray}\nThen, $\\Ncc=1$ and $\\pcPair=(\\Xs{1},\\Xs{2})$.\nSuppose that all subsystems take binary states, and\nthe entropy of each subsystem takes the maximum amount: $\\shaent{\\Xj}=\\shaent{\\Xj\\prm}=\\ln2$ for all $j$.\nWe assume the following correlations:\n\\begin{IEEEeqnarray}{cCc}\n\\muI{\\Xs{1}}{\\Xs{2}} = \\muI{\\Xs{1}}{\\Xs{2}\\prm} = \\muI{\\Xs{2}}{\\Xs{2}\\prm} = \\ln2,\n\\\\\n\\muI{\\Xs{1}}{\\Xs{1}\\prm} = \\muI{\\Xs{2}}{\\Xs{1}\\prm} = \\muI{\\Xs{2}\\prm}{\\Xs{1}\\prm} = 0.\n\\end{IEEEeqnarray}\nThen, we have $\\Delta \\Htot=\\ln2$. Since $\\DelIOnePC=\\ln2$ and $\\DelITwoPC = \\DelIThreePC = \\DelIFourPC = 0$, we obtain $\\DelI=\\ln2$. \nAssuming that the internal energy is constant, our inequality~(\\ref{eq:WQHD4}) implies $\\Wavg \\leq 0$\nalthough the SLT~(\\ref{eq:SLT}) and the {\\SLLI}s~(\\ref{eq:HItype1}) (\\ref{eq:WHtype2}) claim that the net maximum work is $\\ln 2$.\n\n\\textit{Example 3 --}\nTo observe how $\\DelITwoPC, \\DelIThreePC$ and $\\DelIFourPC$ appear, we consider a system consisting of five subsystems with the following dependency:\n\\begin{IEEEeqnarray}{cCr}\n\\DepX{1} = \\{\\Xs{3}\\},\\quad\n\\DepX{2} = \\{\\Xs{3}, \\Xs{5}\\},\\quad\n\\DepX{3} = \\{\\Xs{4}\\},\n\\NonumberNewline\n\\DepX{4} = \\{\\Xs{5}\\},\\quad\n\\DepX{5} = \\emptyset\n\\IEEEeqnarraynumspace\n\\end{IEEEeqnarray}\nThen, $\\Ncc=1$ and the sequence of $\\pcPair$ can be $(\\Xs{3},\\Xs{1})$, $(\\{\\Xs{1},\\Xs{3}\\}, \\Xs{2})$, $(\\Xs{4}, \\{\\Xs{1},\\Xs{2}, \\Xs{3}\\})$, $(\\Xs{5}, \\{\\Xs{1}, \\Xs{2},\\Xs{3}, \\Xs{4}\\})$. \nWe suppose all subsystems take binary states except $\\Xs{1}$, which takes four states.\nWe denote by $\\Xs{1H}$ and $\\Xs{1V}$ the first and second bit of $\\Xs{1}$.\nThe entropy of each subsystem takes the maximum amount: $\\shaent{\\Xj}=\\shaent{\\Xj\\prm}=\\ln2$ for all $j$ except $\\shaent{\\Xs{1}}=\\shaent{\\Xs{1}\\prm}=2\\ln2$.\nThere are no correlations among the subsystems other than the following:\n\\begin{IEEEeqnarray}{rCl}\n\\muI{\\Xs{1H}}{\\Xs{2}} &=& \\muI{\\Xs{2}}{\\Xs{4}} \n= \\muI{\\Xs{1V}}{\\Xs{3}} \n\\NonumberNewline\n&=& \\muI{\\Xs{3}}{\\Xs{5}} = \\ln2.\n\\end{IEEEeqnarray}\nThen, we have $\\Delta \\Htot = 4\\ln2$. \nSince $\\DelIOnePC=\\DelITwoPC = \\DelIThreePC=\\DelIFourPC=0$ other than $\\DelIThree{\\Xs{3},\\Xs{1}}=\\DelITwo{\\{\\Xs{3},\\Xs{1}\\}, \\Xs{2}}=\\DelIFour{\\{\\Xs{3},\\Xs{1}\\}, \\Xs{2}}=\\ln2$, we obtain $\\DelI=3\\ln2$. \nAssuming that the internal energy is constant, our inequality~(\\ref{eq:WQHD4}) implies $\\Wavg \\leq \\ln2$, although the SLT~(\\ref{eq:SLT}) and the {\\SLLI}s~(\\ref{eq:HItype1})(\\ref{eq:WHtype2}) assert that the maximum work is $4\\ln2$. \nIndeed, when the third subsystem performs feedback control, the maximum work $\\ln2$ is achieved. \n\nMeanwhile, if the dependency is provided by \n$\\DepX{1} = \\emptyset,\\quad\\DepX{2} = \\DepX{3} = \\DepX{4} = \\DepX{5} = \\{\\Xs{1}\\}$,\nthen $\\DelI$ is zero, and our \\SLLI claims that the same upper limit $\\Wavg=4\\ln2$ as the SLT and the {\\SLLI}s~(\\ref{eq:HItype1})(\\ref{eq:WHtype2}).\n\n\\textit{Discussion --}\nBy the definition, $\\DelI$ is the decrease in mutual informations between subsystems evolving independently.\nPrecisely, $\\DelIinner{\\Gcomp}$ is the negative sum of $\\DelIOnePC, \\DelITwoPC, \\DelIThreePC$, and $\\DelIFourPC$ that are the mutual information productions through the independent processes conditioned on the directly influencing subsystems.\nFor example, $\\DelITwoPC$ is the mutual information production between $\\p$ and $\\CE$, where $\\p$ evolves independently from $\\CE$ conditioned on $\\D$.\nWe can refer to Fig.~\\ref{fig:pc_depend}, where the arrows represent dependencies, to understand the independencies regarding $\\DelIOnePC, \\DelITwoPC, \\DelIThreePC$, and $\\DelIFourPC$.\nLikewise, $\\DelIinter{}$ is the decrease in the mutual informations through the mutually independent processes, as mentioned earlier.\nHence, $\\DelI$ is varied by changing the dependency and the conditioning event.\n\nIn Example 1, we can interpret that the feedback control reduces $\\DelI$ by turning the independent process not conditioned on other states into the independent process conditioned on the correlated memory's state.\nWhen the controller neither performs the measurement nor influences the controlled system's evolution,\n$\\DelI$ is the decrease in the mutual informations through \\textit{the independent processes that are not conditioned on another state}.\nIn contrast, in the feedback control, the memory state influences the controlled system's evolution, and then\n$\\DelI$ becomes the mutual information decrease through the \\textit{independent process conditioned on the correlated memory state}.\nSince the measurement provides the correlation: $\\muI{\\Xs{1}}{\\Xj}$, this conditioning by the memory state causes less $\\DelI$ in the feedback control.\n\n\n\nAt last, we compare the present study with the exorcise of the Maxwell demon.\nThe latter study recovers the SLT\nfrom a situation in which the feedback control appeared to extract more work than the upper limit by the second law.\nIn contrast, this study illustrates that the feedback control improves the maximum work, which is lower than what the SLT claims in the open-loop control.\nMoreover, our {\\SLLI}s imply that the feedback control does not violate the SLT because our {\\SLLI}s are always tighter than the SLT.\n\n\n\n\n\n\n\n\n\n\n\n\\textit{Conclusion --}\nWe reported the first {\\SLLI}s~(\\ref{eq:HHQD5})-(\\ref{eq:WQHD4}) implying that the feedback control can increase the \\textit{net maximum work}. \nAlthough both the {\\SLLI}s (\\ref{eq:HItype1}) and (\\ref{eq:WHtype2}) do not imply that the feedback control increases the net maximum work, \nour {\\SLLI}s imply it\nbecause the open-loop control causes $\\DelI$, which leads to less net maximum work than what the SLT claims.\nBy the positivity of $\\DelI$,\nout {\\SLLI}s are stronger bounds than the SLT and the previous {\\SLLI}s.\nSince all components of $\\DelI$ are the decreases of the mutual informations among the subsystems, the advantage of the feedback control suggested in this Letter comes from the internal correlations of the controlled system.\nOur work might serve as a theoretical basis for quantifying the usefulness of information processing in physical systems.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\IEEEPARstart{I}{n} recent decades, the rapid development of the digital camera hardware has revolutionized human lives. On one hand, even mid-level mobile devices can easily produce high resolution images and videos. Besides the physical elements, the widespread use of the images and videos also reflects the importance of developing software technology for them. On the other hand, numerous registration techniques for images and video frames have been developed for a long time. The existing registration techniques work well on problems with a moderate size. However, when it comes to the current high quality images and videos, most of the current registration techniques suffer from extremely long computations. This limitation in software seriously impedes fully utilizing the state-of-the-art camera hardware.\n\nOne possible way to accelerate the computation of the registration is to introduce a much coarser grid on the images or video frames. Then, the registration can be done on the coarse grid instead of the high resolution images or video frames. Finally, the fine details can be added back to the coarse registration. It is noteworthy that the quality of the coarse grid strongly affects the quality of the final registration result. If the coarse grid cannot capture the important features of the images or video frames, the final registration result is likely to be unsatisfactory. In particular, for the conventional rectangular coarse grids, since the partitions are restricted in the vertical and horizontal directions, important features such as slant edges and irregular shapes cannot be effectively recorded. By contrast, triangulations allow more freedom in the partition directions as well as the partition sizes. Therefore, it is more desirable to make use of triangulations in simplifying the registration problems.\n\nIn this work, we propose a novel content-aware algorithm to {\\em TR}iangulate {\\em IM}ages, abbreviated as \\emph{TRIM}, that largely simplifies the registration problem for high quality images and video frames. Specifically, given a possibly very high definition image or video frame, we compute a coarse triangulation representation of it. The computation involves a series of steps including subsampling, unsharp masking, segmentation and sparse feature extraction for locating sample points on important features. Then, a high quality triangulation can be created on the set of the sample points using the Delaunay triangulation. With a coarse triangular representation of the images, the registration problem can be significantly accelerated. We apply a landmark-based quasi-conformal registration algorithm \\cite{Lam14} for computing the coarse registration. Finally, the fine detail of the image or video frame is computed with the aid of the coarse registration result. Our proposed framework also serves as a highly efficient and accurate initialization for many different registration approaches.\n\nThe rest of this paper is organized as follows. In Section \\ref{previous}, we review the literature on image and triangular mesh registration. In Section \\ref{contribution}, we highlight the contribution of our work. Our proposed method is explained in details in Section \\ref{main}. In Section \\ref{experiment}, we demonstrate the effectiveness of our approach with numerous real images. The paper is concluded in Section \\ref{conclusion}.\n\n\n\\section{Previous works} \\label{previous}\n\nIn this section, we describe the previous works closely related to our work.\n\nImage registration have been widely studied by different research groups. Surveys on the existing image registration approaches can be found in \\cite{Zitova03,Crum04,Klein09}. In particular, one common approach for guaranteeing the accuracy of the registration is to make use of landmark constraints. Bookstein \\cite{Bookstein78,Bookstein91,Bookstein99} proposed the unidirectional landmark thin-plate spline (UL-TPS) image registration. In \\cite{Johnson02}, Johnson and Christensen presented a landmark-based consistent thin-plate spline (CL-TPS) image registration algorithm. In \\cite{Joshi00}, Joshi et al. proposed the Large Deformation Diffeomorphic Metric Mapping (LDDMM) for registering images with a large deformation. In \\cite{Glaunes04,Glaunes08}, Glaun\\`es et al. computed large deformation diffeomorphisms of images with prescribed displacements of landmarks.\n\nA few works on image triangulations have been reported. In \\cite{Gee94}, Gee et al. introduced a probabilistic approach to the brain image matching problem and described the finite element implementation. In \\cite{Kaufmann13}, Kaufmann et al. introduced a framework for image warping using the finite element method. The triangulations are created using the Delaunay triangulation method \\cite{Shewchuk96} on a point set distributed according to variance in saliency. In \\cite{Lehner07,Lehner08}, Lehner et al. proposed a data-dependent triangulation scheme for image and video compression. Recently, a commercial triangulation image generator called DMesh \\cite{dmesh} has been designed.\n\nIn our work, we handle image registration problems with the aid of triangulations. Numerous algorithms have been proposed for the registration of triangular meshes. In particular, the landmark-driven approaches use prescribed landmark constraints to ensure the accuracy of mesh registration. In \\cite{Wang05,Lui07,Wang07}, Wang et al. proposed a combined energy for computing a landmark constrained optimized conformal mapping of triangular meshes. In \\cite{Lui10}, Lui et al. used vector fields to represent surface maps and computed landmark-based close-to-conformal mappings. Shi et al. \\cite{Shi13} proposed a hyperbolic harmonic registration algorithm with curvature-based landmark matching on triangular meshes of brains. In recent years, quasi-conformal mappings have been widely used for feature-endowed registration \\cite{Zeng11,Zeng14,Lui14,Meng15}. Choi et al. \\cite{Choi15} proposed the FLASH algorithm for landmark aligned harmonic mappings by improving the algorithm in \\cite{Wang05,Lui07} with the aid of quasi-conformal theories. In \\cite{Lam14}, Lam and Lui reported the quasi-conformal landmark registration (QCLR) algorithm for triangular meshes.\n\n\n\\section{Contribution} \\label{contribution}\nOur proposed \\emph{TRIM} approach for accelerating image and video frame registration is advantageous in the following aspects:\n\\begin{enumerate}\n \\item The triangulation algorithm is fully automatic. The important features of the input image are well recorded in the resulting coarse triangulation.\n \\item The algorithm is fast and robust. The coarse triangulation of a typical high resolution image can be computed within seconds.\n \\item The registration of the original images can be simplified by a mapping of the coarse triangulation. The fine details are then restored with the aid of the coarse registration result.\n \\item Our proposed \\emph{TRIM} algorithm is particularly suitable for landmark-based registration as the landmark constraints can be easily included in the coarse triangulations. By contrast, for regular grid-based approaches, accurate landmark correspondences can only be achieved on a very dense grid domain representation.\n \\item Using our proposed approach, the problem scale of the image and video frame registration is significantly reduced. Our method can serve as a fast and accurate initialization for the state-of-the-art image registration algorithms.\n\\end{enumerate}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[height=95mm]{pl.png}\n \\caption{The pipeline of our proposed {\\em TRIM} algorithm for accelerating image registration via coarse triangulation.}\n \\label{fig:pl}\n\\end{figure}\n\n\n\n\\section{Proposed method} \\label{main}\n\nIn this section, we describe our proposed {\\em TRIM} image triangulation approach for efficient registration in details. The pipeline of our proposed framework is described in Figure \\ref{fig:pl}.\n\n\\bigbreak\n\n\\subsection{Construction of coarse triangulation on images}\nGiven two high resolution images $I_1$ and $I_2$, our goal is to compute a fast and accurate mapping $f: I_1 \\to I_2$. Note that directly working on the high resolution images can be inefficient. To accelerate the computation, the first step is to construct a coarse triangular representation of the image $I_1$. In the following, we propose an efficient image triangulation scheme called {\\em TRIM}. Our triangulation scheme is content-aware. Specifically, special objects and edges in the images are effectively captured by a segmentation step, and a suitable coarse triangulation is constructed with the preservation of these features. Our proposed {\\em TRIM} method consists of 6 steps in total.\n\n\\bigbreak\n\n\\subsubsection{Subsampling the input image without affecting the triangulation quality}\nDenote the input image by $I$. To save the computational time for triangulating the input image $I$, one simple remedy is to reduce the problem size by performing certain subsampling on $I$. For ordinary images, subsampling unavoidably creates adverse effects on the image quality. Nevertheless, it does not affect the quality of the coarse triangulation we aim to construct on images.\n\nIn our triangulation scheme, we construct triangulations based on the straight edges and special features on the images. Note that straight edges are preserved in all subsamplings of the images because of the linearity. Hence, our edge-based triangulation is not affected by the mentioned adverse effects. In other words, we can subsample high resolution images to a suitable size for enhancing the efficiency of the remaining steps for the construction of the triangulations. We denote the subsampled image by $\\tilde{I}$.\n\n\\bigbreak\n\n\\subsubsection{Performing unsharp masking on the subsampled image}\nAfter obtaining the subsampled image $\\tilde{I}$, we perform an unsharp masking on $\\tilde{I}$ in order to preserve the edge information in the final triangulation. More specifically, we first transform the data format of the subsampled image $\\tilde{I}$ to the CIELAB standard. Then, we apply the unsharp masking method in \\cite{Polesel00} on the intensity channel of the CIELAB representation of $\\tilde{I}$. The unsharp masking procedure is briefly described as follows.\n\nBy an abuse of notation, we denote $\\tilde{I}(x,y)$ and $\\bar{I}(x,y)$ the intensities of the input subsampled image $\\tilde{I}$ and the output image $\\bar{I}$ respectively, and $G_{\\sigma}(x,y)$ the Gaussian mean of the intensity of the pixel $(x,y)$ with standard derivation $\\sigma$. Specifically, $G_{\\sigma}(x,y)$ is given by\n\n\\begin{equation}\n G_{\\sigma}(x,y) \\triangleq \\frac{1}{\\sigma\\sqrt{2\\pi}}\\int_{(u,v) \\in \\Omega} e^{-\\frac{(u-x)^2 + (v-y)^2}{2\\sigma^2}}.\n\\end{equation}\n\nWe perform an unsharp masking on the image using the following formula\n\\begin{equation}\n \\bar{I}(x,y) = \\tilde{I}(x,y) - \\lambda\\begin{cases} G_{\\sigma}*\\tilde{I}(x,y) & \\text{ if } V_s(x,y)>\\theta, \\\\ 0 & \\text{ if } V_s(x,y) < \\theta, \\end{cases}\n\\end{equation}\nwhere\n\\begin{equation}\n V_s(x,y) \\triangleq \\sqrt{\\frac{1}{Area(M_s)} \\int_{(u,v) \\in M_s} (\\tilde{I}(u,v)-\\tilde{I}_{mean}(x,y))^2 }\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{I}_{mean}(x,y) = \\frac{1}{Area(M_s)} \\int_{(u,v) \\in M_s} \\tilde{I}(u,v).\n\\end{equation}\n\nHere, $M_s$ is the disk with center $(x,y)$ and radius $s$. The effect of the unsharp masking is demonstrated in Figure \\ref{fig:unsharp}. With the aid of this step, we can highlight the edge information in the resulting image $\\bar{I}$ for the construction of the triangulation in the later steps. In our experiment, we choose $\\lambda = 0.5,~\\sigma = 2,~s = 2$, and $\\theta = 0.5$.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.22\\textwidth]{edge_2.png}\n \\includegraphics[width=0.22\\textwidth]{edge_1.png}\n \\caption{An illustration of unsharp masking. Left: the input image. Right: the resulting image. The unsharp masking procedure helps preserving the edge information of the input image to ensure that the vertices in unclear edges can also be extracted.}\n \\label{fig:unsharp}\n\\end{figure}\n\n\\bigbreak\n\n\\subsubsection{Segmenting the image}\nAfter obtaining the image $\\bar{I}$ upon unsharp masking, we perform a segmentation in this step in order to optimally locate the mesh vertices for computing the coarse triangulation. Mathematically, our segmentation problem is described as follows.\n\nSuppose the image $\\bar{I}$ has $L$ intensity levels in each RGB channel. Denote $i$ as a specific intensity level $(i.e. ~ 0 \\leq i \\leq L-1)$. Let $C$ be a color channel of the image $(i.e. ~ C \\in \\lbrace R,G,B \\rbrace)$, and let $h_{i}^{C}$ denote the image histogram for channel $C$, in other words, the number of pixels which correspond to its $i$-th intensity level.\n\nDefine $p_{i}^{C} \\triangleq \\frac{h_{i}^{C}}{N}$, where $N$ represents the total number of pixels in the image $\\bar{I}$. Then we have\n\\begin{equation}\n\\sum\\limits_{\\substack{i = 0, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^L p_{i}^{C} = 1 \\ \\ \\text{ and } \\ \\ \\mu_{T}^{C} = \\sum\\limits_{\\substack{i = 0, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^L ip_{i}^{C}.\n\\end{equation}\n\nSuppose that we want to compress the color space of the image $\\bar{I}$ to $l$ intensity levels. Equivalently, $\\bar{I}$ is to be segmented into $l$ regions $D_{1}^{C},\\cdots,D_{l}^{C}$ by the ordered threshold levels $ x_{j}^{C}, j = 1,\\cdots,l-1$. We define the best segmentation criterion to be maximizing the inter-region intensity-mean variance. More explicitly, we define the cost\n\\begin{equation}\\label{eqt:pso_cost}\n \\sigma^C \\triangleq \\sum\\limits_{\\substack{j = 1, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^l w_{j}^{C}(\\mu_{j}^{C} - \\mu_{T}^{C})^2,\n\\end{equation}\nwhere the probability $w_{j}^{C}$ of occurrence and the intensity-mean $\\mu_{j}^{C}$of the region $D_{j}^{C}$ are respectively given by\n\\begin{equation}\n w_{j}^{C} =\n\\begin{cases}\n\\sum\\limits_{\\substack{i = 0, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^{t_{j}^{C}} p_{i}^{C} & \\text{ if } j = 1, \\\\\n\\sum\\limits_{\\substack{i = t_{j-1}^{C} + 1, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^{t_{j}^{C}} p_{i}^{C} & \\text{ if } 1 < j < l, \\\\\n\\sum\\limits_{\\substack{i = t_{j}^{C} + 1, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^{L - 1} p_{i}^{C} & \\text{ if } j = l,\n\\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n\\mu_{j}^{C} =\n\\begin{cases}\n\\sum\\limits_{\\substack{i = 0, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^{t_{j}^{C}} \\frac{ip_{i}^{C}}{w_{j}^{C}} & \\text{ if } j = 1, \\\\\n\\sum\\limits_{\\substack{i = t_{j-1}^{C} + 1, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^{t_{j}^{C}} \\frac{ip_{i}^{C}}{w_{j}^{C}}\n& \\text{ if } 1 < j < l, \\\\\n\\sum\\limits_{\\substack{i = t_{j}^{C} + 1, \\\\  C \\in \\lbrace R,G,B \\rbrace}}^{L - 1} \\frac{ip_{i}^{C}}{w_{j}^{C}}\n& \\text{ if }  j = l.\n\\end{cases}\n\\end{equation}\n\nHence, we maximize three objective functions of each RGB channel\n\\begin{equation}\n\\underset{1 < x_{1}^{C} < \\cdots < x_{l-1}^{C} < L}{\\arg\\max} \\sigma^C(\\lbrace x_{j}^{C} \\rbrace_{j = 1}^{l-1}),\n\\end{equation}\nwhere $C \\in \\{R, G, B\\}$. Our goal is to find a set of $\\textbf{x} = \\lbrace x_{j}^{C} \\rbrace_{j = 1}^{l-1}$ such that above function is maximized for each RGB channel.\n\nTo solve the aforementioned segmentation optimization problem, we apply the Particle Swarm Optimization (PSO) segmentation algorithm \\cite{Ghamisi12} on the image $\\bar{I}$. The PSO method is used in this segmentation optimization problem for reducing the chance of trapping in local optimums.\n\nAn illustration of the segmentation step is provided in Figure \\ref{fig:pso}. After performing the segmentation, we extract the boundaries of the segments. Then, we can obtain a number of large patches of area in each of which the intensity information is almost the same. They provide us with a reasonable edge base for constructing a coarse triangulation in later steps.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.22\\textwidth]{mnm_1.png}\n \\includegraphics[width=0.22\\textwidth]{mnm_2.png}\n \\caption{An illustration of the segmentation step for compressing the color space to achieve a sparse intensity representation. Left: the original image. Right: the segmentation result.}\n \\label{fig:pso}\n\\end{figure}\n\n\\bigbreak\n\\subsubsection{Sparse feature extraction on the segment boundaries}\nAfter computing the segment boundaries $\\mathcal{B}$ on the image $\\bar{I}$, we aim to extract sparse feature points on $\\mathcal{B}$ in this step. For the final triangulation, it is desirable that the edges of the triangles are as close as possible to the segment boundaries $\\mathcal{B}$, so as to preserve the geometric features of the original image $I$. Also, to improve the efficiency for the computations on the triangulation, the triangulation should be much coarser than the original image. To achieve the mentioned goals, we consider extracting sparse features on the segment boundaries $\\mathcal{B}$ and use them as the vertices of the ultimate triangulated mesh.\n\nConsider a rectangular grid table $G$ on the image $\\bar{I}$. Apparently, the grid table $G$ intersects the segment boundaries $\\mathcal{B}$ at a number of points. Denote $\\mathcal{P}$ as our desired set of sparse features. Conceptually, $\\mathcal{P}$ is made up of the set of points at which $\\mathcal{B}$ intersect the grid $G$, with certain exceptions.\n\nIn order to further reduce the number of feature points for a coarse triangulation, we propose a merging procedure for close points. Specifically, let $g_{i,j}$ be the vertex of the grid $G$ at the $i$-th row and the $j$-th column. We denote $\\mathcal{P}_{i,j}^1$ and $\\mathcal{P}_{i,j}^2$ respectively as the set of points at which $\\mathcal{B}$ intersect the line segment $\\displaystyle \\overline{g_{i,j} g_{i,j+1}}$ and the line segment $\\displaystyle \\overline{g_{i,j} g_{i+1,j}}$.\n\nThere are 3 possible cases for $\\mathcal{P}_{i,j}^k$:\n\\begin{enumerate}[(i)]\n \\item If $|\\mathcal{P}_{i,j}^k|=0$, then there is no intersection point between the line segment and $\\mathcal{B}$ and hence we can neglect it.\n \\item If $|\\mathcal{P}_{i,j}^k|=1$, then there is exactly one intersection point $p_{i,j}^k$ between the line segment and $\\mathcal{B}$. We include this intersection point $p_{i,j}^k$ in our desired set of sparse features $\\mathcal{P}$.\n \\item If $|\\mathcal{P}_{i,j}^k|>1$, then there are multiple intersection points between the line segment and $\\mathcal{B}$. Since these multiple intersection points lie on the same line segment, it implies that they are sufficiently close to each other. In other words, the information they contain about the segment boundaries $\\mathcal{B}$ is highly similar and redundant. Therefore, we consider merging these multiple points as one point.\n\\end{enumerate}\nMore explicitly, for the third case, we compute the centre $m_{i,j}^k$ of the points in $\\mathcal{P}_{i,j}^k$ by\n\\begin{equation}\n\\displaystyle m_{i,j}^k = mean_{\\{p | p \\in \\mathcal{P}_{i,j}^k\\}} p.\n\\end{equation}\nThe merged point $m_{i,j}^k$ is then considered as a desired feature point. In summary, our desired set of sparse features is given by\n\\begin{equation}\n\\displaystyle \\mathcal{P} = \\bigcup_i \\bigcup_j \\{p_{i,j}^1, p_{i,j}^2, m_{i,j}^1, m_{i,j}^2\\}.\n\\end{equation}\n\nAn illustration of the sparse feature extraction scheme is given in Figure \\ref{fig:feature}.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.28\\textwidth]{feature.png}\n \\caption{An illustration of our sparse feature extraction scheme. The chosen sparse feature points are represented by the red dots. If the segment boundary does not intersect an edge, no point is selected. If the segment boundary intersects an edge at exactly one point, the point is selected as a feature point. If the segment boundary intersects an edge at multiple points, the centre of the points is selected as a feature point.}\n \\label{fig:feature}\n\\end{figure}\nHowever, one important problem in this scheme is to determine a suitable size of the grid $G$ so that the sparse feature points are optimally computed. Note that to preserve the regularity of the extracted sparse features, it is desirable that the elements of the grid $G$ are close to perfect squares. Also, to capture the important features as complete as possible, the elements of $G$ should be small enough. Mathematically, the problem can be formulated as follows.\n\nDenote $w$ as the width of the image $\\bar{I}$, $h$ as the height of the image $\\bar{I}$, $w'$ as the number of columns in $G$, $h'$ as the number of rows in $G$, $l_{w}$ as the horizontal length of every element of $G$, and $l_{h}$ as the vertical length of every element of $G$. We further denote $p$ as the percentage of grid edges in $G$ which intersect the segment boundaries $\\mathcal{B}$, and $n$ as the desired number of the sparse feature points. Given the two inputs $p$ and $n$ which respectively correspond to the desired sparse ratio and the desired number of feature points, to find a suitable grid size of $G$, we aim to minimize the cost function\n\n\\begin{equation}\nc(l_{w},l_{h}) = \\left| l_{w} - l_{h} \\right|^2\n\\end{equation}\nsubject to\n\\begin{enumerate}[(i)]\n \\item \\begin{equation}h = h'l_{h},\\end{equation}\n \\item \\begin{equation}w = w'l_{w},\\end{equation}\n \\item \\begin{equation} \\label{eqt:number} p(w'+h'+2w'h') = n.\\end{equation}\n\\end{enumerate}\n\nHere, the first and the second constraint respectively correspond to the horizontal and vertical dimensions of the grid $G$, and the third constraint corresponds to the total number of intersection points. To justify Equation (\\ref{eqt:number}), note that\n\\begin{equation}\n\\begin{split}\n&\\text{Total number of line segments} \\\\\n= \\ &\\text{Total number of horizontal line segments} \\\\\n& + \\text{Total number of vertical line segments}\\\\\n= \\ &h'(w'+1) + w'(h'+1)\\\\\n= \\ &w'+h'+2w'h'.\\\\\n\\end{split}\n\\end{equation}\n\nNote that this minimization problem is nonlinear. To simplify the computation, we assume that $w'$, $h'$ are very large, that is, the grid $G$ is sufficiently dense. Then, from Equation (\\ref{eqt:number}), we have\n\\begin{equation}\n\\begin{split}\n\\frac{p}{n} &= \\frac{1}{w'+h'+2w'h'} \\\\\n& \\approx \\frac{1}{2w'h'} \\\\\n&= \\frac{1}{2\\left(\\frac{w}{l_w}\\right) \\left(\\frac{h}{l_h}\\right)} \\\\\n&= \\frac{l_{w}l_{h}}{2wh}.\n\\end{split}\n\\end{equation}\nBy further assuming that the grid $G$ is sufficiently close to a square grid, we have $l_{w} \\approx l_{h}$. Then, it follows that\n\\begin{equation}\n\\frac{p}{n} \\approx \\frac{l_{w}^2}{2wh}\n\\end{equation}\nand hence\n\\begin{equation}\nl_{w} \\approx \\sqrt{\\frac{2pwh}{n}}.\n\\end{equation}\nSimilarly,\n\\begin{equation}\nl_{h} \\approx \\sqrt{\\frac{2pwh}{n}}.\n\\end{equation}\n\nTo satisfy the integral constraints for $w'$ and $h'$, we make use of the above approximations and set\n\\begin{equation}\nh' = h_0' := \\left\\lfloor \\frac{h}{\\sqrt{\\frac{2pwh}{n}}} \\right\\rfloor = \\left\\lfloor \\sqrt{\\frac{nh}{2pw}} \\right\\rfloor.\n\\end{equation}\nSimilarly, we set\n\\begin{equation}\nw' = w_0' := \\left\\lfloor \\frac{w}{\\sqrt{\\frac{2pwh}{n}}}  \\right\\rfloor = \\left\\lfloor \\sqrt{\\frac{nw}{2ph}} \\right\\rfloor.\n\\end{equation}\nFinally, we take\n\\begin{equation}\nl_{h} = \\frac{h}{h_0'}\n \\end{equation}\nand\n\\begin{equation}\nl_{w} = \\frac{w}{w_0'}.\n \\end{equation}\n\nTo summarize, with the abovementioned strategy for the feature point extraction, we obtain a set of sparse feature points which approximates the segment boundaries $\\mathcal{B}$. Specifically, given the inputs $p$ and $n$, the rectangular grid $G$ we introduce leads to approximately $n$ regularly-extracted sparse feature points. An illustration of the sparse feature extraction scheme is shown in Figure \\ref{fig:delaunay} (left). In our experiments, $p$ is set to be 0.2. A denser triangulated representation can be achieved by increasing the value of $p$.\n\n\\bigbreak\n\n\\subsubsection{Adding landmark points to the vertex set of the desired coarse triangulation}\nThis step is only required when our {\\em TRIM} algorithm is used for landmark-constrained registration. For accurate landmark-constrained registration, it is desirable to include the landmark points in the vertex set of the coarse representations of the input image $I$. One of the most important features of our coarse triangulation approach is that it allows registration with exact landmark constraints on a coarse triangular representation. By contrast, the regular grid-based registration can only be achieved on very dense rectangular grid domains in order to reduce the numerical errors.\n\nWith the abovementioned advantage of our approach, we can freely add a set of landmark points $\\mathcal{P}_{LM}$ to the set of sparse features $\\mathcal{P}$ extracted by the previous procedure. In other words, the landmark points are now considered as a part of our coarse triangulation vertices:\n\\begin{equation}\n\\mathcal{P} = \\bigcup_i \\bigcup_j \\{p_{i,j}^1, p_{i,j}^2, m_{i,j}^1, m_{i,j}^2\\} \\cup \\mathcal{P}_{LM}.\n\\end{equation}\nThen, the landmark-constrained registration of images can be computed by the existing feature-matching techniques for triangular meshes. The existing feature detection approaches such as \\cite{harris88} and \\cite{Lowe04} can be applied for obtaining the landmark points.\n\n\\bigbreak\n\n\\subsubsection{Computing a Delaunay triangulation}\nIn the final step, we construct a triangulation based on the set $\\mathcal{P}$ of feature points. Among all triangulation schemes, the Delaunay triangulation method is chosen since the triangles created by the Delaunay triangulations are more regular. More specifically, if $\\alpha$ and $\\beta$ are two angles opposite to a common edge in a Delaunay triangulation, then they must satisfy the inequality\n\\begin{equation}\n \\alpha + \\beta \\leq \\pi.\n\\end{equation}\n\nIn other words, Delaunay triangulations avoid the formation of sharp and irregular triangles. Note that the regularity does not only enhance the visual quality of the resulting triangulation but also lead to a more stable approximation of the derivatives on the triangles when applying various registration schemes. Therefore, we compute a Delaunay triangulation  on the set $\\mathcal{P}$ of feature points for achieving the ultimate triangulation $\\mathcal{T}$. An illustration of the construction of the Delaunay triangulations is shown in Figure \\ref{fig:delaunay}.\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.23\\textwidth]{mnm_sparse_point.png} \\ \\\n \\includegraphics[width=0.23\\textwidth]{mnm_mesh.png} \\ \\\n \\includegraphics[width=0.23\\textwidth]{mnm_mesh_color.png}\n \\caption{An illustration of extracting the vertices and computing a Delaunay triangulation. Left: the points obtained by the feature extraction step from Figure \\ref{fig:pso}. Middle: a Delaunay triangulation generated on the feature points. Right: the triangulation with a color approximated on each triangle.}\n \\label{fig:delaunay}\n\\end{figure*}\n\nThese 6 steps complete our {\\em TRIM} algorithm as summarized in Algorithm \\ref{algorithm:triangulation}.\n\n\\begin{algorithm}[h]\n\\KwIn{An image $I$, the desired number of image intensity levels $l$ for segmentation, the desired number of feature points $n$, the sparse ratio $p$.}\n\\KwOut{A coarse triangulation $\\mathcal{T}$ that captures the main features of the image.}\n\\BlankLine\n Subsample the input image $I$ to a suitable size and denote the result by $\\tilde{I}$\\;\n Apply an unsharp masking on the subsampled image $\\tilde{I}$ and denote the result by $\\bar{I}$\\;\n Apply the PSO segmentation for compressing the color space of $\\bar{I}$ to $l$ intensity levels, and extract boundaries $\\mathcal{B}$ of the segments\\;\n Extract a set of sparse feature points $\\mathcal{P}$ from the segment boundaries $\\mathcal{B}$ based on the parameters $n$ and $p$\\;\n Add a set of extra landmark points $\\mathcal{P}_{LM}$ to $\\mathcal{P}$ if necessary\\;\n Compute a Delaunay triangulation $\\mathcal{T}$ on the sparse feature points $\\mathcal{P}$.\n\\caption{Our proposed {\\em TRIM} algorithm for triangulating an image.}\n\\label{algorithm:triangulation}\n\\end{algorithm}\n\nIt is noteworthy that our proposed {\\em TRIM} algorithm significantly trims down high resolution images without distorting their important geometric features. Experimental results are shown in Section \\ref{experiment} to demonstrate the effectiveness of the {\\em TRIM} algorithm.\n\n\\subsection{Simplification of the registration using the coarse triangulation}\nWith the above triangulation algorithm for images, we can simplify the image registration problem as a mapping problem of triangulated surfaces. Many conventional image registration approaches are hindered by the long computational time and the accuracy of the initial maps. With the new strategy, it is easy to obtain a highly efficient and reasonably accurate registration of images. Our registration result can serve as a high quality initial map for various algorithms.\n\nIn this work, we apply our proposed {\\em TRIM} algorithm for landmark-aligned quasi-conformal image registration. Ideally, conformal registration are desired as conformal mappings preserve angles and hence the local geometry of the images. However, with the presence of landmark constraints, conformal mappings may not exist. We turn to consider quasi-conformal mappings, a type of mappings which is closely related to the conformal mappings. Mathematically, a \\emph{quasi-conformal mapping} $f: \\mathbb{C} \\to \\mathbb{C}$ satisfies the Beltrami equation\n\\begin{equation}\n\\frac{\\partial f}{\\partial \\bar{z}} = \\mu(z) \\frac{\\partial f}{\\partial z}\n\\end{equation}\nwith $\\mu$ is a complex-valued function with sup norm less than 1. $\\mu$ is called the \\emph{Beltrami coefficient} of $f$. Intuitively, a conformal mapping maps infinitesimal circles to infinitesimal circles, while a quasi-conformal mapping maps infinitesimal circles to infinitesimal ellipses (see Figure \\ref{fig:beltrami}). Readers are referred to \\cite{Gardiner00} for more details.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{beltrami.png}\n \\caption{An illustration of quasi-conformal mappings. The maximal magnification and shrinkage are determined by the Beltrami coefficient $\\mu$ of the mappings.}\n \\label{fig:beltrami}\n\\end{figure}\n\nIn \\cite{Lam14}, Lam and Lui proposed the quasi-conformal landmark registration (QCLR) algorithm for triangular meshes. In this work, we apply the QCLR algorithm on our coarse triangulations of images. More explicitly, to compute a registration mapping $f:I_1 \\to I_2$ between two images $I_1$ and $I_2$ with prescribed point correspondences\n\\begin{equation} \\label{eqt:constraint}\np_i \\longleftrightarrow q_i, i = 1,2,\\cdots,n,\n\\end{equation}\nwe first apply our proposed {\\em TRIM} algorithm and obtain a coarse triangulation $\\mathcal{T}_1$ on $I_1$. Here, we include the feature points $\\{p_i\\}_{i=1}^n$ in the generation of the coarse triangulation, as described in the fifth step of the {\\em TRIM} algorithm. Then, instead of directly computing $f$, we can solve for a map $\\tilde{f}: \\mathcal{T}_1 \\to I_2$. Since the problem size is significantly reduced under the coarse triangulation, the computation for $\\tilde{f}$ is much more efficient than that for $f$.\n\nThe QCLR algorithm \\cite{Lam14} aims to compute the desired quasi-conformal mapping $\\tilde{f}$ by minimizing the energy functional\n\\begin{equation} \\label{eqt:qclr}\nE_{LM}(\\nu) = \\int_{\\mathcal{T}_1} | \\nabla \\nu|^2 + \\alpha \\int_{\\mathcal{T}_1} |\\nu|^2\n\\end{equation}\nsubject to\n\\begin{enumerate}[(i)]\n \\item $\\tilde{f}(p_i) = q_i$ for all $i = 1,2,\\cdots,n$,\n \\item $\\|\\nu\\|_{\\infty} <1$, and\n \\item $\\nu = \\mu(\\tilde{f})$.\n\\end{enumerate}\n\nTo solve the above minimization problem (\\ref{eqt:qclr}), the QCLR algorithm makes use of the penalty splitting method and minimizes\n\n\\begin{equation}\n E_{LM}^{split}(\\nu,\\tilde{f}) = \\int_{\\mathcal{T}_1} |\\nabla \\nu|^2 + \\alpha \\int_{\\mathcal{T}_1} |\\nu|^2 + \\rho \\int_{\\mathcal{T}_1} |\\nu - \\mu(\\tilde{f})|^2\n\\end{equation}\nsubject to\n\\begin{enumerate}[(i)]\n \\item $\\tilde{f}(p_i) = q_i$ for all $i = 1,2,\\cdots,n$ and\n \\item $\\|\\nu\\|_{\\infty} <1$.\n\\end{enumerate}\nThe QCLR algorithm then alternately minimizes the energy $E_{LM}^{split}$ over one of $\\nu$ and $\\tilde{f}$ while fixing the other one.\n\nSpecifically, for computing $\\tilde{f}_n$ while fixing $\\nu_n$ and the landmark constraints, the QCLR algorithm applies the Linear Beltrami Solver proved by Lui et al.\\cite{Lui13}. On the other hand, for computing $\\nu_{n+1}$ while fixing $\\tilde{f}_n$, by considering the Euler-Lagrange equation, it suffices to solve\n\\begin{equation}\n (-\\Delta + 2 \\alpha I + 2\\rho I) \\nu_{n+1} = 2\\rho \\mu(\\tilde{f}_n).\n\\end{equation}\n\nFrom $\\nu_{n+1}$, one can compute the associated quasi-conformal mapping $\\tilde{f}_{n+1}$. However, note that $\\tilde{f}_{n+1}$ may not satisfy the hard landmark constraints \\ref{eqt:constraint} anymore. To enforce the landmark constraints, instead of directly using $\\tilde{f}_{n+1}$ as the updated result, the QCLR algorithm updates $\\nu_{n+1}$ by\n\\begin{equation}\n \\nu_{n+1} \\leftarrow \\nu_{n+1} + t(\\mu(\\tilde{f}_{n+1}) - \\nu_{n+1})\n\\end{equation}\nfor some small $t$. By repeating the abovementioned procedures, we achieve a landmark-constrained quasi-conformal mapping $\\tilde{f}$. For more details of the QCLR algorithm, readers are referred to \\cite{Lam14}.\n\nAfter computing the quasi-conformal mapping $\\tilde{f}$ on the coarse triangulation, we apply an interpolation to retrieve the fine details of the registration. Since the triangulations created by our proposed {\\em TRIM} algorithm preserves the important geometric features and prominent straight lines of the input image, the details of the registration results can be accurately interpolated. Moreover, since the coarse triangulation largely simplifies the input image and reduces the problem size, the computation is largely accelerated.\n\nThe registration procedure is summarized in Algorithm \\ref{algorithm:registration}. Experimental results are illustrated in Section \\ref{experiment} to demonstrate the significance of our coarse triangulation in the registration scheme.\n\n\\begin{algorithm}[h]\n\\KwIn{Two images or video frames $I_1$, $I_2$ to be registered, with the prescribed feature correspondences.}\n\\KwOut{A feature-matching registration mapping $f:I_1 \\to I_2$.}\n\\BlankLine\nCompute a coarse triangulation $\\mathcal{T}_1$ of $I_1$ using our proposed {\\em TRIM} algorithm (Algorithm \\ref{algorithm:triangulation}). Here, we include the prescribed feature points on $I_1$ in the generation of the coarse triangulation $\\mathcal{T}_1$\\;\nSelect landmark correspondences of the coarse triangulation $\\mathcal{T}_1$ and the target image $I_2$. Denote the landmark points on $\\mathcal{T}_1$ and $I_2$ by $\\{p_i\\}_{i=1}^n$ and $\\{q_i\\}_{i=1}^n$ correspondingly\\;\nCompute a landmark based quasi-conformal mapping $\\tilde{f}: \\mathcal{T}_1 \\to \\mathbb{C}$ by the QCLR algorithm in \\cite{Lam14}\\;\nObtain $f$ by $\\tilde{f}$ with a bilinear interpolation between $\\mathcal{T}_j$ and $I_j$.\n\\caption{Feature-based registration via our proposed {\\em TRIM} algorithm.}\n\\label{algorithm:registration}\n\\end{algorithm}\n\n\n\\section{Experimental results} \\label{experiment}\n\nIn this section, we demonstrate the effectiveness of our proposed triangulation scheme. The algorithms are implemented using MATLAB. For solving the mentioned linear systems, the backslash operator ($\\backslash$) in MATLAB is used. The test images are courtesy of the RetargetMe dataset \\cite{Rubinstein10}. The bird image is courtesy of the first author. All experiments are performed on a PC with an Intel(R) Core(TM) i7-4500U CPU @1.80 GHz processor and 8.00 GB RAM.\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.215\\textwidth]{bird.png} \\\n \\includegraphics[width=0.16\\textwidth]{book.png} \\\n \\includegraphics[width=0.32\\textwidth]{pencils.png} \\\n \\includegraphics[width=0.255\\textwidth]{bear_1.png}\n \\center{\\hspace{5mm} Bird \\hspace{27mm}  Book \\hspace{35mm} Pencils \\hspace{45mm}  Teddy}\n \\newline\n \\newline\n \\includegraphics[width=0.215\\textwidth]{bird_mesh.png} \\\n \\includegraphics[width=0.16\\textwidth]{book_mesh.png} \\\n \\includegraphics[width=0.32\\textwidth]{pencils_mesh.png} \\\n \\includegraphics[width=0.255\\textwidth]{bear_mesh.png}\n \\center{Triangulation of Bird \\hspace{3mm}  Triangulation of Book \\hspace{9mm} Triangulation of Pencils  \\hspace{19mm}  Triangulation of Teddy}\n   \\newline\n \\caption{Several images and the triangulations created by our {\\em TRIM} algorithm. Top: the input images. Bottom: the resulting triangulations. The key features of the original images are well represented in our triangulations, and the regions with similar color can be represented by coarse triangulations. }\n \\label{fig:triangulation}\n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.24\\textwidth]{bee.png}\n \\includegraphics[width=0.24\\textwidth]{bee_mesh.png}\n \\includegraphics[width=0.24\\textwidth]{bee_mesh_color.png}\n \\includegraphics[width=0.24\\textwidth]{bee_dmesh.png}\n \\caption{A bee image and the triangulations created by our {\\em TRIM} algorithm and DMesh \\cite{dmesh}. Left: The input image. Middle left: The coarse triangulation created by {\\em TRIM}. Middle right: Our {\\em TRIM} coarse triangulation with a color approximated on each triangle. Right: The triangulation by DMesh \\cite{dmesh}.}\n \\label{fig:triangulation_bee}\n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.24\\textwidth]{butterfly.png}\n \\includegraphics[width=0.24\\textwidth]{butterfly_mesh.png}\n \\includegraphics[width=0.24\\textwidth]{butterfly_mesh_color.png}\n \\includegraphics[width=0.24\\textwidth]{butterfly_dmesh.png}\n \\caption{An butterfly image and the triangulations created by our {\\em TRIM} algorithm and DMesh \\cite{dmesh}. Left: The input image. Middle left: The coarse triangulation created by {\\em TRIM}. Middle right: Our {\\em TRIM} coarse triangulation with a color approximated on each triangle. Right: The triangulation by DMesh \\cite{dmesh}.}\n \\label{fig:triangulation_butterfly}\n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.27\\textwidth]{baseball.png} \\\n \\includegraphics[width=0.27\\textwidth]{baseball_mesh_color.png} \\\n \\includegraphics[width=0.27\\textwidth]{baseball_dmesh.png}\n \\newline\n \\newline\n \\includegraphics[width=0.27\\textwidth]{eagle.png} \\\n \\includegraphics[width=0.27\\textwidth]{eagle_triangulation.png} \\\n \\includegraphics[width=0.27\\textwidth]{eagle_dmesh.png}\n  \\newline\n \\caption{Two more examples created by our {\\em TRIM} algorithm and Dmesh \\cite{dmesh}. Our coarse triangulations capture the important features and closely resemble the original images. Left: The input image. Middle: The coarse triangulation created by {\\em TRIM}. Right: The triangulation by DMesh \\cite{dmesh}.}\n \\label{fig:triangulation_more}\n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.24\\textwidth]{bear_1.png} \\\n \\includegraphics[width=0.24\\textwidth]{bear_trim.png} \\\n \\includegraphics[width=0.24\\textwidth]{bear_noisy.png} \\\n \\includegraphics[width=0.24\\textwidth]{bear_noisy_trim.png}\n \\newline\n \\newline\n \\includegraphics[width=0.24\\textwidth]{cone_1.png} \\\n \\includegraphics[width=0.24\\textwidth]{cone_trim.png} \\\n \\includegraphics[width=0.24\\textwidth]{cone_noisy.png} \\\n \\includegraphics[width=0.24\\textwidth]{cone_noisy_trim.png}\n  \\newline\n \\caption{Two triangulation examples by our {\\em TRIM} algorithm for noisy images. Left: The noise-free images. Middle left: The triangulations computed by {\\em TRIM} based on the noise-free images. Middle right: The noisy images. Right: The triangulations computed by {\\em TRIM} based on the noisy images. It can be observed that the important features of the images are preserved even for noisy images.}\n \\label{fig:noisy}\n\\end{figure*}\n\n\\subsection{The performance of our proposed {\\em TRIM} triangulation scheme}\nIn this subsection, we demonstrate the effectiveness of our triangulation scheme by various examples.\n\nOur proposed algorithm is highly content-aware. Specifically, regions with high similarities or changes in color on an image can be easily recognized. As a result, the triangulations created faithfully preserve the important features by a combination of coarse triangles with different sizes. Some of our triangulation results are illustrated in Figure \\ref{fig:triangulation}. For better visualizations, we color the resulting triangulations by mean of the original colors of corresponding patches. In Figure \\ref{fig:triangulation_bee}, we apply our {\\em TRIM} algorithm on a bee image. It can be observed that the regions of the green background can be effectively represented by coarser triangulations, while the region of the bee and flowers with apparent color differences is well detected and represented by a denser triangulation. Figure \\ref{fig:triangulation_butterfly} shows another example of our triangulation result. The butterfly and the flowers are well represented in our triangulation result. The above examples demonstrate the effectiveness of our triangulation scheme for representing images in a simplified but accurate way. Some more triangulation examples created by our {\\em TRIM} algorithm are shown in Figure \\ref{fig:triangulation_more}. Figure \\ref{fig:noisy} shows some triangulation examples for noisy images. It can be observed that our {\\em TRIM} algorithm can effectively compute content-aware coarse triangulations even for noisy images.\n\nWe have compared our algorithm with the DMesh triangulator \\cite{dmesh} in Figure \\ref{fig:triangulation_bee}, Figure \\ref{fig:triangulation_butterfly} and Figure \\ref{fig:triangulation_more}. It can be observed that our triangulation scheme outperforms DMesh \\cite{dmesh} in terms of the triangulation quality. Our results can better capture the important features of the images. Also, the results by DMesh \\cite{dmesh} may contain unwanted holes while our triangulation results are always perfect rectangles. The comparisons reflect the advantage of our coarse triangulation scheme.\n\nThen, we evaluate the efficiency of our triangulation scheme for various images. Table \\ref{table:trim} shows the detailed statistics. The relationship between the target coarse triangulation size and the computational time is illustrated in Figure \\ref{fig:triangulation_time}. Even for high resolution images, the computational time for the triangulation is only around 10 seconds. It is noteworthy that our {\\em TRIM} algorithm significantly compresses the high resolution images as coarse triangulations with only several thousand triangles.\n\n\\begin{table*}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nImage & Size & Triangulation time (s) & \\# of triangles & Compression rate \\\\ \\hline\nSurfer & 846 $\\times$ 421 & 5.78 & 1043 & 0.1536\\% \\\\ \\hline\nHelicopter & 720 $\\times$ 405 & 5.78 & 1129 & 0.1989\\%\n\\\\ \\hline\nBee & 640 $\\times$ 425 & 7.13 & 1075 & 0.2029\\% \\\\ \\hline\nBird & 1224 $\\times$ 1224 & 7.04 & 1287 & 0.0859\\% \\\\ \\hline\nButterfly & 1024 $\\times$ 700 & 8.00 & 1720 & 0.1232\\% \\\\ \\hline\nBook & 601 $\\times$ 809 & 8.38 & 1629 & 0.3350\\% \\\\ \\hline \nBaseball & 410 $\\times$ 399 & 7.85 & 2315 & 0.7201\\% \\\\ \\hline\nBear & 450 $\\times$ 375 & 7.48 & 2873 & 0.8652\\% \\\\ \\hline\nPencil & 615 $\\times$ 410 & 8.93 & 2633 & 0.5838\\% \\\\ \\hline\nTiger & 2560 $\\times$ 1600 &  13.91 & 3105 & 0.0414\\% \\\\ \\hline\nEagle & 600 $\\times$ 402 & 13.27 & 1952 & 0.4299\\% \\\\ \\hline\n\\end{tabular}\n\\bigbreak\n\\caption{Performance of our {\\em TRIM} algorithm. Here, the compression rate is defined by $\\frac{\\text{\\# of triangle nodes}}{\\text{\\# of pixels}} \\times 100\\%$.}\n\\label{table:trim}\n\\end{table*}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.48\\textwidth]{time_graph.png}\n \\caption{The relationship of the desired coarse triangulation size and the computational time of our proposed {\\it TRIM} algorithm.}\n \\label{fig:triangulation_time}\n\\end{figure}\n\nIt is noteworthy that the combination of the steps in our {\\em TRIM} algorithm is important for achieving a coarse triangulation. More specifically, if certain steps in our algorithm are removed, the triangulation result will become unsatisfactory. Figure \\ref{fig:segmentation} shows two examples of triangulations created by our entire {\\em TRIM} algorithm and by our algorithm with the segmentation step excluded. It can be easily observed that without the segmentation step, the resulting triangulations are extremely dense and hence undesirable for  simplifying further computations. By contrast, the number of triangles produced by our entire {\\em TRIM} algorithm is significantly reduced. The examples highlight the importance of our proposed combination of steps in the {\\em TRIM} algorithm for content-aware coarse triangulation.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.22\\textwidth]{bee_seg_923.png}\n \\includegraphics[width=0.22\\textwidth]{bee_no_seg_3612.png}\n \\includegraphics[width=0.22\\textwidth]{bird_seg_1496.png}\n \\includegraphics[width=0.22\\textwidth]{bird_no_seg_8685.png}\n \\caption{The triangulations created by our entire {\\em TRIM} algorithm (left) and by the algorithm without the segmentation step (Right). It can be observed that the segmentation step is crucial for achieving a coarse triangulation. Number of triangles produced: 923 (top left), 3612 (top right), 1496 (bottom left), 8685 (bottom right).}\n \\label{fig:segmentation}\n\\end{figure}\n\\subsection{The improvement in efficiency of image registration by our {\\em TRIM} algorithm}\nIn this subsection, we demonstrate the effectiveness of our proposed triangulation-based method for landmark-based image registration. In our experiments, the feature points on the images are extracted using the algorithm by Harris and Stephens \\cite{harris88} as landmark constraints. For simplifying the image registration problems, one conventional approach is to make use of coarse regular grids followed by interpolation. It is natural to ask whether our proposed coarse triangulation-based method produces better results. In Figure \\ref{fig:registration_teddy}, we consider a stereo registration problem of two scenes. With the prescribed feature correspondences, we compute the feature-endowed stereo registration via the conventional grid-based approach, the DMesh triangulation approach \\cite{dmesh} and our proposed {\\it TRIM} method. For the grid-based approach and the DMesh triangulation approach \\cite{dmesh}, we take the mesh vertices nearest to the prescribed feature points on the source image as source landmarks. For our proposed {\\it TRIM} method, as the landmark vertices are automatically embedded in the content-aware coarse triangulation, the source landmarks are exactly the feature points detected by the method in \\cite{harris88}.\n\nIt can be observed that our triangulation-based approach produces a much more natural and accurate registration result when compared with both the grid-based approach and the DMesh triangulation approach. In particular, sharp features such as edges are well preserved using our proposed method. By contrast, the edges are seriously distorted in the other two methods. In addition, the geometry of the background in the scenes are well retained via our {\\em TRIM} method but not the other two methods. The higher accuracy of the registration result by our approach can also be visualized by the intensity difference plots. Our triangulation-based approach results in an intensity difference plot with more dark regions than the other two approaches. The advantage of our method over the other two methods is attributed to the geometry preserving feature of our {\\it TRIM} algorithm, in the sense that the triangulations created by {\\it TRIM} are more able to fit into complex features and have more flexibilities in size than regular grids. Also, the triangulations created by DMesh \\cite{dmesh} do not capture the geometric features and hence the registration results are unsatisfactory. They reflect the significance of our content-aware {\\it TRIM} triangulation scheme in computing image registration. Another example is illustrated in Figure \\ref{fig:registration_cone}. Again, it can be easily observed that our proposed {\\em TRIM} triangulation approach leads to a more accurate registration result.\n\nTo highlight the improvement in the efficiency by our proposed {\\em TRIM} algorithm, Table \\ref{table:registration} records the computational time and the error of the registration via the conventional grid-based approach and our {\\it TRIM} triangulation-based approach. It is noteworthy that our proposed coarse triangulation-based method significantly reduces the computational time by over 85\\% on average when compared with the traditional regular grid-based approach. To quantitatively assess the quality of the registration results, we define the matching accuracy by\n\\begin{equation}\n\\frac{\\text{\\# pixels s.t.} \\|\\text{final intensity - original intensity}\\|_1 < \\epsilon}{\\text{Total \\# of pixels}} \\times 100\\%.\n\\end{equation}\nThe threshold $\\epsilon$ is set to be $0.2$ in our experiments. Our triangulation-based method produces registration results with the matching accuracy higher than that of the regular grid-based method by 3\\% on average. The experimental results reflect the advantages of our {\\it TRIM} content-aware coarse triangulations for image registration\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.27\\textwidth]{bear_1.png} \\\n \\includegraphics[width=0.27\\textwidth]{bear_2.png} \\\n \\includegraphics[width=0.27\\textwidth]{bear_registration.png}\n \\newline\n \\newline\n \\includegraphics[width=0.27\\textwidth]{bear_grid_result.png} \\\n \\includegraphics[width=0.27\\textwidth]{dmesh_bear.png} \\\n \\includegraphics[width=0.27\\textwidth]{bear_result.png}\n \\newline\n \\newline\n \\includegraphics[width=0.27\\textwidth]{bear_difference_map_grid.png} \\\n \\includegraphics[width=0.27\\textwidth]{bear_difference_map_dmesh.png} \\\n \\includegraphics[width=0.27\\textwidth]{bear_difference_map_TRIM.png}\n   \\newline\n \\caption{Stereo landmark registration using different algorithms. Top left: The source image. Top middle: The target image. Top Right: The prescribed feature correspondences. Middle left: The registration result by the dense grid-based approach (4 pixels per grid). Middle: The registration result via DMesh \\cite{dmesh}. Middle right: The registration result by our proposed coarse triangulation-based method. Bottom left: The intensity difference after the registration by the dense grid-based approach. Bottom middle: The intensity difference after the registration via DMesh \\cite{dmesh}. Bottom right: The intensity difference after the registration by our proposed coarse triangulation-based method. Because of the greater flexibility of our triangulation scheme, the accuracy of registration by our method is higher than that of the dense grid-based approach.}\n \\label{fig:registration_teddy}\n\\end{figure*}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.27\\textwidth]{cone_1.png} \\\n \\includegraphics[width=0.27\\textwidth]{cone_2.png} \\\n \\includegraphics[width=0.27\\textwidth]{cone_registration.png}\n \\newline\n \\newline\n \\includegraphics[width=0.27\\textwidth]{cone_grid_result.png} \\\n \\includegraphics[width=0.27\\textwidth]{dmesh_cone.png} \\\n \\includegraphics[width=0.27\\textwidth]{cone_result.png}\n  \\newline\n  \\newline\n \\includegraphics[width=0.27\\textwidth]{cone_difference_map_grid.png} \\\n \\includegraphics[width=0.27\\textwidth]{cone_difference_map_dmesh.png} \\\n \\includegraphics[width=0.27\\textwidth]{cone_difference_map_TRIM.png}\n   \\newline\n \\caption{Stereo landmark registration using different algorithms.  Top left: The source image. Top middle: The target image. Top Right: The prescribed feature correspondences. Middle left: The registration result by the dense grid-based approach (4 pixels per grid). Middle: The registration result via DMesh \\cite{dmesh}. Middle right: The registration result by our proposed coarse triangulation-based method. Bottom left: The intensity difference after the registration by the dense grid-based approach. Bottom middle: The intensity difference after the registration via DMesh \\cite{dmesh}. Bottom right: The intensity difference after the registration by our proposed coarse triangulation-based method.}\n \\label{fig:registration_cone}\n\\end{figure*}\n\\begin{table*}[h!]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\nImages & Size & \\multicolumn{4}{c|}{Registration} & Time saving rate \\\\ \\cline{3-6}\n& & \\multicolumn{2}{c|}{Via regular grids} & \\multicolumn{2}{c|}{Via {\\em TRIM}} & \\\\ \\cline{3-6}\n& & Time (s) & Matching accuracy (\\%) & Time (s) & Matching accuracy (\\%) & \\\\ \\hline\nBear & 450 $\\times$ 375   & 102.3 & 59.5 & 13.8 & 70.7 & 13.4897\\% \\\\ \\hline\nCone & 450 $\\times$ 375   & 108.7 & 51.3 & 28.2 & 61.2 & 25.9430\\%\\\\ \\hline\nCloth& 1252 $\\times$ 1110 & 931.0 & 70.7 & 36.0 & 75.4 & 3.8668\\% \\\\ \\hline\nBook & 1390 $\\times$ 1110 & 1204.5 & 59.0 & 51.0 & 63.0 & 4.2341\\% \\\\ \\hline\nBaby & 1390 $\\times$ 1110 &  94.3  & 62.3 & 11.0 & 62.3 & 11.6649\\% \\\\ \\hline\n\n\\end{tabular}\n\n\\bigbreak\n\\caption{The performance of feature-based image registration via our proposed {\\em TRIM} coarse triangulation method and the ordinary coarse grids. Here, the time saving rate is defined by $\\frac{\\text{Registration time via {\\em TRIM}}}{\\text{Registration time via regular grids}} \\times 100\\%$.\n\\label{table:registration}\n\\end{table*}\n\n\n\\section{Conclusion and future work} \\label{conclusion}\nIn this paper, we have established a new framework for accelerating image registration. Using our proposed {\\em TRIM} algorithm, a content-aware coarse triangulation is built on a high resolution image. Then, we can efficiently compute a landmark-based registration using the coarse triangulation. From the coarse registration result, we can obtain the desired fine registration result. Our proposed method is advantageous for a large variety of registration applications with a significant improvement of the computational efficiency and registration accuracy. Hence, our proposed method can serve as an effective initialization for various registration algorithms. In the future, we aim to extend our proposed {\\em TRIM} algorithm in the following two directions.\n\n\\bigbreak\n\n\\subsubsection{n-dimensional registration via {\\em TRIM}}\nOur work can be naturally extended for higher dimensional registration problems. For accelerating $n$-dimensional volumetric registration, one can consider constructing a coarse representation using $n$-simplices. Analogous to our 2-dimensional {\\em TRIM} algorithm, the $n$-dimensional coarse representation should also result in a significant improvement of the efficiency for $n$-dimensional registration problems.\n\n\\bigbreak\n\n\\subsubsection{{\\em TRIM} image compression format}\nRecall that in our proposed {\\em TRIM} algorithm, there are two important parameters, namely, the number of intensity threshold levels $l$ in the image segmentation step and the sparse ratio $p$ for controlling the fineness of the triangular representation. By increasing the values of these two parameters, an image intensity representation becomes more accurate. Hence, given a compression error threshold, we aim to develop an iterative {\\em TRIM} scheme for continuously increasing the two values until the intensity difference between the compressed image and the original image is smaller than the prescribed error threshold. This provides us with a new image compression format using {\\em TRIM}. Since the {\\em TRIM} compression format only consists of the coarse triangular mesh vertices and the corresponding colors, the compression size is much smaller than that of the existing image compression formats.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nNear term quantum devices have a small number of noisy qubits that can support execution of shallow depth circuits (i.e., those with few operational cycles) only. Variational Quantum Algorithms (VQA) aim to leverage the power as well as the limitations imposed by these devices to solve problems of interest such as combinatorial optimization \\cite{farhi2014quantum,wang2018quantum,hadfield2019quantum,cook2020quantum}, quantum chemistry \\cite{mcclean2016theory, grimsley2019adaptive}, and quantum machine learning \\cite{10.1007\/978-3-030-50433-5_45,biamonte2017quantum,torlai2020machine}. VQA divides the entire computation into functional modules, and outsources some of these modules to classical computers. The general framework of VQA can be divided into four steps: (i) encode the problem into a parameterized quantum state $\\ket{\\psi(\\theta)}$ (called the ansatz), where $\\theta = \\{\\theta_1,\\theta_2, \\hdots, \\theta_k\\}$ are $k$ parameters; (ii) prepare and measure the ansatz in a quantum computer, and determine the value of some objective function $C(\\theta)$ (which depends on the problem at hand) from the measurement outcome; (iii) in a classical computer, optimize the set of parameters to find a better set $\\theta' = \\{\\theta'_1,\\theta'_2, \\hdots, \\theta'_k\\}$ such that it minimizes (or maximizes) the objective function; (iv) repeat steps (ii) and (iii) with the new set of parameters until convergence.\n\nQuantum Approximate Optimization Algorithm (QAOA) is a type of VQA that focuses on finding good approximate solutions to combinatorial optimization problems. It has been studied most widely for finding the maximum cut of a (weighted or unweighted) graph (called the Max-Cut problem) \\cite{farhi2014quantum}. For this problem, given a graph $G = (V,E)$ where $V$ is the set of vertices and $E$ is the set of edges, the objective is to partition $V = V_1 \\cup V_2$, such that $V_1 \\cap V_2 = \\phi$, and the number of edges crossing the partition is maximized. Throughout this paper, we shall consider {\\it connected graphs} with $|V| = n$ and $|E| = m$, but the results can be easily extended to disconnected graphs as well.\n\nIn the initial algorithm proposed by Farhi \\cite{farhi2014quantum} for the Max-Cut problem, a depth-$p$ QAOA consists of $p \\geq 1$ layers of alternating operators on the initial state $\\ket{\\psi_0}$ \n\\begin{equation}\n\\label{eq:ansatz}\n    \\ket{\\psi(\\gamma,\\beta)} = ( \\displaystyle \\Pi_{l = 1}^{p} e^{(-i\\beta_l H_M)} e^{(-i\\gamma_l H_P)}) \\ket{\\psi_0}\n\\end{equation}\n\nwhere $H_P$ and $H_M$ are called the Problem and Mixer Hamiltonian respectively, and $\\gamma = \\{\\gamma_1, \\gamma_2, \\hdots, \\gamma_p\\}$ and $\\beta = \\{\\beta_1, \\beta_2, \\hdots, \\beta_p\\}$ are the parameters. It is to be noted that the depth $p$ of the QAOA is not related to the depth of the quantum circuit realizing the algorithm. The problem Hamiltonian describing the Max-Cut can be represented as in Eq.~(\\ref{eq:max_cut}), where $w_{jk}$ is the weight associated with the edge $(j,k)$.\n\\begin{equation}\n\\label{eq:max_cut}\n    H_P = \\frac{1}{2}\\displaystyle \\sum_{(j,k) \\in E} w_{jk} (I - Z_j Z_k)\n\\end{equation}\n\nFurthermore, the mixer Hamiltonian should be an operator that does not commute with the Problem Hamiltonian. In the traditional QAOA, the mixer Hamiltonian is $H_M = \\displaystyle \\sum_{i} X_i$.\n\nVariations to this have been studied to improve the performance of the algorithm --- such as using other mixers \\cite{bartschi2020grover,zhu2020adaptive, yu2021quantum}, training the parameters to reduce the classical optimization cost \\cite{larkin2020evaluation}, and modifying the cost function for faster convergence \\cite{barkoutsos2020improving}. In this paper we stick to the original problem and mixer hamiltonians proposed in the algorithm by Farhi et al. \\cite{farhi2014quantum}. The applicability and effectiveness of our proposed method on the modifications of this algorithm can be looked at as a follow-up work. However, our proposed methods optimize the circuit corresponding to the problem hamiltonian. Since most of the modifications suggested in the literature aim to design more efficient mixers, our proposed optimization should be applicable on those as well.\n\nThe realization of the QAOA circuit for Max-cut requires two CNOT gates for each edge (details given in Sec.~\\ref{sec:ansatz}). Hardware realization of a CNOT gate is, in general, significantly more erroneous than a single qubit gate. Even in the higher end devices of IBM Quantum, such as \\textit{ibmq\\_montreal}, \\textit{ibmq\\_manhattan}, the probability of error for a single qubit gate and a CNOT gate are $\\mathcal{O}(10^{-4})$ and $\\mathcal{O}(10^{-2})$, respectively~\\cite{ibmquantum}. In other words, CNOT gates are $100$ times more likely to be erroneous than single qubit gates. Therefore, we focus primarily on reducing the number of CNOT gates in the design of QAOA ansatz for Max-cut.\n\n\\subsubsection*{Contributions of this paper}\n\nIn this paper, we\n\n\\begin{enumerate}[(i)]\n    \\item propose two optimization methods for reducing the number of CNOT gates in the first layer of the QAOA ansatz based on  (1) an Edge Coloring that can reduce upto $\\lfloor \\frac{n}{2} \\rfloor$ CNOT gates, and (2) a Depth First Search (DFS) that can reduce $n-1$ CNOT gates.\n    \n    \\item prove that there exists no method that can reduce more than $n-1$ CNOT gates while still maintaining a fidelity of 1 with the original QAOA ansatz \\cite{farhi2014quantum}.\n    \n    \\item show that while the Edge Coloring based optimization reduces the depth of the circuit, the DFS based method may increase the depth. We further analytically derive the criteria (involving the increase in the depth and the reduction in the number of CNOT gates) for which the DFS based optimization method still leads to a lower probability of error in the circuit, and show that the IBM Quantum Hardwares \\cite{ibmquantum} conform to that criteria.\n    \n   \n    \n    \\item simulate our proposed optimization methods in Qiskit~\\cite{Qiskit} with the \\textit{ibmq\\_manhattan} noise model and show that for graphs of different sparsity (Erdos-Renyi graphs with the probability of edge varying from 0.4 - 1)\n    \\begin{enumerate}\n        \\item the proposed reduction in the CNOT gate is still retained post transpilation\n        \\item the DFS based method has lower error probability than the Edge Coloring method, which in its turn has lower error probability than the traditional QAOA ansatz.\n    \\end{enumerate}\n\\end{enumerate}\n\nTherefore, for any graph $G = (V,E)$, our proposed method provides reduction in the number of CNOT gates, and hence lowers the error probability of the circuit. Although the DFS method provably surpasses the Edge Coloring method, both in terms of reduction in CNOT gates and lowering the error probability, the latter reduces the depth of the QAOA circuit, and is also used in the DFS based method to arrange the edges which do not form a part of the DFS tree.\n\nFor the rest of this paper, we consider \\emph{unweighted and connected graphs}, $i.e.$, $w_{jk} = 1$, $\\forall$ $(j,k) \\in E$. However, the circuit corresponding to the ansatz does not change if we have a weighted graph \\cite{hadfield2019quantum}. Therefore, every analysis in this paper holds for a weighted graph as well. Furthermore, the analysis of this paper will hold as it is, or with some minimal modification, for disconnected graphs as well.\n\nThe rest of the paper is organized as follows - Section~\\ref{sec:ansatz} briefly discusses the traditional QAOA ansatz design. In Section~\\ref{sec:thm} we provide the proposed optimization and the criteria for it. Section~\\ref{sec:edge_col} and ~\\ref{sec:dfs} describe two methods of optimization based on Edge Coloring and DFS respectively. We provide the respective algorithms and analyze the conditions under which each one reduces the probability of error. We present the results of our simulation in section~\\ref{sec:sim} and conclude in Section~\\ref{sec:con}.\n\n\\section{Traditional ansatz design for QAOA}\n\\label{sec:ansatz}\n\nThe objective function of a depth-$p$ QAOA for Max-Cut~\\cite{farhi2014quantum} can be expressed as\n\\begin{equation}\n\\label{eq:objective}\n    \\max_{\\psi(\\gamma,\\beta)} \\bra{\\psi(\\gamma,\\beta)} H_P \\ket{\\psi(\\gamma,\\beta)}\n\\end{equation}\nwhere $\\gamma = \\{\\gamma_1, \\gamma_2, \\hdots, \\gamma_p\\}$ and $\\beta = \\{\\beta_1, \\beta_2, \\hdots, \\beta_p\\}$ are the parameters. The trial wavefunction $\\ket{\\psi(\\gamma,\\beta)}$ is called the ansatz. The QAOA ansatz has a fixed form as described in Eq.~(\\ref{eq:ansatz}). The initial state $\\ket{\\psi_0}$ is usually the equal superposition of $n$ qubits, where $n = |V|$. Note that the depth of the circuit required to prepare $\\ket{\\psi_0}$ is 1 (Hadamard gates acting simultaneously on all the qubits). Similarly, for each layer of QAOA, the operator $exp(-i \\beta_l H_M)$ can be realized by a depth one circuit of $R_x(\\beta_l)$ gates acting simultaneously on all the qubits.\n\nThe operator $exp(-i \\gamma_l H_P)$ has a more costly implementation. Note that\n\\begin{eqnarray*}\nexp(-i \\gamma_l H_P) = \\displaystyle \\Pi_{(i,j) \\in E} exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right).\n\\end{eqnarray*}\n\nThe operator $exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right)$ acts on each edge $(j,k)$, and is realized as shown below:\n\\begin{figure}[H]\n\\centering\n\t\\begin{quantikz}\n\t\t{q_{j}}&&\\ctrl{1} & \\qw & \\ctrl{1} & \\qw \\\\\n\t\t{q_{k}}&&\\targ{} & \\gate{R_z(2\\gamma_l)} & \\targ{} & \\qw\n\t\\end{quantikz}\n\t\\label{fig:z_jz_k}\n\\end{figure}\n\nHere, $q_j$ and $q_k$ represent qubit indices $j$ and $k$, respectively. Note that Max-Cut is a symmetric problem, and therefore, the selection of control and target from qubits $q_j$ and $q_k$ for the CNOT gate corresponding to the edge $(j,k)$ is irrelevant, $i.e.$ the operator $exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right)$ can be equivalently realized as $CNOT_{kj} (I_k \\otimes R_z(2\\gamma_l)_{j})$ $CNOT_{kj}$. In Fig.~\\ref{fig:depth_qaoa}(a) and (b), we show a 2-regular graph with four vertices and its corresponding QAOA circuit for $p = 1$ respectively.\n\n\\begin{figure}[htb]\n     \\centering\n     \\begin{subfigure}[b]{0.3\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.35]{graph.png}\n         \\caption{A 2-regular graph with four vertices}\n         \\label{col}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.4\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.4]{qaoa_circ.png}\n         \\caption{Max-Cut QAOA circuit for $p=1$  corresponding to the graph}\n         \\label{dfs}\n     \\end{subfigure}\n    \\caption{The Max-Cut QAOA circuit for $p=1$  corresponding to the 2-regular graph with four vertices; the values of $\\gamma$ and $\\beta$ can be arbitrary but those in this figure are the optimum values for this graph when $p = 1$}\n    \\label{fig:depth_qaoa}\n\\end{figure}\n\n\n\\section{Methods for Optimized ansatz design}\n\\label{sec:thm}\n\nSome recent studies have proposed optimization methods for the circuit of the QAOA ansatz with respect to the underlying hardware architecture \\cite{alam2020circuit}. In this paper we propose two \\textit{hardware independent} methods to reduce the number of CNOT gates in the traditional QAOA ansatz. The intuition is that in the circuit realization of the operator $exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right)$ as $CNOT_{jk} (I_j \\otimes R_z(2\\gamma_l)_{k})$ $CNOT_{jk}$, the first CNOT gate can be removed whenever it does not make any contribution to the overall effect of the operator. Our proposed method reduces the number of CNOT gates in the circuit irrespective of the hardware architecture, and hence is applicable for any quantum device.\n\nIn Theorem~\\ref{thm:equiv} we prescribe the condition where the first CNOT gate is irrelevant to the effect of the said operator, and hence may be removed.\n\n\\begin{theorem}\n\\label{thm:equiv}\nLet $\\ket{\\psi}$ be an $n$-qubit state prepared in a uniform superposition (upto relative phase) over all basis states $\\ket{x_1, \\hdots, x_n}$ such that the relative phase on each basis state is a function of a subset $S \\subset$ $\\{1,2,...,n\\}$ of the $n$ qubits (and independent of remaining qubits) $i.e.$\n\\begin{center}\n    $\\ket{\\psi} = \\frac{1}{\\sqrt{2^n}}\\displaystyle \\sum_{x_1,...,x_n} e^{(i \\phi(x_S))}\\ket{x_1,...,x_n}$\n\\end{center}\nwhere $x_S = \\{x_i : i \\in S\\}$ and $\\phi(x_S)$ depicts the relative phase of each superposition state. For any two qubits $\\ket{j}$ and $\\ket{k}$, where $\\ket{k} \\notin S$, and for the two operators $U_1 = CNOT_{jk} (I_j \\otimes R_z(2\\gamma_l)_{k}) CNOT_{jk}$ and $U_2 = (I_j \\otimes R_z(2\\gamma_l)_{k}) CNOT_{jk}$, we have \n\\begin{center}\n    $U_1\\ket{\\psi} = U_2\\ket{\\psi}$.\n\\end{center}\n\\end{theorem}\n\n\\begin{proof}\nLet us consider the action of the operators $U_1$ and $U_2$ on any edge $(j,k)$.\n\\begin{equation}\nU_1\\ket{\\psi} = CNOT_{jk} (I_j \\otimes R_z(2\\gamma_l)_{k}) (CNOT_{jk}) \\ket{\\psi} \\nonumber\n\\end{equation}\n\\begin{align}\n=&\\displaystyle \\sum_{x_1,...,x_n} CNOT_{jk} (I_j \\otimes R_z(2\\gamma_l)_{k}) (CNOT_{jk})\\nonumber \\\\\n& e^{i\\phi(x_S)} \\ket{x_1,...,x_n}  \\\\\n=&\\displaystyle \\sum_{x_1,...,x_n} CNOT_{jk} (I_j \\otimes R_z(2\\gamma_l)_{k}) \\nonumber \\\\\n& e^{i \\phi(x_S)} \\ket{x_1,..,x_k'=x_j \\oplus x_k,.,x_n}  \\\\\n=&\\displaystyle  \\sum_{x_1,...,x_n} e^{i(\\phi(x_S)- \\gamma_l (x_j \\oplus x_k))} CNOT_{jk}  \\nonumber \\\\\n&  \\ket{x_1,..,x_k'=x_j \\oplus x_k,.,x_n}  \\\\\n\\label{eq:u1}\n=& \\displaystyle \\sum_{x_1,...,x_n} e^{i(\\phi(x_S) - \\gamma_l (x_j \\oplus x_k))}\\ket{x_1,...,x_n}\n\\end{align}\nwhere $e^{i \\phi(x_S)}$ is the cumulative effect of operators acting on previous edges (= 0 if $(j,k)$ is the first). We have dropped the normalization constant for brevity.\n\nSimilarly,\n\\begin{equation}\n    U_2\\ket{\\psi} = CNOT_{jk} (I_j \\otimes R_z(2\\gamma_l)_{x_k}) \\ket{\\psi} \\nonumber\n\\end{equation}\n\\begin{equation}\n    = CNOT_{jk} \\displaystyle \\sum_{x_1,...,x_n} e^{i((\\phi(x_S)) - \\gamma_l x_k)}\\ket{x_1,...,x_n} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\label{eq:u2_mid}\n    = \\displaystyle \\sum_{x_1,...,x_n} e^{i((\\phi(x_S)) - \\gamma_l x_k)} \\ket{x_1,..,x_j \\oplus x_k,..,x_n}\n\\end{equation}\n\nwhere the qubit in $k^{\\text{th}}$ position changes to $x_j \\oplus x_k$ due to the $CNOT_{jk}$ operation. Now, substituting $x_k' = x_j \\oplus x_k$ in the above equation, we get\n\n\\begin{equation}\n    U_2\\ket{\\psi} = \\displaystyle \\sum_{x_1,...,x_n} e^{i((\\phi(x_S)) - \\gamma_l x_k)} \\ket{x_1,..,x_j \\oplus x_k,..,x_n} \\nonumber\n\\end{equation}\n\\begin{equation}\n    = \\displaystyle \\sum_{x_1,..,x_k',..,x_n} e^{i((\\phi(x_S)) - \\gamma_l (x_j \\oplus x_k'))} \\ket{x_1,..,x_k',..,x_n} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\label{eq:u2}\n    = \\displaystyle \\sum_{x_1,..,x_k,..,x_n} e^{i((\\phi(x_S)) - \\gamma_l (x_j \\oplus x_k))} \\ket{x_1,..,x_k,..,x_n}\n\\end{equation}\n\nHere since $k \\notin S$, the substitution in second last step, does not change the phase $e^{i \\phi(x_S)}$. The last step is valid since $x_k'$ is a running index and hence can be changed to $x_k$. Thus Eq.~(\\ref{eq:u1}) and Eq.~(\\ref{eq:u2}) are identical.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:cond}\nFor a graph $G$, we can optimize the circuit for the operator $exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right)$ corresponding to an edge $(j,k)$ replacing $U_1$ by $U_2$,  provided that the target vertex does not occur in any of the edge operators run earlier. In other words, the following conditions are sufficient to optimize an edge:-\n\\begin{enumerate}\n    \\item if the vertex $j$ is being operated on for the first time, then it acts either as a control or a target for the CNOT gate corresponding to the operator;\n    \\item the vertex $j$ does not act as a target of the CNOT gate if it occurs as a part of any other edge operators run earlier.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\n\nThe first time we consider an edge adjacent to a vertex $j$, where $j \\notin x_S$, (see Theorem~\\ref{thm:equiv}) the relative phase $\\phi(x_S)$ does not depend on $j$. Thus it satisfies the condition of Theorem~\\ref{thm:equiv} and allows optimization of the operator.\n\n On the other hand, if the vertex $j$ occurs as part of an edge operator already run, the phase on the basis state $\\phi$ can potentially depend on $S$, $i.e.$ $j \\in S$. By not allowing it to act as target, we satisfy the conditions of Theorem~\\ref{thm:equiv}.\n\\end{proof}\n\nFrom the above discussion, it follows that if we arbitrarily choose edges for applying the operator $exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right)$, then it cannot be guaranteed that a large number of edges will conform to  Corollary~\\ref{cor:cond}. The requirement, in fact, imposes a precedence ordering among the edges. In Section~\\ref{sec:edge_col} and ~\\ref{sec:dfs}, we provide two algorithmic procedures for maximizing the number of edges that satisfy the requirement in order to reduce the number of CNOT gates in the ansatz.\n\nFor the rest of the paper, we say that {\\it an edge is optimized} if the operator $U_2$ can be operated on that edge instead of $U_1$.\n\n\\section{Edge Coloring based Ansatz Optimization }\n\\label{sec:edge_col}\nThe total error one incurs in a circuit depends on the number of operators (since a larger number of operators tend to incur more error) and the depth of the circuit (corresponding to relaxation error). In this section, we discuss how one can minimize the depth of the circuit. We also discuss the possibility of reduction in CNOT gates in the depth optimized circuit. \n\nThe operators $H_M$ act on distinct qubits and hence can be run in parallel contributing to a depth of 1 (for each step of the QAOA). On the other hand, the operators in $H_P$ can potentially contribute a lot to depth since the edge operators do not act on disjoint vertices.\nAt a given level of the circuit, we can only apply edge operators corresponding to a vertex disjoint set of edges. Thus the minimum depth of the circuit will correspond to the minimum value $k$ such that we can partition the set of edges $E$ as a disjoint union $\\cup_i E_i$ where each subset $E_i$ consists of a vertex disjoint collection of edges. This in turn corresponds to the edge coloring problem in a graph.\n\nGiven a graph $G = (V,E)$ and a set of colors $\\chi' = \\{\\chi'_1, \\chi'_2, \\hdots, \\chi'_k\\}$, an edge coloring \\cite{west2001introduction} assigns a color to each edge $e \\in E$, such that any two adjacent edges ($i.e.$, edges incident on a common vertex) must be assigned distinct colors. The edge coloring problem comprises of coloring the edges using the minimum number of colors $k$. Note that the operators corresponding to edges having the same color can therefore be executed in parallel.\nMoreover,\n\\begin{enumerate}\n    \\item the number of colors in optimal coloring, called the chromatic index, corresponds to the minimum depth of the circuit;\n    \\item edges having the same color corresponds to the operators $exp\\left(-i \\gamma_l \\left(\\frac{I-Z_j Z_k}{2}\\right)\\right)$ that can be executed simultaneously.\n\\end{enumerate}\n\nOptimal edge coloring is an NP-complete problem \\cite{west2001introduction}.  But it is not practical to allocate exponential time to find the optimal edge-coloring as a pre-processing step for QAOA. Vizing's Theorem states that every simple undirected graph can be edge-colored using at most $\\Delta + 1$ colors, where $\\Delta$ is the maximum degree of the graph \\cite{vizing1964estimate}. This is within an additive factor of 1 since any edge-coloring must use at least $\\Delta$ colors. Misra and Gries algorithm \\cite{misra1992constructive} achieves the above bound constructively in $\\mathcal{O}(n\\cdot m)$ time. Therefore, we use the Misra and Gries edge coloring algorithm.  Algorithm~\\ref{alg:edgecol} below computes the sets of edges having the same color using  Misra and Gries algorithm as a subroutine. It returns the largest set $S_{max}$ of edges having the same color in the coloring computed by Misra and Gries algorithm.\n\n\\begin{algorithm}[H]\n\\caption{Edge Coloring based Ansatz Optimization}\n\\label{alg:edgecol}\n\\begin{algorithmic}[1]\n\\REQUIRE A graph $G = (V,E)$.\n\\ENSURE Largest set $S_{max}$ of edges having the same color.\n\\STATE Use the Misra and Gries algorithm to color the edges of the graph $G$.\n\\STATE $S_i \\leftarrow$ set of edges having the same color $i$, $1 \\leq i \\leq \\chi'$.\n\\STATE $S_{max} \\leftarrow$ $max\\{S_1, S_2, \\hdots, S_{\\chi'}\\}$.\n\\STATE Return $S_{max}$.\n\\end{algorithmic}\n\\end{algorithm}\n\nThis edge coloring approach provides the minimum depth achievable for QAOA ansatz using a polynomial time pre-processing. After reducing the depth, we now try to further reduce errors by decreasing the number of CNOT gates.\nRecall that the operators corresponding to edges with the same color can be executed in parallel. We use the operators corresponding to the edges of $S_{max}$ as the first layer of operators. The other layers can be used in any order.\n\n\\begin{lemma}\nEvery edge in the first layer can be optimized according to Corollary~\\ref{cor:cond}.\n\\end{lemma}\n\n\\begin{proof}\nFor every edge $(u,v)$ in the first layer, both the vertices are adjacent to an edge for the first time, $i.e.$, both $u, v \\notin S$. Therefore, it satisfies the criteria of Corollary~\\ref{cor:cond}, and hence can be optimized. In fact, any one of the qubits corresponding to the two vertices can be selected as the control for the CNOT operation.\n\\end{proof}\n\nSome edges in the corresponding layers may be optimized as well. Nevertheless, it is trivial to come up with examples where this is not the case (e.g., a complete graph of 4-vertices). Therefore, in the worst case scenario, only the edges in the first layer can be optimized. However, since this method does not increase the depth of the circuit, it always leads to a more efficient circuit design than the traditional QAOA circuit with lower depth (by 1 since the first layer of CNOT is absent) and fewer CNOT gates.\n\nFor general graphs, the worst case scenario is, therefore, that only the edges in the first layer can be optimized. In the following subsection we provide an analysis on the number of optimized edges using this method.\n\n\\subsection{Lower and upper bound on the number of optimized edges}\nLet us assume that the chromatic index of a graph $G = (V,E)$ is $\\chi'$. Using the Misra and Gries Theorem \\cite{misra1992constructive} we can find a polynomial time coloring using at most $\\Delta + 1$ colors, where $\\Delta$ is the maximum degree of the graph. Therefore, on an average, $\\lceil \\frac{m}{\\Delta + 1} \\rceil$ edges have the same color.\n\nMore precisely, two extreme cases arise: (i) the colors may be uniformly distributed, and the maximum number of edges having the same color is $\\lceil \\frac{m}{\\Delta + 1} \\rceil$; or (ii) one of the colors is used dominantly for most of the edges. Nevertheless, note that for all the edges adjacent to the same vertex, a particular color can be assigned to one of the edges only. Therefore, the dominant color can be used at most on $\\lfloor \\frac{n}{2} \\rfloor$ edges, where $n = |V|$. Therefore, the possible number of optimized edges that can be obtained via the Edge Coloring method is as shown in Eq.~(\\ref{eq:edge_col}).\n\\begin{equation}\n\\label{eq:edge_col}\n    \\lceil \\frac{m}{\\Delta + 1} \\rceil \\leq ~\\# ~Optimized ~Edges \\leq \\lfloor \\frac{n}{2} \\rfloor.\n\\end{equation}\n\n\\section{Depth First Search based Ansatz Optimization}\n\\label{sec:dfs}\nAs the edge coloring based algorithm can optimize at most $\\lfloor \\frac{n}{2} \\rfloor$ edges, in this section, we present a Depth First Search (DFS) based optimization procedure which can optimize $n-1$ edges. Algorithm~\\ref{alg:dfs}, for obtaining the optimized QAOA ansatz, uses the standard DFS algorithm \\cite{cormen2009introduction}, by returning the tree edges or discovery edges forming the DFS tree.\n\nIn this method, we start from the first vertex of the DFS tree. For every edge $e = (u,v)$ in the DFS tree, the vertex $u$ is made the control and $v$ is made the target for the CNOT gate corresponding to that edge. The edges are operated on sequentially one after another, as in the set $E_{dfs}$ (the tree edges). Once every edge in the DFS tree has been operated on, the remaining edges can be executed in any order. In fact, one may opt to use the Edge Coloring method on the remaining edges to obtain the minimum depth of the circuit corresponding to these edges, although CNOT gates cannot be reduced any further.\n\n\\begin{algorithm}[H]\n\\caption{DFS based Ansatz Optimization }\n\\label{alg:dfs}\n\\begin{algorithmic}[1]\n\\REQUIRE A graph $G = (V,E)$.\n\\ENSURE A list $E_{dfs}$ of $n-1$ edges.\n\\STATE $E_{dfs} = \\{\\}$\n\\STATE $u \\leftarrow$ randomly selected vertex from $V$.\n\\STATE Start DFS from the vertex $u$. For every vertex $v$ discovered from its predecessor $v'$, $E_{dfs} = E_{dfs} \\cup (v',v)$.\n\\STATE Return $E_{dfs}$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{theorem}\nEach edge in the DFS tree can be optimized according to Corollary~\\ref{cor:cond}.\n\\label{thm:dfs}\n\\end{theorem}\n\n\\begin{proof}\nWe prove this by the method of induction. Let $u$ be the vertex from which the DFS tree starts. Then $u$ is being operated on for the first time, and, hence, can act both as a control\/target for the CNOT operation corresponding to the first edge (Corollary~\\ref{cor:cond}). Choose $u$ to be the control.\n\n\\textbf{Base case}: If $v$ is the vertex that is discovered from $u$ via the edge $(u,v)$, then choosing $u$ as the control and $v$ as the target satisfies Corollary~\\ref{cor:cond}. Therefore, the edge $(u,v)$ can be optimized.\n\n\\textbf{Induction hypothesis}: Let the DFS tree has been constructed upto some vertex $j$, and every edge $(e_1, e_2)$ in this DFS tree so far can be optimized, $i.e.$ $e_1$ acts as the control and $e_2$ as the target.\n\n\\textbf{Induction step}: Let the next vertex in the DFS tree, that is discovered from some vertex $i$, is $k$. From DFS algorithm, the vertex $i$ must have been discovered in some previous step. Since vertex $k$ was not previously discovered, so $k \\notin x_S$ and hence the edge $(i,k)$ can be optimized if we select $i$ to be the control and $k$ as the target.\n\\end{proof}\n\nTherefore, the DFS based optimization method provides $n-1$ optimized edges, $i.e.$, a reduction in the number of CNOT gates by $n-1$. We now show in Theorem~\\ref{thm:optimal} that this is the maximum number of edges that can be optimized.\n\n\\begin{figure}[H]\n     \\centering\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.4]{col.png}\n         \\caption{Edge Coloring Based Optimization}\n         \\label{col}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.4]{dfs.png}\n         \\caption{Depth First Search Based Optimization}\n         \\label{dfs}\n     \\end{subfigure}\n    \\caption{Depth of the ansatz circuit when using (a) Edge Coloring and (b) DFS based method; edges having same color can be executed simultaneously. The depth of the spanning tree in the DFS based method is 4, compared to depth 2 for the Edge Coloring based method. However, the number of optimized edges in the Edge Coloring based method is 2, while that by the DFS based method is 3.}\n    \\label{fig:depth}\n\\end{figure}\n\n\\begin{theorem}\nOptimization of ansatz for Max-Cut QAOA with p=1, by deletion of the CNOT gate in the first level for an edge of the graph, can be done for no more than $n-1$ edges.\n\\label{thm:optimal}\n\\end{theorem}\n\n\\begin{proof}\nLet us assume that there is some method by which at least $n$ edges can be optimized. Now, the connected subgraph which contains all the $n$ vertices and at least $n$ optimized edges must contain a cycle. Let $(u,v)$ be an edge of this cycle, $i.e.$, if $(u,v)$ is removed then the residual graph is a tree (in case there are $> n$ edges, the removal of edges can be performed recursively till such an edge $(u,v)$ is obtained whose removal makes the residual graph a tree). For this edge $(u,v)$, both the vertices $u$ and $v$ are endpoints of some other optimized edges as well. Therefore, from Corollary~\\ref{cor:cond} both $u$ and $v$ must act as the control for the CNOT gate corresponding to the edge $(u,v)$ in order for this edge to be optimized, which is not possible. Therefore, it is not possible to optimize more than $n-1$ edges.\n\\end{proof}\n\nTherefore, the DFS method is optimal in the number of optimized edges. However, we note that the DFS based method associates an ordering of the edges, $i.e.$, some of the edges which could have been operated on simultaneously, cannot be done so now.\nThis, in turn, can lead to an increase in the depth of the circuit. Hence, a penalty for this method producing optimal reduction in CNOT gates, is that it increases the depth of the circuit. \n\n\\begin{figure}[htb]\n     \\centering\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.37]{qaoa_col.png}\n         \\caption{Optimized circuit by Edge Coloring based method }\n         \\label{col}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.48\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.37]{qaoa_dfs.png}\n         \\caption{Optimized circuit by DFS based method}\n         \\label{dfs}\n     \\end{subfigure}\n    \\caption{Max-Cut QAOA ansatz with $p=1$ corresponding to (a) Edge Coloring and (b) DFS based optimization. In (a), the first CNOT gates of the operators have been deleted. The operators corresponding to $(q_1,q_2)$ and $(q_3,q_0)$ act in parallel. In (b), the first CNOT gates of three operators have been deleted, but the depth has increased.}\n    \\label{fig:opt}\n\\end{figure}\n\nIn Fig.~\\ref{fig:depth}, we show a 2-regular graph with four vertices. In Fig.~\\ref{fig:depth}(a), the depth of the circuit corresponding to the operator $exp(-i\\gamma_l H_P)$ is 2; the edges of the same color can be operated on simultaneously. If the red (or blue) edges form the first layer, then those two edges are optimized. However, if we use the DFS method, with the DFS tree starting from, say, vertex 1, then the edges $(1,2),(2,3)$ and $(3,4)$ can be optimized (Fig.~\\ref{fig:depth}(b)). Now these three edges must be operated on one after another, followed by the fourth edge. Thus the depth of the circuit corresponding to the operator $exp(-i\\gamma_l H_P)$ becomes 4. The circuits corresponding to these two scenarios are depicted in Fig~\\ref{fig:opt}(a) and (b) respectively.\n\n\nThe question, therefore, is whether this increase in depth is always acceptable, even with the increased reduction in the number of CNOT gates as, with increased depth, the circuit becomes more prone to relaxation error.\nNumerical analysis and simulation (Section~\\ref{sec:sim}) establises that although the depth of the circuit is increased, the overall error probability of the circuit is reduced further.\n\n\\subsection{When is the DFS based method useful?}\n\nIn this subsection, we formulate a relation for which the increase in the depth still leads to a lower probability of error for the reduction in the number of CNOT gates. For this analysis, we make an assumption that the error in the circuit arises only from noisy CNOT gates and the depth of the circuit ($i.e.$, the $T_1$ time). Although this assumption is idealistic, the ansatz primarily consists of layers of CNOT gates. Furthermore, in superconductors, $R_z$ gates are executed virtually \\cite{mckay2017efficient}, and hence does not lead to any gate error. Therefore, CNOT is the primary source of gate error and with increasing depth, the qubits become more prone to relaxation error. Therefore, this assumption allows for a simple but powerful model for analyzing the query at hand.\n\nLet us assume that the time duration and the error probability of each CNOT gate is $t_{cx}$ and $p_{cx}$ respectively. Let there be $N$ layers of CNOT operations. Note that although there can be multiple CNOT gates in each layer, the time duration of each layer is $t_{cx}$ only. Therefore, the probability of no error ($i.e.$, the probability that the circuit remains error free) after $N$ layers of operations, considering only relaxation error, is\n    $exp(-\\frac{N t_{cx}}{T_1})$.\n\nLet there be $k$ CNOT gates in the original circuit. Therefore, the probability of no error after the operation of the CNOT gates, considering only CNOT gate error, is\n    $(1 - p_{cx})^k$.\n\nCombining both the sources of the errors, Eq.~(\\ref{eq:no_err}) gives the probability of success ($i.e.$, the probability of no error) after a single cycle of computation of the QAOA ansatz.\n\\begin{equation}\n    \\label{eq:no_err}\n    P_{success} = (1 - p_{cx})^k \\cdot exp(-\\frac{N t_{cx}}{T_1})\n\\end{equation}\n\nHenceforth, $P_{success}$ will refer to the  probability of success ($i.e.$, how close the noisy outcome is to the noise-free ideal outcome) of the ansatz circuit execution for a single run of the algorithm. Note that in QAOA, the ansatz is computed multiple times for multiple cycles, and the objective is to maximize the expectation value of the outcome.\n\nWe further assume that after the optimization using DFS based method, $k_1$ CNOT gates have been reduced leading to an increase in $N_1$ layers of operations. The probability that this optimized circuit remains error-free is given in Eq.~(\\ref{eq:opt_err}).\n\\begin{equation}\n    \\label{eq:opt_err}\n    P^{opt}_{success} = (1 - p_{cx})^{(k-k_1)} \\cdot exp(-\\frac{(N+N_1) t_{cx}}{T_1})\n\\end{equation}\n\nThe optimization is fruitful only when $P^{opt}_{success} \\geq P_{success}$. Note that\n\\begin{center}\n    $P^{opt}_{success} = P_{success} \\cdot exp(-\\frac{N_1 t_{cx}}{T_1}) \/ (1 - p_{cx})^{k_1}$\n\\end{center}\n\nSince both $P^{opt}_{success}$ and $P_{success} \\leq 1$, the required inequality holds only if $exp(- \\frac{N_1 t_{cx}}{T_1}) \/ (1 - p_{cx})^{k_1} \\geq 1$. In other words,\n\\begin{eqnarray}\n\\label{eq:cond1}\nexp(-\\frac{N_1 t_{cx}}{T_1}) &\\geq& (1 - p_{cx})^{k_1} \\nonumber\\\\\n\\Rightarrow N_1 &\\leq& \\lambda \\times k_1, \\nonumber\\\\\n~where && \\lambda = \\frac{-ln(1 - p_{cx}) \\cdot T_1}{t_{cx}}.\n\\end{eqnarray}\n\nThe constant $\\lambda$ is defined in terms of parameters specific to the quantum device.\n\n\n\\subsubsection{Effect of varying $\\lambda$}\nGiven that $\\lambda = f(t_{cx},p_{cx},T_1)$,  we expect the $T_1$ value to increase, and the $t_{cx}$ and $p_{cx}$ values to decrease as technology improves. The value of $\\lambda$ increases for increasing $T_1$ and\/or decreasing $t_{cx}$, whereas it decreases for decreasing $p_{cx}$. Therefore, \n\n\\begin{itemize}\n    \\item If $p_{cx}$, the probability of error for CNOT gates decreases, the optimization becomes less useful since we are increasing the probability of relaxation error, but the reduction in error probability becomes less. As per this observation, for smaller  $\\lambda$, Eq.~(\\ref{eq:cond1}) is satisfied when the increase in depth is reduced as well.\n    \n    \\item Similarly, if (i) $T_1$ increases, then the qubit can retain its conherence for a longer period of time, or (ii)  $t_{cx}$ decreases, then the overall computation time of the circuit decreases as well, and the circuit can allow some relaxation in the depth even if $T_1$ remains unchanged. We observe that for both of these cases, by Eq.~(\\ref{eq:cond1}),  $\\lambda$ increases, thus allowing more increase in depth for a given reduction in the number of CNOT gates.\n\\end{itemize}\n\n\\subsection{Trade-off between depth and reduction in CNOT gates}\n\nIf the DFS based method is not applied, then the number of layers of CNOT gates is equal to the number of color classes (as in Edge Coloring method). The maximum number of color classes is $\\Delta + 1$ (as discussed in the previous section), and hence the maximum depth of the circuit is $\\Delta + 1$ as well. Now, when the DFS based method is applied, the circuit can be divided into two disjoint sets of edges:\n\\begin{enumerate}\n    \\item The set of edges belonging to the DFS tree which can be optimized. The depth of this portion of the circuit is at most $n-1$ ($i.e.$, the depth of the DFS tree). Each of the operators corresponding to these edges contains a single CNOT gate only, and hence the number of CNOT gate layers is $n-1$ as well.\n    \\item The set of edges that do not belong to the DFS tree and hence are not optimized. The operators corresponding to these edges can be applied in any order, but after all the optimized edges. When removing the edges of the DFS tree, the degree of each vertex is reduced by at least 1. Therefore, the maximum degree of the remaining subgraph is at most $\\Delta - 1$. Therefore, the depth of this portion of the circuit will be at most $\\Delta$ (From Misra and Gries Algorithm). Each of the layer in this portion contains 2 CNOT gates, and hence the number of CNOT gate layers is $2\\Delta$.\n\\end{enumerate}\n\nTherefore, the maximum depth of the circuit after applying the DFS based optimization is $n-1 + \\Delta$. In other words, the increase in depth due to this method is given by Eq.~(\\ref{eq:cond2}).\n\\begin{eqnarray}\n\\label{eq:cond2}\n    n-1 + \\Delta - (\\Delta + 1) = n - 2\n\\end{eqnarray}\n\nRecall that the number of CNOT gates reduced due to the DFS method is always $n-1$. Therefore, from Eq.~(\\ref{eq:cond1}) and $~(\\ref{eq:cond2})$, we get\n\\begin{eqnarray}\n\\label{eq:condition}\nn - 2 &\\leq& \\lambda \\cdot (n-1) \\nonumber \\\\\n\\Rightarrow \\lambda &\\geq& \\frac{n-2}{n-1}\n\\end{eqnarray}\n\nIn Table~\\ref{tab:lambda} we show the average value of $\\lambda$ for some IBM Quantum~\\cite{ibmquantum} devices, ranging from the comparatively more noisy \\textit{ibmq\\_melbourne} to the comparatively less noisy \\textit{ibmq\\_manhattan}.\n\n\\begin{table}[htb]\n    \\centering\n    \\caption{Average value of $\\lambda$ for four IBM Quantum machines ~\\cite{ibmquantum}}\n    \\begin{tabular}{|c|c|}\n    \\hline\n    IBM Quantum devices & Avg value of $\\lambda$\\\\\n    \\hline\n    \\textit{ibmq\\_manhattan} & 3.6\\\\\n    \\hline\n    \\textit{ibmq\\_montreal} & 2.47\\\\\n    \\hline\n    \\textit{ibmq\\_sydney} & 3.35\\\\\n    \\hline\n    \\textit{ibmq\\_melbourne} & 2.03\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:lambda}\n\\end{table}\n\nNote that the lower bound on $\\lambda$, $\\frac{n-2}{n-1}$ (Eq.~(\\ref{eq:condition}), is always less than $1$ for all $n$. In the asymptotic limit, $\\frac{n-2}{n-1} \\rightarrow 1$. Thus, the proposed DFS based optimization method leads to a lower error probability on any quantum device for which $\\lambda \\geq 1$. Table~\\ref{tab:lambda} readily shows that the IBM Quantum hardwares conform to this requirement.\n\n\n\n\n\\section{Results of simulation}\n\\label{sec:sim}\n\n\\begin{table*}[t]\n    \\centering\n    \\caption{Comparison of Max-Cut QAOA ansatz circuits post transpilation on \\textit{ibmq\\_manhattan}: (i) Traditional, (ii) Edge coloring and (iii) DFS based optimization}\n    \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n    \\multirow{2}{*}{Graph Family} & \\multirow{2}{*}{\\# qubits} & \\multicolumn{3}{c|}{\\# CNOT gates in Max-Cut QAOA ansatz circuit}\\\\\n    \\cline{3-5}\n    & & Traditional  & Edge coloring & DFS\\\\\n    \\hline\n    \\multirow{6}{*}{Complete graph} & 10 & 90 & 85 & 81\\\\\n    \\cline{2-5}\n    & 20 & 380 & 370 & 361\\\\\n    \\cline{2-5}\n    & 30 & 870 & 855 & 841\\\\\n    \\cline{2-5}\n    & 40 & 1560 & 1540 & 1521\\\\\n    \\cline{2-5}\n    & 50 & 2450 & 2425 & 2401\\\\\n    \\cline{2-5}\n    & 60 & 3540 & 3510 & 3481\\\\\n    \\hline\n    \\multirow{6}{*}{Erdos-Renyi ($p_{edge}$ = 0.8)} & 10 & 70 & 66 & 61\\\\\n    \\cline{2-5}\n    & 20 & 302 & 292 & 283\\\\\n    \\cline{2-5}\n    & 30 & 698 & 683 & 669\\\\\n    \\cline{2-5}\n    & 40 & 1216 & 1197 & 1177\\\\\n    \\cline{2-5}\n    & 50 & 1956 & 1931 & 1907\\\\\n    \\cline{2-5}\n    & 60 & 2822 & 2792 & 2763 \\\\\n    \\hline\n    \\multirow{6}{*}{Erdos-Renyi ($p_{edge}$ = 0.6)} & 10 & 50 & 46 & 41\\\\\n    \\cline{2-5}\n    & 20 & 234 & 225 & 215\\\\\n    \\cline{2-5}\n    & 30 & 504 & 491 & 475\\\\\n    \\cline{2-5}\n    & 40 & 960 & 940 & 921\\\\\n    \\cline{2-5}\n    & 50 & 1504 & 1479 & 1455\\\\\n    \\cline{2-5}\n    & 60 & 2114 & 2085 & 2055 \\\\\n    \\hline\n    \\multirow{6}{*}{Erdos-Renyi ($p_{edge}$ = 0.4)} & 10 & 36 & 31 & 27\\\\\n    \\cline{2-5}\n    & 20 & 164 & 154 & 145\\\\\n    \\cline{2-5}\n    & 30 & 362 & 348 & 333\\\\\n    \\cline{2-5}\n    & 40 & 586 & 566 & 547\\\\\n    \\cline{2-5}\n    & 50 & 950 & 925 & 901\\\\\n    \\cline{2-5}\n    & 60 & 1468 & 1440 & 1409 \\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:cx_count}\n\\end{table*}\n\nIn this section we show the effect of our optimization methods on reducing the probability of error and the CNOT count of QAOA for Max-Cut. We first show that our proposed reduction is retained in the post transpilation circuit, which is executed on the quantum hardware. Next, we run our simulation with the noise model for \\textit{ibmq\\_manhattan} from IBM Quantum; this noise model corresponds to the actual noise in the IBM Quantum Manhattan device which has $65$ qubits and a Quantum Volume of $32$~\\cite{ibmquantum}. For our simulation purpose, we have considered Erdos-Renyi graphs, where the probability that an edge exists between two vertices, $p_{edge}$, varies respectively from 0.4 to 1 (complete graph). The choice of Erdos-Renyi graph allows us to study the performance of these proposed methods for various sparsity of graphs.\n\nThe circuit that we construct is usually not executed as it is in the IBM Quantum hardware. It undergoes a process called transpilation in which\n\n\\begin{enumerate}[(i)]\n    \\item the gates of the circuit are replaced with one, or a sequence of, basis gates which are actually executed in the quantum hardware. The basis gates of the IBM Quantum devices are \\{$CNOT$, $SX$, $X$, $R_z$ and Identity\\} \\cite{ibmquantum},\n    \n    \\item the circuit is mapped to the underlying connectivity (called the coupling map) of the hardware \\cite{bhattacharjee2020survey},\n    \n    \\item the number of gates in the circuit is reduced using logical equivalence \\cite{burgholzer2020advanced}.\n\\end{enumerate}\n\nA natural question, therefore, is whether the reduction in CNOT gates is retained post transpilation. In Table~\\ref{tab:cx_count} we show the number of CNOT gates in the post transpilation circuit for the \\textit{ibmq\\_manhattan} device as the number of vertices is varied from $10 - 60$ for each of the graph family considered. Our results readily show that the proposed optimization in the number of CNOT gates still hold good even in the transpiled circuit. Since the \\textit{ibmq\\_manhattan} device is a 65-qubits device, we show the results upto 60 qubits, but the results show that the trend will continue for higher qubit devices as well, when they become available.\n\n\\begin{figure*}[t]\n     \\centering\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.3]{erdos04.png}\n         \\caption{Erdos-Renyi graphs with $p_{edge} = 0.4$}\n         \\label{col}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.3]{erdos06.png}\n         \\caption{Erdos-Renyi graphs  with $p_{edge} =  0.6$}\n         \\label{dfs}\n     \\end{subfigure}\n     \\newline\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.3]{erdos08.png}\n         \\caption{Erdos-Renyi graphs  with $p_{edge} = 0.8$}\n         \\label{col}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[scale=0.3]{complete.png}\n         \\caption{Complete graphs}\n         \\label{dfs}\n     \\end{subfigure}\n    \\caption{$|\\braket{\\psi|\\psi_e}|^2$ for graphs of various sparsity: Erdos Renyi graphs ($p_{edge} = 0.4,~0.6,~ 0.8$) and complete graphs}\n    \\label{fig:prob_succ}\n\\end{figure*}\n\nLet $\\ket{\\psi}$ be the state obtained from the noise-free (ideal) computation of the QAOA circuit, and the state obtained from noisy computation be $\\ket{\\psi_e}$. The probability of success of the noisy computation, then, is defined as\n\\begin{equation}\n    \\label{eq:graph_succ}\n    P_{succ} = |\\braket{\\psi|\\psi_e}|^2\n\\end{equation}\nIn Fig.~\\ref{fig:prob_succ}(a) - (d) we plot $P_{succ}$ of the traditional QAOA ansatz, Edge Coloring based and the DFS based optimization method for Erdos-Renyi graphs, where $p_{edge}$, the probability that an edge exists between two vertices, varies from 0.4 to 1 (complete graph). The choice of Erdos-Renyi graph allows us to study the performance of these proposed methods for various sparsity of graphs. For each case we vary the number of vertices $n$ from 4 to 12. For each value of $n$ and $p_{edge}$, the results are averaged over 20 input graph instances, and each instance is an average of 100 randomly generated noisy circuits by the simulator model for \\textit{ibmq\\_manhattan} with noise. Our results readily show that the DFS based method outperforms both the Edge Coloring based method and the traditional QAOA in terms of lower error probability.\n\nFrom Table~\\ref{tab:cx_count}, and our simulation results in Fig.~\\ref{fig:prob_succ}(a)-(d), we can infer that the DFS based optimization outperforms the Edge Coloring based optimization, which again, outperforms the traditional QAOA in the reduction in CNOT count, and the probability of error in the circuit in (i) the actual transpiled circuit that is executed on the quantum devices, as well as (ii) in realistic noisy scenario of quantum devices.\n\n\\section{Conclusion}\n\\label{sec:con}\n\nIn this paper we have proposed two methods to reduce the number of CNOT gates in the traditional QAOA ansatz. The Edge Coloring based method can reduce upto $\\lfloor \\frac{n}{2} \\rfloor$ CNOT gates whereas the DFS based method can reduce $n - 1$ CNOT gates. While the former method provides a depth-optimized circuit, the latter method can increase the depth of the circuit. We analytically derive the constraint for which a particular increase in depth is acceptable given the number of CNOT gates reduced, and show that every graph satisfies this constraint. Therefore, these methods can reduce the number of CNOT gates in the QAOA ansatz for any graph. Finally, we show via simulation, with the \\textit{ibmq\\_manhattan} noise model, that the DFS based method outperforms the Edge Coloring based method, which in its turn, outperforms the traditional QAOA in terms of lower error probability in the circuit. The transpiler procedure of Qiskit maps a circuit to the underlying hardware connectivity graph, and some gates are reduced in this process. This transpiled circuit is executed on the real hardware. We show, with the \\textit{ibmq\\_manhattan} coupling map, that the reduction in the number of CNOT gates still holds post transpilation. Therefore, our proposed methods provide a universal way to an improved QAOA ansatz design. On a final note, all the calculations in this paper considers connected graph, but these carry over easily to disconnected graphs as well.\n\n\\section*{Acknowledgement}\n\nWe acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team. In this paper we have used the noise model of \\textit{ibmq\\_manhattan}, which is one of IBM Quantum Hummingbird r2 Processors. \n\n\\section*{Code Availability}\n\nA notebook providing the code to generate the plots of Fig.~\\ref{fig:prob_succ}(a)-(d) is available open source at \\href{https:\/\/github.com\/RitajitMajumdar\/Optimizing-Ansatz-Design-in-QAOA-for-Max-cut}{https:\/\/github.com\/RitajitMajumdar\/Optimizing-Ansatz-Design-in-QAOA-for-Max-cut}.\n\n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nMagnetic Resonance Fingerprinting (MRF)~\\cite{ref:ma2013} is an emerging Quantitative MRI technology for simultaneous measurement of multiple quantitative bio-properties (e.g. T1 and T2 relaxation times, and proton density (PD)) of tissues in a single and time-efficient scan. However, due to the aggressive spatiotemporal subsampling needed for short scan times, the MRF time-series data and consequently the tissue maps usually contain aliasing artefacts. \n\nCompressed sensing reconstruction algorithms based on using analytical image priors (e.g., sparsity, Total Variation and\/or low-rank) have been proposed to address this problem \\cite{ref:mcgivney2014, ref:asslander2018, ref:golbabaee2021-lrtv_mrfresnet}. Recent works e.g.~\\cite{ref:fang2019oct, ref:chen2020} have been focussed around deep learning approaches that utilise spatiotemporal image priors learned from data for artefact removal. These approaches which are conventionally trained end-to-end on pairs of subsampled and clean data, outperform those using analytical image priors and produce excellent results. However, unlike the traditional compressed sensing algorithms, current trained deep learning models are specific to the subsampling processes used during their training and unable to generalise and remove aliasing artefacts from other subsampling processes given at the testing time.\n\nThe contribution of this work is to propose the first deep Plug-and-Play (PnP) iterative reconstruction algorithm for MRF to address this issue, and to demonstrate that this approach effectively adapts to changing acquisition models, specifically, the MRF k-space subsampling patterns. Iterations of the PnP algorithm~\\cite{ref:venkatakrishnan2013, ref:ahmad2020} follow steps of the Alternating Direction Method of Multipliers (ADMM)~\\cite{ref:boyd2011} which is an optimisation method for model-based compressed sensing reconstruction (for imaging applications of deep learning ADMM and competing methods, see~\\cite{ref:ahmad2020, ref:liang2020, ref:romano2017, ref:sun2019, ref:sun2021}). In our work, the spatiotemporal MRF image priors are learned from data through an image denoiser i.e. a Convolutional Neural Network (CNN), that is pre-trained for removing Additive White Gaussian Noise (AWGN), and not any particular subsampling artefact. This CNN denoiser is used as a data-driven shrinkage operator within the ADMM's iterative reconstruction process. The reconstructed (de-aliased) image time-series are then fed to an MRF dictionary matching step~\\cite{ref:ma2013} for mapping tissues' quantitative parameters.\n\n\n\\section{The MRF Inverse imaging Problem}\n\\label{sec:mrf}\n\nMRF adopts a linear spatiotemporal compressive acquisition model:\n\\begin{equation}\n\t\\label{eq:standard_linear_inverse_problem}\n\t\\bm{y} = \\bm{A}\\bm{x} + \\bm{w}\n\\end{equation}\nwhere $ \\bm{y} \\in \\mathbb{C}^{\\,m\\times T}$ are the k-space measurements collected at $T$ temporal frames and corrupted by some bounded noise $\\bm{w}$. The acquisition process i.e. the linear forward operator $ \\bm{A}: \\mathbb{C}^{\\,n\\times t}\\rightarrow \\mathbb{C}^{\\,m\\times T}$ models Fourier transformations subsampled according to a set of temporally-varying k-space locations in each timeframe combined with a temporal-domain compression scheme~\\cite{ref:mcgivney2014,ref:asslander2018,ref:golbabaee2021-lrtv_mrfresnet} for (PCA subspace) dimensionality reduction i.e. $t\\ll T$. $ \\bm{x}\\in \\mathbb{C}^{\\,n\\times t} $ is the Time-Series of Magnetisation Images (TSMI) across $n$ voxels and $t$ dimension-reduced timeframes (channels). Accelerated MRF acquisition implies working with under-sampled data which makes the inversion of~\\eqref{eq:standard_linear_inverse_problem} i.e. estimating the TSMIs $\\bm{x}$ from the compressed MRF measurements $\\bm{y}$, an ill-posed problem.  \n\\vspace{\\baselineskip}\n\\\\\n\\textbf{The Bloch response model:} \nThe TSMIs magnetisation time-responses are the main source of tissue quantification.  At each voxel $v$, the time responses i.e. the fingerprints, are related to the corresponding voxel's bio-properties, namely, the T1 and  T2 relaxation times, through the solutions of the Bloch differential equations $\\mathcal{B}$ scaled by the proton density (PD)~\\cite{ref:ma2013, jiang2015mr}:\n\\begin{equation}\n\t\\label{eq:magnetisation_response}\n\t\\bm{x}_{v} \\approx \\text{PD}_{v} \\, \\mathcal{B} \\! \\left( \\text{T1}_{v}, \\text{T2}_v \\right)\n\\end{equation}\nWhile this relationship could temporally constrain the inverse problem~\\eqref{eq:standard_linear_inverse_problem}, it turns out to be inadequate to make the problem well-posed~\\cite{ref:golbabaee2021-lrtv_mrfresnet}. The model~\\eqref{eq:magnetisation_response} alone does not capture cross-voxel correlations. Spatial-domain image priors that account for this must be further added to resolve the ill-posedness problem. For this we rely on datasets of anatomical quantitative maps i.e. the T1, T2 and PD maps, to create the TSMIs via \\eqref{eq:magnetisation_response}, and train a denoiser model on them to learn the spatiotemporal structures\/priors for $\\bm{x}$. We then use this trained denoiser to iteratively solve~\\eqref{eq:standard_linear_inverse_problem} for any given forward model $\\bm{A}$. This process is detailed in the next section.\n\n\n\\section{image reconstruction Algorithm}\n\\label{sec:algorithms}\n\nWe describe our algorithm to reconstruct artefact-free TMSIs before feeding them to MRF dictionary-matching for quantitative mapping.\n\n\n\\subsection{The PnP-ADMM algorithm}\n\\label{ssec:pnp_admm}\n\nA model-based reconstruction approach to solve inverse problems like (1) would typically lead to an optimisation of the form\n\\begin{equation}\n\t\\label{eq:opt}\n\t\\arg\\min_{x} \\| \\bm{y} - \\bm{A}\\bm{x}\\|_2^2 + \\phi(\\bm{x})\n\\end{equation}\nwhich can be solved by variety of iterative shrinkage algorithms including ADMM~\\cite{ref:boyd2011} and by iterating between a step to minimise the first term and promote the k-space data consistency according to the tested acquisition process, and another shrinkage step according to a regularisation term $\\phi$ to promote certain structural priors on $\\bm{x}$ to combat the ill-posedness of (1). In the Plug-and-Play (PnP) approach~\\cite{ref:venkatakrishnan2013}, the shrinkage term is an AWGN image denoiser $\\bm{f}$ that builds an implicit regularisation for (1). The denoiser model $\\bm{f}$ could be a trained convolutional neural network (CNN) like~\\cite{ref:ahmad2020} that captures the structural image priors for $\\bm{x}$ by removing generic gaussian noise from $\\bm{x}+\\bm{n}$, where $\\bm{n} \\sim\\mathcal{N}(0,\\sigma)$, and the noise power $\\sigma$ is an experimentally-chosen hyperparameter. \n\nWe use the following ADMM based iterations of the PnP algorithm~\\cite{ref:ahmad2020}: $\\bm{v}_0 =\\bm{A}^Hy$,  $\\bm{u}_0=\\mathbf{0}$, $\\forall k, \\text{iteration number} =1,2,...\\!\\!\\!$   \n\\begin{subequations}\n\t\\begin{align}\n\t\\label{eq:pnp_admm_a}\n\t\\bm{x}_k & = \\bm{h} \\left( \\bm{v}_{k-1} - \\bm{u}_{k-1} \\right) \\\\\n\t\\label{eq:pnp_admm_b}\n\t\\bm{v}_k & = \\bm{f} \\left( \\bm{x}_{k} + \\bm{u}_{k-1} \\right) \\\\\n\t\\label{eq:pnp_admm_c}\n\t\\bm{u}_k & = \\bm{u}_{k-1} + \\left( \\bm{x}_{k} - \\bm{v}_{k} \\right)\n\t\\end{align}\n\\end{subequations}\nwhere\n\\begin{equation}\n\t\\label{eq:h}\n\t\\bm{h} \\left( \\bm{z} \\right) = \\arg\\min_{x}  \\|\\bm{y}-\\bm{Ax}\\|_2^2\n\t + \\gamma \\|\\bm{x}-\\bm{z}\\|_2^2 \n\\end{equation}\nThe step \\eqref{eq:pnp_admm_a} enforces the k-space data consistency, where $ \\bm{h} $ is solved using the conjugate gradient algorithm. Here the ADMM's internal convergence parameter is set to $\\gamma=0.05$ and $ \\bm{z} = \\bm{v}_{k-1} - \\bm{u}_{k-1} $.  The step \\eqref{eq:pnp_admm_b} applies the image denoising shrinkage to promote spatiotemporal TSMI priors, and the final step aggregates the two previous steps to update the next iteration. \n\n\n\\subsection{CNN denoiser}\n\\label{ssec:cnn_denoiser}\nOur PnP algorithm is combined with a pre-trained CNN denoiser which is plugged in as $ \\bm{f} $ in \\eqref{eq:pnp_admm_b} to iteratively restore the TMSI using learned image priors. The denoiser has a U-Net shape architecture following that of~\\cite{ref:zhang2021}. Here we modified the network's input and output dimensions to match the number of TSMI's multiple channels (i.e. we used real-valued TSMIs with $t=10$ in our experiments) to enable multichannel spatiotemporal image processing. A noise level map, filled with $\\sigma$ values, of the same dimensions as other channels was appended to the network's input for multi-noise level denoising, following ~\\cite{ref:zhang2021}. Other hidden layers follow exactly the same specifications as~\\cite{ref:zhang2021}. We trained this model using (multichannel) image patches extracted from $\\{(\\bm{x},\\bm{x}+\\bm{n})\\}$  pairs of clean and noise-contaminated TSMIs for various levels $\\sigma$ of AWGN noise. Further, image patches are patch-wise normalised to the [0,1] range. The clean TSMIs were obtained from a dataset of anatomical quantitative maps via (2).  \n\n\n\\section{Numerical Experiments}\n\\label{sec:experiments}\n\n\\textbf{Dataset:} A dataset of quantitative T1, T2 and PD tissue maps of 2D axial brain scans of 8 healthy volunteers across 15 slices each were used in this study\\footnote{Data was obtained from a 3T GE scanner (MR750w system - GE Healthcare, Waukesha, WI) with 8-channel receive-only head RF coil, $230\\times230$ mm$^2$ FOV, $230\\times230$ image pixels, $5$ mm slice thickness, and used a FISP acquisition protocol with $T=1000$ repetitions, the same flip angles as~\\cite{jiang2015mr}, inversion time of 18 ms, repetition time of 10 ms and echo time of 1.8 ms. The groundtruth tissue maps were obtained by the LRTV algorithm~\\cite{ref:golbabaee2021-lrtv_mrfresnet}. }.\nClean (groundtruth) TSMIs were retrospectively simulated from these maps via~\\eqref{eq:magnetisation_response} using the EPG  Bloch model formalism~\\cite{weigel2015extended} and a truncated (accelerated) variant of the FISP MRF protocol~\\cite{jiang2015mr} with $T=200$ repetitions. PCA was applied to obtain $t=10$ channel dimension-reduced TSMI data~\\cite{ref:mcgivney2014}. The TSMIs were real-valued and their spatial resolution was cropped \nfrom $ 230 \\times 230 $ pixels to $ 224 \\times 224 $ pixels for the U-Net. The dataset was split into 105 slices (from 7 subjects) for training and 15 slices (for 1 held-out subject) for testing.\n\\vspace{\\baselineskip}\n\\\\\n\\textbf{Training the CNN denoiser:} The TSMI training data was augmented by random resizing of the spatial resolution. \nThe CNN is a multichannel image-patch denoiser. For this we extracted patches of size $128\\times128\\times10$ with spatial strides of $17$ from the TSMIs. We augmented the patches by flipping across image axes and used random rotations. The patches were then [0,1] normalised. Random AWGN was then added during each iteration of the training algorithm to create pairs of clean and noisy TSMI patches. The CNN was trained following~\\cite{ref:zhang2021} to 500 epochs using L1 loss, Adam optimiser, batch size $16$, initialised weights using Kaiming uniform with no bias, and the learning rate initialised at $10^{-4}$ and halved every 100,000 iterations. Randomly selected levels of AWGN noise from $\\sigma=10^{\\{-4\\,\\text{to}\\,0\\}}$ were used for training the denoiser, following~\\cite{ref:zhang2021}. For the PnP algorithm, the denoiser was tested with five levels of AWGN noise $\\sigma=10^{\\{-4,-3,-2,-1,0\\}}$ from which $\\sigma=10^{-2}$ yielded the best result. A second denoiser trained solely on the optimum noise level found was utilised for our comparisons.\n\\vspace{\\baselineskip}\n\\\\\n\\textbf{Subsampling models:} We simulated two single-coil Cartesian acquisition processes with distinct k-space subsampling patterns: (i) a spiral readout as in~\\cite{ref:chen2020} with rotating patterns across $T=200$ repetitions was used to subsample the $224\\times224$ Cartesian FFT grid across all timeframes, and (ii) k-space multiple rows subsampling pattern i.e. the multi-shot Echo Planar Imaging (EPI)~\\cite{benjamin2019multi}, with shifted readout lines across the timeframes was used for MRF subsampling. \nBoth MRF acquisitions subsampled 771 k-space locations in each timeframe from a total of $224\\times224$, leading to a compression ratio of 65, and were contaminated with AWGN noise of $30$ dB SNR.\n\\vspace{\\baselineskip}\n\\input{.\/sections\/results_fig_tissue_maps_spiral.tex}\n\\input{.\/sections\/results_fig_tissue_maps_epi.tex}\n\\\\\n\\textbf{Tested reconstruction algorithms:} \nWe compared the performance of the proposed PnP-ADMM algorithm to the SVD-MRF~\\cite{ref:mcgivney2014} and LRTV~\\cite{ref:golbabaee2021-lrtv_mrfresnet} TSMI reconstruction baselines. All algorithms incorporated subspace dimensionality reduction ($t=10$). Further they can all adapt to the MRF subsampled acquisition process at testing time. SVD-MRF is a non-iterative approach based on zero-filling reconstruction $\\hat{\\bm{x}}=\\bm{A}^Hy$. The LRTV is an iterative approach to~\\eqref{eq:opt} based on an analytical Total Variation image (prior) regularisation for $\\phi$. The PnP approach uses data-driven image priors learned by the CNN denoiser. The PnP-ADMM  ran for $100$ iterations, used $\\gamma=0.05$ and its conjugate gradient steps ran to the solution tolerance $10^{-4}$. The LRTV ran for $200$ iterations and used regularisation parameter $\\lambda=4\\times10^{-5}$.\n\\vspace{\\baselineskip}\n\\\\\n\\textbf{Quantitative mapping:} Dictionary matching was used for mapping the reconstructed TSMIs to the T1, T2 and PD images. For this an MRF dictionary of $94'777$ atoms (fingerprints) was created using the EPG Bloch response simulations for a logarithmically-spaced grid of $($T1, T2$)\\in[0.01, 6]\\times [0.004,0.6]$ (s). PCA was applied to compress this dictionary's (i.e. the Bloch responses) temporal dimension to a $t=10$ dimensional subspace~\\cite{ref:mcgivney2014}. The same subspace was used for the TSMIs dimensionality reduction and within all tested reconstruction algorithms.\n\\vspace{\\baselineskip}\n\\input{.\/sections\/results_table.tex}\n\\\\\n\\textbf{Evaluation Metrics:} For evaluation of the TSMI reconstruction performance, the Peak Signal-to-Noise-Ratio (PSNR) and Structural Similarity Index Measure (SSIM) averaged across all 10 temporal channels were used. To evaluate tissue maps, the Mean Absolute Error (MAE), PSNR and SSIM were used. Metrics were calculated for all 15 slices of the held-out test subject and averaged.\n\\vspace{\\baselineskip}\n\\\\\n\\textbf{Results and Discussion:} Fig.\\ref{fig:spiral_results}, Fig.\\ref{fig:epi_results} and Table.\\ref{tab:results_table}, shows PnP-ADMM utilising the same CNN denoiser, trained on generic AWGN, can apply to two different forward models with drastically different subsampling patterns. The output is consistent de-aliasing performance for time-series data and consequently the tissue maps (see supplementary material for a comparison of TSMIs).\n\nPnP-ADMM outperforms tested baselines subjectively (Fig.\\ref{fig:spiral_results}, Fig.\\ref{fig:epi_results}) and objectively (Table.\\ref{tab:results_table}) across all tested metrics, for both tested subsampling patterns. Recovering tissue maps from EPI subsampled data is observed to be generally more challenging than the spiral subsampling scheme (because the centre of k-space is densely sampled by spiral readouts), however, as observed the PnP-ADMM algorithm succeeds while other tested baselines fail. The key to the superior performance of PnP-ADMM lies in its ability to utilise spatiotemporal prior information related to the dataset. SVD-MRF utilises only temporal prior information through the use of PCA, while LRTV utilises generic prior information in the form of temporal priors through PCA and spatial priors through Total Variation. The use of dataset specific spatiotemporal priors, learned by a CNN denoiser, is the crux of PnP-ADMM's superior performance.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nA proof-of-concept is proposed for a PnP-ADMM approach using deep CNN image denoisers for multi-parametric tissue property quantification using MRF compressed sampled acquisitions, which may be useful for cases where the measurement acquisition scheme is unknown during training for deep learning methods and\/or may change during testing. \nThe method was validated on simulated data and consistently outperformed the tested baselines. This was possible due to the use of data-driven spatiotemporal priors learnt by a pre-trained CNN denoiser, which were critical for enhancing the reconstruction of the TSMIs. Future work will include a variety of measurement acquisition settings, the use of non-gridded sampling trajectories and prospective in-vivo scans.\n\n\n\\section{Compliance with ethical standards}\n\\label{sec:ethics}\n\nThis research study was conducted retrospectively using anonymised human subject scans made available by GE Healthcare who obtained informed consent in compliance with the German Act on Medical Devices. Approval was granted by the Ethics Committee of The University of Bath (Date. Sept 2021 \/ No. 6568). \n\n\n\\section{Acknowledgments}\n\\label{sec:acknowledgments}\n\nCarolin M. Pirkl and Marion I. Menzel receive funding from the European Union's Horizon 2020 research and innovation programme, grant agreement No. 952172.\n\n\\bibliographystyle{IEEEbib}\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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edge-orderings}\n\n\\author[Chr.~Reiher]{Christian Reiher}\n\\address{Fachbereich Mathematik, Universit\\\"at Hamburg, Hamburg, Germany}\n\\email{Christian.Reiher@uni-hamburg.de}\n\\email{schacht@math.uni-hamburg.de}\n\\email{kevin.sames@uni-hamburg.de} \n\n\\author[V.~R\\\"{o}dl]{Vojt\\v{e}ch R\\\"{o}dl}\n\\address{Department of Mathematics,\nEmory University, Atlanta, USA}\n\\email{vrodl@emory.edu\n\\email{marcelo.tadeu.sales@emory.edu}\n\n\\thanks{The second author is supported by NSF grant DMS 1764385.}\n\n\\author[M.~Sales]{Marcelo Sales}\n\\thanks{The third author is partially supported by NSF grant DMS 1764385.}\n\n\\author[K.~Sames]{Kevin Sames}\n\n\n\n\n\\author[M.~Schacht]{Mathias Schacht}\n\\thanks{The fifth author is supported by the ERC (PEPCo 724903)}\n\n\\begin{abstract}\n\tFor integers $k\\ge 2$ and $N\\ge 2k+1$ there are $k!2^k$ {\\it canonical orderings} of \n\tthe edges of the complete $k$-uniform hypergraph with vertex set $[N] = \\{1,2,\\dots, N\\}$.\n\tThese are exactly the orderings with the property that any two subsets $A, B\\subseteq [N]$ \n\tof the same size induce isomorphic suborderings. We study the associated {\\it canonisation} problem \n\tto estimate, given $k$ and $n$, the least integer $N$ such that no matter how the $k$-subsets \n\tof $[N]$ are ordered there always exists an $n$-element set $X\\subseteq [N]$ whose $k$-subsets \n\tare ordered canonically. For fixed $k$ we prove lower and upper bounds on these numbers \n\tthat are $k$ times iterated exponential \n\tin a polynomial of $n$. \n\\end{abstract}\n\n\\maketitle\n\n\\@ifstar{\\origsection*}{\\@startsection{section}{1}\\z@{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\normalfont\\scshape\\centering\\S}}{Introduction}\\label{sec:intro}\n\nWe consider numerical aspects of an unpublished result of Klaus Leeb from the early seventies\n(see~\\cites{NPRV85, NR17}). For a natural number $N$ we set $[N]=\\{1,2,\\dots,N\\}$. \nGiven a set $X$ and a nonnegative integer $k$ we write $X^{(k)}=\\{e\\subseteq X\\colon |e|=k\\}$ \nfor the set of all $k$-element subsets of $X$.  \n\n\\subsection{Ramsey theory}\nRecall that for any integers $k, n, r\\ge 1$, Ramsey's theorem~\\cite{Ramsey30} informs us \nthat every sufficiently large integer $N$ satisfies the partition relation \n\\begin{equation}\\label{eq:1500}\n\tN\\longrightarrow (n)^k_r\\,,\n\\end{equation}\nmeaning that for every colouring of $[N]^{(k)}$ with $r$ colours there exists a set $X\\subseteq [N]$\nof size~$n$ such that~$X^{(k)}$ is monochromatic. The least number~$N$ validating~\\eqref{eq:1500}\nis denoted by~$\\mathrm{R}^{(k)}(n, r)$. The {\\it negative} partition relation $N\\longarrownot\\longrightarrow (n)^k_r$\nexpresses the fact that~\\eqref{eq:1500} fails. \n\n\nFor colourings $f\\colon [N]^{(k)}\\longrightarrow {\\mathds N}$ with infinitely many colours, however, \none can, in general, not even hope to obtain a monochromatic set of size $k+1$. \nFor instance, it may happen \nthat every $k$-element subset of $[N]$ has its own private colour. \nNevertheless, Erd\\H{o}s and Rado~\\cites{ER52, ER50}\nestablished a meaningful generalisation of Ramsey's theorem to colourings with infinitely many \ncolours, even if the ground sets stay finite. In this context it is convenient to regard such \ncolourings as equivalence relations, two $k$-sets being in the same equivalence class if and only \nif they are of the same colour. \nNow Erd\\H{o}s and Rado~\\cites{ER52, ER50} call an equivalence relation \non $[N]^{(k)}$ {\\it canonical} if for any two subsets $A, B\\subseteq [N]$ of the same size \nthe equivalence relations induced on~$A^{(k)}$ and~$B^{(k)}$ correspond to each other via \nthe order preserving map between $A$ and $B$. So, roughly speaking, an equivalence relation is \ncanonical if you cannot change its essential properties by restricting your attention to a subset. \nIt turns out that for~$N\\ge 2k+1$ there are always exactly $2^k$ canonical equivalence relations \non $[N]^{(k)}$ that can be parametrised by subsets~$I\\subseteq [k]$ in the following natural way. \nLet $x, y\\in [N]^{(k)}$ be two $k$-sets and let $x=\\{x_1, \\dots, x_k\\}$ \nas well as $y=\\{y_1, \\dots, y_k\\}$ enumerate their elements in increasing order. \nWe write $x\\equiv_I y$ if and only if $x_i=y_i$ holds for all~$i\\in I$.\nNow $\\{\\equiv_I\\colon I\\subseteq [k]\\}$ is the collection of all canonical equivalence \nrelations on~$[N]^{(k)}$. The {\\it canonical Ramsey theorem} of Erd\\H{o}s and Rado \nasserts that given two integers~$k$ and~$n$ there exists an integer $\\ER^{(k)}(n)$ such that \nfor every $N\\ge\\ER^{(k)}(n)$ and every equivalence relation~$\\equiv$ on~$[N]^{(k)}$ there exists a \nset $X\\subseteq [N]$ of size $n$ with the property that $\\equiv$ is canonical on~$X^{(k)}$. \n\nThis result sparked the development of {\\it canonical Ramsey theory} in the seventies. \nLet us commemorate some contributions to the area due to {\\sc Ronald Graham}, \nwho was among the main protagonists of Ramsey theory in the $20^{\\mathrm{th}}$ century: Together \nwith Erd\\H{o}s~\\cite{EG80}, he established a canonical version of van der Waerden's theorem, \nstating that if for $N\\ge \\mathrm{EG}(k)$ we colour $[N]$ with arbitrarily many colours, then there  \nexists an arithmetic progression of length $k$ which is either monochromatic or receives~$k$ \ndistinct colours. With Deuber, Pr\\\"omel, \nand Voigt~\\cite{DGPV} he later obtained a canonical version of the Gallai-Witt theorem, \nwhich is more difficult to state --- in $d$ dimensions the canonical colourings \nfor a given configuration $F\\subseteq {\\mathds Z}^d$ are now parametrised by vector \nsubspaces $U\\subseteq {\\mathds Q}^d$ having a basis of vectors of the form $x-y$ with $x, y\\in F$.\nInterestingly, {\\sc Ronald Graham} was also among the small number of co-authors of Klaus \nLeeb~\\cite{GLR, GLK2} and in joint work with Rothschild they settled a famous conjecture of Rota \non Ramsey properties of finite vector spaces.\n\nThe canonisation discussed here concerns linear orderings. We say that a linear \norder $([N]^{(k)}, \\strictif)$ is {\\it canonical} if for any two sets $A, B\\subseteq [N]$\nof the same size the order preserving map from $A$ to $B$ induces an order preserving \nmap from $(A^{(k)}, \\strictif)$ to $(B^{(k)}, \\strictif)$. It turns out that for $N\\ge 2k+1$ \nthere are exactly $k!2^k$ canonical orderings of $[N]^{(k)}$ that can uniformly be parametrised by \npairs $(\\eps, \\sigma)$ consisting of a {\\it sign vector} $\\eps\\in \\{-1,+1\\}^k$ and a \npermutation $\\sigma\\in \\mathfrak{S}_k$, i.e., a bijective map $\\sigma\\colon [k]\\lra [k]$. \n\n\\begin{definition}\\label{def:canonical}\n\tLet $N, k\\ge 1$ be integers, and let $(\\eps, \\sigma)\\in \\{+1, -1\\}^k\\times \\mathfrak{S}_k$ be a pair \n\tconsisting of a sign vector $\\eps=(\\eps_1, \\dots, \\eps_k)$ and a permutation $\\sigma$ of $[k]$. \n\tThe ordering $\\strictif$ on~$[N]^{(k)}$ \n\t{\\it associated with $(\\eps, \\sigma)$} is the unique ordering with the property that  \n\tif \n\t%\n\t\\[\n\t\t1\\le a_1< \\dots <a_k\\le N\n\t\t\\quad \\text{ and } \\quad 1\\le b_1< \\dots <b_k\\le N\\,,\n\t\\]\n\n\tthen\n\t%\n\t\\[\n\t\t \\{a_1,\\dots,a_k\\} \\strictif \\{b_1,\\dots, b_k\\}\n\t\t \\,\\,\\, \\Longleftrightarrow \\,\\,\\,\n\t\t (\\epsilon_{\\sigma(1)}a_{\\sigma(1)},\\dots,\\epsilon_{\\sigma(k)}a_{\\sigma(k)})\n\t\t <_{\\mathrm{lex}} \n\t\t (\\epsilon_{\\sigma(1)}b_{\\sigma(1)},\\dots,\\epsilon_{\\sigma(k)}b_{\\sigma(k)})\\,,\n\t\\]\n\t%\n\twhere $<_{\\mathrm{lex}}$ denotes the lexicographic order on ${\\mathds Z}^k$.\n\\end{definition}\n\nIt is readily seen that such an ordering $\\strictif$ associated with $(\\eps, \\sigma)$ does \nindeed exist, that it is uniquely determined, and, moreover, canonical. \nConversely, for $N\\ge 2k+1$ every canonical ordering is \nof this form (see Lemma~\\ref{lem:canonical} below). \nWe proceed by stating the aforementioned unpublished theorem due to Klaus Leeb.\n\n\\begin{theorem}[Leeb]\\label{th:leeb}\n\tFor all positive integers $k$ and $n$ there exists an integer $\\mathrm{L}^{(k)}(n)$ such that for \n\tevery~$N\\ge \\mathrm{L}^{(k)}(n)$ and every ordering $([N]^{(k)}, \\strictif)$\n\tthere is a set $X\\subseteq [N]$ of size $n$ such that $(X^{(k)}, \\strictif)$ is canonical. \n\\end{theorem}\n\nIn the sequel, $\\mathrm{L}^{(k)}(n)$ will always refer to the least number with this property. As we shall\nprove, for fixed $k\\ge 2$ the dependence of $\\mathrm{L}^{(k)}(n)$ on $n$ is $k$ times exponential in a \npolynomial of $n$.\n\n\\subsection{Bounds}\nThe study of the Ramsey numbers $\\mathrm{R}^{(k)}(n, r)$ has received extensive \ninterest (see e.g.~\\cites{EHR65, CFS10, GRS90, EH72}). To state some of the currently known bounds, \none recursively defines {\\it tower functions} $t_k\\colon{\\mathds N}\\lra{\\mathds N}$ by \n\\[\n\tt_0(x)=x\n\t\\quad \\text{ and } \\quad \n\tt_{i+1}(x)=2^{t_i(x)}\\,.\n\\]\nArguing probabilistically Erd\\H{o}s~\\cite{Er47} showed \nthat $\\mathrm{R}^{(2)}(n, 2)\\ge 2^{(n+1)\/2}$ and Erd\\H{o}s, Hajnal, and Rado \\cite{EHR65} proved later \nthat $\\mathrm{R}^{(k)}(n, 2)\\ge t_{k-2}(cn^2)$ for $k>2$, where $c$ is a positive constant. \nThey also proved $\\mathrm{R}^{(k)}(n, 4)\\ge t_{k-1}(c_k n)$ for all $k\\ge 2$, \nwhere $c_k>0$ is an absolute constant. In the other direction, it is known \nthat $\\mathrm{R}^{(k)}(n, r)\\le t_{k-1}(C_{k, r}n)$.\n\nEstimates on the canonical Ramsey numbers $\\ER^{(k)}(n)$ were studied in \\cites{LR95, Sh96}. \nNotably, Lefmann and R\\\"{o}dl proved the lower bound $\\ER^{(k)}(n)\\ge t_{k-1}(c_kn^2)$,\nwhile Shelah obtained the complementary upper bound $\\ER^{(k)}(n)\\le t_{k-1}(C_kn^{8(2k-1)})$.\n\nLet us now turn our attention to the Leeb numbers $\\mathrm{L}^{(k)}(n)$. For $k=1$, there exist\nonly two canonical orderings of $[N]^{(1)}$, namely the ``increasing'' and the ``decreasing''\none corresponding to the sign vectors $\\eps=(+1)$ and $\\eps=(-1)$, respectively. \nThus the well-known Erd\\H{o}s-Szekeres theorem~\\cite{ES35} yields the exact \nvalue $\\mathrm{L}^{(1)}(n)=(n-1)^2+1$. Accordingly, Leeb's theorem \ncan be viewed as a multidimensional version of the Erd\\H{o}s-Szekeres theorem and we refer \nto~\\cite{Lang} for further variations on this theme. Our own results can be summarised as follows.\n\n\\begin{theorem}\\label{thm:lower}\n\tIf $n\\ge 4$ and $R\\longarrownot\\longrightarrow (n-1)^2_2$, then $\\mathrm{L}^{(2)}(n)> 2^{R}$. Moreover, if $n\\ge k\\ge 3$, \n\tthen $\\mathrm{L}^{(k)}(4n+k) > 2^{\\mathrm{L}^{(k-1)}(n)-1}$. \n\\end{theorem}\n\nDue to the known lower bounds on diagonal Ramsey numbers this implies \n\\[\n\t\\mathrm{L}^{(2)}(n)\\ge t_2(n\/2)\n\t\\quad \\text {as well as } \\quad \n\t\\mathrm{L}^{(k)}(n)\\ge t_{k}(c_kn) \\text{ for } k\\ge 3\\,. \n\\]\nWe offer an upper bound with the same number of exponentiations.\n\n\\begin{theorem}\\label{thm:upper}\n\tFor every $k\\ge 2$ there exists a constant $C_k$ such that $\\mathrm{L}^{(k)}(n)\\le t_{k}(n^{C_k})$\n\tholds for every $n\\ge 2k$.\n\\end{theorem}\n\nThe case $k=2$ of Theorems~\\ref{thm:lower} and~\\ref{thm:upper} was obtained independently by Conlon, \nFox, and Sudakov in unpublished work~\\cite{Fox}.\n\n\\subsection*{Organisation} We prove Theorem~\\ref{thm:lower} in Section~\\ref{sec:comb}.\nThe upper bound, Theorem~\\ref{thm:upper}, is established in Section~\\ref{sec:upper}. \nLemma~\\ref{lem:2150} from this proof turns out to be applicable to Erd\\H{o}s-Rado \nnumbers as well and Section~\\ref{sec:ERS} illustrates this by showing a variant of \nShelah's result, namely $\\ER^{(k)}(n)\\le t_{k-1}(C_kn^{6k})$. \n\n\\@ifstar{\\origsection*}{\\@startsection{section}{1}\\z@{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\normalfont\\scshape\\centering\\S}}{Lower bounds: Constructions}\\label{sec:comb}\n\n\\subsection{Trees and combs}\nOur lower bound constructions take inspiration from the negative stepping-up lemma due\nto Erd\\H{o}s, Hajnal, and Rado \\cite{EHR65}. Notice that in order to prove an inequality \nof the form $\\mathrm{L}^{(k)}(n)>N$ one needs to exhibit a special ordering $([N]^{(k)}, \\strictif)$\nfor which there is no set $Z\\subseteq [N]$ of size $n$ such that $\\strictif$ orders $Z^{(k)}$ \ncanonically. For us, $N=2^m$ will always be a power of two and for reasons of visualisability \nwe identify the numbers in $[N]$ with the leaves of a binary tree of height $m$. \nWe label the levels of our tree from the bottom to the top with the numbers in $[m]$. \nSo the root is at level $1$ and the leaves are immediately above \nlevel $m$ (see Figure~\\ref{fig:optimalline}).\n\n\\begin{figure}[h]\n\\centering\n{\\hfil \\begin{tikzpicture}[scale=1.0\n    \\coordinate (A) at (0,0);\n    \\coordinate (B) at (-2,1);\n    \\coordinate (C) at (2,1);\n    \\coordinate (D) at (-3,2);\n    \\coordinate (E) at (-1,2);\n\t\\coordinate (F) at (1,2);\n    \\coordinate (G) at (3,2);\n    \\coordinate [label=above:$1$] (H1) at (-3.5,3);\n    \\coordinate [label=above:$2$] (H2) at (-2.5,3);\n    \\coordinate [label=above:$3$] (H3) at (-1.5,3);\n    \\coordinate [label=above:$4$] (H4) at (-0.5,3);\n    \\coordinate [label=above:$5$] (H5) at (0.5,3);\n    \\coordinate [label=above:$6$] (H6) at (1.5,3);\n    \\coordinate [label=above:$7$] (H7) at (2.5,3);\n    \\coordinate [label=above:$8$] (H8) at (3.5,3);\n    \n    \\coordinate [label=left:$1$] (L1) at (-5,0);\n    \\coordinate [label=left:$2$] (L2) at (-5,1);\n    \\coordinate [label=left:$3$] (L3) at (-5,2);\n    \\coordinate (R1) at (5,0);\n    \\coordinate (R2) at (5,1);\n    \\coordinate (R3) at (5,2);\n\n    \\draw[line width=0.5] (C)--(A)--(B);\n    \\draw[line width=0.5] (D)--(B)--(E);\n    \\draw[line width=0.5] (F)--(C)--(G);\n    \\draw[line width=0.5] (H1)--(D)--(H2);\n    \\draw[line width=0.5] (H3)--(E)--(H4);\n    \\draw[line width=0.5] (H5)--(F)--(H6);\n    \\draw[line width=0.5] (H7)--(G)--(H8);\n    \\draw[line width=0.5] (L1)--(L2)--(L3);\n    \\draw[dashed] (L1)--(R1);\n    \\draw[dashed] (L2)--(R2);\n    \\draw[dashed] (L3)--(R3);\n    \\draw (L1) circle [radius=0.05];\n \t\\draw (L2) circle [radius=0.05];\n \t\\draw (L3) circle [radius=0.05];\n    \n    \\end{tikzpicture}\\hfil}\n\\caption{The binary tree and its levels for $m=3$.}\n\\label{fig:optimalline}\n\\end{figure}\n\nAlternatively, we can identify $[N]$ with the set $\\{+1, -1\\}^m$ of all $\\pm1$-vectors of length~$m$\nsuch that the standard ordering on $[N]$ corresponds to the lexicographic one \non~$\\{+1, -1\\}^m$. The literature often works with the number $0$ instead of $-1$ here, but for the \napplication we have in mind our choice turns out to be advantageous.  \n\nNow every set $X\\subseteq [N]$ generates a tree $T_X$, which is the subtree of our binary tree \ngiven by the union of the paths from the leaves in $X$ to the root (see Figure~\\ref{fig:2057}).\n\n\n\\begin{figure}[h]\n\\centering\n{\\hfil \\begin{tikzpicture}[scale=1.0\n    \\coordinate (A) at (0,0);\n    \\coordinate (B) at (-2,1);\n    \\coordinate (C) at (2,1);\n    \\coordinate (D) at (-3,2);\n    \\coordinate (E) at (-1,2);\n\t\\coordinate (F) at (1,2);\n    \\coordinate (G) at (3,2);\n    \\coordinate [label=above:$1$] (H1) at (-3.5,3);\n    \\coordinate [label=above:$2$] (H2) at (-2.5,3);\n    \\coordinate [label=above:$3$] (H3) at (-1.5,3);\n    \\coordinate [label=above:$4$] (H4) at (-0.5,3);\n    \\coordinate [label=above:$5$] (H5) at (0.5,3);\n    \\coordinate [label=above:$6$] (H6) at (1.5,3);\n    \\coordinate [label=above:$7$] (H7) at (2.5,3);\n    \\coordinate [label=above:$8$] (H8) at (3.5,3);\n    \n    \\coordinate [label=left:$1$] (L1) at (-5,0);\n    \\coordinate [label=left:$2$] (L2) at (-5,1);\n    \\coordinate [label=left:$3$] (L3) at (-5,2);\n    \\coordinate (R1) at (5,0);\n    \\coordinate (R2) at (5,1);\n    \\coordinate (R3) at (5,2);\n\n    \\draw[line width=2.5] (C)--(A)--(B);\n    \\draw[line width=2.5] (D)--(B)--(E);\n    \\draw[line width=0.5] (F)--(C);\n    \\draw[line width=2.5] (G)--(C);\n    \\draw[line width=0.5] (H1)--(D);\n    \\draw[line width=2.5] (H2)--(D);\n    \\draw[line width=2.5] (H3)--(E);\n    \\draw[line width=0.5] (H4)--(E);\n    \\draw[line width=0.5] (H5)--(F)--(H6);\n    \\draw[line width=2.5] (H7)--(G);\n    \\draw[line width=0.5] (H8)--(G);\n    \\draw[line width=0.5] (L1)--(L2)--(L3);\n    \\draw[dashed] (L1)--(R1);\n    \\draw[dashed] (L2)--(R2);\n    \\draw[dashed] (L3)--(R3);\n    \\draw (L1) circle [radius=0.05];\n \t\\draw (L2) circle [radius=0.05];\n \t\\draw (L3) circle [radius=0.05];\n \t\\draw[fill] (H2) circle [radius=0.1];\n \t\\draw[fill] (H3) circle [radius=0.1];\n \t\\draw[fill] (H7) circle [radius=0.1];\n \t\\draw[fill] (B) circle [radius=0.1];\n \t\\draw[fill] (A) circle [radius=0.1];\n    \n    \\end{tikzpicture}\\hfil}\n\\caption{The auxiliary tree $T_X$ for $X=\\{2,3,7\\}$.}\n\\label{fig:2057}\n\\end{figure}\n\nAn essential observation on which both the double-exponential lower bound construction \nfor the four-colour Ramsey number of triples and our lower bound on $\\mathrm{L}^{(2)}(n)$ rely \nis that there are two essentially different kinds of triples: those engendering a ``left tree'' \nand those engendering a ``right tree''. Roughly speaking the difference is that when descending\nfrom the leaves to the root, the two elements concurring first are the two smaller \nones for left trees and the two larger ones for right trees. For instance, Figure~\\ref{fig:2057} \ndisplays a left tree.\n\nTo make these notions more precise we introduce the following notation. Given at least two distinct \nvectors $x_1, \\dots, x_t\\in \\{-1, +1\\}^m$ with coordinates $x_i=(x_{i1}, \\dots, x_{im})$ we write \n\\[\n\t\\delta(x_1, \\dots, x_t)=\\min\\{\\mu\\in [m]\\colon x_{1\\mu}= \\dots = x_{t\\mu} \\text{ is not the case}\\}\\,.\n\\]\nfor the first level where two of them differ. \n\nNow let $xyz\\in [N]^{(3)}$ be an arbitrary triple with $x<y<z$ and set $\\delta=\\delta(x, y, z)$. \nThis means that $x$, $y$, and $z$ agree in the first $\\delta-1$ coordinates, while their entries \nin the $\\delta^{\\mathrm{th}}$ coordinate, which we denote by $x_\\delta$, $y_\\delta$, $z_\\delta$,\nfail to coincide. As a consequence of $x<y<z$ we know $x_\\delta\\le y_\\delta\\le z_\\delta$. Thus \n$x_\\delta=-1$, $z_\\delta=+1$, and there remain two possibilities: Depending on \nwhether $y_\\delta=-1$ or $y_\\delta=+1$ we say that $xyz$ forms a {\\it left tree} or \na {\\it right tree}, respectively. Equivalently, $xyz$ yields a left tree \nwhenever $\\delta(x, y)>\\delta(y, z)$ and a right tree whenever $\\delta(x, y)<\\delta(y, z)$.  \n\nWhen the triples are coloured with $\\{\\mathrm{left}, \\mathrm{right}\\}$ depending on whether they form left\ntrees or right trees, the monochromatic sets are called {\\it combs}. \nMore explicitly, for \n\\[\t\n\t1\\le x_1<\\dots < x_t\\le N\n\\]\nthe set $X=\\{x_1, \\dots, x_t\\}$ is a {\\it left comb} if \n$\\delta(x_1, x_2)>\\delta(x_2, x_3)>\\dots >\\delta(x_{t-1}, x_t)$\nand a {\\it right comb} if \n$\\delta(x_1, x_2)<\\delta(x_2, x_3)<\\dots <\\delta(x_{t-1}, x_t)$ (see Figure~\\ref{fig:0001}). \nFor instance, the empty set, every singleton, and every pair are both a left comb and a \nright comb. Since every \ntriple is either a left tree or a right tree, triples are combs in exactly one of the two \ndirections. Some quadruples fail to be combs.   \n\n\\begin{figure}[h]\n\\centering\n{\\hfil \\begin{tikzpicture}[scale=0.7\n    \\coordinate (R1) at (-3,0);\n    \\coordinate (R2) at (-3.5,1);\n    \\coordinate (R3) at (-4,2);\n    \\coordinate (R4) at (-4.5,3);\n    \\coordinate (R5) at (-5,4);\n\t\\coordinate (R6) at (-4,4);\n    \\coordinate (R7) at (-3,4);\n    \\coordinate (R8) at (-2,4);\n    \\coordinate (R9) at (-1,4);\n    \\coordinate (H1) at (3,0);\n    \\coordinate (H2) at (3.5,1);\n    \\coordinate (H3) at (4,2);\n    \\coordinate (H4) at (4.5,3);\n    \\coordinate (H5) at (5,4);\n    \\coordinate (H6) at (4,4);\n    \\coordinate (H7) at (3,4);\n    \\coordinate (H8) at (2,4);\n    \\coordinate (H9) at (1,4);\n    \n\n    \\draw[line width=1] (R1)--(R2)--(R3)--(R4)--(R5);\n    \\draw[line width=1] (R1)--(R9);\n    \\draw[line width=1] (R2)--(R8);\n    \\draw[line width=1] (R3)--(R7);\n    \\draw[line width=1] (R4)--(R6);\n    \n    \\draw[line width=1] (H1)--(H2)--(H3)--(H4)--(H5);\n    \\draw[line width=1] (H1)--(H9);\n    \\draw[line width=1] (H2)--(H8);\n    \\draw[line width=1] (H3)--(H7);\n    \\draw[line width=1] (H4)--(H6);\n    \n    \\draw[fill] (R1) circle [radius=0.1];\n \t\\draw[fill] (R2) circle [radius=0.1];\n \t\\draw[fill] (R3) circle [radius=0.1];\n    \\draw[fill] (R4) circle [radius=0.1];\n \t\\draw[fill] (R5) circle [radius=0.1];\n \t\\draw[fill] (R6) circle [radius=0.1];\n    \\draw[fill] (R7) circle [radius=0.1];\n \t\\draw[fill] (R8) circle [radius=0.1];\n \t\\draw[fill] (R9) circle [radius=0.1];\n    \\draw[fill] (H1) circle [radius=0.1];\n \t\\draw[fill] (H2) circle [radius=0.1];\n \t\\draw[fill] (H3) circle [radius=0.1];\n    \\draw[fill] (H4) circle [radius=0.1];\n \t\\draw[fill] (H5) circle [radius=0.1];\n \t\\draw[fill] (H6) circle [radius=0.1];\n \t\\draw[fill] (H7) circle [radius=0.1];\n \t\\draw[fill] (H8) circle [radius=0.1];\n \t\\draw[fill] (H9) circle [radius=0.1];\n \t\n    \n    \\end{tikzpicture}\\hfil}\n\\caption{A left comb and a right comb.}\n\\label{fig:0001}\n\\end{figure}\n\n\\subsection{Ordering pairs}\nLet us recall that for Ramsey numbers the negative stepping-up lemma shows, e.g., that if \n$m\\longarrownot\\longrightarrow (n-1)^2_2$, then $2^m\\longarrownot\\longrightarrow (n)^3_4$. As we shall reuse the argument, we provide a brief\nsketch. Let $f\\colon [2^m]^{(3)}\\lra \\{\\mathrm{left}, \\mathrm{right}\\}$ be the colouring telling\nus whether a triple generates a left tree or a right tree and \nlet $g\\colon [m]^{(2)}\\lra \\{+1, -1\\}$ be a colouring exemplifying the negative partition \nrelation $m\\longarrownot\\longrightarrow (n-1)^2_2$. Define a \ncolouring \n\\[\n\th\\colon [2^m]^{(3)}\\lra \\{\\mathrm{left}, \\mathrm{right}\\}\\times \\{+1, -1\\}\n\\]\nof the triples with four colours by \n\\[\n\th(xyz)=\\bigl(f(xyz), g\\bigl(\\delta(xy), \\delta(yz)\\bigr)\\bigr)\n\\]\nwhenever $1\\le x<y<z\\le 2^m$.\nIf $Z\\subseteq [2^m]$ is monochromatic with respect to $h$, then, in particular, it needs to be \nmonochromatic for $f$, i.e., $Z$ is a comb. Thus, if $Z=\\{z_1, \\dots, z_{|Z|}\\}$ \nwith $z_1<z_2<\\dots<z_{|Z|}$, then \n$\\{\\delta(z_i, z_{i+1})\\colon i<|Z|\\}$ is a set of size $|Z|-1$ whose pairs are monochromatic \nwith respect to~$g$, for which reason \n$|Z|-1<n-1$. In other words, $h$ establishes the desired negative partition \nrelation $2^m\\longarrownot\\longrightarrow (n)^3_4$.\n\nIn the light of this reasoning it suffices to construct, in the same setup, an ordering \n$([2^m]^{(2)}, \\strictif)$ such that for every triple $e\\in [2^m]^{(3)}$ the colour $h(e)$ \nfrom the foregoing paragraph determines the ``isomorphism type'' of $(e^{(2)}, \\strictif)$ and \nvice versa. Let $xyz$ with $1\\le x<y<z\\le 2^m$ be an arbitrary triple and set $\\xi=\\delta(xy)$\nas well as  $\\eta=\\delta(yz)$. Our goal when constructing $\\strictif$ is that \n \n\\smallskip\n %\n\\begin{center}\n\\begin{tabular}{c|c|c}\nif $xyz$ is a & and & then  \\\\ \\hline \n\\rule{0pt}{3ex}  left tree  & $g(\\eta, \\xi)=+1$ & $xy\\strictif xz\\strictif yz$,\\\\\n\\rule{0pt}{3ex}  left tree  & $g(\\eta, \\xi)=-1$ & $xy\\strictif yz\\strictif xz$, \\\\\n\\rule{0pt}{3ex}  right tree & $g(\\xi, \\eta)=+1$ & $yz\\strictif xy\\strictif xz$, \\\\\n\\rule{0pt}{3ex}  right tree & $g(\\xi, \\eta)=-1$ & $yz\\strictif xz\\strictif xy$.\n\\end{tabular}\n\\end{center}\n\\medskip\n\nIf this can be accomplished, then every set $Z\\subseteq [2^m]$, for which $(Z^{(2)}, \\strictif)$\nis ordered canonically, is monochromatic with respect to the colouring $h$ and, therefore, the \nproof of $\\mathrm{L}^{(2)}(n)>2^m$ will be complete. \n\nLet us introduce one final piece of notation.\nGiven a vector $x=(x_1, \\dots, x_m)\\in \\{-1, +1\\}^m$ and~$\\delta\\in [m]$ we \ndefine the vector $x\\ltimes\\delta\\in\\{-1, +1\\}^{m-\\delta}$ by\n\\[\n\tx\\ltimes\\delta=\\bigl(x_{\\delta+1}g(\\delta, \\delta+1), \\dots, x_mg(\\delta, m)\\bigr)\\,.\n\\]\n\nNow let $\\strictif$ be an arbitrary ordering with the property that for any two distinct \npairs $xy$ and $x'y'$ with $x<y$ and $x'<y'$ the following statements hold.  \n\\begin{enumerate}[label=\\upshape({\\itshape \\arabic*\\,})]\n\t\\item\\label{it:1} If $\\delta(xy)\\ne \\delta(x'y')$, then\n\t\t%\n\t\t\\[\n\t\t\txy\\strictif x'y'\n\t\t\t\\,\\,\\, \\Longleftrightarrow \\,\\,\\,\n\t\t\t\\delta(x'y') < \\delta(xy)\\,.\n\t\t\\]\n\t\t%\n\t\\item\\label{it:2} If $\\delta=\\delta(xy)= \\delta(x'y')$ and at least one \n\t\t\tof $x\\ltimes \\delta \\ne x'\\ltimes \\delta$ or $y\\ltimes \\delta \\ne y'\\ltimes \\delta$\n\t\t\tholds, then  \n\t\t%\n\t\t\\[\n\t\t\txy\\strictif x'y'\n\t\t\t\\,\\,\\, \\Longleftrightarrow \\,\\,\\,\n\t\t\t(x\\ltimes \\delta)\\circ (y\\ltimes \\delta)\n\t\t\t<_{\\mathrm{lex}}\n\t\t\t(x'\\ltimes \\delta)\\circ (y'\\ltimes \\delta)\\,,\n\t\t\\]\n\t\t%\n\t\twhere $\\circ$ indicates concatenation, i.e., that we are comparing $(2m-2\\delta)$-tuples\n\t\tlexicographically.\n\\end{enumerate}\n\nIt remains to show that these two properties of $\\strictif$ imply the four statements in the \nabove table.\nLet us return to this end to our triple $xyz$ with $1\\le x<y<z\\le 2^m$ \nand $\\xi=\\delta(xy)$, $\\eta=\\delta(yz)$.\nIf $xyz$ is a left tree, then $\\delta(xz)=\\delta(yz)<\\delta(xy)$, for which reason~\\ref{it:1} \nimplies $xy\\strictif xz, yz$ and it remains to compare the latter two pairs. \nSince $x\\ltimes \\eta\\ne y\\ltimes \\eta$, the second rule declares that $xz\\strictif yz$ holds if \nand only if $x\\ltimes \\eta<_{\\mathrm{lex}} y\\ltimes \\eta$.\nNow $x$ and $y$ agree in the first $\\xi-1$ coordinates and continue with $x_\\xi=-1$, $y_\\xi=+1$.\nThus $x\\ltimes \\eta$ and $y\\ltimes \\eta$ agree in their first $\\xi-\\eta-1$ coordinates and \nproceed with $x_\\xi g(\\eta, \\xi)$, $y_\\xi g(\\eta, \\xi)$, respectively. So altogether $xz\\strictif yz$\nis equivalent to $g(\\eta, \\xi)=+1$ and we have confirmed the two upper rows of the table. \nThe case where $xyz$ is a right tree is discussed similarly. Summarising, we have thereby proved\nthe first assertion in Theorem~\\ref{thm:lower}.\n\n\\subsection{Negative stepping-up lemma for Leeb numbers}\n\nThroughout this subsection we fix two integers $k\\ge 3$ and $n\\ge k$. We set $m=\\mathrm{L}^{(k-1)}(n)-1$\nand study orderings on $[2^m]^{(k)}$, where~$2^m$ is again identified with $\\{+1, -1\\}^m$. \nWe are to exhibit an ordering $([2^m]^{(k)}, \\strictif)$ without canonically ordered subsets \nof size $4n+k$. Preparing ourselves we consider the following concept. \n\n\\begin{definition}\\label{dfn:imp}\n\tA set $Z\\subseteq [2^m]$ is said to be {\\it impeded} (see Figure~\\ref{fig:impede}) if there exist \n\ttwo elements $x<y$ in~$Z$\n\twith the following properties.\n\t%\n\t\\begin{enumerate}[label=\\upshape({\\itshape \\alph*\\,})]\n\t\t\\item\\label{it:impa} At least $k$ members of $Z$ lie between $x$ and $y$.\n\t\t\\item\\label{it:impb} Setting $\\delta=\\delta(x, y)$ there exist $x', x'', y', y''\\in Z$\n\t\t\tsuch that $x<x'<x''<y''<y'<y$ and $x''_\\delta=-1$ while $y''_\\delta=+1$.\n\t\\end{enumerate}\n\n\\end{definition}\n\n\\begin{figure}[h]\n\\centering\n{\\hfil \\begin{tikzpicture}[scale=0.9\n    \n    \\draw (0,0) ellipse (2cm and 0.7cm);\n    \\draw (6,0) ellipse (2cm and 0.7cm);\n    \\coordinate [label=above:$x$] (x) at (-1.1,0);\n    \\coordinate [label=above:$x'$] (x') at (0,0);\n    \\coordinate [label=above:$x''$] (x'') at (1.1,0);\n    \\coordinate [label=above:$y$] (y) at (7.1,0);\n    \\coordinate [label=above:$y'$] (y') at (6,0);\n    \\coordinate [label=above:$y''$] (y'') at (4.9,0);\n    \\coordinate [label=left:$\\delta+1$] (l1) at (-4,-3);\n    \\coordinate [label=left:$\\delta$] (l2) at (-4,-4);\n    \n    \\filldraw (0,-0.1) circle (1.5pt);\n    \\filldraw (-1.1,-0.1) circle (1.5pt);\n    \\filldraw (1.1,-0.1) circle (1.5pt);\n    \\filldraw (7.1,-0.1) circle (1.5pt);\n    \\filldraw (6,-0.1) circle (1.5pt);\n    \\filldraw (4.9,-0.1) circle (1.5pt);\n    \\draw (-4,-3) circle (1.5pt);\n    \\draw (-4,-4) circle (1.5pt);\n    \n    \n    \\coordinate (r) at (0,-3);\n    \\coordinate (s) at (6,-3);\n    \\coordinate (t) at (3,-4);\n    \\coordinate (a1) at (-3,-1);\n    \\coordinate (a2) at (-3,1);\n    \\coordinate [label=right:$Z$] (a3) at (9,1);\n    \\coordinate (a4) at (9,-1);\n    \\coordinate (e1) at (-1.95,-0.15);\n    \\coordinate (e2) at (1.95,-0.15);\n    \\coordinate (f1) at (4.05,-0.15);\n    \\coordinate (f2) at (7.95,-0.15);\n    \\coordinate (r1) at (10,-3);\n    \\coordinate (r2) at (10,-4);\n    \\coordinate (l3) at (-4,-1.8);\n    \\coordinate (l4) at (-4,-5);\n    \n    \n    \\draw (e1)--(r)--(e2);\n    \\draw (f1)--(s)--(f2);\n    \\draw (r)--(t)--(s);\n    \\draw (a1)--(a2)--(a3)--(a4)--(a1);\n    \\draw[dashed] (l1)--(r1);\n    \\draw[dashed] (l2)--(r2);\n    \\draw (l3)--(l4);  \n    \\end{tikzpicture}\\hfil}\n    \\caption{An impeded set $Z$.}\n    \\label{fig:impede}\n\\end{figure}\n\n\nRoughly speaking, every unimpeded set contains a large comb.\n  \n\\begin{lemma}\\label{lem:2050}\n\tEvery set $Z\\subseteq [2^m]$ is either impeded or possesses two subsets $L, R\\subseteq Z$\n\twith $|L|+|R|\\ge \\tfrac12(|Z|-k+1)$ forming a left comb and a right comb, respectively.  \n\\end{lemma}\n\n\\begin{proof}\n\tOtherwise let $Z$ be a minimal counterexample. Since every set with at most two elements \n\tis a two-sided comb, we have $|Z|\\ge k+2$. Now the vectors $x=\\min(Z)$\n\tand $y=\\max(Z)$ satisfy part~\\ref{it:impa} of Definition~\\ref{dfn:imp}. Let $x'$, $x''$ be the \n\tsecond and third-smallest vector in $Z$ and, similarly, let $y'$ and $y''$ be the second and \n\tthird largest one. If both $x''_\\delta=-1$ and $y''_\\delta=+1$ would hold, \n\twhere $\\delta=\\delta(x, y)$, then the set $Z$ were impeded, contrary to its choice. \n\t\n\tThus at least one of the cases $x''_\\delta=+1$ and $y''_\\delta=-1$ occurs and for reasons \n\tof symmetry we may suppose that $x''_\\delta=+1$. The set $Z_\\star=Z\\setminus \\{x, x'\\}$ \n\tcannot be impeded for then~$Z$ was impeded as well. So by the minimality of $Z$ there \n\texist sets $L, R_\\star\\subseteq Z_\\star$ forming a left comb and a right comb, respectively,\n\tand satisfying $|L|+|R_\\star|\\ge \\tfrac12(|Z_\\star|-k+1)$. For any two elements $r<r'$\n\tfrom $R_\\star$ we have $\\delta(x, r)=\\delta<\\delta(r, r')$ and, consequently, $R=\\{x\\}\\cup R_\\star$\n\tis a right comb. But now $|L|+|R|=|L|+|R_\\star|+1\\ge \\tfrac12(|Z|-k+1)$ yields a \n\tfinal contradiction.\n\\end{proof}\n\nWe proceed with some further notation. Let $<_{\\mathrm{lex}}$ denote the lexicographic ordering \non~$[2^m]^{(k)}$. So explicitly for $X, Y\\in [2^m]^{(k)}$ we have $X<_{\\mathrm{lex}} Y$ if and only \nif $\\sum_{x\\in X}2^{-x}$ is larger than $\\sum_{y\\in Y}2^{-y}$. \n\nFor every set $X\\in [2^m]^{(k)}$ we determine a natural\nnumber $\\phi(X)$ according to the following rule: Consider $x^-=\\min(X)$, \n$x^+=\\max(X)$ and compute $\\delta=\\delta(x^-, x^+)$. Notice that all elements of $X$ agree\nin their first $\\delta-1$ coordinates, while in the $\\delta^{\\mathrm{th}}$ \ncoordinate~$x^-_\\delta=-1$ and~$x^+_\\delta=+1$. \nLet $\\phi(X)$ count the number of members of $X$ having \na $-1$ as their~$\\delta^{\\mathrm{th}}$ coordinate. Notice that if $X$ is a left comb, then \n$\\phi(X)=|X|-1$, while right combs satisfy $\\phi(X)=1$.\n\nNext, if $B$ is either a left or a right comb, we know \nthat $\\pi(B)=\\bigl\\{\\delta(xy)\\colon xy\\in B^{(2)}\\bigr\\}\\subseteq [m]$ is a set of size $|B|-1$ \nwhose elements can be found by applying $\\delta$ to the $|B|-1$ pairs of consecutive members\nof $B$. We call $\\pi(B)$ the {\\it projection} of $B$.   \n\nFinally, instead of a colouring we shall apply an appropriate ordering $([m]^{(k-1)}, \\YY)$ \nsenkrecht.\nBecause of $m<\\mathrm{L}^{(k-1)}(n)$ this ordering can be chosen without canonical subsets of size $n$. \n\nWe now state four desirable properties we would like an ordering $([2^m]^{(k)}, \\strictif)$\nto satisfy for any two distinct sets $X, Y\\in [2^m]^{(k)}$.\n\n\\begin{enumerate}[label=\\upshape({\\itshape \\arabic*\\,})]\n\t\\item\\label{it:11} If $\\phi(X)$ is odd and $\\phi(Y)$ is even, then $X\\strictif Y$. \n\t\\item\\label{it:22} If $X$ and $Y$ are two combs in the same direction having different \n\t\tprojections, then $X\\strictif Y \\Longleftrightarrow \\pi(X)\\YY\\pi(Y)$.\n\t\\item\\label{it:33} If $X$ and $Y$ are left combs with $\\pi(X)=\\pi(Y)$, \n\t\tthen $X\\strictif Y \\Longleftrightarrow Y<_{\\mathrm{lex}} X$.  \n\t\\item\\label{it:44} If $X$ and $Y$ are right combs with $\\pi(X)=\\pi(Y)$, \n\t\tthen $X\\strictif Y \\Longleftrightarrow X<_{\\mathrm{lex}} Y$.  \n\\end{enumerate}       \n\nIt is not hard to see that these properties do not contradict each other. Indeed, in order to \nsatisfy~\\ref{it:11} one partitions $[2^m]^{(k)}$ into two sets depending on the parity of $\\phi$\nand puts the odd class before the even class. The remaining rules~\\ref{it:22}\\,--\\,\\ref{it:44}\nonly concern the internal orderings within these two classes. Similarly,~\\ref{it:22} tells\nus to sub-partition the combs within both classes according to their projections \nand~\\ref{it:33},~\\ref{it:44} only give further instructions how to compare combs within the \nsame sub-class.  \n\nNow it suffices to show for no ordering $([2^m]^{(k)}, \\strictif)$ \nsatisfying~\\ref{it:11}\\,--\\,\\ref{it:44}\nthere is a set $Z\\subseteq [2^m]$ of size~$4n+k$ such that $(Z^{(k)}, \\strictif)$ \nis ordered canonically. Assume contrariwise that~$\\strictif$ and~$Z$ have these properties. \nIf $L\\subseteq Z$ is a left comb, then so is every $X\\in L^{(k)}$ and by~\\ref{it:22} it follows \nthat $\\pi(L)^{(k-1)}$ is ordered canonically by $\\YY$. So our choice of $\\YY$ \nyields $|\\pi(L)|\\le n-1$ and, consequently, $|L|\\le n$. For the same reason, every right \ncomb $R\\subseteq Z$ has at most the size~$n$. But now Lemma~\\ref{lem:2050} tells us that $Z$\nis impeded. Pick $x, x', x'', y'', y', y\\in Z$ as in Definition~\\ref{dfn:imp}. \n\n\\smallskip\n\n{\\it \\hskip2em Special case: $k=3$}\n\n\\smallskip\n\nRecall that every triple is either a left tree or a right tree. Owing to~\\ref{it:11} the \nright trees precede the left trees with respect to $\\strictif$.\n\n\\begin{figure}[h]\n\\centering\n{\\hfil \\begin{tikzpicture}[scale=0.9\n    \n    \\draw (0,0) ellipse (2cm and 0.7cm);\n    \\draw (6,0) ellipse (2cm and 0.7cm);\n    \\coordinate [label=above:$x$] (x) at (-1.1,0);\n    \\coordinate [label=above:$x'$] (x') at (0,0);\n    \\coordinate [label=above:$x''$] (x'') at (1.1,0);\n    \\coordinate [label=above:$y$] (y) at (7.1,0);\n    \\coordinate [label=above:$y'$] (y') at (6,0);\n    \\coordinate [label=above:$y''$] (y'') at (4.9,0);\n    \\coordinate (l1) at (-4,-4);\n    \\coordinate (l2) at (-4,-5);\n    \\coordinate (l3) at (-4,-1.4);\n    \\coordinate (l4) at (-4,-2.6);\n    \\node[] at (-5.3,-5) {$\\delta(x,y)$};\n    \\node[] at (-5.3,-4) {$\\delta(x,y)+1$};\n    \\node[] at (-5.3,-2.6) {$\\delta(y'',y)$};\n    \\node[] at (-5.3,-1.4) {$\\delta(y'',y')$};\n    \n    \\filldraw (0,-0.1) circle (1.5pt);\n    \\filldraw (-1.1,-0.1) circle (1.5pt);\n    \\filldraw (1.1,-0.1) circle (1.5pt);\n    \\filldraw (7.1,-0.1) circle (1.5pt);\n    \\filldraw (6,-0.1) circle (1.5pt);\n    \\filldraw (4.9,-0.1) circle (1.5pt);\n    \\draw (-4,-4) circle (1.5pt);\n    \\draw (-4,-5) circle (1.5pt);\n    \\draw (-4,-1.4) circle (1.5pt);\n    \\draw (-4,-2.6) circle (1.5pt);\n    \n    \n    \\coordinate (r) at (0,-4);\n    \\coordinate (s) at (6,-4);\n    \\coordinate (t) at (3,-5);\n    \\coordinate (w) at (5.45,-1.4);\n    \\coordinate (v) at (6,-2.6);\n    \\coordinate (e1) at (-1.95,-0.15);\n    \\coordinate (e2) at (1.95,-0.15);\n    \\coordinate (f1) at (4.05,-0.15);\n    \\coordinate (f2) at (7.95,-0.15);\n    \\coordinate (r1) at (10,-4);\n    \\coordinate (r2) at (10,-5);\n    \\coordinate (r3) at (10,-1.4);\n    \\coordinate (r4) at (10,-2.6);\n    \\coordinate (l5) at (-4,-0.8);\n    \\coordinate (l6) at (-4,-5.6);\n    \n    \n    \\draw (e1)--(r)--(e2);\n    \\draw (f1)--(s)--(f2);\n    \\draw (r)--(t)--(s);\n    \\draw[dashed] (l1)--(r1);\n    \\draw[dashed] (l2)--(r2);\n    \\draw[dashed] (l3)--(r3);\n    \\draw[dashed] (l4)--(r4);\n    \\draw (l5)--(l6);\n    \\draw (y'')--(w)--(y');\n    \\draw (w)--(v)--(y);\n    \n \t\n    \\end{tikzpicture}\\hfil}\n    \\caption{The case $k=3$.}\n\\end{figure}\n\n\nNow consider the quintuples $A=\\{x, x', x'', y', y\\}$ and $B=\\{x, x', y'', y', y\\}$,\nwhose elements have just been listed in increasing order. By canonicity we know \nthat $(A^{(3)}, \\strictif)$ and $(B^{(3)}, \\strictif)$ are isomorphic via the map that \nsends $x''$ to $y''$ and keeps the four other elements fixed. Since $xx''y'$ and $xx''y$\nare left trees with the same projection,~\\ref{it:33} yields $xx''y\\strictif xx''y'$.\nThis statement in $(A^{(3)}, \\strictif)$ translates to $xy''y\\strictif xy''y'$. \nAs we are now comparing right trees,~\\ref{it:44} discloses that $xy''y$ and $xy''y'$\ncannot have the same projection, for which reason $\\delta(y''y)\\ne \\delta(y''y')$.\nIn other words, $y''y'y$ is a left tree and $\\delta(y''y) = \\delta(y'y)$. \nIt follows that the right trees $xy''y$ and $xy'y$ have the same projection, \nso another application of~\\ref{it:44} leads to $xy''y\\strictif xy'y$. When transferred\nback to $A$ this statement becomes $xx''y\\strictif xy'y$, so we have found a left tree \ncoming before a right tree, contrary to~\\ref{it:11}. This concludes the discussion of the \ncase $k=3$.\n\n\n\\smallskip\n\n{\\it \\hskip2em General case: $k\\ge 4$}\n\n\\smallskip\n\nWe may assume that $x'$, $x''$ are the elements of $Z$ immediately succeeding $x$ and,\nsimilarly, that $y''$, $y'$ are immediately preceeding $y$. According to part~\\ref{it:impa}\nof Definition~\\ref{dfn:imp} there is a set $W\\subseteq Z$ with $|W|=k-2$, $\\min(W)=x''$,\nand $\\max(W)=y''$. Consider the sets $A=W\\cup\\{x, x', y\\}$ and $B=W\\cup\\{x, y', y\\}$\nof size $k+1$. By canonicity we know that the orderings $(A^{(k)}, \\strictif)$ \nand $(B^{(k)}, \\strictif)$ are in the following sense essentially the same. For~$i\\in [k+1]$\nlet $A_i$ and $B_i$ denote the sets of size $k$ arising from $A$ and $B$ by removing \ntheir~$i^{\\mathrm{th}}$ elements, respectively. Now if $A_i\\strictif A_j$ holds for \nsome $i, j\\in [k+1]$,\nthen $B_i\\strictif B_j$ follows. Set $\\phi=\\phi(A)$ and observe that $3\\le \\phi\\le k-1$\ndue to the presence of $x$, $x'$, $x''$, $y$, and $y''$. Moreover \n\\begin{align*}\n\t\\phi(A_1)=\\dots =\\phi(A_\\phi)=\\phi-1, \\qquad \\phi(A_{\\phi+1})=\\dots = \\phi(A_{k+1})=\\phi, \\\\\n\t\\phi(B_1)=\\dots =\\phi(B_{\\phi-1})=\\phi-2, \\quad \\phi(B_{\\phi})=\\dots = \\phi(B_{k+1})=\\phi-1\\,.\n\\end{align*}\nIn view of~\\ref{it:11}, this leads to the equivalences\n\\[\n\tA_1\\strictif A_{k+1}\n\t\\,\\,\\,\\Longleftrightarrow\\,\\,\\,\n\t\\phi\\text{ is even }\n\t\\,\\,\\,\\Longleftrightarrow\\,\\,\\,\n\tB_{k+1}\\strictif B_{1}\\,,\n\\]\nwhich contradict the canonicity of $\\strictif$. \nAltogether, this completes the proof of Theorem~\\ref{thm:lower}. \n\n\\@ifstar{\\origsection*}{\\@startsection{section}{1}\\z@{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\normalfont\\scshape\\centering\\S}}{Upper bounds: the canonisation procedure}\\label{sec:upper}\n\n\\subsection{Preparation}\\label{subsec:31}\nThe proof of Theorem~\\ref{thm:upper} starts with some synchronisation principles for \ncolourings $f\\colon X^{(r)}\\lra\\Gamma$, where $X\\subseteq {\\mathds N}$, $r\\ge 2$, and $\\Gamma$ denotes a \nfinite {\\it set of colours}. A special r\\^{o}le will be played by the set of \ncolours \n\\begin{equation}\\label{eq:Phi}\n\t\\Phi=\\{-1, +1\\}\\,.\n\\end{equation} \n\nGiven a colouring $f\\colon X^{(r)}\\lra\\Phi$ we define the {\\it opposite colouring}\n$(-f)\\colon X^{(r)}\\lra\\Phi$ by $(-f)(x)=-(f(x))$ for all $x\\in X^{(r)}$.  \nFor simplicity we call a colouring $f\\colon X^{(r)}\\lra\\Phi$,\nwhere $r\\ge 1$ and $X\\subseteq {\\mathds N}$, {\\it increasing} if \n$f(x_1, \\dots, x_{r-1}, y)\\le f(x_1, \\dots, x_{r-1}, y')$ holds for all $x_1<\\dots <x_{r-1}<y<y'$\nfrom $X$ and {\\it decreasing} if its opposite colouring is increasing. \nWe refer to $f$ as {\\it monotone} if it is either increasing or decreasing. \nTwo colourings $f\\colon X^{(r)}\\lra\\Phi$ and $g\\colon Y^{(s)}\\lra\\Phi$\nare said to be {\\it monotone in the same direction} if either both are increasing or both are\ndecreasing. Similarly, $f$ and $g$ are {\\it monotone in opposite directions} if $f$ and $-g$ \nare monotone in the same direction.  \n\n\\begin{lemma}\\label{lem:1730}\n\tSuppose that $s, t\\ge 1$ are positive integers and that $X\\subseteq {\\mathds N}$ is a set \n\tof size $|X|\\ge (s-1)(t-1)+1$. If the colouring $f\\colon X^{(2)}\\lra \\Phi$ is monotone, \n\tthen either there is a monochromatic set $A\\in X^{(s)}$ for the colour $-1$ or a \n\tmonochromatic set $B\\in X^{(t)}$ for the colour $+1$.\n\\end{lemma} \n\n\\begin{proof}\n\tDue to the symmetry between $f$ and $-f$ we may suppose that $f$ is increasing. \n\tWe argue by induction on $t$. The base case $t=1$ offers no difficulty, for every \n\tset of size one is monochromatic in both colours. Now suppose that our claim is already known\n\tto be true for $t-1$ in place of $t$, where $t\\ge 2$. Set $z=\\max(X)$\n\tand consider the set $N=\\{x\\in X\\colon f(xz)=-1\\}$. \n\t\t\n\t\\smallskip\n\t\n\t{\\it \\hskip 2em First Case: $|N|\\ge s-1$}\n\t\n\t\\smallskip\n\t\n\tNow it suffices to prove that $A=N\\cup\\{z\\}$ is monochromatic in colour $-1$. \n\tSo let $x<y$ from $A$ be arbitrary and observe that $x\\in N$. If $y=z$, then $f(xy)=-1$\n\tfollows from the definition of $N$. Otherwise we have $y<z$ and the assumption that $f$ \n\tis increasing implies $f(xy)\\le f(xz)=-1$, whence $f(xy)=-1$. \n\t\n\t\\smallskip\n\t\n\t{\\it \\hskip 2em Second Case: $|N|< s-1$}\n\t\n\t\\smallskip\n\t\n\tNow the set $X_\\star=X\\setminus (N\\cup \\{z\\})$\n\tsatisfies $|X_\\star|\\ge |X|-(s-1)\\ge (s-1)(t-2)+1$. Thus the induction hypothesis applies \n\tto $X_\\star$ and yields either a monochromatic set $A$ of size $s$ for $-1$ or a \n\tmonochromatic set $B'$ of size $t-1$ for $+1$. In the former case we are done immediately\n\tand in the latter case the definition of $N$ entails that $B=B'\\cup\\{z\\}$ is monochromatic \n\tfor the colour $+1$.  \n\\end{proof}\n\nThis iterates as follows.\n\n\\begin{lemma}\\label{lem:1731}\n\tGiven a natural number $n$ and a nonempty finite index set $I$ let $X\\subseteq {\\mathds N}$\n\tbe a set with $|X|\\ge |I|!n^{|I|+1}$. If $(f_i)_{i\\in I}$ denotes a family of \n\tmonotone colourings $f_i\\colon X^{(2)}\\lra \\Phi$, then there is a set $Z\\subseteq X$\n\tof size $n$ which is monochromatic with respect to every $f_i$.\n\\end{lemma}\n\n\\begin{proof}\n\tReplacing some of the colourings $f_i$ by their opposites, if necessary, we may assume \n\tthat all of them are increasing. We argue by induction on $|I|$, the base case $|I|=1$ \n\tbeing dealt with in the foregoing lemma. Now suppose that $|I|\\ge 2$ and that \n\tthe lemma has already been established for every set $I'\\subsetneq I$ instead of $I$.\n\tGiven a set $X$ and a family of increasing colourings $(f_i)_{i\\in I}$ of the pairs of \n\telements from $X$ we define an auxiliary colouring $f\\colon X^{(2)}\\lra \\Phi$ by \n\t$f=\\min\\{f_i\\colon i\\in I\\}$, i.e., \n\n\t\\[\n\t\tf(xy)=+1 \n\t\t\\,\\,\\, \\Longleftrightarrow \\,\\,\\,\n\t\t\\forall i\\in I\\,\\, f_i(xy)=+1\\,.\n\t\\]\n\t%\n\t\n\tEvidently, $f$ is increasing as well. So by Lemma~\\ref{lem:1730} there either exists \n\ta set $A\\subseteq X$ with $|A|\\ge |I|!n^{|I|}$ that is monochromatic for $f$ in colour $-1$\n\tor there is a set $Z\\subseteq X$ of size $n$ that is monochromatic for $f$ in colour $+1$. \n\tIn the latter case $Z$ monochromatically has colour~$+1$ with respect to every $f_i$ \n\tand thus it possesses the desired property. \n\t\n\tSo suppose from now on that the former case occurs, i.e., that we have obtained\n\ta set~$A$ of size $|A|\\ge |I|!n^{|I|}$ that is monochromatic with respect to $f$ in colour~$-1$. \n\tPut $z=\\max(A)$ and $X_i=\\{a\\in A\\colon f_i(az)=-1\\}$ for every $i\\in I$. \n\tSince $f(az)=-1$ holds for all $a\\in A\\setminus\\{z\\}$, we know \n\tthat $\\bigcup_{i\\in I}X_i=A\\setminus\\{z\\}$.\n\tOwing to the Schubfachprinzip this implies $|X_{i(\\star)}|\\ge (|I|-1)!n^{|I|}$ for \n\tsome $i(\\star)\\in I$. As in the first case of the proof of Lemma~\\ref{lem:1730} one can \n\tshow that the set $X_{i(\\star)}\\cup\\{z\\}$ monochromatically receives the colour $-1$ with \n\trespect to~$f_{i(\\star)}$. It remains to apply the induction hypothesis to this set and \n\tto $I'=I\\setminus \\{i(\\star)\\}$. \n\\end{proof}\n\nLet us now return to an arbitrary colouring $f\\colon X^{(r)}\\lra \\Gamma$, where $X\\subseteq {\\mathds N}$,\n$r\\ge 2$, and $\\Gamma$ denotes a set of colours. We say that $f$ is {\\it governed} by another \ncolouring $f'\\colon X^{(\\ell)}\\lra\\Gamma$, where $\\ell\\in [r-1]$, \nif $f(x_1, \\dots, x_r)=f'(x_1, \\dots, x_\\ell)$\nholds for all $x_1<\\dots <x_r$ from $X$. \nStandard references call $f$ {\\it end-homogeneous}\nif it is governed by some $(r-1)$-ary function~$f'$ in this sense, but for us it will be useful \nto have words expressing the relationship between~$f$ and~$f'$. \n\n\\begin{lemma}\\label{lem:2150}\n\tGiven integers $n\\ge r\\ge 2$ and $s\\ge 3$, a set of colours $\\Gamma$, and a nonempty finite index \n\tset $I$ let $X\\subseteq {\\mathds N}$ be a set of \n\tsize $|X|\\ge (2|\\Gamma|)^{n^{r-1}} \\cdot (2|I|n^{s-2})^{|I|n^{s-2}}$. \n\tIf $f\\colon X^{(r)}\\lra\\Gamma$ is an arbitrary colouring and $(g_i)_{i\\in I}$ \n\twith $g_i\\colon X^{(s)}\\lra \\Phi$ is a family of \n\tmonotone colourings, then there exist a set $Z\\subseteq X$ of size $n$, a colouring \n\t$f'\\colon Z^{(r-1)}\\lra \\Gamma$ and a family $(g'_i)_{i\\in I}$ of monotone colourings \n\t$g'_i\\colon Z^{(s-1)}\\lra \\Phi$ such that \n\t\\begin{enumerate}[label=\\upshape({\\itshape \\alph*\\,})]\n\t\t\\item\\label{it:aa} $f'$ governs the restriction of $f$ to $Z^{(r)}$,\n\t\t\\item\\label{it:bb} for every $i\\in I$ the colouring $g'_i$ governs the restriction \n\t\t\tof $g_i$ to $Z^{(s)}$,\n\t\t\\item\\label{it:cc} and for every $i\\in I$ the colourings $g_i$, $g'_i$ are monotone in \n\t\t\topposite directions.\n\t\\end{enumerate}\n\\end{lemma}     \n\n\\begin{proof}\n\tAs usual, we may suppose that every $g_i$ is increasing. \n\tAn integer $\\nu\\in [n]$ is said to be {\\it secure} if there exist \n\tsets $Z_\\nu, A_\\nu\\subseteq X$\n\twith $|Z_\\nu|=\\nu$ and $\\max(Z_\\nu)<\\min(A_\\nu)$, \n\ta function $f'\\colon Z_\\nu^{(r-1)}\\lra \\Gamma$, and a family \n\t$(g'_i)_{i\\in I}$ of decreasing colourings $g'_i\\colon Z_\\nu^{(s-1)}\\lra \\Phi$ \n\twith the following properties.\n\t\\begin{enumerate}[label=\\upshape({\\itshape \\roman*\\,})]\n\t\t\\item\\label{it:51} If the numbers $z_1<\\dots<z_{r-1}$ are in $Z_\\nu$ and $z_r>z_{r-1}$ \n\t\t\t\tis in $Z_\\nu\\cup A_\\nu$, then $f(z_1, \\dots, z_r)=f'(z_1, \\dots, z_{r-1})$.\n\t\t\\item\\label{it:52} If $i\\in I$, the numbers $z_1<\\dots<z_{s-1}$ are in $Z_\\nu$, \n\t\t\t\tand $z_{s}>z_{s-1}$ is in $Z_\\nu\\cup A_\\nu$, \n\t\t\t\tthen $g_i(z_1, \\dots, z_{s})=g'_i(z_1, \\dots, z_{s-1})$.\n\t\t\\item\\label{it:53} If $i\\in I$, the numbers $z_1<\\dots<z_{s-1}$ are in $Z_\\nu$, \n\t\t\t\tand $g'_i(z_1, \\dots, z_{s-1})=-1$, then $g_i(z_1, \\dots, z_{s-2}, y, y')=-1$\n\t\t\t\tholds for all $y, y'\\in A_\\nu$.\n\t\t\\item\\label{it:54} Finally, \n\t\t\t$|A_\\nu|\\ge (2|\\Gamma|)^{n^{r-2}(n-\\nu)} \\cdot (2|I|n^{s-2})^{|I|n^{s-2}-\\Omega}$, \n\t\t\twhere $\\Omega$ denotes the number of pairs $(i, Z)$ such that $i\\in I$, \n\t\t\t$\\max(Z_\\nu)\\in Z\\in Z_\\nu^{(s-1)}$, and $g'_i(Z)=-1$.\n\t\\end{enumerate}\n\t\n   Observe that~\\ref{it:51} and~\\ref{it:52} are fairly standard conditions in arguments concerning \n   end-homogeneity. Except for the presence of $\\Omega$ so is~\\ref{it:54}. Clause~\\ref{it:53}\n   interacts well with the demand that the functions $g'_i$ be decreasing. \n  \n\tWe contend that $1$ is secure. If $r\\ge 3$ this can be seen by taking $Z_1=\\{\\min(X)\\}$ \n\tand~$A_1=X\\setminus Z_1$. In the special case $r=2$ we still set~$Z_1=\\{\\min(X)\\}$ but now \n\twe need to pay some attention to condition~\\ref{it:51}. The Schubfachprinzip \n\tyields a colour~$\\gamma\\in \\Gamma$ and a set $A_1\\subseteq X\\setminus Z_1$ of size \n\t$|A_1|\\ge (|X|-1)\/r$ such that $f(\\min(X), a_1)=\\gamma$ holds for all~$a_1\\in A_1$. \n\tSince $A_1$ is at least as large as~\\ref{it:54} demands, $1$ is indeed secure.  \n\t\n\tNotice that if $n$ turns out to be secure, then any set $Z_n$ exemplifying this \n   will have the desired properties. So it suffices to prove that the largest secure number, \n   denoted by $\\nu(\\star)$ in the sequel, is equal to $n$. \n  \n   Assume for the sake of contradiction that $\\nu(\\star)<n$ and let the sets $Z_{\\nu(\\star)}$, \n   $A_{\\nu(\\star)}$, the function  $f'\\colon Z_{\\nu(\\star)}^{(r-1)}\\lra \\Gamma$, and the \n   family $(g'_i)_{i\\in I}$ of decreasing colourings $g'_i\\colon Z_{\\nu(\\star)}^{(s-1)}\\lra \\Phi$\n   witness the security of $\\nu(\\star)$. \n   Due to $\\Omega\\le |I|\\binom{\\nu(\\star)-1}{s-2} < |I|n^{s-2}$ \n   the second factor in~\\ref{it:54} is integral and we may assume without loss of generality \n   that \n   %\n   \\[\n   \t|A_{\\nu(\\star)}|\n\t\t=\n\t\t(2|\\Gamma|)^{n^{r-2}(n-\\nu(\\star))} \\cdot (2|I|n^{s-2})^{|I|n^{s-2}-\\Omega}\\,.\n\t\\]\n\n   \n   Now the plan is to reach a contradiction by establishing the security of $\\nu(\\star)+1$.\n   To this end we shall add a new element to $Z_{\\nu(\\star)}$ and extend the functions $f'$\n   and $g'_i$ appropriately. The conditions that seems hardest to maintain is~\\ref{it:53}. \n   In fact, this task requires some preparations that have to be carried out before one decides which\n   element gets added to~$Z_{\\nu(\\star)}$. \n    \n   Set $z_{\\nu(\\star)}=\\max(Z_{\\nu(\\star)})$ and define \n   %\n   \\[\n \t\tL\t\t\n\t\t=\n\t\t\\bigl\\{(i, Z)\\in I\\times Z_{\\nu(\\star)}^{(s-2)}\n\t\t\t\\colon \\text{ If } z_{\\nu(\\star)}\\not\\in Z, \n\t\t\t\t\\text{ then } g'_i(Z\\cup\\{z_{\\nu(\\star)}\\})=+1\\bigr\\}\\,.\n\t\\]\n\t%\n\tA pair of sets $(W, D)$ with $W\\subseteq L$ and\n\t$D\\subseteq A_{\\nu(\\star)}$ is said to be a {\\it candidate} \n\tif \n\t\\begin{enumerate}\n\t\t\\item[$\\bullet$] for all $(i, Z)\\in W$ and all $dd'\\in D^{(2)}$ \n\t\t\twe have $g_i(Z\\cup\\{d, d'\\})=-1$, \n\t\t\\item[$\\bullet$] \tand $|D|\\ge (2|I|n^{s-2})^{-|W|} |A_{\\nu(\\star)}|$. \n\t\\end{enumerate}\n\tFor instance, $(\\varnothing,  A_{\\nu(\\star)})$ is a candidate and, consequently, we can \n\tpick a candidate $(W_\\star, D_\\star)$ for which $|W_\\star|$ is maximal. \n\tSince $\\Omega+|W_\\star|\\le |I|\\binom{\\nu(\\star)}{s-2}\\le |I|n^{s-2}$,\n\twe may again assume that \n\t%\n\t\\[\n\t\t|D_\\star|\n\t\t= \n\t\t(2|I|n^{s-2})^{-|W_\\star|} |A_{\\nu(\\star)}|\n\t\t=\n\t\t(2|\\Gamma|)^{n^{r-2}(n-\\nu(\\star))} \\cdot (2|I|n^{s-2})^{|I|n^{s-2}-\\Omega-|W_\\star|}\\,.\n\t\\]\n\t%\n\t\n\tNow we partition $D_\\star$ into two consecutive halves, i.e., we write $D_\\star=D'\\charfusion[\\mathbin]{\\cup}{\\cdot} D''$ \n\tsuch that $|D'|=|D''|=\\frac12 |D_\\star|$ and $D'<D''$. Put $d''_\\star=\\min(D'')$ and consider \n\tfor every pair $(i, Z)\\in L \\setminus W_\\star$ the set \n\t%\n\t\\[\n\t\tX(i, Z)=\\bigl\\{d'\\in D'\\colon g_i(Z\\cup \\{d', d''_\\star\\})=-1\\bigr\\}\\,.\n\t\\]\n\t%\n\t\n\tAssume for the sake of contradiction that there exists a pair $(i, Z)\\in L\\setminus W_\\star$\n\tsuch that $|X(i, Z)|\\ge (2|I|n^{s-2})^{-1}|D_\\star|$. Now for all $y<y'$ from $X(i, Z)$\n\tthe assumption that $g_i$ be increasing \n\tyields $g_i(Z\\cup \\{y, y'\\})\\le g_i(Z\\cup \\{y, d''_\\star\\})=-1$, \n\twhence $g_i(Z\\cup \\{y, y'\\})=-1$. In other words, $\\bigl(W_\\star\\cup\\{(i, Z)\\}, X(i, Z)\\bigr)$\n\tis a candidate contradicting the maximality of $|W_\\star|$. \n\t\n\tWe have thereby proved $|X(i, Z)|< (2|I|n^{s-2})^{-1}|D_\\star|$ for all pairs \n\t$(i, Z)\\in L\\setminus W_\\star$. Since $|L\\setminus W_\\star|\\le |L|\\le |I|n^{s-2}$,\n\tthis implies \n\t%\n\t\\[\n\t\t\\bigg|\\bigcup_{(i, Z)\\in L\\setminus W_\\star}X(i, Z)\\bigg|<\\tfrac12|D_\\star|=|D'|\\,.\n\t\\]\n\n\tTherefore, we can pick an element \n\t%\n\t\\[\n\t\tz_{\\nu(\\star)+1}\\in D'\\setminus \\bigcup_{(i, Z)\\in L\\setminus W_\\star}X(i, Z)\n\t\\]\n\n\tand set $Z_{\\nu(\\star)+1}=Z_{\\nu(\\star)}\\cup\\{z_{\\nu(\\star)+1}\\}$. \n\t\n\tWith the help of this set we intend to convince ourselves that $\\nu(\\star)+1$ is secure, \n\twhich will contradict our maximal choice of $\\nu(\\star)$ and thus conclude the proof \n\tof Lemma~\\ref{lem:2150}. Towards this goal, we need to specify an appropriate \n\tset $A_{\\nu(\\star)+1}$, a function $f''\\colon Z_{\\nu(\\star)+1}^{(r-1)}\\lra \\Gamma$, \n\tand a family $(g''_i)_{i\\in I}$ of decreasing \n\tcolourings $g''_i\\colon Z_{\\nu(\\star)+1}^{(s-1)}\\lra \\Phi$ such that the \n\tconditions~\\ref{it:51}\\,--\\,\\ref{it:54} hold in this setting. These choices will be made \n\tin such a way that $A_{\\nu(\\star)+1}\\subseteq D''$, the function $f''$ extends $f'$, and \n\tfor every $i\\in I$ the colouring $g''_i$ extends $g'_i$. \n\t\n\tOur first step consists in determining $A_{\\nu(\\star)+1}$ and $f''$. To this end, we define \n\tfor every $d''\\in D''$ the function $h_{d''}\\colon Z_{\\nu(\\star)}^{(r-2)}\\lra \\Gamma$ by \n\t%\n\t\\[\n\t\th_{d''}(Z)=f(Z\\cup\\{z_{\\nu(\\star)+1}, d''\\})\n\t\t\\quad \\text{ for all } Z\\in Z_{\\nu(\\star)}^{(r-2)}\\,.\n\t\\]\n\t%\n\tSince $|Z_{\\nu(\\star)}^{(r-2)}|\\le n^{r-2}$, there are at most $|\\Gamma|^{n^{r-2}}$ possible \n\tfunctions from $Z_{\\nu(\\star)}^{(r-2)}$ to $\\Gamma$ and the Schubfachprinzip yields \n\ta function $h_\\star\\colon Z_{\\nu(\\star)}^{(r-2)}\\lra \\Gamma$ together with a \n\tset $A_{\\nu(\\star)+1}\\subseteq D''$ satisfying $h_{d''}=h_\\star$ for all $d''\\in A_{\\nu(\\star)+1}$\n\tand $|A_{\\nu(\\star)+1}|\\ge |\\Gamma|^{-n^{r-2}}|D''|$. For later use we remark that \n\t%\n\t\\[\n\t\t |A_{\\nu(\\star)+1}|\n\t\t \\ge \n\t\t \\frac{|D''|}{|\\Gamma|^{n^{r-2}}}\n\t\t =\n\t\t \\frac{|D_\\star|}{2|\\Gamma|^{n^{r-2}}}\n\t\t \\ge\n\t\t (2|\\Gamma|)^{n^{r-2}(n-\\nu(\\star)-1)} \\cdot (2|I|n^{s-2})^{|I|n^{s-2}-\\Omega-|W_\\star|}\\,.\n\t\\]\n\t%\n\t\t  \n\tNow we define the extension $f''\\supseteq f'$ by $f''(Z\\cup \\{z_{\\nu(\\star)+1}\\})=h_\\star(Z)$\n\tfor all $Z\\in Z_{\\nu(\\star)}^{(r-2)}$ and observe that~\\ref{it:51} clearly remains valid. \n\t\n\tProceeding with the extensions $g''_i\\supseteq g'_i$ for $i\\in I$ we set \n\t%\n\t\\[\n\t\tg''_i(Z\\cup \\{z_{\\nu(\\star)+1}\\})=\n\t\t\\begin{cases}\n\t\t\t+1 & \\text{if } (i, Z)\\in L\\setminus W_\\star \\\\\n\t\t\t-1 & \\text{otherwise } \n\t\t\\end{cases}\n\t\\]\n\t%\n\tfor all $i\\in I$ and $Z\\in Z_{\\nu(\\star)}^{(s-2)}$. Let us start the discussion of these \n\tcolourings by verifying that they are indeed decreasing. In other words, we want to check \n\tthat if $i\\in I$ and $Z\\in (Z_{\\nu(\\star)}\\setminus \\{z_{\\nu(\\star)}\\})^{(s-2)}$ satisfy \n\t$g''(Z\\cup \\{z_{\\nu(\\star)}\\})=-1$, then $g''(Z\\cup \\{z_{\\nu(\\star)+1}\\})=-1$ follows.\n\tTo this end we just need to observe that $g''(Z\\cup \\{z_{\\nu(\\star)}\\})=-1$ implies \n\t$(i, Z)\\not\\in L$. \n\t\n\tFor the confirmation of~\\ref{it:52} it suffices to argue that\n\tif $i\\in I$, $Z\\in Z_{\\nu(\\star)}^{(s-2)}$, and $d''\\in D''$, then \n\t$g_i(Z\\cup\\{z_{\\nu(\\star)+1}, d''\\})=g''_i(Z\\cup \\{z_{\\nu(\\star)+1}\\})$. \n\tIf $(i, Z)\\in W_\\star$, then the first bullet in the definition of $(W_\\star, D_\\star)$\n\tbeing a candidate yields indeed $g_i(Z\\cup\\{z_{\\nu(\\star)+1}, d''\\})=-1$. \n\tNext, if $(i, Z)\\in L\\setminus W_\\star$, then $z_{\\nu(\\star)+1}\\not\\in X(i, Z)$\n\tentails $g_i(Z\\cup \\{z_{\\nu(\\star)+1}, d''_\\star\\})=+1$ and thus the assumption that $g_i$ be \n\tincreasing and $d''_\\star=\\min(D'')\\le d''$ reveal $g_i(Z\\cup\\{z_{\\nu(\\star)+1}, d''\\})=+1$,\n\tas desired. Finally, suppose that $(i, Z)\\not\\in L$, whence $z_{\\nu(\\star)}\\not\\in Z$\n\tand $g'_i(Z\\cup\\{z_{\\nu(\\star)}\\})=-1$. Since $g'_i$ satisfies~\\ref{it:53} \n\tand $z_{\\nu(\\star)+1}, d''\\in A_{\\nu(\\star)}$, we have \n\tindeed $g_i(Z\\cup \\{z_{\\nu(\\star)+1}, d''\\})=-1$. Altogether, we have thereby proved \n\tthat the colourings $g''_i$ still obey condition~\\ref{it:52}.\n\t\n\tTo handle the new cases of~\\ref{it:53} we show that if $Z\\in Z_{\\nu(\\star)}^{(s-2)}$ \n\tand either $(i, Z)\\in W_\\star$ or $(i, Z)\\not\\in L$, then $g_i(Z\\cup\\{y, y'\\})=-1$ holds for \n\tall $yy'\\in D_\\star^{(2)}$. For $(i, Z)\\in W_\\star$ this is a direct consequence \n\tof $(W_\\star, D_\\star)$ being a candidate, so we may suppose $(i, Z)\\not\\in L$ from now on,\n\twhich, let us recall, means $z_{\\nu(\\star)}\\not\\in Z$ and $g'_i(Z\\cup\\{z_{\\nu(\\star)}\\})=-1$.  \n\tSo the desired conclusion is already covered by $g'_i$ satisfying~\\ref{it:53}.\n\t\n\tFinally, we need to verify the lower bound on $|A_{\\nu(\\star)+1}|$ promised by~\\ref{it:54}. \n\tDenoting the number of pairs $(i, Z)$ such that $i\\in I$, \n\t$z_{\\nu(\\star)+1}\\in Z\\in Z_{\\nu(\\star)+1}^{(s-1)}$, and $g''_i(Z)=-1$ by $\\Omega'$ it \n\tsuffices to check that $\\Omega' = \\Omega+|W_\\star|$. Since $\\Omega'$ counts the number \n\tof pairs $(i, Z)\\in I\\times Z_{\\nu(\\star)}^{(s-2)}$ \n\tsatisfying $g''_i(Z\\cup\\{z_{\\nu(\\star)+1}\\})=-1$, or in other words $Z\\not\\in L\\setminus W_\\star$,\n\twe have indeed \n\t%\n\t\\begin{align*}\n\t\t\\Omega'\n\t\t&=\n\t\t|I\\times Z_{\\nu(\\star)}^{(s-2)}\\setminus (L\\setminus W_\\star)| \\\\\n\t\t&=\n\t\t|I\\times Z_{\\nu(\\star)}^{(s-2)}\\setminus L|+|W_\\star| \\\\\n\t\t&=\n\t\t\\big|\\bigl\\{(i, Z)\\in I\\times Z_{\\nu(\\star)}^{(s-2)}\\colon \n\t\t\tz_{\\nu(\\star)}\\not\\in Z \\text{ and } g'_i(Z\\cup\\{z_{\\nu(\\star)}\\})=-1 \\bigr\\}\\big|+|W_\\star|\\\\\n\t\t&=\n\t\t\\Omega+|W_\\star|\\,.\n\t\\end{align*}\n\t%\n\t\n\tThis concludes the proof that $\\nu(\\star)+1$ is secure and, as we said before, this contradiction \n\tto the choice of $\\nu(\\star)$ establishes Lemma~\\ref{lem:2150}.   \n\\end{proof}\n\nWe remark that the case $I=\\varnothing$ can be handled similarly but much easier. \nIn this case one can ignore all parts of the proof addressing $I$ and, moreover,\nthere is no need to divide the set $D_\\star=A_{\\nu(\\star)}$ into two halves. Therefore, one \nobtains the following well-known statement. \n\n\\begin{cor}\\label{cor:0053}\n\tIf $n\\ge r\\ge 2$ and $N\\ge |\\Gamma|^{n^{r-1}}$, then for every \n\tcolouring $f\\colon [N]^{(r)}\\lra \\Gamma$ there are a set $X\\subseteq [N]$\n\tof size $n$ and a function $f'\\colon X^{(r-1)}\\lra \\Gamma$ governing the \n\trestriction of $f$ to $X^{(r)}$. \\hfill $\\Box$  \n\\end{cor}\n  \n\\subsection{Definite orderings} \\label{subsec:32}\nResuming the task of proving Theorem~\\ref{thm:upper} we shall now study \norderings $([N]^{(k)}, \\strictif)$ that are ``almost canonical'' in the sense \nthat we can easily extract a sign vector $\\eps=(\\eps_1, \\dots, \\eps_k)\\in \\{+1, -1\\}^k$ and a \npermutation $\\sigma\\in \\mathfrak{S}_k$ from them such that $\\strictif$ behaves in some important ways \nlike the canonical ordering associated with $(\\eps, \\sigma)$. For transparency we shall \nassume $N\\ge 2k+1$ throughout this investigation. \n\nObserve that in the canonical case, an entry $\\eps_i=+1$ in the sign vector indicates that \nwe can increase a set $x\\in [N]^{(k)}$ with respect to $\\strictif$ by increasing its element \nin the $i^{\\mathrm{th}}$ position, whereas an entry $\\eps_i=-1$ signifies that it is the other \nway around. When $\\strictif$ is arbitrary and $i\\in [k]$ we call a \ntriple $S=(X, A, Y)\\in [N]^{(i-1)}\\times [N]^{(2)}\\times [N]^{(k-i)}$ with $X<A<Y$ \nan {\\it $i$-decider}.\nNotice that $i$-deciders are in natural bijective correspondence with $(k+1)$-subsets of $[N]$.\nIf $S=(X, A, Y)$ is such an $i$-decider and $A=\\{c, d\\}$ with $c<d$ we may compare the sets \n$S(c)=X\\cup\\{c\\}\\cup Y$ and $S(d)=X\\cup\\{d\\}\\cup Y$ with respect to~$\\strictif$. \nIf $S(c)\\strictif S(d)$ we set $\\eps_i(S)=+1$ and in case $S(d)\\strictif S(c)$ we put $\\eps_i(S)=-1$.\nSo roughly speaking, $\\eps_i(S)=+1$ means that the $i$-decider $S$ ``thinks'' that $\\eps_i$ should\nhave the value~$+1$. If all $i$-deciders $S$ agree and yield the same sign $\\eps_i(S)$ we say \nthat $\\strictif$ is {\\it $i$-definite} and denote the common value of all $\\eps_i(S)$ by $\\eps_i$. \nIt may happen that $\\strictif$ is $i$-definite for every $i\\in [k]$ and in this case we \ncall the ordering $\\strictif$ {\\it sign-definite} and $\\eps=(\\eps_1, \\dots, \\eps_k)$ is referred \nto as its {\\it sign-vector}.\nClearly, the canonical ordering associated with a pair $(\\eps, \\sigma)\\in \\{+1, -1\\}^k\\times \\mathfrak{S}_k$\nis sign-definite and has sign-vector $\\eps$ in this sense. \n\nNext we explain the extraction of a permutation from an ``almost canonical'' ordering. \nLet $ij\\in [k]^{(2)}$ with $i<j$ be an arbitrary pair of indices. \nObserve that for a permutation~$\\sigma\\in \\mathfrak{S}_k$, the number $i$ occurs before $j$ in the list \n$\\sigma(1), \\dots, \\sigma(k)$ if and only if $\\sigma^{-1}(i) < \\sigma^{-1}(j)$. \nSo for the canonical ordering considered in Definition~\\ref{def:canonical} \n$\\sigma^{-1}(i) < \\sigma^{-1}(j)$ \nindicates that $i$ has in the following sense higher priority than $j$. \nIf we take a set $x\\in [N]^{(k)}$ and change \nboth its $i^{\\mathrm{th}}$ and its $j^{\\mathrm{th}}$ element, while keeping the relative positions \nof the elements fixed, then whether we increase or decrease $x$ with respect to $\\strictif$ is \ndecided by $\\eps_i$ and the direction in which we move the $i^{\\mathrm{th}}$ element alone. \nSimilarly $\\sigma^{-1}(j) < \\sigma^{-1}(i)$ means that $j$ has higher priority than $i$. \nNow suppose that the \nordering $\\strictif$ is not necessarily canonical but still sign-definite with \nsign vector $\\eps=(\\eps_1, \\dots, \\eps_k)$. By an {\\it $ij$-decider}\nwe mean a quintuple \n\\[\n\tP=(X, A, Y, B, Z)\\in [N]^{(i-1)}\\times [N]^{(2)}\\times [N]^{(j-i-1)}\n\t\t\\times [N]^{(2)}\\times [N]^{(k-j)}\n\\]\nwith $X<A<Y<B<Z$. \nSo $ij$-deciders are in natural bijective correspondence with $(k+2)$-subsets of $[N]$. \nGiven such an $ij$-decider $P=(X, A, Y, B, Z)$ and two elements \n$a\\in A$, $b\\in B$ we set $P(a, b)=X\\cup \\{a\\}\\cup Y\\cup \\{b\\}\\cup Z$. \nThe restriction of $\\strictif$ to the four-element set \n\\[\n\t\\langle P\\rangle=\\bigl\\{P(a, b)\\colon (a, b)\\in A\\times B\\bigr\\}\n\\]\nis largely determined by $\\eps_i$ and $\\eps_j$. \nNotably, if we write $A=\\{c_i, d_i\\}$ and $B=\\{c_j, d_j\\}$ such that $\\eps_ic_i<\\eps_id_i$ and \n$\\eps_jc_j<\\eps_jd_j$, then the $\\strictif$-minimum of $\\langle P\\rangle$ is $P(c_i, c_j)$ and \nthe $\\strictif$-maximum is~$P(d_i, d_j)$. So the only piece of information we are lacking \nis how $P(c_i, d_j)$\ncompares to~$P(d_i, c_j)$. We let $\\sigma_{ij}(P)$ denote \n\\begin{center}\n\\begin{tabular}{c|c}\nthe number & provided that  \\\\ \\hline \n\\rule{0pt}{3ex}  $+1$ &  $P(c_i, d_j)\\strictif P(d_i, c_j)$, \\\\\n\\rule{0pt}{3ex}  $-1$ &  $P(d_i, c_j)\\strictif P(c_i, d_j)$.\n\\end{tabular}\n\\end{center}\n\\smallskip\n\nSo intuitively $\\sigma_{ij}(P)=+1$ means that $P$ ``believes'' $i$ to have higher priority \nthan $j$, while in the opposite case it is the other way around. \nIf $\\sigma_{ij}(P)$ stays constant as $P$ varies over \nall $ij$-deciders we say that $\\strictif$ is $ij$-definite and denote the common value of \nall $\\sigma_{ij}(P)$ by $\\sigma_{ij}$. Finally, if $\\strictif$ is {\\it $ij$-definite} for all \npairs $ij\\in[k]^{(2)}$ we call $\\strictif$ a {\\it permutation-definite} ordering. \nWe remark that in the canonical case $\\sigma_{ij}=+1$ is equivalent \nto $\\sigma^{-1}(i)<\\sigma^{-1}(j)$ meaning that the \nstatement $\\sigma_{ij}\\cdot\\bigl(\\sigma^{-1}(j)-\\sigma^{-1}(i)\\bigr)>0$ is always valid.\nLet us check that, conversely, in the permutation-definite case we can define a permutation $\\sigma$ \nhaving this property. \n\n\\begin{lemma}\\label{lem:1518}\n\tSuppose that $N\\ge 2k$. If the ordering $([N]^{(k)}, \\strictif)$ is both sign-definite \n\tand permutation-definite, then there exists a permutation $\\sigma\\in \\mathfrak{S}_k$ such that \n\t%\n\t\\[\n\t\t\\sigma_{ij}\\cdot \\bigl(\\sigma^{-1}(j)-\\sigma^{-1}(i)\\bigr)>0\n\t\\]\n\n\tholds whenever $1\\le i<j\\le k$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $\\eps=(\\eps_1, \\dots, \\eps_k)$ be the sign-vector of $\\strictif$ and write \n\t$\\{2i-1, 2i\\}=\\{c_i, d_i\\}$ such that $\\eps_ic_i<\\eps_id_i$ holds for every $i\\in [k]$.\n\tConsider the sets \n\t%\n\t\\[\n\t\tx_i=\\{c_i\\}\\cup \\{d_j\\colon j\\ne i\\}\n\t\\]\n\t%\n\tand let $\\sigma\\in \\mathfrak{S}_k$ be the permutation \n\tsatisfying \n\t%\n\t\\[\n\t\tx_{\\sigma(1)}\\strictif \\dots \\strictif x_{\\sigma(k)}\\,.\n\t\\]\n\n\tWhenever $1\\le i<j\\le k$ the quintuple \n\t%\n\t\\[\n\t\tP_{ij}=\\bigl(\\{d_r\\colon 1\\le r < i\\}, \\{2i-1, 2i\\}, \\{d_s\\colon i<s<j\\},\n\t\t\t\\{2j-1, 2j\\}, \\{d_t\\colon j<t\\le k\\}\\bigr)\n\t\\]\n\t%\n\tis an $ij$-decider and $\\sigma_{ij}$ is the sign $+1$ if and only if $x_i\\strictif x_j$,\n\twhich is in turn equivalent to $\\sigma^{-1}(i)<\\sigma^{-1}(j)$. In other words, irrespective\n\tof whether $\\sigma_{ij}$ is positive or negative the\n\tstatement $\\sigma_{ij}\\cdot \\bigl(\\sigma^{-1}(j)-\\sigma^{-1}(i)\\bigr)>0$ holds.\n\\end{proof}\n\nThe remainder of this section deals with the question how much a sign-definite and \npermutation-definite ordering $([N]^{(k)}, \\strictif)$ with sign-vector $\\eps$ and \npermutation $\\sigma$ (obtained by means of Lemma~\\ref{lem:1518}) needs to have in \ncommon with the canonical ordering associated \nwith $(\\eps, \\sigma)$ in the sense of Definition~\\ref{def:canonical}. For instance, Lemma~\\ref{lem:1458} below asserts that under these circumstances there\nexists a dense subset $W\\subseteq [N]$ such that $(W^{(k)}, \\strictif)$ is ordered in this \ncanonical way.\n\nThe strategy we use for this purpose is the following. Suppose that $x, y\\in [N]^{(k)}$ \nare written in the form $x=\\{x_1, \\dots, x_k\\}$ and $y=\\{y_1, \\dots, y_k\\}$, \nwhere $x_1<\\dots<x_k$ and $y_1<\\dots<y_k$. Our goal is to prove that in many cases\nwe have \n\\begin{equation}\\label{eq:2718}\n\tx\\strictif y \n\t\\,\\,\\, \\Longleftrightarrow \\,\\,\\,\n\t(\\epsilon_{\\sigma(1)}x_{\\sigma(1)},\\dots,\\epsilon_{\\sigma(k)}x_{\\sigma(k)})\n\t <_{\\mathrm{lex}} \n\t (\\epsilon_{\\sigma(1)}y_{\\sigma(1)},\\dots,\\epsilon_{\\sigma(1)}y_{\\sigma(k)})\\,.\n\\end{equation}\nWe shall say that a pair of $k$-sets $(x, y)$ is {\\it $(\\eps, \\sigma)$-sound} if \nit satisfies the equivalence~\\eqref{eq:2718}. \nMoreover, we set $\\Delta(x, y)=\\{i\\in[k]\\colon x_i\\ne y_i\\}$. \nThe plan towards Lemma~\\ref{lem:1458} is to construct a sequence of sets\n$[N]=W_1\\supseteq\\dots\\supseteq W_k$ such that every pair $(x, y)$ of $k$-sets with\n$x, y\\subseteq W_t$ and $|\\Delta(x, y)|\\le t$ is $(\\eps, \\sigma)$-sound. \nThe statement that follows will assist us in the step of an induction on $t$.  \n\n\\begin{lemma}\\label{lem:1610}\n\tFor $L\\subseteq {\\mathds Z}$ let $(L^{(k)}, \\strictif)$ be an ordering that is sign-definite \n\twith sign-vector~$\\eps$ and permutation-definite with permutation~$\\sigma$. \n\tSuppose $t\\in [k-1]$ has the property that every pair $(x, y)$ of $k$-sets  \n\twith $x, y\\subseteq L$ and $|\\Delta(x, y)|\\le t$ is $(\\eps, \\sigma)$-sound.\n\t\n\t\n\tLet $I\\subseteq [k]$ be a set of $t+1$ indices and let $i\\in [k]$ be the least index \n\twith $\\sigma(i)\\in I$. Further, let $x=\\{x_1, \\dots, x_k\\}$ and $y=\\{y_1, \\dots, y_k\\}$ \n\tbe two $k$-element subsets of $L$ such that $\\Delta(x, y)=I$ and $x_{\\sigma(i)}<y_{\\sigma(i)}$. \n\tIf $a$ denotes the least member of $L$ larger than $x_{\\sigma(i)}$ and the conditions\n\t\\begin{enumerate}\n\t\t\\item[$\\bullet$] $a<y_{\\sigma(i)}$,\n\t\t\\item[$\\bullet$] $\\sigma(i)=\\min(I) \\Longrightarrow a<\\min\\{x_{\\sigma(i)+1}, y_{\\sigma(i)+1}\\}$,\n\t\t\\item[$\\bullet$] and \n\t\t\t$\\sigma(i)=\\max(I) \\Longrightarrow a>\\max\\{x_{\\sigma(i)-1}, y_{\\sigma(i)-1}\\}$\n\t\\end{enumerate} \n\t%\n\thold, then the pair $(x, y)$ is $(\\eps, \\sigma)$-sound.  \n\\end{lemma}\n\n\\begin{proof}\n\tWithout loss of generality we may assume $\\eps_{\\sigma(i)}=+1$. We are to prove \n\tthat $x\\strictif y$ and in all cases this will be accomplished by finding a $k$-set \n\t$z\\subseteq L$ such that the pairs~$(x, z)$ and~$(z, y)$ are $(\\eps, \\sigma)$-sound\n\tand $x\\strictif z\\strictif y$ holds. \n\t\n\tSuppose first that $\\min(I)<\\sigma(i)<\\max(I)$. \n\tDue to $a\\in (x_{\\sigma(i)}, y_{\\sigma(i)})\\subseteq (x_{\\sigma(i)-1}, y_{\\sigma(i)+1})$\n\tthe set $z=\\{x_1, \\dots, x_{\\sigma(i)-1}, a, y_{\\sigma(i)+1}, \\dots, y_k\\}$ has $k$ \n\telements that have just been enumerated in increasing order. Moreover $\\sigma(i)\\ne \\min(I)$\n\tyields $|\\Delta(x, z)|\\le t$ and, therefore, $(x, z)$ is indeed $(\\eps, \\sigma)$-sound, which\n\timplies $x\\strictif z$. Similarly, $\\sigma(i)\\ne \\max(I)$ leads to $z\\strictif y$.\n\t\n\tNext we suppose that $\\sigma(i)=\\min(I)$, which owing to the second bullet entails \n\t%\n\t\\[\n\t\ta<\\min\\{x_{\\sigma(i)+1}, y_{\\sigma(i)+1}\\}\\,.\n\t\\]\n\n\tLet $j\\in [k]$ be the second smallest index with $\\sigma(j)\\in I$. \n\t\n\t\\smallskip\n \t%\n\t\\begin{center}\n\t\\begin{tabular}{c|c|c}\n\n\tIf  & and & then we set $z=$\\\\ \\hline \n\t\\rule{0pt}{3ex}  $x_{\\sigma(j)}< y_{\\sigma(j)}$ & $\\sigma(j)<\\max(I)$ & \n\t\t$\\{x_1, \\dots, x_{\\sigma(i)-1}, a, x_{\\sigma(i)+1}, \\dots, x_{\\sigma(j)}, \n\t\t\ty_{\\sigma(j)+1}, \\dots, y_k\\}$,\\\\\n\t\\rule{0pt}{3ex}  $x_{\\sigma(j)}< y_{\\sigma(j)}$ & $\\sigma(j)=\\max(I)$ & \n\t\t$\\{x_1, \\dots, x_{\\sigma(i)-1}, a, x_{\\sigma(i)+1}, \\dots, x_{\\sigma(j)-1}, \n\t\t\ty_{\\sigma(j)}, \\dots, y_k\\}$,\\\\\n\t\\rule{0pt}{3ex}  $x_{\\sigma(j)}> y_{\\sigma(j)}$ & $\\sigma(j)<\\max(I)$ & \n\t\t$\\{x_1, \\dots, x_{\\sigma(i)-1}, a, y_{\\sigma(i)+1}, \\dots, y_{\\sigma(j)}, \n\t\t\tx_{\\sigma(j)+1}, \\dots, x_k\\}$,\\\\\n\t\\rule{0pt}{3ex}  $x_{\\sigma(j)}> y_{\\sigma(j)}$ & $\\sigma(j)=\\max(I)$ &\n\t\t$\\{x_1, \\dots, x_{\\sigma(i)-1}, a, y_{\\sigma(i)+1}, \\dots, y_{\\sigma(j)-1}, \n\t\t\tx_{\\sigma(j)}, \\dots, x_k\\}$.\n\t\\end{tabular}\n\t\\end{center}\n\\medskip\n\nIn all four cases we have listed the elements of $z$ in increasing order. Furthermore, \nin almost all cases we have $|\\Delta(x, z)|\\le t$ and, consequently, $x\\strictif z$. \nIn fact, the only exception occurs if we are in the second case and $t=2$. But if this happens, \nthen  \n\\[\n\tP=\\bigl(\\{x_1, \\dots, x_{\\sigma(i)-1}\\}, \\{x_{\\sigma(i)}, a\\}, \n\t\t\\{x_{\\sigma(i)+1}, \\dots, x_{\\sigma(j)-1}\\}, \\{x_{\\sigma(j)}, y_{\\sigma(j)}\\}, \n\t\t\\{y_{\\sigma(j)+1}, \\dots, y_k\\}\\bigr)\n\\]\nis an $\\sigma(i)\\sigma(j)$-decider. \nSince the ordering $\\strictif$ is permutation-definite, we know that it orders the four-element \nset $\\langle P\\rangle$ ``correctly'' and, as $x$, $z$ belong to this set, $x\\strictif z$ holds \nin this case as well. \n\nSimilarly, in almost all cases we have $\\Delta(y, z)\\le t$ and $y\\strictif z$, the only \nexception occurring if we are in the fourth case and $t=2$. Under these circumstances \n\\[\n\tQ=\\bigl(\\{x_1, \\dots, x_{\\sigma(i)-1}\\}, \\{a, y_{\\sigma(i)}\\}, \n\t\t\\{y_{\\sigma(i)+1}, \\dots, y_{\\sigma(j)-1}\\}, \\{x_{\\sigma(j)}, y_{\\sigma(j)}\\}, \n\t\t\\{x_{\\sigma(j)+1}, \\dots, x_k\\}\\bigr)\n\\]\nis an $\\sigma(i)\\sigma(j)$-decider and, as before, $y, z\\in \\langle Q\\rangle$ implies $y\\strictif z$.\n\nThis concludes our discussion of the case $\\sigma(i)=\\min(I)$ and it remains to deal with \nthe case $\\sigma(i)=\\max(I)$. Here one can either perform a similar argument, or one argues\nthat this case reduces to the previous one by reversion the ordering of the ground set. That\nis, one considers the ordering $\\bigl((-L)^{(k)}, \\strictif^\\star\\bigr)$ defined by \n\\[\n\tx\\strictif^\\star y  \n\t\\,\\,\\, \\Longleftrightarrow \\,\\,\\,\n\t(-x) \\strictif (-y)\\,,\n\\]\nwhere $-L$ means $\\{-\\ell\\colon \\ell\\in L\\}$ and $-x$, $-y$ are defined similarly. \nIf one replaces $\\strictif$ by $\\strictif^\\star$, \nthen $I^\\star=(k+1)-I$ assumes the r\\^{o}le of $I$, minima correspond to maxima and the second and \nthird bullet in the assumption of our lemma are exchanged. \n\\end{proof}\n\nWe proceed with the existence of ``dense'' canonical subsets promised earlier. \n\n\\begin{lemma}\\label{lem:1458}\n\tIf $N\\ge 2k$ and the ordering $([N]^{(k)}, \\strictif)$ is both sign-definite \n\tand permutation-definite, then there exists a set $W\\subseteq [N]$ of size $|W|\\ge 2^{1-k}N$\n\tsuch that $\\strictif$ is canonical on~$W^{(k)}$.\n\\end{lemma}\n\n\\begin{proof}\n\tFor every $t\\in [k]$ we set \n\t%\n\t\\[\n\t\tW_t=\\bigl\\{n\\in [N]\\colon n\\equiv 1\\pmod{2^{t-1}}\\bigr\\}\\,.\n\t\\]\n\t%\n\tClearly $|W_k|\\ge 2^{1-k}N$, so it suffices to prove that $\\strictif$ is canonical on~$W_k^{(k)}$.\n\t\n\tDenote the sign-vector of $\\strictif$ by $\\eps=(\\eps_1, \\dots, \\eps_k)$ and let $\\sigma\\in \\mathfrak{S}_k$\n\tbe the permutation obtained from $\\strictif$ by means of Lemma~\\ref{lem:1518}. \n\tLet us prove by induction on $t\\in [k]$ that if two $k$-element subsets $x, y\\subseteq W_t$\n\tsatisfy $|\\Delta(x, y)|\\le t$, then $(x, y)$ is $(\\eps, \\sigma)$-sound. \n\t\n\tIn the base case $t=1$ we have $W_1=[N]$ and everything follows from our assumption \n\tthat $\\strictif$ be sign-definite. In the induction step from $t\\in [k-1]$ to $t+1$ \n\twe appeal to Lemma~\\ref{lem:1610}. Since no two elements of $W_{t+1}$ occur in consecutive \n\tpositions of $W_t$, the bulleted assumptions are satisfied and thus there are no problems \n\twith the induction step. \n\t\n\tSince $|\\Delta(x, y)|\\le k$ holds for all $k$-element subsets of $W_k$, the case $t=k$ of \n\tour claim proves the lemma. \n\\end{proof}\n\nNext we characterise the canonical orderings on $([N]^{(k)}, \\strictif)$\nfor $N\\ge 2k+1$. \n\n\\begin{lemma}\\label{lem:canonical}\n\tIf $N\\ge 2k+1$ and the ordering $([N]^{(k)}, \\strictif)$ is canonical, then there exists\n\ta pair $(\\eps, \\sigma)\\in \\{-1, +1\\}^k\\times \\mathfrak{S}_k$ such that $\\strictif$ is the ordering \n\tassociated with $(\\eps, \\sigma)$. \n\\end{lemma}\n\n\\begin{proof}\n\tLet $\\eps$ and $\\sigma$ denote the sign vector and the permutation of $\\strictif$ as introduced\n\tin \\S\\ref{subsec:32}.  \n\tWe shall show that $\\strictif$ coincides with the canonical ordering associated to $(\\eps, \\sigma)$.\n\t\n\tOtherwise there exists a pair $(x, y)$ of subsets of $[N]$ that fails to be $(\\eps, \\sigma)$-sound.\n\tAssume that such a pair $(x, y)$ is chosen with $t+1=|\\Delta(x, y)|$ minimum. Since $\\strictif$ \n\tis sign-definite we know that $t\\in [k-1]$. Set $I=\\Delta(x, y)$ and let $i\\in [k]$ be the \n\tsmallest index with $\\sigma(i)\\in I$. By symmetry we may assume that $x_{\\sigma(i)}<y_{\\sigma(i)}$.\n\tDue to $|I|\\ge 2$ it cannot be the case that~$\\sigma(i)$ is both the minimum and the maximum of \n\t$I$ and as in the proof of Lemma~\\ref{lem:1610} it suffices to study the case $\\sigma(i)\\ne\\max(I)$. \n\t\n\t\\smallskip\n\n\t{\\it \\hskip2em First Case: $\\max(x\\cup y)<N$}\n\n\t\\smallskip\n \n \tDefine a strictly increasing map $\\phi\\colon x\\cup y\\lra [N]$ by $\\phi(z)=z$ \n\tfor $z\\le x_{\\sigma(i)}$ and $\\phi(z)=z+1$ for $z > x_{\\sigma(i)}$. Now by Lemma~\\ref{lem:1610}\n\tthe pair $\\bigl(\\phi[x], \\phi[y]\\bigr)$ is $(\\eps, \\sigma)$-sound. Moreover, the canonicity of\n\t$\\strictif$ tells us that $\\big((x\\cup y)^{(k)}, \\strictif\\bigr)$ \n\tand $\\big(\\phi[x\\cup y]^{(k)}, \\strictif\\bigr)$ are isomorphic via $\\phi$. For these reasons,\n\tthe pair $(x, y)$ is $(\\eps, \\sigma)$-sound as well. \n\n\t\\smallskip\n\n\t{\\it \\hskip2em Second Case: $\\max(x\\cup y)=N$}\n\n\t\\smallskip\n  \n  \tLet $\\eta\\colon x\\cup y\\lra [|x\\cup y|]$ be order-preserving. In view of $|x\\cup y|\\le 2k<N$\n\twe know from the first case that the pair $\\bigl(\\eta[x], \\eta[y]\\bigr)$ is $(\\eps, \\sigma)$-sound\n\tand by canonicity so is $(x, y)$.\n\\end{proof}\n\n\n\n\\begin{remark}\\label{rem:1945}\n\tThe results of this subsection easily yield the weaker upper \n\tbound \n\t%\n\t\\[\n\t\t\\mathrm{L}^{(k)}(n)\n\t\t\\le \n\t\t\\mathrm{R}^{(k+2)}\\left(2^{k-1}n,\\tbinom{k+2}{2}!\\right)\n\t\t\\le \n\t\tt_{k+1}(C_k n)\\,.\n\t\\]\n\n\t\n\tTo see this, let $([N]^{(k)}, \\strictif)$ be any ordering, \n\twhere $N=\\mathrm{R}^{(k+2)}(2^{k-1}n, \\binom{k+2}{2}!)$. For a set $A\\in [N]^{(k+2)}$\n\tthere are $\\binom{k+2}{2}!$ possibilities how the restriction of $\\strictif$ to $A^{(k+2)}$\n\tcan look. So by the definition of Ramsey numbers there is a set $Z\\subseteq [N]$ with \n\t$|Z|=2^{k-1}n$ such that \n\tfor all $A, B\\subseteq Z$ with $|A|=|B|=k+2$ the order-preserving map $\\eta\\colon A\\lra B$\n\tinduces an order preserving map from $(A^{(k)}, \\strictif)$ to $(B^{(k)}, \\strictif)$. \n\tThis implies, in particular, that~$\\strictif$ is sign-definite and permutation-definite \n\ton $Z^{(k)}$. Thus by Lemma~\\ref{lem:1458} there exists a set~$X\\subseteq Z$ \n\twith $|X|\\ge 2^{1-k}|Z|=n$ such that $\\strictif$ orders $X^{(k)}$ canonically. \n\\end{remark}\n\nNow it remains to save one exponentiation, which requires a more careful reasoning.  \n\n\\subsection{Further concepts}\n\nThroughout this subsection, we fix a natural number $k\\ge 2$. \nLet~$\\Gamma$ denote the set of all orderings $\\YY$ that can be imposed on $[k+2]^{(k)}$. \nWe regard $\\Gamma$ as a set of colours having the size $|\\Gamma|=\\binom{k+2}{2}!$. \nAs in Remark~\\ref{rem:1945} we {\\it associate} with every ordering $([N]^{(k)}, \\strictif)$ \nthe colouring $f\\colon [N]^{(k+2)}\\lra \\Gamma$ mapping every $A\\in [N]^{(k+2)}$ to the unique\nordering $\\YY\\in \\Gamma$ with the property that $(A^{(k)}, \\strictif)$ and $([k+2]^{(k)}, \\YY)$\nare isomorphic via the order-preserving map from $A$ to $[k+2]$. Roughly speaking, what \nTheorem~\\ref{thm:upper} asserts is that colourings of this form are so special that one can \nprove a Ramsey theorem for them that requires just $k$ exponentiations rather than the \nexpected $k+1$ exponentiations. \n\nFor an integer $m\\in [k+1]$ we shall say that an ordering $([N]^{(k)}, \\strictif)$ \n{\\it is governed by $m$-sets} if there exists a function $f^m\\colon [N]^{(m)}\\lra \\Gamma$\ngoverning the function $f\\colon [N]^{(k+2)}\\lra \\Gamma$ associated with $\\strictif$.\n\n\\begin{dfn}\n\tIf $([N]^{(k)}, \\strictif)$ is an ordering, $i, m\\in [k]$, and $W\\subseteq [N]$ \n\twe say that $\\eps_i^m\\colon W^{(m)}\\lra \\Phi$ is an {\\it $m$-ary sign function} \n\tfor $(W^{(k)}, \\strictif)$ provided the following holds: If $S=(X, A, Y)$ is an \n\t$i$-decider with $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y\\subseteq W$ and $M$ denotes the set consisting of the $m$ smallest \n\telements of $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y$, then $\\eps_i(S)=\\eps^m_i(M)$.  \n\\end{dfn}\n\nIn other words, the existence of an $m$-ary sign function $\\eps_i^m\\colon W^{(m)}\\lra \\Phi$\nmeans that for $i$-deciders $S=(X, A, Y)$ with $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y\\subseteq W$ the value of \n$\\eps_i(S)$ only depends on the first $m$ elements of $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y$, while the \nremaining $k+1-m$ elements of $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y$ are destitute of any effect. \n\nWe shall frequently utilise the fact that if $m\\in [2, k+1]$ and $([N]^{(k)}, \\strictif)$ is \ngoverned by $m$-sets, then for every $i\\in [k]$ there exists an $(m-1)$-ary sign function\n$\\eps^{m-1}_i\\colon [2, N]^{(m-1)}\\lra \\Phi$. This is because for an $i$-decider $S=(X, A, Y)$ \nwith $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y\\subseteq [2, N]$ the value of $\\eps_i(S)$ can be inferred\nfrom the colour $f(\\{1\\}\\charfusion[\\mathbin]{\\cup}{\\cdot} X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y)$, where $f\\colon [N]^{(k+2)}\\lra \\Gamma$\ndenotes the function associated with $\\strictif$.\n\n\nSuppose that $m\\in [2, k+1]$, $W\\subseteq {\\mathds Z}$ and that $(W^{(k)}, \\strictif)$ denotes an \nordering possessing for every $i\\in [k]$ an $(m-1)$-ary sign \nfunction $\\eps^{m-1}_i\\colon W^{(m-1)}\\lra \\Phi$. If $i<j$ from $[k]$ are two indices \nand $P=(X, A, Y, B, Z)$ is an $ij$-decider, we determine the colour $\\sigma^m_{ij}(P)$\t\nas follows: Let $M$ denote the set consisting of the $m-1$ smallest elements of  \n$X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y\\charfusion[\\mathbin]{\\cup}{\\cdot} B\\charfusion[\\mathbin]{\\cup}{\\cdot} Z$, enumerate $A=\\{c_i, d_i\\}$ as well as $B=\\{c_j, d_j\\}$ \nin such a way that $\\eps^{m-1}_i(M)c_i<\\eps^{m-1}_i(M)d_i$ \nand $\\eps^{m-1}_j(M)c_j<\\eps^{m-1}_j(M)d_j$, and set  \n\n\\begin{center}\n\\begin{tabular}{c|c}\n$\\sigma^m_{ij}(P)=$ & provided that  \\\\ \\hline \n\\rule{0pt}{3ex}  $+1$ &  $P(c_i, d_j)\\strictif P(d_i, c_j)$, \\\\\n\\rule{0pt}{3ex}  $-1$ &  $P(d_i, c_j)\\strictif P(c_i, d_j)$.\n\\end{tabular}\n\\end{center}\n\\smallskip\n\n\n\\begin{dfn}\n\tIf $m\\in [2, k+1]$, $ij\\in [k]^{(2)}$, and the ordering $(W^{(k)}, \\strictif)$ \n\tpossesses $(m-1)$-ary sign \n\tfunctions $\\eps^{m-1}_1, \\dots, \\eps^{m-1}_k$, then we say that \n\t$\\sigma^m_{ij}\\colon W^{(m)}\\lra \\Phi$ is an {\\it $m$-ary permutation colouring} if \n\t$\\sigma^m_{ij}(P)=\\sigma^m_{ij}(M)$ holds, whenever $P=(X, A, Y, B, Z)$ is an $ij$-decider  \n\tand~$M$ denotes the set of the $m$ smallest elements of $X\\charfusion[\\mathbin]{\\cup}{\\cdot} A\\charfusion[\\mathbin]{\\cup}{\\cdot} Y\\charfusion[\\mathbin]{\\cup}{\\cdot} B\\charfusion[\\mathbin]{\\cup}{\\cdot} Z$.\n\\end{dfn}\n\nIt should be clear that if in addition to possessing the $(m-1)$-ary sign \nfunctions $\\eps^{m-1}_1, \\dots, \\eps^{m-1}_k$ the ordering $(W^{(k)}, \\strictif)$\nis governed by $m$-sets, then all $m$-ary permutation colourings $(\\sigma^m_{ij})$\nwith $ij\\in [k]^{(2)}$ exist. Consequently, if $([N]^{(k)}, \\strictif)$ is governed \nby $m$-sets, then for every pair $ij\\in [k]^{(2)}$ there exists a permutation colouring \n$\\sigma^m_{ij}\\colon [2, N]^{(m)}\\lra \\Phi$.\n\n\\subsection{Plan} \\label{subsec:plan}\nLet us now pause to explain our strategy for proving Theorem~\\ref{thm:upper}. \nGiven an arbitrary ordering $([N]^{(k)}, \\strictif)$ with $N=t_{k}(n^{C_k})$\nwe need to find a set $X\\subseteq [N]$ of size $n$ such that $(X^{(k)}, \\strictif)$\nis ordered canonically.\nFollowing the same strategy as in Remark~\\ref{rem:1945} it would suffice to find a set \n$Y\\subseteq [N]$ of size $2^{k-1}n$ which is monochromatic with respect to the \ncolouring $f\\colon [N]^{(k+2)}\\lra \\Gamma$ associated with $\\strictif$. So the problem \nis that we have only~$k$ exponentiations available but hope to synchronise \nsets of size $k+2$. What happens when we nevertheless perform the standard argument producing \nupper bounds on Ramsey numbers is that we obtain \nsets $[N]=Z_k\\supseteq Z_{k-1}\\supseteq \\dots \\supseteq Z_0$\nsuch that for every $m\\in [0, k]$ the ordering~$(Z_m^{(k)}, \\strictif)$ is governed by $(m+2)$-sets \nand the size of $Z_m$ is, roughly, $m$ times exponential. In particular,~$Z_0$ has roughly the size \n$n^{C_k\/2}$ and $(Z_0^{(k)}, \\strictif)$ is governed by pairs. We may assume further that for \nevery pair~$ij\\in [k]^{(2)}$ a permutation colouring $\\sigma^2_{ij}\\colon Z_0^{(2)}\\lra \\Phi$ \nexists. Moreover, it would suffice to find a set $Y\\subseteq Z_0$ that is monochromatic with respect \nto each~$\\sigma^2_{ij}$.\nThis problem still sounds like we need an exponential rather than a polynomial dependence \nbetween~$|Y|$ and~$|Z_0|$, but Lemma~\\ref{lem:1731} suggests the following alternative: if it \nturns out that the colourings $\\sigma^2_{ij}$ are monotone, then we are done (for reasons explained \nin Lemma~\\ref{lem:0127}). \n\nNow for some of the permutation \ncolourings, like $\\sigma^2_{1j}$ with $j\\in [2, k]$, one can indeed prove that they are always monotone\n(see Lemma~\\ref{lem:2222} below), but it seems unlikely that in the setting we have \ndescribed so far one can prove that, for instance, $\\sigma^2_{57}$ is always monotone. To address\nthis issue we exploit that the colourings $\\sigma^2_{ij}$ become known ``gradually'' in the course \nof the argument. When we have just constructed a set $Z_m$ we already know a colouring $\\sigma^{m+2}_{ij}$\nof its $(m+2)$-subsets such that, later, $\\sigma^2_{ij}$ will govern its restriction to~$Z_0^{(m+2)}$.\nSuppose that in the sequence of colourings $\\sigma^k_{57}, \\sigma^{k-1}_{57}, \\dots, \\sigma^2_{57}$ \nthere is at least one that is monotone, say $\\sigma^{m+2}_{57}$, where $m\\ge 1$. \nWhat Lemma~\\ref{lem:2150} tells us is that when selecting $Z_{m-1}\\subseteq Z_{m}$ we can not only \nensure that $(Z_{m-1}^{(k)}, \\strictif)$ is governed by $(m+1)$-sets, but also that $\\sigma^{m+1}_{57}$\nis still monotone. Iterating this observation we see that if some $\\sigma^m_{57}$ is monotone, then\nwe can protect ourselves against ever loosing the monotonicity. Rather, we can arrange that all \nof $\\sigma^m_{57}, \\sigma^{m-1}_{57}, \\dots, \\sigma^2_{57}$ are monotone. We shall see in \nLemma~\\ref{lem:2240} that for~$i\\ge 2$ the permutation colouring~$\\sigma^i_{ij}$ is automatically \nmonotone. \nFor these reasons, we can indeed find a set $Z_0$ of polynomial size on which all pair colourings \n$\\sigma^2_{ij}$ are monotone, which essentially proves Theorem~\\ref{thm:upper}. \n\n\\subsection{Canonisation}\nComing to the details we begin by establishing the monotonicity of~$\\sigma^2_{1j}$ for~$j\\in [2, k]$.   \n\n\\begin{lemma}\\label{lem:2222}\n\tIf the ordering $([N]^{(k)}, \\strictif)$ is sign-definite and governed by pairs, then for \n\tevery $j\\in [2, k]$ the colouring $\\sigma^2_{1j}$ is increasing on $[2, N-k]^{(2)}$. \n\\end{lemma}\n\n\\begin{proof}\n\tGiven three numbers $r<s<t$ from $[2, N-k]$ satisfying~$\\sigma_{1j}^2(rs)=+1$ we are to prove \n\tthat $\\sigma_{1j}^2(rt)=+1$.  \n\tLet $\\eps=(\\eps_1, \\dots, \\eps_k)$ be the sign vector of~$\\strictif$. We partition \n\t%\n\t\\[\n\t\t[N-k+1, N]=Y\\charfusion[\\mathbin]{\\cup}{\\cdot} B\\charfusion[\\mathbin]{\\cup}{\\cdot} Z \\quad \\text{ such that } Y<B<Z\n\t\\]\n\t%\n\tand $|Y|=j-2$,~$|B|=2$,~$|Z|=k-j$\n\tand we enumerate $B=\\{c_j, d_j\\}$ such that $\\eps_jc_j<\\eps_jd_j$. \n\tNotice that $P=(\\varnothing, \\{r, s\\}, Y, B, Z)$ and $Q=(\\varnothing, \\{r, t\\}, Y, B, Z)$ \n\tare $1j$-deciders, whence $\\sigma^2_{1j}(rs)=\\sigma_{1j}(P)$ and~$\\sigma^2_{1j}(rt)=\\sigma_{1j}(Q)$.\n\tThus our task is to derive $\\sigma_{1j}(Q)=+1$ from $\\sigma_{1j}(P)=+1$.\n\t\n\t\\smallskip\n\t\n\t{\\it \\hskip 2em First Case: $\\eps_1=+1$}\n\t\n\t\\smallskip\n\n\tNow $\\sigma_{1j}(P)=+1$ is defined to mean $rYd_jZ\\strictif sYc_jZ$, where $rYd_jZ$\n\tabbreviates $\\{r\\}\\cup Y\\cup \\{d_j\\}\\cup Z$ and $sYc_jZ$ is defined similarly. \n\tMoreover, $S=(\\{s, t\\}, Yc_jZ)$ is a $1$-decider and $\\eps_1(S)=\\eps_1=+1$ \n\timplies $sYc_jZ\\strictif tYc_jZ$.\n\tSo the transitivity of $\\strictif$ leads to $rYd_jZ\\strictif tYc_jZ$, for which reason\n\twe have indeed $\\sigma_{1j}(Q)=+1$. \n\t   \t\n\t\\smallskip\n\t\n\t{\\it \\hskip 2em Second Case: $\\eps_1=-1$}\n\t\n\t\\smallskip\n\t\n\tThis time $\\sigma_{ij}(P)=+1$ translates into $sYd_jZ\\strictif rYc_jZ$\n\tand working with the $1$-decider $(\\{s, t\\}, Yd_jZ)$ we infer $tYd_jZ\\strictif sYd_jZ$.\n\tHence $tYd_jZ\\strictif rYc_jZ$ and $\\sigma_{1j}(Q)=+1$.\n\\end{proof}\n\nVirtually the same proof also yields the following result.\n\n\\begin{lemma}\\label{lem:2240}\n\tIf $2\\le i<j\\le k$ and the ordering $([N]^{(k)}, \\strictif)$ is governed by $i$-sets, \n\tthen the restriction of $\\sigma^i_{ij}$ to $[2, N-k]^{(i)}$ is decreasing.\n\\end{lemma}\n\n\\begin{proof}\n\tSuppose that $X\\subseteq [2, N-k]$ and $r, s\\in [2, N-k]$ satisfy $|X|=i-1$, $X<r<s$, as well as \n\t$\\sigma^i_{ij}(Xr)=-1$. We are to prove that $\\sigma^i_{ij}(Xs)=-1$. To this end we set \n\t$t=N-k+1$ and take three sets $Y, Z, B\\subseteq [N-k+2, N]$ with $Y<B<Z$ of the sizes\n\t$|Y|=j-i-1$, $|B|=2$, and $|Z|=k-j$.\n\tNow $P=(X, \\{r, t\\}, Y, B, Z)$ and $Q=(X, \\{s, t\\}, Y, B, Z)$ are two $ij$-deciders, we have \n\t$\\sigma_{ij}(P)=\\sigma^i_{ij}(Xr)=-1$, and it remains to show $\\sigma_{ij}(Q)=-1$. Recall \n\tthat~$\\strictif$ being governed by $i$-sets implies, in particular, that there are $(i-1)$-ary sign \n\tfunctions~$\\eps^{i-1}_i$ and~$\\eps^{i-1}_j$ defined on $[2, N]^{(k-1)}$. \n\tSo we can enumerate $B=\\{c_j, d_j\\}$ in such a way that \n\t$\\eps^{i-1}_j(X)c_j <\\eps^{i-1}_j(X)d_j$.\n\t\n\t\\smallskip\n\t\n\t{\\it \\hskip 2em First Case: $\\eps^{i-1}_i(X)=+1$}\n\t\n\t\\smallskip\n\t\n\tOur assumption $\\sigma^i_{ij}(P)=-1$ is defined to mean $XtYc_jZ\\strictif XrYd_jZ$. \n\tFurthermore, the triple $S=(X, \\{r, s\\}, Yd_jZ)$ is an $i$-decider \n\tand $\\eps_i(S)=\\eps^{i-1}_i(X)=+1$\n\timplies that $XrYd_jZ \\strictif XsYd_jZ$. So altogether we have $XtYc_jZ\\strictif XsYd_jZ$\n\tor, in other words, $\\sigma^i_{ij}(Q)=-1$. \n\t\n\t\\smallskip\n\t\n\t{\\it \\hskip 2em Second Case: $\\eps^{i-1}_i(X)=-1$}\n\t\n\t\\smallskip\n\n\t This time $\\sigma^i_{ij}(P)=-1$ abbreviates the statement $XrYc_jZ\\strictif XtYd_jZ$\n\t and the case hypothesis $\\eps^{i-1}_i(X)=-1$ implies $XsYc_jZ\\strictif XrYc_jZ$.\n\t For these reasons we have $XsYc_jZ\\strictif XtYd_jZ$, i.e., $\\sigma^i_{ij}(Q)=-1$.\n\\end{proof}\n\nNext we elaborate on the reason why we care about the monotonicity \nof the permutation colourings. \n\n\\begin{lemma}\\label{lem:0127}\n\tThere exists an absolute constant $C'_k$ such that if $n\\ge 2k$, $N\\ge C'_kn^{k^2}$, \n\tthe ordering $([N]^{(k)}, \\strictif)$ is governed by pairs, and all permutation colourings \n\t$\\sigma^2_{ij}$ with $i\\ge 2$ are monotone, then there is a set $Z\\subseteq [N]$ of size $n$ such that \n\t$(Z^{(k)}, \\strictif)$ is ordered canonically.\n\\end{lemma}\n\n\\begin{proof}\n\tSince the ordering $\\strictif$ is governed by pairs, it has unary sign \n\tfunctions $\\eps_1, \\dots, \\eps_k$ defined on $[2, N]^{(1)}$. \n\tIn other words, every $x\\in [2, N]$ has its own ``opinion''\n\t%\n\t\\[\n\t\t\\eps(x)=\\bigl(\\eps_1(x), \\dots, \\eps_k(x)\\bigr)\\in \\Phi^k\n\t\\]\n\t%\n\twhat the sign vector should be,\n\twhere $\\Phi=\\{-1, +1\\}$ was introduced in~\\eqref{eq:Phi}. \n\tDue to the Schubfachprinzip, there are a set $Y\\subseteq [2, N]$ of size $|Y|\\ge 2^{-k}N$\n\tas well as a vector $\\eps\\in\\Phi^k$ such that $\\eps(y)=\\eps$ holds for every $y\\in Y$. \n\tNow~$(Y^{(k)}, \\strictif)$ is sign-definite and by Lemma~\\ref{lem:2222} the set $Y_\\star\\subseteq Y$\n\tobtained from $Y$ by deleting the smallest and the~$k$ largest elements has the following \n\tproperty: for every \n\tpair $ij\\in [k]^{(2)}$ the restriction of~$\\sigma^2_{ij}$ to $Y_\\star^{(2)}$ is monotone. \n\tNotice that~$|Y_\\star|\\ge 2^{-k-1}N$. \n\t\n\tNow Lemma~\\ref{lem:1731} applied to $I=[k]^{(2)}$ and \n\tthe family~$(\\sigma^2_{ij})_{ij\\in I}$\n\tyields a set $Z_\\star\\subseteq Y_\\star$ of size $|Z_\\star|=2^{k-1}n$ that is monochromatic with\n\trespect to every~$\\sigma^2_{ij}$. This means that $(Z_\\star^{(k)}, \\strictif)$ is not only \n\tsign-definite but also permutation-definite, and Lemma~\\ref{lem:1458} leads to the desired set $Z$.   \n\\end{proof}\n\nIt remains to execute the plan laid out in \\S\\ref{subsec:plan}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:upper}]\n\tGiven $k\\ge 2$ we let $C'_k$ denote the constant delivered by the previous lemma.\n\tRecall that $\\Gamma$ denotes the set of linear orderings of $[k+2]^{(k)}$. \n\tNow for an arbitrary number $n\\ge 2k$ we define recursively \n\t%\n\t\\begin{align*}\n\t\tN_0&=C'_k n^{k^2}+k+1\\,, \\\\\n\t\tN_{m}&= (2|\\Gamma|)^{N_{m-1}^{m+1}} \\cdot (2k^2N_{m-1}^{m})^{k^2N_{m-1}^{m}}+k+1 \n\t\t\t\\text{ for }m\\in [k-2] \\,, \\\\\n\t\tN_{k-1}&= |\\Gamma|^{N_{k-2}^{k-2}}\\,,\\\\\n\t\t\\text{ and finally } N_k&= |\\Gamma|^{N_{k-1}^{k-1}}\\,.\n\t\\end{align*}\n\t%\n\t\n\tAn easy induction on $m$ discloses that for every $m\\in [0, k]$ there exists an absolute \n\tconstant $C_{k, m}$ such that $N_m\\le t_m\\bigl(C_{k, m}n^{2k^2}\\bigr)$ holds for all $n\\ge 2k$. \n\tIn particular, there exists an absolute constant $C_k$ such that independently of $n$ \n\twe have $N_k\\le t_{k}(n^{C_k})$.\n\tSo it suffices to show that for every \n\tordering $([N_k]^{(k)}, \\strictif)$ there exists a set $Z\\subseteq [N_k]$ of size $n$ \n\tsuch that $\\strictif$ orders $Z^{(k)}$ canonically. \n\tTwo successive applications of Corollary~\\ref{cor:0053} yield a set $Z_{k-2}\\subseteq [N]$ of \n\tsize $|Z_{k-2}|=N_{k-2}$ such that $({Z}_{k-2}^{(k)}, \\strictif)$ is governed by $k$-sets. \n\t\n\t\\begin{claim}\\label{clm:1632}\n\t\tFor every $m\\in [0, k-2]$ there exists a set $Z_m\\subseteq [N_k]$ of size $|Z_m|=N_m-k-1$\n\t\tsuch that $(Z_{m}^{(k)}, \\strictif)$ is governed by $(m+2)$-sets and \n\t\tfor every $ij\\in [m+2, k]^{(2)}$ there is a monotone permutation \n\t\tcolouring $\\sigma^{m+2}_{ij}\\colon Z_m^{(m+2)}\\lra\\Phi$.\n\t\\end{claim}\n\t\n\n\n\t\n\t\\begin{proof}[Proof of Claim~\\ref{clm:1632}]\n\t\tWe argue by decreasing induction on $m$, the base case $m=k-2$ already being known. \n\t\tNow suppose that $m\\in [k-2]$ and that we already have obtained a set~$Z_m\\subseteq [N_k]$ with \n\t\tthe desired \n\t\tproperty. \n\t\t\n\t\tWe plan to apply Lemma~\\ref{lem:2150}\n\t\tto $N_{m-1}$, $m+2$, $m+2$, and $Z_m$ here in place of~$n$,~$r$,~$s$, and~$X$ there.  \n\t\tInstead of~$f$ there we take the function $f^{m+2}\\colon Z_m^{(m+2)}\\lra \\Gamma$ \n\t\tgoverning the function $f\\colon [N_k]^{(k+2)}\\lra \\Gamma$ associated with $\\strictif$. \n\t\tFinally, we take the index set $I=[m+2, k]^{(2)}$,\n\t\twhich clearly \n\t\tsatisfies $|I|\\le k^2$, and the family $(\\sigma^{m+2}_{ij})_{ij\\in I}$ of monotone \n\t\tpermutation colourings $\\sigma^{m+2}_{ij}\\colon Z_m^{(m+2)}\\lra \\Phi$ provided by the \n\t\tinduction hypothesis. \n\t\tSince \n\t\t%\n\t\t\\[\n\t\t\tN_m-k-1\n\t\t\t\\ge\n\t\t\t(2|\\Gamma|)^{N_{m-1}^{m+1}} \\cdot (2|I|N_{m-1}^{m})^{|I|N_{m-1}^{m}}\\,,\n\t\t\\]\n\t\t%\n\t\tthe intended application of Lemma~\\ref{lem:2150} is justified and we can obtain a set \n\t\t$Z'_{m-1}\\subseteq Z_m$ of size $|Z'_{m-1}|=N_{m-1}$ \n\t\tsuch that $\\bigl({Z'}_{m-1}^{(k)}, \\strictif\\bigr)$ \n\t\tis governed by $(m+1)$-sets and the permutation colourings $(\\sigma^{m+1}_{ij})_{ij\\in I}$ \n\t\tare monotone. \n\t\tThe set $Z_{m-1}$ arising from $Z'_{m-1}$ by cutting off the smallest and the~$k$ \n\t\tlargest elements has all \n\t\tdesired properties, because by Lemma~\\ref{lem:2240} the permutation \n\t\tcolourings $\\sigma^{m+1}_{m+1, j}$\n\t\twith $j\\in [m+2, k]$ are monotone as well. \n\t\\end{proof}\n\t\n\tObserve that the case $m=0$ yields a set $Z_0$ of size $|Z_0|=N_0-(k+1)=C'_k n^{k^2}$. \n\tBy our choice of $C'_k$ according to Lemma~\\ref{lem:0127} there exists a set $Z\\subseteq Z_0$\n\tof size $|Z|=n$ such that $(Z^{(k)}, \\strictif)$ is ordered canonically.  \n\tThis concludes the proof of Theorem~\\ref{thm:upper}. \n\\end{proof}\n\n\\@ifstar{\\origsection*}{\\@startsection{section}{1}\\z@{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}{\\normalfont\\scshape\\centering\\S}}{Shelah's upper bound on Erd\\H{o}s-Rado numbers}\\label{sec:ERS}\nIn this section we show that the results in~\\S\\ref{subsec:31} can also be used for reproving \nShelah's theorem that $k$-uniform Erd\\H{o}s-Rado numbers grow (at most) $k-1$ times exponentially. \nMore precisely, we shall show $\\ER^{(k)}(n)\\le t_{k-1}(C_k n^{6k})$, where, let us recall, \nan estimate of the form $\\ER^{(k)}(n)\\le N$ means that for every equivalence relation~$\\equiv$ \non~$[N]^{(k)}$ there exist an $n$-element subset $X\\subseteq [N]$ and a set $I\\subseteq [k]$ \nsuch that $\\equiv$ and $\\equiv_I$ agree on $X^{(k)}$. Here, for two sets of positive integers \n$x=\\{x_1, \\dots, x_k\\}$ and $y=\\{y_1, \\dots, y_k\\}$ with $x_1<\\dots<x_k$ \nand $y_1<\\dots<y_k$ and for a subset \n$I\\subseteq [k]$ the statement $x\\equiv_I y$ is defined to mean that~$x_i=y_i$ holds for all $i\\in I$. \nThroughout the entire section we fix an integer $k\\ge 2$. Moreover, $\\Gamma$ will always denote the \nset of all equivalence relations on $[k+2]^{(k)}$.\n\n\\subsection{Preparation} \\label{subsec:preparation}\n\nBeginning as in Remark~\\ref{rem:1945} we {\\it associate} with every equivalence relation \n$([N]^{(k)}, \\equiv)$ the colouring $f\\colon [N]^{(k+2)}\\lra \\Gamma$ mapping \nevery set $A\\in [N]^{(k+2)}$ to the unique equivalence relation $\\sim\\in \\Gamma$ with the \nproperty that $(A^{(k)}, \\equiv)$ and $([k+2]^{(k)}, \\sim)$ are isomorphic via the order-preserving \nmap from $A$ to $[k+2]$. For $m\\in [k+2]$ we say that $([N]^{(k)},\\equiv)$ is {\\it governed \nby $m$-sets} if there exists a function $f^m \\colon [N]^{(m)} \\lra \\Gamma$ which governs $f$. \nFor instance, every equivalence relation is governed by $(k+2)$-sets. Next we associate with every \nequivalence relation some auxiliary colourings $g_i$ using the set of colours $\\Phi=\\{-1, +1\\}$\nintroduced in~\\eqref{eq:Phi}.\n\n\\begin{definition} \\label{dfn:pdec}\n\tLet $N\\ge k+1$ be an integer and let $i \\in [k]$ be an index. \n\tThe {\\it $i^{\\mathrm{th}}$ auxiliary function} $g_i \\colon [N]^{(k+1)}\\lra \\Phi$ \n\tof an equivalence relation $([N]^{(k)}, \\equiv)$ is defined by \n\t%\n\t\\[\n\t\tg_i(z_1, \\dots, z_{k+1})= 1 \n\t\t\\,\\,\\,\\Longleftrightarrow\\,\\,\\,\n\t\t\\{z_1, \\dots, z_{i-1}, z_{i+1}, \\dots, z_{k+1}\\} \n\t\t\\equiv \n\t\t\\{z_1, \\dots, z_{i}, z_{i+2}, \\dots, z_{k+1}\\}\n\t\\]\n\t%\n\twhenever $1\\le z_1<\\dots<z_{k+1}\\le N$.\n\\end{definition}\n\nLike the permutation colourings in the previous section, these functions will later be synchronised \nby means of an iterative application of Lemma~\\ref{lem:2150}. Notice that if an equivalence relation \n$([N]^{(k)},\\equiv)$ is governed by $m$-sets, then there exists a function \n$g^{m-1}_i \\colon [2, N]^{(m-1)} \\lra \\Phi$ governing the restriction of $g_i$ to $[2, N]^{(k+1)}$. \nOur next result asserts that at some stage of our argument these functions will automatically \nbecome monotone. \n\n\\begin{lemma}\\label{lem:gov}\n\tIf $N\\ge 2k$, $m \\in [k]$, and $([N]^{(k)},\\equiv)$ is governed by $(m+1)$-sets, \n\tthen the function $g_m^m \\colon [2, N]^{(m)} \\lra \\Phi$ is increasing on $[2,N-k]^{(m)}$.\n\\end{lemma}\n\n\\begin{proof}\nLet some integers $2\\le a_1 <\\dots < a_m < a'_m\\le N-k$ satisfying $g_m^m(a_1,\\dots,a_m)=1$\nbe given. We are to prove that $g_m^m(a_1,\\dots, a_{m-1}, a'_m)=1$ holds as well.\nSet $a_0=1$ and $a_{m+i}=N-k+i$ for every $i\\in [k+1-m]$. Our assumption yields\n$g_m({a_1},\\dots,a_{k+1})=1$, whence\n\\[ \n\t\\{a_1,\\dots,a_m,a_{m+2},\\dots,a_{k+1}\\} \n\t\\equiv \n\t\\{a_1,\\dots,a_{m-1},a_{m+1},\\dots,a_{k+1}\\}\\,. \n\\]\n\nThis piece of information is among the facts encoded in $f(a_0,\\dots,a_{k+1})$\nand, as $([N]^{(k)},\\equiv)$ is governed by $(m+1)$-sets, the same information is known \nto \n\\[\n\tf(a_0,\\dots,a_m,a'_m,a_{m+2},\\dots,a_{k+1})\\,.\n\\]\nConsequently, we also have\n\\[  \n\t\\{a_1,\\dots,a_m,a_{m+2},\\dots,a_{k+1}\\} \n\t\\equiv \n\t\\{a_1,\\dots,a_{m-1},a'_m,a_{m+2},\\dots,a_{k+1}\\}\\,. \n\\]\nBoth displayed equivalences have the same left side, so the transitivity of $\\equiv$ leads to\n\\[\n\t\\{a_1,\\dots,a_{m-1},a'_m,a_{m+2},\\dots,a_{k+1}\\} \n\t\\equiv \n\t\\{a_1,\\dots,a_{m-1},a_{m+1},\\dots,a_{k+1}\\}\\,,\n\\]\nfor which reason \n\\[\n\tg_m^m(a_1, \\dots, a_{m-1}, a'_m)\n\t=\n\tg_m(a_1, \\dots, a_{m-1},a'_m, a_{m+1}, \\dots, a_{k+1})\n\t=\n\t1\\,. \\qedhere\n\\]\n\\end{proof}\n\nLet us observe that for canonical equivalence relations $\\equiv_I$ with $I\\subseteq [k]$ the \nauxiliary functions $g_1, \\dots, g_k$ are constant. Moreover, if $\\eps_1, \\dots, \\eps_k\\in\\Phi$\ndenote the values these functions always attain, then $I=\\{i\\in [k]\\colon \\eps_i=-1\\}$. \nIn the converse direction, if we have an equivalence relation $([N]^{(k)}, \\equiv)$ with the property\nthat such colours $\\eps_1, \\dots, \\eps_k\\in\\Phi$ exist, then we may \ndefine $I=\\{i\\in [k]\\colon \\eps_i=-1\\}$, but it does not follow immediately that~$\\equiv$ \ncoincides with the canonical equivalence relation~$\\equiv_I$. This is why we require an additional \npurging argument. \n\n\\begin{definition}\\label{d:p}\n\tFor $i\\in [k]$ an equivalence relation $([N]^{(k)},\\equiv)$ is said to be {\\it $i$-purged} \n\tif for all integers $a_1, \\dots, a_i, b_1, \\dots, b_i, c_{i+1}, \\dots, c_{k+1}$\n\tsatisfying\n\t%\n\t\\begin{enumerate}[label=\\upshape({\\itshape \\roman*\\,})]\n\t\t\\item\\label{p:1} $1\\le a_1 < \\dots < a_i$ and $ 1\\le b_1 < \\dots < b_i$,\n\t\t\\item\\label{p:2} $\\max(a_i,b_i) < c_{i+1} < \\dots < c_{k+1}\\le N$,\n\t\t\\item\\label{p:3} $g_i(a_1,\\dots,a_i,c_{i+1},\\dots,c_{k+1})\n\t\t\t\t\t\t\t\t\t\t=g_i(b_1,\\dots,b_i,c_{i+1},\t\\dots,c_{k+1})= -1$,\n\t\t\\item\\label{p:4} and $\\{a_1,\\dots,a_i,c_{i+1},\\dots,c_{k}\\}\\equiv \n\t\t\t\t\t\t\t\t\t\t\\{b_1,\\dots,b_i,c_{i+1},\\dots,c_{k}\\}$\n\t\\end{enumerate}\n\t%\n\twe have $a_i = b_i$.\n\\end{definition}\n\n\\begin{lemma}\\label{lem:fin}\n\tSuppose that the equivalence relation $([N]^{(k)},\\equiv)$ is $i$-purged for every $i \\in [k]$. \n\tIf the auxiliary functions $g_1,\\dots,g_k$ are constant, then $([N-1]^{(k)},\\equiv)$ is canonical.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $\\eps_1, \\dots, \\eps_k\\in \\Phi$ be the values which $g_1, \\dots, g_k$ always attain\n\tand set \n\t%\n\t\\[\n\t\tI = \\big\\{i\\in [k] \\colon \\eps_i = -1\\big\\}\\,. \n\t\\]\n\t%\n\tIt suffices to prove that for any two sets $a= \\{ a_1,\\dots,a_k\\}$ and $b = \\{ b_1,\\dots,b_k\\}$\n\tfrom $[N-1]^{(k)}$ whose elements have just been enumerated in increasing order the statement \n\t%\n\t\\begin{equation}\\label{eq:4940}\n\t\t\\{a_1,\\dots,a_k\\} \\equiv \\{b_1,\\dots,b_k\\} \n\t\t\\,\\,\\,\\Longleftrightarrow\\,\\,\\, \n\t\t\\forall i \\in I \\,\\,\\, a_i = b_i \n\t\\end{equation}\n\t%\n\tis valid. To this end we argue by induction on the \n\tnumber $\\mu = |\\{ i \\in [k] \\colon a_i \\neq b_i \\}|$ counting the indices where $a$ and $b$ \n\tdisagree. \n\tIn the base case $\\mu=0$ we have $a=b$ and both sides of the equivalence~\\eqref{eq:4940} are true. \n\tNow suppose that $\\mu\\in [k]$ has the property that~\\eqref{eq:4940} \n\tholds for all pairs of sets from $[N]^{(k)}$ disagreeing less than $\\mu$ times. For the \n\tinduction step we assume that $a, b\\in [N]^{(k)}$ disagree with respect to exactly $\\mu$ indices. \n\t\n\tBeginning with the backwards implication we assume $a_i = b_i$ for all $i \\in I$ \n\tand intend to derive $a\\equiv b$. \n\tNotice that the smallest index $i(\\star) \\in [k]$ satisfying $a_{i(\\star)} \\ne b_{i(\\star)}$ \n\tbelongs to~${[k]\\setminus I}$, whence $\\eps_{i(\\star)} =1$. \n\tBy symmetry we may suppose that $a_{i(\\star)} < b_{i(\\star)}$. \n\tThe induction hypothesis implies that the set \n\t$c=b\\cup\\{a_{i(\\star)}\\}\\setminus\\{b_{i(\\star)}\\}$ is equivalent to $a$ with respect \n\tto~$\\equiv$. Moreover, the definition of  \n\t%\n\t\\[ \n\t\tg_{i(\\star)}(b_1,\\dots,b_{i(\\star)-1},a_{i(\\star)},b_{i(\\star)},\\dots,b_k)\n\t\t=\n\t\t\\varepsilon_{i(\\star)} \n\t\t= 1\n\t\\]\n\t%\n\tyields $b\\equiv c$, so altogether we have indeed $a\\equiv c\\equiv b$.\n\n\tNow suppose conversely that $a\\equiv b$, where $a, b\\in [N-1]^{(k)}$. \n\tDue to $\\mu>0$ there exists a largest \n\tinteger $i(\\star) \\in [k]$ such that $a_{i(\\star)} \\ne b_{i(\\star)}$. \n\tApplying the assumption that $\\equiv$ be $i(\\star)$-purged to \n\t%\n\t\\begin{enumerate}\n\t\t\\item[$\\bullet$] $a_1<\\dots<a_{i(\\star)}$, $b_1<\\dots<b_{i(\\star)}$,\n\t\t\\item[$\\bullet$] and $a_{i(\\star)+1}=b_{i(\\star)+1}, \\dots, a_k=b_k, N$ playing the \n\t\t\tr\\^{o}les of $c_{i(\\star)+1}, \\dots, c_{k+1}$\n\t\\end{enumerate}\n\twe infer $\\varepsilon_{i(\\star)} = +1$, i.e., $i(\\star) \\notin I$. \n\tBy symmetry we may again assume $a_{i(\\star)} < b_{i(\\star)}$. Now \n\t%\n\t\\[\n\t\tg_{i(\\star)}(a_1, \\dots, a_{i(\\star)}, b_{i(\\star)}, a_{i(\\star)+1}, \\dots, a_k)\n\t\t=\n\t\t\\eps_{i(\\star)}\n\t\t=\n\t\t1\n\t\\]\n\t%\n\timplies that the set $c=a\\cup\\{b_{i(\\star)}\\}\\setminus\\{a_{i(\\star)}\\}$ is equivalent to $a$\n\twith respect to $\\equiv$. \n\tSince~$b$ and~$c$ disagree only $\\mu-1$ times, the induction hypothesis yields \n\t$b_i = c_i$ for all~$i\\in I$. Together with $i(\\star)\\not\\in I$ this shows that we have indeed\n\t$a_i = b_i$ for all~$i \\in I$. This completes the induction step and, therefore, the proof \n\tof~\\eqref{eq:4940}. In particular,~$\\equiv$ is canonical. \n\\end{proof}\n\nThe next result shows that from a quantitative point of view purging is accompanied by a  \npolynomial dependence of the involved constants.  \n\n\\begin{lemma}\\label{lem:pur}\n\tLet $m\\in[k]$, $n\\ge k$, and $N\\geq (2kn)^{2k}$.\n\tIf an equivalence relation $([N]^{(k)},\\equiv)$ is governed by $(m+1)$-sets, \n\tthen there exists a set $Z\\subseteq [N]$ of size $n$ such that \n\t$(Z^{(k)},\\equiv)$ is $m$-purged.\n\\end{lemma}\n\n\\begin{proof}\nThe potential counterexamples to $([2, N]^{(k)}, \\equiv)$ being $m$-purged can be regarded \nas $(k+m+1)$-tuples of integers satisfying the conditions~\\ref{p:1}\\,--\\,\\ref{p:4} \nfrom Definition~\\ref{d:p} and $a_m\\ne b_m$. By symmetry it suffices to worry about those \ncounterexamples for which $a_m<b_m$. \nBearing this in mind we define $\\Psi$ to be the set of \nall $(k+m)$-tuples \n\\[\n\t(a_1,\\dots,a_m,b_1,\\dots,b_m,c_{m+1},\\dots,c_k)\\in [2, N]^{k+m}\n\\]\nsuch that \n\\begin{enumerate}\n\t\\item[$\\bullet$] $a_1 < \\dots < a_m$, $b_1 < \\dots < b_m$, \n\t\\item[$\\bullet$] $a_m < b_m$,\n\t\\item[$\\bullet$] $\\max(a_m,b_m) < c_{m+1}<\\dots < c_k$,\n\t\\item[$\\bullet$] $g_m^m(b_1,\\dots,b_m)= -1$,\n\t\\item[$\\bullet$] and $\\{a_1,\\dots,a_m,c_{m+1},\\dots,c_k\\} \n\t\t\t\\equiv \\{b_1,\\dots,b_m,c_{m+1},\\dots,c_k\\}$.\n\\end{enumerate}\nThe key observation is that two members of $\\Psi$ cannot differ in their $(2m)^{\\mathrm{th}}$\nentry alone. \n\n\\begin{claim}\\label{clm:dif}\n\tIf $(a_1,\\dots,a_m,b_1,\\dots,b_m,c_{m+1},\\dots,c_k)$ \n\tand $(a_1,\\dots,a_m,b_1,\\dots,b'_m,c_{m+1},\\dots,c_k)$ belong to $\\Psi$, then $b_m=b'_m$.\n\\end{claim}\n\n\\begin{proof\n\tAssume contrariwise that $b_m < b'_m$. Due to \n\t%\n\t\\[ \n\t\t\\{b_1,\\dots,b'_m,c_{m+1},\\dots,c_k\\} \n\t\t\\equiv \n\t\t\\{a_1,\\dots,a_m,c_{m+1},\\dots,c_k\\} \n\t\t\\equiv \n\t\t\\{b_1,\\dots,b_m,c_{m+1},\\dots,c_k\\}  \n\t\\]\n\t%\n\twe know $g_m(b_1,\\dots,b_m,b_m^\\prime,c_{m+1},\\dots,c_k)= 1$, which contradicts \n\t$g_m^m(b_1,\\dots,b_m)=-1$.\n\\end{proof}\n\nNow we consider the partition \n\\[\n\t\\Psi = \\Psi_0 \\charfusion[\\mathbin]{\\cup}{\\cdot} \\dots \\charfusion[\\mathbin]{\\cup}{\\cdot} \\Psi_{m-1}\n\\]\ndefined by \n\\[\n\t\\Psi_i\n\t=\n\t\\bigl\\{(a_1,\\dots,a_m,b_1,\\dots,b_m,c_{m+1},\\dots,c_k)\\in\\Psi\\colon\n\t\t|\\{ a_1,\\dots,a_m,b_1,\\dots,b_{m-1} \\}|=m+i\\bigr\\}\n\\]\nfor every $i\\in [0, m-1]$. As a consequence of Claim~\\ref{clm:dif} \nevery $(a_1,\\dots,c_k)\\in\\Psi_i$ can be described uniquely \nby specifying \n\\begin{enumerate}[label=\\upshape({\\itshape \\arabic*\\,})]\n\t\\item\\label{it:41} the $(k+i)$-element subset $X=\\{a_1, \\dots, a_m, b_1, \\dots, b_{m-1}, c_{m+1}, \n\t\t\t\\dots, c_k\\}$ of $[2, N]$\n\t\\item\\label{it:42} and telling for each of $a_1, \\dots, a_m, b_1, \\dots, b_{m-1}$\n\t\t\tto which of the $m+i$ members of \n\t\t\t$Y=\\{a_1, \\dots, a_m, b_1, \\dots, b_{m-1}\\}$ they are equal.\n\\end{enumerate}\nThere are $\\binom{N-1}{k+i}$ possibilities for the first decision. The set $Y$ can be obtained \nfrom $X$ by removing the $k-m$ largest elements and, in particular, $Y$ is definable from $X$. \nThus there are at most $(m+i)^{2m-1}$ possibilities for~\\ref{it:42}. \nAltogether, theses considerations establish \n\\[ \n\t|\\Psi_i| \n\t\\le \n\t\\binom{N-1}{k+i}(m+i)^{2m-1} \n\t\\le \n\t(N-1)^{k+i}(2m)^{2m-1} \n\\]\nIn combination with\n\\begin{align*}\n\t\\sum_{Z \\in [2, N]^{(n)}}|\\Psi \\cap Z^{k+m}|\n\t&= \n\t\\sum_{i =0}^{m-1}\\binom{(N-1)-(k+i+1)}{n-(k+i+1)}|\\Psi_i| \\\\\n\t&\\le \n\t\\binom{N-1}{n}\\sum_{i =0}^{m-1}\\Big(\\frac{n}{N-1}\\Big)^{k+i+1}|\\Psi_i|\n\\end{align*}\nthis implies\n\\begin{align*}\n\t\\sum_{Z\\in [2, N]^{(n)}}|\\Psi \\cap Z^{k+m}|\n\t&\\le\n\t\\binom{N-1}{n}\\sum_{i=0}^{m-1}\\frac{n^{k+i+1}}{N-1} (2m)^{2m-1} \\\\\n\t&\\le \n\t\\frac{(2mn)^{2k}}{2(N-1)} \\binom{N-1}{n}\n\t<\n\t\\binom{N-1}{n}\\,.\n\\end{align*}\nConsequently some set $Z\\in [2, N]^{(n)}$ satisfies $\\Psi \\cap Z^{k+m} = \\varnothing$. \nDue to the definition of $\\Psi$ every such set $Z$ is $m$-purged.\n\\end{proof}\n\n\\subsection{Plan} \\label{subsec:erplan}\nSuppose that we have an equivalence relation $([N]^{(k)}, \\equiv)$, where $N$ is, roughly speaking, \n$k-1$ times exponential in $n$, and that we want to find a subset $X\\subseteq [N]$ of size~$n$ \nsuch that $(X^{(k)}, \\equiv)$ is canonical. The material in the previous subsection suggests the \nfollowing strategy. Letting $f\\colon [N]^{(k+2)}\\lra\\Gamma$ denote the function associated \nwith $\\equiv$ we want to select appropriate subsets $[N]=Z_{k-1}\\supseteq\\dots\\supseteq Z_0$ such that \nfor each $m\\in [0, k-1]$ the size of $Z_m$ is, roughly, $m$ times exponential in $n$ and the \nrestriction of $f$ to $Z_m^{(k+2)}$ is governed by $(m+3)$-sets. \nLemma~\\ref{lem:gov} tells us that if we construct such a sequence of \nsets~$(Z_m)_{0\\le m\\le k-1}$ by means of an iterative applications of Lemma~\\ref{lem:2150}, then \nwe can achieve that the restrictions of $g^2_2, \\dots, g^2_k$ to $Z_0^{(2)}$ are monotone. \nNow $|Z_0|$ depends polynomially on $n$, which severely restricts our possibilities as to how \nthe argument can be continued. Certainly Lemma~\\ref{lem:1731} allows us to synchronise \n$g_2, \\dots, g_k$. Moreover, synchronising $g_1$ will only cost an additional square root;\nessentially this is due to the formula $\\ER^{(1)}(n)\\le n^2$. Finally $k$ successive applications of \nLemma~\\ref{lem:pur} allow us to perform a complete purging, and by Lemma~\\ref{lem:fin} we will \nthereby be done. \n\nNow if one completely ignores the last exponent $C_k$ for which one will end up \nproving $\\ER^{(k)}(n)\\le t_{k-1}(n^{C_k})$, then it is entirely immaterial in which order one \ncarries the various steps of the argument out. But from a quantitative point of view it seems \nrecommendable to perform each purging step as early as possible. Illustrating this point \nby means of an example we give a brief sketch as to how one can \nprove $\\ER^{(4)}(n)\\le t_3(Cn^{24})$.\n\\begin{enumerate}\n\t\\item[$\\bullet$] Start with an arbitrary equivalence relation $([N]^{(4)}, \\equiv)$, where \n\t\t\t$N=t_3(Cn^{24})$, and let $f\\colon [N]^{(6)}\\lra \\Gamma$ be its associated \n\t\t\tfunction.\n\t\\item[$\\bullet$] By Corollary~\\ref{cor:0053} there is a set $Z_2\\subseteq Z_3=[N]$\n\t      of size $t_2(C'n^{24})$ on which $f$ is governed by quintuples. \n\t      Now $g^4_4$ is monotone by Lemma~\\ref{lem:gov}. \n\t\\item[$\\bullet$] Before doing anything else we appeal to Lemma~\\ref{lem:pur} and get \n\t      a $4$-purged set $P_2\\subseteq Z_2$ of size $t_2(C''n^{24})$, where $C''$\n\t      is extremely close to $C'$. \n\t\\item[$\\bullet$] Next we apply Lemma~\\ref{lem:2150} and obtain $Z_1\\subseteq P_2$\n\t      of size $t_1(C'''n^{24})$ such that~$g^3_3$,~$g^3_4$ are monotone and \n\t      $f$ is governed by quadruples. \n\t \\item[$\\bullet$] Now Lemma~\\ref{lem:pur} leads us to \n\t \t\ta $3$-purged set $P_1\\subseteq Z_1$ \n\t \t\tof size $t_1(C''''n^{24})$.\n\t\\item[$\\bullet$] Only now we appeal to Lemma~\\ref{lem:2150} again and take $Z_0\\subseteq P_1$\n\t\t\tof size $C'''''n^8$ such that~$g^2_2$,~$g^2_3$,~$g^2_4$ are monotone and $f$ is governed \n\t\t\tby triples. \n\\end{enumerate}\n\nThe next lemma tells us how to complete the argument once $Z_0$ is found. \n\n\\begin{lemma}\\label{lem:end}\n\tLet $n\\ge k$, $N\\ge (k-1)!(2n)^{2k}$, and let $([N]^{(k)},\\equiv)$ be an equivalence relation  \n\twhich is governed by triples and $i$-purged for every $i\\in [3,k]$.  \n\tIf the functions $g_2^2,\\dots,g_k^2$ defined on $[2, N]^{(2)}$ \n\tare monotone, then there exists a set $X\\subseteq [N]$ \n\tof size $n$ such that $(X^{(k)},\\equiv)$ is canonical.\n\\end{lemma}\n\n\\begin{proof}\n\tDue to the monotonicity of $g_2^2,\\dots,g_k^2$ and $N-1\\ge  (k-1)!(4n^2-1)^{k}$ \n\tLemma \\ref{lem:1731} yields a subset $Y\\subseteq [2, N]$ of size $|Y|=4n^2-1$ such that \t\n\tthe functions $g_2^2,\\dots,g_k^2$ are constant on $Y^{(2)}$. \n\tDenote the $k-1$ largest elements of $Y$ by $y_1<\\dots<y_{k-1}$  \n\tand define an equivalence relation $\\sim$\n\ton $Y\\setminus\\{y_1,\\dots,y_{k-1}\\}$ by\n\t%\n\t\\[\n\t\ta \\sim b \n\t\t\\,\\,\\,\\Longleftrightarrow\\,\\,\\, \n\t\t\\{a,y_1,\\dots,y_{k-1}\\} \\equiv \\{b,y_1,\\dots,y_{k-1}\\}\\,.\n\t\\]\n\t%\n\tIn view of $n\\ge k\\ge 2$ we have $|Y\\setminus \\{y_1,\\dots,y_{k-1}\\}| = 4n^2 -k \\ge (2n-k+1)^2+1$ \n\tand, consequently, there exists a set $Z^- \\subseteq Y \\setminus \\{y_1,\\dots,y_{k-1}\\}$ \n\tof size $2n-k+2$ which is either contained in an equivalence class of $\\sim$ or consists \n\tof numbers that are mutually non-equivalent with respect to $\\sim$. In \n\tboth cases  the set $Z= Z^- \\charfusion[\\mathbin]{\\cup}{\\cdot} \\{y_1,\\dots,y_{k-1}\\}$ has the property that the restriction \n\tof $g^2_1$ to $Z^{(2)}$ is constant. Notice that the cardinality of $Z$ is~$2n+1$. \n\tEnumerating the elements of $Z$ in increasing order we write $Z=\\{z_1, \\dots, z_{2n+1}\\}$. \n\t\n\tThe remainder of the proof we shall show that  \n\t%\n\t\\[\n\t\tX_\\star=\\bigl\\{z_1, z_3, \\dots, z_{2n+1}\\bigr\\}\n\t\\]\n\t%\n\thas the property that $(X_\\star^{(k)}, \\equiv)$ is $1$-purged and $2$-purged.\n\tIn the light of Lemma~\\ref{lem:fin} this will imply that the $n$-element set \n\t$X=X_\\star\\setminus\\{z_{2n+1}\\}$ is as desired. \n\t\t\n\tStarting with the former goal we let $\\eps_1\\in\\Phi$ be the constant value that $g^2_1$\n\tattains on $X_\\star^{(2)}$. \n\tBecause of Definition~\\ref{d:p}~\\ref{p:3} we may suppose that $\\eps_1= -1$. \n\tNow let \n\t%\n\t\\[\n\t\ta_1 \\leq b_1 < c_2 < \\dots < c_{k+1}\n\t\\]\n\n\tbe $k+2$ members of $X_\\star$ satisfying \n\t$\\{a_1,c_2,\\dots, c_{k}\\} \\equiv \\{b_1,c_2,\\dots, c_{k}\\}$. \n\tAs the assumption $a_1< b_1$ would yield $\\eps_1=g_1(a_1 ,b_1, c_2, \\dots, c_{k})= 1$, \n\twe have indeed $a_1=b_1$, as required. \n\t\n\tIt remains to show that $(X_\\star^{(k)},\\equiv)$ is $2$-purged as well. \n\tAs before, we may suppose that the constant value $\\eps_2$ attained by $g_2$ on $X_\\star^{(k+1)}$\n\tis~$-1$. Assume for the sake of contradiction that there exist $k+3$ members $a_1<a_2 <b_2$ \n\tand $b_1<b_2<c_3<\\dots<c_{k+1}$ of $X$ such that  \n\t%\n\t\\begin{equation}\\label{eq:4809}\n\t\t\\{a_1,a_2,c_3,\\dots,c_{k}\\} \n\t\t\\equiv \n\t\t\\{ b_1,b_2,c_3,\\dots,c_k\\}\\,.\n\t\\end{equation}\n\t%\n\tPick a number $q\\in Z$ such that $b_2<q<c_3$. Since $f$ is governed by triples, we have\n\t%\n\t\\[ \n\t\tf(a_1,a_2,b_1,b_2,c_3,\\dots,c_{k}) \n\t\t= \n\t\tf(a_1,a_2,b_1,q,c_3,\\dots,c_{k})\\,.\n\t\\]\n\t%\n\tSo~\\eqref{eq:4809} entails \n\t%\n\t\\[\t\n\t\t\\{a_1,a_2,c_3,\\dots,c_{k}\\} \n\t\t\\equiv  \n\t\t\\{b_1,q,c_3,\\dots,c_k\\} \n\t\\]\n\t%\n\tand altogether we obtain \n\t%\n\t\\[ \n\t\t\\{b_1,b_2,c_3,\\dots,c_k\\}\n\t\t\\equiv  \n\t\t\\{b_1,q,c_3,\\dots,c_k\\}\\,,\n\t\\]\n\t%\n\twhich leads us to the contradiction\n\t%\n\t\\[\n\t\t\\eps_2=g_2(b_1,b_2,q,c_3,\\dots,c_k)=1\\,. \\qedhere \n\t\\]\n\t%\n\\end{proof}\n\n\\subsection{Canonisation} \\label{subsec:caners}\nIt remains to execute the plan discussed in the foregoing subsection. \nGiven $n\\ge k\\ge2$ we define positive integers $N_0, P_1, N_1, \\dots, P_{k-2}, N_{k-2}, N_{k-1}$ \naccording to the following recursive rules. \n\\begin{align*}\n\tN_0&=(k-1)!(2n)^{2k}+(k+1)\\,, \\\\\n\tP_m&=(2|\\Gamma|)^{N_{m-1}^{m+2}} \\cdot (2kN_{m-1}^{m})^{kN_{m-1}^{m}} \\quad \\text{ for }m\\in [k-2]\\,, \\\\\n\tN_{m}&= (2kP_m)^{2k}+(k+1) \\quad \\text{ for }m\\in [k-2]\\,,\\\\\n\tN_{k-1}&= |\\Gamma|^{N_{k-2}^{k+1}} \\,.\n\t\\end{align*}\n\t%\n\tWe want to prove that if $N\\ge N_{k-1}$ then for every equivalence relation $([N]^{(k)},\\equiv)$ \t\n\tthere exists a set $X\\subseteq[N]$ of size $n$ such that $(X^{(k)},\\equiv)$ is canonical. \n\tThis will imply \n\t%\n\t\\begin{equation}\\label{eq:ende}\n\t\t\\ER^{(k)}(n)\\le N_{k-1}\\le t_{k-1}(C_kn^{6k})\\,, \n\t\\end{equation}\n\t%\n\twhere $C_k$ depends only on $k$. \n\t\n\tLet $f\\colon[N]^{k+2}\\lra\\Gamma$ be the function associated to a given equivalence relation \n\t$([N]^{(k)}, \\equiv)$, where $N\\ge N_{k-1}$. An initial application of Corollary~\\ref{cor:0053}\n\tleads to a set $Z'_{k-2}\\subseteq [N]$ such that $({Z'}^{(k)}_{k-2}, \\equiv)$ is governed \n\tby $(k+1)$-sets and $|Z'_{k-2}|=N_{k-2}$. \n\tAccording to Lemma~\\ref{lem:gov} the set $Z_{k-2}\\subseteq Z'_{k-2}$\n\tthat arises when one removes the smallest and the $k$ largest elements of~$Z_{k-2}'$ has \n\tthe property that $g^k_k\\colon {Z}^{(k)}_{k-2}\\lra \\Phi$ is monotone. Notice that \n\t$|Z_{k-2}|=N_{k-2}-(k+1)$.\n\n\\begin{claim}\\label{clm:9999}\n\tFor every $m \\in [0,k-2]$ there exists a set $Z_m\\subseteq [N]$ of size $|Z_m|=N_m-(k+1)$ \n\twith the following properties:\n\t%\n\t\\begin{enumerate}[label=\\upshape({\\itshape \\roman*\\,})]\n\t\t\\item\\label{p:5} $(Z_m^{(k)},\\equiv)$ is governed by $(m+3)$-sets.\n\t\t\\item\\label{p:6} For every $i \\in [m+2,k]$ the function $g_i^{m+2}$ is monotone.\n\t\t\\item\\label{p:7} If $i \\in [m+3,k]$, then $(Z_m^{(k)},\\equiv)$ is $i$-purged.\n\t\\end{enumerate}\n\t%\n\\end{claim}\n\nObserve that the case $m=0$ yields a set $Z_0$ of size $|Z_0|=(k-1)!(2n)^{2k}$.  \nThus Lemma~\\ref{lem:end} shows that Claim~\\ref{clm:9999} implies~\\eqref{eq:ende}.\n\n\\begin{proof}[Proof of Claim \\ref{clm:9999}]\n\tWe argue by decreasing induction on $m$. In the base case $m=k-2$ clause~\\ref{p:7} holds \n\tvacuously and the set $Z_{k-2}$ defined above has the desired properties. \n\t\n\tNow suppose that for some $m\\in[k-2]$ we have already found a subset $Z_m\\subseteq [N]$ \n\tof size $N_m-(k+1)=(2kP_m)^{2k}$ fulfilling~\\ref{p:5},~\\ref{p:6}, and~\\ref{p:7}. \n\tOwing to Lemma~\\ref{lem:pur} there exists a set $Q_m\\subseteq Z_m$ \n\tof size $|Q_m|=P_m$ such that $(P_{m}^{(k)},\\equiv)$ is $(m+2)$-purged.\n\tNext we apply Lemma~\\ref{lem:2150} to $f^{m+3} \\colon P_m^{(m+3)} \\lra \\Gamma$ \n\tand the family $(g_i^{m+2})_{m+2\\le i\\le k}$ of monotone colourings. \n\tThis yields a set $Z'_{m-1}\\subseteq P_m$ of size $N_{m-1}$ such that \n\t$({Z'}_m^{(k)},\\equiv)$ is governed by $(m+2)$-sets and for every $i \\in [m+2,k]$ the \n\tfunction~$g_i^{m+1}$ is monotone. 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(3)},\n   volume={2},\n   date={1952},\n   pages={417--439},\n   issn={0024-6115},\n   review={\\MR{65615}},\n   doi={10.1112\/plms\/s3-2.1.417},\n}\n\n\\bib{ES35}{article}{\n   author={Erd\\H{o}s, P.},\n   author={Szekeres, G.},\n   title={A combinatorial problem in geometry},\n   journal={Compositio Math.},\n   volume={2},\n   date={1935},\n   pages={463--470},\n   issn={0010-437X},\n   review={\\MR{1556929}},\n}\n\n\\bib{Fox}{misc}{\n\tauthor={Fox, J.},\n\tnote={Personal communication to V.~R\\\"{o}dl and M.~Sales},\n\tdate={2019},\n}\n\n\\bib{GLR}{article}{\n   author={Graham, R. L.},\n   author={Leeb, K.},\n   author={Rothschild, B. L.},\n   title={Ramsey's theorem for a class of categories},\n   journal={Proc. Nat. Acad. Sci. U.S.A.},\n   volume={69},\n   date={1972},\n   pages={119--120},\n   issn={0027-8424},\n   review={\\MR{306009}},\n   doi={10.1073\/pnas.69.1.119},\n}\n\t\t\n\\bib{GLK2}{article}{\n  author={Graham, R. L.},\n  author={Leeb, K.},\n  author={Rothschild, B. 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J.},\n   author={R\\\"{o}dl, V.},\n   author={Voigt, B.},\n   title={Canonizing ordering theorems for Hales-Jewett structures},\n   journal={J. Combin. Theory Ser. A},\n   volume={40},\n   date={1985},\n   number={2},\n   pages={394--408},\n   issn={0097-3165},\n   review={\\MR{814422}},\n   doi={10.1016\/0097-3165(85)90098-6},\n}\n\t\t\n\\bib{NR17}{article}{\n   author={Ne\\v{s}et\\v{r}il, Jaroslav},\n   author={R\\\"{o}dl, Vojt\\v{e}ch},\n   title={Statistics of orderings},\n   journal={Abh. Math. Semin. Univ. Hambg.},\n   volume={87},\n   date={2017},\n   number={2},\n   pages={421--433},\n   issn={0025-5858},\n   review={\\MR{3696159}},\n   doi={10.1007\/s12188-016-0174-x},\n}\n\n\\bib{Ramsey30}{article}{\n   author={Ramsey, Frank Plumpton},\n   title={On a Problem of Formal Logic},\n   journal={Proceedings London Mathematical Society},\n   volume={30},\n   date={1930},\n   number={1},\n   pages={264--286},\n         doi={10.1112\/plms\/s2-30.1.264},\n}\n\t\t\n\\bib{Sh96}{article}{\n   author={Shelah, Saharon},\n   title={Finite canonization},\n   journal={Comment. Math. Univ. Carolin.},\n   volume={37},\n   date={1996},\n   number={3},\n   pages={445--456},\n   issn={0010-2628},\n   review={\\MR{1426909}},\n}\n\t\t\t\t\t\t\n\\end{biblist}\n\\end{bibdiv}\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}  \\label{sec1}\nDynamical systems~\\cite{strogatz,nayfeh,holmes,birkhoff,len0} capable of having isochronous~\\cite{calogero,Sarkariso2012} oscillations are very important from the point of view of modelling real world systems which exhibit self-sustained oscillations~\\cite{Jenkins_2013}. Isochronicity~\\cite{calogero} is a widely studied subject not only for its relation with stability theory and bifurcation theory~\\cite{strogatz,holmes,rkupadhyay} but also in the discrimination of oscillating orbits in limit cycle, center and center-type focus~\\cite{powerlaw,len3.5} in terms of Li\\'enard equation~\\cite{strogatz,remickens,len4,Saha2019}. Many nonlinear dynamical systems in various scientific disciplines are influenced by the finite propagation time of signals in feedback loops modelled with time-delay~\\cite{goto2007renormalization,atay}. In some systems, such as lasers and electro mechanical models, large varieties of delay appear~\\cite{algaba,hu,jiang,wang,xu} with  van der Pol oscillator~\\cite{strogatz,algaba,limiso} is a standard prototype with delayed feedback have very rich and complex bifurcation~\\cite{jiang,wang,xu}. A classical van der Pol equation with delayed feedback has been extensively investigated having Bogdanov--Taken bifurcation with triple zero and Hopf-zero singularity~\\cite{jiang} as well as transcritical and pitchfork bifurcations~\\cite{wang,Cooke_Grossman_82}. In a similar context stability analysis of a pair of van der Pol oscillators with delayed self-connection, position and velocity couplings are also performed~\\cite{hu}.\n\n{\\color{black}The existence of a limit cycle can be predicted by the Poincar\\'e--Bendixson theorem, but the exact location and size of the limit cycle cannot be predicted in advance~\\cite{strogatz}. This is why perturbation theory is employed to know the finer details about the shape of the limit cycles for various types of Li\\'enard oscillators although the procedure of approaching the problem perturbatively becomes non-trivial. A tremendous progress has been made in the area of dissipative dynamical systems regarding the applicability of the multiscale perturbative methods~\\cite{strogatz,len0,len3.5,remickens,chen1,chen2,len3,Wilson1976,slross} like Krylov--Bogoliubov (K-B), Lindstedt--Poincar\\'e and Renormalisation Group (RG). But, except the RG approach other methods are mostly restricted to weak nonlinearity~\\cite{strogatz,nayfeh,len0,slross}. In a recent development  Sarkar et al.~\\cite{len3.5} have attempted to understand the application of the RG principle to probe the difference between limit cycle and center through problems in dynamics for various 2-D systems.\n\nOne can also characterize limit cycle through the parametric excitation~\\cite{penvo,momeni} as this type of excited system is prepared by time-varying coefficients of the equations of motion which plays a drastically different role compared to the usual direct external driving force through an additive term. Recently, for a van der Pol oscillator the effect of periodically modulating nonlinear term has been investigated as a parametrically excited nonlinearity which can provide the phenomenon of both resonance and antiresonance depending on the frequency of the parametrical drive. Again, time delay even in a simple nonlinear system as a feedback~\\cite{Balanov2005} device induces a large variety of dynamical scenario, like tori and new chaotic attractors along with the fact that the delay modifies the periods and the stabilities of the limit cycles in the system depending on the strength of the feedback and its magnitude. Stability of a periodic orbit, namely a limit cycle or a center using a controlled delay is quite interesting and that too in presence of a parametric excitation leading to resonance and antiresonance is very useful for arbitrary quartic nonlinear system as a generalization of van der Pol system~\\cite{penvo} with a limit cycle.}\n\nInspite of a great deal of investigations on parametric driving, the response function due to a parametric excitation of an arbitrary  periodic orbit is not well understood. Here we have investigated a delayed nonlinear system  to obtain both limit cycle, center and a slowly decaying  center-type~\\cite{powerlaw,len3.5} oscillation for different range of parameters of the proposed model. The model exhibits a very rich dynamics due to the presence of delay which was first introduced by Goto~\\cite{goto2007renormalization} to apply renormalization method~\\cite{len3.5,goto2007renormalization,limiso,chen1,chen2,len3} to compare between perturbative analysis with the exact solution and later Sarkar et al.~\\cite{len3.5,len3} used it in a similar context. The extended version of the model is analysed by Saha et al.~\\cite{powerlaw} through K-B approach exhibiting  a stable limit cycle due to the presence of weak delay. As a further extension with a delayed feedback which becomes a variant of delayed van der Pol system, we have given here an approximate solution from the multiscaled perturbation theory which  is utilised to characterise the  parametric resonance and antiresonance behaviour. This is also more elaborately dealt by performing the bifurcation analysis~\\cite{holmes,rkupadhyay,Cooke_Grossman_82,hassard} of the system due to parametric excitation. As there is no general scheme to handle a delayed system using perturbation theory~\\cite{nayfeh,len0} to study bifurcation or periodic orbit and characterisation of the properties of nonlinear oscillation, in this paper we have analysed the Hopf bifurcation scenario during the parametric excitation  to probe the dynamics due to the interplay of nonlinear damping and delay terms.\n\nThe layout of the paper is as follows. In Section \\ref{sec2}, the nonlinear delay model is introduced along with the K-B analysis. Parametric excitation is studied in Section \\ref{sec3}. In Section \\ref{sec4}, a detailed stability and bifurcation analysis including direction and stability of Hopf bifurcation have been performed. Numerical simulations are executed in Section \\ref{sec5}. Finally, the paper is concluded in Section \\ref{sec7}\n\n\\section{Time Delayed  Nonlinear Feedback Oscillator} \\label{sec2}\nIn this section, we consider a model of delay dynamics  where the oscillation is fed energy through delay term in potential and its total energy increases with time. To control the feedback effect   we have introduced a linear damping term $b$, which gives a center solution similar to simple harmonic oscillator and then in presence of damping and delay we have introduced another quartic nonlinear term $a$. By this way  we have introduced a variant of delayed van der Pol system  which gives both limit cycle and center-type solution by tuning the parameters. The basic equations of the model are\n\\begin{align}\n  \\dot{x}(t) &= y(t), \\nonumber\\\\\n  \\dot{y}(t) &= - \\epsilon \\lbrace x(t-t_d)+(a x^2(t)+b) \\dot{x}(t) \\rbrace-\\omega^2 x(t).      \n  \\label{eq}\n\\end{align}\nHere we have considered, $0<\\epsilon \\ll 1$ is a small perturbative constant and ($0,0$) is the only fixed point. From the above model we can find the delayed  van der Pol case of a limit cycle if we consider $b=-1$ and the sign of the term $\\epsilon x(t-t_d)$ is negative. In absence of $a$ and $b$, the system converts into a  delayed harmonic oscillator which was introduced by Goto~\\cite{goto2007renormalization} to apply conventional renormalization method to show an agreement between  perturbative analysis with the exact solution where the oscillation is fed energy through delay term and its total energy increases with time. Further, in absence of $a$ i.e. $a=0$, a detailed bifurcation analysis in presence of delay for a linear-damped-delayed system has provided by Cooke et al.~\\cite{Cooke_Grossman_82}(Eq.~(10), pp-602) through stability analysis. The extended version i.e. Eq.~\\eqref{eq} has been analysed by Saha et al.~\\cite{powerlaw} through K-B approach which exhibits a stable limit cycle, a center or slowly decaying center-type dynamics due to the presence of weak delay.\n\nIn what follows, the challenges of the Hopf bifurcation analysis for such kind of weak delayed system are investigated in details and we have provided a straight forward way to bifurcation analysis through amplitude equation using K-B averaging, which is much easier than traditional Hopf bifurcation analysis. \n\n\\subsection{Approach through Krylov--Bogoliubov (K-B) Averaging}\n\\label{subsection 2}\nThis time delayed system can be written as,\n\\begin{align}\n\\ddot{x}(t)+\\epsilon h+\\omega^2 x(t) =0;~~~ h=x(t-t_d)+(a x^2+b) \\dot{x}(t).\\label{eq4}\n\\end{align}\n\nAccording to K-B scheme for approximate analytical solution of an ordinary differential equation, we consider, $x(t)=r(t) \\hspace{0.1 cm} \\cos (\\omega t+\\phi(t))$  with $r(t) = \\sqrt{x^2 +\\frac{\\dot{x}^2}{\\omega^2}}$ and $\\phi(t) = -\\omega t+ \\tan^{-1} (- \\frac{\\dot{x}}{\\omega x})$. Then one can obtain $\\dot{r}(t) = \\frac{\\epsilon h}{\\omega} \\sin (\\omega t+\\phi(t))$ and $\\dot{\\phi}(t) = \\frac{\\epsilon h}{\\omega r(t)} \\cos (\\omega t+\\phi(t))$ i.e., the time derivative of  amplitude and phase are of $O(\\epsilon)$.\n\nTaking running average of $\\phi$ dependent functions, $U$ defined as,\n\\begin{align}\n\\overline{U}(t) = \\frac{\\omega}{2 \\pi} \\int_{t-\\frac{\\pi}{\\omega}}^{t+\\frac{\\pi}{\\omega}} U(s) ds,  \\label{3}\n\\end{align} \nwhere $\\omega$ is the natural frequency of the system and considering $\\dot{\\overline{U}}=\\overline{\\dot{U}}$ from the fundamental theorem of calculus,\n one can obtain,\n\\begin{align}\n\\dot{\\overline{r}} &= \\left\\langle \\frac{\\epsilon}{\\omega} \\hspace{.15cm} h(x,y)\\hspace{.15cm} \\sin (\\omega t+\\phi(t)) \\right\\rangle_t,\t\t\\nonumber  \\\\\n\\dot{\\overline{\\phi}} &= \\left\\langle \\frac{\\epsilon \\hspace{.15cm} h(x,y)\\hspace{.15cm}}{\\omega r(t)} \\cos (\\omega t+\\phi(t)) \\right\\rangle_t. \\label{eq7}\n\\end{align}\n\nSince $\\dot{r}(t)$ and $\\dot{\\phi}(t)$ are of $O(\\epsilon)$ then we can set the perturbation on $r$ and $\\phi$ over one cycle as,\n\\begin{align}\nr(t) &=\\overline{r}+O(\\epsilon),\t\\nonumber\\\\\n\\phi(t) &=\\overline{\\phi}+O(\\epsilon),\t\\label{eq8}\n\\end{align}\nwhere $\\overline{r}$ and $\\overline{\\phi}$ are not exactly constant, they are very weakly $t$-dependent so that the error can be negligible.\n\nNow using all the approximations, $h$ reduces to $h= \\overline{r}  \\lbrace \\cos(\\omega t+\\overline{\\phi})\\cos(\\omega t_d)+\\sin(\\omega t+\\overline{\\phi})\\sin(\\omega t_d)-\\omega (a \\overline{r}^2 \\cos^2(\\omega t+\\overline{\\phi})+b)\\sin(\\omega t+\\overline{\\phi}) \\rbrace+O(\\epsilon)$, where one finds $r(t-t_d)$ is approximated as $r(t-t_d)=r(t)-t_d \\dot{r}(t)=r(t)+O(\\epsilon)$ as $\\dot{r}(t)=O(\\epsilon)$ and $0<t_d \\ll1$. Finally one can obtain,\n\\begin{align}\n\\dot{\\overline{r}} &= -\\frac{\\epsilon \\overline{r}}{8} \\left \\lbrace a \\overline{r}^2-4\\left(\\frac{\\sin(\\omega t_d)}{\\omega}-b\\right)\\right \\rbrace, \\nonumber\\\\\n\\dot{\\overline{\\phi}} &= \\frac{\\epsilon}{2 \\omega} \\cos(\\omega t_d),\t\\label{eq10}\n\\end{align}\nwhere $O(\\epsilon^2)$ terms are neglected. Now, if we have $(x_0, y_0)$ as an initial condition of the system and then finding, $r_0$ and $\\phi_0$, and one can find the approximate solution by solving $\\dot{\\overline{r}}$ and $\\dot{\\overline{\\phi}}$. Also, one finds the energy as $E=\\frac{\\omega^2 x^2+\\dot{x}^2}{2}$ to get  the energy consumption over one cycle, which is $\\Delta E=2 \\pi \\overline{r}\\hspace{0.05cm} \\dot{\\overline{r}}$.\n \nNow to focus on the bifurcation point for the above type of delayed feedback oscillator, bifurcation analysis is not meaningful through linear perturbation in $\\epsilon$ around the fixed point ($0,0$), as it truncates the delay. For example, if we put $x=0+\\epsilon X$ and $\\dot{x}=y=0+\\epsilon Y$ to \\eqref{eq} and compare to the $O(\\epsilon)$ to figure out the Jacobian matrix as well as the characteristic equation then it does not include the effect of delay in the characteristic equation. In this situation we get a center solution, where the physical system is something different from the original one. Now coming to the point, which is, the simple delayed feedback case with ($a=0,b=0$), bifurcation is far away from it. Again, when $a=0$, one can find the increasing (when system diverges) or decreasing energy (when system undergoes a focus with a decaying center) according to $b<\\frac{\\sin(\\omega t_d)}{\\omega}$ and $b>\\frac{\\sin(\\omega t_d)}{\\omega}$, respectively. This provides that the value of $b=\\frac{\\sin(\\omega t_d)}{\\omega}$ is the bifurcating point having the center solution. Similarly, for the case of non-zero value of $a$ i.e., in presence of nonlinearity, if $b<\\frac{\\sin(\\omega t_d)}{\\omega}$ then the system has a non-zero radius providing the stable limit cycle solution and if $b>\\frac{\\sin(\\omega t_d)}{\\omega}$ having imaginary radius giving the unstable limit cycle (asymptotically stable fixed point). So,  $b=\\frac{\\sin(\\omega t_d)}{\\omega}$ is the Hopf bifurcating parameter, defining the energy transfer zone between stable and unstable focus connecting a center and slowly decaying center-type solution. Numerical simulation is given below for better understanding.\n\nFig.~\\ref{fig1} shows the corresponding numerical simulations of Eq.~\\eqref{eq} owing to four situations, where (a) is the case for the delay with no nonlinearity and damping, i.e, $a=0,b=0$, which shows a simple feedback oscillator with continuously increasing energy in the system. Further, (b), (c) and (d) shows a center with $a=0,b=\\frac{\\sin(\\omega t_d)}{\\omega}$, a limit cycle(LC) with $a=1,b=0.2<\\frac{\\sin(\\omega t_d)}{\\omega}$ and a slowly decaying center-type orbit or focus with $a=1,b=\\frac{\\sin(\\omega t_d)}{\\omega}$, respectively. Here the finite delay, $t_d$ and $\\epsilon$ are kept fixed at $0.623$ and $0.05$, respectively along with unit frequency.\n \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics*[width=0.75\\linewidth]{f1-phase_protrait.eps}\n\\caption{\\textbf{Time-delayed system:} Phase space plots of Eq.~\\eqref{eq} are shown using approximate amplitude and phase equations with time delay, $t_d=0.623$ where (a) $a=0,b=0$ for a feedback system with increasing phase space area, (b) center with $a=0,b=\\frac{\\sin(\\omega t_d)}{\\omega}$, (c) LC with $a=1,b=0.2<\\frac{\\sin(\\omega t_d)}{\\omega}$ and (d) center-type orbit with $a=1,b=\\frac{\\sin(\\omega t_d)}{\\omega}$ with $\\epsilon=0.05$ and $\\omega=1$.} \\label{fig1}\n\\end{center}\n\\end{figure}\n\n\\section{Parametrically Excited Time Delayed Nonlinear Feedback Oscillator}  \\label{sec3}\nAs an application of the approximate analytical solution and bifurcation situation, in this section, we have investigated the time delayed nonlinear oscillator under parametric excitation. The resonance and antiresonance behaviours for the limit cycle, center and center-type orbits are described both analytically and numerically for weak delayed situation. The stability regions at the resonances of the system under excitation are explored.\n\n\\subsection{Periodic Solutions for Resonance and Antiresonance Cases}\t\\label{subsec1}\nMaking the system parametrically excited~\\cite{penvo,momeni} by  a periodic force $\\cos(\\Omega t)$ with a weighted constant, $\\gamma$, the basic equations become:\n\\begin{align}\n\\dot{x}(t) =& y(t), \t\\nonumber\\\\\n\\dot{y}(t) =& - \\epsilon \\lbrace x(t-t_d)+[1+\\gamma \\cos(\\Omega t)] (a x^2(t)+b) \\dot{x}(t) \\rbrace-\\omega^2 x(t); \\quad \\gamma \\neq 0,  \\Omega \\in \\mathbb{Z}_{\\neq 0},  \\label{7}\n\\end{align}\nwhere $0 < \\epsilon \\ll 1$, $a,b$ are the system parameters and $t_d(0<t_d \\ll 1)$ is indicating the time delay.\n\nTo have an approximate analytical solution of the above system \\eqref{7} one can use the K-B averaging method which can be the asymptotically stable, limit cycle, center and center-type solution depending upon the parameters. But, in this case calculations will be quite harder due to the non zero excitation strength and one can not get it naturally like the previous one due to some singularities as well as heavily coupled nonlinear functional forms of the amplitude and phase equations. K-B averaging gives,\n\\begin{align}\n\\dot{\\overline{r}} =& -\\frac{\\epsilon \\overline{r}}{8} \\left\\lbrace a \\overline{r}^2-4\\left(\\frac{\\sin (\\omega t_d)}{\\omega}-b\\right) \\right\\rbrace +A(\\overline{r},\\overline{\\phi}), \t\\nonumber \\\\\n\\dot{\\overline{\\phi}} =& \\frac{\\epsilon}{2 \\omega} \\cos(\\omega t_d)+B(\\overline{r},\\overline{\\phi}),\t\\label{8}\n\\end{align}\nwhere, $A(\\overline{r},\\overline{\\phi})=A_1(\\overline{r},\\overline{\\phi})+A_2(\\overline{r},\\overline{\\phi})$ and $B(\\overline{r},\\overline{\\phi})=B_1(\\overline{r},\\overline{\\phi})+B_2(\\overline{r},\\overline{\\phi})$ with\n\n{\\scriptsize\n\\begin{align*}\nA_1(\\overline{r},\\overline{\\phi}) =& \\frac{b \\gamma  \\overline{r} \\omega  \\epsilon}{2 \\pi  \\left(8 \\omega ^2 \\Omega -2 \\Omega ^3\\right)}  \\left(\\sin \\left(\\frac{2 \\pi  \\Omega }{\\omega }\\right) \\left(-4 \\omega ^2+\\Omega ^2-\\Omega ^2 \\cos (2 \\overline{\\phi} )\\right)-8 \\omega  \\Omega  \\sin (\\overline{\\phi} ) \\cos (\\overline{\\phi} ) \\sin ^2\\left(\\frac{\\pi  \\Omega }{\\omega }\\right)\\right),\\\\\nA_2(\\overline{r},\\overline{\\overline{\\phi}})= &-\\frac{a \\gamma  \\overline{r}^3 \\omega  \\epsilon}{16 \\pi  \\left(\\Omega ^3-16 \\omega ^2 \\Omega \\right)}  \\left(\\sin \\left(\\frac{2 \\pi  \\Omega }{\\omega }\\right) \\left(-16 \\omega ^2+\\Omega ^2-\\Omega ^2 \\cos (4 \\overline{\\phi} )\\right)-8 \\omega  \\Omega  \\sin (4 \\overline{\\phi} ) \\sin ^2\\left(\\frac{\\pi  \\Omega }{\\omega }\\right)\\right);\\\\\nB_1(\\overline{r},\\overline{\\phi}) =& \\frac{a \\gamma  \\overline{r}^2 \\omega  \\epsilon}{32 \\pi }  \\left(-\\frac{8 \\omega  \\cos (2 \\overline{\\phi} )}{4 \\omega ^2-\\Omega ^2}-\\frac{8 \\omega  \\cos (4 \\overline{\\phi} )}{16 \\omega ^2-\\Omega ^2}+\\frac{2 \\cos \\left(2 \\left(\\frac{\\pi  \\Omega }{\\omega }+\\overline{\\phi} \\right)\\right)}{2 \\omega +\\Omega }+\\frac{\\cos \\left(\\frac{2 \\pi  \\Omega }{\\omega }+4 \\overline{\\phi} \\right)}{4 \\omega +\\Omega }+\\frac{2 \\cos \\left(2 \\overline{\\phi} -\\frac{2 \\pi  \\Omega }{\\omega }\\right)}{2 \\omega -\\Omega }+\\frac{\\cos \\left(4 \\overline{\\phi} -\\frac{2 \\pi  \\Omega }{\\omega }\\right)}{4 \\omega -\\Omega }\\right),\\\\\nB_2(\\overline{r},\\overline{\\phi}) =& \\frac{b \\gamma  \\omega  \\epsilon}{2 \\pi  \\left(8 \\omega ^2-2 \\Omega ^2\\right)}  \\left(\\Omega  \\sin (2 \\overline{\\phi} ) \\sin \\left(\\frac{2 \\pi  \\Omega }{\\omega }\\right)-4 \\omega  \\cos (2 \\overline{\\phi} ) \\sin ^2\\left(\\frac{\\pi  \\Omega }{\\omega }\\right)\\right).\n\\end{align*}}\n\nThere are two singularities in the equation of $\\dot{\\overline{r}}$ and $\\dot{\\overline{\\phi}}$ which are at $\\Omega = 2\\omega, 4\\omega$. One can take limit for amplitude and phase equation near the singularities and get two different sets of equations, are\n\\begin{align}\n\\dot{\\overline{r}}|_{\\Omega \\rightarrow 2\\omega} =& -\\frac{\\epsilon \\overline{r}}{8} \\left \\lbrace a \\overline{r}^2-4\\left(\\frac{\\sin (\\omega t_d)}{\\omega}-b\\right)\\right\\rbrace  +\\frac{1}{4} b \\gamma  \\overline{r} \\epsilon  \\cos (2 \\overline{\\phi} ),\t\\nonumber\\\\\n\\dot{\\overline{\\phi}}|_{\\Omega \\rightarrow 2\\omega} =& \\frac{\\epsilon}{2 \\omega} \\cos(\\omega t_d)-\\frac{1}{8} \\gamma  \\epsilon  \\sin (2 \\overline{\\phi} ) \\left(a \\overline{r}^2+2 b\\right),\t\t\\label{9}\n\\end{align}\nand\n\\begin{align}\n\\dot{\\overline{r}}|_{\\Omega \\rightarrow 4\\omega} =& -\\frac{\\epsilon \\overline{r}}{8} \\left\\lbrace a \\overline{r}^2-4\\left(\\frac{\\sin (\\omega t_d)}{\\omega}-b\\right)\\right\\rbrace  +\\frac{1}{16} a \\gamma  \\overline{r}^3 \\epsilon  \\cos (4 \\overline{\\phi} ),\t\t\\nonumber\\\\\n\\dot{\\overline{\\phi}}|_{\\Omega \\rightarrow 4\\omega} =& \\frac{\\epsilon}{2 \\omega} \\cos(\\omega t_d)-\\frac{1}{16} a \\gamma  \\overline{r}^2 \\epsilon  \\sin (4 \\overline{\\phi} ),\t\t\\label{10}\n\\end{align}\nwhere the non-zero $\\gamma$ carries the additional part. Note that, here the systems are coupled and autonomous. They can be solved analytically by taking some additional conditions in the phase lag or by numerically in a synchronized state. Also, one can get all likely features obtained in the previous section but the distinction of limit cycle, center and center-type situations due to the tuning of system parameters are harder little a bit, which is discussed in the numerical simulation section.\n\nTo have an idea about the excitation strength $\\gamma$ we are now transforming the polar equations into a van der Pol plane for each modes of resonances. It is pre-assumed that the unique trivial fixed point will remain unstable or neutral as we are focussing upon the limit cycle or center or center-type situations. Also, another reason to consider a van der Pol plane is that one can not perform the traditional linear stability analysis for the excited Li\\'enard system~\\eqref{7}. So to perform the same taking a trial solution, $x(t)=u(t) \\cos(\\omega t)+v(t) \\sin(\\omega t)$ and the one can obtain two simplified sets of equations as:\n\n\\begin{align}\n\\dot{u}|_{\\Omega\\rightarrow2\\omega}&= \\frac{\\epsilon}{8 \\omega }  \\left(u \\left(-a \\omega  \\left(u^2+(1-2 \\gamma ) v^2\\right)+2 b (\\gamma -2) \\omega +4 \\sin \\left(\\omega  t_d\\right)\\right)+4 v \\cos \\left(\\omega  t_d\\right)\\right), \\nonumber \\\\\n\\dot{v}|_{\\Omega\\rightarrow2\\omega} &= -\\frac{\\epsilon}{8 \\omega }  \\left(v \\left(a v^2 \\omega +2 b (\\gamma +2) \\omega -4 \\sin \\left(\\omega  t_d\\right)\\right)+a (2 \\gamma +1) u^2 v \\omega +4 u \\cos \\left(\\omega  t_d\\right)\\right),\t\t\\label{eqo2}\n\\end{align}\nand\n\\begin{align}\n\\dot{u}|_{\\Omega\\rightarrow4\\omega}&=  \\frac{\\epsilon}{16 \\omega }  \\left(u \\left(8 \\left(\\sin \\left(\\omega  t_d\\right)-b \\omega \\right)-a (3 \\gamma +2) v^2 \\omega \\right)+a (\\gamma -2) u^3 \\omega +8 v \\cos \\left(\\omega  t_d\\right)\\right),\t\\nonumber \\\\\n\\dot{v}|_{\\Omega\\rightarrow4\\omega}&= \\frac{\\epsilon}{16 \\omega }  \\left(-a (3 \\gamma +2) u^2 v \\omega +a (\\gamma -2) v^3 \\omega +8 v \\left(\\sin \\left(\\omega  t_d\\right)-b \\omega \\right)-8 u \\cos \\left(\\omega  t_d \\right)\\right).\t\\label{eqo4}\n\\end{align}\n \nHere we have approximated $u(t-t_d)=u(t)-t_d \\dot{u}(t)$ and $v(t-t_d)=v(t)-t_d \\dot{v}(t)$ where $O(t_d^2)$, $\\ddot{u}(t), \\ddot{v}(t),\\epsilon \\dot{u}(t), \\epsilon\\dot{v}(t)$ are neglected. Note that, the above sets of equations are the first order autonomous equations and are symmetrical or even in nature with  $(u, v) \\rightarrow (-u,-v)$. Now the traditional stability analysis of Eqs.~\\eqref{eqo2} and \\eqref{eqo4} can be done by linearising around the fixed point $(0,0)$ which is the common fixed point of both the Eqs.~\\eqref{eqo2} and \\eqref{eqo4} as well as the original Eq.~\\eqref{eq}. It is easily deduced from Eqs.~\\eqref{eqo2} and \\eqref{eqo4} that $u = v = 0$ (i.e., $x = 0$) is an equilibrium solution determines the fixed point. The linear stability analysis of Eqs.~\\eqref{eqo2} and \\eqref{eqo4} will help us to find the valid excitation strength under the corresponding state of limit cycle, center  or center-type cases.\n\n\\subsection{Numerical Results for Parametric Excitation}\n\nHere we have numerically characterized the resonance and antiresonance behaviours for the limit cycle, center and center-type cases. Plots of amplitude variations are provided with $\\gamma$ for two different resonances at $\\Omega=2$(black, doted) and $\\Omega=4$(red) in terms of the scaled radius of the orbits. Further, phase space plots, amplitude variations with time and average energy consumption per cycle at their steady state behaviours of each states are also shown. \n\nFigs. \\ref{fig ps}(a), (b) and (c) describe the plots for amplitudes at steady state for the limit cycle, center and center-type cases, respectively, with $\\gamma$ for two different values of $\\Omega$ where the resonances are appearing. In fig.~\\ref{fig ps}(a), for both $\\Omega=2$ and $4$, amplitude increases with $\\gamma$. For $\\Omega=2$, amplitude saturates to a fixed value at steady state, but at $\\Omega=4$, the amplitudes are oscillating at the steady state. The plot has been done by taking averages at  steady state for each $\\gamma$. Similarly, fig.~\\ref{fig ps}(b) shows, amplitude  increases very slowly with $\\gamma$  for $\\Omega=2$ and for $\\Omega=4$, it is fixed and gives the center in nature. Likewise, fig.~\\ref{fig ps}(c) shows, the amplitude increases with $\\gamma$ and gives a limit cycle for $\\Omega=2$, but for $\\Omega=4$, the amplitude increases with a very small amount, showing center-type in nature. So, there is some switching transition at $\\Omega=2$ in each case like center is converting into center-type orbits and center-type orbit is going to limit cycle state due to increase in $\\gamma$.\n\nNext, fig.~\\ref{fig lc}(a) is the phase space plot of the limit cycle due to parametric resonances for $\\Omega=2$ and $\\Omega=4$. Fig.~\\ref{fig lc}(b) describes the amplitudes of the limit cycle oscillators due to excitation, where $\\Omega=2$ case settles down at steady value, but at $\\Omega=4$, amplitude oscillates with two different periods and giving rise to  an oscillatory antiresonance. Plots of energy consumption per cycle for both the cases are given in fig.~\\ref{fig lc}(c) which shows zero value at $\\Omega=2$ as the amplitudes become saturated, but at $\\Omega=4$, the curve oscillates as the amplitudes oscillate and average will be close to zero line. For van der Pol system the resonance and antiresonance behaviours are already studied in ref.~\\cite{penvo}, but here we have obtained an oscillating antiresonance.\n\nFigs.~\\ref{fig c}(a), (b) and (c) are the plots for phase space, amplitude variation with time and change in average energy of each cycle for the center while parametric excitation is acting upon it. Fig.~\\ref{fig c}(b) shows there is no oscillation in amplitude at $\\Omega=4$, but at $\\Omega=2$ the amplitude oscillates and decaying with time and showing the periods are different in each cycle as a footprint. If we take the average of the amplitude in this case then it can be comparable  with the power law of decreasing amplitude and one can say that due to parametric excitation the center case is dramatically converted into center-type case  at the resonating mode, $\\Omega=2$. Fig.~\\ref{fig c}(c) shows that for both resonating modes it touches the zero line whereas at $\\Omega=4$ it settles to a constant value from initial, but at $\\Omega=2$ the dynamics  follows the power law behaviour.\n\nFurther, figs.~\\ref{fig ct}(a), (b) and (c) are the plots for phase space, amplitude-time variation and energy consumption in each cycle for center-type case in presence of parametric excitation. The singularity at $\\Omega=2$ gives a limit cycle oscillation which can be confirmed from fig.~\\ref{fig ct}(b) where the amplitude touches a steady state value. But, at $\\Omega=4$, it follows the power law decay in amplitude with time having a small amount of oscillation that can be fix by taking averages. Fig.~\\ref{fig ct}(c) shows the plots for energy consumption per cycle in each modes where the variation touches the zero value within a small amount of time  for $\\Omega=2$ and at $\\Omega=4$, it shows quite oscillating in nature as amplitude oscillates.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics*[width=\\linewidth]{f2-gama_vs_r_averages.eps}\n\\caption{Plots of scaled radius for different $\\gamma$ with two different values of $\\Omega=2$(black, doted) and $4$(red) where (a) is for the limit cycle situation having $a=1,b=0.2<\\frac{\\sin(\\omega t_d)}{\\omega}$, (b) is the center situation with $a=0,b=\\frac{\\sin(\\omega t_d)}{\\omega}$ and (c) is the center-type situation with $a=1,b=\\frac{\\sin(\\omega t_d)}{\\omega}$. The time delay $t_d, \\epsilon$ and $\\omega$ are fixed at 0.623, 0.05 and 1, respectively.} \t\\label{fig ps}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics*[width=\\linewidth]{f3-LC_case_study.eps}\n\\caption{\\textbf{Limit Cycle Case:} Plots of (a) phase space, (b) scaled radius and (c) energy consumption per cycle for limit cycle due to parametric excitation for $\\Omega=2$(black, doted) and $\\Omega=4$(red). The fixed parameter values for this state are $a=1,~b=0.2<\\frac{\\sin(\\omega t_d)}{\\omega},~\\omega=1,\\gamma=2$ and $ \\epsilon=0.05,$ where time delay $t_d$ is kept fixed in 0.623.}\t\\label{fig lc}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics*[width=\\linewidth]{f4-Centre_case_study.eps}\n\\caption{\\textbf{Center:} Plots of (a) phase space, (b) scaled radius and (c) energy consumption per cycle for limit cycle due to parametric excitation for $\\Omega=2$(black, doted) and $\\Omega=4$(red). The fixed parameter values for this state are $a=0,~b=\\frac{\\sin(\\omega t_d)}{\\omega},~\\omega=1,\\gamma=2,~t_d=0.623$ and $ \\epsilon=0.05.$}\t\\label{fig c}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics*[width=\\linewidth]{f5-Centre_Type_case_study.eps}\n\\caption{\\textbf{Center-type:} Plots of (a) phase space, (b) scaled radius and (c) energy consumption per cycle for limit cycle due to parametric excitation for $\\Omega=2$(black, doted) and $\\Omega=4$(red). The fixed parameter values for this state are $a=1,~b=\\frac{\\sin(\\omega t_d)}{\\omega},~\\omega=1,\\gamma=2,~t_d=0.623$ and $ \\epsilon=0.05.$}\t \\label{fig ct}\n\\end{figure}\n\n\\subsection{Stability Region of Parametrically Excited Limit Cycle}\t\\label{subsec2}\nFor weak time delay as in the earlier numerical and analytical scenario we have shown the stability region of the parametrically excited system. \nRelying on Eqs.~\\eqref{eqo2} and \\eqref{eqo4}, we can determine the steady state solution  and periodic solutions around it, in addition to their stability profile. Assuming that the coefficients $u$ and $v$ are small and vary as $u \\approx e^{st}$ and $v \\approx e^{st}$ one obtain relations for each of the system upon linearising the above Eqs.~\\eqref{eqo2} and \\eqref{eqo4}. At resonance, $\\lbrace s-\\frac{\\epsilon}{2} \\sin(t_d)\\rbrace^2=\\frac{\\epsilon^2}{4}\\left[b^2-\\lbrace\\cos(t_d)+\\frac{b \\gamma}{2}\\rbrace^2\\right]$  leads to $|\\cos(t_d)+\\frac{b \\gamma}{2}|\\le \\cos(t_d)+\\frac{|b \\gamma|}{2}<|b|$, and at antiresonance, $\\lbrace s-\\frac{\\epsilon}{2} \\sin(t_d)\\rbrace^2=\\frac{\\epsilon^2}{4}\\lbrace b^2-\\cos^2(t_d)\\rbrace$  leads to $\\cos(t_d)<|b|$.\n\nThe stability of the fixed point is determined by the eigenvalues of the Jacobian matrix of the vector fields in Eqs.~\\eqref{eqo2} and \\eqref{eqo4}. The characteristic polynomials which give the eigenvalues are given below(for resonance and antiresonance respectively)\n\\begin{align}\n\\lambda^2+ \\epsilon \\lambda (b-\\sin \\left(t_d\\right))-\\frac{1}{16} b^2 \\gamma ^2 \\epsilon ^2+\\frac{b^2 \\epsilon ^2}{4} -\\frac{1}{2} b \\epsilon ^2 \\sin \\left(t_d\\right)+\\frac{\\epsilon ^2}{4}(\\sin^2 (t_d)+\\cos^2 (t_d))=0,\t\\label{eg2}\n\\end{align}\n\\begin{align}\n\\sigma^2+ \\epsilon \\sigma \\left(b-\\sin \\left(t_d\\right)\\right)+\\frac{b^2 \\epsilon ^2}{4}-\\frac{1}{2} b \\epsilon ^2 \\sin \\left(t_d\\right) +\\frac{\\epsilon ^2}{4}(\\sin^2 (t_d)+\\cos^2 (t_d))=0.\t\\label{eg4}\n\\end{align}\n\\normalsize\nFrom Eqs.~\\eqref{eg2} and \\eqref{eg4}, it is clear that the Jacobian matrices of the vector fields at the initial equilibrium solutions both has two complex conjugate eigenvalues, namely $\\lambda_{1,2}=A_1+iB_1$ and $\\sigma_{1,2}=A_2+iB_2$ where $A_1=\\epsilon (\\sin(t_d)-b)\/2=A_2=\\alpha (say), B_1=\\frac{\\epsilon}{2} \\sqrt{\\cos^2(t_d)-\\frac{b^2 \\gamma^2}{4}}$ and $B_2=\\frac{\\epsilon \\cos(t_d)}{2}$. Thereby, solving Eqs.~\\eqref{eqo2} and \\eqref{eqo4} near the origin is amounts to determine a dynamical phase-space trajectory in the form $(u, v) = e^{A_{1,2}t}(\\cos B_{1,2}t, \\sin B_{1,2}t)$. One can find  that the equilibrium state is locally stable for $\\alpha < 0$, while it is unstable for $\\alpha > 0$. For $\\alpha=0$, the eigenvalues are imaginary with close orbital  path will be center or slowly decaying center-type situation. The zero critical value of the parameter, $\\alpha$ determines a Hopf bifurcation point, where one encounters a significant qualitative change in the system's dynamical profile. In our case, one physically assumes $\\alpha > 0$ with an unstable equilibrium state.\n\nFor unit frequency, the eigenvalues for resonance and antiresonance are $$\\lambda_{1,2} \\approx \\pm 0.03125 \\epsilon  \\sqrt{64. b^2 \\gamma ^2-168.847}-0.5 b \\epsilon +0.291737 \\epsilon$$  and  $$\\sigma_{1,2} \\approx -0.5 b \\epsilon +(0.291737\\, \\pm 0.406066 i) \\epsilon$$ (the eigenvalues of all $\\Omega \\neq 2$) respectively, where the values of $a$ and $t_d$ are $1$ and $0.623$ respectively. The respective eigenvalues give the nature of the above autonomous system as well as the original parametrically excited system. From the graphs, we find that $\\Omega = 2$ is the resonant point and at the non-zero singular point $\\Omega = 4$, there exist an oscillating antiresonance. Since, we have chosen $\\omega=1$, the resonance came at $\\Omega = 2$ as a parametric resonance appears at twice the eigen frequency.\n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\resizebox*{2.7in}{!}{\\includegraphics*{f6a-EV_Space_CO_2.eps}} \\label{fig7.1a}} \n\\quad\n\\subfigure[]{\n\\resizebox*{2.7in}{!}{\\includegraphics*{f6b-EV_Space_CO_4.eps}} \\label{fig7.2b}}\n\\caption{Stability diagram for parametrically excited time delayed system for $b$ with time delay ($t_d$) is obtained from linear stability analysis for the cases (a) $\\Omega=2$ and (b) $\\Omega=4$.} \\label{fig es}\n\\end{figure}\nIn fig.~\\ref{fig es} stability diagram for parametrically excited time delayed system for $b$ with time delay ($t_d$) are shown from linear stability analysis for the limiting case (a) $\\Omega=2$ and (b) $\\Omega=4$.\n\n\\section{Stability Analysis and Hopf Bifurcation}  \\label{sec4}\nHere we have proposed stability analysis and Hopf bifurcation of the system for the full range of parameter space. We shall perform local stability analysis of the trivial fixed point, $E_0(0,0)$ and the existence of Hopf bifurcation of the model system \\eqref{eq}. The linearized system at the origin is written as\n\n\\begin{align}\n\\left\\{ \\begin{array}{l}\n\\dot{x}(t) = y(t),\t\\\\\n\\dot{y}(t) =  -\\epsilon x(t - t_d) - \\epsilon b \\dot{x}(t) - {\\omega^2} x(t).\n\\end{array} \\right.  \\label{15}\n\\end{align}\n                                                                                                                                                                             \nThe corresponding characteristic equation of system \\eqref{15} is given by\n\\begin{align*}\n\\lambda^2 + b \\epsilon \\lambda + \\omega^2 + \\epsilon e^{-\\lambda t_d} = 0.\n\\end{align*} \nAbove equation can be rewritten as\n\\begin{align}\n(\\lambda^2 + b \\epsilon \\lambda + \\omega^2) e^{\\lambda t_d} + \\epsilon = 0.  \\label{17}\n\\end{align} \n        \nNow, we have the following cases:\n\n\\textbf{Case 1:} When $t_d =0$, Eq.~\\eqref{17} becomes\n\\begin{align}\n\\lambda^2 + b \\epsilon \\lambda + \\omega^2 + \\epsilon = 0.  \\label{18}\n\\end{align} \n\nWe can see that the conditions for all the roots of Eq.~\\eqref{18} are negative (as all the coefficients are positive) or to have negative real part is given by Routh-Hurwitz criterion as $b \\epsilon>0$ and $\\omega^2 +\\epsilon>0$. Then, the trivial equilibrium point $E_0$ is locally asymptotically stable.\n\n\\textbf{Case 2:} When $t_d \\neq 0$, $\\lambda= i s$ be the root of Eq.~\\eqref{17}, we have\n\\begin{align}\n(-{s^2} + i s b \\epsilon +\\omega^2)(\\cos(t_d s) + i \\sin(t_d s)) +\\epsilon = 0.\t\\label{17a}\n\\end{align}\n\nEquating real and imaginary parts, we have\n\\begin{align}\n(-s^2 + \\omega^2) \\cos(t_d s) -s b \\epsilon \\sin(t_d s)+ \\epsilon =& 0, \\notag \\\\\ns b \\epsilon \\cos(t_d s) +(-s^2 + \\omega^2) \\sin(t_d s) =& 0. \\notag\n\\end{align}\nWe have from above equation\n\\begin{align}\n\\cos(t_d s) =& \\frac{\\epsilon (s^2 -\\omega^2)}{(-s^2 + \\omega^2)^2 +(s b \\epsilon)^2}=\\frac{s^2-\\omega^2}{\\epsilon}, \\notag \\\\\n\\sin(t_d s) =& \\frac{s b \\epsilon^2}{(-s^2 + \\omega^2)^2 +(s b \\epsilon)^2}=sb. \t\\label{19}\n\\end{align}\nNow, from the trigonometrical relation $\\sin^2(t_d s) +\\cos^2(t_d s) =1$, we have\n\\begin{align}\ns^4 +(b^2 \\epsilon^2 -2 \\omega^2) s^2 + \\omega^4 -\\epsilon^2 = 0. \\label{20}\n\\end{align}\nEq.~\\eqref{20} has the following roots $s_j$, which is given by\n\\begin{align}\ns_j = \\pm \\sqrt{ \\frac{ 2 \\omega^2 -b^2 \\epsilon^2 \\pm \\epsilon \\sqrt{b^2 (b^2 \\epsilon^2 - 4 \\omega^2) +4}}{2}}; \\quad j=1,2,3,4.  \\label{21}\n\\end{align}\nThe above Eq.~\\eqref{21} has at least one positive root $s_0$ if the condition $ b < \\frac{\\sqrt{2}}{\\epsilon} \\sqrt{\\omega^2 + \\sqrt{\\omega^4 -\\epsilon^2}}$ satisfies.\n\nWe obtain the corresponding critical value of time delay $t_{d_l}$ for $s_0$ as\n\\begin{align}\nt_{d_l} =  \\left\\lbrace \\begin{array}{ll}\n\\frac{1}{s_0} \\left( \\sin^{-1} {b s_0} +2\\pi l \\right), & \\text{if }  b < \\frac{\\sqrt{2}}{\\epsilon} \\sqrt{\\omega^2 + \\sqrt{\\omega^4 -\\epsilon^2}},  \\\\\n\\frac{1}{s_0} \\left(\\pi -\\sin^{-1} {b s_0} +2\\pi l \\right), & \\text{if }  b > \\frac{\\sqrt{2}}{\\epsilon} \\sqrt{\\omega^2 - \\sqrt{\\omega^4 -\\epsilon^2}},\n\\end{array} \\right. \\quad l= 0, 1, 2, \\cdots.\t\\label{22}\n\\end{align}\n\nDefine $t_d^* = \\min \\lbrace t_{d_l} \\rbrace$, i.e., $t_d^*$ is the smallest positive value of $t_{d_l}$; $l=0,1,2,\\cdots$, given by the above Eq.~\\eqref{22}. Now, we determine whether the roots of Eq.~\\eqref{17} cross the imaginary axis of the complex plane as $t_d$ varies. Let $\\lambda(t_d) = \\xi(t_d) + i s(t_d)$ be a root of the Eq.~\\eqref{17} such that these two conditions $\\lambda(t_d) = \\xi(t_{d_l}) =0$ and $s(t_{d_l})=s_0$ satisfies. \n\n\\begin{lemma}\nTransversality condition satisfies if the following holds\n $$\\left[ \\Re \\left(\\frac{d\\lambda}{dt_d}\\right)^{-1} \\right]_{t_d =t_d^*} \\neq 0.$$\n\\end{lemma}\n\\begin{proof}  \nDifferentiating Eq.~\\eqref{17} with respect to $t_d$, we obtain\n\\begin{align}\n\\left( \\frac{d\\lambda}{dt_d} \\right)^{-1} = \\frac{b \\varepsilon + i 2 s}{s (b \\varepsilon s+ i (s^2-\\omega^2))} +i \\frac{t_d}{s}.\t\\notag\n\\end{align}\nNow collecting the real parts at $t_d = t_d^*$, we have\n\\begin{align}\n\\left[ \\Re \\left( \\frac{d\\lambda}{dt_d} \\right)^{-1} \\right]_{t_d = t_d^*} = \\frac{ (b \\varepsilon)^2 +2 (s^2 - \\omega^2) }{(b \\varepsilon s)^2 + (s^2 - \\omega^2)^2},\t\\label{22a}\n\\end{align}\nwhich shows that \n\\begin{align}\n\\left[ \\Re \\left( \\frac{d\\lambda}{dt_d} \\right)^{-1} \\right]_{t_d = t_d^*} >0, \\text{ if } b < \\frac{\\sqrt{2}}{\\epsilon} \\sqrt{\\omega^2 + \\sqrt{\\omega^4-\\epsilon^2}}.  \\notag\n\\end{align}\nTherefore, the transversality condition is satisfied for each $t_d = t_d^*$ and hence Hopf bifurcation occurs at $t_d = t_d^*$. This completes the proof. \n\\end{proof}\nIf $ \\Re \\left( \\frac{d\\lambda}{dt_d} \\right)>0,$ then all those roots that crosses the imaginary axis with non-zero speed at $(\\lambda=) i s$ from left to right as $t_d$ increases. Now, we state the following result:\n\\begin{theorem}\nIf trivial equilibrium point $E_0$ exists then that point of the model system \\eqref{eq} is locally asymptotically stable when $t_d \\in [0, t_d^*) $ and unstable for $t_d>t_d^* $. Furthermore, the system undergoes Hopf bifurcation at $E_0$ when $t_d = t_d^*,$ provided $b < \\frac{\\sqrt{2}}{\\epsilon} \\sqrt{\\omega^2 + \\sqrt{\\omega^4-\\epsilon^2}}$.\n\\end{theorem}\n\n\\textbf{Note:} For this section and the next section, we have considered $b$ analytically as a constant parameter only.\n\n\\subsection{Direction and Stability of Hopf Bifurcation} \\label{subsec5} \nIn this subsection, we will discuss the direction, stability and period of the bifurcating periodic solutions using normal form and center manifold theory, introduced by Hassard et al.~\\cite{hassard}. We assume that the system \\eqref{eq} about the fixed point $E_0$ undergoes Hopf bifurcation at the critical point $t_{d} =t_d^*$. Then $\\pm is$ are corresponding purely imaginary roots of the characteristic equation at the critical point throughout this subsection. \\\\\nLet $t_d =t_d^* +\\mu$, where $\\mu \\in \\mathbb{R}$. Define the space of continuous real valued functions as $\\mathbb{C}= \\mathbb{C}([-1,0], \\mathbb{R}^2)$. Let $u_{1}(t)=x(t),~ u_{2}(t)=y(t)$ and ${u_i}(t)=u_{i}(t_d t)$ for $i=1,2$; the delay system \\eqref{eq} then converts into functional differential equation in $\\mathbb{C}$ as\n\\begin{align} \n\\dot{u}(t) = L_{\\mu} (u_t) + F(\\mu, u_t),  \\label{23}\n\\end{align} \nwhere $u(t)=(u_1(t)$, $u_2(t))^\\top \\in \\mathbb{C}$, $u_t(\\theta) = u(t+\\theta) = (u_1(t+\\theta)$, $u_2(t+\\theta))^\\top \\in \\mathbb{C}$ and $L_{\\mu}: \\mathbb{C} \\to \\mathbb{R}^2$, $F: \\mathbb{R}\\times \\mathbb{C} \\to \\mathbb{R}^2$ are given respectively by \n\\begin{align}\nL_{\\mu}(\\phi)&=(t_d^* +\\mu) \\begin{pmatrix} \n0 & 1 \\\\ \n-\\omega^2  & -b \\epsilon\n\\end{pmatrix} \\phi(0)+(t_d^*+\\mu) \\begin{pmatrix}\n0 & 0 \\\\ \n-\\varepsilon & 0\n\\end{pmatrix} \\phi(-1), \t\\label{24}  \\\\\nF(\\mu, \\phi) &= (t_d^* +\\mu) \\begin{pmatrix}\n0 \\\\  -a \\epsilon \\phi_1^2(0) \\phi_2(0) \n\\end{pmatrix},\t\t\\label{25}\n\\end{align} \nwhere $\\phi = (\\phi_1, \\phi_2)^\\top \\in \\mathbb{C}$. \\\\\nBy Riesz representation theorem, there exists a matrix $\\eta(\\theta,\\mu)$, $\\theta \\in [0,1]$, whose components are of bounded variation functions such that \n\\begin{align}\nL_{\\mu} \\phi = \\int_{-1}^0 d \\eta(\\theta,\\mu) \\phi(\\theta), \\quad \\text{for } \\phi \\in \\mathbb{C}.\t\\notag\n\\end{align}\nBy considering Eq.~\\eqref{24}, one can choose \n\\begin{align} \n\\eta(\\theta, \\mu) = (t_d^* +\\mu) \\begin{pmatrix} \n0 & 1 \\\\ \n-\\omega^2  & -b \\epsilon\n\\end{pmatrix} \\delta(\\theta)+(t_d^*+\\mu) \\begin{pmatrix}\n0 & 0 \\\\ \n-\\epsilon & 0\n\\end{pmatrix} \\delta(\\theta+1),  \\notag\n\\end{align} \nwhere $\\delta$ is Dirac delta function.                                          \nFor $\\phi \\in \\mathbb{C}$, define\n\\begin{align}\nA(\\mu)\\phi(\\theta) =\\left\\{\\begin{array}{ll}\\displaystyle{\\frac{d \\phi(\\theta)}{d \\theta}},& \\theta \\in [-1,0),\\\\\n\\\\ \\int_{-1}^0 d\\eta(\\theta,\\mu) \\phi(\\theta), & \\theta  = 0, \\end{array}\\right. \\label{26} \\\\\nR(\\mu) \\phi(\\theta)  = \\left\\{\\begin{array}{ll}\n\\begin{pmatrix}\n 0 \\\\ 0 \\end{pmatrix}, & \\theta \\in [-1,0), \\\\\nF(\\mu, \\phi),& \\theta=0.\n\\end{array}\\right. \t\\notag \n\\end{align}\nIn order to convenient study of Hopf bifurcation problem, we transform system \\eqref{23} into an operator equation of the form \n\\begin{align}\n\\dot{u}(t) = A(\\mu) u_t + R(\\mu) u_t,\t\\label{27}\n\\end{align}\nwhere $u_{t}(\\theta)=u(t+\\theta)$, $\\theta \\in [-1,0]$. \\\\\nFor $\\psi \\in \\mathbb{C} ([0,1], (\\mathbb{R}^2)^*)$, the adjoint operator $A^*$ of $A$ is defined by \n\\begin{align}\nA^*(\\mu) \\psi(m) = \\left\\{ \\begin{array}{ll} -\\frac{d\\psi(m)}{dm}, & m\\in(0,1], \\\\ \\int_{-1}^0 \\psi(-t) d\\eta(t,0), & m=0. \\end{array}\\right.  \\notag \n\\end{align} \nFor $\\phi \\in \\mathbb{C} ([-1,0],\\mathbb{R}^2)$ and $\\psi \\in \\mathbb{C} ([0,1],(\\mathbb{R}^2)^*)$, define the bilinear inner product in order to normalize the eigenvectors of operator $A$ and adjoint operator $A^*$. \n\\begin{align}\n\\left\\langle \\psi(m), \\phi(\\theta) \\right\\rangle &= \\bar{\\psi}(0).\\phi(0)-\\int_{-1}^0 \\int_{\\xi = 0}^{\\theta} \\bar{\\psi}^{\\top} (\\xi -\\theta) d\\eta(\\theta) \\phi(\\xi) d\\xi,\t \\label{28} \n\\end{align}\nwhere $\\eta(\\theta)= \\eta(\\theta,0)$. Then $A$ and $A^*$ are adjoint operators. We know that $\\pm i s_{0} t_d^*$ are eigenvalues of $A$. Therefore, they are also eigenvalues of $A^*$. Next we calculate the eigenvector $q(\\theta)$ of $A(0)$ belonging to the eigenvalue $i s_{0} t_d^*$ and eigenvector $q^* (\\theta)$ of  $A^*(0)$ belonging to the eigenvalue $-i s_{0} t_d^*$. \\\\\nThen we have $ A(0) q(\\theta) = i s_{0} t_d^* q(\\theta)$  and $A^*(0) q^*(\\theta) = -i s_{0} t_d^* q^*(\\theta)$. Let $q(\\theta) = (1,\\alpha)^{\\top} e^{i s_{0} t_d^* \\theta}$ and $q^*(\\theta) = P(1,\\beta)^{\\top} e^{-i s_{0} t_d^* \\theta}$.\nThus, we can obtain\n\\begin{align} \n\\alpha =i s_0, \\quad \\beta =\\frac{i s_0}{\\omega^2 +\\epsilon e^{i s_{0} t_d^*}}.  \\label{29} \n\\end{align} \nFrom \\eqref{28}, we have\n\\begin{align}\n\\langle q^*&(m),q(\\theta) \\rangle = \\bar{q^*}(0).q(0)-\\int_{-1}^{0} \\int_{\\xi=0}^{\\theta} \\bar{q^*}^{\\top} (\\xi-\\theta) d\\eta(\\theta) q(\\xi) d\\xi,  \\notag \\\\\n&= \\bar{P} (1+\\alpha \\bar{\\beta}) -\\int_{-1}^{0} \\int_{\\xi=0}^{\\theta} \\bar{P} \\begin{pmatrix} 1 & \\bar{\\beta} \\end{pmatrix}\\times e^{-i t_d^* s_0 (\\xi-\\theta)} d\\eta(\\theta) \\begin{pmatrix} 1 \\\\ \\alpha \\end{pmatrix} e^{i t_d^* s_0 \\xi} d\\xi, \\notag \\\\\n&= \\bar{P} \\bigg[ 1+\\alpha \\bar{\\beta} \\bigg. \\left. -\\int_{-1}^0 \\begin{pmatrix} 1 & \\bar{\\beta} \\end{pmatrix} \\theta e^{i t_d^* s_0 \\theta} \\begin{pmatrix} 1 \\\\ \\alpha \\end{pmatrix} d\\eta(\\theta) \\right],\t\\notag \\\\ \n&= \\bar{P} \\bigg[ 1+\\alpha \\bar{\\beta} \\bigg. \\left. +t_d^* \\begin{pmatrix} 1 & \\bar{\\beta} \\end{pmatrix} \\begin{pmatrix} 0 & 0 \\\\ -\\epsilon & 0 \\end{pmatrix}\\begin{pmatrix} 1 \\\\ \\alpha \\end{pmatrix} e^{-i t_d^* s_0} \\right],  \\notag \\\\\n&= \\bar{P} [ 1+\\alpha \\bar{\\beta} - \\bar{\\beta} \\epsilon t_d^* e^{-i t_d^* s_0} ].\n\\end{align}\nThus, we can choose $\\bar{P}$ as \n\\begin{align} \n\\bar{P} =& \\frac{1}{ 1+\\alpha \\bar{\\beta} -\\bar{\\beta} \\epsilon t_d^* e^{-i t_d^* s_0}},  \\label{30}\n\\end{align} \nthen $ \\langle q^*(m), q(\\theta) \\rangle = 1 $. Furthermore, $ \\langle q^*(m), \\bar{q}(\\theta) \\rangle = 0$. Now we obtain $q$ and $q^{*}$. \\\\\nNext, we use the same notations as in~\\cite{hassard} and we first compute the coordinates to describe the center manifold $C_{0}$ at $\\mu=0$. Here, $u_{t}$ be the solution of \\eqref{23} at $\\mu =0$. Define\n\\begin{align}  \nz(t)= \\langle q^{*}, u_{t} \\rangle, \\label{31} \n\\end{align} \nand then define \n\\begin{align}\nW(t,\\theta) =& u_t (\\theta) -z(t) q(\\theta)-\\bar{z}(t) \\bar{q}(\\theta), \\notag \\\\\n=&  u_t (\\theta) -2 \\Re \\{ z(t) q(\\theta) \\}. \\label{32} \n\\end{align}\nOn center manifold $C_{0}$, we have $W(t, \\theta) = W (z(t), \\bar{z}(t),\\theta)$, where                                                                                                            \n\\begin{align} \nW(z(t),\\bar{z}(t),\\theta) = W_{20} (\\theta) \\frac{z^2}{2} +W_{11} (\\theta) z \\bar{z} + W_{02} (\\theta) \\frac{\\bar{z}^2}{2} + \\cdots,  \\label{33}\n\\end{align} \n$z$ and $\\bar{z}$ are the local coordinates for center manifold $C_{0}$ in the direction of $q$ and $q^{*}$ respectively. Note that $W$ is real if $u_t$ is real. We consider only real solutions.\\\\\nFor the solution $u_t \\in C_{0}$ of \\eqref{23}, since $\\mu =0$, we have\n\\begin{align}\n\\dot{z}(t) =& \\langle q^*, \\dot{u}_t \\rangle,\t\\notag \\\\\n=& \\langle q^*, A(0) u_t +R(0)u_t \\rangle,\t\\notag \\\\\n=& i t_d^* s_0 z + \\bar{q^*}(0).F(0, W(z,\\bar{z},0) +2 \\Re\\{z(t)q(0)\\}), \\notag \\\\\n\\overset{\\Delta}{=} & i t_d^* s_0 z+ \\bar{q^*}(0).F_0(z,\\bar{z}). \\notag\n\\end{align}\nRewrite this equation as\n\\begin{align} \n\\dot{z}(t) = i t_d^* s_0 z + g(z,\\bar{z}), \t \\label{34}\n\\end{align} \nwhere $g(z,\\bar{z}) = \\bar{q^*}(0).F_{0}(z,\\bar{z})$ and expand $g(z,\\bar{z})$ in powers of $z$ and $\\bar{z}$, that is \n\\begin{align} \ng(z, \\bar{z}) = g_{20} \\frac{z^2}{2} + g_{11} z \\bar{z} + g_{02} \\frac{\\bar{z}^2}{2} + g_{21} \\frac{z^2 \\bar{z}}{2}+ \\cdots.   \\label{35}\n\\end{align} \nWe have\n\\begin{align}\ng(z,\\bar{z}) =& \\bar{q^*}(0).F_{0}(z,\\bar{z}),\t\\notag \\\\\n=& \\bar{P} t_d^* \\begin{pmatrix} 1 & \\bar{\\beta} \\end{pmatrix} \\begin{pmatrix} 0 \\\\ -a \\epsilon u_{1t}^2(0) u_{2t}(0) \\end{pmatrix}, \\notag \n\\end{align} \nwhere $u_{t}(\\theta) = (u_{1t}(\\theta), u_{2t}(\\theta))^\\top = W(t,\\theta)+z(t) q(\\theta) +\\bar{z}(t) \\bar{q}(\\theta)$ and $q(\\theta) = (1,\\alpha)^\\top e^{i s_0 t_d^* \\theta}$, then we have\n\n\\begin{align}\n\\begin{pmatrix} u_{1t}(\\theta) \\\\ u_{2t}(\\theta) \\end{pmatrix} = \\begin{pmatrix} W_{20}^{(1)}(\\theta) \\frac{z^2}{2} + W_{11}^{(1)}(\\theta) z \\bar{z}+W_{02}^{(1)}(\\theta) \\frac{\\bar{z}^2}{2} + O(|z, \\bar{z}|^3) \\\\ \nW_{20}^{(2)}(\\theta) \\frac{z^2}{2} + W_{11}^{(2)}(\\theta) z \\bar{z} + W_{02}^{(2)}(\\theta) \\frac{\\bar{z}^2}{2} + O(|z,\\bar{z}|^3) \\end{pmatrix} + z \\begin{pmatrix} 1 \\\\ \n\\alpha \\end{pmatrix} e^{i t_d^* s_0 \\theta} + \\bar{z} \\begin{pmatrix} 1 \\\\ \n\\bar{\\alpha} \\end{pmatrix} e^{-i t_d^* s_0 \\theta},  \\notag\n\\end{align} \nFor $\\theta=0$, we have\n\\begin{align}\nu_{1t}(0) =& z + \\bar{z} + W_{20}^{(1)}(0) \\frac{z^2}{2} + W_{11}^{(1)}(0) z \\bar{z} + W_{02}^{(1)}(0) \\frac{\\bar{z}^2}{2} +O(|z,\\bar{z}|^3),  \\notag \\\\\nu_{2t}(0) =& \\alpha z +\\bar{\\alpha} \\bar{z} +W_{20}^{(2)}(0) \\frac{z^2}{2} +W_{11}^{(2)}(0) z \\bar{z}  +W_{02}^{(2)}(0) \\frac{\\bar{z}^2}{2} +O(|z,\\bar{z}|^3).  \\notag \n\\end{align}\nIt follows that\n\\begin{align}\ng(z,\\bar{z}) =& \\bar{P} t_d^* \\begin{pmatrix} 1, & \\bar{\\beta} \\end{pmatrix} \\begin{pmatrix} 0, & -a \\epsilon \\left(z + \\bar{z} + W_{20}^{(1)}(0)\\frac{z^2}{2} +\\cdots \\right)^2 \\left(\\alpha z + \\bar{\\alpha} \\bar{z} + W_{20}^{(2)}(0) \\frac{z^2}{2} +\\cdots \\right) \\end{pmatrix}^\\top, \\notag \\\\\n=& - a \\epsilon \\bar{\\beta} \\bar{P} t_d^* (2\\alpha +\\bar{\\alpha}) z^{2} \\bar{z} + \\cdots. \\notag\n\\end{align}\n\nComparing the coefficients with \\eqref{35}, we have\n\\begin{align}\ng_{20} =& g_{11} = g_{02} = 0, \\notag \\\\  \ng_{21} =& -2 a \\epsilon \\bar{\\beta} \\bar{P} t_d^* (2 \\alpha + \\bar{\\alpha}).  \\label{36}\n\\end{align} \nThus, we can compute the following quantities:\n\\begin{align}\nC_1(0) =& \\frac{i}{2s_0 t_d^*} \\left(g_{20} g_{11} - 2 |g_{11}|^2 - \\frac{|g_{02}|^2}{3} \\right) + \\frac{g_{21}}{2},  \\notag \\\\\n\\mu_2 =& -\\frac{ \\Re \\{C_1(0) \\}}{\\Re\\{ \\lambda'_{0}(t_d^*) \\}},  \\notag \\\\\n\\beta_2 =& 2 \\Re \\{C_1(0)\\},  \\notag \\\\\nT_2 =& -\\frac{ \\Im \\{C_1(0) \\} + \\mu_2 \\Im \\{ \\lambda'_0(t_d^*) \\}}{s_0 t_d^*}. \\label{37} \n\\end{align}\nAbove formulae give a description of Hopf bifurcation periodic solutions of \\eqref{23} at $t_d = t_d^*$ on the center manifold. Notations $\\mu_2$, $\\beta_2$ and $T_2$ determine respectively the direction of Hopf bifurcation, stability and period of bifurcating periodic solutions~\\cite{hassard}. We summarize the following theorem.\n\\begin{theorem} \\label{4.3}\nFor expressions given in \\eqref{37}, following results hold\n\\begin{enumerate}\n\\item[(i)] If $\\mu_2 > 0, \\, (\\mu_2<0)$, then Hopf bifurcation is supercritical, (subcritical) and the bifurcating periodic solutions exist for $t_d > t_d^*, \\, (t_d<t_d^*)$.\n\\item[(ii)] The bifurcating periodic solutions are stable if $\\beta_2 < 0$ and unstable if $\\beta_2 > 0$.\n\\item[(iii)] The period of the bifurcating periodic solutions increases if $T_2 > 0$ and decreases if $T_2 < 0$.\n\\end{enumerate}\n \\end{theorem}\n\n\\begin{example}\nFor the following model\n\\begin{align} \n\\dot{x}(t) =& y(t), \\notag \\\\ \n\\dot{y}(t) =& -0.05 (x(t-0.623)+(x^{2}(t) +0.58347)y(t))-x(t). \\notag\n\\end{align} \n By taking $a=1, \\omega =1, \\epsilon =0.05, b=0.583474$ for the model system \\eqref{7}. \nWe have from Eq.~\\eqref{37},\n\\begin{align*}\n\\mu_2 =&  0.7563576 > 0,  \\\\\n\\beta_2 =& -0.0308725 < 0, \\\\\nT_2 =& 0.0174092 >0.\n\\end{align*} \nTherefore, from Theorem \\ref{4.3}, we conclude that Hopf bifurcation is supercritical and bifurcating periodic solution is stable with increasing period.\n\\end{example}\n\n\\section{Bifurcation Analysis: Exact Numerical Simulation} \t\\label{sec5}\nTo investigate the effects of nonlinearity ($a$) and damping term ($b$), we carried out detailed bifurcation analysis of the model system \\eqref{7}. Our main objective is to detect the existence of complex system dynamics in the presence of nonlinear damping. The system \\eqref{7} is integrated using Matlab software for different cases of nonlinearity and damping with resonance ($\\Omega=2$) and antiresonance ($\\Omega=4$). For analysing the exact range of stability in details, we have represented bifurcation plots for both the state variables throughout the simulation. System dynamics show symmetric property throughout the simulation for both resonance and antiresonance cases. Time span is $[0,220]$ for fig.~\\ref{fig9} and $[0,500]$ for figs.~\\ref{fig8}, \\ref{fig10} and \\ref{fig11}.\n\n\n\\begin{figure}[!ht]\n\\centering\n\\subfigure[]{\n\\resizebox*{2.2in}{!}{\\includegraphics{f7a-x_I.eps}} \\label{fig8.1a}} \n\\quad\n\\subfigure[]{\n\\resizebox*{2.2in}{!}{\\includegraphics{f7b-y_I.eps}} \\label{fig8.2b}} \n\\caption{Bifurcation diagram for the model system \\eqref{7} with the parameter values $\\epsilon =0.05, \\omega=1$, $a = 0, b = 0$. \\textbf{(Feedback system with increasing phase space area)}} \\label{fig8}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f8a-II_x_2.eps}} \\label{fig9.1a}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f8b-II_y_2.eps}} \\label{fig9.2b}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f8c-II_x_4.eps}} \\label{fig9.3c}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f8d-II_y_4.eps}} \\label{fig9.4d}} \n\\caption{Bifurcation diagram for the model system \\eqref{7} with the parameter values $\\epsilon =0.05, \\gamma = 2, \\omega=1$, $a=0, b = \\frac{\\sin(\\omega t_d)}{\\omega}$. For figures (a) and (b) $\\Omega=2$, and (c) and (d) $\\Omega=4$. \\textbf{(Center)}} \\label{fig9}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f9a-III_x_2.eps}} \\label{fig10.1a}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f9b-III_y_2.eps}} \\label{fig10.2b}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f9c-III_x_4.eps}} \\label{fig10.3c}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f9d-III_y_4.eps}} \\label{fig10.4d}}  \n\\caption{Bifurcation diagram for the model system \\eqref{7} with the parameter values $\\epsilon =0.05, \\gamma = 2, b= \\frac{\\sin(\\omega t_d)}{2 \\omega}, \\omega=1$, $a=1, b < \\frac{\\sin(\\omega t_d)}{\\omega}$. For figures (a) and (b) $\\Omega=2$ and (c) and (d) $\\Omega=4$. \\textbf{(Limit cycle)}} \\label{fig10}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f10a-IV_x_2.eps}} \\label{fig11.1a}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f10b-IV_y_2.eps}} \\label{fig11.2b}}  \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f10c_IV_x_4.eps}} \\label{fig11.3c}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f10d_IV_y_4.eps}} \\label{fig11.4d}}\n\\caption{Bifurcation diagram for the model system \\eqref{7} with the parameter values $\\epsilon =0.05, \\gamma = 2, \\omega=1$, $a=1, b = \\frac{\\sin(\\omega t_d)}{\\omega}$. For figures (a) and (b) $\\Omega=2$, and (c) and (d) $\\Omega=4$. \\textbf{(Center-type orbit)}} \\label{fig11}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f11a-x_b_td.eps}} \\label{fig12.1a}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f11b-y_b_td.eps}} \\label{fig12.2b}}  \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f11c-x_b_td_4.eps}} \\label{fig12.3c}} \n\\quad\n\\subfigure[]{\n\\resizebox*{1.35in}{!}{\\includegraphics{f11d-y_b_td_4.eps}} \\label{fig12.4d}} \n\\caption{Bifurcation diagram of $b$ with the parameter values $\\epsilon =0.05, \\gamma = 1, a=1, \\omega=1, t_d=0.623$. For figures (a) and (b) $\\Omega=2$, and (c) and (d) $\\Omega=4$.} \\label{fig12}\n\\end{figure}\n\nIn fig.~\\ref{fig8}, first we consider the case when $a=0$, $b=0$ i.e., no nonlinearity and damping terms present in the system. In this case, bifurcation depends only on the feedback controller $\\epsilon$ as the frequency $\\omega$ is equal to $1$. Feedback system lies in the range $[-200,200]$, can be observed from fig.~\\ref{fig8}. Initially system is stable and as delay increases system becomes unstable.\n\nWe have investigated the effect of damping term $b = \\frac{\\sin(\\omega t_d)}{\\omega}$ in the absence of nonlinearity for the resonance case in figs.~\\ref{fig9.1a}-\\ref{fig9.2b} and antiresonance in figs.~\\ref{fig9.3c}-\\ref{fig9.4d}. System dynamics exhibits gap dependent bifurcation for both cases. The state variables lie in the domain $[-400,400]$ and $[-20,20]$, and $t_d \\in (0,3]$ and $t_d \\in (0,20]$ for resonance and antiresonance cases respectively. Phase space area of resonance case is larger than the antiresonance case. We observe center solution exists for the system \\eqref{7} as it is highly dependent on damping terms.\n\nIn fig.~\\ref{fig10}, we consider the effect of both nonlinear term  $a=1$ and damping term  $b = \\frac{\\sin(\\omega t_d)}{2 \\omega}$. Bifurcation diagram for resonance and antiresonance cases are executed in figs.~\\ref{fig10.1a} and \\ref{fig10.2b} respectively. System variables lie between $[-4,4]$ and $[-20,20]$, and $t_d \\in (0,20]$ and shows number of stability switching scenario in fig.~\\ref{fig10}. Sequences of period doubling of order 2 and 4 with symmetricity can be remarked for resonance in figs. \\ref{fig10.1a}-\\ref{fig10.2b} whereas for antiresonance, repeated scenario of period doubling and inverse period doubling of order 2, 6 and 12 are perceived in figs. \\ref{fig10.3c}-\\ref{fig10.4d}. Rich period doubling and halving scenario confirms the presence of limit cycles. We observe oscillatory dynamics which converges to a steady state for the system \\eqref{7} from the bifurcation diagrams. \n\nIn fig.~\\ref{fig11}, nonlinearity and damping terms are $a=1$ and $b = \\frac{\\sin(\\omega t_d)}{\\omega}$ respectively. Resonance case is considered in figs.~\\ref{fig11.1a}-\\ref{fig11.2b} and antiresonance in figs.~\\ref{fig11.3c}-\\ref{fig11.4d}. System variables lie between $[-4,4]$ and $t_d \\in [0,10]$  and shows center-type solution in bifurcation plots for both the cases. As $t_d$ increases, $x$ and $y$ also increases for non-resonance case. However, in resonance case $x$ and $y$ does not increases with $t_d$, it attains a maximum saturated peak and repeat it. Sequences of stability switches take places; period-doubling and halving layouts appear or disappear. As $t_d$ increases, the point of projection of period-doubling and halving occurs, exhibits more dense plot; such as it tending towards the chaotic scenario.\n\nIn fig.~\\ref{fig12}, initially system shows limit cycle behaviour with $a=1, ~\\gamma=1$ and in the range of $b \\in [0,0.583]$ and stable for $b>0.583$. Hopf bifurcation occurs for both the resonance and non-resonance cases shown in figs.~\\ref{fig12.1a}-\\ref{fig12.2b} and \\ref{fig12.3c}-\\ref{fig12.4d}, respectively. Hence, we observed that the dynamics of the system bifurcates at the $b=0.583$.\n\n\\section{Discussions and Conclusions}\t\t \\label{sec7}\n\nA delay model in a damped quartic nonlinear oscillator is solved by multiscale perturbation method to obtain various periodic orbits, namely a limit cycle, center and a slowly decaying center with reference to a van der Pol oscillator having a limit cycle. This delay induced periodicity and bifurcation are probed through the parametric resonance and antiresonance. {\\color{black}The calculation of response function due to a parametric excitation of an arbitrary periodic orbit is carried out here through K-B approach which is much handier than RG method specially in the context of  results obtained for the direction of Hopf bifurcation and stability of the bifurcating periodic solutions.} The effect of control parameters such as damping and nonlinear terms are investigated  via bifurcation analysis using normal form and center manifold theory.\n\\begin{enumerate}\n\n\\item We have found the characteristics of resonances due to parametric excitation. The nature of the resonances at $\\Omega=2\\omega$ and $\\Omega=4\\omega$ are investigated for limit cycle, center and center-type cases with $\\omega=1$.\n\n\\item Stability criteria of the parametrically excited system for $\\Omega=2 \\omega$ and $4\\omega$ resonances are investigated  with $\\omega=1$ and they correspond to the approximate solution using K-B method.\n\n\\item  Linear stability analysis and bifurcation scenario for the full range of parameter space of the system \\eqref{eq} is accomplished for the trivial fixed point. Occurrence of Hopf bifurcation of the fixed point ${E_0}(0,0)$ at critical point $t_d^*=0.623$  has been shown at which trivial point looses its stability. The values $\\mu_2 = 0.7563576, \\beta_2 = -0.0308725$ and $T_2 = 0.0174092$ indicates that Hopf bifurcation is supercritical and the bifurcating periodic solutions are stable with increasing periods.  \n\n\\item Stability and direction of Hopf bifurcation have been investigated using center manifold and normal form theory. We have also concluded from the bifurcation diagram, the system is unstable initially only for limit cycle solutions and stable for all the other solutions at the trivial point.\n\n\\item When the periodic orbit is a limit cycle in our system it can be clearly identified for the weakly delayed van der Pol case where the sign of $b$ and coefficient of $x(t-t_d)$ are both $<0$. \n\\end{enumerate}\n\nThus, we can find that the damping is one of most effective parameter which stabilizes the system and delay plays a crucial role to lead the system unstable. In the presence of both damping and nonlinearity with delay stabilizes the system. The possibility of stabilizing a feedback delayed system with parametric excitation have a great effect in control of periodic flows, stabilization of high-speed milling in material formation and cutting process via spindle speed variation~\\cite{stepan}.\n\n\\section*{Acknowledgements}\nSandip Saha acknowledges RGNF, UGC, India for the partial financial support. This work is supported by the Council of Scientific and Industrial Research (CSIR), Govt. of India under grant no. 25(0277)\/17\/EMR-II to R.K. Upadhyay. SS and GG are thankful to Dr. Sagar Chakraborty for useful comments.\n\n\n\\section*{References}\n\n\\bibliographystyle{unsrt}      \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $G$ be a connected semisimple Lie group with finite center. Let $G = NAK$ be a choice of Iwasawa decomposition. Define $\\mathfrak a = \\operatorname{Lie}(A)$, and recall Harish-Chandra's formula for the spherical function of parameter $i\\lambda \\in i\\mathfrak a^*$:\n\\begin{equation}\\label{harishchandraformula}\n\\varphi_{i\\lambda}(g) = \\int_K e^{(\\rho+i\\lambda)(H(kg))} dk \\,,\n\\end{equation}\nwhere $H : G \\to \\mathfrak a$ is the Iwasawa projection and $\\rho \\in \\mathfrak a^*$ the half-sum of positive roots. In \\cite{Michels2022} we are interested in spectral asymptotics in a relative trace formula for maximal flat submanifolds of an associated locally symmetric space. In the analysis one requires asymptotics for the integral\n\\[ \\int_A \\varphi_{i\\lambda}(a) b(a) da \\]\n(with a smooth cutoff $b(a)$)\\,, as well as bounds for twisted versions thereof, as $\\lambda$ grows in $\\mathfrak a^*$. The problems are relative analogues of the problem of bounding spherical functions considered in \\cite{duistermaat1983}.\n\nIn view of \\eqref{harishchandraformula}, one is naturally led to study the behavior of the Iwasawa $\\mathfrak a$-projection along sets of the form $kA$. More specifically, we are interested in the critical points of\n\\[ \\lambda(H(ka)) \\]\nas a function of $a \\in A$.\nThe maximal flat submanifolds of the symmetric space $G\/K$ are the images of the sets $gA$. When $G = \\operatorname{PSL}_2(\\mathbb R)$, then $G\/K$ is the hyperbolic plane, the maximal flats are the geodesics, and the motivating problem of bounding orbital integrals is solved in \\cite{Marshall2016}. It takes as input classical facts about the geometry of geodesics in the hyperbolic plane. For general semisimple $G$ we require analogous results for maximal flat submanifolds, which to our knowledge are not established. This article is concerned with them.\n\nThe article is divided into three parts, which are about the Iwasawa $N$-, $A$- and $K$- projections of the sets $gA$, in each of which we generalize geometric facts that are apparent in the case of $\\operatorname{PSL}_2(\\mathbb R)$.\n\n\\subsection{Statement of results}\n\nConsider the hyperbolic plane $\\mathbb H = \\{x+iy  \\in \\mathbb C : y > 0\\}$ with its hyperbolic metric.\nThe group $\\operatorname{PSL}_2(\\mathbb R)$ acts on $\\mathbb H$ by homographies. We make the standard choice of Iwasawa decomposition: Let $N$ be the subgroup of unipotent upper triangular matrices, $A$ the subgroup of diagonal matrices and $K =  \\operatorname{PSO}_2(\\mathbb R)$. Then $\\operatorname{PSL}_2(\\mathbb R) \\cong N \\times A \\times K$, the diffeomorphism being given by multiplication. The upper half-plane model naturally lends itself to describe the Iwasawa projections geometrically. The element\n\\[ g = \\begin{pmatrix}\n1 & x \\\\\n0 & 1\n\\end{pmatrix}\\begin{pmatrix}\n\\sqrt y & 0 \\\\\n0 & 1\/\\sqrt{y}\n\\end{pmatrix} \\in NA\\] sends $i$ to $x+iy$. That is, the real part of $gi$ can be identified with the $N$-projection of $g$, and the imaginary part with the $A$-projection.\n\nBy a (maximal) geodesic in $\\mathbf H$ we mean the $1$-dimensional submanifold defined by it, although we will sometimes informally speak of geodesics with an orientation. The geodesics in $\\mathbb H$ are the semicircles with centers on the horizontal axis, together with the vertical lines. The action of $\\operatorname{PSL}_2(\\mathbb R)$ on $i$ induces a diffeomorphism between $A$ and the vertical geodesic through $i$. Every geodesic is of the form $gAi$ with $g \\in \\operatorname{PSL}_2(\\mathbb R)$.\n\nObserve that the real part of every geodesic is a bounded set. We generalize this observation to semisimple Lie groups $G$, as follows.\n\n\\begin{theorem}\\label{npartbounded}Let $G$ be a connected semisimple Lie group. For all $g \\in G$, the $N$-projection of $gA$ is a relatively compact set.\n\\end{theorem}\n\nTheorem~\\ref{npartbounded} is proved in \\S\\ref{secNprojection} by diving into the mechanics of the Gram--Schmidt process, and showing that the orthogonalization part can be done with uniformly bounded operations. In fact, we will prove a stronger version with some uniformity, which is Theorem~\\ref{npartboundeduniform}. The uniform version requires to partition the set of all maximal flats, which is naturally identified with $G\/N_G(A)$, into a Zariski open `generic' set and several lower-dimensional `exceptional' sets. This partition generalizes the distinction between semicircles and vertical geodesics in the upper half plane $\\mathbb H$, in which case the semicircles form the generic set. The partition is not dependent on any choice of model for the symmetric space $G\/K$, but inherent to the choice of Iwasawa decomposition of $G$. Some of the lower-dimensional sets come from semistandard Levi subgroups, and it is no surprise that they are exceptional. But in general there are other exceptional sets, and the partition remains quite mysterious.\n\nThe second group of results concerns the Iwasawa $A$-projection. In the case of $G = \\operatorname{PSL}_2(\\mathbb R)$, define the height of a point in $\\mathbb H$ to be its imaginary part. The heights of the points of a geodesic $gAi$ form a bounded-from-above set precisely when the geodesic is a semicircle. In that case, the height is maximized at a unique critical point, which is the midpoint of the semicircle, and the height tends to $0$ at infinity on the geodesic. Such a critical point exists if and only if $gAi$ is not vertical, meaning that $g \\notin N \\cdot N_G(A)$.\n\nFor a general connected semisimple Lie group $G$, we prove the following.\n\n\\begin{theorem}\\label{aresultsallintro}Let $\\lambda \\in \\mathfrak a^*$ be an element that is positive with respect to the choice of Iwasawa decomposition, non-singular and which does not lie in a proper subspace spanned by roots. \n\\begin{enumerate}[label = (\\roman*)]\n\\item For all $g \\in G$ the ``height function''\n\\begin{align*}\nh_{\\lambda, g} : A & \\to \\mathbb R \\\\\na &\\mapsto \\lambda(H(ga))\n\\end{align*}\nhas at most one critical point. If it exists, it is non-degenerate and maximizes $h_{\\lambda, g}$. \n\\item The set of $g \\in G$ for which a given $a \\in A$ is a critical point of $h_{\\lambda, g}$, is a non-empty smooth submanifold of codimension $\\dim(A)$.\nThe set of $g \\in G$ for which $h_{\\lambda, g}$ has a critical point, is open.\n\\end{enumerate}\n\\end{theorem}\n\nWe prove Theorem~\\ref{aresultsallintro} in \\S\\ref{secAprojection}. The proof is quite technical and occupies the larger part of this article. Regarding the second part, note that it is by no means obvious that $h_{\\lambda, g}$ has a critical point for even a single $g$, because the domain $A$ is noncompact. The way in which we prove existence is by varying $g \\in K$, and realizing the elements $g$ with a given critical point as the minima of a certain smooth function on the compact group $K$. It is likely that the set of $g \\in G$ for which $h_{\\lambda, g}$ has a critical point is in fact dense and that this can be proved using a very different argument, which is related to Theorem~\\ref{npartbounded}; see Remark~\\ref{atonegativechamber}.\n\n\\begin{remark}\nThe critical points in Theorem~\\ref{aresultsallintro} can be thought of as giving the midpoints of the flat $gA \\subset S$. It is in general too much to hope that $H(ga)$ has a critical point as a function of $a$. That is, the critical points can depend on $\\lambda \\in \\mathfrak a^*$. (See Example~\\ref{examplemidpointdepends}.) We will not use the term ``midpoint'', all the more because we have found no way to generalize the observation that for $\\operatorname{PSL}_2(\\mathbb R)$, the critical point corresponds to the center of the semicircle $gAi \\subset \\mathbb H$.\\end{remark}\n\n\\begin{remark}In Theorem~\\ref{aresultsallintro}, many things break down when $\\lambda$ is singular, lies in a proper subspace spanned by roots or is nonpositive: the nondegeneracy, uniqueness, and existence of critical points. Regarding non-degeneracy, see Remark~\\ref{hessiancriticalsometimesdef}. For non-uniqueness and non-existence, see \\S\\ref{sectionexistence}.\\end{remark}\n\n\nFinally, we turn our attention to the Iwasawa $K$-projection. The Lie group $\\operatorname{PSL}_2(\\mathbb R)$ is naturally identified with the unit tangent bundle $T^1(\\mathbb H)$ via its action on the vertical vector at the base point, $(i, (0, 1))$. The Iwasawa $K$-projection of an element $g$ corresponds to a choice of direction at the point $gi$. The elements of the Weyl group $N_K(A)$ correspond to the vertical directions, up and down. As we approach infinity along a geodesic, the tangent line to the geodesic tends to a vertical one.\n\nWe generalize this observation as follows.\n\n\\begin{theorem}\\label{KparttoMcentralizer}Let $G$ be a connected semisimple Lie group. Let $g \\in G$ and $H \\in \\mathfrak{a}$. Then the $K$-projection of $g e^{tH}$ tends to $N_K(A) Z_{K}(H)$ as $t \\to + \\infty$.\n\\end{theorem}\n\nTheorem~\\ref{KparttoMcentralizer} is proved by projecting the $K$-projection of the geodesic flow down to the Lie algebra in a specific way and realizing it as the flow of a vector field. The resulting dynamical system is quite mysterious, but the asymptotic behavior of individual orbits can be well understood. It might seem that Theorem~\\ref{KparttoMcentralizer} is a statement about individual geodesics rather than maximal flats, but it is possible to formulate statements with uniformity in the variables $g$ and $H$; see Remark~\\ref{remarkKvsNproofs}.\n\nIn the end the results on the $N$- and $K$-projections were not needed for the analysis in \\cite{Michels2022}. But they complete the picture nicely, might be useful elsewhere, and they barely fail to provide alternative proofs of parts of Theorem~\\ref{aresultsallintro}; see Remark~\\ref{atonegativechamber}. A number of things remain mysterious, in particular the apparent relation between the partition in Theorem~\\ref{npartboundeduniform} on the $N$-projection, and the dynamical system used in the proof of Theorem~\\ref{KparttoMcentralizer}; see Remark~\\ref{remarkKvsNproofs}.\n\n\\section{Preliminaries}\n\n\\label{preliminariesLiegroups}\n\n\\subsection{Lie groups and Lie algebras}\n\n\\label{notationliegroups}\n\nLet $G$ be a reductive Lie group in the sense of Harish-Chandra \\cite{harishchandra1975}; see also \\cite[Chapter VII]{Knapp2002}. Most of the time $G$ will be connected semisimple with finite center and we will then simply say $G$ is semisimple. Let $K \\subset G$ be a maximal compact subgroup and $\\theta$ an involution of $G$ whose fixed point set is $K$. It induces an involution $\\theta$ of the Lie algebra $\\mathfrak g$, whose $+1$ and $-1$ eigenspaces we denote by $\\mathfrak k$ and $\\mathfrak p$ respectively. We denote the exponential of $X \\in \\mathfrak g$ by $\\exp(X)$. Define the semisimple part $\\mathfrak{g}_{ss} = [\\mathfrak g, \\mathfrak g]$. We extend the Killing form on $\\mathfrak{g}_{ss}$ to a nondegenerate symmetric bilinear form $\\langle \\cdot, \\cdot \\rangle$ on $\\mathfrak{g}$ that is positive definite on $\\mathfrak{k}$ and negative definite on $\\mathfrak{p}$, and with respect to which the center $\\mathfrak{z}(\\mathfrak{g})$ is orthogonal to $\\mathfrak{g}_{ss}$. Define $\\langle \\cdot, \\cdot \\rangle_{\\theta} = \\langle \\cdot, - \\theta(\\cdot) \\rangle$, a positive definite symmetric bilinear form. All statements on $\\mathfrak{g}$ involving norms, orthogonality and adjoints will be with respect to $\\langle \\cdot, \\cdot \\rangle_{\\theta}$. Let $\\mathfrak a \\subset \\mathfrak p$ be a maximal abelian subalgebra and $A = \\exp(\\mathfrak a)$. The choices of $\\mathfrak a$ are all conjugate under $K$. Define $P = \\exp(\\mathfrak p)$. The multiplication map $P  \\times K \\to G$ is a diffeomorphism, known as the Cartan decomposition.\n\n\n\\subsection{Symmetric spaces and maximal flats}\n\n\\label{propsmaximalflatssection}\n\nReferences for the following facts about symmetric spaces are \\cite{Eberlein1996, Helgason1978}. \n\nAssume here that $G$ is semisimple. Then $\\theta$ is a Cartan involution. The quotient $S = G\/K$ carries a left-$G$-invariant Riemannian metric induced by the Killing form on $\\mathfrak p$. It is a symmetric space of non-compact type, and every such space arises in this way.\n\nThe maximal flat submanifolds of $S$ are of the form $gAK$ with $g\\in G$. Such $g$ is uniquely determined by the submanifold up to multiplication on the right by $N_G(A)$. When $\\dim(A) = 1$, the maximal flats are precisely the geodesics. The rank of $G$ is defined to be $\\dim(A)$.\n\nLet $p \\in P$. The tangent space $T_{pK}S$ is identified with $\\mathfrak p$, using left multiplication by $p^{-1}$. Take $X \\in \\mathfrak p$ with norm $1$. The geodesic through $pK \\in S$ with tangent vector $X$ has equation $t \\mapsto p e^{tX}K$. A geodesic is called regular when a nonzero tangent vector at any (and hence every) point is a regular element of $\\mathfrak g$. It is called singular otherwise. A geodesic is regular if and only if it lies in a unique maximal flat.\n\n\n\\subsection{Iwasawa decomposition}\n\n\\label{secroots}\n\nLet $\\Sigma$ be the set of restricted roots of $\\mathfrak{a}$ in $\\mathfrak{g}$. By convention, $0  \\notin \\Sigma$. We denote by $\\mathfrak{g}_{\\alpha}$ the root space of a root $\\alpha \\in \\Sigma$, by $m(\\alpha) = \\dim(\\mathfrak g_\\alpha)$ its multiplicity and by $H_\\alpha \\in \\mathfrak a$ the element corresponding to $\\alpha$ under the isomorphism $\\mathfrak a \\cong \\mathfrak a^*$ given by $\\langle \\cdot, \\cdot \\rangle$. Fix a set of positive roots $\\Sigma^{+}$ with basis $\\Pi$. Let $\\mathfrak n \\oplus \\mathfrak a \\oplus \\mathfrak k$ and $N \\times A \\times K$ be the corresponding Iwasawa decompositions of $\\mathfrak g$ and $G$. Define $M = Z_{K}(A)$ and $M' = N_{K}(A)$ and denote by $\\mathfrak{m}$ the Lie algebra of $M$.\n\nDenote the Lie algebra Iwasawa projections by $E_{\\mathfrak{n}}$, $E_{\\mathfrak{a}}$ and $E_{\\mathfrak{k}}$. We have the orthogonal restricted root space decomposition\n\\begin{equation}\\label{rootspacedecomp}\n\\mathfrak{g} = \\mathfrak{a} \\oplus \\mathfrak{m} \\oplus \\bigoplus_{\\alpha \\in \\Sigma} \\mathfrak{g}_{\\alpha}  \\,.\n\\end{equation}\nDenote the projection onto $\\mathfrak g_{\\alpha}$ by $R_\\alpha$.\n\nDenote the Iwasawa projections from $G$ onto $N$, $A$ and $K$ by $n$, $a$ and $\\kappa$. Define the height $H(g) = \\log(a(g)) \\in \\mathfrak{a}$, the logarithm being the Lie logarithm on $A$.\n\nAll choices of the data $(K, A, N)$ are conjugate by an element of $G$. When $G = \\operatorname{GL}_n(\\mathbb R)$ or $\\operatorname{SL}_n(\\mathbb R)$ we make all the standard choices: $K = \\operatorname{O}_n(\\mathbb R)$ respectively $\\operatorname{SO}_n(\\mathbb R)$, $A$ is the connected component of the diagonal subgroup and $N$ is the upper triangular unipotent subgroup.\n\n\n\n\\subsection{Centralizers}\n\\label{defgenericset}\n\n\\label{definitionlevis}\nDenote by $\\mathcal{L}$ the set of centralizers in $G$ of subgroups of $A$. They are the standard Levi subgroups of semistandard parabolic subgroups of $G$. We will denote such a centralizer typically by $L$. It is again reductive and inherits all the data as in the beginning of \\S\\ref{notationliegroups} and \\S\\ref{secroots} from $G$ in the natural way. We allow $G$ to be reductive because we will occasionally need to apply results to Levis $L \\in \\mathcal{L}$. When $L \\in \\mathcal{L}$ with Lie algebra $\\mathfrak{l}$, define $\\mathfrak{a}^{L} = \\mathfrak{l}_{ss} \\cap \\mathfrak{a}$ and $\\mathfrak{a}_{L} = \\mathfrak{z}(\\mathfrak{l}) \\cap \\mathfrak{a}$. Then $\\mathfrak{a} = \\mathfrak{a}^{L} \\oplus \\mathfrak{a}_{L}$ orthogonally.\nThe set $\\mathcal L$ contains $A$ and $G$, and when $G$ is semisimple we have $\\mathfrak a^A = \\mathfrak a_G = 0$ and $\\mathfrak a_A = \\mathfrak a^G = \\mathfrak a$.\n\n\nDefine the positive Weyl chamber $\\mathfrak{a}^{+} = \\{ H \\in \\mathfrak{a} : \\forall \\alpha \\in \\Sigma^{+} : \\alpha(H) > 0 \\}$, the regular set\n\\[ \\mathfrak{a}^{\\operatorname{reg}} = \\mathfrak{a} - \\bigcup_{L \\neq M} \\mathfrak{a}_{L} = \\mathfrak{a} - \\bigcup_{\\alpha \\in \\Sigma} \\ker(\\alpha) \\,, \\]\nand the generic set\n\\[ \\mathfrak{a}^{\\operatorname{gen}} = \\mathfrak{a}^{\\operatorname{reg}} - \\bigcup_{L \\neq G} \\mathfrak{a}^{L} \\,. \\]\nCombined superscripts correspond to intersections: $\\mathfrak{a}^{\\operatorname{gen}, +} = \\mathfrak{a}^{\\operatorname{gen}} \\cap \\mathfrak{a}^{+}$.\n\nWe also define $(\\mathfrak a^*)^{\\operatorname{reg}}$, $(\\mathfrak a^*)^{\\operatorname{gen}}$ and $(\\mathfrak a^*)^{+}$ to be the corresponding subsets under the isomorphism  $\\mathfrak a \\cong \\mathfrak a^*$ defined by the Killing form. When $H \\in \\mathfrak a$ corresponds to $\\lambda \\in \\mathfrak a^*$ under the isomorphism given by the Killing form, then $H \\in \\mathfrak a^{\\operatorname{reg}}$ if and only if $\\lambda$ is not orthogonal to any roots, and $H  \\in \\mathfrak a^{\\operatorname{gen}}$ if and only if $\\lambda$ is in addition not contained in a proper subspace spanned by roots.\n\nWe will frequently use the following lemma, so we take care to properly reference it.\nWe will frequently use the following lemmas, so we take care to properly reference them.\n\n\\begin{lemma} \\label{conjugatetoAimplieslevi}Let $g \\in G$ and $H \\in \\mathfrak{a}$. If $\\operatorname{Ad}_{g}(H) \\in \\mathfrak{a}$, then $g \\in M' Z_{G}(H)$.\\end{lemma}\n\n\\begin{proof}This is stated in \\cite[\\S 5, Lemma~1]{harishchandra1975}. When $g \\in K$, the statement gives precisely the degree of uniqueness in the $KAK$ decomposition of $G$, and a proof can be found in \\cite[Lemma 7.38]{Knapp2002}. The general case can be reduced to $g \\in K$ as follows. Write $g = kp$ in the Cartan decomposition. Then $\\operatorname{Ad}_{p}(H) \\in \\operatorname{Ad}_{k^{-1}}(\\mathfrak{a}) \\subset \\mathfrak{p}$, and \\cite[\\S V.24.C, Proposition 1]{borel1991} implies that $p \\in Z_{G}(H)$. Then $\\operatorname{Ad}_{k}(H) \\in \\mathfrak{a}$, and the conclusion follows from the $g \\in K$ case.\n\\end{proof}\n\n\\begin{lemma}\\label{truelemma}Let $g \\in G$ and $H \\in \\mathfrak{a}$. If $\\operatorname{Ad}_{g}(H) \\in \\mathfrak m \\oplus \\mathfrak{a}$, then $g \\in M' Z_{G}(H)$.\\end{lemma}\n\n\\begin{proof}If we can show that $\\operatorname{Ad}_g(H) \\in \\mathfrak a$, the claim follows from Lemma~\\ref{conjugatetoAimplieslevi}.\n\nConsider the adjoint embedding $\\operatorname{ad} : \\mathfrak{g} \\to \\mathfrak{sl}(\\mathfrak{g})$. Equip $\\mathfrak g$ with any orthonormal basis compatible with the restricted root space decomposition \\eqref{rootspacedecomp}. In such a basis, $\\operatorname{ad}$ sends elements of $\\mathfrak{a}$ to diagonal matrices and elements of $\\mathfrak{k}$ to antisymmetric matrices.\n\nWrite $\\operatorname{Ad}_g(H) = X + H'$ with $X \\in \\mathfrak{m}$ and $H' \\in \\mathfrak a$. In the chosen basis, $\\operatorname{ad}_{ H'}$ is diagonal with real eigenvalues, and the antisymmetric matrix $\\operatorname{ad}_{X}$ is diagonalizable with purely imaginary eigenvalues. Because $[H, X] = 0$, the elements $\\operatorname{ad}_{ H'}$ and $\\operatorname{ad}_{X}$ are simultaneously diagonalizable, so that the eigenvalues of $\\operatorname{ad}_{ H'}+\\operatorname{ad}_{X}$ are those of $\\operatorname{ad}_{ H'}$ plus those of $\\operatorname{ad}_{X}$, in a suitable ordering. If these eigenvalues are real, it must be that $X = 0$.\n\nThis proves that $\\operatorname{Ad}_g(H) \\in \\mathfrak a$, and the lemma follows.\n\\end{proof}\n\n\\begin{lemma}We have $Z_G(A) = MA$ and $N_G(A) = M' A$.\\end{lemma}\n\n\\begin{proof}\nThe first statement follows from \\cite[Proposition 7.25]{Knapp2002}; the second statement follows by combining it with Lemma~\\ref{conjugatetoAimplieslevi}.\n\\end{proof}\n\n\n\\subsection{Derivatives}\n\n\\label{sectionderivatives}\n\nWhen $G$ is any Lie group with Lie algebra $\\mathfrak g$ and $b$ is an element of the universal enveloping algebra $U(\\mathfrak{g})$, we denote by $L_{b}$ the corresponding left invariant differential operator on $C^{\\infty}(G)$\n. When $X \\in \\mathfrak g \\subset U(\\mathfrak g)$, by definition\n\\[ (L_Xf)(g) = \\left.\\frac d{dt}\\right\\rvert_{t=0} f( ge^{tX}) \\,.\\]\nWhen $f : M \\to N$ is a differentiable map between differentiable manifolds, denote its differential at $m \\in M$ by $(Df)_m$.\nUsing left translation we identify all tangent spaces $T_{g} G$ with $\\mathfrak{g}$. When $g \\in G$, denote by $L_{g}$ and $R_{g}$ the left and right multiplication by $g$ on $G$. With our convention on tangent spaces, we then have for all $g, h \\in G$ that\n\\begin{align}\n(D L_{g})_{h} & = \\operatorname{id} \\,,  \\nonumber \\\\\n(D R_{g})_{h} & = \\operatorname{Ad}_{g^{-1}} \\,. \n\\end{align}\nWhen $X, Y \\in \\mathfrak{g}$ we have\n\\begin{align}\nL_{X} \\operatorname{Ad}_{g}(Y) & = \\operatorname{Ad}_{g}([X, Y]) \\,, \\label{derivativeAd} \\\\\nL_{X} \\operatorname{Ad}_{g^{-1}}(Y) & = - [X, \\operatorname{Ad}_{g^{-1}}(Y)] \\label{derivativeAdinverse} \\,.\n\\end{align}\n\nAssume now that $G$ is reductive as in the beginning of \\S \\ref{notationliegroups}.\n\n\n\\begin{lemma} \\label{computationAfirstderivative} \\label{differentialkpart}The differentials of $n$, $a$ and $\\kappa$ at $g \\in G$ are as follows:\n\\begin{align*}\n(D n)_{g} & = \\operatorname{Ad}_{a(g)} \\circ E_{\\mathfrak{n}} \\circ \\operatorname{Ad}_{\\kappa(g)} \\,, \\\\\n(D a)_{g} & = E_{\\mathfrak{a}} \\circ \\operatorname{Ad}_{\\kappa(g)} \\,, \\\\\n(D \\kappa)_{g} & = \\operatorname{Ad}_{\\kappa(g)^{-1}} \\circ E_{\\mathfrak{k}} \\circ \\operatorname{Ad}_{\\kappa(g)} \\,.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}Write $g = nak$ and take $X \\in \\mathfrak{g}$. For the $N$-projection, write\n\\begin{align*}\nn(ge^{X}) & = n \\cdot n(ake^{X}) \\\\\n& = n \\cdot n(a k e^{X}k^{-1}) \\\\\n& = n \\cdot (a \\cdot n(e^{\\operatorname{Ad}_{k}(X)}) \\cdot a^{-1}) \\,.\n\\end{align*}\nIn the last step, we have used that $n(ah) = a n(h) a^{-1}$ for $a \\in A$ and $h \\in G$. Therefore \n\\[ (Dn)_g = (D L_n)_e \\circ (D (\\left.\\operatorname{Ad}_a\\right\\rvert_N))_e \\circ (Dn)_e \\circ \\operatorname{Ad}_k \\,. \\] The first statement now follows from the fact that $(Dn)_{e} = E_{\\mathfrak{n}}$. The other statements are proved similarly.\nFor the $A$-projection we write\n\\[\na(ge^{X}) = a \\cdot a(ke^{X}) = a \\cdot a(e^{\\operatorname{Ad}_{k}(X)}) \n\\]\nand use that $(D a)_{e} = E_{\\mathfrak{a}}$.\nFor the $K$-projection we write\n\\[\n\\kappa(ge^{X})  = \\kappa(k e^{X}) \\\\\n= \\kappa(e^{\\operatorname{Ad}_{k}(X)}) \\cdot k\n\\]\nand use that $(D \\kappa)_{e} = E_{\\mathfrak{k}}$.\n\\end{proof}\n\nThe differential of the $A$-projection is also computed in \\cite[Lemma~6.1]{marshall2015}. Even though the proof there is for $G = \\operatorname{SL}_3(\\mathbb R)$, the argument works in general. Compare also \\cite[Corollary~5.2]{duistermaat1983}, but note that the Iwasawa decomposition used there is $KAN$ instead of $NAK$. \n\n\\begin{lemma} \\label{computationAsecondderivative} For $X, Y \\in \\mathfrak{g}$ and $g \\in G$ we have\n\\begin{align*}\n(L_{X}L_{Y} a)_g & = E_{\\mathfrak{a}}( [E_{\\mathfrak{k}}(\\operatorname{Ad}_{\\kappa(g)}(X)), \\operatorname{Ad}_{\\kappa(g)}(Y)] ) \\\\\n& = E_{\\mathfrak{a}}( [ E_{\\mathfrak{k}}(\\operatorname{Ad}_{\\kappa(g)}(X)), E_{\\mathfrak{n}}( \\operatorname{Ad}_{\\kappa(g)}(Y) ) ] ) \\\\\n& = E_{\\mathfrak{a}}( [ \\operatorname{Ad}_{\\kappa(g)}(X), E_{\\mathfrak{n}}( \\operatorname{Ad}_{\\kappa(g)}(Y) ) ] ) \\\\\n&= \\sum_{\\alpha \\in \\Sigma^+} \\langle\\theta R_{-\\alpha}(\\operatorname{Ad}_{\\kappa(g)}(X)), (R_{\\alpha} - \\theta R_{-\\alpha}) ( \\operatorname{Ad}_{\\kappa(g)}(Y) ) \\rangle_\\theta \\cdot H_\\alpha  \\,.\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nSimilar to the proof of \\cite[Lemma~6.1]{duistermaat1983}. Alternatively, by Lemma~\\ref{computationAfirstderivative} we find\n\\[ L_{Y} a(g) = E_{\\mathfrak{a}} (\\operatorname{Ad}_{\\kappa(g)}(Y) ) \\,. \\]\nUsing the chain rule, \\eqref{derivativeAd} and Lemma~\\ref{differentialkpart} to compute $(D \\kappa)_{g}$ we have\n\\begin{align*}\n L_{X} L_{Y} a(g) & = E_{\\mathfrak{a}} (\\operatorname{Ad}_{\\kappa(g)}([(D \\kappa)_{g} (X), Y]) ) \\\\\n& = E_{\\mathfrak{a}}( [E_{\\mathfrak{k}}(\\operatorname{Ad}_{\\kappa(g)}(X)), \\operatorname{Ad}_{\\kappa(g)}(Y)] ) \\,.\n\\end{align*}\nThis is the first equality. The other equalities follow as in \\cite[Lemma~6.1]{duistermaat1983}.\n\\end{proof}\n\n\\section{The \\texorpdfstring{$N$}{N}-projection and the Gram--Schmidt process}\n\\label{secNprojection}\n\nUnless otherwise specified, $G$ is a semisimple Lie group as in \\S\\ref{notationliegroups}. In this subsection we prove Theorem~\\ref{npartbounded} and the stronger Theorem~\\ref{npartboundeduniform}.\n\n\nRecall that the Iwasawa decomposition for $\\operatorname{SL}_n(\\mathbb R)$ is nothing but the Gram--Schmidt process: Take $g \\in \\operatorname{SL}_n(\\mathbb R)$. There is a unique $n \\in N$ such that the rows of $n^{-1}g$ are orthogonal for the Euclidean inner product on $\\mathbb R^{n}$. There is a unique $a \\in A$ such that the rows of $k := a^{-1}n^{-1}g$ have norm $1$. The Iwasawa decomposition of $g$ is then $nak$.\n\n\\subsection{Intro: \\texorpdfstring{$\\operatorname{SL}_2(\\mathbb R)$}{SL2(R)}}\n\n\\label{npartsltwo}\n\nWe first prove Theorem~\\ref{npartbounded} when $G = \\operatorname{SL}_2(\\mathbb R)$. This is quite trivial but already gives a good idea of what is happening.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{npartbounded} when $G = \\operatorname{SL}_2(\\mathbb R)$] Write $g = \\begin{psmallmatrix}\nv \\\\w\n\\end{psmallmatrix}$, and view $v, w \\in \\mathbb R^2$ as row vectors. Let $y > 0$. Multiplication on the right by $a = \\operatorname{diag}(y, y^{-1}) \\in A$ corresponds to letting the matrix $a$ act on $v$ and $w$. The projection $n(ga)$ is the matrix $\\begin{psmallmatrix}\n1 & x \\\\\n0 & 1\n\\end{psmallmatrix}$ where\n\\[ x = \\frac{\\langle va, wa \\rangle}{\\left\\langle wa, wa \\right\\rangle} \\,. \\]\n It is the unique matrix $n \\in N$ for which the rows of $n^{-1}ga$ are orthogonal. We must show that $x$ is bounded. In the generic case where both coordinates of $w$ in the standard basis are nonzero, the denominator $\\left\\langle wa, wa \\right\\rangle$ is $\\asymp \\max(y^2, y^{-2})$, because $a$ acts by $y$ in one coordinate and by $y^{-1}$ in the other. The numerator is always $\\ll \\max(y^2, y^{-2})$, for the same reason. Therefore in this generic case, $x$ is bounded and the claim follows. When a coordinate of $w$ vanishes, the corresponding term in the denominator $\\left\\langle wa, wa \\right\\rangle$ is nonzero, but the same is then true in the numerator. So the same bounds hold, with $\\max(y^2, y^{-2})$ replaced by $y^{2}$ or $y^{-2}$, depending on which coordinate of $w$ vanishes. The conclusion follows.\n\\end{proof}\n\n\\subsection{\\texorpdfstring{$\\operatorname{SL}_n(\\mathbb R)$}{SLn(R)}, exterior powers}\n\n\\label{npartsln}\n\nFor $\\operatorname{SL}_n(\\mathbb R)$ the same phenomenon occurs, where if a term in a denominator of a certain fraction vanishes, then so does the corresponding term in the numerator. The denominators will here be norms of orthogonal projections onto the orthocomplement of previous vectors, and they are most conveniently expressed using norms on exterior powers\n\nWhen $(V, b)$ is a bilinear space of finite dimension over a field, $b$ can be identified with a linear map $V \\to V^*$. It induces a linear map $\\bigwedge^k V \\to \\bigwedge^k V^* \\cong (\\bigwedge^k V)^*$ for all $k \\geq 0$, where the identification $\\bigwedge^k V^* \\cong (\\bigwedge^k V)^*$ comes from the natural pairing between $\\bigwedge^k V$ and $\\bigwedge^k V^*$. We get a natural bilinear form $\\bigwedge^k b$ on $\\bigwedge^k V$. Assume now that the field is $\\mathbb R$, and that $b$ is symmetric positive definite. Then so is $\\bigwedge^k b$. We denote all induced bilinear forms by $\\langle \\cdot, \\cdot \\rangle$ for convenience. If $(e_i)$ is an orthonormal basis of $V$, then an induced basis of $\\bigwedge^k V$ consisting of elements of the form $e_{j_1} \\wedge \\cdots \\wedge e_{j_k}$ is also orthonormal. As a consequence, if $v, w_1, \\ldots, w_k \\in V$ with the $w_i$ linearly independent, $W = \\operatorname{span}(w_i)$ and $\\operatorname{pr}_{W^\\perp}$ denotes the orthogonal projection onto $W^\\perp$, then\n\\begin{equation}\\label{normprojwedgesingle}\n\\lVert \\operatorname{pr}_{W^\\perp}(v) \\rVert = \\frac{\\left\\lVert v \\wedge \\left(\\bigwedge_{i=1}^k w_i\\right)\\right\\rVert}{\\left\\lVert \\bigwedge_{i=1}^k w_i \\right\\rVert} \\,,\n\\end{equation}\nwhere the norms are those induced on $V$, $\\bigwedge^{k+1} V$ and $\\bigwedge^{k} V$. More generally, for $v_1, v_2 \\in V$ we have\n\\begin{equation}\\label{orthprojectionnormintermsofwedge}\n\\left\\langle \\operatorname{pr}_{W^\\perp}(v_1), \\operatorname{pr}_{W^\\perp}(v_2) \\right\\rangle = \\frac{\\left\\langle v_1 \\wedge \\left(\\bigwedge_{i=1}^k w_i\\right), v_2 \\wedge \\left(\\bigwedge_{i=1}^k w_i\\right)  \\right\\rangle}{\\left\\lVert \\bigwedge_{i=1}^k w_i \\right\\rVert^2} \\,.\n\\end{equation}\nWe also have the $i$th coefficient of $\\operatorname{pr}_W(v)$ in the basis $(w_i)$ is given by\n\\begin{equation}\\label{orthoprojectioncoeffintermsofwedge}\n\\frac{\\left \\langle w_1 \\wedge \\cdots \\wedge \\widehat{w_i} \\wedge v\\wedge \\cdots \\wedge w_{k}, \\bigwedge_{i=1}^k w_i \\right \\rangle}{\\left\\lVert \\bigwedge_{i=1}^k w_i \\right\\rVert^2} \\,,\n\\end{equation}\nwhere the hat denotes omission. \n\n\\begin{proof}[Proof of Theorem~\\ref{npartbounded} when $G = \\operatorname{SL}_n(\\mathbb R)$] Write $g$ as a column of row vectors: $g = (v_1, \\ldots, v_n)^T$ with the $v_i \\in \\mathbb R^n$, and view $a \\in A$ as acting on $\\mathbb R^n$ diagonally. Define $V_i = \\operatorname{span}(v_{i}, \\ldots, v_n)$, so that $V_{n+1} = \\{0\\}$. Define $w_i^a$ to be the $i$th row of $n(ga)^{-1}ga$. Equip $\\mathbb R^n$ with the Euclidean inner product. The first step in the Gram--Schmidt process applied to $ga$ is the recurrence relation\n\\begin{align*}\nw_{n}^a &= v_n a\\,, \\\\\nw_{i-1}^a &= \\operatorname{pr}_{(V_{i}a)^\\perp} (v_{i-1}a) \\\\\n&= v_{i-1}a - \\sum_{j=i}^n \\frac{\\langle v_{i-1}a, w_j^a \\rangle}{\\langle w_j^a, w_j^a \\rangle }w_j^a \\qquad (i = n, \\ldots, 2) \\,.\n\\end{align*}\nIn the above sum, the coefficient in the term with index $j$ is the $(i-1,j)$th entry of $n(ga)$. We must show that those coefficients are bounded independently of $a$. For such a coefficient, with $i \\leq j$, we have\n\\begin{align*}\n\\frac{\\langle v_{i-1}a, w_j^a \\rangle}{\\langle w_j^a, w_j^a \\rangle }\n&= \\frac{\\left\\langle \\operatorname{pr}_{(V_{j+1}a)^\\perp}(v_{i-1}a), \\operatorname{pr}_{(V_{j+1}a)^\\perp}(v_ja)\\right\\rangle}{\\left\\lVert \\operatorname{pr}_{(V_{j+1}a)^\\perp}(v_ja)\\right\\rVert^2}  \\\\\n&= \\frac{\\left\\langle v_{i-1}a \\wedge \\left(\\bigwedge_{k = j+1}^n v_k a\\right), \\bigwedge_{k = j}^n v_k a \\right\\rangle}{\\left\\lVert \\bigwedge_{k = j}^n v_k a \\right\\rVert^2} \\,,\n\\end{align*}\nwhere we have used \\eqref{orthprojectionnormintermsofwedge}.\nThe action of $A$ on $\\bigwedge^{n-j+1} \\mathbb R^n$ is still diagonal in an induced basis. Fix such a basis and write $a = \\operatorname{diag}(a_1, \\ldots, a_K)$ in that basis. If $c_1, \\ldots, c_K$ denote the coordinates of $v_{i-1} \\wedge \\left(\\bigwedge_{k = j+1}^n v_k \\right)$ and $d_1, \\ldots, d_K$ those of $\\bigwedge_{k = j}^n v_k$, then by the triangle inequality\n\\begin{align}\\label{Nprojtriangleinequality}\n\\begin{split}\n\\left\\lvert\\frac{\\langle v_{i-1}a, w_j^a \\rangle}{\\langle w_j^a, w_j^a \\rangle }\\right\\rvert &= \\left\\lvert\\frac{\\sum_{k=1}^K a_k^2 c_k d_k}{\\sum_{k=1}^K a_k^2 d_k^2} \\right\\rvert \\\\\n&\\leq \\max_{d_k \\neq 0} \\left\\lvert \\frac{c_kd_k}{d_k^2}\\right\\rvert \\,.\n\\end{split}\n\\end{align}\nThis bound does not depend on $a$, so that the entries of $n(ga)$ are bounded.\n\\end{proof}\n\n\\subsection{Semisimple groups}\n\n\\label{Npartsemisimplegroup}\n\nNow let $G$ be a semisimple Lie group. There is no real reason to assume that $G$ has finite center, and Theorem~\\ref{npartbounded} is insensitive to central extensions in any case, but we do so because our setup in \\S\\ref{notationliegroups} is under this assumption. Let $\\rho : G \\to \\operatorname{GL}(V)$ be any finite-dimensional representation with discrete kernel, such as the adjoint representation $\\operatorname{Ad} : G \\to \\operatorname{GL}(\\mathfrak g)$, or the standard representation $\\operatorname{Std}$ if $G$ is already linear. The fact that $\\rho$ can be arbitrary is not essential, but it leads to questions about uniformity.\n\n\\begin{remark}Necessarily $\\rho(G) \\subset \\operatorname{SL}(V)$, because $\\mathfrak g = [\\mathfrak g,  \\mathfrak g]$ consists of commutators. Note that $\\rho(G)$ is automatically closed, by \\cite[Proposition 7.9]{Knapp2002} or alternatively by properties of Malcev closure \\cite[\\S1.4.2, Theorem 3]{onishchik1980}.\\end{remark}\n\n\\begin{lemma}\\label{compatibleiwasawa}In a suitable basis of $V$, the groups $\\rho(N)$, $\\rho(A)$ and $\\rho(K)$ are contained in the standard Iwasawa components of $\\operatorname{GL}(V)$.\\end{lemma}\n\n\\begin{proof}As in the proof of \\cite[Proposition 7.9]{Knapp2002}, there is a basis of $V$ such that $\\rho(K) \\subset \\operatorname{SO}_n(\\mathbb R)$ and $\\rho(P)$ consists of symmetric matrices. Because $A \\subset P$ is commutative, by acting on the basis by a suitable element of $\\operatorname{SO}_n(\\mathbb R)$ we may assume that $\\rho(A)$ is diagonal. Then the basis consists of restricted weight vectors. Sorting them by non-increasing weight ensures that $\\rho(N)$ is upper triangular unipotent.\n\\end{proof}\n\nEquip $V$ with an Iwasawa-compatible basis given by Lemma~\\ref{compatibleiwasawa} and denote the corresponding Iwasawa decomposition of $\\operatorname{SL}(V)$ by $N' A' K'$. Denote the $N'$-projection on $\\operatorname{SL}(V)$ by $n'$. \n\n\\begin{proof}[Proof of Theorem~\\ref{npartbounded}] By the case $G = \\operatorname{SL}_n(\\mathbb R)$ proved in \\S\\ref{npartsln}, the projection $n'(\\rho(gA)) \\subset n'(\\rho(g) A')$ is relatively compact in $N'$. It is contained in the closed set $\\rho(N)$, and therefore relatively compact in $\\rho(N)$. Because $\\rho$ has central kernel contained in $K$, it induces an isomorphism $N \\to \\rho(N)$. Therefore $n(gA) \\cong n'(\\rho(gA))$ is relatively compact in $N$.\n\\end{proof}\n\n\\subsection{Uniformity}\n\n\\label{npartpartitionnoncanonical}\n\nFor $G = \\operatorname{SL}_2(\\mathbb R)$ it is apparent from the proof in \\S\\ref{npartsltwo} that uniformity holds in Theorem~\\ref{npartbounded} when the entries of $g$ are bounded and both entries on the second row of $g \\in G$ are bounded away from $0$. We generalize this and partition any semisimple group $G$ into subsets for which uniformity holds on compact subsets. We briefly describe one such partition here and state a result about a more optimal partition in \\S\\ref{npartcanonicalpartitions}.\n\nWe use the notation from \\S\\ref{Npartsemisimplegroup}. Let $d = \\dim(V)$ and let $\\langle \\cdot, \\cdot \\rangle$ be the Euclidean inner product for the chosen basis of $V$. For $\\lambda \\in \\mathfrak a^*$ denote by $V_\\lambda$ the restricted weight space. To prove Theorem~\\ref{npartbounded}, instead of reducing to the case of $\\operatorname{SL}_n(\\mathbb R)$ as in \\S\\ref{Npartsemisimplegroup}, one can also directly use the argument in \\S\\ref{npartsln}, but restricted to elements $g \\in \\rho(G)$, and this leads to a statement with uniformity. For every tuple $\\mathcal S = (\\mathcal S_1, \\ldots,\\mathcal S_d)$  ($\\mathcal S$ for support) of sets of integral weights of $\\mathfrak g$, we may consider the subset $\\Omega_{\\mathcal S}$ of $G$ that consists of the elements $g$ with the following property: For every $j \\in  \\{1, \\ldots, d \\}$, the wedge product of every last $j$ rows of $\\rho(g)$, as an element of $\\bigwedge^j V$, has a nonzero component precisely along the weight spaces $(\\bigwedge^j V)_\\lambda$ with $\\lambda \\in \\mathcal S_j$. It is then apparent that the upper bound analogous to \\eqref{Nprojtriangleinequality} is uniform for $g$ in compact subsets of $\\Omega_{\\mathcal S}$, because the coefficients $d_k$ (which are now norms of weight space projections) are either zero or bounded away from zero.\n\n\\subsection{Canonical partitions}\n\n\\label{npartcanonicalpartitions}\n\nThe sets $\\Omega_{\\mathcal S}$ in \\S\\ref{npartpartitionnoncanonical} depend on the choice of basis of weight vectors of $V$. Moreover, they may form a partition of $G$ that is unnecessarily fine for the uniformity statement to be true; this can happen when the weight spaces of $V$ are not 1-dimensional. It is desirable to construct a basis-independent partition, which is what we do now, and which we expect to be the coarsest possible.\n\nWe continue to use the notation from \\S\\ref{Npartsemisimplegroup} and consider the right action of $\\operatorname{GL}(V)$ on $V$ by transposition: $vg := g^T v$. That is, in the chosen basis of $V$, the rows of $g \\in \\operatorname{GL}(V)$ are the images of the basis elements under the right action of $g$. (This awkward definition is an artifact of our decision to write Iwasawa decompositions as $NAK$ rather than $KAN$.) Note that for $g \\in G$ we have $\\rho(g)^T = \\rho(\\theta(g))^{-1}$, because $\\rho(A)$ consists of diagonal matrices and $\\rho(K) \\subset \\operatorname{SO}_n(\\mathbb R)$. That is, while the right action of $\\operatorname{GL}(V)$ by transposition depends on the choice of basis, the restriction to $G$ depends only on the choice of Iwasawa decomposition of $G$ (and in fact, only on $A$ and $K$).\n\nWe will prove Theorem~\\ref{npartboundeduniform} below in a way similar to the argument sketched in \\S\\ref{npartpartitionnoncanonical}, but using a block-by-block rather than a row-by-row orthogonalization process, where we transform a matrix so that a block of rows (corresponding to a weight space) is orthogonal to previous blocks of rows. That the Iwasawa decomposition corresponds to such a process, is the content of the following lemma.\n\nFor integral weights $\\lambda, \\mu$ of $\\mathfrak g$, we write $\\mu < \\lambda$ if $\\lambda - \\mu$ is nonzero and is a nonnegative integer linear combination of positive roots.\n\n\\begin{lemma}\\label{nprojectionintermsoforthogonalweight}Let $g \\in G$ and $n' = n'(\\rho(g))$. Then for distinct weights $\\lambda, \\mu$ of $V$ we have $V_\\lambda n'^{-1} \\rho(g)\\perp V_\\mu n'^{-1} \\rho(g)$ and $V_\\lambda (n'^{-1}-1) \\perp V_\\lambda$.\\end{lemma}\n\n\\begin{proof}We have that $n'^{-1} \\rho(g) \\in A' K'$. The first statement follows from the facts that $V_\\lambda A' = V_\\lambda$, that $V_\\lambda  \\perp V_\\mu$ and that $K'$ preserves orthogonality. For the second statement, we have for $X \\in \\operatorname{Lie}(N')$ that $X^T V_\\lambda \\in \\bigoplus_{\\mu < \\lambda} V_\\mu$, and exponentiating gives that $V_\\lambda(N' -1)  \\in \\bigoplus_{\\mu < \\lambda} V_\\mu$.\n\\end{proof}\n\nLet $\\Lambda$ be the sets of weights of $\\mathfrak a$ in $V$, denote by $m_\\lambda$ the multiplicity of $\\lambda \\in \\Lambda$, and define $s_\\lambda = \\sum_{\\mu < \\lambda} m_\\mu$. For any tuple $\\mathcal S = (\\mathcal S_\\lambda)_{\\lambda \\in \\Lambda}$ of sets of integral weights of $\\mathfrak g$, consider the subset $\\Omega_{\\mathcal S}$ of $G$ that consists of the elements $g$ with the following property: For every $\\lambda \\in \\Lambda$, the line $\\bigwedge^{s_\\lambda} \\bigoplus_{\\mu < \\lambda} V_\\mu \\rho(g) \\subset \\bigwedge^{s_\\lambda} V$ has nonzero orthogonal projection on the weight-$\\alpha$ spaces $(\\bigwedge^{s_\\lambda} V)_\\alpha$ for $\\alpha \\in S_\\lambda$, and zero projection when $\\alpha \\notin \\mathcal S_\\lambda$.\n\n\\begin{theorem}\\label{npartboundeduniform} Let $\\mathcal S$ be any tuple as above, and $D \\subset \\Omega_{\\mathcal S}$ compact. Then the $N$-projection $n(DA)$ is relatively compact.\n\\end{theorem}\n\nBefore proving Theorem~\\ref{npartboundeduniform}, we state some basic properties of the sets $\\Omega_{\\mathcal S}$.\n\n\\begin{proposition}\\label{coordOmegaspartitionG}The sets of the form $\\Omega_{\\mathcal S}$ (or those that are non-empty) partition $G$. They are stable on the left by $NAM$. \\end{proposition}\n\n\\begin{proof}\nThat the $\\Omega_{\\mathcal S}$ partition $G$ is clear from their definition. Acting on the left by $\\rho(NAM)$ on $g \\in \\operatorname{SL}(V)$ does not change the lines $\\bigwedge^{s_\\lambda} \\bigoplus_{\\mu < \\lambda} V_\\mu g$, so that the individual conditions defining $\\Omega_{\\mathcal S}$ are left-invariant under $NAM$.\n\\end{proof}\n\nFor $\\lambda \\in \\Lambda$, define $\\mathcal S^0_\\lambda$ to be the set of weights $\\alpha$ of $\\mathfrak g$ such that the line\n\\[ \\bigwedge^{s_\\lambda} \\bigoplus_{\\mu < \\lambda} V_\\mu \\rho(g) \\subset \\bigwedge^{s_\\lambda} V \\]\nhas nonzero projection onto the weight-$\\alpha$ subspace for at least one $g \\in G$. Define $\\mathcal S^0 = (\\mathcal S^0_\\lambda)_{\\lambda \\in \\Lambda}$. Then $\\Omega_{\\mathcal S^0}$ contains ``most'' elements of $G$.\n\n\\begin{proposition}\\label{omegaopendense}The set $\\Omega_{\\mathcal S^0}$ is open and dense in $G$. \\end{proposition}\n\n\\begin{proof}\n The image $\\rho(G)$ is the identity component of the real points of a real algebraic closed subgroup of $\\operatorname{SL}(V)$ \\cite[\\S3.3.3]{onishchik1980}. The set $\\rho(\\Omega_{\\mathcal S^0})$ is defined by the non-vanishing of finitely many rational functions (corresponding to projections onto weight spaces) and therefore Zariski open in $\\rho(G)$. (There are no vanishing conditions when $\\mathcal S = \\mathcal S^0$, or rather, they are satisfied on all of $\\rho(G)$.) It follows that $\\Omega_{\\mathcal S^0}$ is open and dense in $G$.\n\\end{proof}\n\n\\begin{example}Take $G = \\operatorname{SL}_2(\\mathbb R)$ and $\\rho = \\operatorname{Std}$ the standard representation. Then $\\Omega_{\\mathcal S^0} = G - NAM'$, because the only defining condition is that both entries on the bottom row are nonzero. The only other non-empty sets of the form $\\Omega_{\\mathcal S}$ are $NAM$ and $NAM \\cdot\\begin{psmallmatrix}\n0 & 1 \\\\\n-1 & 0\n\\end{psmallmatrix}$. The geodesics in $G\/K$ corresponding to elements of $\\Omega_{\\mathcal S^0}$ are semicircles. The other $\\Omega_{\\mathcal S}$ give rise to the vertical geodesics, with both orientations. One can check using an explicit computation that $\\rho = \\operatorname{Ad}$ gives the same partition of $\\operatorname{SL}_2(\\mathbb R)$.\n\\end{example}\n\n\\begin{example}\\label{omegaslthree}Take $G = \\operatorname{SL}_3(\\mathbb R)$ and $\\rho = \\operatorname{Std}$. The sets $\\Omega_{\\mathcal S}$ are determined by their $K$-projection, in view of Lemma~\\ref{coordOmegaspartitionG}. The sets $\\kappa(\\Omega_{S})$ come in all possible dimensions: The $0$-dimensional ones, which are the six right cosets of $M$ in $M'$. The $1$-dimensional ones: the three right $M'$-translates of $G \\cap (\\operatorname{O}(2) \\times \\operatorname{O}(1)) - M'$ and the three right $M'$-translates of $G \\cap (\\operatorname{O}(1) \\times \\operatorname{O}(2)) - M'$. The $2$-dimensional ones: the three right $M'$-translates of the product $(G  \\cap (\\operatorname{O}(2) \\times \\operatorname{O}(1)) - M' )(G \\cap (\\operatorname{O}(1) \\times \\operatorname{O}(2)) - M')$ and the three right $M'$-translates of that product with the two factors interchanged. And finally, the dense open set $\\kappa(\\Omega_{S^0})$.\\end{example}\n\n\\begin{remark}For any $G$, the set $\\kappa(\\Omega_{S^0})$ is disjoint from the sets $LM'$ with $L \\in \\mathcal L$ a standard Levi subgroup of a semistandard parabolic; this follows already from the constraint corresponding to the minimal weights $\\lambda$. But in general it is smaller than just the complement of the sets of the form $K \\cap L M'$, as illustrated in Example~\\ref{omegaslthree}.\n\nIt remains unclear to us whether the partition of $G$ into the $\\Omega_{\\mathcal S}$ depends on the choice of representation $\\rho$, and if not, whether the partition is the coarsest possible (up to splitting into connected components) for uniformity to hold. It would also be desirable to have a concrete description of the partition in general, as we do have when $G = \\operatorname{SL}_3(\\mathbb R)$.\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{npartboundeduniform}] Let $g \\in G$ and $a \\in A$. We view $n'(\\rho(ga))^{-1}$ as a block matrix: For weights $\\lambda \\geq \\mu$ of $V$, define\n\\begin{align*}\nT_{\\lambda, \\mu} : V_\\lambda &\\to V_\\mu \\\\\nv & \\mapsto \\operatorname{pr}_{V_\\mu} \\left( v \\cdot n'(\\rho(ga))^{-1} \\right)\n\\end{align*}\nand define $T_\\lambda = \\sum_{\\mu < \\lambda} T_{\\lambda, \\mu} : V_\\lambda \\to \\bigoplus_{\\mu < \\lambda} V_\\mu$. We must show that each $T_\\lambda$ is a bounded operator, uniformly in $a \\in A$ and $g$ in compact subsets of each $\\Omega_{\\mathcal S}$. By Lemma~\\ref{nprojectionintermsoforthogonalweight}, the map $T_\\lambda$ satisfies\n\\[ ((1+ T_{\\lambda}) V_\\lambda) \\rho(ga) \\perp \\bigoplus_{\\mu < \\lambda} V_\\mu  \\rho(ga) \\,. \\]\nThat is, $(T_{\\lambda} v)  \\rho(ga) = - \\operatorname{pr}_{\\bigoplus_{\\mu < \\lambda} V_\\mu \\rho(ga)} (v \\rho(ga) )$. (These two identities express the block-by-block orthogonalization process.) To show that $T_\\lambda$ is bounded, we may fix any basis $(b_i)_{1 \\leq i \\leq s_\\lambda}$ of $\\bigoplus_{\\mu < \\lambda} V_\\mu $ and we must show that for every $v \\in V_\\lambda$ the coordinates of the orthogonal projection of $v \\rho(ga)$ onto $\\bigoplus_{\\mu < \\lambda} V_\\mu \\rho(ga)$ in the basis $(b_i \\rho(ga))$ are bounded. By \\eqref{orthoprojectioncoeffintermsofwedge}, the $i$th coordinate is equal to\n\\begin{align*}\n\\frac{\\langle (b_1 \\wedge \\cdots \\wedge \\widehat{b_i} \\wedge v\\wedge \\cdots \\wedge b_{s_\\lambda}) \\rho(ga), (b_1 \\wedge \\cdots \\wedge  b_{s_\\lambda}) \\rho(ga) \\rangle}{\\left\\lVert (b_1 \\wedge \\cdots \\wedge b_{s_\\lambda})\\rho(ga) \\right\\rVert^2} \\,,\n\\end{align*}\nwhere the hat denotes omission. We may bound this using an inequality similar to \\eqref{Nprojtriangleinequality}. Concretely, call $d_\\mu$ the projection of $(b_1 \\wedge \\cdots \\wedge b_{s_\\lambda})\\rho(g)$ onto the weight-$\\mu$ subspace of $\\bigwedge ^{s_\\lambda}V$, and $c_\\mu$ that of $(b_1 \\wedge \\cdots \\wedge \\widehat{b_i} \\wedge v\\wedge \\cdots \\wedge b_{s_\\lambda}) \\rho(g)$. Then the fraction above equals\n\\[\n\\frac{\\sum_{\\mu} \\mu(a)^2 \\langle c_\\mu, d_\\mu \\rangle}{\\sum_{\\mu} \\mu(a)^2 \\lVert d_\\mu \\rVert^2}\\,, \\]\nwhere we define the weights on $A$ using the exponential map. By the triangle inequality, this is at most\n\\[ \\max_{d_\\mu \\neq 0} \\frac{\\lvert\\langle c_\\mu, d_\\mu \\rangle \\rvert}{\\lVert d_\\mu \\rVert^2} \\,,\\]\nwhich gives a uniform upper bound for $g$ in compact subsets of each $\\Omega_{\\mathcal S}$, because the $d_\\mu$ are then either zero or are bounded away from zero.\n\\end{proof}\n\n\\section{The \\texorpdfstring{$A$}{A}-projection and extreme points}\n\\label{secAprojection}\n\nIn this subsection we prove Theorem~\\ref{aresultsallintro}. Unless otherwise stated, $G$ is a semisimple Lie group as in \\S\\ref{notationliegroups}. For $H_0 \\in \\mathfrak a$ and $g \\in G$ define\n\\begin{align*}\nh_{H_{0}, g} : A & \\to \\mathbb{R} \\\\\na & \\mapsto \\langle H_{0}, H(g a) \\rangle \\,.\n\\end{align*}\nWhen $H_0$ corresponds to $\\lambda \\in \\mathfrak a^*$ under the isomorphism given by the Killing form, this is precisely the function $h_{\\lambda , g}$ in Theorem~\\ref{aresultsallintro}. Recall from \\S\\ref{defgenericset} that $\\lambda$ is nonsingular if and only if $H_0$ is, $\\lambda$ lies in the positive chamber of $\\mathfrak a^*$ if and only if $H_0 \\in \\mathfrak a^{+}$, and $\\lambda$ does not lie in a proper subspace spanned by roots if and only if $H_0 \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}} \\mathfrak a^L$. Thus the condition on $\\lambda$ in Theorem~\\ref{aresultsallintro} is equivalent to $H_0 \\in \\mathfrak a^{\\operatorname{gen}, +}$.\n\nThe first part of Theorem~\\ref{aresultsallintro}, concerning uniqueness and nondegeneracy, is proved in \\S\\ref{sectionuniqueness}; see Corollary~\\ref{criticalpointshnondegenerate} and Proposition~\\ref{uniquenesscriticalpointcartan}. The second part of Theorem~\\ref{aresultsallintro}, concerning existence, is proved in \\S\\ref{sectionstructurelevelsets} (although the hard work is done in \\S\\ref{sectionexistence}); see Corollary~\\ref{levelsetGcodim} and Proposition~\\ref{regularsetGopen}.\n\n\n\\subsection{Critical points and level sets}\n\nUsing Lemma~\\ref{computationAfirstderivative} we find that the differential of $h_{H_{0}, g}$ at $a \\in A$ is\n\\begin{align} \\label{cartanorbitdifferential}\n\\begin{split}\n(Dh_{H_{0}, g})_{a} : \\mathfrak{a} & \\to \\mathbb{R} \\\\\nH & \\mapsto \\langle H_{0}, \\operatorname{Ad}_{\\kappa(ga)}(H) \\rangle \\,.\n\\end{split}\n\\end{align}\nWhen $G$ is any reductive group with all associated data as in \\S \\ref{notationliegroups}, and $H_{0} \\in \\mathfrak{a}$, define the set $\\mathcal C(G, H_0)$ as follows:\n\\begin{equation}\\label{definitionCreductive}\n\\mathcal{C}(G, H_{0}) = \\{ k \\in K : \\operatorname{Ad}_{k^{-1}} (H_{0}) \\perp \\mathfrak{a} \\} \\,.\n\\end{equation}\nWe allow $G$ to be reductive because we will occasionally need to work with the sets $\\mathcal C(L, H_0)$ for $L \\in \\mathcal L$.\n\nNow let $G$ again be semisimple. The following two lemmas are clear from \\eqref{cartanorbitdifferential} and the definition of $\\mathcal{C}(G, H_{0})$.\n\n\\begin{lemma} \\label{criticalpointsintermsofC}Let $H_{0} \\in \\mathfrak{a}$ and $g \\in G$. Then $a \\in A$ is a critical point of $h_{H_{0}, g}$ if and only if $\\kappa(ga) \\in \\mathcal{C}(G, H_{0})$. \\hfill $\\square$\\end{lemma}\n\n\\begin{lemma}\\label{levelsetszerocritpoint}The set of $k \\in K$ for which $1$ is a critical point of $h_{H_0, k}$ equals $\\mathcal C(G, H_0)$. More generally, the set of $k \\in K$ for which $a \\in A$ is a critical point of $h_{H_0, k}$ equals $\\kappa(\\mathcal C(G, H_0)a^{-1})$.\\hfill $\\square$\\end{lemma}\n\nIn view of Lemma~\\ref{levelsetszerocritpoint}, we refer to the sets $\\kappa(\\mathcal C(G, H_0)a^{-1})$ as level sets. This will be fully justified after we show uniqueness of critical points in Proposition~\\ref{uniquenesscriticalpointcartan}.\n\nBy decomposing $g$ in the Iwasawa decomposition we see that\n\\begin{equation}\\label{heightfunctionreducetokpart}\nh_{H_{0}, g}(a) = \\langle H_{0}, H(g) \\rangle + h_{H_{0}, \\kappa(g)}(a) \\,. \n\\end{equation}\nThe first term in the right hand side being a mere constant, we see that the critical points of $h_{H_0, g}$ coincide with those of $h_{H_0, \\kappa(g)}$. It therefore suffices to understand the critical points of $h_{H_{0}, k}$ for $k \\in K$.\n\n\n\\begin{example} When $G = \\operatorname{PSL}_2(\\mathbb R)$ it is clear from the geometric picture in the introduction that a critical point of $h_{H_{0}, g}$ is a point $a \\in A$ such that the tangent line to $gAi$ at $gai$ is horizontal. This is the content of the above Lemma~\\ref{criticalpointsintermsofC} in this case. The set $\\mathcal{C}(G, H_{0})$ consists of the two elements\n\\begin{align*}\nc_{\\pm} = \\begin{pmatrix}\n\\cos(\\pi \/4) & \\pm \\sin(\\pi \/4) \\\\\n\\mp \\sin(\\pi \/4) & \\cos(\\pi \/4)\n\\end{pmatrix}\n\\end{align*}\ncorresponding to the two horizontal directions. See Figure~\\ref{sltwocpicture} for a picture.\n\\end{example}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\captionsetup{justification=centering}\n\\includegraphics[scale=1,angle=0]{sl2_c.pdf}\n\\caption{Geodesics corresponding to an element $k_1 \\in \\mathcal C(G, H_0)$, and to an element $k_2$ that does not lie in $\\mathcal C(G, H_0)$.}\n\\label{sltwocpicture}\n\\end{center}\n\\end{figure}\n\n\\begin{example}\\label{examplemidpointdepends} Let $G = \\operatorname{SL}_3(\\mathbb R)$. Take elements $H_0, H_0' \\in \\mathfrak a$ that are not proportional, and take $k\\in K$. We claim that $a \\in A$ can never be a critical point of both $h_{H_0, k}$ and $h_{H_0', k}$. Indeed, replacing $k$ by $\\kappa(ka)$ we may assume that $a = 0$, and that $k \\in \\mathcal C(G, H_0) \\cap \\mathcal C(G, H_0')$. That is, the symmetric matrices $\\operatorname{Ad}_{k^{-1}}(H_0)$ and $\\operatorname{Ad}_{k^{-1}}(H_0')$ have zeros on the diagonal. Because $\\mathfrak a$ is $2$-dimensional the same is then true for any $H_0'' \\in \\mathfrak a$. Take now $H_0'' = H_0^2$. The matrix $\\operatorname{Ad}_{k^{-1}}(H_0'') = \\operatorname{Ad}_{k^{-1}}(H_0)\\operatorname{Ad}_{k^{-1}}(H_0)^T$ has zeros on the diagonal on the one hand, but its trace is the sum of squares of the entries of $H_0$ on the other hand. This is a contradiction.\\end{example}\n\n\\subsection{Existence of critical points}\n\n\\label{sectionexistence}\n\nWe want to show that there exists $k \\in K$ with the property that $h_{H_0, k}$ has a critical point. Equivalently, in view of Lemma~\\ref{levelsetszerocritpoint}, that the set $\\mathcal C(G, H_0)$ is nonempty. Before proving that this is indeed the case for generic $H_0 \\in\\mathfrak a$, we need a negative result.\n\nWhen $k \\in K$ centralizes a nonzero subspace $V \\subset \\mathfrak a$, then $a \\mapsto H(ka)$ grows linearly in the directions of $V$. In particular, for $h_{H_0, k}$ to have a critical point, $H_0$ must be orthogonal to $V$. Those critical points behave badly, and this is the reason to impose that $H_0 \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}} \\mathfrak a^{L}$ in Theorem~\\ref{aresultsallintro}. We make this more precise in Lemma~\\ref{mLnocriticalpoints}.\n\n\\begin{lemma} \\label{kpartofmL}When $L \\in \\mathcal{L}$ and $m \\in M'$, we have $\\kappa(mL) \\subset mL$.\\end{lemma}\n\n\\begin{proof}\nIt follows from \\cite[Proposition 7.25, Proposition 7.31]{Knapp2002} that when $L' \\subset G$ is a semistandard Levi subgroup, one has $\\kappa(L') \\subset L'$. We may apply this to $L' = m L m^{-1}$.\n\\end{proof}\n\n\\begin{proposition} \\label{mLnocriticalpoints}When $H_{0} \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}}\\mathfrak{a}^{L}$ and $k \\in \\bigcup_{L \\in \\mathcal L - \\{ G\\}} M' L$, the function $h_{H_{0}, k}$ has no critical point.\n\\end{proposition}\n\n\\begin{proof}Assume $a \\in A$ is a critical point of $h_{H_{0}, k}$, and let $m \\in M'$ and $L \\in \\mathcal L$ be such that $k \\in mL$. By Lemma~\\ref{kpartofmL} we also have $\\kappa(ka) \\in mL$. From \\eqref{cartanorbitdifferential}, letting $H$ vary in $\\mathfrak a_L$, we see that $H_0 \\perp \\operatorname{Ad}_m(\\mathfrak a_L) = \\mathfrak a_{\\operatorname{Ad}_m(L)}$. That is, $H_0 \\in \\mathfrak a^{\\operatorname{Ad}_m(L)}$, which contradicts our assumption.\n\\end{proof}\n\nAn equivalent formulation of Proposition~\\ref{mLnocriticalpoints} is the following.\n\n\\begin{lemma}\\label{CdoesnotmeetML} When $H_{0} \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\} }\\mathfrak{a}^{L}$, the set $\\mathcal{C}(G, H_{0})$ does not meet the set $\\bigcup_{L \\in \\mathcal L - \\{ G\\}} M' L$.\\end{lemma}\n\n\\begin{proof}\nThe elements $k \\in \\mathcal{C}(G, H_{0})$ have the property that $h_{H_0, k}$ has a critical point, namely $1$.\n\\end{proof}\n\nWhen $G$ is reductive, the set $\\mathcal C(G, H_0)$ is trivially empty when $H_{0}$ has a component along $Z(\\mathfrak{g})$. That is, when $H_0 \\notin \\mathfrak g_{ss}$. In particular, when $L$ is a semistandard Levi subgroup of $G$, the set $\\mathcal{C}(L, H_{0})$ is empty if $H_{0} \\notin (\\mathfrak{a} \\cap \\mathfrak{l}_{ss}) = \\mathfrak{a}^{L}$. To study the sets $\\mathcal{C}(G, H_{0})$ with $G$ reductive and $H_{0} \\in \\mathfrak{a} \\cap \\mathfrak{g}_{ss}$, consider the map\n\\begin{align}\\label{definitionfmapC}\n\\begin{split}\nf_{G, H_{0}} : K & \\to \\mathfrak{a} \\cap \\mathfrak{g}_{ss} \\\\\nk & \\mapsto E_{\\mathfrak{a}} (\\operatorname{Ad}_{k^{-1}}(H_{0})) \\,.\n\\end{split}\n\\end{align}\nBy definition, $\\mathcal{C}(G, H_{0}) = f_{G, H_{0}}^{-1}(0)$. By \\eqref{derivativeAdinverse}, the differential of $f_{G, H_{0}}$ at $k \\in K$ is given by\n\\begin{align} \\label{differentialf}\n\\begin{split}\n(Df_{G, H_{0}})_{k} : \\mathfrak{k} & \\to \\mathfrak{a} \\cap \\mathfrak{g}_{ss} \\\\\nX & \\mapsto - E_{\\mathfrak{a}} ([X, \\operatorname{Ad}_{k^{-1}}(H_{0})] ) \\,.\n\\end{split}\n\\end{align}\nFor $G$ semisimple, we want to prove that the sets $\\mathcal{C}(G, H_{0})$ are generically nonempty. To this end, define\n\\begin{align*}\ng_{G, H_{0}} : K & \\to \\mathbb{R}_{\\geq 0} \\\\\nk & \\mapsto \\lVert f_{G, H_{0}}(k) \\rVert^{2} \\,.\n\\end{align*}\nWe have $\\mathcal{C}(G, H_{0}) = g_{G, H_{0}}^{-1}(0)$. Because $g_{G, H_{0}}$ is a continuous function on a compact set, it reaches a minimum. Our aim is to show that the minima of $g_{G, H_{0}}$ satisfy $g_{G, H_{0}}(k) = 0$, it it will follow that $\\mathcal{C}(G, H_{0})$ is nonempty.\n\nUsing Leibniz's rule and \\eqref{differentialf}, the differential of $g_{G, H_0}$ at $k \\in K$ is given by\n\\begin{align}\n(Dg_{G, H_{0}})_{k}(X) & = 2 \\langle (Df_{G, H_{0}})_{k}(X), f_{G, H_{0}}(k) \\rangle \\nonumber \\\\\n& = -2 \\langle [X, \\operatorname{Ad}_{k^{-1}}(H_{0})], f_{G, H_{0}}(k) \\rangle \\label{differentialg} \\,.\n\\end{align}\n\n\\begin{lemma} \\label{criticalsetsubmersion}  Let $G$ be reductive and $H_{0} \\in \\mathfrak{a}^{\\operatorname{reg}} \\cap \\mathfrak{g}_{ss}$. Then $f_{G, H_{0}}$ is a submersion at points $k \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}} M' L$.\n\\end{lemma}\n\n\\begin{proof}Suppose $(Df_{G, H_{0}})_{k}$ is not surjective for some $k \\in K$. Then there exists a nonzero $H \\in \\mathfrak{a} \\cap \\mathfrak{g}_{ss}$ such that $\\langle [X, \\operatorname{Ad}_{k^{-1}}(H_{0})] , H \\rangle = 0$ for all $X \\in \\mathfrak{k}$. Using associativity of the Killing form, this is equivalent to\n\\[ [\\operatorname{Ad}_{k^{-1}}(H_{0}), H] \\perp \\mathfrak{k} \\,. \\]\nBut $[\\operatorname{Ad}_{k^{-1}}(H_{0}), H] \\in [\\mathfrak{p}, \\mathfrak{p}] \\subset \\mathfrak{k}$, so that $[\\operatorname{Ad}_{k^{-1}}(H_{0}), H] = 0$. That is, $\\operatorname{Ad}_{k}(H) \\in Z_{\\mathfrak{p}}(H_{0})$. Because $H_{0}$ is regular, this implies $\\operatorname{Ad}_{k}(H) \\in \\mathfrak{a}$ (see \\cite[Lemma~6.50]{Knapp2002}), and Lemma~\\ref{conjugatetoAimplieslevi} then implies that $k \\in M' L$ with $L = Z_{G}(H)$. Because $H \\notin Z(\\mathfrak{g})$, this is a proper semistandard Levi subgroup. This proves the first statement.\n\\end{proof}\n\n\\begin{corollary} \\label{maintermgenericsubmersionatcriticalsetreductive}  Let $G$ be reductive and $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}} \\cap \\mathfrak{g}_{ss}$. Then $f_{G, H_0}$ is a submersion at the points of $\\mathcal{C}(G, H_{0})$.\\end{corollary}\n\n\\begin{proof}The set $\\mathcal C(G, H_0)$ does not meet any of the sets $M' L$ by Lemma~\\ref{CdoesnotmeetML}, and therefore $f_{G, H_0}$ is a submersion at points of $\\mathcal C(G, H_0)$, by Lemma~\\ref{criticalsetsubmersion}.\n\\end{proof}\n\n\\begin{lemma} \\label{criticalpointgequations}Let $H_{0} \\in \\mathfrak{a}^{\\operatorname{reg}}$. Define $\\mathcal{D}_{H_{0}} = \\{ k \\in K : k \\in Z_{G}(f_{G, H_{0}}(k)) \\}$. The following hold:\n\\begin{enumerate}[label = (\\roman*)]\n\\item The function $g_{G, H_{0}}$ is right invariant under $M'$.\n\\item The set of critical points of $g_{G, H_{0}}$ is $\\mathcal{D}_{H_{0}} M'$.\n\\item Let $k \\in \\mathcal{D}_{H_{0}}$ and define $L = Z_{G}(f_{G, H_{0}}(k))$. Write $H_{0} = H_{L} + H^{L}$ with $H_{L} \\in \\mathfrak{a}_{L}$ and $H^{L} \\in \\mathfrak{a}^{L}$. Then $f_{G, H_{0}}(k) = H_{L}$ and $k \\in \\mathcal{C}(L, H^{L})$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\begin{enumerate}[label = (\\roman*)]\n\\item We show that $f_{G, H_{0}}(km) = \\operatorname{Ad}_{m^{-1}}(f_{G, H_{0}}(k))$ for $m \\in M'$. Because the adjoint action of $M'$ on $\\mathfrak{a}$ is isometric, it will then follow that $g_{G, H_{0}}$ is right invariant under $M'$.\n\nBecause $M'$ normalizes $\\mathfrak{a}$, it normalizes the orthogonal complement $\\mathfrak{a}^{\\perp}$, so that the adjoint action of $M'$ commutes with $E_{\\mathfrak{a}}$. This implies\n\\begin{align*}\nf_{G, H_{0}}(km) & = E_{\\mathfrak{a}}(\\operatorname{Ad}_{m^{-1}k^{-1}}(H_{0})) \\\\\n& = \\operatorname{Ad}_{m^{-1}} (E_{\\mathfrak{a}}( \\operatorname{Ad}_{k^{-1}}(H_{0}))) \\\\\n& = \\operatorname{Ad}_{m^{-1}}(f_{G, H_{0}}(k)) \\,.\n\\end{align*}\n\\item Let $k$ be a critical point of $g$. Using \\eqref{differentialg} we have for all $X \\in \\mathfrak{k}$,\n\\begin{align*}\n0 = (Dg_{G, H_{0}})_{k} & = -2 \\langle E_{\\mathfrak{a}}([X, \\operatorname{Ad}_{k^{-1}}(H_{0})]), f_{G, H_{0}}(k) \\rangle \\,.\n\\end{align*}\nThat is,\n\\[ \\langle \\mathfrak{k}, [\\operatorname{Ad}_{k^{-1}}(H_{0}), f_{G, H_{0}}(k)] \\rangle = 0 \\,. \\]\nBut $[\\operatorname{Ad}_{k^{-1}}(H_{0}), f_{G, H_{0}}(k)] \\in [\\mathfrak{p}, \\mathfrak{p}] = \\mathfrak{k}$, so that this must be zero. Because $H_{0} \\in \\mathfrak{a}^{\\operatorname{reg}}$, this implies that $f_{G, H_{0}}(k) \\in \\operatorname{Ad}_{k^{-1}}(\\mathfrak{a})$. By Lemma~\\ref{conjugatetoAimplieslevi} it follows that $k \\in M' Z_{G}(f_{G, H_{0}}(k))$. Replacing $k$ by an appropriate right translate under $M'$, this becomes $k \\in Z_{G}(f_{G, H_{0}}(k))$. That is, $k \\in \\mathcal{D}_{H_{0}}$.\n\\item Take $k \\in \\mathcal{D}_{H_{0}}$. By definition of $L$ we have $f_{G, H_{0}}(k) \\in \\mathfrak{a}_{L}$. Writing $H_{0} = H_{L} + H^{L}$ and using that $k \\in L$, we have $f_{G, H_{0}}(k) = E_{\\mathfrak{a}}(\\operatorname{Ad}_{k^{-1}}(H_{0})) = H_{L} + E_{\\mathfrak{a}}(\\operatorname{Ad}_{k^{-1}}(H^{L}))$. If we prove that $E_{\\mathfrak{a}}(\\operatorname{Ad}_{k^{-1}}(H^{L})) = 0$, the two statements follow. On the one hand $f_{G, H_{0}}(k) \\in \\mathfrak{a}_{L}$ implies $E_{\\mathfrak{a}}(\\operatorname{Ad}_{k^{-1}}(H^{L})) = f_{G, H_{0}}(k) - H_{L} \\in \\mathfrak{a}_{L}$. On the other hand, $H^{L} \\perp \\mathfrak{a}_{L}$ and $k \\in L$ implies $\\operatorname{Ad}_{k^{-1}}(H^{L}) \\perp \\mathfrak{a}_{L}$, and hence $E_{\\mathfrak{a}}(\\operatorname{Ad}_{k^{-1}}(H^{L})) \\perp \\mathfrak{a}_{L}$. It follows that $E_{\\mathfrak{a}}(\\operatorname{Ad}_{k^{-1}}(H^{L})) \\in \\mathfrak{a}_{L} \\cap \\mathfrak{a}^{L} = \\{ 0 \\}$. \\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{lemma} Let $H_{0} \\in \\mathfrak{a}^{\\operatorname{reg}}$, $\\mathcal{D}_{H_{0}}$ be as in Lemma~\\ref{criticalpointgequations} and $k \\in \\mathcal{D}_{H_{0}}$ a critical point of $g_{G, H_{0}}$. Define $L$, $H_{L}$ and $H^{L}$ as in the same lemma. Then the Hessian of $g_{G, H_{0}}$ at $k$ satisfies\n\\begin{align}\n\\label{existenceHessiansimplified}\n\\begin{split}\n\\frac12 (\\operatorname{Hess}_{k} g_{G, H_0}) (X, X)  & = - \\left\\lVert [X, H_{L}] \\right\\rVert^{2} - \\langle [X, \\operatorname{Ad}_{k^{-1}}(H^{L})], [X, H_{L}] \\rangle \\\\\n& \\mathrel{\\phantom{=}} + \\left\\lVert E_{\\mathfrak{a}}([ X, \\operatorname{Ad}_{k^{-1}}(H^{L})]) \\right\\rVert^{2}\n\\end{split}\n\\end{align}\nfor all $X \\in \\mathfrak{k}$.\n\\end{lemma}\n\n\\begin{proof}\nStarting from \\eqref{differentialg} and using Leibniz's rule and \\eqref{derivativeAdinverse}, we have that the Hessian of $g$ at the critical point $k$ takes the form\n\\begin{align*}\n\\operatorname{Hess}_{k} g : \\mathfrak{k} \\times \\mathfrak{k} & \\to \\mathbb{R} \\\\\n(X, Y) & \\mapsto 2 \\left \\langle E_{\\mathfrak{a}}( [ Y, [ X, \\operatorname{Ad}_{k^{-1}}(H_{0})]]) , f_{G, H_{0}}(k) \\right\\rangle \\\\\n& \\mathrel{\\phantom{\\mapsto}} + 2 \\left \\langle E_{\\mathfrak{a}}( [ Y, \\operatorname{Ad}_{k^{-1}}(H_{0})]) , E_{\\mathfrak{a}}([ X, \\operatorname{Ad}_{k^{-1}}(H_{0})]) \\right\\rangle \\,.\n\\end{align*}\nAs $f_{G, H_{0}}(k) \\in \\mathfrak{a}$, we may drop the projection $E_{\\mathfrak{a}}$ in the first term. Replacing $f_{G, H_{0}}(k)$ by $H_{L}$ and using associativity of the Killing form we obtain\n\\begin{align*}\n& -2 \\left \\langle [ X, \\operatorname{Ad}_{k^{-1}}(H_{0})] , [Y, H_{L}] \\right\\rangle \\\\\n& \\qquad + 2 \\left \\langle E_{\\mathfrak{a}}( [ Y, \\operatorname{Ad}_{k^{-1}}(H_{0})]), E_{\\mathfrak{a}}([ X, \\operatorname{Ad}_{k^{-1}}(H_{0})]) \\right\\rangle \\,.\n\\end{align*}\nThe associated quadratic form on $\\mathfrak{k}$ sends\n\\begin{align*}\nX \\mapsto -2 \\left \\langle [ X, \\operatorname{Ad}_{k^{-1}}(H_{0})] , [X, H_{L}] \\right \\rangle + 2 \\left\\lVert E_{\\mathfrak{a}}([ X, \\operatorname{Ad}_{k^{-1}}(H_{0})]) \\right\\rVert^{2} \\,.\n\\end{align*}\nWriting $H_{0} = H_{L} + H^{L}$ and recalling that $k \\in L$ centralizes $H_{L}$, this becomes\n\\begin{align*}\n& - 2 \\left\\lVert [X, H_{L}] \\right\\rVert^{2} -2 \\langle [X, \\operatorname{Ad}_{k^{-1}}(H^{L})], [X, H_{L}] \\rangle \\\\\n& \\quad + 2 \\left\\lVert E_{\\mathfrak{a}}([ X, H_{L} + \\operatorname{Ad}_{k^{-1}}(H^{L})]) \\right\\rVert^{2} \\,.\n\\end{align*}\nBecause $H_{L} \\in \\mathfrak{a}$, associativity of the Killing form implies $E_{\\mathfrak{a}}([X, H_{L}]) = 0$, so that the third term simplifies and we obtain \\eqref{existenceHessiansimplified}.\n\\end{proof}\n\n\\begin{lemma} \\label{criticalpointgminimum}Let $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}}$ and $k \\in K$ be a critical point of $g_{G, H_{0}}$ with positive semidefinite Hessian. Then $g_{G, H_{0}}(k) = 0$. \\end{lemma}\n\n\\begin{proof}The proof is by contradiction. Assuming that $f_{G, H_{0}}(k) \\neq 0$, we will construct a direction in which the Hessian of $g_{G, H_{0}}$ is negative definite at $k$.\n\nLet $\\mathcal{D}_{H_{0}}$ be as in Lemma~\\ref{criticalpointgequations}. By Lemma~\\ref{criticalpointgequations} and right invariance of $g$ under $M'$, we may assume that $k \\in \\mathcal{D}_{H_{0}}$. Let $L = Z_{G}(f_{G, H_{0}}(k))$ and let $H_{L}$ and $H^{L}$ be as in Lemma~\\ref{criticalpointgequations}, so that $f_{G, H_{0}}(k) = H_{L}$ and $k \\in \\mathcal{C}(L, H^{L})$. \nSuppose that $H_L \\neq 0$. We will construct $X \\in \\mathfrak{k}$ for which the third term in \\eqref{existenceHessiansimplified} is zero, and for which the other terms are nonpositive and not both zero.\n\nSuppose first that $H^{L} \\notin \\mathfrak{a}^{\\operatorname{reg}}$. Then there exists a nonzero element $X \\in \\mathfrak{k}$ with $[X, \\operatorname{Ad}_{k^{-1}}(H^{L})] = 0$. Indeed, if $\\alpha \\in \\Sigma$ is such that $\\alpha(H^{L}) = 0$, take a nonzero $X' \\in (\\mathfrak{g}_{\\alpha} + \\mathfrak{g}_{- \\alpha}) \\cap \\mathfrak{k}$. Then $[X', H^{L}] = 0$, and we can take $X = \\operatorname{Ad}_{k^{-1}}(X')$.\n\nWith this choice of $X$, the second and third terms in \\eqref{existenceHessiansimplified} vanish. We have $[X, H_{L}] = [X, \\operatorname{Ad}_{k^{-1}}(H_{0})] \\neq 0$ by regularity of $H_{0}$, so that\n\\[ (\\operatorname{Hess}_{k} g_{G, H_0}) (X, X) = -2 \\cdot \\left\\lVert [X, H_{L}] \\right\\rVert^{2} < 0 \\,. \\]\nThis is a contradiction.\n\nSuppose now that $H^{L} \\in \\mathfrak{a}^{\\operatorname{reg}}$.\nWe first show that there exists $X \\in \\mathfrak{k}$ for which the second term in \\eqref{existenceHessiansimplified} is strictly negative. After that, we will modify $X$ in such a way that the second term stays the same and such that the third becomes zero.\n\nThe second term in \\eqref{existenceHessiansimplified} equals\n\\[ - \\langle [\\operatorname{Ad}_{k}(X), H^{L}], [\\operatorname{Ad}_{k}(X), H_{L}] \\rangle \\,. \\]\nWrite $\\operatorname{Ad}_{k}(X)$ in the restricted root space decomposition \\eqref{rootspacedecomp} as\n\\[ X = \\sum_{\\alpha \\in \\Sigma} X_{\\alpha} \\,. \\]\nThen the above equals\n\\begin{equation} \\label{secondtermnonemptyproofroots}\n- \\sum_{\\alpha \\in \\Sigma} \\alpha(H^{L}) \\alpha(H_{L}) \\lVert X_{\\alpha} \\rVert_{\\theta}^{2} \\,.\n\\end{equation}\nBecause $H_{L} \\neq 0$ by assumption, there exists $\\beta \\in \\Sigma$ such that $\\beta(H_{L}) \\neq 0$. Because $H^{L} \\in \\mathfrak{a}^{\\operatorname{reg}}$, we have $\\beta(H^{L}) \\beta(H_{L}) \\neq 0$. Using \\cite[Corollary 2.24]{Knapp2002} we have\n\\[ 0 = \\langle H^{L}, H_{L} \\rangle = \\sum_{\\alpha \\in \\Sigma} \\alpha(H^{L}) \\alpha(H_{L}) \\,, \\]\nso that there is at least one strictly positive term in this sum. Let $\\alpha \\in \\Sigma$ be such that $\\alpha(H^{L}) \\alpha(H_{L}) > 0$. Take now a nonzero $X \\in \\mathfrak{k} \\cap (\\mathfrak{g}_{\\alpha} + \\mathfrak{g}_{- \\alpha})$, then \\eqref{secondtermnonemptyproofroots} is strictly negative. That is, the second term in \\eqref{existenceHessiansimplified} is strictly negative.\n\nObserve that when $Y \\in \\mathfrak{k} \\cap \\mathfrak{l}$, adding $Y$ to $X$ does not change the first two terms of \\eqref{existenceHessiansimplified}, because $[Y, H_{L}] = 0$. Thus in order to make the third term in \\eqref{existenceHessiansimplified} zero and obtain a contradiction, it suffices to find $Y \\in \\mathfrak{k} \\cap \\mathfrak{l}$ with\n\\[ E_{\\mathfrak{a}}([Y, \\operatorname{Ad}_{k^{-1}}(H^{L})]) = E_{\\mathfrak{a}}([X, \\operatorname{Ad}_{k^{-1}}(H^{L})]) \\,, \\]\nand it will follow that\n\\[ (\\operatorname{Hess}_{k} g_{G, H_0}) (X - Y, X - Y) = -2 \\left\\lVert [X, H_{L}] \\right\\rVert^{2} - 2 \\langle [X, \\operatorname{Ad}_{k^{-1}}(H^{L})], [X, H_{L}] \\rangle < 0 \\,. \\]\nLet $L' \\subset L$ be the smallest semistandard Levi subgroup with the property that $k \\in L' M'$, say $k  = \\ell'm$. From $k \\in \\mathcal{C}(L, H^{L})$ it follows that $\\operatorname{Ad}_{k^{-1}} (H^{L}) \\perp \\mathfrak{a}$, and therefore $H^{L} \\in \\mathfrak{a}^{L'}$. We also have that $[\\mathfrak{k}, \\operatorname{Ad}_{k^{-1}}(H^{L})] \\perp \\operatorname{Ad}_{k^{-1}}(\\mathfrak{a}) \\supset \\operatorname{Ad}_{m}^{-1}(\\mathfrak{a}_{L'})$, so that $E_{\\mathfrak{a}}([\\mathfrak{k}, \\operatorname{Ad}_{k^{-1}}(H^{L})]) \\subset \\operatorname{Ad}_{m}^{-1}(\\mathfrak{a}^{L'})$. Hence it suffices to show that the map\n\\begin{align} \\label{nonemptycmapsurjective}\n\\begin{split}\n\\mathfrak{k} \\cap \\mathfrak{l} & \\to \\operatorname{Ad}_{m}^{-1}(\\mathfrak{a}^{L'}) \\\\\nY & \\mapsto E_{\\mathfrak{a}}([Y, \\operatorname{Ad}_{k^{-1}}(H^{L})])\n\\end{split}\n\\end{align}\nis surjective. Its restriction to $\\mathfrak{k} \\cap \\mathfrak{l}'$ is precisely $-\\operatorname{Ad}_{m}^{-1} \\circ (Df_{L', H^{L}})_{\\ell'}$ (compare \\eqref{differentialf}). We seek to apply Lemma~\\ref{criticalsetsubmersion} to $L'$.\n\nBy assumption we have $H^{L} \\in \\mathfrak{a}^{\\operatorname{reg}}$, so that $H^{L} \\in (\\mathfrak{a}^{L'})^{\\operatorname{reg}}$. By minimality of $L'$, we have $\\ell' \\notin \\bigcup_{L'' \\subsetneq L'} M' L''$. Therefore Lemma~\\ref{criticalsetsubmersion} shows that $(Df_{L', H^{L}})_{\\ell'}$ is surjective, so that the map \\eqref{nonemptycmapsurjective} is surjective and the conclusion follows.\n\\end{proof}\n\n\\begin{corollary} \\label{Cnonempty}When $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}}$ we have $\\mathcal{C}(G, H_{0}) \\neq \\varnothing$.\\end{corollary}\n\n\\begin{proof}\nThe continuous function $g_{G, H_{0}} : K \\to \\mathbb{R}_{\\geq 0}$ attains a minimum, which must be $0$ by Lemma~\\ref{criticalpointgminimum}.\n\\end{proof}\n\n\n\n\\subsection{Uniqueness of critical points}\n\n\\label{sectionuniqueness}\n\n\\begin{lemma} \\label{hessianquadformnegativedefinite} Let $H_{0} \\in \\mathfrak{a}^{+}$ and $k \\in K - \\bigcup_{L\\in \\mathcal L - \\{ G\\}} M' L$. Then the quadratic form on $\\mathfrak{a}$ defined by\n\\[ H \\mapsto \\langle H_0, [ \\operatorname{Ad}_{k}(H), E_{\\mathfrak n} ( \\operatorname{Ad}_{k}(H) ) ] \\rangle \\]\nis negative definite. In particular, it is nondegenerate.\n\\end{lemma}\n\n\\begin{proof}Take a nonzero $H \\in \\mathfrak{a}$, and write the element $\\operatorname{Ad}_{k}(H) \\in \\mathfrak{p}$ in the restricted root space decomposition \\eqref{rootspacedecomp} as $\\sum_{\\alpha \\in \\Sigma \\cup \\{ 0 \\} } X_{\\alpha}$. Then $\\theta X_{-\\alpha} = -X_\\alpha$. By Lemma~\\ref{computationAsecondderivative}, the quadratic form evaluated at $H$ equals\n\\[ \\sum_{\\alpha \\in \\Sigma^{+}} \\langle -X_{\\alpha}, 2 X_{\\alpha} \\rangle_{\\theta} \\langle H_\\alpha, H_0 \\rangle = -2 \\sum_{\\alpha \\in \\Sigma^{+}} \\lVert X_{\\alpha} \\rVert_{\\theta}^2 \\cdot \\alpha(H_0) \\,, \\]\nwhich is nonpositive because $H_{0}$ lies in the positive Weyl chamber. Therefore the quadratic form is negative semidefinite. The fact that $k \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}} M' L$ implies $\\operatorname{Ad}_{k}(H) \\notin \\mathfrak{a}$ (Lemma~\\ref{conjugatetoAimplieslevi}). So at least one $X_{\\alpha}$ is nonzero, and the statement follows.\n\\end{proof}\n\n\\begin{remark}\\label{hessiancriticalsometimesdef}When $H_{0}$ lies in a mixed Weyl chamber, the quadratic form in Lemma~\\ref{hessianquadformnegativedefinite} can sometimes be degenerate. This happens already for $G = \\operatorname{SL}_{3}(\\mathbb{R})$, even for completely generic values of $H_0$, and there appears to be no structure in the bad pairs $(H_0, k)$.\\end{remark}\n\n\\begin{proposition} \\label{surjectivityorbitalintegralH} Let $H_{0} \\in \\mathfrak{a}^{+}$ and $k \\in K - \\bigcup_{L \\in \\mathcal L - \\{ G\\}} M'L$. Then the Hessian of $h_{H_{0}, k}$ is everywhere negative definite.\\end{proposition}\n\n\\begin{proof}Take $a \\in A$ and define $k_{1} = \\kappa(k a)$. From Lemma~\\ref{kpartofmL} it follows that $k_{1} \\notin \\bigcup_{L\\in \\mathcal L - \\{ G\\}} M' L$. Using Lemma~\\ref{computationAsecondderivative} we have for $H \\in \\mathfrak{a}$ that\n\\begin{align*}\n(\\operatorname{Hess}_{a} h_{H_{0}, k})(H) & = \\langle H_{0}, [\\operatorname{Ad}_{k_{1}}(H), E_{\\mathfrak{n}}(\\operatorname{Ad}_{k_{1}}(H))] \\rangle \\\\\n& = \\langle [H_{0}, \\operatorname{Ad}_{k_{1}}(H)], E_{\\mathfrak n}(\\operatorname{Ad}_{k_{1}}(H)) \\rangle \\,.\n\\end{align*}\nBy Lemma~\\ref{hessianquadformnegativedefinite}, this quadratic form is negative definite.\n\\end{proof}\n\n\\begin{corollary} \\label{criticalpointshnondegenerate}When $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}, +}$ and $k \\in K$, the critical points of $h_{H_{0}, k}$ are nondegenerate.\\end{corollary}\n\n\\begin{proof}If $h_{H_{0}, k}$ has a critical point, then $k \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}} M'L$ by Proposition~\\ref{mLnocriticalpoints}. The Hessian of $h_{H_{0}, k}$ is then everywhere nondegenerate by Proposition~\\ref{surjectivityorbitalintegralH}. In particular, its critical points are nondegenerate.\n\\end{proof}\n\n\\begin{proposition} \\label{uniquenesscriticalpointcartan}When $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}, +}$ and $k \\in K$, the function $h_{H_{0}, k}$ has at most one critical point, which maximizes $h_{H_{0}, k}$.\\end{proposition}\n\n\\begin{proof}By Proposition~\\ref{mLnocriticalpoints}, the existence of one critical point implies that $k \\notin \\bigcup_{L \\in \\mathcal L - \\{ G\\}} M'L$. Proposition~\\ref{surjectivityorbitalintegralH} then implies that the Hessian of $h_{H_{0}, k}$ is everywhere negative definite, so that there can be no other critical point and any critical point maximizes $h_{H_{0}, k}$.\n\\end{proof}\n\n\\subsection{Structure of the level sets}\n\n\\label{sectionstructurelevelsets}\n\nWe have all the necessary information to describe the set $\\mathcal C(G, H_0)$.\n\n\\begin{proposition} \\label{genericcriticalsetstructuresemisimple} When $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}}$, the set $\\mathcal{C}(G, H_{0})$ is a smooth manifold of dimension $\\dim K - \\dim A$ that varies smoothly with $H_{0}$.\n\\end{proposition}\n\n\\begin{proof} By Corollary~\\ref{Cnonempty}, this set is nonempty, and by Lemma~\\ref{maintermgenericsubmersionatcriticalsetreductive}, $f_{G, H_{0}}$ is a submersion at the points of $\\mathcal{C}(G, H_{0})$. Thus $\\mathcal{C}(G, H_{0})$ is a smooth manifold of dimension $\\dim K - \\dim A$, and from the local normal form for submersions it follows that it admits local parametrizations that depend smoothly on $H_{0}$.\n\\end{proof}\n\n\\begin{lemma}\\label{CinvariantbyM} When $H_0 \\in \\mathfrak a$, the set $\\mathcal{C}(G, H_{0})$ is invariant on both sides by $M$.\n\\end{lemma}\n\n\\begin{proof}This is clear from the definition \\eqref{definitionCreductive}, using $\\operatorname{Ad}_G$-invariance of the Killing form.\n\\end{proof}\n\n\\begin{remark}Many arguments simplify when $\\mathcal C(G, H_0)$ is but a finite union of cosets of $M$ in $K$, effectively reducing arguments to the case of $G = \\operatorname{SL}_2(\\mathbb R)$. However, the set $\\mathcal C(G, H_0)$ is usually much bigger; see Proposition~\\ref{Klargecharacterization}.\\end{remark}\n\n\\begin{corollary}\\label{levelsetGcodim}When $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}}$ and $a \\in A$, the set of $g \\in G$ for which $a$ is a critical point of $h_{H_0, g}$ is a smooth submanifold of codimension $\\dim(A)$. It is stable on the left by $NAM$ and on the right by $M$.\\end{corollary}\n\n\\begin{proof}From \\eqref{heightfunctionreducetokpart} we saw that $h_{H_0, g}$ and $h_{H_0, \\kappa(g)}$ have the same critical points. Therefore by Lemma~\\ref{levelsetszerocritpoint} the set in question is equal to $NA \\cdot \\kappa(\\mathcal C(G, H_0)a^{-1})$, which has codimension $\\dim(A)$ in $G = NAK$ by Proposition~\\ref{genericcriticalsetstructuresemisimple}. It is stable on the left by $NA$. It is stable on the left by $M$ because $M$ normalizes $NA$, which implies that $\\kappa(\\cdot)$ commutes with left multiplication by $M$, and $\\mathcal C(G, H_0)$ is stable on the left by $M$ by Lemma~\\ref{CinvariantbyM}. Similarly, invariance on the right follows from Lemma~\\ref{CinvariantbyM}, using that $\\kappa(\\cdot)$ commutes with right multiplication by $K$.\n\\end{proof}\n\nWhen $H_0 \\in \\mathfrak a$, define $\\mathcal{R}_{H_{0}} \\subset G$ to be the set of elements $g$ for which the function $h_{H_{0}, g}$ has a critical point.\n\n\\begin{proposition} \\label{regularsetGopen} When $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}, +}$, the set $\\mathcal{R}_{H_0} \\subset G$ is open and stable on the left by $NAM$.\\end{proposition}\n\n\\begin{proof} Take $g_0 \\in \\mathcal{R}_{H_0}$. By Proposition~\\ref{criticalpointshnondegenerate}, the critical point of $h_{H_{0}, g_0}$ is nondegenerate, call it $\\xi$. We may reformulate this by saying that the map\n\\begin{align*}\nA & \\to \\mathfrak{a}^{*} \\\\\na & \\mapsto (D h_{H_{0}, g_0})_{a}\n\\end{align*}\nhas invertible differential at the level set above $0$, which is the singleton $\\{\\xi \\}$. By the implicit function theorem applied to this smooth map with parameter $g \\in G$, it follows that it has a zero for all $g$ in a neighborhood of $g_0$, which is to say that $\\mathcal{R}_{H_0}$ is open in $G$. The stability under $NAM$ follows from Corollary~\\ref{levelsetGcodim}.\n\\end{proof}\n\n\\begin{remark} \\label{atonegativechamber} Let $\\rho : G \\to \\operatorname{SL}(V)$ be any representation with finite kernel, and let $\\Omega_{\\mathcal S^0} \\subset G$ be the set defined in \\S\\ref{npartcanonicalpartitions}. It is open and dense by Proposition~\\ref{omegaopendense}. It is reasonable to expect that for $g \\in \\Omega_{\\mathcal S^0}$ and $\\lambda \\in (\\mathfrak a^*)^+$ we have $\\lambda(H(ga)) \\to - \\infty$ as $a \\to \\infty$ in $A$. This would then imply that when $H_0 \\in \\mathfrak a^+$ the function $h_{H_0, g}$ has a critical point for all $g \\in \\Omega_{S^0}$, in particular, for all $g$ in an open dense subset of $G$ that does not depend on $H_0$. In fact, when $H_0 \\in \\mathfrak a^{\\operatorname{gen}, +}$ it is reasonable to expect that $\\mathcal R_{H_0} = \\Omega_{\\mathcal S^0}$.\n\nThe statement about the limit and the corollary that $\\mathcal R_{H_0} \\supset \\Omega_{\\mathcal S^0}$ are certainly true when $G = \\operatorname{SL}_n(\\mathbb R)$ and $\\rho = \\operatorname{Std}$. Then the entries of $H(ga)$ can be expressed in terms of quotients of sums of subdeterminants (using \\eqref{normprojwedgesingle}), and when $g \\in \\Omega_{\\mathcal S^0}$ those entries are bounded from above and below by constants times the entries of $a$, but arranged in decreasing order. The general case would require a more careful analysis of the weights of the exterior powers of $V$. This would in particular give an alternative proof of Corollary~\\ref{Cnonempty}, whose only proof we have now is very technical and little intuitive.\\end{remark}\n\n\\subsection{Some dimension bounds}\n\n\\begin{lemma}\\label{complexrootsbigger}Let $\\mathfrak g$ be a complex simple Lie algebra of rank $r$, $\\mathfrak h$ a Cartan subalgebra with roots $\\Sigma$ and a choice of simple roots $\\Pi$. Let $\\alpha \\in \\Pi$. Then there exist at least $r$ roots $\\beta \\in \\Sigma$ with the property that $\\beta \\geq \\alpha$.\\end{lemma}\n\n\\begin{proof}We will show by induction the following statement: for every $1 \\leq m \\leq r$, there exists a set of simple roots $S$ with $|S| = m$ whose span contains $m$ linearly independent roots $\\beta \\geq \\alpha$. For $m = 1$ we take $S = \\{\\alpha\\}$. Assume we have found $S$ of cardinality $m < r$. Because $\\mathfrak g$ is simple, $\\Sigma$ is irreducible, so there exists a simple root $\\gamma \\in \\Pi - S$ which is not orthogonal to all roots in $S$. Thus there exists a positive root $\\beta \\in \\operatorname{span}(S)$ with $\\beta \\geq \\alpha$ which is not orthogonal to $\\gamma$. Because $\\langle \\beta, \\gamma \\rangle_\\theta \\leq 0$ by \\cite[Lemma 2.51]{Knapp2002}, we must have $\\langle \\beta, \\gamma \\rangle_\\theta < 0$. By \\cite[Proposition 2.48]{Knapp2002} this implies that $\\beta + \\gamma$ is a root. We may now take $S' = S \\cup \\{\\gamma\\}$, whose span contains $m$ linearly independent roots $\\geq \\alpha$ in $\\operatorname{span}(S)$ together with the root $\\beta + \\gamma \\geq \\alpha$. This completes the induction.\n\\end{proof}\n\n\\begin{lemma}\\label{simplewithfewroots}Let $\\mathfrak g$ be a real simple Lie algebra with Cartan decomposition $\\mathfrak k \\oplus \\mathfrak p$, maximal abelian subalgebra $\\mathfrak a \\subset \\mathfrak p$ with restricted roots $\\Sigma \\subset \\mathfrak a^*$ and choice of positive roots $\\Sigma^+$. The following are equivalent:\n\\begin{enumerate}\n\\item $\\mathfrak g$ is compact or isomorphic to $\\mathfrak{sl}_2(\\mathbb R)$.\n\\item $\\sum_{\\alpha \\in \\Sigma^+} m(\\alpha) = \\dim(\\mathfrak a)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}Write $r = \\dim(\\mathfrak a)$.\nNote that $\\Sigma$ spans $\\mathfrak a^*$ so that we always have\n\\[ \\sum_{\\alpha \\in \\Sigma^+} m(\\alpha) \\geq \\#\\Sigma^+ \\geq r \\,. \\]\nIt is clear that equality holds if $\\mathfrak g$ is compact or isomorphic to $\\mathfrak{sl}_2(\\mathbb R)$.\n\nNow let $\\mathfrak g$ be any simple Lie algebra. We will show that equality only holds for the examples stated. There are two cases, coming from the classification of simple real Lie algebras.\n\nSuppose first that the complexification $\\mathfrak g_{\\mathbb C}$ is not simple. By \\cite[Theorem 6.94]{Knapp2002} $\\mathfrak g$ is the restriction of scalars of a simple complex Lie algebra. Let $\\mathfrak h \\supset \\mathfrak a$ be a Cartan subalgebra. It follows as in \\cite[\\S VI.11, p.425]{Knapp2002} that all roots of $\\mathfrak h$ have nonzero restriction to $\\mathfrak a$ (and all restrictions are different) and all restricted roots have multiplicity $2$. It follows that\n\\[ \\sum_{\\alpha \\in \\Sigma^+} m(\\alpha) = 2\\#\\Sigma^+ \\geq 2r \\geq r + 1 \\,, \\]\nbecause $r \\geq 1$, meaning that equality does not hold in this case.\n\nSuppose now that the complexification $\\mathfrak g_{\\mathbb C}$ is simple. Let $\\Pi$ be a system of simple restricted roots of $\\mathfrak a$. Let $\\theta$ be a Cartan involution corresponding to $\\mathfrak k \\oplus \\mathfrak p$ and $\\mathfrak h \\subset \\mathfrak g$ a $\\theta$-stable Cartan subalgebra containing $\\mathfrak a$. We may choose an ordering on $\\mathfrak h_{\\mathbb C}^*$ such that the restrictions of positive roots of $\\mathfrak h_{\\mathbb C}$ to $\\mathfrak a$ are either zero or positive. As in \\cite[\\S VI.12, Problem 7]{Knapp2002}, all simple $\\alpha \\in \\Pi$ are restrictions of simple roots of $\\mathfrak h_{\\mathbb C}$. Let $\\alpha_1, \\ldots, \\alpha_r$ be simple lifts to $\\mathfrak h_{\\mathbb C}$ of the simple restricted roots of $\\mathfrak a$. If $\\mathfrak g$ is noncompact, then $r \\geq 1$. Because $\\mathfrak g_{\\mathbb C}$ is simple, by Lemma~\\ref{complexrootsbigger} there are at least $\\dim_{\\mathbb C}(\\mathfrak h_{\\mathbb C}) - 1$ positive roots $\\beta$ of $\\mathfrak h_{\\mathbb C}$ with $\\beta > \\alpha_1$. The restrictions of all these roots are positive. Though the restrictions may coincide, they give the following lower bound for multiplicities:\n\\[ \\sum_{\\alpha \\in \\Sigma^+} m(\\alpha) \\geq r + \\dim_{\\mathbb C}(\\mathfrak h_{\\mathbb C}) - 1 \\,. \\]\nif this is equal to $r$, then $\\dim_{\\mathbb C}(\\mathfrak h_{\\mathbb C}) = 1$. That is, $\\mathfrak g_{\\mathbb C} \\cong \\mathfrak{sl}_2(\\mathbb C)$. Because we are assuming $\\mathfrak g$ is noncompact, it must be the unique noncompact form of $\\mathfrak{sl}_2(\\mathbb C)$, which is $\\mathfrak{sl}_2(\\mathbb R)$.\n\\end{proof}\n\n\\begin{proposition}\\label{Klargecharacterization}Let $G$ be a semisimple Lie group. The following are equivalent:\n\\begin{enumerate}\n\\item $\\dim(K) - \\dim(A) = \\dim(M)$.\n\\item All simple factors of $\\mathfrak g$ are either compact or $\\mathfrak{sl}_2(\\mathbb R)$.\n\\end{enumerate}\n\\end{proposition}\n\nNote that we always have $\\dim(K) \\geq \\dim(M) + \\dim(A)$.\n\n\\begin{proof}\nThe difference $\\dim(K) - \\dim(M)$ is equal to $\\sum_{\\alpha \\in \\Sigma^+} m(\\alpha)$. If this is an equality, then we must have the same equality for all simple factors of $\\mathfrak g$. By Lemma~\\ref{simplewithfewroots}, this is equivalent to saying that  all simple factors of $\\mathfrak g$ are either compact or $\\mathfrak{sl}_2(\\mathbb R)$.\n\\end{proof}\n\n\n\\subsection{Proofs specific to \\texorpdfstring{$\\operatorname{SL}_3(\\mathbb{R})$}{SL3(R)}}\n\nSome of the lemmas that go into the proof of Proposition~\\ref{genericcriticalsetstructuresemisimple} admit more direct and computational proofs when $G = \\operatorname{SL}_{3}(\\mathbb{R})$. We include one of those here, because it features a rather curious inequality.\n\n\\begin{proof}[Direct proof of Corollary~\\ref{Cnonempty} when $G = \\operatorname{SL}(3, \\mathbb{R})$] We use the standard choice of Iwasawa decomposition. Write $H_{0} = \\operatorname{diag}(a, b, c)$. Proving that the set $\\mathcal{C}(G, H_{0})$ is nonempty is equivalent to showing that there exists a symmetric matrix $X = \\operatorname{Ad}_{k^{-1}}(H_{0})$ with zeros on the diagonal, which is isospectral with $H_{0}$. Write $X = \\begin{psmallmatrix}\n0 & x & y \\\\\nx & 0 & z \\\\\ny & z & 0\n\\end{psmallmatrix}$. Then $H_{0}$ and $X$ have the same characteristic polynomial when\n\\begin{align} \\label{XHnoughtequations}\n\\begin{cases}\nx^{2} + y^{2} + z^{2} = - (ab + bc + ca) = \\frac{1}{2}(a^{2} + b^{2} + c^{2}) \\\\\n2 xyz = abc \\,.\n\\end{cases}\n\\end{align}\nBecause the first equation does not see the signs of $x, y, z$, squaring the second does not affect solvability. We may now view \\eqref{XHnoughtequations} as prescribing the arithmetic and geometric mean of $x^{2}, y^{2}, z^{2}$, namely $\\frac{1}{6}(a^{2} + b^{2} + c^{2})$ respectively $(\\frac{1}{4} a^{2}b^{2}c^{2})^{1\/3}$. It is well known that a necessary and sufficient condition for this to have a solution in nonnegative numbers $x^{2}, y^{2}, z^{2}$, is that the arithmetic mean is at least the geometric mean, meaning that\n\\[ \\frac{1}{6}(a^{2} + b^{2} + c^{2}) \\geq \\left( \\frac{1}{4} a^{2}b^{2}c^{2} \\right)^{1\/3} \\,. \\]\nThat this condition is satisfied is Lemma~\\ref{AMGMlemma} below, and it does not require the hypothesis that $H_{0} \\in \\mathfrak{a}^{\\operatorname{gen}}$.\n\\end{proof}\n\n\\begin{lemma}[AM--\\texorpdfstring{$\\sqrt[3]{2}$}{sqrt 3 2}GM] \\label{AMGMlemma}Let $a, b, c \\in \\mathbb{R}$ with $a + b + c = 0$. Then\n\\[ \\left( \\frac{a^{2} + b^{2} + c^{2}}{3} \\right)^{3} \\geq 2 \\cdot a^{2}b^{2}c^{2} \\,. \\]\n\\end{lemma}\n\nThe proof of Lemma~\\ref{AMGMlemma} is an amusing exercise using the AM--GM inequality. Equality holds if and only if $(a, b, c)$ is proportional to $(1, 1, -2)$ or a permutation thereof. With the notations as in Corollary~\\ref{Cnonempty}, this corresponds to the case where $H_{0}$ lies in $\\mathfrak{a}_{L}$ for some semistandard Levi subgroup $L \\in \\mathcal L - \\{ G\\}$.\n\n\n\n\n\\section{The \\texorpdfstring{$K$}{K}-projection and the geodesic flow}\n\\label{secKprojection}\n\nIn this subsection we prove Theorem~\\ref{KparttoMcentralizer}. For the possibility of uniformity, see Remark~\\ref{remarkKvsNproofs}. For $H \\in \\mathfrak{a}$ and $k \\in K$, define a map\n\\begin{align}\n\\label{definitionp}\n\\begin{split}\np_{H, k} : \\mathbb{R} & \\to \\mathfrak{p} \\\\\nt & \\mapsto \\operatorname{Ad}_{\\kappa(k e^{tH})}(H) \\,.\n\\end{split}\n\\end{align}\nUsing \\eqref{derivativeAd} and Lemma~\\ref{differentialkpart} to differentiate $\\kappa$, we find that\n\\[ p_{H, k}' (t) = [E_{\\mathfrak{k}}(p_{H, k}(t)), p_{H, k}(t)] \\,.\\]\nTherefore $p_{H, k}(t)$ is a solution to the homogeneous quadratic differential equation in $\\mathfrak{p}$ given by\n\\begin{equation} \\label{diffequationpt}\nX' = [E_{\\mathfrak{k}}X, X] \n\\end{equation}\nwith initial value $\\operatorname{Ad}_{k}(H)$. Its flow\n\\begin{align} \\label{dynamicalsystem}\n\\begin{split}\np : \\mathbb{R} \\times \\mathfrak{p} & \\to \\mathfrak{p} \\\\\n(t, X) & \\mapsto p_{t}(X) \\,.\n\\end{split}\n\\end{align}\nis given explicitly by $p_{t}(X) = p_{H, k}(t)$ when $X = \\operatorname{Ad}_{k}(H)$, and is in particular defined for all $t \\in \\mathbb{R}$. We will prove Theorem~\\ref{KparttoMcentralizer} by gathering enough information about the dynamical system \\eqref{dynamicalsystem}.\n\n\\begin{example}Let $G = \\operatorname{PSL}_{2}(\\mathbb{R})$ and identify $\\mathfrak{p}$ with $\\mathbb{R}^{2}$ via an isometry that sends $\\mathfrak{a}$ to the horizontal axis $\\mathbb{R} \\times \\{ 0 \\}$. It is clear from \\eqref{diffequationpt} that the points of $\\mathfrak{a}$ are stationary, and it is clear from \\eqref{definitionp} that the norm of $X \\in \\mathfrak{p}$ is invariant under the flow. It is then not hard to see that the phase portrait of the dynamical system \\eqref{dynamicalsystem} is as follows: the points of $\\mathfrak{a}^{+}$ are unstable equilibra, the points of $\\mathfrak{a}^{-}$ are stable equilibria, and apart from the equilibrium at $0$ every other orbit is heteroclinic and describes a Euclidean half circle with endpoints on the horizontal axis, starting at $\\mathfrak{a}^{+}$ and ending at $\\mathfrak{a}^{-}$.\\end{example}\n\nIt is clear from \\eqref{diffequationpt} that all elements of $\\mathfrak{a}$ are equilibria. The key in the proof of Theorem~\\ref{KparttoMcentralizer} is the existence of functions that are monotonic along orbits. For a root $\\alpha \\in \\Sigma$, let $H_{\\alpha} \\in \\mathfrak{a}$ be as in \\S\\ref{secroots}. The set $\\{ H_{\\alpha} : \\alpha \\in \\Pi \\}$ is a basis of $\\mathfrak{a}$. When $\\alpha \\in \\Pi$ and $H \\in \\mathfrak{a}$, define $c_{\\alpha}(H)$ to be the $\\alpha$th coordinate of $H$ in this basis. More generally, define $c_{\\alpha}(X) = c_{\\alpha}(E_{\\mathfrak{a}}(X))$ for $X \\in \\mathfrak{g}$.\n\n\\begin{lemma} \\label{aptderivative}Let $X \\in \\mathfrak{p} - \\mathfrak{a}$. Then $c_{\\alpha}([E_{\\mathfrak{k}}X, X]) \\leq 0$ for all $\\alpha \\in \\Pi$, and at least one is strictly negative. That is, $E_{\\mathfrak{a}}[E_{\\mathfrak{k}}X, X]$ is nonzero and lies in the closure of the negative Weyl chamber.\\end{lemma}\n\n\\begin{proof}Write $X = \\sum_{\\alpha \\in \\Sigma \\cup \\{ 0 \\}} X_{\\alpha}$ in the restricted root space decomposition \\eqref{rootspacedecomp}. Using that $X_{- \\alpha} = - \\theta(X_{\\alpha})$ and $E_{\\mathfrak{k}} X = \\sum_{\\alpha \\in \\Sigma^{+}} (X_{- \\alpha} - X_{\\alpha})$, we find that\n\\begin{align*}\nE_{\\mathfrak{a}}[E_{\\mathfrak{k}}X, X] & = \\sum_{\\alpha \\in \\Sigma^{+}} [X_{- \\alpha} - X_{\\alpha}, X_{- \\alpha} + X_{\\alpha}] \\\\\n& = - 2 \\sum_{\\alpha \\in \\Sigma^{+}} [X_{\\alpha}, X_{- \\alpha}] \\\\\n& = -2 \\sum_{\\alpha \\in \\Sigma^{+}} \\langle X_{\\alpha}, X_{- \\alpha} \\rangle \\cdot H_{\\alpha} \\\\\n& =  -2 \\sum_{\\alpha \\in \\Sigma^{+}} \\lVert X_{\\alpha} \\rVert_{\\theta}^{2} \\cdot H_{\\alpha} \\\\\n& = -2 \\sum_{\\alpha \\in \\Pi} \\sum_{\\beta \\geq \\alpha} \\lVert X_{\\beta} \\rVert_{\\theta}^{2} \\cdot H_{\\alpha}\n\\end{align*}\nwhere in the third equality we have used \\cite[\\S II, Lemma 2.18]{Knapp2002}. Therefore the coefficients of $E_{\\mathfrak{a}}[E_{\\mathfrak{k}}X, X]$ in the basis $\\{ H_{\\alpha} : \\alpha \\in \\Pi \\}$ are nonnegative, and because $X \\notin \\mathfrak{a}$ at least one is nonzero. (This is essentially the same proof as that of Lemma~\\ref{hessianquadformnegativedefinite}.)\n\\end{proof}\n\n\\begin{lemma} \\label{apartdecreasing} Let $X \\in \\mathfrak{p} - \\mathfrak{a}$ and $t > 0$. Then $c_{\\alpha}(p_{t}(X)) \\leq c_{\\alpha}(X)$ for all $\\alpha \\in \\Pi$, and at least one inequality is strict.\n\\end{lemma}\n\n\\begin{proof}By Lemma~\\ref{aptderivative}, every $c_{\\alpha}(p_{t}(X))$ is monotonically decreasing, and near every $t \\in \\mathbb{R}$ at least one is strictly decreasing.\n\\end{proof}\n\n\\begin{lemma} \\label{nonwanderinga}The only non-wandering points for the dynamical system \\eqref{dynamicalsystem} are those of $\\mathfrak{a}$.\n\\end{lemma}\n\n\\begin{proof}Let $X \\in \\mathfrak{p} - \\mathfrak{a}$. By Lemma~\\ref{apartdecreasing} there exists $\\alpha \\in \\Pi$ for which $c_{\\alpha}(p_{1}(X)) < c_{\\alpha}(X)$. By continuity of the flow, there exists a neighborhood $U$ of $X$ such that $c_{\\alpha}(p_{1}(U)) < c_{\\alpha}(U)$. Because $c_{\\alpha}$ is decreasing along orbits, we have $p_{t}(U) \\cap U = \\varnothing$ for all $t \\geq 1$. Thus $X$ is wandering.\n\\end{proof}\n\n\\begin{lemma} \\label{dynamicallimita}Let $X \\in \\mathfrak{p}$. There exists $H \\in \\mathfrak{a}$ such that $\\lim_{t \\to \\infty }p_{t}(X) = H$. \\end{lemma}\n\n\\begin{proof}Write $X = \\operatorname{Ad}_{k}(H)$ with $k \\in K$ and $H \\in \\mathfrak{a}$. Consider its $\\omega$-limit $\\omega(X)$. By Lemma~\\ref{nonwanderinga} we have $\\omega(X) \\subset \\mathfrak{a}$. In view of the explicit expression for $p_{t}(X)$, we have $\\omega(X) \\subset \\operatorname{Ad}_{K}(H)$ by continuity. Thus $\\omega(X) \\subset \\mathfrak{a} \\cap \\operatorname{Ad}_{K}(H)$. By Lemma~\\ref{conjugatetoAimplieslevi}, this intersection is equal to the Weyl group orbit of $H$. In particular, it is discrete. Because $\\omega(X)$ is connected, it must consist of a single point of $\\mathfrak{a}$.\n\\end{proof} \n\n\\begin{proof}[Proof of Proposition~\\ref{KparttoMcentralizer}] The $K$-projection of $ge^{tH}$ does not depend on the triangular part of $g$, so we may assume that $g \\in K$. Conjugation by $k \\in K$ gives a diffeomorphism\n\\[ K \/ Z_{K}(H) \\to \\operatorname{Ad}_{K}(H) \\,. \\]\nTake $k \\in G$. When $t \\to + \\infty$, the image of $\\kappa(k e^{tH})$ under this map tends to an element $\\operatorname{Ad}_{m}(H) \\in \\operatorname{Ad}_{M'}(H)$ by Lemma~\\ref{dynamicallimita}. Therefore $\\kappa(k e^{tH})$ tends to the point $m Z_{K}(H)$ in the quotient $K \/ Z_{K}(H)$. Equivalently, $\\kappa(k e^{tH})$ tends to the set $m Z_{K}(H)$ in $K$.\n\\end{proof}\n\n\\begin{remark}\\label{remarkKvsNproofs}In general the convergence in Lemma~\\ref{dynamicallimita} is not uniform. Already for $G = \\operatorname{SL}_3(\\mathbb R)$ there are heteroclinic orbits that have nearby orbits with totally different limit points, even some that emanate from equilibria in the positive Weyl chamber. Some of these orbits, that come in one-dimensional families, can be seen to lie in Levi subalgebras (Lie algebras of $L \\in \\mathcal L$). But not all of them do and certainly not those that come in two-dimensional families. In fact, it seems likely that these badly behaved orbits correspond exactly to the partition of $K$ into the sets $\\kappa(\\Omega_{\\mathcal S})$ defined in \\S\\ref{npartcanonicalpartitions}; compare Example~\\ref{omegaslthree}.\n\nFurther evidence for this is the following observation. Let $\\rho : G \\to \\operatorname{SL}(V)$ be any representation with finite kernel, and let $\\Omega_{\\mathcal S^0}$ be the set defined in \\S\\ref{npartcanonicalpartitions}. As explained in Remark~\\ref{atonegativechamber} we expect that for $g \\in \\Omega_{\\mathcal S^0}$ and $\\lambda \\in (\\mathfrak a^*)^+$ we have $\\lambda(H(ga)) \\to - \\infty$ as $a \\to \\infty$ in $A$, and we sketched a proof of this when $G =\\operatorname{SL}_n(\\mathbb R)$ and $\\rho = \\operatorname{Std}$. Moreover, when $\\lambda \\in \\Sigma^+$ is merely a positive root, it is still reasonable to expect that $\\lambda(H(ga)) \\to - \\infty$ as $a \\to \\infty$ in $A$ uniformly along regular directions; this is also apparent for $G =\\operatorname{SL}_n(\\mathbb R)$. Now writing $ga = n'a'k'$, we have for $H \\in \\mathfrak a$ that $\\operatorname{Ad}_{k'}(H) = \\operatorname{Ad}_{a'^{-1} n'^{-1}g}(H)$. When $g \\in  \\Omega_{\\mathcal S^0}$ stays in a compact set, then so does $n'$, by Theorem~\\ref{npartboundeduniform}, and if $\\lambda(a') \\to - \\infty$ for all positive roots $\\lambda$, then the positive root space components of $\\operatorname{Ad}_{a'^{-1} n'^{-1}g}(H)$ tend to zero. On the other hand this equals $\\operatorname{Ad}_{k'}(H)$, which lives in a bounded subset of $\\mathfrak p$, so that the negative root space components approach zero as well, meaning that $\\operatorname{Ad}_{k'}(H) \\to \\mathfrak a$ and therefore $k' \\to M' Z_K(H)$. Taking $H \\in \\mathfrak a$ regular, we would obtain that $k' \\to M'$, uniformly for $g$ in compact subsets of $\\Omega_{\\mathcal S^0}$ and $a\\to \\infty$ inside regular cones of $A$.\n\\end{remark}\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe discovery of the 125-{\\rm GeV } SM-like scalar \\cite{LHC} and the present absence of any new physics signals has opened up a whole host of questions as to the true nature of the electroweak symmetry breaking and to what may lie beyond the Standard Model. The sole existence of the 125-{\\rm GeV } particle would leave unanswered several deep questions such as the origin of neutrino masses, the hierarchy of quark and lepton masses among many others. It also implies that the electroweak vacuum is metastable with drastic consequences in the very far-distant future \\cite{metastable}. It remains to be seen whether this most simple picture- albeit one with many question marks- will be the ultimate theory of nature or it is merely an effective theory at current accessible energies whose reality tests are incomplete and more non-SM phenomena will pop up in the not-too-distant future with Run II of the LHC.\n\nDespite the present lack of new physics at the LHC, it does not imply that it is not there. On the contrary, new physics has already appeared in the neutrino sector through neutrino oscillation and its implication on neutrino masses. This evidence, although quite clear, is only indirect and does not show where the new physics that gives rise to the aforementioned phenomena may appear. This difficulty in finding a direct evidence for the new physics involved in generating neutrino masses is compounded by the fact that these masses are so tiny, more than seven orders of magnitude smaller than the lightest lepton: the electron. In the most generic scenario of the elegant seesaw mechanism for generating tiny masses, the right-handed neutrinos are sterile i.e. singlets under the electroweak gauge group. In a nutshell, the two mass eigenvalues are $m_D^2\/M$ and $M_R$ where the Dirac mass $m_D$ is proportional to the electroweak scale while the Majorana mass $M_R$ is $\\gg m_D$.\nIn addition to the fact that $\\nu_R$'s are assumed to be electroweak singlets, the very large values for $M$ in a generic scenario makes it very very difficult to probe the crucial physics, namely that which gives rise to $M_R$ which is responsible for the lightness of the ``active\" neutrinos. Another facet of this new physics is the Majorana nature of the ``active\" neutrinos themselves which could manifest itself through neutrino less double beta decays which so far have not been observed. Through neutrino oscillations, we have a hint of new physics but what it might be  and where to look for it is still a big mystery at the present time. \n\nThe aforementioned uncertainties rest in large part on the assumption that right-handed neutrinos are electroweak singlets. This usually comes from\na certain extension of the SM such as the Left-Right symmetric model $SU(2)_L \\times SU(2)_R \\times U(1)_{B-L}$ \\cite{senjanovic} or the Grand Unified model $SO(10)$, among others. \nThe L-R symmetric version has an advantage over the GUT version in that the right-handed neutrino Majorana mass is proportional to the breaking scale of $SU(2)_R$ which could be much lower than a typical GUT scale and which could open up the possibility for detecting right-handed neutrinos (and $W_R$ as well). A recent search for right-handed neutrinos and $W_R$ at the LHC \\cite{CMSWR} has yielded a lower bound of around 3 TeV for $W_R$. Bounds on right-handed neutrino masses were also presented in \\cite{CMSWR}. We will come back to the aforementioned remarks below.\n\nIf one is however willing to entertain the idea that right-handed neutrinos {\\em are not} sterile, there is an entire panorama of accessible phenomena that can be searched for and studied. A non-sterile right-handed neutrino necessarily interacts with the electroweak gauge bosons and the Majorana mass term is expected to carry the electroweak quantum number and hence is proportional to the electroweak breaking scale. Right-handed neutrinos could then be searched for both from an interaction point of view and from an energetic one. A model of this kind was put forth by one of us (PQH) \\cite{pqnur}  (the EW$\\nu_R$  model).\n\nIn the EW$\\nu_R$  model \\cite{pqnur}, right-handed neutrinos are parts of $SU(2)$ doublets along with their charged partners (the mirror charged leptons). Anomaly freedom dictates the existence of doublets of right-handed mirror quarks. The particle content of the model is listed in the next section. The existence of extra doublets of chiral fermions, the mirror quarks and leptons, is potentially fatal for the model because of their contributions to the electroweak precision parameters, in particular the S-parameter. Those extra chiral doublets would make a ``large\" contribution to the S-parameter, an undesirable outcome. Fortunately, the EW$\\nu_R$  model contains a Higgs triplet which makes opposite contributions to the S-parameter and thus offsetting those of the mirror fermions. An exhaustive study of the electroweak precision parameters within the framework of the EW$\\nu_R$  model has been carried out in \\cite{hung2} with the main result being that there is a large parameter space which satisfies the precision constraints. \n\nThe EW$\\nu_R$  model in its original inception \\cite{pqnur} contains, beside one Higgs doublet which couples to both SM and mirror fermions, two scalar triplets, one (complex) with hypercharge $Y\/2=1$ and another (real) with $Y\/2=0$. Out of the thirteen degrees of freedom (4 for the doublet, 6 for the complex triplet and 3 for the real triplet), three are absorbed by W's and Z and the remaining {\\em ten} become physical degrees of freedom. Can one of those ten physical scalars describe the observed 125-{\\rm GeV } SM-like scalar? If not, what minimal extension would be needed for that purpose? Where and how does one look for the more massive scalars which could be CP even or odd? \n\nThe plan of the paper is as follows. Section~\\ref{sec:summary} will be devoted to a summary of the EW$\\nu_R$  model with its particle content and, in particular for this paper, its scalar sector. For completeness, the electroweak precision parameter constraints will also be summarized. Section~\\ref{sec:extewnur} presents some of the salient points concerning the scalar sector of the original EW$\\nu_R$  model. A particular attention is paid to what this sector has to say about the 125-{\\rm GeV } SM-like scalar. We show why the lightest spin-0 particle has to be CP-odd if one wishes to identify it with the 125-{\\rm GeV } object. This has to do with the fact that the production cross section for the scalar is very large compared with the equivalent SM quantity. This occurs when a single Higgs doublet couples to both SM and mirror fermions. The CP-odd option unfortunately is ruled out by the likelihood analysis which favors the CP-even case \\cite{LHC2}. At the end of this section we present a simple extension of the original model by adding one extra Higgs doublet. In this extension, by imposing a global symmetry, one Higgs doublet is made to couple to SM fermions while the other one couples only to mirror fermions. The scalar mass eigenstates and eigenvalues are shown as well as their couplings to fermions and gauge bosons. Section~\\ref{sec:data} discusses the implications of the extended model in light of the existence of the 125-{\\rm GeV } SM-like scalar. We will show in that section the dual nature of the 125-{\\rm GeV } SM-like scalar and only further measurements can tell whether or not it is an ``impostor\".\n\n\n\\section{The EW$\\nu_R$  model: A summary}\\label{sec:summary}\n\nThe main idea of the EW$\\nu_R$  model \\cite{pqnur} was to search for a model in which right-handed neutrinos {\\em naturally} acquire a mass proportional to the electroweak scale $\\Lambda_{EW} = 246 ~{\\rm GeV }$. For this to occur, the {\\em most natural} way to implement this idea is for right-handed neutrinos to be {\\em non-sterile}. In particular, the simplest way is to put them in doublets along with right-handed mirror charged lepton partners. In this manner, a Majorana mass term of the type $M \\nu_{R}^{T} \\sigma_2 \\nu_R$ necessarily carries an $SU(2) \\times U(1)$ quantum number and transforms like an $SU(2)$ triplet. (Details are summarized below.) As shown in \\cite{pqnur}, a new Higgs sector including triplets is needed and it obviously participates in the symmetry breaking of the electroweak gauge group. The EW$\\nu_R$  model of \\cite{pqnur} is highly testable for the following reasons: 1)  $\\nu_R$'s are sufficiently light; 2) $\\nu_R$'s couple to W and Z and can be produced through these couplings; 3) The presence of an extended Higgs sector.\n\nAt this point, a comparison with the popular Left-Right symmetric model (L-R model) is in order here.  Basically, one would like to probe the physics that governs the right-handed neutrino Majorana mass scale $M_R$ since it is a cornerstone of the seesaw mechanism. \n\nAs we will review below, the gauge structure of  the EW$\\nu_R$  model  \\cite{pqnur} is the same as that of the SM, namely $SU(2) \\times U(1)$. The gauge structure of the L-R model is $SU(2)_L \\times SU(2)_R \\times U(1)_{B-L}$. As a result, the highest mass scale of the EW$\\nu_R$  model is $\\Lambda_{EW} \\sim 246\\; {\\rm GeV }$, while the L-R model is characterized by {\\em two} scales:  $\\Lambda_{L}$ and $\\Lambda_{R}$ with $\\Lambda_{R}$ being generally much larger than $\\Lambda_{L}$.  The $\\nu_R$ Majorana mass is proportional to $\\Lambda_{EW}$ in the EW$\\nu_R$  model and is naturally {\\em ``light\"} while, for the L-R model, it is proportional to $\\Lambda_{R}$ and could be ``heavy\". \n\nThe next difference lies in the production mechanism and detection of $\\nu_R$'s. In the L-R model, $\\nu_R$'s  are produced through the process $\\bar{u} + d \\rightarrow W_R \\rightarrow N + l$ \\cite{keung}. In contrast, the production of $\\nu_R$'s in  EW$\\nu_R$  model proceeds through $q + \\bar{q} \\rightarrow Z \\rightarrow \\nu_R + \\bar{\\nu}_R$ or $\\nu_R + \\nu_R$ ($\\nu_R$ is a Majorana particle). Since $M_Z $ is the mass scale that enters the production cross section for $\\nu_R$ in the EW$\\nu_R$  model, one expects the number of events characteristic of that model to be {\\em significantly} larger than that for the L-R model which is controlled by $M_{W_R}$, making it much easier to probe signals such as like-sign dileptons \\cite{pqnur}. \n\nTo summarize, the main difference between the EW$\\nu_R$  model and the L-R model is the question of energy scales. Right-handed neutrinos in the EW$\\nu_R$  model have masses proportional to $\\Lambda_{EW}$ and hence are accessible experimentally through the direct coupling with the Z-boson at present (and future) colliders. Its physics is {\\em bounded} by the {\\em electroweak scale}. The scale $\\Lambda_{R}$ of the L-R model on the other hand is unknown and is only bounded from below experimentally. It would be much harder to probe $SU(2)_R$ if that scale turns out to be much higher than the present experimental bound of 3 {\\rm TeV }'s. The direct evidence for the seesaw mechanism through the production and decay of $\\nu_R$ is certainly within the reach of the EW$\\nu_R$  model, both from the energetic and production points of view.\n\nNext, concerning the scalar sector, the Higgs triplet of $SU(2)_R$ could be quite heavy, its mass being proportional to $\\Lambda_{R}$. The physics of the Higgs triplet of the EW$\\nu_R$  model is controlled by the electroweak scale and its existence (or non-existence) can hopefully be verified in Run II of the LHC. This is the subject of the present manuscript.\n\nLast but not least, the L-R model can be embedded in a grand unified group $SO(10)$. As mentioned in \\cite{pqnur}, the EW$\\nu_R$  model can be embedded in the group $E_6$. However, we consider the testability of the EW$\\nu_R$  model to be more important than its embedding in a GUT group.\n\n\n\\subsection{Gauge structure and particle content of the EW$\\nu_R$  model }\nBelow is a summary of the gauge structure and particle content of the minimal EW$\\nu_R$  model of \\cite{pqnur}.\nThe notations for the leptons and quarks are generic for any family.\n\n\\begin{itemize}\n\n\\item Gauge group: $SU(3)_C \\times SU(2) \\times U(1)_Y$\n\n\\item Lepton $SU(2)$ doublets (generic notation):\n\nSM: \n\\begin{equation}\nl_{L} = \\left( \\begin{array}{c}\n\\nu_L \\\\\ne_{L}\n\\end{array} \\right)\\;;\n\\end{equation}\n\nMirror: \n\\begin{equation}\nl^{M}_{R} = \\left( \\begin{array}{c}\n\\nu_R \\\\\ne^{M}_{R}\n\\end{array} \\right)\\;.\n\\end{equation}\n\n\\item Lepton $SU(2)$ singlets (generic notation):\n\nSM: $e_R$ ; Mirror: $e^{M}_L$\n\n\\item Quark $SU(2)$ doublets (generic notation):\n\nSM: \n\\begin{equation}\nq_{L} = \\left( \\begin{array}{c}\nu_{L} \\\\\nd_{L}\n\\end{array} \\right)\\;;\n\\end{equation}\n\nMirror: \n\\begin{equation}\nq^{M}_{R} = \\left( \\begin{array}{c}\nu^{M}_{R} \\\\\nd^{M}_{R}\n\\end{array} \\right)\\;.\n\\end{equation}\n\n\\item Quark $SU(2)$ singlets (generic notation):\n\nSM: $u_R$ ,  $d_R$  ;  Mirror: $u^{M}_L$ , $d^{M}_L$.\n\n\\item The Higgs sector:\n\na) One Higgs doublet: $\\Phi$. This Higgs doublet couples to both SM and mirror fermions.\n\nb) One complex Higgs triplet with $Y\/2=1$ containing doubly-charged scalars:\n\\begin{equation}\n\\tilde{\\chi} = \\frac{1}{\\sqrt{2}}\\,\\vec{\\tau}.\\vec{\\chi}=\n\\left( \\begin{array}{cc}\n\\frac{1}{\\sqrt{2}}\\,\\chi^{+} & \\chi^{++} \\\\\n\\chi^{0} & -\\frac{1}{\\sqrt{2}}\\,\\chi^{+}\n\\end{array} \\right)\\;.\n\\end{equation}\n\nc) One real Higgs triplet with $Y\/2=0$:\n\\begin{equation}\n(\\xi^+, \\xi^0, \\xi^-)\\,.\n\\end{equation}\n\nd) One SM singlet Higgs: $\\phi_S$.\n\n\n\\end{itemize}\n\n\\subsection{Symmetry breaking in the EW$\\nu_R$  model}\n$SU(2) \\times U(1)$ is spontaneously broken by the vacuum expectation values (VEV) of the Higgs doublet and triplets. The Higgs potential \\cite{ChanoGold,pqnur} has a global $SU(2)_L \\times SU(2)_R$ symmetry. The triplets transform as $(3,3)$ and the doublet as $(2,2)$ under that global symmetry. Specifically,\n\\begin{equation}\n\\chi = \\left( \\begin{array}{ccc}\n\\chi^{0} &\\xi^{+}& \\chi^{++} \\\\\n\\chi^{-} &\\xi^{0}&\\chi^{+} \\\\\n\\chi^{--}&\\xi^{-}& \\chi^{0*}\n\\end{array} \\right) \\,,\n\\end{equation}\nand\n\\begin{equation}\n\\Phi = \\left( \\begin{array}{cc}\n\\phi^{0} & \\phi^{+} \\\\\n\\phi^{-} & \\phi^{0} \n\\end{array} \\right)  \\, .\n\\end{equation}\nProper vacuum alignment dictates $\\langle \\chi^{0} \\rangle = \\langle \\xi^{0} \\rangle = v_M$ i.e.\n\\begin{equation}\n\\langle \\chi \\rangle = \\left( \\begin{array}{ccc}\nv_M &0&0 \\\\\n0&v_M&0 \\\\\n0&0&v_M\n\\end{array} \\right) \\,,\n\\end{equation}\n\\begin{equation}\n\\langle \\Phi \\rangle = \\left( \\begin{array}{cc}\nv_2\/\\sqrt{2} &0 \\\\\n0&v_2\/\\sqrt{2} \n\\end{array} \\right) \\,.\n\\end{equation}\nThese VEVs leave an unbroken $SU(2)_D$ custodial symmetry i.e. $SU(2)_L \\times SU(2)_R \\rightarrow SU(2)_D$. This ensures that $\\rho = M_{W}^2\/M_{Z}^2 \\cos^2 \\theta_W=1$ at tree level and one now has\n\\begin{equation}\nv= \\sqrt{v_2^2+ 8\\,v_M^2} \\approx 246 \\, {\\rm GeV } \\,.\n\\end{equation}\n\nAs discussed in \\cite{pqaranda,hung2,GeorgMach}, with respect to $SU(2)$, the two triplets (one real and one complex) and one doublet sum up to 13 degrees of freedom, 3 of which are Nambu-Goldstone bosons absorbed by W's and Z  leaving 10 physical degrees of freedom. Under the custodial symmetry $SU(2)_D$, these transform as\n\\begin{eqnarray}\n\t\t\\text{five-plet (quintet)} &\\rightarrow& H_5^{\\pm\\pm},\\; H_5^\\pm,\\; H_5^0;\\nonumber\\\\[0.5em]\n\t\t\\text{triplet} &\\rightarrow& H_3^\\pm,\\; H_3^0;\\nonumber\\\\[0.5em]\n\t\t\\text{two singlets} &\\rightarrow& H_1^0,\\; H_1^{0\\prime}\\,.  \\nonumber\n\\end{eqnarray}  \nThe expressions for the scalar states can be explicitly found in Eq.~(\\ref{eq:higgs}), by setting $s_{2M}\\rightarrow 0$ and $\\Phi_{2M} \\rightarrow 0$.\n\n\\subsection{The seesaw mechanism in the EW$\\nu_R$  model}\nThe main purpose of the EW$\\nu_R$  model was to provide a scenario in which right-handed neutrinos are non-sterile and get their masses out of the symmetry breaking of $SU(2) \\times U(1)$. A Majorana mass term of the form $M_{R} \\, \\nu_{R}^{T}\\,  \\sigma_{2} \\, \\nu_R$ in the EW$\\nu_R$  model comes from the following Yukawa interaction:\n\\begin{equation}\ng_{M} l_R^{M,T} \\,\\sigma_2\\, \\tilde{\\chi} \\, l_R^{M} \\, ,\n\\end{equation}\nwhich gives \n\\begin{equation}\ng_M \\, \\nu_{R}^{T}\\,  \\sigma_{2} \\, \\nu_R \\chi^{0} \\,.\n\\end{equation}\nThe right-handed neutrino Majorana mass is now intrinsically linked to the breaking scale of $SU(2) \\times U(1)$ through the VEV of $\\tilde{\\chi}$ as\n\\begin{equation}\nM_R = g_M\\,v_M  \\, .\n\\end{equation}\nAs stressed in \\cite{pqnur}, $M_R$ is bounded from below because $\\nu_R$ are now members of an $SU(2)$ doublet and would contribute to the Z-boson width leading to the lower bound: \n\\begin{equation}\nM_R \\geq M_Z\/2 \\approx 46 ~{\\rm GeV }\\,.\n\\end{equation}\n\nAs discussed in \\cite{pqnur}, a global symmetry $U(1)_{MF}$ (referred to as $U(1)_M$ in \\cite{pqnur}) was imposed so as to forbid a term of the form $g_{L} l_L^{M,T} \\,\\sigma_2\\, \\tilde{\\chi} \\, l_L^{M}$ which would give a large Majorana mass $g_L v_M$ to the left-handed neutrino, unless $g_L$ is unnaturally fine-tuned to be tiny. As discussed in \\cite{pqnur}, this is accomplished by the following transformation properties: $(l_R^M, ~e_L^M) \\rightarrow e^{\\imath\\alpha_{MF}} (l_R^M, ~e_L^M)$, $\\widetilde{\\chi} \\rightarrow e^{-2\\imath \\alpha_{MF}} \\widetilde{\\chi}$ and $\\phi_S \\rightarrow e^{-\\imath\\alpha_{MF}}\\phi_S$, with all other SM particles being $U(1)_{MF}$ singlets. We will come back to this symmetry and its extended version below.\n\nA Dirac mass term is of the form $m_D (\\nu_{L}^{\\dagger}\\, \\nu_R + h.c.)$. This is a product of two doublets and the simplest choice for the Higgs scalar is an $SU(2)$ singlet with zero hypercharge, namely $\\phi_S$.\n\\begin{equation}\n{\\cal L}_S = g_{Sl} \\, \\bar{l}_{L}\\, \\phi_S \\, l^{M}_{R} + H.c. \\,.\n\\end{equation}\nWith $\\langle \\phi_S \\rangle = v_S$, the Dirac mass is given by\n\\begin{equation}\nm_D = g_{Sl} v_S \\,.\n\\end{equation}\n\nThe magnitude of the light neutrino mass given by\n\\begin{equation}\nm_\\nu = \\frac{m_D^2}{M_R} <O(eV) \\, ,\n\\end{equation}\nimplying $v_S \\sim O(10^5 \\, eV)$ if we assume $g_{Sl} \\sim O(1)$ or $v_S \\sim O(\\Lambda_{EW} \\sim 246 \\,{\\rm GeV })$ for $g_{Sl} \\sim O(10^{-7})$.\n\n\\subsection{Constraints from electroweak precision data}\n\nThe presence of extra $SU(2)$ doublets of chiral fermions in the form of mirror fermions would seriously affect the constraints from electroweak precision data. As first mentioned in \\cite{pqnur}, the positive contribution of mirror fermions to the S-parameter could be compensated by the negative contribution to S from the Higgs triplets. A detailed analysis has been performed in \\cite{hung2} which showed that there is a large parameter space in the model which satisfies the present constraints of the electroweak precision data. A sample of the plots summarizing the scatter plots of the model is given below. In those plots, we had 3500 points inside the $2\\sigma$ constraint. Among them, 100 points fall inside the $1\\sigma$ region. The number of these data points were taken for illustrative purpose and could be larger with a longer program run time. Moreover, it is possible to find any combination in the parameters space, $m_Z\\leq m_H, m_{q^M}\\leq 600\\;{\\rm GeV }, \\;150\\leq m_{l^M} \\leq 600 \\;{\\rm GeV }, \\; m_Z\/2\\leq m_{\\nu_R} \\leq 600\\;{\\rm GeV }$, that can satisfy the constraints. As one can see,  the size of parameter space in the model is {\\em not} fine-tuned. This is illustrated in the figures below. \n\n$\\widetilde{S}$ and $\\widetilde{T}$ are new Physics contributions to the $S$ and $T$ parameters respectively. One can see from Fig.~\\ref{TvsS} that the EW$\\nu_R$  model satisfies very well the constraints from the electroweak precision data and has passed the first (indirect) test. \n\\begin{figure}[H]\n\\centering\n \\includegraphics[scale=0.35]{TS_126.pdf} \n \\caption{\\label{TStotal}{\\small Total $\\tilde{T}$ versus $\\tilde{S}$ with the 1 and 2 $\\sigma$ experimental contours}}\n\\label{TvsS}\n\\end{figure}\nHere and in \\cite{hung2}, $\\tilde{S}$ and $\\tilde{T}$ are defined as the sum of the contributions from the scalar and mirror fermion sectors, namely $\\tilde{S}=\\tilde{S}_S + \\tilde{S}_{MF}$ and $\\tilde{T}=\\tilde{T}_S + \\tilde{T}_{MF}$. For clarity, we also show the plots of $\\tilde{S}_S$ versus $\\tilde{S}_{MF}$ and $\\tilde{T}_S$ versus $\\tilde{T}_{MF}$. \n\\begin{figure}[H]\n\\centering\n    \\includegraphics[scale=0.35]{SsVsSmf.pdf} \n \\caption{{\\small Constrained $\\tilde{S}_S$ versus $\\tilde{S}_{MF}$}}\n\\label{SsvsSmf}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n    \\includegraphics[scale=0.35]{TsVsTmf.pdf} \n \\caption{{\\small Constrained $\\tilde{T}_S$ versus $\\tilde{T}_{MF}$}}\n\\label{TsvsTmf}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{T_lepton.pdf}\n\t\\caption{{\\small Constrained $\\tilde{T}$ versus mass splitting between $e^M$ and $\\nu_R$: $(m_{e^M} - M_R)$}}\n\t\\label{TvsmeM-MR}\n\\end{figure}\nFrom these plots, one can see that the mirror fermions within a doublet generally tend to be non-degenerate although they can as well be degenerate (Fig.~\\ref{TvsmeM-MR}). The mass-splittings within a mirror quark generation yield a similar-looking plot, not shown here.\n(The statement made in \\cite{hung2} which is based on a thorough analysis is more accurate than that made in the original paper \\cite{pqnur} concerning the size of that mass splitting.) The interesting phenomenology of the mirror sector is beyond the scope of this manuscript and is under investigation.\n\nThroughout the analysis, one can see the importance of the negative contributions coming from the scalar sector which provides a counterweigh to the positive ones coming from the mirror fermion sector in such a way that the sum of the two stays within the $2\\sigma$ constraint. This is certainly not a fine-tuning procedure, since we do not require precise cancellations between the two sectors.\nFurther details can be found in \\cite{hung2}.\n\nAn important point which is worth repeating here is the role played by the scalar triplet in regulating the new physics contribution to the $S$ and $T$-parameters. In fact, it has been pointed out in \\cite{hung2} that the contribution to $S$ from the scalar triplet can be made increasingly negative by increasing the mass of the doubly-charged Higgs as an example. This can offset the positive contribution coming from the mirror fermions. One can see the importance of the scalar sector, in particular the Higgs triplets, in making the EW$\\nu_R$  model consistent with precision data.\n\nThe next step is to examine constraints coming from direct searches of the Higgs boson.\n\n\\subsection{Constraint on the ``minimal\" EW$\\nu_R$  model from the 125-GeV SM-like Higgs boson}\\label{sec:minimalH}\n\nBy ``minimal\" we mean that the Higgs structure is as described above: one Higgs doublet and two Higgs triplets. Some phenomenology of these scalars has been investigated in \\cite{pqaranda}. This topic will be revisited in a future publication. For the purpose of this manuscript, we shall focus on the four neutral states: $H_5^0 $, $H_3^0$, $H_1^0$ and $H_1^{0\\prime}$  and in particular $H_3^0$ and $H_1^0$ since the other two do not couple to SM and mirror fermions \\cite{hung2}. $H_3^0$ and $H_1^0$ are CP-odd and CP-even respectively. As shown in \\cite{hung2}, because of the coupling $g_{H_{1}^0\\,q\\bar{q}}=-\\imath \\frac{m_{q} g}{2m_{W} c_{H}}$, the gluon fusion production cross section for $H_1^0$ was estimated to be $\\sigma_{EW\\nu_R} \\agt 49\\, \\sigma_{SM}$ where the factor $49= (1+6)^2$ takes into account the contributions from the top and mirror quarks. This alone practically ruled out $H_1^0$ as the 125-{\\rm GeV } SM-like scalar. Also, since the coupling to fermions are very similar to that of the SM, modulo the factor $1\/\\cos \\theta_H$, the various branching ratios (BR) are expected to be of the order of those of the SM and the signal strengths ($\\mu = (\\sigma \\times BR)\/(\\sigma \\times BR)_{SM}$) will largely exceed observations. \n\nIt was shown in \\cite{hung2} that the CP-odd (pseudoscalar) $H_3^0$ could, with the appropriate choice of parameters, can fit the bill for being the 125-{\\rm GeV } object both in terms of the production cross section and in terms of branching ratios. However, a likelihood analysis ruled this option out by more than 3 $\\sigma$s \\cite{LHC2}. Although a measurement of the spin and parity of the 125-{\\rm GeV } object is yet to be performed, it is fair to assume that it is more likely to be a $0^+$ state.\n\nAs one can see, the reason why the CP-even $H_1^0$ has such a large gluon fusion production cross section (at least 49 times larger than the SM one at at same mass) is because it comes from the Higgs doublet (the real part of the neutral component) which couples to SM fermions {\\em as well as} mirror fermions. The loop controlling the gluon fusion production of $H_1^0$ is dominated by the top quark and the mirror quarks giving rise to the factor of 49 mentioned above while it is dominated only by the top quark contribution in the SM. An extension in the Higgs sector of the minimal EW$\\nu_R$  model is needed. This is shown in the next section.\n\n\\section{Extended EW$\\nu_R$  model}\\label{sec:extewnur}\nThe simplest extension- and, in fact, the most natural one - of the minimal EW$\\nu_R$  model is to have {\\em two} Higgs doublets with one coupled to SM fermions and the other one to mirror fermions. One extra Higgs doublet leads to 4 more degrees of freedom. To prevent cross coupling, a global symmetry will be imposed. Basically, we introduce the following Higgs doublets along with the corresponding global symmetries $U(1)_{SM} \\times U(1)_{MF}$:\n\\begin{eqnarray}\n\\label{smtrans}\n\t\t\tU(1)_{SM}&:&\\Phi_2 \\;\\rightarrow e^{\\imath \\alpha_{SM}} \\; \\Phi_2\\nonumber\\\\\n\t\t\t&&(q_L^{SM}, l_L^{SM}) \\;\\rightarrow e^{\\imath \\alpha_{SM}} \\; (q_L^{SM}, l_L^{SM})\\,,\n\\end{eqnarray}\n\t\t\n\\begin{eqnarray}\n\\label{qmtrans}\n\t\t\tU(1)_{MF}&:&\\;\\Phi_{2M} \\;\\rightarrow e^{\\imath \\alpha_{MF}} \\; \\Phi_{2M}\\nonumber\\\\\n\t\t\t&&\\;(q_R^{M}, l_R^{M}) \\;\\rightarrow e^{\\imath \\alpha_{MF}} \\; (q_R^{M}, l_R^{M})\\,,\n\\end{eqnarray}\n\\begin{equation}\n\t\t\\phi_S \\;\\rightarrow e^{-\\imath(\\alpha_{MF} - \\alpha_{SM})} \\; \\phi_S \\,,\n\\end{equation}\n\\begin{equation}\n\t\t\\label{chitrans}\n\t\t\\tilde{\\chi} \\;\\rightarrow e^{-2\\imath \\alpha_{MF}} \\; \\tilde{\\chi} \\,.\n\\end{equation}\nAll other fields ($SU(2)$-singlet right-handed SM fermions, left-handed mirror fermions) are singlets under  $U(1)_{SM}\\,\\times U(1)_{MF}$. \t\n\t\nThese symmetries will forbid, at tree level, Yukawa couplings of the form $g_Y \\bar{f}_L \\Phi_{2}M f_R$ and $g_Y \\bar{f}^{M}_R \\Phi_2 f^{M}_L$ . Only Yukawa interactions of the type $g_Y \\bar{f}_L \\Phi_2 f_R$ and $g_Y \\bar{f}^{M}_R \\Phi_{2M} f^{M}_L$ are allowed. The Yukawa couplings of the physical states to SM and mirror fermions will involve mixing angles. This is detailed below.\n\n\\par When a $Y = 1$ complex scalar doublet is added to the minimal EW$\\nu_R$  model, under the global $SU(2)_L \\times SU(2)_R$ we have the following scalar fields:\n\\begin{equation}\n\t\\label{Phi1}\n\t\\Phi_{2}=\\left( \\begin{array}{cc}\n\t\\phi_{2}^{0,*} & \\phi_{2}^{+} \\\\\n\t\\phi_{2}^{-} & \\phi_{2}^0\n\t\\end{array} \\right) \\,,\n\\end{equation}\n\\begin{equation}\n\t\\label{Phi2}\n\t\\Phi_{2M}=\\left( \\begin{array}{cc}\n\t\\phi_{2M}^{0,*} & \\phi_{2M}^{+} \\\\\n\t\\phi_{2M}^{-} & \\phi_{2M}^0\n\t\\end{array} \\right) \\,,\n\\end{equation}\nand\n\\begin{equation}\n\t\\label{chi}\n\t\\chi = \\left( \\begin{array}{ccc}\n\t\\chi^{0} &\\xi^{+}& \\chi^{++} \\\\\n\t\\chi^{-} &\\xi^{0}&\\chi^{+} \\\\\n\t\\chi^{--}&\\xi^{-}& \\chi^{0*}\n\t\\end{array} \\right) \\,.\n\\end{equation}\nProper vacuum alignment for $SU(2)_L \\times U(1)_Y \\rightarrow U(1)_{em}$ gives\n\\begin{equation}\n\\label{phi1vev}\n\\langle \\Phi_{2} \\rangle = \\left( \\begin{array}{cc}\nv_2\/\\sqrt{2} &0 \\\\\n0&v_2\/\\sqrt{2} \n\\end{array} \\right) \\,,\n\\end{equation}\n\\begin{equation}\n\\label{phi2vev}\n\\langle \\Phi_{2M} \\rangle = \\left( \\begin{array}{cc}\nv_{2M}\/\\sqrt{2} &0 \\\\\n0&v_{2M}\/\\sqrt{2} \n\\end{array} \\right) \\,,\n\\end{equation}\nand\n\\begin{equation}\n\\label{chivev}\n\\langle \\chi \\rangle = \\left( \\begin{array}{ccc}\nv_M &0&0 \\\\\n0&v_M&0 \\\\\n0&0&v_M\n\\end{array} \\right) \\,.\n\\end{equation}\nThus, the VEVs of real parts of $\\Phi_2$, $\\Phi_{2M}$ and $\\chi$ are $(v_2\/\\sqrt{2})$, $(v_{2M}\/\\sqrt{2})$ and $v_M$ respectively such that\n\\begin{equation}\n\tv_2^2 + v_{2M}^2 + 8\\,v_M^2 = v^2\\,,\n\\end{equation}\nwhere $v \\approx 246 ~{\\rm GeV }$. We define\n\\begin{equation}\\label{eq:sdefs}\n\ts_2 = \\dfrac{v_2}{v}; ~~s_{2M} = \\dfrac{v_{2M}}{v}; ~~s_M = \\dfrac{2 \\sqrt{2}\\; v_M}{v}\\,.\n\\end{equation}\nNotice that apart from $\\xi$, this model contains the same number of triplet ($\\widetilde{\\chi}$) and the same number of doublets (2: $\\Phi_2$, $\\Phi_{2M}$) as the L-R model. The big difference comes from the fact that the VEV of the triplet $\\widetilde{\\chi}$ is bounded from above by the $\\Lambda_{EW} \\sim 246~{\\rm GeV }$, while the VEV of the L-R triplet is only bounded from below by experiment to be $\\gtrsim 3~{\\rm TeV }$ \\cite{CMSWR}.\n\n\\par A generic $SU(2)_L \\times SU(2)_R$ preserving potential for these scalars can now be written as\n\\begin{widetext}\n\t\\begin{flalign}\n\t\\label{pot0}\n\t\t&V(\\Phi_{2},\\Phi_{2M},\\chi) = \\lambda_1 \\Big[Tr\\Phi_{2}^\\dagger \\Phi_{2} - v_2^2 \\Big]^2 + \\lambda_2 \\Big[Tr\\Phi_{2M}^\\dagger \\Phi_{2M} - v_{2M}^2 \\Big]^2 + \\lambda_3 \\Big[Tr\\chi^\\dagger \\chi - 3 v_M^2\\Big]^2&\\nonumber\\\\[0.5em]\n\t\t&\\; +\\lambda_4 \\Big[Tr\\Phi_{2}^\\dagger \\Phi_{2} - v_2^2 + Tr\\Phi_{2M}^\\dagger \\Phi_{2M} - v_{2M}^2 + Tr\\chi^\\dagger \\chi - 3 v_M^2 \\Big]^2&\\nonumber\\\\[0.5em]\n\t\t&\\; + \\lambda_5 \\Big[(Tr\\Phi_{2}^\\dagger \\Phi_{2})\\; (Tr\\chi^\\dagger \\chi) - 2\\; (Tr\\Phi_{2}^\\dagger \\frac{\\tau^a}{2} \\Phi_{2} \\frac{\\tau^b}{2})\\; (Tr\\chi^\\dagger T^a \\chi T^b) \\Big]&\\nonumber\\\\[0.5em]\n\t\t&\\; + \\lambda_6 \\Big[(Tr\\Phi_{2M}^\\dagger \\Phi_{2M})\\; (Tr\\chi^\\dagger \\chi) - 2\\; (Tr\\Phi_{2M}^\\dagger \\frac{\\tau^a}{2} \\Phi_{2M} \\frac{\\tau^b}{2})\\; (Tr\\chi^\\dagger T^a \\chi T^b) \\Big]&\\nonumber\\\\[0.5em]\n\t\t&\\; + \\lambda_7 \\Big[(Tr\\Phi_{2}^\\dagger \\Phi_{2})\\; (Tr\\Phi_{2M}^\\dagger \\Phi_{2M})\\; - (Tr\\Phi_{2}^\\dagger \\Phi_{2M})\\; (Tr\\Phi_{2M}^\\dagger \\Phi_{2}) \\Big]\n\t\n\t\n\t\t +\\lambda_8 \\Big[3\\; Tr\\chi^\\dagger \\chi \\chi^\\dagger \\chi - (Tr\\chi^\\dagger \\chi)^2 \\Big]\\,.&\n\t\\end{flalign}\n\\end{widetext}\nNote that this potential, like the one in the minimal EW$\\nu_R$  model is also invariant under $\\chi \\rightarrow -\\chi$. Now it is also invariant under the global $U(1)_{\\text{SM}} \\times U(1)_{\\text{MF}}$  symmetry. The vacuum alignment given above breaks the global $SU(2)_L \\times SU(2)_R$  down to the custodial $SU(2)_D$. One still has $M_W = g\\,v\/2$ and $M_Z = M_W\/\\cos \\theta_W$, but now $v= \\sqrt{v_2^2 + v_{2M}^2 + 8\\,v_M^2} \\approx 246 \\, {\\rm GeV }$. It is found that three `massless' Nambu-Goldstone Bosons can be obtained after spontaneous breaking of $SU(2)_L \\times U(1)_Y$ to $U(1)_{em}$, when a condition $\\lambda_5 = \\lambda_6 = \\lambda_7$ imposed on the potential above. Thus, the potential that should be used to find the physical Higgs states is\n\\begin{widetext}\n\t\\begin{flalign}\n\t\t\\label{eq:pot}\n\t\t&V(\\Phi_{2},\\Phi_{2M},\\chi) = \\lambda_1 \\Big[Tr\\Phi_{2}^\\dagger \\Phi_{2} - v_2^2 \\Big]^2 + \\lambda_2 \\Big[Tr\\Phi_{2M}^\\dagger \\Phi_{2M} - v_{2M}^2 \\Big]^2 + \\lambda_3 \\Big[Tr\\chi^\\dagger \\chi - 3 v_M^2\\Big]^2&\\nonumber\\\\[0.5em]\n\t\t&\\; +\\lambda_4 \\Big[Tr\\Phi_{2}^\\dagger \\Phi_{2} - v_2^2 + Tr\\Phi_{2M}^\\dagger \\Phi_{2M} - v_{2M}^2 + Tr\\chi^\\dagger \\chi - 3 v_M^2 \\Big]^2&\\nonumber\\\\[0.5em]\n\t\t&\\; + \\lambda_5 \\Big[(Tr\\Phi_{2}^\\dagger \\Phi_{2})\\; (Tr\\chi^\\dagger \\chi) - 2\\; (Tr\\Phi_{2}^\\dagger \\frac{\\tau^a}{2} \\Phi_{2} \\frac{\\tau^b}{2})\\; (Tr\\chi^\\dagger T^a \\chi T^b) + (Tr\\Phi_{2M}^\\dagger \\Phi_{2M})\\; (Tr\\chi^\\dagger \\chi)&\\nonumber\\\\[0.5em]\n\t\t&\\; - 2\\; (Tr\\Phi_{2M}^\\dagger \\frac{\\tau^a}{2} \\Phi_{2M} \\frac{\\tau^b}{2})\\; (Tr\\chi^\\dagger T^a \\chi T^b) + (Tr\\Phi_{2}^\\dagger \\Phi_{2})\\; (Tr\\Phi_{2M}^\\dagger \\Phi_{2M})\\; - (Tr\\Phi_{2}^\\dagger \\Phi_{2M})\\; (Tr\\Phi_{2M}^\\dagger \\Phi_{2}) \\Big]& \\nonumber\\\\[0.5em]\n\t\t&\\; +\\lambda_8 \\Big[3\\; Tr\\chi^\\dagger \\chi \\chi^\\dagger \\chi - (Tr\\chi^\\dagger \\chi)^2 \\Big]\\,.&\n\t\\end{flalign}\n\\end{widetext}\nAfter the spontaneous breaking of $SU(2)_L \\times U(1)_Y \\rightarrow U(1)_{em}$, besides the three Nambu-Goldstone Bosons, there are {\\em twelve} physical scalars grouped into $5 ~+ ~3 ~+ ~3 ~+ ~1$ of the custodial $SU(2)_D$ with 3 custodial singlets.\nTo express the Nambu-Goldstone Bosons and the physical scalars let us adopt the following convenient notation:\n\\begin{eqnarray}\n\tv &=& \\sqrt{v_2^2 + v_{2M}^2 + 8 v_M^2}\\,,\\nonumber\\\\[0.5em]\n\ts_2 &=& \\frac{v_2}{v}\\,,\\; s_{2M} = \\frac{v_{2M}}{v}\\,,\\; s_M = \\frac{2\\sqrt{2}\\;v_M}{v}\\,, \\nonumber\\\\[0.5em]\n\tc_2 &=& \\frac{\\sqrt{v_{2M}^2 + 8 v_M^2}}{v}\\,,\\; c_{2M} = \\frac{\\sqrt{v_2^2 + 8 v_M^2}}{v}\\,,\\nonumber\\\\[0.5em]\n\tc_M &=& \\frac{\\sqrt{v_2^2 + v_{2M}^2}}{v}\\,.\n\\end{eqnarray}\nThus,\n\\begin{equation}\n\ts_2^2 + c_2^2 = s_{2M}^2 + c_{2M}^2 = s_M^2 + c_M^2 = 1\\,.\n\\end{equation}\n[In the limit $s_{2M} \\rightarrow 0$ i.e. the extended EW$\\nu_R \\rightarrow minimal~\\text{EW}\\nu_R$, $s_M \\rightarrow s_H$ and $c_M \\rightarrow c_H$]. Let us also define, like we did for the minimal EW$\\nu_R$  model,:\n\t\\begin{eqnarray}\\label{eq:subsid}\n\t\t\\phi_{2}^0 &\\equiv& \\frac{1}{\\sqrt{2}} \\Big(v_2 + \\phi_{2}^{0r} + \\imath \\phi_{2}^{0\\imath}\\Big),\\nonumber\\\\[0.5em]\n\t\t\\phi_{2M}^0 &\\equiv& \\frac{1}{\\sqrt{2}} \\Big(v_{2M} + \\phi_{2M}^{0r} + \\imath \\phi_{2M}^{0\\imath}\\Big),\\nonumber\\\\[0.5em]\n\t\t\\chi^0 &\\equiv& v_M + \\frac{1}{\\sqrt{2}} \\Big(\\chi^{0r} + \\imath \\chi^{0\\imath}\\Big);\\\\[0.5em]\n\t\t\\psi^\\pm &\\equiv& \\frac{1}{\\sqrt{2}} \\Big(\\chi^\\pm + \\xi^\\pm \\Big),\\hspace{1em}\\zeta^\\pm \\equiv \\frac{1}{\\sqrt{2}} \\Big(\\chi^\\pm - \\xi^\\pm\\Big)\\,,\n\t\\end{eqnarray}\nfor the complex neutral and charged fields respectively. With these fields the Nambu-Goldstone Bosons are given by\n\\begin{eqnarray}\n\tG_3^\\pm &=& s_2 \\phi_{2}^\\pm + s_{2M} \\phi_{2M}^\\pm + s_M \\psi^\\pm,\\nonumber\\\\[0.5em]\n\tG_3^0 &=& \\imath \\Big(- s_2 \\phi_{2}^{0\\imath} - s_{2M} \\phi_{2M}^{0\\imath} + s_M \\chi^{0\\imath}\\Big).\n\\end{eqnarray}\nThe physical scalars can be grouped, as stated in the previous section, based on their transformation properties under $SU(2)_D$ as follows:\n\t\\begin{eqnarray}\n\t\t\\text{five-plet (quintet)} &\\rightarrow& H_5^{\\pm\\pm},\\; H_5^\\pm,\\; H_5^0;\\nonumber\\\\[0.5em]\n\t\t\\text{triplet} &\\rightarrow& H_{3}^\\pm,\\; H_{3}^0;\\nonumber\\\\[0.5em]\n\t\t\\text{triplet} &\\rightarrow& H_{3M}^\\pm,\\; H_{3M}^0;\\nonumber\\\\[0.5em]\n\t\t\\text{three singlets} &\\rightarrow& H_1^0,\\; H_{1M}^0,\\; H_1^{0\\prime}\\,,\n\t\\end{eqnarray}\nwhere\n\t\\begin{eqnarray}\\label{eq:higgs}\n\t\tH_5^{++} &=& \\chi^{++},\\; H_5^+ = \\zeta^+,\\; H_5^0 = \\frac{1}{\\sqrt{6}}\\Big(2\\xi^0 - \\sqrt{2}\\chi^{0r}\\Big),\\nonumber\\\\[0.5em]\n\t\tH_3^+ &=& - \\frac{s_2 s_M}{c_M} \\phi_{2}^+ - \\frac{s_{2M} s_M}{c_M} \\phi_{2M}^+ + c_M \\psi^+,\\nonumber\\\\[0.5em]\n\t\tH_3^0 &=& \\imath \\left(\\frac{s_2 s_M}{c_M} \\phi_{2}^{0i} + \\frac{s_{2M} s_M}{c_M} \\phi_{2M}^{0i} + c_M \\chi^{0i}\\right),\\nonumber\\\\[0.5em]\n\t\tH_{3M}^+ &=& - \\frac{s_{2M}}{c_M} \\phi_{2}^+ + \\frac{s_2}{c_M} \\phi_{2M}^+,\\nonumber\\\\[0.5em]\n\t\tH_{3M}^0 &=& \\imath \\left(- \\frac{s_{2M}}{c_M} \\phi_{2}^{0i} + \\frac{s_2}{c_M} \\phi_{2M}^{0i}\\right),\\nonumber\\\\[0.5em]\n\t\tH_1^0 &=& \\phi_{2}^{0r},\\; H_{1M}^0 = \\phi_{2M}^{0r},\\nonumber\\\\[0.5em]\n\t\tH_1^{0\\prime} &=& \\frac{1}{\\sqrt{3}} \\Big(\\sqrt{2}\\chi^{0r} + \\xi^0\\Big)\\,,\n\t\\end{eqnarray}\nwith phase conventions $H_5^{--} = (H_5^{++})^\\ast$, $H_5^- = -(H_5^+)^\\ast$, $H_{3}^- = -(H_{3}^+)^\\ast$, $H_{3M}^- = -(H_{3M}^+)^\\ast$, $H_{3}^0 = -(H_{3}^0)^\\ast$ and $H_{3}^0 = -(H_{3}^0)^\\ast$. The masses of these physical scalars can easily be obtained from Eq. (\\ref{eq:pot}). Since, the potential preserves the $SU(2)_D$ custodial symmetry, members of the physical scalar multiplets have degenerate masses. These masses are\n\\begin{eqnarray}\\label{eq:multiplet}\n\tm_5^2 &=& 3 ( \\lambda_5 c_M^2 + \\lambda_8 s_M^2 ) v^2\\,,\\nonumber\\\\[0.5em]\n\tm_{3}^2 &=& \\lambda_5 v^2\\,,\\;\\;\\; m_{3M}^2 = \\lambda_5 ( 1 + c_M^2 ) v^2\\,.\n\\end{eqnarray}\nIn general, the $H_1^0$, $H_{1M}^0$ and $H_1^{0 \\prime}$ can mix according to the mass-squared matrix\n\\begin{eqnarray}\\label{eq:m2sing}\n\t&&\\mathcal{M}_{\\text{singlets}}^2 = v^2 \\times\\nonumber\\\\[0.5em]\n\t&&\\left( \n\t\\begin{array}{ccc}\n\t\t8 s_2^2 (\\lambda_1 + \\lambda_4)\t&\t8 s_2 s_{2M} \\lambda_4\t\t\t\t&\t2 \\sqrt{6} s_2 s_M \\lambda_4\\\\[0.5mm]\n\t\t8 s_2 s_{2M} \\lambda_4\t\t\t\t&\t8 s_{2M}^2 (\\lambda_2 + \\lambda_4)\t&\t2 \\sqrt{6} s_{2M} s_M \\lambda_4\\\\[0.5mm]\n\t\t2 \\sqrt{6} s_2 s_M \\lambda_4\t\t&\t2 \\sqrt{6} s_{2M} s_M \\lambda_4\t\t&\t3 s_M^2 (\\lambda_3 + \\lambda_4)\n\t\\end{array}\n\t\\right)\\,.\n\\end{eqnarray}\nHence, after the electroweak symmetry breaking generic $SU(2)_D$ singlet mass eigenstates are given by:\n\\begin{equation}\\label{eq:Hmasseigen}\n\t\\left( \\begin{array}{l}\t\t\\widetilde{H} \\\\ \\widetilde{H}^\\prime \\\\ \\widetilde{H}^{\\prime\\prime} \\end{array}\\right) = \n\t\\left( \\begin{array}{ccc}  \ta_{1,1} \t\t& a_{1,1M} \t\t& a_{1,1'} \\\\\n\t\t\t\t\t\t\t\ta_{1M,1} \t\t& a_{1M,1M} \t\t& a_{1M,1'} \\\\\n\t\t\t\t\t\t\t\ta_{1',1} \t\t& a_{1',1M} \t\t& a_{1',1'}\n\t\\end{array} \\right) ~\n\t\\left( \\begin{array}{l} \t\tH_1^0\\\\ H_{1M}^0\\\\ H_1^{0\\prime}\\end{array}\\right)\\,.\n\\end{equation}\nWe denote the mass eigenstates by $\\widetilde{H}$, $\\widetilde{H}^\\prime$, and $\\widetilde{H}^{\\prime\\prime}$. We adopt a convention of denoting the lightest of the three by $\\widetilde{H} $, the next heavier one by $\\widetilde{H}^\\prime $ and the heaviest state by $\\widetilde{H}^{\\prime\\prime} $. Thus, the $125$-${\\rm GeV }$ candidate in the EW$\\nu_R$  model will be denoted by $\\widetilde{H} $. Here the $3\\times 3$ matrix on the right-hand side lists mixings between $H_1^0$, $H_{1M}^0$ and $H_1^{0\\prime}$. \n\nTo obtain the mass eigenstates, including the 125-{\\rm GeV } candidate in this model, we need to diagonalize the mass matrix in Eq. (\\ref{eq:m2sing}). Due to the complicated dependence of $\\mathcal{M}_{\\text{singlets}}$ on many parameters, it is necessary to diagonalize Eq.~(\\ref{eq:m2sing}) numerically. As a result, the elements of the (numerical) eigenvector matrix $\\{a_{ij}\\}$ depend on:\n\\begin{itemize}\n\t\\item\n\tRatios of VEVs: $s_2$, $s_{2M}$, $s_M$, and\n\t\\item\n\tScalar couplings: $\\lambda_1$, $\\lambda_2$, $\\lambda_3$, $\\lambda_4$.\n\\end{itemize}\nIt should be noted that in the limit $\\lambda_4 \\rightarrow 0$ the off-diagonal elements in $\\mathcal{M}_{\\text{singlets}}$ vanish. \n\nSuch mixed physical scalars have interesting implications for decay properties of the 125-{\\rm GeV } as well as the two heavier mass eigenstates. This is where we will focus on in the next two sections.\n\n\\begin{flushleft}\n{\\bf A note on the pseudo Nambu-Goldstone bosons in the EW$\\nu_R$  model:}\n\\end{flushleft}\n\n\nLast but not least is the important question concerning whether or not Nambu-Goldstone (NG) bosons arise in the spontaneous breakdown of the aforementioned global symmetry $U(1)_{SM} \\times U(1)_{MF}$. This issue has been discussed in \\cite{ChanoGold} as well as in the original EW$\\nu_R$  minimal model \\cite{pqnur}. The conclusion was the fact that there is a term in the potential which is required for proper vacuum alignment and which breaks {\\em explicitly} the global symmetry ($U(1)_{MF}$ for the minimal model) and the would-be NG bosons acquire a mass proportional to the corresponding couplings. For the sake of clarity and also because we now also have $U(1)_{SM}$ in addition to $U(1)_{MF}$, we will briefly review the essential points discussed in \\cite{ChanoGold} and \\cite{pqnur}.\n\nLet us first look at the minimal model of \\cite{pqnur} where $\\Phi_{2M}$ is absent. Eq.~({\\ref{pot0}) without $\\Phi_{2M}$ is of the same form as that used in \\cite{ChanoGold} where it has been shown that the term containing $\\lambda_5$ is needed for proper vacuum alignment of $\\Phi_{2}$ and $\\chi$ so that $SU(2) \\times U(1)_Y \\rightarrow U(1)_{em}$.  In this minimal model, only the mirror fermions, $\\tilde{\\chi}$ and $\\phi_S$ transform non-trivially under $U(1)_{MF}$ as shown in Eq.~(\\ref{qmtrans}) and Eq.~(\\ref{chitrans}) with all other fields being singlets under $U(1)_{MF}$ including the SM fermions, $\\Phi_{2}$ and $\\xi^{\\pm,0}$. The term proportional to $\\lambda_5$ contains mixings such as $\\xi^{0} \\chi^{0}$, $\\xi^{+} \\chi^{-}$ and breaks explicitly the $U(1)_{MF}$ symmetry. The would-be triplet of NG bosons, $H_{3}^{\\pm,0}$, acquire a mass proportional to $\\lambda_5$ as shown in Eq.~(\\ref{eq:multiplet}). The proper vacuum alignment ($\\lambda_5 \\neq 0$) prevents the appearance of NG bosons in the minimal model! \n\nThe above argument can be generalized to the extended model with $U(1)_{SM}\\,\\times U(1)_{MF}$. The various fields transform under $U(1)_{SM}\\,\\times U(1)_{MF}$ as shown in Eq.~(\\ref{smtrans}), Eq.~(\\ref{qmtrans}) and Eq.~(\\ref{chitrans}). Again, one notices that proper vacuum alignment requires the term proportional to $\\lambda_5$ to be present as discussed above. From Eq.~(\\ref{eq:pot}), one can again see that this term contains a mixing between $\\xi$ and $\\tilde{\\chi}$ and now also a mixing between $\\Phi_{2M}$ and $\\Phi_{2}$. As a result, $U(1)_{SM}\\,\\times U(1)_{MF}$ is broken explicitly by the term proportional to $\\lambda_5$. Consequently, the would-be NG bosons, $H_{3}^{\\pm,0}$ and $H_{3M}^{\\pm,0}$, now acquire a mass proportional to $\\lambda_5$ as shown in Eq.~(\\ref{eq:multiplet}). The proper vacuum alignment ($\\lambda_5 \\neq 0$) prevents the appearance of NG bosons in the extended model as well! \n\n\n \n\n\n\\section{125-GeV candidate in EW$\\nu_R$  Model}\n\n\nMeasured properties of the 125-{\\rm GeV } scalar particle that was discovered at the LHC so far tend to be close to the properties of SM Higgs boson. Hence, in every model of BSM Physics it is imperative to (i) have at least one Higgs particle with mass of about 125 {\\rm GeV } having SM-like decay signal strengths, and (ii) study the implications of these properties in the `allowed' parameter space of the model (e.g. allowed masses of any BSM particles in the model, etc.). To check the viability of a model or to search for the model experimentally, decay properties of the 125-{\\rm GeV } Higgs boson candidate in the model must be studied.\n\nWe denote the $125$-${\\rm GeV }$ candidate in the EW$\\nu_R$  model by $\\widetilde{H} $. From Eq.~(\\ref{eq:Hmasseigen}) we see that it is a mixture of $H_1^0$, $H_{1M}^0$ and $H_1^{0\\prime}$. Recall that $H_1^0$ comes from the SM-like scalar doublet $\\Phi_2$, $H_{1M}^0$ comes from doublet $\\Phi_{2M}$ and $H_1^{0\\prime}$ from triplet $\\chi$. \n\nBecause the measured decay signal strengths of the 125-{\\rm GeV } Higgs boson are close to SM predictions, intuitively one might expect that $H_1^0$ has to be the dominant component of $\\widetilde{H} $. But our investigation shows that the 125-{\\rm GeV } $\\widetilde{H} $ can have SM-like decay signal strengths, even if $H_1^0$ is a sub-dominant component in it. Hence the dual-like nature of 125-{\\rm GeV } Higgs boson from perspective of the EW$\\nu_R$  model. In this section we will discuss this dual-like nature and its implications.\n\nIn the first subsection we will explain the methodology used in the analysis; the next subsection presents the analysis and results for $\\widetilde{H} \\sim H_1^0$ case. In the third subsection, we present a more interesting case, where, although $H_1^0$ is a subdominant component of $125$-${\\rm GeV }$ $\\widetilde{H} $, it still satisfies the experimental constraints on the signal strengths.\n\n\\begin{flushleft}\n{\\bf A note on CP of the 125-GeV Higgs in the EW$\\nu_R$  model:}\n\\end{flushleft}\nAs seen in Sec.~\\ref{sec:extewnur}, the EW$\\nu_R$  model has 6 neutral physical scalars, of which 3 are CP-even states ($H^0_1$, $H^0_{1M}$, $H^\\prime_1$) and 3 are CP-odd states ($H^0_3$, $H^0_{3M}$, $H^0_5$). Their couplings to fermions and gauge bosons are listed in Tables~\\ref{table:h_ferm} -~\\ref{table:feyn22n} in Appendix~\\ref{sec:extewnurcoup}. Among these $H^0_5$ does not couple to charged fermions. \n\nIt can be seen from Table~\\ref{table:h_ferm} that decay widths of $H^0_3\/H^0_{3M} ~\\rightarrow f ~\\bar{f}$ can be close to the SM predictions for some combinations of the BSM parameters in the couplings. But as mentioned in Section~\\ref{sec:minimalH}, $H^0_3, ~H^0_{3M}$ are disfavored as $125$-${\\rm GeV }$ candidates as compared to the CP-even hypothesis \\cite{LHC2}.\n\nHence, in this paper while considering 125-{\\rm GeV } candidate in the EW$\\nu_R$  model, we proceed with the hypothesis that this candidate is a CP-even eigenstate \\footnote{The possibility that the 125-{\\rm GeV } Higgs boson is a linear combination of CP-even and CP-odd state has not been thoroughly checked experimentally yet. The spin and parity of the 125-{\\rm GeV } scalar are yet to be measured at CMS and ATLAS. Thus, in this paper, we will stick to CP-eigenstate hypothesis based on the likelihood analysis}. \n\nOut of the 3 CP-even Higgs bosons, only $H^0_1$ can have decay widths to SM fermions similar to the SM predictions. Therefore, one might expect that in the EW$\\nu_R$  model $H^0_1$ is the candidate for 125-{\\rm GeV } Higgs boson. However, in the absence of explicit decay widths and based on current available signal strengths for various decay channels, we will show that the 125-{\\rm GeV } Higgs boson can be very different from the standard expectation. It is in the spirit of our analysis that we may coin the term ``Dr.~Jekyll\" to the Standard Model expectation (a mild impostor) and the term ``Mr.~Hyde\" to the definite ``impostor\" (which mainly comes from the scalar triplet) scenario presented in the paper. \n\n\\subsection{Methodology for comparing the EW$\\nu_R$  model predictions with data}\\label{sec:data}\n\nFor any given decay channel of a Higgs, CMS and ATLAS experiments at the LHC measure the total cross section of the decay process. \nThe cross section of any decay channel of the Higgs boson that is measured at the LHC is given by\n\\begin{equation}\n\t\\sigma(H \\text{-decay}) = \\sigma(H \\text{-production}) \\times BR(H \\text{-decay})\\,,\n\\end{equation}\nwhere $\\sigma(H \\text{-production})$ is the production cross section of $H$ and $BR(H \\text{-decay})$ is the Branching Ratio of the decay channel of $H$ that is under consideration.\n\\begin{equation}\n\tBR(H \\text{-decay}) = \\frac{\\Gamma(H \\text{-decay})}{\\Gamma_H}\\,,\n\\end{equation}\nwhere $\\Gamma(H \\text{-decay})$ is the partial width of the $H$-decay channel, and $\\Gamma_H$ is the total width of $H$. To compare the data with the Standard Model predictions, the ratio of the measured signal strength to its SM-predicted value is presented, denoted by $\\mu$. \n\\begin{equation}\\label{eq:mudef}\n\t\\mu(H \\text{-decay}) = \\frac{\\sigma(H \\text{-decay})}{\\sigma_{SM}(H \\text{-decay})}\\,,\n\\end{equation}\n$\\sigma(H \\text{-decay})$ being measured experimentally or predicted by a model.\n\nTherefore, to compare the EW$\\nu_R$-predicted decay signal strengths with the data, we investigate the agreement between the ratio of EW$\\nu_R$ prediction with the SM prediction $\\mu_{\\text{EW$\\nu_R$ }} = \\sigma_{\\text{EW$\\nu_R$ }}(H \\text{-decay}) \/ \\sigma_{SM}(H \\text{-decay})$, to the ratio of measured decay cross section with the SM-prediction $\\mu_{data} = \\sigma_{data}(H \\text{-decay}) \/ \\sigma_{SM}(H \\text{-decay})$. $\\sigma_{H \\text{-decay}}$ in the EW$\\nu_R$  model with the predictions SM for that decay channel. \n\nHence, we need to calculate \n\\begin{itemize}\n\t\\item\n\tpartial decay widths for these channels, \n\t\\item\n\tthe total width of $\\widetilde{H} $ and\n\t\\item\n\tthe production cross-section of $gg\\rightarrow\\widetilde{H} $.\n\\end{itemize}\n\nThe analysis is done in the following steps:\n\\begin{enumerate}\n\t\\item\n\tIdentify all the decay channels that contribute significantly to the total width of the 125-{\\rm GeV } $\\widetilde{H} $\n\tIdentify the variables on which the aforementioned three quantities depend.\n\t\\item\n\tIdentify the limits on the variables.\n\t\\item\n\tSelect a set of values for the variables within their respective limits.\n\t\\item\n\tCalculate the signal strengths $\\mu$ in various channels for the $125$-${\\rm GeV }$ $\\widetilde{H} $, and compare them with the measured values from CMS.\n\\end{enumerate}\n\n\\subsubsection{Decay channels under consideration}\n\nFor this analysis we calculate signal strengths $\\mu$ for decay channels $\\widetilde{H} \\rightarrow ZZ,~W^+W^-,~\\gamma\\gamma,~b\\bar{b},~\\tau\\bar{\\tau}$. We calculate the production cross-section and partial widths of various decay channels as explained in Appendix~\\ref{sec:hdecay}. The total width of the 125-{\\rm GeV } $\\widetilde{H} $ is calculated by adding individual partial decay widths:\n\\begin{eqnarray}\\label{eq:Hwidth}\n\\Gamma_{\\widetilde{H} } = &&~\\Gamma_{\\widetilde{H} \\rightarrow b\\bar{b}} ~+ ~\\Gamma_{\\widetilde{H} \\rightarrow \\tau\\bar{\\tau}} ~+ ~\\Gamma_{\\widetilde{H} \\rightarrow c\\bar{c}} ~+ ~\\Gamma_{\\widetilde{H} \\rightarrow W^+ W^-}\\nonumber\\\\[0.5em]\n&& + ~\\Gamma_{\\widetilde{H} \\rightarrow ZZ} ~+ ~\\Gamma_{\\widetilde{H} \\rightarrow gg} ~+ ~\\Gamma_{\\widetilde{H} \\rightarrow \\gamma \\gamma}\\,.\n\\end{eqnarray}\nAmong all the partial widths considered above, $\\Gamma_{\\widetilde{H} \\rightarrow b\\bar{b}}$ and $\\Gamma_{\\widetilde{H} \\rightarrow W^+ W^-}$ are the most dominant for the SM-Higgs. Because of the constraint $m_{f^M} > 100~{\\rm GeV }$, the decay channel $\\widetilde{H} \\rightarrow f^M \\bar{f}^M$ does not occur at the leading order, when $f^M$ is on-shell.\n\nIn what follows we identify the relevant variables in the analysis and estimate their allowed ranges.\n\n\n\\subsubsection{Lower limit on the masses of charged mirror fermions}\n\nThe lower limit of $102 ~{\\rm GeV }$ on the masses of charged mirror leptons and mirror quarks is imposed based on the results of search for sequential heavy charged leptons and quarks at LEP3 (refer `Heavy Charged Leptons' and `Heavy Quarks' sections in \\cite{pdg} and references therein). Strictly speaking these constraints apply only to sequential heavy fermions, such as $L' \\rightarrow \\tau Z \\rightarrow \\tau l\\bar{l}, \\tau q \\bar{q}$ or $Q^\\prime\\rightarrow bZ\\rightarrow b~ l\\bar{l}, b~ q \\bar{q}$ or $Q^\\prime\\rightarrow bW^+\\rightarrow b~ l\\bar{\\nu}, ~b~q\\bar{q^\\prime}$ etc.\n\nHowever, charged mirror fermions in the EW$\\nu_R$  model couple to the SM fermions in an altogether different way, through the scalar singlet $\\phi_S$ \\cite{pqnur,pqnur2}: $q^M\\rightarrow q\\phi_S$, $l^M\\rightarrow l\\phi_S$. This $\\phi_S$ would appear as missing energy in the detector. Thus, the signature of final states   in charged mirror fermion decay would involve a lepton + missing $E_T$ or a jet + missing $E_T$. Moreover, at CMS or ATLAS these decays could occur outside the beam-pipe and inside the silicon vertex detector \\cite{pqnur,pqnur2}. Therefore, the constraints from the aforementioned searches do not directly apply to charged mirror fermions. We still impose these constraints on charged mirror fermions, arguing that if these mirror fermions were lighter than $\\sim 100 ~{\\rm GeV }$, they would have been discovered at 200 {\\rm GeV } LEP3 \\cite{pdg}.\n\n\\subsubsection{Limits on VEVs, scalar and Yukawa couplings}\n\nWe consider only the cases where the scalar couplings and Yukawa couplings of mirror fermions are perturbative. The perturbative constraint on scalar and Yukawa couplings are $\\lambda_i\/4\\pi \\lesssim \\mathcal{O}(1)$ and $\\alpha_{f^M} = g_{MF}^2\/4\\pi \\lesssim \\mathcal{O}(1)$ respectively. For numerical analysis we limit ourselves to cases, where $\\lambda_i \/4\\pi\\leq 1.3$ and $\\alpha_{f^M} \\leq 1.5$.\n\nAs discussed towards the end of Sec.~\\ref{sec:extewnur}, the $SU(2)_D$ singlet mass eigenstates depend on $s_2$, $s_{2M}$ and $s_M$. Therefore, they also depend on the vacuum expectation values (VEVs) of the real parts of $\\Phi_2$, $\\Phi_{2M}$ and $\\chi$. While investigating different numerical forms of $\\{a_{ij}\\}$, one needs to vary the VEV's. Hence, it is necessary to estimate the limits on these VEVs before analyzing the 125-{\\rm GeV } candidate in detail. \n\nRecall that the charged SM fermions, the charged mirror fermions and the right handed neutrinos get their masses due to $v_2$, $v_{2M}$, and $v_M$ respectively. Various constraints on these masses constrain the ranges of the VEVs. \n\nIf the pole mass of top quark ($173.5~{\\rm GeV }$), the heaviest SM fermion, is {\\em perturbative} and comes from $v_2$, then $v_2 \\gtrsim 69 ~{\\rm GeV }$ (because $g^2_{top} \\leq 4 \\pi$). We set the lower bound on the masses of all the charged mirror fermions at $102~{\\rm GeV }$, which is the LEP3 \\cite{pdg} bound on the heavy BSM quarks and BSM charged leptons. Hence, considering a constraint of $g^2_{MF}\/4\\pi \\leq 1.5$ on the Yukawa couplings of all the charged mirror fermions, $v_{2M} \\gtrsim 27 ~{\\rm GeV }$, implying $v_M \\lesssim 80 ~{\\rm GeV }$. Thus, for perturbative determination of $M_R$ requires $M_R \\lesssim 283 ~{\\rm GeV }$. We also know that $M_R \\geq M_Z \/ 2\\; \\approx\\; 45.5 ~{\\rm GeV }$ \\cite{pqnur}, and, hence, $v_M \\gtrsim 13 ~{\\rm GeV }$. This implies that $v_2 \\lesssim 241 ~{\\rm GeV }$ and $v_{2M} \\lesssim 233 ~{\\rm GeV }$. This limit on $v_{2M}$ along with the perturbative limit on $g_{MF}$ sets an upper limit on the masses of the mirror fermions: $m_{MF} \\lesssim 715 ~{\\rm GeV }$. The allowed ranges for VEVs and for parameters defined in Eq~(\\ref{eq:sdefs}) are summarized in the table below.\n\\begin{table}[!htb]\n\t\\renewcommand{\\arraystretch}{2}\n\t\\centering\n\t\\caption{\\label{tab:v_ranges}Allowed ranges of VEVs and parameters defined in Eq. (\\ref{eq:sdefs}). All values are given in {\\rm GeV }.}\n\t\\begin{tabular}{l|l}\n\t\\hline\n\t$69 ~\\lesssim ~v_2 ~~\\lesssim ~241$ & $~0.28 ~\\lesssim ~s_2 ~~\\lesssim ~0.98$\\\\\n\t$33 ~\\lesssim ~v_{2M} \\lesssim ~233~~$ & $~0.13 ~\\lesssim ~s_{2M} \\lesssim ~0.95$\\\\\n\t$13 ~\\lesssim ~v_M ~\\lesssim ~~83$ & $~0.15 ~\\lesssim ~s_M ~\\lesssim ~0.95$\\\\\n\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{Common predictions for multiple decay channels}\n\nIn the EW$\\nu_R$  model, predictions for the signal strengths of $\\widetilde{H} \\rightarrow W^+W^-$ and $\\widetilde{H} \\rightarrow ZZ$ are equal. Similarly, predictions for the signal strengths of $\\widetilde{H} \\rightarrow b\\bar{b}$ are equal to those for $\\widetilde{H} \\rightarrow\\tau\\bar{\\tau}$. This is expected, since as seen in Appendix~\\ref{sec:hdecay},\n\\begin{eqnarray}\\label{eq:ww_zz}\n\t\\frac{\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow W^+W^-)}{\\Gamma^{SM}(H_{SM}^0\\rightarrow W^+W^-)} &=& \\frac{\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow ZZ)}{\\Gamma^{SM}(H_{SM}^0\\rightarrow ZZ)}\\,,\\nonumber\\\\[0.5em]\n\t\\frac{\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow b\\bar{b})}{\\Gamma^{SM}(H_{SM}^0\\rightarrow b\\bar{b})} &=& \\frac{\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow \\tau\\bar{\\tau})}{\\Gamma^{SM}(H_{SM}^0\\rightarrow \\tau\\bar{\\tau})}\\,.\n\\end{eqnarray}\n\nKeeping all this in mind, in the next two subsections we analyze in detail the decay properties of the 125-{\\rm GeV } candidate in the EW$\\nu_R$  model.\n\n\\subsubsection{Numerical Analysis}\\label{sec:numerical}\n\nFor this analysis a C++ code was written, also using some functionality of ROOT \\cite{ROOT}. We investigated this case in following steps:\n\\begin{itemize}\n\t\\item\n\tWe generated random combinations of $s_2,~s_{2M},~s_M,~\\lambda_1,~\\lambda_2,~\\lambda_3$ and $\\lambda_4$, using TRandom3 random number generator in ROOT. These parameters were varied over the following ranges:\n\\begin{eqnarray}\\label{eq:hyde125para}\n\t-4 \\pi ~~\\leq ~~\\lambda_1, ~&\\lambda_2, ~\\lambda_3&, ~\\lambda_4 ~~\\leq ~~4 \\pi\\,,\\nonumber\\\\[0.5em]\n\t0.28 ~~\\leq ~~&s_2& ~~\\leq ~~0.98\\,,\\nonumber\\\\[0.5em]\n\t0.13 ~~\\leq ~~&s_{2M}& ~~\\leq ~~0.95\\,,\\nonumber\\\\[0.5em]\n\t0.15 ~~\\leq ~~&s_M& ~~\\leq ~~0.95\\,.\n\\end{eqnarray}\nThe limits $|\\lambda|\/4\\pi \\lesssim 1$ are set so that $\\lambda$'s are perturbative. Limits on $s_2$, $s_{2M}$, $s_M$ are based on Table~\\ref{tab:v_ranges}.\n\t\\item\n\tWe numerically diagonalized the singlet mass matrix in Eq.~(\\ref{eq:m2sing}) formed by every combination of the parameters to find the mass eigenvalues and corresponding eigenvector matrix (mixing matrix) in Eq.~(\\ref{eq:Hmasseigen}). Only those combinations of parameters, which yielded the lightest mass eigenvalue in the range $125.7\\pm 1.0~{\\rm GeV }$, were saved. 4 million such parameter combinations were found. \n\t\\item\n\tFor all the saved combinations we calculated various signal strengths for each of these combinations. The gluon-gluon fusion channel was considered to calculate the predicted production cross section of the $\\widetilde{H} $. The partial decay widths were calculated according to Appendix~\\ref{sec:hdecay}, and the total width was calculated using Eq.~(\\ref{eq:Hwidth}).\n\t\\item\n\tIn addition to the parameters in Eq~(\\ref{eq:hyde125para}), following parameters are required to calculate the partial widths of $\\widetilde{H}\\rightarrow \\gamma\\gamma$ and $\\widetilde{H}\\rightarrow g g$, and the cross section of $g g \\rightarrow \\widetilde{H}$:\n\\begin{eqnarray}\\label{eq:hydemupara}\n\t0 ~~&\\leq& ~~\\lambda_5 ~~\\leq ~~15\\,, \\text{ varied with }\\Delta\\lambda_5 \\sim 1.07\\,,\\nonumber\\\\[0.5em]\n\t\\lambda_8 &=& -1\\,, ~m_{H_3^+} = ~m_{H_{3M}^+} = 500 ~{\\rm GeV }\\,,\\nonumber\\\\[0.5em]\n\tm_{H_5^+} &=& 200 ~{\\rm GeV }\\,, ~m_{H_5^{++}} = 320 ~{\\rm GeV }\\,, ~m_{q_3^M} = 120 ~{\\rm GeV }\\,,\\nonumber\\\\[0.5em]\n\tm_{q_1^M} &=& m_{q_2^M} = m_{l^M} = 102 ~{\\rm GeV }\\,.\n\\end{eqnarray}\n\t\\item\n\tWe checked if the signal strengths $\\mu$'s of the 125-{\\rm GeV } $\\widetilde{H} $ in various decay channels are within the $1\\sigma$ constraints on the signal strengths, as measured by CMS experiment. We did not impose constraints from both the CMS and ATLAS, because for some of the decay channels considered here, the signal strength measurements from CMS and ATLAS do not agree with each other within the $1\\sigma$ constraints. Also, CMS and ATLAS have not published their combined measurements from the recent analyses. We therefore chose to check agreement with the CMS measurements. \n\t\n\tDepending on their $1\\sigma$ constraints, certain combinations out of the 4 million would agree with either only with CMS or with ATLAS results. Thus, imposing the constraints from ATLAS would discard some of the combinations that the CMS constraints would allow and vice versa. However, this would not change any of the conclusions of the paper.\n\t\\item\n\tWe found 1501 out of 4 million combinations of the parameters that satisfy $1\\sigma$ constraints from CMS on the 125-{\\rm GeV } Higgs signal strengths in $WW$, $ZZ$, $b\\bar{b}$, $\\tau\\bar{\\tau}$ and $\\gamma\\gamma$ decay channels. Table~\\ref{tab:hyde} lists 16 examples out of 1501 cases, with the masses of $\\widetilde{H} $, $\\widetilde{H}^\\prime $, $\\widetilde{H}^{\\prime\\prime} $, their mixing-matrix elements, and the signal strengths of the 125-{\\rm GeV } $\\widetilde{H} $ for various decay channels. \n\t\\item\n\tIn the code, there was no constraint imposed as to what is to be the dominant component in $\\widetilde{H} $. Interestingly, hardly any combinations among the 4 million had $H_1^0$ as a dominant component in the 125-{\\rm GeV } $\\widetilde{H} $. This means that either\n\t\\begin{enumerate}\n\t\t\\item\n\t\tat the mass of about 125-{\\rm GeV }, 4 million combinations do not yield enough resolution in the parameter space so as to find the $\\widetilde{H} \\sim H_1^0$ case, OR\n\t\t\\item\n\t\tthe $\\widetilde{H} \\sim H_1^0$ case cannot be found with the imposed limits on the parameters, and it requires at least some of these parameters to have values outside of these limits. \n\t\\end{enumerate}\n\t\\item\n\tThus, this scan of the parameter space only yielded {\\em Mr.~Hyde} cases, where the SM-like $H_1^0$ is a subdominant component in the 125-{\\rm GeV } $\\widetilde{H} $. Implications of these cases will be further discussed in section \\ref{sec:hyde}.\n\t\\item\n\tOn the other hand, to find the combinations of the parameters for which the 125-{\\rm GeV } $\\widetilde{H} $ has a dominant SM-like $H_1^0$ component, and which also satisfy the CMS constraints on the signal strengths, we had to choose some of the scalar couplings to have values outside $[-4\\pi,~4\\pi]$. These {\\em Dr.~Jekyll} cases thus require some interactions within the scalar sector to be in the strong-coupling regime. In the next subsection we discuss this scenario in detail.\n\n\\end{itemize}\n\n\n\\subsection{$\\widetilde{H} $ as 125-GeV Higgs candidate with a dominant SM-like component}\\label{sec:jekyll}\n\n\nWe illustrate the step-by-step process which we followed to analyze this case. \n\\begin{itemize}\n\t\\item\n\tA Mathematica code was written to numerically diagonalize the custodial-singlet mass matrix in Eq.~(\\ref{eq:m2sing}) and obtain its mass eigenvalues and eigenvector matrix i.e. the mixing matrix in Eq.~(\\ref{eq:Hmasseigen}). \n\t\\item\n\tIn this code, the values of $s_2 = 0.92, ~s_{2M} = 0.16$ (and thus, $s_M \\approx 0.36$) were fixed. The analysis was performed for different $s_2$ values, but, for $\\widetilde{H}  \\sim H_1^0$, only the cases with $s_2 \\gtrsim 0.9$ were found to satisfy the experimental constraints on the signal strengths of the 125-{\\rm GeV } Higgs at LHC.\n\t\\item\n\tAfter fixing $s_2$ and $s_{2M}$, the scalar couplings $\\lambda_1$, $\\lambda_2$, $\\lambda_3$ and $\\lambda_4$ were manually varied so that $|\\lambda|\/4\\pi \\leq 1.3$, in order to find the combinations of $\\lambda$'s that yield the lowest eigenvalue of the mass matrix to be $125.7\\pm 1.0 ~{\\rm GeV }$ and the corresponding eigenstate to have dominant $H_1^0$ component.\n\t\\item\n\tRecall (refer to Eq.~(\\ref{eq:pot})) that $\\lambda_1$, $\\lambda_2$ and $\\lambda_3$ are the self-couplings of $\\Phi_2$, $\\Phi_{2M}$ and $\\chi$ respectively. $\\lambda_5$ is the measure of cross couplings of $\\Phi_2$, $\\Phi_{2M}$ and $\\chi$.\n\t\\item\n\tAs stated in section \\ref{sec:numerical}, we found combinations of the parameters which satisfy the CMS constraints on the signal strengths, when $\\lambda_2, ~\\lambda_5 > 4\\pi$. $|\\lambda_1|, ~|\\lambda_4|, ~|\\lambda_8|$ are still $\\leq 4\\pi$, while $\\lambda_3 \\approx 15$. For illustrative purpose, we show below two of many cases which satisfy the CMS constraints.\n\t\\item\nThe calculation of the partial width of the $\\widetilde{H} ~\\rightarrow~\\gamma\\gamma$ channel necessitates fixing the values or ranges for the remaining parameters. In the example cases shown below we fix other parameters as follows:\n\\begin{itemize}\n\t\\item\n\t$m_{H_3^+} = 600 ~{\\rm GeV }, ~m_{H_{3M}^+} = 700 ~{\\rm GeV }$,\n\t\\item\n\tmasses of all three charged mirror leptons $m_{l^M} = 102 ~{\\rm GeV }$,\n\t\\item\n\tmass of lightest two generations of mirror quarks $m_{q_1^M} = m_{q_2^M} = 102 ~{\\rm GeV }$,\n\t\\item\n\tfor the purpose of partial widths of $\\widetilde{H} $-decays in scenarios above, we also fix mass of the third mirror quark generation at $m_{q^M} = 120 ~{\\rm GeV }$. This mass will be varied to analyze constraints on $\\widetilde{H}  \\sim H_{1M}^0$.\n\\end{itemize}\n\t\\item\nThe values of $m_{H_3^+}$ and $m_{H_{3M}^+}$ are chosen so as to have largest allowed ranges for $m_{H_5^+}$ and $m_{H_5^{++}}$. We vary the latter two over the range $\\sim 400 ~- ~730 ~{\\rm GeV }$ for Example 1 and 2. This variation does not affect much the predictions for the signal strengths of the $\\widetilde{H} $ decays to $W^+W^-$, $ZZ$ and $f\\bar{f}$, but only changes that for $\\widetilde{H}  \\rightarrow \\gamma\\gamma$. $m_{H_5^+}$ and $m_{H_5^{++}}$ vary in correlation when the CMS constraints on the signal strength of the diphoton decay channel are imposed. For the numerical calculation of other signal strengths in the following two examples we chose on of these correlated pairs of the two masses.\n\n\t\\item\nExample 1: $\\lambda_1 = -0.077, ~\\lambda_2 = 14.06, ~\\lambda_3 = 15.4, ~\\lambda_4 = 0.1175, ~\\lambda_5 = 15, ~\\lambda_8 = -1$ and $m_{H_5^+} = 500~{\\rm GeV }, ~m_{H_5^{++}} = 540~{\\rm GeV }$. \nFixing these along with $s_2 = 0.92, ~s_{2M} = 0.16,~s_M \\approx 0.36$, fully determines the singlet mass matrix, and hence the mixing matrix, given by:\n\\begin{equation}\\label{eq:jekyll1}\n\t\\left( \\begin{array}{l}\t\t\\widetilde{H} \\\\ \\widetilde{H}^\\prime \\\\ \\widetilde{H}^{\\prime\\prime} \\end{array}\\right) = \n\t\\left( \\begin{array}{ccc}  \t0.998 \t\t& -0.0518 \t\t& -0.0329 \\\\\n\t\t\t\t\t\t\t\t0.0514 \t\t& 0.999 \t\t& -0.0140 \\\\\n\t\t\t\t\t\t\t\t0.0336 \t\t& 0.0123 \t\t& 0.999\n\t\\end{array} \\right) \n\t\\left( \\begin{array}{l} \t\tH_1^0\\\\ H_{1M}^0\\\\ H_1^{0\\prime}\\end{array}\\right)\\,,\n\\end{equation}\nwith $\\widetilde{H}  \\sim H_1^0$, $\\widetilde{H}^\\prime  \\sim H_{1M}^0$, $\\widetilde{H}^{\\prime\\prime}  \\sim H_1^{0\\prime}$ and $m_{\\widetilde{H} } = 125.7 ~{\\rm GeV }$, $m_{\\widetilde{H}^\\prime } = 420 ~{\\rm GeV }$, $m_{\\widetilde{H}^{\\prime\\prime} } = 601 ~{\\rm GeV }$. $a_{1,1M}$ - the (1,2) element of the $3\\times 3$ matrix can actually vary between (-0.0515, -0.05295) and still satisfy CMS constraints. \\\\%We will refer to this as `Example 1'.\nAnother example is Example 2: $\\lambda_1 = 0.0329, ~\\lambda_2 = 14.2, ~\\lambda_3 = 15.4, ~\\lambda_4 = 0.0056, ~\\lambda_5 = 15, ~\\lambda_8 = -1$, and $m_{H_5^+} = 590~{\\rm GeV }, ~m_{H_5^{++}} = 600~{\\rm GeV }$,\n\\begin{flalign}\\label{eq:jekyll2}\n\t&\\left( \\begin{array}{l}\t\t\\widetilde{H} \\\\ \\widetilde{H}^\\prime \\\\ \\widetilde{H}^{\\prime\\prime} \\end{array}\\right) =&\\nonumber\\\\[0.5em]\n\t&\\left( \\begin{array}{ccc}  \t0.99999...\t& -2.49\\times10^{-3}\t\t& -1.60\\times10^{-3} \\\\\n\t\t\t\t\t\t\t\t2.49\\times10^{-3}\t& 0.99999...\t\t\t& -5.30\\times10^{-4} \\\\\n\t\t\t\t\t\t\t\t1.60\\times10^{-3}\t& 5.26\\times10^{-4}\t& 0.99999..\n\t\\end{array} \\right) \n\t\\left( \\begin{array}{l} \t\tH_1^0\\\\ H_{1M}^0\\\\ H_1^{0\\prime}\\end{array}\\right),&\n\\end{flalign}\nwith $\\widetilde{H}  \\sim H_1^0$, $\\widetilde{H}^\\prime  \\sim H_{1M}^0$, $\\widetilde{H}^{\\prime\\prime}  \\sim H_1^{0\\prime}$ and $m_{\\widetilde{H} } = 125.7 ~{\\rm GeV }$, $m_{\\widetilde{H}^\\prime } = 420 ~{\\rm GeV }$, $m_{\\widetilde{H}^{\\prime\\prime} } = 599 ~{\\rm GeV }$. The allowed range for $a_{1,1M}$ - the (1,2) element of the $3\\times 3$ matrix is $(-1.20, -3.40)\\times10^{-3}$.\n\n\\begin{table*}[!htb]\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\centering\n\t\\caption{\\label{tab:sigXBRjekyll}Partial width of $H\\rightarrow gg$ as the measure of the production cross section, partial widths and branching ratios for various channels in SM (for $m_{H_{SM}} = 125.7~{\\rm GeV }$ with total width = $4.17\\times 10^{-3}$ {\\rm GeV }, and the EW$\\nu_R$  model for {\\em Dr.~Jekyll} example 2 scenario: $a_{1,1M} = -0.0025$, where $m_{\\widetilde{H} } = 125.7~{\\rm GeV }$, total width = $4.45\\times 10^{-3}$ {\\rm GeV } and $\\widetilde{H}  \\sim H_1^0$. All the partial widths are given in {\\rm GeV }.}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\t\\hline\n    ~ & \\multicolumn{3}{c|}{SM}   & \\multicolumn{3}{c|}{EW$\\nu_R$ }  &  \\multirow{2}{*}{$\\mu = \\dfrac{(\\sigma_{Hgg} \\times BR)_{\\text{SM}}}{(\\sigma_{\\widetilde{H}  gg} \\times BR)_{\\text{EW$\\nu_R$ }}}$}      \\\\\n     \\cline{2-7}\n    ~ & $\\Gamma_{H\\rightarrow gg}$      & Partial Width   & BR       & $\\Gamma_{\\widetilde{H} \\rightarrow gg}$      & Partial Width   & BR       & ~         \\\\\n    ~ & $\\propto \\sigma_{gg\\rightarrow H}$ & ~ & ~ & $\\propto \\sigma_{gg\\rightarrow H}$ & ~ & ~ & ~ \\\\\n     \\hline\n\t$\\widetilde{H} \\rightarrow W^+W^-$   \t& 3.55E-04 \t\t& 9.42E-04 & 2.26E-01 & 3.46E-04 & 7.63E-04 & 1.72E-01 & 0.74 \\\\\n\t$\\widetilde{H} \\rightarrow ZZ$   \t& 3.55E-04 \t\t& 1.17E-04 & 2.81E-02 & 3.46E-04 & 9.49E-05 & 2.13E-02 & 0.74 \\\\\n\t$\\widetilde{H} \\rightarrow b\\bar{b}$   \t& 3.55E-04 \t\t& 2.36E-03 & 5.66E-01 & 3.46E-04 & 2.79E-03 & 6.26E-01 & 1.07  \\\\\n\t$\\widetilde{H} \\rightarrow \\tau\\bar{\\tau}$\t& 3.55E-04\t& 2.59E-04 & 6.21E-02 & 3.46E-04 & 3.06E-04 & 6.87E-02 & 1.07  \\\\\n\t$\\widetilde{H} \\rightarrow \\gamma\\gamma$\t\t& 3.55E-04\t& 9.51E-06 & 2.28E-03 & 3.46E-04 & 1.26E-05 & 2.82E-03 & 1.21  \\\\\n\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\t\\item\nNotice that, although Examples 1 and 2 have very different values for the off-diagonal elements in $\\{a_{ij}\\}$, they yield comparable numerical signal strength predictions, the reason being principally that in both the cases $\\widetilde{H} \\sim H_1^0$. We can also find other cases having intermediate values for the off-diagonal elements yielding comparable signal strengths.\n\n\t\\item\n\tTable \\ref{tab:sigXBRjekyll} shows cross section of 125-{\\rm GeV } $H \\rightarrow gg$ (as a measure of production cross section), partial widths and branching ratios in the SM and the EW$\\nu_R$  model, for example 2. We see that these partial widths are not very different from those in SM. This is expected as, in this case, the couplings of $H_1^0$ with the SM gauge bosons and fermions are also close to those of the SM Higgs.\n\t\\item\nThe partial widths and the signal strengths for $W^+W^-$ and $ZZ$ decay channels are smaller, whereas those for $b\\bar{b}$, $\\tau\\bar{\\tau}$ and $\\gamma\\gamma$ decay channels are larger, than the corresponding values in SM. \n\nIt is because, for the example in Table \\ref{tab:sigXBRjekyll}, $s_2 < |a_{1,1}| < 1$, and as per Eq.~(\\ref{eq:WWZZwid}) the partial width $\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow W^+W^-, ~ZZ) \\sim |s_2 ~a_{1,1}|^2 \\times \\Gamma^{SM}(H^0_{SM}\\rightarrow W^+W^-, ~ZZ)$ . \n\nOn the other hand, as seen in Eq.~(\\ref{eq:ffwid}), $\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow f\\bar{f}) \\sim |a_{1,1}\/s_2|^2 ~\\Gamma^{SM}(H^0\\rightarrow f ~\\bar{f})>\\Gamma^{SM}(H^0\\rightarrow f ~\\bar{f})$. \n\n$\\Gamma^{EW\\nu_R}(\\widetilde{H} \\rightarrow \\gamma\\gamma)$ is larger than the corresponding SM value, because in the EW$\\nu_R$  model, charged scalars and mirror fermions also contribute to this decay through triangle loops (refer to Appendix \\ref{sec:diphoton}). Recall that in SM this decay is dominated only by the $W$ loop.\n\n\t\\item\nFig.~\\ref{fig:hdecaylim} shows the comparison between the CMS data for signal strengths $\\mu(H\\text{-decay})$ of the 125-{\\rm GeV } Higgs boson, and the corresponding predictions for the 125-{\\rm GeV } $\\widetilde{H} $ in the EW$\\nu_R$  model, for examples 1 and 2 in {\\em Dr.~Jekyll} scenario and examples 1, 2 and 3 in {\\em Mr.~Hyde} scenario, discussed in the next subsection. \n\nFor calculating the EW$\\nu_R$  predictions, we have considered the gluon-gluon fusion production channel $(gg\\rightarrow \\widetilde{H} )$, which is the most dominant Higgs-production channel at the LHC. Calculations of the predictions in the EW$\\nu_R$  model are explained in Appendix~\\ref{sec:hdecay}.\n\t\\item\n\tNotice that the predicted ranges for $\\mu(\\widetilde{H} \\rightarrow W^+W^-, ~ZZ)$ and $\\mu(\\widetilde{H} \\rightarrow b\\bar{b}, \\tau\\bar{\\tau})$ are much narrower than the allowed ranges by the CMS constraints. \n\t\n\tA wider range of $a_{1,1M}$ than shown in Eqs.~(\\ref{eq:jekyll1}), (\\ref{eq:jekyll2}) is allowed if we impose the constraints on only, say, $\\widetilde{H}  \\rightarrow W^+W^-$ decay. However, for a part of the $a_{1,1M}$ range that satisfies the constraints on $\\mu(\\widetilde{H} \\rightarrow W^+W^-)$, the constraints on one or more of the other decay channels are not satisfied, and vice versa. So is true for all the other decay channels. Hence, when we seek the range of $a_{1,1M}$ that satisfies the constraints on {\\em all 4} of the $\\widetilde{H} \\rightarrow W^+W^-, ~ZZ, ~b\\bar{b}, ~\\tau\\bar{\\tau})$ decay channels, the predicted ranges for the signal strengths of these different channels are correlated. This shortens the range of $a_{1,1M}$ and of the signal strength predictions. These correlated predictions are shown in Fig.~\\ref{fig:hdecaylim}.\n\n\t\\item\nThe predicted range for $\\mu(\\widetilde{H} \\rightarrow \\gamma\\gamma)$ spans over $0-2.5$, because over the ranges of $m_{H_5^+}$ and $m_{H_5^{++}}$, $\\mu(\\widetilde{H} \\rightarrow \\gamma\\gamma)$ can easily vary without significantly affecting the predictions for the signal strengths of other decay channels.\n\n\t\\item\n\t{\\bf Conclusions from Fig.~\\ref{fig:hdecaylim}:}\n\tWe see that in the $\\widetilde{H}  \\sim H_1^0$ scenario predictions of the EW$\\nu_R$  model for various signal strengths agree with those of the 125-{\\rm GeV } Higgs boson, as measured by CMS. A slightly, but not very, different mixing matrices can also agree with the ATLAS measurements. Future measurements of partial widths would therefore be required to disentangle this scenario from that the SM.\n\n\\end{itemize}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{hdecay_limits_040715.pdf}\n\t\\caption{\\label{fig:hdecaylim}Figure shows the predictions of $\\mu(\\widetilde{H}  \\rightarrow ~b\\bar{b}, ~\\tau\\bar{\\tau}, ~\\gamma\\gamma, ~W^+W^-, ~ZZ)$ in the EW$\\nu_R$  model for examples 1 and 2 in {\\em Dr.~Jekyll} and example 1, 2 and 3 in {\\em Mr.~Hyde} scenarios, in comparison with corresponding best fit values by CMS \\cite{h_ww_122013, h_zz_4l_122013, h_bb_102013, h_tautau_012014}.}\n\\end{figure}\n\nWe now come to the most interesting part of our analysis, the one in which the 125-{\\rm GeV } Higgs boson is very unlike the SM Higgs.\n\n\n\\subsection{$\\widetilde{H} $ as the 125-GeV Higgs candidate with a sub-dominant SM-like component}\\label{sec:hyde}\n\n\nCan the 125-{\\rm GeV } $\\widetilde{H} $ in the EW$\\nu_R$  model have $H_1^0$ as a subdominant component and still satisfy the experimental constraints on its signal strengths? There are only two CP-even, neutral scalar states other than $H_1^0$, and they are $H_{1M}^0$ and $H_1^{0\\prime}$. The analysis explained in section \\ref{sec:numerical} revealed 1501 out of 4 million parameter combinations, for which $H_1^0$ can, indeed, be a subdominant component in 125-{\\rm GeV } $\\widetilde{H} $ while agreeing with the measured signal strengths of the 125-{\\rm GeV } Higgs at the LHC - the scenario we earlier referred to as {\\em Mr.~Hyde} scenario.\n\n\\subsubsection{Results of the analysis}\n\n\\begin{itemize}\n\t\\item\n\tTable~\\ref{tab:hyde} shows 16 out of the 1501 combinations of the parameters. \n\t\t\\item\n\tIt can be seen from Table~\\ref{tab:hyde} that in {\\em Mr.~Hyde} scenario, the CMS constraints on the signal strength can be satisfied, even when the scalar couplings satisfy $|\\lambda\/4\\pi|<1$. This means that the scalar particles heavier than the 125-{\\rm GeV } Higgs, need not be strongly coupled, and could be potentially detected as {\\em narrow resonances} at the LHC.\n\t\n\t\\item\n\tSimilarly, $s_{2M}$ can be larger than in {\\em Dr.~Jekyll} scenario. The mirror fermion masses are given in terms of $s_{2M}$ by\n\\begin{equation}\n\tm_{f^M} = \\dfrac{g_{MF} s_{2M} v}{\\sqrt{2}}\\,.\n\\end{equation}\n\nConsequently, larger (than in {\\em Dr.~Jekyll} scenario) masses of the mirror fermions are allowed by the perturbative limit on their Yukawa couplings. In other words, for a given mass of the mirror fermions, their Yukawa couplings in {\\em Mr.~Hyde} scenario can be smaller than those in {\\em Dr.~Jekyll} scenario.\n\n\t\\item\n\tTo highlight interesting features of this scenario, we consider three examples listed in Table~\\ref{tab:hyde}.\\\\\n\t\t\n\t\\item\nExample 1 (row 1 of Table~\\ref{tab:hyde}): $s_2 = 0.900$, $s_{2M} = 0.270$, $s_M = 0.341$, $\\lambda_1 = -0.481, ~\\lambda_2 = 6.00, ~\\lambda_3 = 1.46, ~\\lambda_4 = 2.99, ~\\lambda_5 = 2, ~\\lambda_8 = -1$,\n\\begin{equation}\\label{eq:hyde1}\n\t\\left( \\begin{array}{l}\t\t\\widetilde{H} \\\\ \\widetilde{H}^\\prime \\\\ \\widetilde{H}^{\\prime\\prime} \\end{array}\\right) = \n\t\\left( \\begin{array}{ccc}  \t0.300 \t\t& -0.094 \t\t& -0.949 \\\\\n\t\t\t\t\t\t\t\t0.334 \t\t& -0.921\t \t& -0.197 \\\\\n\t\t\t\t\t\t\t\t0.893\t \t& 0.376 \t\t& 0.246\n\t\\end{array} \\right) \n\t\\left( \\begin{array}{l} \t\tH_1^0\\\\ H_{1M}^0\\\\ H_1^{0\\prime}\\end{array}\\right),\n\\end{equation}\nwith $\\widetilde{H}  \\sim H_1^{0\\prime}$, $\\widetilde{H}^\\prime  \\sim H_{1M}^0$, $\\widetilde{H}^{\\prime\\prime}  \\sim H_1^0$; $m_{\\widetilde{H} } = 125.8 ~{\\rm GeV }$, $m_{\\widetilde{H}^\\prime } = 416 ~{\\rm GeV }$, $m_{\\widetilde{H}^{\\prime\\prime} } = 1100 ~{\\rm GeV }$, $M_R \\lesssim 105 ~{\\rm GeV }$, and $\\mu(\\widetilde{H} \\rightarrow W^+W^-\/ZZ) = 0.72$, $\\mu(\\widetilde{H} \\rightarrow \\gamma\\gamma) = 0.91$, $\\mu(\\widetilde{H} \\rightarrow b\\bar{b}\/\\tau\\bar{\\tau}) = 1.00$.\\\\ \n\n\n\t\\item\nExample 2 (row 2 of Table~\\ref{tab:hyde}): $s_2 = 0.514$, $s_{2M} = 0.841$, $s_M = 0.168$, $\\lambda_1 = 6.15, ~\\lambda_2 = 7.68, ~\\lambda_3 = 8.84, ~\\lambda_4 = -2.131502, ~\\lambda_5 = 5, ~\\lambda_8 = -1$,\n\\begin{equation}\\label{eq:hyde2}\n\t\\left( \\begin{array}{l}\t\t\\widetilde{H} \\\\ \\widetilde{H}^\\prime \\\\ \\widetilde{H}^{\\prime\\prime} \\end{array}\\right) = \n\t\\left( \\begin{array}{ccc}  \t0.188 \t\t& 0.091 \t\t& 0.978 \\\\\n\t\t\t\t\t\t\t\t-0.941 \t\t& -0.268\t \t& 0.207 \\\\\n\t\t\t\t\t\t\t\t-0.281\t \t& 0.959 \t\t& -0.035\n\t\\end{array} \\right) \n\t\\left( \\begin{array}{l} \tH_1^0\\\\ H_{1M}^0\\\\ H_1^{0\\prime}\\end{array}\\right),\n\\end{equation}\nwith $\\widetilde{H}  \\sim H_1^{0\\prime}$, $\\widetilde{H}^\\prime  \\sim H_1^0$, $\\widetilde{H}^{\\prime\\prime}  \\sim H_{1M}^{0}$; $m_{\\widetilde{H} } = 125.2 ~{\\rm GeV }$, $m_{\\widetilde{H}^\\prime } = 633 ~{\\rm GeV }$, $m_{\\widetilde{H}^{\\prime\\prime} } = 1427 ~{\\rm GeV }$, $M_R \\lesssim 52.0 ~{\\rm GeV }$, and $\\mu(\\widetilde{H} \\rightarrow W^+W^-\/ZZ) = 0.94$, $\\mu(\\widetilde{H} \\rightarrow \\gamma\\gamma) = 0.89$, $\\mu(\\widetilde{H} \\rightarrow b\\bar{b}\/\\tau\\bar{\\tau}) = 0.65$.\\\\ \n\n\n\t\\item\nExample 3 (row 3 of Table~\\ref{tab:hyde}): $s_2 = 0.401$, $s_{2M} = 0.900$, $s_M = 0.151$, $\\lambda_1 = 4.76, ~\\lambda_2 = 3.41, ~\\lambda_3 = 7.71, ~\\lambda_4 = -1.29, ~\\lambda_5 = 4, ~\\lambda_8 = -1$,\n\\begin{equation}\\label{eq:hyde3}\n\t\\left( \\begin{array}{l}\t\t\\widetilde{H} \\\\ \\widetilde{H}^\\prime \\\\ \\widetilde{H}^{\\prime\\prime} \\end{array}\\right) = \n\t\\left( \\begin{array}{ccc}  \t0.187\t\t& 0.115 \t\t& 0.976 \\\\\n\t\t\t\t\t\t\t\t0.922 \t\t& 0.321\t \t\t& -0.215 \\\\\n\t\t\t\t\t\t\t\t0.338\t \t& -0.940 \t\t& 0.046\n\t\\end{array} \\right) \n\t\\left( \\begin{array}{l} \t\tH_1^0\\\\ H_{1M}^0\\\\ H_1^{0\\prime}\\end{array}\\right),\n\\end{equation}\nwith $\\widetilde{H}  \\sim H_1^{0\\prime}$, $\\widetilde{H}^\\prime  \\sim H_1^0$, $\\widetilde{H}^{\\prime\\prime}  \\sim H_{1M}^{0}$; $m_{\\widetilde{H} } = 125.6 ~{\\rm GeV }$, $m_{\\widetilde{H}^\\prime } = 454 ~{\\rm GeV }$, $m_{\\widetilde{H}^{\\prime\\prime} } = 959 ~{\\rm GeV }$, $M_R \\lesssim 46.4 ~{\\rm GeV }$, and $\\mu(\\widetilde{H} \\rightarrow W^+W^-\/ZZ) = 0.89$, $\\mu(\\widetilde{H} \\rightarrow \\gamma\\gamma) = 1.09$, $\\mu(\\widetilde{H} \\rightarrow b\\bar{b}\/\\tau\\bar{\\tau}) = 1.06$.\n\n\t\\item\n\tIn example 1, $H_{1M}^0$ is the dominant component in $\\widetilde{H}^\\prime $, whereas $H_1^0$ is the dominant in $\\widetilde{H}^\\prime $ in examples 2 and 3. Although the mixing matrices in examples 2 and 3 are not very different, the ratio of VEV's $s_2$, $s_{2M}$ are different enough to result in the signal strengths that are not very similar (especially for $\\widetilde{H} \\rightarrow f\\bar{f}$). As the partial width of $\\widetilde{H} \\rightarrow f\\bar{f}$ is proportional to $|a_{1,1}\/s_2|^2$, it changes rapidly with $s_2$. Also, because we have 6 mirror quarks which contribute to the cross section of gluon-gluon fusion, the production cross section dominantly changes as $\\sim |a_{1,1}\/s_2+6 ~a_{1,1M}\/s_{2M}|^2$. Thus, any change in $a_{1,1M}\/s_{2M}$ is amplified while calculating the signal strengths. \n\n\t\\item\nComparison of the signal strengths for the three examples with the CMS constraints on them can be seen in Fig.~\\ref{fig:hdecaylim}. Notice the agreement between the predictions for the signal strengths with the CMS constraints in the figure. This agreement demonstrates that the {\\em SM-like signal strengths of 125-{\\rm GeV } Higgs at the LHC are not sufficient to conclude that it is a SM-like Higgs, or even if it has a dominant SM-like component}.\n\n\t\\item\nTable~\\ref{tab:sigXBRhyde} shows the partial widths, branching ratios and the signal strengths for {\\em Mr.~Hyde} scenario in the EW$\\nu_R$  model and SM. It can be seen that the partial widths in this scenario are very different from the SM (smaller by a factor of $\\sim 5$ for the example in the table), but it results in similar signal strengths. Measurements of the partial widths are therefore necessary to be able to experimentally distinguish between {\\em Mr.~Hyde} scenario and SM.\n\\end{itemize}\n\\begin{table*}[!htb]\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\centering\n\t\\caption{\\label{tab:sigXBRhyde}Partial width of $H\\rightarrow gg$ as the measure of the production cross section, partial widths and branching ratios for various channels in SM (for $m_{H_{SM}} = 125.6~{\\rm GeV }$ and total width $4.15\\times 10^{-3}$ {\\rm GeV }), and the EW$\\nu_R$  model for row 3 in Table \\ref{tab:hyde}, also in Eq.~(\\ref{eq:hyde3}) where $\\widetilde{H}  \\sim H_1^{0\\prime}$ (with $m_{\\widetilde{H} } = 125.6~{\\rm GeV }$ and total width $1.34\\times 10^{-3}$ {\\rm GeV }). All the partial widths are given in {\\rm GeV }.}\n\t\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\t\\hline\n    ~ & \\multicolumn{3}{c|}{SM}   & \\multicolumn{3}{c|}{EW$\\nu_R$ }  &  \\multirow{2}{*}{$\\mu = \\dfrac{(\\sigma_{Hgg} \\times BR)_{\\text{SM}}}{(\\sigma_{\\widetilde{H}  gg} \\times BR)_{\\text{EW$\\nu_R$ }}}$}      \\\\\n     \\cline{2-7}\n    ~ & $\\Gamma_{H\\rightarrow gg}$      & Partial Width   & BR       & $\\Gamma_{\\widetilde{H} \\rightarrow gg}$      & Partial Width   & BR       & ~         \\\\\n    ~ & $\\propto \\sigma_{gg\\rightarrow H}$ & ~ & ~ & $\\propto \\sigma_{gg\\rightarrow H}$ & ~ & ~ & ~ \\\\\n     \\hline\n\t$\\widetilde{H} \\rightarrow W^+W^-$   \t& 3.54E-04 \t\t& 9.30E-04 & 2.24E-01 & 5.75E-04 & 1.64E-04 & 1.23E-01 & 0.89 \\\\\n\t$\\widetilde{H} \\rightarrow ZZ$   \t\t& 3.54E-04 \t\t& 1.16E-04 & 2.79E-02 & 5.75E-04 & 2.04E-05 & 1.53E-02 & 0.89 \\\\\n\t$\\widetilde{H} \\rightarrow b\\bar{b}$   \t& 3.54E-04 \t\t& 2.35E-03 & 5.67E-01 & 5.75E-04 & 5.07E-04 & 3.79E-01 & 1.06  \\\\\n\t$\\widetilde{H} \\rightarrow \\tau\\bar{\\tau}$\t& 3.54E-04\t& 2.58E-04 & 6.22E-02 & 5.75E-04 & 5.42E-05 & 4.06E-02 & 1.06  \\\\\n\t$\\widetilde{H} \\rightarrow \\gamma\\gamma$\t\t& 3.54E-04\t& 9.46E-06 & 2.28E-03 & 5.75E-04 & 2.04E-06 & 1.53E-03 & 1.09  \\\\\n\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\\subsubsection{Remarks on the $H_{1M}^0$ component in $\\widetilde{H} $}\n\nA few remarks are in order here:\n\\begin{itemize}\n\t\\item\n\tNotice that for all the cases listed in Table~\\ref{tab:hyde} $H_1^{0\\prime}$ is the dominant component in the 125-{\\rm GeV } $\\widetilde{H} $. In all 1501 cases we found, the modulus of the coefficient of $H_{1M}^0$ in the 125-{\\rm GeV } $\\widetilde{H} $ was $\\leq 0.32$.\n\t\\item\n\tIn the gluon-gluon fusion channel $H_{1M}^0$ is produced through triangle loops of 6 mirror quarks. Therefore, if $H_{1M}^0$ is the dominant component in $\\widetilde{H} $, then the production cross section of $\\widetilde{H} $ could become too high to be compensated by small branching ratios. Thus, it makes sense that $H_{1M}^0$ is disfavored to be the dominant component in $\\widetilde{H} $, by the constraints on the signal strengths.\n\t\\item\n\tEven if $H_{1M}^0$ is a sub-dominant component in $\\widetilde{H} $, one should not think that it has decoupled from the other two singlets. In other words, the scalar doublet $\\Phi_{2M}$ does not really decouple from $\\Phi_2$ and $\\chi$. This is because:\n\t\\begin{itemize}\n\t\t\\item\n\t\tEven if $H_{1M}^0$ has a small coefficient in $\\widetilde{H} $, its production amplitude through 6 mirror quarks has a significant contribution to the production cross section of $\\widetilde{H} $.\n\t\t\\item\n\t\tThe real degree of freedom of $\\Phi_{2M}$ leads to $H_{1M}^0$. But its other degrees of freedom also contribute to other physical particles such as $H_3^{0,\\pm}$, $H_{3M}^{0,\\pm}$. These particles contribute to $\\widetilde{H}  \\rightarrow \\gamma\\gamma$ and the total width. Hence, they play a role in ensuring that the branching ratios are in the appropriate range to achieve an agreement with the signal strength constraints.\n\t\\end{itemize}\n\t\\item\n\tThus, although $H_{1M}^0$ is a sub-dominant component in $\\widetilde{H} $, the scalar doublet $\\Phi_{2M}$, newly added to the minimal EW$\\nu_R$  model, plays a crucial role in accommodating the 125-{\\rm GeV } Higgs boson in the EW$\\nu_R$  model, in {\\em Mr.~Hyde} as well as {\\em Dr.~Jekyll} scenario.\n\\end{itemize}\n\n\\begin{turnpage}\n\\begin{table*}[!]\n\t\\renewcommand{\\arraystretch}{1.2}\n\t\\centering\n\t\\caption{\\label{tab:hyde}All the masses and the total width of $\\widetilde{H} $ are given in {\\rm GeV }. Fixed parameters as given in Eq. (\\ref{eq:hydemupara}).\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n~ & $s_2$   & $s_{2M}$  & $s_M$   & $\\lambda_1$   & $\\lambda_2$   & $\\lambda_3$  & $\\lambda_4$   & $\\lambda_5$   & $m_{\\widetilde{H} }$   & $m_{\\widetilde{H}^\\prime }$ & $m_{\\widetilde{H}^{\\prime\\prime} }$ & $a_{1,1}$ & $a_{1,1M}$ & $a_{1,1'}$ & $a_{1M,1}$ & $a_{1M,1M}$ & $a_{1M,1'}$ & $a_{1',1}$ & $a_{1',1M}$ & $a_{1',1'}$ & $\\Gamma_{\\widetilde{H} }$  & $\\mu_{WW\/ZZ}$ & $\\mu_{\\gamma\\gamma}$ & $\\mu_{bb\/\\tau\\tau}$ & $M_R\\lesssim$ \\\\\n\\hline\n1  & 0.90 & 0.27 & 0.34 & -0.48   & 6.00    & 1.46    & 2.99    & 2       & 125.8 & 416  & 1100   & 0.301 & -0.094 & -0.949  & 0.334  & -0.922  & 0.197    & 0.893   & 0.376    & 0.246     & 1.66E-03 & 0.719  & 0.914    & 1.002  & 105.0 \\\\\n2  & 0.51 & 0.84 & 0.17 & 6.15    & 7.68    & 8.84    & -2.13   & 5       & 125.2 & 633  & 1427   & 0.189 & 0.091  & 0.978   & -0.941 & -0.268  & 0.207    & -0.281  & 0.959    & -0.035    & 9.61E-04 & 0.941  & 0.895    & 0.647  & 52.0  \\\\\n3  & 0.41 & 0.90 & 0.15 & 4.76    & 3.41    & 7.71    & -1.29   & 4       & 125.6 & 454  & 959    & 0.187 & 0.115  & 0.976   & 0.922  & 0.321   & -0.215   & 0.338   & -0.940   & 0.046     & 1.34E-03 & 0.891  & 1.089    & 1.062  & 46.4  \\\\\n4  & 0.87 & 0.32 & 0.36 & -0.39   & 4.40    & 1.21    & 4.48    & 1       & 126.2 & 420  & 1382   & 0.303 & -0.087 & -0.949  & 0.357  & -0.913  & 0.197    & 0.884   & 0.398    & 0.246     & 1.12E-03 & 0.753  & 1.108    & 0.849  & 111.9 \\\\\n5  & 0.35 & 0.91 & 0.22 & 8.73    & 5.26    & 4.88    & -1.59   & 9       & 126.1 & 617  & 1237   & 0.143 & 0.090  & 0.986   & 0.966  & 0.204   & -0.159   & 0.216   & -0.975   & 0.057     & 1.12E-03 & 0.994  & 0.995    & 0.682  & 69.2  \\\\\n6  & 0.31 & 0.87 & 0.38 & 4.99    & 9.67    & 2.10    & -1.02   & 1       & 126.4 & 435  & 1786   & 0.238 & 0.041  & 0.970   & 0.970  & 0.035   & -0.239   & 0.044   & -0.999   & 0.031     & 2.62E-03 & 0.953  & 1.128    & 1.092  & 118.1 \\\\\n7  & 0.31 & 0.87 & 0.38 & 4.99    & 9.67    & 2.10    & -1.02   & 2       & 126.4 & 435  & 1786   & 0.238 & 0.041  & 0.970   & 0.970  & 0.035   & -0.239   & 0.044   & -0.999   & 0.031     & 2.62E-03 & 0.954  & 0.894    & 1.093  & 118.1 \\\\\n8  & 0.92 & 0.21 & 0.31 & -0.73   & 9.34    & 1.83    & 9.25    & 5       & 125.6 & 412  & 1988   & 0.239 & -0.096 & -0.966  & 0.210  & -0.966  & 0.148    & 0.948   & 0.239    & 0.211     & 2.91E-03 & 0.882  & 1.341    & 0.708  & 95.7  \\\\\n9  & 0.36 & 0.92 & 0.16 & 6.78    & 3.10    & 7.16    & -1.34   & 4       & 126.1 & 501  & 905    & 0.156 & 0.129  & 0.979   & -0.901 & -0.388  & 0.195    & -0.405  & 0.912    & -0.056    & 1.37E-03 & 1.031  & 1.001    & 1.043  & 49.6  \\\\\n10 & 0.92 & 0.19 & 0.35 & -0.34   & 11.69   & 1.17    & 2.79    & 1       & 126.7 & 428  & 1067   & 0.272 & -0.047 & -0.961  & 0.237  & -0.965  & 0.114    & 0.932   & 0.259    & 0.251     & 9.77E-04 & 0.729  & 1.107    & 0.663  & 108.4 \\\\\n11 & 0.95 & 0.12 & 0.29 & -0.46   & 9.71    & 1.68    & 3.37    & 9       & 126.0 & 248  & 1167   & 0.230 & -0.126 & -0.965  & 0.113  & -0.981  & 0.155    & 0.967   & 0.145    & 0.211     & 1.57E-02 & 0.742  & 1.083    & 0.678  & 89.2  \\\\\n12 & 0.38 & 0.89 & 0.26 & 3.47    & 5.24    & 3.25    & -0.99   & 10      & 125.2 & 409  & 1281   & 0.241 & 0.065  & 0.968   & 0.964  & 0.097   & -0.247   & 0.110   & -0.993   & 0.039     & 1.82E-03 & 0.849  & 0.967    & 1.080  & 79.6  \\\\\n13 & 0.30 & 0.93 & 0.22 & 5.01    & 2.67    & 3.06    & -0.75   & 12      & 125.1 & 415  & 906    & 0.131 & 0.075  & 0.989   & 0.979  & 0.146   & -0.141   & 0.155   & -0.986   & 0.054     & 1.06E-03 & 0.770  & 1.266    & 0.666  & 69.0  \\\\\n14 & 0.36 & 0.89 & 0.28 & 2.34    & 2.56    & 2.04    & -0.52   & 12      & 126.0 & 333  & 890    & 0.211 & 0.071  & 0.975   & 0.971  & 0.104   & -0.217   & 0.117   & -0.992   & 0.047     & 1.74E-03 & 1.006  & 1.267    & 0.984  & 87.0  \\\\\n15 & 0.48 & 0.86 & 0.20 & 2.26    & 2.90    & 4.11    & -0.74   & 6       & 126.0 & 376  & 896    & 0.217 & 0.090  & 0.972   & 0.950  & 0.208   & -0.231   & 0.223   & -0.974   & 0.040     & 1.29E-03 & 1.012  & 0.917    & 0.857  & 60.9  \\\\\n16 & 0.32 & 0.91 & 0.26 & 2.76    & 2.07    & 2.18    & -0.49   & 11      & 126.3 & 323  & 804    & 0.182 & 0.076  & 0.980   & 0.975  & 0.115   & -0.190   & 0.127   & -0.991   & 0.053     & 1.69E-03 & 0.919  & 0.930    & 1.019  & 80.6  \\\\\n   \\hline\n\\end{tabular}\n\\end{table*}\n\\end{turnpage}\nBefore concluding this section, we will briefly discuss some indirect constraints on the next heavier scalar $\\widetilde{H}^\\prime $, in both these scenarios.\n\\subsection{The next heavier neutral scalar $\\widetilde{H}^\\prime $}\n\nIn {\\em Dr.~Jekyll} scenario, examples 1 and 2 that we considered have $H_{1M}^0$ as the dominant component in $\\widetilde{H}^\\prime $, which is the next heavier physical scalar after the 125-{\\rm GeV } $\\widetilde{H} $. Here the total width of $\\widetilde{H}^\\prime $ is also greater than its mass, with the scalar coupling $\\lambda_2>4\\pi$. Thus, it is a strongly coupled scalar, which is difficult to detect as a narrow resonance.\n\nIn example 1 of {\\em Mr.~Hyde} scenario, $H_{1M}^0$ is the dominant component in $\\widetilde{H}^\\prime $, while in examples 2 and 3 $H_1^0$ is the dominant component in $\\widetilde{H}^\\prime $. In all 3 examples, $\\widetilde{H}^\\prime $ has a total width $<10\\%$ of its mass.\n\nThis subsection compares the signal strength of $\\widetilde{H}^\\prime \\rightarrow W^+W^-$ and the $\\sigma \\times BR(\\widetilde{H}^\\prime \\rightarrow \\gamma\\gamma)$ with the CMS constraints on SM-like heavy Higgs, for examples having $m_{\\widetilde{H}^\\prime } \\lesssim 600 ~{\\rm GeV }$. These CMS constraints \\cite{CMS_heavy_Higgs_WW, HWWsearch, ATLAS_diphoton_july2014, CMS_diphoton_july2014} assume the Standard model background, whereas, in the EW$\\nu_R$  model, extra processes involving mirror fermions and extra scalars also contribute to the background in addition to the SM processes. The background in this model is therefore expected to be larger than that in the SM. A detailed study of this background is out of the scope of this paper.\n\nAlthough the SM background does not strictly apply to $\\widetilde{H}^\\prime $ in the EW$\\nu_R$  model, we show how the EW$\\nu_R$  predictions compare with the experimental constraints.\n\nFor our calculations we computed the total width of $\\widetilde{H}^\\prime $ using\n\\begin{eqnarray}\\label{eq:widthH1M}\n\\Gamma_{\\widetilde{H}^\\prime } = &&~\\sum_{i=1}^3\\Gamma_{\\widetilde{H}^\\prime \\rightarrow q_i^M\\bar{q}_i^M} ~+ \\sum^3_{j=1} ~\\times ~\\Gamma_{\\widetilde{H}^\\prime \\rightarrow l_j^M\\bar{l}_j^M}\\nonumber\\\\[0.5em]\n &&+ ~\\Gamma_{\\widetilde{H}^\\prime \\rightarrow t\\bar{t}} ~+ ~\\Gamma_{\\widetilde{H}^\\prime \\rightarrow b\\bar{b}} \\nonumber\\\\[0.5em]\n&& + ~\\Gamma_{\\widetilde{H} \\rightarrow \\tau\\bar{\\tau}} ~+ ~\\Gamma_{\\widetilde{H} \\rightarrow c\\bar{c}} ~+\\Gamma_{\\widetilde{H}^\\prime \\rightarrow W^+ W^-}\\nonumber\\\\[0.5em]\n&& + ~\\Gamma_{\\widetilde{H}^\\prime \\rightarrow ZZ} ~+ ~\\Gamma_{\\widetilde{H}^\\prime \\rightarrow gg} ~+ ~\\Gamma_{\\widetilde{H}^\\prime \\rightarrow \\gamma \\gamma}\\,.\n\\end{eqnarray}\nThe partial decay widths were calculated using the method illustrated in Appendix~\\ref{sec:hdecay}.\n\n\\subsubsection{Constraints on the signal strength of $\\widetilde{H}^\\prime \\rightarrow W^+W^-$}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{muH1MWW_042915.pdf}\n\t\\caption{\\label{fig:h1mww}Predicted signal strength of $\\widetilde{H}^\\prime \\rightarrow W^+W^-$ in 4 example scenarios (blue and purple squares). The results of the search for SM-like Higgs boson up to 600 {\\rm GeV } with the $1\\sigma$ (green band) and $2\\sigma$ (yellow band) limits on the SM background (dotted curve) and CMS data (solid black curve) are also displayed.}\n\\end{figure}\n\\begin{itemize}\n\t\\item\n\tFig.~\\ref{fig:h1mww} shows the signal strength of $\\widetilde{H}^\\prime \\rightarrow W^+W^-$ for examples 1 and 2 in {\\em Dr.~Jekyll} scenario (blue squares) and exmaples 1 and 3 in {\\em Mr.~Hyde} scenario (purple squares). The $1-$ and $2-\\sigma$ SM background bands and the CMS data \\cite{CMS_heavy_Higgs_WW, HWWsearch} are also displayed. The signal strength for example 2 in {\\em Mr.~Hyde} scenario is not displayed as $m_{\\widetilde{H}^\\prime } = 633~{\\rm GeV }$ for this example, but the CMS data for this decay channel are only available up to 600~{\\rm GeV }. \n\t\n\t\\item\n\tIn the figure, notice that the predicted signal strengths for examples 1 and 2 in {\\em Dr.~Jekyll} case and example 1 in {\\em Mr.~Hyde} case are within the $\\pm 1\\sigma$ SM-background bands. Therefore, the CMS data are surely not conclusive for confirming or ruling out these examples.\n\t\n\t\\item\n\tExample 1 in {\\em Mr.~Hyde} scenario predicts a signal strength $\\mu(\\widetilde{H}^\\prime \\rightarrow W^+W^-) \\approx 1.3$, which is certainly larger the SM-background band and the data. $H_1^0$ - the SM-like Higgs is the dominant component in $\\widetilde{H}^\\prime $ in this example. However, the SM-background still does not strictly apply here, since additional background processes can contribute to it. For example, production of $W^+W^-$ from two gluons through a box loop of mirror quarks\n\\end{itemize}\n\\subsubsection{Constraints on $\\widetilde{H}^\\prime \\sim H_{1M}^0$ from $\\gamma\\gamma$-decay channel}\n\nThe constraints on $\\sigma(gg\\rightarrow \\widetilde{H}^\\prime )\\times BR(\\widetilde{H}^\\prime \\rightarrow \\gamma\\gamma)$ from CMS \\cite{CMS_diphoton_july2014} and ATLAS \\cite{ATLAS_diphoton_july2014} are accompanied by assumptions that the total width of the SM-like heavy Higgs is $0.1 ~{\\rm GeV }$ or $10\\%$ of its mass. The total width of $\\widetilde{H}^\\prime $ in our scenarios does not follow either of these patterns. We observed that $\\sigma(gg\\rightarrow\\widetilde{H}^\\prime )\\times BR(\\widetilde{H}^\\prime \\rightarrow \\gamma\\gamma)$ predictions for all the examples in both the scenarios is consistently lower than the CMS and ATLAS constraints.\n\n\\subsubsection{A comment on $\\widetilde{H}^{\\prime\\prime} $}\n\nIn the examples of {\\em Dr.~Jekyll} scenario considered in this paper, $\\widetilde{H}^{\\prime\\prime}  \\sim H_1^{0\\prime}$, and in the examples of {\\em Mr.~Hyde} scenario that we have considered, $\\widetilde{H}^{\\prime\\prime}  \\sim H_1^0$ or $H_{1M}^0$. For all these examples, $m_{\\widetilde{H}^{\\prime\\prime} } \\gtrsim 600~{\\rm GeV }$. So far, the CMS data in Fig~\\ref{fig:h1mww} are not sensitive to the signal strengths of the order of SM predictions in this mass range. A detailed analysis of its signal strengths will be of interest, when more data for this higher mass range are available and are analyzed with full EW$\\nu_R$  background.\n\\subsection{Conclusions about the 125-GeV Higgs candidate in the EW$\\nu_R$  model}\\label{sec:125conclude}\n\n\\begin{itemize}\n\t\\item\n\tIn this section we investigated two very different scenarios of the 125-{\\rm GeV } Higgs boson at the LHC from the perspective of the EW$\\nu_R$  model:\n\t\\begin{enumerate}\n\t\t\\item\n\t\t{\\em Dr.~Jekyll}: $H_1^0$, which is the real part of the SM-like scalar doublet $\\Phi_2$, is the dominant component in the 125-{\\rm GeV } $\\widetilde{H} $, and\n\t\t\\item\n\t\t{\\em Mr.~Hyde}: $H_1^0$ is a sub-dominant component in the 125-{\\rm GeV } $\\widetilde{H} $ - a more interesting scenario.\n\t\\end{enumerate}\n\t\n\t\\item\n\tWe demonstrated that in both these scenarios the signal strengths of the 125-{\\rm GeV } $\\widetilde{H} $ in $W^+W^-, ~ZZ, ~\\gamma\\gamma, ~b\\bar{b}$ and $\\tau\\bar{\\tau}$ decay channels agree with the constraints from CMS (and also ATLAS) data. Thus, from the perspective of the EW$\\nu_R$  model, the present data at the LHC are inconclusive about whether SM-like $H_1^0$ is {\\em the dominant} or {\\em a sub-dominant component} in the 125-{\\rm GeV } particle. Hence ``the dual nature\" of the 125-{\\rm GeV } Higgs in the EW$\\nu_R$  model.\n\t\n\tMore data, measurements of individual partial widths and study of the heavier physical scalars in the EW$\\nu_R$  model are necessary to distinguish between either of these scenarios and the SM-Higgs.\n\n\t\\item\n\tAs expected, the individual partial widths of the 125-{\\rm GeV } $\\widetilde{H} $ in {\\em Dr.~Jekyll} scenario are not very different from those in SM. Here, the scalar couplings $|\\lambda_2|, ~|\\lambda_3|, ~|\\lambda_5|$ need to be greater than  $4\\pi$ to satisfy the constraints on the signal strengths. This means that the heavier scalars in this scenario tend to be strongly coupled and have large widths.\n\n\tDominant SM-like component in the 125-{\\rm GeV } Higgs also leads to $v_2$ (the VEV of $\\Phi_2$) being the dominant part in $v$, and  smaller $v_{2M}$, which gives masses to the mirror fermions. Consequently, the masses of the mirror fermions, allowed by the perturbative limit on their Yukawa couplings, cannot be much greater than $\\sim 120 ~{\\rm GeV }$. We adopt a lower limit of 102~{\\rm GeV } set by the constraints from LEP3 \\cite{pdg}.\n\n\t\\item\n\tHence, if future measurements of the individual decay widths of 125-{\\rm GeV } Higgs result in SM-like widths, then it is more likely to be consistent with {\\em Dr.~Jekyll} scenario. In this case, the heavier scalars would appear not as narrow resonances, but as broad resonances or enhancement in the background in this model.\n\n\tSince the SM-like $H_1^0$ is the dominant component in the 125-{\\rm GeV } $\\widetilde{H} $, the effective theory around this energy looks like SM, in which the heavier scalars are {\\em integrated out}.\n\t\n\t\\item\n\tIn contrast, the individual partial widths of the 125-{\\rm GeV } $\\widetilde{H} $ are very different from those in SM, in {\\em Mr.~Hyde} scenario. In all 1501 combinations of the parameters that we found to agree with the experimental $1\\sigma$ constraints on the signal strengths contain $H_1^{0\\prime}$ as the dominant component in the 125-{\\rm GeV } $\\widetilde{H} $. The predicted signal strengths of this $\\widetilde{H} $ agree with the experimental $1\\sigma$ constraints on the signal strengths even when the scalar couplings $|\\lambda|$'s are smaller than $4\\pi$. The heavier scalars in this case are not strongly coupled, as a result.\n\t\n\tThe $H_{1M}^0$ as a dominant component in $\\widetilde{H} $ is disfavored to agree with the constraints on the signal strengths, due to its large contribution to the cross section of $gg \\rightarrow \\widetilde{H} $. \n\t\n\tBecause $v_{2M}$ is not constrained to be small in this case, the perturbative upper limit on the mirror fermions is about 700~{\\rm GeV }.\n\t\n\t\\item\n\tTherefore, if the partial widths of the 125-{\\rm GeV } $\\widetilde{H} $ are measured to be very different from those in SM, it would point towards {\\em Mr.~Hyde} scenario. The heavier scalars in this case have narrow widths and can be detected as resonances.\n\t\n\tThe SM-like $H_1^0$ is the dominant component in one of the heavier scalars, $\\widetilde{H}^\\prime $ or $\\widetilde{H}^{\\prime\\prime} $. Thus, the effective theory around 125~{\\rm GeV } is very different from SM, while the SM-like $H_1^0$ is {\\em integrated out} with the heavier scalars.\n\n\t\\item\n\tAs can be seen from Eq~(\\ref{eq:hydemupara}) we scanned only a part of the entire parameter space in the EW$\\nu_R$  model, by fixing values or ranges of the parameters. A more thorough scan of the parameter space could be of interest, especially if more data from the LHC Run II show any signs of physics Beyond the Standard Model. \n\n\\end{itemize}\n\nIn the next section we will briefly discuss some of the decay properties of the CP-odd neutral spin-0 states - $H_3^0$ and $H_{3M}^0$.\n\n\\section{Signals of CP-odd spin zero states }\n\nIn addition to the 125-{\\rm GeV } $\\tilde{H}$, the EW$\\nu_R$ ~model also contains CP-odd spin zero states - $H_{3}^0, H_{3M}^0 $ - and two other heavy CP-even spin zero states - $\\widetilde{H}^\\prime $, and $\\widetilde{H}^{\\prime\\prime} $. In this section, we show possibilities to probe the signal of the neutral pseudo-scalars in various decay channels at LHC. To do so, we will investigate the product of production cross section and the branching ratio, a.k.a the absolute signal strength, in $H_{3,3M}^0\\rightarrow \\gamma\\gamma,~\\tau\\tau$ channels. We will also calculate the signal strengths ($\\mu$) for $H_{3}^0,~H_{3M}^0$ and the SM Higgs boson $H_{SM}$ in other channels.\\\\\n\\begin{equation}\\label{eq:muy}\n\\mu \\; = \\; \\dfrac{\\sigma(gg \\rightarrow H_{3,3M}^0) Br( H_{3,3M}^0 \\rightarrow XX )}{\\sigma(gg \\rightarrow H_{SM}) Br( H_{SM} \\rightarrow XX)}\n\\\\\n\\end{equation}\n\nIn this extension of the EW$\\nu_R$  model, the degenerate masses of two $SU(2)_D$ custodial triplets are related by\n\\begin{equation}\\label{eq:mH3H3M}\n \\dfrac{m_{H_3}^2}{m_{H_{3M}}^2} = \\dfrac{1}{1+c_M^2}\n\\end{equation}\nWe assume that the neutral states $H_3^0$ and $H_{3M}^0$ obey this relationship, and use two cases of  $s_M = 0.168;\\;0.36$. The lighter one, $m_{H_3^0}$, is scanned over ranges $130 - 440\\;{\\rm GeV }$, whereas the mass $m_{H_{3M}^0}$ of the heavier one over $182 - 618\\; {\\rm GeV }$ and $177 - 601\\; {\\rm GeV }$, respectively.\n\n\\subsection{Ratio of production cross sections }\n\nAt the LHC, $H_3^0,~H_{3M}^0$ are expected to be produced mainly through gluon-gluon fusion, similar to $H_{SM}$. By using effective coupling approximation, we have\n\\begin{equation}\\label{eq:csection}\n  R=\\dfrac{\\sigma(gg \\rightarrow H_{3,3M}^0)}{\\sigma(gg \\rightarrow H_{SM})} \\approx \\dfrac{\\Gamma(H_{3,3M}^0 \\rightarrow gg)}{\\Gamma(H_{SM} \\rightarrow gg)}\\,.\n\\end{equation}\n$H_{3,3M}^0$ do not couple directly to the gauge bosons $W, ~Z, ~\\gamma$. And triplet couplings, such as $ H_{3,3M}^0 H_{3,3M}^+ H_{3,3M}^-,\\; H_{3,3M}^0 H_5^+ H_5^-,\\; H_{3,3M}^0 H_5^{++} H_5^{--}$, are forbidden by CP conservation. Therefore, only fermionic loops involving the top quark and the mirror quarks contribute to the gluonic decay of $H_3^0$ and $H_{3M}^0$ \\cite{qcd_corr3}:\n \\begin{eqnarray}\\label{eq:H3ogg}\n  \\Gamma \\left( H_{3,3M}^0 \\rightarrow gg \\right) =&& \\dfrac{G_F \\alpha_s^2}{16\\sqrt{2} \\pi^3} m_{H_{3,3M}^0}^3\\nonumber\\\\[1em]\n  &\\times& \\left| \\sum\\limits_{Q} g_Q^{H_{3,3M}^0} F_Q^{H_{3,3M}^0}(\\tau_f)\\right|^2\\,,\n \\end{eqnarray}\n \\begin{equation}\\label{eq:F_H3o} \n F_Q^{H_{3,3M}^0}(\\tau_f) = \\tau_f f(\\tau_f)\\,,\n \\end{equation}\n \\begin{equation}\n \\tau_t = 4 m_f^2\/m_{H_{3,3M}^0}^2\\,,\n \\end{equation}\n\nwhere $g_Q^{H_{3,3M}^0}$ are couplings of $H_3^0$ and $H_{3M}^0$ to the top quark and mirror quarks, listed in Table \\ref{table:h_ferm}.\n\nHere, $\\sum\\limits_{Q}$ is summed over the top quark and mirror quarks. However, the contributions from mirror quarks can be suppressed due to the fact that the mirror-up quarks and the mirror-down quarks couple to $H_{3,3M}^0$ with opposite signs. In this work, we particularly consider degenerate mirror fermion doublets, meaning $m_{u^M} = m_{d^M}$, for simplicity. As a result, the contributions from mirror quarks cancel out. Thus, only the top-quark loop appears in the production cross section of $H_3^0, H_{3M}^0$. Then, the ratios of production cross section are given by\n \\begin{equation}\\label{eq:csectionapprox}\n R_{H_3^0}= \\tau_t^2  \\dfrac{\\left|\\tan\\theta_M f(\\tau_t)\\right|^2}{\\left|\\tau_t+(\\tau_t-1)f(\\tau_t)\\right|^2}\n \\end{equation}\nfor $H_3^0$, and\n \\begin{equation}\\label{eq:csection3oM}\n R_{H_{3M}^0}= \\tau_t^2 \\dfrac{\\left|s_{2M}f(\\tau_t)\\right|^2}{\\left|s_2(\\tau_t+(\\tau_t-1)f(\\tau_t))\\right|^2}\n \\end{equation}\nfor $H_{3M}^0$.\n\n\\subsection{ In \\;$ \\gamma\\gamma$ channel}\n\nATLAS \\cite{ATLAS_diphoton_july2014} and CMS \\cite{CMS_diphoton_july2014} have recently reported their results of the search for narrow resonances in spin-0 state to diphoton decay channel, up to $600~{\\rm GeV }$ for CMS and $840~{\\rm GeV }$ for ATLAS. Both the reports make certain assumptions about the total width of the decaying spin-0 state. They present the upper limit on the production cross section times branching ratio for this channel at $ 95\\% $ confidence level. So far, no significant excess has been found, except for the two $2\\sigma$ resonances above the background at $m = 201\\;{\\rm GeV }$ and $m = 530\\;{\\rm GeV }$ in the ATLAS analysis. We compare our predictions in $\\gamma\\gamma$ channel with those results, even though the assumptions about the total width of resonances are not generally applicable to the EW$\\nu_R$ ~model.\n\nSimilar to the gluonic decay, only fermionic loops contribute to the partial width of $H_{3,3M}^0 \\rightarrow \\gamma \\gamma$, given by \\cite{qcd_corr3}\n\\begin{equation}\\label{eq:H3ogamma}\n\\Gamma\\left( H_{3,3M}^0\\rightarrow \\gamma\\gamma\\right) = \\dfrac{g^2\\;\\alpha^2\\; m^3_{H_{3,3M}^0}}{256\\;\\pi\\; m_W^2}\\left|\\sum\\limits_{i} ~ N_{ci}\\; e_i^2 \\; g_i \\; F_i^{H_{3,3M}^0} \\right|^2\\,.\n\\end{equation}\nHere, $i=$ top quark, six mirror quarks, and three charged mirror leptons. The total widths of $H_{3,3M}^0$ are calculated by summing all the partial widths.\n\\begin{eqnarray}\\label{eq:width33M}\n\\Gamma_{H_{3,3M}}=&&~\\Gamma(H_{3,3M}^0\\rightarrow\\gamma\\gamma)+\\Gamma(H_{3,3M}^0\\rightarrow gg)\\nonumber\\\\\n    &&+\\Gamma(H_{3,3M}^0\\rightarrow W^+W^-)+\\Gamma(H_{3,3M}^0\\rightarrow ZZ)\\nonumber\\\\\n    &&+\\Gamma(H_{3,3M}^0\\rightarrow\\tau\\bar{\\tau})+\\Gamma(H_{3,3M}^0\\rightarrow t\\bar{t})\\nonumber\\\\\n    &&+\\Gamma(H_{3,3M}^0\\rightarrow c\\bar{c})+\\Gamma(H_{3,3M}^0\\rightarrow b\\bar{b})\\nonumber\\\\\n    &&+ \\sum\\limits_{i=1}^{6}\\Gamma(H_{3,3M}^0\\rightarrow q^M_i\\bar{q}^M_i)\\nonumber\\\\\n    && +\\sum\\limits_{j=1}^{3}\\Gamma(H_{3,3M}^0\\rightarrow l^M_j\\bar{l}^M_j)\n\\end{eqnarray}\nThe branching ratio of $H_{3,3M}^0\\rightarrow\\gamma\\gamma$ is\n\\begin{equation}\n \\label{eq:Branching33M}\n  Br(H_{3,3M}^0\\rightarrow\\gamma\\gamma)=\\dfrac{\\Gamma(H_{3,3M}^0\\rightarrow \\gamma\\gamma)}{\\Gamma_{H_{3,3M}^0}}\n\\end{equation}\nThe absolute signal strength of $H_{3,3M}^0\\rightarrow\\gamma\\gamma$ is defined as\\\\\n\\begin{flalign*}\n\\label{eq:Xbranching33M}\n \\sigma \\times BR(H_{3,3M}^0\\rightarrow\\gamma\\gamma) = R\\times\\sigma(gg \\rightarrow H_{SM})\\\\\n  \\times Br(H_{3,3M}^0 \\rightarrow \\gamma \\gamma)\\\\\n\\end{flalign*}\nAt any particular mass, $R_{H_{3,3M}^0}$ and $Br(H_{3,3M}^0 \\rightarrow \\gamma\\gamma)$ are calculated directly, while  $\\sigma(gg \\rightarrow H_{SM})$ is taken from the handbook of Higgs cross section \\cite{HXreport3}. To be consistent with the previous analysis, we also provide two scenarios which correspond to the dual nature of the 125-{\\rm GeV } scalar. For illustrative purposes, we consider up- and down- members of mirror quark doublets to have degenerate masses. The first two generations of mirror quarks and all charged mirror leptons have the same mass, $m_{q^M_1} = m_{q^M_2} = m_{l^M} = 102 \\;{\\rm GeV }$. The heaviest mirror quark generation has a mass $m_{q^M_3} = 120 \\;{\\rm GeV }$. Masses of all right-handed neutrinos are at $M_R = 50 \\;{\\rm GeV }$. \n  \\begin{itemize}\n  \\item\nIn the case of {\\em Dr.~Jekyll}, as $\\widetilde{H} \\sim H_1^0$, the mixing angles are $s_2 = 0.92,\\;s_{2M} = 0.16,\\;s_M = 0.36$, which corresponds to the first example in the section \\ref{sec:jekyll}. In the plot below, we present the dependence of the production cross section times branching ratio of $H_{3,3M}$ on mass. Moreover, the next heavy CP-even state is $\\tilde{H}^{\\prime}$ with the mass of $m_{\\widetilde{H}^\\prime }=420\\;{\\rm GeV }$. So we incorporate the production cross section times branching ration of $\\widetilde{H}^\\prime \\rightarrow\\gamma\\gamma$ also. \n\n  \\begin{figure}[H]\n    \\includegraphics[scale=0.45]{XBranching_gamma_H1oDominant.pdf}\n   \\caption{\\label{fig:gamma_H1oDominant} {\\small The production cross section times branching ratio in $\\gamma\\gamma$ channel of $H_3^0$ and $H_{3M}^0$, in {\\em Dr.~Jekyll} scenario. $m_{H_3^0}=130 - 440 \\;{\\rm GeV }$, $m_{H_{3M}^0}=177 - 601\\;{\\rm GeV }$ } }\n   \\end{figure}\n   \\item\n   In the other case when $H_1^0$ is sub-dominant in $\\widetilde{H} $ or {\\em Mr.~Hyde}, a set of parameters is chosen as $s_2=0.514,\\;s_{2M}=0.841,\\;s_M=0.168$ corresponding to the example 2 in the section \\ref{sec:hyde}. In this scenario, all the heavy CP-even states are above $600 {\\rm GeV }$. So that we just present the dependence of the production cross section times branching ratio of $H_{3,3M}\\rightarrow\\gamma\\gamma$. \n   \\begin{figure}[H]\n    \\includegraphics[scale=0.45]{XBranching_gamma_H1oSubdominant.pdf}\n   \\caption{\\label{fig:gamma_H1oSubdominant} {\\small The production cross section times branching ratio in $\\gamma\\gamma$ channel of $H_3^0$ and $H_{3M}^0$, in {\\em Mr.~Hyde} scenario. $m_{H_3^0}=130 - 440 \\;{\\rm GeV }$, $m_{H_{3M}^0}=182 - 618\\;{\\rm GeV }$ } }\n   \\end{figure}\n\\end{itemize}\nRemarks:\n\\begin{enumerate}\n  \\item\n   Before fermionic thresholds, $2m_{q^M_{1,2}}$ and $2m_{l^M}$, the signal strength for $H_{3,3M}^0$ can be larger than what is measured by ATLAS and CMS for heavy SM-like scalar. To be conservative, we can exclude $m_{H_{3}^0} \\lesssim 150~{\\rm GeV }$ and $m_{H_{3M}^0} \\lesssim 205~{\\rm GeV }$ (FIG.\\ref{fig:gamma_H1oDominant}) and $\\lesssim 210~{\\rm GeV }$ (FIG.\\ref{fig:gamma_H1oSubdominant}). However, for some other set of parameters, the signal strengths could be well below the upper limit set by ATLAS and CMS.\n  \\item\n    As $m_{H_3^0}$ increases, more mirror fermionic decay channels are kinematically allowed. On the other hand, the production cross section decreases. The branching ratios of $H_{3,3M}^0\\rightarrow\\gamma\\gamma$ therefore decrease rapidly beyond the thresholds, $2m_{q^M_{1,2}}, \\;2m_{q^M_3},\\;2m_{l^M},\\;2m_t$. As a result, the signal strengths in both the cases are below the experimental upper limits.\n  \\item\n   At the same mass, the signal strengths of CP-odd spin-0 states are generally larger than those of CP-even scalars. Consequently, it is easier to detect CP-odd spin-0 states than the CP-even ones. \n\\end{enumerate}\n\n\\subsection{In $\\tau\\bar{\\tau}$ channel}\n\nRecently, ATLAS \\cite{ATLAS_tau} and CMS \\cite{CMS_tau} also reported their new results in $\\tau\\bar{\\tau}$ channel. Although, the main aim of their reports is to look for MSSM neutral boson, they provide a model independent limit on the production cross section times branching ratio of a general spin-zero state. Therefore, in this part we investigate the signal strength of our $H_{3,3M}^0\\rightarrow\\tau\\bar{\\tau}$ with two sets of parameters considered in the previous subsection.\n\n  \\begin{itemize}\n   \\item In the {\\em Dr.~Jekyll} case. \n   \\begin{figure}[H]\n    \\includegraphics[scale=0.45]{XBranching_tau_H1oDominant.pdf}\n    \\caption{\\label{fig:tau_H1oDominant} {\\small The production cross section times branching ratio in $\\tau\\bar{\\tau}$ channel of $H_3^0$ and $H_{3M}^0$, in {\\em Dr.~Jekyll} scenario. $m_{H_3^0}=130 - 440 \\;{\\rm GeV }$, $m_{H_{3M}^0}=177 - 601\\;{\\rm GeV }$ } }\n    \\end{figure}\n    \\item In the {\\em Mr.~Hyde} case.\n     \\begin{figure}[H]\n       \\includegraphics[scale=0.45]{XBranching_tau_H1oSubdominant.pdf}\n    \\caption{\\label{fig:tau_H1oSubdominant} {\\small The production cross section times branching ratio in $\\tau\\bar{\\tau}$ channel of $H_3^0$ and $H_{3M}^0$, in {\\em Mr.~Hyde} scenario. $m_{H_3^0}=130 - 440 \\;{\\rm GeV }$, $m_{H_{3M}^0}=182 - 618\\;{\\rm GeV }$ }} \n      \\end{figure}\n\\end{itemize}\nRemarks:\n\\begin{enumerate}\n  \\item\n   In both the cases, the signal strengths can exceed the upper limit from ATLAS and CMS before the thresholds of 2 times mirror fermion masses, which is $204\\;{\\rm GeV }$. It happens because, unlike SM Higgs, the decay processes such as $H_{3,3M}^0\\rightarrow WW\/ZZ$ occur only at the loop level, and their partial widths are relatively small. Consequently, the branching ratios of $H_{3,3M}^0\\rightarrow\\tau\\bar{\\tau}$ are not as small as in SM. Hence, the signal strength for this channel is one order above the upper limits set by ATLAS and CMS. However, we believe that in a wide range of parameter space of the $EW\\nu_R$ model, it is still possible to agree with the limits in this mass region.    \n\t\\item\n\tAfter passing the first threshold, the signal strengths of both $H_{3,3M}^0\\rightarrow\\tau\\bar{\\tau}$ decrease rapidly, because the total widths $\\Gamma_{H_{3,3M}^0}$ are dominated by the fermionic decays. Then they reach another peak at $2m_t$. Over the entire region after the first threshold, the signal strengths for both $H_{3,3M}^0$ are below the limits.\n  \\end{enumerate}\n\n\\subsection{In $WW\/ZZ$  channels}\n\nIn this model, pseudo-scalars $H_{3,3M}^0$ do not couple directly to $W^\\pm$ and $Z$. Decay processes $H_{3,3M}^0 \\rightarrow WW\/ZZ$ occur only at loop levels. It is expected that these processes will be highly suppressed. To prove that, we calculate the signal strengths ($\\mu$) for $H_{3}^0\\rightarrow WW\/ZZ$ and $H_{SM}\\rightarrow WW\/ZZ$. $\\mu$ is defined in Eq.~(\\ref{eq:muy}). \n\\begin{eqnarray}\\label{eq:muyVV}\n\\mu_{VV} &=& \\dfrac{\\sigma(gg \\rightarrow H_3^0) Br(H_3^0 \\rightarrow VV)}{\\sigma(gg \\rightarrow H_{SM}) Br(H_{SM} \\rightarrow VV)}\\nonumber\\\\[1em]\n&=& R_{H_3^0}\\dfrac {Br(H_3^0 \\rightarrow VV)}{ Br(H_{SM} \\rightarrow VV)}\\,,\n\\end{eqnarray}\nwhere $V=W,Z$. Once again, $Br(H_{SM} \\rightarrow VV)$ is taken from \\cite{HXreport3}, while the ratio of production cross sections $R_{H_3^0}$ in Eq.~(\\ref{eq:csectionapprox}) and $Br(H_3^0 \\rightarrow VV)$ are calculated directly. At one loop order, the partial decay width for these processes are \\cite{GunionKao}\n\\begin{itemize}\n\\item $H_3^0 \\rightarrow WW$\n \\begin{equation}\\label{eq:H3oWW}\n \\Gamma( H_3^0 \\rightarrow WW ) = \\dfrac{3^2g^6(m_{H_3^0}^2 - 4m_{W}^2)^{3\/2}}{2^{14}\\pi^5m_W^2}|\\textit{A}_{WW}|^2\n \\end{equation}\n \\begin{equation}\\label{eq:Aww}\n\\textit{A}_{WW} = m_t^2t_MA_t^W - m_b^2t_M^2A_b^W + \\dfrac{m_{l^M}^2}{\\sqrt{2}}A_{l^M}^W + \\dfrac{M_R^2}{\\sqrt{2}c_M}A_{\\nu_R}^W\\,;\n \\end{equation}\n\\item $H_3^0\\rightarrow ZZ$\n \\begin{equation}\\label{eq:H3ozz}\n \\Gamma( H_3^0 \\rightarrow ZZ ) = \\dfrac{3^2g^6(m_{H_3^0}^2 - 4m_{Z}^2)^{3\/2}}{2^{15}\\pi^5m_W^2}|\\textit{A}_{ZZ}|^2\n \\end{equation}\n \\begin{equation}\\label{eq:Azz}\n\\textit{A}_{ZZ} = m_t^2t_MA_t^Z - m_b^2t_MA_b^Z + \\dfrac{m_{l^M}^2}{\\sqrt{2}}A_{l^M}^Z\\,.\n \\end{equation}\n\\end{itemize}\n$A_f^{W\/Z}$ are amplitudes with top and bottom quarks, mirror charged leptons, and right-handed neutrinos in the loops. They have specific forms given in the Appendix~\\ref{sec:AH3}.\n\n \\begin{figure}[H] \n   \\includegraphics[scale=0.45]{MuyH3oHsm_WWZZ_170_450.pdf}\n   \\caption{\\label{fig:muyWW}{\\small Ratio of strength signal in $WW\/ZZ$ channel of $H_3^0$ comparing to $H_{SM}$}}\n  \\end{figure}\nAs expected, the signal strengths of $H_3^0$ in vector boson channels are highly suppressed. \n\n\\subsection{$(H_{3}^0 \\rightarrow \\bar{l}^M l^M )\/ (H_{SM}\\rightarrow WW)$}\n\nAt the LHC, $H\\rightarrow WW$ is an important channel to search for new scalars in the high-mass region. From Fig.~\\ref{fig:muyWW}, we see that the ratio of the signal strengths is suppressed very much for pseudoscalar $H_3^0$ in this model. However, it can also decay through $H_{3}^0 \\rightarrow \\bar{l}^M l^M \\rightarrow \\bar{l} \\phi_S l \\phi_S^* $, where $\\phi_S$ is invisible and considered as missing transverse energy ($E_T^M$). Thus, the signal of this process is identified with 2 leptons and missing $E_T^{M}$, which mimics the signal for $WW$ decay of a scalar such as the SM Higgs boson: $H_{SM} \\rightarrow W^+ W^- \\rightarrow \\bar{l} \\nu l \\bar{\\nu}$. For the case $m_{l^M} = 102~{\\rm GeV }$, $m_H = 210 - 500 {\\rm GeV }$, both the mirror lepton intermediate states are on-shell.\n\\begin{equation}\\label{eq:H3olmlm}\n\\Gamma \\left(H_{3}^0 \\rightarrow \\bar{l}^M l^M\\right) =  \\frac{g^2m_{H_3^0}^3t_M^2}{256\\pi ~m_W^2} \\tau_{l^M}\\sqrt{1 - \\tau_{l^M}}\\,, \n\\end{equation}\nwhere $\\tau_{l^M}= 4m_{l^M}^2\/m_{H_3^0}^2$, and\n\\begin{equation}\\label{eq:lMlphiS}\n \\Gamma \\left(l^M\\rightarrow l\\phi_S^*\\right) = g_{sl}^2\\dfrac{m_{l^M}}{32\\pi}\\,.\n\\end{equation} \nSimilar to the previous comparisons, we define the signal strength\n\\begin{equation}\\label{eq:RH3oHsm} \n  \\mu_{l\\bar{l}}=R_{H_3^0} \\frac{Br\\left(H_{3}^0 \\rightarrow \\bar{l}^M l^M\\right) Br\\left(l^M \\rightarrow l\\phi_S^* \\right)}{Br \\left( H_{SM} \\rightarrow W^+ W^- \\right) Br\\left(  W\\rightarrow l\\bar{\\nu}\\right)}\\,,\n\\end{equation}\nwhere $Br\\left(  W^-\\rightarrow l\\bar{\\nu}\\right) = 0.108 $ \\cite{pdg}. With $M_R=50\\;{\\rm GeV }$, $l^M\\rightarrow \\nu_R\\bar{\\nu} l$ is kinamatically possible.\n\\begin{equation}\\label{eq:BrlMPhiS}\n Br\\left(l^M \\rightarrow l\\phi_S^* \\right) = \\dfrac{\\Gamma \\left(l^M\\rightarrow l\\phi_S^*\\right)}{\\Gamma \\left(l^M\\rightarrow l\\phi_S^*\\right)+\\Gamma \\left(l^M\\rightarrow \\nu_R\\bar{\\nu} l\\right)}\\,.\n\\end{equation}\nIt is clear that the ratio $\\mu_{l\\bar{l}}$ depends on the value of $g_{sl}$. The search for a high-mass Higgs boson in $H\\rightarrow W^+W^-\\rightarrow l\\bar{\\nu} \\bar{l}\\nu$ was carried out at both ATLAS (in the range $260 - 1000 \\;{\\rm GeV }$ \\cite{ATLAS-CONF-2013-067}) and CMS (in the range $145 - 1000\\;{\\rm GeV }$) \\cite{CMS_heavy_Higgs_WW}. No excess over the background was detected in the entire mass region that was scanned. \n\nThe observed $95\\%$ CL upper limit on the ratio of the signal strengths is below $\\mu=1$ all the way up to $m_H\\approx 600\\;{\\rm GeV }$ \\cite{CMS_heavy_Higgs_WW}. Therefore, we can set upper limits as $\\mu_{l\\bar{l}}\\leq 1$ and hence, $g_{sl}\\leq 10^{-3}$. \n\n\n\\section{Conclusions}\n\nThe 125-{\\rm GeV } object has presented us with a challenge to understand its nature: Is it really the SM Higgs boson as it appears to be or is it simply an {\\em impostor}? So far, the only data available to us are given in terms of the so-called signal strengths, $\\mu$, as defined in Eq.~(\\ref{eq:mudef}). The signal strengths for the various decay modes of the SM Higgs boson are consistent with data. However,  It turns out that it might be possible for various BSM models to be consistent with experiment also based solely on such signal strengths. This is what we have shown in this paper in the context of the $EW\\nu_R$ in its extended version.\n\nAs we have described in the beginning of our paper, the $EW\\nu_R$ \\cite{pqnur} was invented with the purpose of realizing the seesaw mechanism at the electroweak scale instead of some GUT scale. As such one can {\\em directly} test the seesaw mechanism at the LHC and at the proposed ILC through the physics associated with the model such as lepton-number violating production of electroweak-scale Majorana right-handed neutrinos and -this is the subject of the present paper- Higgs physics beyond that of the SM. \n\nThe extended $EW\\nu_R$ model discussed in this paper contains three neutral CP-even mass eigenstates, $\\widetilde{H} $, $\\widetilde{H}^\\prime $ and $\\widetilde{H}^{\\prime\\prime} $, which are linear combinations of $H_{1}^{0}$, $H_{1M}^0$ which couple to SM fermions and mirror fermions respectively and $H_1^{0\\prime}$ which couples only to $\\nu_R$'s. The notation for the mass eigenstates $\\widetilde{H} $, $\\widetilde{H}^\\prime $ and $\\widetilde{H}^{\\prime\\prime} $ refers to states with increasing masses. We scanned the parameter space with the following requirements in mind: 1) The mass of the lightest state should be $\\sim 125 \\, {\\rm GeV }$; 2) The mixing angles should be such that the signal strengths fit the data from CMS and ATLAS. We found many combinations of $H_{1}^{0}$, $H_{1M}^0$ and $H_1^{0\\prime}$ which satisfy those requirements. What is interesting here is the dual nature of the 125-{\\rm GeV } scalar that we uncovered in our scanning of the parameter space: 1) There are states with the SM-like scalar $H_{1}^{0}$ as a dominant component; 2) There are states with $H_1^{0\\prime}$ as a dominant component and is thus {\\em very unlike} that of the SM model. In other words, these states are impostors. All of these states - and we are far from exhausting the parameter space - yield the signal strengths compatible with the CMS and ATLAS data.\n\nIt goes without saying that detailed studies of various properties of the 125-{\\rm GeV }  scalar such as the total width, partial widths, ..., are needed to determine if it were indeed the SM Higgs boson or just simply an {\\em impostor}. Of course, a discovery of one or several extra scalars would definitely point toward physics beyond the SM. In the extended $EW\\nu_R$ model, although the aforementioned 125-{\\rm GeV }-like scalars all yield comparable signal strengths, details such as production cross sections, branching ratios, total widths and partial widths can differ quite a bit from one another. States with $H_{1}^{0}$ as a dominant component ({\\em Dr.~Jekyll}) tend to behave more like the SM Higgs boson while the scenario in which $H_{1}^{0}$ as a sub-dominant component ({\\em Mr.~Hyde}) is very different. In other words, we may have discovered a scalar which is involved in the electroweak symmetry breaking but which {\\em may not be} the SM Higgs boson. \n\nIf, in the absence of direct measurements of decay widths, how could one tell {\\em Dr.~Jekyll} from {\\em Mr.~Hyde}? First, it goes without saying that a discovery of one or more extra scalars will definitely point to BSM physics. In the context of the EW$\\nu_R$  model, if the extra states are {\\em broad} and strongly interacting, we would be dealing with {\\em Dr.~Jekyll}'s scenario which is more SM-like in terms of the 125-{\\rm GeV } scalar. On the other hand, if the extra states are narrow resonances, we would be facing a truly interesting scenario, that of {\\em Mr.~Hyde} in which the 125-{\\rm GeV } scalar is truly an {\\em impostor}. Direct measurements of decay widths in this case would confirm whether or not this is the case.\n\n\n\n\n\\section{Acknowledgements}\n\nWe would like to thank Giuseppe Cerati for providing results of the search for SM-like heavy Higgs boson at CMS. This work was supported by US DOE grant DE-FG02-97ER41027. ASK was supported by the Graduate Fellowship of the Department of Physics, University of Virginia.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Intro}\nIn recent years, distributionally robust optimization  formulations\nbased on Wasserstein distances have sparked a substantial amount of\ninterest in the literature. One reason for this interest, as\ndemonstrated by a range of examples in statistical learning\nand operations research,\nis that these formulations provide a flexible way to quantify and\nhedge against the impact of model misspecification.  Motivated by those\napplications, this paper aims to understand the fundamental\nstatistical properties, such as asymptotic normality of the\ndistributionally robust estimators and the associated confidence\nregions deemed optimal in a suitable sense to be described shortly.\n\nBefore providing a review of Wasserstein distributionally robust\noptimization and its connections to several areas, such as artificial\nintelligence, machine learning and operations research, we set the\nstage by first introducing the elements of a typical data-driven\ndistributionally robust estimation problem.\n\n Suppose that\n$\\{X_k: 1 \\leq k \\leq n\\} \\subset \\mathbb{R}^m$ are independent and\nidentically distributed samples\nfrom an unknown distribution $P_\\ast.$ A typical non-robust stochastic\noptimization formulation informed by $P_{n}$ focuses on minimizing\nempirical expected loss of the form,\n$E_{P_{n}}\\left\\{ \\ell (X;\\beta )\\right\\} = n^{-1} \\sum_{i=1}^n\n\\ell(X_i;\\beta)$, over the parameter choices\n$\\beta \\in B \\subseteq \\mathbb{R}^d.$ In this paper, we take $B$ to be\na closed, convex subset of $\\mathbb{R}^d.$\nLet the empirical risk minimization estimators be\n\\begin{equation}\n  \\beta _{n}^{ERM}\\in \\text{arg}\\min_{\\beta \\in B}E_{P_{n}}\\left\\{ \\ell (X;\\beta\n    )\\right\\}.  \\label{ERM_1}\n\\end{equation}%\n\n\nOn the other hand, a distributionally robust formulation recognizes\nthe distributional uncertainty inherent in $P_{n}$ being a noisy\nrepresentation of an unknown distribution. Therefore, it enriches the\nempirical risk minimization (\\ref{ERM_1}) by considering an estimator\nof the form,\n\\begin{equation}\n  \\beta _{n}^{DRO}(\\delta )\\in \\arg \\min_{\\beta \\in B}\\sup_{P\\,\\in \\,\\mathcal{U}%\n    _{\\delta }(P_{n})}E_{P}\\left\\{ \\ell (X;\\beta )\\right\\} ,\n\\label{Wass_DRO_A}\n\\end{equation}%\nwhere the set $\\mathcal{U}_{\\delta }(P_{n})$ is called the\ndistributional uncertainty region and $\\delta $ is the size of the\ndistributional uncertainty. Here, given a measurable function\n$f(\\cdot),$ the notation $E_P\\{f(X)\\}$ denotes expectation\nwith respect to a probability distribution $P.$ Wasserstein\ndistributionally robust formulations advocate choosing,\n\\begin{equation*}\n  \\mathcal{U}_{\\delta }(P_{n})=\\{P \\in \\mathcal{P}(\\Omega):\n  W(P_{n},P)\\leq \\delta^{1\/2}\\},\n\\end{equation*}%\nwhere $W(P_{n},P)$ is the Wasserstein distance between distributions\n$P_{n}$ and $P$ defined below, and $\\mathcal{P}(\\Omega)$ is the set of probability\ndistributions supported on a closed set\n$\\Omega \\subseteq \\mathbb{R}^m.$\n\\begin{definition}[Wasserstein distances]\n  \\textnormal{Given a lower semicontinuous function\n    $c:\\Omega \\times \\Omega \\rightarrow [0,\\infty],$ the optimal\n    transport cost $D_c(P,Q)$ between any two distributions\n    $P,Q \\in \\mathcal{P}(\\Omega)$ is defined as,\n    \\begin{align*}\n      D_c(P,Q) = \\min_{\\pi \\in \\Pi(P,Q)}  E_{\\pi}\\left\\{c(X,Y)\\right\\}\n    \\end{align*}\n    where $\\Pi(P,Q)$ denotes the set of all joint distributions of the\n    random vector $(X,Y)$ with marginal distributions $P$ and $Q,$\n    respectively. If we specifically take $c(x,y) = d(x,y)^2,$ where\n    $d(\\cdot)$ is a metric, we obtain the Wasserstein distance of\n    order 2 by setting\n    $\n      W(P,Q) = \\left\\{D_c(P,Q)\\right\\}^{1\/2}.\n    $\n  }\n  \\label{defn:WD}\n\\end{definition}\n\n\nThe quantity $W(P_{n},P)$ may be interpreted as the cheapest way to\ntransport mass from the distribution $P_{n}$ to the mass of another\nprobability distribution $P,$ while measuring the cost of\ntransportation from location $x \\in \\Omega$ to location $y \\in \\Omega$\nin terms of the squared distance between $x$ and $y.$ In this paper,\nwe shall work with Wasserstein distances of order 2, which explains\nwhy it is natural to use $\\delta^{1\/2}$ to specify the distributional\nuncertainty region $\\mathcal{U}_{\\delta }(P_{n})$ as above.  Since\n$W(P_n,P_n) = 0,$ the empirical risk minimizing estimator in\n(\\ref{ERM_1}) can be seen as a special case of the formulation\n(\\ref{Wass_DRO_A}) by setting $\\delta = 0.$\n\n  The need for selecting model parameters or making decisions using a data driven approach which is robust to model uncertainties has sparked a rapidly growing literature on Wasserstein distributionally robust optimization, via formulations such as  (\\ref{Wass_DRO_A}); see, for example,\n\\citet{MohajerinEsfahani2017,ZHAO2018262,blanchet2016quantifying,gao2016distributionally,gao2018robust,wolfram2018}\nfor applications in operations research and\n\\cite{yang2017convex,yang2018Wasserstein} for examples\nspecifically in stochastic control.\n\nIn principle, the min-max formulation (\\ref{Wass_DRO_A}) is\n``distributionally robust'' in the\nsense that its solution guarantees a uniform performance over all\nprobability distributions in $%\n\\mathcal{U}_{\\delta_{n}}(P_{n}).$ Roughly speaking, for every choice\nof parameter or decision $\\beta,$ the min-max game type formulation in\n(\\ref{Wass_DRO_A}) introduces an adversary that chooses the most\nadversarial distribution from a class of distributions\n$\\mathcal{U}_{\\delta_{n}}(P_{n}).$ The goal of the procedure is to\nthen choose a decision that also hedges against these adversarial\nperturbations, thus introducing adversarial robustness into settings\nwhere the quality of optimal solutions are sensitive to incorrect\nmodel assumptions.\n\n\nInterestingly, the min-max formulation (\\ref{Wass_DRO_A}), which is\nderived from the above robustness viewpoint, has been shown to recover\nmany machine learning estimators when applied to suitable loss\nfunctions $\\ell(\\cdot)$; some examples include the square-root lasso and\nsupport vector machines \\citep{blanchet2016robust}, the group lasso\n\\citep{blanchet2017distributionally2}, adaptive regularization\n\\citep{volpi2018generalizing,blanchet2017data}, among others\n\\citep{shafieezadeh-abadeh_distributionally_2015,gaostatlearn,\n  duchidistributionally,chen2018robust}.\nThe utility of the distributionally robust formulation\n(\\ref{Wass_DRO_A}) has also been explored in adversarial training of\nNeural Networks; see, for example\n\\citet{sinha2018certifiable,staib2017distributionally}.\n\n\nGeneric formulations such as (\\ref{Wass_DRO_A}) are becoming increasingly\ntractable; see, for example, %\n\\citet{MohajerinEsfahani2017,luo2017decomposition} for convex programming\nbased approaches and \\citet{sinha2018certifiable,blanchet2018optimal} for\nstochastic gradient descent based iterative schemes.\n\n\nMotivated by these wide range of applications, we investigate the asymptotic behaviour of the optimal value and optimal solutions of (\\ref{Wass_DRO_A}). In order to\nspecifically describe the contributions, let us introduce the following\nnotation. For any positive integer $n$ and $\\delta _{n}>0,$ let\n\\begin{equation*}\n\\Psi _{n}(\\beta )=\\sup_{P\\,\\in \\,\\mathcal{U}_{\\delta\n_{n}}(P_{n})}E_{P}\\left\\{ \\ell (X;\\beta )\\right\\}\n\\end{equation*}%\ndenote the distributionally robust objective function in\n\\eqref{Wass_DRO_A}. Suppose that $\\beta _{\\ast }$ uniquely minimizes\nthe population risk. According to (\\ref{ERM_1}) - (\\ref{Wass_DRO_A}),\nwe have $\\beta_n^{DRO}$ and $\\beta_n^{ERM}$ minimize, respectively,\nthe distributionally robust loss $\\Psi_n(\\beta)$ and the empirical\nloss in (\\ref{ERM_1}). Next, let\n\\begin{equation}\n\\Lambda _{\\delta _{n}}(P_{n})=\\big\\{ \\beta \\in B:\\beta \\in\n\\arg \\min_{\\beta  \\in B }E_{P}\\left\\{ \\ell (X;\\beta )\\right\\} \\text{ for some }P\\in\n\\mathcal{U}_{\\delta _{n}}(P_{n})\\big\\}\n\\label{NaturalCR}\n\\end{equation}\ndenote the set of choices of $\\beta \\in B$ that are ``compatible'' with the distributional uncertainty\nregion, in the sense that for every\n$\\beta \\in \\Lambda _{\\delta _{n}}(P_{n}),$ there exists a probability\ndistribution $P\\in \\mathcal{U}%\n_{\\delta _{n}}(P_{n})$ for which $\\beta $ is optimal. In other words,\nif $\\mathcal{U}_{\\delta _{n}}(P_{n})$ represents the set of\nprobabilistic models which are, based on the empirical evidence,\nplausible representations  of the\nunderlying phenomena, then each of such representations induces an\noptimal decision and $\\Lambda _{\\delta _{n}}(P_{n})$ encodes the set of plausible decisions. Let\n  $\\Lambda^+ _{\\delta _{n}}(P_{n})$ be the closure of $\\cap_{\\epsilon>0}\\Lambda_{\\delta_n+\\epsilon}(P_n)$. Typically,  $\\Lambda^+_{\\delta _{n}}(P_{n}) = \\Lambda_{\\delta _{n}}(P_{n})$, but this is not always true as illustrated in Example \\ref{eg:Lambda-plus}. Asymptotically, as $\\delta_n$ decreases to zero, the distinction is negligible. However, choosing a set such as $\\Lambda^+ _{\\delta _{n}}(P_{n})$ as a natural set of plausible decisions is sensible because we guarantee that a distributionally robust solution belongs to this region. Our main result also implies that all distributionally robust solutions are asymptotically equivalent; within $o_p(n^{-1\/2})$ distance from each other.\n\n{\nWith the above notation, the key contributions of this article can be\ndescribed as follows.\n\nWe first establish the convergence in distribution of the triplet,\n\\begin{equation}\n\\big( n^{1\/2}\\{\\beta_{n}^{ERM}-\\beta_{\\ast}\\},\\ n^{\\bar{\\gamma}\/2}\\{\\beta\n_{n}^{DRO}(\\delta_n)-\\beta_{\\ast}\\},\\ n^{1\/2}\\big\\{\n\\Lambda^+_{\\delta_{n}}(P_{n} )-\\beta_{\\ast}\\big\\} \\big) ,\n\\label{MLT-triplet}\n\\end{equation}\nfor a suitable $\\bar{\\gamma}\\in(0,1\/2]$ that depends\n  on the rate at which the size of the distributional uncertainty,\n$\\delta_{n}$%\n, is decreased to zero; see Theorem\n  \\ref{thm:levelsets-master}. We identify the joint limiting\n  distributions of the triplet \\eqref{MLT-triplet}. The third\ncomponent of the triplet in \\eqref{MLT-triplet}, namely, $%\nn^{1\/2}\\{\\Lambda^+_{\\delta_{n}}(P_{n})-\\beta_{\\ast}\\}$, considers a\nsuitably scaled and centered version of the choices of $\\beta \\in B$\nwhich are compatible with the respective distributional uncertainty\nregion $\\mathcal{U}_{\\delta_{n}}(P_{n})$ in the sense described\nabove. Therefore,  $\\Lambda^+_{\\delta_{n}}(P_{n})$ is a natural choice of the confidence region. We further develop an approximation for $\\Lambda^+_{\\delta_{n}}(P_{n})$; see Section \\ref{Sec_Confidence_Regions}.\n\nSecond, we utilize\nthe limiting result of \\eqref{MLT-triplet} to examine how the choice\nof the size of distributional ambiguity, $\\delta_{n}$, affects the\nqualitative properties of the distributionally robust estimators and\nthe induced confidence regions. Specifically, choosing\n$\\delta_{n}=\\eta n^{-\\gamma}$, we characterize the behaviour of the\nsolutions for different choices of $\\eta,\\gamma\\in (0,\\infty),$ as\n$n\\rightarrow\\infty$. It emerges that the canonical, $%\nO(n^{-1\/2})$, rate of convergence is achieved only if $\\gamma\\leq1$\nand the limiting distribution corresponding to the distributionally\nrobust estimator and that of the empirical risk minimizer\nare different only if $\\gamma\\geq1$. Hence to both\nobtain the canonical rate and tangible benefits from the distributionally robust optimization\nformulation, we must choose $\\gamma=1$, which corresponds to the\nresulting $\\bar{\\gamma}$ in \\eqref{MLT-triplet} to be equal to 1.\nMoreover, given any $\\alpha\\in(0,1)$, utilizing the limiting distribution of\nthe triplet in \\eqref{MLT-triplet}, we are able to identify a positive\nconstant $\\eta_{\\alpha}\\in(0,+\\infty)$ such that whenever $\\eta\\geq\n\\eta_{\\alpha}$ in the choice $\\delta_{n}=\\eta\/n$, the set $\\Lambda^+\n_{\\delta_{n}}(P_{n})$ is an asymptotic $(1-\\alpha)$-confidence region for $%\n\\beta_{\\ast}$.\n\n\n\n\n\nFinally, we establish the existence of an equilibrium game value. The distributionally robust optimization formulation assumes that the adversary selects a probability model after the statistician chooses a parameter. The equilibrium value of the game is attained if inf-sup equals sup-inf in (\\ref{Wass_DRO_A}), namely, if we allow the statistician to choose a parameter optimally after the adversary selects a probability model. We show in great generality that the equilibrium value of the game exists; see Theorem \\ref{minmax_prop}.}\n\n\n\n\n\n\n\n\n\nWe end the introduction with a discussion of related statistical results.\nThe asymptotic normality of M-estimators which minimize an empirical risk of\nthe form, $E_{P_{n}}\\{\\ell(X;\\beta)\\},$ was first established in the\npioneering work of \\citet{huber1967}.  Subsequent asymptotic\ncharacterizations in the presence of constraints on the choices of\nparameter vector $\\beta$ have been developed in\n\\citet{dupacova1988,shapiro1989,Shapiro1991,shapiromoor,shapiro2000},\nagain in the standard M-estimation setting.  Our work here is\ndifferent because of the presence of the adversarial perturbation to\nthe loss represented by the inner maximization in (\\ref{Wass_DRO_A}).\n\nAsymptotic normality in the related context of regularized estimators for\nleast squares regression has been established in %\n\\citet{knight2000asymptotics}. As mentioned earlier, distributionally robust\nestimators of the form (\\ref{Wass_DRO_A}) recover lasso-type estimators as\nparticular examples \\citep{blanchet2016robust}. In these cases, the\ninner max problem involving the adversary can be solved in closed form,\nresulting in the presence of regularization. However, our results can be\napplied even in the general context in which no closed form solution to the\ninner maximization can be obtained. Therefore, our results in this paper can\nbe seen as extensions of the results by \\citet{knight2000asymptotics}, from\na distributionally robust optimization perspective.\n\nWe comment that some of our results involving convergence of sets may\nbe of interest to applications in the area of empirical likelihood %\n\\citep{owen_empirical_1988,owen_empirical_1990,owen2001empirical}. This\nis because $\\Lambda_{\\delta_{n}}( P_{n}) $ can be characterized in\nterms of a function, namely, the robust Wasserstein profile function,\nwhich resembles the definition of the empirical likelihood profile\nfunction. We refer the reader to \\citet{blanchet2016robust} for more discussion on the robust Wasserstein profile function and its connections to empirical likelihood. We also refer to \\citet{cisneros2019distributionally} for additional applications, including graphical lasso, which could benefit from our results.\n\n\n\\label{sec:assumption_results}\n\n\\section{Preliminaries and Assumptions}\n\n\n\n\n\n\n\\subsection{Convergence of closed sets}\n\\label{SectConv_Sets}\nWe begin with a brief introduction to the notion of convergence of\nclosed sets before introducing the assumptions required to state our\nmain results.  For a sequence $\\{A_k: k \\geq 1\\}$ of closed subsets of\n$\\mathbb{R}^d,$ the inner and outer limits are defined, respectively,\nby\n\\begin{align*}\n\\text{Li}_{n \\rightarrow \\infty}\\ A_n = &\\big\\{ z \\in \\mathbb{R}^d: \\text{\nthere exists a sequence } (a_n)_{n \\geq 1} {\\text{with } a_n \\in A_n} \\text{ convergent to } z\\},\n \\text{ and } \\\\\n\\text{Ls}_{n \\rightarrow \\infty}\\ A_n = &\\big\\{ z \\in \\mathbb{R}^d: \\text{\nthere exist positive integers } n_1 < n_2 < n_3 < \\cdots \\text{ and } a_k\n\\in A_{n_k}  \\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad  \\text{ such that the sequence }\n(a_k)_{k \\geq 1} \\text{ is convergent to } z\\big\\}.\n\\end{align*}\nWe clearly have\n$\\text{Li}_{n \\rightarrow \\infty}\\ A_{n}\\subseteq \\text{Ls}_{n \\rightarrow\n\\infty}\\ A_{n }.$\nThe sequence $\\{A_{n}:n\\geq 1\\}$ is said to converge to a set $A$ in the\nPainlev\\'{e}-Kuratowski (PK) sense if\n\\begin{equation*}\nA= \\text{Li}_{n\\rightarrow \\infty}\\ A_{n}=\\text{Ls}_{n \\rightarrow \\infty} \\\nA_{n},\n\\end{equation*}%\nin which case we write PK-$\\lim_{n}A_{n}=A$. Since $\\mathbb{R}^{d}$ is a\nlocally compact Hausdorff space, the topology induced by Painlev\\'{e}%\n-Kuratowski convergence on the space of closed subsets\n                                                      \nof $\\mathbb{R}^{d}$ is completely metrisable, separable and coincides\nwith the well-known topology of closed convergence, also known as Fell\ntopology; see \\citet[Chapter 1]{molchanov2005theory}. The notion of\nconvergence of sets we utilize here will be the above defined\nPainlev\\'{e}-Kuratowski convergence. { After equipping the space of closed subsets with the Borel $\\sigma$-algebra, we are able to define probability measures and further define the usual weak convergence of measures; see, for example, \\citet[Chapter 1]{billingsley2013convergence}.}\n\n\\subsection{Assumptions and notation}\nThroughout the paper, we use $A\\succ 0$ to denote that a given\nsymmetric matrix $A$ is positive definite and the notation $C^\\circ$ and ${\\rm cl} (C)$\nto denote the interior and closure of a subset $C$ of Euclidean space, respectively. In the case\nof taking expectations with respect to the data-generating distribution\n$P_\\ast,$ we drop the subindex in the expectation operator as in,\n$E_{P_\\ast} \\left\\{ f(X)\\right\\} = E\\left\\{ f(X)\\right\\}.$ We use $\\Rightarrow$ to denote weak convergence and $\\rightarrow$ to denote convergence in probability. We let $\\mathbb{I}(\\cdot)$ be the indicator function. { Let $\\|\\cdot\\|_p$ be the dual norm of  $\\|\\cdot\\|_q$ where $1\/p + 1\/q = 1$ for $q\\in (1,\\infty),$ and $p = \\infty$ or $1$ for $q=1$ or $\\infty$, respectively.}\n\nAs mentioned in Section \\ref{sec:Intro}, suppose that $\\Omega$ is a\nclosed subset of $\\mathbb{R}^m$ and $B$ is a closed, convex subset of\n$\\mathbb{R}^d.$ Assumptions A1 and A2 below are taken to be satisfied\nthroughout the development, unless indicated otherwise.\n\n(A1) The transportation cost\n$c:\\Omega \\times \\Omega \\rightarrow [0,\\infty]$ is of the form\n$c(u,w)=\\Vert u-w\\Vert_{q}^{2}$.\n\n(A2)  The function\n$\\ell:\\Omega \\times B \\rightarrow \\mathbb{R}$ satisfies the following\nproperties:\n\\begin{itemize}\n\\vspace*{-3mm}\n\\item[a)] The loss function $\\ell(\\cdot)$ is twice continuously differentiable,\n  and for each $x$, $\\ell(x,\\cdot)$ is convex.\n\\item[b)] Let $h(x,\\beta)=D_{\\beta}\\ell(x,\\beta)$, and there\n  exists $\\beta_{\\ast}\\in B^\\circ$ satisfying the optimality condition\n  $E\\{ h(X,\\beta_{\\ast })\\} = 0.$ In addition,\n  $E \\{\\Vert h(X,\\beta_{\\ast })\\Vert _{2}^{2}\\} <\\infty,$ the\n   symmetric matrix\n  $C=E\\left\\{ D_{\\beta }h(X,\\beta_{\\ast})\\right\\} \\succ 0,$\n  $E\\left\\{ D_{x}h(X,\\beta_{\\ast})D_{x}h(X,\\beta_{\\ast})^{\\mathrm{\\scriptscriptstyle T} }\\right\\}\n  \\succ 0$, and { $\\mathrm{pr}\\{\\|D_x\\ell(X,\\beta_*)\\|_p >0\\}>0$}\n\n\\item[c)] For every $\\beta \\in \\mathbb{R}^d,$\n  $\\| D_{xx}\\ell(\\,\\cdot\\,;\\beta) \\|_p$ is  uniformly continuous and bounded by a continuous\n  function $%\n  M(\\beta)$. {Further, there exists a positive constant $M'<\\infty$ such that $\\|D_xh(x,\\beta)\\|_q \\leq M'(1+\\|x\\|_q) $  for $\\beta$ in a neighborhood of $\\beta_*$}. In addition, $D_xh(\\cdot)$ and\n  $D_\\beta h(\\cdot)$ satisfy the following locally Lipschitz\n  continuity:\n\\begin{align*}\n\\left\\Vert D_{x}h(x+\\Delta,\\beta_{\\ast}+u)-D_{x}h(x,\\beta_{\\ast})\\right\\Vert\n_{q} & \\leq\\kappa^{\\prime}(x)\\big( \\left\\Vert \\Delta\\right\\Vert\n_{q}+\\left\\Vert u\\right\\Vert _{q}\\big), \\\\\n\\left\\Vert D_{\\beta}h(x+\\Delta,\\beta_{\\ast}+u)-D_{\\beta}h(x,\\beta_{\\ast\n})\\right\\Vert _{q} & \\leq\\bar{\\kappa}(x)\\big(\\left\\Vert \\Delta\\right\\Vert\n                     _{q}+\\left\\Vert u\\right\\Vert _{q}\\big),\n\\end{align*}\nfor $ \\left\\Vert\\Delta\\right\\Vert\n_{q}+\\left\\Vert u\\right\\Vert _{q} \\leq 1$, where $\\kappa^{\\prime},\\bar{\\kappa}:\\mathbb{R}^{m}\\rightarrow\\lbrack0,%\n\\infty) $ are such that $E[\\{\\kappa^{\\prime}(X_{i})\\}^{2}]<\\infty$ and $E\\{%\n\\bar {\\kappa}^{2}(X_{i})\\}<\\infty.$\n\\end{itemize}\n\n\nAssumption A1 covers most of the cases in the literature described in\nSection \\ref{sec:Intro}. One exception that does not immediately\nsatisfy Assumption A1, but which can be easily adapted after a simple\nchange-of-variables, is the weighted $l_{2}$ norm (also known as\nMahalanobis distance), namely\n$c( x,y) =( x-y) ^{{\\ \\mathrm{\\scriptscriptstyle T} }}A( x-y) $, where\n$A \\succ 0,$ see \\citet{blanchet2018optimal}.\nThe requirement that $\\ell (\\cdot )$ is twice differentiable in\nAssumption A2.a is useful in the analysis to identify a second-order\nexpansion for the objective in (\\ref{Wass_DRO_A}), which helps\nquantify the the impact of adversarial perturbations. Convexity of\n$\\ell (x,\\cdot ),$ together with $C$ being positive definite in A2.b,\nimplies uniqueness of $\\beta _{\\ast }$. The uniqueness\n  of $\\beta_*$ is a standard assumption in the derivation of rates of\n  convergence for estimators; see, for example,\n  \\citet{huber1967}, \\citet[Section 3.2.2]{van1996weak}.  Assumption\nA2.b also allows us to rule out redundancies in the underlying source\nof randomness (e.g. colinearity in the setting of linear regression).\n The first part of Assumption A2.c ensures that the inner maximization in  (\\ref{Wass_DRO_A}) is finite by controlling the magnitude of the adversarial perturbations. {\n The local Lipschitz continuity requirement in $x$ arises with the optimal transportation analysis technique in \\cite{blanchet2016robust}, c.f. Assumption A6. Analogous regularity in $\\beta$ is useful in proving the confidence region limit theorem; see the discussion following Theorem \\ref{thm:master}. }\nLimiting results which study the impact of relaxing some of these assumptions are given immediately after describing the main result in Section\n\\ref{Sec_main_limit} below.\n\n\n\\section{Main results}\n\\subsection{The main limit theorem\\label{Sec_main_limit}}\nIn order to state our main results we introduce a few more definitions. Define\n\\begin{equation*}\n  \\varphi (\\xi )= 4^{-1}E\\big[ \\big\\Vert \\big\\{ D_{x}h(X,\\beta _{\\ast\n        })\\big\\} ^{ \\mathrm{\\scriptscriptstyle T} }\\xi \\big\\Vert _{p}^{2}\\big],\n\\end{equation*}%\nand its convex conjugate,\n$\\varphi ^{\\ast }(\\zeta )=\\sup_{\\xi \\in \\mathbb{R}^{d}}\\left\\{ \\xi ^{{\n\\mathrm{\\scriptscriptstyle T} }}\\zeta -\\varphi (\\xi )\\right\\}.$\nIn addition, define\n\\begin{align}\n  S(\\beta )&=\\left[E \\left\\{\\Vert D_{x}\\ell (X;\\beta )\\Vert _{p}^{2}\n             \\right\\}\\right]^{1\/2},\n    \\label{defn:sensitivity-term}\\\\\n  f_{\\eta ,\\gamma }(x) &= x\\mathbb{I}{(\\gamma \\geq 1)}-\\eta^{1\/2} D_{\\beta\n                         }S(\\beta _{\\ast })\\mathbb{I}{(\\gamma \\leq 1)},\n                         \\label{family-f-eta-gamma}\n\\end{align}\nfor $\\eta \\geq 0,\\gamma \\geq 0$. { By Assumption A2.b, we have $S(\\beta)$ is differentiable at $\\beta_*$.} Recall the matrix\n$C=E\\left\\{ D_{\\beta }h(X,\\beta _{\\ast })\\right\\} $ introduced in\nAssumption A2.b and\n\\begin{equation}\n  \\Lambda^+_{\\delta _{n}}(P_{n}) = \\mathrm{cl}\\left\\{\\cap_{\\epsilon>0} \\Lambda_{\\delta _{n}+\\epsilon}(P_{n})\\right\\},\n  \\label{NaturalCRplus}\n\\end{equation}\nwhich is the right limit of $\\Lambda_{\\delta_n}(P_n)$ defined in\n(\\ref{NaturalCR}). Finally, define the sets,\n\\begin{align}\n  \\Lambda_{\\eta } =\\left\\{ u:\\varphi ^{\\ast }(Cu)\\leq \\eta \\right\\},\n  \\qquad\n  \\Lambda_{\\eta ,\\gamma} =\n\\begin{cases}\n  \\Lambda_\\eta \\quad &\n  \\text{ if }  \\gamma =1, \\\\\n  \\mathbb{R}^{d} \\quad  & \\text{ if }\\gamma <1, \\\\\n  \\{0\\} \\quad   &\\text{ if }\\gamma >1.%\n\\end{cases}\n\\label{Lim_Set}\n\\end{align}\n\nWe now state our main result.\n\n\\begin{theorem}\n  \\label{thm:levelsets-master}\n  \\sloppy{Suppose that Assumptions A1 - A2 are satisfied with {$q\\in(1,\\infty)$},\n    $\\Omega = \\mathbb{R}^m$ and $E\\left(\\Vert X \\Vert_2^2\\right) < \\infty.$ If\n    $H\\sim\\mathcal{N}(0,\\mathrm{cov}\\{h(X,\\beta_{\\ast})\\})$ and\n    $\\delta_{n}=n^{-\\gamma}\\eta$ \\ for some\n    $\\gamma,\\eta\\in(0,\\infty),$} then we have the following joint\n  convergence in distribution:\n\\begin{align*}\n  & \\big( n^{1\/2}\\big\\{\\beta_{n}^{ERM}-\\beta_{\\ast}\\big\\},\\ n^{\\bar{\\gamma}\/2}\n    \\big\\{\\beta_{n}^{DRO}(\\delta_n)-\\beta_{\\ast}\\big\\},\\ n^{1\/2}\\big\\{\n    \\Lambda^+_{\\delta_{n}}(P_{n})-\\beta_{\\ast}\\big\\} \\big) \\\\\n  &\\hspace{200pt} \\Rightarrow \\big( C^{-1}H,\\ C^{-1}f_{\\eta,\\gamma}(H),\\ \\Lambda\n    _{\\eta,\\gamma}+C^{-1}H\\big) ,\n\\end{align*}\nwhere $\\bar{\\gamma}=\\min\\{\\gamma,1\\}$ and $\\Lambda_{\\eta,\\gamma}$ is\ndefined as in (\\ref{Lim_Set}).\n\\end{theorem}\n\nThe proof of Theorem \\ref{thm:levelsets-master} is presented in\nSection \\ref{sec:levelsets-master}. { For $q=1$ or $\\infty$, which corresponds to $%\np=\\infty $ or 1, $S(\\beta )$ may not be differentiable at $\\beta_*$, then the limiting\ndistribution  presents a discontinuity which\nmakes it difficult to use in practice.  Hence, we prefer not to cover\nthis here.} Theorem \\ref{thm:levelsets-master}\ncan be used as a powerful conceptual tool. For example, let us examine\nhow a sensible choice for the parameter $\\delta_{n}$ can be obtained\nas an application of Theorem \\ref{thm:levelsets-master} by considering\nthe following cases:\n\n{Case 1, where $\\gamma > 1$:} If $n\\delta_{n} \\rightarrow 0$\ncorresponding to the case $\\gamma > 1,$ we have $f_{0,\\gamma}(H)=H$\nfrom the definition of the parametric family in\n\\eqref{family-f-eta-gamma}. Therefore, from Theorem\n\\ref{thm:levelsets-master},\n\\begin{align*}\n & \\big( n^{1\/2}\\{\\beta_{n}^{ERM}-\\beta_{\\ast}\\},\\ n^{\\bar{\\gamma}\/2}\\{\\beta\n    _{n}^{DRO}(\\delta_n)-\\beta_{\\ast}\\},\\ n^{1\/2}\\big\\{\n    \\Lambda^+_{\\delta_{n}}(P_{n})-\\beta_{\\ast}\\big\\} \\big) \\\\& \\hspace{150pt}\\Rightarrow\n    \\big(C^{-1}H,C^{-1}H,\\{C^{-1}H\\}\\big),\n\\end{align*}\nwhich implies that the influence of the robustification vanishes in\nthe limit when $\\delta_{n}=o(n^{-1})$.\n\n{Case 2, where $\\gamma < 1$:} If $n\\delta_{n}\\rightarrow\\infty$\ncorresponding to the case $\\gamma<1$, the rate of convergence for the\ndistributionally robust estimator is slower than the canonical than\n$O(n^{-1\/2})$ rate:\n\\begin{equation}\n\\beta_{n}^{DRO}(\\delta_n) = \\beta_{\\ast}-\\eta^{1\/2} n^{-\\gamma\/2}\nC^{-1}D_{\\beta}S(\\beta _{\\ast})+o_{p}\\big(\nn^{-\\gamma\/2}\\big),  \\label{BDRO}\n\\end{equation}\nwhere $n^{\\gamma\/2}o_{p}( n^{-\\gamma\/2}) \\rightarrow0,$ in\nprobability, as $n\\rightarrow\\infty$. The relationship (\\ref{BDRO})\nreveals an uninteresting limit,\n$n^{1\/2}\\{ \\Lambda^+_{\\delta_{n}} (P_{n})-\\beta_{\\ast}\\}\n\\Rightarrow\\mathbb{R}^{d}$, exposing a slower than $O(n^{-1\/2})$ rate\nof convergence $\\Lambda^+_{\\delta_n }(P_{n}).$ In fact, (\\ref{BDRO})\nindicates that $O( n^{-\\gamma \/2})$ scaling will result in a\nnon-degenerate limit.\n\n{Case 3, where $\\gamma = 1$:}  when $\\delta_{n}=\\eta\/n$, we have that all components in\nthe triplet in Theorem \\ref{thm:levelsets-master} have non-trivial\nlimits.\n\n  Theorem \\ref{minmax_prop} below provides a geometric\ninsight relating $\\beta_{n}^{DRO}(\\delta_n)$, $\\beta_{n}^{ERM}$ and\n$\\Lambda^+_{\\delta_{n}}(P_{n})$, which justifies a picture describing\n$%\n\\Lambda^+_{\\delta_{n}}(P_{n})$ as a set containing both\n$\\beta_{n}^{DRO}(%\n\\delta_n)$ and $\\beta_{n}^{ERM}$. The observation that\n$\\beta_{n}^{ERM}\\in\\Lambda_{ \\delta_{n}}(P_{n})$ is immediate because\n$\\Lambda_{\\delta}(P_{n})$ is increasing in $\\delta$, so\n$\\beta_{n}^{ERM}\\in\\Lambda_{0}(P_{n})\\subset%\n\\Lambda^+_{\\delta_{n}}(P_{n})$. On the other hand, the observation that\n$\\beta_{n}^{DRO}(\\delta_n)\\in\\Lambda^+_{\\delta_{n}}(P_{n})$ is\nnon-trivial and it relies on the exchangeability of inf and sup in\nTheorem \\ref{minmax_prop} below. An appropriate choice of $\\eta$\nwhich results in the set $\\Lambda_{\\delta_n}^+(P_{n})$ also possessing\ndesirable coverage for $\\beta_\\ast$ is prescribed in Section\n\\ref{Sec_Confidence_Regions}.\n\\begin{theorem}\n  \\label{minmax_prop}\n  { Suppose that Assumption A1 is enforced. We further assume the loss function $\\ell(\\cdot)$ is\n  continuous and non-negative, for each $x$, $\\ell(x,\\cdot)$ is convex, and $E_{P_*}\\{\\ell(X,\\beta)\\}$ has a unique optimizer $\\beta_*\\in B^\\circ$.}\n  Then for any\n  $\\delta >0,$\n\\begin{equation}\n  \\inf_{\\beta \\in B}\\sup_{P \\in\\, \\mathcal{U}_{\\delta }(P_{n}) }\n  E_{P}\\left\\{ \\ell(X;\\beta )\\right\\} =\n  \\sup_{P \\in\\, \\mathcal{U}_{\\delta }(P_{n}) }\\inf_{\\beta \\in B} E_{P}\n  \\left\\{ \\ell (X;\\beta )\\right\\},\n  \\label{minmax_eqn}\n\\end{equation}\nand there exists a distributionally robust estimator choice\n$\\beta_{n}^{DRO}(\\delta) \\in \\Lambda^+_{\\delta}(P_{n})$.\n\\end{theorem}\nThe proof of Theorem \\ref{minmax_prop} is presented in Section\n\\ref{ssec:proof_prop_corr_1} of the supplementary material.  Example\n\\ref{eg:Lambda-plus} below demonstrates that the set of minimizers of\nthe distributionally robust formulation (\\ref{Wass_DRO_A}) is not\nnecessarily unique and that the set $\\Lambda_{\\delta}(P_{n})$ may not contain Distributionally robust solutions.\nTheorem~\\ref{minmax_prop} indicates that the right-limit\n$ \\Lambda^+_{\\delta}(P_{n})$  contains a distributionally robust solution. Theorem\n  \\ref{thm:levelsets-master} implies that the minimizers of\n  (\\ref{Wass_DRO_A}) differ by at most $o_p(n^{-1\/2})$ in magnitude,\n  which indicates that they are asymptotically equivalent and the\n  inclusion of one solution of (\\ref{Wass_DRO_A}) in\n  $\\Lambda^+_{\\delta}(P_{n})$ is sufficient for the scaling\n  considered.\n\n\\begin{example}\n  Let the loss function\nbe\n\\[\\ell(x,\\beta)=f(\\beta)+\\{x^2-\\log(x^2+1)\\} f(\\beta-4),\\]\nwhere $f(\\beta)= 3\\beta^2\/4-1\/8\\beta^4 +3\/8$ for $\\beta\\in [-1,1]$,\nand $f(\\beta)=|\\beta|$, otherwise. $\\ell(x,\\beta)$ is twice-differentiable and convex satisfying Assumptions A1 - A2. Then, if the empirical measure\n$P_n$ is a Dirac measure centered at zero with $n=1$, and\n$\\delta = 1$, we have the distributionally robust estimators\n$\\beta_{n}^{DRO}(\\delta) \\in [1,3]$. Further,\n$[1,3] \\subset \\Lambda^+_{\\delta}(P_{n})$ but\n$[1,3] \\cap \\Lambda_{\\delta}(P_{n}) =  \\varnothing$.\n\\label{eg:Lambda-plus}\n\\end{example}\n\n\nNext, we turn to the relationship between $\\beta _{n}^{ERM}$ and\n$\\beta _{n}^{DRO}(\\delta _{n}),$ when $\\delta _{n}=\\eta \/ n$. From\nthe first two terms in the triplet, we have,\n\\begin{align}\n  \\beta _{n}^{DRO}(\\delta _{n})\\\n  & =\\ \\beta _{n}^{ERM}-\\eta^{1\/2}\n    {C^{-1}D_{\\beta }S(\\beta_{\\ast })}{n^{-1\/2}}\n    + o_{p}\\big( n^{-1\/2}\\big)\n    \\label{Rel_DRO_ERM} \\\\\n  & =\\ \\beta _{n}^{ERM}-\\delta_n^{1\/2}{C^{-1}D_{\\beta }S(\\beta _{n}^{ERM})}\n    + o_{p}\\big( \\delta_n\\big) .  \\notag\n\\end{align}%\nThe right hand side of (\\ref{Rel_DRO_ERM}) points to the canonical\n$O\\left( n^{-1\/2}\\right) $ rate of convergence of the Wasserstein distributionally robust\nestimator and it can readily be used to construct confidence regions,\nas we shall explain in Section \\ref{Sec_Confidence_Regions} below.\n\nRelation (\\ref{Rel_DRO_ERM}) also exposes the presence of an\nasymptotic bias term, namely,\n$S(\\beta)= [E \\{\\Vert D_{x}\\ell(X;\\beta)\\Vert_{p}^{2}\\}]^{1\/2}$, which\npoints towards selection of optimizers possessing reduced sensitivity\nwith respect to perturbations in data.  A precise mathematical\nstatement of this sensitivity-reduction property is given in Corollary\n\\ref{Cor_Sensitivity} below and its proof is presented in Section\n\\ref{ssec:proof_prop_corr_1} of the supplementary material.\n\n\\begin{corollary}\n  \\label{Cor_Sensitivity}Suppose that A1 - A2 are in force and\n  consider\n\\begin{equation}\n\\bar{\\beta}_{n}^{DRO} \\in \\arg\\min_{\\beta \\in B}\\left[E_{P_{n}}\\left\\{ \\ell\n(X;\\beta)\\right\\} +   n^{-1\/2} \\left[ \\eta E_{P_{n}}\\left\\{ \\Vert\n  D_{x}\\ell(X;\\beta )\\Vert_{p}^{2}\\right\\} \\right]^{1\/2}\\right].\n\\label{eq:sensitivity-red}\n\\end{equation}\nThen, if $\\delta_{n}=\\eta\/n$, we have that $\\beta _{n}^{DRO}(\\delta_n) =\n\\bar{\\beta}_{n}^{DRO}+o_{p}(n^{-1\/2})$.\n\\end{corollary}\n\nWhile the formulation on the right-hand side of\n(\\ref{eq:sensitivity-red}) is conceptually appealing, it may not be\ndesirable from an optimization point of view due to the potentially\nnonconvex nature of the objective involved. On the other hand, under\nAssumption A2, the distributionally robust objective\n$\\Psi _{n}(\\beta )$ is convex; see, for example, the reasoning in\n\\citet[Theorem 2a]{blanchet2018optimal} while also enjoying the\nsensitivity-reduction property of the formulation in\n(\\ref{eq:sensitivity-red}).\n\nA similar type of result to Corollary \\ref{Cor_Sensitivity} is given\nin \\citet{gaostatlearn}, but the focus there is on the objective\nfunction of (\\ref{Wass_DRO_A}) being approximated by a suitable\nregularization. The difference between this type of result and\nCorollary \\ref{Cor_Sensitivity} is that our focus is on the asymptotic\nequivalence of the actual optimizers.  Behind a result such as\nCorollary \\ref{Cor_Sensitivity}, it is key to have a more nuanced\napproximation which precisely characterizes the second order term of\nsize $O(\\delta_n)$; see Proposition A1 in the supplementary material.\n\n\n\n\nWe conclude this section with results which examine the effects of\nrelaxing some assumptions made in the statement of Theorem\n\\ref{thm:levelsets-master} above. Proposition\n\\ref{prop:constrained-support} below asserts that convergence of the\nnatural confidence region $\\Lambda^+_{\\delta_n}(P_n),$ as identified in\nTheorem \\ref{thm:levelsets-master}, holds even if the support of the\nprobability distributions in the uncertainty region\n$\\mathcal{U}_{\\delta_n}(P_n)$ is constrained to be a strict subset\n$\\Omega$ of $\\mathbb{R}^d.$ For this purpose, we introduce the\nfollowing notation: For any set $C\\in \\mathbb{R}^{m},$ let\n$C^{\\epsilon }=\\left\\{ x\\in C: B_{\\epsilon }\\left( x\\right) \\subset\n  C\\right\\} ,$ where $B_{\\epsilon }\\left( x\\right) $ is the\nneighborhood around $x$ defined as\n$B_{\\epsilon }\\left( x\\right) =\\left\\{ y: \\left\\Vert y-x\\right\\Vert\n  _{2}\\leq \\epsilon \\right\\} .$ Thus, for any probability measure $P$,\nwe have\n$\\lim_{\\epsilon \\rightarrow 0} {P}\\left( C^{\\epsilon }\\right)\n={P}\\left( C^{\\circ }\\right).$\n\n\\begin{proposition}\n  Suppose that Assumptions A1 - A2 are satisfied with {$q\\in[1,\\infty]$} and\n  $E\\left(\\Vert X \\Vert_2^2\\right) < \\infty$. In addition, suppose that the data\n  generating measure $P_*$ satisfies\n  $P_{\\ast}(\\Omega^\\circ) =1$. If we take\n  $H\\sim\\mathcal{N}(0,\\mathrm{cov}\\{h(X,\\beta_{\\ast})\\})$ and\n  $\\delta_{n}=n^{-\\gamma}\\eta$ \\ for some $\\gamma,\\eta\\in(0,\\infty),$\n  then the following convergence holds as $n \\rightarrow \\infty$:\n  \\begin{align*}\n    n^{1\/2} \\left\\{ \\Lambda_{\\delta_n}(P_n) - \\beta_\\ast \\right\\}\n    \\Rightarrow \\Lambda_{\\eta,\\gamma} + C^{-1}H.\n  \\end{align*}\n  \\label{prop:constrained-support}\n\\end{proposition}\nThe steps involved in proving Proposition\n\\ref{prop:constrained-support} are presented in Section\n\\ref{sec:proof}. A discussion on the validity of a central limit theorem\nfor the estimator $\\beta_n^{DRO},$ in the presence of constraints\nrestricting transportation within the support set $\\Omega$, is\npresented in Section \\ref{sec:discussion}.\n\nIn the case where the unique minimizer $\\beta_\\ast$ may not\nnecessarily lie in interior of the set $B$ (as opposed to the\nrequirement in Assumption A2.b, one may obtain the extension in\nProposition \\ref{prop:DRO-boundary} as the limiting result for the\nestimator $\\beta^{DRO}_n(\\delta_n)$. As in the previous results, we\ntake $h(x,\\beta) = D_\\beta\\ell(x;\\beta).$ The proof of Proposition\n\\ref{prop:DRO-boundary} is in Section \\ref{ssec:proofs_dro_bound_prop}\nof the supplementary material.\n\\begin{proposition}\n  Suppose that Assumptions A1, A2.a, A2.c are satisfied and $\\beta_\\ast$\n  is the unique minimizer of $\\min_{\\beta \\in B} E\\{\\ell(X,\\beta)\\}.$\n  Suppose that the set $B$ is compact and there exist\n  $\\varepsilon > 0$ and twice continuously differentiable functions\n  $g_i(\\beta)$ such that,\n  \\begin{align*}\n    B \\cap B_\\varepsilon(\\beta_\\ast) =\n    \\left\\{ \\beta \\in B_\\varepsilon(\\beta_\\ast): g_i(\\beta) = 0,\n    i \\in I, \\ g_j(\\beta) \\leq 0, j \\in J\\right\\},\n  \\end{align*}\n  where $I,J$ are finite index sets and $g_i(\\beta_\\ast) = 0$ for all\n  $i \\in J.$ With this identification of the set $B,$ suppose that the following so-called Mangasarian-Fromovitz constraint qualification is satisfied at $\\beta_\\ast$: the\n  gradient vectors $\\{D g_i(\\beta_\\ast): i \\in I\\}$ are linearly\n  independent and there exists a vector $w$ such that\n  $w^{\\mathrm{\\scriptscriptstyle T} } D g_i(\\beta_\\ast) = 0$ for all $i \\in I$ and\n  {$w^{\\mathrm{\\scriptscriptstyle T} } D g_j(\\beta_\\ast) < 0$} for all $j \\in J.$\n \n \n \n       \n\n  Suppose that $\\Lambda_0$ is the set of Lagrange multipliers satisfying\n  the first-order optimality conditions and the following\n  second-order sufficient conditions: $\\lambda \\in \\Lambda_0$ if and\n  only if $D_\\beta L(\\beta_\\ast,\\lambda) = 0,$ $\\lambda_i \\geq 0$ for\n  $i \\in J,$ and\n  $\\max_{\\lambda \\in \\Lambda_0} w^{\\mathrm{\\scriptscriptstyle T} } D_{\\beta \\beta}\n  L(\\beta_\\ast,\\lambda)w > 0$ for all $w \\in \\mathcal{C},$ where\n  \\[L(\\beta,\\lambda) = E\\{\\ell(X,\\beta)\\} + \\sum_{i \\in I \\cup J}\n    \\lambda_i g_i(\\beta)\\] is the Lagrangian function associated with\n  the minimization $\\min_{\\beta \\in B} E\\{\\ell(X,\\beta)\\}$ and\n  \\[ \\mathcal{C} = \\left\\{ w: w^{\\mathrm{\\scriptscriptstyle T} } Dg_i(\\beta_\\ast) = 0, i \\in I, \\\n      w^{\\mathrm{\\scriptscriptstyle T} } Dg_j(\\beta_\\ast) \\leq 0, j \\in J, \\ w^{\\mathrm{\\scriptscriptstyle T} } E\\{h(X,\\beta_\\ast)\\}\n      \\leq 0 \\right\\}\\]\n \n  is the non-empty cone of critical directions.  In addition, suppose\n  that $\\omega(\\xi)$ is the unique minimizer of\n  $\\min_{u \\in \\mathcal{C}}\\big\\{\\xi^{\\mathrm{\\scriptscriptstyle T} } u + 2^{-1}q(u) \\big\\},$ where\n \n  $q(u) = \\max \\left\\{u^{\\mathrm{\\scriptscriptstyle T} } D_{\\beta \\beta}L(\\beta_\\ast,\\lambda) u :\n    \\lambda \\in \\Lambda_0 \\right\\}.$ Then, if $\\delta_n = \\eta n^{-1}$\n  for $\\eta \\in (0,\\infty)$,\n  $E \\{\\Vert h(X,\\beta_\\ast) \\Vert_2^2\\} < \\infty$ and\n  $E\\{D_\\beta h(X,\\beta_\\ast)\\} \\succ 0,$ we have the following\n  convergence as $n \\rightarrow \\infty$:\n  \\begin{align*}\n    n^{1\/2}\\left\\{\\beta^{DRO}_n(\\delta_n)  - \\beta_\\ast\\right\\} \\Rightarrow\n    \\omega\\left\\{-H + \\eta^{1\/2}D_\\beta S(\\beta_\\ast)\\right\\},\n  \\end{align*}\n  where $H \\sim \\mathcal{N}(0,\\mathrm{cov}\\{h(X,\\beta_\\ast)\\}).$\n  \\label{prop:DRO-boundary}\n\\end{proposition}\n\nThe Mangasarian-Fromovitz constraint qualification conditions and the\nnecessary and sufficient conditions in the statement of Proposition\n\\ref{prop:DRO-boundary} are standard in the literature if the optimal\n$\\beta_\\ast$ lies on the boundary of the set $B$; see, for example,\n\\cite{shapiro1989}. Please refer the discussion following Theorem 3.1\nin \\cite{shapiro1989} for sufficient conditions under which\n$\\omega(\\xi)$ is unique.\n\n\nProposition \\ref{prop:DRO-property} extends the sensitivity reduction\nproperty in Corollary \\ref{Cor_Sensitivity} to settings where the\nminimizer for $\\min_{\\beta \\in B} E_{P_*}\\{\\ell(X;\\beta)\\}$ is not\nunique. The proof of Proposition \\ref{prop:DRO-property} is presented\nin Section \\ref{ssec:proofs_dro_bound_prop} of the supplementary\nmaterial.\n\n\\begin{proposition}\n  Suppose that Assumptions A1, A2.a and A2.c are satisfied, the set\n  $B$ is compact, and the choice of the radii $(\\delta_n: n \\geq 1)$ is\n  such that $n\\delta_n \\rightarrow \\eta \\in (0,\\infty).$ Let the set $B_* $ be $\\arg\\min_{\\beta \\in B}  E_{P_*}\\{\\ell(X;\\beta)\\}$. Then, the distributionally robust optimization\n  objective $\\Psi_n(\\beta)$ satisfies,\n  \\begin{align}\n    n ^{1\/2} \\left[\\Psi_n(\\beta) - E\\{\\ell(X;\\beta)\\} \\right]\n    \\Rightarrow Z(\\beta) + \\eta^{1\/2} S(\\beta),\n    \\label{rhs-nonunique}\n  \\end{align}\n  where $Z(\\cdot)$ is a zero mean Gaussian process with covariance\n  function $\\text{cov}\\{Z(\\beta_1),Z(\\beta_2)\\}\n  =\\text{cov}\\{\\ell(X,\\beta_1),\\ell(X,\\beta_2)\\}.$ The above\n  weak convergence holds, as $n \\rightarrow \\infty,$ on the space of\n  continuous functions equipped with the uniform topology on compact sets.\n    %\n \n \n \n \n \n     \n \n \n \n \n \n \n \n  Consequently, if\n  $\\arg \\min_{\\beta \\in B_\\ast}\\{Z(\\beta) +\n  \\eta^{1\/2} S(\\beta)\\}$ is singleton with probability one, we have as\n  $n\\rightarrow \\infty,$\n  \\begin{equation*}\n    \\beta_n^{DRO}(\\delta_n) \\Rightarrow \\textnormal{arg}\\,\\textnormal{min}_{\\beta \\in B_\\ast}\\big\\{\n     Z(\\beta )+ \\eta^{1\/2} S(\\beta ) \\big\\}.\n\\end{equation*}%\n\n\\label{prop:DRO-property}\n\\end{proposition}\n\n\n\n\\subsection{Construction of Wasserstein distributionally robust confidence regions}\n  \\label{Sec_Confidence_Regions}\nAs mentioned in the Introduction, for suitably chosen $\\delta_n,$ the\nset $\\Lambda^+_{\\delta_{n}}(P_{n})$ represents a natural confidence\nregion. In particular, $\\Lambda^+_{\\delta_{n}}(P_{n})$ possesses an\nasymptotically desired coverage, say at level at least $1-\\alpha$, if\nand only if\n\\begin{equation*}\n1-\\alpha\\leq\\lim_{n\\rightarrow\\infty}\\mathrm{pr}\\left\\{ \\beta_{\\ast}\\in\n\\Lambda^+_{\\delta_{n}}(P_{n})\\right\\} =\\mathrm{pr}[-C^{-1}H\\in\\{u:\\varphi^{%\n\\ast }(Cu)\\leq\\eta\\}],\n\\end{equation*}\nor, equivalently, if $\\eta\\geq\\eta_{\\alpha}$, where $\\eta_{\\alpha}$ is\nthe $%\n(1-\\alpha)$-quantile of the random variable $\\varphi^{\\ast}(H)$.\n\nRecall the earlier geometric insight describing\n$\\Lambda^+_{\\delta_{n}}(P_{n})$ as a set containing both\n$\\beta_{n}^{DRO}(\\delta_n)$ and $\\beta_{n}^{ERM},$ as a consequence of\nTheorem \\ref{minmax_prop}.  Following this, if we let\n$\\eta\\geq\\eta_{\\alpha},$ we then have,\n\\begin{align*}\n  \\lim_{n\\rightarrow\\infty}\\mathrm{pr}\\big\\{\\beta_{\\ast}\\in\\Lambda^+_{\\delta_{n}}(P_{n}),\\\n  \\beta_{n}^{DRO}\\in\\Lambda^+_{\\delta_{n}}(P_{n}),\\\n  \\beta_{n}^{ERM}\\in\\Lambda^+_{\\delta_{n}}(P_{n})\\big\\}\n  &= \\lim_{n\\rightarrow\\infty}\\mathrm{pr} \\big\\{ \\beta_{\\ast}\\in\\Lambda^+_{\\delta_{n}}(P_{n})\\big\\}\\\\\n  &\\geq  1-\\alpha,\n\\end{align*}\nwhich presents the picture of $\\Lambda^+_{\\delta_n}(P_n)$ as a\nconfidence region simultaneously containing $\\beta_\\ast,\\beta_n^{ERM}$\nand $\\beta_{n}^{DRO}(\\delta_n),$ with a desired level of confidence.\n\nThe function $\\varphi^{\\ast}(H)$ can be computed in closed form in\nsome settings. But, typically, computing $\\varphi^{\\ast}(\\cdot)$ may\nbe challenging. We now describe how to obtain a consistent estimator\nfor $\\eta_{\\alpha}$. Define the empirical version of $\\varphi (\\xi )$,\nnamely%\n\\begin{equation*}\n  \\varphi _{n}(\\xi ) = \\frac{1}{4}E_{P_{n}}\\left[ \\left\\Vert \\left\\{\n        D_{x}h(X,\\beta _{\\ast })\\right\\} ^{ \\mathrm{\\scriptscriptstyle T} }\\xi\n    \\right\\Vert _{p}^{2}\\right] = \\frac{1}{4n}\\sum_{i=1}^{n}\\left\\Vert \\left\\{\n      D_{x}h(X,\\beta _{\\ast })\\right\\} ^{ \\mathrm{\\scriptscriptstyle T} }\\xi\n  \\right\\Vert_{p}^{2},\n\\end{equation*}%\nand the associated empirical convex conjugate,\n$\\varphi _{n}^{\\ast }(\\zeta )=\\sup_{\\xi \\in \\mathbb{R}^{d}}\\left\\{ \\xi\n  ^{{ \\mathrm{\\scriptscriptstyle T} }}\\zeta -\\varphi _{n}(\\xi\n  )\\right\\}.$\nProposition \\ref{Prop_Reg_Plug} below, whose proof is in Section\n\\ref{ssec:proof_prop_reg} of the supplementary material, provides a\nbasis for computing a consistent estimator for $\\eta _{\\alpha }$.\n\n\\begin{proposition}\n  \\label{Prop_Reg_Plug}Let ${\\Xi}_{n}$ be any consistent estimator of\n  $%\n  \\text{cov}\\{ h\\left( X,\\beta\\right) \\} $ and write $\\bar{\\Xi}_{n}$\n  for any factorization of $\\Xi_{n}$ such that\n  $\\bar{\\Xi}_{n}\\bar{\\Xi}_{n}^{{\\mathrm{\\scriptscriptstyle T} }}=\\Xi_{n}$%\n  . Let $Z$ be a $d$-dimensional standard Gaussian random vector\n  independent of the sequence $\\left( X_{n}:n\\geq1\\right) $. Then, i)\n  the distribution of $%\n  \\varphi^{\\ast}(Z)$ is continuous, ii)\n  $\\varphi_{n}^{\\ast }(\\cdot)\\Rightarrow\\varphi^{\\ast}(\\cdot)$ as\n  $n\\rightarrow\\infty$ uniformly on compact sets, and iii)\n  $\\varphi_{n}^{\\ast}(\\bar{\\Xi}_{n}Z)\\Rightarrow\\varphi ^{\\ast}(H)$.\n\\end{proposition}\n\nGiven the collection of samples $\\left\\{ X_{i}\\right\\} _{i=1}^{n}$, we\ncan generate independent and identically distributed copies of $Z$ and\nuse Monte Carlo to estimate the quantile\n$\\left( 1-\\alpha\\right) $-quantile, $%\n\\eta_{\\alpha}\\left( n\\right) $, of\n$\\varphi_{n}^{\\ast}(\\bar{\\Xi}_{n}Z)$. The previous proposition implies\nthat\n$\\eta_{\\alpha}\\left( n\\right) =\\eta_{\\alpha}+o_{p}\\left( 1\\right),$ as\n$n\\rightarrow\\infty$ This is sufficient to obtain an implementable\nexpression for $\\beta_n^{DRO}\\{\\eta_\\alpha(n)\/n\\}$ which is\nasymptotically equivalent to (\\ref{Rel_DRO_ERM}), as it differs only\nby an error of maginutude $o_{p}(n^{-1\/2}).$\n\nNext, we provide rigorous support for the approximation\n\\begin{equation*}\n\\Lambda^+_{\\delta_{n}}(P_{n})\\approx\\beta_{n}^{ERM}+n^{-1\/2}\\Lambda_{\\eta},\n\\end{equation*}\nwhich can be used to approximate $\\Lambda^+_{\\delta_{n}}(P_{n})$, providing we\ncan estimate $\\Lambda_{\\eta}$.\n\n\\begin{corollary}\n\\label{cor:CIcenter-plugin}Under the assumptions of Theorem \\ref%\n{thm:levelsets-master} and $\\gamma = 1$, we have\n\\begin{equation*}\nn^{1\/2}\\left\\{ \\Lambda^+_{\\delta_{n}}(P_{n})-\\beta_{n}^{ERM}\\right \\}\n\\Rightarrow\\Lambda_{\\eta}.\n\\end{equation*}\nMoreover, if $\\eta\\left( n\\right) =\\eta+o\\left( 1\\right) $, and $%\nC_{n}\\rightarrow C$, then\n\\begin{equation*}\n\\Lambda_{\\eta\\left( n\\right) }^{n}=\\{u:\\varphi_{n}^{\\ast}(C_{n}u)\\leq\n\\eta\\left( n\\right) \\}\\rightarrow\\Lambda_{\\eta}.\n\\end{equation*}\n\\end{corollary}\n\n\\begin{proof}[Proof  of Corollary \\ref{cor:CIcenter-plugin}]\nBy Following directly from Theorem \\ref{thm:levelsets-master} and an application\nof continuous mapping theorem as in,\n\\begin{equation*}\nn^{1\/2}\\left\\{ \\Lambda^+_{\\delta_{n}}(P_{n})-\\beta_{n}^{ERM}\\right\\}\n=n^{1\/2}\\left\\{ \\Lambda^+_{\\delta_{n}}(P_{n})-\\beta_{\\ast}\\right\\} -\nn^{1\/2}\\left\\{ \\beta_{n}^{ERM}-\\beta_{\\ast}\\right\\} \\Rightarrow\n\\Lambda_{\\eta}+C^{-1}H-C^{-1}H.\n\\end{equation*}\nThe second part of the result follows from the regularity results in\nProposition \\ref{Prop_Reg_Plug}.\n\\end{proof}\n\nThe next result,  as we shall explain, allows us to obtain\ncomputationally efficient approximations of the set $\\Lambda_{\\eta}.$\nA completely analogous result can be used to estimate\n$\\Lambda_{\\eta\\left( n\\right) }^{n}$, simply replacing\n$\\varphi^{\\ast}(\\cdot)$, $\\varphi(\\cdot)$ and $C$ by $%\n\\varphi_{n}^{\\ast}(\\cdot)$, $\\varphi_{n}(\\cdot)$ and $C_{n}.$\n\n\\begin{proposition}\nThe support function of the convex set $\\Lambda _{\\eta }=\\{u:\\varphi ^{\\ast\n}(Cu)\\leq \\eta \\}$ is,\n\\begin{equation*}\nh_{\\Lambda _{\\eta }}(v)=2\\{\\eta \\varphi (C^{-1}v)\\}^{1\/2},\n\\end{equation*}%\nwhere the support function of a convex set $A$ is defined as $h_{A}(x)=\\sup\n\\{x\\cdot a:a\\in A\\}.$ \\label{prop:level-sets-char}\n\\end{proposition}\nThe proof of Proposition \\ref{prop:level-sets-char} is in Section \\ref{ssec:proof_prop_reg} of the supplementary material.\n\\begin{remark}\n\\textnormal{\\ Proposition \\ref{prop:level-sets-char} can be used to obtain a\ntight envelope of the set $\\Lambda _{\\eta }$ by evaluating an intersection\nof hyperplanes that enclose $\\Lambda _{\\eta }$. Recall from the definition\nof support function that\n\\begin{equation*}\n\\Lambda _{\\eta }=\\cap _{u}\\{v:u\\cdot v\\leq h_{\\Lambda _{\\eta }}\\left(\nu\\right) \\}.\n\\end{equation*}%\nTherefore for any $u_{1},...,u_{m},$ we have\n$\\Lambda _{\\eta }$ is contained in\n$\\cap _{u_{1},...u_{m}}\\{v:u_{i}\\cdot v\\leq h_{\\Lambda _{\\eta }}\\left(\n  u_{i}\\right) \\},$\nand\n$\\Lambda _{\\eta \\left( n\\right) }^{n}$ is contained in\n$\\cap _{u_{1},...u_{m}}\\{v:u_{i}\\cdot v\\leq h_{\\Lambda _{\\eta \\left(\n      n\\right) }^{n}}\\left( u_{i}\\right) \\}.$\n}\n\\end{remark}\n\n\n\n\\section{Numerical examples: Geometry and coverage probabilities}\n\n\\label{sec:geo_insignts}\n\n\\subsection{Distributionally robust linear regression}\nWe first offer a brief introduction to the distributionally robust\nversion of the linear regression problem considered in\n\\citet{blanchet2016robust}. Specifically, the data is generated by\n$Y=\\beta_{\\ast}^{{ \\mathrm{\\scriptscriptstyle T} }}X+\\epsilon,$ where\n$X\\in\\mathbb{R}^{d}$ and $%\n\\epsilon$ are independent, $C=E(XX^{{\\mathrm{\\scriptscriptstyle T%\n    } }}) $ and $\\epsilon\\sim\\mathcal{N}(0,\\sigma^{2})$. We consider\nsquare loss\n$\\ell(x,y;\\beta)= 1\/2(y-\\beta^{{\\mathrm{\\scriptscriptstyle T}\n  }}x)^{2}$ and take the cost function\n$c:\\mathbb{R}^{d+1} \\times \\mathbb{R}%\n^{d+1} \\rightarrow [0,\\infty]$ to be\n\\begin{equation}\nc\\{(x,y),(u,v)\\}=\\left\\{\n\\begin{array}{c}\n\\left\\Vert x-u\\right\\Vert _{q}^{2} \\\\\n\\infty%\n\\end{array}\n\\right.\n\\begin{array}{c}\n\\text{if }y=v, \\\\\n\\text{otherwise.}%\n\\end{array}%\n\\label{tr-cost-LinReg}\n\\end{equation}\nThen, from \\citet[Theorem 1]{blanchet2016robust}, we have\n\\begin{equation}\n  \\min_{\\beta\\in\\mathbb{R}^{d}}\\sup_{P:D_{c}(P,P_{n})\\leq\\delta_{n}}E_{P}\\left[\n    \\ell(X,Y;\\beta)\\right] = \\frac{1}{2}\\min_{\\beta\\in\\mathbb{R}^{d}}\\big[ E_{P_{n}}\n  \\left\\{ (Y-\\beta^{{\\mathrm{\\scriptscriptstyle T} }}X)^{2}\\right\\}^{1\/2}\n  +\\delta_{n}^{1\/2}\\left\\Vert \\beta\\right\\Vert _{p}\\big]^{2},\n\\label{Sqrt_Lasso_Est}\n\\end{equation}\nwhere $p$ satisfies $1\/p+1\/q=1.$ Following Corollary\n\\ref{cor:CIcenter-plugin}, an approximate confidence region is\n\\begin{equation*}\n\\Lambda^+_{\\delta_{n}}(P_{n})\\approx n^{-1\/2}\\Lambda_{\\eta_{\\alpha}}+\\beta\n_{n}^{ERM},\n\\end{equation*}\nwhere\n$\\Lambda_{\\eta_\\alpha}=\\{\\theta:\\varphi^{\\ast}( C\\theta) \\leq\n\\eta_{\\alpha}\\},$\n$\\varphi(\\xi)= 4^{-1}E \\{\\Vert e\\xi-( \\xi^{{\\mathrm{%\n      \\scriptscriptstyle T} }}X) \\beta_\\ast\\Vert_{p}^{2}\\},$ the\nconstant $\\eta_\\alpha$ is such that\n$\\mathrm{pr} \\{\\varphi^\\ast (H)\\leq1-\\alpha\\}=\\eta_{\\alpha}$ for\n$H\\sim\\mathcal{N}(0,C\\sigma^{2}),$ and $\\delta_{n}=\\eta_{\\alpha}\/n.$\nBy performing a change of variables via linear transformation in the\nanalysis of the case $c(x,y) = \\Vert x - y \\Vert_2^2,$ Theorem \\ref%\n{thm:levelsets-master} can be directly adapted to the choice $c(x,y)$\nbeing a Mahalanobis metric as in,\n\\begin{align}\nc\\left( x,y\\right) =\\left( x-y\\right) ^{{\\mathrm{\\scriptscriptstyle T} }%\n}A\\left( x-y\\right),  \\label{Mahalanobis-metric}\n\\end{align}\nfor some matrix $A \\succ 0.$ The respective\n$\\Lambda_{\\eta_\\alpha} = \\{\\theta:\\varphi^{\\ast}( C\\theta) \\leq\n\\eta_{\\alpha}\\}$ is computed in terms of\n\\begin{equation*}\n  \\varphi(\\xi)=4^{-1}E\\big\\{ \\Vert\n  \\xi^{{\\mathrm{\\scriptscriptstyle T} }}\n  D_{x}h(X,\\beta_{\\ast })A^{-1\/2}\\Vert _{2}^{2}\\big\\}.\n\\end{equation*}\nFor the choice $c\\left( x,y\\right) =\\left( x-y\\right) ^{{\\mathrm{%\n\\scriptscriptstyle T} }}A\\left( x-y\\right), $ the relationship between distributionally robust\nand regularized estimators, as in (\\ref{Sqrt_Lasso_Est}), is\n\\begin{equation*}\n  \\min_{\\beta\\in\\mathbb{R}^{d}}\\sup_{P:D_{c}(P,P_{n})\\leq\\delta_{n}}E_{P}\\left\\{\n    l(X,Y;\\beta)\\right\\} = \\frac{1}{2}\\min_{\\beta\\in\\mathbb{R}^{d}}\\left[\n    E_{P_{n}}\n    \\left\\{ (Y-\\beta^{{\\mathrm{\\scriptscriptstyle T} }}X)^{2}\\right\\}^{1\/2}\n    +\\delta_{n}^{-1\/2}\\big\\Vert A^{-1\/2}\\beta\\big\\Vert _{2}\\right] ^{2}.\n\\end{equation*}\nSee \\citet{blanchet2017data} for an account of improved out-of-sample\nperformance resulting from Mahalanobis cost choices.\n\n\n\n\\subsection{Shape of confidence regions}\n\nThe goal of this section is to provide some numerical implementations\nto gain intuition about the geometry of the set $\\Lambda_{\\eta}$ for\ndifferent transportation cost choices. We use the empirical set\n\\begin{equation*}\n  \\Lambda_{\\eta_{\\alpha}}^n=\\{\\theta:{\\varphi}^{\\ast}_n( {C}_n\n  \\theta) \\leq n^{-1\/2}\\tilde{\\eta}_{\\alpha}\\},\n\\end{equation*}\nto approximate the desired confidence region as in Corollary\n\\ref{cor:CIcenter-plugin}. In the above expression,\n${\\varphi}_n(\\xi)= 4^{-1}E_{P_{n}} \\{\\Vert e\\xi-\\left(\n  \\xi^{\\mathrm{\\scriptscriptstyle T} } X\\right)\n\\beta_{n}^{ERM}\\Vert_{p}^{2}\\} $, $\\eta_{\\alpha}(n)$ is such that\n$\\mathrm{pr}(\\tilde{\\varphi}^{\\ast}_n(H)\\leq1-\\alpha)=\\eta_{\\alpha}(n)$ for\n$H\\sim%\n\\mathcal{N}(0,{C}_n{\\sigma}_n^{2}),$\n${C}_n=E_{P_{n}}\\left[ XX^{ \\mathrm{\\scriptscriptstyle T}\n  }\\right] $, and\n${\\sigma}^{2}_n =E_{P_{n}}[ \\{(Y-( \\beta_{n}^{ERM})^{{\\mathrm{\\scriptscriptstyle T} } } X\\}^{2}] $.\n\nIn the following numerical experiments, the data is sampled from a linear\nregression model with parameters $\\sigma^{2}=1$, $\\beta_{*}=[0.5,0.1]^{{\n\\mathrm{\\scriptscriptstyle T} }},n=100$ and\n\\begin{equation}\nX\\sim\\mathcal{N}\\left( 0,\\left[\n\\begin{array}{cc}\n  1 & \\rho \\\\\n  \\rho & 1%\n\\end{array}\n\\right] \\right),\n\\label{eq:cov-matrix}\n\\end{equation}\nwith $\\rho = 0.7.$ In Figures \\ref{1-norm}-\\ref{inf-norm}, we draw the\n$95\\%$ confidence region corresponding to the \\sloppy{choices\n  $p=1,3\/2,2,3,\\infty $,} ($%\nq=\\infty,3,2,3\/2,$ respectively) by means of support functions defined\nin Proposition %\n\\ref{prop:level-sets-char}. In addition, a confidence region for\n$\\beta _{\\ast }$ resulting from the asymptotic normality of the\nleast-squares estimator,\n${n}^{1\/2}(\\beta _{n}^{ERM}-\\beta ^{\\ast })\\Rightarrow \\mathcal{N}%\n(0,C^{-1}\\sigma ^{2}),$\nis\n\\begin{equation*}\n\\Lambda _{CLT}(P_{n})=n^{-1\/2}\\{\\theta :\\theta ^{{\\mathrm{\\scriptscriptstyle %\nT}}}C\\theta \/\\sigma ^{2}\\leq \\chi _{1-\\alpha }^{2}(d)\\}+\\beta _{n}^{ERM},\n\\end{equation*}%\nwhere $\\chi _{1-\\alpha }^{2}(d)$ is the $1-\\alpha $ quantile of the\nchi-squared distribution with $d$ degrees of freedom. One can select the\nmatrix $A$ in the Mahalanobis metric \\eqref{Mahalanobis-metric} such that\nthe resulting confidence region coincides with $\\Lambda _{CLT}(P_{n})$.\nNamely, $A$ is chosen by solving the equation%\n\\begin{equation}\nE\\left[ \\left\\{ e\\xi -\\left( \\xi ^{{\\mathrm{\\scriptscriptstyle T}}}X\\right)\n\\beta _{\\ast }\\right\\} A^{-1}\\left\\{ e\\xi -\\left( \\xi ^{{\\mathrm{%\n\\scriptscriptstyle T} }}X\\right) \\beta _{\\ast }\\right\\} ^{{\\mathrm{%\n\\scriptscriptstyle T} }}\\right] =C\\sigma ^{2}.  \\label{eqn_A2}\n\\end{equation}\n\nFigure \\ref{clt} gives the confidence region for the choice $p=2$ and\n$%\n\\Lambda_{CLT}(P_n)$ superimposed with various distributionally robust\nminimizer along with the empirical risk minimizer.\nIt is evident from the figures that $p=1$ gives a diamond shape, $p=2$\ngives an elliptical shape and $p=\\infty$ gives a rectangular shape.\nFurthermore, we see that the distributionally robust optimization solutions all reside in their\nrespective confidence regions but may lie outside of the confidence\nregions of other norms.\n\\begin{figure}[htbh]\n\\centering\n\\subfigure[$p=1$]{\n\\label{1-norm} \\includegraphics[width=1.81in]{1-norm-std_supp.eps}}\n\\subfigure[$p=1.5$]{\n\\label{2-norm} \\includegraphics[width=1.81in]{15-norm-std_supp.eps}}\n\\subfigure[$p=2$]{\n\\label{2-norm} \\includegraphics[width=1.81in]{2-norm-std_supp.eps}}\n\\subfigure[$p=3$]{\n\\label{2-norm} \\includegraphics[width=1.81in]{3-norm-std_supp.eps}}\n\\subfigure[$p=\\infty$]{\n\\label{inf-norm} \\includegraphics[width=1.81in]{Inf-norm-std_supp.eps}}\n\\subfigure[CLT]{\n\\label{clt} \\includegraphics[width=1.81in]{clt_new.eps}}\n\\caption{Confidence regions for different norm choices and central\n  limit theorem based confidence region plotted together with the\n  respective $\\beta_n^{DRO}$ estimators and $\\beta_n^{ERM}$}.\n\\end{figure}\nWe find the induced confidence regions constructed by the Wasserstein\ndistributionally robust optimization formulations are somewhat similar across the various $l_{p}$\nnorms, but they are all different to the standard central limit\ntheorem based confidence region. As noted, the Mahalanobis cost can be\ncalibrated to exactly match the standard central limit theorem\nconfidence region.\n\\subsection{Coverage probabilities and distributionally robust optimization solutions}\nIn this section, we test the scenario in which the covariates are\nhighly correlated. Specifically, the data is sampled from a linear\nregression model with parameters $\\sigma ^{2}=1$, $n=100$, $p=2$. The\nrandom vector $X$ is taken to be distributed in (\\ref{eq:cov-matrix}),\nconsidering three different values for $\\rho:$ we choose\n$\\rho = 0.95,0,-0.95.$ We consider the following two cases for the\nunderlying parameter $\\beta_\\ast$: $\\beta_{*}=[0.5,0.5]^{\\mathrm{\\scriptscriptstyle T} }$ and\n$\\beta_{*}=[1,0]^{\\mathrm{\\scriptscriptstyle T} }.$ In Table \\ref{tab:coverage} below, we report the\ncoverage probabilities of the underlying $\\beta _{*}$ and\n$\\beta_n^{DRO}(\\delta_n)$ in both the $\\ell_2$-confidence region and\nthe central limit theorem based confidence regions.  Specifically, we\nreport the following four probabilities:\n \\[\n{\\rm pr}\\{\\beta^{DRO}_n \\in \\Lambda^+_{\\delta_{n}}(P_{n})\\}, \\quad {\\rm pr}\\{\\beta_* \\in \\Lambda^+_{\\delta_{n}}(P_{n})\\}, \\quad {\\rm pr}\\{\\beta^{DRO}_n \\in \\Lambda_{CLT}(P_{n})\\}, \\quad {\\rm pr}\\{\\beta_* \\in \\Lambda_{CLT}(P_{n})\\}.\n \\]\n We sample 1000 datasets and report the coverage probabilities in\n Table \\ref{tab:coverage}. From Table \\ref{tab:coverage}, we observe\n that for $\\beta_*$, both the $\\ell_2$ confidence region and the\n central limit theorem based confidence region achieve the target 95\\%\n coverage. Furthermore, the coverage for the distributionally robust\n estimator of the $\\ell_2$ confidence region is 100\\%, which validates\n our theory. However, when $\\rho = -0.95$ and\n $\\beta_* = [0.5,0.5]^{\\mathrm{\\scriptscriptstyle T} }$, the coverage for the distributionally\n robust estimator in the central limit theorem based confidence region\n is only $75.8 \\%$. In this example, the asymptotic results developed\n indicate that this coverage probability converges to zero, when $n$\n tends to infinity.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Coverage Probability}\n    \\begin{tabular}{cccccc}\n      \\toprule\n      $\\beta_0$    &    $\\rho$   & \\multicolumn{2}{c}{$\\ell_2$-confidence region} & \\multicolumn{2}{c}{CLT confidence region} \\\\\n      \\midrule\n                   &       &  Coverage for $\\beta_n^{DRO}$ & Coverage for $\\beta_*$ & \\multicolumn{1}{l}{Coverage for $\\beta_n^{DRO}$} & Coverage for $\\beta_*$ \\\\\n      \\midrule\n      \\multirow{3}[0]{*}{$\\begin{bmatrix} 0.5 \\\\ 0.5 \\end{bmatrix}$}\n                   & 0.95  & 100.0\\% & 94.5\\% & 99.4\\% & 94.6\\% \\\\\n                   & 0     & 100.0\\% & 94.0\\% & 97.1\\% & 93.5\\% \\\\\n                   & -0.95 & 100.0\\% & 94.8\\% & 75.8\\% & 94.4\\% \\\\\n\n          \\midrule\n    \\multirow{3}[0]{*}{$\\begin{bmatrix} 1.0 \\\\ 0.0 \\end{bmatrix}$}\n          & 0.95  & 100.0\\% & 94.6\\% & 93.7\\% & 95.4\\% \\\\\n          & 0     & 100.0\\% & 94.6\\% & 100\\% & 94.1\\% \\\\\n          & -0.95 & 100.0\\% & 95.3\\% & 91.2\\% & 94.9\\% \\\\\n\n          \\bottomrule\n    \\end{tabular}%\n  \\label{tab:coverage}%\n\\end{table}%\nFigures \\ref{fig:scatter_0.5} and \\ref{fig:scatter_1} show the scatter\nplots of the estimators, $\\beta_n^{ERM}$ and $\\beta_n^{DRO},$ when the\nunderlying $\\beta_\\ast$ takes the values $[0.5,0.5]^{\\mathrm{\\scriptscriptstyle T} }$ and\n$[1,0]^{\\mathrm{\\scriptscriptstyle T} }$, respectively. In the near-collinearity\n  cases where $\\rho = 0.95$ or $-0.95$, the lower spreads for the\n  distributionally robust estimators reveal their better performance\n  over the empirical risk minimizing solutions. The utility of the\n  proposed $\\ell_2$-confidence region emerges in light of the better\n  performance of the distributionally robust estimator $\\beta_n^{DRO}$\n  and its aforementioned lack of membership in $\\Lambda_{CLT}(P_n).$\n\\begin{figure}[ptbh]\n\\centering\n\\subfigure[$\\rho=0.95$]{\n \\includegraphics[width=1.81in]{scatter_point_95_5.eps}}\n\\subfigure[$\\rho=0$]{\n \\includegraphics[width=1.81in]{scatter_point_0_5.eps}}\n\\subfigure[$\\rho=-0.95$]{\n\\includegraphics[width=1.81in]{scatter_point_-95_5.eps}}\n\\caption{Scatter plots of $\\beta_n^{ERM}$ (black circles) and\n  $\\beta_n^{DRO}$ (red circles) for $\\beta_0 = [0.5,0.5]^{\\mathrm{\\scriptscriptstyle T} }$. }\n\\label{fig:scatter_0.5}\n\\end{figure}\n\\begin{figure}[ptbh]\n\\centering\n\\subfigure[$\\rho=0.95$]{\n \\includegraphics[width=1.81in]{scatter_point_95_10.eps}}\n\\subfigure[$\\rho=0$]{\n \\includegraphics[width=1.81in]{scatter_point_0_10.eps}}\n\\subfigure[$\\rho=-0.95$]{\n\\includegraphics[width=1.81in]{scatter_point_-95_10.eps}}\n\\caption{Scatter plots of $\\beta_n^{ERM}$ (black circles) and\n  $\\beta_n^{DRO}$ (red circles) for $\\beta_0 = [1.0,0.0]^{\\mathrm{\\scriptscriptstyle T} }$.}\n\\label{fig:scatter_1}\n\\end{figure}\n\\section{Proofs of main results}\n\n\\label{sec:proof} Theorem \\ref{thm:levelsets-master} is obtained by\nconsidering appropriate level sets involving auxiliary functionals\nwhich we define next. Following \\citet{blanchet2016robust}, we define\nthe robust Wasserstein profile function, associated with the\nestimation of $\\beta_\\ast$ by solving\n$E_{P_n}\\{D_\\beta h(X,\\beta)\\} = 0,$ as follows:\n\\begin{equation*}\nR_{n}( \\beta) =\\inf_{P \\in \\mathcal{P}(\\Omega)} \\big[D_c(P,P_{n}) : \\beta \\in\n\\arg \\min_{\\beta \\in B }E_{P}\\left\\{ \\ell (X;\\beta )\\right\\} \\big].\n\\end{equation*}\nThis definition, as noted in \\citet{blanchet2016robust}, allows to\ncharacterize the set $\\Lambda^+_{\\delta}\\left( P_{n}\\right)$ in terms\nof an associated level set; in particular, we have,\n\\begin{equation}\n\\Lambda^+_{\\delta}( P_{n}) = \\mathrm{cl}{\\left\\{\\beta:R_{n}( \\beta) \\leq\\delta \\right\\}},\n\\label{CR_based_on_RWP}\n\\end{equation}\nwhere $\\mathrm{cl}(\\cdot)$ denotes closure. Indeed, this is because\n\\begin{equation*}\n  \\Lambda^+_{\\delta}( P_{n})= \\mathrm{cl} \\big[\\cap_{\\epsilon>0}\\big\\{ \\beta \\in B:\\beta \\in\n  \\arg \\min_{\\beta \\in B }E_{P}\\{ \\ell (X;\\beta )\\} \\text{ for some }P\\in\n  \\mathcal{U}_{\\delta _{n}+\\epsilon}(P_{n})\\big\\} \\big].\n\\end{equation*}\nIf $\\beta \\in B^\\circ$, we have\n$R_{n}( \\beta) =\\inf_{P \\in \\mathcal{P}(\\Omega)}[D_c(P,P_{n}) :E_{P}\n\\left\\{h( X,\\beta)\\right\\} = 0 ].$\n\nNext, for the sequence of radii $\\delta _{n}=n^{-\\gamma }\\eta $, for some\npositive constants $\\eta ,\\gamma $, define functions $V_{n}^{DRO}:\\mathbb{R}%\n^{d}\\rightarrow \\mathbb{R}$ and $V_{n}^{ERM}:\\mathbb{R}^{d}\\rightarrow\n\\mathbb{R}$, as below, by considering suitably scaled versions of the\ndistributionally robust and empirical risk objective functions, namely\n\\begin{align*}\n  V_{n}^{DRO}(u)& =n^{\\bar{\\gamma}}\\big \\{ \\Psi _{n}\\big( \\beta _{\\ast\n                  }+n^{-\\bar{\\gamma}\/2}u\\big) -\\Psi _{n}(\\beta _{\\ast })\\big\\} \\text{ and }\n  \\\\\n  V_{n}^{ERM}(u)& =n\\big[ E_{P_{n}}\\big\\{ \\ell (X;\\beta _{\\ast\n                  }+n^{-1\/2}u)\\big\\} -E_{P_{n}}\\big\\{ \\ell (X;\\beta _{\\ast })\\big\\} \\big],\n\\end{align*}%\nwhere $\\bar{\\gamma}=\\min \\left\\{ \\gamma ,1\\right\\} $ is defined in Theorem %\n\\ref{thm:levelsets-master}. Moreover, define $V:\\mathbb{R}^{d}\\times \\mathbb{%\nR}^{d}\\rightarrow \\mathbb{R}$ via\n\\begin{equation*}\n  V(x,u)=x^{{\\mathrm{\\scriptscriptstyle T} }}u+ {2}^{-1}u^{{\\mathrm{\\scriptscriptstyle T} }}Cu.\n\\end{equation*}%\nThe following result, as we shall see, can be used to establish Theorem \\ref%\n{thm:levelsets-master} directly.\n\n\\begin{theorem}\n  \\label{thm:master}\n  Suppose that the assumptions made in Theorem\n  \\ref{thm:levelsets-master} hold. Then we have,\n\\begin{equation*}\n\\big\\{ V_{n}^{ERM}(\\cdot),\\ V_{n}^{DRO}(\\cdot),\\ nR_{n}\\big( \\beta_{\\ast\n}+n^{-1\/2}\\times\\cdot\\ \\big) \\big\\} \\Rightarrow\\left\\{ V(-H,\\cdot ),\\\nV\\{-f_{\\eta,\\gamma}(H),\\cdot\\},\\ \\varphi^{\\ast}(H-C\\times\\cdot\\,)\\right\\} ,\n\\end{equation*}\non the space $C(\\mathbb{R}^{d};\\mathbb{R})^{3}$ equipped with the topology\nof uniform convergence in compact sets.\n\\end{theorem}\n{ Ensuring  smoothness of $D_{\\beta}h(x+\\Delta,\\beta)$ and $D_{x}h(x+\\Delta,\\beta)$ around $\\beta=\\beta_{\\ast},$ as in Assumption A2.c, is useful towards investigating the behavior of $nR_{n}\\big( \\cdot \\big)$ in the neighborhood of $\\beta^*,$ as required in the third component in the triplet in Theorem \\ref{thm:master}.}\n\n\\subsection{Proof of Theorem \\protect\\ref{thm:master}}\nThroughout this section, we suppose that the assumptions imposed in\nTheorem \\ref{thm:levelsets-master} hold. Let\n\\begin{equation*}\nH_{n} = {n}^{-1\/2}\\sum_{i=1}^{n}h\\left( X_{i},\\beta_{\\ast}\\right)\n\\end{equation*}\nThe following sequence of results will be useful in proving Theorem\n\\ref{thm:master} and Proposition\n\\ref{prop:constrained-support}. Propositions \\ref{prop:ERM} and \\ref{prop:DRO} hold true for $\\Omega = \\mathbb{R}^d$; while propositions \\ref{prop:RWPEq} - \\ref{prop:RWPcont} hold true for general $\\Omega$ under the assumption $P_{\\ast}(\\Omega^\\circ) =1$ in Proposition \\ref{prop:constrained-support}.\n\n\\begin{proposition}\nFix $\\alpha\\in\\lbrack0,1].$ Given $\\varepsilon,\\varepsilon^{\\prime},K>0,$ there exists a\npositive integer $n_{0}$ such that\n\\begin{equation*}\n{\\rm pr} \\left[\\big\\vert n^{\\alpha-1}V_{n}^{ERM}\\{n^{( 1-\\alpha)\n\/2}u\\}-n^{\\alpha\/2}H_{n}^{{\\mathrm{\\scriptscriptstyle T} }}u-2^{-1}u^{{\\mathrm{\\scriptscriptstyle T} }}Cu\\big\\vert \\leq\n\\varepsilon^{\\prime} \\right] \\geq 1-\\varepsilon,\n\\end{equation*}\nfor every $n>n_{0}$ and $\\Vert u\\Vert_2\\leq K.$ Specifically, if $\\alpha=1,$\nwe have%\n\\begin{equation}\n{\\rm pr} \\left\\{\\left\\vert V_{n}^{ERM}(u)-H_{n}^{{\\mathrm{\\scriptscriptstyle T} }}u- 2^{-1}u^{{\\mathrm{\\scriptscriptstyle T} }}Cu\\right\\vert\n\\leq\\varepsilon^{\\prime}\\right\\} \\geq 1-\\varepsilon.\n\\label{eqn:V_ERM}\n\\end{equation}\n\\label{prop:ERM}\n\\end{proposition}\n\\begin{proposition}\nGiven $\\varepsilon,\\varepsilon ^{\\prime },K>0,$ there exists a positive integer $n_{0}$\nsuch that\n\\begin{equation}\n{\\rm pr} \\left\\{ \\left\\vert V_{n}^{DRO}(u)+f_{\\eta ,\\gamma }(-H_{n})^{{\\mathrm{\\scriptscriptstyle T} }}u- 2^{-1}\nu^{{\\mathrm{\\scriptscriptstyle T} }}Cu\\right\\vert \\leq \\varepsilon ^{\\prime } \\right\\} \\geq 1-\\varepsilon,\n\\label{eqn:V_DRO}\n\\end{equation}%\nfor every $n>n_{0}$ and $\\Vert u\\Vert_2 \\leq K.$ \\label{prop:DRO}\n\\end{proposition}\n\\begin{proposition}\n  Define the set $\\Theta \\subset \\mathbb{R}^d$ as\n\\[\n  \\Theta=\\{ \\beta \\in B^\\circ : 0 \\in {\\rm conv} [\\{h(x,\\beta)\\mid x \\in\n  \\Omega\\}]^\\circ \\},\n\\]\nwhere ${\\rm conv} (S)$ denotes the convex hull of the set $S$.\nFor $\\beta_* + n^{-1\/2} u \\in \\Theta$, We have,\n\\begin{align*}\nn R_{n}\\big( \\beta_{\\ast}+ n^{-1\/2}u\\big) = \\max_{\\xi} \\left\\{ -\\xi^{{\\mathrm{\\scriptscriptstyle T} }}\nH_{n} - M_{n}(\\xi,u) \\right\\} ,\n\\end{align*}\nwhere\n\\begin{align*}\n  M_{n}(\\xi,u)\n  &= \\frac{1}{n} \\sum_{i=1}^{n} \\max_{\\Delta:X_i + n^{-1\/2}\\Delta \\in \\Omega} \\left\\{ \\xi^{{\\mathrm{\\scriptscriptstyle T} }} \\int_{0}^{1}\n    D_{x}h\\big( X_{i} + n^{-1\/2}t\\Delta, \\beta_{\\ast}+n^{-1\/2} tu\n    \\big) \\Delta {\\rm d} t \\right .  \\\\\n  &\\qquad\\qquad + \\xi^{{\\mathrm{\\scriptscriptstyle T} }} \\int_{0}^{1}D_{\\beta}h\\big( X_{i} + n^{-1\/2}t\\Delta,\n    \\left. \\beta_{\\ast}+ n^{-1\/2} t u\\big) u{\\rm d} t - \\Vert\\Delta\\Vert\n    _{q}^{2} \\right\\}.\n\\end{align*}\nFurthermore, there exists a neighborhood of $\\beta*$, $B_\\epsilon(\\beta_*)$ such that $B_\\epsilon(\\beta_*) \\subset \\Theta$.\n\\label{prop:RWPEq}\n\\end{proposition}\n\n\n\\begin{proposition}\nConsider any $\\varepsilon ,\\varepsilon ^{\\prime },K>0.$ Then there exist $%\nb_{0}\\in (0,\\infty )$ such that for any $b\\geq b_{0},c_0> 0,\\epsilon_0>0$, we\nhave a positive integer $n_{0}$ such that,\n\\begin{equation*}\n\\mathrm{pr}\\left[ \\sup_{\\Vert u\\Vert _{2}\\leq K}\\big \\{ nR_{n}\\big( \\beta\n_{\\ast }+n^{-1\/2}u\\big) -f_{up}(H_{n},u,b,c)\\big \\} \\leq \\varepsilon\n^{\\prime }\\right] \\geq 1-\\varepsilon ,\n\\end{equation*}%\nfor all $n\\geq n_{0},$ and $f_{up}(H_{n},u,b,c)$ equals\n\\begin{equation*}\n  \\max_{\\Vert \\xi \\Vert _{p}\\leq b}\\big\\{-\\xi ^{{\\mathrm{\\scriptscriptstyle T} }}H_{n}-%\n  E \\big[ 4^{-1}\\Vert \\{ D_{x}h(X,\\beta _{\\ast\n        })\\} ^{{\\mathrm{\\scriptscriptstyle T} }}\\xi \\Vert _{p}^{2}+\\xi ^{{\\mathrm{\\scriptscriptstyle T} }}D_{\\beta }h(X,\\beta _{\\ast\n      })u\\big] \\mathbb{I}(X \\in C_0^{\\epsilon_0}) \\big\\},\n\\end{equation*}\nwith $C_0=\\{x \\in \\Omega: \\|x\\|_p \\leq c_0\\}.$\n\\label{prop:RWPUB}\n\\end{proposition}\n\\begin{proposition}\n  For any $\\varepsilon ,\\varepsilon ^{\\prime },K,b>0,$ there exists a\n  positive integer $n_{0}$ such that,\n\\begin{equation*}\n{\\rm pr}\\left[ \\sup_{\\Vert u\\Vert _{2}\\leq K}\\big\\{ nR_{n}\\big( \\beta\n_{\\ast }+n^{-1\/2}u\\big) -f_{low}(H_{n},u,b)\\big\\} \\geq -\\varepsilon\n^{\\prime }\\right] \\geq 1-\\varepsilon ,\n\\end{equation*}%\nfor all $n>n_{0},$ where\n\\begin{equation*}\n  f_{low}(H_{n},u,b)=\\max_{\\Vert \\xi \\Vert _{p}\\leq b}\\big\\{-\\xi ^{{\\mathrm{\\scriptscriptstyle T} }}H_{n}-%\n  E\\left[ 4^{-1}\\Vert \\left\\{ D_{x}h(X,\\beta _{\\ast })\\right\\}\n    ^{{\\mathrm{\\scriptscriptstyle T} }}\\xi \\Vert _{p}^{2}+\\xi ^{{\\mathrm{\\scriptscriptstyle T} }}D_{\\beta }h(X,\\beta _{\\ast })u\\right] \\big\\}.\n\\end{equation*}\n\\label{prop:RWPLB}\n\\end{proposition}\n\n\\begin{proposition}\n  \\label{prop:tightness-1} For any $\\varepsilon>0,$ there exist\n  constants $a,n_{0} > 0$ such that for every\n  $n\\geq n_{0},$\n\\begin{equation*}\n{\\rm pr}\\left\\{ nR_{n}(\\beta_{\\ast})\\leq a\\right\\} \\geq1-\\varepsilon,\n\\end{equation*}\n\n\\end{proposition}\n\n\\begin{proposition}\nFor any $\\varepsilon,\\varepsilon^{\\prime}, K > 0,$ there exist positive\nconstants $n_{0},\\delta$ such that,\n\\begin{align*}\n\\sup_{\\underset{\\Vert u_{i}\\Vert_{2} \\leq K}{\\Vert u_{1} - u_{2}\\Vert_{2}\n\\leq\\delta}} \\big\\vert n R_{n}\\big( \\beta_{\\ast}+ n^{-1\/2}u_{1}\\big) - n\nR_{n}\\big( \\beta_{\\ast}+ n^{-1\/2}u_{2}\\big) \\big\\vert \\leq\n\\varepsilon^{\\prime},\n\\end{align*}\nwith probability exceeding $1-\\varepsilon,$ for every $n > n_{0}.$ \\label%\n{prop:RWPcont}\n\\end{proposition}\n\nProofs of Propositions \\ref{prop:ERM} - \\ref{prop:RWPcont} are furnished in\nSection \\ref{sec:proofs-props-master-thm} in the supplementary material. With the statements of these\nresults, we proceed with the proof of Theorem \\ref{thm:master} as follows.\n\n\\begin{proof}[Proof  of Theorem \\ref{thm:master}]\n  Since $E\\{h(X,\\beta _{\\ast })\\}=0,$ it follows from central limit\n  theorem that\n$\nH_{n}\\Rightarrow -H,\n$\nwhere\n$H\\sim \\mathcal{N}(0,E\\{h(X,\\beta _{\\ast })h(X,\\beta _{\\ast })^{\n  {\\mathrm{\\scriptscriptstyle T} }}\\}).$ Since inequalities \\eqref{eqn:V_DRO} and \\eqref{eqn:V_ERM}\nare associated with the same $H_n$ , it follows from Propositions\n\\ref%\n{prop:ERM} and \\ref{prop:DRO} that,\n\\begin{equation}\nV_{n}^{ERM}(\\cdot )\\Rightarrow V^{ERM}(\\cdot )=V(-H,\\cdot )\\quad \\text{ and\n}\\quad V_{n}^{DRO}(\\cdot )\\Rightarrow V^{DRO}(\\cdot )=V\\{-f_{\\eta ,\\gamma\n}(H),\\cdot \\}  \\label{inter-mas-thm-pf-0}\n\\end{equation}%\njointly, on the space topologized by uniform convergence on compact sets.\n\nTo prove convergence of the third component of the triplet considered in\nTheorem \\ref{thm:master}, observe from the definitions of $\\varphi ^{\\ast\n}(\\cdot )$ and $C$ that,\n\\begin{equation}\n  \\varphi ^{\\ast }(H-Cu)=\\max_{\\xi }\\big(\\xi ^{{\\mathrm{\\scriptscriptstyle T}}}\n  [ H-E\\{ D_{\\beta }h(X,\\beta _{\\ast })\\} u] - 4^{-1}\n  E\\Vert  \\{ D_{x}h(X,\\beta _{\\ast })\\} ^{{\\mathrm{\\scriptscriptstyle %\n        T}}}\\xi \\Vert _{p}^{2}\\big).  \\label{Expr-psi-HCu}\n\\end{equation}%\nConsider any fixed $K\\in (0,+\\infty ).$ Due to the weak convergence $%\nH_{n}\\Rightarrow -H,$ applications of continuous mapping theorem to the\nbounds in Proposition \\ref{prop:RWPUB},~\\ref{prop:RWPLB} result in the\nconclusions that,\n\\begin{equation}\n  f_{up}(H_{n},u,b,c)\\Rightarrow \\max_{\\Vert \\xi \\Vert _{p}\\leq b}\\big\\{\\xi ^{{{\\mathrm{\\scriptscriptstyle T} }}}H-\n  E\\big[ {4}^{-1}\\Vert \\big\\{D_{x}h(X,\\beta _{\\ast })] ^{{\\mathrm{\\scriptscriptstyle T}}}\\xi\n  \\Vert_{p}^{2}+\\xi ^{{\\mathrm{\\scriptscriptstyle T}}}D_{\\beta }h(X,\\beta _{\\ast})u\\big]\n  \\mathbb{I}(X \\in C_0^{\\epsilon_0}) \\big\\},\n\\label{inter-mas-up}\n\\end{equation}%\n\\begin{equation}\n  f_{low}(H_{n},u,b)\\Rightarrow \\max_{\\Vert \\xi \\Vert _{p}\\leq b}\\big\\{\\xi ^{{{\\mathrm{\\scriptscriptstyle T} }}}H-\n  E\\big[ {4}^{-1}\\Vert  \\{D_{x}h(X,\\beta _{\\ast })\\}^{{\\mathrm{\\scriptscriptstyle T}}}\\xi \\Vert\n  _{p}^{2}+\\xi ^{{\\mathrm{\\scriptscriptstyle T}}}D_{\\beta }h(X,\\beta _{\\ast\n  })u\\big] \\big\\},  \\label{inter-mas-low}\n\\end{equation}%\nfor any $u$ satisfying $\\Vert u\\Vert _{2}\\leq K.$ Since the bounds in\nPropositions \\ref{prop:RWPUB},~\\ref{prop:RWPLB} hold for arbitrarily large\nchoices for constants $b,c,$ and arbitrarily small choice for constant $\\epsilon_0$ combining with the assumption $P_*(\\Omega^\\circ)=1$, we conclude from the observations %\n\\eqref{Expr-psi-HCu}, \\eqref{inter-mas-up}, and \\eqref{inter-mas-low}   that\n\\begin{equation}\n  nR_{n}\\big( \\beta _{\\ast }+n^{-1\/2}u\\big) \\Rightarrow \\varphi ^{\\ast}(H-Cu),  \\label{mas-fdim-conv}\n\\end{equation}%\nfor any $u$ satisfying $\\Vert u\\Vert _{2}\\leq K.$ Finally, we have\nfrom Propositions \\ref{prop:tightness-1} and \\ref%\n{prop:RWPcont} that the collection\n$\\{nR_{n}(\\beta_{\\ast}+n^{-1\/2}\\times%\n\\cdot\\,)\\}$ is tight; see, for example, \\citet[Theorem\n7.4]{billingsley2013convergence}. As a consequence of this tightness\nand the finite dimensional convergence in \\eqref{mas-fdim-conv}, we\nhave that,\n\\begin{equation*}\nnR_{n}\\big( \\beta_{\\ast}+n^{-1\/2}\\times\\cdot\\,\\big) \\Rightarrow\n\\varphi^{\\ast}(H-C\\times\\cdot\\,).\n\\end{equation*}\nCombining this observation with those in \\eqref{inter-mas-thm-pf-0}, we\nobtain the desired convergence result in Theorem \\ref{thm:master}. Furthermore, since $f_{low}(H_{n},u,b)$ and $f_{up}(H_{n},u,b)$ are associated with the same $H_n$ with inequalities \\eqref{eqn:V_DRO} and \\eqref{eqn:V_ERM}, we have the three terms converge jointly.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\protect\\ref{thm:levelsets-master}}\n\n\\begin{proof}[Proof  of Theorem \\ref{thm:levelsets-master}]\n  \\label{sec:levelsets-master} Theorem \\ref{thm:levelsets-master} is\n  proved by considering suitable level sets of the component functions\n  in the triplet, $%\n  \\{V_{n}^{ERM}(\\cdot), \\ V_{n}^{DRO}(\\cdot), \\ nR_{n}(\n    \\beta_{\\ast}+ n^{-1\/2}\\times \\cdot\\ )\\}, $ considered in\n  Theorem \\ref{thm:master}.  To reduce clutter in expressions, from\n  here-onwards we refer the distributionally robust estimator\n  \\eqref{Wass_DRO_A}, simply as $\\beta^{DRO}_n,$ with the dependence\n  on the radius $\\delta_n$ to be understood from the context. To begin,\n  consider the following tightness result whose proof is provided in\n  Section \\ref{ssec:proofs-props-mast-lev-sets}.\n\n\\begin{proposition}\nThe sequences $\\{\\arg\\min\\xspace_{u} \\, V_{n}^{ERM}(u): n \\geq1\\}$ and $%\n\\{\\arg\\min\\xspace_{u} \\, V_{n}^{DRO}(u): n \\geq1\\}$ are tight. \\label%\n{prop:argmintightness}\n\\end{proposition}\n\nObserve that $V_{n}^{ERM}(\\cdot)$ and $V_{n}^{DRO}(\\cdot)$ are\nminimized, respectively, at $n^{1\/2}(\\beta_{n}^{ERM}-\\beta_{\\ast})$\nand $n^{\\bar{\\gamma }\/2}(\\beta_{n}^{DRO}-\\beta_{\\ast}).$ Furthermore,\ndue to the positive definiteness of $C$ in Assumption A2.b, we\nhave that $V^{ERM}(\\cdot)$ and $V^{DRO}(\\cdot)$ are strongly convex\nwith respect to $u$ and have unique minimizers, with probability\n1. Therefore, due to the tightness of the sequences\n$\\{n^{1\/2}(\\beta_{n}^{ERM}-\\beta_{\\ast})\\}_{n\\geq1}$ and $\\{n^{%\n  \\bar{\\gamma}\/2}(\\beta_{n}^{DRO}-\\beta_{\\ast})\\}_{n\\geq1}$; see\nProposition %\n\\ref{prop:argmintightness} and the weak convergence of\n$V_{n}^{ERM}(\\cdot)$ and $V_{n}^{DRO}(\\cdot)$ in Theorem\n\\ref{thm:master}, we have the following convergences:\n\\begin{align}\n  n^{1\/2}(\\beta_{n}^{ERM}-\\beta_{\\ast})&\\Rightarrow\\arg\\min\\xspace%\n  _{u}\\,V(-H,u)=C^{-1}H,  \\label{inter-mas-thm-pf-1}\\\\\n  n^{\\bar{\\gamma}\/2}(\\beta_{n}^{DRO}-\\beta_{\\ast})\n  &\\Rightarrow\\arg\\min \\xspace%\n  _{u}\\,V^{DRO}(u)=C^{-1}f_{\\eta,\\gamma}(H)\n  \\nonumber\n\\end{align}\n\nFinally, to prove the convergence of the sets $\\Lambda^+_{\\delta_{n}}(P_{n}),$\nwe proceed as follows. Define\n\\begin{equation*}\nG_{n}(u) = nR_{n}(\\beta_{\\ast}+ n^{-1\/2}u), \\quad G(u) = \\varphi^{\\ast\n}(H-Cu), \\quad\\text{ and } \\quad\\alpha_{n} = n\\delta_{n}.\n\\end{equation*}\nFor any function $f:B \\rightarrow\\mathbb{R}$ and $\\alpha \\in[%\n0,+\\infty],$ let lev$(f,\\alpha)$ denote the level set\n$\\{ x \\in \\mathbb{R}^d: f(x) \\leq\\alpha\\}.$\n\n\\begin{proposition}\n  If $\\delta_{n} = n^{-1}\\eta,$ then\n  $\\mathrm{cl}\\{$\\textnormal{lev}$(G_{n},\\alpha_{n})\\} \\Rightarrow$\n  \\textnormal{lev}$(G,\\eta).$ \\label{prop-levset-eq1}\n\\end{proposition}\n\n\\begin{proposition}\nIf $\\delta_{n} = n^{-\\gamma}\\eta$ for some $\\gamma> 1,$ then $\\mathrm{cl}\\{$\\textnormal{lev}%\n$(G_{n},\\alpha_{n})\\} \\Rightarrow\\{C^{-1}H\\}.$ \\label{prop-levset-gthan1}\n\\end{proposition}\n\n\\begin{proposition}\n  If $\\delta_{n} = n^{-\\gamma}\\eta$ for some $\\gamma< 1,$ then\n  $\\mathrm{cl}\\{$\\textnormal{lev}$(G_{n},\\alpha_{n})\\} \\Rightarrow$\n  $\\mathbb{R}^{d}.$ \\label{prop-levset-lthan1}\n\\end{proposition}\n\nPropositions \\ref{prop-levset-eq1} - \\ref{prop-levset-lthan1} above, whose\nproofs are furnished in Section \\ref{ssec:proofs-props-mast-lev-sets}, allow\nus to complete the proof of Theorem \\ref{thm:levelsets-master} as follows.\nIt follows from the definition of $R_{n}(\\beta )$ that,\n\\begin{equation*}\n\\Lambda^+_{\\delta _{n}}(P_{n})=\\left\\{ \\beta :R_{n}(\\beta )\\leq \\delta\n_{n}\\right\\} =\\beta _{\\ast }+n^{-1\/2}\\left\\{ u:G_{n}(u)\\leq \\alpha\n_{n}\\right\\} .\n\\end{equation*}%\nWe have from Propositions \\ref{prop-levset-eq1} -\n\\ref{prop-levset-lthan1} that\n\\begin{equation*}\nn^{1\/2}\\left( \\Lambda^+_{\\delta _{n}}(P_{n})-\\beta _{\\ast }\\right) =\\left\\{\nu:G_{n}(u)\\leq \\alpha _{n}\\right\\} \\Rightarrow\n\\begin{cases}\n\\text{lev}(G,\\eta )\\quad & \\text{ if }\\gamma =1, \\\\\n\\mathbb{R}^{d} & \\text{ if }\\gamma <1, \\\\\n\\{C^{-1}H\\} & \\text{ if }\\gamma >1.%\n\\end{cases}\n\\end{equation*}%\nObserve that $\\varphi ^{\\ast }(u)=\\varphi ^{\\ast }(-u).$ Therefore,\nlev$%\n(G,\\eta )=\\{u:\\varphi ^{\\ast }(H-Cu)\\leq \\eta \\}=C^{-1}H+\\{u:\\varphi\n^{\\ast }(Cu)\\leq \\eta \\}.$ Since the three terms in Theorem\n\\ref{thm:master} converge jointly, we have the three terms in Theorem\n\\ref%\n{thm:levelsets-master} also converge jointly. This completes the proof\nof Theorem \\ref%\n{thm:levelsets-master}.\n\\end{proof}\n\nProposition \\ref{prop:constrained-support} follows by adopting exactly\nthe same steps which are used to establish the convergence of\n$n^{1\/2}\\left\\{\\Lambda^+_{\\delta _{n}}(P_{n})-\\beta _{\\ast }\\right\\}$ in the proof\nof Theorem \\ref{thm:levelsets-master}.\n\n\n\n\n\\section{Discussions}\n\\label{sec:discussion}\nWe discuss the subtleties in\nderiving a limit theorem for the distributionally robust estimator\n$\\beta_n^{DRO}$ when the support of the random vector $X,$ denoted by\n$\\Omega,$ is constrained to be a strict subset of $\\mathbb{R}^m.$\nSuppose that the support of $X$ is constrained to be contained in the\nset $\\Omega = \\{x \\in \\mathbb{R}^m: Ax \\leq b\\}$ specified in terms of\nlinear constraints involving an $l \\times m$ matrix $A$ and\n$b \\in \\mathbb{R}^l.$ For the sake of clarity, we discuss here only\nthe non-degenerate case where $\\delta_n = \\eta\/ n.$\n\nConsidering the transportation cost $c(x,y) = \\Vert x- y\\Vert_2^2$ in\nDefinition \\ref{defn:WD}, we demonstrate in Section \\ref{ssec:discussion_proof} of the\nSupplementary material that the central limit theorem,\n$n^{1\/2}\\{\\beta_n^{DRO}(\\delta_n) - \\beta_\\ast\\} \\Rightarrow\nC^{-1}H - \\eta^{1\/2} C^{-1}D_\\beta S(\\beta_\\ast),$ continues to hold,\nfor example, in the elementary\ncase\nwhere the matrix $A$ has linearly independent rows, $X$ has a\nprobability density which is absolutely continuous with respect to the\nLebesgue measure on $\\mathbb{R}^m$ and the support $\\Omega$ is\ncompact.\nA key element which emerges in the verification (offered in\nProposition \\ref{prop:DRO-constrained} in Section \\ref{ssec:discussion_proof} of the supplementary material) is that\nthe fraction of samples which get transported to the boundary of the\nset $\\Omega$ stays $O_p(n^{-1\/2}),$ as $n \\rightarrow \\infty.$\n\n\nOn the other hand, when the set\n$\\Omega = \\{x \\in \\mathbb{R}^m: Ax \\leq b\\}$ has equality constraints\nas in, for example,\n$\\Omega = \\{(x_1,x_2,\\ldots,x_m) \\in \\mathbb{R}^2: x_1 - x_2 = 0\\},$\nthe bias term in the limit theorem gets affected due to the constraint\nbinding all the samples $\\{X_1,\\ldots,X_n\\}$ and the fraction of\nsamples which get transported to the boundary of the set $\\Omega$ is\n1. This is easily seen in the linear regression example in Section\n\\ref{sec:geo_insignts} where $\\ell(x,y;\\beta) = (y - \\beta^{\\mathrm{\\scriptscriptstyle T} } x)^2$ and\nthe support is taken as\n$\\Omega = \\{(x_1,x_2) \\in \\mathbb{R}^2: x_1 = x_2\\}.$ For this\nelementary example, we instead have,\n\\begin{align}\n    n^{1\/2}\\big\\{\\beta_n^{DRO}(\\delta_n) - \\beta_\\ast\\big\\} \\Rightarrow\n  C^{-1}H - \\eta^{1\/2} C^{-1}D_\\beta \\tilde{S}(\\beta_\\ast),\n  \\label{clt-support-constraints}\n\\end{align}\nwhere $\\tilde{S}(\\beta)$ is different from the term $S(\\beta)$ as in,\n$ \\tilde{S}(\\beta_\\ast) = 2^{1\/2-1\/q} { \\vert \\beta^\\T1 \\vert} {\n  \\Vert \\beta \\Vert_p^{-1}} S(\\beta).$\nHere, recall the earlier definition\n$S(\\beta) = [E\\{ \\Vert D_x\\ell(X;\\beta)\\Vert_p^2\\}]^{1\/2}$ in\n\\eqref{defn:sensitivity-term} for the unconstrained support case. The\ncomputations required to arrive at the above conclusion are presented\nin Example A1 in Section A.3 of the supplementary material. {\nIn the presence of general support constraints of the form $\\Omega = \\{x \\in \\mathbb{R}^m:Ax= b\\},$ we show with Example A2 in Section A.3 that \\eqref{clt-support-constraints} holds with $\\tilde{S}(\\beta) = \\Vert P_{\\mathcal{N}(A)} \\beta \\Vert_2$ for quadratic losses of the form $\\ell(x;\\beta) = a + \\beta^{\\mathrm{\\scriptscriptstyle T} } x + \\beta^{\\mathrm{\\scriptscriptstyle T} } C \\beta;$ here $A$ is taken to be a matrix with linearly independent rows and $P_{\\mathcal{N}(A)}$ denotes the projection operator onto the null space of $A.$ The bias term here is again different when compared to the term resulting from $S(\\beta) = \\Vert \\beta \\Vert_2$ exhibited in Theorem \\ref{thm:levelsets-master}.}\nAs reasoned above, the presence of equality constraints for the support\n$\\Omega$ introduces new challenges to be tackled in another study.\n\n\\section*{Acknowledgements}\nMaterial in this paper is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0397. Additional support is gratefully acknowledged from NSF grants 1915967, 1820942 and 1838576 and  MOE SRG ESD 2018 134.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nDeep learning offers a powerful framework for learning increasingly complex\nrepresentations for visual recognition tasks. The work of Krizhevsky \\etal\n\\cite{KSH13} convincingly demonstrated that deep neural networks can be very\neffective in classifying images in the challenging Imagenet benchmark\n\\cite{DDSL+09}, significantly outperforming computer vision systems built on\ntop of engineered features like SIFT \\cite{Lowe04}. Their success spurred a\nlot of interest in the machine learning and computer vision\ncommunities. Subsequent work has improved our understanding and has refined\ncertain aspects of this class of models \\cite{ZeFe13b}. A number of different\nstudies have further shown that the features learned by deep neural networks\nare generic and can be successfully employed in a black-box fashion in other\ndatasets or tasks such as image detection \\cite{ZeFe13b, OuWa13, SEZM+14,\n  GDDM14, RASC14, CSVZ14}.\n\nThe deep learning models that so far have proven most successful in image\nrecognition tasks, e.g.\n\\cite{SzegedyLJSRAEVR14,nin,SiZi14},\nare feed-forward convolutional neural networks trained in a\nsupervised fashion to minimize a regularized training set classification error\nby back-propagation. Their recent success is partly due to the availability of\nlarge annotated datasets and fast GPU computing, and partly due to some\nimportant methodological developments such as dropout regularization and\nrectifier linear activations \\cite{KSH13}. However, the key building blocks of\ndeep neural networks for images have been around for many years \\cite{LBBH98}:\n(1) Deep Convolutional Neural Networks (DCNNs) with small receptive fields that\nspatially share parameters within each layer. (2) Gradual abstraction and\nspatial resolution reduction after each convolutional layer as we ascend the\nnetwork hierarchy, typically via max-pooling \\cite{RiPo99, JKRL09}\n\n\\begin{figure}[!tbp]\n  \\centering\n  \\begin{tabular}{c}\n    \\includegraphics[width=0.9\\columnwidth]{figs\/illustration\/epitome}\n  \\end{tabular}\n  \\caption{Epitomes represent {\\em a set} of small images through a common data\n    structure: The epitome is a single, larger image which can produce\n    several smaller filters by picking a position and cropping a small window.  While\n    epitomes where originally employed for generative image modelling, in this\n    work we propose to use  epitomes as an efficient method for parameter\n    sharing in Deep Learning: Epitomes are used in hierarchical networks,\n    and their parameters are learned discriminatively through\n    back-propagation.}\n  \\label{fig:epitome_bag}\n\\end{figure}\n\nThis paper examines several aspects of invariance to deformations in the context of DCNNs for visual recognition. We treat both locally defined (non-rigid) and global \ndeformations, demonstrating that a more refined treatment yields improved classification performance.\n\n Our first main contribution is\nto build a deep neural network around the epitomic representation\n\\cite{JFK03}. The image epitome, illustrated in Figure~\\ref{fig:epitome_bag},\nis a data structure appropriate for learning translation-aware image\nrepresentations, naturally disentagling appearance and position modeling of\nvisual patterns. While originally developped for generative image modelling, we show here that the epitome data structure can be used to  train DCCNs discrimatively.\n\nFor this, we substitute every pair of  convolution and max-pooling layers\ntypically used in DCNNs with an `Epitomic Convolution' layer. As illustrated in \nFig.~\\ref{fig:epitome_diagram},  Epitomic Convolution in an input-centered dual alternative to the filter-centered standard\nmax-pooling. Namely, rather than searching in the input image (or layer) for the strongest response of a filter, we search across the set of filters encapsuled in a epitome for the strongest response to an image patch.\n\\mycomment{\nfor each regularly-spaced input data patch in the lower layer we search across\nfilters in the epitomic dictionary for the strongest response. In max-pooling\non the other hand, for each filter in the dictionary we search within a window\nin the lower input data layer for the strongest response.}\nOur Epitomic Convolution is designed to have the same computational and model complexity as Max-Pooling, while similar filters to share parameters. \n\nWe use a dictionary of `mini-epitomes' at each network layer: each\nmini-epitome is only slightly larger than the corresponding input data patch,\njust enough to accomodate for the desired extent of position invariance. For\neach input data patch, the mini-epitome layer outputs a single value per\nmini-epitome, which is the maximum response across all filters in the\nmini-epitome. In \\cite{Papa14} we discuss another deep epitomic network\nvariant built on top of large epitomes which learns topographically organized\nfeatures.\n\nOur second main contribution is to explicitly deal with object scale and\nposition when applying DCNNs to image classification. We show\nthat this can be done efficiently using a patchwork data structure in a  principled, consistent manner during both training and testing. \n While\nfusing classification results extracted from multiple image windows is\nstandard practice, we show that when incorporating multiple windows during training in a\nMultiple Instance Learning (MIL) framework we obtain substantially larger\ngains.\n\nFinally, we  explore to what extent we can push scale and position search\n towards developing efficient and effective sliding window object\ndetectors on top of DCNNs. We accelerate sliding window detection by introducing \n a simple method to reduce the effective size and receptive field of a DCNN pre-trained on ImageNet and show that by performing an explicit search over position, scale, and aspect ratios we can obtain  results that are comparable to the current-state-of-the-art while being substantially simpler and easier to train, as well as more efficient. \n \n\nWe quantitatively evaluate the proposed models primarily in image\nclassification experiments on the Imagenet ILSVRC-2012 large-scale image\nclassification task. We train the model by error back-propagation to minimize\nthe classification loss, similarly to \\cite{KSH13}. We have experimented with\ndeep mini-epitomic networks of different architectures (Class A, B, and C). We\nhave carried out the bulk of our comparisons between epitomes and max-pooling\nusing comparable mid-sized networks having 6 convolutional and 3 fully\nconnected layers whose structure closely follows that in \\cite{SEZM+14}\n(Class-A). In these evaluations the deep epitomic network achieves 13.6\\%\ntop-5 error on the validation set, which is 0.6\\% better than the\ncorresponding conventional max-pooled convolutional network whose error rate\nis 14.2\\%, with both networks having the same computational cost. Note that\nthe error rate of the original model in \\cite{KSH13} is 18.2\\%, using however\na smaller network. We have also more recently experimented with larger deep\nmini-epitomic networks (Class-B), which have the same number of levels but\nmore neurons per layer than those in Class-A. The plain deep epitomic net in\nClass-B achieves an error rate of 11.9\\%. When accompanied with explicit scale\nand position search (patchwork variant), its error drops further down to\n10.0\\%. Finally, following \\cite{SiZi14}, we have most recently experimented\nwith a very deep epitomic network with 13 convolutional and 3 fully connected\nlayers (Class-C). This achieves an even lower error rate of 10.0\\% (without\nscale and position search). All these performance numbers refer to\nclassification with a single network. We also find that the proposed epitomic\nmodel converges faster, especially when the filters in the dictionary are\nmean- and contrast-normalized, which is related to \\cite{ZeFe13b}. We have\nfound this normalization to also accelerate convergence of standard max-pooled\nnetworks. \n\nIn \\cite{Papa14} we further report additional sets of experiments. First, we\nshow that a deep epitomic network trained on Imagenet can be effectively used\nas black-box feature extractor for tasks such as Caltech-101 image\nclassification. Second, we report excellent image classification results on\nthe MNIST and CIFAR-10 benchmarks with smaller deep epitomic networks trained\nfrom scratch on these small-image datasets.\n\n\\paragraph{Related work}\n\nOur model builds on the epitomic image representation \\cite{JFK03}, which was\ninitially geared towards image and video modeling tasks. Single-level\ndictionaries of image epitomes learned in an unsupervised fashion for image\ndenoising have been explored in \\cite{AhEl08, BMBP11}. Recently, single-level\nmini-epitomes learned by a variant of K-means have been proposed as an\nalternative to SIFT for image classification \\cite{PCY14}. To our knowledge,\nepitomes have not been studied before in conjunction with deep models or\nlearned to optimize a supervised objective.\n\nThe proposed epitomic model is closely related to maxout networks\n\\cite{GWMCB13}. Similarly to epitomic matching, the response of a maxout layer\nis the maximum across filter responses. The critical difference is that the\nepitomic layer is hard-wired to model position invariance, since filters\nextracted from an epitome share values in their area of overlap. This\nparameter sharing significantly reduces the number of free parameters that\nneed to be learned. Maxout is typically used in conjunction with max-pooling\n\\cite{GWMCB13}, while epitomes fully substitute for it. Moreover, maxout\nrequires random input perturbations with dropout during model training,\notherwise it is prone to creating inactive features. On the contrary, we\nhave found that learning deep epitomic networks does not require dropout in\nthe convolutional layers -- similarly to \\cite{KSH13}, we only use dropout\nregularization in the fully connected layers of our network.\n\nOther variants of max pooling have been explored before. Stochastic pooling\n\\cite{ZeFe13a} has been proposed in conjunction with supervised\nlearning. Probabilistic pooling \\cite{LGRN09} and deconvolutional networks\n\\cite{ZKTF10} have been proposed before in conjunction with unsupervised\nlearning, avoiding the theoretical and practical difficulties associated with\nbuilding probabilistic models on top of max-pooling. While we do not explore\nit in this paper, we are also very interested in pursuing unsupervised\nlearning methods appropriate for the deep epitomic representation.\n\n\\section{Deep Epitomic Convolutional Networks}\n\\label{sec:epitomes}\n\n\\begin{figure}[!tbp]\n  \\centering\n  \\begin{tabular}{c c}\n    \\includegraphics[width=0.45\\columnwidth]{figs\/illustration\/conv_max}&\n    \\includegraphics[width=0.45\\columnwidth]{figs\/illustration\/epitomic_conv}\\\\\n    (a)&(b)\n  \\end{tabular}\n  \\caption{(a) Standard max-pooled convolution: For each filter we look for\n    its best match within a small window in the data layer. (b) Proposed\n    epitomic convolution (mini-epitome variant): For input data patches\n    sparsely sampled on a regular grid we look for their best match in each\n    mini-epitome.}\n  \\label{fig:epitome_diagram}\n\\end{figure}\n\n\\subsection{Mini-Epitomic deep networks}\n\n\n\\begin{figure*}[!tbp]\n  \\centering\n  \\begin{tabular}{c c}\n    \\includegraphics[scale=1.4]{figs\/visualize\/epito2b30\/conv1_0}&\n    \\includegraphics[scale=1.4]{figs\/visualize\/overfeat2b30\/conv1_0}\\\\\n    (a) & (b)\n  \\end{tabular}\n  \\caption{Filters at the first convolutional layer of: (a) Proposed\n    \\textsl{Epitomic} model with 96 mini-epitomes, each having size\n    \\by{12}{12} pixels. (b) Baseline \\textsl{Max-Pool} model with 96 filters\n    of size \\by{8}{8} pixels each.}\n  \\label{fig:visualize_conv1}\n\\end{figure*}\n\n\nWe first describe a single layer of the mini-epitome variant of the proposed\nmodel, with reference to Fig.~\\ref{fig:epitome_diagram}. In standard\nmax-pooled convolution, we have a dictionary of $K$ filters of spatial size\n$\\by{W}{W}$ pixels spanning $C$ channels, which we represent as real-valued\nvectors $\\{\\wv_k\\}_{k=1}^K$ with $W \\cdot W \\cdot C$ elements. We apply each\nof them in a convolutional fashion to every $\\by{W}{W}$ input patch\n$\\{\\xv_i\\}$ densely extracted from each position in the input layer which also\nhas $C$ channels. A reduced resolution output map is produced by computing the\nmaximum response within a small $\\by{D}{D}$ window of displacements $p \\in\n\\mathcal{N}_{input}$ around positions $i$ in the input map which are $D$\npixels apart from each other. \nThe output map $\\{z_{i,k}\\}$ of standard max-pooled convolution has spatial\nresolution reduced by a factor of $D$ across each dimension and will consist\nof $K$ channels, one for each of the $K$ filters. Specifically:\n\\begin{equation}\n  (z_{i,k},p_{i,k}) \\leftarrow \\max_{p \\in \\mathcal{N}_{image}}\n  \\xv_{i+p}^T \\wv_k \\,\n  \\label{eq:conv-max}\n\\end{equation}\nwhere $p_{i,k}$ points to the input layer position where the maximum is\nattained (argmax).\n\nIn the proposed epitomic convolution scheme we replace the filters with larger\nmini-epitomes $\\{\\vv_k\\}_{k=1}^K$ of spatial size $\\by{V}{V}$ pixels, where $V\n= W+D-1$. Each mini-epitome contains $D^2$ filters $\\{\\wv_{k,p}\\}_{k=1}^K$ of\nsize $\\by{W}{W}$, one for each of the $\\by{D}{D}$ displacements $p \\in\n\\mathcal{N}_{epit}$ in the epitome. We \\emph{sparsely} extract patches\n$\\{\\xv_i\\}$ from the input layer on a regular grid with stride $D$ pixels. In\nthe proposed epitomic convolution model we reverse the role of filters and\ninput layer patches, computing the maximum response over epitomic positions\nrather than input layer positions:\n\\begin{equation}\n  (y_{i,k},p_{i,k}) \\leftarrow \\max_{p \\in \\mathcal{N}_{epitome}}\n  \\xv_i^T \\wv_{k,p} \\,\n  \\label{eq:epitomic-conv}\n\\end{equation}\nwhere $p_{i,k}$ now points to the position in the epitome where the maximum is\nattained. Since the input position is fixed, we can think of epitomic matching\nas an input-centered dual alternative to the filter-centered standard max-pooling.\n\nComputing the maximum response over filters rather than image\npositions resembles the maxout scheme of \\cite{GWMCB13}, yet in the proposed\nmodel the filters within the epitome are constrained to share values in their\narea of overlap.\n\nSimilarly to max-pooled convolution, the epitomic convolution output map\n$\\{y_{i,k}\\}$ has $K$ channels and is subsampled by a factor of $D$ across\neach spatial dimension. Epitomic convolution has the same computational cost\nas max-pooled convolution. For each output map value, they both require\ncomputing $D^2$ inner products followed by finding the maximum\nresponse. Epitomic convolution requires $D^2$ times more work per input patch,\nbut this is exactly compensated by the fact that we extract input patches\nsparsely with a stride of $D$ pixels.\n\nSimilarly to standard max-pooling, the main computational primitive is\nmulti-channel convolution with the set of filters in the epitomic dictionary,\nwhich we implement as matrix-matrix multiplication and carry out on the GPU,\nusing the cuBLAS library.\n\nIt is noteworthy that conventional max-pooled convolution can be cast as\nspecial case of epitomic convolution. More specifically, we can exactly\nreplicate max-pooled convolution with filters of size $\\by{W}{W}$ and\n$\\by{D}{D}$ max-pooling using as epitomes zero-padded versions of the original\nfilters (with padding equal to $D-1$ pixels on each side of the filter) and\nepitomic filter size equal to $W+D-1$. In that sense, epitomic convolution is\na generalization of max-pooling.\n\nTo build a deep epitomic model, we stack multiple epitomic convolution layers\non top of each other. The output of each layer passes through a rectified\nlinear activation unit $y_{i,k} \\leftarrow \\max(y_{i,k} + \\beta_k, 0)$ and fed\nas input to the subsequent layer, where $\\beta_k$ is the bias. Similarly to\n\\cite{KSH13}, the final two layers of our network for Imagenet image\nclassification are fully connected and are regularized by dropout. We learn\nthe model parameters (epitomic weights and biases for each layer) in a\nsupervised fashion by error back propagation. We present full details of our\nmodel architecture and training methodology in the experimental section.\n\n\n\\subsection{Optional mean and contrast normalization}\n\\label{sec:mean_con_norm}\n\nMotivated by \\cite{ZeFe13b}, we have explored the effect of filter mean\nand contrast normalization on deep epitomic network training. More\nspecifically, we considered a variant of the model where the epitomic\nconvolution responses are computed as:\n\\begin{equation}\n  (y_{i,k},p_{i,k}) \\leftarrow \\max_{p \\in \\mathcal{N}_{epitome}}\n  \\frac{\\xv_i^T \\bar{\\wv}_{k,p}}{\\norm{\\bar{\\wv}_{k,p}}_\\lambda} \\,\n  \\label{eq:epitomic-conv-norm}\n\\end{equation}\nwhere $\\bar{\\wv}_{k,p}$ is a mean-normalized version of the filters and\n$\\norm{\\bar{\\wv}_{k,p}}_\\lambda \\triangleq (\\bar{\\wv}_{k,p}^T \\bar{\\wv}_{k,p}\n+ \\lambda)^{1\/2}$ is their contrast, with $\\lambda = 0.01$ a small positive\nconstant. This normalization requires only a slight modification of the\nstochastic gradient descent update formula and incurs negligible computational\noverhead. Note that the contrast normalization explored here is slightly\ndifferent than the one in \\cite{ZeFe13b}, who only scale down the filters\nwhenever their contrast exceeds a pre-defined threshold.\n\nWe have found the mean and contrast normalization of\nEq.~\\eqref{eq:epitomic-conv-norm} to significantly accelerate training not\nonly of epitomic but also max-pooled convolutional nets.\n\n\n\\subsection{Image Classification Experiments}\n\n\\begin{table*}[t]\n\\setlength{\\tabcolsep}{3pt}\n\\begin{center}\n\\scalebox{1.00} {\n\\begin{tabular}{|l||c|c|c|c|c|c||c|c||c|}\n  \\hline  \n  Layer           &   1 & 2 & 3 & 4 & 5 & 6 & 7 & 8   & Out \\\\\n  \\hline\n  \\hline\n  Type            & conv +   & conv +   & conv & conv & conv & conv + & full + & full + & full \\\\\n                  & lrn + max& lrn + max&      &      &      & max    & dropout& dropout&      \\\\ \\hline\n  Output channels &  96      & 192      & 256  & 384  & 512  & 512    & 4096   &  4096  & 1000 \\\\ \\hline\n  Filter size     &  8x8     & 6x6      & 3x3  & 3x3  & 3x3  & 3x3    & -    & -    & - \\\\ \\hline\n  Input stride    &  2x2     & 1x1      & 1x1  & 1x1  & 1x1  & 1x1    & -    & -    & - \\\\ \\hline\n  Pooling size    &  3x3     & 2x2      & -    & -    & -    & 3x3    & -    & -    & - \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Architecture of the baseline \\textsl{Max-Pool} convolutional network (Class-A).}\n\\label{tab:max_pool_net}\n\\end{center}\n\\end{table*}\n\n\n\\paragraph{Image classification tasks}\n\nWe have performed most of our experimental investigation with epitomes on the\nImagenet ILSVRC-2012 dataset \\cite{DDSL+09}, focusing on the task of image\nclassification. This dataset contains more than 1.2 million training images,\n50,000 validation images, and 100,000 test images. Each image is assigned to\none out of 1,000 possible object categories. Performance is evaluated using\nthe top-5 classification error. Such large-scale image datasets have proven so\nfar essential to successfully train big deep neural networks with supervised\ncriteria.\n\n\n\\paragraph{Network architecture and training methodology}\n\n\nFor our Imagenet experiments, we compare the proposed deep epitomic networks\nwith deep max-pooled convolutional networks. For fair comparison, we use as\nsimilar architectures as possible, involving in both cases six convolutional\nlayers, followed by two fully-connected layers and a 1000-way softmax\nlayer. We use rectified linear activation units throughout the\nnetwork. Similarly to \\cite{KSH13}, we apply local response normalization\n(LRN) to the output of the first two convolutional layers and dropout to the\noutput of the two fully-connected layers. This is the Class-A of the models\nwe considered.\n\nThe architecture of our baseline \\textsl{Max-Pool} network is specified on\nTable~\\ref{tab:max_pool_net}. It employs max-pooling in the convolutional\nlayers 1, 2, and 6. To accelerate computation, it uses an image stride equal\nto 2 pixels in the first layer. It has a similar structure with the Overfeat\nmodel \\cite{SEZM+14}, yet significantly fewer neurons in the convolutional\nlayers 2 to 6. Another difference with \\cite{SEZM+14} is the use of LRN, which\nto our experience facilitates training.\n\nThe architecture of the proposed \\textsl{Epitomic} network is specified on\nTable~\\ref{tab:mini_epitome_net}. It has exactly the same number of neurons at\neach layer as the \\textsl{Max-Pool} model but it uses mini-epitomes in place\nof convolution + max pooling at layers 1, 2, and 6. It uses the same filter\nsizes with the \\textsl{Max-Pool} model and the mini-epitome sizes have been\nselected so as to allow the same extent of translation invariance as the\ncorresponding layers in the baseline model. We use input image stride equal to\n4 pixels and further perform epitomic search with stride equal to 2 pixels in\nthe first layer to also accelerate computation.\n\n\\begin{table*}[t]\n\\setlength{\\tabcolsep}{3pt}\n\\begin{center}\n\\scalebox{1.00} {\n\\begin{tabular}{|l||c|c|c|c|c|c||c|c||c|}\n  \\hline  \n  Layer           &   1 & 2 & 3 & 4 & 5 & 6 & 7 & 8   & Out \\\\\n  \\hline\n  \\hline\n  Type            & epit-conv& epit-conv& conv & conv & conv & epit-conv & full + & full + & full \\\\\n                  & + lrn    & + lrn    &      &      &      &           & dropout& dropout&      \\\\ \\hline\n  Output channels &  96      & 192      & 256  & 384  & 512  & 512       & 4096   &  4096  & 1000 \\\\ \\hline\n  Epitome size    &  12x12   & 8x8      &  -   &  -   &  -   & 5x5       & -    & -    & - \\\\ \\hline\n  Filter size     &  8x8     & 6x6      & 3x3  & 3x3  & 3x3  & 3x3       & -    & -    & - \\\\ \\hline\n  Input stride    &  4x4     & 3x3      & 1x1  & 1x1  & 1x1  & 3x3       & -    & -    & - \\\\ \\hline\n  Epitome stride  &  2x2     & 1x1      &  -   &  -   &  -   & 1x1       & -    & -    & - \\\\ \\hline\n\\end{tabular}\n}\n\\caption{Architecture of the proposed \\textsl{Epitomic} convolutional network (Class-A).}\n\\label{tab:mini_epitome_net}\n\\end{center}\n\\end{table*}\n\n\n\nWe have also tried variants of the two models above where we activate the\nmean and contrast normalization scheme of Section~\\ref{sec:mean_con_norm} in\nlayers 1, 2, and 6 of the network.\n\nWe followed the methodology of \\cite{KSH13} in training our models. We used\nstochastic gradient ascent with learning rate initialized to 0.01 and\ndecreased by a factor of 10 each time the validation error stopped\nimproving. We used momentum equal to 0.9 and mini-batches of 128 images. The\nweight decay factor was equal to $\\by{5}{10^{-4}}$. Importantly, weight decay\nneeds to be turned off for the layers that use mean and contrast\nnormalization. Training each of the three models takes two weeks using a\nsingle NVIDIA Titan GPU. Similarly to \\cite{CSVZ14}, we resized the training\nimages to have small dimension equal to 256 pixels while preserving their\naspect ratio and not cropping their large dimension. We also subtracted for\neach image pixel the global mean RGB color values computed over the whole\nImagenet training set. During training, we presented the networks with\n$\\by{220}{220}$ crops randomly sampled from the resized image area, flipped\nleft-to-right with probability 0.5, also injecting global color noise exactly\nas in \\cite{KSH13}. During evaluation, we presented the networks with 10\nregularly sampled image crops (center + 4 corners, as well as their\nleft-to-right flipped versions).\n\n\\paragraph{Weight visualization}\n\nWe visualize in Figure~\\ref{fig:visualize_conv1} the layer weights at the first\nlayer of the networks above. The networks learn receptive fields sensitive to\nedge, blob, texture, and color patterns.\n\n\\paragraph{Classification results}\n\nWe report at Table~\\ref{tab:imagenet_results} our results on the Imagenet\nILSVRC-2012 benchmark, also including results previously reported in the\nliterature \\cite{KSH13, ZeFe13b, SEZM+14}. These all refer to the top-5 error\non the validation set and are obtained with a single network. Our best result\nat 13.6\\% with the proposed \\textsl{Epitomic-Norm} network is 0.6\\% better\nthan the baseline \\textsl{Max-Pool} result at 14.2\\% error. \nThe improved performance that we got with the \\textsl{Max-Pool} baseline\nnetwork compared to Overfeat \\cite{SEZM+14} is most likely due to our use of\nLRN and aspect ratio preserving image resizing. \n\n\n\n\\begin{table*}[t]\n\\setlength{\\tabcolsep}{3pt}\n\\begin{center}\n\\scalebox{0.9} {\n\\begin{tabular}{|l||c|c|c||c|c|c|c||c||c|}\n  \\hline\n              & \\multicolumn{3}{|c||}{Previous literature}  & \\multicolumn{4}{|c||}{Class-A}             &  Class-B & Class-C \\\\\n  \\hline\n  Model       & Krizhevsky  & Zeiler-Fergus  & Overfeat     & Max-Pool &  Max-Pool & Epitomic  & Epitomic & Epitomic & Epitomic\\\\\n              & \\cite{KSH13}& \\cite{ZeFe13b} &\\cite{SEZM+14}&          &  + norm   &           & + norm   &  +norm   & +norm   \\\\\n  \\hline\n  Top-5 Error &   18.2\\%    &    16.0\\%      &  14.7\\%      &  14.2\\%  &  14.4\\%   &\\bf{13.7\\%}&\\bf{13.6\\%}& 11.9\\%  & 10.0\\%  \\\\\n  \\hline\n\\end{tabular}\n}\n\\caption{Imagenet ILSVRC-2012 top-5 error on validation set. All performance\n  figures are obtained with a single network, averaging classification\n  probabilities over 10 image crops (center + 4 corners, as well as their\n  left-to-right flipped versions). Classes B and C refer to respectively\n  larger and deeper models.}\n\\label{tab:imagenet_results}\n\\end{center}\n\\end{table*}\n\n\n\n\n\n\nFurther experimental results with the epitomic model on the Caltech-101 task\nand on the MNIST and CIDAR-10 datasets are described in \\cite{Papa14}. \n\n\n\n\\paragraph{Mean-contrast normalization and convergence speed}\n\nWe comment on the learning speed and convergence properties of the different\nmodels we experimented with on Imagenet. We show in\nFigure~\\ref{fig:imagenet_optim} how the top-5 validation error improves as\nlearning progresses for the different models we tested, with or without\nmean+contrast normalization. For reference, we also include a corresponding\nplot we re-produced for the original model of Krizhevsky \\etal\n\\cite{KSH13}. We observe that mean+contrast normalization significantly\naccelerates convergence of both epitomic and max-pooled models, without\nhowever significantly influencing the final model quality. The epitomic models\nconverge faster and are stabler during learning compared to the max-pooled\nbaselines, whose performance fluctuates more.\n\n\\begin{figure}[!tbp]\n  \\centering\n  \\includegraphics[width=0.8\\columnwidth]{figs\/optimization\/val_acc}\n  \\caption{Top-5 validation set accuracy (center non-flipped crop only) for\n    different models and normalization.}\n  \\label{fig:imagenet_optim}\n\\end{figure}\n\n\n\\paragraph{Experiments with larger and deeper epitomic networks}\n\nWe have also experimented with larger (Class-B) and very deep (Class-C)\nversions of the proposed deep epitomic networks. The large Class-B network has\nthe same number of levels but more neurons per layer than the networks in\nClass-A. It achieves an error rate of 11.9\\%. \n\nInspired by the success of the top-performing methods in this year's Imagenet\ncompetition, we have also very recently experimented with a very deep network\nhaving 13 convolutional and 3 fully connected layers, which roughly follows\nthe architecture of the 16 layer net in \\cite{SiZi14}. Our Class-C deep\nepitomic network achieves 10.0\\% error rate in the Imagenet task. The\nstate-of-art 16 layer net in \\cite{SiZi14} (without multi-scale\ntraining\/testing) achieves an even lower 9.0\\% error rate, but using a more\nsophisticated procedure for aggregating the results of multiple image crops (in\nplace of our simple 10-view testing procedure). As extra evidence to the\nimproved robustness of training our deep epitomic networks, it is noteworthy\nto mention that we managed to train our very deep epitomic net starting from a\nrandom initialization, while \\cite{SiZi14} had to bootstrap their very deep\nnetworks from shallower ones.\n\n\\section{Explicit Scale and Position Search in DCNNs}\n\nThe effects of object translation and scaling impede the training of accurate classifiers from datasets that lack bounding box annotations - as is the case in the ImageNet classification Challenge. The Epitomic Convolution\/Max-Pooling modules allow us to extract an image representation that is invariant to signal deformations taking place at the pooling region's scale; when used in alternation with feature downsampling (`striding') this can result in invariance to increasingly large-scale signal transformations, eventually dealing with global position and scale changes. \n\n\n\n\n\n\\newcommand{X}{X}\n\\newcommand{\\mathbf{X}}{\\mathbf{X}}\n\\newcommand{y}{y}\n\n\\newcommand{\\refeq}[1]{Eq.~\\ref{#1}}\n\n\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\t\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\nA better treatment of deformations can be achieved by factoring deformations\ninto local (non-rigid) and global (translation\/scale) changes. We can then\nsimulate the effect of the latter during training and testing, by transforming the input images, obtaining us an additional hold on the problem. We now describe how this idea can be exploited in the setting of Multiple Instance Learning (MIL).\n\n\\subsection{MIL-based training of DCNNs}\nConsidering a binary classification problem with $N$ image-label pairs\n$\\mathcal{S} = \\{(X_i,y_i)\\}, i=1,\\ldots,N$, training aims at minimizing the following criterion:\n\\begin{eqnarray}\nC(f,\\mathcal{S}) = \\sum_{i=1}^{N} l(y_i,f(X_i)) + R(f), \\label{eq:original} \n\\end{eqnarray}\nwhere $f$ is the classifier, $l(y,f(X))$ is the  loss function and $R$ is a regularizer. \n\n{\\em Dataset augmentation} amounts to turning an image  $X_i$ into a set of images $\\mathbf{X}_i = \\{X^1_i,\\ldots,X^K_i\\}$  by transforming $\\mathbf{X}_i$ synthetically; e.g. considering $T$  translations and $S$ scalings yields a set with $K = T S$ elements. \nThe most common approach to using Dataset augmentation consists in   treating each element of $\\mathbf{X}_i$  as a new training sample, i.e.  substituting the loss $l(y,f(X_i))$ in \\label{eq:original} by the sum of the classifier's loss on all images:\n\\begin{eqnarray}\nL(y_i,\\mathbf{X}_i) \\doteq \\sum_{k=1}^K l(y_i,f(X^k_i)).\n\\label{eq:sum1}\n\\end{eqnarray}\nThis corresponds to the dataset augmentation technique use  e.g. in \\cite{Howa13}.\nA recently introduced alternative is the `sum-pooling' technique used in \\cite{SiZi14}, which can be understood as using the following loss:\n\\begin{eqnarray}\nL(y_i,\\mathbf{X}_i) =  l(y_i,\\frac{1}{K}\\sum_{k=1}^K f(X^k_i)), \\label{eq:sum2}\n\\end{eqnarray}\nwhich averages the classifier's score over translated versions of the input image. \nThe summation used in both of these approaches favors classifiers  that consistently score highly on positive samples, irrespective of the object's position and scale - this is when the loss is minimized. As such, these classifiers  can be seen as pursuing the invariance of $f$. \n\nThere is however a  tradeoff between invariance and classification accuracy \\cite{Varma}. Even though pursuing invariance accounts for the effects of transformations, it does not make the classification task any easier: the training objective aims at a classifier that would allow all transformed  images to make it through its `sieve'. Understandably, a classifier that only considers centered objects of a fixed scale can acchieve higher accuracy: this would allow us to devotes all modelling resources to the treatment of local deformations. \n\nFor this, we  let our classifier  `choose' its preferred transformation \nIn particular, we define the loss function to be:\n\\begin{eqnarray}\nL(y_i,\\mathbf{X}_i) =  l(y_i,\\max_{k} f(X^k_i)), \\label{eq:mil}\n\\end{eqnarray}\nwhich amounts to letting the classifier choose the transformation that maximizes its response on a per-sample basis, and then accordingly penalizing that response. In particular the loss function requires that the classifier's response is large on at least one  position for a positive sample - and small everywhere for a negative. \n\nThis idea amounts to the simplest case of  Multiple Instance Learning \\cite{diet97}: $\\mathbf{X}_i$ can be seen as a {\\em bag of features} and the individual elements of $\\mathbf{X}_i$ can be seen as {\\em instances}.  \nFor the particular case of the hinge loss function, this would lead us to the latent-SVM training objective \\cite{FGMR10,AndrewsHT02}.\n\nUsing this loss function during training amounts to treating the object's\nposition and scale as a latent variable, and performing alternating\noptimization over the classifier's score function. During testing we perform\na search over transformations and keep the best classifier score, which can be understood as maximizing\nover the latent transformation variables.\nThe resulting score\n$F(\\mathbf{X}_i) = \\max_{k} f(X^k_i)$\nis transformation-invariant, but is built on top of a classifier tuned for a single scale- and translation- combination. The MIL setting allows us to \ntrain and test our classifiers consistently, using the same set of image\ntranslations and scalings.\n\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l||c||c|}\n  \\hline  \n  Model       & Epitomic   &    Epitomic   \\\\\n              & (Class-B)  &  (patchwork)  \\\\ \\hline\n  Top-5 Error &    11.9\\%  &     10.0\\%    \\\\\n  \\hline\n\\end{tabular}\n\\caption{Imagenet ILSVRC-2012 top-5 error on validation set. We compare the\n  Class-B mean and contrast normalized deep epitomic network of\n  Table~\\ref{tab:imagenet_results} with its Patchwork fine-tuned version that\n  also includes scale and position search.}\n\\label{tab:imagenet_results2}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Efficient Convolutional Implementation}\n\\newcommand{\\mathcal{I}}{\\mathcal{I}}\nWe now turn to practical aspects of integrating  MIL  into DCNN training. DCNNs are commonly trained with input images of a fixed size, $q\\times q$. For an arbitrarily-sized input image  $I$, if  we \ndenote by $\\mathcal{I}(x,y,s)$ its image pyramid, naively computing the maximization in \\refeq{eq:mil} would require to cropping many $q\\times q$ boxes from $\\mathcal{I}(x,y,s)$, evaluating $f$ on them, yielding $f(x,y,s)$, and then penalizing  $l(y,\\max_{x,y,s} f(x,y,s))$ during training (we ignore downsampling and boundary effects for simplicity). Doing this would require a large amount of GPU memory, communication and computation time. Instead, by properly modifying the input and architecture of our network\nwe can  share computation to efficently implement exhaustive search during training and testing. \n\n\\begin{figure}[!tbp]\n\t\\centering\n\t\\begin{tabular}{c}\n\t\t\\includegraphics[width=\\columnwidth]{figs\/illustration\/patchwork}\n\t\\end{tabular}\n\t\\caption{We use the image patchwork technique to efficiently implement scale and position search in DCNN training: an image  pyramid is unfolded into a image `patchwork', where sliding a fixed-size window amounts to search over multiple positions and scales. The maximum of the classifier's score on all such  windows is efficiently gathered by max-pooling the DCNN's top-layer responses, and is used to accommodate scale and position changes during both training and testing.\n\t\t}\n\t\\label{fig:patchwork}\n\\end{figure}\n\n\nFor this, we first  draw inspiration from the `image patchwork' technique\nintroduced in \\cite{Dubout} and exploited in DCNNs by \\cite{iandola}. The\ntechnique consists in embedding a whole image pyramid $\\mathcal{I}(x,y,s)$, into a\nsingle, larger, patchwork image $P(x',y')$; any position $(x',y')$ in $P$\ncorresponds to a $(x,y,s)$ combination in $I$.  This was originally conceived\nas a means of accelerating multi-scale FFT-based convolutions in \\cite{Dubout}\nand convolutional feature extraction in \\cite{iandola}. Instead we view it a\nstepping stone to implementing scale and position search during DCNN training.\n\nIn particular,  we treat the last, fully-connected, layers of a DCNN as\n$1\\times1$ convolution kernels; we can then obtain the $f(x,y,s)$ score\ndescribed  above by providing $P$ as input to our network, since the output of\nour network's final layer at any position $(x',y')$  will correspond to the\noutput corresponding to a $q\\times q$ square cropped around $(x,y,s)$. This\nallows us to incorporate the $\\max$ operation used in MIL's training\ncriterion, \\refeq{eq:mil}, as an additional max-pooling layer situated on top\nof the network's original score function. This makes it possible to\nefficiently incorporate global scale and position search in a seamless manner\nwithin DCNN training.\n\n\n\\subsection{Image Classification Results}\n\nWe have experimented with the scheme outlined above in combination with our\nDeep Epitomic Network (Class-B variant) presented in the previous Section.  We\nuse a \\by{720}{720} patchwork formed from 6 different image scales (square\nboxes with size 400, 300, 220, 160, 120, and 90 pixels). We have resized all\ntrain\/test images to square size, changing their aspect ratio if needed. We\nhave initialized this scale\/position search net with the parameters of our\nstandard Class-B epitomic net. We fine-tuned the network parameters for about\nan epoch on the Imagenet train set, following the training methodology\noutlined earlier.\n\nWe have obtained a substantial further decrease in the testing error rates,\ncutting the top-5 error rate from 11.9\\% down to 10.0\\%, as shown in\nTable~4. This reduction in error rate is competitive with the best reduction obtained by  more complicated techniques involving many views for evaluation \\cite{SiZi14,SiZi14}, while also allowing for consistent end-to-end training and testing. \n\n\nThe network outlined above also provides cues for the\nscale and position of the dominant object in the image. A simple fixed mapping\nof the ``argmax'' patchwork position in the last max-pooling layer (computed\nby averaging the bounding box positions in the training set) yields 48.3\\%\nerror rate in the Imagenet 2012 localization task without incurring any extra\ncomputation.\n\n\\section{Sliding Window Object Detection with DCNNs}\n\\label{sec:detection}\n\n\nThe success of explicit position and scale search in image classification\nsuggests using DCNNs for sliding-window detection; even though for pedestrians\nexcellent results have been reported in \\cite{yangW13} recent works on\ncombining convolutional networks \\cite{SEZM+14}, or sliding window detectors\nwith CNN features \\cite{iandola,savalle} still lag behind the current\nstate-of-the-art techniques \\cite{GDDM14,HeZR014,DeepId}.\n \nStarting from the RCNN work of \\cite{GDDM14}, all such techniques compute DCNN\nfeatures `on-demand' for a set of 1000-2000 image regions delivered by\nselective search \\cite{ssearch}, and apply a separately trained SVM detection\non top. This approach has recently been shown to deliver compelling detection\nresults; most recently, \\cite{GDDM14} have shown that combining RCNNs with the\nnetwork of \\cite{SiZi14} pushes the mean AP performance on Pascal VOC 2007 to\n$66\\%$ ($62\\%$ without bounding box regression), but at the cost of 60 seconds per image (acceleration techniques such\nas Spatial Pyramid Pooling \\cite{HeZR014} can still be applied, though).\n \nDespite the performance gap of sliding window detection to RCNNs, we consider\nsliding window detection simpler, and potentially more amenable to analysis\nand improvement. \n \n\\subsection{Explicit search over aspect ratios}\n\nOur basic object detection system uses  as\ninput to a DCNN an image patchwork that includes 11\nscales logarithmically sampled, from 2 times down to 1\/6 of the image size.\nUnlike recent works  \\cite{iandola,savalle} that only\noperate with the first five, `convolutional' layers of a deeper network,\n in our case we set the fully-connected layers operate convolutionally, and use\nthe DCNN class scores for proposing square bounding boxes as object detection\nproposals. This processes a typical PASCAL VOC 2007 image in less than 1 sec\non a Titan Tesla K40. \n\nThis square bounding box detector (without any bounding box regression\npost-processing) yields a mean Average Precision of $43.0\\%$ on Pascal VOC\n2007. To further analyze which of this system's errors are due to constraining\nthe bounding box proposals to be square, we investigated the system's\nperformance in the presence of an `oracle' that provides the optimal aspect\nratio (using the ground truth annotations) for any given square bounding box\nproposal. We found  this oracle bounding box prediction to increase\nperformance to $56.7\\%$. This indicates that square box prediction is\ninsufficient for achieving competitive object detection results.\n\nOnce again, we pursue an explicit search approach to directly estimate the\naspect ratio of bounding box proposals without needing to resort to an\noracle. We account for aspect ratio variability by scaling the image along a\nsingle dimension. For instance, scaling an image by $.5$ vertically means that\na vertically-elongated \\by{100}{200} region appears as a \\by{100}{100} square\nregion in the transformed image. Our square detector can then find this\npreferable to other, differently scaled versions of the same region, and\nthereby hint at the right object ratio. Aspect ratios that receive a lower\nscore are eliminated at the final nonmaximum suppression stage. We account for\naspect ratio during both testing and training (see below).\n\nWe perform this operation for 5 distinct aspect ratios, spanning the range of\n$[1\/3,3]$ with a geometric progression, as illustrated in\nFigure~\\ref{fig:pw}. This is applied at the whole patchwork level -- sliding\nwindow detection on these patchworks then amounts to a joint search over\nscale, position, and aspect ratio. This is more time-demanding (requiring 5\ntimes more computation) but still quite fast, requiring about 5 secs on a\nTesla K40 for an average Pascal VOC 2007 image, and yields non-square bounding\nbox predictions without resorting to an oracle.\n\nThe related detection results are shown in Table~\\ref{tab:voc2007}. Aspect\nratio search yields a very competitive mean Average Precision score of\n$56.4\\%$ (without any bounding box regression post-processing). This is only\nslightly lower than the oracle-based score $56.7\\%$, indicating that also\nnormalizing for aspect ratio during training leads to better object models.\n\nIt is noteworthy that our $56.4\\%$ mAP result is significantly better than the\n$46.9\\%$ mAP result reported very recently by \\cite{WEF14}. Our better results\nshould be attributed to our different training procedure (detailed below), our\nexplicit search over aspect ratios, and the use of a more powerful DCNN\nclassifier. Our system is also significantly simpler than \\cite{WEF14} (which\nalso integrates deformable part models and includes non maximum suppression\nduring training). We anticipate that integrating these components into our\nsystem could further increase performance at a higher computational cost.\n\nWe observe that our system's average performance ($56.4\\%$ mAP) is  still below the one obtained by \\cite{GDDM14} when using the network of \\cite{SiZi14} ($62.2\\%$ mAP without bounding box regression). This can be attributed to several aspects of our system, including (i) using a smaller number of network parameters (detailed below) (ii) performing the detection with smaller windows (detailed below)\n(iii) not using a retraining stage with the hinge loss and  hard-negative mining, as \\cite{GDDM14} do, and (iv)  missing out on regions found by Selective Search. We are confident that factor (iv) is the least important - having experimentally verified that the recall of bounding boxes is systematically better according to our pyramid's hypothesized positions, rather than the boxes delivered by Selective Search. We are currently investigating the effects of factors (i)-(iii).\nStill, we consider that our system has an edge on the efficiency side: our detector requires approximately 5 seconds to consider all position, scale and aspect ratio combinations, while the  \n system of \\cite{GDDM14} with the network of \\cite{SiZi14} requires approximately 60 seconds, on identical hardware (a Tesla K40 GPU).\n  \n\\begin{table*}[t]\\scriptsize\n\\setlength{\\tabcolsep}{3pt}\n\\begin{center}\n\\begin{tabular}{|c|c*{19}{|c}||c|}\n\\hline\nVOC 2007 test & aero & bike & bird & boat & bottle  & bus & car & cat & chair & cow & table & dog & horse & mbike & person & plant & sheep & sofa & train & tv   & mAP\\\\\n\\hline\\hline\nOur work (VGG)& 64.4 & 72.1 & 54.6 & 40.4 & 46.5    & 66.2& 72.9& 58.2& 31.8  &69.8 & 31.8  & 59.3& 71.1  & 68.3  & 64.7   & 31.0  & 55.0  & 49.8 & 55.3  & 64.4 & 56.4\\\\\n\\hline\nRCNN7 \\cite{GDDM14} (VGG) & 71.6 &  73.5 & 58.1 & 42.2 & 39.4 & 70.7  & 76.0 &  74.5 & 38.7 & 71.0 & 56.9 & 74.5 & 67.9 &  69.6 & 59.3 & 35.7 & 62.1 &  64.0 & 66.5 & 71.2 &  62.2\\\\\\hline\nRCNN7 \\cite{GDDM14} (UoT) &  64.2 & 69.7 & 50.0 & 41.9 & 32.0 & 62.6 & 71.0 & 60.7 & 32.7 & 58.5 & 46.5 & 56.1 & 60.6 & 66.8 & 54.2 & 31.5 & 52.8 & 48.9 & 57.9 & 64.7 & 54.2\\\\\\hline\nDPM \\cite{WEF14} (UoT)& 49.3 & 69.5 & 31.9 & 28.7 & 40.4 & 61.5 & 61.5 & 41.5 & 25.5 & 44.5 & 47.8 & 32.0 & 67.5 &  61.8 & 46.7 & 25.9 & 40.5 & 46.0 & 57.1 & 58.2 & 46.9\\\\\\hline\n \\end{tabular}\n \\caption{Detection average precision (\\%) on the PASCAL VOC 2007 test set,\n   using the proposed CNN sliding window detector that performs explicit\n   position, scale, and aspect ratio search. We compare to the RCNN architecture of \\cite{GDDM14}\n   and the end-to-end trained DPMs of \\cite{WEF14}. In parenthesis we indicate the DCNN used for detection: UoT is the University of Toronto DCNN \\cite{KSH13} and VGG is the DCNN of Oxford's Visual Geometry Group \\cite{SiZi14}.\n   }\n \\label{tab:voc2007}\n \\end{center}\n\\end{table*}\n\n\\newcommand{}{}\n\\newcommand{.62}{.62}\n\\newcommand{.35}{.35}\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ccc}\n\\includegraphics[width=.35\\columnwidth,height=.35\\columnwidth]{figs\/supplement\/input}\n&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_15}&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_16}\\\\\nInput image & Patchwork, $\\alpha = 0.33$ & Patchwork, $\\alpha = 0.57$ \\\\\n&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_s_15}&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_s_16}\n\\\\\n& `car' score, $\\alpha = 0.33$ & `car' score, $\\alpha = 0.57$\\\\\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_19}&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_17}&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_18}\\\\\nPatchwork, $\\alpha = 1$ & Patchwork, $\\alpha = 1.73$ & Patchwork, $\\alpha = 3.00$\\\\\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_s_19}&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_s_17}&\n\\includegraphics[width=.62\\columnwidth,height=.62\\columnwidth]{figs\/supplement\/pw_s_18}\\\\\n `car' score, $\\alpha = 1.00$ & `car' score, $\\alpha = 1.73$ &  `car' score, $\\alpha = 3.00$ \\\\\n\\end{tabular}\n\\caption{Patchwork images (even rows) and car detector scores (odd rows) used for\n  explicit position, scale, and aspect ratio (`$alpha$') search. We observe that the score is maximized at patchwork positions (i.e. image positions, scale, and aspect ratio combinations)  corresponding to  square-shaped cars. }\n \\label{fig:pw}\n\\end{figure*}\n\n\\subsection{DCNN training for sliding window detection}\nWe deviate from the newtork fine-tuning used in the RCNN system \\cite{GDDM14}\nin two ways: firstly, we do not rely on the Selective Search \\cite{ssearch}\nregion proposals to gather training samples; and, secondly, we modify the\nnetwork's structure to  process smaller images, and, resultantly,\ninclude less parameters.  We detail these two modifications below.\n\n\\subsubsection{Model Training}\nThe RCNN system of \\cite{GDDM14}  adapts a network trained for the Imagenet\nclassification task to object detection on Pascal VOC. For this, the authors\nuse the regions proposed by selective search to generate positive and negative\ntraining samples, for each of the 20 categories of Pascal VOC; if a region has\nan Intersection-over-Union  (IoU) above $.5$ for any bounding box of a class\nit is declared as being a positive example for that class, or else it is a\nnegative example. These examples are used in  a network `fine-tuning' stage,\nwhich amounts to running back-propagation with these training samples.\n\nRather than relying on Selective Search to provide training samples, we\nexploit the dense sampling of positions, scales, and aspect ratios of our\nalgorithm. This allows us to use substantially cleaner examples, and train\nwith a higher IoU threshold for positives.\n\nIn particular, as illustrated in Figure~\\ref{fig:posneg}, we keep track of all\nthe windows that would be visited by our sliding window detector; given a\nground-truth bounding box, we randomly pick 30 of those that have an IoU score\nabove $0.7$ with it; if we have less than 30, we decrease the threshold to\n$0.6$ and if we still cannot find as many we set the threshold finally to\n$0.5$.  Similarily, for every positive bounding box we sample 200 negative\nboxes that have an IoU score between $0.2$ and $0.5$ - aiming at `teaching'\nour classifier what a poor localization looks like.\n\nWe have verified that doing this, rather than using selective search windows\ngives us  clearly better detector scores, both visually and quantitatively. We\nconsider this to be one of the advantages of using a sliding window detector,\nnamely that we do not need to rely on an external region proposal module for\ntraining.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{c c c c}\n\\includegraphics[width=.5\\columnwidth]{figs\/supplement\/boxes_pos_pyr}&\n\\includegraphics[width=.5\\columnwidth]{figs\/supplement\/boxes_neg_pyr} &\n\\includegraphics[width=.5\\columnwidth]{figs\/supplement\/boxes_pos_rcnn}&\n\\includegraphics[width=.5\\columnwidth]{figs\/supplement\/boxes_neg_rcnn}\\\\\na&\nb &\nc&\nd\\\\\n\\multicolumn{2}{c}{\nSliding window positives (left)\nand  negatives (right)} & \n\\multicolumn{2}{c}{\nSelective search positives (left)\nand  negatives (right).}\n\\end{tabular}\n\\caption{Bounding boxes used for network finetuning. Our sliding window detector can use many more bounding boxes as positives (a) and negatives (b). The training samples available from selective search (c) are fewer, forcing the training to use poorly localized positives (d). When finetuning our network only with the latter examples performance would deteriorate.}\n\\label{fig:posneg}\n\\end{figure*}\n\n\\subsubsection{Re-purposing Classification Networks for Image Detection}\n\nHerein we describe how we have re-purposed the publicly available state-of-art\n16 layer classification network of \\cite{SiZi14} (VGG-16) into an efficient\nand effective component of our sliding window detector.\n\n\\paragraph{Dense sliding window feature extraction with the hole algorithm}\n\nDense spatial score evaluation is instrumental in the success of our CNN\nsliding window detector. \n\nAs a first step to implement this, we convert the fully-connected layers\nof VGG-16 into convolutional ones and run the network in a convolutional\nfashion on the patchwork. However this is not enough as it yields very\nsparsely computed detection scores (with a stride of 32 pixels). To compute\nscores more densely at our target stride of 8 pixels, we develop a variation\nof the method previously employed by \\cite{GCMG+13, SEZM+14}. We skip\nsubsampling after the last two max-pooling layers in the network of\n\\cite{SiZi14} and modify the convolutional filters in the layers that follow\nthem by introducing zeros to increase their length (\\by{2}{} in the last three\nconvolutional layers and \\by{4}{} in the first fully connected layer). We can\nimplement this more efficiently by keeping the filters intact and instead\nsparsely sample the feature maps on which they are applied on using a stride\nof 2 or 4 pixels, respectively. This approach is known as the `hole algorithm'\n(`atrous algorithm') and has been developed before for efficient computation\nof the undecimated wavelet transform \\cite{Mall99}. We have implemented this\nwithin the Caffe framework by adding to the \\textsl{im2col} function (it\nconverts multi-channel feature maps to vectorized patches) the option to\nsparsely sample the underlying feature map.\n\n\\paragraph{Shrinking the receptive field of neural networks}\n\nMost recent DCNN-based image recognition methods rely on networks pre-trained\non the Imagenet large-scale classification task. These networks typically have\nlarge receptive field size, \\by{224}{224} in the case of the VGG-16 net we\nconsider. We have found this receptive field size to be too large to allow\ngood localization accuracy (unless one uses heavily zoomed-in versions of the\nimage). Moreover, after converting the network to a fully convolutional one,\nthe first fully connected layer has 4,096 filters of large \\by{7}{7} spatial\nsize and becomes the computational bottleneck in our sliding window detector\nsystem.\n\nWe have addressed both of these serious practical problems by spatially\nsubsampling the first FC layer to \\by{4}{4} spatial size. This has reduced the\nreceptive field of the network down to \\by{128}{128} pixels and has reduced\ncomputation time for the first FC layer by 3 times.\n\n\\section{Conclusions}\n\nThis paper examines multiple facets of invariance in the context of deep\nconvolutional networks for visual recognition. First, we have proposed a new\nepitomic convolutional layer which acts as a substitute to a pair of\nconsecutive convolution and max-pooling layers, and shown that it brings\nperformance improvements and exhibits better behavior during training. Second,\nwe have demonstrated that treating scale and position as latent variables and\noptimizing over them during both training and testing yields significant\nimage classification performance gains. Pushing scale and position search\nfurther, we have shown promising  results which\nsuggest that DCNNs can be efficient and effective for dense sliding window\nbased object detection. Further pursuing this topic is the main direction of\nour future work.\n\n\\paragraph{Reproducibility} We implemented the proposed methods by extending\nthe excellent Caffe software framework \\cite{Jia13}. When this work gets\npublished we will publicly share our source code and configuration files with\nexact parameters fully reproducing the results reported in this paper.\n\n\\paragraph{Acknowledgments} We gratefully acknowledge the support of NVIDIA\nCorporation with the donation of GPUs used for this research.\n\n\n\\bibliographystyle{ieee}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and Statement of the Result}\n\nThe curvature of the Weil-Pe\\-ters\\-son\\ metric recently attracted further interest. In this note we will give the precise estimate for the average of the scalar curvature $S_{WP}$ of the Weil-Petersson metric on the moduli space $\\ol{\\mathcal M} _g$ as $g$ tends to infinity. The result is the value\n$$\n \\frac{1}{(g-1)}  \\frac{\\bigintss_{\\ol{\\cM}_g} \\, (-S_{W\\! P})\\,dV_{W \\! P}}{\\bigintss_{\\ol{\\cM}_g} dV_{W\\! P}} =  \\frac{13}{4\\pi} +  \\frac{\\pi}{12} \\frac{1}{g} + \\left( \\frac{1}{4\\pi} +\\frac{\\pi}{12} \\right) \\frac{1}{g^2}+ O\\left(\\frac{1}{g^3}\\right).\n$$\nThe proof of the asymptotics will be based upon methods of Algebraic Geometry. Wolpert showed in \\cite{wo1,wo2} that Mumford's canonical class $\\kappa_1$ (the tautological class obtained from the universal curve) from \\cite{mu1,mu2} (cf.\\ \\cite{ac,ac2}) is the cohomology of the Weil-Pe\\-ters\\-son\\ form extended to the compactification  up to the factor $2\\pi^2$ together with the fact that its restriction to the boundary equals to the Weil-Pe\\-ters\\-son\\ cohomomology of the boundary (interpreted as related to  moduli of punctured surfaces of lower genus).\n\nThe finiteness of the Weil-Pe\\-ters\\-son\\ volume itself is a consequence of Masur's estimates \\cite{ma}, whereas the curvature was computed by Wolpert \\cite{wo3} and Fischer-Tromba \\cite{tro}. These results implied strong negativity properties, in particular the strict negativity of the scalar curvature. It is known that the scalar curvature tends to $-\\infty $ towards the boundary. Precise estimates of the curvature of the Weil-Pe\\-ters\\-son\\ metric towards the boundary are contained in \\cite{s} \\and \\cite{t} with a later developments by Liu-Sun-Yau \\cite{lsy1,lsy2}.\n\nEstimates of the Weil-Pe\\-ters\\-son\\ volume had been given by Mirzakhani \\cite{mi}, Mirzakhani-Zograf \\cite{m-z}, Penner \\cite{pe}, Grushevsky \\cite{gru}, Zograf \\cite{zo2,zo} and previously in \\cite{s-t}. The algebraic aspect is contained in the push-pull formulas by Arbarello and Cornalba \\cite{ac,ac2}.\n\n\nThe Weil-Pe\\-ters\\-son\\ volume of the moduli spaces  $\\ol{\\mathcal M} _{g,n}$ of Riemann surfaces of genus $g$ with $n$ punctures is denoted by\n$$\nV_{g,n}=\\int_{{\\mathcal M}_{g,n}}\\kappa_1^{3g-3+n}.\n$$\nFinally the relationship of intersection numbers and volumes as related to two dimensional gravity ought to pointed out. Pertinent references are \\cite{dij,fp,getz,kon,mz,witt}.\n\n\n\nWe showed the following estimates.\n\\begin{theorem}[\\cite{s-t}]\n\\begin{itemize}\n\\item[(i)]\n Let $g>1$. Then\n\\begin{equation}\\label{eq:upper_est}\nV_{g,0} \\geq \\frac{1}{28} V_{g-1,2} + \\frac{1}{672} V_{g-1,1} +\n\\frac{1}{14} \\sum_{j=2}^{[g\/2]} V_{j,1}V_{g-j,1}\n-\\frac{1}{28} (V_{\\frac{g}{2},1})^2 ,\n\\end{equation}\nwith $V_{\\frac{g}{2},1}=0$, if $g$ is odd.\n\\item[(ii)]\nThere exist constants $0 < c < C$, independent of $n$ such that\n\\begin{equation}\\label{eq:asymp1}\nc^g  (2g)! \\leq \\frac{V_{g,n}}{(3g-3+n)!} \\leq C^g  (2g)!\n\\end{equation}\nfor all fixed $n\\geq 0$ and large $g$.\n\\end{itemize}\n\\end{theorem}\nConcerning \\eqref{eq:asymp1}, a lower estimate for $n=1$ is due to Penner \\cite[Theorem 6.2.2]{pe}, and the upper estimates for $n \\geq 1$ were first shown by Grushevsky in \\cite[Sec.~7]{gru}.\n\nBounds for the curvature of the Weil-Pe\\-ters\\-son\\ metric were proven by Wu and Wolf in \\cite{ww} and Wu in \\cite{wu1,wu2}. A recent result is the following:\n\\begin{theorem}[{Bridgeman-Wu, \\cite{b-w}}]\\label{th:briwu}\nDenote by $S_{W\\!P}$ the scalar curvature of the Weil-Pe\\-ters\\-son\\ form $\\omega_{WP}$, and by $dV_{W\\!P}$ its volume element. There exist constants $0<c<C$ such that\n\\begin{equation}\\label{eq:briwu}\nc\\cdot g \\leq \\frac{\\bigintss_{\\ol{\\cM}_g} \\, (-S_{W\\! P})\\,dV_{W \\! P}}{\\bigintss_{\\ol{\\cM}_g} dV_{W\\! P}} \\leq C\\cdot g\n\\end{equation}\nfor all $g$.\n\\end{theorem}\nWe will show that the Bridgeman-Wu estimate follow in an algebraic way using \\eqref{eq:upper_est}.\n\nThe actual asymptotics of the Weil-Pe\\-ters\\-son\\ volume (containing \\eqref{eq:asymp1}) was computed by Mirzakhani and Zograf. (We use the above normalization for the Weil-Pe\\-ters\\-son\\ volume.)\n\\begin{theorem}[{\\cite[Theorem 1.2]{mz}}]\nThere exists a constant $C\\in(0,\\infty)$ such that for any given $k\\geq 1$, $n\\geq 0$\n\\begin{equation}\\label{eq:mz}\n  V_{g,n}= C_{MZ} \\frac{(3g-3+n)! \\, (2g-3+n)! \\, 2^{g-3+n}}{\\pi^{2g}\\sqrt{g}}\\left( 1+ \\sum_{j=1}^{k} \\frac{c_n^{(j)}}{g^j} + O\\left(\\frac{1}{g^{k+1}}\\right)   \\right)\n\\end{equation}\nas $g\\to\\infty$.\n\\end{theorem}\nThe theorem contains further characterizations of the polynomials $c^{(j)}_n$ -- we will need the case $k\\leq 1$, and $n\\leq 2$. The Mirzakhani-Zograf constant $C_{MZ}$ is conjectured to be $C_{MZ}=1\/\\sqrt\\pi$.\n\nLet again $\\delta$ be the $\\mathbb Q$-divisor related to the boundary components of the moduli space (see also below), and denote by $\\kappa_1$ the Mumford class. Recall that the class $[\\omega_{WP}]$  of the Weil-Pe\\-ters\\-son\\ form was computed in \\cite{wo1,wo2} as $2 \\pi^2 \\kappa_1$. Set $\\eta= [-Ric(\\omega_{WP})\/2\\pi]$.\n\\begin{maintheorem*}\n  The average total scalar curvature of the Weil-Pe\\-ters\\-son\\ metric on the moduli space $\\ol{\\mathcal M} _g$ satisfies the estimate\n  \\begin{eqnarray}\n   \\frac{13}{4\\pi} &\\leq&\n     \\frac{3}{\\pi} \\frac{ \\eta \\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{1}{(g-1)}  \\frac{\\bigintss_{\\ol{\\cM}_g} \\, (-S_{W\\! P})\\,dV_{W \\! P}}{\\bigintss_{\\ol{\\cM}_g} dV_{W\\! P}} =  \\frac{13}{4\\pi} +  \\frac{1}{4\\pi}  E_g  , \\label{eq:Eg}  \\quad \\text{ where } \\\\ \\nonumber\n \\frac{1}{4\\pi} E_g &=&  \\frac{\\pi}{12} \\frac{1}{g} + \\left( \\frac{1}{4\\pi} +\\frac{\\pi}{12} \\right) \\frac{1}{g^2}+ O\\left(\\frac{1}{g^3}\\right) \\text{ for } g\\to \\infty.\\\\\\text{ The precise value is } \\nonumber\\\\\n\\label{eq:kappa}\n E_g &=& \\frac{\\kappa_1^{3g-4} \\cdot \\delta }{\\kappa_1^{3g-3}} \\text{ where }\\delta = \\sum_{j=0}^{[g\/2]} \\delta_j.\n\\end{eqnarray}\n\\end{maintheorem*}\nNote that the ampleness of $\\kappa_1$ implies that $E_g$ is positive.\n\n\n\\section{Proof}\n\nLet $D$ denote the compactifying divisor of ${\\mathcal M} _g$ with components $\\Delta_j$ for $j=0,\\ldots,[g\/2]$. These give rise to $\\mathbb Q$-divisors $\\delta_j$ such that in terms of the generally used notation\n$$\n[\\Delta_1]= 2 \\delta_1 \\text{ and } [\\Delta_j]=  \\delta_j \\text{ for } j \\neq 1.\n$$\nThen $\\delta:= \\sum_{j=0}^{[g\/2]} \\delta_j$ so that\n$$\nD= \\delta +\\delta_1.\n$$\nNote that there exist branched two-sheeted coverings $\\ol{\\mathcal M} _{g-1,2} \\to \\Delta_0$, \\quad $\\ol{\\mathcal M} _{g-1,1}\\times \\ol{\\mathcal M} _{1,1} \\to \\Delta_1$, and $\\ol{\\mathcal M} _{g\/2,1}\\times\\ol{\\mathcal M} _{g\/2,1} \\to \\Delta_{g\/2}$ for even $g$ yielding extra factors $1\/2$ in Lemma~\\ref{le:le1}.\n\n.\n\nTwo classical facts are needed. Mumford's direct image bundle $\\lambda$ satisfies \\cite{mu1}\n$$\n\\kappa_1 = 12 \\lambda - \\delta,\n$$\nand by \\cite{hm}  the canonical bundle $K_{\\ol{\\mathcal M} _{g,0}}$ is equal to\n$$\nK_{\\ol{\\mathcal M} _{g,0}} = 13 \\lambda - 2\\delta - \\delta_1.\n$$\n\n\n\nIn \\cite[Corollary 5.5]{t} it was shown by means of a a singular Mumford good hermitian metric that $\\eta$ is the Chern class of the dual of the logarithmic tangent bundle $T_{{\\overline{\\mathcal M} }_{g,0}}({\\log} D)$. Therefore\n\\begin{equation}\\label{eq:ric}\n  [\\eta] = K_{\\ol{\\mathcal M} _{g,0}} + [D].\n\\end{equation}\nNow \\eqref{eq:Eg} can be shown:\n\\begin{gather*}\n\\frac{ \\eta \\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{ (K_{\\ol{\\mathcal M} _{g,0}}+\\delta+\\delta_1) \\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{(13 \\lambda - \\delta )\\cdot \\kappa_1^{3g - 4}}{  \\kappa_1^{3g - 3}} = \\frac{(\\frac{13}{12} \\kappa_1 + \\frac{1}{12}\\delta)\\cdot \\kappa_1^{3g - 4}}{ \\kappa_1^{3g - 3}}\n\\end{gather*}\nThe main result consists of the estimate for $E_g$. The special value for $V_{1,1}$ etc.\\ are taken from \\cite{s-t}.\n\\begin{lemma}\\label{le:le1}\n\\begin{eqnarray}\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_0 }{\\kappa_1^{3g-3}} &=& \\frac{1}{2} \\frac{V_{g-1,2}}{V_{g,0}}\n  \\\\\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_1 }{\\kappa_1^{3g-3}} &=&   \\frac{1}{2}  \\cdot \\frac{V_{1,1}\\cdot V_{g-1,1}}{V_{g,0}} = \\frac{1}{48} \\cdot \\frac{ V_{g-1,1}}{V_{g,0}}  \\\\\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_j }{\\kappa_1^{3g-3}} &=& \\frac{V_{j,1}\\cdot V_{g-j,1}}{V_{g,0}} \\qquad \\text{ for }\\quad 2\\leq j\\ \\leq [(g-1)\/2]  \\\\\\label{eq:deltaj}\n  \\frac{\\kappa_1^{3g-4} \\cdot \\delta_{g\/2} }{\\kappa_1^{3g-3}} &=&\\frac{1}{2}  \\frac{(V_{g\/2,1})^2}{V_{g,0}}   \\text{ \\ for \\ } g \\text{ \\ even}.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{remark}\n Lemma~\\ref{le:le1}, and \\eqref{eq:Eg} together with \\eqref{eq:upper_est} imply the Bridgeman-Wu Theorem~\\ref{th:briwu}.\n\\end{remark}\n\n\n\nWe will apply the Mirzakhani-Zograf estimate \\eqref{eq:mz}, and the following values contained in \\cite[Remark 1.2]{mz}:\n\\begin{eqnarray}\n  c^{(1)}_0 & = & \\frac{7}{12} -\\frac{17}{6\\pi^2} \\label{eq:c0}\\\\\n  c^{(1)}_1 & = & \\frac{1}{3} - \\frac{5}{6\\pi^2} \\label{e1:c1}\\\\\n  c^{(1)}_2 & = & \\frac{1}{12} + \\frac{1}{6\\pi^2} .\\label{eq:c2}\n\\end{eqnarray}\n\n\n\\begin{lemma}\n  \\begin{eqnarray}\n \n    \\frac{1}{2}\\frac{V_{g-1,2}}{V_{g,0}} &=&\n    \\frac{\\pi^2 }{3g} \\left( 1 + \\left(\\frac{3}{\\pi^2} + 1 \\right)\\frac{1}{g} + O\\left( \\frac{1}{g^2} \\right)     \\right)\n    \\label{eq:g-1_2}\\\\\n    \\frac{1}{48}\\frac{V_{g-1,1}}{V_{g,0}} &=& \\frac{1}{144}\\sqrt{\\frac{g}{g-1}} \\frac{\\pi^2}{(g-1)(3g-4)(2g-3)}\\left(1 + O\\left(\\frac{1}{g}\\right)\\right) = O\\left(\\frac{1}{g^3}\\right)\\\\\n    \\frac{V_{j,1}\\cdot V_{g-j,1}}{V_{g,0}} &=& \\frac{ C_{MZ} }{2}\\sqrt{\\frac{g}{j(g-j)}} \\frac{1}{(3g-3) \\left(3g-4\\atop 3j-2 \\right)(2g-3) \\left(2g-4\\atop 2j-2  \\right) }\\left(1 + O\\left(\\frac{1}{g}\\right)\\right)\\label{eq:Vj1}\\\\ \\nonumber && \\text{ for } 2\\leq j \\leq[(g-1)\/2]\\\\\n    \\frac{1}{2} \\frac{(V_{g\/2,1})^2}{V_{g,0}} &=&  \\frac{ C_{MZ} }{2}{\\frac{1}{\\sqrt{g}}} \\frac{1}{(3g-3) \\left(3g-4\\atop (3g-4)\/2 \\right)(2g-3) \\left(2g-4\\atop g-2  \\right) }\\left(1 + O\\left(\\frac{1}{g}\\right) \\right)\\label{eq:gover2}\\\\ && \\nonumber  \\text{ for g even}.\n  \\end{eqnarray}\n\\end{lemma}\nThe computation of \\eqref{eq:g-1_2} is the following.\n\\begin{gather*}\n  \\frac{1}{2}\\frac{V_{g-1,2}}{V_{g,0}} = \\frac{1}{2} \\frac{(3g-4)!\\,(2g-3)!\\, 2^{g-2} \\pi^{2g}\\, \\sqrt{g}}{(3g-3)!\\,(2g-3)!\\,2^{g-3}\\,\\pi^{2g-2}\\,\\sqrt{g-1}}\\cdot \\widetilde C_g =\\\\ = \\frac{\\pi^2}{3g} \\left(\\frac{g}{g-1} \\right)^{3\/2} \\cdot \\widetilde C_g  = \\frac{\\pi^2}{3 g } \\left( 1 + \\frac{3}{2 g} + O\\left(\\frac{1}{g^2}  \\right)   \\right) \\cdot \\widetilde C_g\n\\end{gather*}\nwhere\n\\begin{gather*}\n  \\widetilde C_g = \\left(1 + c^{(1)}_{2}\\frac{1}{g-1} + O\\left(\\frac{1}{g^2}\\right)\\right) \\Big\/ \\left(1 + c^{(1)}_{0}\\frac{1}{g} + O\\left(\\frac{1}{g^2}\\right)\\right) \\\\\n\\end{gather*}\n\n\n\nThe coefficients $c^{(1)}_n$ do not contribute to the remaining terms up to order three in $1\/g$.\n\nWe need an elementary estimate:\n\nLet $l \\leq k \\leq n$ be positive integers,  then $(n-k)(k-l) \\geq 0,$ i.e. $\\frac{n-k+l}{l} \\geq \\frac{n}{k}.$ So we derive the well known inequality\n \\begin{equation} \\binom{n}{k} = \\prod_{l=1}^{k} \\frac{n-k+l}{l}  \\geq \\left(\\frac{n}{k}\\right)^k.  \\label{binomial} \\end{equation}\n\nThe function\n$$\nf(x) := \\log \\left( \\frac{h}{x} \\right)^x = x ( \\log h - \\log x)\n$$\nattains its maximum value at $x=h\/e$ so that its minimum will be taken at either end of a given interval.\n\nThe binomial coefficients in \\eqref{eq:Vj1} attain asymptotic lower estimates for $2\\leq j\\leq [g\/2]$ at $j=2$ namely\n$$\n \\left(3g-4\\atop 3j-2 \\right)\\geq \\left( \\frac{3g-4}{4}\\right)^4 \\quad \\text{and} \\quad \\left(2g-4\\atop 2j-2  \\right)\\geq (g-2)^2.\n$$\nHence, summing up all terms of the type \\eqref{eq:Vj1} yields a term of growth order at most $O(1\/g^7)$. The last term \\eqref{eq:gover2} has a at most a negative exponential growth with respect to $g$ and can also be disregarded.\n\n\n\n This proves the main theorem. \\qed\n\n{\\it Acknowledgements.} The first named author expresses his thanks to the Department of Mathematics at Tor Vergata for their kind hospitality, and for support by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5, and by DFG SCHU771\/5-1 (Singular Hermitian Metrics\/Analytic Theory of Moduli Spaces). The second named author is supported, by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5, and by MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.\u00a0\n\n\n\\frenchspacing\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Introduction}\nIn this paper we present the Reachable Polyhedral Marching (RPM) algorithm for computing forward and backward reachable sets of deep neural networks with rectified linear unit (ReLU) activation.  This is a critical building block to proving safety properties for autonomous systems with learned perception, dynamics, or control components in the loop.  Specifically, given a set of input values, RPM will compute the set of all output values that can be obtained under the ReLU network (the forward reachable set, or image, of the input set).  Similarly, given a set of output values, RPM will compute the set of all input values that can lead to those output values under the ReLU network (the backward reachable set, or preimage, of the output values).  When the ReLU network is part of a dynamical process, as is common in robots with deep learned perception or control components, RPM can compute such reachable sets for multiple time steps into the future or the past without explicitly iterating over each time step.  Figure \\ref{fig:cell_enum} shows the incremental nature of how RPM enumerates input space polyhedra over which the ReLU network is affine.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width = 0.98\\columnwidth]{incremental.png}\n  \\caption{Snapshots of how RPM incrementally enumerates the 2D input space polyhedra for which a random ReLU network is affine, resulting in the explicit piecewise-affine representation. Polyhedron color is random.}\n  \\label{fig:cell_enum}\n\\end{figure}\n\n\nAll existing algorithms that compute exact reachable sets iterate through the network layer-by-layer \\cite{xiang2017reachable,nnv,yang2020reachability}, which may become intractable if the network is applied iteratively in a feedback loop.  RPM applies a fundamentally different approach to avoid layer-by-layer computation.  Instead, we compute the reachable set one polyhedral cell at a time, where each cell represents a region of the input space over which the network is affine.  Each cell can be computed quickly through a series of linear programs equal to the number of neurons in the network, regardless of width or depth.  Our algorithm computes cells until the explored set of cells fills the desired reachable set. In this way, our method is geometrically similar to fast marching methods in optimal control \\cite{HJBFastMarching}, path planning \\cite{PlanningFastMarching,FMT}, and graphics \\cite{FastMarchingCubes,lei2020analytic}.\n\nWe demonstrate RPM in two examples related to the safety verification of dynamical systems with learned components.  First, we compute forward reachable and backward reachable sets for a learned dynamical model of a pendulum over 50 time steps in the future and past, respectively. Furthermore, we compare with a state of the art reachability method \\cite{yang2020reachability} in a safety verification problem involving ACAS Xu, a neural network policy for aircraft collision avoidance.  Our algorithm finds unsafe inputs for this network many times faster than \\cite{yang2020reachability} and when no unsafe input exists, our algorithm certifies safety approximately 2 times slower.  \n\n\n\n\n\nThe paper is organized as follows. We give related work in Section \\ref{Related Work} and give background and state the problem in Section \\ref{Background}.  Section \\ref{PWA} presents our main algorithm, and explains its derivation. In Section \\ref{reach} we describe how the RPM algorithm can be used to perform forward and backward reachability over multiple time steps. In Section \\ref{Examples} we present the aforementioned examples of RPM.\n\n\n\n\n\\section{Related Work} \\label{Related Work}\nThough the analysis of neural networks is a young field, a broad literature has emerged to address varied questions related to interpretability, verification, and safety. Much work has been dedicated to characterizing the expressive potential of ReLU networks by studying how the number of affine regions scales with network depth and width \\cite{bengio,Hanin_2019,understanding}. Other research includes encoding piecewise-affine (PWA) functions as ReLU networks \\cite{fem,reverseengineering}, learning deep signed distance functions and extracting level sets \\cite{lei2020analytic}, and learning neural network dynamics or controllers that satisfy stability conditions \\cite{zico,jin2020neural}, which may be more broadly grouped with correct-by-construction training approaches \\cite{art,lincolnlab}. \n\n\nSpurred on by pioneering methods such as Reluplex \\cite{reluplex}, the field of neural network verification has emerged to address the problem of analyzing properties of neural networks over continuous input sets. A survey of the neural network verification literature is given by \\cite{survey}. Reachability approaches are subset of this literature and are especially useful for analysis of learned dynamical systems. Reachability methods can be categorized into overapproximate and exact methods. Overapproximate methods often compute neuron-wise bounds either from interval arithmetic or symbolically \\cite{maxsens,fastlip,reluval,neurify,crown}. Optimization based approaches such as (mixed-integer) linear programming are also used to solve for bounds on a reachable output set in \\cite{sherlock,alessio1,alessio2,proveable}. Other approaches include modeling the network as a hybrid system \\cite{verisig}, abstracting the domain \\cite{abstract}, and performing layer-by-layer operations on zonotopes \\cite{AI2}. Further, \\cite{julian2019reachability} demonstrated how \\cite{reluplex, sherlock, reluval, verisig} could be used to perform closed-loop reachability of a dynamical system given a neural network controller.\n\nExact reachability methods have also been proposed, although to a lesser degree \\cite{xiang2017reachable,nnv,yang2020reachability}. These methods generally perform the set operations of affine transformation, intersection\/division, and projection layer-by-layer through a ReLU network. Layer-by-layer approaches have also been proposed to solve for the explicit PWA representation of a ReLU network \\cite{robinson2020dissecting,hanin2019complexity}.\nExact methods, unlike overapproximate methods, can compute backward reachable sets.\n\nOur RPM algorithm inherits all advantages of exact methods, but is unique in that it does not iterate layer-by-layer. This can lead to faster verification decisions when finding unsafe inputs. Layer-by-layer methods must compute the entire reachable set, regardless of whether a network violates a safety property. In contrast, if our algorithm encounters a cell with an unsafe input, it can return that result immediately without computing the entire reachable set. Finally, all intermediate polyhedra and affine map matrices of layer-by-layer methods must be stored in memory until the algorithm terminates (since intermediate polyhedra may be split by later neurons). Conversely, since our method computes the true PWA function incrementally, once a polyhedron-affine map pair is computed it can be sent to external memory and only a binary vector (of the neuron activations) needs to be stored to continue the algorithm. This is especially useful because the number of affine regions is hard to estimate before runtime.\n\n\n\\section{Background and Problem Statement} \\label{Background}\nAn $n$-layer feedforward neural network implements a function $F(\\textbf{x}) : \\mathbb{R}^{k_0} \\rightarrow \\mathbb{R}^{k_n}$ to give a map from inputs $\\mathbf{x}$ to outputs $\\textbf{y} = F(\\textbf{x})$. Each layer in $F$ is a function $F_i(\\textbf{z}_{i-1}) : \\mathbb{R}^{k_{i-1}} \\rightarrow \\mathbb{R}^{k_i}$ where $\\textbf{z}_{i} \\in \\mathbb{R}^{k_i}$ is the hidden layer variable for layer $i$. We assume $\\textbf{z}_0 = \\textbf{x}$ and $\\textbf{z}_n = \\textbf{y}$. The function for layer $i$ is\n\\begin{align}\n    F_i(\\textbf{z}_{i-1}) = \\textbf{z}_{i} = \\sigma_i(\\hat{\\textbf{z}}_{i}) = \\sigma_i(\\textbf{W}_i \\textbf{z}_{i-1} + \\textbf{b}_i),\n\\end{align}\nwhere $\\sigma_i$ is the activation function, $\\hat{\\textbf{z}}_{i}$ is the preactivation value, and $\\textbf{W}_i$ and $\\textbf{b}_i$ are the weights and biases for layer $i$. We assume $\\sigma_{n}$ is an identity map and all hidden layer activations are the rectified linear unit (ReLU)\n\\begin{align}\n    \\sigma_i(\\hat{\\textbf{z}}_{i}) = [max(0,\\hat{z}_{i,1}),...,max(0,\\hat{z}_{i,k_i})]^\\top.\n\\end{align}\n\nFrom the above definition we can augment the weights to construct an equivalent neural network that has no bias terms. This equivalent formulation is such that $\\tilde{F}(\\tilde{\\textbf{x}}) : \\mathbb{R}^{k_0+1} \\rightarrow \\mathbb{R}^{k_n}$ and $\\tilde{\\textbf{x}} = [x_1, x_2, ..., x_n, 1]^\\top$. We then have\n\\begin{subequations}\n    \\begin{align}\n        \\tilde{F}_i(\\tilde{\\textbf{z}}_{i-1}) = \\tilde{\\textbf{z}}_i = \\sigma_i(\\tilde{\\textbf{W}}_i \\tilde{\\textbf{z}}_{i-1}) \\\\\n        \\tilde{\\textbf{W}}_i = \n        \\begin{bmatrix}\n            \\textbf{W}_i & \\textbf{b}_i \\\\\n            \\textbf{0}^\\top & 1\n        \\end{bmatrix} \\\\\n        \\tilde{\\textbf{W}}_n = \n    \\begin{bmatrix}\n        \\textbf{W}_n & \\textbf{b}_n\n    \\end{bmatrix}.\n    \\end{align}\n\\end{subequations}\n\nIt is well-known that the function implemented by a ReLU network is continuous and PWA \\cite{bengio,Hanin_2019,understanding}.  We mathematically characterize PWA functions below.\n\\newtheorem{definition}{Definition}\n\\begin{definition}[Polyhedral Complex]\nA polyhedral complex $\\mathcal{C}$ is a finite set of convex polyhedra where every polyhedron contains its faces and the intersection between two polyhedra is either a face of both $c_1$ and $c_2$ or empty. We henceforth refer to convex polyhedra simply as polyhedra.\n\\end{definition}\n\nA PWA function is a function $M(\\textbf{x}) : \\mathbb{R}^{n} \\rightarrow \\mathbb{R}^{m}$ where\n\\begin{align}\n    M(\\textbf{x}) = \\textbf{C}_i \\textbf{x} + \\textbf{d}_i \\ \\ \\ \\forall \\textbf{x} \\in c_i \\ \\ \\ \\forall c_i \\in \\mathcal{C} \\label{eq:PWA}\n\\end{align}\nfor $\\textbf{C}_i \\in \\mathbb{R}^{m \\times n}$, $\\textbf{d}_i \\in \\mathbb{R}^{m}$, and $c_i$ elements of a polyhedral complex $\\mathcal{C}$. Here we consider only continuous PWA functions. Equation (\\ref{eq:PWA}) is the \\textit{explicit} PWA representation.\n\nThe forward reachability problem is to solve for the image of a specified input set under the neural network map\n\\begin{align}\n    \\mathcal{Y} = \\{F(\\textbf{x}) \\vert \\textbf{x} \\in \\mathcal{X} \\}.\n\\end{align}\nLikewise, the backward reachability problem is to solve for the preimage of a specified output set\n\\begin{align}\n    \\mathcal{X} = \\{x \\vert F(\\textbf{x}) \\in \\mathcal{Y} \\}.\n\\end{align} \n\nOur algorithm transforms a ReLU network into an explicit PWA representation by computing a set of polyhedra ($c_i$) and associated affine maps $(\\textbf{C}_i, \\textbf{d}_i)$.  There exist efficient methods for computing forward and backward reachable sets of polyhedral sets under PWA maps. Accordingly, we restrict our focus to these problems. \n\n\n\\section{Explicit PWA Representation} \\label{PWA}\nFirst, we seek to construct the explicit PWA representation of a ReLU network. Our method constructs a PWA function for a ReLU network by enumerating each polyhedral cell and its associated affine map. In Sec.~\\ref{Cells} we show how cells and affine maps are computed from the \\textit{activation pattern} of a ReLU network. In Sec.~\\ref{Essential} we show how cell representations are reduced to a minimal form, which is used in Sec.~\\ref{Neighbors} to determine neighboring polyhedra given a current polyhedron. This leads to a recursive procedure in which we explore an expanding front of polyhedra, ultimately giving the explicit PWA representation, as explained in Sec.~\\ref{Cell Enumeration}.\n\n\\subsection{Determining Polyhedral Cells from Activation Patterns} \\label{Cells}\nFor a given input to a ReLU network, every hidden neuron has an associated binary neuron activation of zero or one corresponding to whether the preactivation value was nonpositive or positive, respectively\\footnote{This choice is arbitrary and some others use the opposite convention.}. We refer to the vector of all neuron activations as the \\textit{activation pattern} of the network for a given input. For neuron $j$ in layer $i$,\n\\begin{subequations}\n    \\begin{align}\n        AP_{ij}(\\textbf{x}) = \\begin{cases}\n                                 1\\ \\  \\hat{z}_{ij} > 0 \\\\    \n                                 0\\ \\  \\hat{z}_{ij} \\le 0 \n                              \\end{cases} \\label{eq:AP_1} \\\\\n        AP_i(\\textbf{x}) = [ AP_{i,1}(\\textbf{x}), ..., AP_{i,k_i}(\\textbf{x})  ]^\\top \\\\\n        AP(\\textbf{x}) = (AP_1(\\textbf{x}), ..., AP_{n-1}(\\textbf{x})). \\label{eq:AP}\n    \\end{align}\n\\end{subequations}\nFor a network with $N$ hidden neurons there are $2^N$ possible combinations of neuron activations, however not all are realizable by the network. Practically, the number of activation patterns is better approximated by $N^{k_0}$ \\cite{hanin2019deep}.\n\nTo construct an explicit PWA representation we first want to characterize the set of inputs $c$ for which the network reduces to a single affine map. For a fixed activation pattern the ReLU network simplifies to the affine map $\\textbf{C} \\textbf{x} + \\textbf{d}$ where\n\\begin{subequations}\n    \\begin{align}\n        \\begin{bmatrix}\n            \\textbf{C} & \\textbf{d}\n        \\end{bmatrix} = \n        \\tilde{\\textbf{W}}_n \\prod_{i=1}^{n-1} \\tilde{\\textbf{W}}^c_i \\label{eq:linear} \\\\\n        \\tilde{\\textbf{W}}^c_i(\\textbf{x}) = diag(AP_{i}(\\textbf{x}))\\tilde{\\textbf{W}}_i, \\label{eq:diag}\n    \\end{align}\n\\end{subequations}\nand $diag(\\cdot)$ diagonalizes a vector to a matrix. Equation (\\ref{eq:diag}) equivalently sets row $j$ of $\\tilde{\\textbf{W}}_i$ to zero if $AP_{ij} = 0$.\n\nThe activation pattern only changes if some $\\hat{z}_{ij}$ switches from positive to nonpositive (or vice-versa), $c$ is thus given by a set of linear constraints, one for each neuron in the network.  The constraint for neuron $j$ in layer $i$ is\n\\begin{align}\n    \\begin{cases}\n          \\textbf{a}_{ij}^{\\top} \\textbf{x} \\ge b_{ij}\\ \\ \\text{if}\\  AP_{ij} = 1 \\\\    \n          \\textbf{a}_{ij}^{\\top} \\textbf{x} \\le b_{ij}\\ \\ \\text{if}\\  AP_{ij} = 0\n      \\end{cases} \\label{eq:constraints}\n\\end{align}\nwhere\n\\begin{align}\n    \\begin{bmatrix}\n        \\textbf{a}_{ij} & -b_{ij}\n    \\end{bmatrix} =  \n    \\tilde{\\textbf{W}}_i[j,:] \\prod_{l=1}^{i-1} \\tilde{\\textbf{W}}_l, \\label{eq:ab}\n\\end{align}\nand $\\tilde{\\textbf{W}}_i[j,:]$ denotes the $j^{th}$ row of $\\tilde{\\textbf{W}}_i$. Equation (\\ref{eq:ab}) is similar to (\\ref{eq:linear}) but instead simply gives the affine map parameters for a single neuron output. Strict inequalities are not present in (\\ref{eq:constraints}) because the affine map $\\textbf{Cx} + \\textbf{d}$ holds over the interior and boundary of the set where the activation pattern is constant. Since $c$ is an intersection of halfspaces, the set is a polyhedron. A similar formulation is given in \\cite{lei2020analytic} for finding the surface of an object for graphics rendering. Note that it is not uncommon for neurons to have the degenerate linear constraint $\\textbf{0}^{\\top} \\textbf{x} \\le 0$ which is satisfied for all $\\textbf{x} \\in c$, and also for multiple different neurons in the network to have equivalent constraints (perhaps scaled arbitrarily). To address these edge cases we introduce Definitions \\ref{def: hrep}, \\ref{def: duplicate}, \\ref{def: redundant} before describing our fast marching method below.  \n\n\\begin{definition}[H-representation]\nThe halfspace representation (H-representation) of a polyhedron is  \n\\begin{align}\n    P_H = \\{\\textbf{x} \\in \\mathbb{R}^d \\vert \\textbf{A} \\textbf{x} \\le \\textbf{b} \\} \\label{eq:hrep}\n\\end{align}\nwhere $\\textbf{A}$ an $m \\times d$ matrix and $\\textbf{b}$ an $m$-dimensional vector. Equality constraints may also be included if the set $P_H$ is of lower dimension than the ambient dimension. \\label{def: hrep}\n\\end{definition}\n\n\\begin{definition}[Duplicate Constraints]\nA constraint $\\textbf{a}_j^{\\top} \\textbf{x} \\le b_j$ is duplicate if there exists a scalar $\\alpha$ and a prior constraint $\\textbf{a}_i^{\\top} \\textbf{x} \\le b_i$ where $i<j$ such that $\\alpha [\\textbf{a}_j\\ b_j] = [\\textbf{a}_i\\ b_i]$ where $\\alpha > 0$. \\label{def: duplicate}\n\\end{definition}\n\\begin{definition}[Redundant \\& Essential Constraints]\nA constraint $\\textbf{a}_i^{\\top} \\textbf{x} \\le b_i$ is redundant if the feasible set does not change upon its removal or if it is a duplicate constraint. An essential constraint is one which is not redundant. \\label{def: redundant}\n\\end{definition}\n\nWe use (\\ref{eq:constraints}) to construct an H-representation of a cell $c$ by enumerating all constraints associated with hidden layer neurons.  The resulting $\\textbf{a}_{ij}$ vectors and $b_{ij}$ scalars are collected into ($\\textbf{A}$, $\\textbf{b}$) to give the desired H-representation from (\\ref{eq:hrep}).  For an arbitrary ReLU network in an arbitrary activation pattern, we find that the resulting H-representation often has redundant constraints, which need to be removed to give a minimal representation of the polyhedron.  \n\n\n\n\\subsection{Finding Essential Constraints} \\label{Essential}\nWe first normalize all constraints, remove any duplicates, and consider the resulting  H-representation $\\textbf{A}\\textbf{x} \\le \\textbf{b}$. To determine if the $i$th constraint is essential or redundant, define a new set of constraints with the $i$th constraint removed,\n\\begin{subequations}\n    \\begin{align}\n        \\tilde{\\textbf{A}} = [\\textbf{a}_1\\ ...\\ \\textbf{a}_{i-1}\\ \\textbf{a}_{i+1}\\ ...\\  \\textbf{a}_m]^\\top \\\\\n         \\tilde{\\textbf{b}} = [b_1\\ ...\\ b_{i-1}\\ b_{i+1}\\ ...\\  b_m]^\\top,\n    \\end{align}\n\\end{subequations}\n\nand solve the linear program\n\\begin{subequations}\n    \\begin{align}\n        \\max_{\\textbf{x}} \\quad & \\textbf{a}^{\\top}_i \\textbf{x}\\\\\n        \\textrm{subject to} \\quad & \\tilde{\\textbf{A}}\\textbf{x} \\le \\tilde{\\textbf{b}}.\n    \\end{align}\n    If the optimal objective value is less than or equal to $b_i$, constraint $i$ is redundant.\n\\end{subequations}\nIn the worst case, a single LP must be solved for each constraint to determine whether it is essential or redundant. However, heuristics exist to avoid this worst case complexity. We use the bounding box heuristic to quickly find a subset of redundant constraints \\cite{morari}. Empirically, this results in identifying about $90\\%$ of the redundant constraints. We find that other heuristics beyond the bounding box method do not improve performance. Our implementation uses the GLPK open-source LP solver and the JuMP optimization package \\cite{DunningHuchetteLubin2017}.\n\n\n\\subsection{Determining Neighboring Activation Patterns} \\label{Neighbors}\nGiven a cell $c$ in the input space of a ReLU network, we are interested in finding the activation pattern corresponding to a neighboring cell, $c'$. We can then determine the H-representation of $c'$ using the procedure outlined above. The challenge in finding the activation pattern for $c'$ is in determining which individual neuron activations ($AP_{ij}$ for some $ij$) must be flipped when $\\textbf{x}$ transitions from $c$ to $c'$. \n\nThe boundary between $c$ and $c'$ ($c \\cap c'$) is defined by a linear constraint of $c$. We refer to this as the \\textit{neighbor constraint}. Intuitively, to generate $AP^{c'}$ one ought to simply flip the activation of the neuron defining this neighbor constraint (and all neurons defining duplicate constraints). However, this intuitive procedure is incomplete because it does not correctly deal with degenerate constraints $\\textbf{0}^{\\top} \\textbf{x} \\le 0 \\ \\forall \\textbf{x} \\in c$, which are common in practice.  A degenerate constraint holds everywhere over $c$, but it may turn into a non-degenerate constraint for $c'$ due to a neuron activation in an earlier layer being flipped when moving to cell $c'$. Conversely, a non-degenerate constraint in $c$ may turn into a degenerate one when passing to $c'$.  To define a correct procedure for flipping neuron activations we introduce a taxonomy for classifying neurons based on their associated linear constraints.\n\n\\begin{definition}[Type 1, 2, 3 neurons] \\label{types}\nWhen identifying a neighboring cell $c'$ from a current cell $c$, neurons whose linear constraints in $c$ define $c \\cap c'$ are called \\textit{Type 1}. neurons with degenerate linear constraints in $c$, $\\textbf{0}^{\\top} \\textbf{x} \\le 0 \\ \\forall \\textbf{x} \\in c$, are called \\textit{Type 2}. All other neurons are \\textit{Type 3}. \n\\end{definition}\nConstraints labeled by type are illustrated in Figure \\ref{fig:constraints}a. \n\nThough not stated explicitly, the linear constraint parameters from (\\ref{eq:constraints}) are (nonlinear) functions of the input. Given Definition \\ref{types}, for inputs $\\textbf{x} \\in c \\cap c'$,\n\\begin{subequations}\n    \\begin{align}\n        \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} = b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 1} \\\\\n        \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} = b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 2} \\\\\n        \\begin{cases}\n            \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} > b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 3}\\ \\text{and}\\ AP_{ij} = 1 \\\\\n            \\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} < b_{ij}(\\textbf{x})\\ \\ \\text{if}\\  ij \\in \\text{Type 3}\\ \\text{and}\\ AP_{ij} = 0\n        \\end{cases}. \\label{eq:T3}\n    \\end{align}\n\\end{subequations}\nThe function $\\textbf{a}_{ij}^{\\top}(\\textbf{x}) \\textbf{x} - b_{ij}(\\textbf{x})$ is continuous because it is simply the function for computing the preactivation value $\\hat{z}_{ij}$. From this continuity, we know that the strict inequality of (\\ref{eq:T3}) holds for all inputs in $c'$ as well. Therefore, the activation of a Type 3 neuron is not flipped. \n\n\\begin{figure}\n     \\centering\n     \\includegraphics[width=0.95\\columnwidth]{constraints_smaller.png}\n     \\caption{(a) Example $c$ and $c'$ with Type 1 and Type 3 constraints labeled. For (b)-(e), regions in green denote where the nonlinear constraint is strictly satisfied, regions in blue denote where the nonlinear constraint function evaluates to zero, and regions in red denote where the nonlinear constraint is violated. (b) Type 1 constraint where the neuron activation must flip when going from $c$ to $c'$. (c) Type 1 constraint where the neuron activation is set to zero when going from $c$ to $c'$. (d) Type 2 constraint where the neuron activation must flip when going from $c$ to $c'$. (e) Type 2 constraint where the neuron activation remains zero when going from $c$ to $c'$.}\n     \\label{fig:constraints}\n\\end{figure}\n\n \nConversely, inputs in $c'$ may be either feasible or infeasible for Type 1 and Type 2 constraints. Figure \\ref{fig:constraints}b-c illustrates the possible feasible and infeasible regions of a Type 1 constraint. Figure \\ref{fig:constraints}d-e illustrates the possible feasible and infeasible regions of a Type 2 constraint. It follows that the procedure to generate a neighboring activation pattern is to apply (\\ref{eq:constraints}) layer-by-layer updating the activation pattern in place, where Type 1 and Type 2 neuron activations are set to zero if their new linear constraints are $\\textbf{0}^{\\top} \\textbf{x} \\le 0$, otherwise, they are flipped. This procedure is formally defined in Algorithm \\ref{Algo:neighbor_ap}.  We note that applying this neuron flipping algorithm to identify a neighboring cell $c'$ is exceedingly fast, even for networks with 10,000s of neurons, as we only require a single logical check at each neuron. \n\\begin{algorithm}[!h]\n\\SetAlgoLined\n\\DontPrintSemicolon\n \\KwIn{$AP^{c}$, $Type\\ 1$, $Type\\ 2$,  $(W_0, \\ldots, W_N$)}\n \\KwOut{$AP^{c'}$}\n $AP^{c'} \\leftarrow  AP^{c}$ \\algorithmiccomment{Initialize new AP as old AP} \\\\\n \n\\For(\\algorithmiccomment{For each neuron}){$i \\in [1,...,n-1]$, $j \\in [1,...k_i]$}{ \n\\If{$ij \\in Type\\ 1\\ or\\ ij \\in Type\\ 2$}{\n$\\textbf{a}_{ij},\\ b_{ij} \\leftarrow$ Equation \\ref{eq:ab} \\algorithmiccomment{$AP^{c'}_{ij}$ hyperplane} \\\\\n\\uIf{$[\\textbf{a}_{ij}\\ b_{ij}] = \\textbf{0}$}{$AP^{c'}_{ij} \\leftarrow 0$}\n\\Else{$AP^{c'}_{ij} \\leftarrow \\neg AP^{c'}_{ij} $}\n}\n}\n \n\\Return{$AP^{c'}$}\n\\caption{Neighboring Activation Pattern}\n\\label{Algo:neighbor_ap}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Reachable Polyhedral Marching} \\label{Cell Enumeration}\nFrom (\\ref{eq:constraints}) and Algorithm \\ref{Algo:neighbor_ap}, we define our main algorithm, Reachable Polyhedral Marching (RPM) in Algorithm \\ref{Algo:cell_enumeration} for explicitly enumerating all polyhedral cells in the input space of a ReLU network, resulting in the explicit PWA representation. First, we start with an initial point in the input space and evaluate the network to find the activation pattern. From this the H-representation is found using (\\ref{eq:constraints}). Essential constraints are then identified as described in Section \\ref{Essential}. For each essential constraint we generate a neighboring activation pattern using Algorithm \\ref{Algo:neighbor_ap}. The process then repeats with each new neighboring activation pattern being added to a working set. A neighbor activation pattern only gets added to the working set if it has not already been visited and it is not already in the working set. For a given starting cell each neighbor cell is enumerated, and since each is connected to another, Algorithm \\ref{Algo:cell_enumeration} is guaranteed to enumerate every cell in the input space. Figure \\ref{fig:cell_enum} shows the result of applying Algorithm \\ref{Algo:cell_enumeration} to a randomly initialized ReLU network.\n\\begin{algorithm}\n\\SetAlgoLined\n\\DontPrintSemicolon\n \\KwIn{$AP^{c_0}$, $(W_0, \\ldots, W_N$)}\n \\KwOut{Explicit PWA representation}\n $input\\ cells$ = $\\emptyset$;  $visited\\ cells$ = $\\emptyset$ \\\\\n $working\\ set$ = \\{$AP^{c_0}$\\} \\\\\n \\While{$working\\ set \\neq \\emptyset$}{\n $AP^{c} \\leftarrow$ pop element off of $working\\ set$ \\\\\n $\\textbf{C}^{c},\\ \\textbf{d}^{c} \\leftarrow$ Equation \\ref{eq:linear} \\algorithmiccomment{Retrieve affine map} \\\\\n $H^{c}_{rep} \\leftarrow$ Equation \\ref{eq:constraints} \\algorithmiccomment{Retrieve H-representation} \\\\\n $H^{c}_{rep} \\leftarrow$ remove redundant $H^{c}_{rep}$ constraints \\\\\n push ($\\textbf{C}^{c},\\ \\textbf{d}^{c},\\ H^{c}_{rep}$) onto $input\\ cells$ \\\\\n \\For{$\\textbf{a}_k, b_k \\in H^{c}_{rep}$}{\n $AP^{c'} \\leftarrow$ Algorithm \\ref{Algo:neighbor_ap} \\\\\n \\If{$AP^{c'} \\notin visited\\ cells \\cup working\\ set$}{\n push $AP^{c'}$ onto $working\\ set$ \\\\\n }\n }\n push $AP^{c}$ onto $visited\\ cells$ \\\\\n }\n \\Return{input cells}\n \\caption{Reachable Polyhedral Marching}\n \\label{Algo:cell_enumeration}\n\\end{algorithm}\n\n\n\\section{Reachability} \\label{reach}\n\\subsection{Forward Reachability} \\label{Forward}\nThe forward reachable set of a PWA function over some input set is simply the union of forward reachable sets of each individual polyhedron under its associated affine map. The image of a polyhedron under an affine map is\n\\begin{align}\n    P_{out} = \\{\\textbf{y} \\vert \\textbf{y}=\\textbf{Cx}+\\textbf{d}, \\textbf{Ax} \\le \\textbf{b}\\}. \\label{eq:forward reach}\n\\end{align}\nFor $\\textbf{C}$ invertible, the H-representation of the image is\n\\begin{align}\n    P_{out} = \\{\\textbf{y} \\vert \\textbf{AC}^{-1}\\textbf{y} \\le \\textbf{b} + \\textbf{AC}^{-1} \\textbf{d}\\}. \\label{eq:image}\n\\end{align}\nIn the case the affine map is not invertible, more general polyhedral projection methods such as block elimination, Fourier-Motzkin elimination, or parametric linear programming can compute the H-representation of the image \\cite{cdd_manual, mpLP}. Our implementation uses the block elimination projection in the case of a non-invertible affine map \\cite{cdd}.\n\nOur RPM algorithm is used to perform forward reachability as follows. We first specify a polyhedral input set whose image through the ReLU network we want to compute.  This is a set over which to perform the RPM algorithm. For each activation pattern $AP^{c}$ we also compute the image of $H^{c}_{rep}$ under the map $\\textbf{C}^{c}\\textbf{x} + \\textbf{d}^{c}$. To do this, an additional line is introduced between lines 9 and 10 of Algorithm \\ref{Algo:cell_enumeration}\n\\begin{align}\n    H^{c}_{rep,\\ forward} \\leftarrow project(H^{c}_{rep}, \\textbf{C}^{c}, \\textbf{d}^{c})\n\\end{align}\nwhere $project(\\cdot, \\cdot, \\cdot)$ applies (\\ref{eq:image}) if $\\textbf{C}^{c}$ invertible and block elimination otherwise.\n\n\n\\subsection{Backward Reachability} \\label{Backward}\nThe preimage of a polyhedron under an affine map is\n\\begin{subequations}\n    \\begin{align}\n        P_{in} = \\{\\textbf{x} \\vert \\textbf{y}=\\textbf{Cx}+\\textbf{d}, \\textbf{Ay} \\le \\textbf{b}\\} \\\\\n        P_{in} = \\{\\textbf{x} \\vert \\textbf{ACx} \\le \\textbf{b}-\\textbf{Ad}\\}. \\label{eq:backward reach}\n    \\end{align}\n\\end{subequations}\nLike forward reachability, performing backward reachability only requires a small modification to Algorithm \\ref{Algo:cell_enumeration}. We first specify a polyhedral output set whose preimage we would like to compute.  For each activation pattern $AP^{c}$, we also compute the intersection of $H^{c}_{rep}$ with the preimage of the given output set under the map $\\textbf{C}^{c}\\textbf{x} + \\textbf{d}^{c}$. Two additional lines are thus introduced between lines 9 and 10 of Algorithm \\ref{Algo:cell_enumeration}\n\\begin{subequations}\n    \\begin{align}\n    P^{c}_{in} \\leftarrow \\text{Equation}\\ \\ref{eq:backward reach} \\label{eq:p_in} \\\\\n    H^{c}_{rep,\\ backward} \\leftarrow H^{c}_{rep} \\cap P^{c}_{in}. \\label{eq:h_back}\n    \\end{align}\n\\end{subequations}\nMultiple backward reachable sets can be solved for simultaneously at the added cost of repeating (\\ref{eq:p_in}) and (\\ref{eq:h_back}) for each output set argument to Algorithm \\ref{Algo:cell_enumeration}.\n\nFinally, we address the issue of finding forward and backward reachable sets iterated over multiple time steps.  For this, we note that a ReLU network that is applied iteratively over $T$ timesteps, $\\textbf{x}_{t+1} = F(\\textbf{x}_t)$ for $t = 0, \\ldots, T-1$, is mathematically equivalent to a single ReLU network consisting of $T$ copies of the original network concatenated end to end, $\\textbf{x}_T = F_T(\\textbf{x}_0) := F \\circ \\cdots \\circ F(\\textbf{x}_0)$.  If the original network $F(\\textbf{x})$ has $N$ neurons and $L$ layers, the equivalent multi-time step network $F_T(\\textbf{x})$ has $TN$ neurons and $TL$ layers.  We simply perform RPM for forward or backward reachability on the equivalent multi-time step network $F_T(\\textbf{x})$.\n\n\\section{Examples} \\label{Examples}\nAll examples are run on a 2013 Dell Latitude E6430s laptop with Intel Core i7 3GHz processor and 16GB of RAM. The cell coloring in plots is random.\n\\subsection{Damped Pendulum Example} \\label{Pendulum}\n\nThe forward and backward reachability algorithms can be applied to discrete-time dynamical systems represented as ReLU networks. In this example we analyze a ReLU network that approximates the discrete-time dynamics of a damped pendulum. The function learned by the network is \n\\begin{align}\n    \\bm{x}_{t+1} = f_{nn}(\\bm{x}_t)\n    \\label{pendulum}\n\\end{align}\nwhere $\\bm{x} = [\\theta,  \\dot{\\theta}]^\\top$. The time-step used is $0.1$ seconds. The learned dynamics function $f_{nn}$ is a ReLU network with a single hidden layer of 12 neurons. An example trajectory of the learned system is shown in Figure \\ref{fig:damped}. We can compose multiple copies of $f_{nn}$ together to get a neural network which outputs an arbitrary future state. For instance, composing the network with itself 50 times results in a final network with 600 neurons and output $\\bm{x}_{t+50}$. \n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width = 0.6\\columnwidth]{damped_sine_f.png}\n    \\caption{50 time-step trajectory from $\\bm{x}_0 = [30, 0]^\\top$}\n    \\label{fig:damped}\n\\end{figure}\n\n\\begin{figure*}\n     \\centering\n     \\begin{subfigure}[b]{0.95\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{input_50_f.png}\n         \\caption{Input cells.}\n         \\label{fig:pend_in}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.95\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{output_50_f.png}\n         \\caption{Output cells.}\n         \\label{fig:pend_out}\n     \\end{subfigure}\n        \\caption{(a) Input cells of the 50 time-step pendulum network. (b) image of input cells under the neural network map.}\n        \\label{fig:forward}\n\\end{figure*}\n\nFigure \\ref{fig:damped} suggests that states of the learned system tend toward the origin. We can use the forward reachability algorithm to determine where \\textit{all} states within some set of initial conditions lead to over some time interval. In Figure \\ref{fig:pend_in} we show the result of applying the cell enumeration algorithm and the forward reachability algorithm to the 50 time step system for initial conditions $-90 \\degree \\le \\theta \\le 90 \\degree$ and $-90 \\degree \/ s \\le \\dot{\\theta} \\le 90 \\degree \/ s$. The resulting number of regions is 2781 and the forward reachability version of the RPM algorithm took 87s to complete. It is clear from Figure \\ref{fig:pend_in} that all initial conditions are mapped to a subset of the initial input set, proving that the original input set is forward invariant over 50 time step increments, and indicating that trajectories tend to approach the origin over time.\n\nFurther, we perform backward reachability to find the set of inputs that map to a small neighborhood around the origin after 50 time steps. Figure \\ref{fig:back_reach} shows the inputs that map to the output set $-5 \\degree \\le \\theta \\le 5 \\degree$ and $-2 \\degree \/ s \\le \\dot{\\theta} \\le 2 \\degree \/ s$ after 50 time steps. Interestingly, only initial conditions with positive angles map to the target set. In the true dynamics, the initial angles would be symmetric about zero. This analysis can thus help us identify undesirable modeling artifacts to inform retraining or control of the model.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 0.9\\columnwidth]{back_reach_f.png}\n    \\caption{Set of initial states that lead to a neighborhood around the origin after 50 time steps for the pendulum network.}\n    \\label{fig:back_reach}\n\\end{figure}\n\n\n\\subsection{Aircraft Collision Avoidance Example} \\label{ACAS}\nThe ACAS Xu networks are 45 distinct policy networks designed to issue advisory warnings to avoid mid-air collisions for unmanned aircarft. Each network has five inputs, five outputs (advisories), and 300 neurons. Inputs are relative distance, angles, and speeds. See \\cite{acas} for more details. \n\nThe appendix of \\cite{reluplex} lists ten safety properties the networks should satisfy. Here we consider Property 3. This property is satisfied if the network never outputs a ``Clear of Conflict\" advisory when the intruder is directly ahead and moving towards the ownship. We use the backward reachability algorithm to verify whether Property 3 is satisfied. A nonempty backward reachable set implies that some allowable inputs will map to unsafe outputs and the safety property is not satisfied. Table \\ref{table:verification} shows the results of verifying Property 3 for four of the networks. A comparison is given against the fastest existing exact reachability method based on the face lattice representation \\cite{yang2020reachability} as well as the Reluplex and Marabou verification algorithms \\cite{reluplex,marabou}. Each algorithm was run on a single core. We note that a parallelized algorithm is also provided for the face lattice approach \\cite{yang2020reachability}, and our RPM algorithm is also amenable to a parallelized implementation to be explored in future work. \n\nThe results in Table \\ref{table:verification} show the advantage of our proposed method when verifying network properties that are found to not hold. Our RPM algorithm is anytime, in the sense that once an unsafe input polyhedron is found, the algorithm can terminate without completing the full reachability computation. This is in contrast to all existing exact reachability methods that proceed layer-by-layer through the network.  They must solve the entire backward reachability problem to conclude any verification result, whether safe or unsafe.\n\n\\begin{table}\n\\centering\n\\caption{ACAS Xu Property 3 Verification Results}\n \\begin{tabular}{|c| c| c| c| c| c|} \n \\hline\n Network & Result & \\begin{tabular}{@{}c@{}}Time (s) \\\\ RPM\\end{tabular} & \\begin{tabular}{@{}c@{}}Time (s) \\\\ Face Lattice\\end{tabular} & \\begin{tabular}{@{}c@{}}Time (s) \\\\ Reluplex\\end{tabular} & \\begin{tabular}{@{}c@{}}Time (s) \\\\ Marabou\\end{tabular} \\\\ [0.5ex] \n \\hline\n $N_{1,7}$ & Unsafe & \\textbf{0.01} & 6.66 & 2.15 & 0.70 \\\\ \n \\hline\n $N_{1,8}$ & Unsafe & \\textbf{0.01} & 5.45 & 4.32 & 1.49 \\\\\n \\hline\n $N_{3,8}$ & Safe & 19.72 & \\textbf{9.35} & 231.28 & 40.13 \\\\\n \\hline\n $N_{5,6}$ & Safe & 31.92 & \\textbf{17.28} & 366.39 & 54.07 \\\\\n \\hline\n\\end{tabular}\n\\label{table:verification}\n\\end{table}\n\n\n\\section{Conclusion}\nWe proposed the Reachable Polyhedral Marching \n(RPM) algorithm to efficiently construct the exact PWA representation of a ReLU network. RPM computes an explicit PWA function representation for a given ReLU network.  This PWA function can then be used to quickly find forward and backward reachable sets over multiple time steps. Solving for multiple backward reachable sets can be done simultaneously at little added cost. RPM is shown to be especially fast when searching for the existence of unsafe inputs during verification.  In the future, we will investigate parallel implementations of RPM to further improve computational speed.\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\\bibliographystyle{.\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Model  Hamiltonian}\nThere  is a range of values of the strength of the confinement potential\nfor which spontaneous interlayer phase coherence\nis  stable in double quantum dots\\cite{hu}. Outside this stability range\nthe shape of the groundstate is expected to undergo a  reconstruction\\cite{yang0}.\nIt is the\ncompetition between the confinement potential and the\nCoulomb interaction which drives such a reconstruction.  When the confinement\npotential weakens the repulsive Coulomb interaction\npushes electrons from each other, and can create  density depletions inside the\ndroplet. \n\nThe interplay between the reconstruction and the\nformation of merons in the presence of an external potential is\nunclear.  \nWe investigated within a Hartree-Fock (HF) theory how merons\nnucleate in the groundstate when a lateral distortion with a\nreflection symmetry is present  in the confinement potential\\cite{yang0}.  We found that when a sufficiently large distortion\nis present merons can be nucleated.\nBoth density depletions and merons can coexist in the reconstructed states.\nIn the present work we \ndemonstrate by integrating the electron density that these merons have total charge $\\pm1\/2$.\nIn addition we \npresent  numerical evidence for the presence of edge merons, which\nappear to have a half-integer  winding number.\n\n\nAn electron in each dot is in a parabolic potential $\\frac{1}{2}m^*\\Omega^2r^2$.\nIn the strong magnetic field limit the energy of an  electron with the angular momentum $m$ in a  quantum dot is\n$\\epsilon_m=\\frac{\\hbar\\omega_c}{2}+\\hbar\\omega_p(m+1)$, where\n$\\omega_c$ is the cyclotron frequency, $\\hbar\\omega_p=m^*\n\\Omega^2 \\ell^2$, and $\\ell$ is the magnetic length.\nHereafter we will measure $\\hbar\\omega_p$ in units of\n$e^2\/\\varepsilon\\ell$, using\n$\\gamma=\\hbar\\omega_p\/(e^2\/\\varepsilon\\ell)$ which is a dimensionless quantity proportional to\nthe confinement potential strength  (For GaAs $\\gamma=0.131 (\\hbar\\omega_p[meV])^2\/(B[T])^{3\/2}$). The\nreal spins of all electrons are assumed to be frozen aligning with\nthe magnetic field.\nWe consider \na   distortion in the confinement potential of the dot that may be generated by attaching a\nmetallic stripe vertically on the  cylindrical  metallic gate\nsurrounding the vertical quantum dots. The vertical distance between two dots is $d$. \nLet us make a simple model for this  type of distortions.\nWe take a separable form for the distortion potential\n$U_{\\textrm{dis}}(\\vec{r},z)=v g (\\vec{r}) h (z)$, where $v$ is the strength of the distortion.\nThe matrix elements of this potential are\n\\begin{eqnarray}\n\\left<n_1 \\tau_1|U_{\\mathrm{dis}}|n_2 \\tau_2 \\right> = \n\\left(u_{s} \\delta_{\\tau_1 \\tau_2}\\!+\\! u_{d} (1-\\delta_{\\tau_1\n\\tau_2})\\right)<n_1|g (\\vec{r})|n_2>,\n\\end{eqnarray}\nwhere  $\\left|n_j \\tau_j \\right>=c_{n_j \\tau_j}^{\\dagger} \\left|0\\right>$ denotes an electron with the \nangular momentum $n_j$ in the dot labeled by $\\tau_j$.\nThe parameters $u_{s}$ and $u_{d}$ are defined as \n\\begin{eqnarray}\nu_{s} &=& v \\int \\mathrm{d}z h(z) \\left|\n\\varphi\\left(z-\\frac{d}{2}\\right) \\right|^2 \\; ,\\\\\nu_{d} &=& v \\int \\mathrm{d}z h(z) \\varphi^*\\left(z-\\frac{d}{2}\\right) \\varphi \\left(z+\\frac{d}{2}\\right).\n\\end{eqnarray}\nThe parameter $u_{s}$ ($u_{d}$) denotes the strength  of the matrix elements of the disorder\npotential between intra (inter) layer electrons.\nThe wavefunctions\n$\\varphi \\left(z+\\frac{d}{2}\\right)$  and $\\varphi\\left(z-\\frac{d}{2}\\right)$ are, respectively,\nthe left and right dot wavefunctions along the vertical direction.\n\n\nThe total Hamiltonian is $H = H_0 +V$, where\n\\begin{eqnarray}\nH_0 \\!&=&\\! \\sum_{m\\sigma} \\epsilon_m c_{m \\sigma}^\\dagger c_{m\n\\sigma} +\\sum_{m\\sigma}  \\sum_{m'\\sigma'}  \\left< m \\sigma|\nU_{\\mathrm{dis}}|m' \\sigma'\\right> c_{m \\sigma}^\\dagger c_{m'\n\\sigma'},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nV &=& \\frac{1}{2}\\sum_{m_1 m_2 m_3 m_4} \\sum_{\\sigma=\\uparrow \\downarrow} \n\\left( \\left< m_1 m_2 |V_\\mathrm{s}| m_3 m_4 \\right>\nc_{m_1 \\sigma}^\\dagger c_{m_2 \\sigma}^\\dagger c_{m_3 \\sigma} c_{m_4 \\sigma} \\right. \\nonumber \\\\\n&& \\left. + \\left< m_1 m_2 |V_\\mathrm {d} | m_3 m_4 \\right>\nc_{m_1 \\sigma}^\\dagger c_{m_2,-\\sigma}^\\dagger c_{m_3,-\\sigma}\nc_{m_4 \\sigma} \\right).\n\\end{eqnarray}\nThe first term of $H_0$ represents the confining energy due to the\nparabolic potential. In the second term, $U_{\\mathrm{dis}}$ is the\ndistortion potential. In the interaction\nHamiltonian $V$, the term $V_s$ ($V_d$) represents the Coulomb interaction\npotential between the electrons in the same (different) dots. In\nthe spatial coordinates, those interactions are given by\n$V_s(\\vec{r})=e^2\/\\varepsilon r$ and\n$V_d(\\vec{r})=e^2\/\\varepsilon|\\vec{r}+d\\hat{z}|$, respectively,\nwhere $\\varepsilon$ is the dielectric constant.\nThe spontaneous interlayer phase coherent state (maximum density droplet state\\cite{yang1,yang2}),\nand meron excitations  are  well approximated by a HF theory\\cite{moon}.\nThe pseudospin density is computed with the calculated  density\nmatrix.\n\n\n\n\n\n\n\n\\section{Results}\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width= 0.5 \\textwidth]{density3d.g_0.085.vi_0.50b.eps}\n\\includegraphics[width= 0.5 \\textwidth]{density3d.g_0.085.vi_0.60b.eps}\n\\caption{Electron densities in units of $1\/\\ell^2$ for  $(u_{s},u_{d})=(1.5,0.25)(e^2\/\\varepsilon\\ell)$ and\n$(u_{s},u_{d})=(1.8,0.3)(e^2\/\\varepsilon\\ell)$.}\n\\label{fig:density3d.g_0.085.vi_0.50}\n\\end{center}\n\\end{figure}\n\nFig. \\ref{fig:density3d.g_0.085.vi_0.50}\n displays total electron densities\nfor two different strengths of the lateral distortion with the\ncenter at the coordinate $(-5,5)\\ell$ by  a Gaussian lateral potential with the range $a=3\\ell$.\nResults for $\\gamma=0.085$ are shown for two different strengths\nof the distortion\n$(u_{s},u_{d})=(1.5,0.25),(1.8,0.3)(e^2\/\\varepsilon\\ell)$.\nThe maximum\ndensity is bigger than  $1\/2\\pi\\ell^2$, which implies that\nthe total local filling factor is bigger than one where the merons are located.  This is consistent\nwith the presence of merons with negative charges on the background of the total electron\ndensity $1\/2\\pi\\ell^2$.\nAt  $(u_{s},u_{d})=(1.5,0.25)(e^2\/\\varepsilon\\ell)$ a meron is formed\nwith the charge $-e\/2$ and vorticity $-$.\nFor the bigger value $(u_{s},u_{d})=(1.8,0.3)(e^2\/\\varepsilon\\ell)$\ntwo merons appear and the\nvorticities for the lower and upper merons are $-$ and $+$,\nrespectively. \nIntegrating the electron density  over the range that satisfy $n(r)>\\frac{1}{2 \\pi\\ell^2}$,\nwe  find the excess charges\nto be  \n$-0.54 e$ and $-0.98 e$,\nwhich indicates that one meron exists in the upper figure of Fig. \\ref{fig:density3d.g_0.085.vi_0.50}\nand a meron pair exists in the lower figure of Fig. \\ref{fig:density3d.g_0.085.vi_0.50}.\nWe have attempted to evaluate the total excess charge carried by the merons using the expression\\cite{moon} \n\\begin{eqnarray}\n\\Delta N = -\\frac{1}{8 \\pi} \\int {\\mathrm d}^2 \\bf{r} \\\n\\epsilon_{\\mu \\nu} \\bf{m}(\\bf{r}) \\cdot [\\partial_\\mu \\bf{m}(\\bf{r}) \\times \\partial_\\nu \\bf{m}(\\bf{r})],\n\\end{eqnarray}\nwhere the polarization vector $\\vec{m}(r)$ is  $\\vec{\\tau}(r)\/n(r)$ ($\\vec{\\tau}(r)=2\\vec{s}\/\\hbar$ and $\\vec{s}$ is the pseudospin).\nHowever, this method gave inaccurate results.   We believe this is because the expression is valid\nonly for slowly varying pseudospin fields.\n\n\n\\begin{figure}[!htp]\n\\begin{center}\n\\includegraphics[width=1.0  \\textwidth]{meronpair_one_imp.g_0.066.vi_0.08.hmf1.eps}\n\\caption{ Arrows are projections of  $\\vec{\\tau}$ on the $xy$ plane\nand the gray scale indicates the $z$-component of $\\vec{\\tau}$.\nThe dashed circle is the edge of the electron droplet and $+$ and $-$\nindicate the core positions of merons with positive and negative winding numbers.\nThe cross sign is where the impurity is located.\nThe black dot represents the center of an edge meron, shown in the inset, and\nthe small circles show electron density depletions.\n}\n\\label{fig:meronpair_one_imp.g_0.066.vi_0.08}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!htp]\n\\begin{center}\n\\includegraphics[width=0.6  \\textwidth]{halfmeron.one.imp.g_0.066.vi_0.08.a.eps}\n\\caption{\nMagnified view of the spin texture of an edge meron with nearly a half-integer winding number.}\n\\label{fig:lfmeron.one.imp.g_0.066.vi_0.08}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nFig.~\\ref{fig:meronpair_one_imp.g_0.066.vi_0.08} displays how the\npseudospin texture \nfor $\\gamma=0.066$ and $(u_{s},u_{d})=(0.24, 0.04)(e^2\/\\varepsilon\\ell)$.\nWe  observe merons inside the dots.\nThe values of the polarization $m_z(r)=\\tau_z(r)\/n(r)$ at the meron cores are, within numerical accuracy, $+1$ and $-1$.\nThis implies that the\ncharges should be $-e\/2$ since the vorticities of these two objects are opposite.  \nThe total charge of the merons is thus $-e$.\nNote that there are also five density depletions inside the dots (the open circles in the figure).  \nThese  modulations of electron densities are not simply\nrelated to the shape of\n$U_{\\mathrm{para}}(r)+U_{\\mathrm{dis}}(r)$. Since the densities are\ninhomogeneous the HF single particle potential must be included into\nconsideration. It is non-trivial to anticipate the shape of the HF\nsingle particle potential since it reflects the delicate interplay between the\nHartree self energy, exchange self energy, and confinement potential\\cite{yang3}.\n\nWe also observe  an  edge meron near the edge in Fig.~\\ref{fig:meronpair_one_imp.g_0.066.vi_0.08}.\nA magnified view of the pseudospin texture is shown in Fig.~\\ref{fig:lfmeron.one.imp.g_0.066.vi_0.08}. \nWe   draw a circle around the center of this object and see how the in-plane \npseudospins rotate along the circle.\nThe in-plane\npseudospins nearly flips as   the half circle is followed inside the dot.\nThe magnitude of in-plane pseudospin values decreases to zero as the edge region is passed.\nWe believe thus that the edge meron  has approximately a half integer winding number.\nIt is complicated to calculate accurately the excess charge of the edge meron since\nthe electron density varies strongly near the edge even in the absence of an edge meron.\nIt would be worthwhile to develop an analytical theory for edge merons.\n\n\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Geometry}\n     \\subsection{Arrangement}\n        \\subsubsection{Enumeration of all cells in arrangements}\n            \\begin{description}\n                \\item[Input] An arrangement of distinct hyperplanes.\n                \\item[Output] All cells in arrangements.\n                \\item[Complexity] $O(nm\\ell(n,m)N)$ total time and $O(nm)$ space.\n                \\item[Comment] $n$ is the dimension, $m$ is the number of hyperplanes, and $\\ell(n, m)$ is the time for solving an LP with $n$ variables and $m-1$ inequalities.  $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n     \\subsection{Face}\n        \\subsubsection{Enumeration of all arrangements}\n            \\begin{description}\n                \\item[Input] An integer $n$ and $\\mathcal{J} \\subseteq \\{+, -\\}^n$.\n                \\item[Output] The set $\\mathcal{F}$ of faces if $\\mathcal{J}$ is the set of maximal faces of an oriented matroid\n                \\item[Complexity] $O(\\max(n^2 |\\mathcal{J}|^3, n^3 |\\mathcal{J}|^2))$ total time.\n                \\item[Comment] Their paper treats a reconstruction of the location vectors of all faces from the graph.\n                \\item[Reference] \\cite{Fukuda1991}\n            \\end{description}\n     \\subsection{Matching}\n        \\subsubsection{Enumeration of all non-crossing perfect matchings (and convex partitions) in a given points}\n            \\begin{description}\n                \\item[Input] A point sets $P$ in a plane\n                \\item[Output] All non-crossing perfect matchings (and convex partitions) in $P$. \n                \\item[Complexity] After $O(2^nn^4)$ preprocessing time, polynomial delay. \n                \\item[Comment] $n$ is the number of points in $P$\n                \\item[Reference] \\cite{Wettstein2014}\n            \\end{description}\n     \\subsection{Nearest neighbor}\n        \\subsubsection{Enumeration of the $k$ smallest distances between pairs of points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane.\n                \\item[Output] The $k$ smallest distances between pairs of points in non-decreasing order.\n                \\item[Complexity] $O(n \\log n + k \\log k)$ total time and $O(n + k)$ space.\n                \\item[Reference] \\cite{Dickerson1992}\n            \\end{description}\n     \\subsection{Spanning tree}\n        \\subsubsection{Enumeration of the Euclidean $k$ best spanning trees in a plane with $n$ points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane\n                \\item[Output] The Euclidean $k$ best spanning trees of the given points.\n                \\item[Complexity] $O(k^2 n + n \\log n)$ total time\n                \\item[Comment] An Euclidean spanning tree in a plane is a spanning tree of the complete graph whose vertices are all given points and the weight of an edge equal to the Euclidean distance between the corresponding vertices.\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Enumeration of the orthogonal $k$ best spanning trees in a plane with $n$ points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane\n                \\item[Output] The orthogonal $k$ best spanning trees of the given points.\n                \\item[Complexity] $O(k^2 + k n \\log \\log (n\/k) + n \\log n)$ total time.\n                \\item[Comment] An orthogonal spanning tree in a plane is a spanning tree of the complete graph whose vertices are all given points and the weight of an edge equal to the $L_1$ distance between the corresponding vertices.\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees of a plane}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All spanning trees of $P$.\n                \\item[Complexity] $O(|P|^3N)$ total time and $O(|P|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of the Euclidean $k$ best spanning trees in a plane with $n$ points}\n            \\begin{description}\n                \\item[Input] $n$ points in a plane\n                \\item[Output] The Euclidean $k$ best spanning trees of the given points.\n                \\item[Complexity] $O(n \\log n \\log k + k \\min(k, n)^{1\/2})$ total time\n                \\item[Comment] An Euclidean spanning tree in a plane is a spanning tree of the complete graph whose vertices are all given points and the weight of an edge equal to the Euclidean distance between the corresponding vertices.\n                \\item[Reference] \\cite{Eppstein1997}\n            \\end{description}\n     \\subsection{Triangulation}\n        \\subsubsection{Enumeration of all regular triangulations}\n            \\begin{description}\n                \\item[Input] $n$ points.\n                \\item[Output] All regular triangulations of given points in $(d-1)$-dimensional space.\n                \\item[Complexity] $O(dsLP(r, ds)T)$ time and $O(ds)$ space.\n                \\item[Comment] $s$ is the upper bound of the number of simplcies contained in one regular triangulation, $LP(r, ds)$ denotes the time complexity of solving linear programming problem with $ds$ strict inequality constraints in $r$ variables, and $T$ is the number of regular triangulations.\n                \\item[Reference] \\cite{Masada1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning regular triangulations}\n            \\begin{description}\n                \\item[Input] $n$ points.\n                \\item[Output] All spanning regular triangulations of given points in $(d-1)$-dimensional space.\n                \\item[Complexity] $O(dsLP(r, ds)T')$ time and $O(ds)$ space.\n                \\item[Comment] $s$ is the upper bound of the number of simplcies contained in one regular triangulation, $LP(r, ds)$ denotes the time complexity of solving linear programming problem with $ds$ strict inequality constraints in $r$ variables, and $T'$ is the number of regular triangulations.\n                \\item[Reference] \\cite{Masada1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a point set}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All triangulations of $P$.\n                \\item[Complexity] $O(|P|N)$ total time and $O(|P|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations in general dimensions}\n            \\begin{description}\n                \\item[Input] Points.\n                \\item[Output] All triangulations.\n                \\item[Complexity] See the paper.\n                \\item[Comment] This algorithm uses the enumeration algorithm for maximal independent sets.\n                \\item[Reference] \\cite{Takeuchi}\n            \\end{description}\n        \\subsubsection{Enumeration of all regular triangulations in general dimensions}\n            \\begin{description}\n                \\item[Input] Points.\n                \\item[Output] All regular triangulations.\n                \\item[Complexity] See the paper.\n                \\item[Comment] This algorithm uses the enumeration algorithm for maximal independent sets.\n                \\item[Reference] \\cite{Takeuchi}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulation paths of a point set}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All triangulation paths of $P$.\n                \\item[Complexity] $O(N|P|^3\\log |P|)$ time and $O(|P|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Dumitrescu2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all pseudotriangulations of a finite point set}\n            \\begin{description}\n                \\item[Input] A point set $S$ of size $n$.\n                \\item[Output] All pointed pseudotriangulations of $S$.\n                \\item[Complexity] $O(\\log n)$ time per solution with linear space.\n                \\item[Reference] \\cite{Bereg2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all pseudotriangulations of a point set}\n            \\begin{description}\n                \\item[Input] A point set $P$.\n                \\item[Output] All pseudotriangulations of $P$.\n                \\item[Complexity] $O(n)$ time per pseudotriangulation.\n                \\item[Comment] $n$ is the number of points in $P$.\n                \\item[Reference] \\cite{Bronnimann2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a convex polygon of $n$ vertices with numbered}\n            \\begin{description}\n                \\item[Input] A convex polygon $P$ with $n$ vertices that are numbered.\n                \\item[Output] All triangulations of $P$.\n                \\item[Complexity] $O(1)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Parvez2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a convex polygon of $n$ vertices}\n            \\begin{description}\n                \\item[Input] A convex polygon $P$ with $n$ vertices that are not numbered.\n                \\item[Output] All triangulations of $P$.\n                \\item[Complexity] $O(n^2)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Parvez2009}\n            \\end{description}\n\\section{Graph}\n     \\subsection{$k$-degenerate graph}\n        \\subsubsection{Enumerate well-ordered (strongly) $k$-generate with $n$ vertices}\n            \\begin{description}\n                \\item[Input] $n$: the number of vertices.\n                \\item[Output] All well-ordered (strongly) $k$-degenerate graphs with $n$ vertices.\n                \\item[Complexity] $O(nm+m^2)$ time per enumerated and printed graph.\n                \\item[Comment] $m$ is the number of edges of printed graphs.\n                \\item[Reference] \\cite{Bauer2010a}\n            \\end{description}\n        \\subsubsection{Enumerate well-ordered (strongly) $k$-generate with $n$ vertices and $m$ edges}\n            \\begin{description}\n                \\item[Input] $n$: the number of vertices, $m$: the number of edges.\n                \\item[Output] All well-ordered (strongly) $k$-degenerate graphs with $n$ vertices and $m$ edges.\n                \\item[Complexity] $O(n^{3\/2}m^2)$ time per enumerated and printed graph.\n                \\item[Reference] \\cite{Bauer2010a}\n            \\end{description}\n     \\subsection{Bounded union}\n        \\subsubsection{Enumeration of all bounded union}\n            \\begin{description}\n                \\item[Input] A graph with $h$ multiedges each with at most $d$ vertices.\n                \\item[Output] All minimal subset $U$ of at most $k$ vertices that entirely includes at least one set from each multiedge.\n                \\item[Complexity] $O(dc^{k+1}h + \\min(kc^{2k}, hkc^k))$ time.\n                \\item[Reference] \\cite{Damaschke2006}\n            \\end{description}\n     \\subsection{Bridge}\n        \\subsubsection{Enumeration of all bridge-avoiding extensions of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an edge subset $B \\subseteq E$.\n                \\item[Output] All bridge-avoiding extensions of $G$.\n                \\item[Complexity] $O(K^2 \\log(K) |E|^2 + K^2(|V|+|E|)|E|^2)$ total time.\n                \\item[Comment] $K$ is the number of bridge-avoiding extensions.  $X$ is a bridge-avoiding extensions of $G$ if $X$ is a minimal edge set $X \\subseteq E \\setminus B$ such that no edge $b \\in B$ is a bridge in $(V, B\\cup X)$.\n                \\item[Reference] \\cite{Khachiyan2008a}\n            \\end{description}\n     \\subsection{Bubble}\n        \\subsubsection{Enumeration of all bubbles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V, E)$ and a source $s$.\n                \\item[Output] All bubbles with a given source $s$.\n                \\item[Complexity] $O(|V| + |E|)$ delay with $O(|V|(|V| + |E|))$ preprocessing time.\n                \\item[Comment] An $(s,t)$-bubble is two disjoint $(s,t)$-paths that only share $s$ and $t$.\n                \\item[Reference] \\cite{Birmele2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t, \\alpha_1, \\alpha_2)$-bubble in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$, source vertex $s$, and two upper bounds $\\alpha_1$ and $\\alpha_2$.\n                \\item[Output] All $(s, t, \\alpha_1, \\alpha_2)$-bubble in $G$.\n                \\item[Complexity] $O(|V|(|E|+|V|\\log |V|))$ delay.\n                \\item[Comment] A pair of two vertex-disjoint $(s, t)$-paths $p_1$ and $p_2$ is $(s, t, \\alpha_1, \\alpha_2)$-bubble in $G$, if $|p_1| \\le \\alpha_1$ and $|p_2| \\le \\alpha_2$.\n                \\item[Reference] \\cite{Sacomoto2013b}\n            \\end{description}\n     \\subsection{Chord diagram}\n        \\subsubsection{Enumeration of all non-isomorphic chord diagrams}\n            \\begin{description}\n                \\item[Input] An integer $2n$.\n                \\item[Output] All non-isomorphic chord diagrams with $2n$ points.\n                \\item[Comment] A \\textit{chord diagram} is a set of $2n$ points on an oriented circle (counterclockwise) joined pairwise by $n$ chords.  In an experiment, their algorithm runs in almost CAT, but there is no Mathematical proof.\n                \\item[Reference] \\cite{Sawada2002a}\n            \\end{description}\n     \\subsection{Chordal graph}\n        \\subsubsection{Enumeration of all chordal graphs in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All chordal graphs in $G$.\n                \\item[Complexity] $O(1)$ delay with $O(n^3)$ space.\n                \\item[Comment] Using reverse search.\n                \\item[Reference] \\cite{Kiyomi2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordal supergraph that contains a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All chordal supergraphs each of which contain $G$.\n                \\item[Complexity] $O(|V|^3)$ time for each and $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Kiyomi2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordal sandwiches in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All chordal sandwiches in $G$.\n                \\item[Complexity] Polynomial delay with polynomial space.\n                \\item[Reference] \\cite{Kijima2010}\n            \\end{description}\n     \\subsection{Clique}\n        \\subsubsection{Enumeration of all cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cliques in $G$. \n                \\item[Comment] They pointed out Augustson and Minker's algorithm has two errors. \n                \\item[Reference] \\cite{Mulligan1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal clique in a graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G=(V, E)$.\n                \\item[Output] All maximal clique.\n                \\item[Reference] \\cite{Akkoyunlu1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All cliques in $G$.\n                \\item[Reference] \\cite{Bron1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G$. \n                \\item[Output] All cliques in $G$. \n                \\item[Reference] \\cite{Johnston1976}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Gerhards1979}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum cliques in a circle graph}\n            \\begin{description}\n                \\item[Input] A circle graph $G = (V, E)$.\n                \\item[Output] All maximum cliques in $G$.\n                \\item[Complexity] $O(1)$ time per maximum clique.\n                \\item[Comment] A graph is a circle graph if it is the intersection graph of chords in a circle. The definition of a circle graph can be found in Information System on Graph Classes and their Inclusions\\footnote{\\url{http:\/\/www.graphclasses.org\/classes\/gc\\_132.html}}.\n                \\item[Reference] \\cite{Rotem1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a connected graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$ and an order $l \\ge 2$ of cliques.\n                \\item[Output] All cliques in $G$.\n                \\item[Complexity] $O(l\\alpha(G)^{l-2}|E|)$ total time and linear space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a connected graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Complexity] $O(\\alpha(G)|E|)$ total time and linear space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum cliques of a circular-arc graph}\n            \\begin{description}\n                \\item[Input] A circular-arc graph $G =(V, E)$.\n                \\item[Output] All maximum cliques of $G$.\n                \\item[Complexity] $O(|V|^{3.5} + N)$.\n                \\item[Comment] $N$ is the number of maximum cliques of $G$.  For a family $A$ of arcs on a circle, a graph $G = (V, E)$ is called the \\textit{circular-arc graph} for $A$ if there exists a one-to-one correspondence between $V$ and $A$ such that two vertices in $V$ are adjacent if and only if their corresponding arcs in $A$ intersect.\n                \\item[Reference] \\cite{Kashiwabara1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All maximal cliques included in $G$.\n                \\item[Complexity] $O(M(n))$ time delay and $O(n^2)$ space, or $O(\\delta^4)$ time delay and $O(n+m)$ space, after $O(nm)$ preprocessing.\n                \\item[Comment] $M(n)$: the time needed to multiply two $n \\times n$ matrices, $\\delta$: maximum degree of $G=(V,E)$, $n$: total number of vertices, and $m$: total number of edges.\n                \\item[Reference] \\cite{Makino2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G=(V,E)$.\n                \\item[Output] All maximal bicliques included in $G$.\n                \\item[Complexity] $O(M(n))$ time delay and $O(n^2)$ space, or $O(\\delta^4)$ time delay and $O(n+m)$ space, after $O(nm)$ preprocessing.\n                \\item[Comment] $M(n)$: the time needed to multiply two $n \\times n$ matrices, $\\delta$: maximum degree of $G=(V,E)$, $n$: total number of vertices, and $m$: total number of edges.\n                \\item[Reference] \\cite{Makino2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a dynamic graph}\n            \\begin{description}\n                \\item[Input] A graph $G_t = (V, E_t)$.\n                \\item[Output] All maximal cliques in $G_t$.\n                \\item[Reference] \\cite{Stix2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a dynamic graph}\n            \\begin{description}\n                \\item[Input] A dynamic graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Stix2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all bicliques in a graph in lexicographical order}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bicliques in $G$.\n                \\item[Complexity] $O(|V|^3)$ delay and $O(2^{|V|})$ space.\n                \\item[Comment] There is no polynomial-delay enumeration algorithm for all bicliques in reverse lexicographical order unless $P = NP$.\n                \\item[Reference] \\cite{Dias2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E )$.\n                \\item[Output] All maximal Cliques in $G$.\n                \\item[Complexity] $O(\\frac{\\Delta M_c Tri^2}{P})$ delay and $O(|V| + |E|)$ space.\n                \\item[Comment] $\\Delta$ is the maximum degree in $G$, $M_c$ is the size of the maximum clique in $G$, and $Tri$ is the number of triangles in $G$. This algorithm runs on the computer with $P$ processors.\n                \\item[Reference] \\cite{Du2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G$.\n                \\item[Output] All cliques in $G$.\n                \\item[Complexity] $O(1)$ time per solution on average.\n                \\item[Reference] \\cite{Kiyomi2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Complexity] $O(3^{n\/3})$ total time.\n                \\item[Reference] \\cite{Tomita2006}\n            \\end{description}\n        \\subsubsection{Enumeration of $K$-largest maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] $K$-largest maximal cliques in $G$.\n                \\item[Reference] \\cite{Bulo2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Pan2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal bicliques in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (U, V, E)$.\n                \\item[Output] All maximal bicliques in $G$ in lexicographical order on $U$.\n                \\item[Complexity] $O((|U| + |V|)^2)$ delay with exponential space.\n                \\item[Reference] \\cite{Gely2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a comparability graph}\n            \\begin{description}\n                \\item[Input] A comparability graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$ in lexicographical order.\n                \\item[Complexity] $O(|V|)$ delay and $O(|V| + |E|)$ space.\n                \\item[Reference] \\cite{Gely2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximal cliques in $G$ in lexicographical order.\n                \\item[Complexity] $O(|V||E|)$ delay with exponential space.\n                \\item[Reference] \\cite{Gely2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Comment] This algorithm can be paralleled.\n                \\item[Reference] \\cite{Schmidt2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-isolated maximal clique in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a positive real number $c$.\n                \\item[Output] All $c$-isolated maximal clique in $G$.\n                \\item[Complexity] $O(c^42^{2c}|E|)$ total time.\n                \\item[Reference] \\cite{Ito2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-isolated pseudo-clique in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a positive real number $c$.\n                \\item[Output] All $c$-isolated $PC(k-\\log k, \\frac{k}{\\log k})$ in $G$.\n                \\item[Complexity] $O(c^42^{2c}|E|)$ total time.\n                \\item[Comment] $PC(\\alpha, \\beta)$ is a induced subgraph of $G$ with an average degree at least $\\alpha$ and the minimum degree at least $\\beta$.\n                \\item[Reference] \\cite{Ito2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-isolated maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $c$.\n                \\item[Output] All avg-$c$-isolated maximal cliques in $G$.\n                \\item[Complexity] $O(2^cc^5|E|)$ total time.\n                \\item[Comment] A vertex set $S \\subseteq V$ with $k$ vertices is $c$-isolated if it has less than $ck$ outgoing edges, where an outgoing edge is an edge between a vertex in $S$ and a vertex in $V \\setminus S$.\n                \\item[Reference] \\cite{Huffner2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all max-$c$-isolated maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $c$.\n                \\item[Output] All max-$c$-isolated maximal cliques in $G$.\n                \\item[Complexity] $O(2^cc^5|E|)$ total time.\n                \\item[Comment] A vertex set $S \\subseteq V$ with $k$ vertices is max-$c$-isolated if every vertex in $S$ has less than $c$ neighbors in $V \\setminus S$.\n                \\item[Reference] \\cite{Huffner2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal $c$-isolated cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $c$.\n                \\item[Output] All maximal $c$-isolated cliques in $G$.\n                \\item[Complexity] $O(2.89^cc^2|E|)$ total time.\n                \\item[Comment] A vertex set $S \\subseteq V$ with $k$ vertices is $c$-isolated if it has less than $ck$ outgoing edges, where an outgoing edge is an edge between a vertex in $S$ and a vertex in $V \\setminus S$.\n                \\item[Reference] \\cite{Komusiewicz2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all cliques in a graph with degeneracy $d$}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ with degeneracy $d$.\n                \\item[Output] All cliques in $G$.\n                \\item[Complexity] $O(d|V|3^{d\/3})$ time.\n                \\item[Comment] They also show the largest possible number of maximal cliques in $G$. The number is $(|V|-d)3^{d\/3}$.\n                \\item[Reference] \\cite{Eppstein2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all pseudo-cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a threshold $\\theta$.\n                \\item[Output] All pseudo-cliques in $G$, whose densities are no less than $\\theta$.\n                \\item[Complexity] $O(\\Delta |V| + \\mathrm{min}\\{\\Delta^2, |V| + |E|\\})$ delay with $O(|V| + |E|)$ space.\n                \\item[Comment] $\\Delta$ is the maximum degree of $G$. The density of a subgraph $G[S]$ induced by $S \\subseteq V$ is given by the number of edges over the number of all its vertex pairs.\n                \\item[Reference] \\cite{Uno2008a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph with limited memory}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal graphs in $G$.\n                \\item[Complexity] See the paper.\n                \\item[Reference] \\cite{Cheng2012b}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G=(V, E)$.\n                \\item[Output] All maximal clique.\n                \\item[Complexity] $O(\\delta\\cdot H^3)$ time delay and $O(n+m)$ space.\n                \\item[Comment] $H$ satisfies $|\\{v\\in V | \\sigma(v)\\ge H\\}| \\le H$.\n                \\item[Reference] \\cite{Chang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal cliques in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal cliques in $G$.\n                \\item[Reference] \\cite{Henry2013}\n            \\end{description}\n     \\subsection{Coloring}\n        \\subsubsection{Enumeration of all the edge colorings in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (V, E)$.\n                \\item[Output] All edge colorings in $G$.\n                \\item[Complexity] $O(\\Delta|E|)$ time per solution and space.\n                \\item[Comment] $\\Delta$ is the maximum degree in $G$.\n                \\item[Reference] \\cite{Yasuko1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all the edge colorings of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B = (V, E)$.\n                \\item[Output] All the edge colorings pf $B$.\n                \\item[Complexity] $O(T(|V|+|E|+\\Delta) + K \\min \\{|V|^2 + |E|, T(|V|, |E|, \\Delta)\\})$ total time and $O(|E|\\Delta)$ space.\n                \\item[Comment] $\\Delta$ is the number of maximum degree and $T(|V|, |E|, \\Delta)$ is the time complexity of an edge coloring algorithm.\n                \\item[Reference] \\cite{Matsui1996b}\n            \\end{description}\n        \\subsubsection{Enumeration of all the edge colorings of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B = (V, E)$.\n                \\item[Output] All the edge colorings pf $B$.\n                \\item[Complexity] $O(|V|)$ time per solution and $O(|E|)$ space.\n                \\item[Reference] \\cite{Matsui1996a}\n            \\end{description}\n     \\subsection{Connected}\n        \\subsubsection{Enumeration of all minimal strongly connected subgraphs in a strongly connected subgraph}\n            \\begin{description}\n                \\item[Input] A strongly connected subgraph $G$.\n                \\item[Output] All minimal strongly connected subgraph.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Reference] \\cite{Boros2004b}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal 2-vertex connected spanning subgraphs in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All minimal 2-vertex connected spanning subgraphs of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Reference] \\cite{Khachiyan2006a}\n            \\end{description}\n     \\subsection{Cut}\n        \\subsubsection{Enumeration of all cuts between all pair of vertices in a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cuts bewteen all pair of vertices in $G$\n                \\item[Reference] \\cite{Martelli1976a}\n            \\end{description}\n        \\subsubsection{Enumeration of all cutsets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All cutsets in $G$.\n                \\item[Complexity] $O((|V| + |E|)(\\mu + 1))$ total time and $O(|V| + |E|)$ or $O(|V|^2)$ space.\n                \\item[Comment] $\\mu$ is the number of solutions.\n                \\item[Reference] \\cite{Tsukiyama1980}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ best cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] $k$ best cuts in $G$.\n                \\item[Complexity] $O(k|V|^4)$ total time.\n                \\item[Reference] \\cite{Hamacher1984a}\n            \\end{description}\n        \\subsubsection{Enumeration of $K$-best cuts in a network}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $K$-best cuts in $G$.\n                \\item[Complexity] $O(K\\cdot |V|^4)$ time.\n                \\item[Reference] \\cite{Hamacher1984a}\n            \\end{description}\n        \\subsubsection{Enumeration of all articulation pairs in a planar graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G = (V, E)$.\n                \\item[Output] All articulation pairs in $G$.\n                \\item[Complexity] $O(|V|^2)$ total time.\n                \\item[Reference] \\cite{Laumond1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimum-size separating vertex sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimum-size separating vertex sets.\n                \\item[Complexity] $\\Theta(M|V| + C) = O(2^kn^3)$ total time.\n                \\item[Comment] $M$ is the number of solutions, $k$ is the connectivity of $G$, and $C = k|V| \\min(k(|V|+|E|), A)$, where $A$ is the time complexity of the best maximum network flow algorithm for unit network.\n                \\item[Reference] \\cite{Kanevsky1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separators in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal separators in $G$.\n                \\item[Complexity] $O(|V|^6R)$ total time.\n                \\item[Comment] $R$ is the number of solutions.\n                \\item[Reference] \\cite{Kloks1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All $(s, t)$-cuts in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All $(s, t)$-cuts in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-uniformly directed cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All $(s, t)$-uniformly directed cuts in $G$.\n                \\item[Complexity] $O(|V|)$ time per cut.\n                \\item[Comment] An \\textit{undirected directed cut} is also called a UDC. An $(s, t)$-DUC is an $(s, t)$-cut $(X, Y)$ such that $(X, Y) = \\emptyset$, where $(X, Y) = \\{(u, v) \\in E: u\\in X, v\\in Y\\}$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimum weighted $(s, t)$ cuts in an weighted graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All minimum weighted $(s, t)$ cuts in $G$.\n                \\item[Complexity] $O(|V|)$ time per cut.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all semidirected $(s, t)$ cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and two vertices $s, t$ in $G$.\n                \\item[Output] All semidirected $(s, t)$ cuts in $G$.\n                \\item[Complexity] $O(|V||E|)$ time per cut.\n                \\item[Comment] For a subset $D$ of directed edges, a \\textit{semidirected $(s, t)$ cut} with respect to $D$ is an $(s, t)$ cut $(X, Y)$ such that $(X, Y) \\cup (Y, X)$ defnes an undirected $(s, t)$ cutset and such that $(X, Y) \\cap D = \\emptyset$, where a \\textit{cutset} is an minimal cut set.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all strong $(s, K)$ cutsets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All strong $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{strong $(s, K)$ cutset} is minimal cuts of the form $(X, Y)$ where $s \\in X$ and $K \\subseteq Y$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all strong $(s, K)$ cutsets in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All strong $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{strong $(s, K)$ cutset} is minimal cuts of the form $(X, Y)$ where $s \\in X$ and $K \\subseteq Y$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all quasi $(s, K)$ cuts in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All quasi $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{Quasi $(s, K)$ cut} is an edge set that is strong $(s, A)$-cutsets for some $A \\subseteq K$ and $A \\neq \\emptyset$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all quasi $(s, K)$ cuts in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All quasi $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{Quasi $(s, K)$ cut} is an edge set that is strong $(s, A)$-cutsets for some $A \\subseteq K$ and $A \\neq \\emptyset$.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, K)$ cutsets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ , $s\\in V$ and $K \\subseteq V \\setminus \\{s\\}$.\n                \\item[Output] All $(s, K)$ cutsets in $G$.\n                \\item[Complexity] $O(|E|)$ time per cut.\n                \\item[Comment] An \\textit{$(s, K)$-cut} is defined to be any cut $(X, Y)$ for which $s \\in X$ and $K \\cap Y \\neq \\emptyset$. A \\textit{$(s, K)$ cutset} is minimal $(s, K)$ cuts.\n                \\item[Reference] \\cite{Provan1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal $a$-$b$ separators in a graph}\n            \\begin{description}\n                \\item[Input] An undirected connected simple graph $G = (V, E)$, non-adjacent vertices $a$ and $b$ in $G$.\n                \\item[Output] All minimal $a$-$b$ separator of $G$.\n                \\item[Complexity] $O(R_{ab}|V|^3)$ total time.\n                \\item[Comment] $R_{ab}$ is the number of minimal $a$-$b$ separators.\n                \\item[Reference] \\cite{Shen1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal $a$-$b$ separators in a graph}\n            \\begin{description}\n                \\item[Input] An undirected connected simple graph $G = (V, E)$, non-adjacent vertices $a$ and $b$ in $G$.\n                \\item[Output] All minimal $a$-$b$ separator of $G$.\n                \\item[Complexity] $O(R_{ab}|V|\/\\log|V|)$ total time.\n                \\item[Comment] $R_{ab}$ is the number of minimal $a$-$b$ separators. This algorithm needs $O(|V|^3)$ processors on a CREW PRAM.\n                \\item[Reference] \\cite{Shen1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separators of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal separators of $G$.\n                \\item[Complexity] $O(|V|^5R)$ total time.\n                \\item[Comment] $R$ is the number of solutions.\n                \\item[Reference] \\cite{Kloks1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separator of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal separator of $G$.\n                \\item[Complexity] $O(|V|^3)$ time per solution.\n                \\item[Reference] \\cite{Berry2000}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separator of a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All minimal separator of $G$.\n                \\item[Complexity] $O(|V|+|E|)$ total time.\n                \\item[Reference] \\cite{Chandran2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all cut conjunctions for a given set of vertex pairs in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V,E)$, and a collection $B = \\{(s_1,t_1),...,(s_k,t_k)\\}$ of $k$ vertex pairs $s_i,t_i \\in V$.\n                \\item[Output] All cut conjunctions for $B$ in $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] A minimal edge sets $X \\subseteq E$ is a \\textit{cut conjunction} if, for all $i = 1, \\dots, k$, vertices $s_i$ and $t_i$ is disconnected in $G' = (V, E\\setminus X)$. A cut conjunction is a generalization of an $s-t$ cut.\n                \\item[Reference] \\cite{Khachiyan2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal separators of a 3-connected planar graph}\n            \\begin{description}\n                \\item[Input] A 3-connected planar graph $G = (V, E)$.\n                \\item[Output] All minimal separators of $G$.\n                \\item[Complexity] $O(|V|)$ time per solution.\n                \\item[Reference] \\cite{Mazoit2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all cut conjunctions of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a collection $B = \\{(s_1, t_1), \\dots, (s_k, t_k)\\}$.\n                \\item[Output] All cut conjunctions of $G$.\n                \\item[Complexity] $O(K^2 \\log(K) |V||E|^2 + K^2|B|(|V|+|E|)|E|^2)$ total time.\n                \\item[Comment] $K$ is the number of cut conjunctions.  $X$ is a cut conjunctions of $G$ if $X$ is a minimal edge set such that for all $i = 1, \\dots, k$, a pair of vertices $s_i$ and $t_i$ is disconnected in $(V, E\\setminus X)$.\n                \\item[Reference] \\cite{Khachiyan2008a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in an weighted directed graph}\n            \\begin{description}\n                \\item[Input] An weighted directed graph $G = (V, E)$.\n                \\item[Output] All cuts in $G$ by non-decreasing weights ordering.\n                \\item[Complexity] $O(|V||E|\\log(|V|^2\/|E|))$ delay.\n                \\item[Reference] \\cite{Yeh2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-cuts in an weighted undirected graph}\n            \\begin{description}\n                \\item[Input] An weighted undirected graph $G = (V, E)$.\n                \\item[Output] All cuts in $G$ by non-decreasing weights ordering.\n                \\item[Complexity] $O(|V||E|\\log(|V|^2\/|E|))$ delay.\n                \\item[Reference] \\cite{Yeh2010}\n            \\end{description}\n     \\subsection{Cycle}\n        \\subsubsection{Enumeration of all cycles in an $n$-cube, where $n \\le 4$}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All cycles (closed paths) in an $n$-cube. \n                \\item[Reference] \\cite{Gilbert1958}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Welch1965}\n            \\end{description}\n        \\subsubsection{Enumeration of all simple cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All simple cycles in $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Ponstein1966}\n            \\end{description}\n        \\subsubsection{Enumeration of all circuits in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All circuits in $G$. \n                \\item[Reference] \\cite{Welch1966}\n            \\end{description}\n        \\subsubsection{Enumeration of all Hamiltonian circuits in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All Hamiltonian circuits in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed circuits in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G$. \n                \\item[Output] All directed circuits in $G$. \n                \\item[Reference] \\cite{Kamae1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Danielson1968}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Rao1969}\n            \\end{description}\n        \\subsubsection{Enumeration of all elementary circuit in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All elementary circuit in $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Tiernan1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Char1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Reference] \\cite{Weinblatt1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a finite undirected graph}\n            \\begin{description}\n                \\item[Input] A finite undirected graph $G$. \n                \\item[Output] All cycles in $G$. \n                \\item[Comment] He claimed that J. T. Welch, Jr.'s algorithm is wrong. \n                \\item[Reference] \\cite{Weinblatt1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V,E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O((|V|\\cdot |E|)(C+1))$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $C$ is the number of cycles included in $G$.\n                \\item[Reference] \\cite{Tarjan1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$. \n                \\item[Output] All cycles in $G$. \n                \\item[Complexity] $O(|E| + c(|V| \\times |E|))$ total, where $c$ is the number of circuits in $G$. \n                \\item[Reference] \\cite{Ehrenfeucht1973}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V, E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O((|V|+ |E|)(C+1))$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $C$ is the number of cycles included in $G$.\n                \\item[Reference] \\cite{Johnson1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|E|)$ time per cycle with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|E|)$ time per cycle with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G=(V,E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|V|(C+1))$ total time and $O(|V|)$ space.\n                \\item[Comment] $C$ is the number of cycles included in $G$.\n                \\item[Reference] \\cite{Syso1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles of a given length in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$ and an integer $k$.\n                \\item[Output] All cycles of length $k$ in $G$.\n                \\item[Complexity] See paper.\n                \\item[Reference] \\cite{Alon1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All cycles in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all small cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All cycles with length at most $5$ in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All chordless cycles in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all Hamiltonian cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All Hamiltonian cycles in $G$.\n                \\item[Reference] \\cite{Wild2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All cycles in $G$.\n                \\item[Complexity] $O(|E| + \\sum_{c \\in \\mathcal{C}(G)}|c|)$ total time.\n                \\item[Comment] $\\mathcal{C}(G)$ is the set of all cycles in $G$.\n                \\item[Reference] \\cite{Ferreira2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All chordless cycles in $G$.\n                \\item[Complexity] $\\tilde{O}(|E| +|V|\\cdot C )$ total time.\n                \\item[Comment] $C$ is the number of chordless cycles in $G$.\n                \\item[Reference] \\cite{Ferreira2014}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless cycles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All chordless cycles in $G$.\n                \\item[Complexity] $O(|V|+|E|)$ time per chordless cycle.\n                \\item[Reference] \\cite{Unoc}\n            \\end{description}\n     \\subsection{Dominating set}\n        \\subsubsection{Enumeration of all minimal dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(1.7159^{|V|})$ total time.\n                \\item[Reference] \\cite{Fomin2008}\n            \\end{description}\n        \\subsubsection{Enumeration of $z$ smallest weighted edge dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$, and positive integers $k$ and $z$. Each edge of $G$ has a positive weight.\n                \\item[Output] $z$ smallest weighted edge dominating sets in $G$.\n                \\item[Complexity] $O(5.6^{2k}k^4z^2 +4^{2k}k^3z|V|)$ total time.\n                \\item[Reference] \\cite{Wang2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a line graph}\n            \\begin{description}\n                \\item[Input] A line graph $G$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(||G||^5N^6)$ total time.\n                \\item[Comment] $||G||$ is the size of $G$ and $N$ is the number of minimal dominating sets in $G$.\n                \\item[Reference] \\cite{Kante2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a path graph or $(C_4, C_5, craw)$-free graph}\n            \\begin{description}\n                \\item[Input] A line graph or $(C_4, C_5, craw)$-free graph $G$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(||G||^2N^3)$ total time.\n                \\item[Comment] $||G||$ is the size of $G$ and $N$ is the number of minimal dominating sets in $G$.\n                \\item[Reference] \\cite{Kante2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal edge-dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All minimal edge-dominating sets in $G$.\n                \\item[Complexity] $O(||L(G)||^5N^6)$ total time.\n                \\item[Comment] $L(G)$ is the line graph of $G$, $||L(G)||$ is the size of $L(G)$, and $N$ is the number of minimal edge-dominating sets in $G$.\n                \\item[Reference] \\cite{Kante2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in an undirected permutation graph}\n            \\begin{description}\n                \\item[Input] An undirected permutation graph $G=(V, E)$.\n                \\item[Output] All minimal dominating sets.\n                \\item[Complexity] $O(|V|)$ delay with $O(|V|^8)$ pre-processing.\n                \\item[Reference] \\cite{Kante2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in an undirected interval graph}\n            \\begin{description}\n                \\item[Input] An undirected interval graph $G=(V, E)$.\n                \\item[Output] All minimal dominating sets.\n                \\item[Complexity] $O(|V|)$ delay with $O(|V|^3)$ pre-processing.\n                \\item[Reference] \\cite{Kante2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal edge dominating sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal edge dominating sets in $G$.\n                \\item[Complexity] $O(|V|^6|\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a line graph}\n            \\begin{description}\n                \\item[Input] A line graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(|V|^2|E|^2|\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal edge dominating sets in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (V, E)$.\n                \\item[Output] All minimal edge dominating sets in $G$.\n                \\item[Complexity] $O(|V|^4|\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in the line graph of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(|V|^2|E||\\mathcal{L}|)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a graph of girth at least 7}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ of girth at least 7.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] $O(|V|^2|E||\\mathcal{L}|^2)$ delay.\n                \\item[Comment] $\\mathcal{L}$ is the set of already generated solutions.\n                \\item[Reference] \\cite{Golovach2013a}\n            \\end{description}\n        \\subsubsection{Enumeration of all 2-dominating sets in a tree}\n            \\begin{description}\n                \\item[Input] A tree $T = (V, E)$.\n                \\item[Output] All 2-dominating sets of $T$.\n                \\item[Complexity] $O(1.3248^n)$ total time.\n                \\item[Comment] If a subset $U\\subseteq V$ is a 2-dominating set if every vertex $v \\in V \\setminus U$ has at least two neighbors in $U$.\n                \\item[Reference] \\cite{Krzywkowski2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a $P_6$-free chordal graph}\n            \\begin{description}\n                \\item[Input] A $P_6$-free chordal graph $G = (V, E)$.\n                \\item[Output] All minimal dominating sets in $G$.\n                \\item[Complexity] Linear delay with $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Kante2014}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal dominating sets in a chordal bipartite graph}\n            \\begin{description}\n                \\item[Input] A chordal bipartite graph $G$. \n                \\item[Output] All minimal dominating sets in $G$. \n                \\item[Complexity] $O(n^3m|\\mathcal{L}|^2)$ delay and the total running time is $O(n^3m|\\mathcal{L}^*|^2)$. \n                \\item[Comment] $n$ is the number vertices in $G$, \n            $m$ is the number of edges in $G$, \n            $\\mathcal{L}$ is the family of already generated minimal dominating sets, and \n            $\\mathcal{L}^*$ is the family of all minimal dominating sets. \n        \n                \\item[Reference] \\cite{Golovach2015}\n            \\end{description}\n     \\subsection{Drawing}\n        \\subsubsection{Enumeration of all rectangle drawings with $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rectangle drawings with $n$ faces.\n                \\item[Complexity] $O(n)$ time per drawing and space.\n                \\item[Reference] \\cite{Takagi2004}\n            \\end{description}\n     \\subsection{Feedback arc set}\n        \\subsubsection{Enumeration of all minimal feedback arc sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All minimal feedback arc sets in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal feedback arc sets in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V,E)$.\n                \\item[Output] All minimal feedback arc sets in $G$.\n                \\item[Complexity] $O(|V||E|(|V|+|E|))$ time delay.\n                \\item[Reference] \\cite{Schwikowski2002}\n            \\end{description}\n     \\subsection{Feedback vertex set}\n        \\subsubsection{Enumeration of all minimal feedback vertex sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All minimal feedback vertex sets in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all feedback vertex sets in a strongly connected directed graph}\n            \\begin{description}\n                \\item[Input] A strongly connected directed graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] All feedback vertex sets of size $k$ in $G$.\n                \\item[Complexity] $O(|V|^{k-1}|E|)$ total time and $O(|E|)$ space.\n                \\item[Reference] \\cite{Garey1978}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal feedback vertex sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All minimal feedback vertex sets in $G$.\n                \\item[Complexity] $O(|V||E|(|V|+|E|))$ time delay.\n                \\item[Reference] \\cite{Schwikowski2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal feedback vertex sets in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G=(V,E)$.\n                \\item[Output] All minimal feedback vertex sets in $G$.\n                \\item[Complexity] $O(|V|^2(|V|+|E|))$ time delay.\n                \\item[Reference] \\cite{Schwikowski2002}\n            \\end{description}\n     \\subsection{General}\n        \\subsubsection{Enumeration of all graphs in an almost sure first order fimiles}\n            \\begin{description}\n                \\item[Input] A first order language $\\theta$.\n                \\item[Output] All graphs in $\\mathcal{G}_\\theta$.\n                \\item[Complexity] Polynomial space and delay.\n                \\item[Reference] \\cite{Goldberg1993b}\n            \\end{description}\n     \\subsection{Independent set}\n        \\subsubsection{Enumeration of all maximal independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O(|V||E|\\mu)$ total time.\n                \\item[Comment] $\\mu$ is the number of maximal independent sets of $G$. This algorithm can also enumerate all the maximal cliques.\n                \\item[Reference] \\cite{Tsukiyama1977a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a claw-free graph}\n            \\begin{description}\n                \\item[Input] A claw-free graph $G$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Reference] \\cite{Minty1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in an undirected graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal independent sets in $G$ in lexicographically.\n                \\item[Reference] \\cite{Loukakis1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in an interval graph}\n            \\begin{description}\n                \\item[Input] An interval graph $G = (V, E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O(|V|^2 + \\beta)$ total time.\n                \\item[Comment] $\\beta$ is the sum of the vertices in all maximal independent sets of $G$.\n                \\item[Reference] \\cite{Leung1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a circular-arc graph}\n            \\begin{description}\n                \\item[Input] A circular-arc graph $G = (V, E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O(|V|^2 + \\beta)$ total time.\n                \\item[Comment] $\\beta$ is the sum of the vertices in all maximal independent sets of $G$.\n                \\item[Reference] \\cite{Leung1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All maximal independent sets in $G$.\n                \\item[Complexity] $O((|V| + |E|)N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Leung1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All maximal independent sets included in $G$.\n                \\item[Complexity] $O(n(m+n\\log C))=O(n^3)$ delay and exponential space.\n                \\item[Comment] $C$: total number of maximal independent sets, $n$: total number of vertices, and $m$: total number of edges. Is there no polynomial space and delay algorithm?\n                \\item[Reference] \\cite{Johnson1988}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent sets of a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B =(V, E)$.\n                \\item[Output] All maximum independent sets of $B$.\n                \\item[Complexity] $O(|V|^{2.5} + N)$.\n                \\item[Comment] $N$ is the number of maximum independent sets of $B$.\n                \\item[Reference] \\cite{Kashiwabara1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets on a tree in lexicographic order}\n            \\begin{description}\n                \\item[Input] Tree $T = (V, E)$.\n                \\item[Output] All maximal independent sets on $T$ in lexicographic order.\n                \\item[Complexity] $O(|V|^2)$ delay with $O(|V|)$ space.\n                \\item[Reference] \\cite{Chang1994a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.114|E|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ with maximum degree 3.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.171|V|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ with maximum degree 4.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.228|V|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent set of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximum independent set of $G$.\n                \\item[Complexity] $O(2^{0.290|V|})$ total time and polynomial space.\n                \\item[Reference] \\cite{Beigel1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and a position integer $k$.\n                \\item[Output] Enumeration of all maximal independent sets with at most size $k$ of $G$.\n                \\item[Complexity] $O(3^{4k-|V|} 4^{|V|-3k})$ total time.\n                \\item[Reference] \\cite{Eppstein2003a}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All independent sets in $G$.\n                \\item[Complexity] Constant time per solution on average after $O(|V| + |E|)$ time for preprocessing.\n                \\item[Reference] \\cite{Okamoto2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets of size $k$ in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$ and a positive integer $k$.\n                \\item[Output] All independent sets of size $k$ in $G$.\n                \\item[Complexity] Constant time per solution on average after $O((|V| + |E|)|V|^2)$ time for preprocessing.\n                \\item[Reference] \\cite{Okamoto2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent sets in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All maximum independent sets in $G$.\n                \\item[Complexity] Constant time per solution on average after $O((|V| + |E|)|V|^2)$ time for preprocessing.\n                \\item[Reference] \\cite{Okamoto2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets in an input chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$.\n                \\item[Output] All independent sets in $G$.\n                \\item[Complexity] $O(1)$ delay and $O(|V|(|V|+|E|))$ time and space for preprocessing.\n                \\item[Comment] Counting all independent sets in an input chordal graph needs $O(n+m)$ time.\n                \\item[Reference] \\cite{Okamoto2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximum independent sets in an input chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$.\n                \\item[Output] All maximum independent sets in $G$.\n                \\item[Complexity] $O(1)$ delay and $O(|V|(|V|+|E|))$ time and space for preprocessing.\n                \\item[Comment] Counting all maximum independent sets in an input chordal graph needs $O(n+m)$ time.\n                \\item[Reference] \\cite{Okamoto2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all independent sets with $k$ vertices in an input chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G=(V,E)$ and an integer $k$.\n                \\item[Output] All maximum independent sets with $k$ vertices in $G$.\n                \\item[Complexity] $O(1)$ delay and $O(k|V|(|V|+|E|))$ time and space for preprocessing.\n                \\item[Comment] The number of independent sets with $k$ vertices in an input chordal graph needs $O(k^2(|V|+|E|))$ time.\n                \\item[Reference] \\cite{Okamoto2008}\n            \\end{description}\n     \\subsection{Interval graph}\n        \\subsubsection{Enumeration of all interval supergraph that contains a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All interval supergraphs each of which contain $G$.\n                \\item[Complexity] $O(|V|^3)$ time for each and $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Kiyomi2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of all interval graph of a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All interval graphs of $G$.\n                \\item[Complexity] $O((|V|+|E|)^2)$ time for each.\n                \\item[Reference] \\cite{Kiyomi2006a}\n            \\end{description}\n        \\subsubsection{Enumeration of Proper Interval Graphs}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All proper interval graphs with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per proper interval graph and $O(n)$ space, after $O(n^2)$ preprocessing time.\n                \\item[Comment] Preprocessing: generating the complete graph with $n$ vertices.\n                \\item[Reference] \\cite{Saitoh2010a}\n            \\end{description}\n     \\subsection{Matching}\n        \\subsubsection{Enumeration of the $k$ best perfect matchings of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best perfect matchings of $G$ in order.\n                \\item[Complexity] $O(k|V|^3)$ total time.\n                \\item[Reference] \\cite{Chegireddy1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all stable marriage}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All stable marriage of $G$.\n                \\item[Complexity] $O(|V|)$ time per solution and $O(|V|^2)$ space.\n                \\item[Reference] \\cite{Gusfield1989}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimum cost perfect matchings in an weighted bipartite graph}\n            \\begin{description}\n                \\item[Input] An weighted bipartite graph $B = (V, E)$.\n                \\item[Output] All minimum cost perfect matchings in $B$.\n                \\item[Complexity] $O(|V|(|V| + |E|))$ time per solution and $O(|V| + |E|)$ space\n                \\item[Reference] \\cite{Fukuda1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect matchings in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B = (U, V, E)$, where $|U| = |V|$.\n                \\item[Output] All perfect matchings in $B$.\n                \\item[Complexity] $O(c(|V| + |E|))$ total time and $O(|V| + |E|)$ space, after $O(n^{2.5})$ preprocessing time.\n                \\item[Comment] $c$ is the number of solutions.\n                \\item[Reference] \\cite{Fukuda1994}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ best perfect matchings of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best perfect matchings of $G$ in decreasing order.\n                \\item[Complexity] $O(k|V|^3)$ total time with $O(k|V|^2)$ space.\n                \\item[Reference] \\cite{Matsui1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect, maximum, and maximal matchings in bipartite graphs}\n            \\begin{description}\n                \\item[Input] A bipartite graph $B=(V, E)$.\n                \\item[Output] All perfect, maximum, and maximal matching in $B$.\n                \\item[Complexity] $O(|V|)$ time per matching.\n                \\item[Reference] \\cite{Uno1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal matchings in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All maximal matchings in $G$.\n                \\item[Complexity] $O(|V| + |E| + \\Delta N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of solutions and $\\Delta$ is the maximum degree in $G$.\n                \\item[Reference] \\cite{Uno2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal blocker in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (U, V, E)$.\n                \\item[Output] All minimal blocker in $G$.\n                \\item[Complexity] Polynomial delay and space.\n                \\item[Comment] A \\textit{blocker} of $G$ is an edge subset $X$ of $E$ such that $G' = (U, V, E \\setminus X)$ has no perfect matching.\n                \\item[Reference] \\cite{Boros2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all basic perfect 2-matchings in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All basic perfect 2-matchings in $G$.\n                \\item[Complexity] Incremental polynomial delay.\n                \\item[Comment] A \\textit{basic 2-matching} of $G$ is a subset of edges that cover the vertices with vertex-disjoint edges and vertex-disjoint odd cycles.\n                \\item[Reference] \\cite{Boros2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all $d$-factor in a bipartite graph}\n            \\begin{description}\n                \\item[Input] A bipartite graph $G = (U, V, E)$ and any non negative function $d: A \\cup B \\to \\{0, 1, \\cdots, |U| + |V|\\}$.\n                \\item[Output] All $d$-factor in $G$.\n                \\item[Complexity] $O(|E|)$ delay.\n                \\item[Comment] A $d$-factor in $G$ is a subgraph $G' = (U, V, X)$ covering all vertices of $G$, whose each vertex $v$ has degree $d(v)$. If for any $v \\in U \\cup V$, $d(v) = 1$, $G'$ is a perfect matching.\n                \\item[Reference] \\cite{Boros2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal induced matchings in a triangle-free graph}\n            \\begin{description}\n                \\item[Input] A triangle-free graph $G$. \n                \\item[Output] All maximal induced matchings in $G$. \n                \\item[Complexity] $O(1.4423^n)$ total time with polynomial delay. \n                \\item[Comment] $n$ is the number of vertices in $G$. \n                \\item[Reference] \\cite{Basavaraju2014}\n            \\end{description}\n     \\subsection{Matroid}\n        \\subsubsection{Enumeration of all bases of a graphic matroid in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bases of a graphic matroid in $G$.\n                \\item[Complexity] $O(|V| + |E| + N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of solutions.  If $G$ is connected, any base is a spanning tree.\n                \\item[Reference] \\cite{Uno1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all bases of a linear matroid in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bases of a linear matroid in $G$.\n                \\item[Complexity] $O(|V|)$ time per solution and $O(|V|^2|E|)$ preprocessing after time.\n                \\item[Reference] \\cite{Uno1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all bases of a matching matroid in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All bases of a matching matroid in $G$.\n                \\item[Complexity] $O(|V| + |E|)$ time per solution.\n                \\item[Reference] \\cite{Uno1998}\n            \\end{description}\n     \\subsection{Ordering}\n        \\subsubsection{Enumeration of all topological sortings of a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G =(V, E)$.\n                \\item[Output] All topological sortings of $G$.\n                \\item[Complexity] $O(|V| + |E|)$ time per sorting and $O(|V| + |E|)$ space.\n                \\item[Reference] \\cite{Knuth1974}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of a given set in lexicographically}\n            \\begin{description}\n                \\item[Input] An $n$-element set $S$. \n                \\item[Output] All topological sortings of $S$ in lexicographically.\n                \\item[Complexity] $O(m)$ time per solution(?). \n                \\item[Reference] \\cite{Knuth1979}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of a po set}\n            \\begin{description}\n                \\item[Input] A partial order set $P$.\n                \\item[Output] All topological sortings of $P$.\n                \\item[Complexity] $O(|P|)$ time per solution.\n                \\item[Comment] $|P|$ is the number of objects in $P$.\n                \\item[Reference] \\cite{Varol1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings in a directed acyclic graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G$.\n                \\item[Output] All topological sortings in $G$.\n                \\item[Complexity] $O(1)$ amortized time per solution with $O(|G|)$.\n                \\item[Comment] Linear extensions correspond to topological sortings.\n                \\item[Reference] \\cite{Pruesse1991a}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All topological sortings in $G$.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Comment] A topological sorting is also known as a linear extension.\n                \\item[Reference] \\cite{Ruskey1992a}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings}\n            \\begin{description}\n                \\item[Input] A directed acyclic graph $D$.\n                \\item[Output] All topological sortings $D$.\n                \\item[Complexity] $O(1)$ amortized time per topological sorting and $O(|V|)$ space in addition to the space used for $D$.\n                \\item[Comment] Linear sortings correspond to topological sortings.\n                \\item[Reference] \\cite{Pruesse1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all linear extensions of a given poset}\n            \\begin{description}\n                \\item[Input] A poset $P$.\n                \\item[Output] All linear extensions of $P$.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Comment] Their algorithm is a loop-free algorithm.\n                \\item[Reference] \\cite{Canfield1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of an acyclic directed graph}\n            \\begin{description}\n                \\item[Input] An acyclic directed graph $G = (V, E)$.\n                \\item[Output] All topological sortings of $G$.\n                \\item[Complexity] $O(|V|N)$ total time and $O(|V||E|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect elimination orderings}\n            \\begin{description}\n                \\item[Input] A chordal graph $G$.\n                \\item[Output] All perfect elimination orderings of $G$.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Chandran2003}\n            \\end{description}\n        \\subsubsection{Enumeration of all forest extensions of a partially ordered set}\n            \\begin{description}\n                \\item[Input] A partially ordered set $P$.\n                \\item[Output] All forest extensions of $P$.\n                \\item[Complexity] $O(|E|^2)$ delay and $O(|E||R|)$ space.\n                \\item[Comment] $E$ is the set of elements. $R$ is the binary relation on $E$.\n                \\item[Reference] \\cite{Szwarcfiter2003b}\n            \\end{description}\n        \\subsubsection{Enumeration of all topological sortings of a directed acyclic graph}\n            \\begin{description}\n                \\item[Input] A directed acyclic graph $D$.\n                \\item[Output] All topological sortings $D$.\n                \\item[Complexity] $O(1)$ delay per topological sorting.\n                \\item[Comment] Linear extensions correspond to topological sortings.\n                \\item[Reference] \\cite{Ono2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all realizer of a triangulated planar graph}\n            \\begin{description}\n                \\item[Input] A triangulated planar graph $G = (V, E)$.\n                \\item[Output] All realizer of $G$.\n                \\item[Complexity] $O(|V|)$ time per realizer.\n                \\item[Reference] \\cite{Yamanaka2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect elimination orderings of a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All perfect elimination orderings of $G$.\n                \\item[Complexity] $O(1)$ time per solution on average with $O(|V|^2)$ space and $O(|V|^3)$ with $O(|V|^2)$ space pre-computation.\n                \\item[Reference] \\cite{Matsui2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all perfect sequences in a chordal graph}\n            \\begin{description}\n                \\item[Input] A chordal graph $G = (V, E)$.\n                \\item[Output] All perfect sequences of $G$.\n                \\item[Complexity] $O(1)$ time per graph with $O(|V|^2)$ space with $O(|V|^3)$ time and $O(|V|^2)$ space pre-computation.\n                \\item[Reference] \\cite{Matsui2010}\n            \\end{description}\n     \\subsection{Orientation}\n        \\subsubsection{Enumeration of all acyclic orientation of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All acyclic orientation of $G$.\n                \\item[Complexity] $O(N(|V|+|E|))$ total time ($O(|V|(|V|+E|))$ delay) and $O(|V| + |E|)$ space.\n                \\item[Comment] A acyclic orientation of $G$ is an assignment of directions of each edge such that $G$ is acyclic.\n                \\item[Reference] \\cite{Barbosa1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all $(s, t)$-orientations of a biconnected planar graph}\n            \\begin{description}\n                \\item[Input] A biconnected planar graph $G = (V, E)$ and an edge $(s, t)$ in $G$.\n                \\item[Output] All $(s, t)$-orientations of $G$.\n                \\item[Complexity] $O(|V|)$ time per solution.\n                \\item[Reference] \\cite{Setiawan2011}\n            \\end{description}\n     \\subsection{Other}\n        \\subsubsection{Enumeration of all Hamiltonian centers in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All Hamiltonian centers in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all CA-sets of a directed graph}\n            \\begin{description}\n                \\item[Input] Directed graph $G = (V, E)$.\n                \\item[Output] All CA-sets of $G$.\n                \\item[Complexity] $O(|V|^{2.49+} + \\gamma)$.\n                \\item[Comment] $\\gamma$ is the output size. $S \\subset V$ is a \\textit{CA-set} if, for each $v \\in S$, all ancestor of $v$ belongs to $S$.\n                \\item[Reference] \\cite{Kashiwabara1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal induced subgraphs for (connected) hereditary graph properties}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All maximal induced subgraphs in $\\mathcal{P}(G)$.\n                \\item[Complexity] See the paper.\n                \\item[Comment] $\\mathcal{P}$ is a set of subgraphs of $G$ with (connected) hereditary graph properties.\n                \\item[Reference] \\cite{Cohen2008}\n            \\end{description}\n     \\subsection{Path}\n        \\subsubsection{Enumeration of all simple paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.  \n                \\item[Output] All simple paths in $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Ponstein1966}\n            \\end{description}\n        \\subsubsection{Enumeration of all Hamiltonian paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All Hamiltonian paths in $G$. \n                \\item[Reference] \\cite{Yau1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed paths in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G$. \n                \\item[Output] All directed paths in $G$. \n                \\item[Reference] \\cite{Kamae1967}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All paths in $G$. \n                \\item[Reference] \\cite{Kroft1967b}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All paths in $G$. \n                \\item[Reference] \\cite{Danielson1968}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $K$ shortest paths in $G$.\n                \\item[Complexity] $O(|V|^3)$ total time.\n                \\item[Reference] \\cite{Yen1971}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $k$ shortest paths in $G$.\n                \\item[Complexity] $O(k|V|c(|V|))$ total time.\n                \\item[Comment] (?) $c(n)$ is the time complexity to find an optimal solution to a problem with $n$ (0, 1) variables.\n                \\item[Reference] \\cite{Lawler1972}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All paths in $G$.\n                \\item[Complexity] $O(|E|)$ time per path with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of all paths in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All paths in $G$.\n                \\item[Complexity] $O(|E|)$ time per path with $O(|E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E). $\n                \\item[Output] $K$ shortest paths in $G$.\n                \\item[Complexity] $O(k|V|^3)$ total time.\n                \\item[Reference] \\cite{Shier1979}\n            \\end{description}\n        \\subsubsection{Generation of the $k$-th longest path in a tree}\n            \\begin{description}\n                \\item[Input] A tree $T = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$-th longest path in $T$.\n                \\item[Complexity] $O(n\\log^2 n)$ time.\n                \\item[Reference] \\cite{Megiddo1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All shortest paths in $G$.\n                \\item[Reference] \\cite{Florian1981}\n            \\end{description}\n        \\subsubsection{Enumeration of $k$ shortest paths of a directed graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $k$ shortest paths that may contains cycles in $G$.\n                \\item[Reference] \\cite{Martins1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all quickest paths in a network}\n            \\begin{description}\n                \\item[Input] A network $N = (V, E, c, \\ell)$.\n                \\item[Output] All quickest paths in $N$.\n                \\item[Complexity] $O(rS|V||E| + rS|V|^2\\log|V|)$ total time.\n                \\item[Comment] $c$ is a positive edge weight function and $\\ell$ is a nonnegative edge weight function. $r$ is the number of distinct capacity value of $N$. $S$ is the number of solutions.\n                \\item[Reference] \\cite{Rosen1991}\n            \\end{description}\n        \\subsubsection{Counting all acyclic walks in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] The number of acyclic walks in $G$.\n                \\item[Reference] \\cite{Babic1993}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ shortest paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest shortest paths in $G$.\n                \\item[Complexity] $O(k|E|)$ total time.\n                \\item[Reference] \\cite{Azevedo1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all constrained quickest paths in a network}\n            \\begin{description}\n                \\item[Input] Network $N = (V, E)$ and constraints $L$ and $C$.\n                \\item[Output] All quickest paths in $N$.\n                \\item[Complexity] $O(k|V|^2|E|)$ total time.\n                \\item[Comment] $k$ is the number of solutions. A quickest path is a variant of a shortest path.\n                \\item[Reference] \\cite{Gen-Huey1994}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ shortest paths in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest shortest paths in $G$.\n                \\item[Complexity] $O(k|E|)$ total time\n                \\item[Reference] \\cite{Eppstein1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal path conjunctions in a graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$, $s_1, s_2, t_1 \\in V$, $T_2 \\subseteq V$, and $\\mathcal{P} = \\{ (s_1, t_1)\\} \\cup \\{(s_2, t): t\\in T_2\\}$.\n                \\item[Output] All minimal path conjunctions in $G$.\n                \\item[Complexity] Polynomial delay.\n                \\item[Comment] A path conjunction is a edge subset $E' \\subseteq E$ such that for all $(s, t) \\in \\mathcal{P}$, $s$ is connected to $t$ in the graph $G' = (V, E')$.\n                \\item[Reference] \\cite{Boros2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all st-paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$ and $s, v \\in V$.\n                \\item[Output] All st-paths in $G$.\n                \\item[Complexity] $O(|E| + \\sum_{\\pi \\in \\mathcal{P}_{st}(G)}|\\pi|)$ total time.\n                \\item[Comment] $\\mathcal{P}_{st}(G)$ is the set of all st-paths in $G$.\n                \\item[Reference] \\cite{Ferreira2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all $P_3$'s in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$.\n                \\item[Output] All of all $P_3$'s in $G$.\n                \\item[Complexity] $O(|E|^{1.5} + p_3(G))$ total time.\n                \\item[Comment] $P_3$ is a induced path of $G$ with three vertices and $p_3(G)$ is the number of $P_3$ in $G$.\n                \\item[Reference] \\cite{Hoang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all $P_k$'s in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$ and an integer $k \\ge 4$.\n                \\item[Output] All of all $P_k$'s in $G$.\n                \\item[Complexity] $O(|V|^{k-1} + p_k(G) + k\\cdot c_k(G))$ total time.\n                \\item[Comment] $P_k$ and $C_k$ are a induced path and cycle of $G$ with $k$ vertices, respectively. $p_k(G)$ and $c_k(G)$ are the number of $P_k$ and $C_k$ in $G$, respectively.\n                \\item[Reference] \\cite{Hoang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all $C_k$'s in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$ and an integer $k \\ge 4$.\n                \\item[Output] All of all $C_k$'s in $G$.\n                \\item[Complexity] $O(|V|^{k-1} + p_k(G) + c_k(G))$ total time.\n                \\item[Comment] $P_k$ and $C_k$ are a induced path and cycle of $G$ with $k$ vertices, respectively. $p_k(G)$ and $c_k(G)$ are the number of $P_k$ and $C_k$ in $G$, respectively.\n                \\item[Reference] \\cite{Hoang2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless $st$-paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$ and two vertices $s, t \\in V$.\n                \\item[Output] All chordless $st$-paths in $G$.\n                \\item[Complexity] $\\tilde{O}(|E| +|V|\\cdot P )$ total time.\n                \\item[Comment] $P$ is the number of chordless $st$-paths in $G$.\n                \\item[Reference] \\cite{Ferreira2014}\n            \\end{description}\n        \\subsubsection{Enumeration of all chordless st-paths in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$ and $s, t \\in V$.\n                \\item[Output] All chordless st-paths (from $s$ to $t$) in $G$.\n                \\item[Complexity] $O(|V|+|E|)$ time per chordless st-path.\n                \\item[Reference] \\cite{Unoc}\n            \\end{description}\n     \\subsection{Permutation graph}\n        \\subsubsection{Enumeration of all connected bipartite permutation graphs with $n$ vertices}\n            \\begin{description}\n                \\item[Input] A graph size $n$.\n                \\item[Output] All connected bipartite permutation graphs.\n                \\item[Complexity] $O(1)$ time per graph with $O(n)$ space.\n                \\item[Reference] \\cite{Saitoh2010b}\n            \\end{description}\n     \\subsection{Pitch}\n        \\subsubsection{Enumeration of all stories in a graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E, S, T)$ that has the set of source vertices $S$ and the set of target vertices of $T$.\n                \\item[Output] All stories in $G$.\n                \\item[Comment] A \\textit{pitch} $P$ of $G$ is a set of arcs $E' \\subseteq E$, such that the subgraph $G' =(V', E')$ of $G$, where $V' \\subseteq V$ is the set of vertices of G having at least one out-going or in-coming arc in $E'$, is acyclic and for each vertex $w \\in V' \\setminus S$, $w$ is not a source in $G'$, and for each vertex $w \\in V' \\setminus T$ ,$w$ is not a target in $G'$. $P$ is a \\textit{story} if $P$ is maximal.\n                \\item[Reference] \\cite{Acuna2012}\n            \\end{description}\n        \\subsubsection{Enumeration of all pitches}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E, S, T)$ that has the set of source vertices $S$ and the set of target vertices of $T$.\n                \\item[Output] All pitches in $G$.\n                \\item[Complexity] $O(|V| + |E|)$ delay with $O(|V| + |E|)$ space.\n                \\item[Comment] A pitch $P$ of $G$ is a set of arcs $E' \\subseteq E$, such that the subgraph $G' =(V', E')$ of $G$, where $V' \\subseteq V$ is the set of vertices of G having at least one out-going or in-coming arc in $E'$, is acyclic and for each vertex $w \\in V' \\setminus S$, $w$ is not a source in $G'$, and for each vertex $w \\in V' \\setminus T$ ,$w$ is not a target in $G'$. Enumeration of all stories (maximal pitches) is still open.\n                \\item[Reference] \\cite{Borassi2013}\n            \\end{description}\n     \\subsection{Planar graph}\n        \\subsubsection{Enumeration of all maximal planar graphs with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All maximal planar graph with $n$ vertices.\n                \\item[Complexity] $O(n^3)$ time per graph with $O(n)$ space.\n                \\item[Comment] A planar graph with $n$ vertices is \\textit{maximal} if it has exactly $3n -6$ edges.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all based floorplans with at most $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All based floorplans with at most $n$ faces.\n                \\item[Complexity] $O(1)$ time per solution with $O(n)$ space.\n                \\item[Comment] A planar graph is called a \\textit{floorplan} if every face is a rectangle. A \\textit{based floorplan} is a floorplan with one designated base line segment on the outer face.\n                \\item[Reference] \\cite{Nakano2001a}\n            \\end{description}\n        \\subsubsection{Enumeration of all based floorplans with exactly $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All based floorplans with exactly $n$ faces.\n                \\item[Complexity] $O(1)$ time per solution with $O(n)$ space.\n                \\item[Comment] A planar graph is called a \\textit{floorplan} if every face is a rectangle. A \\textit{based floorplan} is a floorplan with one designated base line segment on the outer face.\n                \\item[Reference] \\cite{Nakano2001a}\n            \\end{description}\n        \\subsubsection{Enumeration of all floorplans with exactly $n$ faces}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All floorplans with exactly $n$ faces.\n                \\item[Complexity] $O(1)$ time per solution with $O(n)$ space.\n                \\item[Comment] A planar graph is called a \\textit{floorplan} if every face is a rectangle.\n                \\item[Reference] \\cite{Nakano2001a}\n            \\end{description}\n        \\subsubsection{Enumeration of all internally triconnected planar graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All internally triconnected planar graphs with exactly $n$ vertices such that $\\kappa(G) = 2$ and the size of each inner face is at most $g$.\n                \\item[Complexity] $O(n^3)$ time per solution on average with $O(n)$ space.\n                \\item[Reference] \\cite{Zhuang2010a}\n            \\end{description}\n     \\subsection{Plane graph}\n        \\subsubsection{Enumeration of all plane straight-line graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A point set $P$ in the plane.\n                \\item[Output] All plane straight-line graphs on $P$.\n                \\item[Complexity] $O(|P|\\log|P|)$ time per solution.\n                \\item[Comment] Use gray code.\n                \\item[Reference] \\cite{Aichholzer2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane and connected straight-line graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A point set $P$ in the plane.\n                \\item[Output] All plane and connected straight-line graphs on $P$.\n                \\item[Complexity] $O(|P|\\log|P|)$ time per solution.\n                \\item[Comment] Use gray code.\n                \\item[Reference] \\cite{Aichholzer2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane spanning trees on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A point set $P$ in the plane.\n                \\item[Output] All plane spanning trees on $P$.\n                \\item[Complexity] $O(|P|\\log|P|)$ time per solution.\n                \\item[Comment] Use gray code.\n                \\item[Reference] \\cite{Aichholzer2007}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane graphs}\n            \\begin{description}\n                \\item[Input] $m$: the maximum number of edges.\n                \\item[Output] All connected rooted plane graphs with at most $m$ edges.\n                \\item[Complexity] amortized $O(1)$ time per graph with $O(m)$ space.\n                \\item[Comment] This algorithm does not outputs the entire graph but the difference from previous one.\n                \\item[Reference] \\cite{Yamanaka2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane graphs}\n            \\begin{description}\n                \\item[Input] $m$: the maximum number of edges.\n                \\item[Output] All connected non-rooted plane graphs with at most $m$ edges.\n                \\item[Complexity] $O(m^3)$ time per graph with $O(m)$ space.\n                \\item[Comment] This algorithm does not outputs the entire graph but the difference from previous one.\n                \\item[Reference] \\cite{Yamanaka2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All plane graphs on $P$.\n                \\item[Complexity] $O(N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing spanning connected graphs on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing spanning connected graphs on $P$.\n                \\item[Complexity] $O(N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing spanning trees on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing spanning trees on $P$.\n                \\item[Complexity] $O(N + |P|tri(P))$ total time.\n                \\item[Comment] $N$ is the number of solutions and $tri(P)$ is the number of triangulations of $P$.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing minimally rigid frameworks on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing minimally rigid frameworks on $P$.\n                \\item[Complexity] $O(|P|^2N)$ total time.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all non-crossing perfect matchings on a given point set in the plane}\n            \\begin{description}\n                \\item[Input] A fixed point set $P$.\n                \\item[Output] All non-crossing perfect matchings on $P$.\n                \\item[Complexity] $O(|P|^{3\/2}tri(P) + |P|^{5\/2}N)$ total time.\n                \\item[Comment] $N$ is the number of solutions and $tri(P)$ is the number of triangulations of $P$.\n                \\item[Reference] \\cite{Katoh2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all triconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All triconnected rooted plane graphs with $n$ vertices, whose each inner face has the length at most $g$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space after $O(n)$ time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all triconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected rooted plane graphs with $n$ vertices.\n                \\item[Complexity] $O(n^3)$ delay with $O(n)$ space after $O(n)$ time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All biconnected rooted plane graphs with exactly $n$ vertices such that each inner face is of length at most $g$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space, after an $O(n)$ time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010c}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All biconnected plane graphs with at most $n$ vertices such that each inner face is of length at most $g$.\n                \\item[Complexity] $O(n^3)$ time per solution on average with $O(n)$ space.\n                \\item[Reference] \\cite{Zhuang2010c}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected rooted plane graphs}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All biconnected rooted plane graphs with at most $n$ vertices such that each inner face is of length at most $g$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space.\n                \\item[Reference] \\cite{Zhuang2010c}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane graphs in $\\mathcal{G}_{\\text{int}}(n, g) - \\mathcal{G}_3(n, g)$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $g$.\n                \\item[Output] All rooted plane graphs in $\\mathcal{G}_{\\text{int}}(n, g) - \\mathcal{G}_3(n, g)$\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space and time preprocessing.\n                \\item[Reference] \\cite{Zhuang2010a}\n            \\end{description}\n     \\subsection{Polytope}\n        \\subsubsection{Enumeration of all 3-polytopes of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All 3-polytopes of $G$.\n                \\item[Reference] \\cite{Deza1993}\n            \\end{description}\n     \\subsection{Quadrangle}\n        \\subsubsection{Enumeration of all quadrangles in a graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$.\n                \\item[Output] All quadrangles in $G$.\n                \\item[Complexity] $O(\\alpha(G)|E|)$ total time and $O(|E|)$ space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n     \\subsection{Quadrangulation}\n        \\subsubsection{Enumeration of all based biconnected plane quadrangulations with at most $f$ faces}\n            \\begin{description}\n                \\item[Input] An integer $f$.\n                \\item[Output] All based biconnected plane quadrangulations with at most $f$ faces.\n                \\item[Complexity] $O(1)$ time per quadrangulation and $O(f)$ space.\n                \\item[Comment] A \\textit{plane quadrangulation} is a plane graph such that each inner face has exactly four edges on its contour. A \\textit{based plane quadrangulation} is a plane quadrangulation with one designated edge on the outer face.\n                \\item[Reference] \\cite{Li2003}\n            \\end{description}\n     \\subsection{Regular graph}\n        \\subsubsection{Enumeration of all cubic graphs with less than or equal to $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All cubic graphs with less than or equal to $n$ vertices. \n                \\item[Reference] \\cite{Brinkmann2011}\n            \\end{description}\n     \\subsection{Series-parallel}\n        \\subsubsection{Enumeration of all series-parallel graphs with at most $m$ edges}\n            \\begin{description}\n                \\item[Input] An integer $m$.\n                \\item[Output] All series-parallel graphs with at most $m$ edges.\n                \\item[Complexity] $O(m)$ time per graph.\n                \\item[Reference] \\cite{Kawano2005a}\n            \\end{description}\n     \\subsection{Spanning subgraph}\n        \\subsubsection{Enumeration of all minimal $k$-vertex connected spanning subgraphs in a $k$-connected graph.}\n            \\begin{description}\n                \\item[Input] A $k$-connected graph $G$.\n                \\item[Output] All minimal $k$-vertex connected spanning subgraphs in $G$.\n                \\item[Complexity] $O(K^3|E|^3|n| + K^2|E|^5|V|^4 + K|V|^k|E|^2)$ total time.\n                \\item[Comment] $K$ is the number of solutions.  A graph $G$ is $k$-connected if a subgraph of $G$ obtained by removing at most $k-1$ vertices is still connected.\n                \\item[Reference] \\cite{Boros2007}\n            \\end{description}\n     \\subsection{Spanning tree}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All spanning trees in  $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Hakimi1961}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All spanning trees of $G$.\n                \\item[Comment] In this paper, 'trees' indicate 'spanning trees'. \n                \\item[Reference] \\cite{Minty1965}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All spanning trees of $G$. \n                \\item[Reference] \\cite{Mayeda1965a}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V||E|^2)$ time per spanning tree with $O(|V||E|)$ space.\n                \\item[Reference] \\cite{Read1975}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(k|E|\\alpha (|E|,|V|) + |E|\\log |E|)$ total time and $O(k + |E|)$ space, $\\alpha(\\cdot)$ is Tarjan's inverse of Ackermann's function.\n                \\item[Reference] \\cite{Gabow1977}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The all spanning trees in $G$ in order.\n                \\item[Complexity] $O(N|V|)$ total time and $O(N + |E|)$ space, $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Gabow1977}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V| + |E| + |V|N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Gabow1978}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V| + |E| + |E|N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Gabow1978}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(k|E| + \\min (|V|^2 ,|E|\\log \\log |V|))$ total time and $O(k + |E|)$ space\n                \\item[Reference] \\cite{Katoh1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Comment] They analized Char's enumeration algorithm.\n                \\item[Reference] \\cite{Jayakumar1984}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph in increasing order}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(|E|\\log\\log_{(2+|E|\/|V|)} n + k^2\\sqrt{|E|})$ total time and $O(|E| + k\\sqrt{|E|})$ space.\n                \\item[Reference] \\cite{Frederickson1985}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a planar graph in increasing order}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(|V| + k^2(\\log |V|)^2)$ total time and $O(|V| + k(\\log |V|)^2)$ space.\n                \\item[Reference] \\cite{Frederickson1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all undirected minimum spanning trees in an undirected graph}\n            \\begin{description}\n                \\item[Input] An undirected graph $G = (V, E)$.\n                \\item[Output] All undirected minimum spanning trees in $G$.\n                \\item[Complexity] $O(|E|\\log \\beta(|E|, |V|))$ total time.\n                \\item[Comment] $\\beta(|E|, |V|) = \\min\\{i | \\log^{(i)}|V| \\le |E|\/|N|\\}$.\n                \\item[Reference] \\cite{Gabow1986a}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed minimum spanning trees in an directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All directed minimum spanning trees in $G$.\n                \\item[Complexity] $O(|V|\\log \\beta(|E|, |V|))$ total time.\n                \\item[Comment] $\\beta(|E|, |V|) = \\min\\{i | \\log^{(i)}|V| \\le |E|\/|N|\\}$.\n                \\item[Reference] \\cite{Gabow1986a}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning tree in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning tree of $G$.\n                \\item[Complexity] $O(|V|+|E|+N)$ total time and $O(|V||E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Kapoor1991}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning tree in an weighted graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$.\n                \\item[Output] All spanning tree of $G$ in increasing order of weight.\n                \\item[Complexity] $O(N\\log |V|+|V||E|)$ total time and $O(N + |V|^2|E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Kapoor1991}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning tree in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning tree of $G$.\n                \\item[Complexity] $O(N|V|+|V|^3)$ total time and $O(|V|^2)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Kapoor1991}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ best spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best spanning trees of $G$.\n                \\item[Complexity] $O(m\\log\\beta(|E|, |V|) + k^2)$ total time.\n                \\item[Comment] $\\beta(|E|, |V|) = \\min \\{\\log^{(i)}|V| \\le |E|\/|V|\\}$.\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ best spanning trees in a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ best spanning trees of $G$.\n                \\item[Complexity] $O(n + k^2)$ total time\n                \\item[Reference] \\cite{Eppstein1992}\n            \\end{description}\n        \\subsubsection{Generation of the $k$-th minimum spanning tree in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$-th minimum spanning tree of $G$.\n                \\item[Complexity] $O((|V||E|)^{k-1})$ time.\n                \\item[Reference] \\cite{Mayr1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(N + |V| + |E|)$ total time and $O(|V||E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees in $G$.\n                \\item[Reference] \\cite{Shioura1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] INCORRECT: $O(N \\log |V| + |V|^2\\alpha(V, V) + |V||E|)$\n                \\item[Comment] $\\alpha$: the inverse Ackermann's function.  [KR2000] gives this result is wrong.\n                \\item[Reference] \\cite{Hariharan1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|E|+ND(|V|, |E|))$ total time and $O(|E|+DS(|V|, |E|))$ space.\n                \\item[Comment] $D(|V|, |E|)$ and $DS(|V|, |E|)$ are the time and space complexities of the data structure for updating the minimum spanning tree in an undirected graph with $|V|$ vertices and $|E|$ edges. Here $N$ denotes the number of directed spanning trees in $G$.\n                \\item[Reference] \\cite{Uno1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V,E)$.\n                \\item[Output] All spanning trees included in $G$.\n                \\item[Complexity] $O(N+|V|+|E|)$ total time and $O(|V|+|E|)$ space.\n                \\item[Comment] $N$ = number of spanning trees in $G$.\n                \\item[Reference] \\cite{Shioura1997}\n            \\end{description}\n        \\subsubsection{Enumeration of the $k$ smallest weight spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] The $k$ smallest weight spanning trees in $G$.\n                \\item[Complexity] $O(m \\log \\log* n + k \\min(n, k)^{1\/2})$ total time, or a randomized version taking $O(m + k \\min(n, k)^{1\/2})$ total time.\n                \\item[Reference] \\cite{Eppstein1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$.\n                \\item[Complexity] $O(|V| + |E| + \\tau)$ time and $O(|V| + |E|)$ space (depth first manner) or $O(\\tau|V| + |E|)$ space (breadth first manner).\n                \\item[Comment] By using breadth first manner, the proposed algorithm can be used in a parallel computer.\n                \\item[Reference] \\cite{Matsui1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees in a graph in non-decreasing order}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All spanning trees in $G$ in non-decreasing order.\n                \\item[Complexity] $O(|V| + |E| + \\tau)$ time and $O(\\tau|V| + |E|)$ space.\n                \\item[Comment] Using breadth first manner.\n                \\item[Reference] \\cite{Matsui1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all directed spanning trees in a directed graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$.\n                \\item[Output] All directed spanning trees in $G$.\n                \\item[Complexity] $O(|E| \\log |V| + |V| + N \\log^2 |V|)$ total time and $O(|E| + |V|)$ space.\n                \\item[Comment] $N$ is the number of directed spanning trees.\n                \\item[Reference] \\cite{Uno1998a}\n            \\end{description}\n        \\subsubsection{Enumeration of all spanning trees of an weighted graph in order of increasing cost}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$.\n                \\item[Output] All spanning trees of $G$ in order of increasing cost.\n                \\item[Complexity] $O(N|E|\\log |E| + N^2)$ total time and $O(N|E|)$ space.\n                \\item[Comment] $N$ is the number of spanning trees of $G$.\n                \\item[Reference] \\cite{Sorensen2005}\n            \\end{description}\n        \\subsubsection{Enumeration of all the minimum spanning trees in a graph}\n            \\begin{description}\n                \\item[Input] An weighted graph $G = (V, E)$.\n                \\item[Output] All the minimum spanning trees in $G$.\n                \\item[Complexity] $O(N|E|\\log|V|)$ total time and $O(|E|)$ space.\n                \\item[Comment] $N$ is the number of the minimum spanning trees in $G$.\n                \\item[Reference] \\cite{Yamada2010}\n            \\end{description}\n     \\subsection{Steiner tree}\n        \\subsubsection{Enumeration of all Steiner $W$-trees in a connected graph}\n            \\begin{description}\n                \\item[Input] A connected graph $G = (V, E)$, a vertex set $W \\subseteq V$ such that $|W| = k$, for a fixed integer $k$.\n                \\item[Output] Enumeration of all Steiner $W$-trees in $G$.\n                \\item[Complexity] $O(|V|^2(|V|+|E|) + |V|^{k-2} + N|V|)$ total time with $O(|V|^{k-2} + |V|^2(|V| + |E|))$ space.\n                \\item[Comment] A connected subgraph $T$ of $G$ is a Steiner $W$-tree if $W \\subseteq V(T)$ and $|E(T)$ is minimum.\n                \\item[Reference] \\cite{Dourado2014}\n            \\end{description}\n     \\subsection{Subforest}\n        \\subsubsection{Enumeration of all $k$-trees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$. \n                \\item[Output] All $k$-trees in $G$. \n                \\item[Comment] A $k$-tree is a forest with $k$ connected components. \n                \\item[Reference] \\cite{Hakimi1964}\n            \\end{description}\n     \\subsection{Subgraph}\n        \\subsubsection{Enumeration of all connected induced subgraph of a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All connected induced subgraph of $G$.\n                \\item[Complexity] $O(|V||E|N)$ total time and $O(|V| + |E|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all connected common maximal subgraphs in two graphs}\n            \\begin{description}\n                \\item[Input] Two graphs $G$ and $G'$.\n                \\item[Output] All connected common maximal subgraphs in $G$ and $G'$.\n                \\item[Reference] \\cite{Koch2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$, $S \\subseteq V$, and requirements $r(u, v)$ for all $(u, v) \\in V \\times V$.\n                \\item[Output] All minimal spanning graph $H$ of $G$ satisfying $\\lambda^S_H \\ge r(u, v) \\; \\forall (u, v) \\in V \\times V$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] The $S$-connectivity $\\lambda^S_G(u, v)$ of $(u, v)$ in $G$ is the maximum number of $uv$-paths such that no two of them have an edge or a node in $S \\setminus \\{u, v\\}$ in common. This complexity holds for edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-outconnected minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$, a vertex $s \\in V$, and an integer $k$.\n                \\item[Output] All minimal $k$-outconneted from $s$ spanning subgraph of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] A graph is $k$-outconnected from $s$ if it contains $k$ internally-disjoint $st$-paths for every $t \\in V$. This complexity holds for both vertex and edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-outconnected minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$, a vertex $s \\in V$, and an integer $k$.\n                \\item[Output] All minimal $k$-outconneted from $s$ spanning subgraph of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] A graph is $k$-outconnected from $s$ if it contains $k$ internally-disjoint $st$-paths for every $t \\in V$. This complexity holds for both vertex and edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-connected minimal spanning graph}\n            \\begin{description}\n                \\item[Input] A directed graph $G = (V, E)$ and an integer $k$.\n                \\item[Output] All minimal $k$-connected spanning subgraph of $G$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] This complexity holds for both vertex and edge-connectivity.\n                \\item[Reference] \\cite{Nutov2009}\n            \\end{description}\n     \\subsection{Subtree}\n        \\subsubsection{Enumeration of all subtrees in an input tree}\n            \\begin{description}\n                \\item[Input] A tree $T=(V, E)$.\n                \\item[Output] All subtrees included in $T$.\n                \\item[Complexity] $O(|V|)$ delay and $O(|V|)$ space.\n                \\item[Reference] \\cite{Ruskey1981}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-noded subtrees in a tree}\n            \\begin{description}\n                \\item[Input] A tree $T$ and an integer $k$.\n                \\item[Output] All $k$-noded subtrees in $T$.\n                \\item[Reference] \\cite{Hikita1983}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-subtrees in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G=(V, E)$ and a positive integer $k$.\n                \\item[Output] All $k$-subtrees included in $G$.\n                \\item[Complexity] $O(sk)$ total time, $O(k)$ amortized time per solution, and $O(|E|)$ space.\n                \\item[Comment] $s$ = number of $k$-subtrees in $G$, a $k$-subtree means a connected, acyclic, and edge induced subgraph with $k$ vertices.\n                \\item[Reference] \\cite{Ferreira2011}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-subtrees in an input tree}\n            \\begin{description}\n                \\item[Input] A tree $T=(V, E)$ and an integer $k$.\n                \\item[Output] All $k$-subtrees included in $T$.\n                \\item[Complexity] $O(1)$ delay and $O(|V|)$ space after $O(|V|)$ time preprocessing.\n                \\item[Comment] A $k$-subtree is a connected, acyclic, and edge induced subgraph with $k$ vertices.\n                \\item[Reference] \\cite{Wasa2012b}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-cardinarity subtrees of a tree with $w$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $k$ and a tree $T$ with $w$ element, where $k \\le w$.\n                \\item[Output] All subtrees with $k$ vertices of $T$.\n                \\item[Complexity] $O(Nw^5)$ total time.\n                \\item[Comment] $N$ is the number of ideals.\n                \\item[Reference] \\cite{Wild2013}\n            \\end{description}\n        \\subsubsection{Enumeration of all induced subtrees in a $k$-degenerate graph}\n            \\begin{description}\n                \\item[Input] A $k$-degenerate graph $G = (V, E)$.\n                \\item[Output] All induced subtrees in $G$.\n                \\item[Complexity] $O(k)$ amortized time per solution with $O(|V| + |E|)$ space and preprocessing time.\n                \\item[Reference] \\cite{Wasa2010}\n            \\end{description}\n     \\subsection{Tour}\n        \\subsubsection{Enumeration of $k$ best solutions to the Chinese postman problem solutions}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] $K$ best solutions to the Chinese postman problem.\n                \\item[Complexity] $O(S( n,m ) + K( n + m + \\log k + nT( n + m,m ) ) )$ where $S( s,t )$ denotes the time complexity of an algorithm for ordinary Chinese postman problems and $T( s,t )$ denotes the time complexity of a post-optimal algorithm for non-bipartite matching problems defined on a graph with s vertices and t edges.\n                \\item[Reference] \\cite{Saruwatari1993}\n            \\end{description}\n     \\subsection{Tree}\n        \\subsubsection{Enumeration of all binary trees with fixed number leaves in lexicographically}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All binary trees with $n$ leaves in lexicographically. \n                \\item[Complexity] $O(1)$ time per binary tree. \n                \\item[Reference] \\cite{Ruskey1977}\n            \\end{description}\n        \\subsubsection{Enumeration of all $t$-ary trees with fixed number leaves in lexicographically}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All $t$-ary trees with $n$ leaves in lexicographically. \n                \\item[Complexity] $O(t)$ time per binary tree. \n                \\item[Reference] \\cite{Ruskey1978a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-ary trees with $n$ vertices\n                \\item[Complexity] $O(1)$ time per solution\n                \\item[Reference] \\cite{Trojanowski1978}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All binary trees with $n$ vertices. \n                \\item[Reference] \\cite{Rotem1978a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-ary trees with $n$ vertices\n                \\item[Reference] \\cite{Zaks1979}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Beyer1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Reference] \\cite{Proskurowski1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees in lexicographically}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All $k$-ary trees with $n$ internal vertices in lexicographically\n                \\item[Complexity] $O(1- (k-1)^{k-1}\/k^k)^{-1}$ time per solution. This limit is $4\/3$ for the binary case. \n                \\item[Reference] \\cite{Zaks1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all regular $k$-ary trees with $n$ ndoes}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$.\n                \\item[Output] All regular $k$-ary trees with $n$ ndoes.\n                \\item[Reference] \\cite{Zaks1982}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices in the lexicographic ordering}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices in the lexicographic ordering\n                \\item[Complexity] $O(1)$ time per solution on average.\n                \\item[Reference] \\cite{Zerling1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(n)$ time per solution.\n                \\item[Reference] \\cite{Proskurowski1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all ordered trees with $n$ internal vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All ordered trees with $n$ internal vertices.\n                \\item[Reference] \\cite{Er1985}\n            \\end{description}\n        \\subsubsection{Enumeration of all free trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All free trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Reference] \\cite{Wright1986}\n            \\end{description}\n        \\subsubsection{Enumeration of all $t$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Integers $t$ and $n$.\n                \\item[Output] All $t$-ary trees with $n$ vertices.\n                \\item[Reference] \\cite{Er1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Reference] \\cite{Lucas1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all trees with $n$ vertices and $m$ leaves}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $m$.\n                \\item[Output] All trees with $n$ vertices and $m$ leaves\n                \\item[Reference] \\cite{Pallo1987}\n            \\end{description}\n        \\subsubsection{Enumeration of all $t$-ary trees in A-order}\n            \\begin{description}\n                \\item[Input] Integers $t$ and $n$.\n                \\item[Output] All $t$-ary trees with $n$ vertices in A-order.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{VanBaronaigien1988}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ leaves.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Comment] A strong Gray code can be listed in constant average time per solution.\n                \\item[Reference] \\cite{Ruskey1990}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Comment] A loopless generation algorithm is an algorithm where the amount of computation to go from one object to the next is $O(1)$.\n                \\item[Reference] \\cite{VanBaronaigien1991}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees in natural order}\n            \\begin{description}\n                \\item[Input] Two integers $k$ and $n$.\n                \\item[Output] All $k$-ary trees with $n$ vertices.\n                \\item[Reference] \\cite{Er1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Reference] \\cite{Lucas1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All binary trees with $n$ vertices.\n                \\item[Reference] \\cite{Bapiraju1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary tree}\n            \\begin{description}\n                \\item[Input] An integer $k$.\n                \\item[Output] All $k$-ary trees.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Reference] \\cite{Korsh1994}\n            \\end{description}\n        \\subsubsection{Enumeration of all binary trees}\n            \\begin{description}\n                \\item[Output] All binary trees.\n                \\item[Complexity] $O(1)$ time per tree.\n                \\item[Reference] \\cite{Xiang1997}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] Two integers $k$ and $n$.\n                \\item[Output] All $k$-ary trees with $n$ vertices.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Comment] Shifts and loopless algorithm.\n                \\item[Reference] \\cite{Korsh1998}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with at most $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted plane trees with at most $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with exactly $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted plane trees with exactly $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with at most $n$ vertices and the maximum degree $D$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $D$.\n                \\item[Output] All rooted plane trees with at most $n$ vertices and the maximum degree $D$.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted plane trees with exactly $n$ vertices and exactly $c$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted plane trees with exactly $n$ vertices and exactly $c$ leaves .\n                \\item[Complexity] $O(n-c)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{rooted plane tree} is a rooted tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all plane trees with exactly $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All plane trees with exactly $n$ vertices.\n                \\item[Complexity] $O(n^3)$ time per tree with $O(n)$ space.\n                \\item[Comment] A \\textit{plane tree} is a tree with a left-to-right ordering specified for the children of each vertex.\n                \\item[Reference] \\cite{Nakano2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted trees with at most $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All rooted tree with at most $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree and $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2003}\n            \\end{description}\n        \\subsubsection{Enumeration of all $n$-trees}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All $n$-trees.\n                \\item[Complexity] $O(n^4N)$ total time.\n                \\item[Comment] Reverse search. $N$ is the number of solutions.\n                \\item[Reference] \\cite{Aringhieri2003}\n            \\end{description}\n        \\subsubsection{Enumeration of all trees with $n$ vertices and $d$ diameter}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $d$.\n                \\item[Output] All trees with $n$ vertices and $d$ diameter.\n                \\item[Complexity] $O(1)$ time per tree with $O(n)$ space.\n                \\item[Comment] By using the algorithm for each $d = 2, \\dots, n-1$, all trees can be enumerated.\n                \\item[Reference] \\cite{Nakano2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all $c$-tree with at most $v$ vertices and diameter $d$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $d$\n                \\item[Output] All $c$-tree with at most $v$ vertices and diameter $d$.\n                \\item[Complexity] $O(1)$ time per tree.\n                \\item[Comment] A tree is a $c$-tree if each vertex has a color $c \\in \\{c_1, \\dots, c_m\\}$.\n                \\item[Reference] \\cite{Nakano2005a}\n            \\end{description}\n        \\subsubsection{Enumeration of all nonisomorphic rooted plane trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All nonisomorphic rooted plane trees with $n$ vertices.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Sawada2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all nonisomorphic free plane trees with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All nonisomorphic free plane trees with $n$ vertices.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Sawada2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all multitrees satisfying given constraints}\n            \\begin{description}\n                \\item[Input] A set $\\Sigma$ of labels, a function $val: \\Sigma \\to \\mathbb{Z}^+$, and a feature vector $g$ of level $K$.\n                \\item[Output] All $(\\Sigma, val)$-labeled multitrees $T$ such that $f_K(T) = g$ and $deg(v;T) = val(\\ell(v))$ for all vertices $v \\in T$.\n                \\item[Comment] This algorithm is for chemical graphs.\n                \\item[Reference] \\cite{Ishida2008}\n            \\end{description}\n        \\subsubsection{Enumeration of all ordered trees with $n$ vertices and $k$ leaves}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $k$.\n                \\item[Output] All ordered trees with $n$ vertices and $k$ leaves.\n                \\item[Complexity] $O(1)$ delay and $O(n)$ space.\n                \\item[Reference] \\cite{Yamanaka2009a}\n            \\end{description}\n        \\subsubsection{Enumeration of all trees with specified degree sequence}\n            \\begin{description}\n                \\item[Input] A degree sequence $D$.\n                \\item[Output] All trees with $D$.\n                \\item[Complexity] $O(1)$ time per tree.\n                \\item[Reference] \\cite{Nakano2009}\n            \\end{description}\n     \\subsection{Triangle}\n        \\subsubsection{Enumeration of all minimal triangle graphs with a fixed number vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All minimal triangle graphs with $n$ vertices. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Bowen1967b}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangles in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All triangles in $G$.\n                \\item[Complexity] $O(\\alpha(G)|E|)$ total time and linear space.\n                \\item[Comment] $\\alpha(G)$ is the minimum number of edge-disjoint spanning forests into which $G$ can be decomposed.  If $G$ is planar, then the time complexity becomes $O(|V|)$.\n                \\item[Reference] \\cite{Chiba1985}\n            \\end{description}\n     \\subsection{Triangulation}\n        \\subsubsection{Enumeration of all triangulations of 2-sphere}\n            \\begin{description}\n                \\item[Input] A 2-sphere $G$.  \n                \\item[Output] All triangulations of $G$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Bowen1967a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $r$-rooted $2$-connected triangulations of a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $r$.\n                \\item[Output] All $r$-rooted $2$-connected triangulations of $G$.\n                \\item[Complexity] $O(|V|^2N)$ total time and $O(|V|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996b}\n            \\end{description}\n        \\subsubsection{Enumeration of all $r$-rooted $3$-connected triangulations of a planar graph}\n            \\begin{description}\n                \\item[Input] A planar graph $G = (V, E)$ and an integer $r$.\n                \\item[Output] All $r$-rooted $3$-connected triangulations of $G$.\n                \\item[Complexity] $O(|V|^2N)$ total time and $O(|V|)$ space.\n                \\item[Comment] $N$ is the number of solutions.\n                \\item[Reference] \\cite{Avis1996b}\n            \\end{description}\n        \\subsubsection{Enumeration of all based plane triangulations with $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All based plane triangulations.\n                \\item[Complexity] $O(1)$ time per based plane triangulation with $O(n)$ space.\n                \\item[Comment] A \\textit{based plane triangulation} is a plane triangulation with one designated edge on the outer face. The algorithm does not output entire solution but output the difference from the previous solution.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected based plane triangulations with $n$ vertices and $r$ vertices on the outer face}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $r$.\n                \\item[Output] All biconnected based plane triangulations with $n$ vertices and $r$ vertices on the outer face.\n                \\item[Complexity] $O(1)$ time per based plane triangulation with $O(n)$ space.\n                \\item[Comment] A \\textit{based plane triangulation} is a plane triangulation with one designated edge on the outer face. The algorithm does not output entire solution but output the difference from the previous solution.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected plane triangulations with $n$ vertices and $r$ vertices on the outer face}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $r$.\n                \\item[Output] All biconnected based plane triangulations with $n$ vertices and $r$ vertices on the outer face.\n                \\item[Complexity] $O(r^2n)$ time per based plane triangulation with $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2004a}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted triconnected plane triangulations with at most $n$ vertices}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected rooted plane triangulations with at most $n$ vertices.\n                \\item[Complexity] $O(1)$ time per tree and $O(n)$ space.\n                \\item[Comment] A \\textit{rooted} plane triangulation is a plane triangulation with one designated vertex on the outer face.\n                \\item[Reference] \\cite{Nakano2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all rooted triconnected plane triangulations with exactly $n$ vertices and $r$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected rooted plane triangulations with exactly $n$ vertices and $r$ leaves.\n                \\item[Complexity] $O(r)$ time per tree and $O(n)$ space.\n                \\item[Comment] A \\textit{rooted} plane triangulation is a plane triangulation with one designated vertex on the outer face.\n                \\item[Reference] \\cite{Nakano2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all triconnected plane triangulations with exactly $n$ vertices and $r$ leaves}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All triconnected plane triangulations with exactly $n$ vertices and $r$ leaves.\n                \\item[Complexity] $O(r^n)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2001}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations}\n            \\begin{description}\n                \\item[Input] A set $S$ of $n$ points in the general position in the plane.\n                \\item[Output] all the triangulations whose vertex set is $S$ and edge set includes the convex hull of $S$.\n                \\item[Complexity] $O(\\log\\log n)$ time per triangulation and linear space.\n                \\item[Comment] Whether there is the algorithm that outputs all triangulations in constant time delay?\n                \\item[Reference] \\cite{Bespamyatnikh2002}\n            \\end{description}\n        \\subsubsection{Enumeration of all biconnected plane triangulations with $n$ vertices and $r$ vertices on the outer faces}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $r$.\n                \\item[Output] All biconnected plane triangulations with $n$ vertices and $r$ vertices on the outer faces.\n                \\item[Complexity] $O(rn)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Nakano2004b}\n            \\end{description}\n        \\subsubsection{Enumeration of all triangulations of a triconnected plane graph of $n$ vertices}\n            \\begin{description}\n                \\item[Input] A triconnected planar graph $G$ with $n$ vertices.\n                \\item[Output] All triangulations of $G$.\n                \\item[Complexity] $O(1)$ time per triangulation and $O(n)$ space.\n                \\item[Reference] \\cite{Parvez2009}\n            \\end{description}\n     \\subsection{Vertex cover}\n        \\subsubsection{Enumeration of all minimal vertex covers in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal vertex covers of size up to $k$ in $G$.\n                \\item[Complexity] $O^*(1.6181^k)$ total time.\n                \\item[Comment] This algorithm also lists some non-minimal vertex covers. This algorithm uses compact representation technique.\n                \\item[Reference] \\cite{Fernau2006}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal vertex covers of size at most $k$ in a graph}\n            \\begin{description}\n                \\item[Input] A graph $G = (V, E)$.\n                \\item[Output] All minimal vertex covers of size at most $k$ in $G$.\n                \\item[Complexity] $O(|E| + k^2 2^k)$ total time.\n                \\item[Reference] \\cite{Damaschke2006}\n            \\end{description}\n\\section{Hypergraph}\n     \\subsection{Acyclic subhypergraph}\n        \\subsubsection{Enumeration of all maximal $\\alpha$-acyclic subhypergraphs in a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $H = (V, \\mathcal{E})$.\n                \\item[Output] All maximal $\\alpha$-acyclic subhypergraphs in $H$.\n                \\item[Complexity] $O(|\\mathcal{E}|^2(|V|+|\\mathcal{E}|))$ delay and $O(|\\mathcal{E}|)$space.\n                \\item[Comment] The name of their algorithm is $\\mathtt{GenMAS}$. This algorithm uses the algorithm $\\mathtt{FindMAS}$ that outputs a maximal $\\alpha$-acyclic subhypergraph.\n                \\item[Reference] \\cite{Daigo2009}\n            \\end{description}\n        \\subsubsection{Enumeration of all Berge acyclic subhypergraphs in a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H}$.\n                \\item[Output] All Berge acyclic subhypergraphs in $\\mathcal{H}$.\n                \\item[Complexity] $O(rd\\tau(m))$ time per subhypergraph.\n                \\item[Comment] $r$ and $d$ are the rank and the degree of $\\mathcal{H}$ and $\\tau(m) = O((\\log\\log m)^2 \/ \\log \\log \\log (m))$.\n                \\item[Reference] \\cite{Wasa2013}\n            \\end{description}\n     \\subsection{Independent set}\n        \\subsubsection{Enumeration of all maximal independent set of a hypergraph of bounded dimension}\n            \\begin{description}\n                \\item[Input] A hypergraph $H$ of bounded dimension.\n                \\item[Output] All maximal independent set of a hypergraph of bounded dimension. \n                \\item[Comment] The proposed algorithm runs in parallel.\n                \\item[Reference] \\cite{Boros2000a}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a $c$-conformal hypergraph}\n            \\begin{description}\n                \\item[Input] A $c$-conformal hypergraph $\\mathcal{H} \\in A(k, r)$, where $c \\le$ constant and $k + r \\le c$.\n                \\item[Output] All maximal independent sets in $\\mathcal{H}$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Comment] $A(k, r)$ is the class of hyperedges with $(k, r)$-bounded intersections, i.e. in which the intersection of any $k$ distinct hyperedges has size at most $r$.\n                \\item[Reference] \\cite{Boros2004}\n            \\end{description}\n        \\subsubsection{Enumeration of all maximal independent sets in a hypergraph of bounded intersections}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H} \\in A(k, r)$, where $k + r \\le$ constant.\n                \\item[Output] All maximal independent sets in $\\mathcal{H}$.\n                \\item[Complexity] Incremental polynomial time with polynomial space.\n                \\item[Comment] $A(k, r)$ is the class of hyperedges with $(k, r)$-bounded intersections, i.e. in which the intersection of any $k$ distinct hyperedges has size at most $r$.\n                \\item[Reference] \\cite{Boros2004}\n            \\end{description}\n     \\subsection{Transversal}\n        \\subsubsection{Enumeration of all minimal transversal of a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H}$.\n                \\item[Output] All minimal transversal of $\\mathcal{H}$.\n                \\item[Complexity] $(n + N)^{O(\\log n)}$ total time with $O(n \\log n)$ words.\n                \\item[Comment] $n = \\sum_{X \\in \\mathcal{H}}|X|$ and $N$ is the number of solutions.\n                \\item[Reference] \\cite{Tamaki2000}\n            \\end{description}\n        \\subsubsection{Enumeration of all minimal transversal in a hypergraph}\n            \\begin{description}\n                \\item[Input] A hypergraph $\\mathcal{H}\\in A(k, r)$.\n                \\item[Output] All minimal transversal in $\\mathcal{H}$.\n                \\item[Complexity] $O(n^{k+r+1}|\\mathcal{H}^d|^{r+1})$ total time and $O(N^{r+1})$ total space.\n                \\item[Comment] $A(k, r)$ is the class of hyperedges with $(k, r)$-bounded intersections, i.e. in which the intersection of any $k$ distinct hyperedges has size at most $r$.  Minimal transversals of hypergraphs in some restricted classes can be enumerating in polynomial delay and space.\n                \\item[Reference] \\cite{Boros2004}\n            \\end{description}\n\\section{Matroid}\n     \\subsection{Basis}\n        \\subsubsection{Enumeration of all common bases in two matroids}\n            \\begin{description}\n                \\item[Input] Two matroids  $M_1 = (E, \\beta_1)$ and $M_2 = (E, \\beta_2)$.\n                \\item[Output] All $B \\in \\beta_1 \\cap \\beta_2$.\n                \\item[Complexity] $O(|E| ( |E|^2 + t)\\lambda)$ total time and $O(|E|^2)$ space.\n                \\item[Comment] $\\lambda$ is the number of common bases and $t$ is time complexity of one pivot operation.\n                \\item[Reference] \\cite{Fukuda1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all basis of a matroid}\n            \\begin{description}\n                \\item[Input] A matroid $M$ on the ground set $P$ with rank $m$.\n                \\item[Output] All basis of $M$.\n                \\item[Complexity] $O((m|P| + t(Piv))N)$ total time and space complexity independent of $N$.\n                \\item[Comment] $N$ is the number of solutions. $t(Piv)$ is the time necessary to do one pivot operation.\n                \\item[Reference] \\cite{Avis1996}\n            \\end{description}\n     \\subsection{Spanning}\n        \\subsubsection{Enumeration of all minimal spanning and connected subsets in a matroid}\n            \\begin{description}\n                \\item[Input] A matroid $M$.\n                \\item[Output] All minimal spanning and connected subsets in $M$.\n                \\item[Complexity] Incremental quasi-polynomial time.\n                \\item[Comment] $f(x)$ is a quasi-polynomial if $f(x) \\in O(2^{polylog (n)})$.\n                \\item[Reference] \\cite{Khachiyan2006a}\n            \\end{description}\n     \\subsection{Subset}\n        \\subsubsection{Enumeration of all maximal subset}\n            \\begin{description}\n                \\item[Input] A binary matroid $M$ on ground set $S$ and $B = \\{b_1, b_2\\} \\subseteq S$.\n                \\item[Output] All maximal subsets $X$ of $A := S \\setminus B$ that span neither $b_1$ nor $b_2$.\n                \\item[Complexity] Incremental polynomial time.\n                \\item[Reference] \\cite{Khachiyan2008a}\n            \\end{description}\n\\section{Order}\n     \\subsection{Ideal}\n        \\subsubsection{Enumeration of all $k$-cardinarity ideals of a $w$-element poset}\n            \\begin{description}\n                \\item[Input] An integer $k$ and a poset $P$ with $w$ element, where $k \\le w$.\n                \\item[Output] All $k$-cardinarity ideals of $P$.\n                \\item[Complexity] $O(Nw^3)$ total time.\n                \\item[Comment] $N$ is the number of ideals.\n                \\item[Reference] \\cite{Wild2013}\n            \\end{description}\n\\section{Other}\n     \\subsection{Assignment}\n        \\subsubsection{Enumeration of all assignment}\n            \\begin{description}\n                \\item[Input] An integer $n$ and $n \\times n$ cost matrix $C = (c_{ij})$.\n                \\item[Output] All assignments that minimizes $\\sum^{i=n}_{i=1}\\sum^{j=n}_{j=1} c_{ij}x_{ij}$ subject to $\\sum^{i=n}_{i=1} x_{ij} = 1 \\; (j = 1, \\dots, n)$, $\\sum^{j=n}_{j=1} x_{ij} = 1\\; (i = 1, \\dots, n)$, and $x_{ij} \\ge 0$.\n                \\item[Reference] \\cite{Murty1968}\n            \\end{description}\n     \\subsection{Full disjunction}\n        \\subsubsection{Enumeration of all full disjunction in an acyclic set}\n            \\begin{description}\n                \\item[Input] An acyclic set of relation $R$ with $N$ tuples.\n                \\item[Output] All full disjunctions of $R$.\n                \\item[Complexity] $O(N)$ delay.\n                \\item[Reference] \\cite{Cohen2006}\n            \\end{description}\n     \\subsection{Matrix}\n        \\subsubsection{Enumeration of all minimal sets of at most $k$ rows the deletion of which leaves a PP matrix}\n            \\begin{description}\n                \\item[Input] A binary $n \\times m$ matrix $B$ and a positive integer $k$, where $n > 4k$.\n                \\item[Output] All minimal sets of at most $k$ rows the deletion of which leaves a PP matrix.\n                \\item[Complexity] $O(3^k nm)$ time.\n                \\item[Comment] A PP matrix is a perfect phylogeny matrix.\n                \\item[Reference] \\cite{Damaschke2006}\n            \\end{description}\n     \\subsection{Round-robin tournament score}\n        \\subsubsection{Enumeration of all round-robin tournament scores of $n$ players}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All round-robin tournament scores of $n$ players. \n                \\item[Reference] \\cite{Narayana1971}\n            \\end{description}\n\\section{Permutation}\n     \\subsection{Arrangements}\n        \\subsubsection{Enumeration of all arrangements with $n$ marks}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All arrangements with $n$ marks. \n                \\item[Complexity] $O(n!)$ total time. \n                \\item[Reference] \\cite{Wells1961}\n            \\end{description}\n        \\subsubsection{Enumeration of all arrangements with $n$ marks}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All arrangements with $n$ marks. \n                \\item[Complexity] $O(n!)$ total time. \n                \\item[Reference] \\cite{Johnson1963}\n            \\end{description}\n     \\subsection{Ladder lottery}\n        \\subsubsection{Enumeration of all optimal ladder lotteries}\n            \\begin{description}\n                \\item[Input] A permutation.\n                \\item[Output] All optimal ladder lotteries with satisfying the permutation.\n                \\item[Complexity] $O(1)$ time per solution on average, $O(n^2)$ space, and $O(n^2)$ time preprocessing.\n                \\item[Comment] $n$ is the length of the input permutation.  We call a ladder lottery is an optimal when the number of horizontal lines in the ladder lottery is minimum. Ladder lotteries are also known as arrangements of pseudolines.\n                \\item[Reference] \\cite{Yamanaka2010}\n            \\end{description}\n        \\subsubsection{Enumeration of all ladder lotteries with $k$ bars}\n            \\begin{description}\n                \\item[Input] A permutation $\\pi$ and integer $k$.\n                \\item[Output] All ladder lotteries of $\\pi$ with $k$ bars.\n                \\item[Complexity] $O(1)$ time per ladder lottery.\n                \\item[Comment] Ladder lotteries are also known as \\textit{Amida kuji} in Japan.\n                \\item[Reference] \\cite{Yamanaka2014}\n            \\end{description}\n     \\subsection{Set}\n        \\subsubsection{Enumeration of all permutations of a set of elements}\n            \\begin{description}\n                \\item[Input] A set $S$. \n                \\item[Output] All permutations of $S$\n                \\item[Comment] He proposed the general algorithm for some combinatorial problems. \n                \\item[Reference] \\cite{Ehrlich1973a}\n            \\end{description}\n        \\subsubsection{Enumeration of all permutations of a set of elements}\n            \\begin{description}\n                \\item[Input] A set $S$. \n                \\item[Output] All permutations of $S$\n                \\item[Reference] \\cite{Dershowitz1975}\n            \\end{description}\n\\section{SAT}\n     \\subsection{Boolean CSP}\n        \\subsubsection{Enumeration of all models of $\\phi$ by non-decreasing weight}\n            \\begin{description}\n                \\item[Input] A $\\Gamma$-formula $\\phi$.\n                \\item[Output] All models of $\\phi$ by non-decreasing weight.\n                \\item[Complexity] If $\\Gamma$ is Horn or width-2 affine, there exists a polynomial delay algorithm.\n                \\item[Comment] Otherwise, such an algorithm does not exist unless $P \\neq NP$.\n                \\item[Reference] \\cite{Creignou2011}\n            \\end{description}\n\\section{Set}\n     \\subsection{Bitstring}\n        \\subsubsection{Enumeration of all bitstrings of length $n$ that contains exactly $k$ 1's}\n            \\begin{description}\n                \\item[Input] An even integer $n$ and odd integer $k$.\n                \\item[Output] All bitstrings of length $n$ that contains exactly $k$ 1's.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Ruskey1988}\n            \\end{description}\n     \\subsection{Ideals}\n        \\subsubsection{Enumeration of all ideals in a poset}\n            \\begin{description}\n                \\item[Input] A poset $\\mathcal{P}$.\n                \\item[Output] All ideals in $\\mathcal{p}$.\n                \\item[Complexity] $O(1)$ delay.\n                \\item[Reference] \\cite{Koda1993}\n            \\end{description}\n     \\subsection{Partition}\n        \\subsubsection{Enumeration of all paritions in natural order}\n            \\begin{description}\n                \\item[Input] An integer $n$. \n                \\item[Output] All partitions of $n$ in natural order\n                \\item[Reference] \\cite{McKay1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all paritions with restriction}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All partitions of $n$ whose the smallest part is greater than or equal to $k$. \n                \\item[Reference] \\cite{White1970}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-partitions of $n$}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-partitions of $n$. \n                \\item[Reference] \\cite{Narayana1971}\n            \\end{description}\n        \\subsubsection{Enumeration of all paritions of an integer}\n            \\begin{description}\n                \\item[Input] An integer $n$.  \n                \\item[Output] All paritions of $n$. \n                \\item[Complexity]  \n                \\item[Reference] \\cite{Fenner1980}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of a set}\n            \\begin{description}\n                \\item[Input] A set $S$.\n                \\item[Output] All partitions of $S$.\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Semba1984}\n            \\end{description}\n        \\subsubsection{Enumeration of all partions of $n$ into integers of size at most $k$}\n            \\begin{description}\n                \\item[Input] Integers $n$ and $k$.\n                \\item[Output] All partions of $n$ into integers of size at most $k$.\n                \\item[Reference] \\cite{Savage1989}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of a set into a fixed number of blocks}\n            \\begin{description}\n                \\item[Input] An integer $k$.\n                \\item[Output] All partitions of a set into $k$ blocks\n                \\item[Complexity] $O(1)$ amortized time per solution.\n                \\item[Reference] \\cite{Ruskey1993}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of an integer $n$}\n            \\begin{description}\n                \\item[Input] Three integers $n$, $k$, and $\\sigma$.\n                \\item[Output] All partitions $P_\\sigma(n, k)$ of $n$ into parts of size at most $k$ in which parts are congruent to $1$ modulo $\\sigma$.\n                \\item[Complexity] $O(N)$ total time. E.g., $P_3(11, 8) = P_3(11, 7) = \\left\\{ \\{7, 4\\}, \\{7, 1, 1, 1, 1\\}, \\{4, 4, 1, 1, 1\\}, \\{4, 1, 1, 1, 1, 1, 1, 1\\}, \\{1, \\dots, 1\\}\\right\\}$.\n                \\item[Comment] $N$ is the number of partitions.\n                \\item[Reference] \\cite{Rasmussen1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all partitions of an integer $n$}\n            \\begin{description}\n                \\item[Input] Two integers $n$ and $k$.\n                \\item[Output] All partitions $D(n, k)$ of $n$ into distinct parts of size at most $k$.\n                \\item[Complexity] $O(N)$ total time. E.g., $D(10, 5) = \\{\\{5, 4, 1\\}, \\{5, 3, 2\\}, \\{4, 3, 2, 1\\}\\}$ and $D(11, 4) = \\emptyset$.\n                \\item[Comment] $N$ is the number of partitions.\n                \\item[Reference] \\cite{Rasmussen1995}\n            \\end{description}\n        \\subsubsection{Enumeration of all partition of $\\{1, \\dots, n\\}$ into $k$ non-empty subsets}\n            \\begin{description}\n                \\item[Input] An integer $k$.\n                \\item[Output] All partition of $\\{1, \\dots, n\\}$ into $k$ non-empty subsets.\n                \\item[Complexity] $O(1)$ time per solution.\n                \\item[Comment] The number of such partitions is known as the Stirling number of the second kind.\n                \\item[Reference] \\cite{Kawano2005b}\n            \\end{description}\n        \\subsubsection{Enumeration of all integer partitions in (anti-)lexicographical order}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All integer partitions of $n$.\n                \\item[Complexity] $O(1)$ time per solution on average.\n                \\item[Reference] \\cite{Stojmenovic2007}\n            \\end{description}\n\\section{String}\n     \\subsection{Binary string}\n        \\subsubsection{Enumeration of all binary string with fixed number ones}\n            \\begin{description}\n                \\item[Input] Two integers $n$ and $k$, where $n \\ge k$.  \n                \\item[Output] All binary string with length $n$ and $k$ ones. \n                \\item[Complexity] $O(1)$ time per binary string. \n                \\item[Reference] \\cite{Bitner1976}\n            \\end{description}\n     \\subsection{Bracelet}\n        \\subsubsection{Enumeration of all $k$-ary bracelets}\n            \\begin{description}\n                \\item[Input] $n$: a length of a bracelet, $k$: a number of alphabet size.\n                \\item[Output] All $k$-ary bracelets.\n                \\item[Complexity] $O(1)$ amortized per output and $O(n)$ space.\n                \\item[Comment] A bracelet is the lexicographically smallest element of an equivalence class of $k$-ary strings under string rotation and reversal.\n                \\item[Reference] \\cite{Sawada2001}\n            \\end{description}\n     \\subsection{Lyndon word}\n        \\subsubsection{Enumeration of all $k$-ary Lyndon brackets of length $n$}\n            \\begin{description}\n                \\item[Input] Two integers $k$ and $n$.\n                \\item[Output] All $k$-ary Lyndon brackets of length $n$.\n                \\item[Complexity] $O(n)$ time per solution.\n                \\item[Reference] \\cite{Sawada2003}\n            \\end{description}\n     \\subsection{Necklace}\n        \\subsubsection{Enumeration of all $k$-color necklaces with $n$ beads}\n            \\begin{description}\n                \\item[Input] Integers $k$ and $n$. \n                \\item[Output] All $k$-color necklaces with $n$ beads\n                \\item[Reference] \\cite{Fredricksen1978a}\n            \\end{description}\n        \\subsubsection{Enumeration of all necklaces of length $n$ with two colors}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All necklaces of length $n$ with two colors.\n                \\item[Reference] \\cite{Fredricksen1986b}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces}\n            \\begin{description}\n                \\item[Input] $n$: a length of a necklace, $k$: a number of alphabet size.\n                \\item[Output] All $k$-ary necklaces with length $n$.\n                \\item[Complexity] $O(1)$ amortized per output and $O(n)$ space.\n                \\item[Comment] A $k$-ary necklace is an equivalence class of k-ary strings under rotation.\n                \\item[Reference] \\cite{Ruskey1992}\n            \\end{description}\n        \\subsubsection{Enumeration of all $n$-bit necklaces with fixed density $d$}\n            \\begin{description}\n                \\item[Input] Two integer $n$ and $d$.\n                \\item[Output] All $n$-bit necklaces with fixed density $d$.\n                \\item[Complexity] $O(nN)$ total time and $O(n)$ space.\n                \\item[Comment] $N$ is the number of solutions. A density of a $n$-bit necklace $T$ is $d$ if $T$ has $d$ ones.\n                \\item[Reference] \\cite{Wang1996}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces with fixed density}\n            \\begin{description}\n                \\item[Input] $n$: a length of a necklace, $k$: a number of alphabet size, $d$: a number of nonzero characters.\n                \\item[Output] All $k$-ary necklaces with fixed density $d$.\n                \\item[Complexity] $O(1)$ amortized per output and $O(n)$ space.\n                \\item[Comment] The set of $3$-ary necklace with $2$-density and $4$-length is $N_3(4,2) = \\{0011, 0012, 0021, 0022, 0101, 0102, 0202\\}$.\n                \\item[Reference] \\cite{Ruskey1999}\n            \\end{description}\n        \\subsubsection{Enumeration of all strings of some family}\n            \\begin{description}\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Comment] This algorithm can list not only all necklaces but also all strings in other some family with CAT.\n                \\item[Reference] \\cite{Cattell2000}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces with fixed content of length $n$}\n            \\begin{description}\n                \\item[Input] Two integer $k$ and $n$, and a content.\n                \\item[Output] All $k$-ary necklaces with fixed content of length $n$.\n                \\item[Complexity] Constant amortized time per solution.\n                \\item[Reference] \\cite{Sawada2003a}\n            \\end{description}\n     \\subsection{Parenthesis}\n        \\subsubsection{Enumeration of all well-formed parenthesis with length $2n$}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All well-formed parenthesis with length $2n$ in lexicographical ordering.\n                \\item[Reference] \\cite{Er1983}\n            \\end{description}\n        \\subsubsection{Enumeration of all well-formed parenthesis strings of size $n$}\n            \\begin{description}\n                \\item[Input] An integer $n$.\n                \\item[Output] All well-formed parenthesis strings of size $n$.\n                \\item[Complexity] $O(1)$ delay with $O(n)$ space, or $O(n)$ delay with $O(1)$ space.\n                \\item[Reference] \\cite{Walsh1998}\n            \\end{description}\n     \\subsection{Substring}\n        \\subsubsection{Enumeration of all $k$-ary strings of length $n$ that have no substring equal to $f$}\n            \\begin{description}\n                \\item[Input] A $k$-ary string $f$ with length $m$, and a positive integer $n$.\n                \\item[Output] All $k$-ary strings of length $n$ that have no substring equal to $f$.\n                \\item[Complexity] $O(1)$ time per string.\n                \\item[Reference] \\cite{Ruskey2000a}\n            \\end{description}\n        \\subsubsection{Enumeration of all circular $k$-ary strings of length $n$ that have no substring equal to $f$}\n            \\begin{description}\n                \\item[Input] A $k$-ary string $f$ with length $m$, and a positive integer $n$.\n                \\item[Output] All circular $k$-ary strings of length $n$ that have no substring equal to $f$.\n                \\item[Complexity] $O(1)$ time per string.\n                \\item[Reference] \\cite{Ruskey2000a}\n            \\end{description}\n        \\subsubsection{Enumeration of all $k$-ary necklaces of length $n$ that have no substring equal to $f$}\n            \\begin{description}\n                \\item[Input] A $k$-ary string $f$ with length $m$, and a positive integer $n$.\n                \\item[Output] All $k$-ary necklaces of length $n$ that have no substring equal to $f$.\n                \\item[Complexity] $O(1)$ time per string.\n                \\item[Comment] $f$ is an aperiodic necklace.\n                \\item[Reference] \\cite{Ruskey2000a}\n            \\end{description}\n\\section{Survey}\n     \\subsection{Enumeration}\n        \\subsubsection{Enumeration of $k$-best enumeration}\n            \\begin{description}\n                \\item[Comment] This is the full version of the Springer Encyclopedia of Algorithms, 2014.\n                \\item[Reference] \\cite{Eppstein2014}\n            \\end{description}\n     \\subsection{Graph}\n        \\subsubsection{Alglrotihms for enumeration of all cycles in a given graph}\n            \\begin{description}\n                \\item[Input] A graph $G$. \n                \\item[Output] All cycles belonging to $G$. \n                \\item[Comment] 26 cycle enumeration algorithms are introduced. \n                \\item[Reference] \\cite{Mateti1976a}\n            \\end{description}\n        \\subsubsection{Enumeration of clique enumeration}\n            \\begin{description}\n                \\item[Comment] In Sec.3.\n                \\item[Reference] \\cite{Pardalos1994}\n            \\end{description}\n        \\subsubsection{Enumeration of gray code algorithms}\n            \\begin{description}\n                \\item[Reference] \\cite{Savage1997}\n            \\end{description}\n     \\subsection{Logic}\n        \\subsubsection{}\n            \\begin{description}\n                \\item[Comment] The author of this paper investigate the class of databases with constant delay the answers to a query.\n                \\item[Reference] \\cite{Segoufin2013}\n            \\end{description}\n\n\\section*{Acknowledgements}\nI am deeply grateful to \nKomei Fukuda, \nLeslie Goldberg, \nChristian Komusiewicz, \nAndrea Marino, \nYasuko Matsui, \nValia Mitsou, \nShin-ichi Nakano, \nRyuhei Uehara, \nand Katsuhisa Yamanaka for enumerating.\n\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEinstein challenged the Copenhagen interpretation of quantum mechanics by proposing with Podolsky and Rosen \\cite{EPR35}\nGedanken-Experiments, briefly called EPR experiments.\nThese experiments were to demonstrate that quantum mechanics is\nincomplete, having missed in its description of physical reality some elements of that reality.\n\nAbout 30 years after the EPR paper, Bell \\cite{BELL93} derived an inequality for functions of elements of physical reality,\nknown since as Bell's inequality, that in his opinion had to be obeyed by all of classical physics, meaning in essence by the\nframework of Einstein's relativity. Wigner transformed Bell's derivation into a set theoretical approach \\cite{WIGN70}.\n\nThe work of Bell and his followers had very important consequences for the views on the foundations of physics.\nA majority of physicists believe that either (i) any physical\ntheory using counterfactually definite functions and obeying Einstein's local realism (see \\ref{separation}) must obey Bell-type inequalities and\/or\n(ii) Wigner's derivation of Bell-type inequalities is only based on set theory (not involving counterfactual reasoning) and\nEinstein's local realism. There is a considerable body of serious mathematical-physical work that has raised objections against\nboth (i) and (ii) (see particularly \\cite{KHRE08b} and references therein as well as \\cite{ROSI14} and \\cite{GEUR98}). We have\nshown in \\cite{HESS16b} that (i) is false and it is the purpose of this paper to show that (ii) is also false.\n\nThe belief in (ii) arose mainly from the work of Wigner \\cite{WIGN70} and a popularized version of Wigner's work by\nd'Espagnat \\cite{ESPA79}. (d'Espagnat, however, also uses counterfactual reasoning that raises additional questions\n\\cite{HESS16b}.) The fact that many EPR-types of experiments have been published that violate Bell's and Wigner's inequalities,\nis therefore presented as proof against the validity of Einstein's local realism, because it appears inconceivable that set\ntheory is incorrect.\n\nThere exists, however, the following problem with this latter assessment. Set theory represents a mathematical framework based\non axioms that may be regarded as definitions. As such, set theory is not related to any measurements and experiments and,\ntherefore, also not to EPR experiments. It is thus necessary to find a connection between the sets of elements of physical reality\nthat the experiments provide and the sets of mathematical abstractions that Wigner uses in his proof. We analyze this\nconnection in great detail and show that Wigner's sets of mathematical abstractions contain a topological-combinatorial ordering\nthat is incompatible with the actual ordering of measurement-events in space and time (or space-time). We suggest that it is\nthis incompatibility that leads to the contradictions of Wigner's work with actual EPR measurements and not any failure of\nEinstein's local realism. An analogous situation was discussed by Poincar\\'e and Einstein, who resolved the experimental\ncontradictions to Euclidean geometry by pointing to the connection of Euclidean geometry with the physical reality, a connection\nachieved by the introduction of the concept of rigid bodies \\cite{EINS82}.\n\n\nAs we will show, there exists an analogy to the Einstein-Poincar\\'e discussions, because Wigner connected his mathematical\nsets to the actual measurements in a way that is inconsistent with the space-time physics of the actual EPR experiments. Key to\nour reasoning is the fact that the set of data of the actual measurements is necessarily involving space- and time- (or\nspace-time-) coordinates to designate the correlated pairs. These sets of space and time coordinates are inconsistent with\nWigner's sets and subsets that are ordered according to location in three-dimensional space and equipment settings in that space,\nbut do not involve measurement times.\n\nWe show in detail that Wigner's derivation of Bell type inequalities (that restrict the possible correlations exhibited by EPR\nmeasurements) does actually not use Einstein's local realism at all. Instead Wigner assumes in a hidden and unjustified way\ncertain topological combinatorial properties of his mathematical sets that are inconsistent with the macroscopic properties of\nthe measurement equipment in space and time as determined by clocks, protractors and meter-measures. The contradictions between\nWigner's (Bell-type) inequalities and well known experiments (see \\ref{exp}) must, therefore be blamed on Wigner's\ntopological-combinatorial assumptions and not on Einstein's local realism.\n\nTo prove this finding, we first analyze the elements of physical reality used in EPR experiments and form the sets of physical\nelements that must be dealt with in Wigner's mathematical proof (see section~\\ref{separation}). We then review the essential portion of\nWigner's mathematical proof and point to his unwarranted assumptions (see section~\\ref{add}). Finally, in section~\\ref{sec4}\nwe deduce from these latter\nassumptions the topological-combinatorial problems of Wigner's work and conclude that the well known line ``death by experiment\"\napplies to Wigner's theory not to Einstein's.\nOur conclusions are summarized in section~\\ref{sec5}.\n\n\n\n\\section{Einstein's realism, EPRB experiments and their resulting data} \\label{separation}\n\n\n\nWe consider in the following only a variation of EPR experiments as suggested by Bohm and, therefore called EPRB experiments\n\\cite{ASPE82b,WEIH98}. This type of experiment involves the measurement of the two single entities of correlated pairs\nof particles. Each single entity is measured at a different set of space-coordinates. These space-coordinates must, from the\nviewpoint of physics, describe the entire local measurement equipment for both measurement locations. For brevity and because\nimportant measurements have actually been performed there, we will denote the two measurement locations and corresponding space\ncoordinates just by ``Tenerife\" and ``La Palma\". The space-like separation of the measurements is a necessity to invoke\nEinstein's separation principle, which is an important factor of the EPR argument and focus of the EPR discussion. Einstein's\nseparation principle excludes all influences between the two measurement stations that are faster than the speed of light in\nvacuum.\nThis principle guarantees a certain independence of the two separated measurement stations and represents\nthe core of what is commonly called Einstein's local realism. A broader discussion and detailed definition of Einstein's local\nrealism has been given by Fine~\\cite{Fine96}.\n\nIt is, therefore, of utmost importance for the discussions of Bell and Wigner to exclude in the actual experiments\ninfluences from the other location that propagate with speed slower or equal to that light. How can one make sure that there exist no\nsuch other influences and in addition that one indeed measures correlated pairs? This goal has been achieved in \\cite{ASPE82b},\n\\cite{WEIH98} and other experiments.\nThese measurements relate to the Clauser-Horn-Shimony-Holt (CHSH) inequality.\nWe note here, however, that all of our\nreasoning applies, as will become evident below, also to this type of inequality. We just discuss the Bell inequality in our\nanalysis, because Wigner's proof referred only to this type of inequality.\n\n\\bigskip\n\\subsection{Switching instruments and pairing particles}\n\nKey to Bell's and Wigner's reasoning is the rapid switching of instruments on both Tenerife and La Palma, just before a\nmeasurement of the spin of an incoming particle is performed. In this way one ensures that the instrument settings on the other\nisland cannot influence a given measurement with the speed of light or lower speeds. What Bell and Wigner wished to find out was\nwhether Einstein's local realism is correct or incorrect. The importance of this fast switching is emphasized over and over by\nBell \\cite{BELL93} in his discussions, but no trace of it can be found in either Bell's or Wigner's formalism and algebra of\ntheir actual proofs. The reason for this obvious lack of correspondence between theory and experiment is the assumption of Bell\nand all his followers that the possible outcomes of pair-measurements are just a given for their mathematical abstractions\n(functions, sets) and the fact how the pairing experimentally occurred is not included in the theory.\n\nHowever, it is precisely this pairing and timing problem, where Einstein embedded the necessity of using a space-time system,\nhis space-time system, as the basis for his Gedanken-Experiments and the theoretical approach with Podolsky and Rosen,\n\\cite{EPR35}, \\cite{HESS15}. There also exists no other known way to pair and correlate space like separated measurement\noutcomes than by space and time measurements.%\n\\footnote{We do not wish to use quantum-physics arguments in our reasoning. However, it must be said that the nature of\nthe experiments and the possibility of quantum fluctuations excludes any other method of pairing and leaves only space and time\nmeasurements. Suppose, for example, that we just count on each island the number of measurements and then assume that those with\nequal number (deduced from the counting) will be selected as correlated pairs (post measurement, of course). Then a single\ncounting error (caused for example by a quantum fluctuation in the measurement equipment) will completely mess up the ordering\nand will lead to pairing of the wrong entities. One must, therefore, use synchronized clocks in the space-like separated\nmeasurement stations, as well as knowledge about the space-like separation in order to identify which detector clicks represent,\nwith high likelihood, measurements of a correlated pair.}\n\nThe timing of the measurements is, thus, a most important element of the physical reality, of the data, and we denote the\nclock-time of the $n$th measurement in Tenerife by $t_n$ and that in La Palma by $t_n'$. Note that subsequent measurement pairs\nare necessarily time-like separated because the switching of the macroscopic settings cannot be instantaneous.  To identify\nactual correlated pairs, some rule needs to be employed that selects the pairs. An example of such a rule would be that the time\ndifference of the measurements at the two different locations (islands) must be smaller than a certain value $W$. We emphasize\nthat through any such rule an instantaneous logical and physical connection is established between the measurements at the two\ndifferent locations that has nothing to do with instantaneous influences, much less with information transfer. Without such a\nconnection, no correlations between the space-like separated measurements can be established. Only with such a connection can\nintricate correlations due to physical law be revealed that now may involve also the many-body dynamics of the measurement\nequipment of both locations.\nIt also must be emphasized that the analysis of the raw data of three different\nEPRB experiments~\\cite{WEIH98,AGUE09,ADEN12,VIST12}\ndo show a significant $W$-dependence of the amount of violations of Bell-type inequalities (actually the CHSH inequality),\nfor values of $W$ that are much smaller than the average time between the detection of photons~\\cite{ZHAO08,RAED13a}.\n\nFrom what follows in this paper, it appears imperative to perform measurements that involve dependencies on the\ntiming and on the instrument settings in relation to that timing.\n\nWigner ignored the measurement times and corresponding connections when considering possible and actual outcomes in his theory\nand we will see that this negligence has serious consequences for his set theoretic calculations of correlations.\n\n\n\\subsection{Elements of physical reality: Notation and Notebook-entries} \\label{exp}\n\n\n\nActual EPRB measurements (e.g. \\cite{ASPE82b}, \\cite{WEIH98}) monitor, in addition to the clock-times of the detector events,\ntwo equipment orientations that determine the spin (polarization) of incoming particles. These registrations, for photons often\nobtained with polarizers and avalanche photodiodes, have been denoted by a variety of symbols such as $horizontal$ and\n$vertical$ etc. and Wigner uses $+$ and $-$ for any particles that he considers.\nWe denote the measurement result for the $n$th\nmeasurement in Tenerife by $s_n$ and that in La Palma by $s_n'$ each being either $-$ or $+$.\n\nAs mentioned, the atoms and molecules constituting the measurement equipment have many body interactions with the incoming\nparticles, from both the view of Einstein's physics and also from the most modern view of quantum physics. Nevertheless, both\nBell, Wigner and all their followers describe this equipment, as the Copenhagen school of quantum mechanics does, just by a\nthree-dimensional unit vector of space, indicating the direction of a polarizer or of a Stern-Gerlach magnet. We denote this unit\nvector for the nth experiment by ${\\bf j}_n$ in Tenerife and ${\\bf j}_n'$ in La Palma, respectively. Wigner assumed that this\nunit vector may assume precisely the same three directions $\\bf a, b, c$ in both Tenerife and La Palma and that these directions\nare randomly chosen on each island in the actual experiments. Using these assumptions and notational conventions we arrive at\nEPRB notebook-entry pairs of the kind:\n\\begin{equation} [s_n, {\\bf j}_n, t_n; s_n', {\\bf j}_n', t_n' ] ,\n\\label{25septn1}\n\\end{equation}\nwhere, as mentioned, both $s_n$ and $s_n'$ may (exclusively) assume the value of either $+$ or $-$ and both ${\\bf\nj}_n$ and ${\\bf j}_n'$ are randomly chosen from the vectors $\\bf a, b, c$. {\\it The parenthesis $[ \\cdot]$ indicates the\nmeasurement of a correlated pair} and the symbol ``;\" denotes the separation of entries for Tenerife and La Palma. The number 9\nof the time-selected pairs thus could be represented by the following actual notebook entry $[+, {\\bf a}, t_9; -, {\\bf b},\nt_9']$. As far as EPRB experiments are concerned such entries are collections of elements of physical reality (in the sense of\nMach) and represent the only physical reality that we need to consider to discuss Wigner's proof.\n\nAs mentioned, both Bell and Wigner use Einstein locality as a tool of reasoning: For carefully designed EPRB experiments they\nreason, that the measurement outcome $s_n$ does not depend on what vector ${\\bf j}_n'$ is chosen in La Palma and $s_n'$ does not\ndepend on ${\\bf j}_n$ in Tenerife.\n\nWigner discussed in his paper three pairs of measurements with the given settings $[{\\bf a, b}], [{\\bf a, c}]$ and $[{\\bf b,\nc}]$, exactly those that Bell had chosen for his inequality. Such a selection gives us the following six-tuple that we call\nhenceforth Bell's six-tuple:\n\n\\begin{equation} [s_k, {\\bf a}, t_k; s_k', {\\bf b}, t_k' ] \\text{ }[s_l, {\\bf a}, t_l; s_l', {\\bf\nc}, t_l' ] \\text{ }[s_m, {\\bf b}, t_m; s_m', {\\bf c}, t_m' ] .\\label{25septn2}\n\\end{equation}\nHere $k=1,\\ldots,M$, $l=M+1,\\ldots,2M$, and $m=2M+1,\\ldots,3M$\nwhere $M$ is a large number determined by the total number of\nmeasurements. The brackets $[ \\cdot]$ indicates again the measurement of a correlated pair.  If we assume that these pairs are derived from\nthe notebook entries (\\ref{25septn1}), then Einstein's local realism does not put any constraint on the possible values of the\npairs $s_k, s_k'$, $s_l, s_l'$ and $s_m, s_m'$. For given measurement times, they may assume any of the possible $2^6$\ncombinations of values of $+$ and $-$, because from both a physical and mathematical point of view there are at this point no restrictions to the possible outcomes except that they are two-valued.\n\nFor each pair, we are interested with Wigner in only two different possibilities when measuring on the individual particles:\neither equal outcomes or unequal outcomes.\nTherefore, for a string of $M$ measurements of pairs, we have $2^M$ different possible outcomes. If we have three such strings\nof pairs we have $2^{3M}$ possible different correlations of the pair outcomes. Wigner reduces the number of correlations that he\nconsiders essential significantly further, by counting only those that yield a different fraction of the final count of equal\nrelative to  unequal outcomes. Simple counting leads then to the conclusion that for each string of $M$ measurements of\npairs we have $M+1$ different possible outcomes that lead to a different fraction of equal and unequal outcomes\n(see Table~\\ref{tab1} for an illustration of the counting procedure).\nTherefore we obtain for the six-tuple (2) $(M+1)^3$, possible outcomes that Wigner counts as different.%\n\\footnote{The mathematical proof for\nthis fact is easily performed by writing out the possible different outcomes in matrix-form (first column all equal, last all\ndifferent) and using the method of complete induction assuming correctness for $M$ and proving the result for $M+1$. We have\npreviously given a very transparent proof by using the numbers $+1$ and $-1$ for the possible equal and different pair-outcomes\nrespectively [8].}\n\n\\begin{table}[ht]\n\\caption{\nFictive example of all possible notebook entries of an experiment illustrating Wigner's method of counting.\nThe ``e'' (``u'') in columns $C_1$, $C_2$, and $C_3$ indicate that a pair yields outcomes that are equal (unequal)\nfor $M=3$ different measurements with the same setting pair and the $2^3 = 8$ possibilities given.\nThe last column labeled ``e\/u'' gives the fraction of equal and unequal correlations $C_1$, $C_2$, and $C_3$.\nThe maximum number of different items in the last column is given by $M+1=4$ in this particular example.\n}\n\\setlength{\\tabcolsep}{10pt}\n\\begin{center}\n    \\begin{tabular}{ | c | c | c | c |}\n    \\hline\n $C_1$ & $C_2$ & $C_3$ & e\/u \\\\\n    \\hline\ne& e  &e  & 3\/0 \\\\\nu& u  &e  & 1\/2 \\\\\nu& u  &u  & 0\/3 \\\\\ne& u  &u  & 1\/2 \\\\\ne& e  &u  & 2\/1 \\\\\nu& e  &u  & 1\/2 \\\\\ne& u  &e  & 2\/1 \\\\\nu& e  &e  & 2\/1 \\\\\n    \\hline\n    \\end{tabular}\n\\end{center}\n\\label{tab1}\n\\end{table}\n\nAs we have mentioned, Einstein's local realism has not been used up to now. We show below that it also is not used in Wigner's\nfurther considerations that lead to his variation of Bell's inequality. Wigner's theory does thus derive, without reference to\nEinstein locality, how many different correlations of equal and different outcomes may occur for the $3M$\npairs of measurements. Nevertheless, Wigner's number of possible different correlations is significantly reduced below the above\nmentioned value of $(M+1)^3$ and is only proportional to $M^2$.\n\nWe ask, therefore, the question: What caused the reduction of the possible number of correlations in Wigner's proof? Our answer\nis that it was not any assumption regarding Einstein's local realism. Instead, the reduction of the number of possible\ncorrelations arises from Wigner's unwarranted mathematical assumption that is related to set theoretic probability theory.\nThe reduction of possible correlations is, therefore, artificial and does not apply to EPR experiments.\nAs mentioned, the celebrated line\n``death by experiment\" is correct but it applies to Wigner's theory and not, as reported in so many sensationalist articles, to Einstein's.\n\n\n\\section{Wigner's additional assumptions} \\label{add}\n\nWigner did not consider any dependence of the measurement outcomes on the measurement times and just assumed that the possible\nor actual outcomes $s_n$ and $s_n'$ involve automatically, so to speak per fiat, a correlated pair. The measurement locations\nand times have, however, topological-combinatorial consequences for the sets of elements of physical reality that are\nrepresented by Wigner's sets of mathematical abstractions. It is just these latter consequences that we discuss here.\n\\footnote{Bell's reasoning starts to deviate here from Wigner because of his introduction of counterfactually\ndefinite functions. We have dealt with this approach in connection with a many body dynamics of particle-equipment interactions\nin \\cite{HESS16b}.}\n\nWigner defines (p. 1007, left column last paragraph of \\cite{WIGN70}) ``domains of the space of the hidden variables\" and claims\n(in agreement with our thinking about possible outcomes outlined above) that ``we have only $2^6$ essentially different domains.\nThese can be characterized by symbols:\n \\begin{equation} (\\sigma_1, \\sigma_2, \\sigma_3; \\tau_1, \\tau_2, \\tau_3)\n , \\label{22octn2}\n\\end{equation}\nall $\\sigma$ and $\\tau$ assuming two possible values: $+$ or $-$, and the $\\sigma$ referring to the first, the\n$\\tau$ to the second, particle.\" Wigner's detailed explanations also point out that the domains characterized by\n$(\\sigma_1,\\sigma_2, \\sigma_3; \\tau_1, \\tau_2, \\tau_3)$\ncorrespond precisely to the respective equipment-settings\n$(\\bf a, b, c; a, b, c)$ which leads to the 9 possible pairings (see Eq.~(6) below)\nfrom which the possible pairs of Bell's six-tuple (2) and possible outcomes\n($s_k, s_k'$, $s_l, s_l'$ and $s_m, s_m'$ in our notation) ) can be selected.\n\nIn the next paragraph Wigner states his most crucial assumption by a definition: ``Let $(\\sigma_1, \\sigma_2, \\sigma_3; \\tau_1,\n\\tau_2, \\tau_3)$ denote henceforth the probability that the hidden parameters assume, for the singlet state of the two spins, a\nvalue lying in the domain which was denoted by this symbol\". As innocuous as this definition looks, it contains a very big\nassumption, the assumption of the existence of a joint probability for the six-tuples (\\ref{22octn2}) of possible outcomes of\nthe measurements, while the actual measurements are only performed in pairs.\n\nWigner's assumption about the joint probability of the domains of variables (parameters) that determine the possible outcomes\nimplies the existence of consistent joint probabilities for the possible and actual outcomes of measurements such as\n$(+, -, -; -, +, -)$. Note, however, that these possible outcomes are now assumed to be listed in six-tuples with the\nsetting-sequence of $\\bf a, b, c$ on each island, in spite of the fact that actually only pairs are measured. From a set\ntheoretic viewpoint we must ask ourselves: Which sets is Wigner discussing? Without doubt these must be sets of measurements of\ntime-correlated pairs.\n\nAs an aside, Bernard d'Espagnat \\cite{ESPA79} presented a counterfactual interpretation of Wigner's work by introducing\nmultiple hypothetical values that  every single particle ``possesses\" and that are assumed to exist in addition to the possible measurement outcomes\nfor particle pairs. We do not make these additional unwarranted assumptions. Our reasoning below does, however, also apply to d'Espagnat's work, with the addition of our findings about\ncounterfactual approaches in \\cite{HESS16b}.%\n\nWigner himself intended, as did Bell originally, to use exclusively elements of physical reality in the sense of Mach and we discuss his\nwork from this point of view only.\n\n\n\\section{Problems with Wigner's set-theoretical assumptions}\\label{sec4}\n\nAs an illustration of how far reaching Wigner's assumptions are, consider Bell's six-tuple (\\ref{25septn2}) which contains three\ncorrelated pairs and add to each of the pairs an arbitrary (artificial) third notebook entry for the setting that is not already\ncontained in the pair-outcomes: setting $\\bf c$ for the first pair, $\\bf b$ for the second and $\\bf a$ for the third.\nAs an example we obtain then for the first pair-column a triple-column:\\footnote{This is precisely what d'Espagnat did\nin  his variation on the theme of Wigner (see p162 of his essay in Scientific American \\cite{ESPA79})}\n\n\\begin{equation} [s_k, {\\bf a}, t_k; s_k', {\\bf b}, t_k' ] \\text{ }\\rightarrow\\text{ } (s_k, {\\bf a}, t_k ; s_k', {\\bf b}, t_k'\n\/ s_?,{\\bf c}, t_?) .\\label{24octn1}\n\\end{equation}\nThe $?$ as subscripts indicate the arbitrariness of choice of the additions.\nWe have separated these additions by $``\/\"$ instead of $``;\"$, because the latter indicates the separation of the measurements\non two islands and it is not determined from which island the additions originate. The three entries are also not all correlated\nin time, because $t_?$ cannot equal any of the other measurement times of the triple. Different settings must correspond to\ndifferent times in Einstein's physics. We did not use the symbol $[ \\cdot ]$ on the right side of (\\ref{24octn1}) but used\ninstead $( \\cdot )$, because the pairing by measurement times cannot be guaranteed for all of them. Together with the other two\npairs of the Bell six-tuple, we obtain thus three triples each with settings $\\bf a, b, c$. All the followers of Bell and\nWigner, not just d'Espagnat but also Leggett-Garg \\cite{LEGG85} and others, assume explicitly or implicitly that the joint\ntriple probabilities exist and are common to all three triples.\n\nHowever, it is a well known fact of probability theory, particularly the set theoretic probability theory of Kolmogorov, that\nthe existence of joint triple (and higher order) probabilities is not guaranteed \\cite{KHRE08b}, \\cite{HESS15}. In the above\nexample, each column of triples may, for example, have different joint triple probabilities and a joint triple probability that\nis common to all three columns of triples may not exist. The proof of the validity of Bell type inequalities requires an\nexistence proof of joint triple (quadruple etc.) probabilities from the underlying physics. Wigner just assumed these\nprobabilities to exist and, therefore, assumed what he had to prove.\n\nThis is an enormously important point that has been overlooked also by Bell and all his followers \\cite{BELL93}. If one claims\nthat Wigner's approach is a set theoretic approach, we need to use a set theoretic definition of probability measures. Such a\ndefinition indeed exists and is, as mentioned, given by the probability framework of Kolmogorov \\cite{FELL68}. That framework\nteaches us that, for a countable number of considered possible outcomes such as $(\\sigma_1, \\sigma_2, \\sigma_3; \\tau_1, \\tau_2,\n\\tau_3)$, one can define consistent joint probabilities if and only if the actual measurements can also be listed or taken in\nform of such six-tuples so that we can establish a one to one correlation of any of Wigner's $\\sigma, \\tau$ with the (possible)\nmeasurement outcomes $s_n$ and $s_n'$. Wigner has thus made a mathematical assumption of serious consequences without having any\nphysical or mathematical reason.\n\n\nSeveral different cases need to be considered to present the full proof that Wigner's approach must not be applied to EPRB\nexperiments. We just present one example that is typical and list the following six-tuple using Wigner's subsets related to\nsettings $\\bf a, b, c$ for both Tenerife and La Palma with a possible choice of measurement times included:\n\n\\begin{equation} (s_h, {\\bf a}, t_h \/\ns_i, {\\bf b}, t_i \/s_j, {\\bf c} ,t_j \\text{ } ; \\text{ }s_i', {\\bf a}, t_i' \/ s_j', {\\bf b}, t_j' \/ s_h', {\\bf c}, t_h').\n\\label{26septn1}\n\\end{equation}\n\nThe symbol $``\/\"$ separates the three Wigner subsets on one given island. The indexes $h, i, j$,\nrepresent natural numbers $1, 2, 3...$ that are now different from those used previously $(k, l, m)$ in the six-tuple\n(\\ref{25septn2}). The reason for using a different notation is that six-tuples (\\ref{25septn2}) and (\\ref{26septn1}) cannot be\ntransformed into each other without violating the space and time correlations of the actual pairing arising from the original\nmeasurements. The particular chosen pairing of six-tuple (\\ref{26septn1}) is determined by the measurement times and thus is\ngiven by $[{\\bf a; c}]$, $[{\\bf b; a}]$ and $[{\\bf c; b}]$, while the pairing of six-tuple (\\ref{25septn2}) is $[{\\bf a; b}]$,\n$[{\\bf a; c}]$ and $[{\\bf b; c}]$ and the pairing by measurement times must be different whenever the topology (location and setting) of the\nmeasurements of the pairs is different.\n\nWigner never included considerations of both measurement times and topology and used six-tuple (\\ref{26septn1}) to\nobtain the possible outcomes for all the following 9 possible pairings:\n\\begin{equation} ({\\bf a; a}), ({\\bf a; b}), [{\\bf a;\nc}], [{\\bf b; a}], ({\\bf b; b}), ({\\bf b; c}), ({\\bf c; a}), [{\\bf c; b}], ({\\bf c; c}). \\label{22octn1}\n\\end{equation}\nWe use again $[ \\cdot ]$ only for the guaranteed correlated pairs (through measurement times).\nFor all other pairs we use $( \\cdot )$.\nFor the pairings of Bell's six-tuple (\\ref{25septn2}) and using (\\ref{22octn2}) Wigner thus obtains:\n\n\\begin{equation} (\\sigma_1,\n{\\bf a}; \\tau_2, {\\bf b}) \\text{ }[\\sigma_1, {\\bf a}; \\tau_3, {\\bf c} ] \\text{ }(\\sigma_2, {\\bf b}; \\tau_3, {\\bf\nc}) .\\label{22octn3}\n\\end{equation}\n\nThe $\\sigma$ and $\\tau$ denote here a certain given outcome of either$+$ or $-$. For\nexample, we may have $\\sigma_1 = +$ , $\\tau_3 = -$, $\\tau_2 = +$ and $\\sigma_2 = -$. This innocently looking fact represents a\nbig restriction for the possible outcomes. The two pairs with settings $(\\bf a; b)$ and $[{\\bf a; c}]$ must now have the same\noutcome $\\sigma_1=+$ in Tenerife and the same is true for the two pairs $[{\\bf a; c}]$ and $({\\bf b; c})$ which now must have\nidentical outcome $\\tau_3= -$ in La Palma in spite of the fact that they must, in principle, correspond to two different pairs\nof actual measurement.\n\nWe consider now the possible correlations for the outcomes of the $M$ six-tuples of Bell (corresponding to $3M$ measurements of pairs) that lead to different ratios of equal and different outcomes of the pairs.\nThe number of these correlations is reduced from the original $(M+1)^3$ to only $2(M+1)^2$ different possibilities of correlations as can be seen from six-tuple (\\ref{22octn3}). If we also\nrequire that $\\tau_2 = \\sigma_2$ as Wigner did, because he argued that equal settings need to have equal outcomes, then we\nobtain only $(M+1)^2$ different possibilities of correlating equal $++,--$ and different $+-,-+$ outcomes for the pairs in\nBell's six-tuples (\\ref{25septn2}). This latter reduction is identical to the reduction obtained by Bell in his inequality.\n\nAs seen from the measurement times, only 3 of the pairings of (\\ref{22octn1}) do correspond to originally correlated pairs.\nNote that $(\\bf a; b)$ and $[\\bf b; a]$ cannot be treated on the same footing, although they have only interchanged settings. It\nis also very important to note that only one pair of Wigner's (\\ref{22octn3}) is actually correlated. Each of the equal settings\nchosen on the same island must appear with different measurement time in Bell's six-tuple (\\ref{25septn2}). In contrast, any\nsetting that appears twice on the same island must be associated with the same measurement and, therefore, be related to the\nsame measurement time because of Wigner's procedure involving six-tuple (\\ref{26septn1}) . {\\it However, it is physically\nimpossible to actually measure two different pairs which exhibit the same setting and the same measurement time at one given\nlocation}.\n\nThe ordering of the mathematical abstractions that form the subsets that Wigner chose are simply incompatible with\nthe ordering of the original measurements in space and time. The measurement time is the only guarantee for the correct pairing\nof the original measurements. These facts demonstrate that Wigner oversimplified complex topological-combinatorial factors that\nare vital for the outcomes of his considerations. Another important fact, for the incorrectness of Wigner's pairings that arise\nfrom his 9 possibilities (\\ref{22octn1}), is that the pairs with equal settings are not derived from originally correlated pairs\nand therefore may randomly assume all of the value-pairs $++$, $--$, $+-$ and $-+$. They contribute to the equal outcome\n$++, --$ count, that is so important for Wigner's reasoning, only half of what Wigner actually assumed.\n\nThus, Wigner reduced the number of possible correlations of EPRB measurements by assuming without justification the\nexistence of certain joint probabilities. He did not realize this fact and was, therefore, faced with the problem to explain\nthe contradictions with actual experiments if they occurred\n(and as indeed were found~\\cite{ASPE82b,WEIH98,AGUE09,ADEN12,VIST12}).\nBecause Wigner was convinced that he used only set theory and Einstein's local realism, he mentioned that a violation of Einstein's\nseparation principle would lead to ``$4^9$ domains .....of the nine measurements\" corresponding to Eq.~( \\ref{22octn1}). He thus\nintroduced violations of Einstein's realism to explain possible future disagreements of his theory with experiments and\nmeasurements, so to speak as a deus ex machina, that would resolve contradictions; not as essential part of his proof.\n\nWe note in passing that all of our (and Wigner's) reasoning that is related to the triple of settings $\\bf a, b, c$ and Bell's\ninequality can be repeated with the same findings for quadruples $\\bf a, b, c, d$ and the CHSH inequality.\n\n\\section{Conclusion}\\label{sec5}\n\nThese facts show that Wigner's procedure to derive Bell's inequality is set-theoretically neither general nor sound.\nWigner's work, taken in conjunction with experiments such as presented in \\cite{ASPE82b,WEIH98,AGUE09,ADEN12,VIST12} does not prove\nviolations of Einstein's realism. It proves only that Wigner's assumptions about the existence of certain joint probabilities\nare incorrect.  As discussed above, the essential part of Wigner's proof does not use Einstein's local realism at all.\nThus, we believe to have shown beyond any reasonable doubt that Wigner derived his\nreduction of possible correlations and his Bell type inequality from set theoretically unjustified assumptions about the\nexistence of joint probabilities.\n\n\\section*{Aknowledgement}\nThe authors thank Michael Revzen for bringing to our attention that the caption of Table I was lacking clarity.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\uppercase{Introduction}}\n\n\\noindent\nThere is a generalized consensus that knowledge repositories are a key\ningredient in the whole process of Knowledge Management,\ncf.~\\cite{Duhon98,Konig2012}.\nFurthermore, being able to rely upon the consistency of the information they\nprovide is paramount to any business whatsoever.\nDatabases and database management systems, by far the most common framework for\nknowledge storage and retrieval, have been around for many years now,\nand have evolved substantially, at pace with information technology.\nIn this paper, we are focusing on the important aspect of database consistency.\n\nTypical database management systems allow the user to specify\nintegrity constraints on the data as logical statements that are\nrequired to be satisfied at any given point in time.\nThe classical problem is how to guarantee that such constraints still\nhold after updating databases~\\cite{Abiteboul1988}, and what repairs\nhave to be made when the constraints are violated~\\cite{Katsuno1991},\nwithout making any assumptions about how the inconsistencies came about.\nRepairing an inconsistent database~\\cite{Eiter1992} is a highly\ncomplex process; also, it is widely accepted that human\nintervention is often necessary to choose an adequate repair.\nThat said, every progress towards automation in this field is nevertheless important.\n\nIn particular, the framework of active integrity\nconstraints~\\cite{Flesca2004,Caroprese2011} was introduced more recently with the\ngoal of giving operational mechanisms to compute repairs of\ninconsistent databases.\nThis framework has subsequently been extended to consider\npreferences~\\cite{Caroprese2007} and to find ``best'' repairs\nautomatically~\\cite{CEGN2013} and efficiently~\\cite{lcf:14}.\n\nActive integrity constraints (AICs) seem to be a promising framework\nfor the purpose of achieving reliability in information retrieval:\n\\begin{itemize}\n\\item AICs are expressive enough to encompass the majority of\n  integrity constraints that are typically found in practice;\n\\item AICs allow the definition of preferred ways to calculate\n  repairs, through specific actions to be taken in specific\n  inconsistent situations;\n\\item AICs provide mechanisms to resolve inconsistencies while the\n  database is in use;\n\\item AICs can enhance databases to provide a basis for self-healing\n  autonomic systems.\n\\end{itemize}\nTo the best of our knowledge, no real-world implementation of an\nAIC--enhanced database system exists today.\nThis paper presents a prototype tool that implements the tree--based\nalgorithms for computing repairs presented\nin~\\cite{Caroprese2011,CEGN2013}.\nWhile not yet ready for productive deployment, this implementation can\nwork successfully with database management systems working in the SQL\nframework, and is readily extendible to other (nearly arbitrary)\ndatabase management systems thanks to its modular design.\n\nThis paper is structured as follows.\nSection~\\ref{sec:background} recapitulates previous work on active\nintegrity constraints and repair trees.\nSection~\\ref{sec:tool} introduces our tool, \\texttt{repAIrC}, and describes its\nimplementation, focusing on the new theoretical results that were\nnecessary to bridge the gap between theory and practice.\nSection~\\ref{sec:parallel} then discusses how parallel computation\ncapabilities are incorporated in \\texttt{repAIrC}\\ to make the search for repairs\nmore efficient.\nSection~\\ref{sec:concl} summarizes our achievements and gives a brief\noutlook into future developments.\n\n\\section{\\uppercase{Active integrity constraints}}\n\\label{sec:background}\n\n\\noindent\nActive integrity constraints (AICs) were introduced in~\\cite{Flesca2004} and\nfurther explored in~\\cite{Caroprese2009,Caroprese2011}, which define the basic\nconcepts and prove complexity bounds for the problem of repairing inconsistent\ndatabases.\nThese authors introduce declarative semantics for different types of repairs,\nobtaining their complexity results by means of a translation into revision\nprogramming.\nIn practice, however, this does not yield algorithms that are applicable to\nreal-life databases; for this reason, a direct operational semantics\nfor AICs was proposed in~\\cite{CEGN2013}, presenting database-oriented\nalgorithms for finding repairs.\nThe present paper describes a tool that can actually execute these algorithms in\ncollaboration with an SQL database management system.\n\n\\subsection{Syntax and Declarative Semantics}\n\nFor the purpose of this work, we can view a database simply as a set of atomic\nformulas over a typed function-free\nfirst-order signature $\\Sigma$, which we will assume throughout to be fixed.\nLet $\\mathcal At$ be the set of closed atomic formulas over $\\Sigma$.\nA database $\\I$ \\emph{entails} literal $L$, $\\I\\models L$, if $L\\in\\mathcal At$ and\n$L\\in\\I$, or if $L$ is $\\mathsf{not}\\  a$ with $a\\in\\mathcal At$ and $a\\notin\\I$.\n\nAn integrity constraint is a clause $$L_1,\\ldots,L_m \\supset \\bot$$ where each\n$L_i$ is a literal over $\\Sigma$, with intended semantics that\n$\\forall(L_1\\wedge\\ldots\\wedge L_m)$ should not hold.\nAs is usual in logic programming, we require that if $L_i$ contains a negated variable $x$, then $x$\nalready occurs in $L_1,\\ldots,L_{i-1}$.\nWe say that $\\I$ \\emph{satisfies} integrity constraint $r$, $\\I\\models r$, if,\nfor every instantiation $\\theta$ of the variables in $r$, it is the case that\n$\\I\\not\\models L\\theta$ for some $L$ in $r$; and $\\I$ satisfies a set $\\eta$ of\nintegrity constraints, $\\I\\models\\eta$, if it satisfies each integrity\nconstraint in $\\eta$.\n\nIf $\\I\\not\\models\\eta$, then $\\I$ may be updated through \\emph{update actions}\nof the form $+a$ and $-a$, where $a\\in\\mathcal At$, stating that $a$ is to be inserted\nin or deleted from $\\I$, respectively.\nA set of update actions $\\mathcal U$ is \\emph{consistent} if it does not contain both\n$+a$ and $-a$, for any $a\\in\\mathcal At$; in this case, $\\I$ can be updated by $\\mathcal U$,\nyielding the database\n$$\\I\\circ\\mathcal U = (\\I\\cup\\left\\{a\\mid{+a}\\in\\mathcal U\\right\\})\\setminus \\left\\{a\\mid{-a}\\in\\mathcal U\\right\\}\\,.$$\nThe problem of database repair is to find $\\mathcal U$ such that $\\I\\circ\\mathcal U\\models\\eta$.\n\n\\begin{definition}\n  Let $\\I$ be a database and $\\eta$ a set of integrity constraints.\n  A \\emph{weak repair} for $\\langle\\mathcal I,\\eta\\rangle$ is a consistent set $\\mathcal U$ of update actions\n  such that:\n  (i)~every action in $\\mathcal U$ changes $\\I$; and\n  (ii)~$\\I\\circ\\mathcal U\\models\\eta$.\n  A \\emph{repair} for $\\langle\\mathcal I,\\eta\\rangle$ is a weak repair $\\mathcal U$ for $\\langle\\mathcal I,\\eta\\rangle$ that is minimal w.r.t.\\ set\n  inclusion.\n\\end{definition}\nThe distinction between weak repairs and repairs embodies the\nstandard principle of \\emph{minimality of change}~\\cite{Winslett1990}.\n\nThe problem of deciding whether there exists a (weak) repair for an\ninconsistent database is\n$NP$-complete~\\cite{Caroprese2011}.\nFurthermore, simply detecting that a\ndatabase is inconsistent does not give any information on how it can be\nrepaired.\nIn order to address this issue, those authors proposed active integrity constraints (AICs),\nwhich guide the process of selection of a repair by pairing literals with the\ncorresponding update actions.\n\nIn the syntax of AICs, we extend the notion of update action by allowing\nvariables.\nGiven an action $\\alpha$, the literal corresponding to it is $\\mathsf{lit}(\\alpha)$,\ndefined as $a$ if $\\alpha={+a}$ and $\\mathsf{not}\\  a$ if $\\alpha={-a}$; conversely, the\nupdate action corresponding to a literal $L$, $\\mathsf{ua}(L)$, is $+a$ if $L=a$ and\n$-a$ if $L=\\mathsf{not}\\  a$.\nThe \\emph{dual} of $a$ is $\\mathsf{not}\\  a$, and conversely; the dual of $L$ is denoted\n$L^D$.\nAn \\emph{active integrity constraint} is thus an expression $r$ of the form\n$$L_1,\\ldots,L_m \\supset \\alpha_1\\mid\\ldots\\mid\\alpha_k$$\nwhere the $L_i$ (in the \\emph{body} of $r$, $\\body r$) are literals and the\n$\\alpha_j$ (in the \\emph{head} of $r$, $\\head r$) are update actions, such that\n\\[\\left\\{\\mathsf{lit}(\\alpha_1)^D,\\ldots,\\mathsf{lit}(\\alpha_k)^D\\right\\}\\subseteq\\left\\{L_1,\\ldots,L_m\\right\\}\\,.\\]\nThe set $\\mathsf{lit}(\\head r)^D$ contains the \\emph{updatable} literals of $r$.\nThe \\emph{non-updatable} literals of $r$ form the set\n$\\mathsf{nup}(r)=\\body r\\setminus\\mathsf{lit}\\left(\\head r\\right)^D$.\n\nThe natural semantics for AICs restricts the notion of weak repair.\n\n\\begin{definition}\n  Let $\\I$ be a database, $\\eta$ a set of AICs and $\\mathcal U$ be a (weak) repair for\n  $\\langle\\mathcal I,\\eta\\rangle$.\n  Then $\\mathcal U$ is a \\emph{founded (weak) repair} for $\\langle\\mathcal I,\\eta\\rangle$ if, for every action\n  $\\alpha\\in\\mathcal U$, there is a closed instance $r'$ of $r\\in\\eta$ such that\n  $\\alpha\\in\\head{r'}$ and $\\I\\circ\\mathcal U\\models L$ for every\n  $L\\in\\body{r'}\\setminus\\left\\{\\mathsf{lit}(\\alpha)^D\\right\\}$.\n\\end{definition}\n\nThe problem of deciding whether there exists a weak founded repair for an\ninconsistent database is again $NP$-complete, while the similar problem for\nfounded repairs is $\\Sigma^P_2$-complete.\nDespite their natural definition, founded repairs can include circular support\nfor actions, which can be undesirable; this led to the introduction of justified\nrepairs~\\cite{Caroprese2011}.\n\nWe say that a set $\\mathcal U$ of update actions is \\emph{closed} under $r$ if\n$\\mathsf{nup}(r)\\subseteq\\mathsf{lit}(\\mathcal U)$ implies $\\head r\\cap\\mathcal U\\neq\\emptyset$, and it is\nclosed under a set $\\eta$ of AICs if it is closed under every closed instance of\nevery rule in $\\eta$.\nIn particular, every founded weak repair for $\\langle\\mathcal I,\\eta\\rangle$ is by definition\nclosed under~$\\eta$.\n\nA closed update action $+a$ (resp.\\ $-a$) is a \\emph{no-effect} action w.r.t.\\\n$(\\I,\\I\\circ\\mathcal U)$ if $a\\in\\I\\cap(\\I\\circ\\mathcal U)$\n(resp.\\ $a\\notin\\I\\cup(\\I\\circ\\mathcal U)$).\nThe set of all no-effect actions w.r.t.\\ $(\\I,\\I\\circ\\mathcal U)$ is denoted by\n$\\neff{\\U}$.\nA set of update actions $\\mathcal U$ is a justified action set if it coincides with the\nset of update actions forced by the set of AICs and the database before and\nafter applying $\\mathcal U$~\\cite{Caroprese2011}.\n\n\\begin{definition}\n  Let $\\I$ be a database and $\\eta$ a set of AICs.\n  A consistent set $\\mathcal U$ of update actions is a \\emph{justified action set} for\n  $\\langle\\mathcal I,\\eta\\rangle$ if it is a minimal set of update actions containing $\\neff{\\U}$ and\n  closed under $\\eta$.\n  If $\\mathcal U$ is a justified action set for $\\langle\\mathcal I,\\eta\\rangle$, then $\\mathcal U\\setminus\\neff{\\U}$ is a\n  justified weak repair for $\\langle\\mathcal I,\\eta\\rangle$.\n\\end{definition}\nIn particular, it has been shown that justified repairs are always\nfounded~\\cite{Caroprese2011}.\nThe problem of deciding whether there exist justified weak repairs or justified repairs\nfor $\\langle\\mathcal I,\\eta\\rangle$ is again a\n$\\Sigma^P_2$-complete problem, becoming $NP$-complete if one restricts the AICs\nto contain only one action in their head (\\emph{normal} AICs).\n\n\\subsection{Operational Semantics}\n\nThe declarative semantics of AICs is not very satisfactory, as it does not\ncapture the operational nature of rules.\nIn particular, the quantification over all no-effect actions in the definition\nof justified action set poses a practical problem.\nTherefore, an operational semantics for AICs was proposed in~\\cite{CEGN2013},\nwhich we now summarize.\n\n\\begin{definition}\n  Let $\\I$ be a database and $\\eta$ be a set of AICs.\n  \\begin{itemize}\n  \\item The \\emph{repair tree} for $\\langle\\mathcal I,\\eta\\rangle$, $T_{\\langle\\mathcal I,\\eta\\rangle}$, is a labeled tree\n    where: nodes are sets of update actions; each edge is labeled with a closed\n    instance of a rule in $\\eta$;\n    the root is $\\emptyset$; and for each consistent node $n$ and closed\n    instance $r$ of a rule in $\\eta$, if $\\I\\circ n\\not\\models r$ then for each\n    $L\\in\\body{r}$ the set $n'=n\\cup\\left\\{\\mathsf{ua}(L)^D\\right\\}$ is a child of $n$,\n    with the edge from $n$ to $n'$ labeled by $r$.\n  \\item The \\emph{founded repair tree} for $\\langle\\mathcal I,\\eta\\rangle$, $T^f_{\\langle\\mathcal I,\\eta\\rangle}$, is\n    constructed as $T_{\\langle\\mathcal I,\\eta\\rangle}$ but requiring that $\\mathsf{ua}(L)$ occur in the head of\n    some closed instance of a rule in $\\eta$.\n  \\item The \\emph{well-founded repair tree} for $\\langle\\mathcal I,\\eta\\rangle$, $T^{wf}_{\\langle\\mathcal I,\\eta\\rangle}$, is\n    also constructed as $T_{\\langle\\mathcal I,\\eta\\rangle}$ but requiring that $\\mathsf{ua}(L)$ occur in the\n    head of the rule being applied.\n  \\item The \\emph{justified repair tree} for $\\langle\\mathcal I,\\eta\\rangle$, $T^j_{\\langle\\mathcal I,\\eta\\rangle}$, has nodes\n    that are \\emph{pairs} of sets of update actions $\\langle\\mathcal U,\\mathcal J\\rangle$, with\n    root $\\langle\\emptyset,\\emptyset\\rangle$.\n    For each node $n$ and closed instance $r$ of a rule in $\\eta$, if\n    $\\I\\circ\\mathcal U_n\\not\\models r$, then for each $\\alpha\\in\\head{r}$ there is a\n    descendant $n'$ of $n$, with the edge from $n$ to $n'$ labeled by $r$,\n    where:\n    $\\mathcal U_{n'} = \\mathcal U_n\\cup\\left\\{\\alpha\\right\\}$;\n    and $\\mathcal J_{n'} = \\left(\\mathcal J_n\\cup\\{\\mathsf{ua}(\\mathsf{nup}(r))\\}\\right)\\setminus\\mathcal U_n$.\n  \\end{itemize}\n\\end{definition}\n\nThe properties of repair trees are summarized in the following results, proved\nin~\\cite{CEGN2013}.\n\\begin{theorem}\n  \\label{thm:trees}\n  Let $\\I$ be a database and $\\eta$ be a set of AICs.  Then:\n  \\begin{enumerate}\n  \\item $T_{\\langle\\mathcal I,\\eta\\rangle}$ is finite.\n  \\item Every consistent leaf of $T_{\\langle\\mathcal I,\\eta\\rangle}$ is labeled by a weak repair for $\\langle\\mathcal I,\\eta\\rangle$.\n  \\item If $\\mathcal U$ is a repair for $\\langle\\mathcal I,\\eta\\rangle$, then there is a branch of $T_{\\langle\\mathcal I,\\eta\\rangle}$\n    ending with a leaf labeled by $\\mathcal U$.\n  \\item If $\\mathcal U$ is a founded repair for $\\langle\\mathcal I,\\eta\\rangle$, then there\n    is a branch of $T^f_{\\langle\\mathcal I,\\eta\\rangle}$ ending with a leaf\n    labeled by $\\mathcal U$.\n  \\item If $\\mathcal U$ is a justified repair for $\\langle\\mathcal I,\\eta\\rangle$, then there\n    is a branch of $T^j_{\\langle\\mathcal I,\\eta\\rangle}$ ending with a leaf\n    labeled by $\\mathcal U$.\n  \\item If $\\eta$ is a set of normal AICs and $\\langle\\mathcal U,\\mathcal J\\rangle$ is a leaf of\n    $T^j_{\\langle\\mathcal I,\\eta\\rangle}$ with $\\mathcal U$ consistent and $\\mathcal U\\cap\\mathcal J=\\emptyset$, then $\\mathcal U$ is a\n    justified repair for $\\langle\\mathcal I,\\eta\\rangle$.\n  \\end{enumerate}\n\\end{theorem}\nNot all leaves will correspond to repairs of the desired kind; in particular,\nthere may be weak repairs in repair trees.\nAlso, both $T^f_{\\langle\\mathcal I,\\eta\\rangle}$ and $T^j_{\\langle\\mathcal I,\\eta\\rangle}$ typically contain leaves that do not\ncorrespond to founded or justified (weak) repairs -- otherwise the problem of\ndeciding whether there exists a founded or justified weak repair for $\\langle\\mathcal I,\\eta\\rangle$\nwould be solvable in non-deterministic polynomial time.\nThe leaves of the well-founded repair tree for $\\langle\\mathcal I,\\eta\\rangle$ correspond to a new type of weak repairs,\ncalled \\emph{well-founded weak repairs}, not\nconsidered in the original works on AICs.\n\n\\subsection{Parallel Computation of Repairs}\n\\label{ssec:par}\n\nThe computation of founded or justified repairs can be improved by dividing the\nset of AICs into independent sets that can be processed\nindependently, simply merging the computed repairs at the end~\\cite{lcf:14}.\nHere, we adapt the definitions given therein to the first-order scenario.\nTwo sets of AICs $\\eta_1$ and $\\eta_2$ are independent if the same atom does not\noccur in a literal in the body of a closed instance of two distinct rules\n$r_1\\in\\eta_1$ and $r_2\\in\\eta_2$.\nIf $\\eta_1$ and $\\eta_2$ are independent, then repairs for\n$\\langle I,\\eta_1\\cup\\eta_2\\rangle$ are exactly the unions of a repair for\n$\\langle\\I,\\eta_1\\rangle$ and $\\langle\\I,\\eta_2\\rangle$; furthermore,\nthe result still holds if one considers founded, well-founded or\njustified repairs.\n\nIf an atom occurs in a literal in the body of a closed instance of a rule in\n$\\eta_2$ and in an action in the head of a closed instance of a rule in\n$\\eta_1$, but not conversely, then we say that $\\eta_1$ \\emph{precedes}\n$\\eta_2$.\nFounded\/justified (but not well-founded) repairs for $\\eta_1\\cup\\eta_2$ can be\ncomputed in a stratified way, by first repairing $\\I$ w.r.t.~$\\eta_1$, and then\nrepairing the result w.r.t.~$\\eta_2$.\n\nSplitting a set of AICs into independent sets or stratifying it can be solved\nusing standard algorithms on graphs, as we describe in\nSection~\\ref{sec:parallel}.\n\n\\section{\\uppercase{The tool}}\n\\label{sec:tool}\n\n\\noindent\nThe tool \\texttt{repAIrC}\\ is implemented in Java, and its simplified UML class diagram can\nbe seen in Figure~\\ref{fig:classes}.\nStructurally, this tool can be split into four main separate components,\ncentered on the four classes marked in bold in that figure.\n\\begin{itemize}\n\\item Objects of type \\class{AIC} implement active integrity constraints.\n\\item Implementations of interface \\class{DB} provide the necessary tools to\n  interact with a particular database management system; currently, we provide\n  functionality for SQL databases supported by JDBC.\n\\item Objects of type \\class{RepairTree} correspond to concrete repair trees;\n  their exact type will be the subclass corresponding to a particular kind of\n  repairs.\n\\item Class \\class{RunRepairGUI} provides the graphical interface to interact\n  with the user.\n\\end{itemize}\n\n\\begin{figure*}[!ht]\n  \\centering\n  \\scriptsize\\input{class.pdf_tex}\n  \\caption{Class diagram for \\texttt{repAIrC}.}\n  \\label{fig:classes}\n\\end{figure*}\n\nAn important design aspect has to do with extensibility and modularity.\nA first prototype focused on the construction of repair trees, and used simple\ntext files to mimick databases as lists of propositional atoms, in the style\nof~\\cite{Caroprese2011,CEGN2013}.\nLater, parallelization capabilities were added (as explained in\nSection~\\ref{sec:parallel}), requiring changes only to \\class{RepairController}\n-- the class that controls the execution of the whole process.\nLikewise, the extension of \\texttt{repAIrC}\\ to SQL databases and the addition of the\nstratification mechanism only required localized changes in the classes directly\nconcerned with those processes.\n\nThe next subsections detail the implementation of the classes \\class{AIC}, \\class{DB},\n\\class{RepairTree} and \\class{RunRepairTreeGUI}.\n\n\\subsection{Representing Active Integrity Constraints}\n\n\\noindent\nIn the practical setting, it makes sense to diverge a little from the\ntheoretical definition of AICs.\n\\begin{itemize}\n\\item Real-world tables found in DBs contain many columns, most of which are typically irrelevant for a given integrity constraint.\n\\item The columns of a table are not static, i.e., columns are usually added or removed during a database's lifecycle.\n\\item The order of columns in a table should not matter, as they are identified by a unique column name.\n\\end{itemize}\nTo deal pragmatically with these three aspects, we will write atoms using a more database-oriented notation, allowing\nthe arguments to be provided in any order, but requiring that the column names\nbe provided.\nThe special token \\verb+$+ is used as first character of a variable.\nSo, for example, the literal \\verb+hasInsurance(firstName=$X, type='basic')+\nwill match any entry in table \\verb+hasInsurance+ having value \\verb+basic+ in\ncolumn \\verb+type+ and any value in column \\verb+firstName+; this table may\nadditionally have other columns.\nNegative literals are preceded by the keyword \\verb+NOT+, while actions must\nbegin with \\verb-+- or \\verb+-+.\nLiterals and actions are separated by commas, and the body and head of an AIC\nare separated by \\verb+->+.\nThe AIC is finished when \\verb+;+ is encountered, thus allowing constraints to\nspan several lines.\n\nAICs are provided in a text file, which is parsed by a parser generated\nautomatically using JavaCC and transformed into objects of type \\class{AIC}.\nThese contain a body and a head, which are respectively\n\\class{List<Literal>} and \\class{List<Action>};\nfor consistency with the underlying theory, \\class{Literal} and \\class{Action}\nare implemented separately, although their objects are isomorphic:\nthey contain an object of type \\class{Clause} (which consists of the\nname of a table in the database and a list of pairs column name\/value)\nand a flag indicating whether they are positive\/negated (literals) or additions\/removals\n(actions).\n\n\\begin{example}\n\\label{ex:AIC1}\nConsider the following active integrity constraints for an employee database.\nThe first states\nthat the boss (as specified in the \\textnormal{\\sf category} table) cannot be a junior employee (i.e., have an entry in the \\textnormal{\\sf junior} table);\nthe second states that every junior employee must have some basic insurance (as specified in the \\textnormal{\\sf insured} table).\n\\[\n  \\mathsf{junior}(X), \\mathsf{category}(\\mathsf{boss},X) \\supset\n  -\\mathsf{junior}(X)\n\\]\n\\begin{multline*}\n  \\label{AICex2}\n  \\mathsf{junior}(X), \\mathsf{not\\ insured}(X,\\mathsf{basic})\n  \\\\ \\supset +\\mathsf{insured}(X,\\mathsf{basic})\n\\end{multline*}\n\nThese are written in the concrete text-based syntax of the \\texttt{repAIrC}\\ tool as\n{\\small\n\\begin{verbatim}\njunior(id = $X),\n  category(type = boss, empId = $X)\n  -> - junior(id = $X);\n\njunior(id = $X),\n  NOT insured(empId = $X, type = basic)\n  -> + insured(empId = $X, type = basic);\n\\end{verbatim}}\n\\noindent respectively, assuming the corresponding column names for\nthe atributes.\nNote that, thanks to our usage of explicit column naming, the column names for the same variable need not have\nidentical designations.\n\\end{example}\n\n\\subsection{Interfacing with the Database}\n\\label{ssec:interface}\n\nDatabase operations (queries and updates) are defined in the\n\\class{DB} interface, which contains the following methods.\n\\begin{itemize}\n\\item \\verb+getUpdateActions(AIC aic)+: queries the database for all the instances\n  of \\verb+aic+ that are not satisfied in its current state, returning a\n  \\verb+Collection<Collection<Action>>+ that contains the corresponding\n  instantiations of the head of \\verb+aic+.\n\\item \\verb+update(Collection<Action> actions)+: applies all update actions in\n  \\verb+actions+ to the database (void).\n\\item \\verb+undo(Collection<Action> actions)+: undoes the effect of all update\n  actions in \\verb+actions+ (void).\n\\item \\verb+aicsCompatible(Collection<AIC> aics)+: checks that all the elements of\n  \\verb+aics+ are compatible with the structure of the database.\n\\item \\verb+disconnect()+: disconnects from the database (void). The\n  connection is established when the object is originally constructed.\n\\end{itemize}\n\nSome of these methods require more detailed comments.\nThe construction of the repair tree also requires that the database be changed\ninteractively, but upon conclusion the database should be returned to its\noriginal state.\nIn theory, this would be achievable by applying the \\verb+update+ method with\nthe duals of the actions that were used to change the database; but this turns\nout not to be the case for deletion actions.\nSince the AICs may underspecify the entries in the database (because some fields\nare left implicit), the implementation of \\verb+update+ must take care to\nstore the values of all rows that are deleted from the database.\nIn turn, the \\verb+undo+ method will read this information every time it has to\nundo a deletion action, in order to find out exactly what entries to re-add.\n\nThe method \\verb+aicsCompatible+ is necessary because the AICs are given\nindependently of the database, but they must be compatible with its structure --\notherwise, all queries will return errors.\nIncluding this method in the interface allows the AICs to be tested before any queries are made,\nthus significantly reducing the number of exceptions that can occur during\nprogram execution.\n\n\n\n\n\n  \n\n\nCurrently, \\texttt{repAIrC}\\ includes an implementation \\verb+DBMySQL+ of \\verb+DB+, which\nworks with SQL databases.\nThe interaction between \\texttt{repAIrC}\\ and the database is achieved by means of JDBC, a\nJava database connectivity technology able to interface with nearly all existing\nSQL databases.\nIn order to determine whether an AIC is satisfied by a database, method\n\\verb+getUpdateActions+ first builds a single SQL query corresponding to the body of\nthe AIC.\nThis method builds two separate \\verb+SELECT+ statements, one for the positive\nand another for the negative literals in the body of the AIC.\nEach time a new variable is found, the table and column where it occurs are\nstored, so that future references to the same variable in a positive literal can\nbe unified by using inner joins.\nThe \\verb+select+ statement for the negative literals is then connected to the\nother one using a \\verb+WHERE NOT EXISTS+ condition.\nVariables in the negative literals must necessarily appear first in a positive\nliteral in the same AIC; therefore, they can then be connected by a \\verb+WHERE+\nclause instead of an inner join.\n\n\\begin{example}\nThe bodies of the integrity constraints in Example~\\ref{ex:AIC1} generate the\nfollowing SQL queries.\n{\\small\n\\begin{verbatim}\nSELECT * FROM junior\n  INNER JOIN dept_emp\n  ON junior.id=category.empId\n  WHERE category.type=`boss'\n\nSELECT * FROM junior\n  WHERE NOT EXISTS\n (SELECT * FROM insured\n    WHERE insured.empId=junior.id\n    AND insured.type=`basic')\n\\end{verbatim}}\n\\end{example}\n\n\\subsection{Implementing Repair Trees}\n\nThe implementation of the repair trees directly follows the algorithms described\nin Section~\\ref{sec:background}.\nDifferent types of repair trees are implemented using inheritance, so that most\nof the code can be reused in the more complex trees.\nThe trees are constructed in a breadth-first manner, and all non-contradictory\nleaves that are found are stored in a list.\nAt the end, this list is pruned so that only the minimal elements (w.r.t.~set\ninclusion) remain -- as these are the ones that correspond to repairs.\n\nWhile constructing the tree, the database has to be temporarily updated and\nrestored.\nIndeed, to calculate the descendants of a node, we first need to evaluate all\nAICs at that node in order to determine which ones are violated; this requires\nquerying a modified version of the database that takes into account the update\nactions in the current node.\n\nIn order to avoid concurrency issues, these updates are performed in a transaction-style way,\nwhere we update the database, perform the necessary SQL queries, and rollback to\nthe original state, guaranteeing that other threads interacting with the\ndatabase during this process neither see the modifications nor lead to inconsistent repair trees.\nThis becomes of particular interest when the parallel processing tools described\nin Section~\\ref{sec:parallel} are put into place.\nAlthough this adds some overhead to the execution time,\nat the end of that section we discuss why scalability is not a practically relevant concern.\n\nAfter finding all the leaves of the repair tree, a further step is needed in the\ncase one is looking for founded or justified repairs, as the corresponding trees\nmay contain leaves that do not correspond to repairs with the desired property.\nThis step is skipped if all AICs are normal, in view of the results\nfrom~\\cite{CEGN2013}.\nFor founded repairs, we directly apply the definition: for each action $\\alpha$,\ncheck that there is an AIC with $\\alpha$ in its head and such that all other\nliterals in its body are satisfied by the database.\n\nFor justified repairs, the validation step is less obvious.\nDirectly following the definition requires constructing the set of\nno-effect actions, which is essentially as large as the database, and iterating\nover subsets of this set.\nThis is obviously not possible to do in practical settings.\nTherefore, we use some criteria to simplify this step.\n\n\\begin{lemma}\n  If a rule $r$ was not applied in the branch leading to $\\mathcal U$, then $\\mathcal U$ is\n  closed under $r$.\n\\end{lemma}\n\\begin{proof}\n  Suppose that $r$ was never applied and assume $\\mathsf{nup}(r)\\subseteq\\neff\\mathcal U$.\n  Then necessarily $\\head r\\cap\\neff\\mathcal U\\neq\\emptyset$, otherwise $r$ would be\n  applicable and $\\mathcal U$ would not be a repair.\n\\end{proof}\n\nBy construction, $\\mathcal U$ is also closed for all rules applied in the branch leading\nto it.\n\nLet $\\mathcal U$ be a candidate justified weak repair.\nIn order to test it, we need to show that $\\mathcal U\\cup\\neff\\mathcal U$ is a justified\naction set (see~\\cite{CEGN2013}), which requires iterating over all subsets of\n$\\mathcal U\\cup\\neff\\mathcal U$ that contain $\\neff\\mathcal U$.\nClearly this can be achieved by iterating over subsets of $\\mathcal U$.\n\nBut if $\\mathcal U^\\ast\\subseteq \\mathcal U$, then $\\mathsf{nup}(r)\\cap \\mathcal U^\\ast=\\emptyset$; this\nallows us to simplify the closedness condition to: if $\\mathsf{nup}(r)\\subseteq\\neff\\mathcal U$,\nthen $\\mathcal U^\\ast\\cap\\head r=\\emptyset$.\nThe antecedent needs then only be done once (since it only depends on $\\mathcal U$),\nwhereas the consequent does not require consulting the database.\n\nThe following result summarizes these properties.\n\\begin{lemma}\n  A weak repair $\\mathcal U$ in a leaf of the justified repair tree for $\\langle\\mathcal I,\\eta\\rangle$ is a\n  justified weak repair for $\\langle\\mathcal I,\\eta\\rangle$ iff, for every set $\\mathcal U^\\ast\\subseteq\\mathcal U$, \n  if $\\mathsf{nup}(r)\\subseteq\\neff\\mathcal U$, then $\\mathcal U^\\ast\\cap\\head r=\\emptyset$.\n\\end{lemma}\n\nThe different implementations of repair trees use different subclasses of the\nabstract class \\verb+Node+; in particular, nodes of \\verb+JustifiedRepairTree+s\nmust keep track not only of the sets of update actions being constructed, but\nalso of the sets of non-updatable actions that were assumed.\nThese labels are stored as \\verb+Set<Action>+ using \\verb+HashSet+ from the\nJava library as implementation, as they are repeatedly tested\nfor membership everytime a new node is generated.\n\nFor efficiency, repair trees maintain internally a set of the sets of update\nactions that label nodes constructed so far as a \\verb+Set<Node>+.\nThis is used to avoid generating duplicate nodes with the same label.\nSince this set is used mainly for querying, it is again implemented as a\n\\verb+HashSet+.\nNodes with inconsistent labels are also immediately eliminated, since they can\nonly produce inconsistent leaves.\n\n\\subsection{Interfacing with the User}\n\n\\begin{figure}[b]\n  \\centering\n  \\resizebox{.9\\columnwidth}!{\\includegraphics{scrshot3.png}}\n  \\caption{The initial screen for \\texttt{repAIrC}.}\n  \\label{fig:menu}\n\\end{figure}\n\n\\noindent\nThe user interface for \\texttt{repAIrC}\\ is implemented using the standard Java GUI widget\ntoolkit \\verb+Swing+, and is rather straightforward.\nOn startup, the user is presented with the dialog box depicted in\nFigure~\\ref{fig:menu}.\n\nThe user can then provide credentials to connect to a database, as well as enter\na file containing a set of AICs.\nIf the connection to the database is successful and the file is successfully\nparsed, \\texttt{repAIrC}\\ invokes the \\verb+aicsCompatible+ method required by the\nimplementation of the \\verb+DB+ interface (see Section~\\ref{ssec:interface}) and\nverifies that all tables and columns mentioned in the set of AICs are valid\ntables and columns in the database.\nIf this is not the case, then an error message is generated and the user is\nrequired to select new files; otherwise, the buttons for configuration and\ncomputation of repairs become active.\n\nOnce the initialization has succeeded, one can check the database for\nconsistency and obtain different types of repairs, computed using the repair\ntree described above.\nAs it may be of interest to obtain also weak repairs, the user is given the\npossibility of selecting whether to\nsee only the repairs computed, or all valid leaves of the repair tree -- which\ntypically include some weak repairs.\nIn both cases the necessary validations are performed, so that leaves that do not\ncorrespond to repairs (in the case of founded or justified repairs) are never\npresented.\n\nAn example output screen after successful computation of the repairs for an\ninconsistent database can be seen in Figure~\\ref{fig:output}.\n\n\\begin{figure}[b]\n  \\centering\n  \\resizebox{.9\\columnwidth}!{\\includegraphics{scrshot2.png}}\n  \\caption{Possible repairs of an inconsistent database.}\n  \\label{fig:output}\n\\end{figure}\n\n\\section{\\uppercase{Parallelization and Stratification}}\n\\label{sec:parallel}\n\nAs described in Section~\\ref{ssec:par}, it is possible to parallelize the search for\nrepairs of different kinds by splitting the set of AICs into independent sets;\nin the case of founded or justified repairs, this parallelization can be taken\none step further by also stratifying the set of AICs.\nEven though finding partitions and\/or stratifications is asymptotically not\nvery expensive (it can be solved in linear time by the well-known graph\nalgorithms described below), it may still take noticeable time if the set of\nAICs grows very large.\n\nSince, by definition, partitions and stratifications are independent of the\nactual database, it makes sense to avoid repeating their computation unless the\nset of AICs changes.\nFor this reason, parallelization capabilities are implemented in \\texttt{repAIrC}\\ in a\ntwo-stage process.\nInside \\texttt{repAIrC}, the user can switch to the \\verb+Preprocess+ tab, which provides\noptions for computing partitions and stratifications of a set of AICs.\nThis results in an annotated file which still can be read by the parser; in the\nmain tab, parallel computation is automatically enabled whenever the input file\nis annotated in a proper manner.\n\n\\subsection{Implementation}\n\nComputing optimal partitions in the spirit of~\\cite{lcf:14} is not feasible in a\nsetting where variables are present, as this would require considering all\nclosed instances of all AICs -- but it is also not desirable, as it would also\nresult in a significant increase of the number of queries to the database.\nInstead, we work with the adapted definition of dependency given in\nSection~\\ref{sec:background}.\nGiven a set of AICs, \\texttt{repAIrC}\\ constructs the adjacency matrix for the undirected\ngraph whose nodes are AICs and such that there is an edge between $r_1$ to $r_2$\niff $r_1$ and $r_2$ are not independent.\nA partition is then computed simply by finding the connected components in this\ngraph by a standard graph algorithm.\n\nThe partitions computed are then written to a file, where each partition begins\nwith the line\n\\begin{verbatim}\n#PARTITION_BEGIN_[NO]#\n\\end{verbatim}\nwhere \\verb+[NO]+ is the number of the current partition, and ends with\n\\begin{verbatim}\n#PARTITION_END#\n\\end{verbatim}\nand the AICs in each partition are inserted in between, in the standard format.\n\nTo compute the partitions for stratification, we need to find the strongly connected components of a\nsimilar graph.\nThis is now a directed graph where there is an edge from $r_1$ to $r_2$ if $r_1$\nprecedes $r_2$.\nThe implementation is a variant of Tarjan's algorithm~\\cite{Tarjan72}, adapted to\ngive also the dependencies between the connected components.\n\nThe computed stratification is then written to a file with a similar syntax to\nthe previous one, to which a dependency section is added, between the special\ndelimiters\n\\begin{verbatim}\n#DEPENDENCIES_BEGIN#\n\\end{verbatim}\nand\n\\begin{verbatim}\n#DEPENDENCIES_END#\n\\end{verbatim}\n\\noindent The dependencies are included in this section as a sequence of strings\n\\verb+X -> Y+, one per line, where \\verb+X+ and \\verb+Y+ are the numbers of two\npartitions\nand \\verb+Y+ precedes \\verb+X+.\n\n\\begin{example}\n\\label{ex:strat}\n  The two AICs from Example~\\ref{ex:AIC1} cannot be parallelized, as they both\n  use the \\verb+junior+ table, but they can be stratified, as only the first one\n  makes changes to this table.\n  Preprocessing this example by \\texttt{repAIrC}\\ would return the following output.\n\n{\\small\n\\begin{verbatim}\n#PARTITION_BEGIN_1#\njunior(id = $X),\n  category(type = boss, empId = $X)\n  -> - junior(id = $X);\n#PARTITION_END#\n#PARTITION_BEGIN_2#\njunior(id = $X),\n  NOT insured(empId = $X, type = basic)\n  -> + insured(empId = $X, type = basic);\n#PARTITION_END#\n#DEPENDENCIES_BEGIN#\n2 -> 1\n#DEPENDENCIES_END#\n\\end{verbatim}}\n\nImagine a simple scenario where the \\verb+junior+ table contains a single entry.\nThen, computing repairs for this set of AICs can be achieved by first repairing\npartition $1$ (which will generate a tree with only one node) and then repairing\nthe resulting database w.r.t.~partition $2$ (which builds another tree, also\nwith only one node).\nBy comparison, processing the two AICs simultaneously would potentially give a\ntree with $4$ nodes, as both AICs would have to be considered at each stage.\n\\end{example}\n\nIn general, if there are $n$ entries in the \\verb+junior+ table, the stratified\napproach will construct at most $n+1$ trees with a total of $n^2+n$ nodes (one\ntree with $n$ nodes for the first AIC, at most $n$ trees with at most $n$ nodes\nfor the second AIC).\nBy contrast, processing both AICs together will construct a tree with\npotentially $(2n)!$ leaves, which by removing duplicate nodes may still contain\n$2^{2n}$ nodes.\n\nThis example shows that, by stratifying AICs, we can actually get an exponential\ndecrease on the size of the repair trees being built -- and therefore also on\nthe total runtime.\n\nIn addition to alleviating the exponential blowup of the repair trees, parallelization\nand stratification also allow for a multi-threaded implementation, where repair trees\nare built in parallel in multiple concurrent threads. To ensure that the dependencies between the\npartitions are respected, the threads are instructed to wait for other threads that compute preceding\npartitions. In Example~\\ref{ex:strat}, the thread processing partition 2 would be instructed\nto first wait for the thread processing partition 1 to finish.\n\nOur empirical evaluation of \\texttt{repAIrC}\\ showed that speedups of a factor of $4$ to $7$\nwere observable even when processing small\nparallelizable sets of only two or three AICs. For larger sets of AICs,\nparallelization and stratification are necessary to obtain feasible runtimes. In one application,\nwhich allowed for $15$ partitions to be processed independently, the stratified version computed\nthe founded repairs in approximately $1$ second, whereas the sequential version did not terminate\nwithin a time limit of $15000$ seconds. This corresponds to a speedup of at least four orders of magnitude,\ndemonstrating the practical impact of the contributions of this section.\n\n\\subsection{Practical Assessment}\n\nIn the worst case, parallelization and stratification will have no impact on the\nconstruction of the repair tree, as it is possible to construct a set of AICs\nwith no independent subsets.\nHowever, the worst case is not the general case, and it is reasonable to believe\nthat real-life sets of AICs will actually have a high parallelization potential.\n\nIndeed, integrity constraints typically reflect high-level consistency\nrequirements of the database, which in turn capture the hierarchical nature of\nrelational databases, where more complex relations are built from simpler ones.\nThus, when specifying \\emph{active} integrity constraints there will naturally\nbe a preference to correct inconsistencies by updating the more complex tables\nrather than the most primitive ones.\n\nFurthermore, in a real setting we are not so much interested in repairing a\ndatabase once, but rather in ensuring that it remains consistent as its\ninformation changes.\nTherefore, it is likely that inconsistencies that arise will be localized to a\nparticular table.\nThe ability to process independent sets of AICs separately guarantees that we\nwill not be repeatedly evaluating those constraints that were not broken by\nrecent changes, focusing only on the constraints that can actually become\nunsatisfied as we attempt to fix the inconsistency.\n\nFor the same reason, scalability of the techniques we implemented is not a relevant issue:\nthere is no practical need to develop a tool that is able to fix hundreds of inconsistencies\nefficiently simultaneously, since each change to the database will likely only impact a few AICs.\n\n\\section{\\uppercase{Conclusions and Future Work}}\n\\label{sec:concl}\n\n\\noindent\nWe presented a working prototype of a tool, called \\texttt{repAIrC}, to check integrity of\nreal-world SQL databases with respect to a given set of\nactive integrity constraints, and to compute different\ntypes of repairs automatically in case inconsistency is detected, following the ideas and algorithms\nin~\\cite{Flesca2004,Caroprese2007,Caroprese2011,CEGN2013,lcf:14}.\nThis tool is the first implementation of a concept we believe to have\nthe potential to be integrated in current database management systems.\n\nOur tool currently does not automatically apply repairs to the database, rather presenting them to the user.\nAs discussed in \\cite{Eiter1992}, such a functionality is not likely to be obtainable, as human intervention in the process of database repair is generally accepted to be necessary.\nThat said, automating the generation of a small and relevant set of repairs is a first important step in\nensuring a consistent data basis in Knowledge Management.\n\nIn order to deal with real-world heterogenous knowledge management systems, we\nare currently working on extending and generalizing the notion of (active)\nintegrity constraints to encompass more complex knowledge repositories such as\nontologies, expert reasoning systems, and distributed knowledge bases.\nThe design of \\texttt{repAIrC}\\ has been with this extension in mind, and we believe that\nits modularity will allow us to generalize it to work with such knowledge\nmanagement systems once the right theoretical framework is developed.\n\nOn the technical side, we are planning to speed up the system by integrating a local\ndatabase cache for peforming the many update and undo actions during exploration\nof the repair trees without the overhead of an external database connection.\n\n\\section*{\\uppercase{Acknowledgments}}\n\nThis work was supported by the Danish Council for Independent Research, Natural Sciences, and by FCT\/MCTES\/PIDDAC under centre grant to BioISI (Centre Reference: UID\/MULTI\/04046\/2013).\nMarta Ludovico was sponsored by a grant ``Bolsa Universidade de Lisboa \/ Funda\\c c\\~ao Amadeu Dias''.\n\n\\vfill\n\\bibliographystyle{apalike}\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe use of non-polynomial basis functions in the context of finite element methods dates back at least to the early 1980s \\cite{MR806382}.\nIn recent years, there has been much interest in using non-polynomial basis functions to discretize wave propagation problems in the frequency domain \\cite{CesD98,MoiHP}.\nA prominent example is the use of plane wave bases  to solve the Helmholtz equation at high frequencies. The motivation behind this approach is to reduce the number of degrees of freedom per wavelength required to obtain accurate results. Thus obtained Trefftz methods have proved very successful in practice, hence  it is a natural question to ask whether they can be extended and whether they can be equally successful in the time-domain. \n\nThe most natural way of including space-time Trefftz basis functions is within the confines of a space-time discontinuous Galerkin (dG) method. \nIn this work, we construct and analyse a new space-time interior penalty dG method for the second order wave equation, that can utilize Trefftz polynomials as local basis functions. The method discretizes the wave equation in primal form and is defined using space-time slabs to ensure solvability on each time-step, as well as to aid the presentation and the analysis. However, with minor modifications, completely unstructured space-time meshes are, in principle, possible in the proposed space-time dG framework. This results in a stable, dissipative scheme for general polynomial bases. For Trefftz basis, we prove quasi-optimality in the dG energy norm, which we show to be an upper bound for a standard space-time energy norm. Numerical results in the dG norm show that the \\reply{theoretical estimates of the convergence order are optimal}. Furthermore, the numerical results show that higher order schemes have excellent energy conservation properties and work well for systems with energy at high frequencies. Comparison with standard polynomial bases show that the same convergence order and approximation properties is obtained with considerably fewer degrees of freedom. Furthermore, the implementation is less expensive due to integration being restricted to the space-time skeleton.\n\n\n\n\nSpace-time variational methods for wave-type problems have appeared in the late 1980's with the works of Hughes and Hulbert \\cite{HulH88,HulH90} and Johnson \\cite{Joh93}.\nFirst numerical experiments with  Trefftz space-time dG methods were performed in \\cite{PetFT}. Currently there is significant activity on the topic \\cite{Egger1,KreMPS,Egger2,moiola_proc,KreSTW}.  \\reply{ In particular in \\cite{Egger1,KreMPS} a Trefftz space-time local dG method for the Maxwell equations, written as a first order system, resulting in a two-field formulation, has been analysed. The dG method designed and analyzed in the present work involves interior-penalty-type numerical fluxes in space, resulting in one-field approximation in space, as well as a one-field dG method in time, with very similar computational stencil widths compared to the local dG method introduced in \\cite{KreMPS}.}\n\n\n\nTypically,  finite element methods for linear  (and some spatially nonlinear) wave problems are based on a continuous or discontinuous finite element discretisation of the spatial variables complemented with standard time-stepping schemes, the most popular of which are explicit, such as the leap-frog scheme, due to the acceptable CFL restrictions. Though even the low order methods can be conservative, for acceptable accuracy when energies at high frequencies are excited, higher order methods are  essential \\cite{CohBook,AinMM,AguDE}. Compared to these methods, introduction of higher order approximations is much more straightforward in the context of space-time dG methods. \\reply{Moreover, space-time dG methods, such as the one presented below, do \\emph{not} require any CFL-type restrictions, owing to their implicit time-stepping interpretation.}  If a general space-time mesh can be used, a judicious choice of the mesh can result in a quasi-explicit method where only small local systems need to be solved \\cite{FalR,MonR}.\n\nThe remainder of this work is structured as follows. In the next section, we introduce the model problem. In Sections~\\ref{sec:space} and \\ref{sec:method} we construct the space-time interior penalty method and we show its stability. We proceed in Section~\\ref{sec:Tspace} to analyse polynomial Trefftz spaces and prove quasi-optimality. Finally in Section~\\ref{apriori}, we prove convergence rates for the $d$-dimensional method in space, $d=1,2,3$; moreover, we also provide $hp$-version a priori bounds for $d=1$. A series of numerical experiments in Section \\ref{sec:numerics}, illustrates the theoretical findings and highlights the good performance of the proposed method in practice.\n\n\n\n\n\\section{Model problem}\\label{sec:model}\n\nWe consider the wave equation\n\\begin{equation}\n  \\label{eq:wave}  \n\\begin{aligned}\n \\ddot u -\\nabla \\cdot \\left(a\\, \\nabla u\\right) &= 0 & &\\text{in } \\Omega \\times [0,T], \\\\\n u &= 0 & &\\text{on } \\partial\\Omega \\times [0,T],\\\\\n u(x,0) = u_0(x), \\;  \\dot u(x,0) &= v_0(x), & &\\text{in } \\Omega,\n\\end{aligned}\n\\end{equation}\nwhere $\\Omega$ is a bounded Lipschitz domain in $\\mathbb{R}^d$,  $\\partial\\Omega$ its boundary and $0 < c_a < a(x) < C_a$ a  piecewise constant function. If $\\Omega_j$ and $\\Omega_k$ are two subsets of $\\Omega$ with the boundary $\\Gamma_{jk}$ separating them and with $a \\equiv a_k$ in $\\Omega_k$ and $a \\equiv a_j$ in $\\Omega_j$, then if we denote by $u_j = u|_{\\Omega_j}$ and $u_k = u|_{\\Omega_k}$ we further have the transmission conditions\n \\begin{equation}\n   \\label{eq:transmission}\nu_j = u_k, \\quad \na_j\\partial_{\\mathbf{n}} u_j = a_k\\partial_{\\mathbf{n}} u_k, \\qquad \\text{ on } \\Gamma_{jk},   \n \\end{equation}\nwhere $\\mathbf{n}$ is the exterior normal to $\\Omega_j$ (or $\\Omega_k$).\n\nWe denote by $L^p(\\Sigma)$, $1\\le p\\le +\\infty$, the standard Lebesgue spaces, $\\Sigma\\subset\\mathbb{R}^d$, $d \\in \\{ 2,3,4\\}$, with corresponding norms $\\|\\cdot\\|_{p,\\Sigma}$; the norm of $L^2(\\Sigma)$ will be denoted by $\\ltwo{\\cdot}{\\Sigma}$. Further, $( \\cdot,\\cdot )_{\\Sigma}$ denotes the standard $L^2$-inner product on $\\Sigma$; when the arguments are vectors of $L^2$-functions, the $L^2$-inner product is modified in the standard fashion. \nWe denote by $H^s(\\Sigma)$ the standard Hilbertian Sobolev space of index $s\\in\\mathbb{R}$ of real-valued functions defined on $\\Sigma\\subset\\mathbb{R}^d$; in particular $H^1_0(\\Sigma)$ signifies the space of functions in $H^1(\\Sigma)$ whose traces onto the boundary $\\partial\\Sigma$ vanish.  For $1\\le p\\le +\\infty$, we denote the standard Bochner spaces by $L^p(0,T;X)$, with $X$ being a Banach space with norm $\\|\\cdot\\|_X$.\nFinally, we denote by $C(0,T;X)$ the space of continuous functions $v: [0,T]\\to X$ with norm $\\|v\\|_{C(0,T;X)}:=\\max_{0\\le t\\le T}\\|v(t)\\|_{X}<+\\infty$.\n\nLet $u_0 \\in H^1_0(\\Omega)$ and $v_0 \\in L^2(\\Omega)$, then \\eqref{eq:wave} has a unique (weak) solution $u$  with \n\\begin{equation}\n  \\label{eq:smoothness}\nu \\in L^2([0,T]; H^1_0(\\Omega)), \\quad  \n\\dot u \\in L^2([0,T]; L^2(\\Omega)), \\quad  \n\\ddot u \\in L^2([0,T]; H^{-1}(\\Omega)), \n\\end{equation}\nsee \\cite[Theorem 8.1]{LioMI}.  Furthermore, see \\cite[Theorem 8.2]{LioMI}, the solution is continuous in time with\n\\begin{equation}\n  \\label{eq:smoothness1}\nu \\in C([0,T]; H^1_0(\\Omega)), \\quad  \n\\dot u \\in C([0,T]; L^2(\\Omega)).\n\\end{equation}\n\\our{We denote the space of all solutions by\n\\begin{equation}\n  \\label{eq:soln_space}\n  \\mathcal{X} = \\left\\{ u \\;|\\; u \\text{ weak solution of \\eqref{eq:wave}  with } u_0 \\in H^1_0(\\Omega), v_0 \\in L^2(\\Omega)\\right\\}.\n\\end{equation}}\n\n\\section{Space-time finite element space}\\label{sec:space}\nWe aim to discretize this problem by a new space-time interior penalty discontinuous Galerkin method. In principle, this could be done on a general space-time mesh, however for the simplicity of presentation (and implementation) we construct a time discretization $0 = t_0 < t_1 < \\dots<t_n\\dots < t_N = T$ and locally quasi-uniform spatial-meshes $\\mathcal{T}_n$ of $\\Omega$ consisting of open $d$-dimensional simplices or $d$-box-type elements such that $\\reply{\\overline\\Omega} = \\cup_{K\\in  \\mathcal{T}_n} \\overline K$, \\reply{ with $K \\cap \\widetilde{K} = \\emptyset$ for $K, \\widetilde{K} \\in \\mathcal{T}_n$ and $K \\neq \\widetilde{K}$}. Therefore the space-time mesh consists of time-slabs  $\\mathcal{T}_n \\times I_n$, where $I_n = (t_n, t_{n+1})$ \\our{and $\\tau_n:=t_{n+1}-t_n$}. \n\n\n\nThe discrete space-time mesh will consist of piecewise polynomials on each time-slab, given by the local space-time finite element space:\n\\[\n\\dspace{p} := \\left\\{u \\in L^2(\\Omega \\times I_n)\\;:\\; u|_{K \\times I_n} \\in \\mathcal{P}_{p}(\\mathbb{R}^{d+1}),\\; K \\in \\mathcal{T}_n\\right\\},\n\\]\nwhere $\\mathcal{P}_{p}$ is the space of polynomials of total degree $p$; the complete space-time finite element space on $\\Omega \\times [0,T]$, will be denoted by\n\\[\n\\dspaceT{p} := \\{ u \\in L^2(\\Omega \\times [0,T]) \\;:\\; u|_{\\Omega \\times I_n} \\in \\dspace{p}, \\; n  =0,1,\\dots, N-1\\}.\n\\]\n\nWe require some notation. The {\\em skeleton} of the mesh is defined by $\\Gamma_n := \\cup_{K \\in \\mathcal{T}_n} \\partial K$ and the interior skeleton by $\\Gamma_n^{\\text{int}} = \\Gamma_n \\setminus \\partial\\Omega$. Moreover, we define the union of two skeletons of two subsequent meshes by $\\hat{\\Gamma}_n:=\\Gamma_{n-1} \\cup\\Gamma_n$; see Figure~\\ref{figure:mesh}.\n\n\\begin{figure}[htbp]\n \\centering\n\\scalebox{0.6}{\n\\input{mesh.pdf_t}\n}\n \\caption{\\small An example of a space-time mesh. Skeletons $\\Gamma_n$ and $\\Gamma_{n-1}$ are highlighted by black dots. The union of the two we denote by $\\hat{\\Gamma}_n$.}\n \\label{figure:mesh}\n\\end{figure}\n\n\\begin{remark}\n\\reply{The numerical method presented below naturally admits variable spatial meshes from one space-time slab to the next. This flexibility will be crucial for future developments in the context of space-time adaptive dG methods, aiming to use local spatio-temporal resolution to resolve sharp moving fronts. If the spatial mesh remains fixed, all the formulas hold with $\\Gamma = \\Gamma_n = \\hat{\\Gamma}_n$ for all $n$.}\n\\end{remark}\n\n  Let $K^+$ and $K^-$ be two spatial elements sharing a face $e = \\bar{K}^+\\cap \\bar{K}^- \\reply{\\subset} \\Gamma_n^{\\text{int}}$, with respective outward normal vectors $\\mathbf{n}^+$ and $\\mathbf{n}^-$ on $e$. For $u : \\Omega \\rightarrow \\mathbb{R}$ and $\\mathbf{v} : \\Omega \\rightarrow \\mathbb{R}^d$,  let $u^{\\pm} : e \\rightarrow \\mathbb{R}$ and $\\mathbf{v}^\\pm : e \\rightarrow \\mathbb{R}^d$ be the traces on $e$ with limits taken from $K^\\pm$. We define the respective jumps and averages across each face $e\\in \\Gamma_n^{\\text{int}}$ by\n\\[\n\\begin{aligned}\n  \\sa{u}|_e &= \\tfrac12(u^++u^-), & \\sa{\\mathbf{v}}|_e &= \\tfrac12(\\mathbf{v}^++\\mathbf{v}^-),\\\\\n  \\sj{u}|_e &= u^+\\mathbf{n}^++u^-\\mathbf{n}^-, & \\sj{\\mathbf{v}}|_e &= \\mathbf{v}^+\\cdot\\mathbf{n}^++\\mathbf{v}^-\\cdot\\mathbf{n}^-;\n\\end{aligned}\n\\]\nif $e \\reply{\\subset  \\overline{K^+} }\\cap \\partial\\Omega$, we set $\\sa{\\mathbf{v}}|_e = \\mathbf{v}^+$ and $\\sj{u}|_e = u^+\\mathbf{n}^+$. Further, we define the temporal jump by\n\\[\n\\tj{u(t_n)} = u(t_n^+)-u(t_n^-), \\qquad \\tj{u(t_0)} = u(t_0^+).\n\\]\nWe will denote the spatial meshsize by $h : \\Omega \\times [0,T] \\rightarrow \\mathbb{R}$, defined by $h(x,t) = \\operatorname{diam}(K)$ if $x \\in K$ for $K \\in {\\mathcal T}_n$ and $t \\in I_n$; when $x\\in e = \\bar{K}^+\\cap \\bar{K}^- $, we set $h(x,t):= \\sa{h}$ to be the average. Finally we assume that there exists $c_{\\mathcal T} > 0$ such that\n\\begin{equation}\n  \\label{eq:mesh_reg}\n\\diam(K)\/\\rho_K \\leq c_{\\mathcal T}, \\qquad \\forall K \\in \\mathcal{T}_n, \\; n = 0,1,\\dots, N-1,\n\\end{equation}\nwhere $\\rho_K$ is the radius of the inscribed circle of $K$.\n\n\\our{For simplicity of the presentation \\emph{only} of the a priori error bounds below, we shall later make a shape-regularity assumption on the space-time  mesh  (cf., Assumption \\ref{as:st_reg}).  We stress, however, that the stability results presented below do not depend on Assumption \\ref{as:st_reg} and, therefore, the numerical method proposed below is unconditionally stable for any choice of spatial and temporal meshsizes. Indeed, an important advantage in using such space-time methods is that they do \\emph{not} require any CFL-type restrictions.}\n\n\n\nFinally, the broken spatial gradient will be denoted by $\\nabla_n v$, given by $(\\nabla_n v)|_K:= (\\nabla v)|_K$ for all $K\\in\\mathcal{T}_n$ and a $v\\in C(I_n;H^1_0(\\Omega))+\\dspace{p}$; collectively, we shall denote the broken gradient by $\\widetilde{\\nabla} v$  defined as\n$(\\widetilde{\\nabla} v)|_{\\Omega\\times I_n}:= (\\nabla_n v)|_{\\Omega\\times I_n}$, $n=0,\\dots,N-1,$\nfor $v\\in C(\\prod_{n=0}^{N-1}\\mathring{I}_n;H^1_0(\\Omega))+\\dspaceT{p}$, i.e., $v$ is allowed to be  discontinuous both in space and in time.\n\n\n\n\\section{A space-time discontinuous Galerkin method}\\label{sec:method}\nTo derive the weak form suitable for dG discretisation we will follow an energy argument. We start by assuming that $u$ is a smooth enough solution of \\eqref{eq:wave} and  let  $v \\in \\mathcal{X} + \\dspaceT{p}$. The standard symmetric interior penalty dG weak formulation on the time-slab $I_n$ when tested with $\\dot v$ is given by\n\\begin{equation}\\label{weak_IP_one}\n\\begin{split}\n   (\\ddot u,\\dot v)_{\\Omega \\times I_n} &+ (a \\widetilde{\\nabla} u,\\widetilde{\\nabla} \\dot v)_{\\Omega \\times I_n}\n- (\\sa{a \\nabla u}, \\sj{\\dot v})_{\\Gamma_n \\times I_n}\\\\ \n&- (\\sj{u},\\sa{a \\nabla \\dot v})_{\\Gamma_n \\times I_n}+( \\sigma_0 \\sj{u} , \\sj{\\dot v} )_{\\Gamma_n \\times I_n} = 0,\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\n  \\label{eq:sigma_defn}\n\\sigma_0(x,t) = C_{\\sigma_0} p^2(h(x,t))^{-1}  ,\n\\end{equation}\nfor a positive constant $C_{\\sigma_0}$ to be made precise later.\nThis motivates the following definition of \\emph{discrete energy} $E_h(t,v)$ at time $t \\in I_n$, for \\reply{$v\\in \\mathcal{X} + \\dspaceT{p}$}:\n\\[\nE_h(t,v) := \\tfrac12\\|\\dot v(t)\\|^2_{\\Omega}+\\tfrac12\\|\\sqrt{a}\\widetilde{\\nabla} v(t)\\|^2_{\\Omega}\n+\\tfrac12\\|\\sqrt{\\sigma_0}\\sj{v(t)}\\|^2_{\\Gamma_n}-\\lip{\\sa{a \\nabla v(t)}}{ \\sj{ v(t)}}{\\Gamma_n}.\n\\]\nUsing the classical inverse inequality\n$\n  \\|v\\|^2_{\\partial K} \\leq C_{\\text{inv}} p^2 |\\partial K|\/|K| \\|v\\|^2_K$, for all $ v \\in \\mathcal{P}^p(K)$,  (see, e.g., \\cite{PieE,SchwabBook},) we see that\n\\[\n\\begin{split}  \n2\\int_{\\Gamma_n} |\\sa{a \\nabla v(t,\\reply{s})}|^2 ds &\\leq  C_a^2 \\sum_{K \\in \\mathcal{T}_n} \\int_{\\partial K} |\\nabla v(t,\\reply{s})|^2 ds\\\\\n&\\leq  \\sum_{K \\in \\mathcal{T}_n} \\frac{C_a^2C_{\\text{inv}}p^2|\\partial K|}{|K|}\\int_{ K}  |\\nabla v(t,\\reply{x})|^2 dx\\\\\n&\\leq  \\sum_{K \\in \\mathcal{T}_n} \\frac{c_{\\mathcal T} C_a^2 C_{\\text{inv}}p^2}{c_a h_K}\\int_{ K} |\\sqrt{a}  \\nabla v(t,\\reply{x})|^2 dx.\n\\end{split}\n\\]\nHence,  if the penalisation parameter $C_{\\sigma_0}$ is chosen large enough: in particular \n\\begin{equation}\n  \\label{eq:Cstab_ineq}\nC_{\\sigma_0} \\geq c_\\mathcal{T} C_a^2 C_{\\text{inv}}\/c_a,  \n\\end{equation}\nsuffices we have that\n\\begin{equation}\\label{coercivity_inverse}\n  \\begin{split}    \n|\\lip{\\sa{a \\nabla v(t)}}{ \\sj{ v(t)}}{\\Gamma_n}| &\\le \\tfrac12  \\|\\sqrt{\\sigma_0}\\sj{v(t)}\\|^2_{\\Gamma_n}+\\tfrac12 \\int_{\\Gamma_n}\\sigma_0^{-1} |\\sa{a \\nabla v(t,\\reply{s})}|^2 ds\\\\\n\\\\&\\le \\tfrac12\\|\\sqrt{\\sigma_0}\\sj{v(t)}\\|^2_{\\Gamma_n} +\\tfrac{1}{\\reply{4}}\\|\\sqrt{a}\\widetilde{\\nabla} v(t)\\|^2_{\\Omega},\n  \\end{split}\n\\end{equation}\nensuring the \\reply{non-negativity} of the energy $E_h(t,v)$ for functions in \\reply{$\\dspaceT{p}$}:\n\\[\nE_h(t,v) \\geq   \\tfrac12\\|\\dot v(t)\\|^2_{\\Omega}+\\tfrac14\\|\\sqrt{a}\\widetilde{\\nabla} v(t)\\|^2_{\\Omega}, \\qquad \\text{for all } v \\in \\mathcal{X}+\\dspaceT{p}.\n\\]\n\n Choosing as test function $v = u$ in \\eqref{weak_IP_one} and summing over $n$, we obtain\n\\[\n\\begin{split}\n 0 &= \\sum_{n = 0}^{N-1} \\int_{I_n}\\frac{d}{dt} \\Big(\\tfrac12 \\ltwo{\\dot u}{\\Omega}^2+\\tfrac12 \\ltwo{\\sqrt{a} \\widetilde{\\nabla} u}{\\Omega}^2- \\lip{\\sa{a\\nabla u}}{\\sj{ u}}{\\Gamma_n} + \\tfrac12 \\ltwo{\\sqrt{\\sigma_0} \\sj{u}}{\\Gamma_n}^2 \\Big) dt\n\\\\\n&= E_h(t_N^-,u)-E_h(t_0^+,u)-\\sum_{n = 1}^{N-1}\\tj{E_h(t_n,u)}.\n\\end{split}\n\\] \nIn order to allow for discontinuities in time, the formulation \\eqref{weak_IP_one} needs to be modified (in a consistent fashion) to control the terms $\\tj{E_h(t_n,u)}$ that have no sign. To this end, we shall use the elementary algebraic identity\n\\begin{equation}\n  \\label{eq:upwind}\n  \\begin{split}    \n -\\tj{ f(u(t_n))g(u(t_n))} +\\tj{ f(u(t_n))}g(u(t_n^+))\n&+ \\tj{ g(u(t_n))}f(u(t_n^+)) \\\\\n  &= \\tj{f(u(t_n))}\\tj{g(u(t_n))},\n  \\end{split}\n\\end{equation}\n\\reply{for some scalar quantities $f,g$ which may, in general, be discontinuous across $t_n$. The idea here is to add terms that change the jump of a product to the product of jumps in the above energy identity, without compromising its consistency. For instance, to $\\lip{\\ddot u}{ \\dot v}{\\Omega \\times I_n} $  in \\eqref{weak_IP_one} we add the extra term $\\lip{\\tj{\\dot u(t_n)}}{\\dot v(t_n^+)}{\\Omega}$.\nFor $n > 0$, this does not change the consistency of \\eqref{weak_IP_one} with respect to \\eqref{eq:wave} in weak form, as the smoothness assumptions on the initial data ensure that the exact solution satisfies $\\dot u \\in C([0,T]; L^2(\\Omega))$;  for $n = 0$, we have by the initial condition $\\dot u(t_0) = v_0$ that\n\\[\n\\lip{\\tj{\\dot u(t_0)}}{\\dot v(t_0^+)}{\\Omega}\n= \\lip{\\dot u(t_0^+)}{\\dot v(t_0^+)}{\\Omega}\n= \\lip{v_0}{\\dot v(t_0^+)}{\\Omega}.\n\\]\nHence, for consistency, we add the term $\\lip{\\dot u_0}{\\dot v(t_0^+)}{\\Omega}$ to the right-hand side also.\nUsing \\eqref{eq:upwind} now, for $u=v=w$, with $w$ piecewise sufficiently smooth function, we have\n\\[\n\\begin{split}  \n\\sum_{n = 0}^{N-1}\\lip{\\ddot w}{ \\dot w}{\\Omega \\times I_n}  +\\lip{\\tj{\\dot w(t_n)}}{\\dot w(t_n^+)}{\\Omega}\n=& \\tfrac12\\ltwo{\\dot w(t^-_N)}{\\Omega}^2+\\sum_{n = 0}^{N-1}\\lip{\\tj{\\dot w(t_n)}}{\\dot w(t_n^+)}{\\Omega} -\\tj{\\tfrac12\\ltwo{\\dot w(t_n)}{\\Omega}^2} \\\\\n=&\\tfrac12\\ltwo{\\dot w(t_N^-)}{\\Omega}^2+\\tfrac12\\ltwo{\\dot w(t_0^+)}{\\Omega}^2+\n\\tfrac12\\sum_{n = 1}^{N-1}\\ltwo{\\tj{\\dot w(t_n)}}{\\Omega}^2,\n\\end{split}\n\\]\nwith the additional terms contributing to energy dissipation leading to a stable method.\nCompletely analogous considerations lead to addition of corresponding terms to treat the second and the last terms on the left-hand side of \\eqref{weak_IP_one}. For the remaining third and fourth terms on the left-hand side of \\eqref{weak_IP_one}, we include the additional terms $ - \\lip{\\tj{\\sa{a\\widetilde{\\nabla} u(t_n)}}}{\\sj{ v(t_n^+)}}{\\hat{\\Gamma}_n}$ and $- \\lip{\\tj{\\sj{u(t_n)}}}{\\sa{a\\nabla  v(t_n^+)}}{\\hat{\\Gamma}_n}$; note that these again do not change the consistency due to both $u\\in C([0,T]; H^1_0(\\Omega))$ and \\eqref{eq:transmission}. The use of $\\hat{\\Gamma}_n = \\Gamma_{n-1} \\cup \\Gamma_n$ in the last terms also merits a brief explanation: since we assume that both the solution and its spatial flux are continuous within a space-time element $K \\times I_n$, $K \\in \\mathcal{T}_n$ it follows that\n\\[\n \\lip{\\sa{a \\nabla u}}{\\sj{\\dot v}}{\\Gamma_n \\times I_n} =  \\lip{\\sa{a \\nabla u}}{\\sj{\\dot v}}{\\hat{\\Gamma}_n \\times I_n}\\ \\text{ and }\\ \n \\lip{\\sj{u}}{\\sa{a \\nabla \\dot v}}{\\Gamma_n \\times I_n}\n= \\lip{\\sj{u}}{\\sa{a \\nabla \\dot v}}{\\hat{\\Gamma}_n \\times I_n}.\n\\]\n}\n\\reply{For consistency, the terms $- \\lip{ \\sa{a\\nabla u_0}}{\\sj{ v(t_0^+)}}{\\Gamma_0}- \\lip{ \\sj{u_0}}{\\sa{a\\nabla  v(t_0^+)}}{\\Gamma_0}$ are also added to the right-hand side of \\eqref{weak_IP_one}. The above identity can again be used to show that we will have terms of the type $\\lip{\\tj{\\sa{a\\widetilde{\\nabla} u(t_n)}}}{\\tj{\\sj{ u(t_n)}}}{\\hat{\\Gamma}_n}$ in the energy identity; these do not have a sign but can be bounded by the other terms in the energy using \\eqref{coercivity_inverse}.}\n\nIn view of the above considerations, we can now state the space-time weak formulation of our method:\n\\begin{equation}\n  \\label{eq:formulation}\n  \\begin{aligned}\n\\sum_{n = 0}^{N-1}\\lip{\\ddot u}{ \\dot v}{\\Omega \\times I_n} &+\\lip{\\tj{\\dot u(t_n)}}{\\dot v(t_n^+)}{\\Omega} \n+\\lip{a \\widetilde{\\nabla} u}{ \\widetilde{\\nabla} \\dot v}{\\Omega \\times I_n} + \\lip{\\tj{a \\widetilde{\\nabla} u(t_n)}}{\\widetilde{\\nabla} v(t_n^+)}{\\Omega}\\\\\n&- \\lip{\\sa{a \\nabla u}}{\\sj{\\dot v}}{\\Gamma_n \\times I_n} - \\lip{\\tj{\\sa{a\\widetilde{\\nabla} u(t_n)}}}{\\sj{ v(t_n^+)}}{\\hat{\\Gamma}_n}  \n\\\\ &- \\lip{\\sj{u}}{\\sa{a \\nabla \\dot v}}{\\Gamma_n \\times I_n}  \n- \\lip{\\tj{\\sj{u(t_n)}}}{\\sa{a\\nabla  v(t_n^+)}}{\\hat{\\Gamma}_n} \n\\\\ &+ \\lip{\\sigma_0  \\sj{u}}{\\sj{\\dot v}}{\\Gamma_n \\times I_n} + \\lip{\\sigma_0 \\tj{ \\sj{u(t_n)}}}{ \\sj{ v(t_n^+)}}{\\hat{\\Gamma}_n} \n\\\\ &+  \\lip{\\sigma_1 \\sj{u}}{\\sj{v}}{\\Gamma_n \\times I_n}\n+\\lip{\\sigma_2 \\sj{a\\nabla u}}{\\sj{a\\nabla v}}{\\Gamma_n \\times I_n}  = b^{\\text{init}}(v),\n  \\end{aligned}\n\\end{equation}\n\\reply{where  $b^{\\text{init}}$ is given by\n\\begin{equation}\n  \\label{eq:rhsform_init}\n  \\begin{aligned}    \n  b^{\\text{init}}(v) := & \n\\lip{ v_0}{ \\dot v(t_0^+)}{\\Omega} + \\lip{a \\widetilde{\\nabla} u_0}{ \\widetilde{\\nabla} v(t_0^+)}{\\Omega}\n- \\lip{ \\sa{a\\nabla u_0}}{\\sj{ v(t_0^+)}}{\\Gamma_0}\\\\\n&- \\lip{ \\sj{u_0}}{\\sa{a\\nabla  v(t_0^+)}}{\\Gamma_0}\n+ \\lip{\\sigma_0   \\sj{u_0}}{\\sj{ v(t_0^+)}}{\\Gamma_0}.\n  \\end{aligned}  \n\\end{equation}}\nNote that the last two terms in the definition of $b^{\\text{init}}$ are zero if the initial data is continuous in space.\nThe terms with positive penalty parameters $\\sigma_1$ and $\\sigma_2$ in the weak formulation \\eqref{eq:formulation} do not affect the consistency of the weak formulation; the need for their inclusion in the method will become apparent in the convergence analysis.\n\n\n\\reply{ Although the choice of $\\sigma_0$, $\\sigma_1$, and $\\sigma_2$ will only become evident in the analysis, for easy referral we state the choice of the stabilisation parameters $\\sigma_0$, $\\sigma_1$, and $\\sigma_2$ here also: $\\sigma_0$ is chosen as \\eqref{eq:sigma_defn}, (see also \\eqref{eq:Cstab_ineq},) and we set\n\\[\n\\sigma_1|_{\\partial K\\cap \\Gamma_n\\times I_n}= C_a\\frac{ p^3}{h\\tau_n} , \\qquad\n\\sigma_2 = \\frac{h}{C_a\\tau_n}.\n\\]\n}\n\nThus, we arrive at a space-time discrete method, which can be thought of in two ways: as a method for obtaining a discrete solution on a fixed space-time domain $\\Omega \\times [0,T]$, or as a time-stepping method. The former viewpoint will be  useful in obtaining convergence estimates, while the latter in implementing the method. Consequently, we define the following three bilinear forms to describe these two viewpoints:\n\\begin{equation}\n  \\label{eq:lhsform}\n  \\begin{aligned}    \n  a_n(u,v) &:=\\\n   \\lip{\\ddot u}{\\dot v}{\\Omega \\times I_n}+ \\lip{\\dot u(t^+_n)}{ \\dot v(t_n^+)}{\\Omega} \\\\\n   &+\\lip{a\\widetilde{\\nabla} u}{\\widetilde{\\nabla} \\dot v}{\\Omega \\times I_n}+ \\lip{ a\\widetilde{\\nabla} u(t_n^+)}{ \\widetilde{\\nabla} v(t_n^+)}{\\Omega}\\\\\n   &- \\lip{\\sa{a\\nabla u}}{\\sj{\\dot v}}{\\Gamma_n \\times I_n}- \\lip{\\sa{a\\nabla u(t_n^+)}}{\\sj{ v(t_n^+)}}{\\Gamma_n}  \\\\\n   &- \\lip{\\sj{u}}{ \\sa{a\\nabla \\dot v}}{\\Gamma_n \\times I_n} - \\lip{\\sj{u(t_n^+)}}{\\sa{a\\nabla  v(t_n^+)}}{\\Gamma_n}  \\\\\n   &+ \\lip{\\sigma_0   \\sj{u}}{ \\sj{\\dot v}}{\\Gamma_n \\times I_n} + \\lip{\\sigma_0  \\sj{u(t_n^+)}}{\\sj{ v(t_n^+)}}{\\Gamma_n} \\\\\n&+ \\lip{\\sigma_1   \\sj{u}}{ \\sj{ v}}{\\Gamma_n \\times I_n}\n+ \\lip{\\sigma_2   \\sj{a \\nabla u}}{ \\sj{a \\nabla v}}{\\Gamma_n \\times I_n},\n  \\end{aligned}\n\\end{equation}\n\\begin{equation}\n  \\label{eq:rhsform}\n  \\begin{aligned}    \n  b_n(u,v) :=&\\  \n\\lip{ \\dot u(t^-_n)}{ \\dot v(t_n^+)}{\\Omega} + \\lip{a\\widetilde{\\nabla} u(t_n^-)}{ \\widetilde{\\nabla} v(t_n^+)}{\\Omega}\n- \\lip{ \\sa{a\\nabla u(t_n^-)}}{\\sj{ v(t_n^+)}}{\\Gamma_n}\\\\\n&- \\lip{ \\sj{u(t_n^-)}}{\\sa{a\\nabla  v(t_n^+)}}{\\Gamma_{n-1}}\n+ \\lip{\\sigma_0   \\sj{u(t_n^-)}}{\\sj{ v(t_n^+)}}{\\hat{\\Gamma}_n},\n  \\end{aligned}  \n\\end{equation}\n and\n\\begin{equation}\n  \\label{eq:aform}  \na(u,v) := \\sum_{n = 0}^{N-1} a_n(u,v)-\\sum_{n = 1}^{N-1}b_n(u,v),\n\\end{equation}\nwhich just gives the left-hand side in \\eqref{eq:formulation}. \n\\begin{definition}\\label{defn:time_stepping}\nGiven subspaces $X_n \\subseteq \\dspace{p}$, the time-stepping method is described by: find $u^n \\in X_n$, $n = 1, 2, \\dots, N-1$,  such that\n\\begin{equation}\n  \\label{eq:timestep}\na_n(u^n,v) = b_n(u^{n-1},v), \\qquad  \\text{ for all }v \\in X_n\n\\end{equation}\n\\reply{and\n\\begin{equation}\n  \\label{eq:timestep_initial}\n  a_0(u^0,v) = b^{\\text{init}}(v), \\qquad  \\text{ for all }v \\in X_0.\n\\end{equation}}\n Equivalently, given a subspace $X \\subseteq \\dspaceT{p}$, the full space-time discrete system can be presented as: find $u \\in X$  such that\n\\begin{equation}\n  \\label{eq:fullsys}\n  a(u,v) = b^{\\text{init}}(v), \\qquad \\text{for all } v \\in X.\n\\end{equation}  \n\\end{definition}\n\\begin{lemma}\n  The following identities hold for \\reply{any $w \\in \\mathcal{X}+ \\dspaceT{p}$}:\n\\begin{equation}\n  \\label{eq:a_nuu}\na_n(w,w) = E_h(t_{n+1}^-,w)+E_h(t_n^+,w)+\\ltwo{\\sqrt{\\sigma_1}   \\sj{w}}{\\Gamma_n \\times I_n}^2\n+ \\ltwo{\\sqrt{\\sigma_2}   \\sj{a \\nabla w}}{\\Gamma_n \\times I_n}^2,    \n\\end{equation}\nfor $n = 0,1,\\dots,N-1$, and\n\\begin{equation}\n  \\label{eq:dg_norm_expr}  \n\\begin{split}  \na(w,w) = E_h(t_N^-,w)&+E_h(t_0^+,w) +\\sum_{n = 1}^{N-1} \\Big(\\tfrac12 \\ltwo{ \\tj{\\dot w(t_n)}}{\\Omega}^2+\\tfrac12\\ltwo{\\sqrt{a}\\tj{ \\widetilde{\\nabla} w(t_n)}}{\\Omega}^2 \\\\\n&-\\lip{\\tj{\\sa{a \\widetilde{\\nabla} w(t_n)}}}{\\tj{\\sj{w(t_n)}}}{\\hat{\\Gamma}_n}+\\tfrac12\\ltwo{ \\tj{\\sqrt{\\sigma_0}\\sj{w(t_n)}}}{\\hat{\\Gamma}_n}^2\\Big)\\\\\n&+ \\sum_{n = 0}^{N-1}\\Big(\\ltwo{\\sqrt{\\sigma_1} \\sj{w}}{\\Gamma_n \\times I_n}^2\n+ \\ltwo{\\sqrt{\\sigma_2} \\sj{a \\nabla w}}{\\Gamma_n \\times I_n}^2\\Big).\n\\end{split}\n\\end{equation} \n\\end{lemma}\n\\begin{proof}\n  The identities follow from the definitions of the bilinear forms and the energy $E_h(t,w)$.\n\\end{proof}\n\nNext we investigate the consistency and stability of the discrete scheme.\n\n\\begin{theorem}[Consistency and Stability]\\label{thm:const_stab} Let the space $\\dspaceT{p}$ be given. Then, the following statements hold:\n  \\begin{enumerate}\n  \\item  \\label{thm:consistency} Let $u$ be the weak solution of \\eqref{eq:wave}, with $u_0 \\in H^1_0(\\Omega)$ and $v_0 \\in L^2(\\Omega)$. Then $u$ satisfies \\eqref{eq:fullsys}.\n\\item  \\label{thm:stability} \\reply{For $C_{\\sigma_0}$  satisfying} \\eqref{eq:Cstab_ineq} and for any $v \\in \\dspaceT{p}$ and $t \\in (0,T)$, the energy $E_h(t,v)$ \nis bounded from below by\n\\begin{equation}\\label{energy_positive}\nE_h(t,v) \\geq  \\tfrac12\\ltwo{\\dot v(t)}{\\Omega}^2+\\tfrac14\\ltwo{\\sqrt{a}\\widetilde{\\nabla} v(t)}{\\Omega}^2. \n\\end{equation}\nFurther, if $X$ is a subspace of $\\dspaceT{p}$ and $U \\in X$ is the discrete solution, i.e., satisfies \\eqref{eq:fullsys}, then \n$\nE_h(t_N^-,U) \\leq E_h(t_1^-,U).\n$\n  \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nStatement \\ref{thm:consistency} follows from the derivation of the formulation and the regularity of the unique solution $u$; see \\eqref{eq:smoothness} and \\eqref{eq:smoothness1}.  We have already shown the positivity of the energy under the condition on $C_{\\sigma_0}$; see \\eqref{coercivity_inverse}. \n\nTo prove the remaining statement we proceed as follows. \\reply{Combining \\eqref{eq:fullsys} with \\eqref{eq:dg_norm_expr} gives} the energy identity \n\\begin{equation}\n  \\label{eq:dis_energy}\n  \\begin{split}    \nE_h(t_N^-,U) = \\ &  b^{\\text{init}}(U)-E_h(t_0^+,U)-\\sum_{n = 1}^{N-1} \\tfrac12 \\ltwo{\\tj{\\dot U(t_n)}}{\\Omega}^2+\\tfrac12\\ltwo{\\sqrt{a}\\tj{\\widetilde{\\nabla} U(t_n)}}{\\Omega}^2 \\\\\n&+\\sum_{n = 1}^{N-1} \\lip{\\tj{\\sa{a \\widetilde{\\nabla} U(t_n)}}}{\\tj{\\sj{U(t_n)} }}{\\hat{\\Gamma}_n}-\\tfrac12\\ltwo{\\sqrt{\\sigma_0} \\tj{\\sj{U(t_n)}}}{\\hat{\\Gamma}_n}^2\\\\\n&- \\sum_{n = 0}^{N-1}\\ltwo{\\sqrt{\\sigma_1}\\sj{U}}{\\Gamma_n \\times I_n}^2\n- \\sum_{n = 0}^{N-1}\\ltwo{\\sqrt{\\sigma_2}\\sj{a \\nabla U}}{\\Gamma_n \\times I_n}^2.\n  \\end{split}\n\\end{equation}\n\nExpression \\eqref{eq:a_nuu}\nimplies that\n\\[\na_0(U,U) = E_h(t_{1}^-,U)+E_h(t_0^+,U)+\\ltwo{\\sqrt{\\sigma_1}   \\sj{U}}{\\Gamma_0 \\times I_0}^2\n+ \\ltwo{\\sqrt{\\sigma}_2   \\sj{a \\nabla U}}{\\Gamma_0 \\times I_0}^2 = b^{\\text{init}}(U).\n\\]\nHence, the energy identity \\eqref{eq:dis_energy} can be written as\n\\begin{equation}\n  \\label{eq:dis_energy1}\n  \\begin{split}    \nE_h(t_N^-,U) =&\\  E_h(t_1^-,U)-\\sum_{n = 1}^{N-1} \\Big(\\tfrac12 \\ltwo{\\tj{\\dot U(t_n)}}{\\Omega}^2+\\tfrac12\\ltwo{\\sqrt{a}\\tj{\\widetilde{\\nabla} U(t_n)}}{\\Omega}^2 \\\\\n&-\\lip{\\tj{\\sa{a\\widetilde{\\nabla} U(t_n)}}}{\\tj{\\sj{U(t_n)} }}{\\hat{\\Gamma}_n}+\\tfrac12\\ltwo{\\sqrt{\\sigma_0} \\tj{\\sj{U(t_n)}}}{\\hat{\\Gamma}_n}^2\\\\\n&+ \\ltwo{\\sqrt{\\sigma_1}\\sj{U}}{\\Gamma_n \\times I_n}^2\n+ \\ltwo{\\sqrt{\\sigma_2}\\sj{a \\nabla U}}{\\Gamma_n \\times I_n}^2\\Big).\n  \\end{split}\n\\end{equation}\n Arguments used to prove the non-negativity of the discrete energy \\eqref{coercivity_inverse}, also show that the above equality implies that the discrete energy decreases at each time-step.\n\\end{proof}\n\n\\section{Polynomial Trefftz spaces}\\label{sec:Tspace}\n\nWe shall consider the discrete space of local polynomial solutions to the wave equation, where we make an additional assumption on the mesh and on $a(x)$ that allows us to define the Trefftz spaces. \\reply{Such polynomial spaces have already appeared in literature; see for example \\cite{KreMPS,MacJ,Whi03}.}\n\n\\begin{assumption}\nLet the diffusion coefficient $a(\\cdot)$ and the mesh be such that $a(\\cdot)$ is constant  in each element $K \\in \\mathcal{T}_n$ for each $n$.  \n\\end{assumption}\n\n\\begin{definition}[Polynomial Trefftz spaces]\\label{defn:trefftz_space}\nLet $S_{n,\\text{Trefftz}}^{h,p} \\subseteq \\dspace{p}$ be a subspace of functions satisfying the homogeneous wave equation on any space-time element $K \\times I_n$:\n\\[\nS_{n,\\text{Trefftz}}^{h,p}: = \\left \\{ v \\in \\dspace{p} \\;:\\; \\ddot v(t,x) -\\nabla \\cdot \\left( a \\,\\nabla v \\right)(t,x) = 0, \\quad t \\in I_n, x \\in K,\\,  K\\in \\mathcal{T}_n\\right\\}.\n\\]  \nThe space on $\\Omega \\times [0,T]$ is then defined as\n\\[\nV_{\\text{Trefftz}}^{h,p} = \\left\\{ u \\in L^2(\\Omega \\times [0,T]) \\;:\\; u|_{\\Omega \\times I_n} \\in S_{n,\\text{Trefftz}}^{h,p}, \\; n = 0,1\\dots, N-1\\right\\} \\subseteq \\dspaceT{p}.\n\\]\n\\end{definition}\nPolynomial plane waves are examples of functions in this space\n\\[\n(t+ a^{-1\/2}\\alpha \\cdot  x )^j, \\qquad |\\alpha| = 1, \\alpha \\in \\mathbb{R}^d, j \\in \\{0,\\dots,p\\}.\n\\]\n\n\\begin{proposition}\\label{prop:trefftz_poly}\n  The local dimension of the Trefftz space in $\\mathbb{R}^d$ is given by\n\\[\n\\operatorname{dim}(S_{n,\\text{Trefftz}}^{h,p}(K)) = \\left\\{\n  \\begin{array}{cc}\n    2p+1 & d = 1\\\\\n    (p+1)^2 & d = 2\\\\\n    \\tfrac16 (p+1)(p+2)(2p+3) & d = 3\n  \\end{array}\n\\right..\n\\]\n\\end{proposition}\n\\begin{proof}\n  The proof for $d = 1$ is clear. For $d = 3$ in \\cite{Whi03} it is shown that the dimension of \\reply{the space of  Trefftz, homogeneous polynomials of degree}  $j$ is $(j+1)^2$, hence the total dimension is given by\n\\[\n\\sum_{j = 0}^p (j+1)^2 = \\tfrac16 (p+1)(p+2)(2p+3).\n\\]\nThe case $d = 2$ is proved similarly by noticing that the dimension of \\reply{the space of Trefftz, homogeneous polynomials} of  degree $j$ is $2j+1$.\n\\end{proof}\n\n\\begin{example}\nIn two dimensions, \\reply{with a constant, scalar diffusion coefficient $a > 0$},  the following is a  basis for $S_{n,\\text{Trefftz}}^{h,p}$ with $p = 3$\n\\[\n\\begin{split}  \n\\{ &1,t,x,y,tx,ty,xy,at^2+x^2,at^2+y^2, \\\\ &xyt, at^3+3x^2t, x^3+3at^2x, y^3+3at^2y,(at^2+x^2)y, (at^2+y^2)x,(x^2-y^2)t\\}.\n\\end{split}\n\\]\n\\end{example}\n\\subsection{Existence and uniqueness}\nNext we prove that the discontinuous Galerkin energy norm is indeed a norm on the subspace of Trefftz polynomials. This also includes piecewise linear polynomials as $V_{\\text{Trefftz}}^{h,p} = \\dspaceT{p}$ for $p = 1$.\n\n\n\n\\begin{proposition}\\label{prop:seminorms}\nWith the choice of $C_{\\sigma_0}$ as in \\eqref{eq:Cstab_ineq} and $\\sigma_1, \\sigma_2 > 0$,\nbilinear forms $a_n(\\cdot,\\cdot)$ and $a(\\cdot,\\cdot)$ give rise to two semi-norms\n\\[\n\\dgnorm{v}_n := \\big(a_n(v,v)\\big)^{1\/2}, \\qquad v \\in \\dspace{p}\n\\] \nand\n\\[\n\\dgnorm{v} := \\big(a(v,v)\\big)^{1\/2}, \\qquad v \\in \\dspaceT{p}.\n\\] \nThese are in fact norms on Trefftz subspaces $S_{n,\\text{Trefftz}}^{h,p}$ and $V_{\\text{Trefftz}}^{h,p}$.\n\\end{proposition}\n\\begin{proof}\nRecalling \\eqref{eq:a_nuu}  and using  \\eqref{energy_positive}, we deduce that \n$\\dgnorm{v}_n^2 \\geq 0$ and is hence a semi-norm.\n\nSuppose $\\dgnorm{v}_n = 0$ for $v \\in S_{n,\\text{Trefftz}}^{h,p}$. Then, $a \\nabla v$  and $v$ have no jumps across the space skeleton and hence $v$ is a weak solution of the homogeneous wave equation on $\\Omega \\times I_n$ with zero initial and boundary conditions. Uniqueness implies $v \\equiv 0$ and hence that $\\dgnorm{\\cdot}_n$ is a norm on this space.\n\nThe analysis of $\\dgnorm{\\cdot}$ is similar recalling \\eqref{eq:dg_norm_expr}, which shows that $\\dgnorm{\\cdot}$ is a semi-norm if the stabilization parameter is chosen correctly. Proceeding as in the first case, shows that it is in fact a norm on the Trefftz spaces.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:ex_un}\nUnder the conditions of the above proposition and with initial data $u_0 \\in H^1_0(\\Omega)$, $v_0 \\in L^2(\\Omega)$, the discrete system \\eqref{eq:fullsys} with $X = V_{\\text{Trefftz}}^{h,p}$ has a unique solution.\n\\end{corollary}\n\\begin{proof}\n  The uniqeness of the solution to \\eqref{eq:fullsys} over the Trefftz space $X = V_{\\text{Trefftz}}^{h,p}$ follows from $a(\\cdot,\\cdot)$ being a norm on this space. Existence of the solution to the linear system follows from uniqueness.\n\\end{proof}\n\nNext, we present the convergence analysis of the Trefftz based method.\n\n\n\\subsection{Convergence analysis}\nWe shall now establish the quasi-optimality of the proposed method.\n\n\\begin{proposition}\nLet \\our{$w \\in \\mathcal{X} + V_{\\text{Trefftz}}^{h,p}$} and $v \\in V_{\\text{Trefftz}}^{h,p}$, then \n\\[\n|a(w,v)| \\leq C_\\star \\dgnorml{w}\\dgnorm{v},\n\\]  \nfor some constant $C_\\star > 0$ and \n\\[\n\\begin{split}\n\\dgnorml{w}^2 =& \\tfrac12\\sum_{n = 1}^{N} \\Big(\\|\\dot w(t_n^-)\\|^2_{\\Omega}+\\|\\sqrt{a}\\nabla w(t_n^-)\\|^2_{\\Omega}\n+\\|\\sqrt{\\sigma_0}\\sj{w(t_n^-)}\\|^2_{\\Gamma_n}+\\|\\sigma_0^{-1\/2}\\sa{a\\nabla w(t_n^-)}\\|^2_{\\Gamma_n}\\Big)\\\\\n&+\\sum_{n = 0}^{N-1}\\left(\\ltwo{\\sqrt{\\sigma_1}\\sj{w}}{\\Gamma_n \\times I_n}^2\n+\\ltwo{\\sqrt{\\sigma_2}\\sj{a\\nabla w}}{\\Gamma_n \\times I_n}^2\n+\\ltwo{\\sigma_2^{-1\/2}\\sa{\\dot w}}{\\Gamma_n^{\\rm int} \\times I_n}^2\\right.\\\\\n&\\qquad\\qquad+\\left.\\ltwo{\\sigma_1^{-1\/2} \\sa{a\\nabla \\dot w}}{\\Gamma_n \\times I_n}^2+\\ltwo{\\sigma_0\\sigma_1^{-1\/2}\\sj{\\dot w}}{\\Gamma_n \\times I_n}^2\\right).\n\\end{split}\n\\]\n\\end{proposition}\n\\begin{proof}\nIntegration by parts gives\n\\[\n\\begin{split}\n \\lip{\\ddot w}{ \\dot v}{\\Omega\\times I_n} + \\lip{a\\nabla w}{ \\nabla \\dot v}{\\Omega\\times I_n} \n =\\  &-\\lip{\\dot w}{ \\ddot v}{\\Omega\\times I_n} - \\lip{a\\nabla \\dot w}{ \\nabla  v}{\\Omega\\times I_n} \\\\\n&+\\lip{\\dot w(t_{n+1}^-)}{ \\dot v(t_{n+1}^-)}{\\Omega}-\\lip{\\dot w(t_{n}^+)}{ \\dot v(t_{n}^+)}{\\Omega}\\\\\n& + \\lip{a\\nabla w(t_{n+1}^-)}{ \\nabla v(t_{n+1}^-)}{\\Omega}-\\lip{a\\nabla w(t_n^+)}{ \\nabla v(t_n^+)}{\\Omega}\\\\\n=\\ &-\\lip{\\sj{\\dot w}}{\\sa{a \\nabla v}}{\\Gamma_n\\times I_n}-\\lip{\\sa{\\dot w}}{\\sj{a \\nabla v}}{\\Gamma_n^{\\text{int}}\\times I_n}\\\\\n&+\\lip{\\dot w(t_{n+1}^-)}{ \\dot v(t_{n+1}^-)}{\\Omega}-\\lip{\\dot w(t_{n}^+)}{ \\dot v(t_{n}^+)}{\\Omega}\\\\\n& + \\lip{a\\nabla w(t_{n+1}^-)}{ \\nabla v(t_{n+1}^-)}{\\Omega}-\\lip{a\\nabla w(t_n^+)}{ \\nabla v(t_n^+)}{\\Omega},\n\\end{split}\n\\]\nsince $v\\in V_{\\text{Trefftz}}^{h,p}$ and by using the (elementary) identity\n\\[\n-\\lip{a\\nabla \\dot w}{ \\nabla  v}{\\Omega\\times I_n}  = \\lip{\\dot w}{ \\nabla\\cdot a\\nabla  v}{\\Omega\\times I_n} \n-\\lip{\\sj{\\dot w}}{\\sa{a \\nabla v}}{\\Gamma_n\\times I_n}-\\lip{\\sa{\\dot w}}{\\sj{a \\nabla v}}{\\Gamma_n^{\\text{int}}\\times I_n},\n\\]\nin the second step. Further applications of integration by parts in time yield\n\\[\n\\begin{split}\n&-\\lip{\\sj{\\dot w}}{\\sa{a \\nabla v}}{\\Gamma_n\\times I_n} \\\\\n=\\ & \\lip{\\sj{ w}}{\\sa{a \\nabla \\dot v}}{\\Gamma_n\\times I_n}-\\lip{\\sj{ w(t_{n+1}^-)}}{\\sa{a \\nabla  v(t_{n+1}^-)}}{\\Gamma_n} +\\lip{\\sj{ w(t_{n}^+)}}{\\sa{a \\nabla  v(t_{n}^+)}}{\\Gamma_n},\n\\end{split}\n\\]\nand\n\\[\n\\begin{split}  \n&-\\lip{ \\sa{a\\nabla w} }{ \\sj{\\dot v}}{\\Gamma_n\\times I_n} +\\lip{\\sigma_0 \\sj{w}}{ \\sj{\\dot v}}{\\Gamma_n\\times I_n}\\\\\n= \\ & \\lip{\\sa{a\\nabla \\dot w}}{ \\sj{v}}{\\Gamma_n\\times I_n} -\\lip{\\sigma_0 \\sj{\\dot w}}{\\sj{ v}}{\\Gamma_n\\times I_n} \\\\\n&-\\lip{ \\sa{a\\nabla w(t_{n+1}^-)}}{ \\sj{ v(t_{n+1}^-)}}{\\Gamma_n}\n+\\lip{ \\sa{a\\nabla w(t_{n}^+)}}{ \\sj{ v(t_{n}^+)}}{\\Gamma_n}\\\\\n&+\\lip{\\sigma_0\\sj{w(t_{n+1}^-)}}{ \\sj{ v(t_{n+1}^-)}}{\\Gamma_n}\n-\\lip{\\sigma_0 \\sj{w(t_{n}^+)}}{ \\sj{ v(t_{n}^+)}}{\\Gamma_n}.\n\\end{split}\n\\]\nSubstituting these int\n\\eqref{eq:lhsform}, we obtain\n\\begin{equation}\n  \\label{eq:trefftz_bfn}  \n  \\begin{aligned}    \n  a_n(w,v) =& \\ \n    \\lip{\\sa{a\\nabla \\dot w}}{\\sj{ v}}{\\Gamma_n \\times I_n}- \\lip{\\sa{a\\nabla w(t_{n+1}^-)}}{\\sj{ v(t_{n+1}^-)}}{\\Gamma_n} \\\\\n&- \\lip{\\sigma_0   \\sj{\\dot w}}{\\sj{v}}{\\Gamma_n \\times I_n}+ \\lip{\\sigma_0  \\sj{w(t_{n+1}^-)}}{\\sj{ v(t_{n+1}^-)}}{\\Gamma_n}\\\\\n&- \\lip{ \\sa{\\dot w}}{ \\sj{a \\nabla v}}{\\Gamma_n^{\\rm int}\\times I_n}- \\lip{\\sj{w(t_{n+1}^-)}}{\\sa{a\\nabla  v(t_{n+1}^-)}}{\\Gamma_n} \\\\\n&+ \\lip{\\dot w(t^-_{n+1})}{\\dot v(t_{n+1}^-)}{\\Omega} + \\lip{a\\nabla w(t_{n+1}^-)}{ \\nabla v(t_{n+1}^-)}{\\Omega}\\\\\n&\n+ \\lip{\\sigma_1  \\sj{w}}{\\sj{ v}}{\\Gamma_n \\times I_n}\n+ \\lip{\\sigma_2  \\sj{a\\nabla w}}{\\sj{a \\nabla v}}{\\Gamma_n \\times I_n}.\n  \\end{aligned}\n\\end{equation}\nTherefore, upon adopting the notational convention \n$\n\\tj{f(t_N^-)} := f(t_N^-), \n$\nwe have\n\\begin{equation}\n  \\label{eq:trefftz_bf}  \n\\begin{split}\n  a(w,v) =& \\sum_{n = 0}^{N-1}a_n(w,v) - \\sum_{n = 1}^{N-1}b_n(w,v)\\\\\n=&  \\sum_{n = 0}^{N-1}\\Big( \\lip{\\sa{a\\nabla\\dot  w}}{ \\sj{ v}}{\\Gamma_n\\times I_n}\n- \\lip{\\sigma_0   \\sj{\\dot w}}{ \\sj{v}}{\\Gamma_n\\times I_n} \n- \\lip{ \\sa{\\dot w}}{ \\sj{a\\nabla v}}{\\Gamma_n^{\\rm int}\\times I_n}  \\\\\n&+ \\lip{\\sigma_1  \\sj{w}}{ \\sj{ v}}{\\Gamma_n \\times I_n}\n+\\lip{\\sigma_2  \\sj{a\\nabla w}}{ \\sj{a \\nabla v}}{\\Gamma_n \\times I_n}\\Big)\\\\\n&- \\sum_{n = 1}^{N}\\Big(\\lip{ \\dot w(t_n^-)}{ \\tj{\\dot v(t_n)}}{\\Omega} + \\lip{a\\nabla w(t_n^-)}{ \\tj{\\nabla v(t_n)}}{\\Omega}\\\\\n&- \\lip{\\sa{a\\nabla w(t_n^-)}}{\\tj{\\sj{ v(t_n)}}}{\\Gamma_n} \n- \\lip{ \\sj{w(t_n^-)}}{\\tj{\\sa{a\\nabla  v(t_n)}}}{\\Gamma_n}\\\\\n&+ \\lip{\\sigma_0  \\sj{w(t_n^-)}}{\\tj{\\sj{ v(t_n)}}}{\\Gamma_n}\\Big).\n\\end{split}\n\\end{equation}\n\n\nIt is now clear how to estimate most of the terms to obtain the stated result using the Cauchy-Schwarz inequality. \nThe first two terms on the right hand side in the above sum are estimated as follows\n\\[\n \\lip{ \\sigma_1^{-1\/2}(\\sa{a\\nabla\\dot  w}- \\sigma_0   \\sj{\\dot w})}{\\sqrt{\\sigma_1}\\sj{ v}}{\\Gamma_n \\times I_n}\n\\leq \\ltwo{\\sigma_1^{-1\/2}(\\sa{a\\nabla\\dot  w}-  \\sigma_0 \\sj{\\dot w})}{\\Gamma_n \\times I_n}\n\\ltwo{\\sqrt{\\sigma_1} \\sj{v}}{\\Gamma_n \\times I_n};\n\\]\nfor the third term, we have \n\\[\n\\begin{split}\n\\lip{ \\sa{\\dot w}}{ \\sj{a\\nabla v}}{\\Gamma_n^{\\rm int}\\times I_n}   \n& \\le \\ltwo{\\sigma_2^{-1\/2} \\sa{\\dot w}}{\\Gamma_n^{\\rm int} \\times I_n}   \\ltwo{\\sqrt{\\sigma_2} \\sj{a\\nabla v}}{\\Gamma_n^{\\rm int} \\times I_n}.\n\\end{split}\n\\]\n\\end{proof}\n\n\\begin{remark}\\label{rem:cheaper}\n Note that \\eqref{eq:trefftz_bfn} shows that for Trefftz functions the bilinear form can be evaluated without computing integrals over the volume terms $\\Omega \\times I_n$. This can bring considerable savings, especially in higher spatial dimensions; see Figure~\\ref{fig:Conv_Time}.\n\\end{remark}\n\n\\reply{\nIt is possible to define the space-time dG method above with the classical (discontinuous) space of \\emph{all} polynomials of degree $p$ (total degree or of degree $p$ on each variable). The resulting method then appears to work in practice also. Some numerical experiments and comparison with the smaller polynomial Trefftz space are given Section \\ref{sec:numerics}. The error analysis of the method with the full polynomial space of degree $p$ is not straightforward, though. In particular, it is not immediately clear how to treat the volume terms which do not vanish in this case. This would be essential in completing an error analysis for such spaces also.}\n\n\n\n\\begin{theorem}\\label{th:best_approx}\n  Let $U \\in V_{\\text{Trefftz}}^{h,p}$ be the discrete solution of the Trefftz time-space discontinuous Galerkin method and let \\our{$u \\in \\mathcal{X}$} be the exact solution. Then, we have\n\\[\n\\dgnorm{U-u} \\leq\\inf_{V \\in V_{\\text{Trefftz}}^{h,p}}\\big (  C_\\star \\dgnorml{V - u}+\\dgnorm{V-u} \\big). \n\\]\n\n\\end{theorem}\n\\begin{proof}\nBy Galerkin orthogonality\n\\[\na(V -U, v) = a(V-u,v),\n\\]\nfor any $V,v \\in V_{\\text{Trefftz}}^{h,p}$. Hence,\n\\[\n\\dgnorm{V - U}^2 = a(V - U, V - U)\n = a(V - u, V - U) \\leq C_\\star \\dgnorml{V - u}\\dgnorm{V - U},\n\\]\ngiving\n\\[\n\\dgnorm{U-u} \\leq \\dgnorm{V-U}+\\dgnorm{V-u} \\leq  C_\\star\\dgnorml{V - u}+\\dgnorm{V-u}.\n\\]\n\\end{proof}\n\nTo conclude this section we show that in the case of Trefftz polynomials, the discrete norm can be bounded below by an $L^2$-temporal norm.  For simplicity of the presentation \\emph{only}, we shall, henceforth, make the following assumption.\n \\our{\n\\begin{assumption} \\label{as:st_reg}\nWe assume that $\\diam(K\\times I_n)\/\\rho_{K\\times I_n} \\leq c_{\\mathcal T}$, for all $K \\in \\mathcal{T}_n$, $n = 0,1,\\dots, N-1$.\n\\end{assumption}}\n\n\\our{In the simplifying context of space-time meshes consisting of prismatic meshes without local time-stepping, Assumption \\ref{as:st_reg} implies global quasi-uniformity. Given the tensor-product\/pris\\-matic structure of the space-time elements $K\\times I_n$, it is by all means possible to extend the error analysis below to space-time meshes not satisfying Assumption \\ref{as:st_reg}. Moreover, with minor modifications only, it is possible to extend the findings of this work to meshes with space-time elements with variable temporal dimension lengths; the notational overhead for such a development is deemed excessive given the a-priori error analysis point of view of this work.}\n\n\\begin{proposition}\\label{prop:positiveE}\n  For any $v \\in S_{n,\\text{Trefftz}}^{h,p}$ it holds\n\\[\n\\ltwo{\\dot v}{\\Omega \\times I_n}^2+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} v}{\\Omega \\times I_n}^2 \n\\leq (t_{n+1}-t_n)e^{\\widetilde C(t_{n+1}-t_n)\/\\underline{h}}\\left(\\ltwo{\\dot v(t_n^+)}{ \\Omega}^2+\\ltwo{\\sqrt{a} \\widetilde{\\nabla} v(t_n^+)}{ \\Omega}^2 \\right),\n\\]\nwhere\n$\n\\reply{\\widetilde C} := c_{\\mathcal T}C_{\\text{inv}}p^2 C_a$ and\n$ \\underline{h} := \\min_{x \\in \\Omega} h(x,t)$, $t \\in I_n$.\nThe same estimate holds with $t_n^+$ replaced by $t_{n+1}^-$.\n\n\\reply{Consequently, under Assumption ~\\ref{as:st_reg},\n\\[\n\\|\\dot V\\|^2_{\\Omega \\times (0,T)}+\\|\\sqrt{a}\\widetilde{\\nabla} V\\|^2_{\\Omega \\times (0,T)}\n\\leq C e^{\\widetilde C c_{\\mathcal T}}T\\dgnorm{V}^2,\n\\]\nfor all $V \\in V_{\\text{Trefftz}}^{h,p}$ with a constant $C>0$ independent of the meshsize.}\n\\end{proposition}\n\\begin{proof}\nNote that for an element $K$ with exterior normal $\\nu$\n\\[\n\\begin{split}  \n\\frac{d}{dt}  \\left(\\tfrac12\\ltwo{\\dot v(t)}{K}^2+\\tfrac12 \\ltwo{\\sqrt{a} \\nabla v(t)}{K}^2\\right)\n &= \\lip{\\ddot v(t)}{ \\dot v(t)}{K} +  \\lip{a\\nabla v(t)}{ \\nabla \\dot v(t)}{K}\\\\\n&= \\lip{\\nu \\cdot a \\nabla  v(t)}{\\dot v(t)}{\\partial K}\\\\\n&\\leq \\tfrac12 \\ltwo{\\nu \\cdot a \\nabla  v(t)}{\\partial K}^2 +\\tfrac12 \\ltwo{\\dot v(t)}{\\partial K}^2\\\\\n&\\leq  C_{\\text{inv}}p^2 |\\partial K|\/|K|\\left(\\tfrac12\\ltwo{a \\nabla v(t)}{ K}^2+ \\tfrac12\\ltwo{\\dot v(t) }{ K}^2 \\right),\\\\\n&\\leq C_a C_{\\text{inv}}p^2 c_{\\mathcal T} h_K^{-1}\\left(\\tfrac12\\ltwo{\\sqrt{a}\\nabla v(t)}{ K}^2+ \\tfrac12\\ltwo{\\dot v(t) }{ K}^2 \\right),\n\\end{split}\n\\]\nwhere we have used the discrete trace inequality.\nGronwall inequality now gives us \n\\[\n\\tfrac12\\ltwo{\\sqrt{a}\\nabla v(t)}{K}^2+ \\tfrac12\\ltwo{\\dot v(t)}{ K}^2 \n\\leq  e^{\\widetilde C(t-t_n)\/{h_K}}\\left(\\tfrac12\\ltwo{\\sqrt{a}\\nabla v(t_n^+)}{ K}^2+ \\tfrac12\\ltwo{\\dot v(t_n^+)}{ K}^2 \\right),\n\\]\nas well as\n\\[\n\\tfrac12\\ltwo{\\sqrt{a}\\nabla v(t)}{K}^2+ \\tfrac12\\ltwo{\\dot v(t)}{ K}^2 \n\\leq  e^{\\widetilde C(t_{n+1}-t_n)\/h_K}\\left(\\tfrac12\\ltwo{\\sqrt{a}\\nabla v(t_{n+1}^-)}{ K}^2+ \\tfrac12\\ltwo{\\dot v(t_{n+1}^-)}{ K}^2 \\right),\n\\]\nfor all $t \\in [t_n,t_{n+1}]$.  Integrating in time and summing over all $K$ gives the required result. \\reply{The final inequality follows from the definition of the discrete dG norm.}\n\\end{proof}\n\nThe above two results allow us to conclude that we can also bound the error in a more standard norm.\n\\begin{corollary}\\label{cor:positiveE}\nUnder the hypothesis of Theorem~\\ref{th:best_approx} and under Assumption \\ref{as:st_reg}, we have\n\\[\n\\begin{split}  \n\\ltwo{\\dot U -\\dot u}{\\Omega \\times [0,T]}+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} (U-u)}{\\Omega \\times [0,T]} \n\\leq C &\\inf_{V \\in V_{\\text{Trefftz}}^{h,p}}\\big (  C_\\star \\dgnorml{V - u}\\\\ &+\\ltwo{\\dot V -\\dot u}{\\Omega \\times [0,T]}+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} (V-u)}{\\Omega \\times [0,T]} \\big),\n\\end{split}\n\\]\nfor some positive constant $C$, independent of the mesh parameters and of $u$ and $U$. \n\\end{corollary}\n\\begin{proof}\n\\reply{\nTriangle inequality and Proposition \\ref{prop:positiveE} imply that, for any $V \\in V_{\\text{Trefftz}}^{h,p}$, \n\\[\n\\begin{split}  \n& \\ltwo{\\dot U -\\dot u}{\\Omega \\times [0,T]}+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} (U-u)}{\\Omega \\times [0,T]}\\\\\n\\leq & \\ltwo{\\dot U -\\dot V}{\\Omega \\times [0,T]}+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} (U-V)}{\\Omega \\times [0,T]}\n\\\\\n&+ \\ltwo{\\dot V -\\dot u}{\\Omega \\times [0,T]}+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} (V-u)}{\\Omega \\times [0,T]}\\\\\n\\leq & C\\dgnorm{U-V}\n+ \\ltwo{\\dot V -\\dot u}{\\Omega \\times [0,T]}+\\ltwo{\\sqrt{a}\\widetilde{\\nabla} (V-u)}{\\Omega \\times [0,T]}.\n\\end{split}\n\\]\nAs in the proof of Theorem~\\ref{th:best_approx}, we also have that \n$\n\\dgnorm{V-U} \\leq C_\\star \\dgnorml{V-u},\n$\nwhich completes the proof.\n}\n\\end{proof}\n\n\\section{A priori error bounds}\\label{apriori}\nWe shown next  the Trefftz basis \nis sufficient to deliver the expected rates of convergence for the proposed method.\n\n\n\\begin{lemma}\\label{lemma:abstract_error_bound}\nLet the setting of Theorem~\\ref{th:best_approx} hold, let $\\our{v}_h \\in V_{\\text{Trefftz}}^{h,p}$ be an arbitrary function in the discrete space, and let $\\eta = u-\\our{v}_h$. Then\n\\begin{equation}\\label{eq:abstract_error_bound}\n\\begin{aligned}\n\\dgnorm{U-u}^2 \\leq \\ &  C\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} \n\\our{C_a}\\bigg( \\frac{p^2}{\\our{\\tau_n}}\\Big( \\our{\\max\\big\\{1,\\frac{\\tau_n^2}{h_K^2}\\big\\}}\\|\\dot{\\eta}\\|_{ K\\times I_n}^2+ \\|\\nabla\\eta\\|_{ K\\times I_n}^2\\Big)\\\\\n&\\qquad\\quad\\qquad+\\our{\\tau_n}\\Big( \\|\\nabla\\dot{\\eta}\\|_{ K\\times I_n}^2+ \\our{\\max\\big\\{1,\\frac{h_K^2}{\\tau_n^2}\\big\\}}p^{-1}\\|D^2\\eta\\|_{ K\\times I_n}^2\\Big)\\\\\n&\\qquad\\quad\\qquad+\\frac{\\our{h_K^2\\tau_n}}{p^4} \\|D^2\\dot{\\eta}\\|_{ K\\times I_n}^2+\\our{\\frac{p^4}{h_K^2\\tau_n} \\|\\eta\\|_{ K\\times I_n}^2}\\bigg),\n\\end{aligned}\n\\end{equation}\nwhere $U \\in V_{\\text{Trefftz}}^{h,p}$ is the discrete solution. \n\\end{lemma}\n\\begin{proof}\n Theorem \\ref{th:best_approx} implies \n\\[\n\\dgnorm{U-u} \\leq  C_\\star \\dgnorml{\\eta}+\\dgnorm{\\eta}.\n\\]\nWe shall now estimate each term of the norms on the right-hand side. We shall repeatedly use the standard trace estimate \n$\n \\| v \\|_{\\partial \\omega}^2 \\le C \\| v \\|_{ \\omega} ( \\| v \\|_{ \\omega}^2+ \\| \\nabla v \\|_{ \\omega}^2)^{1\/2},\n$\nfor $v\\in H^1(\\omega)$, where $\\omega$ is a subset of \\reply{$\\mathbb{R}^k$, $k=1,\\dots,d+1$}.\nWe proceed as follows:\n\\[\n\\begin{aligned}\n\\sum_{n = 1}^{N} \\|\\dot \\eta(t_n^-)\\|^2_{\\Omega} \n&=   \\sum_{n = 1}^{N} \\sum_{K\\in \\mathcal{T}^{n-1}} \\|\\dot \\eta\\|^2_{K\\times \\{t_n^-\\}}\n\\le C\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} \\Big( \\frac{C_a p}{\\our{\\tau_n} }\\|\\dot \\eta\\|^2_{K\\times I_n}+ \\frac{\\our{\\tau_n} }{C_a p}\\|\\ddot \\eta\\|^2_{K\\times I_n}\\Big)\\\\\n& \\le C\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} \\Big( \\frac{C_a p}{\\our{\\tau_n} }\\|\\dot \\eta\\|^2_{K\\times I_n}+ \\frac{\\our{\\tau_n}}{C_a p}\\|\\nabla\\cdot a(\\cdot)\\nabla \\eta\\|^2_{K\\times I_n}\\Big)\\\\\n& \\le C\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p}{\\our{\\tau_n} }\\|\\dot \\eta\\|^2_{K\\times I_n}+ \\frac{\\our{\\tau_n}}{p}\\|\\Delta \\eta\\|^2_{K\\times I_n}\\Big);\n\\end{aligned}\n\\]\nWe prefer to retain an explicit dependence on the polynomial degree $p$ at this point, as it will be of relevance in the error analysis for $d=1$.\nIn analogous fashion, we also have\n\\[\n\\begin{aligned}\n\\sum_{n = 1}^{N} \\| \\sqrt{a}\\nabla \\eta(t_n^-)\\|^2_{\\Omega} \n&\n\\le C\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p}{\\our{\\tau_n}}\\|\\nabla \\eta\\|^2_{K\\times I_n}+ \\frac{\\our{\\tau_n}}{ p}\\|\\nabla\\dot \\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nNext, we estimate the penalty term:\n\\[\n\\begin{aligned}\n\\sum_{n = 1}^{N}  \\|\\sqrt{\\sigma_0^{}}\\sj{\\eta(t_n^-)}\\|^2_{\\Gamma_n} \n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p^3}{\\our{\\tau_n h_K}}\\| \\eta\\|^2_{\\partial K\\times I_n}+ p\\our{\\frac{\\tau_n}{h_K}}\\|\\dot \\eta\\|^2_{\\partial K\\times I_n}\\Big)\\\\\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p^4}{\\our{\\tau_n h_K^2}}\\| \\eta\\|^2_{K\\times I_n}+\\frac{p^2}{\\our{\\tau_n} }\\|\\nabla \\eta\\|^2_{K\\times I_n}\\\\\n   &\\qquad \\qquad\\qquad\\quad+ \\frac{p^2\\our{\\tau_n}}{\\our{h_K^2}}\\|\\dot \\eta\\|^2_{K\\times I_n}+ \\our{\\tau_n} \\| \\nabla\\dot\\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nSimilarly, we also have\n\\[\n\\begin{aligned}\n\\sum_{n = 1}^{N}  \\|\\sigma_0^{-1\/2}\\sa{a\\nabla \\eta(t_n^-)}\\|^2_{\\Gamma_n}\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{\\our{h_K}}{p\\,\\our{\\tau_n}}\\| \\nabla \\eta\\|^2_{\\partial K\\times I_n}+ \\frac{\\our{h_K\\tau_n}}{p^3}\\|\\nabla\\dot \\eta\\|^2_{\\partial K\\times I_n}\\Big)\\\\\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{1}{\\our{\\tau_n} }\\| \\nabla \\eta\\|^2_{K\\times I_n}+\\frac{\\our{h_K^2} }{p^2\\our{\\tau_n}}\\|D^2 \\eta\\|^2_{K\\times I_n}\\\\\n   &\\qquad \\qquad\\qquad\\quad+ \\frac{\\our{\\tau_n}  }{p^2}\\|\\nabla\\dot \\eta\\|^2_{K\\times I_n}+\\frac{\\our{h_K^2\\tau_n} }{p^4}\\| D^2\\dot\\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nNext, recalling that $\\sigma_1|_{\\partial K\\cap \\Gamma_n\\times I_n}= C_a p^3\/(\\our{h_K\\tau_n} )$, we estimate\n\\[\n\\begin{aligned}\n\\sum_{n = 0}^{N-1}\\ltwo{\\sqrt{\\sigma_1}\\sj{\\eta}}{\\Gamma_n \\times I_n}^2\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p^4}{\\our{\\tau_n h_K^2}}\\|  \\eta\\|^2_{ K\\times I_n}+ \\frac{p^2}{\\our{\\tau_n} }\\|\\nabla \\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nFurther, since $\\sigma_2= \\our{h_K}\/(C_a\\our{\\tau_n})$,  we have\n\\[\n\\begin{aligned}\n\\sum_{n = 0}^{N-1} \\ltwo{\\sqrt{\\sigma_2}\\sj{a\\nabla \\eta}}{\\Gamma_n \\times I_n}^2\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p^2}{\\our{\\tau_n} }\\|\\nabla  \\eta\\|^2_{ K\\times I_n}+ \\frac{\\our{h_K^2}}{p^2\\our{\\tau_n}}\\|D^2 \\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nThe next term is treated as follows:\n\\[\n\\begin{aligned}\n\\sum_{n = 0}^{N-1} \\ltwo{\\sigma_2^{-1\/2}\\sa{\\dot \\eta}}{\\Gamma_n^{\\rm int} \\times I_n}^2\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p^2\\our{\\tau_n}}{\\our{h_K^2} }\\| \\dot \\eta\\|^2_{ K\\times I_n}+ \\frac{\\our{\\tau_n} }{p^2}\\|\\nabla \\dot \\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nContinuing, we have\n\\[\\begin{aligned}\n\\sum_{n = 0}^{N-1}\\ltwo{\\sigma_1^{-1\/2} \\sa{a\\nabla \\dot \\eta}}{\\Gamma_n \\times I_n}^2\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{\\our{\\tau_n} }{p^2}\\| \\nabla \\dot\\eta\\|^2_{ K\\times I_n}+ \\frac{\\our{\\tau_n h_K^2}}{p^4}\\| D^2 \\dot \\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nFinally, we estimate\n\\[\n\\begin{aligned}\n\\sum_{n = 0}^{N-1}\\ltwo{\\sigma_0\\sigma_1^{-1\/2}\\sj{\\dot u}}{\\Gamma_n \\times I_n}^2\n&\n\\le C  \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} C_a \\Big( \\frac{p^2\\our{\\tau_n}}{\\our{h_K^2} }\\|  \\dot\\eta\\|^2_{ K\\times I_n}+ \\our{\\tau_n} \\| \\nabla \\dot \\eta\\|^2_{K\\times I_n}\\Big).\n\\end{aligned}\n\\]\nThe remaining terms in $\\dgnorm{\\eta}$ are treated completely analogously.\n\\end{proof}\n\n\\our{We have kept the explicit dependence on the local spatial and temporal meshsizes to further emphasize that the proposed method does not require any CFL-type restrictions for stability and convergence: as long as $h_K\\sim \\tau_n$ the above bound is sufficient to show optimal convergence, as we shall see below.}\nTo complete the error analysis, we need to prove the existence of an appropriate approximation in $V_{\\text{Trefftz}}^{h,p}$ of the exact solution. We first show how to obtain\nsuch an approximation locally.\n\n\\begin{proposition}\nLet $J \\subset \\mathbb{R}^{d+1}$ be star-shaped with respect to a ball $B \\subset J$. Then, there exists a projector \n\\[\n\\Pi^{p}: H^{p+1}(J) \\rightarrow \\mathcal{P}_p(J)\n\\]\nsuch that for any $v \\in H^{p+1}(J)$\n\\[\n\\|D^\\beta (v-\\Pi^{p}v)\\|_{J} \\leq  C(\\diam(J))^{p+1-|\\beta|} \\|v\\|_{H^{p+1}(J)}, \\qquad\n|\\beta| \\leq p\n\\]\nand further if $v$ satisfies the wave equation $\\ddot u - \\nabla \\cdot a \\nabla v = 0$ in $J$ then so does $\\Pi^pv$. The constant $C$ depends on $p$ and on the shape of $J$\\footnote{By the shape of the domain, is here meant the chunkiness parameter $\\diam(J)\/\\rho_{\\text{max}}$ where $\\rho_{\\text{max}} = \\sup\\{\\rho : J \\text{ is star-shaped with respect to a ball of radius } \\rho\\}$}.\n\\end{proposition}\n\\begin{proof}\nWe can define $\\Pi^p v$ to be the averaged Taylor polynomial of order $p$,\n\\[\n\\Pi^p v(x) = \\sum_{|\\alpha| \\leq p} \\frac1{\\alpha!} \\int_B D^\\alpha v(y) (x-y)^\\alpha \\phi(y) dy,\n\\]\nwhere $\\phi \\in C_0^\\infty(\\mathbb{R}^{d+1})$ is an arbitrary cut-off function satisfying $\\int_B \\phi = 1$ and $\\operatorname{supp} \\phi = \\overline{B}$; for details see \\cite{brenner_scott}. Then Bramble-Hilbert lemma gives us the approximation property required, see \\cite[Lemma~4.3.8]{brenner_scott}, and so it only remains to show that $\\Pi^p v$ satisfies the wave equation if $v$ does. For $p \\leq 1$ this is clear. For $p \\geq 2$, the result follows from the linearity of $\\Pi^p$, the fact that the wave equation contains only operators of second order, and the following property of averaged Taylor polynomials:\n\\[\nD^\\alpha \\Pi^p v = \\Pi^{p-|\\alpha|} D^\\alpha v, \\qquad |\\alpha| \\leq p.\n\\]\n\\end{proof}\n\nApplying such a projector to the exact solution and combining this with Lemma~\\ref{lemma:abstract_error_bound} gives us a proof of the convergence order of the discrete scheme.\n\n\\begin{theorem}\\label{thm:conv_order}\n  Let the exact solution $u \\in \\mathcal{X}$ be such that for each space time element $K \\times I_n$, $u|_{K \\times I_n} \\in H^{s+1}(K \\times I_n)$ for some $0 \\leq s \\leq p$. \n Then\n\\begin{equation}\\label{eq:gen_error_bound}\n\\dgnorm{U-u} \\leq C\\left( \\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} (h_K^{(n)})^{2s-1}  \\|u\\|^2_{H^{s+1}(K \\times I_n)}\\right)^{1\/2} \\leq C(\\reply{u}) h^{s-1\/2},\n\\end{equation}\nwhere $h = \\max_{K,n} h^{(n)}_K$ and \n\\[\nC(\\reply{u}) = \\left(\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n}  \\|u\\|^2_{H^{s+1}(K \\times I_n)}\\right)^{1\/2}.\n\\]\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\subsection{$hp$-version error analysis for $d=1$}\\label{sec:approx_one}\nAs we shall now show, the Trefftz basis is sufficient to deliver the expected $hp$-version a priori error bounds for $d=1$, along with a proof of the exponential convergence of the $p$-version space-time dG method for the case of analytic exact solutions. \\reply{See \\cite{KreMPS} for another $hp$-analysis in 1D of a different Trefftz based method.}\n\nTo discuss the Trefftz-basis case for $d=1$, we let $K = [x_0,x_1]$ and start from the basic observation that the exact solution to the wave equation on each space time element is of the form\n\\begin{equation}\\label{eq:exact_sol_form}\nu(x,t)|_{K\\times I_n} = F^1_{n,K}(a^{-1\/2}x+t)+F^2_{n,K}(a^{-1\/2}x-t),\n\\end{equation}\nwhere we can define $F^1$ and $F^2$ by\n\\[\nF^1_{n,K}(a^{-1\/2}x+t) = \\tfrac12 u(x,t)+\\tfrac12v(x,t), \\qquad\n(x,t) \\in K \\times I_n\n\\]\nand\n\\[\nF^2_{n,K}(a^{-1\/2}x-t) = \\tfrac12 u(x,t)-\\tfrac12 v(x,t), \\qquad\n(x,t) \\in K \\times I_n,\n\\]\nwhere\n\\[\nv(x,t) = a^{1\/2}\\int_{t_n}^t u_x(x,\\tau) d\\tau+a^{-1\/2}\\int_{x_0}^xu_t(x',t_n)dx'.\n\\]\nIt is not difficult to see that these are well-defined, i.e., that the right-hand sides indeed depend only on $a^{-1\/2}x\\pm t$ by virtue of satisfying the equations $a^{1\/2} f_x \\mp f_t = 0$ respectively.\n\n\nFor $\\hat{I}:=(-1,1)$, we define\nthe $H^1$-projection operator $\\hat{\\lambda}_p:H^1(\\hat{I})\\to\n\\mathcal{P}_p(\\hat{I})$, $p\\ge 1$, defined by\nsetting, for $\\hat{u}\\in H^1(\\hat{I})$,\n\\[(\\hat{\\lambda}_p\\hat{u})(x):=\\int_{-1}^{x}\\hat{\\pi}_{p-1}(\\hat{u}')(\\eta)\\,\\mathrm{d}\n\\eta +\\hat{u}(-1),\\qquad x\\in \\hat{I},\\] with $\\hat{\\pi}_{p-1}$\nbeing the $L^2$-orthogonal projection operator onto\n$\\mathcal{P}_{p-1}(\\hat{I})$. \n\nNow, upon considering the linear scalings $\\psi^1_{n,K}:\\hat{I}\\to J^1_{n,K}$,  $K\\in\\mathcal{T}_{n}$, such that\n\\[\nJ^1_{n,K}:=( \\min_{(x,t)\\in K\\times I_n} \\{x+ct\\}, \\max_{(x,t)\\in K\\times I_n} \\{x+ct\\} ),\n\\]\nand $\\psi^2_{n,K}:\\hat{I}\\to J^2_{n,K}$,  $K\\in\\mathcal{T}_{n}$, such that\n\\[\nJ^2_{n,K}:=( \\min_{(x,t)\\in K\\times I_n} \\{x-ct\\}, \\max_{(x,t)\\in K\\times I_n} \\{x-ct\\} ),\n\\]\nwe define the univariate space-time elemental projection operators $\\lambda^i_p$, $i=1,2$, piecewise by\n\\[\n(\\lambda^i_p F)|_{J^i_{n,K}} := \\hat{\\lambda}^i_p ((F\\circ \\psi^i_{n,K})|_{\\hat{I}}), \\quad K\\in\\mathcal{T}_{n},\\ n=0,1,\\dots, N-1.\n\\]\nUsing these, we can now define the \\emph{Trefftz projection} $\\Pi_p u$ of a function $u$ of the form \\eqref{eq:exact_sol_form} element-wise by\n\\begin{equation}\\label{trefftz_proj}\n(\\Pi_p u)|_{K\\times I_n} := \\lambda^1_p F^1_{n,K}(x+ct) + \\lambda^2_p F^2_{n,K}(x-ct),\n\\end{equation}\n$K\\in\\mathcal{T}_{n}$, $n=0,1,\\dots, N-1$. The approximation properties of $\\Pi_p$ follow from the respective properties of $\\lambda^i_p$, $i=1,2$. Space-time shape regularity implies $J^i_{n,K}\\sim h_K^{(n)}$, $i=1,2$. \n\nWe denote by $\\Phi(p,s)$ the quantity\n$\n \\Phi(p,s):=(\\Gamma(p-s+1)\/\\Gamma(p+s+1))^{\\frac{1}{2}},\n$\nwith $p,s$ real numbers such that $0\\le s\\le p$\nand $\\Gamma(\\cdot)$ being the Gamma function; we also adopt the standard convention\n$\\Gamma(1)=0!=1$. Making use of \\emph{Stirling's formula},\n$\\Gamma(n)\\sim\\sqrt{2\\pi}n^{n-\\frac{1}{2}}e^{-n}$, $n>0$,\nwe have, \n$\n  \\Phi(p,s) \\le C p^{-s}\n$, for $p\\ge 1$,\nwith $0\\le s\\le p$ and $C>0$ constant depending only on $s$.\n\nWe have the following $hp$-approximation results for $\\lambda^i_p$, $i=1,2$.\n\n\\begin{lemma}\\label{approx_theorem}\nLet $v \\in H^{k+1}(J)$, for $k\\ge 1$, and let $h=\\diam(J)$ for \\reply{an open, bounded interval $J\\subset\\mathbb{R}$}; finally let $\\lambda_p$ be any of the $\\lambda^i_p$, $i=1,2$. Then the following error bounds hold:\n\n\\begin{equation}\\label{ltwobounis}\n   \\ltwo{v - \\lambda_p v}{J} \\le Cp^{-1}\\Phi(p,s)h^{s+1}|v|_{s+1,J},\n\\end{equation}\nand\n\\begin{equation}\\label{dxbounis}\n   \\ltwo{v' - (\\lambda_p v)'}{J} \\le C\\Phi(p,s)h^{s}|v|_{s+1,J},\n\\end{equation}\nwith $0\\le s\\le \\min\\{p,k\\}$, $p\\ge 1$. \nAlso, let $v\\in H^{k+1}(J)$, with $k\\ge 2$. Then, the following bound holds:\n\\begin{equation}\\label{htwo_bound}\n  \\ltwo{v''- (\\lambda_p v)''}{J}\\le C p^{3\/2}\\Phi(p,m)h^{m-1}|v|_{m+1,J},\n\\end{equation}\nwith $1\\le m\\le \\min\\{p,k\\}$. Finally, let $v\\in H^{k+1}(J)$, with $k\\ge 3$. Then, the following bound holds:\n\\begin{equation}\\label{hthree_bound}\n  \\ltwo{v'''- (\\lambda_p v)'''}{J}\\le C p^{7\/2}\\Phi(p,l)h^{l-2}|v|_{l+1,J},\n\\end{equation}\nwith $2\\le l\\le \\min\\{p-1,k\\}$.\n\\end{lemma}\n\\begin{proof}\n The proof \nof (\\ref{ltwobounis}) and (\\ref{dxbounis}) for the $H^1$-projection $\\lambda_p^i$ can be found, e.g., in \\cite{SchwabBook}. \nThe proof of \\eqref{htwo_bound} can be found in \\cite{GH}, while the proof of \\eqref{hthree_bound} follows along the same lines as in the proof of  \\eqref{htwo_bound} from \\cite{GH}.\n\\end{proof}\n\nThese $hp$-approximation estimates imply the following bound.\n \n\\begin{theorem}\\label{approx_theorem_dg_norms}\nLet $u|_{K\\times I_n} \\in H^{k+1}(K\\times I_n)$, for $k\\ge 3$ be the exact solution to \\eqref{eq:wave}. Then, \\our{for space-time meshes satisfying Assumption \\ref{as:st_reg},} the following error\nbounds hold:\n\\begin{equation}\\label{eq:aprioribound_hp}\n\\dgnorm{U-u}^2 \\leq \\reply{C}  p^{3} \\Phi^2 (p,s)\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n} \\our{ \\diam(K\\times I_n)}^{2s-1} |u|_{s+1,K\\times I_n}^2,\n\\end{equation}\nfor $3\\le s\\le \\min\\reply{\\{}p+1,k\\reply{\\}}$ and $h=\\max_{K,n}\\{ \\our{ \\diam(K\\times I_n)}\\}$, with $C>0$ constant, independent of $p$, $h$, \\reply{$u$, and of} $U$. Moreover, if $u$ is analytic on a neighbourhood of $\\Omega$, there exists $r>0$, depending on the analyticity region of $u$ \\reply{in a neighbourhood of $\\Omega\\times(0,T)$}, such that\n\\begin{equation}\\label{eq:aprioribound_hp_exponential}\n\\dgnorm{U-u}^2 \\leq  C(u) p^{3} \\exp(-r p)\\sum_{n = 0}^{N-1} \\sum_{K\\in \\mathcal{T}^n}  |K\\times I_n|  \\our{\\diam(K\\times I_n)}^{2s-1}.\n\\end{equation} \n\\end{theorem}\n\n\\begin{proof}\nThe proof of \\eqref{eq:aprioribound_hp} follows by combining the $hp$-approximation bounds from \\eqref{approx_theorem} with Lemma \\ref{lemma:abstract_error_bound}.\n\nFor \\eqref{eq:aprioribound_hp_exponential}, we work as follows. Analyticity of $u$ implies that there exists a $d>0$, \\reply{such that for all $s\\ge 0$,}\n\\begin{equation}\\label{analyticity}\n|u|_{s,K\\times I_n} \\le C d^s\\Gamma(s+1)|K\\times I_n|^{1\/2},\n\\end{equation}\n\\reply{cf., e.g., \\cite[Theorem 1.9.3]{davis}}.\nUsing this \\eqref{analyticity}, setting $s=\\reply{\\gamma} p$ for some $0<\\reply{\\gamma}<1$, along with Stirling's formula, we can arrive to the bound\n\\[\n\\Phi^2 (p,\\gamma p) |u|_{\\gamma p+1,K\\times I_n}^2 \\le C \\bigg( (2\\gamma d)^{2\\gamma}\\frac{(1-\\gamma)^{1-\\gamma}}{(1+\\gamma)^{1+\\gamma}} \\bigg)^p|K\\times I_n|,\n\\]\nwith the precise choice of $\\gamma$ remaining at our disposal. The function \n\\[\nF(\\gamma) := (2\\gamma d)^{2\\gamma}\\frac{(1-\\gamma)^{1-\\gamma}}{(1+\\gamma)^{1+\\gamma}} ,\n\\]\nhas a minimum at $\\gamma_{\\min}:=(1+4d^2)^{-1\/2}$, giving $F(\\gamma_{\\min})<1$. Setting, now $r = 1\/2|\\log F(\\gamma_{\\min})|$, the result follows.\n\\end{proof}\n\n\\begin{remark}\nThe bound \\eqref{eq:aprioribound_hp} is suboptimal in $p$ by one order. This is a standard feature of $hp$-version dG methods whose analysis requires the use of $hp$-type inverse estimates. It is possible to slightly improve on this result and obtaining only $1\/2$ order $p$-suboptimal bounds, using the classical $hp$-approximation results from \\cite{Can_Quar,BS}, instead of the $H^1$-projection operator as done above. These results, however, are not suitable for the proof of the exponential rate of $p$-convergence. \n\\end{remark}\n\n\\section{Numerical experiments}\\label{sec:numerics} \\reply{We present a series of numerical experiments aiming to highlight the performance of the proposed method above. In each experiment, the spatial meshes are kept fixed $\\mathcal{T} = \\mathcal{T}_n$ and uniform time-step is used. In the one-dimensional ($d=1$) examples the spatial mesh is a uniform set of intervals, whereas for $d=2$, the spatial mesh is a quasiuniform triangulation. The resulting linear systems at each timestep are solved by standard sparse direct solvers.}\n\\subsection{Experiments in one dimension}\nWe consider the wave equation with constant diffusion coefficient $a\\equiv 1$, spatial domain $\\Omega = (0,1)$  and initial data \n\n\\begin{equation}\n  \\label{eq:init_data}  \nu(x,0) = e^{-\\left(\\frac{x-5\/8}{\\delta}\\right)^2}, \\qquad \\dot u(x,0) = 0,\n\\end{equation}\nwhere $\\delta \\leq \\delta_0 = 7.5 \\times 10^{-2}$. Note that the initial data is not exactly zero at the boundary, but is less than $10^{-11}$ in the range of parameter $\\delta$ that we consider. This slight incompatibility with the boundary condition does not influence in any visible way our numerical results. Since the energy of the exact solution stays constant it is given for all times by\n\\[\n\\text{exact energy} = \\tfrac12\\ltwo{u_x(x,0)}{\\Omega}^2 \\approx 2 \\delta^{-1}\\int_{-\\infty}^\\infty y^2 e^{-2y^2}dy = \\delta^{-1} \\frac{\\sqrt{\\pi}}{2\\sqrt{2}},\n\\]\nwhere the approximation in the second step is of the order of $10^{-11}$ for reasons given above and the final equality is obtained by using integration by parts to reduce it to the Gaussian integral \\cite{GraR2000}.\n\nAs the problem is in one spatial dimension, the exact solution is not difficult to obtain. The error will be computed in the dG norm \n\\[\n\\text{error } = \\dgnorm{u- u_h}.\n\\]\nSince the exact solution is smooth, note that, see \\eqref{eq:dg_norm_expr}, \n\\[\n\\dgnorm{u}^2 = a(u,u) = 2 \\times \\text{exact energy}.\n\\]\n\n\\subsubsection{Convergence order}\n\nWe first investigate the convergence order of the numerical method. Though we did not analysed full polynomial spaces,  we also give numerical experiments for these as it is interesting to compare the two sets of results.\n\nIn this subsection, we choose $\\delta_0 = 7.5 \\times 10^{-2}$ and $T = 1\/4$. Note that we choose such a small time interval in order to reach the asymptotic regime earlier -- this will especially be important for lower orders. In Figure~\\ref{fig:conv} and Tables~\\ref{tab:conv_order} and \\ref{tab:linear_order}, the convergence curves and numerically computed convergence orders are given. These confirm the theoretical results. Note that the errors obtained by the full and the Trefftz spaces are very similar for the same order, but the Trefftz spaces require fewer degrees of freedom and cheaper implementation; see Remark~\\ref{rem:cheaper} and Figure~\\ref{fig:Conv_Time}. We have also found that higher order approximation converges without the two extra stabilisation terms, i.e., with $\\sigma_1 = \\sigma_2 = 0$, but with the piecewise linear functions it stagnates.\n\n\n\n\\begin{figure}[hbtp]\n \\centering\n\\includegraphics[scale=0.45]{convTreftz1d5c.pdf}  \n\\includegraphics[scale=.45]{convPolyNew1d5c.pdf} \n\\caption{\\small Convergence of the Trefftz, on the left, and polynomial, on the right, time-space dG method of order $p$. The error is plotted  against the uniform mesh width in time and space $h = T\/N$. }\n \\label{fig:conv}\n\\end{figure}\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=0.450]{TimeTreftz1d5c.pdf}\n\\includegraphics[scale=0.450]{PolyTime1d5c.pdf} \n\\caption{\\small Error against computational time for Trefftz (on the left) and polynomial spaces (on the right). The much lower times for Trefftz spaces are due both to the smaller number of degrees of freedom for the same accuracy and to the cheaper construction of matrices; see Remark~\\ref{rem:cheaper}.}\n\\label{fig:Conv_Time}\n\\end{figure}\n\n\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=.4]{Error_orderTt2c.pdf} \n\\caption{\\small Convergence of the Trefftz method with fixed mesh width $h=1\/40$ and increasing polynomial order $p$.}\n\\label{fig:Convp}\n\\end{figure}\n\n\n\n\n\\begin{table}\n  \\centering\n   \\begin{tabular}{c|cccc}\n  $N$  & $p=2$ & $p=3$ & $p=4$  & $p=5$\\\\\\hline\n$5$ & $0.98$ & $1.85$ & $3.64$  & $5.07$\\\\\n$10$ & $1.37$ & $2.10$ & $3.57$ & $5.06$\\\\\n$20$ & $1.38$ & $2.28$ & $3.52$ & $4.77$\\\\\n$40$ & $1.46$ & $2.42$ & $3.51$ & $4.76$\\\\\n$80$ & $1.49$ & $2.51$ & $3.51$ & $4.63$ \n \\end{tabular}\n \\hspace{.8cm}\n  \\begin{tabular}{c|cccc}\n$N$  & $p = 2$& $p = 3$ & $p=4$ & $p=5$\\\\\\hline\n5 & 0.90 & 1.85  & 3.74 & 5.56 \\\\\n10 & 1.17 & 2.11 & 3.39 & 5.05\\\\\n20 & 1.34 & 2.26 & 3.41 & 4.31\\\\\n40 & 1.44 & 2.38 & 3.41 & 4.91\\\\\n80 & 1.45 & 2.54 & 3.46 & 4.79\n\\end{tabular}\n  \\caption{\\small Numerically obtained orders of convergence of the error in the dG norm $  \\dgnorm{\\cdot}$  for Trefftz spaces on the left and for polynomial space on the right}\n\\label{tab:conv_order}\n\n\\end{table}\n\n\n\n\n\n\n\\begin{table}  \n\\centering\n\\begin{tabular}{| l | l | l | l | l | l | l | l | l |}\n\\hline\n$N$ &  80 & 160 & 320 & 640 & 1280 & 2560 & 5120   \\\\ \\hline\n$p = 1$     &  0.08 & 0.11 & 0.19 & 0.29 & 0.38 & 0.44 & 0.47 \\\\ \\hline\n\\end{tabular}\n\\caption{\\small Numerically obtained orders of convergence of the error in the dG norm $\\dgnorm{\\cdot}$ for linear elements.}\\label{tab:linear_order}\n\\end{table}\n\\subsubsection{Long-time energy behaviour}\n\nThe time-space dG method that we developed is dissipative, so we expect the energy to decay over time. However, if the accuracy of the approximation is high we expect this decay to be very slow. This is indeed reflected in the numerical experiments shown in Figure~\\ref{fig:energy_decaylog}, where we compute the energy\n\\[\nE(t) = \\tfrac12\\|\\dot u_h (t)\\|^2+\\tfrac12\\|\\nabla u_h (t)\\|^2\n\\]\nup-to time $T = 5$ with $\\delta = \\delta_0\/4$.\n\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=0.45]{semilogTreftzzc.pdf}\n\\includegraphics[scale=0.45]{semilogpolyyc.pdf}\n\\caption{\\small Plot of the energy $E(t) = \\tfrac12\\|\\dot u_h (t)\\|^2+\\tfrac12\\|\\nabla u_h (t)\\|^2$ against time for the Trefftz spaces on the left and polynomial spaces on the right.\n\\label{fig:energy_decaylog}\n\\end{figure}\n\n\n\n\n\n\\subsubsection{Waves with energy at high-frequences}\n\nNote  that if we decrease the parameter $\\delta > 0$ in the definition of the initial data \\eqref{eq:init_data}, the Gaussian becomes narrower and energy at higher frequences is excited. In the following set of experiments we investigate the error while decreasing both $\\delta > 0$ and  the mesh-width $h > 0$. In an ideal scenario, $h \\propto \\delta$ would be sufficient to obtain a constant relative error which we define as\n\\begin{equation}\n  \\label{eq:error_delta}  \n\\text{error}_\\delta  = \\left(\\frac\\delta2\\ltwo{\\dot u(\\cdot,T)-\\dot u_h(\\cdot,T^-)}{\\Omega}^2+\\frac\\delta2\\ltwo{\\nabla u(\\cdot,T)-\\nabla u_h(\\cdot,T^-)}{\\Omega}^2\\right)^{1\/2}.\n\\end{equation}\nThe results given in Figure~\\ref{fig:error_delta} show that already for $p = 3$ the performance is good and for larger $p$ for this range of $\\delta$, the error does not visibly change for different $\\delta$ as long as $h \\propto \\delta$.\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=0.45]{scaledErr_Trefftz5c.pdf} \n\\caption{\\small The plot of the scaled error see \\eqref{eq:error_delta}, against $h\/\\delta$, for Trefftz space.The final time is chosen to be $T=1$.\n\\label{fig:error_delta}}\n\\end{figure}\n\n\n\\subsection{Experiments in two dimensions}\n\nWe conclude the section on numerical experiments by considering the wave equation \n\\[\n\\ddot u -\\Delta u = 0\n\\]\non the square $[0,1]^2 \\subset \\mathbb{R}^2$ with homogeneous Dirichlet boundary condition and initial data\n\\[\nu(x,y,0) =  \\sin \\pi x \\sin \\pi y, \\qquad \\dot u(\\cdot, 0) = 0.\n\\]\nThe analytical solution is given by\n\\[\nu(x,y,t) = \\cos(\\sqrt{2}\\pi t) \\sin \\pi x \\sin \\pi y.\n\\]\nWe investigate the convergence of the error in the energy norm at the final time-step\n\\begin{equation}\n  \\label{eq:2d_error}\n\\text{error}  = \\left(\\tfrac12\\ltwo{\\dot u(\\cdot,T)-\\dot u_h(\\cdot,T^-)}{\\Omega}^2+\\tfrac12\\ltwo{\\nabla u(\\cdot,T)-\\nabla u_h(\\cdot,T^-)}{\\Omega}^2\\right)^{1\/2}.  \n\\end{equation}\nThe convergence plots are given in Figure~\\ref{fig:2d_energyconv} and the computed convergence orders in Table~\\ref{tab:2d_convorder}. Note that, \\reply{for the weaker error notion \\eqref{eq:2d_error},} we do not lose half an order of convergence as when computing the error in the discrete norm $\\dgnorm{\\cdot}$. The theory presented above, does not predict this behaviour. The observed rate of convergence is not, however, surprising as, unlike the dG norm, this error measure does not accumulate the errors over all time-steps.\n\n\\begin{figure}\n  \\centering\n\\includegraphics[width=.4\\textwidth]{EnergyErrT2New2c.pdf}\n\\includegraphics[width=.4\\textwidth]{EnergyErrPolyNew2D.pdf}  \n  \\caption{\\small Convergence of the error \\eqref{eq:2d_error} at the final time for Trefftz basis on the left and full polynomial basis on the right.}\n  \\label{fig:2d_energyconv}\n\\end{figure}\n\n\n\\begin{table}[hbtp]\n  \\centering\n   \\begin{tabular}{c|cccc}\n  $N$ & $p=1$ & $p=2$ & $p=3$ & $p=4$\\\\\\hline\n$10$ &$0.046$ & $2.039$ & $2.219$ & $3.935$ \\\\\n$20$ &$0.115$ & $1.990$ & $2.539$ & $3.977$ \\\\\n$40$ &$0.160$ & $1.979$ & $2.819$ & $3.996$ \\\\\n \\end{tabular}\n \\hspace{.8cm}\n  \\begin{tabular}{c|cccc}\n$N$ & $p=1$ & $p = 2$& $p = 3$ & $p=4$ \\\\\\hline\n10 & 0.046 & 1.897 & 2.133 & 3.956 \\\\\n20 & 0.115 &1.896 & 2.248 & 3.967\\\\\n40 & 0.160 & 1.935 & 2.612 & 3.982 \\\\\n\\end{tabular}\n\\caption{\\small Numerically obtained convergence orders of the error \\eqref{eq:2d_error} at the final time for Trefftz  on the left and for the full polynomial space on the right. \\label{tab:2d_convorder}}\\vspace{-.5cm}\n\\end{table}\n\\FloatBarrier\n\n\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nDuring the last decade, there has been a major interest in using nonlinearity\nfor achieving non-reciprocal transmission, aiming at designing efficient wave\ndiodes in particular with applications within the optical domain. The basic\nidea, that transmission between two waveguides with intensity-dependent (Kerr)\nrefractive index and different propagation constants becomes asymmetric in the\nnonlinear regime, was probably first put forward by Trillo and Wabnitz\n\\cite{TW86}. A similar mechanism was later analyzed for a different setup\nby Lepri and Casati \\cite{11,lepri}, who considered the nonreciprocal\ntransmission through\na layered photonic (or phononic) system, with a small central segment of\nnonlinear, nonmirror symmetric layers embedded in an infinite linear lattice.\nIn both cases, the systems were modeled using a discrete nonlinear\nSchr{\\\"o}dinger (DNLS) equation with cubic on-site nonlinearity, with\nparticular focus\non the case with two nonlinear sites (DNLS dimer) which can be solved exactly.\nAs is well known \\cite{DLS86,WS,btm}, stationary transmission through DNLS\nchains with\non-site nonlinearity generically exhibits multistability and hysteresis\neffects when using {\\em input} intensity $|R_0|^2$ as independent parameter,\nwhile the\nexistence of a backward transfer map guarantees that the stationary\ntransmission\ncoefficient is a single-valued function of the {\\em output} intensity $|T|^2$ .\n\n\nLater, in the context of asymmetric transmission of signals through\nnon-linear asymmetric dimer layers, it was shown that multistability could be\nenhanced by weak saturation of the (cubic) on-site nonlinearity;\n  this favors the nonreciprocal transmission and also yields opposite effects\nof the rectifying action for short\/long wavelength signals\n\\cite{assuncao,Erik}.\nA similar study was carried out for saturable nonlinear oligomer DNLS segments\n($N=1,2,3$)\nembedded in a linear Schr{\\\"o}dinger chain \\cite{jd}. Other relevant works\nconcern the asymmetric wave transmission through oligomers with cubic-quintic\non-site nonlinearity \\cite{wasay18}, and the enhancement of the non-reciprocal\ntransmission under saturable cubic-quintic nonlinear responses for\ndimers \\cite{wasay3}.\n A cubic on-site nonlinearity was also shown to yield\nnon-reciprocal transmission if combined with an asymmetric geometric shape of\nthe nonlinear part \\cite{LiRen}, and moreover, for a system with two nonlinear\nsites separated by a number of linear sites, the transmission was shown to\nbe generically asymmetric unless certain resonance conditions were fulfilled\n\\cite{RenPRB}\n(analogous resonance conditions were also obtained for the continuous\nSchr{\\\"o}dinger equation with nonlinear $\\delta$-function scatterers\n\\cite{RenPRB,RenPRE})\n\nA common feature of the above mentioned earlier works is, that the nonlinearity\n(cubic, cubic-quintic or saturable) resided only in the on-site terms, which\nsimplifies the analytical as well as the numerical treatment due to the\nexistence of a unique backward transfer map, which thus never yields more than\none solution for a given output $|T|$. However, although on-site nonlinearities\ntypically dominate in most physical applications of DNLS lattices, there are\ncertain situations where the additional effects of inter-site nonlinearities\nmay be important, e.g., for optical waveguides embedded in a nonlinear medium\n\\cite{Oster03}, Bose-Einstein condensates in optical lattices \\cite{ST03}\n(dipolar condensates in particular \\cite{Serbia,Chile}), or,\nmore generally, when the DNLS equation is considered as a rotating-wave type\napproximation of anharmonically coupled oscillators \\cite{MJ06}, or as a\ntight-binding approximation of the nonlinear Schr{\\\"o}dinger equation with\nspatial periodicity in both the linear potential and nonlinearity\ncoefficient \\cite{Abd08}. It is thus also of interest to investigate, whether\nthe presence of a non-negligible inter-site nonlinearity in the central\nsegment may have any major\nqualitative effects on the non-reciprocal wave transmission. A first step in\nthis direction was taken in \\cite{wasay2}, where transmission through an\nasymmetric DNLS dimer with cubic on-site as well as inter-site nonlinearity was\nstudied for a special situation, assuming a particular relation between the\ncomplex amplitudes of the two dimer sites, that allowed the derivation of\na unique backward transfer map and thus a unique solution for a given $|T|$.\nHowever, as we will show in the present work, this is {\\em not} the case for\nthe\ngeneric situation in presence of inter-site nonlinearities.\n\n\nIn this work, we generalize the model introduced in \\cite{wasay2} to have a\n{\\em saturable inter-site} nonlinearity between the two dimer sites,  leaving\nthe on-site nonlinearity cubic.\nAs we will see, due to the absence of a unique backward transfer map,\na small inter-site saturability typically yields {\\em more\nthan one solution} in the regime of small $|T|$.\nThese solutions are distinguished by\ntheir different relative phases between the dimer sites, and may be obtained\nnumerically by solving an additional equation for this phase difference.\nMost importantly, the general distinctive feature of this work in view of all\nprevious works is that we will present the scenario of how a gradual\ntransition from the non-saturated to saturated case takes place, i.e., to\nanalyse in detail how the saturated inter-site case connects to the\nnon-saturated inter-site case.\n\nAlthough we consider here a specific form of\nsaturable inter-site nonlinearities, appropriate e.g. for\nphotvoltaic-photorefractive materials \\cite{Valley94},\nthe method that we present should be\napplicable for transmission through generic inter-site nonlinear dimer\nsegments. Notably, a somewhat different form of saturable nonlinear\ncoupling within a dimer was recently proposed \\cite{Hadad17} and\nimplemented through an electric circuit ladder \\cite{Hadad18}.\n\nThe outline of this paper is as follows. In Sec.\\ \\ref{sec:Model} we introduce\nthe dynamical model and set up the corresponding stationary transmission\nproblem. We derive the corresponding backward transfer map, point out the\nreason for its general non-uniqueness, and analyze its solutions\nanalytically in some limiting cases. In Sec.\\ \\ref{sec:mtc} we investigate\nin more detail, by numerical means, the transitions  between regimes\nwith two, three, or one distinct solutions of the backward transfer map,\nfor increasing saturation strength. Results for the multi-channel\nstationary transmission coefficients\nas a function of wave number and transmitted intensity, for given parameter\nvalues and varying saturation strength, are reported in\nSec.\\ \\ref{sec:transmission}. The asymmetric stationary transmission\nproperties are investigated in Sec.\\ \\ref{sec:asymmetric}, where also the\nrectification factor is calculated in various regimes of saturability for\nthe different solution branches. Section \\ref{sec:stability} reports\nthe linear stability analysis of the different branches of stationary\nscattering solutions, with illustrations of instability-induced dynamics.\nIn Sec.\\ \\ref{sec:Gaussian} we perform dynamical simulations with\nGaussian wavepackets and discuss the transmission and rectification\nproperties. Concluding remarks are made in Sec.\\ \\ref{sec:conclusions}, and\nsome results for different parameter values than those used in the\nmain paper are shown in Appendix.\n\n \\section{Model}\n\\label{sec:Model}\n We introduce the dynamical equation of the model as follows,\n\n\n\n \\bea{}\n i\\frac{dA_n}{dt}\\!=\\!V_nA_n\\!-\\!C\\!\\left(A_{n+1}\\!+\\!A_{n-1}\\right)\\!+\\!\\gamma_n|A_n|^2A_n\\!+\\!\n\\left(\\!\\frac{\\epsilon_{n}|A_{n+1}|^2}{1\\!+\\!\\beta|A_{n+1}|^2|A_n|^2}\\!+\\!\\frac{\\epsilon_{n-1}|A_{n-1}|^2}{1\\!+\\!\\beta|A_{n-1}|^2|A_n|^2}\\!\\right)\\!A_n .\n \\label{dynamical}\n \\end{eqnarray}\nIn Eq.\\ \\eqref{dynamical}, $n$ is the lattice site counter,\n$A_n$ is the amplitude at site $n$, and $V_n$ is the linear on-site energy of\neach\nsite inside the one-dimensional lattice. The parameter $\\gamma_n$ determines\nthe strength of the local nonlinearity,\nwhich we take to have a standard cubic (Kerr) form, while\n$\\epsilon_n$ represents the nonlocal (inter-site) saturable nonlinearity.\nWe saturate only the inter-site nonlinearity as the effect of saturating the\non-site cubic nonlinearity has been addressed in earlier work\n\\cite{jd,assuncao,wasay3,Erik}. $\\beta$ is the saturation parameter;\nin the unsaturated limit $\\beta =0$ we recover the DNLS model with\ncubic inter-site\nnonlinearity studied in \\cite{wasay2} (relevant e.g.\\ for dipolar Bose-Einstein\ncondensates \\cite{Chile}), while in the strongly saturated limit\n$\\beta \\rightarrow \\infty$ the coupling becomes governed only by the linear\ncoupling constant $C$, and the model reduces to the standard cubic on-site\nDNLS model \\cite{11,lepri}.\nLike the standard DNLS model \\cite{EJ03}, the system is Hamiltonian,\nwith the saturable inter-site terms arising from additional terms\n$\\frac{\\epsilon_n}{\\beta} \\ln \\left(1+\\beta|A_n|^2|A_{n+1}|^2\\right)$\nin the Hamiltonian.\nWithout loss of generality, the parameter $C$\nwill be chosen to be unity. Focusing our attention on a dimer\nsituated at lattice sites 1 and 2 with nonlinear inter-site interactions\nonly between the two dimer sites, the site\ndependent parameters $\\gamma$ and $V$ have non-zero\ncontributions only in the\nregion $1\\leq n\\leq2$, and $\\epsilon$ only for $n=1$.\nThis means that waves can propagate freely outside the dimer.\n\nThe set of dynamical equations \\eqref{dynamical} has stationary solutions of\nthe form $A_n(t)=A_ne^{-i\\omega t}$.\nWhen a signal (incoming or outgoing wave) is away from the dimer, the system\nis linear, and these solutions satisfy the dispersion relation\n$\\omega=-2$cos$k$, with $k$ being the wave vector of some specific harmonic\ncomponent of the wave and $0\\leq k\\leq\\pi$. These solutions render\nEq.\\ \\eqref{dynamical} stationary. The resulting stationary equation can be\nwritten\nin the form of a backward-map analogous to \\cite{11,btm},\n \\bea{}\nA_{n-1}=-A_{n+1}+\\left(V_n-\\omega+\\gamma_n|A_n|^2+\\frac{\\epsilon_{n}|A_{n+1}|^2}{1+\\beta|A_{n+1}|^2|A_n|^2}+\\frac{\\epsilon_{n-1}|A_{n-1}|^2}{1+\\beta|A_{n-1}|^2|A_n|^2}\\right)A_n .\n\\label{stationary}\n \\end{eqnarray}\nIn the absence of inter-site nonlinearities ($\\epsilon_n \\equiv 0$), this\nrelation allows one to immediately\nconstruct the amplitudes by a backward iteration, assuming that the solution\nis known at $n\\rightarrow \\infty$. By contrast, with nonzero $\\epsilon_1$\nan additional relation between the complex amplitudes\n$A_1$ and $A_2$ at the nonlinear dimer sites is needed.\n We will focus on the scattering properties of plane wave solutions of the\nfollowing form,\n\\bea{}\nA_n=\n\\Bigg\\{\n  \\begin{array}{c}\n  R_0 e^{ikn}+Re^{-ikn}             \\qquad n\\leq 1\n  \\\\\n  Te^{ikn}                          ~~\\qquad\\qquad\\qquad n\\geq 2 \\\\\n  \\end{array} ,\n  \\label{planewave}\n\\end{eqnarray}\n where $R_0$, $R$ and $T$ are the amplitudes of incoming, reflected and\ntransmitted wave, respectively. Applying the ansatz in Eq.\\ \\eqref{planewave}\nsite-by-site,  we get at site $n=0$,\n \\bea{}\n A_0=R_0+R ,\n \\label{site0}\n \\end{eqnarray}\n  and at site $n=1$,\n \\bea{}\n A_1=R_0e^{ik}+Re^{-ik} .\n \\label{site1}\n \\end{eqnarray}\n With $A_0$ and $A_1$, the amplitudes of the reflected and incident waves can\nbe computed as\n\\bea{}\nR=\\frac{A_0 e^{ik}-A_1}{e^{ik}-e^{-ik}} ,\n\\label{R}\n\\end{eqnarray}\nand\n\\bea{}\nR_0=\\frac{A_0 e^{-ik}-A_1}{e^{-ik}-e^{ik}} .\n\\label{R0}\n\\end{eqnarray}\n\n We can rewrite the backward map \\eqref{stationary} for the dimer\n($n=2$), with the inter-site nonlinear interactions considered only between\nthe two dimer sites, as\n\\bea{}\nA_1=-A_3+\\left(V_2-\\omega+\\gamma_2|A_2|^2+\\frac{\\epsilon_1|A_{1}|^2}{1+\\beta|A_{1}|^2|A_2|^2}\\right)A_2 .\n\\label{n2}\n\\end{eqnarray}\nWith Eq.\\ \\eqref{planewave}, the wave amplitudes at the dimer interface\nfor the outgoing, right-propagating waves ($k>0$) is\n$A_2=Te^{2ik}$ and $A_3=Te^{3ik}$. In addition, we may obtain an expression for\n$|A_1|^2$ from the general current conservation law for stationary solutions,\n\\bea{}\n\\textmd{Im}[A_n^\\ast A_{n+1}]=|T|^2 \\sin k,\n\\label{cl}\n\\end{eqnarray}\nfor all $n$.\nWithout loss of generality, we may choose the arbitrary overall phase\nsuch that the amplitude $A_1$ at site 1 is real.\nFrom \\eqref{cl} with $n=1$ and \\eqref{planewave} with $n=2$ it then follows\nstraightforwardly that $A_1 \\textmd{Im}(Te^{2ik}) = |T|^2 \\sin k$, i.e.,\n\\bea{}\n|A_1|^2=\\frac{|T|^4\\textmd{sin}^2k}{[\\textmd{Im}(Te^{2ik})]^2} .\n\\label{f1}\n\\end{eqnarray}\nThe fact that $T$ is not real in general then leads to the following relation,\n\\bea{}\n|A_1|^2=\\frac{|T|^2\\sin^2k}{[\\sin(2k+\\varphi)]^2},\n\\end{eqnarray}\nwhere $\\varphi=\\textmd{arg}(T)$. Thus, Eq.\\ \\eqref{n2} becomes\n\\bea{}\nA_1=-Te^{3ik}+\\left(V_2-\\omega+\\gamma_2|T|^2+\\frac{\\epsilon_1|T|^2\\textmd{sin}^2k}{\\left[\\textmd{sin}(2k+\\textmd{arg}(T))\\right]^2+\\beta|T|^4\\textmd{sin}^2k}\\right)Te^{2ik} ,\n\\label{n3}\n\\end{eqnarray}\nwhich can be re-written as\n \\bea{}\n A_1=Te^{2ik}(\\delta_2-e^{ik}) ,\n \\label{n5}\n \\end{eqnarray}\n with $\\delta_2=V_2-\\omega+\\gamma_2|T|^2+\\frac{\\epsilon_1|T|^2\\textmd{sin}^2k}{\\left[\\textmd{sin}(2k+\\textmd{arg}(T))\\right]^2+\\beta|T|^4\\textmd{sin}^2k}$.\n\n Now in a similar way, for $n=1$ with the nonlinear inter-site interactions\nonly between the two dimer sites, from Eq.\\eqref{stationary} we get\n\\bea{}\nA_0=-A_2+\\left(V_1-\\omega+\\gamma_1|A_1|^2+\\frac{\\epsilon_1|A_{2}|^2}{1+\\beta|A_1|^2|A_{2}|^2}\\right)A_1 ,\n\\label{n21}\n\\end{eqnarray}\nwhich together with Eq.\\eqref{n5} leads to\n\\bea{}\nA_0=-Te^{2ik}+\\left(V_1-\\omega+\\gamma_1|T|^2|\\delta_2-e^{ik}|^2+\\frac{\\epsilon_1|T|^2}{1+\\beta|T|^4|\\delta_2-e^{ik}|^2}\\right)\nTe^{2ik}(\\delta_2-e^{ik}) ,\n\\label{n31}\n\\end{eqnarray}\nwhich can be rewritten as\n\\bea{}\nA_0=Te^{2ik}\\left[\\delta_1(\\delta_2-e^{ik})-1\\right] ,\n\\label{n6}\n\\end{eqnarray}\nwith $\\delta_1=V_1-\\omega+\\gamma_1|T|^2|\\delta_2-e^{ik}|^2+\\frac{\\epsilon_1|T|^2}{1+\\beta|T|^4|\\delta_2-e^{ik}|^2}$.\n\n This implies that the transmission coefficient $t(k,|T|^2; \\arg(T))$ can now\nbe computed by using Eq.\\ \\eqref{n6} and Eq.\\ \\eqref{n5} in Eq.\\ \\eqref{R0}.\nThe result is\n\\bea{}\nt(k,|T|^2;\\arg(T))=\\frac{|T|^2}{|R_0|^2}=\\left|\\frac{e^{-ik}-e^{ik}}{(\\delta_2-e^{ik})(\\delta_1-e^{ik})-1}\\right|^2 .\n\\label{t}\n\\end{eqnarray}\n For the left-propagating waves (with $k<0$), a similar computation yields\nthe transmission coefficients, i.e., we only need to exchange the\nsubscripts 1 \\& 2.\n\nNote that, in contrast to previously studied DNLS-type transmission problems\n(e.g., \\cite{11,lepri,DLS86,WS,btm,assuncao,Erik,jd,wasay18,wasay3,LiRen,RenPRB}),\nEq.\\ \\eqref{t} does {\\em not} necessarily determine the transmission\ncoefficient for a plane wave of wave vector $k$ as a unique function of\nthe transmitted intensity $|T|^2$, since in presence of intersite\nnonlinearities, there may be multiple solution\nbranches $i$ corresponding to the same $|T|^2$ but with different phases\n$\\varphi_i\\equiv \\arg(T)$. Since Eq.\\ \\eqref{n3} was obtained by assuming\n$A_1$ real,\nthe interpretation of the quantity ``$\\arg(T)$\" is an additional phase shift\nbetween the two dimer sites 1 and 2, due to the internal properties of the\ndimer. The computation of this additional phase factor $\\arg(T)$ is\nnontrivial. Imposing the \"reality'' assumption on $A_1$ by putting the\nimaginary part of the right-hand side of Eq.\\ \\eqref{n3} to zero,\nleads to the following equation,\n \\bea{}\n \\frac{\\sin\\left[3k+\\arg(T)\\right]}{\\sin\\left[2k+\\arg(T)\\right]}=V_2-\\omega+\\gamma_2|T|^2+\\frac{\\epsilon_1|T|^2\\sin^2k}{\\left[\\sin(2k+\\arg(T))\\right]^2+\\beta|T|^4\\sin^2k}.\n \\label{argT}\n \\end{eqnarray}\nSolving Eq.\\eqref{argT} analytically for $\\arg(T)=\\varphi_i(|T|,k)$ is\nnontrivial in the general case, but we may immediately note that adding\nany multiple of $\\pi$ to a solution gives another solution, so we may restrict\nto the interval $0\\leq \\varphi < \\pi$ (adding $\\pi$ just switches an overall\nsign).\nFor the case with pure on-site nonlinearity ($\\epsilon_1=0$), we recover the\nunique solution\n \\bea{}\n \\varphi_{\\epsilon_1=0}(|T|,k)  = -2k + \\arctan\\left[\\frac{\\sin k}\n{V_2+\\cos k+\\gamma_2|T|^2}\\right].\n \\label{argT1}\n \\end{eqnarray}\nNote that in the linear limit ($|T|\\to 0$) and non-scattering case\n($V_2 = 0$), Eq.\\ \\eqref{argT1} yields\n$\\arg(T)=-k$ (due to the choice of origin in  \\eqref{planewave}).\n\nWith unsaturated inter-site nonlinearity ($\\epsilon_1\\neq 0$ but $\\beta=0$),\nEq.\\ \\eqref{argT} can be written as a quadratic equation for\n$y\\equiv \\sin^2(2k+\\varphi)$:\n \\bea{}\n y^2(e^2+\\sin^2k)+y (2e \\epsilon_1|T|^2-1) \\sin^2k  + \\epsilon_1^2|T|^4 \\sin^4k = 0,\n \\label{quadratic}\n \\end{eqnarray}\nwhere $e\\equiv V_2+\\cos k + \\gamma_2 |T|^2$.\nThus, it is clear from \\eqref{quadratic} that for unsaturated inter-site\nnonlinearities\nthere are generally {\\em two distinct\nsolutions} for {\\em small} $|T|$, and {\\em no} solutions for {\\em large} $|T|$.\nFor small $|T|$ we may express the solutions to $O(|T|^4)$ as\n\\bea{}\n\\varphi_1 \\simeq -2k +  \\arctan\\left[\\frac{\\sin k}{V_2+\\cos k}\n\\left\\{1-\\frac{\\gamma_2+\\epsilon_1[(V_2+\\cos k)^2+\\sin^2k]} {V_2+\\cos k}|T|^2\n\\right\\}\n\\right],\n \\label{unsat1}\n \\end{eqnarray}\nand\n\\bea{}\n\\varphi_2 \\simeq -2k ,\n \\label{unsat2}\n \\end{eqnarray}\nrespectively. Thus we note that the solution $\\varphi_1$ in \\eqref{unsat1}\ncoincides with the small-$|T|$ limit of the pure on-site solution \\eqref{argT1}\nwhen $\\epsilon_1 \\rightarrow 0$, while the solution $\\varphi_2$ in\n\\eqref{unsat2} yields a new possible transmission channel not existing for\npure on-site nonlinearity (the existence of this additional channel was suppressed in \\cite{wasay2} due to the choice of a specific relation between the complex amplitudes of the dimer sites).\n\nFor the general case with saturable inter-site nonlinearity\n($\\beta \\neq 0$), Eq.\\ \\eqref{argT} instead becomes a cubic equation for $y$\n($0\\leq y \\leq 1$),\n \\bea{}\n(y+\\beta |T|^4\\sin^2 k)^2 (1-y) \\sin^2 k = y\n\\left[e(y+\\beta|T|^4 \\sin^2 k)+\\epsilon_1|T|^2 \\sin^2 k \\right]^2 .\n \\label{cubic}\n \\end{eqnarray}\nThus, depending on the parameter values, there are {\\em either one or three}\ndistinct solutions $\\varphi_i$, corresponding to different possible\ntransmission channels.\n By analyzing the coefficients in \\eqref{cubic}, we find that in the limit of\nsmall $|T|$ there is a transition at $\\beta\/\\epsilon_1^2 =1\/4$, so that for\n$\\beta\/\\epsilon_1^2 > 1\/4$ there is only one real and positive solution for\n$y$,\ncorresponding to\nthe phase factor $\\varphi_1$ in \\eqref{unsat1} for the unsaturated case.\nOn the other hand, for $\\beta\/\\epsilon_1^2 < 1\/4$ two additional solutions\nappear, both originating from the solution $\\varphi_2$ in \\eqref{unsat2}\nfor the unsaturated\ncase, with corrections of order $|T|^4$ and higher. Below,\nthe value $\\beta\/\\epsilon_1^2 = 1\/4$ will be taken as indicating the\ntransition between regimes of ``low saturation'' and ``medium saturation''.\n\n\nMoreover, in the limit of large  $|T|$, $y=0$ is the only real solution,\nthus yielding a single channel with phase factor $\\varphi_2$ given by \\eqref{unsat2}. Note that this agrees also with the large-$|T|$ limit of the pure on-site\nsolution \\eqref{argT1}, as it should since the coupling becomes effectively\nlinear due to strong saturation.\nFor general nonzero $|T|$, the phase factor $\\varphi_i(|T|,k)$ is computed\nnumerically,\nand we will illustrate the typical scenario for different regimes\nof saturability in the following section. As we will see, there are\nsignificant regimes with three distinct channels for small and intermediate\nvalues\nof $|T|$ when $\\beta$ is not too large.\n\n\\section{Multiple transmission channels}\n\\label{sec:mtc}\n\n\nAs described above, one of our goals is to investigate how the saturated case\nconnects to the non-saturated case, which hinges on the detailed investigation\nof scenarios occurring at different levels of saturation. The analysis is thus\ndivided into\nthree distinct saturation regimes with carefully chosen representative\nsaturation values. To be specific, we fix the value of $\\epsilon_1$ to\n$\\epsilon_1=0.5$, consider\nan ``ultra-low saturation'' regime for\n$\\beta=0.01$ (i.e., $\\beta\/\\epsilon_1^2 =0.04 \\ll 1\/4$), a\nlow saturation regime for $\\beta=0.05$\n(i.e., $\\beta\/\\epsilon_1^2 =0.2 < 1\/4$),  and a medium saturation\nregime $\\beta=0.5$ (i.e., $\\beta\/\\epsilon_1^2 =2 > 1\/4$) (the regime\nof stronger saturations is less interesting since it essentially reproduces\nwell known results for pure on-site nonlinearity \\cite{11,lepri}).\nUnless otherwise noted, we will also fix the on-site nonlinearity strength\non the dimer symmetric as $\\gamma_1=\\gamma_2=1$, and express the asymmetric\non-site potential as $V_{1,2}=V^{(0)}(1\\pm\\varepsilon_V)$, with specifically\nchosen $V^{(0)}=-2.50$ and $\\varepsilon_V=0.05$\n(some results for different values\nof $\\varepsilon_V, V^{(0)}$ are discussed in Appendix \\ref{AppB}).\n\n\n\n  \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB00T5new.jpg}\n   \n  \\end{minipage}\n \n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB01T5new.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB05T5new.jpg}\n   \n  \\end{minipage}\n \n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB50T5new.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB00T1new.jpg}\n   \n  \\end{minipage}\n \n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB01T1new.jpg}\n   \n  \\end{minipage}\n \n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB05T1new.jpg}\n   \n  \\end{minipage}\n \n  \\begin{minipage}[h]{0.23\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnContourB50T1new.jpg}\n   \n  \\end{minipage}\n  \\caption{Contour plot of solutions of Eq.\\eqref{argT} as a function of $k$\nand $\\arg(T)=\\varphi_i(|T|,k)$ for $V_2=-2.3750$, $\\gamma_2=1$ and\n$\\epsilon_1=0.5$. Upper row:  $|T|=0.5$;\nlower row: $|T|=1$. Columns from\nleft to right:  $\\beta=0$,  $\\beta=0.01$,  $\\beta=0.05$,  $\\beta=0.5$. }\n \\label{fargTb0}\n  \\end{figure}\nThe solutions to Eq.\\ \\eqref{argT} for two distinct nonzero\n$|T|$ values ($|T|$=0.5 and $|T|$=1) in the unsaturated and the three\nsaturation regimes are illustrated in Fig.\\ \\ref{fargTb0} for $0\\leq k\\leq\\pi$\nversus $0\\leq\\varphi_i(|T|,k)\\leq\\pi$. The ''dotted'' lines in these\nfigures represent the singularity in the left-hand side of  \\eqref{argT}\nat $y=0$ ($\\varphi = -2k \\mod \\pi$), which lies very close to the actual\nsolution for some cases.\nIn Fig.\\ \\ref{multisols}, the corresponding solutions ($\\varphi_i(|T|,k)$) are\npresented  as a function of $|T|$ for various (fixed) values of $k$.\n\nFirstly, for the non-saturated case ($\\beta=0$, left columns in\nFigs.\\ \\ref{fargTb0}-\\ref{multisols}), as predicted from \\eqref{quadratic}\nthere is a two-solution regime which exists for all $k$ for small $|T|$,\nand no valid solutions for higher $|T|$ above a certain cut-off, which differs\nslightly for different $k$.\nFor the chosen set of parameter values, in the case of relatively small\n$k\\leq1.4$ this cut-off value is approximately $|T|=1.2$,\nand slightly increases for larger $k$ values, i.e., for $2.8\\leq k\\leq\\pi$, the cut-off is at $|T|=1.8$ approximately.\nThus, there are either two or zero transmission channels in the case of zero\nsaturation.\n  \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB00K02new.jpg}\n   \n  \\end{minipage}\n    \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB01K02.jpg}\n   \n  \\end{minipage}\n    \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB05K02.jpg}\n   \n  \\end{minipage}\n    \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB50K02new.jpg}\n   \n    \\end{minipage}\n    \\\\\n   \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB00K15new.jpg}\n   \n  \\end{minipage}\n   \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB01K15new1.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB05K15.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB50K15new.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB00K25new.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB01K25.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB05K25.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.24\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SolnsB50K25new.jpg}\n   \n  \\end{minipage}\n  \\caption{Plots of $\\varphi_i(|T|,k)$ as a function of $|T|$ with three fixed\n    values of $k$: $k=0.2$ (upper row), $k=1.5$\n    (middle row), $k=2.5$\n    (bottom row). Columns from left to right:\n      unsaturated case ($\\beta=0$),\nultra-low ($\\beta=0.01$), low ($\\beta=0.05$), and medium saturation\n($\\beta=0.5$) strengths. All other parameter values as before. Blue, orange\nand green curves correspond to the first, second and third solution branch,\nrespectively.}\n\\label{multisols}\n  \\end{figure}\n\n   As soon as we switch from the unsaturated to the ultra-low saturation\n(second column in  Figs.\\ \\ref{fargTb0}-\\ref{multisols}),\na third solution immediately appears in the small-$|T|$ solution regime,\nas predicted from \\eqref{cubic}. Note that this additional solution\n(green curve in Fig.\\ \\ref{multisols}) as expected almost coincides with the\n``singularity line'' $\\varphi = -2k \\mod \\pi$ in Fig.\\ \\ref{fargTb0}.\nAbove a certain cut-off depending upon $k$\n(which is essentially the same as for the unsaturated case),\nonly this new third solution persists throughout the parameter space.\nSo there is either a three-solution regime (for small $|T|$) or a\nsingle-solution regime for higher $|T|$.\n  \n\n\n\nFor the case of low saturations ($\\beta=0.05$, third column in\nFigs.\\ \\ref{fargTb0}-\\ref{multisols}), the three-solution regime as predicted\nalways persists for small $|T|$ (orange and green branches in\nFig.\\ \\ref{multisols} almost coincide) and also for small $k$ (the\napparent gap close to $k=0$ in Fig.\\ \\ref{fargTb0} is due to graphics\nlimitations).\nHowever, for slightly larger $k$ ($k>0.6$), the scenario is different as\ncompared to the ultra-low case:  the three-solution regime at small $|T|$\nis interrupted by a single-solution regime\n(blue curve only in Fig.\\  \\ref{multisols}) for some interval of $|T|$, and\nthen there is a  second (small) three-solution regime which exists upto\na cut-off as in the ultra-low case. For all higher $|T|$, there is a\nsingle solution (the third solution, green curve in Fig.\\  \\ref{multisols}).\nNote however that the single-solution regime sandwiched between the two\nthree-solution regimes \\textit{does not} belong to the\n'third solution branch', but rather it belongs to the 'first solution branch'.\n\nFinally, in the medium saturation range ($\\beta=0.5$, right\n  column in\nFigs.\\ \\ref{fargTb0}-\\ref{multisols})), the multi-solutions are\nentirely suppressed leaving behind only a single-solution regime for all $k$\nand $|T|$.\n\n\n \n\n\n\n\n\nAlso, it is to be noted (see Appendix \\ref{AppB})\nthat the stretch of intensities that exhibit the\nmulti-channel regime strongly depends on the energy at site 2, represented by\nthe parameter $V_2$. For a smaller $V_2$ the multi-channel regime persists for\na longer stretch of intensities (i.e., the cut-off is higher) and vice versa.\n\n\n\n\\section{Effects of saturation on multi-channel transmission}\n\\label{sec:transmission}\n\nIn this section, we present in Fig.\\  \\ref{u00001} the transmission scenario\nvia density plots of the\ntransmission coefficient $t(k,|T|^2)$ from Eq.\\ \\eqref{t}, for the parameter values\ncorresponding to the different\nsaturation regimes discussed above. The first, second and third transmission\nchannels correspond to blue, orange and green curves from\nFig.\\ \\ref{multisols},\nrespectively. The results for the two transmission channels in the\nunsaturated case ($\\beta=0$) are visually identical to those of the\nfirst and second channels for the ultra-low saturation ($\\beta=0.01$),\nand thus we do not show these figures below.\n \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpd11B01.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.2cm}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpd22B01.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.2cm}\n  \\begin{minipage}[h]{0.34\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpd33B01.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpd1B05Sol1.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpd2B05Sol2.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpd3B05Sol3.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n  \\includegraphics[width=\\linewidth]{dpdB55.jpg}\n   \n  \\end{minipage}\n  \\caption{Color plots of transmission coefficient as a function of\n$k$ and $|T|$ for the three different saturation regimes.\nUpper row: $\\beta=0.01$; middle row: $\\beta=0.05$; lower row:\n$\\beta=0.5$. Left, middle and right panels in two upper rows correspond to\nthe 1st, 2nd and 3rd transmission channel, respectively. In all figures,\n$\\epsilon_{1}=0.5$, $\\varepsilon_V=0.05$, $V^{(0)}=-2.5$, and $\\gamma_{1,2}=1$.\n}\n \\label{u00001}\n   \\end{figure}\n\n\n\\subsubsection*{Ultra Low Saturation Regime, $\\beta=0.01$}\n\n\n\n\n\n\nFrom the upper row in Fig.\\ \\ref{u00001}, we see that the major regimes\nof good transmission occur along the first channel, and that transmission\nalong the third channel is essentially negligible. Note however that there\nis a narrow peak with close to perfect transmission also along the second\nchannel, most apparent around $k=0.2$ and $k=2.5$. This second peak appears\nalso for pure on-site nonlinearities \\cite{11,wasay18}; what is important\nto note here is thus that with inter-site nonlinearity (unsaturated or with\nultra-low saturation), the second transmission peak moves over to the second\nbranch of solution, and thus there is only one peak in each of the two first\nchannels.\n\n\n\n\n\\subsubsection*{Low Saturation Regime, $\\beta=0.05$}\n\nHere, we see from the second row of Fig.\\ \\ref{u00001} that the transmission\nalong the first channel is essentially the same as for smaller saturation,\nbut that of the second channel narrows down as the existence region for the\ncorresponding solution shrinks, as discussed above.\nNote that when $\\beta=0.05$ there are two distinct three-solution regimes\nfor $k>0.6$ (see Fig.\\ \\ref{multisols}), but the\ntransmssion along the second and third channels is always negligible in the\nsmall-$|T|$ region. Note also that since the third solution now connects\nto the second solution when $k>0.6$, it also picks up some noticeable, but\nstill small, transmission close to the connection point (seen for\n$1.4\\lesssim |T| \\lesssim 1.8$ in Fig.\\ \\ref{u00001}).\n\n\n\n\n\n\n\n\n\\subsubsection*{Medium Saturation Regime, $\\beta=0.5$}\n\nAs discussed above, in the medium saturation regime\nonly single solutions persist\nthroughout the parameter space.\nThe transmission coefficient shown in the lowest part of Fig.\\  \\ref{u00001}\nis qualitatively similar to analogous plots for the pure on-site nonlinearity\ncase \\cite{11,wasay18}: two separate transmission peaks for small $k$ which\nmerge to a single, broad peak around $k=\\pi\/2$, and then split up again,\neventually yielding four distinct peaks close to $k=\\pi$. Note also that\nalthough a propagating solution exists for arbitrarily large $|T|$ and all $k$,\nthe transmission coefficient is essentially negligible for $|T|\\gtrsim 2.1$.\n\n\n\n\n \\section{Asymmetric multi-channel transmission}\n\\label{sec:asymmetric}\n\n\nHaving identified regimes of multi-solutions, we now consider the\npossibility for multi-channel asymmetric transmission\ndue to the presence of a small nonzero asymmetry between on-site energies.\nTo determine the efficiency of nonreciprocal transmission, i.e.,\ntransmission at diode-like modes, we define the {\\em rectifying factor}\n$\\mathcal{R}$ as\n\\cite{11}\n\\bea{}\n\\mathcal{R}=\\frac{t(k,|T|^2)-t(-k,|T|^2)}{t(k,|T|^2)+t(-k,|T|^2)} ,\n\\label{rf}\n\\end{eqnarray}\nwhere $-1\\leq\\mathcal{R}\\leq+1$. A perfect diode-like transmission occurs at\n$\\mathcal{R}=\\pm1$. In Fig.\\ \\ref{Rultralow3sol} we\npresent the results of the rectifying factor along the different\ntransmission channels for the three different saturability regimes\n(as in previous section, the results for the unsaturated case is visually\nidentical to those of the first two channels for the ultra-low saturation, and\nthus not shown).\n \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpr11B01.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpr22B01.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpr33B01.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpr11B05.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpr22B05.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{dpr33B05.jpg}\n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n \n    \\includegraphics[width=\\linewidth]{dprB5.jpg}\n  \\end{minipage}\n  \\caption{Color plots of rectifying action for $\\beta=0.01$ (upper row),\n$\\beta=0.05$ (middle row) and  $\\beta=0.5$ (lower row). In two upper rows,\nthe three panels from left to right correspond to the first, second and third\nsolution, respectively. Site dependent parameter strengths $V^{(0)}=-2.5$,\n$\\varepsilon_V=0.05$, $\\epsilon_{1}=0.5$, $\\gamma_{1,2}=1$. In two upper rows,\nregimes where the channel transmits only right-propagating (left-propagating)\nmodes are coded white (black), while regimes where the channel has no\nsolutions for any propagation direction are coded orange.}\n \\label{Rultralow3sol}\n   \\end{figure}\nThe\ntransmission scenario is presented in more detail below via plots of the\ntransmission coefficient $t(k,|T|^2)$ in Eq.\\ \\eqref{t} as a function of\ntransmitted {\\em intensity} $|T|^2$ for the case of ultra-low saturation\n($\\beta=0.01$), low saturation ($\\beta=0.05$) and medium saturation\n($\\beta=0.5$) in Fig.\\ \\ref{vv1}, Fig.\\ \\ref{k151} and Fig.\\ \\ref{vv1111},\nrespectively. Note that the blue curves in Figs.\\ \\ref{vv1} - \\ref{vv1111}\ncorrespond to the right-propagating case ($k>0$), while the red dotted curves\nrepresent the left-propagation ($k<0$), both plotted in their respective\nexistence regimes. Note also that the first, second and third solution branches\nin these figures correspond to blue, orange and green curves from\nFig.\\ \\ref{multisols}, respectively.\n\n\n \\subsubsection*{Ultra-Low Saturation, $\\beta=0.01$}\n\nFor the first transmission channel, we see from upper left\nFig.\\ \\ref{Rultralow3sol} that there are essentially two regimes with\nconsiderable rectification action that also correspond to large transmission\ncoefficients according to Fig.\\ \\ref{u00001}: a regime for small $k$ and\n$T \\approx 0.5$, and another regime for large $k$ around $T \\approx 1$. As seen\nin upper and lower left Fig.\\ \\ref{vv1}, these regimes originate in shifts\nof large-amplitude transmission peaks, and are analogous to large-rectification\nregimes existing for pure on-site nonlinearities \\cite{11,wasay18}.\nOn the other hand, in the regime of intermediate $k$ where transmission peaks\nare broad, transmission is close to symmetric as seen in middle left\nFig.\\ \\ref{vv1}. Moreover, the black band with $\\mathcal{R}=-1$ in upper left\nFig.\\ \\ref{Rultralow3sol} for $1.3 \\lesssim |T|\\lesssim 1.9$ arises since\nthe existence regime for the first transmission channel is always\nsmaller for the right-propagating wave than for the left-propagating for\nthe corresponding set of parameter values\n(see left vertical panel in  Fig.\\ \\ref{vv1}).\n  \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K02Sol1.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K02Sol2.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K02Sol3.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K15Sol1.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K15Sol2.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K15Sol3.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K25Sol1.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K25Sol2.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB01K25Sol3.jpg}\n   \n  \\end{minipage}\n\n   \\caption{Transmission coefficient $t(k,|T|^2)$ as a function of $|T|^2$\n     for the ultra-low saturated case\n$\\beta=0.01$\n     for fixed $k$ values, from top to bottom\n     $k=0.2,1.5,2.5$. Left, middle and right vertical panels\n  correspond to first, second and third solution branches, respectively.\n \n    \n     All parameter values as before.}\n \\label{vv1}\n  \\end{figure}\n\nFor the second transmission channel, the most interesting effect is the shift\nof the narrow transmission peak with close to perfect transmission, leading to\nthe bright band with $\\mathcal{R}$ close to 1  for\n$1.1 \\lesssim |T|\\lesssim 1.8$ in upper middle Fig.\\ \\ref{Rultralow3sol}.\nThis is seen more\nexplicitly in the middle vertical panel of Fig.\\ \\ref{vv1}.\nNote also the small regime with $\\mathcal{R}$ close to -1 for $k$ close to\n$\\pi$ and $1.8 \\lesssim |T| \\lesssim 1.9$, corresponding to\nthe peak of almost perfect left-propagating transmission appearing\nwhile the right-propagating transmission is declining. As the existence\nregimes for the first and second solutions are identical in the regime of\nultra-low saturability, the black band with $\\mathcal{R}=-1$ for the second\nsolution in  upper middle Fig.\\ \\ref{Rultralow3sol} will be the same as that\nfor the first solution.\n\nFinally, as seen in upper right Fig.\\ \\ref{u00001} the transmission along the\nthird channel (corresponding to the green\nsolution branch in Fig.\\ref{multisols}) is negligibly small, and\nessentially symmetric (upper right Fig.\\ \\ref{Rultralow3sol} and\nright vertical panel of Fig.\\ \\ref{vv1}).\nSo the system behaves as a nearly perfect mirror for this channel, in both\ndirections.\n\n\\subsubsection*{Low Saturation, $\\beta=0.05$}\n\n\\begin{figure}[!htbp]\n\\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K02Sol1.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K02Sol2.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K02Sol3.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K15Sol1.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K15Sol2.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K15Sol3.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K25Sol1.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K25Sol2.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n   \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB05K25Sol3.jpg}\n   \n  \\end{minipage}\n\n   \\caption{\n    \n Same as Fig. \\ref{vv1} but\n     for the low saturated case\n     $\\beta=0.05$.\nAll other parameters same as before.}\n \\label{k151}\n  \\end{figure}\n\nFor the first transmission channel (middle left Fig.\\ \\ref{Rultralow3sol}\nand left vertical panel in Fig.\\ \\ref{k151}), there are only minor\ndifferences compared to the ultra-low saturation regime.\n\nFor the second channel (middle central Fig.\\ \\ref{Rultralow3sol}\nand middle vertical panel in Fig.\\ \\ref{k151}), the major effects appear due to\nthe shrinking of its existence region, for both propagation directions. As\na result, for some $k$-values its existence regimes for left and right\npropagation are fully disjoint, leading to a splitting of the  black band\nin the rectification plot for $\\beta=0.01$ into a ``triplet band'' with\nwhite (only right-propagation),\norange (no propagation in any direction), and black (only left-propagation)\nregions appearing in order as $|T|$ is increased. Moreover, the upper cut-off\nfor the low-$|T|$ regime when $k \\gtrsim 0.6$ is also slightly larger for\nthe right-propagating wave, leading to the narrow white stripe around\n$|T|\\simeq 0.5$ in middle central Fig.\\ \\ref{Rultralow3sol}. However,\ntransmission is essentially negligible in both directions in the low-$|T|$\nregime of the second channel (see middle vertical panel in Fig.\\ \\ref{k151}).\n\nFor the third channel (middle right Fig.\\ \\ref{Rultralow3sol}\nand right vertical panel in Fig.\\ \\ref{k151}), regimes of non-negligible\ntransmission appear only close to its lower cut-off for larger $k$ (see\nmiddle right Fig.\\ \\ref{u00001}), which is seen to be somewhat larger for\nthe left-propagating wave. The result is the upper white band in the\nrectification plot corresponding to only right-propagation, followed for\nslightly larger $|T|$ by a band with $\\mathcal{R}$ close to -1. The lower\nwhite stripe is the same as for the second solution since their\nsmall-$|T|$ existence regimes are identical.\n\n\\subsubsection*{Medium Saturation, $\\beta=0.5$}\n\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB50K02.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB50K15.jpg}\n   \n  \\end{minipage}\n  \\hspace{-0.7cm}\n  \\begin{minipage}[h]{0.35\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{TranscoeffB50K25.jpg}\n   \n  \\end{minipage}\n\n  \\caption{Transmission coefficient $t(k,|T|^2)$ as a function of $|T|^2$\n    for the medium saturated case $\\beta=0.5$,  for fixed $k$ values\n   \n    $k=0.2$ (left), $k=1.5$ (middle), $k=2.5$ (right)}.\n \\label{vv1111}\n  \\end{figure}\nIn this regime, the multi-solution regime is entirely suppressed and thus\nthere is only a single solution for all $k$ and $|T|$. We\npresent the rectifying action in lower Fig.\\ \\ref{Rultralow3sol} and\ncorresponding plots of transmission coefficient in Fig.\\ \\ref{vv1111}.\nThe plots are qualitatively similar to those for pure on-site\nnonlinearity \\cite{11,wasay18}. Note that the previous black bands in the\nrectification plots for\nthe first and second solutions in regimes of smaller saturability now have\nturned into a dark band with $\\mathcal{R}$ close to -1, as the narrow\ntransmission peaks in the high-$|T|$ regime\nnow appear for the same solution branch (left Fig.\\ \\ref{vv1111}). The\nadditional transmission peaks appearing for large $k$\n(lower Fig.\\ \\ref{u00001} and right Fig.\\ \\ref{vv1111}) also lead to\na more complicated pattern of regimes with $\\mathcal {R}$ alternating\nbetween values close to +1 and -1 for increasing $|T|$, when $k$ is close to\n$\\pi$.\n\n\n\n\n\\section{Stability Analysis}\n\\label{sec:stability}\n\nIn this section, we investigate the dynamical stability of the stationary\nsolutions used in previous sections. The stability analysis will closely\nfollow as done for instance in \\cite{lepri,jd} for pure on-site\nnonlinearities, and it is performed by first perturbing the solutions as\n\\bea{}\nA_n(t)=P_n(t)+\\delta R_n(t) .\n\\label{perturbation}\n\\end{eqnarray}\nThis will linearize the equation of motion \\eqref{dynamical}. The resulting\nlinearized set of dynamical equations of order $\\delta$ is\n\\bea{}\ni\\dot{R_n}-V_nR_n+R_{n+1}+R_{n-1}\n=\n\\gamma_n\\left(2R_n\\!\\mid\\!P_n\\!\\mid^2\\!+~P_n^2R_n^\\ast \\right)+~~~~~~~~~~~~~\\nonumber\n\\\\\n\\epsilon_{n}\\left[\\frac{W_{n+1} P_n}{(1+\\beta|P_{n+1}|^2|P_{n}|^2)^2}-\\frac{|P_{n+1}|^4 P_n\\beta Z}{(1+\\beta|P_{n+1}|^2|P_{n}|^2)^2}+\\frac{R_n|P_{n+1}|^2}{1+\\beta|P_{n+1}|^2|P_{n}|^2}\\right]+~~~~~~\\nonumber\n\\\\\n\\epsilon_{n-1}\\left[\\frac{W_{n-1} P_n}{(1+\\beta|P_{n-1}|^2|P_{n}|^2)^2}-\\frac{|P_{n-1}|^4 P_n\\beta Z}{(1+\\beta|P_{n-1}|^2|P_{n}|^2)^2}+\\frac{R_n|P_{n-1}|^2}{1+\\beta|P_{n-1}|^2|P_{n}|^2} \\right] ,\n\\label{linearized}\n\\end{eqnarray}\nwhere $W_{n+1}=P^\\ast_{n+1}R_{n+1}+R^\\ast_{n+1}P_{n+1}$, $W_{n-1}=P^\\ast_{n-1}R_{n-1}+R^\\ast_{n-1}P_{n-1}$ and $Z=P^\\ast_nR_n+R^\\ast_nP_n$.\n\nWith the ansatz $P_n(t)=A_ne^{-i\\omega t},~A_n\\neq A_n(t)$ being the\ncomplex amplitudes of a stationary solution as before,\nand $R_n(t)=e^{-i\\omega t}(a_n e^{i\\nu t}+b_n e^{-i\\nu^\\ast t})$,\nthe linearized set of equations \\eqref{linearized} yields an eigenvalue\nproblem of the following form\n\\begin{gather}\n \\nu\\begin{bmatrix} a_n \\\\ b^\\ast_n \\end{bmatrix}\n =\n   \\begin{bmatrix}\n   M_1  &\n   M_2 \\\\\n   M_3 &\n   M_4\n   \\end{bmatrix}.\\begin{bmatrix} a_n \\\\ b^\\ast_n \\end{bmatrix} .\n   \\label{evals}\n\\end{gather}\nSpecifying to the two nonlinear sites (dimer), the resulting matrices are\n\\begin{gather}\n\\hspace{-1cm} M_1\n =\n   \\begin{bmatrix}\n   \\omega-V_1-2\\gamma_1|A_1|^2+\\frac{\\epsilon_1|A_2|^4|A_1|^2\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}-\\frac{\\epsilon_1|A_2|^2}{1+\\beta|A_2|^2|A_1|^2}  &\n   1-\\frac{\\epsilon_1A_2^\\ast A_1}{(1+\\beta|A_2|^2|A_1|^2)^2} \\\\\n   1-\\frac{\\epsilon_1A_1^\\ast A_2}{(1+\\beta|A_1|^2|A_2|^2)^2} &\n   \\omega-V_2-2\\gamma_2|A_2|^2+\\frac{\\epsilon_1|A_1|^4|A_2|^2\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}-\\frac{\\epsilon_1|A_1|^2}{1+\\beta|A_1|^2|A_2|^2}\n   \\end{bmatrix}\\nonumber\n\\end{gather}\n\\begin{gather}\n M_2\n =\n   \\begin{bmatrix}\n   -\\gamma_1 A_1^2+\\frac{\\epsilon_1|A_2|^4 A_1^2\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}  &\n   \\frac{-\\epsilon_1A_2 A_1}{(1+\\beta|A_2|^2|A_1|^2)^2} \\\\\n   \\frac{-\\epsilon_1A_1 A_2}{(1+\\beta|A_1|^2|A_2|^2)^2} &\n   -\\gamma_2 A_2^2+\\frac{\\epsilon_1|A_1|^4 A_2^2\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}\n   \\end{bmatrix}\\nonumber\n\\end{gather}\n\\begin{gather}\n M_3\n =\n   \\begin{bmatrix}\n   \\gamma_1 A_1^{\\ast 2}-\\frac{\\epsilon_1|A_2|^4 A_1^{\\ast 2}\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}  &\n   \\frac{\\epsilon_1A^\\ast_2 A^\\ast_1}{(1+\\beta|A_2|^2|A_1|^2)^2} \\\\\n   \\frac{\\epsilon_1A^\\ast_1 A^\\ast_2}{(1+\\beta|A_1|^2|A_2|^2)^2} &\n   \\gamma_2 A_2^{\\ast 2}-\\frac{\\epsilon_1|A_1|^4 A_2^{\\ast 2}\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}\n   \\end{bmatrix}\\nonumber\n\\end{gather}\nand\n\\begin{gather}\n\\hspace{-1cm} M_4\n \\!=\\!\n   \\begin{bmatrix}\n   -\\omega\\!+\\!V^\\ast_1\\!+2\\gamma_1|A_1|^2-\\frac{\\epsilon_1|A_2|^4|A_1|^2\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}+\\frac{\\epsilon_1|A_2|^2}{1+\\beta|A_2|^2|A_1|^2}  &\n   -1+\\frac{\\epsilon_1A_2 A^\\ast_1}{(1+\\beta|A_2|^2|A_1|^2)^2} \\\\\n   -1+\\frac{\\epsilon_1A_1 A^\\ast_2}{(1+\\beta|A_1|^2|A_2|^2)^2} &\n   -\\omega\\!+\\!V^\\ast_2\\!+2\\gamma_2|A_2|^2-\\frac{\\epsilon_1|A_1|^4|A_2|^2\\beta}{(1+\\beta|A_1|^2|A_2|^2)^2}+\\frac{\\epsilon_1|A_1|^2}{1+\\beta|A_1|^2|A_2|^2}\n   \\end{bmatrix} .\n\\nonumber\n\\end{gather}\n\n For lattices of size $m$ (with $m=2n+1$ where $n$ is the site counter for\neach linear side) with this type of nonlinear dimer embedded inside the linear\nchains to its right and left, the matrices ($M_{1,..,4}$) will be higher\ndimensional, i.e., $m\\times m$. However, apart from\n the dimer sites the site-dependent coefficients are zero, therefore, in the\nlinear region all entries in $M_2$ and $M_3$ will be zero, and\n \\bea{}\n M_1&=&diag(\\omega)+G \\nonumber\n \\\\\n M_4&=&diag(-\\omega)-G\\nonumber ,\n \\end{eqnarray}\n where $G$ is an $m\\times m$ sparse matrix with ones on both super- and\nsub-diagonals. The eigenvalue problem corresponding to Eq.\\ \\eqref{evals} is\nthen essentially the eigenvalue computation of the resulting $2m\\times 2m$\nsparse matrix. This is done by direct numerical computation of the eigenvalues\nfor a finite-sized chain with the nonlinear dimer embedded in the center.\n\n We will present below sample results of this stability analysis for all\nmulti-solution saturation regimes. The eigenvalue\/eigenvector computations\nwill be shown for a lattice of 201 sites, and the corresponding time\npropagation computations have been performed for lattices of up to 2001 sites,\nin\norder to accommodate for large $t$-values and avoid boundary errors. As pointed\nout in \\cite{lepri}, extended eigenvectors corresponding to a continuous\nspectrum in the infinite-chain limit may cause spurious instabilities due to\nboundary effects, of the order $1\/m$ ($\\sim 5\\times 10^{-3}$ below). Solutions\nexhibiting only such unstable eigenmodes thus correspond to linearly\nstable scattering solutions for the original set-up.\n\n \\subsubsection*{Ultra Low Saturation, $\\beta=0.01$}\n\nWe have chosen $|T|=1$ and $k=0.2,1.5,2.5$ to represent solutions\ncorresponding to small, medium and large wavenumbers respectively, in order to\nconnect with our earlier discussion.\nThe stationary solutions\nare depicted in Fig.\\ \\ref{ssb01}, for the three solution branches.\n\\begin{figure}[!htbp]\n\\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB01K02.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB01K15.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB01K25.jpg}\n   \n  \\end{minipage}\n\n   \\caption{Stationary solutions at $\\beta=0.01$, $|T|=1$, real (red circles)\nand imaginary (green squares) part of the solutions: left vertical panel for\n$k=0.2$, middle vertical panel for $k=1.5$ and right vertical panel for\n$k=2.5$. In each vertical panel,\ntop plot corresponds to the first solution branch, middle to the second\nbranch and lower to the third branch.}\n \\label{ssb01}\n  \\end{figure}\nNote that in this regime, the transmission coefficient is considerable only\nfor the first solution, very small for the second and essentially negligible\nfor the third, for all values of $k$.\n \n\nResults from the stability analysis\nare presented in Fig.\\ \\ref{b01evals}.\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K02T1Sol1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K02T1Sol2a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K02T1Sol3a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K15T1Sol1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K15T1Sol2a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K15T1Sol3a.jpg}\n   \n  \\end{minipage}\n\\\\\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K25T1Sol1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K25T1Sol2a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B01K25T1Sol3a.jpg}\n   \n  \\end{minipage}\n     \\caption{Stability analysis for $|T|=1$ at $\\beta=0.01$:\n$k=0.2$ (first and second row), $k=1.5$ (third and fourth row),\n$k=2.5$ (fifth and sixth row).\nLeft vertical panel shows the eigenvalues and most unstable\neigenvectors for first solution,\nmiddle panel for the second solution and right panel for the third solution.}\n \\label{b01evals}\n  \\end{figure}\nFor the first solution (left vertical panel), we find that it is always\nunstable for these parameter values, with an unstable eigenvector localized\nat the dimer. For small and intermediate $k$, the unstable\neigenvalues are complex\ncorresponding to a rather weak oscillatory instability, while a stronger\npurely exponential instability appears for larger $k$. The second solution\n(middle vertical panel)\nis essentially stable for small and intermediate $k$ (the observed tiny\nimaginary parts of eigenvalues correspond to extended eigenvectors and are\nspurious due to boundary effects as discussed above), while a very weak,\noscillatory instability appears for larger $k$ due to a resonance between\na mode localized at the dimer and the continuous spectrum. The third solution\n(right vertical panel) is strongly unstable with a purely imaginary\neigenvalue, and an eigenmode strongly localized at site 2.\n\n\nThe dynamics resulting from the instabilities of the first solution is\nillustrated in Fig.\\ref{tpropb01sol1}.\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB01K02T1Sol1t400.jpg}\n  \\end{minipage}\n   \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n   \\includegraphics[width=\\linewidth]{tpropB01K15T1Sol1t400.jpg}\n  \\end{minipage}\n   \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB01K25T1Sol1t400.jpg}\n  \\end{minipage}\n   \\caption{Time propagation corresponding to the first solution\nin Figs.\\ \\ref{ssb01}-\\ref{b01evals};\n$k=0.2$, $k=1.5$, $k=2.5$ from left to right.\nSnapshot at time 400 with a small $\\mathcal{O}(10^{-3})$ arbitrary perturbation\ninserted at site 1.\nUpper figures: Real (blue) and imaginary (red) parts. Lower: $|A_n|^2$.\n}\n \\label{tpropb01sol1}\n  \\end{figure}\n \nFor $k=0.2$ we observe that, after an initially oscillatory dynamics with\nexponentially increasing amplitudes at the center, the solution settles down\ninto an almost stationary transmission regime corresponding to a\n{\\em larger} $|T|$ ($|A_2|\\approx 1.52$ at time 400 in left\nFig.\\ \\ref{tpropb01sol1}). However, this value of\n$|T|$ is above the existence threshold for the first solution at $k=0.2$, and\nthe solution is not strictly stationary but shows small-amplitude oscillations\nfor $|A|$ at the dimer sites, as well as a weak long-wavelength spatial\nmodulation. At $k=1.5$ (middle Fig.\\ \\ref{tpropb01sol1}) the oscillatory\ninstability is stronger and yields persistent spatiotemporal oscillations,\nwhile at  $k=2.5$ (right Fig.\\ \\ref{tpropb01sol1}) the solution instead, after\nthe initial strong non-oscillatory exponential instability, settles down into\nan essentially stationary state with considerably {\\em smaller} $|T|$\n($|A_2|\\approx 0.27$ at time 400).\n\nThe second solution is essentially stable for $|T|=1$, and no significant\nchanges are observed in the time evolution for any of the considered values\nof $k$.\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB01K02T1Sol3t40Sites401.jpg}\n  \\end{minipage}\n   \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n   \\includegraphics[width=\\linewidth]{tpropB01K15T1Sol3t40Sites401.jpg}\n  \\end{minipage}\n   \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB01K25T1Sol3t40Sites401.jpg}\n  \\end{minipage}\n   \\caption{\n    \n     Similar as  Fig.\\ \\ref{tpropb01sol1} but for\n       the third solution\nin Figs.\\ \\ref{ssb01}-\\ref{b01evals},\nwith snapshots at time 40.\nOnly sites 2-30 of a 401-site system are shown.\n}\n \\label{tpropb01sol3}\n  \\end{figure}\nFor the third solution, instability-induced dynamics is\nillustrated in Fig.\\ \\ref{tpropb01sol3}, where the amplitudes\nof the initial portion of the transmitting side only (amplitudes at the\nincoming side are of the order of $10^5$ as in Fig.\\ \\ref{ssb01}) are\nshown at time 40\n(the instability develops rapidly due to the large eigenvalue).\nTypically,\nthe instability results in an initial decrease of $|A_2|$ (where the unstable\neigenmode is localized) to values close to zero, followed by recurring\noscillations between small and larger amplitudes at this site. As a result of\nthese oscillations the transmitted intensity will start to deviate from 1 as\nseen previously for the first solution branch; note however from\nFig.\\ \\ref{tpropb01sol3} that now $|A_n|<1$ for $k=0.2$ and $|A_n|>1$ for\n$k=1.5$ and $k=2.5$ at the initial portion of the transmitting side\n(the transmission coefficient evidently remains very small\ndue to the huge amplitudes on the left side).\n\n\n \n\n\n\n\n \n\n\n  \\subsubsection*{Low saturation, $\\beta=0.05$}\n\nWe will here pick $|T|=0.4$ as representative for the small-$T$ regime giving\nthree different solutions for each of the sample $k$ values\n($k=0.2, 1.5, 2.5$),\nand show the stationary solution plots in Fig.\\ \\ref{ssb05}.\n  \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB05K02T04.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB05K15T04.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB05K25T04.jpg}\n   \n  \\end{minipage}\n  \\caption{\n   \nSame as Fig.\\ \\ref{ssb01} but with $\\beta=0.05$ and $|T|=0.4$.\n  }\n \\label{ssb05}\n  \\end{figure}\nNote that in this regime, the transmission coefficient is non-negligible only\nfor the first solution.\n\n\nThe stability of each solution in this regime at the respective $k$ values is\nillustrated in Fig.\\ \\ref{b05evals}.\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K02T04Sol1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K02T04Sol2a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K02T04Sol3a.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K15T04Sol1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K15T04Sol2a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K15T04Sol3a.jpg}\n   \n  \\end{minipage}\n\\\\\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K25T04Sol1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K25T04Sol2a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05K25T04Sol3a.jpg}\n   \n  \\end{minipage}\n  \\\\\n\n  \\caption{\n   \n    Same as Fig.\\ \\ref{b01evals} but with $\\beta=0.05$ and\n      $|T|=0.4$.\n}\n \\label{b05evals}\n  \\end{figure}\nComparing to Fig.\\ \\ref{b01evals}, we note that the instabilities for the first\nsolution here are generally weaker, and to our numerical accuracy it is\nlinearly stable at $k=1.5$ (i.e., for wave numbers close to $\\pi\/2$). The\nsecond solution is still stable for small and intermediate wave numbers, but\nnow destabilizes with a purely imaginary eigenvalue for larger $k$, where it\nis close to its bifurcation point with the third solution\n(see Fig.\\ \\ref{multisols}). The third solution remains strongly unstable.\n\nTo illustrate the outcome of these instabilities, we show in Fig.\\\n\\ref{tpropb05} snapshots\nof solutions only for unstable cases that differ significantly to those\npreviously shown for $\\beta=0.01$ and $|T|=1$. For the first solution at\n$k=0.2$ (left Fig.\\ \\ref{tpropb05})\nthe instability is now non-oscillatory, and results in a slight\ndecrease of the amplitude on the transmitting side\n($|A_2|\\approx 0.39$ at time 400). At $k=1.5$ the solution is stable, and\nat $k=2.5$ the unstable dynamics is analogous to that of\nFig.\\ \\ref{tpropb01sol1}. The unstable dynamics of the second solution\nat $k=2.5$ is illustrated in right Fig.\\ \\ref{tpropb05} and is similar to\nthat previously described for the third solution, resulting after some\ntransient in amplitudes $|A_n|>0.4$ at the initial portion of the transmitting\nside (the transmission coefficient remaining small with maximal amplitudes\n $|A_n|\\sim 2000$ on the left side).\nFor the third solution, dynamics is analogous\nto that observed in Fig.\\ \\ref{tpropb01sol3}.\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB05K02T04Sol1t400.jpg}\n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB05K25T04Sol2t400Sites2001.jpg}\n   \n  \\end{minipage}\n   \\caption{Time propagation of unstable stationary solutions\nfrom Figs.\\ \\ref{ssb05}-\\ref{b05evals};\nLeft: First solution, $k=0.2$.\nRight: Second solution, $k=2.5$ (only sites 2-30 are shown).\nSnapshots at time 400\nwith a small $\\mathcal{O}(10^{-3})$ arbitrary perturbation\ninserted at site 1. Upper figures: Real (blue) and imaginary (red) parts.\nLower: $|A_n|^2$.\n}\n \\label{tpropb05}\n  \\end{figure}\n\n\n  \\subsubsection*{Medium Saturation, $\\beta=0.5$}\n\nAt this saturation strength, only a single-solution regime persists for all\n$k$ and $|T|$ values. We will pick two representative $|T|$ cases to present\nthe results below, $|T|=1$ and $|T|=2$, to see how the scenario differs\nfor relatively small versus larger intensities.\nStationary solution plots are shown in Fig.\\ref{ssb5T1}.\n  \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB5K02T1.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB5K15T1.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB5K25T1.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB5K02T2.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB5K15T2.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.32\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{SSB5K25T2.jpg}\n   \n  \\end{minipage}\n  \\caption{Stationary solutions at $\\beta=0.5$, real (red circles) and\nimaginary (green squares) part of the solutions: Left, middle and right\nvertical panels for $k=0.2$, $k=1.5$ and $k=2.5$, respectively. Upper\nhorizontal panel corresponds to $|T|=1$, lower to $|T|=2.$.}\n \\label{ssb5T1}\n  \\end{figure}\nNote that for $|T|=2$, only the case $k=2.5$ corresponds to a considerable\ntransmission coefficient (cf.\\ Figs.\\ \\ref{u00001}, \\ref{vv1111})\n\nThe stability analysis with corresponding examples of unstable time\npropagation for the case of $|T|=1$ and the three sample $k$ values is\npresented in Fig.\\ref{b5evalsK02SSR}.\n  \\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5K02T1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5K15T1a.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5K25T1a.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB5K02T1t200Sites2001.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB5K15T1t100Sites2001.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB5K25T1t100Sites2001.jpg}\n   \n  \\end{minipage}\n  \\caption{Top two horizontal panels: Stability analysis for $|T|=1$ and\n$\\beta=0.5$, with eigenvalues at top and most unstable eigenvectors at\nsecond row. Left, middle and right plots correspond to $k=0.2$, $k=1.5$ and\n$k=2.5$, respectively. Lower two horizontal panels: snap shots of time\npropagation for the corresponing values of $k$ (at time 200 for $k=0.2$ and\n100 for $k=1.5, 2.5$). Third row: real (blue) and imaginary (red) parts;\nfourth row:  $|A_n|^2$.}\n \\label{b5evalsK02SSR}\n  \\end{figure}\nComparing with Figs.\\ \\ref{b01evals}-\\ref{tpropb01sol1}, we note that the\nscenario is very similar to that of the first solution for the same value of\n$|T|$ when $\\beta=0.01$; the main qualitative difference is seen for\n$k=2.5$ where the instability now is oscillatory and slightly weaker. Thus,\nwe may conclude that as long as the amplitudes at the dimer sites are\nmoderate, which will typically be the case when $|T|$ is relatively small and\ntransmission coefficient $t$ is relatively large, the scenario in the regime\nof medium saturation is qualitatively similar to that for the first\nsolution branch for weaker saturability.\n\n  The case of various sample $k$ at higher intensities ($|T|=2$)\nis shown in Fig.\\ \\ref{b5evalsK15SSR}.\n\\begin{figure}[!htbp]\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5K02T2.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5K15T2.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5K25T2.jpg}\n   \n  \\end{minipage}\n  \\\\\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB5K02T2t300Sites2001.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB5K15T2t300Sites2001.jpg}\n   \n  \\end{minipage}\n  \\begin{minipage}[h]{0.3\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{tpropB5K25T2t300Sites2001.jpg}\n   \n  \\end{minipage}\n   \\caption{Similar as Fig.\\ \\ref{b5evalsK02SSR} but for $|T|=2$. The\nsnapshots for time propagation plots are all at time 300.\n}\n \\label{b5evalsK15SSR}\n  \\end{figure}\nFor $k=0.2$ (left Fig.\\ \\ref{b5evalsK15SSR}) and $k=1.5$ (middle\nFig.\\ \\ref{b5evalsK15SSR}) the solution is now stable (but with very small\ntransmission coefficient), while for $k=2.5$ (right Fig.\\ \\ref{b5evalsK15SSR})\nthe instability scenario is\nsimilar as above for $|T|=1$, resulting after some time in an almost\nstationary transmission with considerably smaller amplitudes at the\ntransmitted side of the dimer ($|A_2|\\approx 0.57$ at time 300).\n\nSummarizing this section, we note some similarities and differences\ncompared to previously reported stability\nresults for pure on-site nonlinearities\n\\cite{lepri,jd}. As in \\cite{lepri,jd}, we observe that the\nexactly stationary propagating solutions are unstable in regimes where the\ntransmission coefficient is significant. Stationary waves having small\ntransmission\ncoefficient appear typically as stable as noted in \\cite{lepri,jd}, except\nin the multisolution regimes where the third\nsolution branch (which has no counterpart neither for systems with pure on-site\nnonlinearity, nor with unsaturated inter-site nonlinearities) is generally\nunstable, and the second solution branch destabilizes close to its\nbifurcation with the third solution. As long as the instability is weak,\nit is typically\noscillatory (complex eigenvalues) as originally reported in \\cite{lepri}.\nIn regimes of stronger instabilities eigenvalues become purely imaginary, as\nalso seen for the saturable on-site nonlinarity in \\cite{jd}. However, a\nmajor difference is that in all cases reported in \\cite{lepri,jd} for\non-site nonlinearities, the\ninstability resulted in a trapped, localized defect mode at the central\nnonlinear sites. As seen above, none of the here considered instability regimes\nresulted in any significant trapping at the dimer. Thus, it appears that\ninter-site nonlinearities generally\ncounteract the creation of a localized dimer mode. Another difference is that\nin \\cite{jd}, it was concluded that instabilities generically\n(for on-site saturable oligomers) appeared to transport power to the right\npart of the lattice for $k>0$ (thus decreasing the power of the part\nimmediately to the left of the dimer). Here, we observe this scenario in some\ncases (mainly for first solution branch with small $k$ when instability is\noscillatory, and third branch  at larger $k$), while in other cases the\nscenario is opposite with a decrease of power at the right side\n(e.g., for large $k$ in the single-solution regime and for first solution\nin multi-solution regime, and for small $k$ for third solution). Thus, even\nthough the transmission coefficient for stationary transmission\nin the regime of medium saturation\n(Figs.\\ \\ref{u00001}, \\ref{vv1111}) looks qualitatively similar to the\non-site nonlinearity cases, the instability-induced dynamics may be quite\ndifferent.\n \n\n  \\section{Propagation of an initial Gaussian}\n\\label{sec:Gaussian}\n\n  As an example different from the stationary plane wave solutions we\ninvestigate, by direct numerical integration of\nEq.\\ \\eqref{dynamical}, the time propagation of an initial Gaussian wavepacket\nthrough the chain with the dimer defect having saturated inter-site\nnonlinear interactions between the two dimer sites.\nThe initial Gaussian data is\n\\bea{}\nA_n(0)= I~ \\textmd{exp}\\left[-\\frac{\\left(n-n_0\\right)^2}{w^2}+ik_in\\right] ,\n\\label{gwp}\n\\end{eqnarray}\nwhere $I$ and $p$ are the amplitude and width of the initial wave-packet taken\nto be $\\sqrt{3}$ and $56$, respectively. Typical results for this initial\ncondition with $|k_0|=\\pi\/2$ (corresponding\nto maximum propagation speed and minimum dispersion) in the regimes of low\n($\\beta=0.05$) and medium ($\\beta=0.5$) saturation are shown in\nFig.\\ \\ref{gwpb00001}. (Results in the ultra-low saturation regime are\nvery similar to those for $\\beta=0.05$.)\n\\begin{figure}[!htbp]\n   \n   \n   \\begin{minipage}[h]{0.4\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B05tpL.jpg}\n   \n  \\end{minipage}\n  \\hspace{0.5cm}\n  \\begin{minipage}[h]{0.4\\linewidth}\n    \\centering\n   \\includegraphics[width=\\linewidth]{B05tpR.jpg}\n   \n  \\end{minipage}\n \\begin{minipage}[h]{0.4\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5tpL.jpg}\n   \n  \\end{minipage}\n  \\hspace{0.5cm}\n  \\begin{minipage}[h]{0.4\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{B5tpR.jpg}\n   \n  \\end{minipage}\n   \\caption{Color plots of $|A_n(t)|^2$ as a function of $n$ (lattice sites)\nand $t$ (time), with the Gaussian initial data of Eq.\\ \\eqref{gwp}, saturation\n$\\beta=0.05$ (upper horizontal panel) and\n$\\beta=0.5$ (lower horizontal panel); wavenumber $k_0=\\pm\\frac{\\pi}{2}$,\nand all other parameters as before. Left (right) vertical panel corresponds to\nthe cases with left (right) incidence. The dimer is located at sites\n501 and 502 in a 1000 site lattice.\n}\n \\label{gwpb00001}\n   \\end{figure}\n\nThe transmission coefficient for the wavepacket is defined\nas in \\cite{11} to be the ratio of transmitted power to the initial input\npower. For the case of right propagating signal, it is given by\n  \\bea{}\n  \\tau_+=\\frac{\\sum_{i>\\frac{n}{2}+2}|A_i(t_f)|^2}{\\sum_{i<\\frac{n}{2}+1}|A_i(0)|^2} ,\n  \\label{tcR}\n  \\end{eqnarray}\nwhere $n$ is the total number of sites, the dimer is located at\nsites $n\/2+1$ and $n\/2+2$, and $t_f$ the final time for the\nnumerical integration. Equation \\eqref{tcR} can be understood by a direct\nanalogy with Eq.\\ \\eqref{t}. For $\\beta=0.05$,\nthe transmission coefficient for the left\nincidence (right propagating case; top left plot in Fig.\\ \\ref{gwpb00001})\nis found to be $\\tau_+=0.474$. In an analogous way, the transmission\ncoefficient for the right incidence (left propagating) case depicted in\nthe top right plot in Fig.\\ \\ref{gwpb00001} turns out to be\n$\\tau_-=0.603$. The rectifying factor is computed by the formula\n$f=\\frac{\\tau_+-\\tau_-}{\\tau_++\\tau_-}$, in analogy with Eq.\\ \\eqref{rf},\nwhich in this case is found to be $f=-0.120$. The distinct left and right\ntransmission coefficients reflect the fact that the parity symmetry of the\ndimer defect is indeed broken.\nThe lower horizontal panel in Fig.\\ \\ref{gwpb00001} corresponds to the regime\nof medium $\\beta$ strength ($\\beta=0.5$). The transmission coefficients are\nfound to be $\\tau_+=0.797$ and $\\tau_-=0.819$, and the corresponding\nrectifying factor is found to be $f=-0.013$.\nThus, increasing saturation implies that the transmission increases but the\nasymmetry decreases significantly, as was also found in previous\nworks for on-site saturability \\cite{Erik,wasay3}. This is also consistent\nwith results for the stationary transmission, e.g., by comparing middle left\nplot in Fig.\\ \\ref{k151}  and middle plot in Fig.\\ \\ref{vv1111}.\nFor $\\beta=0.05$ and $|k|$ close to $\\pi\/2$ (Fig.\\ \\ref{k151})\nthere is essentially no transmission\nwith $|T|^2 > 2$, and thus the peak intensity of the Gaussian with $I^2 = 3$\ncannot be transmitted, while for $\\beta=0.5$ (Fig.\\ \\ref{vv1111}) there\nis almost complete transmission around $|T|^2 = 2.5$, allowing for a larger\nportion of the Gaussian to be transmitted. On the other hand, the main\ntransmission around $|T|^2 = 2.5$ for $\\beta=0.5$ is almost symmetric for\nleft- and right-propagation, while for  $\\beta=0.05$ the stationary\ntransmission for right-propagation (blue curve) dips sharply around\n$|T|^2 = 1.8$, while for left-propagation (red curve) the dip appears later,\naround $|T|^2 = 2$. Thus, a considerable part of the Gaussian  with $I^2 = 3$\nmay be transmitted to the left but not to the right. (Weak instabilites of\nthe stationary transmission modes described in previous section will\nnot significantly affect the transmission of a rapidly moving and\nnot too wide Gaussian, if the time for the Gaussian to pass the dimer will\nbe shorter than the time for the instability to develop.)\n\nIt should also be noted from Fig.\\ \\ref{gwpb00001} that, in addition to\npartial transmission\/reflection of the Gaussian and creation of\nsmall-amplitude radiation waves, a rather small part will remain trapped\nat the dimer sites. For the cases shown in Fig.\\ \\ref{gwpb00001}, we find\nat time $t_f=250$ the trapped intensity $P_{trap}\\equiv |A_{501}|^2+|A_{502}|^2$ to\nbe:  $P_{trap}(\\beta=0.05, k=+\\pi\/2)=0.77$,\n$P_{trap}(\\beta=0.05, k=-\\pi\/2)=2.17$,\n$P_{trap}(\\beta=0.5, k=+\\pi\/2)=2.79$\n$P_{trap}(\\beta=0.5, k=-\\pi\/2)=2.29$.\nThus, in these cases considerably less trapping appears in the case when the\nmain part is reflected, compared to when it is mainly transmitted. This is\nopposite to the example considered in \\cite{11}, where trapping was enhanced\nwhen the main part was reflected. We may also note that for on-site\nsaturabilites, increasing saturation strength typically decreases trapping\n\\cite{Erik}, while we here observed an opposite tendency (which might be\nintuitively understood as increasing the saturability of the inter-site\nnonlinearity implies that the on-site nonlinearity will be relatively more\nimportant, and trapping is in general mainly associated with on-site\nnonlinearities).\n\n\n\nNote also that the transmission coefficient typically is largest\nwhen the incoming Gaussian first encounters the dimer site with smallest\n$|V_2|$, i.e., the site where the deviation from the linear chain with\nzero on-site potential is smallest. Intuitively this seems reasonable, and an\nanalogous remark was made for a saturable on-site potential in \\cite{jd}.\nWe checked a few other parameter values and obtained analogous results.\nKeeping $V_1=-2.625$ and changing the on-site amplitude on the right\ndimer site to $V_2=-1.875$ (i.e., decreasing its magnitude), we obtained\nfor $\\beta=0.05$ that $\\tau_+=0.594$ and $\\tau_-=0.746$ with\nrectifying factor $f=-0.113$, i.e., transmission increases while rectification\nremains almost the same (and with the same sign). On the other hand, changing\nthe on-site amplitude at the right dimer site to $V_2=-3.125$\n(i.e., increasing its magnitude to have $|V_2|>|V_1|$) resulted in\n $\\tau_+=0.722$ and $\\tau_-=0.542$ with $f=+0.143$. This means that the\ndiode-like transmission in the low-saturation regime, as expected, gets\nreversed when $|V_2|>|V_1|$.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nThe main aim of this work has been to provide a clear and comprehensive\ndescription of qualitatively novel effects that appear in the\ntransmission scenario of a DNLS-type dimer when nonlinear coupling between\nthe dimer sites is taken into account, in addition to a standard onsite\n(cubic) nonlinearity. For the generality of the description, we considered\na saturable intersite nonlinearity, with a parameter $\\beta$ interpolating\nbetween a purely cubic intersite + onsite nonlinearity at $\\beta=0$, and\na cubic pure onsite nonlinearity at large $\\beta$. A major novel result is\nthat, in contrast to the commonly studied cases with pure onsite nonlinearity,\nthe transmission coefficient for stationary transmission is in general no\nlonger a single-valued function\nof the transmitted intensity $|T|$, as the standard backward transfer map in\nregimes of small and moderate\n$|T|$ and low saturation will have three distinct solutions\n(two in the case of strictly zero saturation). These solutions\ndiffer in their relative phase shift across the dimer.\nAs the saturation\nincreases, solutions disappear through bifurcations at critical saturation\nstrengths (depending on the wave number), leaving a single solution\nbranch in regimes of medium and strong saturation.\n\nWe analyzed numerically the transmission coefficient of the three different\nbranches, and showed how these merged into the single-valued (as a function\nof $|T|$) transmission picture, known from previous works, as saturation\nstrength increased. We also performed a linear stability analysis\nof the stationary\nscattering solutions. In low-saturation regimes with three solution branches,\none branch, with very small transmission, exhibited strong instabilities,\nwhile from the other two branches solutions with small transmission\ncoefficients were typically stable, and those with larger transmission\nexhibited rather weak instabilites, growing larger when approaching\ntransmission peaks. Qualitatively the results for the latter two\nbranches agree with previous work for pure onsite DNLS. Studying by direct\nnumerical simulations the effect of the instabilities on the transmission, two\nnotable differences to previous works were found as a result of the\nintersite nonlinearities: (i) we did not observe any\nsignificant trapping at the dimer sites; (ii) while previous works always\nfound transport of power towards the transmission side, in some regimes we\ninstead found transport of power towards incoming\/reflected side.\n\nWe also analyzed the left\/right asymmetries in the transmission coefficient\nwith different linear onsite potentials at the dimer sites, both for\nstationary plane waves and for\nrather wide and rapidly moving Gaussian excitations with amplitiude near the\nmain transmission threshold. In addition to shifting the location of\ntransmission peaks, the multiple transmission branches for low saturability\nleads to a novel rectification effect for stationary transmission, where a\ntransmission peak in one direction, for a given branch and a given $|T|$,\nmay correspond to non-existence of solutions in the opposite propagation\ndirection for this branch. For the Gaussian propagation, we found that\nincreasing saturation strength typically would increase the transmission\ncoefficient but decrease the rectifying factor, as the stationary transmission\nspectrum also became broader and more symmetric for wave vectors close to\n$\\pi\/2$.\n\nFinally, we also comment on possible applications of our work.\nOur choice of model arose from the interest in studying the gradual transition\nfrom a system with non-saturated\nto saturated intersite nonlinearities, keeping onsite nonlinearities\nnon-saturated. The saturability is typically in the form\nappearing from photovoltaic-photorefractive materials, which may be a suitable\nclass of systems where the phenomenology of multichannel asymmetric\ntransmission described here may be observed. A relevant topic for future\nresearch would be to perform a similar analysis for the type of saturable\nnonlinear couplings proposed in \\cite{Hadad17}, with direct application to\nelectric circuit ladders \\cite{Hadad18}.\n\n\\section*{Acknowledgments}\n\nM.A.W would like to thank Jennie D'Ambroise for several fruitful\ndiscussions and to Byoung S. Ham for the facilities. M.A.W acknowledges financial support by the ICT R\\&D program of MSIT\/IITP (1711073835: Reliable crypto-system standards and core technology development for secure quantum key distribution network) and GRI grant funded by GIST in 2018. M.J. thanks Erik Johansson for discussions during his thesis work\n\\cite{Erik}, which served as a source of inspiration for the present work.\n\n\n\n \\bibliographystyle{unsrt}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA major open problem in differential geometry is to determine whether a closed almost complex manifold of dimension at least six always admits an integrable complex structure. By a celebrated theorem of Newlander and Nirenberg, an almost complex structure is integrable if and only if it satisfies a certain system of first order partial differential equations codified by the vanishing of the Nijenhuis tensor. An understanding of the topology of the space of all almost complex structures may be useful in the search for those which are integrable. We take this as motivation for our study of the topology of the space of almost complex structures on a six--manifold. The present work is a continuation of \\cite{FGM21} which focused on the six--sphere and touched upon the case of almost complex structures on six--manifolds with vanishing first Chern class.\n\nLet $M$ denote an oriented six--manifold and $\\mathcal J(M)$ be the space of \nall almost complex structures on $M$ inducing the given orientation.\nFixing a Riemannian metric on $M$, the inclusion $J(M) \\hookrightarrow \\mathcal J(M)$ of the subspace of almost complex structures which are orthogonal with respect to the metric is a homotopy equivalence, allowing us to analyze $\\mathcal J(M)$ via a space of sections of a fiber bundle with compact fibers. Generally, over an oriented Riemannian $2n$--manifold $M$ one can consider the $SO(2n)\/U(n)$ bundle of pointwise linear orthogonal complex structures $Z_+(M) \\to M$ compatible with the orientation on $M$; the space $Z_+(M)$ is known as the \\emph{(positive) twistor space} of $M$. Sections of this bundle correspond to orthogonal almost complex structures on $M$ compatible with the orientation.\n\nThe twistor space construction can be done in any dimension, but six--manifolds enjoy the special property that the fiber of the twistor space is likewise six--dimensional. Hence two orthogonal almost complex structures give embeddings of the manifold into its twistor space which generically intersect at a finite number of points. One of our main results (\\Cref{intersection}) computes this intersection number in terms of the Chern classes of the two almost complex structures. Employing a theorem of Michelsohn and Salamon, we observe that negative self-intersection restricts the possible deformations of an orthogonal complex structure (\\Cref{deform}).\n\nIn order to study the homotopy type of $J(M)$, in Section 3 we use the existence of spin$^c$ structures on oriented four--manifolds and almost complex six--manifolds to describe the integral cohomology of their twistor spaces. This will include both the positive and the \\emph{negative} twistor space $Z_-(M)$, i.e. the total space of the bundle of linear complex structures inducing the opposite orientation on the tangent space. Presumably these results are well known, but we were not able to find them in the literature in this generality.\n\n\nWe take particular care with orientations throughout, due to varying historical conventions (e.g. the original \\cite{AHS78}, and \\cite{Hi81}, refer to the negative twistor space as the ``twistor space'', while this terminology is reserved for the positive twistor space for instance in \\cite{LM}\\footnote{Despite this, the statement of \\cite[Theorem 4.1]{AHS78} in \\cite[Theorem IV.9.14]{LM} uses the convention of \\cite{AHS78}.} and \\cite{Sa84}), and to the authors' difficulty in verifying some formulas in \\cite[p.135]{Hi81}; see \\Cref{hitchin}.\n\nThe twistor spaces $Z_+(M)$ and $Z_-(M)$ (whose homotopy types generally differ when the dimension of $M$ is divisible by four) carry two natural almost complex structures, differing by the choice of induced orientation on the fiber (see e.g. \\cite[Proposition 3.1]{Sa84}), known as the \\emph{Atiyah--Hitchin--Singer} and \\emph{Eells--Salamon} almost complex structures. Even though the standard complex structure and its negative on the fibers $SO(4)\/U(2) \\cong \\mathbb{CP}^1$ and $SO(6)\/U(3) \\cong \\mathbb{CP}^3$ are biholomorphic via complex conjugation, the Eells--Salamon almost complex structure is never integrable \\cite[Proposition 3.4]{Sa95}, in contrast to the Atiyah--Hitchin--Singer structure (this phenomenon is not unique to dimensions four and six).\n\nOur second main result is a description of the rational homotopy type of (a given component of) the space of almost complex structures on six--manifolds, under the additional assumption that $b_1(M) = 0$ and $\\int_M c_1c_2 - c_3 \\neq 0$ (\\Cref{Qhomotopy6}). For this we use an expression of $J(M)$ as the quotient of the space of sections of an $S^7$--bundle by an $S^1$--action (\\Cref{fibration}), which one can compare with the result obtained for $c_1=0$ in \\cite[Proposition 5.1]{FGM21}.\n\nThroughout we provide examples for all of the mentioned results.\n\n\n\n\\subsection*{Notation and conventions}\n\n The projectivization of a complex vector bundle $E$ is denoted by $\\mathbb{P}(E)$. We denote the Chern classes of a complex vector bundle by $c_i$, and the Pontryagin classes of a real vector bundle by $p_i$; if the vector bundle is complex, by its Pontryagin classes we mean those of the underlying real bundle. The Euler class is denoted by $\\mathrm{e}$. The Euler characteristic of a space is denoted by $\\chi$, and $\\sigma$ is the signature of an oriented closed manifold. All manifolds are assumed connected unless otherwise stated.  \n\nWe denote by $\\Gamma(E)$ the space of sections of a bundle $E \\to M$ (where $E$ is not necessarily a vector bundle). If $s \\colon M \\to E$ is a section, $\\Gamma(E)_s$ denotes the connected component of $s$ in $\\Gamma(E)$.\nFor $E \\to M$ a vector bundle with a metric, we write $S(E) \\to M$ for the associated sphere bundle. We denote by $J(M)$ the connected component of a specific point in $\\Gamma(Z_+(M))$, which will be clear from context. For a vector space $V$ with an inner product, $J(V)$ will denote the space of all orthogonal (linear) complex structures on $V$. If $V$ is oriented, $J_+(V)$ (respectively $J_-(V)$) denotes the connected component of $J(V)$ consisting of elements which induce the given orientation on $V$ (respectively the opposite orientation). By $\\Map(X,Y)$ we denote the space of unbased maps $X \\to Y$ with the compact-open topology.\n\nCohomology is singular cohomology with integral coefficients unless another choice of coefficients is indicated. The evaluation of a cohomology class $\\alpha$ on a  homology class $A$ is denoted by $\\langle \\alpha, A \\rangle$ or by $\\int_A \\alpha$. In the context of rational homotopy theoretic models, $\\Lambda(x_i)$ denotes the free graded--commutative algebra on generators $x_i$. \n\nWe will make frequent use of the projective bundle formula, which states that for a complex vector bundle $E \\to M$ of complex rank $k$, the cohomology of the projectivized bundle $\\mathbb{P}(E) \\xrightarrow{p} M$ is the free $H^*(M)$--module on one generator $x$ subject only to the relation $x^k + c_1(E)x^{k-1} + \\cdots + c_{k-1}(E)x + c_k(E) = 0$, where $x$ is the first Chern class of the line bundle $\\mathcal{O}_{E^*}(1)$ (see \\cite[Definition 15.13]{D12}). This line bundle is dual\nto the tautological (Hopf) line sub-bundle of $p^*(E)$ whose fiber over $\\ell \\in \\mathbb{P}(E_x)$ is $\\ell \\subset E_x$. It restricts to $\\mathcal{O}(1)$ on each fiber $\\mathbb{CP}^{k-1}$. \n\n\n\n\\subsection*{Acknowledgements} We thank Luis Fernandez and Scott Wilson for numerous helpful discussions. The first author was partially supported by FCT\/Portugal through CAMGSD, IST-ID, projects UIDB\/04459\/2020 and UIDP\/04459\/2020. The second author would like to thank the Max Planck Institute for Mathematics in Bonn for its support, along with the Mittag-Leffler Institute in Djursholm for its hospitality during a visit to the ``Higher algebraic structures in algebra, topology and geometry'' program, where part of this work was carried out.\n\n\\section{Preliminaries}\nIn this section we review some of the models for the space of orthogonal complex structures on a euclidean space following \\cite[Section IV.9]{LM}, to which we refer for further details. \n\nLet $(V, (\\cdot, \\cdot ))$ be a $2n$--dimensional real inner product space. We will write $$J(V)=\\{ J \\in \\End(V) \\colon J^2=-I, J \\text{ is orthogonal} \\}$$\nfor the space of orthogonal complex structures on $V$. This space naturally has the structure of a complex algebraic variety, given by the isomorphism\n$$ J(V) \\xrightarrow{\\phi} \\Gr_n^{Iso}(V\\otimes \\mathbb{C}) $$\nwith the Grassmannian of $n$--dimensional complex subspaces of $V\\otimes \\mathbb{C}$ which are isotropic with respect to the complex bilinear form on $V\\otimes \\mathbb{C}$ obtained from the inner product on $V$ by linear extension. The map $\\phi$ is defined by\n$$ \\phi(J) = \\{ v\\otimes 1 + Jv \\otimes i \\colon v \\in V\\} $$\n(it assigns to $J$ the plane $V_J^{0,1} \\subset V\\otimes \\mathbb{C}$, which is the $-i$ eigenspace of the complex linear extension of $J$ to $V\\otimes \\mathbb{C}$). \n\nThere is a tautologous complex vector bundle on $J(V)$ defined by \n$$J(V) \\times V \\xrightarrow{\\pi_1} J(V),$$\nwhere the fiber over $J\\in J(V)$ is given the complex structure determined by $J$.\nThe isomorphism $\\phi$ is covered by a bundle map to the dual (or conjugate) tautological bundle over the Grassmannian. \n\n\\begin{lemma}\n\\label{positchern}\nConsider the universal bundle \n$$\nJ(\\mathbb{R}^{2n})=O(2n)\/U(n) \\xrightarrow{i} BU(n) \\to BO(2n).\n$$\nThen\n\\begin{enumerate}[(i)]\n\\item $i$ classifies the tautologous complex vector bundle on $J(\\mathbb{R}^{2n})$.\n\\item $i^*(c_1)$ generates a subgroup of index $2$ in the infinite cyclic second cohomology group of \neach of the two components of $J(\\mathbb{R}^{2n})$.\n\\item $i^*(c_1)$ is positive. In particular, $i^*(c_1)^{\\frac{n^2-n}{2}}$ evaluates on the orientation class of each of the two components of $J(\\mathbb{R}^{2n})$ to a positive integer.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\\begin{enumerate}[(i)]\n\\item The universal bundle can be realized by taking $BU(n)=EO(2n)\/U(n)$. The universal bundle over $BU(n)$ is then given by $EO(2n)\\times_{U(n)} \\mathbb{R}^{2n} \\to EO(2n)\/U(n)$, and pulls back along the inclusion of the fiber $i$ to $O(2n) \\times_{U(n)} \\mathbb{R}^{2n} \\to O(2n)\/U(n)$, which is the homogeneous expression of the tautologous bundle. \n\\item The inclusion-induced map $BU(n) \\to BO(2n)$ factors through the\ndouble cover $BSO(2n)\\to BO(2n)$, and the two components of $O(2n)\/U(n)$ are, \nrespectively, the fiber $SO(2n)\/U(n)$ of $BU(n) \\to BSO(2n)$ and the image \nof this fiber under a lift to $BU(n)$ of the non-trivial deck transformation of \n$BSO(2n)$. Therefore it suffices to prove the statement for the component\n$SO(2n)\/U(n)$. This follows from the Serre spectral sequence of the bundle\n$BU(n) \\to BSO(2n)$ (see for instance \\cite[Theorem III.6.11]{MiTo}).\n\n\\item $i^*(c_1)$ is the pullback of $c_1$ of the dual tautological bundle under the embedding \n$$J(\\mathbb{R}^{2n}) \\xrightarrow{\\cong} \\Gr_n^{Iso}(\\mathbb{C}^{2n}) \\subset \\Gr_n(\\mathbb{C}^{2n}).$$\nThe $n^{\\mathrm{th}}$ exterior power of the dual tautological bundle on $\\Gr(\\mathbb{C}^{2n})$ gives the \nPl\\\"ucker embedding of the Grassmannian in projective space. Hence $i^*(c_1)$ is the first\nChern class of a line bundle on $J(\\mathbb{R}^{2n})$ whose sections embed $J(\\mathbb{R}^{2n})$ in projective space and is therefore positive.\\qedhere \n\\end{enumerate}\n\\end{proof}\n\nThe above identification of orthogonal complex structures with isotropic planes in \n$V\\otimes \\mathbb{C}$ leads to another description of the space $J(V)$ which will play an \nimportant role in this paper. \n\nLet $\\mathcal{C}l(V)$ denote the Clifford algebra determined by the quadratic form $q(v)=-\\|v\\|^2$, and let $\\mathbb{C}l(V)=\\mathcal{C}l(V)\\otimes_\\mathbb{R} \\mathbb{C}$ \ndenote its complexification. Let $S_{\\mathbb{C}}$ denote an irreducible ($2^n$--dimensional) $\\mathbb{C}l(V)$-module.\n\nFor each element, i.e. \\emph{spinor}, $\\sigma \\in S_{\\mathbb{C}}$ we have the map $V\\otimes_\\mathbb{R} \\mathbb{C} \\to S_{\\mathbb{C}}$ given by right Clifford multiplication by $\\sigma$. A spinor is said to be \\emph{pure} if the associated kernel is half--dimensional. Let $PS_{\\mathbb{C}} \\subset S_\\mathbb{C}$ denote the subset of pure spinors. As the subspace $\\ker (\\cdot \\sigma) \\subset V\\otimes \\mathbb{C}$ is isotropic\nthere is a natural map from the projectivization of the set of pure spinors to the isotropic Grassmannian\n$$\n\\mathbb{P}( PS_\\mathbb{C} ) \\xrightarrow{\\psi}  \\Gr^{Iso}_{2n}(V\\otimes \\mathbb{C})\n$$\ndefined by $[\\sigma] \\mapsto \\ker (\\cdot \\sigma)$. This is a map of algebraic varieties and is in fact an $SO(2n)$-equivariant isomorphism \\cite[Proposition IV.9.7]{LM}.\n\nThe space $J(V)$ has two connected components corresponding to the two possible orientations induced by the complex structure. A choice of orientation for $V$ leads to a decomposition\n$$J(V) = J_+(V) \\coprod J_-(V)$$\nwith $J_+(V)$ the component corresponding to the given orientation. In turn this decomposes the Grassmannian of isotropic subspaces into the components of positive and negative isotropic subspaces. \n\nOn the level of spinors this decomposition takes the following form.\nThe complex volume element in $\\mathbb{C}l(V)$ is the element \n$\\omega_\\mathbb{C} = i^n e_1 \\cdots e_{2n}$, where \n$\\{e_i\\}$ is any oriented orthonormal basis. As $\\omega_{\\mathbb{C}}^2=1$, the volume element decomposes\nthe module $S_C$ into a direct sum of $+1$ and $-1$ eigenspaces \n$$\nS_\\mathbb{C} = S_\\mathbb{C}^+ \\oplus S_\\mathbb{C}^-.\n$$\nThe elements of the summands are called the positive and negative spinors, respectively. \nNote that, since $\\omega_\\mathbb{C}$ anti-commutes with the action of $V \\subset \\mathbb{C}l(V)$, both \n$S_\\mathbb{C}^+$ and $S_\\mathbb{C}^-$ are spin$^c$ representations, called the positive and negative spinor representations respectively. \n\nEvery pure spinor is necessarily either positive or negative and the isomorphisms $\\phi,\\psi$ give rise to isomorphisms of algebraic varieties\n$$\n\\mathbb{P}(PS_\\mathbb{C}^\\pm) \\cong J_\\pm(V).\n$$\nIn dimensions $2n=4,6$ all non-zero spinors are pure \\cite[Remark IV.9.12]{LM} and the isomorphisms identify the spaces\nof orthogonal complex structure with the disjoint union of the projectivization of positive \nand negative spinors. \n\nA point $J \\in J(V)$ gives rise to a useful description (cf. \\cite[Chapter I (5.27)]{LM})\nof the irreducible module $S_\\mathbb{C}$.\nWe will write $\\langle \\cdot, \\cdot \\rangle$ for the Hermitian inner product on the complex vector space $(V,J)$ whose real part is $ (\\cdot, \\cdot )$.\nLet $\\Lambda_\\mathbb{C}^*(V)$ denote the exterior algebra of the complex vector space $(V,J)$.\nContraction of a vector $v \\in V$ with $\\omega \\in \\Lambda_\\mathbb{C}^k(V)$ is determined by the expression\n$$\nv \\lrcorner (w_1 \\wedge \\cdots \\wedge w_k) = \\sum_{j=1}^k (-1)^{j+1}\\langle v, w_j \\rangle w_1 \\wedge \\cdots \\wedge \\widehat{w_j} \\wedge \\cdots \\wedge w_k.\n$$\nThe action $V \\otimes_\\mathbb{R} \\Lambda_\\mathbb{C}^*(V) \\to \\Lambda_\\mathbb{C}^*(V)$ defined by \n\\begin{equation*}\nv \\cdot \\omega = v \\wedge \\omega - v \\lrcorner \\,  \\omega\n\\end{equation*}\nsatisfies\n\\begin{equation*}\nv\\cdot(v \\cdot \\omega) = - \\|v\\|^2 \\omega,\n\\end{equation*}\nas one easily checks on an orthonormal basis for $\\Lambda_\\mathbb{C}^*$ having a \nreal multiple of a non-zero $v$ as its first element. The complex linear extension of this action gives $\\Lambda_\\mathbb{C}^*(V)$ the structure of a $\\mathbb{C}l(V)$-module of complex dimension $2^n$. This is the dimension of the irreducible complex $\\mathbb{C}l(V)$-module, so $\\Lambda_\\mathbb{C}^*(V)$ is a convenient description of this module. \n\n\\iffalse\nIndeed, one can consider an orthonormal basis $v_1,Jv_1, \\ldots, v_n, Jv_n$ for $V$ \nwith $v=\\alpha v_1$ and check that \\eqref{clifalg} holds for each $\\omega$ in the standard orthonormal basis for $\\Lambda_\\mathbb{C}^*(V)$:\n$$\nv \\cdot v_{i_1} \\wedge \\cdots \\wedge v_{i_k} = \\begin{cases}\n- \\alpha \\ v_{i_2} \\wedge \\cdots \\wedge v_{i_k}  & \\text{ if } i_1=1\\\\\n\\alpha \\ v_{1} \\wedge v_{i_1} \\wedge \\cdots \\wedge v_{i_k} & \\text{ if } i_1 \\neq 1 \n\\end{cases}\n$$\nand hence\n$$\nv \\cdot( v \\cdot v_{i_1} \\wedge \\cdots \\wedge v_{i_k} ) =\n- \\alpha^2 \\ v_{i_1} \\wedge v_{i_2} \\wedge \\cdots \\wedge v_{i_k}  = - \\|v\\|^2 v_{i_1} \\wedge \\cdots \\wedge v_{i_k} \n$$\n\\fi\n\n\n\n\n\n\\begin{lemma}\n\\label{plusmin}\nLet $(V,J)$ be an oriented $2n$--dimensional euclidean real vector space with an orthogonal complex structure $J$ compatible with the orientation and let $S_\\mathbb{C}=\\Lambda^*_\\mathbb{C}(V)$ be the irreducible $\\mathbb{C}l(V)$-module described above. Then \n$$\nS_\\mathbb{C}^+ = \\bigoplus_{k \\text{ even }} \\Lambda_\\mathbb{C}^k(V), \\quad \\quad S_\\mathbb{C}^- = \\bigoplus_{k \\text{ odd }} \\Lambda_\\mathbb{C}^k(V).\n$$\n\\end{lemma}\n\\begin{proof}\nLet $e_1, \\ldots, e_n$ be an orthonormal basis for $(V,J)$. Then we can write the complex volume element as \n$$\n\\omega_\\mathbb{C} = i^n e_1 Je_1 \\cdots e_n Je_n .\n$$\nConsider the basis $e_{i_1} \\wedge \\cdots \\wedge e_{i_k}$ for $\\Lambda^k_{\\mathbb{C}}$ with $1\\leq i_1 < \\cdots < i_k \\leq n$. Let $1 \\leq m \\leq n$\nand assume that $m \\not \\in \\{i_1, \\ldots, i_k \\}$. Then\n\\begin{eqnarray*}\n(e_m Je_m) \\cdot e_{i_1} \\wedge \\cdots \\wedge e_{i_k}  & = & e_m \\cdot \\left( Je_m \\wedge e_{i_1} \\wedge \\cdots \\wedge e_{i_k} \\right) \\\\\n& = & -\\langle e_m, Je_m \\rangle e_{i_1} \\wedge \\cdots \\wedge e_{i_k} = -i \\ e_{i_1} \\wedge \\cdots \\wedge e_{i_k}.\n\\end{eqnarray*}\nOn the other hand, if $m \\in \\{i_1, \\ldots, i_k\\}$, let $l$ be such that $m=i_l$. Then\n\\begin{eqnarray*}\n(e_m Je_m) \\cdot e_{i_1} \\wedge \\cdots \\wedge e_{i_k}  & = & e_m \\cdot \\left(  - (-1)^{l-1} \\langle Je_m, e_{i_l} \\rangle e_{i_1} \\wedge \\cdots \\wedge \\widehat{e_{i_l}} \\wedge \\cdots \\wedge e_{i_k} \\right)\\\\\n& = & e_m \\wedge\\left(  (-1)^l (-i)  e_{i_1} \\wedge \\cdots \\wedge \\widehat{e_{i_l}} \\wedge \\cdots \\wedge e_{i_k} \\right) \\\\\n& = & (-1)^{l-1} (-1)^l(-i)  e_{i_1} \\wedge \\cdots \\wedge e_{i_k} = i  e_{i_1} \\wedge \\cdots \\wedge e_{i_k}.\n\\end{eqnarray*}\nHence \n$$\n\\omega_\\mathbb{C} \\cdot  e_{i_1} \\wedge \\cdots \\wedge e_{i_k} = i^n i^k (-i)^{n-k}  e_{i_1} \\wedge \\cdots \\wedge e_{i_k} = (-1)^{2n-k} e_{i_1} \\wedge  \\cdots \\wedge e_{i_k},\n$$\nwhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\n\\label{lift}\nFor $(V,J)$ an oriented $2n$--dimensional euclidean real vector space with an orthogonal complex structure $J$ compatible with the orientation, the line in $S^+_\\mathbb{C}$ corresponding to $J\\in J_+(V)$ is $\\Lambda^0_\\mathbb{C}(V)\\subset S_\\mathbb{C}^+$. \nIndeed,\n$$(v \\otimes 1 + w \\otimes i) \\cdot 1 = v+Jw = 0 \\ \\Leftrightarrow \\ w=Jv. $$\n\\end{remark}\n\n\\section{Twistor spaces of four-- and six--manifolds}\n\nIn this section we describe the integral cohomology of the twistor space of (Riemannian) almost complex six--manifolds, along with the integral cohomology of the twistor space of arbitrary oriented (Riemannian) four--manifolds. Finally we compute the Chern classes of the Atiyah--Hitchin--Singer almost complex structure on the latter. \n\n\\subsection{Spinors on almost complex four-- and six--manifolds}\n\nLet $M$ be an oriented Riemannian manifold and let $Z(M)$ denote the space of orthogonal almost complex structures on $M$, called the \\emph{twistor space}. The space \n$Z(M)$ has two components corresponding to the structures which induce the given orientation on $M$ or the opposite orientation. Recall, we \ndenote these by $Z_{\\pm}(M)$, and refer to them as the \\emph{positive} and \\emph{negative} twistor spaces.\n\nThe Atiyah--Hitchin--Singer almost complex structure on $Z(M)$ is defined as follows: the Levi--Civita connection induces a connection on the bundle $Z(M) \\to M$ splitting $TZ(M)$ into\na direct sum of horizontal and vertical subspaces\n$$\nTZ(M) =T^h Z(M) \\oplus T^vZ(M).\n$$\nThe almost complex structure on $T^v Z(M)$ is determined by the algebraic variety structure of the fibers $J(T_x M)$ of $Z(M) \\to M$ explained in Section 2. At a point $J_x$ in the fiber\nover $x\\in M$ the complex structure on $T^h_{J_x} Z(M) = T_x M$ is given\\footnote{The Eells--Salamon almost complex structure is obtained replacing the vertical component of the Atiyah--Hitchin--Singer almost complex structure by its negative; see \\cite[Section 3]{Sa84} for details.} by $J_x$.\n\n\n\nA spin$^c$ structure on $M$ \\cite[Appendix D]{LM} yields a complex spinor \nbundle $S_\\mathbb{C}(M)\\to M$ together with an action\n$$\n\\mathbb{C}l(TM) \\otimes S_\\mathbb{C}(M) \\to S_\\mathbb{C}(M).\n$$\nThe discussion in the previous section then provides a one-to-one correspondence between projectivized pure spinors on $M$ and orthogonal complex structures inducing a given orientation: \n$$\nZ_{\\pm}(M) \\cong \\mathbb{P}( PS_{\\mathbb{C}}^{\\pm}(M)).\n$$\nNow suppose $M$ is given an orthogonal almost complex structure $J$ compatible with the orientation. Then $J$ gives rise to a canonical spin$^c$ structure on $M$ via the group homomomorphism $U(n) \\xrightarrow{\\kappa} \\Spin^c(2n)$, which is the unique\nlift as a group homomorphism in the following diagram\n$$\n\\begin{tikzcd} \n& \\Spin^c(2n) = (\\Spin(2n) \\times S^1)\/\\{\\pm(1,1)\\} \\ar[d, \"\\pi \\times \\delta\"] \\\\\nU(n) \\ar[ur,\"\\kappa\"] \\ar[r,\"i \\times \\det\"] & SO(2n) \\times S^1\n\\end{tikzcd}\n$$\nHere $i$ denotes the inclusion, $\\pi([h, e^{i\\theta}]) = [h]$, and $\\delta([h,e^{i\\theta}])=e^{2i\\theta}$.\n\nAn element $g\\in U(n)$ is diagonalized by some orthonormal basis $(e_1,Je_1, \\ldots, e_n, Je_n)$. If \n$$g=\\mathrm{diag}(e^{i\\theta_1},\\ldots, e^{i\\theta_n}),$$ then $\\kappa(g)= \\left[  \\prod_{k=1}^n \\left( \\cos \\frac{\\theta_k} 2  + i \\sin \\frac{\\theta_k}{2} e_k J e_k\\right), \\  e^{i \\frac{\\sum \\theta_k}{2}} \\right]$ (cf. \\cite[(D.10)]{LM}). Using \nthe formulas for the action of $e_i Je_i$ in the proof of \\Cref{plusmin} we see that the spinor action of $\\kappa(g)$ \non $\\Lambda^*_{\\mathbb{C}}(\\mathbb{R}^{2n})$, where $\\mathbb{R}^{2n}$ is equipped with the complex structure given by $\\mathrm{diag}\\left( \\left(\\begin{smallmatrix} 0 & -1 \\\\ 1 & 0 \\end{smallmatrix}\\right), \\ldots, \\left(\\begin{smallmatrix} 0 & -1 \\\\ 1 & 0 \\end{smallmatrix}\\right) \\right)$, agrees with the standard action of $g$ on this space. We obtain the following result:\n\n\\begin{prop}\n\\label{identification}\nLet $M$ be an oriented Riemannian manifold with an orthogonal almost complex structure $J$ compatible with the orientation. Then we have the following isomorphisms of smooth fiber bundles over $M$:\n\\begin{enumerate}\n\\item If $\\dim(M)=4$, then\n$$ \nZ_+(M) \\cong \\mathbb{P}( \\mathbb{C} \\oplus \\Lambda^2_\\mathbb{C}(TM)), \\quad \\quad Z_{-}(M) \\cong \\mathbb{P}( TM).\n$$\n\\item If $\\dim(M)=6$, then \n$$\nZ_+(M) \\cong \\mathbb{P}( \\mathbb{C} \\oplus \\Lambda^2_\\mathbb{C}(TM)), \\quad \\quad Z_{-}(M) \\cong \\mathbb{P}( TM \\oplus \\Lambda_\\mathbb{C}^3(TM)). \n$$\n\\end{enumerate}\n\\end{prop}\n\n\\begin{remark}\n The Atiyah--Hitchin--Singer and Eells--Salamon almost complex structures on (either) twistor space do not agree with the natural almost complex structure on the projectivization. Indeed, the projection map from the projectivization is complex linear whilst this is not the case for the twistor projection.\n\\end{remark}\n\nThe projective bundle formula now gives the following formulas for the cohomology rings of the components of the twistor space in terms of the \nChern classes of $(TM, J)$. \n\n\\begin{cor}\n\\label{cohomtwis}\nLet $M$ be an oriented Riemannian manifold with an orthogonal almost complex structure $J$ compatible with the orientation. We have the following isomorphisms as algebras over $H^*(M)$:\n\\begin{enumerate}\n\\item If $\\dim(M)=4$, then\n\\begin{align*}\nH^*(Z_+(M)) &\\cong H^*(M)[x]\/(x^2+c_1 x), \\\\ H^*(Z_{-}(M)) &\\cong H^*(M)[x]\/(x^2 + c_1 x + c_2). \\end{align*}\n\\item If $\\dim(M)=6$, then \n\\begin{align*}\nH^*(Z_+(M)) &\\cong H^*(M)[x]\/(x^4 +2c_1x^3 +(c_1^2+c_2) x^2 + (c_1c_2-c_3)x), \\\\ \nH^*(Z_{-}(M)) &\\cong H^*(M)[x]\/(x^4 + 2c_1x^3+ (c_1^2+c_2)x^2 +  (c_1c_2+c_3)x). \\end{align*}\n\\end{enumerate}\n\\end{cor}\n\n\\begin{remark} \\begin{itemize} \\item The formulas above in dimension four agree with \\cite[Theorem 11.2]{ESa85}, where it is established that over a K\\\"ahler surface $M$, the positive twistor space is the projectivization of $T^{2,0}M \\oplus \\mathbb{C}$ (note that $T^{2,0}M \\cong \\Lambda_{\\mathbb{C}}^2(TM)$), and the negative twistor space is the projectivization of the holomorphic tangent bundle.\n\n\\item In \\cite[Section 6]{Ev14}, Evans describes the cohomology ring of the positive twistor space of a (not necessarily almost complex) six--manifold, with complex coefficients. The description is in terms of the first Chern class of $T^h(Z_+(M))$. Using \\Cref{chernclasses}, one can check that Evans' description agrees with the one in \\Cref{cohomtwis} in the case of almost complex manifolds. We remark that Evans' computation can quickly be reproduced using naturality in the pullback diagram $$\\begin{tikzcd}\nZ_+(M) \\arrow[d] \\arrow[r] & BU(3) \\arrow[d] \\\\\nM \\arrow[r]              & BSO(6)         \n\\end{tikzcd}$$\nand expressing $H^*(BU(3);\\mathbb{Q})$ as an $H^*(BSO(6);\\mathbb{Q})$-algebra using the known behavior of the right-hand side vertical map on cohomology.\n\\end{itemize}\n\\end{remark}\n\n\\begin{remark}\\label{cp2bar} In order for the formulas of \\Cref{cohomtwis} (1) to hold, it is crucial to have an almost complex structure \\emph{inducing the given orientation} on the base manifold. For instance, the positive twistor space of $\\overline{\\mathbb{CP}^2}$ (i.e. the complex projective plane with the opposite orientation) with the Fubini--Study metric is the full flag variety $U(3)\/\\left( U(1)\\times U(1) \\times U(1) \\right)$ \\cite[p.217, Example 1]{Sa84}, whose cohomology is given by $$\\mathbb{Z}[x,y,z]\/(x+y+z, xy+xz+yz, xyz) \\cong \\mathbb{Z}[x,y]\/(x^2+xy+y^2, x^2y+xy^2),$$ where $x,y,z$ are in degree two \\cite[Proposition 31.1]{B53}. This ring is not isomorphic to $$\\mathbb{Z}[x,y]\/(y^3, x^2 + kyx)$$ for any integer $k$, so the cohomology of $Z_+(\\overline{\\mathbb{CP}^2})$ can not be expressed as in the first item of \\Cref{cohomtwis} (1). This corresponds to the fact that $\\overline{\\mathbb{CP}^2}$ does not admit an almost complex structure compatible with its orientation (which can of course be seen by other means as well, e.g. Hirzebruch's congruence $\\chi + \\sigma \\equiv 0 \\bmod 4$ for closed almost complex four--manifolds). \\end{remark}\n\n\\subsection{$\\Spin^c(4)$-bundles and twistor spaces of four--dimensional manifolds}\n\nTwistor spaces of oriented Riemannian four--manifolds give interesting examples of almost complex six--manifolds which we will consider in the following sections.  In this sub-section we will compute their integral cohomology as well as the Chern classes of\ntheir Atiyah--Hitchin--Singer almost complex structures. \n\nOur computation requires knowledge of the integral cohomology of $B\\Spin^c(4)$. The cohomology of $B\\Spin^c(n)$ is described in detail in \\cite{Du18}. We need only the simple case when $n=4$, so we include an elementary treatment of this case. \n\nOur aim is to compute the cohomology ring of the ``universal twistor space\" over $B\\Spin^c(4)$, namely the bundle of orthogonal complex structures on the fibers of the $4$--plane bundle over $B\\Spin^c(4)$ classified by the canonical map $B\\Spin^c(4) \\to BSO(4)$. Since every oriented four--manifold admits a spin$^c$ structure, naturality will \nyield the cohomology ring of the twistor space.\n\nWe will use unit quaternions to describe the four--dimensional spin groups and orthogonal complex structures. We consider the quaternions $\\mathbb{H}$ as a vector space over $\\mathbb{C}$ via\nright multiplication by $\\mathbb{C} \\subset \\mathbb{H}$, i.e. we consider the identification of $\\mathbb{C}^2$ with $\\mathbb{H}$ given by\n$$  (z_1, z_2) \\mapsto z_1 + jz_2. $$\nThe standard complex structure on $\\mathbb{H}= \\mathbb{R}^4$ is then right multiplication by $i$,\nand $(1,i,j,-k)$ is an oriented basis. \n\nThe unit quaternion $q= w_1 + j w_2$ acts on $\\mathbb{H}$ by left multiplication as \n$$ (w_1 + jw_2) (z_1+jz_2) = (w_1z_1 -\\overline{w_2} z_2) + j(w_2 z_1+z_2 \\overline{w_1} )$$\nwhich corresponds via the identification with $\\mathbb{C}^2$ to the matrix\n$$\n\\left[ \\begin{array}{cc}\nw_1 & -\\overline{w_2} \\\\\nw_2 & \\overline{w_1}\n\\end{array}\n\\right]\n$$\nwith $|w_1|^2 +|w_2|^2=1$. This will be our identification of $Sp(1)$ with $SU(2)$.\n\nThe universal cover of $SO(4)$ is modelled by the map $Sp(1) \\times Sp(1) \\to SO(4)$\ngiven by \n$$ (q_1, q_2) \\mapsto \\left( v \\mapsto q_1 v \\overline{q_2} \\right) $$\nand we will identify\n$$\nSO(4) = (Sp(1) \\times Sp(1))\/\\{\\pm (1,1)\\}.\n$$\nIn these terms, the subgroup $U(2) \\subset SO(4)$ is \n$$\nU(2) = (Sp(1) \\times S^1)\/\\{\\pm (1,1)\\}.\n$$\n\nThe group $\\Spin^c(4)$ is defined as \n$$\n\\Spin^c(4) = (Sp(1) \\times Sp(1) \\times S^1)\/\\{\\pm(1,1,1)\\}.\n$$\nThere are two canonical homomorphisms \n\\begin{equation}\n\\label{canonichom}\nSO(4)  \\xleftarrow{\\pi} \\Spin^c(4) \\xrightarrow{\\delta} S^1\n\\end{equation}\ngiven by \n$$ \n\\pi( [q_1,q_2,e^{i\\theta}]) = [q_1,q_2],\n\\quad \\quad \n\\delta( [q_1,q_2, e^{i\\theta}]) = e^{2i\\theta}. \n$$\nThe canonical map $U(2) \\xrightarrow{\\kappa} \\Spin^c(4)$ is the unique lift as a group \nhomomorphism in the diagram\n$$\n\\begin{tikzcd}\n& \\Spin^c(4) \\ar[\"\\pi \\times \\delta\",d] \\\\\nU(2) \\ar[\"\\kappa\",ur] \\ar[\"i\\times \\det\", r] & SO(4)\\times S^1\n\\end{tikzcd}\n$$\nIn terms of the coordinates above, this is given by $\\kappa([q_1,e^{i\\theta}])=[q_1,e^{i\\theta},e^{-i\\theta}]$.\n\nLet $x \\in H^2(BS^1)=H^2(\\mathbb{CP}^\\infty)$ denote the standard generator (the Chern class\nof the dual tautological line bundle on $\\mathbb{CP}^\\infty$) and\n$$\n\\alpha= (B\\delta)^*(x) \\in H^2(B\\Spin^c(4)).\n$$\nWe will write $p_1$ and $\\mathrm{e}$ respectively for the universal Pontryagin class and Euler class in \n$H^*(BSO(4))$.\n\n\\begin{lemma}\n\\label{cohspin}\nLet $\\pi\\colon \\Spin^c(4) \\to SO(4)$ be\nas in \\eqref{canonichom}. There are unique classes $S_1,S_2 \\in H^4(B\\Spin^c(4))$ such\nthat \n\\begin{align*}\n4S_1 &=B\\pi^* p_1 - 2B\\pi^* \\mathrm{e} - \\alpha^2, \\\\ \n4S_2  &=B\\pi^* p_1 + 2B\\pi^* \\mathrm{e} - \\alpha^2.\n\\end{align*}\nMoreover the cohomology ring of $B\\Spin^c(4)$ is a polynomial algebra,\n$$\nH^*(B\\Spin^c(4)) \\cong \\mathbb{Z}[\\alpha,S_1,S_2].\n$$\n\\end{lemma}\n\\begin{proof}\nThe short exact sequence of groups \n$$\nSp(1) \\times Sp(1) \\xrightarrow{\\iota} \\Spin^c(4) \\xrightarrow{\\delta} S^1,\n$$\nwhere $\\iota(q_1,q_2)=[q_1,q_2,1]$, leads to a fiber sequence \n\\begin{equation}\n\\label{fs}\nBSp(1)\\times BSp(1) \\xrightarrow{B\\iota} B\\Spin^c(4) \\xrightarrow{B\\delta} BS^1.\n\\end{equation}\nLet $A,B \\in H^4(BSp(1) \\times BSp(1))$ be the generators which map to \n$x^2 \\in H^4(BS^1)$ under the maps induced by the\nnatural inclusions $e^{i\\theta} \\mapsto (e^{i\\theta},1)$ and $e^{i\\theta} \\mapsto \n(1,e^{i\\theta})$ of $S^1$ in $Sp(1) \\times Sp(1)$.\nBy the Serre spectral sequence of the fibration \\eqref{fs}, any elements in \n$H^4(B\\Spin^c(4))$ mapping to $A,B$ under $B\\iota^*$ together with $\\alpha$ will\nfreely generate the cohomology ring of $B\\Spin^c(4)$.\n\nThe effect of the composition \n$$ BSp(1) \\times BSp(1) \\xrightarrow{B\\iota} B\\Spin^c(4) \\xrightarrow{B\\pi} BSO(4) $$\non degree $4$ cohomology is determined by the homomorphism\n$$\nS^1 \\times S^1 \\xrightarrow{ \\begin{bmatrix} 1 & -1 \\\\ -1 &  -1 \\end{bmatrix} } S^1 \\times S^1\n$$\ninduced by $\\pi \\circ \\iota$ on the standard maximal tori.\nWriting $a,b \\in H^2(BS^1 \\times BS^1)$ for the standard generators corresponding to the maximal torus of $SO(4)$, and $x,y$ for \nthe corresponding generators for $Sp(1)\\times Sp(1)$, we have\n$$ a^2+b^2 \\mapsto (x-y)^2+ (-x-y)^2= 2x^2 +2y^2, \\quad \\quad ab \\mapsto -x^2+y^2, $$\nand therefore $p_1,\\mathrm{e} \\in H^4(BSO(4))$ map to $2A + 2B$ and $-A+B$ in $H^4(BSp(1)\\times BSp(1))$ respectively. Therefore\n$$ B\\iota^* B\\pi^* (p_1 -2\\mathrm{e}) = 4A, \\quad B\\iota^* B\\pi^* (p_1+2\\mathrm{e}) = 4B.$$\nIt follows from the short exact sequence \n$$\n0 \\to \\mathbb{Z}\\alpha^2 \\to H^4(B\\Spin^c(4)) \\xrightarrow{B\\iota^*} H^4(BSp(1)\\times BSp(1)) \\to 0 \n$$\nthat there exists $\\lambda \\in \\mathbb{Z}$ such that $B\\pi^*(p_1-2\\mathrm{e}) + \\lambda \\alpha^2$\nis (uniquely) divisible by $4$. \n\nThe composition \n$$\n\\mathbb{CP}^2 \\xrightarrow{\\tau_{\\mathbb{CP}^2}} BU(2) \\xrightarrow{B\\kappa} B\\Spin^c(4) \n\\xrightarrow{B\\delta} BS^1\n$$\nclassifying the second exterior power of the tangent bundle of $\\mathbb{CP}^2$ induces multiplication by $3$ on $H^2$, and hence $(B\\kappa \\circ \\tau_{\\mathbb{CP}^2})^*$ provides a $2$--local splitting of the map $B\\delta^*$ on $H^4$.  \n\nThe pullback to $H^4(\\mathbb{CP}^2)$ of \n$B\\pi^*(p_1 - 2\\mathrm{e}) + \\lambda \\alpha^2 \\in H^4(B\\Spin^c(4))$ is\n$(3-6+9\\lambda)$ times the orientation class, and the smallest value of \n$\\lambda$ for which this number is a multiple of $4$ is $\\lambda=-1$. \n\nSince $B\\pi^*(p_1) - 2B\\pi^*(\\mathrm{e}) - \\alpha^2 \\in H^4(B\\Spin^c(4))$ is divisible\nby $4$, we conclude that\n$$\nB\\iota^*\\left( \\frac{ B\\pi^*(p_1) - 2B\\pi^*(\\mathrm{e}) - \\alpha^2 }{4} \\right)=A,\n$$\nand similarly \n$$\nB\\iota^*\\left( \\frac{ B\\pi^*(p_1) + 2B\\pi^*(\\mathrm{e}) - \\alpha^2 }{4} \\right)=B,$$ which completes the proof.\n\\end{proof}\n\nFor four--dimensional $M$, a choice of spin$^c$ structure gives us the following diagram of pullback squares:\n$$\n\\begin{tikzcd}\nZ(M) \\ar[r] \\ar[d] & Z(B\\Spin^c(4)) \\ar[r] \\ar[d] & Z(BSO(4)) \\ar[d] \\\\ M \\ar[r] &\nB\\Spin^c(4) \\ar[r,\"B\\pi\"] & BSO(4) \n\\end{tikzcd}\n$$\nwhere $Z(BG)$ denotes the bundle of orthogonal complex structures on the universal oriented $4$--plane bundle over $BG$ for $G=SO(4)$ and its pullback for $G=\\Spin^c(4)$.\n\nWe will see that each of the two components of $Z(B\\Spin^c(4))$ is the projectivization of a complex plane bundle over $B\\Spin^c(4)$. The projective bundle formula together with naturality will give us the following description of the cohomology ring of $Z(M)$ as an $H^*(M)$-algebra.\n\n\\begin{prop}\n\\label{cohtwist4}\nLet $M$ be an oriented Riemannian four--manifold and $\\alpha \\in H^2(M)$ be an integral\nlift of $w_2(M)$ (classifying the complex line bundle associated to a spin$^c$-structure on $M$). Then there is an isomorphism of $H^*(M)$-algebras\n\\begin{equation}\n\\label{cohtwis4}\nH^*(Z_\\pm(M)) \\cong  H^*(M)[x]\/\\left( x^2 + \\alpha x - \\frac{ p_1 \\pm 2\\mathrm{e} - \\alpha^2}{4} \\right).\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nUnder our identification of $\\mathbb{R}^4$ with $\\mathbb{H}$,\nthe orthogonal complex structures on $\\mathbb{R}^4$ compatible with the orientation (respectively, opposite orientation) are given by right (respectively, left) multiplication by a unit imaginary quaternion. The element $[q_1,q_2] \\in SO(4)$ acts on $J_+(\\mathbb{R}^4)$ by \n$$\n[q_1,q_2] u = q_2 u \\overline q_2,\n$$\nand on $J_-(\\mathbb{R}^4)$ by \n$$\n[q_1,q_2] u = q_1 u \\overline q_1.\n$$\nThat is, $SO(4)$ acts on orthogonal complex structures via the two projections to $SO(3)$.\nThe commutative diagrams\n$$\n\\begin{tikzcd}\n\\Spin^c(4)  \\arrow[d,\"\\pi\"] \\arrow[r,\"\\chi_+\"] &  U(2) \\arrow[d] & &   \\left[q_1,q_2,e^{i\\theta}\\right] \\arrow[d, mapsto] \\arrow[r, mapsto] &  \\left[q_2, e^{-i\\theta}\\right] \\arrow[d, mapsto] \\\\\nSO(4) \\arrow[r, \"\\pi_{\\mathrm{right}}\"] & SO(3) & & \\left[q_1,q_2\\right] \\arrow[r, mapsto] & \\left[q_2\\right]\n\\end{tikzcd}\n$$\nand\n$$\n\\begin{tikzcd}\n\\Spin^c(4)  \\arrow[d,\"\\pi\"] \\arrow[r,\"\\chi_-\"] &  U(2) \\arrow[d] & &   \\left[q_1,q_2,e^{i\\theta}\\right] \\arrow[d, mapsto] \\arrow[r, mapsto] &  \\left[q_1, e^{-i\\theta}\\right] \\arrow[d, mapsto] \\\\\nSO(4) \\arrow[r, \"\\pi_{\\mathrm{left}}\"] & SO(3) & & \\left[q_1,q_2\\right] \\arrow[r, mapsto] & \\left[q_1\\right]\n\\end{tikzcd}\n$$\nallow us to express each component of the bundle of orthogonal complex structures \nassociated to an oriented $4$--plane bundle with a spin$^c$ structure as the projectivization of a complex plane bundle. The formula \\eqref{cohtwis4} will follow from the projective bundle formula once we compute the Chern classes of these plane bundles. By naturality it suffices to compute the images of $c_1, c_2$ under $B\\chi_{\\pm}^*$.\n\nAs the composition $\\Spin^c(4) \\xrightarrow{\\chi_\\pm} U(2) \\xrightarrow{\\det} S^1 $\nequals $\\delta$, we have $B\\chi_{\\pm}^*(c_1) = \\alpha$.\n\nWe will now show that \n$$  \\quad \\quad  B\\chi_+^*(c_2) =-S_2,  \\quad \\quad B\\chi_-^*(c_2)=-S_1.$$\nIn view of \\Cref{cohspin}, this will complete the proof. The maps \n$$\nSp(1) \\times Sp(1) \\xrightarrow{\\iota} \\Spin^c(4) \\xrightarrow{\\chi_\\pm} U(2)\n$$\nare given respectively by $(q_1,q_2) \\mapsto [q_2,1]$ and $(q_1,q_2) \\mapsto [q_1,1]$ (i.e. they correspond respectively to\nthe inclusions of the right and left copies of $SU(2)$ in $U(2)$). Therefore, with respect to the standard basis for $H^4 (BSp(1)\\times BSp(1))$ used in the proof of \\Cref{cohspin}, we\nhave $B(\\iota \\circ \\chi_+)^*c_2 = -B$ and $B(\\iota\\circ \\chi_-)^*c_2 =-A$. \n\nHence \n$$\nB\\chi_+^*(c_2) = -S_2 + \\lambda \\alpha^2, \\quad \\quad B\\chi_-^*(c_2) = -S_2 + \\mu \\alpha^2, \\quad \\text{ for some } \\lambda, \\mu \\in \\mathbb{Z}.\n$$\nWe can determine the coefficients $\\lambda$ and $\\mu$ by mapping to $H^4(\\mathbb{CP}^2)$ under $\\mathbb{CP}^2 \\xrightarrow{B\\kappa \\circ \\tau_{\\mathbb{CP}^2}}  B\\Spin^c(4)$. \nThe compositions\n$$ U(2) \\xrightarrow{\\kappa} \\Spin^c(4) \\xrightarrow{\\chi_{\\pm}} U(2) $$\nare given by \n$$\n[q, e^{i\\theta}] \\mapsto [e^{i\\theta},e^{i\\theta}]  \\quad \\text{ and } \\quad [q, e^{i\\theta}] \\mapsto [q,e^{i\\theta}]\n$$\nrespectively. The second map is the identity, while the first map is the representation $U(2) \\xrightarrow{ 1\\oplus \\det } U(2)$ (cf. \\Cref{identification} (1)).\nIt follows that the composition \n$$ \\mathbb{CP}^2 \\to B\\Spin^c(4) \\xrightarrow{B\\chi_+} BU(2) $$\nclassifies the bundle $\\mathbb{C} \\oplus \\Lambda^2 T\\mathbb{CP}^2$, and \n$$ \\mathbb{CP}^2 \\to B\\Spin^c(4) \\xrightarrow{B\\chi_-} BU(2) $$\nclassifies the tangent bundle of $\\mathbb{CP}^2$. Thus $c_2 \\in H^4(BU(2))$ must go to $0 \\in H^4(\\mathbb{CP}^2)$ under the composition $\\mathbb{CP}^2 \\to B\\Spin^c(4) \\xrightarrow{B\\chi_+} BU(2)$, and to $3\\in H^4(\\mathbb{CP}^2)$ \nunder the composition $\\mathbb{CP}^2 \\to B\\Spin^c(4) \\xrightarrow{B\\chi_-} BU(2)$, i.e. \n\\begin{align*}\n- (3 + 6 - 9)\/4 + 9 \\lambda = 0 \\quad \\Leftrightarrow \\quad \\lambda=0, \\quad \\mathrm{and} \\quad -(3-6-9)\/4 + 9\\mu &=  3 \\quad \\Leftrightarrow \\quad \\mu = 0. \\qedhere  \\end{align*}\n\\end{proof}\n\nRecall that $T^v Z_{\\pm}(M)$ and $T^hZ_{\\pm}(M)$ denote the vertical and horizontal subbundles of $TZ_{\\pm}(M)$ which are complex vector bundles via the Atiyah--Hitchin--Singer almost complex structure. \n\n\\begin{prop}\\label{chernclasses}\nLet $M$ be an oriented Riemannian four--manifold and $\\alpha \\in H^2(M)$ a choice of lift of $w_2(M)$. Then, in terms of the expression for $H^*(Z_\\pm(M))$ in \\Cref{cohtwist4}, the total Chern class of the standard almost complex structure on $Z_\\pm(M)$ is\n\\begin{equation}\n\\label{chernform}\n1 + (4x+2\\alpha) + (p_1 \\pm 3\\mathrm{e}) \\pm (\\alpha + 2x)\\mathrm{e}.\n\\end{equation}\nMoreover, $c_1(T^vZ_\\pm(M))=c_1(T^hZ_\\pm(M))=\\alpha + 2x$.\n\\end{prop}\n\\begin{proof}\nIt suffices to compute the total Chern classes of the vertical and horizontal sub-bundles of $T Z_{\\pm}(M)$. We saw in the proof of \\Cref{cohtwist4} that $Z_\\pm(M)$ is the projectivization\nof a rank two complex vector bundle $E$ with total Chern class \n$$\n1+ \\alpha - \\frac{p_1 \\pm 2\\mathrm{e} - \\alpha^2}{4}.$$\n\nAs $T^v(\\mathbb{P}(E))\\oplus \\mathbb{C} \\cong \\mathcal{O}_{E^*}(1) \\otimes p^*E$ (where $\\mathbb{P}(E) \\xrightarrow{p} M$ is the projection and $\\mathcal{O}_{E^*}(1)$ \nis the canonical line bundle restricting to $\\mathcal{O}(1)$ on each fiber \\cite[Definition 15.13]{D12}) we \nhave that \n$$\nc_1(T^v(Z_\\pm(M))) = c_1(E) + 2 c_1(\\mathcal{O}_{E^*}(1)) = \\alpha + 2x.\n$$\n\nAs for the horizontal bundle, consider the pullback diagrams \n$$\n\\begin{tikzcd}\nZ_+(M) \\ar[r] \\ar[d] & Z_+(B\\Spin^c(4)) \\ar[r,\"\\tau\"] \\ar[d] & Z_+(BSO(4))=BU(2)\n\\ar[d,\"Bi\"] \\\\\nM \\ar[r] & B\\Spin^c(4) \\ar[r,\"B\\pi\"] & BSO(4) \n\\end{tikzcd}\n$$\nNote that $Z_+(BSO(4))= ESO(4) \\times_{SO(4)} SO(4)\/U(2)=BU(2)$, and under this identification the projection $Z_+(BSO(4)) \\to BSO(4)$ is the map $Bi$ induced by the inclusion $U(2) \\subset SO(4)$. Moreover, the universal bundle $ESO(4) \\times_{U(2)} \\mathbb{R}^4 \\to ESO(4)\/U(2)$\npulls back to $T^h Z_+(M)$ under the composite map $Z_+(M) \\to Z_+(B\\Spin^c(4)) \\xrightarrow{\\tau} BU(2)$ (cf. \\Cref{positchern}).\n\nWe saw in the proof of \\Cref{cohtwist4} that\n$$\nH^*(Z_+(B\\Spin^c(4)))\\cong \\mathbb{Z}[\\alpha,S_1,S_2,x]\/(x^2+\\alpha x - S_2).\n$$\nIn terms of this identification we have $\\tau^*(c_2)=B\\pi^*\\mathrm{e} = -S_1+S_2$\nand \n$$\n\\tau^*(c_1) = \\gamma x+\\lambda \\alpha \\text{ for some } \\gamma, \\lambda \\in \\mathbb{Z}.\n$$\nAs the bundle classified by $\\mathbb{CP}^1 \\cong SO(4)\/U(2) \\to BU(2)$ has $c_1$ equal to two times the orientation class (see \\Cref{positchern}), we have $\\gamma=2$.\nSince $p_1\\in H^4(BSO(4))$ maps to $c_1^2-2c_2$ in $H^4(BU(2))$, and to $2S_1+2S_2 + \\alpha^2$ in $H^4(B\\Spin^c(4))$ we see that \n$$\n(2x+\\lambda \\alpha)^2 +2S_1-2S_2 = 2S_1+2S_2+ \\alpha^2  \\quad \\Leftrightarrow \\quad 4x^2 + 4\\lambda \\alpha x + \\lambda^2 \\alpha^2 = 4S_2+\\alpha^2.\n$$\nHence $\\lambda=1$, yielding the formula \\eqref{chernform} in the case of $Z_+(M)$. \n\nLet $k \\in O(4)$ be an element with determinant $-1$ and let \n$\\varphi \\colon SO(4) \\to SO(4)$ denote conjugation by $k$. As a homogeneous space\nof $SO(4)$ we have $J_-(\\mathbb{R}^4)=SO(4)\/\\varphi(U(2))$. Hence\n$Z_-(BSO(4))= ESO(4) \\times_{SO(4)}\\varphi(U(2)) = B(\\varphi(U(2))$ \nand we have a commutative diagram \n$$\n\\begin{tikzcd}\nZ_-(M) \\ar[r] \\ar[d] & Z_-(B\\Spin^c(4)) \\ar[r,\"\\tau'\"] \\ar[d] \\ar[rr,bend left, \"\\tau\"] & B(\\varphi(U(2)) \\ar[r, \"B\\varphi^{-1}\"] \\ar[d] & BU(2) \\ar[d] \\\\\nM \\ar[r] & B\\Spin^c(4) \\ar[r, \"B\\pi\"] & BSO(4) \\ar[r,\"B\\varphi^{-1}\"] & BSO(4) \n\\end{tikzcd}\n$$\nwith the middle and left-hand squares both pullback squares \nand the right hand square an \nisomorphism of fiber bundles induced by the automorphism $\\varphi^{-1}$.\nNote that $B\\varphi^{-1}$ is covered by an isomorphism between the (complex) universal\nbundles $ESO(4) \\times_{\\varphi(U(2))} \\mathbb{R}^4 \\to ESO(4)\\times_{U(2)} \\mathbb{R}^4$,\nso that the composition $Z_-(M) \\to Z_-(B\\Spin^c(4)) \\xrightarrow{\\tau} BU(2)$\nclassifies $T^h Z_-(M)$.\n\nConsidering the action of $B\\varphi$ on the maximal torus we see that $B\\varphi \\colon\nBSO(4) \\to BSO(4)$ has the following effect on cohomology:\n$$B\\varphi^*(p_i)=p_i, \\quad B\\varphi^* (\\mathrm{e}) = -\\mathrm{e}.$$\n\nArguing as before, we conclude that $\\tau^*(c_2)=-B\\pi^*(\\mathrm{e})=S_1-S_2$ and \n$\\tau^*(c_1)=2x+\\alpha$,\nleading to the formula \\eqref{chernform} in the case of $Z_-(M)$.\n\\end{proof}\n\n\\begin{remark}\\label{hitchin} \\begin{itemize}\n\n\\item The formulas in \\Cref{chernclasses} for $c_2$ and $c_3$ of $Z_-(M)$ differ by a sign from those given in \\cite[p.135]{Hi81}; however, the fundamental class being used in loc. cit. seems to also differ by a sign from the one induced by the Atiyah--Hitchin--Singer almost complex structure on $Z_-(M)$. Hence the values of the Chern numbers given in \\cite[(1.5)]{Hi81} coincide with those obtained with the Chern classes in \\Cref{chernclasses}.\n\n\\item In \\cite[(1.4)]{Hi81}, there is a description of the real cohomology ring of the negative twistor space of a four--manifold $X$, as the free $H^*(X;\\mathbb{R})$--module generated by $h=\\frac{1}{2}c_1(T^v Z_-(X))$ subject only to the relation $h^2 = \\tfrac{1}{2}\\mathrm{e}(X) - \\tfrac{1}{4}p_1(X)$. \nHowever, this description is at odds with the identification of the negative twistor space of $\\mathbb{CP}^2$ as the full flag variety $U(3)\/\\left( U(1)\\times U(1) \\times U(1) \\right)$ \\cite[p.133]{Hi81}, \\cite[p.217, Example 1]{Sa84}. Namely, $H^*(\\mathbb{CP}^2;\\mathbb{R}) \\cong \\mathbb{R}[x]\/(x^3)$, where $\\langle x^2, [\\mathbb{CP}^2] \\rangle = 1$, so $\\mathrm{e}(\\mathbb{CP}^2) = p_1(\\mathbb{CP}^2) = 3x^2$. Therefore the real cohomology of the negative twistor space is,\naccording to \\cite[(1.4)]{Hi81}, $\\mathbb{R}[x,h]\/(x^3, h^2 - \\tfrac{3}{4}x^2)$. One can check directly, however, that this (graded) ring is not isomorphic to the cohomology of the flag variety, i.e. to $\\mathbb{R}[x,h]\/(x^2+xh+h^2, x^2 h + h x^2)$. \n\nIn terms of \\Cref{cohtwist4} and \\Cref{chernclasses}, we have $h=x+\\frac{\\alpha}{2}$ and therefore $h^2=\\frac{1}{4} p_1(x) - \\frac{1}{2} \\mathrm{e}(X)$.\n\\end{itemize}\n\\end{remark}\n\n\\begin{remark} \nLet $M$ be an almost complex Riemannian six--manifold.\nThe fiberwise diffeomorphism $a\\colon Z_+(M) \\to Z_-(M)$ sending\n$J_x \\mapsto -J_x \\in Z(T_xM)$ induces an anti-holomorphic map\nbetween the fibers and hence \n$$\na^*(c_1(T_v(Z_-(M)))) = -c_1(T_v(Z_+(M))).\n$$\nThe spaces $Z_\\pm(M)$ are projectivizations of rank 4 complex vector bundles with first Chern class $2c_1(M)$ (see \\Cref{cohomtwis}), and so as in \\Cref{chernclasses} we see that $c_1(T_v Z_\\pm(M))=2c_1(M)+4x$. Therefore \n$$\na^*(2c_1 + 4x) = -2c_1 - 4x \\Rightarrow a^*(4x) = -4x-4c_1.\n$$\nAs there is no torsion in $H^2$ of the universal example $Z(BU(3))$, it follows that $a^*(x)=-x-c_1$. One can check that $a^*$ yields an isomorphism between the $H^*(M)$-algebras of \\Cref{cohomtwis}.\n\\end{remark}\n\nIn view of the previous remark, from now on we will restrict our attention to the positive twistor space of almost complex Riemannian six--manifolds. \n\n\\section{On the homotopy type of the space of almost complex structures on six--manifolds}\n\nIn this section we study the homotopy type of the components of the space of almost complex structures on connected six-manifolds $M$, complementing our previous treatment \\cite{FGM21} of the case when $c_1=0$. \n\n\\begin{thm}\\label{fibration}\nLet $M$ be an oriented Riemannian manifold of dimension four or six, $J$ an orthogonal almost complex structure on $M$ compatible with the orientation, and $J(M)$ its component in the space of orthogonal almost complex structures. Let $S_{\\mathbb{C}}^+(M)$ be the positive spinor bundle on $M$ determined by $J$. Then the map\n$$\n\\Gamma( S(S_\\mathbb{C}^+(M))) \\to J(M)\n$$ \nis surjective, and for any lift $s$ of $J$,\n$$\n\\Map(M,S^1) \\to \\Gamma( S(S_\\mathbb{C}^+))_s \\to J(M)\n$$\nis a fiber sequence.\n\\end{thm}\n\\begin{proof}\nIdentifying $S_\\mathbb{C}^+(M)$ with $\\oplus_{k \\text{ even}} \\Lambda^k_\\mathbb{C}(TM)$, \\Cref{lift} shows that\nthe constant section $s \\colon M \\to \\Lambda^0_\\mathbb{C}(TM) \\subset S_\\mathbb{C}^+(M)$ defined by \n$s(x)=1$ lifts the section $J$. As $S(S_\\mathbb{C}^+) \\to Z_+(M)$ is a fiberwise fibration over $M$, the map induced on sections is a \nfibration. Since the fiber over $J$ is non-empty, any choice of lift of $J$ identifies the\nfiber with $\\Map(M,S^1)$. \n\\end{proof}\n\n\\begin{remark} The above argument remains valid in all dimensions if one replaces $S(S_{\\mathbb{C}}^+(M))$ with $S(S_{\\mathbb{C}}^+(M)) \\cap PS^+_{\\mathbb{C}}(M)$. \\end{remark}\n\n\nNow let $(M,J)$ be a closed almost complex Riemannian six--manifold with $H^1(M;\\mathbb{Z}) = 0$ satisfying $c_1c_2 - c_3 \\neq 0$. By \\Cref{fibration}, we have the fibration $\\Map(M,S^1) \\to \\Gamma( S(S_C^+(M)))_s \\to J(M)$. Since $H^1(M;\\mathbb{Z}) = 0$, evaluation at a chosen basepoint gives a homotopy equivalence $\\Map(M,S^1) \\xrightarrow{ev} S^1$, so we have a principal fiber sequence $S^1 \\to \\Gamma(S(S_C^+(M)))_s \\to J(M)$.\n\nWe can thus consider instead the fiber sequence \\begin{equation}\\label{fibration2} \\Gamma(S(S_{\\mathbb{C}}^+))_s \\to J(M) \\to BS^1 \\simeq \\mathbb{CP}^\\infty. \\end{equation} Now, $S(S_{\\mathbb{C}}^+)$ is an oriented fiber bundle over $M$ with fiber $S^7$, and hence it is classified by a map $M \\to BAut^+(S^7)$ to the classifying space of orientation-preserving homotopy automorphisms of $S^7$. It is known that $BAut^+(S^7)_{\\mathbb{Q}} \\simeq K(\\mathbb{Q}, 8)$, where the subscript of $\\mathbb{Q}$ denotes (Sullivan) rationalization \\cite[\\textsection 11]{S77}. By \\cite[Theorem 5.3]{M87}, the rationalization of $\\Gamma(S(S_{\\mathbb{C}}^+))_s$ is homotopy equivalent to the space of sections of the fiberwise rationalized $S^7$ bundle over $M$. Since $H^8(M;\\mathbb{Q}) = 0$, this latter bundle is trivial, and hence $\\Gamma(S_{\\mathbb{C}}^+)_{\\mathbb{Q}} \\simeq \\Map(M, S^7_{\\mathbb{Q}})$. Denoting the Betti numbers of $M$ by $b_i$, by Thom's theorem on the space of maps into an Eilenberg--Maclane space, the latter has the homotopy type of $S^1_{\\mathbb{Q}} \\times S^7_{\\mathbb{Q}} \\times K(\\mathbb{Q}, 3)^{b_4} \\times K(\\mathbb{Q}, 4)^{b_3} \\times K(\\mathbb{Q},5)^{b_2}$.\n\nWe can describe the fundamental group of $J(M)$ by using a theorem of Crabb and Sutherland:\n\n\\begin{thm}\\cite[Theorem 2.12(i)]{CS84} Let $M$ be a closed connected $2n$--manifold and $\\xi$ a complex $(n+1)$--plane bundle over $X$. Denote by $N\\xi$ any component of the space of sections of $\\mathbb{P}\\xi$ whose elements lift to sections of $\\xi$. Then $\\pi_1(N\\xi)$ is a central extension $$0 \\to \\mathbb{Z}\/\\left(\\langle c_n(\\xi), [M] \\rangle \\right) \\to \\pi_1(N\\xi) \\to H^1(M) \\to 0.$$ \\end{thm}\n\nBy \\Cref{fibration} we can apply this to $\\xi = S_{\\mathbb{C}}^+(M) \\cong \\mathbb{C} \\oplus \\Lambda_{\\mathbb{C}}^2 TM$, which satisfies $N \\xi = J(M)$. Therefore, the fundamental group of $J(M)$ is given by $\\mathbb{Z}$ modulo $\\int_M c_1c_2-c_3$ (where $c_i = c_i(TM)$), see \\Cref{cohomtwis}. By \\cite{M87}, $J(M)$ is a nilpotent space as it is the space of sections of a fibration with nilpotent fiber (namely $\\mathbb{CP}^3$) over a finite--dimensional base. We may thus rationalize the fibration (\\ref{fibration2}) to obtain the fibration $$(\\Gamma(S(S_{\\mathbb{C}}^+))_s)_{\\mathbb{Q}} \\to J(M)_{\\mathbb{Q}} \\to K(\\mathbb{Q}, 2).$$ Consider the degree one rational class corresponding to the factor $S^1_{\\mathbb{Q}}$ in the fiber. If $c_1c_2 - c_3 \\neq 0$, then since $\\pi_1(J(M)_\\mathbb{Q}) = 0$, this class must hit (a non-zero multiple of) the degree two generator in $K(\\mathbb{Q},2)$ in the Serre spectral sequence. This tells us that a model for $J(M)$ is of the form $$(\\Lambda(x_2, z_1, z_7, z_3^{i}, z_4^{j}, z_5^{k}), dz_1 = x_2, dz_7 \\in (x_2), dz_3^i \\in (x_2), dz_4^j \\in (x_2), dz_5^k \\in (x_2)),$$ where $(x_2)$ denotes the ideal generated by $x_2$, and $i,j,k$ range over sets of size $b_4,b_3,b_2$ respectively.\n\nNotice that such a model is not minimal, and in fact a minimal model is obtained by quotienting out the differential ideal generated by $z_1$. Indeed, by the argument in \\cite[Proposition 2]{VS76} we see that the map $(\\Lambda, d) \\to (\\Lambda\/(z_1, dz_1)\\Lambda, \\bar{d})$ induces an isomorphism on cohomology.\nHere by $(\\Lambda,d)$ we denote the differential graded algebra displayed above, and by $\\bar{d}$ the induced differential on the quotient. We see that $$(\\Lambda\/(z_1, x_2)\\Lambda, \\bar{d}) \\cong (\\Lambda(z_7, z_3^{i}, z_4^{j}, z_5^{k}), \\bar{d}=0),$$ where the right-hand side is minimal. To summarize, we have the following:\n\n\\begin{thm}\\label{Qhomotopy6} Let $M$ be a connected closed six--manifold with $b_1 = 0$ equipped with an almost complex structure $J$ such that $\\int_M c_1c_2-c_3 \\neq 0$. Then the space of almost complex structures on $M$ in the component of $J$ is a nilpotent space with finite cyclic fundamental group and minimal model given by $(\\Lambda(z_7, z_3^{i}, z_4^{j}, z_5^{k}), d = 0)$; here $i,j,k$ range over sets of size $b_4(M),b_3(M),b_2(M)$ ($=b_4(M)$) respectively. In particular, this space is formal. \\qed \\end{thm}\n\nWe emphasize that the above makes no nilpotency assumption on $M$. Alternatively, a model for (a given component of) the space of almost complex structures can be obtained with the Haefliger--Sullivan model for the space of sections of a fibration \\cite[\\textsection 11]{S77}, \\cite{H82}, applied to $\\mathbb{CP}^3 \\to Z_+(M) \\to M$. Our approach above lets one quickly describe the minimal model of $J(M)$ using the geometry of the setup. \n\n\n\\begin{example}\\label{examplesQhomotopy}  Using \\Cref{Qhomotopy6}, we immediately obtain the following:\n\n\\begin{enumerate}\n\\item Observe that the space of almost complex structures inducing a given orientation on the connected sum $g(S^3 \\times S^3)$ is connected. By \\Cref{Qhomotopy6}, for $g \\neq 1$, the rationalization of this space is $K(\\mathbb{Q}, 7) \\times K(\\mathbb{Q}, 4)^{2g}$. In particular, the space of almost complex structures on $S^6$ has the same rational homotopy type as $S^7$, i.e. $K(\\mathbb{Q},7)$; cf. \\cite{FGM21} where it is shown that a certain natural inclusion $\\mathbb{RP}^7 \\hookrightarrow J(S^6)$ induces an isomorphism on rational homotopy groups and fundamental groups. (Note that the covering $S^7 \\to \\mathbb{RP}^7$ induces an isomorphism on rational homotopy groups.) \n\nFor $g=1$, one can calculate directly using the Haefliger--Sullivan model that the space of almost complex structures on $S^3 \\times S^3$ has the same rational homotopy type as $S^1 \\times \\mathbb{CP}^3 \\times (\\mathbb{HP}^\\infty)^2$, cf. \\cite[Example 5.4]{FGM21}.\n\n\\item The components of almost complex structures on $\\mathbb{CP}^3$ are parametrized by $c_1 = 2kx$, $c_2 = (2k^2-2)x^2$, where $x$ is the generator of $H^2(\\mathbb{CP}^3)$ such that $\\langle x^3, [\\mathbb{CP}^3]\\rangle = 1$ (i.e. $x = c_1(\\mathcal{O}(1))$ for the standard complex structure). Hence $c_1c_2 - c_3 = 4k^3 - 4k - 4$, which is never zero. Hence every component of almost complex structures has rationalization $K(\\mathbb{Q},7) \\times K(\\mathbb{Q},5) \\times K(\\mathbb{Q},3)$. Contrast this with the trivial $\\mathbb{CP}^3$ bundle over $\\mathbb{CP}^3$, whose (infinitely many) components of sections exhibit two distinct rational homotopy types \\cite[Example 3.4]{MR85}; the space of maps $\\mathbb{CP}^3 \\to \\mathbb{CP}^3$ homotopic to any fixed essential map has rationalization $K(\\mathbb{Q},7)\\times K(\\mathbb{Q},5)\\times K(\\mathbb{Q},3)$.  \\end{enumerate} \\end{example}\n\n\n\n\n\n\n\\section{Intersections of almost complex structures on six--manifolds as sections of the twistor space}\n\nWe will now use the results of Section 3 to calculate the homological intersection of two almost complex structures on a given six--manifold $M$, after identifying them with the image of $M$ under the corresponding section of the twistor bundle.\n\n\n\n\\begin{lemma}\\label{poincaredual}\nLet $E \\xrightarrow{\\pi} M$ be a complex vector bundle of rank $n$ over a closed oriented manifold $M$. Let $s \\colon M \\to \\mathbb{P}(E \\oplus \\mathbb{C})$ denote the canonical section given pointwise by $[0: \\cdots : 0 : 1]$.\nThen the Poincar\\'e dual of $s_*[M]$ in  $H^*(\\mathbb{P}(E\\oplus \\mathbb{C})) \\cong  H^*(M)[x]\/(x^{n+1} + c_1 x^n + \\cdots + c_n x)$ is\n$$\nx^n + c_1x^{n-1} + \\cdots + c_n,\n$$\nwhere $c_i$ denote the Chern classes of $E$.\\footnote{Here the orientation on $\\mathbb{P}(E\\oplus \\mathbb{C})$ is understood to be the one induced by the orientation on $M$ and the canonical orientation on the fiber $\\mathbb{CP}^n$ corresponding to $\\langle x^n, [\\mathbb{CP}^n] \\rangle = 1$.}\n\\end{lemma}\n\\begin{proof}\nThe Poincar\\'e dual to $s_*[M]$ is the image of the Thom class of the normal bundle to the submanifold $s(M) \\subset \\mathbb{P}(E \\oplus \\mathbb{C})$ under the first map below in the long exact sequence of a pair,\n$$\nH^{2n}(\\mathbb{P}(E \\oplus \\mathbb{C}), \\mathbb{P}(E \\oplus \\mathbb{C}) \\setminus s(M)) \\to H^{2n}(\\mathbb{P}(E \\oplus \\mathbb{C})) \\to H^{2n} (\\mathbb{P}(E \\oplus \\mathbb{C}) \\setminus s(M)),\n$$\n\nwhere $H^{2n}(\\mathbb{P}(E \\oplus \\mathbb{C}), \\mathbb{P}(E \\oplus \\mathbb{C}) \\setminus s(M))$ has been identified with $H^{2n}(U, U \\setminus s(M))$ for some tubular neighborhood $U$ of $s(M)$ via excision; see \\cite[\\textsection 6.11]{Br}. This is a non-zero class, as $s_*[M]$ is non-zero since it projects to $[M]$.  \n\nNow notice that $\\mathbb{P}(E \\oplus \\mathbb{C}) \\setminus s(M)$ deformation retracts to $\\mathbb{P}(E)$. Therefore the effect of the second map on cohomology is the natural quotient\n$$  H^{2n}(M)[x]\/(x^{n+1} + c_1 x^n + \\cdots + c_n x)  \\to H^{2n}(M)[x]\/(x^n + c_1 x^{n-1} + \\cdots + c_n), $$\nand thus the kernel is generated by $x^n + c_1x^{n-1} + \\cdots + c_n$. Hence the Thom class maps to $x^n + c_1x^{n-1} + \\cdots + c_n$ up to sign. To check the sign, we restrict to a fiber and evaluate against the fundamental class $[\\mathbb{CP}^n]$. Since the Thom class is the orientation of the normal bundle, which is given by the orientation of the fibers, and $\\langle x^n, [\\mathbb{CP}^n] \\rangle = 1$, we see that the sign is positive. \\end{proof}\n\n\n\\begin{thm}\\label{selfintersection}\n\nLet $s$ be a section of the positive twistor bundle over a closed Riemannian six--manifold $M$, and give $M$ the orientation determined by $s$. Then the homological self-intersection number $s_*[M] \\cdot s_*[M]$ of $s(M)$ in $Z_+(M)$ is given by $\\int_M c_1c_2 - c_3$, where the $c_i$ are the Chern classes of the almost complex structure determined by $s$.\n\n\\end{thm}\n\n\\begin{proof}  In terms of the identification in \\Cref{identification}, $s$ is the canonical section of $\\mathbb{P}(S_{\\mathbb{C}}^+) = \\mathbb{P}(\\Lambda_\\mathbb{C}^2 TM \\oplus \\mathbb{C})$ (see the proof of \\Cref{fibration}), where $TM$ is equipped with the complex structure determined by $s$. To compute the homological self-intersection of $s(M)$, we integrate the square of the expression obtained in \\Cref{poincaredual} for its Poincar\\'e dual in $Z_+(M)$. We use \\Cref{cohomtwis}, and observe that since $x(x^3 + c_1(S_{\\mathbb{C}}^+)x^2 + c_2(S_{\\mathbb{C}}^+)x + c_3(S_{\\mathbb{C}}^+)) = 0$ in $H^*(Z_+(M))$, we have \\begin{align*} (x^3 + c_1(S_{\\mathbb{C}}^+)x^2 + c_2(S_{\\mathbb{C}}^+)x + c_3(S_{\\mathbb{C}}^+))^2 &= c_3(S_{\\mathbb{C}}^+)(x^3 + c_1(S_{\\mathbb{C}}^+)x^2 + c_2(S_{\\mathbb{C}}^+)x + c_3(S_{\\mathbb{C}}^+)) \\\\ &= c_3(S_{\\mathbb{C}}^+) x^3 = (c_1c_2 - c_3)x^3. \\qedhere \\end{align*} \\end{proof} \n\nUsing \\Cref{selfintersection}, we obtain the following:\n\n\\begin{example} \n\\label{intersectiontwistor4d} \\begin{enumerate} \\item\nFor any closed oriented Riemannian four--manifold $M$, by \\Cref{chernclasses}, we have the following Chern numbers of the Atiyah--Hitchin--Singer almost complex structure on $Z_+(M)$:\n\\begin{align*} \n\\int_{Z_+(M)} c_1(Z_+(M))c_2(Z_+(M)) &= \\int_{Z_+(M)} (4x+2\\alpha)(p_1+3\\mathrm{e}) \\\\ &= \\left( \\int_{\\mathbb{CP}^1}4x \\right)\\left( \\int_{M} (p_1(M)+3\\mathrm{e}(M)) \\right) = 12(\\sigma(M)+\\chi(M)), \\\\\n\\int_{Z_+(M)} c_3(Z_+(M)) & = \\int_{Z_+(M)} (2x+\\alpha)\\mathrm{e} = 2 \\chi(M).\n\\end{align*}\n\nHence the self-intersection of the Atiyah--Hitchin--Singer almost complex structure on $Z_+(M)$ in $Z_+(Z_+(M))$ is $12\\sigma(M) + 10\\chi(M)$. \n\nBy \\cite[Lemma 3]{Ar97}, closed oriented four--manifolds admitting a maximally non-integrable almost complex structure satisfy $5\\chi + 6\\sigma = 0$. Hence for these manifolds, the corresponding self-intersection number is zero.\n\n\\item By a theorem of Taubes \\cite[Theorem 1.1]{T92}, given any closed oriented four--manifold $M$, for every large enough $k$ the manifold $N = \\overline{M} \\# k \\mathbb{CP}^2$ carries a self-dual metric. Hence its negative twistor space $X = Z_-(N)$ is a complex manifold \\cite[Theorem 4.1]{AHS78}. It satisfies \\begin{align*} \\int_X c_1(X)c_2(X) = 12(\\chi(N) - \\sigma(N)), \\quad \\int_X c_3(X) = 2\\chi(N).\\end{align*} \n\nIndeed, we obtain this again from \\Cref{chernclasses}; namely \\begin{align*} \\int_X c_1(X)c_2(X) &= \\int_X 2(\\alpha +2x)(p_1(N)-3\\mathrm{e}(N)) = \\left( \\int_{\\mathbb{CP}^1}4x \\right) \\left( \\int_{\\overline{N}} (p_1(N)-3\\mathrm{e}(N)) \\right) \\\\ &= -12(\\sigma(N)-\\chi(N)).\\end{align*} This agrees with  \\cite[(1.5)]{Hi81}. Alternatively, from \\cite[Section 4]{EaSi93} we see that the holomorphic Euler characteristic of $X$ is given by $1 - b_1(N) + b_2^+(N)$. Since $\\chi(N) = 2+b_2^+(N) +b_2^-(N) - 2b_1(N)$ and $\\sigma(N) = b_2^+(N) - b_2^-(N)$, we have $\\tfrac{1}{2}(\\chi(N) - \\sigma(N)) = 1 - b_1+b_2^-$, which equals the Todd genus $\\int_X \\tfrac{1}{24}c_1c_2(X)$. The Euler characteristic of $X$ is twice that of $N$, as seen immediately e.g. using the Leray--Hirsch theorem.\n\nFrom here we obtain $$\\int_X c_1(X)c_2(X) - c_3(X) = 10\\chi(M) + 12\\sigma(M) - 2k.$$ Hence for any $M$, for large enough $k$ we obtain compact complex threefolds $X$ with negative homological self-intersection in their own positive twistor space. \n\\end{enumerate}\n\\end{example}\n\n\n\\begin{example}\\label{exampleselfintersection}\n\\begin{enumerate}\n\\item For the connected sum $g(S^3 \\times S^3)$, the self-intersection number of any almost complex structure in $Z_+(g(S^3 \\times S^3))$ is $2g-2$. In particular, for $S^6$ we have $-2$.\n\\item $S^2 \\times S^4$ admits a unique homotopy class of almost complex structures for every choice of $c_1 = 2k\\alpha$, where $\\alpha$ generates $H^2$. Since $p_1 = 0$ we see that $c_2 = 0$ for any $k \\in \\mathbb{Z}$, and so the self-intersection is $-4$ for any homotopy class of almost complex structure. Likewise, for $g\\left( S^2 \\times S^4 \\right)$, the self-intersection of any almost complex structure is $-2g-2$.\n\\item Recall that the homotopy classes of almost complex structures on $\\mathbb{CP}^3$ are parametrized by $c_1 = 2kx$, $c_2 = (2k^2-2)x^2$. The self-intersection numbers are given by $4k^3 - 4k - 4$; in particular, for the standard complex structure it is 20.\n\n\\item To obtain more examples of compact complex threefolds with negative self-intersection, one can blow up any given compact complex threefold $X$ at a point sufficiently many times. Indeed, the blowup at one point, diffeomorphic to $X \\# \\overline{\\mathbb{CP}^3}$, satisfies $\\int_{X \\# \\overline{\\mathbb{CP}^3}} c_1c_2(X \\# \\overline{\\mathbb{CP}^3}) = \\int_X c_1c_2(X)$ by invariance of the Todd genus under bimeromorphisms, and the Euler characteristic is two larger than that of $X$. Note that if $X$ is K\\\"ahler, this gives examples of K\\\"ahler threefolds with negative self-intersection.\n\\item In \\cite{Le99}, LeBrun builds examples of compact complex threefolds $X_m$, $m>0$, each diffeomorphic to $K3\\times S^2$, with $\\int_{X_m} c_1c_2(X_m) = 48m$. Hence the homological self-intersection number is given by $48(m-1)$.\n\\item By \\cite[Corollary 3.1]{GS14}, for a connected symplectic six--manifold $M$ with a Hamiltonian circle action with isolated fixed points, we have $\\int_M c_1c_2 = 24$ and so $\\int_M c_1c_2 - c_3 = 24-\\chi(M)$. In particular, this formula applies to toric six--manifolds.\n\n\\end{enumerate} \\end{example}\n \nRecall the following theorem of Michelsohn, Salamon, Atiyah--Hitchin--Singer:\n\n\\begin{thm}(\\cite[Theorem 9.11]{LM}\\label{sectiontheorem}, \\cite[Theorem 1]{Sa95}) A section $s$ of the positive twistor space $Z_+(M) \\to M$ is pseudoholomorphic, with respect to the almost complex structure $J$ corresponding to $s$ on $M$ and the Atiyah--Hitchin--Singer almost complex structure on $Z_+(M)$, if and only if $J$ is integrable. \\end{thm}\n\nNow suppose we have an integrable orthogonal $J$ on a Riemannian six--manifold $M$. Then $s$ gives us an embedding of $M$ into $Z_+(M)$ as an almost complex submanifold. If we were able to perturb $s(M)$ to another almost complex submanifold that intersects $s(M)$ transversally, then the homological intersection number $s_*[M] \\cdot s_*[M]$ would have to be $\\geq 0$. A way to obtain perturbations of $s(M)$ to almost complex submanifolds is by translating $J$ by an isometry of $(M,g)$, but any perturbation to a transverse representative would do. Note that it is unique to dimension six that $s_*[M] \\cdot s_*[M]$ inside $Z_+(M)$ gives an integer. \n\n\\begin{cor}\\label{deform} Suppose a closed Riemannian six--manifold $M$ carries an integrable complex structure, corresponding to a section $s$ of $Z_+(M)$, with $\\int_M c_1c_2 - c_3 < 0$. Then there is no representative of the homology class $s_*[M]$ given by an almost complex submanifold of $Z_+(M)$ which intersects $s(M)$ transversally. \\end{cor}\n\n\\begin{example} Consider $S^6$ with the round metric. Take the canonical almost complex structure induced by the octonions, and consider its orbit under the action of the isometry group $SO(7)$, diffeomorphic to $\\mathbb{RP}^7$. After identifying the twistor space of round $S^6$ with the Grassmannian of oriented 2--planes in $\\mathbb{R}^8$, Calabi and Gluck \\cite{CG} show that these ``octonion'' $J$'s send $S^6$ to the family of 2--planes of the form $(o, v)$, where $o$ is a fixed unit octonion depending on $J$ (so, the image of $S^6$ is the unit sphere in the hyperplane orthogonal to a given line in $\\mathbb{R}^8$.) Two such nearby families intersect transversally. Therefore, the $-2$ homological self-intersection obtained in \\Cref{exampleselfintersection} shows the octonion $J$'s are not integrable. Of course, this also follows by a direct computation of the Nijenhuis tensor, and the result is a special case of the classical theorem of Blanchard \\cite{Bl53} and LeBrun \\cite{Le87} that there is no integrable complex structure on $S^6$ orthogonal with respect to the round metric.  \\end{example}\n\nIn order to study intersections of almost complex structures in different components of the space of sections of the twistor space we will now analyze the dependence of our formula for the cohomology (\\Cref{cohomtwis}) on the choice of almost complex structure. \n\nConsider the map of fibrations\n\\begin{equation}\n\\label{comparison}\n\\begin{tikzcd}\n\\mathbb{CP}^3 \\ar[r,\"j\"] \\ar[d] & \\mathbb{CP}^\\infty \\ar[d] \\\\\nBU(3) \\ar[r,\"B\\kappa\"] \\ar[d] & B\\Spin^c(6)\\ar[d] \\\\\nBSO(6) \\ar[r,\"=\"] & BSO(6) \n\\end{tikzcd}\n\\end{equation}\nAs the canonical map $\\kappa \\colon U(3) \\to \\Spin^c(6)$ is an isomorphism on $\\pi_1$, we see \nthat $j$ is an isomorphism on $\\pi_2$ and hence the $5$--lemma implies that $B\\kappa$ is a \n$7$--equivalence. Therefore the set of isomorphism classes of almost complex structures on six--manifolds is in natural bijective correspondence with the set of isomorphism classes of spin$^c$ structures via the canonical map from the former to the latter.\n\nThe set of isomorphism classes of spin$^c$-structures on a manifold $M$ is a torsor over the group $H^2(M)$\nof isomorphism classes of complex line bundles \n(corresponding to the action of the fiber on\nthe principal fibration along the right-hand column of \\eqref{comparison}). Letting \n$[s]$ denote the spin$^c$ structure associated to the almost complex structure $s\\colon \nM \\to Z_+(M)$, and writing $a \\cdot [s]$ for the action of the cohomology class \n$a \\in H^2(M)$ on $[s]$, we have the following result:\n\n\\begin{lemma}\nLet $M$ be an oriented Riemannian six--manifold and let $s,s'\\colon M \\to Z_+(M)$ be \ntwo almost complex structures on $M$ compatible with the orientation of $M$. \nLet $a \\in H^2(M)$ be (the unique cohomology class) such that $[s']=a\\cdot [s]$.\nThen the canonical isomorphism between the expressions for $H^*(Z_+(M))$ given in \\Cref{cohomtwis} in terms of $s,s'$ is given by\n$$\nx' \\mapsto x - a. \n$$\n\\end{lemma}\n\\begin{proof}\nLet $E$ and $E'$ be the positive spinor bundles associated to the spin$^c$ structures\n$[s]$ and $[s']$ respectively, and let $L$ denote a complex line bundle with $c_1(L)=a$.\n\nThe relation $a \\cdot [s]=[s']$ implies that there is an isomorphism \n$$\\psi \\colon L \\otimes E \\to E'.$$\nThe kernel in $T_x M \\otimes \\mathbb{C}$ of right multiplication by a positive spinor \n$v \\in E_x$ is the same as the kernel of right multiplication by $\\lambda \\otimes v \\in \nL_x \\otimes E_x$ for any $\\lambda \\neq 0$, so the canonical isomorphism given by the \ncomposition\n$$\n\\mathbb{P}(L\\otimes E) \\xrightarrow{\\phi'} Z_+(M) \\xrightarrow{\\phi^{-1}} \\mathbb{P}(E)\n$$\nis the obvious one sending the line $L \\otimes (\\mathbb{C} v)$ to $\\mathbb{C} v$. \n\nWe conclude that on $Z_+(M)$ the tautological (Hopf) line bundles $H$ and $H'$ coming \nrespectively from the identifications with $\\mathbb{P}(E)$ and $\\mathbb{P}(E')$ satisfy\nthe relation \n$$\nH' \\cong H \\otimes L.\n$$\nSince $x'=c_1(H'^*)$ and $x=c_1(H^*)$, this completes the proof.\n\\end{proof}\n\nGiven a spin$^c$ structure $[s]$ we will write $c_1([s])$ for the Chern class of \nthe line bundle classified by the projection $B\\Spin^c(2n) \\xrightarrow{B\\delta} BS^1$. \nNote that $c_1( a\\cdot [s]) = 2a + c_1([s])$ so the difference class $a \\in H^2(M)$ \nbetween two almost complex structures $s,s'$ is a specific ``square root'' of the \ndifference $c_1'-c_1$. We can suggestively write \n\\begin{equation}\n\\label{div2}\nx' = x + \\frac{c_1-c_1'}{2},\n\\end{equation}\nwhich is unambiguous on rational cohomology (or when there is no $2$--torsion in $H^2(M)$).\n\n\\begin{remark}\nIf we use $s\\colon M \\to Z_+(M)$ to write \n$$\nH^*(Z_+(M)) = H^*(M)[x]\/(x^4 + 2c_1 x^3 + (c_1^2+c_2)x^2 + (c_1c_2-c_3) x),\n$$\nthen $s^*(x)=0$. This follows from \\Cref{lift} which implies that \nthe pullback by $s$ of the Hopf bundle over $\\mathbb P(E)$ is trivial. Using the isomorphism $H^2(Z_+(M)) \\xrightarrow{i^*\\oplus s^*} H^2(\\mathbb{CP}^3) \\oplus H^2(M)$ one sees that the first Chern class of the horizontal\nplane bundle on $Z_+(M)$ is\\footnote{This computation also follows from the arguments used in the proof of \\Cref{chernclasses}.} $c_1(M)+2x$ (for any choice of almost complex structure) leading again to \\eqref{div2}.\n\\end{remark}\n\n\n\\iffalse\nBefore considering intersections, we first comment on the various possible presentations of this cohomology. First we record the following result:\n\n\\begin{prop}\\label{pullbackx}\nLet $M$ be a Riemannian oriented six--manifold and $Z_+(M) \\to M$ be its positive twistor space. Let $s \\colon M \\to Z_+(M)$ be an almost \ncomplex structure and use it to write \n$$\nH^*(Z_+(M)) = H^*(M)[x]\/(x^4 + 2c_1 x^3 + (c_1^2+c_2)x^2 + (c_1c_2-c_3) x).\n$$\nThen $s^*(x)=0$.\n\\end{prop}\n\\begin{proof}\nIn the projective bundle formula, $x$ is $c_1$ of the dual tautological line bundle over the projectivized positive spinors. Its pullback under $s$\nis $c_1$ of the line bundle on $M$ whose fiber over $x$ is the dual of the line of positive spinors determined by the almost complex \nstructure on $x$. It follows from \\Cref{lift} that this line bundle is trivial (this is why $s$ always lifts to a section of the positive spinor bundle).\n\\end{proof}\n\nConsider the pullback square \n$$ \\begin{tikzcd}\n\\mathbb{CP}^3 \\arrow[r, \"=\"] \\arrow[d, \"i\"']                                                           & \\mathbb{CP}^3 \\arrow[d, \"j\"] \\\\\nZ_+(M) \\arrow[d, \"\\pi\"'] \\arrow[r, \"\\tau\"]                                                          & BU(3) \\arrow[d]      \\\\\nM \\arrow[r] \\arrow[ru, \"\\tilde{s}\"'] \\arrow[u, \"s\" description, shift right=2] & BSO(6)              \n\\end{tikzcd} $$\n\ngiven by a choice of almost complex structure on $M$.\nLet us write $C_1$ for the first Chern class in $H^2(BU(3))$ to distinguish it from the Chern class $c_1$ of the almost complex structure on $M$\ndetermined by the lift $\\tilde{s} = fs$. Let $\\hat{x}$ denote the class $c_1(\\mathcal{O}(1)) \\in H^2(\\mathbb{CP}^3)$. Recall that $\\tau$ classifies the horizontal bundle on $Z_+(M)$, see the proof of \\Cref{cohomtwis}.\n\nWe show that $\\tau^*(C_1) = c_1 + 2x$; we emphasize that $x$ depends on the chosen almost complex structure on $M$. Notice that $i^*x = \\hat{x}$ and $i^*\\tau^*(C_1) = j^*(C_1) = 2\\hat{x}$ by \\Cref{positchern}. We have $\\tau^*(C_1) = 2x + \\pi^*(b)$ for some $b \\in H^2(M)$\nand $s^*\\tau^*(C_1) = \\tilde{s}^*C_1 = c_1$. Hence $c_1 = s^*\\tau^*(C_1) = s^*(2x) + s^*\\pi^*(b)$. By \\Cref{pullbackx}, we have $s^*(x) = 0$ and hence $b = c_1$, giving us $\\tau^*(C_1) = 2x + c_1$ (here, as usual, we write $c_1$ instead of $\\pi^*(c_1)$). \n\n\nSince this is true for any choice of almost complex structure, given another choice $s'$ we must have $c_1 + 2x = \\tau^*(C_1) = c_1' + 2x'$, where $x'$ is the class analogous to $x$ for the almost complex structure determined by $s'$. Hence, $2(x'-x) = c_1 - c_1'$, which we choose to capture by denoting $x' - x$ by $\\tfrac{1}{2}(c_1-c_1')$ (note that on its own, the latter expression is not necessarily well-defined, as there might be two--torsion in the cohomology). This allows us to suggestively write\n\\begin{equation}\\label{x'x}\nx' = x + \\tfrac{1}{2}(c_1 - c_1').\n\\end{equation}\n\nDenoting for simplicity $A = x'-x = \\tfrac{1}{2}(c_1-c_1')$, by \\Cref{cohomtwis} the formula for $H^*(Z_+(M))$ can be written in terms of the new generator $x'=x+A$ as\n$$\nH^*(M)[x']\/((x'-A)^4 + 2c_1(x'-A)^3 + (c_1^2+ c_2)(x'-A)^2 + (c_1c_2 - c_3)(x'-A).\n$$\nOn the other hand, the cohomology must equal\n$$\nH^*(M)[x']\/((x')^4+ 2c_1' (x')^3 + (c_1'^2 + c_2') (x')^2 + (c_1' c_2' -c_3') x' .\n$$\n\nLooking at the equalities between the coefficients of powers of $x'$ in the expressions above gives us $A = \\frac 1 2 (c_1-c_1')$ (true by definition), $c_1^2 - 2c_2 = c_1'^2 - 2c_2'$ (both equal the first Pontryagin class of $M$), and $c_3 = c_3'$ (both equal the Euler class).\n\n\\begin{remark}\nTwo different almost complex structures give rise to different positive spinor bundles with the same projectivization, as we have been using above. This is explained on the level of structure groups by the commutativity of the following diagram (given the formula obtained in \\Cref{plusmin} for the positive spinor bundle):\n$$\n\\xymatrix{\nU(3)  \\ar[r]^{\\Lambda^2 \\oplus \\Id} \\ar[d] & U(4) \\ar[d] \\\\\nSO(6) \\ar[r] &  PSO(6) \\cong PU(4)\n}\n$$\nwhere the identification on the bottom-right comes from the exceptional isomorphism $SU(4)\\cong \\Spin(6)$. \n\\end{remark}\n\\fi \n\n\\begin{thm}\\label{intersection} Let $J$ and $J'$ be two orthogonal almost complex structures inducing the same orientation on a closed Riemannian six--manifold $M$, corresponding to sections $s$ and $s'$ of the twistor bundle. Denote by $c_i, c_i'$ their respective Chern classes. Then the intersection number $s_*[M] \\cdot s'_*[M]$ in $Z_+(M)$ is given by $$\\int_M \\tfrac{1}{8}\\left( c_1^3 + c_1^2 c_1' - c_1c_1'^2 - c_1'^3 \\right) + \\tfrac{1}{2} \\left( c_1 c_2' + c_1' c_2' \\right) - c_3.$$\n\\end{thm}\n\n\\begin{proof}\nBy \\Cref{cohomtwis}, the presentation of $H^*(Z_+(M))$ with reference to $J$ is given by $$H^*(M)[x]\/(x^4 + 2c_1x^3 + (c_1^2+c_2)x^2 + (c_1c_2-c_3)x),$$ and with reference to $J'$ it is given by $$H^*(M)[x']\/((x')^4 + 2c_1'(x')^3 + ((c_1')^2 + c_2')(x')^2 + (c_1'c_2'-c_3')x').$$ Combining \\Cref{cohomtwis} with \\Cref{poincaredual}, we have that the Poincar\\'e dual of $s_*[M]$ is given by $$x^3 + 2c_1x^2 + (c_1^2+c_2)x + (c_1c_2-c_3),$$ and the Poincar\\'e dual of $s'_*[M]$ is given by $$(x')^3 + 2c_1'(x')^2 + ((c_1')^2 + c_2')x' + c_1'c_2'-c_3'.$$ Recall from \\eqref{div2} that $x' = x + \\tfrac{1}{2}(c_1 - c_1')$. (Our calculation will be insensitive to torsion, so we may in fact work in the rational cohomology ring where there is no ambiguity in writing $\\tfrac{1}{2}(c_1 - c_1')$.) Using this to rewrite the expression for $s'_*[M]$, and multiplying with $s_*[M]$, yields $$x^3\\left( \\tfrac{1}{8}\\left( c_1^3 + c_1^2 c_1' - c_1c_1'^2 - c_1'^3 \\right) + \\tfrac{1}{2} \\left( c_1 c_2' + c_1' c_2' \\right) - c_3' \\right).$$ Pairing with the fundamental class of $Z_+(M)$ gives the result. \\end{proof}\n\n\n\\begin{remark} The expression for the intersection number in \\Cref{intersection} is symmetric in $J$ and $J'$ as expected, which can be verified by using $c_1^2 - 2c_2 = p_1 = c_1'^2 - 2c_2'$ and $c_3 = c_3'$. \\end{remark}\n\n\\begin{example} Let $J$ and $J'$ be two almost complex structures on $\\mathbb{CP}^3$, with Chern classes $c_i$ and $c_i'$. Then $c_1 = 2kx, c_2 = (2k^2 - 2)x^2, c_3 = 4x^3$ and $c_1' = 2\\ell x, c_2' = (2\\ell^2 - 2)x^2, c_3' = 4x^3$, where $k, \\ell$ are integers. From \\Cref{intersection} we get that $J$ and $J'$, thought of as sections of the twistor bundle, have homological self-intersection given by $$k^3 + k^2 \\ell + k \\ell^2 + \\ell^3 - 2k - 2\\ell - 4.$$ In particular, taking $J$ to be the standard complex structure, lying in the component determined by $k = 2$, we have that the homological intersection with any $J'$ with $\\ell < 0$ is negative.\n\\end{example}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPresently, there exists an interest in the study of the effects of the spin in\nthe trajectory of test particles in rotating gravitational fields. The\nimportance of this topic increases when dealing with phenomena of astrophysics\nsuch as accretion discs in rotating black holes \\cite{tanaka 1996},\nGravitomagnetics Effects \\cite{faruque2004} or gravitational waves induced by\nspinning particles orbiting a rotating black hole \\cite{mino}. Therefore, we\nshall work the equations of motion for test particles in a weak Kerr metric\nwhich will be integrated numerically in the particular case when the test\nparticles are orbiting circularlly with the purpose of studying the effects of\nspin in the trajectories of test particles in rotating gravitational fields.\n\nThe motion of particles in a gravitational field is given by the geodesics\nequation. The solution to this equation depends on the problem, and therefore\nthere are different methods for its solution \\cite{carter} \\cite{abramowicz}.\nBasically, we take two cases in motion of test particles in a gravitational\nfield of a rotating mass. The first case describes the trajectory of a\nspinless test particle, and the second one the trajectory of a spinning test\nparticle in a weak Kerr metric. For the second case a representation is used\nthat does not include the third-order derivatives of the coordinates, and\nyields the equations of motion for a spinning test particle in a gravitational\nfield without any restrictions on its velocity and spin orientations\n\\cite{plyatsko}.\n\nFor the first case, authors as Tanaka \\textit{et al. } \\cite{tanaka 1996}\nyield the set of equations of motion for orbiting spinless test particles. In\nthis case the equations of motion for the spinless test particle are\nconsidered both in the equatorial \\cite{bardeen}, \\cite{wilkins},\n\\cite{calvani}, and the non-equatorial plane \\cite{wilkins}, \\cite{stog},\n\\cite{teo} (Kheng, L., Perng, S., Sze Jackson, T.: Massive Particle Orbits\nAround Kerr Black Holes. Unpublished).\n\nFor the study of test particles in a rotating field, some authors have solved\nthe equations of motion for spinless and spinning test particles in the\nparticular case of circular orbits in the equatorial plane of a Kerr metric\n\\cite{tanaka 1996}, \\cite{bardeen}, \\cite{suzuki}, \\cite{mash}, \\cite{dadhich\n, \\cite{bini 2005}, \\cite{tod}, \\cite{tartaglia}. In addition, Plyastko, R.\n\\textit{et. al. }yield the full set of Mathisson-Papapetrou-Dixon Equations\n(MPD equations) for spinning test particles in the Kerr gravitational field\n\\cite{plyatsko}. These authors integrate numerically the MPD equations for the\ncase of the Schwarzschild metric. In this paper, we use the method of MPD\nEquations given by Plyastko, R. \\textit{et. al. }for calculating the\ntrajectories of spinless and spinning test particles in equatorial planes for\ncircular orbits, i.e., constant radius in a weak Kerr metric. In the\nliterature, there are not works that study via MPD equations the trajectories\nof spining test particles in weak fields.\n\nWith the purpose to prove the equations of motion that we worked, we shall\nsolve numerically the set of equations of motion obtained via MPD Equations in\nthe case when the spinless test particle is in the equatorial plane and will\ncompare the results with works that involve astronomy, especially the study of\nsatellites which orbit around the Earth. We take the same initial conditions\nin the two cases for describing the trajectory both a spinless particle and a\nspinning particle in a weak Kerr metric. Then, we compare the cartesian\ncoordinates ($x,y,z$) for the trajectory of two particles that travel in the\nsame orbit but in opposite directions. We shall take both for a spinless test\nparticle and for a spinning test particle orbiting in a weak Kerr field.\n\nThis work is organized as follows. In Section 2 we give a brief introduction\nto the MPD\\ Equations that work the set of equations of motion for test\nparticles both spinless and spinning in a rotating gravitational field. From\nthe MPD equations of motion, we yield the equations of motion for spinless and\nspinning test particles and will study the spinless test particles. Also, we\nwill give the set of MPD equations given by Plyatsko \\textit{et al.\n}\\cite{plyatsko} in a schematic form for working the case of a weak Kerr\nmetric. In Section 3, we present the Gravitomagnetic Clock Effect in order to\nprove our set of equations for spinless and spinning test particles. Then, in\nSection 4, we make a numerical comparison for spinless and spinning test\nparticle via MPD equations in the equatorial plane. We take the initial values\nfrom a satellite that is orbiting around on the Earth; then, we substitute\nthese values in the MPD equations both for spinless particles and for spinning\nparticles, and finally we make a numerical comparison of the trajectory in\ncartesian coordinates for two particles that travel in the same orbit, but in\nopposite directions. In the last section, the conclusions and some future\nworks are formulated in order to describe spinning test particles in a weak\nKerr metric.\n\n\\section{Brief introduction to the Mathisson-Papapetrou-Dixon Equations}\n\nIn general the MPD equations \\cite{mathisson}, \\cite{papapetrou}, \\cite{dixon}\nare given by the dynamics of extended bodies in the general theory of\nrelativity which includes any gravitational background. For the solution of\nour problem, we take the case of a distribution of mass ($m$) with a spin\ntensor ($S^{\\rho\\sigma}$) around a rotating central source ($M$) which has a\nmetric tensor $g_{\\mu\\nu}$. These equations of motion for a spinning test\nparticle are obtained in terms of an expansion that depends on the derivatives\nof the metric and the multipole moments of the energy-momentum tensor\n($T^{\\mu\\nu}$) \\cite{dixon} which describe the motion of an extended body. In\nthis work, we shall take a body sufficiently small so that all higher\nmultipoles can be neglected. According to this restriction the MPD equations\nare given b\n\\begin{equation}\n\\frac{D}{ds}\\left(  mu^{\\lambda}+u_{\\lambda}\\frac{DS^{\\lambda\\mu}}{ds}\\right)\n=-\\frac{1}{2}u^{\\pi}S^{\\rho\\sigma}R_{\\pi\\rho\\sigma}^{\\lambda},\\label{mov1\n\\end{equation}\n\n\\begin{equation}\n\\frac{D}{ds}S^{\\mu\\nu}+u^{\\mu}u_{\\sigma}\\frac{DS^{\\nu\\sigma}}{ds}-u^{\\nu\n}u_{\\sigma}\\frac{DS^{\\mu\\sigma}}{ds}=0,\\label{mov2\n\\end{equation}\nwhere $D\/ds$ means the covariant derivative, and the antisymmetric tensor\n$S^{\\mu\\nu}$ are the linear and spin angular momenta, respectively.\n$R^{\\lambda}{}_{\\pi\\rho\\sigma}$ is the curvature tensor, and $u^{\\mu}=dz^{\\mu\n}\/ds$. But we do not have the evolution equation for $u^{\\mu}$ and it is\nneccesary to single out the center of mass which determines the world line as\na representing path and specifies a point about which the momentum and spin of\nthe particle are calculated. This world line can be determined from physical\nconsiderations \\cite{karpov}. In general, two conditions are usually imposed.\nThe Mathisson-Pirani supplementary condition is \\cite{mathisson} \\cite{pirani\n\\begin{equation}\nu_{\\sigma}S^{\\mu\\sigma}=0\\label{mp\n\\end{equation}\nand the Tulczyjew-Dixon condition \\cite{dixon\n\n\\begin{equation}\np_{\\sigma}S^{\\mu\\sigma}=0\\label{cond1\n\\end{equation}\nwher\n\\begin{equation}\np^{\\sigma}=mu^{\\sigma}+u_{\\lambda}\\frac{DS^{\\sigma\\lambda}}{ds\n\\end{equation}\nis the four momentum.\n\nFor to obtain the set of MPD equations, we take the MP condition (\\ref{mp})\nwhich has three independent relationships between $S^{\\mu\\sigma}$ and\n$u_{\\sigma}$. By this condition $S^{i4}$ is given b\n\\begin{equation}\nS^{i4}=\\frac{u_{k}}{u_{4}}S^{ki\n\\end{equation}\nwith this expression we can deal the independent components $S^{ik}$.\nSometimes it is more convenient the vector spin which is given by\n$\\ S_{i}=\\frac{1}{2u_{4}}\\sqrt{-g}\\epsilon_{ikl}S^{kl}$, where $\\epsilon\n_{ikl}$ is the spatial L\\'{e}vi-Civit\\`{a} symbol.\n\nOn the other hand, when the space-time admits a Killing vector $\\xi^{\\upsilon\n}$, there exists a property that includes the covariant derivative and the\nspin tensor, which gives a constant and is given by the expression \\cite{dixon\n1979\n\\begin{equation}\np^{\\nu}\\xi_{\\nu}+\\frac{1}{2}\\xi_{\\nu,\\mu}S^{\\nu\\mu}=\\text{ constant,\n\\label{16\n\\end{equation}\nwhere $p^{\\nu}$ is the linear momentum, $\\xi_{\\nu,\\mu}$ is the covariant\nderivative of the Killing vector, and $S^{\\nu\\mu}$ is the spin tensor of the\nparticle. In the case of the Kerr metric, there are two Killing vectors, owing\nto its stationary and axisymmetric nature. In consequence, Eq. (\\ref{16})\nyields two constants of motion: $E$ is the total energy and $L$ is the\ncomponent of its angular momentum along the axis of symmetry \\cite{iorio}.\n\n\\subsection{MPD Equations for a spinning test particle in a metric of rotating\nbody}\n\nGiven that the spinning body test is sufficiently small in regard to the\ncharacteristic length the equations of motion (Eqs. \\ref{mov1} and \\ref{mov2})\nare reduced to the case when the test particles are orbiting a metric of\nrotating body. Then, we will give the equations of motion for the case of a\nspinning test particle for a weak Kerr metric (Appendix A).\n\nFirst of all, we take the paper by R.M. Plyatsko \\textit{et al. \n\\cite{plyatsko} for obtaining the full set of the exact MPD equations for the\nmotion of a spinning test particle in the Kerr field if the MP condition\n(\\ref{mp}) is taken into account and obtain a general scheme for the set of\nequations of motion for a spinning test particle in a rotating field. Plyatsko\n\\textit{et al. }use the dimensionless quantities \\textit{y}$_{i}$ with\nparticl\n\\'{\ns coordinates b\n\\begin{equation}\ny_{1}=\\frac{r}{M}\\text{, \\ \\ \\ \\ \\ \\ }y_{2}=\\theta\\text{, \\ \\ \\ \\ \ny_{3}=\\varphi\\text{, \\ \\ \\ \\ \\ }y_{4}=\\frac{t}{M}\\label{y1\n\\end{equation}\nfor its 4-velocit\n\\begin{equation}\ny_{5}=u^{1}\\text{, \\ \\ \\ \\ \\ }y_{6}=Mu^{2}\\text{, \\ \\ \\ }y_{7}=Mu^{3}\\text{,\n\\ \\ \\ }y_{8}=u^{4}\\label{y5\n\\end{equation}\nand the spin components \\cite{plyatsko 2010\n\\begin{equation}\ny_{9}=\\frac{S_{1}}{mM}\\text{, \\ \\ \\ \\ \\ \\ \\ }y_{10}=\\frac{S_{2}}{mM^{2\n}\\text{, \\ \\ \\ \\ \\ }y_{11}=\\frac{S_{3}}{mM^{2}}\\label{y9\n\\end{equation}\n\n\nIn addition, they introduce another dimensionless quantities with regard to\nthe proper time $s$ and the constant of motion $E$, $J_{z}\n\\begin{equation}\nx=\\frac{s}{M}\\text{, \\ \\ \\ \\ \\ }\\widehat{E}=\\frac{E}{m}\\text{, \\ \\ \\ \n\\widehat{J}=\\frac{J_{z}}{mM\n\\end{equation}\n\n\nThe set of the MPD equations for a spinning particle in the Kerr field is\ngiven by eleven equations. The first four equations ar\n\\begin{equation}\n\\overset{\\bullet}{y}_{1}=y_{5}\\text{, \\ \\ \\ \\ \\ }\\overset{\\bullet}{y\n_{2}=y_{6}\\text{, \\ \\ \\ \\ \\ }\\overset{\\bullet}{y}_{3}=y_{7}\\text{,\n\\ \\ \\ \\ }\\overset{\\bullet}{y}_{4}=y_{8\n\\end{equation}\nwhere a dot denotes the usual derivative with respect to $x$.\n\nThe fifth equation is given by the first three equations of (\\ref{mov1}) with\nthe indexes $\\lambda=1,2,3$. The result is multiplied by $S_{1,}S_{2},S_{3}$\nand with the MP condition (\\ref{mp}) we have the relationship: $S^{i4\n=\\frac{u_{k}}{u_{4}}S^{ki}$ and $S_{i}=\\frac{1}{2u_{4}}\\sqrt{-g\n\\varepsilon_{ikl}S^{kl}$, we obtai\n\\begin{equation}\nmS_{i}\\frac{Du^{i}}{ds}=-\\frac{1}{2}u^{\\pi}S^{\\rho\\sigma}S_{j}R_{\\pi\\rho\n\\sigma}^{j\n\\end{equation}\nwhich can be written a\n\\begin{equation}\ny_{9}\\overset{\\bullet}{y}_{5}+y_{10}\\overset{\\bullet}{y}_{6}+y_{11\n\\overset{\\bullet}{y}_{7}=A-y_{9}Q_{1}-y_{10}Q_{2}-y_{11}Q_{3\n\\end{equation}\nwher\n\\begin{equation}\nQ_{i}=\\Gamma_{\\mu\\nu}^{i}u^{\\mu}u^{\\nu}\\text{, \\ \\ \\ \\ }A=\\frac{u^{\\pi}\n{\\sqrt{-g}}u_{4}\\epsilon^{i\\rho\\sigma}S_{i}S_{j}R_{\\pi\\rho\\sigma}^{j\n\\end{equation}\n\n\nThe sixth equation is given by\n\\begin{equation}\nu_{\\nu}\\frac{Du^{\\nu}}{ds}=0\n\\end{equation}\nwhich can be written a\n\\begin{equation}\np_{1}\\overset{\\bullet}{y}_{5}+p_{2}\\overset{\\bullet}{y}_{6}+p_{3\n\\overset{\\bullet}{y}_{7}+p_{4}\\overset{\\bullet}{y}_{8}=-p_{1}Q_{1}-p_{2\nQ_{2}-p_{3}Q_{3}-p_{4}Q_{4\n\\end{equation}\nwher\n\\begin{equation}\np_{\\alpha}=u_{\\alpha}=g_{\\mu\\alpha}u^{\\alpha\n\\end{equation}\n\n\nThe seventh equation is given b\n\\begin{equation}\nE=P_{4}-\\frac{1}{2}g_{4\\mu,\\nu}S^{\\mu\\nu\n\\end{equation}\nwhich can be written a\n\\begin{equation}\nc_{1}\\overset{\\bullet}{y}_{5}+c_{2}\\overset{\\bullet}{y}_{6}+c_{3\n\\overset{\\bullet}{y}_{7}=C-c_{1}Q_{1}-c_{2}Q_{2}-c_{3}Q_{3}+\\text{\\ \n\\widehat{E\n\\end{equation}\nwher\n\\[\nd=\\frac{1}{\\sqrt{-g}\n\\]\n\n\\begin{align}\nc_{1}  &  =-dg_{11}g_{22}g_{44}u^{2}S_{3}-d\\left(  g_{34}^{2}-g_{33\ng_{44}\\right)  g_{11}u^{3}S_{2}\\nonumber\\\\\nc_{2}  &  =dg_{11}g_{22}g_{44}u^{1}S_{3}+d\\left(  g_{34}^{2}-g_{33\ng_{44}\\right)  g_{22}u^{3}S_{1}\\nonumber\\\\\nc_{3}  &  =d\\left(  g_{34}^{2}-g_{33}g_{44}\\right)  g_{11}u^{1}S_{2}-d\\left(\ng_{34}^{2}-g_{33}g_{44}\\right)  g_{22}u^{2}S_{1\n\\end{align}\n\n\\begin{equation}\nC=g_{44}u^{4}-dg_{44}u^{4}g_{43,2}S_{1}+d\\left(  g_{44}u^{4}g_{43,1\n-g_{33}u^{3}g_{44,1}\\right)  S_{2}+dg_{22}u^{2}g_{44,1}S_{3\n\\end{equation}\n\n\nThe eighth equation is given b\n\\begin{equation}\nJ_{z}=-P_{3}+\\frac{1}{2}g_{3\\mu,\\nu}S^{\\mu\\nu\n\\end{equation}\nwhich can be written a\n\\begin{equation}\nd_{1}\\overset{\\bullet}{y}_{5}+d_{2}\\overset{\\bullet}{y}_{6}+d_{3\n\\overset{\\bullet}{y}_{8}=D-d_{1}Q_{1}-d_{2}Q_{2}-d_{3}Q_{4}-\\widehat{J\n\\end{equation}\nwher\n\\begin{align}\nd_{1}  &  =-dg_{11}g_{22}g_{34}u^{2}S_{3}+dg_{11}g_{33}g_{34}u^{3\nS_{2}+dg_{11}g_{34}^{2}u^{4}S_{2}-dg_{11}g_{33}g_{44}u^{4}S_{2}\\nonumber\\\\\nd_{2}  &  =-dg_{11}g_{22}g_{34}u^{1}S_{3}-dg_{22}g_{33}g_{34}u^{3\nS_{1}-dg_{22}g_{34}^{2}u^{4}S_{1}+dg_{22}g_{33}g_{44}u^{4}S_{1}\\nonumber\\\\\nd_{3}  &  =-dg_{11}g_{34}^{2}u^{1}S_{2}+dg_{22}g_{34}^{2}u^{2}S_{1\n+dg_{22}g_{33}g_{44}u^{2}S_{1}-dg_{11}g_{33}g_{34}u^{1}S_{2\n\\end{align}\n\n\\begin{align}\nD  & =g_{33}u^{3}-dg_{22}u^{2}g_{34,2}S_{1}+d\\left(  g_{44}u^{4\ng_{33,1}+g_{11}u^{1}g_{34,1}-g_{33}u^{3}g_{34,1}\\right)  S_{2}\\nonumber\\\\\n& -dg_{11}u^{1}g_{34,1}S_{3\n\\end{align}\n\n\nFinally, the last three equations are given b\n\\begin{equation}\nu^{4}\\overset{\\bullet}{S}_{i}+2\\left(  \\overset{\\bullet}{u}_{[4}u_{i]}-u^{\\pi\n}u_{\\rho}\\Gamma_{\\pi\\lbrack4}^{\\rho}u_{i]}\\right)  S_{k}u^{k}+2S_{n\n\\Gamma_{\\pi\\lbrack4}^{n}u_{i]}u^{\\pi}=0\n\\end{equation}\nwhich give the derivatives of three components of vector spin\n($\\overset{\\bullet}{S}_{i}$): $\\overset{\\bullet}{y}_{9}$, $\\overset{\\bullet\n}{y}_{10}$ and $\\overset{\\bullet}{y}_{11}$.\n\nAfter achieving the system of equations of motion for spinning test particles,\nwe numerically solve it. We used the fourth-order Runge Kutta method. First we\ntake the case when a test particle is orbiting far away from the central\nsource and is in the equatorial plane. For our numerical calculations, we take\nthe parameters both of the central source and the test particle such as the\nradio, the energy, the angular momentum and the components tangential and\nradial of the four-velocity ($u^{\\mu}$). We calculate the orbit of a test\nparticle both spinless and spinning a weak Kerr metric in cartesian\ncoordinates ($x$, $y$, \\thinspace$z$). Then, we make a comparison of the time\nthat a test particle takes to do a lap in the two cases and give some conclusions.\n\n\\subsection{Equations of motion for a spinning test particle in a weak Kerr\nmetric}\n\nIn the last section, we obtained the general scheme for the set of equations\nof motion of a spinning test particle in the gravitational field of a rotating\nbody. Now, we yield the set of equations for the case of a spinning test\nparticle in the equatorial plane of a weak metric Kerr (Appendix A). This set\nof equations is given b\n\n\\begin{equation}\nr^{\\prime}[s]=\\frac{dr}{ds}\\text{; \\ \\ \\ }\\theta^{\\prime}[s]=\\frac{d\\theta\n}{ds}=0\\text{; \\ \\ \\ }\\varphi^{\\prime}[s]=\\frac{d\\varphi}{ds}\\text{;\n\\ \\ \\ \\ }\n\\acute{\n[s]=\\frac{dt}{ds\n\\end{equation}\n\n\\begin{align}\n\\frac{d^{2}r}{ds^{2}}  &  =\\left(  \\frac{c_{3}d_{3}}{R_{1}}\\right)  \\left(\n\\frac{R_{2}}{c_{3}}+\\frac{R_{3}}{d_{3}}+R_{4}\\right) \\\\\n\\frac{d^{2}\\varphi}{ds^{2}}  &  =\\left(  \\frac{-c_{1}d_{3}}{R_{1}}\\right)\n\\left(  \\frac{R_{2}}{c_{3}}+\\frac{R_{3}}{d_{3}}+R_{4}\\right)  +\\frac{\\left(\nB-c_{1}Q_{1}+\\overset{\\wedge}{E}\\right)  }{c_{3}}-Q_{3}\\\\\n\\frac{d^{2}t}{ds^{2}}  &  =\\left(  \\frac{-d_{1}c_{3}}{R_{1}}\\right)  \\left(\n\\frac{R_{2}}{c_{3}}+\\frac{R_{3}}{d_{3}}+R_{4}\\right)  +\\frac{\\left(\nF-d_{1}Q_{1}-\\overset{\\wedge}{J}\\right)  }{d_{3}\n\\end{align}\n\n\\[\nz:=\\left(  r[s]\\right)  ^{2}\\text{; \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }q:=r[s]\\left(\nr[s]-2\\right)  \\text{; \\ \\ \\ }\\psi:=\\left(  r[s]\\right)  ^{2\n\\]\n\n\\[\n\\eta:=3\\left(  r[s]\\right)  ^{2}\\text{; \\ \\ \\ \\ }\\chi:=\\left(  r[s]\\right)\n^{2}\\text{; \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\xi:=\\left(  r[s]\\right)  ^{2\n\\]\n\n\\[\np:=2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)  ^{2\nr[s]\\frac{d\\varphi}{ds}+\\left(  \\left(  r[s]\\right)  ^{2}-2r[s]\\right)\n\\frac{dt}{ds}\\text{; \n\\]\n\n\\[\n\\text{\\ }p_{1}:=-\\left(  1-\\frac{2}{r[s]}\\right)  ^{-1}\\frac{dr}{ds}\\text{;\n\\ \\ }p_{2}:=0;\n\\]\n\n\\[\n\\text{\\ }p_{3}:=\\frac{2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)\n\\right)  ^{2}r[s]\\frac{dt}{ds}-\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)\n\\right)  ^{2}\\left(  r[s]\\right)  ^{4}\\frac{d\\varphi}{ds}}{\\left(\nr[s]\\right)  ^{2}}\\text{;\n\\]\n\n\\[\np_{4}:=\\frac{2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)\n^{2}r[s]\\frac{d\\varphi}{ds}+\\left(  \\left(  r[s]\\right)  ^{2}-2r[s]\\right)\n\\frac{dt}{ds}}{\\left(  r[s]\\right)  ^{2}\n\\]\n\n\\[\nc_{1}:=S_{2}\\sin\\left(  \\frac{\\pi}{2}\\right)  \\frac{d\\varphi}{ds}\\text{;\n\\ \\ \\ \\ \\ }c_{2}:=0\\text{; \\ \\ \\ \\ \\ }c_{3}:=-S_{2}\\sin\\left(  \\frac{\\pi\n{2}\\right)  \\frac{dr}{ds\n\\]\n\n\\[\nd_{1}:=-S_{2}\\sin\\left(  \\frac{\\pi}{2}\\right)  \\frac{dt}{ds}\\text{;\n\\ \\ \\ \\ }d_{2}:=0\\text{; \\ \\ \\ \\ }d_{3}:=S_{2}\\sin\\left(  \\frac{\\pi\n{2}\\right)  \\frac{dr}{ds\n\\]\n\n\\begin{align*}\nQ_{1}  &  :=\\frac{\\left(  \\left(  r[s]-2\\right)  -r[s]+2\\right)  }{\\left(\nr[s]\\right)  \\left(  r[s]-2\\right)  +\\alpha^{2}}\\left(  \\frac{dr}{ds}\\right)\n^{2}-\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)  ^{2}\\left(\nr[s]-2\\right)  \\left(  \\frac{d\\varphi}{ds}\\right)  ^{2}\\\\\n&  +\\frac{\\left(  r[s]-2\\right)  }{\\left(  r[s]\\right)  ^{3}}\\left(  \\frac\n{dt}{ds}\\right)  ^{2}-\\frac{2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)\n\\right)  ^{2}\\left(  r[s]-2\\right)  }{\\left(  r[s]\\right)  ^{3}}\\frac\n{d\\varphi}{ds}\\frac{dt}{ds\n\\end{align*}\n\n\\begin{align*}\nQ_{2}  &  :=0\\\\\nQ_{3}  &  :=\\frac{2}{\\left(  r[s]\\right)  }\\frac{dr}{ds}\\frac{d\\varphi\n{ds}+\\frac{2\\alpha}{\\left(  r[s]\\right)  ^{3}\\left(  r[s]-2\\right)  }\\frac\n{dr}{ds}\\frac{dt}{ds}\\\\\nQ_{4}  &  :=-\\frac{6\\alpha}{\\left(  r[s]\\right)  \\left(  r[s]-2\\right)  \n\\frac{dr}{ds}\\frac{d\\varphi}{ds}+\\frac{2}{\\left(  r[s]\\right)  \\left(\nr[s]-2\\right)  }\\frac{dr}{ds}\\frac{dt}{ds\n\\end{align*}\n\n\\[\nB:=-\\left(  1-\\frac{2}{\\left(  r[s]\\right)  }\\right)  \\frac{dt}{ds\n-\\frac{2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)  ^{2}}{\\left(\nr[s]\\right)  ^{2}}\\frac{dr}{ds}\\frac{d\\varphi}{ds}+\\frac{S_{2}\\sin\\left(\n\\frac{\\pi}{2}\\right)  }{\\left(  r[s]\\right)  ^{2}}\\frac{d\\varphi}{ds\n-\\frac{\\alpha S_{2}\\sin\\left(  \\frac{\\pi}{2}\\right)  }{\\left(  r[s]\\right)\n^{4}}\\frac{dt}{ds\n\\]\n\n\\begin{align*}\nF  &  :=-\\frac{3\\alpha S_{2}\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)\n^{3}}{\\left(  r[s]\\right)  ^{2}}\\frac{d\\varphi}{ds}-\\frac{S_{2}\\left(  \\left(\nr[s]\\right)  -2\\right)  }{\\left(  r[s]\\right)  ^{2}}\\frac{dt}{ds}\\\\\n&  +\\frac{\\left(\n\\begin{array}\n[c]{c\n2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)  ^{2}\\left(\nr[s]\\right)  ^{2}\\frac{dr}{ds}\\left(  \\frac{d\\varphi}{ds}\\right)  ^{2}\\\\\n+\\left(  r[s]\\right)  ^{3}\\left(  r[s]-2\\right)  \\frac{dr}{ds}\\frac{dt\n{ds}-2\\alpha\\left(  \\left(  r[s]\\right)  -2\\right)  \\left(  \\frac{dt\n{ds}\\right)  ^{2\n\\end{array}\n\\right)  \\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)  ^{2}\n{2\\alpha\\left(  \\sin\\left(  \\frac{\\pi}{2}\\right)  \\right)  ^{2}r[s]\\frac\n{d\\varphi}{ds}+\\left(  \\left(  r[s]\\right)  ^{2}-2r[s]\\right)  \\frac{dt}{ds}\n\\end{align*}\n\n\\begin{align*}\nR_{1}  &  :=c_{3}d_{3}p_{1}-d_{3}p_{3}c_{1}-p_{4}d_{1}c_{3}\\\\\nR_{2}  &  :=p_{3}\\left(  c_{1}Q_{1}-B-\\overset{\\wedge}{E}\\right) \\\\\nR_{3}  &  :=p_{4}\\left(  d_{1}Q_{1}-F+\\overset{\\wedge}{J}\\right) \\\\\nR_{4}  &  :=-p_{1}Q_{1\n\\end{align*}\n\n\n\\subsection{MPD Equations for spinless test particle in a weak Kerr metric}\n\nThe traditional form of MP equations is \\cite{mathisson\n\\begin{equation}\n\\frac{D}{ds}\\left(  mu^{\\lambda}+u_{\\lambda}\\frac{DS^{\\lambda\\mu}}{ds}\\right)\n=-\\frac{1}{2}u^{\\pi}S^{\\rho\\sigma}R_{\\pi\\rho\\sigma}^{\\lambda}\\label{mp1\n\\end{equation}\n\n\nFirst of all, we consider the case of the motion of a spinning test particle\nin equatorial circular orbits ($\\theta=\\pi\/2$) from the weak Kerr source, that\nis, $a\/r\\ll1$ and $MG\/c^{2}$. For this case we take \\cite{plyatsko 2013\n\\begin{equation}\nu^{1}=0\\text{, \\ }u^{2}=0\\text{, \\ }u^{3}=\\text{const, \\ }u^{4}=\\text{ const\n\\end{equation}\nwhen the spin is perpendicular to this plane and the MP condition (\\ref{mp}),\nwit\n\\begin{equation}\nS_{1}\\equiv S_{r}=0\\text{, \\ }S_{2}\\equiv S_{\\theta}\\neq0\\text{, \\ \nS_{3}\\equiv S_{\\varphi}=0.\n\\end{equation}\n\n\nThe equation is given b\n\\[\n-y_{1}^{3}\\ast y_{7}^{2}-2\\alpha\\ast y_{7}y_{8}+y_{8}^{2}-3\\ast\\alpha\n\\ast\\varepsilon_{0}y_{7}^{2}+3\\varepsilon_{0}y_{7}y_{8}-3\\alpha\\varepsilon\n_{0}y_{8}^{2}y_{1}^{-2}+3\\alpha\\varepsilon_{0}y_{1}^{2}y_{7}^{4\n\\\n\\[\n-\\alpha\\varepsilon_{0}\\left(  1-\\frac{2}{y_{1}}\\right)  y_{8}^{4}y_{1\n^{-3}+\\varepsilon_{0}\\left(  y_{1}^{6}-3y_{1}^{5}\\right)  y_{7}^{3}y_{8\ny_{1}^{-3}+\\alpha\\varepsilon_{0}\\left(  3y_{1}^{3}-11y_{1}^{2}\\right)\ny_{7}^{2}y_{8}^{2}y_{1}^{-3\n\\\n\\begin{equation}\n+\\varepsilon_{0}\\left(  -y_{1}^{3}+3y_{1}^{2}\\right)  y_{7}y_{8}^{3}y_{1\n^{-3}=0\\label{c\n\\end{equation}\n\n\nThen, for the case when the particle does not have spin the set of equations\n(\\ref{c}) with the dimensionless quantities \\textit{y}$_{i}$ (\\ref{y1}) and\n(\\ref{y5}) is reduced t\n\\begin{equation}\n-y_{1}^{3}\\ast y_{7}^{2}-2\\alpha\\ast y_{7}y_{8}+y_{8}^{2}=0\\label{a\n\\end{equation}\nwhere $\\alpha=a\/M$.\n\nIn addition to Eq. (\\ref{a}), we take the condition $u_{\\mu}u^{\\mu}=1$ and\nobtai\n\\begin{equation}\n-y_{1}^{2}\\ast y_{7}^{2}+4\\alpha\\frac{y_{7}y_{8}}{y_{1}}+\\left(  1-\\frac\n{2M}{y_{1}}\\right)  y_{8}^{2}=1\\label{b\n\\end{equation}\n\n\nWe solve the system of equations (\\ref{a}) and (\\ref{b}) for the case of a\ncircular orbit and obtain the values of \\ $y_{7}=Mu^{3}$ and\\ $y_{8}=u^{4}$.\n\n\\subsection{Constants of motion for a weak Kerr metric}\n\nWith the Tulczyjew-Dixon condition (\\ref{cond1}), we determine the center of\nmass of the particle and let $u^{\\mu}$ be its four-velocity; also, the MPD\nequations (\\ref{mov1}) and (\\ref{mov2}) yield a unique $u^{\\mu}$, namely\n\\cite{kyrian\n\n\\begin{equation}\nu^{\\mu}=V^{\\mu}+\\frac{1}{2}\\left(  \\frac{S^{\\mu\\nu}R_{\\nu\\rho\\sigma\\kappa\n}V^{\\rho}S^{\\sigma\\kappa}}{m^{2}+\\frac{1}{4}R_{\\chi\\xi\\zeta\\eta}S^{\\chi\\xi\n}S^{\\zeta\\eta}}\\right)  .\\label{condition\n\\end{equation}\n$u^{\\mu}$ and $V^{\\mu}$ are named by Dixon as kinematical four velocity and\ndynamical four velocity, respectively \\cite{dixon 1979}.\n\nSince $p_{\\mu}p^{\\mu}=$ constant and $S_{\\rho\\sigma}S^{\\rho\\sigma}=$ constant\nalong the particle trajectory \\cite{wald}, we may se\n\n\\begin{equation}\nu_{\\mu}u^{\\mu}=-1\\text{, \\ \\ \\ \\ \\ \\ }S_{0}^{2}=S_{\\mu}S^{\\mu}=\\frac{1\n{2m^{2}}S_{\\mu\\nu}S^{\\mu\\nu}\\text{,}\\label{7a\n\\end{equation}\nand with these expressions, we obtain the center of mass condition and the\nrelation between the spin tensor and the vector spin.\n\nNext, we reduced the set of MPD equations given by Plyatsko \\textit{et al.\n}\\cite{plyatsko} for the case when a spinning test particle is a weak Kerr\nmetric in the equatorial plane and follows a circular orbit. Then, for the\ninitial conditions we need the values of both the energy ($E$) and the\ncomponent $z$ of the angular momentum ($J$) for a weak Kerr metric. In this\ncase, the constants of motion are given b\n\n\\begin{align}\nE  &  =m\\left(  g_{44}+g_{34}\\right)  V^{4}+\\frac{Ma}{r^{2}}S^{13}-\\frac\n{M}{r^{2}}\\frac{g_{33}V^{3}}{g_{44}V^{4}}S^{13}\\label{E}\\\\\nJ_{z}  &  =-m\\left(  g_{33}+g_{34}\\right)  V^{3}+r\\sin^{2}\\theta S^{13\n-\\frac{Ma}{r^{2}}\\frac{\\left(  g_{33}+g_{34}\\right)  V^{3}}{\\left(\ng_{44}+g_{34}\\right)  V^{4}}S^{13}\\label{J\n\\end{align}\nwhere $V^{3}$ and $V^{4}$ are components of the dynamical 4-velocity and\n$S^{13}$ is the perpendicular component of the spin vector.\n\nWe yield the components of the 4-velocity $u^{\\lambda}$ (\\ref{condition}) for\nthe case of a spinning test particle in a weak Kerr metric and in the\nequatorial plane $\\theta=\\pi\/2$ when spin is orthogonal to this plane and has\na constant radius ($x^{1}=r=$ constant). We use the Boyer-Lindquist\ncoordinates ($x^{1}=r,$ $x^{2}=\\theta$, $x^{3}=\\varphi$, $x^{4}=t$). We hav\n\\begin{align}\nu^{1}  &  =0\\text{, \\ \\ \\ \\ \\ }u^{2}=0\\text{, \\ \\ \\ \\ \\ }u^{3}\\neq0\\text{,\n\\ \\ \\ \\ \\ }u^{4}\\neq0\\text{,}\\label{29}\\\\\nS^{12}  &  =0\\text{, \\ \\ \\ \\ \\ }S^{23}=0\\text{, \\ \\ \\ \\ \\ }S^{13\n\\neq0\\label{30\n\\end{align}\n\n\nIn addition to (\\ref{30}) by Tulczyjew-Dixon condition (\\ref{cond1}) we writ\n\\begin{equation}\nS^{14}=-\\frac{P_{3}}{P_{4}}S^{13}\\text{, \\ \\ \\ \\ \\ }S^{24}=0\\text{,\n\\ \\ \\ \\ \\ }S^{34}=\\frac{P_{1}}{P_{4}}S^{13}\\label{31\n\\end{equation}\n\n\nUsing (\\ref{7a}), (\\ref{29})-(\\ref{31}) and taking the components of the\nRiemann tensor for the weak Kerr metric in the equatorial plane, from\n(\\ref{condition}) we obtai\n\\begin{align}\nu^{1} &  =NV^{1}\\left(  1+\\frac{3M}{r^{3}}V_{3}V^{3}\\frac{S_{0}^{2}\n{m^{2}\\Delta}+a\\frac{S_{0}^{2}}{m^{2}\\Delta}k_{1}\\frac{V_{3}}{V_{4}}\\right)\n\\nonumber\\\\\nu^{2} &  =V^{2}=0\\nonumber\\\\\nu^{3} &  =NV^{3}\\left(  1+\\frac{3M}{r^{3}}\\left(  V_{3}V^{3}-1\\right)\n\\frac{S_{0}^{2}M}{m^{2}\\Delta}+a\\frac{S_{0}^{2}M}{m^{2}\\Delta}k_{3}\\frac\n{V^{3}}{V^{4}}\\right) \\nonumber\\\\\nu^{4} &  =NV^{4}\\left(  1+\\frac{3M}{r^{3}}V_{3}V^{3}\\frac{S_{0}^{2}\n{m^{2}\\Delta}+a\\frac{S_{0}^{2}M}{m^{2}\\Delta}k_{4}\\left(  \\frac{V^{3}}{V^{4\n}\\right)  ^{2}\\right) \\label{32\n\\end{align}\nwhere the constants $k_{1}$, $k_{3}$ and $k_{4}$ are given b\n\\begin{align*}\nk_{1} &  =\\frac{3M\\left(  1-\\frac{4M}{3r}\\right)  }{g_{11}g_{44}}\\\\\nk_{3} &  =k_{1}\\\\\nk_{4} &  =\\frac{k_{1}}{rg_{44}\n\\end{align*}\nand the expression $\\Delta=1+\\frac{1}{4m^{2}}R_{\\chi\\xi\\zeta\\eta}S^{\\chi\\xi\n}S^{\\zeta\\eta}$\\ for the weak Kerr metric is given b\n\\begin{equation}\n\\Delta=1+\\frac{S_{0}^{2}M}{m^{2}r^{3}}\\left(  1-3V_{3}V_{3}-aA\\frac{V^{3\n}{V^{4}}\\right) \\label{33\n\\end{equation}\nwhere $a=J\/Mc$ is the angular density of the central mass and\n\\[\nA=\\frac{3M\\left(  1-\\frac{4M}{3r}\\right)  g_{33}}{g_{44}\n\\]\n\n\nWe insert (\\ref{33}) into (\\ref{32}), we ge\n\\begin{align}\nu^{1}  &  =\\frac{NV^{1}}{\\Delta}\\left(  1+\\frac{S_{0}^{2}M}{m^{2}r^{3}\n-a\\frac{S_{0}^{2}}{m^{2}}\\left(  MA-k_{1}\\right)  \\frac{V^{3}}{V^{4}}\\right)\n\\nonumber\\\\\nu^{3}  &  =\\frac{NV^{3}}{\\Delta}\\left(  1-\\frac{2S_{0}^{2}M}{m^{2}r^{3\n}-a\\frac{S_{0}^{2}}{m^{2}}\\left(  MA-k_{3}\\right)  \\frac{V^{3}}{V^{4}}\\right)\n\\nonumber\\\\\nu^{4}  &  =\\frac{NV^{4}}{\\Delta}\\left(  1+\\frac{2S_{0}^{2}M}{m^{2}r^{3\n}-a\\frac{S_{0}^{2}}{m^{2}}\\left(  MA-k_{4}\\left(  \\frac{V^{3}}{V^{4}}\\right)\n^{2}\\right)  \\frac{V^{3}}{V^{4}}\\right) \\label{34\n\\end{align}\n\n\nWe introduc\n\\begin{equation}\n\\varepsilon=\\frac{\\left\\vert S_{0}\\right\\vert }{mr\n\\end{equation}\nand obtain the expression for $N$ from the condition\n\\begin{align*}\nV_{\\lambda}V^{\\lambda}  &  =0\\\\\nV_{\\lambda}S^{\\lambda\\nu}  &  =0\n\\end{align*}\n\n\n$N$ is given b\n\\begin{equation}\nN=\\frac{\\Delta}{R}\\label{36a\n\\end{equation}\nwher\n\\begin{equation}\nR=\\left(  m^{2}\\Delta^{2}+S_{0}^{4}R^{\\mu\\tau\\rho\\delta}R_{\\mu\\tau\\rho\\delta\n}\\right)  ^\n\\frac12\n}\\label{36\n\\end{equation}\n\n\nWe insert (\\ref{36a}) into (\\ref{34}) and obtain the components from\n$V^{\\lambda}\n\\begin{align}\nV^{1}  &  =\\frac{Ru^{1}}{\\left(  1+\\frac{S_{0}^{2}M}{m^{2}r^{3}}\\left(\n1+a\\left(  A+k_{1}\\right)  \\right)  \\right)  }\\nonumber\\\\\nV^{3}  &  =\\frac{Ru^{3}}{\\left(  1-\\frac{S_{0}^{2}M}{m^{2}r^{3}}\\left(\n1+a\\left(  A-k_{3}\\right)  \\right)  \\right)  }\\nonumber\\\\\nV^{4}  &  =\\frac{Ru^{4}}{\\left(  1+\\frac{S_{0}^{2}M}{m^{2}r^{3}}\\left(\n1-a\\left(  A+3\\left(  u^{3}\\right)  ^{2}k_{4}\\right)  \\right)  \\right)\n}\\label{37\n\\end{align}\n\n\nWe replace the components of the dynamical 4-velocity (\\ref{37}) in the\nconstants of motion (\\ref{E}) and (\\ref{J}) for the case of a spinning test\nparticle in a weak Kerr field.\n\n\\section{Gravitomagnetic clock effect for spinning test particles}\n\nFor cheking our results, we review the papers in regarding to Gravitomagnetic\nclock effect \\cite{tsoubelis} and compare their numerical results with ours.\nThere is a phenomenon called the gravitomagnetic clock effect which consists\nof a difference in the time it takes for two test particles to travel around a\nrotating massive body in the equatorial plane and in opposite directions\n\\cite{faruque2004}. This difference is given by $t_{+}-t_{-}=4\\pi a\/c$, where\n$a=J\/Mc$ is the angular density of the central mass. Tartaglia has studied the\ngeometrical aspects of this phenomenon \\cite{tartaglia}, \\cite{tartaglia2001}\nand Faruque yields the equation of the gravitomagnetic clock effect with spin\na\n\\begin{equation}\nt_{+}-t_{-}=4\\pi a-6\\pi S_{0}\\text{,}\\label{gm1\n\\end{equation}\nwhere $S_{0}$ is the magnitude of the spin.\n\nIn true units this relation is given b\n\\begin{equation}\nt_{+}-t_{-}=\\frac{4\\pi J_{M}}{Mc^{2}}-\\frac{6\\pi J}{mc^{2}},\\label{gm\n\\end{equation}\nwhere the first relation of the right could be used to measure $J\/M$ directly\nfor an astronomical body; in the case of the Earth $t_{+}-t_{-}\\simeq10^{-7\n\\operatorname{s\n$, while for the Sun $t_{+}-t_{-}\\simeq10^{-5\n\\operatorname{s\n$ \\cite{mash 1999}.\n\n\\section{Numerical comparison for spinless and spinning test particle via MPD\nequations}\n\nIn this section we give the numerical results for the case of a spinning\nsatellite orbiting around the Earth \\cite{iorio 2001}. We took the data of the\nAriane-5 satellite which is a space European vehicle that is part of the\nAriane family \\cite{ariane}. For our calculations we assume the satellite\nfollows a circular orbit with radius equal to $3.5\\times10^{6\n\\operatorname{m\n$ and travels in the equatorial plane. Of course, the satellite has an orbit\nthat is more complex. The initial conditions are given by geometrized units,\nwhere the gravitational constant $G$, and speed of light $c$, are set equal to one.\n\nAccording with the features of the Ariane-5 satellite, its initial conditions\nar\n\\[\n\\text{Mass}(m)_{\\text{satellite}}=3\\times10^{3\n\\operatorname{kg\n\\text{, \\ \\ \\ \\ \\ \\ \\ }\\varepsilon_{0}=\\frac{S_{0}}{mr}=6.0976\\times10^{-11\n\\]\n\n\nThe fundamental frequency in the longitudinal axis equals 30 Hz and the\ncomponents of the four velocity of the satellite are given by a set of\nequations (\\ref{c}) and (\\ref{b}) for a orbit of $3.5\\times10^{6\n\\operatorname{m\n$. For this case, in ordinary units, the azimuthal component is $u^{3\n=2.42294\\times10^{3}$ \n\\operatorname{m\n\n\\operatorname{s\n$\\ .\n\nWe take the set of MPD equations for a spinning test particle in a weak Kerr\nmetric (Appendix A) and write the initial conditions for this satellite. With\nthe Runge-Kutta method of order 4 \\cite{press}, we obtain the cartesian\ncoordinates for a circular orbit when the satellite orbits in the same sense\nof rotation of the central source ($a$). The program code is in the Appendix\nB. We register the time that satellite takes for doing a lap. Then we take the\nsame data in the case when the satellite orbits with the sense of rotation\ncontrary to that of the central source ($a$). Finally, we take the difference\nof time in these two orbits and obtai\n\\begin{equation}\n\\Delta\\tau_{\\text{spinning}}=\\tau_{+}-\\tau_{-}=7.275957\\times10^{-7\n\\operatorname{s\n\\end{equation}\n\n\nNow we take the case when the test particle does not have spin and calculate\nthe cartesian coordinates ($x$, $y$, \\thinspace$z$) for a circular orbit of a\nspinless test particle around a rotating body mass both in the same sense of\nrotation of the central mass and in opposite direction. In this case, we take\nthe set of MPD equations for a spinless particle, Eqs. (\\ref{a}) - (\\ref{b}).\nThere is a spinless particle in the equatorial plane ($u^{2}=0$) and with a\nradius constant ($u^{1}=0$). We assume the same initial conditions as in the\nprevious case. As the above part, we calculate the difference of time of two\nparticles travel in the same orbit, but in opposite directions, and the result\ni\n\\begin{equation}\n\\Delta\\tau_{\\text{spinless}}=\\tau_{+}-\\tau_{-}=9.01062\\times10^{-7\n\\operatorname{s\n\\end{equation}\n\n\nThis result is according to the literature. In some papers this difference of\ntime is called Effect Gravitomagnetic and is given by the expression\n\\cite{iorio 2001\n\\begin{equation}\n\\left(  \\tau_{+}-\\tau_{-}\\right)  _{\\phi=2\\pi}\\simeq4\\pi\\frac{J_{\\oplus\n}{M_{\\oplus}c^{2}}\\simeq10^{-7\n\\operatorname{s\n\\end{equation}\nwhere $M_{\\oplus}$ and $J_{\\oplus}$ are the values of the mass of Earth and\nthe angular momentum respect\n\nAccording to the results, the spinless test particle in a positive sense\ncompletes a full orbit before the particle with the sense of rotation\ncontrary. This phenomenon is due to drag of the inertial frames with respect\nto infinity and is called the Lense-Thirring effect \\cite{mash 1984}. In the\ncase of the spinning test particles, not only there is a difference in the\ntime given by the Lense-Thirring effect, but also by a coupling between the\nangular momentum of the central body with the spin of the particle\n\\cite{chandra}. The features change if the test particle rotates in one\ndirection or the other; therefore, the period is different for one sense and\nfor the other, and if the particle has spin or not. The difference of time\nbetween the spinless particles and the spinning particles is so small that the\nresult is the same order of the shift ($10^{-7\n\\operatorname{s\n$). In other words, when the spinning test particle is very small compared\nwith the central mass, the influence of the value of spin in the shift of time\nis insignificant in regard to lapse of time.\n\n\\section{Conclusions}\n\nIn this paper, we take the Mathisson-Papapetrou-Dixon (MPD) equations given by\nPlyatsko \\textit{et al.} and obtained explicitly the MPD equations for the\ncase when the spinning test particle is orbiting in a rotating weak field.\nWork that was not in the literature. In addition, we gave a scheme for the\neleven equations of the full set of equations of motion when the particle is\norbiting a rotating gravitational field. In the second part, we worked the\nconstants of motion such as the energy ($E$) and the angular momentum ($J_{z\n$) of the spinning test particle in a weak Kerr metric. Finally, we calculated\nthe trajectories in cartesian coordinates ($x$, $y$, \\thinspace$z$) of test\nparticles both spinless and spinning orbiting in a weak Kerr metric and\ncompared the time of two circular orbits in the equatorial plane for two test\nparticles that travel in the same orbit but in opposite directions. In the\ncase of the Earth, both for the spinless particles and the spinning particles\nthere is a difference of time in their trajectories when they describe a full\nrevolution with respect to an asymptotically inertial observer. This\nphenomenum is called Gravitomagnetic Effect. From this situation, we concluded\nthat this shift, in the case of the spinless test particles, is given by the\nangular momentum from the central source which drags the inertial systems in\nthe same sense of the rotation of the rotating massive body. For the case of\nthe spinning test particles, this time lapse is given not only by the angular\nmomentum from the central mass, but also by the couple between the angular\nmomentum from the massive rotating body and the parallel component of the spin\nof the test particle. In the MPD equations, this couple is given by the\nrelationship between the components of the Riemman tensor ($R^{\\mu}$\n$_{\\nu\\rho\\sigma} $) and the spin tensor ($S^{\\rho\\sigma}$).\n\nIn the future we will work in the set of Equations of motion of a test\nparticle both spinless and spinning for spherical orbits, that is, with\nconstant radius and out of the equatorial plane in a weak Kerr metric. In\naddition, we are interesting in relating these equations with the experiments\ntype Michelson and Morley.\n\n\\begin{acknowledgement}\nOne of the authors is grateful with Pontificia Universidad Javeriana at\nBogot\\'{a} and with Professor Roman Plyastsko for his helpful suggestions.\n\\end{acknowledgement}\n\n\\bigskip\n\n\\begin{center}\n\\textbf{Appendix A}\n\n\\textbf{Weak Kerr Metric}\n\\end{center}\n\nThe components of a weak Kerr Metric are given b\n\n\\[\ng_{\\mu\\nu}=\\left(\n\\begin{array}\n[c]{cccc\n1-\\frac{2M}{r} & 0 & 0 & \\frac{2Ma\\sin^{2}\\theta}{r}\\\\\n0 & -1-\\frac{2M}{r} & 0 & 0\\\\\n0 & 0 & -r^{2} & 0\\\\\n\\frac{2Ma\\sin^{2}\\theta}{r} & 0 & 0 & -r^{2}\\sin^{2}\\theta\n\\end{array}\n\\right)\n\\]\n\n\n\\bigskip\n\n\\begin{center}\n\\textbf{Appendix B}\n\n\\textbf{Program code}\n\\end{center}\n\n$TF=1\\ast10^{6};\\epsilon_{0}=6.0976\\ast10^{-11};y1=3.1085\\ast10^{4\n;y10=y1\\ast\\epsilon_{0};$\n\n$y5=0;\\alpha=1.9765\\ast10^{-16};$\n\n$M=4.431948\\ast10^{-3};m=2.228\\ast10^{-24};y2=\\frac{\\pi}{2};$\n\nSetPrecision$[$NSolve[\\{$-(y1)^{3}(y_{7})^{2}-2\\ast\\alpha\\ast y_{7\ny_{8}+(y_{8})^{2}-3\\ast\\alpha\\ast\\epsilon_{0}(y1)^{2}\\ast\\frac{(y_{7})^{2\n}{(y1)^{2}}+3\\ast\\epsilon_{0}\\ast y_{7}y_{8}-3\\ast\\alpha\\ast\\epsilon_{0\n\\ast\\frac{(y_{8})^{2}}{(y1)^{2}}+$\n\n$3\\ast\\alpha\\ast\\epsilon_{0}\\ast(y1)^{2}(y_{7})^{4}-\\alpha\\ast\\epsilon_{0\n\\ast(1-\\frac{2}{y1})\\frac{(y_{8})^{4}}{(y1)^{3}}+\\epsilon_{0}((y1)^{6\n-3(y1)^{5})\\frac{(y_{7})^{3}y_{8}}{(y1)^{3}}+\\alpha\\ast\\epsilon_{0\n(3(y1)^{3}-11(y1)^{2})\\ast\\frac{(y_{7})^{2}(y_{8})^{2}}{(y1)^{3}}+$\n\n$\\epsilon_{0}\\ast(-(y1)^{3}+3(y1)^{2})\\frac{y_{7}(y_{8})^{3}}{(y1)^{3\n}==0,-(y1)^{2}(y_{7})^{2}+\\frac{4\\ast\\alpha\\ast y_{7}\\ast y_{8}}{y1}$\n\n$+\\left(  1-\\frac{2}{y1}\\right)  (y_{8})^{2}==1\\},\\{y_{7},y_{8}\\}],10]$\n\nsystem1 $=\\{y_{3\n\\acute{\n[s]==y_{7},y_{4\n\\acute{\n[s]==y_{8},y_{3}[0]==0,y_{4}[0]==0\\};$\n\n$sol1=$ NDSolve$[$system1$,\\{y_{3},y_{4},y_{3\n\\acute{\n,y_{4\n\\acute{\n\\},\\{s,0,TF\\},$\n\n$Method\\rightarrow\"Automatic\",MaxSteps\\rightarrow1\\ast10^{10}]$\n\n$graph1=$ ParametricPlot3D$[$Evaluate$[\\{(y1)\\ast\\sin\\frac{\\pi}{2}\\ast\n\\cos[{{{{{y_{3}[s]}}}}}],(y1)\\ast\\sin\\frac{\\pi}{2}\\ast\\sin[{{{{{y_{3}[s]}}}\n}],(y1)\\ast\\cos\\frac{\\pi}{2}\\}\/.sol1],\\{s,0,TF\\},$\n\n$AxesLabel->\\{\"x\",\"y\",\"z\"\\},PlotStyle->\\{Blue\\}]$\n\nSetPrecision$[Table[\\{s,y1\\ast\\sin[y2]\\ast\\sin[{{{{{y_{3}[s]}}}\n}]\\}\/.sol1,\\{s,0,TF,9.53674316406\\ast10^{-7}\\}],20]$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nNGC 5548 is a bright, low-redshift (z $=$ 0.0172) Seyfert 1 galaxy that has \nreceived a considerable amount of attention over the past decade. In particular, \nNGC 5548 has been the subject of a number of intensive spectroscopic monitoring \ncampaigns in the optical and UV (Korista et al. 1995). \nThese efforts have yielded important results on the nature of the continuum \nsource and the size, geometry, and kinematics of the broad-line region (BLR), \nfor which the responsivity peaks at about 20 light days from the nucleus\n(Peterson et al. 1994). Little attention has been paid to the narrow-line region \n(NLR) in NGC 5548, however, although it could lead to a better understanding of \nthe circumnuclear environment at much greater distances from the nucleus of this\notherwise well-studied active galaxy.\n\nIn general, studies of the NLR in active galaxies are important for \nunderstanding the nature of the NLR clouds and the interaction of the central \ncontinuum source with the surrounding galaxy on large scales. Emission-line \nstudies and detailed photoionization modeling are particularly useful \nfor determining the range of physical conditions and reddening amongst the NLRs \nin active galaxies. A comparison of the NLR properties in Seyfert \n1 and Seyfert 2 galaxies should be helpful in testing unified theories, which \npostulate that the two types are the same object viewed from \ndifferent perspectives, such that the continuum source and BLR are ``hidden'' in \nSeyfert 2 galaxies (Miller \\& Goodrich 1990; Antonucci 1993). If this basic \nhypothesis is correct, then the intrinsic properties of the NLRs in Seyfert 1 \nand Seyfert 2 galaxies should {\\it not} show large systematic differences.\n\nThe narrow lines in Seyfert 1 galaxies are more difficult to measure \nthan in Seyfert 2 galaxies, due to blending with the broad lines. However, given \nspectra with sufficient signal-to-noise ratio and spectral resolution, these \ncomponents can be isolated and measured reasonably well\n(Crenshaw \\& Peterson 1986). Measurements of the narrow lines in the optical are \ngiven by Cohen (1983) for a large number of Seyfert 1 galaxies. In the UV, these \nmeasurements were difficult with the low spectral resolution of the \n{\\it International Ultraviolet Explorer} ({\\it IUE}), but are possible with \ninstruments on \nthe {\\it Hubble Space Telescope} ({\\it HST}), such as the Faint Object \nSpectrograph (FOS).\n\nThe FOS UV spectra of NGC 5548 presented in a previous paper (Crenshaw, Boggess, \n\\& Wu 1993; hereafter Paper I) provide a good starting point for detailed \nstudies of the NLR in a Seyfert 1 galaxy. These observations happened to occur\nat a time when the broad emission lines (and continuum fluxes) were at an \nhistoric low in the UV, and the contrast between the broad and narrow \ncomponents is thereby enhanced. Paper I gives the UV spectrum and measurements \nof the broad and narrow lines in NGC 5548. Relative to the other narrow lines, \nC~IV $\\lambda$1549 is much stronger in NGC 5548 than in Seyfert 2 galaxies, \nindicating a higher ionization parameter and\/or harder continuum in the NLR of \nNGC 5548. Narrow Mg II $\\lambda$2800 emission is very weak or absent in NGC \n5548, and Paper I presents two possible explanations: 1) the NLR clouds lack the \npresence of a partially-ionized zone (i.e., they are optically thin to ionizing \nradiation), and\/or 2) dust grains are present in the NLR clouds, and the Mg II \nflux is weak due to depletion and\/or destruction from multiple scatterings and \neventual absorption of the photons by dust (Kraemer \\& Harrington 1986; Ferland \n1992).\n\nWe now have the opportunity to investigate the preliminary results from Paper I \nin more detail, by including ground-based optical spectra and photoionization \nmodels. From the ground-based monitoring campaigns, we have selected \nspectra that cover the full optical range (3000 -- 10,000 \\AA) and were\nobserved around the same time period as the UV data, when the broad-line fluxes \nwere very low. The combination of optical and UV lines provides a wide range of \nemission-line diagnostics, as well as an opportunity to deredden the lines using \nthe He~II recombination lines. We can then use multicomponent \nphotoionization models to match the dereddened line \nratios and probe the physical conditions in the NLR of NGC 5548.\n\n\\section{Observations and Data Analysis}\n\n\\subsection{UV and Optical Spectra}\n\nWe obtained the FOS UV spectra of NGC 5548 through a 1\\arcsecpoint0 circular \naperture on 1992 July 5 UT. Paper I gives the details of the observations and \nmeasurements, along with the UV spectrum and emission-line fluxes (with \nassociated errors). The observations were made prior to the installation of \nCOSTAR on {\\it HST}, so near-simultaneous {\\it IUE} spectra were used to adjust \nthe absolute flux levels of the FOS spectra. As noted in Paper I, \nthe scale factors needed to bring the FOS continuum fluxes up to the {\\it IUE} \nlevels are around 1.4 -- 1.5, which are somewhat higher than the \nvalues of 1.1 -- 1.3 for our other Seyfert observations. We concluded that the \nSeyfert nucleus may not have been accurately centered in the aperture. Koratkar \net al. (1996) suggest that another possible explanation for the discrepancy in \nabsolute fluxes is nonlinearity in the {\\it IUE} detectors. However, we have \nseen no evidence for this possibility in other observations at these flux \nlevels, so we have no reason to distrust the {\\it IUE} fluxes. In addition, \nreprocessed versions of these spectra that we obtained from the {\\it HST} and \n{\\it IUE} archives have not changed the original fluxes by more than 10\\%, so we \ncontinue to use the values from Paper I. Some of the emission lines in Paper I \nhave only a single number quoted for the flux (as opposed to separate values for \nthe broad and narrow components); a single value represents the narrow-line \ncontribution, since the broad component is either not present or too weak to be \ndetected in these cases.\n\nWe selected two optical spectra obtained during a four-year monitoring campaign \non NGC 5548 (Peterson et al. 1994), from a time interval of $\\sim$30 days when \nthe H$\\beta$ and continuum light curves were at their lowest levels to date.\nThe spectra were chosen on the basis of their large wavelength coverage\n(3000 - 10000 \\AA), high signal-to-noise ratio ($\\geq$50 per resolution \nelement in the continuum at 5200 \\AA), and acceptable resolution ($\\sim$8 \n\\AA). The spectra were obtained through a 4\\arcsecpoint0 x \n10\\arcsecpoint0 aperture with the 3.0-m Shane telescope $+$ Kast spectrograph on \n1992 April 21 and 1992 May 23 UT. Additional details on the observations are \ngiven by Peterson et al. (1994). The absolute flux levels were \nadjusted by scaling the optical spectra so that the [O~III] $\\lambda$5007 flux \nis 5.58 x 10$^{-13}$ ergs s$^{-1}$ cm$^{-2}$, a value determined from \nobservations through large apertures on spectrophotometric nights (Peterson et \nal. 1991). The scale factors we used are 1.36 for the 1992 April 24 spectrum and \n1.01 for the 1992 May 23 spectrum.\n\nPlots of the optical spectra are shown in Figure 1 (the UV spectrum is shown in \nPaper I). The contrast between the broad and narrow components of the permitted \nemission lines is most clearly seen in H$\\beta$. The 1992 April 21 spectrum was \nobtained at an historic low level, with a continuum flux of  \nF$_{\\lambda}$(5100~\\AA) $=$ 5.5 x 10$^{-15}$ ergs s$^{-1}$ \ncm$^{-2}$ \\AA$^{-1}$ and total H$\\beta$ flux of F(H$\\beta$) $=$ 3.2 x 10$^{-13}$ \nergs s$^{-1}$ cm$^{-2}$. The flux levels are a little higher for the 1992 May 23 \nspectrum with F$_{\\lambda}$(5100 \\AA) $=$ 6.0 x 10$^{-15}$ ergs s$^{-1}$ \ncm$^{-2}$ \\AA$^{-1}$ and F(H$\\beta$) $=$ 3.7 x 10$^{-13}$ ergs s$^{-1}$ \ncm$^{-2}$.  At the time of the FOS \nobservations on 1992 July 5, the continuum and H$\\beta$ fluxes were close to the \nsame levels as those from the second optical spectrum, according to the light \ncurves of Peterson et al. (1994).\n\nAlthough the optical aperture is much larger than the one used for the UV \nobservations, we have substantial evidence that it does not contain much \nadditional NLR flux. Unfortunately, there are no {\\it HST} narrow-band images in \n[O~III] or other strong lines that could be used to directly determine the \ndistribution of narrow-line emission close to the nucleus. However, there is \nsignificant evidence that the apparent size of the NLR is very small in NGC \n5548. Peterson et al. (1995) find from a ground-based image that the [O~III] \nemission is pointlike, given a point-spread function that is characterized by a \nwidth of 2\\arcsecpoint0 (FWHM). In addition, Wilson \\& Ulvestad (1982) show that \nin an aperture that is 4\\arcsecpoint2 in diameter, the [O~III] $\\lambda$5007 \nfluxes at positions offset from the nucleus by 4\\arcsecpoint5 -- 6$''$\nare about 100 times weaker than the nuclear flux.\nMore importantly, in Paper I we found that the strongest UV lines in the {\\it \nIUE} 20$''$ x 10$''$ aperture have fluxes that are only $\\sim$ 20\\% higher than \nthose in the FOS 1\\arcsecpoint0 aperture. Thus, the observed UV to optical line \nratios that we quote are at most 20\\%  too low, which has little effect on our \ncomparisons with the model results.\n\nIn order to measure the flux of each narrow optical line, we used\na local baseline determined by linear interpolation between adjacent \ncontinuum regions or broad profile wings (in the case of profiles \nconsisting of broad and narrow components). For severely blended lines like \nH$\\alpha$ and [N~II] $\\lambda\\lambda$6548, 6584, we used the [O~III] \n$\\lambda$5007 profile as a template to deblend the lines (see Crenshaw \\& \nPeterson 1986). The adopted flux for each narrow component is the average of the \nvalues from each of the two spectra.\n\nWe determined the reddening of the narrow emission lines from the He~II \n$\\lambda$1640\/$\\lambda$4686 ratio and the Galactic reddening curve of Savage \\& \nMathis (1979). For the temperatures and densities typical of the NLR, the He II \nlines are due to recombination, and this particular ratio only varies from 6.3 \nto 7.6 (Seaton 1978); we adopt an intrinsic value of 7.2, consistent with our \nmodel values (Section 3). The observed He~II $\\lambda$1640\/$\\lambda$4686 ratio \nis 5.5 $\\pm$ 1.6, which yields a reddening of E$_{B-V}$ $=$ 0.07 mag \n$^{+0.09}_{-0.06}$. The portion of the reddening that is due to our own Galaxy \nis E$_{B-V}$ $=$ 0.03 mag, determined from a neutral hydrogen column density of \nN$_{HI}$ $=$ 1.6 x 10$^{20}$ cm$^{-2}$ (Murphy et al. 1996) and the relationship \nE$_{B-V}$ $=$N$_{HI}$\/5.2 x 10$^{21}$ cm$^{-2}$ (Shull \\& Van Steenburg 1985).\nWe note that the intrinsic reddening of the narrow emission lines in this \nSeyfert 1 galaxy, E$_{B-V}$ $\\approx$ 0.04 mag, is much smaller than typical \nvalues of 0.2 -- 0.4 mag obtained for Seyfert 2 galaxies (MacAlpine 1988; \nFerland \\& Osterbrock 1986; Kraemer et al. 1994). \n\nWe determined errors in the dereddened ratios from the sum in quadrature of the \nerrors from three sources: photon noise, different reasonable continuum \nplacements, and reddening. Errors in the optical ratios are dominated by \ncontinuum placement, whereas errors in the UV to optical ratios are due to both \ncontinuum placement and uncertainties in the reddening correction. Errors in the \nweak lines in both regions also have a significant contribution from photon \nnoise. As we discussed earlier in this section, there are some possible sources \nof systematic error in the UV to optical line ratios, on which we placed upper \nlimits of $\\sim$20\\%.\n\nTable 1 gives the observed and dereddened narrow-line ratios relative to \nH$\\beta$, and errors in the dereddened ratios. Cohen (1983) gives the next \nmost comprehensive list of optical line ratios; in general, Cohen's observed \nratios agree with ours to within the errors. A number of investigators have \nindependently determined the narrow H$\\beta$\/[O~III] $\\lambda$5007 ratio in NGC \n5548 (Cohen 1983; Crenshaw \\& Peterson 1986; Peterson 1987; Wamsteker et al. \n1990; Rosenblatt et al. 1992; Wanders \\& Peterson 1996): these values range from \n0.10 to 0.15, compared to our value of 0.12 $\\pm$ 0.01.\n\n\\subsection{The Ionizing Continuum}\n\nEstimates of the ionizing continuum are needed as input values for the \nphotoionization models of the NLR. We choose the continuum data points given by \nKrolik et al. (1991), since they represent the historic mean levels for this \nobject. As always, the greatest uncertainty is the shape of the extreme \nultraviolet (EUV) continuum. Figure 2 \ngives the UV continuum point closest to the EUV region, at 1340 \\AA, and the \nX-ray continuum points from Krolik et al. (cf. Turner \n\\& Pounds 1989; Clavel et al. 1991) in terms of luminosity (ergs \ns$^{-1}$ Hz$^{-1}$), which we have adjusted for a Hubble constant of H$_{0}$ $=$ \n75 km s$^{-1}$ Mpc$^{-1}$. The dotted line in Figure 1 gives Krolik et al.'s \ncontinuum fit in the EUV, which is a power law determined from the UV data along \nwith an exponential cutoff designed to meet the first X-ray point. We prefer a \nfit with two power laws (in the form L$_{\\nu}$ $=$ K$\\nu^{\\alpha}$), given the \nevidence for an upturn in the spectrum at energies smaller than 1 -- 2 keV \n(i.e., a soft X-ray excess). A fit to these data yields  $\\alpha$ $=$ \n$-$1.40$\\pm$0.03 in the EUV and soft X-rays, and $\\alpha$ $=$ $-$0.40$\\pm$0.03 \nin the hard X-rays; the break point is at $\\nu$ $=$ 10$^{17.1}$ Hz$^{-1}$ (1.3 \nkeV).\n\nNGC 5548 was monitored by the {\\it Extreme Ultraviolet Explorer} ({\\it EUVE}) \nover a two month period during 1993 March -- May \n(Marshall et al. 1997).  During this time, the EUV flux varied by a factor of \nfour from peak to minimum, and the average flux (corrected for Galactic \nneutral hydrogen absorption) was 135 $\\mu$Jy at $\\sim$76 \\AA.\nThese observations provide an important constraint on the EUV ionizing \ncontinuum, but were not used directly in our continuum fit for two reasons. \nFirst, the neutral hydrogen absorption due to our Galaxy is well known (see the \nprevious section), but there could be additional absorption along the \nline of sight. Second, the {\\it EUVE} flux, averaged over two months, may not \nbe representative of the average flux over many years. Given these caveats, we \nplot the {\\it EUVE} continuum point in Figure 2 for comparison with our adopted \ncontinuum; the error bar was determined from an estimate of $\\pm$10\\% \nuncertainty in \nGalactic N$_{HI}$ (see Murphy et al. 1996). The {\\it EUVE} point is slightly \nhigher than \nthe continuum fits in Figure 2, but appears to be consistent with our and Krolik \net al.'s \nadoption of a relatively steep continuum. If we use Krolik et al.'s continuum or \na continuum formed by joining the UV, {\\it EUVE}, and X-ray points with line \nsegments \n(in log space), the total EUV flux increases by factors of only 1.19 and 1.30, \nrespectively, and the flux at the frequency of the {\\it EUVE} observation \nincreases by \nfactors of 1.40 and 1.84, respectively. The effects of adopting these other \ncontinua are small, and will be discussed later in the paper.\n\n\\section{Photoionization Models}\n\nIn modeling the narrow-line emission of NGC 5548, we have adhered to our basic \nphilosophy of keeping the number of free parameters to a minimum, by using the \navailable observational constraints and the simplest assumptions possible.\nThe parameters are varied until the best agreement is obtained with the \nobserved line ratios, and additional input parameters are only included if \nthey are needed to provide a reasonable match to the majority of the  \nlines. Discrepancies between the model predictions and specific lines are then \ninvestigated to provide further insight into the physical conditions. In some \ncases we generate variations on the standard model using additional parameters \n(such as dust) or nonstandard values of the initial parameters (such as nonsolar \nabundances) to illustrate our ideas for resolving the discrepancies.\n\n\\subsection{Methodology}\n\nThe basic modeling methodology that we employ is described in \nKraemer et al. (1994) and the details of the photoionization code are given in \nKraemer (1985). To review the major points, we assume plane parallel \ngeometry, which is reasonable if the gas is ionized by radiation from a central \nsource at a distance that is large compared to the extent of the cloud. The gas \nis assumed to be atomic (i.e., there is no molecular component). For radiation \nbounded models, we stop the integration into the slab when the electron \ntemperature falls below 5000 K and there is no longer any significant line \nemission. The emission line photon escape is through the ionized face of the \nslab. Details of the treatment of dust in the models are described in Kraemer \n(1985). Since the work of Kraemer et al. (1994), we have added iron to the \nelements modeled in the code. The atomic data that we used can be found in \nPradhan \\& Peng (1994), and references contained therein, as well as through \nFerland's ``Cloudy and Associates'' World Wide Web site \n(http:\/\/www.pa.uky.edu\/$\\sim$gary\/cloudy).\nThe final output of these models is an emission line spectrum. The line\nstrengths are tabulated relative to H$\\beta$. In addition, the model\ngives the volume emissivity of H$\\beta$, from which we can determine\nthe mass of gas required to produce the observed line emission, the\nefficiency of production of H$\\beta$ photons, and an estimate of the covering \nfactor.\n\nIn order to keep the input parameters to a minimum, we kept two of them fixed \nfor our standard model: the shape of the ionizing continuum and the abundances. \nWe used the simplest possible ionizing continuum consistent with the \nobservations, as described in Section 2.2. In addition, we have assumed \nsolar elemental abundances for the standard model as follows (see Lambert \\& \nLuck 1978): He = 0.1, C = 3.4 x 10$^{-4}$, O = 6.8 x 10$^{-4}$, N = 1.2 x \n10$^{-4}$, Ne = 1.1 x 10$^{-4}$, S = 1.5 x 10$^{-4}$, Si = 3.1 x 10$^{-5}$, Mg = \n3.3 x 10$^{-5}$, Fe = 4 x 10$^{-5}$ (relative to hydrogen by number).\n\nOur photoionization models are parameterized in terms of the density of\natomic hydrogen (N$_{H}$) and the dimensionless ionization parameter at the \nilluminated face of the cloud:\n\n\\begin{equation}\nU = \\int^{\\infty}_{\\nu_0} ~\\frac{L_\\nu}{h\\nu}~d\\nu ~\/~ (4\\pi~D^2~N_{H}~c),\n\\end{equation}\n\n\\noindent where {\\it L$_{\\nu}$} is the frequency-dependent luminosity of \nthe ionizing continuum, {\\it D} is the distance between the cloud and the \nionizing source, and h$\\nu_{0}$ = 13.6 eV. \n\nWe show that we must add two enhancements to our standard model to obtain an \nacceptable match to the observations. First, we need two components of gas,  \ncharacterized by different ionization parameters and densities. Second, we show \nthat the inner component must be optically thin (i.e., radiation bounded) at the \nLyman edge (13.6 eV). We are then able to vary the ionization parameter and \ndensity of each component to match the observations. Of course, the resulting \nstandard model is an oversimplification, since it is likely that the NLR clouds \nare characterized by a number of different ionization parameters, densities, and \noptical depths. We have effectively averaged the initial conditions for each of \nthese components to fit the largest selection of line ratios. Given this \nsimplification, the two-component model gives a surprisingly good fit \nto the observations.\n\nThe narrow-line spectrum of NGC 5548 is dominated by high ionization lines,\nsuch as C~IV $\\lambda$1549, N V $\\lambda$1240, and [Ne V] $\\lambda\\lambda$\n3346, 3426, as well as the coronal lines of [Fe~VII] and [Fe~X]. This indicates \nthat there is a high ionization component relatively near the central source, \nbut presumably outside the BLR, since these lines are much narrower\n($\\leq$~500~km~s$^{-1}$ FWHM) than the broad lines ($\\sim$ 5000 km s$^{-1}$ \nFWHM).\nAlso present in the spectrum are relatively strong lines of [N II] \n$\\lambda\\lambda$6584, 6548 and [O II] $\\lambda$3727. These are typical of the \nnarrow emission line regions of many Seyfert 2 galaxies (Koski 1978; Shuder\n\\& Osterbrock 1981), and presumably arise in a component of relatively low \nionization and low density.\n\nThe choice of density for these models is based on the relative strengths\nof certain forbidden emission lines in the spectrum of NGC 5548. \nAn emission line will not be an important coolant in gas with sufficiently \nlarge electron density that collisional de-excitation of the line will dominate\nover radiative transition (see Osterbrock 1989).  For example, the observed \n[Ne~V]~$\\lambda$3426\/H$\\beta$\nratio indicates that the density of the component in which that emission\noriginates must be less than $\\sim$5~x~10$^{7}$ cm$^{-3}$ (DeRobertis \\& \nOsterbrock 1984). Given the hardness of the ionizing continuum, one would expect \nhigh temperatures in this highly ionized gas.\nThe lower limit on the density of this component can be estimated from the\nrelative strength of the [O~III] $\\lambda$5007 emission. The ratio of [O III] \n$\\lambda$5007\/ [Ne V] $\\lambda$3426\nis less than 5, which indicates either low density, highly ionized gas, as\nis often seen in the extended NLR of Seyfert 2 galaxies (Storchi-\nBergmann et al. 1996), or density greater than 10$^{6}$ cm$^{-3}$ and \nsome collisional supression of the $\\lambda$5007 line. The ratio of [O~III] \n$\\lambda$4363\/$\\lambda$5007 is very high ($\\sim$0.09). In the low density \nlimit, this ratio is\n$\\sim$7 x 10$^{-3}$ at a temperature of 10$^{4}$~K (Osterbrock 1989). \nAt this low density, it is unlikely that \ntemperatures consistent with photoionization equilibrium can increase this\nratio to the observed value, and probable that the large \n$\\lambda$4363\/$\\lambda$5007\nratio is due to the fact that some of this emission arises in gas of high\ndensity (i.e., $>$ 10$^{6}$ cm$^{-3}$). \nAt this density and level of ionization, the \n[N~II]~ $\\lambda\\lambda$6548, 6584 emission from this component will be \nnegligible, so there must be a lower ionization and lower density component \npresent. In order for the [N~II] lines to be among the principal coolants from \nthis component, its density must be less than 1 x 10$^{5}$ cm$^{-3}$. Other \nstudies (cf. Filippenko \\& Halpern 1984; Filippenko 1985; Kraemer et al. 1994) \nhave shown that a range of densities in the NLR of Seyfert galaxies is likely, \nso it is not surprising that this condition exists in NGC 5548.\n\nTo summarize, the range in density, along with the presence of emission lines \nfrom a wide range of ionization states, indicates that more than one model \ncomponent is needed to fit the NLR spectrum of NGC 5548. \nThe existence of strong high ionization lines such as C IV \n$\\lambda$1549, N V $\\lambda$1240, and [Ne V] $\\lambda\\lambda$3346, 3426, and our \nevidence for high densities in the region that they are produced, requires a \ncomponent of gas relatively close to the central source. The weakness of Mg II \n$\\lambda$2800 and [O I] $\\lambda\\lambda$6300, 6364 indicate that these gas \nclouds lack a significant partially ionized zone, and therefore must be\n{\\it optically thin} to the ionizing radiation (or ``matter bounded''). \nWe further investigate the conditions in these two components below.\n\n\\subsection {Model Results and Comparison to Observations}\n\nOur approach in modeling NGC 5548 was to fit the high ionization\ncomponent first and then add components as needed to fit the lower\nionization lines (in the end, only one additional component was needed). Given \nthe constraints and assumptions described in the\nprevious section, we arrived at values of N$_{H}$ $=$ 1 x 10$^{7}$ cm$^{-3}$ and\nU $=$ 10$^{-1.5}$ for the high ionization component. Substantially lower \ndensities would result in \n[O~III]~$\\lambda$5007 being too strong, and higher densities would quench the \n[Ne V] emission. A higher ionization parameter is possible but, given our EUV \ncontinuum, would not increase the relative strengths of any of the high \nionization lines other than [Fe X] $\\lambda$6374, at the expense of putting this \ncomponent at distances much closer than $\\sim$1 pc from the continuum source \n(see Section 4). Models were run to varying optical depth at the Lyman limit \n$\\tau_{0}$, with the constraint that the Mg II and [O I] lines could not become \ntoo strong. After comparing the results of models run with $\\tau_{0}$ $=$ 1.5 to \n10, we found that $\\tau_{0}$ $=$ 2.5 gave the best fit. The emission line \nspectrum from this model, INNER, is given in Table 2.\n\nA second component, OUTER, was needed to fit the lower ionization lines. \nWe found that U $=$ 10$^{-2.5}$ and N$_{H}$ $=$ 2 x 10$^{4}$ cm$^{-3}$ gave a \ngood simultaneous fit to the [O~III]~$\\lambda$5007\/[O~II]~$\\lambda$3727 and \n[N~II] $\\lambda$6584\/H$\\beta$ ratios. Unlike our models for Mrk 3 and I~Zw~92 \n(Kraemer \\& Harrington 1986; Kraemer et al. 1994), there was no need to add a \nthird component for NGC 5548, since there is no obvious contribution from a \ncomponent of very low density ($<$ 10$^{3}$ cm$^{-3}$) low \nionization gas, such as very strong [O II] $\\lambda$3727 and [N~I] $\\lambda$5200 \nlines. The resulting emission line spectrum from OUTER is also included in \nTable 2. (We will discuss two variations on INNER and OUTER in the next \nsection.)\n\nIn order to fit the observed (and dereddened) narrow-line spectrum of NGC 5548, \nwe combined the output spectrum of the two standard components INNER and OUTER. \nIn previous studies, we attempted to weigh the contributions from each component \nto fit specific emission line ratios. For the model of NGC 5548, we simply took \nan equal contribution from INNER and OUTER. The relative simplicity\nof the narrow-line spectrum and lack of a strong contribution from very \nlow-ionization gas makes such a simple fit possible. The combined spectrum\nis given, along with the dereddened observed spectrum for comparison, in\nTable 3 (horizontal lines in the table indicate that the models do not predict \nthe strengths of these emission lines). \n\nComparison of the model predictions to the dereddened observed spectrum in Table \n3 shows agreement, to within the errors, for most of the lines. In particular, \nthese include C~IV~$\\lambda$1549, He II \n$\\lambda$1640, [O~II] $\\lambda$3727, [Ne III] $\\lambda$3869, [N II] \n$\\lambda$6584, and the Balmer decrement.\nIn radiation bounded gas,\nthe ratio of the He II lines to H$\\beta$ is strongly dependent on the shape of \nthe ionizing continuum, because neutral hydrogen is the \ndominant absorber of ionizing radiation between 13.6 eV and 54.4 eV, while above \n54.4 eV, singly ionized helium dominates.\nIf there is a component of matter bounded gas, He II\/H$\\beta$ is less easily \npredicted. The accuracy of our fit to this ratio indicates that the relative \ncontributions of the matter and radiation bounded components are approximately \ncorrect, given the observational constraints on the ionizing continuum.\nThe fact that we have a reasonable fit \nfor lines that span a wide range of ionization and critical densities supports \nour values for density and ionization parameter. Most of the \ndiscrepancies between the observations and models are in the lowest and highest \nionization lines, which we will address below. \n\n\\subsection{Discrepancies and Possible Explanations}\n\nFirst, we address differences between the predicted and observed ratios for the \nlow ionization lines. The Mg II $\\lambda$2800 and [O I] $\\lambda\\lambda$6300, \n6364 lines are still predicted to be too strong by our standard model, by \nfactors $\\geq$ 6 and 2.5, respectively. Nearly all this emission is coming from \nOUTER. Two factors determine the strength of the [O I] lines: the hardness of \nthe ionizing continuum and the physical depth of the emission line clouds. Since \nit appears that we have a good fit for the ionizing continuum, the weakness in \nthe observed [O~I] lines gives a limit on the depth of the clouds. Truncating \nthe integration of OUTER at $\\tau_{0}$ $\\approx$ 1000 would give a better fit to \nthe [O I] without affecting the other important line ratios. This results in a \ncloud depth of $\\sim$2.5 x 10$^{15}$ cm.\nThe overprediction of the Mg II $\\lambda$2800 line strength presents a somewhat\ndifferent problem. In order to reduce the contribution of this line from INNER, \nwe assumed a matter bounded model for this component. To provide a better match \nto the observation of little or no Mg II emission, we can reduce the model \ncontribution by modifying OUTER. The Mg$^{+}$ emissivity is greatest near the \nH$^{+}$\/H$^{0}$ transition zone in OUTER, so a \nsimple truncation at much lower optical depths is not feasible, as it would have\na much greater effect on the other line ratios. \n\nAn obvious explanation for the weak observed Mg II is depletion of the magnesium \ninto dust grains, along with suppression of the resonance photons by multiple \nscatterings and eventual absorption by dust. This was suggested in Paper I (cf. \nKraemer \\& Harrington 1986; Ferland 1992). For comparison with our standard \nmodel, we generated a version of OUTER that includes dust, assuming \na dust to gas ratio that is 30\\% of that found in the Galactic interstellar \nmedium, with equal amounts of graphite and silicate grains and accompanying \ndepletions. These assumptions were made to avoid biasing our results by simply \nhaving all of the Mg depleted into dust grains. We assumed relative element \ndepletions as calculated by Seab \\& \nShull (1983), and the grain size distributions determined by Mathis et al. \n(1977) and Draine \\& Lee (1984); details of the treatment of dust in the code \nare given by Kraemer (1985). The results of the model are given in Table 2, and \nnot only show a significant drop in the relative strength of Mg II \n$\\lambda$2800, but also a drop in the Ly$\\alpha$ strength, as is expected due to \nthe preferential dust absorption of multiply scattered UV resonance lines. The \nlower Ly$\\alpha$\/H$\\beta$ ratio is a better fit to the observations. A \nsubstantially larger dust-to-gas ratio than we assumed would result in a \nLy$\\alpha$\/H$\\beta$ ratio that is lower than observed. Therefore, \nit is likely that there is some dust mixed in with the low-ionization gas, \nalthough with a lower dust to gas ratio than found in the ISM, and that \ndepletion coupled with the resonance line suppression explain the weak Mg II. \n\nSecond, we address discrepancies in the high ionization lines.\nSpecifically, the lines of N~V, [Ne~V], [Fe~VII], and [Fe~X] are too weak by \nfactors of 2 to 4 compared to the observations.\nAs we mentioned earlier, the density is well constrained, so increasing the \nionization parameter brings the gas well within 1 pc, into the realm of \nthe BLR. However, these lines are relatively narrow (FWHM $\\leq$ 500 km \ns$^{-1}$, see Moore et al. 1996) and they are not likely to arise very close to \nthe BLR. The strengths of the high ionization lines can also be enhanced \nrelative to H$\\beta$ by truncating the integration of INNER at a lower optical \ndepth. However, this has the problem of enhancing the He II emission relative to \nH$\\beta$ in the model. More likely solutions to the problem of underpredicting \nthe high ionization lines include 1) shock ionization, 2) a large ``blue \nbump'' in the EUV continuum, or 3) supersolar abundances.\n\nPredicting the strengths of the coronal lines has always been a problem with \nsimple photoionization models, as Viegas-Aldrovandi \\& Contini (1989) discuss in \nsome detail for the Fe lines. They suggest that there may be shocked gas mixed \nin with the photoionized clouds and that these high ionization lines may arise \nthere. Although this is certainly a possible factor, there may be other \nplausible explanations which avoid adding another level of complexity.\n\nAn obvious way in which the coronal lines might be enhanced is if there\nwere a component of ionizing radiation that contributed significantly\nat energies between 100 and 500 eV. Although there has been some\nspeculation about the presence of a ``blue bump'' in the EUV, recent work\nby Zheng et al. (1997) on low-redshift quasars shows that the {\\it near} EUV \ncontinuum is likely to be much steeper than previously \nsupposed. In NGC 5548, the {\\it EUVE} continuum point in Figure 2 is further \nconfirmation that a large blue bump is not present in the spectrum \nof NGC 5548. Another possibility may be diffuse radiation from the intercloud \nmedium. Tran (1995) has shown that there may be a contribution to the continuum \nradiation in some Seyfert 2 galaxies from thermal emission from the intercloud\nmedium responsible for the scattering of the hidden BLR emission into\nthe observer's line of sight. If a similar medium with temperatures\n$\\approx$ 5~x~10$^{5}$~K exists in the NLR of NGC 5548, it is possible that \nfree-free radiation and line emission that arise within it may contribute to the \nionization of the inner narrow-line gas. Although this component would be weak \ncompared to the continuum radiation emitted by the central source, it could have\na significant local contribution to the ionization balance of clouds existing\nwithin this inner region. However, recent observations suggest that the extended \nUV continuum seen in some Seyferts may be due to starbursts (Heckman et al. \n1997), which would not contribute to the high ionization lines.\n\nIn studies of medium redshift QSOs, Ferland et al. (1996) found\nevidence of supersolar abundances. There is no direct evidence of\nelemental enhancements in Seyfert galaxies, but it is certainly not\nimplausible, particularly near the nucleus, where the most intense activity\noccurs. As Oliva (1996) points out, the coronal line emission will be enhanced \nproportionally to the abundance of the atomic species.\nAs a comparison, we ran a version of INNER with a heavy element\nabundance that is twice solar, and the results are shown in Table 3. In \nparticular,\nthe relative strengths of the [Fe VII] and [Ne V] lines have increased,\nwhile many lines, such as C IV $\\lambda$1549 and C III] $\\lambda$1909, show \nlittle change. It is possible, then, that the observed strength of \nsome of these lines is in part due to enhanced abundances. Note that we have not \nattempted to adjust the increase in abundances to fit assumptions about the\ntype of star formation that might be expected.\n\n\\section{Discussion}\n\nFrom our standard model, we can estimate several global properties of the NLR in \nNGC 5548, including the covering factor and physical size, in addition to more \nlocal properties, including optical depth and presence of dust.\nWe determine the covering factor of the NLR gas from the observed and model \nvalues of the ``conversion efficiency'', $\\eta$ , which is the ratio of H$\\beta$ \nphotons to ionizing photons. The covering factor is given by\nC~$=$~$\\eta$(observed)\/$\\eta$(model). A value of C $>$ 1 would indicate that the \nionizing radiation is anisotropic, which we would not expect to be the case \nfor a Seyfert 1 galaxy, since the central source is seen directly in such \nobjects. Assuming Ho= 75 km s$^{-1}$ Mpc$^{-1}$, the observed H$\\beta$ \nflux corresponds to a luminosity of 3.8 x 10$^{40}$ ergs s$^{-1}$, or 9.3 x \n10$^{51}$  H$\\beta$ photons s$^{-1}$. From the continuum observations described \nin section 2.2, we calculate a total luminosity of ionizing photons of 1.09 x \n10$^{54}$ s$^{-1}$. This yields an observed $\\eta$ = 0.009. Our fit to the \nobserved emission line spectrum assumed that each component in our model \ncontributed 50\\% of the H$\\beta$ emission. The resulting values of $\\eta$ were \n0.06 for INNER and 0.11 for OUTER, and the covering factors are 0.07 and 0.04, \nrespectively, so C(NLR) = 0.11. The value is small compared to those found for \nSeyfert 2 galaxies, which are often $>$ 1 (Kinney et al. 1991; Kraemer et al. \n1994), and there is no evidence for anisotropic radiation. Note that this \nestimate of covering factor does not include the BLR clouds, which contribute at \nleast 50\\% of the flux in many of the strong lines, even when NGC 5548 is in its \nlowest state (Paper I). \n\nGiven the ionization parameters and densities of the two components from our \nstandard model, as well as the ionizing luminosity, the \ncharacteristic sizes (i.e., radii) for the two emitting regions are 1 pc for \nINNER, and 70 pc for OUTER. \n(Using the higher continuum luminosity given by the {\\it EUVE} point in Figure 2 \nwould increase these values by a factor of only $\\sqrt{1.30}$, or 1.14.)\nThus, the NLR of NGC 5548 is {\\it physically} \ncompact, since the size of OUTER is much smaller than typical values of 200 -- \n1000 pc determined for Seyfert 2 galaxies, using the same methods that we have \ndescribed in this paper (Kraemer \\& Harrington 1986; Kraemer et al. 1994).\nAlthough there have been reports of extended emission from NGC 5548 (Wilson et \nal. 1989), the contribution to the integrated emission line spectrum from this \nregion is small (Peterson et al. 1995); we estimated the contribution to the \nnarrow UV lines outside of a 1$''$ aperture (330 pc for H$_{0}$ $=$ 75 km \ns$^{-1}$ Mpc$^{-1}$) to be only $\\sim$20\\%. This is consistent with our finding \nthat the majority of the narrow emission must arise in a region with a \n``diameter'' of 140 pc.\n\nPogge (1989) and Schmitt \\& Kinney (1996) claim that the\n{\\it apparent} size of the NLR in a Seyfert 1 galaxy is typically much \nsmaller than that of a Seyfert 2 galaxy, although this may be due to a selection \neffect, since most of the Seyfert 2s in the {it HST} archive were selected on \nthe basis of their extended emission (Wilson 1997).\nNevertheless, it is clear that the majority of the Seyfert 1 galaxies in these \nstudies are apparently compact. This cannot be explained \nby viewing angle alone, since the opening angles \nof the presumed ionization cones are large, and in the simplest version of the \nunified model, the apparent extent of most Seyfert galaxies should be much \nlarger, even if viewed ``pole-on'' (Schmitt \\& Kinney 1996).\nOur results show that a possible explanation for the small apparent size of the \nNLRs in many Seyfert 1 galaxies is that they are truly (i.e., physically) \ncompact. Schmitt \\& Kinney explain this phenomenon as a result of\nthe orientation of the obscuring torus with respect\nto the plane of the galaxy. However, their model does not explain the dominance\nof the high ionization lines in NGC 5548 and many other Seyfert 1 galaxies, a \nfeature not generally seen in Seyfert 2 galaxies \n(Koski 1978; Shuder \\& Osterbrock 1981). Thus, if the unified model applies we \nmight expect that the narrow emission line spectrum of this Seyfert 1 galaxy \nwould resemble that of Seyfert 2 galaxies, and, further, we would expect \nto see a noticeable contribution to the spectrum from gas hundreds of parsecs \nfrom the central source. It is possible that the high ionization region in \nSeyfert 2 galaxies is obscured by a torus, but this does not explain the absence \nof a low ionization component in NGC 5548. Not only do our models show that the \nnarrow-line spectrum can be well fitted without such a component, but it is \nclear that the NLR of NGC 5548 is dominated by high ionization gas that must \nbe located close to the central source.\n\nA few Seyfert 1 galaxies do indeed show significant NLR emission at\nlarge distances from the central source. For example, {\\it HST} observations of \nNGC 4151 (Evans et al. 1993; Hutchings et al. 1997) reveal an array of knots and \nfilaments out to a kiloparsec that are almost certainly ionized by the central \ncontinuum. It may be that some Seyfert 1 galaxies, such as NGC 4151, have \nextended emission line regions and narrow lines in their spectra that closely\nresemble Seyfert 2 galaxies. Cohen (1983) studied the optical narrow\nline spectra of a group of Seyfert 1 galaxies and found some resemblance\nalthough, as a group, they appeared to be of somewhat higher ionization.\nCertainly, some of the galaxies in his study have spectra that are\nindistinguishable from those of type 2 Seyferts, but others, like NGC 5548, \nappear to be dominated by high ionization lines, a condition that appears to be \nrare among Seyfert 2 galaxies. It would be extremely interesting if one could \ndetermine if these high ionization objects are as compact as NGC 5548 appears to \nbe. Unfortunately, optical spectra alone are insufficient for detailed \nmodeling of the NLR, and thus far, accurate measurements of the narrow-line \nstrengths in the UV have only been obtained for NGC 5548\nand NGC 4151 (Ferland \\& Mushotzky 1982).\n\nAs we stated above, our finding that the inner component of gas is\noptically thin at the Lyman limit is based both on the absence of strong\nMg II emission and the relative strength of the high ionization lines.\nIf this is indeed true, not only for NGC 5548 but for other Seyfert 1\ngalaxies, it may be a clue to the origin of the inner narrow-line\ngas. Thin filaments or knots of reasonably high density ($\\sim$ 10$^{7}$ \ncm$^{-3}$) could be the result of outflow from the BLR, either as condensations\nin an expanding intercloud medium or as ``tails'' of BLR clouds, driven\nout by radiation pressure. If so, the small physical depths of the clouds\ninferred from our modeling ($\\sim$ 10$^{14}$ cm) may constrain the sizes of \ntheir BLR progenitors. Note that there has been some recent success in \nmodels of the BLR using a component of\noptically thin clouds (Shields, Ferland, \\& Peterson 1995).\nThere have been other studies suggesting that the \nclouds in the NLR are matter bounded or that some mix of radiation bounded and \nmatter bounded clouds exist (Wilson et al. 1997; Viegas-Aldrovandi 1988). Such a \nmix might give rise to the filamentary structure seen in some [O III] images of \nSeyfert galaxies rather than the typical molecular clouds found in spiral \ngalaxies, and may give a clue about the origin of the NLR gas.\n\nOne other point regarding the NLR gas is that it appears that there\nmay be some dust present within the outer components of the emitting\nregion. Dust is certainly able to exist in clouds at this proximity\nto the central source, with dust temperatures reaching a few hundred\nK (Kraemer \\& Harrington 1986). Nevertheless, the history of the dust\nin this gas is unknown. Our finding that the outer clouds\nmay not be purely radiation bounded would indicate that they are not\nsimply interstellar molecular clouds that have an outer shell ionized\nby the central source. Our finding that there is probably dust mixed in with\nthis gas will be important in determining the origin of this component.\n\nFinally, Moore et al. (1996) find a correlation of ionization potential of the \nnarrow emission lines with velocity width in NGC 5548. Combined with our finding \nthat, to first order, the ionization level decreases with distance, this \nindicates that the radial velocities decrease with increasing distance. This \ntrend is also seen in a more direct fashion in spatially resolved spectra of the \ninner NLR in NGC 4151 (Hutchings et al. 1997).\n\n\\section{Conclusions}\n\nWe have analyzed UV and optical spectra of the Seyfert 1 galaxy NGC 5548 that \nwere obtained when it was at an historical minimum. We were able to \nisolate the narrow emission line components (which do not vary in flux on short \ntime scales), due to the relative weakness of the more rapidly variable broad \nemission lines that are usually blended with the narrow components.\nWe have constructed photoionization models of the narrow-line region of this \ngalaxy, and are able to successfully match the observed dereddened ratios of a \nlarge number of emission lines to within the errors, with the \nexceptions noted in Section 3.3. Since we used the best direct observational \nevidence of the shape of the ionizing continuum, rather than making adjustments \nbased on fitting emission line ratios, the quality of this fit is particularly \nsatisfying. The fact that a good fit was obtained for the permitted emission \nlines, such as C~IV~$\\lambda$1549, and the forbidden lines, such as [Ne III] \n$\\lambda$3869, [O III] $\\lambda$5007, [O II] $\\lambda$3727, [N II] \n$\\lambda$6584, etc., indicates that the range of physical conditions assumed \nin these models is approximately correct.\n\nFrom our analysis and modeling of these spectra, we can make several statements \nregarding the physical conditions in the NLR of NGC 5548. First, it is clear \nthat the principal source of ionization in the NLR of NGC 5548 is the central \ncontinuum source. This conclusion is borne out by the quality of the fit to the \nemission line spectrum. The NLR covering factor is reasonably small (C $=$ \n0.11), so the NLR gas does not need to intercept much of the ionizing continuum \nto produce the observed emission lines fluxes. A second conclusion is that the \nhighly ionized gas in the inner part of the NLR appears to be optically thin at \nthe Lyman limit ($\\tau_{0}$ $\\approx$ 2.5), which yields constraints on the \nphysical depths of these clouds and may provide a clue to their origin. We have \nalso presented evidence for supersolar abundances in the inner portion of the \nNLR, and dust in the outer portion. \n\nThe most important conclusion that we have reached in this study is\nthat the NLR of NGC 5548 is physically compact, with the majority of emission \ncoming from a distance $\\leq$~70 pc from the nucleus. By contrast, the unified \nmodel of Seyfert galaxies suggests that the {\\it physical} dimensions of the NLR \nin Seyfert 1 and 2 galaxies should be similar. Additional studies of the type \nthat we have presented in this paper, particularly of other Seyfert 1 galaxies, \nare important for testing this aspect of the unified model.\n\n\\acknowledgments\n\nWe thank the referee, Andrew Wilson, for helpful comments and suggestions.\nWe are grateful to Fred Bruhweiler and Pat Harrington for helpful discussions on \nthe physical properties of iron and the availability of atomic data.\nS.B.K. and D.M.C. acknowledge support from NASA grant NAG~5-4103.\nA.V.F acknowledges support from NASA grant NAG~5-3556, and B.M.P. \nacknowledges support from NSF grant AST-9420080.\n\n\\clearpage\t\t\t       \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{SecIntro}\nA crucial step in the history of General Relativity (GR) was Einstein's adoption of the principle of general covariance which states that the form of our physical laws should be independent of any choice of coordinate systems. The conceptual benefits of writing a theory in a coordinate-free way are immense. A generally covariant formulation of a theory has at least two major benefits: 1) it more clearly exposes the theory's geometric background structure, and 2) it thereby helps clarify our understanding of the theory's symmetries (i.e., its structure\/solution preserving transformations). It does both of these by disentangling the theory's substantive content from representational artifacts which arise in particular coordinate representations~\\cite{Pooley2015,Norton1993,EarmanJohn1989Weas}. Thus, general covariance is an indispensable tool for a modern understanding of spacetime theories. \n\nMotivated by quantum gravity, one may wish to extend these notions to quantum spacetime theories (whatever those are). Relatedly, one might want to extend these notions to discrete spacetime theories (i.e., lattice theories\\footnote{Given the results of this paper and of \\cite{DiscreteGenCovPart1}, calling these ``lattice theories'' can be misleading. This would be analogous to referring to continuum spacetime theories as ``coordinate theories''. As I will discuss, in both cases the coordinate systems\/lattice structure are merely representational artifacts and so do not deserve ``first billing'' so to speak. All lattice theories are best thought of as lattice-representable theories. Similarly, the term ``discrete spacetime theories'' ought to be here read as ``discretely-representable spacetime theories''. As discussed here (and in \\cite{DiscreteGenCovPart1}), the defining feature of such theories is that they have a finite density of degrees of freedom, see the work of Achim Kempf~\\cite{UnsharpKempf,Kempf2003,Kempf2004a,Kempf2004b,Kempf2006}}.). In \\cite{DiscreteGenCovPart1} I developed two analogs of general covariance for such discrete spacetime theories in a non-Lorentzian setting. The aim of this paper is to extend these results to a Lorentzian setting. Indeed, the analysis provided here is nearly identical to the one carried out in \\cite{DiscreteGenCovPart1}, although each paper is self-contained.\n\nIn either setting these discrete analogs of general covariance reveal that lattice structure is rather less like a fixed background structure or a fundamental part of some underlying manifold and rather more like a coordinate system, i.e., merely a representational artifact. Indeed, these discrete analogs are built upon a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. \n\nThis paper is largely inspired by the brilliant work of mathematical physicist Achim Kempf~\\cite{Kempf_1997,UnsharpKempf,Kempf2000b,Kempf2003,Kempf2004a,Kempf2004b,Kempf2006,Martin2008,Kempf_2010,Kempf2013,Pye2015,Kempf2018} among others~\\cite{PyeThesis,Pye2022,BEH_2020}. A key feature present both here and in Kempf's work is the sampling property of bandlimited function revealed by the Nyquist-Shannon sampling theory~\\cite{GARCIA200263,SamplingTutorial,UnserM2000SyaS}. I review sampling theory in more detail in Sec.~\\ref{SecSamplingTheory}, but let me overview here. Bandlimited functions are those with have a limited extent in Fourier space (i.e., compact support). Bandlimited functions have the following sampling property: they can be exactly reconstructed knowing only the values that they take on any sufficiently dense sample lattice. What ``sufficiently dense'' means is fixed in terms of the size of the function's support in Fourier space.\n\nNyquist-Shannon sampling theory was first discovered in the context of information processing as a way of converting between analog and digital signals (i.e., between continuous and discrete information). Sampling theory found its first application in fundamental spacetime physics with Kempf's \\cite{Kempf_1997,UnsharpKempf}, ultimately leading to his thesis that ``Spacetime could be simultaneously continuous and discrete, in the same way that information can be'' \\cite{Kempf_2010}. Kempf's thoughts on these topics is the primary inspiration for this paper and deserves wider appreciation by the philosophy of physics community. For an overview of Kempf's works on this topic see~\\cite{Kempf2018}.\n\nMy thesis in \\cite{DiscreteGenCovPart1} is in broad agreement with Kempf's with one crucial alteration. I stress that the sampling property of bandlimited functions indicates that bandlimited physics can be simultaneously \\textit{represented as} continuous and discrete, (i.e., on a continuous or discrete spacetime). However, I further argue (both here and in \\cite{DiscreteGenCovPart1}) that when one investigates these two representations one finds substantial issue with taking the discrete representation as fundamental. These issues stem from the rich analogy between the lattice structures and coordinate systems mentioned above.\n\nThis analogy is supported here (and in \\cite{DiscreteGenCovPart1}) by the three lessons each of which tell against an intuitions one is likely to have regarding lattice structure. To motivate these (wrong) intuitions, consider the following situation.\n\nSuppose that after substantial empirical investigation of our micro-physical reality we find what appear to be ``lattice artifacts''. For instance, we may find ourselves restricted to only quarter rotation, or one-sixth rotation symmetries. Intuitively, this would suggest that the world is fundamentally set on a lattice of the kind shown in Fig.~\\ref{FigLat}, i.e., a square or hexagonal lattice.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/FigLat.pdf}\n\\caption{Three of the lattice structures in space considered throughout this paper. The arrows indicate the indexing conventions for the lattice sites. Repeated in time, these give us something like a square 2D lattice, a cubic 3D lattice, and a hexagonal 3D lattice respectively. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\\label{FigLat}\n\\end{figure}\n\nSuppose that we have great predictive success when modeling the world as being set on (for instance) a square lattice with next-to-nearest neighbor interactions. Would this in any way prove that the world is fundamentally set on such a lattice? No, all this would prove is that the world can be \\textit{faithfully represented} on such a lattice with such interactions, at least empirically. Anything can be faithfully represented in any number of ways, this is just mathematics. Some extra-empirical work must be done to know if we should take the lattice structure appearing in this representation seriously. That is, we must ask which parts of the theory are substantive and which parts are merely representational? The discrete analogs of general covariance developed here (and in \\cite{DiscreteGenCovPart1}) answer this question: lattice structures are coordinate-like representational artifacts and so ultimately have no physical content.\n\nTo flesh out the contrary received position however, let us proceed without this analogy for the moment. We can ask: beyond merely appearing in our hypothetical empirically successful theory, what reason do we have to take the lattice structures which appear in this theory seriously? Well, intuitively the lattice structures appear to play a very substantial role in these theories, not merely a representational one. One likely has the following three interconnected first intuitions regarding the role that the lattice and lattice structure play in discrete spacetime theories:\n\\begin{itemize}\n\\item[1.] They restrict our possible symmetries. Taking the lattice structure to be a part of the theory's fixed background structure, our possible symmetries are limited to those which preserve this fixed structure. Intuitively a theory set on a square lattice can only have the symmetries of that lattice. Similarly for a hexagonal lattice, or even an unstructured lattice.\n\\item[2.] Differing lattice structures distinguishes our theories. Two theories with different lattice structures (e.g., square, hexagonal, irregular, etc.) cannot be identical. As suggested above they have different fixed background structures and as therefore have different symmetries.\n\\item[3.] The lattice is fundamentally ``baked-into'' the theory. Firstly, it is what the fundamental fields are defined over: they map lattice sites (and in \\cite{DiscreteGenCovPart1} times) into some value space. Secondly, the bare lattice is what the lattice structure structures. Thirdly, it is what limits us to discrete permutation symmetries in advance of further limitations from the lattice structure.\n\\end{itemize}\nThese intuitions will be fleshed out and made more concrete in Sec.~\\ref{SecKlein1}. However, as this paper demonstrates, each of the above intuitions are doubly wrong and overhasty.\n\nWhat goes wrong with the above intuitions is that we attempted to directly transplant our notions of background structure and symmetry from continuous to discrete spacetime theories. This is an incautious way to proceed and is apt to lead us astray. Recall that, as discussed above, our notions of background structure and symmetry are best understood in light of general covariance. It is only once we understand what is substantial and what is merely representational in our theories that we have any hope of properly understanding them. Therefore, we ought to instead first transplant a notion of general covariance into our discrete spacetime theories and then see what conclusions we are led to regarding the role that the lattice and lattice structure play in our discrete spacetime theories. This transplant has been done in a non-Lorentzian setting in \\cite{DiscreteGenCovPart1}. Here I extend these results to a Lorentzian setting.\n\nThis paper will teach us three lessons each of which negates one of the above intuitions about the role that lattice structure plays in discrete spacetime theories.\n\nFirstly, as I will show, taking any lattice structure seriously as a fixed background structure systematically under predicts the symmetries that discrete theories can and do have. Indeed, as I will show neither the bare lattice itself nor its lattice structure in any way restrict a theory's possible symmetries. In \\cite{DiscreteGenCovPart1}, for non-Lorentzian theories I have shown that there is no conceptual barrier to having a theory with continuous translation and rotation symmetries formulated on a discrete lattice. Indeed, in \\cite{DiscreteGenCovPart1} I presented a perfectly rotation invariant lattice theory. As I discuss in \\cite{DiscreteGenCovPart1}, this is analogous to the familiar fact that there is no conceptual barrier to having a continuum theory with rotational symmetry formulated on a Cartesian coordinate system. Here, I repeat this analysis in a Lorentzian context. In Sec.~\\ref{PerfectLorentz}, I present a perfectly Lorentzian lattice theory. \n\nSecondly, as I will show, discrete theories which are initially presented to us with very different lattice structures (i.e., square vs. hexagonal) may nonetheless turn out to be completely equivalent theories or to be overlapping parts of some larger theory. Moreover, given any discrete theory with some lattice structure we can always re-describe it using a different lattice structure. As I will discuss, this is analogous to the familiar fact that our continuum theories can be described in different coordinates, and moreover we can switch between these coordinate systems freely.\n\nThirdly, as I will show, in addition to being able to switch between lattice structures, we can also reformulate any discrete theory in such a way that it has no lattice structure whatsoever. Indeed, we can always do away with the lattice altogether. As I will discuss, this is analogous to the familiar fact that any continuum theory can be written in a generally covariant (i.e., coordinate-free) way.\n\nThese three lessons combine to give us a rich analogy between lattice structures and coordinate systems. It is from this rich analogy that the central claims of this paper follow. Namely, from this analogy it follows that the lattice structure supposedly underlying any discrete ``lattice'' theory has the same level of physical import as coordinates do, i.e., none at all. Thus, as I argued in \\cite{DiscreteGenCovPart1}, the world cannot be ``fundamentally set on a square lattice'' (or any other lattice) any more than it could be ``fundamentally set in a certain coordinate system''. Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly merely representational. Spacetime cannot be a lattice (even when it might be representable as such). Specifically, I claimed that properly understood, there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories. This paper extends these conclusions to a Lorentzian context.\n\nOnce one begins thinking of lattices as a merely representational structure, a path opens for perfectly Lorentzian lattice theories. As proponents of causal set theory correctly point out, no single fixed spacetime lattice is Poincar\\'e invariant. This (apparently) spells big trouble for any lattice-based Lorentzian theories. They, however, avoid this issue by considering instead a random Poisson sprinkling of lattice points which does not pick out any preferred direction and hence does not explicitly break Poincar\\'e symmetry, at least on average. However, given the deflationary position this paper takes towards lattices, I claim there is no issue to be avoided. Like coordinate systems, lattice structures are just a representational tool for helping us express our theory. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Cartesian coordinates are distorted under Lorentz boosts, but we can still use them to describe our Lorentzian theories without issue. The same is true of lattices. Indeed, in Sec.~\\ref{PerfectLorentz} I will present a perfectly Lorentzian lattice theory.\n\n\\subsection{Outline of the Paper}\nIn Sec.~\\ref{SecSevenKG}, I will introduce seven discrete Klein Gordon equations in an interpretation-neutral way and solve their dynamics. Then, in Sec.~\\ref{SecKlein1}, I will make a first attempt at interpreting these theories. I will (ultimately wrongly) identify their underlying manifold, locality properties, and symmetries. Among other issues, a central problem with this first attempt is that it takes the lattice itself to be the underlying spacetime manifold and thereby unequivocally cannot support continuous translation and rotation symmetries. This systematically under predicts the symmetries that these theories can and do have.\n\nIn Sec.~\\ref{SecKlein2}, I will provide a second attempt at interpreting these theories which fixes this issue (albeit in a slightly unsatisfying way). In particular, in this second attempt I deny that the lattice is the underlying spacetime manifold. Instead, I ``internalize'' it into the theory's value space. Fruitfully, this second interpretation does allow for continuous translation and rotation symmetries and even a (limited) Lorentz boost symmetry. However, the key move here of ``internalization'' has several unsatisfying consequences. For instance, the continuous symmetries we find here are all classified as internal (i.e, associated with the value space) whereas intuitively they ought to be external (i.e, associated with the manifold).\n\nWe thus will need a third attempt at interpreting these theories which externalizes these symmetries. Sec.~\\ref{SecExtPart1} - Sec.~\\ref{SecExtPart2} lay the groundwork for this third interpretation. In particular, they describe a principled way of 1) inventing a continuous spacetime manifold for our formerly discrete theories to live on and 2) embedding our theory's states\/dynamics onto this manifold as a new dynamical field. In the middle of this, in Sec.~\\ref{SecSamplingTheory}, I will provide an informal overview of the primary mathematical tools used in the latter half of this paper. Namely, I will review the basics of Nyquist-Shannon sampling theory and bandlimited functions. \n\nWith this groundwork complete, in Sec.~\\ref{SecKlein3} and Sec.~\\ref{SecKlein3Extra} I will provide a third attempt at interpreting these seven theories which fixes all issues arising in the previous two interpretations. For instance, like in my second attempt, this third interpretation can support continuous translation and rotation symmetries as well as a (limited) Lorentz boost symmetry. However, unlike the second attempt it realizes them as external symmetries (i.e., associated with the underlying manifold, not the theory's value space).\n\nIn Sec.~\\ref{SecDisGenCov}, I will review the lessons learned in these three attempts at interpretation. As I will discuss, the lessons learned combine to give us a rich analogy between lattice structures and coordinate systems. As I will discuss, there are actually two ways of fleshing out this analogy: one internal and one external. This section spells out these analogies in detail, each of which gives us a discrete analog of general covariance. I find reason to prefer the external notion, but either is likely to be fruitful for further investigation\/use. Sec.~\\ref{PerfectLorentz} provides us with a perfectly Lorentzian lattice theory as promised.\n\nFinally, in Sec. \\ref{SecConclusion} I will summarize the results of this paper, discuss its implications, and provide an outlook of future work.\n\nFor comments on what this means for the the dynamical vs geometrical spacetime debate~\\cite{EarmanJohn1989Weas,TwiceOver,BelotGordon2000GaM,Menon2019,BrownPooley1999,Nonentity,HuggettNick2006TRAo,StevensSyman2014Tdat,DoratoMauro2007RTbS,Norton2008,Pooley2013} see \\cite{DiscreteGenCovPart1}. Here I will focus on the implications this work has for quantum gravity especially causal set theory~\\cite{Surya2019}.\n\n\\section{Seven Discrete Klein Gordon Equations}\\label{SecSevenKG}\nIn this section I will introduce seven discrete Klein Gordon equations (KG1-KG7) in an interpretation-neutral way and solve their dynamics. These theories are all describable as being set on a lattice in both space and time. In each of these theories the lattice in space will simply repeat itself in time. I consider the following three cases for the lattice in space: a uniform 1D lattice, a square 2D lattice, and a hexagonal 2D lattice, see Fig.~\\ref{FigLat}. Repeated in time, these give us something like a square 2D lattice, a cubic 3D lattice, and a hexagonal 3D lattice respectively.\n\nAs harmful as these choices seem to be to Lorentz invariance (as well as continuous translation and rotation invariance) as I will show they ultimately pose no barrier to our theories having these symmetries. As I will argue, these choices of lattice are ultimately merely choices of representation which have absolutely nothing to do with the thing being represented. In particular, there is no need for our representational structure to have the same symmetries as the thing being represented. There is no issue with using Cartesian coordinates to describe a rotationally invariant state\/dynamics. I claim that analogously there is no issue with using a lattice to describe a state\/dynamics with continuous translation and rotation invariance and even Lorentz invariance. This claim has already been demonstrated in \\cite{DiscreteGenCovPart1} for states\/dynamics with continuous translation and rotation invariance. This paper extends this claim about lattice structures to Lorentz invariance as well.\n\n\\subsection{Introducing KG1-KG7}\nTo begin, let us consider the continuum Klein Gordon equations in $1+1$ and $2+1$ dimensions:\n\\begin{align}\\label{KG00}\n&\\text{\\bf Continuum Klein Gordon Eq. (KG00):}\\\\\n\\nonumber\n&\\partial_{t}^2\\varphi(t,x) = (\\partial_x^2-M^2) \\, \\varphi(t,x)\\\\\n\\label{KG0}\n&\\text{\\bf Continuum Klein Gordon Eq. (KG0):}\\\\\n\\nonumber\n&\\partial_{t}^2\\varphi(t,x,y) = (\\partial_x^2+\\partial_y^2-M^2) \\, \\varphi(t,x,y)\n\\end{align}\nwith some mass $M\\geq0$. For a generally covariant (i.e., coordinate-free) view of these theories, see Sec.~\\ref{SecFullGenCov}. \n\nFor our first three discrete Klein Gordon theories, let us consider the theories with only nearest-neighbor (N.N.) interactions on the above discussed lattices which best approximate KG0 and KG00. Namely:\n\\begin{align}\\label{KG1Long}\n&\\text{\\bf 1D N.N. Klein Gordon Eq. (KG1):}\\\\\n\\nonumber\n&[\\phi_{j-1,n}-2\\phi_{j,n}+\\phi_{j+1,n}]\\\\\n\\nonumber\n&\\quad=[\\phi_{j,n-1}-2\\phi_{j,n}+\\phi_{j,n+1}]\n-\\mu^2 \\phi_{j,n}\\\\\n&\\text{\\bf Square N.N. Klein Gordon Eq. (KG4):}\\label{KG4Long}\\\\\n\\nonumber\n&[\\phi_{j-1,n,m}-2\\phi_{j,n,m}+\\phi_{j+1,n,m}]\\\\\n\\nonumber\n&\\quad = [ \\ \\phi_{j,n-1,m}-2\\phi_{j,n,m}+\\phi_{j,n+1,m}\\\\\n\\nonumber\n&\\quad \\ \\ +\\phi_{j,n,m-1}-2\\phi_{j,n,m}+\\phi_{j,n,m+1}]\n-\\mu^2 \\phi_{j,n,m}\\\\\n&\\text{\\bf Hexagonal N.N. Klein Gordon Eq. (KG5):}\\label{KG5Long}\\\\\n\\nonumber\n&[\\phi_{j-1,n,m}-2\\phi_{j,n,m}+\\phi_{j+1,n,m}]\\\\\n\\nonumber\n&\\quad = \\frac{2}{3}[ \\ \\phi_{j,n-1,m}-2\\phi_{j,n,m}+\\phi_{j,n+1,m}\\\\\n\\nonumber\n&\\qquad \\quad +\\phi_{j,n,m-1}\\!-\\!2\\phi_{j,n,m}\\!+\\!\\phi_{j,n,m+1}\\\\\n\\nonumber\n&\\qquad \\quad +\\phi_{j,n+1,m-1}-2\\phi_{j,n,m}+\\phi_{j,n-1,m+1}]\n-\\mu^2 \\phi_{j,n,m}\n\\end{align}\nwith $j\\in\\mathbb{Z}$ indexing time and $n\\in\\mathbb{Z}$ and $m\\in\\mathbb{Z}$ indexing space. See Fig.~\\ref{FigLat} for the indexing convention. Here $\\mu\\in\\mathbb{R}$ is a dimensionless number playing the role of the field's mass. The terms in square brackets in the above expressions are the best possible approximations of the second derivative on each lattice which make use of only nearest neighbor interactions. These theories are named KG1, KG4, and KG5 in correspondence with the discrete heat equations considered in \\cite{DiscreteGenCovPart1} and in anticipation of their further treatment later in this section. \n\nThis section has promised to introduce these theories in an interpretation neutral way. As such, some of the above discussion needs to be hedged. In particular, in introducing these theories I have made casual comparison between parts of these theories' dynamical equations and various approximations of the second derivative. While, as I will discuss, such comparisons can be made, to do so immediately is unearned. It comes dangerously close to imagining the spacetime lattices discussed above as being embedded in a continuous manifold. This may be something we want to do later (see Sec.~\\ref{SecExtPart1}), but it is a non-trivial interpretational move which ought not be done so casually. \n\nCrucially, in this paper I will begin by analyzing these theories as discrete-native theories. As such, it's important to think of these discrete spacetime theories as self-sufficient theories in their own right. We must not begin by thinking of them as various discretizations or bandlimitations of the continuum theories. While, as I will discuss, these discrete theories have some notable relationships to various continuum theories it is important to resist any temptation to see these continuum theories as ``where they came from''. Rather, let us pretend these theories ``came from nowhere'' and let us see what sense we can make of them.\n\nAnother bit of hedging: in introducing the above three theories I casually associated them with the lattice structures shown in Fig.~\\ref{FigLat} (each repeated in time). Making such associations ab initio is unwarranted. While we may eventually associate these theories with those lattice structures we cannot do so immediately. Such an association would need to be made following careful consideration of the dynamics. (Such an exercise is carried out in Sec.~\\ref{SecKlein1}.) Beginning here in an interpretation-neutral way these theories ought to be seen as being defined over a completely unstructured lattice. \n\nI will reflect this concern in my notation as follows. The labels for the lattice sites are presently too structured (e.g., $(j,n)\\in\\mathbb{Z}^2$ and $(j,n,m)\\in\\mathbb{Z}^3$). Instead we ought to think of the lattice sites as having labels $\\ell\\in L$ for some set $L$. Crucially, at this point the set of labels for the lattice sites, $L$, is just that, an unstructured set.\n\nUp to isomorphism (here, generic bijections, i.e. generic relabelings), sets are uniquely specified by their cardinality. The set of labels for the lattice sites is here countable, $\\ell\\in L\\cong\\mathbb{Z}\\cong\\mathbb{Z}^2\\cong\\mathbb{Z}^3$. Reframed this way the above discussed theories each consider the same discrete variables $\\phi_\\ell\\in\\mathbb{R}$. In particular, KG1 considers variables $\\phi_\\ell$ which under some convenient relabeling of the lattice sites, $\\ell\\in L\\mapsto (j,n)\\in\\mathbb{Z}^2$, satisfies Eq.~\\eqref{KG1Long}. Similarly, KG4 and KG5 consider variables $\\phi_\\ell$ which under some convenient relabeling of the lattice sites, $\\ell\\in L\\mapsto (j,n,m)\\in\\mathbb{Z}^3$, satisfy Eq.~\\eqref{KG4Long} and Eq.~\\eqref{KG5Long} respectively.\n\nIt's important to stress that the mere existence of these convenient relabelings by itself has no interpretative force. The fact that our labels $(j,n)\\in\\mathbb{Z}^2$ and $(j,n,m)\\in\\mathbb{Z}^3$ in some sense form a square 2D lattice and cubic 3D lattice in no way forces us to think of $L$ as being structured in this way (indeed, we might later like to think of $L$ as a hexagonal 3D lattice). In particular, the fact that these labels are in a sense equidistant from each other does not force us to think of the lattice sites as being equidistant from each other. Nor are we forced to think that ``the distance between lattice sites'' to be meaningful at all. Dynamical considerations may later push us in this direction, but the mere convenience of this labeling should not.\n\nI have above introduced three out of seven discrete Klein Gordon theories. In order to introduce the other four theories, it is convenient (but not necessary) to first reformulate things. In particular, let us reorganize the $\\phi_\\ell$ variables into a vector, namely,\n\\begin{align}\\label{PhiDef}\n\\bm{\\Phi}= \\sum_{\\ell\\in L} \\phi_\\ell \\, \\bm{b}_\\ell. \n\\end{align}\nwhere $\\bm{b}_\\ell$ is a linearly-independent basis vector for each $\\ell\\in L$ and \\mbox{$\\bm{\\Phi}$} is a vector in the vector space \\mbox{$\\mathbb{R}^L\\coloneqq\\text{span}(\\{\\bm{b}_\\ell\\}_{\\ell\\in L})$}. For later reference, it should be noted that $\\phi_\\ell$ is also a vector in a vector space: namely, $F_L$ the space of functions $f:L\\to\\mathbb{R}$. Note that Eq.~\\eqref{PhiDef} is an vector space isomorphism between these vector spaces, $\\mathbb{R}^L\\cong F_L$. Everything which follows concerning $\\bm{\\Phi}\\in\\mathbb{R}^L$ has an isomorphic description in terms of $\\phi_\\ell\\in F_L$.\n\nRecall that for KG1 the lattice sites $\\ell\\in L$ have a convenient relabeling in terms of two integer indices, \\mbox{$\\ell\\in L\\mapsto (j,n)\\in\\mathbb{Z}^2$}. We can use this relabeling to grant the vector space a tensor product structure as \\mbox{$\\mathbb{R}^L\\mapsto\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} by taking \\mbox{$\\bm{b}_\\ell\\mapsto \\bm{e}_j\\otimes\\bm{e}_n$} where \n\\begin{align}\n\\bm{e}_m = (\\dots,0,0,1,0,0,\\dots)^\\intercal\\in\\mathbb{R}^\\mathbb{Z}\n\\end{align}\nwith the 1 in the $m^\\text{th}$ position. Under this restructuring of KG1 we have, \n\\begin{align}\\label{PhiVec1}\n\\bm{\\Phi}= \\sum_{j,n\\in\\mathbb{Z}} \\phi_{j,n} \\, \\bm{e}_j\\otimes\\bm{e}_n. \n\\end{align}\nIn these terms the dynamics of KG1 is given by,\n\\begin{align}\n\\label{DKG1}\n&\\text{\\bf Klein Gordon Equation 1 (KG1):}\\\\\n\\nonumber\n&\\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi}=\\Delta_{(1),\\text{n}}^2 \\, \\bm{\\Phi}\n-\\mu^2\\bm{\\Phi}\n\\end{align}\nwhere the notation $A_\\text{j}\\coloneqq A\\otimes\\openone$ and $A_\\text{n}\\coloneqq\\openone\\otimes A$ mean $A$ acts only on the first or second tensor space respectively. The linear operator $\\Delta_{(1)}^2$ appearing twice in the above expression is the following bi-infinite Toeplitz matrix:\n\\begin{align}\\label{Delta12}\n\\Delta_{(1)}^2=\\{\\Delta^+,\\Delta^-\\}\n&=\\text{Toeplitz}(1,\\,-2,\\,1)\\\\\n\\nonumber\n\\Delta^+&=\\text{Toeplitz}(0,-1,\\,1)\\\\\n\\nonumber\n\\Delta^-&=\\text{Toeplitz}(-1,\\,1,\\,0)\n\\end{align}\nwhere the curly brackets indicate the anticommutator, $\\{A,B\\}= \\frac{1}{2}(A\\,B + B\\,A)$. Recall that Toeplitz matrices are so called diagonal-constant matrices with \\mbox{$[A]_{i,j}=[A]_{i+1,j+1}$}. Thus, the values in the above expression give the matrix's values on either side of the main diagonal.\n\nAlthough above I warned about thinking in terms of derivative approximations prematurely, a few comments are here warranted. Note that $\\Delta^+$ is associated with the forward derivative approximation, $\\Delta^-$ is be associated with the backwards derivative approximation, and $\\Delta_{(1)}^2$ is associated with the nearest neighbor second derivative approximation,\n\\begin{align}\n\\nonumber\n\\Delta_{(1)}^2\/\\epsilon^2:\\ \\partial_x^2 f(x)\n&\\approx\\frac{f(x+\\epsilon)-2 f(x)+f(x-\\epsilon)}{\\epsilon^2}.\n\\end{align}\nAs stressed above, we ought to be cautious not to lean too heavily on these relationships when interpreting these discrete theories. \n\nIn addition to KG1, I will also consider two more theories with ``improved derivative approximations''. Namely,\n\\begin{align}\n\\label{DKG2}\n&\\text{\\bf Klein Gordon Equation 2 (KG2):}\\\\\n\\nonumber\n&\\Delta_{(2),\\text{j}}^2\\,\\bm{\\Phi}=\\Delta_{(2),\\text{n}}^2 \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\\\\\n\\label{DKG3}\n&\\text{\\bf Klein Gordon Equation 3 (KG3):}\\\\\n\\nonumber\n&D_{\\text{j}}^2\\,\\bm{\\Phi}=D_{\\text{n}}^2 \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nwhere\n\\begin{align}\\label{BigToeplitz}\n\\Delta_{(2)}^2&=\\text{Toeplitz}(\\frac{-1}{12},\\,\\frac{4}{3},\\,\\frac{-5}{2},\\,\\frac{4}{3},\\,\\frac{-1}{12})\\\\\n\\nonumber\nD&=\\text{Toeplitz}(\\dots,\\!\\frac{-1}{5},\\!\\frac{1}{4},\\!\\frac{-1}{3},\\!\\frac{1}{2},\\!-1,\\!0,\\!1,\\!\\frac{-1}{2},\\!\\frac{1}{3},\\!\\frac{-1}{4},\\!\\frac{1}{5},\\!\\dots)\\\\\n\\nonumber\nD^2&=\\text{Toeplitz}(\\dots,\\!\\frac{-2}{16},\\!\\frac{2}{9},\\!\\frac{-2}{4},\\!\\frac{2}{1},\\!\\frac{-2\\pi^2}{6},\\!\\frac{2}{1},\\!\\frac{-2}{4},\\!\\frac{2}{9},\\!\\frac{-2}{16},\\!\\dots).\n\\end{align}\nNote that $\\Delta_{(2)}^2$ is related to the next-to-nearest-neighbor approximation to the second derivative. Obviously, the longer range we make our derivative approximations the more accurate they can be. The infinite-range operator $D$ (and its square $D^2$) in some sense are the best discrete approximations to the derivative (and second derivative) possible. The defining property of $D$ is that it is diagonal in the (discrete) Fourier basis with spectrum,\n\\begin{align}\\label{LambdaD}\n\\lambda_D(k)=-\\mathrm{i}\\,\\underline{k}    \n\\end{align}\nwhere $\\underline{k}=k$ for $k\\in[-\\pi,\\pi]$ repeating itself cyclically with period $2\\pi$ outside of this region. This is in tight connection with the continuum derivative operator $\\partial_x$ which is diagonal in the (continuum) Fourier basis with spectrum \\mbox{$\\lambda_{\\partial_x}(k)=-\\mathrm{i} \\, k$} for $k\\in[-\\infty,\\infty]$. \n\nAlternatively, one can construct $D^2$ in the following way: generalize $\\Delta_{(1)}^2$ and $\\Delta_{(2)}^2$ to $\\Delta_{(n)}^2$ namely the best second derivative approximation which considers up to $n^\\text{th}$ neighbors to either side. Taking the limit $n\\to\\infty$ gives $D^2=\\lim_{n\\to\\infty}\\Delta_{(n)}^2$. Other aspects of $D$ will be discussed in Sec.~\\ref{SecSamplingTheory} (including its related derivative approximation Eq.~\\eqref{ExactDerivative}) but enough has been said for now.\n\nWhile these connections to derivative approximations allow us to export some intuitions from the continuum theories into these discrete theories, we must resist this (at least for now). In particular, I should stress again that we should not  be thinking of any of KG1, KG2 and KG3 as coming from the continuum theory under some approximation of the derivative.\n\nLet's next reformulate KG4 and KG5 in terms of \\mbox{$\\bm{\\Phi}\\in\\mathbb{R}^L$}. In these cases we have a convenient relabeling of the lattice sites in terms of three integer indices, $\\ell\\mapsto (j,n,m)$. As before we can use this relabeling to grant the vector space a tensor product structure as $\\mathbb{R}^L\\mapsto\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ by taking by taking \\mbox{$\\bm{b}_\\ell\\mapsto \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$}. Under this restructuring we have,\n\\begin{align}\\label{PhiVec2}\n\\bm{\\Phi}= \\sum_{j,n,m\\in\\mathbb{Z}} \\phi_{j,n,m} \\, \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m. \n\\end{align}\nIn these terms the dynamics of KG4 given above (namely, Eq.~\\eqref{KG4Long}) is now given by,\n\\begin{align}\\label{DKG4}\n&\\text{\\bf Klein Gordon Equation 4 (KG4):}\\\\\n\\nonumber\n&\\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi}=(\\Delta_{(1),\\text{n}}^2+\\Delta_{(1),\\text{m}}^2) \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nA similar treatment of the dynamics of KG5 (namely, Eq.~\\eqref{KG5Long}) gives us,\n\\begin{align}\\label{DKG5}\n&\\text{\\bf Klein Gordon Equation 5 (KG5):}\\\\\n\\nonumber\n&\\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi} =\\frac{2}{3} \\, \\Big[\\Delta^2_{(1),\\text{n}}+ \\Delta^2_{(1),\\text{m}}\\\\\n\\nonumber\n&\\qquad\\qquad\\quad \\ +\\big\\{\\Delta^+_\\text{m}-\\Delta^+_\\text{n},\\Delta^-_\\text{m}-\\Delta^-_\\text{n}\\big\\}\\Big] \\, \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nWhile the third term in the square brackets looks complicated, it is just the analog of $\\Delta^2_{(1),\\text{n}}$ and $\\Delta^2_{(1),\\text{m}}$ but in the $m-n$ direction. See Eq.~\\eqref{Delta12}.\n\nFinally, in addition to KG4 and KG5 I consider the following two theories:\n\\begin{align}\n\\label{DKG6}\n&\\text{\\bf Klein Gordon Equation 6 (KG6):}\\\\\n\\nonumber\n&D_{\\text{j}}^2\\,\\bm{\\Phi}=(D_{\\text{n}}^2+D_{\\text{m}}^2) \\ \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\\\\\n\\label{DKG7}\n&\\text{\\bf Klein Gordon Equation 7 (KG7):}\\\\\n\\nonumber\n&D_{\\text{j}}^2\\,\\bm{\\Phi}= \\frac{2}{3}\\left(D^2_\\text{n}+D^2_\\text{m}+(D_\\text{m}-D_\\text{n})^2\\right) \\, \\bm{\\Phi}-\\mu^2 \\, \\bm{\\Phi}\n\\end{align}\nwhich resemble KG4 and KG5 but which make use of an infinite range coupling between lattice sites. Having introduced these seven theories, let us next solve their dynamics.\n\n\\subsection{Solving Their Dynamics}\nConveniently, each of KG1-KG7 admit planewave solutions. Moreover, in each case these planewave solutions form a complete basis of solutions.\n\nConsidering first KG1-KG3 we have solutions of the form,\n\\begin{align}\\label{PlaneWave123}\n\\phi_{j,n}(\\omega,k)=e^{-\\mathrm{i}\\, \\omega \\, j - \\mathrm{i}\\, k \\, n}.\n\\end{align}\nwith $\\omega,k\\in\\mathbb{R}^2$. It should be noted however, that outside of the range $\\omega,k\\in[-\\pi,\\pi]$ these planewaves repeat themselves with period $2\\pi$ due to Euler's identity, $\\exp(2\\pi\\mathrm{i})=1$. In terms of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ these planewaves are:\n\\begin{align}\n\\bm{\\Phi}(\\omega,k)=\\sum_{j,n\\in\\mathbb{Z}} \\phi_{j,n}(\\omega,k) \\, \\bm{e}_j\\otimes\\bm{e}_n. \n\\end{align}\nFrom this planewave basis we can recover the $\\bm{e}_j\\otimes\\bm{e}_n$ basis as:\n\\begin{align}\n\\bm{e}_j\\otimes\\bm{e}_n\n&=\\frac{1}{(2\\pi)^2}\\int\\!\\!\\!\\!\\int_{-\\pi}^\\pi e^{\\mathrm{i}\\,\\omega \\,j+\\mathrm{i}\\, k\\, n}\\, \\bm{\\Phi}(\\omega,k)\\,\\d\\omega\\d k. \n\\end{align}\nThese planewaves are only a solution if $\\omega$ and $k$ satisfy the theory's dispersion relation which can be straight-forwardly calculated from the theory's dynamics:\n\\begin{align}\n\\text{KG1:}& \\quad \\!\\! 2-2\\,\\cos(\\omega)= \\mu^2+2-2\\,\\cos(k)\\\\\n\\text{KG2:}& \\quad \\!\\! \\frac{1}{6}\\,(\\cos(2\\,\\omega)-16\\,\\cos(\\omega)+15)\\\\\n\\nonumber\n&= \\mu^2+\\frac{1}{6}\\,(\\cos(2\\,k)-16\\,\\cos(k)+15)\\\\\n\\text{KG3:}& \\quad \\!\\! \\underline{\\omega}^2= \\mu^2+\\underline{k}^2\n\\end{align}\nwhere $\\underline{\\omega}=\\omega$ and $\\underline{k}=k$ for $\\omega,k\\in[-\\pi,\\pi]$ repeating themselves cyclically with period $2\\pi$ outside of this region. Note that the dispersion relation for KG3 follows from Eq.~\\eqref{LambdaD}, essentially from the definition of $D$.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.45\\textwidth]{PaperFigures\/FigKleinDisp.pdf}\n\\caption{The dispersion relations for the planewave solutions to the discrete Klein Gordon equations are plotted as a function of wavenumber for KG1, KG2 and KG3 with $\\mu=1.25$.}\\label{FigKleinDisp}\n\\end{figure}\n\nFig.~\\ref{FigKleinDisp} shows these dispersion relations restricted to the region $\\omega,k\\in[-\\pi,\\pi]$ with a field mass of $\\mu=1.25$. Qualitatively, KG1-KG3 all seem to agree with each other at low wavenumbers. They appear to mostly differ with respects to the rate at which high wavenumber planewaves oscillate. Let's investigate how these theories behave for planewaves with periods and wavelengths which span many lattice sites, that is with $\\vert\\omega\\vert,\\vert k\\vert\\ll\\pi$.\n\\begin{align}\n\\nonumber\n\\text{KG1:}& \\quad \\!\\! \\omega^2= \\mu^2+k^2+\\frac{\\omega^4-k^4}{12}+\\mathcal{O}(\\omega^6)+\\mathcal{O}(k^6)\\\\\n\\nonumber\n\\text{KG2:}& \\quad \\!\\! \\omega^2= \\mu^2+k^2-\\frac{\\omega^6-k^6}{90}+\\mathcal{O}(\\omega^8)+\\mathcal{O}(k^8)\\\\\n\\text{KG3:}& \\quad \\!\\! \\omega^2 = \\mu^2+k^2.\n\\end{align}\nNote that the dispersion relation for KG3 exactly matches that of the continuum theory, not only within this regime but for all $\\omega,k\\in[-\\pi,\\pi]$. In the $\\vert\\omega\\vert,\\vert k\\vert\\ll\\pi$ regime, KG2 gives a better approximation of the continuum theory than KG1 does. This is due to its longer range coupling giving a better approximation of the derivative. \n\nIf we consider only solutions with all or most of their planewave support with $\\vert\\omega\\vert,\\vert k\\vert\\ll\\pi$, we have an approximate one-to-one correspondence between the solutions to these theories. This is roughly why each of these theories have the same continuum limit, namely KG00 defined above. In terms of the rate at which these theories converge to the continuum theory in the continuum limit, one can expect KG3 to outpace KG2 which outpaces KG1. (As I discussed in \\cite{DiscreteGenCovPart1}, this is in a way counter-intuitive: why does the most non-local discrete theory give the best approximation of our perfectly local continuum theory?)\n\nHowever, while interesting in their own right, these relationships with the continuum theory are not directly helpful in helping us understand KG1-KG3 in their own terms as discrete-native theories.\n\nMoving on to KG4-KG7, their planewave solutions are of the form, \n\\begin{align}\n\\phi_{j,n,m}(\\omega,k_1,k_2)=e^{-\\mathrm{i}\\,\\omega \\,j-\\mathrm{i}\\, k_1\\, n-\\mathrm{i}\\, k_2\\, m}\n\\end{align}\nwith $\\omega,k_1,k_2\\in\\mathbb{R}^2$. Again, it should be noted however, that outside of the range \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} these planewaves repeat themselves with period $2\\pi$ due to Euler's identity, $\\exp(2\\pi\\mathrm{i})=1$. In terms of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} these planewaves are:\n\\begin{align}\n\\nonumber\n\\bm{\\Phi}(\\omega,k_1,k_2)=\\sum_{j,n,m\\in\\mathbb{Z}} \\phi_{j,n,m}(\\omega,k_1,k_2) \\, \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m. \n\\end{align}\nFrom this planewave basis we can recover the $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$ basis as:\n\\begin{align}\n&\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\\\\n\\nonumber\n&=\\frac{1}{(2\\pi)^3}\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_{-\\pi}^\\pi e^{\\mathrm{i}\\,\\omega \\,j+\\mathrm{i}\\, k_1 n+\\mathrm{i}\\, k_2 m}\\,\\bm{\\Phi}(\\omega,k_1,k_2)\\,\\d\\omega\\d k_1 \\d k_2. \n\\end{align}\n\nThe dispersion relation for each of these theories is given by:\n\\begin{align}\n\\text{KG4:}& \\ 2-2\\cos(\\omega)= \\mu^2\\!+\\!4\\!-\\!2\\cos(k_1)\\!-\\!2\\cos(k_2)\\\\\n\\nonumber\n\\text{KG5:}& \\  2-2\\,\\cos(\\omega)= \\mu^2 \\\\\n\\nonumber\n&+\\frac{4}{3}\\big[3-\\cos(k_1)-\\cos(k_1)-\\cos(k_2-k_1)\\big]\\\\\n\\nonumber\n\\text{KG6:}& \\ \\underline{\\omega}^2= \\mu^2 +\\underline{k_1}^2 +\\underline{k_2}^2\\\\\n\\nonumber\n\\text{KG7:}& \\ \\underline{\\omega}^2= \\mu^2 +\\frac{2}{3}\\big[\\underline{k_1}^2 +\\underline{k_2}^2 +(\\underline{k_2}-\\underline{k_1})^2\\big].\n\\end{align}\nNote that the dispersion relation for KG6 and KG7 follow from Eq.~\\eqref{LambdaD}, essentially from the definition of $D$.\n\nUnlike KG1-KG3, these theories do not all agree with each other in the small $\\omega,k_1,k_2$ regime. KG4 and KG6 agree that for $\\vert\\omega\\vert, \\vert k_1\\vert, \\vert k_2\\vert \\ll \\pi$ we have \\mbox{$\\omega^2= \\mu^2 +k_1^2 +k_2^2$}. Moreover, KG5 and KG7 agree with each other in this regime, but not with KG4 and KG6. Do we have two different results in the continuum limit here?\n\nCloser examination reveals that we do not. The key to realizing this is to note that under the transformation, \n\\begin{align}\\label{SkewKG7KG6}\n\\omega\\mapsto \\omega,\\quad\nk_1\\mapsto k_1,\\quad\nk_2\\mapsto \\frac{1}{2}\\, k_1+\\frac{\\sqrt{3}}{2}\\,k_2.\n\\end{align}\nwe have the dispersion relation for KG7 mapping exactly onto the one for KG6. The inverse of this map is\n\\begin{align}\\label{SkewKG6KG7}\n\\omega\\mapsto \\omega,\\quad\nk_1\\mapsto k_1,\\quad\nk_2\\mapsto \\frac{2k_2-k_1}{\\sqrt{3}}.\n\\end{align}\nTechnically, when acting on the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ these transformations are only each other's inverses when we have \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} both before and after the transformation. This is due to the $2\\pi$ periodicity of these planewaves. Fortunately however, all of KG7's planewave solutions with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} remain in this region after applying Eq.~\\eqref{SkewKG7KG6}. The same is true of KG6: its planewave solutions with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} also remain in this region after applying Eq.~\\eqref{SkewKG6KG7}. As I will soon discuss, this means we have an exact one-to-one correspondence between KG6 and KG7's solutions (much ado will be made about this later.) Applying this transformation to KG5 does not map it onto KG4, but it does bring their $\\vert\\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ behavior into agreement. \n\nThus, if we consider only solutions with all or most of their planewave support with $\\vert\\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ (or the appropriately transformed regime for KG5 and KG7) then we have an approximate one-to-one correspondence between the solutions to these theories. Within this regime we can define their common continuum limit, KG0. Repeating our analysis of the convergence rates of KG1-KG3 here, we expect KG6 and KG7 to converge in the continuum limit faster than KG4 and KG5 do.\n\n\\vspace{0.25cm}\n\nThis paper will make three attempts at interpreting these seven discrete theories. Allow me to identify in advance three important points of comparison between these interpretations. \n\nThe first important point of comparison is what sense they make of these different convergence rates in the continuum limit. As discussed above, in terms of this convergence rate we expect \\mbox{$\\text{KG3}>\\text{KG2}>\\text{KG1}$} and similarly \\mbox{$\\text{KG6}, \\text{KG7} > \\text{KG4}, \\text{KG5}$} with higher rated theories converging more quickly. This is in tension with our intuitive sense of locality for these theories: judging locality by the number of lattice sites coupled together we have\n\\mbox{$\\text{KG1} > \\text{KG2} > \\text{KG3}$} with higher rated theories being more local and similarly\n\\mbox{$\\text{KG4},\\text{KG5} > \\text{KG6},\\text{KG7}$}. How is it that our most non-local theories are somehow the nearest to our perfectly local continuum theory?\n\nRegarding how these three interpretations deal with this tension, not much changes between the discrete heat equations considered in \\cite{DiscreteGenCovPart1} and the discrete Klein Gordon equations considered here. As such, I will leave any detailed discussion of this issue to \\cite{DiscreteGenCovPart1} and direct the interested reader there. Roughly, the second and third interpretations deal with this tension by negating or reversing all of the above intuitive locality judgements.\n\nA second important point of comparison between these three interpretations will be what sense they make of the  above-noted exact one-to-one correspondence between KG6 and KG7's solutions. (More will be said about this in Sec.~\\ref{SecKlein2}.) It is important to note that the mere existence of such a one-to-one correspondence does not automatically mean that these theories are identical or even equivalent; All it means technically is that their solution spaces have the same cardinality. As I will discuss, some of the coming interpretations recognize KG6 and KG7 as being equivalent whereas others do not.\n\nA third important point of comparison between the coming interpretations will be what sense they make of these theories having continuous symmetries. For instance, the dispersion relation for KG6 appears to be in some sense rotation invariant and even Lorentz invariant (at least in Fourier space and staying inside of the region \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}). In a sense, KG7 might have these symmetries too: given the above-noted one-to-one correspondence between the solutions of KG6 and KG7, there may be some (skewed) sense in which KG7 is rotation invariant and Lorentz invariant as well. All of this will be made precise later on. As I will discuss, some interpretations consider KG6 and KG7 to have a rotation symmetry and even (limited) Lorentz boost symmetries whereas others do not. As I will discuss in Sec.~\\ref{PerfectLorentz}, KG6 and KG7 can be seen as representationally-limited parts of a larger perfectly Lorentzian lattice theory.\n\nHaving introduced these theories and solved their dynamics in an interpretation-neutral way. We can now make a first (ultimately misled) attempt at interpreting them.\n\n\\section{A First Attempt at Interpreting KG1-KG7}\\label{SecKlein1}\nNow that we have introduced these seven discrete theories and solved their dynamics, let's get on to interpreting them. Let us begin by following our first intuitions and analyze these seven discrete theories concerning their underlying manifold, locality properties and symmetries. Ultimately however, as I will discuss later, much of the following is misled and will need to be revisited and revised later. Luckily, retracing where we went wrong here will be instructive later.\n\nLet's start by taking the initial formulation of the above theories in terms of $\\phi_\\ell$ seriously, i.e. Eq.~\\eqref{KG1Long}, Eq.~\\eqref{KG4Long} and Eq.~\\eqref{KG5Long}. Taken literally as written, what are these theories about? Intuitively these theories are about a field $\\phi_\\ell$ which maps lattice sites ($\\ell\\in L\\cong\\mathbb{Z}\\cong\\mathbb{Z}^2\\cong\\mathbb{Z}^3$) into field amplitudes ($\\phi_\\ell\\in\\mathbb{R}$). That is a field $\\phi:Q\\to \\mathcal{V}$ with a discrete manifold $Q=L$ and value space $\\mathcal{V}=\\mathbb{R}$. Thus, taking $\\phi:Q\\to \\mathcal{V}$ seriously as a fundamental field leads us to thinking of $Q=L$ as the theory's underlying manifold and $\\mathcal{V}=\\mathbb{R}$ as the theory's value space. It is important to note that here, $Q$ is the entire manifold, it is not being thought of as embedded in some larger manifold. (However, a view like this will be considered in Sec.~\\ref{SecExtPart1}.) \n\nTaking $Q$ to be these theories' underlying manifold has consequences for our understanding of the locality of these theories. In a highly intuitive sense, theory KG1 is the most local in that it couples together the fewest lattice sites (only nearest neighbors). Following this KG2 is the next most local in the same sense: it couples only next-to-nearest neighbors. Finally, in this sense KG3 is the least local, it has an infinite range coupling. As mentioned above, there is some tension however with the rate we expect each of these theories to converge at in the continuum limit. How is it that our most non-local theories are somehow the nearest to our perfectly local continuum theory? This first interpretation can do little to resolve this tension, I refer the interested reader to \\cite{DiscreteGenCovPart1} for further discussion.\n\n\\subsection{Intuitive Symmetries}\nWith this manifold $Q=L$ and value space $\\mathcal{V}=\\mathbb{R}$ picked out, what can we expect of these theories' symmetries? For any spacetime theory there are roughly three kinds of symmetries: 1) external symmetries associated with automorphisms of the manifold, here $\\text{Auto}(Q)$, 2) internal symmetries associated with automorphisms of the value space, here $\\text{Auto}(\\mathcal{V})$, and gauge symmetries which result from allowing these internal symmetries to vary smoothly across the manifold. But what are the relevant notions of automorphism here?\n\nAnswering this question for $\\text{Auto}(Q)$ will require us to distinguish what structures are ``built into'' $Q$ and what are ``built on top of'' $Q$. The analogous distinction in the continuum case is that we generally take the manifold's differentiable structure to be built into it while the Minkowski metric, for instance, is something additional built on top of the manifold. In this paper, I am officially agnostic on where we draw this line in the discrete case. However, for didactic purposes I will here be as conservative as possible giving $Q$ as little structure as is sensible. Note that the less structure we associate with $Q$ the larger the class of relevant automorphisms $\\text{Auto}(Q)$ will be. Thus, I am taking $\\text{Auto}(Q)$ to be as large as it can reasonably be. \n\nHere the minimal structure we can reasonably associate with $Q=L$ is that of a set. As such the largest $\\text{Auto}(Q)$ could reasonably be is permutations of the lattice sites, \\mbox{$\\text{Auto}(Q)=\\text{Perm}(L)$}. \n\nIn addition to $\\text{Auto}(Q)$ we might also have internal symmetries $\\text{Auto}(\\mathcal{V})$ and gauge symmetries. While in general there may be abundant internal or gauge symmetries, for the present cases there are not many. In particular, for all of the above-mentioned theories we only have $\\mathcal{V}=\\mathbb{R}$. As mentioned following Eq.~\\eqref{PhiDef}, our (potentially off-shell) discrete fields are themselves vectors $\\phi_\\ell\\in F_L$. Namely, they are closed under addition and scalar multiplication and hence form a vector space. This addition and scalar multiplication is carried out lattice-site-by-lattice-site. Thus, the field's value space $\\mathcal{V}=\\mathbb{R}$ is also structured like a vector space. \n\nThe value space $\\mathcal{V}=\\mathbb{R}$ may additionally have more structure than this. However, as above, for didactic purposes I will here minimize the assumed structure in order to maximize possible symmetries. We can even drop the zero vector from our consideration taking $\\mathcal{V}=\\mathbb{R}$ to be an affine vector space.\nTherefore, I will take \\mbox{$\\text{Auto}(\\mathcal{V})=\\text{Aff}(\\mathbb{R})$} such that our internal symmetries are linear-affine rescalings of $\\phi_\\ell$, namely $\\phi_\\ell\\mapsto c_1\\phi_\\ell+c_2$. We can find the theory's gauge symmetries by letting \\mbox{$c_1,c_2\\in\\mathbb{R}$} vary smoothly across $Q$. That is, $\\phi_\\ell\\mapsto c_{\\ell,1}\\phi_\\ell+c_{\\ell,2}$.\n\nThus, in total, for KG1-KG7 the widest scope of symmetry transformations available to us (at least on this interpretation) are:\n\\begin{align}\\label{PermutationLong}\ns:\\quad \\phi_\\ell\\mapsto c_{P(\\ell),1} \\phi_{P(\\ell)}+c_{P(\\ell),2}\n\\end{align}\nfor some permutation $P:L\\to L$.\n\nFor later reference it will be convenient to translate these potential symmetry transformations in terms of the vector, $\\bm{\\Phi}\\in\\mathbb{R}^L$, as \n\\begin{align}\\label{Permutation}\ns:\\quad\\bm{\\Phi}\\mapsto C_1\\,P\\,\\bm{\\Phi}+\\bm{c}_2,\n\\end{align}\nfor some permutation matrix, $P$, a diagonal matrix $C_1$ and a vector $\\bm{c}_2$. Here $P$ captures the theory's possible external symmetries:  the possibility of permuting lattice sites. The diagonal matrix $C_1$ and the vector $\\bm{c}_2$ capture the theory's possible gauge symmetries: the possibility of linear-affine rescalings of $\\phi_\\ell$ which vary smoothly across $Q$.\n\nI will next discuss which transformations of this form preserve the dynamics of KG1-KG7. It should be clear from the outset however that (at least on this interpretation) these theories cannot have continuous spacial translation and rotation, let alone Lorentzian boost symmetries. Indeed, I have been charitable considering the lattice sites structured only as a set (perhaps artificially) increasing the size of $\\text{Auto}(Q)$. Given this, it would be highly surprising if we found KG1-KG7 to have symmetries outside of this set. (Such a surprise is coming in the Sec.~\\ref{SecKlein2}.) \n\nAs I will show in Sec.~\\ref{SecKlein2}, this first interpretation of these theories systematically under predicts the symmetries that discrete spacetime theories can and do have. Fixing this issue will lead one to a discrete analog of general covariance. We here under-predict symmetries because we are taking these theories' lattice structures too seriously. Properly understood, they are merely a coordinate-like representational artifact and so do not limit our symmetries. Before that however, let's see the symmetries these theories have on this interpretation.\n\n\\subsubsection*{Symmetries of KG1-KG7: First Attempt}\nWhat then are the symmetries of KG1-KG7 according to this interpretation? A technical investigation of the symmetries of KG1-KG7 on this interpretation is carried out in Appendix~\\ref{AppA}, but the results are the following. For KG1-KG3 the dynamical symmetries of the form Eq.~\\eqref{PermutationLong} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] discrete shifts which map lattice site \\mbox{$(j,n)\\mapsto (j-d_1,n-d_2)$} for some integers $d_1,d_2\\in\\mathbb{Z}$,\n    \\item[2)] two negation symmetries which map lattice site $(j,n)\\mapsto (-j,n)$ and $(j,n)\\mapsto (j,-n)$ respectively,\n    \\item[3)] global linear rescaling which maps \\mbox{$\\phi_\\ell\\mapsto c_1\\phi_\\ell$} for some $c_1\\in\\mathbb{R}$,\n    \\item[4)] local affine rescaling which maps \\mbox{$\\phi_\\ell\\mapsto \\phi_\\ell + c_{2,\\ell}(t)$} for some $c_{2,\\ell}(t)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are the symmetries of a uniform 1D lattices in space  and a uniform 1D lattice in time, \\mbox{$z_{j,n}=(j,n)\\in\\mathbb{R}^2$} (plus linear-affine rescalings). These are two independent 1D lattices (rather than a single square 2D lattice) because we do not have quarter rotations between space and time among our symmetries. Previously I had warned against prematurely interpreting the lattice sites underlying KG1-KG3 as being organized into a square lattice. As it turns out, having investigated these theories' dynamical symmetries this warning was warranted.\n\nWhat about KG4 and KG6? For KG4 and KG6 the dynamical symmetries of the form Eq.~\\eqref{PermutationLong} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] discrete shifts which map lattice site \\mbox{$(j,n,m)\\mapsto (j-d_1,n-d_2,m-d_3)$} for some integers $d_1,d_2,d_3\\in\\mathbb{Z}$,\n    \\item[2)] three negation symmetries which map lattice site $(j,n,m)\\mapsto (-j,n,m)$ and $(j,n,m)\\mapsto (j,-n,m)$ and $(j,n,m)\\mapsto (j,n,-m)$ respectively,\n    \\item[3)] a 4-fold symmetry which maps lattice site \\mbox{$(j,n,m)\\mapsto (j,m,-n)$}, \n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\ell\\mapsto c_1\\phi_\\ell$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\ell\\mapsto \\phi_\\ell + c_{2,\\ell}(t)$} for some $c_{2,\\ell}(t)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are the symmetries of a square 2D lattice in space and a uniform 1D lattice in time, \\mbox{$z_{j,n,m}=(j,n,m)\\in\\mathbb{R}^3$} (plus linear-affine rescalings). The above 4-fold symmetry corresponds to quarter rotation in space. These are two independent lattices (rather than a single cubic 3D lattice) because we do not have quarter rotations between space and time among our symmetries. Previously I had warned against prematurely interpreting the lattice sites underlying KG4 and KG6 as being organized into a cubic lattice. As it turns out, having investigated these theories' dynamical symmetries this warning was warranted.\n\nWhat about KG5 and KG7? For KG5 and KG7 the dynamical symmetries of the form Eq.~\\eqref{PermutationLong} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] discrete shifts which map lattice site \\mbox{$(j,n,m)\\mapsto (j-d_1,n-d_2,m-d_3)$} for some integers $d_1,d_2,d_3\\in\\mathbb{Z}$,\n    \\item[2)] an exchange symmetry which maps lattice site $(j,n,m)\\mapsto (j,m,n)$,\n    \\item[3)] a 6-fold symmetry which maps lattice site \\mbox{$(j,n,m)\\mapsto (j,-m,n+m)$}. (Roughly, this permutes the three terms in the right hand side of Eq.~\\eqref{DKG5} for KG5 and Eq.~\\eqref{DKG7} for KG7.),\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\ell\\mapsto c_1\\phi_\\ell$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\ell\\mapsto \\phi_\\ell + c_{2,\\ell}(t)$} for some $c_{2,\\ell}(t)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are the symmetries of a hexagonal 2D lattice in space and a uniform 1D lattice in time, \\mbox{$z_{j,n,m}=(j,n+m\/2,\\sqrt{3}m\/2)\\in\\mathbb{R}^3$} (plus linear-affine rescalings). The above 6-fold symmetry corresponds to one-sixth rotation in space. Previously I had warned against prematurely interpreting the lattice sites underlying KG5 and KG7 as being organized into a cubic 3D lattice, prompted by the convenient relabeling $\\ell\\mapsto (j,n,m)$. As it turns out, having investigated these theories' dynamical symmetries this warning was well warranted.\n\nThus, by investigating these theories' dynamical symmetries we were able to find what sort of lattice structure the assumed-to-be unstructured lattice $L$ actually has for each theory (e.g. in space a uniform 1D lattice, a square lattice, and a hexagonal lattice each together with a uniform 1D lattice in time).\n\n\\vspace{0.25cm}\n\nFinally, in this interpretation what sense can be made of KG6 and KG7 having a nice one-to-one correspondence between their solutions discussed at the end of Sec.~\\ref{SecSevenKG}? While this correspondence between solutions certainly exists, little sense can be made of it here in support of the equivalence of these theories. As the above discussion has revealed this interpretation associates very different symmetries to KG6 and KG7 and correspondingly very different lattice structures. While there is nothing technically wrong per se with this assessment our later interpretations will make better sense of this correspondence.\n\n\\vspace{0.25cm}\n\nTo summarize, this interpretation has the benefit of being highly intuitive. Taking the fields given to us, \\mbox{$\\phi:Q\\to\\mathbb{R}$}, seriously we identified the underlying manifold as $Q=L$. From this we got some intuitive notions of locality. Moreover, by finding these theories' dynamical symmetries we were able to grant some more structure to their lattice sites. By and large, the interpretation seems to validate all of the first intuitions laid out in Sec.~\\ref{SecIntro}. On this interpretation, the lattice seems to play a substantive role in the theory: it seems to restrict our symmetries, it seems to distinguish our theories from one another, be essentially ``baked-into'' the formalism. (As I will discuss in the next section, none of this is right.)\n\nHowever, there are three major issues with this interpretation which will become clear in light of our later interpretations. Firstly, our locality assessments are in tension with the rates at which these theories converge to the (perfectly local) continuum theory in the continuum limit, see \\cite{DiscreteGenCovPart1} for further discussion. Secondly, despite the niceness of the one-to-one correspondence between the solutions to KG6 and KG7, this interpretation regards them as significantly different theories: with different lattice structures and (here consequently) different symmetries. The final issue (which will become clear in the next section) is that this interpretation drastically under predicts the kinds of symmetries which KG1-KG7 can and do have. In fact, each of KG1-KG7 have a hidden continuous translation symmetry. Moreover, KG6 and KG7 have a hidden continuous rotation symmetry. Moreover still, KG3, KG6, and KG7 all have a hidden (limited) Lorentzian boost symmetry.\n\nAs I will discuss, the root of all of these issues is taking the theory's lattice structure too seriously. As I will argue, when properly understood, they are merely a coordinate-like representational artifact. Indeed, as I will show in the next section they do not limit our theory's symmetries. Moreover, theories appearing initially with different lattice structures may nonetheless be equivalent. Finally, these theories can always be reformulated to refer to no lattice structure at all. These three points establish a strong analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum theories. Ultimately, spelling out this analogy in detail in Sec.~\\ref{SecDisGenCov} will give us a discrete analog of general covariance (now extended to a Lorentzian context). Indeed, this will lead us to a perfectly Lorentzian lattice theory in Sec.~\\ref{PerfectLorentz}.\n\n\\section{A Second Attempt at Interpreting KG1-KG7}\\label{SecKlein2}\nIn the previous section, I claimed that KG1-KG7 have hidden continuous translation and rotation symmetries and even (limited) Lorentzian boost symmetries. But how can this be? How can discrete spacetime theories have such continuous symmetries? As I discussed in the previous section, if we take our underlying manifold to be \\mbox{$Q=L$} then these theories clearly cannot support continuous translation and rotation symmetries let alone Lorentzian boosts.\n\nIn order to avoid this conclusion we must deny the premise, $Q$ must not be the underlying manifold. What led us to believe $Q$ was the underlying manifold? We arrived at this conclusion by focusing on $\\phi_\\ell\\in F_L$ and thereby taking the real scalar field $\\phi:Q\\to\\mathcal{V}$ to be fundamental. $Q$ is the underlying manifold because it is where our fundamental field maps from. In order to avoid this conclusion we must deny the premise, the field $\\phi:Q\\to\\mathcal{V}$ must not be fundamental. \n\nBut if $\\phi:Q\\to\\mathcal{V}$ is not fundamental then what is? Fortunately, our above discussion has already provided us with another object which we might take as fundamental. Namely, $\\bm{\\Phi}$ defined in Eq.~\\eqref{PhiDef}. These vectors $\\bm{\\Phi}\\in\\mathbb{R}^L$ are in a one-to-one correspondence with the discrete fields $\\phi_\\ell\\in F_L$, Moreover, these vector spaces are isomorphic $\\mathbb{R}^L\\cong F_L$ with Eq.~\\eqref{PhiDef} being a vector space isomorphism between them.\n\nOn this second interpretation I will be taking the formulations of KG1-KG7 in terms of $\\bm{\\Phi}$ seriously: namely Eq.~\\eqref{DKG1}, Eq.~\\eqref{DKG2}, Eq.~\\eqref{DKG3}, and Eqs.~\\eqref{DKG4}-\\eqref{DKG7}. Taken literally as written, what are these theories about? These theories are intuitively about an infinite dimensional vector ($\\bm{\\Phi}\\in\\mathbb{R}^L$) which satisfies some dynamical constraint. \n\nThere are two ways in which one might try to make sense of $\\bm{\\Phi}$ as a field $\\bm{\\Phi}:\\mathcal{M}\\to\\mathcal{V}$ with some manifold $\\mathcal{M}$ and some value space $\\mathcal{V}$. Firstly, one might consider a one-point manifold $\\mathcal{M}=p$ with $\\bm{\\Phi}$ simply being the field value there $\\bm{\\Phi}\\coloneqq \\bm{\\Phi}(p)$. Alternatively, one might try to make sense of $\\bm{\\Phi}$ as a field ``from nowhere'' with an empty manifold $\\mathcal{M}=\\emptyset$. On this view $\\bm{\\Phi}$ is a vector field which takes in no input and returns the vector $\\bm{\\Phi}$.\n\nIn any case, on this interpretation $Q$ is no longer the underlying manifold for KG1-KG7. Indeed, on this interpretation the lattice sites, $L$, are no longer our spacetime manifold (there may not even be a manifold). Rather, they have been ``internalized'' into the value space $\\mathcal{V}=\\mathbb{R}^L$. In particular, in defining this vector space we have associated with each lattice site $\\ell\\in L$ a basis vector $\\bm{b}_\\ell$. See Eq.~\\eqref{PhiDef}. However, as I will discuss, these particular basis vectors play no special role in these theories. Indeed, looking back at the dynamics for each of KG1-KG7 written in terms of $\\bm{\\Phi}$, one can see that in each case it can be made basis-independent.\n\nLet's see what consequences this interpretive stance has for these theories' locality and symmetry. To preview: this second interpretation either dissolves or resolves all of our issues with the first interpretation. Firstly, the tension is dissolved between our theories' differences in locality and their differences in convergence rate in the continuum limit. With no spacetime manifold we no longer have access to any spaciotemporal notion of locality. There simply are no differences in locality anymore. I refer the interested reader to \\cite{DiscreteGenCovPart1} for further discussion.\n\nSecondly, KG6 and KG7 are seen to be equivalent in a stronger sense. And thirdly, perhaps most importantly, this interpretation reveals KG1-KG7's hidden continuous translation and rotation symmetries and even (limited) Lorentzian boost symmetries. However, as I will discuss, this interpretation has some issues of its own which will ultimately require us to make a third attempt at interpreting these theories in Sec.~\\ref{SecKlein3}.\n\n\\subsection{Internalized Symmetries}\nHow does this internalization move affect our theory's capacity for symmetry? How can we now have continuous translation and rotation symmetries as well as a Lorentzian boost symmetry? At first glance, this may appear to have made things worse. Without a manifold (or even if the manifold is a single point) we no longer have any possibility for external symmetries. However, while there are certainly less possible external symmetries, we are now open to a wider range of internal symmetries. It is among these internal symmetries that we will find KG1-KG7's hidden continuous translation and rotation symmetries and even (limited) Lorentzian boost symmetries. As I will argue these symmetries can reasonably be given these names despite being internal symmetries. (In Sec.~\\ref{SecKlein3} I will present a third attempt at interpreting these theories which ``externalizes'' these symmetries, making them genuinely spacial translations, rotations, and Lorentzian boosts.)\n\nWith our focus now on $\\bm{\\Phi}\\in\\mathbb{R}^L$, let us consider its possibilities for symmetries. As discussed above, we have no possibility of external symmetries associated with the manifold. However, we do have possible internal symmetries associated with the value space (i.e., an infinite dimensional vector space) we now have the full range of invertible linear-affine transformations over $\\mathbb{R}^L$, namely $\\text{Auto}(\\mathcal{V})=\\text{Aff}(\\mathbb{R}^L)$. There are no gauge symmetries here as there is no longer any manifold for them to smoothly vary across. Thus, in total the possibly symmetries for our theories under this interpretation are,\n\\begin{align}\\label{GaugeVR}\ns:\\quad\\bm{\\Phi}\\mapsto \\Lambda\\,\\bm{\\Phi}+\\bm{c}\n\\end{align}\nfor some invertible linear transformation $\\Lambda\\in\\text{GL}(\\mathbb{R}^L)$ and some vector $\\bm{c}\\in \\mathbb{R}^L$. \n\nContrast this with the symmetries available to us on our first interpretation, namely Eq.~\\eqref{Permutation}. We can role this new class of possible symmetries back onto our first interpretation as follows: Note that because $\\mathbb{R}^L \\cong F_L$ we have $\\text{Aff}(\\mathbb{R}^L)\\cong \\text{Aff}(F_L)$. The set of transformations $\\text{Aff}(F_L)$ acting on $\\phi_\\ell\\in F_L$ is much larger than what we previously considered: namely, \\mbox{$\\text{Auto}(\\mathbb{V})=\\text{Aff}(\\mathbb{R})$} varying smoothly over \\mbox{$\\text{Auto}(Q)=\\text{Perm}(L)$}. Indeed, the present interpretation has a significantly wider class of symmetries than before. \n\nMoving back to our second interpretation, our previous class of transformations (i.e, \\mbox{$\\text{Auto}(\\mathbb{V})=\\text{Aff}(\\mathbb{R})$} varying smoothly over \\mbox{$\\text{Auto}(Q)=\\text{Perm}(L)$}) corresponds to only a subset of our present consideration: $\\text{Aff}(\\mathbb{R}^L)$. The difference is that before we could only apply a permutation matrix $P$ followed by a diagonal matrix $C_1$ whereas now we are allowed a general linear transformation $\\Lambda$. \n\nNote that permutation and diagonal matrices are basis-dependent notions. Our first interpretation took the lattice sites $\\ell\\in L$ seriously as a part of the manifold $Q=L$ and this is reflected in its conception of symmetries. Converted into $\\mathbb{R}^L$ this first conception of these theories' possible symmetries gives special treatment to the basis associated with the lattice sites, namely $\\{\\bm{b}_\\ell\\}_{\\ell\\in L}$. In particular, on our first interpretation, our possible symmetries are those of the form Eq.~\\eqref{GaugeVR} which \\textit{additionally} preserve this basis (up to rescaling, and reordering).\n\nThis basis receives no special treatment on this second interpretation. While it is true that $\\{\\bm{b}_\\ell\\}_{\\ell\\in L}$ were used in the initial construction of $\\bm{\\Phi}$, after this they no longer play any special role. We are always free to redescribe $\\bm{\\Phi}$ in a different basis if we wish. Indeed, here any change of basis transformation is of the form Eq.~\\eqref{GaugeVR} and hence a symmetry.\n\nWith this attachment to the basis $\\{\\bm{b}_\\ell\\}_{\\ell\\in L}$ dropped, we now have a wider class of symmetries. Indeed, everything which was previously considered a symmetry will be here as well and possibly more. Perhaps among this more general class of symmetries we may find new continuous symmetries. Let's see.\n\n\\subsubsection*{Symmetries of KG1-KG7: Second Attempt}\nWhich of the above transformations are symmetries for KG1-KG7? A non-exhaustive  investigation of the symmetries of KG1-KG7 on this interpretation is carried out in Appendix~\\ref{AppA}, but the results are the following. For KG1 and KG2 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)]  action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$ where \\mbox{$T^\\epsilon_\\text{j}=T^\\epsilon\\otimes\\openone$} with $T^\\epsilon$ defined below. Similarly for $T^\\epsilon_\\text{n}=\\openone\\otimes T^\\epsilon$,\n    \\item[2)] two negation symmetries which map basis vectors as  $\\bm{e}_j\\otimes\\bm{e}_n\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_n$ and $\\bm{e}_j\\otimes\\bm{e}_n\\mapsto \\bm{e}_j\\otimes\\bm{e}_{-n}$ respectively,\n    \\item[3)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k)\\mapsto \\tilde{f}(\\omega,k)\\,\\bm{\\Phi}(\\omega,k)$} for some non-zero complex function $\\tilde{f}(\\omega,k)\\in\\mathbb{C}$ with $\\omega,k\\in[-\\pi,\\pi]$,\n    \\item[4)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation with two differences: Firstly, global rescaling $\\phi_\\ell\\mapsto c_1 \\phi_\\ell$ has been refined to a local Fourier rescaling. Note that the discrete Fourier transform itself is in $\\text{Aff}(\\mathbb{R}^L)$ and so is in the class of potential symmetries considered here.\n\nSecondly, discrete shifts have been replaced with action by \n\\begin{align}\\label{TDef}\nT^\\epsilon\\coloneqq\\text{exp}(-\\epsilon D)\n\\end{align}\nwith $\\epsilon\\in\\mathbb{R}$ acting on each tensor factor. A straight-forward calculation shows that $T^\\epsilon$ acts on the planewave basis \\mbox{$\\bm{\\Phi}(k)\\coloneqq \\sum_{n\\in\\mathbb{Z}}e^{-\\mathrm{i} k n}\\bm{e}_n$} with $k\\in[-\\pi,\\pi]$ as\n\\begin{align}\n\\nonumber\nT^\\epsilon:\\bm{\\Phi}(k)\\mapsto \\exp(\\mathrm{i} k \\epsilon)\\,\\bm{\\Phi}(k).\n\\end{align}\nUsing \\mbox{$\\bm{e}_m\n=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi e^{\\mathrm{i}\\, k\\, m}\\, \\bm{\\Phi}(k)\\,\\d k$} we can recover how $T^\\epsilon$ acts on the basis $\\bm{e}_m$ as\n\\begin{align}\nT^\\epsilon: \\bm{e}_m \\mapsto \\sum_{b\\in\\mathbb{Z}} S_m(b+\\epsilon) \\, \\bm{e}_b\n\\end{align}\nwhere \n\\begin{align}\\label{SincDef}\nS(y)=\\frac{\\sin(\\pi y)}{\\pi y}, \\quad\\text{and}\\quad\nS_m(y)=S(y-m),       \n\\end{align}\nare the normalized and shifted sinc functions. Note that $S_n(m)=\\delta_{nm}$ for integers $n$ and $m$.\n\nAs I will now discuss, $T^\\epsilon$ can be thought of as a continuous translation operator for three reasons despite it being here classified as an internal symmetry. Note that none of these reasons depend on $T^\\epsilon$ being a symmetry of the dynamics.\n\nFirst note that $T^\\epsilon$ is a generalization of the discrete shift operation in the sense that taking $\\epsilon=d_1\\in\\mathbb{Z}$ reduces action by $T^\\epsilon$ to the map $T^{d_1}:\\bm{e}_m\\mapsto \\bm{e}_{m-d_1}$ on basis vectors and relatedly the map $m\\mapsto m-d_1$ on lattice sites. \n\nSecond note that $T^\\epsilon$ is additive in the sense that $T^{\\epsilon_1}\\,T^{\\epsilon_2}=T^{\\epsilon_1+\\epsilon_2}$. In the language or representation theory $T^\\epsilon$ is a representation of the translation group on the vector space $\\mathbb{R}^\\mathbb{Z}\\cong\\mathbb{R}^L$. In particular, this means \\mbox{$T^{1\/2}\\,T^{1\/2}=T^1$}: there is something we can do twice to move one space forward. The same is true for all fractions adding to one. \n\nThird, recall from the discussion following Eq.~\\eqref{LambdaD} that $D$ is closely related to the continuum derivative operator $\\partial_x$, exactly matching its spectrum for $k\\in[-\\pi,\\pi]$. Recall also that the derivative is the generator of translation, i.e. $h(x-\\epsilon)=\\text{exp}(-\\epsilon\\, \\partial_x) h(x)$. In this sense also \\mbox{$T^\\epsilon=\\text{exp}(-\\epsilon D)$} is a translation operator. More will be said about $T^\\epsilon$ in Sec.~\\ref{SecSamplingTheory}.\n\nThus we have our first big lesson: despite the fact that KG1-KG2 can be represented on a lattice, they nonetheless have a continuous translation symmetry. This continuous translation symmetry was hidden from us on our first interpretation because we there took the lattice to be hard-wired in as a part of the manifold. Here, we do not take the lattice structure so seriously. We have internalized it into the value space where it then disappears from view as just one basis among many.\n\nBefore KG3 had the same symmetries as KG1 and KG2, now it does not. However, allow me to skip over KG3 temporarily. In the previous interpretation the symmetries of KG4 and KG6 matched each other, both being associated with a 2D square lattice. Moreover, the symmetries of KG5 and KG7 matched each other, both being associated with a hexagonal 2D lattice. Here however, these pairings are broken up and a new matching pair is formed between KG6 and KG7. More will be said about this momentarily.\n\nLet's consider KG4 first. For KG4 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$,\n    \\item[2)] three negation symmetries which map basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{-m}$} respectively,\n    \\item[3)] a 4-fold symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{m}\\otimes\\bm{e}_{-n}$},\n    \\item[4)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\tilde{f}(\\omega,k_1,k_2)\\,\\bm{\\Phi}(\\omega,k_1,k_2)$} for some non-zero complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$ with $\\omega,k_1,k_2\\in[-\\pi,\\pi]$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the symmetries which we found on our first interpretation (plus local Fourier rescaling) but with action by $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ replacing the discrete shifts. The same discussion following Eq.~\\eqref{TDef} applies here, justifying us calling these continuous translation operations. Thus, KG4 has three continuous translation symmetries despite being initially represented on a lattice, Eq.~\\eqref{KG4Long}.\n\nLet's next consider KG5. For KG5 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$,\n    \\item[2)] a negation symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}$} and an exchange symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{m}\\otimes\\bm{e}_{n}$},\n    \\item[3)] a 6-fold symmetry which maps basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-m}\\otimes\\bm{e}_{n+m}$}. (Roughly, this permutes the three terms in Eq.~\\eqref{DKG5}),\n    \\item[4)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\tilde{f}(\\omega,k_1,k_2)\\,\\bm{\\Phi}(\\omega,k_1,k_2)$} for some non-zero complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$ with $t\\in\\mathbb{R}$ and $\\omega,k_1,k_2\\in[-\\pi,\\pi]$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the symmetries which we found on our first interpretation (plus local Fourier rescaling) but with action by $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ replacing the discrete shifts. The same discussion following Eq.~\\eqref{TDef} applies here, justifying us calling these continuous translation operations. Thus, KG5 has three continuous translation symmetries despite being initially represented on a lattice, Eq.~\\eqref{KG5Long}.\n\nBefore moving on to analyze the symmetries of KG6 and KG7, let's first see what this interpretation has to say about them being equivalent to one another. As noted at the end of Sec.~\\ref{SecSevenKG}, KG6 and KG7 have a nice one-to-one correspondence between their solutions. Allow me to spell this out in detail now.\n\nBefore this, however, it is worth briefly noting a rather weak sense in which each of KG4-KG7 are equivalent to each other. As noted following Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7} there is an approximate one-to-one correspondence between each of these theories in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ regime as they approach their common continuum limit, KG0. By contrast,  as I will show, KG6 and KG7 have an \\textit{exact} one-to-one correspondence over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2}<\\pi$ and indeed more. This includes all of their solutions but not all of $\\omega,k_1,k_2\\in[-\\pi,\\pi]$.\n\nThis one-to-one correspondence is mediated by the transformations Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7}. Let's first rewrite these in terms of \\mbox{$\\bm{\\Phi}\\in\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} as follows. Consider first the transformation which maps the dispersion relation for KG7 onto the one for KG6 (namely, Eq.~\\eqref{SkewKG7KG6}). Consider its action on the planewave basis \n$\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}, namely\n\\begin{align}\n\\nonumber\n\\Lambda_{\\text{KG7}\\to\\text{KG6}}:\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\bm{\\Phi}\\left(\\omega,k_1,\\frac{1}{2}k_1+\\frac{\\sqrt{3}}{2}k_2\\right)  \n\\end{align}\nA straight-forward calculation shows this acts on the \\mbox{$\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$} basis as:\n\\begin{align}\\label{SkewKG7KG6Basis}\n&\\Lambda_{\\text{KG7}\\to\\text{KG6}}:\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\mapsto \\\\\n\\nonumber\n&\\bm{e}_j\\otimes\\left(\\sum_{b_1,b_2\\in\\mathbb{Z}}S_n(b_1+b_2\/2) \\,S_m(\\sqrt{3}\\, b_2\/2)\\, \\bm{e}_{b_1}\\otimes\\bm{e}_{b_2}\\right).  \n\\end{align}\nConsider also the transformation which maps the dispersion relation for KG6 onto the one for KG7 (namely, Eq.~\\eqref{SkewKG6KG7}). Consider its action on the planewave basis \n$\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}, namely as\n\\begin{align}\n\\nonumber\n\\Lambda_{\\text{KG6}\\to\\text{KG7}}:\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\bm{\\Phi}\\left(\\omega,k_1,\\frac{2k_2-k_1}{\\sqrt{3}}\\right)  \n\\end{align}\nA straight-forward calculation shows this acts on the \\mbox{$\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$} basis as:\n\\begin{align}\n&\\Lambda_{\\text{KG6}\\to\\text{KG7}}:\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\mapsto \\\\\n\\nonumber\n&\\bm{e}_j\\otimes\\left(\\sum_{b_1,b_2\\in\\mathbb{Z}}S_n(b_1-b_2\/\\sqrt{3}) \\,S_m(2\\, b_2\/\\sqrt{3})\\, \\bm{e}_{b_1}\\otimes\\bm{e}_{b_2}\\right).  \n\\end{align}\n\nIt should be noted however that despite the fact that Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7} are each other's inverses, $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ and $\\Lambda_{\\text{KG6}\\to\\text{KG7}}$ are not each other's inverses (at least not on the whole of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}). This is due to the $2\\pi$ periodic identification of the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$. Indeed, when viewed as acting on $\\mathcal{V}=\\mathbb{R}^L$, the transformation $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ is not even invertible. They are only each other's inverses when we have \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} both before and after these transformations.\n\nFor these reasons we need to consider the following two subspaces of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}: \n\\begin{align}\n\\mathbb{R}^L_\\text{KG7}\\!\\coloneqq\\text{span}(&\\bm{\\Phi}(\\omega,k_1,k_2)\\vert\\text{with }\\omega,k_1,k_2\\in[-\\pi,\\pi]\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG7KG6}})\\\\\n\\mathbb{R}^L_\\text{KG6}\\!\\coloneqq\\text{span}(&\\bm{\\Phi}(\\omega,k_1,k_2)\\vert\\text{with }\\omega,k_1,k_2\\in[-\\pi,\\pi]\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG6KG7}}).\n\\end{align}\nFor later reference it should be noted that the rotation invariant subspace,\n\\begin{align}\\label{RLrotinv}\n\\mathbb{R}^L_\\text{rot.inv}\\coloneqq\\text{span}\\left(\\bm{\\Phi}(\\omega,k_1,k_2)\\Big\\vert\\,\\sqrt{k_1^2+k_2^2}<\\pi\\right)\n\\end{align}\nis a subspace of $\\mathbb{R}^L_\\text{KG6}$, that is $\\mathbb{R}^L_\\text{rot.inv}\\subset \\mathbb{R}^L_\\text{KG6}$.\n\nRestricted to $\\mathbb{R}^L_\\text{KG6}$ and $\\mathbb{R}^L_\\text{KG7}$ these transformations are invertible and indeed are each other's inverses. Fortunately, all of KG6's solutions are in $\\mathbb{R}^L_\\text{KG6}$ (and moreover they are in $\\mathbb{R}^L_\\text{rot.inv}$ as well). Similarly, all of KG7's solutions are in $\\mathbb{R}^L_\\text{KG7}$. Thus, $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ maps generic solutions to KG7 onto generic solutions for KG6 in an invertible way. Therefore, $\\Lambda_{\\text{KG6}\\to\\text{KG7}}$ and $\\Lambda_{\\text{KG7}\\to\\text{KG6}}$ give us not only a one-to-one correspondence between the solutions to KG6 and KG7 but a solution-preserving vector-space isomorphism between KG6 and KG7. One can gloss this situation saying: the KPMs of KG6 and KG7 are not isomorphic, but their DPMs are.\n\nThe fact that this is solution-preserving vector-space isomorphism rather than merely a one-to-one correspondence has substantial consequences for these theories' symmetries. Namely, this forces KG6 and KG7 to have the same symmetries. This is because these transformations are both of the form Eq.~\\eqref{GaugeVR} (but notably not of the form Eq.~\\eqref{Permutation}) for any symmetry transformation for KG6 there is a corresponding symmetry transformation for KG7 and vice versa. \n\nThis is in strong contrast to the results of our previous analysis in Sec.~\\ref{SecKlein1}. There KG6 was seen to have symmetries associated with a square 2D lattice and KG7 was seen to have the symmetries associated with a hexagonal 2D lattice. By contrast, in the present interpretation KG6 and KG7 are thoroughly equivalent: We have a solution-preserving vector-space isomorphism between them. Thus, on this interpretation the only difference between KG6 and KG7 is a change of basis.\n\nThus we have our second big lesson: despite the fact that KG6 and KG7 can be represented with very different lattice structures (i.e., a square lattice versus a hexagonal lattice) they have nonetheless turned out to be completely equivalent to one another. This equivalence was hidden from us on our first interpretation because we there took the lattice too seriously. As I will now discuss, this reduced their continuous rotation symmetries down to quarter rotations and one-sixth rotations respectively and thereby made them inequivalent. Here, we do not take the lattice structure so seriously. We have here internalized it into the value space where it subsequently disappears from view as just one basis among many.\n\nIn addition to switching between lattice structures, in this interpretation we can also do away with them altogether. In this interpretation, a lattice structure is associated with a choice of basis for $\\mathcal{V}=\\mathbb{R}^L$. A choice of basis (like a choice of coordinates) is ultimately a merely representational choice, reflecting no physics. We can always choose, if we like, to work within a basis-free formulation of these theories. That is, ultimately, a lattice-free formulation of these theories. Thus we have our third big lesson: given a discrete spacetime theory with some lattice structure we can always reformulate it in such a way that it has no lattice structure whatsoever.\n\nIn the rest of this subsection I will only discuss the symmetries KG6; analogous conclusions are true for KG7 after applying $\\Lambda_{\\text{KG6}\\to\\text{KG7}}$. For KG6 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$,\n    \\item[2)] three negation symmetries which map basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-n}\\otimes\\bm{e}_{m}$}, and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{-m}$} respectively,\n    \\item[3)] action by $R^\\theta$  sending $\\bm{\\Phi}\\mapsto R^\\theta \\bm{\\Phi}$ with $R^\\theta$ defined below. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[4)] action by $\\Lambda^w_\\text{j,n}$ sending $\\bm{\\Phi}\\mapsto \\Lambda^w_\\text{j,n} \\bm{\\Phi}$ with $\\Lambda^w_\\text{j,n}$ defined below. Similarly for $\\Lambda^w_\\text{j,m}$ which is defined below as well. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[5)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto \\tilde{f}(\\omega,k_1,k_2)\\,\\bm{\\Phi}(\\omega,k_1,k_2)$} for some non-zero complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$ with $\\omega,k_1,k_2\\in[-\\pi,\\pi]$,\n    \\item[6)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nAs with KG4 and KG5, we have here gained local Fourier rescaling and action by $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ has replaced the discrete shifts from before. The same discussion following Eq.~\\eqref{TDef} applies here, justifying us calling these continuous translation operations. Thus, KG6 (and KG7) have three continuous translation symmetries despite being initially represented on a lattice.\n\nAdditionally, we have the quarter rotation symmetry from our first interpretation replaced with action by $R^\\theta$, which as I will argue is essentially a continuous rotation transformation. Before that, it is worth noting a rather weak sense in which each of KG4-KG7 are rotation invariant. Each of these theories is approximately rotation invariant in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ regime as they approach the continuum limit. By contrast, as I will show, KG6 is \\textit{exactly} rotation invariant over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2}<\\pi$, that is the whole of $\\mathbb{R}^L_\\text{rot.inv.}$. Since all of KG6's solutions lie inside of $\\mathbb{R}^L_\\text{rot.inv.}$, $R^\\theta$ will always map its solutions to solutions in an invertible way, and will hence be a symmetry.\n\nThis alleged continuous rotation transformation $R^\\theta$ is given by \n\\begin{align}\\label{RthetaDef}\nR^\\theta \\coloneqq \\exp(-\\theta (N_\\text{n} D_\\text{m}-N_\\text{m} D_\\text{n}))\n\\end{align}\nwith $\\theta\\in\\mathbb{R}$ and where $N$ is the ``position operator'' which acts on the basis $\\bm{e}_m$ as $N\\bm{e}_m=m\\,\\bm{e}_m$ for $m\\in\\mathbb{Z}$. Thus \\mbox{$N_\\text{n}=\\openone\\otimes N\\otimes\\openone$} and \\mbox{$N_\\text{m}=\\openone\\otimes \\openone\\otimes N$} return the second and third index respectively when acting on $\\bm{e}_j\\otimes \\bm{e}_n\\otimes \\bm{e}_m$.\n\nA straight-forward calculation shows that $R^\\theta$ acts on the planewave basis $\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} as\n\\begin{align}\nR^\\theta:\\,&\\bm{\\Phi}(\\omega,k_1,k_2) \\mapsto\\\\\n\\nonumber\n&\\bm{\\Phi}(\\omega,\\cos(\\theta)k_1-\\sin(\\theta)k_2,\\sin(\\theta)k_1+\\cos(\\theta)k_2).\n\\end{align}\nand acts on the basis $\\bm{e}_j\\otimes \\bm{e}_n\\otimes \\bm{e}_m$ as\n\\begin{align}\n&R^\\theta: \\bm{e}_j\\!\\otimes\\bm{e}_n\\!\\otimes\\bm{e}_m \\mapsto\\bm{e}_{j}\\otimes\\!\\!\\!\\!\\sum_{b_1,b_2\\in\\mathbb{Z}} R_{nm}^{b_1 b_2}(\\theta)\\,\n\\bm{e}_{b_1}\\!\\!\\otimes\\bm{e}_{b_2}\\\\\n\\nonumber\n&R_{nm}^{b_1 b_2}\\!(\\theta)\\!=\\!S_n(\\cos(\\theta) b_1 \\!-\\! \\sin(\\theta) b_2) S_m(\\sin(\\theta) b_1\\!+\\! \\cos(\\theta)b_2).\n\\end{align}\n\nIt should be noted that $R^\\theta$ is not invertible (at least not on the whole of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}). As I will soon discuss, this is due to the $2\\pi$ periodic identification of the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$. However, $R^\\theta$ is invertible over $\\mathbb{R}^L_\\text{rot.inv.}$ which contains all of KG6's solutions. Thus, for KG6 $R^\\theta$ always maps solutions to solutions in an invertible way, and is hence a symmetry.\n\nTo see why $R^\\theta$ is not invertible over all of $\\mathcal{V}=\\mathbb{R}^L$ note that $R^{\\pi\/4}$ maps two different planewaves to the same place: Firstly note,\n\\begin{align}\nR^{\\pi\/4}\\bm{\\Phi}(\\omega,\\pi,\\pi)&=\\bm{\\Phi}(\\omega,0,\\sqrt{2}\\pi)\\\\\n\\nonumber\n&=\\bm{\\Phi}(\\omega,0,\\sqrt{2}\\pi-2\\pi)\n\\end{align}\nsince the planewaves repeat themselves with period $2\\pi$. Secondly note, \n\\begin{align}\n\\nonumber\nR^{\\pi\/4}\\bm{\\Phi}(\\omega,\\pi-\\sqrt{2}\\pi,\\pi-\\sqrt{2}\\pi)&=\\bm{\\Phi}(\\omega,0,\\sqrt{2}\\pi-2\\pi).\n\\end{align}\nSuch issues do not arise when $\\sqrt{k_1^2+k_2^2}<\\pi$. Thus, when we restrict our attention to $\\mathbb{R}^L_\\text{rot.inv}$ (which contains all of KG6's solutions) then $R^\\theta$ is invertible and indeed a symmetry. One can gloss this situation saying: rotation does not map the KPMs of KG6 onto themselves in an invertible way, but it does for the DPMs of KG6. If we cut the KPMs of KG6 down to $\\mathbb{R}^L_\\text{rot.inv}$ this minor issue is fixed.\n\nAs I will now discuss, $R^\\theta$ can be thought of as a continuous rotation operator for three reasons despite it being here an internal symmetry. First note that $R^\\theta$ is a generalization of quarter rotation operation in the sense that taking $\\theta=\\pi\/2$ reduces action by $R^\\theta$ to the map \\mbox{$R^{\\pi\/2}:\\bm{e}_{j}\\otimes\\bm{e}_{n}\\otimes\\bm{e}_{m}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{m}\\otimes\\bm{e}_{-n}$} on basis vectors and relatedly the map $(j,n,m)\\mapsto (j,m,-n)$ on lattice sites.\n\nSecond, note that restricted to $\\mathbb{R}^L_\\text{rot.inv.}$, $R^\\theta$ is cyclically additive in the sense that $R^{\\theta_1}\\,R^{\\theta_2}=R^{\\theta_1+\\theta_2}$ with $R^{2\\pi}=\\openone$. In the language of representation theory, $R^\\theta$ is a representation of the rotation group on the vector space $\\mathbb{R}^L_\\text{rot.inv.}$. In particular, this means \\mbox{$R^{\\pi\/4}\\,R^{\\pi\/4}=R^{\\pi\/2}$}.  There is something we can do twice to make a quarter rotation. Similarly for all fractional rotations. Moreover, note that together with the above discussed translations, these constitute a representation of the Euclidean group over $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nThird, recall from the discussion following Eq.~\\eqref{LambdaD} that $D$ is closely related to the continuum derivative operator $\\partial_x$, exactly matching its spectrum for $k\\in[-\\pi,\\pi]$. Recall also that rotations are generated through the derivative as \\mbox{$h(R^\\theta(x,y))= \\exp(-\\theta (x \\partial_y-y \\partial_x))h(x,y)$}. In this sense also $R^\\theta$ is a rotation operator. More will be said about $R^\\theta$ in Appendix~\\ref{AppA}.\n\nThis adds to our first big lesson: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a continuous rotation symmetry. This, in addition to their continuous translation symmetries. These continuous translation and rotations symmetries were hidden from us on our first interpretation because we there took the lattice representations too seriously. Here, we do not take the lattice structure so seriously. Instead, we have internalized it into the value space where it then disappears from view as just one basis among many.\n\nLet's next discuss KG6's (limited) Lorentzian boost symmetry. In addition to generalizing the 4-fold symmetry into to a continuous rotation symmetry, KG6 also has a brand new symmetry on this interpretation, namely ``action by $\\Lambda^w_\\text{j,n}$ and\/or $\\Lambda^w_\\text{j,m}$''. As I will argue these are essentially Lorentz boost transformations. Before that, it is worth noting a rather weak sense in which each of KG4-KG7 are Lorentz invariant. Each of these theories is approximately Lorentz invariant as they approach the continuum limit regime $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll\\pi$ at least boosts parameters $w$ which keep them in this regime. By contrast, as I will show, KG6 is \\textit{exactly} Lorentz invariant over \\textit{a finite-sized region} around $\\omega,k_1,k_2=0$ and boost parameter $w=0$.\n\nThis alleged Lorentz boost transformations are given by\n\\begin{align}\\label{LambdaWDef}\n\\Lambda^w_\\text{j,n}\\coloneqq\\exp(-w (&N_\\text{j} D_\\text{n} + N_\\text{n} D_\\text{j})),\\\\\n\\nonumber\n\\Lambda^w_\\text{j,m}\\coloneqq\\exp(-w (&N_\\text{j} D_\\text{m} + N_\\text{m} D_\\text{j}))\n\\end{align}\nwith $w\\in\\mathbb{R}$. Note that $\\Lambda^w_\\text{j,n}$ acts only on the first and second tensor factor, whereas $\\Lambda^w_\\text{j,m}$ acts only on the first and third factors. In what follows I will focus on $\\Lambda^w_\\text{j,n}$, with similar results following for $\\Lambda^w_\\text{j,m}$.\n\nA straight-forward calculation shows that $\\Lambda^w_\\text{j,n}$ acts on the planewave basis $\\bm{\\Phi}(\\omega,k_1,k_2)$ with \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} as\n\\begin{align}\n\\Lambda^w_\\text{j,n}:\\,&\\bm{\\Phi}(\\omega,k_1,k_2) \\mapsto\\\\\n\\nonumber\n&\\bm{\\Phi}(\\cosh(w)\\omega+\\sinh(w)k_1,\\sinh(w)\\omega+\\cosh(w)k_1,k_2).\n\\end{align}\nThis is a Lorentz boost in discrete Fourier space. It follows from this that $\\Lambda^w_\\text{j,n}$ acts on the basis $\\bm{e}_j\\otimes \\bm{e}_n\\otimes\\bm{e}_m$ as\n\\begin{align}\n\\nonumber\n&\\Lambda^w_\\text{j,n}: \\bm{e}_j\\otimes\\!\\bm{e}_n\\otimes\\!\\bm{e}_m\\mapsto\\!\\!\\!\\!\\sum_{b_1,b_2\\in\\mathbb{Z}} \\Lambda_{jn}^{b_1 b_2}(w)\\,\n\\bm{e}_{b_1}\\otimes\\!\\bm{e}_{b_2}\\otimes\\!\\bm{e}_m\\\\\n&\\Lambda_{jn}^{b_1 b_2}(w)=S_j(\\cosh(w) b_1 + \\sinh(w) b_2)\\\\ \n\\nonumber\n&\\qquad\\qquad\\times S_n(\\sinh(w) b_1+\\cosh(w) b_2).\n\\end{align}\nLike with $R^\\theta$ discussed above, $\\Lambda^w_\\text{j,n}$ is a symmetries of the dynamics in a qualified sense: namely, $\\Lambda^w_\\text{j,n}$ is not invertible (at least not on the whole of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}). Indeed, each of these transformations is only invertible for a subspace of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$}. As before, this is due to the $2\\pi$ periodic identification of the planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$. To see this note that $\\Lambda^{\\ln(2)}_\\text{j,n}$ maps two different planewaves to the same place: Firstly note,\n\\begin{align}\n\\Lambda^{\\ln(2)}_\\text{j,n}\\bm{\\Phi}(\\pi,\\pi,k_2)&=\\bm{\\Phi}(2\\pi,2\\pi,k_2)\\\\\n\\nonumber\n&=\\bm{\\Phi}(0,0,k_2)\n\\end{align}\nsince the planewaves repeat themselves with period $2\\pi$. Secondly note, \n\\begin{align}\n\\nonumber\n\\Lambda^{\\ln(2)}_\\text{j,n}\\bm{\\Phi}(0,0,k_2)&=\\bm{\\Phi}(0,0,k_2)\n\\end{align}\nSuch issues do not arise when we have \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} both before and after the Lorentz transformation. \n\nThus, $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ are invertible over the portion of \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$} spanned by planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ which satisfy the following two conditions:\n\\begin{align}\\label{BoostCond}\n\\vert\\cosh(w)\\,\\omega+\\sinh(w)\\, k_1\\,\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\sinh(w)\\,\\omega+\\cosh(w)\\, k_2\\,\\vert<&\\pi.\n\\end{align}\n\nUnfortunately, unlike with $R^\\theta$ this issue cannot be so easily contained: the region where both $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ are invertible depends on $w$ and indeed, shrinks to nothing as $w\\to\\pm\\infty$. When boosted enough, any planewave except $\\omega=k_1=k_2=0$ will leave the region \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. Moreover, for any $w$ there is some planewave in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} which when boosted by $w$ leaves this region. However, despite this, for every planewave in $\\omega,k_1,k_2\\in(-\\pi,\\pi)$ there is some small enough boost $w\\neq0$ which keeps it in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. Moreover, for every \\mbox{$w$} there is some neighborhood around \\mbox{$\\omega,k_1,k_2=0$} where boosting by $w$ leaves us in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. When restricted to acting on the span of these planewaves $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,n}$ are both invertible.\n\nIt is in the following sense that $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ are symmetries of KG6. Consider solutions to KG6 which are supported only over planewaves in some finite neighborhood of $\\omega,k_1,k_2=0$. Consider $\\Lambda^{w}_\\text{j,n}$ and $\\Lambda^{w}_\\text{j,m}$ with $w$ in some finite neighborhood of $w=0$. Choose the neighborhoods such that Eq.~\\eqref{BoostCond} is satisfied throughout. Restricting our attention to this regime, all of these transformations map exactly map solutions to solutions in an invertible way. As I claimed above, KG6 is \\textit{exactly} Lorentz invariant over \\textit{a finite-sized region} around $\\omega,k_1,k_2=0$ and boost parameter $w=0$.\n\nRecall as mentioned above that for all planewaves in $\\omega,k_1,k_2\\in(-\\pi,\\pi)$ there is some small enough boost \\mbox{$w\\neq0$} which keeps it in \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. Thus, KG6 has a differential Lorentz boost invariance over all of \\mbox{$\\omega,k_1,k_2\\in(-\\pi,\\pi)$}. Note that the same is true of translations and rotations. Including translations and rotations, KG6 has a differential Poincar\\'e invariance over $\\omega,k_1,k_2\\in(-\\pi,\\pi)$. Concretely, taken all together $\\Lambda^w_\\text{j,n}$, $\\Lambda^w_\\text{j,m}$, $R^\\theta$, $T^\\epsilon_\\text{j}$, $T^\\epsilon_\\text{n}$ and $T^\\epsilon_\\text{m}$ form a representation of the Poincar\\'e algebra over the space spanned by \\mbox{$\\omega,k_1,k_2\\in(-\\pi,\\pi)$}. In particular, this means that for every algebraic fact about the differential Poincar\\'e transformations there is an analogous fact here with the group action being replaced by matrix multiplication. Exponentiating this representation of the Poincar\\'e algebra we recover a finite part of the Poincar\\'e group. Namely, the finite-sized collection of states and transformations satisfying:\n\\begin{align}\\label{PoincareCond}\n\\vert\\cosh(w)\\,\\omega+\\sinh(w)\\,\\vert k\\vert\\,\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\sinh(w)\\,\\omega+\\cosh(w)\\,\\vert k\\vert\\,\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\cos(\\theta)\\,k_1-\\sin(\\theta) k_2\\vert<&\\pi\\\\\n\\nonumber\n\\vert\\sin(\\theta)\\,k_1+\\cos(\\theta) k_2\\vert<&\\pi.\n\\end{align}\nIndeed, KG6 is \\textit{exactly} Poincar\\'e invariant over \\textit{a finite-sized region} around $\\omega,k_1,k_2=0$ and $w=\\theta=0$.\n\nAs I will now discuss, in this regime $\\Lambda^w_\\text{j,m}$ and $\\Lambda^w_\\text{j,n}$ can be thought of as implementing Lorentzian boosts for two reasons despite being here categorized as an internal symmetry. Firstly, recall the close relationship noted above between $D$ and $\\partial_x$. Recall also that Lorentz boosts are generated through the derivative as \\mbox{$h(\\Lambda^w(t,x))= \\exp(-w (x \\partial_t + t \\partial_x))h(t,x)$}. \n\nSecondly, as I have already mentioned, together with our above discussed translations and rotations,  $\\Lambda^w_\\text{j,m}$ and $\\Lambda^w_\\text{j,n}$ give us a representation of the Poincar\\'e algebra and even a finite portion of the Poincar\\'e group.\n\nThus we have yet another addendum to our first big lesson: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a Lorentzian boost symmetries (in a finite but limited regime). This, in addition to their continuous translation and rotation symmetries. As impressive as this admittedly limited Lorentz symmetry is, we can do better: In Sec.~\\ref{PerfectLorentz} I will provide a perfectly Lorentzian lattice theory.\n\nFinally, let's consider KG3. For KG3 the dynamical symmetries of the form Eq.~\\eqref{GaugeVR} include:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] action by $T^\\epsilon_\\text{j}$ sending $\\bm{\\Phi}\\mapsto T^\\epsilon_\\text{j} \\bm{\\Phi}$. Similarly for $T^\\epsilon_\\text{n}$.\n    \\item[2)] two negation symmetries which map basis vectors as \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\mapsto \\bm{e}_{-j}\\otimes\\bm{e}_{n}$} and \\mbox{$\\bm{e}_{j}\\otimes\\bm{e}_{n}\\mapsto \\bm{e}_{j}\\otimes\\bm{e}_{-n}$} respectively,\n    \\item[3)] action by $\\Lambda^w_\\text{jn}$ sending $\\bm{\\Phi}\\mapsto \\Lambda^w_\\text{jn} \\bm{\\Phi}$ where \\mbox{$\\Lambda^w_\\text{jn}$} is defined above,\n    \\item[4)] a local Fourier rescaling symmetry which maps \\mbox{$\\bm{\\Phi}(\\omega,k)\\mapsto \\tilde{f}(\\omega,k)\\,\\bm{\\Phi}(\\omega,k)$} for some non-zero complex function $\\tilde{f}(\\omega,k)\\in\\mathbb{C}$ with $\\omega,k\\in[-\\pi,\\pi]$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\bm{\\Phi}\\mapsto \\bm{\\Phi} + \\bm{c}_2$} for some $\\bm{c}_2$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nAs with KG1 and KG2 we have gained local Fourier rescaling and discrete shifts have been replaced with action by $T^\\epsilon$. As discussed above we are justified in calling $T^\\epsilon$ a continuous translation operation. For KG3 we also have a new symmetry, namely action by $\\Lambda^w_\\text{jn}$. For the reasons discussed above, this can be thought of as a (limited) Lorentzian boost symmetry despite it being here classified as an internal symmetry.\n\n\\vspace{0.25cm}\nTo summarize: this second attempt at interpreting KG1-KG7 has fixed all of the issues with our previous interpretation. Firstly, there is no longer any tension between these theories' differing locality properties and the rates at which they converge to the (perfectly local) continuum theory in the continuum limit. (There are no longer any differences in locality.) See \\cite{DiscreteGenCovPart1} for further discussion. Secondly, the fact that we have a nice one-to-one correspondence between the solutions to KG6 and KG7 is now more satisfyingly reflected in their matching symmetries. Finally, this interpretation has exposed the fact that KG1-KG7 have hidden continuous translation and rotation symmetries as well as a (limited) Lorentzian boost symmetries.\n\nBy and large, the interpretation invalidates all of the first intuitions laid out in Sec.~\\ref{SecIntro}. As this interpretation has revealed, the lattice seems to play a merely representational role in the theory: it does not restrict our symmetries. Moreover, theories initially appearing with different lattice structures may nonetheless turn out to be completely equivalent. The process for switching between lattice structures is here a change of basis in the value space. Indeed, we have a third lesson: there is no sense in which these lattice structures are essentially ``baked-into'' these theories; on this interpretation our theories make no reference to any lattice structure if we work in a basis-independent way. No basis is dynamically favored. \n\nAs I discussed in \\cite{DiscreteGenCovPart1} these three lessons lay the foundation for a strong analogy between the lattice structures which appear in our discrete spacetime theories and the coordinate systems which appear in our continuum spacetime theories. This analogy here extended to Lorentzian theories.\n\nThese are substantial lessons, but ultimately this interpretation has a few issues of its own. Firstly, the way that the tension is dissolved between locality and convergence in the continuum limit is unsatisfying. Intuitively, we ought to be able to extract intuitions about locality from the lattice sites. \n\nMoreover, while this interpretation has indeed exposed KG1-KG7's hidden symmetries, the way it classifies them seems wrong. They are here classified as internal symmetries (i.e., symmetries on the value space) whereas intuitively they should be external symmetries (i.e., symmetries on the manifold).\n\nThe root of all of these issues is taking the theory's lattice structure to be internalized into the theory's value space. Our third attempt at interpreting these theories will fix this by externalizing these symmetries. As I will discuss, this gives us access to a perspective within which we can find a perfectly Lorentzian lattice theory (i.e., not limited as the above theories are).\n\n\\section{Externalizing these theories - Part 1}\\label{SecExtPart1}\nIn the previous section it was revealed that KG1-KG7 have hidden continuous symmetry transformations which intuitively correspond to spacial translations and rotation and even (limited) Lorentzian boosts. In our first attempt at interpreting KG1-KG7 the possibility of such symmetries were outright denied, see Sec.~\\ref{SecKlein1}. In our second attempt, these hidden symmetries were exposed, but they were classified (unintuitively) as internal symmetries, see Sec.~\\ref{SecKlein2}. This is due to an ``internalization'' move made in our second interpretation. This move also had the unfortunate consequence of undercutting our ability to use the lattice sites to reason about locality.\n\nIn this section I will show how we can externalize these symmetries by in a principled way 1) inventing a continuous manifold for our formerly discrete theories to live on and 2) embedding our theory's states\/dynamics onto this manifold as a new dynamical field.\n\n\\subsection{A Principled Choice of Spacetime Manifold}\\label{SecKlein3A}\nIf we are going to externalize these symmetries then we need to have a big enough manifold on which to do the job. What spacetime manifold $\\mathcal{M}$ might be up to the task? The first thing we must do is pick out which of our theory's symmetries we would like to externalize (there may be some symmetries we want to keep internal). For KG1-KG7 we want to externalize the following symmetries: continuous translations, mirror reflections, as well as  discrete rotations for KG4 and KG5 and continuous rotations for KG6 and KG7. For each theory we can collect these dynamical symmetries together in a group $G^\\text{dym}_\\text{to-be-ext}$. Clearly, our choice of spacetime manifold $\\mathcal{M}$ needs to be big enough to have $G^\\text{dym}_\\text{to-be-ext}$ as a subgroup of $\\text{Diff}(\\mathcal{M})$. Let us call this the symmetry-fitting constraint\\footnote{As I will soon discuss, there may be transformations which we want to externalize even when they are not symmetries.}. \n\nOf course, symmetry-fitting alone doesn't uniquely specify which manifold we ought to use. If $\\mathcal{M}$ works, then so does any larger $\\mathcal{M}'$ with $\\mathcal{M}$ as a sub-manifold. For standard Occamistic reasons, it is natural to go with the smallest manifold which gets the job done. The larger the gap between the groups $G^\\text{dym}_\\text{to-be-ext}$ and $\\text{Diff}(\\mathcal{M})$ the more fixed spacetime structures will need to be introduced later on (see Sec.~\\ref{SecFullGenCov}).\n\nIn principle, we are free to pick any large-enough manifold which we like to embed KG1-KG7 onto. However, perhaps surprisingly, if we make natural choices about how the translation operations we have already identified (see Eq.~\\eqref{TDef}) fit onto the new spacetime manifold then our choice of $\\mathcal{M}$ is actually fixed up to diffeomorphism. In particular, I demand the following: Certain translation operations on $\\mathbb{R}^L$ are to correspond (perhaps in a complicated way) to parallel transport on the new spacetime manifold $\\mathcal{M}$. For KG1-KG3 these to-be-externalized translation operations are $T^{\\epsilon_1}_\\text{j}$ and $T^{\\epsilon_2}_\\text{n}$. For KG4-KG7 these to-be-externalized translation operations are $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and \n$T^{\\epsilon_3}_\\text{m}$. Let us call this the translation-matching constraint. As I will soon show, this constraint fixes the new spacetime manifold $\\mathcal{M}$ up to diffeomorphism.\n\nBefore fleshing this out, however, it's worth reflecting on two questions focusing on KG4-KG7: What exactly makes $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ translation operations? Moreover, what motivation do we have to externalize these particular translation operations? To answer: firstly, these can be thought of as translation operations for the reasons discussed following Eq.~\\eqref{TDef}. Note these reasons are unrelated to the fact that these are dynamical symmetries of KG4-KG7. Suppose the dynamics of KG4 given by Eq.~\\eqref{KG4Long} was modified to have explicit dependence on the index $n$. In this case, $T^{\\epsilon_2}_\\text{n}$ would no longer be a dynamical symmetry but it would still be a translation operation (and moreover, one worth externalizing).\n\nSecondly, why externalize these translation operations in particular? As I will now discuss, any motivation for externalizing these particular translation operations must come from the dynamics. Forgoing any dynamical considerations, all we can say about KG1-KG7 is that they concern vectors $\\bm{\\Phi}\\in\\mathbb{R}^L$ (or alternatively functions \\mbox{$\\phi:L\\to\\mathbb{R}$} in $F_L$). Recall that pre-dynamics the set of labels for lattice sites $L$ is uncountable but otherwise unstructured; We might index it using any number of indices we like, \\mbox{$L\\cong\\mathbb{Z}\\cong\\mathbb{Z}^2\\cong\\dots\\cong\\mathbb{Z}^{17}\\cong\\dots$}. Using each of these re-indexings we can grant $\\mathbb{R}^L$ different tensor product structures, \\mbox{$\\mathbb{R}^L\\cong\\mathbb{R}^Z\\cong\\mathbb{R}^Z\\otimes\\mathbb{R}^Z\\cong\\dots$}. In each tensor factor we can define a translation operation $T^\\epsilon$ as in Eq.~\\eqref{TDef}. Thus, the vector space $\\mathbb{R}^L$ supports representations of: the 1D translation group, the 2D translation group, $\\dots$, the 17D translation group, etc. Given all these possibilities, why externalize $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ in particular? The answer must come from the dynamics.\n\nOne good reason to consider $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ worthy of externalization is that they are dynamical symmetries of KG4-KG7. However, as mentioned above, something might be a translation operation (and moreover, one worthy of externalizing) even if it's not a dynamical symmetry. Unfortunately, I don't have a perfect rule for how to identify such cases. The best I can offer is to check whether its associated derivative (or some function thereof) appears in the dynamical equations.\n\nRegardless, its sufficiently clear for KG4-KG7 that $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and \n$T^{\\epsilon_3}_\\text{m}$ are among the translation operations worthy of externalizing. Our translation-matching constraint suggests that these ought to correspond (perhaps in a complicated way) to parallel transport on the new spacetime manifold $\\mathcal{M}$. I claim that (at least in this case) this constraint fixes the spacetime manifold $\\mathcal{M}$ up to diffeomorphism.\n\nTo show this I will first pick out at each point \\mbox{$p\\in\\mathcal{M}$} on the manifold three independent directions in the tangent space at $p$. To realize the translation-matching constraint, I demand that differential translation by $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ is then to be carried out on the manifold as parallel transport in these three directions. Note that this already implies that the dimension of $\\mathcal{M}$ is at least three.\n\nIndeed, in this case we have good reason to take $\\mathcal{M}$ to be exactly three dimensional. To see this, note that \n\\begin{align}\\label{RLCover}\n\\mathbb{R}^L&\\cong \\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\\\\n&=\\text{span}(T^{\\epsilon_1}_\\text{j}  T^{\\epsilon_2}_\\text{n}  T^{\\epsilon_3}_\\text{m} \\, \\bm{e}_0\\!\\otimes\\bm{e}_0\\!\\otimes\\bm{e}_0\\vert (\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3)\n\\end{align}\nThat is, beginning from $\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0$ these translations cover all of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. That is, they cover all of our state space in the second interpretation. The three translations $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$ are thus already enough to cover every kinematic possibility. We need no further dimensions and so by Occam's razor we may then take the spacetime manifold $\\mathcal{M}$ to be three dimensional. \n\nBut how does the translation-matching constraint (which I have yet to state technically) fix $\\mathcal{M}$ up to diffeomorphism? As I will now discuss, (at least in this case) $\\mathcal{M}$ it is fixed up to diffeomorphism by the group theoretic properties of $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$. Let $G_\\text{trans.}$ be the group formed by all compositions of $T^{\\epsilon_1}_\\text{j}$, $T^{\\epsilon_2}_\\text{n}$, and $T^{\\epsilon_3}_\\text{m}$. Note that the group $G_\\text{trans.}$ is a Lie group, and hence is also a differentiable manifold.\n\nRecall that so far the translation-matching constraint associated to each of our differential translations, a direction in the tangent space of each point $p\\in\\mathcal{M}$. Since $G_\\text{trans.}$ contains nothing but these translations, this gives us a one-to-one correspondence between the algebra $g_\\text{trans.}$ of $G_\\text{trans.}$ and (at least a subspace of) the of the tangent space of each point on the manifold. Let's take these tangent vectors to very smoothly across the manifold. Let $\\mathcal{M}_\\text{trans.}(p)$ be the submanifold of points on the spacetime manifold reachable from $p\\in\\mathcal{M}$ by following these tangent vectors. The algebra $g_\\text{trans.}$ is related to the group $G_\\text{trans.}$ in the same way that this tangent space it related to $\\mathcal{M}_\\text{trans.}(p)$, namely by repeated application of the exponential map. Hence, I take translation-matching to demand that $\\mathcal{M}_\\text{trans.}(p)\\cong G_\\text{trans.}$ as differentiable manifolds for every $p\\in\\mathcal{M}$. \n\nAs discussed above, we have reason to take $\\mathcal{M}$ to be three dimensional. Hence (assuming that $\\mathcal{M}$ is connected) any three dimensional submanifold (e.g. $\\mathcal{M}_\\text{trans.}(p)$) must in fact be the entire manifold. Therefore we have $\\mathcal{M}\\cong G_\\text{trans.}$ as differentiable manifolds.\n\nBut what can we say about this translation group $G_\\text{trans.}$ viewed as a manifold? Notice that these three continuous translation operations comprising $G_\\text{trans.}$ all commute with each other. As such for any combination of them $T_\\text{generic}\\in G_\\text{trans.}$ we always have a unique factorization of the form,\n\\begin{align}\\label{Tgeneric}\nT_\\text{generic} = T^{\\epsilon_1}_\\text{j} \\ T^{\\epsilon_2}_\\text{n} \\ T^{\\epsilon_3}_\\text{m}\n\\end{align}\nMoreover, this factorization represents all of $G_\\text{trans.}$ without redundancy. Indeed, we can smoothly parameterize all of $G_\\text{trans.}$ using parameters \\mbox{$(\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3$} in a one-to-one way. That is, $G_\\text{trans.}\\cong\\mathbb{R}^3$ as differentiable manifolds. Therefore, we have $\\mathcal{M}\\cong\\mathbb{R}^3$ as well. Note this means that we have access to a global coordinate system for $\\mathcal{M}$.\n\nThus, from the translation-matching constraint alone we are forced to take $\\mathcal{M}\\cong\\mathbb{R}^3$ for KG4-KG7. For KG1-KG3 the same translation-matching constraint forces us to take $\\mathcal{M}\\cong\\mathbb{R}^2$. Note that in both cases these manifolds satisfy the symmetry-fitting constraint: $\\text{Diff}(\\mathcal{M})$ is large enough to contain $G^\\text{dyn}_\\text{to-be-ext}$.\n\n\\subsection{A Principled Choice of Embedding}\nNow that we have a spacetime manifold selected, we need to somehow embed $\\bm{\\Phi}$ (or equivalently $\\phi_\\ell$) into it. Here I will begin with $\\bm{\\Phi}$ with the approach from $\\phi_\\ell$ being addressed in Sec.~\\ref{SecExtPart2}. The goal in either case is to construct in a principled way from either $\\bm{\\Phi}$ or $\\phi_\\ell$ a new field \\mbox{$\\phi:\\mathcal{M}\\to\\mathbb{R}$} defined over this manifold. \n\nBefore this, however, recall that our move from the first to the second interpretation was mediated by means of a vector-space isomorphism $F_L\\cong\\mathbb{R}^L$ namely Eq.~\\eqref{PhiDef}. Here too, our reinterpretation will be mediated by a vector-space isomorphism.\n\nLet $F(\\mathcal{M})$ be the set of all real scalar functions \\mbox{$f:\\mathcal{M}\\to\\mathbb{R}$}. Notice that this is a vector space as it is closed under addition and scalar multiplication. Our goal here is to in a principle way construct an isomorphism between \\mbox{$\\mathbb{R}^L$} and some vector subspace of \\mbox{$F(\\mathcal{M})$}. Indeed, $F(\\mathcal{M})$ is much larger than $\\mathbb{R}^L$ (having an uncountably infinite dimension). As such we will only ever map $\\mathbb{R}^L$ onto some subset of it, $F\\subset F(\\mathcal{M})$. In order for us to have a vector space isomorphism between $\\mathbb{R}^L$ and $F$ the following conditions must be satisfied. $F$ must be: 1) closed under addition and scalar multiplication and hence a vector space, and 2) countably infinite dimensional and hence isomorphic to $\\mathbb{R}^L$, i.e., \\mbox{$F\\cong\\mathbb{R}^\\mathbb{Z}\\cong\\mathbb{R}^L$}. With the target subset characterized, the embedding of $\\bm{\\Phi}$ onto $\\mathcal{M}$ is then accomplished by picking a vector-space isomorphism $E:\\mathbb{R}^L\\to F$. The embedded field is then gives to us as $\\phi\\coloneqq E(\\bm{\\Phi})$.\n\nIt appears we have a great deal of freedom here. Suppose that, in line with the translation-matching constraint discussed above, we take $\\mathcal{M}\\cong\\mathbb{R}^2$ for KG1-KG3 and $\\mathcal{M}\\cong\\mathbb{R}^3$ for KG4-KG7. In this case, we still have great freedom in picking both the target vector space $F\\subset F(\\mathcal{M})$ and the isomorphism $E:\\mathbb{R}^L\\to F$. However, as I will now discuss, translation-matching also drastically limits these choices as well. Allow me to demonstrate with KG4-KG7.\n\nFirst, allow me to explicitly define a coordinate system for $\\mathcal{M}\\cong G_\\text{trans.}\\cong\\mathbb{R}^3$. As discussed above we already have an explicit smooth parametrization of $G_\\text{trans.}$ via \\mbox{$(\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3$}, see Eq.~\\eqref{Tgeneric}. In addition to this, concretely realizing our translation-matching constraint requires us to fix a smooth map from $G_\\text{trans.}$ to $\\mathcal{M}$. Combined these two maps give us a global coordinate system $(\\epsilon_1,\\epsilon_2,\\epsilon_3)\\in\\mathbb{R}^3$ for $\\mathcal{M}$. We can then rescale these to give new coordinates $(t,x,y)\\in\\mathbb{R}^3$ by adding in a length scale $a>0$ with $t=\\epsilon_1 a$, $x=\\epsilon_2 a$, and $y=\\epsilon_3 a$ for some fixed length scale $a$. \n\nIn total this picks out a diffeomorphism \\mbox{$d_\\text{trans.}:\\mathbb{R}^3\\to\\mathcal{M}$} which assigns global coordinates $(t,x,y)\\in\\mathbb{R}^3$ to $\\mathcal{M}$. For much of the next sections I will work with $\\phi$ in these coordinates. Namely I will work with the pull back, $\\phi\\circ d_\\text{trans.}:\\mathbb{R}^3\\to\\mathbb{R}$. One may worry that this choice of coordinates is arbitrary. Indeed, if one changes the smooth map from $G_\\text{trans.}$ to $\\mathcal{M}$ referenced above we end up with a different coordinate system $d_\\text{trans.}$ related to the original by some diffeomorphism. While this is true, no matter how one realizes the translation-matching constraint there will be some coordinate system would result from the above construction. In what follows, nothing depends on how the translation-matching constraint is realized.\n\nA word of warning. The coordinates used above might end up lying on the manifold in a curvilinear way. Crucially, the naive distance and metric structure associated with these coordinates do not automatically have any spaciotemporal significance. At present, our manifold $\\mathcal{M}$ still has no metric. When a metric arises later, it will be due to dynamical considerations, not from any choice of coordinates. A coordinate-independent view of these theories will be given in Sec.~\\ref{SecFullGenCov}.\n\nWith this said, by construction we have the following in correspondences in these coordinate:\n\\begin{flushleft}\\begin{enumerate}\n    \\item Action by $T_\\text{j}^\\epsilon$ on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ acts on the manifold $\\mathcal{M}$ in these coordinates as $(t,x,y)\\mapsto(t-\\epsilon\\,a,x,y)$,\n    \\item Action by $T_\\text{n}^\\epsilon$ on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ acts on the manifold $\\mathcal{M}$ in these coordinates as $(t,x,y)\\mapsto(t,x-\\epsilon\\,a,y)$,\n    \\item Action by $T_\\text{m}^\\epsilon$ on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ acts on the manifold $\\mathcal{M}$ in these coordinates as $(t,x,y)\\mapsto(t,x,y-\\epsilon\\,a)$.\n\\end{enumerate}\\end{flushleft}\nIn particular, matching up differential translations on $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ with those on $\\mathcal{M}$ requires\n\\begin{align}\nE(D_\\text{j}\\,\\bm{\\Phi})&=a\\,\\partial_t E(\\bm{\\Phi})\\\\\n\\nonumber\nE(D_\\text{n}\\bm{\\Phi})&=a\\,\\partial_x E(\\bm{\\Phi})\\\\\n\\nonumber\nE(D_\\text{m}\\bm{\\Phi})&=a\\,\\partial_y E(\\bm{\\Phi})\n\\end{align}\nin these coordinates for all $\\bm{\\Phi}\\in\\mathbb{R}^L$.\n\nEvaluating these conditions in the planewave basis for $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$ (namely $\\bm{\\Phi}(\\omega,k_1,k_2)$ for \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}) we see that $E(\\bm{\\Phi}(\\omega,k_1,k_2))$ must be simultaneously an eigenvector of $\\partial_t$, $\\partial_x$, and $\\partial_y$ with eigenvalues $-\\mathrm{i} \\omega\/a$, $-\\mathrm{i} k_1\/a$, and $-\\mathrm{i} k_2\/a$ respectively. This uniquely picks out the continuum planewaves:\n\\begin{align}\\label{PlaneWaveCont}\n\\text{KG1-KG3:}\\quad&\\phi(t,x;\\omega,k)\n\\coloneqq e^{-\\mathrm{i} \\omega t -\\mathrm{i} k x}\\\\\n\\nonumber\n\\text{KG4-KG7:}\\quad&\\phi(t,x,y;\\omega,k_1,k_2)\n\\coloneqq e^{-\\mathrm{i} \\omega t -\\mathrm{i} k_1 x-\\mathrm{i} k_2 y}.\n\\end{align}\nIn particular, we are forced to take\n\\begin{align}\\label{Eftilde}\nE:\\,&\\bm{\\Phi}(\\omega,k_1,k_2)\\\\\n\\nonumber\n&\\mapsto\\tilde{f}(\\omega,k_1,k_2)\\, \\phi(t,x,y;\\omega\/a,k_1\/a,k_2\/a)\n\\end{align}\nfor some complex function $\\tilde{f}(\\omega,k_1,k_2)\\in\\mathbb{C}$. That is, each discrete planewave must map onto something proportional to the corresponding continuous planewaves (rescaled by $a$)\n\nNote that up to a local rescaling of Fourier space (which recall is a symmetry of the dynamics) our embedding is uniquely fixed by translation-matching. Even without this consideration however, our translation-matching constraint still fixes the vector space $F\\subset F(\\mathcal{M})$ we can embed \\mbox{$\\bm{\\Phi}\\in\\mathbb{R}^L$} onto, note the following. We must have \\mbox{$\\tilde{f}(\\omega,k_1,k_2)\\neq0$} for all \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$} otherwise $E$ will not be invertible an hence not an isomorphism. Therefore $F$ is fixed as,\n\\begin{align}\n&F=\\text{span}(E(\\bm{\\Phi}(\\omega,k_1,k_2))\\vert \\omega,k_1,k_2\\in[-\\pi,\\pi])\\\\\n\\nonumber\n&=\\!\\text{span}(\\tilde{f}(\\omega,k_1,k_2)\\,\\phi(t,x,y;\\frac{\\omega}{a},\\frac{k_1}{a},\\frac{k_2}{a})\\vert \\omega,k_1,k_2\\in\\![-\\pi,\\pi])\\\\\n\\nonumber\n&=\\text{span}(\\phi(t,x,y;\\frac{\\omega}{a},\\frac{k_1}{a},\\frac{k_2}{a})\\vert \\omega,k_1,k_2\\in[-\\pi,\\pi])\\\\\n\\nonumber\n&=\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert \\omega,k_1,k_2\\in[-\\pi\/a,\\pi\/a]).\n\\end{align}\nIn light of this, let us define\n\\begin{align}\n&\\text{KG1-KG3:}\\\\\n\\nonumber\n&F^K\\coloneqq\\text{span}(\\phi(t,x;\\omega,k)\\vert \\omega,k\\in[-K,K])\\\\\n&\\text{KG4-KG7:}\\\\\n\\nonumber\n&F^K\\coloneqq\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert \\omega,k_1,k_2\\in[-K,K])\n\\end{align}\nwhere $K=\\pi\/a$. These are the spaces of bandlimited functions with bandwidth $\\omega,k\\in[-K,K]$ and \\mbox{$\\omega,k_1,k_2\\in[-K,K]$} respectively. Thus, demanding translation-matching we are forced to map $\\bm{\\Phi}\\in\\mathbb{R}^L$ onto some bandlimited $\\phi(t,x,y)\\in F=F^K$ (at least in the coordinates $d_\\text{trans.}$).\n\nThe next section will discuss in detail some remarkable properties of bandlimited functions, namely their sampling property. Before that however, let's find an interpretation for the $\\tilde{f}(\\omega,k_1,k_2)$ function appearing in Eq.~\\eqref{Eftilde}. First note that applying $E$ to the basis vector $\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0$ we have\n\\begin{align}\\label{Embedf}\nE(\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0)=f\\!\\left(\\frac{t}{a},\\frac{x}{a},\\frac{y}{a}\\right)\n\\end{align}\nwhere $f(t,x,y)$ is the inverse Fourier transform of $\\tilde{f}(\\omega,k_1,k_2)$. Next note that by applying $T_\\text{j}^\\epsilon$, $T_\\text{n}^\\epsilon$, and $T_\\text{m}^\\epsilon$ with integer arguments we can get from $\\bm{e}_0\\otimes\\bm{e}_0\\otimes\\bm{e}_0$ to any other basis vector, $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$. In these coordinates this means,\n\\begin{align}\n&E(\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m)=f\\left(\\frac{t}{a}-j,\\frac{x}{a}-n,\\frac{y}{a}-m\\right)\n\\end{align}\nThus, we can understand a choice of $\\tilde{f}$ as picking a profile $f(t,x,y)\\in F^\\pi$. A translated and rescaled copy of this profile is then associated with each basis vector.\n\nTo review: Our translation-matching considerations have greatly constrained our choice of both the spacetime manifold $\\mathcal{M}$ and the embedding of $\\bm{\\Phi}$ onto $F(\\mathcal{M})$. In particular, $\\mathcal{M}$ was forced to be diffeomorphic to $\\mathbb{R}^2$ for KG1-KG3 and to $\\mathbb{R}^3$ for KG4-KG7. Moreover, the vector space $F\\subset F(\\mathcal{M})$ which we map $\\bm{\\Phi}\\in\\mathbb{R}^L$ into is forced to be the space of bandlimited functions with some bandwidth (i.e., bandlimited in coordinates $d_\\text{trans.}$ with $\\omega,k_1,k_2\\in[-K,K]$).\n\nBeyond this, our only freedom left is in picking a profile $f$ to associate with each basis vector $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$. In Sec.~\\ref{SecExtPart2} I will motivate a principled choice for $f$. Before that however, let's discuss bandlimited function in some detail.\n\n\\section{Brief Review of Bandlimited Functions and Nyquist-Shannon Sampling Theory}\\label{SecSamplingTheory}\n The previous section has given us reason to care about bandlimited functions. A bandlimited function is one whose Fourier transform has compact support. The bandwidth of such a function is the extent of its support in Fourier space. As I will now discuss, such functions have a remarkable sampling property: they can be exactly reconstructed knowing only their values at a (sufficiently dense) set of sample points. The study of such functions constitutes Nyquist-Shannon sampling theory. For a selection of introductory texts on sampling theory see ~\\cite{GARCIA200263,SamplingTutorial,UnserM2000SyaS}.\n\nTo introduce the topic I will at first restrict our attention to the one-dimensional case with uniform sample lattice before generalizing to higher dimensions and non-uniform samplings later on.\n\n\\subsection{One Dimension Uniform Sample Lattices}\\label{Sec1DUniform}\nConsider a generic bandlimited function, $f_\\text{B}(x)$, with a bandwidth of $K$. That is, a function $f_\\text{B}(x)$ such that its Fourier transform,\n\\begin{align}\n\\mathcal{F}[f_\\text{B}(x)](k)\\coloneqq\\int_{-\\infty}^\\infty f_\\text{B}(x) \\, e^{-\\mathrm{i} k x} \\d x,\n\\end{align}\nhas support only for wavenumbers $\\vert k\\vert< K$.\n\nSuppose that we know the value of $f_\\text{B}(x)$ only at the regularly spaced sample points, $x_n=n\\,a+b$, with some spacing, \\mbox{$0\\leq a\\leq a^*\\coloneqq\\pi\/K$}, and offset, $b\\in\\mathbb{R}$. Let \\mbox{$f_n=f_\\text{B}(x_n)$} be these sample values. Having only the discrete sample data, $\\{(x_n,f_n)\\}_{n\\in\\mathbb{Z}}$, how well can we approximate the function? \n\nThe Nyquist-Shannon sampling theorem~\\cite{ShannonOriginal} tells us that from this data we can reconstruct $f_\\text{B}$ exactly everywhere! That is, from this discrete data, $\\{(x_n,f_n)\\}_{n\\in\\mathbb{Z}}$, we can determine everything about the function $f_\\text{B}$ everywhere. In particular, the following reconstruction is exact, \n\\begin{align}\\label{SincRecon}\nf_\\text{B}(x) \n= S_n\\!\\left(\\frac{x-b}{a}\\right) f_n,\n\\end{align}\nwhere\n\\begin{align}\nS(y)=\\frac{\\sin(\\pi y)}{\\pi y}, \\quad\\text{and}\\quad\nS_n(y)=S(y-n), \n\\end{align}\nare the normalized and shifted sinc functions. Note that $S_n(m)=\\delta_{nm}$ for integers $n$ and $m$. Moreover, note that each $S_n(x)$ is both $L_1$ and $L_2$ normalized and that taken together the set $\\{S_n(x)\\}_{n\\in\\mathbb{Z}}$ forms an orthonormal basis with respects to the $L_2$ inner product. The fact that any bandlimited function can be reconstructed in this way is equivalent to the fact that this orthonormal basis spans the space of bandlimited functions with bandwidth of $K=\\pi$.\n\n\\begin{figure}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesa.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesb.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesc.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplesd.pdf}\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/Fig1DSamplese.pdf}\n\\caption{Several different (but completely equivalent) graphical representations of the bandlimited function \\mbox{$f_\\text{B}(x)=1+S(x-1\/2)+x\\,S(x\/2)^2$} with bandwidth of $K=\\pi$ and consequently a critical spacing of $a^*=\\pi\/K=1$. Subfigure a) shows the function values for all $x$. b) shows the values of $f_\\text{B}$ at $x_n=n\/2$. Since $1\/2<a^*=1$ this is an instance of oversampling. c) shows the values of $f_\\text{B}$ at $x_n=n$. This is an instance of critical sampling. d) shows the values of $f_\\text{B}$ at $x_n=n+1\/3$. This too is an instance of critical sampling. e) shows a non-uniform sampling of $f_\\text{B}$. From any of these samplings we can recover the function $f_\\text{B}$ exactly everywhere. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\\label{Fig1DSamples}\n\\end{figure}\n\nAs a concrete example, let us consider the function $f_\\text{B}(x)=1+S(x-1\/2)+x\\,S(x\/2)^2$, shown in Fig.~\\ref{Fig1DSamples}a). This function has a bandwidth of $K=\\pi$ and so has a critical sample spacing of $a^*=\\pi\/K=1$. Thus, we can fully reconstruct $f_\\text{B}(x)$ knowing only its values at $x_n=n\\,a+b$ for any spacing $a\\leq a*=1$. In particular the sample values at $x_n=n\/2$ are sufficient to exactly reconstruct the function, see Fig.~\\ref{Fig1DSamples}b). So too are the sample values at the integers $x_n=n$ and at $x_n=n+1\/3$, see Fig.~\\ref{Fig1DSamples}c) and Fig.~\\ref{Fig1DSamples}d). In each of these cases the reconstruction is given by Eq.~\\eqref{SincRecon}.\n\nEverything about this function can be reconstructed from any uniform sample lattice with $a\\leq a^*=1$. In particular, the value of $f_\\text{B}$ two third's of the way between sample point, $f_\\text{B}(2\/3)$, is fixed by $\\{(n,f_\\text{B}(n))\\}_{n\\in\\mathbb{Z}}$ even though we have no sample at or even near $x=2\/3$. The derivative of $f_\\text{B}$ at zero, $f_\\text{B}'(0)$, is fixed by $\\{(n,f_\\text{B}(n))\\}_{n\\in\\mathbb{Z}}$ even though the only sample point we have in this neighborhood is $f_\\text{B}(0)$. Moreover, the derivative at $x=2\/3$, namely $f_\\text{B}'(2\/3)$, is fixed by $\\{(n,f_\\text{B}(n))\\}_{n\\in\\mathbb{Z}}$ even we have no sample points in the neighborhood.\n\nOn first exposure this may be shocking: how can a function's behavior everywhere be fixed by its value at a discrete set of points? When $f_\\text{B}$ is represented discretely, where has all of the information gone? Where is the information about the derivative at $x=2\/3$ stored in the discrete representation? \n\nTo see this, it is convenient (but not necessary) to organize our sample values $f_n=f(x_n)$ into a vector as,\n\\begin{align}\n\\bm{f}&=(\\dots,f_{-1},f_0,f_1,f_2,\\dots)^\\intercal\n= \\sum_{n\\in\\mathbb{Z}} f_{n} \\, \\bm{w}_n\n\\end{align}\nwhere $\\bm{w}_n = (\\dots,0,0,1,0,0,\\dots)^\\intercal\\in\\mathbb{R}^\\mathbb{Z}$ with the 1 in the $n^\\text{th}$ position. \n\nThe values that $f_\\text{B}$ takes at the sample points $x_n$ can be recovered from $\\bm{f}$ as\n\\begin{align}\n\\nonumber\nf_\\text{B}(x_n) \n&= f_n\n= \\bm{w}_n^\\intercal\\bm{f},\n\\end{align}\nNext notice that translating a bandlimited function preserves its bandwidth. As such both $f_\\text{B}(x)$ and $f_\\text{B}(x+\\epsilon a)$ can be represented as Eq.~\\eqref{SincRecon}. Using this fact, we can recover the values that $f_\\text{B}$ takes away from the sample points (i.e., at \\mbox{$x=x_n+\\epsilon\\,a$} for $\\epsilon\\in\\mathbb{R}$) as\n\\begin{align}\\label{TBdef}\nf_\\text{B}(x_n+\\epsilon\\,a) \n&= \\sum_{m=-\\infty}^\\infty S_m(n+\\epsilon) \\ f_m\n=\\bm{w}_n^\\intercal \\, T_\\text{B}^\\epsilon \\, \\bm{f}\n\\end{align}\nwhere the entries of the matrix $T^\\epsilon_\\text{B}$ are $[T^\\epsilon_\\text{B}]_{i,j}=S_i(j+\\epsilon)$. Note $T_\\text{B}^\\epsilon$ acts as the translation operator for this representation of bandlimited functions. If $\\bm{f}$ represents $f_\\text{B}(x)$ then $T^\\epsilon_\\text{B}\\bm{f}$ represents $f_\\text{B}(x+\\epsilon \\,a)$.\n\nFrom this translation operator we can identify the derivative operator for bandlimited functions, $D_\\text{B}$, as \n\\begin{align}\n\\nonumber\nD_\\text{B}\\coloneqq\\lim_{\\epsilon\\to0} \\frac{T^\\epsilon_\\text{B}-\\openone}{\\epsilon}. \\end{align}\nIt should be noted that $D_\\text{B}$ and $T_\\text{B}^\\epsilon$ commute and moreover we have the usual relationship between derivatives and translations, $T_\\text{B}^\\epsilon=\\exp(\\epsilon\\, D_\\text{B})$.\n\nFrom the above definition of $D_\\text{B}$ one can easily work out its matrix entries as \\mbox{$[D_\\text{B}]_{i,j}=(-1)^{i-j}\/(i-j)$} when $i\\neq j$ and 0 when $i=j$. That is, \n\\begin{align}\\label{DBMatrix}\nD_\\text{B}&=\\text{Toeplitz}(\\dots,\\!\\frac{1}{4},\\!\\frac{-1}{3},\\!\\frac{1}{2},\\!-1,\\!0,\\!1,\\!\\frac{-1}{2},\\!\\frac{1}{3},\\!\\frac{-1}{4},\\!\\dots)\n\\end{align}\nNote that $D_\\text{B}$ acts as the derivative operator for this representation of bandlimited functions. If $\\bm{f}$ represents $f_\\text{B}(x)$ then $\\frac{1}{a}D_\\text{B}\\bm{f}$ represents $f_\\text{B}'(x)$.\n\nComparing this with the $D$ operator introduced in Eq.~\\eqref{BigToeplitz} we see that they are numerically identical. Indeed, $D_\\text{B}=D$ and moreover $T_\\text{B}^\\epsilon=T^\\epsilon$. If we were to extend our discussion to two-dimensional functions we could find a discrete representation of the rotation operator for bandlimited functions, $R_\\text{B}^\\theta$. This would come out numerically equal to the $R^\\theta$ operator introduced earlier in Eq.~\\eqref{RthetaDef}, namely $R_\\text{B}^\\theta=R^\\theta$. See Appendix~\\ref{AppA} for further discussion. Thus, the discrete notions of derivative, translation, and rotation that we have been using up until now are intimately connected with bandlimited functions.\n\nIt should be noted that $D=D_\\text{B}$ gives us the following remarkable derivative approximation (which is exact for bandlimited functions):\n\\begin{align}\\label{ExactDerivative}\n\\partial_x f(x)\n&\\approx2\\sum_{m=1}^\\infty (-1)^{m+1} \\frac{f(x+m\\,a)-f(x-m\\,a)}{2\\,m\\,a}.\n\\end{align}\nRelatedly, we have the second derivative approximation (which is exact for bandlimited functions):\n\\begin{align}\\label{ExactDerivative2}\n\\partial_x^2 f(x)\n&\\approx2\\sum_{m=1}^\\infty (-1)^{m+1} \\frac{f(x+m\\,a)+f(x-m\\,a)}{m^2\\,a^2}\\\\\n\\nonumber\n&-\\frac{\\pi^2}{3}f(x).\n\\end{align}\nNamely, when $f$ is bandlimited with bandwidth of $K$ and $a\\leq\\pi\/K$ then these formulas are exact. Moreover, if the Fourier transform of $f$ is mostly supported in $[-K,K]$ with thin tails (e.g, Gaussian tails) outside this region, then these are very good derivative approximations. \n\nUltimately we can compute any derivative of $f_\\text{B}$ anywhere from our sample data as,\n\\begin{align}\n\\nonumber\n\\partial_x^r\\,f_\\text{B}(x_n+\\epsilon\\,a) \n=\\frac{1}{a^r}\\,\\bm{w}_n^\\intercal \\, D_\\text{B}^r T_\\text{B}^\\epsilon \\, \\, \\bm{f}.\n\\end{align}\nThus, we can recover any value or derivative of $f_\\text{B}$ from its values on any sufficiently dense uniform sample lattice.\n\nNote that each $f_\\text{B}$ is representable in this way on a wide number of sample lattices with differing spacings $a<\\pi\/K$ and differing offsets $b$. Translating between these different descriptions of $f_\\text{B}$ is equivalent to a change of basis on the vector of sample values $\\bm{f}$.\n\n\\subsection{Non-Uniform Sample Lattices}\\label{Sec2DSampling}\nThe previous subsection showed how any value or derivative of $f_\\text{B}$ can be recovered from its values on any sufficiently dense uniform sample lattice. Moreover, it showed how changing between representing $f_\\text{B}$ with different uniform sample lattices is ultimately just a change of basis on $\\bm{f}$. \n\nAs I will now discuss, we can also represent $\\bm{f}$ on any sufficiently dense non-uniform lattice. To motivate this, consider first an oversampling of $f_\\text{B}$. For example, figure Fig.~\\ref{Fig1DSamples}b) shows $f_\\text{B}$ sampled at twice the necessary frequency. This is a representation of $\\bm{f}$ in an overcomplete basis. Imagine oversampling by a factor of ten with a spacing of $a=a^*\/10$. Intuitively, this sample lattice has ten times the information needed to recover the function exactly. If we were to delete all but every tenth data point we would still be able to recover the function. But what if we just half of the sample points, but did so randomly? This would result in a non-uniform sample lattice. See for instance Fig.~\\ref{Fig1DSamples}e). Hopefully, the reader has some intuition that at least some non-uniform sample lattices are sufficient to exactly reconstruct $f_\\text{B}$.\n\nThe scope of such non-uniform samplings is established by various non-uniform sampling theorems~\\cite{GARCIA200263,SamplingTutorial}. The details of these theorems are not important here; They can all be summarized as saying that reconstruction is possible when our non-uniform sample points are ``sufficiently dense'' in some technical sense. The sampling shown in Fig. \\ref{Fig1DSamples}e) is sufficiently dense. The reconstruction in the non-uniform case is significantly more complicated than it is in the uniform case. Rather than Eq.~\\eqref{SincRecon}, in the non-uniform case our reconstruction is of the form,\n\\begin{align}\\label{GenRecon}\nf_\\text{B}(x)=\\sum_{m=-\\infty}^\\infty G_m(x;\\{z_n\\}_{n\\in\\mathbb{Z}}) \\, f_\\text{B}(z_m) \n\\end{align}\nfor some reconstruction functions, $G_m$, which depend in a complicated way on the location of all of the other sample points, $\\{z_n\\}_{n\\in\\mathbb{Z}}$.\n\n\n\\subsection{Higher Dimensional Sampling}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{PaperFigures\/Fig2DSamples.pdf}\n\\caption{Several different (but completely equivalent) graphical representations of the bandlimited function given by Eq.~\\eqref{J1}. This function has a bandwidth of $\\sqrt{k_x^2+k_y^2}<K=\\pi$ and so has a critical spacing of $a^*=\\pi\/K=1$ in every direction. The scale of each subfigure is 5x5. In each subfigure, the colored regions are the Voronoi cells around the sample points (black). Subfigure a) shows the function values for all $x$. b) shows $f_\\text{B}$ sampled on a square lattice with $z_{n,m}=(n\/2,m\/2)$. Since $1\/2<a^*=1$ this is an instance of oversampling. c) shows $f_\\text{B}$ sampled on a square lattice with $z_{n,m}=(n,m)$. This is an instance of critical sampling since $a=a^*=1$. d) shows $f_\\text{B}$ sampled on a square lattice with $z_{n,m}=(n+m,n-m)\/\\sqrt{2}$. e) $f_\\text{B}$ sampled on a hexagonal lattice of with \\mbox{$z_{n,m}=(n+m\/2,\\sqrt{3}m\/2)\\in\\mathbb{R}^2$}. f) shows $f_\\text{B}$ sampled on an irregular lattice. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\n\\label{Fig2DSamples}\n\\end{figure*}\n\nThe same story about bandlimited functions is largely true in higher dimensions as well. A two-dimensional function $f_\\text{B}(x,y)$ is bandlimited if is Fourier transform $\\mathcal{F}[f_\\text{B}(x,y)](k_x,k_y)$ is compactly supported in the \\mbox{($k_x$, $k_y$)-plane}. Specifying the value of the bandwidth is less straightforward in the high dimensional case as the Fourier transform's support may have different extents in different directions. However, any compact region can be bounded in a square. We can thus always imagine $f_\\text{B}(x,y)$ as being bandlimited with \\mbox{$k_x,k_y\\in[-K,K]$} for some $K>0$. As such, we can represent $f_\\text{B}(x,y)$ with a (sufficiently dense) uniform sample lattice in both the $x$ and $y$ directions. That is, we can represent any bandlimited $f_\\text{B}(x,y)$ in terms of its sample values on a sufficiently dense square lattice. Once we have such a uniform sampling, the reasoning carried out above applies unchanged. We can represent $f_\\text{B}(x,y)$ on any sufficiently dense non-uniform lattice.\n\nFor a concrete example consider the bandlimited function shown shown in Fig.~\\ref{Fig2DSamples}a), namely,\n\\begin{align}\\label{J1}\nf_\\text{B}(x,y)=J_1(\\pi\\,r)\/(\\pi\\,r) \n\\end{align}\nwhere $J_1$ is the first Bessel function and $r=\\sqrt{x^2+y^2}$. This function is bandlimited with $\\sqrt{k_x^2+k_y^2}<K=\\pi$ and hence critical spacing $a^*=\\pi\/K=1$. Moreover, note that this function is rotation invariant.\n\nGiven this function's bandwidth of $K=\\pi$, we can represent it via its sample values taken on a square lattice with spacing $a=1\/2\\leq a^*=1$, see Fig. \\ref{Fig2DSamples}b). We can also use a coarser square lattice with a spacing of $a=a^*=1$, see Fig. \\ref{Fig2DSamples}c). We could also use a rotated square lattice, see Fig. \\ref{Fig2DSamples}d). Sampling the function on a hexagonal lattice also works, see Fig. \\ref{Fig2DSamples}e). Finally we can use a non-uniform lattice of sample points, see Fig. \\ref{Fig2DSamples}f). From each of these discrete representations, we could recover the original bandlimited function everywhere exactly via some generalization of Eq.~\\eqref{SincRecon} in the uniform cases and Eq.~\\eqref{GenRecon} in the irregular case.\n\nThus, there is no conceptual barrier to representing a rotationally invariant bandlimited function on a square lattice. Indeed, there is no issue with representing such a function on any sufficiently dense lattice. In light of the analogy proposed in this paper, we can see this as analogous to the unsurprising fact that there is no conceptual barrier to representing rotationally invariant functions in Cartesian coordinates. There is no requirement that our representation (be it a choice of coordinates or a choice of sample points) latches onto the symmetries of what is being represented. \n\nThus we have a non-uniform sampling theory for higher dimensions. But what about a sampling theory on curved spaces? While such things are not relevant for the aims of this paper, recently notable progress has been made on developing a sampling theory for curved manifolds~\\cite{CurvedSampling,Martin2008}.\n\n\\section{Externalizing these theories - Part 2}\\label{SecExtPart2}\nHaving reviewed the sampling property of bandlimited functions, let's return to the task of embedding $\\bm{\\Phi}$ into our spacetime manifold $\\mathcal{M}$. As discussed in Sec.~\\ref{SecExtPart1}, for KG4-KG7 our translation-matching constraint forces the new field \\mbox{$\\phi:\\mathcal{M}\\to\\mathbb{R}$} to be bandlimited in the coordinate system $d_\\text{trans.}$ with bandwidth $\\omega,k_1,k_2\\in[-K,K]$ where $K=\\pi\/a$. Similarly for KG1-KG3. In either case, let's note this by giving the field a subscript $\\text{B}$ as \\mbox{$\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$}. \n\nBefore continuing on, it should be noted that, the above discussed translation-matching constraint guarantees that $\\phi_\\text{B}\\circ d_\\text{trans.}$ will have the sampling property discussed in the previous section.\n\n\\subsection{A Principled Choice of Profile}\nWhat freedom remains for our choice of embedding? Focusing on KG4-KG7, in coordinate $d_\\text{trans.}$ our remaining freedom is a choice of profile $f(t,x,y)\\in F^\\pi$ appearing in Eq.~\\eqref{Embedf}. A translated and rescaled copy of this profile is associated with each basis vector $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$. As noted following Eq.~\\eqref{Eftilde}, up to a local recaling of Fourier space (which is a symmetry of the dynamics) our embedding is fixed. However, in order to completely fix this embedding, we need to pick a profile $f(t,x,y)$.\n\nThe discussion in the previous section offers us a promising candidate for this profile namely \\mbox{$f(t,x,y)=S_0(t)S_0(x)S_0(y)$}. This profile is the unique one in $F^\\pi$ with the following property\\footnote{To give the technical details, this $f$ is the unique function in $F^\\pi$ which evaluates to zero at all integer arguments except $(0,0,0)$ where it returns $1$. That is, \\mbox{$f(j,n,m)=\\delta_{j0}\\delta_{n0}\\delta_{m0}$} for $j,n,m\\in\\mathbb{Z}$. This uniqueness is a direct consequence of the Nyquist-Shannon sampling theorem.}: it makes $\\phi_\\text{B}(t,x,y)$ evaluated at $z_{j,n,m}=(j\\,a,n\\,a,m\\,a)$ take the value of $\\phi_{j,n,m}$. That is, it makes our original discrete variables $\\phi_{j,n,m}$ sample values of $\\phi_\\text{B}$ at the sample points $z_{j,n,m}$. Let us call this choice of profiles the lattice-as-sample-points constraint.\n\nTo be clear, what hinges on our choice of profile $f$ is not whether or not $\\phi_\\text{B}$ has the sampling property discussed in the previous section; This property is guaranteed independent of our choice of $f$. Rather, what is at stake is just whether our original discrete variables $\\phi_{j,n,m}$ \\mbox{\\textit{themselves}} are sample values.\n\nIt follows from the lattice-as-sample-points constraint that $E$ acts on the planewave basis as\n\\begin{align}\\label{Embed0}\nE:\\bm{\\Phi}(\\omega,k_1,k_2)\\mapsto\\phi(t,x,y;\\omega\/a,k_1\/a,k_2\/a)\n\\end{align}\nThat is, discrete planewaves are mapped onto continuous planewaves (rescaled by $a$). Moreover, this constraint forces $E$ to act on the basis $\\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m$ as\n\\begin{align}\\label{Embed}\nE: \\bm{e}_j\\otimes\\bm{e}_n\\otimes\\bm{e}_m\\mapsto\nS_{j}(t\/a) \\, S_{n}(x\/a) \\, S_{m}(y\/a).\n\\end{align}\nIn particular, applying this to  Eq.~\\eqref{PhiVec2} gives us\n\\begin{align}\\label{PhiSincRecon1}\n&\\text{KG1-KG3:}\\\\\n\\nonumber\n&\\quad\\phi_\\text{B}(t,x)\n=\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\,S_{n}(x\/a) \\ \\phi_{j,n}\\\\\n\\label{PhiSincRecon2}\n&\\text{KG4-KG7:}\\\\\n\\nonumber\n&\\quad\\phi_\\text{B}(t,x,y)\n=\\sum_{j,n,m\\in\\mathbb{Z}} S_{j}(t\/a) \\, S_{n}(x\/a) \\, S_{m}(x\/a) \\, \\phi_{j,n,m}.\n\\end{align}\nwhere analogous reasoning applies for KG1-KG3.\n\nThus from these translation-matching and lattice-as-sample-points constraints it follows that our new field must be the result of a certain bandlimited reconstruction. Namely, for KG4-KG7 it must come from a reconstruction using sample values $\\phi_{j,n,m}$ at sample point $z_{j,n,m}$. Similarly for KG1-KG3 it must come from using sample values $\\phi_{j,n}$ at sample points $z_{j,n}=(j\\,a,n\\,a)$.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{PaperFigures\/RowABC.pdf}\n\\includegraphics[width=0.95\\textwidth]{PaperFigures\/RowDEF.pdf}\n\\caption{Subfigure b) shows a solution to KG3 formulated as a bandlimited function in the coordinates given by $d_\\text{trans.}$.\nThis field in these coordinates obeys Eq.~\\eqref{DKG3bandlimited}. Subfigure a) shows the values that this function takes on a square lattice in spacetime.  These discrete field values obey Eq.~\\eqref{DKG3}. From this embedding we can uniquely reconstruct the bandlimited field $\\phi_\\text{B}$. However, these sample points play no special role in the theory. We are free to redescribe the state and its dynamics by resampling it on any sufficiently dense collection of sample points. For instance, we might describe $\\phi_\\text{B}$ using the boosted sample points shown in subfigure c). Not all states will be describable on both of these lattices, but some are. If we boost both the sample values and the field as shown in subfigure d) e) and f) nothing changes. Both sets of sample values still obey Eq.~\\eqref{DKG3}. Moreover, in the bandlimited field still obeys Eq.~\\eqref{DKG3bandlimited}.}\\label{FigKleinSpaceTime}\n\\end{figure*}\n\nFig.~\\ref{FigKleinSpaceTime}a) shows for KG1-KG3 these sample points \\mbox{$z_{j,n}=(j\\,a,n\\,a)$} and sample values as they lie on the spacetime manifold $\\mathcal{M}\\cong\\mathbb{R}^2$ in the coordinates $d_\\text{trans.}$ Fig.~\\ref{FigKleinSpaceTime}b) shows for KG3 what the reconstructed bandlimited function $\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$ might look like in these coordinates. One can imagine an analogous figure for KG4-KG7 with the sample points forming a cubic lattice in the coordinates $d_\\text{trans.}$.\n\nOne may worry that we are here taking a cubic 3D lattice for each of KG4-KG7 whereas for KG5 and KG7 we naturally ought to embed onto a hexagonal 3D lattice. This was after all, the lattice structure picked out by our symmetry analysis in the first interpretation. However, one need not worry for two reasons: Firstly, there is no real sense in which the sample points $z_{j,n}$ and $z_{j,n,m}$ form a square and cubic lattice respectively. These sample lattices only appear square and cubic in this particular coordinate system; in other coordinate systems they won't, see Fig.~\\ref{FigKleinSpaceTime}d). Indeed, the coordinates we are using here may lie on the spacetime manifold $\\mathcal{M}$ in a curvilinear way. As such one ought not to think of these sample points as being arranged in a square or cubic lattice on the spacetime manifold. Indeed, at this point the manifold still does not have a metric.\n\nSecondly, after we have used the sample points $z_{j,n}$ and $z_{j,n,m}$ to build $\\phi_\\text{B}$ they no longer play any special role in the theory. As discussed in the previous section, we are always free to resample a bandlimited function on any sufficiently dense sample lattice. Recall Fig.~\\ref{Fig2DSamples}. Thus even staying in this coordinate system (where our initial sample points do form a cubic 3D lattice) we are free to resample $\\phi_\\text{B}$ on a hexagonal 3D lattice. Moreover, we also might be able to resample $\\phi_\\text{B}$ on a boosted lattice as shown in Fig.~\\ref{FigKleinSpaceTime}c).\n\nThus, in a sense the lattice sites $L$ from our first interpretation still exist here they are embedded onto our new spacetime manifold, $\\ell\\in L\\mapsto z_\\ell\\in\\mathcal{M}$. However, once embedded, these lattice sites no longer play any special role in the theory; We can always resample.\n\n\\vspace{0.25cm}\n\nTo review: In Sec.~\\ref{SecExtPart1} we saw how the translation-matching constraint forced us to embed $\\bm{\\Phi}$ onto $\\mathcal{M}$ as a some bandlimited function, $\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$ (i.e., bandlimited in coordinates $d_\\text{trans.}$ with $\\omega,k_1,k_2\\in[-K,K]$). In this section, we further demanded the lattice-as-sample-points constraint. This allowed us to think of our initial discrete variables $\\phi_\\ell$ as being the values that $\\phi_\\text{B}$ takes at some sample points $z_\\ell\\in\\mathcal{M}$. Taken together these two constraints completely fix how $\\phi_\\text{B}:\\mathcal{M}\\to\\mathbb{R}$ is built from $\\bm{\\Phi}$ (or equivalently $\\phi_\\ell$). In particular, in the global coordinate system given by $d_\\text{trans.}$ we must have Eq.~\\eqref{PhiSincRecon1} and Eq.~\\eqref{PhiSincRecon2}.\n\nIn the next subsection, I will rewrite the dynamics of each of KG1-KG7 in terms of this new field $\\phi_\\text{B}$ in the coordinates $d_\\text{trans.}$. Following this, in Secs.~\\ref{SecKlein3} and \\ref{SecKlein3Extra}, I will begin interpreting this theory independently of how they were constructed. In Sec.~\\ref{SecFullGenCov} I will give a coordinate-independent formulation of these theories.\n\n\\subsection{Bandlimited Dynamics}\\label{SecKlein3B}\nThe previous subsections has given us a principled way to embed $\\phi_\\ell$ onto  our new spacetime manifold $\\mathcal{M}$ as a bandlimited function. Namely, Eq.~\\eqref{PhiSincRecon1} and Eq.~\\eqref{PhiSincRecon2}. Equivalently we can think of embedding $\\bm{\\Phi}$ onto our new spacetime manifold $\\mathcal{M}$ as a bandlimited function. Namely, Eq.~\\eqref{Embed}. Either of these perspectives allow us to translate the kinematics of KG1-KG7 into this new continuous setting. This subsection will additionally translate over the dynamics into this new continuous setting.\n\nLet's begin with KG1. This translation is aided by the fact that the derivative is the generator of translations, i.e.,  $h(x+a)=\\text{exp}(a\\, \\partial_x) h(x)$. Mapping the left hand side of the dynamics of KG1 (i.e., Eq.~\\eqref{KG1Long}) onto a bandlimited function we have\n\\begin{align}\n&\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\, S_{n}(x\/a)\\, \\left(\\text{L.H.S. of Eq.~\\eqref{KG1Long}}\\right)\\\\\n\\nonumber\n&=\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\,S_{n}(x\/a) \\big[\\phi_{j+1,n}-2\\phi_{j,n}+\\phi_{j-1,n}\\big]\\\\\n\\nonumber\n&=\\phi_\\text{B}(t - a,x) - 2\\phi_\\text{B}(t,x) + \\phi_\\text{B}(t + a,x)\\\\\n\\nonumber\n&=[\\exp(-a\\,\\partial_t)-2+\\exp(a\\,\\partial_t)] \\phi_\\text{B}(t,x)\\\\\n\\nonumber\n&=[2\\cosh(a\\,\\partial_t)-2] \\ \\phi_\\text{B}(t,x).\n\\end{align}\nSimilarly, for the right hand side of Eq.~\\eqref{KG1Long} we have\n\\begin{align}\n&\\sum_{j,n\\in\\mathbb{Z}} S_{j}(t\/a)\\, S_{n}(x\/a)\\, \\left(\\text{R.H.S. of Eq.~\\eqref{KG1Long}}\\right)\\\\\n\\nonumber\n&=[\\mu^2+2\\cosh(a\\,\\partial_t)-2] \\ \\phi_\\text{B}(t,x).\n\\end{align}\nThus, when the sample values $\\phi_{j,n}=\\phi_\\text{B}(z_{j,n})$ obey Eq.~\\eqref{KG1Long}, the bandlimited field obeys,\n\\begin{align}\\label{DKG1bandlimited}\n\\text{KG1:}\\,[2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}\\!=\\![\\mu^2+2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}\n\\end{align}\nSimilarly for the other six theories we have:\n\\begin{align}\\label{DKG2bandlimited}\n\\text{KG2:}&\\ \\frac{1}{6}[-\\cosh(2a\\partial_t)\\!+\\!16\\cosh(a\\partial_t)\\!-\\!15]\\phi_\\text{B}\\\\\n\\nonumber\n&=\\left(\\mu^2+\\frac{1}{6} [-\\cosh(2a\\partial_x)\\!+\\!16\\cosh(a\\partial_x)\\!-\\!15]\\right)\\phi_\\text{B}\\\\\n\\label{DKG3bandlimited}\n\\text{KG3:}&\\ \\partial_t^2\\phi_\\text{B}\n=(\\mu_0^2+\\partial_x^2) \\, \\phi_\\text{B}\\\\\n\\label{DKG4bandlimited}\n\\text{KG4:}&\\  [2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}\\\\\n\\nonumber\n&=[\\mu^2+2\\cosh(a\\,\\partial_x)+2\\cosh(a\\,\\partial_y)\\!-\\!4]\\phi_\\text{B}\\\\\n\\label{DKG5bandlimited}\n\\text{KG5:}&\\ [2\\cosh(a\\,\\partial_t)\\!-\\!2] \\phi_\\text{B}=\\Big[\\mu^2+\\frac{4}{3}\\cosh(a\\,\\partial_x)\\\\\n\\nonumber\n&\\qquad\\quad+\\frac{4}{3}\\cosh(a\\,\\partial_y) +\\frac{4}{3}\\cosh(a\\,(\\partial_y-\\partial_x))\\!-\\!4\\Big]\n\\phi_\\text{B},\\\\\n\\label{DKG6bandlimited}\n\\text{KG6:}&\\  \n\\partial_t^2\\phi_\\text{B}\n=(\\mu_0^2+\\partial_x^2+\\partial_y^2) \\, \\phi_\\text{B}\\\\\n\\label{DKG7bandlimited}\n\\text{KG7:}&\\  \n\\partial_t^2\\phi_\\text{B}\n\\!=\\!\\Big(\\mu_0^2\\!+\\!\\frac{2}{3}\\big[\\partial_x^2\\!+\\!\\partial_y^2\\!+\\!(\\partial_x\\!-\\!\\partial_y)^2\\big]\\!\\Big) \\phi_\\text{B}\n\\end{align}\nwhere $\\mu_0\\coloneqq\\mu\/a$. \n\nNote that the dynamics of KG3 and KG6 are exactly the same as those of the continuum Klein Gordon equations KG00 and KG0 with a field mass \\mbox{$M=\\mu_0$}. Moreover, after a coordinate transformation, \n\\begin{align}\\label{CoorKG7KG6}\nt\\mapsto t,\\quad\nx\\mapsto x + \\frac{1}{2} y,\\quad\ny\\mapsto \\frac{\\sqrt{3}}{2}y\n\\end{align}\nthe dynamics of KG7 exactly maps onto the dynamics of KG6 (and consequently, KG0). Note that this change of coordinates is equivalent to Eq.~\\eqref{SkewKG7KG6} applied in continuum Fourier space. Moreover, note that Eq.~\\eqref{CoorKG7KG6} maps a square lattice to a hexagonal one, see Fig.~\\ref{FigSkew}. As I will discuss later, this means that KG6 and KG7 can be seen as describing the same bandlimited theory, just using different sample points.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.4\\textwidth]{PaperFigures\/FigSkew.pdf}\n\\caption{As this figure shows, a certain linear transformation of coordinates (namely Eq.~\\eqref{CoorKG7KG6}) maps a square lattice to a hexagonal one. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\\label{FigSkew}\n\\end{figure}\n\nApplying this coordinate transformation to KG5's bandlimited dynamics gives us:\n\\begin{align}\\label{DKG5bandlimitedSkew}\n\\text{KG5:}\\ &[2\\cosh(a\\partial_t)\\!-\\!2] \\phi_\\text{B}=\\Big[\\mu^2-4+\\frac{4}{3}\\cosh(a\\,\\partial_x)\\\\\n\\nonumber\n&\\qquad\\qquad\\qquad+\\frac{4}{3}\\cosh(a(\\sqrt{3} \\partial_y+\\partial_x)\/2)\\\\\n\\nonumber\n&\\qquad\\qquad\\qquad+\\frac{4}{3}\\cosh(a(\\sqrt{3} \\partial_y-\\partial_x)\/2)\\Big]\n\\phi_\\text{B}.\n\\end{align}\nNote that this dynamics now manifestly has a one-sixth rotational symmetry. Sampling the above dynamics on a hexagonal lattice gives us our original dynamics for KG5. Sampling the above dynamics on a square lattice gives us the following. Given Eq.~\\eqref{ExactDerivative} sampling on a square lattice effectively means taking $a\\,\\partial_t\\to D_\\text{j}$, $a\\,\\partial_x\\to D_\\text{n}$ and $a\\,\\partial_y\\to D_\\text{m}$ and replacing $\\phi_\\text{B}$ with $\\bm{\\Phi}$. For the above dynamics of KG5 this gives:\n\\begin{align}\\label{DKG5Skew}\n\\text{KG5:}\\ \\Delta_{(1),\\text{j}}^2\\,\\bm{\\Phi}&=\\Big[\\mu^2-4+\\frac{4}{3}\\cosh(D_\\text{n})\\\\\n\\nonumber\n&+\\frac{4}{3}\\cosh((\\sqrt{3} D_\\text{m}+D_\\text{n})\/2)\\\\\n\\nonumber\n&+\\frac{4}{3}\\cosh((\\sqrt{3} D_\\text{m}-D_\\text{n})\/2)\\Big]\n\\phi_\\text{B}.\n\\end{align}\nNote that before this resampling KG5 was local in the intuitive sense nearest-neighbor couplings only, see Eq.~\\eqref{KG5Long}. After resampling it is infinite range. Thus, the intuitive notion of locality in terms of nearest neighbors is unstable under resampling. For this and other reasons (see \\cite{DiscreteGenCovPart1}) the notion of locality arising in the bandlimited formulation is preferable. Sample values can obey what appears to be non-local dynamics, because the sample values themselves are non-local objects: they ought to be associated with sinc profiles not spacetime points, see \\cite{DiscreteGenCovPart1}.\n\nWe can easily solve each of these dynamical equations. Just as in Sec. \\ref{SecSevenKG} these dynamics admit a complete basis of planewave solutions, given by Eq.~\\eqref{PlaneWaveCont}. A few key differences should be noted between these continuum planewaves and the discrete planewaves considered earlier, namely $\\bm{\\Phi}(\\omega,k)$ and $\\bm{\\Phi}(\\omega,k_1,k_2)$. Firstly, there is a difference of scale: note how our embedding Eq.~\\eqref{Embed0} rescales the planewaves by a factor of $1\/a$. Secondly, the discrete planewaves $\\bm{\\Phi}(\\omega,k)$ and $\\bm{\\Phi}(\\omega,k_1,k_2)$ repeated themselves cyclically with period $2\\pi$ outside of the regions \\mbox{$\\omega,k\\in[-\\pi,\\pi]$} and \\mbox{$\\omega,k_1,k_2\\in[-\\pi,\\pi]$}. The continuum planewaves do not do this. Now each planewave with $\\omega,k_1,k_2\\in\\mathbb{R}$ is distinct. Here, however, by its construction $\\phi_\\text{B}$ only has support over planewaves with \\mbox{$\\omega,k_1,k_2\\in[-K,K]$} with $K=\\pi\/a$.\n\nEach of these planewaves are solutions only when they obey the following dispersion relations: \n\\begin{align}\n\\text{KG1:}& \\quad \\!\\! 2-2\\,\\cos(\\omega\\,a)= \\mu^2+2-2\\,\\cos(k\\,a)\\\\\n\\text{KG2:}& \\quad \\!\\! \\frac{1}{6}\\,(\\cos(2\\,\\omega\\,a)-16\\,\\cos(\\omega\\,a)+15)\\\\\n\\nonumber\n&= \\mu^2+\\frac{1}{6}\\,(\\cos(2\\,k\\,a)-16\\,\\cos(k\\,a)+15)\\\\\n\\text{KG3:}& \\quad \\!\\! \\omega^2= \\mu_0^2+k^2\\\\\n\\text{KG4:}& \\ 2\\!-\\!2\\cos(\\omega a)\\!=\\! \\mu^2\\!+\\!4\\!-\\!2\\cos(k_1 a)\\!-\\!2\\cos(k_2 a)\\\\\n\\text{KG5:}& \\  2-2\\,\\cos(\\omega\\,a)= \\mu^2 \\\\\n\\nonumber\n&+\\frac{4}{3}\\big[3-\\cos(k_1\\,a)-\\cos(k_1\\,a)-\\cos(k_2\\,a-k_1\\,a)\\big]\\\\\n\\text{KG6:}& \\ \\omega^2 = \\mu_0^2 +k_1^2 +k_2^2\\\\\n\\text{KG7:}& \\ \\omega^2= \\mu_0^2 +\\frac{2}{3}\\big[k_1^2 +k_2^2 +(k_2-k_1)^2\\big].\n\\end{align}\nNote that the dispersion relation of KG3 and KG6 are exactly the same as those of the continuum Klein Gordon equations KG00 and KG0 with a field mass $M=\\mu_0$. Moreover, after the coordinate transformation, Eq.~\\eqref{SkewKG6KG7}, \nthe dispersion relation of KG7 exactly maps onto that of KG6 (and consequently, KG0). More will be said about this later.\n\nWe are now ready to make a third attempt at interpreting these theories. \n\n\\section{A Third Attempt at Interpreting KG1-KG7 - Part 1}\\label{SecKlein3}\nThe previous three sections have (after much effort) given us a new formulation of KG1-KG7 to interpret. In particular, this new formulation has aimed to externalize many of the internal symmetries which we discovered in Sec.~\\ref{SecKlein2}. \n\nOn this third interpretation I will be taking the formulations of KG1-KG7 in terms of $\\phi_\\text{B}$ seriously: namely Eq.~\\eqref{DKG1bandlimited}-\\eqref{DKG7bandlimited}. Taken literally as written, what are these theories about? \nIntuitively these theories are about a field $\\phi_\\text{B}$ which maps points on a manifold ($p\\in\\mathcal{M}$) into field amplitudes ($\\phi_\\text{B}(p)\\in\\mathbb{R}$). That is a field $\\phi_\\text{B}:\\mathcal{M}\\to \\mathcal{V}$ with a manifold $\\mathcal{M}$ and value space $\\mathcal{V}=\\mathbb{R}$. Thus, taking $\\phi_\\text{B}:\\mathcal{M}\\to \\mathcal{V}$ seriously as a fundamental field leads us to thinking of $\\mathcal{M}$ as the theory's underlying manifold and $\\mathcal{V}=\\mathbb{R}$ as the theory's value space. Indeed, on this interpretation\nKG1-KG7 are continuum spacetime theories of the sort we\nare used to interpreting (albeit ones which consider only bandlimited fields). \n\nNote that here we have $\\mathcal{M}\\cong\\mathbb{R}^d$ with $d=2$ for KG1-KG3 and $d=3$ for KG4-KG7. As such, in either case we have access to global coordinate systems. The fields $\\phi_\\text{B}$ considered by this interpretation are bandlimited in the following sense. There exists a fixed special diffeomorphism $d_\\text{coor.}:\\mathbb{R}^d\\to\\mathcal{M}$ which giving us a fixed special global coordinate system for $\\mathcal{M}$. In these special coordinates the field is bandlimited with $\\omega,k_1,k_2\\in[-K,K]$ where $K=\\pi\/a$. That is $\\phi_\\text{B}\\circ d_\\text{trans.}\\in F^K$. It is in these special coordinates that the fields $\\phi_\\text{B}$ obey the dynamical equations Eq.~\\eqref{DKG1bandlimited}-Eq.~\\eqref{DKG7bandlimited}.\n\nIn our construction of $\\phi_\\text{B}$, this special coordinate system is the one which is associated to the translation operations $T^\\epsilon$ by our translation-matching condition, namely $d_\\text{coor.}=d_\\text{trans.}$. Here however, we are trying to interpret KG1-KG7 as formulated above independent of how we arrived here. Thus, presently, this special coordinate system $d_\\text{coor.}$ is just something promised to us by God. Of course, we know from the lessons of general covariance to be skeptical of ``special coordinate systems''. Coordinate systems are representational artifacts, reflecting no physics. We can and should reformulate our theories in a generally covariant way so as to avoid them. A coordinate independent view of them will be given in Sec.~\\ref{SecFullGenCov}.\n\nAs I will discuss, taking $\\mathcal{M}$ to be these theories' underlying manifold has substantial consequences for these theories' locality properties and symmetries. To preview: this third interpretation either fixes all of our issues with the first and second interpretations. \n\nIn our first interpretation there was some tension between our theories' differences in locality and their differences in convergence rate in the continuum limit. The second interpretation addressed this tension in a hamfistedly way: denying that there are differences in locality in the first place. This move also had the unfortunate consequence of undercutting our ability to use the lattice sites to reason about locality. This interpretation improves on this by instead bringing the locality of these theories into harmony with their convergence rates in the continuum limit. Not much changes between the discrete heat equations considered in \\cite{DiscreteGenCovPart1} and the discrete Klein Gordon equations considered here. As such, I will leave discussion of this issue to \\cite{DiscreteGenCovPart1} and direct the interested reader there.\n\nIn our first interpretation, KG6 and KG7 were seen as distinct theories with radically different symmetries despite there being a nice one-to-one correspondence between their solutions. In our second interpretation, KG6 and KG7 were more satisfyingly seen to be completely equivalent, differing only by a change of basis in the value space. Here too KG6 and KG7 will turn out to be completely equivalent differing only by a change of coordinates. Moreover, as I will discuss, we can here see KG6 and KG7 as parts of a larger unified theory viewed in two ways with differently limited representational capacities.\n\nOur first interpretation outright denied the possibility that KG1-KG7 could have continuous symmetries (e.g., translations, rotations, Lorentz boosts). Our second interpretation then fixed this oversight by revealing KG1-KG7's hidden continuous translation and rotation symmetries and even their (limited) Lorentzian boost symmetries. However, these were unintuitively categorized as internal symmetries. This interpretation maintains all of these hidden symmetries, but more satisfyingly categorizes them as external symmetries. Moreover, seeing KG6 and KG7 as parts of a larger unified theory will guide us towards a perfectly Lorentzian lattice theory.\n\nAs I discussed in Sec.~\\ref{SecKlein2}, the improvements that the second interpretation made over the first one all centered around the following realization. The lattice structures appearing in KG1-KG7 are merely representational artifacts and as such should not be taken seriously as a substantive part of the theory. As I will discuss, this realization is deepened and clarified on the third interpretation.\n\n\\subsection{Bandlimited Symmetries}\nHow does this externalization move affect our theory's capacity for symmetry? Now that we have a continuous spacetime manifold $\\mathcal{M}$ underlying these theories, we have a greatly improved capacity for external symmetries, namely $\\text{Auto}(\\mathcal{M})=\\text{Diff}(\\mathcal{M})$. By construction all of the symmetries which we wished to externalize $G^\\text{dym}_\\text{to-be-ext}$ fit inside of $\\text{Diff}(\\mathcal{M})$. \n\nIn addition to these external symmetries, we also have some capacity for internal symmetries, namely $\\text{Auto}(\\mathcal{V})$. Note that the space of (potentially off-shell) bandlimited fields are themselves vectors $\\phi_\\text{B}(t,x,y)\\in F^K$. Namely, they are closed under addition and scalar multiplication and hence a vector space. This addition and scalar multiplication is carried out in parallel at each spacetime point. Thus, the field's value space $\\mathcal{V}=\\mathbb{R}$ is also structured like a vector space. Therefore, $\\text{Auto}(\\mathcal{V})=\\text{Aff}(\\mathbb{R})$ such that our internal symmetries are linear-affine rescalings of $\\phi_\\text{B}$. We might also find gauge symmetries by allowing these to vary smoothly across the manifold. Thus, in total the possibly symmetries for our theories under this interpretation are,\n\\begin{align}\\label{BandlimitedSymmetries}\ns:\\quad\\phi_\\text{B}\\mapsto C \\, \\phi_\\text{B}\\circ d +c\n\\end{align}\nwhere $C:\\mathcal{M}\\to\\mathbb{R}$ and $c:\\mathcal{M}\\to\\mathbb{R}$ are some real scalar functions and $d:\\mathcal{M}\\to\\mathcal{M}$ is some  diffeomorphism \\mbox{$d\\in\\text{Diff}(\\mathcal{M})$}. Note that \n\nAre these more or less possible symmetries than we had on our second interpretation? $\\text{Diff}(\\mathcal{M})$ is a tremendously large group and is in fact bigger than $\\text{Aff}(\\mathbb{R}^L)$. One might expect that we have strictly more possible symmetries here. However, looks can be deceiving. \n\nWe can translate our old potential symmetries into the new continuum setting as follows. Note that because $\\mathbb{R}^L \\cong F^K$ we have $\\text{Aff}(\\mathbb{R}^L)\\cong \\text{Aff}(F^K)$. Using $d_\\text{coor.}$ we can think of $\\text{Aff}(F^K)$ as a subgroup of $\\text{Aff}(F(\\mathcal{M}))$. Note that we can also see the above discussed transformations (i.e., \\mbox{$\\text{Aff}(\\mathbb{R})$} varying smoothly over \\mbox{$\\text{Diff}(\\mathcal{M})$}) as a subgroup of $\\text{Aff}(F(\\mathcal{M}))$. \n\nViewed this way, neither is strictly larger. Indeed, $\\text{Aff}(F^K)$ includes transformations in Fourier space which are inaccessible from $\\text{Aff}(\\mathbb{R})$ and $\\text{Diff}(\\mathcal{M})$. In particular, the local Fourier rescaling symmetry from Sec.~\\ref{SecKlein2} is no longer available to us. On the other hand, our new symmetries contain transformations which are not closed over $F^K$. However, this comparison seems unfair, what if we restrict our attention transformations closed over $F^K$? With this restriction, our new range of possible symmetries is a strict subset of $\\text{Aff}(F^K)$. \n\nThus, in moving from the second to the third interpretation, we have actually lost some capacity for symmetry. The source of this reduction in capacity is us imposing structure on the theory by our choice of manifold and value space.\n\nAs I will now discuss, besides the local Fourier rescaling symmetry, all of our previous symmetries are maintained on this third interpretation. Moreover, these remaining symmetries are reformulated and recategorized. As I will now show, on this interpretation the continuous translation, rotation and (limited) Lorentz boost symmetries identified earlier are now honest-to-goodness manifold symmetries, represented by diffeomorphisms $d\\in\\text{Diff}(\\mathcal{M})$.\n\n\\subsubsection*{Symmetries of KG1-KG7: Third Attempt}\nWhich transformations of the form Eq.~\\eqref{BandlimitedSymmetries} are symmetries for KG1-KG7? A technical investigation of the symmetries of KG1-KG7 on this interpretation is carried out in Appendix~\\ref{AppA}, but the results are the following. For KG1 and KG2 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\n\\begin{enumerate}\n    \\item[1)]  continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0)$} for $t_0,x_0\\in\\mathbb{R}$,\n    \\item[2)] two negation symmetries which map \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(-t,x)$} and \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t,-x)$} respectively,\n    \\item[3)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[4)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\n\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation and reflection symmetries are here external symmetries. Translation here is generated by \\mbox{$\\exp(-t_0\\partial_t-x_0\\partial_x)$} for some $t_0,x_0\\in\\mathbb{R}$. Notice that translating a bandlimited function by any amount preserves its bandwidth. Thus, $\\phi_\\text{B}(t,x)$ remains in $F^K$ upon translation.\n\nThus, the first big lesson from Sec.~\\ref{SecKlein2} is repeated here: despite the fact that our discrete theories KG1 and KG2 can be represented on a lattice, they nonetheless have a continuous translation symmetry. This continuous translation symmetry was hidden on our first interpretation because we there took the lattice to be a fundamental part of the manifold. Here, we do not take the lattice structure so seriously. We have embedded it onto the manifold where it then disappears from view as just one of many possible sample lattices.\n\nAllow me to skip over KG3 temporarily. For KG4 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0,y-y_0)$} for some $t_0,x_0,y_0\\in\\mathbb{R}$,\n    \\item[2)] three negation symmetries which map\n    \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(-t,x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,-x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,x,-y)$} respectively,\n    \\item[3)] a 4-fold symmetry which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,y,-x)$},\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and 4-fold symmetries are here external symmetries. Thus, KG4 has three continuous translation symmetries despite initially being represented on a lattice, Eq.~\\eqref{KG4Long}.\n\nLet's next consider KG5. For KG5 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0,y-y_0)$} for some $t_0,x_0,y_0\\in\\mathbb{R}$,\n    \\item[2)] a negation symmetry which maps\n    \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(-t,x,y)$}, and an exchange symmetry which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,y,x)$},\n    \\item[3)] a 6-fold symmetry which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,-y,x+y)$}. (Roughly, this permutes the three terms in Eq.~\\eqref{DKG5bandlimited}),\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and 6-fold symmetries are here external symmetries. Thus, KG5 has three continuous translation symmetries despite initially being represented on a lattice, Eq.~\\eqref{KG5Long}. Note that after a coordinate transformation Eq.~\\eqref{CoorKG7KG6} the above noted 6-fold symmetry is realized as one-sixth rotations, see Eq.~\\eqref{DKG5Skew}.\n\nBefore moving on to analyze the symmetries of KG6 and KG7, let's first see what this interpretation has to say about them being equivalent to one another. As discussed in Sec.~\\ref{SecKlein2} KG6 and KG7 have a solution-preserving vector space isomorphism between a substantial portion of their models, including all of their solutions. Since our second and third interpretations are related by a solution-preserving vector space isomorphism, \\mbox{$E:\\mathbb{R}^L\\to F^K$}, the same is broadly true here. However, some of the details change.\n\nAs I noted in Sec.~\\ref{SecKlein2} each of KG4-KG7 are equivalent to each other in a weak sense: approximately in the continuum limit regime, $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll \\pi$. Here KG4-KG7 are approximately equivalent in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll K=\\pi\/a$ regime. Note however that this is not the continuum limit regime, we are already in the continuum. Rather this is the regime which is significantly below the bandwidth $K$. While each of KG4-KG7 are equivalent in this weak sense, KG6 and KG7 are equivalent in a much stronger sense: KG6 and KG7 is in \\textit{exact} one-to-one correspondence over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2} < K$ and indeed more. This includes all of their solutions but not all of $\\omega,k_1,k_2\\in[-\\pi,\\pi]$.\n\nThis exact correspondence is mediated by Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7} in Fourier space or equivalently the coordinate transformations Eq.~\\eqref{CoorKG7KG6} and its inverse. On our second interpretation, we ran into technical trouble here due to the $2\\pi$ periodic identification of the discrete planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ stemming from Euler's identity. This caused our maps between KG6 and KG7 not to be each other's inverses over the whole of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. We were led to consider instead the subspaces $\\mathbb{R}^L_\\text{KG6}$ and $\\mathbb{R}^L_\\text{KG7}$ where these transformations were each other's inverse.\n\nIn the present interpretation, something similar happens but with some key differences. Unlike before, here our maps between KG6 and KG7 given wide scope are invertible and indeed each other's inverses. Wide scope here meaning seen as acting on $F(\\mathbb{R}^3)$ the set of all functions $f(t,x,y)$. However, as before, seen as acting on the relevant vector space $F^K$ we have issues. Namely, Eq.~\\eqref{SkewKG7KG6} and Eq.~\\eqref{SkewKG6KG7}, understood as acting on $F^K$ are not closed: a function supported over $\\omega,k_1,k_2\\in[-K,K]$ may have support outside of here after these transformations.\n\nIn Sec.~\\ref{SecKlein2} we fixed the non-invertible issue by focusing on the subspaces $\\mathbb{R}^L_\\text{KG6}$ and $\\mathbb{R}^L_\\text{KG7}$ where these transformations are each other's inverses. We might overcome our current not-closed-over-$F^K$ issue here in the same way defining \n\\begin{align}\nF^K_\\text{KG7}\\coloneqq&\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert\\,\\text{in }F^K\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG7KG6}})\\\\\nF^K_\\text{KG6}\\coloneqq&\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert\\,\\text{in }F^K\\\\\n\\nonumber\n&\\text{ before and after applying Eq.~\\eqref{SkewKG6KG7}})\\\\\n\\label{FKrotinv}\nF^K_\\text{rot.inv.}\\!\\!\\coloneqq&\\text{span}(\\phi(t,x,y;\\omega,k_1,k_2)\\vert k_1^2+k_2^2 < K^2).\n\\end{align}\nNote that under our embedding $E$ these subspaces are isomorphic to $\\mathbb{R}^L_\\text{KG6}$, $\\mathbb{R}^L_\\text{KG7}$ and $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nRestricted to $F^K_\\text{KG6}$ and $F^K_\\text{KG7}$ these transformations are invertible and indeed are each other's inverses. All of KG6's solutions are in $F^K_\\text{KG6}$ (and moreover they are in $F^K_\\text{rot.inv}$ as well). Similarly, all of KG7's solutions are in $F^K_\\text{KG7}$. Thus, Eq.~\\eqref{CoorKG7KG6} maps generic solutions to KG7 onto generic solutions for KG6 in an invertible way. Therefore, Eq.~\\eqref{CoorKG7KG6} and its inverse give us not only a one-to-one correspondence between the solutions to KG6 and KG7 but a solution-preserving vector-space isomorphism between KG6 and KG7. One can gloss this situation saying: the KPMs of KG6 and KG7 are not isomorphic, but their DPMs are.\n\nThe fact that these transformations (namely, Eq.~\\eqref{CoorKG7KG6} and its inverse) are of the form Eq.~\\eqref{BandlimitedSymmetries} means that for any symmetry transformation for KG6 there is a corresponding symmetry transformation for KG7 and vice versa. In the present interpretation KG6 and KG7 are thoroughly equivalent: We have a solution-preserving vector-space isomorphism between them. Indeed, on this interpretation the only difference between KG6 and KG7 is a change of coordinates. \n\nThus our second big lesson from Sec.~\\ref{SecKlein2} is repeated here: despite KG6 and KG7 being initially presented to us with very different lattice structures (i.e., a square lattice in space versus a hexagonal lattice in space) they have nonetheless turned out to be completely equivalent to one another. This equivalence was hidden from us on our first interpretation because we there took the lattice too seriously. As I discuss in Sec.~\\ref{SecKlein2}, this reduced their continuous rotation symmetries down to quarter rotations and one-sixth rotations respectively and thereby made them inequivalent. Here, we do not take the lattice structure so seriously. We have embedded it onto the manifold where it then disappears from view as just one of many possible sample lattices.\n\nOne substantial difference from our second interpretation should be noted here. We are not forced to shrink $F^K$ down to $F^K_\\text{KG6}$ and $F^K_\\text{KG7}$ to see KG6 and KG7 as equivalent; we have another option. Namely, rather than shrinking $F^K$ we could also expand it. Consider a theory just like KG6 on this interpretation, but with a bandwidth of $2\\,K$ instead of $K$. Clearly, KG6 is a subtheory of this expanded theory. Note that the coordinate transformation which maps KG7 onto KG6, namely Eq.~\\eqref{SkewKG7KG6}, skews its support in Fourier space. However, its support remains inside of $k_1,k_2\\in[-2K,2K]$. Thus, KG7 is also a subtheory of our expanded theory. We can here see KG6 and KG7 as parts of a larger theory. As I will discuss in Sec.~\\ref{PerfectLorentz}, KG6 and KG7 are the parts of this extended theory which is visible to us when we restrict ourselves to only certain sets of representational tools. This line of thought will lead us to a perfectly Lorentzian lattice theory.\n\nFor the rest of this subsection I will only discuss the symmetries of KG6; analogous conclusions are true for KG7 after applying Eq.~\\eqref{CoorKG7KG6}. For KG6 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\\begin{enumerate}\n    \\item[1)] continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0,y-y_0)$} for some $t_0,x_0,y_0\\in\\mathbb{R}$,\n    \\item[2)] three negation symmetries which map\n    \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(-t,x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,-x,y)$}, and \\mbox{$\\phi_\\text{B}(t,x,y)\\mapsto \\phi_\\text{B}(t,x,-y)$} respectively,\n    \\item[3)] continuous rotation which maps \\mbox{$\\phi_\\text{B}(t,x,y)$} to \\mbox{$\\phi_\\text{B}(t,x\\cos(\\theta)-y\\sin(\\theta),x\\sin{\\theta}+y\\cos(\\theta))$} for some $\\theta\\in\\mathbb{R}$. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[4)] two Lorentz boosts which map \\mbox{$\\phi_\\text{B}(t,x,y)$} to \\mbox{$\\phi_\\text{B}(t\\cosh(w)+x\\sinh(w),t\\sinh(w)+x\\cosh(w),y)$} and map \\mbox{$\\phi_\\text{B}(t,x,y)$} to \\mbox{$\\phi_\\text{B}(t\\cosh(w)+y\\sinh(w),x,t\\sinh(w)+y\\cosh(w))$} respectively for some $w\\in\\mathbb{R}$. (This being a symmetry requires some qualification as I will discuss below.),\n    \\item[5)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[6)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and rotation, and Lorentz boost symmetries are here external symmetries. Thus, KG6 (and KG7) each have three continuous translation symmetries despite initially being represented on a lattice.\n\nIn addition to this, KG6 has a (qualified) continuous rotation symmetry. But in what sense is this only a qualified symmetry? Much of the discussion from the previous section following Eq.~\\eqref{RthetaDef} applies here as well with some key differences.\n\nAs I noted in Sec.~\\ref{SecKlein2}, each of KG4-KG7 are rotation invariant in a weak sense: approximately in the continuum limit regime, $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll \\pi$. Here KG4-KG7 are approximately rotation invariant in the $\\vert \\omega\\vert,\\vert k_1\\vert,\\vert k_2\\vert\\ll K=\\pi\/a$ regime. Note however that this is not the continuum limit regime, we are already in the continuum. Rather, this is the regime which is significantly below the bandwidth $K$. While each of KG4-KG7 are rotation invariant in this weak sense, KG6 is rotation invariant in a much stronger sense: KG6 is \\textit{exactly} rotation over \\textit{the whole of} $\\sqrt{k_1^2+k_2^2} < K$.\n\nRotation is here generated by \\mbox{$\\exp(-\\theta (x \\partial_y - y \\partial_x))$} for some $\\theta\\in\\mathbb{R}$. Acting on the continuum planewave basis this transformation merely rotates their wavenumbers in Fourier space. On our second interpretation, we ran into technical trouble here due to the $2\\pi$ periodic identification of the discrete planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ stemming from Euler's identity. This caused $R^\\theta$ not to be invertible over the whole of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. We were led to consider only the rotation invariant subspace $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nIn the present interpretation, something similar happens but with some key differences. Unlike before, here given wide scope rotations are always invertible. Wide scope here meaning seen as acting on $F(\\mathbb{R}^3)$ the set of all functions $f(t,x,y)$. However, as before, seen as acting on the relevant vector space $F^K$ we have issues. Namely, understood as acting on $F^K$ rotation is not closed: some $f$ supported over $\\omega,k_1,k_2\\in[-K,K]$ may have support outside of here after being rotated.\n\nIn Sec.~\\ref{SecKlein2} we fixed the non-invertible issue by focusing on the rotation invariant subspace $\\mathbb{R}^L_\\text{rot.inv.}\\subset \\mathbb{R}^L$. We might overcome our current not-closed-over-$F^K$ issue here in the same way. While rotation is not closed over $F^K$ it is closed over $F^K_\\text{rot.inv.}$ defined in Eq.~\\eqref{FKrotinv}. Note that under our embedding $E$ this subspace is isomorphic to the vector space $\\mathbb{R}^L_\\text{rot.inv.}$ defined in Eq.~\\eqref{RLrotinv}.\n\nNote that all of KG6's solutions are within $F^K_\\text{rot.inv.}$. Within $F^K_\\text{rot.inv.}$ the above discussed rotation transformation is of the form Eq.~\\eqref{BandlimitedSymmetries} and maps solutions to KG6 onto solution to KG6 in an invertible way, and is hence a symmetry. One can gloss this situation saying: rotation does not map the KPMs of KG6 onto themselves in an invertible way, but it does for the DPMs of KG6. If we cut the KPMs of KG6 down to $F^K_\\text{rot.inv}$ this minor issue is fixed.\n\nThis adds to our first big lesson from Sec.~\\ref{SecKlein2}: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a continuous rotation symmetry. This, in addition to their continuous translation symmetries. These continuous translation and rotations symmetries were hidden from us on our first interpretation because we there took the lattice representations too seriously. Here, we do not take the lattice structure so seriously. We have embedded it onto the manifold where it then disappears from view as just one of many possible sample lattices.\n\nOne substantial difference from our second interpretation should be noted here. We are not forced to shrink $F^K$ down to $F^K_\\text{rot.inv.}$ to make KG6 rotation invariant; we have another option. Rather than shrinking $F^K$ down to its rotation invariant core, we could also expand it by adding in every state reachable from $F^K$ by rotation yielding $F^{\\sqrt{2}K}_\\text{rot.inv.}$. Consider a theory just like KG6 on this interpretation, but defined over $F^{\\sqrt{2}K}_\\text{rot.inv.}$ instead of $F^K$. Clearly, KG6 is a subtheory of this expanded theory. As I will discuss in Sec.~\\ref{PerfectLorentz}, KG6 is the part of this extended theory which is visible to us when we restrict ourselves to only a certain set of representational tools. This line of thought will lead us to a perfectly Lorentzian lattice theory.\n\nLet's next discuss how KG6's (limited) Lorentzian boost symmetry is realized on this interpretation. The Lorentz boosts of KG6 are here generated by \\mbox{$\\exp(-w (x \\partial_t + t \\partial_x))$} and \\mbox{$\\exp(-w (y \\partial_t + t \\partial_y))$} for some boost parameter $w\\in\\mathbb{R}$. Notice that boosting a bandlimited function also boosts its support in Fourier space. On our second interpretation, we ran into technical trouble here due to the $2\\pi$ periodic identification of the discrete planewaves $\\bm{\\Phi}(\\omega,k_1,k_2)$ stemming from Euler's identity. This caused $R^\\theta$ not to be invertible over the whole of $\\mathbb{R}^L\\cong\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}\\otimes\\mathbb{R}^\\mathbb{Z}$. We were led to consider only the rotation invariant subspace $\\mathbb{R}^L_\\text{rot.inv.}$.\n\nIn the present interpretation, something similar happens but with some key differences. Unlike before, here given wide scope Lorentz boosts are always invertible. Wide scope here meaning seen as acting on $F(\\mathbb{R}^3)$ the set of all functions $f(t,x,y)$. However, as before, seen as acting on the relevant vector space $F^K$ we have issues. Namely, understood as acting on $F^K$ Lorentz boosts are not invertible nor are they even functions: some $f$ supported over $\\omega,k_1,k_2\\in[-K,K]$ may have support outside of here after being boosted. \n\nIn Sec.~\\ref{SecKlein2} we fixed the non-invertible issue by focusing on the finite-sized region around $\\omega,k_1,k_2=0$ and boost parameter $w=0$ where $\\Lambda^w_\\text{j,n}$ and $\\Lambda^w_\\text{j,m}$ are invertible. Moreover, we were even able to find a representation of the Poincar\\'e algebra. We might overcome our current not-closed-over-$F^K$ issue here in the same way. There is some finite-sized region around $\\omega,k_1,k_2=0$ and boost parameter $w=0$ where Lorentz boosts keep us inside $F^K$. Moreover, as above, we could find a representation of the Poincar\\'e algebra over $F^K$ as well as a finite portion of the Poincar\\'e group.\n\nSo far the situation here is very much like it was in Sec.~\\ref{SecKlein2}. Our first big lesson is here repeated: despite the fact that KG6 and KG7 can be represented on a cubic 3D lattice and a hexagonal 3D lattice respectively, they nonetheless both have a Lorentzian boost symmetries (in a finite but limited regime).\n\nHowever, there is one substantial difference here which ultimately points the way towards a perfectly Lorentzian lattice theory. As I will discuss in Sec.~\\ref{PerfectLorentz}, we can see KG6 as the part of some extended theory which is visible to us when we restrict ourselves to only a certain set of representational tools. This extended theory is perfectly Lorentz invariant. Viewed this way, KG6's failure to be Poincar\\'e invariant in an unqualified way ought to be seen as a problem of representational capacity and not of physics.\n\nFinally, let's consider KG3. For KG3 the dynamical symmetries of the form Eq.~\\eqref{BandlimitedSymmetries} are:\n\\begin{flushleft}\n\\begin{enumerate}\n    \\item[1)]  continuous translation which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t-t_0,x-x_0)$} for $t_0,x_0\\in\\mathbb{R}$,\n    \\item[2)] two negation symmetries which map \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(-t,x)$} and \\mbox{$\\phi_\\text{B}(t,x)\\mapsto \\phi_\\text{B}(t,-x)$} respectively,\n    \\item[3)] a Lorentz boosts which maps \\mbox{$\\phi_\\text{B}(t,x)$} to \\mbox{$\\phi_\\text{B}(t\\cosh(w)+x\\sinh(w),t\\sinh(w)+x\\cosh(w))$},\n    \\item[4)] global linear rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)$} for some $c_1\\in\\mathbb{R}$,\n    \\item[5)] local affine rescaling which maps \\mbox{$\\phi_\\text{B}(t,x)\\mapsto c_1\\,\\phi_\\text{B}(t,x)+c_2(t,x)$} for some $c_2(t,x)$ which is also a solution of the dynamics.\n\\end{enumerate}\n\\end{flushleft}\nThese are exactly the same symmetries that we found on the previous interpretation (sans local Fourier rescaling) just recategorized: the translation, reflection, and (limited) Lorentz boost symmetries are here external symmetries. \n\n\\vspace{0.25cm}\n\nTo summarize: this third attempt at interpreting KG1-KG7 has fixed all of the issues with our first and second interpretations. Firstly, the tension between our theories' differences in locality and their differences in convergence rate in the continuum limit has been harmoniously resolved. This improves upon the hamfisted dissolution of tension given by our second interpretation, see \\cite{DiscreteGenCovPart1}. Secondly, like on the second interpretation, here KG6 and KG7 are seen to be completely equivalent. Here however, we can moreover see KG6 and KG7 in their entirety as parts of a larger unified theory. Finally, like on the second interpretation, we have here exposed KG1-KG7's hidden continuous translation and rotation symmetries and even limited Lorentz symmetries. Here however, these are more satisfyingly categorized as external symmetries.\n\nLike the previous interpretation, this interpretation by and large invalidates all of the first intuitions laid out in Sec.~\\ref{SecIntro}. Indeed here the lattice seems to play a merely representational role in the theory: it does not restrict our symmetries. Moreover, theories initially appearing with different lattices may nonetheless turn out to be substantially equivalent. The process for switching between lattice structures is here done by Nyquist-Shannon resampling. Indeed, the third big lesson from Sec.~\\ref{SecKlein2} is repeated here: there is no sense in which these lattice structures are essentially ``baked-into'' these theories; our bandlimited theories make no reference to any lattice structure. \n\nAs I will discuss in Sec.~\\ref{SecDisGenCov}, these three lessons lay the foundation for a strong analogy between the lattice structures which appear in our discrete spacetime theories and the coordinate systems which appear in our continuum spacetime theories. This ultimately gives rise to a discrete analog of general covariance.\n\nBefore that however, a bit more must be said about this third interpretation.\n\n\\section{A Third Attempt at Interpreting KG1-KG7 - Part 2}\\label{SecKlein3Extra}\nThe previous section presented a third interpretation of our seven discrete theories KG1-KG7. This section will flesh out this interpretation in two ways. Firstly, I will provide some more explicit demonstrations of these theories' symmetries. In particular, I will explicitly demonstrate the fact that the symmetries of these theories are independent of which sample lattice we use to represent them. Secondly, I will write our above formulations of KG1-KG7 in terms of $\\phi_\\text{B}$ in the coordinate-independent language of differential geometry. This will help clarify their symmetries and assumed background structures.\n\n\\subsection{Demonstration of Bandlimited Symmetries}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow1.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow2.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow3.pdf}\n\\caption{The dynamics of KG4 is here shown being carried out in a variety of lattice representations. In the left most column the initial condition is shown in its bandlimited representation, given by Eq.~\\eqref{InCond}. In the rightmost column the final evolved state is shown in its bandlimited representation. Here the evolution time is $t=2$ and the field mass is $\\mu=0$. This state can be found in four different ways. Firstly by applying the bandlimited dynamics Eq.~\\eqref{DKG4bandlimited} to the bandlimited initial condition. The other three ways are shown in the three rows of this figure. The first row shows the initial condition being sampled onto a square lattice. This is then evolved forward in time via Eq.~\\eqref{KG4Long}. The bandlimited representation of the final state is then recovered through the methods discussed in Sec.~\\ref{SecSamplingTheory}. The second and third rows show the same process carried out on a hexagonal lattice and an irregular lattice. Notice that the final state has a 4-fold symmetry regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. We can represent any bandlimited dynamics on any (sufficiently dense) lattice.}\\label{FigEvolutionKG4}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow4.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow5.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow6.pdf}\n\\caption{The dynamics of KG5 is here shown being carried out in a variety of lattice representations. In the left most column the initial condition is shown in its bandlimited representation, given by Eq.~\\eqref{InCond}. In the rightmost column the final evolved state is shown in its bandlimited representation. Here the evolution time is $t=108$ and the field mass is $\\mu=0$. This state can be found in four different ways. Firstly by applying the bandlimited dynamics Eq.~\\eqref{DKG5bandlimitedSkew} to the bandlimited initial condition. The other three ways are shown in the three rows of this figure. The second row shows the initial condition being sampled onto a hexagonal lattice. This is then evolved forward in time via Eq.~\\eqref{KG5Long}. The bandlimited representation of the final state is then recovered through the methods discussed in Sec.~\\ref{SecSamplingTheory}. The second and third rows show the same process carried out on a square lattice and an irregular lattice. Notice that the final state has a 6-fold symmetry regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. We can represent any bandlimited dynamics on any (sufficiently dense) lattice.}\\label{FigEvolutionKG5}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow7.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow8.pdf}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{PaperFigures\/RRow9.pdf}\n\\caption{The dynamics of KG6 is here shown being carried out in a variety of lattice representations. In the left most column the initial condition is shown in its bandlimited representation, given by Eq.~\\eqref{InCond}. In the rightmost column the final evolved state is shown in its bandlimited representation. Here the evolution time is $t=6$ and the field mass is $\\mu=0$. This state can be found in four different ways. Firstly by applying the bandlimited dynamics Eq.~\\eqref{DKG6bandlimited} to the bandlimited initial condition. The other three ways are shown in the three rows of this figure. The first row shows the initial condition being sampled onto a square lattice. This is then evolved forward in time via Eq.~\\eqref{DKG6}. The bandlimited representation of the final state is then recovered through the methods discussed in Sec.~\\ref{SecSamplingTheory}. The second and third rows show the same process carried out on a hexagonal lattice and an irregular lattice. Notice that the final state is rotation invariant regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. We can represent any bandlimited dynamics on any (sufficiently dense) lattice.}\\label{FigEvolutionKG6}\n\\end{figure*}\n\nIn this subsection I will provide some more explicit demonstrations of these theories' symmetries. In particular, I will explicitly demonstrate the fact that the symmetries of some theory's dynamics has nothing to do with the symmetries of the lattice it is represented on. Just as we can represent any bandlimited state on any sufficiently dense lattice, so too can we represent any bandlimited dynamics on any sufficiently dense lattice. To see this consider Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6}. \n\nIn each of these figures we begin from some initial state with a bandlimited representation at $t=0$ of,\n\\begin{align}\\label{InCond}\n\\phi_\\text{B}(0,x,y)=\\frac{J_1(\\pi r_2)}{\\pi r_2}\n+\\frac{J_0(\\pi r_2)-J_2(\\pi r_2)}{2}\n\\end{align}\nwhere $J_n(r)$ is the $n^\\text{th}$ Bessel function and $r=\\sqrt{x^2+y^2}$ and $r_2=r\/\\sqrt{2}$. This function is shown in the first columns of Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6}. Note that this function is rotationally invariant.\n\nThis function is bandlimited with bandwidth of $K=\\pi\/\\sqrt{2}$. We can therefore represent this function with sampling on a square 2D lattice with $a=1<\\sqrt{2}$. We could equivalently represent this function on a hexagonal 2D lattice or even an irregular 2D lattice. For each of KG4, KG5 and KG6=KG7 Such representations are shown in the second columns of Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6}. \n\nFor each of these theories, we then have a choice of which representation to carry out the dynamics in. I here consider four options: as a bandlimited function, as samples on a square lattice, as samples on a hexagonal lattice or as samples on an irregular lattice. The various options for KG4, KG5 and KG6=KG7 are shown in Figs. \\ref{FigEvolutionKG4}, \\ref{FigEvolutionKG5} and \\ref{FigEvolutionKG6} respectively. \n\nLet's begin with the dynamics of KG5 represented on a hexagonal lattice. This is shown in the middle row of Fig.~\\ref{FigEvolutionKG5}. The bandlimited representation of the initial distribution is shown Fig.~\\ref{FigEvolutionKG5}b1). The initial sample points on the hexagonal lattice are shown Fig.~\\ref{FigEvolutionKG5}b2). These can be evolved forward in time using Eq.~\\eqref{KG5Long}. The resulting time-evolved sample points are shown in Fig.~\\ref{FigEvolutionKG5}b3). From these we can reconstruct a bandlimited representation for the state using the techniques discussed in Sec.~\\ref{SecSamplingTheory}. The resulting reconstruction is shown in Fig.~\\ref{FigEvolutionKG5}b4).\n\nAlternatively, we could have carried out this evolution with no lattice representation at all. That is, we could have skipped from Fig.~\\ref{FigEvolutionKG5}b1) directly to Fig.~\\ref{FigEvolutionKG5}b4). We could do this by applying the dynamics Eq.~\\eqref{DKG5bandlimitedSkew} directly to the bandlimited initial condition Eq.~\\eqref{InCond}. It is in this sense that the bandlimited and discrete representations of our dynamics are equivalent.\n\nThe first and third rows of Fig.~\\ref{FigEvolutionKG5} show the exact same evolution via KG5 represented on different lattices, namely a square lattice and an irregular lattice. In the first row the evolution is carried out by a resampled version of Eq.~\\eqref{KG5Long}, namely Eq.~\\eqref{DKG5Skew}. In the third row the evolution is carried out by whatever resampling of Eq.~\\eqref{KG5Long} corresponds to this irregular lattice. \n\nNotice that the final state has a 6-fold symmetry regardless of how the dynamics is represented. Moreover, notice that the final state is the same regardless of how the dynamics is represented. Just as we can represent any bandlimited state on any lattice, so too can we represent any bandlimited dynamics on any lattice.\n\nFig.~\\ref{FigEvolutionKG4} makes the same demonstration for KG4. Notice that the final state has a 4-fold symmetry regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented.\n\nLikewise, Fig.~\\ref{FigEvolutionKG6} makes the same demonstration for KG6. Notice that the final state is rotation invariant regardless of how the dynamics is represented. Notice that the final state is the same regardless of how the dynamics is represented. \n\nThese figures demonstrate clear as can be that a theory's lattice structure has nothing to do with its dynamical symmetries. We can represent any bandlimited dynamics on any lattice.\n\n\\subsection{Bandlimited General Covariance}\\label{SecFullGenCov}\nAs the discussion throughout Sec.~\\ref{SecKlein3} has shown, giving our discrete theory a bandlimited representation has had many of the same benefits one expects from a generally covariant formulation. Namely, we have exposed certain parts of our theory as merely representational artifacts and in the process we have come to a better understanding of our theory's symmetries and background structures. This is the work of the titular discrete analog of general covariance. This analogy was originally introduced in \\cite{DiscreteGenCovPart1} and has here been expanded to a Lorentzian context. The details of this analogy will be spelled out in detail in Sec.~\\ref{SecDisGenCov}.\n\nBefore that, however, I will show how to combine this discrete analog with our usual continuum notion of general covariance. Note that on this interpretation KG1-KG7 are ultimately continuum spacetime theories of the sort we are used to interpreting (albeit ones which consider only bandlimited fields). Hence we can apply to these theories the standard techniques of general covariance. (For a review see Appendix A of \\cite{DiscreteGenCovPart1}).\n\nAs I discussed at the beginning of Sec.~\\ref{SecKlein3} on this third interpretation KG1-KG7 are about a bandlimited field \\mbox{$\\phi_\\text{B}:\\mathcal{M}\\to \\mathcal{V}$} with a value space $\\mathcal{V}=\\mathbb{R}$ and a manifold $\\mathcal{M}\\cong\\mathbb{R}^d$ with $d=2$ for KG1-KG3 and $d=3$ for KG4-KG7. As such, in either case we have access to global coordinate systems for $\\mathcal{M}$. The fields $\\phi_\\text{B}$ considered by this interpretation are bandlimited in the following sense. There exists a fixed special diffeomorphism \\mbox{$d_\\text{coor.}:\\mathbb{R}^d\\to\\mathcal{M}$} which giving us a fixed special global coordinate system for $\\mathcal{M}$. In these special coordinates the field is bandlimited. That is, $\\phi_\\text{B}\\circ d_\\text{trans.}\\in F^K$. It is in these special coordinates that the fields $\\phi_\\text{B}$ obey the dynamical equations Eq.~\\eqref{DKG1bandlimited}-Eq.~\\eqref{DKG7bandlimited}.\n\nOf course, we know from the lessons of continuum general covariance that one ought to be suspicious of any ``special coordinates'' appearing in a supposedly fundamental formulation of a theory. This section will remove any reference to these (or any) coordinates from our bandlimited formulation of KG1-KG7. As is expected of such a generally covariant reformulation, this will reveal these theories' underlying geometric background structures.\n\nFor simplicity, however, I will just consider KG4 and KG6 here. To start let us first write the continuum theory KG0 in the coordinate-free language of differential geometry. After substantial work (see \\cite{Pooley2015} or Appendix A of \\cite{DiscreteGenCovPart1}) one can rewrite Eq.~\\eqref{KG0} in the coordinate-free language of differential geometry as follows:\n\\begin{align}\\label{KG0GenCov}\n\\text{KG0:}\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},\\eta_\\text{ab},\\varphi\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\varphi = 0,   \n\\end{align}\nThe geometric objects used in this formulation are as follows. $\\mathcal{M}$ is a 2+1 dimensional differentiable manifold. $\\eta_\\text{ab}$, is a fixed Lorentzian metric field with signature $(-1,1,1)$. Here $\\nabla_\\text{a}$ is the unique covariant derivative operator compatible with the metric, (i.e. with $\\nabla_\\text{c}\\,\\eta_\\text{ab}=0$). $\\varphi:\\mathcal{M}\\to\\mathbb{R}$ is a dynamical real scalar field. \n\nMathematical structures satisfying the above conditions (independent of whether it satisfies the dynamics) are the theory's kinematically possible models (KPMs). This theory's dynamically possible models (DPMs) are the subset of the KPMs which additionally satisfy the theory's dynamics.\n\nNext let's consider KG4. Rewritten in the coordinate-free language of differential geometry, KG4 becomes:\n\\begin{align}\\label{DKG4GenCov}\n\\text{KG4:}\\ \\text{KPMs:}\\ &\\langle \\mathcal{M}, \\eta_\\text{ab},T^\\text{a},X^\\text{a},Y^\\text{a},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\ &\nF(T^\\text{a}\\nabla_\\text{a})\\phi_\\text{B}\n= \\big[-\\mu^2+F(X^\\text{b}\\nabla_\\text{b})\\\\\n\\nonumber\n&\\qquad\\qquad\\qquad\\qquad \\ +F(Y^\\text{c}\\nabla_\\text{c})\\big]\\,\\phi_\\text{B}.\n\\end{align}\nwhere \n\\begin{align}\nF(z)=2\\text{cosh}(a\\,z)-2.\n\\end{align}\nAt the level of dynamics, KG4 and KG0 are radically different. Note that the dynamics of KG0 is local, only involving finite order of derivative. By contrast, the dynamics of KG4 is highly non-local.\n\nThese theories are also substantially different at the level of KPMs. The three key differences between KG4 and the continuum theory KG0 are as follows. Firstly KG4 resticts us to manifolds $\\mathcal{M}\\cong\\mathbb{R}^3$ whereas KG0 does not. Secondly, the field $\\phi_\\text{B}\\in F^K$ in KG4 is bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$ whereas the field $\\psi$ in KG0 is not. I owe the reader a geometric explanation of what this means.\n\nFinally, KG4 has three extra pieces of spacetime structure which KG0 lacks. Namely, $T^\\text{a}$, $X^\\text{a}$ and $Y^\\text{a}$ are three fixed constant unit vectors which are orthogonal to each other. $T^\\text{a}$ is timelike whereas $X^\\text{a}$ and $Y^\\text{a}$ are spacelike. That is, \n\\begin{align}\n\\nabla_\\text{a}T^\\text{b}&=0 &\n\\nabla_\\text{a}Y^\\text{b}&=0 &\n\\nabla_\\text{a}Y^\\text{b}&=0\\\\\n\\nonumber\n\\eta_\\text{ab} T^\\text{a}T^\\text{b}&=-1 &\n\\eta_\\text{ab} X^\\text{a}X^\\text{b}&=1 &\n\\eta_\\text{ab} Y^\\text{a}Y^\\text{b}&=1\\\\\n\\nonumber\n\\eta_\\text{ab} T^\\text{a}X^\\text{b}&=0 &\n\\eta_\\text{ab} X^\\text{a}Y^\\text{b}&=0 &\n\\eta_\\text{ab} Y^\\text{a}T^\\text{b}&=0\n\\end{align}\nRoughly, $X^\\text{a}$ and $Y^\\text{a}$ here serve to pick out the directions for the rotational anomalies appearing in Fig.~\\ref{FigEvolutionKG4}. $T^\\text{a}$ picks out a standardized way of moving forward in time (i.e, translation generated by $T^\\text{a}\\nabla_\\text{a}$). That is, $T^\\text{a}$ provides a rest frame.\n\nIn terms of these spacetime structures, what does it here mean to say that $\\phi_\\text{B}\\in F^K$ is  ``bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$''. Consider the following eigen-problem for functions $h:\\mathcal{M}\\to\\mathbb{R}$ on the manifold: \n\\begin{align}\nT^\\text{a}T^\\text{b}\\nabla_\\text{a}\\nabla_\\text{b} h = -\\omega^2 h\\\\\n\\nonumber\nX^\\text{a}X^\\text{b}\\nabla_\\text{a}\\nabla_\\text{b} h = -k_1^2 h\\\\\n\\nonumber\nY^\\text{a}Y^\\text{b}\\nabla_\\text{a}\\nabla_\\text{b} h = -k_2^2 h\n\\end{align}\nA function $f:\\mathcal{M}\\to\\mathbb{R}$ is bandlimited if and only if it is the sum of these eigensolutions over a compact range of Fourier space. The extent of this range is its bandwidth. In these terms we can state the requirement that $\\phi_\\text{B}$ is bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$.\n\nLet's move on to KG6. Rewritten in the coordinate-free language of differential geometry, KG6 becomes:\n\\begin{align}\\label{DKG6GenCov}\n\\text{KG6:}\\qquad\\text{KPMs:}\\quad&\\langle \\mathcal{M}, \\eta_\\text{ab},T^\\text{a},X^\\text{a},Y^\\text{a},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-\\mu_0^2)\\,\\phi_\\text{B} = 0\n\\end{align}\nwith spacetime structures as defined following Eq.~\\eqref{KG0GenCov}, $\\mathcal{M}\\cong\\mathbb{R}^3$ and $\\phi_\\text{B}\\in F^K$ in the same sense as KG4. \n\nAt the level of dynamics, KG6 and KG0 are nearly identical. The dynamics for KG0 has a field mass of $M$ whereas KG6 has $\\mu_0=\\mu\/a$. This can be disregarded by setting $\\mu_0=M$.\n\nHowever, at the level of KPMs, KG6 and KG0 are substantially different. The three key differences between KG6 and the continuum theory KG0 are as follows. Firstly KG6 resrticts us to manifolds $\\mathcal{M}\\cong\\mathbb{R}^3$ whereas KG0 does not. Secondly, the field $\\phi_\\text{B}$ in KG6 is bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$ in the sense defined above whereas the field $\\psi$ in KG0 is not. Finally, KG6 has the same extra spacetime structures $T^a$, $X^a$ and $Y^a$ that KG4 does. Indeed, at the level of KPMs KG6 and KG4 are identical.\n\nOne might notice that these extra pieces of spacetime structure don't play any role in the dynamics of KG6. Why then are they there? $T^a$, $X^a$, and $Y^a$ are needed to spell out what it means for $\\bm{\\phi}_\\text{B}$ to be bandlimited with bandwidth $\\omega,k_1,k_2\\in[-K,K]$. This region in Fourier space is not Poincar\\'e invariant, we need some extra structure to point out in which directions the corners go.\n\nThis is a very strange sort of spacetime structure. It doesn't participate in the dynamics, all it does is allows us to articulate a certain restriction on the space of KPMs. As I will discuss in Sec.~\\ref{PerfectLorentz}, it is best to think of KG6 as being the part of an extended theory which is visible to us when we restrict our set of representational tools. On this view, while $X^a$ and $Y^a$ are real spacetime structures for KG4, for KG6 they are merely representational artifacts. \n\nBefore that, however, allow me to spell out in detail the discrete analogs of general covariance which have been developed throughout this paper.\n\n\\section{Two Discrete Analogs of General Covariance}\\label{SecDisGenCov}\nThree lessons have been repeated throughout this paper. Each of these lessons is visible in both our second and third attempts at interpreting KG1-KG7. Combined these lessons give us a rich analogy between lattice structures and coordinate systems: Lattice structure is rather less like a fixed background structure and rather more like a coordinate system, i.e., merely a representational artifact.\n\nThese three lessons run counter to the three first intuitions one is likely to have regarding lattice structure discussed in Sec.~\\ref{SecIntro}. Namely, that lattices and lattice structure: restrict our symmetries, distinguish our theories, and are fundamentally ``baked-into'' the theory. As we have seen, they do not restrict our symmetries, they do not distinguish our theories and they are merely representational not fundamental. In particular, we have learned the following three lessons.\n\nOur first lesson was that taking the lattice structure seriously as a fixed background structure or as the underlying spacetime manifold systematically under predicts the symmetries that discrete spacetime theories can and do have. Indeed, discrete theories can have significantly more symmetries than our first intuitions might allow for. As Sec.~\\ref{SecKlein2} and Sec.~\\ref{SecKlein3} have shown each of KG1-KG7 has a continuous translation symmetry despite being introduced with explicit lattice structures. Moreover, KG6 and KG7 even have a continuous rotational symmetry and a (limited) Lorentzian boost symmetry. As I will discuss in Sec.~\\ref{PerfectLorentz}, these theories can be seen as part of a larger lattice theory which is perfectly Lorentzian. Thus, the fact that a lattice structure was used in the initial statement of these theories' dynamics does not in any way restrict their symmetries. There is no conceptual barrier to having a theory with continuous symmetries represented on a discrete lattice.\n\nIn light of the proposed analogy between lattice structure and coordinate systems this first lesson is not mysterious. Coordinate systems are neither background structure nor a fundamental part of the manifold. The use of a certain coordinate system does not in any way restrict a theory's symmetries. Indeed, it is a familiar fact that there is no conceptual barrier to having a rotationally invariant theory formulated on a Cartesian coordinate system.\n\nOur second lesson was that discrete theories which are initially presented to us with very different lattice structures may nonetheless turn out to be completely equivalent theories. Indeed, as we have seen, two of our discrete theories (KG6 and KG7) have a nice one-to-one correspondence between their solutions. This despite the fact that these theories were initially presented to us with different lattice structures (i.e., a square lattice in space and a hexagonal lattice in space respectively). However, despite this nice correspondence, when in Sec.~\\ref{SecKlein1} we took these lattice structures seriously as a fixed background structure, we found that KG6 and KG7 were inequivalent; namely, they were here judged to have different symmetries. \n\nOnly in Sec.~\\ref{SecKlein2} and Sec.~\\ref{SecKlein3} when stopped taking the lattice structure so seriously did we ultimately see KG6 and KG7 as having the same symmetries. Indeed, in these later two interpretations KG6 and KG7 were seen to be completely equivalent. They are simply re-descriptions of a single theory with slightly different limitations in representational capacities. In Sec.~\\ref{SecKlein2} this re-description is a change of basis in the theory's value space, whereas in Sec.~\\ref{SecKlein3} this re-description is merely a change of coordinates. What we thought were distinct lattice theories are really just different sample points being used to describe one-and-the-same bandlimited field theory.\n\nIn light of the proposed analogy between lattice structure and coordinate systems this second lesson is not mysterious. Unsurprisingly, continuum theories presented to us in different coordinate systems may turn out to be equivalent. Moreover, we can always re-describe any continuum theory in any coordinates we wish.\n\nOur third lesson was that, in addition to being able to switch between lattice structures, we can also reformulate any discrete theory in such a way that it has no lattice structure whatsoever. I have shown two ways of doing this. In Sec.~\\ref{SecKlein2} this was done by internalizing the lattice structure into the theory's value space. In Sec.~\\ref{SecKlein3} this was done by embedding the discrete theory onto a continuous manifold using bandlimited functions. Adopting a lattice structure and switching between them was then handled using Nyquist-Shannon sampling theory discussed in Sec.~\\ref{SecSamplingTheory}.\n\nIn light of the proposed analogy between lattice structure and coordinate systems this third lesson is not mysterious. This is analogous to the familiar fact that any continuum theory can be written in a generally covariant (i.e., coordinate-free) way. Thus, the two above-discussed ways of reformulating a discrete theory to be lattice-free are each analogous to reformulating a continuum theory to be coordinate-free (i.e., a generally covariant reformulation). Thus we have not one but two discrete analogs of general covariance. See Fig.~\\ref{FigTwoAnalogies}.\n\n\\begin{figure}[t!]\n\\begin{flushleft}\n\\text{{\\bf Internal Discrete General Covariance:}}\n\\end{flushleft}\n$\\begin{array}{rcl}\n\\text{Coordinate Systems} & \\!\\!\\leftrightarrow\\!\\! & \\text{Lattice Structure}\\\\\n\\text{Changing Coordinates} & \\!\\!\\leftrightarrow\\!\\! & \\text{Changing Lattice Structures}\\\\\n\\ & \\ & \\text{by changing basis in value space}\\\\\n\\text{Gen. Cov. Formulation} & \\!\\!\\leftrightarrow\\!\\! & \\text{Basis-Free Formulation}\\\\\n\\text{(i.e., coordinate-free)} & \\ & \\text{(i.e., lattice-free)}\\\\\n\\end{array}$\\\\\n\\begin{flushleft}\n\\text{{\\bf External Discrete General Covariance:}}\n\\end{flushleft}\n$\\begin{array}{rcl}\n\\text{Coordinate Systems} & \\!\\!\\leftrightarrow\\!\\! & \\text{Lattice Structure}\\\\\n\\text{Changing Coordinates} & \\!\\!\\leftrightarrow\\!\\! & \\text{Changing Lattice Structures}\\\\\n\\ & \\ & \\text{by Nyquist-Shannon resampling}\\\\\n\\text{Gen. Cov. Formulation} & \\!\\!\\leftrightarrow\\!\\! & \\text{Bandlimited Formulation}\\\\\n\\text{(i.e., coordinate-free)} & \\ & \\text{(i.e., lattice-free)}\\\\\n\\end{array}$\n\\caption{A schematic of the two notions of discrete general covariance introduced \\cite{DiscreteGenCovPart1}. The internal strategy is applied to KG1-KG7 in Sec.~\\ref{SecKlein2} whereas the external strategy is applied to KG1-KG7 in Sec.~\\ref{SecKlein3}. These are compared in Sec.~\\ref{SecDisGenCov}. [\\textit{Reproduced with permission from} \\cite{DiscreteGenCovPart1}.]}\n\\label{FigTwoAnalogies}\n\\end{figure}\n\nBefore contrasting these two analogies, let's recap what they agree on. In either case, as one would hope, our discrete analog helps us to disentangle a discrete theory's substantive content from its merely representational artifacts. In particular, in both cases, lattice structure is revealed to be non-substantive and merely representational as is the lattice itself. Lattice structure is no more attached or baked-into to our discrete spacetime theories than coordinate systems are to our continuum theory. In either case, getting clear about this has helped us to expose our discrete theory's hidden continuous symmetries.\n\nWhat distinguishes these two notions of discrete general covariance is how they treat the lattice structure after it has been revealed as being coordinate-like and so merely representational. The approach in Sec.~\\ref{SecKlein2} was to internalize the lattice structure into the theory's value space. By contrast, the approach in Sec.~\\ref{SecKlein3} was to keep the lattice structure external, but to flesh it out into a continuous manifold such that it is no longer fundamental. Let us therefore call these two notions of discrete general covariance internal and external respectively.\n\nThese two approaches pick out very different underlying manifolds for our discretely-representable spacetime theories. As a consequence, they license very different conclusions about locality and symmetries. In particular, while for KG1-KG7 these internal and external approaches have more-or-less agreed as to what symmetries there are, they have disagreed about how they are to be classified.\n\nIn each of these differences I find reason to favor the external approach. To briefly overview my feelings: It is more natural for the continuous Poincar\\'e symmetries of KG1-KG7 to be classified as external. Moreover, keeping the lattice structure external as a part of the manifold, allows us to draw intuitions about locality from it. In particular, the external approach has a better way  of resolving some puzzles about locality and convergence in the continuum limit. However, neither of these reasons are decisive and I think either the internal or external approach is likely to be fruitful for further investigation\/use.\n\n\\section{A Perfectly Lorentzian Lattice Theory}\\label{PerfectLorentz}\nAs I will now discuss, we can leverage this analogy between lattice structures and coordinate systems to construct a perfectly Lorentz invariant lattice theory.\n\nAs discussed in Sec.~\\ref{SecFullGenCov} KG6 is not Lorentz invariant in an unqualified way due to its KPMs not being closed under boosts. Stating the condition that $\\phi_\\text{B}$ be bandlimited with $\\omega,k_1,k_2\\in[-K,K]$ requires the use of two space-like fields $X^a$ and $Y^a$ and a time-like field $T^a$ which break Lorentz invariance.\n\nAs discussed in Sec.~\\ref{SecKlein3}, we can see KG6 as a part of an extended theory. In terms of rotations, we can see KG6 as a part of an larger rotation invariant theory defined over $F^{\\sqrt{2}K}_\\text{rot.inv.}$ that is with $\\sqrt{k_1^2+k_2^2}<\\sqrt{2}K$. As I will now discuss, we can do the same thing here with Poincar\\'e invariance. Namely, we can find a Poincar\\'e invariant extension of KG6.\n\nIn particular, we can expand $F^K$ by adding to it every state reachable from $F^K$ by any Poincar\\'e transformation and then closing under addition. This defines $F^K_\\text{Poin.inv.}$ defined as follows: \\mbox{$f(t,x,y)\\in F^K_\\text{Poin.inv.}$} if and only if \n\\begin{enumerate}\n    \\item[1)] there exists some compact region $\\mathcal{K}(f)$ where the Fourier transform of $f$ is supported and \n    \\item[2)] this compact region $\\mathcal{K}(f)$ itself is contained within the portion of Fourier space satisfying \\mbox{$- d\\, K^2 \\leq w^2 - \\vert k\\vert^2 \\leq K^2$}\n\\end{enumerate}\nwith $d$ being the number of spacial dimensions and $\\vert k\\vert$ being the norm of the wavenumber, i.e., $d=2$ and \\mbox{$\\vert k\\vert=\\sqrt{k_1^2+k_2^2}$}. The first condition comes from the fact that Poincar\\'e transformations map compact regions in Fourier space to compact regions in Fourier space. The second condition comes from first closing $F^K$ under rotations in space and then Lorentz boosts then finite sums.\n\nTo get away from coordinates, let's now define $F^K_\\text{Poin.inv.}$ geometrically in terms of the metric $\\eta_\\text{ab}$. This definition comes in two parts. First let's define the space $F_\\text{B}$ which satisfies the first constraint. Consider the eigen-problem for functions $h:\\mathcal{M}\\to\\mathbb{R}$\n\\begin{align}\\label{FBdef}\n\\eta^\\text{ab} \\nabla_\\text{a}\\nabla_\\text{b} h= -\\lambda h\n\\end{align}\nA function $f:\\mathcal{M}\\to\\mathbb{R}$ is in $F_\\text{B}$ if and only if it is the sum of these eigensolutions over a compact range of $\\lambda\\in\\mathbb{R}$ of Fourier space. A function is further within $F^K_\\text{Poin.inv.}$ if and only if this compact range of $\\lambda$s is contained within \\mbox{$- d\\, K^2 \\leq \\lambda \\leq K^2$}.\n\nWe can thus define following perfectly Lorentz invariant lattice theory:\n\\begin{align}\\label{KGPoininvKGenCov}\n\\text{KG}_\\text{Poin.inv.}^K\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},\\eta_\\text{ab},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\phi_\\text{B} = 0,\n\\end{align}\nwith spacetime structures as defined following Eq.~\\eqref{KG0GenCov}, $\\mathcal{M}\\cong\\mathbb{R}^3$ and $\\phi_\\text{B}\\in F^K_\\text{Poin.inv.}$ in the sense discussed above. Note that KG6 is a subtheory of $\\text{KG}_\\text{Poin.inv.}^K$. \n\nHow does $\\text{KG}_\\text{Poin.inv.}^K$ differ from KG0? Their only difference is that we are here limited to manifolds $\\mathcal{M}\\cong\\mathbb{R}^3$ and that the field $\\phi_\\text{B}$ in $\\text{KG}_\\text{Poin.inv.}^K$ is bandlimited in $F_\\text{Poin.inv.}^K$, whereas the field $\\psi$ in KG0 is not. In fact, for either theory the dynamics guarantees that if the field amplitude starts off bandlimited it will stay bandlimited. Thus this restriction of the allowed dynamical fields is really just a restriction on the allowed initial conditions. Thus, ultimately the only difference between $\\text{KG}_\\text{Poin.inv.}^K$ and KG0 is a restriction on the initial conditions.\n\nAs innocent as this restriction on initial conditions may seem, I believe it has serious implications for the nature of the underlying manifold. As I will discuss later, $\\text{KG}_\\text{Poin.inv.}^K$ has a nice sampling property and therefore has access to discrete descriptions which make no reference to the manifold. By contrast, the manifold in KG0 seems descriptively essential. Moreover, as discussed in \\cite{DiscreteGenCovPart1}, this restriction on initial conditions has serious implications for counterfactual reasoning and locality.\n\nBut what makes $\\text{KG}_\\text{Poin.inv.}^K$ a lattice theory? Note that because every compact region is bounded in a large enough cube, every $f\\in F_\\text{B}$ is representable by a fine-enough cubic lattice. Doing so one would find its sample values on this lattice obey our original discrete equation for KG6, Eq.~\\eqref{DKG6}, with some rescaling of $\\mu$. Hence, as each state in this theory is representable on some lattice, this is a perfectly Lorentzian lattice theory. \n\nThis, should be compared with compared with the Lorentz-covariant sampling theory for fields developed in~\\cite{Pye2022}. There, only the second condition defining $\\text{KG}_\\text{Poin.inv.}^K$ is applied to the fields. Let us call this a Lorentz-covariant bandlimitation. Lorentz-covariant UV cutoffs have been considered across the physics literature recently, see~\\cite{Kempf2004a,Kempf2004b,Kempf_2010,Kempf2013,Pye2015,Kempf2018,PyeThesis,BEH_2020,Pye2022} among others.\n\nSince \\mbox{$- d\\, K^2 \\leq w^2 - \\vert k\\vert^2 \\leq K^2$} is not a compact region in Fourier space, the resulting fields are not bandlimited in the usual sense; as their support in Fourier space may be non-compact. Rather these functions are Lorentz-bandlimited. Such functions don't have the sampling property discussed in Sec.~\\ref{SecSamplingTheory}, but may have a new Lorentz sampling property. Indeed, \\cite{Pye2022} develops a new Lorentz-covariant sampling theory. Here none of that is necessary as our fields are each bandlimited in the original sense. Each field can be represented exactly on some lattice in the standard way discussed in Sec.~\\ref{SecSamplingTheory}.\n\nOne may still complain that this should not be called a lattice theory. For each state here we need a different lattice to represent it, there is no single ontologically significant lattice on which we can represent all of our states. As I will soon discuss, there is in fact such an all-representing lattice (albeit a non-fundamental one), but first let me argue against the spirit of this complaint.\n\nGiven the deflationary attitude taken in this paper towards lattice structures, the lack of an all-representing lattice wouldn't bother me. As I have argued throughout this paper, lattice structures are merely representational artifacts playing no substantive role in our theories: they do not limit our symmetries, they do not distinguish our theories, and they are not fundamentally ``baked-into'' our theories. Indeed, as I have argued (here and in \\cite{DiscreteGenCovPart1}), there is a rich analogy between the lattice structures appearing in our discrete spacetime theories and the coordinate systems appearing in our continuum spacetime theories. We wouldn't refer to our continuum spacetime theories as ``coordinate theories'', they are rather coordinate-representable theories. Analogously, I claim that properly understood, there are no such thing as lattice-fundamental theories, rather there are only lattice-representable theories. Hence, the above theory is a lattice theory in the strongest sense available.\n\nAccepting this, one might still feel there ought to be an all-representing lattice (albeit a non-fundamental one). As I will discuss, there is one but if there weren't this wouldn't be an issue. Note that we are often forced to represent our continuum theories with multiple overlapping complementary coordinate systems. That is, we often lack an all-representing coordinate system. Given the central analogy of this paper, why should the lack of an all-representing lattice bother us?\n\nReturning to answering the complaint directly however, there is such an all-representing lattice. Consider a cubic lattice with some fixed spacing. Consider another cubic lattice with twice the resolution, offset from the first lattice by some rational amount. Consider an infinite sequence of lattices each doubling in resolution with some rational offset. At some point any function $f\\in F_\\text{B}$ will be representable on one of these lattices. Consider the \\textit{deep lattice} which results from taking the infinite union of all of these lattices (but not their closure under the limit). Every function $f\\in F_\\text{B}$ is representable on this deep lattice\\footnote{Indeed any $f\\in F_\\text{B}$ can be represented on some finite length initial segment on this.}. Moreover, note that the deep lattice has a countable number of lattice sites, by construction it is a subset of the rationals $\\mathbb{Q}^3$.\n\nOne final comment on $F_\\text{Poin.inv.}^K$. Note that this space is closed under (finite) addition and scalar multiplication and is hence a vector space. As the deep lattice shows, this is a vector space with a countably infinite dimension. Thus, given its translation, rotation and Lorentz boost invariances $F_\\text{Poin.inv.}^K$ supports a countably infinite dimensional representation of the Poincar\\'e group.\n\nViewing KG6 as a part of $\\text{KG}_\\text{Poin.inv.}^{K}$ sheds some light on its failure to be Lotentz invariant in an unqualified way. To see how, it is instructive to restate this in terms of sample lattices. KG6 is a theory about all of the bandlimited functions which are representable on a fixed cubic sample lattice with spacing $a$. By contrast, $\\text{KG}_\\text{Poin.inv.}^{K}$ is a theory about\\footnote{Technically, $\\text{KG}_\\text{Poin.inv.}^{K}$ is about this collection of bandlimited functions closed under addition.} all of the bandlimited functions representable on a fixed cubic sample lattice with spacing $a$ \\textit{or any Poincar\\'e transformation thereof.}\n\nThus, KG6 is the part of $\\text{KG}_\\text{Poin.inv.}^{K}$ which is visible to us when we restrict ourselves to only a certain set of representational tools. We can thus see KG6's failure to be Lorentz invariant in an unqualified way as a problem of representational capacity and not of physics. Sure, if we limit ourselves to functions which are representable on a fixed cubic lattice, we lose rotation symmetry. But what physical reason do we have to limit our representational tools in this way?\n\nConsider an analogous situation involving coordinate systems. Almost all manifolds cannot be covered with a single global coordinate system. For instance, the 2-sphere needs at least two coordinate systems to cover it. Consider the sphere under arbitrary rotations. Note that no coordinate system is closed under these transformations. If we limit ourselves to functions which are supported entirely within the scope of a single fixed coordinate system, we lose rotation symmetry. Would it be right to say that no theory which is set on a sphere can be rotation invariant? Of course not, this just reflects the fact that each of our coordinate systems individually have a limited representational capacity. We can regain rotation invariance by using multiple representational tools, i.e., multiple coordinate systems. \n\nApplying this lesson to KG6, we ought to view it as being able to represent a Lorentz non-invariant part of a wider Lorentz invariant theory. As discussed above, making use of multiple sample lattices (i.e., all Poincar\\'e transformed cubic lattices with spacing $a$) we can describe a rotation invariant theory $\\text{KG}_\\text{Poin.inv.}^{K}$ over $F^{K}_\\text{Poin.inv.}$.\n\nThe equivalence between KG6 and KG7 can also be seen this way. Note that the coordinate transformation which maps KG7 onto KG6, namely Eq.~\\eqref{CoorKG7KG6}, maps a cubic 3D lattice to a hexagonal 3D lattice maintaining the lattice spacing $a$, see Fig.~\\ref{FigSkew}. Thus, KG7 is a theory about all bandlimited functions representable on a hexagonal 3D sample lattice with spacing $a$. The subsets of KG6 and KG7's KPMs which are equivalent to each other ($F^K_\\text{KG6}\\cong F^K_\\text{KG7}$) are exactly those parts which are representable on both a square and hexagonal lattice. All of KG6 and KG7's DPMs are representable in both ways, but not all of their KPMs. We can here see KG6 and KG7 as parts of a larger unified theory viewed in two ways with differently limited representational capacities.\n\nNote that even the extended theory $\\text{KG}_\\text{Poin.inv.}^{K}$ has an artificially limited representational capacity. What space do we find if we use all of our representational tools (i.e., every possible sample lattice)? The answer is $F_\\text{B}$ defined above.\n\nContrast $F_\\text{B}$ with $F(\\mathbb{R}^3)$ the space of all functions $f(t,x,y)$. The Gaussian distribution is in $F(\\mathbb{R}^3)$ but not $F_\\text{B}$. One might gloss this difference as follows: $F(\\mathbb{R}^3)$ contains actually infinite frequencies, whereas $F_\\text{B}$ contains arbitrarily large but always finite frequencies. That is, $F_\\text{B}$ contains only a potential infinity of frequencies. Note that while $F_\\text{B}$ is closed under finite sums it is not closed under infinite sums.\n\nUsing $F_\\text{B}$ we can thus define following perfectly Lorentz invariant lattice theory:\n\\begin{align}\\label{KGBGenCov}\n\\text{KG}_\\text{B}\\text{:}\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},\\eta_\\text{ab},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(\\eta^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\phi_\\text{B} = 0,\n\\end{align}\nwith spacetime structures as defined following Eq.~\\eqref{KG0GenCov}, $\\mathcal{M}\\cong\\mathbb{R}^3$ and $\\phi_\\text{B}\\in F_\\text{B}$ in the sense defined following Eq.~\\eqref{FBdef}.\n\nWe can even give $\\text{KG}_\\text{B}$ a diffeomorphism invariant reformulation as:\n\\begin{align}\\label{SR2like}\\text{KG}_\\text{B}'\\text{:}\\qquad\\text{KPMs:}\\quad&\\langle\\mathcal{M},g_\\text{ab},\\phi_\\text{B}\\rangle\\\\\n\\nonumber\n\\text{DPMs:}\\quad&(g^\\text{ab}\\nabla_\\text{a}\\nabla_\\text{b}-M^2)\\,\\phi_\\text{B} = 0\\\\\n\\nonumber\n&R^\\text{a}{}_\\text{bcd}=0\n\\end{align}\nHere $\\mathcal{M}\\cong\\mathbb{R}^3$. The fixed Lorentzian metric field, $\\eta_\\text{ab}$, has been replaced with a dynamical metric field, $g_\\text{ab}$. The dynamical metric field varies from model to model. $\\nabla_\\text{a}$ is still the unique covariant derivative operator compatible with the metric, (i.e. with $\\nabla_\\text{c}\\,g_\\text{ab}=0$). The dynamical metric field obey a new dynamical equation, $R^\\text{a}{}_\\text{bcd}=0$, where $R^\\text{a}{}_\\text{bcd}$ is the Riemann tensor associated with the unique compatible derivative $\\nabla_\\text{a}$. The dynamical field $\\phi_\\text{B}$ is bandlimited in the same sense as above (i.e., $\\phi_\\text{B}\\in F_\\text{B}$) with two differences. Firstly, $g_\\text{ab}$ replaces $\\eta_\\text{ab}$ in Eq.~\\eqref{FBdef}. Secondly, this condition ought to be viewed as a compatibility constraint acting jointly on the metric $g_\\text{ab}$, the covariant derivative $\\nabla_\\text{a}$, and the field $\\phi_\\text{B}$. Comparing with \\cite{Pooley2015}, $\\text{KG}_\\text{B}$ is like SR1 and $\\text{KG}_\\text{B}'$ is like SR2. \n\nNote that the above discussion is suggestive of a possible procedure by which any spacetime theory can be given a bandlimited formulation and so gain the sampling property discussed in Sec.~\\ref{SecSamplingTheory}. \n\nPerhaps surprisingly, all of the above claims about $\\text{KG}_\\text{Poin.inv.}^K$ hold for $\\text{KG}_\\text{B}$ (and relatedly $\\text{KG}_\\text{B}'$). Its state space $F_\\text{B}$ is closed under Poincar\\'e transformation. Moreover, it is closed under generic diffeomorphisms. Every function $f\\in F_\\text{B}$ is representable on some fine-enough cubic lattice. Doing so one would find its sample values on this lattice obey Eq.~\\eqref{DKG6} with some rescaling of $\\mu$.\n\nNote that this space is closed under (finite) addition and scalar multiplication and is hence a vector space. Moreover, $F_\\text{B}$ has a deep lattice just like $F^K_\\text{Poin.inv.}$ does. Indeed, it has exactly the same deep lattice. Hence $F_\\text{B}$ is a countably infinite dimensional vector space. It supports a countably infinite dimensional representation of the Poincar\\'e group.\n\nAs I see it, the proper view of KG6 (even beginning from $\\phi_\\ell$) is the one given by Eq.~\\eqref{DKG6GenCov} understood as a part of the maximally extended theory $\\text{KG}_\\text{B}$ (or $\\text{KG}_\\text{B}'$ depending on taste). In particular, KG6 is the part of $\\text{KG}_\\text{B}$ which is visible to us if we restrict ourselves to only certain representational tools.\n\nHow does $\\text{KG}_\\text{B}$ differ from the continuum theory KG0? KG0 assents to the existence of actually infinite frequencies whereas $\\text{KG}_\\text{B}$ does not. To me $\\text{KG}_\\text{B}$ appears to be on better ground empirically speaking than KG0: I have never and will never see a literally infinite frequency photon. As small as the difference is between $\\text{KG}_\\text{B}$ and KG0, I expect there to be radical differences in views on spacetime, but this is a topic for another paper.\n\n\\section{Conclusion}\\label{SecConclusion}\nThis paper has given an exhaustive study of the seven discrete Klein Gordon equations presented in Sec.~\\ref{SecSevenKG}. These theories were initially formulated in terms of a real valued function over lattice sites, $\\phi_\\ell\\in F_L$. Next, as an infinite dimensional vector, $\\bm{\\Phi}\\in \\mathbb{R}^L$. And finally, as a bandlimited field $\\phi_\\text{B}\\in F^K$ over a continuous spacetime manifold $\\mathcal{M}$. Each of these redescriptions was carried out by a vector space isomorphism, $F_L\\cong\\mathbb{R}^L\\cong F^K$.\n\nAlong the way, we have learned three substantial lessons about the role that lattice structures play in our discrete spacetime theories. These lessons serve to undermine the three first intuitions about lattice structure laid out in Sec.~\\ref{SecIntro}. As I have shown, lattice structures don't restrict symmetries, they don't distinguish our theories, and they are not fundamentally ``baked-into'' these theories. As I have discussed, these lessons lay the foundation for a rich analogy between the lattice structures which appear in our discrete spacetime theories and the coordinate systems which appear in our continuum spacetime theories. Indeed, my analysis has shown that lattice structure is rather less like a fixed background structure or a fundamental part of some underlying manifold and rather more like a coordinate system, i.e., merely a representational artifact. This extends the results of \\cite{DiscreteGenCovPart1} to a Lorentzian setting.\n\nBased upon this analogy, we have two discrete analogs of general covariance (see Fig.~\\ref{FigTwoAnalogies}) each of which are useful for exposing hidden symmetries and background structures in our discrete spacetime theories (i.e., lattice theories). In either case, as hoped, when applied to such theories this discrete analog helps us disentangle the theory's substantive content from its representational artifacts. \n\nThese results are significant as they tell strongly against the intuitions laid out in Sec.~\\ref{SecIntro}. One might have an intuition that the world could be fundamentally set on a lattice. This lattice might be square or hexagonal and we might discover which by probing the world at the smallest possible scales looking for violations of rotational symmetry, or other lattice artifacts. Many serious efforts at quantum gravity assume that the world is set on something like a lattice at the smallest scales (although these are often substantially more complicated than the lattices assumed here). However, as this paper clearly demonstrates this just cannot be the case. \n\nThe world cannot be ``fundamentally set on a square lattice'' (or any other lattice) any more than it could be ``fundamentally set in a certain coordinate system''. Like coordinate systems, lattice structures are just not the sort of thing that can be fundamental; they are both thoroughly merely representational. Spacetime cannot be a lattice (even when it might be representable as such). Specifically, I claim that properly understood, there are no such things as lattice-fundamental theories, rather there are only lattice-representable theories.\n\nThe primary inspiration for this paper has been the work of mathematical physicist Achim Kempf~\\cite{Kempf_1997,UnsharpKempf,Kempf2000b,Kempf2003,Kempf2004a,Kempf2004b,Kempf2006,Martin2008,Kempf_2010,Kempf2013,Pye2015,Kempf2018} among others~\\cite{PyeThesis,Pye2022,BEH_2020}. For an overview of Kempf's works on this topic and his closely related thesis that ``Spacetime could be simultaneously continuous and discrete, in the same way that information can be''~\\cite{Kempf_2010}, see~\\cite{Kempf2018}. My thesis here inserts a ``representable as'' and then analyzes which parts of these representations should be thought of as substantial. Lattice structures turn out to be coordinate-like representational artifacts.\n\nFor a discussion of the impact I anticipate these results having on the philosophy of space and time generally, see the conclusion of \\cite{DiscreteGenCovPart1}. This includes some discussion of the dynamical versus geometric debate regarding how we should view spacetime structures~\\cite{EarmanJohn1989Weas,TwiceOver,BelotGordon2000GaM,BrownPooley1999,Nonentity,HuggettNick2006TRAo,StevensSyman2014Tdat,DoratoMauro2007RTbS,Norton2008,Pooley2013}. As we are now in a Lorentzian context, I will focus my attention on the aspects of this work which relate to quantum gravity and Lattice QFT.\n\nPeople often have arguments (e.g., from quantum gravity) that spacetime must be a lattice of some sort, are such arguments in disagreement with my conclusion? This would need to be checked on a case-by-case basis, but there is room for compatibility here. Upon closer investigation, such arguments may only show that spacetime must be \\textit{representable as} a lattice. In this case there is no disagreement with my conclusion. For instance, bandlimited functions can always be represented as a lattice of sample values; it's just that upon closer philosophical investigations these lattice representations cannot possibly be fundamental, as they are merely coordinate-like representational tools. However, if their arguments carry ontological force as well as representational force, then there is disagreement.\n\nWhat does my conclusion mean for those who model spacetime as a lattice (e.g., some approaches to quantum gravity)? There is no issue per se with modeling spacetime as a lattice: e.g., as I just mentioned bandlimited physics is discretely representable. My conclusion only speaks to how such models should be interpreted. Indeed, those who describe their physics on a lattice for reasons of approximation (e.g. the Lattice QFT community, people studying crystalline solid state systems) are free to ignore my conclusion. However, they ought not ignore the central analogy or the three lessons which support it. \n\nFor them my lesson is this: ``lattice artifacts'' arise under the following two conditions. They arise when 1) we represent our continuum physics on a lattice \\textit{2) and then modify the dynamics to be simple in this representation}. As this paper has shown, the blame is to be placed entirely on the second step. Square-shaped lattice artifacts do not come from using a square lattice, see Fig.~\\ref{FigEvolutionKG4}. These artifacts come from using dynamics which are relatively simple (i.e., nearest neighbor interactions) when represented on a square lattice. One can define rotation invariant dynamics on a square lattice, Fig.~\\ref{FigEvolutionKG6}, but not with nearest neighbor interactions.\n\nSo how should lattice-based approaches to quantum gravity be interpreted then? Namely, what about loop quantum gravity (LQG) and causal set theory~\\cite{Surya2019}? I am not an authority on either of these approaches, but I can offer some comments. The conclusion of this paper does not directly apply to LQG. This is because the spacetime considered in LQG consists not only of lattice sites but also links\/edges between lattice sites. In order to more directly apply, the results of this paper would need to be extended to more complicated gauge theories first.\n\nRegarding causal set theory~\\cite{Surya2019}, its proponents often note that no fixed spacetime lattice is Poincar\\'e invariant. This (apparently) spells big trouble for any lattice-based Lorentzian theories. They, however, avoid this issue by considering instead a random Poisson sprinkling of lattice points which does not pick out any preferred direction and hence does not explicitly break Poincar\\'e symmetry, at least on average. However, given the deflationary position this paper takes towards lattices, I claim there is no issue to be avoided. Like coordinate systems, lattice structures are just a representational tool for helping us express our theory in a certain way. There is no need for the symmetries of our representational tools to latch onto the symmetries of the thing being represented. Cartesian coordinates are distorted under Lorentz boosts, but we can still use them to describe our Lorentzian theories without issue. Indeed, in Sec.~\\ref{PerfectLorentz} I have presented a perfectly Lorentzian lattice theory.\n\n\\begin{acknowledgments}\nThe authors thanks James Read, Achim Kempf, Jason Pye, and Nick Menicucci and the Barrio RQI for their helpful feedback.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nLyapunov control has been proven successful in controlling highly chaotic non-linear oscillators \\cite{Ref1}\\cite{Ref2}\\cite{Ref3} . One of the fundamentals that contribute to the success or failure of any type of control strategy is the controller and system parameters. Therefore, researchers have explored different methods to find the precise parameters that would lead to achieving the best system results \\cite{Ref4}\\cite{Ref5}. \\\\\nOne of the methods that was utilized to achieve the previously mentioned goals is Genetic algorithm (GA). GAs have been successful in cases where all the system dynamics are clearly defined and known to some extent or with systems where limited system disturbances are introduced and minor parameter tuning is required \\cite{Ref6}\\cite{Ref7}. In some cases, several system assumptions are needed in order to allow the GA to run successfully.\\\\ \n\nDue to some of the limitations found in using GAs such as inability to quickly converge to the final solution or adapt to unknown system dynamics or unknown disturbances. Researchers though after different approaches that wouldn't reduce the system agility and at the same time would be able to handle unknown system characteristics. A hybrid approach of Fuzzy Control and GAs was researched \\cite{Ref8} but system linearization is a requirement in order to use the previously mentioned method .\\\\ \n\nAnother approach that is recently being researched is the use of Machine Learning to enhance the controller performance. For example through the use of Episodic learning \\cite{Ref9}\\cite{Ref10}. Most recently, there is the introduction neural lyapunov control which proposes the use of deep learning to find the control and Lyapunov functions. The approach mentioned in \\cite{Ref9}\\cite{Ref10} is suitable for find the best system parameters that would initially lead the system to stability and reduced the system error. The problem with approach in \\cite{Ref9}\\cite{Ref10} is that it assumes that the system is deterministic, time invariant, and affine in the control input. while in real life situation external perturbations might occur resulting in system failure at any moment while the system is running \\cite{Ref11}\\cite{Ref12}. The approach proposed in \\cite{Ref25} \\cite{Ref26} and \\cite{Ref27}attempts to predict the control and Lyapunov functions that would lead to system stability but under specific conditions where the system dynamics are deterministic in nature. The approach proposed in this research is novel to the best of our knowledge in that it discards the assumption of an ideal environment or fully known system dynamics and seeks continuous enhancement of the controller outcome through continuous monitoring of the system error, reference signal, system dynamics and control signal and accordingly adjust the system and controller parameters to improve the controller performance without the need to disrupt the system output.\n\n The focus of the research is to allow the Deep Learning Algorithm to learn the system from a continuously improving dataset and according to the slope of the output error the algorithm relearns the system and collects the needed information.\\\\   \n\nThe proposed method is applied to a non-linear chaotic combined system of Duffing and Van der pol oscillators\\cite{Ref24}. The aforementioned system was chosen to test the Deep learning algorithm response to unpredicted system disturbances and unknown system dynamics\\cite{Ref21}\\cite{Ref22}\\cite{Ref23}. An algorithm was developed to aid and trigger the Deep Neural Network when needed to adapt to new system dynamics and according to preset conditions. The algorithm records and feeds an updated data set to the DNN in order to relearn the system dynamics if certain conditions are detected to be true. Once the process of retraining is complete, the algorithm generates a random array of parameters and the array of parameters is fed to the DNN to predict the output error. If the DNN predicts that the error slope will be reduced from the current value improving the performance of the system, then the suggested controller parameters are allowed to be set as the new controller parameters. Otherwise the algorithm would create a new array set of randomly generated parameters and re-feed them to the DNN to provide its predictions in a continuous loop. Once the controller parameters are updated the algorithm monitors the actual error compared to the predicted error to determine the network viability and accordingly if the error difference is greater than a set value the algorithm would trigger a retraining request of the DNN. \\\\\n\\section{LYAPUNOV CONTROL ON DUFFING - VANDERPOL OSCILLATOR MODEL}\n\\label{Section II}\nThis section provides an overview of the Duffing-Van der pol system dynamics. \nDuffing-Van der pol mathematical model is  \n\\begin{equation}\n\\label{eq:1}\n\\ddot{x}+ \\delta\\dot{x}+ \\epsilon_1( x^2\\ddot{x}+x\\dot{x}^2)+\\epsilon_2x^3 = \\mu(t) + \\gamma cos(\\omega t)\t\n\\end{equation} \nIt combines both non-linear stiffness and non-linear damping [1].\nSetting\n\\begin{equation}\n\\label{eq:2}\n\\dot{x} = x_2  \n\\end{equation} \nand\n\\begin{equation}\n\\label{eq:3}\n\t x = x_1\n\\end{equation}\ntherefore\n\\begin{equation}\n\\label{eq:4}\n\t \\dot{x_1} = x_2\n\\end{equation} \nwe get the state space model\n \\begin{equation}\n\\label{eq:5}\n\t \\dot{x_2} = \\frac {1} {1+\\epsilon_1 X^2} (\\delta \\dot{x} - \\epsilon_1 x \\dot{x^2} -  \\epsilon_2 x^3 +\\mu(t) +\\gamma cos(\\omega t)) \n\\end{equation}   \n\n\\begin{equation}\n\\label{eq:6}\n\\begin{bmatrix}\n\\dot{x_1}  \\\\ \n\\dot{x_2}  \n\\end{bmatrix}\n =\n  \\begin{bmatrix}\n   x_2 \\\\\n  \\frac{1}{1+ \\epsilon_1 x^2} (\\delta x_2 - \\epsilon_1 x x_2^2 - \\epsilon_2x^3)  \n   \\end{bmatrix}\n\\end{equation}\n\\begin{equation}\n\\label{eq:7} \ny  = \\begin{bmatrix}  x_1 \\\\  x_2     \\end{bmatrix}\n\\end{equation}  \n\\section{LYAPUNOV CONTROLLER DESIGN}\n\\label{Section III}\nIn this section we give an overview of the controller design. The controller is designed with the purpose of achieving Duffing- van der pol oscillator system stability at an increased forcing frequency. Desired state q\\textsubscript{d} and actual state q.\n\\begin{equation}\n\\label{eq:8}\ne=q_d - q\n\\end{equation} \nA control LYAPUNOV candidate is then chosen to be\n\\begin{equation}\n\\label{eq:9}\nV = \\frac{1}{2} (\\gamma_1 \\dot{e} + \\gamma_2 e)\n\\end{equation} \nAs can be seen from \\cite{Ref11}, the transient behaviour of the error dynamics can be influenced by a suitable choice of the CLF and gamma parameters, respectively \\cite{Ref10}. Therefore the CLF has been chosen to be positive definite for all \\begin{equation} x,\\dot{x} \\end{equation} taking the time derivative of V\n\\begin{equation}\n\\label{eq:11}\n\\dot{V} = (\\gamma_1 \\dot{e} + \\gamma_2 e)(\\gamma_1 \\ddot{e} + \\gamma_2 \\dot{e})\n\\end{equation} \n\\begin{equation}\n\\label{eq:12}\n\\dot{V} = -kV\n\\end{equation} \nwhich is globally exponentially stable as V is globally positive definite. Hence, by substitution in Eqn.\\ref{eq:11} we find the control law\n\\begin{equation}\n\\label{eq:13}\nu = \\frac{1+\\epsilon_1 x^2}{\\gamma_2} (\\frac{k}{2} \\gamma_1 e+ \\frac {k}{2} \\gamma_2  \\dot{e} + \\gamma_1  \\dot{e} + \\gamma_2 \\ddot{x_des}) - pCos(\\omega t)+\\delta \\dot{x} + x + \\epsilon_1  \\dot{x} x^2 + \\epsilon_2 x^3\n\\end{equation} \n\\section{SIMULATION RESULTS FOR UNOPTIMIZED SYSTEM}\n\\label{Section IV}\nA Duffing- van der pol oscillator model with \\textdelta=0.5, \\textepsilon\\textsubscript{1}=1.6, \\textepsilon\\textsubscript{2}=-0.8, P=3, \\textomega=10 and the control parameters \\textgamma\\textsubscript{1} =12, \\textgamma\\textsubscript{2} =4, k=115. The parameters were manually adjusted and tuned to give low error \\ref{fig:1} and high stability \\ref{fig:2}.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3in]{1.png}\n\\caption{Reference signal (dotted green) $5^{th}$ harmonic function with amplitudes of 0.1, 0.5 and 1 and Lyapunov controller output (red) for t=2.5s}\n\\label{fig:1}      \n\\end{figure}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3in]{2.png}\n\\caption{Phase portrait showing a stable system with no interference under the harmonic reference signal t=2.5s}\n\\label{fig:2}      \n\\end{figure}\n\\section{LYAPUNOV CONTROLLER PARAMETER OPTIMIZATION}\n\\label{Section V}\n\\subsection{OVERVIEW}\nThe process of selecting the proper Sigmas to reduce the error of the model can be modelled as an exhaustive search problem. Where several Sigmas are tried and the one that results in a reduction in the system error is selected. Though this process results in the best Sigmas, yet it is time consuming. To overcome this problem, we developed a deep learning model that researches different system and controller parameters. A given sigma is good if it reduces the error of the system. The pipeline is presented in Fig. \\ref{fig:6}. Initially, the model is run, and the system error is monitored. If the error surpasses a curtain threshold the neural network is queried for a sigma that would reduce the system error given the current state of the system. The algorithm for the entire process is defined below.\nThe network proposed in Fig.\\ref{fig:5} is initially trained and then used to predict new Sigmas; Sigma 1 ($s_1$), and Sigma 2 ($s_2$) if the system error (sys err) exceeds a given threshold (0.8). The network is then given the current state of the system in terms of time (t), position (p), velocity (v), and the randomly generated ($s_1$) and its corresponding ($s_2$) and it predicts the system error. If the predicted error is less than the current system error, then the system is updated with the newly generated Sigmas that are speculated to reduce the error. Every iteration the newly generated Sigmas are their corresponding system parameters are stored. After each 100 iterations the system is probed for perturbations by computing the average of the system error throughout the previous 100 iterations (avg\\_sys\\_err) if this error is larger than the previous system average (prev sys avg) the network is retrained on a portion of the old data (Memo) used in previous training as well as the data from the previous 100 iterations (new data). The motive behind using the old data as well as the new data is to avoid catastrophic forgetting as mentioned in \\cite{Ref14}. \\\\\n The idea of using a memory is usually known as self-refreshing memory and was initially introduced and tested in \\cite{Ref15}\\cite{Ref16}\\cite{Ref17}. The idea here is that even when there are perturbations they might not last for so long and consequently we don't want the network to lose its ability to predict the System error if the system parameters return back to norm after the network was retrained several times.\n\\begin{algorithm}\n\\SetAlgoLined\nMemory = 10\\% of the old training data prev\\_sys\\_avg = system error average of the training data\n initialization\\;\n \\While{n $>$ 2000}{\n sys\\_err = current\\_system\\_error\\;\n\\If{sys\\_err  $>$ 0.8}\n\t           { \\While {True}\n\t                       {\n\t\t\t\t$s_1$ = randomly generated\\;\n\t                        $s_2$ = - $s_1$\/8\\;\n\t                        predicted\\_sys\\_err = predict (t, V, p, v, $s_1$, $s_2$)\\;\n\t\t\t\t\\If {predicted\\_sys\\_err  $<$  sys\\_err}\n\t                             {\n\t\t\t\t\tBreak\\;\n\t\t\t\t      \t\t}\n\t\t\t\t\t}\n\t\t\t\tupdate\\_controller ($s_1$, $s_2$)\\;\n\t                \tnew\\_data = Save (t, V, p, v, $s_1$, $s_2$, new\\_sys\\_err)\\;\n\t                \tavg\\_sys\\_err = avg\\_sys\\_err + new\\_sys\\_err\\;}\n\\If {n == 0}{\n\t              \\If { avg\\_sys\\_err \/ 100 $>$ prev\\_avg}\n\t                 { retrain\\_network(Memo + new\\_data)\\;\n\t                   Memo = Memo + 10\\% of the new\\_data\\;\n\t                   prev\\_sys\\_avg = avg\\_sys\\_err\\;\n\t                   Clear new\\_data\\;\n\t                   Clear avg\\_sys\\_err\\;}\n}\n}\n \\caption{Adaptive Algorithm Steps for Sigma prediction}\n\\end{algorithm}\n\\subsection{DATASET}\n\\label{sub1}\nThe Time, Error, velocity, position and control signal were collected every 1 millisecond and added into an array of values. The dataset was split into 60\\% training and 40\\% test. A continuous update to the dataset is done according to the calculated error slope every 10ms (100 iterations) as explained above.\n\\subsection{NETWORK ARCHITECTURE}\n\\label{sub2}\nThe network architecture is in Fig. \\ref{fig:3} The network is composed of 5 blocks (15 layers). The number of layers 15 was chosen for its efficiency and high performance. It was found that if a lower number of layers is used the DNN performance was affected and when a higher number of layers was used it had no effect on the performance but reduced the DNN efficiency.  A block is composed of 3 layers: a fully connected convolution layer followed by batch normalization layer and rectified linear unit (RELU) in Eqn. \\ref{eq:14}.\n\\begin{equation}\n\\label{eq:14} \noutput = max (0.0, input)\n\\end{equation} \nThe batch normalization layer allows us to use higher learning rates as it makes ensures that activations are not too high or low. The RELU is a nearly linear function in the sense that it is a piece wise linear function with two linear pieces. This feature increases its capability to preserve many of the properties that make linear models easy to optimize with gradient-based methods as well as the properties that make linear models generalize well. The last block has a dropout layer aside from the fully connected convolution layer and a regression layer to predict the error. Dropout \\cite{Ref18} is a regularization method that was developed to solve the problem of overfitting. The overfitting problem occurs when the neural network learns every minor detail in the training data. The effect of such a behaviour is that the network will have high accuracy on the training data but very low accuracy on the test data. This means that the network is unable to generalize on unforeseen data. The proposed solution for this problem is the dropout technique. The dropout technique ignores a number of layer outputs randomly. Such a behaviour makes the layer appear as and be treated-as a layer with a different number of nodes and connectivity to the prior layer. Consequently, each update to a layer during training is performed with a different \"perspective\" of the configured layer.\n The regression layer computes the mean squared error loss. The network is trained to learn a function F in Eqn. \\ref{eq:15}.\n\\begin{equation} \n\\label{eq:15}\nF : (t, x, v, s_1, s_2) \\rightarrow e\n\\end{equation} \n\nWhere \\textit{s\\textsubscript{1}} and \\textit{s\\textsubscript{2}} are the two Sigmas, \\textit{t} is the time and \\textit{e} is the error of the model.\n\\subsection{NETWORK TRAINING AND TESTING}\\label{sub3}\nThe network is initially trained for 5 epochs using Adam optimization algorithm. The learning rate is initially set to 0.001 and is reduced by a factor of 0.2 every 5 epochs. This allows large weight changes in the beginning of the learning process and small changes or fine-tuning towards the end of the learning process. This gives more time for fine-tuning. While the model is running if the system error passes a given threshold the Neural network is used to choose two Sigmas that would lower the error. This is done by selecting a uniformly distributed random number in the interval [-50, 50] for $s_1$. The restriction of $s_1$ and $s_2$ was added later in the research to avoid the genetic algorithm and DNN going into a continuous loop of changing the Sigmas to find the perfect candidate. It was also found that the smaller the sigma the better the system outcome but in order to maintain controller flexibility the range was set to [-50, 50] to account for unexpected changes in the system behaviour. \nThe network does not pass the proposed Sigma 1 and Sigma 2 to the model and controller unless Sigma 1 and Sigma 2 are predicted to reduce the system error. This helps in tuning the entire pipeline to take small\/large steps to change the behaviour of the duffing van der pol model given that sudden large changes in the Sigmas may result in system failure. \n The idea behind predicting the error and not the Sigmas is the fact that we can have more than one error for a given Sigma, which gives more flexibility for choosing the range of sigmas to consider.  The root of the mean square error graph for the first 5 epochs are plated in Fig. \\ref{fig:5}. The results show that the neural network presented in Fig.  \\ref{fig:6} was able to predict the error with an error of 0.03564.\n \\begin{center}\n \\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.7in]{4.png}\n\\caption{Neural network used for predicting the best Sigmas that would result in a decrease in the system error. The input to the network is the time (\\textit{t}), displacement (\\textit{x}), velocity (\\textit{v}), Sigma 1 (\\textit{$s_1$}), Sigma 2 (\\textit{$s_2$}) and the output is the expected system error (\\textit{Error}) for the system. $s_2$ = - $s_1$\/8}\n\\label{fig:3}      \n\\end{figure}\n \\end{center}\n\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{5.png}\n\\caption{Neural network used for predicting the best Sigmas that would result in a decrease in the system error. The input to the network is the time (\\textit{t}), displacement (\\textit{x}), velocity (\\textit{v}), Sigma 1 (\\textit{$s_1$}), Sigma2 (\\textit{$s_2$}) and the output is the expected system error (\\textit{Error}) for the system}\n\\label{fig:5}      \n\\end{figure}\n\n\\section{RESULTS AND DISCUSSION}\n\\label{Section V}\nIn Fig. \\ref{fig:5} the system is shown to go into instability and fails 0.3 seconds from start due to the use of unfit Sigma 1 = 100 and Sigma 2 = 0.6 while in Fig. \\ref{fig:6} we show that using the proposed deep neural network to predict the appropriate Sigma1 and Sigma 2 lead to stabilizing the system and maintaining the system error under the set threshold. The results show the efficiency of the neural network in predicting the error given the sigma, as well as the efficiency of the algorithm in preventing the system from failing likewise in continuously enhancing its performance by keeping the system error as low as possible. On the other hand, maintaining a constant sigma results in large system error and the system might eventually fails.\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{7.png}\n\\caption{The system error goes to infinity as shown when the algorithm is not applied}\n\\label{fig:6}      \n\\includegraphics[width=3in]{6.png}\n\\caption{The system error after using the neural network in Fig. 7 to get a good estimate of $s_1$ and $s_2$.}\n\\end{figure}\n\\subsection{Algorithm Reaction to parameter sabotage}\n\\label{sub4}\nTo elaborate on the ability of the proposed solution to tackle mid system instability or sudden changes. The system parameters are manually overwritten while the system is running to measure the NN and Algorithm effectiveness in returning the system to stability.\\\\\nAs demonstrated by Figs \\ref{fig:7} and \\ref{fig:8} the deep network was able to predict the best parameters to re-establish stability of the system and controller within 0.4 ms. \n\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{8.png}\n\\caption{Error uptick at t = 0.8867}\n\\label{fig:7}\n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{9.png}\n\\caption{The Algorithm reacting to the sudden change by adjusting $s_1$}\n\\label{fig:8}  \n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{10.png}\n\\caption{The Algorithm reacting to the sudden change by adjusting $s_2$}\n\\label{fig:9}  \n\\end{figure}\n\n\\subsection{Improving the system performance}\n\\label{sub5}\nIn this section we show the ability of the proposed method in finding the best system parameters while the system and controllers are running.\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{11.png}\n\\caption{Phase diagram of the algorithm before and after finding the optimal system parameters for the conditions which shows major improvement compared to Fig.\\ref{fig:5}}\n\\label{fig:10}  \n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{12.png}\n\\caption{$s_1$ and $s_2$ are changed while the system is running to improve performance and reduce error}\n\\label{fig:11}  \n\\end{figure}\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{13.png}\n\\caption{Error is shown to be reduced from the starting point by updating the system parameters and the algorithm is successful in maintain the error within bounds}\n\\label{fig:12}  \n\\end{figure}\n\\subsection{Network Retraining}\n\\label{sub5}\nDuring the re-training phase of the neural network we noticed that without the use of the memory it might take longer and, in some cases, go into an infinite state without finding the optimum solution. The speed with which the system returns back to it stable state affects the retraining phase as we only consider the change in the average of the system error. If the difference between the previous system error average and the new system error average is not large enough then the condition for retraining won't be met and the system will get stuck trying to find a pair of Sigmas that would result in error reduction. This won't be possible without memory because the moment the network is retrained the new data overwrites the information learned by the network. The use of memory allows the network to remember the stable and perturbed history of the system parameters.\n\\begin{figure}[!t] \\centering\n\\includegraphics[width=3in]{14.png}\n\\caption{Re-training to improve the deep neural network performance in order to adapt to the introduction of new system dynamics}\n\\label{fig:13}  \n\\end{figure}\n\\section{CONCLUSION AND FUTURE WORK}\n\\label{Section VII}\nLyapunov control was applied to Duffing- van der pol\noscillator model that is experiencing chaotic behaviour. The\nstudy shows the effectiveness of deep learning combination\nwith nonlinear Lyapunov control in finding the best\nparameters to maintain system stability. The study shows\nthat deep learning with the proposed algorithm enables the\nuser to effortlessly find the best parameters for the controller\nand the system initially and recalibrate the parameters if any\ndisturbances or new dynamics are introduced to the system.\nIn future work we would like to investigate the effect of\nchanging the control strategy according to the type of\ninstability detected. We speculate that depending on the type\nof perturbations and system dynamics, changing the control\nstrategy might be more effective than changing the\nparameters only. Future considerations include a network\nthat can determine which control strategy would have the\nhighest impact on returning the system to stability and cause\nerror reduction.\n\n\n \\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nParticle physics candidates for dark matter abound, the most popular being the lightest neutralino of supersymmetric models such as the Minimal Supersymmetic Standard Model (MSSM) and its extensions. On the other hand, non-supersymmetric models also have their own dark matter candidate, the most popular one being the axion. Introduced by Peccei and Quinn \\cite{Peccei:1977ur} long ago to solve the strong CP problem, the axion has required an extra global $U(1)$ symmetry, attached to the fields of the Standard Model (SM).\n\nIts inclusion in a supersymmetric lagrangean has also always required the introduction of an extra global symmetry, together with a supersymmetric partner (the axino). To bring both components (the axion and the neutralino) under the spell of the gauge principle requires, quite likely, a gauging of the axionic symmetries.  \n\nThe study of axionic symmetries and of their gauging is an important aspect of string theory at lower energy, being connected with the presence, in these effective models, of several moduli fields, deprived of a potential and derived from geometrical compactifications. The attempt to match these descriptions with effective models (with or without supersymmetry and gravity) based on ordinary quantum field theory and supersymmetry (in flat space) is pursued in several works \n\\cite{Coriano:2007fw,Coriano:2007xg} \\cite{Anastasopoulos:2008jt}, where \nthe role of the anomalous interactions and of light pseudoscalars is investigated under the tenets of gauge invariance and unitarity \\cite{Armillis:2008bg}. \nIn general, these effective field theories are characterized by the presence of higher dimensional operators to correct for the exposed (axial) anomaly (in the form of Wess-Zumino terms with St\\\"uckelberg axions) \\cite{Coriano:2007fw}. Their gauge structure, in fact, requires at least one extra anomalous $U(1)$ interaction.\n  \n   The presence of gauge couplings and of a mass for this axion not related by the same suppression scale ($f_a$) - which is of the order of $10^{10}$ GeV for a traditional (ungauged) axion - makes this new axion a very attractive dark matter candidate. However, building axion models with gauged axionic symmetries is rather challenging from the field theory point of view since it requires a grasp of the unusual features of the chiral anomaly, from the organization of the effective action(s) to the presence of anomaly poles that challenge the consistency of the S-matrix  in some of their scattering amplitudes \\cite{Coriano:2008pg}.\n\\section{Supersymmetric extensions}\nOne of the realizations of field theories which contain axion-like particles in their spectrum is the Minimal Low-Scale Orientifold Model (MLSOM) \\cite{Coriano':2005js}, based on a construction involving charge assignments obtained from intersecting branes. These models introduce one St\\\"uckelberg axion for each anomalous $U(1)$ present in the gauge structure. In \\cite{Coriano':2005js} it is shown that the physical spectrum of the MLSOM contains always one physical axion, independently of the number of anomalous U(1)'s. \n\nSupersymmetric extensions of these class of models have been discussed \nrather recently  \\cite{Coriano:2008xa}, using a superpotential which is the one typical of the USSM \\cite{Cvetic:1997ky}, extending previous studies Respect to the MSSM, the USSM contains an extra singlet superfield and an extra U(1) gauge symmetry, which is anomaly free. \nIn the USSM-A model of  \\cite{Coriano:2008xa}, which is the theory proposed as a possible supersymmetric extension of the MLSOM, this symmetry has been left anomalous, and the bosonic mechanism of cancellation of these anomalies, which is \nenforced via the inclusion of Wess-Zumino (counter)terms,  has been generalized. These models contain a combined Higgs-St\\\"uckelberg mechanism for the generation of the mass of the anomalous gauge bosons. \n\nA physical axion state (the axi-Higgs) appears quite naturally, together with its supersymmetric partner, the axino. \nThis second state becomes a component of the fermionic neutralinos, after diagonalization of the mass matrix of the neutral fermion sector. \n\nCompared to other constructions, in which the axion disappeares from the physical spectrum being just a goldstone mode, in the USSM-A the presence of the extra singlet superfield allows a physical projection of the St\\\"uckelberg axion of the model on the axi-Higgs ($\\chi$). Therefore this physical state inherits direct axion-like couplings (such as $\\chi F\\tilde{F}$) to the gauge fields, becoming a gauged supersymmetric axion \\cite{Coriano:2008aw}. The particle is massless in the absence of Peccei-Quinn-breaking potentials, in which the axion appears as a phase, while the instanton vacuum can naturally lift its mass up to \n$10^{-4}$ eV as in the Peccei Quinn (invisible axion) case.\n\nIn the non-supersymmetric version of these theories, i.e. in the MLSOM, one of the special features of these theories is the presence of anomalous trilinear gauge interactions, which are totally absent in the Standard Model (SM) due to anomaly cancellation \\cite{Armillis:2007tb}. We just recall that in the SM these interactions are suppressed and appear only away from the chiral limit in $AVV$ diagrams  (for instance in $Z \\gamma \\gamma$ vertices). We show in Fig. 1 results for the production of two axions $(\\chi)$ in the MLSOM at the Tevatron (left panel) and at the LHC (right panel), as a function of the invariant mass $Q$ of the final state, obtained from a recent analysis. The gluon fusion chnnel is the most important one for their study. The processes involve the usual triangle diagrams as for the production of the Higgs sector. In our case we have considered two Higgs doublets $(h_0, H_0)$ in the CP-even sector. The s-channel exchange involves either an $h_0$, a $H_0$ or a $\\chi$. \nWe have selected a mass of the $h_0=15$ GeV and of $120$ GeV for the $H_0$, with $m_\\chi=5$ GeV.\n\n \\section{Conclusions}\n The search for light (gauged) pseudoscalars is for sure a challenge for the experimental acvitity at the LHC \\cite{Armillis:2008vp}, requiring precise evaluation of the QCD background. However, while the detection of anomalous extra Z prime which modify the neutral current sector of these theories remains difficult, the identification of a light pseudoscalar (in the mass range of 1 to 10 GeV), which is their second signature, is more favoured.  \n\\begin{figure}[t]\n\\includegraphics[width=5cm, angle=-90]{multiax_Teva_2par.ps}\n\\includegraphics[width=5cm, angle=-90]{multiax_LHC_2par.ps}\n\\caption{\\small $gg\\rightarrow$ 2-scalar processes mediated by a three-linear vertex at the Tevatron and at the LHC.}\n\\label{2scalar_prod_LHC}\n\\end{figure}\n\\newpage\n\\centerline {\\bf Acknowledgments}\nWe thank Roberta Armillis and Luigi Delle Rose for discussions.\nThis work was supported (in part) by the European Union through the Marie Curie Research and Training Network ``Universenet'' (MRTN-CT-2006-035863). \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\begin{quote}\n{\\em One ought to recognize that the present political chaos is connected\nwith the decay of language, and that one can probably bring about some\nimprovement by starting at the verbal end.}  --- Orwell,\n``Politics and the English language''\n\\end{quote}\n\n\\medskip\n\n\n\n\nWe have entered an era where very large amounts of politically\noriented text are now available online.\nThis\nincludes both official documents, such as the full text of \nlaws and the proceedings of legislative bodies, and unofficial\ndocuments, such as postings on weblogs (blogs) devoted to politics.\nIn some sense, \nthe availability of such data is simply a manifestation of \na general trend of ``everybody putting their records on the\nInternet''.\\footnote{It is worth pointing out that the United States'\nLibrary of Congress was an extremely early adopter of Web technology:\nthe THOMAS database (http:\/\/thomas.loc.gov) of congressional bills and related data was\nlaunched in January 1995, when Mosaic was not quite two years old and\nAltavista did not yet exist.}\nThe online accessibility of \npolitically oriented texts in particular, however,\nis a phenomenon that some have gone so far as to say will have a potentially\nsociety-changing effect.\n\nIn the \nUnited States, \nfor example,\ngovernmental bodies are providing and soliciting political\ndocuments via the Internet,\nwith lofty goals in mind: \n{\\em electronic rulemaking} (eRulemaking)\ninitiatives\ninvolving the\n``electronic collection,\ndistribution, synthesis, and analysis of public commentary in the\nregulatory rulemaking process'', may ``[alter] the citizen-government\nrelationship'' \\cite{Shulman+Schlosberg:02a}.  Additionally, much media attention\nhas been focused  recently on the potential impact that\nInternet sites \nmay have on \npolitics\\footnote{E.g., ``Internet injects sweeping change\ninto U.S. politics'', Adam Nagourney, {\\em The New York Times}, April\n2, 2006.}, or at least on political journalism\\footnote{E.g., ``The\nEnd of News?'', Michael Massing, {\\em The New York Review of Books}, \nDecember 1, 2005.}. \nRegardless of whether one views such claims as clear-sighted prophecy\nor mere hype,\nit is obviously important to help people understand and analyze\npolitically oriented text,\ngiven the importance of enabling\ninformed participation in the political process.\n\nEvaluative and persuasive documents,\nsuch as a politician's speech regarding a bill or \na blogger's commentary on a\nlegislative proposal, form a particularly interesting type of\npolitically oriented text.  \nPeople are much more likely to consult such \nevaluative statements \nthan the actual text of a bill or law\nunder discussion, \ngiven the dense nature of legislative language and the fact that (U.S.) bills\noften reach several hundred pages in length\n\\cite{Smith+Robert+VanderWielen:05a}. Moreover, \npolitical opinions are explicitly solicited in the eRulemaking scenario.\n\nIn the analysis of \nevaluative language, it is fundamentally\nnecessary to determine whether the author\/speaker supports or\ndisapproves of the topic of discussion.  \nIn this paper, we investigate\nthe following specific instantiation of this problem: \nwe seek to determine\nfrom the transcripts of U.S. Congressional floor debates whether each\n``speech'' (continuous single-speaker segment of text) represents\nsupport for or opposition to a proposed piece of legislation.  Note\nthat from an experimental point of view, this is a very convenient\nproblem to work with because we can automatically determine ground\ntruth (and thus avoid the need for manual annotation) simply by\nconsulting publicly available \nvoting records.\n\n\\paragraph{\n\\mbox{Task properties}}\n\nDetermining whether or not a speaker \nsupports\na \nproposal falls within the realm of {\\em\nsentiment analysis}, an extremely active research area devoted to the\ncomputational treatment of subjective or opinion-oriented language\n(early work includes Wiebe and Rapaport \\shortcite{Wiebe+Rapaport:88a},\nHearst \\shortcite{Hearst:92a}, Sack \\shortcite{Sack:94a}, and \\newcite{Wiebe:94a};\nsee Esuli \\shortcite{Esuli:sentbib} for an active bibliography).  In\nparticular, since we treat \neach individual speech\nwithin a \ndebate as a single ``document'', \nwe are considering \na version of\n{\\em document-level sentiment-polarity classification}, \nnamely,  automatically distinguishing between \npositive and negative\ndocuments\n\\cite{Das+Chen:01a,Pang+Lee+Vaithyanathan:02a,Turney:02a,Dave+Lawrence+Pennock:03a}.\n\n\n\n\n\n\n\nMost sentiment-polarity classifiers proposed in the recent literature  categorize\neach document independently.  A few others  incorporate\nvarious measures of inter-document \nsimilarity between the texts to be labeled\n\\cite{Agarwal+Bhattacharyya:05a,Pang+Lee:05a,Goldberg+Zhu:06a}.  \nMany interesting \nopinion-oriented documents, however, can be linked through certain relationships \nthat occur in the context of evaluative {\\em discussions}.\nFor example, we may find textual\\footnote{\nBecause we are most interested in techniques applicable across\ndomains, we\nrestrict consideration \nto \nNLP aspects of the problem, \nignoring external problem-specific information.\nFor\nexample, although most votes \nin our corpus were almost completely along party lines (and despite\nthe fact that same-party information is easily incorporated via\nthe methods we propose), \nwe did not use party-affiliation data.\nIndeed, \nin\nother settings (e.g., a movie-discussion listserv) one \nmay not be able to determine the participants' political\nleanings,  \nand such information may not lead to significantly improved results\neven if it were available.\n}\nevidence of a high likelihood of {\\em agreement} between two speakers, \n such as explicit assertions (``I second\nthat!'') or quotation of messages in emails or postings (see\n\\newcite{Mullen+Malouf:06a} but cf. \\newcite{Agrawal+al:03a}).\nAgreement evidence can be a powerful aid in our classification\ntask: for example,  we can\neasily categorize a complicated (or overly terse) document if we find\nwithin it indications of agreement with a clearly positive text.\n\n\n\n\n\n\n\n\n\n\nObviously, incorporating agreement information provides additional\nbenefit only when the \ninput\ndocuments \nare relatively difficult to classify\nindividually.  Intuition suggests that this is true of the\ndata with which we experiment, for several reasons.  \nFirst, U.S. congressional debates contain\nvery rich language and cover an extremely wide variety of topics, \nranging from flag burning to international policy to the federal budget. \nDebates are also subject to digressions, some fairly natural \nand others less so (e.g., ``Why\nare we discussing this bill when the plight of my constituents\nregarding this other issue is being ignored?'')\n\nSecond, an important characteristic of persuasive language is that \nspeakers may spend more time presenting evidence in support of their positions\n(or attacking the evidence presented by others) \n than directly stating their attitudes.  An extreme example will\nillustrate the problems involved. Consider a speech that \ndescribes the U.S. flag as \ndeeply inspirational, \nand thus\ncontains only positive language.  \nIf the bill under discussion is\na proposed flag-burning ban, then the speech is {\\em supportive}; but\nif the bill under discussion is aimed at rescinding an existing flag-burning\nban, the speech may represent {\\em opposition} to the legislation.  \nGiven the \ncurrent state of the art in sentiment analysis, it is doubtful that one \ncould determine the (probably topic-specific) relationship between \npresented evidence and speaker opinion.\n\n\n\n\n\n\n\n\n\\begin{table*}[th]\n\\begin{center}\n\\begin{tabular}{|l||r|rrl|} \\hline\n                      &total &train &test &development \\\\ \\hline\nspeech segments\\xspace              &3857  &2740     &860  &257 \\\\\n{\\debate}s\\xspace              &53    &38       &10   &5 \\\\\naverage number of speech segments\\xspace per debate\\xspace  &72.8  &72.1  &86.0  &51.4  \\\\\naverage number of speakers per debate\\xspace  &32.1  &30.9    &41.1   &22.6   \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:data}\nCorpus statistics. }\n\\end{center}\n\\end{table*}\n\n\n\n\\paragraph{Qualitative summary of results}\n\n\nThe above difficulties underscore the importance of enhancing standard\nclassification techniques with new information sources that \npromise to improve accuracy, \nsuch as inter-document relationships between the documents to be\nlabeled.\nIn this paper, we demonstrate that the incorporation of agreement modeling \ncan provide substantial improvements over \nthe application of\nsupport vector machines\n(SVMs) in isolation, which \nrepresents the state of the art in the individual classification \nof documents.  \nThe enhanced accuracies are obtained\nvia a fairly primitive automatically-acquired \n``agreement detector'' and a\nconceptually simple method for integrating isolated-document and\nagreement-based information.  We thus view our results as\ndemonstrating the potentially large benefits of exploiting \nsentiment-related discourse-segment relationships in\nsentiment-analysis tasks.\n\n\n\n\n\n\n\n\n\\section{Corpus}\n\\label{sec:data}\n\n\nThis section outlines the main steps of the process by which we\ncreated \nour corpus (download site:\nwww.cs.cornell.edu\/home\/llee\/data\/convote.html).\n\n\\medskip\n\nGovTrack \n(\\mbox{http:\/\/govtrack.us})\nis an independent website \nrun by Joshua Tauberer\nthat\ncollects publicly available\ndata on \nthe legislative and fund-raising activities of U.S. congresspeople.  Due to its\nextensive cross-referencing and collating of information, it \nwas nominated for a 2006 ``Webby'' award.\nA crucial characteristic of GovTrack from our point of\nview is that the information is provided in a very convenient format;\nfor instance, the floor-debate transcripts are broken into\nseparate HTML files according to the subject of the debate, so we\ncan trivially derive long sequences of speeches guaranteed to cover the\nsame topic.\n\n\n\nWe extracted from GovTrack all available transcripts of U.S.  floor\ndebates in the House of Representatives for the year 2005 (3268 pages\nof transcripts in total), together with voting records for all\nroll-call\nvotes during that year.\nWe concentrated on {\\debate}s\\xspace regarding ``controversial'' bills (ones in which the\nlosing side generated at least 20\\% of the speeches) because\nthese debates should presumably exhibit more interesting\ndiscourse structure.\n\nEach \ndebate consists of a series of {\\em speech segments\\xspace}, where each \nsegment is a\nsequence of uninterrupted utterances by a single speaker.\nSince \nspeech segments\\xspace represent\nnatural  discourse units, we \ntreat them as the basic \nunit to be classified.\nEach speech segment\\xspace was labeled by the vote (``yea''\\xspace or ``nay''\\xspace) cast for the proposed bill\nby the person who uttered the speech segment\\xspace.\n\nWe automatically discarded those speech segments\\xspace belonging to a class of \nformulaic, generally one-sentence utterances \nfocused \non the yielding of time on the house floor (for example, \n``Madam Speaker, I am pleased to yield 5 minutes to the gentleman from Massachusetts''), as such speech segments\\xspace are\nclearly off-topic.\nWe also removed speech segments\\xspace \ncontaining the term ``amendment'', since we found \nduring initial inspection that\n these speeches \ngenerally reflect a speaker's opinion on an amendment, and this\nopinion may \ndiffer from the speaker's opinion \non the underlying bill under discussion.\n\nWe randomly split the data into training, test, and development\n(parameter-tuning) sets \nrepresenting\nroughly 70\\%, 20\\%, and 10\\% of our data, respectively (see Table \\ref{tab:data}).  \nThe speech segments\\xspace remained grouped by debate\\xspace, with 38 \n{\\debate}s\\xspace\nassigned to the training set, \n10 to the test set, and 5 to the development set; \nwe \nrequire that the speech segments\\xspace from an individual debate\\xspace\nall appear in the same \nset because our goal is to\nexamine classification of speech segments\\xspace in the context of the\nsurrounding discussion.\n\\section{Method}\n\\label{sec:method}\n\nThe support\/oppose \nclassification \nproblem can be approached through \nthe use of standard classifiers such as support vector machines (SVMs), \nwhich consider each text unit in isolation.\nAs discussed in Section \\ref{sec:intro}, however, the conversational\nnature of our data implies the existence of various relationships that\ncan be exploited \nto improve cumulative classification accuracy for \nspeech segments\\xspace belonging to the same debate\\xspace.\nOur classification framework,\ndirectly inspired by \\newcite{Blum+Chawla:01a},\n integrates both perspectives, optimizing its\nlabeling of speech segments\\xspace based on \nboth individual \nspeech-segment\\xspace classification scores and preferences for groups of speech segments\\xspace to \nreceive the same label.\nIn this section, we discuss \nthe specific classification framework that we adopt\nand the set of mechanisms that we propose for  \nmodeling specific types of relationships.\n \n\n\\subsection{Classification framework}\n\\label{sec:method:graph}\n\n\nLet $s_1,s_2,\\ldots, s_{n}$ be the sequence\nof speech segments\\xspace within a given debate, and let ${\\cal Y}\\xspace$ and ${\\cal N}\\xspace$ stand\nfor the ``yea'' and ``nay'' class, respectively. Assume we have a\nnon-negative function $\\mbox{{\\it ind}}\\xspace(s,C)$ indicating the\ndegree of preference that an individual-document classifier, such as an SVM,\nhas for placing speech-segment\\xspace $s$ in class $C$.  Also, assume that \nsome pairs of speech segments\\xspace have {\\em\nweighted links} between them, where the non-negative {\\em strength} (weight)\n${\\mathit str}(\\ell)$ for a link $\\ell$ indicates the degree to which it\nis preferable that the linked speech segments\\xspace receive the same label.\nThen, any class assignment $c = c(s_1),\nc(s_2),\\ldots, c(s_{n})$\ncan be assigned a {\\em cost}\n$$\n\\sum_{s}\n\\mbox{{\\it ind}}\\xspace(s,\\overline{c}(s)) +\n\\sum_{s,s':\\: c(s) \\neq c(s')}\n\\,\n  \\sum_{\\ell\\; {\\rm between}\\,  {s, s'} }{\\mathit str}(\\ell),\n$$\nwhere $\\overline{c}(s)$ is the ``opposite'' class from ${c}(s)$.\nA {\\em minimum-cost} assignment thus represents an optimum way to classify the\nspeech segments\\xspace \nso that \neach one tends not to be put into the class that the\nindividual-document classifier disprefers, but at the same time, highly associated\nspeech segments\\xspace tend not to be put in different classes.\n\nAs has been previously observed and exploited in the NLP literature\n\\cite{Pang+Lee:04a,Agarwal+Bhattacharyya:05a,Barzilay+Lapata:05a},\nthe above optimization function, unlike many others that have been proposed\nfor \ngraph or set partitioning, can be solved {\\em exactly} in\nan \nprovably\nefficient manner via methods for finding minimum cuts in\ngraphs.  In our view, the contribution of our work is the examination\nof new types of relationships, not the method by which such\nrelationships are incorporated into the classification decision.\n\n\n\n\\subsection{Classifying speech segments\\xspace in isolation}\n\\newcommand{${\\rm  SVM}^{light}$\\xspace}{${\\rm  SVM}^{light}$\\xspace}\nIn our experiments, we \nemployed\nthe well-known \nclassifier \n${\\rm  SVM}^{light}$\\xspace\n to obtain individual-document \nclassification scores, treating ${\\cal Y}\\xspace$ as the positive \nclass and using plain unigrams as features.\\footnote{${\\rm  SVM}^{light}$\\xspace is available at\nsvmlight.joachims.org.  \nDefault parameters were\nused,\nalthough experimentation with different parameter settings is an\nimportant direction for future work\n\\cite{Daelemans+Hoste:02a,Munson+Cardie+Caruana:05a}.}\nFollowing standard practice in sentiment analysis \\cite{Pang+Lee+Vaithyanathan:02a}, the input to ${\\rm  SVM}^{light}$\\xspace consisted \nof normalized presence-of-feature (rather\n  than frequency-of-feature) vectors.  The $\\mbox{{\\it ind}}\\xspace$ value for each speech segment\\xspace\n $s$ \nwas based on\nthe signed distance $\\refconfvar(\\spvar)\\xspace$ from the vector representing\n $s$ to the trained SVM decision \nplane:\n$$\n\\mbox{{\\it ind}}\\xspace(s,{\\cal Y}\\xspace) \\stackrel{{\\rm def}}{=} \n\\begin{cases}\n1 & \\refconfvar(\\spvar)\\xspace > 2\\sigma_{s}; \\cr\n\\left(1 + \\frac{\\refconfvar(\\spvar)\\xspace}{2\\sigma_{s}}\\right)\/2 & \n\\vert \\refconfvar(\\spvar)\\xspace \\vert \\leq 2\\sigma_{s};\n\\cr\n0 & \\refconfvar(\\spvar)\\xspace < -2\\sigma_{s}\n\\end{cases}\n$$\nwhere $\\sigma_{s}$ is the standard deviation of  $d(s)$ over all\nspeech segments\\xspace $s$ in the\ndebate in question,\nand $\\mbox{{\\it ind}}\\xspace(s,{\\cal N}\\xspace) \\stackrel{{\\rm def}}{=} 1 - \\mbox{{\\it ind}}\\xspace(s,{\\cal Y}\\xspace)$.\n\n\nWe now turn to the more interesting problem of representing the\npreferences that\nspeech segments\\xspace may have for being assigned to the same class.\n\n\n\\subsection{Relationships between speech segments\\xspace}\n\\label{sec:method:relationship}\n\nA wide range of relationships between text segments \ncan be modeled as positive-strength links.\nHere we discuss two types of constraints that are considered in this work. \n\n\n\n\n\n\n\\paragraph{Same-speaker constraints:} \nIn Congressional debates and in general \nsocial-discourse contexts, \na single speaker may make \na number of comments regarding \na\ntopic.  \nIt is reasonable to expect that in many settings, the participants in\na discussion may be convinced\nto change their opinions midway through a debate.\nHence,\nin the general case we wish to be able \nto express\n``soft'' preferences for all of an author's statements\nto receive the same label, \nwhere\nthe strengths of such constraints could, \nfor instance, vary according to the time elapsed between the statements.\nWeighted links are an appropriate means to express such variation.\n\n\n\nHowever, if\nwe assume that most speakers do not change their positions in the \ncourse of a discussion, we can conclude that all \ncomments made by the same speaker\nmust\nreceive the same label.\nThis assumption\nholds by fiat for the ground-truth labels in our dataset because these\nlabels were derived from the single vote cast by the speaker on the\nbill being discussed.\\footnote{\nWe are attempting to\ndetermine whether a speech segment\\xspace represents support or not.  This differs\nfrom the problem of determining what the speaker's actual opinion is,\na problem that, as an anonymous reviewer \nput it, is complicated\nby ``grandstanding, backroom deals, or, more innocently, plain change\nof mind \n(`I voted for it before I voted against it')''.}\nWe can implement this assumption\nvia links whose weights are essentially infinite.  \nAlthough\none can also implement this assumption via concatenation of\nsame-speaker speech segments\\xspace (see\nSection \\ref{sec:eval:global}), we view the fact that our graph-based\nframework  incorporates both hard and soft constraints in a principled\nfashion as an\nadvantage of our approach.\n\n\n\\paragraph{Different-speaker agreements} In House discourse, it is common for one \nspeaker to make reference to another\nin the context of an\nagreement or disagreement over the topic of discussion.\nThe systematic identification of instances of \nagreement \ncan, as we have discussed, \nbe a powerful tool \nfor the development of intelligently\nselected weights\nfor links between speech segments\\xspace.  \n\n\n\nThe problem of agreement identification \ncan be decomposed into two\nsub-problems: identifying references and their \ntargets, and deciding whether\neach reference represents an instance of agreement.\nIn our case, the first task is straightforward\nbecause we focused solely on by-name references.\\footnote {One subtlety\n  is that for the purposes of mining agreement cues \n(but {\\em not} for\n  evaluating overall support\/oppose classification accuracy), we \ntemporarily\nre-inserted into\nour dataset previously filtered speech segments\\xspace containing the term\n``yield'', since the yielding of time on the House floor typically\nindicates agreement even though the yield statements contain little\nrelevant text on their own.} \nHence, we will now concentrate on the second, more interesting task.\n\nWe approach the problem of classifying references by \nrepresenting each reference with a word-presence vector derived from a \nwindow of text surrounding \nthe reference.\\footnote{\nWe found good \ndevelopment-set performance using\nthe 30 tokens before,\n20 tokens after,\nand the name itself.}  \nIn the training set, we classify each reference connecting two speakers \nwith a positive or negative label \ndepending on whether the two\nvoted the same way on the bill under discussion\\footnote{Since we are \nconcerned with references that potentially represent relationships \nbetween speech segments\\xspace, we ignore\nreferences for which the target of the reference did not speak in the\ndebate\\xspace in which the reference was made.}.  These labels are then used to\ntrain an SVM classifier, the output of which is subsequently used to\ncreate weights\non {\\em agreement links} in the test set as follows.\n\n Let $\\agrconfvar(\\refvar)$ denote the distance from the vector representing\n reference $r$ to the agreement-detector SVM's decision plane,\n and let\n$\\sigma_\\refvar$ be the standard deviation of  $d(r)$ over all\n references in the debate\\xspace in question.\nWe then define the strength $\\mathit{agr}\\xspace$ of the {\\em agreement link} corresponding\nto the reference\nas:\n$$\n\\mathit{agr}\\xspace(r) \\stackrel{{\\rm def}}{=} \\begin{cases}\n0           & \\agrconfvar(\\refvar) < \\theta_{\\mbox{agr}}\\xspace; \\cr\n\\alpha\\xspace \\cdot \\agrconfvar(\\refvar)\/ 4\\sigma_\\refvar & \\theta_{\\mbox{agr}}\\xspace \\le \\agrconfvar(\\refvar) \\le 4\\sigma_\\refvar; \\cr\n\\alpha\\xspace & \\agrconfvar(\\refvar) > 4\\sigma_\\refvar.\n\\end{cases}\n$$\nThe free parameter $\\alpha\\xspace$ specifies the relative importance of the\n$\\mathit{agr}\\xspace$ scores.\nThe threshold $\\theta_{\\mbox{agr}}\\xspace$ controls \nthe precision of\nthe agreement\nlinks,\nin that values of  $\\theta_{\\mbox{agr}}\\xspace$ greater than zero mean that greater\nconfidence is required before an agreement link can be added.\n\\footnote{Our implementation\nputs a link between\njust one arbitrary pair of speech segments\\xspace among all those uttered by a given\npair of\napparently agreeing speakers.  The ``infinite-weight'' same-speaker\nlinks propagate the agreement information to all other such pairs.\n} \n\n\n\n\n\n\n\n\n\n\\section{Evaluation}\n\\label{sec:eval}\n\nThis section presents experiments testing the utility  of \nusing\nspeech-segment\\xspace relationships,\nevaluating against a number of baselines.\nAll reported results use values for the\nfree parameter $\\alpha\\xspace$ \nderived via tuning on the development set. \nIn the tables, {\\bf boldface} indicates the development- and\ntest-set results for the {\\em development-set-optimal} parameter\nsettings, as one would make algorithmic choices based on\ndevelopment-set performance.\n\n\n\\subsection{Preliminaries: Reference classification}\n\\label{sec:eval:agr}\n\n\nRecall that to gather inter-speaker agreement information, the\nstrategy employed in this paper is to classify \nby-name \nreferences to other speakers as to whether they\nindicate agreement or not.\n\n\n\n\n\\begin{table}[t]\n\\begin{tabular}{|l|cc|}\n \\hline\n\\parbox[b]{4.5cm}{\\mbox{Agreement classifier} \\\\ \\hspace*{.2cm}(``{reference}$\\Rightarrow$agreement?'')}      &\n\\parbox[b]{1cm}{Devel. set}  &\\parbox[b]{1cm}{Test set} \\\\\n\\hline\\hline\n\nmajority baseline\\xspace & 81.51 & 80.26 \\\\\n\\mbox{Train: no \\amendabbrev}; \\useposref & 84.25 & 81.07 \\\\\n\\mbox{Train: with \\amendabbrev}; \\useposref & \\bestdev{86.99} & \\testbestdev{80.10} \\\\\n\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:agr} \nAgreement-classifier accuracy, in percent.  ``Amdmts''=``speech segments\\xspace containing the\nword {`amendment'}''.\nRecall that boldface indicates results for\n  development-set-optimal settings.\n}\n\\end{table}\n\n\nTo train \nour agreement classifier,\nwe\nexperimented with undoing the deletion of amendment-related speech segments\\xspace\nin the training set.  Note that such speech segments\\xspace were {\\em never}\nincluded in the development or test set, since, as discussed in \nSection\n \\ref{sec:data}, their labels are probably noisy; however, including them in\nthe {\\em training} set allows the classifier to examine more instances even\nthough some of them are labeled incorrectly.  As Table\n\\ref{tab:agr} shows, using more, if noisy, data yields better\nagreement-classification results on the {development} set, and so we use\nthat policy in all subsequent experiments.\n\\footnote{\nUnfortunately, this policy leads to inferior {\\em test-set}\nagreement classification.  Section \\ref{sec:eval:devtest}\ncontains further discussion.\n}\n\n\n\nAn important observation is that precision may be more important than\naccuracy in deciding which agreement links to add: false positives\nwith respect to \nagreement\ncan cause speech segments\\xspace to be\nincorrectly assigned the same label, whereas false negatives mean only\nthat agreement-based information about other speech segments\\xspace is not employed.  As described above, we\ncan raise agreement precision by \nincreasing the threshold $\\theta_{\\mbox{agr}}\\xspace$,\nwhich specifies \nthe required confidence for the addition of an agreement link.\nIndeed, Table \\ref{tab:agr-highprec} shows that we can \nimprove \nagreement precision\nby \nsetting  $\\theta_{\\mbox{agr}}\\xspace$  to the (positive) mean agreement score $\\mu\\xspace$ assigned by the SVM\nagreement-classifier over all references in the given\n{debate\\xspace}\\footnote{We elected not to explicitly tune the value of\n$\\theta_{\\mbox{agr}}\\xspace$ in order to minimize the number of free parameters to deal\nwith.}.  However, this comes at the cost of\ngreatly reducing agreement accuracy (development: 64.38\\%; test:\n66.18\\%) due to \nlowered recall levels.  Whether or not better\nspeech-segment\\xspace classification \nis ultimately achieved \nis discussed\nin the next sections.\n\n\n\n\n\\begin{table}[t]\n\\begin{tabular}{|l|cc|}\n \\hline\n{{Agreement classifier}}      &\n\\multicolumn{2}{|c|}{Precision (in percent):} \\\\\n& \n{Devel. set}  &\n{Test set} \\\\\n\\hline\\hline\n\n$\\refthres = 0$ & 86.23 & 82.55 \\\\\n$\\refthres = \\refmean$ & \\bestdev{89.41} & \\testbestdev{88.47} \\\\\n\\hline\n\n\\end{tabular}\n\\caption{\\label{tab:agr-highprec} \nAgreement-classifier precision.\n}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{\\Localsvmsetting}\n\\label{sec:eval:local}\n\n\n\n\\paragraph{Baselines} The first two data rows of Table\n\\ref{tab:results-local} depict baseline performance results.  \nThe \n$\\#(\\mbox{``support''}) - \\#(\\mbox{``oppos''})$\\xspace}\\newcommand{\\ressvmbl}{SVM\\xspace\n baseline is meant to explore whether the\n speech-segment\\xspace classification task can be reduced to simple\nlexical checks.  Specifically, this method uses the signed difference\nbetween the number of words containing the stem ``support'' and the\nnumber of words containing the stem ``oppos'' (returning the\nmajority class if the difference is 0).  No better than 62.67\\%\ntest-set accuracy is obtained by either baseline.\n\n\n\\providecommand{\\per}{}\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|cc|} \\hline\n\\parbox[b]{4.3cm}{Support\/oppose classifer \\\\ \\hspace*{.2cm}(``{speech segment\\xspace}$\\Rightarrow$yea?'')}      & \\parbox[b]{1cm}{Devel. set}  &\\parbox[b]{1cm}{Test set} \\\\ \\hline\\hline\nmajority baseline\\xspace  &54.09\\per        &58.37\\per \\\\\n$\\#(\\mbox{``support''}) - \\#(\\mbox{``oppos''})$\\xspace}\\newcommand{\\ressvmbl}{SVM\\xspace      &59.14\\per        &62.67\\per \\\\ \\hline\n\\ressvmbl [{speech segment\\xspace}]      &70.04\\per        &66.05\\per \\\\\nSVM + same-speaker links\\xspace &79.77\\per        &67.21\\per \\\\ \nSVM + same-speaker links\\xspace $\\ldots$  & & \\\\\n{~~~} \\spaceplus \\refs,  \\useposref \\xspace        &\\bestdev{89.11\\per}        &\\testbestdev{70.81\\per} \\\\ \n{~~~} \\spaceplus \\refs, \\useavgref \\xspace     &87.94\\per        &71.16\\per \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{tab:results-local}\n\\Localsvmsetting accuracy, in percent.}\n\\end{center}\n\\end{table}\n\n\n\n\\providecommand{\\per}{}\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|l|cc|} \\hline\n\\parbox[b]{4.3cm}{Support\/oppose classifer \\\\ \\hspace*{.2cm}(``{speech segment\\xspace}$\\Rightarrow$yea?'')}      & \\parbox[b]{1cm}{Devel. set}  &\\parbox[b]{1cm}{Test set} \\\\ \\hline\\hline\n\\ressvmbl [speaker]      &71.60\\per        &70.00\\per \\\\\nSVM + agreement links\\xspace $\\ldots$ & & \\\\\n{~~~} with $\\refthres = 0$\n     &\\bestdev{88.72}\\per        &\\testbestdev{71.28}\\per \\\\ \n{~~~} with $\\refthres = \\refmean$\n &84.44\\per        &76.05\\per \\\\ \n\\hline\n\\end{tabular}\n\\caption{\\label{tab:results-global}\n\\Globalclass \\spunith classification\\xspace accuracy,\nin percent.  \nHere,\nthe\ninitial SVM is run on the concatenation of all of a given speaker's\nspeech segments\\xspace,\nbut \nthe results \nare computed over speech segments\\xspace\n(not speakers),\nso that \nthey can be\ncompared to those in Table \\ref{tab:results-local}.\n}\n\\end{center}\n\\end{table}\n\n\n\\paragraph{Using relationship information}\nApplying an SVM to classify each speech segment\\xspace in isolation leads to clear\nimprovements over the two baseline methods, as demonstrated in \nTable \\ref{tab:results-local}.\nWhen we impose the constraint that all speech segments\\xspace uttered by the same speaker\nreceive the same label\nvia ``same-speaker links'', \nboth test-set\nand development-set accuracy increase even more, in the latter case\nquite substantially so.\n\n\n\nThe last two lines of Table \\ref{tab:results-local} show that the best\nresults are obtained by incorporating agreement information as well.\nThe highest test-set result, 71.16\\%, is obtained by using a high-precision\nthreshold to determine which agreement links to add.  While the\ndevelopment-set results would induce us to utilize the standard threshold\nvalue of 0, which is sub-optimal on the test set, the\n$\\refthres = 0$\\xspace \nagreement-link\npolicy still achieves noticeable improvement \nover  not using  agreement links\n(test set: 70.81\\% vs. 67.21\\%).\n\n\n\n\n\n\\subsection{\\Globalclass \\spunith classification\\xspace}\n\\label{sec:eval:global}\n\nWe \nuse speech segments\\xspace as the  unit of classification \nbecause\nthey represent natural discourse \nunits.\nAs a  consequence, \nwe \nare able to\nexploit \nrelationships \nat the speech-segment\\xspace\nlevel.  \nHowever, it is interesting to consider whether we really need to\nconsider relationships specifically between speech segments\\xspace themselves, or whether it\nsuffices to simply consider relationships between the {\\em speakers} of\nthe speech segments\\xspace.\nIn particular, as an alternative to using same-speaker links, \nwe tried a {\\em speaker-based\\xspace} approach wherein \nthe way we determine\nthe initial\nindividual-document classification score for \neach \nspeech segment\\xspace uttered by a\nperson $p$ in a given debate \nis to run\nan SVM \non the\nconcatenation of\n{\\em all} of $p$'s speech segments\\xspace\nwithin that debate.\n(We also ensure that agreement-link information is propagated from\nspeech-segment\\xspace \nto speaker pairs.)\n\n\n\n\n\nHow \ndoes the use of \nsame-speaker links compare to the concatenation of\neach speaker's speech segments\\xspace?   \nTables  \\ref{tab:results-local} and \\ref{tab:results-global} \nshow that, not surprisingly, \nthe SVM individual-document\nclassifier works better on the concatenated speech segments\\xspace than on the\nspeech segments\\xspace in isolation.\nHowever, the effect on overall\nclassification accuracy is less clear: the development set favors\nsame-speaker links over concatenation, \nwhile\nthe test set does not.\n\n\n\n\n\n\n\n\n\nBut we stress that\nthe most important observation we can make from \n Table \\ref{tab:results-global} is  that once again, the\naddition of agreement information leads to substantial improvements in\naccuracy.  \n\n\n\\subsection{``Hard'' agreement constraints}  Recall that in \nin our experiments, \nwe created \nfinite-weight\nagreement links, so that speech segments\\xspace appearing in pairs flagged by our (imperfect)\nagreement detector can potentially receive different labels.  We also\nexperimented with \n{\\em forcing} such speech segments\\xspace to receive the same label,\neither through infinite-weight agreement links or through a speech-segment\\xspace\nconcatenation strategy similar to that described in the previous\nsubsection.  Both strategies resulted in clear degradation in\nperformance on both the development and test sets, a finding that \nvalidates our encoding of agreement information as ``soft'' preferences.\n\n\n\\subsection{On the development\/test set split}\n\\label{sec:eval:devtest}\n\nWe have seen several cases in which the\nmethod that performs best on the development set does not yield the\nbest test-set performance.  However, we felt that it would be\nillegitimate to change the train\/development\/test \nsets \nin a post hoc\nfashion, that is, after seeing the experimental results.  \n\nMoreover, and crucially, it is very clear that using agreement\ninformation, encoded as preferences within our graph-based approach\nrather than as hard constraints, yields substantial improvements on\nboth the development and test set; this, we believe, is our most\nimportant finding.\n\n\n\n\\section{Related work}\n\n\n\\paragraph{Politically-oriented text}  \nSentiment analysis has specifically been proposed as a key enabling\ntechnology in eRulemaking, allowing the  automatic analysis of\nthe opinions that people submit\n\\cite{Shulman+al:05a,Cardie+al:06a,Kwon+Shulman+Hovy:06a}.\nThere has also been work focused upon\ndetermining the political leaning (e.g.,\n``liberal'' vs. ``conservative'') of a document or \nauthor, \nwhere most\npreviously-proposed methods make no direct use of relationships\nbetween the documents to be classified (the ``unlabeled'' texts)\n\\cite{Laver+Benoit+Garry:03a,Efron:04a,Mullen+Malouf:06a}.\nAn exception is \\newcite{Grefenstette+Qu+Shanahan+Evans:04a}, who experimented with\n  determining the political orientation of websites essentially by\n  classifying the concatenation of all the documents found on that site.\n\n\n\n\n\nOthers have applied \nthe NLP technologies of near-duplicate detection and\ntopic-based text categorization to politically oriented text\n\\cite{Yang+Callan:05a,Purpura+Hillard:06a}.\n\n\\paragraph{Detecting agreement} We used a simple method to learn to\nidentify \ncross-speaker\nreferences \nindicating\nagreement.  More sophisticated approaches\nhave been proposed\n\\cite{Hillard+Ostendorf+Shriberg:03a}, including an extension that, in\nan interesting reversal of our problem, makes use of\nsentiment-polarity indicators within \nspeech segments\\xspace \\cite{Galley+McKeown+Hirschberg+Shriberg:04a}.  \nAlso relevant is work on the general problems of dialog-act tagging\n\\cite{Stolcke+al:00a},  citation analysis\n\\cite{Lehnert+Cardie+Riloff:90}, and computational rhetorical analysis \\cite{Marcu:00a,Teufel+Moens:02a}.\n\nWe currently do not have an efficient means to encode {\\em\ndisagreement} information as hard constraints; we plan to\ninvestigate incorporating\nsuch information in future work.\n\n\n\\paragraph{Relationships between the unlabeled items}  \n\\newcite{Carvalho+Cohen:05a} consider sequential relations between different\ntypes of emails (e.g.,\nbetween requests and satisfactions thereof) to classify\nmessages, and thus also explicitly exploit the structure of \nconversations.\n\nPrevious\nsentiment-analysis work in different domains\nhas considered inter-document similarity\n\\cite{Agarwal+Bhattacharyya:05a,Pang+Lee:05a,Goldberg+Zhu:06a} or \nexplicit inter-document references in the form of hyperlinks  \\cite{Agrawal+al:03a}.\n\n\n\nNotable early papers on graph-based semi-supervised learning include\nBlum and Chawla \\shortcite{Blum+Chawla:01a}, Bansal, Blum, and Chawla\n\\shortcite{Bansal+Blum+Chawla:02a}, Kondor and Lafferty \\shortcite{Kondor+Lafferty:02a}, and\nJoachims \\shortcite{Joachims:03a}.  Zhu \\shortcite{Zhu:semi-survey} maintains a survey\nof this area.\n\nRecently, several alternative, often quite sophisticated approaches to {\\em collective\n  classification} have been proposed\n\\cite{Neville+Jensen:00a,Lafferty+McCallum+Pereira:01a,Getoor+al:02a,Taskar+al:02a,Taskar+Guestrin+Koller:03a,Taskar+Chatalbashev+Koller:04a,McCallum+Wellner:04a}.\nIt would be interesting to investigate the application of such methods\nto our problem.  However,  we also believe that our approach has\nimportant advantages, including  conceptual simplicity and \nthe fact that it\nis based on an  underlying optimization problem that is provably\nand \nin practice\neasy to solve.\n\n\n\n\\section{Conclusion and future work}\n\\label{sec:conc}\n\n\nIn this study, we focused on very general \ntypes\nof cross-document\nclassification preferences, \nutilizing constraints based only on speaker\nidentity and on direct textual references between statements.  \nWe showed that the integration of \neven very limited information regarding\ninter-document relationships\ncan significantly increase the accuracy\nof support\/opposition classification.\n\nThe simple constraints \nmodeled in our study, however, represent \njust a small \nportion \nof the rich network of relationships that connect\nstatements \nand speakers\nacross the political universe and in the wider realm of\nopinionated social discourse.  \nOne intriguing possibility is to take advantage of (readily identifiable)\ninformation regarding interpersonal relationships, making use of \nspeaker\/author affiliations, positions\nwithin a social hierarchy, and so on.\nOr, we could even  attempt to model relationships\nbetween\ntopics or concepts, \nin a kind of extension of collaborative filtering.\nFor example, \nperhaps we could infer that \ntwo\nspeakers sharing a common opinion on evolutionary biologist Richard\nDawkins (a.k.a. ``Darwin's rottweiler'') will be likely to agree\nin a debate centered on Intelligent Design.  While such functionality is\nwell beyond the scope of our current study, we are optimistic that we\ncan develop methods to exploit\nadditional types of relationships in future work.\n\n\n\n\n\n\n\\paragraph*{Acknowledgments}  We thank \nClaire Cardie, \nJon Kleinberg, Michael Macy, Andrew\nMyers, \nand the six anonymous EMNLP referees for valuable discussions\nand comments.  We also thank Reviewer 1 for generously providing\nadditional {\\em post hoc} feedback, and the EMNLP chairs Eric Gaussier\nand Dan Jurafsky for facilitating the process (as well as for\nallowing authors an extra proceedings page$\\ldots$).\nThis paper is based upon work supported in part by the National\nScience Foundation under grant no. IIS-0329064 and an Alfred P. Sloan\nResearch Fellowship. Any opinions,\nfindings, and conclusions or recommendations expressed are those of\nthe authors and do not necessarily reflect the views or official\npolicies, either expressed or implied, of any sponsoring institutions,\nthe U.S. government, or any other entity.\n\n\\vspace*{-.1in}\n\n{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\nA new liquid crystal equilibrium phase was recently discovered \\cite{cestari:phase,chen:chiral,borshch:nematic,chen:twist-bend}. Any such discovery is \\emph{per se} a rare event, but this was even more striking as in some specific materials an achiral phase which had long been known was shown to conceal a chiral mutant, attainable upon cooling through a weakly first-order transition. This is known as the twist-bend nematic ($\\mathrm{N_{tb}}$) phase.\\footnote{The name \\emph{twist-bend} was introduced by Dozov~\\cite{dozov:spontaneous} together with \\emph{splay-bend} to indicate the two alternative nematic distortions which, unlike pure bend and pure splay, can fill the three-dimensional space, as previously observed by Meyer~\\cite{meyer:structural}.} Molecular bend seems to be necessary for such a phase to be displayed, but it is not sufficient, as most bent-core molecules fail to exhibit it \\cite{jakli:liquid}. In the molecular architecture capable of inducing the $\\mathrm{N_{tb}}$ phase, dimers with rigid cores are connected by sufficiently flexible linkers.\\footnote{Very recently, experimental evidence has also been provided for \\TBN phases arising from \\emph{rigid} bent-core molecules \\cite{chen:twist-bend}.} The molecular effective curvature, while inducing \\emph{no} microscopic \\emph{twist}, allegedly favors a \\emph{chiral} collective arrangement in which bow-shaped molecules uniformly precess along an ideal cylindrical \\emph{helix}.\n\nFigure~\\ref{fig:spiral_arc} sketches the picture envisaged here.\n\\begin{figure}\n  \\centering\n  \\subfigure[]{\\includegraphics[width=.35\\linewidth,angle=0]{arc.eps}}\n  \\hspace{.05\\linewidth}\n  \\subfigure[]{\\includegraphics[width=.2\\linewidth,angle=0]{spiral_new.eps}}\n  \\caption{(a) Molecular achiral model with two symmetry axes, one polar, $\\bm{p}$, and one apolar, $\\bm{m}$. (b) One variant of the helical nematic phase with helix axis $\\twist$. In the other variant, not shown here, the helix winds downwards, in the direction opposite to $\\twist$.}\n\\label{fig:spiral_arc}\n\\end{figure}\nA single bow-shaped molecule exhibits two symmetry axes, represented by the unit vectors $\\bm{p}$ and $\\bm{m}$, polar the former and apolar the latter.\\footnote{Despite a visual illusion caused by the curved arc, in Fig.~\\ref{fig:spiral_arc}(a) the lengths of $\\bm{p}$ and $\\bm{m}$ are just the same.} The local symmetry point-group is $\\mathsf{C}_{2v}$, but, as explained by Lorman and Mettout~\\cite{lorman:unconventional,lorman:theory}, by combining the symmetries broken in the helical arrangement in Fig.~\\ref{fig:spiral_arc}(b), namely, the continuous translations along the helix axis and the continuous rotations around that axis, a symmetry is recovered which involves any given translation along the helix axis, provided it is accompanied by an appropriately tuned rotation. This forbids any smectic modulation in the mass density, rendering the helical phase purely nematic. While the nematic director $\\n$ is defined as the ensemble average $\\n:=\\langle\\bm{m}\\rangle$, no polar order survives in a helical phase, as $\\langle\\bm{p}\\rangle=\\bm{0}$.\\footnote{For this reason, no macroscopic analogue of $\\bm{p}$ will be introduced in the theory, and the phase will be treated as macroscopically \\emph{apolar}.} Fig.~\\ref{fig:spiral_arc}(b) shows only one of the chiral variants that symmetry allows in a \\TBN phase; the other winds in the direction opposite to $\\twist$. In both cases, $\\n$ makes a fixed \\emph{cone angle} $\\ca$ with $\\twist$. For definiteness, we shall call each chiral variant a \\emph{helical} nematic phase. In a different language, each helical nematic phase is precisely the $C$-phase predicted by Lorman and Mettout \\cite{lorman:unconventional,lorman:theory}, which breaks spontaneously the molecular chiral symmetry, producing two equivalent macroscopic variants with opposite chiralities (see Fig.~\\ref{fig:spiral_arc}). This paper is intended to study separately each helical nematic variant hosted in a \\TBN phase. How opposite variants may be brought in contact within a purely elastic theory will be the subject of a forthcoming paper \\cite{virga:double-well}.\n\nThough, in retrospect, a number of experimental studies had already anticipated \\TBN phases (see, for example, \\cite{sepelj:intercalated,imrie:liquid,panov:spontaneous}, to cite just a few), a clear identification of the new phase was achieved in \\cite{cestari:phase} by a combination of methods (see also \\cite{henderson:methylene,cestari:crucial,panov:microsecond,panov:field-induced,beguin:chirality}), and an impressive direct evidence for it has been provided in \\cite{chen:chiral,borshch:nematic,chen:twist-bend} (see also \\cite{copic:nematic}), with accurate measurements of both the helix pitch ($\\approx10\\,\\mathrm{nm}$) and cone angle ($\\approx20\\degree$).\nTheoretically, $\\mathrm{N_{tb}}$ phases had already been predicted by Meyer~\\cite{meyer:structural} and Dozov~\\cite{dozov:spontaneous}, from two different perspectives, the former inspired by the symmetry of polar electrostatic interactions (a line of thought recently further pursued in \\cite{shamid:statistical}) and the latter starting from purely elastic (and steric) considerations. A helical molecular arrangement was also seen in the molecular simulations of Memmer~\\cite{memmer:liquid}, who considered bent-core Gay-Berne molecules with \\emph{no} polar electrostatic interactions, though featuring an effective shape polarity.\n\nDozov's $\\mathrm{N_{tb}}$ theory requires a \\emph{negative} bend elastic constant, which is compatible with the boundedness (from below) of the total energy only if appropriate \\emph{quartic} corrections are introduced in the energy density. However, a large number of these terms are allowed by symmetry \\cite{dozov:spontaneous}, and the theory may realistically be applied only by choosing just a few of them and neglecting all the others \\cite{meyer:flexoelectrically}. Moreover, recent experiments \\cite{adlem:chemically} have reported an increase in the (positive) bend constant measured in the nematic phase near the transition to the $\\mathrm{N_{tb}}$ phase. In an attempt to justify the negative elastic bend constant required by Dozov's theory, a recent study has replaced it with an \\emph{effective} bend constant resulting from the coupling with the polarization characteristic of flexo-elasticity \\cite{shamid:statistical}. As a consequence, however, the twist-bend modulated phase envisaged in \\cite{shamid:statistical} is locally \\emph{ferroelectric}, whereas, as explained by the symmetry argument of Lorman and Mettout~\\cite{lorman:theory,lorman:unconventional}, each helical nematic phase is expected to be \\emph{apolar}.\n\nThe theories in \\cite{dozov:spontaneous} and \\cite{shamid:statistical} are not in contradiction with one another, the only difference being that the latter justifies a negative bend elastic constant as commanded by the very bend-polarization coupling that gives rise to a modulated polar phase. There is still a conceptual difference between these theories: the former is purely elastic but quartic, whereas the latter is quadratic but flexo-electric. Being highly localized, the ferroelectricity envisaged in \\cite{shamid:statistical} does not produce any average macroscopic polarization, and so it would be compatible with the current experimental observations which have not found so far any trace of macroscopic ferroelectricity. This, however, leaves the question unanswered as to whether an intrinsically quadratic, purely elastic theory could also explain \\TBN phases.\nIn this work, we propose such a theory; it will feature the same elastic constants as Frank's classical theory \\cite{frank:theory}, with a positive bend constant and an extra director field. This theory reduces to Frank's for ordinary nematics, for which the extra director becomes redundant.\n\nThis paper is organized as follows. In Section~\\ref{sec:helical}, we shall formally introduce a helical nematic phase, defined as each \\TBN variant with a prescribed \\emph{helicity} in its ground state.\\footnote{See \\eqref{eq:helicity} for a precise definition.} A quadratic elastic free energy will be considered for each helical nematic variant, under the (temporary) assumption that they can be thought of as separate manifestations of one and the same \\TBN phase. For a given sign of the helicity, negative for definiteness, we shall apply the proposed elastic theory to two classical instabilities, namely, the helix unwinding first encountered in chiral nematics (in Section~\\ref{sec:helix}), and the Freedericks transition that has long made it possible to measure the elastic constants of ordinary nematics (in Section~\\ref{sec:freedericks}). Both these applications will acquire some new nuances within the present theory. In Section~\\ref{sec:Double-Well}, we first derive the quadratic elastic energy density for a helical nematic phase with \\emph{positive} helicity. The helical nematic variants with both helicities are then combined together in a \\TBN phase; the corresponding elastic energy density need to attain one and the same minimum in two separate \\emph{wells}. There are several ways to construct such an energy, which by necessity will \\emph{not} be \\emph{convex}; we shall build upon the quadratic elastic energy for a single helical phase arrived at in Section~\\ref{sec:helical}. In the two ways that we consider in detail, the elastic energy density for a \\TBN phase features only four elastic constants. Finally, in Section~\\ref{sec:conclusion}, we collect the conclusions reached in this work and comment on some possible avenues for future research.\n\n\n\\section{Helical nematic phases}\\label{sec:helical}\nA \\TBN differs from classical nematics in its ground state, the state relative to which the elastic cost of all distortions is to be accounted for. The ground state of a classical nematic is the uniform alignment (in an arbitrary direction) of the nematic director $\\n$. The $\\mathrm{N_{tb}}$ ground state is a \\emph{heliconical} twist, in which $\\n$ performs a uniform precession, making everywhere the angle $\\ca$ with a helix axis, $\\twist$, arbitrarily oriented in space.\nSuch a ground state should reflect the intrinsically less distorted molecular arrangement that results from minimizing the interaction energy of the achiral, bent molecules that comprise the medium.\n\nBy symmetry, there are indeed \\emph{two} such states, distinguished by the sense of precession (either clockwise or anticlockwise around $\\twist$). In general, borrowing a definition from Fluid Dynamics (see, for example, \\cite{moffatt:degree}), we call the pseudoscalar\n\\begin{equation}\\label{eq:helicity}\n\\eta:=\\n\\cdot\\curl\\n,\n\\end{equation}\nthe \\emph{helicity} of the director field $\\n$. We shall see now that it is precisely the sign of $\\eta$ that differentiates the two variants of the ground state of a $\\mathrm{N_{tb}}$.\n\nLetting $\\twist$ coincide with the unit vector $\\e_z$ of a Cartesian frame $(x,y,z)$, the fields $\\ground^\\pm$ representing the ground states can be written in the form\n\\begin{equation}\\label{eq:ground}\n\\begin{split}\n\\ground^\\pm=&\\sin\\ca\\cos(\\pm qz+\\az_0)\\,\\e_x\\\\+&\\sin\\ca\\sin(\\pm qz+\\az_0)\\,\\e_y+\\cos\\ca\\,\\e_z,\n\\end{split}\n\\end{equation}\nwhere $\\az_0$ is an arbitrary phase angle, $q$ is a prescribed \\emph{wave parameter}, taken to be non-negative, as characteristic of the condensed phase as the \\emph{cone angle} $\\ca$ (see Fig.~\\ref{fig:spherical}).\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\linewidth,angle=0]{Spherical.eps}\n\\caption{The angles $\\ca$ and $\\az=\\pm qz+\\az_0$ that in \\eqref{eq:ground} represent the fields $\\ground^\\pm$ are illustrated in a Cartesian frame $(x,y,z)$. Both $\\ca$ and $q$ are fixed parameters characteristic of the phase.}\n\\label{fig:spherical}\n\\end{figure}\nThe \\emph{pitch} $p$ corresponding to $q$ is given by\\footnote{No confusion should arise here between the pitch $p$ of the \\TBN phase and the polar vector $\\bm{p}$ mentioned in the Introduction. The former is macroscopic in nature, whereas the latter is microscopic.}\n\\begin{equation*}\\label{eq:p}\np:=\\frac{2\\pi}{q}.\n\\end{equation*}\nOn every two planes orthogonal to $\\e_z$ and $p$ apart, each field $\\ground^\\pm$ in \\eqref{eq:ground} delivers one and the same nematic director. A simple computation shows that\n\\begin{equation}\\label{eq:helicity_ground_state}\n\\eta^\\pm:=\\ground^\\pm\\cdot\\curl\\ground^\\pm=\\mp q\\sin^2\\ca.\n\\end{equation}\n\nWe shall call \\emph{helical nematic} each of the two phases for which either $\\ground^+$ or $\\ground^-$ is the nematic field representing the ground state. Here, for definiteness, we shall develop our theory as if only $\\ground^+$ represented the ground state. By \\eqref{eq:helicity_ground_state}, such a state has \\emph{negative} helicity. The corresponding case of positive helicity will be studied in Section~\\ref{sec:positive}. Until then we shall drop the superscript $^+$ from $\\ground^+$ to avoid clatter.\n\n\\subsection{Negative helicity}\\label{sec:negative}\nIt readily follows from \\eqref{eq:ground} that\\footnote{Here and below $\\times$ and $\\otimes$ denote the vector and tensor products of vectors. In particular, for any two vectors, $\\bm{a}$ and $\\bm{b}$, $\\bm{a}\\otimes\\bm{b}$ is the second-rank tensor that acts as follows on a generic vector $\\bm{v}$, $(\\bm{a}\\otimes\\bm{b})\\bm{v}:=(\\bm{b}\\cdot\\bm{v})\\bm{a}$, where $\\cdot$ denotes the inner product of vectors. An alternative way of denoting the dyadic product $\\bm{a}\\otimes\\bm{b}$ would be simply $\\bm{a}\\bm{b}$. In Cartesian components, they are both represented as $a_ib_j$.}\n\\begin{equation}\\label{eq:grad_ground}\n\\nabla\\ground=q\\left(\\e_z\\times\\ground\\right)\\otimes\\e_z.\n\\end{equation}\nMore generally, for $\\n$ prescribed at a point in space, the tensor\n\\begin{equation}\\label{eq:natural_distortion}\n\\Twist:=q(\\twist\\times\\n)\\otimes\\twist\n\\end{equation}\nexpresses the \\emph{natural} distortion\\footnote{A natural distortion is a distortion present in the ground state, when the latter fails to be the uniform nematic field $\\n$ for which $\\N\\equiv\\zero$.} that would be associated at that point with the preferred helical configuration agreeing with the prescribed nematic director $\\n$ and having $\\twist$ as helix axis. We imagine that in the absence of any frustrating cause, given $\\n$ at a point, the director field would attain in its vicinity a spatial arrangement such that\n\\begin{equation}\\label{eq:local_and_natural_distortions}\n\\nabla\\n=\\Twist,\n\\end{equation}\nwhere $\\Twist$ is as in \\eqref{eq:natural_distortion} and $\\twist$ is any unit vector such that\n\\begin{equation}\\label{eq:cone_constraint}\n\\n\\cdot\\twist=\\cos\\ca.\n\\end{equation}\nThis would make \\eqref{eq:grad_ground} locally satisfied, even though $\\n$ does not coincide with $\\ground$ in the large. Correspondingly, the elastic energy that would locally measure the distortional cost should be measured relative to the whole class of natural distortions, vanishing whenever any of the latter is attained. With this in mind, we write the elastic energy density $f^-_e$ in the form\\footnote{The superscript $^-$ will remind us that the ground state of the helical nematic phase we are considering here has a negative helicity $\\eta$,}\n\\begin{equation}\\label{eq:elastic_energy_density_definition}\nf^-_e(\\twist,\\n,\\N)=\\frac12(\\N-\\Twist)\\cdot\\elan[\\N-\\Twist],\n\\end{equation}\nwhere $\\elan$ is the most general positive-definite, symmetric fourth-order tensor invariant under rotations about $\\n$. Clearly, if for given $\\n$ and $\\N$, $\\twist$ can be chosen in \\eqref{eq:elastic_energy_density_definition} so that \\eqref{eq:local_and_natural_distortions} is satisfied, $f^-_e$ vanishes, attaining its absolute minimum. On the contrary, if  there is no such $\\twist$, then minimizing $f^-_e$ in $\\twist$ would identify the natural, undistorted state closest to the nematic distortion represented by $\\N$ in the metric induced by $\\elan$. For this reason, here both $\\n$ and $\\twist$ are to be considered as unknown fields linked by \\eqref{eq:cone_constraint}: at equilibrium, the free-energy functional that we shall construct is to be minimized in both these fields.\n\nCombining the general representation formula for $\\elan$ and the identities\\footnote{The superscript $\\T$ means tensor transposition.}\n\\begin{equation*}\\label{eq:tensor_identities}\n(\\N)\\T\\n=\\zero,\\qquad\\Twist\\T\\n=\\zero,\n\\end{equation*}\nwe can reduce $\\ela$ in \\eqref{eq:elastic_energy_density_definition} to the following equivalent form\n\\begin{equation}\\label{eq:K_reduced_representation}\n\\ela_{ijhk}=k_1\\delta_{ih}\\delta_{jk}+k_2\\delta_{ij}\\delta_{hk}+k_3\\delta_{ih}n_jn_k\n+k_4\\delta_{ik}\\delta_{jh},\n\\end{equation}\nwhere $k_1$, $k_2$, $k_3$, and $k_4$ are elastic moduli (as in \\cite[p.\\,114]{virga:variational}). By use of \\eqref{eq:K_reduced_representation}, of \\eqref{eq:cone_constraint}, and\n\\begin{equation*}\\label{eq:scalar_identities}\n\\tr\\Twist=0,\\qquad \\N\\cdot\\Twist=q(\\twist\\times\\n)\\cdot(\\N)\\twist,\n\\end{equation*}\nwe transform \\eqref{eq:elastic_energy_density_definition} into\n\\begin{equation}\\label{eq:elastic_energy_density_transformed}\n\\begin{split}\nf^-_e(\\twist,\\n,\\N)\n&=\\frac12\\Bigl\\{K_{11}(\\diver\\n)^2+K_{22}(\\n\\cdot\\curl\\n+q|\\twist\\times\\n|^2)^2\\\\\n&+K_{33}|\\n\\times\\curl\\n+q(\\n\\cdot\\twist)\\,\\twist\\times\\n|^2\\\\\n&+K_{24}[\\tr(\\N)^2-(\\diver\\n)^2]\\Bigr\\}\n-K_{24}q\\,\\twist\\times\\n\\cdot(\\N)\\T\\twist,\n\\end{split}\n\\end{equation}\nwhere $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$, which are analogous to the classical Frank's constants \\cite{frank:theory}, are related to the moduli $k_1$, $k_2$, $k_3$, and $k_4$ through the equations\n\\begin{equation*}\\label{eq:K_and_k}\nK_{11}=k_1+k_2+k_4,\\quad K_{22}=k_1,\\quad\nK_{33}=k_1+k_3,\\quad K_{24}=k_1+k_4.\n\\end{equation*}\nFor $f^-_e$ in \\eqref{eq:elastic_energy_density_transformed} to be positive definite, the elastic constants $K_{11}$, $K_{22}$, $K_{33}$, and $K_{24}$ must obey the inequalities,\n\\begin{equation*}\\label{eq:Ericksen_inequalities}\n2K_{11}\\geqq K_{24},\\quad 2K_{22}\\geqq K_{24},\\quad\nK_{33}\\geqq0,\\quad K_{24}\\geqq0,\n\\end{equation*}\nwhich coincide with the classical Ericksen's inequalities for ordinary nematics \\cite{ericksen:inequalities}.\n\nThe total elastic energy $\\free^-_e$ is represented by the functional\n\\begin{equation*}\\label{eq:elastic_energy_functional}\n\\free^-_e[\\twist,\\n]:=\\int_\\body f^-_e(\\twist,\\n,\\N)dV,\n\\end{equation*}\nwhere both $\\twist$ and $\\n$ are subject to the pointwise constraint \\eqref{eq:cone_constraint} and either of them is prescribed on the boundary $\\boundary$ of the region in space occupied by the medium.\nIt is worth recalling that both $\\twist$ and $\\n$ are fields that need to be determined so as to obey \\eqref{eq:cone_constraint} and to minimize $\\free^-_e$. In this theory, the helix axis of the preferred conical state and the nematic director of the actual molecular organization equally participate in the energy minimization with the objective of reducing the discrepancy between natural and actual nematic distortions.\nPhysically, $\\twist$ represents the optic axis of the medium, likely to be the only optic observable when the pitch $p$ ranges in the the nanometric domain.\n\nSuch an abundance of state variables is a direct consequence of the degeneracy in the ground state admitted for helical nematics. The \\emph{vacuum manifold}, as is often called the set of distortions that minimize $f^-_e$, is indeed a three-dimensional orbit of congruent cones, identifiable with $\\sphere\\times\\Circle$, where $\\sphere$ is the unit sphere and $\\Circle$ the unit circle in three space dimensions. By \\eqref{eq:natural_distortion}, all natural distortions $\\Twist$ are represented by $\\twist\\in\\sphere$ and any $\\n$ in the cone of semi-amplitude $\\ca$ around $\\twist$. This shows, yet in another language, how rich in states is the single well where $f^-_e$ vanishes.\n\nA number of remarks are suggested by formula \\eqref{eq:elastic_energy_density_transformed}. First, it reduces to the elastic free-energy density of ordinary nematics, which features only $\\n$, when either the wave parameter $q$ or the cone angle $\\ca$ vanish, thus indicating two possible mechanisms to induce helicity in an ordinary nematic. Second, for $\\ca=\\frac\\pi2$, it delivers an alternative energy density for chiral nematics, which is positive-definite for all $K_{24}\\geqq0$ (whereas, to ensure energy positive-definiteness, the classical theory requires that $K_{24}=0$, see \\cite[p.\\,127]{virga:variational}). Finally, for arbitrary $q>0$ and $0<\\ca<\\frac{\\pi}{2}$, $f^-_e$ in \\eqref{eq:elastic_energy_density_transformed} is invariant under the reversal of either $\\twist$ or $\\n$, showing that both fields enjoy the nematic symmetry.\n\n\\section{Helix unwinding}\\label{sec:helix}\nIn the presence of an external field, say an electric field $\\field$, the free-energy density acquires an extra term, which we write as\n\\begin{equation*}\\label{eq:electric_energy density}\nf_E(\\n)=-\\frac12\\ez\\ea E^2(\\n\\cdot\\e)^2,\n\\end{equation*}\nwhere $\\ez$ is the vacuum permittivity, $\\ea$ is the (relative) dielectric anisotropy, and we have set $\\field=E\\e$, with $E>0$ and $\\e$ a unit vector. Accordingly, the total free-energy functional $\\free^-$ is defined as\n\\begin{equation}\\label{eq:free_energy_total}\n\\free^-[\\twist,\\n]:=\\int_\\body\\{f^-_e(\\twist,\\n,\\N)+f_E(\\n)\\}dV.\n\\end{equation}\n\nTo minimize $\\free^-$ when $\\body$ is the whole space, we shall assume that $\\twist$ is uniform and $\\n$ is spatially periodic with period $L$ and we shall compute the average $F^-$ of $\\free^-$ over an infinite slab of thickness $L$ orthogonal to $\\twist$. Letting $\\twist$ coincide with the unit vector $\\e_z$ of a given Cartesian frame $(x,y,z)$, we represent $\\n$ in precisely the same form adopted in \\eqref{eq:ground} for $\\ground$, but with $qz+\\az_0$ replaced by a function $\\az=\\az(z)$ such that\n\\begin{equation*}\\label{eq:periodicity_azimut}\n\\az(0)=0\\quad\\text{and}\\quad\\az(L)=2\\pi.\n\\end{equation*}\nWith no loss of generality, we may choose $\\e$ in the $(y,z)$ plane and represent it as\n\\begin{equation*}\\label{eq:a_representation}\n\\e=\\cos\\psi\\,\\e_z+\\sin\\psi\\,\\e_y.\n\\end{equation*}\nComputing $F^-$ on the ground state $\\az=2\\pi z\/p$ (where $L=p$), one easily sees that the average energy is smaller for $\\psi=\\frac\\pi2$ than for $\\psi=0$, whenever either\n\\begin{enumerate}[(a)]\n\\item\\label{case:1} $\\ea<0$ and $\\ca<\\cac:=\\arctan\\sqrt{2}\\doteq54.7\\degree$, or\n\\item\\label{case:2} $\\ea>0$ and $\\ca>\\cac$,\n\\end{enumerate}\nwhich are the only cases we shall consider here. In case \\eqref{case:1} (and for $\\psi=\\frac\\pi2$), $F^-$ reduces to\n\\begin{equation*}\\label{eq:F_unwinding}\n\\begin{split}\nF^-[L,\\az]=K\\sin^2\\ca\\left\\{\\frac{1}{2L}\\int_0^L\\left[\\az'^2+\\frac{1}{\\cohe^2}\\sin^2\\az\\right]dz -\\frac{2\\pi q}{L} \\right\\},\n\\end{split}\n\\end{equation*}\nwhere\n\\begin{equation*}\\label{eq:K_effective}\nK:=K_{22}\\sin^2\\ca+K_{33}\\cos^2\\ca\n\\end{equation*}\nis an effective \\emph{twist-bend} elastic constant and\n\\begin{equation}\\label{eq:coherence_length}\n\\cohe:=\\frac{1}{E}\\sqrt{\\frac{K}{\\ez|\\ea|}}\n\\end{equation}\nis a field \\emph{coherence length}.\n\nMinimizing $F^-$ in both $L$\\ and $\\az$ is a problem formally akin to the classical problem of unwinding the cholesteric helix \\cite{degennes:calul,meyer:effects,meyer:distortion}. The minimizing $\\az$ is determined implicitly by\n\\begin{equation*}\\label{eq:phi_unwinding}\n\\frac{z}{L}=\\frac14\\left(\\frac{\\ElF(\\az+\\frac\\pi2,k)}{\\ElK(k)}-1\\right),\n\\end{equation*}\nwhere $\\ElF$ and $\\ElK$ are the elliptic and complete elliptic integrals of the first kind, respectively, and $k$ is the root in the interval $[0,1]$ of the equation\n\\begin{equation}\\label{eq:root_k}\n\\frac{\\ElE(k)}{k}=\\pi^2\\frac{\\cohe}{p},\n\\end{equation}\nwhere $\\ElE$ is the complete elliptic integral of the second kind \\cite[p.\\,486]{olver:NHMF}. Equation \\eqref{eq:root_k}\nhas a (unique) root only for\n\\begin{equation*}\n\\cohe\\geqq\\cohec:=\\frac{p}{\\pi^2},\n\\end{equation*}\nfor which $L$ is correspondingly delivered by\n\\begin{equation*}\\label{eq:root_L}\nL=4\\cohe k\\ElK(k).\n\\end{equation*}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{LLL_Graph.eps}\n\\caption{\nThe graph of the spatial period $L$ of the distorted helix under field (scaled to the undistorted pitch $p$) against the field coherence length $\\cohe$ in \\eqref{eq:coherence_length}\n(equally scaled to $p$). The complete unwinding takes place for $\\cohe\/p=1\/\\pi^2\\doteq0.101$, where the graph diverges.}\n\\label{fig:L_graph}\n\\end{figure}\nFigure~\\ref{fig:L_graph} shows the graph of $L$ against $\\cohe$, which diverges as $\\cohe$ approaches $\\cohec$. As for ordinary chiral nematics, a field with coherence length $\\cohec$ unwinds completely the helix of a $\\mathrm{N_{tb}}$, but its actual strength now depends on the elastic constants $K_{22}$ and $K_{33}$, and the cone angle $\\ca$.\nThe measured values of $p$ range in the domain of tens of nanometers. Thus, the steepness of the graph in Fig.~\\ref{fig:L_graph} along with \\eqref{eq:coherence_length} suggest that in actual terms the field should be very strong for any unwinding to be noticed.\\footnote{A simple estimate based on \\eqref{eq:root_k} would show that a field strength larger than $100\\,\\mathrm{V}\/\\mu\\mathrm{m}$ is needed to unwind the \\TBN helix even if we assume $\\ea$ as large as $10$ and $K$ as small as $1\\,\\mathrm{pN}$.}\n\n\\section{Freedericks transition}\\label{sec:freedericks}\nIn ordinary nematics, the Freedericks transition is an instability that enables one to measure the classical Frank's elastic constants. For a $\\mathrm{N_{tb}}$, the setting is complicated by the role played by the additional field $\\twist$.\n\nWe consider a $\\mathrm{N_{tb}}$ liquid crystal confined between two parallel plates, placed in a Cartesian frame $(x,y,z)$ at $z=0$ and $z=d$, respectively. On both plates, we subject $\\n$ to a \\emph{conical} anchoring with respect to the plates' normal $\\e_z$ at precisely the cone angle $\\ca$, so that, by the constraint \\eqref{eq:cone_constraint}, $\\twist$ is there prescribed to coincide with $\\e_z$. Within the infinite cell bounded by these plates we allow $\\twist$ to vary in the $(y,z)$ plane, so that\n\\begin{equation*}\\label{eq:FrederikS_e_representation}\n\\twist=\\cos\\psi\\,\\e_z+\\sin\\psi\\,\\e_y,\n\\end{equation*}\nwhere $\\psi=\\psi(z)$. Letting\n\\begin{equation*}\n\\twist_\\perp:=\\sin\\psi\\,\\e_z-\\cos\\psi\\,\\e_y,\n\\end{equation*}\nwe represent the nematic director as\n\\begin{equation}\\label{eq:Freedericks_n_representation}\n\\n=\\cos\\ca\\,\\twist+\\sin\\ca\\cos\\az\\,\\twist_\\perp+\\sin\\ca\\sin\\az\\,\\e_x,\n\\end{equation}\nwhere $\\az=\\az(z)$ is the precession angle. The function $\\psi$ is subject to the conditions\n\\begin{equation}\\label{eq:Freedericks_psi_boundary_conditions}\n\\psi(0)=\\psi(d)=0,\n\\end{equation}\nwhile $\\az$ is free in the whole of $[0,d]$. The system is further subjected to an external field, $\\field=E\\e_z$. In the following analysis, we shall assume that either of the aforementioned  cases \\eqref{case:1} or \\eqref{case:2} occurs here.\n\nTaking $p\/d\\ll1$ and treating $\\psi$ as a perturbation, of which we only retain quadratic terms in the energy, when the precession angle represents the undistorted helix, $\\az=2\\pi z\/p$, we obtain the average $F^-$ of the free-energy functional $\\free^-$ in \\eqref{eq:free_energy_total}\n(per unit area of the plates) as\n\\begin{equation}\\label{eq:Freedericks_average_free_energy}\nF^-[\\psi]=\\frac12 K_{33}\\int_0^d\\left\\{A\\psi'^2+Bq^2\\psi^2\\right\\}dz,\n\\end{equation}\nwhere\n\\begin{gather*}\nA:=\\frac12\\Big[\\sin^2\\ca(k_{11}+k_{22}\\cos^2\\ca)+\\cos^2\\ca(1+\\cos^2\\ca)\\Big]\n\\\\\nB:=\\frac12\\Big[\\sin^2\\ca(k_{11}+k_{22}\\cos^2\\ca+\\sin^2\\ca\n-\\frac{1}{(\\cohe q)^2}|2\\cos^2\\ca-\\sin^2\\ca|\\Big]\n\\\\\nk_{11}:=\\frac{K_{11}}{K_{33}},\\quad k_{22}:=\\frac{K_{22}}{K_{33}},\\quad\\cohe:=\\frac{1}{E}\\sqrt{\\frac{K_{33}}{\\ez|\\ea|}}.\n\\end{gather*}\nIt is now easily seen that $\\psi\\equiv0$ is a locally stable extremum of the functional $F^-$\nin \\eqref{eq:Freedericks_average_free_energy} subject to \\eqref{eq:Freedericks_psi_boundary_conditions}\nfor $\\cohe>\\cohec$, whereas it is locally unstable for $\\cohe<\\cohec$, where\n\\begin{equation}\\label{eq:Freedericks_critical_xi_E}\n\\frac{\\cohec}{p}:=\\frac{\\sqrt{|2-t^2|(1+t^2)}}{\\pi\\sqrt{4[(1+k_{11})t^2+k_{11}+k_{22}]t^2+ \\frac{p^2}{d^2}[k_{11}t^4+(1+k_{11}+k_{22})t^2+2]}}\n\\end{equation}\nand $t:=\\tan\\ca$. Both this and the helix unwinding treated in Section~\\ref{sec:helix} are continuous transitions.\n\nSince in this theory $p\/d$ is small, $\\cohec$ is only weakly dependent on the cell thickness $d$. It should be noted however that for $\\ca=0$, \\eqref{eq:Freedericks_critical_xi_E} reduces to $\\cohec=d\/\\pi$, which reproduces the classical Freedericks threshold \\cite[p.\\,179]{virga:variational}.\n\nIn the special, hypothetical case of equal elastic constants (for which $k_{11}=k_{22}=1$), $\\cohec$ acquires a simpler form, which retains the qualitative features of \\eqref{eq:Freedericks_critical_xi_E}:\n\\begin{equation}\\label{eq:Freedericks_critical_xi_E_equal_constants}\n\\frac{\\cohec}{p}=\\frac{\\sqrt{|2-t^2|}}{\\pi\\sqrt{8t^2+\\frac{p^2}{d^2}\\frac{t^4+3t^2+2}{1+t^2}}}.\n\\end{equation}\n\\begin{figure}\n\\centering\n\\includegraphics[width=.9\\linewidth]{F_Graph.eps}\n\\caption{\nThe graphs of $\\cohec$ (scaled to the pitch $p$ of the undistorted helix) against the cone angle $\\ca$ (expressed in degrees), as delivered by \\eqref{eq:Freedericks_critical_xi_E_equal_constants} under the assumption of equal elastic constants, for $p\/d=0$ (solid line) and $p\/d=1$ (dotted line). For $\\ca=0$, the former graph diverges, while the latter reaches the value $\\cohec\/p=1\/\\pi\\doteq0.318$. For $0<p\/d<1$, the graphs of $\\cohec$, none of which diverges, fill the region bounded by the graphs shown here.}\n\\label{fig:F_Graph}\n\\end{figure}\nThe graph of $\\cohec$\nas delivered by \\eqref{eq:Freedericks_critical_xi_E_equal_constants}\nis plotted in Fig.~\\ref{fig:F_Graph} for $p\/d=0$ and $p\/d=1$, the graphs for all intermediate values of $p\/d$ being sandwiched between them. These graphs show that for $\\ca$ sufficiently large, in particular for $\\ca>\\cac$ (and $\\ea>0$), $\\cohec$ is virtually independent of $d$, whereas it is not so for $\\ca$ small. Moreover, for moderate values of $\\ca$ and $d$ much larger than $p$, $\\cohec$ can easily be made equal to several times the pitch of the undistorted helix.\nIf, as contemplated by \\eqref{eq:Freedericks_critical_xi_E}, the actual field required to ignite the Freedericks transition is weakly dependent on the cell thickness $d$, the corresponding critical potential $U_\\mathrm{c}$ would scale almost linearly with $d$.\\footnote{In the experiments performed in \\cite{borshch:nematic}, it was found that $U_\\mathrm{c}\\varpropto\\sqrt{d}$, but the boundary conditions imposed there seem to differ from the conical boundary conditions considered here. Moreover, in \\cite{borshch:nematic} the transition nucleated locally from the inside of the cell instead of happening uniformly, as presumed here. This might suggest that in the real experiment both helical variants present in a \\TBN are participating in the transition. It would then be advisable taking with a grain of salt any direct comparison of our theory for helical nematics with experiments available for the whole \\TBN phase.}\n\n\\section{Double-well energy}\\label{sec:Double-Well}\nA \\TBN phase can be regarded as a mixture of two helical nematic phases with opposite helicities. The ground  states of these phases corresponding to all admissible natural distortions are the members of two symmteric energy wells. In a way, this is reminiscent of the mixture of martensite twins in some solid crystals, which are equi-energetic variants with symmetrically sheared lattices (see, for example, \\cite[p.\\,129]{muller:entropy}). A thorough mathematical theory of these solid phases is based on a non-convex energy functional in the elastic deformation, featuring a multiplicity of energy wells \\cite{ball:fine,ball:proposed}. Below, adapting these ideas to the new context envisaged here, in which the energy depends on the local distortion of the molecular arrangement, and no deformation from a reference configuration is involved, we show how to construct a double-well elastic energy density $\\dens$ for a \\TBN phase starting from the energy densities for helical nematic phases of opposite helicities. To this end we need first supplement $\\dens^-$ in \\eqref{eq:elastic_energy_density_transformed} with the appropriate energy density $\\dens^+$ for a helical nematic phase of positive helicity.\n\n\\subsection{Positive helicity}\\label{sec:positive}\nPrescribing the helicity of the ground state of a helical nematic phase to be positive, instead of negative as above, would amount to replacing \\eqref{eq:elastic_energy_density_definition} by\n\\begin{equation}\\label{eq:elastic_energy_density_definition_positive}\nf^+_e(\\twist,\\n,\\N)=\\frac12(\\N+\\Twist)\\cdot\\elan[\\N+\\Twist],\n\\end{equation}\nstill with $\\Twist$ as in \\eqref{eq:natural_distortion} and $q>0$. Our development following \\eqref{eq:elastic_energy_density_definition} could be repeated verbatim here and it would lead us to the same conclusion obtained by subjecting \\eqref{eq:elastic_energy_density_transformed} to the \\emph{formal} change of $q$ into $-q$:\n\\begin{equation}\\label{eq:elastic_energy_density_transformed_positive}\n\\begin{split}\nf^+_e(\\twist,\\n,\\N)\n&=\\frac12\\Bigl\\{K_{11}(\\diver\\n)^2+K_{22}(\\n\\cdot\\curl\\n-q|\\twist\\times\\n|^2)^2\\\\\n&+K_{33}|\\n\\times\\curl\\n-q(\\n\\cdot\\twist)\\,\\twist\\times\\n|^2\\\\\n&+K_{24}[\\tr(\\N)^2-(\\diver\\n)^2]\\Bigr\\}\n+K_{24}q\\,\\twist\\times\\n\\cdot(\\N)\\T\\twist,\n\\end{split}\n\\end{equation}\nwhere $q$ remains a \\emph{positive} parameter.\n\nClearly, the energy well of \\eqref{eq:elastic_energy_density_definition_positive} and \\eqref{eq:elastic_energy_density_transformed_positive} is formally obtained from the corresponding well of \\eqref{eq:elastic_energy_density_definition} and \\eqref{eq:elastic_energy_density_transformed} by a sign inversion. \n\n\\subsection{\\TBN free energy density}\\label{sec:TBN}\nFollowing \\cite{truskinovsky:ericksen}, which studied systematically how to extend the non-convex energy first proposed by Ericksen~\\cite{ericksen:equilibrium} for a one-dimensional elastic bar, we consider two possible choices for the elastic energy density $\\dens$ of a \\TBN phase:\n\\begin{subequations}\\label{eq:f_e_TBN}\n\\begin{equation}\\label{eq:f_e_TBN_quadratic}\nf_e(\\twist,\\n,\\N)=\\min\\{f_e^+(\\twist,\\n,\\N),f_e^-(\\twist,\\n,\\N)\\},\n\\end{equation}\n\\begin{equation}\\label{eq:f_e_TBN_quartic}\nf_e(\\twist,\\n,\\N)=\\frac{1}{f_0}f_e^+(\\twist,\\n,\\N)f_e^-(\\twist,\\n,\\N),\n\\end{equation}\nwhere\n\\begin{equation*}\nf_0:=\\frac12\\sin^2\\ca(K_{22}\\sin^2\\ca+ K_{33}\\cos^2\\ca)\n\\end{equation*}\nis a normalization constant chosen so as to ensure that $\\dens(\\twist,\\n,\\zero)=\\dens^\\pm(\\twist,\\n,\\zero)$.\n\\end{subequations}\nWhile $\\dens$ in \\eqref{eq:f_e_TBN_quadratic} is quadratic around each well, it fails to be smooth for $\\N=\\zero$. On the other hand, $\\dens$ in \\eqref{eq:f_e_TBN_quartic} is everywhere smooth, but it is quartic. The differences between these energy densities are illustrated pictorially in Fig.~\\ref{fig:Free_Plots}.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.7\\linewidth,angle=0]{Free_plots.eps}\n  \\caption{One-dimensional pictures for $\\dens$ in \\eqref{eq:f_e_TBN_quadratic} (solid line) and \\eqref{eq:f_e_TBN_quartic} (dashed line). Here $\\az=\\az(z)$ would represent the precession angle in a molecular arrangement such as that described by \\eqref{eq:Freedericks_n_representation} and $\\az'$ is its spatial derivative. Each minimum is representative for a three-dimensional well described by \\eqref{eq:natural_distortion} and its mirror image (with $q$ replaced by $-q$).}\n\\label{fig:Free_Plots}\n\\end{figure}\n\nWe shall not explore other possible forms $\\dens$. We only heed that both \\eqref{eq:f_e_TBN_quadratic} and \\eqref{eq:f_e_TBN_quartic} inherit the simple structure of the elastic energy density of a helical nematic phase, which features only four positive elastic constants, as in the classical Frank's theory of ordinary nematics. General considerations on how to match ground states extracted from two different wells of $\\dens$ (at zero energy cost) are independent of the peculiar form assumed for this function, as they are only consequences of the structure of each well. A study of the geometric compatibility conditions that arise in this case will be presented elsewhere \\cite{virga:double-well}.\n\n\\section{Conclusion}\\label{sec:conclusion}\nThe elastic energy density proposed in \\eqref{eq:elastic_energy_density_transformed} and \\eqref{eq:elastic_energy_density_transformed_positive} to describe the equilibrium distortions of each helical variant of a \\TBN phase featured just the classical four elastic constants and introduced the helix axis $\\twist$ in addition to the nematic director $\\n$. The instabilities studied above illustrated two second-order transitions that differ also qualitatively from their classical analogues. Only experiments may decide at this stage whether a quadratic elastic theory based on either \\eqref{eq:f_e_TBN_quadratic} or \\eqref{eq:f_e_TBN_quartic} above\\footnote{Even if globally quartic, the energy density in \\eqref{eq:f_e_TBN_quartic} is quadratic about each energy well. Moreover, it features only the four elastic constants required by the quadratic energy density of a helical phase with prescribed helicity.} is better suited to describe the novel \\TBN phases than the quartic theory proposed in \\cite{dozov:spontaneous}. The instabilities described in this paper for each \\TBN variant  just provide a theoretical means to set the quadratic theory to the test.\n\nTwo director fields, namely, $\\n$ and $\\twist$, were deemed necessary here to describe the local distortion of a \\TBN phase.\nThis poses the question as to which defects both fields may exhibit and how they are interwoven, in view of the constraint \\eqref{eq:cone_constraint}.\nAn extra field also requires extra boundary conditions. The question is how to set general boundary conditions for both fields to grant existence to the energy minimizers.\n\nFinally, no hydrodynamic considerations have entered our study, but the question should be asked as to whether the relaxation in time of $\\twist$ represents a further source of dissipation.\n\n\n\n\\section*{Acknowledgements}\nI wish to thank Oleg D. Lavrentovich for his encouragement to pursue this study and for his kindness in providing me with some of his results prior to their publication. I am also greatful to Mikhail A. Osipov for having suggested studying the work of Lorman and Mettout \\cite{lorman:theory} on the symmetry of helical nematics. I am finally indebted to the kindness of two anonymous Reviewers whose critical remarks and constructive suggestions improved considerably this manuscript.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Background and motivation}\n\nThe notion of Maslov indices arised from the study of semi-classical\napproximations in quantum mechanics \\cite{maslov1972theorie} in the\nphase space $(\\mathbb{R}^{2n},dx_{i}\\wedge dy_{i}$). It plays an\nimportant role in the problem of quantization as well as gives insights\ninto the geometric aspect of classical dynamics \\cite{dullin2005maslov}.\nIn \\cite{arnold1972classe} Arnold proved that the Maslov index of\na closed path $\\tilde{\\gamma}$ of Lagrangian planes in $\\mathbb{R}^{2n}$\ncan be characterized either as the (signed) number of intersections\nof $\\tilde{\\gamma}$ with the Maslov cycle $\\mathbb{\\mathfrak{M}}_{0}$\ninduced by the Lagrangian subspace $\\mathbb{E}_{0}=\\mathbb{R}^{n}\\times\\{0\\}$,\nor, equivalently, as the degree of the Maslov-Arnold map for $\\tilde{\\gamma}$:\n\\begin{equation}\n\\mathfrak{m}_{\\tilde{\\gamma}}:S^{1}\\xrightarrow{\\tilde{\\gamma}}\\Lambda(n)\\cong\\mathbb{U}(n)\\big\/\\mathbb{O}(n)\\xrightarrow{det_{\\mathbb{C}}^{2}}S^{1},\\label{eq:def_Maslov-Arnold}\n\\end{equation}\nwhere $\\Lambda(n)$ is the Lagrangian Grassmannian of $\\mathbb{R}^{2n}$.\nLet $\\frac{\\partial}{\\partial\\theta}=\\frac{d}{d\\theta}e^{2\\pi i\\cdot\\theta}$\nbe the standard vector field on $S^{1}$ and $d\\theta$ be its dual\n$1$-form. When $\\tilde{\\gamma}$ is smooth, $\\mathfrak{m}_{\\tilde{\\gamma}}$\ncan be calculated as \n\\begin{equation}\n\\mathfrak{m}_{\\tilde{\\gamma}}=\\int_{\\tilde{\\gamma}}\\tilde{\\eta}=\\int_{\\gamma}d\\theta,\\label{eq:maslov index by integration}\n\\end{equation}\nwhere $\\tilde{\\eta}$ is the pullback of the canonical $1$-form $d\\theta$\non $S^{1}$ via the Maslov-Arnold map\\cite{contreras2003asymptotic},\nand $\\gamma=det_{\\mathbb{C}}^{2}(\\tilde{\\gamma})$ is a loop in $S^{1}$. \n\nIn the general case where $(M,\\omega)$ is a symplectic manifold,\nthe Lagrangian Grassmannian will be replaced by the bundle of Lagrangian\nplanes $\\Lambda_{pl}$, which consists of all the Lagrangian subspaces\nof the tangent spaces of $M$ \\cite{contreras2003asymptotic,butler2009maslov}.\nNamely:\n\\begin{equation}\n\\Lambda_{pl}=\\bigcup_{p\\in M}\\{Lagrangian\\ subspaces\\ of\\ T_{p}M\\}.\\label{eq:Lambda_pl_over_M}\n\\end{equation}\nA Maslov cycle $\\mathbb{\\mathfrak{M}}$ consists of all the Lagrangian\nspaces that have nontrivial intersection with a Lagrangian subbundle\n$\\mathfrak{L}$, and the Maslov index of a loop in $\\Lambda_{pl}$\nis then defined in the same spirit as the number of intersections\nwith $\\mathbb{\\mathfrak{M}}$.\n\nAt the same time, $\\Lambda_{pl}$ can also be viewed as a structure\nassociated to an almost complex structure $J$ compatible with $\\omega$.\nIn the case $(M,\\omega)=(\\mathbb{R}^{2n},dx_{i}\\wedge dy_{i}$) the\nbundle of Lagrangian planes is a trivial bundle\n\\[\n\\mathbb{R}^{2n}\\times\\Lambda(n)=\\mathbb{R}^{2n}\\times\\mathbb{U}(n)\\big\/\\mathbb{O}(n),\n\\]\nwhere the last space can be interpreted as the quotient space of $\\mathbb{R}^{2n}\\times\\mathbb{U}(n)$\nby the $\\mathbb{O}(n)$ action. $\\mathbb{R}^{2n}\\times\\mathbb{U}(n)$\nis the unitary frame bundle consisting of all the unitary frames of\nthe tangent spaces of $\\mathbb{R}^{2n}$ with respect to the canonical\nalmost complex structure\n\\[\nJ_{0}=\\left[\\begin{array}{cc}\n0 & -I_{n}\\\\\nI_{n} & 0\n\\end{array}\\right]\n\\]\non the tangent bundle $T\\mathbb{R}^{n}$. The $S^{1}$ appearing at\nthe end of the sequence (\\ref{eq:def_Maslov-Arnold}) as the codomain\nof $\\mathfrak{m}_{\\tilde{\\gamma}}$ should be thought of as the $S^{1}$\nprincipal bundle \n\\[\n\\mathbb{R}^{2n}\\times\\mathbb{U}(n)\\big\/\\mathbb{EU}(n)\\cong\\mathbb{R}^{2n}\\times S^{1},\n\\]\nwhere $\\mathbb{EU}(n)$ is the subgroup of $\\mathbb{U}(n)$ consisting\nof all the matrices with complex determinants equal to $1$ or $-1$.\nFor an arbitrary symplectic manifold $(M,\\omega)$, an almost complex\nstructure $J$ compatible with the symplectic structure $\\omega$\ncan always be chosen, and the corresponding unitary frame bundle (associated\nto $J$)\n\\[\n\\pi_{Fr_{J}^{u}}:Fr_{J}^{u}\\rightarrow M\n\\]\nis a $\\mathbb{U}(n)-$principal bundle (with $\\mathbb{U}(n)$ acting\nfrom the right in our setting). Therefore we can define the associated\nfiber bundle \n\\[\n\\pi_{\\Lambda_{J}}:\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)\\rightarrow M\n\\]\nas well as the $S^{1}$-principal bundle\n\\[\n\\pi_{\\Gamma_{J}^{2}}:\\Gamma_{J}^{2}=Fr_{J}^{u}\\big\/\\mathbb{EU}(n)\\rightarrow M,\n\\]\nwhich are the counterparts of $\\mathbb{R}^{2n}\\times\\Lambda(n)$ and\n$\\mathbb{R}^{2n}\\times\\mathbb{U}(n)\\big\/\\mathbb{EU}(n)$, respectively,\nin the general setting. \n\nThe space $\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)$ is indeed in\none-one correspondence to $\\Lambda_{pl}$. This is because each fiber\n$\\pi_{\\Lambda_{J}}^{-1}(p)$ of $\\Lambda_{J}$ with $p\\in M$ is exactly\nthe Lagrangian Grassmannian of the symplectic vector space $(T_{p}M,\\omega_{p})$\nwith respect to the compatible linear complex structure $J\\big|_{T_{p}M}$.\nIn the text we will use interchangably the spaces $\\Lambda_{J}$ and\n$\\Lambda_{pl}$. From the set-theoretic point of view, the advantage\nof thinking about $\\Lambda_{pl}$ is that for any symplectic flow\n$\\varphi$ on $M$, the tangent maps $\\varphi_{*}^{t}$ induces a\nflow on $\\Lambda_{pl}$ in a natural way. However, the advantage of\nconsidering the space $\\Lambda_{J}$ is that its relation with $\\Gamma_{J}^{2}$\nis straigthforward. \n\nSince $\\mathbb{O}(n)\\subset\\mathbb{EU}(n)$ and they are both closed\nsubgroups of $\\mathbb{U}(n)$, each $\\mathbb{O}(n)$ orbit in $Fr_{J}^{u}$\nlies in a single $\\mathbb{EU}(n)$ orbit, and there is a natural quotient\nmap $det_{J}^{2}$ from $Fr_{J}^{u}\\big\/\\mathbb{O}(n)$ to $Fr_{J}^{u}\\big\/\\mathbb{EU}(n)$,\nsending each $\\mathbb{O}(n)$ orbit to its containing $\\mathbb{EU}(n)$\norbit. Viewed in proper local trivializations of the bundles $\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)\\rightarrow M$\nand $\\Gamma_{J}^{2}=Fr_{J}^{u}\\big\/\\mathbb{EU}(n)\\rightarrow M$,\nsuch a quotient map takes the form\n\\[\nU\\times\\mathbb{U}(n)\\big\/\\mathbb{O}(n)\\ni(p,[A])\\mapsto(p,det_{\\mathbb{C}}^{2}A)\\in U\\times S^{1}\n\\]\nwith respect to some local charts over some open set $U$ of $M$.\nDue to this reason, we denote this quotient map by\n\\[\ndet_{J}^{2}:\\Lambda_{J}\\rightarrow\\Gamma_{J}^{2}.\n\\]\nand the following composition\n\\[\ndet_{J}^{2}\\circ\\tilde{\\gamma}:S^{1}\\rightarrow\\Gamma_{J}^{2}\n\\]\nis then the Malsov-Arnold map for a closed path $\\tilde{\\gamma}$\nin $\\Lambda_{J}$. When the bundle admits a trivialization $\\Gamma_{J}^{2}\\cong M\\times S^{1}$\nwith respect to some global section $\\mathfrak{s}$, such a trivialization\nleads to the definition of an index. To be precise, with the isomorphism\nfrom $\\Gamma_{J}^{2}$ to $M\\times S^{1}$ being denoted by $tr_{\\mathfrak{s}}$,\nan index $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma})$ for a closed\npath $\\tilde{\\gamma}$ of Lagrangian planes can be defined as the\ndegree of the map defined in the following composition:\n\\begin{equation}\nS^{1}\\xrightarrow{\\tilde{\\gamma}}\\Lambda_{J}\\xrightarrow{det_{J}^{2}}\\Gamma_{J}^{2}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}\\xrightarrow{pr_{S^{1}}}S^{1}.\\label{eq:ind_=00007B=00005Cpsi=00007D}\n\\end{equation}\nNote that a globally defined Lagrangian vector subbundle gives a global\nsection $\\sigma$ to $\\Lambda_{J}$(or $\\Lambda_{pl}$), and then\nthe composition $\\mathfrak{s}=det_{J}^{2}\\circ\\sigma$ is a global\nsection to $\\pi_{\\Gamma_{J}^{2}}$, which gives a trivialization $\\Gamma_{J}^{2}\\cong M\\times S^{1}$.\nAlso note that this is always the case where $M$ is the cotangent\nbundle of some manifold, and the Lagrangian subbundle can be chosen\nas the vertical distribution. In this case, the index $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma})$\ndefined above is exactly the Maslov index (with respect to $\\mathfrak{s}$)\nin the usual sense. \n\nThe discussion above shows why we become interested in the principal\n$S^{1}$ bundle $\\Gamma_{J}^{2}$ and the other related structures,\n$Fr_{J}^{u}$ and $\\Lambda_{J}$, over a symplectic manifold $(M,\\omega)$.\nIn some cases it would be more convenient to consider the principal\n$S^{1}$ bundle $\\Gamma_{J}\\cong Fr_{J}^{u}\\big\/\\mathbb{SU}(n)$,\nwhich is related to $\\Gamma_{J}^{2}$ by the relation $\\Gamma_{J}^{2}\\cong\\Gamma_{J}\\big\/\\{\\pm1\\}$.\nIn the following text, we call $\\pi_{\\Gamma_{J}}:\\Gamma_{J}\\rightarrow M$\n(or simply $\\Gamma_{J}$) and $\\pi_{\\Gamma_{J}^{2}}:\\Gamma_{J}^{2}\\rightarrow M$\nMaslov $S^{1}$ bundles. Note that although from the set-theoretic\npoint of view, the definitions of $Fr_{J}^{u}$, $\\Lambda_{J}$ and\n$\\Gamma_{J}$ depend on the choice of $J$, their bundle structures\nare independent of the choice of the compatible almost complex structure. \n\n\\subsection{Maslov data: a nonintegrable version of Maslov indices}\n\nGenerally speaking, the principal $S^{1}$ bundles $\\Gamma_{J}$ and\n$\\Gamma_{J}^{2}$ are not necesarrily trivial, and the mapping \\ref{eq:ind_=00007B=00005Cpsi=00007D}\nis not available (this is the case for $S^{2}$, as we will explain\nlater). However, the integration \\ref{eq:maslov index by integration}\ncan still be defined by treating $\\tilde{\\eta}$ as the pullback of\nany connection $1$-form of $\\Gamma_{J}^{2}$ (or alternatively $\\Gamma_{J}$)\non $\\Lambda_{J}$. Following such a perspective, we note that a global\nsection of $\\Gamma_{J}^{2}$ is just an integral manifold of a flat\nconnection (which is an integrable horizontal distribution) on $\\Gamma_{J}^{2}$.\nTo generalize the notion of Maslov indices for the case where $\\Gamma_{J}^{2}$\nis a nontrivial bundle, we can replace the ``integrable horizontal\ndistribution'' with simply ``a connection''. We introduce a nonintegrable\nversion of Maslov indices for smooth loops in $\\Lambda_{pl}$ (or\n$\\Lambda_{J}$) with respect to an arbitrary connection $1$-form\n$\\beta=\\eta\\cdot\\frac{\\partial}{\\partial\\theta}$ on $\\Gamma_{J}^{2}$. \n\\begin{defn}\n\\label{def:Nonintegrable-Maslov-Quantity for =00005CLambda} For any\nsmooth loop $\\tilde{\\gamma}:[0,1]\\rightarrow\\Lambda_{J}$, define\nits Maslov data with respect to $\\beta$ to be\n\\[\n\\tilde{\\mathfrak{m}}_{\\beta}(\\tilde{\\gamma})=\\int_{\\tilde{\\gamma}}\\tilde{\\eta}\n\\]\nwith $\\tilde{\\eta}$ being the pullback of $\\eta$ on $\\Lambda_{J}$\nvia $det_{J}^{2}$. \n\\end{defn}\n\nNote that whenever $\\tilde{\\gamma}$ is a smooth loop in $\\Lambda_{J}$,\n\\[\n\\gamma=det_{J}^{2}\\circ\\tilde{\\gamma}\n\\]\nis a smooth loop in $\\Gamma_{J}^{2}$. Moreover, it holds\n\n\\begin{equation}\n\\int_{\\gamma}\\eta=\\int_{\\tilde{\\gamma}}\\tilde{\\eta}=\\tilde{\\mathfrak{m}}_{\\beta}(\\tilde{\\gamma}),\\label{eq:Maslov-quantity for a path\/loop}\n\\end{equation}\nand hence it will be convenient to consider $\\gamma$ instead of $\\tilde{\\gamma}$\nand think of $\\tilde{\\mathfrak{m}}_{\\beta}(\\tilde{\\gamma})$ as a\nquantity of $\\gamma$. We will use the symbol $\\mathfrak{m}_{\\beta}(\\gamma)$\nfor the integration $\\int_{\\gamma}\\eta$ and call it the Maslov data\nof $\\gamma$ with respect to $\\beta$. \n\nWhen the horizontal distribution $\\ker\\beta$ has an integral manifold\n$\\mathcal{S}_{0}$ being a one-sheet covering over $M$, Definition\n\\ref{def:Nonintegrable-Maslov-Quantity for =00005CLambda} gives the\nMaslov indices in the usual sense with respect to the Malsov cycle\n$\\mathfrak{M}_{0}=(det_{J}^{2})^{-1}(\\mathcal{S}_{0})$ (or to the\nsecion $\\mathfrak{s}$). Note that this is the case in \\cite{contreras2003asymptotic}\nwhere a dynamical system is considered in a vicinity of a connected\nsubmanifold $\\Sigma$ in $M$ on which (the restriction of) the bundle\nof Lagrangian planes $\\Lambda_{pl}$ can be trivialized as $\\Sigma\\times\\Lambda(n)$.\nAlso note that in general, $\\ker\\beta$ being flat does not necessary\nmean that the integral manifolds are single-sheeted coverings. A discussion\ntaking care of a nontrivial bundle $\\Gamma_{J}^{2}$ with flat connections\nwould however be beyond the scope this work, and therefore we will\nnot go into details for this issue. \n\\begin{rem}\nWe avoid using the term $index$ just for emphasizing that the Maslov\ndata is not necessarily a topological quantity, even when $\\tilde{\\gamma}$\nis a loop. \n\\end{rem}\n\n\n\\subsection{Purposes, Main Results and Layout}\n\nThis work is intended to be an exploration of the interplay between\ndynamics on $M$ and the geometry of $\\Gamma_{J}$(and\/or $\\Gamma_{J}^{2}$),\nas well as an investigation into the notion of Maslov data proposed\nin Definition \\ref{def:Nonintegrable-Maslov-Quantity for =00005CLambda}\nas a generalized\/nonintegrable version of ordinary Maslov indices. \n\nThe dynamics that we are interested in on the manifolds will be symplectic\ngroup actions. Note that given the flow of an arbitrary symplectic\nvector field $X$, $\\varphi:\\mathbb{R}\\times M\\rightarrow M$, the\ntangent map $\\varphi_{*}^{t}$ maps each Lagrangian space to another.\nNamely, $\\varphi_{*}$ defines a flow on $\\Lambda_{pl}$ which covers\n$\\varphi$. Since each trajectory $\\tilde{\\gamma}$ of $\\varphi_{*}$\nis mapped to a path $\\gamma$ in $\\Gamma_{J}^{2}$ by $\\gamma=det_{J}^{2}\\circ\\tilde{\\gamma}$,\nit is natural to expect that, under some conditions, the composition\n\\[\ndet_{J}^{2}\\circ\\varphi_{*}:\\mathbb{R}\\times\\Lambda_{pl}\\rightarrow\\Gamma_{J}^{2}\n\\]\nfactors through the space $\\mathbb{R}\\times\\Gamma_{J}^{2}$ and induces\na flow $\\varphi_{\\Gamma^{2}}$ on $\\Gamma_{J}^{2}$. Such a factorization\nindeed exists when the vector field $X$ is an infinitesimal generator\nof a symplectic group action by a compact Lie group $G$. Based on\nthis reason, we focus our study on compact and connected Lie group\nactions. \n\nIn the following we introduce the main results of this work. The simpliest\ncase where the Maslov bundles are non-trivial is given by $M=S^{2}$.\nNote that $S^{2}$ is a symplectic homogeneous space with a transitive\nsymplectic $\\mathbb{SO}(3)$ action. Also note that $H_{dR}^{1}(S^{2})=0$\nand hence any symplectic vector field on $S^{2}$ is Hamiltonian.\nA discussion of the $\\mathbb{SO}(3)$ action on $S^{2}$ (Section\n\\ref{sec:compact-group-actions}) shows that the lifted $\\mathbb{SO}(3)$\nactions on the Maslov $S^{1}$ bundles $\\Gamma_{S^{2}}$ and $\\Gamma_{S^{2}}^{2}$\nare transitive (Propersition \\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}),\nand the Hamiltonians for the infinitesimal generators can be expressed\nin terms of an invariant connection $1$-form of the bundle $\\Gamma_{S^{2}}\\rightarrow S^{2}$\n(Proposition \\ref{prop:symplectic=00003DHamiltonian on S^2}). These\nproperties are then extended to more general settings. To be specific,\nwe obtain the following result for symplectic homogeneous spaces:\n\\begin{thm}\n\\label{thm:=00005C=00007BGamma=00007D_J-homogeneous}Let $G$ be a\ncompact Lie group acting transitively and symplectically on $M$.\nIf the first real Chern class of the Maslov $S^{1}$ bundle $\\Gamma_{J}$\nis nonvanishing, then $\\Gamma_{J}$ is also a homogeneous $G$-space.\n\\end{thm}\n\n\\noindent We moreover get the following theorem which exitends the\nresult obtained in \\cite{lupton1995cohomologically,kaoru1992obstruction,cho2012chern}\nabout symplectic circle actions being Hamiltonian on monotone manifolds:\n\\begin{thm}\n\\label{thm:=00005C=00007Bomega=00007D=00003D=00003Dr=00005Bc=00005D:Symplectic--Hamiltonian}Let\n$(M,\\omega)$ be a symplectic manifold, and let $\\pi_{\\Gamma}:\\Gamma_{J}\\rightarrow M$\nbe its Maslov $S^{1}$ bundle with the first real Chern class $c_{\\Gamma}\\in H_{dR}^{2}(M)$.\nIf it satisfies $[\\omega]=r\\cdot c_{\\Gamma}$ for some real number\n$r$, then any symplectic action $\\Phi$ on $M$ by a compact Lie\ngroup $G$ is Hamiltonian. More specifically, for any $v\\in\\mathfrak{g}$,\nthere exists a Hamiltonian $H_{v}$ of $X_{v}$ such that $\\pi_{\\Gamma}^{*}(H_{v})=r\\cdot f_{\\beta}(\\mathcal{X}_{v})$\nfor some real number $r\\neq0$, with $\\beta=f_{\\beta}\\cdot\\frac{\\partial}{\\partial\\theta}$\nbeing a connection $1$-form invariant under the lifted $G$-action\n$\\Phi_{\\Gamma}$.\n\\end{thm}\n\n\\begin{rem}\n$X_{v}$ and $\\mathcal{X}_{v}$ in the statement of Theorem \\ref{thm:=00005C=00007Bomega=00007D=00003D=00003Dr=00005Bc=00005D:Symplectic--Hamiltonian}\nabove stand for the infinitesimal generators associated to $v$ of\nthe action $\\Phi$ on $M$ and the lifted action $\\Phi_{\\Gamma}$\non $\\Gamma_{J}$, respectively.\n\\end{rem}\n\nWe also discuss the Maslov data of symplectic circle actions. When\n$G=S^{1}$, the orbits of $\\Phi_{\\Gamma}$ in $\\Gamma_{J}$ covering\nthe same orbit of $\\Phi$ in $M$ have the same Maslov data. In other\nwords, for $w,w'\\in\\pi_{\\text{\\ensuremath{\\Gamma}}}^{-1}(p)$ with\n$\\gamma_{w}(z)=\\Phi_{\\Gamma}^{z}(w)$ and $\\gamma_{w'}(z)=\\Phi_{\\Gamma}^{z}(w')$,\n$\\mathfrak{m}_{\\beta}(\\gamma_{w})=\\mathfrak{m}_{\\beta}(\\gamma_{w'})$\nfor any connection $1$-form $\\beta$ on $\\Gamma_{J}$. Therefore,\nit defines a smooth function $Q_{\\theta}$ (which we call the $\\beta$-Maslov\ndata of $\\Phi$) on $M$ with $Q_{\\theta}(p)=\\mathfrak{m}_{\\beta}(\\gamma_{w})$.\nAlthough with a different connection $\\beta$ the function $Q_{\\theta}$\nmay be different, its values at the fixed points of the action $\\Phi$\nturn out to be independent of $\\beta$. When $[\\omega]=r\\cdot c_{\\Gamma}$,\n$Q_{\\theta}$ (after scaling) is a Hamiltonian of the action $\\Phi$.\n\nThis paper is organized as follows.\n\nIn Section \\ref{sec:Basics of Maslov-Index}, we discuss some basic\nproperties of Maslov indices from the perspective from the geometry\nof $\\Gamma_{J}^{2}$. In Section \\ref{sec:Basics-of-Principal S^1 bundles},\nwe recall some basics about principal $S^{1}$ bundles which will\nbe used in the later discussion. In Section \\ref{sec:compact-group-actions},\nwe discuss the lift of the $G$-action $\\Phi$ and study the case\nof $M=S^{2}$, which serves as an example in which the Maslov $S^{1}$\nbundle $\\Gamma_{S^{2}}$ is non-trivial. We explain that the lifted\n$\\mathbb{SO}(3)$ action acts transitively on $\\Gamma_{S^{2}}$ (Proposition\n\\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}) and\nshow that the Hamiltonians of its infinitesimal generators can be\nwritten down in terms of a connection $1$-form which is invariant\nunder the lifted $G$-action on $\\mathbb{SO}(3)\\cong\\Gamma_{S^{2}}$\n(Proposition \\ref{prop:symplectic=00003DHamiltonian on S^2}). In\nSection \\ref{sec:dynamics-on Maslov S^1 bundles}, we extend Proposition\n\\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D} and\n\\ref{prop:symplectic=00003DHamiltonian on S^2} to Theorem \\ref{thm:=00005C=00007BGamma=00007D_J-homogeneous}\nand \\ref{thm:=00005C=00007Bomega=00007D=00003D=00003Dr=00005Bc=00005D:Symplectic--Hamiltonian}.\nIn Section \\ref{sec:Beta-Maslov-Quantity for S^1-action}, we define\nthe $\\beta$-Maslov data of a symplectic $S^{1}$ action and discuss\nits properties, especially its values at the fixed points of the action.\n\n\\subsection{Notations}\n\nIn this paper we always assume $G$ to be a compact Lie group acting\nsympletically on $(M,\\omega)$. Denote the Lie algebra of $G$ by\n$\\mathfrak{g}$. The action of $G$ on $M$ is denoted by $\\Phi$.\nThe lifted actions on $\\Gamma_{J}$ and $\\Gamma_{J}^{2}$ are denoted\nby $\\Phi_{\\Gamma}$ and $\\Phi_{\\Gamma^{2}}$, respectively. For $v\\in\\mathfrak{g}$,\n$X_{v}$ is the corresponding infinitesimal generator of $\\Phi$ on\n$M$ , $\\mathcal{X}_{v}$ stands for the generator on $\\Gamma_{J}$(or\n$\\Gamma_{J}^{2}$), and $\\mathbb{X}_{v}$ is the generator of the\naction $\\Phi_{*}$ on $\\Lambda_{pl}$ which is induced naturally by\nthe tangent maps.\n\nWe denote by $\\frac{\\partial}{\\partial\\theta}$ the vector field on\n$S^{1}$ defined by $\\frac{d}{d\\theta}e^{2\\pi i\\cdot\\theta}$. Note\nthat, on a principal $S^{1}$ bundle $P$, there is a\/an structural\/inherent\n$S^{1}$ action. We also use the symbol $\\frac{\\partial}{\\partial\\theta}$\nto denote the corresponding infinitesimal generator of this $S^{1}$\naction.\n\n\\section{\\label{sec:Basics of Maslov-Index}Basic properties of Maslov indices}\n\n\\subsection{\\label{subsec:Lagrangian-subbundles and Maslov-indices}Lagrangian\nsubbundles and Maslov indices}\n\nLet $\\mathcal{N}$ be a submanifold in $M$ and $\\mathfrak{L}_{\\mathcal{N}}\\subset TM$\nbe a Lagrangian subbundle of $TM$ over $\\mathcal{N}$. That is, $\\mathfrak{L}_{\\mathcal{N}}\\big|_{b}$\nis a Lagrangian subspace for any $b\\in\\mathcal{N}$. It is straightward\nto see that $\\mathfrak{L}_{\\mathcal{N}}$ always induces a smooth\nsection $\\sigma_{\\mathcal{N}}$ from $\\mathcal{N}$ to $\\Lambda_{pl}\\cong\\Lambda_{J}=Fr_{J}^{u}\\big\/\\mathbb{O}(n)$\nwhich assigns to each $b\\in\\mathcal{N}$ the Lagrangian plane $\\mathfrak{L}_{\\mathcal{N}}\\big|_{b}$.\nMoreover, when $\\mathfrak{L}_{\\mathcal{N}}$ is orientable, it also\ninduces a section $\\tilde{\\sigma}_{\\mathcal{N}}$ from $\\mathcal{N}$\nto $Fr_{J}^{u}\\big\/\\mathbb{SO}(n)$. We give a brief account of the\nlatter case.\n\nWhen $\\mathfrak{L}_{\\mathcal{N}}$ is orientable, we can first fix\nan orientation. For any $b\\in\\mathcal{N}$, there is a neighbourhood\n$U_{b}$ in $\\mathcal{N}$ on which it admits an ordered local frame\n$f_{U_{b}}^{*}=(e_{1},...,e_{n})$ of $\\mathfrak{L}_{\\mathcal{N}}$\nwhich fits the orientation. With some modification, it can be extended\nto a(n) (ordered) unitary frame\n\\[\nf_{U_{b}}=(e_{1},...,e_{n},s_{1},...,s_{n})\n\\]\non $U_{b}$ for $TM$ with $s_{i}=Je_{i}$. Then \n\\[\n\\sigma_{b}:U_{b}\\ni x\\mapsto f_{U_{b}}(x)\\in Fr_{J}^{u}\n\\]\nis a local section for $Fr_{J}^{u}$. By composing with the quotient\nmap $q_{\\mathbb{SO}(n)}$ from $Fr_{J}^{u}$ to $Fr_{J}^{u}\\big\/\\mathbb{SO}(n)$\nwe get a local section $\\tilde{\\sigma}_{\\mathcal{S},b}$ from $U_{b}$\nto $Fr_{J}^{u}\\big\/\\mathbb{SO}(n)$. Namely,\n\\[\n\\tilde{\\sigma}_{\\mathcal{S},b}=q_{\\mathbb{SO}(n)}\\circ\\sigma_{b}.\n\\]\nThe collection $\\{U_{b}\\big|b\\in\\mathcal{N}\\}$ then constitutes an\nopen cover of $\\mathcal{N}$ over each element of which there is a\n(smooth) section $\\sigma_{\\mathcal{N},b}$. It remains to check that\nwhen $U_{b}\\cap U_{b'}\\neq\\emptyset$, $\\tilde{\\sigma}_{\\mathcal{S},b}$\nagrees with $\\tilde{\\sigma}_{\\mathcal{S},b'}$ on $U_{b}\\cap U_{b'}$.\nSince both $f_{U_{b}}^{*}$ and $f_{U_{b'}}^{*}$($=(e'_{1},...,e'_{n})$)\ncan be taken as orthonormal frames with respect to the Euclidean metric\n$g_{J}$ that fit the same orientation, at each $x\\in U_{b}\\cap U_{b'}$,\nthey are related by a matrix $C$ from $\\mathbb{SO}(n)$ via\n\\[\n(e'_{1},...,e'_{n})=(e_{1},...,e_{n})\\cdot C.\n\\]\nFor $f_{U_{b'}}=(e'_{1},...,e'_{n},s'_{1},...,s'_{n})$, it holds\n\\[\ns'_{i}=J\\cdot e'_{i},\n\\]\nand then it is straightforward to check that\n\\[\n(e'_{1},...,e'_{n},s'_{1},...,s'_{n})=(e_{1},...,e_{n},s_{1},...,s_{n})\\cdot C.\n\\]\nIt implies $\\tilde{\\sigma}_{\\mathcal{N},b'}(x)=\\tilde{\\sigma}_{\\mathcal{N},b}(x)$.\nThen we get $\\tilde{\\sigma}_{\\mathcal{N}}$ by simply piecing the\nlocal sections $\\tilde{\\sigma}_{\\mathcal{N},b}$ together.\n\nNow we consider the case $\\mathcal{N}=M$. Suppose that $\\mathcal{S}\\subset M$\nis a Lagrangian submanifold, and that $\\Gamma_{J}^{2}$ admits a global\nsection $\\mathfrak{s}^{2}$ which induces a trivialization\n\\[\ntr_{\\mathfrak{s}^{2}}:\\Gamma_{J}^{2}\\rightarrow M\\times S^{1}.\n\\]\nLet $\\gamma$ be a loop on $\\mathcal{S}$. Then $\\sigma_{\\mathcal{S}}\\circ\\gamma:t\\mapsto T_{\\gamma(t)}\\mathcal{S}$\nis a loop in $\\Lambda_{J}$. Composed with the map $det_{J}^{2}$\nit becomes a loop in $\\Gamma_{J}^{2}$, and the Maslov index $\\mathfrak{m}_{\\mathfrak{s}^{2}}(\\gamma)$\nis the degree of the following map\n\\[\nS^{1}\\stackrel{\\sigma_{\\mathcal{S}}\\circ\\gamma}{\\longrightarrow}\\Lambda_{J}\\stackrel{det_{J}^{2}}{\\longrightarrow}\\Gamma_{J}^{2}\\stackrel{tr_{\\mathfrak{s}^{2}}}{\\longrightarrow}M\\times S^{1}\\stackrel{pr_{S^{1}}}{\\longrightarrow}S^{1}.\n\\]\nNow suppose that $\\mathcal{S}$ is an orientable Lagrangian submanifold\nand that $\\Gamma_{J}$ admits a global section $\\mathfrak{s}$. Then\nwe have a global section of $\\Gamma_{J}^{2}$, which is given by $\\mathfrak{s}^{2}=q_{\\pm1}(\\mathfrak{s})$.\nHere $q_{\\pm1}:\\Gamma_{J}\\rightarrow\\Gamma_{J}^{2}\\big\/\\{\\pm1\\}$\nis the quotient map. As a consequence, we have the following commutative\ndiagram, which suggests that the Maslov index $\\mathfrak{m}_{\\mathfrak{s^{2}}}(\\gamma)$\nshould be an even number:\n\\begin{equation}\n\\text{\\ensuremath{\\begin{CD}S^{1}@>\\tilde{\\sigma}_{\\mathcal{S}}\\circ\\gamma>>Fr_{J}^{u}\\big\/\\mathbb{SO}(n)@>det_{J}>>\\Gamma_{J}@>tr_{\\mathfrak{s}}>>M\\times S^{1}@>pr_{S^{1}}>>S^{1}\\\\\n@|@Vq_{\\pm1}VV@Vq_{\\pm1}VV@Vq_{\\pm1}VV@V{\\scriptscriptstyle square}VV\\\\\nS^{1}@>\\sigma_{\\mathcal{S}}\\circ\\gamma>>\\Lambda_{J}@>det_{J}^{2}>>\\Gamma_{J}^{2}@>tr_{\\mathfrak{s}^{2}}>>M\\times S^{1}@>pr_{S^{1}}>>S^{1}\n\\end{CD}}}.\\label{Diagram:oriented Maslov indices and Maslov indices}\n\\end{equation}\n\n\\begin{rem}\nIf $\\mathfrak{L}$ is a Lagrangian subbundle of $TM$ over $M$, it\ninduces a global section $\\sigma:M\\rightarrow\\Lambda_{J}$ by sending\neach $b\\in M$ to the fiber $\\mathfrak{L}_{b}$ of $\\mathfrak{L}$\nabove it. Then \n\\[\n\\mathfrak{s}^{2}:=det_{J}^{2}\\circ\\sigma\n\\]\nis a global section to the bundle $\\Gamma_{J}^{2}$. Furthermore,\nif $\\mathfrak{L}$ is orientable, then it also induces a trivialization\nfor $\\Gamma_{J}$.\n\\end{rem}\n\n\n\\subsection{\\label{subsec:Simply connectedness and Local Maslov indices}Maslov\nindices on simply connected spaces}\n\nGenerally speaking, a loop $\\tilde{\\gamma}$ of Lagrangian planes\n(or its image $\\gamma$ in $\\Gamma_{J}^{2}$) will have different\nMaslov indices with respect to different global sections of $\\Gamma_{J}^{2}$.\nHowever, when the manifold $M$ is simply connected, Malsov indices\nof a loop with respect to different sections\/Maslov cycles will be\nidentical. We give a brief explanation of this fact from the perspective\nof the geometry of the bundle $\\Gamma_{J}^{2}$. \n\nSuppose that $\\mathfrak{s}$ and $\\mathfrak{s}'$ are sections of\n$\\text{\\ensuremath{\\Gamma^{2}}}$. In terms of the trivialization\n$\\text{\\ensuremath{\\Gamma^{2}}}\\xrightarrow{tr_{\\mathfrak{s}'}}M\\times S^{1}$\nwith respect to $\\mathfrak{s}'$, the section $\\mathfrak{s}:M\\rightarrow M\\times S^{1}$\ntakes the form $\\mathfrak{s}(p)=(p,\\theta_{p})$ with $\\theta:p\\mapsto\\theta_{p}$\nbeing a smooth map from $M$ to $S^{1}$. For a loop $\\gamma\\subset\\text{\\ensuremath{\\Gamma^{2}}}$,\nit takes the form $S^{1}\\ni z\\mapsto\\big(\\lambda(z),\\tau'(z)\\big)\\in M\\times S^{1}$\nin the trivialization $\\text{\\ensuremath{\\Gamma^{2}}}\\xrightarrow{tr_{\\mathfrak{s}'}}M\\times S^{1}$,\nand the form $S^{1}\\ni z\\mapsto\\big(\\lambda(z),\\tau(z)\\big)\\in M\\times S^{1}$\nin $\\text{\\ensuremath{\\Gamma^{2}}}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}$,\nwith the maps $\\lambda:S^{1}\\rightarrow M$, $\\tau',\\tau:S^{1}\\rightarrow S^{1}$.\nThe corresponding Maslov indices $\\mathfrak{m}_{\\mathfrak{s}'}(\\gamma)$\nand $\\mathfrak{m}_{\\mathfrak{s}}(\\gamma)$ are then the degrees of\nthe maps $\\tau'$ and $\\tau$, respectively. It holds that $\\tau'(z)=\\tau(z)\\cdot\\theta_{\\lambda(z)}$.\nSince $M$ is simply connected, the mapping $z\\rightarrow\\theta_{\\lambda(z)}$\nwhich is subject to the following factorization\n\\[\nS^{1}\\xrightarrow{\\lambda}M\\xrightarrow{\\theta}S^{1}\n\\]\nhas degree $0$, and hence $\\tau$ and $\\tau'$ have the same degree.\n\n\\section{\\label{sec:Basics-of-Principal S^1 bundles}Basics on Principal $S^{1}$-Bundles}\n\nFor later purposes, we recall in the section basic properties of a\nprincipal $S^{1}$ bundle. A principal $S^{1}$-bundle consists of\na total space $P$ on which there is a free $S^{1}$ action, its orbifold\n$B=P\\big\/S^{1}$, and the quotient map $\\pi_{P}:P\\rightarrow B$.\nTo distinguish it from other group actions, in the rest of the paper\nwe will call this $S^{1}$ action the inherent\/structural $S^{1}$\naction. For simplicity, we will also call $P$ the principal bundle\nwhen it is clear from the context.\n\nA connection $\\mathcal{H}$ on the bundle $\\pi_{P}:P\\rightarrow B$\nis a horizontal distribution invariant under the inherent $S^{1}$\naction, and the tangent bundle $TP$ is split as a direct sum $\\mathcal{H}\\oplus\\mathcal{V}$\nwith $\\mathcal{V}=\\ker\\pi_{P,*}$ being the vertical distribution.\nThe Lie algebra of $S^{1}$ is $T_{1}S^{1}=\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta}$\nwith $\\frac{\\partial}{\\partial\\theta}=\\frac{d}{d\\theta}\\bigg|_{\\theta=0}e^{i\\cdot2\\pi\\theta}.$\nThe following map\n\\[\n\\eta:P\\times\\bigg(\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta}\\bigg)\\rightarrow\\mathcal{V}\n\\]\nwith \n\\[\n\\bigg(p,r\\frac{\\partial}{\\partial\\theta}\\bigg)\\mapsto\\frac{d}{d\\theta}\\bigg|_{\\theta=0}(p\\cdot e^{i\\cdot2\\pi r\\cdot\\theta})\n\\]\nis a vector bundle isomorphism. For simplicity we also denote the\nvector $\\frac{d}{d\\theta}\\bigg|_{\\theta=0}(p\\cdot e^{i\\cdot2\\pi r\\cdot\\theta})$\nby $r\\frac{\\partial}{\\partial\\theta}$. The connection $1$-form associated\nto $\\mathcal{H}$ is\n\\[\n\\alpha:TP\\xrightarrow{pr_{\\mathcal{V}}}\\mathcal{V}\\cong P\\times(\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta})\\rightarrow\\mathbb{R}\\cdot\\frac{\\partial}{\\partial\\theta}\n\\]\nwith $pr_{\\mathcal{V}}$ being the projection associated to the splitting\n$TP=\\mathcal{H}\\oplus\\mathcal{V}$ onto $\\mathcal{V}$. $\\alpha$\nis $S^{1}$ invariant, and there is an $S^{1}$-invariant $1$-form\n$f_{\\alpha}$ such that $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$.\nBy abusing the terminology we also call $f_{\\alpha}$ a connection\n$1$-form on $P$.\n\nNote that $f_{\\alpha}(\\frac{\\partial}{\\partial\\theta})\\equiv1$. The\n$S^{1}$-invariance of $f_{\\alpha}$ implies $\\mathcal{L}_{\\frac{\\partial}{\\partial\\theta}}f_{\\alpha}=0$.\nThen by Cartan's formula, we get\n\\[\n0=\\mathcal{L}_{\\frac{\\partial}{\\partial\\theta}}f_{\\alpha}=\\iota_{\\frac{\\partial}{\\partial\\theta}}df_{\\alpha}+d\\bigg(f_{\\alpha}\\big(\\frac{\\partial}{\\partial\\theta}\\big)\\bigg)=\\iota_{\\frac{\\partial}{\\partial\\theta}}df_{\\alpha}.\n\\]\nWith $pr_{\\mathcal{H}}$ being the projection of $\\mathcal{H}\\oplus\\mathcal{V}$\nonto $\\mathcal{H}$, the equation above implies that \n\\[\ndf_{\\alpha}(pr_{\\mathcal{H}}\\cdot,pr_{\\mathcal{H}}\\cdot)=df_{\\alpha}.\n\\]\nSince $df_{\\alpha}$, $\\mathcal{H}$ and $\\mathcal{V}$ are all $S^{1}$-invariant,\nthis implies that there is a closed $2-$form $\\Omega_{\\alpha}$ on\n$B$ such that $\\pi_{P}^{*}(\\Omega_{\\alpha})=df_{\\alpha}$. $[\\Omega_{\\alpha}]\\in H_{dR}^{2}(B)$\nis independent of $\\alpha$ and is called the characteristic class\nof the bundle $P\\rightarrow B$.\n\nBy the definition of $f_{\\alpha}$, $\\ker f_{\\alpha}=\\mathcal{H}$.\nAccording to the Frobenius integrability theorem, $\\mathcal{H}$ is\nan integrable distribution if and only if $f_{\\alpha}\\wedge df_{\\alpha}=0$.\nIn this case, such an identity actually requires $df_{\\alpha}$ to\nbe $0$. This is because if $df_{\\alpha}\\neq0$, then there exist\n$u,v\\in\\mathcal{H}_{p}=\\ker f_{\\alpha}\\big|_{p}$ such that $df_{\\alpha}(u,v)\\neq0$,\nand then \n\\[\nf_{\\alpha}\\wedge df_{\\alpha}(\\frac{\\partial}{\\partial\\theta},u,v)=f_{\\alpha}(\\frac{\\partial}{\\partial\\theta})\\cdot df_{\\alpha}(u,v)=df_{\\alpha}(u,v)\\neq0,\n\\]\nyielding a contradiction. As a result, the corresponding $2$-form\n$\\Omega_{\\alpha}=0$, and hence $[\\Omega_{\\alpha}]=0$.\n\nConversely, if $[\\Omega_{\\alpha}]=0$, then there exists a $1$-form\n$\\tau$ on $B$ such that $d\\tau=\\Omega_{\\alpha}$. Let $\\alpha'=\\alpha-\\pi_{\\Gamma_{J}}^{*}(\\tau)\\cdot\\frac{\\partial}{\\partial\\theta}$.\nThen $\\alpha'$ is also $S^{1}$-invariant with\n\\[\n\\alpha'(r\\frac{\\partial}{\\partial\\theta})=\\alpha(r\\frac{\\partial}{\\partial\\theta})=r\\frac{\\partial}{\\partial\\theta},\n\\]\nnamely, it is also a connection $1$-form, and $\\mathcal{H}'=\\ker\\alpha'$\ndefines another connection on $P\\rightarrow B$. With $f_{\\alpha'}\\cdot\\frac{\\partial}{\\partial\\theta}=\\alpha'$,\nit holds\n\\[\nf_{\\alpha'}=f_{\\alpha}-\\pi_{\\Gamma_{J}}^{*}(\\tau).\n\\]\nThen \n\\[\ndf_{\\alpha'}=\\pi_{\\Gamma_{J}}^{*}(\\Omega_{\\alpha})-\\pi_{\\Gamma_{J}}^{*}(d\\tau)=0,\n\\]\nimplying $\\mathcal{H}'$ to be integrable. Therefore, the characterstic\nclass of a principal $S^{1}$-bundle is zero if and only if the bundle\nhas an integrable connection.\n\n\\section{\\label{sec:compact-group-actions}Compact group actions on Maslov\n$S^{1}$ bundles}\n\nLet\n\\[\n\\Phi:G\\times M\\rightarrow M\n\\]\nbe a symplectic left action on $M$ by a compact Lie group $G$ .\nNamely, $\\Phi^{h'h}=\\Phi^{h'}\\circ\\Phi^{h}$ and $\\Phi^{h}{}^{*}(\\omega)=\\omega$\nfor $h,h'\\in G$. In this secion, we first show that $\\Phi$ can be\nlifted to a $G$-action on $\\Gamma_{J}$. Namely, there is a $G$-action\n$\\Phi_{\\Gamma}$ on $\\Gamma_{J}$ that covers $\\Phi$. Moreover, $\\Phi_{\\Gamma}$\ncommutes with the inherent $S^{1}$ action on $\\Gamma_{J}$. Then\nwe study such a lifted $G$-action for the case where $M$ is a homogeneous\n$G$-space.\n\\begin{defn}\n\\label{def:group action on the bundle}Let $P\\rightarrow B$ be a\nprincipal $S^{1}$-bundle. A group action on the bundle $P\\rightarrow B$\n(or simply saying the bundle $P$) by $G$ is a $G$-action on the\nmanifold $P$ such that it commutes with the inherent $S^{1}$-action.\n\\end{defn}\n\n\n\\subsection{$G$-actions on $Fr_{J}^{u}$ and $\\Gamma_{J}$}\n\nThe $G$-action $\\Phi$ on $M$ can be lifted to a $G$-action on\nthe bundle $Fr_{J}^{u}\\rightarrow M$ by resorting to a $G$-equivariant\nalmost complex structure $J$. In the following we briefly explain\nhow this is done.\n\nLet $g$ be an arbitrary Riemannian metric on $M$, and let $dh$\nbe a probability measure on $G$ which is invariant under the right\ntranslations. Define a new Riemannian metric $\\bar{g}$ with\n\\[\n\\bar{g}(u,v)=\\int_{G}\\Phi_{h}{}^{*}(g)(u,v)dh\n\\]\nwith $u,v\\in T_{x}M$ and $x\\in M$. Then $\\bar{g}$ is invariant\nunder the (left) $G$-action $\\Phi$. Let $\\mathcal{A}$ be the vector\nbundle isomorphism on $TM$ defined by \n\\begin{equation}\n\\omega(u,\\cdot)=\\bar{g}(\\mathcal{A}u,\\cdot)\\label{eq:def-=00005Cmathcal=00007BA=00007D}\n\\end{equation}\nand let \n\\begin{equation}\n\\bar{J}=\\mathcal{A}^{-1}\\sqrt{-\\mathcal{A}^{2}}.\\label{eq:def-=00005Cbar=00007BJ=00007D}\n\\end{equation}\nThen $\\text{\\ensuremath{\\bar{J}}}$ is an almost complex structure\ncompatible with $\\omega$, and $g_{\\bar{J}}(\\cdot,\\cdot)=\\omega(\\bar{J}\\cdot,\\cdot)$\ndefines a Riemannian metric. For convenience we denote by $h_{*}$\nthe pushforward $\\Phi_{h,*}$. It is a standard result that $\\bar{J}$\ncommutes with $h_{*}$ for all $h\\in G$, and $g_{\\bar{J}}$ is then\ninvariant under the $G$-action. For completeness we give an argument\nin the following.\n\\begin{lem}\n\\label{lem:G-invariant-unitary-structure}Suppose that $\\bar{g}$\nis a Riemannian metric invariant under the $G$-action $\\Phi$. Let\n$\\mathcal{A}$ and $\\bar{J}$ be endomorphisms on $TM$ defined respectively\nby (\\ref{eq:def-=00005Cmathcal=00007BA=00007D}) and (\\ref{eq:def-=00005Cbar=00007BJ=00007D})\nabove. Then for any $h\\in G$, it holds\n\\begin{equation}\nh_{*}\\circ\\bar{J}=\\bar{J}\\circ h_{*}.\\label{eq:h^*J=00003DJh^*}\n\\end{equation}\nand\n\\begin{equation}\nh^{*}g_{\\bar{J}}(\\cdot,\\cdot)=g_{\\bar{J}}(\\cdot,\\cdot).\\label{eq:h^*g=00003Dg}\n\\end{equation}\nAs a consequence, $h_{*}$ maps the unitary frame bundle $Fr_{\\bar{J}}^{u}$\nto itself. \n\\end{lem}\n\n\\begin{proof}\nSuppose that (\\ref{eq:h^*J=00003DJh^*}) holds, then \n\\[\nh^{*}g_{\\bar{J}}(\\cdot,\\cdot)=\\omega(\\bar{J}\\circ h_{*}\\cdot,h_{*}\\cdot)=\\omega(h_{*}\\circ\\bar{J}\\cdot,h_{*}\\cdot)=g_{\\bar{J}}(\\cdot,\\cdot)\n\\]\ngives (\\ref{eq:h^*g=00003Dg}).\n\nWe start to prove (\\ref{eq:h^*J=00003DJh^*}) by looking at the bundle\nisomorphism $\\mathcal{A}$. Since $G$ acts symplectically on $M$,\nit holds\n\\[\n\\begin{aligned}\\bar{g}(\\mathcal{A}h_{*}u,h_{*}\\cdot) &  & = & \\omega(h_{*}u,h_{*}\\cdot)\\\\\n &  & = & \\omega(u,\\cdot)\\\\\n &  & = & \\bar{g}(\\mathcal{A}u,\\cdot)\\\\\n &  & = & \\bar{g}(h_{*}\\mathcal{A}u,h_{*}\\cdot)\n\\end{aligned}\n.\n\\]\nNow that $\\mathcal{A}$ is bijective and $u$ runs over $TM$, this\nimplies \n\\[\n\\mathcal{A}h_{*}=h_{*}\\mathcal{A}.\n\\]\nConsequently, $h_{*}$ also commutes with $\\mathcal{A}^{-1}$ and\n$-\\mathcal{A}^{2}$. For proving (\\ref{eq:h^*J=00003DJh^*}), it remains\nto show that \n\\[\n\\mathcal{B}h_{*}=h_{*}\\mathcal{B}\n\\]\nwith $\\mathcal{B}=\\sqrt{-\\mathcal{A}^{2}}$, i.e. $\\mathcal{B}^{2}=-\\mathcal{A}^{2}$. \n\nNote that, at each $x$ in $M$, $\\mathcal{B}_{x}^{2}=\\mathcal{B}^{2}\\big|_{T_{x}M}$\nand $\\mathcal{B}_{x}$ are both self-adjoint positive operators with\nrespect to $\\bar{g}$, and a vector $v\\in T_{x}M$ is an eigenvector\nof $\\mathcal{B}_{x}^{2}$ with eigenvalue $\\lambda^{2}$ if and only\nif it is an eigenvector of $\\mathcal{B}_{x}$ with eigenvalue $\\lambda>0$,\ni.e.\n\\[\n\\mathcal{B}_{x}^{2}(v)=\\lambda^{2}v\\iff\\mathcal{B}_{x}(v)=\\lambda v.\n\\]\nResorting to the commutativity, we have\n\\[\n\\mathcal{B}_{h\\cdot x}^{2}h_{*}(v)=h_{*}\\mathcal{B}_{x}^{2}(v)=\\lambda^{2}h_{*}(v),\n\\]\nand then \n\\[\n\\mathcal{B}_{h\\cdot x}h_{*}(v)=\\lambda h_{*}(v).\n\\]\nThe discussion above amounts to\n\\[\n\\mathcal{B}_{x}(v)=\\lambda v\\iff\\mathcal{B}_{h\\cdot x}h_{*}(v)=\\lambda h_{*}(v).\n\\]\nNow let $\\{e_{1},...,e_{2n}\\}$ be a basis of $T_{x}M$ with each\n$e_{i}$ being an eigenvector of $\\mathcal{B}_{x}$ with eigenvalue\n$\\lambda_{i}$. Then\n\\[\nh_{*}\\mathcal{B}_{x}(e_{i})=\\lambda_{i}h_{*}(e_{i})=\\mathcal{B}_{h\\cdot x}h_{*}(e_{i}).\n\\]\nSince $\\{e_{1},...,e_{2n}\\}$ is a basis, this concludes the proof.\n\\end{proof}\nThe tangent maps of the actions by the elements in $G$ gives the\nfollowing $G$-action on $Fr_{\\bar{J}}^{u}$:\n\n\\[\n\\Phi_{\\#}:G\\times Fr_{\\bar{J}}^{u}\\rightarrow Fr_{\\bar{J}}^{u}\n\\]\n\\[\n\\Phi_{\\#}^{h}(u_{i},v_{i})=\\big(h_{*}(u_{i}),h_{*}(v_{i})\\big).\n\\]\nFor simplicity, we write $\\Phi_{\\#}^{h}(u_{i},v_{i})$ as $h_{\\#}(u_{i},v_{i})$.\nFor any $C\\in\\mathbb{U}(n)$, it can also be checked that\n\\begin{equation}\n\\big(h_{*}(u_{1}),...,h_{*}(v_{n})\\big)\\cdot C=h_{\\#}\\big((u_{1},...,v_{n})\\cdot C\\big),\\label{eq:h=000023_is_equivariant}\n\\end{equation}\nand hence $\\Phi_{\\#}$ is a $G$-action on $Fr_{\\bar{J}}^{u}\\rightarrow M$.\nDue to (\\ref{eq:h=000023_is_equivariant}), $\\Phi_{\\#}$ induces a\nsmooth $G$-action $\\Phi_{\\Gamma}$ on $\\Gamma_{\\bar{J}}=Fr_{\\bar{J}}^{u}\\big\/\\mathbb{SU}(n)$.\nMoreover, $\\Phi_{\\Gamma}$ on $\\Gamma_{\\bar{J}}$ commutes with the\n$S^{1}$ action. To see this, note that \n\\[\n\\mathbb{U}(n)\\big\/\\mathbb{SU}(n)\\ni[C]\\mapsto det_{\\mathbb{C}}(C)\\in S^{1}\n\\]\nis an isomorphism, and the $S^{1}$ action on $\\Gamma_{J}$ is given\nby\n\\[\n[u_{1},...,v_{n}]\\cdot e^{i\\theta}:=[(u_{1},...,v_{n})]\\cdot[C]=[(u_{1},...,v_{n})\\cdot C]\n\\]\nwith $det_{\\mathbb{C}}(C)=e^{i\\theta}$. Then \n\\[\n\\begin{aligned}\\Phi_{\\Gamma}^{h}\\big([u_{1},...,v_{n}]\\cdot e^{i\\theta}\\big) &  & = & \\Phi_{\\Gamma}^{h}[(u_{1},...,v_{n})\\cdot C]\\\\\n &  & = & [(h_{*}u_{1},...,h_{*}v_{n})\\cdot C]\\\\\n &  & = & \\Phi_{\\Gamma}^{h}\\big([u_{1},...,v_{n}]\\big)\\cdot e^{i\\theta}\n\\end{aligned}\n.\n\\]\nSince the unitary structure on $M$ is unique up to isomorphism, the\ndiscussion in this subsection amounts to the following proposition:\n\\begin{prop}\n\\label{prop:Existence of =00005CPhi=000023}Suppose that $\\Phi$ is\na symplectic group action on $(M,\\omega)$ by a compact Lie group\n$G$. Then there exists a $G$-action $\\Phi_{\\#}$ on $Fr_{J}^{u}\\rightarrow M$\nthat covers $\\Phi$. As a consequence, it induces $G$-actions $\\Phi_{\\Gamma}$\nand $\\Phi_{\\Gamma^{2}}$ on the bundles $\\Gamma_{J}$ and $\\Gamma_{J}^{2}$,\nrespectively. These actions covers $\\Phi$ and are covered by $\\Phi_{\\#}$.\n\\end{prop}\n\n\n\\subsection{\\label{subsec:An-example: S^2}An example: $S^{2}$}\n\nWe can get the idea of the main result in this section by looking\ninto the case where $M=S^{2}$. This might be the simplest example\nin which the Maslov $S^{1}$ bundles are non-trivial.\n\nConsider $S^{2}$ as an embedded submanifold in $\\mathbb{R}^{3}$.\nAt each point $p\\in S^{2}$, the tangent space $T_{p}S^{2}$ is a\nsubspace of $T_{p}\\mathbb{R}^{3}$. Let $\\bar{n}$ be the restriction\nof the vector field $x\\frac{\\partial}{\\partial x}+y\\frac{\\partial}{\\partial y}+z\\frac{\\partial}{\\partial z}$\non $S^{2}$. With $T_{p}\\mathbb{R}^{3}$ being identified with $\\mathbb{R}^{3}$,\nit holds $\\bar{n}_{p}=p$. Then\n\\[\n\\omega_{S^{2}}=\\iota_{\\bar{n}}dx\\wedge dy\\wedge dz.\n\\]\nis a symplectic structure on $S^{2}$. Viewing $u,v\\in T_{p}S^{2}$\nand as vectors in $\\mathbb{R}^{3}$, it holds\n\\[\n\\omega_{S^{2}}(u,v)=\\bar{n}\\cdot(u\\times v).\n\\]\nWith respect to $\\omega_{S^{2}}$, \n\\[\nJ_{S^{2}}(u):=-\\bar{n}\\times u\n\\]\ndefines a compatible almost complex structure, and \n\\[\ng_{S^{2}}(\\cdot,\\cdot):=\\omega_{S^{2}}(J_{S^{2}}\\cdot,\\cdot)\n\\]\nis the restriction of the standard Riemannian metric on $\\mathbb{R}^{3}$\nto $S^{2}$. \n\nDenote by $\\Gamma_{S^{2}}$ and $Fr_{S^{2}}^{u}$, respectively, the\nMaslov $S^{1}$ bundle ($\\Gamma_{J}$) and the unitary frame bundle\n($Fr_{J}^{u}$) of $(S^{2},\\omega_{S^{2}})$. Over each $p\\in S^{2}$,\neach unitary frame can be denoted uniquely and distinctly as\n\\[\n\\big(u,J_{S^{2}}(u);p\\big)=(u,\\bar{n}_{p}\\times u;p)=(u,p\\times u;p)\n\\]\nwith $u\\in\\mathbb{R}^{3}$ tangent to $S^{2}$ at $p$. When written\nas a matrix $[u,p\\times u,p]$ with $u$ , $p\\times u$ and $p$ being\nthe columns, this is an element in $\\mathbb{SO}(3)$. More precisely,\nthe following map\n\\[\n\\mathfrak{mat}:Fr_{S^{2}}^{u}\\ni\\big(u,-J_{S^{2}}(u);p\\big)\\mapsto[u,p\\times u,p]\\in\\mathbb{SO}(3)\n\\]\nis a diffeomorphism from $Fr_{S^{2}}^{u}$ to $\\mathbb{SO}(3)$. Since\n$\\mathbb{SU}(1)=\\{1\\}$, $\\Gamma_{S^{2}}=Fr_{S^{2}}^{u}$. Since $\\mathbb{SO}(3)$\nacts symplectically on $(S^{2},\\omega_{S^{2}})$, there is an $\\mathbb{SO}(3)$-action\non the bundle $\\pi_{\\Gamma_{S^{2}}}:\\Gamma_{S^{2}}\\rightarrow S^{2}$\nwhich covers the action on $S^{2}$. In fact, this action is exactly\ngiven by \n\\[\n\\mathbb{SO}(3)\\times\\Gamma_{S^{2}}\\xrightarrow{id\\times\\mathfrak{mat}}\\mathbb{SO}(3)\\times\\mathbb{SO}(3)\\xrightarrow{multiply}\\mathbb{SO}(3)\\xrightarrow{\\mathfrak{mat}^{-1}}\\Gamma_{S^{2}}\n\\]\nwith\n\\[\n\\big(A,[u,p\\times u,p]\\big)\\mapsto[Au,Ap\\times Au,Ap].\n\\]\n\nHere $\\mathbb{SO}(3)\\times\\mathbb{SO}(3)\\xrightarrow{multiply}\\mathbb{SO}(3)$\nis simply the group multiplication $(A,B)\\mapsto A\\cdot B$, and hence\nthe lifted group action is recognized as the left action of $\\mathbb{SO}(3)$\non itself (identified as $\\Gamma_{S^{2}}$). As a result, we have\nthe following proposition.\n\\begin{prop}\n\\label{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}The\nlifted $\\mathbb{SO}(3)$ action is transitive on $\\Gamma_{S^{2}}$.\n\\end{prop}\n\n\\begin{proof}\nThis is because the left action of $\\mathbb{SO}(3)$ on itself is\ntransitive.\n\\end{proof}\nLet $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$ be a\nconnection $1$-form on $\\Gamma_{S^{2}}$. Since $S^{2}$ is simply\nconnected and the principal bundle $\\pi_{\\Gamma_{S^{2}}}:\\Gamma_{S^{2}}\\rightarrow S^{2}$\nis not trivial, there is no integrable connection on $\\Gamma_{S^{2}}$\nand hence $df_{\\alpha}\\neq0$. Note that $df_{\\alpha}$ descends to\na nondegenerate closed $2-$form $\\Omega_{\\alpha}$ on $S^{2}$, and\nsince $H_{dR}^{2}(S^{2})=\\mathbb{R}$ it satisfies $[\\Omega_{\\alpha}]=r\\cdot[\\omega]$\nfor some real number $r\\neq0$. Moreover, we have the following proposition.\n\\begin{lem}\n\\label{lem:symp-potential on S^2}There exists an $\\mathbb{SO}(3)$-invariant\nconnection $1$-form $\\bar{\\alpha}$ on $\\Gamma_{S^{2}}$ such that\n$df_{\\bar{\\alpha}}=r\\cdot\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)$ with $r$\nbeing some non-zero real number. \n\\end{lem}\n\n\\begin{proof}\nStarting with an arbitrary connection $1$-form $\\alpha$, for each\nelement $v$ of the tangent bundle $T\\Gamma_{S^{2}}$, define\n\\[\n\\bar{f}(v):=\\int_{\\mathbb{SO}(3)}f_{\\alpha}(h_{*}v)dh\n\\]\nwith $dh$ being a right invariant probability measure on $\\mathbb{SO}(3)$.\nThen $\\bar{f}$ is both $S^{1}$-invariant and $\\mathbb{SO}(3)$ invariant,\nand $\\bar{f}(\\frac{\\partial}{\\partial\\theta})\\equiv1$. Therefore,\n$\\bar{\\alpha}:=\\bar{f}\\cdot\\frac{\\partial}{\\partial\\theta}$ defines\na connection $1$-form which is invariant under the $\\mathbb{SO}(3)$-action.\nThen $d\\bar{f}$ is also $\\mathbb{SO}(3)$ invariant, and so is $\\Omega_{\\bar{\\alpha}}$\nwith $d\\bar{f}=\\pi_{\\Gamma_{S^{2}}}^{*}(\\Omega_{\\bar{\\alpha}})$.\nSince $\\Omega_{\\bar{\\alpha}}\\neq0$ and $S^{2}$ is $2$ dimensional,\nthere exists a non-zero real number $r$ such that $\\Omega_{\\bar{\\alpha}}=r\\cdot\\omega$.\nNote that $\\bar{f}=f_{\\bar{\\alpha}}$ and this concludes the proof.\n\\end{proof}\n\\begin{prop}\n\\label{prop:symplectic=00003DHamiltonian on S^2}There exists a connection\n$1$-form $\\bar{\\beta}:=f_{\\bar{\\beta}}\\cdot\\frac{\\partial}{\\partial\\theta}$\nsuch that, for each $v\\in\\mathfrak{so}(3)$, the corresponding infinitesimal\ngenerator $X_{v}$ has a Hamiltonian function $H_{v}$ on $S^{2}$\nsatisfying $H_{v}\\circ\\pi_{\\Gamma_{S^{2}}}=-\\frac{1}{r}\\cdot f_{\\bar{\\beta}}(\\mathcal{X}_{v})$\nwith some non-zero constant $r$.\n\\end{prop}\n\n\\begin{proof}\nLet $\\bar{\\alpha}$ be the connection $1$-form in Lemma \\ref{lem:symp-potential on S^2}.\nFor simplicity, we write the symplectic form $\\omega_{S^{2}}$ as\n$\\omega$ in the following discussion. Applying the averaging method\nagain to $\\bar{\\alpha}$ with the $G$-action on $\\Gamma_{S^{2}}$:\n\\[\n\\bar{\\beta}(v):=\\int_{G}\\bar{\\alpha}\\circ h_{*}(v)dh\n\\]\n with $dh$ being a right invariant probability measure on $G$. $f_{\\bar{\\beta}}$\nis then invariant under the $G$-action, and\n\\[\n\\begin{aligned}df_{\\bar{\\beta}}(u,v) &  & = & \\int_{G}h^{*}(df_{\\bar{\\alpha}})(u,v)dh\\\\\n &  & = & r\\cdot\\int_{G}h^{*}\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)(u,v)dh\\\\\n &  & = & r\\cdot\\int_{G}\\pi_{\\Gamma_{S^{2}}}^{*}h^{*}(\\omega)(u,v)dh\\\\\n &  & = & r\\cdot\\int_{G}\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)(u,v)dh\\\\\n &  & = & r\\cdot\\pi_{\\Gamma_{S^{2}}}^{*}(\\omega)(u,v)\n\\end{aligned}\n.\n\\]\nBy Cartan's formula,\n\\[\n\\begin{aligned}d\\big(f_{\\bar{\\beta}}(\\mathcal{X}_{v})\\big)(\\cdot) &  & = & \\mathcal{L}_{\\mathcal{X}_{v}}f_{\\bar{\\beta}}-\\iota_{\\mathcal{X}_{v}}df_{\\bar{\\beta}}\\\\\n &  & = & -r\\cdot\\omega\\big(\\pi_{\\Gamma_{S^{2}},*}(\\mathcal{X}_{v}),\\pi_{\\Gamma_{S^{2}},*}\\cdot\\big)\\\\\n &  & = & -r\\cdot\\omega(X_{v},\\pi_{\\Gamma_{S^{2}},*}\\cdot)\n\\end{aligned}\n.\n\\]\nNote that the $G$-action commutes with the $S^{1}$-action on $\\Gamma_{S^{2}}$,\nand hence $\\mathcal{X}_{v}$ is $S^{1}$-invariant. As a result, $-\\frac{1}{r}\\cdot f_{\\bar{\\beta}}(\\mathcal{X}_{v})$\nis also $S^{1}$-invariant, and there is a fucntion $H_{v}$ on $M$\nsuch that\n\\[\n-\\frac{1}{r}\\cdot f_{\\bar{\\beta}}(\\mathcal{X}_{v})=H_{v}\\circ\\pi_{\\Gamma_{S^{2}}}.\n\\]\nWith the identity deduced above it yields \n\\[\ndH_{v}\\circ\\pi_{\\Gamma_{S^{2}},*}=\\omega\\big(X_{v},\\pi_{\\Gamma_{S^{2}},*}\\cdot\\big)\n\\]\nand then \n\\[\ndH_{v}=\\omega(X_{v},\\cdot).\n\\]\n\\end{proof}\n\n\\section{\\label{sec:dynamics-on Maslov S^1 bundles}Dynamics on the Maslov\n$S^{1}$ Bundles}\n\nIn this section we extend the properities obtained for $S^{2}$, Proposition\n\\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D} and\n\\ref{prop:symplectic=00003DHamiltonian on S^2}, to more general settings.\n\n\\subsection{Extension for Proposition \\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}\nto symplectic homogeneous $G$-spaces}\n\nIn this subsections we extend Proposition \\ref{prop:SO(3) tansitive on =00005CGamma_=00007BS^2=00007D}\nto the case where the group action $\\Phi$ by $G$ is transitive on\n$M$ . Namely, $M$ is a symplectic homogeneous $G$-space. For each\n$p\\in M$ and $w\\in\\Gamma_{J}$, denote by $G_{p}$ the isotropy group\nof the action $\\Phi$ at $p$, and by $G_{w}$ the isotropy group\nof $\\Phi_{\\Gamma}$ at $w$.\n\nBy the transitivity of the $G$-action, $M$ is diffeomorphic to $G\\big\/G_{p_{0}}$\nfor any fixed point $p_{0}$ through the map\n\\[\n\\mathcal{F}_{p_{0}}:G\\big\/G_{p_{0}}\\ni[h]\\mapsto h\\cdot p_{0}\\in M.\n\\]\nLet $w_{0}$ be a point in $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$. Then $G_{w_{0}}\\subset G_{p_{0}}$.\nSince $\\Phi_{\\Gamma}$ on $\\Gamma_{J}$ covers $\\Phi$ on $M$, $G_{p_{0}}$\nacts on the fiber $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$. That is, $h\\cdot w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$\nfor any $h\\in G_{p_{0}}$ and $w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$.\nIt turns out that $G_{w}=G_{w_{0}}$ for any $w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$,\nand that the $G_{p_{0}}$-action on $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$\ninduces a homomorphism to $S^{1}$ with $G_{w_{0}}$ being the kernel. \n\\begin{prop}\n\\label{prop:=00005Cphi_p}There exists a Lie group homomorphism $\\phi_{p}$\nfrom $G_{p}$ to $S^{1}$ with $G_{w}$ being the kernel such that,\nfor $w\\in\\pi_{\\Gamma_{J}}^{-1}(p)$ and $h\\in G_{p}$, it holds\n\\[\nh\\cdot w=w\\cdot\\phi_{p}(h).\n\\]\nMoreover, the family $\\{\\mathcal{\\phi}_{p}\\}$ is $G$-related in\nthe sense that\n\\begin{equation}\n\\phi_{h'\\cdot p}\\circ Ad_{h'}=\\phi_{p}\\label{eq:adjoint action on =00007B=00005Cphi=00007D}\n\\end{equation}\nwith $p\\in M$ and $h'\\in G$. \n\\end{prop}\n\n\\begin{proof}\nWe start by arguing with the fixed points $p_{0}$ and $w_{0}$. Since\n$S^{1}$ acts transitively and freely on $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$,\nfor each $h\\in G_{p_{0}}$ there exists a unique $z_{h}\\in S^{1}$\nsuch that $h\\cdot w_{0}=w_{0}\\cdot z_{h}$. Then \n\\[\nw\\cdot z_{h'h}=(h'h)\\cdot w_{0}=w_{0}\\cdot z_{h}\\cdot z_{h'}=w_{0}\\cdot(z_{h'}z_{h}),\n\\]\nand hence $z_{h'h}=z_{h'}z_{h}.$ Define $\\phi_{p_{0}}$with $\\phi_{p_{0}}(h)=z_{h}.$\nThen this is a homomorphism, and $h\\cdot w_{0}=w_{0}$ if and only\nif $\\phi_{p_{0}}(h)=1_{G}$. Hence $\\ker\\phi_{p_{0}}=G_{w_{0}}$.\nNote that $\\pi_{\\Gamma_{J}}^{-1}(p_{0})$ is diffeomorphic to $S^{1}$\nvia $\\bar{\\theta}_{w_{0}}:w_{0}\\cdot z\\mapsto z$. Since $\\phi_{p_{0}}(h)=\\bar{\\theta}_{w_{0}}\\circ\\Phi_{\\Gamma}^{h}(w_{0})$,\n$\\phi_{p_{0}}$ is a smooth map, and therefore it is a Lie group homomorphism.\n\nThe assignment of $z_{h}$ to $h$ is actually independent of the\nchoice of $w_{0}$. Namely, the identity $h\\cdot w=w\\cdot z_{h}$\nholds for all $w\\in\\pi_{\\Gamma_{J}}^{-1}(p_{0})$. To see this, first\nnote that there existis $z\\in S^{1}$ such that $w_{0}\\cdot z=w$,\nand then\n\\[\nh\\cdot w=h\\cdot(w_{0}\\cdot z)=(h\\cdot w_{0})\\cdot z=w_{0}\\cdot z_{h}\\cdot z=w\\cdot z_{h}.\n\\]\nMoreover, the argument above holds for any point $p$, and then the\nhomomorphism $\\mathcal{\\phi}_{p}$ can be constructed in the same\nway. \n\nIt remains to show that $G$ acts on the family $\\{\\mathcal{\\phi}_{p}\\}$\naccording to Eq.\\ref{eq:adjoint action on =00007B=00005Cphi=00007D}.\nLet $w$ be a point on the fiber $\\pi_{\\Gamma}^{-1}(p)$. Then $w'=h'\\cdot w\\in\\pi_{\\Gamma}^{-1}(h'\\cdot p)$.\nFor any $h\\in G_{p}$,\n\\[\n\\begin{aligned}h'hh'^{-1}\\cdot w' &  & = & h'hw\\\\\n &  & = & h'\\cdot\\big(w\\cdot\\phi_{p}(h)\\big)\\\\\n &  & = & (h'\\cdot w)\\cdot\\phi_{p}(h)\\\\\n &  & = & w'\\cdot\\phi_{p}(h),\n\\end{aligned}\n\\]\nwhich implies $\\phi_{h'\\cdot p}(h'hh'^{-1})=\\phi_{p}(h)$ and concludes\nthe proof.\n\\end{proof}\nDue to (\\ref{eq:adjoint action on =00007B=00005Cphi=00007D}) in Proposition\n\\ref{prop:=00005Cphi_p} above, for any $p,p'\\in M$, $\\text{Im}\\phi_{p}=\\text{Im}\\phi_{p'}$,\nand we denote this subgroup by $S_{\\phi}^{1}$. Since $G_{p}$ is\ncompact, $S_{\\phi}^{1}=\\phi_{p}(G_{p})$ is a compact subgroup of\n$S^{1}$. Therefore it is either a finite cyclic group $\\{e^{i\\frac{2\\pi}{k}\\cdot l}\\big|l=0,....,k-1\\}$,\nor $S^{1}$ itself.\n\nFor each element $v$ of the Lie algebra $\\mathfrak{g}$ of $G$,\ndenote by $\\mathcal{X}_{v}$ the infinitesimal generator of $\\Phi_{\\Gamma}$\nin the direction $v$. That is, $\\mathcal{X}_{v}(p)=\\frac{d}{dt}exp(tv)\\cdot p$\nwith $t\\mapsto exp(tv)$ being the one parameter subgroup of $G$\ngenerated by $v$. We claim:\n\\begin{prop}\n\\label{prop:If:S_=00005Cphi-finite}$S_{\\phi}^{1}$ is either $S^{1}$\nor a finite subgroup of $S^{1}$. If $S_{\\phi}$ is finite, then the\ninifinitesimal generators $\\mathcal{X}_{v}$ span an integrable connection\n$\\mathcal{D}$ on the bundle $\\Gamma_{J}$ with the orbits $G\\cdot w$\nbeing the maximal connected integral manifolds.\n\\end{prop}\n\n\\begin{proof}\nAs an orbit of a compact Lie group action, $G\\cdot w$ is an embedded\nsubmanifold in $\\Gamma_{J}$, and then its dimension is no larger\nthan $2n+1$. If $\\dim G\\cdot w=2n+1$, $G\\cdot w$ is an open set\nin $\\Gamma_{J}$ and then $G\\cdot w\\cap\\pi_{\\Gamma_{J}}^{-1}(p)$\nis open in $\\pi_{\\Gamma_{J}}^{-1}(p)$ (and then it is the whole $\\pi_{\\Gamma_{J}}^{-1}(p)$\nsince it is open and compact for any $p\\in M$). Now suppose that\n$\\dim G\\cdot w\\leq2n+1$. Since $G$ acts transitively on $M$ and\n$\\pi_{\\Gamma}\\circ\\Phi_{\\#}=\\Phi\\circ\\pi_{\\Gamma}$, it holds\n\\[\n\\pi_{\\Gamma,*}\\big(T_{w}(G\\cdot w)\\big)=T_{\\pi_{\\Gamma}(w)}M,\n\\]\nand hence $\\dim G\\cdot w\\geq2n$. Together this yields $\\dim G\\cdot w=2n$.\nDue to the commutativity of $\\Phi_{\\Gamma}$ and the $S^{1}$action\non $\\Gamma_{J}$, $(G\\cdot w)\\cdot z=G\\cdot(w\\cdot z)$ with $z\\in S^{1}$.\nThis means $(G\\cdot w)\\cdot z$ is exactly the $G$-orbit through\n$w\\cdot z$, and the $S^{1}$ action maps orbits to orbits. Since\n\\[\nT_{w}(G\\cdot w)=span\\{\\mathcal{X}_{v}\\big|v\\in\\mathfrak{g}\\}=\\mathcal{D}_{w},\n\\]\n$\\mathcal{D}$ is a $2n$ dimensional distribution invariant under\nthe $S^{1}$ action, and $\\pi_{\\Gamma,*}(\\mathcal{D}_{w})=T_{\\pi_{\\Gamma}(w)}M$.\n\\end{proof}\nTherefore, from Proposition \\ref{prop:If:S_=00005Cphi-finite} we\ncan deduce:\n\\begin{thm}\nLet $G$ be a compact Lie group acting transitively and symplectically\non $M$. If the characteristic class of the Maslov $S^{1}$ bundle\n$\\Gamma_{J}$ is non-zero, then $\\Gamma_{J}$ is also a homogeneous\n$G$-space.\n\\end{thm}\n\n\\begin{proof}\nIf $\\Phi_{\\Gamma}$ does not act transitively on $\\Gamma_{J}$, then\n$S_{\\phi}^{1}$ should be a finite subgroup of $S^{1}$. According\nto Proposition \\ref{prop:If:S_=00005Cphi-finite}, $\\mathcal{D}$\nwould be an integrable connection, and then the characteristic class\nwould be zero, which violates the condition.\n\\end{proof}\n\n\\subsection{Extension for Proposition \\ref{prop:symplectic=00003DHamiltonian on S^2}\nto the case $[\\omega]=r\\cdot c_{\\Gamma}$}\n\nThe argument for Proposition \\ref{lem:symp-potential on S^2} can\nactually be applied to any symplectic action on a symplectic manifold\n$(M,\\omega)$ satisfying $[\\omega]=r\\cdot c_{\\Gamma}$ for $r\\in\\mathbb{R}$.\nInstead of directly giving a proof, we introduce some notions and\nformalize the demonstration in such a way that the relevant structures\nare better illustrated.\n\nRecall that a symplectic $G$-action on $M$ is Hamiltonian if and\nonly if there is a smooth map (called a momentum map)\n\\[\nF:M\\rightarrow\\mathfrak{g}^{*}\n\\]\nsuch that for each $h\\in G$ and $v\\in\\mathfrak{g}$, $F$ is $G$-coadjoint\nequivariant, i.e.\n\\[\nF_{h\\cdot x}=Ad_{h^{-1}}^{*}F_{x},\n\\]\nand the mapping \n\n\\[\nF^{v}:M\\ni x\\mapsto F_{x}(v)\\in\\mathbb{R}\n\\]\ndefines a Hamiltonian for $\\mathcal{X}_{v}$ with $v\\in\\mathfrak{g}$.\n\\begin{rem}\nIn the rest of this subsection the symbol ``$\\Gamma$'' stands for\nboth the spaces $\\Gamma_{J}$ and $\\Gamma_{J}^{2}$, since the same\nargment works for both of them.\n\\end{rem}\n\n\\begin{defn}\nAn $S^{1}$-invariant $1$ form $\\eta$ on the bundle $\\pi_{\\Gamma}:\\Gamma\\rightarrow M$\nis called a symplectic potential if $d\\eta=-\\pi_{P}^{*}(\\omega)$.\n\\end{defn}\n\nIt is straighforward to check that averging a symplectic potential\n$\\eta$ with the (lifted) $G$ action $\\Phi_{\\Gamma}$ gives a $G$-invariant\nsymplectic potential $\\bar{\\eta}$. We have the following lemma.\n\\begin{lem}\n\\label{lem:coadjoint-equivariant map from a symplectic potential}Let\n$\\bar{\\eta}$ be a $G$-invariant symplectic potential. For each $p\\in\\Gamma$,\nlet $\\mu_{p}$ be the element in $\\mathfrak{g}^{*}$ with $\\mu_{p}(v)=\\bar{\\eta}(\\mathcal{X}_{v})(p)$\nfor $v\\in\\mathfrak{g}$, where $\\mathcal{X}_{v}$ is the infinitesimal\ngenerator of $\\Phi_{\\Gamma}$ corresponding to $v$. Then the map\n$\\mu$ from $\\Gamma$ to $\\mathfrak{g}^{*}$ defined by \n\\[\n\\mu:p\\mapsto\\mu_{p}\n\\]\nis a momentum map for $G$.\n\\end{lem}\n\n\\begin{proof}\nFor the smoothness of $\\mu$, it suffices to show that the map $\\bar{\\mu}:\\Gamma\\times\\mathfrak{g}\\rightarrow\\mathbb{R}$\ndefined by $\\bar{\\mu}(p,v)=\\mu_{p}(v)$ is smooth. Observe that $\\bar{\\mu}$\nfactors as\n\\[\n\\bar{\\mu}:\\Gamma\\times\\mathfrak{g}\\stackrel{\\mathcal{X}}{\\longrightarrow}T\\Gamma\\stackrel{\\bar{\\eta}}{\\longrightarrow}\\mathbb{R}\n\\]\nwith $\\mathcal{X}$ being the map from $P\\times\\mathfrak{g}$ to $TP$\nsending $(p,v)$ to $\\mathcal{X}_{v}(p)$. The factorization below\nshows smoothness of $\\mathcal{X}$:\n\n\\[\n\\mathcal{X}:\\Gamma\\times\\mathfrak{g}\\stackrel{\\sigma}{\\longrightarrow}T\\Gamma\\times TG\\cong T(\\Gamma\\times G)\\stackrel{D\\Phi_{\\Gamma}}{\\longrightarrow}T\\Gamma.\n\\]\nHere $\\sigma$ is the map sending $(p,v)\\in\\Gamma\\times\\mathfrak{g}$\nto $(\\bar{\\boldsymbol{0}}_{p},v)\\in T_{p}\\Gamma\\times T_{1}G$ with\n$\\bar{\\boldsymbol{0}}$ being the zero section from $\\Gamma$ to $T\\Gamma$,\nand $D\\Phi_{\\Gamma}$ is the tangent map of $\\Phi_{\\Gamma}$. Hence\n$\\mathcal{X}$ is smooth. As a consequence, $\\bar{\\mu}$ is also smooth.\n\nSince $\\bar{\\eta}$ and $\\mathcal{X}_{v}$ are both $S^{1}$-invariant\non $\\Gamma$, the function $\\bar{\\eta}(\\mathcal{X}_{v})$ is constant\nalong each $S^{1}$-fiber, and hence there exists a function $H_{v}$\non $M$ such that $H_{v}\\circ\\pi=\\bar{\\eta}(\\mathcal{X}_{v})$. Check\nthat $0=\\mathcal{L}_{\\mathcal{X}_{v}}(\\bar{\\eta})=d\\iota_{\\mathcal{X}_{v}}\\bar{\\eta}+\\iota_{\\mathcal{X}_{v}}d\\bar{\\eta}$\nand then it holds\n\\[\nd\\big(\\bar{\\eta}(\\mathcal{X}_{v})\\big)=-\\iota_{\\mathcal{X}_{v}}d\\bar{\\eta}=\\iota_{\\mathcal{X}_{v}}\\pi_{P}^{*}(\\omega),\n\\]\nimplying $\\iota_{X_{v}}\\omega=dH_{v}$, where $X_{v}=\\pi_{\\Gamma,*}(\\mathcal{X}_{v})$\nis the generator of $\\Phi$ corresponding to $v$.\n\nIt remains to show that $\\mu$ is $G$-coadjoint equivariant. Note\nthat for $h\\in G$ and $p\\in\\Gamma$,\n\\[\nh_{*}(\\mathcal{X}_{v}\\big|_{p})=\\mathcal{X}_{Ad_{h,*}(v)}(h\\cdot p).\n\\]\nThen \n\\[\n\\begin{aligned}\\mu_{h\\cdot p}(v) &  & = & \\bar{\\eta}(\\mathcal{X}_{v})(h\\cdot p)\\\\\n &  & = & \\bar{\\eta}\\big|_{h\\cdot p}\\big(\\mathcal{X}_{v}(h\\cdot p)\\big)\\\\\n &  & = & \\bar{\\eta}\\big|_{h\\cdot p}\\big(h_{*}\\mathcal{X}_{Ad_{h^{-1},*}(v)}(p)\\big)\\\\\n &  & = & h^{*}(\\bar{\\eta}\\big|_{h\\cdot p})\\big(\\mathcal{X}_{Ad_{h^{-1},*}(v)}(p)\\big)\\\\\n &  & = & \\bar{\\eta}\\big|_{p}\\big(\\mathcal{X}_{Ad_{h^{-1},*}(v)}(p)\\big)\\\\\n &  & = & \\mu_{p}\\circ Ad_{h^{-1},*}(v)\n\\end{aligned}\n\\]\nand this concludes the proof.\n\\end{proof}\nWe give the following remark for later reference.\n\\begin{thm}\n\\label{thm:Symplectic=00003D=00003DHamiltonian}If $[\\omega]=r\\cdot c_{\\Gamma}$\nwith $c_{\\Gamma}$ being the first real Chern class of $\\Gamma$ and\n$r\\in\\mathbb{R}$, then any symplectic action $\\Phi$ on $M$ by a\ncompact Lie group $G$ is Hamiltonian. Moreover, if $r\\neq0$, then\nthere exists a connection $1$-form $\\beta=f_{\\beta}\\cdot\\frac{\\partial}{\\partial\\theta}$\nwith some nonzero real number $r$ such that $\\frac{1}{r}f_{\\beta}$\nis a $G$-invariant symplectic potential, and then $\\frac{1}{r}f_{\\beta}(\\mathcal{X}_{v})$\nis an Hamiltonian for $X_{v}$.\n\\end{thm}\n\n\\begin{proof}\nIf the homology class $[\\omega]=0$, then there exists a $1$-form\non $M$ with $d\\tau=\\omega$. It is straightforward to check that\n$\\pi_{P}^{*}(\\tau)$ is an $S^{1}$-invariant $1$-form on $\\Gamma$\nand is a symplectic potential. If $[\\omega]\\neq0$, then there exists\na non-zero constant $r$ such that $[r\\cdot\\omega]=[\\Omega_{\\alpha}]$\nwith $\\pi_{\\Gamma}^{*}(\\Omega_{\\alpha})=df_{\\alpha}$ for a connection\n$1$-form $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$.\nThen it holds $r\\cdot\\omega=\\Omega_{\\alpha}+d\\tau$ for some $1$-form\n$\\tau$ on $M$. Let $f_{\\beta}=f_{\\alpha}+\\pi_{\\Gamma_{J}}^{*}(\\tau)$\nand then $df_{\\beta}=r\\cdot\\pi_{\\Gamma}^{*}(\\omega)$. Hence $\\frac{1}{r}f_{\\beta}$\nis a symplectic potential on the bundle $\\Gamma\\rightarrow M$. \n\nThe conclusion then follows from Lemma \\ref{lem:coadjoint-equivariant map from a symplectic potential}.\n\\end{proof}\n\n\\subsection{Conservation laws}\n\nWhen $\\eta$ (or $\\beta$) is invariant under the action $\\Phi_{\\Gamma^{2}}$,\nit is always true that $\\eta(\\mathcal{X}_{v})$ is constant along\nthe flow of $\\mathcal{X}_{v}$, i.e. $\\mathcal{L}_{\\mathcal{X}_{v}}\\eta(\\mathcal{X}_{v})=0$.\nThis is because $\\mathcal{X}_{v}$ and $\\eta$ are invariant under\nthe pushforward and pullback of the flow of $\\mathcal{X}_{v}$, respectively.\nMoreover, since $\\mathcal{X}_{v}$ and $\\eta$ are also invariant\nunder the inherent $S^{1}$ action of the bundle $\\Gamma^{2}$, there\nalways exists a function $Q_{v}$ on $M$ such that $\\eta(\\mathcal{X}_{v})=Q_{v}\\circ\\pi_{\\Gamma^{2}}$.\nSince $\\pi_{\\Gamma^{2}}$ is a submersion, the smoothness of $\\eta(\\mathcal{X}_{v})$\nimplies the smoothness of $Q_{v}$. Check that $d\\eta(\\mathcal{X}_{v})=dQ_{v}\\circ\\pi_{\\Gamma^{2},*}$\nand then $0=dQ_{v}\\circ\\pi_{\\Gamma^{2},*}(\\mathcal{X}_{v})=dQ_{v}(X_{v})$,\nmeaning $Q_{v}$ is invariant under the flow $\\varphi_{v}$ of $X_{v}$.\nThis implies in particular that, if the trajectory of the one-parameter\nsubgroup $\\exp(tv)$ is dense in $G$, then $Q_{v}$ is invariant\nunder the $G$-action.\n\n\\section{\\label{sec:Beta-Maslov-Quantity for S^1-action}$\\beta$-Maslov data\nfor Symplectic $S^{1}$ Actions}\n\nIn this section we consider the case $G=S^{1}$. For a connection\n$1$-form $\\beta=\\eta\\cdot\\frac{\\partial}{\\partial\\theta}$ on $\\Gamma^{2}$,\nthe $\\beta$-Maslov data $Q_{\\beta}$ of the $S^{1}$ action $\\Phi$\nis the function on $M$ defined by\n\\begin{equation}\nQ_{\\beta}(p)=\\mathfrak{m}_{\\beta}(\\gamma_{w})=\\int_{\\gamma_{w}}\\eta\\label{eq:def: Maslov-Quantity Q_=00005Ctheta}\n\\end{equation}\nwith $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$ and $\\gamma_{w}(z)=\\Phi_{\\Gamma^{2}}^{z}(w)$\nfor $z\\in S^{1}$. Note that the integral in (\\ref{eq:def: Maslov-Quantity Q_=00005Ctheta})\nis independent of the choice of $w$ on $\\pi_{\\Gamma^{2}}^{-1}(p)$\nand hence $Q_{\\beta}$ is a well-defined function on $M$. To see\nthis, note that for $w,w'\\in\\pi_{\\Gamma^{2}}^{-1}(p)$, there exists\nan element $z\\in S^{1}$ such that $w'=w\\cdot z$ and then $\\gamma_{w'}=\\gamma_{w}\\cdot z$,\nand $\\int_{\\gamma_{w}}\\eta=\\int_{\\gamma_{w'}}\\eta$ follows from the\nfact that $\\eta$ is invariant under the inherent $S^{1}$ action\non $\\Gamma^{2}$ (since $\\beta$ is a connection $1$-form).\n\nFor simplicity we denote by $\\varphi$ the period-$1$ flow of the\ncircle action, namely, $\\frac{d}{dt}\\varphi^{t}=X_{\\theta}$. \n\n\\subsection{\\label{subsec:The-Maslov-index of the S^1 action}The Maslov index\nof the $S^{1}$ action}\n\nIn this subsection we look at the case when $\\Gamma^{2}$ is a trivial\nbundle. \n\nLet $\\mathfrak{s}$ be a global section of $\\Gamma^{2}$. For each\nLagrangian plane $w$, $\\tilde{\\gamma}_{w}(z)=\\Phi_{*}^{z}(w)$ with\n$z\\in S^{1}$ is a loop in $\\Lambda_{pl}$ and hence has a Maslov\nindex $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma}_{w})$ with respect\nto $\\mathfrak{s}$. Note that $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma}_{w})$\nequals to the degree of the map obtained by the following composition\n\n\\begin{equation}\nS^{1}\\cong[0,1]\\big\/\\{0,1\\}\\xrightarrow{\\Phi_{*}(w)}\\Lambda_{pl}\\xrightarrow{det_{J}^{2}}\\Gamma_{J}^{2}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}\\xrightarrow{pr_{S^{1}}}S^{1},\\label{eq:Maslov index for circle actions (Lambda)}\n\\end{equation}\nwhich, together with the connectedness of $\\Lambda_{pl}$, implies\nthat $\\mathfrak{m}_{\\mathfrak{s}}(\\tilde{\\gamma}_{w})$ is actually\nindependent of $w$. Namely, all the orbits in $\\Lambda_{pl}$ of\nthe action $\\Phi_{*}$ have the same Maslov index. \n\\begin{rem}\nNote that the construction above is equivalent to looking at the degree\n$\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})$ of the following map\n\\begin{equation}\nS^{1}\\cong[0,1]\\big\/\\{0,1\\}\\xrightarrow{\\Phi_{\\Gamma^{2}}^{(\\cdot)}(w)}\\Gamma_{J}^{2}\\xrightarrow{tr_{\\mathfrak{s}}}M\\times S^{1}\\xrightarrow{pr_{S^{1}}}S^{1}\\label{eq:Maslov index for circle actions (Gamma)}\n\\end{equation}\nwith $\\gamma_{w}(z)=\\Phi_{\\Gamma^{2}}^{z}(w)$.\n\\end{rem}\n\nWe show further that when the action has a fixed point, then $\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})$\nis also independent of the choice of $\\mathfrak{s}$, and hence we\ncan simply talk about the Maslov index of the flow with respect to\n$\\mathfrak{s}$ and denote it by $\\mathfrak{m}_{\\mathfrak{s}}(\\Phi)$.\n\nLet $p$ be an arbitrary fixed point. Then on the fiber $\\pi_{\\Gamma^{2}}^{-1}(p)$,\nthe vector field takes the form as $\\mathcal{X}_{\\theta}(w)=a_{w}\\frac{\\partial}{\\partial\\theta}$\nfor all $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$. Since $\\mathcal{X}_{\\theta}$\nis invariant under the inherent $S^{1}$ action, it holds $a_{w}\\equiv k_{p}$\non $\\pi_{\\Gamma^{2}}^{-1}(p)$. Let $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$\nbe the connection $1$-form that has (the image of) $\\mathfrak{s}$\nas an integral manifold of its kernel. Now that $\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})$\nis independent of the choice of $w$, we particularly choose $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$.\nThen it yields\n\\begin{equation}\n\\mathfrak{m}_{\\mathfrak{s}}(\\gamma_{w})=\\int_{0}^{1}f_{\\alpha}(\\mathcal{X}_{\\theta})\\circ\\gamma_{w}(t)dt=k_{p}.\\label{eq:def_local-Maslov}\n\\end{equation}\nand our arugment is concluded by the fact that $k_{p}$ is independent\nof $\\mathfrak{s}$. \n\\begin{rem}\n\\label{rem:X_=00005Ctheta=00003Dkp}According to the discussion above\nwe have $\\mathcal{X}_{\\theta}=k_{p}\\frac{\\partial}{\\partial\\theta}$\non the fiber $\\pi_{\\Gamma^{2}}^{-1}(p)$.\n\\end{rem}\n\nWe call $k_{p}$ defined by \\ref{eq:def_local-Maslov} the local Maslov\nindex of the $S^{1}$ action at the fixed point $p$. Note that, on\nthe one hand this definition does not depend on the triviality of\n$\\Gamma_{J}^{2}$, and on the other hand it equals to the Maslov index\nof the restricted action on an invariant Daboux chart of $p$. The\ndiscussion above leads to the following proposition.\n\\begin{prop}\n\\label{prop:local maslov index=00003D=00003Dglobal maslov index}When\nthe bundle $\\Gamma_{J}^{2}$ is trivial and the $S^{1}$ action $\\Phi$\nhas fixed points, then for any fixed points $p,p'$ and sections $\\mathfrak{s},\\mathfrak{s}'$,\nit holds\n\\[\nk_{p}=k_{p'}=\\mathfrak{m}_{\\mathfrak{s}}(\\Phi)=\\mathfrak{m}_{\\mathfrak{s}'}(\\Phi).\n\\]\n\\end{prop}\n\n\n\\subsection{The $\\beta$-Maslov data $Q_{\\beta}$}\n\nNow we take a look that the function $Q_{\\beta}$ defined in \\ref{eq:def: Maslov-Quantity Q_=00005Ctheta}.\nAs is noted in Remark \\ref{rem:X_=00005Ctheta=00003Dkp}, on the fiber\nover a fixed point $p$, it holds $\\mathcal{X}_{\\theta}=k_{p}\\frac{\\partial}{\\partial\\theta}$\nand then $\\eta(\\mathcal{X}_{\\theta})\\equiv k_{p}$ on the loop the\nloop $\\gamma_{w}(z)=\\Phi_{\\Gamma^{2}}^{z}(w)$ with $w\\in\\pi_{\\Gamma^{2}}^{-1}(p)$.\nHence\n\\begin{equation}\nQ_{\\beta}(p):=\\int_{\\gamma_{w}}\\eta=\\int_{0}^{1}\\eta(\\mathcal{X}_{\\theta})_{\\gamma_{w}(t)}dt=k_{p}.\\label{eq:Q(p)=00003Dk_p}\n\\end{equation}\nIn particular, if $\\beta$ is invariant under $\\Phi_{\\Gamma_{J}^{2}}$,\nthen $\\eta(\\mathcal{X}_{\\theta})$ is constant along any orbit of\n$\\Phi_{\\Gamma_{J}^{2}}$ and it holds that $\\pi_{\\Gamma^{2}}^{*}(Q_{\\beta})=\\eta(\\mathcal{X}_{\\theta})$.\n\nThe following remark comes as an observation on Eq.(\\ref{eq:Q(p)=00003Dk_p}). \n\\begin{rem}\nAlthough the construction of $Q_{\\beta}$ depends on the choice of\n$\\beta$, its values at a fixed point of the $S^{1}$ action do not.\nIn particular, if the action has fixed points with different local\nMaslov indices, then $Q_{\\beta}$ is never constant.\n\\end{rem}\n\nNote that for some invariant neighbourhood a fixed point $p$ of the\nsymplectic $S^{1}$ action, a Daboux chart\n\\[\n(U,dq_{i}\\wedge dp_{i})\\xrightarrow{\\phi}(M,\\omega)\n\\]\nwith $\\phi(0)=p$ can be chosen such that viewed in the chart the\nrestricted $S^{1}$ action is linearized as $(e^{2m_{1}\\pi i\\cdot t},...,e^{2m_{n}\\pi i\\cdot t}).$\nWe denote this linearized $S^{1}$ action on $U$ by $L_{\\Phi}$.\nWe call $(m_{1},...,m_{n})\\in\\mathbb{Z}^{n}$ the resonance type of\nthe fixed point $p$. From the structure of $Q_{\\beta}$ we have the\nfollowing proposition. \n\\begin{prop}\n\\label{prop:Maslov data at the fixed points}At the fixed point $p$\nof resonant type $(m_{1},...,m_{n})$, the local Maslov index is\n\\begin{equation}\nk_{p}=Q_{\\beta}(p)=2\\cdot\\sum_{i=1}^{n}m_{i}.\\label{eq:Maslov data=000026resonant type}\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nSince $\\phi$ is a symplectomorphism, it induces a bundle isomorphism\n$\\phi_{*}$ from $\\Gamma_{U}^{2}$ to $\\Gamma_{J}^{2}\\big|_{\\tilde{U}}$\nwith $\\tilde{U}=\\phi(U)$. Here $\\Gamma_{U}^{2}$ is the Maslov $S^{1}$\nbundle for $(U,dq_{i}\\wedge dp_{i})$ and it is isomorphic to $U\\times S^{1}$.\nThe lifted $S^{1}$ action $L_{\\Phi}$ on $\\Gamma_{U}^{2}$ and the\naction $\\Phi_{\\Gamma^{2}}$ on $\\Gamma_{J}^{2}\\big|_{\\tilde{U}}$\nare topologically conjugate via $\\phi_{*}$. Then it is straigtforward\nto check that the local Maslov index of $L_{\\Phi}$ at $0\\in U$ is\nthe same as that of $\\Phi_{\\Gamma^{2}}$ at $p$ (both equal to $k_{p})$.\nDue to Proposion \\ref{prop:local maslov index=00003D=00003Dglobal maslov index},\nthis index equals the ordinary Maslov index of the linearized flow\n$(e^{2m_{1}\\pi i\\cdot t},...,e^{2m_{n}\\pi i\\cdot t})$ on $(U,dq_{i}\\wedge dp_{i})$,\nwhich can be computed directly via the Maslov-Arnold map and is equal\nto $2\\cdot\\sum_{i=1}^{n}m_{i}$.\n\\end{proof}\nThe following corollary is a consequence of Theorem \\ref{thm:Symplectic=00003D=00003DHamiltonian}\nand Proposition \\ref{prop:Maslov data at the fixed points}.\n\\begin{cor}\nIf $[\\omega]=r\\cdot c_{\\Gamma}$ and $\\beta$ is invariant under $\\Phi_{\\Gamma_{J}^{2}}$,\n$h_{\\theta}=r\\cdot Q_{\\beta}$ is an Hamiltonian for $\\Phi$. Then\n$h_{\\theta}$ takes the same value at all its critical points of the\nsame resonance type. Moreover, the critical values of $h_{\\theta}$\nlie in the lattice $r\\cdot\\mathbb{Z}$.\n\\end{cor}\n\nWhile we can tell directly from Eq.(\\ref{eq:Maslov data=000026resonant type})\nthat the local Maslov index at a fixed point is an even number, it\ncan also be observed from the bundle structures of $\\Gamma$ and $\\Gamma^{2}$.\nAt a fixed point $p$, since $\\dot{\\gamma}_{w}(t)=\\mathcal{X}_{\\theta}=k_{p}\\frac{\\partial}{\\partial\\theta}$\nand $\\gamma_{w}(0)=\\gamma_{w}(1)=w$, the number $k_{p}$ is an integer\nand it counts how many rounds the orbit $\\gamma_{w}$ winds around\nthe fiber $\\pi_{\\Gamma^{2}}^{-1}(p)\\cong S^{1}$. Note that $\\Phi$\nis also lifted to the action $\\Phi_{\\Gamma}$ on $\\Gamma_{J}$ and\nit satisfies\n\\begin{equation}\nq_{\\pm}\\circ\\Phi_{\\Gamma}=\\Phi_{\\Gamma^{2}}\\circ q_{\\pm},\\label{eq:=00005CPhi-q=00003Dq-=00005CPhi}\n\\end{equation}\nwhere $q_{\\pm}:\\Gamma_{J}\\rightarrow\\Gamma_{J}^{2}$ is the map in\nthe commutative diagram \\ref{Diagram:oriented Maslov indices and Maslov indices}\n(in Subsection \\ref{subsec:Lagrangian-subbundles and Maslov-indices}).\nFrom Eq.(\\ref{eq:=00005CPhi-q=00003Dq-=00005CPhi}) we can tell that\nthe closed orbit(s) of $\\Phi_{\\Gamma}$ has (have) nontrivial winding\nalong the fiber $\\pi_{\\Gamma}^{-1}(p)$ if and only if the closed\norbit(s) of $\\Phi_{\\Gamma^{2}}$ has (have) nontrivial winding along\nthe fiber $\\pi_{\\Gamma^{2}}^{-1}(p)$, and that when $\\Phi_{\\Gamma}$\ngoes one round, $\\Phi_{\\Gamma^{2}}$ goes twice. Therefore, $k_{p}$\nis always an even number.\n\nIt is then straightforward to deduce the following corollary from\nthe results and discussion above, as well as the theorem of Delzant's\npolytopes (for references about Delzant's polytopes, see \\cite{mcduff2017introduction}\nor \\cite{da2008lectures}).\n\\begin{cor}\n(A Delzant picture) Suppose that $G=\\mathbb{T}^{l}$. Then there is\na smooth map $\\mathcal{Q}_{\\beta}=(Q_{\\beta}^{(1)},...,Q_{\\beta}^{(l)})$\nfrom $M$ to $\\mathbb{R}^{l}$ such that $\\mathcal{Q}$ maps the fixed\npoints of the $\\mathbb{T}^{l}$ action to the lattice $2\\cdot\\mathbb{Z}^{l}\\subset\\mathbb{R}^{l}$.\nIf $M$ is compact and $[\\omega]=r\\cdot c_{\\Gamma}$, then the image\n$\\text{Im}\\mathcal{Q}_{\\beta}$ is a polytope with the vertices taking\nthe form $(k_{p}^{1},...,k_{p}^{l})$ with $k_{p}^{j}$ being local\nMaslov indices at $p$.\n\\end{cor}\n\n\\begin{prop}\nSuppose that $c_{\\Gamma}=0$ and $\\Phi$ is a symplectic $S^{1}$\naction on $M$. Then all the local Maslov indices at the fixed points\nare equal.\n\\end{prop}\n\n\\begin{proof}\nSince $c_{\\Gamma}=0$, there is a connection $1$-form $\\alpha=f_{\\alpha}\\cdot\\frac{\\partial}{\\partial\\theta}$\non $\\Gamma_{J}^{2}$ such that $df_{\\alpha}=0$. For any two fixed\npoints $p_{0},p_{1}$ of $\\Phi$, choose $w_{0}\\in\\pi_{\\Gamma^{2}}^{-1}(p_{0})$\nand $w_{1}\\in\\pi_{\\Gamma^{2}}^{-1}(p_{1})$. Let $\\lambda:[0,1]\\rightarrow\\Gamma_{J}^{2}$\nbe a smooth path from $w_{0}$ to $w_{1}$. Define a map $h$ from\n$S^{1}\\times[0,1]$ to $\\Gamma_{J}^{2}$ by\n\\[\nh(z,t)=\\Phi_{\\Gamma^{2}}^{z}\\circ\\lambda(t).\n\\]\nLet $\\tilde{f}_{\\alpha}$ be the pulled-back $1$-form of $f_{\\alpha}$\non $S^{1}\\times[0,1]$, i.e. $\\tilde{f}_{\\alpha}=h^{*}(f_{\\alpha})$.\nThen it holds that $d\\tilde{f}_{\\alpha}=h^{*}(df_{\\alpha})=0$. By\nStokes' formula,\n\\begin{equation}\n\\int_{S^{1}\\times\\{0\\}}\\tilde{f}_{\\alpha}=\\int_{S^{1}\\times\\{1\\}}\\tilde{f}_{\\alpha}.\\label{eq:boundary integration}\n\\end{equation}\nNote that for $i\\in\\{0,1\\}$, it holds $h_{*}(\\frac{\\partial}{\\partial\\theta}\\bigg|_{(z,i)})=\\mathcal{X}_{\\theta}\\big(\\Phi_{\\Gamma^{2}}^{z}(w_{i})\\big)$.\nAs a consequence, we have\n\\[\n\\int_{S^{1}\\times\\{i\\}}\\tilde{f}_{\\alpha}=\\int_{0}^{1}\\tilde{f}_{\\alpha}(\\frac{\\partial}{\\partial\\theta})\\bigg|_{(e^{2\\pi i\\cdot t},i)}dt=\\int_{0}^{1}f_{\\alpha}(\\mathcal{X}_{\\theta})\\bigg|_{\\gamma_{w_{i}}(t)}dt=k_{p_{i}}\n\\]\nwith $\\gamma_{w_{i}}(t)=\\Phi_{\\Gamma^{2}}^{e^{2\\pi i\\cdot t}}(w_{i})$.\nTogether with Eq.(\\ref{eq:boundary integration}) it concludes the\nproof.\n\\end{proof}\nThe following two remarks can be obtained by averaging the Liouville\n$1$-form of a cotangent bundle, but they also appear as consequences\nof the discussion above about the local Maslov indices. Note that\nwhen $M$ is a cotangent bundle, it holds that $c_{\\Gamma}=[\\omega]=0$,\nand the Maslov $S^{1}$ bundles are trivial.\n\\begin{rem}\nSuppose that $M$ is a connected cotangent bundle and $h_{\\theta}$\nis a Hamiltonian of the $S^{1}$ action. Since $k_{p}=k_{p'}$ for\nany fixed points, $h_{\\theta}$ has at most $1$ critical value.\n\\end{rem}\n\n\\begin{rem}\nConsider the case where $M$ is a cotangent bundle and $\\dim M=4$.\nNote that in a neighbourhood of a fixed point $p$, the $S^{1}$ action\ncan be linearized in a Daboux chart as $t\\mapsto(e^{2\\pi i\\cdot mt},e^{-2\\pi i\\cdot nt})$.\nWhen $k_{p}=0$, we have $m-n=0$, and hence the fixed points are\nall $1:-1$ resonances.\n\\end{rem}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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The in-plane directions are bulk-like\nand can be separated within the 3D Schr\\\"odinger equation. The variation over the different layers can then be described by a 1D model, most commonly by a square quantum well, where the eigenenergies increase quadratically and scale with $1\/d^2$, $d$ being the well width. In many cases, however, a deviation from this standard\ncase is observed.\nFor epitaxially grown SnO$_2$, InSb, and Pb films with low charge densities, measurements with exponents about $1.5$ have been reported~\\cite{SciRep.5.17424, PhysStatSolB.33.425, PhysRevB.66.233408}.\nFurthermore, Kr\\\"oger et al.~\\cite{PhysRevB.97.045403} and Hirahara et al.~\\cite{PhysRevB.75.035422} investigated Bi films \n\\cite{PhysRevLett.115.106803, PhysRevLett.93.046403, PhysRevLett.117.236402, JElectronSpec.201.98, NanoLett.12.1776}\nand demonstrated that these exhibit equidistant energy levels with a $1\/d$ thickness dependence\n\\cite{PhysRevB.97.045403, PhysRevB.75.035422}.\nFor this, Kr\\\"oger et al.\\ prepared epitaxially grown Bi(111) films on Si(111) samples.\nThe thicknesses varied between 20 and 100 bilayers (i.e.\\ $8\\,\\text{nm}$ to $40\\,\\text{nm}$).\nThe conductance $G$ was measured as a function of temperature $T$.\nA general dependence was fitted to this data and an effective bandgap $E_\\text{g}$ was extracted by describing the temperature dependence of the conductance by thermal excitation, $G\\propto\\text{exp}(E_\\text{g}\/k_\\text{B}T)$, where $k_\\text{B}$ is the Boltzmann constant.\nHirahara et al.\\ prepared Bi(001) films grown on Si(111) samples.\nThe thicknesses varied from 7 to 40 bilayers (i.e.\\ $2.8\\,\\text{nm}$ to $16\\,\\text{nm}$).\nThe energy levels were measured with angle-resolved photoemission spectroscopy (ARPES) and were found to be equidistant.\nThis motivates a description by the well-known harmonic oscillator potential in contrast to the usually found square well potential, which was also stated by Kr\\\"oger et al.\nAn explanation given by Hirahara et al.\\ is the phase shift accumulation model, originating from the Bohr-Sommerfeld quantization condition, which yields the $1\/d$-dependence if the dispersion near the Fermi energy is linear~\\cite{PhysRevB.75.035422}.\nNevertheless, the question must be asked whether equidistant energy levels actually necessitate a harmonic potential or whether alternative potential shapes with equidistant levels exist, especially considering that only a limited energy range is examined and a finite measurement uncertainty is present.\nThe general answer is ``no''~\\cite{CommMathPhys.82.471}, which was demonstrated by previous analytic derivations resulting in different potentials by generalizing the creation\/annihilation operator.\nFor this purpose, the factorization method \\cite{JMathPhys.25.3387, TheorChemAcc.110.403} or the more general shift-operator approach~\\cite{Chaos.4.47, SovPhysJETP.75.446} was used.\n\nFirst principle studies of any material are expected to result in a potential which oscillates with the atomic positions of the real structure which is put in.\nAs the harmonic oscillator is a valid model, an expectation could be that the potential of thin Bi films would on average and at a scale larger than the atomic structure be quadratic with additional spatial variations at smaller scale.\nWithin this manuscript we focus on the model point of view and calculate model potentials satisfying the previous expectation.\nWe hope that this publication will serve to interpret experimental results more carefully and clarify that even when the energy levels are known, a wide range of potentials must still be considered.\nIn the following, we assume equidistant energy levels and investigate what follows for the potential.\nIn \\sec{Experiments} the experimental results are modeled by a truncated harmonic oscillator, yielding good agreement of the energy levels with the analytical harmonic oscillator case up to a certain energy and providing a more physical description including the specific layer thickness.\n\\Sec{Shift_operator} treats the 1D  Schr\\\"odinger equation as an inverse problem by calculating the potentials from the given equidistant energy spectrum.\nThis is done using the shift-operator approach.\nWe present some previous analytical results and some new numerical calculations and possible interpretations concerning real structures.\nIn \\sec{Perturbation} we show by perturbation theory that polynomial potentials apart from the quadratic harmonic oscillator potential cannot exhibit an infinite number of \\textbf{exactly} equidistant energy levels.\n\n\\section{Comparison with experiments}\\label{sec:Experiments}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{graphics\/experiment.pdf}\n\t\\caption{Experimental data of Kr\\\"oger et al.~\\cite{PhysRevB.97.045403} (3 different Bi films) and Hirahara et al.~\\cite{PhysRevB.75.035422} in comparison with a fit to the harmonic oscillator.}\n\t\\label{fig:HO:experiment}\n\\end{figure}\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/oscillator_truncated.pdf}\n\t\\caption{Left: energy levels. Center: energy level spacings. Right: thickness-dependent energy level. Colored lines denote the 1D harmonic oscillator. Black dots denote the truncated 1D harmonic oscillator of 16\\,nm thickness.}\n\t\\label{fig:HO:truncated}\n\\end{figure}\n\nIn this section, we concentrate on the interpretation of experimental results for Bismuth films from the literature.\nThe experimental results of Kr\\\"oger et al.~\\cite{PhysRevB.97.045403} and Hirahara et al.~\\cite{PhysRevB.75.035422}, i.e.\\ the energy level spacings $\\Delta E$ as a function of the Bi film thickness $d$, are depicted in \\fig{HO:experiment} (colored symbols).\n$d_\\text{BL}$ is the thickness of a Bi bilayer, which form a stable unit in (111) direction.\nFor the Kr\\\"oger data, the measured bandgap $E_\\text{g}$ is interpreted as energy spacing $\\Delta E$.\nThe inverse thickness dependence can be clearly seen. The solid line shows a fit with a $d^{-1}$ thickness dependence.\n\nTo describe the experimental data and to connect the eigenenergies to a physical situation, the model of a truncated quadratic potential\n\\begin{flalign}\n\tV(x) &= V_0\\begin{cases}\n\t\t1 & |x|\\geq\\frac{d}{2}\\\\\n\t\t\\left(\\frac{2x}{d}\\right)^2 & |x|\\leq\\frac{d}{2}\n\t\\end{cases} &\n\\end{flalign}\nis used, where $d$ is the thickness of the material and the potential at the well edge $x=\\pm\\frac{d}{2}$ equals the well depth $V_0$.\nExemplary results are shown in \\fig{HO:truncated} for $V_0\/m^\\ast=8\\,\\text{eV}\/m_\\text{e}$ and $d=16\\,\\text{nm}$, where $m^\\ast$ is the effective mass and $m_\\text{e}$ is the electron mass.\nThe energy eigenvalues $E_n$ and the energy level spacings $\\Delta E$ of the truncated potential are in very good agreement with the non-truncated potential (i.e.\\ the harmonic oscillator) for $V<0.9V_0$, yielding\n\\begin{align}\n\tE_n = \\Delta E\\left(n+\\dfrac{1}{2}\\right) \\quad,\\quad \\Delta E = \\frac{2\\hbar}{d}\\sqrt{\\frac{2V_0}{m^\\ast}} \\label{eqn:fit}\n\\end{align}\nwith equidistant energy spacing $\\Delta E$, which scales like $1\/d$.\n\nA regression according to~\\eqn{fit} for reproducing the thickness dependence of the experimental data in \\fig{HO:experiment} yields $\\Delta E = 3.34\\,\\text{eV}\\cdot d_\\text{BL}\/d$ and $V_0\/m^\\ast=2.93\\,\\text{eV}\/m_\\text{e}$.\nAll data are in good agreement with the regression. We conclude that the experimental data may be well fitted with a truncated harmonic oscillator potential which yields the desired equidistant energy states. Nevertheless, the question remains whether this is the only possible shape for the potential. The next section thus deals with finding alternative potential shapes with the same energy spectrum.\n\n\\section{Anharmonic oscillators with equidistant energy levels}\\label{sec:Shift_operator}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{graphics\/L2.pdf}\n\t\\caption{Potential and lowest states of the second-order shift operator for $A=1$. The offset of the states are the corresponding eigenenergies.}\n\t\\label{fig:L2}\n\\end{figure}\n\nIn this section we show how the creation\/annihilation operators of the harmonic oscillator can be generalized with the aim of finding further potentials with equidistant spectra.\nAn approach to model equidistant energy levels is using the shift operator to treat the Schr\\\"odinger equation as an inverse problem, explained in the following.\nWe first give a general introduction and show previous analytical results by Dubov et al.~\\cite{Chaos.4.47, SovPhysJETP.75.446}.\nAfterwards we present new numerical calculations and possible interpretations concerning real structures.\n\n\\subsection{Method}\n\nThe general idea is to define for an arbitrary potential the shift operator $\\mathcal{L}$, which shifts the eigenstates $\\psi$ in energy:\n\\begin{align}\n\t\\mathcal{L}\\psi(\\xi,\\epsilon) &= \\psi(\\xi,\\epsilon+1) \\quad.\n\\end{align}\nHere, $\\xi=\\sqrt{m\\omega\/\\hbar}x$ and $\\epsilon=E\/\\hbar\\omega$ are the dimensionless real space and energy.\nIf the states are equidistant, $\\mathcal{L}$ transforms state $\\psi_n$ into state $\\psi_{n+1}$, resulting in the operator equation\n\\begin{align}\n\t[\\mathcal{H},\\mathcal{L}] &= \\mathcal{L} \\quad.\n\\end{align}\nIt can be solved by applying\n\\begin{align}\n\t\\mathcal{L} = \\sum_{k=0}^K \\alpha_k(\\xi)(\\text{i}\\mathcal{P})^k \\quad \\text{with}\\quad K\\geq 1 \\quad.\n\\end{align}\n$\\mathcal{P}$ is the momentum operator.\n$\\alpha_k(\\xi)$ are free parameters.\n\nThe first-order case ($K=1$) leads to the harmonic potential $U_0=\\frac{1}{2}\\xi^2$ and $\\mathcal{L}$ equals the creation operator.\nThe derivations can be found in the Supplementary Material.\nThe solution of the second-order shift operator gives the isotonic oscillator~\\cite{PhysLettA.70.177, IntJQuantumChem.110.1317, JPhysAMathGen.20.4331, JPhysAMathTheo.41.085301}\n\\begin{align}\n\tU_1 = \\frac{1}{8}\\xi^2 + \\frac{A}{\\xi^2} \\label{eqn:U_2}\n\\end{align}\nwith either $\\xi>0$ or $\\xi<0$, depicted in \\fig{L2} (black curve).\nIt can be interpreted as a half harmonic potential with a smooth wall, where the smoothness can be adjusted with $A$.\nFor the half harmonic oscillator (with a hard wall) only the odd solutions remain, resulting in a doubled energy level difference.\nThe potential in \\eqn{U_2} yields very similar eigenstates and the same eigenenergies as the half harmonic oscillator, but slightly shifted in energy.\nThe eigenstates have a similar form to the harmonic potential:\n\\begin{align}\n\t\\epsilon_n &= \\frac{1}{2} + n + \\frac{1}{4}\\sqrt{1+8A} \\quad,\\\\\n\t\\psi_n(\\xi) &= C_n \\xi^{(1+\\sqrt{1+8A})\/2} \\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\sum_{k=0}^n a_k \\xi^{2k} \\quad.\n\\end{align}\nThe eigenstates are also shown in \\fig{L2}.\nIt can be concluded that with this potential it is possible to model asymmetric minima in contrast to the pure harmonic potential.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_U.pdf}\n\t\\caption{Potentials of the third-order shift operator $U_0 \\ldots U_{21}$.}\n\t\\label{fig:L3}\n\\end{figure}\n\nThe third-order shift operator results in the differential equation\n\\begin{gather}\n\t\\frac{3}{2}\\left(W^2\\right)'' - \\frac{1}{4}W'''' + \\xi^2W'' + 3\\xi W' = 0 \\label{eqn:L3:U:DGL}\n\\end{gather}\nwith $U = W + \\frac{1}{2}\\xi^2$.\nThe prime denotes the first derivative.\n The equation can be solved as an initial value problem or a boundary value problem with any given conditions, resulting in many different solutions for the potential.\nPossible solutions obtained by a Darboux transformation~\\cite{Chaos.4.47, OsakaJMath.36.949} are\n\\begin{align}\n\tU_m &= -\\frac{1}{2}\\xi^2 - \\frac{2}{3}(2m+1) + \\left( \\frac{P_m'(\\xi)}{P_m(\\xi)} + \\xi \\right)^2 \\quad,\\\\\n\tP_{2m} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k)!}\\xi^{2k} \\quad,\\\\\n\tP_{2m+1} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k+1)!}\\xi^{2k+1} \\quad.\n\\end{align}\nThe index $m$ numbers the solutions.\nThey are drawn in \\fig{L3}.\nA further shift of the solutions in $\\xi$ and $U$ is valid, but not a superposition due to the non-linearity of the constituting equations.\n$U_{2m}$ are quadratic-like potentials with a dip at the minimum.\n$U_0$ is the harmonic potential.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=1$.\nThe ground state of potential $U_{2m}$ is additionally lowered by $2m$.\n$U_{2m+1}$ are singular potentials at ${\\xi=0}$.\n$U_1$ is one solution of the second-order shift operator.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=2$ ($\\Delta\\epsilon=1$ can be achieved by rescaling $\\xi=\\tilde{\\xi}\/2$).\nThe eigenstates of these anharmonic potentials are also similar to the ones of the harmonic potential.\nEspecially the potentials $U_1$ and $U_{2n+1}$ allow to construct asymmetric 1D models which describe equidistant energy levels, providing a much more flexible tool than a restriction to the analytical harmonic oscillator offers.\n\nFurther examples and their derivations can be found in the Supplementary Material.\n\n\\subsection{Numerical results}\n\n\\Eqn{L3:U:DGL} or its first integrals\n\\begin{gather}\n\t\\frac{3}{2}\\xi\\left(W^2\\right)' - \\frac{3}{2}W^2 - \\frac{1}{4}\\xi W''' + \\frac{1}{4}W'' + \\xi^3W' = A \\quad,\\label{eqn:L3:U:DGL:2}\\\\\n\t-\\frac{1}{2}\\left( \\frac{A + \\frac{3}{2}W^2 - \\frac{1}{4}W''}{\\xi} \\right)^2 - \\frac{1}{2}W^3 + \\frac{1}{8}\\left(W'\\right)^2 = AW + B\\label{eqn:L3:U:DGL:3}\n\\end{gather}\ncan also be solved numerically to achieve further types of potentials.\nWe calculated the potential using the explicit Runge--Kutta method of 4th order or higher as implemented in the software package Mathematica~\\cite{Mathematica.12.1}.\nWe quantify the numerical error of the solution using the local relative residuum, i.e. the residuum normalized to the maximum absolute additive contribution to the differential equation, evaluated as a function of $\\xi$.\nWe get values below $10^{-4}$, which is sufficiently small to consider the solution correct.\nThe Schr\\\"odinger equation is solved by discretizing the differential operator and the potential on an equidistant grid ($\\Delta x<0.01\\,\\text{nm}$) and solving the resulting matrix eigenvalue problem as implemented in Mathematica.\nFor this, the calculated spatial dimension is much larger than presented in the following, i.e. large enough for the potential at the boundary to be higher than the highest considered energy level, to prevent numerical errors from limiting the spatial dimension.\n\nThree different potentials and corresponding solutions of the Schr\\\"odinger equation are depicted in Fig.~\\ref{fig:L3:num_1}, Fig.~\\ref{fig:L3:num_2}, and Fig.~\\ref{fig:L3:num_3}.\nThey represent three different types of solutions, named type-1, type-2, and type-3 potentials in the following.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_numerisch1.pdf}\n\t\\caption{Numerical solution of \\eqn{L3:U:DGL:2} with $A=-0.4$ and $W(1)=W'(1)=W''(1)=0$. Left: type-1 potential and lowest states. The offset of the states are the corresponding eigenenergies. Right: energy levels. The inset shows the energy level spacings.}\n\t\\label{fig:L3:num_1}\n\\end{figure}\n\n\\begin{figure}[!b]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_numerisch2.pdf}\n\t\\caption{Numerical solution of \\eqn{L3:U:DGL:2} with $A=-0.001$ and $W(1)=W'(1)=W''(1)=0$. Left: type-2 potential and selected states (quantum numbers 1, 11, 22, 32, 42, 52, 63, 73, 84, 95). The offset of the states are the corresponding eigenenergies. Two different classes of states are denoted by color (class 1: red, class 2: blue). Right: energy levels. Arrows mark the states shown in (a). The insets show the energy level spacings.}\n\t\\label{fig:L3:num_2}\n\\end{figure}\n\n\\fig{L3:num_1} shows a potential with two singularities and a minimum in between.\nThe singularities behave like $1\/x^2$, similar to the $U_{2n+1}$ potential.\nThe wave functions are also of similar shape as the ones of the former potentials.\nThe energy levels are roughly equidistant in the depicted range with $\\Delta E\\approx 1040\\,\\text{meV}$.\nThe higher states increase quadratically in energy instead of linearly.\nThis could be due to the finite discretization and the resulting bad energetic resolution or due to a badly converged potential near the singularities.\nHowever, the $1\/x^2$ dependence near the singularities acts like a square box, resulting in quadratically increasing energy levels.\n\nThe second type of potentials, see example in \\fig{L3:num_2}, also has a $1\/x^2$ singularity for positive values of $x$, similar to $U_{2n+1}$.\nYet for $x<0$, the potential oscillates and the strength of these oscillations increases with rising absolute value of $x$.\nThe states of this potential can be separated into two classes:\n(1) States localized in the oscillation minima (red states in \\fig{L3:num_2}).\n(2) Delocalized states oscillating between the smooth wall and the potential oscillation peaks of the corresponding energy range (blue states in \\fig{L3:num_2}).\nThe depicted example exhibits nearly linearly increasing energy levels.\nAdditional small irregular variations are due to the 2 classes of states.\nThe insets in \\fig{L3:num_2} show the energy level spacings of these distinct classes.\nThe class-1 states (upper inset) show linearly increasing energy levels with $\\Delta E\\approx 358\\,\\text{meV}$.\nThe class-2 states (lower inset) vary between $\\Delta E=300\\,\\text{meV}$ and $\\Delta E=450\\,\\text{meV}$.\nThe superposition of these two classes of states with different energy level spacings leads to the variations in the combined picture and although the class-2 states are not truly equidistant, the deviations from the overall linearity of all states are small.\nThe overall average energy level spacing is $\\Delta E\\approx 180\\,\\text{meV}$.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/L3_numerisch3.pdf}\n\t\\caption{Numerical solution of \\eqn{L3:U:DGL:3} with $A=0$, $B=-1$, and $W(1)=W'(1)=0$. Left: type-3 potential and selected states (quantum numbers 5, 13, 24, 33, 44, 53, 62, 72, 82). The offset of the states are the corresponding eigenenergies. Two different classes of states are denoted by color (class 1: red and green, class 2: blue). Right: energy levels. Arrows mark the states shown in (a). The insets show the energy level spacings (red data points are behind green data points).}\n\t\\label{fig:L3:num_3}\n\\end{figure}\n\n\\Fig{L3:num_3} depicts the third type of potentials with no singularities, but oscillations in both directions and also displaying two different\nclasses of states.\nClass 1 are states mostly localized within the potential oscillations at the left or right (red and green in \\fig{L3:num_3}).\nThere are also states similar to the depicted ones, but located at the other side.\nClass 2 are delocalized states oscillating between the left and right  potential oscillation peaks of corresponding energy (blue in \\fig{L3:num_3}).\nEnergy levels increase almost linearly; the small periodic variations are due to the two classes of states.\nThe insets in \\fig{L3:num_3} show the energy level spacings of these distinct classes.\nIt can be clearly seen that for each class linearly increasing energy levels are present.\nClass 1 (upper inset in \\fig{L3:num_3}) has $\\Delta E=358\\,\\text{meV}$ for states located at the left (green) as well as for states located at the right (red).\nThe states are just shifted by a small energy due to the slightly asymmetry of the potential.\nClass 2 (lower inset) has $\\Delta E\\approx 267\\,\\text{meV}$.\nThe class-2 energy level spacings vary slightly (about 10\\%).\nBut as this is a differential value, deviations from linearity are sufficiently small and in good agreement with experimental results.\nThe superposition of these two classes of states with different energy level spacings leads to the irregular variations in the combined picture.\nIn total, the energy levels show a linear dependence with an overall average energy level spacing of $\\Delta E\\approx 108\\,\\text{meV}$.\nThe additional irregular variation on the large scale are small.\nWith regard to experimental results with some error bars these irregularities may not be visible.\n\nThe presented 1D potentials allow more interpretations concerning the comparison with real structures than a pure 1D harmonic oscillator model.\nThey can be related to different possible cases occurring in reality.\nFor example, a thin, free-standing slab, surrounded by vacuum, has states which are trapped within the slab in the direction perpendicular to the slab and exponentially decaying into vacuum.\nThis could be modeled by the type-1 potential of the numerical calculations, which diverges at the slab-vacuum interface.\nA thin layer on a substrate could be modeled by the type-2 potential.\nThis diverges at the layer-vacuum interface and oscillates into the bulk-substrate, representing the periodic structure.\nA thin sandwich-layer between two other substrates could be modeled by the type-3 potential, which oscillates into both directions to describe the penetration into the periodic bulk material.\nAnother interpretation could be to use the type-2 or type-3 potential to describe the oscillating atomic structure within the layer itself.\nThe experimental results from literature~\\cite{PhysRevB.97.045403, PhysRevB.75.035422, PhysRevLett.115.106803} have the structure silicon--bismuth--vacuum and fall into these categories.\nThe $\\Delta E=358\\,\\text{meV}$ achieved in \\fig{L3:num_2} is in the range of possible experimental $\\Delta E$ of \\fig{HO:experiment}.\nUsing the result of the fit in \\eqn{fit}, this corresponds to a thickness of $9.3$ bismuth bilayers, i.e. $3.7\\,\\text{nm}$.\nWithin the potential in \\fig{L3:num_2} this approximately matches the width of the non-oscillating part at the right near the singularity.\nThe oscillating part represents the transition into the bulk-like part with wave lengths between $0.2\\,\\text{nm}$ (for $-10\\,\\text{nm}<x<-8\\,\\text{nm}$) and $0.6\\,\\text{nm}$ (for $-4\\,\\text{nm}<x<-2\\,\\text{nm}$).\nThis nicely matches the periodicity of possible atomic structures (the bismuth bilayer thickness is approximately $0.4\\,\\text{nm}$, the silicon lattice constant is $0.54\\,\\text{nm}$).\n\nIn this section spectra have been studied which are equidistant in a limited energy range or contain a subset of equidistant states.\nIn the next section we show what can be learned from perturbation theory.\nThe difference to the former shift-operator approach is that, in the following, searching for equidistant eigenstates means an infinite number of exactly equidistant states.\nThe formerly shown results are less strict and do not belong to this for the following reasons:\nPotentials $U_{2m+1}$ and the numerical type-1 potential have non-polynomial shapes and quadratic non-integrable singularities.\nPotentials $U_{2m}$ consider parts of the spectrum being non-equidistant (i.e. the ground state).\nThe numerical type-2 and type-3 potentials lead to states which split into two classes with separate energy spacing each.\n\n\\section{Perturbed 1D harmonic oscillator}\\label{sec:Perturbation}\n\nIn this section, we use perturbation theory to show that an \\textbf{exactly} equidistant energy spectrum \\textbf{over the whole energy range} cannot be achieved by a polynomial potential other than the harmonic (quadratic) potential.\nTo this end, the 1D harmonic oscillator is perturbed with an arbitrary polynomial perturbation potential ${u(\\xi)=\\sum_{j=0}^Nu_j\\xi^j}$,\n\\begin{align}\n\t\\left[ -\\dfrac{\\partial^2}{\\partial\\xi^2} + \\xi^2 + \\lambda u(\\xi) \\right] \\psi_n(\\xi) &= \\epsilon_n^{(0)}\\psi_n(\\xi).\n\\end{align}\nAs the equations are linear, we consider each monomial $\\xi^j$ individually.\nThe perturbation potential ${u(\\xi)=\\xi^j}$ in the basis of the harmonic oscillator eigenstates gives the perturbation coefficients\n\\begin{align}\n\tu_{mn}^{(j)} &= \\frac{1}{\\sqrt{\\pi 2^{m+n}m!n!}} \\int\\limits_{-\\infty}^\\infty \\xi^j\\text{e}^{-\\xi^2}H_m(\\xi)H_n(\\xi) \\,\\text{d}\\xi \\quad,\n\\end{align}\nwhere $H_n(\\xi)$ are the Hermite polynomials.\n\nThe diagonal elements $u_{kk}^{(j)}$ equal the first-order energy correction $\\epsilon_k^{(1)}$.\nThey are equal to zero for odd values of $j$.\nFor even numbers $j$, $u_{kk}^{(j)}$ is a polynomial in $k$ of order $j\/2$ as shown in \\fig{u_kk_j} for different monomial perturbations ${u(\\xi)=\\xi^j}$.\nFor the smallest values of $j$ one gets\n\\begin{align}\n\tu_{kk}^{(0)} = 1 \\quad,\\quad\n\tu_{kk}^{(2)} = \\frac{1}{2} + k \\quad,\\quad\n\tu_{kk}^{(4)} = \\frac{3}{4} + \\frac{3}{2}k + \\frac{3}{2}k^2 \\quad.\n\\end{align}\nConsequently, if the harmonic oscillator is perturbed with a quadratic potential, $\\epsilon_k^{(1)}$ is linear in $k$, reproducing the equidistant energy levels of the harmonic oscillator.\nIf the harmonic oscillator is perturbed with a non-quadratic polynomial potential, $\\epsilon_k^{(1)}$ is a superposition of different powers of $k$, yielding a different spectrum.\nThus, only the quadratic polynomial gives equidistant first-order energy corrections.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{graphics\/u_kk.pdf}\n\t\\caption{$u_{kk}^{(j)}$ for different perturbation potentials $u(\\xi)=\\xi^j$.}\n\t\\label{fig:u_kk_j}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\TwoColumnWidth]{graphics\/u_kk-d.pdf}\n\t\\caption{$u_{k,k-l}^{(j)}$ for different $l$ and for different perturbation potentials $u(\\xi)=\\xi^j$.}\n\t\\label{fig:u_mn_j}\n\\end{figure}\n\nThe non-diagonal elements $u_{mn}^{(j)}$ are needed to calculate the higher order energy corrections $\\epsilon_k^{(\\geq 2)}$.\nFor $p$th order a $(p-1)$-fold sum over a $p$-fold product of $u_{mn}^{(j)}$ with different indices has to be calculated.\nIf $m+n+j$ is odd or if $|m-n|>j$, then ${u_{mn}^{(j)}=0}$.\nThis leads to finite summations over no more than $j+1$ elements $u_{k,k+l}^{(j)}$ with $-j\\leq l\\leq j$.\nThe $u_{k,k+l}^{(j)}$ are shown in \\fig{u_mn_j} for different $l$ and for different polynomials ${u(\\xi)=\\xi^j}$.\nThe non-zero elements for a specific $l$ can be written as\n\\begin{align}\n\tu_{k,k-l}^{(j)} = \\sqrt{\\frac{k!}{2^l(k-l)!}}p_{kl}^{(j)}\n\\end{align}\nwith a polynomial $p_{kl}^{(j)}$ in $k$ of order $(j-l)\/2$.\nThus, $(u_{k,k-l}^{(j)})^2$ is a polynomial in $k$ of order $j$ for all $l$.\nThe previous diagonal case $l=0$ is included.\nIn particular, for the smallest values of $j$ one gets\n\\begin{flalign}\n\tp_{k,1}^{(1)} &= 1 \\quad,\n\t& p_{k,1}^{(3)} &= \\frac{3}{2}k \\quad,\n\t& p_{k,1}^{(5)} &= \\frac{5}{4}(1+2k^2) \\quad,\\\\\n\tp_{k,2}^{(2)} &= 1 \\quad,\n\t& p_{k,2}^{(4)} &= -1+2k \\quad,\n\t& p_{k,2}^{(6)} &= \\frac{15}{4}(1-k+k^2) \\quad.&&\\hspace{15em}\n\\end{flalign}\n\nWith this, the second order energy correction\n\\begin{align}\n\t\\epsilon_k^{(2)} &= -\\sum_{n\\neq k}\\frac{u_{nk}u_{kn}}{\\epsilon_n^{(0)}-\\epsilon_k^{(0)}} = -\\frac{1}{2} \\sum_{\\substack{1\\leq l\\leq j\\\\l-j~\\text{even}}}\\frac{(u_{k,k+l}^{(j)})^2-(u_{k,k-l}^{(j)})^2}{l} = \\mathcal{O}(k^{j-1})\n\\end{align}\nis a polynomial in $k$ of order $j-1$, as the highest order in $k$ cancels out with the difference $(u_{k,k+l}^{(j)})^2-(u_{k,k-l}^{(j)})^2$.\nConsequently, the second order energy corrections scale linearly with $k$ only for the quadratic potential ($j=2$).\nTreating the higher-order perturbations similarly, in the $k^{3j\/2}$-scaling two hierarchic differences lead to a cancellation of the two highest orders of $k$.\nWith this, $\\epsilon_k^{(3)}$ is a polynomial in $k$ of order $3j\/2-2$ and also linear scaling only for the quadratic potential.\n\nAnother argumentation can be done using the $p$th order perturbation corrections, written as a recursion:\n\\begin{align}\n\t\\epsilon_k^{(p)} &= \\sum_nu_{kn}c_{kn}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{kk}^{(p-q)} \\\\\n\tc_{km}^{(p)} &= \\frac{1}{\\epsilon_k^{(0)}-\\epsilon_m^{(0)}}\\left( \\sum_nu_{mn}c_{kn}^{(p-1)} - u_{kk}c_{km}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{km}^{(p-q)} \\right) \\quad.\n\\end{align}\nThe general argumentation is as follows:\nFor the harmonic oscillator, $c_{km}^{(0)}$ is independent of $k$ and $\\epsilon_k^{(0)}$ is linear in $k$.\nIf $c_{km}^{(p-1)}$ is independent of $k$ and $\\epsilon_k^{(p-1)}$ is linear in $k$, which is fulfilled for $p=1$, then $c_{km}^{(p)}$ is also independent of $k$ and $\\epsilon_k^{(p)}$ is also linear in $k$.\nFor the case $j\\neq 2$ yields: if $c_{km}^{(p-1)}$ scales like $k^x$ and $\\epsilon_k^{(p-1)}$ scales like $k^y$, then $c_{km}^{(p)}$ scales like $k^{x+j\/2-1}+k^{x+y-1}$ and $\\epsilon_k^{(p)}$ scales like $k^{x+j\/2}+k^{x+y}$.\nConsequently, a linear dependence of $\\epsilon_k$ on $k$ can only be achieved if $y=1$, thus $x=0$, giving $j=2$, which is the harmonic oscillator.\nIn total, for a general perturbation potential ${u(\\xi)=\\sum_{j=0}^Nu_j\\xi^j}$, the perturbed energy levels are a sum of linear functions of $k$ for the quadratic case and a sum of functions of $k$ with different exponents for the non-quadratic cases, which does not sum up to a linear dependence.\n\nThough it may seem so, \\Sec{Shift_operator} and \\Sec{Perturbation} do not contradict each other. \nThe previous argumentation holds only for a polynomial potential which produces an equidistant spectrum for all eigenstates and with no tolerance in the level spacing.\nIt does not apply to power series and spectra which are only equidistant in a specific energy range or approximately equidistant spectra, as required to model experimental spectra.\n\nThe derivations of this section and some more results can be found in the Supplementary Material.\n\n\\section{Summary and conclusions}\n\nIn this publication, we discuss the implications that may be drawn from the experimental observation of an equidistant energy level spacing. \nThough a harmonic potential comes to mind immediately, this shape of potential is sufficient, but not necessary for the equidistant spacing, especially considering that experimental\nresults cover a finite energy range with a finite measurement uncertainty. The question is thus more involved than is visible at first sight.\nIn this publication, we discuss different aspects of the problem.\n \nIn the first part we compare a 1D harmonic oscillator model with experimental results, looking at the analytical harmonic potential as well as on\na truncated version. \nThe energy levels measured in thin Bi films of different thicknesses and the resulting thickness-dependence fit the theoretical model with comparable energy values and dimensions.\nThus, the 1D harmonic oscillator is a valid model to describe the quantum well states in thin films as shown for Bi films.\n\nHowever, this still leaves the question whether it is the only possible potential to explain the measurements.\nTherefore, in the second part of the paper, we discuss anharmonic potentials of non-polynomial shape and apply the shift-operator \napproach to explicitly calculate potentials with an equidistant spectrum.\nWe present a number of analytic and numeric examples representing different types of potentials including asymmetric minima, singularities, and oscillations.\nThese potentials seem well-suited 1D models to describe the potential of thin slabs, thin films on substrates or thin sandwich films between two substrates.\nThey also fit the expectation that first principle calculations of real materials should result in oscillating potentials which are in average quadratic to fit the harmonic oscillator model.\n\nIn the last part, we apply $n$th-order perturbation theory with a polynomial perturbation potential to the quantum-mechanical harmonic oscillator problem.\nWe calculate and discuss the perturbation integrals $u_{k,k-l}$ and the resulting $n$th order energy corrections as a function of $k$ for single monomials $\\xi^j$,\nshowing that $u_{k,k-l}^2$ is a polynomial in $k$ of order $j$ and arguing that every non-quadratic perturbation monomial leads to a non-quadratic $k$-dependence of the $n$th-order energy corrections, leaving only the harmonic term to produce an equidistant spectrum.\n\nThus we conclude that all possible anharmonic potentials producing exactly equidistant spectra must have a non-polynomial form.\nAnharmonic non-polynomial potentials exist, which could describe real structures and produce spectra which mirror the experimental results.\n\n\\section*{Acknowledgment}\n\nWe thank C. Tegenkamp and H.-R. Berger for helpful discussions.\n\n\n\\section{Shift-operator approach}\n\n\\subsection{Definition and basic operator equation}\n\n\\noindent The Schr\\\"odinger equation in dimensionless coordinates\n\\begin{align}\n\\begin{aligned}\n\t\\xi &= \\sqrt{\\frac{m\\omega}{\\hbar}}x \\quad,& \\epsilon_n &= \\frac{E_n}{\\hbar\\omega} \\quad,& \\psi(\\xi) &= \\sqrt[4]{\\frac{\\hbar}{m\\omega}}\\Psi(x)\\quad,& U(\\xi) &= \\frac{V(x)}{\\hbar\\omega}\n\\end{aligned}\n\\end{align}\nis\n\\begin{align}\n\t\\left[ -\\dfrac{1}{2}\\dfrac{\\partial^2}{\\partial\\xi^2} + U(\\xi) \\right] \\psi_n(\\xi) = \\epsilon_n\\psi_n(\\xi) \\quad .\n\\end{align}\nThe shift operator\n\\begin{align}\n\t\\mathcal{L}\\psi(\\xi,\\epsilon) = \\psi(\\xi,\\epsilon+1) \\quad,\\quad \\mathcal{L}^\\dagger\\psi(\\xi,\\epsilon) = \\psi(\\xi,\\epsilon-1)\n\\end{align}\nshifts each state in energy by $\\hbar\\omega$.\nApplying the Hamiltonian operator gives\n\\begin{align}\n\\begin{aligned}\n\t\\mathcal{H}\\mathcal{L}\\psi(\\xi,\\epsilon) &= (\\epsilon+1)\\psi(\\xi,\\epsilon+1) \\quad,& \\mathcal{H}\\mathcal{L}^\\dagger\\psi(\\xi,\\epsilon) &= (\\epsilon-1)\\psi(\\xi,\\epsilon-1) \\quad,\\\\\n\t\\mathcal{L}\\mathcal{H}\\psi(\\xi,\\epsilon) &= \\epsilon\\psi(\\xi,\\epsilon+1) \\quad,& \\mathcal{L}^\\dagger\\mathcal{H}\\psi(\\xi,\\epsilon) &= \\epsilon\\psi(\\xi,\\epsilon-1) \\quad.\n\\end{aligned}\n\\end{align}\nFor an equidistant spectrum $\\epsilon_{n+1}=\\epsilon_n+1$ this gets\n\\begin{align}\n\\begin{aligned}\n\t\\mathcal{H}\\mathcal{L}\\psi_n(\\xi) &= \\epsilon_{n+1}\\psi_{n+1}(\\xi) \\quad,& \\mathcal{H}\\mathcal{L}^\\dagger\\psi_n(\\xi) &= \\epsilon_{n-1}\\psi_{n-1}(\\xi) \\quad,\\\\\n\t\\mathcal{L}\\mathcal{H}\\psi_n(\\xi) &= \\epsilon_n\\psi_{n+1}(\\xi) \\quad,& \\mathcal{L}^\\dagger\\mathcal{H}\\psi_n(\\xi) &= \\epsilon_n\\psi_{n-1}(\\xi) \\quad.\n\\end{aligned}\n\\end{align}\nWith this, a resulting operator equation can be set up using commutators:\n\\begin{align}\n\t[\\mathcal{H},\\mathcal{L}] = \\mathcal{L} \\quad,\\quad [\\mathcal{H},\\mathcal{L}^\\dagger] &= -\\mathcal{L}^\\dagger \\quad.\\label{eqn:shift_operator_equation}\n\\end{align}\nIt follows $[\\mathcal{H},\\mathcal{L}\\mathcal{L}^\\dagger] = [\\mathcal{H},\\mathcal{L}^\\dagger\\mathcal{L}] = 0$.\nConsequently, $\\mathcal{L}\\mathcal{L}^\\dagger$, $\\mathcal{L}^\\dagger\\mathcal{L}$, and $[\\mathcal{L},\\mathcal{L}^\\dagger]$ are functions of $\\mathcal{H}$. Thus, $\\mathcal{L}$ can be written as a polynomial of the dimensionless momentum operator $\\mathcal{P}=-\\text{i}\\frac{\\partial}{\\partial\\xi}$:\n\\begin{align}\n\t\\mathcal{L} = \\sum_{k=0}^K \\alpha_k(\\xi)(\\text{i}\\mathcal{P})^k \\quad.\\label{eqn:L:general}\n\\end{align}\n$K$ is the order of the shift operator.\n$\\alpha_k(\\xi)$ are free parameters.\nInserting \\eqn{L:general} into \\eqn{shift_operator_equation} and comparing the coefficients yields $K+2$ differential equations for $K+1$ unknown functions $\\alpha_k(\\xi)$ and the additional unknown function $U(\\xi)$.\n\n\\subsection{First-order shift operator}\n\n\\noindent Inserting the first-order shift operator\n\\begin{align}\n\t\\mathcal{L} = \\alpha_0(\\xi) + \\alpha_1(\\xi)\\text{i}\\mathcal{P}\n\\end{align}\ninto \\eqn{shift_operator_equation} yields\n\\begin{alignat}{4}\n\\begin{aligned}\n\t0 &= \\alpha_0 + \\frac{1}{2}\\alpha_0'' + U'\\alpha_1 \\quad,\\\\\n\t0 &= \\alpha_0' + \\alpha_1 + \\frac{1}{2}\\alpha_1'' \\quad,\\\\\n\t0 &= \\alpha_1' \\quad.\n\\end{aligned}\n\\end{alignat}\nThe solution for the $\\alpha_k$ yields the condition\n\\begin{align}\n\tU' = \\xi \\quad.\n\\end{align}\nIntegrating this yields the harmonic oscillator potential\n\\begin{align}\n\tU_0 = \\frac{1}{2}\\xi^2 \\quad.\n\\end{align}\nThe corresponding shift operator\n\\begin{align}\n\t\\mathcal{L} = -\\xi + \\text{i}\\mathcal{P} \\quad,\\quad \\mathcal{L}^\\dagger = \\xi + \\text{i}\\mathcal{P}\n\\end{align}\nis the creation resp.~the annihilation operator.\nThe corresponding solution of the Schr\\\"odinger equation is\n\\begin{align}\n\t\\epsilon_n &= \\frac{1}{2} + n \\quad,\\quad \\psi_n(\\xi) = \\frac{1}{\\sqrt{2^nn!\\sqrt{\\pi}}} H_n(\\xi)\\exp\\left(-\\frac{1}{2}\\xi^2\\right) \\quad,\n\\end{align}\nwhere $H_n(\\xi)$ are the Hermite polynomials.\nThe potential and the states of the first-order shift operator are depicted in \\fig{L1}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{supplement_graphics\/L1.pdf}\n\t\\caption{Potential and lowest states of the first-order shift operator.}\n\t\\label{fig:L1}\n\\end{figure}\n\n\\subsection{Second-order shift operator}\n\n\\noindent Inserting the second-order shift operator\n\\begin{align}\n\t\\mathcal{L} = \\alpha_0(\\xi) + \\alpha_1(\\xi)\\text{i}\\mathcal{P} + \\alpha_2(\\xi)(\\text{i}\\mathcal{P})^2\n\\end{align}\ninto \\eqn{shift_operator_equation} yields\n\\begin{alignat}{5}\n\\begin{aligned}\n\t0 &= \\alpha_0 + \\frac{1}{2}\\alpha_0'' + U'\\alpha_1 + U''\\alpha_2 = 0 \\quad,\\\\\n\t0 &= \\alpha_0' + \\alpha_1 + \\frac{1}{2}\\alpha_1'' + 2U'\\alpha_2 \\quad,\\\\\n\t0 &= \\alpha_1' + \\alpha_2 + \\frac{1}{2}\\alpha_2'' \\quad,\\\\\n\t0 &= \\alpha_2' \\quad.\n\\end{aligned}\n\\end{alignat}\nThe solution for the $\\alpha_k$ yields the condition\n\\begin{align}\n\t\\xi U' + 2U = \\frac{1}{2}\\xi^2 \\quad.\n\\end{align}\nMultiplying with $\\xi$ and summarizing using the product rule yields\n\\begin{align}\n\t(\\xi^2U)' = \\frac{1}{2}\\xi^3 \\quad.\n\\end{align}\nIntegrating and dividing by $\\xi^2$ yields\n\\begin{align}\n\tU_1 = \\frac{1}{8}\\xi^2 + \\frac{A}{\\xi^2} \\quad.\n\\end{align}\nThe shift operator is\n\\begin{align}\n\\begin{aligned}\n\t\\mathcal{L} &= \\frac{1}{4}\\xi^2 - \\frac{2A}{\\xi^2} - \\frac{1}{2} - \\xi\\text{i}\\mathcal{P} - (\\text{i}\\mathcal{P})^2 \\quad,\\quad \\mathcal{L}^\\dagger = \\frac{1}{4}\\xi^2 - \\frac{2A}{\\xi^2} + \\frac{1}{2} + \\xi\\text{i}\\mathcal{P} - (\\text{i}\\mathcal{P})^2 \\quad.\n\\end{aligned}\n\\end{align}\nThe solution of the Schr\\\"odinger equation can be achieved with the asymptotic behaviour\n\\begin{align}\n\t\\xi &\\rightarrow\\infty: & \\psi_\\infty''-\\frac{1}{2}\\xi^2\\psi_\\infty &= 0 \\quad, & \\psi_\\infty &\\sim \\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\quad,\\hspace{20em}\\\\\n\t\\xi &\\rightarrow 0: & \\psi_0''-\\frac{4A}{\\xi^2}\\psi_0 &= 0 \\quad, & \\psi_0 &\\sim \\xi^{(1+\\sqrt{1+8A})\/2}\n\\end{align}\nand the consequential approach\n\\begin{align}\n\t\\psi_n(\\xi) &= \\xi^{(1+\\sqrt{1+8A})\/2}\\sum_{k=0}^na_k\\xi^k\\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\quad.\n\\end{align}\nIt follows\n\\begin{align}\n\t\\epsilon_n &= \\frac{1}{2} + n + \\frac{1}{4}\\sqrt{1+8A} \\quad,\\\\\n\t\\psi_n(\\xi) &= C_n J_n(\\xi) \\left(\\frac{\\xi^2}{2}\\right)^{(1+\\sqrt{1+8A})\/4} \\exp\\left(-\\frac{1}{4}\\xi^2\\right) \\quad,\\\\\n\tJ_n(\\xi) &= \\sum_{k=0}^n (-1)^k \\begin{pmatrix}n\\\\k\\end{pmatrix} \\frac{\\Gamma\\left(\\frac{\\sqrt{1+8A}}{2}+1\\right)}{\\Gamma\\left(\\frac{\\sqrt{1+8A}}{2}+1+k\\right)} \\left(\\frac{\\xi^2}{2}\\right)^k \\quad,\\\\\n\tC_n &= \\sqrt{\\frac{\\sqrt{2}\\Gamma\\left(\\frac{\\sqrt{1+8A}}{2}+1+n\\right)}{n!\\Gamma^2\\left(\\frac{\\sqrt{1+8A}}{2}+1\\right)}} \\quad.\n\\end{align}\nThe potential and the states of the second-order shift operator are depicted in \\fig{L2}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\OneColumnWidth]{supplement_graphics\/L2.pdf}\n\t\\caption{Potential and lowest states of the second-order shift operator.}\n\t\\label{fig:L2}\n\\end{figure}\n\n\\subsection{Third-order shift operator}\n\n\\noindent Inserting the thirt-order shift operator\n\\begin{align}\n\t\\mathcal{L} = \\alpha_0(\\xi) + \\alpha_1(\\xi)\\text{i}\\mathcal{P} + \\alpha_2(\\xi)(\\text{i}\\mathcal{P})^2 + \\alpha_3(\\xi)(\\text{i}\\mathcal{P})^3\n\\end{align}\ninto \\eqn{shift_operator_equation} yields\n\\begin{alignat}{6}\n\t0 &= \\alpha_0 + \\frac{1}{2}\\alpha_0'' + U'\\alpha_1 + U''\\alpha_2 + U'''\\alpha_3 \\quad,\\\\\n\t0 &= \\alpha_0' + \\alpha_1 + \\frac{1}{2}\\alpha_1'' + 2U'\\alpha_2 + 3U''\\alpha_3 \\quad,\\\\\n\t0 &= \\alpha_1' + \\alpha_2 + \\frac{1}{2}\\alpha_2'' + 3U'\\alpha_3 \\quad,\\\\\n\t0 &= \\alpha_2' + \\alpha_3 + \\frac{1}{2}\\alpha_3'' \\quad,\\\\\n\t0 &= \\alpha_3' \\quad.\n\\end{alignat}\nThe solution for the $\\alpha_k$ yields the shift operator\n\\begin{align}\n\t\\mathcal{L} &= \\left( \\frac{3}{2}\\left(U^2\\right)' - \\frac{1}{4}U''' - \\frac{\\xi^2+3}{2}U' + \\frac{1}{2}\\xi \\right) + \\left( \\frac{1}{2}\\xi^2 - 3U - 1 \\right) \\text{i}\\mathcal{P} - \\xi(\\text{i}\\mathcal{P})^2 + (\\text{i}\\mathcal{P})^3 \\quad,\\\\\n\t\\mathcal{L}^\\dagger &= -\\left( \\frac{3}{2}\\left(U^2\\right)' - \\frac{1}{4}U''' - \\frac{\\xi^2+3}{2}U' - \\frac{1}{2}\\xi \\right) + \\left( \\frac{1}{2}\\xi^2 - 3U + 1 \\right) \\text{i}\\mathcal{P} + \\xi(\\text{i}\\mathcal{P})^2 + (\\text{i}\\mathcal{P})^3\n\\end{align}\nand the condition\n\\begin{align}\n\t\\frac{3}{2}\\left(W^2\\right)'' - \\frac{1}{4}W'''' + \\xi^2W'' + 3\\xi W' = 0\n\\end{align}\nwith $U = W + \\frac{1}{2}\\xi^2$.\nMultiplying with $\\xi$ and summarizing using the product rule yields the first integral\n\\begin{align}\n\t\\frac{3}{2}\\xi\\left(W^2\\right)' - \\frac{3}{2}W^2 - \\frac{1}{4}\\xi W''' + \\frac{1}{4}W'' + \\xi^3W' = A \\quad.\\label{eqn:DGL:W}\n\\end{align}\nMultiplying with $W'$, using the identities of \\eqn{DGL:W}\n\\begin{gather}\n\t\\frac{3}{2}\\xi\\left(W^2\\right)' - \\frac{1}{4}\\xi W''' + \\xi^3W' = A + \\frac{3}{2}W^2 - \\frac{1}{4}W'' \\quad,\\\\\n\t \\xi^3W' = -\\xi^2\\left( \\frac{A + \\frac{3}{2}W^2 - \\frac{1}{4}W''}{\\xi} \\right)' \\quad,\n\\end{gather}\nand summarizing using the chain rule yields another first integral\n\\begin{align}\n\t&-\\frac{1}{2}\\left( \\frac{A + \\frac{3}{2}W^2 - \\frac{1}{4}W''}{\\xi} \\right)^2 - \\frac{1}{2}W^3 + \\frac{1}{8}\\left(W'\\right)^2 = AW + B \\quad.\n\\end{align}\nUsing a Darboux transformation, a possible solution can be written in the form\n\\begin{align}\n\tW &= -\\xi^2 + \\frac{4}{3}\\mu + \\left( \\frac{\\chi_\\mu'}{\\chi_\\mu} \\right)^2\n\\end{align}\nwith\n\\begin{align}\n\t-\\frac{1}{2}\\chi_\\mu'' + \\frac{1}{2}\\xi^2\\chi_\\mu = \\mu\\chi_\\mu \\quad.\n\\end{align}\nWith $\\mu=-\\left(m+\\dfrac{1}{2}\\right)$ two possible classes of solutions are\n\\begin{align}\n\tU_m &= -\\frac{1}{2}\\xi^2 - \\frac{2}{3}(2m+1) + \\left( \\frac{P_m'(\\xi)}{P_m(\\xi)} + \\xi \\right)^2 \\quad,\\\\\n\tP_{2m} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k)!}\\xi^{2k} \\quad,\\\\\n\tP_{2m+1} &= \\sum_{k=0}^m\\frac{4^k}{(m-k)!(2k+1)!}\\xi^{2k+1} \\quad.\n\\end{align}\n$U_{2n}$ are quadratic potentials with a dip at the minimum.\n$U_0$ is the harmonic potential.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=1$.\nThe ground state of potential $U_{2n}$ is additionally lowered by $2n$.\n$U_{2n+1}$ are singular potentials at $\\xi=0$.\n$U_1$ is the one of the second-order shift operator.\nThe corresponding energy levels are equidistant with $\\Delta\\epsilon=2$ ($\\Delta\\epsilon=1$ can be achieved by rescaling $\\xi=\\xi'\/2$).\nThe potentials of the third-order shift operator are depicted in \\fig{L3:U}. The potentials $U_2$ and $U_4$ and corresponding lowest states are shown in \\fig{L3:U2} and \\fig{L3:U4}.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\TwoColumnWidth]{supplement_graphics\/L3_U.pdf}\n\t\\caption{Potentials of the third-order shift operator obtained by a Darboux transformation.}\n\t\\label{fig:L3:U}\n\\end{figure}\n\n\\begin{figure}[!b]\n\t\\centering\n\t\\begin{minipage}{\\OneColumnWidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{supplement_graphics\/L3_U2.pdf}\n\t\t\\caption{Potential $U_2$ and corresponding lowest states of the third-order shift operator.}\n\t\t\\label{fig:L3:U2}\n\t\\end{minipage}\\hfill\n\t\\begin{minipage}{\\OneColumnWidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{supplement_graphics\/L3_U4.pdf}\n\t\t\\caption{Potential $U_4$ and corresponding lowest states of the third-order shift operator.}\n\t\t\\label{fig:L3:U4}\n\t\\end{minipage}\n\\end{figure}\n\n\\section{Perturbed 1D harmonic oscillator}\n\n\\noindent The Schr\\\"odinger equation for the 1D harmonic oscillator is\n\\begin{align}\n\t\\left[ -\\frac{\\hbar^2}{2m}\\dfrac{\\partial^2}{\\partial x^2} + \\frac{m\\omega^2}{2}x^2 \\right] \\Psi_n(x) &= E_n^{(0)}\\Psi_n(x) \\quad.\n\\end{align}\nSubstituting with the dimensionless variables\n\\begin{align}\n\tx &= \\sqrt{\\frac{\\hbar}{m\\omega}}\\xi \\quad,\\quad E_n^{(0)} = \\hbar\\omega\\epsilon_n^{(0)} \\quad,\\quad \\Psi(x) = \\sqrt[4]{\\frac{m\\omega}{\\hbar}}\\psi(\\xi)\n\\end{align}\nyields the dimensionless formula\n\\begin{align}\n\t\\left[ -\\frac{1}{2}\\dfrac{\\partial^2}{\\partial\\xi^2} + \\frac{1}{2}\\xi^2 \\right] \\psi_n(\\xi) &= \\epsilon_n^{(0)}\\psi_n(\\xi) \\quad.\n\\end{align}\nIts solution is\n\\begin{align}\n\t\\epsilon_n^{(0)} &= n+\\frac{1}{2} \\quad,\\quad \\psi_n(\\xi) = \\frac{1}{\\sqrt{2^nn!\\sqrt{\\pi}}} \\exp\\left(-\\frac{1}{2}\\xi^2\\right) H_n\\left(\\xi\\right) \\quad,\n\\end{align}\nwhere $H_n$ are the Hermite polynomials. They can be expressed the following:\n\\begin{align}\n\tH_n(\\xi) &= 2\\xi H_{n-1}(\\xi) - 2(n-1)H_{n-2}(\\xi) \\quad.\\label{eqn:Hermite:recursive}\n\\end{align}\n\n\\noindent The Schr\\\"odinger equation of a system, which is perturbed with the perturbation potential $v(x) = \\frac{1}{2}\\hbar\\omega u(\\xi)$, respective the corresponding dimensionless perturbation potential $u(\\xi)$, is\n\\begin{align}\n\t\\left[ \\mathcal{H}_0 + \\lambda u(\\xi) \\right] \\phi_k(\\xi) &= \\epsilon_k\\phi_k(\\xi) \\quad.\n\\end{align}\nTransforming the eigenstates into the basis of unperturbed eigenstates and expanding the eigenenergies and coefficients in Taylor series in $\\lambda$,\n\\begin{align}\n\t\\phi_k &= \\sum_nc_{kn}\\psi_n \\quad,\\quad \\epsilon_k = \\sum_p\\lambda^p\\epsilon_k^{(p)} \\quad,\\quad c_{km} = \\sum_p\\lambda^pc_{km}^{(p)} \\quad,\\quad u_{mn} = \\int\\limits_{-\\infty}^\\infty\\psi_m^\\ast(\\xi)u(\\xi)\\psi_n(\\xi)\\,\\text{d}\\xi \\quad,\n\\end{align}\nyields\n\\begin{align}\n\t&\\left(\\epsilon_m^{(0)}-\\epsilon_k^{(0)}\\right)c_{km}^{(0)} + \\sum_{p=1}^\\infty \\left[ \\vphantom{\\sum_{q=1}^{p-1}} \\left(\\epsilon_m^{(0)}-\\epsilon_k^{(0)}\\right)c_{km}^{(p)} - \\epsilon_k^{(p)}\\delta_{km} - \\sum_{q=1}^{p-1}\\epsilon_k^{(q)}c_{km}^{(p-q)} + \\sum_nc_{kn}^{(p-1)}u_{mn} \\right]\\lambda^p = 0 \\quad.\n\\end{align}\nThis gives recursion equations for the $p$th order corrections:\n\\begin{align}\n\t\\epsilon_k^{(p)} &= \\sum_{n\\neq k}u_{kn}c_{kn}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{kk}^{(p-q)} \\\\\n\tc_{km}^{(p)} &= \\frac{1}{\\epsilon_k^{(0)}-\\epsilon_m^{(0)}}\\left( \\sum_nu_{mn}c_{kn}^{(p-1)} - u_{kk}c_{km}^{(p-1)} - \\sum_{q=2}^{p-1}\\epsilon_k^{(q)}c_{km}^{(p-q)} \\right) \\quad.\n\\end{align}\nEspecially for the 1st, 2nd, and 3rd order corrections, one gets\n\\begin{align}\n\t\\epsilon_k^{(1)} &= u_{kk} \\quad,\\\\\n\t\\epsilon_k^{(2)} &= -\\sum_{n\\neq k}\\frac{u_{nk}u_{kn}}{\\epsilon_n^{(0)}-\\epsilon_k^{(0)}} \\quad,\\\\\n\t\\epsilon_k^{(3)} &= -\\sum_{n\\neq k}\\frac{u_{nk}u_{kk}u_{kn}}{\\left(\\epsilon_n^{(0)}-\\epsilon_k^{(0)}\\right)^2} + \\sum_{n\\neq k}\\sum_{l\\neq k}\\frac{u_{kn}u_{nl}u_{lk}}{\\left(\\epsilon_n^{(0)}-\\epsilon_k^{(0)}\\right)\\left(\\epsilon_l^{(0)}-\\epsilon_k^{(0)}\\right)} \\quad.\n\\end{align}\n\n\\noindent Expanding $u(\\xi)$ in a Taylor series in $\\xi$,\n\\begin{align}\n\tu(\\xi) &= \\sum_j u_j\\xi^j \\quad,\n\\end{align}\ngives\n\\begin{align}\n\tu_{mn} &= \\sum_j u_j u_{mn}^{(j)} \\quad,\\\\\n\tu_{mn}^{(j)} &= \\frac{1}{\\sqrt{2^{m+n}m!n!}\\sqrt{\\pi}} \\int\\limits_{-\\infty}^\\infty \\xi^j\\text{e}^{-\\xi^2}H_m(\\xi)H_n(\\xi) \\,\\text{d}\\xi \\quad.\n\\end{align}\nInserting~\\eqn{Hermite:recursive} gives the recursion\n\\begin{align}\n\tu_{mn}^{(j)} &= \\sqrt{\\frac{n+1}{2}}u_{m,n+1}^{(j-1)} + \\sqrt{\\frac{n}{2}}u_{m,n-1}^{(j-1)} \\quad.\n\\end{align}\n\n\\noindent The diagonal elements $u_{kk}^{(j)}$ are polynomials in $k$ of order $j\/2$.\nFor the smallest $j$ one gets\n\\begin{align}\n\tu_{kk}^{(0)} &= 1 \\quad,\\\\\n\tu_{kk}^{(2)} &= \\frac{1}{2} + k \\quad,\\\\\n\tu_{kk}^{(4)} &= \\frac{3}{4} + \\frac{3}{2}k + \\frac{3}{2}k^2 \\quad,\\\\\n\tu_{kk}^{(6)} &= \\frac{15}{8} + 5k + \\frac{15}{4}k^2 + \\frac{5}{2}k^3 \\quad,\\\\\n\tu_{kk}^{(8)} &= \\frac{105}{16} + \\frac{35}{2}k + \\frac{175}{8}k^2 + \\frac{35}{4}k^3 + \\frac{35}{8}k^4 \\quad.\n\\end{align}\nThe nondiagonal elements can be written as\n\\begin{align}\n\tu_{k,k-l}^{(j)} = \\sqrt{\\frac{k!}{2^l(k-l)!}}p_{kl}^{(j)} \\quad ,\n\\end{align}\nwhere $p_{kl}^{(j)}$ is a polynomial in $k$ of order $(j-l)\/2$.\nThus, $(u_{k,k-l}^{(j)})^2$ is a polynomial in $k$ of order $j$ for all $l$.\nFor the smallest $j$  and $l$ one gets\n\\begin{align}\n\tp_{k,1}^{(1)} &= 1 \\quad,\\\\\n\tp_{k,1}^{(3)} &= \\frac{3}{2}k \\quad,\\\\\n\tp_{k,1}^{(5)} &= \\frac{5}{4}(1+2k^2) \\quad,\\\\\n\tp_{k,1}^{(7)} &= \\frac{35}{8}k(2+k^2) \\quad,\\\\\n\tp_{k,2}^{(2)} &= 1 \\quad,\\\\\n\tp_{k,2}^{(4)} &= -1+2k \\quad,\\\\\n\tp_{k,2}^{(6)} &= \\frac{15}{4}(1-k+k^2) \\quad,\\\\\n\tp_{k,2}^{(8)} &= \\frac{7}{2}(-1+2k)(3-k+k^2) \\quad,\\\\\n\tp_{k,3}^{(3)} &= 1 \\quad,\\\\\n\tp_{k,3}^{(5)} &= \\frac{5}{2}(-1+k) \\quad,\\\\\n\tp_{k,3}^{(7)} &= \\frac{21}{4}(2-2k+k^2) \\quad,\\\\\n\tp_{k,3}^{(9)} &= \\frac{21}{4}(-1+k)(9-4k+2k^2) \\quad,\\\\\n\tp_{k,4}^{(4)} &= 1 \\quad,\\\\\n\tp_{k,4}^{(6)} &= \\frac{3}{2}(-3+2k) \\quad,\\\\\n\tp_{k,4}^{(8)} &= \\frac{7}{2}(7-6k+2k^2) \\quad,\\\\\n\tp_{k,4}^{(10)} &= \\frac{15}{4}(-3+2k)(13-6k+2k^2) \\quad.\n\\end{align}\nThe results are shown in \\fig{u_mn_j}.\n\n\\begin{figure}[tbp]\n\t\\includegraphics{supplement_graphics\/u_kk-d.pdf}\n\t\\caption{$u_{k,k-l}^{(j)}$ for different $l$ and for different perturbation potentials $u(\\xi)=\\xi^j$.}\n\t\\label{fig:u_mn_j}\n\\end{figure}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt  is long known that  the forward Compton scattering (CS) amplitudes can, by unitarity, causality and crossing, be expressed\nthrough integrals of the photoabsorption cross sections~\\cite{GellMann:1954db}. \nThe low-energy expansions of these expressions lead to a number of useful sum rules, most notably \nthose of Baldin \\cite{Baldin}, and Gerasimov, Drell and Hearn (GDH)~\\cite{Gerasimov:1965et,Drell:1966jv}. Given the photoabsorption cross sections,\none can thus provide a reliable assessment of some of the static electromagnetic \nproperties of the nucleon and nuclei, as well as of the forward CS amplitudes in general.\nFor the proton, the first such assessments was performed in the early \n1970s \\cite{Damashek,Armstrong}. Since then, the knowledge of the photoabsorption cross sections\nappreciably improved, and yet for the unpolarized case only the Baldin\nsum rule has been updated~\\cite{Schroder,Babusci,Olmos}. \nIn this work, we provide a re-assessment of the forward spin-independent amplitude of proton CS, and evaluate the associated sum rules involving the\n{\\em dipole and quadrupole polarizabilities of the proton}.\n\n\nSum rules are essentially the only way to gain empirical knowledge of \nthe forward CS amplitudes.  It is impossible to access the forward kinematics \nin real CS experiments. The measurement of the forward spin-independent CS amplitude can be done indirectly through the process of dilepton photoproduction ($\\gamma \\,p \\to  p \\,e^+ e^-$)~\\cite{Alvensleben}. The timelike CS, involved\nin the process of dilepton photoproduction, yields access to real CS given \nthe small virtuality of the outgoing photon, or equivalently,  the nearly vanishing invariant\nmass of the produced pair. The experimental result~\\cite{Alvensleben} \ncompared well with the aforementioned evaluations \\cite{Damashek,Armstrong}. \nDespite the substantial\nadditions to the database of total photoabsorption cross sections, the works of Damashek and Gilman (DG) \\cite{Damashek} as well as Armstrong {\\it et al.}~\\cite{Armstrong} remained to be, until now, the only evaluations\nof the full amplitude.\n\nThe newer data were used, however, in the most recent evaluations of the Baldin sum rule \\cite{Babusci,Olmos}, which \nyields the sum of the electric and magnetic dipole polarizabilities,\n\\Eqref{BaldinSR}. These recent analyses obtained somewhat lower value for the sum than DG, cf.~Table \\ref{sumruletest}. In this work we find that the difference between the early and the recent evaluations arises from systematic inconsistencies in the experimental database. \nWe also obtain the sum rule value for a combination of higher-order quadrupole polarizabilities and \ncompare it with several theoretical predictions.\n\nThis paper is organized as follows.\nIn Sect.\\ \\ref{sec:overview} we give a brief overview of \nthe Kramers-Kronig relation and sum rules for polarizabilities.\nIn Sect.\\ \\ref{sec:fitting} we discuss the fitting procedure\nfor the unpolarized total proton photoabsorption cross section data.\nThe sum rule evaluations of scalar polarizabilities\nand of the spin-independent forward CS amplitude are presented \nin Sect.\\ \\ref{sec:sumrules}.\nConclusions are given in Sect.\\ \\ref{sec:summary}. The Appendix\ndemonstrates the elastic-channel contribution to the sum rules and polarizabilities on the example of one-loop scalar QED.\n\n\n\n\\begin{figure*}[hbt]\n\\includegraphics[width=0.8\\textwidth]{figs\/cs_tot.pdf}\n\\caption{\n(Color online)\nFits of experimental data for the total photoabsorption cross section on the proton.  Fit I is obtained using MAID \\cite{MAID} results below the $2\\pi$-production, and data from LEGS \\cite{BNL} and Armstrong {\\it et al.}~\\cite{Armstrong} above it. Fit II uses SAID \\cite{SAID} and the data of MacCormick {\\it et al.}~\\cite{MacCormick}. Both fits use Bartalini {\\it et al.} \\cite{Bartalini} and the high-energy data \\cite{Caldwell,Aid,Chekanov} displayed in the insert.\n}\n\\label{fig:CS_tot}\n\\end{figure*}\n\\section{Forward Compton Amplitude and Sum Rules}\n\\label{sec:overview}\n\n\n\nFor a spin-1\/2 target, such as the proton, \nthe forward CS amplitude is given by\n\\begin{equation} \nT_{fi} = f(\\nu) \\, \\boldsymbol{\\varepsilon}^{\\prime\\,\\ast }\\cdot \\boldsymbol{\\varepsilon} \\, + \\, g(\\nu) \\,i \\, (\\boldsymbol{\\varepsilon}^{\\prime\\,\\ast }\\! \\times \\boldsymbol{\\varepsilon})\\cdot\\boldsymbol{\\sigma},\n\\eqlab{definition}\n\\end{equation} \nwhere $f$ and $g$ are scalar functions of the photon lab energy $\\nu$;\nvectors $\\boldsymbol{\\varepsilon}$ and $\\boldsymbol{\\sigma}$ represent the photon and proton polarizations, respectively. The crossing symmetry implies that the spin-independent amplitude $f$ is an even\nand the spin-dependent amplitude $g$ is an odd function of $\\nu$.\n\nThe optical theorem (unitarity) relates the imaginary\npart of the amplitudes to the total photoabsorption cross sections:\n\\begin{subequations}\n\\begin{eqnarray}\n\\im f(\\nu) & = & \\frac{\\nu}{8\\pi} \\left[\\sigma_{1\/2}(\\nu)+\\sigma_{3\/2}(\\nu)\\right] , \\\\\n\\im g(\\nu) & = & \\frac{\\nu}{8\\pi}\n\\left[\\sigma_{1\/2}(\\nu)-\\sigma_{3\/2}(\\nu)\\right] .\n\\eqlab{unitarity}\n\\end{eqnarray}\n\\end{subequations}\nHere $\\sigma_{\\lambda} \\def\\La{{\\it\\Lambda}}(\\nu) $ is the doubly-polarized cross section\nwith $\\lambda$ representing the\ncombined helicity of the initial $\\gamma} \\def\\Ga{{\\it\\Gamma} p$ state.  Averaging over the\npolarization of initial particles gives the  unpolarized\nphotoabsorption cross section, $\\sigma =\\mbox{\\small{$\\frac{1}{2}$}} ( \\sigma_{1\/2}+\\sigma_{3\/2})$.\n\n\n\n\nIn the present article we focus on relations involving  the spin-independent amplitude $f$, and the unpolarized cross section\n$\\sigma} \\def\\Si{{\\it\\Sigma}$.\nThe Kramers-Kronig relation between these quantities exploits the optical theorem, causality and crossing symmetry, to yield for the proton~\\cite{GellMann:1954db}:\n\\begin{equation}\n\\eqlab{KK}\n\\re f(\\nu) =-\\,\\frac{\\alpha}{M_p} + \\frac{\\nu^2}{2\\pi^2}\\fint_0^\\infty\\! \\frac{{\\rm d} \\nu'}{\\nu^{\\prime \\, 2}-\\nu^2}\\, \\sigma} \\def\\Si{{\\it\\Sigma}(\\nu'),\\end{equation}\nwhere $\\alpha=e^2\/4\\pi$ is the fine-structure constant and $M_p$ is the proton mass; the slashed integral denotes the principal-value integration. \n\nWe next would like to consider the low-energy expansion of $f$.\nAt this point it is important to note that the elastic scattering, i.e.\\ the CS process itself, \nis one of the photoabsorption processes. The total CS cross section does\nnot vanish for $\\nu\\to 0$ but goes to a constant --- the Thomson cross section: \n\\begin{equation} \n\\sigma} \\def\\Si{{\\it\\Sigma}(0 ) = \\frac{8\\pi \\alpha^2}{3 M_p^2}.\n\\end{equation} \nThis means \\Eqref{KK} does not admit a Taylor-series expansion\naround $\\nu=0$ (each coefficient in that expansion is infrared divergent, cf. Appendix).\nSuch expansion is nonetheless important for establishing the polarizability sum rules.\nWe hence prefer to take the CS out of the total cross section, i.e.:\n\\begin{equation} \n\\sigma} \\def\\Si{{\\it\\Sigma}(\\nu) = \\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{CS}}(\\nu)+\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu), \n\\end{equation} \nwhere $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}$ can be assumed to be dominated\nby hadron-production processes, for which there is a threshold \nat some $\\nu_0 > m_\\pi$. \n\n\n\nThe amplitude $f$ can be decomposed accordingly into the elastic\nand inelastic terms,\n\\begin{eqnarray} \nf(\\nu) & = & f_{\\mathrm{el}}(\\nu) + f_{\\mathrm{inel}}(\\nu)  , \\\\\nf_{\\mathrm{el}}(\\nu) &=& -\\,\\frac{\\alpha}{M_p} + \\frac{\\nu^2}{2\\pi^2}\\fint_0^\\infty\\! \\frac{{\\rm d} \\nu'}{\\nu^{\\prime \\, 2}-\\nu^2}\\, \\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{CS}}(\\nu'),\n\\eqlab{fel}\\\\\nf_{\\mathrm{inel}}(\\nu) &=& \\frac{\\nu^2}{2\\pi^2}\\fint_{\\nu_0}^\\infty\\! \\frac{{\\rm d} \\nu'}{\\nu^{\\prime \\, 2}-\\nu^2}\\, \\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu').\n\\eqlab{fbar}\n\\end{eqnarray} \nThe details on dealing with $f_{\\mathrm{el}}$ can be found in \nAppendix \\ref{sec:scalarQEDExample}. In what follows, however, we neglect\nthe contribution from $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{CS}}$, as it is suppressed by an extra order of $\\alpha$. Hence we set $f_{\\mathrm{el}}(\\nu) = -\\alpha\/M_p$, as is usually done. \n\nConsidering $f_{\\mathrm{inel}}$,\nthe low-energy expansion of both sides of \\Eqref{fbar} leads to the sum rules for polarizabilities. At the leading order [$O(\\nu^2)$], one\nobtains the Baldin sum rule \\cite{Baldin} for the\nsum of electric ($\\alpha_{E1}$) and magnetic ($\\beta_{M1}$) dipole polarizabilities:\n\\begin{equation}\n\\eqlab{BaldinSR}\n\\alpha_{E1} + \\beta_{M1} =\\frac{1}{2\\pi^2} \\int_{\\nu_0}^\\infty \\!{\\rm d}\\nu \\,\\frac{\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu)}{\\nu^2}.\n\\end{equation}\nAt $O(\\nu^4)$ we obtain `the 4\\textsuperscript{th}-order sum rule': \n\\begin{eqnarray}\n\\eqlab{4thSR}\n\\alpha_{E\\nu} + \\beta_{M\\nu} + \\frac{1}{12}\\,(\\alpha_{E2} + \\beta_{M2}) =\\frac{1}{2\\pi^2} \\int_{\\nu_0}^\\infty \\!{\\rm d}\\nu \\,\\frac{\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu)}{\\nu^4},\\nonumber\\\\\n\\end{eqnarray}\nwhich involves the quadrupole  polarizabilities\n$\\alpha_{E2}$, $\\beta_{M2}$, as well as the leading dispersive\ncontribution to the dipole polarizabilities denoted as $\\alpha_{E\\nu}$, $\\beta_{M\\nu}$, see \\cite{Babusci:1998ww} for more details.\n\n\nOur aim here is to provide an empirical fit of the available\ndata for $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}$ and evaluate the various sum rules. \n\n\\section{Fits of the Photoabsorption Cross Section}\n\\label{sec:fitting}\n\nThe presently available experimental data, together with\nthe results of the empirical analyses MAID and SAID, as well as our fits are displayed in \\Figref{CS_tot}. In our fitting we\ndistinguish the following \nthree regions:\n\\begin{itemize}\n \\item {\\it low-energy},   $\\nu \\in [\\nu_0   \\,,\\;  \\nu_1  ) $;\n \\item {\\it medium-energy},   $\\nu \\in [\\nu_1   \\,,\\;  2\\;\\mathrm{GeV}) $;\n \\item {\\it high-energy}, $\\nu \\in [2\\;\\mathrm{GeV} \\,,\\;  \\infty ) $;\n\\end{itemize}\nwhere $\\nu_0$ ($\\simeq 0.145$ GeV) and $\\nu_1$ ($\\simeq 0.309$ GeV) are respectively the thresholds for the single- and double-pion photoproduction on the proton.\n\nIn the {\\it low-energy} region we use the pion production \n($\\pi^+  n$ and $\\pi^0 p$) cross sections from the \nMAID \\cite{MAID} and SAID \\cite{SAID} partial-wave analyses. \nIn our error estimate we assign a 2\\% uncertainty on these values.\n\nIn the {\\it medium-energy} region we fit the actual experimental data, using a sum of Breit-Wigner resonances and a background. \nFollowing \\cite{Babusci}, we take \nsix Breit-Wigner resonances, each parameterized \nas\n\\begin{equation}\n\\label{res_fit}\n\\sigma} \\def\\Si{{\\it\\Sigma}_R(W)=A \\cdot \\frac{\\Gamma^2\/4}{(W-M)^2 + \\Gamma^2\/4},\n\\end{equation}\nwhere $W=\\sqrt{s}$ is the total energy of the $\\gamma p$ system.\nThe background function is from \\cite{Armstrong}:\n\\begin{equation}\n\\label{bgd_fit}\n\\sigma} \\def\\Si{{\\it\\Sigma}_B(W)=\\sum_{k=-2}^2 C_k (W-W_0)^k,\n\\end{equation}\nwhere $W_0=M_p+m_{\\pi}$ corresponds with\nthe pion photoproduction threshold.\n\n\n\n\\begin{table}[bth]\n{\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n&$M$ [MeV] & $\\Gamma$ [MeV] & $A$ [$\\upmu$b]\\\\\n\\hline\n\\hline\n\\multirow{6}{*}{\\begin{sideways} \\hspace{-0.4cm} fit I\\end{sideways}}&$1213.6 \\pm  0.1$ & $117.6 \\pm  1.9$ &  $522.7 \\pm  17.0$\\\\\n&$1412.8 \\pm  5.9$ & $ 82.8 \\pm 26.8$ &  $ 40.1 \\pm 33.8$\\\\\n&$1496.0 \\pm  2.8$ & $136.5 \\pm 11.1$ &  $161.8 \\pm 32.4$\\\\\n&$1649.4 \\pm  4.1$ & $135.3 \\pm 15.3$ &  $ 83.2 \\pm 22.7$\\\\\n&$1697.5 \\pm  2.6$ & $ 18.8 \\pm 12.6$ &  $ 18.2 \\pm 26.0$\\\\\n&$1894.3 \\pm 15.6$ & $302.0 \\pm 41.3$ &  $ 31.5 \\pm  8.7$\\\\\n\\hline\n\\hline\n\\multirow{6}{*}{\\begin{sideways}\\hspace{-0.4cm}  fit II\\end{sideways}}&$1214.8 \\pm  0.1$ & $99.0 \\pm  1.1$ & $502.3 \\pm 12.3$\\\\\n&$1403.9 \\pm  6.2$ & $118.2 \\pm 19.6$ & $ 51.8 \\pm 23.8$\\\\\n&$1496.9 \\pm  2.1$ & $133.4 \\pm  9.4$ & $162.0 \\pm 29.2$\\\\\n&$1648.0 \\pm  4.4$ & $135.2 \\pm 15.9$ & $ 83.6 \\pm 23.8$\\\\\n&$1697.2 \\pm  2.7$ & $ 21.2 \\pm 13.2$ & $ 18.7 \\pm 25.9$\\\\\n&$1893.7 \\pm 17.4$ & $323.5 \\pm 45.3$ & $ 31.7 \\pm  9.1$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Fitting parameters for the resonances (\\ref{res_fit})\nobtained for fit I and II.}\n\\label{table:res_par_2}\n\\end{table}\n\n\n\\begin{table}[bth]\n{\n\\begin{tabular}{|c|c|c|}\n\\hline\n& fit I & fit II \\\\\n\\hline\n$C_{-2}$, [$\\upmu$b $\\cdot$ GeV$^2$] & $ 0.44  \\pm  0.22$ & $ 0.26 \\pm  0.17$\\\\\n$C_{-1}$, [$\\upmu$b $\\cdot$ GeV]     & $-11.06 \\pm  3.69$ & $-7.97 \\pm  2.89$\\\\\n$C_0$, [$\\upmu$b]                    & $ 74.38 \\pm 20.16$ & $57.27 \\pm 16.09$\\\\\n$C_1$, [$\\upmu$b $\\cdot$ GeV$^{-1}$] & $ 22.18 \\pm 37.71$ & $54.26 \\pm 31.07$\\\\\n$C_2$, [$\\upmu$b $\\cdot$ GeV$^{-2}$] & $ 37.69 \\pm 21.48$ & $19.51 \\pm 18.17$\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Fitting parameters for the background (\\ref{bgd_fit})\nobtained for fit I and II in the resonance region.}\n\\label{table:bgd_par}\n\\end{table}\n\n\n\nObserving a significant discrepancy between SAID and MAID \naround the $\\De$(1232)-resonance peak and a similar discrepancy\nbetween two sets of experimental data, we have made two\ndifferent fits:\n\\begin{itemize}\n\\item[I.] MAID\\cite{MAID} + LEGS \\cite{BNL} + Armstrong {\\it et al.}~\\cite{Armstrong}, \n\\item[II.] SAID \\cite{SAID} + MacCormick {\\it et al.}~\\cite{MacCormick}.\n\\end{itemize}\nThey are shown in \\Figref{CS_tot} by the red solid and blue dashed lines, respectively. The corresponding values of parameters are\ngiven in Tables~\\ref{table:res_par_2} and \\ref{table:bgd_par}.\nIn both fits we have also made\nuse of the GRAAL 2007 data \\cite{Bartalini}, shown in the figure by\nlight-blue squares. These data had not been available at the time of the previous sum rule evaluations. \n\n\nFinally, for the {\\it high-energy region} we use the\nstandard Regge form \\cite[p.~191]{Barnett}:\n\\begin{equation}\n\\label{regge_fit}\n\\sigma} \\def\\Si{{\\it\\Sigma}_\\mathrm{Regge}(W)=c_1 \\, W^{p_1} + c_2 \\, W^{p_2}.\\\\\n\\end{equation}\nFor $W$ in GeV and the cross section in $\\upmu$b, we obtain\nthe following parameters (for both of our fits):\n\\begin{gather*}\nc_1 = 62.0 \\pm 8.1, \\quad c_2 = 126.3 \\pm 4.3, \\\\\np_1 = 0.184 \\pm 0.032, \\quad p_2 = -0.81 \\pm 0.12.\n\\end{gather*}\nWe also tried the high-energy parameterization used\nin \\cite{Babusci}, but obtained a worse fit and abandoned it. \n\n\n\n\n\n\\begin{figure*}[bht]\n\t\\includegraphics[width=0.8\\textwidth]{figs\/re_f.pdf}\n\t\\caption{(Color online) Our evaluation of $\\re f$ based on the two fits\nof the photoabsorption cross section, compared with previous evaluations \\cite{Damashek,Olmos,Armstrong}. The experimental data point is from Ref.~\\cite{Alvensleben}.}\n\t\\label{fig:Re_f}\n\\end{figure*}\n\n\\begin{table*}[tbh]\n\\begin{ruledtabular}\n\\centering \n\\caption{\n\tEmpirical evaluations of sum rules \n    and verification of the\n    Kramers-Kronig relation for the proton.}\n    \\begin{tabular}{|r|l|l|l|l|}\n\t&  Baldin& 4\\textsuperscript{th} order &  6\\textsuperscript{th} order\\footnote{$ \\int_{\\nu_0}^\\infty \\!{\\rm d}\\nu\\, \\nu^{-6} \\,\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}(\\nu)\/(2\\pi^2)$} & $\\re f$($2.2$ GeV) \\\\\n    \t&[$10^{-4}$ fm$^3$]&[$10^{-4}$ fm$^5$]&[$10^{-4}$ fm$^7$]&[$\\upmu$b $\\cdot$ GeV]\\\\\n\t\\hline\n\tDamashek--Gilman \\cite{Damashek}      & $14.2\\pm 0.3$  &  &  & $-11.5$\\footnote{Interpolated value.}\\\\\n    Armstrong {\\it et al.} \\cite{Armstrong} &              &  &  & $-10.8$\\\\\n\tSchr{\\\"o}der \\cite{Schroder}            & $14.7\\pm0.7$  &$6.4$&  & \\\\\n\tBabusci {\\it et al.} \\cite{Babusci}     & $13.69\\pm0.14$ &  &  & \\\\\n\tA2 Collaboration \\cite{Olmos} & $13.8\\pm0.4$   &  &  & $-10.5$\\footnote{Based on the cross-section parametrization from \\cite{Ahrens}.}\\\\\n\tMAID ($\\pi$ chan.)~\\cite{MAID}  & $11.63$\\footnote{Integrated from threshold to $\\nu_\\mathrm{max}=1.663$ GeV.}     &  &  & \\\\\n\tSAID ($\\pi$ chan.)~\\cite{SAID}             & $11.5$\\footnote{Integrated from threshold to $\\nu_\\mathrm{max}=2$ GeV.}           &  &  & \\\\\n\t\\hline\n    This work &&&&\\\\\n    \t fit I & $14.29\\pm 0.27$ & $6.08 \\pm 0.12 $ & $ 4.36 \\pm 0.09 $ & $-10.35$ \\\\\n\t fit II  & $13.85\\pm 0.22$ & $6.01 \\pm 0.11 $ & $ 4.42 \\pm 0.08 $ & $-9.97 $ \\\\\n     \t\\hline\n    Experiment &&&&\\\\\n      Alvensleben {\\it et al.} \\cite{Alvensleben} &          &  &  & $-12.3\\pm2.4$\\\\\n\t\\end{tabular}\n\t\\label{sumruletest}\n   \\end{ruledtabular}\n\\end{table*}\n\n\\twocolumngrid \n\nThe fitting was done with the help of the {\\it SciPy} package for Python.\nThe resulting chi-square, evaluated as\n\\begin{equation}\n\\eqlab{chisq}\n\\chi^2 = \\sum_i \\frac{\\left(\\sigma_i^{\\mathrm{fit}}  - \\sigma_i^{\\mathrm{exp}}\\right)^2}{(\\De\\sigma_i^{\\mathrm{exp}})^2},\n\\end{equation}\nis of about the same quality for the two fits. In the intermediate region,\nwe obtain $\\chi^2\/\\mbox{point}=0.7$ for fit I, and $\\chi^2\/\\mbox{point}=0.6$\nfor fit II. In the high-energy region $\\chi^2\/\\mbox{point}=1.2$ in both cases.\nAgain, the low-energy region is not fitted but is borrowed from, respectively, the MAID and SAID\nanalyses.\n\n\n\\section{Sum Rule Evaluations}\n\\label{sec:sumrules}\n\n\n\n\nHaving obtained the fits of the total photoabsorption cross section $\\sigma} \\def\\Si{{\\it\\Sigma}_{\\mathrm{abs}}$, we evaluate the integrals \nin Eqs.~\\eref{fbar}, \\eref{BaldinSR} and \\eref{4thSR}; the\nresults are presented in \\Figref{Re_f} and Table~\\ref{sumruletest}.\n\nTables \\ref{BaldinSR_regions} and \\ref{4thSR_regions} show contributions of each region to the Baldin and the 4\\textsuperscript{th}-order sum rule, respectively. \nThe uncertainty in calculating an integral\n$I_n = \\int{\\rm d}\\nu\\,  \\nu^{-n} \\sigma(\\nu) $ has been evaluated as follows:\n\\begin{equation}\n\\Delta I_n = \\sum_i \\frac{\\Delta \\nu_i}{\\nu_i^n} \\chi_i^2 \\,\\Delta\\sigma_i^{\\mathrm{exp}},\n\\end{equation}\nwhere $\\chi_i^2$ is the chi-square at the point $i$, cf.~\\Eqref{chisq}.\n\n\n\n\\begin{table}[tbh]{\n\\caption{Contributions of different regions to the Baldin sum rule\nfor the two fits in \\Figref{CS_tot}.}\n   \\label{BaldinSR_regions}\n$\\alpha_{E1} + \\beta_{M1}$\\; [$10^{-4}$ fm$^3$]\\\\\n\t\\centering \n\t\\begin{tabular}{|c|c|c|c|}\n\t\\hline\n\\diagbox{Fit}{Region}& {\\it low-energy} & {\\it medium-energy}  & {\\it high-energy}  \\\\\n\t\\hline\n\tI & $6.12 \\pm 0.12$ & $7.53 \\pm 0.13$ & $0.64 \\pm 0.02$ \\\\\n\tII& $6.06 \\pm 0.12$ & $7.15 \\pm 0.08$ & $0.64 \\pm 0.02$ \\\\\n\t\\hline\n\t\\end{tabular}\n\t\\par}\n \n\\end{table}\n\n\\begin{table}[tbh]{\n\\caption{ Contributions of different regions to the 4\\textsuperscript{th}-order sum rule\nfor the two fits in \\Figref{CS_tot}.} \\label{4thSR_regions}\n$\\alpha_{E\\nu} + \\beta_{M\\nu} + \\frac{1}{12}(\\alpha_{E2} + \\beta_{M2})$ \\; [$10^{-4}$ fm$^5$]\\\\\n\t\\centering \n\t\\begin{tabular}{|c|c|c|c|}\n\t\\hline\n\t\\diagbox{Fit}{Region} & {\\it low-energy} & {\\it medium-energy}   & {\\it high-energy}  \\\\\n\t\\hline\n\tI  & $4.50 \\pm 0.09$ & $1.58 \\pm 0.03$ & $(219 \\pm 8)\\times 10^{-5}$ \\\\\n\tII  & $4.53 \\pm 0.09$ & $1.48 \\pm 0.01$ & $(219 \\pm 8)\\times 10^{-5}$ \\\\\n\t\\hline\n\t\\end{tabular}\n\t\\par}\n\\end{table}\n\nThe corresponding full results (sum of the three regions) \nare given in \nTable~\\ref{sumruletest}, and compared with the results of previous works. In this table we also give the result for the 6th-order\nintegral, and for the full amplitude $f$ at $\\nu=2.2$ GeV.\nThe real part of $f$ is plotted in \\Figref{Re_f} over a broad\nenergy range and compared with previous evaluations and \nthe experimental number from the 1973 DESY experiment at \n2.2 GeV. Although none of the evaluations really contradicts the experiment, \nthere is a clear tendency to a higher central value. \n\n\nThe new\ndilepton photoproduction experiments planned  at the Mainz Microtron (MAMI) could perhaps provide experimental values in the lower energy range. Obviously, the regions of the extrema (e.g., the $\\De(1232)$ region or the interval between 0.6 and 0.7 GeV) are most interesting as the different evaluations seem to differ there the most. In\nthe region around 0.6 GeV, for example, one of our evaluations (fit I) is nearly \nidentical with Armstrong's~\\cite{Armstrong}, while the other one (fit II) is aligning with DG~\\cite{Damashek} and A2 Coll.~\\cite{Olmos}. An appropriately precise\nexperiment could tell which of the groups is correct, if any.\n\n\\begin{figure}[htb]\n\t\\includegraphics[width=0.95\\columnwidth]{figs\/f_chipt_hw.pdf}\n\t\\caption{(Color online) Our evaluations of $f(\\nu)$ compared to the $\\chi$PT calculation\nof Ref.~\\cite{Lensky:2009uv}.}\n\t\\label{fig:ReIm_f}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\t\\includegraphics[width=0.92\\columnwidth]{figs\/nosr_hw.pdf}\n\t\\caption{(Color online) The 4\\textsuperscript{th}-order sum rule constraint for $\\alpha_{E\\nu} +\\, \\frac{1}{12} \\alpha_{E2}$ and $\\beta_{M\\nu} +\\, \\frac{1}{12} \\beta_{M2}$ combinations of polarizabilities, compared to results from dispersion relation approaches (DR) \\cite{Babusci:1998ww,Drechsel:2002ar}, baryon chiral perturbation theory (B$\\chi$PT) \\cite{Lensky:2015}, and heavy baryon chiral perturbation theory (HB$\\chi$PT) \\cite{Holstein:1999uu}. The errors of the $\\chi$PT derive from our  crude estimate of the next-order corrections.}\n\t\\label{fig:4thSR}\n\\end{figure}\n\n\nFigure~\\ref{fig:ReIm_f} shows both the real and imaginary\npart of $f$ at lower energies, where it can be compared\nwith a calculation done within chiral perturbation theory ($\\chi$PT) \\cite{Lensky:2009uv}. A rather nice agreement between theory and empirical evaluations is observed for energies up to\nabout the pion-production threshold.  \n\n\nFor very low energy this comparison can\nbe made more quantitative by looking at the polarizabilities. While for the Baldin\nsum rule the situation was extensively discussed in the literature (cf.~\\cite{Griesshammer:2012we} for a recent review), the \n4\\textsuperscript{th}-order sum rule was not studied at all. \nIt can,\nhowever, be very useful in unraveling the higher-order polarizabilities,\nas illustrated by \\Figref{4thSR}. This is the plot of a combination of proton magnetic\npolarizabilities versus electric, where the various theory predictions are compared with our 4\\textsuperscript{th}-order sum rule evaluation. The band representing the sum rule covers the \ninterval between the two values given in \nTable~\\ref{sumruletest} (rows `fit I' and `fit II'). The sum rule clearly provides a\nmodel-independent constraint on these polarizabilities and a rather stringent test for the theoretical approaches.  \n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:summary}\nThe fundamental relation between the photon absorption and scattering, encompassed in the Kramers-Kronig type of relations, allows us\nto evaluate the forward Compton scattering off protons \nusing the empirical knowledge of the total photoabsorption cross sections.\nThe present database of the unpolarized photoabsorption cross section is not entirely\nconsistent and so as to reflect that we obtain two distinct fits to it.\nThe two fits yield slightly different results for the spin-independent\namplitude $f(\\nu)$, and hence for its low-energy expansion characterized\nby the scalar  polarizabilities of the proton. Our two results for the\nsum of dipole polarizabilities (or, Baldin sum rule) correspond nicely\nwith the  results of previous evaluations, which too can be separated into\ntwo groups: \nthe {\\it old} \\cite{Damashek,Schroder}, with the value slightly above 14 (in units of $10^{-4}$ fm\\textsuperscript{3}), and \nthe {\\it new}~\\cite{Babusci,Olmos}, with the value slightly below 14.\nThe 1996 DAPHNE@MAMI experiment \\cite{MacCormick}, superseding\nthe 1972 experiment of \nArmstrong {\\it et al.}~\\cite{Armstrong}, is clearly responsible\nfor this difference. Neglecting the older data in favor of the newer ones, yields the lower value of the Baldin sum rule, and vice versa. \nWhile one can take a preference in one of the two fits and corresponding\nresults, we prefer to think of their difference\nas a systematic uncertainty in the present \nevaluation of the polarizabilities  and of the forward spin-independent amplitude of the proton. \n\nAs far as polarizabilities are concerned,\nonly the Baldin sum rule is appreciably affected by the inconsistency\nin the photoabsorption database. Nevertheless, the two results\n(fit I and II in Table~\\ref{sumruletest}) are not in conflict with\neach other,\ngiven the overlapping error bars. It is customary to take a statistical average in such cases.\nTaking a weighted average\\footnote{For the weighted average, $\\bar x\\pm \\bar \\sigma} \\def\\Si{{\\it\\Sigma}$, over \na set $\\{ x_i \\pm \\sigma} \\def\\Si{{\\it\\Sigma}_i \\}$, we use \\cite[p.~120]{Parratt:1961}: \n$$\\bar x = \\frac{\\sum_i x_i\/\\sigma} \\def\\Si{{\\it\\Sigma}_i^2}{\\sum_j 1\/\\sigma} \\def\\Si{{\\it\\Sigma}_j^2},\\; \n\\bar \\sigma} \\def\\Si{{\\it\\Sigma} = \\Big(\\frac{\\sum_i (x_i-\\bar x)^2\/\\sigma} \\def\\Si{{\\it\\Sigma}_i^2}{\\sum_j 1\/\\sigma} \\def\\Si{{\\it\\Sigma}_j^2}\\Big)^{1\/2}.$$ } over our two values for the Baldin sum rule  we obtain:\n$\\alpha_{E1}^{(\\mathrm{p})}+\\beta_{M1}^{(\\mathrm{p})}=(14.0\\pm 0.2)\\times 10^{-4}\\,\\mathrm{fm}^3.$\nThe error bar here does directly not include\nthe aforementioned systematic uncertainty of the cross section database.\nHowever, since the two results are fairly well surmised by the weighted\naverage, the latter should be less prone to \nthe systematic uncertainty of the database.\n\nWe have presented a first study of the sum rule\ninvolving the quadrupole polarizabilities, \\Eqref{4thSR},\nhere referred to as 'the 4\\textsuperscript{th}-order sum rule'. \nOur weighted average value for this sum rule, in the proton case, \nis $6.04(4) \\times 10^{-4}\\,\\mathrm{fm}^5$. It agrees\nvery nicely with the state-of-the-art calculations \nof these polarizabilities based\non fixed-$t$ dispersion relations and chiral perturbation theory,\nsee \\Figref{4thSR}. We note that, while the calculations\ndemonstrate significant differences in the values of individual\nhigher-order polarizabilities, these differences apparently \ncancel out from the\nforward combination of these polarizabilities which enters the sum rule.\n\n\nIn the subsequent paper we will discuss the evaluation  of the forward {\\it spin-dependent} amplitude\n$g(\\nu)$ and  related sum rules for the forward spin polarizabilities\nof the proton. The knowledge of the two amplitudes will allow us\nto reconstruct the observables for the proton Compton scattering\nat zero angle.\n\n\n\\begin{acknowledgments}\nWe thank J\\\"urgen Ahrens for kindly supplying us with a database of total photoabsorption cross sections.\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SFB 1044 [The Low-Energy Frontier of the Standard Model], and the Graduate School DFG\/GRK 1581\n[Symmetry Breaking in Fundamental Interactions].\n\\end{acknowledgments}\n\n\n\\onecolumngrid\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{5sec:Intro}Introduction}\n\nA quantum point contact (QPC) is the simplest mesoscopic device that\ndirectly shows quantum mechanical properties. It is a short\nballistic transport channel between two electron reservoirs, which\nshows quantized conductance as a function of the width of the\nchannel \\cite{5Wees,5Wharam}.  A widely applied approach for\nimplementing QPCs is using a split-gate structure on the surface of\na heterostructure with a two-dimensional electron gas (2DEG) at\nabout 100~nm beneath its surface. The conventional design of such a\nsplit-gate QPC has two metallic gate fingers\n(Fig.~\\ref{fig1SEMpic}a). Operating this device with a negative gate\nvoltage $V_g$ results in the formation of a barrier with a small\ntunable opening between two 2DEG reservoirs, because the 2DEG below\nthe gate fingers gets depleted over a range that depends on $V_g$.\nFor electrons in the 2DEG, this appears as an electrostatic\npotential $U$ that is a large barrier with a small opening in the\nform of a saddle-point potential (Fig.~\\ref{fig3SaddlePotential}).\nThe saddle-point potential gives transverse confinement in the\nchannel that is roughly parabolic, which results for this transverse\ndirection in a discrete set of electronic energy levels. For\nelectron transport along the channel this gives a discrete set of\nsubbands with one-dimensional character. Quantized conductance\nappears because each subband contributes $G_{0} = 2e^2\/h$ to the\nchannel's conductance \\cite{5Wees,5Wharam}, where $e$ is the\nelectron charge and $h$ is Planck's constant.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{SEMimage.pdf}\n\\caption{(a) SEM image of a conventional split-gate quantum point\ncontact (QPC). It has two gate fingers (QPC$_{\\rm 2F}$ device). The\nlength of the QPC channel is fixed and can be parameterized by the\nlithographic length $L_{litho}$ of the gate structure. The diagram\nalso illustrates the measurement scheme (see main text for details).\n(b) SEM image of a length tunable QPC with 6 gate fingers (QPC$_{\\rm\n6F}$ device). Here the effective length $L_{eff}$ of the QPC can be\ntuned by changing the ratio of the gate voltages on the central\ngates ($V_{g1}$) and side gates ($V_{g2}$).} \\label{fig1SEMpic}\n\\end{figure}\n\n\n\n\n\n\nWe present here the design and experimental characterization of QPCs\nwhich offer additional control over the shape of the saddle-point\npotential. We focused on developing devices for which the effective\nlength of the saddle-point potential (along the transport direction)\ncan be tuned \\textit{in situ}. The additional control is implemented\nwith a symmetric split-gate design based on 6 gate fingers\n(Fig.~\\ref{fig1SEMpic}b). Such devices will be denoted as QPC$_{\\rm\n6F}$, and conventional devices with 2 gate fingers\n(Fig.~\\ref{fig1SEMpic}a) as QPC$_{\\rm 2F}$. These QPC$_{\\rm 6F}$ are\noperated with the gate voltage on the outer fingers ($V_{g2}$) less\nnegative than the gate voltage on the central fingers ($V_{g1}$) to\navoid quantum dot formation. Sweeping $V_{g1}$ from more to less\nnegative values opens the QPC$_{\\rm 6F}$. By co-sweeping $V_{g2}$ at\nfixed ratio $V_{g2}\/V_{g1}$ it behaves as a QPC with a certain\nlength for the saddle-point potential, and this length can be chosen\nby setting $V_{g2}\/V_{g1}$: It is shortest for $V_{g2}\/V_{g1}\n\\approx 0$ and longest for $V_{g2}\/V_{g1} \\lessapprox 1$. For our\ndesign the effective length could be tuned from about 200~nm to\n600~nm.\n\n\nThe motivation for developing these length-tunable QPCs comes from\nstudies of electron many-body effects in QPCs. A well-known\nmanifestation of these many-body effects is the so-called\n0.7~anomaly \\cite{5Thomas1996}, which is an additional shoulder at\n$0.7~G_{0}$ in quantized conductance traces. These many-body effects\nare, despite many experimental and theoretical studies since 1996\n\\cite{5Review0.7}, not yet fully understood. Recent theoretical work\n\\cite{5Meir} suggested that many-body effects cause the formation of\none or more self-consistent localized states in the QPC channel, and\nthat these effects result in the 0.7~anomaly and the other\nsignatures of many-body physics. This theoretical work predicted a\nclear dependence on the length of the QPC channel, and testing this\ndirectly requires experiments where this length is varied.\n\nThe work by Koop \\textit{et al.} \\cite{5Koop} already explored the\nrelation between the device geometry and parameters that describe\nthe many-body effects in a large set of QPC$_{\\rm 2F}$ devices. This\nwork compared nominally identical devices, and devices for which the\nlithographic length $L_{litho}$ (see Fig.~\\ref{fig1SEMpic}a) and\nwidth of the channel in the split-gate structure was varied. These\nresults were, however, not conclusive. The parameters that describe\nthe many-body effects showed large, seemingly random variation, not\ncorrelated with the device geometry. At the same time, the devices\nshowed (besides the 0.7 anomaly) clean quantized conductance traces,\nand the parameters that reflect the non-interacting electron physics\ndid show the variation that one expects when changing the geometry\n(for example, the channel pinch-off gate voltage $V_{po}$ and\nsubband spacing $\\hbar \\omega_{12}$). This confirms that these QPCs\nhad saddle-point potentials that were smooth enough for showing\nquantized conductance, while it also shows that the many-body\neffects are very sensitive to small static fluctuations on these\nsaddle-point potentials or to nanoscale device-to-device variations\nin the dimensions of the potentials. This picture was confirmed by\nshifting the channel position inside a particular QPC$_{\\rm 2F}$\ndevice. This can be implemented by operating a QPC$_{\\rm 2F}$ with a\ndifference $\\Delta V_g$ between the values of $V_g$ on the two gate\nfingers in Fig.~\\ref{fig1SEMpic}a. Such a channel shift did not\nchange the quantized conductance significantly, but did cause strong\nvariation in the signatures of many-body physics. Earlier work had\nestablished that such device-to-device fluctuations can be due to\nremote defects or impurities, a slight variation in electron\ndensity, or due to the nanoscale variation in devices that is\ninherent to the nanofabrication process\n\\cite{5Koop,5DefectsInteractions,5JohnAltDavies}. Consequently,\nstudying how the many-body effects depend on the length of the QPC\nchannel requires QPCs for which the channel length can be tuned\ncontinuously \\textit{in situ}, and where this can be operated\nwithout a transverse displacement of the QPC channel in the\nsemiconductor material. The work that we report here aimed at\nrealizing such devices.\n\n\n\n\nThis article is organized as follows:\nSection~\\ref{5sec:designConsids} starts with a short overview of the\noptions and the choices we made for realizing the QPC$_{\\rm 6F}$\ndevices. Next, in Section~\\ref{5sec:Simulations}, we present the\nresults of electrostatic simulations. In\nSection~\\ref{5sec:sampleFabAndTech}, we describe the sample\nfabrication and measurement techniques. This is followed by\ncomparing results from simulations and experiments for QPC$_{\\rm\n6F}$ devices in Section \\ref{5sec:ExperimentalRelization}, and\nSection~\\ref{5sec:Conclusion} summarizes our conclusions.\n\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{DesignQPC6F.pdf}\n\\caption{(a) Design of the geometry of the 6 gate fingers for a\nQPC$_{\\rm 6F}$ device with 6 rectangular gate fingers. (b) Design of\na QPC$_{\\rm 6F}$ device with the 4 outer gates in a shape that\nexplicitly induces a funnel shape for the entry and exit of the QPC\ntransport channel.} \\label{fig2DesignQPC6F}\n\\end{figure}\n\n\n\n\n\\section{\\label{5sec:designConsids}Design considerations}\nWe designed our QPC$_{\\rm 6F}$ devices with 6 rectangular gate\nfingers, in a symmetric layout with two sets of 3 parallel gate\nfingers (Fig.~\\ref{fig1SEMpic}b and Fig.~\\ref{fig2DesignQPC6F}a).\nSEM inspection of fabricated devices yields that the central gate\nfinger is 200~nm wide (as measured along the direction of channel\nlength $L_{eff}$). The outer gate fingers are 160~nm wide, and the\nnarrow gaps between gate fingers are 44~nm wide. This yields $(200 +\n2 \\cdot 160 + 2 \\cdot 44)~{\\rm nm} = 608~{\\rm nm}$ for the total\ndistance between the outer sides of the 3 parallel gate fingers. The\nlithographic width of the QPC channel (distance between the two sets\nof 3 gate fingers) is 350~nm.\n\nAn example of alternative designs for the gate geometries that we\nconsidered  is in Fig.~\\ref{fig2DesignQPC6F}b. This design has a\ntwo-sided funnel shape for the channel and this could result in\nlength-tunable QPC operation that better maintains a regular shape\nfor the saddle-point potential. However, the electrostatic\nsimulations in Section~\\ref{5sec:Simulations} show that the\nrectangular gate fingers as in Fig.~\\ref{fig2DesignQPC6F}a also give\na length-tunable saddle-point potential that maintains a regular\nshape while tuning the length. This observation holds for a range of\ndevice dimensions similar to our design. For our particular design,\nthe lithographic length and width (350~nm) of the channel are\ncomparable, and the 2DEG is as far as 110~nm distance below the\nsurface (and the part in the center of the channel that actually\ncontains electrons is very narrow, about 20~nm). In this regime, the\nsaddle-point potential is strongly rounded with respect to the\nlithographic shapes of the gates (see for example\nFig.~\\ref{fig3SaddlePotential}c,d). An important advantage of the\nrectangular design is that it provides two clear points for\ncalibrating the effective channel length $L_{eff}$: Operating at\n$V_{g2}\/V_{g1} = 0$ gives $L_{eff}=L_{litho}$ for the central gate\nfinger alone (200~nm, see Fig.~\\ref{fig1SEMpic}b), while operating\nat $V_{g2}\/V_{g1} = 1$ gives $L_{eff}$ equal to the lithographic\ndistance between the outer sides of the 3 parallel gate fingers\n(608~nm).\n\n\n\nA point of concern for this design that deserves attention is\nwhether the narrow gaps between the 3 parallel gate fingers induce\nsignificant structure on the saddle-point potential. The\nelectrostatic simulations show that this is not the case (see again\nthe examples in Fig.~\\ref{fig3SaddlePotential}c,d). The part of the\nchannel that contains electrons is relatively far away from the gate\nelectrodes, and the potential $U$ at this location is strongly\nrounded. Notably, the full height of the potentials in\nFig.~\\ref{fig3SaddlePotential} is about 1~eV, while the occupied\nsubbands are at a height of only about 10~meV above the stationary\npoint of the saddle-point potential (in the center of the channel).\nSuch gaps between parallel gate electrodes can be much narrower when\ndepositing a wider gate on top of the central gate, with an\ninsulating layer between them. We chose against applying this idea\nsince we also aimed to have devices with a very low level of noise\nand instabilities from charge fluctuations at defect and impurity\nsites in the device materials. In this respect, we expect better\nbehavior when all gate fingers are deposited in a single fabrication\ncycle, and when deposition of an insulating oxide or polymer layer\ncan be omitted.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{\\label{5sec:Simulations}Electrostatic simulations}\n\nThis section presents results of electrostatic simulations of the\nsaddle-point potentials that define the QPC channel. The focus is on\nthe design with 6 rectangular gate fingers\n(Fig.~\\ref{fig2DesignQPC6F}a), with gate dimensions as mentioned in\nthe beginning of Section~\\ref{5sec:designConsids}. The simulations\nare based on the modeling approach that was introduced by Davies\n\\textit{et al.} \\cite{5Davies}.\n\n\n\\subsection{Davies' method for simulating 2DEG electrostatics}\n\nDavies \\textit{et al.} \\cite{5Davies} introduced a method for\nmodeling the electrostatics of gated 2DEG. It calculates the\nelectrostatic potential $U$ for electrons in the 2DEG regions around\nthe gates (the approach only applies to the situation where the 2DEG\nunderneath the gates is depleted due to a negative voltage on gate\nelectrodes). There are other models and approaches\n\\cite{5JohnAltDavies,5SniderAltDavies,5MinhanAltDavies,5LauxAltDavies,5John1AlteDavies}\nfor calculating such potential landscapes, but these are all more\ncomplicated and computationally more demanding. The approach by\nDavies \\textit{et al.} is relatively simple. It does not account for\nelectrostatic screening effects, and, notably, it does not account\nfor the electron many-body interactions that were mentioned earlier.\nStill, it was shown that it is well suited for calculating a valid\npicture of a QPC saddle-point potential near the channel pinch-off\nsituation \\cite{5Koop}.\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Saddlepotential.pdf}\n\\caption{(a),(b) Saddle-point potentials that represent the\nelectrostatic potential $U$ felt by electrons in the 2DEG plane. The\nplots represent an area of 1000$\\times$1000~${\\rm nm^2}$, centered\nat the middle of a QPC channel with a length $L_{litho}$ of 200~nm\n(a) and 600~nm (b) of a QPC$_{\\rm 2F}$ device with a lithographic\nchannel width of 350~nm. It is calculated for the material\nparameters that are valid for the measured devices. See\nFig.~\\ref{fig1SEMpic}b for relating the $x$- and $y$-direction to\nthe gate geometry. (c),(d) Similar saddle-point potentials $U$\ncalculated for QPC$_{\\rm 6F}$ devices (with material parameters and\ngeometry as the measured devices). The effective channel length is\nshorter for the case that is calculated for $V_{g2}\/V_{g1}=0.2$ (c)\nthan for the case $V_{g2}\/V_{g1}=0.8$ (d) (also note that QPC$_{\\rm\n6F}$ results for $V_{g2}\/V_{g1}=0$ are the same as plot (a)). Panel\n(c) and (d) also show that the narrow gaps between 3 parallel gate\nfingers do not induce significant structure at low energies in the\nsaddle-point passage (it only induces a weak fingerprint off to the\nside in the channel, at energies that are much higher than the\noccupied electron levels, see panel (d)).}\n\\label{fig3SaddlePotential}\n\\end{figure}\n\n\n\n\n\nThe negative voltage on a gate that is needed to exactly deplete\n2DEG underneath a large gate is called the threshold voltage\n$V_{t}$, and it is to a good approximation given by\n\\begin{equation}\nV_{t} = \\frac{-e n_{2D}d}{\\epsilon_r\\epsilon_0}. \\label{5equ:Vt}\n\\end{equation}\nHere $n_{2D}$ is the electron density in the 2DEG (at zero gate\nvoltage), $d$ is the depth of the 2DEG, $\\epsilon_r$ is the relative\ndielectric constant of the material below the gate, and $\\epsilon_0$\nis the dielectric constant of vacuum (for details see\nRef.~\\onlinecite{5Koop,5Davies}). The value of $V_t$ for a certain\n2DEG material defines the value $U_0$ where the electrostatic\npotential $U$ for electrons in the 2DEG becomes higher than the\nchemical potential of the 2DEG. In turn, this can be used to define\nin an arbitrary potential landscape $U$ (for arbitrary gates shapes\nand for arbitrary gate voltages) the positions where $U=U_0$. That\nis, one can calculate the positions in a gated device structure\nwhere there is a boundary between depleted and non-depleted 2DEG,\nand also calculate the electrostatic potential $U$ around such\npoints. When the center of the QPC has $U=U_0$, the channel is at\npinch-off and no electrons can pass through the QPC. The gate\nvoltage at which this happens is called the pinch-off voltage\n$V_{po}$. Notably, the calculated value of $U$ at a certain position\nis simply the superposition of all the contributions to $U$ from\ndifferent gate electrodes, and it is linear in the gate voltage on\neach of these electrodes \\cite{5Davies}.\n\nFigure~\\ref{fig3SaddlePotential} presents examples of saddle-point\npotentials $U$ that are calculated with Davies' method, both for\nQPC$_{\\rm 2F}$ and QPC$_{\\rm 6F}$ devices. The calculations are for\nmaterial parameters and geometries of measured devices (as described\nin detail in the next sections).\nFigures~\\ref{fig3SaddlePotential}c,d show that the length of the\ntransport channel depends on the applied ratio $V_{g2}\/V_{g1}$, and\nthat the narrow gaps between 3 parallel electrodes in QPC$_{\\rm 6F}$\ndevices do not give significant structure on the saddle-point\npotential in the operation regime that we consider.\n\n\n\n\n\\subsection{Definition and tuning of the effective length $L_{eff}$}\n\nThe focus of this work is on realizing QPC channels with a tunable\nlength. The channels are in fact saddle-point potentials (see\nFig.~\\ref{fig3SaddlePotential}), and it is for such a smooth shape\nnot obvious what the value is of the channel length. We therefore\ncharacterize this channel length with the parameter $L_{eff}$, which\ncorresponds to the value of the lithographic length $L_{litho}$ of a\nQPC$_{\\rm 2F}$ type device (with rectangular gate electrodes, see\nFig.~\\ref{fig1SEMpic}a) that gives effectively the same saddle-point\npotential.\n\n\n\\begin{figure}[!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{Lalpha.pdf}\n\\caption{(a) Schematic representation of a QPC$_{\\rm 6F}$ device,\nillustrating length variables that are introduced in the main text.\n(b) The calculated length $L_{\\alpha}$ for a range of values of the\nlithographic length of QPC$_{\\rm 2F}$ devices, for three values of\n$\\alpha$. (c) The calculated effective length $L_{\\alpha}$ for\n$\\alpha=1.1$ for a QPC$_{\\rm 6F}$ device, as a function of the ratio\n$V_{g2}\/V_{g1}$.} \\label{5fig:Lalpha}\n\\end{figure}\n\n\n\n\nWe implemented this as follows. We calculated the saddle-point\npotential $U(x,y)$ for the pinch-off situation (see\nFig.~\\ref{fig1SEMpic}b and Fig.~\\ref{fig3SaddlePotential}a for how\nthe $x$- and $y$-directions are defined). The transverse confinement\nin the middle of the QPC (defined as $x=0$, $y=0$) is parabolic to a\nvery good approximation. When moving out of the channel along the\n$x$-direction, the transverse confinement becomes weaker, but\nremains at first parabolic. Notably, the energy eigenstates for\nconfinement in such a parabolic potential, described as\n\\begin{equation}\nU(y)= \\frac12 m^* \\omega_0^2 y^2,\n\\label{5equ:parabollaUy}\n\\end{equation}\nhave a width that is (for all levels) proportional to\n$\\omega_0^{-1\/2}$. In this expression $m^*$ is the effective mass of\nthe electron and $\\omega_0$ is the angular frequency of natural\noscillations in this potential. The parameter $\\omega_0$ defines\nhere the steepness of $U(y)$, and we obtain $\\omega_0 (x)$ values\nfrom fitting Eq.~\\ref{5equ:parabollaUy} to potentials $U(x,y)$\nobtained with Davies's method. We use this and investigate the width\n$\\Delta y(x)$ in $y$-direction for the lowest energy eigenstate, at\nall positions $x$ along the channel (see Fig.~\\ref{5fig:Lalpha}a).\nFor parabolic confinement this wavefunction in $y$-direction has a\nGaussian shape and has a width\n\\begin{equation}\n\\Delta y(x) = \\sqrt{\\frac{h}{4 \\pi \\; m^* \\; \\omega_0(x)}}.\n\\label{5equ:widthDeltaY}\n\\end{equation}\n\n\n\n\n\nWith this approach we analyzed that the distance from $x=0$ to the\n$x$-position $x_{\\alpha}$ where the value $\\Delta y(x)$ increased by\na factor $\\alpha \\approx 1.1$ defines a suitable point for defining\nthe value of $L_{eff}$. That is, we define\n\\begin{equation}\nL_{\\alpha} = 2 \\; x_{\\alpha},\n\\label{5equ:DEFLalpha}\n\\end{equation}\nand find $x_{\\alpha}$ by solving\n\\begin{equation}\n\\Delta y(x=x_{\\alpha}) = \\alpha \\cdot \\Delta y(x=0)\n\\label{5equ:SolveXalpha}\n\\end{equation}\nfor a certain $\\alpha$. Subsequently, $L_{eff}$ is defined by using the suitable $\\alpha$ value,\n\\begin{equation}\nL_{eff} = L_{\\alpha} \\; \\;{\\rm for} \\; \\; \\alpha=1.1.\n\\label{5equ:DEFLeffalpha}\n\\end{equation}\nWe came to this parameterization as follows. We used this\n\\textit{ansatz} first in simulations of QPC$_{\\rm 2F}$ devices.\nHere, we explored for different values of $\\alpha$ the relation\nbetween $L_{litho}$ and $L_{\\alpha}$. Results of this for $\\alpha =\n1.05$, 1.1 and 1.2 are presented in Fig.~\\ref{5fig:Lalpha}b. For the\nrange of $L_{litho}$ values that is of interest to our study\n($\\sim$100~nm to $\\sim$500~nm), we find the most reasonable overall\nagreement between the actual value for $L_{litho}$ (input to the\nsimulation) and the value $L_{\\alpha}$ (derived from the simulation)\nfor $\\alpha=1.1$. The agreement is not perfect, but we analyzed that\nthe deviation is within an uncertainty that we need to assume\nbecause the exact shapes of saddle-point potentials in different\ndevice geometries do show some variation, and because the limited\nvalidity of Davies' method. Nevertheless, it provides a reasonable\nrecipe for assigning a value $L_{eff}$ to any saddle-point\npotential, with at most 20\\% error.\n\nFig.~\\ref{5fig:Lalpha}c presents results of calculating\n$L_{\\alpha}=L_{eff}$ for $\\alpha=1.1$ from simulations of a\nQPC$_{\\rm 6F}$ device, operated at different values for\n$V_{g2}\/V_{g1}$. The results show a clear monotonic trend, with\n$L_{eff}=210~{\\rm nm}$ for $V_{g2}\/V_{g1}=0$ to $L_{eff}=525~{\\rm\nnm}$ for $V_{g2}\/V_{g1}=1$. This is for a QPC$_{\\rm 6F}$ device for\nwhich we expect $L_{eff}=200~{\\rm nm}$ for $V_{g2}\/V_{g1}=0$ and\n$L_{eff}=608~{\\rm nm}$ for $V_{g2}\/V_{g1}=1$ (see\nSection~\\ref{5sec:designConsids}). In\nSection~\\ref{5sec:ExperimentalRelization} we discuss how this latter\npoint is used for applying a small correction to the simulated\nvalues for $L_{eff}$. These simulations show that the QPC$_{\\rm 6F}$\nthat we consider allows for tuning $L_{eff}$ by about a factor 3.\n\n\n\nIt is worthwhile to note that our current design showed optimal\nbehavior in the sense that it can tune $L_{eff}$ from about 200~nm\nto 600~nm, while the dependence of $L_{eff}$ on $V_{g2}\/V_{g1}$ is\nclose to linear. We also simulated QPC$_{\\rm 6F}$ devices with wider\ngate electrodes for the outer gates, and (as mentioned in\nSection~\\ref{5sec:designConsids}) devices with gate geometries as in\nFig.~\\ref{fig2DesignQPC6F}b. These devices showed a steeper slope\nfor part of the relation between $V_{g2}\/V_{g1}$ and $L_{eff}$,\nwhich is not desirable.\n\n\n\n\n\\section{\\label{5sec:sampleFabAndTech}Sample fabrication and measurement techniques}\n\nWe fabricated QPC devices with a GaAs\/${\\rm Al}_{0.35}{\\rm\nGa}_{0.65}{\\rm As}$ MBE-grown heterostructure, which has a 2DEG at\n110~nm depth below its surface from modulation doping. The layer\nsequence and thickness of the materials from top to bottom\n(\\textit{i.e.} going into the material) starts with a $5$ nm GaAs\ncapping layer, then a 60~nm ${\\rm Al}_{0.35}{\\rm Ga}_{0.65}{\\rm As}$\nlayer with Si doping at about $1 \\times 10^{18} \\text{cm}^{-3}$,\nwhich is followed by an undoped spacer layer of 45~nm. The 2DEG is\nlocated in a heterojunction quantum well at the interface with the\nnext layer, which is a 650~nm undoped GaAs layer. This\nheterostructure was grown on a commercial semi-insulating GaAs\nwafer, after first growing a sequence of 10 GaAs\/AlAs layers for\nsmoothing the surface and trapping impurities. The 2DEG had an\nelectron density $n_{2D}=1.6 \\cdot 10^{15}~{\\rm m^{-2}}$ and a\nmobility $\\mu = 118~{\\rm m^2 V^{-1} s^{-1}}$. We fabricated both\nconventional QPC$_{\\rm 2F}$ devices and QPC$_{\\rm 6F}$ devices by\nstandard electron-beam lithography and clean-room techniques. The\ngate fingers were deposited using 15~nm Au on top of a 5~nm Ti\nsticking layer. For measuring transport through the QPCs we realized\nohmic contacts to the 2DEG reservoirs by annealing of a AuGe\/Ni\/Au\nstack that was deposited on the wafer surface\n\\cite{5OhmicKoopIqbal}. The geometries of the fabricated devices\nwere already described in the beginning of\nSection~\\ref{5sec:designConsids}.\n\nThe measurements were performed in a He-bath cryostat and in a\ndilution refrigerator, thus getting access to effective electron\ntemperatures from 80~mK to 4.2~K. We used standard lock-in\ntechniques with an a.c.~excitation voltage $V_{bias}=10~{\\rm \\mu V}$\nRMS at $387$~Hz. Fig.~\\ref{fig1SEMpic}a shows the 4-probe\nvoltage-biased measurement scheme, where both the current and the\nactual voltage drop $V_{sd}$ across the QPC channel are measured\nsuch that any influence of series resistances could be removed\nunambiguously. The gate voltages are applied with respect to a\nsingle grounded point in the loop that carries the QPC current.\n\n\n\n\n\n\\section{\\label{5sec:ExperimentalRelization}Experimental realization of length-tunable QPCs}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{IsoGfinal.pdf}\n\\caption {QPC linear conductance as a function of $V_{g1}$ and\n$V_{g2}$ for a QPC$_{\\rm 6F}$ device, presented in the form of\niso-conductance lines at integer $G_0$ levels. The conductance was\nmeasured at 4.2~K where the quantized conductance is nearly fully\nwashed out by temperature. The two operational regimes above and\nbelow the line $V_{g1}=V_{g2}$ yield QPC and quantum dot behavior,\nrespectively.} \\label{5fig:Isoconductance2D}\n\\end{figure}\n\nThis section presents an experimental characterization of the\nQPC$_{\\rm 6F}$ devices that we designed (Fig~\\ref{fig1SEMpic}b) and\nwe compare the results to our simulations.\nFigure~\\ref{5fig:Isoconductance2D} presents measurements of the\nconductance $G$ as a function of $V_{g1}$ and $V_{g2}$. Several\nlabels in the plot illustrate relevant concepts, which were partly\ndiscussed before. For the area in this plot with $V_{g2}$ more\nnegative than $V_{g1}$ we expect some quantum-dot like localization\nin the middle of the channel and this regime should therefore be\navoided in studies of QPC behavior. Further, the plot illustrates\nthat operation for a particular value of $L_{eff}$ requires\nco-sweeping of $V_{g1}$ and $V_{g2}$ from a particular point below\npinch-off in a straight line to the pivot point. This corresponds to\nopening the QPC at a fixed ratio for $V_{g2}\/V_{g1}$. The pivot\npoint is the point where the gate voltages do not alter the original\nelectron density of the 2DEG. For this measurement this is for\n$V_{g1}=V_{g2}= 0~{\\rm V}$, but this is different for the case of\nbiased cool downs. We carried out biased cool downs for suppressing\nnoise from charge instabilities in the donor layer\n\\cite{5PRBBCool,5PRLBCool}. For such experiments the QPCs were\ncooled down with a positive voltage on the gates. We typically used\n+0.3~V, and observed indeed better stability with respect to charge\nnoise. The effect of such a cool down can be described as a\ncontribution to the gate voltage of -0.3~V that is frozen into the\nmaterial \\cite{5PRBBCool,5PRLBCool}. Consequently, co-sweeping of\n$V_{g1}$ and $V_{g2}$ for maintaining a fixed channel length must\nnow be carried out with respect to the pivot point $V_{g1}=V_{g2}= +\n0.3~{\\rm V}$ instead of $V_{g1}=V_{g2}= 0~{\\rm V}$.\n\n\n\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.85\\columnwidth]{LsimuVsMeasured.pdf}\n\\caption{(a) Experimentally determined relation between the\npinch-off gate voltage $V_{po}$ and the lithographic length of\nQPC$_{\\rm 2F}$ devices. Points are experimental results. The solid\nline is a phenomenological expression that was used for\nparameterizing the relation between $V_{po}$ and the lithographic\nlength. (b) Comparison between measured and simulated values of the\neffective channel length for a QPC$_{\\rm 6F}$ device.}\n\\label{5fig:LalphaVsmeasured}\n\\end{figure}\n\n\n\nThe theory behind the Davies method illustrates why operation at\nfixed effective lenght requires a fixed ratio $V_{g2}\/V_{g1}$. All\npoints in the potentials landscapes $U$ for QPC$_{\\rm 2F}$ devices\nas in Fig.~\\ref{fig3SaddlePotential}a,b have a height that scales\nlinear with the gate voltage $V_g$. Thus, when opening the QPC the\nfull saddle-point potential changes height at a fixed shape.\nMimicking this situation with QPC$_{\\rm 6F}$ devices requires a\nfixed ratio $V_{g2}\/V_{g1}$, again because $V_{g1}$ and $V_{g2}$\ninfluence $U$ in a linear manner. The plot also illustrates the two\nspecial operation lines where the effective length of the channel is\nunambiguous, and we used these points to better calibrate the\nrelation between $V_{g2}\/V_{g1}$ and $L_{eff}$. The first case is\nthe line at $V_{g2}=0$, which yields $L_{eff}=200~{\\rm nm}$, as\ndefined by the central gates alone. The second case is the line\n$V_{g1}=V_{g2}$. Here $L_{eff}$ is 608~nm, as defined by the full\nlithographic length of the 3 gate fingers.\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{LinearG.pdf}\n\\caption{The linear conductance $G$ as a function of $V_{g1}$ for a\nQPC$_{\\rm 6F}$ device, measured at 80~mK. Different traces are for\nfixed ratio $V_{g2}\/V_{g1}=0$ to $V_{g2}\/V_{g1}=1$ (left to right,\ntraces \\textit{not} offset), which corresponds to increasing the\neffective channel length $L_{eff}$ from 200~nm to 608~nm.}\n\\label{5fig:LinearG}\n\\end{figure}\n\n\n\nWe improved and further checked our calibration of the relation\nbetween $V_{g2}\/V_{g1}$ and $L_{eff}$ as follows. We used the trend\nthat came out of the simulations (Fig.~\\ref{5fig:Lalpha}c), but\npinned the curve at 200~nm for $V_{g2}\/V_{g1}=0$ and at 608~nm for\n$V_{g2}\/V_{g1}=1$ (black line in Fig.~\\ref{5fig:LalphaVsmeasured}b).\nThis trace shows good agreement with results from an independent\ncheck (dashed line) that used the pinch-off gate voltage $V_{po}$ as\nan identifier for the effective length. This independent check used\ndata from a set of QPC$_{\\rm 2F}$ devices for calibrating the\nrelation between $L_{litho}$ and $V_{po}$\n(Fig.~\\ref{5fig:LalphaVsmeasured}a). This shows the trend that\nshorter QPC$_{\\rm 2F}$ devices require a more negative gate voltage\nto reach pinch-off \\cite{5VpvsL}. We related this to the pinch-off\nvalues in QPC$_{\\rm 6F}$ devices. In particular, we analyzed the\npinch-off points on the $V_{g1}$ axis, and its dependence on\n$V_{g2}\/V_{g1}$ (see also Fig.~\\ref{5fig:LinearG}). The results of\nusing this for assigning a certain $L_{eff}$ to each $V_{g2}\/V_{g1}$\nis the dashed line in Fig.~\\ref{5fig:LalphaVsmeasured}b, and shows\ngood agreement with the values that were obtained from simulations.\nWe can thus assign a value to $L_{eff}$ for each $V_{g2}\/V_{g1}$\nwith an absolute error that is at most 50~nm. Notably, the relative\nerror when describing the increase in $L_{eff}$ upon increasing\n$V_{g2}\/V_{g1}$ is much smaller.\n\n\n\nThe results in Fig.~\\ref{5fig:LinearG} provide an example of linear\nconductance measurements on a QPC$_{\\rm 6F}$ device at 80~mK. The\ntraces show clear quantized conductance plateaus for all settings of\n$L_{eff}$. Several of these linear conductance traces also show the\n0.7~anomaly, and the strength of its (here rather weak) expression\nshows a modulation as a function of $L_{eff}$ over about 3 periods.\nA detailed study of this type of periodicity can be found in\nRef.~\\onlinecite{5IqbalPerio}. This example illustrates the validity\nand importance of our type of QPCs in studies of length-dependent\ntransport properties.\n\n\n\n\n\n\n\n\n\\section{\\label{5sec:Conclusion}Conclusions}\n\nWe have developed and characterized length-tunable QPCs that are\nbased on a symmetric split-gate geometry with 6 gate fingers. Gate\nstructures with different shapes and dimensions can be designed\ndepending upon the required range for length tuning and for\noptimizing the tuning curve. For our purpose (QPCs with an effective\nchannel length between about 200~nm and 600~nm, and 350~nm channel\nwidth) we found that simple rectangular gate fingers are an\nattractive choice. Our simulations and experimental results are in\nclose agreement. We were able to tune the effective length by about\na factor 3, from 200~nm to 608~nm. QPCs are the simplest devices\nthat show clear signatures of many-body physics, as for example the\n$0.7$ anomaly and the Zero-Bias Anomaly (ZBA) \\cite{Cronenwett2002}.\nOur length-tunable QPCs provide an interesting platform for\nsystematically investigating these many-body effects. In particular,\nthese QPCs provide a method for studying the influence of the QPC\ngeometry without suffering from device-to-device fluctuations that\nhamper such studies in conventional QPCs with 2 gate fingers.\nStudies in this direction are presented in\nRef.~\\onlinecite{5IqbalPerio}.\n\n\\section*{ACKNOWLEDGMENTS}\nWe thank Y.~Meir for discussions, B.~H.~J.~Wolfs for technical\nassistance, and the German programs DFG-SPP 1285, Research school\nRuhr-Universit\\\"{a}t Bochum and BMBF QuaHL-Rep 01BQ1035 for\nfinancial support. MJI acknowledges a scholarship from the Higher\nEducation Commission of Pakistan.\n\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\IEEEPARstart{D}{igital} pulse interval modulation (DPIM) \\cite{Ghassemlooy2012} is an energy-efficient modulation technique for optical wireless communications based on intensity modulation and direct detection (IM\/DD). Compared with conventional on-off keying (OOK), pulse amplitude modulation (PAM), pulse width modulation (PWM) and pulse position modulation (PPM), it utilizes pulse intervals to carry data and does not require accurate symbol-level synchronization.\nSince proposed, it has been widely investigated in spectral evaluation \\cite{Cariolaro2001,Ono2001} and error performance over various channels \\cite{Zhang2000,Ghassemlooy2001,Ghassemlooy2003,Ghassemlooy2005,Popoola2006,Ghassemlooy2008,Ma2010,Jin2010,Jiang2014,Liu2014,Xu2015,Dabiri2016,Das2019}. Many enhanced modulation schemes have been proposed\n\\cite{Ghassemlooy2006,Aldibbiat2001,Ono2002,Ono2004,Sethakaset2005,Numata2011,He2013,Abdullah2014,Liao2014,Mi2016}. For instance, the authors of \\cite{Ghassemlooy2006}  and \\cite{Aldibbiat2001} proposed multilevel digital pulse interval modulation (MDPIM) and dual-header pulse interval modulation (DHPIM), which improved the information-carrying capability by additionally varying  pulse amplitudes and  pulse widths, respectively.\n\n\n\\begin{figure*}[t]\n  \\centering\n \n    \\includegraphics[width=0.9\\textwidth]{system_model_new}\\\\\n  \\caption{System model of DPIM}\n  \\label{System_Model}\n\\end{figure*}\n\nDPIM symbols do not have equal duration. The fact complicates the analysis and application of DPIM.  Note that a single-chip error not only corrupts the bits directly associated with that chip but also shifts the bits that follow those bits \\cite{Shiu1999}. Another distinct property resulting from the nonuniform symbol length is that the symbol boundaries are\nnot known before detection. Conventional soft decoding approaches for invariant symbol duration such as the Viterbi algorithm are not applicable \\cite{Shiu1999}. The optimal soft decoding of\nDPIM requires the use of high-complexity maximum likelihood sequence detector (MLSD). Therefore, most practical implementations of\nDPIM would probably employ hard-decision decoding, i.e., sample-by-sample optimal threshold detection (OTD). Even though OTD has low complexity, it suffers from more severe bit error propagation, which leads to high bit error ratio (BER) in an erroneous packet. When using forward error correction (FEC) codes to correct the erroneous packet, lots of redundancy must be added in codewords because of the high BER thus resulting in a low rate. In other words, because of the high BER in an erroneous packet, DPIM can not benefit from the association with high-rate FEC codes for packet correction. To fix this problem, the error propagation of DPIM should be mitigated. \nMotivated by improving the system reliability, this paper proposes a low-complexity detection method that achieves the same performance as MLSD leveraging the sparsity of  DPIM sequences. Moreover, the paper proposes a barrier signal design for DPIM to limit the error propagation. It should be mentioned that\nwe adopt the average BER metric to measure the system reliability in this paper, which is defined by the packet error rate (PER) multiplying the expected BER in an erroneous packet.  Compared with the PER metric used in  \\cite{Zhang2000,Ghassemlooy2001,Ghassemlooy2003,Ghassemlooy2005,Popoola2006,Ghassemlooy2008,Ma2010,Jin2010,Jiang2014,Liu2014,Xu2015,Dabiri2016,Das2019},\nthe average BER metric can measure both the PER and the error propagation level in an erroneous packet. The contributions of this paper are summarized as follows.\n \\begin{itemize}\n\\item At the receiver side, the optimal MLSD of DPIM is formulated as a sparsity-constrained least square minimization problem. An ordered sequence detection (OSD) is proposed to solve the problem with low complexity.  The computational complexity and detection delay of OSD are analyzed. The relationship between OSD and the orthogonal matching pursuit (OMP) algorithm in \\cite{Tropp2007} is revealed, which explains why OSD can achieve the same performance as MLSD.\n\\item At the transmitter side, a barrier signal-aided digital pulse interval modulation (BDPIM) is proposed to limit the error propagation and jointly utilized with OSD.  For large packet detection, we propose to use a combination of OTD and OSD to reduce the storage delay.  The parameter optimization of BDPIM is investigated by simulations.  \n\n\\item Approximate upper bounds of the average BER performance of DPIM with OTD (DPIM-OTD), DPIM with OSD (DPIM-OSD), BDPIM with OSD (BDPIM-OSD) and BDPIM with a combination of OTD and OSD (BDPIM-OTD-OSD) are derived.  Compared with \\cite{Guo2018d}, the approximate BER upper bounds provided in this paper are applicable to all SNR regimes. Simulations are conducted to corroborate our theoretical analysis.\n\n\\item Comparisons are made among coded and uncoded  DPIM-OTD, DPIM-OSD, BDPIM-OSD,  BDPIM-OTD-OSD, etc. For extension, we investigate the performance evaluation over Gamma-Gamma turbulence channels. \n \\end{itemize} \n\n \n The remainder of the paper is organized as follows.  Section II describes the system model and the proposed designs. Section III presents the approximate upper bounds of the average BER performance of all described systems.\nIn Section IV, we show the superiority of the proposed schemes by simulations, investigate the impact of parameter settings on performance and discuss the extension to Gamma-Gamma  turbulence channels. Conclusions are drawn in the last section.\n\nIn this paper, the normal letter $x$ represents a scalar. The boldface letter $\\textbf{x}$ stands for a vector. $||\\textbf{x}||_0$ and $||\\textbf{x}||_2$ denote the $l_0$ norm and $l_2$ norm of vector $\\textbf{x}$, respectively. $\\mathbb{R}$ represents the real-number field. $\\mathbb{E}(\\cdot)$ represents the expectation operation. $\\lfloor x\\rceil$ denotes the nearest integer to $x$. $\\ln$ and $\\log$ stand for the logarithm functions of base $e$ and base $2$, respectively. $\\mathcal{Q}(\\cdot)$ denotes the tail distribution function of the standard normal distribution. $\\mathrm{erf}(\\cdot)$ is the Gaussian error function. $n~\\mathrm{mod}~K=0$ when $n$ is a multiple of $K$ and otherwise $n~\\mathrm{mod}~K\\neq0$. \n\n \n\n\\begin{figure}[t]\n  \\centering\n \n \\includegraphics[width=0.42\\textwidth]{signal_form_new}\\\\\n  \\caption{The signal forms of DPIM}\n  \\label{Signal_form}\n  \\end{figure}\n  \n\\section{System Model and Proposed Designs}\nIn this section, the system model of conventional DPIM is first presented and then followed by an introduction to existing detection methods (i.e., MLSD, OTD) and the proposed OSD.  Also,   the new signal design, i.e., BDPIM is  described. \n\\subsection{System Model of DPIM}\nWe consider a DPIM system as illustrated in Fig. \\ref{System_Model}, where  $N_s$-symbol packets are transmitted by using $M$-ary DPIM.  The signal forms for DPIM has two types with and without guard intervals (GIs).\nTo demonstrate their differences,  we illustrate the signal forms of $4$-DPIM  in Fig. \\ref{Signal_form}.  The data bits $\\{00,~01,~10,~11\\}$ are mapped to  $\\{\\mathrm{A},~ \\mathrm{A}0,~\\mathrm{A}00,~\\mathrm{A}000\\}$ with no GI, and to $\\{\\mathrm{A}0,~\\mathrm{A}00,~\\mathrm{A}000,~\\mathrm{A}0000\\}$ with $1$ GI, where $\\mathrm{A}$ represents the peak amplitude and is proportional to the optical power. For the sake of simplicity, we assume that the proportional coefficient is $1$. In this paper, a general case with $g$ ($g\\in\\mathbb{N}$) GIs is considered\nand the average symbol duration is given by\n\\begin{equation}\nL_s=\\frac{1}{M}\\sum_{l=1}^{M}l+g=\\frac{M+2g+1}{2}.\n\\end{equation}\nNote that because the symbol duration of DPIM is variant, the total sequence length denoted by $L$ is also variant and the expected sequence length of $N_s$-symbol packets is $\\mathbb{E}\\{L\\}=N_sL_s$.\n\n Let $h$ denote the channel coefficient.  The received signal vector $\\textbf{y}\\in\\mathbb{R}^{L}$ can be written as\n\\begin{equation}\\label{sig_model}\n\\textbf{y}=h\\textbf{x}+\\textbf{n},\n\\end{equation}\nwhere $\\textbf{x}\\in\\mathbb{R}^{L}$ represents the transmit signal vector, and $\\textbf{n}\\in\\mathbb{R}^{L}$ is real-valued additive white Gaussian noise (AWGN) with zero mean and variance $\\sigma_n^2\\textbf{I}_L$. Moreover, the electrical SNR is denoted as $\\gamma$ which is expressed as $\\gamma=\\mathrm{A}^2\/\\sigma_n^2$.\n\\subsection{Detection Methods}\n\\subsubsection{MLSD}\nIt is assumed that $h$ is known at the receiver. With full knowledge of $h$, $N_s$ and $L$, MLSD is given by\n\\begin{equation}\n\\begin{split}\n(\\textbf{P1}):\\hat{\\textbf{x}}&=\\arg\\max_{||\\textbf{x}||_0=N_s}p_{\\textbf{Y}}(\\textbf{y}|\\textbf{x},h)\\\\\n&=\\arg\\min_{||\\textbf{x}||_0=N_s}||\\textbf{y}-h\\textbf{x}||_2^2,\n\\end{split}\n\\end{equation}\nsince the probability density function (PDF) of \\textbf{y} conditioned\non \\textbf{x} and $h$ is\n\\begin{equation}\np_{\\textbf{Y}}(\\textbf{y}|\\textbf{x},h) \\propto \\exp(-||\\textbf{y}-h\\textbf{x}||_2^2).\n\\end{equation}\nThere are a total of $\\left(L\\atop N_s\\right)$ feasible solutions for  $\\hat{\\textbf{x}}$. For each feasible solution, $L$ real-number multiplications are needed to compute the $l_2$ norm. Therefore, MLSD based on exhaustive search requires $\\left(L\\atop N_s\\right)L$ multiplications. Due to the high computational complexity,  it  is prohibitive especially for large packet detection.\n\n\\subsubsection{OTD} OTD detects the sequence using an optimal threshold $h\\mathrm{A}_\\mathrm{T}$. The key of OTD is finding $\\mathrm{A}_\\mathrm{T}$ based on the minimizing chip-error probability criterion. Specifically, the chip error probability can be expressed as\n\\begin{equation}\n\\begin{split}\nP_c=&\\mathrm{Pr}(0)\\mathrm{Pr}(0\\rightarrow \\mathrm{A})+\\mathrm{Pr}(\\mathrm{A})\\mathrm{Pr}(\\mathrm{A} \\rightarrow 0)\\\\\n=&\\mathrm{Pr}(0)\\int_{\\mathrm{A}_\\mathrm{T}}^{\\infty}\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2x^2}{2\\sigma_n^2}}dx\n\\\\&~+\\mathrm{Pr}(\\mathrm{A})\\int^{\\mathrm{A}_\\mathrm{T}}_{-\\infty}\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2(x-\\mathrm{A})^2}{2\\sigma_n^2}}dx.\n\\end{split}\n\\end{equation}\nTaking the first derivative of $P_c$ with respect to $A_{T}$, we have\n\\begin{equation}\n\\frac{\\partial P_c}{\\partial\\mathrm{A}_\\mathrm{T}}=\\mathrm{Pr}(\\mathrm{A})\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2(\\mathrm{A}-\\mathrm{A}_{\\mathrm{T}})^2}{2\\sigma_n^2}} -\\mathrm{Pr}(0)\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2\\mathrm{A}_{\\mathrm{T}}^2}{2\\sigma_n^2}}.\n\\end{equation}\nBy solving the equation $\\frac{\\partial P_c}{\\partial\\mathrm{A}_\\mathrm{T}}=0$, \nwe obtain the optimal threshold $\\mathrm{A}_{\\mathrm{T}}$ as\n\\begin{equation}\n\\mathrm{A}_{\\mathrm{T}}=\\frac{\\mathrm{A}}{2}-\\frac{\\mathrm{A}}{h^2\\gamma}\\ln\\frac{\\mathrm{Pr}(\\mathrm{A})}{\\mathrm{Pr}(0)}.\n\\end{equation}\nFrom  (\\ref{eqAt}), it is observed that $\\mathrm{A}_{\\mathrm{T}}$ is determined by $\\mathrm{Pr}(\\mathrm{A})$ and $\\mathrm{Pr}(\\mathrm{0})$, which are expressed as\n\\begin{equation}\\label{PrA}\n\\mathrm{Pr}(\\mathrm{A})=\\frac{N_s}{L},\n\\end{equation}\nand\n\\begin{equation}\\label{Pr0}\n\\mathrm{Pr}(0)=\\frac{L-N_s}{L},\n\\end{equation}\nrespectively. From (\\ref{PrA}) and (\\ref{Pr0}), it is observed that $N_s$ and $L$ should be be known for the determination of $\\mathrm{A}_\\mathrm{T}$. In practical implementation, the exact $\\mathrm{Pr}(\\mathrm{A})$, $\\mathrm{Pr}(0)$ can be approximately given by their expectations which are $\\mathbb{E}\\{\\mathrm{Pr}(\\mathrm{A})\\}=1\/L_s$ and $\\mathbb{E}\\{\\mathrm{Pr}(\\mathrm{A})\\}=1-1\/L_s$. Based on these approximations, $\\mathrm{A}_{\\mathrm{T}}$ can be determined without the exact knowledge of $N_s$ and $L$ as\n\\begin{equation}\\label{eqAt}\n\\mathrm{A}_{\\mathrm{T}}\\approx\\frac{\\mathrm{A}}{2}+\\frac{ \\mathrm{A}}{h^2\\gamma}\\ln{(L_s-1)}.\n\\end{equation}\n\\subsubsection{OSD} The optimal MLSD expressed as (\\textbf{P1}) is indeed a sparsity-constrained least square minimization problem and the average sparsity level of DPIM sequences is $\\mathbb{E}\\{N_s\/L\\}=1\/L_s$. According to \\cite{Tropp2007}, the problem can be well solved by the OMP algorithm.\nIn use of the OMP algorithm,  the sequence can be iteratively detected as\n\\begin{equation}\n\\hat{t}_k=\\arg \\max_{t=1,2,\\cdots,L} |\\langle\\textbf{r}_{k},\\textbf{e}_t\\rangle|\n\\end{equation}\nwith $\\textbf{r}_1=\\textbf{y}$ and  $\\textbf{r}_{k}$ being updated by\n\\begin{equation}\n\\textbf{r}_{k+1}=\\textbf{r}_{k}-\\langle\\textbf{r}_{k},\\textbf{e}_{\\hat{t}_k}\\rangle \\textbf{e}_{\\hat{t}_k},\n\\end{equation}\nwhere $\\hat{t}_k$ is the position of $\\mathrm{A}$ detected in the $k$th iteration, $\\textbf{e}_t\\in\\mathbb{R}^L$ denotes the $t$th basis vector with $t$th entry being one and others being zeros.  It is observed from the procedure that OMP detects the $k$th largest receive signal as $\\mathrm{A}$ in the $k$th iteration and repeats the detection until the number of iterations  equals to $N_s$.\n Instead of using the iterative procedure, we propose a simplified equivalent OSD in this paper. In OSD, a sorting algorithm is used to sort $\\mathbf{y}=[y(1),y(2),\\cdots,y(L)]$, the $N_s$ largest of which are detected as $\\mathrm{A}$ and the others as $0$. Let $\\{i(t)\\}$ represent the order of $\\{y(t)\\}$, where $i(t)=1$ means that $y(t)$ is the largest. Mathematically,  the detected signal sequence can be written as\n\\begin{equation}\nd(t)=\\begin{cases}\n\\mathrm{A}, & 1\\leq i(t) \\leq N_s,\\\\\n0, &N_s+1\\leq i(t) \\leq L.\n\\end{cases}\n\\end{equation}\nThen by inputting $\\mathbf{d}=[d(1),d(2),\\cdots,d(L)]$ into conventional DPIM demodulator, we finally obtain the data bits. Moreover, it should be mentioned that $h$ is not used in the OSD procedure, which means that channel estimation is not required.\n\\begin{table*}[t]\n\\centering\n\\caption{Comparisons of Three Detection Methods}\\label{tab1}\n\\footnotesize\n    \\begin{tabular}{ | c | c | c | c|c|c|}\n    \\hline\n    Method & Detection manner & \\tabincell{c}{Computation \\\\ complexity\/delay} & Storage delay  &Performance & \\tabincell{c}{Required information} \\\\ \\hline\n    MLSD & packet-by-packet &  $O\\left(\\left(L\\atop N_s\\right)L\\right)$,  High & Low if $L$ is small &Optimal&$h$, $L$, $N_s$\\\\ \\hline\n   OTD & sample-by-sample &  $O\\left(L\\right)$, Low & Low&Suboptimal&$h$, $L_s$\\\\ \\hline\n   OSD & packet-by-packet &  $O\\left(L\\log L\\right)$, Low  & Low if $L$ is small&Optimal&$L$, $N_s$\\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\\begin{table*}[]\n\\caption{Data mapping of PPM, DPIM, MDPIM, DHPIM and BDPIM}\n\\center\n\\begin{tabular}{lllllll}\nData& PPM & DPIM & MDPIM\\cite{Ghassemlooy2006} & $\\textrm{DHPIM}$\\cite{Aldibbiat2001}&  $\\textrm{BDPIM}_{n~\\mathrm{mod}~K\\neq0}$&$\\textrm{BDPIM}_{n~\\mathrm{mod}~K=0}$\\\\\n 000&$\\mathrm{A}$0000000&$\\mathrm{A}$0&$\\mathrm{A}_{\\mathrm{L}}0$ &$\\mathrm{A}$0& $\\mathrm{A}_{\\mathrm{L}}$0&$\\mathrm{A}_{\\mathrm{H}}$0\\\\\n 001&0$\\mathrm{A}$000000&$\\mathrm{A}$00 &$\\mathrm{A}_{\\mathrm{L}}$00  &$\\mathrm{A}$00 &$\\mathrm{A}_{\\mathrm{L}}$00&$\\mathrm{A}_{\\mathrm{H}}$00\\\\\n 010&00$\\mathrm{A}$00000&$\\mathrm{A}$000&$\\mathrm{A}_{\\mathrm{L}}$000& $\\mathrm{A}$000&$\\mathrm{A}_{\\mathrm{L}}$000&$\\mathrm{A}_{\\mathrm{H}}$000\\\\\n 011&000$\\mathrm{A}$0000&$\\mathrm{A}$0000&$\\mathrm{A}_{\\mathrm{L}}$0000&$\\mathrm{A}$0000&$\\mathrm{A}_{\\mathrm{L}}$0000&$\\mathrm{A}_{\\mathrm{H}}$0000\\\\\n100&0000$\\mathrm{A}$000&$\\mathrm{A}$00000&$\\mathrm{A}_{\\mathrm{H}}$0&$\\mathrm{A}$$\\mathrm{A}$0& $\\mathrm{A}_{\\mathrm{L}}$00000&$\\mathrm{A}_{\\mathrm{H}}$00000\\\\\n 101&00000$\\mathrm{A}$00&$\\mathrm{A}$000000 &$\\mathrm{A}_{\\mathrm{H}}$00  &$\\mathrm{A}$$\\mathrm{A}$00 &$\\mathrm{A}_{\\mathrm{L}}$000000&$\\mathrm{A}_{\\mathrm{H}}$000000\\\\\n 110&000000$\\mathrm{A}$0&$\\mathrm{A}$0000000& $\\mathrm{A}_{\\mathrm{H}}$000 &$\\mathrm{A}$$\\mathrm{A}$000 &$\\mathrm{A}_{\\mathrm{L}}$0000000&$\\mathrm{A}_{\\mathrm{H}}$0000000\\\\\n 111&0000000$\\mathrm{A}$&$\\mathrm{A}$00000000&$\\mathrm{A}_{\\mathrm{H}}$0000&$\\mathrm{A}$$\\mathrm{A}$0000  &$\\mathrm{A}_{\\mathrm{L}}$00000000&$\\mathrm{A}_{\\mathrm{H}}$00000000\\\\\n\\end{tabular}\n\\end{table*}\n\n\\emph{Computational Complexity and Delay Analysis of Three Detection Methods:} As analyzed in Section II-B-1,  MLSD based on exhaustive search requires $\\left(L\\atop N_s\\right)L$ multiplications, which can be expressed as $\\mathcal{O}\\left(\\left(L\\atop N_s\\right)L\\right)$.\nOTD requires $L$ comparisons, the complexity of which can be given by $\\mathcal{O}\\left(L\\right)$. OSD can sort $L$ received signals by using the QuickSort algorithm, whose complexity is $\\mathcal{O}(L\\log L)$.  \nThe detection delay includes the storage latency and the computation delay. Since MLSD and OSD are operated packet-by-packet, the storage delay of an $L$-chip packet is\n\\begin{equation}\\label{tau1}\n\\tau_{1}=\\frac{L}{R_{c}},\n\\end{equation}\nwhere $R_{c}$ is the transmission rate in chips per second. Differently, OTD is performed sample-by-sample, and thus the storage delay of OTD is\n\\begin{equation}\n\\tau_{2}=\\frac{1}{R_{c}}.\n\\end{equation}\nFor high-rate optical wireless communications, the storage latency of OTD only depends on the transmission rate and thus it is low regardless of the sequence length. The latency of OSD and that of MLSD are also dependent on $L$, and therefore they are low only if $L$ is small.  The computational delay is proportional to the computational complexity. Since the complexity of OTD and OSD is quite low, the computation delay can be omitted. On the contrary, the complexity of MLSD is high thus resulting in a computation delay of $O\\left(\\left(L\\atop N_s\\right)L\\right)$ unit times (UTs), where a UT is the amount of time it takes for a multiplication. For clearly viewing the differences of these detection methods, we list the comparison results in terms of complexity, storage delay, performance as well as the required information in Table I.\n\n\n\n\\subsection{Barrier Signal Design}\nThe severe error propagation of DPIM mainly results from unknown symbol boundaries. To overcome it, we propose a barrier signal design, i.e., BDPIM.  As shown in  Fig. \\ref{Packet}, BDPIM signals have two amplitudes. The amplitude of the $n$th symbol is $\\mathrm{A}_{\\mathrm{H}}$, when $n~\\mathrm{mod}~K=0$  and otherwise $\\mathrm{A}_{\\mathrm{L}}$ in BDPIM, where $\\mathrm{A}_{\\mathrm{H}}>\\mathrm{A}_{\\mathrm{L}}$ and $K$ is a positive integer.  For fair comparison with DPIM under the same optical power constraint, the power allocation in every $K$ symbols of BDPIM  satisfies \n\\begin{equation}\\label{Con1}\n(K-1)\\mathrm{A}_{\\mathrm{L}}+\\mathrm{A}_{\\mathrm{H}}=K\\mathrm{A}.\n\\end{equation}\n To show the difference of the proposed design from existing signal designs, we list the data mapping of PPM, DPIM with 1 GI, MDPIM \\cite{Ghassemlooy2006}, DHPIM\\cite{Aldibbiat2001}, the proposed BDPIM in Table II. It is shown that BDPIM  has the same average symbol duration as DPIM, which is larger than that of MDPIM \\cite{Ghassemlooy2006} and that of DHPIM \\cite{Aldibbiat2001}.\nFor simplicity, we assume that the number of symbols $N_s$ in a packet can be divided by $K$, which means $N_s$ can be expressed as $N_s=KQ$ and $Q$ is a positive integer.\n\\begin{figure}[t]\n  \\centering\n \n    \\includegraphics[width=0.4\\textwidth]{frame_structure_new}\\\\\n  \\caption{The signal forms of BDPIM}\n  \\label{Packet}\n\\end{figure}\n\n\nAt the receiver, OSD for BDPIM has two phases. In the first OSD phase, $\\mathbf{y}$ is sorted and the $Q$ largest are detected as $\\mathrm{A}_{\\mathrm{H}}$, the positions of which is recorded as $t_1, t_2,\\cdots, t_Q$.  Let $t_1'< t_2' <\\cdots< t_Q'$ denote the sorted positions in an ascent order and $t'_0=0$. In the second OSD phase,  $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ is sorted, the $(K-1)$ largest values of which are detected as $\\mathrm{A}_{\\mathrm{L}}$ and others as $0$ for any $i=0,1,\\cdots,Q-1$.  At last, by inputting the detected signal vector into conventional DPIM demodulator, we finally obtain the data bits.  The detailed OSD algorithm for BDPIM is listed in Algorithm 1.\n\\begin{algorithm}[t] \n\\caption{OSD Algorithm for BDPIM}\n\\label{alg:osd}\n\\begin{algorithmic} \n\\STATE  \\textbf{Input:} $\\mathbf{y}$, $L$, $N_s$ and $Q$\n\\STATE Sort $\\mathbf{y}$ and output the order $\\{i(t)\\}$.\n\\STATE Detect $1\\leq i(t)\\leq Q$ as $\\mathrm{A}_{\\mathrm{H}}$ and output the indices $t_1, t_2,\\cdots, t_Q$.\n\\STATE Sort $t_1, t_2,\\cdots, t_Q$ in an ascent order $t_1'< t_2' <\\cdots< t_Q'$.\n\\STATE $t_0=0$.\n\\FOR{$i=0:Q-1$}\n\\STATE Sort $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ and detect the $(K-1)$ largest as $\\mathrm{A}_{\\mathrm{L}}$ and others as $0$.\n\\ENDFOR\n\\end{algorithmic}\n \\end{algorithm}\n\nAs described, OSD for BDPIM includes two phases. The first involves a sorting algorithm operating on $L$ received signals, whose complexity is $\\mathcal{O}(L\\log L)$. The second involves sorting algorithms operating on $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ for all $i=0,1,\\cdots,Q-1$. The average length of \n $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ is around $L\/Q$ and the sorting operations are conducted for $Q$ times. Therefore,  the complexity of the second detection phase is around $\\mathcal{O}\\left(L\\log(L\/Q)\\right)$. In summary, the total complexity of OSD for BDPIM is also around $\\mathcal{O}(L\\log L)$.  \nThe storage delay is in the same order of OSD for DPIM and dependent on the packet size. For large packet detection, the delay may cause unbearable latency. To address this issue, we propose to use a combination of OTD and OSD. OTD is first used to detect $\\mathrm{A}_{\\mathrm{H}}$ according to an optimal threshold $h\\mathrm{A}_{\\mathrm{T}}'$. If a sample is not detected as $\\mathrm{A}_{\\mathrm{H}}$, we store it in a buffer. If it is, we activate OSD to detect the signal sequence in the buffer. In this way, long sequences are split into small pieces. Specifically, the sequence of a packet is divided into the pieces of average length $L\/Q$.\nTherefore, both the storage delay and  processing time can be reduced  by a factor of $Q$. \n\\section{Main Results on BER Performance}\nThis section presents the main results on the approximate upper bounds of the average BER performance of all schemes, including DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD, where the average BER performance is defined by\n\\begin{equation}\nP_e=\\overline{\\mathsf{BER}}P_p,\n\\end{equation}\nwhere $\\overline{\\mathsf{BER}}$ denotes the expected BER in an erroneous packet reflecting the error propagation level and $P_p$ represents the PER. \n\\subsection{Approximate BER Upper Bounds of DPIM-OTD}\n The PER of OTD can be written as\n\\begin{equation}\\label{Pp1}\nP_{p_1}=1-(1-P_c)^L,\n\\end{equation}\nbased on which and the approximate expression of $\\mathrm{A}_{\\mathrm{T}}$ in (\\ref{eqAt}) we have the following theorem:\n\\begin{theorem} \nThe average BER of DPIM-OTD\nis approximately upper bounded by\n\\begin{equation}\\label{Pe1}\n\\begin{split}\n P_{e_1}^\\mathcal{U}\n\\approx\\frac{2-2(1-P_c)^L-LP_c(1-P_c)^{L-1}}{4},\n\\end{split}\n\\end{equation}\nwhere the chip error probability is approximated as\n\\begin{equation}\\label{Pc}\n\\begin{split}\nP_c&\\approx\\frac{L_s-1}{L_s} \\mathcal{Q}\\left(\\frac{h\\sqrt{\\gamma}}{2}+\\frac{1}{h\\sqrt{\\gamma}}\\ln(L_s-1)\\right)\n\\\\&~~~~~+\\frac{1}{L_s} \\mathcal{Q}\\left(\\frac{h\\sqrt{\\gamma}}{2}-\\frac{1}{h\\sqrt{\\gamma}}\\ln(L_s-1)\\right).\n\\end{split}\n\\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix A.\n\\end{IEEEproof}\n\\emph{Remark and Observation:} From (\\ref{Pe1}) and (\\ref{Pc}), it is observed that $P_{e_1}^\\mathcal{U}$ is a function of $\\gamma$, $L$, $h$ and $L_s$.\nBased on the  following inequalities of derivatives\n\\begin{equation}\n\\frac{\\partial P_{e_1}^\\mathcal{U}}{\\partial P_c}>0,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial P_c}{\\partial\\gamma}<0,\n\\end{equation}\nand the chain rule, we have \n\\begin{equation}\n\\frac{\\partial P_{e_1}^\\mathcal{U}}{\\partial \\gamma}=\\frac{\\partial P_{e_1}^\\mathcal{U}}{\\partial P_c}\\cdot\\frac{\\partial P_c}{\\partial\\gamma}<0.\n\\end{equation}\nThis indicates that $P_{e_1}^\\mathcal{U}$ decreases as $\\gamma$ increases. \nTaking the first derivative of $P_{e_1}^\\mathcal{U}$ with respect to $L$, we obtain\n\\begin{equation}\n\\begin{split}\n\\frac{\\partial P_e^\\mathcal{U}}{\\partial L}\\geq0,\n\\end{split}\n\\end{equation}\nwhich shows that $P_{e_1}^\\mathcal{U}$ increases as $L$ increases. \n\n\nWe demonstrate the simulation result of the average BER performance and the derived approximate upper bound of DPIM-OTD in Fig. \\ref{2a1}. In the simulations, $h=1$, $N_s=100$ and $4$-DPIM with $1$ GI are used. \nThe results are averaged over $10^5$ Monte Carlo simulations. Results show that the derived approximate upper bound matches the simulation result well especially in the low and high SNR regime. \n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a1}\\\\\n \\caption{Approximate BER upper bound of DPIM-OTD where $h=1$, $N_s=100$ and $4$-DPIM with $1$-GI are employed.}\n  \\label{2a1}\n\\end{figure}\n\n\n\n\n\\subsection{Approximate BER Upper Bounds of DPIM-OSD}\nIn use of OSD, chip errors occur in pairs in an erroneous packet. In detail, as long as a \\emph{false alarm error} occurs,  an \\emph{erasure error} occurs, because the number of $\\mathrm{A}$s by OSD is strictly equal to $N_s$. Based on the fact, the PER of DPIM-OSD is indeed the probability of there being at least one pair of chip errors and can be written as\n\\begin{equation}\\label{eqPp2}\nP_{p_2}=\\mathrm{Pr}\\{U_{1:L-N_s}>V_{N_s:N_s}\\},\n\\end{equation}\nwhere $U_{1:L-N_s}$ denotes the $1$st largest of $L-N_s$ received signals when $0$ is transmitted and $V_{N_s:N_s}$ denotes the $N_s$th largest (i.e., the smallest) of $N_s$ received signals when $\\mathrm{A}$ is transmitted.  \nFor any random variables $U$ and $V$, the probability of $U>V$ can be calculated by\n\\begin{equation}\n\\begin{split}\n\\mathrm{Pr}\\{U>V\\}\n&=\\int_{-\\infty}^{+\\infty}[1-F_U(v)]f_V(v) dv,\\\\\n\\end{split}\n\\end{equation}\nwhere $F_U(u)$ is the cumulative distribution function (CDF) of variable $U$ and  $f_V(v)$ is the PDF of variable $V$.\nGiven $U_{k_1:n_1}$ as the $k_1$th largest of $n_1$ real Gaussian variables with mean $\\mu_1$ and variance $\\sigma_n^2\/h^2$,  $V_{k_2:n_2}$ as the $k_2$th largest of $n_2$ real Gaussian variables with mean $\\mu_2$ and variance $\\sigma_n^2\/h^2$, we define an $\\mathcal{OR}$ function to represent $\\mathrm{Pr}\\{U_{k_1:n_1}>V_{k_2:n_2}\\}$ as\n\\setlength{\\arraycolsep}{0.1pt}\n\\begin{equation}\\label{ORF}\n\\begin{split}\n&\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n\\mu_1, & k_1, & n_1 \\\\\n\\mu_2,&k_2, & n_2 \\\\\n\\end{array}\\right.}\\triangleq\\mathrm{Pr}\\{U_{k_1:n_1}>V_{k_2:n_2}\\}\\\\\n& =\\int_{-\\infty}^{+\\infty}[1-F_{U_{k_1:n_1}}(v)]f_{V_{k_2:n_2}}(v) dv.\\\\\n\\end{split}\n\\end{equation}\nAccording to \\cite{Yang2011},  the CDF $F_{U_{k_1:n_1}}(v)$ in (\\ref{ORF}) is given by\n\\begin{equation}\n\\begin{split}\nF_{U_{k_1:n_1}}(v)=\\sum_{k=n_1+1-k_1}^{n_1}\\left(n_1\\atop k\\right)F_U(v)^k(1-F_U(v))^{n_1-k},\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\nF_U(v)=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{hv-h\\mu_1}{\\sqrt{2}\\sigma_n}\\right)\\right].\n\\end{equation}\nThe PDF $f_{V_{k_2:n_2}}(v)$ is written as \n\\begin{equation}\n\\begin{split}\n&f_{V_{k_2:n_2}}(v)=\\\\\n&\\frac{n_2!}{(n_2-k_2)!(k_2-1)!}F_V(v)^{n_2-k_2}[1-F_V(v)]^{k_2-1}f_{V}(v),\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\nf_V(v)=\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2(v-\\mu_2)^2}{2\\sigma_n^2}},\n\\end{equation}\nand \n\\begin{equation}\nF_V(v)=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{hv-h\\mu_2}{\\sqrt{2}\\sigma_n}\\right)\\right].\n\\end{equation}\n\n\n Based on the definition of the $\\mathcal{OR}$ function,\nthe PER of DPIM-OSD can be given by\n\\begin{equation}\\label{Pp2}\nP_{p_2}=\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.},\n\\end{equation}\naccording to which, we derive the following theorem:\n\\begin{theorem}\nThe average BER of DPIM-OSD is approximately upper bounded by\n\\begin{equation}\\label{Pe2a}\n\\begin{split}\n P_{e_2}^\\mathcal{U}\\approx\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0,  & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 2, & L-N_s \\\\\n\\mathrm{A},&N_s-1, & N_s \\\\\n\\end{array}\\right.}\\right),\n\\end{split}\n\\end{equation}\nand the approximation holds when $L$ is large.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix B.\n\\end{IEEEproof}\n\n\\emph{Remark and Observation:} The approximate upper bound ${P_{e_2}^\\mathcal{U}}$ in (\\ref{Pe2a}) is a function of  $h$, $\\mathrm{A}$, $\\sigma_n$, $L$ and $N_s$. It has more theoretical significance than practical significance since the defined $\\mathcal{OR}$ function involves complicated integrals thus offering no clear insights into the system performance. To simplify the analytical result, we  propose an approximation of the $\\mathcal{OR}$ function as\n\\begin{equation}\n\\begin{split}\n\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n\\mu_1,& k_1, & n_1 \\\\\n\\mu_2,&k_2, & n_2 \\\\\n\\end{array}\\right.}&\\approx\\mathrm{Pr}\\{U_{k_1:n_1}>v_{k_2:n_2}\\}\\\\&=1-F_{U_{k_1:n_1}}(v_{k_2:n_2}),\n\\end{split}\n\\end{equation}\nwhere $v_{k_2:n_2}=\\alpha\\mathbb{E}(V_{k_2:n_2})$ and $\\alpha$ is a constant. We use $\\alpha=0.82$ in this paper and the accuracy of the approximation  is investigated by simulations.  Based on the approximation, we have \n\\begin{equation}\n\\begin{split}\n\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0,  & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}&\\approx 1-F_U(v_1)^{L-N_s}\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\n&\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0,  & 2, & L-N_s \\\\\n\\mathrm{A}, &N_s-1, & N_s \\\\\n\\end{array}\\right.}\n\\approx 1-F_{U_{2:L-N_s}}(v_2)\n\\approx 1-F_{U_{2:L-N_s}}(v_1)\n\\\\&=1-F_U(v_1)^{L-N_s}-\n(L-N_s)F_U(v_1)^{L-N_s-1}[1-F_U(v_1)],\n\\end{split}\n\\end{equation}\nwhere $v_{1}=\\alpha\\mathbb{E}(V_{N_s:N_s})$ and $v_2=\\alpha\\mathbb{E}(V_{N_s-1:N_s})$. Based on these approximations and Theorem 2, we deduce a tractable approximate BER upper bound as\n\\begin{equation}\\label{Pe2b}\n\\begin{split}\n \\overline{P_{e_2}^\\mathcal{U}}\\approx&\\frac{1}{2}\\left[1-F_U(v_1)^{L-N_s}\\right]\\\\&-\\frac{1}{3}(L-N_s)F_U(v_1)^{L-N_s-1}[1-F_U(v_1)].\n\\end{split}\n\\end{equation}\n The new approximate upper bound $\\overline{P_{e_2}^\\mathcal{U}}$ is a function of \n$h$, $\\mathrm{A}$, $v_1$, $\\sigma_n$, $L$ and $N_s$. It is noted that $\\mathbb{E}(V_{N_s:N_s})$ in the expression of $v_1$ is the expectation of the least order statistics of $N_s$ Gaussian samples with mean $\\mathrm{A}$ and covariance $\\sigma_n^2\/h^2$, which can be approximately computed by \\cite{Chen1999}\n\\begin{equation}\n\\mathbb{E}(V_{N_s:N_s})=\\mathrm{A}-\\frac{\\sigma_n\\phi^{-1}\\left(0.5264^{1\/N_s}\\right)}{h},\n\\end{equation}\nwhere $\\phi^{-1}(\\cdot)$ is the inverse of the Gaussian CDF \\cite{Chen1999}. $\\overline{P_{e_2}^\\mathcal{U}}$ is much simpler than (\\ref{Pe2a}) and can be regarded as a closed-form expression because it only involves the $\\mathrm{erf}$ function, which can be further approximated by \\cite{Chiani2003}\n\\begin{equation}\n\\mathrm{erf}(v)\\approx1-\\frac{1}{6}e^{-v^2}-\\frac{1}{2}e^{-\\frac{4}{3}v^2}.\n\\end{equation}\n\n\n\n\nUnder the same simulation setups as that in Section III-A, we simulate the BER performance and compute the derived bounds numerically as illustrated in Fig. \\ref{2a2}.  Results demonstrate that the approximate upper bound in (\\ref{Pe2a}) matches the BER performance well, but is too complicated to analyze.  The tractable approximate upper bound in (\\ref{Pe2b}) has a gap to the simulation result, but it enjoys low complexity.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a2}\\\\\n \\caption{Approximate BER upper bounds of DPIM-OSD where $h=1$, $N_s=100$ and $4$-DPIM with $1$-GI are employed.}\n  \\label{2a2}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\setcounter{equation}{42}\n\\begin{equation}\\label{Pe3a}\n\\begin{split}\n P_{e_3}^\\mathcal{U}\n\\approx&\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  2, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q-1, & Q \\\\\n\\end{array}\\right.}\\right)\\left(1-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\\right)\\\\\n&+\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}\\right)\\left(1-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.}\\right)\\\\\n&+\\frac{1}{2}\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.}\\times\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\\label{Pe3b}\n\\begin{split}\n\\overline{P_{e_3}^\\mathcal{U}}\n\\approx&\\frac{1}{2}-\\frac{1}{2}F_U(v_3)^{N_s-Q}F_U(v_4)^{\\lfloor KL_s\\rceil-K}-\\frac{1}{3}(N_s-Q)F_U(v_3)^{N_s-Q-1}[1-F_U(v_3)]F_U(v_4)^{\\lfloor KL_s\\rceil-K}\\\\\n&-\\frac{1}{3}(\\lfloor KL_s\\rceil-K)F_U(v_4)^{\\lfloor KL_s\\rceil-K-1}[1-F_U(v_4)]F_U(v_3)^{N_s-Q}\n\\end{split}\n\\end{equation}\n\\hrule\n\\end{figure*}\n\n\n\n\\subsection{Approximate BER Upper Bounds of BDPIM-OSD}\nIn use of BDPIM-OSD, the error probability in the first OSD phase of detecting $\\mathrm{A}_{\\mathrm{H}}$ can be expressed as \n\\setcounter{equation}{40}\n\\begin{equation}\\label{PLH}\nP_{\\mathrm{LH}}=\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.},\n\\end{equation}\nand that in the second OSD phase of detecting $\\mathrm{A}_{\\mathrm{L}}$ is\n\\begin{equation}\\label{P0L}\nP_{0\\mathrm{L}}=\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}.\n\\end{equation}\nBased on these, we have the following theorem:\n\n\n\n\\begin{theorem}\nThe BER of BDPIM-OSD is \napproximately upper bounded by (\\ref{Pe3a}) when $L$ is large.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix C.\n\\end{IEEEproof}\n\n\\emph{Remark and Observation:} It is observed from (\\ref{Pe3a}) that $ P_{e_3}^\\mathcal{U}$ is affected by various system parameters such as $\\mathrm{A}_\\mathrm{L}$, $\\mathrm{A}_\\mathrm{H}$, $K$ as well as the channel condition $h$ and $\\sigma_n$. To simplify its expression, based on the approximation of the $\\mathcal{OR}$ function used in Section III-B, we obtain a tractable approximate BER upper bound as (\\ref{Pe3b}), where $v_3=\\alpha\\mathbb{E}(V_{Q:Q})$ and $v_4=\\alpha\\mathbb{E}(V_{K-1:K-1})$.\n\n\n\nTo corroborate our studies, we simulate a system using BDPIM with 1 GI and OSD. The parameters are set as: $h=1$, $N_s=100$, $K=10$, $\\mathrm{A}_{\\mathrm{H}}=2.3$ and $\\mathrm{A}_{\\mathrm{L}}=0.86$. The simulation result and the bounds are given in  Fig. \\ref{2a3}. It is observed that the approximate upper bound in (\\ref{Pe3a}) is tight over all SNR regimes and the tractable  approximate upper bound in (\\ref{Pe3b}) is tight only in the low and high SNR regime. From these results, the accuracy in approximating the $\\mathcal{OR}$ function can also be validated. \n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a3}\\\\\n \\caption{Approximate BER upper bounds of BDPIM-OSD where $h=1$, $N_s=100$, $K=10$, $\\mathrm{A}_{\\mathrm{H}}=2.3$, $\\mathrm{A}_{\\mathrm{L}}=0.86$ and $4$-BDPIM with $1$GI are employed.}\n  \\label{2a3}\n\\end{figure}\n\n\n\\subsection{Approximate BER Upper Bounds of BDPIM-OTD-OSD}\n\n\nFor BDPIM-OTD-OSD, the first phase is detecting the barrier chip of amplitude $\\mathrm{A}_{\\mathrm{H}}$ by using a  threshold $h\\mathrm{A}_\\mathrm{T}'$. Based on the similar analysis in Section III-A, $\\mathrm{A}_\\mathrm{T}'$ can be computed by\n\\setcounter{equation}{44}\n\\begin{equation}\\label{eqAt1}\n\\mathrm{A}_{\\mathrm{T}}'=\\frac{\\mathrm{A}_\\mathrm{H}+\\mathrm{A}_\\mathrm{L}}{2}+\\frac{\\sigma_n^2}{h^2(\\mathrm{A}_\\mathrm{H}-\\mathrm{A}_\\mathrm{L})}\\ln(K-1).\n\\end{equation}\nUsing $h\\mathrm{A}_{\\mathrm{T}}'$ for detection, the PEP in the first OTD phase  can be calculated by \n\\begin{equation}\\label{PLHN}\nP_{\\mathrm{LH}}'=1- (1-P_{c}')^{N_s},\n\\end{equation}\nwhere $P_c'$ writes \n\\begin{equation}\\label{Ps}\n\\begin{split}\nP_{c}'=&\\frac{1}{K} \\mathcal{Q}\\left(\\frac{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}{2\\sigma_n}+\\frac{\\sigma_n}{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}\\ln(K-1)\\right)\n\\\\&+\\frac{K-1}{K} \\mathcal{Q}\\left(\\frac{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}{2\\sigma_n}-\\frac{\\sigma_n}{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}\\ln(K-1)\\right).\n\\end{split}\n\\end{equation}\nThe error detection probability in the second OSD phase is the same as that in (\\ref{P0L}). Based on these, we have the following theorem:\n\\begin{theorem}\nThe BER of BDPIM-OTD-OSD is approximately upper bounded by (\\ref{Pe4a}) when $L$ is large.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix D.\n\\end{IEEEproof}\n\\begin{figure*}\n\\setcounter{equation}{47}\n\\begin{equation}\\label{Pe4a}\n\\begin{split}\n P_{e_4}^\\mathcal{U}\n\\approx&\\frac{2-2(1-P_c')^{N_s}-LP_c'(1-P_c')^{N_s-1}}{4}\\left(1-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\\right)\\\\\n&+\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}\\right)(1-P_c')^{N_s}\\\\\n&+\\frac{1}{2}\\left[1-(1-P_c')^{N_s}\\right]\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\\label{Pe4b}\n\\begin{split}\n\\overline{P_{e_4}^\\mathcal{U}}\n\\approx&\\frac{1}{2}-\\frac{1}{2}(1-P_c')^{N_s}F_U(v_4)-\\frac{1}{4}LP_c'(1-P_c')^{N_s-1}F_U(v_4)^{\\lfloor KL_s\\rceil-K}\\\\\n&-\\frac{1}{3}(\\lfloor KL_s\\rceil-K)F_U(v_4)^{\\lfloor KL_s\\rceil-K-1}[1-F_U(v_4)](1-P_c')^{N_s}\n\\end{split}\n\\end{equation}\n\\hrule\n\\end{figure*}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a4}\\\\\n \\caption{Approximate BER upper bounds of BDPIM-OTD-OSD where $h=1$, $N_s=100$, $K=10$, $\\mathrm{A}_{\\mathrm{H}}=2.3$, $\\mathrm{A}_{\\mathrm{L}}=0.86$ and $4$-BDPIM with $1$GI are employed.}\n  \\label{2a4}\n\\end{figure}\n\n\\emph{Remark and Observation:}\nSimilarly, using the approximation of the $\\mathcal{OR}$ function used in Section III-B, we obtain a tractable approximate BER upper bound as (\\ref{Pe4b}).\n Under the same simulation setups as that in Section III-C, we simulate BDPIM-OTD-OSD. The simulation result and the bounds are given in  Fig. \\ref{2a4}. Similarly, it is also observed that the upper bound in (\\ref{Pe4a}) are quite tight over all SNR regimes and that in (\\ref{Pe4a}) is close to the BER performance in the low and high SNR regime.\n\n\n\n\n\\section{Simulation and Discussion}\nThis section presents the simulation results and has four subsections. In the first subsection, we show the superiority of the proposed OSD over OTD and MLSD.  In the second subsection,  we show the superiority of BDPIM in uncoded and coded systems. In the third subsection, we study the impact of the parameters on the performance of the proposed BDPIM. In the last subsection, we investigate the extension to Gamma-Gamma turbulence channels.\n\n\n\\subsection{Superiority of OSD}\nFirstly, we compare the proposed OSD with OTD and MLSD using exhaustive search in terms of BER and time complexity. Because MLSD using exhaustive search has prohibitive high complexity when the sequence is long, we compare these schemes in a system transmitting small packets containing $4$ symbols per packet. In the comparison, $h=1$ and $4$-DPIM with $1$ GI are employed. We run $10^5$ Monte Carlo simulations and evaluate the BER performance as well as the average time complexity in UTs. The BER performance comparison is shown in Fig. \\ref{MLSDa}, from which we observe that OSD exhibits the same performance as MLSD using exhaustive search and considerably outperforms OTD over a wide SNR regime. The average time complexity  is included in Fig. \\ref{MLSDb}, which shows that OSD is much more computationally efficient than MLSD using exhaustive search.\n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig0_a}\\\\\n \\caption{Comparisons among OTD, OSD and MLSD with exhaustive search in terms of BER  where $N_s=4$ and $4$-DPIM with $1$ GI are employed.}\n  \\label{MLSDa}\n\\end{figure}\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig0_b}\\\\\n \\caption{Comparisons among OTD, OSD and MLSD in terms of time complexity, where a UT is the amount of time it takes for a multiplication.}\n  \\label{MLSDb}\n\\end{figure}\n\n\n\\subsection{Superiority of BDPIM}\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2b}\\\\\n \\caption{Comparisons among MDPIM-OTD \\cite{Ghassemlooy2006}, DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD in terms of uncoded BER performance.}\n  \\label{BDPIM}\n\\end{figure}\n\nSecondly, we compare BDPIM with DPIM and MDPIM \\cite{Ghassemlooy2006} using various detectors in uncoded and coded systems. The parameters $h=1$, $N_s=100$, $K=10$, and $4$-ary BDPIM with $1$ GI are employed. In coded systems, an interleaver of length $K\\log M=20$ is used. To show the proposed BDPIM in association with high-rate FEC codes, a rate-$1\/2$ convolution code is adopted and its code generator can be expressed as $[g_0,g_1]^T=[1+d+d^2,1+d^2]$. Additionally, the Viterbi hard-decision decoder is used for decoding. For $\\mathrm{A}_{\\mathrm{H}}$ and $\\mathrm{A}_{\\mathrm{L}}$, we adopt the optimal solution obtained in Section IV-C. The uncoded simulation results are included Fig. \\ref{BDPIM}. It demonstrates that BDPIM-OSD exhibits the best BER performance in all depicted SNR regimes. It slightly outperforms BDPIM-OTD-OSD, which is applicable in large packet cases. Compared with DPIM-OSD, BDPIM-OSD offers a significant gain at the medium SNR regime. The gain shrinks as SNR goes larger. The reason is as follows. The BER performance is affected by two factors, the PER and the expected BER (affected by error propagation) in an erroneous packet. BDPIM uses some power to offer built-in barriers, which slightly increases PER and greatly reduces the expected BER in an erroneous packet. At the medium SNR regime, the second factor affects the system performance more. As SNR increases, the impacts of two factors on BER performance become comparable. Therefore, the gain shrinks as SNR increases. From Fig. \\ref{BDPIM}, we also observe that BDPIM-OSD greatly outperforms DPIM-OTD by $2$ dB at a BER of $10^{-2}$. Additionally, we compare BDPIM with the MDPIM in \\cite{Ghassemlooy2006}, which uses multiple amplitudes to additionally transmit information. Note that as the average symbol duration of MDPIM is shorter than BDPIM (shown in Table II), we simulate MDPIM under the same average power per chip for fairness. It is observed that BDPIM-OSD considerably outperforms MDPIM by more than $4$ dB. This is reasonable because MDPIM targets to improve the transmission rate while BDPIM aims to enhance the system reliability.\n\n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2c}\\\\\n \\caption{Comparisons among MDPIM-OTD \\cite{Ghassemlooy2006}, DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD in terms of coded BER performance.}\n  \\label{codedBDPIM}\n\\end{figure}\n\nThe comparison made in coded systems is demonstrated in Fig. \\ref{codedBDPIM}.  It shows that BDPIM-OSD still achieves the best performance and is slightly better than BDPIM-OTD-OSD. Both of them greatly outperform the others. Note that the gain brought by BDPIM-OSD does not shrink as SNR increases in coded systems compared with DPIM-OSD. On the contrary, the gain goes larger as SNR increases. This is because the length of block error is short in BDPIM by limiting the error propagation which enables the functionality of the rate-1\/2 FEC code in improving BER performance. For DPIM, even though its PER is low in the high SNR regime (shown in uncoded systems), the error bits in an erroneous packet are too many to be corrected. As a result, the gain becomes more significant. Moreover, Fig. \\ref{codedBDPIM} also demonstrates that the gains brought by BDPIM-OSD  are also more considerable than that in Fig. \\ref{BDPIM} compared with DPIM-OTD and MDPIM-OTD.\nTherefore, we conclude that the gain brought by BDPIM is more considerable in coded systems. \n\n\\subsection{Parameter Impact on the Performance of BDPIM}\nGiven the number of symbols contained in a packet $N_s$, the parameters of BDPIM include $K$, $\\mathrm{A}_{\\mathrm{L}}$ and $\\mathrm{A}_{\\mathrm{H}}$.  The parameter optimization problems of BDPIM with different detectors can be expressed as\n\\begin{equation}\\label{Problem}\n\\begin{split}\nK^*,\\mathrm{A}_{\\mathrm{L}}^*, \\mathrm{A}_{\\mathrm{H}}^*=&\\arg\\min_{K,\\mathrm{A}_{\\mathrm{L}},\\mathrm{A}_{\\mathrm{H}}} P_{e_i}^{\\mathcal{U}}(K,\\mathrm{A}_{\\mathrm{L}},\\mathrm{A}_{\\mathrm{H}}), ~i=3,4\\\\\n\\mathrm{subject~to:~}& 0<\\mathrm{A}_{\\mathrm{L}}<\\mathrm{A}_{\\mathrm{H}}\\textrm{~and~} (\\ref{Con1}).\\\\\n\\end{split}\n\\end{equation}\nGiven $K$ and $\\mathrm{A}$, the constraint in (\\ref{Con1}) can be expressed as \n\\begin{equation}\n\\mathrm{A}_{\\mathrm{H}}=K\\mathrm{A}-(K-1)\\mathrm{A}_{\\mathrm{L}}.\n\\end{equation}\nThus, the problem in (\\ref{Problem}) becomes a one-dimensional search problem of finding the optimal $\\mathrm{A}_{\\mathrm{L}}$ from $(0,\\mathrm{A})$. As the first derivative of $P_{e_i}^{\\mathcal{U}}(K,\\mathrm{A}_{\\mathrm{L}},\\mathrm{A}_{\\mathrm{H}}), ~i=3,4$ with respect to $\\mathrm{A}_{\\mathrm{L}}$ has no simple expression, it is difficult to derive the closed-form solution. Therefore, we resort to using numerical search methods such as the bisection search algorithm \\cite{Boyd2004} in this paper. \n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig3a_new}\\\\\n \\caption{SNR thresholds for a BER of $10^{-3}$ versus different $\\mathrm{A}_{\\mathrm{L}}\/\\mathrm{A}$.}\n  \\label{3a_new}\n\\end{figure}\n\n\n\n\nTo show how these parameters affect the performance of BDPIM, we simulate the system with a varying parameter. By setting $\\mathrm{A}_{\\mathrm{L}}\/\\mathrm{A}$ to vary from $0$ to $1$ by a step of $0.01$, $h=1$, $N_s=100$ and $K=10$, we obtain the SNR thresholds for a BER of $10^{-3}$ in coded and uncoded BDPIM systems as shown in Fig. \\ref{3a_new}.   It shows that the optimal value of $\\mathrm{A}_{\\mathrm{L}}$ in coded systems is smaller than that in uncoded systems. That is, the optimal value of $\\mathrm{A}_{\\mathrm{H}}$ in coded systems is larger than that in uncoded systems. It can be explained as follows. The FEC code can correct the errors of the second phase conditioned the first phase is correctly detected. Therefore, the larger $\\mathrm{A}_{\\mathrm{H}}$ should be adopted in coded systems to ensure the correctness of the first detection phase.\nAdditionally, by setting $K$ to vary in $\\{5,10,20,25,50\\}$, $h=1$, $N_s=100$ and $\\mathrm{A}_{\\mathrm{L}}$, $\\mathrm{A}_{\\mathrm{H}}$ as the searched optimal values, we obtain the SNR thresholds for a BER of $10^{-3}$ in coded and uncoded BDPIM systems as shown in Fig. \\ref{3b_new}. It shows that the values of $K$ affect coded systems more than uncoded systems and the optimal ones for all systems are $10$. Specifically, the SNR thresholds vary from $16.5$ dB to $17.3$ dB in uncoded systems as $K$ varies, while the SNR thresholds vary in a much larger range from $14.8$ dB to $16.8$ dB in coded systems as $K$ varies. This is because $K$ affects the length of block error and determines whether the block error can be corrected or not.\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig3b_new}\\\\\n \\caption{SNR thresholds for a BER of $10^{-3}$ versus different $K$.}\n  \\label{3b_new}\n\\end{figure}\n\\subsection{Extension to Gamma-Gamma Turbulence Channels}\nIn this part, we consider the extension of the proposed designs to other channels. The Gamma-Gamma turbulence channels are taken as an example. The ergodic  approximate  upper bound for any given  channel coefficient distribution can be expressed as\n\\begin{equation}\\label{EPe}\n\\widetilde{P_{e_{i}}^{\\mathcal{U}}}=\\int_{0}^{+\\infty}P_{e_{i}}^{\\mathcal{U}}(h)f_{h}{(h)} dh, ~i=1,2,3,4.\n\\end{equation}\nFor Gamma-Gamma turbulence channels, the channel coefficient distribution can be expressed as\\cite{Jaiswal2017}\n\\begin{equation}\\label{pdfH}\nf_{h}{(h)}=\\frac{2(\\lambda\\mu)^{\\frac{\\lambda+\\mu}{2}}}{\\Gamma(\\lambda)\\Gamma(\\mu)}h^{\\frac{\\lambda+\\mu}{2}}K_{\\lambda-\\mu}\\left(\\sqrt{\\lambda\\mu h}\\right)\n\\end{equation}\nwhere $\\Gamma(\\cdot)$ and $K_{\\lambda-\\mu}(\\cdot)$ are the Gamma function and the modified Bessel function, respectively. In (\\ref{pdfH}),  $\\lambda$ and $\\mu$ are the turbulence parameters that characterize the irradiance fluctuations.\n By inserting (\\ref{pdfH}) into (\\ref{EPe}), one can readily extend to the analytical results given in Theorems 1-4 to Gamma-Gamma turbulence channels. \n\nWe simulate the system under weak turbulence condition, where $\\lambda$ and $\\mu$ are set as $11.6$ and $10.1$ \\cite{Jaiswal2017}, respectively. The  results are included in Fig. \\ref{GGBDPIM}. Results show that the proposed BDPIM and OSD can bring about $4$-dB gain compared with the widely adopted DPIM-OTD over a wide SNR regime.\n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig4a}\\\\\n \\caption{Comparisons among MDPIM-OTD \\cite{Ghassemlooy2006}, DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD in terms of BER performance over Gamma-Gamma weak turbulence channel.}\n  \\label{GGBDPIM}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\nThis paper proposed an OSD and a BDPIM to improve the system reliability of DPIM. With  low complexity, OSD achieves the same system performance as the optimal MLSD in terms of BER. Through specific power allocation, BDPIM makes the error propagation bounded and enables the association with high-rate FEC codes, which greatly improves the system reliability. The  approximate BER upper bounds of four schemes including DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD were derived. The approximate upper bounds were shown tight over all SNR regimes, but they are expressed with complicated integrals. To fix this, this paper also provided tractable approximate upper bounds, which are expressed in closed-form expressions. We studied how parameter settings affect the performance by simulations.  We compared all schemes over AWGN channel and over Gamma-Gamma turbulence channels. Result showed that the proposed BDPIM and OSD can bring considerable performance gain and enhance the system reliability significantly.\n\n\n\n\n\\appendices\n\n\n\n\\section{Proof of Theorem 1}\nFor DPIM-OTD, the expected BER in an erroneous packet  can be expressed as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_1=\\beta_{1}\\overline{\\mathsf{BER}}_{1,1}+(1-\\beta_{1})\\overline{\\mathsf{BER}}_{1,o}.\n\\end{equation}\nwhere $\\beta_1$ is the probability of there being a chip error conditioned a packet is wrongly detected, $\\overline{\\mathsf{BER}}_{1,1}$ represents the expected BER in an erroneous packet with a single-chip error and $\\overline{\\mathsf{BER}}_{1,o}$ stands for the expected BER in an erroneous packet with more than one chip error.\nFor either a \\emph{false alarm error} or an \\emph{erasure error} in OTD, the error not only corrupts the bits directly associated with that chip, but also shifts the bits that follow those bits, such that\n\\begin{equation}\\label{BER1}\n\\overline{\\mathsf{BER}}_{1,1}=\\sum_{t=1}^{L}\\frac{N_s-\\mathcal{N}(t)}{2LN_s},\n\\end{equation}\nwhere $t$ is the error position and $\\mathcal{N}(t)$ stands for the number of symbols (i.e., the number of $\\mathrm{A}$s)  between the $1$st and the $t$th chip. Since the average symbol duration is $L_s$, $\\mathcal{N}(t)\\approx t\/L_s$ and $L\\approx N_sL_s$. Substituting these into (\\ref{BER1}), we can get $\\overline{\\mathsf{BER}}_{1,1}\\approx (L-1)\/4L\\approx1\/4$. \nFor the other cases,  $\\overline{\\mathsf{BER}}_{1,o}$ is less than $1\/2$ (the BER of random guess).\nAs to the probability $\\beta_1$, it can be calculated by\n\\begin{equation}\\label{beta1}\n\\begin{split}\n\\beta_1=\\frac{LP_c(1-P_c)^{L-1}}{1-(1-P_c)^L}.\n\\end{split}\n\\end{equation}\nSummarizing above, $\\overline{\\mathsf{BER}}_1\\leq \\frac{2-\\beta_1}{4}$ and $P_{e_1}\\leq \\frac{2-\\beta_1}{4}P_{p_1}$. By substituting $\\beta_1$ in (\\ref{beta1}) and $P_{p_1}$ in (\\ref{Pp1}), Theorem 1 is proved.\n\n\n\n\n\n\n\\section{Proof of Theorem 2}\nAs stated in Section III-B, the chip errors of OSD occur in pairs.\nFor DPIM-OSD, the expected BER in an erroneous packet  can be expressed as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_2=\\beta_{2}\\overline{\\mathsf{BER}}_{2,1}+(1-\\beta_{2})\\overline{\\mathsf{BER}}_{2,o}.\n\\end{equation}\nwhere $\\beta_2$ represents the probability of there being a pair of chip errors conditioned a packet is wrongly detected, $\\overline{\\mathsf{BER}}_{2,1}$ denotes the expected BER in an erroneous packet with a pair of chip errors and $\\overline{\\mathsf{BER}}_{2,o}$ is the expected BER in an erroneous packet with more than one pair of chip errors.\nWhen a single pair of chip errors occur in OSD, the errors will corrupt the bits between these two chips and $\\overline{\\mathsf{BER}}_{2,1}$ can be expressed as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_{2,1}=\\sum_{t_1=1}^{L}\\sum_{t_2=t_1+1}^{L}\\frac{\\mathcal{N}(t_2)-\\mathcal{N}(t_1)}{L(L-1)N_s},\n\\end{equation}\nwhere $t_1$ and $t_2$ are the error positions. Based on the approximations $\\mathcal{N}(t_2)\\approx t_2\/L_s$, $\\mathcal{N}(t_1)\\approx t_1\/L_s$ and $L\\approx N_sL_s$, the BER of DPIM-OSD can be obtained as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_{2,1}\\approx \\frac{2L^2-7L+5}{12L^2}\\approx \\frac{1}{6}.\n\\end{equation}\nThis indicates that averagely, one sixth of bits are in error in an erroneous packet for this case. \nFor the other cases, $\\overline{\\mathsf{BER}}_{2,o}$ is less than $1\/2$ (the BER of random guess), i.e., $\\overline{\\mathsf{BER}}_{2,o}\\leq \\frac{1}{2}$.\nAs to the probability $\\beta_2$, it can be expressed as\n\\begin{equation}\\label{beta2}\n\\begin{split}\n\\beta_2&=\\frac{P_{p_2}-\\mathrm{Pr}\\{U_{2:L-N_s}>V_{N_s-1:N_s}\\}}{P_{p_2}}\\\\\n&\\approx\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 2, & L-N_s \\\\\n\\mathrm{A},&N_s-1, & N_s \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}},\n\\end{split}\n\\end{equation}\nbecause $P_{p_2}$ in (\\ref{eqPp2}) is the probability that there are at least one pair of chip errors in an erroneous packet, $\\mathrm{Pr}\\{U_{2:L-N_s}>V_{N_s-1:N_s}\\}$ represents the probability that there are at least two pairs of chip errors in an erroneous packet and the difference between the two terms is the probability of a single pair of chip errors.\nSummarizing above, $\\overline{\\mathsf{BER}}_2\\leq \\frac{(3-2\\beta_2)}{6}$ and $P_{e_2}\\leq \\frac{(3-2\\beta_2)}{6}P_{p_2}$. By substituting $\\beta_2$ in (\\ref{beta2}) and $P_{p_2}$ in (\\ref{Pp2}), Theorem 2 is proved.\n\n\n\n\n\n\n\\section{Proof of Theorem 3}\nThere are 5 events for all error cases of BDPIM-OSD.\n\nEvent 1: The first OSD phase is wrongly detected, the second OSD phase is correctly detected and there are only a single pair of chip errors in the first OSD phase. The probability  of this event is $\\beta_{3,a}P_{\\mathrm{LH}}(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,1}$. According to the similar analysis in Appendix B, $\\overline{\\mathsf{BER}}_{3,1}\\approx1\/6$ and $\\beta_{3,a}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{3,a}=\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, & 1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},&Q, & Q \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, & 2, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},&Q-1, & Q \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, & 1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},&Q, & Q \\\\\n\\end{array}\\right.}}.\\\\\n\\end{split}\n\\end{equation}\n\n\nEvent 2: The first OSD phase is correctly detected, the second OSD phase is wrongly detected and there are only a single pair of of chip errors in the second OSD phase. The probability of this event is $\\beta_{3,b}(1-P_{\\mathrm{LH}})P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,2}$. According to the similar analysis in Appendix B, $\\overline{\\mathsf{BER}}_{3,2}\\approx1\/6$ and $\\beta_{3,b}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{3,b}=\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}},\\\\\n\\end{split}\n\\end{equation}\n\nEvent 3: The first OSD phase is wrongly detected, the second OSD phase is correctly detected and there are more than one pair of chip errors in the first OSD phase. The probability  of this event is $(1-\\beta_{3,a})P_{\\mathrm{LH}}(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,3}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 4: The first OSD phase is correctly detected, the second OSD phase is wrongly detected and there are more than one pair of chip errors in the second OSD phase. The probability  of this event is $(1-\\beta_{3,b})(1-P_{\\mathrm{LH}})P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,4}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 5: Both OSD phases are wrongly detected. The probability  of this event is $P_{\\mathrm{LH}}P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,5}$, which is less than $1\/2$ (the BER of random guess). \n\nSummarizing all above and substituting $P_{\\mathrm{LH}}$ in (\\ref{PLH}) and, $P_{0\\mathrm{L}}$ in (\\ref{P0L}), Theorem 3 can be derived.\n\n\n\\section{Proof of Theorem 4}\nThere are 5 events for all error cases of BDPIM-OTD-OSD.\n\nEvent 1: The first OTD phase is wrongly detected, the second OSD phase is correctly detected and there is only a single chip error in the first OTD phase. The probability  of this event is $\\beta_{4,a}P_{\\mathrm{LH}}'(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,1}$. According to the similar analysis in Appendix A, $\\overline{\\mathsf{BER}}_{4,1}\\approx1\/4$ and $\\beta_{4,a}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{4,a}=\\frac{N_sP_c'(1-P_c')^{N_s-1}}{1-(1-P_c')^N_s}.\n\\end{split}\n\\end{equation}\n\n\nEvent 2: The first OTD phase is correctly detected, the second OSD phase is wrongly detected and there are only a single pair of of chip errors in the second OSD phase. The probability of this event is $\\beta_{4,b}(1-P_{\\mathrm{LH}}')P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,2}$. According to the similar analysis in Appendix B, $\\overline{\\mathsf{BER}}_{4,2}\\approx1\/6$ and $\\beta_{4,b}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{4,b}=\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}},\\\\\n\\end{split}\n\\end{equation}\n\nEvent 3: The first OTD phase is wrongly detected, the second OSD phase is correctly detected and there are more than one chip error in the first OTD phase. The probability  of this event is $(1-\\beta_{4,a})P_{\\mathrm{LH}}'(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,3}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 4: The first OTD phase is correctly detected, the second OSD phase is wrongly detected and there are more than one pair of chip errors in the second OSD phase. The probability  of this event is $(1-\\beta_{4,b})(1-P_{\\mathrm{LH}}')P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,4}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 5: Both OTD and OSD phases are wrongly detected. The probability  of this event is $P_{\\mathrm{LH}}'P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,5}$, which is less than $1\/2$ (the BER of random guess). \n\nSummarizing all above and substituting $P_{\\mathrm{LH}}'$ in (\\ref{PLHN}) and, $P_{0\\mathrm{L}}$ in (\\ref{P0L}), Theorem 4 can be derived.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nProduction of heavy quarks is of substantial and ongoing interest in high energy hadronic collisions. This statement had not changed in nearly 40 years, when the charm and bottom flavoured particles were discovered. The energy scale for the production of charm and bottom quarks is significantly higher than the typical Quantum Chromodynamics (QCD) scale, $\\Lambda_{QCD}\\sim 0.2$ GeV. This gives the value of strong coupling of the order of $\\alpha_S \\sim 0.2 - 0.3$, which is small enough to apply perturbative QCD techniques. At the present level of knowledge and current experimental abilities, heavy flavours are known as one of the best testing grounds of the theory of hard QCD interactions. Theoretical analyses of charm and bottom cross sections in proton-proton interactions provide\nan unique precision tool in this context. Due to their mass, heavy quarks are also belived to be a special probe of the medium created in heavy ion collisions.\nSince heavy quarks are only produced in the initial stage of the heavy ion collisions, heavy quark distributions from proton-proton interactions\nsupply a well defined initial state. Their further propagation through the hot and dense medium probes its interesting characteristics.\n \nMeasurements of charm and bottom cross sections at hadron colliders can be performed in the so-called indirect way.\nThis method is based on measurement of leptons from semileptonic decays of open charm and bottom mesons, which are often called non-photonic. The semileptonic decay modes allow for an indirect measurement of the D and B meson cross sections and by further extrapolation the charm and bottom quark cross sections. The indirect methodologies to measure non-photonic electrons\/muons have been used since the early 1970's \\cite{Appel:1974yu}.\n\nThe decay of hadrons by the weak interaction can be viewed as a process of decay of their constituent quarks. \nThe charm and bottom flavours are not preserved in weak interactions, so their weak decays are possible. Within the semileptonic decays, the allowed (according to electric charge conservation) transitions $b \\to c,u$ and $c \\to s,d$ are involved by the emission of charged $W$ boson, which further creates a lepton and the corresponding antineutrino $W \\to l \\bar{\\nu}_l$. The semileptonic decay widths are proportional to the square of the appropriate elements of the Cabbibo-Kobayashi-Maskawa (CKM) quark-mixing matrix \\cite{Kobayashi:1973fv}, which contains information on the strength of flavour-changing\nweak decays. The decays within the same quark generation are strongly favoured over decays between generations.\n\nMain virtues of the semileptonic modes come from the fact that they have bigger branching fractions than the hadronic $D$ meson decay channels. Moreover, in the former the effects of strong interactions can be isolated and thus they are better accessible\nexperimentally. In addition, direct lepton production through weak and electromagnetic interactions in hadronic collisions is suppressed relative to the strong interaction by twelve and four orders of magnitude, respectively \\cite{Halzen:1984mc}.\n\nHowever, electrons and muons are certainly not rare paticles because they are abundantly produced in light hadron decays.\nThese rather problematic background must be accounted for and eliminated through experimental techniques. If the primary sources of background are well understood and substracted, the remaining events can then be attributed to the heavy flavour signal.\n\nAnother method for experimental investigation of charm and bottom quark production at hadron colliders is the direct procedure\nbased on full reconstruction of all decay products of open charm and bottom mesons, for instance in the \n$D^0 \\to K^- \\pi^+ $, $D^+ \\to K^- \\pi^+ \\pi^+$ or $B^+ \\to J\/\\psi K^+ \\to K^+ \\mu^+ \\mu^-$ channels. The hadronic decay products can be used to built\ninvariant mass distributions, permitting direct observation of $D$ or $B$ meson as a peak in the experimental invariant mass spectrum. Open charm $D$ and $B$ mesons are characterized by rather long lifetimes, of the order of $\\sim 10^{-13}$ and $\\sim 10^{-12}$ seconds, respecitvely. Charm and bottom quarks decay essentially at the collision vertex, while heavy flavour mesons decay from a secondary vertex offset by the boosted lifetime of the paticle. In the direct approach the charm and bottom contributions can be well separated, which is not the case in the indirect method. In the latter case, it can be achieved only within the analysis of lepton-meson (e.g. $e$-$D$) correlations \\cite{Mischke:2008af}, which are easily available e.g. in the $k_t$-factorization approach.\n \nThe STAR and PHENIX collaborations have measured transverse momentum distributions of electrons coming from the semileptonic decays of\nheavy flavoured hadrons in proton-proton scattering at the RHIC energy $\\sqrt{s} = 200$ GeV with lepton transverse momenta up to $10$ GeV in the midrapidity region \\cite{Adare:2006hc,Abelev:2006db}. In addition, the STAR collaboration was able to separate the charm and bottom contributions to the spectra of heavy flavour electrons \\cite{Agakishiev:2011mr}.\nThe PHENIX collaboration has also measured non-photonic dilepton invariant mass spectrum from $0$ to $8$ GeV in proton-proton collisions at $\\sqrt{s}=200$ GeV \\cite{Adare:2008ac}. On the theoretical side, the cross sections for inclusive production of the non-photonic electrons at RHIC have been studied theoretically up to the next-to-leading order pQCD collinear approximation within the FONLL approach in Ref.~\\cite{Cacciari:2005rk}. The first theoretical investigation within the competitive QCD $k_t$-factorization framework was done in Refs.~\\cite{Luszczak:2008je,Maciula:2010yw}, including more exclusive studies of kinematical correlations.\n\nVery recently, the STAR collaboration measured for the first time transverse momentum spectra of $D^*$ and $D^0$ mesons up to $6$ GeV at $\\sqrt{s} = 200$ \\cite{Adamczyk:2012af} and $500$ GeV \\cite{STAR-Dmeson500} in proton-proton collisions. Before, studies of charm production at RHIC through hadronic decay channels were performed only in Cu-Cu collisions \\cite{Baumgart:2008zz} and in proton-proton scattering but for $D^*$ mesons produced in jets \\cite{Abelev:2009aj}.\nUp to now, the new STAR proton-proton data on open charm production were studied only in the context of high energy pA collisions in the Color Glass Condensate framework with the unintegrated gluon densities from the solution of rcBK equation \\cite{Fujii:2013yja}.   \n\nOur aim here is to make first theoretical analysis of the measured hadron-level charm differential cross section within the $k_t$-factorization approach. The open charm meson data allow us to make more direct comparison of the theoretical predictions and RHIC experimental results on heavy flavour production without including additional step related to the semileptonic decays. Recently, the formalism of the $k_t$-factorization approach has been found to give very good description of open charm \\cite{Maciula:2013wg} and bottom \\cite{Jung:2011yt,Karpishkov:2014epa} production rates and kinematical correlations in proton-proton scattering at $\\sqrt{s} = 7$ TeV measured by the ALICE, ATLAS, CMS and LHCb experiments. However, a significant sensitivity of theoretical predictions on the model of unintegrated (transverse momentum dependent) gluon distributions (UGDFs) used in calculations has been also reported.  \n\nTherefore, it is very interesting to make similar study for the STAR experimental data on open heavy flavour production at $\\sqrt{s} = 200$ and $500$ GeV. This         \nmay be a good test of different models of UGDFs in the RHIC kinematical regime where one can probe parton (gluon) distributons at intermediate longitudinal momentum fractions $x_1\/x_2 \\sim 10^{-2} - 10^{-1}$. Here, we wish to pay particular attention on UGDF models favoured by the LHC data and on a new up-to-date parametrizations based on the HERA collider DIS high-precision data. The present study is an important extension of our previous paper, where charm and bottom cross section at RHIC has been considered in the context of semileptonic decays of open heavy mesons \\cite{Luszczak:2008je}.\n \nPrecise predictions for charmed mesons may also shed new light on the theoretical understanding of non-photonic lepton production at RHIC.\nOur previous studies of these processes within the $k_t$-factorization approach were based on rather older models of UGDFs which may be the reason of the reported missing strength in description of the RHIC experimental data. Similar problem was also noticed within the NLO collinear calculations in the FONLL model, where only upper limits of the theoretical predictions are consistent with the relevant STAR and PHENIX data \\cite{Cacciari:2005rk}. Therefore, in the following paper we will also revise theoretical cross sections of the non-photonic lepton production at RHIC wihtin the $k_t$-factorizaton approach, taking as a reference point the results obtained in the analysis of the new hadron-level STAR data. In the following calculation, except of updated UGDFs, we also take into account the effect of transformation of semileptonic decay functions between laboratory ($e^{+}e^{-}$ center-of-mass system) and rest frames of decaying $D$ or $B$ mesons.\n\n\\section{Theoretical formalism}\n\nSeveral different mechanisms play a role in heavy quark hadroproduction. In general, in the framework of QCD there are two types of the $\\mathcal{O}(\\alpha_S^2)$ leading-order (LO) $2 \\to 2$ subprocesses: $q\\bar q \\to Q \\overline Q$ and $gg \\to Q \\overline Q$ \\cite{Combridge:1978kx}, often referred to as heavy quark-antiquark pair creation. The first mechanism, $q\\bar q$-annihilation, is important only near the threshold and at very large invariant masses of $Q\\overline Q$ system or extremely forward rapidities. This contribution is therefore especially important in the case of top quark production, however, for charm and bottom production, starting from RHIC, through Tevatron, up to the LHC it can be safely neglected. At high energies, production of charm and bottom flavours is dominated by the gluon-gluon fusion, which is the starting point of the following analysis. In the case of heavy quark production the $\\mathcal{O}(\\alpha_S^3)$ next-to-leading order (NLO) perturbative contributions have been found to be of special importance (see e.g. \\cite{Beenakker:1990maa}) and have to be included in order to describe the heavy flavour high energy experimental data.\n\nThe QED contributions, with one or two photons initiated reactions, such as $\\gamma g \\to Q \\overline Q$, $g \\gamma \\to Q \\overline Q$, $\\gamma \\gamma \\to Q \\overline Q$ have been carefully studied in Ref.~\\cite{Luszczak:2011uh} together with other sub-leading contributions to production of charm and were found to be negligibly small at high energies.\n\nIn the following, the cross sections for charm and bottom quark production in proton-proton collisions are calculated in the framework of the $k_t$-factorization approach. This framework has been successfully applied for different high energy processes, including heavy quark production (see e.g.~\\cite{Maciula:2013wg} and references therein). According to this approach, the transverse momenta $k_{t}$'s (virtualities) of partons which initiate reaction are taken into account and the sum of transverse momenta of the final $Q$ and $\\overline Q$ no longer\ncancels. Then the LO differential cross section for the $Q \\overline Q$ pair production reads:\n\\begin{eqnarray}\\label{LO_kt-factorization} \n\\frac{d \\sigma(h_1 h_2 \\to Q \\overline Q \\, X)}{d y_1 d y_2 d^2p_{1,t} d^2p_{2,t}} &=& \\sum_{i,j} \\;\n\\int \\frac{d^2 k_{1,t}}{\\pi} \\frac{d^2 k_{2,t}}{\\pi}\n\\frac{1}{16 \\pi^2 (x_1 x_2 s)^2} \\; \\overline{ | {\\cal M}^{off-shell}_{g^* g^* \\to Q \\overline Q} |^2}\n \\\\  \n&& \\times  \\; \\delta^{2} \\left( \\vec{k}_{1,t} + \\vec{k}_{2,t} \n                 - \\vec{p}_{1,t} - \\vec{p}_{2,t} \\right) \\;\n{\\cal F}_g(x_1,k_{1,t}^2) \\; {\\cal F}_g(x_2,k_{2,t}^2) \\; \\nonumber ,   \n\\end{eqnarray}\nwhere ${\\cal F}_g(x_1,k_{1,t}^2)$ and ${\\cal F}_g(x_2,k_{2,t}^2)$\nare the UGDFs for the both colliding hadrons. The extra integration is over transverse momenta of the initial\npartons. Explicit treatment of the transverse part of momenta makes the approach very efficient in studies of correlation observables. The two-dimensional Dirac delta function assures momentum conservation.\nThe unintegrated (transverse momentum dependent) gluon distributions must be evaluated at:\n\\begin{equation}\nx_1 = \\frac{m_{1,t}}{\\sqrt{s}}\\exp( y_1) \n     + \\frac{m_{2,t}}{\\sqrt{s}}\\exp( y_2), \\;\\;\\;\\;\\;\\;\nx_2 = \\frac{m_{1,t}}{\\sqrt{s}}\\exp(-y_1) \n     + \\frac{m_{2,t}}{\\sqrt{s}}\\exp(-y_2), \\nonumber\n\\end{equation}\nwhere $m_{i,t} = \\sqrt{p_{i,t}^2 + m_Q^2}$ is the quark\/antiquark transverse mass. In the case of heavy quark production at RHIC energies, especially in the central rapidity region, one test kinematical regime of $x > 10^{-2}$.  \n\nThe LO matrix element squared for off-shell gluons is taken in the analytic form proposed by Catani, Ciafaloni and Hautmann (CCH) \\cite{Catani:1990eg}. This analytic formula was basically derived within the standard QCD framework and can be adopted to the numerical analyses. It was also checked that the CCH expression is consistent with those presented later in Refs.~\\cite{Collins:1991ty,Ball:2001pq} and in the limit of $k_{1,t}^2 \\to 0$, $k_{2,t}^2 \\to 0$ it converges to the on-shell formula.\n\nThe calculation of higher-order corrections in the $k_t$-factorization is much more complicated than in the case of collinear approximation.\nHowever, the common statement is that actually in the $k_{t}$-factorization approach with LO off-shell matrix elements some part of higher-order corrections is effectively included. This is due to emission of extra gluons encoded\nin the unintegrated gluon densities. More details of the theoretical formalism adopted here can be found in Ref.\\cite{Maciula:2013wg}. \n  \nIn the numerical calculation below we have applied several unintegrated gluon densities which are based on different theoretical assumptions. The Kimber-Martin-Ryskin (KMR) UGDF is derived from a modified DGLAP-BFKL evolution equation \\cite{Kimber:2001sc,Watt:2003mx} and has been found recently to work very well in the case of charm and bottom production at the LHC. A special emphasis here is put on the UGDF models obtained as a solution of CCFM evolution equation. Here we use an older Jung set$B0$ parametrization \\cite{Jung:2004gs} and up-to-date JH2013 distributions \\cite{Hautmann:2013tba} determined from the fits to HERA high-precision DIS measurements. The JH2013 set1 is obtained from the fit to inclusive $F_{2}$ data only while JH2013 set2 is derived from the fit to both $F^{charm}_{2}$ and $F_{2}$ data. The UGDFs based on BFKL and BK equations are not applied in the following analysis since they are dedicated to smaller-$x$ values.    \n\nIn the calculation of charm and bottom quark cross sections the central value of numerical results is obtained with the renormalization and factorization scales $\\mu^2 = \\mu_{R}^{2} = \\mu_{F}^{2} = \\frac{m^{2}_{1,t} + m^{2}_{2,t}}{2}$ and quark mass $m_{c} = 1.5$ and $m_{b} = 4.75$ GeV for charm and bottom, respectively. The uncertainties of the predictions are estimated by changing quark mass by $\\pm 0.25$ GeV and by varying scales by a factor $2$.\nThe gray shaded bands drawn in the following figures represent these both sources of uncertainties summed in quadrature. The MSTW08LO \\cite{Martin:2009iq} collinear parton distribution function (PDF) is used for the calculation of the KMR unintegrated gluon density.\n\nThe transition from quarks and gluons to hadrons, called hadronization or parton fragmentation, can be so far approached only through phenomenological models. In principle, in the case of multi-particle final states the Lund string model \\cite{Andersson:1983ia} and the cluster fragmentation model \\cite{Webber:1983if} are often used. However, the hadronization of heavy quarks in non-Monte-Carlo calculations is usually done with the help of fragmentation functions (FF). The latter are similar objects as the parton distribution functions (PDFs) and provide the probability for finding a hadron produced from a high energy quark or gluon.\n\nConsidering fragmentation of a high energy quark (parton) $q$ with zero transverse momentum $p_t$ into a hadron $q \\to h + X$ one usually assumes that the transition is soft and does not add any transverse momentum. In consequence it is a delta function in transverse momentum.\n\nDefining $D(z)dz$ as the probability for the quark $q$ fragmenting into a hadron $h$ which carries a fraction $z$ of its longitudinal momentum, the spectrum of hadrons is given by\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\delta^{(2)}(\\vec{p}_{t,h}) \\int dz dx D(z) \\frac{d\\sigma_q}{dx}\\delta(x_h - zx) \\;\n\\end{equation}\nor \n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h} = \\int \\frac{dz}{z} D(z) \\frac{d\\sigma_q}{dx}\\Bigg\\vert_{x = x_h\/z} \\;.\n\\end{equation}\n\nThis can be generalized to the fragmentation at finite $p_t$ with intrinsic transverse momentum $\\kappa_t$. Thus the incoming quark has transverse momentum $p_{t,q}$ and the outgoing hadron transverse momentum can be decomposed as $p_{t,h} = \\kappa_t + z p_{t,q}$. Moving back to the situation where $ p_{t,q} = 0$ one can argue that the intrinsic transverse momentum $\\kappa_t$ must be small.\n\nIntroducing the transverse momentum dependent fragmentation probablity $D(z,\\kappa_t)dzd\\kappa_t$ one obtains\nthe hadron spectum as\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz d^2\\kappa_t D(z,\\kappa_t) dx_q d^2 p_{t,q} \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\n\\delta(x_h - zx_q) \\delta^{(2)}(\\vec{p}_{t,h} - \\vec{\\kappa_t} - z\\vec{p}_{t,q}) \\;.\n\\end{equation}\nHowever, one often neglects the intrinsic transverse momentum assuming:\n\\begin{equation}\nD(z,\\kappa_t) = D(z)\\delta^{(2)}(\\vec{\\kappa_t}).\n\\end{equation}\nThen the general formula reads\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz d x_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\n\\delta(x_h - zx_q) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}) \\;,\n\\end{equation}\nor integrating out the $\\delta$-functions\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int \\frac{dz}{z^2} D(z) \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\\Bigg\\vert_{x_q = x_h\/z \\atop p_{t,q} = p_{t,h}\/z } \\;.\n\\end{equation}\nIt is belived that this procedure provides correct implementation of small intrinsic transverse momentum into the splitting. Since the hadron on-shell four momentum is fully specified by $x_h$ and $p_{t,h}$, starting from the above formula one can calculate many different distributions. Especially conversion to the rapidity distributions is very simple because of the trivial jacobian:\n\\begin{equation}\n\\frac{xd\\sigma}{dx d^2 p_t} = \\frac{d\\sigma}{dy d^2 p_t}.\n\\end{equation}\nIt can be further written\n\\begin{eqnarray}\n\\frac{d\\sigma_h}{dy_h d^2 p_{t,h}}\\!&=&\\!\\frac{x_h d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz dx_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\nx_h \\cdot \\delta(x_h - z x_q) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q})\\nonumber \\\\\n&=& \\int dz \\frac{dx_q}{x_q} d^2 p_{t,q} D(z) \\frac{x_q d\\sigma_q}{dx_q d^2 p_{t,q}}\nx_h \\cdot \\delta(x_h - z x_q) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}).\n\\end{eqnarray}\nNeglecting masses, one has\n\\begin{equation}\nx_h = \\frac{p_{t,h}}{\\sqrt{s}} e^{y_h}, \\;\\;\\;\\;\\;\\; x_q = \\frac{p_{t,q}}{\\sqrt{s}} e^{y_q} \\;,\n\\end{equation}\nso that,\n\\begin{eqnarray}\n\\frac{d\\sigma_h}{dy_h d^2 p_{t,h}}\\!&=&\\!\\frac{x_h d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz dy_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\n\\delta(1 - e^{(y_q - y_h)}) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q})\\nonumber \\\\\n&=& \\int dz dy_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\n\\delta(y_q - y_h) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}) \\nonumber \\\\\n&=& \\int \\frac{dz}{z^2} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}} \\Bigg\\vert_{y_q = y_h \\atop p_{t,q} = p_{t,h}\/z} \\;.\n\\end{eqnarray}\nThis way one gets (reproduces) the standard formula for the fragmentation in the case of light hadrons.\n\nWhen going to more general case, taking masses into account and introducing $m_t = \\sqrt{p_t^2 + m^2}$, one gets\n\\begin{equation}\nx_h = \\frac{m_t^h}{\\sqrt{s}} e^{y_h}, \\;\\;\\;\\;\\;\\; x_q = \\frac{m_t^q}{\\sqrt{s}} e^{y_q} \\;,\n\\end{equation}\nor\n\\begin{equation}\ny_h = \\log \\left( \\frac{x_h\\sqrt{s}}{m_t^h} \\right), \\;\\;\\;\\;\\;\\; y_q = \\log \\left( \\frac{x_q\\sqrt{s}}{m_t^q} \\right), \n\\end{equation}\nthen, the $z$-dependent rapidity shift between quark and hadron reads\n\\begin{equation}\n\\delta y = y_q - y_h = \\log \\left( \\frac{x_q m_t^h}{x_h m_t^q} \\right) = \\log \\left( \\frac{m_t^h}{z m_t^q} \\right).\n\\end{equation}\nThe delta function now becomes\n\\begin{eqnarray}\nx_h \\delta(x_h - x_q z) &=& \\delta \\left( 1 - \\frac{z x_q}{x_h} \\right)= \\delta \\left( 1- \\frac{z m_t^q}{m_t^h} e^{(y_q - y_h)} \\right) \\nonumber \\\\\n&=&\n\\delta \\left( 1 - e^{(y^q - \\delta y - y_h)} \\right) = \\delta(y_q - \\delta y - y^h).\n\\end{eqnarray}\nFinally,\n\\begin{eqnarray}\n\\frac{d\\sigma_h}{dy_h d^2 p_{t,h}} &=& \\int dz dy_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\n\\delta(y_h - y_q + \\delta y)\\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}) \\nonumber \\\\\n&=& \\int \\frac{dz}{z^2} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\\Bigg\\vert_{y_q = y_h + \\delta y \\atop p_{t,q} = p_{t,h}\/z} \\;. \n\\end{eqnarray}\nIn turn, using $p_{t,q} = p_{t,h}\/z$, the rapidity shift $\\delta y$ can be rewritten\n\\begin{equation}\n\\delta y = \\frac{1}{2}\\log \\left( \\frac{p^2_{t,h} + m_h^2}{p^2_{t,h} + z^2 m_q^2} \\right).\n\\end{equation}\n\nIt is clear that the rescalling of the transverse momentum is the most important effect. This is because one deals with very steep functions of transverse momenta. From the reason that rapidity spectra are usually flat, or slowly varying, the shift $\\delta y$ is not so important. In fact, it is entirely negligible, if $p^2_{t,h} \\gg m_q^2, m_h^2$. The shift is most important at very small $p^2_{t,h} \\ll m_q^2, m_h^2$, where it becomes\n\\begin{equation}\n\\delta y \\sim \\log\\left( \\frac{m_h}{zm_q}  \\right) \\approx \\log\\left( \\frac{1}{z} \\right). \n\\end{equation}\nIt is worth to notice, that at finite $p_{t,h}$ it should never be really large: small $z$ is damped by the fact that the quark spectrum drops rapidly as a function of $p_{t,q} = p_{t,h}\/z$. However, at $p_{t,h}$ = 0, that suppression causes an effect and the whole integral over $z$ becomes important, with very small $z$ causing large rapidity shifts. Fortunately, for heavy quarks, the fragmentation function is peaked at large $z$ (see e.g. \\cite{Luszczak:2014cxa}). Moreover, one has to remember, that taking into account the small-$z$ region in the fragmentation function is theoretically not warranted, since the standard DGLAP approach to fragmentation breaks down in this region. \n\nTaking all together, according to the above formalism, in the following numerical calculations\nthe inclusive distributions of open charm and bottom hadrons $h =D, B$ are obtained through a convolution of inclusive distributions of heavy quarks\/antiquarks and $Q \\to h$ fragmentation functions:\n\\begin{equation}\n\\frac{d \\sigma(pp \\rightarrow h \\bar{h} X)}{d y_h d^2 p_{t,h}} \\approx\n\\int_0^1 \\frac{dz}{z^2} D_{Q \\to h}(z)\n\\frac{d \\sigma(pp \\rightarrow Q \\overline{Q} X)}{d y_Q d^2 p_{t,Q}}\n\\Bigg\\vert_{y_Q = y_h \\atop p_{t,Q} = p_{t,h}\/z} \\;,\n\\label{Q_to_h}\n\\end{equation}\nwhere $p_{t,Q} = \\frac{p_{t,h}}{z}$ and $z$ is the fraction of\nlongitudinal momentum of heavy quark $Q$ carried by a hadron $h$.\nThe origin why the approximation typical for light hadrons assuming that $y_{Q}$ is\nunchanged in the fragmentation process, i.e. $y_h = y_Q$, is also applied in the case of heavy hadrons was carefully clarified in the previous paragraph and is commonly accepted. \n\nAs a default set in all the following numerical calculations the standard Peterson model of fragmentation function \\cite{Peterson:1982ak} with the parameters $\\varepsilon_{c} = 0.02$ and $\\varepsilon_{b} = 0.001$ is applied. This choice of fragmentation function and parameters is based on our previous theoretical studies of open charm production at the LHC \\cite{Maciula:2013wg}, where detailed analysis of uncertainties related with application of different models of FFs was done. Here, we decided not to repeat all the previously analyzed issues and take into consideration only the most data-favoured scenario\\footnote{This is also consistent with prescription applied in the FONNL framework, where rather harder fragmentation functions are suggested \\cite{Cacciari:2005rk}.}. The main conclusions should not change when moving from LHC to RHIC energies and the uncertainties due to the fragmentation effects may be neglected with respect to those related to the perturbative part of the calculation. \n  \nIn the calculations of the cross sections for $D^{0}$ and $D^{*}$ mesons the fragmentation functions should be normalized to the relevant branching fractions $\\textrm{BR}(c \\to D)$, e.g. from Ref.~\\cite{Lohrmann:2011np}. However, the measured by STAR differential distributions for $D^{0}$ and $D^{*}$ meson are normalized to the parton-level $c\\bar c$ cross section which simply means that the $\\textrm{BR}(c \\to D) = 1$ should be taken in numerical calculations.\n\nTheoretical predictions for production of non-photonic leptons in proton-proton scattering is a three-step process. The whole procedure can be written in the following schematic way:\n\\begin{equation}\n\\frac{d \\sigma(pp \\to l^{\\pm} X )}{d y_e d^2 p_{t,e}} =\n\\frac{d \\sigma(pp \\to Q X)}{d y_Q d^2 p_{t,Q}} \\otimes\nD_{Q \\to h} \\otimes\nf_{h \\to l^{\\pm}} \\; ,\n\\label{whole_procedure}\n\\end{equation}\nwhere the symbol $\\otimes$ denotes a generic convolution. Thus, the cross section for non-photonic leptons is a convolution of the cross section for heavy quarks with fragmentation function $D_{Q \\to h}$ and with semileptonic decay function\n$f_{h \\to l^{\\pm}}$ for heavy mesons.\n\nIn principle, the semileptonic decays can be calculated \\cite{Artuso:2008vf}.\nThe simplest approach to describe the decays of $D$ and $B$ mesons is given by the spectator model \\cite{Artuso:2008vf}, where the QCD effects from the higher-order corrections between heavy $Q$ and light $q$ quarks are neglected. This model works better for bottom quarks since there the mass sufficiently suppresses these corrections. In the case of charm, the QCD effects become more important but they can be also qualitatively modelled.\n\nSince there are many decay channels with different\nnumber of particles the above procedure is not easy and rather labor-intensive. It introduces some model \nuncertainties and requires inclusion of all final state channels explicitly.\n\nAn alternative way to incorporate semileptonic decays into theoretical model is to take relevant experimental input. For example, the CLEO \\cite{Adam:2006nu} and BABAR \\cite{Aubert:2004td} collaborations have measured very precisely the momentum spectrum of electrons\/positrons coming from the decays of $D$ and $B$ mesons, respectively. This is done by producing resonances: $\\Psi(3770)$\nwhich decays into $D$ and $\\bar D$ mesons, and $\\Upsilon(4S)$ which decays into $B$ and $\\bar B$ mesons.\n\nThis less ambitious but more pragmatic approach is based on purely empirical fits to (not absolutely normalized) CLEO and BABAR experimental data points. These electron decay functions should account for the proper\nbranching fractions which are known experimentally (see e.g. \\cite{Beringer:1900zz,Adam:2006nu,Aubert:2004td}).\nThe branching fractions for various species of $D$ mesons are different:\n\\begin{eqnarray}\n&&\\mathrm{BR}(D^+\\to~e^+ \\nu_e X)=16.13\\pm 0.20(\\mathrm{stat.})\\pm 0.33(\\mathrm{syst.})\\%,\\nonumber \\\\\n&&\\mathrm{BR}(D^0\\to~e^+ \\nu_e X)=6.46\\pm 0.17(\\mathrm{stat.})\\pm 0.13(\\mathrm{syst.})\\%. \n\\end{eqnarray}\nBecause the shapes of positron spectra for both decays are \nidentical within error bars we can take the average value of BR($D\\!\\to\\!e \\, \\nu_e \\, X) \\approx 10 \\%$ and \nsimplify the calculation. In turn, the branching fraction of open bottom is found to be:\n\\begin{equation}\n\\mathrm{BR}(B\\to e \\, \\nu_e \\, X) = 10.36 \\pm 0.06(\\mathrm{stat.}) \\pm 0.23(\\mathrm{syst.}) \\% .\n\\end{equation} \n\nAfter renormalizing to experimental branching fractions the adjusted decay functions are then use to generate leptons \nin the rest frame of the decaying $D$ and $B$ mesons in a Monte Carlo approach. This way one can avoid all uncertainties associated with explicit calculations of semileptonic decays of mesons.\n\nIn both cases the heavy mesons are almost at rest, so in practice one measures the meson rest frame\ndistributions of electrons\/positrons. With this assumption one can find a good fit to the CLEO and BABAR data with:\n\\begin{eqnarray}\nf^{Lab}_{CLEO}(p) &=& 12.55 (p+0.02)^{2.55} (0.98-p)^{2.75}\\;, \\label{CLEO_fit_function} \\\\\nf^{Lab}_{BABAR}(p) &=& \\left( 126.16+14293.09 \\exp(-2.24 \\log(2.51-0.97 p)^2 \\right) \\nonumber \\\\\n&&\\times \\left( -41.79+42.78 \\exp(-0.5(|p-1.27|) \/1.8 )^{8.78} \\right )\\;.\n \\label{BABAR_fit_function}\n\\end{eqnarray}\nIn these purely empirical parametrizations $p$ must be taken in GeV.\n\nIn order to take into account the small effect of the non-zero motion of the $D$ mesons in \nthe case of the CLEO experiment and of the $B$ mesons in the case of the BABAR experiment, the above parametrizations of the fits in the laboratory frames have to be modified. The improvement can be achieved by including the boost of the new modified rest frame functions to the CLEO and BABAR laboratory frames. The quality of fits from Eqs.~(\\ref{CLEO_fit_function}) and (\\ref{BABAR_fit_function}) will be reproduced. The $D$ and $B$ rest frame decay functions take the following form:\n\\begin{eqnarray}\nf^{Rest}_{CLEO}(p) &=& 12.7 (p+0.047)^{2.72} (0.9-p)^{2.21}\\;, \\label{CLEO_fit_boost}  \\\\\nf^{Rest}_{BABAR}(p) &=& \\left( 126.16+14511.2 \\exp(-1.93 \\log(2.7-1.0825 p)^2 \\right) \\nonumber \\\\\n&&\\times \\left( -41.79+42.78 \\exp(-0.5(|p-1.27|) \/1.8 )^{8.78} \\right )\\;.\n \\label{BABAR_fit_boost}\n\\end{eqnarray}\n\n\\begin{figure}[tb]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{CLEO_decay.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{BABAR_decay.eps}}\n\\end{minipage}\n   \\caption{\n\\small Fits to the CLEO (left) and BABAR (right) data. The solid lines correspond to the parametrizations in the laboratory frames and the dashed lines to the meson rest frames, which represent incorporation of effects related to the non-zero motion of decaying mesons.}\n \\label{fig:p-decay-1}\n\\end{figure}\n\nBoth, laboratory and rest frame parametrizations of the semileptonic decay functions for $D$ and $B$ mesons are drawn in Fig.~\\ref{fig:p-decay-1} together with the CLEO (left panel) and BABAR (right panel) experimental data. Some small differences between the different parametrizations appear only at larger values of electron momentum. The influence of this effect on differential cross sections of non-photonic leptons is expected to be negligible and will be shown when presenting numerical results. Our analytical formulas for the rest frame decay functions only slightly differ from those obtained in Ref.~\\cite{Bolzoni:2012kx}.\n\nThe theoretical model for non-photonic lepton production in hadronic reactions described here has been recently found to give a very good description of the experimental data collected with the ALICE detector at the LHC \\cite{Abelev:2014hla,Abelev:2014gla}.\n\n\\section{Numerical results}\n\nThe total cross section for charm production in $pp$ scattering extracted from the STAR measurement of $D^{0}$ and $D^{*}$ mesons at $\\sqrt{s} = 200$ GeV is $\\sigma_{c\\bar{c}} = 797\\pm210^{208}_{295}$ $\\mu$b. Corresponding calculated total cross sections is $\\sigma_{c\\bar{c}}^{\\mathrm{setB}0} = 541$ with the CCFM Jung set$B0$ UGDF. The calculated value is consistent with the measured value taking into account large experimental uncertainties. The total cross section at $\\sqrt{s} = 500$ GeV using the same UGDF is predicted to be $\\sigma_{c\\bar{c}}^{\\mathrm{setB}0} = 1006$ $\\mu$b.\n\nThe STAR collaboration also carried out a measurement of charm production cross sections at midrapidity $\\frac{d\\sigma_{c\\bar{c}}}{dy}|_{y = 0}$. Comparison of the experimental results and the theoretical ones is presented in Table \\ref{tableRHIC-Dmesons}.  \nHere again the CCFM Jung set$B0$ UGDF give results consistent with the measurements.\n\n\\begin{table}[tb]%\n\\caption{The midrapidity $\\frac{d\\sigma_{c\\bar{c}}}{dy}|_{y = 0}$ cross section for charm production in proton-proton scattering at $\\sqrt{s} = 200$ and $500$ GeV: the STAR results versus results of calculation with the Jung set$B0$ UGDF.}\n\\newcolumntype{Z}{>{\\centering\\arraybackslash}X}\n\\label{tableRHIC-Dmesons}\n\\centering %\n\\begin{tabularx}{1.0\\textwidth}{ZZZZ}\n\\toprule[0.1em] %\n\\\\[-3.4ex] \n\\multicolumn{2}{c}{Experiment: STAR, $\\frac{d\\sigma_{c\\bar{c}}}{dy}|_{y = 0}$}  & Theory: Jung set$B0$ \\\\ [+0.4ex]\n   \n\\bottomrule[0.1em]\n     & \\\\ [-3.0ex]\n $\\sqrt{s} = 200$ GeV  &  $170\\pm45^{+38}_{-59}$ $\\mu$b  & $130$ $\\mu$b \\\\ [+0.4ex]\n $\\sqrt{s} = 500$ GeV  &  $217\\pm86\\pm73$ $\\mu$b         & $191$ $\\mu$b \\\\ [+1.4ex]\n\n\\bottomrule[0.1em]\n\n\\end{tabularx}\n\n\\end{table}\n\n\\subsection{Open charm mesons}\n\nFigure \\ref{fig:pt-star-D1} presents transverse momentum distributions of charmed mesons in proton-proton collisions at $\\sqrt{s} = 200$ GeV for $|y_{D}| < 0.5$ (left panel) and at $\\sqrt{s} = 500$ GeV for $|y_{D}| < 1.0$ (right panel) together with the STAR data points. Both experimental data and theoretical results are normalized to the $c\\bar c$ parton-level cross section dividing by the $c \\to D^*, D^0$ fragmentation fractions. Results of numerical calculations obtained with the KMR (dotted line), the JH2013 set1 (long-dashed-dotted line), set2 (long-dashed line) and the Jung setB$0$ (solid line) UGDFs are shown. At both energies very good description of the experimental data is obtained with the Jung setB$0$ UGDF. Results calculated with the JH2013 set1 UGDF overestimate the data points in the whole range of measured $p_t$'s.\nThe JH2013 set2 and the KMR UGDFs significantly underestimate the distribution measured at $\\sqrt{s} = 200$ GeV and the situation is only slightly improved in the case of the higher energy where the results of both of them reach the two last data points at larger transverse momenta.    \n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_updfs_eps02_200.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_updfs_eps02_500.eps}}\n\\end{minipage}\n   \\caption{\n\\small The transverse momentum distribution of $D$ mesons normalized to the parton-level cross section at $\\sqrt{s} = 200$ (left) and $500$ GeV (right). The STAR experimental data points are compared to the results of the $k_{t}$-factorization calculations with the KMR (dotted line), the JH2013 set1 (long-dashed-dotted line), set2 (long-dashed line) and the Jung setB$0$ (solid line) UGDFs. }\n \\label{fig:pt-star-D1}\n\\end{figure}\n\nMain uncertainties of the theoretical calculations coming from the perturbative part are shown in Fig.~\\ref{fig:pt-D-star-uncert}.\nThe shaded bands represent uncertainties of the calculations with the Jung setB$0$ UGDF related to the choice of the factorization and\/or renormalization scales and those due to the charm quark mass. The result from the FONLL approach is also drawn for comparison. The uncertainties are larger at lower transverse momenta, where the effects of quark mass uncertainties are more important, and decrease with increasing $p_t$. The FONLL predictions underestimate the experimental data almost in the whole measured range. Their central value coincides with the lower-limit of the $k_{t}$-factorization predictions with the Jung setB$0$ UGDF.\n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_setB0_uncert_eps02_200.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_setB0_uncert_eps02_500.eps}}\n\\end{minipage}\n   \\caption{\n\\small The uncertainties of the theoretical predictions for Jung setB$0$ UGDF at $\\sqrt{s} = 200$ (left) and $500$ GeV (right). Uncertainties due to the choice of the factorization and\/or renormalization scales and those related to the charm quarks mass are summed in quadrature. In the case of $\\sqrt{s} = 200$ GeV data the results obtained within the FONLL framework are drawn for comparison. Details are specified in the plots.}\n \\label{fig:pt-D-star-uncert}\n\\end{figure}\n\n\n\\subsection{Non-photonic electrons}\n\nA first theoretical investigation of the non-photonic electron production at RHIC within the framework of the $k_t$-factorization was performed in Ref.~\\cite{Luszczak:2008je}. Some missing strength in the description of the measured differential distributions has been reported there, especially in the region of small transverse momenta. In the meantime, the STAR collaboration has published new measurements of non-photonic electrons with separated charm and bottom contributions \\cite{Agakishiev:2011mr}. Therefore, it is very interesting to make a revision of theoretical cross sections taking into account the new\nresults from the hadron-level analysis of charm production at RHIC. Here, our previous results from Ref.~\\cite{Luszczak:2008je} are updated by the application of the Jung setB$0$ UGDF which as was shown in the previous subsection works very well for the STAR data on open charm meson production.\n\nThe experimental cross sections for charm and bottom flavoured electrons measured at RHIC are collected in Table \\ref{table-NPE-RHIC}. The values calculated with the Jung setB$0$ UGDF are consistent with the measurements.\n\n\\begin{table}[!h]\\small%\n\\caption{The experimental and theoretical cross sections $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$ for non-photonic electron production in proton-proton scattering at $\\sqrt{s} = 200$ GeV.}\n\\newcolumntype{Z}{>{\\centering\\arraybackslash}X}\n\\label{table-NPE-RHIC}\n\\centering %\n\\begin{tabularx}{1.0\\textwidth}{ZZZZ}\n\\toprule[0.1em] %\n\\\\[-2.4ex] \n                  \\multicolumn{2}{c}{Experiment}  & Theory,  Jung setB0 \\\\ [+0.4ex]\n                  \n\\bottomrule[0.1em]\n     & \\\\ [-2.0ex]\n STAR, $pp\\rightarrow c\\bar{c} X \\rightarrow e X'$                               &  \\multirow{2}*{$6.2\\pm0.7\\pm1.5$ nb}         & \\multirow{2}*{$7.55$ nb } \\\\ [+0.4ex]\n        $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$,  $3<p_{\\perp}<10$ GeV          &                                                 &       \\\\ [+1.4ex]\n        \n  STAR,  $pp\\rightarrow b\\bar{b} X \\rightarrow e X'$                               &  \\multirow{2}*{$4.0\\pm0.5\\pm1.1$ nb}   & \\multirow{2}*{$6.65$ nb}    \\\\ [+0.4ex]       \n   $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$,  $3<p_{\\perp}<10$ GeV                  &                                     &       \\\\ [+1.4ex] \n\n PHENIX, $pp\\rightarrow c\\bar{c} X \\rightarrow e X'$                               &  \\multirow{2}*{$5.95\\pm0.59\\pm2.0$ $\\mu$b}         & \\multirow{2}*{$5.09$ $\\mu$b } \\\\ [+0.4ex]\n        $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$,  $p_{\\perp}>0.4$ GeV          &                                                 &       \\\\ [+1.4ex]\n\n\\bottomrule[0.1em]\n\n\\end{tabularx}\n\n\\end{table}\n\nFigure~\\ref{fig:pt-rhic-npe-1} shows the transverse momentum distributions of electrons from semileptonic decays of charm flavoured hadrons $H_c$ (left panel) and from bottom hadrons $H_b$ (right panel) measured by STAR. The experimental data is compared to the numerical results for the Jung setB$0$ UGDF. The theoretical uncertainties coming from the perturbative part of calculations are also shown for completeness. The rest frame semileptonic decay functions from Eqs.~(\\ref{CLEO_fit_boost}) and (\\ref{BABAR_fit_boost}) are used. It is also assumed that the charm and bottom baryons decay semileptonically in the same way as $D$ and $B$ mesons, and therefore baryonic contributions may be effectively included by treating the baryons as mesons and taking BR$(c \\to D$; $b \\to B) = 1$ . The numerical results very well descibe the experimental data. The central value of the Jung setB$0$ UGDF give distributions that are sligthly above the predictions of the FONLL central value, especially in the small-$p_t$ region. In this case also the JH2013 set1 UGDF reasonably describes the data points taking into account experimental uncertainties.\nAs in the case of open charm data, the lines that corespond to the KMR and JH2013 set2 UGDFs lie much below the measured lepton distributions for both, charm and bottom components.    \n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_charm.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_bottom.eps}}\n\\end{minipage}\n   \\caption{\n\\small Transverse momentum distributions for electrons from semileptonic decays of charm (left) and bottom hadrons (right) measured in $pp$ scattering at $\\sqrt{s} = 200$ GeV. The STAR experimental data are compared to the $k_t$-factorization theoretical predictions obtained with different UGDFs as well as to the FONLL results. Theoretical uncertainties due to quark mass and scales variation are also shown. Further details are specified in the figures. }\n \\label{fig:pt-rhic-npe-1}\n\\end{figure}\n\n\n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_sum.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_sum_uncert.eps}}\n\\end{minipage}\n   \\caption{\n\\small Transverse momentum distributions of electrons coming from both charm and bottom hadrons summed together $H_{c+b}$, measured in $pp$-scattering at $\\sqrt{s} = 200$ GeV. The STAR and PHENIX experimental data are compared to the theoretical predictions obtained with the Jung setB$0$ UGDF. Separated charm and bottom contributions (left) and theoretical uncertainties due to quark mass and scales variation (right) are also shown. The FONLL predictions are drawn for comparison. Further details are specified in the figures. }\n \\label{fig:pt-rhic-npe-2}\n\\end{figure}\n\nIn Fig.~\\ref{fig:pt-rhic-npe-2} the results for summed contributions of charm and bottom flavours are shown.\nHere, the results of calculations are compared to the experimental distributions of heavy flavour electrons, that contain both charm and bottom components.\nThe left panel presents separately charm and bottom contributions as well as the sum of them. In the right panel, the uncertainties of the predictions for the Jung setB$0$ UGDF are drawn together with the lines corresponding to the FONLL results.\nIn contrast to the previous studies in Ref.~\\cite{Luszczak:2008je}, here the $k_t$-factorization results give excellent description of the STAR and PHENIX data, sligthly better than those from the FONLL approach, which are almost identical to the lower for the Jung setB$0$ UGDF. The crossing point between charm and bottom components is found to lies roughly at $p_t = 4$ GeV, which is in agreement with other theoretical investigations (see e.g. Ref.~\\cite{Cacciari:2005rk}).  \n\n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_charm_fdecay.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_bottom_boost.eps}}\n\\end{minipage}\n   \\caption{\n\\small The effect of using two different sets of the semileptonic decays functions, discussed in the present paper, at $\\sqrt{s} = 200$ GeV. The results obtained with the laboratory frame fits from Eqs.~(\\ref{CLEO_fit_function}) and (\\ref{BABAR_fit_function}) are compared to the meson rest frame fits from Eqs.~(\\ref{CLEO_fit_boost}) and~(\\ref{BABAR_fit_boost}). }\n \\label{fig:pt-rhic-npe-ratio-1}\n\\end{figure}\n\nFinally, the effects related to the fitting procedure of the CLEO and BABAR semileptonic data are depicted in Fig.~\\ref{fig:pt-rhic-npe-ratio-1}.\nHere, both sets of the semileptonic decay functions are used. As can be observed from the figure these effects do not really affect the electron spectra at RHIC. The difference is very small and sligthtly increases at higher transverse momenta. In the case of charm flavour (left panel), the application of the boosted decay function leads to a damping of the cross section by about $20\\%$ at $p_t = 10$ GeV. For the bottom flavour (right panel) the corresponding suppresion is only about $5\\%$, which is completely negligible.  \n\n\n\\section{Conclusions}\n\nIn this paper we have dicussed production of charm mesons and non-photonic\nelectrons in proton-proton scattering at the BNL RHIC.\nThe calculation of the charm quark-antiquark pairs has been performed\nin the framework of $k_t$-factorization approach which effectively\nincludes higher-order pQCD corrections.\n\nWe have used different models of unintegrated gluon distributions\nfrom the literature, including those that were applied recently to describe\ncharm data at the LHC and others used to descibe HERA deep-inelastic scattering data.\nThe hadronization of heavy quarks to mesons has been done by means of fragmentation\nfunction technique. The theoretical transverse momentum distributions of charmed\nmesons has been compared with recent experimental data of the STAR collaboration\ncollected at $\\sqrt{s}$ = 200 and 500 GeV.\nWe have carefully quantified uncertainties related to the choice of factorization\/renormalization\nscales as well as quark\/antiquark masses.\nWe have obtained very good agreement with the measured cross sections\nfor the Jung setB$0$ UGDF.\nFurthermore, our results have been compared with the results of the FONLL model.\nThe two approaches give rather similar results.\n\nSemileptonic decays of charmed and bottom mesons have been included\nvia empirical decay functions fitted to the CLEO and BABAR ($e^+ e^-$) data for vector meson decays.\nWe have shown that the inclusion of kinematical boost from meson ($D$ or $B$) \nrest frame to the $e^+ e^-$ center of mass (laboratory) system leads to only \nsmall modifications of the resulting decay functions and as a consequence \nalso for the distributions of non-photonic electrons in proton-proton collisions \nat the RHIC energies.\nConsequently we have obtained a rather good description of the electron\/positron \ntransverse momentum distributions of the STAR collaboration with the same UGDF as \nfor the charmed mesons.\nThis also demonstrates indirectly consistency of the meson and \nnon-photonic electron data.\n\n\n\\vspace{1cm}\n\n{\\bf Acknowledgments}\n\nWe are particularly indebted to Jaro Bielcik for a discussion of experimental aspects and to Wolfgang Sch{\\\"a}fer for a detailed discussion of effects related to quark-to-meson transition. This study was partially supported by the Polish National Science Centre grant DEC-2013\/09\/D\/ST2\/03724.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe hard--exclusive photo electroproduction off a nucleon, among a\nclass of hard exclusive but inelastic processes, is generally\nconsidered as the theoretic cleanest process to gain access to the\nso--called generalized parton distributions (GPDs)\n\\cite{MueRobGeyDitHor94,Ji96}. These non--perturbative\ndistributions contain manifold information that cannot be gained\nfrom measurements of deep (semi--)inelastic or elastic and other\nexclusive processes \\cite{BroLep80}. Based on a partonic\ninterpretation, GPDs allow us to reveal the internal structure of\nthe probed hadron, especially of the nucleon, from a new\nperspective, for comprehensive reviews see Ref.\\\n\\cite{Die03aBelRad05}.\n\nWe recall that the process in question has two interfering\nsubprocesses, namely, the Bethe--Heitler brems\\-strahlung and\ndeeply virtual Compton scattering (DVCS). The former is known in\nterms of the electromagnetic form factors and so the latter can be\nmeasured in several asymmetries, see e.g., Ref.\\ \\cite{DVCS}, that\nappear in leading order of the expansion with respect to the\ninverse photon virtuality, i.e., single and double spin and charge\nasymmetries\n\\cite{DieGouPirRal97,BelMueNieSch00,BelMueKir01,KirMue03}. Relying\non the validity of the operator product expansion (OPE) of two\nelectromagnetic currents, the DVCS amplitude factorizes to leading\npower, i.e., twist--two, in short-- and long--distance physics, where\nthe former and latter are incorporated in (resummed)\nWilson--coefficients and GPDs, respectively\n\\cite{MueRobGeyDitHor94}. Equivalently, in the partonic framework,\nthe DVCS amplitude to leading power accuracy is represented as\nconvolution of hard--scattering amplitudes, systematically\ncalculable in powers of the strong coupling, and GPDs. The\nfactorization of collinear singularities was shown to all orders\nin perturbation theory \\cite{ColFre98}.\n\nLet us remind that for experimental accessible scales, the\nperturbative description of elastic exclusive processes is\ncontroversially debated \\cite{IsgLle89a}. With respect to this\nfact one might wonder whether the perturbative GPD framework is\njustified. For the GPD phenomenology it is essential to clarify\nthis issue; unfortunately, this is an interlaced task.  The\nrelative size of perturbative and non--perturbative, i.e., power\nsuppressed contributions,  might serve as criteria for the\nreliability of the perturbative framework. The DVCS\nhard--scattering amplitudes (Wilson--coefficients) were\nperturbatively evaluated up to next--to--leading order (NLO) in the\nstrong coupling \\cite{BelMue97a,ManPilSteVanWei98}. It turned out\nthat these radiative corrections  can be of the order of 30\\% to\n50\\% for fixed target kinematics \\cite{BelMueNieSch99}. Even much\nlarger corrections have been reported for the kinematics of\ncollider experiments \\cite{FreMcD01b}. Mainly, but not only, this\nis due the appearances of gluonic GPDs at NLO, see also discussion\nin Sect.\\ 6.2.3 of Ref.\\ \\cite{BelMueKir01}. The problem arises\nwhether these sizeable corrections are related to the appearance\nof gluons, are induced by an awkward choice of GPD ans\\\"atze, or\nindicate that at experimental accessible scales the perturbative\nregime has not be fully set in.\n\nTo get a deeper insight into this issue,  we study in this letter\nperturbative corrections to the DVCS process beyond NLO accuracy\nwithin a framework that avoids a cumbersome diagrammatic\ncalculation. Based on the conformally covariant OPE (COPE),  the\nperturbative corrections up to  next--to--next--to--leading order\n(NNLO) can be economically evaluated  \\cite{Mue97a,BelMue97a}, see\nalso  review \\cite{BraKorMue03}. Thereby we restrict ourselves to\nthe flavor non--singlet part of the helicity conserved twist--two\namplitude in the parity even sector, which is the dominant\ncontribution in several DVCS observables. In Sect.\\\n\\ref{Sec-GenFor} we outline the  evaluation of this amplitude by\nmeans of the COPE \\cite{Mue97a,BelMue97a,MelMuePas02}. This yields\na divergent series of conformal GPD moments, which is resummed\nwithin a Mellin--Barnes integral. In Sect.\\ \\ref{Sec-NumRes} we\nnumerically study the radiative corrections, mainly due to the\nWilson--coefficients, at NLO and NNLO. Especially, we explore the\nnumerical differences of the NLO corrections in the modified\nminimal subtraction ($\\overline{\\rm MS}$) and conformal\nsubtraction (CS) schemes. We then provide  estimates to NNLO\naccuracy. Finally, in Sect.\\ \\ref{Sec-Con} we conclude.\n\n\n\\section{General formalism}\n\\label{Sec-GenFor}\n\nLet us first recall the standard perturbative QCD framework.\nUsually, one employs the $\\overline{\\rm MS}$ scheme, which is\nbased on dimensional regularization and the removal of the poles\nwith respect to the dimensional parameter $(4-n)\/2$. This scheme\nis used twofold: (i) to define the renormalized strong coupling\nand (ii) for the factorization of collinear singularities or if\none wishes  for the renormalization of (leading twist) composite\noperators\\footnote{To leading twist approximation there is an\none--to--one cross talk between the ultra--violet and collinear\nsingularities in the partonic matrix elements of these\noperators.}, which are labelled by the good quantum number spin.\nIn exclusive reactions such operators with given spin are also be\nbuild within total derivatives. In general operators with the same\nspin will mix under renormalization. To get rid of this mixing\nphenomenon, one changes at some stage of the full calculation in an\nexplicit or implicit manner to a basis of multiplicatively\nrenormalizable operators, i.e., one diagonalizes the evolution\nequation \\cite{EfrRad80}. For vanishing $\\beta(\\alpha_s)$ function\nthis  transformation leads to conformal\noperators \\cite{Mue94}.  Such operators  transform covariantly\nunder the so--called collinear conformal transformation\n$SL(2,\\mathbb R)$ and are members of infinite dimensional\nconformal multiplets (towers) that are labelled by the conformal\nspin. With other words one takes explicitly or implicitly\nadvantages from the underlying conformal symmetry of the classical\nQCD Lagrangian. If the trace anomaly of the energy--momentum\ntensor, proportional to $\\beta(\\alpha_s)$, is absent, conformal\nsymmetry is present in perturbative QCD and its predictive power\ncan be employed at any order.\n\nIn the following we evaluate the scattering amplitude for the DVCS\nprocess, which is expressed in terms of the time ordered product\nof two electromagnetic currents\n\\begin{eqnarray}\n\\label{Def-HadTen} T_{\\mu\\nu} (q, P_1, P_2) = \\frac{i}{e^2} \\int\ndx e^{i x\\cdot q} \\langle P_2, S_2 | T j_\\mu (x\/2) j_\\nu (-x\/2) |\nP_1, S_1 \\rangle,\n\\end{eqnarray}\nwhere $q = (q_1 + q_2)\/2$ ($\\mu$  and $q_2$ refers to the outgoing\nreal photon). The incoming photon has a large virtuality\n$q_1^2=-{\\cal Q}^2$ and requiring that in the limit $-q^2 = Q^2\\to\n\\infty$ the scaling variables\n\\begin{eqnarray}\n\\xi = \\frac{Q^2}{P\\cdot q}\\,,\\qquad \\eta = -\\frac{\\Delta\\cdot\nq}{P\\cdot q}\\,,\\qquad P = P_1 + P_2\\,, \\qquad \\Delta=P_2-P_1\\,.\n\\end{eqnarray}\nand the momentum transfer squared $\\Delta^2$  are fixed, the\ndominant contributions arise from the light--cone singularities of\nthe time ordered product. In this generalized Bjorken limit one\ncan now employ the OPE to evaluate the hadronic tensor in terms of\nthe leading twist--two operators, where for DVCS kinematics\n$\\eta\\simeq \\xi$ and $Q^2 \\simeq {\\cal Q}^2\/2$.\n\nBefore we outline this step let us introduce a parameterization of\nthe hadronic tensor\n\\begin{eqnarray}\n\\label{decom-T} T_{\\mu\\nu} (q,P,\\Delta) &=& - \\tilde{g}_{\\mu\\nu}\n\\frac{q_\\sigma V^\\sigma}{P\\cdot q} - i \\tilde{\\epsilon}_{\\mu \\nu \\rho\n\\sigma} \\frac{q^\\rho A^\\sigma}{P\\cdot q} + \\cdots.\n\\end{eqnarray}\nTo ensure current conservation, the metric and Levi--Civita tensors\nare contracted here with projection operators, for explicit\ndefinitions of $\\tilde g_{\\mu\\nu}$ and $\\tilde\n\\epsilon_{\\mu\\nu\\rho\\sigma}$ see, e.g., Ref.\\\n\\cite{BelMueNieSch00}. The ellipsis indicates terms that are\nfinally power suppressed in the DVCS amplitude or a determined by\nthe gluon transversity GPD, which is not considered here. We note\nthat in the forward limit $\\Delta\\to 0$ the first and second term\non the r.h.s.\\ are expressed by  the deep inelastic structure\nfunctions $F_1$ and $g_1$. In the parity even sector the vector\n\\begin{eqnarray}\n\\label{dec-FF-V} V^{\\sigma} = \\bar U (P_2, S_2) \\left( {\\cal H}\n\\gamma^\\sigma + {\\cal E} \\frac{i\\sigma^{\\sigma\\rho}\n\\Delta_\\rho}{2M} \\right) U (P_1, S_1) + \\cdots\\, ,\n\\end{eqnarray}\nis decomposed in Compton form factors (CFFs) ${\\cal H}$ and ${\\cal\nE}$, similar for the axial--vector $A^{\\sigma}$ in terms of\n$\\widetilde{\\cal H}$ and $\\widetilde{\\cal E}$, where again higher\ntwist contributions are neglected.\n\nNow we are in the position to employ the OPE.  Let us first\nsuppose that the trace anomaly is absent and so conformal symmetry\ncan be employed. Formally, this can be achieved by assuming that a\nhypothetical fixed point exist, i.e., $\\beta(\\alpha_s^\\ast)=0$. We\ncan then use the COPE, which tells us how the total derivatives\nare arranged. Moreover, it can be considered as a partial wave\nexpansion with respect to the conformal spin $j+2$. For the\ntime--ordered product of two electromagnetic currents it reads in\nthe flavor non--singlet sector as \\cite{FerGriGat71,Mue97a}\n\\begin{eqnarray}\n\\label{Def-COPE}\n \\left\\{T j_\\mu (x) j_\\nu (0)\\right\\}^{\\rm NS} &\\!\\!\\! =\\!\\!\\! &\n\\frac{1}{\\pi^2} \\frac{-\\tilde g_{\\mu \\nu}}{(-x^2 + i 0)^2}\n\\sum_{j=1}^\\infty \\left[1- (-1)^j\\right]\n\\frac{\\Gamma(2-\\gamma_j\/2)(2+j+\\gamma_j\/2)}{\n2^{\\gamma_j\/2}\\Gamma(j+1+\\gamma_j\/2)}\n C_j^{\\rm NS}([-x^2 + i 0] \\mu^2)\n\\nonumber\\\\\n&&\\hspace{3cm}\\times (-2 i x_-)^{j+1} \\int_{0}^{1}\\!du\\,\n[u(1-u)]^{j+1+\\gamma_j\/2} {\\cal O}^{\\rm NS}_j(u x_-) + \\cdots\\,,\n\\end{eqnarray}\nwhere $x_- = \\widetilde n \\cdot x $ with $\\widetilde n^2=0$ is the\nprojection of the vector $x$ on the light cone.  The anomalous\ndimensions of the multiplicatively renormalizable operators ${\\cal\nO}^{\\rm NS}_j$ are denoted as $\\gamma_j(\\alpha_s^\\ast)$. The advantage\nof the COPE is that the Wilson--coefficients (electrical charge\nfactors will be omitted)\n\\begin{eqnarray}\n\\label{Def-WilCoeC} C_j^{\\rm NS}(-x^2 \\mu^2) =\n\\left(-\\mu^2 x^2\\right)^{\\gamma_j\/2}\n\\frac{2^{j+1+\\gamma_j\/2}\\Gamma(5\/2+j+\\gamma_j\/2)}{\n\\Gamma(3\/2)\\Gamma(3+j+\\gamma_j\/2)}\\,\nc_j(\\alpha_s^\\ast)\\,,\n\\end{eqnarray}\nare up to the normalization $c_j(\\alpha_s^\\ast)$ known. The conformal\noperators\n\\begin{eqnarray}\n\\label{Def-ConOpe} {\\cal O}^{\\rm NS}_j(u x_-) =\n\\frac{\\Gamma(3\/2)\\Gamma(1+j)}{2^{j} \\Gamma(3\/2+j)}\\,  (i\n\\partial_+)^j \\bar{\\psi}(u x_-) { \\gamma_+}\\, C_j^{3\/2}\\!\\left(\\!\n\\frac{\\!\\stackrel{\\leftrightarrow}{D}_{+}}{\\partial_+}\\!\n\\right)\\psi(u x_-)\\,,\n\\end{eqnarray}\nare defined with a non--standard normalization. Here\n$\\stackrel{\\leftrightarrow}{D}_+ = \\stackrel{\\rightarrow}{D}_+\n-\\stackrel{\\leftarrow}{D}_+$ and $\\partial_+ =\n\\stackrel{\\rightarrow}{D}_+ + \\stackrel{\\leftarrow}{D}_+$ are the\ncovariant and total derivatives, contracted with the light-like\nvector $n$, i.e., $n^2=0$ and $n\\cdot\\widetilde n =1$, $C_j^{3\/2}$\nare the Gegenbauer polynomials of order $j$ with index $3\/2$. The\nnormalization is chosen so that in the forward limit $\\Delta \\to\n0$ the reduced matrix elements of the conformal operators\n\\begin{eqnarray}\n\\label{Def-ConMomVec}\n\\frac{1}{P_+^{j+1}} \\langle P_2, S_2 \\big|  {\\cal O}_j(0) \\big|P_1,\nS_1 \\rangle= \\frac{1}{P_+}\\bar U (P_2, S_2)\\! \\left(\\! H_j(\\eta,\\Delta^2,\\mu^2)\n\\gamma_+ + E_j(\\eta,\\Delta^2,\\mu^2) \\frac{i\\sigma_{+\\nu}\n\\Delta^\\nu}{2M}\\! \\right) \\! U (P_1, S_1)\n\\end{eqnarray}\ncoincide with the Mellin moments of parton densities. Especially,\nfor odd values of $j$ the forward limit of $H_j$ is the sum of the\nquark and anti--quark  density moments, i.e., $\\lim_{\\Delta\\to\n0}H_j=q_j+\\overline{q}_j$. Moreover, in this limit the\n$u$--integration in Eq.\\ (\\ref{Def-COPE}) can be performed and\nleads to the well--known OPE that is used for the deep inelastic\nscattering structure function $F_1$.  Hence, $c_j(\\alpha_s^\\ast)$\nare identified as the Wilson--coefficients of $F_1$, known to NNLO\norder.\n\n\nEmploying the COPE (\\ref{Def-COPE}), we can now achieve the\nfactorization of the CFFs (\\ref{dec-FF-V}) in a straightforward\nmanner. Plugging Eq.\\ (\\ref{dec-FF-V}) into the definition of the\nhadronic tensor (\\ref{Def-HadTen}), performing Fourier transform,\nand form factor decomposition (\\ref{dec-FF-V}), we arrive for the\nDVCS kinematics $\\eta=\\xi$ at \\cite{Mue97a}\n\\begin{eqnarray}\n\\label{Def-ConParDecInt} \\left\\{{{\\cal H}^{\\rm NS} \\atop {\\cal\nE}^{\\rm NS}} \\right\\}\n= \\sum_{j=1}^\\infty \\frac{1-(-1)^j}{2} {\\xi}^{-j-1} C^{\\rm\nNS}_j(\\mu^2\/Q^2) \\left\\{{H_j^{\\rm NS} \\atop E_j^{\\rm NS}} \\right\\}\n(\\eta, \\Delta^2,\\mu^2)\\Big|_{\\eta=\\xi}\\,.\n\\end{eqnarray}\nHere $H_j^{\\rm NS}$ and $E_j^{\\rm NS}$, cf.\\ Eq.\\ (\\ref{Def-ConMomVec}),\nare given by the conformal moments of the corresponding GPDs\n\\begin{eqnarray}\n\\label{Def-ConGPDmom}\n\\left\\{{H_j^{\\rm NS} \\atop E_j^{\\rm NS}} \\right\\} (\\eta,\n\\Delta^2,\\mu^2)= \\frac{\\Gamma(3\/2)\\Gamma(1+j)}{2^{j}\n\\Gamma(3\/2+j)} \\int_{-1}^{1 }\\! dx\\,\\eta^j\nC_j^{3\/2}\\!\\left(\\frac{x}{\\eta}\\right) \\left\\{{H^{\\rm NS} \\atop\nE^{\\rm NS}} \\right\\} (x,\\eta, \\Delta^2,\\mu^2)\\,.\n\\end{eqnarray}\nNote that these expectation values are\nmeasurable on the lattice \\cite{Hagetal03}.\n\n\nUnfortunately, the series (\\ref{Def-ConParDecInt}) is divergent\nfor $|\\xi| < 1$ and must be resummed, e.g., by means of the\nSommerfeld--Watson transformation \\cite{MueSch05}. Alternatively,\nusing the trick that $\\eta$ is not equated to $\\xi$ allows to deal\nwith Eq.\\ (\\ref{Def-ConParDecInt}) in the analogous manner  as it\nis known from deep inelastic scattering. Namely, a dispersion\nrelation allows to express the Mellin moments of the imaginary\npart by the partial waves, appearing in the COPE\n(\\ref{Def-ConParDecInt}), and thus the inverse  Mellin transform\nprovides the imaginary part of the CFFs. The real\npart can be restored from the dispersion relation, too, and\nfinally equating $\\eta=\\xi$ leads to the same representation as\npresented in Ref.\\ \\cite{MueSch05}:\n\\begin{eqnarray}\n\\label{Def-MelBar} \\left\\{{{\\cal H}^{\\rm NS} \\atop {\\cal E}^{\\rm\nNS}} \\right\\}(\\xi,\\Delta^2,\\mu^2)\n=  \\frac{1}{2 i}\\int_{c-i \\infty}^{c+ i \\infty}\\! dj\\, \\xi^{-j-1}\n\\left[i +\\tan\\left(\\frac{\\pi j}{2}\\right) \\right] C_j^{\\rm\nNS}(Q\/\\mu) \\left\\{{H_j^{\\rm NS} \\atop E_j^{\\rm NS}} \\right\\} (\\xi,\n\\Delta^2,\\mu^2)\\,,\n\\end{eqnarray}\nwhere all singularities of the conformal GPD moments and conformal\nmoments lie on the l.h.s.\\ of the integration path $(c<1)$. Note,\nhowever, that the analytic continuation of  the\nWilson--coefficients (\\ref{Def-WilCoeC}) with respect to the\nconformal spin, analogously done as in deep inelastic scattering,\nleads essentially to an exponential  $2^j$ growing with increasing\n$j$. For $\\xi> 1$, this must be weighed down by a suppression\nfactor that comes from the conformal moments\n(\\ref{Def-ConGPDmom}). This is a rather nontrivial requirement on\ntheir analytic continuation. It should be done in such a way that\nthe integral (\\ref{Def-MelBar}) remains unchanged if the\nintegration contour is closed by an infinite arc, surrounding  the\nfirst and forth quadrant.  The residue theorem states then that\nthis Mellin--Barnes integral is for $\\xi> 1$ equivalent to the\nseries (\\ref{Def-ConParDecInt}). We note that the form of the\nintegrand (\\ref{Def-MelBar}) does in fact not rely on conformal\nsymmetry and so it can be used in any scheme within the\ncorresponding Wilson--coefficients.\n\n\n\nIn the $\\overline{\\rm MS}$ scheme the conformal symmetry is not\nmanifestly implemented in the OPE, however, this failure can be\ncured by a finite renormalization \\cite{Mue97a,BelMue97a}. As\nexplained above, in such a CS scheme and for $\\beta=0$ we can\nsimply borrow the Wilson--coefficients $c_j(\\alpha_s)$ and\nanomalous dimensions $\\gamma_j(\\alpha_s)$ from deep inelastic\nscattering. The inclusion of $\\beta$ proportional terms in Eq.\\\n(\\ref{Def-MelBar}) is, as factorization by itself, conventional.\nTwo possibilities have been discussed in Ref.\\ \\cite{MelMuePas02}.\nNamely, in the CS scheme, one might add the $\\beta$ proportional\nterm from the $\\overline{\\rm MS}$ scheme, evaluated to NNLO in\nRef.\\ \\cite{BelSch98}, while in the $\\overline{\\rm CS}$ scheme the\nrunning of the coupling is implemented in the form of the COPE\n(\\ref{Def-COPE}) in such a way that the Wilson--coefficients\nautonomously evolve in the considered order \\cite{MelMuePas02}. In\nthis letter we prefer the latter convention and, moreover, will\nequate the factorization and renormalization scales, i.e,\n$\\mu=\\mu_f=\\mu_r$. After expansion with respect to $\\alpha_s$, we\nwrite, discarding in the following the superscript NS,  the\nWilson--coefficients (\\ref{Def-WilCoeC}) as\n\\begin{eqnarray}\n\\label{Res-WilCoe-Exp-CS}\nC_\n&\\!\\!\\! =\\!\\!\\!  & \\frac{2^{j+1}\n\\Gamma(j+5\/2)}{\\Gamma(3\/2)\\Gamma(j+3)} \\left[{\\cal C}_j^{(0)} +\n\\frac{\\alpha_s(\\mu)}{2\\pi} {\\cal C}_j^{\n(1)}({\\cal Q}\/\\mu)\n+ \\frac{\\alpha^2_s(\\mu)}{(2\\pi)^2} {\\cal C}_j^{\n(2)}({\\cal Q}\/\\mu) + {\\cal O}(\\alpha_s^3) \\right]\\,,\n\\\\\n\\label{Res-WilCoe-CS-NLO}\n{\\cal C}_j^{\n(0)} &\\!\\!\\! =\\!\\!\\! & 1\\,,\\qquad\n {\\cal C}_j^{\n(1)}\n= c_j^{\n(1) }+\n\\frac{\\gamma_j^{\n(0)}}{2} \\left[s^{(1)}_j+\\ln\\frac{\\mu^2}{{\\cal Q}^2}\\right]\\,,\n\\\\\n\\label{Res-WilCoe-CS-NNLO}\n{\\cal C}_j^\n(2)} &\\!\\!\\! =\\!\\!\\!  & c_j^{\n(2)}+\n\\frac{\\gamma_j^\n(1)}}{2} \\left[ s^{(1)}_j +\n\\ln\\frac{\\mu^2}{{\\cal Q}^2}\\right] + \\frac{\\gamma_j^{\n(0)}}{2} \\left[ c_j^{\n(1)} \\left\\{s^{(1)}_j +\n\\ln\\frac{\\mu^2}{{\\cal Q}^2}\\right\\} + \\frac{\\gamma_j^{\n(0)}}{4} \\right.\n\\\\\n&&\n \\times\\left.\\left\\{s^{(2)}_j + 2 s^{(1)}_j \\ln\\frac{\\mu^2}{{\\cal\nQ}^2}+ \\ln^2\\frac{\\mu^2}{{\\cal Q}^2} \\right\\}\\right]\n-\\frac{\\beta_0}{2}\n \\left[ c_j^{\n(1)}+\n\\frac{\\gamma_j^{\n(0)}}{2} s_j^{(1)} + \\frac{\\gamma_j^{\n(0)}}{4} \\ln\\frac{\\mu^2}{{\\cal Q}^2} \\right]\\ln\\frac{\\mu^2}{{\\cal\nQ}^2}\\,,\n \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\gamma_j^\n(0)}&\\!\\!\\!=\\!\\!\\!& C_F \\left(4S_1(j+1)- 3\n-\\frac{2}{(j+1)(j+2)}  \\right)\\,, \\quad C_F= \\frac{4}{3}\\,,\\\n\\\\\nc_j^\n(1)}&\\!\\!\\!=\\!\\!\\!& C_F \\left[S^2_{1}(1 + j) +\n\\frac{3}{2} S_{1}(j + 2) - \\frac{9}{2}  + \\frac{5-2S_{1}(j)}{2(j +\n1)(j + 2)} -   S_{2}(j + 1)\\right]\\,,\n\\\\\ns_j^{(1)}&\\!\\!\\!=\\!\\!\\!& S_1(j+3\/2)-S_1(j+2) + 2 \\ln(2)\\,, \\quad\ns_j^{(2)}= \\left(s_j^{(1)}\\right)^2 -S_2(j+3\/2)+ S_2(j+2) \\,.\n\\end{eqnarray}\nThe analytic continuation of the harmonic sums are defined by\n$S_1(z)= d\\ln \\Gamma(z+1)\/dz +\\gamma_E$ and $S_2(z)= -d^2\\ln\n\\Gamma(z+1)\/dz^2 + \\zeta(2)$, where $\\gamma_E$ is the Euler\nconstant. The first expansion coefficient of $\\beta(g)\/g =\n(\\alpha_s\/4\\pi) \\beta_0 + {\\cal O}(\\alpha_s^2)$ is $\\beta_0 =\n(2\/3) n_f-11$, where $n_f$ is the number of active quarks. The\ntwo--loop quantities $c_j^{(2)}$ and $\\gamma_j^{(1)}$ are lengthy\nand can be obtained from Ref.\\ \\cite{CurFurPet80,ZijNee92}. The\nevolution of the flavor non--singlet (integer) conformal moments in\nthis $\\overline{\\rm CS}$ scheme is  governed by\n\\begin{eqnarray}\n\\label{Def-RGE} \\mu\\frac{d}{d\\mu} \\left\\{{ H_{j}^{\\rm NS}\\atop\nE_{j}^{\\rm NS}}\\right\\}(\\eta, \\Delta^2,\\mu^2) &\\!\\!\\!=\\!\\!\\!& -\\Bigg[\n\\frac{\\alpha_s(\\mu)}{2\\pi} \\gamma_j^{(0)} +\n\\frac{\\alpha_s^2(\\mu)}{(2\\pi)^2} \\gamma_j^{(1)}+\n\\frac{\\alpha_s^3(\\mu)}{(2\\pi)^3} \\gamma_j^{(2)} +{\\cal\nO}(\\alpha_s^4) \\Bigg] \\left\\{{ H_{j}^{\\rm NS}\\atop E_{j}^{\\rm NS}}\\right\\}(\\eta,\n\\Delta^2,\\mu^2)\n\\nonumber\\\\\n&&\\hspace{0.5cm} -\\frac{\\beta_0}{2}\n\\frac{\\alpha_s^3(\\mu)}{(2\\pi)^3}\\sum_{k=0}^{j-2} \\eta^{j-k}\n\\left[\\Delta_{jk}^{\\overline{{\\rm CS}}}+{\\cal O}(\\alpha_s)\n\\right]\\left\\{{ H_{k}^{\\rm NS}\\atop E_{k}^{\\rm NS}}\\right\\}(\\eta, \\Delta^2,\\mu^2)\\,,\n\\end{eqnarray}\nwhere the mixing matrix $\\Delta_{jk}^{\\overline{{\\rm CS}}}$ is not\ncompletely known.\n\n\n\nWe remark that the NLO corrections in the $\\overline{\\rm MS}$\nscheme can be easily evaluated from the conformal moments of the\nhard--scattering amplitude, e.g., given in Ref.\\\n\\cite{BelMueNieSch99}. All integrals, which are needed, are given\nin Appendix C of Ref.\\ \\cite{MelMuePas02} for integer conformal\nspin. The analytic continuation is straightforward and so in the\n$\\overline{\\rm MS}$ scheme Eq.\\ (\\ref{Res-WilCoe-CS-NLO}) is to\nreplace by\n\\begin{eqnarray}\n\\label{Res-WilCoe-MS-NLO} {\\cal C}_j^{ {\\overline{\\rm MS}}\n(1)}&\\!\\!\\!=\\!\\!\\!& C_F \\left[2 S^2_{1}(1 + j)- \\frac{9}{2}  +\n\\frac{5-4S_{1}(j+1)}{2(j + 1)(j + 2)} +\n\\frac{1}{(j+1)^2(j+2)^2}\\right] + \\frac{\\gamma_j^{\n(0)}}{2}\n\\ln\\frac{\\mu^2}{{\\cal Q}^2} \\,.\n\\end{eqnarray}\nIn this scheme also the complete anomalous dimension matrix is\nknown to two--loop accuracy \\cite{Mue94}.\n\n\n\n\n\\section{Numerical results}\n\\label{Sec-NumRes}\n\nIn this Section we numerically study the   radiative corrections\nto the CFF\n\\begin{eqnarray}\n\\label{Def-ComForFacH}\n{\\cal H}^{{\\rm N}^P{\\rm LO}\n=\n\n\n \\!\\sum_{p=0}^P\n \\left(\\!\\frac{\\alpha_s(\\mu)}{2\\pi}\\!\\right)^p\n\\frac{1}{2 i}\\!\\int_{c-i\\infty }^{c+i\\infty }\\!\\!\\!dj\\, \\xi^{-j-1}\n\\frac{2^{j+1}\\Gamma(5\/2+j)}{\\Gamma(3\/2)\\Gamma(3+j)} \\left[\\!\ni+\\tan\\!\\left(\\!\\frac{\\pi j}{2}\\!\\right)\\!\\right]\n {\\cal C}^{(p)}_j\\!\\!\\left(\\!\\frac{{\\cal\nQ}}{\\mu}\\!\\right)\nH_j^{\\rm NS}(\\xi,\\Delta^2,\\mu^2 )\n\\end{eqnarray}\nat NLO $(P=1)$ and NNLO $(P=2)$ accuracy. Certainly, the size of\nthe radiative corrections depend on the GPD distribution itself.\nAs a GPD model assumption, we suppose that the expansion of\n$H_j^{\\rm NS}$ in powers of $\\xi^2$  induces a systematic\nexpansion of the CFF (\\ref{Def-ComForFacH}). A closer look to this\nissue has been given in Sect.\\ 4 of Ref.\\ \\cite{MueSch05}. The GPD\nmoments are generically parameterized as\n\\begin{eqnarray}\n\\label{Def-BuiBlo} H^{\\rm NS}_j(\\Delta^2,\\xi,\\mu_0^2)= F^{\\rm NS}(\\Delta^2)\n\\frac{B(1-\\alpha(\\Delta^2)+j,\\beta+1)}{B(1-\\alpha(\\Delta^2),\\beta+1)}\n+ {\\cal O}(\\xi^2)\\,.\n\\end{eqnarray}\nIn the forward limit, i.e., $\\Delta\\to 0$,  these moments reduce\nto the Mellin moments of the unpolarized parton density. Hence,\nthe parameters $\\alpha$ and $\\beta$ characterize the small and\nlarge $x$ behavior, respectively, i.e., $q(x,\\mu_0)\\propto\nx^{-\\alpha} (1-x)^\\beta$. For non-singlet parton densities the\ngeneric values are $\\alpha(\\Delta^2=0)=1\/2$ and $\\beta=3$.  In the\noff--forward kinematics we consider $\\alpha(\\Delta^2) = \\alpha(0) +\n\\alpha^\\prime(0)\\Delta^2 $ as a linear meson Regge trajectory with\n$\\alpha(0) = 1\/2$ and $\\alpha^\\prime(0) = 0.9\\, \\mbox{\\rm\nGeV}^{-2}$. Note also that the conformal GPD moment with $j=0$\nreduces to the elastic form factor $F^{\\rm NS}(\\Delta^2)$.\nCertainly, a more realistic ansatz of GPD moments is formed by a\nlinear combination of the building blocks (\\ref{Def-BuiBlo}).\n\n\\begin{figure}[t]\n\\begin{center}\n\\mbox{\n\\begin{picture}(450,240)(0,0)\n\\put(175,240){(a)} \\put(10,130){\\insertfig{7.1}{FigNNLO1a}}\n\\put(-10,145){\\rotatebox{90}{$K_{\\rm abs}({\\cal Q}^2= 2.5\\,\n{\\rm GeV}^2)$}}\n \\put(430,240){(b)}\n\\put(260,130){\\insertfig{7.1}{FigNNLO1b}}\n \\put(175,110){(c)}\n\\put(10,0){\\insertfig{7.1}{FigNNLO1c}} \\put(205,-5){$\\xi$}\n\\put(430,110){(d)} \\put(260,0){\\insertfig{7.1}{FigNNLO1d}}\n\\put(-10,15){\\rotatebox{90}{$K_{\\rm arg}({\\cal Q}^2= 2.5\\,\n{\\rm GeV}^2)$}} \\put(455,-5){$\\xi$} \\put(66,107){\\tiny NNLO,\\,\n$\\beta_0=-9$} \\put(66,94){\\tiny NNLO,\\, $\\beta_0=0$}\n\\put(66,82){\\tiny NLO, $\\overline{\\rm CS}$} \\put(66,69){\\tiny NLO, $\\overline{\\rm\nMS}$}\n\\end{picture}\n}\n\\end{center}\n\\caption{ \\label{FigNNLO} The relative radiative corrections,\ndefined in Eq.\\ (\\ref{Def-Rrat}), are plotted versus $\\xi$ for the\nmodulus [(a) and (b)] and phase [(c) and (d)] of the  CFF\n(\\ref{Def-ComForFacH}) with $\\alpha=0.5$ [(a) and (c)] and\n$\\alpha= -0.1$ [(b) and (d)]: NNLO in full (solid) and $\\beta_0=0$\n(dash--dotted) as well as in NLO for the $\\overline{\\rm CS}$ (dashed) and\n$\\overline{\\rm MS}$ (dotted) scheme. We  always set $\\mu=\n{\\cal Q}$ and  $\\alpha_s({\\cal Q}^2= 2.5\\, {\\rm GeV}^2)\/\\pi = 0.1$. }\n\\end{figure}\nLet us first compare the radiative corrections in the\n$\\overline{\\rm MS}$ and $\\overline{\\rm CS}$ scheme to NLO accuracy. Strictly\nspoken, at a given input scale ${\\cal Q}_0$ there is no difference\nbetween both predictions, if the non--perturbative quantities are\ntransformed, too,\nand a consequent expansion in $\\alpha_s$ is performed.  Usually,\nthe GPD moments are taken from some non--perturbative (model)\ncalculation or ansatz, where the matching with the perturbative\nprescriptions has its own uncertainties. So let us take the same\ninput in both schemes and study the relative changes\n\\begin{eqnarray}\n\\label{Def-Rrat} K^P_{\\rm abs}=\\frac{\\left|{\\cal H}^{{\\rm N}^P{\\rm\nLO}}\\right|}{\\left|{\\cal H}^{{\\rm N}^{P-1}{\\rm LO}}\\right|}\\,,\n\\qquad {K}^P_{\\rm arg}= \\frac{{\\rm arg}\\!\\left(\\! {\\cal H}^{{\\rm\nN}^P{\\rm LO}}\\!\\right)}{{\\rm arg}\\!\\left( {\\cal H}^{{\\rm\nN}^{P-1}{\\rm LO}}\\right)}\n\\end{eqnarray}\nto the modulus and phase of the CFF (\\ref{Def-ComForFacH}). These\nquantities do not suffer from large radiative corrections as it is\nartificially the case for the real part of the\namplitude\\footnote{\\label{Foo-DefRadCor}The real part possesses a\nzero in the valence quark region, which position depends on  the\napproximation. Hence, in the vicinity of this zero the relative\nradiative corrections blow up.}. One should bear in mind that\nthese factors are a measure for the necessary reparameterization\nof the GPD when one includes the next order in a given scheme. In\nFig.\\ \\ref{FigNNLO} we depict for the typical kinematics in fixed\ntarget experiments, i.e., $0.05\\lesssim \\xi \\lesssim 0.3 $, the\n$K$ factors to NLO as dashed and dotted lines for the\n$\\overline{\\rm CS}$ and $\\overline{\\rm MS}$ schemes, respectively.\nWe set $\\mu={\\cal Q }$ and independent of the considered\napproximation we choose $\\alpha_s(\\mu_r^2= 2.5\\,{\\rm GeV}^2 ) =\n0.1 \\pi$. From the panels (a) and (b) it can been realized that in\nthe $\\overline{\\rm MS}$ scheme the radiative corrections to the\nmodulus are up to 20\\% and 30\\% for $\\alpha=0.5$ ($\\Delta^2=0$)\nand $\\alpha=-0.1$ ($\\Delta^2=-0.\\overline{6}\\; {\\rm GeV}^2 $),\nrespectively. In the $\\overline{\\rm CS}$ scheme these radiative\ncorrections  are reduced by 30\\%. Note that such a reduction has\nbeen observed in a quite different processes, namely, for the\nphoton--to--pion transition form factor\\footnote{This process is\nalso evaluated within the OPE and the  reduction of NLO\ncorrections has a common origin. Namely, in the $\\overline{\\rm\nCS}$, compared to the $\\overline{\\rm MS}$, scheme the first few\nWilson--coefficients are smaller, while the stronger logarithmic\ngrowing with increasing $j$ is anyway suppressed by the\nnon--perturbative input, see Eqs.\\ (\\ref{Res-WilCoe-CS-NLO}) and\n(\\ref{Res-WilCoe-MS-NLO}).}. The relative radiative corrections to\nthe phase is in both cases of about 15\\% at $\\xi=0.6$ and\ndiminishes with decreasing $\\xi$. These findings qualitatively\nagree with previous ones in which the Radyushkin ansatz for GPDs\nwas used \\cite{BelMueNieSch99}.\n\n\nWe study now the radiative corrections to NNLO accuracy. To\nsimplify their evaluation, we take for $c_j^{(2)}$ a fit, given in\nRef.\\ \\cite{NeeVog99}, rather than the exact expression. For the\n$\\beta_0$ proportional term we have checked that within\n$|c|=|\\Re{\\rm e} j| \\le 1\/2$, see Mellin--Barnes integral\n(\\ref{Def-ComForFacH}), the accuracy is on the level of one per\nmill or better. Outside this region, the deviation can be larger\nand one might use Fortran routines \\cite{BluMoc05}.  For three\nquark flavors $n_f=3$, the same scale settings, and the initial\nvalue of $\\alpha_s$ as specified above for NLO,  we plot in Fig.\\\n\\ref{FigNNLO} the ratios (\\ref{Def-Rrat}) in the $\\overline{\\rm\nCS}$ scheme for $P=2$ as solid line. The $\\xi$--dependence of the\nmodulus $K$--factor, see panels (a) and (b), is rather flat and\nthe modulus decreases on the 5\\% level. The radiative correction\nto the phase is again negligible for smaller values of $\\xi$ and\nincreases now only to 5\\% at $\\xi=0.6$. We note that the $\\beta_0$\ninduced corrections are about twice times larger than the\nremaining ones and are opposite in sign, see dash--dotted line.\nRemarkably, for $\\beta_0=0$ we find then  an opposite behavior of\nthe $K$ factors as in NLO. This arise from a sign alternating\nseries and as for the photon--to--pion transition form factor, this\nmight be  considered as a reminiscence on Sudakov double logs\n\\cite{MusRad97}.\n\n\nLet us finally address the modification of the scale dependence\ndue to the higher order corrections. Note that the analysis about scheme\ndependence in Ref.\\ \\cite{Mue98} suggests that the discrepancy\nbetween the $\\overline{{\\rm CS}}$ and $\\overline{{\\rm MS}}$\nschemes at NLO accuracy are mainly induced by the\nWilson--coefficients while the evolution yields minor differences.\nSo we only consider here the $\\overline{{\\rm CS}}$ scheme and\nanalogously  as in  Eq.\\ (\\ref{Def-Rrat}), we quantify the\nrelative changes of $d{\\cal H}\/d\\ln{\\cal Q}^2$ by the ratios\n\\begin{eqnarray}\\label{Def-Rrat-dot}\n\\dot{K}^P_{\\rm abs}= \\left|\\frac{d {\\cal H}^{\\rm {N}^P{\\rm\nLO}}}{d\\ln{\\cal Q}^2}\\right|\\Bigg\/\\left|\\frac{d {\\cal H}^{\\rm\n{N}^{P-1}{\\rm LO}}}{d\\ln{\\cal Q}^2}\\right|\\,, \\; \\dot{K}^P_{\\rm\narg}=\n  \\left[\\pi +{\\rm arg}\\!\\left(\\!\n  \\frac{d\\, {\\cal H}^{\\rm {N}^P{\\rm LO}}}{d\\ln{\\cal Q}^2}\\!\\right)\n  \\right]\\Bigg\/\n  \\left[\\pi +{\\rm arg}\\!\n  \\left(\\! \\frac{d\\, {\\cal H}^{\\rm {N}^{P-1}{\\rm LO}}}{d\\ln{\\cal\n  Q}^2}\\!\\right)\\right].\n\\end{eqnarray}\nBoth the numerator and denominator in the latter $\\dot{K}$--factor\nare now defined in the interval $[0,2\\pi]$ and so the appearance\nof a zero in the denominator is avoided, cf.\\ footnote\n\\ref{Foo-DefRadCor}. We take the same scale setting and initial\ncondition for the (exact) evolution of $\\alpha_s({\\cal Q})$ as\nabove. The conformal moments (\\ref{Def-BuiBlo}) are evolved in the\n$\\overline{\\rm CS}$ scheme.  The input scale is now $\\mu_0^2= 0.5\\,\n{\\rm GeV}^2$, which is typical for the matching of perturbative\nand non--perturbative QCD. The non--leading logs in the solution of\nthe evolution equation (\\ref{Def-RGE}) are expanded with respect\nto $\\alpha_s$ and are consistently combined with the\nWilson--coefficients (\\ref{Res-WilCoe-Exp-CS}) in the considered\norder. Here the forward anomalous dimensions are taken from Ref.\\\n\\cite{MocVerVog04} and the unknown NNLO mixing term\n$\\Delta_{jk}^{\\overline{{\\rm CS}}}$ in Eq.\\ (\\ref{Def-RGE}) is\nneglected. This mixing can be suppressed at the input scale within\nan appropriate initial condition  and so we expect only a minor\nnumerical effect; see also Ref.\\\nRef.\\ \\cite{Mue98}.\n\\begin{figure}[t]\n\\begin{center}\n\\mbox{\n\\begin{picture}(450,120)(0,0)\n \\put(175,100){(a)}\n\\put(10,0){\\insertfig{7.2}{FigNNLO2a}} \\put(205,-5){$\\xi$}\n\\put(425,100){(b)} \\put(260,0){\\insertfig{7.1}{FigNNLO2b}}\n\\put(-15,20){\\rotatebox{90}{$\\dot{K}^{\\rm abs}({\\cal Q}^2= 4\\,\n{\\rm GeV}^2 ) $}} \\put(235,20){\\rotatebox{90}{$\\dot{K}^{\\rm arg}{\\cal\nQ}^2=4\\,{\\rm GeV}^2 ) $}} \\put(455,-5){$\\xi$} \\put(315,116){\\tiny\nNNLO,\\, $\\alpha=0.5$} \\put(315,104){\\tiny NLO,\\,\\phantom{N}\n$\\alpha=0.5$} \\put(315,92){\\tiny NNLO,\\,\n $\\alpha=-0.1$} \\put(315,81){\\tiny NLO,\\,\\phantom{N}\n$\\alpha=-0.1$}\n\\end{picture}\n}\n\\end{center}\n\\caption{ \\label{Fig-ScaDep} The relative change of scale\ndependence, cf.\\ Eq.\\ (\\ref{Def-Rrat-dot}), in the $\\overline{\\rm\nCS}$ scheme at NLO (dashed, dotted)  and NNLO (solid, dash--dotted)\nversus $\\xi$ is depicted for the modulus (a) and phase (b) of the\nCFF (\\ref{Def-ComForFacH}) with $\\alpha=0.5$ (dashed, solid)  and\n$\\alpha= -0.1$ (dotted, dash--dotted) and ${\\cal Q}^2 =4\\, {\\rm GeV}^2$.\nWe set $\\mu = {\\cal Q}$, $\\alpha_s(\\mu_r^2= 2.5\\, {\\rm GeV}^2 ) \/\\pi =\n0.1$ and took the input (\\ref{Def-BuiBlo}) at the scale ${\\cal\nQ}^2_{0} = 0.5\\, {\\rm GeV}^2 $. }\n\\end{figure}\nThe dashed and dotted lines in Fig.\\ \\ref{Fig-ScaDep} show that in\nNLO the scale dependence changes of about 30\\% to 50\\% for the\nmodulus and up to 20\\% for the phase. The latter $\\dot{K}$--factor\nbecomes close to one for smaller values of $\\xi$. The NNLO\ncorrections, compared to the full NLO result, are milder. We\nobserve (solid and dash--dotted line) a 20\\% or less and a maximal\n10\\% correction for the modulus and phase, respectively.\n\n\n\n\\section{Conclusions}\n\\label{Sec-Con}\n\nIn this letter we have taken the first steps in the full investigation\nof radiative NNLO corrections to the DVCS scattering\namplitude. Thereby, we employed the COPE, which allows an\neconomical treatment of perturbative corrections, and restricted\nourselves to the CFF $\\cal H$ in the flavor non--singlet sector.\nThis study can be immediately adopted for $\\cal E$. The extension\nto the axial vector case, i.e., to $\\widetilde {\\cal H}$ and\n$\\widetilde {\\cal E}$, is straightforward, however, here the anomalous\ndimensions to three loop order cannot be borrowed from the\npolarized DIS results. The conformal  approach can be also\nstraightforwardly extended to the flavor singlet sector, which\nwill be presented somewhere else.\n\nWe relied on GPDs for which the expansion of the conformal moments\nin powers of $\\xi^2$ is meaningful and used in our analysis only\nthe leading term. Whether this assumption is  too restrictive is\nan open problem, which might be resolved by lattice calculations\n\\cite{Hagetal03}. At least there is a warning from perturbative\nQCD. Namely, in the $\\overline{\\rm MS}$ scheme $\\xi^2$--suppressed\ncontributions in the Wilson--coefficients are resummed and lead to\nlarger radiative corrections than in the $\\overline{\\rm CS}$ one.\nLet us summarize our numerical findings for fixed target\nkinematics, i.e., $0.05 \\lesssim\\xi \\lesssim 0.3$:\n\\begin{itemize}\n\\item Compared to\nthe $\\overline{\\rm MS}$ scheme, in the $\\overline{\\rm CS}$ one the\nradiative NLO corrections to the modulus of the CFF are reduced by\n30\\% or so, while for the phase no significant differences appear.\n\n\\item The relative NNLO corrections compared to the NLO ones are\nof the 5\\% level or below, where the $\\beta_0$ proportional\nones dominate and determine the sign.\n\n\\item The change of the  scale dependence due to the NLO corrections\nis rather large and is for the modulus (phase) of the CFF of about\n30\\% to 40\\% (15\\% or smaller). Comparing NNLO to NLO, the\ntwo--loop corrections are reduced to less than 20\\% (10\\%) .\n\\end{itemize}\nLet us add that the\nWilson--coefficients in the vector and axial--vector channel  possess a\nsimilar $j$ dependence. So we might expect that for\n$\\widetilde {\\cal H}$  and $\\widetilde {\\cal E}$ (axial--vector\ncase) comparable radiative corrections appear.\nWe conclude that our findings support the perturbative description\nof the DVCS process at scales which are accessible in the\nkinematics of present fixed target experiments.\n\n{\\ }\n\n\\noindent\n{\\bf Acknowledgement}\n\n\\noindent\nFor discussions I am indebted to A.\\ Manashov, B.~Meli{\\' c}, K.~Kumeri{\\v c}ki,\nK.~Passek--Kumeri{\\v c}ki,  and A.\\ Sch\\\"afer. This project has been\nsupported by the Deutsche Forschungsgemeinschaft and the\nU.S. National Science Foundation under grant no. PHY--0456520.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nModeling materials in the electronic ground-state requires an accurate description of the Born-Oppenheimer potential energy surface (PES). First-principles models based on density functional theory (DFT) have good predictive power, but high computational complexity. Empirical force fields are more efficient but often less accurate than DFT, and are typically not applicable when chemical bonds change within a system of interest. \nPES models based on machine-learning (ML) offer a promising route to overcome the dilemma of accuracy versus efficiency~\\cite{behler2007generalized,bartok2010gaussian,rupp2012fast,bartok2013representing,thompson2015spectral,shapeev2016moment,schutt2017quantum,smith2017ani,chmiela2017machine,chmiela2018towards,yao2018tensormol,han2018deep,zhang2018deep,zhang2018end,lubbers2018hierarchical,unke2019physnet}. \nThese models combine force field efficiency with first-principles accuracy and have been successful \nin a number of applications that could not be tackled with direct\nfirst principles studies. \nHere, we focus on one such method, the deep potential (DP) scheme~\\cite{zhang2018deep,zhang2018end}, which, upon training with DFT data, can achieve \naccuracy comparable to that of \\textit{ab initio} molecular dynamics (AIMD). On high-performance computer platforms, deep potential molecular dynamics (DPMD) simulations can model millions of atoms on timescales of tens of nanoseconds or even longer~\\cite{lu202186,jia2020pushing}. \nSuccessful applications to date include studies of\nthe signatures of a liquid-liquid phase transition in supercooled water~\\cite{gartner2020signatures}, nucleation of crystalline Si from the melt~\\cite{bonati2018silicon}, \nphase diagrams of Ga~\\cite{niu2020ab} and  water~\\cite{zhang2021phase}, \nstructural motifs in the growth of quasicrystals~\\cite{han2020dynamic}, the reactive uptake of \\chem{N_2O_5} in interfacial processes~\\cite{galib2021reactive},\nreactive processes in combustion~\\cite{zeng2020complex}, \nthe dissociative chemisorption of water at the water-titania interface~\\cite{andrade2020free}, and of the 1D cooperative diffusion in a simple cubic crystal~\\cite{wang2021electronically}.\n\n\nMost ML models, including DP, assume that the PES can be represented by a sum of atomic terms describing the many-body interactions among the atoms within some fixed cutoff distance. This construction is computationally advantageous as its complexity scales linearly with system size. On physical grounds, however, it entails a severe approximation, because electrons and nuclei are charged particles that experience long-range Coulomb forces. In spite of that, local effective interaction models work well in many cases. In metals, this can be attributed to complete electronic screening. In insulators, screening is incomplete, yet, the interatomic interactions are often represented\nsatisfactorily by short-range ML models. These models are trained on DFT data, which include long-range electrostatics, suggesting\nthat some features of the long-range interactions may be mimicked by short-range models. \nWhile this treatment is sufficient in many cases, particularly when dealing with systems that are, on average, homogeneous and neutral at the molecular scale, failures of the short-range models in cluster, interfacial and vapor phase properties are well known~\\cite{yue2021short,niblett2021learning}.\nA quintessential example of a property\nthat cannot be simulated by short-range models is the longitudinal-transverse splitting of the long-wavelength\noptical phonon modes in polar crystals. Long-range electrostatics should be even more important in heterogeneous systems, where \ncharge separation at large scale may be induced by external fields or by chemical potentials. Examples include electrolyte solutions \nin contact with electrodes~\\cite{scalfi2021molecular},\ndipolar surfaces~\\cite{patra2003molecular,wohlert2004range}, protein folding problems~\\cite{zhou2018electrostatic}, and the binding affinity of ligands to proteins~\\cite{rocklin2013calculating}. \n\n\nA possible way of incorporating long-range electrostatics within ML models is to assume \nthat the PES has two contributions. One accounts \nfor short-range interactions and is constructed as in standard ML models. The other accounts for\nlong-range interactions and is approximated by the electrostatic energy of a system of partial point charges located at the atom sites, \nwith the added constraint of global charge conservation. \nThe effective charges can be either defined empirically, as commonly done in the context of \nforce-field models ~\\cite{deng2019electrostatic}, or can be found by machine learning~\\cite{artrith2011high,bereau2015transferable,yao2018tensormol,bereau2018non,unke2019physnet},\nin which case the partial charges are matched to reference charges extracted from DFT calculations by\npartitioning the electronic charge density among the atoms~\\cite{artrith2011high,bereau2015transferable,bereau2018non}. \nA difficulty with this approach is that an atom in a molecule, or a material, does not have a well defined charge, because the charge densities of neighboring atoms overlap.\nAs a consequence, different partitioning schemes may lead to different results~\\cite{nebgen2018transferable,sifain2018discovering}. \nTo reduce this ambiguity, charge partitioning schemes have been combined with the concepts of electronegativity and hardness~\\cite{ghasemi2015interatomic}, \nwhich, however, also lack a rigorous definition. \nInterestingly, the accuracy and transferability of molecular ML models with effective atomic charges \nwere found to improve significantly when the molecular dipole moment was added to energy and forces in the training data~\\cite{ghasemi2015interatomic,ko2021fourth,ko2021general}. \n\\recheck{This condition is implemented in PhysNet~\\cite{unke2019physnet}, a model in which the partial charges are trained to best reproduce total energy and dipole of a system.}\nA more flexible representation than the point charge models \nis provided by the long-distance equivariant (LODE) framework, \nin which the charge density is approximated by a sum\nof atom centered spherical Gaussians, or other localized functions~\\cite{grisafi2019incorporating,grisafi2021multi}.\nLODE successfully describes the mutual interaction of charged molecular species at large separation, a \nproperty beyond the reach of short-range ML models.\n   \nThe above schemes can be seen as different realizations of a coarse-graining transformation that approximates   \nthe electronic density with a sum of atom centered spherical contributions. \n\\recheck{This representation can provide a good overall fit of the charge density and of the total dipole of a molecule, but, in general, does not describe correctly the dipolar fluctuations that couple with the electric field generated by distant charges in extended systems within periodic boundary conditions.\nThe reason is that the dipole of a periodic crystal can only be defined modulo a quantum, and the polarization fluctuations typically include dipolar fluxes across the cell boundary, originating from the delocalized nature of the quantum mechanical electronic charge distribution. \nThese fluxes are related to the phase of the wavefunction and cannot be described, in general, by partial atomic charges (see e.g., Ref.~\\citen{vanderbilt2018berry}). \nInterestingly, an exception to this rule occurs when a crystal is made of nonoverlapping molecular units, because, in this case, the total dipole is the sum of the molecular dipoles.}\n\n\n\n\nIn this work, we propose an alternative deep learning model for insulating systems that overcomes this limitation.\nAs in previous approaches, we assume that the PES has short- and long-range contributions. \nThe short-range contribution has the standard form of the DP model. The long-range contribution is the electrostatic energy of a system of spherical Gaussian charges, associated with the ions (nuclei+frozen core electrons) and the valence electrons, located, respectively, at the ionic and electronic sites. The latter are defined by the averages of the positions of the maximally localized Wannier centers~\\cite{marzari2012maximally} associated with specific atoms~\\cite{zhang2020deep}. \nIn general, ionic and electronic sites do not coincide. The electronic site coordinates are not independent dynamical variables, but are fixed by the atomic configuration, as required by the Born-Oppenheimer adiabatic separation of ionic and electronic dynamics.\n\\recheck{We assume that the electronic sites, hereafter also called Wannier centroids, depend only on the atoms in their neighborhood, a conjecture that can be rationalized in terms of the so-called nearsightedness of electronic matter~\\cite{prodan2005nearsightedness}. \nWe found that this condition is well satisfied in all systems we have studied to date.}\nThus, the environmental dependence of the centroids can be modeled accurately by a deep neural network like deep Wannier (DW), \nwhich was introduced in Ref.~\\citen{zhang2020deep} to describe the dielectric response of insulators.  \nBy combining DP and DW we construct a model of the PES that includes explicit long-range electrostatic interactions. \nThe new scheme, called deep potential long-range (DPLR), has several advantages relative to atom centered models. \nFirst, the centroid distributions have integer charges and automatically conserve the total charge of the system. Second,\ntheir positions derive from a unitary transformation in the subspace of the occupied Kohn-Sham orbitals. \n\\recheck{By construction, this representation rigorously describes molecular dipoles in finite systems and dipolar fluctuations in condensed media (see e.g.~Ref.~\\citen{vanderbilt2018berry}).}\nThe spherical Gaussian distributions adopted in DPLR differ from the maximally localized Wannier distributions in the quadrupole and higher moments. This results in a potential energy error that converges well with size. In contrast,\nthe errors of atom centered distributions start at the dipole level, and the potential energy error is only conditionally convergent with \nsize. \nThe DPLR model is physically \nmeaningful, symmetry preserving, and smooth, so that forces and virial can be analytically calculated, preserving the conservative properties of the adopted molecular dynamics scheme. \nThe approach is also computationally efficient, as fast algorithms \nfor calculating Ewald sums for Gaussian charges under periodic boundary conditions are well established~\\cite{ewald1921die,hockney1988computer,darden1993pme,essmann1995spm}. DPLR describes polarization fluctuations in finite and extended systems. Thus, it also allows us to model the response of a system to an externally applied field. \n\n\nWhile preparing this manuscript, we became aware of a\nrecent preprint by Gao and Remsing~\\cite{gao2021self}, in which the\nelectronic structure information encoded in the Wannier centers is used to construct a ML model of the PES including long-range electrostatics. This model, called self-consistent field neural network (SCFNN), differs from DPLR because the separation between short- and long-range\ncontributions is not done in the way in which the training data are handled, but in the way in which they are generated. \nThe model requires short-range DFT data obtained, in principle, from calculations with a truncated Coulomb potential. \nA drawback of this formulation is that it introduces a self-consistency condition for the Wannier center positions. \nAs a consequence, the simplicity of the ML construction is lost, and the dynamics of the model is no longer conservative, unless\nformulations such as those proposed by Car and Parrinello are introduced~\\cite{car1985unified}. In practice, separating standard DFT data into data\nfor a truncated Coulomb potential and data for its long-range counterpart is not straightforward, and the authors overcome this difficulty by invoking linear response conditions. \n \n\nIn this manuscript, we focus on DPLR and\nhow it compares to the original DP model.\nTheory and method are presented in\nSect.~II. In Sect.~III we construct the DP, DW, and DPLR models for water that we use in the applications discussed in Sect.~IV.\nIn that section we compare DPLR, DP, and direct DFT calculations for the binding energy curve of the water dimer and for the free energy profile of a water molecule as a function of the distance from a liquid water slab.\nIn Sect.~V we construct DP and DPLR models for small atomic displacements around the classical ground-state in crystalline NaCl, and use these models to compute the phonon dispersion curves.\nWe show that the non-analytic contribution to the dynamical matrix, arising from long-range dipole-dipole interactions, can be calculated with DW and its polarizability extension~\\cite{sommers2020raman}. \nWe then study the evolution of the analytical phonon dispersion curves calculated with finite supercells of increasing size. \nWhile the modes calculated with the DP model do not exhibit size dependence, those calculated with DPLR depend on size, and partially recover the longitudinal-transverse splitting of the optical modes, within the limitations of the finite supercells used in the calculations. \\recheck{In Sect. VI we discuss how the model can be used to study the response of a system to an externally applied electric field.}  Finally, Sect.~VII is devoted to our conclusions.  \n\n\n\n\\section{Theory and method}\n\nWe focus on extended systems modelled with a periodically repeated supercell. Finite systems can be investigated in this way, provided the supercell is large enough that interactions between periodic images can be neglected. With periodic boundary conditions the system must be electrically neutral. However, the scheme could be easily extended to different choices of boundary conditions.     \n\n\\subsection{Electrostatic energy}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{fig-wc.pdf}\n    \\caption{The Wanner centroid (WC) associated with a water molecule is represented by the purple dot. The maximally localized Wannier Centers (MLWCs) corresponding to bond pairs and lone pairs are represented by green and blue dots, respectively. The WC is the geometric center of the 4 MLWCs. In molecular dynamics trajectories of water systems the WCs are uniquely associated with the oxygen atoms.}\n    \\label{fig:wc}\n\\end{figure}\n\nIn the reference DFT model we adopt a pseudopotential framework. Thus, in what follows, electrons stand for valence electrons, and ions for nuclei plus frozen core electrons.\nHowever, the method would be applicable to all-electron DFT methods as well, in which case ions would stand for nuclei and electrons would include valence and core. \nWe indicate the charge density of ions and electrons by $\\rho_{\\textrm{ion}}(r)$ and $\\rho_e(r)$, respectively. When the system is in the ionic configuration\n$\\{R_I\\}$, $\\rho_{\\textrm{ion}}(r)$ is given by: \n\\begin{align}\n    \\rho_{\\textrm{ion}}(r) = \\sum_I \\rho_I (r-R_I),\n\\end{align}\nwhere $R_I$ indicates ion coordinates, and $\\rho_I (r) = q_I \\delta (r)$, in terms of $q_I$, the charge of the ion $I$, and of $\\delta (r)$, the Dirac delta distribution. In insulators, the density of the electrons, $\\rho_e(r)$, can be represented by a sum of local distributions \nvia a unitary transformation of the occupied Bloch orbitals onto maximally localized Wannier functions~\\cite{marzari2012maximally}. The centers of the local distributions are called maximally localized Wannier centers (MLWC). \n\\recheck{Often, during molecular evolution}, the same MLWCs can be uniquely\nassigned to the same atom along a trajectory.\nFor example, in water systems the same four MLWCs are always nearest \nto the same oxygen atom irrespective of liquid diffusion and molecular dissociation. It is then convenient to define the $n$-th Wannier centroid (WC), $W_n$, as the average of\nthe MLWCs assigned to a given atom, as illustrated in Fig.~\\ref{fig:wc} for a water molecule. \n\\recheck{Lumping into a single centroid associated to the O atom the four Wannier centers of a water molecule has the following advantages: (i) it makes straightforward to satisfy, within the deep neural network (DNN) model, the Born-Oppenheimer constraint of parametric dependence of the ground-state electronic properties on the atomic coordinates; (ii) it eliminates the need for imposing permutational symmetry between equivalent Wannier centers in a molecule; (iii) it is computationally efficient.\nWe note, however, that a unique association of the Wannier centers to specific atoms is not possible in presence of electron transfer between different atoms. Dealing with these events requires a significant generalization of the method. Ignoring this possibility,}\n$\\rho_e (r)$ takes the form:           \n\\begin{align}\n    \\rho_e(r) = \\sum_n \\rho_n(r-W_n),\n\\end{align}\nwhere $\\rho_n$ is the sum of the maximally localized Wannier distributions associated with the $n$-th WC. \nFor simplicity, we consider\nspin saturated systems, in which each maximally localized Wannier distribution carries a charge of 2 electrons ($-2e$). Thus, $\\rho_n$ carries an integer charge $q_n$, equal to $-2e$ times the number of Wannier functions associated with the $n$-th centroid. For instance, in the water examples of the manuscript, $q_n = -8e$. The charge neutrality condition implies that the total charge density, i.e., $\\rho_t (r) = \\rho_{\\textrm{ion}} (r) + \\rho_e (r)$, satisfies:\n\\begin{align}\n    \\int dr \\rho_t(r) = \\int dr ( \\rho_{\\textrm{ion}} (r) + \\rho_e (r) ) = 0\n\\end{align}\nThus, the total electrostatic energy is well defined and given by:\n\\begin{align}\\label{eqn:total-e}\n    E_t = \\frac 12 \\int dr dr' \\rho_t(r) v(r-r') \\rho_t(r')\n\\end{align}\nHere $v(r) = 1\/\\vert r\\vert$ is the Coulomb interaction, and it is understood that the (infinite) self-energy of the ionic point charges should be subtracted from the integral. The energy in Eq.~\\eqref{eqn:total-e}  is calculated exactly in the DFT reference model, while it is approximated, in the DPLR model, by the electrostatic energy of a system of spherical Gaussian charge distributions $g_I$ and $g_n$ located at ionic ($R_I$) and WC ($W_n$) sites, respectively. In DPLR one adds to this energy a short-range contribution described by the DP part of the model, which takes care of all the additional many-body effects within $r_c$, the cutoff radius of the model. Thus, all residual errors of the approximation for the electrostatic energy should be associated with long-range effects, for which $r > r_c$.    \nThe Gaussian distributions $g_I$ and $g_n$ integrate to $q_I$ and $q_n$, respectively, but are assumed to have the same spread $1 \/ (\\sqrt 2 \\beta)$, with $\\beta$ an adjustable parameter. \nThe true distributions, $\\rho_I$ and $\\rho_n$, differ from $g_I$ and $g_n$ according to: \n\\begin{align}\\label{eqn:gi-di}\n    &\\rho_I(r) = g_I(r) + \\rho_I(r) - g_I(r) = g_I(r) + \\Delta_I(r)\\\\\\label{eqn:gn-dn}\n    &\\rho_n(r) = g_n(r) + \\rho_n(r) - g_n(r) = g_n(r) + \\Delta_n(r)\n\\end{align}\nThe distributions $\\Delta_I$ and $\\Delta_n$, in Eqs.~\\eqref{eqn:gi-di} and ~\\eqref{eqn:gn-dn}, integrate to zero. $\\Delta_I$ is known, having all its mutipole moments equal to zero, while $\\Delta_n$ has zero dipole moment, but is otherwise unknown. We choose the spread parameter $\\beta$ so that the Gaussian distributions are at the same time smooth on the atomic scale and \nwell localized within $r_c$. More details on the choice of $\\beta$ will be given when discussing specific examples later in the manuscript.\nWhile the electrostatic potential generated by $\\Delta_I$, which has spherical symmetry, decays to 0 exponentially for $r > r_c$, the potential generated by $\\Delta_n$ decays to 0 algebraically for $r > r_c$ because, in general, the distribution $\\Delta_n$ has non-zero quadrupole and higher multipole moments. Thus, the potential generated by $\\Delta_n$ decays at least as fast as $r^{-3}$ for $r > r_c$ and has magnitude controlled by the quadrupole moments specific to the system. Typically, we expect quadrupoles on the order of 1 a.u., which would correspond to $\\Delta_n(r=r_c)$ being approximately equal to 1 meV for $r_c = 6$ A. \nGiven that the DP part of the model takes care of the effects within $r_c$, the error of the above procedure is associated with effects \nbeyond $r_c$, which are controlled primarily by $\\Delta_n$. \n\nThe total ionic and electronic Gaussian distributions are given by:\n\\begin{align}\n    & G_{\\textrm{ion}}(r) = \\sum_I g_I(r-R_I)\\\\\n    & G_e(r) = \\sum_n g_n (r-W_n)\n\\end{align}\nThe  corresponding total Gaussian charge distribution is $G_t(r) = G_{\\textrm{ion}}(r) + G_e(r)$. \nThe electrostatic energy associated with $G_t$ is easily calculated in Fourier space~\\cite{ewald1921die}\n\\begin{align}\\label{eqn:e-dplr}\n    E_{G_t} & = \\frac{1}{2\\pi V} \\sum_{m\\neq 0, \\vert m\\vert \\leq L} \\frac{\\exp(-\\pi^2 m^2\/\\beta^2)}{m^2}  S^2(m),\n\\end{align}\nwhere $L$ is the cutoff in Fourier space, and $S(m)$, the structure factor, is given by:\n\\begin{align}\\label{eqn:sm}\n    S(m) = \\sum_I q_I e^{-2\\pi i m  R_I} + \\sum_n q_n e^{-2\\pi i m W_n}\n\\end{align}\nDue to the smoothness of the distributions, the Fourier sum in Eq.~\\eqref{eqn:e-dplr} converges rapidly. \nFast algorithms for calculating the electrostatic energy are available, such as the smooth particle mesh Ewald (SPME)~\\cite{darden1993pme,essmann1995spm} and the particle-particle-particle-mesh (PPPM)~\\cite{hockney1988computer} methods. The latter is adopted in the applications discussed in this manuscript.\nThe electrostatic energy $E_{G_t}$ in Eq. \\eqref{eqn:e-dplr} contains also short-range contributions. To avoid double counting, $E_{G_t}$ is subtracted from the total potential energy when training the DP part of the DPLR model.  \n\n\nThe error $\\mathcal E$, due to neglecting the contribution of $\\Delta_n$ to the electrostatic energy, is the sum of two terms, $\\mathcal E_1$ and $\\mathcal E_2$, given respectively by:\n\\begin{align}\\label{eqn:error-e1}\n    &\\mathcal E_1 = \\int drdr' G_t(r) v(r-r') \\sum_n\\Delta_n (r') \\\\ \\label{eqn:error-e2}\n    &\\mathcal E_2 = \\int drdr' \\sum_m\\Delta_m(r) v(r-r') \\sum_n\\Delta_n(r)\n\\end{align}\nThe two integrals above are well defined for extended systems as the corresponding charge distributions are neutral.\nSince the lowest non-zero multipole of $\\Delta_n$ is the quadrupole, $\\mathcal E_2$ in Eq.~\\eqref{eqn:error-e2} is (at worst) a sum of quadrupole-quadrupole interactions, decaying as $r^{-5}$. \nOnly the part of that sum involving terms with (roughly) $r>r_c$ cannot be learned by the DP model. \nThese terms contribute to the error. Taking 6~\\AA\\  for a typical $r_c$ value, the sum would include terms smaller than 1~meV.\nDue to the fast rate of decay the double sum should converge rapidly. \nWe could estimate $\\mathcal E_1$ in Eq.~\\eqref{eqn:error-e1} in a similar fashion by writing $G_t$ as a sum of local dipole contributions, resulting in a sum of dipole-quadrupole interactions that decay as $r^{-4}$, i.e., still rapidly but less rapidly than the terms in $\\mathcal E_2$.\nThis estimate is, however, affected by the arbitrariness of the local dipole definition. \nA better way of estimating $\\mathcal E_1$ is the following. Let $E_\\Delta(r)$ be the electric field generated by the sum of the $\\Delta_n$ distributions:\n\\begin{align}\\label{eqn:error-efield}\n    E_\\Delta(r) = \\int dr' \\frac{r - r'}{\\vert r - r'\\vert^3} \\sum_n\\Delta_n(r)\n\\end{align}\nTo leading order, the electric field in Eq.~\\eqref{eqn:error-efield} is a sum of quadrupole contributions that decay like $r^{-4}$ for $r > r_c$. This field is essentially uniform on the molecular scale. Let us call the corresponding average field $\\bar E_\\Delta$. Then, the average energy error can be written in terms of the cell dipole (polarization times volume) $M$ as:\n\\begin{align}\\label{eqn:error-e1-me}\n    \\mathcal E_1 = - M \\cdot \\bar E_\\Delta\n\\end{align}\nIn extended systems $M$ is defined modulo a quantum\\recheck{\\cite{vanderbilt2018berry}}. Thus, the error in Eq. \\eqref{eqn:error-e1-me} is only well defined for a change of ionic coordinates that leads to a change of $M$. \n\nIn summary, the error in the LR contribution to the electrostatic energy, when the true electronic distribution $\\rho_e$ is replaced by \nthe Gaussian distribution $G_e$, is given by a sum of small terms that decay at most as $r^{-4}$ for $r > r_c$. \nIt would be possible to further reduce these errors by representing $G_e$ as a sum of non-spherical Gaussian distributions\nhaving the spreads of the Wannier centroid distributions, which could be learned by an extended DW approach. In this way \nthe lowest non-zero multipole of $\\Delta_n$ would be the octupole. At $r_c$=6~\\AA, such treatment would reduce the error \nby approximately an order of magnitude. However, this formulation would also result in a significant increase of the computational\ncomplexity, which was deemed unnecessary based on our tests.\n\nWe remark that formulations of LR electrostatics based on effective atomic charges cannot reproduce the fluctuations of $M$ \\recheck{in condensed phase}. In these formulations, the atom centered \nelectronic charge distribution differ locally from the true distribution $\\rho_e$ at the dipole rather than at the quadrupole \nlevel. \nTherefore, \nthe corresponding error would be associated with dipole-dipole interactions that are only conditionally convergent. \n\\recheck{It is worth noting that, in finite systems, the error is reduced by adopting partial charge models trained to reproduce the molecular dipole.}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Deep Potential Long-Range model} \n\nIn DPLR the PES is given by\n\\begin{align}\\label{eqn:e-decomp}\n    E = E_\\mathrm{sr} + E_{G_t},\n\\end{align}\nwhere the electrostatic energy $E_{G_t}$ was introduced in Eq.~\\eqref{eqn:e-dplr},\nand $E_\\mathrm{sr}$ is a short-range\ncontribution constructed as in the standard DP model (see Appendix~\\ref{sec:app-srdp} for more details). However, $E_\\mathrm{sr}$\nis not equal to the standard DP energy, because the Gaussian electrostatic energy $E_{G_t}$ is subtracted from the DFT training data to avoid double counting.  \n\nThe forces and the virial are derivatives of the energy with respect to the atomic positions and the simulation cell tensor, respectively.\nWithin DPLR, the energy $E$ depends on the atomic coordinates, both explicitly and implicitly, via the dependence of the WCs on the atomic coordinates. The implicit dependence makes the calculations nontrivial.\n\nThe force on atom $I$ is given by\n\\begin{align}\\label{eq:tmp-a1}\n    F_I = \n    - \\frac{\\partial E}{\\partial R_I} \n    =\n    - \\frac{\\partial E_\\mathrm{sr}}{\\partial R_I} \n    - \\frac{\\partial E_{G_t}}{\\partial R_I} \n    - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_n} \n    \\frac{\\partial W_n}{\\partial R_I}. \n\\end{align}\nHere the environmental dependence of $W_n$ is given by the DW model. We recall that there is a one-to-one correspondence \nbetween the Wannier centroid $n$ and its nearest atom, the atom $I$. This correspondence establishes a bijective mapping between the indices of the WCs and the indices of a subset of the ions, that we indicate, equivalently, by $n = n(I)$ or by $I = I(n)$. Due to the nearsightedness of the electronic matter~\\cite{}, the position of the WC associated with atom $I$ depends on the local environment\nof that atom within a cutoff radius $r_c$ that can be safely assumed to be equal to the cutoff radius of the DP model. \nIndicating with $D_n$ the displacement of $W_n$ relative to $R_I$, we have: \n\\begin{align}\n    W_n = R_{I(n)} + D_n,\n\\end{align}\nwhich relates the environmental dependence of $D_n$ to that of $W_n$. The latter \nis smooth by construction in the DW model.\nThus,\n\\begin{align}\n    \\frac{\\partial W_n}{\\partial R_J}\n    = \n    \\delta_{I(n),J} + \\frac{\\partial D_n}{\\partial R_J},\n\\end{align}\nwhere $\\delta_{I,J}$ is the Kronecker delta, and the force on atom $I$ in Eq. \\eqref{eq:tmp-a1} becomes\n\\begin{align}\\label{eq:app-f}\n     F_I = \n     - \\frac{\\partial E_\\mathrm{sr}}{\\partial R_I} \n    - \\frac{\\partial E_{G_t}}{\\partial R_I} \n    - \\frac{\\partial E_{G_t}}{\\partial W_{n(I)}} \n    - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_n} \n    \\frac{\\partial D_n}{\\partial R_I}. \n\\end{align}\nThe first term on the right-hand-side (RHS) of Eq. \\eqref{eq:app-f} is the standard DP force, the second and third terms give \nthe electrostatic force on atom $I$ and its associated WC, respectively, while\nthe fourth term originates from the environmental dependence of the WC.\nThe derivatives $\\partial D_n\/\\partial R_I$ are calculated by back-propagating the DW model.\n\nWe indicate the cell tensor by $h = \\{h_{\\alpha\\beta}\\}$, with $h_{\\alpha\\beta}$ being the $\\beta$-th component of $\\alpha$-th cell vector. \nThe virial is defined by\n\\begin{align}\\label{eq:app-v0}\n\\Xi_{\\alpha\\beta} \n= -\\frac{\\partial E}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta},\n\\end{align}\nwhere summation over repeated indices is assumed.\nBy using Eq.~\\eqref{eqn:e-decomp} we have\n\\begin{align}\\label{eq:app-v1}\n\\Xi_{\\alpha\\beta} = -\\frac{\\partial E_\\mathrm{sr}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}  -\\frac{\\partial E_{G_t}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}\n= \\Xi^\\mathrm{sr}_{\\alpha\\beta} -\\frac{\\partial E_{G_t}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta},  \n\\end{align}\nThe first term on the RHS of Eq.~\\eqref{eq:app-v1} is the short-range virial contribution, $\\Xi^\\mathrm{sr}_{\\alpha\\beta}$,\nwhich is calculated as in the standard DP model. The remaining term is the electrostatic virial contribution, whose calculation is non-trivial as we detail below. \nFirst, it is convenient to define the reciprocal cell tensor $h^{-1} = \\{h^{-1}_{\\beta\\alpha}\\}$, in which $h^{-1}_{\\beta\\alpha}$ is the $\\beta$-th component of $\\alpha$-th reciprocal cell vector. The direct and reciprocal cell tensors define linear scaling transformations in direct and\nreciprocal space that are useful in MD simulations with variable cell.\nIn direct space, a scaled coordinate $s$ is related to its non-scaled counterpart $R$ by $R = sh$, and, in reciprocal space, \nthe lattice vector $m$ is related to its scaled counterpart $k$ by $m = h^{-1}k$.\nAs a consequence, the scalar product \n$mR = sk$ is independent of the cell tensor $h$, and so is the factor $e^{2\\pi imR_I}$ in Eq.~\\eqref{eqn:sm}. However, the factor $e^{2\\pi imW_I}$ in the same equation does depend on the cell vector $h$, because in general \nthe positions of the WCs do not change linearly with a cell deformation.\nThen, we can write for the electrostatic contribution to the virial:\n\\begin{align}\\label{eq:tmp-a4}\n    -\\frac{\\partial E_{G_t}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}\n    =\n    \\Xi^{\\mathrm{rec}}_{\\alpha\\beta}\n    +\n    \\Xi^{c}_{\\alpha\\beta},\n\\end{align}\nwith\n\\begin{align}\\label{eq:tmp-a5}\n    \\Xi^{\\mathrm{rec}}_{\\alpha\\beta} & = \n    -\\frac{\\partial}{\\partial h_{\\gamma\\alpha}} \\Big(\\frac1{2\\pi V} \\Big)\n    h_{\\gamma\\beta} \\sum_{m\\neq0}f(m)S^2(m)\n    -\n    \\frac1{2\\pi V} \\sum_{m\\neq0}\\frac{\\partial f(m)}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta} S^2(m) \\\\ \\label{eq:tmp-a6}\n    \\Xi^{c}_{\\alpha\\beta} &= \n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m)\n    S(m) \\frac{\\partial S(m)} {\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}, \n\\end{align}\nwhere we used the definition $f(m) = \\frac1{m^2}{\\exp(-\\pi^2m^2\/\\beta)}$.\n$\\Xi^\\mathrm{rec}_{\\alpha\\beta}$ in Eq.~\\eqref{eq:tmp-a5} is the standard reciprocal virial contribution of the Ewald method, while the correction term in Eq.~\\eqref{eq:tmp-a6} accounts for the nonlinear dependence of the WCs on the cell tensor. \nThe correction term $ \\Xi^{c}_{\\alpha\\beta}$ is calculated as follows\n\\begin{align}\\label{eq:tmp-a8}\n    \\Xi^{c}_{\\alpha\\beta}\n    =\n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m) S(m)\n    \\sum_n q_n e^{2\\pi iW_n m} 2\\pi i \n    \\Big(\n    \\frac{\\partial W_{n\\delta}}{\\partial h_{\\gamma\\alpha}} m_\\delta +\n    W_{n\\delta} \\frac{\\partial m_\\delta}{\\partial h_{\\gamma\\alpha}}\n    \\Big)\n    h_{\\gamma\\beta}, \n\\end{align}\nwhere the terms in parentheses are given by\n\\begin{align*}\n    \\frac{\\partial W_{n\\delta}}{\\partial h_{\\gamma\\alpha}} m_\\delta h_{\\gamma\\beta}\n    &= \n    \\sum_J \\frac{\\partial W_{n\\delta}}{\\partial R_{J\\epsilon}}\\frac{\\partial R_{J\\epsilon}}{\\partial h_{\\gamma\\alpha}} m_\\delta h_{\\gamma\\beta}\n    =\n    \\sum_J \\frac{\\partial W_{n\\delta}}{\\partial R_{J\\alpha}} R_{J\\beta} m_\\delta\n    = \n    R_{n\\beta} m_\\alpha\n    +\n    \\sum_J \\frac{\\partial D_{n\\delta}}{\\partial R_{J\\alpha}} R_{J\\beta} m_\\delta\\\\\n    W_{n\\delta} \\frac{\\partial m_\\delta}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}\n    & =\n    W_{n\\delta} \\frac{\\partial h^{-1}_{\\delta\\epsilon}}{\\partial h_{\\gamma\\alpha}} k_\\epsilon h_{\\gamma\\beta}  \n    =\n    -W_{n\\delta}h^{-1}_{\\delta\\eta} \\frac{\\partial h_{\\eta\\zeta}}{\\partial h_{\\gamma\\alpha}} h^{-1}_{\\zeta\\epsilon} k_\\epsilon h_{\\gamma\\beta}\n    =\n    -W_{n\\delta}h^{-1}_{\\delta\\gamma}  h^{-1}_{\\alpha\\epsilon} k_\\epsilon h_{\\gamma\\beta}\n    = - W_{n\\beta} m_\\alpha\n\\end{align*}\nThus, the correction virial term in Eq.~\\eqref{eq:tmp-a8} becomes\n\\begin{align}\n    \\Xi^{c}_{\\alpha\\beta}\n    =\n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m) S(m)\n    \\sum_n q_n e^{2\\pi iW_n m} 2\\pi i \n    \\Big(\n    D_{n\\beta}m_\\alpha \n    +\n    \\sum_J \\frac{\\partial D_{n\\delta}}{\\partial R_{J\\alpha}} R_{J\\beta} m_\\delta\n    \\Big)\n\\end{align}\nGiven that \n\\begin{align}\n    -\\frac{\\partial E_{G_t}}{\\partial W_{n\\alpha}}\n    =\n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m) S(m)\n    2\\pi i m_\\alpha q_n e^{2\\pi iW_n m}, \n\\end{align}\nthe correction virial simplifies to\n\\begin{align}\\label{eq:tmp-a11}\n     \\Xi^{c}_{\\alpha\\beta}\n     =\n     - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_{n\\alpha}} D_{n\\beta} \n     -\n     \\sum_{nJ} \\frac{\\partial E_{G_t}}{\\partial W_{n}}\n     \\frac{\\partial D_{n}}{\\partial R_{J\\alpha}} R_{J\\beta}\n\\end{align}\nFinally, using Eqs.~\\eqref{eq:app-v1}, \\eqref{eq:tmp-a4}, \\eqref{eq:tmp-a5} and  \\eqref{eq:tmp-a11}, the virial takes the more compact form\n\\begin{align}\n    \\Xi_{\\alpha\\beta} = \\Xi^\\mathrm{sr}_{\\alpha\\beta} + \n    \\Xi^\\mathrm{rec}_{\\alpha\\beta}\n    - \\sum_{nJ} \\frac{\\partial E_{G_t}}{\\partial W_{n}}\n     \\frac{\\partial D_{n}}{\\partial R_{J\\alpha}} R_{J\\beta}\n    - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_{n\\alpha}} D_{n\\beta} \n\\end{align}\n\n\nWe checked that the above formulae are correct by comparing the analytical derivatives in the formulae to the numerical    \nderivatives calculated with finite differences. As a consequence,\nin molecular dynamics simulations the conservation laws associated with the equations of motion should be satisfied within the accuracy of the adopted numerical implementation. \nWe find, for instance, that in NVE trajectories of liquid water with 128 molecules, the total energy shows a small drift of approximately $0.4$~meV\/\\chem{H_2O} ($\\sim$1~K) per 100 ps with the DPLR approach, when using a mesh spacing $\\eta=0.98$~\\AA\\ in the calculation of the Ewald sum. As shown in Fig.~\\ref{fig:ener_drift}, this is small but greater than the drift of a DP trajectory for the same system, which is almost not observable in a 100~ps trajectory.\nThe total energy drift in the DPLR trajectory is controlled by the numerical accuracy of the fast algorithm for computing the Ewald sum, here the PPPM method, and can be further reduced by using a finer mesh.\nAs shown in Fig.~\\ref{fig:ener_drift}, the drift of DPLR is not observable when the mesh spacing $\\eta$ is set equal to $0.49$~\\AA.\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{fig-ener-drift.pdf}\n    \\caption{Drift of the total energy of DP and DPLR models in a 100~ps NVE simulation of 128 water molecules. \n    The time-step is set to 0.5~fs.\n    Two mesh spacings of the PPPM method, i.e.~$\\eta = 0.98$~\\AA\\ and $\\eta = 0.49$~\\AA, are used in the DPLR model.\n    }\n    \\label{fig:ener_drift}\n\\end{figure}\n\n\n\\section{Deep Potential Long-Range model for water}\n\nWe construct a DPLR model for water based on the PBE functional approximation of DFT~\\cite{perdew1996generalized}. \nIt is well known that PBE substantially overestimates the strength of the hydrogen-bonds in water. \nAs a consequence, it fails to describe correctly some important thermodynamic properties, like the relative density of ice and water at ambient pressure~\\cite{gaiduk2015density}.\nHowever, here we are not interested in constructing a state-of-the-art water model, but rather we intend to demonstrate that DPLR can capture long-range electrostatic effects missing in the DP model.\n\\recheck{For this purpose, the choice of the exchange-correlation functional is not a critical issue. It would be straightforward to adopt more accurate functionals than PBE by simply replacing PBE with any other functional of choice, in the labeling steps whereby energy, force and virial tensor are calculated at selected atomic configurations.\n}\nIn the next three subsections, we construct DP, DW, and DPLR models based on the PBE functional approximation.  \n\n\n\\subsection{Training data and DP model}\n\nWe use the concurrent learning scheme Deep Potential GENerator (DP-GEN)~\\cite{zhang2019active,zhang2020dp}. DP-GEN iteratively enlarges the training dataset of labeled DFT data and refines a representative ensemble of DP models by learning from a growing dataset. In this work the representative ensemble consists of four models that differ in the random initialization of the network parameters but are otherwise trained on the same data. In the DP-GEN protocol \nthe four DP models are used to sample the relevant thermodynamic space with deep potential molecular dynamics (DPMD) trajectories. The DP construction includes three stages.\nIn the first one (iteration 1 to 7), bulk configurations in the thermodynamics space $200 \\leq T \\leq 400$~K and $1\\leq P \\leq 10^4$~Bar are explored with isothermal-isobaric (NPT) DPMD. \nThe initial configurations for bulk NPT DPMD simulations are snapshots of DPMD simulations with 128 molecules in the liquid state obtained with the DP model developed in Ref.~\\citen{zhang2018deep}.\nThe second stage extends from iteration 8 to iteration 15. In this stage slab configurations are explored with canonical (NVT) DPMD simulations in the temperature range $200 \\leq T \\leq 400$~K.\nThe initial configurations are prepared by adding a vacuum region on top of bulk liquid configurations. \nDifferent surface structures with thickness of the vacuum region ranging from 0.5~\\AA\\ to  7.0~\\AA\\ are considered.\nIn the third stage, from iteration 16 to 23, low density bulk configurations are explored by NVT simulations for $200 \\leq T \\leq 400$~K. Initial configurations are created by randomly removing 64, 96, 112 and 120 molecules from the initial configurations of stage 1.\nAt each stage, the time span of the DPMD simulations is gradually increased over the iterations, from 1~ps to 10~ps.  \nThe maximal deviation of the forces predicted by the ensemble of DP models is monitored along the trajectories and used to classify the explored configurations. \nConfigurations with deviation between 0.15 to 0.30~eV\/\\AA\\ are labelled as training data, following the protocol detailed in Ref.~\\citen{zhang2020dp}.\nSingle-shot DFT calculations are performed at these configurations.\n\nDFT calculations are carried out with the Vienna {\\it ab initio} simulation package (VASP) version 5.4.4~\\cite{kresse1996efficiency,kresse1996efficient},\nusing the PBE exchange-correlation energy. \nThe kinetic energy cut-off for the planewave basis is set to 1500~eV. \nThe self-consistent field procedure is stopped when the energy difference between the last two iterations is less than $10^{-6}$~eV.  \n\nThe DP models are trained with the DeePMD-kit package~\\cite{wang2018deepmd}. \nThe cut-off radius $r_c$ for the atomic environments is set to 6~\\AA. \nThe size of the embedding and that of the fitting network are set to (25, 50, 100) and (240, 240, 240), respectively. \nThe DP models are trained with the Adam stochastic gradient descent scheme~\\cite{kingma2015} using $10^6$ steps, during which the learning rate exponentially decays from $10^{-3}$ to $3.5\\times 10^{-8}$.\nAt each training step, a subset of the training dataset, referred to as the mini-batch, is used to calculate the gradient of the loss function. \nThe size of the minibatch is 1, i.e.~only 1 configuration is used for calculating the gradient.\n\nAt the end of each training stage the fraction of configurations with gradient deviation larger than 0.15~eV\/\\AA\\ is reduced to less than 0.1\\%, indicating satisfactory convergence of the DP-GEN cycle.\nA total of 448 configurations are selected as training data during the DP-GEN protocol.\nThey include 323 bulk,  94 surface, and 31 low density configurations.\nThe dataset used for training the initial DP models has 135 configurations, thus the total training data includes 583 configurations. \n\n\n\\subsection{DW model}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{fig-accuracy_corr.pdf}\n    \\caption{Training accuracy of the DW model for water. The three panels report the three Cartesian displacements of the Wannier centroids (WC) relative to their nearest oxygen atom predicted by DW (vertical axis) and by DFT (horizontal axis).}\n    \\label{fig:cwc-train-err}\n\\end{figure}\n\nFor each water configuration, the DW model gives the position of the WC associated with a given molecule relative to the O atom of that molecule. \nThe training data are the same data used to generate the DP model.\nEach WC is uniquely associated with a particular O atom and depends on the atomic environment of that atom within a cutoff radius, which is set here to 6~\\AA, i.e., it is the same cutoff $r_c$ of the DP model. \nWe use the standard DP descriptor $\\mathcal D = \\mathcal D^a$ (see Appendix for details).\nThe sizes of the embedding and fitting networks are (25, 50, 100) and (100, 100, 100), respectively. \nThe model is trained with the Adam stochastic gradient descent method~\\cite{kingma2015} using $10^6$ steps. \nThe batch size is set equal to 1. The learning rate exponentially decays from $1.0 \\times 10^{-2}$ to $5.6\\times 10^{-8}$.\n\nThe training accuracy of the DW model is $1.7\\times 10^{-3}$~\\AA, which is much smaller than the typical distance $D$ of a WC from its reference O atom. \nThe test accuracy is $1.9\\times 10^{-3}$~\\AA. \nThus, the generalization gap between test and training errors is almost negligible.\nThe correlation between labelled and predicted $D$ is graphically illustrated in  Fig.~\\ref{fig:cwc-train-err}.\n\n\n\\subsection{DPLR model}\\label{sec:lrdp-train}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig-error-rc4-6-e.pdf}\n    \\caption{Root mean square error (RMSE) of the energy predicted by the DPLR model as a function of the spread parameter $\\beta$. The training error of bulk (solid squares connected by a red solid line) and surface configurations (solid circles connected by a red dashed line), and the test error of surface configurations (open circles connected by a red dotted line) are shown (see text for more details).\n    The corresponding errors for the DP model are plotted as a black solid line (bulk training error), a black dashed line (surface training error), and a black dotted line (test error).\n   \n    }\n    \\label{fig:error}\n\\end{figure}\n\nThe DPLR model is trained on the dataset generated by DP-GEN.\nIn DPLR, the spread parameter $\\beta$, i.e.~the inverse spread of the Gaussian charge distribution, needs to be fixed. \nIn the limit of $\\beta$ going to 0, the Gaussian width is infinite and DPLR reduces to the standard DP model.\nOn the other hand, for $\\beta$ going to infinity, the Gaussian charges become point charges. \nIn this limit, the magnitude of the intramolecular Coulomb force between the oxygen ion ($q_I=6e$) and its associated WC ($q_n=-8e$), for a typical separation distance ($D\\approx 0.07$\\AA), is about $10^4$~eV\/\\AA\\ , which is about four orders of magnitude larger than the average atomic force. \nThis situation would create serious numerical problems due to the difficulty of resolving a labeled force from the strong intramolecular Coulomb force.\nTherefore, there should be an optimal $\\beta$, neither too small to avoid reduction to the standard DP model, nor too large to avoid distributions too close to point-charges. \n\nTo study the effect of the spread parameter on the accuracy of the DPLR models, we train different DPLR models with\ndifferent spread parameters ($\\beta = \\{0.1, 0.2,\\dots,0.6\\}$\\AA$^{-1}$). \nTo ensure proper convergence of the Ewald sum in Eq.~\\eqref{eqn:e-dplr}, the mesh spacing $\\eta$ is set equal to the largest value compatible with the constraint $\\eta\\cdot\\beta \\leq 0.4$.  \nIn DP and in the short-range part of the DPLR models we adopt a hybrid descriptor (see Appendix for details) $\\mathcal D = (\\mathcal D^a, \\mathcal D^r)$, and the cut-off of $\\mathcal D^a$ is set to $r_c^a = 4$~\\AA, while the cut-off of $\\mathcal D^r$ is set to $r_c = 6$~\\AA. The size of the embedding net of  $\\mathcal D^a$ is (25, 50, 100), and that of  $\\mathcal D^r$ is (10, 20, 40).\nTraining and test errors, as a function of $\\beta$, are shown in Fig.~\\ref{fig:error}.\nThe training dataset is divided into two parts, one consisting only of bulk configurations and the other consisting only of surface configurations. \nThe test dataset includes surface configurations with 128 molecules with a vacuum region of 7 or 12~\\AA\\ , generated along a 100-ps DPLR molecular dynamics  \n(DPLRMD) trajectory for each of the two widths of the vacuum region. Configurations are recorded every 10~ps, and are labeled with \\emph{ab initio} DFT calculations. Thus, the test dataset includes 20 surface configurations in total. \n\nIn Fig.~\\ref{fig:error}, one can see that the energy training error for the bulk configurations is smaller than the corresponding error for the surface configurations in both DP and DPLR models, reflecting \nthe more difficult task of interpolating the PES for surface than for bulk configurations, due to the larger variance of the local environments in the former case. For $\\beta < 0.5$~\\AA$^{-1}$, the DPLR training errors are close to those of DP, but for $\\beta \\geq 0.5$~\\AA$^{-1}$ they gradually\nincrease and become notably larger than the corresponding DP errors, a behavior that we attribute to numerical difficulties with\nGaussian charges that are too localized.\n\nThe test dataset is composed of independent configurations not included in the training dataset, and serves to assess the capability of the\nthe models to deal with unforeseen situations. \nIn the case of the standard DP model, the test error (2.0~meV\/\\chem{H_2O}) is larger than the surface training error (1.2~meV\/\\chem{H_2O}). \nThe gap, usually called generalization gap, between test and training errors indicates that a model overfits the training data and looses accuracy when generalized to cases not included in the training dataset. \nThe DPLR model with a small $\\beta$ has a generalization gap comparable to that of the standard DP model, as expected because for $\\beta \\rightarrow 0$~\\AA$^{-1}$ DPLR reduces to DP.\nHowever, for $\\beta \\approx 0.4$~\\AA$^{-1}$, the generalization gap is minimal. In this case \nthe training error and the test error are 1.3 and 1.5~meV\/\\chem{H_2O}, respectively,\nindicating that inclusion of long-range electrostatics contributes to the model's generalization ability.\nFor $\\beta \\geq 0.5$~\\AA$^{-1}$ the test error increases with $\\beta$, as expected from numerical difficulties \nwith charge distributions that are too strongly localized.\n\n\nBased on the above analysis, the optimal spread parameter $\\beta$ should have a value of about 0.4~\\AA$^{-1}$. It is interesting to note that, with this value of the spread parameter, the radial extent of the Gaussian distribution $g_n(r)$ about the oxygen atom is close to the physical extent of the (valence) electron density of a water molecule, as measured by the radius of the surface on which the electron density is approximately one order of magnitude smaller than its maximum value. Thus, with the optimal $\\beta$ the spherical Gaussian distribution $g_n(r)$ approximates (roughly) the physical WC distribution $\\rho_n(r)$ and the error $\\Delta_n(r)$ in Eq.~\\eqref{eqn:gn-dn} is minimized.  \n\nIn the rest of the manuscript, we set $\\beta = 0.4$~\\AA${}^{-1}$ , unless stated otherwise.\nWe note that simply increasing the cut-off radius of the standard DP model to 8~\\AA\\ only marginally reduces training and test errors. Indeed, when using $r_c=8$~\\AA\\ in place of $r_c=6$~\\AA,  \nthe bulk and surface training errors are reduced from 1.02~meV\/\\chem{H_2O} and 1.26~meV\/\\chem{H_2O} to 1.01~meV\/\\chem{H_2O} and 1.18~meV\/\\chem{H_2O}, respectively, while the\ntest error is reduced from 2.14~meV\/\\chem{H_2O} to 2.08~meV\/\\chem{H_2O}.\nThis indicates that the long-range effect is non-trivial and can only be captured by including explicitly \nthe long-range electrostatic interaction.\n\n\n\n\n\\section{Application to two water systems}\n\nWe compare the performance of DPLR and DP in two test cases. In one we compute the potential energy of interaction of two water molecules \nas a function of the relative distance. \nIn the other we compute the free energy profile a water molecule versus its distance from a liquid water slab. \nThe second example requires the calculation of a thermal property of a relatively complex system. \nIn both cases, long-range electrostatic interactions\namong the electric dipoles of the water molecules play a role. \nThese interactions, absent in the DP model, are described with sufficient accuracy by DPLR. \n\n\n\\subsection{Water dimer}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{fig-dimer-aligned-1.pdf}\n    \\caption{Potential energy of the water dimer as a function of the separation distance between the two water molecules. \n    The standard deviation of four independently trained DPLR models with the same spread parameter $\\beta = 0.4$~\\AA$^{-1}$ is shown by the error bars in the plot. The top panel compares DFT with DP and DPLR models with cutoff radii $r_c^a = r_c=4$\\AA.\n    \\recheck{The bottom panel compares DFT with DP and DPLR models with cutoff radii $r_c^a = 4$~\\AA\\ and $r_c=6$\\AA .\n    All energies are referred to the potential energy minimum $E_{min}$.}\n   \n    }\n    \\label{fig:dimer}\n\\end{figure}\n\nThe potential energy curve of a water dimer as a function of the separation distance $d_O$ between the two molecules is reported in Fig.~\\ref{fig:dimer}. \n\\recheck{Both DP and DPLR models describe equally well the potential energy at short distance.\nThe DP model does not correctly predict the energy curve at a distance beyond the cut-off radius. \nDue to the finite range of the DP model, the corresponding potential energy curve misses the\n$1\/d_O^3$ tail of the DFT reference, which is due to dipole-dipole interactions. \nBy contrast, the long-range Coulomb tail is recovered accurately by the DPLR model, as shown in both panels of Fig.~\\ref{fig:dimer}.\nThe accuracy at intermediate distance improves by increasing the cutoff in all the models, as expected. \nAlthough DPLR has a higher ability of representing the energy surface, it is marginally less accurate than the DP model between 3.5 and 4.0~\\AA. This can be explained by the fact that dimer configurations were not included in the training dataset.\nFrom Fig.~\\ref{fig:dimer} one concludes that DPLR is a superior model with both cutoffs.\n}\n\n\n\n\n\n\n\\subsection{Free energy profile of a molecule at varying distance from a liquid slab}\n\n\nIn this subsection, we test the performance of DPLR in a calculation of the\nfree energy profile of a water molecule as a function of its distance from a liquid water slab. The model has cut-off radii of $r_c^a = 4$~\\AA\\ and $r_c = 6$~\\AA, spread parameter of 0.4~\\AA$^{-1}$, and Ewald mesh spacing of 1~\\AA. We simulate the slab and the adjacent vacuum region on a periodic box of size 50\\AA$\\times$ 11\\AA $\\times$ 11\\AA. The slab contains 64 molecules, let to equilibrate with a long canonical MD run at a temperature of 300~K starting from an initial bulk liquid configuration. After equilibration, the half width of the slab is $\\sim 7.9$~\\AA ~and the adjacent vacuum region has a width of $\\sim 34.2$~\\AA.\nWe insert an additional molecule to this system, initially near the center of the vacuum region where it interacts weakly with the slab. Then, we vary the distance of the molecule from the slab and we calculate the free energy profile as a function of the distance, while keeping the whole system in equilibrium at 300~K. The distance of the molecule from the slab is defined by the distance of the respective centers of mass. A typical configuration is illustrated in the inset of Fig.~\\ref{fig:fe} for a distance $s$ between the molecule and the liquid slab. In the atomic configuration $\\{R\\}$, the distance of the molecule from the slab is represented by the collective variable $\\sigma(\\{R\\})$. In MD simulations an holonomic constraint, such as $\\sigma(\\{R\\}) = s$, is easily imposed by a Lagrange multiplier. We set the initial distance to $s_{\\textrm{far}} = 17$~\\AA. At this distance the molecule experiences weak long-range dipolar interactions with the slab. We indicate by $F$ the projection of the force on the molecule along the direction connecting the center of mass of the molecule and that of the slab. When the molecule is displaced from $s_{\\textrm{far}}$ to $s$, the corresponding free energy change, $A(s)$, is given by: \n\\begin{align}\\label{eq:pomf}\n    A(s) = \\int_s^{s_{\\textrm{far}}} \\langle F \\rangle^\\tau d \\tau,\n    \\quad\n    \\langle F \\rangle^\\tau = \\frac1Z \\int F e^{-\\beta H(\\{R\\})} \\delta(\\sigma(\\{R\\}) - \\tau) d\\{R\\}.\n\\end{align}\nHere $H(\\{R\\})$ and $Z$ are the Hamiltonian and the partition function of the system. \nFor each value of $\\tau$ the ensemble average in Eq. \\eqref{eq:pomf} is calculated with canonical MD at 300~K for configurations that satisfy the constraint $\\sigma(\\{R\\}) = \\tau$. We compute the ensemble average at a discrete set of $\\tau$ values using 1000~ps long MD trajectories, the first 200 ps of which serve for equilibration and are not included in the observation time. During observation configurations are recorded every 10~ps.\nBy varying $s$ from $s_{\\textrm{far}}$ to $s_{\\textrm{near}} = 9$~\\AA, we obtain the blue free energy profile reported in panel (a) of Fig.~\\ref{fig:fe} for the DPLR model. We divide the interval $[s_{\\textrm{near}}, s_{\\textrm{far}}]$ into 8 equal intervals and calculate the corresponding free energy changes by\nevaluating the integrals in Eq. \\eqref{eq:pomf} numerically with the trapezoidal rule.   \nFor $s$ in the vicinity of $s_{\\textrm{near}}$ the tagged molecule forms hydrogen bonds with neighboring molecules in the slab, suggesting that incorporation into the slab is taking place. \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig-fe.pdf}\n    \\caption{The free energy of water molecule absorption to a water slab.\n    The panel (a) presents the free energy calculated by DFT, DPLR and DP models. \n    The panel (b) presents the error in free energy of the DPLR and DP models with respect to DFT.\n    The error bar in the plot is given by the standard deviation of four independently trained models.\n    The insert is a schematic plot of the system configuration.\n    }\n    \\label{fig:fe}\n\\end{figure}\n\n\nWe use a simple approximation to estimate the deviation of the calculated DPLR profile from the DFT reference. Instead of re-weighting with the DFT energy the configurations along DPLR trajectories, which would be expensive, given the large number of configurations needed for statistical accuracy, we keep the DPLR weights and approximate the ensemble average with the following expression, which only requires DFT calculations at a relatively small number of configurations:\n\\begin{align}\\label{eq:approx}\n    \\langle F_\\mathrm{DFT} \\rangle^\\tau_\\mathrm{DFT} \\approx \\langle F_\\mathrm{DFT} \\rangle^\\tau_\\mathrm{DPLR}.\n\\end{align}\nThis procedure can be justified because DPLR reproduces accurately DFT energies with an average error of $\\sim 1$~meV\/\\chem{H_2O}. \nWe find that 80 configurations, equally spaced in time along an equilibrated 800~ps long trajectory, are sufficient to compute the average force at each $\\tau$ value. \nThe results are reported in panel (b) of Fig.~\\ref{fig:fe}, where the baseline DFT reference corresponds to $\\Delta A(s) = 0$. \nThe deviations of DPLR from the baseline, calculated at discrete distances in the interval $[s_{\\textrm{near}}, s_{\\textrm{far}}]$, appear in the same figure as blue points connected by blue lines. \nThe blue points represent the average of four independent DPLR models trained on the same data with same hyper-parameters but different random initialization of the model parameters.\nThe blue error bars are the corresponding standard deviations of the predictions of the four models. Overall, the deviation of DPLR from DFT is quite small and the uncertainty of the model, estimated from the standard deviation of the four models, is even smaller.     \n\nWe now consider a DP model with same cutoff radius of DPLR but without explicit long-range Coulomb interactions. To estimate the deviation of this model from DPLR  we adopt a similar approximation to the one adopted in Eq.~\\eqref{eq:approx} , i.e., we assume:\n\\begin{align}\n    \\langle F_{\\textrm{DP}} \\rangle^\\tau_{\\textrm{DP}} \\approx \\langle F_{\\textrm{DP}}\\rangle^\\tau_\\mathrm{DPLR},\n\\end{align}\nand use the same configurations as before to calculate the ensemble averages and the free energy differences. The deviations of DP from DPLR and from the baseline DFT reference are reported as red points with error bars connected by red lines in panel (b) of Fig.~\\ref{fig:fe}. As before, the DP points are averages of four independent models trained on the same data with different random initialization. The average free energy profile obtained in this way is reported as a red line in panel (a) of the same figure, showing that when the molecule approaches the slab the gain in free energy is smaller for DP than for DPLR, a result consistent with the missing attractive dipolar interactions in the DP model. However, the most important outcome of this analysis is that the DP model is affected by a large uncertainty. When evaluated along DPLR trajectories the standard deviation of the independent DP models grows rapidly as the tagged molecule approaches the slab to become almost an order of magnitude larger than the uncertainty of the corresponding DPLR model when the tagged molecule is near the slab. We attribute this behavior to the absence of long-range electrostatic interactions in the DP model as opposed to DPLR and DFT.        \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Phonons in crystalline sodium chloride}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{fig-gamma-all.pdf}\n    \\caption{The LO-TO splitting of the phonon spectra of NaCl crystal.\n    DPLR, DP, DFT and experiment (Expt.)~\\cite{raunio1969phonon} are compared. \n    (a) Spectra obtained from the analytical dynamical matrix (see text) on a $2\\times2\\times2$ (64 atoms) supercell.\n    (b) Spectra obtained from the full dynamical matrix, including  analytical and non-analytical correction (NAC) terms (see text), on a $2\\times2\\times2$ supercell.\n    (c-e) Spectra obtained from the analytical dynamical matrix on a $3\\times3\\times3$ (216 atoms) (c), $5\\times5\\times5$ (1000 atoms) (d), and $6\\times6\\times6$ (1728 atoms) (e) supercell, respectively.\n    }\n    \\label{fig:nacl}\n\\end{figure}\n\nIn this section we consider the phonon dispersion curves in the sodium-chloride (NaCl) crystal, as a further example to illustrate the importance of long-range electrostatic interactions. \nFor the reference DFT model we adopt the PBE exchange-correlation functional and use pseudopotentials for the electron-ion interactions. DP and DPLR models for NaCl are trained on DFT data generated in a 200~ps DPMD NPT trajectory of a $2\\times 2\\times 2$ supercell (64 atoms) at 100~K and 1~Bar. In the DFT data we use  Brillouin Zone (BZ) sampling with  \na $4\\times 4\\times 4 $ Monkhorst Pack mesh to calculate the interatomic potential and the maximally localized Wannier distributions. The hyper-parameters of the DP and DPLR models are the same as in the water models discussed earlier in the manuscript. Using the same arguments of \nSect.~\\ref{sec:lrdp-train}, we find that the optimal value of the spread parameter for NaCl is $\\beta = 0.2$~\\AA$^{-1}$. The smaller $\\beta$ of NaCl compared to water reflects the fact that the electron density distribution about the Cl atom in NaCl is more delocalized than the one about the O atom in the water molecule. The training accuracy of the DP model is $7.6\\times 10^{-4}$~eV\/atom for energy and $2.0\\times 10^{-3}$~eV\/\\AA\\ for atomic forces,  similar to that of  the DPLR model, which is $7.7\\times 10^{-4}$~eV\/atom for energy and $2.6\\times 10^{-3}$~eV\/\\AA\\ for atomic forces. \n\n\n\nTo compute the phonons, we calculate numerically the force constants from the second derivatives of the PES with respect to atomic displacements. We then obtain the phonon frequencies at a generic point $\\bm q$ of the BZ of the crystal with the \\texttt{phonopy} package~\\cite{phonopy}, by diagonalizing the dynamical matrix given by a Fourier expansion of the force constants at wavevector $\\bm q$. This approach would be adequate in non-polar materials where the force constants have finite range. In polar materials, like NaCl, long-range electrostatic interactions yield a non-analytic contribution to the dynamical matrix, which should be added to the analytic contribution calculated with the above procedure. We report in panel (a) of Fig.~\\ref{fig:nacl} the phonon dispersion curves along the $\\Gamma$-$X$ segment of the BZ obtained from the analytical part of the dynamical matrix by using a $2\\times 2\\times 2$ supercell for the force constants. DP and DPLR curves coincide within the accuracy of the calculation, and are very close to the corresponding DFT curves, as one could have expected given that a $2\\times 2\\times 2$ supercell was used for training the models. Experimental frequencies are also reported in the same panel, from which we see that, while the calculated acoustic and transverse optical (TO) modes agree with experiment, the calculated longitudinal optical (LO) modes deviate considerably from experiment, particularly in the long wavelength limit, where the calculated modes fail to exhibit an LO-TO splitting. Adding the non-analytic contribution to the dynamical matrix restores agreement between theory and experiment, as illustrated in panel (b) of Fig.~\\ref{fig:nacl}. The non-analytic contribution to the dynamical matrix is given by~\\cite{baroni2001phonons}:\n\\begin{align}\\label{eqn:nac}\n\\begin{aligned}\n     \\tilde{D}^{\\textrm{NA}}_{\\kappa\\alpha,\\kappa'\\beta}(\\bm q)\n    &=\n    \\frac{4\\pi}{\\Omega \\sqrt{M_\\kappa M_{\\kappa'}}}\n    \\frac{ \n    (\\sum_\\gamma q_\\gamma Z^\\ast_{\\kappa,\\gamma\\alpha})\n    (\\sum_{\\gamma'} q_{\\gamma'} Z^\\ast_{\\kappa',\\gamma'\\beta})\n    }{\n    \\sum_{\\alpha\\beta} q_\\alpha \\epsilon_{\\alpha\\beta}^\\infty q_\\beta\n    }\n\\end{aligned}\n\\end{align}\nHere, $\\Omega$ is the volume, $\\kappa,\\kappa'$ are atom indices, $\\alpha,\\beta,\\gamma$ are Cartesian directions, ${M_\\kappa},{M_{\\kappa'}}$ are atomic masses, $Z^\\ast_\\kappa,Z^\\ast_{\\kappa'}$ are Born dynamical charge tensors, and $\\epsilon^\\infty$ is the dielectric tensor that describes the static response of the electrons at fixed ions. We recall that the dynamical Born charges are the derivatives of the polarization with respect to atomic displacements, and are therefore directly accessible from the DW network within DPLR. The dielectric tensor is related to the dielectric susceptibility or polarizability tensor $\\chi$ via $\\epsilon^\\infty = 1+4\\pi\\chi$. The susceptibility is defined by the derivatives of the polarization with respect to an applied electric field and is accessible within a DW extension introduced to model the polarizability surface~\\cite{sommers2020raman}. Thus, all the quantities needed for calculating the phonons in polar materials are accessible within the extended DP methodology. \n\nIt is interesting to monitor the evolution of the analytical phonon modes calculated within DP and DPLR when the size of the supercell used to compute the force constants is increased. This is shown in panel (c), (d), and (e) of Fig.~\\ref{fig:nacl} for a $3\\times3\\times3$ supercell (216 atoms), a $5\\times5\\times5$ supercell (1000 atoms), and a $6\\times6\\times6$ supercell (1728 atoms), respectively. The DP modes are essentially independent of size, indicating that, with the chosen cutoff radius ($r_c=6$~\\AA), the force constants for atomic distances larger than those probed in a $2\\times2\\times2$ supercell are negligible. By contrast, the LO mode of DPLR is strongly size dependent, reflecting the appearance of not negligible force constants on larger supercells due to long-range dipolar interactions. With larger supercells the analytical spectra from DPLR recover a larger portion of the correct LO modes. However, the convergence is slow as the non-analytic behavior for $\\bm q \\rightarrow 0$ cannot be recovered from numerical Fourier interpolation. At small wavevectors we observe an upward bump in the LO and TO modes that becomes sharper and moves to smaller $\\bm q$ values as the size of the supercell increases. The bump is a manifestation of the Gibbs oscillations that are expected to occur in place of the LO-TO splitting when attempting to reproduce the non-analytic behavior of the dynamical matrix with a Fourier sum~\\cite{gonze1994interatomic}.\n\nThe long range contributions that are responsible for the spectral changes in panels (c), (d), and (e) derive from the Ewald contribution to the PES within DPLR. This example illustrates an essential advantage of DPLR over DP in materials where long-range electrostatic effects are important. While the DP model can describe well such materials on the supercell used for training, it is unable to extrapolate the PES to larger supercells, a drawback that is absent in the DPLR model.   \n\n\n\n\n\n\\section{External fields}\n\nIn this manuscript we have considered the electrostatic fields originating from internal charges, but, in the linear regime, it is straightforward to include the effect of an external time dependent field $\\mathcal E_\\textrm{ext}(t)$ that couples with the polarization $P(\\{R\\})$. \nIn this situation the PES takes the form:\n\\begin{align}\\label{eq:concl-1}\n    E(\\{R\\}) = \n    E_{\\textrm{DPLR}\/\\textrm{DP}}(\\{R\\}) -\n    P(\\{R\\}) \\cdot \\mathcal E_\\textrm{ext}(t)\n\\end{align}\nIn Eq.~\\eqref{eq:concl-1} the PES in absence of external field can be provided either by DPLR or by DP, depending on whether the size dependence of the electrostatic energy should be included explicitly or not. \nEq.~\\eqref{eq:concl-1} makes possible non-equilibrium MD simulations of an insulating system driven by an external electric field. The response of the system to the external field can also be studied with equilibrium MD using the Kubo formalism, as it was done in Ref.~\\citen{zhang2020deep} to study the infrared absorption spectra of liquid water. \nIn that work the standard DP model for the PES was used, which was adequate because at the infrared frequencies long-range electrostatic effects can be ignored as the dominant correlations are between neighboring molecular dipoles~\\cite{chen2008role}. Long-range correlations among the molecular dipoles become important in the static ionic limit~\\cite{ballenegger2004structure}.\nIn that limit, which is relevant to study the static dielectric constant of water, one should use the DPLR model for the PES in Eq.~\\eqref{eq:concl-1}.              \n \nEq.~\\eqref{eq:concl-1} can be further extended by including the quadratic response of the electrons via the susceptibility tensor $\\chi$:\n\\begin{align}\n    E(\\{R\\}) = \n    E_{\\textrm{DPLR}\/\\textrm{DP}}(\\{R\\}) - \n    P(\\{R\\}) \\cdot \\mathcal E_\\textrm{ext}(t) + \n    \\frac12\\mathcal E_\\textrm{ext}(t) \\cdot \\chi(\\{R\\}) \\cdot \\mathcal E_\\textrm{ext}(t)\n\\end{align}\nThis approach was used in the context of the Kubo formalism to compute the Raman spectra of water in Ref.~\\citen{sommers2020raman}, a study that required the environmental dependence of the electric susceptibility.\nThe latter was modeled by a DNN extending the DW approach to the polarizability tensor. \n\\recheck{The polarizability tensor is defined by the derivatives (i.e. the response) of the Wannier centroid positions with respect to an applied electric field in the limit of zero field. These derivatives are described by an extension of the DW model trained on DFT calculations in presence of a small, i.e. numerically infinitesimal, electric field, as discussed in Ref.~\\citen{sommers2020raman}.}\nThe Raman scattering response is dominated by short range correlations between molecular polarizabilities, and the standard DP model was sufficient for the PES.\n \n\n\\section{Conclusions}\n\nIn this manuscript we examined a fundamental limitation of the local representation of the PES common to most ML models. These models are trained on quantum mechanical data, typically DFT data, on relatively small systems. They are then used in molecular simulations on larger systems that become accessible in view of the computational efficiency of the models. While successful in many cases, this procedure may lead to errors for physical properties affected by long-range electrostatic interactions. We have shown that these effects can be modeled accurately by the DPLR model, which extends the DP methodology using information on the centers of the electronic charge. The latter is encoded in the location of the Wannier centroids, which are averages of the Wannier centers uniquely asssociated to specific atoms. The environmental dependence of the Wannier centroids is described by DW, a DNN model, which we introduced in previous work to describe the dielectric polarization of insulators. Using the information from DW, DPLR augments the local representation of the PES of the standard DP model with the long-range electrostatic energy of ions and electrons, modeled by spherical Gaussian charge distributions centered at ions and Wannier centroids, respectively. The width of the Gaussian distributions is fixed by the value of the spread parameter $\\beta$. Interestingly, the optimal $\\beta$ that minimizes the generalization gap in the ML procedure, is close to the physical value expected from the spatial delocalization of the reference Wannier centroid distribution within DFT, as illustrated in our water and NaCl examples. The variation with size of the long-range electrostatic contribution, ignored in the standard DP model, is approximated accurately in the DPLR model, under the assumption that the electrostatic fields from which this contribution originates are a weak perturbation that can be treated within linear response theory. Because of that, it is possible to learn the environmental dependence of the Wannier centroids ignoring the size dependence of the electrostatic energy as done in the DW scheme.         \n\n\nDPLR uses two DNN models to represent the PES: the standard DP network for the short-range interactions and the DW network for computing long-range electrostatic interactions with the Ewald method.\nBecause of the complex architecture of the DPLR network, whereby force calculations require backward propagation within both DP and DW, and because of the need for Ewald calculations, DPLR simulations are more expensive than standard DP calculations.\nBased on our experience, an increase of the computational burden of \napproximately a factor of 5 should be expected\nwhen using DPLR in place of DP. The decision on which of the two models should be used in a given application should be guided by the physical properties of interest, whether long-range correlations originating by Coulomb forces are important or not. \n\nIn developing the DPLR model we assumed that long-range electrostatic effects beyond the cutoff radius of the DP model originate a weak dependence on size that can be described within the linear response regime. This seems a reasonable assumption based on our experience and on the examples discussed in this manuscript. We cannot exclude, however, that effects of long-range electrostatics beyond the linear response regime may manifest in some circumstances. In these cases, the assumption made here that the environmental dependence of the Wannier centroids is not affected by size should be revisited, and one should envision a self-consistent condition for the centroids under the action of the long-range electrostatic field along lines similar to those developed in Ref.~\\citen{gao2021self}. We expect that a better understanding of all these issues should emerge from widespread application of the DPLR model.\n\n\\recheck{\nThe major simplifying assumption of the current DPLR model is that the WCs are uniquely associated to specific atoms. Thus, they cannot split or recombine along molecular dynamics trajectories. This assumption limits the capability of the model to deal with chemical reactions: proton transfer reactions like those discussed, for example, in Ref.~\\citen{zhang2020deep} and in Ref.~\\citen{andrade2020free}, are allowed, but not, in general, electron transfer reactions, in which an electron is donated by one atomic entity to another. The centroid assumption shall be relaxed,\nwithin the limits of adiabatic ground-state dynamics, in a future generalization of the DPLR model.  \n}\n\n\n\n\\section*{Data availability}\n\\recheck{All data and codes needed to reproduce this work, including DP, DW, and DPLR models, are publicly available at \\url{https:\/\/doi.org\/10.5281\/zenodo.6024644}.}\n\n\\section*{Acknowledgement}\nThis work was supported by the Computational Chemical Sciences Center ``Chemistry in Solution and at Interfaces\" funded by the US Department of Energy under Award No. DE-SC0019394.  \nThe work of H.W.~was supported by the National Science Foundation of China under Grant No.11871110 and 12122103.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $E$ be an elliptic curve defined over the rational field $\\mathbb{Q}$.\nFor a prime $p$ where $E$ has good reduction, we let $E_p$\ndenote the reduced curve modulo $p$ and $\\#E_p(\\mathbb{F}_p)$\nthe number of $\\mathbb{F}_p$-rational points.\nThen  the trace of the Frobenius morphism at $p$, $a_p(E)$, satisfies\nthe well-known identity\n$\\#E_p(\\mathbb{F}_p)=p+1-a_p(E)$ and the Hasse bound\n$|a_p(E)|<2\\sqrt p$.\n\nLet $N$ be a positive integer. We are interested in the number of primes for\nwhich $\\#E_p(\\mathbb{F}_p)=N$.  In particular, we are interested in the behavior of the prime\ncounting function\n\\begin{equation*}\nM_E(N):=\\#\\{p: \\#E_p(\\mathbb{F}_p)=N\\}.\n\\end{equation*}\nNote that if $\\#E_p(\\mathbb{F}_p)=N$, then the Hasse bound implies\n$(\\sqrt p-1)^2<N<(\\sqrt p+1)^2$,\nwhich in turn implies that\n\\begin{equation*}\\label{p range}\nN^-:=(\\sqrt N-1)^2<p<(\\sqrt N+1)^2=:N^+.\n\\end{equation*}\nHence, $M_E(N)$ is a finite number, and we have the trivial bound\n\\begin{equation}\\label{trivial bound}\nM_E(N)\\ll \\frac{\\sqrt N}{\\log(N+1)}.\n\\end{equation}\nIn~\\cite{Kow:2006}, Kowalski shows that if $E$ possesses complex multiplication (CM), then\n\\begin{equation}\\label{Kowalski's question}\nM_E(N)\\ll_{E,\\varepsilon}N^\\varepsilon\n\\end{equation}\nfor any $\\varepsilon > 0$.\nHe asks if the same might be true for curves without CM.\nHowever, no bound between~\\eqref{trivial bound}\nand~\\eqref{Kowalski's question} is known for curves without CM.\n\nGiven an integer $N$, it is always possible through a Chinese Remainder Theorem\nargument to find an elliptic curve $E$ that achieves the upper bound~\\eqref{trivial bound}, i.e.,\nsuch that $\\#E_p(\\mathbb{F}_p)=N$ for every prime $p$ in the interval $(N^-,N^+)$.\nYet, for a fixed curve, one expects $M_E(N)$ to be quite small.\nConsider the following na\\\"ive probabilistic model for $M_E(N)$.\nIf we suppose that the values of $\\# E(\\mathbb{F}_p)$ are uniformly distributed, i.e., that\n\\begin{equation} \\label{naivemodel}\n\\mbox{Prob} \\left( \\# E(\\mathbb{F}_p) = N \\right) =\n\\begin{cases}\n\\frac{1}{4\\sqrt{p}} &\\text{if } N^-<p<N^+, \\\\\n 0 & \\text{otherwise,}\n\\end{cases}\n\\end{equation}\nthen we expect that\n\\begin{equation}\\label{expect M}\n\\begin{split}\nM_E(N)\n&\\approx\\sum_{p} \\mbox{Prob} \\left( \\# E(\\mathbb{F}_p) = N \\right) = \\sum_{N^-<p<N^+} \\frac{1}{4 \\sqrt{p}} \\\\\n&\\approx\\frac{1}{4 \\sqrt{N}} \\int_{N^-}^{N^+} \\frac{\\mathrm{d} t}{\\log{t}} \\sim \\frac{1}{\\log{N}}.\n\\end{split}\n\\end{equation}\nMoreover, it is quite easy to show (see~\\cite{Kow:2006} for example) that\n\\begin{equation}\\label{reg avg of M}\n\\sum_{N\\le X}M_E(N)=\\pi(X)+O(\\sqrt X)\\sim\\frac{X}{\\log X},\n\\end{equation}\nwhere as usual, $\\pi(X):=\\#\\{p\\le X: p \\text{ is prime}\\}$.\nTherefore, the average order of $M_E(N)$ is $1\/\\log N$\nin accordance with the above model.\nPerhaps the correct way to interpret these statements is to say that\n$M_E(N)$ must be equal to zero on a density one subset of the integers for the mere reason that\nthe primes are a subset of the integers of density zero.\nFinally, we note that while it is not difficult to see that $\\liminf M_E(N)=0$, \nnumerical computations~\\cite{Kow:2006} are consistent with the possibility that $\\limsup M_E(N)=\\infty$.\nIn fact, using the model~\\eqref{naivemodel} as in~\\cite{Kow:2006}, it is possible to predict this.\n\n\\section{Statement of results}\n\nIn this paper, we study the average for $M_E(N)$  \nover all elliptic curves over $\\mathbb{Q}$ (and not over $N$ as in (\\ref{reg avg of M})).\nGiven integers $a$ and $b$, let $E_{a,b}$ be the elliptic curve defined by the\nWeierstrass equation\n\\begin{equation*}\nE_{a,b}: y^2=x^3+ax+b.\n\\end{equation*}\nFor $A,B>0$, we define a set of Weierstrass equations by\n\\begin{equation*}\n\\mathscr{C}(A,B):=\\{E_{a,b}: |a|\\le A, |b|\\le B, \\Delta(E_{a,b})\\ne 0\\}.\n\\end{equation*}\nThe following is our first main result.\n\n\\begin{thm}\\label{upper bound thm}\nIf $A,B\\ge \\sqrt N\\log N$ and $AB\\ge N^{3\/2}(\\log N)^2$, then\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\\ll \\frac{\\log{\\log{N}}}{\\log{N}}\n\\end{equation*}\nholds uniformly for $N\\ge 3$.\n\\end{thm}\n\\begin{rmk}\nWe refer to the expression on the left hand side of the above inequality as the average order of\n$M_E(N)$ taken over the family $\\mathscr{C}(A,B)$.  \n\\end{rmk}\n\n\nUnder an additional hypothesis  concerning the\nshort interval distribution of primes in arithmetic progressions, we can prove an asymptotic\nformula for the average order of $M_E(N)$ over $\\mathscr{C}(A,B)$.\nIn particular, we note that all of the primes counted\nby $M_E(N)$ are of size $N$ lying in an interval of length $4\\sqrt N$.\nTherefore, we require an appropriate\nshort interval version of the Barban-Davenport-Halberstam Theorem.\n\nGiven real parameters $X,Y>0$ and integers $q$ and $a$,\nwe let $\\theta(X,Y;q,a)$ denote the\nweighted prime counting function\n\\begin{equation*}\n\\theta(X,Y;q,a):=\\sum_{\\substack{X<p\\le X+Y\\\\ p\\equiv a\\pmod q}}\\log p,\n\\end{equation*}\nand we let $E(X,Y;q,a)$ be the error in approximating $\\theta(X,Y;q,a)$ by $Y\/\\varphi(q)$.\nThat is,\n\\begin{equation*}\nE(X,Y;q,a):=\\theta(X,Y;q,a)-\\frac{Y}{\\varphi(q)}.\n\\end{equation*}\n\n\\begin{conj}\\label{bdh conj}\n(Barban-Davenport-Halberstam for intervals of length $X^\\eta$)\n\\label{shortBDH} Let $0 < \\eta \\leq 1$, and let $\\beta > 0$ be arbitrary. Suppose\nthat $X^\\eta \\leq Y \\leq X$, and that $Y \/ (\\log{X})^\\beta \\leq Q \\leq Y.$\nThen\n\\begin{align*}\n\\sum_{q \\leq Q}\\sum_{\\substack{a=1\\\\ (a,q)=1}}^{q}\\left| E(X,Y;q,a)\\right|^2 \\ll YQ \\log{X}.\n\\end{align*}\n\\end{conj}\n\\begin{rmk}\nIf $\\eta=1$, this is essentially the classical Barban-Davenport-Halberstam Theorem.\nSee for example~\\cite[p.~196]{Dav:1980}.\nThe best results known are due to Languasco, Perelli, and Zaccagnini~\\cite{LPZ:2010} who\nshow that for any $\\epsilon>0$, Conjecture~\\ref{shortBDH} holds unconditionally for\n$\\eta = 7\/12 + \\epsilon$ and for $\\eta=1\/2+\\epsilon$ under the\nGeneralized Riemann Hypothesis.\nFor our application, we essentially need $\\eta=1\/2-\\epsilon$.\n\\end{rmk}\n\n\\begin{thm}\\label{main thm}\nLet $\\gamma>0$,\nand assume that Conjecture~\\ref{bdh conj} holds with\n$\\eta=\\frac{1}{2}-(\\gamma+2)\\frac{\\log\\log N}{\\log N}$.\n Suppose further that $A,B\\ge \\sqrt N(\\log N)^{1+\\gamma}\\log\\log N$ and that\n$AB\\ge N^{3\/2}(\\log N)^{2+\\gamma}\\log\\log N$.\nThen for any odd integer $N$, we have\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n=K(N)\\frac{N}{\\varphi(N)\\log N}+O\\left(\\frac{1}{(\\log N)^{1+\\gamma}}\\right),\n\\end{equation*}\nwhere\n\\begin{equation*}\nK(N):=\n\t\\prod_{\\ell \\nmid N}\\left(\n\t1-\\frac{\\leg{N-1}{\\ell}^2\\ell+1}{(\\ell-1)^2(\\ell+1)}\n\t\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{1}{\\ell^{\\nu_\\ell(N)}(\\ell-1)}\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{\\ell-\\leg{-N_\\ell}{\\ell}}\n\t{\\ell^{\\nu_\\ell(N)+1}(\\ell-1)}\\right),\n\\end{equation*}\n$\\nu_\\ell$ denotes the usual $\\ell$-adic valuation, and\n$N_\\ell:=N\/\\ell^{\\nu_\\ell(N)}$ denotes the $\\ell$-free part of $N$.\n\\end{thm}\n\\begin{rmk}\nWe note that $K(N)$ is uniformly bounded as a function of $N$.\nWe also note that $N\/\\varphi(N)\\ll\\log\\log N$ (see~\\cite[Theorem 328]{HW:1979} for example), which\ngives the upper bound of Theorem \\ref{upper bound thm}.\nWorking with V. Chandee and D. Koukoulopoulos, the authors have recently shown that the upper bound \nimplicit in Theorem~\\ref{main thm} holds unconditionally.  That is, Theorem~\\ref{upper bound thm} holds \nwith $\\log\\log N$ replaced by $N\/\\varphi(N)$.\n\\end{rmk}\n\n\nThe average of Theorem~\\ref{main thm} displays some\ninteresting characteristics that are not present in the average order~\\eqref{reg avg of M}.\nIn particular, the main term of the average in Theorem~\\ref{main thm}\ndoes not depend solely on the size of the integer $N$ but also on some arithmetic properties of\n$N$ as it involves the factor $K(N)N\/\\varphi(N)$.\nThe occurrence of the weight $\\varphi(N)$ appearing in the denominator  seems to suggest that\nthis is another example of the Cohen-Lenstra Heuristics~\\cite{CL:1984,CL:1984-2},\nwhich predict that random groups $G$ occur with probability weighted by $1\/\\#\\mathrm{Aut}(G)$.\nNotice that if as an additive group $E(\\mathbb{F}_p) \\simeq\\mathbb{Z}\/N\\mathbb{Z}$,\nthen $\\#\\mathrm{Aut}(E(\\mathbb{F}_p)) = \\varphi(N)$.  Indeed, the Cohen-Lenstra Heuristics predict that\nrelative to other groups of same size, the cyclic groups are the most likely to occur since they have the \nfewest number of automorphisms.\n\nIn some recent work, the authors explored this connection further by\nconsidering the average of\n\\begin{equation*}\nM_E(G) := \\# \\left\\{ p : E(\\mathbb{F}_p) \\simeq G \\right\\}\n\\end{equation*}\nfor those Abelian groups $G$ which may arise as the group of $\\mathbb{F}_p$-rational points of\nan elliptic curve.  \nThis is the subject of a forthcoming paper~\\cite{DS2}.\nGiven an elliptic curve $E$, it is well-known that\n\\begin{equation*}\nE(\\mathbb{F}_p) \\simeq \\mathbb{Z}\/N_1 \\mathbb{Z} \\times \\mathbb{Z} \/ N_1 N_2 \\mathbb{Z},\n\\end{equation*}\nfor some positive integers $N_1, N_2$ satisfying the Hasse bound: $|p+1-N_1^2 N_2| \\leq 2 \\sqrt{p}$.\nUnder Conjecture~\\ref{bdh conj}, it is shown in~\\cite{DS2} that\nfor every odd order group $G = \\mathbb{Z}\/N_1 \\mathbb{Z} \\times \\mathbb{Z}\/N_1 N_2 \\mathbb{Z}$, we have that\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_E(G) \\sim\n\t K(G) \\frac{\\# G}{\\# \\mathrm{Aut}(G)\\log(\\#G)},\n\\end{equation*}\nprovided that $A,B$, and the exponent of $G$ (the size of the largest cyclic subgroup) are large enough with \nrespect to $\\#G=N_1^2 N_2$.  The function $K(G)$ is explicitly computed and shown to be \nnon-zero and absolutely bounded as a function of $G$.\n\n\nWe can express the results of Theorem~\\ref{main thm} as stating that for a ``random curve\"\n$E\/\\mathbb{Q}$ and a ``random prime\" $p \\in (N^-, N^+)$,\n\\begin{eqnarray*}\n\\mathrm{Prob}\\left(\\#E(\\mathbb{F}_p) = N \\right)\n\\approx \\frac{K(N) \\frac{N}{\\varphi(N) \\log(N)}}{ \\frac{4 \\sqrt{N}}{\\log{N}}}\n= \\frac{K(N) N}{\\varphi(N)} \\frac{1}{4\\sqrt{N}}\n\\end{eqnarray*}\nrefining the na\\\"ive model given by~\\eqref{naivemodel}.\nHere as in~\\eqref{naivemodel}, we make the assumption that there are about\n$4\\sqrt N\/\\log N$ primes in the interval $(N^-,N^+)$ though we can not justify such an\nassumption even under the Riemann Hypothesis.\n\nThere are many open conjectures about the distributions of invariants associated with the \nreductions of a fixed elliptic curve over the finite fields $\\mathbb{F}_p$ such as the famous conjectures \nof Koblitz~\\cite{Kob:1988} and of Lang and Trotter~\\cite{LT:1976}.\nThe Koblitz Conjecture concerns the number of primes $p\\le X$ such that $\\#E(\\mathbb{F}_p)$ is prime.\nThe fixed trace Lang-Trotter Conjecture concerns the number of primes $p\\le X$ such that\nthe trace of Frobenius $a_p(E)$ is equal to a fixed integer $t$.  Another \nconjecture of Lang and Trotter (also called the Lang-Trotter Conjecture) \nconcerns the number of primes\n$p\\le X$ such that the Frobenius field $\\mathbb{Q}(\\sqrt{a_p(E)^2-4p})$ is a fixed \nimaginary quadratic field $K$.\n\nThese conjectures are all completely open.\nTo gain evidence, it is natural to consider the averages for these conjectures over \nsome family of elliptic curves.\nThis has been done by various authors originating with the work of Fouvry and \nMurty~\\cite{FM:1996} for the number of supersingular primes (i.e., the fixed trace Lang-Trotter \nConjecture for $t=0$).  See~\\cite{DP:1999,DP:2004,Jam:2004,BBIJ:2005,JS:2011,CFJKP:2011} \nfor other averages regarding the fixed trace Lang-Trotter Conjecture.\nThe average order for the Koblitz Conjecture was considered in~\\cite{BCD:2011}.\nVery recently, the average has been successfully carried out for the Lang-Trotter Conjecture\non Frobenius fields~\\cite{CIJ:pp}.\nThe average order that we consider in this paper displays a very different character than \nthe above averages.   This is primarily because the size of primes considered varies with the \nparameter $N$. Moreover, they all must lie in a very short interval.\nThis necessitates the use of a short interval version of the Barban-Davenport-Halberstam Theorem \n(Conjecture~\\ref{bdh conj} above).\nThis is also the first time that one observes a Cohen-Lenstra phenomenon governing the \ndistribution of the average.\n\n\n\\section{Acknowledgement}\nThe authors would like to thank K. Soundararajan for pointing out how to improve\nthe average upper bound of Theorem \\ref{upper bound thm} by using short Euler products\nholding for almost all characters. They also thank Henri Cohen, Andrew Granville and\nDimitris Koukoulopoulos for useful discussions related to this work and Andrea  Smith for a\ncareful reading of the manuscript.\n\n\n\\section{Reduction to an average of class numbers}\n\nGiven a (not necessarily fundamental) discriminant $D<0$, we follow Lenstra~\\cite{Len:1987}\nin defining the \\textit{Kronecker class number} of discriminant $D$ by\n\\begin{equation}\\label{defn of K-class no}\nH(D):=\\sum_{\\substack{f^2|D\\\\ \\frac{D}{f^2}\\equiv 0,1\\pmod{4}}}\\frac{h(D\/f^2)}{w(D\/f^2)},\n\\end{equation}\nwhere $h(d)$ denotes the (ordinary) class number of the unique imaginary quadratic\norder of discriminant $d<0$ and $w(d)$ denotes the cardinality of its unit group.\n\\begin{thm}[Deuring]\\label{deuring}\nLet $p>3$ be a prime and $t$ an integer such that $t^2-4p<0$.\nThen\n\\begin{equation*}\n\\sum_{\\substack{\\tilde{E}\/\\mathbb{F}_p\\\\ a_p(E)=t}}\\frac{1}{\\#\\mathrm{Aut}(E)}\n=H(t^2-4p),\n\\end{equation*}\nwhere the sum is over the $\\mathbb{F}_p$-isomorphism classes of elliptic curves.\n\\end{thm}\n\\begin{proof}\nSee~\\cite[p.~654]{Len:1987}.\n\\end{proof}\n\nThe first step in computing the average order of $M_E(N)$ over $\\mathscr{C}(A,B)$ is to reduce to an\naverage of class numbers by using Deuring's Theorem. The following estimate\nwill then be crucial to obtain the upper bound of Theorem~\\ref{upper bound thm}, and is also used\nin getting an optimal average length in Theorem~\\ref{main thm}.\n\n\\begin{prop}\\label{upper bound for H on avg}\nFor primes $p$ in the range $N^-<p<N^+$, we define the quadratic polynomial\n\\begin{equation}\\label{defn of D}\nD_N(p):=(p+1-N)^2-4p=p^2-2(N+1)p+(N-1)^2.\n\\end{equation}\nThen\n\\begin{equation} \\label{sumofthethm}\n\\sum_{N^-<p<N^+}H\\left(D_N(p)\\right)\\ll \\frac{N \\log{\\log{N}}}{\\log{N}}.\n\\end{equation}\n\\end{prop}\n\nBefore giving the proof of this result, we define\nsome notation that we will use throughout the remainder of the article.\nGiven a negative discriminant $d$, we write $\\chi_d$ for the Kronecker symbol $\\leg{d}{\\cdot}$.\nSince $d$ is not a perfect square, the Dirichlet $L$-series defined by\n\\begin{equation*}\nL(s,\\chi_d):=\\sum_{n=1}^\\infty\\frac{\\chi_d(n)}{n^s}\n\\end{equation*}\nconverges at $s=1$.\nFinally, given a positive integer $f$, we let\n\\begin{equation}\\label{defn of d}\nd_{N,f}(p):=\\frac{D_N(p)}{f^2},\n\\end{equation}\nwhere $D_N(p)$ is defined by~\\eqref{defn of D}.\n\n\\begin{proof}[Proof of Proposition~\\ref{upper bound for H on avg}.]\nThe definition of the Kronecker class number~\\eqref{defn of K-class no} and the\nclass number formula~\\cite[p.~515]{IK:2004}\n\\begin{equation}\\label{class no formula}\n\\frac{h(d)}{w(d)}=\\frac{\\sqrt{|d|}}{2\\pi}L(1,\\chi_d)\n\\end{equation}\ngive us the identity\n\\begin{equation*}\n\\sum_{N^-<p<N^+} H(D_N(p))\n= \\sum_{N^-<p<N^+}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{\\sqrt{|D_N(p)|}}{2\\pi f}L(1,\\chi_{d_{N,f}(p)}).\n\\end{equation*}\nSince $|D_N(p)|\\le 4N$, this yields\n\\begin{equation}\\label{use bound for D}\n\\begin{split}\n\\sum_{N^-<p<N^+} H(D_N(p))\n&\\ll\\sqrt N\\sum_{N^-<p<N^+}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}.\n\\end{split}\n\\end{equation}\n\n\nIn order to obtain an optimal bound for this expression, we will use the fact that\nfor almost all primitive Dirichlet characters $\\psi$, $L(1,\\psi)$ is well-approximated by a very\nshort Euler product.  More precisely, fix any integer $\\alpha\\ge1$.\nThen by~\\cite[Proposition 2.2]{GS:2003}, we know that\n\\begin{equation} \\label{shortEP}\nL(1, \\psi) = \\prod_{\\ell \\leq (\\log{Q})^\\alpha} \\left( 1 - \\frac{\\psi(\\ell)}{\\ell} \\right)^{-1}\n\t \\left( 1 + \\underline{o}(1) \\right)\n\\end{equation}\nfor all but at most $Q^{2\/\\alpha + 5 \\log{\\log{\\log{Q}}}\/\\log{\\log{Q}}}$ of the primitive characters of conductor less than $Q$. We remark that this gives a good upper bound for\n$L(1,\\chi)$ whenever $\\chi$ is a Dirichlet character modulo $q\\le Q$ which is induced by\na primitive character $\\psi$ satisfying~\\eqref{shortEP}.\nIndeed, let $\\chi$ be such a Dirichlet character, and let $\\psi$ be the primitive character that\ninduces $\\chi$. Then by~\\eqref{shortEP}, we have\n\\begin{equation*}\n\\begin{split}\nL(1, \\chi)\n&= \\prod_{\\ell \\mid q} \\left( 1 - \\frac{\\psi(\\ell)}{\\ell} \\right)\n \\prod_{\\ell\\le (\\log{Q})^\\alpha}\\left( 1-\\frac{\\psi(\\ell)}{\\ell}\\right)^{-1}\\left(1+\\underline{o}(1) \\right)\\\\\n&=\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1-\\frac{\\psi(\\ell)}{\\ell} \\right)\n\\prod_{\\ell \\leq (\\log{Q})^\\alpha} \\left( 1 - \\frac{\\chi(\\ell)}{\\ell} \\right)^{-1}\n\\left(1+\\underline{o}(1) \\right)\\\\\n&\\ll\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1+\\frac{1}{\\ell} \\right)\n\\prod_{\\ell \\leq (\\log{Q})^\\alpha} \\left( 1 - \\frac{1}{\\ell} \\right)^{-1}\\\\\n&\\ll\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1+\\frac{1}{\\ell} \\right)\\log\\log Q,\n\\end{split}\n\\end{equation*}\nwhere the last line follows by Mertens' formula~\\cite[p.~34]{IK:2004} since $\\alpha$ is fixed.\nFor the remaining product, we observe that\n\\begin{equation*}\n\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1+\\frac{1}{\\ell} \\right)\n\\le\\exp\\left\\{\\sum_{\\substack{\\ell\\mid q\\\\ \\ell>(\\log Q)^\\alpha}}\\frac{1}{\\ell}\\right\\}\n\\le\\exp\\left\\{\\frac{\\omega(q)}{(\\log Q)^{\\alpha}}\\right\\}\n\\le\\exp\\left\\{\\frac{(\\log q)^{1-\\alpha}}{\\log 2}\\right\\},\n\\end{equation*}\nwhere $\\omega(q)$ denotes the number of distinct prime factors of $q$.\nTherefore, since $\\alpha\\ge1$,  we may conclude that\nif $\\chi$ is a character of modulus $q\\le Q$ and~\\eqref{shortEP} holds for the primitive\ncharacter inducing $\\chi$, then\n\\begin{equation}\\label{nexcept-L bound}\nL(1,\\chi)\\ll\\log\\log Q.\n\\end{equation}\n\nWe make use of this fact in~\\eqref{use bound for D} as follows.\nLet $d_N^*(p)$ be the discriminant of the imaginary\nquadratic field $\\mathbb{Q}(\\sqrt{D_N(p)})$.  Then $d_N^*(p)$ is a fundamental discriminant,\nand $\\chi_{d_N^*(p)}$ is the primitive character inducing every character of the set\n$\\{\\chi_{d_{N,f}(p)}: f^2\\mid D_N(p)\\}$.  Furthermore, $|d_N^*(p)|$ is the conductor of\neach of these characters, and $3\\le |d_N^*(p)|\\le 4N$.\nNow fix some $\\alpha>100$, and\nlet $\\mathscr E(Q)$ be the set of primitive characters of conductor less than or equal to $Q$\nfor which~\\eqref{shortEP} does not hold.  Then $\\# \\mathscr E(4N)\\ll N^{1\/50}$.\nWe now divide the outer sum over $p$ on the right-hand side of ~\\eqref{use bound for D} \naccording to whether or not the primitive character $\\chi_{d_N^*(p)}$ is in the\nexceptional set $\\mathscr E(4N)$.  For those $p$ for which $\\chi_{d_N^*(p)}$ is not\nexceptional, we use~\\eqref{nexcept-L bound}, writing\n\\begin{equation*}\n\\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)} \\not\\in \\mathscr{E}(4N)}}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}\n\\ll\\log{\\log{N}}\\sum_{f\\le 2\\sqrt N}\\frac{1}{f}\\sum_{\\substack{N^-<p<N^+\\\\  {f^2\\mid D_N(p)}}} 1.\n\\end{equation*}\nTo bound the sum over $p$, we apply the Brun-Titchmarsh inequality~\\cite[p.~167]{IK:2004},\nwhich gives that\n\\begin{equation*}\n\\#\\{N^-<p<N^+ : p\\equiv a\\pmod f\\}\n\\ll\\frac{\\sqrt N}{\\varphi(f)\\log(4\\sqrt N\/f)}.\n\\end{equation*}\nFor the sum over $a$, we use the bound\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod f\\}\\ll\\sqrt f,\n\\end{equation*}\nwhich is Lemma~\\ref{bound quad cong soln count} of\nSection~\\ref{proofs of lemmas}.\nCombining these two estimates, we have that\n\\begin{equation}\\label{nexcept-sum bound}\n\\begin{split}\n\\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)} \\not\\in \\mathscr{E}(4N)}}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}\n&\\ll \\frac{\\sqrt N\\log{\\log{N}}}{\\log{N}} \\sum_{f\\ge 1}\\frac{\\log f}{f^{1\/2}\\varphi(f)}\\\\\n&\\ll\\frac{\\sqrt N\\log{\\log{N}}}{\\log{N}}.\n\\end{split}\n\\end{equation}\n\nIt remains to estimate the sum over primes $p$ such that\n$\\chi_{d_N^*(p)} \\in \\mathscr{E}(4N)$.\nIn that case, we simply need the standard bound\n\\begin{equation*}\nL(1, \\chi) \\ll \\log{q},\n\\end{equation*}\nwhich is valid for all Dirichlet characters of conductor $q$ \n(see~\\cite[p.~96]{Dav:1980} for example).\nWe note that\n\\begin{equation*}\n\\#\\{N^-<p<N^+: \\chi_{d_N^*(p)}\\in\\mathscr E(4N)\\}\\le\\# \\mathscr E(4N)\\tau(4N),\n\\end{equation*}\nwhere $\\tau(n)$ denotes the number of positive divisors of $n$.\nTherefore, we obtain the bound\n\\begin{equation}\\label{except-sum bound}\n\\begin{split}\n\\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)}\\in \\mathscr{E}(4N)}}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}\n&\\ll\\log{N} \\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)}\\in\\mathscr{E}(4N)}}\n\t\\sum_{f^2 \\mid D_N(p)} \\frac{1}{f}\\\\\n&\\ll N^{1\/50+\\varepsilon} \n\\end{split}\n\\end{equation}\nfor any $\\varepsilon > 0$.\nCombining~\\eqref{use bound for D},~\\eqref{nexcept-sum bound}, and~\\eqref{except-sum bound}\ncompletes the proof of Proposition~\\ref{upper bound for H on avg}.\n\\end{proof}\n\n\\begin{prop}\\label{average order in terms of class numbers}\nLet $D_N(p)$ be as defined by~\\eqref{defn of D}. Then\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n=\\sum_{N^-<p<N^+}\\frac{H(D_N(p))}{p}+\\mathcal E(N;A,B),\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\mathcal E(N;A,B)\\ll\n\\frac{\\log\\log N}{N\\log N}+\\left(\\frac{1}{A}+\\frac{1}{B}\\right)\\sqrt N\\log\\log N+\\frac{N^{3\/2}\\log N\\log\\log N}{AB}\n\\end{equation*}\nuniformly for $A,B>0$.\n\\end{prop}\n\\begin{rmk}\nThe above holds without assuming that $N$ is odd.\n\\end{rmk}\n\n\\begin{proof}[Proof of Proposition~\\ref{average order in terms of class numbers}]\nFirst, we write $M_E(N)$ as a sum over primes and interchange sums\nto obtain\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\n\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n&=\n\\frac{1}{\\#\\mathscr{C}(A,B)}\n\\sum_{N^-<p<N^+}\n\\sum_{\\substack{E\\in\\mathscr{C}(A,B)\\\\ \\#E_p(\\mathbb{F}_p)=N}}1.\\\\\n\\end{split}\n\\end{equation*}\nFor each prime $p$, we group the $E\\in\\mathscr{C}(A,B)$ according to which isomorphism\nclass they reduce modulo $p$, writing\n\\begin{equation*}\n\\sum_{\\substack{E\\in\\mathscr{C}(A,B)\\\\ \\#E_p(\\mathbb{F}_p)=N}}1\n=\\sum_{\\substack{\\tilde E\/\\mathbb{F}_p\\\\ \\#\\tilde E(\\mathbb{F}_p)=N}}\n\t\\#\\{E\\in\\mathscr{C}(A,B): E_p\\cong\\tilde E\\}.\n\\end{equation*}\nFor $N$ large enough ($N\\ge 8$), the primes $2$ and $3$ will not enter into\nthe sum over $p$.  Thus, we will assume\nthat $p>3$ throughout the remainder of the article.\nTherefore, given a elliptic curve defined over $\\mathbb{F}_p$, we may associate\na Weierstrass equation, say $E_{s,t}: y^2=x^3+sx+t$ with $s,t\\in\\mathbb{F}_p$.\nUsing a character sum argument as in~\\cite[pp.~93-95]{FM:1996}, we have\n\\begin{equation*}\n\\begin{split}\n\\#\\{E\\in\\mathscr{C}(A,B): E_p\\cong E_{s,t}\\}\n=\\frac{4AB}{\\#\\mathrm{Aut}(E_{s,t})p}\n&+O\\left(\\frac{AB}{p^2}+\\sqrt p(\\log p)^2\\right)\\\\\n&+\\begin{cases}\nO\\left(\\frac{A\\log p}{\\sqrt p}+\\frac{B\\log p}{\\sqrt p}\\right)&\\text{if }st\\ne 0,\\\\\nO\\left(\\frac{A\\log p}{\\sqrt p}+B\\log p\\right)&\\text{if }s=0,\\\\\nO\\left(A\\log p+\\frac{B\\log p}{\\sqrt p}\\right)&\\text{if }t=0.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nHere $\\mathrm{Aut}(E_{s,t})$ denotes the size of the automorphism group of $E_{s,t}$ over $\\mathbb{F}_p$.\nSubstituting this estimate and applying Theorem~\\ref{deuring}, we find that\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\n\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n=\n\\sum_{N^-<p<N^+}\\frac{H(D_N(p))}{p}+\\mathcal E(N;A,B),\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\begin{split}\n\\mathcal E(N;A,B)\n&\\ll\\left\\{\\frac{1}{N^2}\n\t+\\frac{\\log N}{\\sqrt N}\\left(\\frac{1}{A}+\\frac{1}{B}\\right)\n\t+\\frac{\\sqrt N(\\log N)^2}{AB}\\right\\}\n\t\\sum_{N^-<p<N^+}H(D_N(p))\\\\\n&+\\left(\\frac{1}{A}+\\frac{1}{B}\\right)\\log N\\sum_{N^-<p<N^+}1.\n\\end{split}\n\\end{equation*}\nIn estimating the error, we have used the facts that $\\#\\mathscr{C}(A,B)=4AB+O(A+B)$, $p=N+O(\\sqrt N)$ for every\nprime $p$ in the interval $(N^-,N^+)$, and there are at most $10$ isomorphism classes $E_{s,t}$ over \n$\\mathbb{F}_p$ with $st=0$.\nThe result now follows by applying Proposition~\\ref{upper bound for H on avg} and the \nupper bound $\\{N^-<p<N^+\\}\\ll\\sqrt N\/\\log N$.\n\\end{proof}\n\n\nTheorem~\\ref{upper bound thm} now follows immediately upon combining\nProposition ~\\ref{upper bound for H on avg} and\nProposition~\\ref{average order in terms of class numbers}, and noting again that\n$p=N+O(\\sqrt N)$ for every prime $p$ in the interval\n$(N^-,N^+)$.\nFurthermore, we have reduced the proof of Theorem~\\ref{main thm} to computing\nthe sum\n\\begin{equation*}\n\\sum_{N^-<p<N^+}\\frac{H(D_N(p))}{p}.\n\\end{equation*}\nThis computation requires several intermediate results.\nTherefore, we delay it until Section~\\ref{proof of main thm}.\n\n\n\\section{A short average of special values of Dirichlet $L$-functions}\n\nSince Theorem~\\ref{main thm} holds only for odd $N$, we assume for the\nthe remainder of the article that $N$ is an odd integer except during\nthe proof of Lemma~\\ref{bound quad cong soln count}.  Recall that\nLemma~\\ref{bound quad cong soln count} was used in the proof of\nProposition~\\ref{upper bound for H on avg}, which holds for all positive integers $N$.\n\nIn this section, we prove a short average result for special values of Dirichlet\n$L$-functions that is needed to compute our average over elliptic curves. As the average\nis very short, we need that the equivalent of the Barban-Davenport-Halberstam Theorem holds\nfor intervals of that size.  \n\n\n\\begin{thm} \\label{thmaverageSV}\nLet $\\gamma>0$.\nSuppose that $N^-\\le X<X+Y\\le N^+$ with $Y\\gg \\sqrt N\/(\\log N)^\\upsilon$ for some choice of \n$\\upsilon\\ge 0$.\nAssume that Conjecture~\\ref{shortBDH} holds for intervals of length $Y$.\nThen for odd integers $N$,\n\\begin{equation*}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p = K_0(N)Y + O \\left( \\frac{Y}{(\\log{N})^\\gamma} \\right),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{K(N) as a double infinite sum}\nK_0(N):=\n\\sum_{\\substack{f=1\\\\ (f,2)=1}}^\\infty\\frac{1}{f}\\sum_{n=1}^\\infty\\frac{1}{n\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\t\\#C_N(a,n,f),\n\\end{equation}\nand\n\\begin{equation}\\label{defn of C}\nC_N(a,n,f):=\\left\\{z\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*: D_N(z)\\equiv af^2\\pmod{4nf^2}\\right\\}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nLet $U$ be a real parameter to be determined.\nUsing partial summation and Burgess' bound for character sums~\\cite[Theorem 2]{Bur:1963} to\nbound the tail of the $L$-series, we have\n\\begin{equation*}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\n=\\sum_{n\\ge 1}\\leg{d_{N,f}(p)}{n}\\frac{1}{n}\n=\\sum_{n\\le U}\\leg{d_{N,f}(p)}{n}\\frac{1}{n}\n\t+O\\left(\\frac{|d_{N,f}(p)|^{\\frac{7}{32}}}{\\sqrt U}\\right).\n\\end{equation*}\nFor $N^-<p<N^+$, we have $|d_{N,f}(p)|\\le 4N\/f^2$, and hence\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\\log p\n\\sum_{n> U}\\frac{1}{n}\n\\leg{d_{N,f}(p)}{n}\n\\ll\\frac{YN^{7\/32}}{\\sqrt U}.\n\\end{split}\n\\end{equation*}\nNow let $V$ be a real parameter to be determined.\nUsing Lemma~\\ref{bound quad cong soln count}, we obtain\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{V<f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{n\\le U}&\n\\frac{1}{n}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\leg{d_{N,f}(p)}{n}\\log p\\\\\n&\\ll\\log U\\log N\\sum_{\\substack{V<f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\n\t\\frac{1}{f}\\sum_{\\substack{X<k\\le X+4\\sqrt N\\\\ f|D_N(p)}}1\\\\\n&\\ll\\log U\\log N\\sum_{\\substack{V<f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\n\\frac{\\sqrt N\\#\\{z\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(z)\\equiv 0\\pmod f\\}}{f^2}\\\\\n&\\ll\\sqrt N\\log U\\log N\\sum_{f>V}\\frac{1}{f^{3\/2}}\\\\\n&\\ll\\frac{\\sqrt N\\log U\\log N}{\\sqrt V},\n\\end{split}\n\\end{equation*}\nand therefore,\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n&=\\sum_{\\substack{f\\le V\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{n\\le U}\n\\frac{1}{n}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\leg{d_{N,f}(p)}{n}\\log p\\\\\n&\\quad+O\\left(\\frac{YN^{7\/32}}{\\sqrt U}+\\frac{\\sqrt N\\log U\\log N}{\\sqrt V}\\right).\n\\end{split}\n\\end{equation*}\nWith $C_N(a,n,f)$ as defined by~\\eqref{defn of C},\nwe regroup terms on the right hand side above, writing\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\leg{d_{N,f}(p)}{n}\\log p\n&=\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}\\theta(X,Y;4nf^2,b)\\\\\n&\\quad+O\\left(\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{\\substack{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}\\\\ D_N(p)\\equiv af^2\\\\ p\\mid 4nf^2}}\\log p\\right).\n\\end{split}\n\\end{equation*}\nIf $p$ satisfies the congruence\n$D_N(p)\\equiv af^2\\pmod{4nf^2}$ and $p$ divides $4nf^2$, then $p$ divides\n$(4nf^2,(N-1)^2-af^2)$.  It follows that such a $p$ must divide $n(N-1)$.\nHence, the error term\n\\begin{equation*}\n\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{\\substack{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}\\\\ D_N(p)\\equiv af^2\\\\ p\\mid 4nf^2}}\\log p\n\\ll U\\log N\\log V + U\\log U\\log V,\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n&=\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}\\theta(X,Y;4nf^2,b)\\\\\n&\\quad+O\\left(\n\t\t\\frac{YN^{7\/32}}{\\sqrt U}+\\frac{\\sqrt N\\log U\\log N}{\\sqrt V}\n\t\t+U\\log(UN)\\log V\\right).\n\\end{split}\n\\end{equation*}\nWe approximate $\\theta(X,Y;4nf^2,b)$ by $Y\/\\varphi(4nf^2)$, incurring an error of\n\\begin{equation}\\label{error from approximating sum over primes}\n\\sum_{\\substack{f\\le V,n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}E(X,Y; 4nf^2,b).\n\\end{equation}\nFor any given value of $b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*$, there is at most one value of\n$a\\in\\mathbb{Z}\/4n\\mathbb{Z}$ satisfying the congruence $D_N(b)\\equiv af^2\\pmod{4nf^2}$.  Hence, interchanging the two inner sums shows that\n\\begin{equation*}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}E(X,Y; 4nf^2,b)\n\\ll \\sum_{b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*}\\left|E(X,Y;4nf^2,b)\\right|.\n\\end{equation*}\nBy Cauchy-Schwarz,\nthe error term~\\eqref{error from approximating sum over primes} is\n\\begin{equation*}\n\\begin{split}\n&\\ll\n\\sum_{f\\le V}\\frac{1}{f}\n\\sum_{n\\le U}\n\\frac{1}{n}\n\\sum_{b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*}\\left|E(X,Y;4nf^2,b)\\right|\\\\\n&\\le\n\\sum_{f\\le V}\\frac{1}{f}\n\\left[\n\\sum_{n\\le U}\\frac{\\varphi(4nf^2)}{n^2}\n\\right]^{1\/2}\n\\left[\n\\sum_{n\\le U}\n\\sum_{b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*}\n\\left|E(X,Y;4nf^2,b)\\right|^2\n\\right]^{1\/2}\\\\\n&\\ll\nV\\sqrt{\\log U}\n\\left[\n\\sum_{q\\le 4UV^2}\\sum_{\\substack{a=1\\\\ (a,q)=1}}^q\n\\left|E(X,Y;q,a)\\right|^2\\right]^{1\/2}.\n\\end{split}\n\\end{equation*}\nAssuming Conjecture~\\ref{shortBDH} for an appropriate value of $\\eta$, we obtain the bound\n\\begin{equation*}\nV\\sqrt{\\log U}\n\\left[\n\\sum_{q\\le 4UV^2}\\sum_{\\substack{a=1\\\\ (a,q)=1}}^q\n\\left|E(X,Y;q,a)\\right|^2\\right]^{1\/2}\n\\ll\\frac{YV\\sqrt{\\log U\\log N}}{(\\log N)^{2\\upsilon+3\\gamma+5}}\n\\end{equation*}\nwhenever\n\\begin{equation}\\label{bdh restriction on U and V}\nUV^2\\le \\frac{Y}{(\\log N)^{4\\upsilon+6\\gamma+10}}.\n\\end{equation}\nThus, we have\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n&=Y\\sum_{\\substack{f\\le V,n\\le U\\\\ (f,2)=1}}\\frac{1}{fn\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\t\\#C_N(a,n,f) \\\\\n&\\quad+O \\left( \\frac{YN^{7\/32}}{\\sqrt U}+\\frac{\\sqrt N\\log U\\log N}{\\sqrt V}+U\\log(UN)\\log V\\right)\\\\\n&\\quad+O\\left(\\frac{YV\\sqrt{\\log U\\log N}}{(\\log N)^{2\\upsilon+3\\gamma+5}}\\right).\n\\end{split}\n\\end{equation*}\n\n\\begin{lma}\\label{complete K sum}\nFor any $U,V,\\epsilon>0$, we have\n\\begin{equation*}\n\\begin{split}\nK_0(N)=\n\\sum_{\\substack{f\\le V,n\\le U\\\\ (f,2)=1}}\\frac{1}{fn\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\\#C_N(a,n,f)\t\n\t+O\\left(\\frac{N^\\epsilon}{\\sqrt U}+\\frac{\\log\\log N}{V}\\right).\n\\end{split}\n\\end{equation*}\n\\end{lma}\nWe delay the proof of Lemma~\\ref{complete K sum} until Section~\\ref{proofs of lemmas}.\nApplying the lemma and choosing\n\\begin{equation*}\nU=\\frac{Y}{(\\log N)^{4\\upsilon+10\\gamma+18}},\\quad V=(\\log N)^{2\\upsilon+2\\gamma +4},\n\\end{equation*}\nwe have\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n=K_0(N)Y&\n+O\\left(\\frac{Y}{(\\log N)^\\gamma}\\right)\n\\end{split}\n\\end{equation*}\nprovided that $Y\\gg\\sqrt N\/(\\log N)^{\\upsilon}$.\nNote that our choice of $U,V$ satisfies the condition~\\eqref{bdh restriction on U and V}.\n\\end{proof}\n\n\\section{Computing the ``almost constant\"}\n\nRecall that $C_N(a,n,f)$ was defined by\n\\begin{equation*}\nC_N(a,n,f)=\\left\\{z\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*: D_N(z)\\equiv af^2\\pmod{4nf^2}\\right\\},\n\\end{equation*}\nwhere $D_N(z)=z^2-2(N+1)z+(N-1)^2$.\nThe following is the main result of this section.\n\\begin{prop}\\label{product formula}\nWith $K(N)$ as defined in Theorem~\\ref{main thm} and $K_0(N)$ as defined in\nTheorem~\\ref{thmaverageSV}, we have\n\\begin{equation*}\n\\frac{N}{\\varphi(N)}K(N)=K_0(N).\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nBy the Chinese Remainder Theorem and the definition of $C_N(a,n,f)$,\n\\begin{equation*}\n\\#C_N(a,n,f)=\n\\prod_{\\ell\\mid 4nf^2}\\#C_N^{(\\ell)}(a,n,f),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{crt pieces of C}\nC_N^{(\\ell)}(a,n,f):=\n\\{z\\in(\\mathbb{Z}\/\\ell^{\\nu_\\ell(4nf^2)}\\mathbb{Z})^*:\n\tD_N(z)\\equiv af^2\\pmod{\\ell^{\\nu_\\ell(4nf^2)}}\\}.\n\\end{equation}\nWe require the following lemma whose proof we delay until Section~\\ref{proofs of lemmas}.\n\n\\begin{lma}\\label{counting solutions}\nSuppose that $N$ and $f$ are odd and that $a\\equiv 1\\pmod 4$.\nLet $\\ell$ be any odd prime dividing $nf$, and let\n$e=\\nu_\\ell(4nf^2)=\\nu_\\ell(nf^2)$.\nIf $\\ell\\nmid 4N+af^2$, then\n\\begin{equation*}\n\\#C_N^{(\\ell)}(a,n,f)=\n\\begin{cases}\n1+\\leg{4N+af^2}{\\ell}&\\text{if }\\ell\\nmid (N-1)^2-af^2,\\\\\n1&\\text{if }\\ell\\mid (N-1)^2-af^2.\n\\end{cases}\n\\end{equation*}\nIf $\\ell\\mid 4N+af^2$, then with $s=\\nu_\\ell(4N+af^2)$, we have\n\\begin{equation*}\n\\begin{split}\n\\#C_N^{(\\ell)}(a,n,f)\n&=\\begin{cases}\n2\\leg{N+1}{\\ell}^2\\ell^{s\/2}\n\t&\\text{if }1\\le s<e,\\ 2\\mid s, \\text{ and }\\leg{(4N+af^2)\/\\ell^s}{\\ell}=1,\\\\\n\\leg{N+1}{\\ell}^2\\ell^{\\floor{e\/2}}\n\t&\\text{if }s\\ge e,\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nIn particular, if $\\ell\\mid f$, then\n\\begin{equation*}\n\\#C_N^{(\\ell)}(1,1,f)=\\begin{cases}\n1+\\leg{N(N-1)^2}{\\ell}&\\text{if }\\ell\\nmid N,\\\\\n2\\ell^{\\nu_\\ell(N)\/2}&\\text{if }1\\le\\nu_\\ell(N)<2\\nu_\\ell(f),\\ 2\\mid\\nu_\\ell(N),\n\t\\text{ and }\\leg{N_\\ell}{\\ell}=1,\\\\\n\\ell^{\\nu_\\ell(f)}&\\text{if }2\\nu_\\ell(f)\\le\\nu_\\ell(N),\\\\\n0&\\text{otherwise},\n\\end{cases}\n\\end{equation*}\nwhere $N_\\ell=N\/\\ell^{\\nu_\\ell(N)}$ is the $\\ell$-free part of $N$.\nFurthermore,\n\\begin{equation*}\n\\#C_N^{(2)}(a,n,f)\n=\\begin{cases}\n2&\\text{if }\\nu_2(4nf^2)=2+\\nu_2(n)=2,\\\\\n4&\\text{if }\\nu_2(4nf^2)=2+\\nu_2(n)\\ge 3\\text{ and }a\\equiv 5\\pmod 8,\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{equation*}\n\\end{lma}\n\nBy Lemma~\\ref{counting solutions},\nwe may write\n\\begin{equation*}\n\\#C_N^{(2)}(a,n,f)\n=2\\mathcal S_2(n,a),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{defn of S_2}\n\\mathcal S_2(n,a):=\\begin{cases}\n1&\\text{if } 2\\nmid n,\\\\\n2&\\text{if }2\\mid n\\text{ and } a\\equiv 5\\pmod 8,\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\nNote that if $\\ell$ is a prime dividing $f$ and not dividing $n$, then\n$af^2\\equiv 0\\pmod{\\ell^{\\nu_\\ell(4nf^2)}}$ as $\\nu_\\ell(4nf^2)=\\nu_\\ell(f^2)$.\nHence, in this case, $\\#C_N^{(\\ell)}(a,n,f)$ does not depend on the value of $a$, and so we write\n$\\#C_N^{(\\ell)}(a,n,f)=\\#C_N^{(\\ell)}(1,1,f)$ if $\\ell\\mid f$ and $\\ell\\nmid n$.\nTherefore, letting $n'$ denote the odd part of $n$ and\n\\begin{equation*}\nc_{N,f}(n):=\\sum_{\\substack{a\\in(\\mathbb{Z}\/4n\\mathbb{Z})^*\\\\ a\\equiv 1\\pmod 4}}\n\\leg{a}{n}\n\\mathcal S_2(n,a)\n\\prod_{\\ell\\mid n'}\n\\#C_N^{(\\ell)}(a,n,f),\n\\end{equation*}\nwe may write\n\\begin{equation}\\label{separate n and f}\n\\begin{split}\nK_0(N)\n&=\\sum_{\\substack{f=1\\\\ (f,2)=1}}^\\infty\\frac{1}{f}\\sum_{n=1}^\\infty\\frac{1}{n\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\t\\#C_N(a,n,f)\\\\\n&=\\sum_{\\substack{f=1\\\\ (f,2)=1}}^\\infty\\frac{1}{f^2\\varphi(f)}\n\t\\sum_{n=1}^\\infty\\frac{2\\varphi((n,f))}{(n,f)n\\varphi(4n)}\n\t\\left[\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\nmid n}}\\#C_N^{(\\ell)}(1,1,f)\\right]c_{N,f}(n)\\\\\n&=\\sum_{f=1}^\\infty\\strut^\\prime\n\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n=1}^\\infty\\frac{2\\varphi((n,f))}{(n,f)n\\varphi(4n)}\n\t\\left[\\prod_{\\ell\\mid (f,n)}\\#C_N^{(\\ell)}(1,1,f)\\right]^{-1}c_{N,f}(n),\n\\end{split}\n\\end{equation}\nwhere the prime on the outer sum indicates that the sum is to be restricted to those $f$ which are\nodd and are not divisible by any prime for which $\\#C_N^{(\\ell)}(1,1,f)=0$.\n\n\nIn order to proceed further, we must show how to compute the function $c_{N,f}(n)$.\nWe summarize the computation in the following lemma whose proof we also delay\nuntil Section~\\ref{proofs of lemmas}.\n\n\\begin{lma}\\label{compute factors of c}\nSuppose that $N$ and $f$ are odd.\nThe function $c_{N,f}(n)$ is multiplicative in $n$.\nLet $\\alpha$ be a positive integer and $\\ell$ an odd prime.\nThen\n\\begin{equation*}\n\\frac{c_{N,f}(2^\\alpha)}{2^{\\alpha-1}}=(-1)^\\alpha 2.\n\\end{equation*}\nIf $\\ell\\mid f$ and $\\ell\\nmid N$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)\n\\begin{cases}\n\\ell-1&\\text{if }2\\mid\\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\nIf $\\ell\\mid N$ and $\\ell\\nmid f$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}=\\ell-2.\n\\end{equation*}\nIf $\\ell\\nmid Nf$, then\n\\begin{equation*}\n\\begin{split}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n&=\\begin{cases}\n\\ell-2-\\leg{N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n+\\leg{N+1}{\\ell}^2\n\t&\\text{if }2\\mid\\alpha,\\\\\n-1-\\leg{-N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n+\\leg{-N(N+1)^2}{\\ell}\n\t&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\begin{cases}\n\\ell-1-\\leg{N}{\\ell}-\\leg{N-1}{\\ell}^2\n\t&\\text{if }2\\mid\\alpha,\\\\\n-1-\\leg{N-1}{\\ell}^2\n\t&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nIf $\\ell\\mid (f,N)$ and $2\\nu_\\ell(f)<\\nu_\\ell(N)$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)(\\ell-1).\n\\end{equation*}\nIf $\\ell\\mid (f,N)$ and $\\nu_\\ell(N)<2\\nu_\\ell(f)$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n\\ell-1&\\text{if }2\\mid \\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\nIf $\\ell\\mid (f,N)$ and $\\nu_\\ell(N)=2\\nu_\\ell(f)$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n\\left(\\ell-1-\\leg{N_\\ell}{\\ell}+\\leg{-N_\\ell}{\\ell}\\right)&\\text{if }2\\mid\\alpha,\\\\\n\\left(\\leg{-N_\\ell}{\\ell}-1\\right)&\\text{if }2\\nmid\\alpha,\\\\\n\\end{cases}\n\\end{equation*}\nwhere $N_\\ell=N\/\\ell^{\\nu_\\ell(N)}$ denotes the $\\ell$-free part of $N$.\nFurthermore, for any $n$, we have the bound\n\\begin{equation*}\nc_{N,f}(n)\\ll\\frac{n\\prod_{\\ell\\mid (f,n)}\\#C_N(1,1,f)}{\\kappa_{2N}(n)},\n\\end{equation*}\nwhere for any integer $m$, $\\kappa_m(n)$ is the multiplicative function defined on prime\npowers by\n\\begin{equation}\\label{defn of kappa}\n\\kappa_m(\\ell^\\alpha)\n:=\\begin{cases}\n\\ell&\\text{if }2\\nmid\\alpha\\text{ and }\\ell\\nmid m,\\\\\n1&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\\end{lma}\n\nRecalling the restrictions on $f$ in~\\eqref{separate n and f} and applying\nLemma~\\ref{compute factors of c}, the sum over $n$ in~\\eqref{separate n and f} may be factored as\n\\begin{equation*}\n\\begin{split}\n&\\sum_{n=1}^\\infty\\frac{2\\varphi((n,f))}{(n,f)n\\varphi(4n)}\n\t\\left[\\prod_{\\ell\\mid (f,n)}\\#C_N^{(\\ell)}(1,1,f)\\right]^{-1}c_{N,f}(n)\\\\\n&\\quad=\\left\\{\\sum_{\\alpha\\ge 0}\\frac{2c_{N,f}(2^\\alpha)}{2^\\alpha\\varphi(2^{\\alpha+2})}\\right\\}\n\t\\prod_{\\substack{\\ell\\nmid f\\\\ \\ell\\ne 2}}\n\t\t\\left\\{\\sum_{\\alpha\\ge 0}\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^\\alpha\\varphi(\\ell^\\alpha)}\\right\\}\n\t\\prod_{\\ell\\mid f}\n\t\t \\left\\{1+ \\sum_{\\alpha\\ge 1}\\frac{\\varphi((\\ell^\\alpha,f))c_{N,f}(\\ell^\\alpha)}\n \t\t {(\\ell^\\alpha,f)\\ell^\\alpha\\varphi(\\ell^\\alpha)\\#C_N^{(\\ell)}(1,1,f)}\\right\\}\n\\\\\n &\\quad\n =\\frac{2}{3}\n\t\\prod_{\\substack{\\ell\\nmid f\\\\ \\ell\\mid N}}F_0(\\ell)\t\n\t\\prod_{\\substack{\\ell\\nmid f\\\\ \\ell\\nmid N\\\\ \\ell\\ne 2}}F_1(\\ell)\n\t\\prod_{\\ell\\mid f}F_2(\\ell,f),\n\\end{split}\n\\end{equation*}\nwhere for any odd prime $\\ell$, we make the definitions\n\\begin{align*}\nF_0(\\ell)&:=\\left(1+\\frac{\\ell-2}{(\\ell-1)^2}\\right),\\\\\nF_1(\\ell)\n&:=\\left(1-\n\t\\frac{\\leg{N-1}{\\ell}^2\\ell\n\t+\\leg{N}{\\ell}+\\leg{N-1}{\\ell}^2+1}{(\\ell-1)(\\ell^2-1)}\n\\right),\\\\\nF_2(\\ell,f)&:=\\begin{cases}\n\\left(1+\\frac{1}{\\ell(\\ell+1)}\\right)&\\text{if }\\nu_\\ell(N)<2\\nu_\\ell(f),\\\\\n\\left(1+\\frac{1}{\\ell}\\right)&\\text{if }\\nu_\\ell(N)>2\\nu_\\ell(f),\\\\\n\\left(1+\n\t\\frac{\\leg{-N_\\ell}{\\ell}\\ell+\\leg{-N_\\ell}{\\ell}-\\leg{N_\\ell}{\\ell}-1}{\\ell(\\ell^2-1)}\n\t\\right)\n\t&\\text{if }\\nu_\\ell(N)=2\\nu_\\ell(f).\n\\end{cases}\n\\end{align*}\nSubstituting this back into equation~\\eqref{separate n and f}, we have\n\\begin{equation}\\label{f sum remaining}\nK_0(N)\n=\\frac{2}{3}\n\t\\prod_{\\ell\\mid N}F_0(\\ell)\n\t\\prod_{\\substack{\\ell \\nmid N\\\\ \\ell\\ne 2}} F_1(\\ell)\n\t\\sum_{\\substack{f=1\\\\ (f,2)=1}}^{\\infty}\\strut^\\prime\n\t\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{\\varphi(f)f^2}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\mid N}}\\frac{F_2(\\ell,f)}{F_0(\\ell)}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\nmid N}}\\frac{F_2(\\ell,f)}{F_1(\\ell)}.\n\\end{equation}\nThe sum over $f$ may be factored as\n\\begin{equation*}\n\\begin{split}\n&\\sum_{\\substack{f=1\\\\ (f,2)=1}}^{\\infty}\\strut^\\prime\n\t\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{\\varphi(f)f^2}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\mid N}}\\frac{F_2(\\ell,f)}{F_0(\\ell)}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\nmid N}}\\frac{F_2(\\ell,f)}{F_1(\\ell)}\\\\\n&\\quad=\\prod_{\\ell\\mid N}\n\t\\left\\{1+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\n\t\t\\right\\}\n\\prod_{\\substack{\\ell\\nmid N\\\\ \\ell\\ne 2}}\n\t\\left\\{1+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_1(\\ell)}\n\t\\right\\}.\n\\end{split}\n\\end{equation*}\nWhen $\\ell\\nmid 2N$, the factor simplifies as\n\\begin{equation*}\n\\begin{split}\n1+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_1(\\ell)}\n&=1+\\frac{\\ell C_N^{(\\ell)}(1,1,\\ell)F_2(\\ell,\\ell)}{F_1(\\ell)(\\ell-1)}\n\t\\sum_{\\alpha\\ge 1}\\frac{1}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\left(1+\\leg{N(N-1)^2}{\\ell}\\right)(\\ell^2+\\ell+1)}{(\\ell^2-1)(\\ell^3-1)F_1(\\ell)}\\\\\n&=1+\\frac{1+\\leg{N(N-1)^2}{\\ell}}{(\\ell^2-1)(\\ell-1)F_1(\\ell)}.\n\\end{split}\n\\end{equation*}\nWhen $\\nu_\\ell(N)$ is odd, the factor \nsimplifies as\n\\begin{equation*}\n\\begin{split}\n1&+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\\\\\n\\quad&=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha\\ge 1}\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\n\t\\sum_{\\alpha=1}^{\\floor{\\nu_\\ell(N)\/2}}\n\t\\frac{\\ell^{\\alpha}\\left(1+\\frac{1}{\\ell}\\right)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{(\\ell+1)(1-\\ell^{1-\\nu_\\ell(N)})}{F_0(\\ell)(\\ell-1)(\\ell^2-1)}\\\\\n&=1+\\frac{1-\\ell^{1-\\nu_\\ell(N)}}{F_0(\\ell)(\\ell-1)^2}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}.\n\\end{split}\n\\end{equation*}\nWhen $\\nu_\\ell(N)$ positive, even, and $\\leg{N_\\ell}{\\ell}=-1$, the factor simplifies as\n\\begin{equation*}\n\\begin{split}\n1&+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\\\\\n\\quad&=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha\\ge 1}\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell\\#C_N^{(\\ell)}(1,1,\\ell^{\\nu_\\ell(N)\/2})F_2(\\ell,\\ell^{\\nu_\\ell(N)\/2})}\n\t{F_0(\\ell)(\\ell-1)\\ell^{3\\nu_\\ell(N)\/2}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell^2-\\ell-\\leg{-1}{\\ell}}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell-\\leg{-1}{\\ell}}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}.\n\\end{split}\n\\end{equation*}\nWhen $\\nu_\\ell(N)$ positive, even, and $\\leg{N_\\ell}{\\ell}=1$, the factor simplifies as\n\\begin{equation*}\n\\begin{split}\n1&+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\n=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha\\ge 1}\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell\\#C_N^{(\\ell)}(1,1,\\ell^{\\nu_\\ell(N)\/2})F_2(\\ell,\\ell^{\\nu_\\ell(N)\/2})}\n\t{F_0(\\ell)(\\ell-1)\\ell^{3\\nu_\\ell(N)\/2}}\\\\\n&\\quad+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha=\\frac{\\nu_\\ell(N)}{2}+1}^\\infty\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell(\\ell^2-1)+\\leg{-1}{\\ell}\\ell+\\leg{-1}{\\ell}-2}\n\t{F_0(\\ell)(\\ell-1)(\\ell^2-1)\\ell^{\\nu_\\ell(N)}}\\\\\n&\\quad+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha=\\frac{\\nu_\\ell(N)}{2}+1}^\\infty\n\t\\frac{2\\ell^{\\nu_\\ell(N)\/2}\\left(1+\\frac{1}{\\ell(\\ell+1)}\\right)}{\\ell^{3\\alpha}}\\\\\n\n\n\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell^2-\\ell+\\leg{-1}{\\ell}}\n\t{F_0(\\ell)(\\ell-1)^2\\ell^{\\nu_\\ell(N)}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}{F_0(\\ell)(\\ell-1)^2\\ell^{\\nu_\\ell(N)}}.\n\\end{split}\n\\end{equation*}\n\nSubstituting this back into~\\eqref{f sum remaining}, we find that\n\\begin{equation*}\n\\begin{split}\nK_0(N)\n&=\\frac{2}{3}\n\t\\prod_{\\ell \\nmid 2N}\\left(F_1(\\ell)+\\frac{1+\\leg{N(N-1)^2}{\\ell}}{(\\ell^2-1)(\\ell-1)}\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(F_0(\\ell)+\\frac{(\\ell^{\\nu_\\ell(N)}-\\ell)}{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&\\quad\\times\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(F_0(\\ell)\n\t+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&=\\frac{2}{3}\n\t\\prod_{\\ell \\nmid 2N}\\left(\n\t1-\\frac{\\leg{N-1}{\\ell}^2\\left[\\ell+1-\\leg{N}{\\ell}\\right]+\\leg{N}{\\ell}}{(\\ell-1)(\\ell^2-1)}\n\t\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(1+\\frac{\\ell^{\\nu_\\ell(N)+1}-\\ell^{\\nu_\\ell(N)}-\\ell}\n\t{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&\\quad\\times\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(1+\\frac{\\ell^{\\nu_\\ell(N)+1}-\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}\n\t{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&=\\frac{N}{\\varphi(N)}\n\t\\prod_{\\ell \\nmid N}\\left(\n\t1-\\frac{\\leg{N-1}{\\ell}^2\\ell+1}{(\\ell-1)(\\ell^2-1)}\n\t\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{1}{\\ell^{\\nu_\\ell(N)}(\\ell-1)}\\right)\\\\\n&\\quad\\times\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{\\ell-\\leg{-N_\\ell}{\\ell}}\n\t{\\ell^{\\nu_\\ell(N)+1}(\\ell-1)}\\right).\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{main thm}}\\label{proof of main thm}\n\nWe are now ready to give the proof of our main result.\n\\begin{proof}[Proof of Theorem~\\ref{main thm}.]\nBy Proposition~\\ref{average order in terms of class numbers},\nwe see that Theorem~\\ref{main thm} follows if we show that\n\\begin{equation*}\n\\sum_{N^-<p< N^+}\\frac{H(D_N(p))}{p}\n=K(N)\\frac{N}{\\varphi(N)\\log N}+O\\left(\\frac{1}{(\\log N)^{1+\\gamma}}\\right).\n\\end{equation*}\nWe begin by dividing the interval $(N^-,N^+)$ into intervals of length\n$Y:=\\frac{\\sqrt N}{\\floor{(\\log N)^{\\gamma+2}}}$.  For each integer $k$ in\n$I:=[-2\\sqrt N\/Y, 2\\sqrt N\/Y)\\cap\\mathbb{Z}$,\nwe write $X=X_k:=N+1+kY$.\nThus,\n\\begin{equation}\\label{break into shorter averages}\n\\sum_{N^-<p< N^+}\\frac{H(D_N(p))}{p}\n=\\sum_{k\\in I}\\sum_{X_k<p\\le X_k+Y}\\frac{H(D_N(p))}{p}.\n\\end{equation}\n\n\n\nRecalling the definition of the Kronecker class number, the definition of $d_{N,f}(p)$, and the\nclass number formula\n(see~\\eqref{defn of K-class no},~\\eqref{defn of d}, and~\\eqref{class no formula}),\nwe have the identity\n\\begin{equation}\\label{analytic Kronecker class number formula}\nH(D_N(p))=\\frac{1}{2\\pi}\\sum_{\\substack{f^2|D_N(p)\\\\ d_{N,f}(p)\\equiv 0,1\\pmod 4}}\n\\sqrt{|d_{N,f}(p)|}L\\left(1,\\chi_{d_{N,f}(p)}\\right).\n\\end{equation}\nWe assume that $N$ is large enough so that there are only odd primes $p$ satisfying\nthe condition $N^-<p<N^+$.\nTherefore, since we assumed that $N$ is odd,\n\\begin{equation*}\nD_N(p)=(p+1-N)^2-4p\\equiv 1\\pmod 4,\n\\end{equation*}\nand it follows that there are only odd $f$ in the above sum and that\n$d_{N,f}(p)\\equiv 1\\pmod 4$ for each such $f$.\nHence, summing~\\eqref{analytic Kronecker class number formula} over all primes $p$\nin the range $X<p\\le X+Y$ and switching the order of summation, we have\n\\begin{equation}\\label{reduce to short average of L-values}\n\\begin{split}\n\\sum_{X<p\\le X+Y}\\frac{H(D_N(p))}{p}\n&=\\frac{1}{2\\pi}\\sum_{X<p\\le X+Y}\\frac{1}{p}\n\\sum_{f^2|D_N(p)}\n\\sqrt{|d_{N,f}(p)|}L\\left(1,\\chi_{d_{N,f}(p)}\\right)\\\\\n&=\\frac{1}{2\\pi}\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\frac{\\sqrt{|D_N(p)|}}{p}L\\left(1,\\chi_{d_{N,f}(p)}\\right).\n\\end{split}\n\\end{equation}\n\nWe now change ``weights,\" approximating $\\frac{\\sqrt{|D_N(p)|}}{p}$ by\n$\\frac{\\sqrt{|D_N(X)|}\\log p}{N\\log N}$.  If $p$ is a prime in the interval $(X,X+Y]$,\nthen $p=X+O(Y)$, and hence\n\\begin{equation*}\nD_N(p)\n=D_N(X)+O(Y\\sqrt N).\n\\end{equation*}\nLet $X^*$ be the value minimizing the function $\\sqrt{|D_N(t)|}$ on the interval $[X,X+Y]$.\nSince it is also true that $p=N+O(\\sqrt N)$, we have that\n\\begin{equation*}\n\\left|\\frac{\\sqrt{|D_N(p)|}}{p}-\\frac{\\sqrt{|D_N(X)|}\\log p}{N\\log N}\\right|\n\\ll\\begin{cases}\n\\frac{\\sqrt{|D_N(X)|}}{N^{3\/2}}+\\frac{\\sqrt{Y}}{N^{3\/4}}&\\text{if }N^{\\pm}\\in [X,X+Y],\\\\\n\\frac{\\sqrt{|D_N(X)|}}{N^{3\/2}}+\\frac{Y}{N^{1\/2}\\sqrt{|D_N(X^*)|}}&\\text{otherwise.}\n\\end{cases}\n\\end{equation*}\nHence, by Theorem~\\ref{thmaverageSV} and Proposition~\\ref{product formula}, the right hand side\nof~\\eqref{reduce to short average of L-values} is equal to\n\\begin{equation}\\label{change weights}\n\\begin{split}\n\\frac{K(N)Y\\sqrt{|D_N(X)|}}{2\\pi \\varphi(N)\\log N}\n+\\begin{cases}\nO\\left(\\frac{Y\\sqrt{|D_N(X)|}}{N(\\log N)^{\\gamma+1}}\n\t+\\frac{Y\\sqrt{|D_N(X)|}\\log N}{N^{3\/2}}+\\frac{Y^{3\/2}\\log N}{N^{3\/4}}\\right)\n\t&\\text{ if }N^\\pm\\in [X,X+Y],\\\\\nO\\left(\n\\frac{Y\\sqrt{|D_N(X)|}}{N(\\log N)^{\\gamma+1}}\n\t+\\frac{Y\\sqrt{|D_N(X)|}\\log N}{N^{3\/2}}\n\t+\\frac{Y^2\\log N}{N^{1\/2}\\sqrt{|D_N(X^*)|}} \\right)\n\t&\\text{otherwise}.\n\\end{cases}\n\\end{split}\n\\end{equation}\nSince $D_N(X_k)=0$ for $k$ on the endpoints of the interval $[-2\\sqrt N\/Y,2\\sqrt N\/Y]\\supset I$,\nby the Euler-Maclaurin summation formula, we have\n\\begin{equation*}\n\\begin{split}\n\\sum_{k\\in I}\\sqrt{|D_N(X_k)|}\n&=\\int_{-2\\sqrt N\/Y}^{2\\sqrt N\/Y}\\sqrt{4N-(tY)^2}\\mathrm{d} t\n+O\\left(\\int_{-2\\sqrt N\/Y}^{2\\sqrt N\/Y}\\frac{\\left| tY^2\\right|}{\\sqrt{4N-(tY)^2}}\\mathrm{d} t\\right)\\\\\n&=\\frac{2\\pi N}{Y}\n+O\\left(\\sqrt N\\right),\n\\end{split}\n\\end{equation*}\nand furthermore,\n\\begin{equation*}\n\\sum_{k\\in I}\\frac{1}{\\sqrt{|D_N(X_k^*)|}}\n\\ll\\int_{-2\\sqrt N\/Y}^{2\\sqrt N\/Y}\\frac{\\mathrm{d} t}{\\sqrt{4N-(tY)^2}}\n=\\frac{1}{Y}\\int_{-2\\sqrt N}^{2\\sqrt N}\\frac{\\mathrm{d} u}{\\sqrt{4N-u^2}}\n=\\frac{\\pi}{Y}.\n\\end{equation*}\nWhence, summing~\\eqref{change weights} over $k\\in I$, we have\n\\begin{equation*}\n\\sum_{N^-<p< N^+}\\frac{H(D_N(p))}{p}\n=K(N)\\frac{N}{\\varphi(N)\\log N}\n\t+O\\left( \\frac{1}{(\\log N)^{\\gamma+1}}\n\t+\\frac{Y\\log N}{\\sqrt N}\n\t\\right).\n\\end{equation*}\n\\end{proof}\n\n\n\\section{Proofs of lemmas}\\label{proofs of lemmas}\n\n\nIn this section, we give the proofs of the technical lemmas needed in the rest of the paper.\n\n\\begin{lma}\\label{bound quad cong soln count}\nFor every positive integer $f$,\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod f\\}\\ll\\sqrt f.\n\\end{equation*}\n\\end{lma}\n\\begin{rmk}\nRecall that since this lemma was used in the proof of Proposition~\\ref{upper bound for H on avg},\nwe do not assume that $N$ is odd here.\n\\end{rmk}\n\\begin{proof}[Proof of Lemma~\\ref{bound quad cong soln count}.]\nFirst, we use the Chinese Remainder Theorem to write\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{f}\\}\n=\\prod_{\\ell\\mid f}\\#\\{a\\in\\mathbb{Z}\/\\ell^{\\nu_\\ell(f)}\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{\\ell^{\\nu_\\ell(f)}}\\}.\n\\end{equation*}\nWe will show that\n\\begin{equation}\\label{prime power mod bound}\n\\#\\{a\\in\\mathbb{Z}\/\\ell^{e}\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{\\ell^{e}}\\}\n\\le\\begin{cases}\n\\max\\{\\ell^{\\floor{e\/2}},2\\ell^{(e-1)\/2}\\}&\\text{if }\\ell>2,\\\\\n\\max\\{\\ell^{\\floor{e\/2}},4\\ell^{(e-1)\/2}\\}&\\text{if }\\ell=2.\n\\end{cases}\n\\end{equation}\nFrom this, we readily deduce that\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{f}\\}\\le 8\\sqrt f,\n\\end{equation*}\nwhich is a more precise result than stated in the lemma.\n\nWe now give the proof of~\\eqref{prime power mod bound}.\nSince\n\\begin{equation*}\nD_N(a)=a^2-2(N+1)a+(N-1)^2=(a-N-1)^2-4N,\n\\end{equation*}\nit suffices to consider the number of solutions to the congruence\n\\begin{equation}\\label{change of var}\nZ^2\\equiv 4N\\pmod{\\ell^e}.\n\\end{equation}\nSuppose $z$ is an integer solution to~\\eqref{change of var} and\nwrite $4N=\\ell^sN_0$ with $(\\ell,N_0)=1$.\nIf $s\\ge e$, it follows that $z\\equiv 0\\pmod{\\ell^{\\ceil{e\/2}}}$, and hence there are at most\n$\\ell^{\\floor{e\/2}}$ solutions to~\\eqref{change of var}.\nThus, we may assume that $s<e$ and write\n\\begin{equation*}\nz^2= \\ell^s(N_0+\\ell^{e-s}k)\n\\end{equation*}\nfor some integer $k$.\nSince $(\\ell,N_0)=1$, it follows that $s$ must be even.\nWriting $s=2s_0$, we see that $z=\\ell^{s_0}x$, where\n$x$ is an integer solution to the congruence\n\\begin{equation}\\label{reduced cong}\nX^2\\equiv N_0\\pmod{\\ell^{e-s}}.\n\\end{equation}\nSo, in particular, $z\\equiv\\ell^{s_0}x\\pmod{\\ell^{e-s_0}}$.\nSince $(\\ell,N_0)=1$, it is a classical exercise as in~\\cite[p.~98]{HW:1979} to show that\n\\begin{equation*}\n\\#\\{X\\in\\mathbb{Z}\/\\ell^{e-s}\\mathbb{Z}: X^2\\equiv N_0\\pmod{\\ell^{e-s}}\\}\n\\le\\begin{cases}\n2&\\text{if }\\ell>2,\\\\\n4&\\text{if }\\ell=2.\n\\end{cases}\n\\end{equation*}\nTherefore, there are at most $2\\ell^{e-(e-s_0)}=2\\ell^{s\/2}\\le 2\\ell^{(e-1)\/2}$ solutions\nto~\\eqref{change of var} when $\\ell$ is odd, and there are at most\n$4\\ell^{e-(e-s_0)}=4\\ell^{s\/2}\\le 4\\ell^{(e-1)\/2}$\nsolutions when $\\ell=2$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{complete K sum}]\nBy Lemma~\\ref{compute factors of c},\n$c_{N,f}(n)\\ll \\frac{n\\prod_{\\ell\\mid(f,n)}\\#C_N^{(\\ell)}(1,1,f)}{\\kappa_{2N}(n)}$,\nwhere for any positive integer $m$,\n$\\kappa_{m}(n)$ is the multiplicative function defined on the prime powers\nby~\\eqref{defn of kappa}.\nTherefore,\n\\begin{equation}\\label{diff bn K and truncated K}\n\\begin{split}\nK_0(N)&-\\sum_{\\substack{f\\le V\\\\ (f,2)=1}}\\frac{1}{f}\\sum_{n\\le U}\\frac{1}{n\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\\#C_N(a,n,f)\\\\\n&\\quad\\ll\\sum_{f\\le V}\\strut^\\prime\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n>U}\\frac{2\\varphi((n,f))c_{N,f}(n)}{(n,f)n\\varphi(4n)\n\t\\prod_{\\ell\\mid (n,f)}\\#C_N^{(\\ell)}(1,1,f)}\\\\\n\t&\\quad\\quad+\\sum_{f> V}\\strut^\\prime\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n\\ge 1}\\frac{2\\varphi((n,f))c_{N,f}(n)}{(n,f)n\\varphi(4n)\n\t\\prod_{\\ell\\mid (n,f)}\\#C_N^{(\\ell)}(1,1,f)}\\\\\n&\\quad\\ll\\sum_{\\substack{f\\le V\\\\ (f,2)=1}}\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n>U}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\\\\\n\t&\\quad\\quad+\\sum_{\\substack{f> V\\\\ (f,2)=1}}\n\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n\\ge 1}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)},\n\\end{split}\n\\end{equation}\nwhere the primes on the sums on $f$ are meant to indicate that the sums are to be restricted\nto odd $f$ such that $\\#C_N^{(\\ell)}(1,1,f)\\ne 0$ for all primes $\\ell$ dividing $f$.\n\n\nIn~\\cite[Lemma 3.4]{DP:1999}, we find that\n\\begin{equation*}\n\\sum_{n>U}\\frac{1}{\\kappa_1(n)\\varphi(n)}\n\\sim\\frac{c_0}{\\sqrt U}\n\\end{equation*}\nfor some positive constant $c_0$.\nIn particular, this implies that the full sum converges.\nFrom this we obtain a crude bound for the tail of the sum over $n$\n\\begin{equation*}\n\\begin{split}\n\\sum_{n>U}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\n&=\\sum_{\\substack{km>U\\\\ (m,2N)=1\\\\ \\ell\\mid k\\Rightarrow\\ell\\mid 2N}}\n\t\\frac{1}{\\kappa_1(m)\\varphi(m)\\varphi(k)}\n=\\sum_{\\substack{k\\ge 1\\\\ \\ell\\mid k\\Rightarrow\\ell\\mid 2N}}\\frac{1}{\\varphi(k)}\n\t\\sum_{\\substack{m>U\/k\\\\ (m,2N)=1}}\\frac{1}{\\kappa_1(m)\\varphi(m)}\\\\\n&\\ll\\sum_{\\substack{k\\ge 1\\\\ \\ell\\mid k\\Rightarrow\\ell\\mid 2N}}\\frac{1}{\\varphi(k)}\n\t\\frac{\\sqrt k}{\\sqrt U}\n\\ll\\frac{1}{\\sqrt U}\\prod_{\\ell\\mid N}\\left(1+\\frac{\\ell}{(\\ell-1)(\\sqrt\\ell-1)}\\right)\\\\\n&=\\frac{1}{\\sqrt U}\\frac{N}{\\varphi(N)}\n\t\\prod_{\\ell\\mid N}\\left(1+\\frac{1}{\\sqrt\\ell(\\ell-1)}\\right)\\left(1+\\frac{1}{\\sqrt\\ell}\\right)\\\\\n&\\ll\\frac{1}{\\sqrt U}\\frac{N}{\\varphi(N)}\n\t\\prod_{\\ell\\mid N}\\left(1+\\frac{1}{\\sqrt\\ell}\\right).\n\\end{split}\n\\end{equation*}\nWe have already noted that $N\/\\varphi(N)\\ll\\log\\log N$.\nIt is a straightforward exercise as in~\\cite[p.~63]{MV:2007} to show that\n\\begin{equation*}\n\\prod_{\\ell\\mid N}\\left(1+\\frac{1}{\\sqrt\\ell}\\right)\n<\\exp\\left\\{O\\left(\\frac{\\sqrt{\\log N}}{\\log\\log N}\\right)\\right\\}.\n\\end{equation*}\nThus, we conclude that\n\\begin{equation}\\label{n tail}\n\\sum_{n>U}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\n\\ll\\frac{N^\\epsilon}{\\sqrt U}\n\\end{equation}\nfor any $\\epsilon>0$.\nFor the full sum over $n$, we need a sharper bound in the $N$-aspect, which we obtain\nby writing\n\\begin{equation}\\label{full n sum bound}\n\\begin{split}\n\\sum_{n\\ge 1}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\n&=\\prod_{\\ell\\mid 2N}\\left(1+\\frac{\\ell}{(\\ell-1)^2}\\right)\n\t\\sum_{\\substack{n\\ge 1\\\\ (n,2N)=1}}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\\\\\n&\\le\\frac{2N}{\\varphi(2N)}\\prod_{\\ell\\mid 2N}\\left(1+\\frac{1}{\\ell(\\ell-1)}\\right)\n\t\\sum_{n\\ge 1}\\frac{1}{\\kappa_1(n)\\varphi(n)}\\\\\n&\\ll\\log\\log N.\t\n\\end{split}\n\\end{equation}\n\nFor any odd prime $\\ell$ dividing $f$, we obtain the bounds\n\\begin{equation*}\n\\#C_N^{(\\ell)}(1,1,f)\n\\le\\begin{cases}\n2\\ell^{\\nu_\\ell(N)\/2}&\\text{if }\\nu_\\ell(f)>\\nu_\\ell(N)\/2\\text{ and }2\\mid\\nu_\\ell(N),\\\\\n\\ell^{\\nu_\\ell(f)}&\\text{otherwise}\n\\end{cases}\n\\end{equation*}\nfrom Lemma~\\ref{counting solutions}.\nHowever, if $\\nu_\\ell(f)>\\nu_\\ell(N)\/2$ and $2\\mid\\nu_\\ell(N)$,\nit follows that $\\nu_\\ell(f)\\ge 1+\\nu_\\ell(N)\/2$, and hence\n\\begin{equation*}\n2\\ell^{\\nu_\\ell(N)\/2}\\le\\ell^{1+\\nu_\\ell(N)\/2}\\le\\ell^{\\nu_\\ell(f)}\n\\end{equation*}\nsince $\\ell>2$.\nTherefore, for every odd integer $f$, we have that\n\\begin{equation*}\n\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)\\le f,\n\\end{equation*}\nand hence\n\\begin{equation}\\label{f tail}\n\\sum_{\\substack{f> V\\\\ (f,2)=1}}\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n<\\sum_{f>V}\\frac{1}{f\\varphi(f)}\\ll\\frac{1}{V}.\n\\end{equation}\nSubstituting the bounds~\\eqref{n tail},~\\eqref{full n sum bound},\nand~\\eqref{f tail} into~\\eqref{diff bn K and truncated K}, the lemma follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{counting solutions}]\nUpon completing the square, we have that\n\\begin{equation*}\nz^2-2(N+1)z+(N-1)^2-af^2=(z-N-1)^2-(4N+af^2).\n\\end{equation*}\nSince $N$ is odd, the number of invertible solutions to the congruence\n\\begin{equation*}\n(z-N-1)^2\\equiv 4N+af^2\\pmod{2^\\alpha}\n\\end{equation*}\nis the same as the number of invertible solutions to the congruence\n$Z^2\\equiv 4N+af^2\\pmod{2^\\alpha}$.\nSince $4N+af^2\\equiv 4+a\\pmod{8}$, the computation of\n$C_N^{(2)}(a,n,f)$ is thus reduced to a standard exercise.\nSee~\\cite[p.~98]{HW:1979} for example.\n\nIf $\\ell\\nmid 4N+af^2$, then there are exactly $1+\\leg{4N+af^2}{\\ell}$\nsolutions to the congruence\n\\begin{equation}\\label{odd prime power cong}\n(z-N-1)^2\\equiv 4N+af^2\\pmod{\\ell^\\alpha}.\n\\end{equation}\nHowever,\nif $\\ell$ divides the constant term, $(N-1)^2-af^2$, then we have exactly one\ninvertible solution and exactly one noninvertible solution.\n\nIt remains to treat the case when $\\ell\\mid 4N+af^2$.\nWe write $4N+af^2=\\ell^sm$ with $(m,\\ell)=1$.\nFirst, we observe that any solution $z$ to~\\eqref{odd prime power cong} must\nsatisfy $z\\equiv N+1\\pmod{\\ell}$.  Therefore, if $\\ell\\mid N+1$, then there\nare no invertible solutions; if $\\ell\\nmid N+1$, then every solution is invertible.\nHence, we assume that $\\ell\\nmid N+1$.  Now, the number of invertible\nsolutions to~\\eqref{odd prime power cong} is equal to the number of\n(noninvertible) solutions to\n\\begin{equation}\\label{odd prime power cong with vanishing disc}\nZ^2\\equiv \\ell^sm\\pmod{\\ell^e}.\n\\end{equation}\nIf $s\\ge e$, then $Z$ is a solution if and only if\n$Z\\equiv 0\\pmod{\\ell^{\\ceil{e\/2}}}$.\nThere are exactly $\\ell^{e-\\ceil{e\/2}}=\\ell^{\\floor{e\/2}}$ such values for $Z$\nmodulo $\\ell^e$.\nNow, suppose that $0<s<e$.\nThen $Z$ is a solution to~\\eqref{odd prime power cong with vanishing disc} if and only if\n\\begin{equation*}\nZ^2=\\ell^sm+\\ell^e k=\\ell^s\\left(m+\\ell^{e-s}\\right)\n\\end{equation*}\nfor some integer $k$.\nSince $\\ell\\nmid m$ and $s<e$, we see that there can be no such $Z$ if\n$s$ is odd or if $\\leg{m}{\\ell}=-1$.\nThus, we assume that $s=2s_0$, $\\leg{m}{\\ell}=1$, and we write\n$r^2=m+\\ell^{e-s}$.\nUnder this assumption, there are exactly two (distinct modulo $\\ell^{e-s}$)\nsolutions, say $r_1$ and $r_2$, to the\ncongruence $r^2\\equiv m\\pmod{\\ell^{e-2s_0}}$.\nTherefore, if $Z$ is a solution to~\\eqref{odd prime power cong with vanishing disc},\nthen either $Z=\\ell^{s_0}(r_1+k_1\\ell^{e-s})$ for some integer $k_1$ or\n$Z=\\ell^{s_0}(r_2+k_2\\ell^{e-s})$ for some integer $k_2$.\nIn other words, $Z\\equiv\\ell^{s_0}r_1\\pmod{\\ell^{e-s_0}}$ or\n$Z\\equiv\\ell^{s_0}r_2\\pmod{\\ell^{e-s_0}}$.\nIt is not hard to check that if $Z$ satisfies either of these two conditions, then\n$Z$ is a solution to~\\eqref{odd prime power cong with vanishing disc}.\nThere are exactly $2\\ell^{e-(e-s_0)}=2\\ell^{s_0}$ such values\nfor $Z$ modulo $\\ell^e$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{compute factors of c}]\nIt is easily checked that $c_{N,f}(1)=1$.\nIf $n$ is odd, we observe that\n\\begin{equation*}\nc_{N,f}(n)=\\sum_{a\\in(\\mathbb{Z}\/n\\mathbb{Z})^*}\n\\leg{a}{n}\n\\prod_{\\ell\\mid n}\n\\#C_N^{(\\ell)}(a,n,f).\n\\end{equation*}\nNow, suppose that $(m,n)=1$.  It follows that at least one of $m$ and $n$ must be odd.\nWithout loss of generality,\nwe assume that $n$ is odd.  Choose integers $m_0$ and $n_0$ so that\n$4mm_0+nn_0=1$.\nThen\n\\begin{equation*}\n\\begin{split}\nc_{N,f}(n)c_{N,f}(m)&=\n\\sum_{a_1\\in(\\mathbb{Z}\/n\\mathbb{Z})^*}\n\\leg{a_1}{n}\n\\prod_{\\ell\\mid n}\n\\#C_N^{(\\ell)}(a_1,n,f)\\\\\n&\\quad\\times\n\\sum_{\\substack{a_2\\in(\\mathbb{Z}\/4m\\mathbb{Z})^*\\\\ a_2\\equiv 1\\pmod 4}}\n\\leg{a_2}{m}\n\\mathcal S_2(m,a_2)\n\\prod_{\\ell\\mid m'}\n\\#C_N^{(\\ell)}(a_2,m,f)\\\\\n&=\n\\sum_{\\substack{a_1\\in(\\mathbb{Z}\/n\\mathbb{Z})^*,\\\\ a_2\\in(\\mathbb{Z}\/4m\\mathbb{Z})^*\\\\ a_2\\equiv 1\\pmod 4}}\n\\leg{a_14mm_0+a_2nn_0}{nm}\n\\mathcal S_2(nm,a_14mm_0+a_2nn_0)\\\\\n&\\quad\\times\\prod_{\\ell\\mid nm'}\n\\#C_N^{(\\ell)}(a_14mm_0+a_2nn_0,nm,f)\\\\\n&=c_{N,f}(nm).\n\\end{split}\n\\end{equation*}\nHence $c_{N,f}(n)$ is multiplicative in $n$.\n\nBy~\\eqref{defn of S_2}, we find that\n\\begin{equation*}\nc_{N,f}(2^\\alpha)\n=\\sum_{\\substack{a\\in(\\mathbb{Z}\/2^{\\alpha+2}\\mathbb{Z})^*\\\\ a\\equiv 1\\pmod 4}}\n\\leg{a}{2^\\alpha}\\mathcal S_2(2^\\alpha,a)\n=2\\sum_{\\substack{a\\in(\\mathbb{Z}\/2^{\\alpha+2}\\mathbb{Z})^*\\\\ a\\equiv 5\\pmod 8}}\n\\leg{a}{2}^\\alpha\n=(-1)^\\alpha 2^{\\alpha}\n\\end{equation*}\nfor $\\alpha\\ge 1$.\n\nWe now consider the case when $\\ell^\\alpha$ is an odd prime power.\nIn view of Lemma~\\ref{counting solutions}, it is natural to split\n$c_{N,f}(\\ell^\\alpha)$ into two parts, writing\n\\begin{align*}\nc_{N,f}^{(1)}(\\ell^\\alpha)&:=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f),\\\\\nc_{N,f}^{(0)}(\\ell^\\alpha)&:=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n\\end{align*}\nso that\n\\begin{equation}\\label{split c for odd ell}\nc_{N,f}(\\ell^\\alpha)\n=c_{N,f}^{(1)}(\\ell^\\alpha)\n+c_{N,f}^{(0)}(\\ell^\\alpha).\n\\end{equation}\n\nWe concentrate on $c_{N,f}^{(1)}(\\ell^\\alpha)$ first.\nApplying Lemma~\\ref{counting solutions}, we have\n\\begin{equation}\\label{nonvanishing disc part of c}\n\\begin{split}\nc_{N,f}^{(1)}(\\ell^\\alpha)\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)+\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\left[1+\\leg{4N+af^2}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\\\\\n&=\\ell^{\\alpha-1}\\left\\{\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\left[1+\\leg{4N+af^2}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\right\\}.\n\\end{split}\n\\end{equation}\nIn order to finish evaluating this sum, we split into cases.\nFirst, assume that $\\ell\\mid f$.  Then the sum defining $c_{N,f}^{(1)}(\\ell^\\alpha)$ is\nempty unless $\\ell\\nmid N$.\nIn this case, $\\ell\\mid (N-1)^2-af^2$ if and only if $\\ell\\mid N-1$.\nTherefore, if $\\ell\\mid f$ and $\\ell\\nmid N$, then\n\\begin{equation}\\label{c1 ell divides f}\n\\begin{split}\n\\frac{c_{N,f}^{(1)}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n&=\\begin{cases}\\displaystyle\n\\left[1+\\leg{N}{\\ell}\\right]\\sum_{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*}\\leg{a}{\\ell}^\\alpha\n\t&\\text{if }\\ell\\nmid N-1,\\\\\n\\displaystyle\n\\sum_{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*}\\leg{a}{\\ell}^\\alpha\t\n\t&\\text{if }\\ell\\mid N-1\n\\end{cases}\\\\\n&=\\begin{cases}\n(\\ell-1)\\left[1+\\leg{N}{\\ell}\\right]&\\text{if }2\\mid\\alpha\\text{ and }\\ell\\nmid N-1,\\\\\n\\ell -1&\\text{if }2\\mid\\alpha\\text{ and }\\ell\\mid N-1,\\\\\n0&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\begin{cases}\n(\\ell-1)\\left(1+\\leg{N}{\\ell}\\leg{N-1}{\\ell}^2\\right)\n\t&\\text{if }2\\mid\\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n(\\ell-1)&\\text{if }2\\mid\\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha\n\\end{cases}\n\\end{split}\n\\end{equation}\nas $\\#C_N^{(\\ell)}(1,1,f)=\\left(1+\\leg{N}{\\ell}\\leg{N-1}{\\ell}^2\\right)$ in this case.\n\nNow, suppose that $\\ell\\nmid f$.\nUnder this assumption, we note that $\\leg{a}{\\ell}=\\leg{af^2}{\\ell}$.\nPicking up with equation~\\eqref{nonvanishing disc part of c} and dividing through\nby $\\ell^{\\alpha-1}$, we have\n\\begin{equation}\\label{make c1 sum full}\n\\begin{split}\n\\frac{c_{N,f}^{(1)}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{af^2}{\\ell}^\\alpha\n\\left[1+\\leg{4N+af^2}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{af^2}{\\ell}^\\alpha\\\\\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+a\\\\ \\ell\\nmid (N-1)^2-a}}\n\\leg{a}{\\ell}^\\alpha\n\\left[1+\\leg{4N+a}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+a\\\\ \\ell\\mid (N-1)^2-a}}\n\\leg{a}{\\ell}^\\alpha\\\\\n&=\n\\sum_{\\substack{a\\in\\mathbb{Z}\/\\ell\\mathbb{Z}\\\\ a\\not\\equiv -4N\\pmod\\ell}}\n\\leg{a}{\\ell}^\\alpha\n+\\sum_{\\substack{a\\in\\mathbb{Z}\/\\ell\\mathbb{Z}\\\\  a\\not\\equiv (N-1)^2\\pmod\\ell}}\n\\leg{a}{\\ell}^\\alpha\\leg{4N+a}{\\ell}\\\\\n&=\\left[\\sum_{b=1}^\\ell\\left(1+\\leg{b}{\\ell}\\right)\\leg{b-4N}{\\ell}^\\alpha\\right]\n-\\leg{-N}{\\ell}^\\alpha-\\leg{N^2-1}{\\ell}^{2}\n\\end{split}\n\\end{equation}\nupon completing the sums and making the change of variables $b=4N+a$.\nOne easily computes that\n\\begin{equation*}\n\\sum_{b=1}^\\ell\\left(1+\\leg{b}{\\ell}\\right)\\leg{b-4N}{\\ell}^\\alpha\n=\\begin{cases}\n\\ell-1 &\\text{if }\\ell\\mid N,\\\\\n\\ell-1-\\leg{N}{\\ell} &\\text{if }\\ell\\nmid N\\text{ and }2\\mid\\alpha,\\\\\n-1 &\\text{if }\\ell\\nmid N\\text{ and }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\nFor example, the third case is Exercise 1(b) of~\\cite[p.~301]{MV:2007}.\nTherefore,\n\\begin{equation*}\n\\frac{c_{N,f}^{(1)}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\begin{cases}\n\\ell-2\n\t&\\text{if } \\ell\\mid N\\text{ and }\\ell\\nmid f,\\\\\n\\ell-2-\\leg{N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n\t&\\text{if }\\ell\\nmid Nf\\text{ and }2\\mid\\alpha,\\\\\n-1-\\leg{-N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n\t&\\text{if }\\ell\\nmid Nf\\text{ and }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\n\n\nIt remains to compute $c_{N,f}^{(0)}(\\ell^\\alpha)$.  We first consider the case when $\\ell\\nmid f$.\nNote that the sum defining $c_{N,f}^{(0)}(\\ell^\\alpha)$ is empty unless $\\ell\\nmid N$ as well.\nThus, under this assumption, we have that\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{af^2}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+a}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1)\\\\\n&=\\leg{-4N}{\\ell}^\\alpha\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+a}}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1).\n\\end{split}\n\\end{equation*}\nIn order to evaluate this last sum, for each $a\\in(\\mathbb{Z}\/\\ell^{\\alpha}\\mathbb{Z})^*$,\n we choose an integer representative in the range $-4N< a\\le\\ell^\\alpha-4N$.  This ensures that\n$0\\le\\nu_\\ell(4N+a)\\le\\alpha$.\nThere is exactly one such choice of $a$ so that $\\nu_\\ell(4N+a)=\\alpha$, namely\n$a=\\ell^\\alpha-4N$.\nFor $1\\le t\\le\\alpha-1$,  the number of such $a$ with $\\nu_\\ell(4N+a)=t$ is\n$(\\ell-1)\\ell^{\\alpha-t-1}$, but for only half of those values is\n$\\leg{(4N+a)\/\\ell^t}{\\ell}=1$.  Perhaps, the easiest way to see this is to consider the\nbase-$\\ell$ expansion of $4N+a$ for each $a$ in this range.\nTherefore, applying Lemma~\\ref{counting solutions} again, we have\n\\begin{equation*}\n\\begin{split}\n\\sum_{t=1}^\\alpha\n\\sum_{\\substack{0< 4N+a\\le\\ell^\\alpha\\\\ \\nu_\\ell(4N+a)=t}}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1)\n&=\\leg{N+1}{\\ell}^2\\ell^{\\floor{\\alpha\/2}}\n+\\sum_{t=1}^{\\floor{(\\alpha-1)\/2}}\n\\sum_{\\substack{0<4N+a<\\ell^\\alpha\\\\ \\nu_\\ell(4N+a)=2t}}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1)\\\\\n&=\\leg{N+1}{\\ell}^2\\ell^{\\floor{\\alpha\/2}}\n+\\sum_{t=1}^{\\floor{(\\alpha-1)\/2}}\\frac{(\\ell-1)\\ell^{\\alpha-2t-1}}{2}\n\t2\\leg{N+1}{\\ell}^2\\ell^t\\\\\n&=\\leg{N+1}{\\ell}^2\\ell^{\\floor{\\alpha\/2}}\n\t+\\leg{N+1}{\\ell}^2\\ell^{\\alpha-1}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\\\\\n&=\\leg{N+1}{\\ell}^2\\left(\n\\ell^{\\floor{\\alpha\/2}}+\n\\ell^{\\alpha-1}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\n\\right)\\\\\n&=\\leg{N+1}{\\ell}^2\\ell^{\\alpha-1}.\n\\end{split}\n\\end{equation*}\nTherefore, if $\\ell\\nmid fN$, then\n\\begin{equation*}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n=\\leg{-N}{\\ell}^\\alpha\\leg{N+1}{\\ell}^2\\ell^{\\alpha-1}.\n\\end{equation*}\n\nWe now compute $c_{N,f}^{(0)}(\\ell^\\alpha)$ in the case that $\\ell\\mid f$.\nNote that, in this case, the sum defining $c_{N,f}^{(0)}(\\ell^\\alpha)$ is empty unless\n$\\ell\\mid N$.\nNote that $\\nu_\\ell(4N+af^2)\\ge\\min\\{\\nu_\\ell(N),2\\nu_\\ell(f)\\}$ with equality holding if\n$\\nu_\\ell(N)\\ne 2\\nu_\\ell(f)$.\n\nFirst, suppose that $2\\nu_\\ell(f)<\\nu_\\ell(N)$.\nNoting that $e=\\nu_\\ell(4\\ell^\\alpha f^2)>2\\nu_\\ell(f)$,\nby Lemma~\\ref{counting solutions} we have that\n\\begin{equation*}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n=2\\ell^{\\nu_\\ell(f)}=2\\#C_N^{(\\ell)}(1,1,f)\n\\end{equation*}\nif and only if $\\leg{a}{\\ell}=\\leg{(4N+af^2)\/\\ell^{2\\nu_\\ell(f)}}{\\ell}=1$.\nHence,\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=2\\#C_N^{(\\ell)}(1,1,f)\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\leg{a}{\\ell}=1}}\\leg{a}{\\ell}^\\alpha\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\ell^{\\alpha-1}(\\ell-1)\n\\end{split}\n\\end{equation*}\nif $1< 2\\nu_\\ell(f)<\\nu_\\ell(N)$.\n\nNow, suppose that $\\nu_\\ell(N)<2\\nu_\\ell(f)$. Since $e=\\nu_\\ell(4\\ell^\\alpha f^2)>\\nu_\\ell(N)$,\nby Lemma~\\ref{counting solutions}, we have that\n\\begin{equation*}\n\\begin{split}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n&=\\begin{cases}\n2\\ell^{\\nu_\\ell(N)\/2}&\\text{if } 2\\mid\\nu_\\ell(N)\\text{ and }\\leg{N_\\ell}{\\ell}=1,\\\\\n0&\\text{otherwise}\n\\end{cases}\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\n\\end{split}\n\\end{equation*}\nfor every $a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*$.\nHence,\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\sum_{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*}\n\\leg{a}{\\ell}^\\alpha\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n\\ell^{\\alpha-1}(\\ell-1)&\\text{if }2\\mid \\alpha,\\\\\n0&\\text{if }2\\nmid \\alpha\n\\end{cases}\n\\end{split}\n\\end{equation*}\nif $1\\le\\nu_\\ell(N)<2\\nu_\\ell(f)$.\n\nFinally, consider the case when $2\\nu_\\ell(f)=\\nu_\\ell(N)$.\nLet $r=\\nu_\\ell(f)$ and $s=\\nu_\\ell(N)$ and write\n$f=\\ell^rf_\\ell$ and $N=\\ell^sN_\\ell$ with $(\\ell,f_\\ell N_\\ell)=1$.\nThen\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n=\\sum_{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*}\n\\leg{af_\\ell^2}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,\\ell^rf_\\ell)\\\\\n&=\\sum_{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,\\ell^r).\n\\end{split}\n\\end{equation*}\nTo evaluate this last sum, for each $a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*$, we choose an\ninteger representative in the range $-4N_\\ell< a\\le\\ell^\\alpha-4N_\\ell$.\nThis ensures that $0\\le\\nu_\\ell(4N_\\ell+a)\\le\\alpha$, and hence that\n$2r\\le\\nu_\\ell(4N+a\\ell^{2r})\\le2r+\\alpha$.\nSimilar to before, there is exactly one choice of $a$ such that\n$\\nu_\\ell(4N_\\ell+a)=\\alpha$, namely $a=\\ell^\\alpha-4N_\\ell$.\nFor $1\\le t\\le\\alpha-1$, there are\n$(\\ell-1)\\ell^{\\alpha-t-1}$ choices with $\\nu_\\ell(4N_\\ell+a)=t$, but for only half of\nthose values is $\\leg{(4N_\\ell+a)\/\\ell^t}{\\ell}=1$.\nNote that if $\\ell\\mid 4N_\\ell+a$, then $\\leg{a}{\\ell}=\\leg{-N_\\ell}{\\ell}$.\nTherefore, if $1<\\nu_\\ell(N)=2\\nu_\\ell(f)$, then\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{t=0}^\\alpha\\sum_{\\substack{0< 4N_\\ell+a\\le \\ell^\\alpha\\\\ \\nu_\\ell(4N_\\ell+a)=t}}\n\t\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,\\ell^r)\\\\\n&=\\leg{-N_\\ell}{\\ell}^\\alpha\\ell^{\\floor{(2r+\\alpha)\/2}}\n\t+\\sum_{t=1}^{\\floor{(\\alpha-1)\/2}}\n\t\\leg{-N_\\ell}{\\ell}^\\alpha\\frac{(\\ell-1)\\ell^{\\alpha-2t-1}}{2}2\\ell^{r+t}\\\\\n&\\quad+\\sum_{\\substack{0<4N_\\ell+a<\\ell^\\alpha\\\\ \\leg{4N_\\ell+a}{\\ell}=1}}\n\t\\leg{a}{\\ell}^\\alpha 2\\ell^r\\\\\n&=\\leg{-N_\\ell}{\\ell}^\\alpha\\ell^{r+\\floor{\\alpha\/2}}\n\t+\\leg{-N_\\ell}{\\ell}^\\alpha\\ell^{\\alpha-1+r}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\\\\\n&\\quad+\\ell^{r+\\alpha-1}\n\t\\sum_{a\\in\\mathbb{Z}\/\\ell\\mathbb{Z}}\\leg{a}{\\ell}^\\alpha\\left(1+\\leg{4N_\\ell+a}{\\ell}\\right)\\\\\n&=\\leg{-N_\\ell}{\\ell}\\ell^r\\left[\\ell^{\\floor{\\alpha\/2}}\n\t+\\ell^{\\alpha-1}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\\right]\\\\\n&\\quad+\\ell^{r+\\alpha-1}\n\t\\sum_{b\\in\\mathbb{Z}\/\\ell\\mathbb{Z}}\\left(1+\\leg{b}{\\ell}\\right)\\leg{b-4N_\\ell}{\\ell}^\\alpha\\\\\n&=\n\\ell^{r+\\alpha-1}\n\\begin{cases}\n\\leg{-N_\\ell}{\\ell}+\\ell-1-\\leg{N_\\ell}{\\ell}&\\text{if }2\\mid\\alpha,\\\\\n\\leg{-N_\\ell}{\\ell}-1&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\ell^{\\alpha-1}\n\\begin{cases}\n\\leg{-N_\\ell}{\\ell}+\\ell-1-\\leg{N_\\ell}{\\ell}&\\text{if }2\\mid\\alpha,\\\\\n\\leg{-N_\\ell}{\\ell}-1&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nThe lemma now follows by combining our computations for $c_{N,f}^{(0)}(\\ell^\\alpha)$\nand $c_{N,f}^{(1)}(\\ell^\\alpha)$.\n\n\\end{proof}\n\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec I}\nThe concept of frustration  plays an important role\nin the search for novel quantum states of condensed matter, see,\ne.g., \\cite{lnp04,buch2,moessner01,frust1,frust2}. The\ninvestigation of frustrating quantum spin systems is a challenging\ntask. Exact statements about the properties of  quantum spin\nsystem are known only in exceptional cases. The simplest known\nexact eigenstate is the fully polarized ferromagnetic state.\nFurthermore the one- and two-magnon excitations above the fully\npolarized ferromagnetic state  also can be calculated exactly,\nsee, e.g., \\cite{mattis81,Kuzian07,Zhitomirsky10,nishimoto2011}. An\nexample for non-trivial eigenstates is Bethe's famous solution for\nthe one-dimensional (1D) Heisenberg antiferromagnet (HAFM)\n\\cite{bethe}.\nThe investigation of strongly frustrated magnetic systems\nsurprisingly led to the discovery of several new exact\neigenstates. Some of the eigenstates found for  frustrated quantum\nmagnets are  of quite simple nature and for several physical\nquantities, e.g., the spin correlation functions, analytical\nexpressions can be found. Hence such exact eigenstates may play an\nimportant role either as groundstates of real quantum magnets or\nat least as groundstates of idealized models which can be used as\nreference states for more complex quantum spin systems.\nA well-known class of exact eigenstates are dimerized singlet\nstates, where a direct product of pair singlet states is an\neigenstate of the quantum spin system.  Such states become\ngroundstates for certain values\/regions of frustration. The most\nprominent examples are the Majumdar-Gosh state of the 1D $J_1-J_2$\nspin-half HAFM \\cite{majumdar} and the orthogonal dimer state of\nthe Shastry-Sutherland  model, see, e.g.,\n\\cite{shastry81,Mila,Miyahara,Lauchli,uhrig2004,darradi2005}. Many other frustrated spin\nmodels in one, two or three dimensions are known which have also\ndimer-singlet product states as groundstates, see, e.g.,\n\\cite{pimpinelli,ivanov97,japaner3d,koga,schul02}. A systematic\ninvestigation of systems with dimerized eigenstates can be found\nin  \\cite{schmidt05}. Note that these dimer-singlet product\ngroundstates  have gapped magnetic excitations and lead therefore\nto a plateau in the magnetization at $m=0$.\nRecently it has been demonstrated for the 1D counterpart of the\nShastry-Sutherland model \\cite{ivanov97,koga,schul02}, that more\ngeneral product eigenstates containing chain fragments of finite\nlength can lead to an infinite series of magnetization plateaus\n\\cite{schul02}.\n\n\nOther examples of product ground states are single-spin product\nstates of 1D XYZ model \\cite{mueller85} and the the highly\ndegenerate ground-state manifold of localized-magnon states found\nfor antiferromagnetic quantum spin systems on various frustrated\nlattices \\cite{lm}. Finally, we mention the so-called central-spin\nmodel or Heisenberg star where also exact statements on the\ngroundstate are known \\cite{starI}.\n\nAlthough, at first glance such singlet-product states seem to\nexist only for 'exotic' lattice models, it turned out that such\nmodels are not only a playground of theoreticians but may become\nrelevant for experimental research. The most prominent example is\nthe above mentioned Shastry-Sutherland model introduced in 1981\n\\cite{shastry81} for which only in 1999 the corresponding\nquasi-two-dimen\\-sional compound SrCu$_2$(BO$_3$)$_2$ was found\n\\cite{Kage,srcubo}. Other examples are the quasi-1D spin-Peierls\ncompound $CuGeO_3$, see, e.g., \\cite{cugeo},  or the star-lattice\ncompound\n[Fe$_3$($\\mu_3$-O)($\\mu$T-OAc)$_6$-(H$_2$O)$_3$][Fe$_3$($\\mu_3$-O)($\\mu$-OAc)$_{7.5}$]$_2\n\\cdot$7 H$_2$O.\\cite{star-exp,star-theor}\n\nIn the present paper we combine the ideas of Shastry and\nSutherland \\cite{shastry81} and our recent findings on exact\ntrimerized singlet product ground states (TSPGS's) for 1D\ninteger-spin Heisenberg systems \\cite{schmidt10} and discuss such\nTSPGS's on a  two-dimensional modified Shastry-Sutherland\nsquare-lattice model. Section \\ref{sec_egs} shortly recapitulates\nthe theory of TSPGS's and section \\ref{sec_model} defines the\nmodified  Shastry-Sutherland model and its finite realizations\nthat will be analyzed in what follows. We have concentrated in our\nnumerical studies on the size of the gap for the exact ground\nstate for finite lattices of $N=12$ (for spin  quantum numbers\n$s=1$, $s=2$), as well as  $N=18$ and $N=24$ (for $s=1$) and on\nthe magnetization curves for selected values of $J_2$, see section\n\\ref{sec_nr}. The analytical results in section \\ref{sec ar}\nmainly concern upper and lower bounds of the gap function. These\nresults depend on a slightly generalized statement and proof of\nthe gap theorem, first formulated in \\cite{schmidt10}, which is\ndone in appendix~\\ref{app}. Finally, appendix~\\ref{class} contains exact results on\nclassical magnetization curves for the model under consideration.\n\n\n\n\\section{Exact ground states}\n\\label{sec_egs} The anti-ferromagnetic uniform spin trimer\n\\begin{equation}\\label{egs1}\nH_1=J(\\op{\\bf{s}}_0\\cdot\\op{\\bf{s}}_1+\n\\op{\\bf{s}}_0\\cdot\\op{\\bf{s}}_2+\\op{\\bf{s}}_1\\cdot\\op{\\bf{s}}_2)\n\\end{equation}\nhas, for $J>0$ and integer $s$, a unique $S=0$ ground state, denoted\nby $[0,1,2]$, with ground state energy\n\\begin{equation}\\label{egs2}\nE_0=-\\frac{3}{2}J s(s+1) \\;.\n\\end{equation}\nThe corresponding product state\n\\begin{equation}\\label{egs3}\n\\Phi=\\bigotimes_{i=1}^{\\mathcal N}[i0,i1,i2]\n\\end{equation}\nwill be an eigenstate of a system of ${\\mathcal N}$ coupled spin\ntrimers indexed by $i=1,\\ldots,{\\mathcal N}$ with Hamiltonian\n\\begin{equation}\\label{egs4}\nH=\\sum_{i\\epsilon\nj\\delta}J_{i\\epsilon,j\\delta}\\,\\op{\\bf{s}}_{i\\delta}\\cdot\\op{\\bf{s}}_{j\\epsilon}\n\\;,\n\\end{equation}\nif and only if the coupling between different trimers is ``balanced\"\nin the following sense:\n\\begin{equation}\\label{egs5}\nJ_{i\\delta,j\\delta}+J_{i\\epsilon,j\\epsilon}=J_{i\\delta,j\\epsilon}+J_{i\\epsilon,j\\delta}\n\\end{equation}\nfor all $1\\le i<j\\le{\\mathcal N}$ and $\\delta,\\epsilon=0,1,2$, see\n\\cite{schmidt10}. Moreover, (\\ref{egs3}) will be a ground state of\n(\\ref{egs4}), a TSPGS, if the intra-trimer coupling is almost\nuniform and the inter-trimer coupling is not too strong\n\\cite{schmidt10}.\nThe domain of coupling constants where this is the case will be called the ``TSPGS-region\".\\\\\n\nIf the system of trimers has a periodic lattice structure, the\ndifference $\\Delta E$ between the energy of the first excited state\nand that of the ground state can be shown \\cite{schmidt10} to be\nbounded from below independently of the system size. In other\nwords, the TSPGS is ``gapped\".\n\n\n\\begin{figure}\n\\resizebox{0.9\\columnwidth}{!}{\n\\includegraphics{24_2d_trimer_a.eps}\n} \\caption{ The modified Shastry-Sutherland model on the decorated\nsquare lattice for  $N= 24$ sites (periodic conditions imposed) used for exact diagonalization.}\n\\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{12_18.eps}\n} \\caption{Two finite decorated square lattices of  $N= 12$  and\n$N=18$ sites used for exact diagonalization.} \\label{fig2}\n\\end{figure}\n\n\n\n\\section{The model}\n\\label{sec_model}\n\nWe consider the inter-spin Heisenberg Hamiltonian on a decorated\nsquare-lattice (see figure~\\ref{fig1}). It results from the\nwell-known Shastry-Sutherland model by replacing its diagonals by\nequilateral triangles with uniform intra-trimer interaction strength\n$J_1>0$. The set of triangles is divided in a bi-partite fashion\ninto two disjoint subsets of triangles of type I and type II,\ncorresponding to diagonals with positive slope resp.~negative ones,\nsee figure~\\ref{fig1}. Each triangle of, say, type I is surrounded\nby four\ntriangles of type II and connected to each of them with three bonds of strength $J_2$.\\\\\nIt follows that the inter-trimer coupling satisfies the balance\ncondition (\\ref{egs5}) and hence the theory of TSPGS's applies. In\nparticular, two questions arise which will be addressed in the\nfollowing sections: What is the size of the TSPGS-region and of what\nkind are the lowest excitations? The latter question is also\nconnected to the issue of magnetization plateaus which will be\nshortly discussed below.\n\n\n\n\n\\section{Results}\n\n\\subsection{Numerical results}\n\\label{sec_nr} In what follows we set $J_1=1$ and consider $J_2$ as\nthe variable bond strength. To study the region where the TSPGS is\nthe ground state of the model (\\ref{egs4}) we use the Lanczos exact\ndiagonalization (ED) technique. Since for spin quantum numbers $s >\n1\/2$ considered here the size of the Hamiltonian matrix grows much\nfaster with system size $N$ than for $s=1\/2$, we are restricted to\nfinite lattices of $N= 12,18$ and $24$ for $s=1$ and $N=12$ for\n$s=2$. The largest lattice is shown in figure~\\ref{fig1}, whereas\nthe smaller lattices are shown in figure~\\ref{fig2}. Although the\ncriterion for the existence of TSPGS's (see section \\ref{sec_model})\nare fulfilled, we have to mention that for the small lattices of\n$N=12$ and $N=18$ the exchange pattern of the $J_1$ diagonal bonds\nin the squares do not match to the infinite system. Nevertheless, we\nhave included the data for $N= 12$ and $18$ to get an impression on\nfinite-size effects and on the influence of the spin quantum number\n$s$.\n\nAccording to \\cite{schmidt10} the TSPGS is gapped. Hence we use the\nspin gap, see figure~\\ref{fig3}, to detect the critical points\n$J^{c1}_2$ and $J^{c2}_2$, where the  TSPGS gives way for other\nground states. We find for $s=1$ the values $J^{c1}_2\n=-0.570,-0.578$, and $-0.587$ and  $J^{c1}_2 =0.434, 0.446$, and\n$0.454 $  for $N=12,18$, and $24$, respectively (cf.\nfigure~\\ref{fig3}(a)). For $s=2$ and $N=12$ we have $J^{c1}_2\n=-0.400$ and  $J^{c2}_2 =0.322$, cf. figure~\\ref{fig3}(b). These\nvalues lie between the upper and lower bounds which will be derived\nfor $J^{c1}_2$ and  $J^{c2}_2$ in the next section for $N \\to\n\\infty$. The nature of the lowest excited state depends on $J_2$.\nAround  $J_2=0$ it is a triplet state with strong antiferromagnetic\ncorrelations along the trimer bonds and weak correlations between\nthe trimers. Near $J^{c1}_2$ the lowest excitation is a\nferrimagnetic state, i.e. the total spin is $S=Ns\/3$ and the system\nsplits into two ferromagnetically correlated sublattices containing\non the one hand the $2N\/3$ square-lattice sites (i.e. sites\n$0,1,\\ldots,15$ in figure~\\ref{fig1}) and on the other hand the\n$N\/3$ additional sites (i.e. sites $16,17,\\ldots,23$ in\nfigure~\\ref{fig1}). The spin correlations between both sublattices\nare anti-ferromagnetic. The ferrimagnetic state is the ground state\nfor $-1.5 < J_2 < J^{c1}_2$. Near $J^{c2}_2$ the lowest excitation\nis a collective singlet state with strong correlations along all\nbonds, and, this state becomes\nthe ground state at $J_2=J^{c2}_2$.\n\n\n\n\n\n\\begin{figure}\n\\vspace*{5cm}\n\\scalebox{0.7}{\\includegraphics{gap_multi_mit_schranken_vert.eps}}\n\\caption{Numerical exact data for $N=12$, $18$, and\n$24$ (symbols) as well as upper (black solid line) and lower bounds\n(red solid line) for the excitation gap $\\Delta E $. (a) spin\nquantum number $s=1$;\n(b) spin quantum number $s=2$.\nNote that the labels $S=1$, $S=0$, $S=2N\/3$ (ferri), and $S=8$\n(ferri) characterize the total spin of the  excited state.\n}\n\\label{fig3}      \n\\end{figure}\n\n\n\n\\begin{figure}\n\\vspace*{5cm}\n\\scalebox{0.7}{\\includegraphics{m_h_plat_multi_vertical.eps}}\n\\caption{(a) Magnetization curve $m(h)$ for selected\nvalues of $J_2$ and $s=1$ (thick lines $N=24$, thin lines $N=18$);\n(b) Plateau widths $\\Delta h$  of the $m=1\/3$ and the  $m=2\/3$ plateaus  as\na function of $J_2$ for $N=24$ and $N=18$ and $s=1$.\n}\n\\label{fig4}      \n\\end{figure}\n\n\n\nIt is well  known that the magnetization curve of the\nShastry-Sutherland model (as well as that of the corresponding\nmaterial SrCu$_2$(BO$_3$)$_2$) possesses a series of pla\\-teaus, see,\ne.g., \\cite{Kage,kodama,misguich,mila}. Motivated by this, we study\nnow briefly the  magnetization curve $M(h)$ (where $M$ is the total\nmagnetization  and  $h$ is the strength of the external magnetic\nfield) for the considered model for $s=1$ using ED\nfor $N=18$ and $N=24$ sites. ED results  for the\nrelative magnetization $m=M\/M_{sat}$ versus magnetic field $h$ for\n$N=18$ and $N=24$ sites are shown in figure~\\ref{fig4}a. Again the\nfinite-size effects seem to be small. Trivially, in the limit\n$J_2=0$ the $m(h)$ curve consists of three equidistant plateaus and\njumps according to the magnetization  curve of an individual\ntriangle. Switching on a ferromagnetic inter-triangle bond $J_2 < 0$\nthe general shape of the magnetization curve is preserved. However,\nthe  saturation field as well as the end points of the plateaus\ndecrease almost lineraly with $J_2$ and become zero at $J_2=1.5$,\nwhere the ground state becomes  the fully polarized ferromagnetic\nstate.\n\nIn case of a moderate antiferromagnetic inter-triangle bond $J_2 >\n0$ the plateaus at  $m=1\/3$ and $m=2\/3$ still exist, however the\ndiscontinuous transition between plateaus becomes smooth. Note that\na $m=1\/3$ plateau was also found for the standard Shastry-Sutherland\nmodel \\cite{misguich,mila}. The plateau widths $\\Delta h$ of the $m=1\/3$\nand $m=2\/3$ plateaus in dependence on $J_2$ is shown in\nfigure~\\ref{fig4}b. Obviously, both  widths shrink monotonously with\nincreasing of $|J_2|$.\nIf $J_2$ approaches the critical value   $J^{c1}_2$ we find\nindications for additional plateaus, e.g., at $m=5\/6$. Note, however,\nthat our finite-size analysis of the plateaus naturally could miss\nother plateaus present in infinite systems, see, e.g., the discussion\nof the ED data of the $m(h)$ curve of the standard\nShastry-Sutherland model in \\cite{wir04}. Hence, the study of\nthe magnetization process of the considered quantum spin model needs\nfurther attention based on alternative methods.\n\nOne might expect that the presence of these plateaus and jumps may\nbe linked purely to quantum effects because they are often not\nobserved in equivalent classical models at $T=0$\n\\cite{lm,kawamura,zhito,cabra}. However, for the present model the\nplateau at $m=1\/3$ survives in the classical limit for $J_2 < 0$ as\nwe will show in appendix~\\ref{class}.\n\n\n\n\\subsection{Analytical results}\\label{sec ar}\n\\subsubsection{$s=1$}\n\\begin{figure}\n\\vspace*{5cm}\n\\scalebox{0.85}{\\includegraphics{HR69_vertical.eps}}\n\\caption{Two possible subsystems of the modified\nShastry-Sutherland lattice, see e.~g.~figure \\ref{fig1}. The upper\none, $H_6$, consists of two coupled triangles; the lower one, $H_9$\nof three triangles.\n}\n\\label{figH69}      \n\\end{figure}\nIn order to obtain analytical results about the TSPGS-region we have\nadapted theorem $3$ of \\cite{schmidt10} to the present situation. A\nslightly more general version of this theorem is stated and proven\nin appendix~\\ref{app}. It yields lower bounds for the gap $\\Delta E$ of the\nform $\\Delta E\\ge f(4J)$ and the TSPGS-region in terms of properties\nof simpler spin systems of which the lattice can be composed, see\nfigure~\\ref{fig3}. These subsystems are chosen here as systems\nisomorphic to $H_6$, see figure \\ref{figH69}, consisting of two\nneighboring triangles. For $s=1$ the gap function $x\\equiv \\delta_6\nE=f(J),\\,J\\equiv\\frac{J2}{J1}$ of $H_6$ is obtained as a special\ncase of equation (\\ref{ar2a}) given below.\nThis yields the corresponding bounds for the\nTSPGS-region $(J^{c1},J^{c2})$\n\\begin{equation}\\label{ar3}\nJ^{c1}<\\frac{3-\\sqrt{73}}{16}\\approx -0.3465<\\frac{1}{4}<J^{c2}\\quad\n\\mbox{for  }s=1\\;.\n\\end{equation}\nThe function $\\delta_6 E=f(J)$ according to (\\ref{ar2a}) also\nprovides an upper bound for the gap function of the lattice, since it\nrepresents the energy of a state orthogonal to the TSPGS, albeit not\nan eigenstate of $H$. This bound is very close to the numerically\ndetermined gap function in the case of $N=12$, see figure \\ref{fig3}a,\nbut considerably deviates in the cases of $N=18$ and $N=24$. This\nindicates that, in general, the lowest excitations of the lattice\nare different from the excitations of $H_6$.\\\\\n\n\\subsubsection{General $s$}\n\nIt is possible to analytically calculate the energy of the lowest\nexcitations of $H_6$ for general integer $s$. The corresponding gap\n$\\delta_6 E=x=f(J)$ is obtained as the lowest root of the following\ncubic equation\n\\begin{eqnarray}\\nonumber\n&-&(x-4) (x-2) (x-1 )\n -(x-1 ) (2x-5 ) J\\\\ \\label{ar2a}\n&+&( 1 - 3 r - x + r x)J^2 +r J^3=0\n\\end{eqnarray}\nwhere we have set $r\\equiv s(s+1)$. From this result one derives the\nlower bound\n\\begin{equation}\\label{ar2b}\n\\Delta E\\ge f(4J) \\quad \\mbox{for general }s\n\\end{equation}\nand a lattice of arbitrary size, see theorem $1$ in appendix~\\ref{app}\nadapted to the system under consideration.\nThe corresponding curves are shrinking in $J$-direction with\nincreasing $s$ and yield inner bounds for the TSPGS-region\n$(J^{c1},J^{c2})$ of the form\n\\begin{equation}\\label{ar2b1}\nJ^{c1} < J_{L}^{(1)}<0<J_{L}^{(2)}<J^{c2}\n\\;,\n\\end{equation}\nsee figure \\ref{figG1} (green curves).\nUpon scaling w.~r.~t.~the new variable\n$j\\equiv\\sqrt{r}J$ the graphs of (\\ref{ar2a}) asymptotically approach the curve given by\n\\begin{equation}\\label{ar2c}\nj^2=\\frac{(x-4)(x-2)(x-1)}{16(x-3)}\\;,\n\\end{equation}\nwith Taylor expansion\n\\begin{equation}\\label{ar2d}\nx=1-\\frac{32}{3}j^2+{\\mathcal O}(j^3)\\;,\n\\end{equation}\nsee figure \\ref{figG3b}.\nHence $J_{L}^{(i)}$ assumes for $s\\to\\infty$ asymptotically the form\n\\begin{equation}\\label{ar2d1}\n|J_{L}^{(i)}|\\sim\\frac{1}{\\sqrt{6s(s+1)}},\\;\\;i=1,2\n\\;.\n\\end{equation}\n\\\\\n\n\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{FIGLB.eps}\n} \\caption{Lower bounds of the scaled gap function\n$\\delta_6 E(j),\\;j=\\sqrt{s(s+1)}J$ of the modified\nShastry-Sutherland spin lattice for $s=1,\\ldots,10$ (thin curves)\nobtained from Eq.~(\\ref{ar2a}). The curves approach the asymptotic\n(\\ref{ar2c}) for $s\\rightarrow\\infty$  (thick red curve) which has a\nsimple quadratic approximation (\\ref{ar2d}) (thick green curve).} \\label{figG3b}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{FIGUB.eps}\n} \\caption{Upper bounds of the scaled gap function\n$\\delta_0 E(j),\\;j=\\sqrt{s(s+1)}J$ of the modified\nShastry-Sutherland spin lattice for $s=1,\\ldots,10$ (thin curves)\nobtained from Eq.~(\\ref{ar6}). The curves approach the asymptotic\n(\\ref{ar7}) for $s\\rightarrow\\infty$  (thick red curve) which has a\nsimple quadratic approximation (\\ref{ar8}) (thick green curve).} \\label{figG3a}\n\\end{figure}\n\n\nIn order to obtain close upper bounds $g(J)$ of the gap $\\Delta(E)$\nin the case $N\\ge 18$ we calculate\nthe energy of a certain (degenerate) state that involves three\ntriangles for arbitrary integer $s$, say, one triangle of type $I$\nand two neighboring triangles of type $II$, see figure \\ref{figH69}.\nThis state is obtained as an exact eigenstate of $H_0$, which is the\nfull Hamiltonian $H$, restricted to a $4^3=64-$dimensional subspace\nspanned by product states of the form\n\\begin{equation}\\label{ar4}\n\\phi_i\\otimes\\phi_j\\otimes\\phi_k,\\quad i,j,k=0,\\ldots,3\\;.\n\\end{equation}\nThe $\\phi_n$ live in the $(2s+1)^3$-dimensional Hilbert spaces\nbelonging to one of the three triangles. $\\phi_0=[0,1,2]$ denotes\nthe TSPGS of the corresponding triangle and\n\\begin{equation}\\label{ar5}\n\\phi_i\\equiv \\frac{\\op{s}_0^{(i)}\\phi_0}{||\n\\op{s}_0^{(i)}\\phi_0||},\\;i=1,2,3\\;,\n\\end{equation}\nwhere $\\op{s}_0^{(i)}$ is the $i$-th component of the spin operator\n$\\op{\\bf s}_0$ pertaining to the spin site number $0$, an\narbitrarily chosen spin site of the corresponding triangle. The gap\nfunction of $H_0$ will be denoted by $x\\equiv\\delta_0 E=g(J)$ and\nconstitutes an upper bound for $\\Delta(E)$.\nIt has the following implicit form, using $r\\equiv s(s+1)$:\n\\begin{eqnarray}\\nonumber\n0&=& -12(x-3)^2(x-2)(x-1)-6J(x-3)(x-1)(4x-9)\\\\ \\nonumber\n && +J^2\n(x-3)(9-9x+4r(7x-15))+16 J^3 r (2x-5)\\;.\\\\\n&& \\label{ar6}\n\\end{eqnarray}\nAgain, the function $g$ belongs to the lowest branch of (\\ref{ar6}).\nThe corresponding curves are shrinking in $J$-direction with\nincreasing $s$ and yield outer bounds for the TSPGS-region\n$(J^{c1},J^{c2})$ of the form\n\\begin{equation}\\label{ar6a}\nJ_{U}^{(1)}<J^{c1} < 0<J^{c2}<J_{U}^{(2)}\n\\;,\n\\end{equation}\nsee figure \\ref{figG1} (red curves).\nUpon scaling w.~r.~t.~the new variable\n$j\\equiv\\sqrt{r}J$ the graphs of (\\ref{ar6})\nasymptotically approach the curve given by\n\\begin{equation}\\label{ar7}\nj^2=\\frac{3(x-3)(x-2)(x-1)}{7x-15}\\;,\n\\end{equation}\nwith Taylor expansion\n\\begin{equation}\\label{ar8}\nx=1-\\frac{4}{3}j^2+{\\mathcal O}(j^3)\\;,\n\\end{equation}\nsee figure \\ref{figG3a}.\nHence $J_{U}^{(i)}$ assumes for $s\\to\\infty$ asymptotically the form\n\\begin{equation}\\label{ar8a}\n|J_{U}^{(i)}|\\sim\\sqrt{\\frac{6}{5s(s+1)}},\\;\\;i=1,2\n\\;.\n\\end{equation}\n\\\\\n\n\n\\begin{figure}\n\\resizebox{1.0\\columnwidth}{!}{\n\\includegraphics{FIGG1.eps}\n} \\caption{Exact bounds for the TSPGS-region $(J^{c1},J^{c2})$} for $s=1,\\ldots,10$\nof the form $J_{U}^{(1)}<J^{c1}< J_{L}^{(1)}< 0<J_{L}^{(2)}<J^{c2}<J_{U}^{(2)}$.\nThese are derived\nfrom (\\ref{ar2a}) (green curves, inner bounds) and (\\ref{ar6}) (red curves, outer bounds).\nIn the classical limit $s\\rightarrow\\infty$\nthe TSPGS-region shrinks to zero according to (\\ref{ar2d1}) and (\\ref{ar8a}).\n\\label{figG1}\n\\end{figure}\n\n\n\n\n\nAlthough these curves constitute only upper\nbounds of the true gap functions, the comparison with the numerical\nresults for $N=18$ and $N=24$ reveals a close approximation to both\ncurves, see figure \\ref{fig3}. This supports our conjecture that\n(\\ref{ar6}) indeed may serve as an analytical approximation of the\ngap functions for large $N$ and arbitrary integer $s$. This would\nmean that the excitations from the TSPGS can be viewed as local\nexcitations essentially concentrated on three neighboring triangles.\nNumerically determined spin correlation functions seem to be in accordance with\nthis conjecture. Of course, the corresponding excited state will be\nlargely degenerate due to the translational symmetry of the lattice.\nWe expect an almost flat $\\bf{k}$-dependance of the energy band\n$E(\\bf{k})$. This expectation is also supported by our numerical\nresults. We have found that the lowest excitations close to $J=0$\nhave the total spin quantum number $S=1$ in accordance with our\nmodel.\\\\\n\nIn the case $N=12$ where we have performed numerical calculations for\n$s=1$ and $s=2$ it is not possible to put a subsystem of type $H_9$  into the\nlattice and the above results do not apply. However,\nan analogous method can be applied to two\ncoupled triangles of type $H_6$ and yields an upper bound of the gap function of the form\n\\begin{equation}\\label{ar2e}\n\\Delta E\\le \\frac{1}{12} (18 - 3 J - \\sqrt{9(J-2)^2  + 96 J^2\ns(s+1)}).\n\\end{equation}\nThe numerically determined gap together with the bounds (\\ref{ar2b})\nand  (\\ref{ar2e}) is represented in figure \\ref{fig3} b.\\\\\n\n\n\\section*{Acknowledgement}\nThe numerical calculations were performed using J.~Schulenburg's\n{\\it spinpack}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Glossary}\n\n\\begin{itemize}[leftmargin=3cm]\n    \\item[Network]  A \\emph{network} is a collection of nodes (also called vertices) and edges (also called links) linking pair of nodes. Mathematically, it is represented by a graph ${G}=({V},{E})$ where ${V}$ is the set of nodes and ${E}\\subseteq V\\times V$ is the set of edges. Additional information can be attached to each node or edge, for example edges can have different weights. Edges can be undirected or directed.\n    \n    \\item[Adjacency matrix] The \\emph{adjacency matrix},  $\\matr{A}$,  of a network is a $N\\times N$ matrix ($N=\\left|{V}\\right|$) with element $A_{ij}=1$ if there is an edge from node $i$ and to node $j$ and $A_{ij}=0$ otherwise. If the network is weighted, $A_{ij}=w_{ij}$ where $w_{ij}\\in\\mathbb{R}$ is the weight associated with the edge between nodes $i$ and $j$ if it exists and $A_{ij}=0$ otherwise.\n    For undirected network, $A_{ij}=A_{ji}$, i.e. $\\matr{A}$ is symmetric.\n    \n    \\item[Degree of a node]  The \\emph{degree}, $k_i$, of node $i$ in an undirected network is equal to its number of connections, i.e. $k_i=\\sum_j A_{ij}$. For directed network, we differentiate the \\emph{in-degree}, $k^{in}_i=\\sum_j A_{ji}$ and the \\emph{out-degree}, $k^{out}_i=\\sum_j A_{ij}$, i.e. the number of edges in-coming to node $i$ and out-coming from node $i$, respectively. In weighted undirected networks, the degree of a node is replaced by its \\emph{strength}, $s_i=\\sum_j w_{ij}$ or \\emph{in-strength} and \\emph{out-strength} for weighted directed networks.\n    \n    \\item[Walk] A \\emph{walk} is an alternating sequence of vertices and edges in which every vertex is incident to both the edges that come before and after it in the sequence.\n    \\item[Path] A \\emph{path} on a graph is a walk in which all vertices and edges are distinct.\n    \\item[Connected components] A \\emph{connected component} of an undirected graph $G(V,E)$ is a subgraph of $G$, made of a subset of $V$ and all the edges connecting nodes of the subset toghether,  where there exist a path between each pair of nodes. In directed graphs, we differentiate \\emph{strongly-connected components}, where there exist a path in both directions between all pairs of nodes, and \\emph{weakly-connected components}, where there exist a path in at least one direction between all pairs of nodes.\n\\end{itemize}\n\n\n\n\n\\section{Why study networks?}\n\n\nIn \\index{network science} complex systems are represented as a mathematical \\index{graphs} consisting of a set of nodes representing the components and a set of edges representing their interactions.\nThe framework of networks has led to significant advances in the understanding of the \nstructure, formation and function of complex systems\\,\\cite{newman2003structure,barrat2008dynamical,boccaletti2006complex,albert2002statistical}.\nSocial and biological processes \nsuch as the dynamics of epidemics\\,\\cite{pastor2015epidemic}, \nthe diffusion of information in social media\\,\\cite{zhang2016dynamics},\nthe interactions between species in ecosystems\\,\\cite{montoya2006ecological} or the  communication between neurons in our brains\\,\\cite{bullmore2009complex} are all actively studied using dynamical models on complex networks.\nIn all of these systems, the patterns of connections at the individual level play a fundamental role on the global dynamics and finding the most important nodes allows one to better understand and predict their behaviors.\n\n\n\n\n\n\n\\section{Definition of the Subject}\nReal-world \\index{complex networks} are characterized by a number of structural features differentiating them from regular networks, such as lattices, but also making them different than completely random graphs, such as Erd\\H{o}s-R\\'enyi graphs\\,\\cite{newman2003structure}.\nThe properties that can be found in complex networks include, for example, a modular organization, also called community structure\\,\\cite{Fortunato2010}, or the so-called small-world effect, i.e. the fact that most pairs of nodes are connected by a very short path compared to the sizes of the networks\\,\\cite{Watts1998}.\nA characteristic of real-world complex networks that interests us here and that has been intensively studied is the fact that their degree distributions are usually very heterogeneous\\,\\cite{albert2002statistical,newman2003structure}, indicating that there is usually large differences in the number of connections that nodes have.\nMany real world networks have been found to have degree distributions resembling power laws and are sometimes referred to as scale-free\\,\\cite{barabasi2009scale}.\nWhether these real-world systems are really scale-free and whether their degree distributions are really power laws is still disputed\\,\\cite{Holme2019}.\nHowever, what is undeniable is the fact that many real world networks have degree distributions that are heavy-tailed with a minority of the nodes concentrating the majority of the connections.\n\nIn these systems, a small set of essential nodes can shape the collective dynamics of the entire systems.\nFor example, during epidemic outbreaks of infectious diseases, some individuals, known as \\emph{super-spreaders}, infect disproportionately more secondary contacts, as compared to most others\\,\\cite{kitsak2010identification,stein2011super},\nkeystone species in ecology are responsible for the integrity and stability of ecosystems\\,\\cite{may1972will,scheffer2012anticipating,mills1993keystone,morone2018kcore}\nand specific regions in brain networks are more important than others in the formation of memory\\,\\cite{bullmore2009complex,del2018finding,reis2014avoiding,zamora2010cortical}.\nIn social networks, a small set of \\index{influencers} can drive the global dynamics of the system\\,\\cite{bovet2019influence} and opinion leaders are capable of influencing the public viewpoint on certain trending topics \\,\\cite{watts2007influentials}.\nAn important research effort in network science has therefore been dedicated to the development of methods allowing to find the most important nodes in networks. \nIntuitively, nodes with a large degree are likely to be more successful to trigger large-scale propagations or to control a large number of nodes. The degree centrality ranks nodes in terms of their degree and allows to identify highly connected \\emph{\\index{hubs}} present in most real-world complex networks that play an essential role in controlling their dynamics and maintaining their integrity\\,\\cite{barabasi1999emergence,albert2000error,cohen2001breakdown}.\nWhile the \\index{degree centrality} is arguably the simplest \\index{centrality measure}, it only uses local information about each node to rank its centrality and more complex centrality measures have been developed in order to capture the collective network effects impacting the influence of a node.\nMost centrality measures are based on a notion of distance between nodes or on a way to traverse the network that allows to compute how \"central\" a node is compared to other nodes while capturing the heterogeneous structural patterns of complex networks.\nIn the following, we describe centrality measures based on the notions of \\index{network traversal} they rely on.\nThis short entry aims at being an introduction to this extremely vast topic, with many contributions from several fields, and is by no means an exhaustive review of all the literature about network centralities (see, for example, \\,\\cite{freeman1978centrality,Borgatti2005,Perra2008,landherr2010critical,rodrigues2019network} for more exhaustive reviews).\n\n\n\\subsection{Centrality measures based on shortest paths}\n\n\\index{Closeness centrality}\\,\\cite{bavelas1950communication, beauchamp1965improved,sabidussi1966centrality,dangalchev2006residual} and \\index{betweenness centrality}\\,\\cite{freeman1978centrality,friedkin1991theoretical} are two dual measures of centrality initially developed in social sciences to assess the importance in terms of ease of access to others nodes (closeness) and brokering power (betweenness) of individuals in social networks.\nThese centralities have since then been used in many other contexts.\nThey are based on the assumption that information, or influence, propagates between two nodes in the most efficient way, i.e. by following the shortest path between them.\n\nConsidering an unweighted and undirected graph $G=(V,E)$, a walk on the graph $G$ is an alternating sequence of vertices and edges in which every vertex is incident to both the edges that come before and after it in the sequence. A path on a graph is a walk in which all vertices and edges are distinct.\nA graph is said to be connected if there exists a path from any node node to any other node in the graph.\nConsidering the distance $\\text{dist}(s,r)$ between two nodes $s$ and $r$ in a connected undirected and unweighted graph $G=(V,E)$ as the number of edges in the \\index{shortest path} between them, the total distance of vertex $v$ is defined as the sum of its distance to all other vertices\n\n$$\\text{dist}(s) = \\sum_{r\\in V} \\text{dist}(s,r),$$\n\nwhich is larger for vertices that are the farther away from other vertices.\nTherefore, the \\emph{closeness centrality} of a node $s$ on a connected and undirected network is usually defined as\\,\\cite{bavelas1950communication, beauchamp1965improved}\n\n\\begin{equation}\n    c_C(s) = \\frac{1}{\\sum_{r\\in V} \\text{dist}(s,r)} = \\frac{1}{\\text{dist}(s)}.\n\\end{equation}\n\nThe betweenness centrality captures the brokering power of node as the opportunity it has to intercept or influence the communications happening between pairs of other nodes. \nLet $\\sigma(s,r)$ be the number of shortest paths between $s$ and $r$ and let $\\sigma(s,r|b)$ be the number of shortest paths between $s$ and $r$ passing by a brokering node $b\\in V\\setminus \\{s,r\\}$. We consider $\\sigma(s,s)=1$ and $\\sigma(s,r|b)=0$ if $b\\in\\{s,r\\}$.\nWe call the quantity $\\delta(s,b,r)=\\frac{\\sigma(s,r|b)}{\\sigma(s,r)}$ the \\emph{dependency} of a sender $s$ and a receiver $r$ on a broker $b$\\,\\cite{Brandes2016}.\nThe \\emph{betweenness centrality} in a connected and undirected graph $G=(V,E)$ is defined as the sum of dependencies of all communicating pairs on a broker $b$\\,\\cite{freeman1980gatekeeper}\n\n\\begin{equation}\n    c_B(b) = \\sum_{s,r\\in V} \\frac{\\sigma(s,r|b)}{\\sigma(s,r)}= \\sum_{s,r\\in V} \\delta(s,b,r),\n    \\label{eq:betweenness}\n\\end{equation}\n\nand can be seen as the overall potential control of $b$ on the communications in G.\n\nBetweenness and closeness centrality can be seen as being dual to each other conceptually expressing either the independence from the control of others (closeness) or the potential control over others (betweenness)\\,\\cite{freeman1980gatekeeper,Brandes2016}.\nIndeed, by noting that when considering shortest paths between $s$ and $r$ with different brokers $b$ at a fixed distance $d$ from $s$ ($1 \\leq d < \\text{dist}(s,r)$) each shortest path pass through exactly one broker and therefore \n$\\sum_{b\\in V : \\text{dist}(s,b)=d} \\frac{\\sigma(s,r|b)}{\\sigma(s,r)} = 1$.\nThis implies that the sum of dependencies between a sender $s$ and a receiver $r$ taken over of possible brokers $b$ is proportional to the distance between $s$ and $r$: $\\sum_b \\delta(s,b,r)= \\sum_{d=1}^{\\text{dist}(s,r)-1} \\sum_{b\\in V : \\text{dist}(s,b)=d} \\frac{\\sigma(s,r|b)}{\\sigma(s,r)} =\\text{dist}(s,r)-1$.\nThis observation allows one to define the closeness centrality as $c_C(s)^{-1} = (N-1) + \\sum_{b,r\\in{V}}\\delta(s,b,r)$ revealing its mathematical duality with betweenness centrality as a different partial sum, over $b$ and $r$ instead of $s$ and $r$, of the dependencies $\\delta(s,b,r)$ showing its interpretation as a measure of lack of independence on others\\,\\cite{Brandes2016}.\nFigure \\ref{fig:closness_vs_betweenness} shows the kite graph introduced by  Krackhardt in 1990 as an illustration of the different ranking obtained by these centrality measures\\,\\cite{krackhardt1990assessing}.\n\n\\begin{figure}[ht]\n\\centering\n \\includegraphics[width=0.5\\linewidth]{closness_vs_betweenness}\n \\caption{Krackhardt kite graph showing the different ranking obtained using degree, closeness and betweenness centralities\\,\\cite{krackhardt1990assessing}. \n The closeness of a node is represented by its size and the betweenness of a node is indicated by its color with nodes with larger betweenness being of a lighter shade.\n The node with largest degree centrality is 3.\n Nodes 5 and 6 have the largest closeness centrality and node 7 has the largest betweenness centrality.\n }\n \\label{fig:closness_vs_betweenness}\n\\end{figure}\n\nIn directed networks, closeness and betweenness centrality can be generalized by considering the notion of reachability instead of distance. In weighted networks where edge weights typically represent a distance or a lag in the connection between adjacent nodes, the length of a path is usually taken as being the sum of the weights of its edges.\nWe refer the reader to \\,\\cite{Brandes2016} for a discussion about the generalizations of the closeness and betweenness centrality to directed and weighted networks, and to their interpretations in these cases.\n\nMany variants of closeness and betweenness have been developed\\,\\cite{brandes2008variants}. The concept of betweenness centrality has also been extended to edges by considering the number of shortest paths containing an edge instead of a node in eq. (\\ref{eq:betweenness})\\,\\cite{brandes2008variants} or, for example, by considering the fraction of minimum spanning trees of a graph that contain a given edge\\,\\cite{teixeira2015not}.\nVersions of the closeness and betweenness centralities based on random walks instead of shortest paths have also been developed\\,\\cite{white2003algorithms,Newman2005}.\n\n\n\\subsection{Centrality measures based on walks}\n\nSeveral widely used centrality measures can be seen as being based on the concept of \\index{graph walks}. \nWalks, alternating sequences of vertices and edges in which every vertex is incident to both the edges that come before and after it in the sequence, are useful to count the number of possible ways there is to reach a given node starting from another node.\nCentrality measures based on walks usually try to find the nodes from which there are the largest number of walks reaching other nodes.\nFor unweighted networks, the number of walks of length $\\ell$ existing between two nodes $i$ and $j$ is given by the element $(i,j)$ of the $l^{\\text{th}}$ power of the adjacency matrix: $\\left(\\matr{A}^\\ell\\right)_{ij}$.\nThe number of walks of length $\\ell$ starting from node $i$ is then given by $w(\\ell)_i=\\sum_j\\left(\\matr{A}^\\ell\\right)_{ij}$ or in matrix notation $\\matr{w}(\\ell)=\\matr{A}^\\ell \\matr{1}$, where $\\matr{1}$ is the unit vector.\nThe \\emph{degree centrality} of a node $i$ can be expressed as the number of walks of length 1 starting from it:\n\\begin{equation}\n    c_{\\text{deg}}(i) = w(1)_i = \\sum_j A_{ij}.\n\\end{equation}\n\nIn order to take into account information about the surrounding of a node in its centrality score, several researchers have developed centrality measures that consider longer walks. The idea being that if a node is close to a node with a high centrality, its centrality should also be high\\,\\cite{katz1953new,bonacich1972factoring}.\nStarting from an initial guess for the centrality of each node given by the vector $\\matr{x}(0)$ (e.g. $\\matr{x}(0)=\\matr{1}$), we can propagate this initial centrality through walks of a given length. \nThe centrality vector for walks of length $\\ell$ is given by \n\\begin{equation}\n    \\matr{x}(\\ell) = \\matr{A}^\\ell \\matr{x}(0).\n    \\label{eq:eig_cen_it}\n\\end{equation}\n\nWriting the initial centrality as a linear combination of the eigenvector of the adjacency matrix, one can see that as $\\ell\\xrightarrow{}\\infty$ the centrality will converge to a vector proportional to the leading eigenvector of $\\matr{A}$, the one corresponding to its largest eigenvalue, and that takes into account global information about the network structure\\,\\cite{Newman2010networks}.\nAs $\\matr{A}$ is non-negative and if $G$ is connected, the \\index{Perron-Frobenius theorem} ensures that the leading eigenvector is positive and that the associated eigenvalue is positive and simple.\nThe resulting vector is called the \\emph{\\index{eigenvector centrality}} whose element $i$ is defined as \n\n\\begin{equation}\n    c_\\text{eig}(i) \\propto \\sum_j A_{ij} c_\\text{eig}(j).\n\\end{equation}\n\nIn weighted networks, the eigenvector centrality is propagated along walks on the graph that are weighted by the edge weights.\nThe eigenvector centrality can be computed similarly on directed and undirected networks with the difference that for directed networks, the adjacency matrix is in general asymmetric and has therefore two different sets of left- and right-eigenvectors. Depending on the application, one may prefer to use the largest left-eigenvector or the largest right-eigenvector if one is more interested in nodes having a large number of incoming walks or outgoing walks, respectively.\nHowever, other types of centralities are usually preferred for directed networks as the eigenvector centrality of nodes outside of a strongly connected component (a subgraph where there exist a path in both directions between all pairs of nodes) may be equal to zero. In particular, in directed graphs without cycles (closed directed paths), the eigenvector centrality is zero for all nodes.\nA comparison of the degree centrality with the eigenvector centrality is shown in Fig. \\ref{fig:deg_vs_eigen} for an undirected network of American football teams\\,\\cite{girvan2002community,evans2010clique}.\n\n\\begin{figure}[ht]\n\\centering\n \\includegraphics[width=0.8\\linewidth]{football_deg_vs_eigenvec}\n \\caption{Network of American football games between Division IA colleges during regular season Fall 2000\\,\\cite{girvan2002community,evans2010clique}.\n Degree centrality for this undirected graph is shown on the left and eigenvector centrality is shown on the right. Lighter shades indicate a higher centrality.\n Several nodes have the same degree in this network making the degree centrality unable to disantangle the importance of nodes with the same degree.\n Eigenvector centrality has a tendency of being localized in regions where many high degree nodes are close to each other.\n }\n \\label{fig:deg_vs_eigen}\n\\end{figure}\n\nThe \\emph{\\index{subgraph centrality}} is based on the idea of counting the number of times that a node takes part in the different connected subgraphs of a network, with smaller subgraphs having higher importance\\,\\cite{Estrada2005}.\nThe subgraphs are identified by counting closed walks of different lengths that start and end on a given node. The subgraph centrality of node $i$ is defined as\n\n\\begin{equation}\n    c_{\\text{Sub}}(i) = \\sum_{\\ell=0}^{\\infty}\\frac{\\left(\\matr{A}^\\ell\\right)_{ii}}{\\ell!}.\n\\end{equation}\n\nBy noticing that $c_{\\text{Sub}}(i)=\\left(e^\\matr{A}\\right)_{ii}$, where $e^\\matr{A}$ denotes the matrix exponential of $\\matr{A}$,\nthe subgraph centrality can be obtained from the spectrum of the adjacency matrix as\\,\\cite{Estrada2005}\n\n\\begin{equation}\n    c_{\\text{Sub}}(i) = \\sum_{i=1}^{N}(v_j^i)^2e^{\\lambda_j}, \n\\end{equation}\n\nwhere $v_1,v_2, \\ldots, v_N$ form an orthonormal basis of $\\mathbb{R}^N$ and are eigenvectors of $\\matr{A}$ associated with the eigenvalues $\\lambda_1, \\lambda_2, \\ldots, \\lambda_N$.\nThis centrality is more discriminative than the degree, betweenness, closeness or eigenvector centralities and gives a distinctly different ranking of nodes in real-world complex networks\\,\\cite{Estrada2005}.\n\nAnother centrality measures based on walks that tries to solve the issue of the eigenvector centrality in directed networks is the \\index{Katz centrality}\\,\\cite{katz1953new}.\nThe Katz centrality of node $i$ is the weighted sum of all walks emanating from $i$, with the count for walks of length $\\ell$ weighted by a factor $\\alpha^\\ell$ where $0<\\alpha<1$. In this way the importance of longer walks is diminished compared to shorter walks.\nThe \\emph{Katz centrality} of node $i$ can be written as\\,\\cite{katz1953new}\n\n\\begin{equation}\n    c_{\\text{Katz}}(i) = 1 + \\sum_{\\ell=1}^{\\infty}\\sum_{j} \\alpha^\\ell \\left(\\matr{A}^\\ell\\right)_{ij},\n\\end{equation}\n\nwhere the unit shift does not alter the ranking of the nodes and may be seen as arising from a single walk of length zero.\nIn order for the centrality to converge, the factor $\\alpha$ must be smaller than the reciprocal of the absolute value of the largest eigenvalue of $\\matr{A}$\\,\\cite{katz1953new,bonacich1972factoring,bonacich1987power}.\nIn this case, we can also note that $\\matr{I}+\\alpha\\matr{A}+\\alpha^2\\matr{A}^2+\\alpha^3\\matr{A}^3 + \\cdots = \\left(\\matr{I}-\\alpha\\matr{A}\\right)^{-1}$.\nThus, the vector $\\matr{x}$ giving the Katz centrality of each node is also solution of the matrix equation $\\matr{x}=\\left(\\matr{I}-\\alpha\\matr{A}\\right)^{-1}\\matr{1}$ which can be rewritten as $\\matr{x}=\\alpha\\matr{A}\\matr{x}+\\matr{1}$.\nThe Katz centrality can therefore be computed iteratively in a similar manner than the eigenvector centrality (with eq. (\\ref{eq:eig_cen_it})) using \n\\begin{equation}\n    \\matr{x}(t+1) = \\alpha\\matr{A}\\matr{x}(t)+1.\n    \\label{eq:katz_it}\n\\end{equation}\n\nA unit value is added at each iteration guaranteeing that the Katz centrality will never be equal to zero, even in directed networks.\nThe factor $\\alpha$ can also be seen as balancing the importance of the eigenvector term compared to the constant term added at each iteration.\nIn the limit as $\\alpha$ approaches the reciprocal of the absolute value of the largest eigenvalue of $\\matr{A}$, the Katz centrality approaches the eigenvector centrality on strongly connected graphs\\,\\cite{bonacich1972factoring,bonacich1987power}.\n\nWalk-based centralities have been generalized to hypergraphs (e.g. \\,\\cite{bonacich2004hyper,benson2019three}), multiplex networks (e.g. \\,\\cite{sola2013eigenvector,de2015ranking}) and temporal networks (e.g. \\,\\cite{nicosia2013graph,praprotnik2015spectral,taylor2017eigenvector}).\n\n\n\\subsection{Centrality measures based on random walks}\n\n\\index{Random walks} on graphs have been extensively studied and have many applications in the study of diffusive processes on networks and for the characterization of the structure of complex networks \\,\\cite{Rosvall2008,Delvenne2010,durrett2010some,Masuda2016}.\nA random walk on a graph is defined by the trajectory of a walker that, at each time step, jumps to a neighbors of $i$ with equal probability. On an unweighted and undirected network, the transition probability for a walker on node $i$ to jump to node $j$ is equal to \n\n\\begin{equation}\n    T_{ij} = \\frac{A_{ij}}{k_i},\n\\end{equation}\n\nor in matrix notation $\\matr{T}=\\matr{D}^{-1}\\matr{A}$, where $\\matr{D}=\\text{diag}(\\matr{k})$ is the diagonal degree matrix.\nIn directed networks, the degree is replaced by the out-degree and in weighted networks with positive weights, the probability transition is proportional to the weight of edge $(i,j)$.\nCentrality measures based on random walks exploit the fact that,\nif a centrality value propagates as a random walk, at each iteration, its value is divided across all out-going edges of a node.\nThis is different than for eigenvector and Katz centralities, based on walks, where the centrality tends to concentrate on a few hubs in the network, leaving other nodes almost indistinguishable with very low scores due to the repeated reflection of influence along the mutual connections between hubs during iterations of eqs. (\\ref{eq:eig_cen_it}) \\& (\\ref{eq:katz_it}) (see Figs. \\ref{fig:deg_vs_eigen} \\& \\ref{fig:pr_vs_katz}).\n\nThe distribution probability of finding a walker at time step $n$ on any node of the network can be written as a row-vector $\\matr{p}(n)$ with $\\sum_i p_i = 1$.\nThe evolution of the distribution probability is given by \n\n\\begin{equation}\n    \\matr{p}(n+1) = \\matr{p}(n)\\matr{T}.\n    \\label{eq:rw_update}\n\\end{equation}\n\nOn connected undirected network, the random walk reaches a stationary distribution satisfying $\\matr{p}^\\star=\\matr{p}^\\star\\matr{T}$ where the probability at each node is proportional to its degree:\n\\begin{equation}\n    p^\\star_i = \\frac{k_i}{\\sum_j k_j}.\n\\end{equation}\n\nOn undirected networks, the degree centrality can therefore also be seen as the stationary state of the random walk process.\nOn directed networks, random walks are not guaranteed to converge to a unique stationary state.\nA stationary distribution only exists on strongly connected components of a directed network\\,\\cite{aldous2002reversible,Masuda2016}.\n\n\n\n\nTo overcome this issue the \\emph{\\index{PageRank}} centrality modifies the classical random walk by introducing a \"teleportation\" probability, i.e. at each step, the walkers have a given probability to teleport uniformly at random to any other nodes of the network. The update equation for the PageRank probability density is given by \\,\\cite{brin1998anatomy,page1999pagerank,Masuda2016}\n\n\\begin{equation}\n    \\matr{p}(n+1) = \\alpha\\matr{p}(n)\\matr{S} + \\frac{1-\\alpha}{N}\\matr{1}^T,\n    \\label{eq:pg_it}\n\\end{equation}\n\nwhere $\\matr{S}$ is a transition matrix constructed from $\\matr{A}$ such that $S_{ij}=A_{ij}\/k_\\text{out}(i)$ when $k_\\text{out}(j)>0$.\nFor \"dangling nodes\", that have no out-going edges (i.e. $k_\\text{out}(i)=0$),\n$S_{ij}=1\/N$, meaning that a walker on such a node has a uniform probability to jump to any other nodes in the network.\nOn non-dangling nodes, at each step, walkers have a probability $\\alpha<1$ to follow an out-going edge and a probability $1-\\alpha$ to teleport to any node in the network.\nThis modified random walks is now ergodic also on networks that are not strongly connected and converges to the probability density vector giving the \\emph{PageRank} centrality of node $i$ as\n\n\\begin{equation}\n    c_\\text{PR}(i) = \\alpha \\sum_j c_\\text{PR}(j) S_{ji}  + \\frac{1-\\alpha}{N}\n\\end{equation}\n\nwhich is equal to the normalized eigenvector corresponding to the largest positive eigenvalue of the so-called Google matrix $\\matr{G}=\\alpha \\matr{S}+\\frac{1-\\alpha}{N}\\matr{1}\\matr{1}^T$\\,\\cite{ermann2015google}.\nPageRank was famously developed for ranking websites for the search engine Google considering a \"random surfer model\" navigating through webpages by randomly clicking on hyperlinks\\,\\cite{page1999pagerank}.\nPageRank also found many applications in other aspects, such as ranking scientists and academic papers \\,\\cite{ding2009pagerank}, images \\,\\cite{jing2008visualrank} and proteins \\,\\cite{ivan2010web}.\nFigure \\ref{fig:pr_vs_katz} shows a comparison of the PageRank and Katz centralities on a directed network of hyperlinks between weblogs\\,\\cite{adamic2005political}.\n\nThe similarities between equations (\\ref{eq:eig_cen_it}) and (\\ref{eq:rw_update}) as well as equations (\\ref{eq:katz_it}) and (\\ref{eq:pg_it}) reveal that the degree and the PageRank centralities can be seen as similar to the eigenvector and Katz centralities, but for random walks instead of regular graph walks. In the case of random walk based centralities, one looks for eigenvectors of transition matrices instead of adjacency matrices.\n\n\n\\begin{figure}[ht]\n\\centering\n \\includegraphics[width=0.8\\linewidth]{polblogs_pagerank_vs_katz}\n \\caption{A directed network of hyperlinks between weblogs on US politics, recorded in 2005 by Adamic and Glance\\,\\cite{adamic2005political}.\n PageRank centrality is shown on the left and Katz centrality is shown on the right. Lighter shades indicate a higher centrality.\n The Katz centrality has a tendency of being \"localized\" in regions with many close-by hubs.\n }\n \\label{fig:pr_vs_katz}\n\\end{figure}\n\n\n\\subsection{Centrality measures based on non-backtracking walks}\n\nIn order to address the issue of \"localization\" of the eigenvector centrality mentioned above, i.e. when most of the weight of centrality vector concentrates around one or a few nodes in the network, several authors have focused on using \\index{non-backtracking walks} instead of regular or random walks\\,\\cite{Martin2014,Arrigo2018,Arrigo2020}.\nNon-backtracking walks are graph walks where backtracking steps, i.e. steps where the walk comes back to an immediately preceding node, are not permitted.\nUsing this type of walks can sometimes solve the problem of localization as they decrease reflections between hubs during iterations of the centrality\\,\\cite{Martin2014}.\nNon-backtracking walks on unweighted networks are usually described using the $2M\\times2M$ matrix $\\matr{B}$ where rows and columns correspond to the directed edges of the network ($M=\\left|{E}\\right|$).\nIf the network is undirected, an equivalent directed network is considered by replacing each edge by a pair of directed edges in both directions.\nElement ($i\\rightarrow{}j,\\ell\\rightarrow{}h$) of $\\matr{B}$ is equal to one only if $j=\\ell$ and $i\\ne h$, i.e. there is a non-backtracking path of length 2 $i\\rightarrow{}j\\rightarrow{}h$ in the network. \nMore succinctly, \n\n\\begin{equation}\n    B_{i\\rightarrow{}j,\\ell\\rightarrow{}h} = \\delta_{j\\ell}(1-\\delta_{ih}),\n\\end{equation}\n\nwhere $\\delta_{ij}$ is the Kronecker delta.\nThe matrix $\\matr{B}$ is referred to as the \\emph{\\index{non-backtracking matrix}} or the \\emph{\\index{Hashimoto matrix}}\\,\\cite{hashimoto1989zeta}.\nPowers of $\\matr{B}$ enumerate non-backtracking walks similarly to powers of the adjacency enumerate walks\\,\\cite{Arrigo2020}.\nThe non-backtracking matrix is in general asymmetric with eigenvalues that are in general complex, however, \nsince its entries are non-negatives, by the \\index{Perron-Frobenius theorem}, its leading eigenvalue, $\\lambda$, is real and non-negative, and there exists a corresponding leading eigenvector, $\\matr{v}$, whose elements are also non-negative real numbers.\nIf $G$ is connected and not a tree, i.e. it has at least one cycle, $\\lambda$ is positive\\,\\cite{lin2019non}.\nThey satisfy the eigenvector equation\n\n\\begin{equation}\n    \\lambda\\matr{v}=\\matr{B}\\matr{v}\n\\end{equation}\n\nand the \\emph{\\index{non-backtracking eigenvector centrality}} of node $j$ is defined by \\,\\cite{Martin2014}\n\n\\begin{equation}\n    c_\\text{NBeig}(j) = \\sum_i A_{ij}v_{i\\rightarrow{}j}.\n\\end{equation}\n\nFinding the leading eigenvector of $\\matr{B}$ can however be computationally demanding for large graphs as the size of $\\matr{B}$ is usually much larger than the size of $\\matr{A}$.\nHowever, the computation can be made much faster by computing directly $c_\\text{NBeig}$ as the first $N$ elements of a $2N\\times2N$ matrix called the Ihara-Bass matrix\\,\\cite{krzakala2013spectral}.\n\nNon-backtracking walks have also been used to modify other centrality measures, such as the Katz\\,\\cite{Arrigo2018}, random walk\\,\\cite{lin2019non} or PageRank\\,\\cite{aleja2019non} centralities.\nResearch on centrality measures based on non-backtracking walks is very active with recent results showing that they can also suffer from localization issues on some networks\\,\\cite{barucca2016centrality,pastor2020localization}.\\\\\n\nNon-backtracking walks are also used in the problem of \\index{influence maximization}: finding the minimal set of nodes, the \\emph{influencers},\nwhich, if activated, would cause the spread of information to the whole network, or, if immunized, would prevent the diffusion\nof a large scale epidemic\\,\\cite{kempe2003maximizing}.\nInfluence maximization can be mapped to the problem of \\index{optimal percolation} of random networks\\,\\cite{morone2015influence} which consists in identifying the minimal set of nodes whose removal would dismember the network in many disconnected and non-extensive components. The fragmentation of the network is measured by the size of the largest connected component, called the giant component of the network.\n\nThe intuition behind the usage of non-backtracking walks in the optimal percolation problem comes from the fact that the giant component is held together by long paths and that powers of the non-backtracking matrix allows to quickly find them.\nThe removal of nodes is represented with the vector $\\matr{n} =(n_1,\\ldots,n_N)$ where $n_i =0$ if $i$ is removed (influencer) or $n_i =1$ otherwise.\nConsidering undirected locally tree-like random graphs,\na modified $2M\\times 2M$ non-backtracking matrix $\\matr{M}$ is then defined as \n$M_{i\\rightarrow{}j,\\ell\\rightarrow{}h} = n_\\ell B_{i\\rightarrow{}j,\\ell\\rightarrow{}h}$.\nGiven an initial arbitrary positive vector $\\matr{w}(0)$, repeated iterations with $\\matr{M}$,\n\n\\begin{equation}\nw(n)_{i\\rightarrow{}j}=\\sum_{kl}M_{i\\rightarrow{}i,k\\rightarrow{}l}w(n-1)_{k\\rightarrow{}l},\n\\end{equation}\n\nincrease the norm of the vector as the influence of nodes on non-backtracking walks of larger and larger length \nare included.\nThe growth rate of the vector's norm is determined by the value of the largest eigenvalue, $\\lambda(\\matr{n})$ of the modified non-backtracking matrix.\nAs the influencer nodes are removed, the value of $\\lambda(\\matr{n})$ decreases until the giant component is reduced to a tree (a graph without cycles) plus only one cycle when $\\lambda(\\matr{n})$=1.\nThe removal of supplementary nodes quickly destroys the giant component and $\\lambda(\\matr{n})$ falls to zero\\,\\cite{morone2015influence}.\nThe optimal influence problem can therefore be rephrased as finding the optimal configuration $\\matr{n}$ that minimizes the largest eigenvalue of $\\matr{M}$\\,\\cite{morone2015influence}, which consists in removing the nodes that are on the most non-backtracking walks in the network and that are keeping the giant component connected.\nThe \\emph{\\index{collective influence}}\\,\\cite{morone2015influence} of a node, defined as \n\n\\begin{equation}\n    \\text{CI}_\\ell(i) = (k_i-1)\\sum_{j\\in \\partial \\text{Ball}(i,\\ell)}(k_j - 1),\n\\end{equation}\nwhere $\\partial \\text{Ball}(i,\\ell)$ is the set of the nodes at a distance $\\ell$ from $i$, is a measure of which node to remove in order to produce the largest diminution of the largest eigenvalue of non-backtracking matrix.\nThe ranking of the nodes in term of collective influence is produced by removing the node with the largest CI value, recomputing the CI values of its neighbors and repeating the operation until the giant component of the network disappears.\nAn advantage of CI is that it gives a high rank to seemingly \"weak nodes\" with a small number of connections that are in fact surrounded by highly connected nodes and therefore on the path of many non-backtracking walks.\nThe problem of influence maximization in networks is actively researched using many different approaches (see, for example, \\,\\cite{braunstein2016network,mugisha2016identifying,zdeborova2016fast,radicchi2016beyond,pan2016influence,pei2020influencer}).\n\n\n\n\n\n\n\n\n\n\n\\section*{Conclusion and Future Directions}\n\nWe have seen that there is not a single measure of centrality in networks and that one should carefully choose an appropriate measure depending on the subject of investigation. \nIn practice, centrality measures should be chosen depending on whether the network is directed or not and whether, for example, one is looking for highly connected nodes, nodes with the most brokering power, nodes that share the most influence with each others or the minimal set of nodes that can spread information to the whole network. The differences in ranking obtained from different centrality measures should also be examined in order to better understand the signification of each ranking in a given context.\n\nThe research landscape on centrality measures for complex networks is much vaster than what has been presented in this short entry.\nResearch is very active with recent and future directions including, for example, the properties of centrality measures based on non-backtracking walks\\,\\cite{pastor2020localization},\ncentralities in signed networks (networks where weights can be positive or negative)\\,\\cite{liu2020simple} and \nmultilayer networks\\,\\cite{sola2013eigenvector,de2015ranking,wang2018new,wu2019tensor}.\nResearch is also active on the development of centralities for specific applications based on network dynamics not necessarily captured by network traversals. Examples include centralities for social networks based on game theoretical concepts\\,\\cite{Shapley1953,Gomez2003,michalak2013efficient},  centralities for detecting vulnerabilities in power-grids based on networks of oscillators\\,\\cite{gutierrez2013vulnerability,tyloo2018robustness,Tyloo2019} or centralities for identifying key genes in gene regulatory networks based on the propagation of biological signals\\,\\cite{zotenko2008hubs,missiuro2009information,kim2011identifying,cowen2017network}.\n\nResearch is also ongoing about the development of centralities in network models that extend the concept of network as an ensemble of static pairwise relations, namely, centralities in temporal networks, i.e. time-evolving networks, \\,\\cite{nicosia2013graph,praprotnik2015spectral,taylor2017eigenvector,flores2018eigenvector,lv2019pagerank} and in hypergraphs or simplicial complexes, i.e. generalizations of graphs where an edge can join more than two nodes\\,\\cite{bonacich2004hyper,estrada2018centralities,benson2019three,aksoy2020hypernetwork,serrano2020centrality}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Renormalization constants for the leptonic vector-boson decay at \\texorpdfstring{$\\order{\\alphas\\alpha}$}{O(as a)}}\n\\label{app:ff}\n\nIn this appendix we provide the expressions for the finite\n$\\mathcal{O}(\\alphas\\alpha)$ counterterms to the leptonic vertices of\nthe $\\PW$ and $\\PZ$ bosons in the on-shell renormalization\nscheme following the conventions of Ref.~\\cite{Denner:1991kt}.\n\n\n\\subsection{Vector-boson self-energies}\n\nThe transverse and longitudial parts of the vector-boson self-energy,\n$\\Sigma^{V_aV_b}_\\rT$ and $\\Sigma^{V_aV_b}_\\rL$, are defined by the decomposition of the\nirreducible two-point function $\\Gamma^{V_aV_b}_{\\mu\\nu}$ as\n\\begin{equation}\n  \\Gamma^{V_aV_b}_{\\mu\\nu}(q)=-\\ri \\delta_{ab} g_{\\mu\\nu} (q^2-M_{V_a}^2)-\n\\ri  \\left(g_{\\mu\\nu}-\\frac{q_\\mu q_\\nu}{q^2}\\right)\\Sigma^{V_aV_b}_\\rT(q^2)-\\ri\n  \\frac{q_\\mu q_\\nu}{q^2}\\Sigma^{V_aV_b}_\\rL(q^2),\n\\end{equation}\nwhere $q$ is the momentum carried by the vector bosons $V_{a,b}$.\nThe $\\mathcal{O}(\\alphas\\alpha)$ corrections to the vector-boson\nself-energies are given in Ref.~\\cite{Djouadi:1993ss} in  terms of\nscalar functions $\\Pi^{V,A}_\\rT$.\\footnote{%\nNote that in the expression given in Eq.~(5.4) of Ref.~\\cite{Djouadi:1993ss} for\n$\\Pi^{V,A}_\\rT$ in the special case of one vanishing fermion mass, the sign of the term \n$1\/3(2+\\alpha)(\\alpha-1)G(x)$ should be reversed in agreement with Ref.~\\cite{Djouadi:1987di}. We thank Paolo Gambino\nfor communication on this point.\n} \nTreating all quarks apart from the top quark as massless, the transverse parts of the vector-boson self-energies can be expressed as follows,\n\\begin{align}\n\\Sigma_\\rT^{WW, (\\alphas\\alpha)}(s) & =\\frac{\\alphas\\alpha}{8\\pi^2 \\sw^2}\n   \\left[2({\\Pi}^V_\\rT(s,0,0)+{\\Pi}^A_\\rT(s,0,0))\n   +   ({\\Pi}^V_\\rT(s,m_\\Pqt^2,0)+{\\Pi}^A_\\rT(s,m_\\Pqt^2,0))\n   \\right], \\nonumber\\\\\n\\Sigma_\\rT^{ZZ,(\\alphas\\alpha)}(s)&\n=\\frac{\\alphas\\alpha}{4\\pi^2 \\sw^2 \\cw^2}\n   \\left[\\left(\\frac{44}{9}\\sw^4-\\frac{14}{3}\\sw^2+\\frac{5}{4} \\right)\n     {\\Pi}^V_\\rT(s,0,0)+\\tfrac{5}{4}{\\Pi}^A_\\rT(s,0,0) \\right. \\nonumber\\\\\n&\\quad\\left.   +  \\left(\\frac{1}{2}-\\frac{4}{3} \\sw^2\\right)^2{\\Pi}^V_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\n+\\tfrac{1}{4}{\\Pi}^A_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\n   \\right], \\nonumber \\\\\n\\Sigma_\\rT^{AA,(\\alphas\\alpha)}(s)&\n=\\frac{\\alphas\\alpha}{\\pi^2}\n\\left[\\frac{11}{9}{\\Pi}^V_\\rT(s,0,0) + \\frac{4}{9} {\\Pi}^V_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\\right],\n\\nonumber \\\\\n\\Sigma_\\rT^{AZ,(\\alphas\\alpha)}(s)\n&=-\\frac{\\alphas\\alpha}{2\\pi^2\\sw\\cw}\\left[\n\\left(\\frac{7}{6}-\\frac{22}{9}\\sw^2\\right)\\Pi^V_\\rT(s,0,0)\n+\\left(\\frac{1}{3}-\\frac{8}{9} \\sw^2\\right)\\Pi^V_\\rT(s,m_\\Pqt^2,m_\\Pqt^2)\\right] ,\n\\end{align}\nwhere the explicit expressions for the scalar functions $\\Pi^{V,A}_\\rT$\ngiven in Ref.~\\cite{Djouadi:1993ss} include the colour factor $N_c=3$.\nThe top-quark mass renormalization is performed in the on-shell\nscheme.\n\n\n\n\\subsection{Vertex counterterms}\nAt $\\order{\\alphas\\alpha}$, the vertex counterterms for the leptonic\nvector-boson decay receive only contributions from the vector-boson\nself-energies, \n\\begin{align}\n  \\delta_{\\PW\\Pl_1\\Pal_2}^{\\mathrm{ct},(\\alphas\\alpha)}&=\n  \\delta Z_e^{(\\alphas\\alpha)} -\\frac{\\delta \\sw^{(\\alphas\\alpha)}}{\\sw}\n  +\\frac{1}{2}\\delta Z_W^{(\\alphas\\alpha)},\\nonumber\\\\\n  \\delta^{\\mathrm{ct},\\tau_\\ell,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal} &=\n  \\frac{\\delta g_\\Pl^{\\tau_\\ell,(\\alphas\\alpha)}}{g_\\Pl^{\\tau_\\ell}}  \n  +\\frac{1}{2}\\delta Z^{(\\alphas\\alpha)}_{ZZ} - \\frac{Q_\\Pl}{2g_\\Pl^{\\tau_\\ell}}\n\\delta Z_{AZ}^{(\\alphas\\alpha)},\n \\end{align}\nwhere $\\tau_\\ell=\\pm$ denotes the lepton chirality.\nThe coupling constants entering the $\\PZ$-boson vertex are given by\n\\begin{equation}\n  \\begin{aligned}\n    g_\\Pl^+&=-\\frac{\\sw}{\\cw}Q_\\Pl, & \n    \\delta g_\\Pl^{+}&=-\\frac{\\sw}{\\cw}Q_\\Pl\n    \\left[\\delta Z_e\n        +\\frac{1}{\\cw^2}\\frac{\\delta \\sw}{\\sw}\\right],  \\\\\n    g_\\Pl^-&=\\frac{1}{\\sw\\cw}\\left(I^3_{\\rw,\\Pl}-\\sw^2 Q_\\Pl\\right),&\n    \\delta g_\\Pl^{-}&=\\frac{I^3_{\\rw,\\Pl}}{\\sw\\cw}\n    \\left[\\delta Z_e\n        +\\frac{\\sw^2-\\cw^2}{\\cw^2}\\frac{\\delta \\sw}{\\sw} \\right]+ \n\\delta g_\\Pl^{+},\n  \\end{aligned}\n\\end{equation}\nwhere $I^3_{\\rw,\\Pl}=-\\frac{1}{2}$ is the third component of the weak\nisospin of the charged lepton $\\Pl$.\n\nThe vector-boson wave-function renormalization constants can be expressed in terms of the self-energies as follows,\n\\begin{align}\n  \\delta Z_{AA}&=-\\left.\\frac{\\partial\\Sigma^{AA}_\\rT(k^2)}{\\partial k^2}\n  \\right|_{k^2=0}, &&\n  \\nonumber\\\\\n  \\delta Z_{W}&=-\\left.\n\\Re\\frac{\\partial \\Sigma_\\rT^{WW}(s)}{\\partial s}\\right|_{s=\\MW^2}, & \n \\delta Z_{ZZ}&=-\\left. \\Re \\frac{\\partial \\Sigma_\\rT^{ZZ}(s)}{\\partial s}\n   \\right|_{s=\\MZ^2},\n\\nonumber \\\\\n \\delta Z_{ZA}&=2\\frac{\\Sigma_\\rT^{AZ}(0)}{\\MZ^2}, &\n \\delta Z_{AZ}&=-2\\Re\\frac{\\Sigma_\\rT^{AZ}(\\MZ^2)}{\\MZ^2},\n\\end{align}\nwhere the $\\order{\\alphas\\alpha}$ contribution to $ \\delta Z_{ZA}$ vanishes~\\cite{Djouadi:1993ss}.\nThe charge-renormalization constant $\\delta Z_e$ in the  $\\alpha(0)$ input-parameter scheme\n is given by\n\\begin{align}\n\\label{eq:dZe0}\n  \\delta Z_e^{\\alpha(0)}\n&=-\\frac{1}{2}\\delta Z_{AA}-\\frac{\\sw}{\\cw}\\frac{1}{2}\\delta Z_{ZA}.\n\\end{align}\nThe transition to the  $G_\\mu$-scheme is performed according to\nEq.~\\eqref{eq:dZeGmu}. The renormalization constant for the weak mixing angle is given by\n\\begin{equation}\n  \\frac{\\delta \\sw}{\\sw}\n=-\\frac{\\cw^2}{2\\sw^2}\n\\left(\\frac{\\Re\\Sigma_\\rT^{WW}(\\MW^2)}{\\MW^2}-\\frac{\\Re\\Sigma_\\rT^{ZZ}(\\MZ^2)}{\\MZ^2}\\right).\n\\end{equation}\n\nThe final expressions of the  $\\order{\\alphas\\alpha}$ counterterms in the $G_\\mu$-scheme in terms of self-energies are then given by\n\\begin{align}\n\\delta_{\\PW\\Pl_1\\Pal_2}^{\\mathrm{ct},(\\alphas\\alpha)}=&\n  -\\frac{1}{2}\\left(\n    \\Re\\frac{\\partial \\Sigma_\\rT^{WW,(\\alphas\\alpha)}(s)}{\\partial s}|_{s=\\MW^2} \n    +\\frac{\\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(0)\n      -\\Re \\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(\\MW^2)}{\\MW^2} \\right),\\nonumber \\\\\n   \\delta^{\\mathrm{ct},+,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal} =&\n-\\frac{1}{2}\n    \\frac{\\Re\\partial \\Sigma_\\rT^{ZZ,(\\alphas\\alpha)}(s)}{\\partial s}|_{s=\\MZ^2}\n    -\\left(1-\\frac{2}{\\sw^2}\\right)\n    \\frac{\\Re\\Sigma_\\rT^{ZZ,(\\alphas\\alpha)}(\\MZ^2)}{2\\MZ^2}\n\\nonumber\\\\\n&\n-\\frac{\\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(0)}{2\\MW^2}\n+ \\left(1-\\frac{1}{\\sw^2}\\right)\n  \\frac{\\Re\\Sigma^{WW,(\\alphas\\alpha)}_{\\rT}(\\MW^2)}{\\MW^2}\n  -\\frac{\\cw}{\\sw}\\frac{\\Re\\Sigma_\\rT^{AZ,(\\alphas\\alpha)}(\\MZ^2)}{\\MZ^2},\\nonumber\\\\\n  \\delta^{\\mathrm{ct},-,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}=& \\,\n  \\delta^{\\mathrm{ct},+,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}\n+\\frac{I^3_{\\rw,f}}{\\sw\\cw g_{\\ell}^-}\n\\left[-\\frac{2\\delta \\sw^{(\\alphas\\alpha)}}{\\sw}\n+\\frac{\\cw}{\\sw}\\frac{\\Sigma_\\rT^{AZ,(\\alphas\\alpha)}(\\MZ^2)}{\\MZ^2}\\right].\n\\label{eq:dZVNNLO}\n\\end{align}\n\n\n\\section{Explicit form of the IR-safe contributions to the factorizable initial--final corrections}\n\\label{app:ir-safe}\n\nIn this section we provide the explicit expressions for each of the contributions to the factorizable initial--final corrections in our master formula~\\eqref{eq:IF:final-master}.\n\nThe double-virtual corrections~\\eqref{eq:IF:master-VV} and the (virtual\nQCD)$\\times$(real photonic) corrections~\\eqref{eq:IF:master-VR}\nare obtained by dressing the virtual part of the NLO QCD corrections with the factorizable final-state EW corrections,\n\\begin{subequations}\n\\label{eq:IF:master-VX-qq}\n\\begin{align}\n \n  \\label{eq:IF:master-VV-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Vew} \n  &=\n  \\int_2 \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\;\\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0\\;\n  \\diptimes\\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] , \n  \\\\\n \n  \\label{eq:IF:master-VR-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}\n  &=\n  \\iint\\limits_{2+\\Pgg}\n  \\biggl\\{ \n  \\rd\\sigma_\\dec^\\Rew \n  -4\\pi\\alpha \\biggl[Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}\\Bigr)\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\n  \\biggr] \n  \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr]\n \n  \\biggr\\} ,\n\\end{align}\n\\end{subequations}\nwhere the integrated counterpart of the QED dipoles $\\Iew$ is defined in Eq.~\\eqref{eq:nlo:dec:master:Iew}.\nHere $\\lips[IJ]{2}$ denotes the set of momenta of the two-particle phase\nspace after applying the momentum mapping associated to  the dipole $\\gsub{IJ}$\nor $\\dsub{IJ}$. \nAs in the NLO QCD corrections, only the quark--anti-quark induced\nchannel receives a non-vanishing contribution.\n\n\nThe IR-regularized (real QCD)$\\times$(virtual EW) corrections~\\eqref{eq:IF:master-RV} and the double-real corrections~\\eqref{eq:IF:master-RR} are obtained by dressing the real-emission part of the NLO QCD cross sections with the final-state factorizable corrections.\nNote the two-fold application of the dipole subtraction formalism in case of of the double-real corrections.\nFor the quark--anti-quark channel, the explicit expressions are\n\\begin{subequations}\n\\label{eq:IF:master-RX-qq}\n\\begin{align} \n \n  \\label{eq:IF:master-RV-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  &=\n  \\int_3 \\;\\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\nonumber\\\\\\MoveEqLeft\\times\\biggl\\{ \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs}\n \n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Paq_a\\Pg)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pq_b\\Pg)\\Paq_a]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}  \n  \\biggr\\}, \n  \\\\\n \n  \\label{eq:IF:master-RR-qq} \n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew} \n  &=\n  \\iint\\limits_{3+\\Pgg}\\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  \\nonumber\\\\&\\quad\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew \\Bigl(\\lips[(\\Paq_a\\Pg)\\Pq_b]{2+\\Pgg}\\Bigr)\n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew \\Bigl(\\lips[(\\Pq_b\\Pg)\\Paq_a]{2+\\Pgg}\\Bigr)\n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1V]{3}\\Bigr)\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1\\Pal_2]{3}\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\nonumber\\\\&\\quad\n  +4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Paq_a\\Pg)\\Pq_b}_{2,\\Pl_1V}\\Bigr)\n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pq_b\\Pg)\\Paq_a}_{2,\\Pl_1V}\\Bigr)\n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Paq_a\\Pg)\\Pq_b}_{2,\\Pl_1\\Pal_2}\\Bigr)\n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pq_b\\Pg)\\Paq_a}_{2,\\Pl_1\\Pal_2}\\Bigr)\n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\} .\n\\end{align}  \n\\end{subequations}\nHere the phase-space kinematics obtained by the successive application\nof both QCD and EW dipole mappings  is denoted by $\\dtilde{\\Phi}^{(ab)c}_{2,IJ}$.  \nA detailed discussion of the behaviour of the individual terms of the double-real corrections in the various singular regions is given in Sect.~\\ref{sec:IF:master}.\nThe corresponding expressions for the quark--gluon-initiated subprocesses read\n\\begin{subequations}\n\\label{eq:IF:master-RX-gq}\n\\begin{align}\n \n  \\label{eq:IF:master-RV-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  &=\n  \\int_3 \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\biggl\\{ \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pg\\Pq_a)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\biggr\\}, \n  \\\\\n \n  \\label{eq:IF:master-RR-gq} \n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew} \n  &=\n  \\iint\\limits_{3+\\Pgg}\\biggl\\{\n  \\rd\\sigma_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew \\Bigl(\\lips[(\\Pg\\Pq_a)\\Pq_b]{2+\\Pgg}\\Bigr)\n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1V]{3}\\Bigr)\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs} \\Bigl(\\lips[\\Pl_1\\Pal_2]{3}\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\nonumber\\\\&\\quad\n  +4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pg\\Pq_a)\\Pq_b}_{2,\\Pl_1V}\\Bigr)\n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\nonumber\\\\&\\qquad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\dtilde{\\Phi}^{(\\Pg\\Pq_a)\\Pq_b}_{2,\\Pl_1\\Pal_2}\\Bigr)\n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\} .\n\\end{align}\n\\end{subequations}\nThe contribution to the $\\Pg\\bar q_a$ channel is given in an analogous manner,\nbut is not spelled out explicitly.\n\nThe collinear counterterms with additional virtual EW~\\eqref{eq:IF:master-CV} \nand real-photonic~\\eqref{eq:IF:master-CR} corrections \nare constructed from the corresponding term of the NLO QCD corrections by dressing \nthem with the respective factorizable final-state corrections,\n\\begin{subequations}\n\\label{eq:IF:master-CX-qq}\n\\begin{align}\n \n  \\label{eq:IF:master-CV-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x\\int_2\n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_a,p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Paq_a,\\Paq_a} \n  \\nonumber\\\\&\\quad\n  +\\int_0^1 \\rd x\\int_2\n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (p_a,xp_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pq_b,\\Pq_b} ,\n  \\\\\n \n  \\label{eq:IF:master-CR-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes \\Rew}\n  &=\n  \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew(xp_a,p_b)\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}(xp_a,p_b)\\Bigr)\n  \\nonumber\\\\&\\quad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}(xp_a,p_b)\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\}\\diptimes(\\CSK+\\CSP)^{\\Paq_a,\\Paq_a}  \n  \\nonumber\\\\&\n  + \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew(p_a,xp_b)\n  \\nonumber\\\\&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}(p_a,xp_b)\\Bigr)\n  \\nonumber\\\\&\\quad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}(p_a,xp_b)\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\}\\diptimes(\\CSK+\\CSP)^{\\Pq_b,\\Pq_b}  \n  .\n\\end{align}\n\\end{subequations}\nThe corresponding formulae for the gluon--quark channel read\n\\begin{subequations}\n\\label{eq:IF:master-CX-gq}\n\\begin{align}\n \n  \\label{eq:IF:master-CV-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x\\int_2\n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_{\\Pg},p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pg,\\Paq_a}  ,\n  \\\\\n \n  \\label{eq:IF:master-CR-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes \\Rew}\n  &=\n  \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\biggl\\{\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew(xp_{\\Pg},p_b)\n  \\nonumber\\\\*&\\quad\n  -4\\pi\\alpha \\biggl[\n  Q_{\\Pl_1}(Q_{\\Pl_1}-Q_{\\Pl_2}) \\; \\dsub{\\Pl_1V} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1V]{2}(xp_{\\Pg},p_b)\\Bigr)\n  \\nonumber\\\\*&\\quad\n  +Q_{\\Pl_1}Q_{\\Pl_2} \\; \\gsub{\\Pl_1\\Pal_2} \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{0} \\Bigl(\\lips[\\Pl_1\\Pal_2]{2}(xp_{\\Pg},p_b)\\Bigr)\n  +(\\Pl_1\\leftrightarrow\\Pal_2)\\biggr] \n  \\biggr\\}\\diptimes(\\CSK+\\CSP)^{\\Pg,\\Paq_a}  \n\\end{align}\n\\end{subequations}\nand analogous expressions for the $\\Pg\\bar q_a$ channel.\nHere we have made the dependence on the momenta of the incoming partons explicit in order to indicate which particle undergoes a collinear splitting with the momentum fraction given by the convolution variable $x$.\n\n\n\n\\section{Non-collinear-safe observables}\n\\label{app:non-collinear-safe}\nIn order to treat non-collinear-safe observables with respect to the final-state leptons \n$i=\\Pl_1,\\Pal_2$ following Ref.~\\cite{Dittmaier:2008md}, \nthe $n$-particle kinematics in the phase space of the subtraction function\nis treated as an $(n+1)$-particle event with a collinear lepton--photon pair,\nwhere the momentum shared between the two collinear particles is controlled\nby the variable $z_{iJ}$,\n\n\\begin{subequations}\n\\label{eq:thetacut:kabelschacht:nnlo}\n\\begin{align}\n \n  \\rd\\sigma^0_{\\PA}\\Bigl(\\lips[iJ]{2}\\Bigr) \n \n  &\\;\\longrightarrow\\;\n  \\rd\\sigma^0_{\\PA}\\Bigl(\\lips[iJ]{2}\\Bigr)\\;\n  \\Theta_\\cut\\Bigl(\\lips[iJ]{2} \\;\\Big\\vert\n  \\;k_{i}=z_{iJ}\\,\\tilde{k}_{i}, \n  \\;k=(1-z_{iJ})\\,\\tilde{k}_{i} \\Bigr) ,\n  \\\\\n \n  \\rd\\sigma^\\Rs_{\\PA}\\Bigl(\\lips[iJ]{3}\\Bigr) \n \n  &\\;\\longrightarrow\\;\n  \\rd\\sigma^\\Rs_{\\PA}\\Bigl(\\lips[iJ]{3}\\Bigr)\\;\n  \\Theta_\\cut\\Bigl(\\lips[iJ]{3} \\;\\Big\\vert\n  \\;k_{i}=z_{iJ}\\,\\tilde{k}_{i}, \n  \\;k=(1-z_{iJ})\\,\\tilde{k}_{i} \\Bigr) ,\n  \\\\\n \n  \\rd\\sigma^0_{\\PA}\\Bigl(\\dtilde{\\Phi}^{(ab)c}_{2,iJ}\\Bigr)\n \n  &\\;\\longrightarrow\\;\n  \\rd\\sigma^0_{\\PA}\\Bigl(\\dtilde{\\Phi}^{(ab)c}_{2,iJ}\\Bigr)\\;\n  \\Theta_\\cut\\Bigl(\\dtilde{\\Phi}^{(ab)c}_{2,iJ} \\;\\Big\\vert\n  \\;\\tilde{k}_{i}=z_{iJ}\\,\\dtilde{k}_{i}, \n  \\;\\tilde{k}=(1-z_{iJ})\\,\\dtilde{k}_{i} \\Bigr) ,\n\\end{align}\n\\end{subequations}\nwhere we have made explicit the cut function for the computation of observables in the notation.\nThis modification induces additional convolution terms over the distribution $[\\cIbarew(z)]_+$ with\n\\begin{align}\n  \\label{eq:nlo:dec:master:cIbarew}\n  \\cIbarew(z)\n  &=\n  \\frac{\\alpha}{2\\pi}\\, Q_{\\Pl_1}\\biggl[\n  (Q_{\\Pl_1}-Q_{\\Pl_2})\\; \\cDbarsub{\\Pl_1 V}(z)\n  +Q_{\\Pl_2}\\; \\cGbarsub{\\Pl_1\\Pal_2}(z)\n  \\biggr]\n  \\nonumber\\\\&\\quad\\times\n  \\Theta_\\cut\\left(\\lips[]{2} \\;\\Big\\vert\n  \\;k_{\\Pl_1}=z\\,\\tilde{k}_{\\Pl_1}, \n  \\;k=(1-z)\\,\\tilde{k}_{\\Pl_1} \\right)\n \n  +(\\Pl_1\\leftrightarrow\\Pal_2) ,\n\\end{align}\nwhich we indicate by the label ``$\\Rbarew$''.\nThe contribution with virtual QCD corrections is given by\n\\begin{align}\n  \\label{eq:IF:master-VRbar-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Rbarew} \n  &=\n  \\int_0^1\\rd z \\int_2 \\left[\\cIbarew(z)\\right]_+ \n  \\;\\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0\\;\n  \\diptimes\\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] , \n\\end{align}\nand the real-emission corrections for the quark--anti-quark and gluon--quark induced contributions read\n\\begin{align} \n  \\label{eq:IF:master-RRbar-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rbarew}\n  &=\n  \\int_0^1\\rd z \\int_3 \\left[\\cIbarew(z)\\right]_+ \n  \\biggl\\{ \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Paq_a\\Pg)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Paq_a,\\Paq_a}\n  \\nonumber\\\\&\\quad\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pq_b\\Pg)\\Paq_a]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pq_b,\\Pq_b}  \n  \\biggr\\} ,\n\\end{align}\n\\begin{align}\n  \\label{eq:IF:master-RRbar-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Rbarew}\n  &=\n  \\int_0^1\\rd z \\int_3 \\left[\\cIbarew(z)\\right]_+ \n  \\biggl\\{ \\rd\\sigma_{\\Pg\\Pq_b,\\PA}^{\\Rs}\n  - \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \\Bigl(\\lips[(\\Pg\\Pq_a)\\Pq_b]{n}\\Bigr) \n  \\diptimes\\CSV^{\\Pg,\\Paq_a}\n  \\biggr\\} \n\\end{align}\nand analogous terms for the $\\Pg\\bar q_a$ channel.\nThe $\\CSK$ and $\\CSP$ operators contain an additional convolution over the momentum \nfractions~$x$ of the incoming partons and can be written in the following form,\n\\begin{align}\n  \\label{eq:IF:master-CRbar-qq}\n  \\sigreg_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Rbarew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x \\int_0^1\\rd z  \\int_2  \\left[\\cIbarew(z)\\right]_+\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_a,p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Paq_a,\\Paq_a} \n  \\nonumber\\\\&\\quad\n  +\\int_0^1 \\rd x \\int_0^1\\rd z  \\int_2 \\left[\\cIbarew(z)\\right]_+\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (p_a,xp_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pq_b,\\Pq_b} ,\n\\end{align}\n\\begin{align}\n  \\label{eq:IF:master-CRbar-gq}\n  \\sigreg_{\\Pg\\Pq_b,\\pro\\times\\dec}^{\\Cs\\otimes\\Rbarew}\n  &=\n  \\phantom{+}\\int_0^1 \\rd x \\int_0^1\\rd z  \\int_2 \\left[\\cIbarew(z)\\right]_+ \n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 (xp_{\\Pg},p_b)\n  \\diptimes(\\CSK+\\CSP)^{\\Pg,\\Paq_a} \n\\end{align}\nand analogous terms for the $\\Pg\\bar q_a$ channel.\nNote that the $\\CSK$ and $\\CSP$ operators, in general, contain plus distributions with respect to the variable $x$, and the above equations need to be properly evaluated in combination with the plus distribution $[\\cIbarew(z)]_+$ that acts on the integration variable~$z$.\n\n\n\n\n\n\n\n\\section{PHOTOS settings}\n\\label{app:photos}\n\nThe results using the PHOTOS parton shower shown in Figs.~\\ref{fig:distNNLO-IF-FSR-Wp} and \\ref{fig:distNNLO-IF-FSR-Z} of Sect.~\\ref{sec:FSR} were obtained with version 2.15 of the program and the following options:\n\\begin{tabbing}\n  \\qquad\\=\\hspace{.35\\textwidth}\\=\\hspace{.35\\textwidth}\\=\\kill\n  \\> \\texttt{ISEC=.FALSE.}, \n  \\> \\texttt{ITRE=.FALSE.}, \n  \\> \\texttt{IEXP=.FALSE.}, \\\\\n  \\> \\texttt{IFTOP=.FALSE.}, \n  \\> \\texttt{XPHCUT=0.01D0} \\; (default value).\n\\end{tabbing}\nThese settings restrict the parton shower to \\emph{at most one} additional photon emission in order to simulate the impact of $\\order{\\alpha}$ corrections. \nFurther settings which differ for the charged-current and neutral-current processes are as follows:\n\\begin{tabbing}\n  \\qquad\\=\\hspace{.35\\textwidth}\\=\\hspace{.35\\textwidth}\\=\\kill\n  \\> \\textbf{\\boldmath{\\large \\PWpm production:}} \\> \\textbf{\\boldmath{\\large \\PZ production:}} \\\\\n  \\> \\texttt{IFW=.TRUE.},              \\> \\texttt{IFW=.FALSE.}, \\\\\n  \\> \\texttt{INTERF=.FALSE.},          \\> \\texttt{INTERF=.TRUE.}, \\\\\n  \\> \\texttt{ALPHA=}\\;$\\alpha_{G_\\mu}$, \\> \\texttt{ALPHA=}\\;$\\alpha(0)$.\n\\end{tabbing}\nNote that the electromagnetic coupling constant $\\alpha$ is adjusted to the respective value used in our calculation of the initial--final corrections as described in Sect.~\\ref{sec:IF:numerics}.\n\n\n\\section*{Acknowledgement}\nThis project is supported by the German Research Foundation (DFG) via grant DI 784\/2-1 and the German Federal Ministry for Education and Research (BMBF). \nMoreover, A.H.\\ is supported via the ERC Advanced Grant MC@NNLO (340983).\nC.S.\\ is supported by the Heisenberg Programme of the DFG.\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nThe Drell--Yan-like $\\PW$- and $\\PZ$-boson production processes are among the most precise probes of the Standard Model and do \nnot only serve as key benchmark or ``standard candle'' processes, \nbut further allow for precision measurements of the $\\PW$-boson mass and the effective weak mixing angle.\nThis task of precision physics requires a further increase in the accuracy of the theoretical predictions, where the mixed QCD--electroweak corrections of \\order{\\alphas\\alpha} currently represent the largest unknown component of radiative corrections in terms of fixed-order predictions.\n\nIn our previous paper~\\cite{Dittmaier:2014qza} we have established a framework for evaluating the \\order{\\alphas\\alpha} corrections to Drell--Yan processes in the resonance region using the pole approximation and presented the calculation of the so-called non-factorizable corrections.  They turned out to be phenomenologically negligible, so that the \\order{\\alphas\\alpha} corrections almost entirely result from \nfactorizable corrections that can be separately attributed to production and decay of the $\\PW$\/$\\PZ$ boson (up to spin correlations).\n\n\nIn this paper we have presented the calculation of the \nso-called factorizable corrections of ``initial--final'' and ``final--final''\ntypes.  The latter were calculated in Sect.~\\ref{sec:calc-fact-nnlo-ff} and\nonly comprise finite  counterterm contributions which were found to be numerically very small ($< 0.1\\%$) and therefore can be safely neglected for all phenomenological purposes.  The former, on the other hand, combine large QCD corrections to the production with large EW corrections to the decay subprocesses and are expected to be the dominant contribution of the $\\order{\\alphas\\alpha}$ corrections.  Their calculation has been presented in Sect.~\\ref{sec:calc-fact-nnlo-if}, and we have shown numerical results in Sect.~\\ref{sec:if-results} for the most important observables for the $\\PW$-boson mass measurement: the transverse-mass and lepton-transverse-momentum distributions for \\PW production.  The results for the neutral-current process comprise the invariant-mass and the lepton-transverse-momentum distributions.  In the framework of the pole approximation, the only missing \\order{\\alphas\\alpha} corrections are now those of ``initial--initial'' type. Based on the results of the NLO electroweak calculation, these are expected to be numerically small.\n\nWe have used our results for the dominant $\\order{\\alphas\\alpha}$ corrections to test the validity of simpler approximate combinations of EW and QCD corrections: \nFirstly, we use a naive product ansatz multiplying the NLO QCD and EW\ncorrection\n factors, and secondly, we approximate the $\\order{\\alphas\\alpha}$ contribution by combining leading-logarithmic approximations of QED final-state radiation with the NLO QCD corrections.\n\nWe have demonstrated  in Sect.~\\ref{sec:if-results} that naive products fail to capture the factorizable initial--final corrections in distributions such as in\nthe transverse momentum of the lepton, which are sensitive to QCD\ninitial-state radiation and therefore require  a correct treatment of the double-real-emission part of the NNLO corrections. Naive products also fail to capture observables that are strongly affected by a\n redistribution of events due to final-state real-emission corrections,\n such as the invariant-mass distribution of the neutral-current process.  On the other hand, if an observable is less affected by such a redistribution of events or is only affected by it in the vicinity of the resonance, such as the transverse-mass distribution of the charged-current process, the naive products are able to reproduce the factorizable initial--final corrections to a large extent. \n\n In Sect.~\\ref{sec:FSR} we have investigated to which extent the factorizable initial--final corrections calculated in this paper  can be approximated by a combination of the NLO QCD corrections and a collinear approximation of  real-photon emission through a  QED structure function approach\nor a  QED parton shower such as PHOTOS.  For the invariant-mass distribution in $\\PZ$-boson production we observe a significant improvement in the agreement compared to the naive product ansatz,\nsince\nboth PHOTOS and the QED structure functions model  the redistribution of events due to final-state radiation, which\n is responsible for the bulk of the corrections in this observable. \nOur results can furthermore be used to validate Monte Carlo event generators where  $\\order{\\alphas\\alpha}$ corrections are approximated by  a combination of NLO matrix elements and parton showers. \n\nFinally, \nin Section~\\ref{sec:mw} we have illustrated the phenomenological impact of the $\\order{\\alphas\\alpha}$ corrections by estimating the mass shift induced by the \n factorizable initial--final corrections as  $\\approx-14~\\MeV$  for the case\n of bare muons  and $\\approx-4~\\MeV$ \nfor dressed leptons.\nThese corrections therefore have to be properly taken into account in the $\\PW$-boson mass measurements at the LHC, which aim at a precision of about $10~\\MeV$.\nIt will be interesting to investigate the impact of the $\\order{\\alphas\\alpha}$ corrections on the measurement of the effective weak mixing angle as well in the future.\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe class of Drell--Yan-like processes \nis one of the most prominent types of particle reactions\nat hadron colliders and describes the production of a lepton pair through an intermediate gauge-boson decay,\n\\begin{equation*}\n  \\Pp\\Pp \/ \\Pp\\Pap \\;\\to\\; V \\;\\to\\; \\Pl_1\\Pal_2 \\;+\\; X .\n\\end{equation*}\nDepending on the electric charge of the colour-neutral gauge boson $V$, the process can be further classified into the neutral-current ($V=\\PZ\/\\Pgg$) and the charged-current ($V=\\PWpm$) processes. \nThe large production rate in combination with the clean experimental signature of the leptonic vector-boson decay allows this process to be measured with great precision.\nMoreover, the Dell--Yan-like production of $\\PW$ or $\\PZ$ bosons is one of the theoretically best understood and most precisely predicted processes.\nAs a consequence, electroweak (EW) \ngauge-boson production is among the most important \n``standard-candle'' processes at the LHC (see, e.g.\\ Refs.~\\cite{Abdullin:2006aa,Gerber:2007xk}).\nIts cross section can be used as a luminosity monitor, and the measurement of\nthe mass and width of the $\\PZ$ boson represents a powerful tool for detector\ncalibration.\nFurthermore, the \\PW charge asymmetry and the rapidity distribution of the $\\PZ$ boson deliver important constraints in the fit of the \nparton distribution functions (PDFs)~\\cite{Boonekamp:2009yd}, which represent crucial ingredients for almost all predictions at the LHC.\n\nOf particular relevance for precision tests of the Standard Model \nis the potential of the Drell--Yan process at the LHC  for high-precision measurements in the resonance regions, where the effective weak mixing angle, quantified by $\\sin^2\\theta_\\text{eff}^{\\Pl}$,\nmight be extracted from data with LEP precision~\\cite{Haywood:1999qg}.\nThe $\\PW$-boson mass can be determined from a fit to the distributions of the lepton transverse momentum ($p_{\\rT,\\Pl}$) and the transverse mass of the lepton pair ($M_{\\rT,\\Pl\\Pgn}$) which exhibit Jacobian peaks around $\\MW$ and $\\MW\/2$, respectively, and allow for a precise extraction of the mass\nwith a sensitivity below \\SI{10}{\\MeV}~\\cite{Besson:2008zs,Baak:2013fwa} provided that PDF uncertainties can be reduced\\cite{Rojo:2013nia,Quackenbush:2015yra,Bozzi:2015hha,Bozzi:2015zja}.\n\nTo fully exploit the potential of the extraordinary experimental precision that is achievable for the Drell--Yan process, it is necessary to have theoretical predictions that match or even surpass the \nexpected accuracy.\nThe current state of the art includes\nQCD corrections at next-to-next-to-leading-order (NNLO)\naccuracy~\\cite{Hamberg:1990np,Harlander:2002wh,Anastasiou:2003ds,Melnikov:2006di,Melnikov:2006kv,Catani:2009sm,Gavin:2010az,Gavin:2012sy}\nsupplemented by leading higher-order soft-gluon\neffects~\\cite{Moch:2005ba,Laenen:2005uz,Idilbi:2005ky,Ravindran:2007sv}\n and soft-gluon resummation for  small transverse momenta~\\cite{Balazs:1997xd,Landry:2002ix,Bozzi:2010xn,Mantry:2010mk,Becher:2011xn,Guzzi:2013aja,Catani:2015vma}.\nFor event generation, next-to-leading-order (NLO) \ncalculations  have been matched to parton showers~\\cite{Frixione:2006gn,Alioli:2008gx,Hamilton:2008pd}, with a recent effort to include NNLO corrections in a parton-shower framework~\\cite{Hoeche:2014aia,Karlberg:2014qua,Alioli:2015toa}.\nConcerning EW effects, the \nNLO corrections~\\cite{Baur:1997wa,Zykunov:2001mn,Baur:2001ze,Dittmaier:2001ay,Baur:2004ig,Arbuzov:2005dd,CarloniCalame:2006zq,Zykunov:2005tc,CarloniCalame:2007cd,Arbuzov:2007db,Brensing:2007qm,Dittmaier:2009cr}\nas well as leading higher-order effects from multiple photon emission and\nuniversal weak effects~\\cite{Baur:1999hm,Placzek:2003zg,CarloniCalame:2003ux,Brensing:2007qm,Dittmaier:2009cr} are known. The sensitivity to the photon PDF through \nphoton-induced production channels has been studied in \nRefs.~\\cite{Arbuzov:2007kp,CarloniCalame:2007cd,Dittmaier:2009cr,Boughezal:2013cwa}.\n \nIn addition to the N$^3$LO QCD corrections, the next frontier in theoretical fixed-order computations is given by the calculation of the mixed QCD--EW corrections of \\order{\\alphas\\alpha}~\\cite{Andersen:2014efa}.\nThese corrections can affect observables relevant for the \\MW determination at the percent level~\\cite{Balossini:2009sa} and therefore must be under theoretical control.\nUp to now, QCD and EW corrections have been combined in various\napproximations~\\cite{Cao:2004yy,Richardson:2010gz,Bernaciak:2012hj,Barze:2012tt,Li:2012wna,Barze:2013yca}.  \nHowever, a full NNLO calculation at \\order{\\alphas\\alpha} is necessary \nfor a proper combination of  QCD and EW corrections without ambiguities.\nHere some partial results for two-loop amplitudes~\\cite{Kotikov:2007vr,Kilgore:2011pa,Bonciani:2011zz} as well as the  full \\order{\\alphas\\alpha} corrections to the \\PW and \\PZ decay widths~\\cite{Czarnecki:1996ei,Kara:2013dua} are known. \nA complete calculation of the \\order{\\alphas\\alpha} corrections requires to combine the double-virtual corrections with \n the \\order{\\alpha} EW corrections to $\\PW\/\\PZ+\\jet$ production~\\cite{Kuhn:2005az,Kuhn:2007cv,Hollik:2007sq,Denner:2009gj,Denner:2011vu,Denner:2012ts,Hollik:2015pja}, the \\order{\\alphas} QCD corrections to $\\PW\/\\PZ+\\Pgg$ \nproduction~\\cite{Smith:1989xz,Ohnemus:1992jn,Ohnemus:1994qp,Dixon:1998py,Campbell:1999ah,DeFlorian:2000sg,Hollik:2007sq,Campbell:2011bn,Denner:2014bna,Denner:2015fca}, and the double-real corrections using a method to regularize \ninfrared (IR) singularities.\n\n\n In a previous paper~\\cite{Dittmaier:2014qza}, we have initiated the\n calculation of the \\order{\\alphas\\alpha} corrections to Drell--Yan\n processes in the resonance region via the so-called \\emph{pole\n   approximation}~(PA)~\\cite{Stuart:1991xk}, which has been\n successfully applied to the EW corrections to \n $\\PW$~production~\\cite{Wackeroth:1996hz,Baur:1998kt,Dittmaier:2001ay,Dittmaier:2014qza}\n and $\\PZ$~production~\\cite{Dittmaier:2014qza} at NLO.\n It is based on a systematic expansion of the cross section about the\n resonance pole and is suitable for theoretical predictions in the\n vicinity of the gauge-boson resonance, where the higher precision is\n especially relevant.  The PA splits the corrections into distinct\n well-defined subsets, which can be calculated separately. This allows\n to assess the numerical impact of different classes of corrections\n and to identify the dominant contributions.  More precisely, the\n contributions can be classified into two types: the factorizable and\n the non-factorizable corrections.  In the former, the\ncorrections can be separately attributed to the production and\nthe subsequent decay of the gauge boson, whereas in the latter \nthe production and decay subprocesses are\n linked by the exchange of soft photons.  \nAt \\order{\\alpha}, the PA shows agreement with the known \nNLO EW corrections up to fractions of $1\\%$ near the resonance, i.e.\\ at a phenomenologically satisfactory level~\\cite{Dittmaier:2014qza}.\nIn particular, the bulk of the NLO EW corrections near the resonance can be attributed to the factorizable corrections to the \\PW\/\\PZ~decay subprocesses, while\nthe factorizable corrections to the production process are mostly suppressed below the \npercent level,\nand the non-factorizable contributions being even smaller.\n\nBased on the quality of the PA at NLO \nwe are confident that this approach is suitable to calculate the  $\\order{\\alphas\\alpha}$ corrections with sufficient accuracy for the description of observables that are dominated by the resonances.\n The non-factorizable\n corrections comprise the conceptually most challenging contribution\n to the PA and have been computed at \\order{\\alphas\\alpha} in\n Ref.~\\cite{Dittmaier:2014qza}.  They turn out to be very small and,\n thus, demonstrate that for phenomenological purposes the\n \\order{\\alphas\\alpha} corrections can be factorized into terms\n associated with initial-state and\/or final-state corrections and combinations of the two types.\n In this paper we calculate the factorizable corrections of the type\n ``initial--final'', which combine large QCD corrections to the\n production with the large EW corrections to the decay of the $\\PW$\/$\\PZ$\n boson. \n Therefore we expect to capture  the dominant contribution at\n \\order{\\alphas\\alpha} to observables relevant for precision physics\n dominated by the \\PW and \\PZ resonances.  We also compute the corrections of ``final--final'' type, which are given only by finite counterterms to the leptonic vector-boson decay. \n The remaining factorizable ``initial--initial''\n corrections are expected to deliver only a small contribution \nand would further require $\\order{\\alphas\\alpha}$-corrected PDFs for a consistent evaluation, which are however not available.\nIt is all the more important to isolate this contribution in a well-defined\nmanner, as it is accomplished by the PA.\n\n\n\nA technical aspect of higher-order calculations involving massless particles is the proper treatment of \nIR singularities that are associated with configurations involving soft and\/or collinear particles.\nTo this end, we use the dipole subtraction \nformalism~\\cite{Catani:1996vz,Dittmaier:1999mb,Catani:2002hc,Dittmaier:2008md} \nand its extension for decay processes presented in Ref.~\\cite{Basso:2015gca}\nfor the analytic cancellation of all IR singularities.\nAlthough the cancellation of IR singularities in the \\order{\\alphas\\alpha} corrections presented in this work is accomplished by using a combined approach of the techniques developed for NLO calculations, it represents one of the main technical difficulties in the calculation and we devote special attention to its discussion.\n\n\nThis paper is organized as follows:\nIn Section~\\ref{sec:nnlo} we present the calculation of the initial--final and\nfinal--final factorizable corrections.\nWe discuss the construction of an IR-finite final result for the\ninitial--final corrections in detail with a special focus on the treatment of\nthe combined IR singularities of the QCD and EW corrections. \nOur numerical results are presented in Section~\\ref{sec:IF:numerics}, where we compare them to different versions of a naive product ansatz obtained by multiplying NLO QCD and EW correction \nfactors, and to a leading-logarithmic treatment of photon radiation \nas provided by the structure-function approach or QED parton showers such as PHOTOS~\\cite{Golonka:2005pn}.\nWe further perform a $\\chi^2$ fit in order to estimate the effect of the NNLO\n\\order{\\alphas\\alpha} corrections on the measurement of the  $\\PW$-boson mass.\nA summary is given in Sect.~\\ref{sec:concl}.\n\n\n\n\n\n\\section{Calculation of the dominant \\texorpdfstring{$\\order{\\alphas\\alpha}$}{O(as a)} corrections in pole approximation}\n\\label{sec:nnlo}\n\n\nIn this section we identify and calculate the dominant $\\order{\\alphas\\alpha}$ corrections to the charged-current and neutral-current Drell--Yan processes in the vicinity of an intermediate vector-boson resonance. \nIn Sect.~\\ref{sec:method} we describe the classification of the  $\\order{\\alphas\\alpha}$ corrections in the framework of the PA~\\cite{Dittmaier:2014qza}. We identify  factorizable contributions of ``initial--final'' type---i.e.\\ the combination of QCD corrections to vector-boson production with EW corrections to vector-boson decay---as dominant source for corrections to distributions dominated by the vector-boson resonance.\nThe calculation of the building blocks contributing to \nthe initial--final factorizable corrections is performed in Sect.~\\ref{sec:calc-fact-nnlo-if}.\nIn Sect.~\\ref{sec:IF:master} the different building blocks of the initial--final contributions are combined into a formula suitable for numerical evaluation, where all IR singularities are cancelled explicitly.\nFinally, in Sect.~\\ref{sec:calc-fact-nnlo-ff} we calculate corrections of ``final--final'' type, which are given by pure counterterm contributions and are numerically small.\n\n\\subsection{Survey of types of \\texorpdfstring{$\\order{\\alphas\\alpha}$}{O(as a)} corrections in pole approximation}\n\\label{sec:method}\n\n\n\\begin{figure}[b]\n  \\centering\n  %\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_ii_gen}\n    \\subcaption{Factorizable initial--initial corrections}\n  \\end{subfigure}\n  %\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_if_gen}\n    \\subcaption{Factorizable initial--final corrections}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_ff_gen}\n    \\subcaption{Factorizable final--final corrections}\n  \\end{subfigure}\n  %\n  \\begin{subfigure}[m]{.48\\linewidth}\n    \\centering\n    \\includegraphics{images\/diags\/DY_MIX_nf_gen}\n    \\subcaption {Non-factorizable corrections}\n  \\end{subfigure}\n  %\n  \\caption{The four types of corrections that contribute to the mixed QCD--EW corrections in the PA illustrated in terms of generic two-loop amplitudes. Simple circles symbolize tree structures, double circles one-loop corrections, and triple circles two-loop contributions.}\n  \\label{fig:NNLOcontrib}\n\\end{figure}\nThe PA for Drell--Yan\nprocesses~\\cite{Stuart:1991xk,Wackeroth:1996hz,Baur:1998kt,Dittmaier:2001ay,Dittmaier:2014qza}\nprovides a systematic classification of contributions to Feynman\ndiagrams that are enhanced by the resonant propagator of a vector\nboson~$V=\\PW,\\PZ$.\nThe leading corrections\nin the expansion around the resonance pole arise from factorizable\ncorrections to $\\PW$\/$\\PZ$ production and decay subprocesses, and\nnon-factorizable corrections that link production and decay by\nsoft-photon exchange.  The PA separates corrections to production and\ndecay stages in a consistent and gauge-invariant way.  \nThis is\nparticularly relevant for the charged-current Drell-Yan process, where\nphoton radiation off the intermediate $\\PW$~boson contributes\nsimultaneously to the corrections to production and decay of a\n$\\PW$~boson, and to the non-factorizable contributions.  Applications\nof different variants of the PA to NLO EW\ncorrections~\\cite{Wackeroth:1996hz,Baur:1998kt,Dittmaier:2001ay,Dittmaier:2014qza}\nhave been validated by a comparison to the complete EW NLO\ncalculations and show excellent agreement at the order of some $0.1\\%$\nin kinematic distributions dominated by the resonance region.\n\nThe structure of the PA for \nthe \\order{\\alphas\\alpha} correction has\nbeen worked out in Ref.~\\cite{Dittmaier:2014qza}, where details of the\nmethod and our setup can be found.  \nThe corrections can be classified into the four types of\ncontributions shown in Fig.~\\ref{fig:NNLOcontrib} for the case of the\ndouble-virtual corrections. For each class of contributions with\nthe exception of the final--final corrections (c), also the\nassociated real--virtual and \ndouble-real corrections have to be\ncomputed, obtained by replacing one or both of the labels $\\alpha$ and $\\alphas$ in\nthe blobs in Fig.~\\ref{fig:NNLOcontrib} by a real\nphoton or gluon, respectively. The corresponding crossed partonic channels, e.g.\\ with\nquark--gluon initial states have to be included in addition.\n\nIn detail, the four types of corrections are characterized as follows:\n\\begin{enumerate}[(a)]\n\\item The initial--initial factorizable corrections are given by\n  two-loop \\order{\\alphas\\alpha} corrections to on-shell $\\PW\/\\PZ$\n  production and the corresponding one-loop real--virtual and\n  tree-level double-real contributions, i.e.\\ $\\PW\/\\PZ+\\jet$\n  production at \\order{\\alpha}, $\\PW\/\\PZ+\\Pgg$ production at\n  \\order{\\alphas}, and the processes $\\PW\/\\PZ+\\Pgg+\\jet$ at tree\n  level. Results for individual ingredients of the initial--initial\n  part are known, such as partial two-loop\n  contributions~\\cite{Kotikov:2007vr,Bonciani:2011zz} and the full\n  \\order{\\alpha} EW corrections to W\/Z+jet production including the\n  W\/Z decays~\\cite{Denner:2009gj,Denner:2011vu,Denner:2012ts}.\n  However, a consistent combination of these building blocks requires\n  also a subtraction scheme for IR singularities at ${\\cal\n    O}(\\alphas\\alpha)$ and has not been performed yet.  Note that\n  currently no PDF set including \\order{\\alphas\\alpha} corrections is\n  available, which is required to absorb IR singularities of the\n  initial--initial corrections from photon radiation collinear to the\n  beams.\n\n  Results of the PA at \\order{\\alpha} show that observables such as\n  the transverse-mass distribution in the case of \\PW production or\n  the lepton-invariant-mass distributions for \\PZ production are\n  extremely insensitive to initial-state photon\n  radiation~\\cite{Dittmaier:2014qza}. Since these distributions also\n  receive relatively moderate QCD corrections, we do not expect\n  significant initial--initial NNLO\n  \\order{\\alphas\\alpha}\n  corrections to such distributions. For observables sensitive to\n  initial-state recoil effects, such as the transverse-lepton-momentum\n  distribution, \nthe \\order{\\alphas\\alpha} corrections should be larger, but still\nvery small compared to the huge QCD \ncorrections.%\n\\footnote{Note that for such observables\n  a fixed-order QCD description is not adequate near the\n  Jacobian peak, so that in this case the initial--initial corrections need\n  to be combined with a resummation of multiple gluon emissions.  At\n  present, such a resummation is available in the POWHEG framework in\n  combination with an approximation to the double-real and\n  real--virtual part of the initial--initial corrections where the\n  first emitted photon or gluon is treated exactly, while further\n  emissions are generated in the collinear\n  approximation~\\cite{Barze:2012tt,Barze:2013yca}.}\n\n\\item  The factorizable initial--final corrections \nconsist of the \\order{\\alphas} corrections to $\\PW\/\\PZ$ production combined with the \\order{\\alpha} corrections to the leptonic  $\\PW\/\\PZ$ decay.\nBoth types of corrections are large and have a sizable impact on the shape of differential distributions at  NLO, so that we\n expect this class of the factorizable corrections to capture the dominant  \\order{\\alphas\\alpha} effects.\nThe computation of these contributions is the main result of this paper and is discussed in Sect.~\\ref{sec:calc-fact-nnlo-if}.  \n Preliminary numerical results of these corrections were presented in\n Refs.~\\cite{Dittmaier:2014koa,Huss:2014eea}.\n\n\\item Factorizable final--final corrections arise from the\n  $\\order{\\alphas\\alpha}$ counterterms of the\n  lepton--$\\PW\/\\PZ$-vertices, which involve only QCD corrections to\n  the vector-boson self-energies. There are no corresponding real\n  contributions, so that\nthe final--final corrections have practically no\n  impact on the shape of distributions. We compute these corrections\n  in Sect.~\\ref{sec:calc-fact-nnlo-ff} below and confirm the\n  expectation that they are phenomenologically negligible.\n\n\\item  The non-factorizable $\\order{\\alphas\\alpha}$ corrections are given by soft-photon corrections\nconnecting the initial state, the intermediate vector boson, and the\nfinal-state leptons, combined with QCD corrections to $V$-boson production. \nAs shown in detail in Ref.~\\cite{Dittmaier:2014qza}, these corrections\ncan be expressed in terms of soft-photon\ncorrection factors to squared tree-level or one-loop QCD matrix\nelements by using gauge-invariance arguments.\nThe numerical impact of these corrections was found to be below the\n$0.1\\%$ level and is therefore negligible for all phenomenological purposes.\n\\end{enumerate}\n\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{\\linewidth}\n    \\centering\n\\includegraphics[scale=1]{images\/diags\/DY_NNLO_R_propid_diag} \n\\caption{Decomposition of a diagram with photon emission off a $V$-boson line into initial--initial and initial--final corrections}\n\\label{fig:amp_NNLO:propagator}\n      \\end{subfigure}\n  \\begin{subfigure}[b]{.5\\linewidth}\n  \\centering\n  \\includegraphics[scale=1.]{images\/diags\/DY_MIXif_Rfac_p_OS}\n  \\caption{Photon emission from the production subprocess}\n  \\label{fig:amp_NNLO:production}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}[b]{.475\\linewidth}\n  \\centering\n  \\includegraphics[scale=1.]{images\/diags\/DY_MIXif_Rfac_d_OS}\n  \\caption{Photon emission from the decay subprocess}\n    \\label{fig:amp_NNLO:decay}\n  \\end{subfigure}\\hfill\n  \\caption{\nDecomposition of the double-real corrections $\\Paq_a(p_a)\\Pq_b(p_b)\\to\\Pl_1(k_1)\\Pal_2(k_2)\\Pg(k_{\\Pg})\\Pgg(k)$  \ninto initial---initial~(\\subref{fig:amp_NNLO:production}) and initial--final~(\\subref{fig:amp_NNLO:decay}) parts,\nillustrated for an example in part~(\\subref{fig:amp_NNLO:propagator}). \n  The momentum $p_V$ of the intermediate vector boson $V$ is given by $p_V=p_a+p_b-k_{\\Pg}-k=k_1+k_2$.  A double line on a $V$ propagator indicates on-shellness while a gauge boson attached to an  encircled subdiagram indicates all possible insertions.}\n \\label{fig:amp_NNLO}\n\\end{figure}\nThe definition of the factorizable corrections and the separation of\ninitial- and final-state corrections is illustrated in\nFig.~\\ref{fig:amp_NNLO} for the case of the double-real\ncorrections. An example\ndiagram for the charged-current process is given in Fig.~\\ref{fig:amp_NNLO:propagator}, which\ncannot be attributed uniquely to the vector-boson production or decay\nsubprocess and displays an overlapping resonance structure due to the\npropagator poles at $p_V^2=\\mu_V^2$ and $(p_V+k)^2=\\mu_V^2$.\nHere $\\mu_V$ combines the real mass and width parameters of $V$,\n$M_V$ and $\\Gamma_V$, to a complex mass value,\n$\\mu_V^2=M_V^2-\\ri M_V\\Gamma_V$.\nHowever,\na simple partial-fractioning identity for the two $V$-boson\npropagators allows us\nto disentangle the two resonance structures and to\ndecompose such diagrams into contributions associated with photon\nemission from the production or decay subprocesses of an on-shell\n$V$~boson (see Eq.~(2.11) in Ref.~\\cite{Dittmaier:2014qza}). This is\nillustrated in Fig.~\\ref{fig:amp_NNLO:propagator}, where the double\nslash on a propagator line indicates that the corresponding momentum\nis set on its mass shell in the rest of the diagram (but not on the\nslashed line itself).  Using this decomposition, the double-real\ncorrections can be divided consistently into initial--initial and\ninitial--final contributions, as shown in\nFig.~\\ref{fig:amp_NNLO:production} and Fig.~\\ref{fig:amp_NNLO:decay},\nrespectively.  Here a diagrammatic notation is used where an encircled\ndiagram with an attached photon or gluon stands for all possibilities\nto attach the photon\/gluon to the fermion line and the gauge boson\n$V$ (see Eq.~(2.12) in Ref.~\\cite{Dittmaier:2014qza} for an\nexample). The initial--final (virtual QCD)$\\times$(real EW) corrections\nare treated analogously.\nAll different contributions to the\nfactorizable initial--final corrections are diagrammatically\ncharacterized in terms of interference diagrams in\nFig.~\\ref{fig:NNLO-if-graphs}.\n\n\\begin{figure}\n  \\centering\n  %\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_vv1}\n    \\qquad\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_vv2}\n    \\caption{Factorizable initial--final double-virtual corrections}\n    \\label{fig:NNLO-if-graphs-vv}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rv1}\n    \\qquad\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rv2}\n    \\caption{Factorizable initial--final (real QCD)$\\times$(virtual EW) corrections}\n    \\label{fig:NNLO-if-graphs-rv}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_vr1_OS}\n    \\caption{Factorizable initial--final (virtual QCD)$\\times$(real photonic) corrections}\n    \\label{fig:NNLO-if-graphs-vr}\n  \\end{subfigure}\n  %\n  \\\\[1.5em]\n  \\begin{subfigure}[m]{\\linewidth}\n    \\centering\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rr1_OS}\n    \\qquad\n    \\includegraphics[scale=0.95]{images\/diags\/DY_MIXif_rr2_OS}\n    \\caption{Factorizable initial--final double-real corrections}\n    \\label{fig:NNLO-if-graphs-rr}\n  \\end{subfigure}\n  %\n  \\caption{Interference diagrams for the various contributions to the factorizable initial--final corrections of \\order{\\alphas\\alpha}, with blobs representing all relevant tree structures.\n  The blobs with ``\\alpha'' inside represent one-loop corrections of \\order{\\alpha}.}\n  \\label{fig:NNLO-if-graphs}\n\\end{figure}\n\n\n\n\\subsection{Calculation of the factorizable initial--final corrections}\n\\label{sec:calc-fact-nnlo-if}\n\nIn this section we calculate the various contributions to the\nfactorizable initial--final corrections of \\order{\\alphas\\alpha} shown\nin Fig.~\\ref{fig:NNLO-if-graphs}.  Most contributions can be expressed in terms of reducible products\nof NLO QCD and NLO EW building blocks. For details on the notation\nused for these NLO results we refer to Ref.~\\cite{Dittmaier:2014qza}.\n\n\n\n\n\\subsubsection{Double-virtual corrections}\n\\label{sec:IF:VV}\n\nThe double-virtual \\order{\\alphas\\alpha} initial--final corrections to the\nsquared $\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2$ amplitude are illustrated in\nFig.~\\ref{fig:NNLO-if-graphs}(a) in terms of interference diagrams.\nThey arise in two ways: from the interference of the tree amplitude\nwith the two-loop \\order{\\alphas\\alpha} amplitude \nand from the interference\nbetween the one-loop amplitudes with \\order{\\alphas} corrections to\n$V$-boson production and \\order{\\alpha} corrections to the decay, respectively,\n\\begin{align}\n    \\bigl\\lvert \\cM^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} \\bigr\\rvert^2 \n    \\;\\Bigr\\rvert_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew} &= \n    \\hphantom{+} 2\\Re\\left\\{\n    \\delta\\cM_{\\Vs\\otimes\\Vew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\n    \\left(\\cM_{0,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\\right)^* \\right\\}\n    \\nonumber\\\\&\\quad\n    + 2\\Re\\left\\{\n    \\delta\\cM_{\\Vew,\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\n    \\left(\\delta\\cM_{\\Vs,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\\right)^* \\right\\} .\n    \\label{eq:IF:NNLO-vv}\n\\end{align}\nThe LO amplitude in PA, $\\M_{0,\\PA}$, differs from the full LO\nmatrix element by the absence of the non-resonant photon diagram in\ncase of the neutral-current Drell--Yan process. \n The first term on the\nright-hand side in Eq.~\\eqref{eq:IF:NNLO-vv} involves the factorizable\ninitial--final contribution to the two-loop amplitude, which takes the form\nof reducible (one-loop)$\\times$(one-loop) diagrams and is defined\nexplicitly as\n\\begin{equation}\n  \\label{eq:IF:mix_vv1}\n    \\delta\\cM_{\\Vs\\otimes\\Vew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} =\n  \\sum_{\\lambda_V} \n  \\frac{\n    \\delta\\cM_\\Vs^{\\Paq_a\\Pq_b\\to V}({\\lambda_V})\\;\n    \\delta\\cM_\\Vew^{V\\to\\Pl_1\\Pal_2}({\\lambda_V})\n  }{p_V^2-\\mu_V^2} \n= \\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\; \\deltadec \\;  \n  \\cM_{0,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} ,\n\\end{equation}\nwhere a sum over the physical polarization states of the\nvector boson $V$, labelled by $\\lambda_V$, is performed. In the second\nstep in Eq.~\\eqref{eq:IF:mix_vv1} the fact is used that the one-loop\ncorrections to the production and decay factorize off the\ncorresponding LO matrix elements,\n\\begin{align}\n  \\label{eq:factQCD:pro:virt}\n   \\delta \\cM_{\\Vs}^{\\Paq_a\\Pq_b\\to V} &=\\delta_{\\Vs}^{V \\Paq_a\\Pq_b}\n \\;\\cM_{0}^{\\Paq_a\\Pq_b\\to V},  \\\\\n  \\label{eq:factEW:dec:virt}\n  \\delta \\cM_{\\Vew,\\dec}^{V\\to\\Pl_1\\Pal_2} &=\n  \\deltadec \n  \\;\\cM_{0,\\PA}^{V\\to\\Pl_1\\Pal_2} .\n\\end{align}\nThe virtual QCD corrections are well known and are quoted explicitly in \nEq.~(2.35) of Ref.~\\cite{Dittmaier:2014qza}.\nThe explicit expressions for the NLO EW correction factors can be found, \ne.g., in Refs.~\\cite{Dittmaier:2001ay,Dittmaier:2009cr}, and are quoted in \nAppendix B.2 of Ref.~\\cite{Huss:2014eea}.\nIn order to maintain gauge invariance, the NLO production \nand decay subamplitudes in Eq.~\\eqref{eq:IF:mix_vv1}, \nand in particular\nthe correction factor $\\deltadec$, are evaluated for on-shell $V$ bosons. \nWe keep the QCD correction factor $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ off shell,\ni.e.\\ without setting $s\\to M_V^2$ there to be closer to the full calculation,\nwhich is possible, because $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ does not depend on\n$M_V$ at all.\nThe on-shell projection $s\\to M_V^2$ in the EW correction\ninvolves some freedom, but numerical\neffects from different implementations are of the same order as the\nintrinsic uncertainty of the PA.  However, the choice of the mappings\nin the virtual and real corrections has to match properly in order to\nensure the correct cancellation of IR singularities. \n\nThe expressions~\\eqref{eq:factQCD:pro:virt}\nand~\\eqref{eq:factEW:dec:virt} also enter the one-loop interference\nterms in the second line of Eq.~\\eqref{eq:IF:NNLO-vv}.  The final\n  result for the double-virtual corrections to the cross section is\n  therefore given by\n\\begin{align}\n  \\label{eq:IF:mix_vv}\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Vew} &= \n  2 \\biggl[\n  \\Re\\Bigl\\{ \\deltadec \\; \\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr\\}\n  + \\Re\\Bigl\\{ \\deltadec \\Bigl(\\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr)^* \\Bigr\\}\n  \\biggr] \\; \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 \n  \\nonumber\\\\&=\n  4\\Re\\Bigl\\{ \\deltadec \\Bigr\\}\n  \\Re\\Bigl\\{ \\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr\\}  \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^0 .\n\\end{align}\nBoth the EW and QCD correction factors contain soft and collinear\nsingularities, which take the form of $\\frac{1}{\\epsilon^2}$ poles for\nmassless fermions. Therefore, in principle, Eq.~\\eqref{eq:IF:mix_vv}\nrequires the evaluation of the correction factors up to\n\\order{\\epsilon^2} in order to obtain all finite \\order{\\epsilon^0}\nterms. %\nHowever, after applying the subtraction formalism, which we describe\nin detail in Sect.~\\ref{sec:IF:master}, the poles are cancelled before\nperforming the full expansion in $\\epsilon$ and, thus, the results up\nto order \\order{\\epsilon^0} turn out to be sufficient.  This result is obvious if\nthe soft and collinear singularities are not regularized in\n$D=4-2\\epsilon$ dimensions,\nbut by small mass parameters, where no rational terms \nfrom the multiplication of $1\/\\epsilon$ poles with $D$-dimensional quantities\nexist at all.\n\n\\subsubsection{(Real QCD)\\texorpdfstring{${\\times}$}{x}(virtual EW) corrections}\n\\label{sec:IF:RV}\n\nThe (real QCD)$\\times$(virtual EW) contributions to the factorizable initial--final corrections shown in Fig.~\\ref{fig:NNLO-if-graphs}(\\subref{fig:NNLO-if-graphs-rv}) arise by \nincluding the virtual corrections to the leptonic $\\PW$\/$\\PZ$ decays  to the various partonic subprocesses of $V+\\text{jet}$ production, \n\\begin{subequations}\n\\label{eq:rQCD}\n\\begin{align}\n  \\Paq_a(p_a) \\,+\\, \\Pq_b(p_b) &\\,\\to\\, V(p_V) \\,+\\, \\Pg(k_{\\Pg}) ,\\\\\n  \\Pg(p_{\\Pg}) \\,+\\, \\Pq_b(p_b) &\\,\\to\\, V(p_V) \\,+\\, \\Pq_a(k_a) ,\\\\\n  \\Pg(p_{\\Pg}) \\,+\\, \\Paq_a(p_a) &\\,\\to\\, V(p_V)  \\,+\\, \\Paq_b(k_b) .\n\\end{align}\n\\end{subequations}\nFor the quark-induced channel, the corrections are given by replacing the virtual QCD amplitude in Eq.~\\eqref{eq:IF:mix_vv1} by the corresponding amplitude for real-gluon emission,\n\\begin{align}\n  \\label{eq:IF:mix-rv-gen}\n  \\delta\\cM_{\\Rs\\otimes\\Vew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pg} &=\n  \\sum_{\\lambda_V} \n  \\frac{\n    \\cM_{\\Rs}^{\\Paq_a\\Pq_b\\to \\Pg V}({\\lambda_V})\\;\n    \\delta\\cM_\\Vew^{V\\to\\Pl_1\\Pal_2}({\\lambda_V})\n  }{p_V^2-\\mu_V^2} .\n\\end{align}\nAnalogously to the double-virtual case, the EW decay\nsubamplitude is evaluated for on-shell vector bosons,\nwhile the QCD correction is kept off shell.\nUsing the factorization property of the EW one-loop decay corrections~\\eqref{eq:factEW:dec:virt}, the (real QCD)$\\times$(virtual EW) correction to the cross section in the quark--anti-quark channel is proportional to the real NLO QCD corrections $\\rd\\sigma^{\\Rs}$,  \n\\begin{equation}\n\\label{eq:IF:mix-rv}\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  = 2\\Re\\Bigl\\{ \\deltadec \\Bigr\\}\\rd\\sigma_{\\Paq_a\\Pq_b,\\PA}^{\\Rs}  .\n\\end{equation}\nAs for the Born amplitude, the label PA in the real-emission corrections indicates that all non-resonant terms, i.e.\\ the photon-exchange diagrams in case of the neutral-current process, are omitted in the QCD real-emission amplitudes. \nAnalogous expressions hold for the gluon--quark and gluon--anti-quark initiated subprocesses to $V+\\text{jet}$ production.\n\n\n\n\\subsubsection{(Virtual QCD)\\texorpdfstring{${\\times}$}{x}(real photonic) corrections}\n\\label{sec:IF:VR}\n\nThe (virtual QCD)$\\times$(real photonic) factorizable corrections of\ninitial--final type arise from the generic interference diagram shown\nin Fig.~\\ref{fig:NNLO-if-graphs}(\\subref{fig:NNLO-if-graphs-vr}). They\nare obtained by combining the real-photon corrections to on-shell\n$V$-boson decay with the virtual QCD corrections to $V$-boson\nproduction,\n\\begin{equation}\n\\cM_{\\Vs\\otimes\\Rew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg}\n =\\sum_{\\lambda_V} \\frac{\\delta \\cM_\\Vs^{\\Paq_a\\Pq_b\\to V}({\\lambda_V})\\;\n  \\cM_{\\Rew}^{V\\to\\Pl_1\\Pal_2\\Pgg}({\\lambda_V})}{(p_V+k)^2-\\mu_V^2} = \n\\delta_{\\Vs}^{V \\Paq_a\\Pq_b}\\cM_{\\Rew,\\fact,\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg}.\n\\end{equation}\nIn the second step, Eq.~\\eqref{eq:factQCD:pro:virt} has been used to factorize the virtual QCD correction factor from the  matrix element $ \\cM_{\\Rew,\\fact,\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg}$ for the factorizable NLO decay corrections (see Eq.~(2.14) in Ref~\\cite{Dittmaier:2014qza}).\nAgain the matrix elements for the EW \ndecay subprocess is evaluated for on-shell vector bosons,\nwhile the QCD correction factor is kept off shell. \nAs illustrated in Fig.~\\ref{fig:amp_NNLO}\nand discussed in detail in Ref.~\\cite{Dittmaier:2014qza}, the splitting of photon-emission \neffects off the intermediate $V$-boson into parts corresponding to initial- or final-state\nradiation separates the two resonance propagator factors\n$1\/(p_V^2-\\MV^2)$ and $1\/[(p_V^2+k)^2-\\MV^2]$, respectively, where $p_V=k_1+k_2$.\nFor factorizable EW decay correction we, thus, have to perform the on-shell projection\n$(p_V^2+k)^2\\to \\MV^2$.\nThe resulting contribution \nof the (virtual QCD)$\\times$(real photonic) corrections to the cross section therefore assumes the form\n\\begin{align}\n  \\label{eq:IF:mix_vr}\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\pro\\times\\dec}^{\\Vs\\otimes\\Rew} &= \n  2 \\Re\\Bigl\\{ \\delta_{\\Vs}^{V\\Paq_a\\Pq_b} \\Bigr\\}  \\;\n  \\rd\\sigma_{\\Paq_a\\Pq_b,\\dec}^\\Rew .\n\\end{align}\n\n\n\\subsubsection{Double-real corrections}\n\\label{sec:IF:RR}\n\nThe double-real emission corrections  are illustrated by interference diagrams in Fig.~\\ref{fig:NNLO-if-graphs}(d) and are defined by the real-emission matrix elements for the $V+\\text{jet}$ production subprocesses~\\eqref{eq:rQCD} with the subsequent decay $V\\to \\Pl_1\\Pal_2\\Pgg$,\n\\begin{align}\n  \\label{eq:IF:mix-rr-gen}\n \\cM_{\\Rs\\otimes\\Rew,\\pro\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2\\Pgg\\Pg}\n =\\sum_{\\lambda_V} \\frac{\\cM_\\Rs^{\\Paq_a\\Pq_b\\to V\\Pg}({\\lambda_V})\\;\n  \\cM_{\\Rew}^{V\\to\\Pl_1\\Pal_2\\Pgg}({\\lambda_V})}{(p_V+k)^2-\\mu_V^2}, \n\\end{align}\nwith analogous expressions for the $\\Pg q$ and $\\Pg\\bar q$ channels.\nThe non-resonant contribution arising from the case $V=\\Pgg$ in the neutral-current process is again not included.\nCompact explicit results for the helicity amplitudes of the double-real corrections can be found in Ref.~\\cite{Huss:2014eea}.\nThe double-real contribution to the cross section, $\\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}$, \nis defined in terms of the square of the matrix element~\\eqref{eq:IF:mix-rr-gen}\nwhere the decay subamplitudes are evaluated for on-shell $V$~bosons. \nDue to the spin correlations of the production and decay matrix elements and\nthe full kinematics of the $2\\to 4$ scattering process, the double-real\ncorrections do not factorize further into separate EW and QCD correction\nfactors, in contrast to the other classes of\n factorizable initial--final corrections.\n\n\n\n\n\\subsection{Treatment of infrared singularities for the factorizable initial--final corrections}\n\\label{sec:IF:master}\n\nThe NNLO \\order{\\alphas\\alpha} contributions to the cross section due\nto the factorizable initial--final corrections are\nobtained by integrating the four contributions discussed in the\nprevious section over the respective phase spaces,\n\\begin{align}\n  \\label{eq:IF:generic-master-first}\n  \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} \n  &=\n  \\int_{2} \\rd\\sigma_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew}\n  + \\iint\\limits_{2+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}\n   + \\int_{3} \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  + \\iint\\limits_{3+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  \\nonumber\\\\&\\quad\n  + \\int_{2} \\rd\\sigma_{\\pro\\times\\dec}^{\\Cs\\otimes\\Vew} \n  + \\iint\\limits_{2+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} ,\n\\end{align}\nwhere the additional QCD collinear counterterms in the last line were\nintroduced to absorb the collinear singularities associated with the\nquarks and gluons in the initial state into the NLO PDFs.  Note that\nthe EW corrections are completely confined to the decay subprocess,\nand consequently, there are \nno singularities from initial-state collinear\nquark--photon splittings.  This allows us\nto obtain the collinear\ncounterterms in the last line of\nEq.~\\eqref{eq:IF:generic-master-first} from the customary NLO QCD\ncollinear counterterms $\\rd\\sigma^{\\Cs}$~\\cite{Catani:1996vz} by\nreplacing the LO cross sections by the appropriate real or virtual EW\ndecay corrections in the PA. Using the results of\nSect.~\\ref{sec:calc-fact-nnlo-if} we can write\n\\begin{align}\n  \\label{eq:IF:generic-master}\n  \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} \n  &=\n  \\int_{2} 4\\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}\\;\n  \\Re\\Bigl\\{\\deltadec\\Bigr\\}\\;\n  \\rd\\sigma_\\PA^0\n  + \\iint\\limits_{2+\\Pgg} 2\\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}\\;\n  \\rd\\sigma_\\dec^{\\Rew} \n  \\nonumber\\\\&\\quad\n  + \\int_{3} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\}\\; \\rd\\sigma_\\PA^{\\Rs} \n  + \\iint\\limits_{3+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  \\nonumber\\\\&\\quad\n  + \\int_{2} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \\; \\rd\\sigma_\\PA^{\\Cs} \n  + \\iint\\limits_{2+\\Pgg} \\rd\\sigma_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} .\n\\end{align}\n\n\nApplying the QCD dipole subtraction formalism~\\cite{Catani:1996vz} \n in order to cancel the IR\nsingularities associated with the QCD corrections,\nEq.~\\eqref{eq:IF:generic-master} can be written in the following form,\n\\begin{align}\n  \\label{eq:IF:dipole-qcd-master}\n  \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} &= \n  \\int_{2} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \\;\n  \\rd\\sigma_\\PA^0 \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr]\n  \\nonumber\\\\&\\quad\n  + \\iint\\limits_{2+\\Pgg}  \n  \\rd\\sigma_\\dec^\\Rew \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr]\n  \\nonumber\\\\&\\quad\n  + \\int_{3} 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \n  \\biggl\\{ \\rd\\sigma_\\PA^{\\Rs} \n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\\rd\\sigma_\\PA^0\\diptimes\\CSV \\biggr\\}\n  \\nonumber\\\\&\\quad\n  + \\iint\\limits_{3+\\Pgg} \\biggl\\{\n  \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\\rd\\sigma_\\dec^\\Rew\\diptimes\\CSV \\biggr\\}\n  \\nonumber\\\\&\\quad\n  + \\int_0^1\\rd x\\int_2\n  2\\Re\\Bigl\\{\\deltadec\\Bigr\\} \\;\n  \\rd\\sigma_\\PA^0 \\diptimes(\\CSK+\\CSP) \n  \\nonumber\\\\&\\quad\n  + \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n  \\rd\\sigma_\\dec^\\Rew\\diptimes(\\CSK+\\CSP) .\n\\end{align}\nThe explicit expressions of the dipole operators  $\\CSV$ and the insertion operators $\\CSI$, \n$\\CSK$, and $\\CSP$ can be found in Ref.~\\cite{Catani:1996vz}.\nThe symbol  $\\otimes$ denotes possible additional helicity and colour correlations,\nand it is implicitly assumed that the cross sections multiplying the dipole operators $\\CSV$ are evaluated on the respective dipole-mapped phase-space point.\nThe explicit expressions associated with the NLO QCD corrections were given in Ref.~\\cite{Dittmaier:2014qza}.\n\n\nAll individual integrals appearing in Eq.~\\eqref{eq:IF:dipole-qcd-master} are now free of QCD singularities, but remain IR divergent owing to the singularities contained in the EW corrections which still need to be cancelled between the virtual corrections and the corresponding real-photon-emission parts.\nFor this purpose we employ the dipole subtraction formalism for photon radiation~\\cite{Dittmaier:1999mb,Dittmaier:2008md}, in particular  the extension of the formalism to treat decay kinematics described in detail in Ref.~\\cite{Basso:2015gca}.\nAs a result, we are able to arrange the six contributions in Eq.~\\eqref{eq:IF:dipole-qcd-master} into a form where all IR divergences are cancelled in the integrands explicitly,\n\\begin{align}\n   \\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}\n  &=\\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}\n  +\\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} ,\n \\label{eq:IF:final-master}\n\\end{align}\nwhere each term is an IR-finite object and its phase-space integration can be performed numerically in four dimensions.\nEquation~\\eqref{eq:IF:final-master} is our master formula for the numerical evaluation discussed in Sect.~\\ref{sec:IF:numerics}.\nExplicit expressions for all contributions for the quark--anti-quark\nand quark--gluon induced channels are given in\nAppendix~\\ref{app:ir-safe}.\n\n\nThe first two terms in Eq.~\\eqref{eq:IF:final-master} arise from the sum of the double-virtual and the (virtual QCD)$\\times$(real photonic) corrections, including the insertion operators from the QCD dipole formalism, and correspond to the sum of the first two lines in Eq.~\\eqref{eq:IF:dipole-qcd-master}.\nApplying the dipole formalism to rearrange the IR singularities of photonic origin between the virtual and real EW corrections, we obtain the following expressions for the IR-finite virtual QCD contributions to the cross section, \n\\begin{align} \n  \\label{eq:IF:master-VV}\n  \\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Vew}&= \n  \\int_{2} \n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr] \\;\n  \\rd\\sigma_\\PA^0 \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] , \\\\\n  \\label{eq:IF:master-VR}\n  \\sigreg_{\\pro\\times\\dec}^{\\Vs\\otimes\\Rew}&=\n  \\iint\\limits_{2+\\Pgg}  \\biggl\\{ \\rd\\sigma_\\dec^\\Rew \n  -\\sum_{\\substack{I,J\\\\I\\ne J}} \\rd\\sigma_\\PA^0 \\diptimes \\dVew[IJ] \\biggr\\}\n  \\diptimes\n  \\Bigl[ 2 \\Re\\Bigl\\{\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}\\Bigr\\}+\\CSI \\Bigr] ,\n\\end{align}\nwhere the sum over the emitter--spectator pairs ($I,J$) in Eq.~\\eqref{eq:IF:master-VR} extends over all particles of the decay subprocess, i.e.\\ $I,J=\\Pl_1,\\Pal_2,V$.\nWe have introduced a compact notation for the QED dipoles,\n\\begin{align}\n  \\label{eq:definition:dVew}\n  \\dVew[IJ] &= 4\\pi\\alpha\\; \\eta_IQ_I\\,\\eta_JQ_J\\,\n  \\begin{cases}\n  \\dsub{IV} , & \\text{for } (I=\\Pl_1,\\Pal_2) \\land (J=V) , \\\\\n  \\gsub{IJ} , & \\text{for } (I=\\Pl_1,\\Pal_2) \\land (J=\\Pl_1,\\Pal_2) , \\\\\n  0 , & \\text{for } I=V ,\n  \\end{cases}\n\\end{align}\nwhere $\\eta_i=1$ for incoming particles and outgoing antiparticles and $\\eta_i=-1$ for incoming antiparticles and outgoing particles. \nThe corresponding endpoint contributions are given by\n\\begin{align}\n  \\label{eq:nlo:dec:master:Iew}\n  \\Iew\n  &=\n  \\frac{\\alpha}{2\\pi}\\, Q_{\\Pl_1}\\biggl[\n  (Q_{\\Pl_1}-Q_{\\Pl_2})\\; \\Dsub{\\Pl_1 V}\n  +Q_{\\Pl_2}\\; \\Gsub{\\Pl_1\\Pal_2} \n  \\biggr]\n  +(\\Pl_1\\leftrightarrow\\Pal_2) ,\n\\end{align}\nwhere the functions $\\gsub{}$ and $\\Gsub{}$ are given in Ref.~\\cite{Dittmaier:1999mb}, while  $\\dsub{}$ and $\\Dsub{}$ are the decay dipoles and their integrated counterparts constructed in Ref.~\\cite{Basso:2015gca}.\nWhenever we write $\\Pl_1\\leftrightarrow\\Pal_2$, this implies the interchange\n$Q_{\\Pl_1}\\leftrightarrow Q_{\\Pl_2}$ of the electric charges of the respective fermions,\nirrespective of their particle or antiparticle nature.\n\nAs anticipated in Sect.~\\ref{sec:IF:VV}, all IR singularities contained in $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ cancel exactly against the corresponding poles of the $\\CSI$ operator within the second square bracket of Eq.~\\eqref{eq:IF:master-VV}.\nSimilarly, all singularities in $\\deltadec$ cancel against the corresponding poles in $\\Iew$ in the first square bracket of Eq.~\\eqref{eq:IF:master-VV}.\nAs a consequence, it is sufficient to use the correction factors $\\deltadec$ and $\\delta_{\\Vs}^{V\\Paq_a\\Pq_b}$ up to \\order{\\epsilon^0}.\nFurthermore, we recall that the correction factors $\\deltadec$ \nare evaluated at the on-shell point $p_V^2=\\MV^2$ and, thus, are\nindependent of the phase-space kinematics.%\n\n\nThe contributions involving real QCD corrections are given by the third and forth term in Eq.~\\eqref{eq:IF:final-master}.\nThey are obtained by applying the QED dipole subtraction formalism to the sum of the third and forth line of Eq.~\\eqref{eq:IF:dipole-qcd-master} and result in the following expressions for the IR-finite real-gluon contributions to the cross section,\n\\begin{align}\n  \\label{eq:IF:master-RV}\n  \\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Vew}&= \n  \\int_{3} \n  \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\n  \\biggl\\{ \\rd\\sigma_\\PA^{\\Rs} \n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\\rd\\sigma_\\PA^0\\diptimes\\CSV \\biggr\\} ,\n  \\\\\n  \\label{eq:IF:master-RR}\n  \\sigreg_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}&=\n  \\iint\\limits_{3+\\Pgg} \\biggl\\{\n  \\rd\\sigma_{\\pro\\times\\dec}^{\\Rs\\otimes\\Rew}\n  - \\sum_{\\substack{\\QCD\\\\\\text{dipoles}}} \\rd\\sigma_\\dec^\\Rew\\diptimes\\CSV \n  - \\sum_{\\substack{I,J\\\\I\\ne J}}\\rd\\sigma_\\PA^\\Rs \\diptimes \\dVew[IJ]\n  \\nonumber\\\\&\\qquad\n  +\\sum_{\\substack{\\QCD\\\\\\text{dipoles}}}\n  \\sum_{\\substack{I,J\\\\I\\ne J}}\n  \\rd\\sigma_\\PA^0 \\diptimes\\CSV \\diptimes \\dVew[IJ]\n  \\biggr\\}  .\n\\end{align}\n\nIt is instructive to  examine the local cancellation of the IR singularities in Eq.~\\eqref{eq:IF:master-RR} in more detail.\nThe second term inside the curly brackets of Eq.~\\eqref{eq:IF:master-RR} acts as a local counterterm to the double-real emission cross section $\\rd\\sigma^{\\Rs\\otimes\\Rew}$ in all regions of phase space where the additional QCD radiation becomes unresolved, i.e.\\ soft and\/or collinear to the beam. \nThe third term inside the curly brackets of Eq.~\\eqref{eq:IF:master-RR} analogously  ensures the cancellation of IR singularities in the phase-space regions where the photon becomes soft and\/or collinear to a final-state lepton.\nA subtlety arises in the double-unresolved cases, where the cross sections $\\rd\\sigma_\\dec^\\Rew$ and $\\rd\\sigma_\\PA^\\Rs$ become singular as well, and both subtraction terms above will simultaneously act as a local counterterm, leading to the twofold subtraction of the IR singularities.\nThis disparity in the double-unresolved limits is exactly compensated by the last term inside the curly brackets of Eq.~\\eqref{eq:IF:master-RR}, which therefore has the opposite sign.\nNote that the evaluation of this last term  involves the successive application of two dipole phase-space mappings.\nOwing to the property of the factorizable initial--final corrections where the emissions in the production and decay stages of the $V$ boson proceed independently, the two dipole mappings do not interfere with each other and\nthe order in which they are applied is irrelevant.\nA related property is the factorization of the dipole phase space, where the two one-particle subspaces associated with the two unresolved emissions can be isolated simultaneously.  \nThis has the important consequence\nthat the analytic integration over the \ngluon and photon momenta can be carried out in the same manner as  at\nNLO, which allows \nus to reuse the known results for the integrated\ndipoles without modification.\n\n\n\nFinally, we consider the convolution terms with additional virtual or real EW corrections given by the last two terms in Eq.~\\eqref{eq:IF:final-master}.\nSince these contributions are essentially given by the lower-order (in \\alphas) cross sections, convoluted with the insertion operators $\\CSK$ and $\\CSP$, they pose no additional complications, and the resulting IR-finite contributions to\nthe cross section can be written as\n\\begin{align}\n \\label{eq:IF:master-CV}\n \\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Vew}&=\n \\int_0^1\\rd x\\int_2\n \\Bigl[ 2\\Re\\Bigl\\{\\deltadec\\Bigr\\} + \\Iew \\Bigr]\\;\n \\rd\\sigma_\\PA^0 \\diptimes(\\CSK+\\CSP)  , \\\\\n \\label{eq:IF:master-CR}\n \\sigreg_{\\pro\\times\\dec}^{\\Cs\\otimes\\Rew} &= \n \\int_0^1\\rd x\\iint\\limits_{2+\\Pgg}\n \\biggl\\{ \\rd\\sigma_\\dec^\\Rew \n -\\sum_{\\substack{I,J\\\\I\\ne J}} \n \\rd\\sigma_\\PA^0 \\diptimes \\dVew[IJ] \\biggr\\}\n \\diptimes(\\CSK+\\CSP) .\n\\end{align}\nOwing to the Lorentz invariance of the dipole formalism, no special treatment is required in contrast to our calculation of the non-factorizable corrections discussed in Ref.~\\cite{Dittmaier:2014qza}, which was carried out with the slicing method to isolate soft-photon singularities.\n\n\nThe results presented so far are appropriate for the case of IR-safe observables, i.e.\\ for the case where\ncollinear photons and leptons are recombined to a ``dressed'' lepton carrying their total momentum.\nFor non-collinear-safe observables with respect to the final-state leptons, i.e.\\ the treatment of bare muons without photon recombination, we use the method of Ref.~\\cite{Dittmaier:2008md} and its extension to decay kinematics described in Ref.~\\cite{Basso:2015gca}.\nThe required modifications are described \nin Appendix~\\ref{app:non-collinear-safe}.\n\n\n\n\n\n\n\\subsection{Factorizable final--final corrections}\n\\label{sec:calc-fact-nnlo-ff}\n\nThe factorizable  NNLO corrections of final--final type arise purely from the counterterms to \nthe $V\\Pl_1\\Pal_2$ vertex and therefore factorize from the LO matrix element,\n\\begin{equation}\n  \\label{eq:FF:mix_vv1}\n  \\delta\\cM_{\\Vs\\otimes\\Vew,\\dec\\times\\dec}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2}\n  = \\delta^{\\mathrm{ct}, (\\alphas\\alpha)}_{V \\Pl_1\\Pal_2}\\;  \\;  \n  \\cM_{0,\\PA}^{\\Paq_a\\Pq_b\\to\\Pl_1\\Pal_2} .\n\\end{equation}\nThe counterterms for the leptonic vector-boson decay only receive\ncontributions from the vector-boson self-energies at\n$\\order{\\alphas\\alpha}$~\\cite{Chang:1981qq,Djouadi:1987gn,Djouadi:1987di,Kniehl:1988ie,Kniehl:1989yc,Djouadi:1993ss},\nwhich enter the counterterms through the vector-boson wave-function\nrenormalization constants and through the renormalization constants of the\nelectromagnetic coupling and the weak-mixing angle. \nThere is only one type of\ncontribution from one-loop diagrams with insertions of one-loop\n$\\order{\\alphas}$ or $\\order{\\alpha}$ counterterms.\nIt results from massive quark loops in the vector-boson self-energies \nwhere the QCD mass renormalization\nconstant has to be taken into account. \nWe make use of the expressions\nfor the vector-boson self-energies of Ref.~\\cite{Djouadi:1993ss},\nwhich include the QCD quark mass counterterm in the on-shell scheme.  \nThe expressions  for the self-energies in terms of the scalar integrals computed in Ref.~\\cite{Djouadi:1993ss} are given in Appendix~\\ref{app:ff}.\n\nThe vertex counterterms in the on-shell renormalization scheme are\nobtained from the expressions for the corresponding NLO EW\ncounterterms~\\cite{Denner:1991kt} upon replacing the one-loop\nvector-boson self-energies by the two-loop\n$\\mathcal{O}(\\alphas\\alpha)$ results and dropping lepton wave-function\nrenormalization constants, which receive no correction at this\norder.   \nWe employ the $G_\\mu$ input-parameter scheme where the\n electromagnetic\n coupling constant\nis derived from the Fermi constant $G_\\mu$ via the relation\n\\begin{equation}\n  \\label{eq:G_mu-scheme}\n  \\alpha_{G_\\mu} \\;=\\; \\frac{\\sqrt{2}}{\\pi} G_\\mu \\MW^2 \n  \\left( 1-\\frac{\\MW^2}{\\MZ^2} \\right) .\n\\end{equation}\nThe counterterm $\\delta Z_e$ for\nthe electromagnetic charge in the $G_\\mu$ scheme  is related to the one\nin the $\\alpha(0)$ input-parameter scheme as follows,\n\\begin{equation}\n\\label{eq:dZeGmu}\n   \\delta Z_e^{G_\\mu}= \\delta Z_e^{\\alpha(0)}\n-\\frac{1}{2}\\Delta r .\n\\end{equation}\nThe quantity $\\Delta r$ comprises all higher-order corrections to muon\ndecay excluding the contributions that constitute QED corrections in\nthe Fermi model, which are included in the definition of the muon\ndecay constant $G_\\mu$~\\cite{Sirlin:1980nh},\n\\begin{align}\n\\Delta r &=\n\\left.\\frac{\\partial\\Sigma^{AA}_{\\rT}(k^2)}{\\partial k^2}\\right\\vert_{k^2=0} \n- 2\\frac{\\delta \\sw}{\\sw}+\n2\\frac{\\cw}{\\sw}\\;\\frac{\\Sigma_{\\rT}^{AZ}(0)}{\\MZ^2} +\n\\frac{\\Sigma^{WW}_{\\rT}(0)- \\Re\\, \\Sigma^{WW}_{\\rT}(\\MW^2)}{\\MW^2} +\\delta r ,\n\\end{align}\nwith the renormalization constant $\\delta \\sw$ \nof the weak-mixing angle and the transverse parts of the vector-boson self-energies,\n$\\Sigma^{VV'}_{\\rT}$.  The $\\mathcal{O}(\\alphas\\alpha)$\ncontribution to $\\Delta r$ simplifies due to the fact that there is no contribution\nto the finite remainder\n$\\delta r$ at this order and the photon--$\\PZ$-boson mixing\nself-energy $\\Sigma_\\rT^{AZ}$ vanishes at zero\nmomentum~\\cite{Djouadi:1993ss}.  \nMoreover, since there are no loop corrections\nto the leptonic vector-boson decay at $\\mathcal{O}(\\alphas\\alpha)$,\nthe vertex counterterms are finite. The expressions for the\ncounterterms $\\delta^{\\mathrm{ct}, (\\alphas\\alpha)}_{V \\Pl_1\\Pal_2}$\nin terms of vector-boson self-energies are explicitly given in\nAppendix~\\ref{app:ff}.  \n\n\nThe contribution of the final--final corrections to the cross-section prediction are obtained by a simple phase-space integration over the Born kinematics,\n\\begin{equation}\n  \\sighat_{\\dec\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} = \n  \\int_{2} 2  \\Re\\Bigl\\{\\delta^{\\mathrm{ct}, (\\alphas\\alpha)}_{V \\Pl_1\\Pal_2}\\Bigr\\}\\;\n  \\rd\\sigma_\\PA^0 .\n\\end{equation}\n\n Using the values\nof the input parameters given in Eq.~\\eqref{eq:params} below, the\nnumerical value of the counterterm for the  $\\PW\\Pgnl\\Pal$ vertex\nis given by\n\\begin{equation}\n  \\delta_{\\PW\\Pgnl\\Pal}^{\\mathrm{ct},(\\alphas\\alpha)}=\n  \\frac{\\alphas\\,\\alpha}{\\pi^2}\\times 0.93.\n\\end{equation}\nThe final--final correction to the cross section for the\ncharged-current cross section is therefore below the $0.1\\%$ level and\nphenomenologically negligible.  This can be partially attributed to the choice of the $G_\\mu$-scheme where universal corrections to charged-current processes are absorbed in the value of $\\alpha_{G_\\mu}$.\nThe numerical values of the counterterms $\\delta^{\\mathrm{ct},\\tau,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}$\nfor the $\\PZ\\Pl\\Pal$ vertices with lepton chiralities $\\tau=\\pm$\nare somewhat larger, but of opposite sign:\n\\begin{subequations}\n\\begin{align}\n   \\delta^{\\mathrm{ct},+,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}& =\n  \\frac{\\alphas\\,\\alpha}{\\pi^2}\\times (-49.3), \\\\\n   \\delta^{\\mathrm{ct},-,(\\alphas\\alpha)}_{\\PZ\\Pl\\Pal}&  =\n  \\frac{\\alphas\\,\\alpha}{\\pi^2}\\times (+31.8).\n\\end{align}\n\\end{subequations}\nThe resulting corrections to the neutral-current Drell--Yan\nprocess are, however, suppressed\nfar below the $0.1\\%$ level due to cancellations between the right-\nand left-handed production channels and are therefore also negligible for all phenomenological purposes.\n\n\n\n\n\n\\section{Numerical results}\n\\label{sec:IF:numerics}\n\n\nIn this section we present the numerical results for the \ndominant mixed QCD--EW corrections to the Drell--Yan process at the LHC for a centre-of-mass energy of $\\sqrt{s}=14~\\TeV$. We consider the two processes \n\\begin{eqnarray}\n  \\label{eq:nlo-num-procs}\n\\nonumber\n  \\Pp+\\Pp \\;\\to&\\; \\PWp \\;&\\to\\; \\Pgnl+\\Plp +X, \\\\\n  \\Pp+\\Pp \\;\\to&\\; {\\PZ} \\;&\\to\\; \\Plm+\\Plp +X\n\\nonumber\n\\end{eqnarray}\nwith electrons or muons in the final state ($\\Pl=\\Pe,\\Pgm$). \nWe further distinguish two\nalternative treatments of photon radiation: In the ``dressed-lepton''\ncase, collinear photon--lepton configurations are treated inclusively\nusing a photon-recombination procedure.  As a result, the numerical\npredictions do not contain large logarithms of the lepton mass, which\ncan be set to zero. The dressed-lepton results are appropriate\nmostly for electrons in the final state. In the ``bare-muon'' case, no such\nrecombination is performed, reflecting the experimental situation\nwhich allows for the detection of isolated muons.  We perform a comparison\nto naive factorization prescriptions of QCD and EW corrections, as\nwell as to a modelling of photonic final-state radiation~(FSR) by\nstructure functions or a photon shower. Moreover, we estimate the impact of the NNLO QCD--EW\ncorrections on the measurement of the $\\PW$-boson mass.\n\n\\subsection{Input parameters and event selection}\n\\label{sec:input-cuts}\n\nThe setup for the calculation is analogous to the one used in\nRef.~\\cite{Dittmaier:2014qza}.  \nThe choice of input parameters closely follows\nRef.~\\cite{Beringer:1900zz},\n\\begin{equation}\n\\label{eq:params}\n\\begin{aligned}\n  \\MW^\\OS \\;=&\\; 80.385~\\GeV ,\n  &\\GW^\\OS \\;=&\\; 2.085~\\GeV , \\\\\n  \\MZ^\\OS \\;=&\\; 91.1876~\\GeV ,\n  &\\Gamma_\\PZ^\\OS \\;=&\\; 2.4952~\\GeV , \\\\\n  M_\\PH \\;=&\\; 125.9~\\GeV , \n  &m_\\Pqt \\;=&\\; 173.07~\\GeV , \\\\\n  G_\\mu \\;=&\\; 1.1663787\\times 10^{-5} ~\\GeV^{-2} , \n  &\\alpha(0) \\;=&\\; 1\/137.035999074 , \\\\\n  \\alphas(\\MZ) \\;=&\\; 0.119 .\n\\end{aligned}\n\\end{equation}\n We convert the on-shell masses\nand decay widths of the vector bosons to the corresponding pole masses\nand widths as spelled out in Ref.~\\cite{Dittmaier:2014qza}.\n\nThe electromagnetic coupling constant used in the LO predictions is\nobtained from the Fermi constant by Eq.~\\eqref{eq:G_mu-scheme}.\nIn the charged-current process, all relative electroweak corrections are computed using $\\alpha_{G_\\mu}$.\nIn the neutral-current process, however, we follow  Ref.~\\cite{Dittmaier:2009cr} and use $\\alpha(0)$  consistently in the relative photonic corrections while the remaining relative weak corrections are proportional to $\\alpha_{G_\\mu}$.\nThe same prescription is applied to the relative $\\order{\\alpha_s\\alpha}$ corrections. \n\n\nThe masses of the light quark flavours (\\Pqu, \\Pqd, \\Pqc, \\Pqs, \\Pqb) and of the leptons are neglected throughout, with the only exception in case of non-collinear-safe observables, where the final-state collinear singularity is regularized by the mass of the muon,\n\\begin{equation}\n  \\label{eq:muon-mass}\n  m_{\\Pgm} \\;=\\; \\SI{105.658369}{\\MeV} .\n\\end{equation}\nThe CKM matrix is chosen diagonal in the third generation and the mixing between the first two generations is parametrized by the following values for the entries of the quark-mixing matrix,\n\\begin{equation}\n  \\label{eq:ckm}\n  \\lvert V_{\\Pqu\\Pqd} \\rvert \\,=\\, \n  \\lvert V_{\\Pqc\\Pqs} \\rvert \\,=\\, 0.974, \\qquad\n  \\lvert V_{\\Pqc\\Pqd} \\rvert \\,=\\, \n  \\lvert V_{\\Pqu\\Pqs} \\rvert \\,=\\, 0.227. \n\\end{equation}\nFor the PDFs we consistently use the NNPDF2.3 sets~\\cite{Ball:2012cx}, where the NLO and NNLO \nQCD--EW corrections are evaluated using the NNPDF2.3QED NLO set~\\cite{Ball:2013hta}, which also includes \\order{\\alpha} corrections.\nThe value of the strong coupling $\\alphas(\\MZ)$ quoted in Eq.~\\eqref{eq:params} is dictated by the choice of these PDF sets.\nFor the evaluation of the full NLO\nEW corrections entering the naive products below, we employ the DIS factorization scheme to absorb the mass singularities into the PDFs.\nThe renormalization and factorization scales are set equal, with a fixed value given by the respective gauge-boson mass, \n\\begin{equation}\n  \\label{eq:scale}\n  \\mur \\;=\\; \\muf \\;\\equiv\\; \\mu \\;=\\; \\MV ,\n\\end{equation}\nof the process under consideration.\n\nFor the experimental identification of the Drell--Yan process we impose the following cuts on the transverse momenta and rapidities of the charged leptons,\n\\begin{align}\n  \\label{eq:cut-lep}\n  p_{\\rT,\\Plpm} &> 25~\\GeV , &\n  \\lvert y_{\\Plpm} \\rvert &< 2.5 , \n\\end{align}\nand an additional cut on the missing transverse energy\n\\begin{equation}\n  \\label{eq:cut-miss}\n  E_\\rT^\\miss > 25~\\GeV ,\n\\end{equation}\nin case of the charged-current process.\nFor the neutral-current process we further require a cut on the invariant mass of the lepton pair,\n\\begin{equation}\n  \\label{eq:cut-mll}\n  M_{\\Pl\\Pl} > 50~\\GeV ,\n\\end{equation}\nin order to avoid the photon pole at $M_{\\Pl\\Pl}\\to0$.\n\nFor the dressed-lepton case, in addition, a photon recombination procedure analogous to the one used in Refs.~\\cite{Dittmaier:2001ay,Dittmaier:2009cr} is applied:\n\\begin{enumerate}\n\\item Photons close to the beam with a rapidity $\\lvert \\eta_{\\Pgg} \\rvert > 3$ are treated as beam remnants and are not further considered in the event selection.\n\\item For the photons that pass the first step, the angular distance to the charged leptons $R_{\\Plpm\\Pgg}=\\sqrt{(\\eta_{\\Plpm}-\\eta_{\\Pgg})^2 + (\\phi_{\\Plpm}-\\phi_{\\Pgg})^2}$ is computed, where $\\phi$ denotes the azimuthal angle in the transverse plane. \n  If the distance $R_{\\Plpm\\Pgg}$ \nbetween the photon and the closest lepton is smaller than \n$0.1$, the photon is recombined with the lepton by adding the respective four-momenta, $\\Plpm(k_i)+\\Pgg(k)\\to\\Plpm(k_i+k)$.\n\\item Finally, the event selection cuts from Eqs.~\\eqref{eq:cut-lep}--\\eqref{eq:cut-mll} are applied to the resulting event kinematics.\n\\end{enumerate}\n\n\n\n\\subsection{Results for the dominant factorizable corrections}\n\\label{sec:if-results}\n\nThe NNLO QCD--EW corrections to the hadronic Drell--Yan cross section \nare dominated by\nthe factorizable initial--final \\order{\\alphas\\alpha} corrections,\n$\\Delta\\sigma_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}$,\nwhich are obtained\nby convoluting the corresponding partonic corrections  \n$\\sighat_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}$ calculated in\nSection~\\ref{sec:nnlo} with the PDFs.\n Our default prediction for Drell--Yan processes is then obtained by\n adding these NNLO corrections to  the sum of \nthe full NLO QCD and EW corrections, \n\\begin{align} \n\\label{eq:def:nnlo:if}\n  \\sigma^{\\NNLO_{\\rs\\otimes\\rew}} &=\n  \\sigma^0 +\\Delta\\sigma^{\\NLO_\\rs} + \\Delta\\sigma^{\\NLO_\\rew}\n  +\\Delta\\sigma_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}} ,\n\\end{align}\nwhere all terms are consistently evaluated with the NNPDF2.3QED NLO PDFs.\nThe non-factorizable corrections computed in \nRef.~\\cite{Dittmaier:2014qza} were found to have a negligible impact on the cross section and are therefore not included here.\nSimilarly, the factorizable corrections of ``final--final'' type discussed in Sect.~\\ref{sec:calc-fact-nnlo-ff} turn out to have a negligible impact on the cross-section prediction and are therefore not included in Eq.~\\eqref{eq:def:nnlo:if} either.\n\nOur result allows to validate estimates of the NNLO QCD--EW corrections based on a naive product ansatz. For this purpose, we define the naive product of the NLO QCD cross section and the relative EW corrections,\n\\begin{align}\n  \\sigma_{\\text{naive fact}}^{\\NNLO_{\\rs\\otimes\\rew}} &=\n  \\sigma^{\\NLO_\\rs} (1+\\delta_{\\alpha})\n  \\nonumber\\\\&=\n  \\sigma^0 +\\Delta\\sigma^{\\NLO_\\rs} + \\Delta\\sigma^{\\NLO_\\rew}\n  +\\Delta\\sigma^{\\NLO_\\rs}\\;\\delta_{\\alpha}  ,\n  \\label{eq:def:nnlo:naive:fact}\n\\end{align}\nwhere the relative EW corrections are defined as the ratio of the NLO EW contribution $\\Delta\\sigma^{\\NLO_\\rew}$ with respect to the LO contribution $\\sigma^0$ according to\n\\begin{align}\n  \\label{eq:def:deltaew}\n  \\delta_{\\alpha} &\\equiv \n  \\frac{\\Delta\\sigma^{\\NLO_\\rew}}{\\sigma^0} ,\n\\end{align}\nwhere both denominator and numerator are evaluated with the same NLO PDFs, so that the EW correction factors are practically independent of the PDFs.\nIn order to compare the factorized expression to the NNLO corrections, we define two different versions of the NLO EW corrections in Eq.~\\eqref{eq:def:deltaew}: First, based on the full \\order{\\alpha} correction ($\\delta_\\alpha$), and second, based on the dominant EW final-state correction of the PA ($\\delta_\\alpha^\\dec$).\n\nDefining the correction factors,%\n\\footnote{\n  Note that the correction factor $\\delta_{\\alphas}'$ differs from that in the standard QCD $K$~factor $K_{\\mathrm{NLO}_{\\mathrm{s}}}=\\sigma_{\\mathrm{NLO}_{\\mathrm{s}}}\/\\sigma_{\\mathrm{LO}}\\equiv 1+\\delta_{\\alpha_s}$ due to the use of different PDF sets in the Born contributions.\n  See Ref.~\\cite{Dittmaier:2014koa} for further discussion.\n} \n\\begin{align}\n  \\label{eq:def:deltas'}\n  \\delta^{\\pro\\times\\dec}_{\\alphas\\alpha} &\\equiv\n  \\frac{\\Delta\\sigma_{\\pro\\times\\dec}^{\\NNLO_{\\rs\\otimes\\rew}}}{\\sigma^\\LO} , &\n  \\delta_{\\alphas}' &\\equiv \n  \\frac{\\Delta\\sigma^{\\NLO_\\rs}}{\\sigma^\\LO} ,\n\\end{align}\nwe can cast the relative difference of our best prediction~\\eqref{eq:def:nnlo:if} and the \nproduct ansatz~\\eqref{eq:def:nnlo:naive:fact} into the following form,\n\\begin{align}\n\\label{eq:diff-naive}\n  \\frac{\\sigma^{\\NNLO_{\\rs\\otimes\\rew}}\n    -\\sigma_{\\text{naive fact}}^{\\NNLO_{\\rs\\otimes\\rew}}}\n  {\\sigma^\\LO}\n  &=\n  \\delta^{\\pro\\times\\dec}_{\\alphas\\alpha} - \\delta_{\\alphas}' \\delta_\\alpha ,\n\\end{align}\nwhere the LO prediction $\\sigma^\\LO$ in the denominators is evaluated with the LO PDFs.\nThe difference of the relative NNLO\ncorrection $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ and the naive\nproduct $\\delta_{\\alphas}' \\delta_\\alpha^{(\\dec)}$ therefore allows to\nassess the validity of a naive \nproduct ansatz.  As observed in\nSect.~\\ref{sec:calc-fact-nnlo-if}, most contributions to the\nfactorizable initial--final corrections take the reducible form of a\nproduct of two NLO corrections, with the exception of the double-real\nemission corrections which are defined with the full kinematics of the\n$2\\to 4$ phase space.\nNote that the double-real contributions are the only ones where the final-state leptons receive recoils from both QCD and photonic radiation, an effect that cannot be captured by naively multiplying NLO QCD and EW corrections. \nAny large deviations between\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ and\n$\\delta_{\\alphas}'\\delta_\\alpha^{(\\dec)}$ can therefore be attributed\nto this type of contribution. The difference of the naive product\ndefined in terms of $\\delta_\\alpha^{\\dec}$ and $\\delta_\\alpha$\nallows us to assess the  impact of the missing $\\order{\\alphas\\alpha}$ corrections beyond the initial--final\ncorrections considered in our calculation and therefore also provides\nan error estimate of the PA, and in particular of the omission of the\ncorrections of initial--initial type.\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIF+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIF+rec}\n  \\end{minipage}\n  \\caption{Relative factorizable corrections of \\order{\\alphas\\alpha} induced by initial-state QCD and final-state EW contributions to the transverse-mass~(left) and transverse-lepton-momentum~(right) distributions for \\PWp production at the LHC.\n  The naive products of the NLO correction factors $\\delta_{\\alphas}'$ and $\\delta_\\alpha$ are shown for comparison.}\n  \\label{fig:distNNLO-IF-Wp}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIF+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIF+rec}\n  \\end{minipage}\n  \\caption{Relative factorizable corrections of \\order{\\alphas\\alpha} induced by initial-state QCD and final-state EW contributions to the lepton-invariant-mass distribution~(left) and a transverse-lepton-momentum \ndistribution~(right) for \\PZ production at the LHC.\n  The naive products of the NLO correction factors $\\delta_{\\alphas}'$ and $\\delta_\\alpha$ are shown for comparison.}\n  \\label{fig:distNNLO-IF-Z}\n\\end{figure}\n\nFigure~\\ref{fig:distNNLO-IF-Wp} shows the numerical results for the\nrelative \\order{\\alphas\\alpha} initial--final factorizable corrections $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ to the transverse-mass ($M_{\\rT,\\Pgn\\Pl}$) and the\ntransverse-lepton-momentum ($p_{\\rT,\\Pl}$) distributions for \\PWp\nproduction at the LHC.  For \\PZ production,\nFigure~\\ref{fig:distNNLO-IF-Z} displays the results for the\nlepton-invariant-mass ($M_{\\Pl\\Pl}$) distribution and a\ntransverse-lepton-momentum ($p_{\\rT,\\Pl^+}$) distribution.  In both\nfigures, the upper\nplots show the results for bare muons, the lower panels correspond to\nthe corrections with photon recombination. \nIn Figs.~\\ref{fig:distNNLO-IF-Wp} and~\\ref{fig:distNNLO-IF-Z} we\nalso compare to the two different implementations of a naive\nproduct of correction factors discussed after\nEq.~\\eqref{eq:diff-naive}.  \n In the following, we\nmainly focus on the results for bare muons.  The respective results\nwith photon recombination display the same general features as those\nfor bare muons, but the relative corrections\nare reduced by approximately a factor of two.\nThis reduction is familiar from NLO EW results and is induced by the\ncancellation of the collinear singularities by restoring the level of\ninclusiveness required for the KLN theorem.\nOne observes that the NNLO\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ corrections are in general better\napproximated by the simple product ansatz for the case of bare muons\nthan for dressed leptons. This can\nbe understood from the fact that the dominant part of the corrections\nstem from the collinear logarithms $\\ln(m_{\\Pgm})$ which are known to\nfactorize.\n\n\nFor the $M_{\\rT,\\Pgn\\Pl}$ distribution for $\\PWp$ production (upper left plot in\nFig.~\\ref{fig:distNNLO-IF-Wp}), the mixed NNLO QCD--EW corrections\nfor bare muons are moderate and amount to approximately $\\SI{-1.7}{\\%}$ around the\nresonance, which is about an order of magnitude smaller than the NLO\nEW corrections.\\footnote{The structure observed in the correction\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ around\n$M_{\\rT,\\Pgn\\Pl}\\approx\\SI{62}{\\GeV}$ can be attributed to the\ninterplay of the kinematics of the double-real emission\ncorrections and the event selection. It arises close to the kinematic\nboundary $M_{\\rT,\\Pgn\\Pl}>\\SI{50}{\\GeV}$ for the back-to-back kinematics of the non-radiative process implied by the  cut\n$p_{\\rT,\\Plpm},E_{\\rT}^\\text{miss}>\\SI{25}{\\GeV}$.} \nBoth variants of the naive product provide a good approximation to the\nfull result in the region around and below the Jacobian peak, which is\ndominated by resonant $\\PW$ production.  For larger values of\n$M_{\\rT,\\Pgn\\Pl}$, the product $\\delta_{\\alphas}' \\delta_\\alpha $\nbased on the full NLO EW correction factor deviates from the other curves, which signals\nthe growing importance of effects beyond the PA. However, the\ndeviations amount to only few per-mille for\n$M_{\\rT,\\Pgn\\Pl}\\lesssim\\SI{90}{\\GeV}$.\nThe overall good agreement between the\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ corrections and both naive\nproducts can be attributed to  well-known insensitivity of the\nobservable $M_{\\rT,\\Pgn\\Pl}$ to initial-state radiation effects\nalready seen in the case of NLO corrections in Ref~\\cite{Dittmaier:2014qza}.\n \nFor the $p_{\\rT,\\Pl}$ distributions in the case of bare muons (upper right plots\nin Figs.~\\ref{fig:distNNLO-IF-Wp} and \\ref{fig:distNNLO-IF-Z},\nrespectively) we observe corrections that are small far below the\nJacobian peak, but which rise to about $15\\%$ ($20\\%$) on the Jacobian\npeak at $p_{\\rT,\\Pl}\\approx\\MV\/2$ for the case of the \\PWp boson (\\PZ\nboson) and then display a steep drop reaching almost $-50\\%$ at\n$p_{\\rT,\\Pl}=\\SI{50}{\\GeV}$.  This enhancement stems from the large\nQCD corrections above the Jacobian peak familiar from the NLO QCD\nresults (see e.g.\\ Fig.~8 in Ref.~\\cite{Dittmaier:2014qza}) where\nthe recoil due to real\nQCD radiation shifts events with resonant $\\PW$\/$\\PZ$ bosons above the\nJacobian peak.\nThe\nnaive product ansatz fails to provide a good description of the full\nresult $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ and deviates by\n5--10\\% at the Jacobian peak, where the PA is expected to be the most\naccurate.  This can be attributed to the strong influence of the\nrecoil induced by initial-state radiation on the transverse momentum,\nwhich implies a larger effect of the double-real emission corrections\non this distribution that are not captured correctly by the naive\nproducts.  The two versions of the naive products display larger\ndeviations than in the $M_{\\rT,\\Pgn\\Pl}$ distribution discussed above,\nwhich signals a larger impact of the missing \\order{\\alphas\\alpha}\ninitial--initial corrections.  However, these deviations should be\ninterpreted with care, \nsince a fixed-order\nprediction is not sufficient to describe this distribution around the\npeak region $p_{\\rT,\\Pl}\\approx\\MV\/2$, which corresponds to the\nkinematic onset for $V+\\text{jet}$ production and is known to\nrequire QCD resummation for a proper description.\n\nIn case of the $M_{\\Pl\\Pl}$\ndistribution for $\\PZ$ production (left-hand plots in\nFig.~\\ref{fig:distNNLO-IF-Z}),  corrections up to $10\\%$  are\nobserved below the resonance for the case of bare muons. This is consistent with the large EW\ncorrections at NLO in this region, which arise from final-state photon\nradiation that shifts the reconstructed value of the invariant\nlepton-pair mass away from the resonance to lower values.\nThe naive product approximates the\nfull initial--final corrections\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ reasonably well at the resonance\nitself ($M_{\\Pl\\Pl}=\\MZ$) and  above, but completely fails already a\nlittle below the resonance where the naive products do not even\nreproduce the sign of the full\n$\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ correction.\nThis deviation occurs although  the invariant-mass distribution is \nwidely unaffected by initial-state radiation effects. \nThe fact that we obtain almost identical corrections \nfrom the two versions of the product $\\delta_{\\alphas}'\\delta_\\alpha^\\dec$ \nand $\\delta_{\\alphas}'\\delta_\\alpha$ demonstrates the insensitivity of this\nobservable to photonic initial-state radiation.\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relNLOZ}\n  \\caption{}\n  \\label{fig:distNNLO-IFZ:NLO-breakdown}\n  \\end{subfigure}\n  \\hfill\n  \\begin{subfigure}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIFZ}\n  \\caption{}\n  \\label{fig:distNNLO-IFZ:more}\n  \\end{subfigure}\n  \\caption{(\\subref{fig:distNNLO-IFZ:NLO-breakdown})~The correction factors \nto Z-boson production (as shown in the upper left plot of Fig.\\ref{fig:distNNLO-IF-Z})\nentering the naive product ansatz broken down into its individual contributions with the QCD corrections further divided into the $\\Pq\\Paq$- and $\\Pq\\Pg$-induced contributions.\n  (\\subref{fig:distNNLO-IFZ:more})~A comparison of the corrections shown in the upper left panel of Fig.~\\ref{fig:distNNLO-IF-Z} with the modified product using the value of the QCD corrections at the resonance $\\delta_{\\alphas}'(M_{\\Pl\\Pl}=\\MZ)\\approx 6.5\\%$.}\n  \\label{fig:distNNLO-IFZ}\n\\end{figure}\n\nIn order to locate the source \nof this large discrepancy we examine\nthe individual correction factors in the naive product in more detail.\nWe restrict ourselves to the case of bare muons and  the full NLO EW\ncorrection factor $\\delta_\\alpha$ for definiteness, which does not\naffect our conclusions.\nIn Fig.~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:NLO-breakdown}),\n we separately\n plot the two correction factors that enter the naive \nproduct $\\delta_{\\alphas}'\\times\\delta_\\alpha$ and further divide the QCD corrections into the \n$\\Pq\\Paq$- and the $\\Pq\\Pg$-induced contributions.\nWe observe that the two different $\\Pq\\Paq$- and $\\Pq\\Pg$-induced channels individually receive large QCD corrections, however, they differ in sign, \nso that large cancellations take place in the sum $\\delta_{\\alphas}'$.\nA small mismatch in the corrections of the individual channels can therefore quickly lead to a large effect in the QCD corrections which is then further enhanced by the large EW corrections in the product ansatz $\\delta_{\\alphas}' \\times \\delta_{\\alpha}$.\nMoreover,\nFigure~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:NLO-breakdown})\nreveals that the QCD correction factor $\\delta_{\\alphas}'$ is\nresponsible for the sign change at $M_{\\Pl\\Pl}\\approx\\SI{83}{\\GeV}$\nwhich is the most striking  disagreement of the naive product ansatz\nwith the full factorizable initial--final corrections.\nThis zero crossing happens more than three widths below the resonance where the cross section is reduced by almost two orders of magnitudes compared to the resonance region and, furthermore, will be very sensitive to event selection cuts, since $\\delta_{\\alphas}'$ arises from the cancellation of two large corrections as we have seen above.\nIn Fig.~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:NLO-breakdown}) we\nobserve the large EW corrections below the resonance mentioned above, which \narise due to the redistribution of  events near the $\\PZ$\npole to lower lepton invariant masses by final-state photon radiation.\nThe similar form of the factorizable NNLO initial--final\ncorrections indicates that they mainly stem from an analogous mechanism.\nThis suggests that it is more appropriate to  replace\nthe QCD correction factor $\\delta_{\\alphas}'$ in the naive product by its value at the resonance\n$\\delta_{\\alphas}'(M_{\\Pl\\Pl}=\\MZ)\\approx 6.5\\%$, which corresponds to\nthe location of the events that are responsible for the bulk of the\nlarge EW corrections below the resonance. \nIn contrast, the naive product ansatz simply multiplies the\ncorrections locally on a bin-by-bin basis. This causes a mismatch in the correction factors and fails to account for the\nmigration of events due to FSR.\nThe comparison of the previous results and the modified product is shown in Fig.~\\ref{fig:distNNLO-IFZ}(\\subref{fig:distNNLO-IFZ:more}) and clearly shows an improvement despite its very crude construction.\n\nContrary to the lepton-invariant-mass distribution, the\ntransverse-mass distribution is dominated by events with resonant \\PW\nbosons even in the range below the Jacobian peak,\n$M_{\\rT,\\Pgn\\Pl}\\lesssim\\MW$, so it is less sensitive to the\nredistribution of events to lower $M_{\\rT,\\Pgn\\Pl}$.\nThis explains why the naive product can provide a good approximation\nof the full initial--final NNLO corrections.  It should be emphasized,\nhowever, that even in the case of the $M_{\\rT,\\Pgn\\Pl}$ distribution\nany event selection criteria that deplete events with resonant \\PW bosons\nbelow the Jacobian peak will result in increased sensitivity to the effects of\nFSR and can potentially lead to a failure of a naive\nproduct ansatz. \n\nIn conclusion, simple approximations in terms of products of\ncorrection factors have to be used with care and require a careful\ncase-by-case investigation of their validity.\n\n\\subsection{Leading-logarithmic approximation for final-state photon radiation}\n\\label{sec:FSR}\n\nAs is evident from Figs.~\\ref{fig:distNNLO-IF-Wp} and~\\ref{fig:distNNLO-IF-Z}, a naive \nproduct of QCD and EW correction factors~\\eqref{eq:def:nnlo:naive:fact} is not adequate to approximate the NNLO QCD--EW corrections for all observables.  \nA promising approach to a factorized approximation for the dominant initial--final corrections can be obtained by combining the full NLO QCD corrections to vector-boson production with the leading-logarithmic~(LL) approximation for the final-state corrections.\nThe benefit in this approximation lies in the fact that the interplay of the recoil effects from jet and photon emission is properly taken into account.\nOn the other hand, the logarithmic approximation neglects certain (non-universal) finite contributions, which are, however, suppressed with respect to the dominating radiation effects.\n\nIn the structure-function approach~\\cite{Kuraev:1985hb}, the\nleading-logarithmic approximation of the {pho\\-to\\-nic} decay\ncorrections is\ncombined with the NLO QCD corrections to the production by a convolution,\n\\begin{align}\n  \\label{eq:LL1FSR}\n  \\Delta\\sigma_{\\Pp\\Pp, \\llog\\FSR}^{\\NNLO_{\\rs\\otimes\\rew}} &=\n  %\n  \\int \\rd\\sigma^{\\mathrm{NLO}_\\rs}(p_1,p_2;k_1,k_2) \n  \\int^1_0 \\rd z_1 \\, \\int^1_0 \\rd z_2 \\,\n  \\;\\Theta_\\cut\\bigl(z_1k_1\\bigr)\n  \\;\\Theta_\\cut\\bigl(z_2k_2\\bigr)\n  \\nonumber\\\\* &\\quad\\times \n\\biggl[\\,\\Gamma_{\\Pl_1\\Pl_1}^{\\llog}(z_1,Q^2)\\Gamma_{\\Pl_2\\Pl_2}^{\\llog}(z_2,Q^2)\n-\\delta(1-z_1)\\,\\delta(1-z_2)\\biggr]\n  \\nonumber\\\\ &=\n  %\n  \\int \\rd\\sigma^{\\mathrm{NLO}_\\rs}(p_1,p_2;k_1,k_2) \\int^1_0 \\rd z_1 \\, \\int^1_0 \\rd z_2 \\,\n  \\;\\Theta_\\cut\\bigl(z_1k_1\\bigr)\n  \\;\\Theta_\\cut\\bigl(z_2k_2\\bigr)\n  \\nonumber\\\\*\n  &\\quad \\times\\biggl[\n \\delta(1-z_2)\\,\\Gamma_{\\Pl_1\\Pl_1}^{\\llog,1}(z_1,Q^2)\n  +\\delta(1-z_1)\\,\\Gamma_{\\Pl_2\\Pl_2}^{\\llog,1}(z_2,Q^2)\n  +\\order{\\alpha^2} \\biggr] \n  ,\n\\end{align}\nwhere  $\\rd\\sigma^{\\mathrm{NLO}_\\rs}$ includes the virtual and real QCD corrections.\nThe step function $\\Theta_{\\cut}(z_i k_i)$ is equal to 1 if the event\npasses the cut on the rescaled lepton momentum $z_i k_i$ and 0\notherwise.  The variables $z_i$ are the momentum fractions describing the\nrespective \nlepton energy loss by collinear photon emission. For the charged-current process only one of the convolutions is present.\nThe $\\order{\\alpha}$ contribution to the structure function $\\Gamma_{\\Pl\\Pl}^{\\llog}$ reads\n\\begin{eqnarray}\n\\label{eq:LLll}\n  \\Gamma_{\\Pl\\Pl}^{\\llog,1}(z,Q^2) &=&\n  Q_{\\Pl}^2\\,\\frac{\\beta_\\Pl}{4} \\left(\\frac{1+z^2}{1-z}\\right)_+ ,\n\\end{eqnarray}\nwhere the large mass logarithm appears in the variable\n\\begin{equation}\n\\label{eq:beta_l}\n\\beta_\\Pl = \\frac{2\\,\\alpha}{\\pi}\n\\left[\\ln\\biggl(\\frac{Q^2}{\\Ml^2}\\biggr)-1\\right]\n\\end{equation}\nand \n$Q_{\\Pl}$ denotes the relative electric charge of the lepton $\\ell$. \nIn order to be consistent in the comparison with our calculation as described in Sect.~\\ref{sec:input-cuts}, the electromagnetic coupling constant $\\alpha$ appearing in Eq.~\\eqref{eq:beta_l} is set to $\\alpha_{G_\\mu}$ and $\\alpha(0)$ for the charged-current and neutral-current processes, respectively.\nThe scale $Q$ is chosen as the gauge-boson mass,\n\\begin{equation}\n\\label{eq:FSR_scale}\nQ = M_{V} ,\n\\end{equation}\nand the scale-variation bands shown in the numeric results are obtained by varying the scale by a factor of two up and down from the central scale choice,\n\\begin{equation}\nQ = \\xi \\cdot M_V , \\qquad \\xi = \\tfrac{1}{2}, 1, 2 .\n\\label{eq:ll1fsr:variation}\n\\end{equation}\nSince the mass logarithms cancel in observables where photon emission collinear to the final-state charged leptons is treated fully inclusively, \nthe structure-function approach is only applicable to non-collinear-safe observables, i.e.\\ to the bare-muon case. \n\nIn contrast, in \nparton-shower approaches to photon radiation (see e.g.\\ Refs.~\\cite{Placzek:2003zg,CarloniCalame:2003ux,CarloniCalame:2005vc})\nthe photon momenta transverse to the lepton momentum are \ngenerated as well, following the differential factorization formula, so that\nthe method is also applicable to the case of collinear-safe observables,\ni.e.\\ to the dressed-lepton case. For this purpose, we have\nimplemented the combination of the exact NLO QCD prediction for\nvector-boson production with the simulation of final-state photon radiation\nusing PHOTOS~\\cite{Golonka:2005pn}. Since we are interested in\ncomparing  to the\n\\order{\\alphas\\alpha} corrections in our setup, we only generate a\nsingle photon emission using PHOTOS and use the same scheme for\n$\\alpha$ as described in Sect.~\\ref{sec:input-cuts}. \nDetails on the specific settings within the PHOTOS parton shower are given in Appendix~\\ref{app:photos}.\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIFCompGF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_mt2_relIFCompGF+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIFCompGF} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/W+_ptl+2_relIFCompGF+rec}\n  \\end{minipage}\n  \\caption{Comparison of the approximation obtained\n    from PHOTOS for the relative \\order{\\alphas\\alpha} initial-state\n    QCD and final-state EW corrections to our best prediction\n    $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ for the case of the\n    transverse-mass~(left) and transverse-lepton-momentum~(right)\n    distributions for \\PWp production at the LHC, as in\n    Fig.~\\ref{fig:distNNLO-IF-Wp}.  In the bare-muon case, the\n    result~\\eqref{eq:LL1FSR} of the structure-function approach is\n    also shown.  }\n  \\label{fig:distNNLO-IF-FSR-Wp}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIFComp} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_mll2_relIFComp+rec}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[b]{.48\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIFComp} \\\\\n  \\includegraphics[width=\\textwidth]{.\/images\/plots\/Z_ptl+2_relIFComp+rec}\n  \\end{minipage}\n  \\caption{Comparison of the approximation obtained from PHOTOS for the\n    relative \\order{\\alphas\\alpha} initial-state QCD and final-state\n    EW corrections to our best prediction\n    $\\delta^{\\pro\\times\\dec}_{\\alphas\\alpha}$ for the case of the\n    lepton-invariant-mass distribution~(left) and a\n    transverse-lepton-momentum distribution~(right) for \\PZ production at\n    the LHC, as in Fig.~\\ref{fig:distNNLO-IF-Z}. In the bare-muon case, the\n    result~\\eqref{eq:LL1FSR} of the structure-function approach is\n    also shown.}\n  \\label{fig:distNNLO-IF-FSR-Z}\n\\end{figure}\n\nIn Figs.~\\ref{fig:distNNLO-IF-FSR-Wp} and \\ref{fig:distNNLO-IF-FSR-Z}\nwe compare our best prediction~\\eqref{eq:def:nnlo:if} for the\nfactorizable initial--final \\order{\\alphas\\alpha} corrections to the\ncombination of NLO QCD corrections with the approximate FSR\nobtained from PHOTOS for the case of $\\PWp$ production and $\\PZ$\nproduction, respectively. For the bare-muon case also the result of\nthe structure-function approach according to Eq.~\\eqref{eq:LL1FSR} is\nshown. The combination of the NLO QCD corrections and \napproximate FSR leads to a clear improvement compared to the\nnaive product approximations investigated in\nSection~\\ref{sec:if-results}.  This is particularly apparent in the\nneutral-current process where the $M_{\\Pl\\Pl}$ distribution is correctly modelled\nby both FSR approximations, whereas the naive products shown in\nFigs.~\\ref{fig:distNNLO-IF-Z} and~\\ref{fig:distNNLO-IFZ} completely failed to\ndescribe this distribution. \n In the $M_{\\rT,\\Pgn\\Pl}$ spectrum of the charged-current\nprocess in Fig.~\\ref{fig:distNNLO-IF-FSR-Wp} one also finds good\nagreement of the different results below the Jacobian peak and an\nimprovement over the naive product approximations in\nFig.~\\ref{fig:distNNLO-IF-Wp}. \nThe description of the $p_{\\rT,\\Pl}$ distributions is also improved compared to the naive product approximations, but some differences remain in the charged-current process.\n\n\nIn spite of the good agreement of the two versions of incorporating final-state-radiation effects, the intrinsic uncertainty of the leading-logarithmic approximations should be kept in mind.\nFor the structure-function approach, this uncertainty is illustrated by the band width resulting from the variation~\\eqref{eq:ll1fsr:variation} of the QED scale $Q$.\nWe remark that the multi-photon corrections obtained by employing the\nun-expanded\nstructure-functions $\\Gamma_{\\Pl\\Pl}^{\\llog}(z,Q^2)$  in Eq.~\\eqref{eq:LL1FSR}\nlie well within the aforementioned scale bands, which shows that a\nproper matching to the full NLO EW calculation is needed to\nremove the dominant uncertainty of the LL approximation and to \n predict the higher-order effects reliably.\nFor PHOTOS the intrinsic uncertainty is not shown and not easy to\nquantify.\nThe good quality of the PHOTOS approximation results from the fact that the finite terms in the photon emission probability are specifically adapted to $\\PW\/\\PZ$-boson decays.\nThe level of agreement with our ``full prediction'', thus, cannot be taken over to other processes.\n\n\n\\subsection{Impact on the \\texorpdfstring{$\\PW$}{W}-boson mass extraction}\n\\label{sec:mw}\n\nIn order to estimate the effect of the \\order{\\alphas \\alpha}\ncorrections on the extraction of the $\\PW$-boson mass at the LHC we\nhave performed a $\\chi^2$ fit of the $M_{\\rT,\\nu \\ell}$ distribution.\nWe treat the $M_{\\rT,\\nu \\ell}$ spectra calculated in various\ntheoretical approximations for a reference mass $\\MW^\\OS= 80.385~\\GeV$\nas ``pseudo-data'' that we fit with \n ``templates'' calculated\nusing the LO predictions $\\sigma^0$ (with NLO PDFs)\nfor different values of $\\MW^\\OS$.\nSpecifically, we have generated results for $27$ transverse-mass bins in the\ninterval $M_{\\rT,\\nu \\ell}= [64,\\,91]~\\GeV$ in steps of $1~\\GeV$, varying the $\\PW$-boson mass in the interval $\\MW=[80.085,\\,80.785]~\\GeV$ with steps of $\\Delta \\MW=10~ \\MeV$ (steps of $\\Delta\n\\MW=5~\\MeV$ in the interval $\\MW=[80.285,\\,80.485]~\\GeV$).  \nUsing a linear interpolation between neighbouring $\\MW$ values, we obtain the integrated cross sections in the $i$th $M_{\\rT,\\nu \\ell}$ bin, $\\sigma_i^0(\\MW)$, as a continuous function of $\\MW$. \nThe best-fit value $\\MW^{\\mathrm{fit}, \\mathrm{th}}$ \nquantifying the impact of a higher-order correction\nin the theoretical cross section $\\sigma^{\\mathrm{th}}$ is\nthen obtained from the minimum of the function\n\\begin{equation}\n\\label{eq:chi2}\n  \\chi^2(\\MW^{\\mathrm{fit},\\mathrm{th}})=\\sum_{i}\n  \\frac{\\left[\\sigma_i^{\\mathrm{th}}(\\MW^\\OS)\n      -\\sigma_i^0(\\MW^{\\mathrm{fit},\\mathrm{th}})\\right]^2}{\n    2 \\Delta\\sigma_i^2}\\,,\n\\end{equation}\nwhere the sum over $i$ runs over the transverse-mass bins.\nHere $\\sigma_i^{\\mathrm{th}}$ and $\\sigma_i^0$ are the\nintegrated cross sections in the $i$-th bin, uniformly rescaled so\nthat the sum over all $27$ bins is identical for all considered cross\nsections.  We assume a statistical error of the pseudo-data and take\n$\\Delta\\sigma_i^2\\propto \\sigma_i^{\\mathrm{th}}$.  We have also\nperformed a two-parameter fit where the normalization of the templates\nis fitted simultaneously, leading to identical results.  Similarly,\nallowing the $\\PW$-boson width in the templates to float and fitting\n$\\MW$ and $\\GW$ simultaneously does not\nsignificantly affect the estimate of the effect of the\n\\order{\\alphas\\alpha} corrections on the \\MW measurement.\n\n \\begin{figure}\n   \\centering\n   \\includegraphics[width=.48\\linewidth]{images\/mwfit\/plot_dchi2_nlo_int}\n   \\includegraphics[width=.48\\linewidth]{images\/mwfit\/plot_dchi2_nnlo_int}\n   \\caption{The $\\Delta\\chi^2=\\chi^2-\\chi^2_{\\mathrm{min}}$\n     distributions obtained from the fit of the NLO EW corrections\n     (left) and the NNLO production--decay corrections (right) to LO\n     templates in arbitrary units for $\\chi^2$. The NLO mass shift $\\Delta\\MW^{\\mathrm{NLO}_\\rew}$ is given relative to the reference mass  $\\MW^\\OS= 80.385~\\GeV$, the NNLO shift $\\Delta\\MW^{\\mathrm{NNLO}}$ is given relative to the output mass of the fit to the sum of the EW and QCD corrections as defined in Eq~\\eqref{eq:dmwnnlo}.}\n   \\label{fig:chi2}\n \\end{figure}\n\nIn the experimental measurements of the transverse-mass distribution,\nthe Jacobian peak is washed out due to the finite energy and momentum\nresolution of the detectors. In our simple estimate of the impact of\nhigher-order corrections on the extracted value of the $\\PW$-boson\nmass, we do not attempt to model such effects.  We expect the detector\neffects to affect the different theory predictions in a similar way\nand to cancel to a large extent in our estimated mass shift, which is\nobtained from a difference of mass values extracted from pseudo-data\ncalculated using different theory predictions.  This assumption is\nsupported by the fact that our estimate of the effect of the NLO EW\ncorrections is similar to the one obtained in\nRef.~\\cite{CarloniCalame:2003ux} using a Gaussian smearing of the\nfour-momenta to simulate detector effects.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{@{} l r@{\\enspace}l r@{\\enspace}l @{}}\n  \\toprule\n  & \\multicolumn{2}{c}{bare muons} & \\multicolumn{2}{c}{dressed leptons} \\\\\n  \\cmidrule(r){2-3} \\cmidrule(l){4-5}\n  & $M^\\mathrm{fit}_\\mathrm{W}\\;[\\mathrm{GeV}]$ & $\\qquad\\Delta M_\\mathrm{W}$ \n  & $M^\\mathrm{fit}_\\mathrm{W}\\;[\\mathrm{GeV}]$ & $\\qquad\\Delta M_\\mathrm{W}$ \\\\\n  \\midrule\n  LO          & $80.385$ & \\multirow{2}*{$\\bigg\\}\\;-90~\\mathrm{MeV}$} \n              & $80.385$ & \\multirow{2}*{$\\bigg\\}\\;-40~\\mathrm{MeV}$} \\\\\n         $\\NLO_\\rew$ & $80.295$ &                   \n              & $80.345$ &                   \\\\\n  \\midrule\n   $\\NLO_{\\rs\\oplus\\rew}$\n        & $80.374$ & \\multirow{2}*{$\\biggr\\}\\;-14~\\mathrm{MeV}$} \n              & $80.417$ & \\multirow{2}*{$\\biggr\\}\\;-4~\\mathrm{MeV}$} \\\\\n  NNLO        & $80.360$ &                   \n              & $80.413$ &                   \\\\\n  \\bottomrule\n\\end{tabular}\n\\caption{Values of the $\\PW$-boson Mass in  \\GeV obtained from the $\\chi^2$ fit\nof the  $M_{\\rT,\\nu \\ell}$ distribution in  different theoretical\napproximations  to LO templates and the resulting mass shifts.}\n  \\label{tab:mass_shift}\n\\end{table}\n\nThe  fit results for several NLO approximations and our best NNLO prediction~\\eqref{eq:def:nnlo:if} are given in Table~\\ref{tab:mass_shift}.\nTo validate our procedure we estimate the mass shift due to the NLO EW\ncorrections by using the prediction\n$\\sigma^{\\NLO_\\rew}=\\sigma^0+\\Delta\n\\sigma^{\\NLO_\\rew}$\n as the pseudo-data $\\sigma^{\\mathrm{th}}$\nin~\\eqref{eq:chi2}. The $\\chi^2$ distribution is shown on the left-hand side of\nFig.~\\ref{fig:chi2} as a function of the mass shift\n$\\Delta\\MW^{\\NLO_\\rew}$ for the dressed-lepton and bare-muon cases.\nFrom the minima of the distributions one finds a mass shift of\n$\\Delta \\MW^{\\NLO_\\rew}\\approx -90~\\MeV$ for bare muons and $\\Delta\n\\MW^{\\NLO_\\rew}\\approx -40~\\MeV$ for dressed muons.  These values are\ncomparable to previous results reported in \nRef.~\\cite{CarloniCalame:2003ux}.\\footnote{\n  In Ref.~\\cite{CarloniCalame:2003ux} the values $\\Delta\\MW=110~\\MeV$\n  ($20~\\MeV$) are obtained for the bare-muon (dressed-lepton)\n  case. These values are obtained using the\n  $\\order{\\alpha}$-truncation of a LL shower and for\n  lepton-identification criteria appropriate for the Tevatron taken\n  from Ref.~\\cite{Baur:1998kt}, so they cannot be compared directly to\n  our results. In particular, in the dressed-lepton case, a looser\n  recombination criterion $R_{\\Plpm\\Pgg}< 0.2$ is applied, which is\n  consistent with a smaller impact of the EW corrections.  Note that\n  the role of pseudo-data and templates is reversed in\n  Ref.~\\cite{CarloniCalame:2003ux} so that the mass shift has the\n  opposite sign.}  \nAlternatively, the effect of the EW corrections can\nbe estimated by comparing the value of $\\MW$ obtained from a fit to\nthe naive product of EW and QCD\ncorrections~\\eqref{eq:def:nnlo:naive:fact} to the result of a fit to\nthe NLO QCD cross section. The results are consistent with the shift\nestimated from the NLO EW corrections alone. \n\nWe have also estimated the effect of multi-photon radiation on the\n$\\MW$ measurement in the bare-muon case using the\nstructure-function approach given in  Eq.~\\eqref{eq:LL1FSR}. As\ndiscussed in detail in Ref.~\\cite{Brensing:2007qm} we match the exponentiated LL-FSR\ncorrections evaluated in the  $\\alpha(0)$-scheme to\nthe NLO calculation in the $\\alpha_{G_\\mu}$-scheme, avoiding double-counting. \nWe obtain a mass shift $\\Delta\\MW^{\\mathrm{FSR}}\\approx 9~\\MeV$\nrelative to the result of the fit to the NLO EW prediction, which\nis in qualitative agreement with the result of\nRef.~\\cite{CarloniCalame:2003ux}.\n\nTo estimate the impact of the  initial--final \\order{\\alphas\\alpha}\ncorrections  we consider the mass shift relative to the full NLO result,\n\\begin{equation}\n  \\label{eq:dmwnnlo}\n  \\Delta\\MW^{\\NNLO}=\\MW^{\\mathrm{fit},\\NNLO_{\\rs\\otimes\\rew}^{\\pro\\times\\dec}}\n  -\\MW^{\\mathrm{fit},\\NLO_{\\rs\\oplus\\rew}}\n\\end{equation}\nwhere $\\MW^{\\mathrm{fit},\\NNLO_{\\rs\\otimes\\rew}^{\\pro\\times\\dec}}$ is the\nresult of using our best\nprediction~\\eqref{eq:def:nnlo:if} to generate the pseudo-data, while\nthe sum of the NLO QCD and EW corrections is used for \n$\\Delta\\MW^{\\mathrm{fit},\\NLO_{\\rs\\oplus\\rew}}$. The resulting $\\Delta\\chi^2$\ndistributions for the mass shift are shown in the right-hand plot in\nFig.~\\ref{fig:chi2}.  \nIn the bare-muon case, we obtain\na mass shift due to \\order{\\alphas\\alpha} corrections of $\\Delta\n\\MW^{\\mathrm{NNLO}}\\approx -14~\\MeV$ while for the dressed-lepton case we\nget $\\Delta \\MW^{\\mathrm{NNLO}}\\approx -4~\\MeV$.  \n\nIdentical shifts result from replacing the NNLO prediction by the\nnaive product~\\eqref{eq:def:nnlo:naive:fact}, which is expected from\nthe good agreement for the $M_{\\rT,\\nu \\ell}$-spectrum in\nFig.~\\ref{fig:distNNLO-IF-Wp}.  Using instead the leading-logarithmic\napproximation of the final-state photon radiation obtained using\nPHOTOS to compute the \\order{\\alphas\\alpha} corrections, we obtain a\nmass shift of $\\Delta\\MW^{\\NNLO}=-11~\\MeV$ ($-4~\\MeV$) for the\nbare-muon (dressed-lepton) case.  The effect of the\n\\order{\\alphas\\alpha} corrections on the mass measurement is therefore\nof a similar or larger magnitude than the effect of multi-photon\nradiation.\n  We emphasize that \nthe result $\\Delta\\MW^{\\NNLO}\\approx -14~\\MeV$ is a simple estimate of the\n impact of the full \\order{\\alphas\\alpha} corrections\non the $\\MW$ measurement.  The order of magnitude shows that these\ncorrections must \nbe taken into account properly in order to reach the $10~\\MeV$ accuracy\ngoal of the LHC experiments. \nIt is beyond the scope of this paper to\nvalidate the accuracy of the \nprevious and current theoretical modelling used by the\nexperimental collaborations in the $\\MW$ measurements, which includes\nthe \\order{\\alphas\\alpha} corrections in some approximation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection*{Singularities of energy minimizing harmonic maps}\n\nEnergy minimizing harmonic maps between manifolds may have singularities if the domain dimension is $3$ or higher. The most well-known example is the map \n\\[\n{\\ensuremath{\\mathbb R}}^3 \\times {\\ensuremath{\\mathbb R}}^{n-3} \\ni (x,y) \\mapsto x\/|x| \\in {\\ensuremath{\\mathbb S}}^2. \n\\]\nIn general, any energy minimizer $u$ is smooth outside the closed singular set $\\operatorname{sing} u$ of Hausdorff dimension $n-3$ or less, $n$ being the dimension of the domain (Schoen, Uhlenbeck \\cite{SchUhl82,SchUhl83cor}). The phenomenon of singularities is now well-understood in dimension $3$, when singularities form a discrete set. In recent years, there has been a~substantial progress concerning the case $n \\ge 4$. Naber and Valtorta \\cite{NabVal17} have proved that the singular set has locally finite $(n-3)$-dimensional Hausdorff measure and is $(n-3)$-rectifiable, i.e., can be essentially covered by countably many Lipschitz images of ${\\ensuremath{\\mathbb R}}^{n-3}$; the latter was already known (due to Simon \\cite{Sim95}) in the case when the target manifold is real-analytic. \n\nThe results cited above are mostly concerned with the size of the singular set, but do not imply \\textit{lower bounds} on the singular set. In particular, the possibility that the singular set is an arbitrary subset of an $(n-3)$-dimensional manifold (with many small gaps) is not excluded by \\cite{NabVal17,SchUhl82,SchUhl83cor,Sim95}. \n\nLower bounds on the size are indeed possible in the presence of a~topological obstruction; the following example is simple but instructive. \n\n\\begin{ex}\nConsider the smooth boundary map $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\times {\\ensuremath{\\mathbb S}}^1 \\to {\\ensuremath{\\mathbb S}}^2$ given by $\\varphi(x,y) = x$ and (some) $u \\colon \\B^3 \\times {\\ensuremath{\\mathbb S}}^1 \\to {\\ensuremath{\\mathbb S}}^2$ minimizing the energy in the class of maps equal to $\\varphi$ on the boundary. Restricting $u$ to a slice $\\B^3 \\times \\{ y \\}$ and applying Brouwer's theorem, we see that each such slice contains a singular point. This shows that $\\mathcal{H}^1(\\operatorname{sing} u) \\ge \\mathcal{H}^1({\\ensuremath{\\mathbb S}}^1) = 2 \\pi$. In this particular case one can actually prove that $u(x,y)=x\/|x|$, but the presented reasoning applies also to any $\\varphi$ in the same homotopy class. \n\\end{ex}\n\nIn the special case of maps $u \\colon \\B^4 \\to {\\ensuremath{\\mathbb S}}^2$, Hardt and Lin \\cite{HarLin90} obtained the following remarkable result. \n\\begin{thm}\n\\label{thm:HarLin90}\nThe singular set of an energy minimizer $u \\colon \\B^4 \\to {\\ensuremath{\\mathbb S}}^2$ is locally a~union of a~finite set and a~finite family of H{\\\"o}lder continuous closed curves with a~finite number of crossings. \n\\end{thm}\nThe same claim was obtained also for maps $u \\colon \\B^5 \\to {\\ensuremath{\\mathbb S}}^3$ (Lin-Wang \\cite{LinWan06}). To the author's knowledge, these are the only two cases where $\\operatorname{sing} u$ was shown to be essentially a~manifold.  \n\nThe above theorem relies on the classification of tangent maps from ${\\ensuremath{\\mathbb R}}^3$ into ${\\ensuremath{\\mathbb S}}^2$ carried out by Brezis, Coron and Lieb \\cite{BreCorLie86}; for ${\\ensuremath{\\mathbb S}}^3$, a similar classification was obtained by Nakajima \\cite{Nak06}. These maps describe the infinitesimal behavior of $u$ at a typical point of $\\operatorname{sing} u$. \n\n\\subsection*{Main results}\n\nThe present paper aims to extract the topological obstruction responsible for preventing gaps in the singular set of maps into ${\\ensuremath{\\mathbb S}}^2$. To this end, we distinguish particular homotopy classes of tangent maps ${\\ensuremath{\\mathbb R}}^3 \\to {\\ensuremath{\\mathcal N}}$ (called here \\textit{indecomposable classes}) for any closed Riemannian manifold ${\\ensuremath{\\mathcal N}}$. \n\nTo each homotopy class $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ we assign its lowest energy level $\\Theta(\\alpha)$ and call $\\alpha$ indecomposable if $\\Theta(\\alpha) < \\infty$ and $\\alpha$ cannot be represented as a sum of homotopy classes $\\alpha_j \\in \\pi_2({\\ensuremath{\\mathcal N}})$ with strictly smaller energy levels $\\Theta(\\alpha_j)$. We then restrict our attention to singularites with fixed topological type $\\alpha$ -- we define $\\operatorname{sing}_\\alpha u$ to be the set of points at which some tangent map of $u$ has type $\\alpha$. Rigorous definitions are given in Section~\\ref{ch:HCMs}. \n\nAnother goal is to generalize the result of Hardt and Lin \\cite{HarLin90} to higher dimensional domains. The difficulty lies in the fact that the singular set is stratified -- it decomposes into parts of different dimensions. For $u \\colon \\B^4 \\to {\\ensuremath{\\mathbb S}}^2$, there are only two strata: one is formed by H{\\\"o}lder continuous curves and the other by their crossing points and some isolated points. In the theorem below, we were only able to study the top-dimensional part $\\operatorname{sing}_* u$ of the singular set. Again, the necessary notions are introduced in Section \\ref{ch:tangent-maps}. \n\nFor simplicity, we only consider the standard Euclidean ball $\\B^n$ as the domain, but the results hold true for any manifold. This is due to the fact that we only consider the infinitesimal behavior of maps. A detailed explanation can be found in \\cite{NabVal17} and \\cite[Sec.~8]{Sim95}. \n\n\\begin{thm}\n\\label{thm:Holder-regularity}\nLet $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ be an energy minimizing map into a closed Riemannian manifold ${\\ensuremath{\\mathcal N}}$, $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ be an indecomposable homotopy class, and $\\Theta(\\alpha)$ be its lowest energy level. Then for each exponent $0 < \\gamma < 1$ there is $\\delta(\\gamma,n,\\alpha,{\\ensuremath{\\mathcal N}}) > 0$ such that the set \n\\[\n\\left \\{\nx \\in \\operatorname{sing}_\\alpha u : \n\\lim_{r \\to 0} r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2 < \\Theta(\\alpha) + \\delta\n\\right \\}\n\\]\nforms an open subset of $\\operatorname{sing} u$ and it is a topological $(n-3)$-dimensional manifold of H{\\\"o}lder class $C^{0,\\gamma}$. \n\\end{thm}\n\nIn the case when ${\\ensuremath{\\mathcal N}}$ is a real-analytic manifold, Simon \\cite[Lemma~4.3]{Sim95} showed that the set of possible energy densities $\\lim_{r \\to 0} r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2$ is discrete. This allows us to slightly strenghten the statement above. The same conclusion holds also if ${\\ensuremath{\\mathcal N}}$ satisfies the integrability assumption introduced in \\cite[Ch.~3.13]{Sim96}. \n\n\\begin{cor}\n\\label{cor:discrete-levels}\nIf $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ is an energy minimizing map into a real-analytic manifold ${\\ensuremath{\\mathcal N}}$ and $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ is an indecomposable homotopy class, then \n\\[\n\\left \\{\nx \\in \\operatorname{sing}_\\alpha u : \n\\lim_{r \\to 0} r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2 = \\Theta(\\alpha) \n\\right \\}\n\\]\nforms an open subset of $\\operatorname{sing} u$ and it is a topological $(n-3)$-dimensional manifold of H{\\\"o}lder class $C^{0,\\gamma}$ with any $0 < \\gamma < 1$. \n\\end{cor}\n\nSpecializing to the case ${\\ensuremath{\\mathcal N}} = {\\ensuremath{\\mathbb S}}^2$ and recalling the classification of tangent maps \\cite{BreCorLie86}, we obtain a partial generalization of Theorem \\ref{thm:HarLin90} \\cite{HarLin90} to arbitrary dimensions: \n\n\\begin{cor}\n\\label{cor:sphere-target}\nIf $u \\colon \\B^n \\to {\\ensuremath{\\mathbb S}}^2$ is an energy minimizing map, then the top-dimensional part $\\operatorname{sing}_* u$ forms an open subset of $\\operatorname{sing} u$ and it is a topological $(n-3)$-dimensional manifold of H{\\\"o}lder class $C^{0,\\gamma}$ with any $0 < \\gamma < 1$. \n\\end{cor}\n\n\\subsection*{An outline}\n\nSection \\ref{ch:preliminaries} recalls all notions and results needed in the sequel. These mostly come from the seminal work of Schoen and Uhlenbeck \\cite{SchUhl82}, but the presentation here follows Simon's lecture notes \\cite{Sim96}. We also define indecomposable homotopy classes of maps from ${\\ensuremath{\\mathbb S}}^2$ into ${\\ensuremath{\\mathcal N}}$.\n\nSince Theorem \\ref{thm:Holder-regularity} only concerns the singularities of indecomposable types, it is worthwhile to investigate the existence of such classes, which we do in Section \\ref{ch:good-classes-exist}. Indeed, we show that for any ${\\ensuremath{\\mathcal N}}$ the second homotopy group $\\pi_2({\\ensuremath{\\mathcal N}},p)$ is generated (up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$ on $\\pi_2({\\ensuremath{\\mathcal N}},p)$) by indecomposable homotopy classes. This is very close to the classical (slightly weaker) result due to Sacks and Uhlenbeck \\cite{SacUhl81} (see also \\cite{Str85}): smooth harmonic maps from ${\\ensuremath{\\mathbb S}}^2$ into ${\\ensuremath{\\mathcal N}}$ generate the whole group $\\pi_2({\\ensuremath{\\mathcal N}},p)$ (up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$). \n\nTo obtain bi-H{\\\"o}lder-equivalence with a Euclidean ball, we employ Reifenberg's topological disc theorem \\cite{Rei60} (see also~\\cite{Sim-notes}). We recall its statement and the so-called Reifenberg flatness condition in Section \\ref{ch:flatness}. We also introduce a flatness condition for an energy minimizer $u$ which includes Reifenberg flatness for $\\operatorname{sing} u$, but also forces $u$ to be close to a tangent map. \n\nThe main results are proved in Section \\ref{ch:proof}. The difficulty in applying Reifenberg's theorem to $\\operatorname{sing} u$ lies in showing that this set has no gaps. This is done in Lemma \\ref{lem:no-drop}; this is also the point where our topological assumptions play a role. Then we are able to show that if $u$ satisfies the flatness condition on the ball $\\B_2(0)$, it also satisfies the same condition on each smaller ball $\\B_r(0)$ (Corollary \\ref{cor:all-scales}) and on each ball $\\B_1(z)$ centered at a point $z \\in \\B_1$ with enough energy density (Proposition \\ref{prop:all-balls}). Combining these results, we check the hypotheses of Reifenberg's theorem and establish Theorem \\ref{thm:Holder-regularity}.\n\nSome interesting observations not needed for the proof of Theorem \\ref{thm:Holder-regularity} are gathered in Section \\ref{ch:additional}. \n\n\\section{Preliminaries}\n\\label{ch:preliminaries}\n\n\\subsection*{Regularity of energy minimizers}\n\nIn what follows, $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ is an energy minimizing map into a closed Riemannian manifold ${\\ensuremath{\\mathcal N}}$. Here we recall the basic properties of such maps \\cite{SchUhl82}. \n\nDenote the rescaled energy \n\\[\n\\theta_u(x,r) := r^{2-n} \\int_{\\B_r(x)} |\\nabla u|^2 \n\\quad \\text{for } \\B_r(x) \\subseteq \\B_1, \n\\]\nwhich is monotone in $r$: \n\\[\n\\tfrac{\\partial}{\\partial r} \\theta_u(x,r) = 2 \\int_{\\partial \\B_r(x)} \\frac{|\\nabla u \\cdot (y-x)|^2}{|y-x|^n} \\ge 0.\n\\]\nThis enables us to define the energy density at $x$: \n\\[\n\\theta_u(x,0) := \\lim_{r \\to 0} \\theta_u(x,r),\n\\]\nwhich is by definition an upper semicontinuous function (in both $x \\in \\B^n$ and $u \\in W^{1,2}$) \\cite[2.11]{Sim96}. Obviously, $\\theta_u(x,0)=0$ at regular points. \n\nThe main regularity statement is the following $\\varepsilon$-regularity theorem: \n\\begin{align}\n\\label{eq:eps-regularity}\n\\text{there is } \\varepsilon(n,{\\ensuremath{\\mathcal N}})>0 \\text{ s.t. } \\theta_u(x,2r) < \\varepsilon \n& \\Rightarrow u \\text{ is smooth on } \\B_r(x), \\\\\n\\nonumber\n\\text{in particular } \\theta_u(x,0) < \\varepsilon & \\Rightarrow x \\notin \\operatorname{sing} u.\n\\end{align}\n\nWe also note two compactness theorems for a sequence $u_k$ of energy minimizers: \n\\begin{itemize}\n\\item\nif $u_k \\rightharpoonup u$ in $H^1$, then $u$ is an energy minimizer and the convergence is actually strong in $H^1$ \\cite{Luc88}, \n\\item\nif $u_k \\to u$ as above, then the convergence is uniform on compact sets disjoint from $\\operatorname{sing} u$ \\cite{SchUhl82}. \n\\end{itemize}\n\n\\subsection*{Tangent maps}\n\\label{ch:tangent-maps}\n\nGiven an energy minimizer $u \\colon \\B^n \\to {\\ensuremath{\\mathcal N}}$ and a point $x \\in \\B^n$, consider the family of rescaled maps $u_r(y) = u(x+ry)$. By the results from the previous section, each sequence $r_j \\to 0$ has a subsequence for which $u_{r_j}$ converges in $W^{1,2}_\\mathrm{loc}({\\ensuremath{\\mathbb R}}^n)$ to some energy minimizer $\\varphi$, called a tangent map of $u$ at $x$ (possibly non-unique). This limit map is homogeneous, i.e., $\\varphi(\\lambda x) = \\varphi(x)$ for all $\\lambda > 0$, $x \\in {\\ensuremath{\\mathbb R}}^n$. \n\nFor a homogeneous energy minimizer $\\varphi \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$, the energy density $\\theta_\\varphi(y,0)$ is maximal at $y=0$; moreover, equality $\\theta_\\varphi(y) = \\theta_\\varphi(0)$ at some other point $y$ leads to higher symmetry: $\\varphi(x+ty) = \\varphi(x)$ for all $t \\in {\\ensuremath{\\mathbb R}}$, $x \\in {\\ensuremath{\\mathbb R}}^n$. Let $S(\\varphi)$ be defined by \n\\[\nS(\\varphi) = \\left\\{ y \\in {\\ensuremath{\\mathbb R}}^n : \\theta_\\varphi(y) = \\theta_\\varphi(0) \\right\\}. \n\\]\nThen $S(\\varphi)$ is a linear subspace of ${\\ensuremath{\\mathbb R}}^n$ describing the symmetries of $\\varphi$: \n\\[\n\\varphi(x+y) = \\varphi(x) \\quad \\text{for all } x \\in {\\ensuremath{\\mathbb R}}^n, \\ y \\in S(\\varphi). \n\\]\nFor non-constant $\\varphi$, we have $S(\\varphi) \\subseteq \\operatorname{sing} \\varphi$. If $\\dim S(\\varphi) = n-3$, this is necessarily an equality. \n\n\\subsection*{Top-dimensional part of the singular set}\n\\label{ch:HCMs}\n\nIf $u$ is an energy minimizer, for each $j = 0,1,2,\\ldots,n-1$ we define \n\\[\nS_j = \\left\\{ y \\in \\operatorname{sing} u : \\dim S(\\varphi) \\le j \\text{ for all tangent maps $\\varphi$ of $u$ at $y$} \\right\\},\n\\]\nwhich leads to the classical stratification of the singular set \n\\[\nS_0 \\subseteq S_1 \\subseteq \\ldots \\subseteq S_{n-3} = S_{n-2} = S_{n-1} = \\operatorname{sing} u.\n\\]\nIt is known \\cite{SchUhl82} that each $S_j$ has Hausdorff dimension at most $j$, in particular \n\\[\n\\dim_H \\operatorname{sing} u \\le n-3.\n\\] \n\n\\medskip\n\nIn what follows, we are interested in the top-dimensional part of the singular set: \n\\begin{align*}\n\\operatorname{sing}_* u & = S_{n-3} \\setminus S_{n-4} \\\\\n& = \\left\\{ y \\in \\operatorname{sing} u : \\dim S(\\varphi) = n-3 \\text{ for some tangent map $\\varphi$ of $u$ at $y$} \\right\\}. \n\\end{align*}\nNote that \n\\[\n\\dim_H (\\operatorname{sing} u \\setminus \\operatorname{sing}_* u) \\le n-4. \n\\]\nFollowing \\cite{Sim96}, we shall call any homogeneous energy minimizing $\\varphi \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $\\dim S(\\varphi) = n-3$ a \\textit{homogeneous cylindrical map} (abbreviated HCM). \n\n\\medskip\n\nConsider now such a HCM $\\varphi_0 \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $S(\\varphi_0) = {\\ensuremath{\\mathbb R}}^{n-3} \\times \\mathbf{0}$. This map actually depends only on $3$ variables, i.e. $\\varphi_0(x,y) = \\varphi_1(y)$ for some homogeneous $\\varphi_1 \\colon {\\ensuremath{\\mathbb R}}^3 \\to {\\ensuremath{\\mathcal N}}$. By \\cite[Lemma~2.1]{HarLin90}, the map $\\varphi_1$ defined in this way is energy minimizing if and only if $\\varphi_0$ is. Since $\\varphi_1$ is homogeneous, it is uniquely determined by its restriction to the unit sphere $\\varphi_2 \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$, which is a smooth harmonic map. \n\nFrom now on, we shall abuse the notation and use the same symbol for all three maps $\\varphi_0,\\varphi_1,\\varphi_2$; the precise meaning should be clear from the context. Note that their energies differ by a multiplicative constant: \n\\[\n\\int_{{\\ensuremath{\\mathbb S}}^2} |\\nabla \\varphi_2|^2 = \\int_{\\B_1^3} |\\nabla \\varphi_1|^2 = C(n) \\int_{\\B_1^n} |\\nabla \\varphi_0|^2, \n\\]\nso energy comparison does not lead to confusion. \n\n\\textit{Homotopy type} of a HCM always refers to the map $\\varphi_2 \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ (as $\\varphi_0,\\varphi_1$ are discontinuous and defined on contractible domains). For a general HCM $\\varphi_0$ we may choose a rotation $q$ that maps $S(\\varphi_0)$ to ${\\ensuremath{\\mathbb R}}^{n-3} \\times \\mathbf{0}$ and thus reduce to the previous case. We then say that $\\varphi_0$ has homotopy type $\\alpha$ if $\\varphi_0 \\circ q^{-1}$ restricted to $\\mathbf{0} \\times {\\ensuremath{\\mathbb S}}^2$ has type $\\alpha$. \n\n\\begin{rem}\nThere is a subtle ambiguity here. Depending on the choice of $q$, we may obtain two homotopy types that differ by a composition with the antipodal map, i.e. both $[\\varphi_2(x)]$ and $[\\varphi_2(-x)]$. \n\\end{rem}\n\nUsing this terminology, singular points in $\\operatorname{sing}_* u$ can be classified according to their energy density and the homotopy type of a tangent map. Since we only consider basepoint-free homotopies, we denote by $\\pi_2({\\ensuremath{\\mathcal N}})$ the set of homotopy classes of continuous maps ${\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$. Note that in general it does not carry a group structure, as it is the quotient of the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$ on $\\pi_2({\\ensuremath{\\mathcal N}},p)$. \n\\begin{df}\n\\label{df:notation}\nFor any homotopy type $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ we let \n\\[\n\\operatorname{sing}_{\\alpha} u = \n\\left\\{ y \\in \\operatorname{sing} u : \\text{ some tangent map of $u$ at $y$ is a HCM of type } \\alpha \\right\\}. \n\\]\nWe also denote its lowest energy level by \n\\[\n\\Theta(\\alpha) := \\inf \\left \\{ \\int_{\\B_1^n} |\\nabla \\varphi|^2 : \\varphi \\text{ is a HCM of type } \\alpha \\right \\}.\n\\]\nA simple compactness argument shows that this infimum is either infinite (if no HCM has type $\\alpha$) or achieved by some minimal HCM. We also let \n\\[\n\\operatorname{sing}_{\\ge \\Theta} u = \n\\left\\{ y \\operatorname{sing} u : \\theta_u(y,0) \\ge \\Theta \\right\\}. \n\\]\nwhich is a closed set by upper semicontinuity of $\\theta_u(\\cdot,0)$. \n\\end{df}\n\nAt this point we cannot exclude the case when there are many homotopically different tangent maps at one point. However, this cannot happen under an additional assumption described below (see Remark \\ref{rem:unique-type}). Again, since $\\pi_2({\\ensuremath{\\mathcal N}})$ is not necessarily a group, the decomposition in Definition \\ref{df:decomposition} is to be understood up to the action of $\\pi_1({\\ensuremath{\\mathcal N}})$, as described in Section \\ref{ch:good-classes-exist} (see also the formulation of \\cite[Thm.~5.9]{SacUhl81}). \n\n\\begin{df}\n\\label{df:decomposition}\nConsider $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ with $\\Theta(\\alpha) < \\infty$. This homotopy class is called decomposable if there is a decomposition \n\\[\n\\alpha = \\alpha_1 + \\ldots + \\alpha_k \\quad \\text{in } \\pi_2({\\ensuremath{\\mathcal N}}),\n\\]\nwhere $\\Theta(\\alpha_j) < \\Theta(\\alpha)$ for each $j=1,\\ldots,k$. Otherwise $\\alpha$ is called indecomposable. \n\\end{df}\n\nNote that the above criterion does not depend on the dimension $n$, but only on the manifold ${\\ensuremath{\\mathcal N}}$. \n\nAs a special case, $\\alpha$ is indecomposable if $\\Theta(\\alpha)$ is the smallest among all non-trivial homotopy types. In this case the proof is much easier (see the remark below Lemma \\ref{lem:no-drop}).\n\n\\medskip\n\nSimilar decompositions of this type appear naturally as a result of the \\textit{bubbling phenomenon} when one tries to minimize the energy in a given homotopy class. More precisely, recall that by \\cite{SacUhl81} (see also \\cite{Str85}) any smooth map $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ can be decomposed as $[\\varphi] = [\\varphi_1] + \\ldots + [\\varphi_k]$, where each $\\varphi_j$ is a harmonic map and \n\\[\n\\sum_{j=1}^k \\int_{{\\ensuremath{\\mathbb S}}^2} |\\nabla \\varphi_j^2| \\le \\int_{{\\ensuremath{\\mathbb S}}^2} |\\nabla \\varphi^2|.\n\\]\nMotivated by these decompositions, one could replace the condition $\\Theta(\\alpha) > \\max_j \\Theta(\\alpha_j)$ in Definition \\ref{df:decomposition} by $\\Theta(\\alpha) \\ge \\sum_j \\Theta(\\alpha_j)$, thus enlarging the set of indecomposable classes. \nA natural conjecture here would be that Theorem \\ref{thm:Holder-regularity} continues to hold in this case, but the author was not able to verify it. \n\n\\begin{ex}\nBy the classification from \\cite{BreCorLie86}, the only HCMs into the sphere ${\\ensuremath{\\mathbb S}}^2$ are isometries $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathbb S}}^2$. Thus, for $\\alpha \\in \\pi_2({\\ensuremath{\\mathbb S}}^2)$ we have \n\\[\n\\Theta(\\alpha) = \n\\begin{cases}\n0 & \\text{ for } \\alpha = 0, \\\\\n4 \\pi & \\text{ for } \\alpha = [\\pm \\operatorname{id}], \\\\\n\\infty & \\text{ otherwise.}\n\\end{cases}\n\\]\nBy Definition \\ref{df:decomposition}, the indecomposable classes here are $0,[\\operatorname{id}],[-\\operatorname{id}]$. Note that these classes generate the whole group $\\pi_2({\\ensuremath{\\mathbb S}}^2)$ (see Proposition \\ref{prop:indecomposable-types} for the general case). \n\\end{ex}\n\n\\section{Existence of indecomposable homotopy classes}\n\\label{ch:good-classes-exist}\n\nWe show that the set of all indecomposable homotopy classes generates $\\pi_2({\\ensuremath{\\mathcal N}})$. Similarly to \\cite[Thm.~5.9]{SacUhl81}, we only consider basepoint-free homotopies, so this statement should be understood as generating $\\pi_2({\\ensuremath{\\mathcal N}},p)$ up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$. In other words, for any $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ there are indecomposable homotopy classes $\\alpha_1,\\ldots,\\alpha_k \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and a continuous map \n\\[\nu \\colon \\B^3 \\setminus \\bigcup_{j=1}^k \\B_j \\to {\\ensuremath{\\mathcal N}} \n\\]\nsuch that $u|_{\\partial \\B} \\in \\alpha$ and $u|_{\\partial \\B_j} \\in \\alpha_j$, where $\\B_j \\Subset \\B$ are smaller disjoint balls. \n\n\\medskip\n\nThis can be divided into two steps as follows. \n\n\\begin{prop}\n\\label{prop:indecomposable-types}\nLet ${\\ensuremath{\\mathcal N}}$ be a closed Riemannian manifold. Then \n\\begin{enumerate}[(a)]\n\\item\nthe set of all HCMs $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ generates $\\pi_2({\\ensuremath{\\mathcal N}})$, \n\\item\neach HCM $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ as an element of $\\pi_2({\\ensuremath{\\mathcal N}})$ can be decomposed into indecomposable homotopy classes. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nTo show part (a), fix $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and choose a smooth map $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ of this type. Then there exists (possibly non-unique) $u \\in W^{1,2}(\\B_1^3,{\\ensuremath{\\mathcal N}})$ such that \n\\[\n\\int_{\\B_1} |\\nabla u|^2 = \\min \\left \\{ \\int_{\\B_1} |\\nabla v|^2 : v \\in W^{1,2}(\\B_1,{\\ensuremath{\\mathcal N}}), \\ v = \\varphi \\text{ on } {\\ensuremath{\\mathbb S}}^2 \\right \\}.\n\\]\nSuch a minimizer has at most a~finite number of interior singularities $p_1,\\ldots,p_k \\in \\B_1$. At each $p_j$ there is a~(possibly non-unique) tangent map $\\varphi_j$, which is necessarily a~HCM; by uniform convergence away from the singularity, $u$ restricted to $\\partial \\B_r(p_j)$ is homotopic to $\\varphi_j$ for some arbitrary small $r$ (in consequence, also for all sufficiently small $r$). This yields the decomposition \n\\[\n[\\varphi] = [\\varphi_1] + \\ldots + [\\varphi_k] \\quad \\text{ in } \\pi_2({\\ensuremath{\\mathcal N}}). \n\\]\n\nPart (b) follows from the definition by a compactness argument, which allows us to exclude infinite decompositions. Consider any homotopy type $\\alpha \\in {\\ensuremath{\\mathcal N}}$ represented by a HCM, i.e. with $\\Theta(\\alpha) < \\infty$. First let us show that there are only finitely many homotopy types $\\beta \\in \\pi_2({\\ensuremath{\\mathcal N}})$ with $\\Theta(\\beta) \\le \\Theta(\\alpha)$. Indeed, otherwise we would have an infinite sequence of HCMs $\\varphi_k \\colon \\B_1^3 \\to {\\ensuremath{\\mathcal N}}$ with distinct homotopy types and uniformly bounded energy. Without loss of generality, $\\varphi_k$ converges to some HCM $\\varphi$ in $W^{1,2}(\\B_1)$, but also in $C^0({\\ensuremath{\\mathbb S}}^2)$. This shows that almost all $\\varphi_k$ have the same homotopy type as $\\varphi$, which is a contradiction. \n\nIf $\\alpha$ is decomposable, we have $\\alpha = \\alpha_1 + \\ldots + \\alpha_k$, where $\\Theta(\\alpha_j) < \\Theta(\\alpha)$ for each $j$. Decomposing further each $\\alpha_j$ whenever possible, and iterating this procedure until all obtained homotopy types are indecomposable, we arrive at claim (b). One only needs to note that this procedure stops after at most $N$ steps, where $N$ is the number of homotopy types from the last paragraph. Indeed, any branch of the decomposition tree is a sequence $\\beta_0, \\beta_1, \\beta_2, \\ldots$ with $\\beta_0 = \\alpha$ and $\\Theta(\\beta_{j+1}) < \\Theta(\\beta_j)$ for each $j$, so it contains at most $N$ elements. \n\\end{proof}\n\nWe remark that a similar decomposition was obtained by Sacks and Uhlenbeck \\cite{SacUhl81} (see also \\cite{Str85}): smooth harmonic maps from ${\\ensuremath{\\mathbb S}}^2$ into ${\\ensuremath{\\mathcal N}}$ generate the whole group $\\pi_2({\\ensuremath{\\mathcal N}},p)$ up to the action of $\\pi_1({\\ensuremath{\\mathcal N}},p)$. It may be that some homotopy classes in $\\pi_2({\\ensuremath{\\mathcal N}})$ do not contain any harmonic map. Since here we only consider harmonic maps $\\varphi \\colon {\\ensuremath{\\mathbb S}}^2 \\to {\\ensuremath{\\mathcal N}}$ for which the homogeneous extension $\\varphi \\colon \\B^3 \\to {\\ensuremath{\\mathcal N}}$ is energy minimizing, our result is a slight generalization. \n\n\\section{Notions of flatness and Reifenberg's topological disc theorem}\n\\label{ch:flatness}\n\nH{\\\"o}lder regularity of the singular set will be obtained by an application of Reifenberg's topological disc theorem \\cite{Rei60} (see also~\\cite{Sim-notes}). To state it, we first need the following notion of flatness (for our purposes restricted to codimension $3$). \n\n\\begin{df}\n\\label{df:set-flatness}\nA set $A \\subseteq {\\ensuremath{\\mathbb R}}^n$ is said to be $\\varepsilon$-Reifenberg flat in the ball $\\B_r(x)$ (with respect to $L$) if \n\\[\nA \\cap \\B_r(x) \\subseteq \\B_{r \\varepsilon} L \n\\quad \\text{and} \\quad\nL \\cap \\B_r(x) \\subseteq \\B_{r \\varepsilon} A \n\\]\nfor some $(n-3)$-dimensional affine plane $L$ through $x$.\n\\end{df}\n\nThe above condition means exactly that the normalized Hausdorff distance on $\\B_r(x)$ from $A$ to some $(n-3)$-dimensional affine plane through $x$ is not larger than~$\\varepsilon$. \n\n\\begin{thm}[Reifenberg's topological disc theorem]\n\\label{thm:Reifenberg}\nFor each H{\\\"o}lder exponent $0 < \\gamma < 1$ there is $\\varepsilon(n,\\gamma) > 0$ such that the following holds. If a closed set $A \\subseteq {\\ensuremath{\\mathbb R}}^n$ containing the origin is $\\varepsilon$-Reifenberg flat in each ball $\\B_r(x)$ with $x \\in A \\cap \\B_1$ and $r < 1$, then the set $A \\cap \\B_1$ is bi-H{\\\"o}lder equivalent to the closed unit ball $\\B^{n-3} \\subseteq {\\ensuremath{\\mathbb R}}^{n-3}$ with exponent $\\gamma$. \n\\end{thm}\n\n\\medskip\n\nWe shall also make repeated use of the following condition for energy minimizing maps. \n\n\\begin{df}\n\\label{df:flatness}\nFix an indecomposable homotopy class $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and let $\\Theta = \\Theta(\\alpha)$ be its lowest energy level (as in Definition \\ref{df:notation}). We say that an energy minimizer $u$ is $\\delta$-flat in the ball $\\B_r(x)$ (of type $\\alpha$) if \n\\begin{enumerate}\n\\item\n\\label{df:flatness-energy}\n$x$ is a singular point of $u$ and $\\Theta \\le \\theta_u(x,0) \\le \\theta_u(x,r) \\le \\Theta + \\delta$, \n\\item\n\\label{df:flatness-Reifenberg}\n$\\operatorname{sing} u$ is $\\frac{1}{10}$-Reifenberg flat in $\\B_r(x)$ with respect to some $L$, and $u$ restricted to $(x+L^\\perp) \\cap \\partial \\B_r(x)$ has homotopy type $\\alpha$. \n\\end{enumerate}\n\\end{df}\n\nFrom now on, we consider a non-trivial indecomposable class $\\alpha$ and its lowest energy level $\\Theta = \\Theta(\\alpha)$ to be fixed. \n\nThe main feature of this definition is that $\\delta$-flatness in a ball trivially ensures that condition \\ref{df:flatness-energy} is satisfied in all smaller concentric balls, and one only needs to check condition \\ref{df:flatness-Reifenberg} (see Corollary \\ref{cor:all-scales}). This is why the constant $\\tfrac{1}{10}$ in condition \\ref{df:flatness-Reifenberg} was chosen reasonably weak. \n\n\\section{Regularity of the singular set}\n\\label{ch:proof}\n\n\\subsection*{Stability of indecomposable singularities}\n\nAs a first step, we show that if an energy minimizer $u$ restricted to some sphere has homotopy type $\\alpha$, then $\\operatorname{sing} u$ satisfies the flatness condition of Definition \\ref{df:flatness} and the energy density of $u$ cannot drop in a smaller ball. Note that the claim of Lemma \\ref{lem:no-drop} is essentially stronger than the condition $L \\cap \\B_1 \\subseteq \\B_{\\varepsilon} (\\operatorname{sing}_{\\ge \\Theta} u)$ from Definition \\ref{df:set-flatness}. \n\n\\medskip\n\nObserve that some tubular neighborhood $\\B_{\\eta} {\\ensuremath{\\mathcal N}} \\subseteq {\\ensuremath{\\mathbb R}}^M$ admits a continuous retraction $\\pi_{\\ensuremath{\\mathcal N}}$ onto ${\\ensuremath{\\mathcal N}}$. As a consequence, if two continuous functions $f,g$ into ${\\ensuremath{\\mathcal N}} \\subseteq {\\ensuremath{\\mathbb R}}^M$ are close enough in supremum norm, then \n\\[\n(t,x) \\mapsto \\pi_{{\\ensuremath{\\mathcal N}}}(tf(x) + (1-t)g(x)) \n\\]\nyields a homotopy between them. \n\n\\begin{lem}\n\\label{lem:no-drop}\nAssume that $\\operatorname{sing} u \\cap \\B_1 \\subseteq \\B_{\\varepsilon} L$ for some $0 < \\varepsilon < \\tfrac 12$ and some $(n-3)$-dimensional plane $L$ through $0$. Assume further that $u$ restricted to $L^\\perp \\cap \\partial \\B_1$ has homotopy type $\\alpha$. Then \n\\[\nL \\cap \\B_{1-\\varepsilon} \\subseteq \\pi_L( \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1 ),\n\\]\nwhere $\\pi_L$ denotes the orthogonal projection onto $L$. In particular, $\\operatorname{sing}_{\\ge \\Theta} u$ is $\\varepsilon$-Reifenberg flat in $\\B_1$. \n\\end{lem}\n\nBefore giving the full proof, let us consider the special case when $\\Theta(\\alpha)$ is the lowest among all non-trivial homotopy types. In this case, the proof is simpler and does not depend on the deep results of Naber and Valtorta \\cite{NabVal17}.\n\nFor each $y \\in L \\cap \\B_{1-\\varepsilon}$, $u$ restricted to the sphere $(y + L^\\perp) \\cap \\partial \\B_\\varepsilon(y)$ has homotopy type $\\alpha$, hence cannot be continuously extended to the ball $(y + L^\\perp) \\cap \\B_\\varepsilon(y)$. This shows the weaker inclusion $L \\cap \\B_{1-\\varepsilon} \\subseteq \\pi_L( \\operatorname{sing} u \\cap \\B_1 )$. Recall that $\\mathcal{H}^{n-3}$-a.e. point $z \\in \\operatorname{sing} u$ belongs to $\\operatorname{sing}_* u$ and hence $\\theta_u(z,0) \\ge \\Theta$ due to our additional assumption. Since $\\operatorname{sing}_{\\ge \\Theta} u$ is a closed set, we obtain the stronger inclusion. \n\n\\begin{proof}[Proof of Lemma \\ref{lem:no-drop}]\nFor simplicity, let us rotate so that $L = {\\ensuremath{\\mathbb R}}^{n-3} \\times \\mathbf{0}$. Assume for the contrary that $L \\cap \\B_{1-\\varepsilon}$ is not covered by the projection of $\\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$. Since the latter is a compact set, it has to be disjoint with some cylinder $\\B^{n-3}_\\delta(z) \\times {\\ensuremath{\\mathbb R}}^3$. \n\nRecall that by the recent important work of Naber and Valtorta \\cite{NabVal17}, the set $\\operatorname{sing} u \\cap \\B_1$ has finite $\\mathcal{H}^{n-3}$ measure. Moreover, the set $\\operatorname{sing} u$ is $(n-3)$-rectifiable and for $\\mathcal{H}^{n-3}$-a.e. $y \\in \\operatorname{sing} u$ there exists an $(n-3)$-dimensional tangent plane $\\operatorname{Tan}(\\operatorname{sing} u,y)$ coinciding with $S(\\varphi)$ for every tangent map $\\varphi$ of $u$ at $y$. Let us temporarily assume that these tangent planes are transversal to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^{n-3}$, i.e. \n\\begin{equation}\n\\label{eq:transverse}\n\\operatorname{Tan}(\\operatorname{sing} u,y) \\pitchfork \\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3 \\quad \\text{for $\\mathcal{H}^{n-3}$-a.e. } y \\in \\operatorname{sing} u \\cap \\B_1. \n\\end{equation}\n\nWe shall need Eilenberg's inequality, a Fubini type inequality valid for any $\\mathcal{H}^{n-3}$-measurable set $A$ with finite measure (see \\cite[7.7,~7.8]{Mat95}): \n\\[\n\\int_{\\B^{n-3}_\\delta(z)} \\mathcal{H}^0(A \\cap \\pi_L^{-1}(y)) \\phantom{.}\\mathrm{d} y \\le \\omega_{n-3} \\mathcal{H}^{n-3}(A) \n\\]\nApplying the above inequality twice -- once with $A$ as the singular set and once with $A$ as its exceptional part of measure zero -- we learn that for a.e. $y \\in \\B^{n-3}_\\delta(z)$ the slice $\\operatorname{sing} u \\cap \\B_1 \\cap \\pi_L^{-1}(y)$ consists of finitely many points, at each of them the tangent plane exists and is transverse to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3$ (i.e. the direction of slicing). \n\nLet us choose one such $y$ and denote these singular points by $p_1,\\ldots,p_k$. Let also \n\\[\nL_j := \\operatorname{Tan}(\\operatorname{sing} u, p_j), \n\\quad \nr_0 := \\tfrac 12 \\min_{i,j} \\left ( |p_i-p_j|, \\varepsilon - |p_i-y| \\right ).\n\\]\nFor each $j=1,\\ldots,k$, there is a HCM $\\varphi_j \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $S(\\varphi_j) = L_j$ such that the sequence of rescaled maps $u_{r_i}(x) = u(p_j+r_i x)$ converges to $\\varphi_j$ in $W^{1,2}(\\B_1)$ for some sequence $r_i \\to 0$. Note that by our assumption, $\\varphi_j$ has energy density strictly less than $\\Theta$. Since this convergence is uniform away from $L_j$, maps $u_{r_i}$ and $\\varphi_j$ are homotopic on $L_j^\\perp \\cap \\partial \\B_1$ for large enough $i$. Tilting $L_j^\\perp$ to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3$ and rescaling, we get that maps $u$ and $\\varphi_j$ are homotopic on $\\pi_L^{-1}(y) \\cap \\partial \\B_{r_j}(p_j)$ for some small $r_j < r_0$. Recalling that $u$ restricted to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3 \\cap \\partial \\B_1$ (and hence also to $\\pi_L^{-1}(y) \\cap \\partial \\B_\\varepsilon(y)$) has homotopy type $\\alpha$, we conclude \n\\[\n\\alpha = [\\varphi_1] + \\ldots + [\\varphi_k] \\quad \\text{in } \\pi_2({\\ensuremath{\\mathcal N}}),\n\\]\nwhere each $\\varphi_j$ has energy density smaller than $\\Theta$, which is a contradiction with the assumption that $\\alpha$ is indecomposable. \n\nTo finish the proof, we need to get rid of the additional assumption \\eqref{eq:transverse}. This is done by using the following simple transversality lemma. \n\n\\begin{lem}\n\\label{lem:transverse}\nLet $n = a+b$, consider the Grassmannian $G(n,a)$ with the standard volume measure $\\lambda$ and $G(n,b)$ with a finite positive Borel measure $\\mu$. Then the set \n\\[\n\\{ E \\in G(n,a) : \\mu(\\{ F \\in G(n,b) : E \\not\\pitchfork F \\}) > 0 \\})\n\\]\nhas zero $\\lambda$ measure. \n\\end{lem}\n\nPostponing its proof for the moment, we complete the reasoning as follows. Choose $a = 3$, $b = n-3$, and let $\\mu$ be the measure $\\mathcal{H}^{n-3} \\llcorner \\operatorname{sing} u \\cap \\B_1$ pushed-forward by the map $\\operatorname{Tan}(\\operatorname{sing} u, \\cdot)$, i.e.\n\\[\n\\mu(U) = \\mathcal{H}^{n-3} (\\{ y \\in \\operatorname{sing} u \\cap \\B_2 : \\operatorname{Tan}(\\operatorname{sing} u, y) \\in U \\}). \n\\]\nThen the set in Lemma \\ref{lem:transverse} has measure zero, in particular its complement is dense. Hence we can choose $E \\in G(n,3)$ so that $E \\pitchfork F$ for $\\mu$-a.e. $F \\in G(n,n-3)$, with $E$ arbitrarily close to $\\mathbf{0} \\times {\\ensuremath{\\mathbb R}}^3$. This amounts to satisfying \\eqref{eq:transverse} with a slightly tilted direction of slicing. Recall that $\\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$ is disjoint with the cylinder $\\B^{n-3}_\\delta(z) \\times {\\ensuremath{\\mathbb R}}^3$, in consequence it is also disjoint with a smaller cylinder in direction $E^\\perp$. It is easy to see that the rest of the proof remains unchanged. \n\\end{proof}\n\n\\begin{proof}[Proof of Lemma \\ref{lem:transverse}]\nFirst note that for each $F \\in G(n,b)$ the set of all $E \\in G(n,a)$ non-transversal to $F$ is a finite sum of smooth submanifolds of $G(n,a)$ of positive codimension \n\\[\n\\{ E \\in G(n,a) : E \\not\\pitchfork F \\} = \\bigcup_{c=1}^{\\min(a,b)} \\{ E \\in G(n,a) : \\dim E \\cap F = c \\},\n\\]\nhence it has zero $\\lambda$ measure. Applying Fubini theorem, we get \n\\begin{align*}\n\\int_{G(n,a)} \\mu(\\{ F \\in G(n,b) : E \\not\\pitchfork F \\}) \\phantom{.}\\mathrm{d} \\lambda(E) \n& = \\int_{G(n,a)} \\int_{G(n,b)} \\mathbf{1}_{E \\not\\pitchfork F} \\phantom{.}\\mathrm{d} \\mu(F) \\phantom{.}\\mathrm{d} \\lambda(E) \\\\\n& = \\int_{G(n,l)} \\lambda(\\{ E \\in G(n,a) : E \\not\\pitchfork F \\}) \\phantom{.}\\mathrm{d} \\mu(F) \\\\\n& = 0, \n\\end{align*}\nso the integrand has to be zero for $\\lambda$-a.e. $E \\in G(n,a)$. \n\\end{proof}\n\n\\subsection*{Propagation of $\\delta$-flatness to finer scales}\n\nIn this section we investigate some important consequences of Definition \\ref{df:flatness}. Assuming that an energy minimizing map $u$ is $\\delta$-flat in $\\B_1$ (with small $\\delta > 0$), we shall see that $\\operatorname{sing} u$ is actually more flat than a priori assumed (Lemma \\ref{lem:fund}), $u$ is also $\\delta$-flat in all smaller concentric balls (Corollary \\ref{cor:all-scales}), and that $0 \\in \\operatorname{sing}_\\alpha u$ (Corollary \\ref{cor:top-stratum}). \n\n\\begin{lem}\n\\label{lem:fund}\nFor every $\\varepsilon > 0$ there is $\\delta_1(\\varepsilon) > 0$ such that if $u$ is $\\delta_1$-flat in $\\B_1$, then $\\operatorname{sing} u$ is $\\varepsilon$-Reifenberg flat in $\\B_1$ and $\\| u - \\varphi \\|_{W^{1,2}(\\B_1)} \\le \\varepsilon$ for some HCM $\\varphi$ of homotopy type $\\alpha$ with energy density $\\Theta$. Moreover, $\\operatorname{sing} u$ is $\\varepsilon$-Reifenberg flat in $\\B_1$ with respect to the $(n-3)$-dimensional plane $S(\\varphi)$. \n\\end{lem}\n\n\\begin{proof}\nWe employ the usual contradiction argument. Let $u_k$ be a sequence of minimizing harmonic maps such that $u_k$ is $1\/k$-flat in $\\B_1$, with $\\operatorname{sing} u$ $\\tfrac{1}{10}$-Reifenberg flat with respect to a fixed plane $L$. \nChoosing a subsequence, we have $u_k \\to \\varphi$ in $W^{1,2}(\\B_1)$ for some energy minimizing $\\varphi$. By condition \\ref{df:flatness-energy} in Definition \\ref{df:flatness}, $\\varphi$ is homogeneous with energy density $\\Theta$. By Lemma \\ref{lem:no-drop}, for each $k$ the set $\\operatorname{sing}_{\\ge \\Theta} u_k$ is $\\frac{1}{10}$-Reifenberg flat in $\\B_1$ with respect to $L$. Taking the limit and exploiting the upper semicontinuity of $\\theta_\\cdot(\\cdot,0)$ with respect to both the map and the point, we conclude that the set \n\\[\nS(\\varphi) \\equiv \\operatorname{sing}_{\\ge \\Theta} \\varphi\n\\]\nis not contained in any $(n-4)$-dimensional plane. On the other hand, it is itself a~linear subspace of dimension at most $n-3$, so we learn that $\\varphi$ is a HCM of homotopy type $\\alpha$ (by uniform convergence away from $L$). For large enough $k$, $u_k$ is $\\varepsilon$-close to $\\varphi$ in $W^{1,2}(\\B_1)$ and its singular set is contained in $\\B_\\varepsilon S(\\varphi)$ (this is a~consequence of upper semicontinuity of $\\theta_\\cdot(\\cdot,0)$ and $\\varepsilon$-regularity \\eqref{eq:eps-regularity}), which finishes the proof by another application of Lemma \\ref{lem:no-drop}. \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:all-scales}\nIf $\\delta \\le \\delta_1(\\frac{1}{20})$ and $u$ is $\\delta$-flat in $\\B_1$, then $u$ is also $\\delta$-flat in any smaller ball $\\B_r$ centered at $0$ with $0 < r \\le 1$. \n\\end{cor}\n\n\\begin{proof}\nCondition \\ref{df:flatness-energy} of Definition \\ref{df:flatness} is trivially satisfied. As for condition \\ref{df:flatness-Reifenberg}, it follows from Lemma \\ref{lem:fund} that $\\operatorname{sing} u$ is $\\frac{1}{20}$-Reifenberg flat in $\\B_1$, hence $\\frac{1}{10}$-flat in any ball $\\B_r$ with $\\frac 12 \\le r \\le 1$. In consequence, $u$ is $\\delta$-flat in each of these balls. Then the claim follows by iteration of Lemma \\ref{lem:fund} rescaled to smaller and smaller balls. \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:top-stratum}\nIf $\\delta \\le \\delta_1(\\frac{1}{20})$ and $u$ is $\\delta$-flat in $\\B_1$, then every tangent map to $u$ at $0$ is a HCM of type $\\alpha$. In particular, $0 \\in \\operatorname{sing}_{\\alpha} u$. \n\\end{cor}\n\n\\begin{proof}\nLet $\\varphi$ be any tangent map to $u$ at $0$, i.e. a $W^{1,2}(\\B_1)$-limit of rescaled functions $u_k(x) = u(r_k x)$ for some sequence $r_k \\to 0$; any thus obtained $\\varphi$ is homogeneous. By Corollary \\ref{cor:all-scales}, each $u_k$ is $\\delta$-flat in $\\B_1$ (with $\\operatorname{sing} u_k$ $\\tfrac{1}{10}$-Reifenberg flat with respect to some $L_k$), so the claim follows from Lemma \\ref{lem:no-drop} as in the proof of Lemma \\ref{lem:fund}. The only difference is that the planes $L_k$ may change, but without loss of generality $L_k \\to L$ in $G(n,n-3)$, which is enough to conclude that $\\operatorname{sing}_{\\ge \\Theta} \\varphi$ spans an $(n-3)$-dimensional plane. \n\\end{proof}\n\n\\subsection*{Moving the ball center}\n\n\\begin{prop}\n\\label{prop:all-balls}\nFor every $\\varepsilon > 0$ there is $\\delta_2(\\varepsilon) > 0$ such that if $u$ is $\\delta_2$-flat in $\\B_2$, then $u$ is $\\delta_1(\\varepsilon)$-flat in each of the balls $\\B_r(z)$ with $z \\in \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$ and $0 < r \\le 1$. Moreover, the sets $\\operatorname{sing}_\\alpha u$ and $\\operatorname{sing}_{\\ge \\Theta} u$ restricted to the ball $\\B_1$ coincide. \n\\end{prop}\n\n\\begin{proof}\nChoose $\\delta_2 := \\min(\\delta_1(\\varepsilon),\\delta_1(\\eta\/2))$ according to Lemma \\ref{lem:fund}, where $\\eta>0$ is to be fixed in a moment. Applying Lemma \\ref{lem:fund} rescaled to the ball $\\B_2$, denote by $\\varphi$ the approximating HCM and let $L = S(\\varphi)$. To obtain the first claim, we first show that $\\theta_u(z,1) \\le \\Theta + \\delta_1(\\varepsilon)$ for each $z \\in \\B_1 \\cap \\B_{\\eta} L$. First, \n\\[\n\\int_{\\B_1(z)} |\\nabla u|^2 \\le \\int_{\\B_1(z)} |\\nabla \\varphi|^2 + C \\eta^2,  \n\\]\nby Lemma \\ref{lem:fund}. If $z' = \\pi_L(z)$, then $|z-z'| < \\eta$ and \n\\[\n\\int_{\\B_1(z)} |\\nabla \\varphi|^2 \\le \\int_{\\B_{1+\\eta}(z')} |\\nabla \\varphi|^2 = (1+\\eta)^{n-2} \\Theta\n\\]\nby $L$-invariance of $\\varphi$ in $z'$-direction. If $\\eta$ is chosen small enough (depending on $\\delta_1(\\varepsilon)$), we obtain $\\theta_u(z,1) \\le \\Theta + \\delta_1(\\varepsilon)$. \n\n\\medskip\n\nSince each point $z \\in \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_1$ lies in $\\B_\\eta L$ by Lemma \\ref{lem:fund}, the above reasoning shows that condition \\ref{df:flatness-energy} of Definition \\ref{df:flatness} holds for the ball $\\B_1(z)$. Condition \\ref{df:flatness-Reifenberg} is satisfied by our assumptions, this ball is $\\delta_1(\\varepsilon)$-flat. Then Corollary \\ref{cor:all-scales} implies $\\delta_1(\\varepsilon)$-flatness of $u$ also in all smaller balls $\\B_r(z)$. \n\nBy Corollary \\ref{cor:top-stratum} we now have $z \\in \\operatorname{sing}_\\alpha u$ for each $z \\in \\operatorname{sing}_{\\ge \\Theta} \\cap \\B_1$. The inverse inclusion $\\operatorname{sing}_\\alpha u \\subseteq \\operatorname{sing}_{\\ge \\Theta} u$ is evident from the definition of $\\Theta(\\alpha)$. \n\\end{proof}\n\n\\begin{cor}\n\\label{cor:only-top}\nUnder the assumptions of Proposition \\ref{prop:all-balls}, the whole singular set $\\operatorname{sing} u$ restricted to the ball $\\B_{1\/2}$ coincides with $\\operatorname{sing}_{\\ge \\Theta} u$ (and hence with $\\operatorname{sing}_\\alpha u$). \n\\end{cor}\n\n\\begin{proof}\nAssume that the ball $\\B_{1\/2}$ contains a point $p \\in \\operatorname{sing} u \\setminus \\operatorname{sing}_{\\ge \\Theta} u$. We may choose a point $z \\in \\operatorname{sing}_{\\ge \\Theta} u$ closest to $p$ (as it is a closed set) and set $r = 2|p-z|$. Clearly $z \\in \\B_1$ and $0 < r \\le 1$, so $u$ is $\\delta_1(\\varepsilon)$-flat in $\\B_r(z)$. Choose $L = L(z,r)$ according to Definition \\ref{df:set-flatness}. Then by Lemma \\ref{lem:no-drop} there is a point $z' \\in \\operatorname{sing}_{\\ge \\Theta} u \\cap \\B_r(z)$ such that $\\pi_L(z')=\\pi_L(p)$. Since both $|\\pi_L(p)-p|$ and $|\\pi_L(z')-z|$ are less than $\\tfrac{r}{10}$, the triangle inequality yields a~contradiction with minimality of $z$. \n\\end{proof}\n\nIn order to apply the above results, one needs to know that $u$ is $\\delta$-flat in at least one ball. \n\n\\begin{lem}\n\\label{lem:some-flatness}\nLet $\\delta > 0$. If $0 \\in \\operatorname{sing}_\\alpha u$ and $\\theta_u(0,0) < \\Theta + \\delta$, then there is $r > 0$ such that $u$ is $\\delta$-flat in $\\B_r$. \n\\end{lem}\n\n\\begin{proof}\nNote that condition \\ref{df:flatness-energy} of Definition \\ref{df:flatness} is trivially satisfied for small enough~$r$. \n\nBy definition of $\\operatorname{sing}_\\alpha u$, some sequence of rescaled functions $u_k(x) = u(r_k x)$ converges in $W^{1,2}(\\B_1)$ to a HCM $\\varphi$ of homotopy type $\\alpha$ for some sequence $r_k \\to 0$. For large enough $k$, we have $\\operatorname{sing} u_k \\cap \\B_1 \\subseteq \\B_{1\/10} S(\\varphi)$. Since the convergence is uniform away from $S(\\varphi)$, $u_k$ restricted to $S(\\varphi)^\\perp \\cap \\partial \\B_1$ has homotopy type $\\alpha$, so condition \\ref{df:flatness-Reifenberg} follows from Lemma \\ref{lem:no-drop}. Rescaling, we see that $u$ is $\\delta$-flat in $\\B_{r_k}$ for large enough $k$. \n\\end{proof}\n\n\\begin{rem}\n\\label{rem:unique-type}\nCombining Lemma \\ref{lem:some-flatness} with Corollary \\ref{cor:top-stratum}, we see that \\textit{some} can be changed to \\textit{any} in the definition of $\\operatorname{sing}_\\alpha$, if only we restrict ourselves to points with energy density close to optimal. \nThat is, if $y \\in \\operatorname{sing}_\\alpha u$ and $\\theta_u(y,0) < \\Theta + \\delta_1(\\tfrac{1}{20})$, then every tangent map of $u$ at $y$ is a HCM of type $\\alpha$.\n\\end{rem}\n\nWe are now ready to prove the main theorem. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:Holder-regularity}]\nFix the H{\\\"o}lder exponent $0 < \\gamma < 1$ and choose $\\varepsilon = \\varepsilon(\\gamma,n) > 0$ according to Reifenberg's topological disc theorem (Theorem \\ref{thm:Reifenberg}), then fix $\\delta$ to be $\\delta_2(\\varepsilon)$ from Proposition \\ref{prop:all-balls}. \n\nChoose a point $p \\in \\operatorname{sing}_\\alpha u$ such that $\\theta_u(p,0) < \\Theta + \\delta$. According to Lemma \\ref{lem:some-flatness}, $u$ is $\\delta_2(\\varepsilon)$-flat in some ball $\\B_{2r}(p)$. By Proposition \\ref{prop:all-balls}, the set $\\operatorname{sing}_\\alpha u \\cap \\B_{r}(p)$ is closed and $\\varepsilon$-flat in each ball $\\B_s(z)$ centered at $z \\in \\operatorname{sing}_\\alpha u \\cap \\B_{r}(p)$ with radius $0 < s < r$. Applying Theorem \\ref{thm:Reifenberg}, we conclude that $\\operatorname{sing}_\\alpha u \\cap \\B_{r}(p)$ is bi-H{\\\"o}lder equivalent (with exponent $\\gamma$) to an $(n-3)$-dimensional ball. \n\nBy upper semicontinuity of $\\theta_u(\\cdot,0)$ we can ensure $\\theta_u(y,0) < \\Theta + \\delta$ for all $y \\in \\B_{r}(p)$ (jsut by taking $r$ small enough), which together with Corollary \\ref{cor:only-top} shows that the set in question forms an open subset of $\\operatorname{sing} u$. \n\\end{proof}\n\n\\section{Additional results}\n\\label{ch:additional}\n\nIn this subsection we discuss two elementary observations that give a better description of $\\delta$-flatness, but were not needed for the proof of Theorem \\ref{thm:Holder-regularity}. We fix an indecomposable homotopy type $\\alpha \\in \\pi_2({\\ensuremath{\\mathcal N}})$ and its lowest energy level $\\Theta = \\Theta(\\alpha)$. \n\n\\medskip\n\nThe following lemma shows that condition \\ref{df:flatness-Reifenberg} in Definition \\ref{df:flatness} can be dropped if one assumes a priori that $x \\in \\operatorname{sing}_\\alpha u$. This gives us an equivalent condition for $\\delta$-flatness. \n\n\\begin{lem}\n\\label{lem:only-energy-assumption}\nAssume that $0 \\in \\operatorname{sing}_\\alpha u$. If $\\delta \\le \\delta_1(\\frac{1}{20})$ and $\\theta_u(0,1) \\le \\Theta + \\delta$, then $u$ is $\\delta$-flat in $\\B_1$. \n\\end{lem}\n\n\\begin{proof}\nInspecting the proof of Lemma \\ref{lem:fund}, we see that condition \\ref{df:flatness-Reifenberg} of Definition \\ref{df:flatness} was only needed to ensure required symmetry of approximating homogenous minimizer $\\varphi$. Hence it would be enough to assume condition \\ref{df:flatness-Reifenberg} of Definition \\ref{df:flatness} in a~smaller ball $\\B_{1\/2}$, and $\\delta$-flatness in $\\B_1$ follows as in Lemma \\ref{lem:fund}. \n\nBy Lemma \\ref{lem:some-flatness}, there is $r > 0$ (possibly very small) such that $u$ is $\\delta$-flat in $\\B_r$. Applying the reasoning above, we see it is also $\\delta$-flat in every ball $\\B_s$ with $r \\le s \\le \\min(1,2r)$. An iteration of this argument (as in Corollary \\ref{cor:all-scales}, but in the opposite direction) leads to the claim. \n\\end{proof}\n\n\\medskip\n\nThe last lemma gives a uniform bound (independent of $u$) for the rate of convergence $\\theta_u(x,r) \\to \\theta_u(x,0)$ when $r \\to 0$, assuming $\\theta_u(x,r)$ is already close to $\\theta_u(x,0)$. This assumption cannot be dropped, if only there exist tangent maps $\\varphi \\colon {\\ensuremath{\\mathbb R}}^n \\to {\\ensuremath{\\mathcal N}}$ with $\\dim_H \\operatorname{sing} \\varphi = n-3$ which are not HCMs. \n\nAn additional assumption is needed to ensure that the energy density is not greater than $\\Theta$. This assumption is automatically satisfied if ${\\ensuremath{\\mathcal N}}$ is real-analytic of integrable in the sense of \\cite[Ch.~3.13]{Sim96}; see the remark preceding Corollary \\ref{cor:discrete-levels}. \n\n\\begin{lem}\n\\label{lem:rate-of-convergence}\nAssume additionally that $\\Theta$ is an isolated energy level for HCMs of type $\\alpha$. Assume that $0 \\in \\operatorname{sing}_\\alpha u$ and $\\theta_u(0,1) \\le \\Theta + \\delta_4$ with $\\delta_3(n,\\alpha,{\\ensuremath{\\mathcal N}}) > 0$ sufficiently small. Then for every $\\delta > 0$ there is $r(\\delta,n,{\\ensuremath{\\mathcal N}}) > 0$ such that $\\theta_u(0,r) \\le \\Theta + \\delta$ (in consequence, $u$ is $\\delta$-flat in $\\B_r$). \n\\end{lem}\n\n\\begin{proof}\nWe choose $\\delta_3 > 0$ smaller than $\\delta_1(\\frac{1}{20})$ from Lemma \\ref{lem:fund} and such that \n\\[\n\\int_{\\B_1^n} |\\nabla \\varphi|^2 \\notin (\\Theta,\\Theta+\\delta_3]\n\\]\nfor each HCM $\\varphi$ of type $\\alpha$. \n\nFor the sake of contradiction, assume there is a sequence of such energy minimizing maps $u_k$ with \n\\[\n\\Theta + \\delta \\le \\theta_{u_k}(0,1\/k) \\le \\theta_{u_k}(0,1) \\le \\Theta + \\delta_3.\n\\]\nTaking a subsequence, we obtain a limit map $u$ such that \n\\[\n\\Theta + \\delta \\le \\theta_u(0,0) \\le \\theta_{u}(0,1) \\le \\Theta + \\delta_3.\n\\]\nIt follows from Lemma \\ref{lem:only-energy-assumption} that each $u_k$ is $\\delta_1(\\frac{1}{20})$-flat in $\\B_1$, hence so is $u$ and by Corollary \\ref{cor:top-stratum} we infer $0 \\in \\operatorname{sing}_\\alpha u$. In particular the energy density $\\theta_u(0,0)$ is either $\\Theta$ or greater than $\\Theta+\\delta_3$, a contradiction.  \n\\end{proof}\n\n\\section*{Acknowledgments}\n\nThe author would like to thank Maciej Borodzik for fruitful discussions and helpful suggestions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\nTo build applications that can cope with a continuous and dynamic threat landscape, it is vital to empower developers with methodologies and tools for the development of effective defences \\cite{green2016}. It is also essential that these methodologies and tools can handle the fast-paced nature of modern development practices, such as Continuous Integration (CI) and Continuous Delivery (CD).  These practices are often applied within a DevOps environment in which collaboration between software development and IT operations, including security, is encouraged\\cite{ebert2016}. If the current tools and methodologies remain unchanged, security will be perceived as slowing down the development process, making it less appreciated by developers and thus leading to insecure software applications.  \n\nOrganizations which utilize a DevOps culture typically have an overarching view regarding what is deployed in their environment.  However, this usually focuses on the performance of applications, or errors caused within applications. To gain similar insights on security-related events, it is necessary to integrate attack awareness into an application. Otherwise, ongoing attacks may be missed, and there is the potential for a compromise or data breach to go unnoticed. Data breaches can have a substantial financial and reputational impact on an organization- according to IBM Security\\cite{ibmsecurity2019}, a breach which takes 200 days to resolve can cost a company \\$4.56 million. \n\nThis paper discusses a taxonomy of approaches regarding how to integrate attack awareness into applications, providing a guide for developers and security researchers.  The remainder of the paper is structured as follows: Section \\ref{problemstatement} describes the need for attack aware applications.  Current approaches are described in Section \\ref{relatedwork}.  Section \\ref{analysis} provides an analysis of these solutions before conclusions are drawn in Section \\ref{conclusion}.\n \n\\section{Problem Statement} \\label{problemstatement}\nMany existing applications cannot provide real-time intelligence regarding their security state \\cite{Watson2011}. This lack of information means developers do not know whether an attack has taken place, or which parts of an application attracts the most attention from those with malicious intentions.  Integrating attack awareness can address this knowledge gap, providing developers with actionable insights based on the context of the application. \n\nIntegration is generally performed by the developer or by utilizing an agent which makes an application attack-aware at runtime. While a fully automated approach is desirable, it is only feasible for application components developed using established practices and technologies \\cite{zhu2016}.  Additionally, off-the-shelf protection methods implemented by agents are limited to specific attack classes \\cite{noderasp20182}\\cite{hawkins2017} and known attacks \\cite{openrasp20182}. A taxonomy of approaches can guide researchers and developers in their choice of an appropriate integration method, taking into consideration the usability of the integration method.\n\n\\section{Related Work} \\label{relatedwork}\nApproaching intrusion detection from within an application has been hypothesized by Sielken \\cite{Sielken1999} to detect attackers that abuse target applications without exposing anomalous behavior, and by avoiding the use of detectable attack patterns. Detection of these attacks can be achieved by utilizing contextual information such as the methods invoked while using a certain application feature. There are two approaches proposed to collect contextual application information.  The first approach regularly scans internal values of the target application, such as the authorization level of an user, and documents any changes of these values. The second approach embeds code triggers directly into the target applications' code. These can be conditional checks such as whether a predefined option that has been selected by a user has been modified.  Work by Kerschbaum et al. \\cite{kerschbaum2002} provides an example of code triggers usage to detect network-based attacks within the OpenBSD kernel, and application-specific attacks from within Sendmail. The AppSensor framework \\cite{Watson2011}, and BlackWatch \\cite{Hall2019} are further examples which apply the code trigger variant.\n\nThe concept of reference monitoring, as defined by Anderson \\cite{Anderson1972}, enforces an access policy upon user programs in execution that access references such as other programs, data or peripherals. Originally, reference monitors were implemented at operating system (OS) level to enforce a security policy on applications interacting with the OS kernel. However, there are two further approaches- the reference monitor is either embedded in an interpreter that runs the target application or it is embedded in the target application directly \\cite{Erlingsson2003}.\n\nSelf-protecting software systems are another closely related field of research, focusing on systems which autonomously defend themselves against attacks, as envisioned by Kephart and Chess\\cite{Kephart2003}. Furthermore, these systems can also anticipate security risks and mitigate them by proactively enabling countermeasures. In the nine dimensional taxonomy of self-protecting software systems by Yuan and Malek \\cite{Yuan2012}, the authors dedicate one dimension for research which focuses on achieving self-protection at a specific phase within the Software Development Lifecycle (SDLC). These phases are classified as design, development, testing and implementation, but are simplified in Yuan and Malek's \\cite{Yuan2012} work as self-protection achieved at development-time and runtime. \n\n\\section{Analysis} \\label{analysis}\nThis section provides an analysis of previous research and existing solutions, summarized in Figure \\ref{fig:intfig}, creating a taxonomy of approaches for integrating attack-awareness in applications. Based on the research in section \\ref{relatedwork}, the integration is performed by a developer or by an agent.  In the proposed taxonomy, it is assumed there is access to the source code of the target application required for the developer-driven approaches.\n\n\\textbf{Developer-Driven: }Developers of an application can integrate attack awareness by implementing security controls in the target application, or by configuring existing security controls within the target application. The main body of work in developer-driven attack awareness is conducted manually, and more detail is provided in subsections \\ref{manualintegration} and \\ref{aspectorientedprogramming}.\n\n\\textbf{Agent-Driven: }An agent (a software component or a software application which autonomously acts on behalf of its user) can also be designed to integrate attack awareness. Though the focus is on autonomous integration, the agent can provide an interface to enable manual configuration, generating security controls at runtime. An agent that is part of the interpreter can provide attack awareness through the runtime environment and thus to any application running in this environment. Subsection \\ref{runtimeinstrumentation} describes agents integrating attack awareness through instrumentation. \n\n\\begin{figure*}\n    \\center\n    \\includegraphics[height=4.5cm]{integration-approaches.pdf}\n    \\caption{Developer and agent-driven approaches to integrate attack awareness into an application}\n    \\label{fig:intfig}\n\\end{figure*} \n\n\\subsection{Manual Integration} \\label{manualintegration}\nSolutions like the AppSensor framework \\cite{Watson2011} and BlackWatch \\cite{Hall2019} rely on the hypothesis that an attacker or malicious behavior can be detected by knowing the `normal' behavior of the target application, and by being able to monitor deviations from such behavior. Application developers are therefore ideal candidates to apply this integration approach as they have specified, designed and implemented the application, and know where to strategically place security controls.\n\nWhile this approach is effective against attackers targeting the business logic of an application, it is less so when it comes to attacks such as injection attacks, which require developers to have expertise in their execution. This, however, is often not the case for security expertise among developers in general as argued by Wurster and Oorschot \\cite{Wurster2008} - developers may lack security expertise.\n\nIt is the developers responsibility to put the security controls in place, however there is a chance that the placement is neglected due to a lack of priority or time spent on non-security tasks, leading to areas of the application which are not covered. In the worst case, this approach will not be accepted by the developers as it can be seen as an additional effort which needs to be completed on top of others tasks \\cite{Hall2019}\\cite{zhu2016}. Due to this issue, integration approaches should utilize proven development techniques e.g. Kim et al. \\cite{kim2012} utilizes the dependency injection technique to make the integration of attack awareness scalable and reusable. This technique is part of modern application frameworks and while it is familiar to many developers, it can also address the issue of accidentally omitting areas of the application. From the perspective of framework developers, it is suggested to build the security controls into the framework.  According to an empirical study by Peguero et al. \\cite{peguero2018}, applications using built-in security controls derived from a framework are less susceptible to vulnerabilities than applications where developers were responsible for implementing security controls. \n\n\\subsection{Aspect-Oriented Programming} \\label{aspectorientedprogramming}\nAspect-Oriented Programming (AOP) is a technique to separate cross-cutting concerns, such as logging and monitoring, which are used throughout an application but which do not represent any business functionality \\cite{Kiczales1997}. These cross-cutting concerns, also referred to as aspects, are weaved in to the target functionality of the application by intercepting function or method calls at runtime.\n\nSecurity controls can also be defined as aspects as they are not tied to the target application's business functionality. Serme et al.\\cite{Serme2014} present an AOP approach on integrating input validation controls, whereas Phung et al. \\cite{Phung2009} demonstrate aspectized security policies which prevent malicious behaviour in the target application.  \n\nAOP reduces the manual effort required when initially implementing the security policies that are processed by aspects. An aspect can be implemented as code in the programming language of the target application \\cite{Serme2014}\\cite{Phung2009} but it can also be implemented in a modeling language such as Unified Modeling Language (UML)\\cite{zhu2009}, or in a Program Query Language (PQL) \\cite{livshits2006}. The various implementation methods can address the usability requirements of the different roles involved with the development of an application. A developer may prefer to implement an aspect as code whereas an architect may rather implement an aspect in a modeling language. Regardless of the role, the implementation method of choice should be usable for its target audience as highlighted by Viega et al.\\cite{Viega2001}, to prevent developer-induced errors and to reduce the required expertise to implement security controls.  \n\nInstead of manually implementing an aspect from scratch, it can also be generated using, for example, the output of a static code analysis tool \\cite{simic2013}. While this method seems to automate the integration process at first, in reality it shifts the manual effort to the usage of a tool or application which generates the output required for the aspect generation. Whether the integration can be fully automated depends on the degree of automation supported by the tool or application in question.  \n\nIraqi and Bakkali \\cite{iraqi2019} developed a framework that can learn to detect outlier behavior in method invocations without supervision. The method invocation features required for the learning process are implemented as a feature extractor aspect, which demonstrates another use case for integrating attack awareness with AOP.  \n\n\\subsection{Runtime Environment and Binary Instrumentation} \\label{runtimeinstrumentation}\nThe approaches analysed in the previous sections are effective when integrating security controls into a single application at one time. However, integrating security controls from within the runtime environment can affect any application running in the environment. Existing solutions are based on a modified interpreter as in ZenIDS \\cite{hawkins2017} or Node RASP \\cite{noderasp2018}. Another variant is also known as a Runtime Application Self-Protection (RASP) agent \\cite{gartner2020}, a library which can be loaded by an interpreter at runtime e.g. OpenRASP \\cite{openrasp2018} or Sqreen \\cite{jeanbaptiste2019}.\n\nIn the ideal case, RASP agents can be deployed in a plug and play manner, requiring only an initial configuration as Haupert et al. \\cite{haupert2018} describes regarding the deployment of Promon SHIELD RASP \\cite{promon2020}. In cases where an agent does not require any configuration or a learning phase, attacks are detected using techniques that, e.g., combine taint-tracking with lexical analysis \\cite{noderasp20182} or that monitor common input sinks and output sources for known malicious behavior and signatures \\cite{openrasp20182}. Depending on the RASP implementation, the aforementioned techniques may only cover a limited set of sinks and sources, and the detection of known attacks relies on the completeness of the malicious behavior or signature database available. As RASP solutions are platform specific by design, it can also be the case that specific technologies used within a target application, or the entire application, might not be able to benefit from this integration approach.\n\nWhile an instrumented runtime environment may be suitable for applications written in interpreted languages, those written in compiled languages require individual instrumentation.  With binary instrumentation, security controls can be integrated into an applications binary code to guarantee control-flow integrity (CFI) at runtime \\cite{abadi2009}. The effectiveness of CFI is limited to attacks attempting to hijack the control flow of an application, such as buffer overflows.  Although it is feasible to integrate CFI in closed source applications, many available CFI implementations require access to source code\\cite{goktas2014}, making it impractical for legacy or third-party applications.\n\n\\subsection{Summary}\nThe previous subsections have shown that all attack awareness approaches require some form of manual interaction, either at the setup or configuration phase. This is particularly pertinent for the detection of attacks against the business logic of an application, as these are unique for every application. The runtime environment and binary instrumentation approaches are no exception and also require manual intervention by the developers in the form of placing function calls which can track authenticated users\\cite{hawkins2017} or custom events\\cite{sqreen2020}. As with AOP approaches, it is then up to the agent implementation to provide a method of specifying policies, acting on the tracked events. This will influence how usable that method is in the context of the application.\n\nAlthough the agent-driven approach has the most potential for automation, there are other aspects which make manual integration worthwhile, e.g. the insertion of code triggers can be completed in a few lines of non-fragmented code\\cite{kerschbaum2002}. Compared to instrumented monitoring, code triggers are only called when they lie in the execution path of an attack and thus cause no performance overhead during non-malicious interactions. The code triggers are an inherent part of the application, providing constant attack-awareness in any environment in which the application is deployed.\n\nThe programming language can also be relevant when choosing AOP for attack awareness integration. To implement AOP in JavaScript, one can overwrite built-in functions, which makes self-protection possible \\cite{Phung2009}. In contrast, PHP does not support the same functionality for built-in functions- a language extension must be installed first. Extension usage might not be feasible in certain environments due to restrictions in what can be deployed, or the lack of control over the environment. This would also be an exclusion criteria for choosing a RASP agent in the previously described environment. \n\nThe approaches thus far focused on using attack awareness for application protection. However, the same concept can be utilized for security testing e.g. the insights provided by an agent could be used for the generation of security tests. Since the tester is an established role in modern development environments, enhancing the capabilities of such individuals could be a method to improve the adoption of attack awareness. \n\n\\section{Conclusion and Future Work} \\label{conclusion}\nWork presented in this paper explored different methods of including attack awareness within applications. The taxonomy highlights that while there are a number of potential solutions, they do not necessarily meet the needs of developers. \nResearch currently being undertaken by \u00dcnl\u00fc \\cite{Unlu2019} seeks to enhance the field of attack awareness and software security. As part of this process, the work will involve the investigation of techniques which aim to enhance the process by which developers include attack awareness within applications. By achieving an enhancement in the adoption of attack awareness techniques, the proposed research will contribute to bringing application security closer to developers by creating the necessary tools for them.  Such tools will be tailored to the needs of developers, empowering them to build secure applications.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and motivation}\\label{Intro}\n\n$\\;\\;\\;$\n\nIn general, the electromagnetic  field\ntensor ${F}$, expressed by a $4\\times 4$  antisymmetric matrix, is used to describe\nthe electromagnetic field intensity. This description involves 6 parameters.  An alternative way to describe an\nelectromagnetic  field is by use of the 4-potential. In a chosen gauge, the 4-potential\ntransforms as a 4-vector. The electromagnetic field tensor is then recovered by differentiating the 4-potential.\nChoice of a gauge can reduce the degrees of freedom of this description to from 4 to 3.\n\nIn 1904 E. T. Whittaker introduced \\cite{Whittaker} two scalar potential functions. Thus, he was able to reduce\nthe degrees of freedom of the electromagnetic  field description to 2.\nHe showed that the electromagnetic field can be expressed in terms of the\nsecond derivatives of these functions. However, he was not able to find the covariance of his scalar potentials.\nH. S. Ruse \\cite{Ruse} improved  the result of Whittaker. In 2009 Y. Friedman  and S. Gwertzman  showed  \\cite{FGW} that it is possible to combine these two scalar potential functions into one complex-valued function ${S}(x)$ on Minkowski space, which we call the \\textit{prepotential of the electromagnetic field}. Moreover, they showed that this prepotential is invariant under a certain spin half representation of the Lorentz group. Thus, this prepotential provides a covariant description of an electromagnetic field with minimal degrees of freedom.\n\nIn 1953, influenced by M. Born, who was looking for the connection of the wave-function description of elementary particles in quantum mechanics and the electromagnetic field they generate, H. S. Green and E. Wolf introduced \\cite{GreenWolf53} a complex scalar potential for an electromagnetic field.  They described the similarity of the expressions for energy and momentum densities between their potential and the wave function. They were unable, however, to find the connection between their potential and the Whittaker potentials.\n\nAn electromagnetic field is generated by a collection of moving charges.  Thus, a description of an electromagnetic field can be obtained by integrating the fields of moving charges. For a moving charge and observer at\npoint $x$, the position of the charge at the retarded time relative to the observer is a null-vector in spacetime.  We show that for each such vector, there is a complex dimensionless scalar which is invariant under a certain representation of the Lorentz group. The prepotential $S(x)$ is defined to be the logarithm of this scalar.\n\nThe Aharonov-Bohm effect indicates that there is a multiple-valued prepotential of the 4-potential of an electromagnetic field.\n In their paper \\cite{AharonovBohm59}, a scalar function $S$  such that $\\nabla S=(e\\hbar\/c)A$, where $A$ is the 4-potential of the electromagnetic field,  was introduced.\nIt has been shown that if  $\\psi_0$ is the solution of the Schr\\\"odinger equation in the absence of an electromagnetic field, then the function $\\psi=\\psi_0e^{-iS\/\\hbar}$ is the solution of the equation in the presence\nof the field, at least in a simply connected region in which the electromagnetic field vanishes.\nIn the magnetic Aharonov-Bohm experiment, however, the region outside the solenoid is not simply connected. This leads to a multi-valued $\\psi$ in this experiment. As it was shown in \\cite{FOst}, our prepotential is of the same type, but is defined for any electromagnetic field. Our prepotential is also multi-valued.\n\nIn classical mechanics, the negative of the gradient  of a scalar potential equals the force.\n This is true for forces which generate linear acceleration. Such forces are defined by a one-form\n(since their line integral gives the work). The derivative of a scalar function\nis also a one-form. Hence, the derivative of a potential can equal the negative of the\nforce. But in classical mechanics we also have rotating forces, which are described by\ntwo-forms. Such forces cannot be expressed as derivatives of a scalar potential. For example, the electromagnetic field is not conservative in general. It generates both linear and rotational acceleration, and the electromagnetic tensor is a two-form. Hence, it is natural to assume that a kind of second derivative of a scalar potential  will define\nthe force tensor. Note that the usual differential of a gradient of a real-valued function is zero.\nTherefore, the prepotential must be complex-valued, and we will need to define a Lorentz invariant conjugation of the gradient of the prepotential in order to obtain the 4-potential of the field.\n\nAnother important property of a prepotential of an electromagnetic field is its \\textit{locality}. Note that the electromagnetic tensor $F_{\\mu\\nu}$ of a field of a moving charge depends on the position, velocity and acceleration, while the 4-potential $A_\\mu$ depends only on the position and velocity of the source. Our prepotential $S$ depends only on the position of the source.\n\nIn section 2, we obtain a Lorentz group representation based on the complex electromagnetic field tensor. We will show that the regular representation $\\pi$ of the Lorentz group can be decomposed as a product of two commuting representations $\\tilde{\\pi}$ and  $\\tilde{\\pi}^*$. In section 3, we will introduce the prepotential of an electromagnetic field and show its geometric meaning. We will also show the covariance of this prepotential under the action of the representation $\\tilde{\\pi}$. In section 4, we will study the connection between the prepotential and the Faraday vector of a field.\nWe will find the gauge of the prepotential. Explicit solutions for the prepotential and field of a charge at rest and a charged infinite rod will be found. Using computer algebra, we will derive the field of an arbitrary moving charge from its prepotential. The Maxwell equations for the prepotential will be derived in section 5 and we will demonstrate their use for deriving the prepotential and the field of current in straight wire. In section 6, we will show that the representation $\\tilde{\\pi}$ coincides with the spin half representation of the Lorentz group on the spinors. We also show that the matrices occurring in the description of the connection of the prepotential to the field are a representation of the Dirac $\\alpha$-matrices.\n\n\n\n\\section{Lorentz group representation based on complex electromagnetic  field tensor}\\label{secLor}\n\nIn flat Minkowski space $M$, the spacetime coordinates of an event are denoted by\n$x^{\\mu}\\;(\\mu=0,1,2,3)$, with $x^0=ct$. The Minkowski inner product is\n$x\\cdot y=\\eta_{\\mu\\nu}x^{\\mu}y^{\\nu}$, where\nthe Minkowski metric is $\\eta_{\\mu\\nu}=\\operatorname{diag}(1,-1,-1,-1)$.\nThe usual Lorentz group representation  $\\pi$ can be associated with an electromagnetic  field tensor $F_\\alpha^\\beta(\\mathbf{E},\\mathbf{B})$\n\\begin{equation}\\label{EMMixedTensor}\nF=F_\\mu^\\nu = \\left(\\begin{array}{cccc}\n0 & E_1 & E_2 & E_3 \\\\\nE_1 & 0 & cB_3 & -cB_2 \\\\\nE_2 & -cB_3 & 0 & cB_1 \\\\\nE_3 & cB_2 & -cB_1 & 0 \\\\\n\\end{array}\\right),\n\\end{equation}\n where $\\mathbf{E}$ denote the electric field intensity\n and  $\\mathbf{B}$ the magnetic field intensity, as follows.\n\nSince a magnetic field generates a rotation,\na generator of a rotation about the direction $\\mathbf{n}\\in \\mathbb{R}^3$ is defined by  $F_\\alpha^\\beta(0,\\mathbf{B})$, with $c\\mathbf{B}=-\\mathbf{n}$. Thus, the representation of the rotation\nabout the direction $\\mathbf{n}$ with angle $\\omega$ is given by the operator $\\exp(F_\\alpha^\\beta(0,-\\mathbf{n})\\omega).$ For example, a\nrotation $\\mathfrak{R}^1$ about the $x^1$-axis (rotation in the $x^2x^3$ plane) is represented as\n\\begin{equation}\\label{RepPiR1}\n\\pi(\\mathfrak{R}^1)(\\omega)=\\Lambda_{23}(\\omega)=\\exp (\\left(\n                          \\begin{array}{cccc}\n                            0 & 0 & 0 & 0 \\\\\n                            0 & 0 & 0 & 0 \\\\\n                            0 & 0 & 0 & -1 \\\\\n                            0& 0 & 1 & 0 \\\\\n                          \\end{array}\n                        \\right)\\omega)=\\left(\n                                       \\begin{array}{cccc}\n                                         1 & 0 & 0 & 0 \\\\\n                                        0 &  1 & 0 & 0 \\\\\n                                         0 & 0 & \\cos\\omega &  -\\sin\\omega  \\\\\n                                         0 & 0 &  \\sin\\omega  &  \\cos\\omega  \\\\\n                                       \\end{array}\n                                     \\right).\n\\end{equation}\nSimilarly, since an electric field generate boosts, a generator of a boost in the direction $\\mathbf{n}$ can be identified with $F_\\alpha^\\beta(\\mathbf{n},0)$. Thus, the representation of the boosts in the direction $\\mathbf{n}$ with rapidity $\\omega$ is given by the operator $\\exp(F_\\alpha^\\beta(\\mathbf{n},0 )\\omega).$ For example,\nboosts $\\mathfrak{B}^1$ in the $x$-axis are represented as\n\\begin{equation}\\label{RepPiB1}\n\\pi(\\mathfrak{B}^1)(\\omega)=\\Lambda_{01}(\\omega)=\\exp (\\left(\n                          \\begin{array}{cccc}\n                            0 & 1 & 0 & 0 \\\\\n                            1 & 0 & 0 & 0 \\\\\n                            0 & 0 & 0 & 0 \\\\\n                            0& 0 & 0 & 0 \\\\\n                          \\end{array}\n                        \\right)\\omega)=\\left(\n                                       \\begin{array}{cccc}\n                                         \\cosh\\omega &  \\sinh\\omega & 0 & 0 \\\\\n                                        \\sinh\\omega  &  \\cosh\\omega  & 0 & 0 \\\\\n                                         0 & 0 & 1&0  \\\\\n                                         0 & 0 & 0&1  \\\\\n                                       \\end{array}\n                                     \\right).\n\\end{equation}\n\nOur complex prepotential $S(x)$ is a function $M \\rightarrow \\mathbb{C}$ on Minkowski space $M$.\nIts gradient at any spacetime point  belongs to the complexified cotangent space at that\npoint, which we identify with $\\mathbb{C}^4$.  Denote by $M_c$ the complex space $\\mathbb{C}^4$\nendowed with the bilinear complex-valued form $x\\cdot y=\\eta_{\\mu\\nu} x^\\mu y^\\nu$, which can\nbe considered as a complexification of Minkowski space. The bilinear form on $M_c$ is an extension of the Minkowski inner product.\n\nObviously, the representation $\\pi$ of the Lorentz group on $M$ can be extended linearly to\na representation on $M_c$. We claim that this representation can be decomposed as follows.\n\\begin{claim}\\label{pi-decomposition}\nThe representation $\\pi$ of the Lorentz group on $M_c$ can be decomposed  into a product\n\\begin{equation}\\label{DecLorents}\n    \\pi=\\tilde{\\pi}\\tilde{\\pi}^*\n\\end{equation}\nof two commuting representations $\\tilde{\\pi}$ and $\\tilde{\\pi}^*$ on $M_c$.\n\\end{claim}\nIn order to prove this claim, we will decompose the generator $F^\\alpha_\\beta $ of the representation $\\pi$ into a sum\n\\begin{equation}\\label{DecGenLorents}\n    F^\\alpha_\\beta=\\frac{1}{2}\\mathcal{F}^\\alpha_\\beta +\\frac{1}{2}\\bar{\\mathcal{F}}^\\alpha_\\beta ,\n\\end{equation}\nwhere  $\\mathcal{F}^\\alpha_\\beta$ is  the complex electromagnetic tensor and its complex adjoint $\\bar{\\mathcal{F}}^\\alpha_\\beta$, defined below.\n\nAn electromagnetic field can be defined by the \\textit{Faraday vector} $\\mathbf{F}=\\mathbf{E}+ic\\mathbf{B}$. Note that since $i$ is a pseudo-scalar and $\\mathbf{B}$\nis a pseudo-vector, the expression $i\\mathbf{B}$ is a vector which is independent of\nthe chosen  orientation of the space. The Faraday vector is used\nto describe the Lorentz invariant field constants, see for example \\cite{Landau}.\nIn \\cite{FD}, Friedman and Danziger introduced a \\textit{complex electromagnetic tensor} $\\mathcal{F}^\\alpha_\\beta$  for the\ndescription of an electromagnetic field, similar to the one introduced by Silberstein in \\cite{Silberstein1} and\n\\cite{Silberstein2}. Complexified electromagnetic fields were also studied by A. Gersten \\cite{Gerst} and played an important role in obtaining explicit solutions in \\cite{FS} and \\cite{F04} for motion of a charge in constant electromagnetic field.\n\nLet $F^\\alpha_\\beta (\\mathbf{E},\\mathbf{B})$ be the usual electromagnetic tensor. The\ncomplex electromagnetic tensor $\\mathcal{F}^\\alpha_\\beta(\\mathbf{E},\\mathbf{B})$ is defined by $\\mathcal{F}^\\alpha_\\beta(\\mathbf{E},\\mathbf{B})=F^\\alpha_\\beta (\\mathbf{F},-i\\mathbf{F})$\nor\n\\begin{equation}\\label{complex tensor}\n\\mathcal{F}^\\alpha_\\beta=\\left(\n                          \\begin{array}{cccc}\n                            0 & F_1 & F_2& F_3 \\\\\n                            F_1 & 0 & -iF_3 & iF_2 \\\\\n                            F_2& iF_3 & 0 & -iF_1 \\\\\n                            F_3& -iF_2 & iF_1 & 0 \\\\\n                          \\end{array}\n                        \\right)=\\sum_{j=1}^3 (\\rho^j)^\\alpha_\\beta F_j\\,,\n\\end{equation}\nwhere $(\\rho_j)^\\alpha_\\beta$ are the Majorana-Oppenheimer matrices (see \\cite{Dvoeglasov})\n   \\[\n(\\rho^1)^\\alpha_\\beta=\\left(\n\\begin{array}{cccc}\n0 & 1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -i \\\\\n0 & 0 & i & 0 \\\\\n\\end{array}\\right),\\;\n(\\rho^2)^\\alpha_\\beta=\\left(\n\\begin{array}{cccc}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & i \\\\\n1 & 0 & 0 & 0 \\\\\n0 & -i & 0 & 0 \\\\\n\\end{array}\\right),\\]\n\n\\begin{equation}\\label{rho3}\n (\\rho^3)^\\alpha_\\beta=\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 1 \\\\\n0 & 0 & -i & 0 \\\\\n0 & i & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n\\end{array}\\right).\n\\end{equation}\n\nNote that the electromagnetic tensor $F^\\alpha_\\beta$ is the real part of the complex tensor $\\mathcal{F}^\\alpha_\\beta$. Equation (\\ref{DecGenLorents}) holds now if we define the complex conjugate tensor to be $\\bar{\\mathcal{F}}^\\alpha_\\beta=\\sum_{j=1}^3 (\\bar{\\rho}_j)^\\alpha_\\beta \\bar{F}_j$.\n\nThe Majorana-Oppenheimer matrices can be derived from the connection of the 4-potential and the Faraday vector, as follows. It is known that if $A_\\mu$ is the 4-potential of the electromagnetic field, then $E_j=F_{0j}=\\partial_jA_0-\\partial_0A_j= A_{0,j}-A_{j,0}$, where $\\partial_\\mu=\\frac{\\partial}{\\partial x^\\mu}$  and $B^j=(\\nabla\\times\\mathbf{A})^j$, where $\\nabla\\times$ is the curl which is applied to the vector potential. Thus, the the $j$ component of the Faraday vector is connected to the potential as\n\\begin{equation}\\label{FaradayAnd4potent}\n F^j= ( {\\rho}^j)^{\\alpha\\beta}\\partial_\\alpha A_\\beta=( {\\rho}^j)^{\\alpha\\beta} A_{\\beta,\\alpha} \\,,\n\\end{equation}\nshowing that the matrices $\\rho^j$ connect the 4-potential to the Faraday vector of the electromagnetic field.\nUsing (\\ref{complex tensor}), we can define a differential operator which connects the 4-potential with the the complex electromagnetic tensor $\\mathcal{F}^\\alpha_\\beta$:\n\\begin{equation}\\label{AtoCalF}\n    \\mathcal{F}^\\alpha_\\beta=\\sum_{j=1}^3 (\\rho^j)^\\alpha_\\beta ( {\\rho}^j)^{\\nu\\mu}\\partial_\\nu A_\\mu=\\nabla\\times A\\,,\n\\end{equation}\nwhere the \\textit{curl} $\\nabla\\times$ on $M_c$ is defined as\n\\begin{equation}\\label{curlDef}\n   \\nabla\\times=\\sum_{j=1}^3 (\\rho_j)^\\alpha_\\beta ( {\\rho}^j)^{\\nu\\mu}\\partial_\\nu\\,.\n\\end{equation}\n\nWe will now derive a Lorentz group representation on $M_c$ based on the Majorana-Oppenheimer matrices. In Section 6 we will show that this representation can be identified with the known spin-half representation of the Lorentz group. To define this representation, it is enough to define the representation on the generators of boosts and rotations.\n\n\\begin{defn}\\label{pi-tildeDefn}\nDefine a representation $\\tilde{\\pi}$ on $M_c$ by defining generators $\\tilde{B}^j$ of boosts $\\mathfrak{B}^j$ in the direction of $x^j$  to be $\\frac{1}{2}\\rho^j$ and generators $\\tilde{R}^j$ of rotations $\\mathfrak{R}^j$ about the direction $x^j$ to be $\\frac{i}{2}\\rho^j$ .\n\\end{defn}\n\\begin{claim}\\label{pi-tildeRepr}\nThe representation $\\tilde{\\pi}$ is a Lorentz group representation  on $M_c$.\n\\end{claim}\nDirect calculation shows that the operators $\\tilde{R}^j=\\frac{i}{2}\\rho^j$ obey the same commutator relations as the generators of the rotation group $SO(3)$:\n\\[[\\tilde{R}^j,\\tilde{R}^k]=-\\varepsilon^{jk}_l\\tilde{R}^l,\\]\nwhere $\\varepsilon^{ijk}$ is the Levi-Civita tensor.\nMoreover, the $\\tilde{B}^j=\\frac{1}{2}\\rho^j$ matrices obey the same commutator relations as the generators of boosts in the Lorentz group:\n   \\[ [\\tilde{B}^j,\\tilde{B}^k]=\\varepsilon^{jk}_l\\tilde{R}^l,\\;\\;\\;\\;\n   [\\tilde{R}^j,\\tilde{B}^k]=\\varepsilon^{jk}_l\\tilde{B}^l\\]\n   This proves Claim \\ref{pi-tildeRepr}.\n\nThe complex adjoints $\\bar{\\sigma}^j=\\frac{-i}{2}\\bar{\\rho}^j$ and $\\frac{1}{2}\\bar{\\rho}^j$ satisfy similar relations. This leads to a second Lorentz group representation on $M_c$.\n\\begin{defn}\\label{pi-tildeDefnStar}\nDefine a representation $\\tilde{\\pi}^*$ on $M_c$ by defining generators $\\tilde{B}^j$ of boosts $\\mathfrak{B}^j$ in the direction of $x^j$  to be $\\frac{1}{2}\\bar{\\rho}^j$ and generators $\\tilde{R}^j$ of rotations $\\mathfrak{R}^j$ about the direction $x^j$ to be $\\frac{i}{2}\\bar{\\rho}^j$.\n\\end{defn}\nMoreover, the two sets of operators $\\{\\rho^j\\}$ and $\\{\\bar{\\rho}^k\\}$ commute:\n\\begin{equation}\\label{CommutinfMO}\n   [\\rho^j,\\bar{\\rho}^k]=0.\n\\end{equation}\nThus, the representations  $\\tilde{\\pi}$ and $\\tilde{\\pi}^*$ commute.\n Since \\[\\exp F^\\alpha_\\beta =\\exp(\\frac{1}{2}\\mathcal{F}^\\alpha_\\beta +\\frac{1}{2}\\bar{\\mathcal{F}}^\\alpha_\\beta)=\\exp(\\frac{1}{2}\\mathcal{F}^\\alpha_\\beta) \\exp(\\frac{1}{2}\\bar{\\mathcal{F}}^\\alpha_\\beta),\\]\n  the representation $\\pi$, associated with the tensor $F^\\alpha_\\beta$, can be decomposed as a product (\\ref{DecLorents}) of representations $\\tilde{\\pi}$ and $\\tilde{\\pi}^*$. This proves Claim \\ref{pi-decomposition}.\n\n\nIn addition to the above commutation relations, Majorana-Oppenheimer matrices also satisfy \\textit{anti-commutation relations}, which are very helpful for the calculation of the exponents of these matrices. The anti-commutation relations are\n\\begin{equation}\\label{CAR}\n  \\{\\rho^j ,\\rho^k\\}=\\frac{1}{2}(\\rho^j\\rho^k+\\rho^k \\rho^j)=\\delta^{jk}I\\,,\n\\end{equation}\nwhich implies that $(\\rho^j)^2=I$. Using these relation and the power series expansion of the exponent function, the exponents of these matrices are\n\\[\\exp (\\rho^j\\frac{\\omega}{2})= \\cosh \\frac{\\omega}{2} I +\\rho^j\\sinh \\frac{\\omega}{2}\\]\n\\[\\exp (\\rho^j\\frac{i\\omega}{2})= \\cos \\frac{\\omega}{2} I +i\\rho^j\\sin \\frac{\\omega}{2}\\,.\\]\n\n\nWe can define now explicitly the  representation under $\\tilde{\\pi}$ of the rotations and the boosts of the Lorentz group. For example, the rotation $\\mathfrak{R}^1$ about the $x^1$-axis  or rotation in the $x^2x^3$ plane by an angle $\\omega$ is represented by\n\\begin{equation}\\label{RotTildePi}\n \\tilde{\\pi}(\\mathfrak{R}^1)(\\omega)=\\Upsilon_{23}(\\omega)=\\exp (i\\rho^1\\frac{\\omega}{2})= \\cos \\frac{\\omega}{2} I -i\\rho^1\\sin\\frac{\\omega}{2}\n\\end{equation}\n\\[=\\left(\n                                       \\begin{array}{cccc}\n                                         \\cos \\omega\/2 & -i\\sin \\omega\/2 & 0 & 0 \\\\\n                                        -i\\sin \\omega\/2  &  \\cos \\omega\/2 & 0 & 0 \\\\\n                                         0 & 0 &  \\cos \\omega\/2 & -\\sin \\omega\/2 \\\\\n                                        0&0& \\sin \\omega\/2  &  \\cos \\omega\/2\\\\\n                                       \\end{array}\n                                     \\right)\\,.\\]\nThe representation $\\tilde{\\pi}^*$ of this rotation is given by $\\bar{\\Upsilon}_{23}(\\omega)$, the complex conjugate of the above matrix. In the $x^2x^3$ plane, both $\\tilde{\\pi}(\\mathfrak{R}^1)$ and $\\tilde{\\pi}^*(\\mathfrak{R}^1)$ define a rotation by an angle $\\omega\/2$, and their product $\\Upsilon_{23}(\\omega)\\bar{\\Upsilon}_{23}(\\omega)=\\Lambda_{23}(\\omega)$, defined by (\\ref{RepPiR1}), is a rotation by an angle $\\omega$ in this plane. However, in the plane $x^0x^1$, the rotation $\\Lambda_{23}(\\omega)$ under $\\pi$ is the identity, while both $\\Upsilon_{23}(\\omega)$ and $\\bar{\\Upsilon}_{23}(\\omega)$ define a ``complex rotation\" in the plane $\\{x^0+ix^1:\\;\\;x^0,x^1\\in \\mathbb{R}\\}$ by angles $-\\omega\/2$ and $\\omega\/2$, respectively, indicating the there is a rotation around the $x^1$ axis. This observation needs further understanding.\n\n\nSimilarly, the boost in the $x^1$-direction with parameter $\\omega$ is\n\\begin{equation}\\label{Boosttildepi}\n \\tilde{\\pi}(\\mathfrak{B}^1)(\\omega)=\\Upsilon_{01}(\\omega)=\\exp (\\rho^1\\frac{\\omega}{2})= \\cosh \\frac{\\omega}{2} I +\\rho^1\\sinh \\frac{\\omega}{2}\\end{equation}\n\\[=\\left(\n                                       \\begin{array}{cccc}\n                                         \\cosh \\omega\/2 & \\sinh \\omega\/2 & 0 & 0 \\\\\n                                        \\sinh \\omega\/2  &  \\cosh \\omega\/2 & 0 & 0 \\\\\n                                         0 & 0 &  \\cosh \\omega\/2 &-i \\sinh \\omega\/2 \\\\\n                                        0&0& i\\sinh \\omega\/2  &  \\cosh \\omega\/2\\\\\n                                       \\end{array}\n                                     \\right).\\]\n The representation $\\tilde{\\pi}^*$ of this boost is given by $\\bar{\\Upsilon}_{01}(\\omega)$, the complex conjugate of the above matrix. In the $x^0x^1$ plane, both $\\tilde{\\pi}(\\mathfrak{B}^1)$ and $\\tilde{\\pi}^*(\\mathfrak{B}^1)$ define boosts in the $x^1$ direction with rapidity $\\omega\/2$, and their product $\\Upsilon_{01}(\\omega)\\bar{\\Upsilon}_{01}(\\omega)=\\Lambda_{01}(\\omega)$, defined by (\\ref{RepPiB1}), is a boost with rapidity $\\omega$ in the $x^1$ direction. However, in the $x^2x^3$ plane, the boost $\\Lambda_{01}(\\omega)$ under $\\pi$ is the identity, while both $\\Upsilon_{01}(\\omega)$ and $\\bar{\\Upsilon}_{01}(\\omega)$ define ``complex boosts\" in the plane $\\{x^2+ix^3:\\;\\;x^2,x^3\\in \\mathbb{R}\\}$, with rapidity  $-\\omega\/2$ and $\\omega\/2$, respectively, and the product of these boosts is the identity.\n\nNote that the operators $\\Upsilon_{\\alpha\\beta}$ are isometries on the space $M_c$. The representation  $\\tilde{\\pi}$, defined by the operators $\\Upsilon_{\\alpha\\beta}$, is a spin 1\/2 representation of the Lorentz group on $M_c$. Similarly, the representation $\\tilde{\\pi}^*$ is defined by the operators $\\bar{\\Upsilon}_{\\alpha\\beta}$, which are obtained by taking the complex conjugate of $\\Upsilon_{\\alpha\\beta}$. This is also a spin 1\/2 representation of the Lorentz group on $M_c$, and it commutes with the representation $\\tilde{\\pi}$. In section \\ref{SecSpinors}, we will show that the representation $\\tilde{\\pi}$ can be identified with the usual representation of the Lorentz group on the \\textit{spinors}.\n\nLet  $F_\\alpha ^\\beta$ be the electromagnetic tensor of some electromagnetic field. This tensor is covariant under the representation $\\pi$ of the Lorentz group. What is the covariance of the complex tensor $\\mathcal{F}_\\alpha ^\\beta$ associated with $F_\\alpha ^\\beta$? We will show the following.\n\\begin{claim}\\label{covarftildeF}\nThe covariance of the electromagnetic tensor $F_\\alpha ^\\beta$ under the representation $\\pi$ is equivalent to the covariance of the corresponding tensor $\\mathcal{F}_\\alpha ^\\beta$ under the representation $\\tilde\\pi$.\n\\end{claim}\nLet $\\Lambda$ be any element of the representation $\\pi$ of the Lorentz group. By (\\ref{DecLorents}), there is an element $\\Upsilon$ of $\\tilde{\\pi}$ such that $\\Lambda=\\Upsilon\\bar\\Upsilon$. Using that $\\mathcal{F},\\Upsilon \\in \\operatorname{span} \\rho^j$, these operators commute with any $\\bar{\\rho}^k$ and thus with $\\mathcal{\\bar F}, \\bar\\Upsilon$.  Thus,\n\\[ F' = \\Lambda F\\Lambda^{-1}= \\frac{1}{2} (\\Upsilon\\bar\\Upsilon\\mathcal{F}\\bar\\Upsilon^{-1}\\Upsilon^{-1} + \\Upsilon\\bar\\Upsilon\\mathcal{\\bar F}\\bar\\Upsilon^{-1}\\Upsilon^{-1})\\]\\[ = \\frac{1}{2} (\\Upsilon\\mathcal{F}\\Upsilon^{-1} + \\bar\\Upsilon\\mathcal{\\bar F}\\bar\\Upsilon^{-1}) = \\frac{1}{2} (\\mathcal{F'}+\\mathcal{\\bar F'}).\\]\nThis proves Claim \\ref{covarftildeF}.\n\n\n\n\\section{Definition of the prepotential}\n\nDenote by $P$ a point in Minkowski space $M$ at which we want to define\nthe prepotential. We will call $P$ the observer's point and denote his coordinates by $x=x^\\mu$. Denote by $\\check{x}(\\tau)=\\check{x}^\\mu (\\tau): \\mathbb{R}\\rightarrow M$ the worldline of the charge $q$ generating our electromagnetic field, as a function of its proper time. Let the point $Q=\\check{x}(\\tau(x))$ be the unique point of intersection of the past light cone at $P$ with the\nworldline $\\check{x}(\\tau)$ of the charge. The time on the worldline of the charge corresponding to this intersection is uniquely determined by the point $x$. It is called the \\textit{retarded time} and will be denoted by $\\tau(x)$.  Note that radiation emitted at $Q$ at the retarded time will reach $P$ at time $t=x^0\/c$ corresponding to this point. Our prepotential $S(x)$ will depend on the relative position\n\\begin{equation}\\label{PositionDef}\n   r(x)=x-\\check{x}(\\tau(x))\n\\end{equation}\nof the charge at retarded time $\\tau(x)$, see Figure \\ref{chargePotent}.\n\\begin{figure}[h!]\n  \\centering\n\\scalebox{0.4}{\\includegraphics{pointChargePoten.pdf}}\n  \\caption{The four-vectors associated with an observer and a moving charge.}\\label{chargePotent}\n\\end{figure}\n\n The relative position $r$ is a null vector. Hence,\n  \\begin{equation}\\label{NullDecomp}\n   (r^0+r^3)(r^0-r^3)=(r^1+ir^2)(r^1-ir^2).\n\\end{equation}\nIn this decomposition, we have split the four coordinates of $r$ into two groups: the first group contains the time component $r^0$ and the third spatial component $r^3$. The second group contains the first two spatial components $r^1$ and $r^2$.\n  We can now define a dimensionless complex constant $\\zeta$, as follows.\n   \\begin{defn}\\label{zetaDef}\n For any null-vector $r$ in $M_c$, we define a dimensionless complex scalar $\\zeta$ by\n \\begin{equation}\\label{zetaDefEq}\n    \\zeta(r)=\\frac{r^0+r^3}{r^1-ir^2}=\\frac{r^1+ir^2}{r^0-r^3}.\n \\end{equation}\n  For any $x\\in M$ and any world-line $\\check{x}(\\tau)$, we define $\\zeta (x)= \\zeta(r(x))$, where the relative position $r(x)$ is defined by (\\ref{PositionDef}).\n \\end{defn}\n\n  The scalar $\\zeta(r)$ can be identified as a simple function of the stereographic projection of the direction of the vector part $\\mathbf{r}$ of $r$ from the celestial sphere to the Argand plane, as follows. Since $r$ is a null vector, it can be decomposed as $(|\\mathbf{r}|,\\mathbf{r})$ and is defined uniquely by its vector part $\\mathbf{r}$. Define $\\hat{\\mathbf{r}}=\\mathbf{r}\/|\\mathbf{r}|$, a unit vector in the direction vector of $\\mathbf{r}$, and a null vector $\\hat{r}=(|\\hat{\\mathbf{r}}|,\\hat{\\mathbf{r}})$. Since $\\zeta (r) $ is the same for any multiple of $r$, we have $\\zeta (r)=\\zeta (\\hat{r})$. Then, as shown in \\cite{PeR} p. 11, $\\zeta (r)=\\zeta (\\hat{r})=e^{i\\varphi} cot\\frac{\\theta}{2}$, where $\\varphi,\\theta$ are the angles of standard spherical coordinates of $\\hat{r}$. This shows that $\\zeta (r) $ depends only on the direction from the observer to the charge at the retarded time.\n\nThere is no way to associate to every null vector $r$ a non-trivial scalar which is invariant under the usual representation $\\pi$. However, for the representation $\\tilde{\\pi}$ we have:\n\\begin{claim}\\label{zetaInvar}\nFor any null vector $r$, the complex constant $\\zeta(r)$, defined by (\\ref{zetaDefEq}), is invariant under the representation $\\tilde{\\pi}$ on $M_c$.\n\\end{claim}\nThe boost $\\Upsilon _{01}$ of $\\tilde{\\pi}$ in the $x^1$-direction is defined by (\\ref{Boosttildepi}). Applying this boost to the vector $r$ yields\n\\[\\Upsilon _{01} (r)=\\cosh(\\omega\/2)(r^0,r^1,r^2,r^3)+\\sinh(\\omega\/2)(r^1,r^0,-ir^3,ir^2).\\]\nUsing (\\ref{zetaDefEq}), we have\n\\[\\Upsilon _{01} (\\zeta)= \\zeta(\\Upsilon _{01} (r))=   \\frac\n{\\cosh(\\omega\/2)(r^0+r^3)+\\sinh(\\omega\/2)(r^1+ir^2)}{\\cosh(\\omega\/2)(r^1-ir^2)+\\sinh(\\omega\/2)(r^0-r^3)}=\\]\n\\[\\frac{r^0+r^3}{r^1-ir^2}\\cdot\n\\frac{1+\\tanh(\\omega\/2)(r^1+ir^2)\/(r^0+r^3)}{1+\\tanh(\\omega\/2)(r^0-r^3)\/(r^1-ir^2)}\n=\\frac{r^0+r^3}{r^1-ir^2}\\cdot\n\\frac{1+\\tanh(\\omega\/2)\/\\bar{\\zeta}}{1+\\tanh(\\omega\/2)\/\\bar{\\zeta}}\n=\\zeta,\\]\nshowing that $\\zeta$ is invariant under the boost $\\Upsilon _{01}$. Similarly, one may show the\ninvariance of $\\zeta$ under any boost and rotation of the representation $\\tilde{\\pi}$. This proves Claim \\ref{zetaInvar}.\n\n\\begin{defn}\\label{SdefDef} We define the \\textit{prepotential} $S(x)$ at  point $x$ of an electromagnetic field generated by a moving charge $q$ as\n\\begin{equation}\\label{Sdef}\n   S(x)=\\frac{q}{2}\\ln \\zeta (x)=\\frac{q}{2}\\ln \\zeta(r (x)),\n\\end{equation}\nwhere $\\zeta (r)$ is defined by (\\ref{zetaDefEq}) and the relative position of the charge $r(x)$ is defined by (\\ref{PositionDef}).\n\\end{defn}\nFrom Claim \\ref{zetaInvar}, it follows that the prepotential is invariant under the representation $\\tilde{\\pi}$ on $M_c$.\nConsider an electromagnetic field  generated by any charge distribution $\\sigma (x)$. The scalar prepotential for this field is defined by\n \\begin{equation}\\label{scalPotGenera}\n    S(x) =\\frac{1}{2}\\int\\limits_{K^-(x)}\\zeta(r)\\sigma(x+r)d^3r,\n\\end{equation}\nwhere $K^-(x)$ denotes the backward light-cone at $x$, and the integration is by the 3D volume on this cone.\n\n\nThe geometric meaning of this prepotential can be understood if we introduce \\textit{relativistic bipolar coordinates} $(\\varrho^0,\\varrho^1,\\theta,\\varphi)$:\n\\[x^0=\\varrho^0\\cosh\\theta,\\;\\;x^1=\\varrho^1\\cos \\varphi,\\;\\;x^2=\\varrho^1\\sin \\varphi,\\;\\;x^3=\\varrho^0\\sinh \\theta.\\]\nThe angle $\\varphi$ in these coordinates is the same angle $\\varphi$ occurring in polar and spherical coordinates. However, the angle $\\theta$ differs from the angle $\\theta$ of spherical coordinates. These coordinates fit better to our model as they give a simple form for the prepotential. The light-cone in these coordinates has the simple form $\\varrho^0=\\varrho^1=\\varrho$, a hyperplane. For any null-vector $r$ on the light-cone, the invariant constant $\\zeta(r)$, defined by (\\ref{zetaDefEq}), in relativistic bipolar coordinates is\n\\[\\zeta(r)=\\frac{r^0+r^3}{r^1-ir^2}=\\frac{\\varrho e^{\\theta}}{\\varrho e^{-i\\varphi}}=e^{\\theta+i\\varphi}.\\]\nThe prepotential of a moving charge is\n\\[  S(x)=\\frac{q}{2}(\\theta+i\\varphi),\\] which is a multiple of the complex angle $\\theta+i\\varphi$ of the relative position $r(x)$ of the charge.\n\n\\section{ The prepotential and the Faraday vector of the field}\n\n\\subsection{Connection between the prepotential and the Faraday vector of a field}\n\nThe 4-potential will be defined as a conjugate of the gradient of the prepotential $S(x)$.\nSince the prepotential $S(x)$ is invariant under the representation $\\tilde{\\pi}$, we want our\nconjugation to commute with this representation. By Definition \\ref{pi-tildeDefn} of the representation $\\tilde{\\pi}$, the conjugation needs to commute with any operator $\\rho^j$. Using (\\ref{CommutinfMO}), we can choose any operator $\\bar{\\rho}^k$ to define the conjugation. Since, in (\\ref{NullDecomp}), and in the definition of $\\zeta$, we choose the third spatial component to join the time component, also here we chose the $\\tilde{\\pi}$ \\textit{covariant conjugation }$\\mathcal{C}$ on $M_c$ to be given by multiplication by $\\bar{\\rho}^3$:\n\\begin{equation}\\label{ConjOperDef}\n  \\mathcal{C}=\n (\\bar{\\rho}^3)^\\alpha_\\beta=\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 1 \\\\\n0 & 0 & i & 0 \\\\\n0 & -i & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n\\end{array}\\right)\\,.\n\\end{equation}\n\nThe following diagram shows the derivation of the complex tensor $\\mathcal{F}$ of the field from its prepotential $S(x)$:\n \\[\\begin{CD} S @>\\nabla>> \\nabla S @>\\mathcal{C}>>A= \\mathcal{C}(\\nabla S) @>\\nabla\\times>> \\mathcal{F}.\n \\end{CD}\\]\nBy differentiating the prepotential and taking its conjugate, we obtain the 4-potential $A$. Then we use (\\ref{AtoCalF}) to get from $A$ to the complex tensor $\\mathcal{F}$ of the field. The components $F_j$ of the Faraday vector can be found by (\\ref{FaradayAnd4potent}) as\n\\begin{equation}\\label{FfromS}\n F_j={\\partial}_\\nu (\\rho^j)^{\\mu\\nu}(\\bar{\\rho}^3)_\\mu^\\lambda{\\partial}_\\lambda S\\,.\n\\end{equation}\n\\begin{claim}\\label{ClCovStoF}\nThe Faraday vector $ F_j$ defined by (\\ref{FfromS}) is invariant under the representation $\\tilde{\\pi}$\n\\end{claim}\nThis claim follows from the following facts. Claim \\ref{zetaInvar} implies that $S(x)$ is $\\tilde{\\pi}$ invariant. By definition of the conjugation, $(\\bar{\\rho}^3)_\\mu^\\lambda$ is  $\\tilde{\\pi}$  invariant. The remaining operators together form a  $\\tilde{\\pi}$ invariant operator.\n\nThe last expression can be simplified, if for each $j=1,2,3$, we introduce a new tensor $((\\alpha^j)^{\\mu\\lambda}=(\\rho^j)^{\\mu\\nu}(\\bar{\\rho}^3)_\\mu^\\lambda$. Then we can rewrite (\\ref{FfromS}) as\n\\begin{equation}\\label{FfromSAlf}\n F_j={(\\alpha^j)^{\\mu\\lambda}\\partial}_\\nu {\\partial}_\\lambda S\\,.\n\\end{equation}\nThe new tensors are\n\\[(\\alpha^1)^{\\mu\\lambda}=\\left(\n\\begin{array}{cccc}\n0 & 0 & -i & 0 \\\\\n0 & 0 & 0 & -1 \\\\\n-i & 0 & 0 & 0 \\\\\n0 & -1& 0 & 0 \\\\\n\\end{array}\\right),\\;\\;(\\alpha^2)^{\\mu\\lambda}=\\left(\n\\begin{array}{cccc}\n0 & i & 0 & 0 \\\\\ni & 0 & 0 & 0\\\\\n0 & 0 & 0 & -1  \\\\\n0 & 0& -1& 0 \\\\\n\\end{array}\\right)\\]\n\\[(\\alpha^3)^{\\mu\\lambda}=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0& 0 & -1 \\\\\n\\end{array}\\right)\\,.\\]\nThus, (\\ref{FfromSAlf}) defines the following explicit formulas for the connection between the prepotential $S(x)$ and the Faraday vector.\n\\begin{equation}\n                               \\begin{array}{l}\n                              F_1=-2(S_{,13}+iS_{,02})  \\\\\n                                F_2=-2(S_{,23}-iS_{,01}) \\\\\n                               F_3= S_{,00}+S_{,11}+S_{,22}-S_{,33}\\\\\n                             \\end{array}\\label{Fas derS}\n\\end{equation}\n\nBased on these formulas, we obtain the following gauges for the prepotential.\n\\begin{claim}\\label{claimGauge}\nThe prepotential $S(x)$ is not determined uniquely by the electromagnetic field. Let $g(x)$ be a function on $M_c$ from the following list, or a combination of such functions:\n\\begin{enumerate}\n  \\item $g(x)=g(x^1,x^2)$ is harmonic, that is, $g_{,11}+g_{,22}=0$,\n  \\item $g(x)=g(x^0,x^3)$ satisfies the wave equation, that is, $g_{,00}-g_{,33}=0$,\n  \\item $g(x)$ is of order 1 in $x^\\mu$.\n\\end{enumerate}\nThen the transformation of the prepotential\n\\begin{equation}\\label{gagueTrans}\n    S'(x)=S(x)+g(x)\n\\end{equation}\ndoes not affect the field. A function $g(x)$ of this type will be called the \\textit{prepotential gauge}.\n\\end{claim}\nTo prove the claim, it is enough to check that the field of the gauge $g(x)$ defined by (\\ref{Fas derS}) is zero. In case $(i)$, we have $g_{,0}=g_{,3}=0$. Since each term of the components $F_1,F_2$ of the field in\n(\\ref{Fas derS}) involves differentiation by $x^0$ or by $x^3$, these components are zero. The third component $F_3$ is zero since $g_{,00}=g_{,33}=0$ and $g_{,11}+g_{,22}=0$. Similarly, for case $(ii)$, we have $g_{,1}=g_{,2}=0$. Since each term of the components $F_1,F_2$ of the field in\n(\\ref{Fas derS}) involve differentiation by $x^1$ or by $x^2$, these components are zero. The third component $F_3$ is zero since $g_{,11}=g_{,22}=0$ and $g_{,00}-g_{,33}=0$. Since each component of the field involves second derivatives, it will vanish for any function $g(x)$ of order 1 in $x^\\mu$. This completes the proof of the claim.\n\n\nWe demonstrate now the use of  formulas  (\\ref{Fas derS}) and Claim \\ref{claimGauge} in calculating the field of a rest charge at the origin from its prepotential.\nConsider a rest charge $q$ at the origin. The worldline of this charge is $\\check{x}(\\tau)=(\\tau,0,0,0)$. Thus, the relative position of the charge is\n $r=(|x|,x^1,x^2,x^3)$, where $|x|=\\sqrt{(x^1)^2+(x^2)^2+(x^3)^2}$.\nFrom (\\ref{zetaDefEq}) and (\\ref{Sdef}),\n\\[S'(x)=\\frac{q}{2}\\ln\\frac{|x|+x^3}{x^1-ix^2}=\\frac{q}{2}\\ln(|x|+x^3)-\\frac{q}{2}\\ln(x^1-ix^2)\\,.\\]\nThe function $\\ln(x^1-ix^2)$ is harmonic, and, by Claim \\ref{claimGauge}$(i)$, is a prepotential gauge.\n\\begin{exam} For a rest charge $q$ at the origin, we define the prepotential to be\n\\begin{equation}\\label{ResTPrepot}\nS(x)=\\frac{q}{2}\\ln(|x|+x^3)\\,.\n\\end{equation}\n\\end{exam}\nNote that in this case the prepotential is well defined for all $x\\ne0$ and could be chosen to be real valued.\n\\begin{claim}\\label{ClResPoint}\nThe prepotential $S(x)$ defined by (\\ref{ResTPrepot}) defines the electromagnetic field of the rest charge $q$ at the origin to be $\\mathbf{E}=\\frac{\\mathbf{r}}{|\\mathbf{r}|^3}$ and $\\mathbf{B}=0$. Moreover, $S(x)$ satisfies the wave equation $\\square  S(x)=0$.\n\\end{claim}\nSince $S(x)$ is independent of $x^0$, we have $S_{,0}=0$. Direct calculation shows that $\\frac{\\partial}{\\partial x^j}|x|=\\frac{x^j}{|x|}$. Thus, we get\n\\[S_{,1}=\\frac{q}{2|x|}\\frac{x^1}{|x|+x^3},\\;\\;\\;S_{,2}=\\frac{q}{|x|2}\\frac{x^2}{|x|+x^3}\\,,\\]\nand \\[ S_{,3}=\\frac{q}{2}\\frac{1}{|x|+x^3}\\left(\\frac{x^3}{|x|}+1\\right)=\\frac{q}{2\\vert x\\vert}\\,.\\]\nThe 4-potential $A$ in this case is\n\\[A=\\mathcal{C}\\nabla S= \\frac{q}{2\\vert x\\vert}(1, -i\\frac{x^2}{|x|+x^3},i\\frac{x^1}{|x|+x^3},0)\\,.\\]\nThis potential behaves similarly to the usual 4-potential: when $|x|$ approaches infinity, it behaves like $\\frac{1}{|x|}$.\n\nBy use of (\\ref{Fas derS}), we get\n\\[ F_1=-2(S_{,13}+iS_{,02})=-2\\frac{\\partial}{\\partial x^1}S_{,3}=-2\\frac{\\partial}{\\partial x^1}\n\\frac{q}{2|x|}=\\frac{q x^1}{|x|^3}\\,,\\]\n\\[ F_2=-2(S_{,23}-iS_{,01})=-2\\frac{\\partial}{\\partial x^2}S_{,3}=-2\\frac{\\partial}{\\partial x^2}\n\\frac{q}{2|x|}=\\frac{q x^2}{|x|^3}\\,.\\]\n\nTo calculate $F_3$ using (\\ref{Fas derS}), we need first to calculate $S_{,\\mu\\mu}$.\n\\[S_{,11}=\\frac{\\partial}{\\partial x^1}\\frac{q}{2}\n\\frac{x^1}{|x|(|x|+x^3)}=\\frac{q}{2}\\frac{|x|^2+x^3|x|-2(x^1)^2-x^3(x^1)^2\/|x|}{|x|^2(|x|+x^3)^2},\\]\n\\[S_{,22}=\\frac{\\partial}{\\partial x^2}\\frac{q}{2}\n\\frac{x^2}{|x|(|x|+x^3)}=\\frac{q}{2}\\frac{|x|^2+x^3|x|-2(x^2)^2-x^3(x^2)^2\/|x|}{|x|^2(|x|+x^3)^2}.\\]\nThis implies that\n\\[S_{,11}+S_{,22}=\\frac{q x^3}{2|x|^3}\\,.\\]\nSince $S_{,00}=0$ and $S_{,33}=\\frac{-q x^3}{2|x|^3}$, equation (\\ref{Fas derS}) yields\n\\[F_3=\\frac{q x^3}{|x|^3}\\]\nand $\\square  S(x)=S_{,00}-(S_{,11}+S_{,22})-S_{,33}=0$. This proves the claim.\n\nAs a second example, we will find now the prepotential of an infinitely long charged rod. We start with a prepotential\nof a finite rod on the $x^3$ axis, from $x^3=-L$ to $x^3=L$, with charge density $\\sigma$. Denote by $l$ the position of the charge on the rod, so that $-L<l<L$. The worldline of the charge is $\\check{x}(\\tau)=(\\tau, 0,0,l)$, implying that $r=(\\xi,x^1,x^2,x^3-l)$, with\n  \\begin{equation}\\label{xiDef}\n \\xi(l)=\\sqrt{(x^1)^2+(x^2)^2+(x^3-l)^2}=\\sqrt{\\varrho^2+(x^3-l)^2}\\,,\n\\end{equation}\nwhere $\\varrho=\\sqrt{(x^1)^2+(x^2)^2}.$ Thus, from (\\ref{zetaDefEq}), the zeta factor for the charge is \\[\\zeta(r)=\\frac{\\xi(l)+x^3-l}{x^1-ix^2},\\]\nand the prepotential of the rod is\n\\begin{equation}\\label{PrepRod1}\n    S(x)=\\frac{\\lambda}{2}\\int_{-L}^L \\ln\\frac{\\xi(l)+x^3-l}{x^1-ix^2}dl=\\frac{\\lambda}{2}\\int_{-L}^L \\ln(\\xi(l)+x^3-l)dl-\\frac{\\lambda}{2}\\int_{-L}^L \\ln(x^1-ix^2)dl\\,,\n\\end{equation}\nwith $\\xi$ defined by (\\ref{xiDef}).\n\nThe second integral is equal to $2L \\ln(x^1-ix^2)$, which is a harmonic function in $x^1,x^2$, and, by Claim \\ref{claimGauge}$(i)$, is a gauge and will be removed from the prepotential. For the first integral, we use a substitution $z=x^3-l$, implying $dl=-dz$. After the substitution, the first integral is\n  \\[-\\int_{x^3+L}^{x^3-L} \\ln(\\sqrt{\\varrho^2+z^2}+z)dz=\\sqrt{\\varrho^2+z^2}-z\\ln(\\sqrt{\\varrho^2+z^2}+z),\\]\n   and the prepotential (\\ref{PrepRod1}) is equivalent to\n\\[\n    S(x)=\\frac{\\lambda}{2}\\left( (x^3+L)\\ln(\\xi(-L)+x^3+L)-(x^3-L)\\ln(\\xi(L)+x^3-L) +\\xi(L)-\\xi(-L)\\right)\\,.\\]\n\nUsing that $\\xi(L)+x^3-L=\\frac{\\varrho^2}{\\xi(L)-x^3+L}$ and $\\xi(L)-\\xi(-L)=\\frac{-4x^3L}{\\xi(L)+\\xi(-L)}$, we get\n\\[S(x)=\\frac{\\lambda}{2}\\left(-x^3\\ln\\varrho^2+L \\ln\\varrho^2-\\frac{4x^3L}{\\xi(L)+\\xi(-L)}\\right)\\]\\[ +\\frac{\\lambda}{2}\\left((x^3+L)\\ln(\\xi(-L)+x^3+L)+(x^3-L)\\ln(\\xi(L)-x^3+L)\\right).\\]\nThe expression $L\\ln \\varrho^2=L \\ln((x^1)^2+(x^2))$ is a harmonic function in $x^1,x^2$, and, by Claim \\ref{claimGauge}$(i)$, is a gauge and will be removed from the prepotential.\n\nIn order to be able to pass to the limit $L\\rightarrow\\infty$, we will subtract from the prepotential a gauge function with a behavior similar to $S(x)$ for large $L$. An example of such a function is $2x^3\\ln 2L-2x^3$, which by Claim \\ref{claimGauge}$(iii)$ is a gauge. We obtain\n\\[S(x)=\\frac{\\lambda}{2}\\left(-x^3\\ln\\varrho^2-\\{\\frac{4x^3L}{\\xi(L)+\\xi(-L)}-2x^3\\}\\right)\\]\n\\[ +\\frac{\\lambda}{2}\\left((x^3+L)\\ln(\\xi(-L)+x^3+L)+(x^3-L)\\ln(\\xi(L)-x^3+L)-2x^3\\ln 2L\\right).\\]\nSince $\\lim_{L\\to\\infty}\\xi(\\pm L)\/L=1$, we have\n\\[\\lim_{L\\to\\infty}\\frac{4x^3L}{\\xi(L)+\\xi(-L)}-2x^3=\\lim_{L\\to\\infty}\\frac{4x^3}{\\xi(L)\/L+\\xi(-L)\/L}-2x^3=0\\]\nand\n\\[\\lim_{L\\to\\infty}(x^3+L)\\ln(\\xi(-L)+x^3+L)+(x^3-L)\\ln(\\xi(L)-x^3+L)-2x^3\\ln 2L=0\\]\nThus, the prepotential of an infinite rod is\n\\begin{equation}\\label{RodInfPrep}\n   S(x)=-\\frac{\\lambda}{2}x^3\\ln\\varrho^2=-\\lambda x^3\\ln\\sqrt{(x^1)^2+(x^2)^2}\\,.\n\\end{equation}\n\nBy applying formulas (\\ref{Fas derS}) to this prepotential, we get\n\\[\n                               \\begin{array}{l}\n                              F_1=-2(S_{,13}+iS_{,02})=\\lambda \\frac{2x^1}{\\varrho^2} \\\\\n                                F_2=-2(S_{,23}-iS_{,01})=\\lambda \\frac{2x^2}{\\varrho^2} \\\\\n                               F_3= S_{,00}+S_{,11}+S_{,22}-S_{,33}=0\\,,\n                             \\end{array}\\]\nwhich implies that the electric field is $\\mathbf{E}(x)=\\frac{2\\lambda }{\\varrho^2}(x^1,x^2,0)$ and that there is no magnetic field, as is known for the field of a charged rod.\n\n\\subsection{The  Faraday vector  $F$ from the prepotential of a moving charge}\n\nConsider now any charge $q$ with worldline $\\check{x}(\\tau)$. In order to be able to calculate the field of this charge from its prepotential $S(x)$ defined by (\\ref{Sdef}), we will use formulas (\\ref{Fas derS}). To apply this formula, we have to calculate some second partial derivatives of  $S(x)$. The difficulty is that $S$ is a function of $\\zeta(r(x))$, and $r(x)$ depends also on the worldline $\\check{x}(\\tau)$. The idea is to present these derivatives in covariant form. This means that we want to express them by covariant vectors, namely, the relative position $r$ of the observer with respect to the charge at the retarded time $\\tau(x)$, the 4-velocity $u=\\frac{d\\check{x}}{d\\tau}(\\tau(x))$ and the 4-acceleration $a=\\frac{du}{d\\tau}(\\tau(x))$ at the retarded time.\n\nFrom the definition (\\ref{Sdef}) of the prepotential $S(x)$ its derivative by $x^\\mu$ is\n\\begin{equation}\\label{PotDerivat1}\n    S(x)_{,\\mu}=\\frac{q}{2}\\frac{\\zeta_{,\\mu}}{\\zeta}=\\left( \\frac{r^0_{,\\mu} +  r^3_{,\\mu}}{r^1 -ir^2} -\\frac{(r^0+r^3)(r^1_{,\\mu}-ir^2_{,\\mu})}{(r^1 -ir^2)^2}\\right)\\frac{r^1 -ir^2}{r^0+r^3}\n=\\frac{ r^0_{,\\mu} +  r^3_{,\\mu}}{r^0+r^3} -\\frac{r^1_{,\\mu}-ir^2_{,\\mu}}{r^1 -ir^2} \\,.\n\\end{equation}\nThus, to be able to calculate the derivatives of  $S(x)$, we need a formula for the derivatives of the relative position $r(x)$ defined by (\\ref{PositionDef}). The partial derivative $r_{,\\mu}$ of the position vector $r(x)$ by $x^\\mu$ is\n\\[r_{,\\mu}=x_{,\\mu}-y_{,\\mu}(\\tau(x))= x_{,\\mu}-\\partial y\/\\partial \\tau \\,\\tau_{,\\mu}= x_{,\\mu}-u \\tau_{,\\mu}.\\]\nSince $r(x)$ is null, we have $r\\cdot r_{,\\mu}=0$, where $a\\cdot b=\\eta_{\\mu\\nu}a^\\mu b^\\nu$ denote the Minkowski inner product. Thus,\n\\[0=r\\cdot r_{,\\mu}=r\\cdot x_{,\\mu}- (r\\cdot u) \\tau_{,\\mu},\\]\nwhich imply that the derivative of the retarded time is\n\\begin{equation}\\label{tauDer}\n   \\tau(x)_{,\\mu}=\\frac{r_\\mu}{r\\cdot u}.\n\\end{equation}\nThis mean that varying the observers position by $dx^\\mu$ will cause the parameter $\\tau$ to change infinitesimally by $\\frac{r_\\mu}{r\\cdot u}dx^\\mu$. Since $(r)^2=0$ and $(u)^2=1$, from the Cauchy-Schwartz inequality it follows that $r\\cdot u$ is always non-zero. So, equation (\\ref{tauDer}) is always well defined.\n\n\nSubstituting this into the derivative of the position vector we get\n\\begin{equation}\\label{PositionDer}\n   r^\\nu_{,\\mu}=x^\\nu_{,\\mu}-\\frac{r_\\mu u^\\nu}{r\\cdot u}=\\delta^\\nu_\\mu-\\frac{r_\\mu u^\\nu}{r\\cdot u},\n\\end{equation}\nwhere $\\delta^\\nu_\\mu$ denotes the Kronecker delta function. This formula could be understood as follows. An infinitesimal variation  $dx^\\mu$ of the observers position causes also a change of the charge position at the retarded time. The relative position $r$ under such variation is effected by the two changes. The first term in (\\ref{PositionDer}) the effect of infinitesimal variation of the observers position, while the second one express change of the charge position at the retarded time.\n\nSubstituting derivative of the position vector (\\ref{PositionDer}) in (\\ref{PotDerivat1}) we obtain a covariant form for the derivative of the prepotential $S(x)$ as a function of the the relative position $r$ and the 4-velocity of the charge $u$ at the retarded time.\nTo calculate the electromagnetic field of of a moving charge from its prepotential $S(x)$, according (\\ref{Fas derS})\nwe need to calculate some second derivatives of $S(x)$. This could be done by differentiating $S_{,\\mu}$. The expression of $S_{,\\mu}$ involve involve only non-constant vectors $r$ and $u$. The derivatives of $r$ are given by (\\ref{PositionDer}). The derivatives of $u$, by use of (\\ref{tauDer}), are\n\\begin{equation}\\label{uDer}\n    u_{,\\mu}=\\frac{\\partial u}{\\partial \\tau}\\tau (x)_{,\\mu}=\\frac{a r_\\mu}{r\\cdot u}\\,.\n\\end{equation}\nSince an infinitesimal change of the observers position by $dx^\\mu$ causes the change of $\\tau$  by amount $\\frac{r_\\mu}{r\\cdot u}dx^\\mu$, the variation of the 4-velocity $u$ of the charge caused by this change is a multiple of the acceleration $a$ of the charge by this amount.\n\nTo simplify the result of the differentiation by use of computer algebra we can  introduce for any $j=1,2,3$  an anti-symmetric scalar product on vectors of $M_c$ by\n\\begin{equation}\\label{antisysmpriductDef}\n   [a,b]_j=(\\rho^j)_{\\alpha\\beta}a^\\alpha b^\\beta=\\rho^j a\\cdot b,\\;\\;\\;\\;a,b\\in M_c \\,.\n\\end{equation}\nBy use of computer algebra (\\ref{Fas derS}) and using the above formulas for the derivatives we get\n\\begin{equation}\\label{FmovigChargeFronSfinal}\nF^j=q\\left(\\frac{\\left[r,u\\right]_j}{(r\\cdot u)^3}+\\frac{\\left[r,a\\right]_j(r\\cdot u)-\\left[r,u\\right]_j(r\\cdot a)}{2(r\\cdot u)^3}\\right)\n\\end{equation}\n\nUsing $1+3$ decomposition of 4-vectors it can be shown that these formulas coincide with the know field of a moving charge as derived from the Lienard-Wiecher potential, see \\cite{Jackson} p.664. Thus, we have shown that the prepotential $S(x)$ of a moving charge defined by (\\ref{Sdef}) produces by use of (\\ref{Fas derS}) the known electromagnetic field of such charge. Moreover, the superposition principle and that the connection of the prepotential to the field is linear, for a general electromagnetic field, which is produce by a collection of moving charges, by applying  (\\ref{Fas derS}) to the prepotential $S(x)$ defined by formula (\\ref{scalPotGenera}) will recover properly this field.\n\n\n\n\n\n\\section{The Maxwell equations of the prepotential}\n\nIn this section  we define the  Maxwell equations for the prepotential $S(x)$ of the electromagnetic field.\nConsider an electromagnetic field generated by sources $J^\\mu=(\\rho,j^1,j^2,j^3)$.\nThe pair of the scalar Maxwell equations\n\\[\\nabla\\cdot \\mathbf{E}=\\rho,\\;\\;\\nabla\\cdot \\mathbf{B}=0\\]\nfor the Faraday vector $\\mathbf{F}=\\mathbf{E}+ic\\mathbf{B}$ can be rewritten as one equation\n\\begin{equation}\\label{Max scal}\n    \\nabla\\cdot \\mathbf{F}=\\rho\n\\end{equation}\nSubstitute the values of $\\mathbf{F}$ from (\\ref{Fas derS})  to get\n\\[F_{1,1} +F_{2,2}+F_{3,3}=-2S_{,113}-2iS_{,102}-2S_{,223}+2iS_{,201}+S_{,300}+S_{,311}+S_{,322}-S_{,333}\\]\\[=\n  S_{,003}- S_{,113}-S_{,223}-S_{,333} =\\rho\\,.\\]\n So, the first Maxwell equation for $S(x)$ is:\n\\begin{equation}\\label{Max S 0}\n   \\partial_3( \\square  S)=\\rho\n\\end{equation}\n\nThe vector Maxwell equations are\n\\begin{equation}\\label{Max vec}\n    \\nabla\\times \\mathbf{E}+\\dot{\\mathbf{B}}=0,\\;\\;\\nabla\\times \\mathbf{B}-\\dot{\\mathbf{E}}=\\mathbf{j}.\n\\end{equation}\nThey can be rewritten by use of the Faraday vector as one equation\n\\begin{equation}\\label{Max scal}\n    \\mathbf{F}_{,0}+i \\nabla\\times \\mathbf{F}=-\\mathbf{j}.\n\\end{equation}\n\nSubstitute the values of $\\mathbf{F}$ from (\\ref{Fas derS}))  for the first component to get\n\\[F_{1,0} +iF_{3,2}-iF_{2,3}=-2S_{,013}-2iS_{,002}+i(S_{,200}+S_{,211}+S_{,222}-S_{,233})+2i(S_{,323}-iS_{,301})\\]\\[=\n   i(-S_{,002}+S_{,211}+S_{,222}+S_{,233})=-j^1\\,.\\]\n So, the second Maxwell equation for $S(x)$ is:\n\\begin{equation}\\label{Max S 1}\n    -i\\partial _2( \\square  S)=-j^1\\,.\n\\end{equation}\n For the second component we get\n\\[F_{2,0} +iF_{1,3}-iF_{3,1}=-2S_{,023}+2iS_{,001}-2i(S_{,313}+iS_{,302}-i(S_{,100}+S_{,111}+S_{,122}-S_{,133})\\]\n\\[= i(S_{,100}-S_{,111}-S_{,122}-S_{,133})=-j^2\\,.\\]\n So, the third Maxwell equation for $S(x)$ is:\n\\begin{equation}\\label{Max S 2}\n   i\\partial _1( \\square  S)=-j^2\\,.\n\\end{equation}\n For the third component we get\n\\[F_{3,0} +iF_{2,1}-iF_{1,2}=S_{,000}+S_{,011}+S_{,022}-S_{,033}-2i(S_{,123}-iS_{,101})+2i(S_{,213}+iS_{,202})\\]\n\\[=  S_{,000}-S_{,011}-S_{,022}-S_{,033}=-j^3\\,.\\]\n So, the forth Maxwell equation for $S(x)$ is:\n\\begin{equation}\\label{Max S 3}\n   \\partial_0( \\square  S)=-j^3\\,.\n\\end{equation}\n\n\nNote since the left-hand side of these equations is a co-vector we lower the index of the sources of the field by $J_\\mu=(\\rho,-j^1,-j^2,-j^3)$. We will also apply the conjugation $\\mathcal{C}$ defined by (\\ref{ConjOperDef}) to the gradient $\\partial$ to get\n\\[ \\mathcal{C}^\\nu_\\mu \\partial_\\nu =(\\partial_3,-i\\partial_2,i\\partial_1,\\partial_0)\\,.\\]\nWith this notation we can rewrite our 4 Maxwell equations (\\ref{Max S 0}), (\\ref{Max S 1}),(\\ref{Max S 2}) and\n(\\ref{Max S 3}), as\n\\begin{equation}\\label{Maxwell eqn}\n  \\mathcal{C}^\\nu_\\mu \\partial_\\nu ( \\square  S(x))=J_\\mu(x)\\,,\n\\end{equation}\nor in operator form\n\\begin{equation}\\label{Maxwell eqnOper}\n  \\mathcal{C}\\nabla( \\square  S(x))=J(x)\\,.\n\\end{equation}\nOur Maxwell equations (\\ref{Maxwell eqnOper})have some similarity to the ones obtained by use of Clifford algebras, see \\cite{Baylis}.\n\nThe Maxwell equation(s) for the prepotential we obtained is a third order system of equations. But this system can be decomposed into two systems of equations. The first system is a first order one $\\mathcal{C}\\nabla \\Phi=J$. Using that\n$\\mathcal{C}$ is a symmetry, meaning $\\mathcal{C}^2=I$, this system can be rewritten as $\\nabla \\Phi(x)=\\mathcal{C}J(x)$. The meaning of such system is looking for a scalar potential $\\Phi(x)$ for a vector-field $\\mathcal{C}J(x)$. A similar formula is obtained in \\cite{Barut} p.98 for the spinor form of Maxwell equation describing the connection of the sources of the field to the Faraday vector. The second system is a  known second order system $\\square  S(x)=\\Phi(x)$ for $\\Phi(x)$, solved in the previous step. Thus, equation (\\ref{Maxwell eqnOper}) is equivalent to\n\\begin{equation}\\label{MaxTwoEqua}\n    \\nabla \\Phi(x)=\\mathcal{C}J(x),\\;\\;\\square  S(x)=\\Phi(x)\\,.\n\\end{equation}\n\n\nClaim \\ref{ClResPoint} show that $S(x)$ satisfy the wave equation $\\square  S(x)=0$ outside the rest charge. By using the covariance of such property, this will hold for the prepotential of any moving charge.  Using the superposition principle the prepotential of any electromagnetic field will satisfy the wave equation\n\\begin{equation}\\label{waveS}\n    \\square  S(x)=0\n\\end{equation}\nfor $x$ outside the sources of the field.\n\nWe will demonstrate now the use of the Maxwell equations (\\ref{Maxwell eqnOper}) for derivation of the prepotential and the field of a current $I$ in a thin long straight wire. We will assume that the wire is positioned on the $x^3$-axis.  The field of such current is generated by negatively charged electrons with line density $\\tilde{\\lambda}$ moving with velocity $\\mathbf{v}$ in the negative direction of the $x^3$-axis. These charges are compensated by positive charges of the same density resting in the wire. To express the fact that the wire is thin we will use 2D Dirac-delta $\\delta(\\varrho)$ for $\\varrho=\\sqrt{(x^1)^2+(x^2)^2}$ with the property that the integral of this function over a domain $D$ in $x^1,x^2$ plane is equal 1 if the point $(0,0)$ is in $D$ and 0 otherwise.\n\nThe current density of the electron sources of the field are given by\n$ j(e)^\\mu=-\\tilde{\\lambda}\\delta(\\varrho)\\gamma(1,0,0,-v\/c),$\nwhere $\\gamma=\\gamma(v)=(1-v^2\/c^2)^{-1\/2}$. So, the effective density will be $\\lambda=\\tilde{\\lambda}\\gamma$. For the positive charges\n$ j(p)^\\mu=\\lambda\\delta(\\varrho)(1,0,0,0).$\nSo, the total current of the sources of the field will be $ j^\\mu=\\lambda\\delta(\\varrho)(0,0,0, \\frac{v}{c})$ and using that $I=\\lambda v$,  $ j^\\mu=\\delta(\\varrho)(0,0,0, \\frac{I}{c}).$\nThe equation for $\\Phi(x)$ from (\\ref{MaxTwoEqua}) become\n\\[\\Phi_{,0}=- \\frac{I}{c}\\delta(\\varrho),\\;\\;\\Phi_{,1}= \\Phi_{,2}=\\Phi_{,3}=0\\,.\\]\nIntegrating the first equation and using that the average value of the Dirac delta is zero we get $\\Phi(x)=-\\frac{I}{c}\\delta(\\varrho)x^0 .$\n\nNow we need to solve the equation  (\\ref{MaxTwoEqua}) $\\square  S(x)=\\Phi(x)$ for the prepotential $S(x)$. From \\cite{Landau2} it follows that the electromagnetic field must be normal to the wire, implying $F_3=0$ and by use of (\\ref{Fas derS}) that $ S_{,00}+S_{,11}+S_{,22}-S_{,33}=0$. Subtracting from the last  equation $\\square  S(x)=\\Phi(x)$ we get\n\\begin{equation}\\label{prepEqWire1}\n   S_{,11}+S_{,22}=-\\frac{1}{2}\\Phi(x)=\\frac{I}{2c}\\delta(\\varrho)x^0\n\\end{equation}\nand adding these equations we get\n\\begin{equation}\\label{prepEqWire2}\n   S_{,00}-S_{,33}=\\frac{1}{2}\\Phi(x)=-\\frac{I}{2c}\\delta(\\varrho)x^0\\,.\n\\end{equation}\n\nWe can look for a solution for  equations (\\ref{prepEqWire1}) and (\\ref{prepEqWire2}). We will decompose the prepotential $S(x)$ as a sum of two components $S(x)=S_1(x)+S_2(x)$, where each component will satisfy one of the equations and vanish by the differential operator of the second one. We will look  for $S_1(x)$  of the form $S_1(x)=\\frac{I}{2c}x^0f(x^1,x^2)$ for some function $f(x^1,x^2)$. The equation (\\ref{prepEqWire1}) for $f$  become $\\nabla ^2f(x^1,x^2)=\\delta(\\varrho)$. It is known (\\cite{arfken} p.760) that the solution of such equation is $f(x^1,x^2)=\\frac{-1}{2\\pi}\\ln\\varrho$. Thus, $S_1(x)=\\frac{I}{4\\pi c}x^0\\ln\\varrho$ satisfy equation (\\ref{prepEqWire1}) and vanish by the differential operator of equation (\\ref{prepEqWire2}).\n\nIf we take  $S_2(x)$ to be of the form $S_2(x)=\\frac{I}{2c}\\delta(\\varrho)x^0g(x^3)$, then such function will vanish differential operator of (\\ref{prepEqWire1}) and for (\\ref{prepEqWire2}) will give $g(x^3)=-(x^3)^2\/2$.\nThus the prepotential $S(x)$ for a straight wire is\n\\begin{equation}\\label{PrepStWire}\n   S(x)=\\frac{I}{4c}x^0\\left(\\frac{1}{\\pi}\\ln\\varrho- (x^3)^2\\delta(\\varrho)\\right)\\,.\n\\end{equation}\n\nBy used of (\\ref{Fas derS}) we can now calculate the Faraday vector of the field outside the wire. Since $\\ln\\varrho$ is harmonic outside the wire we have $F_3= S_{,00}+S_{,11}+S_{,22}-S_{,33}=0.$ Direct calculation give\n$ F_1=-2(S_{,13}+iS_{,02})=-i\\frac{I}{2\\pi c}\\frac{x^2}{\\varrho^2}$ and\n$ F_2=-2(S_{,23}-iS_{,01})=i\\frac{I}{2\\pi c}\\frac{x^1}{\\varrho^2}.$\n Thus, the electric field is zero and  the magnetic field is\n \\[B_1=\\frac{ I}{2\\pi c^2} \\frac{-x^2}{\\varrho^2},\\;\\; B_2=\\frac{ I}{2\\pi c^2} \\frac{x^1}{\\varrho^2},\n \\;\\; B_3=0.\\]\n This corresponds to the known formulas for such field (see \\cite{Feynman} p.14-4).\n\n\\section{Prepotential, spinors and the Dirac matrices}\\label{SecSpinors}\n\nThe prepotential $S(x)$ of a moving charge that we introduced was based on the decomposition (\\ref{NullDecomp}) of any null vector in $M_c$. This decomposition involve four terms, which could be considered as coordinated of the null vector $r$ in a new basis. This basis on  $M_c$ is called the \\textit{ Bondi null tetrad} and is called also Newman-Penrose basis. It  is defined as\n\\[\\mathbf{ n}_0=\\mathbf{n}=\\frac{1}{\\sqrt{2}}(\\mathbf{e}_0+\\mathbf{e}_3),\\;\n    \\mathbf{ n}_1=\\bar{\\mathbf{m}}=\\frac{1}{\\sqrt{2}}(\\mathbf{e}_1-i\\mathbf{e}_2),\\;\\]\n\\begin{equation}\\label{NPbasisNotaion}\n      \\mathbf{ n}_2=\\mathbf{m}=\\frac{1}{\\sqrt{2}}(\\mathbf{e}_1+i\\mathbf{e}_2),\\;\n      \\mathbf{ n}_3=\\mathbf{l}=\\frac{1}{\\sqrt{2}}(\\mathbf{e}_0-\\mathbf{e}_3).\n\\end{equation}\nFor the significance of the Bondi tetrad see \\cite{PeR}, \\cite{ODonnell} and\n\\cite{FriedmanNP}. All vectors $ \\mathbf{ n}$ are null vectors in $M_c$. Note that application of complex conjugation on $M_c$, which is equivalent\nto replacing $i$ with $-i$, maps the Bondi tetrad into itself, but exchanges $\\mathbf{ n}_1$ with\n$\\mathbf{ n}_2$. This mean that also here $i$ is a pseudo-scalar which with the change\nof orientation, that may be expressed by change of the order of the basis vectors, changes its sign.\n\n\nWe will denote  by $y^\\mu$ the coordinates of a vector in $M$ with respect\nto the Bondi tetrad, meaning $x^\\mu \\mathbf{e}_\\mu=y^\\mu \\mathbf{n}_\\mu.$\nThen, the relation between the coordinates is\n\\begin{equation}\\label{r in term q}\n   x^0=\\frac{1}{\\sqrt{2}}(y^0+y^3), \\; x^1=\\frac{1}{\\sqrt{2}}(y^1+y^2),\\;\n    x^2=i\\frac{1}{\\sqrt{2}}(y^2-y^1),\\;\n    x^3=\\frac{1}{\\sqrt{2}}(y^0-y^3),\n\\end{equation}\nor inversely,\n\\begin{equation}\\label{q in term r}\n   y^0=\\frac{1}{\\sqrt{2}}(x^0+x^3), \\;\n   y^1=\\frac{1}{\\sqrt{2}}(x^1+ix^2),\\; y^2=\\frac{1}{\\sqrt{2}}(x^1-ix^2),\\;\n   y^3=\\frac{1}{\\sqrt{2}}(x^0-x^3).\n\\end{equation}\n\nThe coordinate transformation could be expressed by the transfer\nmatrix $L=L_j^k$ given by\n\\begin{equation}\\label{trasfer matrix}\n    L=\\frac{1}{\\sqrt{2}}\\left(\n        \\begin{array}{cccc}\n          1 & 0& 0 & 1 \\\\\n           0 & 1 & i & 0\\\\\n          0 & 1 & -i & 0 \\\\\n           1 & 0 & 0 & -1\\\\\n        \\end{array}\n      \\right),\\;\\; L^{-1}=\\frac{1}{\\sqrt{2}}\\left(\n        \\begin{array}{cccc}\n          1 & 0& 0 & 1 \\\\\n          0 & 1 & 1 & 0 \\\\\n          0 & -i & i & 0 \\\\\n          1 & 0 & 0 & -1 \\\\\n        \\end{array}\n      \\right).\n\\end{equation}\nThus,\n\\begin{equation}\\label{coord tranf}\n    y^\\nu=L_\\mu^\\nu x^\\mu,\\; \\;\n    x^\\nu=(L^{-1})_\\mu^\\nu y^\\mu\\,.\n\\end{equation}\n\n\nThe metric tensor $\\eta$ in the Bondi tetrad is given by\n\\begin{equation}\\label{NPmetric tensor}\n    \\tilde{\\eta}=\\left(\n        \\begin{array}{cccc}\n          0 & 0 & 0 & 1 \\\\\n          0 & 0 & -1 & 0 \\\\\n          0 &  -1 & 0 & 0 \\\\\n          1 & 0 & 0& 0 \\\\\n        \\end{array}\n      \\right)= \\tilde{\\eta}^{-1}.\n\\end{equation}\n\nThe bilinear symmetric scalar product of two  4-vectors  $\\mathbf{a}=a^\\mu \\mathbf{n}_\\mu$ and\n$\\mathbf{b}=b^\\mu \\mathbf{n}_\\mu$ is given by\n\\[\\mathbf{a}\\cdot \\mathbf{b}=\\tilde{\\eta} _{\\mu\\nu}a^\\mu b^\\nu. \\]\nThis, for example, implies that\n\\begin{equation}\\label{Zero Int NP}\n    (\\mathbf{a})^2=\\mathbf{a}\\cdot \\mathbf{a}=0\\;\\;\\Leftrightarrow\\;\\; a^0a^3=a^1a^2 \\;\\;\\Leftrightarrow\n    \\;\\; \\frac{a^2}{a^0}=\\frac{a^3}{a^1}\\;\\;\\Leftrightarrow\\;\\; \\frac{a^1}{a^0}=\\frac{a^3}{a^2}\\,.\n\\end{equation}\nIn case $a^0=0$, the last identities need to be reversed $\\frac{a^0}{a^2}=\\frac{a^1}{a^3},\\;\\frac{a^0}{a^1}=\\frac{a^2}{a^3}.$\nIn these coordinates, the lowering of indices is denoted by $a_\\mu=\\tilde{\\eta}_{\\mu\\nu} a^\\nu$. For example,\n\\begin{equation}\\label{LowerinIndexinNP}\n   y_0=y^3,\\;\\;y_1=-y^2,\\;\\;y_2=-y^1,\\;\\;y_3=y^0.\n\\end{equation}\n\nTo understand the connection with the spinors of the representation $\\tilde{\\pi}$, which was based on the complex electromagnetic tensor  $\\mathcal{F}$, defined by (\\ref{complex tensor}) we  will present  this  tensor in Bondi tetrad. We will denote by $\\widetilde{\\mathcal{F}}$\nthe matrix of $\\mathcal{F}$ in this representation. By the usual formula of basis transformation we get\n\\begin{equation}\\label{FNP0}\n    \\widetilde{\\mathcal{F}}^\\beta_\\alpha=L\\mathcal{F}^\\beta_\\alpha L^{-1}=\\frac{1}{2}\\left(\n                                         \\begin{array}{cccc}\n                                           F_3 & F_1-iF_2 & 0 & 0 \\\\\n                                           F_1+iF_2 & -F_3 & 0 & 0 \\\\\n                                           0 & 0 & F_3 &F_1-iF_2 \\\\\n                                           0 & 0 & F_1+iF_2 & -F_3 \\\\\n                                         \\end{array}\n                                       \\right)=\\sum_{j=1}^3 (\\tilde{\\rho}^j)^\\alpha_\\beta F_j.\n\\end{equation}\nThis show that the complex electromagnetic tensor become decomposable in Bondi tetrad. This tensor has two invariant subspaces $M_0=span_\\mathbb{C}\\{\\mathbf{n}_0,\\mathbf{n}_1\\}$ and  $M_1=span_\\mathbb{C}\\{\\mathbf{n}_2,\\mathbf{n}_3\\}.$\n\n\\begin{claim}\\label{SpinRepPiTilde}\nThe representation $\\tilde{\\pi}$ of the Lorentz group on $M_c$ introduced in Definition \\ref{pi-tildeDefn} acts as a spin half representation on each of subspaces  $M_0=span_\\mathbb{C}\\{\\mathbf{n}_0,\\mathbf{n}_1\\}$ and  $M_1=span_\\mathbb{C}\\{\\mathbf{n}_2,\\mathbf{n}_3\\}$\n\\end{claim}\nFrom the decomposition (\\ref{FNP0}) of the complex electromagnetic tensor  $\\widetilde{\\mathcal{F}}$, we see that in Bondi tetrad Majorana-Oppenheimer matrices $(\\tilde{\\rho}^j)^\\alpha_\\beta$ act on each $M_k$ as the the usual Pauli matrices $(\\tilde{\\rho}^j)^\\alpha_\\beta M_k=\\sigma^jM_k$, with\n\\begin{equation}\\label{PauliMatDef}\n   \\sigma^1=\\left(\n              \\begin{array}{cc}\n                0   & 1 \\\\\n                1 &0 \\\\\n              \\end{array}\n            \\right),\\;\\; \\sigma^2=\\left(\n              \\begin{array}{cc}\n                0   & -i \\\\\n                i &0 \\\\\n              \\end{array}\n            \\right),\\;\\; \\sigma^3=\\left(\n              \\begin{array}{cc}\n                1   & 0 \\\\\n                0 &-1 \\\\\n              \\end{array}\n            \\right)\\,.\n\\end{equation}\n Generators $\\tilde{B}^j$ of boosts $\\mathfrak{B}^j$ in direction of $x^j$ in representation $\\tilde{\\pi}$ were defined (Definition \\ref{pi-tildeDefn}) by $\\frac{1}{2}\\rho^j$. So, on the subspaces $M_k$ they will act by $\\tilde{\\pi}(\\tilde{B}^j)=\\frac{1}{2}\\sigma^k$. Similarly,  generators $\\tilde{R}^j$ of rotation $\\mathfrak{R}^j$ about the direction $x^j$ in representation $\\tilde{\\pi}$ were defined by $\\frac{i}{2}\\rho^j.$  So, on the subspaces $M_k$ they will act by $\\tilde{\\pi}(\\tilde{R}^j)=\\frac{i}{2}\\sigma^k$. This identify the subspace $M_k$ with spinors and the representation $\\tilde{\\pi}$ with the spin half representation of the Lorentz group on the spinors. This proves the Claim.\n\nThe following tensor decomposition can help us to understand the connection between the representations $\\tilde{\\pi}$ and $\\tilde{\\pi}^*$. We define a tensor decomposition of a $4\\times 4$ matrix as a\ntensor product of $2\\times 2$ matrices  by use of the binary representation of numbers. Each\nof our indices $\\mu=0,1,2,3$ can be considered as a pair of indices $(\\mu_0,\\mu_1)$ with value in $\\mu_k\\in\\{0,1\\}$ by\n\\[0\\longmapsto 00,\\;\\;1\\longmapsto 01,\\;\\;2\\longmapsto 10,\\;\\;3\\longmapsto 11\\,.\\]\nThe \\textit{tensor decomposition} of a two tensor $D=D_{jk}$ is defined by\n\\begin{equation}\\label{tesnor deconposition}\n  D=a\\otimes b, \\;\\;  D_{\\mu\\nu}=D_{(\\mu_0,\\mu_1)(\\nu_0,\\nu_1)}=a_{\\mu_0\\nu_0}\\,b_{\\mu_1\\nu_1}\\,.\n\\end{equation}\nFor example, the tensor $\\tilde{\\eta}$ can be decomposed as $\\tilde{\\eta}=\\left(\n                                                     \\begin{array}{cc}\n                                                       0   & 1 \\\\\n                                                       -1 & 0 \\\\\n                                                     \\end{array}\n                                                   \\right)\\otimes \\left(\n                                                     \\begin{array}{cc}\n                                                       0   & 1 \\\\\n                                                       -1 & 0 \\\\\n                                                     \\end{array}\n                                                   \\right):=\\tilde{\\eta}_2\\otimes \\tilde{\\eta}_2\\,.$\n\nThe following properties of the tensor decomposition can be verified directly from the definition:\n\\[ a\\otimes (b+c)=(a\\otimes b)+( a\\otimes c), \\;\\;k(a\\otimes b)=(ka)\\otimes b=a\\otimes kb\\]\nand\n\\begin{equation}\\label{tensor prop prod}\n   ( a\\otimes b)(c\\otimes d)=ac \\otimes bd,\n\\end{equation}\nwhere $a,b,c,d$ are $2\\times 2$ matrices and $k$ is a constant.\n\nWith this notation,the matrices $\\rho^j$ in Bondi tetrad therefore become\n\\begin{equation}\\label{KjinNP}\n    (\\widetilde{\\rho}^j)^\\mu_\\nu=I_2\\otimes \\frac{1}{2}\\sigma^j.\n\\end{equation}where $\\sigma^j$ denote the  Pauli matrices, and $I_2$ is the  $2\\times 2$ identity matrix.\nIn Bondi tetrad,\nthe tensor $(\\mathcal{F}^*)^\\alpha_\\beta$ becomes\n\\[(\\widetilde{\\mathcal{F}}^*)^\\alpha_\\beta=L\\mathcal{F}^* L^{-1}=\\sum_j\\bar{F}_j\\sigma^*_j\\otimes I_2\\,.\n\\]\nand\n \\[ (\\widetilde{\\bar{\\rho}}^k)^\\mu_\\nu=\\sigma^*_j\\otimes I_2\\,.\\]\nThis and (\\ref{tensor prop prod}) explain why the representations $\\tilde{\\pi}$ and $\\tilde{\\pi}^*$\ncommute.\n\nThe connection  (\\ref{FfromSAlf}) of the prepotential to the Faraday vector of the field is expressed by operators $\\alpha^j:=\\rho^j \\mathcal{C}$. Like the Majorana-Oppenheimer matrices $\\rho^j$, also these matrices satisfy the canonical anti-commutation relations (\\ref{CAR}). In the  Bondi tetrad, see \\cite{FGW}), the matrices $\\alpha^j$ take the usual form  of the Dirac's $\\alpha$-matrices\n$\\left(                \\begin{array}{cc}\n                  \\sigma _j & 0 \\\\\n                  0 & -\\sigma _j \\\\\n                \\end{array}\n              \\right)$\n   where $\\sigma ^j$ are the Pauli matrices (see, for example \\cite{Berestetskii}).\n\n   Note that the matrices $\\rho^j$, which define the representation $\\tilde{\\pi}$,\nalso satisfy the canonical anti-commutation relations (\\ref{CAR}). However, the set $\\rho_1,\\rho_2,\\rho_3$  cannot be completed by a forth anticommuting matrix, needed for the Dirac equation. Only after the conjugation they become Dirac's $\\alpha$-matrices.\n\n\n\nParts of this paper were done in final undergraduate projects at Applied Physics Department at Jerusalem College of Technology by students S. Gwertzman, D. C. Gootvilig and M. Eliyahu under the supervision of the author. The author would like to thank  T. Scarr for editorial proof and E. Yudkin for help with the computer algebra  use of Mathematica.\n\n\\\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe recent discovery of superconductivity in F-doped LaFeAsO with $T_c$ of $\\sim$ 26 K created a flurry of excitement in condensed matter research.~\\cite{Kamihara08} Rapidly, numerous families of Fe-based superconductors such as $RE$FeAs(O$_{1-x}$F$_{x}$) (1111-family, $RE$ = rare earth), $M$Fe$_{2}$As$_{2}$ (122-family, $M$ = Ba, Ca, Sr, K, Cs ...), LiFeAs\/NaFeAs (111-family) and Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ (11-family)~\\cite{Takahashi08,Ren08,Rotter08,Tapp08,Hsu08, Yeh_EPL2008,Fang08} have been found and investigated.\nAlthough the Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ chalcogenides share with their pnictogen-based cousins similar structure, based on planar layers of edge sharing [Fe(Se,Te)]$_4$ tetrahedra, some significant differences were rapidly revealed.\nAmong them are the large magnetic moment ($\\sim$ 2.0-2.5 $\\mu_B$) and the double stripe magnetic ordering of the parent Fe$_{1+y}$Te compound.~\\cite{Fruchart_1975,Li09} Magnetic stripes are found in other Fe-based parent compounds such as BaFe$_2$As$_2$ or LaFeAsO stripes in pnictides, but they are rotated by 45$^{\\circ}$ around the $c$-axis with respect to Fe$_{1+y}$Te,~\\cite{Bao09,Li09, Ma_PRL2009, Han_PRL2009} with significantly lower magnetic moments on Fe in these compound (typically 0.4-1.0 $\\mu_B$\/iron~\\cite{delaCruz_Nature2008, Singh08}).\n\nThe magnetic ordering is accompanied by a structural transition ($T_N \\sim 67 K$),~\\cite{Fruchart_1975} which can be progressively suppressed on substitution of isovalent Se at the Te 2b Wyckoff position. This also results in emergence of superconductivity with a $T_c^{max} \\sim$ 14 K at ambient pressure at the optimum doping.~\\cite{Hsu08, Yeh_EPL2008}\nHowever, numerous reports have recently demonstrated that the excess of interstitial Fe between the chalcogenide layers, even after doping with Se, could affect both superconducting and magnetic properties,  \\textit{e.g.} suppression of transition temperature $T_c$ or lowering shielding fraction, as well as leading to the appearance of weakly localized magnetic states.~\\cite{Martinelli_PRB2010,Khasanov09,Zhang09,Wen09,Liu09,Bendele10,Xu_PRB2011}\n\n\nRaman spectroscopy is an ultimate noninvasive tool and can allow systematic studies of the temperature dependence of the phonon spectrum as function of Se and excess of interstitial Fe. For example, Raman spectroscopy has proved useful in the understanding of the structural, magnetic and electronic properties of superconducting pnictides. ~\\cite{Hanh_PRB2009,Litvinchuk08,Reznik09,Rahlenbeck09,Chauviere09, Fukuda_JPCS2008, LeTacon_PRB2008, Le Tacon09,Choi10, Gallais08, Choi_PRB2008, Fukuda_PRB2011, Um_PRB2012,Litvinchuk_PRB2011}\nAvailable Raman data on single crystals of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ is to date limited.\nIn refs.~\\onlinecite{Xia09} and~\\onlinecite{Okazaki11}, comparison between Se-substituted and parent Fe$_{1+y}$Te samples is made, but neither temperature dependence nor influence of the excess iron concentration on the lattice dynamics are discussed.\nIn ref.~\\onlinecite{Gnezdilov}, the authors study the evolution with temperature of the Raman spectra of Fe$_{1.05}$Te, with a particular emphasis on the anomalously large lineshape of the Te $A_{1g}$ phonon. It is argued to originate from a peculiar spin-orbital frustration effect, that leaves unaffected the $B_{1g}$ phonon after symmetry considerations.\n\nIn this paper, we focus mainly on this Raman active $c$-axis polarized optical phonon in Fe$_{1+y}$Te$_{1-x}$Se$_{x}$, which is common to all the iron-based superconductors families, and discuss its evolution with temperature for various Fe, $y$, and Se content, $x$. We can demonstrate that contrarily to the $A_{1g}$ mode, the observed temperature dependence is strongly affected by composition. To some extent, the phonon behavior through the various phases transitions (depending of the Se content) is consistent with those of the $c$-axis polarized Fe modes reported in the other families of Fe-based superconductors (\\textit{e.g.} 122 and 111). The narrowing of the phonon lineshape through the magnetic transition of the parent compound or the absence of renormalization through the superconducting one are for instance reported.\nOn the other hand, we also show that in some specific conditions, the behavior of the $B_{1g}$ mode seriously deviates from the aforementioned one. The phonon linewidth shows an anomalous broadening in the paramagnetic state of Fe$_{1+y}$Te parent compounds and an unusually strong dependence with the Se concentration in the doped compounds. Further anomalies, indicative of additional decay channels, are found when increasing the concentration of excess Fe $y$.\n\n\nTo try to get some insights about the influence of excess iron on the Raman phonon, we have carried out a Density Functional Theory (DFT) \\textit{ab initio} calculation within the local density approximation (LDA) of the phonon frequencies, including the effects of magnetism and Fe nonstoichiometry in the virtual crystal approximation (VCA). The measured frequencies are in good agreement with the predicted ones, including a softening with increasing Fe content in Fe deficient samples, but the effects of Fe nonstoichiometry are poorly reproduced.\n\nRecent studies of magnetic properties have revealed that excess Fe induces local magnetism and low energy magnetic fluctuations,~\\cite{Xu_PRB2011, Stock} but these can not be treated within our DFT approaches.\nGenerally speaking, whenever low-energy magnetic fluctuations are at play, the $B_{1g}$ phonon behavior deviates from the conventional harmonic picture and cannot be reproduced within our theoretical framework. This suggests that the reported anomalies originate from coupling between the $B_{1g}$ mode and these excitations.\n\n\\section{Experimental Details}\n\n\\begin{table}\n\\caption{\\label{tab:TableSample}A summary of the chemical compositions obtained by EDX and characteristic transitions temperatures of the various Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ samples used in this study.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\nSample Composition & T$_{N}$ &T$_{c}$\\\\\n\\hline\nFe$_{1.02}$Te  &   67 K &~~~---~~\\\\\nFe$_{1.09}$Te  &   65 K &~~~---~~ \\\\\n\\hline\nFe$_{1.00}$Te$_{0.78}$Se$_{0.22}$    &  ---~~ & 11.5 K  \\\\\nFe$_{0.99}$Te$_{0.69}$Se$_{0.31}$   &   ---~~ & 11 K    \\\\\nFe$_{0.98}$Te$_{0.66}$Se$_{0.34}$   &   ---~~ & 10.5 K  \\\\\nFe$_{0.95}$Te$_{0.56}$Se$_{0.44}$  &   ---~~ & 14 K    \\\\\n\n\\hline\nFe$_{1.05}$Te$_{0.58}$Se$_{0.42}$ & ---  & 11.5 K \\\\\nFe$_{1.08}$Te$_{0.73}$Se$_{0.27}$ & ---  & 9 K \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig_eigen.pdf}\n\\caption{(Color online) Room temperature Raman spectra of the Fe$_{1.02}$Te, Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$, and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples (see Table~\\ref{tab:TableSample}). Spectra have been shifted vertically for clarity. Eigendisplacements corresponding to the $A_{1g}$(Te\/Se) and $B_{1g}$(Fe) are represented schematically.}\n\\label{FigEigen}\n\\end{figure}\n\nIn order to discriminate the effects of Fe excess and of Se substitution, we have studied different groups of crystals, listed in Table~\\ref{tab:TableSample}. In the first group, no Se was present and only the Fe concentration was changed Fe$_{1+y}$Te (Fe$_{1.02}$Te, Fe$_{1.09}$Te). In the second group we changed the Se concentration while keeping the Fe concentration as close from 1 as possible (Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$). Finally, we also studied Se-substituted samples containing sizable excess iron (Fe$_{1.05}$Te$_{0.58}$Se$_{0.42}$, Fe$_{1.08}$Te$_{0.73}$Se$_{0.27}$).\nFe$_{1+y}$Te$_{1-x}$Se$_{x}$ single crystals were grown in sealed quartz tube as described elsewhere~\\cite{Liu10, Klein_PRB2010, Tamai_PRL2010, Gresty_JACS2009}.\nThe chemical compositions listed in Table~\\ref{tab:TableSample} were determined using energy dispersive x-ray spectroscopy (EDX), and the antiferromagnetic (AF) and superconducting (SC) transition temperatures $T_N$ and $T_c$ were measured by use of a superconducting quantum interference device magnetometer.\n\n\\begin{figure*}\n\\includegraphics[width=0.9\\linewidth]{Fig_summary.pdf}\n\\caption{(Color online) a) $B_{1g}$ phonon of the Fe$_{1.02}$Te sample for selected temperatures (room temperature, $T \\sim T_N$ and base temperature). Black squares are the data, and the red line is the fit following the procedure described in the text. Phonon intensity have been normalized and the spectra have been shifted vertically for clarity. b) Same plot for the Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ sample. c) same plot for the Fe$_{1.05}$Te$_{0.58}$Se$_{0.42}$ sample.}\n\\label{Fig1}\n\\end{figure*}\n\n\nAll Raman light scattering experiments were performed on freshly cleaved surface of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ single crystals. All the samples were mounted in a helium-flow cryostat allowing measurements between 5 K and room temperature. Spectra were taken in backscattering geometry through a JobinYvon LabRam 1800 single grating spectrometer equipped with a razor-edge filter and a Peltier-cooled charge-coupled-device camera.\nWe used a linearly polarized He$^{+}$\/Ne$^{+}$ mixed gas laser with $\\lambda$ = 632.817 nm for excitation.\nThe laser beam was focused through a 50$\\times$ microscope objective to $\\sim$5 $\\mu$m diameter spot on the sample surface. The power of the incident laser was kept less than 1 mW to avoid laser-induced heating. In order to determine the precise frequency of phonons for each temperature, Neon emission lines were recorded between each measurements. For data analysis, all phonon peaks were fitted by Lorentzian profiles, convoluted with the spectrometer resolution function (a Gaussian line of 2 cm$^{-1}$ full width at half maximum (FWHM)).\n\n\n\\section{Experimental Results}\n\\label{Results}\n\\subsection{Influence of Se doping}\n\\label{ResultsSe}\n\n\\begin{figure*}\n\\includegraphics[width=1\\linewidth]{A1g_no_excess.pdf}\n\\caption{(Color online) Upper panel: Temperature dependence of the $A_{1g}$(Te\/Se) mode frequency of the a) Fe$_{1.02}$Te, b) Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, c) Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, d) Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$, and e) Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples. Lower panel: Temperature dependence of the $A_{1g}$(Te\/Se) mode FWHM of the f) Fe$_{1.02}$Te, g) Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, h) Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, i) Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$, and j) Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples.}\n\\label{Fig_A1g}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=1\\linewidth]{B1g_no_excess.pdf}\n\\caption{(Color online) Upper panel: Temperature dependence of the $B_{1g}$(Fe) mode frequency of the a) Fe$_{1.02}$Te, b) Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, c) Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, d) Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$, and e) Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples. Lower panel: Temperature dependence of the $B_{1g}$(Fe) mode full-width-half-maximum of the f) Fe$_{1.02}$Te, g) Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, h) Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, i) Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$, and j) Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples. Solid lines are fits of the temperature dependence of the $B_{1g}$ phonon frequency and linewidth in the various samples using the anharmonic model described in the text.}\n\\label{Fig_B1g}\n\\end{figure*}\n\nIn Fig.~\\ref{FigEigen} we show the Raman spectrum measured at room temperature on the Se-free Fe$_{1.02}$Te sample, together with the Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$, Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ superconducting samples that all have a Fe stoichiometry close to 1. From symmetry consideration, one expects four Raman active modes:  $A_{1g}$(Te\/Se), $B_{1g}$(Fe), $E_{g}$(Te), $E_{g}$(Fe).\nRaman measurement have been performed in backscattering geometry, with the incident light polarization along the $a$-axis of the single crystal.\nAn analyzer has been used to check the phonon selection rules, but most of the measurements presented here have been performed without in order to maximize the phonon peak intensities.\n\nIn this scattering geometry, only the $A_{1g}$(Te\/Se) and $B_{1g}$(Fe) modes, sketched in Fig.~\\ref{FigEigen}, are present. As seen in Fig.~\\ref{FigEigen}, these modes are found at $\\sim$ 155 cm$^{-1}$ and $\\sim$ 197 cm$^{-1}$ at room temperature, respectively, in agreement with previous reports~\\cite{Xia09,Okazaki11,Gnezdilov}. These two peaks are much broader than in any of the other iron pnictides~\\cite{Rahlenbeck09, Chauviere09, Gallais08}. The large $A_{1g}$ mode linewidth ($\\sim$ 20 cm$^{-1}$ at room temperature in Fe$_{1.02}$Te, almost 3 times larger than the $A_{1g}$ As mode in BaFe$_2$As$_2$~\\cite{Rahlenbeck09}) has  been attributed to spin-orbital frustration effects.~\\cite{Gnezdilov} In the parent Fe$_{1.02}$Te single crystal, an additional peak was observed around 136 cm$^{-1}$. The origin of this mode remains unclear. It is temperature independent (and cannot, therefore, be attributed to the lowering of the crystal symmetry induced by the structural transition), and has been observed irrespective of the Fe excess concentration (see Fig.~\\ref{Fig3a}-a). It is not observed in the Se-rich compounds (see Fig.~\\ref{FigEigen}).\nIn contrast with the results of Xia \\textit{et al.} in ref.~\\onlinecite{Xia09}, claiming the disappearance of the $A_{1g}$ mode in the Raman spectra of FeTe$_{0.92}$ and Fe$_{1.03}$Te$_{0.7}$Se$_{0.3}$ with increasing Se concentration, the mode is clearly visible in all the investigated compounds.\n\nThe frequency of the two phonons is weakly dependent on Se contents: at the lowest recorded temperatures (5 K) it remains essentially constant for Se contents between 22$\\%$ and 34$\\%$ (Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$ and Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$), while a small hardening ($\\sim$ 2 cm$^{-1}$) is observed for the sample with 44$\\%$ of Se (Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$).\nThis latter effect may be caused by the significant Fe deficiency in the Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ sample (see Secs.~\\ref{DFTcalc} and ~\\ref{disc}). Within our experimental errors, this seems to be also the case for the $A_{1g}$ phonon, as trivially expected from the substitution of Te with lighter Se (the mode frequency going, in first approximation, as $M^{-1\/2}$, with $M$ denoting the reduced mass of the considered oscillator). The only noticeable exception, is the Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$ sample, where the $A_{1g}$ mode is broader and softer than in any other compounds. This may originate from an overlapping of the $A_{1g}$ mode with the 136 cm$^{-1}$ peak observed in the parent compounds. It seems that for this particular doping level the modes energies are still separated enough to cause an apparent broadening and shift to lower frequency of the total envelope, but not enough to allow to resolve them individually.\n\nIn Figs.~\\ref{Fig1}-a and -b, we show details of the fitting for the Fe$_{1.02}$Te and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples, that illustrates one of the main finding of our study, \\textit{i.e.} the strong changes in the temperature dependence of the $B_{1g}$ phonon on doping. In the undoped Fe$_{1.02}$Te sample, the $B_{1g}$ mode hardens and broaden with decreasing temperature down to $T_N$, and then softens and narrows down to base temperature. For the Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ sample with the highest Se concentration, we observed a continuous hardening and narrowing of the mode the whole way down to 5 K.\nIn Figs.~\\ref{Fig_A1g} and~\\ref{Fig_B1g}, respectively, we report the full temperature dependence of the frequency of the two $c$-axis polarized modes for the five samples with Fe concentration close to 1. The temperature dependencies of the linewidths for the same samples are given in the lower panels of these figures.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\linewidth]{Fig3a.pdf}\n\\caption{(Color online) a) Room temperature Raman spectra of the parent Fe$_{1.02}$Te and Fe$_{1.09}$Te samples (vertically shifted for clarity). b) Example of the aging effect on the parent single crystals. The spectrum of powdered TeO$_2$ from ref.~\\cite{Pine71} has been added for comparison.}\n\\label{Fig3a}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig3b.pdf}\n\\caption{(Color online) a) and b) Temperature dependence of the frequency the $A_{1g}$ and $B_{1g}$ modes in these samples. c) and d) Temperature dependence of the linewidth the $A_{1g}$ and $B_{1g}$ modes in these samples.}\n\\label{Fig3b}\n\\end{figure}\n\nWhile the temperature is decreased, a hardening of both the $A_{1g}$ and $B_{1g}$ modes in all the systems is observed, as expected from the lattice contraction. No noticeable differences between the samples are seen.\nAs we go through the magnetic transition in the Fe$_{1.02}$Te parent compound, a clear softening of the $B_{1g}$ mode is observed (see also Fig.~\\ref{Fig3b}), while no changes across $T_c$ occur in the samples containing Se. Within our error bars, the $A_{1g}$ mode frequency remains essentially unaffected by these transitions.\nAs shown in Fig.~\\ref{Fig_A1g} weak narrowing of the $A_{1g}$ line with decreasing temperature is observed for Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples, while its broad linewidth remains essentially temperature independent in the Fe$_{1.02}$Te and Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$ sample. In parallel to this, an unusual evolution on Se doping of the temperature dependence of the $B_{1g}$ mode linewidth can be see in the lower panels of Fig.~\\ref{Fig_B1g}.\nStarting from the almost half-doped Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ compound, a conventional behavior is observed: as in most of the materials, this mode narrows with decreasing temperature (the phonon width is inversely proportional to its lifetime, which is expected to increase as the phonon-phonon interaction is reduced when decreasing the temperature~\\cite{Klemens66,Menendez84}). Decreasing Se concentration toward the parent Fe$_{1.02}$Te sample, a smooth evolution from this regular behavior is observed: in Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$ and Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  samples only a weak narrowing of the phonon is observed between room and base temperature, while in Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$ and Fe$_{1.02}$Te the phonon essentially broadens as temperature is decreased. Below the magnetic ordering transition in the Fe$_{1.02}$Te sample, in addition to the softening mentioned above, a narrowing of the $B_{1g}$ phonon is observed, in agreement with recent report on Fe$_{1.05}$Te.~\\cite{Gnezdilov}\n\nThe temperature dependence of both frequency and FWHM of the $B_{1g}$ phonon of the Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples can been well fitted assuming a symmetric anharmonic decay of this optical phonon, \\textit{i.e.} decay into two acoustic modes with identical frequencies and opposite momenta,~\\cite{Klemens66, Menendez84}\n\\begin{equation}\\label{e1}\n\\omega_{ph}(\\textit{T}) = \\omega_{0} - C\\bigg[1+\\frac{2}{e^{\\frac{\\hbar\\omega_{0}}{2k_{B}T}}-1}\\bigg]\n\\end{equation}\n\\begin{equation}\\label{e2}\n\\Gamma_{ph}(\\textit{T}) = \\Gamma_{0} + \\Gamma\\bigg[1+\\frac{2}{e^{\\frac{\\hbar\\omega_{0}}{2k_{B}T}}-1}\\bigg]\n\\end{equation}\nwhere  $C$ and $\\Gamma$ are positive constants, $\\omega_{0}$ is the bare phonon frequency, and $\\Gamma_{0}$ a residual (temperature-independent) linewidth originating from sample imperfections or electron-phonon interactions.\nThe fitting parameters for these three samples are summarized in Table~\\ref{tab:TableFitB1g}.\nFor the Fe$_{1.02}$Te and Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$ samples, as the FWHM increases with decreasing temperature, we can simply not use the latter expression to fit the experimental data.\n\n\\begin{table}\n\\caption{\\label{tab:TableFitB1g}Fitting parameters for the temperature dependence of the $B_{1g}$ phonon linewidth in Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$, Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  and Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nSample               &        $\\omega_0$ (cm$^{-1}$)      & $\\Gamma_0$ (cm$^{-1}$) & $\\Gamma$ (cm$^{-1}$) \\\\\n\\hline\nFe$_{0.99}$Te$_{0.69}$Se$_{0.31}$               &    203.9         &  9.9 & 0.78 \\\\\nFe$_{0.98}$Te$_{0.66}$Se$_{0.34}$                &     203.8        &  9.8 & 0.78 \\\\\nFe$_{0.95}$Te$_{0.56}$Se$_{0.44}$               &       205.6      &  3.44 & 2.1 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{Effect of the iron excess}\n\\subsubsection{Undoped compounds}\n\\label{Exp_Fe_undoped}\nBefore discussing the possible origin of this unusual evolution of the temperature dependence of the $B_{1g}$ mode FWHM, it is interesting to discuss its dependence with the concentration of Fe excess.\nWe show in Fig.~\\ref{Fig3a}-a the raw Raman data obtained at room temperature on the two Se-free samples Fe$_{1.02}$Te and Fe$_{1.09}$Te where the excess iron concentration has been measured to be 2\\%, and 9\\%. No strong differences can be found at first glance.\nThe strong aging effect reported in ref.~\\cite{Xia09} has also been observed in our Fe$_{1+y}$Te samples, as seen in Fig.~\\ref{Fig3a}-b, as well as on the Se-substituted samples (not shown here).\nThe origin of this strong Raman signal is not clear. In ref.~\\cite{Xia09}, the authors attribute it to the formation of amorphous Te as a decomposition product of Fe$_{1+y}$Te$_{1-x}$Se$_{x}$ on the basis of earlier reports~\\cite{Pine71,Pine72}. In particular, in Fig. 2-d from ref.~\\cite{Pine71}, the authors say that the room temperature spectra of amorphous Te is plotted, but in a note added in proof, they recognize that it rather originates from tetragonal TeO$_2$ (the Raman spectra of single crystalline paratellurite TeO$_2$ reported in ref.~\\cite{Pine72} shows a much complex structure).\nWe have added these data to our Fig.~\\ref{Fig3a}-a and found them not inconsistent with the measurements from our aged surface. The main differences are the relative intensities and widths of the two features of TeO$_2$ at 120 and 145 cm$^{-1}$ that may originate either from different texturing and strain of TeO$_2$ and\/or from the presence of Fe in the decomposition product.\n\nThe excess iron induces a small softening of the $A_{1g}$ mode (Fig.~\\ref{Fig3b}-a) but does not affect its already broad linewidth (Fig.~\\ref{Fig3b}-c). No changes in the temperature dependence are seen.\nSimilarly, the case of the $B_{1g}$ mode appears to be more interesting. In the excess-Fe rich sample Fe$_{1.09}$Te, a small softening through the magnetic transition can still be observed (Fig.~\\ref{Fig3b}-b), but the signature of this transition in the FWHM is clearly suppressed (Fig.~\\ref{Fig3b}-d).\nWe note finally that the unusual broadening of the mode with decreasing temperature in the parent compounds is observed irrespective to the measured excess iron concentration (fig.~\\ref{Fig3b}-d), ruling out a disorder origin for this phenomena.\n\n\n\\subsubsection{Se-substituted compounds}\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig4.pdf}\n\\caption{(Color online) a) and c) Effect of the Fe excess on the temperature dependence of the $B_{1g}$ mode frequency in Se-substituted samples. b) and d)  Effect of the Fe excess on the temperature dependence of the $B_{1g}$ mode linewidth in Se-substituted samples.}\n\\label{Fig4}\n\\end{figure}\n\nWe now discuss the effect of the excess iron on the vibrational properties for the Se-rich compounds. As seen directly by comparing the data in Figs.~\\ref{Fig1}-b and -c, the low temperature behavior of the phonons is strongly affected by the Fe concentration.\nIn the case of Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$, as already discussed, regular hardening and narrowing of the phonon are observed as temperature is lowered. This is true between room temperature and 60 K for the Fe$_{1.05}$Te$_{0.58}$Se$_{0.42}$ sample, with a very close Se content to those of Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ but where $\\sim$ 5\\% of excess Fe has been measured. A sudden upturn in this behavior is observed for lower temperature, since at 5K the mode is softer and broader than at 60K.\nThis can be better seen on Figs~\\ref{Fig4}-a and b, where we compare the temperature dependence of frequencies and linewidth of the $B_{1g}$ mode for these two samples. Clearly, as temperature decreases, the phonon hardens and narrows regularly, just as in absence of excess iron, but at a temperature of about 35 K the mode starts to broaden ($\\sim$ 3 cm$^{-1}$) and softens ($\\sim$ 5 cm$^{-1}$).\nIn samples with lower Se contents a similar, although weaker, softening of the phonon is observed in the same temperature range, but with no change in the linewidth. None of these effects were observed on the $A_{1g}$ phonon (not shown here).\n\n\n\\subsection{Summary of the experimental results}\nTo summarize our main experimental findings, we observe the following:\n\\begin{enumerate}[label=\\roman{*})]\n \\item{For null or low Se concentration, an unusual broadening of the $B_{1g}$ phonon FWHM as temperature is decreased. This progressively turns into a conventional anharmonic narrowing as Se is substituted to Te.}\n\\item{Clear softening and narrowing of this $B_{1g}$ phonon through the magnetic transition in the parent compound, suppressed in presence of Fe excess.}\n\\item{Strong softening and broadening of the $B_{1g}$ mode at low temperature in Se substituted samples.}\n\\end{enumerate}\n\n\n\\section{LDA DFT Calculations}\n\\label{DFTcalc}\n\nIn order to gain further insights on the effects of magnetism and Fe concentration on the lattice dynamics in these systems, we have performed DFT calculations of the frequency of the $A_{1g}$ and $B_{1g}$ modes in both paramagnetic (non-spin-polarized, nsp hereafter) and double stripe ordered (spin-polarized, sp) phases.\nWe start with the results obtained for stoichiometric FeTe system using the frozen phonon approach, summarized in Table~\\ref{tab:TableDFe$_{1.02}$Te}.~\\cite{DFT:details} Consistently with previous calculations,~\\cite{Subedi08, singh2} we find a value of the double-stripe magnetic moment at $y=0$ (m =$ 2.2 \\mu_B$), which is close to the experimental one.\n\n\\begin{table}[h]\n\\caption{\\label{tab:TableDFe$_{1.02}$Te} $A_{1g}$ and $B_{1g}$ frequencies of FeTe, from\n non-spin-polarized ($\\omega_{nsp}$) and spin-polarized ($\\omega_{sp}$)\nDFT calculations.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\nmode  & $\\omega_{nsp}$ (cm$^{-1}$) & $\\omega_{sp}$ (cm$^{-1}$) & exp. at 10 K (cm$^{-1}$) \\\\\n  & & & Fe$_{1.02}$Te sample \\\\\n\\hline\n$A_{1g}$ &    135 &        175 & 159.7 \\\\\n$B_{1g}$      &    200.9 &     197.5 & 200.5\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nWe then consider the effect of Fe excess and deficiencies on the $B_{1g}$ mode. Non-stoichiometry was considered within the VCA, which amounts to\n replacing Fe with a ``virtual'' atom with charge $Z_{Fe}^{\\pm y}$ in the self-consistent DFT calculations.\nThis means that the  average potential due to doping is treated self-consistently, but the effects of randomness are disregarded.\nSince, within VCA, it is  impossible to model the isovalent Se\/Te substitution, we do not address the issue of the dependence of the $A_{1g}$ mode on Se\/Te concentration in this work.\nA proper description would require large supercell calculations, beyond the scope of the present paper. We used the FeTe experimental lattice parameters and Te height for all Fe concentrations.~\\cite{Bendele10}\nFor each value of the Fe excess $y$, the frequencies were calculated for nsp and sp configurations.~\\cite{DFT:details} The frequencies we obtained are given in Table~\\ref{tab:TableDFT}, together with the self-consistent value of the magnetic moment at equilibrium.\n\\begin{table}\n\\caption{\\label{tab:TableDFT}$B_{1g}$ frequencies of Fe$_{1+y}$Te, from\n non-spin-polarized ($\\omega_{nsp}$) and spin-polarized ($\\omega_{sp}$)\nDFT calculations. $m$ is the value of the self-consistent\ndouble-stripe moment at zero displacement, in $\\mu_B$.}\n\\begin{ruledtabular}\n\\begin{tabular}{p{2cm}ccc}\n  & $\\omega_{nsp}$ (cm$^{-1}$) & $\\omega_{sp}$ (cm$^{-1}$)& $m$ ($\\mu_B$) \\\\\n\\hline\nFe$_{0.98}$Te  &    216.76 &        207.36 &    2.34\\\\\nFe$_{0.99}$Te  &    207.21 &        199.92 &  2.28\\\\\nFeTe      &    200.94 &        197.54 &    2.20\\\\\nFe$_{1.02}$Te  &    191.30 &        192.40 &    2.06\\\\\nFe$_{1.06}$Te  &    173.00 &        182.00 &    1.60\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nThe values of the calculated frequencies are extremely sensitive to the Fe excess content, which also affects strongly the magnetic moment.\nIn the VCA, the magnetic moments and frequencies decrease monotonically with increasing Fe content, up to the highest doping we calculated ($y = 0.06$).\nWe find an almost linear increase of the $B_{1g}$ frequency with doping, with a linear slope of $523$ and $292$ cm$^{-1}$\/$y$ for nsp and sp calculations, respectively.\nAs discussed in the next section, the experimentally observed effects of Fe non stoichiometry are much weaker.\n\n\\section{Discussion}\n\n\\subsection{Phonon Renormalization through $T_N$: comparison with other families}\n\\label{throughTN}\nIn parent Fe$_{1.02}$Te, we have observed a clear softening and a narrowing of the $B_{1g}$ phonon through the magnetic ordering transition at $T_N$.\nSuch narrowing has also been reported in parent BaFe$_2$As$_2$~\\cite{Chauviere09, Rahlenbeck09} and CaFe$_2$As$_2$~\\cite{Choi10} through the spin-density wave (SDW) transition. In the latter case only, it is accompanied by a jump of the phonon frequency that can, at least partially, be explained by the sudden collapse of the unit cell along the $c$-axis through the transition (such collapse does not take place in BaFe$_2$As$_2$).\n\nOn the contrary, in Fe$_{1+y}$Te, the $c$-axis lattice parameter has been found to expand slightly through the coupled structural-magnetic transition.~\\cite{Martinelli_PRB2010} This certainly favors the observed softening, although one would in this case expect an abrupt jump in the phonon frequency at $T_N$ (the structural transition is first-order) rather than the observed smooth softening between $T_N$ and 10 K.\n\nRegarding the agreement with the calculated phonon frequencies, it has been shown for the 122 and 1111 systems that the experimental phonon frequencies are closer from those obtained with a magnetic calculation than from those where magnetism is not included, even in the paramagnetic state.~\\cite{Boeri_PRB2010, Hanh_PRB2009, Fukuda_PRB2011, Reznik09}\nThis is also the case here as we get a better match with our experimental data using the sp calculation rather than the nsp one (see Sect.~\\ref{DFTcalc}).\n\nFinally, as the phonon linewidth is inversely proportional to the lifetime of the excitation, its renormalization through $T_N$ reflects the changes in the coupling of the phonons to some decay channels.\nIn the 122 arsenides, as the SDW gap opens,~\\cite{Akrap_PRB2009, Boris_PRL2009} a significant reduction of the electronic density of states at the Fermi level occurs, leading to a decrease of electron-phonon coupling that reasonably accounts for the observed sharpening of the phonons.~\\cite{Rahlenbeck09}\nSuch an SDW gap opening has not yet been reported in the 11 compounds,~\\cite{Chen_PRB2009, Xia_PRL09} but has recently been observed in ARPES experiments~\\cite{ARPES_gap} and is very likely responsible for the $B_{1g}$ mode narrowing in the Fe$_{1.02}$Te sample.\n\n\\subsection{Se-substituted systems}\n\\subsubsection{Absence of superconductivity induced anomalies}\nAs mentioned in the Introduction, the $B_{1g}$ Fe mode that shows here the most striking doping dependence is active in all the families of Fe-based superconductors.\nThe absence of renormalization of this phonon as well as of the $A_{1g}$ one through the superconducting transition in the doped compounds is consistent with experiments carried out on the doped 122 arsenide compounds~\\cite{Rahlenbeck09} (with the notable exception being the electron-doped Pr$_{x}$Ca$_{1-x}$Fe$_2$As$_2$,~\\cite{Litvinchuk_PRB2011} where a small hardening of the $B_{1g}$ phonon through $T_c$ has been observed) or on the 111 compound LiFeAs~\\cite{Um_PRB2012}, where no signature of the superconducting transition is seen either in the frequency or in the linewidth of these two modes. This is due to the fact that the superconducting gap amplitude is smaller than the phonon frequencies, hence his opening let them unaffected. The situation is similar for the 11 compounds, as the phonon frequency is much larger than the energies reported for the superconducting gap(s) in various experiments [2$\\Delta \\sim$ 2~meV (17 cm$^{-1}$)~\\cite{Noat_JPCS2010}, 3.4~meV (27.4 cm$^{-1}$)~\\cite{Hanaguri_Science2010} or 4.6~meV (37 cm$^{-1}$)~\\cite{Kato_PRB2009} from STM, 2$\\Delta \\sim$ 3~meV (24 cm$^{-1}$) from NMR,~\\cite{Arcon_PRB2010} and 2$\\Delta \\sim$6~meV (48 cm$^{-1}$)from specific heat~\\cite{Klein_PRB2010}].\n\n\\subsubsection{Se-substitution induced evolution of the $B_{1g}$ mode FWHM temperature dependence}\n\\label{disc}\nWe now turn to the influence of Se concentration on the behavior of the $B_{1g}$ phonon.\nAs it is substituted into the system, we observe a weak increase of the $B_{1g}$ phonon frequency at the lowest temperatures (see Figs.~\\ref{Fig_B1g} b to e), that goes along with the reduction of the $c$-axis parameter reported from x-ray and neutron diffraction experiments.~\\cite{Yeh_EPL2008, Li09, Martinelli_PRB2010}\nThe room temperature linewidth for this mode is only weakly dependent on the Se concentration, although the FWHM clearly tends to sharpen with increasing Se content. This fact is already surprising as one may have rather expected the Se-substitution induced disorder to favor a broadening of the phonon lineshape instead of a narrowing. The normal state temperature dependence of the mode FWHM for different doping levels is even more puzzling. As noted in Sec.~\\ref{ResultsSe}, the situation appears conventional in the Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ samples close to half doping, with a continuous narrowing of the phonon with decreasing temperature.\nOn the contrary, as Se contents are reduced the phonon broadens with decreasing temperature. This is particulary clear for the two Se-free samples (Fe$_{1.02}$Te and Fe$_{1.09}$Te), as seen in Fig.~6-d.\n\nThe temperature dependence of the three samples with the highest Se content have successfully been fitted using a conventional anharmonic decay model (Eq.~\\ref{e2}). Interestingly, looking at the fitting parameters of Table~\\ref{tab:TableFitB1g}, the residual - or temperature-independent - linewidth $\\Gamma_0$ (3.4 cm$^{-1}$) in Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ is comparable with the one of the temperature dependent parameter $\\Gamma$ (2.1 cm$^{-1}$), whereas\nin Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$  and Fe$_{0.99}$Te$_{0.69}$Se$_{0.31}$ we have $\\Gamma_0 \\sim$ 10 cm$^{-1} \\gg \\Gamma$.\n\nThe situation is thus much closer to the one reported for Pr$_{x}$Ca$_{1-x}$Fe$_2$As$_2$~\\cite{Litvinchuk_PRB2011} than for LiFeAs, where the residual linewidth was always vanishingly small.~\\cite{Um_PRB2012} This shows that in Fe$_{1+y}$Te$_{1-x}$Se$_x$ systems, the contribution of anharmonicity to the $B_{1g}$ phonon lifetime is not the dominant one.\nAn additional decay channel for this phonon must therefore take over the usual anharmonic effects, and its contribution increases strongly as Se content is decreased.\nSuch behavior can have at least two possible origins: electron-phonon coupling and spin-phonon coupling.\nIn the first scenario, the increasing relative weight\n$\\Gamma_0$ with decreasing Se content can be trivially related to the increase of the electronic density of states at the Fermi level $N(E_F)$~\\cite{Subedi08} [as $\\Gamma_0 \\propto N(E_F)$)].\nThis may, however, not be sufficient to account for the reported effect: Electron-phonon coupling is, in principle, temperature independent and can, therefore, hardly explain the increasing linewidth of the $B_{1g}$ phonon with decreasing temperature in the systems with the lowest Se contents (Fe$_{1.02}$Te, Fe$_{1.00}$Te$_{0.78}$Se$_{0.22}$).\nHaving in mind the increasing weight of magnetic excitations as Se concentration decreases toward the parent compound,~\\cite{Wen09} we are naturally led to suggest that spin-phonon coupling may be the additional decay channel for the phonons.\nThe effects of excess Fe discussed in the next section indeed confirm an interplay between the lattice and spin degrees of freedom in these systems.\n\n\\subsection{Influence of the iron concentration}\n\n\\subsubsection{Comparison of experimental data with LDA DFT calculation}\nAccording to our calculation in Sec.~\\ref{DFTcalc}, increasing the Fe concentration induces a softening of the $B_{1g}$ mode frequencies in both sp and nsp calculations. The softening rates are found to be $523$ and $292$ cm$^{-1}$\/$y$ for nsp and sp, respectively.\nExperimentally, at low temperature, a small hardening is instead observed in the parent compounds, when going from Fe$_{1.02}$Te to Fe$_{1.09}$Te\n~In Se-substituted samples on the other hand, a $\\sim$7 cm$^{-1}$ softening between Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ and Fe$_{1.05}$Te$_{0.58}$Se$_{0.42}$ $B_{1g}$ phonon frequencies is indeed observed ($\\sim$70 cm$^{-1}$\/$y$).\nAn effect of comparable size is found when comparing the $B_{1g}$ peak frequencies of Fe$_{0.95}$Te$_{0.56}$Se$_{0.44}$ ($\\omega_{B_{1g}} = 205.6 cm^{-1}$) and Fe$_{0.98}$Te$_{0.66}$Se$_{0.34}$ ($\\omega_{B_{1g}} = 203.8 cm^{-1}$), with significantly different Fe deficiencies (even though the Se concentrations in this case slightly differ). This gives a softening rate of $\\sim$ 66 cm$^{-1}$\/$y$.\n\nIn any case, the experimentally observed effects of Fe nonstoichiometry are much weaker than those theoretically calculated, even considering the sp calculation where they are the smallest. One has to keep in mind that accurate comparison to the calculation is tough as the presence of excess ($y > 0$) Fe complicates the situation. In fact, in the VCA there is no qualitative difference between Fe excess and deficiency, as the excess (deficient) charge are both\nlocated around the Fe site. Experimentally, however, it is known that the excess Fe ions are located in the Te planes, and\nthis has qualitatively different effects, the most important being that the effective Fe magnetic moment is enhanced and not reduced,\ndue to the formation of local moments on the excess Fe in the Te planes. This cannot be taken into account by the VCA approach of treating the doping in our LDA calculations. Furthermore, it is important to point out that mean-field LDA DFT calculations cannot reproduce the renormalization of the $B_{1g}$ frequency and linewidth observed at low temperature in the Fe-rich systems, which are discussed in the next paragraph.\n\n\\subsubsection{Excess Fe-induced magnetic fluctuation}\nWe have seen in Sec.~\\ref{Exp_Fe_undoped} that in the Fe$_{1.09}$Te crystal, we are not able to observe clearly the effect of the magnetic transition on the phonons. A small softening has been seen (see Fig.~\\ref{Fig3b}-d), but no narrowing of the line shape.\nIn the Se-substituted system, as shown in Fig.~\\ref{Fig4}, excess iron, in addition to a decrease in $T_c$, induces large effects on both frequency and lineshape of the phonons. The strongest effect occurs close to 50\\% Se content, with a large softening and broadening of the $B_{1g}$ phonon below $T\\sim$35 K, well above $T_c$. To our knowledge, no phase transition has been reported in this temperature range for this doping level, but the occurrence of short-range magnetic fluctuations has been reported.~\\cite{Khasanov09}\nIn the undoped case with low excess iron concentration, it has been shown that a low energy gap in the spin-wave excitation spectrum opens when entering the magnetic state.~\\cite{Stock} Increasing the excess iron concentration, this gap is filled up with low-energy spin fluctuations.~\\cite{Stock}\n\nIn both doped and undoped cases, one effect of excess iron is, therefore, to induce low energy magnetic fluctuations in a temperature range at which we also observe a relative broadening of the $B_{1g}$ phonon, \\textit{i.~e.}, a decrease of its lifetime.\nThis reinforces the point we made at the end of Sec.~\\ref{disc}, indicating that the additional damping for the $B_{1g}$ mode may actually originate from its coupling to magnetic excitations.\n\n\\section{Conclusions}\nWe have carried out a systematic study of the lattice dynamics in the Fe$_{1+y}$Te$_{1-x}$Se$_x$ system, focusing more particularly on the $c$-axis polarized Fe $B_{1g}$ mode. In parent compounds, unlike other systems such as BaFe$_2$As$_2$ or LiFeAs, a nonconventional broadening of this mode is observed as temperature decreases, and a clear signature of the SDW gap opening is observed.\nAs Se is substituted to Te, the temperature dependence of this modes smoothly evolves toward a more regular situation, with the $B_{1g}$ phonon showing conventional anharmonic decay. A good agreement between the observed phonon frequencies and a first-principles calculation including the effects of magnetic ordering is found.\nThe temperature dependence of the phonon linewidth, as well as the effects induced by the Fe nonstoichiometry in these compounds, revealed a peculiar coupling of this mode to magnetic fluctuations in the Fe$_{1+y}$Te$_{1-x}$Se$_x$ system and can, to date, not be satisfactorily reproduced within  state-of-the-art computational approaches.\n\n\\section{Acknowledgement}\n\nWe thank A. Schulz for technical support and A.C. Walters for useful suggestions. This work has been supported by the European project SOPRANO (Grant No. PITN-GA-2008-214040), by the French National Research Agency, (Grant No. ANR-09-Blanc-0211 SupraTetrafer), and by the UK Engineering and Physical Sciences Research Council (MJR, EP\/C511794).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Review of Geometrically Non-Higgsable Seven-branes}\n\\label{sec:review}\n\nWe will study seven-branes using their geometric description in\nF-theory.  There the axiodilaton $\\tau = C_0 + i\\, e^{-\\phi}$ of the type IIB\ntheory is considered to be the complex structure modulus of an\nauxiliary elliptic curve which is fibered over the compact extra dimensional space\n$B$. Such a structure is determined by a Calabi-Yau fourfold $X$\nwhich is elliptically fibered $\\pi:X\\rightarrow B$. An elliptic fibration with section\nis birationally equivalent \\cite{Na88} to a Weierstrass model\n\\eqn{\ny^2 = x^3 + f\\, x + g\n}[]\nwhere $f$ and $g$ are sections of $K_B^{-4}$ and $K_B^{-6}$, respectively, with\n$K_B$ the anticanonical bundle on $B$. The fibers $\\pi^{-1}(p)$ are smooth\nelliptic curves for any point $p$ which is not in the discriminant\nlocus\n\\eqn{\n\\Delta = 4\\,f^3 + 27\\, g^2 = 0.\n}[]\nOn the other hand if $p$ is a generic point in the codimension one locus \n$\\Delta = 0$, then $\\pi^{-1}(p)$ is one of the singular fibers classified by\nKodaira \\cite{MR0187255,MR0205280,MR0228019}.\n\nSeven-branes are located along $\\Delta = 0$. Their precise nature\ndepends on the structure of $f$ and $g$ and therefore also $\\Delta$, which may be an irreducible\neffective divisor or comprised of components\n\\begin{equation}\n\\Delta = \\prod_i \\Delta_i.\n\\end{equation}\nTaking a loop around $\\Delta$ or any component $\\Delta_i=0$ induces an\n$SL(2,\\mathbb{Z})$ monodromy on the associated type IIB supergravity\ntheory. The action on $\\tau$ is\n\\begin{equation}\n\\tau \\mapsto \\frac{a\\tau + b}{c\\tau + d}, \\qquad \\qquad M = \\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix}\\in SL(2,\\mathbb{Z}).\n\\end{equation}\nSeven-brane structure is determined by the Weierstrass model according\nto the order of vanishing of $f$, $g$, and $\\Delta$ along the\nseven-brane. Often in this paper some $\\Delta_i=z^N$, and therefore we\nwill denote the associated orders of vanishing as $ord_z(f,g,\\Delta)$\nas a three-tuple or in terms of the individual orders $ord_z(f)$,\n$ord_z(g)$, and $ord_z(\\Delta)$. From this data the singularity type,\n$SL(2,\\mathbb{Z})$ monodromy, and non-abelian symmetry algebra (up to outer\nmonodromy) can be determined; see Table \\ref{tab:fibs}.  This is the\ngeometric symmetry group, henceforth symmetry group or gauge group, along the\nseven-brane in the absence of symmetry-breaking $G$-flux.  The\nstructure of $\\Delta$ is determined by $f, g$ and there is a moduli space of such choices that corresponds to\nthe complex structure of $X$. Gauge sectors along seven-branes can be\nengineered by tuning $f$ and $g$ relative to their generic structures.\n\n\\begin{table}[thb]\n  \\centering\n\\scalebox{.8}{\n \\begin{tabular}{|c c c c c c c c|}\n\\hline\nType &\n$ord_z(f)$ &\n$ord_z(g)$ &\n$ord_z(\\Delta)$ &\nsingularity & nonabelian symmetry algebra & monodromy & order \\\\ \\hline \\hline \n$I_0$&$\\geq $ 0 & $\\geq $ 0 & 0 & none & none  & $\\begin{pmatrix} 1 & 0 \\\\ 0 & 1\\end{pmatrix}$ & $1$\\\\ \n$I_n$ &0 & 0 & $n \\geq 2$ & $A_{n-1}$ & ${\\mathfrak{su}}(n)$  or ${\\mathfrak{sp}}(\\lfloor\nn\/2\\rfloor)$& $\\begin{pmatrix} 1 & n \\\\ 0 & 1\\end{pmatrix}$ & $\\infty$\\\\\n$II$ & $\\geq 1$ & 1 & 2 & none & none & $\\begin{pmatrix} 1 & 1 \\\\ -1 & 0\\end{pmatrix}$ & $6$\\\\\n$III$ &1 & $\\geq 2$ &3 & $A_1$ & ${\\mathfrak{su}}(2)$ & $\\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix}$ & $4$\\\\\n$IV$ & $\\geq 2$ & 2 & 4 & $A_2$ & ${\\mathfrak{su}}(3)$  or ${\\mathfrak{su}}(2)$& $\\begin{pmatrix} 0 & 1 \\\\ -1 & -1\\end{pmatrix}$ & $3$\\\\\n$I_0^*$&\n$\\geq 2$ & $\\geq 3$ & $6$ &$D_{4}$ & ${\\mathfrak{so}}(8)$ or ${\\mathfrak{so}}(7)$ or ${\\mathfrak{g}}_2$ & $\\begin{pmatrix} -1 & 0 \\\\ 0 & -1\\end{pmatrix}$ &$2$\\\\\n$I_n^*$&\n2 & 3 & $n \\geq 7$ & $D_{n -2}$ & ${\\mathfrak{so}}(2n-4)$  or ${\\mathfrak{so}}(2n -5)$ & $\\begin{pmatrix} -1 & -n \\\\ 0 & -1\\end{pmatrix}$ & $\\infty$\\\\\n$IV^*$& $\\geq 3$ & 4 & 8 & $E_6$ & ${\\mathfrak{e}}_6$  or ${\\mathfrak{f}}_4$& $\\begin{pmatrix} -1 & -1 \\\\ 1 & 0\\end{pmatrix}$ & $3$\\\\\n$III^*$&3 & $\\geq 5$ & 9 & $E_7$ & ${\\mathfrak{e}}_7$ & $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0\\end{pmatrix}$ & $4$\\\\\n$II^*$& $\\geq 4$ & 5 & 10 & $E_8$ & ${\\mathfrak{e}}_8$ & $\\begin{pmatrix} 0 & -1 \\\\ 1 & 1\\end{pmatrix}$ & $6$\\\\\n\\hline\nnon-min &$\\geq 4$ & $\\geq6$ & $\\geq12$ & \\multicolumn{4}{c|}{does not\nappear for supersymmetric vacua} \\\\\n\\hline\n \\end{tabular}}\n  \\caption{The Kodaira fibers, along with their orders of vanishing in a Weierstrass model, singularity type, possible nonabelian\nsymmetry algebras, $SL(2,\\mathbb{Z})$ monodromy, and monodromy order.}\n  \\label{tab:fibs}\n\\end{table}\n\n\n\\vspace{1cm} Let us now turn to geometrically non-Higgsable\nseven-branes. Physically, this means that there are no directions in the\nsupersymmetric moduli space that break the gauge group on the\nseven-branes by splitting them up.  Mathematically, a geometrically\nnon-Higgsable seven-brane along $z=0$ exists when\n\\begin{equation}\n\\Delta = z^N \\, \\tilde \\Delta\n\\end{equation}\nfor any choice of $f$ and $g$, i.e. for a generic point in the complex\nstructure moduli space of $X$, henceforth $\\cs{X}$. For $N>2$ the seven-brane\ncarries a non-trivial gauge group $G$. It is often possible that by tuning $f,g$ to a\nsubvariety $L\\subset \\cs{X}$ the discriminant $\\Delta$ is proportional\nto $z^{M>N}$ and the gauge group along the seven-brane at $z=0$ is\nenhanced to $G'\\supset G$. There may be many such loci $L_i$ in $\\cs{X}$. The statement that a\nnon-Higgsable seven-brane exists for generic complex structure moduli is the statement\nthat it exists for any complex structure in $\\cs{X}\\setminus\n\\{\\bigcup_i L_i\\}$, which is the bulk of $\\cs{X}$ since each $L_i$\nhas non-trivial codimension. Often the discriminant is of the form\n\\begin{equation}\n\\Delta = \\tilde \\Delta \\,\\,\\prod_i z_i^{N_i}\n\\end{equation}\nfor generic complex structure, in which case there is a non-Higgsable\nseven-brane along each locus $z_i=0$. They may intersect, giving rise\nto product group gauge sectors with jointly charged matter that arise\nfrom clusters of intersecting seven-branes. These are referred to as\nnon-Higgsable clusters \\cite{Morrison:2012np,Morrison:2012js}. \n\nThe\npossible gauge groups that may arise along a non-Higgsable\nseven-brane are $E_8, E_7, E_6, F_4, SO(8), SO(7), G_2, SU(3),$ and\n$SU(2)$ and there are five two-factor products with jointly charged matter\nthat may arise. In particular, note that $SU(5)$ and $SO(10)$, which arise\nfrom $I_5$ and $I_1^*$ fibers, may never be non-Higgsable; more generally,\nthis is true of seven-branes with fibers $I_n$ and $I_{n>0}^*$. This is easy\nto see in the $I_n$ case. Such a model has $ord_z(f,g,\\Delta)=(0,0,n)$,\nand $f\\mapsto (1+\\epsilon) f$ for $\\epsilon \\in \\mathbb{C}^*$ is a symmetry breaking complex\nstructure deformation that always exists, by virtue of the model existing in the first place.\nSimilar arguments exist for $I_{n>0}^*$ fibers.\n\n\nThe name ``non-Higgsable clusters'' is a suitable name in\nsix-dimensional compactifications of F-theory, since there the\nassociated six-dimensional gauge sectors do not have any symmetry\nbreaking flat directions in the supersymmetric moduli space, as\ndetermined by $\\cs{X}$ and also the low energy degrees of\nfreedom. However, in four dimensional compactifications there are\nother effects such as T-branes \\cite{Cecotti:2010bp} that may break\nthe gauge group, so that \\emph{geometrically non-Higgsable} is a more\naccurate name.  Furthermore, if $\\Delta \\sim z^2$ for a generic $p\\in\n\\cs{X}$ then $G=\\emptyset$ even though there is a divisor $z=0$ in $B$\nthat is singular, and sometimes a codimension two locus $C$ may be\nsingular for generic moduli even if it is not contained in a\nnon-Higgsable seven-brane. Both have been referred to as\n``non-Higgsable structure'' \\cite{Halverson:2015jua} even though there\nis no associated gauge group. The general feature is the existence of\nsingular structure for generic complex structure moduli, and aside\nfrom these two caveats there is a gauge group on a seven-brane that\ncannot be spontaneously broken by a complex structure deformation.\n\nThough not named as such at the time, the first F-theory\ncompactifications with non-Higgsable seven-branes appeared in\n\\cite{Morrison:1996pp}. These examples have six non-compact dimensions\nand four compact dimensions $B_2$ with $B_2=\\mathbb{F}_n$, and there is a\nnon-Higgsable seven-brane on the $-n$ curve in $\\mathbb{F}_n$ for $n>2$. The\ncomplete set of non-Higgsable clusters and seven-branes that may arise\nin six-dimensional compactifications were classified in\n\\cite{Morrison:2012np} and the examples with toric $B_2$ were\nclassified in \\cite{Morrison:2012js}.  In the latter, all but $16$ of\nthe $61,539$ examples exhibit non-Higgsable clusters or seven-branes,\nand the $16$ that do not are weak Fano varieties, i.e. varieties\nsatisfying $-K\\cdot C\\geq 0$ for any holomorphic curve $C$. In all cases in\nsix dimensions the reason for geometric non-Higgsability is immediately evident\nin the low energy gauge theory: either there is no matter or there is not enough\nmatter to allow for Higgsing consistent with supersymmetry, due to having a half hypermultiplet\nin a pseudoreal representation.\n\nIn examples with four non-compact dimensions the extra dimensions of\nspace are a complex threefold $B_3$ and there are additional\nnon-Higgsable clusters and structures that do not appear in six\ndimensions, including for example loops \\cite{Morrison:2014lca} and\nthe gauge group $SU(3)\\times SU(2)$\n\\cite{GrassiHalversonShanesonTaylor2014}. In the latter case the\nmatter content matches the non-abelian structure of the standard\nmodel. A classification \\cite{Halverson:2015jua} of $B_3$ that are\n$\\mathbb{P}^1$-bundles over certain toric surfaces has non-Higgsable clusters for\n$98.3\\%$ of the roughly $100,000$ examples with over $500$ examples with an $SU(3)\\times SU(2)$\nsector. A broader exploration of toric $B_3$ using Monte Carlo techniques \\cite{Taylor:2015ppa}\nhas non-Higgsable structure for all $B_3$ after an appropriate ``thermalization,'' and approximately $76\\%$\nof the examples have a non-Higgsable $SU(3)\\times SU(2)$ sector. Non-Higgsable clusters also appear in\nthe F-theory geometry with the largest number of currently known flux vacua \\cite{Taylor:2015xtz},\nwhere vacuum counts were estimated using techniques of Ashok, Denef, and Douglas \\cite{Ashok:2003gk,Denef2004}. \nIt is not clear whether cosmological evolution prefers the special vacua associated with a typical $B_3$, perhaps\ncharacterized by \\cite{Taylor2015}, or the typical vacua associated with a special base $B$ that gives the largest number\nof flux vacua, which may be $B_{max}$ of \\cite{Taylor:2015xtz}. Needless to say, this is a fascinating question moving forward.\n\nWhat is becoming clear is that non-Higgsable clusters and structure\nplay a very important role in the landscape of F-theory\ncompactifications. It has become common to say that non-Higgsable\nclusters are doubly generic. The first is a strong sense: for fixed\n$B$, having a non-Higgsable cluster means that there is a non-trivial seven-brane\nfor generic points in $\\cs{X}$. The second is in a weaker, but still compelling, sense:\nthere is growing evidence that generic extra dimensional spaces $B$ give rise to\nnon-Higgsable clusters or structure. One line of evidence is in the large datasets cited above.\nAnother is the argument of \\cite{Halverson:2015jua}: if there is a curve $C \\subset B$ with $-K\\cdot C<0$\nthen $-K$ contains $C$ and $C$ sits inside the discriminant locus, giving non-Higgsable structure on $C$.\nSuch $B$ are ones that are not weak Fano, and it is expected that a generic algebraic surface or threefold\nis of this type. In particular, there are only $105$ topologically distinct Fano threefolds.\n\n\n\\vspace{1cm}\nIn this work we will study the strongly coupled physics associated to fibers that\ncan give rise to geometrically non-Higgsable seven-branes. As such, the analyses of this\npaper include, but are not limited to, F-theory compactifications with non-Higgsable\nseven-branes. These fibers are\n\\begin{equation}\nII, III, IV, I_0^*, IV^*, III^*, II^*,\n\\end{equation}\nand any seven-brane with one of these has an associated $SL(2,\\mathbb{Z})$ monodromy\nmatrix $M$ that is nilpotent, i.e. $M^k=1$ for some $k$. $M_{I_0^*}=-1$ which acts trivially on\n$\\tau$, indicating that this configuration is uncharged, in agreement with the fact that it is\n$4$ $D7$-branes on top of an $O7$ plane from the type IIB point of view. The rest act non-trivially on\n$\\tau$ but the theory comes back to itself after taking $k$ loops around the seven-brane; the seven-brane\ncharges are nilpotent. Though our analyses apply more broadly,\nthey are of particular interest given the prevalence of non-Higgsable clusters in the landscape.\n\n\n\\section{Axiodilaton Profiles and Strong Coupling}\n\\label{sec:axiodilaton}\n\n\nThe primary difference between F-theory and the weakly coupled type\nIIB string is that the axiodilaton $\\tau = C_0 + i e^{-\\phi}$ varies\nover $B$ in F-theory, and therefore so does the string coupling $g_s =\ne^{\\langle \\phi \\rangle}$. The behavior of $\\tau$ near seven-branes affects gauge theories on seven-branes,\nas well as three-brane gauge theories or string scattering in the vicinity of seven-branes.\nIn his seminal works \\cite{MR0187255,MR0205280,MR0228019} Kodaira computed $\\tau$ locally near\nseven-branes in elliptic surfaces.\n\nIn this section we will study axiodilaton profiles via their relation to the Klein $j$-invariant\nof an elliptic curve for elliptic fibrations of arbitrary dimension.\nWe will normalize the $j$-invariant in a standard way by $J:= j \/1728$, and in the\ncase of a Weierstrass model we have\n\\begin{equation}\nJ = \\frac{4f^3}{\\Delta} \\qquad \\text{where} \\qquad \\Delta = 4f^3 + 27g^2.\n\\end{equation}\nIn this formulation the $J$-invariant depends on base coordinates\naccording to the sections $f$ and $g$ of the Weierstrass\nmodel. However, $J$ also depends on the ratio of periods of the\nelliptic curve $\\tau = \\frac{\\omega_1}{\\omega_2}$ where $\\tau$ is the\nvalue of the axiodilaton field at each point in $B$. Thus, if $z$ is a\nbase coordinate we compute $J=J(z)$ directly from the Weierstrass\nmodel, but this can also be thought of as $J = J(\\tau(z))$. By\ninverting the $J$-function, we will determine the axiodilaton profile\nand study it in the vicinity of geometrically non-Higgsable seven-branes.\nWe will also demonstrate that F-theory compactifications generically exhibit\nregions with $O(1)$ string coupling and recover classic results from the perturbative type IIB theory.\n\n\\begin{table}[thb]\n  \\centering\n  \\begin{tabular}{|ccc|} \\hline\n    Fiber & $J$ & $J|_{z=0}$ \\\\ \\hline \\hline\n $II$ & $\\frac{z}{A+z}$& $0$ \\\\ \\hline\n $III$ & $\\frac{1}{1+Az}$& $1$ \\\\ \\hline\n $IV$ & $\\frac{z^2}{A+z^2}$& $0$ \\\\ \\hline\n $I_0^*$ & $\\frac{1}{1+A}$& $\\frac{1}{1+A}$ \\\\ \\hline\n $IV^*$ & $\\frac{z}{A+z}$& $0$ \\\\ \\hline\n $III^*$ & $\\frac{1}{1+Az}$& $1$ \\\\ \\hline\n $II^*$ & $\\frac{z^2}{A+z^2}$& $0$ \\\\ \\hline\n  \\end{tabular}\n  \\caption{The $J$-invariant for seven-branes associated to geometrically non-Higgsable clusters, expressed in a way that is particularly useful for a local analysis near the seven-brane. Here $f=z^n\\, F$ and  $g=z^m\\, G$ with $A= 27G^2\/4F^3$.}\n  \\label{tab:Jw}\n\\end{table}\n\n\nThough there are seven Kodaira fiber types that may give rise to geometrically\nnon-Higgsable seven-branes, $\\{II,III,IV,I_0^*,IV^*,III^*,II^*\\}$, some\nhave the same $J$-invariant and $\\tau$ in the vicinity\nof the brane. In each case the Weierstrass model takes the form\n\\begin{equation}\nf = z^n \\, F, \\qquad g=z^m\\, G, \\qquad \\Delta = z^{min(3n,2m)}\\, \\tilde \\Delta,\n\\end{equation}\nand the $J$-invariant takes a simple form. Near a generic region of the\nseven-brane on $z=0$ both $F$ and $G$\nare non-zero, and therefore so is $A \\equiv 27G^2 \/ 4F^3$. The possibilities\nare computed in Table \\ref{tab:Jw} and the redundancies are \\cite{MR0187255,MR0205280,MR0228019}\n\\begin{equation}\nJ_{II} = J_{IV^*}, \\qquad J_{III} = J_{III^*}, \\qquad J_{IV} = J_{II^*}.\n\\end{equation}\nThis result may seem at odds with the monodromy order for these\nKodaira fibers displayed in Table \\ref{tab:fibs}, since the type $II$\nand $II^*$ fibers have order $6$ whereas the type $IV$ and $IV^*$\nfibers have order $3$. The resolution is that, though the monodromy\nassociated with type $II$ and $II^*$ fibers is $6$, $M_{II}^3 =\nM_{II^*}^3=-I$, where $I$ is the identity matrix, so that the type $II$,\n$II^*$, $IV$, and $IV^*$ fibers all induce an order $3$ action on $\\tau$.\n\n\n\nThere are some special values for $\\tau(J)$ that we will see arise in inverting\n$J$,\n\\begin{equation}\n\\tau(0)=e^{\\pi i\/3}, \\qquad \\tau(1) = i,\n\\end{equation}\nup to an $SL(2,\\mathbb{Z})$ transformation. These values correspond to $g_s = \\frac{2}{\\sqrt{3}}$\nand $g_s=1$, and it is important physically that these cannot be lowered\nby an $SL(2,\\mathbb{Z})$ transformation. Mapping $\\tau\\mapsto \\tau':=\\frac{a\\tau + b}{c\\tau + d}$\nby an arbitrary $SL(2,\\mathbb{Z})$ transformation for $\\tau=e^{\\pi i\/3}$ and $\\tau =i$, respectively,\nwe have new string coupling constants\n\\begin{equation}\ng_s'=(c^2+cd+d^2)\\frac{2}{\\sqrt{3}}\\geq \\frac{2}{\\sqrt{3}}, \\qquad g_s'=(c^2+d^2)\\geq 1,\n\\label{eq:cantgoweak}\n\\end{equation}\nshowing that the string couplings\nwith these two values of $\\tau$ cannot be lowered by a global $SL(2,\\mathbb{Z})$ transformation.\n\nFor each case in Table \\ref{tab:Jw} we will invert $J$ to solve for\n$\\tau$.\n\n\\subsection{Inverting the $J$-function and Ramanujan's Alternative Bases}\n\nThere is a nineteenth century procedure for inverting the $J$-function that is due\nto Jacobi. Recall that the $j$-invariant satisfies\n\\begin{equation}\nj(q) = \\frac{1}{q} + 744 + 196884 \\,\\, q + \\dots \n\\end{equation}\nin terms of $q=e^{2\\pi i  \\tau}$. Jacobi's result relates $j$ to $q$ via hypergeometric\nfunctions, which then allows for the computation of $\\tau$ by taking a logarithm.\nThe result is\n\\begin{equation}\n\\tau = i\\,\\,\\, \\frac{_2F_1(\\frac12,\\frac12;1;1-x)}{_2F_1(\\frac12,\\frac12;1;x)}, \\qquad J = \\frac{4\\, (1-x(1-x))^3}{27\\, x^2(1-x)^2}, \n\\label{eq:tauJJacobi}\n\\end{equation}\nin terms of the hypergeometric function $_2F_1(a,b,c;x)$. For $|x|<1$ it satisfies\n\\begin{equation}\n_2F_1(a,b;c;x) = \\sum_{n=0}^\\infty \\frac{(a)_n(b)_n}{(c)_n \\, n!} x^n,\n\\end{equation}\nwhere $(a)_n=a(a+1)(a+2)\\dots(a+n-1)$ for $n\\in \\mathbb{Z}^+$ is the\nPochhammer symbol. For a particular value of $J$, then, six values of\n$\\tau$ are obtained by solving the sextic in $x$, and these are\nrelated to one another by $SL(2,\\mathbb{Z})$ transformations.\n\nMuch progress was made in the theory of elliptic functions at the beginning of the\n$20^{\\text{th}}$ century by Ramanujan, who recorded his theorems in notebooks \\cite{MR0099904}\nthat were dense with results. In one, he claimed that there are similar inversion formulas\nwhere the base $q$ is not\n\\begin{equation}\nq = exp\\left({-\\pi \\frac{_2F_1(1\/2,1\/2,1,1-x)}{_2F_1(1\/2,1\/2,1,x)}}\\right)\n\\end{equation}\nas it was for Jacobi, but is instead one of\n\\begin{align}\nq &= exp\\left({-\\frac{2\\pi}{\\sqrt{3}} \\frac{_2F_1(1\/3,2\/3,1,1-x)}{_2F_1(1\/3,2\/3,1,x)}}\\right), \\nonumber \\vspace{2cm} \\\\ \nq &= exp\\left({-\\frac{2\\pi}{\\sqrt{2}} \\frac{_2F_1(1\/4,3\/4,1,1-x)}{_2F_1(1\/4,3\/4,1,x)}}\\right), \\nonumber \\\\\nq &= exp\\left({-2\\pi \\frac{_2F_1(1\/6,5\/6,1,1-x)}{_2F_1(1\/6,5\/6,1,x)}}\\right).\n\\label{eq:altbaseinit}\n\\end{align}\nThere has been significant progress \\cite{MR1311903,MR1117903,MR1071759,MR1010408,MR1237931,MR1243610,MR1825995,MR3107523,Cooper2009} in the study of Ramanujan's theories of elliptic functions\nto these alternative bases in recent years, including rigorous proofs of many of Ramanujan's results.\nPractically, these different theories give different ways to study $\\tau$.\n\nIn studying the relationship between $J$, $\\tau$, and Ramanujan's alternative theories, we will\nutilize the notation of Cooper \\cite{Cooper2009}. The alternative bases satisfy\n\\begin{equation}\nq_r := exp\\left(\\frac{-2\\pi}{\\sqrt{r}}\\frac{_2F_1(a_r, 1-a_r;1;1-x_r)}{_2F_1(a_r,1-a_r;1;x_r)}\\right)\n\\label{eq:altq}\n\\end{equation}\nwhere $a_1=\\frac16$, $a_2=\\frac14$, and $a_3=\\frac13$ for $r=1,2,3$ reproduce \\eqref{eq:altbaseinit},\nwhere the $J$ invariant satisfies\n\\begin{equation}\nJ = \\frac{1}{4\\,x_1(1-x_1)} = \\frac{(1+3x_2)^3}{27\\,x_2(1-x_2)^2}=\\frac{(1+8x_3)^3}{64\\,x_3(1-x_3)^3}.\n\\label{eq:JRam}\n\\end{equation}\nFor any value of $J$ one may then solve either Jacobi's sextic (\\ref{eq:tauJJacobi}) or the quadratic,\ncubic, or quartic in \\eqref{eq:JRam}. Other inversion methods also exist, but we will not use them.\n\n\\vspace{.5cm}\nWe utilize these methods to study elliptic fibrations,  beginning with general statements\nand then proceeding to the study of examples near geometrically non-Higgsable seven-branes.\n\nConsider a Weierstrass model, where $J=4f^3\/\\Delta$. In a neighborhood of a seven-brane\non $z=0$ one can compute the $SL(2,\\mathbb{Z})$ monodromy $M$ of the seven-brane by taking a small\nloop around the seven-brane, which includes an action on $\\tau$. Recalling that\n\\begin{equation}\nM^k(\\tau) = \\tau\n\\end{equation} \nfor some $k$ for any geometrically non-Higgsable seven-brane, one would like to verify $k$\ndirectly by inverting the $J$-function. We will see that some of Ramanujan's theories give\nrise to $k$ element sets of $\\tau$ values that are permuted by the monodromy, where $k$ is the\norder of $x_i$ in \\eqref{eq:JRam}.\n\nSince each solution for $x_i$ determines a value of $\\tau$ directly\nvia $q$ in \\eqref{eq:altq}, let us solve for $x_i$ in terms of $J$.\nIn the quadratic case we have\n\\begin{equation}\nx_1 = \\frac{1\\pm \\sqrt{1-1\/J}}{2}\n\\label{eq:xinJquad}\n\\end{equation}\nwhich is degenerate for $J=1$ and is not well defined for $J=0$. Aside from $I_0^*$, these\nare $J$-invariants associated with geometrically non-Higgsable seven-branes. \n\nIn the cubic case we solve the equation\n\\begin{equation}\n27(J-1) \\, x_2^3-27(J+2) \\, x_2^2+ 9(3J-1) \\, x_2-1 = 0\n\\end{equation}\nto obtain $x_2$. \nRather than solving this cubic exactly, let study the behavior of this\ncubic near $J=0$ and $J=1$, since this is the relevant structure near non-Higgsable seven-branes.\nExpanding around $J=0$ by taking $J=\\delta J_0 \\ll 1$ the three solutions for $x_2$ near $J=0$ are \n\\begin{equation}\nx_2 = -\\frac{1}{3}-\\frac{2\\sqrt[3]{2}\\,}{3} e^{2\\pi i n\/3} \\,\\delta J_0^{1\/3}+O(\\delta J_0^{2\/3}), \\qquad n\\in\\{0,1,2\\}  \n\\label{eq:x2nearJis0}\n\\end{equation}\nand we see that the three roots are permuted by an order three monodromy upon taking a small\nloop around $J=0$. Near $J=1$ we take $J = 1+\\delta J_1$ with $\\delta J_1 \\ll 1$ and compute\n\\begin{equation}\nx_2 = \\frac{3}{\\delta J_1} + \\frac{16}{9} - \\frac{16\\, \\delta J_1}{81} + O(\\delta J_1^2), \\qquad\nx_2 = \\frac{1}{9} \\pm \\frac{8\\sqrt{\\delta J_1}}{27\\sqrt{3}} + \\frac{8 \\delta J_1}{81} + O(\\delta J_1^{3\/2}) \n\\label{eq:x2nearJis1}\n\\end{equation}\nfrom which we see that one of the solutions goes to infinity at $J=1$ (as is expected since the cubic\nreduces to a quadratic when $J=1$), whereas the other two are permuted by an order two monodromy around $J=0$.\n\nTherefore, we will study seven-brane theories with $J=1$ $(J=0)$ with Ramanujan's theory where\n$J$ is quadratic (cubic) in $x_1$ ($x_2$), solving for $\\tau$.\n\n\\subsection{Warmup: Reviewing Weakly Coupled Cases}\n\nBefore proceeding to the interesting seven-brane structures that may be non-Higgsable, all of which have\nfinite $J$-invariant, let us consider the seven-branes that may appear in the weakly coupled type IIB\ntheory, which have $J=\\infty$.\n\nLet us begin with the case of $n$ coincident $D7$-branes, which in F-theory language have a Kodaira\nfiber $I_n$.  The Weierstrass model takes takes the form\n\\begin{equation}\nf = F, \\qquad g = G, \\qquad \\Delta = z^n \\tilde \\Delta,\n\\end{equation}\nwhere we have used our common notation of inserting $F$ in $f$ (and $G$ in $g$) even though $f,g\\sim z^0$\nin this case, and $z$ does not divide $\\tilde \\Delta$. Instead, $F$ and $G$ must be tuned to ensure the form\nof $\\Delta$. The $J$-invariant is\n\\begin{equation}\nJ(I_n) = \\frac{4F^3}{z^n\\tilde \\Delta}=:\\frac{C}{z^n},\n\\end{equation}\nand we can see that, indeed, $J=\\infty$ at $z=0$. Solving the theory where $J$ is a quadratic\nin $x_1$ gives\n\\begin{equation}\nx_1 = \\frac12 \\pm \\frac12 \\sqrt{1-\\frac{z^n}{C}}=: \\alpha_\\pm,\n\\end{equation}\nand then using equation (\\ref{eq:altq}) to compute $\\tau(\\alpha_-)$ and Taylor expanding we\nfind\n\\begin{equation}\n\\tau(\\alpha_-) = \\frac{n\\, log(z)}{2\\pi i} + \\cdots,\n\\end{equation}\nwhich induces a monodromy $\\tau \\mapsto \\tau+n$ upon encircling\n$z=0$. This is the expected monodromy of a stack of $n$ D7-branes.\nThe other solution $\\tau(\\alpha_+)$ is S-dual to\n$\\tau(\\alpha_-)$.\n\nNow consider $n\\geq 4$ D7-branes that are on top of an O7-plane, which in F-theory language corresponds to an $I_{n-4}^*$ fiber.\nIn this case the Weierstrass model is\n\\begin{equation}\nf = z^2 F, \\qquad g = z^3 G, \\qquad \\Delta = z^{2+n} \\tilde \\Delta,\n\\end{equation}\nwith $J$-invariant\n\\begin{equation}\nJ(I_{n-4}^*) = \\frac{4F^3}{z^{n-4}\\tilde \\Delta}=: \\frac{C}{z^{n-4}}.\n\\end{equation}\nThen again solving the theory where $J$ is a quadratic in $x_1$ we obtain\n(with similar $\\alpha_\\pm$)\n\\begin{equation}\n\\tau(\\alpha_-) = \\frac{(n-4)\\, log(z)}{2\\pi i} + \\cdots,\n\\end{equation}\nand there is a monodromy $\\tau \\mapsto \\tau + n-4$ upon encircling $z=0$.\nThis is the monodromy expected for $n$ D7-branes on top of an O7-plane, and \nfamously there is no monodromy in the case $n=4$, since the 4 D7-branes\ncancel the charge of the O7-plane.\n\n\n\n\\subsection{Axiodilaton Profiles Near Seven-Branes with $J=1$}\n\nFrom Tables \\ref{tab:fibs} and \\ref{tab:Jw} we see that the seven-branes with \n$J=1$ have fiber of Kodaira type $III$ and $III^*$, which carry gauge\nsymmetry $SU(2)$ and $E_7$, respectively. In both cases we have the same local\nstructure of the $J$-invariant\n\\begin{equation}\n  J(III) = J(III^*) = \\frac{1}{1+Az}.\n\\end{equation}\nUsing equation (\\ref{eq:xinJquad}) we see \n\\begin{equation}\nx_1 = \\frac{1\\pm i \\sqrt{A z}}{2} =: \\alpha_\\pm,\n\\end{equation}\nwhich exhibits a $\\mathbb{Z}_2$ monodromy around $z=0$ that induces a monodromy on $\\tau$. \nUsing the relationship \\eqref{eq:altq} between $q^{2\\pi i \\tau}$ and $x_1$ we compute two\nvalues for $\\tau$\n\\begin{equation}\n\\tau_{\\pm} = i \\, \\frac{_2F_1(\\frac16,\\frac56,1,\\alpha_+)}{_2F_1(\\frac16,\\frac56,1,\\alpha_-)}.\n\\label{eq:tauexactJis1}\n\\end{equation}\nSince the $\\mathbb{Z}_2$ monodromy swaps $\\alpha_\\pm$, it also swaps $\\tau_\\pm$ and noting $\\tau_- = -1\/\\tau_+$\nwe see\n\\begin{equation}\n\\tau_\\pm \\mapsto \\tau_\\mp= - \\frac{1}{\\tau_\\pm}\n\\end{equation}\nunder the monodromy. This matches the behavior associated with the monodromy matrices\n\\begin{equation}\nM_{III}\\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix}, \\qquad M_{III^*} = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0\\end{pmatrix} \n\\end{equation}\nand we have seen the result by explicitly solving for the axiodilaton $\\tau$. Note that this monodromy\nis precisely an $S$-duality, which therefore also swaps electrons and monopoles represented by $(p,q)$-strings\nto D3-brane probes.\n\nHow does the physics, in particular the string coupling, change upon moving away from the seven-brane? Expanding\nthe exact solution (\\ref{eq:x2nearJis1}) near $z=0$ we obtain\n\\begin{equation}\n\\tau_\\pm = i \\pm B\\sqrt{Az}-\\frac{i}{2} B^2 A z + O(z^{3\/2})\n\\end{equation}\nwhere \n\\begin{equation}\nB = \\frac{5\\, \\Gamma(\\frac{7}{12}) \\Gamma(\\frac{11}{12})}{36\\, \\Gamma(\\frac{13}{12}) \\Gamma(\\frac{17}{12})} \\simeq .2638,\n\\end{equation}\nis a constant that depends on values of the Euler $\\Gamma$ function but $A = 27G^2\/4F^3$ depends on the location\nin the base. We see that $\\tau(z)$ satisfies $\\tau(0)=i$ and \n\\begin{equation}\ng_{s,\\pm} \\simeq \\frac{1}{1 \\pm B \\,Im(\\sqrt{Az}) - \\frac12 B^2 Re(Az)}. \n\\end{equation}\nClose to $z=0$ we have\n\\begin{equation}\ng_{s,\\pm} \\simeq \\frac{1}{1 \\pm B \\,Im(\\sqrt{Az})}, \n\\end{equation}\nand we see that the monodromy exchanges a more weakly coupled theory with a more strongly coupled theory, where the deviation from\n$g_s=1$ depends on the model-dependent factor $A$ and the separation $z$ from the brane. This was also implicit from $\\tau\\mapsto -1\/\\tau$.\n\nWe see directly that the string coupling $g_s\\simeq O(1)$ in the\nvicinity of the type $III$ and type $III^*$ seven-branes carrying\n$SU(2)$ and $E_7$ gauge symmetry, respectively, and that Ramanujan's\ntheory where $J$ is a quadratic in $x_1$ gives a set of $\\tau$ values\npermuted by the brane-sourced monodromy. From (\\ref{eq:cantgoweak}),\nan $SL(2,\\mathbb{Z})$ transformation cannot make the theory weakly coupled in\nthis region.  \\vspace{.5cm}\n\nIn the\nprevious section we saw that this method is more natural than the\ntheory where $J$ is cubic in $x_2$ since the $SL(2,\\mathbb{Z})$ monodromy of\nthe type $III$ and $III^*$ Kodaira fibers is $\\mathbb{Z}_2$ rather than\n$\\mathbb{Z}_3$. For completeness, though, inverting using the cubic theory gives\nthree solutions for $x_2$\n\\begin{align}\nx_2 = \\left\\{-\\frac{3}{A z}-\\frac{11}{9}+O\\left(z\\right),\\frac{1}{9}-\\frac{8 i \\sqrt{A^3} \\sqrt{z}}{27 \\sqrt{3}\n   A}+O\\left(z\\right),\\frac{1}{9}+\\frac{8 i \\sqrt{A^3}\n   \\sqrt{z}}{27 \\sqrt{3} A}+O\\left(z\\right)\\right\\},\n\\end{align}\nwhere we see that the first solution decouples near $z=0$ and one is left with the latter two\nsolutions, which we will call $\\beta_\\pm$. There is a $\\mathbb{Z}_2$ monodromy that exchanges $\\beta_\\pm$\nupon encircling $z=0$. Defining $\\tau_\\pm$ for the cubic theory as \n$\\tau_\\pm \\equiv \\tau(\\beta_\\pm)$ we have\n\\begin{equation}\n\\tau_\\pm = \\frac{i}{\\sqrt{2}} \\frac{_2F_1(\\frac14,\\frac34,1,1-\\beta_\\pm)}{_2F_1(\\frac14,\\frac34,1,\\beta_\\pm)},\n\\end{equation}\nand, though direct evaluation gives $\\tau_\\pm=i$ at $z=0$, the monodromy\n$\\tau \\mapsto -\\frac{1}{\\tau}$ is not immediate, instead requiring the use of some identities for the\nhypergeometric function for an exact expression.\nWe have verified via Taylor expansion that $\\tau_\\pm$ are swapped by a $\\mathbb{Z}_2$ monodromy, however.\nThus, the theory quadratic in $x_1$ seems better suited to study $III$ and $III^*$ fibers.\n\n\\subsection{Axiodilaton Profiles Near Seven-Branes with $J=0$}\n\nWe now turn to the study of seven-branes with Kodaira fibers satisfying $J=0$. \nFrom Tables \\ref{tab:fibs} and \\ref{tab:Jw} we see that the seven-branes with \n$J=0$ have general fiber of Kodaira type $II$, $II^*$, $IV$, and $IV^*$. The\nseven-branes of the first two types carry no geometric gauge symmetry and $E_8$,\nrespectively, whereas the latter two exhibit $SU(3)$ ($SU(2)$) and $E_6$ ($F_4$)\ngeometric gauge symmetry respectively, if the geometry does not (does) exhibit\nouter-monodromy that reduces the rank of the gauge group. Recall that\n\\begin{align}\n  J(II) = J(IV^*) = \\frac{z}{A+z}, \\qquad J(II^*) = J(IV) = \\frac{z^2}{A+z^2},\n\\label{eq:localJforJis1}\n\\end{align}\nand that there is no discrepancy in this matching because, though the $SL(2,\\mathbb{Z})$ monodromy\nassociated with these fibers are either order $3$ or $6$,  they are only order\n $3$ on $\\tau$.\n\nLet us utilize the theory where $J$ is a cubic in $x_2$ to study $\\tau$ near these\nseven-branes, beginning with the cases of a type $II$ and $IV^*$ fiber since these\nhave the same local structure of $J$-invariant. The expansion of the three solutions\nto the cubic in $x_2$ expanded around $z=0$ are\n\\begin{equation}\nx_2 =  -\\frac13 - \\frac{2(2A^2)^{1\/3}}{3A} e^{\\frac{2\\pi i n}{3}} \\, z^{1\/3} -\\frac{4}{3(2A^2)^{1\/3}} e^{\\frac{4\\pi i n}{3}} \\, z^{2\/3} - \\frac{1}{A} z + O(z^{4\/3}), \\qquad n\\in\\{0,1,2\\},\n\\end{equation}\nfrom which we see a $\\mathbb{Z}_3$ monodromy upon encircling the seven-brane\nat $z=0$. Letting $\\beta_n$ be the $x_2$ solution for each $n\\in\n\\{0,1,2\\}$, we have\n\\begin{equation}\n\\tau_n := \\tau(\\beta_n) = \\frac{i}{\\sqrt{2}} \\frac{_2F_1(\\frac14, \\frac34, 1,1-\\beta_n)}{_2F_1(\\frac14, \\frac34, 1,\\beta_n)}.\n\\end{equation}\nIf the $\\tau$ values are distinct then there is an order $3$ $SL(2,\\mathbb{Z})$ monodromy on $\\tau$, but its determining that\nits precise action is $\\tau \\mapsto \\frac{\\tau-1}{\\tau}$\nwould require using some identities of hypergeometric functions, unlike in the case of the type $III$ and $III^*$ seven-branes\nwhere its action $\\tau\\mapsto -1\/\\tau$ was immediately clear. Instead we will prove the monodromy numerically in a power series in $z$.\nNumerically at leading order in $z$ and keeping four significant figures in $z^{1\/3}$, we have\n\\begin{align}\n\\tau_0 &\\simeq e^{\\frac{\\pi i}{3}} -\\frac{.3355i}{A^{2\/3}} \\, z^{1\/3} + O(z^{2\/3}) \\nonumber \\\\\n\\tau_1 &\\simeq e^{\\frac{\\pi i}{3}} +\\frac{.2906+.1678 i}{A^{2\/3}} \\, z^{1\/3} + O(z^{2\/3})  \\nonumber \\\\\n\\tau_2 &\\simeq e^{\\frac{\\pi i}{3}} -\\frac{.2906-.1678i }{A^{2\/3}} \\, z^{1\/3} + O(z^{2\/3}).\n\\label{eq:tau012forIIIVs}\n\\end{align}\nWe encircle $z=0$ by writing $z=re^{i\\theta}$ where $r\\in \\mathbb{R}$ is a\nsmall positive number and then varying $\\theta$.  There is a choice of\ndirection: encircling by taking $\\theta$ from $0$ to $2\\pi$ we see\nthat $\\tau_0 \\mapsto \\tau_1$, $\\tau_1\\mapsto \\tau_2$,\n$\\tau_2\\mapsto\\tau_0$, whereas going in the other direction by taking\n$\\theta$ from $0$ to $-2\\pi$ gives the inverse action $\\tau_0 \\mapsto\n\\tau_2$, $\\tau_1\\mapsto \\tau_0$, $\\tau_2\\mapsto\\tau_1$. One can verify that\nthis latter action corresponds to the monodromy $\\tau \\mapsto \\frac{\\tau-1}{\\tau}$;\ni.e. $\\frac{\\tau_i - 1}{\\tau_i} = \\tau_{i-1}$ where $\\tau_{-1}:=\\tau_2$.\n\nNow consider the cases of seven-branes with a\ntype $II^*$ and type $IV$ fiber. From (\\ref{eq:localJforJis1}) we see that this differs \nfrom the analysis we just performed by the replacement $z\\mapsto z^2$, as can be verified\nby direct computation. The solutions to the cubic are\n\\begin{equation}\nx_2 =  -\\frac13 - \\frac{2(2A^2)^{1\/3}}{3A} e^{\\frac{2\\pi i n}{3}} \\, z^{2\/3} -\\frac{4}{3(2A^2)^{1\/3}} e^{\\frac{4\\pi i n}{3}} \\, z^{4\/3} - \\frac{1}{A} z^2 + O(z^{8\/3}), \\qquad n\\in\\{0,1,2\\},\n\\end{equation}\ndefining $\\beta_n$ to be these three solutions the function form of $\\tau$, and therefore $\\tau_n:=\\tau(\\beta_n)$, remains the same.\nNumerically at leading order in $z$, keeping four significant figures in $z^{2\/3}$\nwe have \n\\begin{align}\n\\tau_0 &\\simeq e^{\\frac{\\pi i}{3}} -\\frac{.3355i}{A^{2\/3}} \\, z^{2\/3} + O(z^{4\/3}) \\nonumber \\\\\n\\tau_1 &\\simeq e^{\\frac{\\pi i}{3}} +\\frac{.2906+.1678 i}{A^{2\/3}} \\, z^{2\/3} + O(z^{4\/3})  \\nonumber \\\\\n\\tau_2 &\\simeq e^{\\frac{\\pi i}{3}} -\\frac{.2906-.1678i }{A^{2\/3}} \\, z^{2\/3} + O(z^{4\/3}).\n\\label{eq:tau012forIIIVs}\n\\end{align}\nNote that the monodromy action has changed, though: upon taking $\\theta$ from $0$ to $2 \\pi$\nwe have $\\tau_0\\mapsto \\tau_2$, $\\tau_1\\mapsto \\tau_0$, $\\tau_2\\mapsto \\tau_1$. In the type $II$, $IV^*$\ncase this was the monodromy associated with taking $\\theta$ from $0$ to $-2\\pi$. We see explicitly\nthat the $SL(2,\\mathbb{Z})$ monodromy of $II\/IV^*$ fibers induce the inverse action on $\\tau$ compared to\n$II^*\/IV$ fibers. This is as expected since\n\\begin{equation}\nM_{II}=M_{II^*}^{-1}, \\qquad M_{IV}=M_{IV^*}^{-1}\n\\end{equation}\nand the fact\n\\begin{equation}\nM_{II}=-M_{IV^*}. \\qquad M_{II^*}=-M_{IV}\n\\end{equation}\nimplies that $II$ and $IV^*$ induce the same action on $\\tau$, and similarly for $II^*$ and $IV$.\n\nGiven these explicit solutions for $\\tau$ one can solve for $g_s$ as a function of the local coordinate\n$z$ near the seven-branes with type $II$, $IV^*$, $II^*$, or $IV$ fibers and determine its falloff\nfrom the central value $g_s=2\/\\sqrt{3}$ at $z=0$.\n\n\n\\subsection{The Genericity of Strongly Coupled Regions in F-theory and the Sen Limit}\n\nIn this section we would like to demonstrate the genericity of\nstrongly coupled regions in F-theory. This statement is immediately\nplausible since F-theory is a generalization of the weakly coupled\ntype IIB string with varying axiodilaton, but we would like to argue\nthat there are regions with $O(1)$ $g_s$ for nearly all of the moduli\nspace of F-theory using two concrete lines of evidence, one that utilizes\nnon-Higgsable clusters and one that does not.\n\nRecall from section \\ref{sec:review} that there is growing evidence and argumentation\nthat nearly all extra dimensional topologies $B$ give rise to geometrically non-Higgsable\nstructure, and it is typically the case that those have geometrically\nnon-Higgsable seven-branes. The latter have a Kodaira fiber in the set\n\\begin{equation}\\{II,III,IV,I_0^*,IV^*,III^*,II^*\\}\\end{equation} and\nfor all but\\footnote{It would also be interesting to understand\n  whether a non-Higgsable seven-brane with $I_0^*$ fiber, which could\n  have gauge group $SO(8), SO(7),$ or $G_2$, necessarily forbids a Sen\n  limit; certainly if it is Higgsable, such a limit can exist.}\n$I_0^*$ the axiodilaton is strongly coupled in the vicinity of the\nseven-brane, as seen using the explicit solutions of section\n\\ref{sec:axiodilaton}. If the argumentation regarding the genericity of non-Higgsable\nclusters is correct and a typical compactification has a non-Higgsable\nseven-brane with fiber $II,III,IV,IV^*, III^*,$ or $II^*$, then there are\nregions with $O(1)$ $g_s$ for most of the moduli space of F-theory.\n\n\n\nHowever, there are regions with $O(1)$ $g_s$\nquite generally even in the absence of non-Higgsable clusters.\nConsider a completely general Weierstrass model, which has a\n$J$-invariant\n\\begin{equation}\nJ = \\frac{4f^3}{4f^3+27g^2}.\n\\end{equation}\nIn some cases $f$ and $g$ may be reducible (e.g. if there are\nnon-Higgsable clusters), but it need not be so and it will not affect\nthe following calculation. There are two special loci in this generic\ngeometry, $f=0$ and $g=0$, and on these loci\n\\begin{equation}\nJ|_{f=0} = 0, \\qquad J|_{g=0}=1.\n\\end{equation}\nSince we have seen its utility for studying loci with $J=1$, consider Ramanujan's theory in\nwhich $J$ is a quadratic in $x_1$. Then we have\n\\begin{equation}\nx_1 = \\frac12 \\pm \\frac{3i\\sqrt{3} g}{4f^{3\/2}}=: \\alpha_\\pm\n\\end{equation}\nand solving for $\\tau$ we have\n\\begin{equation}\n\\tau_{\\pm} = i \\, \\frac{_2F_1(\\frac16,\\frac56,1,\\alpha_+)}{_2F_1(\\frac16,\\frac56,1,\\alpha_-)}.\n\\label{eq:tauonfis0}\n\\end{equation}\nwhich is the same form as for seven-branes with type $III$ or $III^*$ Kodaira fibers\nas in \\eqref{eq:tauexactJis1}, though $\\alpha_\\pm$ have different forms which critically\nchange the physics. For example, the change in $\\alpha_\\pm$ changes the local structure of $\\tau$ to\n\\begin{equation}\n\\tau_\\pm = i \\pm B \\frac{g}{f^{3\/2}}+O(g^2)\n\\end{equation} where here $B = \\frac{5\\, \\Gamma(\\frac{7}{12}) \\Gamma(\\frac{11}{12})}{8\\sqrt{3}\\, \\Gamma(\\frac{13}{12}) \\Gamma(\\frac{17}{12})} \\simeq .68542$, and we see (at leading order\\footnote{This is\n  extended to all orders by the fact that the power series expansion for $_2F_1(a,b;c;x)$ holds for $|x|<1$, which occurs for sufficiently\n  small $g$ away from $f=0$.}) that there is no monodromy associated with taking a loop around $g=0$ for $f\\ne0$.\nThis makes physical sense, because such a locus has no seven-branes, and therefore no source for $\\tau$-monodromy! \nNevertheless, the region $g=0$ is strongly coupled: at $g=0$, $\\alpha_+=\\alpha_-$, $\\tau=i$, and therefore\n$g_s=1$. Similar results can be obtained using the $x_2$ theory to solve for $\\tau$ near $f=0$,\nand in that case $x_2$ has three solutions that give $\\tau=e^{\\pi i\/3}$, and therefore $g_s=\\frac{2}{\\sqrt{3}}$ at $f=0$.\n\nThese facts about strong coupling are quite general, independent of\nthe existence or non-existence of any non-trivial seven-brane\nstructure (i.e. with fiber other than $I_1$).  They hold even for\ncompletely smooth models, such as the generic Weierstrass model over\n$\\mathbb{P}^3$.\n\n\n\\vspace{.5cm}\nWhat happens to these strongly coupled regions in the Sen limit?\nRoughly, the Sen limit \\cite{Sen:1996vd,Sen:1997gv} is a weakly\ncoupled limit in moduli space where $J\\mapsto \\infty$ and therefore\n$\\tau \\mapsto i \\infty$, so $g_s\\mapsto 0$. This occurs because the\nWeierstrass model takes the form $f=C \\eta -3 h^2$, $g=h \\left(C \\eta -2 h^2\\right)$\nwith discriminant\n\\begin{equation}\n\\Delta = C^2 \\eta^2(4C\\eta - 9h^2)\n\\end{equation}\nwhere $\\eta$ and $h$ are sections and $C$ is a parameter. Sen's weak coupling limit is the $C\\mapsto 0$\nlimit. However, note that if $C$ is very small but non-zero\n\\begin{equation}\nJ = \\frac{4f^3}{C^2 \\eta^2(4C\\eta - 9h^2)} = \\frac{4 \\left(C \\eta -3 h^2\\right)^3}{C^2 \\eta ^2 \\left(4 C \\eta -9 h^2\\right)}\n\\end{equation}\nis made very large by $1\/C$, but not infinite. Therefore unless it is strictly true that $C=0$,\nthe loci $f=0$ and $g=0$ (i.e. $C\\eta=3h^2$ and $\\{h=0\\}\\cup\\{C\\eta=2h^2\\}$) have $J=0$ and $J=1$, and therefore\n$g_s=2\/\\sqrt{3}$ and $g_s=1$, respectively. In the limit of $C$ becoming very small but non-zero one expects the string coupling to become lower \nin the vicinity of $f=0$ and $g=0$, though $O(1)$ on the loci. \n\nLet us see this explicitly. First, solving quadratic theory for $\\tau$ near $h=0$ (a component of $g=0$) we obtain\n\\begin{equation}\n\\tau_\\pm = i \\pm B\\, (C\\eta)^{-1\/2} \\, h+O(h^2)\n\\end{equation}\nwhere $B = \\frac{5\\, \\Gamma(\\frac{7}{12}) \\Gamma(\\frac{11}{12})}{8\\sqrt{3}\\, \\Gamma(\\frac{13}{12}) \\Gamma(\\frac{17}{12})} \\simeq .68542$.\nAs $C$ becomes large, the $g_s$ associated with one of these solutions for a small $h$ becomes weaker, but $g_s=1$\nat $h=0$ if $C$ is finite. Studying the other component of $g=0$ where $C\\eta=2h^2$ is more difficult because the locus\nitself moves as $C$ is taken to $0$. This component intersects a disc centered at $h=0$ at $h=\\pm \\sqrt{C\\eta\/2}$,\nand solving the quadratic theory near $h=\\sqrt{C\\eta\/2}$ we have\n\\begin{equation}\n\\tau_\\pm = i \\pm i\\,4\\sqrt{2}B\\,\\left(h-\\sqrt{\\frac{C\\eta}{2}}\\right)+O\\left(\\left[h-\\sqrt{\\frac{C\\eta}{2}}\\right]^2\\right).\n\\end{equation}\nIn the strict limit $C=0$ all of these loci collapse to $h=0$ and the theory is weakly\ncoupled, but any $C\\ne 0$ gives $g_s=1$ on the two\ncomponents of $g=0$. Similarly, the locus $f=0$ gives rise to $g_s=2\/\\sqrt{3}$ \npoint at the loci $h=\\pm \\sqrt{\\frac{C\\eta}{3}}$ in an $h$-disc, and this could\nbe studied explicitly using the theory where $J$ is cubic in $x_2$.\n\n\nWhat is going on physically for these components? We have three: one for $f=0$\nand two for $g=0$, and unless $g_s=0$ they are strongly coupled loci.\nExamining the discriminant we see that as $C$ becomes small two\nseven-branes approach the locus $h=0$ and collide in the limit. This\nis the $O7$-plane, and therefore for $C\\ne 0$ the region $h=0$ is the\nstrongly coupled region between the two seven-branes that become the\n$O7$ in the Sen limit. The $h$-disc is useful for studying $f=0$ and the\nother component of $g=0$. The picture is\n\\begin{equation} \n\\begin{tikzpicture}[scale=1]\n    \\draw[xshift=7cm,thick,color=black] (60mm,0mm) circle (1mm);\n    \\draw[xshift=7cm,thick,color=black] (-60mm,0mm) circle (1mm);\n    \\fill[xshift=7cm,thick,color=blue] (0mm,0mm) circle (1mm);\n    \\fill[xshift=7cm,thick,color=blue] (28.2843mm,0mm) circle (1mm);\n    \\fill[xshift=7cm,thick,color=blue] (-28.2843mm,0mm) circle (1mm);\n    \\fill[xshift=7cm,thick,color=red] (23.094mm,0mm) circle (1mm);\n    \\fill[xshift=7cm,thick,color=red] (-23.094mm,0mm) circle (1mm);\n\\end{tikzpicture}\n\\label{eqn:pixpiypiz}\n\\end{equation}\nwhere the hollowed circles are the only places where $\\Delta$\nintersects the $h$-disc. These are the two $(p,q)$ seven-branes that\nbecome the $O7$ in the Sen limit, and in this limit the whole pictures\ncollapse to the central blue dot at $h=0$. The red (blue) dots are\nwhere the $f=0$ ($g=0$) locus intersects the $h$-disc, and they have\n$g_s=2\/\\sqrt{3}$ and $g_s=1$, respectively. They are strongly coupled\nregions that are separated from the branes for finite $C$.\n\n\n\\vspace{.3cm}\n\nThe one caveat that we have not yet discussed involves configurations with\nconstant coupling. They were studied in the K3 case by Dasgupta and\nMukhi \\cite{Dasgupta:1996ij}, with the result (which generalizes beyond\nK3) that seven-branes with fiber $II$, $III$, $IV$, $I_0^*$, $IV^*$,\n$III^*$, and $II^*$ may gave rise to constant coupling\nconfigurations. When this occurs, all but the $I_0^*$ case gives\nrise to constant coupling configurations with $O(1)$ $g_s$. In the\n$I_0^*$ case there are a continuum of possible $g_s$ values that may\nbe constant cross the base $B$. In such a case there may be multiple $I_0^*$ seven-branes\nand the Weierstrass model\ntakes the form\n\\begin{equation}\nf = F \\prod_i z_i^2, \\qquad g = G \\prod_i z_i^3,\n\\end{equation}\nwith $F$ and $G$ necessarily constants, rather than non-trivial sections of a line bundle. In such a case\nthey do not define vanishing loci in $B$ with $O(1)$ $g_s$. Instead, all of the factors of $z_i$ drop out\nof the $J$ invariant,\n\\begin{equation}\nJ = \\frac{4F^3}{4F^3+27G^2}\n\\end{equation}\nwhich is just a constant, not varying over the base. To our knowledge,\nthis is the only possible way to obtain an F-theory compatification\nwith non-zero $g_s\\ll 1$ which is weakly coupled everywhere in $B$.\n\nIn summary, aside from compactifications realizing this $I_0^*$ caveat\nor the strict $g_s=0$ limit, there is a region $f=0$ or $g=0$ (or a component thereof) in every\nF-theory compactification with $O(1)$ $g_s$, and it is not necessarily near\nany seven-brane.  This may have interesting implications for moduli\nstabilization or cosmology in the landscape.\n\n\\section{Non-Perturbative $SU(3)$ and $SU(2)$ Theories}\n\\label{sec:su3su2}\n\n\\subsection{Comparison to D7-brane Theories}\n\nIn this section we would like to compare the non-perturbative\nrealizations of $SU(3)$ and $SU(2)$ theories from type $IV$ and $III$\nfibers to the $SU(3)$ and $SU(2)$ theories\\footnote{The type $IV$ and $I_3$ fibers\nrealize $SU(2)$ ($SU(3)$) theories in six and four-dimensional models if the \nfiber is non-split (split).} that may be realized by\nstacks of three and two $D7$-branes at weak string coupling. The\nlatter are realized by type $I_3$ and $I_2$ fibers, and in some cases\n(but not if the $IV$ or $III$ is non-Higgsable) they are related by\ndeformation.\n\n\nSuch deformations are slightly unusual. In many cases a deformation of\na geometry spontaneously breaks the theory on the seven-brane, but in\nthese cases the deformation leaves the gauge group\nintact\\footnote{This is always true for a $III$ to $I_2$ deformation,\n  and is also true for a $IV$ to $I_3$ deformation provided the deformation preserve\n  the split or non-split property related to non-simply laced groups.}. However, the deformation is\nnon-trivial, since the Kodaira fiber, the number of branes in the stack, the $SL(2,\\mathbb{Z})$ monodromy,\nand the axiodilaton profile all change due to the deformation.\n\nWe begin by considering the relationship between $SU(2)$ theories\nrealized on seven-branes with type $I_2$ and $III$ Kodaira fibers.\nWe expand $f$ and $g$ as\n\\begin{equation}\nf = f_0 + f_1 z \\qquad \\qquad g = g_0 + g_1 z + g_2 z^2,\n\\end{equation}\nwhere $f_0$, $g_0$, and $g_1$ do not depend on $z$ but $f_1$ and $g_2$\ncan contain terms that are both constant in $z$ and depend on $z$. To engineer\nan $I_2$ singularity we move to a sublocus in complex structure moduli where\n\\begin{equation}\nf_0 = -3a^2 \\qquad g_0 = 2a^3 \\qquad f_1 = b \\qquad g_1 = -a b,\n\\end{equation}\nin which case\n\\begin{equation}\n\\Delta = z^2 \\left(108 a^3 g_2-9 a^2 b^2\\right)+ z^3(4b^3-54abg_2)+O(z^4).\n\\end{equation}\nHere $a$ and $b$ are global sections of line bundles $a\\in\n\\Gamma(\\mathcal{O}(-2K))$ and $b \\in \\Gamma(\\mathcal{O}(-4K-Z))$ where $Z$ is the class of the divisor\n$z=0$. Specifying in moduli such that $a=0$, we see that $f_0$, $g_0$, and $g_1$ vanish\nidentically, and the resulting fibration has $(ord f, ord g, ord \\Delta) = (1,2,3)$; thus\nin the limit $a\\mapsto 0$ the $I_2$ fiber over $z=0$ becomes type $III$.\n\nIn a generic disc $D$ containing $z=0$ where $z$ is also the coordinate of\nthe disc, $a$ is a constant since $f_0$, $g_0$, and $g_1$ \ndo not depend on $z$. We can study the geometry\nof the elliptic surface over the disc that is naturally induced from\nrestriction of the elliptic fibration. From the point of view of the\nelliptic surface, then, the $a\\mapsto 0$ limit is just a limit in a\nconstant complex number $a$ (technically $a|_D$, but we abuse\nnotation). Then $a=0$ realizes a type $III$ fiber along $z=0$, and a\nsmall deformation $a\\ne 0$ reduces it to an $I_2$ fiber; both\ngeometrically give rise to $SU(2)$ gauge theories on the\nseven-brane.\n\n We would also like to consider a second deformation\nparameter $\\epsilon$, set $b=g_2=1$ for simplicity, and consider a small\nenough disc that we can drop higher order terms in $z$; then the\ntwo-parameter family of elliptic fibrations over $D$ is given by\n\\begin{align}\nf = -3a^2 + z \\qquad \\qquad g = \\epsilon + 2 a^3 - a z + z^2  \\nonumber \\\\\n\\Delta = 108 a^3 z^2+108 a^3 \\epsilon -9 a^2 z^2-54 a z^3-54 a z \\epsilon +27 z^4+4 z^3+54 z^2\n   \\epsilon +27 \\epsilon ^2.\n\\end{align}\nWe note the behavior of the discriminant in the relevant limits:\n\\begin{align}\n\\epsilon=0&:  \\Delta = z^2 \\left(108 a^3-9 a^2-54 a z+27 z^2+4 z\\right)\\\\\n\\epsilon=a=0&: \\Delta = z^3 \\left(27 z+4\\right),\n\\end{align}\nwhere the $\\epsilon=0$ limit is the limit of $SU(2)$ gauge enhancement with a type\n$I_2$ fiber, and the $\\epsilon=a=0$ limit maintains the $SU(2)$ group but realizes\nthe theory instead with a type $III$ fiber.\n\n\\vspace{.5cm}\n\\noindent \\emph{Analysis of the Two Deformations}\n\nLet us study two different deformations of the type $III$\ntheory that uncover its differences from the $I_2$ theory, and also\nthe relationship between the two via deformation. The first\ndeformation is to take $\\epsilon \\ne 0$ but small; specifically,\n$\\epsilon = .001$. This is a small breaking of the type $III$ $SU(2)$\ntheory, where the deformation causes the three branes comprising the\ntype $III$ theory to split into three $(p,q)$ seven-branes in a smooth\ngeometry with no non-abelian gauge symmetry. For these parameters the\ngeometry appears in Figure \\ref{fig:a0}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.75]{figures\/zIIII2azero.eps} \\qquad \\qquad\n\\includegraphics[scale=.75]{figures\/xIIII2azero.eps}\n\\end{center}\n\\caption{Two figures of the geometry deformed by $\\epsilon=.001$, $a=0$, with\n$(p,q)$ seven-branes on the left and ramification points of the elliptic curve on the right.}\n\\label{fig:a0}\n\\end{figure}\nwhere we have displayed the branes in the $z$-plane as red dots and we\nhave also displayed the $x$-plane above $z=0$, where the blue dots\nrepresent three ramificaton points of the torus described as a double\ncover (as is natural in a Weierstrass model). Any straight line\nbetween two of the blue dots determines a one-cycle in the elliptic\nfiber above $z=0$, and by following straight line paths from $z=0$ to\nthe seven-branes two of the ramification points will collide,\ndetermining a vanishing cycle.\n\nObtaining a consistent picture of the W-boson degrees of freedom throughout\nthe moduli space requires that the geometry provides a mechanism that turns the\nmassive W-bosons of the slightly deformed type $III$ theory to the massive W-bosons \nof the slightly deformed type $I_2$ theory. From the weak coupling limit, we know that\nthe latter are represented by fundamental open strings. From the deformation of the\ntype $III$ singularity performed in \\cite{Grassi:2014sda} we know that the former are \nthree-pronged string junctions. Thus, for $\\epsilon\\ne 0$ small, the continual\nincrease of $a$ from $0$ must turn string junctions into fundamental strings.\n\nFor the deformation of a type $III$ singularity the vanishing cycles were\nderived in \\cite{Grassi:2014sda}; they can be read off by taking\nstraight line paths from the origin. Beginning with the left-most brane\nand working clockwise about $z=0$, the ordered set of vanishing cycles are $Z_{III}=\\{\\pi_2,\\pi_1,\\pi_3\\}$\nwhere these cycles are defined as\n\\begin{equation} \n\\begin{tikzpicture}[scale=1]\n    \\fill[xshift=7cm,thick] (180:10mm) circle (1mm);\n    \\fill[xshift=7cm,thick] (180-120:10mm) circle (1mm);\n    \\fill[xshift=7cm,thick] (180+120:10mm) circle (1mm);\n    \\node at (9.2cm,1.3cm) {$x$};\n    \\draw[xshift=7cm,thick,->] (180:10mm)+(30:1.3mm) -- +(30:16mm);\n    \\draw[xshift=7cm,thick,->] (180-120:10mm)+(-90:1.3mm) -- +(-90:16mm);\n    \\draw[xshift=7cm,thick,->] (180+120:10mm)+(150:1.3mm) -- +(150:16mm);\n    \\node at (6.6cm,0.7cm) {$\\pi_1$};\n    \\node at (8cm,0cm) {$\\pi_2$};\n    \\node at (6.6cm,-0.7cm) {$\\pi_3$};\n    \\draw[xshift=9cm,thick,yshift=1.0cm] (90:0mm) -- (90:4mm);\n    \\draw[xshift=9cm,thick,yshift=1.0cm] (0:0mm) -- (0:4mm);\n \\end{tikzpicture}\n\\label{eqn:pixpiypiz}\n\\end{equation}\nand the massive W-bosons of the broken theory are three-pronged string\njunctions $J_\\pm=(\\pm 1,\\pm 1,\\pm 1)$ which, topologically, are two\nspheres in the elliptic surface due to having asymptotic  charge zero \\cite{Grassi:2014ffa}.\nPictorially, they appear as\n\\begin{equation}\n  \\begin{tikzpicture}\n  \\begin{scope}[rotate=-30]  \n  \\fill[xshift=-20mm] (90:8mm) circle (1mm);\n  \\fill[xshift=-20mm] (210:8mm) circle (1mm);\n  \\fill[xshift=-20mm] (330:8mm) circle (1mm);\n  \\draw[xshift=-20mm,thick] (0:0mm) -- (90:7mm);\n  \\draw[xshift=-20mm,thick] (90:3.5mm) -- (75:3mm);\n  \\draw[xshift=-20mm,thick] (90:3.5mm) -- (105:3mm);\n  \\draw[xshift=-20mm,thick] (0:0mm) -- (210:7mm);\n  \\draw[xshift=-20mm,thick] (210:3.5mm) -- (225:3mm);\n  \\draw[xshift=-20mm,thick] (210:3.5mm) -- (195:3mm);\n  \\draw[xshift=-20mm,thick] (0:0mm) -- (330:7mm); \n  \\draw[xshift=-20mm,thick] (330:3.5mm) -- (345:3mm);\n  \\draw[xshift=-20mm,thick] (330:3.5mm) -- (315:3mm);\n  \\end{scope}\n  \\begin{scope}[yshift=2cm,rotate=-30]\n  \\fill[xshift=20mm] (90:8mm) circle (1mm);\n  \\fill[xshift=20mm] (210:8mm) circle (1mm);\n  \\fill[xshift=20mm] (330:8mm) circle (1mm);\n  \\draw[xshift=20mm,thick] (90:0mm) -- (90:7mm);\n  \\draw[xshift=20mm,thick] (90:3.5mm) -- (78:4.1mm);\n  \\draw[xshift=20mm,thick] (90:3.5mm) -- (102:4.1mm);\n  \\draw[xshift=20mm,thick] (0:0mm) -- (210:7mm);\n  \\draw[xshift=20mm,thick] (210:3.5mm) -- (222:4.1mm);\n  \\draw[xshift=20mm,thick] (210:3.5mm) -- (198:4.1mm);\n  \\draw[xshift=20mm,thick] (0:0mm) -- (330:7mm); \n  \\draw[xshift=20mm,thick] (330:3.5mm) -- (342:4.1mm);\n  \\draw[xshift=20mm,thick] (330:3.5mm) -- (318:4.1mm);\n  \\end{scope}\n  \\end{tikzpicture}\n\\end{equation}\nwhere the black dots represent seven-branes. See \\cite{Grassi:2014sda} for more details of the intersection\ntheory of these particular junctions, as well as their reproduction of the\nDynkin diagram.\n\n\\vspace{.3cm} The first deformation should be thought of as a small\ndeformation of an $SU(2)$ seven-brane theory associated with a type $III$\nsingularity, that is, a Higgsing.  We would now like to study a deformation that\ncorresponds to a large deformation from a type $III$ $SU(2)$ theory to a \ntype $I_2$ $SU(2)$ theory (which does not Higgs the theory), and then a small\ndeformation that Higgses the $SU(2)$ theory on the seven-brane with an $I_2$\nsingular fiber. Heuristically, the third brane of the type $III$ singularity \nshould be far away relative to the distance between the two branes of the deformed\ntype $I_2$ singularity, which in type IIB language are two D7-branes with a small\nsplitting.\n\nWe take $a=.3i$ and $\\epsilon=.001$. The difference between\nthis deformation and the deformation of the previous paragraph is that\nin this case one of the three seven-branes that was originally at $z=0$ is\nmuch further away due to taking $a\\ne 0$, and the question is whether\nthe $(p,q)$ labels of the three-branes changed in the process. The geometry\nappears in Figure \\ref{fig:an0}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=.75]{figures\/zIIII2anonzero.eps} \\qquad \\qquad\n\\includegraphics[scale=.75]{figures\/xIIII2anonzero.eps}\n\\end{center}\n\\caption{Two figures of the geometry deformed by $\\epsilon=.001$, $a=.3i$, with\n$(p,q)$ seven-branes on the left and ramification points of the elliptic curve on the right.}\n\\label{fig:an0}\n\\end{figure}\nwhere the brane on the left is the one that has been moved further away\nfrom the origin by turning on $a$. If one were to maintain this value of $a$\nbut tune $\\epsilon\\mapsto 0$, the two branes close to the origin would collide\nto give an  $SU(2)$ theory with an $I_2$ singular fiber.\nIndeed, with this deformation the two-branes closer\nto the origin both have vanishing cycle $\\pi_1$ and the brane\ndisplaced by the $a$ deformation has vanishing cycle $\\pi_3$, so that\nnow the ordered set of vanishing cycles (beginning with the left-most brane and working\nclockwise about $z=0$) is $Z=\\{\\pi_3,\\pi_1,\\pi_1\\}$. The\nW-bosons of the broken $I_2$ $SU(2)$ theory are the strings between the\n$\\pi_1$ branes, as expected since $I_2$ is the F-theory lift of two\nD7-branes.\n\nWhat has happened? The natural W-boson of the broken type $III$ SU(2)\ntheory (associated to the $a=0, \\epsilon=.001$ deformation) is the\nthree-pronged string junction, but tuning $a$ from $a=0$ to $a=.3i$\nthe natural W-boson of the broken type $I_2$ theory is the fundamental\nstring. The change between the two different descriptions of the W-boson\nis a deformation induced Hanany-Witten move; for related ideas in a different\ngeometry, see \\cite{Cvetic:2010rq}. This can actually be seen via\ncontinuous deformation from $a=0$ to $a=.3i$, during which the leftmost (bottom) seven-brane\nin Figure \\ref{fig:a0} becomes the bottom (leftmost) seven-brane of Figure \\ref{fig:an0}.\nIn the process the straight line path from $z=0$ to the (moving) leftmost seven-brane in Figure \\ref{fig:a0}\nis crossed via the movement of the bottom seven-brane in Figure \\ref{fig:a0}, changing the $(p,q)$ labels\nof the former. This matches the changes in the ordered sets of vanishing cycles.\n\n\n\nThough we have explicitly seen the natural state change using a\ndeformation, let us check the possibilities using the usual algebraic\ndescription of Zwiebach and Dewolfe \\cite{DeWolfeZwiebach} where the\n$(p,q)$ seven-branes are arranged in a line with branch cuts pointing\ndownward. Technically, the branch cuts represent the mentioned\nstraight line paths from the origin, the latter being the point whose\nassociated fiber $E_0$ appears in the relative homology $H_2(X_d,E_0)$\nthat defines the string junctions. $X_d$ is the elliptic surface \nof the disc. The\nW-boson of the broken $SU(2)$ theory associated to a type $III$\nsingularity is\n\\begin{center}\n\\begin{tikzpicture}\n  \\draw[thick,dotted] (0cm,-.25cm) -- (0cm,-1.75cm);\n  \\draw (0cm,0cm) circle (.25cm);\n  \\node at (0cm,0cm) {$\\pi_3$};\n  \\draw[thick,dotted] (1cm,-.25cm) -- (1cm,-1.75cm);\n  \\draw (1cm,0cm) circle (.25cm);\n  \\node at (1cm,0cm) {$\\pi_2$};\n  \\draw[thick,dotted] (-1cm,-.25cm) -- (-1cm,-1.75cm);\n  \\draw (-1cm,0cm) circle (.25cm);\n  \\node at (-1cm,0cm) {$\\pi_1$};\n  \\draw[thick] (-1cm,.25cm) arc (135:45:1.4cm);\n  \\draw[thick] (0cm,.25cm) -- (0cm,.65cm);\n  \\draw[thick] (-.8cm,.5cm) -- (-.65cm,.5cm);\n  \\draw[thick] (-.66cm,.36cm) -- (-.66cm,.5cm);\n  \\draw[thick] (.8cm,.5cm) -- (.65cm,.5cm);\n  \\draw[thick] (.66cm,.36cm) -- (.66cm,.5cm);\n  \\draw[thick] (-.1cm,.4cm) -- (0cm,.5cm);\n  \\draw[thick] (.1cm,.4cm) -- (0cm,.5cm);\n\\end{tikzpicture}\n\\end{center}\nwhere for concreteness we choose a basis such that $\\pi_1 = (1,0)$, $\\pi_2=(0,1)$ and\n$\\pi_3=(-1,-1)$ with the convention that we cross branch cuts by moving to the right. Then\nthe monodromy matrix associated to a $(p,q)$ seven-brane is \n\\begin{equation}\n  \\begin{pmatrix}\n    1-pq && p^2 \\\\\n    -q^2 && 1+pq\n  \\end{pmatrix}\n  \\label{eq:Mpq}\n\\end{equation}\nand one can check that the monodromy of these three branes reproduces the monodromy of the\ntype $III$ singularity, as they must after deformation. We see\n\\begin{equation}\nM_{\\pi_2}M_{\\pi_3}M_{\\pi_1}=M_{III}=\n\\begin{pmatrix}\n  0 && 1 \\\\ -1 && 0\n\\end{pmatrix},\n\\end{equation}\nwhich is the expected behavior.\n\n\nIn this description, the continuous motion of the seven-branes\ndescribed above amounts to the $\\pi_3$ brane crossing the branch cut\nof the $\\pi_2$ brane, changing its vanishing cycle to\n$M_{\\pi_3}^{-1}\\begin{pmatrix}0\\\\1\\end{pmatrix}=\\begin{pmatrix}-1\\\\0\\end{pmatrix}$\nseven-brane; but this is just $-\\pi_1$ and we are free to call it a\n$\\pi_1$ seven-brane instead if we reverse the sign of any junction\ncoming out of the brane. With this movement and relabeling, the above\njunction becomes\n\\begin{center}\n\\begin{tikzpicture}\n  \\draw[thick,dotted] (0cm,-.25cm) -- (0cm,-1.75cm);\n  \\draw (0cm,0cm) circle (.25cm);\n  \\node at (0cm,0cm) {$\\pi_1$};\n  \\draw[thick,dotted] (1cm,-.25cm) -- (1cm,-1.75cm);\n  \\draw (1cm,0cm) circle (.25cm);\n  \\node at (1cm,0cm) {$\\pi_3$};\n  \\draw[thick,dotted] (-1cm,-.25cm) -- (-1cm,-1.75cm);\n  \\draw (-1cm,0cm) circle (.25cm);\n  \\node at (-1cm,0cm) {$\\pi_1$};\n  \\draw[thick] (1cm,.25cm) -- (1cm,.76cm);\n  \\draw[thick] (.9cm,.4cm) -- (1cm,.5cm);\n  \\draw[thick] (1.1cm,.4cm) -- (1cm,.5cm);\n  \\draw[thick] (-.52cm,.61cm) -- (-.37cm,.59cm);\n  \\draw[thick] (-.39cm,.44cm) -- (-.37cm,.59cm);\n  \\draw[thick] (.65cm,-.5cm) -- (.56cm,-.58cm);\n  \\draw[thick] (.63cm,-.7cm) -- (.56cm,-.58cm);\n  \\draw[thick] (-1cm,.25cm) .. controls (0cm,1cm) and (2cm,1cm) .. (2cm,0cm) .. controls (2cm,-1cm) and (.2cm,-.75cm) .. (.15cm,-.2cm);\n\\end{tikzpicture}\n\\end{center}\nand now the branes are in a position to do the relevant Hanany-Witten move. Prior to crossing the branch\ncut from the right, the $(p,q)$ charge of the piece of string that ends on the middle brane (in this picture) is $\\pi_1+\\pi_3$, and after the branch cut it is $\\pi_1$ due\nto the monodromy action $M_{\\pi_3}^{-1}$. If the monodromy action is replaced by a prong by pulling the string\nthrough the brane (that is, performing a Hanany-Witten move), how many extra prongs $k$ are picked up on the right-most brane?\nCharge conservation requires \n\\begin{equation} \\pi_1 + \\pi_3 + k \\,\\pi_3 = \\pi_1 \\end{equation} \nand we see $k=-1$. That is, the Hanany-Witten move replaces the\nportion of the string crossing the branch cut with a prong going into the $\\pi_3$ brane. This cancels against the prong that is already there, leaving\n\\vspace{.3cm}\n\\begin{center}\n\\begin{tikzpicture}\n  \\draw[thick,dotted] (0cm,-.25cm) -- (0cm,-1.75cm);\n  \\draw (0cm,0cm) circle (.25cm);\n  \\node at (0cm,0cm) {$\\pi_1$};\n  \\draw[thick,dotted] (1cm,-.25cm) -- (1cm,-1.75cm);\n  \\draw (1cm,0cm) circle (.25cm);\n  \\node at (1cm,0cm) {$\\pi_3$};\n  \\draw[thick,dotted] (-1cm,-.25cm) -- (-1cm,-1.75cm);\n  \\draw (-1cm,0cm) circle (.25cm);\n  \\node at (-1cm,0cm) {$\\pi_1$};\n  \\draw[thick] (-.75cm,0cm) -- (-.25cm,0cm);\n  \\draw[thick] (-.55cm,.1cm) -- (-.45cm,0cm);\n  \\draw[thick] (-.55cm,-.1cm) -- (-.45cm,0cm);\n\\end{tikzpicture}\n\\end{center}\nwhich, since we have chosen $\\pi_1=(1,0)$, is just a fundamental string.\nIf we had chosen $\\pi_1$ to be some other $(p,q)$ value this would be a $(p,q)$\nstring, but there is always a choice of $SL(2,\\mathbb{Z})$ frame what would turn\nit back into a fundamental string.\n\nIn summary, we see using algebraic techniques that the natural W-boson\nof the Higgsed type $III$ $SU(2)$ theory, which is a three-pronged\nstring junction, is related to the the natural W-boson of the Higgsed\ntype $I_2$ $SU(2)$ theory. The relationship is a brane rearrangement\ntogether with a Hanany-Witten move, which we have seen explicitly\nabove via two deformations of the elliptic surface with a type $III$\nsingular fiber.\n\n\\vspace{.3cm}\n\nThe natural W-bosons of the type $I_3$\ntheory are (in an appropriate $SL(2,\\mathbb{Z})$ duality frame) fundamental\nstrings and the natural W-bosons of the type $IV$ theory include\nstring junctions (see e.g. \\cite{Grassi:2014sda} for an explicit analysis).  Via deformation they must be related to one\nanother, and given the result we have just seen it is natural to expect a brane rearrangement\nand a Hanany-Witten move. This expectation is correct, though we will not\nexplicitly show it since the techniques are so similar to the previous case.\n\n\nWe would like to present the deformation for the interested reader, though. Take \n\\begin{equation}\n  f=-3 a^4+a b\\, z+c\\, z^2+f_3\\, z^3\\qquad \\qquad\n  g=2 a^6-a^3 b\\, z+ \\left(\\frac{b^2}{12}-a^2 c\\right)\\, z^2+g_3 z^3\n\\end{equation}\nwhere $a$, $b$, and $c$ are global section $a\\in \\mathcal{O}(-K)$, $b\\in\n\\mathcal{O}(-3K-Z)$ and $c \\in \\mathcal{O}(-4K-Z)$ with $Z$ is the class of $z=0$.\nHere $f_3$ and $g_3$ contain constant terms in $z$ as well as higher\norder terms.  The discriminant is\n\\begin{align}\n  \\Delta &= \\frac{1}{16} z^3 \\left(1728 a^8 f_3+1728 a^6 g_3-288 a^5 b c-8 a^3 b^3\\right)\\nonumber \\\\ & +\\frac{1}{16} z^4 \\left(-1152 a^5\n   b f_3-144 a^4 c^2-864 a^3 b g_3+120 a^2 b^2 c+3 b^4\\right)+O(z^5)\n\\end{align}\nand we see that there is a gauge theory with $I_3$ fiber on the seven-brane at $z=0$. In the $a=0$ limit we see that\n\\begin{align}\nf = z^2 \\left(c+f_3 z\\right) \\qquad \\qquad g=\\frac{1}{12} z^2 \\left(b^2+12 g_3 z\\right) \\nonumber \\\\\n\\Delta = \\frac{1}{16} z^4 \\left(3 b^4+72 b^2 g_3 z+64 c^3 z^2+192 c^2 f_3 z^3+192 c\n   f_3^2 z^4+64 f_3^3 z^5+432 g_3^2 z^2\\right)\n\\end{align}\nwhich has a gauge theory with type $IV$ Kodaira fiber on the seven-brane at $z=0$. In these cases we have not\nimposed the absence of outer monodromy (i.e. we have no imposed a split fiber), so they are $SU(2)$ gauge theories. If the form is further restricted so\nthat outer monodromy is imposed, it is an $SU(3)$ gauge theory.\n\n\\newsec{Extra Branes at Seven-Brane Intersections}\n\\label{sec:extra branes}\n\nTwo seven-branes that intersect along a codimension two locus $C\\subset B$ are typically only seven-branes that\nintersect along $C$. However, a counterexample giving rise to $SU(3)\\times SU(2)$ gauge symmetry \nwas studied in \\rcite{GrassiHalversonShanesonTaylor2014} where an additional brane with an $I_1$ singular fiber also intersected\nthe curve of $SU(3)\\times SU(2)$ intersection. Though one might have expected a\n$IV$-$III$ collision, a $IV$-$III$-$I_1$ collision occurs automatically. The Weierstrass model is\n $f= z^1 t^2\\, F$ and $g=z^2 t^2\\, G$ with\n\\eqn{\\Delta = z^3 t^4\\,(4t^2 F^3 + 27 z G^2 )=: z^3t^4\\tilde \\Delta,}\nfrom which it can be seen that the brane along $\\tilde \\Delta = 0$ intersects $\\{z=t=0\\}=:C$. Three\nstacks of branes intersect $C$, and we call $\\tilde \\Delta=0$ the ``extra brane'' at $C$.\n\nThere is a natural guess regarding the associated physics:\nwhere there is an extra brane there should be extra string (or string junction) states.\nIn the mentioned example it was found that such a collision gives\nrise not only to the expected bifundamentals of $SU(3)\\times SU(2)$, but also \\cite{GrassiHalversonShanesonTaylor2014}\nfundamental hypermultiplets of $SU(3)$ and $SU(2)$ which could come in\nchiral multiplets if flux is turned on. Such configurations may be non-Higgsable;\nfor example geometries, see \\cite{GrassiHalversonShanesonTaylor2014,Halverson:2015jua,Taylor:2015ppa}.\n\n\\vspace{.3cm}\n\nUnder which circumstances is there necessarily an extra brane?  We take $F_1$ and $F_2$ to be the\nKodaira singular fibers of seven-branes that collide along $C$, with \n\\begin{equation}\n  F_{1,2}\\in\\{II,III,IV,I_0^*,IV^*,III^*,II^*\\}.\\end{equation}\nThese are the possible fibers of non-Higgsable seven-branes.\nLet\n$J_1$ and $J_2$ be the $J$-invariants associated with $F_1$ and $F_2$. By direct calculation we find:\n\\begin{itemize}\n\t\\item If either $F_1$ or $F_2$ is $I_0^*$ there is no additional brane.\n\t\\item If neither $F_1$ nor $F_2$ are $I_0^*$ then there is an additional brane if and only if $J_1\\ne J_2$.\n\t\\item If there is an additional brane then precisely one of the fibers is type $III^*$ or $III$.\n\t\\item If there is an additional brane and the Weierstrass\n          model does not have $(f,g)$ vanishing to order $\\geq (4,6)$\n          along $C$, then the fiber types are either $(IV,III)$ or $(III,II)$.\n\\end{itemize}\nTo see this, let the seven-brane with fibers $F_1$\nand $F_2$ be localized on the divisors $z=0$ and $t=0$ respectively. Then write \n\\begin{equation}\nf = z^a t^b \\, F \\qquad g = z^c t^d\\, G\n\\end{equation}\nwhere $a,b,c,d$ can be determined from Table \\ref{tab:fibs} given the knowledge of $F_1$\nand $F_2$. The discriminant takes the form\n\\begin{equation}\n\\Delta = z^{min(3a,2c)}\\, t^{min(3b,2d)} \\,\\, \\tilde \\Delta\n\\end{equation}\nand there is an extra brane along $C$ whenever $\\tilde\\Delta|_{z=t=0}=0$.\nIn the cases where there are no $(4,6)$ curves, the above conclusions can all be seen directly from Table\n\\ref{table:minimal models with extra brane}, where ``$NF$'' denotes that the $J$ invariant\nof $I_0^*$ is not fixed.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccccccc}\n  $F_1$ & $F_2$ & $J_1$ & $J_2$ & Minimal on $C$& $\\Delta$ & Additional Brane? \\\\ \\hline \\hline\n$IV^*$ & $II$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t\\, z + 27 \\tilde g^2)\\, t^2\\, z^8$ & No \\\\\n$I_0^*$ & $IV$ & $NF$ & $0$ & Yes & $(4 \\tilde f^3\\, t^2 + 27 \\tilde g^2)\\, t^4\\, z^6$ & No \\\\\n$I_0^*$ & $III$ & $NF$ & $1$ & Yes & $(4 \\tilde f^3 +  27\\tilde g^2 \\, t ) \\, t^3\\, z^6$ & No \\\\\n$I_0^*$ & $II$ & $NF$ & $0$ & Yes & $(4 \\tilde f^3\\, t + 27 \\tilde g^2)\\, t^2\\, z^6$ & No \\\\\n$IV$ & $IV$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t^2\\, z^2 + 27 \\tilde g^2)\\, t^4\\, z^4$ & No \\\\\n$IV$ & $III$ & $0$ & $1$ & Yes & $(4 \\tilde f^3\\, z^2 + 27 \\tilde g^2\\, t)\\, t^3\\, z^4$ & Yes \\\\\n$IV$ & $II$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t\\, z^2 + 27 \\tilde g^2)\\, t^2\\, z^4$ & No \\\\\n$III$ & $III$ & $1$ & $1$ & Yes & $(4\\tilde f^3 + 27\\tilde g^2 \\, t\\, z)\\, t^3\\, z^3$ & No \\\\\n$III$ & $II$ & $1$ & $0$ & Yes & $(4 \\tilde f^3\\, t + 27 \\tilde g^2\\, z)\\, t^2\\, z^3$ & Yes \\\\\n$II$ & $II$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t\\, z + 27 \\tilde g^2)\\, t^2\\, z^2$ & No \\\\ \\hline \\hline\n\\end{tabular}\n\\caption{The possible intersecting elliptic seven-branes that may arise, according to their singular fibers $F_1$ and $F_2$, and some of their\nproperties.}\n\\label{table:minimal models with extra brane}\n\\end{table}\n\n\n\n\\subsection{Associated Matter Representations}\n\n\nThere are two possible fiber intersections that force the existence\nof an extra brane, $IV$-$III$ and $III$-$II$. Due to the effects of outer\nmonodromy, the $IV$-$III$ collision allows for the possibility of\nintersecting non-abelian seven-branes with $SU(3)\\times SU(2)$ or\n$SU(2) \\times SU(2)$ gauge symmetry, while intersecting seven-branes\nwith a fiber collision $IV$-$III$ necessarily have $SU(2)$ gauge\nsymmetry. The Lie algebra representations of matter at the $IV$-$III$-$I_1$\n$SU(3)\\times SU(2)$ collision was determined in \\rcite{GrassiHalversonShanesonTaylor2014}. They are\nhypermultiplets of \n\\begin{equation}\n(3,2), (3,1), (1,2)\n\\end{equation} \nin the absence of flux, but can become the chiral non-abelian $SU(3)\\times SU(2)$\nrepresentations of the standard model if chirality inducing G-flux is turned on.\n\n \nLet us determine the Lie algebra representations in the\ncase of the other collision, which is $III$-$II$-$I_1$ via\nanomaly cancellation in six dimensions. \nConsider a six-dimensional F-theory compactification with $B_2=\\mathbb{P}^2$\nand a seven-brane with $SU(2)$ gauge symmetry from a type $III$ fiber on\n$Z=\\{z=0\\}$, a divisor in the hyperplane class. Take also a seven-brane with\nno gauge symmetry and a type $II$ singular fiber on $T=\\{t=0\\}$, also\nin the hyperplane class. Such a Weierstrass model takes the form\n\\eqn{f_{12} = z \\, t\\, f_{10} \\qquad \\qquad g_{18} = z^2 \\, t \\,g_{15}\n  \\qquad \\qquad \\Delta = z^3 t^2\\, (4f_{10}^3\\, t+27g_{15}^2\\,\n  z)\\equiv z^3 \\, \\tilde \\Delta} and we would like to study matter at\nthe intersection $z = \\tilde \\Delta = 0$. These intersections are of\ntwo types: the single point $z=t=0$ and the $10$ points $z=f_{10}=0$\nwith $t\\ne 0$. The latter are all points where seven-branes with a\ntype $III$ and $I_1$ fiber collide; there are $2$ fundamentals of\n$SU(2)$ for each such point \\cite{Grassi:2000we,Grassi:2011hq}. Thus, the $10$ points contribute $20$\nfundamental hypermultiplets. Since anomaly cancellation for an $SU(2)$\nseven-brane on the hyperplane in $\\mathbb{P}^2$ requires $22$ hypermultiplets\n(see e.g. section 2.5 of \\cite{Johnson:2014xpa}), anomaly cancellation\nrequires that the $III$-$II$-$I_1$ intersection also contribute two\nfundamental hypermultiplets.\n\n\n\\newsec{Argyres-Douglas Theories on D3-branes and BPS Dyons as Junctions}\n\\label{sec:D3 probes}\n\nD3-branes in the vicinity of seven-branes realize non-trivial $\\mathcal{N}=2$\nquantum field theories on their worldvolume \\cite{Banks:1996nj}, which\nare broken to $\\mathcal{N}=1$ theories by the background. The position of the\nD3-brane relative to a seven-brane determines a point on the Coulomb\nbranch of the D3-brane theory, and the seven-brane determines a singular\npoint on the Coulomb branch at which additional particle states become\nmassless. This point is reached when the D3-brane collides with the\nseven-brane, and the additional light states are string junctions that\nstretch from the seven-brane to the D3-brane. \nPrevious works \\cite{Mikhailov:1998bx,DeWolfe:1998bi}\ndetermined the string junction representations of BPS particles\nfor well-known $\\mathcal{N}=2$ theories, including $N_f=0,1,2,3,4$\nSeiberg-Witten theory \\cite{Seiberg:1994rs,Seiberg:1994aj}.\n\nIn this section we will do the same for D3-branes probing the type\n$II$, $III$, and $IV$ singularities. Recall that the latter two are\nassociated with the non-perturbative $SU(2)$ and $SU(3)$ theories on\nseven-branes that were discussed in section \\ref{sec:su3su2}. Relative\nto weakly coupled realizations of $SU(2)$ and $SU(3)$ from\n$2$ $D7$-branes and 3 $D7$-branes, which have an $I_2$ and $I_3$ fiber in\nF-theory, the type $III$ and $IV$ theories have an extra seven-brane. \nVia studying local axiodilaton profiles we have seen that in the vicinity of the brane\nthe string coupling is $O(1)$ and that these seven-branes source a nilpotent\n$SL(2,\\mathbb{Z})$ monodromy. Therefore, the worldvolume theories of nearby D3-branes necessarily\ndiffer from the $N_f=2,3$ Seiberg-Witten theories realized on D3-branes near $I_2$ and $I_3$ seven-branes.\n\nThe worldvolume theory on a D3-brane when it has collided with a type $III$ or type $IV$ seven-brane is the\n$N_f=2$ or $3$ Seiberg-Witten theory at its Argyres-Douglas point, respectively. \nWe will see that the characteristic \\cite{Argyres:1995jj} Argyres-Douglas (AD) phenomenon is\nrealized by string junctions. This phenomenon is\nexistence of points in the moduli space where electrons and dyons\ncharged under the $U(1)$ of the $\\mathcal{N}=2$ theory simultaneously become massless. In\nF-theory this occurs when the D3-brane collides with a type $III$ or\ntype $IV$ seven-brane. The AD phenomenon was originally realized in\n$G=SU(3)$ theories, which have genus two Seiberg-Witten curves, and thus one\nmight expect it to not exist in an elliptically fibered setup such as F-theory.\nFor specially tuned $G=SU(2)$ Seiberg-Witten theories Argyres-Douglas\npoints do exist \\cite{Argyres:1995xn}, though; the tuning brings in an extra singularity\nthat turns a type $I_2$ ($I_3$) fiber into a type $III$ ($IV$) fiber. Somewhat ironically,\nF-theory models with non-Higgsable seven-branes of type $III$ and $IV$ do not require any\nsuch tuning; the topology of the base $B$ forces the local structure of the Seiberg-Witten\ncurve that would appear tuned from an $\\mathcal{N}=2$ point of view. \n\n\nWe will use the BPS junction criterion\\footnote{See\n  \\cite{Mikhailov:1998bx} for another study of BPS junctions.} of\n\\cite{DeWolfe:1998bi} to determine the BPS states on three-branes near\nthe type $II$, $III$, and $IV$ seven-branes. This constraint is\n\\begin{equation}\n\\label{eq:BPS}\n(J,J) \\geq -2 + gcd(a(J))\n\\end{equation}\nwhere $J$ is the string junction and $a(J)$ is its asymptotic charge,\nand it is derived from the requirement that in the M-theory picture\n$J$ is a holomorphic curve with boundary, with the boundary a\nnon-trivial one-cycle $a(J)$ in the smooth elliptic curve above the\nD3-brane. This one-cycle $a(J)$ is the asymptotic charge, and it\ndetermines the electric and magnetic charge of the associated junction\nunder the $U(1)$ of the D3-brane theory. Junctions are relative\nhomology cycles \\cite{Grassi:2014ffa}, i.e. two-cycles that may have\nboundary in a smooth elliptic curve above a particular point $p$. This\npoint is given physical meaning if $p$ is the location of a D3-brane.\n\nThroughout this section we will need a basis choice in order to represent the one-cycle $a(J)$ as a vector in\n$\\mathbb{Z}^2$. We choose\n\\begin{equation}\n\\pi_1 = \\begin{pmatrix}1\\\\ 0 \\end{pmatrix}, \\qquad \\pi_3 = \\begin{pmatrix}0 \\\\ 1\\end{pmatrix},\n\\end{equation}\nwhich determines $\\pi_2$ via $\\pi_1+\\pi_2+\\pi_3=0\\in H_1(E_p,\\mathbb{Z})$, where $E_p$ is the elliptic fiber above\nthe D3-brane in the F-theory picture, or alternatively the torus that defines the electric and magnetic\ncharges.\n\n\\subsection{The $N_f=1$ Argyres-Douglas Point and the Type $II$ Kodaira Fiber}\n\nLet us study the F-theory realization of BPS states on a D3-brane near a slightly deformed type $II$ Kodaira fiber. From \\cite{Grassi:2014sda},\nthe vanishing cycles are $Z=\\{\\pi_3,\\pi_1\\}$ and so the $I$-matrix that determines topological intersections is\n\\begin{equation}\nI = (\\cdot,\\cdot) = \\begin{pmatrix} -1 & -\\frac12 \\\\ -\\frac12 & -1\\end{pmatrix}.\n\\end{equation}\nDefining $J=(Q_1,Q_2)$ we have\n\\begin{equation}\n(J,J) = -Q_1Q_2-Q_1^2-Q_2^2 \\qquad \\text{with}\\qquad a(J) = \\begin{pmatrix} Q_2 \\\\ Q_1 \\end{pmatrix}.\n\\end{equation}\nThe string junctions satisfying the BPS particle condition \\eqref{eq:BPS} are\n\\begin{center}\n\\begin{tabular}{c|c}\n$a(J)$ & Junctions \\\\ \\hline\n$(1,0)$ & $(0,1)$ \\\\\n$(1,-1)$ & $(-1,1)$ \\\\\n$(0,1)$ & $(1,0)$\n\\end{tabular}\n\\end{center}\nThese BPS particles arising from string junctions also have BPS\nanti-particles via the action $J\\mapsto -J$, which preserves $(J,J)$\nand $gcd(a(J))$ and therefore the associated junctions $-J$ satisfy\n\\eqref{eq:BPS}. In the M-theory description the geometric object $J$\ncorresponds to a two-cycle, and the associated BPS particle arises\nfrom wrapping an M2-brane on that cycle; the anti-particle associated\nto $-J$ arises from wrapping an anti M2-brane.  Note that there is no\njunction $J$ with $a(J)=0$ and $(J,J)=-2$; as explained in\n\\cite{Grassi:2014sda}, this demonstrates via deformation that the type\nII singularity does not carry a gauge algebra.\n\nOne might be tempted to think that the seven-brane associated to a type II singularity has little impact on the \nlow energy physics of an F-theory compactification, since it does not carry any gauge algebra. However, this\nis not true, and we would like to emphasize:\n\\begin{itemize}\n\\item Locally near the type II seven-brane (or precisely in the 8d theory) the worldvolume theory on the D3-brane is\n  $N_f=1$ Seiberg-Witten theory near its Argyres-Douglas point. At that point BPS electrons, monopoles, and dyons \n  become massless.\n\\item Since $J=0$ for the type $II$ Kodaira fiber, $g_s$ is $O(1)$ in the vicinity of the seven-brane,\n  with $\\tau$ profile solved via Ramanujan's alternative bases in section \\ref{sec:axiodilaton}..\n\\item Even though both can split into two mutually non-local seven-branes, the seven-brane associated to a \ntype II Kodaira fiber is not an orientifold. First, because their $SL(2,\\mathbb{Z})$ monodromies are different, and second because the orientifold famously \\emph{must} split due\nto instanton effects in F-theory, which is not true of the type II seven-brane.\n\\end{itemize}\n\n\n\\subsection{The $N_f=2$ Argyres-Douglas Point and the Type $III$ Kodaira Fiber}\n\nIn this section we study the F-theory realization of BPS states on D3-branes near a slightly deformed type $III$ Kodaira fiber.\nIn the coincident limit this theory is the Argyres-Douglas theory obtained by tuning $N_f=2$ $G=SU(2)$\nSeiberg-Witten theory to its Argyres-Douglas point. \nUtilizing an explicit deformation of \\cite{Grassi:2014sda}, the vanishing cycles are $Z=\\{\\pi_2,\\pi_1,\\pi_3\\}$ and the \n$I$-matrix that determines topological intersections is \n\\begin{equation}\nI = (\\cdot,\\cdot) = \\begin{pmatrix} -1 & \\frac12 & -\\frac12 \\\\ \\frac12 & -1 & \\frac12 \\\\ -\\frac12 & \\frac12 & -1  \\end{pmatrix}.\n\\end{equation}\nDefining a string junction by $J=(Q_1,Q_2,Q_3)$ we have \n\\begin{equation}\n(J,J) = Q_1Q_2+Q_1Q_3-Q_2Q_3 - \\sum_i Q_i^2 \\qquad \\text{with}\\qquad a(J) = \\begin{pmatrix}-Q_1+Q_2 \\\\ -Q_1+Q_3 \\end{pmatrix}.\n\\end{equation}\nThees\nBPS particles arising from string junctions also have BPS anti-particles via the action $J\\mapsto -J$, which\npreserves $(J,J)$ and $gcd(a(J))$, leaving (\\ref{eq:BPS}) invariant.\n\nUsing the constraint \\eqref{eq:BPS} the possible BPS string junctions\ncan be computed directly. The states of self-intersection $-2$ have\n$a(J)=0$ and were identified in \\cite{Grassi:2014sda}. They are $J_+ = (1,1,1)$ and $J_-=(-1,-1,-1)$,\nwhere the $\\pm$ denote a choice of positive and negative root for the associated\n$SU(2)$ algebra that is the gauge algebra on the seven-brane and the flavor\nalgebra of the three-brane theory. The rest of the junctions satisfying \\eqref{eq:BPS} have\n$(J,J)=-1$ and are given by\n\\begin{center}\n\\begin{tabular}{c|c}\n$a(J)$ & Junctions \\\\ \\hline\n$(0,1)$ & $(0,0,1)$,$(-1,-1,0)$ \\\\\n$(1,0)$ & $(0,1,0)$,$(-1,0,-1)$ \\\\\n$(1,1)$ & $(-1,0,0)$ \\\\\n$(1,-1)$ & $(0,1,-1)$\n\\end{tabular}\n\\end{center}\nwhere the sets of junctions in the first two lines are doublets of\n$SU(2)$ since they differ by $J_\\pm$.  This spectrum matches the known\nBPS states in the maximal chamber of the deformed theory with masses\nturned on; see e.g. \\cite{Maruyoshi:2013fwa}.\n\nWe have seen the relationship between the ordered sets of\nvanishing cycles $Z=\\{\\pi_2,\\pi_1,\\pi_3\\}$ of \\cite{Grassi:2014sda}\nand\\footnote{The set of vanishing cycles $\\{A,A,C\\}$ should be compared to the set $\\{\\pi_1,\\pi_1,\\pi_3\\}$ of section \\ref{sec:su3su2}.} $Z=\\{A,A,C\\}$ of \\cite{DeWolfeZwiebach,DeWolfe:1998bi} that can be associated to the\ndeformed type III Kodaira fiber by explicit\ndeformation in section \\ref{sec:su3su2}. Let us study the BPS states with the\nlatter set for the sake of completeness, taking\n\\begin{equation}\nA=\\begin{pmatrix}1 \\\\ 0\\end{pmatrix}, \\qquad C = \\begin{pmatrix}1 \\\\ 1\\end{pmatrix},\n\\end{equation}\nas in \\cite{DeWolfe:1998zf}.\n The $I$-matrix is\n\\begin{equation}\nI = (\\cdot,\\cdot) = \\begin{pmatrix} -1 & 0 & \\frac12 \\\\ 0 & -1 & \\frac12 \\\\ \\frac12 & \\frac12 & -1  \\end{pmatrix}\n\\end{equation}\nand taking $J=(Q_1,Q_2,Q_3)$ we compute\n\\begin{equation}\n(J,J) = Q_3(Q_1+Q_2) - \\sum_i Q_i^2 \\qquad \\text{with}\\qquad a(J) = \\begin{pmatrix}Q_1+Q_2+Q_3\\\\ Q_3 \\end{pmatrix}.\n\\end{equation}\nThere are junctions $J_\\pm$ satisfying the BPS constraint \\eqref{eq:BPS} with $a(J)=0$ and $(J,J)=-2$. They are\n$J_+=(1,-1,0)$ and $J_-=(-1,1,0)$, and they are the $W_\\pm$ bosons of the broken $SU(2)$ theory. The rest of the\njunctions satisfying the BPS condition have $(J,J)=-1$ and satisfy\n\\begin{center}\n\\begin{tabular}{c|c}\n$a(J)$ & Junctions \\\\ \\hline\n$(3,1)$ & $(1,1,1)$ \\\\\n$(2,1)$ & $(1,0,1)$,$(0,1,1)$ \\\\\n$(1,1)$ & $(1,0,1)$ \\\\\n$(1,0)$ & $(1,0,0)$,$(0,1,0)$.\n\\end{tabular}\n\\end{center}\nWe again see two $SU(2)$ doublets and two $SU(2)$ singlets before taking into account the $-J$ junctions.\nThese junctions with electromagnetic charge \nmust be related to those of $Z=\\{\\pi_2,\\pi_1,\\pi_3\\}$ by a Hanany-Witten move, as we explicitly showed for the simple roots\n$J_\\pm$ in section \\ref{sec:su3su2}.\n\n\n\\subsection{The $N_f=3$ Argyres-Douglas Point and the Type $IV$ Kodaira Fiber}\n\nIn this section we study the F-theory realization of BPS states on D3-branes near a slightly deformed type\nIV Kodaira fiber. In the coincident limit the worldvolume theory on the D3-brane is the\nArgyres-Douglas theory obtained by tuning $N_f=3$ $G=SU(2)$ Seiberg-Witten theory to\nits Argyres-Douglas point.\n\nThe ordered set of vanishing cycles derived in \\cite{Grassi:2014sda} is $Z_{IV}=\\{\\pi_1,\\pi_3,\\pi_1,\\pi_3\\}$\nand\nthe associated I-matrix that determines the topological intersections of junctions is\n\\begin{equation}\nI = (\\cdot,\\cdot) = \\begin{pmatrix} -1 & \\frac12 & 0 & \\frac12 \\\\ \\frac12 & -1 & -\\frac12 & 0 \\\\ 0 & -\\frac12 & -1 & \\frac12 \n\\\\ \\frac12 & 0 & \\frac12 & -1 \\end{pmatrix}.\n\\end{equation}\nWriting the junction as a vector $J=(Q_1,Q_2,Q_3,Q_4)$ we have \n\\begin{equation}\n(J,J) = Q_1Q_2 + Q_1Q_4-Q_2Q_3+Q_3Q_4 - \\sum_i Q_i^2 \\qquad \\text{with}\\qquad a(J) = \\begin{pmatrix}Q_1+Q_3 \\\\ Q_2+Q_4 \\end{pmatrix}.\n\\end{equation}\nThese\nBPS particles arising from string junctions also have BPS anti-particles via the action $J\\mapsto -J$, which\npreserves $(J,J)$ and $a(J)$ and therefore the associated junctions $-J$ satisfy \\eqref{eq:BPS}. The latter arise from\nwrapped anti M2-branes in the M-theory picture.\n\nLet us derive the possible BPS string junctions using the constraint \\eqref{eq:BPS}.\nThe junctions of self-intersection $-2$ have\n$a(J)=0$ and were identified in \\cite{Grassi:2014sda}. They are the roots of an $SU(3)$ algebra,\nand we take $J_1=(1,0,-1,0)$\nand $J_2=(0,1,0,-1)$ as the simple roots; then the full set of root junctions is simply $J_1$,$J_2$,\n$J_1+J_2$ and their negatives. Solving the condition for BPS junctions \\eqref{eq:BPS} we find that the\npossible BPS junctions are\n\\begin{center}\n\\begin{tabular}{c|c}\n$a(J)$ & Junctions \\\\ \\hline\n$(-1,-1)$ & $(0,0,-1,-1)$  $(-1,0,0,-1)$  $(-1,-1,0,0)$ \\\\\n$(1,0)$ & $(1,0,0,0)$  $(0,0,1,0)$  $(0,-1,1,1)$ \\\\\n$(0,1)$ & $(1,1,-1,0)$ $(0,1,0,0)$ $(0,0,0,1)$ \\\\\n$(2,1)$ & $(1,0,1,1)$ \\\\\n$(-1,-2)$ & $(-1,-1,0,-1)$ \\\\\n$(-1,1)$ & $(0,1,-1,0)$   \n\\end{tabular}\n\\end{center}\nand we have chosen the ordering of the junctions in the first three\ncolumns to show that they are fundamentals of $SU(3)$. Namely, in each\nof the first thee columns the second junction subtracted from the\nfirst is $J_1$ and the third junction from the second is $J_2$. The\nnegatives of these are anti-fundamental, completing the non-trivial\nflavor hypermultiplets.\n\n\n\\acknowledgments \\vspace{.25cm} I would like to thank Philip Argyres,\nShaun Cooper, Andreas Malmendier, Brent Nelson, Daniel Schultz,\nWashington Taylor, Yi-Nan Wang and Wenbin Yan for helpful discussions\nand correspondence. I am particularly grateful to Antonella Grassi and\nJulius Shaneson for discussions and comments on a draft, and to\nJ.L. Halverson for support and encouragement.  This work is generously\nsupported by startup funding from Northeastern University and the\nNational Science Foundation under Grant No. PHY11-25915.\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccccccc}\n  $F_1$ & $F_2$ & $J_1$ & $J_2$ & Minimal on $C$& $\\Delta$ & Additional Brane? \\\\ \\hline \\hline \n$II^*$ & $II^*$ & $0$ & $0$ & No & $(4 \\tilde f^3\\, t^2\\, z^2 + 27 \\tilde g^2)\\, t^{10}\\, z^{10}$ & No \\\\\n$II^*$ & $III^*$ & $0$ & $1$ & No & $(4 \\tilde f^3\\, z^2 + 27 \\tilde g^2\\, t)\\, t^9\\, z^{10}$ & Yes \\\\\n$II^*$ & $IV^*$ & $0$ & $0$ & No & $(4 \\tilde f^3\\, t\\, z^2 + 27 \\tilde g^2)\\, t^8\\, z^{10}$ & No \\\\\n$II^*$ & $I_0^*$ & $0$ & $NF$ & No & $(4 \\tilde f^3\\, z^2 + 27 \\tilde g^2)\\, t^6\\, z^{10}$ & No\\\\\n$II^*$ & $IV$ & $0$ & $0$ & No & $(4 \\tilde f^3\\, t^2\\, z^2 + 27 \\tilde g^2)\\, t^4\\, z^{10}$ & No \\\\\n$II^*$ & $III$ & $0$ & $1$ & No & $(4 \\tilde f^3\\, z^2 + 27 \\tilde g^2\\, t)\\, t^3\\, z^{10}$ & Yes \\\\\n$II^*$ & $II$ & $0$ & $0$ & No & $(4 \\tilde f^3\\, t\\, z^2 + 27 \\tilde g^2)\\, t^2\\, z^{10}$ & No \\\\\n$III^*$ & $III^*$ & $1$ & $1$ & No & $(4 \\tilde f^3 + 27\\tilde g^2 \\, t)\\, t^9\\, z^9$ & No \\\\\n$III^*$ & $IV^*$ & $1$ & $0$ & No & $(4 \\tilde f^3\\, t + 27 \\tilde g^2\\, z)\\, t^8\\, z^9$ & Yes \\\\\n$III^*$ & $I_0^*$ & $1$ & $NF$ & No & $(4 \\tilde f^3 + 27\\tilde g^2 \\, z)\\, t^6 \\,z^9$ & No \\\\\n$III^*$ & $IV$ & $1$ & $0$ & No & $(4 \\tilde f^3\\, t^2 + 27 \\tilde g^2\\, z)\\, t^4\\, z^9$ & Yes \\\\\n$III^*$ & $III$ & $1$ & $1$ & No & $(4\\tilde f^3 + 27\\tilde g^2 \\, t\\, z)\\, t^3\\, z^9$ & No \\\\\n$III^*$ & $II$ & $1$ & $0$ & No & $(4 \\tilde f^3\\, t + 27 \\tilde g^2\\, z)\\, t^2\\, z^9$ & Yes \\\\\n$IV^*$ & $IV^*$ & $0$ & $0$ & No & $(4 \\tilde f^3\\, t\\, z + 27 \\tilde g^2)\\, t^8\\, z^8$ & No \\\\\n$IV^*$ & $I_0^*$ & $0$ & $NF$ & No & $(4 \\tilde f^3\\, z + 27 \\tilde g^2)\\, t^8 z^8$ & No \\\\\n$IV^*$ & $IV$ & $0$ & $0$ & No & $(4 \\tilde f^3\\, t^2\\, z + 27 \\tilde g^2)\\, t^4\\, z^8$ & No \\\\\n$IV^*$ & $III$ & $0$ & $1$ & No & $(4\\tilde f^3\\, z + 27\\tilde g^2 \\, t) \\, t^3\\, z^8$ & Yes \\\\\n$IV^*$ & $II$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t\\, z + 27 \\tilde g^2)\\, t^2\\, z^8$ & No \\\\\n$I_0^*$ & $I_0^*$ & $NF$ & $NF$ & No & $(4 \\tilde f^3 + 27 \\tilde g^2) \\, t^6\\, z^6$ & No \\\\\n$I_0^*$ & $IV$ & $NF$ & $0$ & Yes & $(4 \\tilde f^3\\, t^2 + 27 \\tilde g^2)\\, t^4\\, z^6$ & No \\\\\n$I_0^*$ & $III$ & $NF$ & $1$ & Yes & $(4 \\tilde f^3 +  27\\tilde g^2 \\, t ) \\, t^3\\, z^6$ & No \\\\\n$I_0^*$ & $II$ & $NF$ & $0$ & Yes & $(4 \\tilde f^3\\, t + 27 \\tilde g^2)\\, t^2\\, z^6$ & No \\\\\n$IV$ & $IV$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t^2\\, z^2 + 27 \\tilde g^2)\\, t^4\\, z^4$ & No \\\\\n$IV$ & $III$ & $0$ & $1$ & Yes & $(4 \\tilde f^3\\, z^2 + 27 \\tilde g^2\\, t)\\, t^3\\, z^4$ & Yes \\\\\n$IV$ & $II$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t\\, z^2 + 27 \\tilde g^2)\\, t^2\\, z^4$ & No \\\\\n$III$ & $III$ & $1$ & $1$ & Yes & $(4\\tilde f^3 + 27\\tilde g^2 \\, t\\, z)\\, t^3\\, z^3$ & No \\\\\n$III$ & $II$ & $1$ & $0$ & Yes & $(4 \\tilde f^3\\, t + 27 \\tilde g^2\\, z)\\, t^2\\, z^3$ & Yes \\\\\n$II$ & $II$ & $0$ & $0$ & Yes & $(4 \\tilde f^3\\, t\\, z + 27 \\tilde g^2)\\, t^2\\, z^2$ & No \\\\ \\hline\\hline\n\\end{tabular}\n\\caption{Properties of all intersections of Kodaira fibers with finite order monodromy, including\nthose that give rise to $(4,6)$ curves. There is an additional brane if and only if $J_1+J_2=1$, and therefore\nit may play an important role in local axiodilaton profiles.}\n\\end{table}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe interpretation of chest radiographs is an essential task in the practice of a radiologist, enabling the early detection of thoracic diseases~\\cite{Rajpurkar2018,Wang2017}. To accelerate and improve the assessment of the continuously increasing number of radiographs, several deep learning solutions have been recently proposed for the automatic classification of radiographic findings~\\cite{Wang2017,Guendel2018,Yao2018}. Due to large variations in image quality or subjective definitions of disease appearance, there is a large inter-rate variability which leads to a high degree of label noise~\\cite{Rajpurkar2018}. Modeling this variability when designing an automatic system for assessing this type of data is essential; an aspect which was not considered in previous work.\n\nUsing principles of information theory and subjective logic~\\cite{Josang2016} based on the Dempster-Shafer framework for modeling of evidence~\\cite{Dempster1968}, we present a method for training a system that generates both an image-level label and a classification uncertainty measure. We evaluate this system for classification of abnormalities on chest radiographs. The main contributions of this paper include:\n\\begin{enumerate}\n    \\item describing a system for jointly learning classification probabilities and classification uncertainty in a parametric model;\n    \\item proposing uncertainty-driven bootstrapping as a means to filter training samples with highest predictive uncertainty to improve robustness and accuracy;\n    \\item comparing methods for generating stochastic classifications to model classification uncertainty; \n    \\item presenting an application of this system to identify cases with uncertain classification, yielding more accurate classification on the remaining cases;\n    \\item showing that the uncertainty measure can distinguish radiographs with correct and incorrect labels according to a multi-radiologist-consensus study.\n    \n\\end{enumerate}\n\n\\section{Background and Motivation}\n\\subsection{Machine Learning for the Assessment of Chest Radiographs}\nThe open access to the ChestX-Ray8 dataset~\\cite{Wang2017} of chest radiographs has led to a series of recent publications that propose machine learning based systems for disease classification. With this dataset, Wang et al.~\\cite{Wang2017} also report a first performance baseline of a deep neural network at an average area under receiver operating characteristic curve (ROC-AUC) of 0.75. These results have been further improved by using multi-scale image analysis~\\cite{Yao2018}, or by actively focusing the attention of the network on the most relevant sub-regions of the lungs~\\cite{Guan2018}. State-of-the-art results on the official split of the ChestX-Ray8 dataset are reported in~\\cite{Guendel2018} (avg. ROC-AUC of 0.81), using a location-aware dense neural network. In light of these contributions, a recent study compares the performance of such an AI system and 9 practicing radiologists~\\cite{Rajpurkar2018}. While the study indicates that the system can surpass human performance, it also highlights the high variability among different expert radiologists for the reading of chest radiographs. The reported average specificity of the readers is very high (over 95\\%), with an average sensitivity of 50\\%$\\,\\pm$\\,8\\%. With such a large inter-rater variability, one may ask: How can real 'ground truth' data be obtained? Does the label noise affect the training? Current solutions do not consider this variability, which leads to models with overconfident predictions and limited generalization.\\medskip\\\\\n\\textbf{Principles of Uncertainty Estimation:}\\, One way to handle this challenge is to explicitly estimate the classification uncertainty from the data. Recent methods for uncertainty estimation in the context of deep learning rely on Bayesian estimation theory~\\cite{Molchanov2017} or ensembles~\\cite{Laks2017} and demonstrate increased robustness to out-of-distribution data. However, these approaches come with significant computational limitations; associated with the high complexity of sampling parameter spaces of deep models for Bayesian risk estimation; or associated with the challenge of managing ensembles of deep models. Sensoy et al.~\\cite{Sensoy2018} propose an efficient alternative based on the theory of subjective logic~\\cite{Josang2016}, training a deep neural network to estimate the sample uncertainty based on observed data.\n\n\\section{Proposed Method}\n\nFollowing the work of Sensoy et al.~\\cite{Sensoy2018} based on the Dempster-Shafer theory of evidence~\\cite{Dempster1968}, we apply principles of subjective logic~\\cite{Josang2016} to derive a binary classification model that can support the joint estimation of per-class probabilities ($\\hat{p}_+; \\hat{p}_-$) and predictive uncertainty $\\hat{u}$. In this context, a decisional framework is defined through the assignment of so called belief masses from evidence collected from observed data to individual attributes, e.g., membership to a class~\\cite{Dempster1968,Josang2016}. Let us denote $b^+$ and $b^-$ the belief values for the positive and negative class, respectively. The uncertainty mass $u$ is defined as: $u = 1 - b^+ - b^-$, where $b^+ = e^+\/E$ and $b^- = e^-\/E$ with $e^+; e^- \\ge 0$ denoting the per-class collected evidence and total evidence $E = e^+ + e^- + 2$. For binary classification, we propose to model the distribution of such evidence values using the beta distribution, defined by two parameters $\\alpha$ and $\\beta$ as: $f(x;\\alpha,\\beta) = \\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}x^{\\alpha - 1}(1 - x)^{\\beta - 1}$, where $\\Gamma$ denotes the gamma function and $\\alpha, \\beta > 1$ with $\\alpha = e^+ + 1$ and $\\beta = e^- + 1$. In this context, the per-class probabilities can be derived as $p^+ = \\alpha\/E$ and $p^- = \\beta\/E$. Figure~\\ref{fig:beta} visualizes the beta distribution for different $\\alpha, \\beta$ values.\n\nA training dataset is provided: $\\mathcal{D} = \\{\\vec{I}_k,y_k\\}_{k=1}^{N}$, composed of $N$ pairs of images $\\vec{I}_k$ with class assignment $y_k\\in\\{0,1\\}$. To estimate the per-class evidence values from the observed data, a deep neural network parametrized by $\\vec{\\theta}$ can be applied, with: $[e^+_k, e^-_k] = \\mathcal{R}(\\vec{I}_k;\\vec{\\theta})$, where $\\mathcal{R}$ denotes the network response function. Using maximum likelihood estimation, one can learn the network parameters $\\hat{\\vec{\\theta}}$ by optimizing the Bayes risk with a beta distributed prior:\n\\begin{equation}\n    \\mathcal{L}^{data}_k = \\int \\|y_k - p_k\\|^2 \\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}p_k^{\\alpha - 1}(1 - p_k)^{\\beta - 1} dp_k,\n\\label{eq:lossmse}\n\\end{equation}\nwhere $k\\in\\{1,\\ldots,N\\}$ denotes the index of the training example from dataset~$\\mathcal{D}$, $p_k$ the predicted probability on the training sample $k$, and $\\mathcal{L}^{data}_k$ defines the goodness of fit. Using linearity properties of the expectation, Eq.~\\ref{eq:lossmse} becomes:\n\\begin{equation}\n    \\mathcal{L}^{data}_k = (y_k - \\hat{p}_k^{\\,+})^2 + (1 - y_k - \\hat{p}_k^{\\,-})^2 + \\frac{\\hat{p}_k^{\\,+}(1 - \\hat{p}_k^{\\,+}) + \\hat{p}_k^{\\,-}(1 - \\hat{p}_k^{\\,-})}{E_k + 1},\n\\end{equation}\nwhere $\\hat{p}_k^{\\,+}$ and $\\hat{p}_k^{\\,-}$ denote the network's probabilistic prediction. The first two terms measure the goodness of fit, and the last term encodes the variance of the prediction~\\cite{Sensoy2018}.\n\n\\begin{figure*}[t]\n\\centering\n\\subfloat[Confident negative]{\n\\includegraphics[height=2.8cm]{b1.pdf}\n}\n\\subfloat[Confident positive]{\n\\includegraphics[height=2.8cm]{b2.pdf}\n}\n\\subfloat[High uncertainty]{\n\\includegraphics[height=2.8cm]{b3.pdf}\n}\n\\qquad\n\\caption{Probability density function of the beta distribution: example parameters ($\\alpha, \\beta$) modeling confident and uncertain predictions.\\label{fig:beta}}\n\\end{figure*}\n\nTo ensure a high uncertainty value for data samples for which the gathered evidence is not conclusive for an accurate classification, an additional regularization term $\\mathcal{L}^{reg}_k$ is added to the loss. Using information theory, this term is defined as the relative entropy, i.e., the Kullback-Leibler divergence, between the beta distributed prior term and the beta distribution with total uncertainty ($\\alpha, \\beta = 1$). In this way, cost deviations from the total uncertainty state, i.e., $u = 1$, which do not contribute to the data fit are accounted for~\\cite{Sensoy2018}. With the additional term, the total cost becomes $\\mathcal{L} = \\sum_{k=1}^{N}\\mathcal{L}_k$ with:\n\\begin{equation}\n    \\mathcal{L}_k = \\mathcal{L}^{data}_k + \\lambda\\,\\text{KL}\\left (f(\\hat{p}_k;\\tilde{\\alpha}_k,\\tilde{\\beta}_k)\\|f(\\hat{p}_k;\\langle 1,1\\rangle)\\right),\n\\end{equation}\nwhere $\\lambda\\in[0,1]$, $\\hat{p}_k = \\hat{p}_k^{\\,+}$, with $(\\tilde{\\alpha}_k, \\tilde{\\beta}_k)=(1, \\beta_k)$ for $y_k = 0$ and $(\\tilde{\\alpha}_k, \\tilde{\\beta}_k)=(\\alpha_k, 1)$ for $y_k = 1$. Removing additive constants and using properties of the logarithm function, one can simplify the regularization term to the following:\n\\begin{equation}\n    \\mathcal{L}^{reg}_k = \\log\\frac{\\Gamma(\\tilde{\\alpha}_k+\\tilde{\\beta}_k)}{\\Gamma(\\tilde{\\alpha}_k)\\Gamma(\\tilde{\\beta_k})} + \\sum_{x\\in\\{\\tilde{\\alpha}_k, \\tilde{\\beta}_k\\}} (x - 1)\\left(\\psi(x) - \\psi(\\tilde{\\alpha}_k + \\tilde{\\beta}_k)\\right),\n\\end{equation}\nwhere $\\psi$ denotes the digamma function and $k\\in\\{1,\\ldots,N\\}$. Using stochastic gradient descent, the total loss $\\mathcal{L}$ is optimized on the training set~$\\mathcal{D}$.\\smallskip\n\n\\textbf{Sampling the Data Distribution:}\\, An important requirement to ensure training stability and to learn a robust estimation of evidence values is an adequate sampling of the data distribution. We empirically found dropout~\\cite{Srivastava2014} to be a simple and very effective strategy to address this problem. In practice, dropout emulates an ensemble model combination driven by the random deactivation of neurons. Alternatively, one may use an explicit ensemble of $M$ models $\\{\\vec{\\theta}_k\\}_{k=1}^{M}$, each trained independently. Following the principles of deep ensembles~\\cite{Laks2017}, the per-class evidence can be computed from the ensemble estimates $\\{e^{(k)}\\}_{k=1}^{M}$ via averaging. In our work, we found dropout to be as effective as deep ensembles.\\smallskip\n\n\\textbf{Uncertainty-driven Bootstrapping:}\\, Given the predictive uncertainty measure $\\hat{u}$, we propose a simple and effective algorithm for filtering the training set with the target of reducing label noise. A fraction of training samples with highest uncertainty are eliminated and the model is retrained on the remaining data. Instead of sample elimination, robust M-estimators may be applied, using a per-sample weight that is inversely proportional to the predicted uncertainty. The hypothesis is that by focusing the training on 'confident' labels, we increase the robustness of the classifier and improve its performance on unseen data.\n\n\\section{Experiments}\n\n\\textbf{Dataset and Setup:}\\, The evaluation is based on two datasets, the ChestX-Ray8~\\cite{Wang2017} and PLCO~\\cite{PLCO}. Both datasets provide a series of AP\/PA chest radiographs with binary labels on the presence of different radiological findings, e.g., granuloma, pleural effusion, or consolidation. The ChestX-Ray8 dataset contains 112,120 images from 30,805 patients, covering 14 findings extracted from radiological reports using natural language processing (NLP)~\\cite{Wang2017}. In contrast, the PLCO dataset was constructed as part of a screening trial, containing 185,421 images from 56,071 patients and covering 12 different abnormalities.\n\nFor performance comparison, we selected location-aware dense networks~\\cite{Guendel2018} as baseline. This method achieves state-of-the-art results on this problem, with a reported average ROC-AUC of \\textbf{0.81} (significantly higher than that of competing methods: 0.75~\\cite{Wang2017} and 0.77~\\cite{Yao2018}) on the official split of the ChestX-Ray8 dataset and a ROC-AUC of \\textbf{0.88} on the official split of the PLCO dataset. To evaluate our method, we identified testing subsets with higher confidence labels from multi-radiologist studies. For PLCO, we randomly selected 565 test images and had 2 board-certified expert radiologists read the images -- updating the labels to the majority vote of the 3 opinions (incl. the original label). For ChestX-Ray8, a subset of 689 test images was randomly selected and read by 4 board-certified radiologists. The final label was decided following a consensus discussion. For both datasets, the remaining data was split in 90\\% training and 10\\% validation. All images were down-sampled to $256\\times 256$ using bilinear interpolation.\\smallskip\n\n\\textbf{System Training:}\\, We constructed our learning model from the DenseNet-121 architecture~\\cite{Huang2017}. A dropout layer with a dropout rate of 0.5 was inserted after the last convolutional layer. We also investigated the benefits of using deep ensembles to improve the sampling ($M=5$ models trained on random subsets of 80\\% of the training data; we refer to this with the keyword \\textbf{[ens]}). A fully connected layer with ReLU activation units maps to the two outputs $\\alpha$ and $\\beta$. We used a systematic grid search to find the optimal configuration of training meta-parameters: learning rate ($10^{-4}$), regularization factor ($\\lambda=1$; decayed to $0.1$ and $0.001$ after 1\/3, respectively 2\/3 of the epochs), training epochs (around 12, using an early stop strategy with a patience of 3 epochs) and a batch size of 128. The low number of epochs is explained by the large size of the dataset.\\smallskip\n\n\\textbf{Uncertainty-driven Sample Rejection:}\\, Given a model trained for the assessment of an arbitrary finding, one can directly estimate the prediction uncertainty $\\hat{u} = 2\/(\\alpha + \\beta) \\in [0, 1]$. This is an orthogonal measure to the predicted probability, with increased values on out-of-distribution cases under the given model. One can use this measure for sample rejection, i.e., set a threshold $u_t$ and steer the system to not output its prediction on all cases with an expected uncertainty larger than $u_t$. Instead, these cases are labeled with the message \\textit{\"Do not know for sure; process case manually\"}. In practice this leads to a significant increase in accuracy compared to the state-of-the-art on the remaining cases, as reported in Table~\\ref{tab:results} and Figure~\\ref{fig:samplereject}. For example, for the identification of granuloma, a rejection rate of 25\\% leads to an increase of over 20\\% in the micro-average F1 score. On the same abnormality, a 50\\% rejection rate leads to a F1 score over 0.99 for the prediction of negative cases. We found no significant difference in average performance when using ensembles (see Figure~\\ref{fig:samplereject}).\\smallskip\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=3.75cm]{granuloma.pdf}\n\\includegraphics[height=3.75cm]{fibrosis.pdf}\n\\caption{Evolution of the F1-scores for the positive (+) and negative (--) classes relative to the sample rejection threshold - determined using the estimated uncertainty. We show the performance for granuloma and fibrosis based on the PLCO dataset~\\cite{PLCO}. The baseline (horizontal dashed lines) is determined using the method from~\\cite{Guendel2018} (working point at max. average of per-class F1 scores). Decision threshold for our method is fixed at 0.5. \\label{fig:samplereject}}\n\\end{figure}\n\\begin{table}[t]\n\\centering\n\\caption{Comparison between the reference method~\\cite{Guendel2018} and several versions of our method calibrated at sample rejection rates of 0\\%, 10\\%, 25\\% and 50\\% (based on the PLCO dataset~\\cite{PLCO}). Lesion refers to lesions of the bones or soft tissue.\\label{tab:results}}\n\\begin{tabular}{L{1.8cm} C{2.5cm} C{1.7cm} C{1.9cm} C{1.9cm} C{1.8cm}}\n&\\multicolumn{5}{c}{\\textbf{ROC-AUC}}\\\\\n\\cmidrule{2-6}\n\\textbf{Finding}&Guendel et al.~\\cite{Guendel2018}&\\textbf{Ours} [0\\%]&\\textbf{Ours} [10\\%]&\\textbf{Ours} [25\\%]&\\textbf{Ours} [50\\%]\\\\\n\\midrule\nGranuloma&0.83&0.85&0.87&\\textbf{0.90}&\\textbf{0.92}\\\\\nFibrosis&0.87&0.88&0.90&\\textbf{0.92}&\\textbf{0.94}\\\\\nScaring&0.82&0.81&0.84&\\textbf{0.89}&\\textbf{0.93}\\\\\nLesion&0.82&0.83&0.86&\\textbf{0.88}&\\textbf{0.90}\\\\\nCardiac Ab.&0.93&0.94&0.95&\\textbf{0.96}&\\textbf{0.97}\\\\\n\\midrule\\midrule\n\\textbf{Average}&0.85&0.86&0.89&\\textbf{0.91}&\\textbf{0.93}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=3cm]{human.pdf}\n\\includegraphics[height=3cm]{catch.pdf}\n\\caption{\\textbf{Left}: Predictive uncertainty distribution on 689 ChestX-Ray test images; a higher uncertainty is associated with cases of the critical set, which required a label correction according to expert committee. \\textbf{Right}: Plot showing the capacity of the algorithm to eliminate cases from the critical set via sample rejection. Bars indicate the percentage of critical cases for each batch of 5\\% rejected cases.\\label{fig:corr}}\n\\end{figure}\n\n\\textbf{System versus Reader Uncertainty:}\\, To provide an insight into the meaning of the uncertainty measure and its correlation with the difficulty of cases, we evaluated our system on the detection of pleural effusion (excess accumulation of fluid in the pleural cavity) based on the ChestX-Ray8 dataset. In particular, we analyzed the test set of 689 cases that were relabeled using an expert committee of 4 experts. We defined a so called \\textit{critical set}, that contains only cases for which the label (positive or negative) was changed after the expert reevaluation. According to the committee, this set contained not only easy examples for which probably the NLP algorithm has failed to properly extract the correct labels from the radiographic report; but also difficult cases where either the image quality was limited or the evidence of effusion was very subtle. In Figure~\\ref{fig:corr} (left), we empirically demonstrate that the uncertainty estimates of our algorithm correlate with the committee's decision to change the label. Specifically, for unchanged cases, our algorithm displayed very low uncertainty estimates (average 0.16) at an average AUC of 0.976 (rejection rate of 0\\%). In contrast, on cases in the critical set, the algorithm showed higher uncertainties distributed between 0.1 and the maximum value of 1 (average 0.41). This empirically demonstrates the ability of the algorithm to recognize the cases where annotation errors occurred in the first place (through NLP or human reader error). In Figure~\\ref{fig:corr} (right) we show how cases of the critical set can be effectively filtered out using sample rejection. Qualitative examples are shown in Figure~\\ref{fig:examples}.\\smallskip\n\n\\textbf{Uncertainty-driven Bootstrapping:}\\, We also investigated the benefit of using bootstrapping based on the uncertainty measure on the example of plural effusion (ChestX-Ray8). \nWe report performance as [\\textit{AUC}; \\textit{F1-score} (pos. class); \\textit{F1-score} (neg. class)]. After training our method, the baseline performance was measured at $[0.89; 0.60; 0.92]$ on testing. We then eliminated 5\\%, 10\\% and 15\\% of training samples with highest uncertainty, and retrained in each case on the remaining data. The metrics improved to $[0.90; 0.68; 0.92]_{5\\%}$, $[0.91; 0.67; 0.94]_{10\\%}$ and $[\\textbf{0.93}; \\textbf{0.69}; \\textbf{0.94}]_{15\\%}$ on the test set. This is a significant increase, demonstrating the potential of this strategy to improve the robustness of the model to the label noise. We are currently focused on further exploring this method.\n\n\\begin{figure*}[t]\n\\centering\n\\subfloat[$\\hat{u},\\hat{p}=0.90,0.45$]{\n\\includegraphics[height=2.8cm]{ex1.pdf}\n\\label{subfig:1}\n}\n\\subfloat[$\\hat{u},\\hat{p}=0.93,0.48$]{\n\\includegraphics[height=2.8cm]{ex2.pdf}\n\\label{subfig:2}\n}\n\\subfloat[$\\hat{u},\\hat{p}=0.54,0.65$]{\n\\includegraphics[height=2.8cm]{ex3.pdf}\n\\label{subfig:3}\n}\n\\subfloat[$\\hat{u},\\hat{p}=0.11,0.05$]{\n\\includegraphics[height=2.8cm]{00002048_000_11negative.png}\n\\label{subfig:4}\n}\n\\caption{ChestX-Ray8 test images assessed for pleural effusion ($\\hat{u}$: est. uncertainty, $\\hat{p}$: output probability; with affected regions circled in red). Figures~\\ref{subfig:1}, \\ref{subfig:2} and \\ref{subfig:3} show positive cases of the critical set with high predictive uncertainty -- possibly explained by the atypical appearance of accumulated fluid in~\\ref{subfig:1}, and poor quality of image~\\ref{subfig:2}. Figure~\\ref{subfig:4} shows a high confidence case with no pleural effusion.\\label{fig:examples}}\n\\end{figure*}\n\n\\section{Conclusion}\nIn conclusion, this paper presents an effective method for the joint estimation of class probabilities and classification uncertainty in the context of chest radiograph assessment. Extensive experiments on two large datasets demonstrate a significant accuracy increase if sample rejection is performed based on the estimated uncertainty measure. In addition, we highlight the capacity of the system to distinguish radiographs with correct and incorrect labels according to a multi-radiologist-consensus user study, using the uncertainty measure only.\\smallskip\n\nThe authors thank the National Cancer Institute for access to NCI's data collected by the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial. The statements contained herein are solely those of the authors and do not represent or imply concurrence or endorsement by NCI.\n\n\\ifx1\\undefined\n\\textbf{Disclaimer} The concepts and information presented in this paper are based on research results that are not commercially available.\n\\fi\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{secintro}\n\nThe goal of these lecture notes, as the title says, is to give a basic introduction to the theory of large deviations at three levels: theory, applications and simulations. The notes follow closely my recent review paper on large deviations and their applications in statistical mechanics \\cite{touchette2009}, but are, in a way, both narrower and wider in scope than this review paper.\n\nThey are narrower, in the sense that the mathematical notations have been cut down to a minimum in order for the theory to be understood by advanced undergraduate and graduate students in science and engineering, having a basic background in probability theory and stochastic processes (see, e.g., \\cite{grimmett2001}). The simplification of the mathematics amounts essentially to two things: i) to focus on random variables taking values in $\\mathbb{R}$ or $\\mathbb{R}^D$, and ii) to state all the results in terms of probability densities, and so to assume that probability densities always exist, if only in a weak sense. These simplifications are justified for most applications and are convenient for conveying the essential ideas of the theory in a clear way, without the hindrance of technical notations.\n\nThese notes also go beyond the review paper \\cite{touchette2009}, in that they cover subjects not contained in that paper, in particular the subject of numerical estimation or \\textit{simulation} of large deviation probabilities. This is an important subject that I intend to cover in more depth in the future.\n\nSections~\\ref{secnum1} and \\ref{secnum2} of these notes are a first and somewhat brief attempt in this direction. Far from covering all the methods that have been developed to simulate large deviations and rare events, they concentrate on the central idea of large deviation simulations, namely that of exponential change of measure, and they elaborate from there on certain simulation techniques that are easily applicable to sums of independent random variables, Markov chains, stochastic differential equations and continuous-time Markov processes in general.\n\nMany of these applications are covered in the exercises contained at the end of each section. The level of difficulty of these exercises is quite varied: some are there to practice the material presented, while others go beyond that material and may take hours, if not days or weeks, to solve completely. For convenience, I have rated them according to Knuth's logarithmic scale.\\footnote{00 = Immediate; 10 = Simple; 20 = Medium; 30 = Moderately hard; 40 = Term project; 50 = Research problem. See the superscript attached to each exercise.}\n\nIn closing this introduction, let me emphasize again that these notes are not meant to be complete in any way. For one thing, they lack the rigorous notation needed for handling large deviations in a precise mathematical way, and only give a hint of the vast subject that is large deviation and rare event simulation. On the simulation side, they also skip over the subject of error estimates, which would fill an entire section in itself. In spite of this, I hope that the material covered will have the effect of enticing readers to learn more about large deviations and of conveying a sense of the wide applicability, depth and beauty of this subject, both at the theoretical and computational levels.\n\n\\section{Basic elements of large deviation theory}\n\\label{sectheory}\n\n\\subsection{Examples of large deviations}\n\nWe  start our study of large deviation theory by considering a sum of real random variables (RV for short) having the form\n\\begin{equation}\nS_n=\\frac{1}{n}\\sum_{i=1}^n X_i.\n\\end{equation}\nSuch a sum is often referred to in mathematics or statistics as a \\textbf{sample mean}. We are interested in computing the \\textbf{probability density function} $p_{S_n}(s)$ (pdf for short) of $S_n$ in the simple case where the $n$ RVs are mutually \\textbf{independent and identically distributed} (IID for short).\\footnote{The pdf of $S_n$ will be denoted by $p_{S_n}(s)$ or more simply by $p(S_n)$ when no confusion arises.} This means that the joint pdf of $X_1,\\ldots,X_n$ factorizes as follows:\n\\begin{equation}\np(X_1,\\ldots,X_n)=\\prod_{i=1}^n p(X_i),\n\\end{equation}\nwith $p(X_i)$ a fixed pdf for each of the $X_i$'s.\\footnote{To avoid confusion, we should write $p_{X_i}(x)$, but in order not to overload the notation, we will simply write $p(X_i)$. The context should make clear what RV we are referring to.} \n\nWe compare next two cases for $p(X_i)$:\n\\begin{itemize}\n\\item \\textbf{Gaussian pdf:}\n\\begin{equation}\np(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\mathrm{e}^{-(x-\\mu)^2\/(2\\sigma^2)},\\quad x\\in\\mathbb{R},\n\\end{equation}\nwhere $\\mu=E[X]$ is the \\textbf{mean} of $X$ and $\\sigma^2=E[(X-\\mu)^2]$ its \\textbf{variance}.\\footnote{The symbol $E[X]$ denotes the expectation or expected value of $X$, which is often denoted by $\\langle X\\rangle$ in physics.}\n\n\\item \\textbf{Exponential pdf:}\n\\begin{equation}\np(x)=\\frac{1}{\\mu} \\mathrm{e}^{-x\/\\mu},\\quad x\\in [0,\\infty),\n\\label{eqexpd1}\n\\end{equation}\nwith mean $E[X]=\\mu>0$. \n\\end{itemize}\n\nWhat is the form of $p_{S_n}(s)$ for each pdf?\n\nTo find out, we write the pdf associated with the event $S_n=s$ by summing the pdf of all the values or \\textbf{realizations} $(x_1,\\ldots,x_n)\\in\\mathbb{R}^n$ of $X_1,\\ldots,X_n$ such that $S_n=s$.\\footnote{Whenever possible, random variables will be denoted by uppercase letters, while their values or realizations will be denoted by lowercase letters. This follows the convention used in probability theory. Thus, we will write $X=x$ to mean that the RV $X$ takes the value $x$.} In terms of Dirac's delta function $\\delta(x)$, this is written as\n\\begin{equation}\np_{S_n}(s)=\\int_\\mathbb{R} \\mathrm{d} x_1\\cdots \\int_\\mathbb{R} \\mathrm{d} x_n\\ \\delta\\big(\\textstyle\\sum_{i=1}^n x_i -ns\\big)\\ p(x_1,\\ldots,x_n).\n\\label{eqrep1}\n\\end{equation}\nFrom this expression, we can then obtain an explicit expression for $p_{S_n}(s)$ by using the method of generating functions (see Exercise~\\ref{excf1}) or by substituting the Fourier representation of $\\delta(x)$ above and by explicitly evaluating the $n+1$ resulting integrals. The result obtained for both the Gaussian and the exponential densities has the general form\n\\begin{equation}\np_{S_n}(s)\\approx \\mathrm{e}^{-nI(s)},\n\\label{eqldt1}\n\\end{equation}\nwhere \n\\begin{equation}\nI(s)=\\frac{(x-\\mu)^2}{2\\sigma^2},\\qquad s\\in\\mathbb{R}\n\\label{eqrfg1}\n\\end{equation}\nfor the Gaussian pdf, whereas\n\\begin{equation}\nI(s)=\\frac{s}{\\mu}-1-\\ln\\frac{s}{\\mu},\\qquad s\\geq 0\n\\label{eqldexp1}\n\\end{equation}\nfor the exponential pdf.\n\n\\begin{figure}[t]\n\\resizebox{\\textwidth}{!}{\\includegraphics{gaussianldt1.pdf}}\n\\caption{(Left) pdf $p_{S_n}(s)$ of the Gaussian sample mean $S_n$ for $\\mu=\\sigma=1$. (Right) $I_n(s)=-\\frac{1}{n}\\ln p_{S_n}(s)$ for different values of $n$ demonstrating a rapid convergence towards the rate function $I(s)$ (dashed line).}\n\\label{figgaussianld1}\n\\end{figure}\n\nWe will come to understand the meaning of the approximation sign ($\\approx$) more clearly in the next subsection. For now we just take it as meaning that the dominant behaviour of $p(S_n)$ as a function of $n$ is a decaying exponential in $n$. Other terms in $n$ that may appear in the exact expression of $p(S_n)$ are sub-exponential in $n$.\n\nThe picture of the behaviour of $p(S_n)$ that we obtain from the result of (\\ref{eqldt1}) is that $p(S_n)$ decays to $0$ exponentially fast with $n$ for all values $S_n=s$ for which the function $I(s)$, which controls the rate of decay of $p(S_n)$, is positive. But notice that $I(s)\\geq 0$ for both the Gaussian and exponential densities, and that $I(s)=0$ in both cases only for $s=\\mu=E[X_i]$. Therefore, since the pdf of $S_n$ is normalized, it must get more and more concentrated around the value $s=\\mu$ as $n\\rightarrow\\infty$ (see Fig.~\\ref{figgaussianld1}), which means that $p_{S_n}(s)\\rightarrow \\delta(s-\\mu)$ in this limit. We say in this case that $S_n$ converges in probability or in density to its mean. \n\nAs a variation of the Gaussian and exponential samples means, let us now consider a sum of \\textbf{discrete random variables} having a probability distribution $P(X_i)$ instead of continuous random variables with a pdf $p(X_i)$.\\footnote{Following the convention in probability theory, we use the lowercase $p$ for continuous probability densities and the uppercase $P$ for discrete probability distributions and probability assignments in general. Moreover, following the notation used before, we will denote the distribution of a discrete $S_n$ by $P_{S_n}(s)$ or simply $P(S_n)$.} To be specific, consider the case of IID \\textbf{Bernoulli RVs} $X_1,\\ldots,X_n$ taking values in the set $\\{0,1\\}$ with probability $P(X_i=0)=1-\\alpha$ and $P(X_i=1)=\\alpha$. What is the form of the probability distribution $P(S_n)$ of $S_n$ in this case?\n\nNotice that we are speaking of a probability distribution because $S_n$ is now a discrete variable taking values in the set $\\{0,1\/n,2\/n,\\ldots,(n-1)\/n,1\\}$. In the previous Gaussian and exponential examples, $S_n$ was a continuous variable characterized by its pdf $p(S_n)$. \n\nWith this in mind, we can obtain the exact expression of $P(S_n)$ using methods similar to those used to obtain $p(S_n)$ (see Exercise~\\ref{exbs1}). The  result  is different from that found for the Gaussian and exponential densities, but what is remarkable is that the distribution of the Bernoulli sample mean also contains a dominant exponential term having the form\n\\begin{equation}\nP_{S_n}(s)\\approx \\mathrm{e}^{-nI (s)},\n\\label{eqldbern1}\n\\end{equation} \nwhere $I(s)$ is now given by\n\\begin{equation}\nI(s)=s\\ln \\frac{s}{\\alpha} +(1-s)\\ln \\frac{1-s}{1-\\alpha},\\qquad s\\in [0,1].\n\\label{eqbern1}\n\\end{equation}\n\nThe behaviour of the exact expression of $P(S_n)$ as $n$ grows is shown in Fig.~\\ref{figbernld1} together with the plot of $I(s)$ as given by Eq.~(\\ref{eqbern1}). Notice how $P(S_n)$ concentrates around its mean $\\mu=E[X_i]=\\alpha$ as a result of the fact that $I(s)\\geq 0$ and that $s=\\mu$ is the only value of $S_n$ for which $I=0$. Notice also how the support of $P(S_n)$ becomes ``denser'' as $n\\rightarrow\\infty$, and compare this property with the fact that $I(s)$ is a continuous function despite $S_n$ being discrete. Should $I(s)$ not be defined for discrete values if $S_n$ is a discrete RV? We address this question next.\n\n\\begin{figure}[t]\n\\resizebox{\\textwidth}{!}{\\includegraphics{bernoullildt1.pdf}}\n\\caption{(Top left) Discrete probability distribution $P_{S_n}(s)$ of the Bernoulli sample mean for $\\alpha=0.4$ and different values of $n$. (Top right) Finite-$n$ rate function $I_n(s)=-\\frac{1}{n}\\ln P_{S_n}(s)$. The rate function $I(s)$ is the dashed line. (Bottom left). Coarse-grained pdf $p_{S_n}(s)$ for the Bernoulli sample mean. (Bottom right) $I_n(s)=-\\frac{1}{n}\\ln p_{S_n}(s)$ as obtained from the coarse-grained pdf.}\n\\label{figbernld1}\n\\end{figure}\n\n\\subsection{The large deviation principle}\n\nThe general exponential form $\\mathrm{e}^{-nI(s)}$ that we found for our three previous sample means (Gaussian, exponential and Bernoulli) is the founding result or property of large deviation theory, referred to as the large deviation principle. The reason why a whole theory can be built on such a seemingly simple result is that it arises in the context of many stochastic processes, not just IID sample means, as we will come to see in the next sections, and as can be seen from other contributions to this volume; see, e.g., Engel's.\\footnote{The volume referred to is the volume of lecture notes produced for the summer school; see \\texttt{http:\/\/www.mcs.uni-oldenburg.de\/}}\n\nThe rigorous, mathematical definition of the large deviation principle involves many concepts of topology and measure theory that are too technical to be presented here (see Sec.~3.1 and Appendix B of \\cite{touchette2009}). For simplicity, we will say here that a random variable $S_n$ or its pdf $p(S_n)$ satisfies a \\textbf{large deviation principle} (LDP) if the following limit exists:\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty} -\\frac{1}{n}\\ln p_{S_n}(s)=I(s)\n\\label{eqldlim1}\n\\end{equation}\nand gives rise to a function $I(s)$, called the \\textbf{rate function}, which is not everywhere zero.\n\nThe relation between this definition and the approximation notation used earlier should be clear: the fact that the behaviour of $p_{S_n}(s)$ is dominated for large $n$ by a decaying exponential means that the exact pdf of $S_n$ can be written as\n\\begin{equation}\np_{S_n}(s)=\\mathrm{e}^{-nI(s)+o(n)}\n\\end{equation}\nwhere $o(n)$ stands for any correction term that is sub-linear in $n$. Taking the large deviation limit of (\\ref{eqldlim1}) then yields\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty} -\\frac{1}{n}\\ln p_{S_n}(s)=I(s)-\\lim_{n\\rightarrow\\infty} \\frac{o(n)}{n} = I(s),\n\\end{equation}\nsince $o(n)\/n\\rightarrow 0$. We see therefore that the large deviation limit of Eq.~(\\ref{eqldlim1}) is the limit needed to retain the dominant exponential term in $p(S_n)$ while discarding any other sub-exponential terms. For this reason, large deviation theory is often said to be concerned with estimates of probabilities on the logarithmic scale. \n\nThis point is illustrated in Fig.~\\ref{figbernld1}. There we see that the function \n\\begin{equation}\nI_n(s)=-\\frac{1}{n}\\ln p_{S_n}(s)\n\\end{equation} \nis not quite equal to the limiting rate function $I(s)$, because of terms of order $o(n)\/n=o(1)$; however, it does converge to $I(s)$ as $n$ gets larger. Plotting $p(S_n)$ on this scale therefore reveals the rate function in the limit of large $n$. This convergence will be encountered repeatedly in the sections on the numerical evaluation of large deviation probabilities.\n\nIt should be emphasized again that the definition of the LDP given above is a simplification of the rigorous definition used in mathematics, due to the mathematician Srinivas Varadhan.\\footnote{Recipient of the 2007 Abel Prize for his ``fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.''} The real, mathematical definition is expressed in terms of probability measures of certain sets rather than in terms of probability densities, and involves upper and lower bounds on these probabilities rather than a simple limit (see Sec.~3.1 and Appendix B of \\cite{touchette2009}). The mathematical definition also applies a priori to any RVs, not just continuous RVs with a pdf.\n\nIn these notes, we will simplify the mathematics by assuming that the random variables or stochastic processes that we study have a pdf. In fact, we will often assume that pdfs exist even for RVs that are not continuous but ``look'' continuous at some scale.\n\nTo illustrate this point, consider again the Bernoulli sample mean. We noticed that $S_n$ in this case is not a continuous RV, and so does not have a pdf. However, we also noticed that the values that $S_n$ can take become dense in the interval $[0,1]$ as $n\\rightarrow\\infty$, which means that the support of the discrete probability distribution $P(S_n)$ becomes dense in this limit, as shown in Fig.~\\ref{figbernld1}. From a practical point of view, it therefore makes sense to treat $S_n$ as a continuous RV for large $n$ by interpolating $P(S_n)$ to a continuous function $p(S_n)$ representing the ``probability density'' of $S_n$.\n\nThis pdf is obtained in general simply by considering the probability $P(S_n\\in [s,s+\\Delta s])$ that $S_n$ takes a value in a tiny interval surrounding the value $s$, and by then dividing this probability by the ``size'' $\\Delta s$ of that interval:\n\\begin{equation}\np_{S_n}(s)=\\frac{P(S_n\\in [s,s+\\Delta s])}{\\Delta s}.\n\\end{equation}\nThe pdf obtained in this way is referred to as a \\textbf{smoothed}, \\textbf{coarse-grained} or \\textbf{weak density}.\\footnote{\\label{fnldp1}To be more precise, we should make clear that the coarse-grained pdf depends on the spacing $\\Delta s$ by writing, say, $p_{S_n,\\Delta s}(s)$. However, to keep the notation to a minimum, we will use the same lowercase $p$ to denote a coarse-grained pdf and a ``true'' continuous pdf. The context should make clear which of the two is used.} In the case of the Bernoulli sample mean, it is simply given by\n\\begin{equation}\np_{S_n}(s)=nP_{S_n}(s)\n\\end{equation}\nif the spacing $\\Delta s$ between the values of $S_n$ is chosen to be $1\/n$.\n\nThis process of replacing a discrete variable by a continuous one in some limit or at some scale is known in physics as the \\textbf{continuous} or \\textbf{continuum limit} or as the \\textbf{macroscopic limit}. In mathematics, this limit is expressed  via the notion of weak convergence (see \\cite{dupuis1997}).\n\n\\subsection{The G\\\"artner-Ellis Theorem}\n\nOur goal in the next sections will be to study random variables and stochastic processes that satisfy an LDP and to find analytical as well as numerical ways to obtain the corresponding rate function. There are many ways whereby a random variable, say $S_n$, can be shown to satisfy an LDP:\n\\begin{itemize}\n\\item \\textbf{Direct method:} Find the expression of $p(S_n)$ and show that it has the form of the LDP;\n\\item \\textbf{Indirect method:} Calculate certain functions of $S_n$, such as generating functions, whose properties can be used to infer that $S_n$ satisfies an LDP;\n\\item \\textbf{Contraction method:} Relate $S_n$ to another random variable, say $A_n$, which is known to satisfy an LDP and derive from this an LDP for $S_n$.\n\\end{itemize}\n\nWe have used the first method when discussing the Gaussian, exponential and Bernoulli sample means. \n\nThe main result of large deviation theory that we will use in these notes to obtain LDPs along the indirect method is called the \\textbf{G\\\"artner-Ellis Theorem} (GE Theorem for short), and is based on the calculation of the following function:\n\\begin{equation}\n\\lambda(k)=\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\ln E[\\mathrm{e}^{nk S_n}],\n\\label{eqscgf1}\n\\end{equation}\nknown as the \\textbf{scaled cumulant generating function}\\footnote{The function $E[\\mathrm{e}^{kX}]$ for $k$ real is known as the \\textbf{generating function} of the RV $X$; $\\ln E[\\mathrm{e}^{kX}]$ is known as the \\textbf{log-generating function} or \\textbf{cumulant generating function}. The word ``scaled'' comes from the extra factor $1\/n$.} (SCGF for short). In this expression $E[\\cdot]$ denotes the expected value, $k$ is a real parameter, and $S_n$ is an arbitrary RV; it is not necessarily an IID sample mean or even a sample mean. \n\nThe point of the GE Theorem is to be able to calculate $\\lambda(k)$ without knowing $p(S_n)$. We will see later that this is possible. Given $\\lambda(k)$, the GE Theorem then says that, if $\\lambda(k)$ is differentiable,\\footnote{The statement of the GE Theorem given here is a simplified version of the full result, which is essentially the result proved by G\\\"artner \\cite{gartner1977}. See Sec.~3.3 of \\cite{touchette2009} for a more complete presentation of GE Theorem and \\cite{ellis1984} for a rigorous account of it.} then \n\\begin{itemize}\n\\item $S_n$ satisfies an LDP, i.e.,\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty}-\\frac{1}{n}\\ln p_{S_n}(s)=I(s);\n\\end{equation}\n\\item The rate function $I(s)$ is given by the \\textbf{Legendre-Fenchel transform} of $\\lambda(k)$:\n\\begin{equation}\nI(s)=\\sup_{k\\in\\mathbb{R}}\\{ks-\\lambda(k)\\},\n\\label{eqlf1}\n\\end{equation}\nwhere ``$\\sup$'' stands for the \\textbf{supremum}.\\footnote{In these notes, a $\\sup$ can be taken to mean the same as a $\\max$.}\n\\end{itemize}\n\nWe will come to calculate $\\lambda(k)$ and its Legendre-Fenchel transform for specific RVs in the next section. It will also be seen in there that $\\lambda(k)$ does not always exist (this typically happens when the pdf of a random variable does not admit an LDP). Some exercises at the end of this section illustrate many useful properties of $\\lambda(k)$ when this function exists and is twice differentiable.\n\n\\subsection{Varadhan's Theorem}\n\nThe rigorous proof of the GE Theorem is too technical to be presented here. However, there is a way to justify this result by deriving in a  heuristic way another result known as Varadhan's Theorem.\n\nThe latter theorem is concerned with the evaluation of a functional\\footnote{A function of a function is called a \\textbf{functional}.} expectation of the form\n\\begin{equation}\nW_n[f]=E[\\mathrm{e}^{nf(S_n)}]=\\int_\\mathbb{R} p_{S_n}(s)\\, \\mathrm{e}^{nf(s)}\\, \\mathrm{d} s,\n\\end{equation} \nwhere $f$ is some function of $S_n$, which is taken to be a real RV for simplicity. Assuming that $S_n$ satisfies an LDP with rate function $I(s)$, we can write\n\\begin{equation}\nW_n[f]\\approx\\int_\\mathbb{R} \\mathrm{e}^{n[f(s)-I(s)]}\\, \\mathrm{d} s.\n\\end{equation}\nwith sub-exponential corrections in $n$. This integral has the form of a so-called \\textbf{Laplace integral}, which is known to be dominated for large $n$ by its largest integrand when it is unique. Assuming this is the case, we can proceed to approximate the whole integral as\n\\begin{equation}\nW_n[f]\\approx \\mathrm{e}^{n \\sup_s [f(s)-I(s)]}.\n\\end{equation}\nSuch an approximation is referred to as a \\textbf{Laplace approximation}, the \\textbf{Laplace principle} or a \\textbf{saddle-point approximation} (see Chap.~6 of \\cite{bender1978}), and is justified in the context of large deviation theory because the corrections to this approximation are sub-exponential in $n$, as are those of the LDP. By defining the following functional:\n\\begin{equation}\n\\lambda[f]=\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\ln W_n(f)\n\\end{equation}\nusing a limit similar to the limit defining the LDP, we then obtain\n\\begin{equation}\n\\lambda[f]=\\sup_{s\\in\\mathbb{R}}\\{f(s)-I(s)\\}.\n\\label{eqvaradhan1}\n\\end{equation}\n\nThe result above is what is referred to as \\textbf{Varadhan Theorem} \\cite{varadhan1966,touchette2009}. The contribution of Varadhan was to prove this result for a large class of RVs, which includes not just IID sample means but also random vectors and even random functions, and to rigorously handle all the heuristic approximations used above. As such, Varadhan's Theorem can be considered as a rigorous and general expression of the Laplace principle.\n\nTo connect Varadhan's Theorem with the GE Theorem, consider the special case $f(s)=ks$ with $k\\in\\mathbb{R}$.\\footnote{Varadhan's Theorem holds in its original form for bounded functions $f$, but another stronger version can be applied to unbounded functions and in particular to $f(x)=kx$; see Theorem 1.3.4 of \\cite{dupuis1997}.} Then Eq.~(\\ref{eqvaradhan1}) becomes\n\\begin{equation}\n\\lambda(k)=\\sup_{s\\in\\mathbb{R}}\\{ks-I(s)\\},\n\\end{equation}\nwhere $\\lambda(k)$ is the same function as the one defined in Eq.~(\\ref{eqscgf1}). Thus we see that if $S_n$ satisfies an LDP with rate function $I(s)$, then the SCGF $\\lambda(k)$ of $S_n$ is the Legendre-Fenchel transform of $I(s)$. This in a sense is the inverse of the GE Theorem, so an obvious question is, can we invert the equation above to obtain $I(s)$ in terms of $\\lambda(k)$? The answer provided by the GE Theorem is that, a sufficient condition for $I(s)$ to be the Legendre-Fenchel transform of $\\lambda(k)$ is for the latter function to be differentiable.\\footnote{The differentiability condition, though sufficient, is not necessary: $I(s)$ in some cases can be the Legendre-Fenchel transform of $\\lambda(k)$ even when the latter is not differentiable; see Sec.~4.4 of \\cite{touchette2009}.}\n\nThis is the closest that we can get to the GE Theorem without proving it. It is important to note, to fully appreciate the importance of that theorem, that it is actually more just than an inversion of Varadhan's Theorem because the existence of $\\lambda(k)$ implies the existence of an LDP for $S_n$. In our heuristic inversion of Varadhan's Theorem we assumed that an LDP exists.\n\n\\subsection{The contraction principle}\n\nWe mentioned before that LDPs can be derived by a contraction method. The basis of this method is the following: let $A_n$ be a random variable $A_n$ known to have an LDP with rate function $I_A(a)$, and consider another random variable $B_n$ which is a function of the first $B_n=f(A_n)$. We want to know whether $B_n$ satisfies an LDP and, if so, to find its rate function.\n\nTo find the answer, write the pdf of $B_n$ in terms of the pdf of $A_n$:\n\\begin{equation}\np_{B_n}(b)=\\int_{\\{a: f(a)=b\\}} p_{A_n}(a)\\, \\mathrm{d} a\n\\end{equation}\nand use the LDP for $A_n$ to write\n\\begin{equation}\np_{B_n}(b)\\approx \\int_{\\{a: f(a)=b\\}} \\mathrm{e}^{-nI_A(a)}\\, \\mathrm{d} a.\n\\end{equation}\nThen apply the Laplace principle to approximate the integral above by its largest term, which corresponds to the minimum of $I(a)$ for  $a$ such that $b=f(a)$. Therefore,\n\\begin{equation}\np_{B_n}(b)\\approx \\exp\\left(-n{\\inf_{\\{a: f(a)=b\\}} I_A(a)}\\right).\n\\end{equation}\nThis shows that $p(B_n)$ also satisfies an LDP with rate function $I_B(b)$ given by\n\\begin{equation}\nI_B(b)=\\inf_{\\{a:f(a)=b\\}} I_A(a).\n\\end{equation}\n\nThis formula is called the \\textbf{contraction principle} because $f$ can be many-to-one, i.e., there might be many $a$'s such that $b=f(a)$, in which case we are ``contracting'' information about the rate function of $A_n$ down to $B_n$. In physical terms, this formula is interpreted by saying that an improbable fluctuation\\footnote{In statistical physics, a deviation of a random variable away from its typical value is referred to as a \\textbf{fluctuation}.} of $B_n$ is brought about by the most probable of all improbable fluctuations of $A_n$.\n\nThe contraction principle has many applications in statistical physics. In particular, the maximum entropy and minimum free energy principles, which are used to find equilibrium states in the microcanonical and canonical ensembles, respectively, can be seen as deriving from the contraction principle (see Sec.~5 of \\cite{touchette2009}).\n \n\n\\subsection{From small to large deviations}\n\nAn LDP for a random variable, say $S_n$ again, gives us a lot of information about its pdf. First, we know that $p(S_n)$ concentrates on certain points corresponding to the zeros of the rate function $I(s)$. These points correspond to the most probable or \\textbf{typical values} of $S_n$ in the limit $n\\rightarrow\\infty$ and can be shown to be related mathematically to the \\textbf{Law of Large Numbers} (LLN for short). In fact, an LDP always implies some form of LLN (see Sec.~3.5.7 of \\cite{touchette2009}).\n\nOften it is not enough to know that $S_n$ converges in probability to some values; we may also want to determine the likelihood that $S_n$ takes a value away but close to its typical value(s). Consider one such typical value $s^*$ and assume that $I(s)$ admits a Taylor series around $s^*$:\n\\begin{equation}\nI(s)=I(s^*)+I'(s^*) (s-s^*)+\\frac{I''(s^*)}{2} (s-s^*)^2+\\cdots.\n\\end{equation}\nSince $s^*$ must correspond to a zero of $I(s)$, the first two terms in this series vanish, and so we are left with the prediction that the \\textbf{small deviations} of $S_n$ around its typical value are Gaussian-distributed:\n\\begin{equation}\np_{S_n}(s)\\approx \\mathrm{e}^{- n I''(s^*) (s-s^*)^2\/2}.\n\\end{equation}\n\nIn this sense, large deviation theory contains the \\textbf{Central Limit Theorem} (see Sec.~3.5.8 of \\cite{touchette2009}). At the same time, large deviation theory can be seen as an extension of the Central Limit Theorem because it gives information not only about the small deviations of $S_n$, but also about its \\textbf{large deviations} far away from its typical value(s); hence the name of the theory.\\footnote{A large deviation is also called a large fluctuation or a \\textbf{rare and extreme event}.}\n\n\\subsection{Exercises}\n\\label{secex1}\n\n\\begin{exercise}\n\\item \\label{excf1}\\exdiff{10} (Generating functions) Let $A_n$ be a sum of $n$ IID RVs:\n\\begin{equation}\nA_n=\\sum_{i=1}^n X_i.\n\\end{equation} \nShow that the generating function of $A_n$, defined as\n\\begin{equation}\nW_{A_n}(k)=E[\\mathrm{e}^{k A_n}],\n\\end{equation}\nsatisfies the following factorization property:\n\\begin{equation}\nW_{A_n}(k)=\\prod_{i=1}^n W_{X_i}(k)=W_{X_i}(k)^n,\n\\end{equation}\n$W_{X_i}(k)$ being the generating function of $X_i$. \n\n\\item \\label{exgauss1}\\exdiff{12} (Gaussian sample mean) Find the expression of $W_X(k)$ for $X$ distributed according to a Gaussian pdf with mean $\\mu$ and variance $\\sigma^2$. From this result, find $W_{S_n}(k)$ following the previous exercise and $p(S_n)$ by inverse Laplace transform. \n\n\\item \\label{exexp1}\\exdiff{15} (Exponential sample mean) Repeat the previous exercise for the exponential pdf shown in Eq.~(\\ref{eqexpd1}). Obtain from your result the approximation shown in (\\ref{eqldexp1}).\n\n\\item \\label{exbs1}\\exdiff{10} (Bernoulli sample mean) Show that the probability distribution of the Bernoulli sample mean is the binomial distribution:\n\\begin{equation}\nP_{S_n}(s)=\\frac{n!}{(sn)![(1-s)n]!}\\ \\alpha^{sn}(1-\\alpha)^{(1-s)n}.\n\\end{equation}\nUse Stirling's approximation to put this result in the form of (\\ref{eqldbern1}) with $I(s)$ given by Eq.~(\\ref{eqbern1}).\n\n\\item \\label{exmulti1}\\exdiff{12} (Multinomial distribution) Repeat the previous exercise for IID RVs taking values in the set $\\{0,1,\\ldots, q-1\\}$ instead of $\\{0,1\\}$. Use Stirling's approximation to arrive at an exponential approximation similar to (\\ref{eqldbern1}). (See solution in \\cite{ellis1999}.)\n\n\\item \\label{exscgf1}\\exdiff{10} (SCGF at the origin) Let $\\lambda(k)$ be the SCGF of an IID sample mean $S_n$. Prove the following properties of $\\lambda(k)$ at $k=0$:\n\\begin{itemize}\n\\item~$\\lambda(0)=0$\n\\item~$\\lambda'(0)=E[S_n]=E[X_i]$\n\\item~$\\lambda''(0)=\\text{var}(X_i)$\n\\end{itemize}\nWhich of these properties remain valid if $S_n$ is not an IID sample mean, but is some general RV?\n\n\\item \\label{excon1}\\exdiff{12} (Convexity of SCGFs) Show that $\\lambda(k)$ is a convex function of $k$.\n\n\\item \\label{exlf1}\\exdiff{12} (Legendre transform) Show that, when $\\lambda(k)$ is everywhere differentiable and is strictly convex (i.e., has no affine or linear parts), the Legendre-Fenchel transform shown in Eq.~(\\ref{eqlf1}) reduces to the \\textbf{Legendre transform}:\n\\begin{equation}\nI(s)=k(s) s-\\lambda(k(s)),\n\\end{equation}\nwhere $k(s)$ is the unique of $\\lambda'(k)=s$. Explain why the latter equation has a unique root.\n\n\\item \\label{excon2}\\exdiff{20} (Convexity of rate functions) Prove that rate functions obtained from the GE Theorem are strictly convex. \n\n\\item \\label{exvar1}\\exdiff{12} (Varadhan's Theorem) Verify Varadhan's Theorem for the Gaussian and exponential sample means by explicitly calculating the Legendre-Fenchel transform of $I(s)$ obtained for these sample means. Compare your results with the expressions of $\\lambda(k)$ obtained from its definition.\n\n\\end{exercise}\n\n\\section{Applications of large deviation theory}\n\\label{secapp}\n\nWe study in this section examples of random variables and stochastic processes with interesting large deviation properties. We start by revisiting the simplest application of large deviation theory, namely, IID sample means, and then move to Markov processes, which are often used for modeling physical and man-made systems. More information about applications of large deviation theory in statistical physics can be found in Secs~5 and 6 of \\cite{touchette2009}, as well as in the contribution of Engel contained in this volume.\n\n\\subsection{IID sample means}\n\\label{secsanov}\n\nConsider again the sample mean\n\\begin{equation}\nS_n=\\frac{1}{n}\\sum_{i=1}^nX_i\n\\end{equation}\ninvolving $n$ real IID variables $X_1,\\ldots,X_n$ with common pdf $p(x)$. To determine whether $S_n$ satisfies an LDP and obtain its rate function, we can follow the GE Theorem and calculate the SCGF $\\lambda(k)$ defined in Eq.~(\\ref{eqscgf1}). Because of the IID nature of $S_n$, $\\lambda(k)$ takes a simple form:\n\\begin{equation}\n\\lambda(k)=\\ln E[\\mathrm{e}^{kX}].\n\\end{equation}\nIn this expression, $X$ is any of the $X_i$'s (remember they are IID). Thus to obtain an LDP for $S_n$, we only have to calculate the log-generating function of $X$, check that it is differentiable and, if so, calculate its Legendre-Fenchel transform (or in this case, its Legendre transform; see Exercise~\\ref{exlf1}).  Exercise~\\ref{exiid1} considers different distributions $p(x)$ for which this calculation can be carried out, including the Gaussian, exponential and Bernoulli distributions studied before.\n\nAs an extra exercise, let us attempt to find the rate function of a special sample mean defined by\n\\begin{equation}\nL_{n,j}=\\frac{1}{n}\\sum_{i=1}^n \\delta_{X_i,j}\n\\label{eqempvec1}\n\\end{equation}\nfor a sequence $X_1,\\ldots,X_n$ of $n$ discrete IID RVs with finite state space $\\mathcal{X}=\\{1,2,\\ldots,q\\}$. This sample mean is called the \\textbf{empirical distribution} of the $X_i$'s as it ``counts'' the number of times the value or symbol $j\\in\\mathcal{X}$ appears in a given realization of $X_1,\\ldots,X_n$. This number is normalized by the total number $n$ of RVs, so what we have is the \\textbf{empirical frequency} for the appearance of the symbol $j$ in realizations of $X_1,\\ldots,X_n$.\n\nThe values of $L_{n,j}$ for all $j\\in\\mathcal{X}$ can be put into a vector $\\mathbf{L}_n$ called the \\textbf{empirical vector}. For the Bernoulli sample mean, for example, $\\mathcal{X}=\\{0,1\\}$ and so $\\mathbf{L}_n$ is a two-dimensional vector $\\mathbf{L}_n=(L_{n,0},L_{n,1})$ containing the empirical frequency of $0$'s and $1$'s appearing in a realization $x_1,\\ldots,x_n$ of $X_1,\\ldots,X_n$:\n\\begin{equation}\nL_{n,0}=\\frac{\\#\\text{ 0's in } x_1,\\ldots,x_n}{n},\\quad L_{n,1}=\\frac{\\#\\text{ 1's in } x_1,\\ldots,x_n}{n}.\n\\end{equation}\n\nTo find the rate function associated with the random vector $\\mathbf{L}_n$, we apply the GE Theorem but adapt it to the case of random vectors by replacing $k$ in $\\lambda(k)$ by a vector $\\mathbf{k}$ having the same dimension as $\\mathbf{L}_n$. The calculation of $\\lambda(\\mathbf{k})$ is left as an exercise (see Exercise~\\ref{exsanov1}). The result is\n\\begin{equation}\n\\lambda(\\mathbf{k})=\\ln \\sum_{j\\in\\mathcal{X}} P_j\\, \\mathrm{e}^{k_j}\n\\label{eqsanov1}\n\\end{equation}\nwhere $P_j=P(X_i=j)$. It is easily checked that $\\lambda(\\mathbf{k})$ is differentiable in the vector sense, so we can use the GE Theorem to conclude that $\\mathbf{L}_n$ satisfies an LDP, which we write as\n\\begin{equation}\np(\\mathbf{L}_n={\\bm\\mu})\\approx \\mathrm{e}^{-n I({\\bm\\mu})},\n\\end{equation}\nwith a vector rate function $I({\\bm\\mu})$ given by the Legendre-Fenchel transform of $\\lambda(\\mathbf{k})$. The calculation of this transform is the subject of Exercise~\\ref{exsanov1}. The final result is\n\\begin{equation}\nI({\\bm\\mu})=\\sum_{j\\in\\mathcal{X}}\\mu_j\\ln\\frac{\\mu_j}{P_j}.\n\\label{eqsanov2}\n\\end{equation}\n\nThis rate function is called the \\textbf{relative entropy} or \\textbf{Kullback-Leibler divergence} \\cite{cover1991}. The full LDP for $\\mathbf{L}_n$ is referred to as \\textbf{Sanov's Theorem} (see \\cite{sanov1961} and Sec.~4.2 of \\cite{touchette2009}). \n\nIt can be checked that $I({\\bm\\mu})$ is a convex function of $\\bm\\mu$ and that it has a unique minimum and zero at ${\\bm\\mu}=\\mathbf{P}$, i.e., $\\mu_j=P_j$ for all $j\\in\\mathcal{X}$. As seen before, this property is an expression of the LLN, which says here that the empirical frequencies $L_{n,j}$ converge to the  probabilities $P_j$ as $n\\rightarrow\\infty$. The LDP goes beyond this result by describing the fluctuations of $\\mathbf{L}_n$ around the typical value ${\\bm\\mu}=\\mathbf{P}$.\n\n\\subsection{Markov chains}\n\\label{subsecmc1}\n\nInstead of assuming that the sample mean $S_n$ arises from a sum of IID RVs $X_1,\\ldots,X_n$, consider the case where the $X_i$'s form a \\textbf{Markov chain}. This means that the joint pdf $p(X_1,\\ldots,X_n)$ has the form\n\\begin{equation}\np(X_1,\\ldots,X_n)=p(X_1)\\prod_{i=1}^{n-1} \\pi(X_{i+1}|X_i),\n\\label{eqmark1}\n\\end{equation}\nwhere $p(X_i)$ is some initial pdf for $X_1$ and $\\pi(X_{i+1}|X_i)$ is the \\textbf{transition probability density} that $X_{i+1}$ follows $X_i$ in the Markov sequence $X_1\\rightarrow X_2\\rightarrow\\cdots\\rightarrow X_n$ (see \\cite{grimmett2001} for background information on Markov chains). \n\nThe GE Theorem can still be applied in this case, but the expression of $\\lambda(k)$ is more complicated. Skipping the details of the calculation (see Sec.~4.3 of \\cite{touchette2009}), we arrive at the following result: provided that the Markov chain is homogeneous and ergodic (see \\cite{grimmett2001} for a definition of ergodicity), the SCGF of $S_n$ is given by\n\\begin{equation}\n\\lambda(k)=\\ln \\zeta(\\tilde\\Pi_k),\n\\label{eqscgfmc1}\n\\end{equation}\nwhere $\\zeta(\\tilde\\Pi_k)$ is the \\textbf{dominant eigenvalue} (i.e., with largest magnitude) of the matrix $\\tilde\\Pi_k$ whose elements $\\tilde\\pi_k(x,x')$ are given by $\\tilde\\pi_k(x,x')=\\pi(x'|x)\\mathrm{e}^{k x'}$. We call the matrix $\\tilde\\Pi_k$ the \\textbf{tilted matrix} associated with $S_n$. If the Markov chain is finite,  it can be proved furthermore that $\\lambda(k)$ is analytic and so differentiable. From the GE Theorem, we then conclude that $S_n$ has an LDP with rate function\n\\begin{equation}\nI(s)=\\sup_{k\\in\\mathbb{R}}\\{ks-\\ln \\zeta(\\tilde\\Pi_k)\\}.\n\\end{equation}\nSome care must be taken if the Markov chain has an infinite number of states. In this case, $\\lambda(k)$ is not necessarily analytic and may not even exist (see, e.g., \\cite{harris2006,harris2007,rakos2008} for illustrations).\n\nAn application of the above result for a simple Bernoulli Markov chain is considered in Exercise~\\ref{exbmc1}. In Exercises~\\ref{excurr1} and \\ref{excurr2}, the case of random variables having the form\n\\begin{equation}\nQ_n=\\frac{1}{n}\\sum_{i=1}^{n-1} q(x_i,x_{i+1})\n\\label{eqcurr1}\n\\end{equation}\nis covered. This type of RV arises in statistical physics when studying currents in stochastic models of interacting particles (see \\cite{harris2007}). The tilted matrix $\\tilde\\Pi_k$ associated with $Q_n$ has the form $\\tilde\\pi_{k}(x,x')=\\pi(x'|x)\\mathrm{e}^{k q(x,x')}$.\n\n\n\\subsection{Continuous-time Markov processes}\n\nThe preceding subsection can be generalized to ergodic Markov processes evolving continuously in time by taking a continuous time limit of discrete-time ergodic Markov chains. \n\nTo illustrate this limit, consider an ergodic continuous-time process described by the state $X(t)$ for $0\\leq t \\leq T$. For an infinitesimal \\textbf{time-step} $\\Delta t$, time-discretize this process into a sequence of $n+1$ states $X_0,X_1,\\ldots,X_n$ with $n=T\/\\Delta t$ and $X_i=X(i\\Delta t)$, $i=0,1,\\ldots,n$. The sequence $X_0,X_1,\\ldots,X_n$ is an ergodic Markov chain with infinitesimal transition matrix $\\Pi(\\Delta t)$ given by\n\\begin{equation}\n\\Pi(\\Delta t)=\\pi (x(t+\\Delta t)|x(t))=\\mathrm{e}^{G\\Delta t}=I+G \\Delta t +o(\\Delta t),\n\\end{equation}\nwhere $I$ is the identity matrix and $G$ the \\textbf{generator} of $X(t)$. With this discretization, it is now possible to study LDPs for processes involving $X(t)$ by studying these processes in discrete time at the level of the Markov chain $X_0,X_1,\\ldots,X_n$, and transfer these LDPs into continuous time by taking the limits $\\Delta t\\rightarrow 0$, $n\\rightarrow\\infty$. \n\nAs a general application of this procedure, consider a so-called \\textbf{additive process} defined by \n\\begin{equation}\nS_T=\\frac{1}{T}\\int_0^T X(t)\\, \\mathrm{d} t.\n\\end{equation}\nThe discrete-time version of this RV is the sample mean\n\\begin{equation}\nS_n=\\frac{1}{n\\Delta t}\\sum_{i=0}^n X_i\\, \\Delta t=\\frac{1}{n}\\sum_{i=0}^n X_i.\n\\end{equation}\nFrom this association, we find that the SCGF of $S_T$, defined by the limit\n\\begin{equation}\n\\lambda(k)=\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\ln E[\\mathrm{e}^{Tk S_T}],\\qquad k\\in\\mathbb{R},\n\\label{eqscgfct1}\n\\end{equation}\nis given by\n\\begin{equation}\n\\lambda(k)=\\lim_{\\Delta t\\rightarrow 0}\\lim_{n\\rightarrow\\infty}\\frac{1}{n \\Delta t}\\ln E[\\mathrm{e}^{k \\Delta t\\sum_{i=0}^n X_i}]\n\\end{equation}\nat the level of the discrete-time Markov chain. \n\nAccording to the previous subsection, the limit above reduces to $\\ln\\zeta(\\tilde\\Pi_{k}(\\Delta t))$, where $\\tilde\\Pi_{k}(\\Delta t)$ is the matrix (or operator) corresponding to\n\\begin{eqnarray}\n\\tilde\\Pi_{k}(\\Delta t) &=& \\mathrm{e}^{k\\Delta t x'} \\Pi(\\Delta t)\\nonumber\\\\\n&=& \\big(1+kx'\\Delta t+o(\\Delta t)\\big)\\big(I+G'\\Delta t+o(\\Delta t)\\big)\\nonumber\\\\\n&=& I+\\tilde G_k \\Delta t+o(\\Delta t),\n\\label{eqttm1}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\tilde G_k=G+k x' \\delta_{x,x'}.\n\\end{equation} \nFor the continuous-time process $X(t)$, we must therefore have\n\\begin{equation}\n\\lambda(k)=\\zeta(\\tilde G_k),\n\\label{eqscgfmc2}\n\\end{equation}\nwhere $\\zeta(\\tilde G_k)$ is the dominant eigenvalue of the \\textbf{tilted generator} $\\tilde G_k$.\\footnote{As for Markov chains, this result holds for continuous, ergodic processes with finite state-space. For infinite-state or continuous-space processes, a similar result holds provided $G_k$ has an isolated dominant eigenvalue.} Note that the reason why the logarithmic does not appear in the expression of $\\lambda(k)$ for the continuous-time process\\footnote{Compare with Eq.~(\\ref{eqscgfmc1}).} is because $\\zeta$ is now the dominant eigenvalue of the generator of the infinitesimal transition matrix, which is itself the exponential of the generator.\n\nWith the knowledge of $\\lambda(k)$, we can obtain an LDP for $S_T$ using the GE Theorem: If the dominant eigenvalue is differentiable in $k$, then $S_T$ satisfies an LDP in the long-time limit, $T\\rightarrow\\infty$, which we write as $p_{S_T}(s)\\approx \\mathrm{e}^{-T I(s)}$, where $I(s)$ is the Legendre-Fenchel transform $\\lambda(k)$. Some applications of this result are presented in Exercises~\\ref{exrt1} and \\ref{excurr2}. Note that in the case of current-type processes having the form of Eq.~(\\ref{eqcurr1}), the tilted generator is not simply given by $\\tilde G_k=G+kx'\\delta_{x,x'}$; see Exercise~\\ref{excurr2}.\n\n\\subsection{Paths large deviations}\n\\label{subsecpathldp}\n\nWe complete our tour of mathematical applications of large deviation theory by studying a different large deviation limit, namely, the low-noise limit of the following \\textbf{stochastic differential equation} (SDE for short):\n\\begin{equation}\n\\dot x(t)=f(x(t))+\\sqrt{\\varepsilon}\\, \\xi(t),\\qquad x(0)=0,\n\\label{eqsde1}\n\\end{equation} \nwhich involves a \\textbf{force} $f(x)$ and a \\textbf{Gaussian white noise} $\\xi(t)$ with the properties $E[\\xi(t)]=0$ and $E[\\xi(t')\\xi(t)]=\\delta(t'-t)$; see Chap.~XVI of \\cite{kampen1992} for background information on SDEs.\n\nWe are interested in studying for this process the pdf of a given \\textbf{random path} $\\{x(t)\\}_{t=0}^T$ of duration $T$ in the limit where the \\textbf{noise power} $\\varepsilon$ vanishes. This abstract pdf can be defined heuristically using path integral methods (see Sec.~6.1 of \\cite{touchette2009}). In the following, we denote this pdf by the functional notation $p[x]$ as a shorthand for $p(\\{x(t)\\}_{t=0}^T)$.\n\nThe idea behind seeing the low-noise limit as a large deviation limit is that, as $\\varepsilon\\rightarrow 0$, the random path arising from the SDE above should converge in probability to the \\textbf{deterministic path} $x(t)$ solving the ordinary differential equation\n\\begin{equation}\n\\dot x(t)=f(x(t)),\\qquad x(0)=0.\n\\label{eqdet1}\n\\end{equation}\nThis convergence is a LLN-type result, and so in the spirit of large deviation theory we are interested in quantifying the likelihood that a random path $\\{x(t)\\}_{t=0}^T$ ventures away from the deterministic path in the limit $\\varepsilon\\rightarrow 0$. The functional LDP that characterizes these path fluctuations has the form\n\\begin{equation}\np[x]\\approx \\mathrm{e}^{-I[x]\/\\varepsilon},\n\\end{equation}\nwhere\n\\begin{equation}\nI[x]=\\int_0^T [\\dot x(t)-f(x(t))]^2\\, \\mathrm{d} t.\n\\label{eqldpf1}\n\\end{equation}\nSee \\cite{freidlin1984} and Sec.~6.1 of \\cite{touchette2009} for historical sources on this LDP.\n\nThe rate functional $I[x]$ is called the \\textbf{action}, \\textbf{Lagrangian} or \\textbf{entropy} of the path $\\{x(t)\\}_{t=0}^T$. The names ``action'' and ``Lagrangian'' come from an analogy with the action of quantum trajectories in the path integral approach of quantum mechanics (see Sec.~6.1 of \\cite{touchette2009}). There is also a close analogy between the low-noise limit of SDEs and the semi-classical or WKB approximation of quantum mechanics \\cite{touchette2009}. \n\nThe path LDP above can be generalized to higher-dimensional SDEs as well as SDEs involving state-dependent noise and correlated noises (see Sec.~6.1 of \\cite{touchette2009}). In all cases, the minimum and zero of the rate functional is the trajectory of the deterministic system obtained in the zero-noise limit. This is verified for the 1D system considered above: $I[x]\\geq 0$ for all trajectories and $I[x]=0$ for the unique trajectory solving Eq.~(\\ref{eqdet1}).\n\nFunctional LDPs are the most refined LDPs that can be derived for SDEs as they characterize the probability of complete trajectories. Other ``coarser'' LDPs can be derived from these by contraction. For example, we might be interested to determine the pdf $p(x,T)$ of the state $x(T)$ reached after a time $T$. The contraction in this case is obvious: $p(x,T)$ must have the large deviation form $p(x,T)\\approx \\mathrm{e}^{-V(x,T)\/\\varepsilon}$ with\n\\begin{equation}\nV(x,T)=\\inf_{x(t):x(0)=0,x(T)=x} I[x].\n\\end{equation}\nThat is, the probability of reaching $x(T)=x$ from $x(0)=0$ is determined by the path connecting these two endpoints having the largest probability. We call this path the \\textbf{optimal path}, \\textbf{maximum likelihood path} or \\textbf{instanton}. Using variational calculus techniques, often used in classical mechanics, it can be proved that this path satisfies an Euler-Lagrange-type equation as well as a Hamilton-type equation (see Sec.~6.1 of \\cite{touchette2009}). Applications of these equations are covered in Exercises~\\ref{exoup1}--\\ref{expol1}.\n\nQuantities similar to the additive process $S_T$ considered in the previous subsection can also be defined for SDEs (see Exercises~\\ref{exrt1} and \\ref{excurr1}). An interesting aspect of these quantities is that their LDPs can involve the noise power $\\varepsilon$ if the low-noise limit is taken, as well as the integration time $T$, which arises because of the additive (in the sense of sample mean) nature of these quantities. In this case, the limit $T\\rightarrow 0$ must be taken before the low-noise limit $\\varepsilon\\rightarrow 0$. If $\\varepsilon\\rightarrow 0$ is taken first, the system considered is no longer random, which means that there can be no LDP in time.\n\n\\subsection{Physical applications}\n\nSome applications of the large deviation results seen so far are covered in the contribution of Engel to this volume. The following list gives an indication of these applications and some key references for learning more about them. A more complete presentation of the applications of large deviation theory in statistical physics can be found in Secs.~5 and 6 of ~\\cite{touchette2009}.\n\\begin{itemize}\n\\item \\textbf{Equilibrium systems:} Equilibrium statistical mechanics, as embodied by the ensemble theory of Boltzmann and Gibbs, can be seen with hindsight as a large deviation theory of many-body systems at equilibrium. This becomes evident by realizing that the thermodynamic limit is a large deviation limit, that the entropy is the equivalent of a rate function and the free energy the equivalent of a SCGF. Moreover, the Legendre transform of thermodynamics connecting the entropy and free energy is nothing but the Legendre-Fenchel transform connecting the rate function and the SCGF in the GE Theorem and in Varadhan's Theorem. For a more complete explanation of these analogies, and historical sources on the development of large deviation theory in relation to equilibrium statistical mechanics, see the book by Ellis \\cite{ellis1985} and Sec.~3.7 of \\cite{touchette2009}. \n \n\\item \\textbf{Chaotic systems and multifractals:} The so-called \\textbf{thermodynamic formalism} of dynamical systems, developed by Ruelle \\cite{ruelle2004} and others, can also be re-interpreted with hindsight as an application of large deviation theory for the study of chaotic systems.\\footnote{There is a reference to this connection in a note of Ruelle's popular book, Chance and Chaos \\cite{ruelle1993} (see~Note 2 of Chap.~19).} There are two quantities in this theory playing the role of the SCGF, namely, the \\textbf{topological pressure} and the \\textbf{structure function}. The Legendre transform appearing in this theory is also an analogue of the one encountered in large deviation theory. References to these analogies can be found in Secs.~7.1 and 7.2 of \\cite{touchette2009}.\n\n\\item \\textbf{Nonequilibrium systems:} Large deviation theory is becoming the standard formalism used in studies of nonequilibrium systems modelled by SDEs and Markov processes in general. In fact, large deviation theory is currently experiencing a sort of revival in physics and mathematics as a result of its growing application in nonequilibrium statistical mechanics. Many LDPs have been derived in this context: LDPs for the current fluctuations or the occupation of interacting particle models, such as the exclusion process, the zero-range process (see \\cite{harris2005}) and their many variants \\cite{spohn1991}, as well as LDPs for work-related and entropy-production-related quantities for nonequilibrium systems modelled with SDEs \\cite{chetrite2008,chetrite2007a}. Good entry points in this vast field of research are \\cite{derrida2007} and \\cite{harris2007}. \n\n\\item \\textbf{Fluctuation relations:} Several LDPs have come to be studied in recent years under the name of fluctuation relations. To illustrate these results, consider the additive process $S_T$ considered earlier and assume that this process admits an LDP with rate function $I(s)$. In many cases, it is interesting not only to know that $S_T$ admits an LDP but to know how probable positive fluctuations of $S_T$ are compared to negative fluctuations. For this purpose, it is common to study the ratio\n\\begin{equation}\n\\frac{p_{S_T}(s)}{p_{S_T}(-s)},\n\\end{equation}\nwhich reduces to\n\\begin{equation}\n\\frac{p_{S_T}(s)}{p_{S_T}(-s)}\\approx \\mathrm{e}^{T[I(-s)-I(s)]}\n\\end{equation}\nif we assume an LDP for $S_T$. In many cases, the difference $I(-s)-I(s)$ is linear in $s$, and one then says that $S_T$ satisfies a \\textbf{conventional fluctuation relation}, whereas if it nonlinear in $s$, then one says that $S_T$ satisfies an \\textbf{extended fluctuation relation}. Other types of fluctuation relations have come to be defined in addition to these; for more information, the reader is referred to Sec.~6.3 of \\cite{touchette2009}. A list of the many physical systems for which fluctuation relations have been derived or observed can also be found in this reference.\n\\end{itemize}\n\n\\subsection{Exercises}\n\\label{secex2}\n\n\\begin{exercise}\n\\item \\label{exiid1}\\exdiff{12} (IID sample means) Use the GE Theorem to find the rate function of the IID sample mean $S_n$ for the following probability distribution and densities:\n\\begin{itemize}\n\\item~Gaussian:\n\\begin{equation}\np(x)=\\frac{1}{\\sqrt{2\\pi \\sigma^2}} \\mathrm{e}^{-(x-\\mu)^2\/(2\\sigma^2)},\\qquad x\\in\\mathbb{R}.\n\\end{equation}\n\\item~Bernoulli: $X_i\\in\\{0,1\\}$, $P(X_i=0)=1-\\alpha$, $P(X_i=1)=\\alpha$.\n\\item~Exponential: \n\\begin{equation}\np(x)=\\frac{1}{\\mu} \\mathrm{e}^{-x\/\\mu},\\qquad x\\in [0,\\infty),\\quad \\mu>0.\n\\end{equation}\n\\item~Uniform:\n\\begin{equation}\np(x)=\\left\\{\n\\begin{array}{lll}\n\\frac{1}{a} & & x\\in[0,a]\\\\\n0 & & \\text{otherwise}.\n\\end{array}\n\\right.\n\\end{equation}\n\\item~Cauchy:\n\\begin{equation}\np(x)=\\frac{\\sigma}{\\pi(x^2+\\sigma^2)},\\qquad x\\in\\mathbb{R},\\quad \\sigma>0.\n\\end{equation}\n\\end{itemize}\n\n\\item \\label{exnc1}\\exdiff{15} (Nonconvex rate functions) Rate functions obtained from the GE Theorem are necessarily convex (strictly convex, in fact; see Exercise~\\ref{excon2}), but rate functions in general need not be convex. Consider, as an example, the process defined by\n\\begin{equation}\nS_n=Y+\\frac{1}{n}\\sum_{i=1}^n X_i,\n\\end{equation}\nwhere $Y=\\pm1$ with probability $\\frac{1}{2}$ and the $X_i$'s are Gaussian IID RVs. Find the rate function for $S_n$ assuming that $Y$ is independent of the $X_i$'s. Then find the corresponding SCGF. What is the relation between the Legendre-Fenchel transform of $\\lambda(k)$ and the rate function $I(s)$? How is the nonconvexity of $I(s)$ related to the differentiability of $\\lambda(k)$? (See Example~4.7 of \\cite{touchette2009} for the solution.)\n\n\\item \\label{exexp2}\\exdiff{15} (Exponential mixture) Repeat the previous exercise by replacing the Bernoulli $Y$ with $Y=Z\/n$, where $Z$ is an exponential RV with mean $1$. (See Example~4.8 of \\cite{touchette2009} for the solution.)\n\n\\item \\label{exprod1}\\exdiff{12} (Product process) Find the rate function of \n\\begin{equation}\nZ_n=\\left(\\prod_{i=1}^n X_i\\right)^{1\/n}\n\\end{equation}\nwhere $X_i\\in\\{1,2\\}$ with $p(X_i=1)=\\alpha$ and $p(X_i=2)=1-\\alpha$, $0<\\alpha<1$.\n\n\\item \\label{exsp1}\\exdiff{15} (Self-process) Consider a sequence of IID Gaussian RVs $X_1,\\ldots,X_n$. Find the rate function of the so-called \\textbf{self-process} defined by \n\\begin{equation}\nS_n=-\\frac{1}{n}\\ln p(X_1,\\ldots,X_n).\n\\end{equation}\n(See Sec.~4.5 of \\cite{touchette2009} for information about this process.) Repeat the problem for other choices of pdf for the RVs.\n\n\\item \\label{exild1}\\exdiff{20} (Iterated large deviations~\\cite{lowe1996}) Consider the sample mean\n\\begin{equation}\nS_{m,n}=\\frac{1}{m}\\sum_{i=1}^m X_{i}^{(n)}\n\\end{equation}\ninvolving $m$ IID copies of a random variable $X^{(n)}$. Show that if $X^{(n)}$ satisfies an LDP of the form\n\\begin{equation}\np_{X^{(n)}}(x)\\approx \\mathrm{e}^{-nI(x)},\n\\end{equation}\nthen $S_{m,n}$ satisfies an LDP having the form\n\\begin{equation}\np_{S_{m,n}}(s)\\approx \\mathrm{e}^{-mn I(x)}.\n\\end{equation}\n\n\\item \\label{exprw1}\\exdiff{40} (Persistent random walk) Use the result of the previous exercise to find the rate function of \n\\begin{equation}\nS_n=\\frac{1}{n}\\sum_{i=1}^n X_i,\n\\end{equation}\nwhere the $X_i$'s are independent Bernoulli RVs with non-identical distribution $P(X_i=0)=\\alpha^i$ and $P(X_i=1)=1-\\alpha^i$ with $0<\\alpha<1$.\n\n\\item \\label{exsanov1}\\exdiff{15} (Sanov's Theorem) Consider the empirical vector $\\mathbf{L}_n$ with components defined in Eq.~(\\ref{eqempvec1}). Derive the  expression found in Eq.~(\\ref{eqsanov1}) for\n\\begin{equation}\n\\lambda(\\mathbf{k})=\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\ln E[\\mathrm{e}^{n\\, \\mathbf{k}\\cdot\\mathbf{L}_n}],\n\\end{equation} \nwhere $\\mathbf{k}\\cdot\\mathbf{L}_n$ denotes the scalar product of $\\mathbf{k}$ and $\\mathbf{L}_n$. Then obtain the rate function found in (\\ref{eqsanov2}) by calculating the Legendre transform of $\\lambda(\\mathbf{k})$. Check explicitly that the rate function is convex and has a single minimum and zero.\n\n\\item \\label{excp1}\\exdiff{20} (Contraction principle) Repeat the first exercise of this section by using the contraction principle. That is, use the rate function of the empirical vector $\\mathbf{L}_n$ to obtain the rate function of $S_n$. What is the mapping from $\\mathbf{L}_n$ to $S_n$? Is this mapping many-to-one or one-to-one?\n \n\\item \\label{exbmc1}\\exdiff{17} (Bernoulli Markov chain) Consider the Bernoulli sample mean \n\\begin{equation}\nS_n=\\frac{1}{n}\\sum_{i=1}^n X_i,\\qquad X_i\\in\\{0,1\\}.\n\\end{equation}\nFind the expression of $\\lambda(k)$ and $I(s)$ for this process assuming that the $X_i$'s form a Markov process with symmetric transition matrix \n\\begin{equation}\n\\pi (x'|x)=\\left\\{\n\\begin{array}{lll}\n1-\\alpha & & x'=x\\\\\n\\alpha & & x'\\neq x\n\\end{array}\n\\right.\n\\end{equation}\nwith $0\\leq \\alpha\\leq 1$. (See Example~4.4 of \\cite{touchette2009} for the solution.)\n\n\\item \\label{exrt1}\\exdiff{20} (Residence time) Consider a Markov process with state space $\\mathcal{X}=\\{0,1\\}$ and generator\n\\begin{equation}\nG=\\left(\n\\begin{array}{cc}\n-\\alpha & \\beta\\\\\n\\alpha & -\\beta\n\\end{array}\n\\right).\n\\end{equation}\nFind for this process the rate function of the random variable\n\\begin{equation}\nL_T=\\frac{1}{T} \\int_0^T \\delta_{X(t),0}\\, \\mathrm{d} t,\n\\end{equation} \nwhich represents the fraction of time the state $X(t)$ of the Markov chain spends in the state $X=0$ over a period of time $T$.\n\n\\item \\label{excurr1}\\exdiff{20} (Current fluctuations in discrete time) Consider the Markov chain of Exercise~\\ref{exbmc1}. Find the rate function of the \\textbf{mean current} $Q_n$ defined as \n\\begin{equation}\nQ_n=\\frac{1}{n}\\sum_{i=1}^n f(x_{i+1},x_i),\n\\end{equation} \n\\begin{equation}\nf (x',x)=\\left\\{\n\\begin{array}{lll}\n0 & & x'=x\\\\\n1 & & x'\\neq x.\n\\end{array}\n\\right.\n\\end{equation}\n$Q_n$ represents the mean number of jumps between the states $0$ and $1$: at each transition of the Markov chain, the current is incremented by 1 whenever a jump between the states $0$ and $1$ occurs.\n\n\\item \\label{excurr2}\\exdiff{22} (Current fluctuations in continuous time) Repeat the previous exercise for the Markov process of Exercise~\\ref{exrt1} and the current $Q_T$ defined as\n\\begin{equation}\nQ_T=\\lim_{\\Delta t\\rightarrow 0}\\frac{1}{T}\\sum_{i=0}^{n-1} f(x_{i+1},x_i),\n\\end{equation}\nwhere $x_i=x(i\\Delta t)$ ad $f(x',x)$ as above, i.e., $f(x_{i+1},x_i)=1-\\delta_{x_i,x_{i+1}}$. Show for this process that the tilted generator is\n\\begin{equation}\n\\tilde G_k=\\left(\n\\begin{array}{cc}\n-\\alpha & \\beta\\,\\mathrm{e}^k\\\\\n\\alpha\\, \\mathrm{e}^k & -\\beta\n\\end{array}\n\\right).\n\\end{equation}\n\n\\item \\label{exoup1}\\exdiff{17} (Ornstein-Uhlenbeck process) Consider the following linear SDE:\n\\begin{equation}\n\\dot x(t)=-\\gamma x(t)+\\sqrt{\\varepsilon}\\, \\xi(t),\n\\end{equation}\noften referred to as the \\textbf{Langevin equation} or \\textbf{Ornstein-Uhlenbeck process}. Find for this SDE the solution of the optimal path connecting the initial point $x(0)=0$ with the fluctuation $x(T)=x$. From the solution, find the pdf $p(x,T)$ as well as the stationary pdf obtained in the limit $T\\rightarrow\\infty$. Assume $\\varepsilon\\rightarrow 0$ throughout but then verify that the results obtained are valid for all $\\varepsilon>0$.\n\n\\item \\label{excons1}\\exdiff{20} (Conservative system) Show by large deviation techniques that the stationary pdf of the SDE \n\\begin{equation}\n\\dot \\mathbf{x} (t)=-\\nabla U(\\mathbf{x}(t))+\\sqrt{\\varepsilon}\\, \\xi(t)\n\\end{equation}\nadmits the LDP $p(\\mathbf{x})\\approx \\mathrm{e}^{-V(\\mathbf{x})\/\\varepsilon}$ with rate function $V(\\mathbf{x})=2U(\\mathbf{x})$. The rate function $V(\\mathbf{x})$ is called the \\textbf{quasi-potential}.\n\n\\item \\label{extrans1}\\exdiff{25} (Transversal force) Show that the LDP found in the previous exercise also holds for the SDE\n\\begin{equation}\n\\dot \\mathbf{x} (t)=-\\nabla U(\\mathbf{x}(t))+\\mathbf{A}(\\mathbf{x})+\\sqrt{\\varepsilon}\\, \\xi(t)\n\\end{equation}\nif $\\nabla U\\cdot \\mathbf{A}=0$. (See Sec.~4.3 of \\cite{freidlin1984} for the solution.)\n\n\\item \\label{expol1}\\exdiff{25} (Noisy Van der Pol oscillator) The following set of SDEs describes a noisy version of the well-known Van der Pol equation:\n\\begin{eqnarray}\n\\dot x&=&v\\nonumber\\\\\n\\dot v&=&-x+v(\\alpha-x^2-v^2)+\\sqrt{\\varepsilon}\\, \\xi.\n\\end{eqnarray}\nThe parameter $\\alpha\\in\\mathbb{R}$ controls a bifurcation of the zero-noise system: for $\\alpha\\leq 0$, the system with $\\xi=0$ has a single attracting point at the origin, whereas for $\\alpha>0$, it has a stable limit cycle centered on the origin. Show that the stationary distribution of the noisy oscillator has the large deviation form $p(x,v)\\approx \\mathrm{e}^{-U(x,v)\/\\varepsilon}$ as $\\varepsilon\\rightarrow 0$ with rate function \n\\begin{equation}\nU(x,v)=-\\alpha (x^2+v^2)+\\frac{1}{2}(x^2+v^2)^2.\n\\end{equation}\nFind a different set of SDEs that has the same rate function. (See \\cite{graham1989} for the solution.)\n\n\\item \\label{exbmp1}\\exdiff{30} (Dragged Brownian particle \\cite{zon2003a}) The stochastic dynamics of a tiny glass bead immersed in water and pulled with laser tweezers at constant velocity can be modelled in the overdamped limit by the following reduced Langevin equation:\n\\begin{equation}\n\\dot x=-(x(t)-\\nu t)+\\sqrt{\\varepsilon}\\, \\xi(t),\n\\end{equation}\nwhere $x(t)$ is the position of the glass bead, $\\nu$ the pulling velocity, $\\xi(t)$ a Gaussian white noise modeling the influence of the surrounding fluid, and $\\varepsilon$ the noise power related to the temperature of the fluid. Show that the \\textbf{mean work} done by the laser as it pulls the glass bead over a time $T$, which is defined as\n\\begin{equation}\nW_T=-\\frac{\\nu}{T}\\int_0^T (x(t)-\\nu t)\\, \\mathrm{d} t,\n\\end{equation}\nsatisfies an LDP in the limit $T\\rightarrow\\infty$. Derive this LDP by assuming first that $\\varepsilon\\rightarrow 0$, and then derive the LDP without this assumption. Finally, determine whether $W_T$ satisfies a conventional or extended fluctuation relation, and study the limit $\\nu\\rightarrow 0$. (See Example~6.9 of \\cite{touchette2009} for more references on this model.)\n\\end{exercise}\n\n\\section{Numerical estimation of large deviation probabilities}\n\\label{secnum1}\n\nThe previous sections might give the false impression that rate functions can be calculated explicitly for many stochastic processes. In fact, exact and explicit expressions of rate functions can be found only in few simple cases. For most stochastic processes of scientific interest (e.g,, noisy nonlinear dynamical systems, chemical reactions, queues, etc.), we have to rely on analytical approximations and numerical methods to evaluate rate functions.\n\nThe rest of these notes attempts to give an overview of some of these numerical methods and to illustrate them with the simple processes (IID sample means and Markov processes) treated before. The goal is to give a flavor of the general ideas behind large deviation simulations rather than a complete survey, so only a subset of the many methods that have come to be devised for numerically estimating rate functions are covered in what follows. Other methods not treated here, such as the transition path method discussed by Dellago in this volume, are mentioned at the end of the next section.\n\n\\subsection{Direct sampling}\n\nThe problem addressed in this section is to obtain a numerical estimate of the pdf $p_{S_n}(s)$ for a real random variable $S_n$ satisfying an LDP, and to extract from this an estimate of the rate function $I(s)$.\\footnote{The literature on large deviation simulation usually considers the estimation of $P(S_n\\in A)$ for some set $A$ rather than the estimation of the whole pdf of $S_n$.} To be general, we take $S_n$ to be a function of $n$ RVs $X_1,\\ldots,X_n$, which at this point are not necessarily IID. To simplify the notation, we will use the shorthand $\\bm\\om=X_1,\\ldots,X_n$. Thus we write $S_n$ as the function $S_n(\\bm\\om)$ and denote by $p(\\bm\\om)$ the joint pdf of the $X_i$'s.\\footnote{For simplicity, we consider the $X_i$'s to be real RVs. The case of discrete RVs follow with slight changes of notations.}\n\nNumerically, we cannot of course obtain $p_{S_n}(s)$ or $I(s)$ for all $s\\in\\mathbb{R}$, but only for a finite number of values $s$, which we take for simplicity to be equally spaced with a small step $\\Delta s$. Following our discussion of the LDP, we thus attempt to estimate the coarse-grained pdf\n\\begin{equation}\np_{S_n}(s)=\\frac{P(S_n\\in [s,s+\\Delta s])}{\\Delta s}=\\frac{P(S_n\\in\\Delta_s)}{\\Delta s},\n\\label{eqcgpdf1}\n\\end{equation} \nwhere $\\Delta_s$ denotes the small interval $[s,s+\\Delta s]$ anchored at the value $s$.\\footnote{Though not explicitly noted, the coarse-grained pdf depends on $\\Delta s$; see Footnote~\\ref{fnldp1}.} \n\nTo construct this estimate, we follow the \\textbf{statistical sampling} or \\textbf{Monte Carlo method}, which we broke down into the following steps (see the contribution of Katzgraber in this volume for more details):\n\\begin{enumerate}\n\\item Generate a \\textbf{sample} $\\{\\bm\\om^{(j)}\\}_{j=1}^L$ of $L$ copies or realizations of the sequence $\\bm\\om$ from its pdf $p(\\bm\\om)$.\n\\item Obtain from this sample a sample $\\{s^{(j)}\\}_{j=1}^L$ of values or realizations for $S_n$:\n\\begin{equation}\ns^{(j)}=S_n(\\bm\\om^{(j)}),\\qquad j=1,\\ldots,L.\n\\end{equation}\n\\item Estimate $P(S_n\\in\\Delta_s)$ by calculating the sample mean\n\\begin{equation}\n\\hat P_L(\\Delta_s)=\\frac{1}{L}\\sum_{j=1}^L 1{\\hskip -2.9pt}\\hbox{I}_{\\Delta_s}(s^{(j)}),\n\\end{equation}\nwhere $1{\\hskip -2.9pt}\\hbox{I}_A(x)$ denotes the \\textbf{indicator function} for the set $A$, which is equal to $1$ if $x\\in A$ and $0$ otherwise. \n\n\\item Turn the \\textbf{estimator} $\\hat P_L(\\Delta_s)$ of the probability $P(S_n\\in\\Delta_s)$ into an estimator $\\hat p_L(s)$ of the probability density $p_{S_n}(s)$:\n\\begin{equation}\n\\hat p_L(s)=\\frac{\\hat P_L(\\Delta_s)}{\\Delta s}=\\frac{1}{L\\Delta s}\\sum_{j=1}^L 1{\\hskip -2.9pt}\\hbox{I}_{\\Delta_s}(s^{(j)}).\n\\end{equation}\n\\end{enumerate}\n\nThe result of these steps is illustrated in Fig.~\\ref{figdirsamp1} for the case of the IID Gaussian sample mean. Note that $\\hat p_L(s)$ above is nothing but an empirical vector for $S_n$ (see Sec.~\\ref{secsanov}) or a \\textbf{density histogram} of the sample $\\{s^{(j)}\\}_{j=1}^L$.\n\n\\begin{figure}[t]\n\\resizebox{\\textwidth}{!}{\\includegraphics{directsamplinggauss1.pdf}}\n\\caption{(Left) Naive sampling of the Gaussian IID sample mean ($\\mu=\\sigma=1$) for $n=10$ and different sample sizes $L$. The dashed line is the exact rate function. As $L$ grows, a larger range of $I(s)$ is sampled. (Right) Naive sampling for a fixed sample size $L=10\\, 000$ and various values of $n$. As $n$ increases, $I_{n,L}(s)$ approaches the expected rate function but the sampling becomes inefficient as it becomes restricted to a narrow domain.}\n\\label{figdirsamp1}\n\\end{figure}\n\nThe reason for choosing $\\hat p(s)$ as our estimator of $p_{S_n}(s)$ is that it is an \\textbf{unbiased estimator} in the sense that\n\\begin{equation}\nE[\\hat p_L(s)]=p_{S_n}(s)\n\\end{equation}\nfor all $L$. Moreover, we know from the LLN that $\\hat p_L(s)$ converges in probability to its mean $p_{S_n}(s)$ as $L\\rightarrow\\infty$. Therefore, the larger our sample, the closer we should get to a valid estimation of $p_{S_n}(s)$. \n\nTo extract a rate function $I(s)$ from $\\hat p_L(s)$, we simply compute\n\\begin{equation}\nI_{n,L}(s)=-\\frac{1}{n}\\ln \\hat p_L(s)\n\\label{eqrfe1}\n\\end{equation}\nand repeat the whole process for larger and larger integer values of $n$ and $L$ until $I_{n,L}(s)$ converges to some desired level of accuracy.\\footnote{Note that $\\hat P_L(\\Delta_s)$ and $\\hat p_L(s)$ differ only by the factor $\\Delta s$. $I_{n,L}(s)$ can therefore be computed from either estimator with a difference $(\\ln \\Delta s)\/n$ that vanishes as $n\\rightarrow\\infty$.}\n\n\\subsection{Importance sampling}\n\nA basic rule of thumb in statistical sampling, suggested by the LLN, is that an event with probability $P$ will appear in a sample of size $L$ roughly $LP$ times. Thus to get at least one instance of that event in the sample, we must have $L>1\/P$ as an approximate lower bound for the size of our sample (see Exercise~\\ref{exerrest1} for a more precise derivation of this estimate).\n\nApplying this result to $p_{S_n}(s)$, we see that, if this pdf satisfies an LDP of the form $p_{S_n}(s)\\approx \\mathrm{e}^{-nI(s)}$, then we need to have $L>\\mathrm{e}^{n I(s)}$ to get at least one instance of the event $S_n\\in \\Delta_s$ in our sample. In other words, our sample must be exponentially large with $n$ in order to see any large deviations (see Fig.~\\ref{figdirsamp1} and Exercise~\\ref{exds1}).\n\nThis is a severe limitation of the sampling scheme outlined earlier, and we call it for this reason \\textbf{crude Monte Carlo} or \\textbf{naive sampling}. The way around this limitation is to use \\textbf{importance sampling} (IS for short) which works basically as follows (see the contribution of Katzgraber for more details):\n\\begin{enumerate}\n\\item Instead of sampling the $X_i$'s according to the joint pdf $p(\\bm\\om)$, sample them according to a new pdf $q (\\bm\\om)$;\\footnote{The pdf $q$ must have a support at least as large as that of $p$, i.e., $q(\\bm\\om)>0$ if $p(\\bm\\om)>0$, otherwise the ratio $R_n$ is ill-defined. In this case, we say that $q$ is \\textbf{relatively continuous} with respect to $p$ and write $q\\gg p$.}\n\\item Calculate instead of $\\hat p_L(s)$ the estimator\n\\begin{equation}\n\\hat q_L(s)=\\frac{1}{L\\Delta s}\\sum_{j=1}^L 1{\\hskip -2.9pt}\\hbox{I}_{\\Delta_s} \\big(S_n(\\bm\\om^{(j)})\\big)\\, R(\\bm\\om^{(j)}),\n\\label{eqqest1}\n\\end{equation}\nwhere \n\\begin{equation}\nR(\\bm\\om)=\\frac{p(\\bm\\om)}{q(\\bm\\om)}\n\\end{equation}\nis the called the \\textbf{likelihood ratio}. Mathematically, $R$ also corresponds to the \\textbf{Radon-Nikodym derivative} of the measures associated with $p$ and $q$.\\footnote{The Radon-Nikodym derivative of two measures $\\mu$ and $\\nu$ such that $\\nu\\gg\\mu$ is defined as $d\\mu\/d\\nu$. If these measures have densities $p$ and $q$, respectively, then $d\\mu\/d\\nu=p\/q$.}\n\\end{enumerate}\n\nThe new estimator $\\hat q_L(s)$ is also an unbiased estimator of $p_{S_n}(s)$ because (see Exercise~\\ref{exuie1})\n\\begin{equation}\nE_q[\\hat q_L(s)]= E_p[\\hat p_L(s)].\n\\label{equb1}\n\\end{equation}\nHowever, there is a reason for choosing $\\hat q_L(s)$ as our new estimator: we might be able to come up with a suitable choice for the pdf $q$ such that $\\hat q_L(s)$ has a smaller variance than $\\hat p_L(s)$. This is in fact the goal of IS: select $q(\\bm\\om)$ so as to minimize the variance\n\\begin{equation}\n\\text{var}_q(\\hat q_L(s))=E_q[(\\hat q_L(s)-p(s))^2].\n\\end{equation}\nIf we can choose a $q$ such that $\\text{var}_q(\\hat q_L(s))<\\text{var}_p(\\hat p_L(s))$, then $\\hat q_L(s)$ will converge faster to $p_{S_n}(s)$ than $\\hat p_L(s)$ as we increase $L$.\n\nIt can be proved (see Exercise~\\ref{exoptq1}) that in the class of all pdfs that are relatively continuous with respect to $p$ there is a unique pdf $q^*$ that minimizes the variance above. This \\textbf{optimal} IS pdf has the form\n\\begin{equation}\nq^*(\\bm\\om) =1{\\hskip -2.9pt}\\hbox{I}_{\\Delta_s}(S_n(\\bm\\om))\\frac{p(\\bm\\om)}{p_{S_n}(s)} =p(\\bm\\om|S_n(\\bm\\om)=s)\n\\label{eqoippdf1}\n\\end{equation}\nand has the desired property that $\\text{var}_{q^*}(\\hat q_L(s))=0$. It does seem therefore that our sampling problem is solved, until one realizes that $q^*$ involves the unknown pdf that we want to estimate, namely, $p_{S_n}(s)$. Consequently, $q^*$ cannot be used in practice as a sampling pdf. Other pdfs must be considered.\n\nThe next subsections present a more practical sampling pdf, which can be proved to be optimal in the large deviation limit $n\\rightarrow\\infty$. We will not attempt to justify the form of this pdf nor will we prove its optimality. Rather, we will attempt to illustrate how it works in the simplest way possible by applying it to IID sample means and Markov processes. For a more complete and precise treatment of IS of large deviation probabilities, see \\cite{bucklew2004} and Chap.~VI of \\cite{asmussen2007}. For background information on IS, see Katzgraber (this volume) and Chap.~V of \\cite{asmussen2007}.\n\n\\subsection{Exponential change of measure}\n\nAn important class of IS pdfs used for estimating $p_{S_n}(s)$ is given by\n\\begin{equation}\np_k(\\bm\\om)=\\frac{\\mathrm{e}^{nk S_n(\\bm\\om)}}{W_n(k)}p(\\bm\\om),\n\\end{equation}\nwhere $k\\in\\mathbb{R}$ and $W_n(k)$ is a normalization factor given by\n\\begin{equation}\nW_n(k)=E_p[\\mathrm{e}^{nkS_n}]=\\int_{\\mathbb{R}^n}\\mathrm{e}^{nkS_n(\\bm\\om)}\\, p(\\bm\\om)\\, \\mathrm{d}\\bm\\om.\n\\end{equation}\nWe call this class or family of pdfs parameterized by $k$ the \\textbf{exponential family}. In large deviation theory, $p_k$ is also known as the \\textbf{tilted pdf} associated with $p$ or the \\textbf{exponential twisting} of $p$.\\footnote{In statistics and actuarial mathematics, $p_k$ is known as the \\textbf{associated law} or \\textbf{Esscher transform} of $p$.} The likelihood ratio associated with this change of pdf is \n\\begin{equation}\nR(\\bm\\om)=\\mathrm{e}^{-nkS_n(\\bm\\om)}\\, W_n(k),\n\\end{equation}\nwhich, for the purpose of estimating $I(s)$, can be approximated by\n\\begin{equation}\nR(\\bm\\om)\\approx \\mathrm{e}^{-n[k S_n(\\bm\\om)-\\lambda(k)]}\n\\end{equation}\ngiven the large deviation approximation $W_n(k)\\approx\\mathrm{e}^{n\\lambda(k)}$.\n\nIt is important to note that a single pdf of the exponential family with parameter $k$ cannot be used to efficiently sample the whole of the function $p_{S_n}(s)$ but only a particular point of that pdf. More precisely, if we want to sample $p_{S_N}(s)$ at the value $s$, we must choose $k$ such that\n\\begin{equation}\nE_{p_k}[\\hat q_L(s)]=p_{S_n}(s)\n\\label{eqecm1}\n\\end{equation}\nwhich is equivalent to solving $\\lambda'(k)=s$ for $k$, where\n\\label{eqscgfis1}\n\\begin{equation}\n\\lambda(k)=\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\ln E_p[\\mathrm{e}^{nk S_n}]\n\\end{equation}\nis the SCGF of $S_n$ defined with respect to $p$ (see Exercise~\\ref{exisscgf1}). Let us denote the solution of these equations by $k(s)$. For many processes $S_n$, it can be proved that the tilted pdf $p_{k(s)}$ corresponding to $k(s)$ is asymptotically optimal in $n$, in the sense that the variance of $\\hat q_L(s)$ under $p_{k(s)}$ goes asymptotically to $0$ as $n\\rightarrow\\infty$. When this happens, the convergence of $\\hat q_L(s)$ towards $p_{S_n}(s)$ is fast for increasing $L$ and requires a sub-exponential number of samples to achieve a given accuracy for $I_{n,L}(s)$. To obtain the full rate function, we then simply need to repeat the sampling for other values of $s$, and so other values of $k$, and scale the whole process for larger values of $n$.\n\nIn practice, it is often easier to reverse the roles of $k$ and $s$ in the sampling. That is, instead of fixing $s$ and selecting $k=k(s)$ for the numerical estimation of $I(s)$, we can fix $k$ to obtain the rate function at $s(k)=\\lambda'(k)$. This way, we can build a parametric representation of $I(s)$ over a certain range of $s$ values by covering or ``scanning'' enough values of $k$.\n\nAt this point, there should be a suspicion that we will not achieve much with this \\textbf{exponential change of measure} method (ECM method short) since the forms of $p_k$ and $\\hat q_L(s)$ presuppose the knowledge of $W_n(k)$ and in turn $\\lambda(k)$. We know from the GE Theorem that $\\lambda(k)$ is in many cases sufficient to obtain $I(s)$, so why taking the trouble of sampling $\\hat q_L(s)$ to get $I(s)$?\\footnote{To make matters worst, the sampling of $\\hat q_L(s)$ does involve $I(s)$ directly; see Exercise~\\ref{exisi1}.}\n\nThe answer to this question is that $I(s)$ can be estimated from $p_{k}$ without knowing $W_n(k)$ using two insights:\n\\begin{itemize} \n\\item Estimate $I(s)$ indirectly by sampling an estimator of $\\lambda(k)$ or of $S_n$, which does not involve $W_n(k)$, instead of sampling the estimator $\\hat q_L(s)$ of $p_{S_n}(s)$;\n\\item Sample $\\bm\\om$ according to the a priori pdf $p(\\bm\\om)$ or use the Metropolis algorithm, also known as the Markov chain Monte Carlo algorithm, to sample $\\bm\\om$ according to $p_k(\\bm\\om)$ without knowing $W_n(k)$ (see Appendix~\\ref{appmh} and the contribution of Katzgraber in this volume).\n\\end{itemize}\nThese points will be explained in the next section (see also Exercise~\\ref{exmetis1}).\n\nFor the rest of this subsection, we will illustrate the ECM method for a simple Gaussian IID sample mean, leaving aside the issue of having to know $W_n(k)$. For this process \n\\begin{equation}\np_k(\\bm\\om)=p_k(x_1,\\ldots,x_n)=\\prod_{i=1}^n p_k(x_i),\n\\end{equation}\nwhere\n\\begin{equation}\np_k(x_i)=\\frac{\\mathrm{e}^{k x_i}\\, p(x_i)}{W(k)},\\qquad W(k)=E_p[\\mathrm{e}^{k X}],\n\\end{equation}\nwith $p(x_i)$ a Gaussian pdf with mean $\\mu$ and variance $\\sigma^2$. We know from Exercise~\\ref{exiid1} the expression of the generating function $W(k)$ for this pdf:\n\\begin{equation}\nW(k)=\\mathrm{e}^{k\\mu+\\frac{\\sigma^2}{2}k^2}.\n\\end{equation}\nThe explicit expression of $p_k(x_i)$ is therefore\n\\begin{equation}\np_k(x_i)=\\mathrm{e}^{k(x_i-\\mu)-\\frac{\\sigma^2}{2}k^2} \\frac{\\mathrm{e}^{-(x_i-\\mu)^2\/(2\\sigma^2)}}{\\sqrt{2\\pi \\sigma^2}}=\\frac{\\mathrm{e}^{-(x_i-\\mu-\\sigma^2 k)^2\/(2\\sigma^2)}}{\\sqrt{2\\pi\\sigma^2}}.\n\\end{equation}\n\nThe estimated rate function obtained by sampling the $X_i$'s according to this Gaussian pdf with mean $\\mu+\\sigma^2 k$ and variance $\\sigma^2$ is shown in Fig.~\\ref{figexpsamp1} for various values of $n$ and $L$. This figure should be compared with Fig.~\\ref{figdirsamp1} which illustrates the naive sampling method for the same sample mean. It is obvious from these two figures that the ECM method is more efficient than naive sampling.\n\n\\begin{figure}[t]\n\\resizebox{\\textwidth}{!}{\\includegraphics{expsamplinggauss1.pdf}}\n\\caption{IS for the Gaussian IID sample mean ($\\mu=\\sigma=1$) with the exponential change of measure.}\n\\label{figexpsamp1}\n\\end{figure}\n\nAs noted before, the rate function can be obtained in a parametric way by ``scanning'' $k$ instead of ``scanning'' $s$ and fixing $k$ according to $s$. If we were to determine $k(s)$ for a given $s$, we would find\n\\begin{equation}\n\\lambda'(k)=\\mu+\\sigma^2 k =s,\n\\end{equation}\nso that $k(s)=(s-\\mu)\/\\sigma^2$. Substituting this result in $p_k(x_i)$, then yields\n\\begin{equation}\np_{k(s)}(x_i)=\\frac{\\mathrm{e}^{-(x_i-s)^2\/(2\\sigma^2)}}{\\sqrt{2\\pi\\sigma^2}}.\n\\end{equation}\nThus to efficiently sample $p_{S_n}(s)$, we must sample the $X_i$'s according to a Gaussian pdf with mean $s$ instead of the a priori pdf with mean $\\mu$. The sampling is efficient in this case simply because $S_n$ concentrates in the sense of the LLN at the value $s$ instead of $\\mu$, which means that $s$ has become the typical value of $S_n$ under $p_{k(s)}$. \n\nThe idea of the ECM method is the same for processes other than IID sample means: the goal in general is to change the sampling pdf from $p$ to $p_k$ in such a way that an event that is a rare event under $p$ becomes a typical under $p_k$.\\footnote{Note that this change is not always possible: for certain processes, there is no $k$ that makes certain events typical under $p_k$. For examples, see \\cite{asmussen2011}.} For more information on the ECM method and IS in general; see \\cite{bucklew2004} and Chaps.~V and VI of \\cite{asmussen2007}. For applications of these methods, see Exercise~\\ref{exis2}.\n\n\n\\subsection{Applications to Markov chains}\n\nThe application of the ECM method to Markov chains follows the IID case closely: we generate $L$ IID realizations $x_1^{(j)},\\ldots,x_n^{(j)}$, $j=1,\\ldots,L$, of the chain $\\bm\\om=X_1,\\ldots, X_n$ according to the tilted joint pdf $p_k(\\bm\\om)$, using either $p_{k}$ directly or a Metropolis-type algorithm. The value $k(s)$ that achieves the efficient sampling of $S_n$ at the value $s$ is determined in the same way as in the IID case by solving $\\lambda'(k)=s$, where $\\lambda(k)$ is now given by Eq.~(\\ref{eqscgfmc1}). In this case, $S_n=s$ becomes the typical event of $S_n$ in the limit $n\\rightarrow\\infty$. A simple application of this procedure is proposed in Exercise~\\ref{exmcis1}. For further reading on the sampling of Markov chains, see \\cite{bucklew2004,sadowsky1989,sadowsky1990,bucklew1990b,sadowsky1996}.\n \nAn important difference to notice between the IID and Markov cases is that the ECM does not preserve the factorization structure of $p(\\bm\\om)$ for the latter case. In other words, $p_k(\\bm\\om)$ does not describe a Markov chain, though, as we noticed, $p_k(\\bm\\om)$ is a product pdf when $p(\\bm\\om)$ is itself a product pdf. In the case of Markov chains, $p_k(\\bm\\om)$ does retain a product structure that looks like a Markov chain, and one may be tempted to define a tilted transition pdf as\n\\begin{equation}\n\\pi_k(x'|x)=\\frac{\\mathrm{e}^{kx'}\\pi(x'|x)}{W(k|x)},\\qquad W(k|x)=\\int_{\\mathbb{R}} \\mathrm{e}^{kx'} \\pi(x'|x)\\, \\mathrm{d} x'.\n\\end{equation}\nHowever, it is easy to see that the joint pdf of $\\bm\\om$ obtained from this transition matrix does not reproduce the tilted pdf $p_k(\\bm\\om)$.\n\n\n\\subsection{Applications to continuous-time Markov processes}\n\nThe generalization of the results of the previous subsection to continuous-time Markov processes follows from our discussion of the continuous-time limit of Markov chains (see Sec.~\\ref{subsecmc1}). In this case, we generate, according to the tilted pdf of the process, a sample of time-discretized trajectories. The choice of $\\Delta t$ and $n$ used in the discretization will of course influence the precision of the estimates of the pdf and rate function, in addition to the sample size $L$.\n\nSimilarly to Markov chains, the tilted pdf of a continuous-time Markov process does not in general describe a new ``tilted'' Markov process having a generator related to the tilted generator $\\tilde G_k$ defined earlier. In some cases, the tilting of a Markov process can be seen as generating a new Markov process, as will be discussed in the next subsection, but this is not true in general.\n\n\n\\subsection{Applications to stochastic differential equations}\n\nStochastic differential equations (SDEs) are covered by the continuous-time results of the previous subsection. However, for this type of stochastic processes the ECM can be expressed more explicitly in terms of the path pdf $p[x]$ introduced in Sec.~\\ref{subsecpathldp}. The tilted version of this pdf, which is the functional analogue of $p_k(\\bm\\om)$, is written as\n\\begin{equation}\np_k[x]=\\frac{\\mathrm{e}^{TkS_T[x]}\\, p[x]}{W_T(k)},\n\\label{eqtppdf1}\n\\end{equation}\nwhere $S_T[x]$ is some functional of the trajectory $\\{x(t)\\}_{t=0}^T$ and \n\\begin{equation}\nW_T(k)=E_p[\\mathrm{e}^{TkS_T}].\n\\end{equation}\nThe likelihood ratio associated with this change of pdf is\n\\begin{equation}\nR[x]=\\frac{p[x]}{p_k[x]}=\\mathrm{e}^{-TkS_T[x]}\\, W_T(k)\\approx \\mathrm{e}^{-T[k S_T[x]-\\lambda(k)]}.\n\\end{equation}\n\nAs a simple application of these expressions, let us consider the SDE\n\\begin{equation}\n\\dot x(t)=\\xi(t),\n\\end{equation}\nwhere $\\xi(t)$ is a Gaussian white noise, and the following additive process:\n\\begin{equation}\nD_T[x]=\\frac{1}{T}\\int_0^T \\dot x(t)\\, \\mathrm{d} t,\n\\end{equation}\nwhich represents the average velocity or \\textbf{drift} of the SDE. What is the form of $p_k[x]$ in this case? Moreover, how can we use this tilted pdf to estimate the rate function $I(d)$ of $D_T$ in the long-time limit?\n\nAn answer to the first question is suggested by recalling the goal of the ECM. The LLN implies for the SDE above that $D_T\\rightarrow 0$ in probability as $T\\rightarrow\\infty$, which means that $D_T=0$ is the typical event of the ``natural'' dynamic of $x(t)$. The tilted dynamics realized by $p_{k(d)}[x]$ should change this typical event to the fluctuation $D_T=d$; that is, the typical event of $D_T$ under $p_{k(d)}[x]$ should be $D_T=d$ rather than $D_T=0$. One candidate SDE which leads to this behavior of $D_T$ is\n\\begin{equation}\n\\dot x(t)=d+\\xi(t),\n\\label{eqbsde1}\n\\end{equation} \nso the obvious guess is that $p_{k(d)}[x]$ is the path pdf of this SDE.\n\nLet us show that this is a correct guess. The form of $p_k[x]$ according to Eq.~(\\ref{eqtppdf1}) is\n\\begin{equation}\np_k[x]\\approx\\mathrm{e}^{-T I_k[x]},\\qquad I_k[x]=I[x]-k D_T[x]+\\lambda(k),\n\\end{equation}\nwhere\n\\begin{equation}\nI[x]=\\frac{1}{2T}\\int_0^T \\dot x^2\\, \\mathrm{d} t\n\\end{equation}\nis the action of our original SDE and\n\\begin{equation}\n\\lambda(k)=\\lim_{T\\rightarrow\\infty}\\frac{1}{T}\\ln E[\\mathrm{e}^{TkD_T}]\n\\end{equation}\nis the SCGF of $D_T$. The expectation entering in the definition of $\\lambda(k)$ can be calculated explicitly using path integrals with the result $\\lambda(k)=k^2\/2$. Therefore,\n\\begin{equation}\nI_k[x]=I[x]-k D_T[x]-\\frac{k^2}{2}.\n\\end{equation}\n\nFrom this result, we find $p_{k(d)}[x]$ by solving $\\lambda'(k(d))=d$, which in this case simply yields $k(d)=d$. Hence\n\\begin{equation}\nI_{k(d)}[x]=I[x]-d D_T[x]+\\frac{d^2}{2}=\\frac{1}{2 T}\\int_0^T (\\dot x-d)^2\\, \\mathrm{d} t,\n\\end{equation}\nwhich is exactly the action of the \\textbf{boosted SDE} of Eq.~(\\ref{eqbsde1}).\n\nSince we have $\\lambda(k)$, we can of course directly obtain the rate function $I(d)$ of $D_T$ by Legendre transform:\n\\begin{equation}\nI(d)=k(d) d-\\lambda(k(d))=\\frac{d^2}{2}.\n\\end{equation}\nThis shows that the fluctuations of $D_T$ are Gaussian, which is not surprising given that $D_T$ is up to a factor $1\/T$ a Brownian motion.\\footnote{Brownian motion is defined as the integral of Gaussian white noise. By writing $\\dot x=\\xi(t)$, we therefore define $x(t)$ to be a Brownian motion.} Note that the same result can be obtained following Exercise~\\ref{exisi1} by calculating the likelihood ratio $R[x]$ for a path $\\{x_d(t)\\}_{t=0}^T$ such that $D_T[x_d]=d$, i.e.,\n\\begin{equation}\nR[x_d]=\\frac{p[x_d]}{p_{k(d)}[x_d]}\\approx \\mathrm{e}^{-T I(d)}.\n\\end{equation}\n\nThese calculations give a representative illustration of the ECM method and the idea that the large deviations of an SDE can be obtained by replacing this SDE by a boosted or \\textbf{effective SDE} whose typical events are large deviations of the former SDE. Exercises~\\ref{exaecm1} and \\ref{exnecm1} show that effective SDEs need not correspond exactly to an ECM, but may in some cases reproduce such a change of measure in the large deviation limit $T\\rightarrow\\infty$. The important point in all cases is to be able to calculate the likelihood ratio between a given SDE and a boosted version of it in order to obtain the desired rate function in the limit $T\\rightarrow\\infty$.\\footnote{The limit $\\varepsilon\\rightarrow 0$ can also be considered.}\n\nFor the particular drift process $D_T$ studied above, the likelihood ratio happens to be equivalent to a mathematical result known as \\textbf{Girsanov's formula}, which is often used in financial mathematics; see Exercise~\\ref{exgir1}.\n\n\\subsection{Exercises}\n\\label{secex3}\n\n\\begin{exercise}\n\n\\item \\label{exbernrv2}\\exdiff{5} (Variance of Bernoulli RVs) Let $X$ be a Bernoulli RV with $P(X=1)=\\alpha$ and $P(X=0)=1-\\alpha$, $0\\leq\\alpha\\leq 1$. Find the variance of $X$ in terms of $\\alpha$.\n\n\\item \\label{exds1}\\exdiff{17} (Direct sampling of IID sample means) Generate on a computer a sample of size $L=100$ of $n$ IID Bernoulli RVs $X_1,\\ldots,X_n$ with bias $\\alpha=0.5$, and numerically estimate the rate function $I(s)$ associated with the sample mean $S_n$ of these RVs using the naive estimator $\\hat p_L(s)$ and the finite-size rate function $I_{n,L}(s)$ defined in Eq.~(\\ref{eqrfe1}). Repeat for $L=10^3,10^4, 10^5$, and $10^6$ and observe how $I_{n,L}(s)$ converges to the exact rate function given in Eq.~(\\ref{eqbern1}). Repeat for exponential RVs.\n\n\\item \\label{exuie1}\\exdiff{10} (Unbiased estimator) Prove that $\\hat q_L(s)$ is an unbiased estimator of $p_{S_n}(s)$, i.e., prove Eq.~(\\ref{equb1}).\n\n\\item \\label{exvarest1}\\exdiff{15} (Estimator variance) The estimator $\\hat p_L(s)$ is actually a sample mean of Bernoulli RVs. Based on this, calculate the variance of this estimator with respect to $p(\\bm\\om)$ using the result of Exercise~\\ref{exbernrv2} above. Then calculate the variance of the importance estimator $\\hat q_L(s)$ with respect to $q(\\bm\\om)$.\n\n\\item \\label{exerrest1}\\exdiff{15} (Relative error of estimators) The real measure of the quality of the estimator $\\hat p_L(s)$ is not so much its variance $\\text{var}(\\hat p_L(s))$ but its relative error $\\text{err}(\\hat p_L(s))$ defined by\n\\begin{equation}\n\\text{err}(\\hat p_L(s))=\\frac{\\sqrt{\\text{var}(\\hat p_L(s))}}{p_{S_n}(s)}.\n\\end{equation}\nUse the previous exercise to find an approximation of $\\text{err}(\\hat p_L(s))$ in terms of $L$ and $n$. From this result, show that the sample size $L$ needed to achieve a certain relative error grows exponentially with $n$. (Source: Sec.~V.I of \\cite{asmussen2007}.)\n\n\\item \\label{exoptq1}\\exdiff{17} (Optimal importance pdf) Show that the density $q^*$ defined in (\\ref{eqoippdf1}) is optimal in the sense that\n\\begin{equation}\n\\text{var}_q(\\hat q_L(s))\\geq\\text{var}_{q^*}(\\hat q_L(s))=0\n\\end{equation}\nfor all $q$ relatively continuous to $p$. (See Chap.~V of \\cite{asmussen2007} for help.)\n\n\\item \\label{exisscgf1}\\exdiff{15} (Exponential change of measure) Show that the value of $k$ solving Eq.~(\\ref{eqecm1}) is given by $\\lambda'(k)=s$, where $\\lambda(k)$ is the SCGF defined in Eq.~(\\ref{eqscgfis1}).\n\n\\item \\exdiff{40} (LDP for estimators) We noticed before that $\\hat p_L(s)$ is the empirical vector of the IID sample $\\{S_n^{(j)}\\}_{j=1}^L$. Based on this, we could study the LDP of $\\hat p_L(s)$ rather than just its mean and variance, as usually done in sampling theory. What is the form of the LDP associated with $\\hat p_L(s)$ with respect to $p(\\bm\\om)$? What is its rate function? What is the minimum and zero of that rate function? Answer the same questions for $\\hat q_L(s)$ with respect to $q(\\bm\\om)$. (Hint: Use Sanov's Theorem and the results of Exercise~\\ref{exild1}.)\n\n\\item \\label{exis2}\\exdiff{25} (Importance sampling of IID sample means) Implement the ECM method to numerically estimate the rate function associated with the IID sample means of Exercise~\\ref{exiid1}. Use a simple IID sampling based on the explicit expression of $p_k$ in each case (i.e., assume that $W(k)$ is known). Study the convergence of the results as a function of $n$ and $L$ as in Fig.~\\ref{figexpsamp1}.\n\n\\item \\label{exisi1}\\exdiff{15} (IS estimator) Show that $\\hat q_L(s)$ for $p_{k(s)}$ has the form\n\\begin{equation}\n\\hat q_L(s)\\approx\\frac{1}{L\\Delta s}\\sum_{j=1}^L 1{\\hskip -2.9pt}\\hbox{I}_{\\Delta_s}(s^{(j)})\\ \\mathrm{e}^{-nI(s^{(j)})}.\n\\end{equation}\nWhy is this estimator trivial for estimating $I(s)$?\n\n\\item \\label{exmetis1}\\exdiff{30} (Metropolis sampling of IID sample means) Repeat Exercise~\\ref{exis2}, but instead of using an IID sampling method, use the Metropolis algorithm to sample the $X_i$'s in $S_n$. (For information on the Metropolis algorithm, see Appendix~\\ref{appmh}.)\n\n\\item \\exdiff{15} (Tilted distribution from contraction) Let $X_1,\\ldots,X_n$ be a sequence of real IID RVs with common pdf $p(x)$, and denote by $S_n$ and $L_n(x)$ the sample mean and empirical functional, respectively, of these RVs. Show that the ``value'' of $L_n(x)$ solving the minimization problem associated with the contraction of the LDP of $L_n(x)$ down to the LDP of $S_n$ is given by\n\\begin{equation}\n\\mu_k(x)=\\frac{\\mathrm{e}^{kx}p(x)}{W(k)},\\qquad W(k)=E_p[\\mathrm{e}^{kX}],\n\\end{equation}\nwhere $k$ is such that $\\lambda'(k)=(\\ln W(k))'=s$. To be more precise, show that this pdf is the solution of the minimization problem\n\\begin{equation}\n\\inf_{\\mu:f(\\mu)=s} I(\\mu),\n\\end{equation}\nwhere\n\\begin{equation}\nI(\\mu)= \\int_\\mathbb{R}\\mathrm{d} x\\, \\mu(x)\\ln\\frac{\\mu(x)}{p(x)}\n\\end{equation}\nis the continuous analogue of the relative entropy defined in Eq.~(\\ref{eqsanov2}) and\n\\begin{equation}\nf(\\mu)=\\int_\\mathbb{R} x\\, \\mu(x)\\, \\mathrm{d} x\n\\end{equation}\nis the contraction function. Assume that $W(k)$ exists. Why is $\\mu_k$ the same as the tilted pdf used in IS?\n\n\\item \\exdiff{50} (Equivalence of ensembles) Comment on the meaning of the following statements: (i) The optimal pdf $q^*$ is to the microcanonical ensemble what the tilted pdf $p_k$ is to the canonical ensemble. (ii) Proving that $p_k$ achieves a zero variance for the estimator $\\hat q_L(s)$ in the limit $n\\rightarrow\\infty$ is the same as proving that the canonical ensemble becomes equivalent to the microcanonical ensemble in the thermodynamic limit.\n\n\\item \\label{exmcis1}\\exdiff{25} (Tilted Bernoulli Markov chain) Find the expression of the tilted joint pdf $p_k(\\bm\\om)$ for the Bernoulli Markov chain of Exercise~\\ref{exbmc1}. Then use the Metropolis algorithm (see Appendix~\\ref{appmh}) to generate a large-enough sample of realizations of $\\bm\\om$ in order to estimate $p_{S_n}(s)$ and $I(s)$. Study the converge of the estimation of $I(s)$ in terms of $n$ and $L$ towards the expression of $I(s)$ found in Exercise~\\ref{exbmc1}. Note that for a starting configuration $x_1,\\ldots,x_n$ and a target configuration $x'_1,\\ldots,x'_n$, the acceptance ratio is\n\\begin{equation}\n\\frac{p_k(x_1',\\ldots,x_n')}{p_k(x_1,\\ldots,x_n)}=\\frac{\\mathrm{e}^{k x_1'} p(x_1')\\prod_{i=2}^n \\mathrm{e}^{k x_i'} \\pi(x_i'|x_{i-1}')}{\\mathrm{e}^{k x_1} p(x_1)\\prod_{i=2}^n \\mathrm{e}^{k x_i} \\pi(x_i|x_{i-1})},\n\\end{equation}\nwhere $p(x_i)$ is some initial pdf for the first state of the Markov chain. What is the form of $\\hat q_L(s)$ in this case?\n\n\\item \\label{exgir1}\\exdiff{20} (Girsanov's formula) Denote by $p[x]$ the path pdf associated with the SDE \n\\begin{equation}\n\\dot x(t)=\\sqrt{\\varepsilon}\\, \\xi(t),\n\\end{equation}\nwhere $\\xi(t)$ is Gaussian white noise with unit noise power. Moreover, let $q[x]$ be the path pdf of the boosted SDE\n\\begin{equation}\n\\dot x(t)=\\mu +\\sqrt{\\varepsilon}\\, \\xi(t).\n\\end{equation}\nShow that\n\\begin{equation}\nR[x]=\\frac{p[x]}{q[x]}=\\exp\\left(-\\frac{\\mu}{\\varepsilon} \\int_0^T \\dot x\\, \\mathrm{d} t+\\frac{\\mu^2}{2\\varepsilon} T\\right)=\\exp\\left(-\\frac{\\mu}{\\varepsilon} x(T) +\\frac{\\mu^2}{2\\varepsilon} T\\right).\n\\end{equation}\n\n\\item \\label{exaecm1}\\exdiff{27} (Effective SDE) Consider the additive process,\n\\begin{equation}\nS_T=\\frac{1}{T}\\int_0^T x(t)\\, \\mathrm{d} t,\n\\end{equation}\nwith $x(t)$ evolving according to the simple Langevin equation of Exercise~\\ref{exoup1}. Show that the tilted path pdf $p_{k(d)}[x]$ associated with this process is \\textit{asymptotically} equal to the path pdf of the following boosted SDE:\n\\begin{equation}\n\\dot x(t)=-\\gamma (x(t)-s)+\\xi(t).\n\\label{eqbsde2}\n\\end{equation} \nMore precisely, show that the action $I_{k(s)}[x]$ of $p_{k(s)}[x]$ and the action $J[x]$ of the boosted SDE differ by a boundary term which is of order $O(1\/T)$.\n\n\\item \\label{exnecm1}\\exdiff{22} (Non-exponential change of measure) Although the path pdf of the boosted SDE of Eq.~(\\ref{eqbsde2}) is not exactly the tilted path pdf $p_{k}[x]$ obtained from the ECM, the likelihood ratio $R[x]$ of these two pdfs does yield the rate function of $S_T$. Verify this by obtaining $R[x]$ exactly and by using the result of Exercise~\\ref{exisi1}.\n\n\\end{exercise} \n\n\\section{Other numerical methods for large deviations}\n\\label{secnum2}\n\nThe use of the ECM method appears to be limited, as mentioned before, by the fact that the tilted pdf $p_k(\\bm\\om)$ involves $W_n(k)$. We show in this section how to circumvent this problem by considering estimators of $W_n(k)$ and $\\lambda(k)$ instead of estimators of the pdf $p({S_n})$, and by sampling these new estimators according to the tilted pdf $p_k(\\bm\\om)$, but with the Metropolis algorithm in order not to rely on $W_n(k)$, or directly according to the a priori pdf $p(\\bm\\om)$. At the end of the section, we also briefly describe other important methods used in numerical simulations of large deviations. \n\n\\subsection{Sample mean method}\n\nWe noted in the previous subsection that the parameter $k$ entering in the ECM had to be chosen by solving $\\lambda'(k)=s$ in order for $s$ to be the typical event of $S_n$ under $p_k$. The reason for this is that\n\\begin{equation}\n\\lim_{n\\rightarrow\\infty }E_{p_k}[S_n]=\\lambda'(k).\n\\end{equation}\nBy the LLN we also have that $S_n\\rightarrow\\lambda'(k)$ in probability as $n\\rightarrow\\infty$ if $S_n$ is sampled according to $p_{k}$.\n\nThese results suggest using\n\\begin{equation}\ns_{L,n}(k)=\\frac{1}{L}\\sum_{j=1}^L S_n(\\bm\\om^{(j)})\n\\end{equation}\nas an estimator of $\\lambda'(k)$ by sampling $\\bm\\om$ according to $p_k(\\bm\\om)$, and then integrating this estimator with the boundary condition $\\lambda(0)=0$ to obtain an estimate of $\\lambda(k)$. In other words, we can take our estimator for $\\lambda(k)$ to be\n\\begin{equation}\n\\hat\\lambda_{L,n}(k)=\\int_0^k s_{L,n}(k')\\, \\mathrm{d} k'.\n\\end{equation}\nFrom this estimator, we then obtain a parametric estimation of $I(s)$ from the GE Theorem by Legendre-transforming $\\hat\\lambda_{L,n}(k)$:\n\\begin{equation}\nI_{L,n}(s)=k s-\\hat\\lambda_{L,n}(k),\n\\label{eqplf1}\n\\end{equation}\nwhere $s=s_{L,n}(k)$ or, alternatively,\n\\begin{equation}\nI_{L,n}(s)=k(s)\\, s-\\hat\\lambda_{L,n}(k(s)),\n\\label{eqplf2}\n\\end{equation}\nwhere $k(s)$ is the root of $\\hat\\lambda_{L,n}'(k)=s$.\\footnote{In practice, we have to check numerically (using finite-scale analysis) that $\\lambda_{n,L}(k)$ does not develop non-differentiable points in $k$ as $n$ and $L$ are increased to $\\infty$. If non-differentiable points arise, then $I(s)$ is recovered from the GE Theorem only in the range of $\\lambda'$; see Sec.~4.4 of \\cite{touchette2009}.}\n\nThe implementation of these estimators with the Metropolis algorithm is done explicitly via the following steps: \n\\begin{enumerate}\n\\item For a given $k\\in\\mathbb{R}$, generate an IID sample $\\{\\bm\\om^{(j)}\\}_{j=1}^L$ of $L$ configurations $\\bm\\om=x_1,\\ldots,x_n$ distributed according to $p_{k}(\\bm\\om)$ using the Metropolis algorithm, which only requires the computation of the ratio\n\\begin{equation}\n\\frac{p_{k}(\\bm\\om)}{p_{k}(\\bm\\om')}=\\frac{\\mathrm{e}^{nkS_n(\\bm\\om)}\\, p(\\bm\\om)}{\\mathrm{e}^{nkS_n(\\bm\\om')}\\, p(\\bm\\om')}\n\\end{equation}\nfor any two configurations $\\bm\\om$ and $\\bm\\om'$; see Appendix~\\ref{appmh};\n\n\\item Compute the estimator $s_{L,n}(k)$ of $\\lambda'(k)$ for the generated sample;\n\\item Repeat the two previous steps for other values of $k$ to obtain a numerical approximation of the function $\\lambda'(k)$;\n\\item Numerically integrate the approximation of $\\lambda'(k)$ over the mesh of $k$ values starting from $k=0$. The result of the integration is the estimator $\\hat\\lambda_{L,n}(k)$ of $\\lambda(k)$;\n\\item Numerically compute the Legendre transform, Eq.~(\\ref{eqplf1}), of the estimator of $\\lambda(k)$ to obtain the estimator $I_{L,n}(s)$ of $I(s)$ at $s=s_{L,n}(k)$;\n\\item Repeat the previous steps for larger values of $n$ and $L$ until the results converge to some desired level of accuracy.\n\\end{enumerate}\n\n\\begin{figure}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\\includegraphics{gaussiansmlambda1.pdf}}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{gaussiansmrate1.pdf}}\n\\caption{(Top left) Empirical sample mean $s_L(k)$ for the Gaussian IID sample mean ($\\mu=\\sigma=1$). The spacing of the $k$ values is $\\Delta k=0.5$. (Top right) SCGF $\\hat\\lambda_L(k)$ obtained by integrating $s_L(k)$. (Bottom) Estimate of $I(s)$ obtained from the Legendre transform of $\\hat\\lambda_L(k)$. The dashed line in each figure is the exact result. Note that the estimate of the rate function for $L=5$ is multi-valued because the corresponding $\\hat\\lambda_L(k)$ is nonconvex.}\n\\label{figgaussiansm1}\n\\end{figure}\n\n\nThe result of these steps are illustrated in Fig.~\\ref{figgaussiansm1} for our test case of the IID Gaussian sample mean for which $\\lambda'(k)=\\mu+\\sigma^2 k$. Exercise~\\ref{exsmm1} covers other sample means studied before (e.g., exponential, binary, etc.).\n\nNote that for IID sample means, one does not need to generate $L$ realizations of the whole sequence $\\bm\\om$, but only $L$ realizations of one summand $X_i$ according the tilted marginal pdf $p_k(x)=\\mathrm{e}^{kx} p(x)\/W(k)$. In this case, $L$ takes the role of the large deviation parameter $n$. For Markov chains, realizations of the whole sequence $\\bm\\om$ must be generated one after the other, e.g., using the Metropolis algorithm.\n\n\\subsection{Empirical generating functions}\n\nThe last method that we cover is based on the estimation of the generating function\n\\begin{equation}\nW_n(k)=E[\\mathrm{e}^{nk S_n}]=\\int_{\\mathbb{R}^n}p(\\bm\\om)\\, \\mathrm{e}^{nkS_n(\\bm\\om)}\\, \\mathrm{d}\\bm\\om.\n\\end{equation}\nConsider first the case where $S_n$ is an IID sample mean. Then $\\lambda(k)=\\ln W(k)$, where $W(k)=E[\\mathrm{e}^{kX_i}]$, as we know from Sec.~\\ref{secsanov}, and so the problem reduces to the estimation of the SCGF of the common pdf $p(X)$ of the $X_i$'s. An obvious estimator for this function is\n\\begin{equation}\n\\hat\\lambda_{L}(k)=\\ln \\frac{1}{L}\\sum_{j=1}^L \\mathrm{e}^{k X^{(j)}},\n\\end{equation} \nwhere $X^{(j)}$ are the IID samples drawn from $p(X)$. From this estimator, which we call the \\textbf{empirical SCGF}, we build a parametric estimator of the rate function $I(s)$ of $S_n$ using the Legendre transform of Eq.~(\\ref{eqplf1}) with\n\\begin{equation}\ns(k)=\\hat\\lambda_L'(k)=\\frac{\\sum_{j=1}^L X^{(j)}\\, \\mathrm{e}^{k X^{(j)}}}{\\sum_{j=1}^L \\mathrm{e}^{k X^{(j)}}}.\n\\end{equation}\n\n\\begin{figure}[t]\n\\centering\n\\resizebox{\\textwidth}{!}{\\includegraphics{gaussianegflambda1.pdf}}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{gaussianegfrate1.pdf}}\n\\caption{(Top left) Empirical SCGF $\\hat\\lambda_L(k)$ for the Gaussian IID sample mean ($\\mu=\\sigma=1$). (Top right)  $s(k)=\\hat\\lambda_L'(k)$. (Bottom) Estimate of $I(s)$ obtained from $\\hat\\lambda_L(k)$. The dashed line in each figure is the exact result.}\n\\label{figgaussianegf1}\n\\end{figure}\n\nFig.~\\ref{figgaussianegf1} shows the result of these estimators for the IID Gaussian sample mean. As can be seen, the convergence of $I_L(s)$ is fast in this case, but is limited to the central part of the rate function. This illustrates one limitation of the empirical SCGF method: for unbounded RVs, such as the Gaussian sample mean, $W(k)$ is correctly recovered for large $|k|$ only for large samples, i.e., large $L$, which implies that the tails of $I(s)$ are also recovered only for large $L$. For bounded RVs, such as the Bernoulli sample mean, this limitation does not arise; see \\cite{duffield1995,duffy2005} and Exercise~\\ref{exegf1}.\n\nThe application of the empirical SCGF method to sample means of ergodic Markov chains is relatively direct. In this case, although $W_n$ no longer factorizes into $W(k)^n$, it is possible to ``group'' the $X_i$'s into blocks of $b$ RVs as follows:\n\\begin{equation}\n\\underbrace{X_1+\\cdots +X_b}_{Y_1} + \\underbrace{X_{b+1}+\\cdots+X_{2b}}_{Y_2}+\\cdots+\\underbrace{X_{n-b+1}+\\cdots+X_n}_{Y_m}\n\\end{equation}\nwhere $m=n\/b$, so as to rewrite the sample mean $S_n$ as\n\\begin{equation}\nS_n=\\frac{1}{n}\\sum_{i=1}^n X_i=\\frac{1}{bm}\\sum_{i=1}^m Y_i.\n\\end{equation}\nIf $b$ is large enough, then the blocks $Y_i$ can be treated as being independent. Moreover, if the Markov chain is ergodic, then the $Y_i$'s should be identically distributed for $i$ large enough. As a result, $W_n(k)$ can be approximated for large $n$ and large $b$ (but $b\\ll n$) as\n\\begin{equation}\nW_n(k)=E[\\mathrm{e}^{k\\sum_{i=1}^m Y_i}]\\approx E[\\mathrm{e}^{k Y_i}]^m,\n\\end{equation}  \nso that\n\\begin{equation}\n\\lambda_n(k)=\\frac{1}{n}\\ln W_n(k)\\approx \\frac{m}{n}\\ln E[\\mathrm{e}^{k Y_i}]=\\frac{1}{b}\\ln E[\\mathrm{e}^{k Y_i}].\n\\end{equation}\nWe are thus back to our original IID problem of estimating a generating function; the only difference is that we must perform the estimation at the level of the $Y_i$'s instead of the $X_i$'s. This means that our estimator for $\\lambda(k)$ is now\n\\begin{equation}\n\\hat\\lambda_{L,n}(k)=\\frac{1}{b}\\ln \\frac{1}{L}\\sum_{j=1}^L \\mathrm{e}^{k Y^{(j)}},\n\\end{equation}\nwhere $Y^{(j)}$, $j=1,\\ldots, L$ are IID samples of the block RVs. Following the IID case, we then obtain an estimate of $I(s)$ in the usual way using the Legendre transform of Eq.~(\\ref{eqplf1}) or (\\ref{eqplf2}).\n\nThe application of these results to Markov chains and SDEs is covered in Exercises~\\ref{exegf1} and \\ref{exegf2}. For more information on the empirical SCGF method, including a more detailed analysis of its convergence, see \\cite{duffield1995,duffy2005}.\n\n\\subsection{Other methods}\n\nWe briefly describe below a number of other important methods used for estimating large deviation probabilities, and give for each of them some pointer references for interested readers. The fundamental ideas and concepts related to sampling and ECM that we have treated before also arise in these other methods.\n\\begin{itemize}\n\\item \\textbf{Splitting methods:} Splitting or cloning methods are the most used sampling methods after Monte Carlo algorithms based on the ECM. The idea of splitting is to ``grow'' a sample or ``population'' $\\{\\bm\\om\\}$ of sequences or configurations in such a way that a given configuration $\\bm\\om$ appears in the population in proportion to its weight $p_k(\\bm\\om)$. This is done in some iterative fashion by ``cloning'' and ``killing'' some configurations. The result of this process is essentially an estimate for $W_n(k)$, which is then used to obtain $I(s)$ by Legendre transform. For mathematically-oriented introductions to splitting methods, see \\cite{lecuyer2007,lecuyer2009b,dean2009,morio2010}; for more physical introductions, see \\cite{giardina2006,lecomte2007a,tailleur2007,tailleur2009}.\n\n\\item \\textbf{Optimal path method:} We briefly mentioned in Sec.~\\ref{subsecpathldp} that rate functions related to SDEs can often be derived by contraction of the action functional $I[x]$. The optimization problem that results from this contraction is often not solvable analytically, but there are several techniques stemming from optimization theory, classical mechanics, and dynamic programming that can be used to solve it. A typical optimization problem in this context is to find the optimal path that a noisy system follows with highest probability to reach a certain state considered to be a rare event or fluctuation. This optimal path is also called a \\textbf{transition path} or a \\textbf{reactive trajectory}, and can be found analytically only in certain special systems, such as linear SDEs and conservative systems; see Exercises~\\ref{exop1}--\\ref{exop3}. For nonlinear systems, the Lagrangian or Hamiltonian equations governing the dynamics of these paths must be simulated directly; see \\cite{graham1989} and Sec.~6.1 of \\cite{touchette2009} for more details.\n\n\\item \\textbf{Transition path method:} The transition path method is essentially a Monte Carlo sampling method in the space of configurations $\\bm\\om$ (for discrete-time processes) or trajectories $\\{x(t)\\}_{t=0}^T$ (for continuous-time processes and SDEs), which aims at finding, as the name suggests, transition paths or optimal paths. The contribution of Dellago in this volume covers this method in detail; see also \\cite{dellago2003,dellago2006,dellago2009,metzner2006,vanden-eijnden2006b}. A variant of the transition method is the so-called \\textbf{string method} \\cite{vanden-eijnden2002} which evolves paths in conservative systems towards optimal paths connecting different local minima of the potential or landscape function of these systems.\n\n\\item \\textbf{Eigenvalue method:} We have noted that the SCGF of an ergodic continuous-time process is given by the dominant eigenvalue of its tilted generator. From this connection, one can attempt to numerically evaluate SCGFs using numerical approximation methods for linear functional operators, combined with numerical algorithms for finding eigenvalues. A recent application of this approach, based on the renormalization group, can be found in \\cite{gorissen2009,gorissen2011}.\n\\end{itemize}\n\n\n\n\\subsection{Exercises}\n\\label{secex4}\n\n\\begin{exercise}\n\\item \\label{exsmm1}\\exdiff{25} (Sample mean method) Implement the steps of the sample mean method to numerically estimate the rate functions of all the IID sample means studied in these notes, including the sample mean of the Bernoulli Markov chain.\n\n\\item \\exdiff{25}\\label{exegf1} (Empirical SCGF method) Repeat the previous exercise using the empirical SCGF method instead of the sample mean method. Explain why the former method converges quickly when $S_n$ is bounded, e.g., in the Bernoulli case. Analyze in detail how the method converges when $S_n$ is unbounded, e.g., for the exponential case. (Source: \\cite{duffield1995,duffy2005}.)\n\n\\item \\exdiff{30}\\label{exegf2} (Sampling of SDEs) Generalize the sample mean and empirical mean methods to SDEs. Study the additive process of Exercise~\\ref{exaecm1} as a test case. \n\n\\item \\exdiff{30}\\label{exop1} (Optimal paths for conservative systems) Consider the conservative system of Exercise~\\ref{excons1} and assume, without loss of generality, that the global minimum of the potential $U(\\mathbf{x})$ is at $\\mathbf{x}=0$. Show that the optimal path which brings the system from $\\mathbf{x}(0)=0$ to some fluctuation $\\mathbf{x}(T)=\\mathbf{x}\\neq 0$ after a time $T$ is the time-reversal of the solution of the noiseless dynamics $\\dot \\mathbf{x} =-\\nabla U(\\mathbf{x})$ under the terminal conditions $\\mathbf{x}(0)=\\mathbf{x}$ and $\\mathbf{x}(T)=0$. The latter path is often called the \\textbf{decay path}. \n\n\\item \\exdiff{25}\\label{exop2} (Optimal path for additive processes) Consider the additive process $S_T$ of Exercise~\\ref{exaecm1} under the Langevin dynamics of Exercise~\\ref{exoup1}. Show that the optimal path leading to the fluctuation $S_T=s$ is the constant path\n\\begin{equation}\nx(t)=s,\\quad t\\in [0,T].\n\\end{equation}\nExplain why this result is the same as the asymptotic solution of the boosted SDE of Eq.~(\\ref{eqbsde2}).\n\n\\item \\exdiff{25}\\label{exop3} (Optimal path for the dragged Brownian particle) Apply the previous exercise to Exercise~\\ref{exbmp1}. Discuss the physical interpretation of the results.\n\n\\end{exercise} \n\n\\section*{Acknowledgments}\n\nI would like to thank Rosemary J. Harris for suggesting many improvements to these notes, as well as Alexander Hartmann and Reinhard Leidl for the invitation to participate to the 2011 Oldenburg Summer School.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nThere remains no consensus as to the mechanism behind the anomalous normal state pseudogap, or its connection, if any, to superconductivity in the underdoped cuprates.\\cite{timusk:1999,kohsaka:2012,  valla:2006}\nHowever, one  common viewpoint is that the pseudogap stems from strong correlation physics, for which the simplest model is the 2D Hubbard model.\nWhile it may be that the answer to the pseudogap problem is to compute the exact electronic correlations throughout the phase diagram,\\cite{gull:2012, leblanc:2013} what the community at large strives for is instead the identification of a simple mechanism, or solvable effective Hamiltonian which contains the essence of the correlations responsible for the pseudogap.  There have been many such proposals, such as, to name a few: loop currents,\\cite{varma:2010} formation of spin singlets,\\cite{lee:2006,  ogata:2008, yrz:2006} charge or spin density waves (d-density wave),\\cite{chakravarty:2001} the spin-fermion model,\\cite{abanov:2000} quasi 1D spin liquids\\cite{tsvelik:2007} and more recent charge density fields emerging from fluctuating antiferromagnetic orders.\\cite{sachdev:2013, hayward:2013}\n While each model is interesting in its own regard, it is not clear which, if any, is describing the physics which gives rise to the pseudogap.\n\nDistinguishing between various phenomenologies requires direct experimental evidence.  A monumental experimental observation was the work of Hashimoto et al.\\cite{hashimoto:2010} who identified that the pseudogap is particle-hole asymmetric.  \n  This has been verified through the subsequent observation   of new Bogoliubov quasiparticle bands which appear below the superconducting transition temperature $T_c$ only in the presence of a particle-hole asymmetric pseudogap.\\cite{he:2011, leblanc:2011:bqp} \nIf the pseudogap is not particle-hole symmetric, then any representation of the pseudogap along the Fermi momenta will not accurately depict the momentum dependence of the pseudogap order.  Such a conclusion has been hinted at by recent experimental and computational work.\\cite{sakai:2010,sakai:2013,sakai:2013:2,sacuto:2012,sacuto:2013}\n\nThe goals of this paper are two-fold: 1) show that there exists at least one model where the inclusion of a momentum independent (s-wave) order-parameter results in an apparent d-wave pseudogap at the Fermi level. 2) Identify signatures of a momentum independent gap in momentum averaged observable quantities. \nTo accomplish these goals, we employ a hybridization phenomenology consistent with one of the most successful models of the pseudogap phase.\\cite{yrz:2006, rice:2012, comin:2014}\nWe examine momentum resolved quantities, such as spectral density and energy dispersion, as well as momentum averaged quantities.  We present results for the Raman B$_{1g}$ and B$_{2g}$ spectra in order to identify the features of the onset of a pseudogap order with either $s-$ or $d-$wave symmetry.  We also compute the single particle density of states in which we identify the most clear signatures of a momentum independent pseudogap from a momentum averaged quantity.\n\nIn Section \\ref{sec:theory} we will build the relevant theoretical framework.  Section \\ref{sec:resolved} will contain results of the phenomenological band structure, Section \\ref{sec:averaged} will focus on momentum averaged quantities, and Section \\ref{sec:conc} will contain a brief summary.\n\n\n\n\n\n\\section{Theory}\\label{sec:theory}\n\\subsection{Hybridization Phenomenology}\\label{sec:phenom}\n\n\nWithin a BCS (Bardeen-Cooper-Schrieffer) theory the superconducting gap  originates at the intersection of two  field operators, namely the fermionic electron (hole) creation (annihilation) operators.  In the standard problem the dispersion takes the simplified form $E_{\\rm sc}(\\vect{k})=\\pm\\sqrt{\\epsilon_{\\vect{k}}^2+\\Delta_{\\rm sc}^2(\\vect{k})}$,  which represents a superconducting gap, $\\Delta_{\\rm sc}(\\vect{k})$, opening at the Fermi level of the electronic dispersion, $\\epsilon_{\\vect{k}}$. \nTherefore, the energy dispersion at a point along the Fermi momentum, $\\vect{k}_F$, reduces to $E_{\\rm sc}(\\vect{k}_F)=\\pm \\Delta_{\\rm sc}(\\vect{k}_F)$.  We will see that this collapse at $\\vect{k}=\\vect{k}_F$ is not relevant for a particle-hole asymmetric pseudogap, and identify instead the momenta about which the dispersion of the pseudogap state collapses.\n\nTo accomplish this, we consider a general hybridization phenomenology of electron operators, $c_{\\vect{k}\\sigma}$, coupling with an excitation defined by the operator, $D_{\\vect{k}\\sigma}$, and we specify the dispersions of these particles throughout the Brillouin zone as $\\epsilon_{\\vect{k}}^{c(D)}$ respectively.  These two operators will share a constant energy contour in momentum space where $\\epsilon_{\\vect{k}}^c=\\epsilon_{\\vect{k}}^D$ along which hybridization can occur, reducing the overall energy of the system, forming a gap.  \nThe hybridization scenario follows as:\\cite{imada:2013, yamaji:2011}\n\n\\begin{align}\n\\mathcal{H} = \\sum\\limits_{\\vect{k}\\sigma}\\epsilon_{\\vect{k}}^c c_{\\vect{k}\\sigma}^\\dagger c_{\\vect{k}\\sigma} + \\epsilon_{\\vect{k}}^D D_{\\vect{k}\\sigma}^\\dagger D_{\\vect{k}\\sigma} - \\lambda \\left(c_{\\vect{k}\\sigma}^\\dagger D_{\\vect{k}\\sigma}+ D_{\\vect{k}\\sigma}^\\dagger c_{\\vect{k}\\sigma}\\right).\n\\end{align}\n\nThe choice of $D_{\\vect{k}\\sigma}$ operators may differ depending on the model.\\cite{smith:2010} For example in most charge, spin or d-density wave models the operator  $D_{\\vect{k}\\sigma}=c^\\dagger_{\\vect{k}+\\vect{Q} \\sigma}$, an electron creation operator translated through a $\\vect{Q}$ scattering momentum.\nTo continue, the Hamiltonian can be rewritten as\n\\begin{align}\\label{eqn:hamiltonian}\n\\mathcal{H}=\\sum\\limits_{\\vect{k}\\sigma} \\Psi^\\dagger(\\vect{k}) \\begin{pmatrix} \\epsilon_{\\vect{k}}^c & -\\lambda \\\\ -\\lambda^* & \\epsilon_{\\vect{k}}^D   \\end{pmatrix}    \\Psi(\\vect{k}) \n\\end{align}\nwhere $\\Psi^\\dagger(\\vect{k}) = \\begin{pmatrix}\nc_{\\vect{k}\\sigma}^\\dagger & D_{\\vect{k}\\sigma}^\\dagger\n\\end{pmatrix}  $.  The problem is then diagonalized by defining $\\epsilon_{\\vect{k}}^-=\\frac{\\epsilon_{\\vect{k}}^c-\\epsilon_{\\vect{k}}^D}{2}$  and $\\epsilon_{\\vect{k}}^+=\\frac{\\epsilon_{\\vect{k}}^c+\\epsilon_{\\vect{k}}^D}{2}$, bonding and anti-bonding bands respectively. \nThe result is a two-band system with energies given by $E_{\\vect{k}}^\\pm = \\epsilon^+_{\\vect{k}} \\pm E_{\\vect{k}}$, where $E_{\\vect{k}}= \\sqrt{{\\epsilon_{\\vect{k}}^-}^2 +\\lambda^2  }$. \nFrom the Hamiltonian of Eq.~(\\ref{eqn:hamiltonian}) we can write the Green's function using $G(\\vect{k},i\\omega_n) ^{-1} = i\\omega_n - \\mathcal{H}$.  While the Hamiltonian and Green's function are matrices, we consider only the coherent electronic element,  $\\left\\langle  c_{\\vect{k}\\sigma}^\\dagger c_{\\vect{k}\\sigma} \\right\\rangle$, of the Green's function which can be written as \n\\begin{eqnarray}\nG(\\boldsymbol{k},\\omega)=\n\\sum_{\\alpha=\\pm}  \\frac{{|W^{\\alpha}_{\\boldsymbol{k}}|^2\t}}{\\omega-E^{\\alpha}_{\\boldsymbol{k}}+ i\\Gamma},\n\\end{eqnarray}\n where $|W_{\\vect{k}}^\\pm|^2= \\frac{1}{2}(1 \\pm \\frac{\\epsilon^-_{\\vect{k}}}{E_{\\vect{k}}})$ and are assumed to be real.\n Off diagonal elements can be treated as well\\cite{valenzuela:2005}, however we will see in the following section that the model presented there neglects off diagonal contributions.\n\nThe resulting spectral density, $A(\\vect{k},\\omega)= -2 {\\rm Im} G(\\vect{k}, \\omega )$, will show a hybridization gap, $\\lambda$, which opens along the momenta corresponding to zeros of the bonding dispersion.  If we define the momenta where this occurs to be $\\vect{k}_B$, such that $\\epsilon_{\\vect{k}_B}^-=0$, then the energy dispersion reduces to $E_{\\vect{k}_B}^\\pm=\\epsilon_{\\vect{k}_B}^+ \\pm \\lambda$.\nThe distinction between a gap opening at the Fermi momentum, $\\vect{k}_F$, (as was the case for a superconducting gap) from a gap opening away from the Fermi momentum, in this case at $\\vect{k}_B$, is the energy of the antibonding dispersion.\n We can therefore obtain a pseudogap in the hybridization scenario without resorting to a complicated momentum dependence of the order parameter, $\\lambda$. Instead, the momentum dependence of the pseudogap has been moved off of $\\lambda$, and is an element of the bonding and antibonding dispersions, $\\epsilon_{\\vect{k}}^-$ and $\\epsilon_{\\vect{k}}^+$ respectively, which contain elements of both the $c_{\\vect{k}\\sigma}$ and $D_{\\vect{k}\\sigma}$ fermion dispersions.  This will be central to understanding the s-wave picture of the pseudogap.\n  \n\n\n\\subsection{Doping dependence through a Yang, Rice, Zhang Ansatz}\nAn optimal starting point for including a doping dependent phase diagram is the rather successful ansatz for the pseudogap self energy due to Yang, Rice and Zhang (YRZ)\\cite{yrz:2006, yrz:2009, leblanc:2009, illes:2009, carbotte:2010:kag, leblanc:2010, ashby:2013} which was recently reviewed.\\cite{rice:2012}  Recent work has reproduced this ansatz from a strong theoretical footing by mapping to a slave boson mean field theory  in which fermions are factored into spinon and holon constituent particles which can then bind together forming a gap at the particle boundary.\\cite{james:2012}  This is therefore one of several phenomenologies which maps to the hybridization formulation of Sec.~\\ref{sec:phenom}.\n\nThe spirit of the YRZ ansatz is to start from an overdoped Fermi liquid system and build in a mechanism for the suppression of electronic states at the Fermi level originating from Mott\u00ad insulating physics as the system is underdoped.  What results is a simple non\u00adinteracting set of bands which contain the key elements needed to describe the basic qualitative features of experiments in the pseudogap state.\\cite{rice:2012}  Naturally this approach has an intrinsic success related to low-\u00adenergy 'hot\u00adspot' physics, which will break down under extreme underdoping and will never correctly capture effects due to scattering.\n The original YRZ ansatz, produced through comparison to calculations of 2 and 4 leg Hubbard ladders,\\cite{konik:2006} enforced a d-wave symmetry to the pseudogap order parameter in the standard form $\\Delta_{pg}^d(\\vect{k})=0.5 \\Delta_{pg}(\\cos k_x - \\cos k_y)$.  \nHere we will compare this standard assumption of the d-wave pseudogap form factor to a momentum independent order parameter  which we will abbreviate throughout as $\\Delta_{\\rm pg}^s=\\Delta_{\\rm pg}$.  In both cases we define the pseudogap scale as $\\Delta_{\\rm pg}=A_0 \\Delta_{\\rm pg}^0(x)$  where $x$ is doping, $\\Delta_{\\rm pg}^0(x)$ is a doping dependent energy scale, and $A_0$ is a prefactor representing temperature dependence such that $A_0=1$ at $T=0$ and $A_0=0$ at the pseudogap critical temperature, $T^*$. \\cite{sakai:2010,sakai:2013} \n\nThe YRZ model can be phrased in terms of a hybridization process by taking the  electronic dispersion, $\\epsilon_{\\vect{k}}^c= - 2t(x)(\\cos\nk_xa  + \\cos k_ya ) - 4t^{\\prime}(x) \\cos k_xa \\cos k_ya - 2t''(x)(\\cos\n2k_xa  + \\cos 2k_ya )-\\mu_p$ and $\\epsilon_{\\vect{k}}^D =   2t(x)(\\cos k_xa  + \\cos k_ya)$.\\cite{yrz:2009}  The electronic dispersion is the third\nnearest-neighbor tight-binding energy from a Gutzwiller projected t-J Hamiltonian which acts on an RVB wavefunction.  This produces an effective tight binding dispersion which contains Gutzwiller renormalization factors, hopping amplitudes $t$-$t^\\prime$-$t^{\\prime\\prime}$ and an electronic chemical potential, $\\mu_p$, which sets the filling of the electronic dispersion.  \nThe $\\epsilon_{\\vect{k}}^D$ dispersion is fundamental to the YRZ ansatz and represents a particle which has a Fermi surface that is independent of doping, related to the half-filled Mott insulator, and which is perfectly susceptible to a vector nesting of $Q=(\\pi,\\pi)$ at the Fermi level.\nTo continue, we construct the coherent piece of the single particle Green's function which results in\n\\begin{eqnarray}\nG(\\boldsymbol{k},\\omega,x)=\n\\sum_{\\alpha=\\pm}  \\frac{{g_t(x) |W^{\\alpha}_{\\boldsymbol{k}}|^2\t}}{\\omega-E^{\\alpha}_{\\boldsymbol{k}}+ i\\Gamma},\n\\label{eqn:sc}\n\\end{eqnarray}\nwhere $g_t(x)= \\frac{2x}{1+x}$.\n The energy dispersions and weighting factors now contain doping dependent hopping parameters, the details of which are located in \\ref{sec:app:yrz}.  It is this doping dependent YRZ Green's function, Eq.~(\\ref{eqn:sc}), which we employ in the following sections.\n \n\n\n\n\\section{Momentum Resolved Quantities}\\label{sec:resolved}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.8\\linewidth]{fig1.eps}\n \\caption{(Color Online) (a) An example at $x=0.16$ of the relevant $k$-space contours, $\\vect{k}_B$ and $\\vect{k}_{na}$, at the Fermi level. (b) Spectral density $A(\\vect{k}_B, \\omega, x=0.12)$ where $\\vect{k}_B$ is the length along the green dashed curve of frame (a).  (c)  Spectral density $A(\\vect{k}_{na}, \\omega, x=0.12)$ for $\\vect{k}_{na}$ along the nearest approach contour, the red curve of frame (a). All quantities were evaluated with $A_0=0.5$ using a momentum independent order, $\\Delta_{\\rm pg}^s$.}\n\\label{fig:nacont}\n\\end{figure*}\n\n\\subsection{Hybridized Spectral Density}\n\n\nWe explore the impact of a momentum independent pseudogap on perceived ARPES spectra.  In Fig.~\\ref{fig:nacont}(a) we trace the momentum and energy dependence of the spectral density $A(\\vect{k},\\omega,x=0.16)$ along two contours in momentum space.  These contours are labelled $\\vect{k}_{B}$ and $\\vect{k}_{na}$.  The first contour, $\\vect{k}_B$, shown as a green-dashed curve  represents the contour in momentum space along which the bonding dispersion, $\\epsilon_{\\vect{k}}^-$, is equal to zero.  The second contour, shown in red, is the contour of nearest-approach (or minimum gap locus), which in this phenomenology is equivalent to the zeros of the bonding dispersion, except where there is a Fermi surface near the nodal region. \nIn Fig.~\\ref{fig:nacont}(b) and (c) we show the spectral density along $\\vect{k}_B$ and $\\vect{k}_{na}$ respectively for an example case at $x=0.12$.  In Fig.~\\ref{fig:nacont}(b) one can see the momentum independent pseudogap and the collapse of the hybridized dispersion, \\emph{i.e.} $E_{\\vect{k}_B}^\\pm \\to \\epsilon_{\\vect{k}_B}^+ \\pm \\Delta_{\\rm pg}^0$.  The splitting of the band is not symmetric about the Fermi level, but is instead symmetric about the antibonding dispersion, $\\epsilon_{\\vect{k}}^+$, traced as the dashed line in Fig.~\\ref{fig:nacont}(b).\nIf one was instead to follow the set of $\\vect{k}_{na}$  the result obtained is Fig.~\\ref{fig:nacont}(c).  Indeed, at negative $\\omega$ these two cases are essentially identical save for the existence of a Fermi surface in the nodal region in Fig.~\\ref{fig:nacont}(c).  What is apparent is that the gap at the antinodes in Fig.~\\ref{fig:nacont}(c) is not caused by a momentum dependent pseudogap, but instead has a simpler explanation in the dispersive nature of the antibonding band, $\\epsilon_{\\vect{k}}^+$.\n\nTo further emphasize this point, we show a common\n treatment of ARPES experimental spectra which is to plot the negative energy spectral peak, here $-E^-_{\\vect{k}_{na}}$, as a function of the amplitude of the d-wave form as  done in Fig.~\\ref{fig:coskxy}.  The momentum independent gaps, $\\Delta_{\\rm pg}^s$ (solid lines), produce what is an apparent d-wave pseudogap at the Fermi level in the antinodal region.  This is purely caused by the antibonding dispersion, $\\epsilon_{\\vect{k}}^+$, which we can see from Fig.~\\ref{fig:nacont}(b) has a cosine-like behavior.  \nThe inclusion of a d-wave form factor, $\\Delta_{\\rm pg}^d$ (red dashed), only acts to increase the apparent slope of the curve for the same parameter values and can be absorbed into the scaling factor of $t_0$.  The inclusion of such a scaling factor $t_0 \\to t_0\/C$ is also shown in Fig.~\\ref{fig:coskxy} (blue double-dashed-dotted lines), illustrating that the $s-$ and $d-$wave cases are qualitatively identical.  \nIncreasing $\\Delta_{\\rm pg}$ or decreasing the doping, $x$, results in a shift of the apparent gap onset to a lower value, which is a generic feature of models with Fermi arcs or pockets.\n Comparison to experimental data in the absence of superconductivity,  shown in the inset of Fig.~\\ref{fig:nacont}, reveals that both $\\Delta_{\\rm pg}^s$ and $\\Delta_{\\rm pg}^d$ are possible candidates to explain the momentum dependence of ARPES spectra along the minimum gap locus since neither deviate strongly from the experimentally suggested linear trend. \n   The details of such a calculation in the superconducting state have been presented previously which adds only a nodal $d-$wave component.\\cite{leblanc:2010, valenzuela:2007}\n\nTo summarize, within this phenomenology  a momentum independent gap opens along the bonding dispersion which results in symmetric bands relative to the energy of the antibonding dispersion and this will appear as a false d-wave pseudogap along the Fermi surface and at negative energies where ARPES can probe.  As a result, both the $\\Delta_{\\rm pg}^s$ and $\\Delta_{\\rm pg}^d$ scenarios are consistent with ARPES observations.\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=.6\\linewidth]{fig2.eps}\n    \\caption{(Color Online) The dominant negative energy spectral peak with of a momentum independent order, $\\Delta_{\\rm pg}^s$ (solid lines), plotted as a function of the amplitude of the standard d-wave gap $\\vert \\cos(k_x) -\\cos(k_y)  \\vert$ compared with the standard $\\Delta_{\\rm pg}^d$ (dashed lines).  For illustrative purposes, an arbitrary scaling factor $\\rm C$ can be applied to each $\\Delta_{\\rm pg}^d$ case to fit the $s-$wave order (blue double-dashed-dotted lines).  $E_{\\vect{k}}^-$ is evaluated along $\\vect{k}_{na}$ as the nearest approach contour is traversed from the nodal point (left) to the antinodal point (right).  Inset: Data on Bi-2212 from Vishik et al. Ref.~\\cite{vishik:2012} for a range of doping, $x$, above the superconducting transition temperature.}\n\\label{fig:coskxy}\n\\end{figure}\n\n\\section{Impact on Momentum Averaged Quantities}\\label{sec:averaged}\n\\subsection{$B_{1g}$ and $B_{2g}$ Raman Spectra}\nRaman spectroscopy has been an influential technique for examining the underdoped cuprates due to an ability to control which region of momentum space is sampled.  For example, the B1g (antinodal) and B2g (nodal) Raman shifts have been used in the past to examine the normal state\\cite{venturini:2002, chen:1997, nemetschek:1997} but also to separate the pseudogap and superconducting energy scales.\\cite{tacon:2006}\nIn the normal state, experiments could be understood through the inclusion of an anisotropic insulating gap originating from a 'hot\u00adspot' near the antinode with an angular $d-$\u00adwave order parameter with a $\\cos(2\\phi)$ dependence in the Brillouin zone angle, $\\phi$.  In the present formulation of a hybridization phenomenology, and given \nGiven the difficulty distinguishing $\\Delta_{\\rm pg}^s$ and $\\Delta_{\\rm pg}^d$ in momentum resolved ARPES, it is not at all understood how the Raman spectra with $s-$ or $d-$wave pseudogap are distinct.  We are therefore looking to start with the simplest model system where a distinction between an $s\u00ad $ or $d\u00ad-$wave structure of the pseudogap might be seen.  For this reason, we compute the Raman shift at the one-loop level and employ the effective mass approximation for the B1g and B2g polarization vertices, $\\gamma_{\\vect{k}}^{B_{1g}}$ and $\\gamma_{\\vect{k}}^{B_{2g}}$ respectively. \nWe also fix the scattering rate to be constant and all other parameters to be the same between the $s-$\u00adwave and $d-$\u00adwave cases.  This should result in the clearest signatures of $s-$\u00ad or $d-$\u00adwave order that are possible from a momentum averaged Raman spectra.\n The calculation of Raman spectra follows from what are now standard treatments from Devereaux et al.\\cite{devereaux:1996}.  The extension of such a treatment to include multiband systems \\cite{boyd:2009} results primarily in extra transitions between bands (interband) in addition to the low energy Drude\u00adlike (intraband) transitions which are sensitive to the band structure at the Fermi level.  \n For what follows, a minimal set of relevant details can be found in \\ref{sec:app:raman}.\n\nA representative case at $x=0.12$ is shown in Fig.~\\ref{fig:raman} employing a momentum independent order, $\\Delta_{\\rm pg}^s$ (solid lines).  Shown in black is the unhybridized case, where $\\Delta_{\\rm pg}^s=0$.  In that case, the band structure is a single band, and the Raman spectra represent only an intraband Drude-like response.  At the onset of $\\Delta_{\\rm pg}$, the single band breaks into two, $E_{\\vect{k}}^\\pm$.  The Raman spectra will then be comprised of a low energy intraband response and a higher energy interband transition, whose onset at $2\\Delta_{\\rm pg}$ is marked by coloured arrows above each curve.  The connection to a Drude response is important\nbecause the low energy spectra is dominated by this response, which gives a peak in both $\\chi^{\\prime \\prime}_{B1g}$ and $\\chi^{\\prime \\prime}_{B2g}$ near $\\omega=\\Gamma$, where $\\Gamma$ is the elastic scattering rate.  \n\nAs the pseudogap increases, the B$_{1g}$ spectra is extremely suppressed at low energy due to the erosion of Fermi surface which occurs in the antinodal region and is strongly sampled by the $\\gamma_{\\vect{k}}^{B1g}$ vertex.\nTo contrast, in the B2g spectra the slope in the limit of $\\omega \\to 0$ is determined by the scattering rate and is essentially unmodified for increasing pseudogap.  As a result there is only very weak suppression of the Drude response whose peak location does not shift with the value of $\\Delta_{\\rm pg}^s$.  It is important to point out that this suppression of the intraband response in either polarization contains no information regarding the scales of the pseudogap, since it contains only intraband ($E_{\\vect{k}}^- \\to E_{\\vect{k}}^-$) transitions.  In the case of $\\chi_{B2g}^{\\prime \\prime}$ this intraband suppression is weak due to the loss of states at the Fermi level, which occurs primarily in the antinodal region, and which is sampled only very weakly by the $\\gamma_{\\vect{k}}^{B_{2g}}$ polarization vertex.\n\nAt larger values of $\\Delta_{\\rm pg}$, both the $\\chi^{\\prime \\prime}_{B1g}$ and $\\chi^{\\prime \\prime}_{B2g}$ spectra exhibit clear interband transitions at the same frequency of $\\omega=2\\Delta_{\\rm pg}^s$.  These corresponding peaks are the lowest energy feature which is evidence for an s-wave pseudogap.  \nA final point of interest is that the highest energy scale would occur in the nodal region for the transition $\\omega= \\Delta_{\\rm max}$ shown in Fig.~\\ref{fig:nacont}(c).  This is the end of the interband transitions and is the least distinct indication of an ordering in the nodal region.\nWe show also for $A_0=0.2$ and $0.5$ the case with the $\\Delta_{\\rm pg}^d$ form factor (dashed lines).  What is evident is that the low energy Raman response with $\\Delta_{\\rm pg}^d$ is not qualitatively distinct from the momentum independent case in either the $\\chi^{\\prime \\prime}_{B1g}$ or $\\chi^{\\prime \\prime}_{B2g}$ spectra.  Both the $s-$ and $d-$wave cases show a similar low energy suppression. With $\\Delta_{\\rm pg}^d$ the interband transitions occur at lower energy in the nodal region due to having a smaller value of the gap at the edge of the Fermi pocket for the same input parameters, a previously understood effect.\\cite{leblanc:2010}  Finally, the high energy feature at $\\Delta_{max}$ does not occur in the d-wave case, since the gap would vanish in the nodal region, and therefore cannot result in the transition shown in Fig.~\\ref{fig:nacont}(c).\n\nTo summarize, examining the loss of low energy Raman signal due to the onset of a pseudogap does not allow for a clear distinction between an $s-$ or $d-$wave pseudogap structure factor.  Instead, both cases present a similar loss of Fermi surface in the antinodal region which reduces the Drude response, and this does not appear to be an indication of a nodal pseudogap ordering.  A momentum independent pseudogap would be best inferred from the coincidence of higher energy interband transitions in the nodal $B2g$ spectra to those in the $B1g$ spectra.\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=.55\\linewidth]{fig3.eps}\n    \\caption{(Color Online) (a)The B1g Raman response, $\\chi_{B1g}^{\\prime \\prime}(\\omega)$, for a representative case of $x=0.12$.  (b)The corresponding B2g Raman response, $\\chi_{B2g}^{\\prime \\prime}(\\omega)$.  Calculations for both $\\Delta_{\\rm pg}^s$ (solid) and $\\Delta_{\\rm pg}^d$ (dashed) are shown.}\n\\label{fig:raman}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Density of States}\nGiven the difficulty of distinguishing $s$- and $d$-wave orders at low energy in a two-particle quantity, such as Raman spectroscopy, we instead compute the simplest measurable single particle quantity, the density of states, $N(\\omega)=\\sum\\limits_{\\vect{k}\\sigma} A(\\vect{k},\\omega)$, which can be related to the tunnelling conductance, $dI\/dV$ through\n\\begin{equation}\n\\frac{dI}{dV}=-\\int d\\omega f^\\prime(\\omega-{\\rm V})N(\\omega).\n\\end{equation}\nThe density of states has been examined several times using the YRZ ansatz with the inclusion of a d-wave pseudogap.\\cite{borne:2010,storey:2013}  \nSuch a calculation can be fit to experimental results of the density of states which show a single asymmetric suppression of states near to the Fermi level.\\cite{borne:2010, pushp:2009}  \nHere we consider the impact of an $s$-wave pseudogap, which does not vanish at the node and is therefore expected to produce a reduction of $N(\\omega)$ away from the Fermi level.\n\nIn the cuprates, STM has been most successful for observing the occurrence of spatial gaps near to the Fermi surface, identified by sharp quasiparticle peaks at $\\pm \\Delta$ centered at the origin of the superconducting gap, the Fermi level.  It is also well known that the inclusion of a bosonic density of states as is the case for phonon-mediated superconductivity creates an imprint of the bosonic couplings at energies away from the Fermi level.\\cite{carbotte:2010}  Such couplings are thought to be responsible for peak-dip-hump structures seen in the cuprates.\\cite{fischer:2007}  In this case, we will see the impact of a simplified hybridization coupling and show that there are relevant structures in $N(\\omega)$ at frequencies away from the Fermi level, which are indeed sensitive to the structure of the pseudogap order.\n\nWe present in Fig.~\\ref{fig:dos}(b) a representative case of the density of states for $x=0.12$ to illustrate the effects of a gradually increasing gap near the pseudogap onset.\n  In our present formulation, the particle-hole  asymmetry results in two separate effects; a suppression of $N(\\omega)$ near to zero bias, and a secondary depressed structure at positive bias.  Separating the two features is a hump structure caused by weight shifted from below (antinodal region) and above (nodal region) in energy.\\cite{fischer:2007, gabovich:2013} \nIn order to understand the necessity of two distinct gaps from a single order, we show for the $A_0=0.5$ case the spectral density, $A(\\vect{k},\\omega)$, as a function of $\\vect{k}_B$ in Fig.~\\ref{fig:dos}(a) using light-blue shading for the region of width $\\Delta_{\\rm pg}$ in both the nodal and anti-nodal sectors and light-red shading for the region in-between.  Considering these regions in Fig.~\\ref{fig:dos}(b), the light-red region shows an increase in $N(\\omega)$ as compared to the normal state, while the light-blue regions are suppressed as compared to the normal state.  This means that a single momentum independent order parameter has resulted in three separate energy features due to the particle-hole asymmetry of the antibonding dispersion.\nWe also provide a contrasting example with a $d-$wave order (dashed-green).  The primary distinction is that the density of states is no longer suppressed at positive energies, and instead has a single suppression near the Fermi level.  This is the most clear distinction between $s-$ and $d-$wave ordering from a momentum averaged quantity.\nThe information regarding the value of the pseudogap order parameter in the nodal region is located at finite energies, not near to the Fermi level, where STM experiments generally focus.\n\nIt is worth remarking that such a positive bias suppression may have been seen experimentally (See for example, Fig.3(e) in Ref.~\\cite{hanaguri:2004}) in connection with static `checkerboard' charge ordering.  Such charge ordering models can be phrased as a hybridization phenomenology with one or more scattering $Q$ vectors and if the spatial modulation is correctly accounted for, the resulting spectra exhibit both a zero bias and positive bias suppression separated by a hump, suggestive of an isotropic in momentum electronic gap with a strongly particle\u00ad-hole asymmetric dispersion.\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=.85\\linewidth]{fig4.eps}\n    \\caption{(Color Online) (a)  Reproduction of Fig.~\\ref{fig:nacont}(b) with shading to identify regions of suppressed (blue) and enhanced (red) density of states.  (b)Normalized density of states, $N(\\omega)$ for increasing $\\Delta_{\\rm pg}^s$ at fixed doping of $x=0.12$.  Marked with arrows are the suppression of $N(\\omega)$ near the Fermi level due to a loss of antinodal (AN) states and a new feature at positive bias due to the loss of nodal (N) states at energies above the Fermi level. Each curve is offset by 0.2 for clarity.  The dashed black curve is $A_0=0$, with no offset to emphasize the regions of enhanced and suppressed $N(\\omega)$. A single case for $\\Delta_{\\rm pg}^d$ at $A_0=0.5$ is shown (dashed-green).}\n\\label{fig:dos}\n\\end{figure}\n\n\n\n\\section{Conclusions}\\label{sec:conc}\nWe have modified a successful ansatz to include a momentum independent hybridization gap and have shown how such a phenomenology results in a false $d$-wave pseudogap along the Fermi surface.   \nWe then compute the electronic Raman response at the onset of a pseudogap and identify the features of a momentum independent order.  From this we assert that a suppression of Raman signal at low energies does not indicate a nodal pseudogap, but is a generic sign of the loss of antinodal Fermi surface states which reduces the intraband Drude response.  A nodal pseudogap can only be suggested experimentally by an interband transition in $\\chi_{B2g}^{\\prime \\prime}(\\omega)$ which would correlate with a similar transition in the $\\chi_{B1g}^{\\prime \\prime}(\\omega)$ spectra.\nFurther, we identify a two gap behavior in the density of states which occurs as an artifact of the dispersive nature of the bonding band, $\\epsilon_{\\vect{k}}^+$, which has separate energies in the antinodal and nodal regions of the Brillouin zone.  This two scale behaviour in $N(\\omega)$ is the clearest signature of a momentum independent pseudogap order.\n\n   \n   There exists now a significant literature comparing results of the YRZ ansatz to experimental work.\\cite{rice:2012}\nPrevious works were focused on low energy probes in the presence of a d-wave superconducting gap which dominates low energy behaviour.\\cite{illes:2009,leblanc:2009,borne:2010,james:2012, comin:2014}  We emphasize that in this work, the inclusion of a momentum independent pseudogap would not significantly impact the low energy nodal region where d-wave superconductivity dominates. This need not be the case at higher energies, as described in Fig.~\\ref{fig:dos}.\n This is a main conclusion of this work, where we have demonstrated that the density of states at energies away from the Fermi level will be sensitive to the details of the zeros of $\\epsilon_{\\vect{k}}^-$ which occur at momenta $\\vect{k}_B$.  This second gap at positive bias, and the resulting hump in between,  should be observable in tunnelling experiments which would support the assertion of a momentum independent pseudogap.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nAn identification code of a graph~\\(G\\) is a set of vertices of~\\(G\\) that form both a dominating and a separating set (Karpovsky, Chakrabarty and Levitin~\\cite{karpo}). Bounds for~\\(\\gamma(G),\\) the minimum size of an identification code for~\\(G,\\) have been evaluated for different classes of graphs: Gravier and Moncel~\\cite{grav} describe graphs whose minimum size identification code contains all vertices apart from exactly one vertex. Foucaud, Klasing, Kosowski and Raspaud~\\cite{flor2} studied constructions of identification codes for triangle-free graphs, with size at most a constant fraction of the number of vertices. Later Foucaud and Perarnau~\\cite{flor1} obtained bounds for~\\(\\gamma(G)\\) in terms of the minimum vertex degree~\\(\\delta\\) for graphs~\\(G\\) whose girth is at least five and also study the size of the identification codes for random regular graphs. Recently,~\\cite{balb} have investigated the use of graph spectra as a tool to characterize the identification codes.\n\n\n\nIn this paper, we study strong identification codes of a graph~\\(G\\) with an index~\\(r \\geq 1,\\) satisfying the additional constraint that each vertex contains at least~\\(r\\) unique neighbours in the code. (For formal definitions, we refer to Section~\\ref{sec_str_code}).  We use the probabilistic method to estimate the minimum size of an identification code and also to determine existence of graphs that allow strong identification codes with a given index.\n\n\n\nThe paper is organized as follows. In Section~\\ref{sec_str_code}, we state and prove our main result, Theorem~\\ref{thm_iden}, regarding the minimum size of a strong identification code with index~\\(r.\\) Next in Section~\\ref{sec_str_neigh}, we state and prove Theorem~\\ref{prop1} regarding existence of graphs that allow strong idenfication codes with a given index.\n\n\n\n\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Strong Identification Codes} \\label{sec_str_code}\nLet~\\(G = (V,E)\\) be a graph with vertex set~\\(V = \\{1,2,\\ldots,n\\}\\) and edge set~\\(E.\\) For a vertex~\\(v \\in G,\\) we define~\\(N(v)\\) to be the set of all vertices adjacent to~\\(v,\\) not including~\\(v\\) and let~\\(N[v] = N(v) \\cup \\{v\\}.\\) We also denote the elements of~\\(N(v)\\) as \\emph{neighbours} of~\\(v.\\) Letting~\\(\\#A\\) denote the number of elements in the set~\\(A,\\) we have the following definition.\n\\begin{Definition}\nA set of vertices~\\({\\cal C}\\) is said to be an \\emph{identification code} for~\\(G\\) with \\emph{index}~\\(r\\) if for any two distinct vertices~\\(v,u \\in G\\) we have\n\\begin{equation}\\label{ghj}\n\\#\\left(\\left(N[v] \\setminus N[u]\\right) \\bigcap {\\cal C}\\right) \\geq r.\n\\end{equation}\n\\end{Definition}\nFor~\\(r=1,\\) the above definition reduces to the concept of identification codes  as studied in Gravier and Moncel~\\cite{grav} and Foucaud and Perarnau~\\cite{flor1}.\n\nClearly, one necessary condition for~\\(G\\) to have an identification code with index~\\(r\\) is that the neighbourhood of any vertex contains at least~\\(r\\) distinct vertices not present in any other vertex, i.e., for any two distinct vertices~\\(u,v \\in G,\\) we must have that\n\\begin{equation}\\label{iden_eq}\n\\#\\left(N[v] \\setminus N[u] \\right) \\geq r.\n\\end{equation}\nIf~(\\ref{iden_eq}) holds, we say that~\\(G\\) has the~\\(r-\\)strong neighbourhood property. In the next Section~\\ref{sec_str_neigh}, we use the probabilistic method to establish the existence of graphs satisfying~(\\ref{iden_eq}).\n\n\nIf the~\\(r-\\)strong neighbourhood property holds, then the whole vertex set of~\\(G\\) is itself an identification code with index~\\(r.\\) For graphs with stronger neighbourhoods, we have the following result regarding the minimum size of an identification code with a given index.\n\\begin{Theorem}\\label{thm_iden} Let~\\(G\\) be a graph on~\\(n\\) vertices satisfying the~\\((r+d+1)-\\)strong neighbourhood property for some integer~\\(d \\geq 1.\\) If~\\(\\theta_n = \\theta_n(G,r,d)\\) is the minimum size of an identification code of~\\(G\\) with index~\\(r,\\) then\n\\begin{equation}\\label{theta_bounds}\n\\frac{1}{\\Delta+1} \\leq \\frac{\\theta_n}{n} \\leq 1-\\frac{c(d,r)}{(\\Delta+1)^{(r+2)\/d}},\n\\end{equation}\nwhere~\\(\\Delta \\geq 2\\) is the maximum degree of a vertex in~\\(G\\) and\\\\\\(c(d,r) := d(d+1)^{-1-1\/d}(2r)^{-1\/d}.\\)\n\\end{Theorem}\nThus if~\\(G\\) has~\\((r+d+1)-\\)strong neighbourhood property for some integer~\\(d \\geq 1,\\) then there exists a \\emph{nontrivial} fraction of vertices of~\\(G\\) that act as an identification code with index~\\(r.\\)\n\n\n\\subsection*{Proof of Theorem~\\ref{thm_iden}}\nThe lower bound in~(\\ref{theta_bounds}) is obtained as follows. If~\\({\\cal C}\\) is an identification code with index~\\(r,\\) then~\\({\\cal C}\\) is also a dominating set in the sense that~\\(\\bigcup_{v \\in {\\cal C}} N[v] = \\{1,2,\\ldots,n\\}.\\) Since~\\(\\#N[v] \\leq \\Delta+1,\\) it follows that~\\(\\#{\\cal C} \\geq \\frac{n}{\\Delta+1}.\\)\n\n\nTo prove the upper bound in~(\\ref{theta_bounds}), we use the probabilistic method similar to Theorem~1.2.2, pp. 4, Alon and Spencer~\\cite{alon}, that concerns the minimum size of a dominating set. We select each vertex of~\\(G\\) independently with probability~\\(q\\) to be determined later and let~\\(Z\\) be the random set of vertices selected.\nSay that a vertex~\\(v \\in G\\) is \\emph{bad} if one of the following two conditions hold: either~\\((a)\\) \\(\\#\\left(N[v] \\bigcap Z \\right) \\leq r-1\\) or~\\((b)\\)~\\(\\#\\left(N[v] \\bigcap Z\\right) \\geq r\\) but there exists a vertex~\\(w\\) satisfying\n\\begin{equation}\\label{gen_h}\n\\#\\left((N[v]\\setminus N[w]) \\bigcap Z\\right) \\leq r-1.\n\\end{equation}\nA vertex of~\\(G\\) which is not bad is said to be \\emph{good}.\n\nIf~\\(Y_b\\) is the set of all bad vertices and~\\(Z_b = \\bigcup_{v \\in Y_b} N[v],\\) then the set~\\(Y := Z \\cup Z_b\\) is an identification code for~\\(G\\) with index~\\(r:\\) Indeed, if~\\(v\\) is a good vertex, then for any vertex~\\(u \\neq v\\) we have\n\\[\\#\\left((N[v] \\setminus N[u]) \\cap Y\\right) \\geq \\#\\left((N[v] \\setminus N[u])\\cap Z\\right) \\geq r.\\]\nOn the other hand if~\\(v\\) is a bad vertex, then for any vertex~\\(u \\neq v,\\) we have by the~\\((r+d+1)-\\)strong neighbourhood property that\n\\begin{eqnarray}\n\\#\\left((N[v] \\setminus N[u]) \\cap Y\\right) &\\geq& \\#\\left((N[v] \\setminus N[u]) \\cap Z_b\\right)  \\nonumber\\\\\n&=& \\#\\left(N[v] \\setminus N[u]\\right) \\nonumber\\\\\n&\\geq& r+d+1. \\nonumber\n\\end{eqnarray}\n\n\nWe now estimate the size of~\\(Z_b\\) by bounding the probability that a vertex~\\(v \\in G\\) is bad. If~\\(\\delta(v) := \\#N[v],\\) then the probability that case~\\((a)\\) occurs is\n\\begin{equation}\\label{case_a}\nf_1(q,v) := \\sum_{l=0}^{r-1} {\\delta(v) \\choose l} q^{l}(1-q)^{\\delta(v)-l}.\n\\end{equation}\nNow if possibility~\\((b)\\) occurs and some vertex~\\(w\\) satisfies~(\\ref{gen_h}), then either~\\(w\\) is a neighbour of~\\(v\\) or~\\(w\\) is at distance two from~\\(v\\) and shares a common neighbour with~\\(v.\\) Letting~\\(\\delta(v,w) := \\#(N[v] \\setminus N[w]),\\) the probability that case~\\((b)\\) occurs is then bounded above by\n\\begin{equation}\nf_2(q,v) := \\max_{w} \\sum_{l=0}^{r-1} {\\delta(v,w) \\choose l} q^{l}(1-q)^{\\delta_1(v,w)-l},\n\\label{case_b_one}\n\\end{equation}\nwhere the maximum is taken over all vertices~\\(w\\) at a distance of at most two from~\\(v.\\) Since the maximum degree of any vertex is at most~\\(\\Delta,\\) there are at most~\\(\\Delta + \\Delta(\\Delta-1) = \\Delta^2\\) neighbours at a distance of at most two from~\\(v.\\) Consequently, the probability that vertex~\\(v\\) is bad is upper bounded by~\\(f_1(q,v) + \\Delta^2 f_2(q,v)\\) and so the expected number of vertices in~\\(Y_b\\) is at most~\\(n( f_1(q,v) + \\Delta^2 f_2(q,v)).\\) This in turn implies that the expected number of vertices in~\\(Z_b =\\bigcup_{v \\in Y_b}N[v]\\) is at most~\\(n(\\Delta f_1(q,v) + \\Delta^3 f_2(q,v)).\\)\n\n\n\n\nSummarizing, we get from the discussion in the previous paragraph that the expected number of vertices in~\\(Z \\cup Z_b\\) is at most\n\\begin{equation}\\label{bad_exp}\nn\\left(q + \\Delta \\max_{v} f_1(q,v) + \\Delta^3 \\max_{v} f_2(q,v)\\right).\n\\end{equation}\nTo find suitable bounds for~\\(f_1(q,v)\\) and~\\(f_2(q,v),\\) we first let~\\[\\delta := \\min_{v} \\#N(v) \\geq r+d\\] be the minimum degree of a vertex in~\\(G.\\) From~(\\ref{case_a}) we have that\n\\[f_1(q,v) \\leq (1-q)^{\\delta-r+1} \\sum_{l=0}^{r-1} {\\delta(v) \\choose l} q^{l} \\leq (1-q)^{d+1} \\sum_{l=0}^{r-1} {\\delta(v) \\choose l} q^{l}\\] and using~\\({\\delta(v) \\choose l} \\leq \\delta^{l}(v) \\leq (\\Delta+1)^{l}\\) and~\\(q \\leq 1,\\) we get that\n\\begin{equation}\\label{f_one_up}\nf_1(q,v)\\leq (1-q)^{d+1} \\sum_{l=0}^{r-1}(\\Delta+1)^{l} \\leq r(\\Delta+1)^{r-1}  (1-q)^{d+1}.\n\\end{equation}\n\nFor estimating~(\\ref{case_b_one}), we argue similarly and use~\\(q \\leq 1\\) and the fact that~\\[\\delta(v,w)  = \\#(N[v] \\setminus N[w]) \\geq r+d+1\\] by the~\\((r+d+1)-\\)strong neighbourhood property, to get that\n\\begin{equation}\\label{f_2_up}\nf_2(q,v) \\leq (1-q)^{d+2} \\sum_{l=0}^{r-1} {\\delta(v,w) \\choose l} \\leq r(\\Delta+1)^{r-1} (1-q)^{d+2}.\n\\end{equation}\nPlugging the estimates~(\\ref{f_one_up}) and~(\\ref{f_2_up}) into~(\\ref{bad_exp}), we get that the expected number of vertices in the identification code~\\(Z \\cup Z_b\\) is at most~\n\\begin{eqnarray}\n\\mathbb{E} \\left(\\#(Z \\cup Z_b)\\right) &\\leq& n q+ r(\\Delta+1)^{r} (1-q)^{d+1} + r(\\Delta+1)^{r+2} (1-q)^{d+2} \\nonumber\\\\\n&\\leq& n(q+ 2r(\\Delta+1)^{r+2}(1-q)^{d+1}) \\nonumber\\\\\n&=:& n \\Gamma(q).\n\\end{eqnarray}\n\nThe solution~\\(q_0\\) to the equation~\\(\\Gamma'(q) = 0\\) satisfies\n\\[1-2r(\\Delta+1)^{r+2}(d+1)(1-q_0)^{d} =0\\] and so~\\[\\Gamma(q_0) =q_0 + \\frac{1-q_0}{d+1} =  1-\\frac{d}{d+1}  \\left(\\frac{1}{2r(\\Delta+1)^{r+2}(d+1)}\\right)^{\\frac{1}{d}}.\\]\nConsequently, there exists at least one identification code with index~\\(r\\) and size at most~\\(n \\Gamma(q_0).\\)~\\(\\qed\\)\n\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Strong Neighbourhood Graphs} \\label{sec_str_neigh}\nIn this section, we use the probabilistic method to obtain~\\(r-\\)strong neighbourhood graphs for any integer~\\(r \\geq 1.\\)\n\nThe cycle~\\(C_n\\) on~\\(n \\geq 4\\) vertices satisfies the~\\(1-\\)strong neighbourhood property. The following result establishes the existence of bounded degree graphs with~\\(w-\\)strong neighbourhoods, for any integer~\\(w\\geq 2.\\)\n\\begin{Theorem}\\label{prop1} Let~\\(w \\geq 2\\) be fixed. There exists an absolute constant~\\(C > 0\\) not depending on~\\(w\\) such that if~\\(n \\geq \\max(C,160(w+1)^2+1) := M(w),\\) then there exists a connected graph~\\(G\\) on~\\(n\\) vertices satisfying the~\\(w-\\)strong neighbourhood property and having maximum vertex degree~\\[\\Delta(G) \\leq \\max(10\\log(2M(w)),8(w+1))+1.\\]\n\\end{Theorem}\nThus for any~\\(w \\geq 2\\) and all~\\(n\\) large, there are bounded degree graphs on~\\(n\\) vertices satisfying the~\\(w-\\)strong neighbourhood property.\n\nWe use the probabilistic method to prove Theorem~\\ref{prop1}, beginning with the following auxiliary Lemma.\n\\begin{Lemma}\\label{deg_lem}\nLet~\\(y \\geq 3\\) be fixed. There exists an absolute constant~\\(C > 0\\) not depending on~\\(y\\) such that if~\\(n \\geq \\max(C,160y^2+1),\\) then there exists a connected graph~\\(G\\) satisfying the~\\((y-1)-\\)strong neighbourhood property and having maximum vertex degree of at most~\\(\\max(32\\log{n},8y).\\)\n\\end{Lemma}\n\nTo prove Lemma~\\ref{deg_lem}, we use the following standard deviation estimate. Let~\\(\\{X_j\\}_{1 \\leq j \\leq L}\\) be independent Bernoulli random variables with~\\(\\mathbb{P}(X_j = 1) = 1-\\mathbb{P}(X_j = 0) > 0.\\) If~\\(T_L = \\sum_{j=1}^{L} X_j,\\theta_L = \\mathbb{E}T_L\\) and~\\(0 < \\epsilon \\leq \\frac{1}{2},\\) then\n\\begin{equation}\\label{conc_est_f}\n\\mathbb{P}\\left(\\left|T_L - \\theta_L\\right| \\geq \\theta_L \\epsilon \\right) \\leq 2\\exp\\left(-\\frac{\\epsilon^2}{4}\\theta_L\\right).\n\\end{equation}\nFor a proof of~(\\ref{conc_est_f}), we refer to Corollary A.1.14, pp. 312 of Alon and Spencer (2008).\n\n\\emph{Proof of Lemma~\\ref{deg_lem}}: We use the probabilistic method. Let~\\(H = G(n,p)\\) be the random graph obtained by setting each edge of the complete graph~\\(K_n\\)\non~\\(n\\) vertices to be independently present with probability~\\(p,\\) where\n\\begin{equation}\\label{p_def}\np = \\frac{\\max(16\\log{n},4y)}{n-1}\n\\end{equation}\nand absent otherwise.\n\n\nLetting~\\(N(1)\\) denote the neighbours of the vertex~\\(1,\\) we have that\\\\\\(\\mathbb{E}\\#N(1) = (n-1)p\\) and so using~(\\ref{conc_est_f}) with~\\(\\epsilon = \\frac{1}{2},\\) we get\n\\begin{equation}\\label{n_one_est}\n\\mathbb{P}\\left(\\frac{(n-1)p}{2} \\leq \\#N(1) \\leq \\frac{3(n-1)p}{2}\\right) \\geq 1-\\exp\\left(-\\frac{(n-1)p}{4}\\right) \\geq 1-\\frac{1}{n^4},\n\\end{equation}\nby~(\\ref{p_def}). Letting\n\\[A_{tot} := \\bigcap_{1 \\leq i \\leq n} \\left\\{\\frac{(n-1)p}{2} \\leq \\#N(1) \\leq \\frac{3(n-1)p}{2}\\right\\},\\] we get by the union bound and~(\\ref{n_one_est}) that\n\\begin{equation}\\label{a_tot_est}\n\\mathbb{P}(A_{tot}) \\geq 1-\\frac{1}{n^3}.\n\\end{equation}\nThus, with high probability, i.e., with probability converging to one as~\\(n~\\rightarrow~\\infty,\\) each vertex in the random graph~\\(H\\) contains at most~\\(2(n-1)p = \\max(32\\log{n}, 8y)\\) neighbours.\n\n\n\n\nNext, we estimate the number of common neighbours between the two vertices~\\(i\\)\nand~\\(j\\) as follows. Letting~\\(T_{ij} := \\#(N(i) \\cap N(j))\\) and using the Chernoff bound, we have for~\\(s,t >0\\)\nthat\n\\begin{equation}\\label{eq_one}\n\\mathbb{P}\\left(T_{ij} > t\\right) \\leq \\left(\\mathbb{E}e^{sT_{ij}}\\right)^{n-2}e^{-st}.\n\\end{equation}\nEach vertex~\\(z \\neq i,j\\) is adjacent to both~\\(i\\) and~\\(j\\) with probability~\\(p^2,\\) independent of the other vertices and so\n\\begin{equation}\\label{eq_two}\n1 < \\mathbb{E}e^{sT_{ij}} = 1 + (e^{s}-1)p^2 \\leq \\exp\\left((e^{s}-1) p^2\\right).\n\\end{equation}\nSubstituting~(\\ref{eq_two}) into~(\\ref{eq_one}) and using~\\((\\mathbb{E}e^{sT_{ij}})^{n-2} \\leq (\\mathbb{E}e^{sT_{ij}})^{n-1},\\) we get\n\\begin{equation}\\label{eq_3}\n\\mathbb{P}\\left(T_{ij} > t\\right) \\leq \\exp\\left((e^{s}-1) (n-1)p^2\\right)e^{-st}.\n\\end{equation}\n\nFor~\\(s=2,\\) we have\n\\begin{eqnarray}\n(e^{s}-1)(n-1)p^2 &=& (e^{2}-1)\\frac{\\max(256(\\log{n})^2,16y^2)}{n-1} \\nonumber\\\\\n&\\leq& \\frac{1}{n-1}\\max(2560(\\log{n})^2,160y^2), \\nonumber\n\\end{eqnarray}\nwhich is at most one, provided that~\\(n \\geq 1+\\max(160y^2,n_0),\\) where~\\(n_0\\) is the smallest integer~\\(x\\) such that~\\(x > 2560(\\log{x})^2.\\)\nFor all~\\(n \\geq 1+\\max(160y^2,n_0),\\) we therefore have from~(\\ref{eq_3}) that\n\\begin{equation}\\label{eq_32}\n\\mathbb{P}\\left(T_{ij} > \\frac{(n-1)p}{4}\\right) \\leq e\\exp\\left(-\\;\\frac{(n-1)p}{2}\\right) \\leq \\frac{e}{n^{8}},\n\\end{equation}\nsince~\\((n-1)p \\geq 16\\log{n}\\) by~(\\ref{p_def}). Thus,\n\\[\\mathbb{P}\\left(\\#\\left(N(i) \\cap N(j)\\right) \\leq \\frac{(n-1)p}{4}\\right)  \\geq 1-\\frac{e}{n^{8}}\\]\nand, using the union bound, we have\n\\begin{equation}\\label{hu}\n\\mathbb{P}\\left(\\bigcap_{i \\neq j} \\left\\{\\#\\left(N(i) \\cap N(j)\\right) \\leq \\frac{(n-1)p}{4}\\right\\}\\right)  \\geq 1-\\frac{e}{n^{6}}.\n\\end{equation}\n\nFrom~(\\ref{a_tot_est}) and~(\\ref{hu}), we get that with high probability, the maximum degree of the random graph~\\(H\\) is at most~\\(2(n-1)p = \\max(32\\log{n},8y).\\)\nIn addition, for any two vertices~\\(v \\neq u,\\) the corresponding neighbourhood sets satisfy\n\\[\\#\\left(N[v] \\setminus N[u]\\right) \\geq \\frac{(n-1)p}{4}-1 \\geq y-1.\\]\nConsequently, with high probability, the graph~\\(H\\) satisfies the~\\((y-1)-\\)strong neighbourhood property\nand because~\\(p\\geq \\frac{16\\log{n}}{n}\\) (see~(\\ref{p_def})), the graph~\\(H\\) is also connected with high probability (pp. 164--165, Bollob\\'as~\\cite{boll}). Summarizing, there exists an absolute constant~\\(C > 0\\) not depending on~\\(y\\) such that if\\\\\\(n \\geq \\max(C,160y^2~+~1),\\) then with high probability~\\(H\\) is connected,\nsatisfies the~\\((y-1)-\\)strong neighbourhood property and has a maximum vertex degree of at most~\\(\\max(32\\log{n},8y).\\)~\\(\\qed\\)\n\n\n\\emph{Proof of Theorem~\\ref{prop1}}: Let~\\(w \\geq 2\\) and~\\(M(w)\\) be as in the statement of Theorem~\\ref{prop1}. For~\\(n \\geq M(w),\\) split~\\(\\{1,2,\\ldots,n\\}\\)\ninto~\\(T\\) sets~\\(V_1,\\ldots,V_T\\) such that~\\(V_i, 1 \\leq i \\leq T-1,\\) has size exactly~\\(M(w)\\) and~\\(V_T\\) has size at least~\\(M(w)\\)\nand at most~\\(2M(w).\\) Using Lemma~\\ref{deg_lem} with~\\(w = y-1,\\) we know that there exists a connected graph~\\(G_i\\) with vertex set~\\(V_i\\) satisfying the~\\(w-\\)strong neighbourhood property and having a maximum vertex degree of at most\\\\\\(\\Delta_0 = \\max(32\\log(2M(w)),8(w+1)).\\) We pick one vertex~\\(u_i \\in V_i\\) and connect~\\(u_i\\) to~\\(u_{i+1}\\) for~\\(1 \\leq i \\leq T-1.\\) The resulting graph~\\(G\\) is connected, satisfies the~\\(w-\\)strong neighbourhood property and has a maximum vertex degree of at most~\\(\\Delta_0+1.\\)~\\(\\qed\\)\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgement}\nI thank Professors Rahul Roy, C. R. Subramanian and the referees for crucial comments that led to an improvement of the paper. I also thank IMSc for my fellowships.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\n\\label{sec:introduction\nModulation-doped $p$-type (Cd,Mn)Te\/(Cd,Mg,Zn)Te quantum wells (QWs)\nremain a unique medium allowing to probe carrier-induced Ising-like ferromagnetism in the two-dimensional (2D) case,\\cite{Haury:1997,Kossacki:2000,Kossacki:2002,Kossacki:2006,Maslana:2006} as in this system the mean free path is longer than the QW width.\nThe presence of ferromagnetism was revealed\nby the observation of spontaneous splitting of the photoluminescence (PL) spectrum,\nwhich indicated that local spin ordering is stable, below the Curie temperature $T_{\\mathrm{C}}$,\nfor times longer than the exciton lifetime.\\cite{Kossacki:2000,Boukari:2002}\nTheoretical analyzes have shown that the magnitude of $T_{\\mathrm{C}}$ as a function of the Mn concentration $x$ and the hole areal density $p$\ncan be quantitatively described provided that in addition to carrier-mediated ferromagnetic spin-spin couplings,\nthe presence of competing short-range antiferromagnetic superexchange interactions as well as\nhole correlation effects are taken into account.\\cite{Dietl:1997,Haury:1997,Dietl:1999,Boukari:2002,Kechrakos:2005}\nThese experiments and their quantitative interpretations were carried out prior to later theoretical studies.\\cite{Lee:2000,Brey:2000,Priour:2005}\n\nA surprising result of both optical and magnetic measurements is the absence of hysteresis loops and,\nhence, of macroscopic spontaneous magnetization below $T_{\\mathrm{C}}$.\\cite{Kossacki:2000,Kossacki:2002}\nThis finding questions our understanding of the actual ground state of the system.\nIt has been suggested, in particular, that the formation of spin density waves,\ndriven by the $q$-dependent carrier susceptibility, may account for the absence of spontaneous magnetization.\\cite{Kossacki:2000}\nThis possibility has recently been considered in the context of\ntheoretical search for carrier-mediated nuclear magnetism in 2D systems.\\cite{Simon:2008}\n\nIn order to determine the importance of various phenomena that control magnetism in such reduced-dimensionality magnetically disordered systems\nwe have carried out time-resolved magneto-optical measurements.\nFurthermore, with the goal to obtain information on mechanisms controlling spin dynamics,\nwe have extended our previous\\cite{Kechrakos:2005} Monte Carlo (MC) simulations of magnetization in\n$p-\\mathrm{Cd}_{1-x}\\mathrm{Mn}_{x}\\mathrm{Te}$ QW.\nThe experimental findings provide a dependence of magnetization relaxation on temperature and carrier density.\nAbove $T_{\\mathrm{C}}$ the relaxation time is found to be shorter than 20~ns,\nwhile below $T_{\\mathrm{C}}$ magnetization persists up to a few microseconds,\na time scale consistent with the absence of spontaneous magnetization in the static measurements.\nInterestingly, we have obtained a similar relative prolongation of the relaxation time below $T_{\\mathrm{C}}$\nfrom MC simulations based on the Metropolis algorithm and on the determination of the one-particle hole eigenfunctions at each MC sweep.\nThe simulations explain also the rate of increase of the relaxation time with the carrier density.\nWe conclude that the absence of magnetic hysteresis in this 2D Ising system can be explained without taking carrier-carrier correlation into account.\n\nAt the same time, by analyzing results of our Monte Carlo simulations,\nwe are able to show that short range spin-spin antiferromagnetic (AFM) interactions  play a crucial role in accelerating magnetization dynamics.\nThis corroborates the outcome of the previous results showing a narrowing of the hysteresis loop by AFM interactions.\\cite{Kechrakos:2005}\nHowever, we realize now, based on much more extensive simulations, that the effect of AFM interactions becomes much reduced\nif the thickness of the layer containing Mn spins is taken to be smaller than the extent of the hole wave function.\nThis finding, from the one hand, substantiates theoretical considerations of Boudinet and Bastard,\\cite{Boudinet:1993}\nsuggesting that magnetization relaxation of bound magnetic polarons in $p$-type CdTe\/(Cd,Mn)Te QWs occurs\nowing to the AFM coupling to the Mn spins located outside the relevant Bohr radius.\nOn the other, this result implies that macroscopic spontaneous magnetization and magnetic hysteresis should be recovered in quantum wells,\nin which the thickness of the Mn layer would be smaller than the region visited by the holes.\n\nFurthermore, we study how a spin-independent part of the potential introduced by Mn impurities affects $T_{\\mathrm{C}}$.\nOur simulations show that alloy disorder tends to reduce $T_{\\mathrm{C}}$.\nThe effect is particularly dramatic for the attractive alloy potential  which, if sufficiently large, leads to a strong hole localization.\nThis result substantiates the notion that delocalized or weakly localized carriers are necessary\nto generate a sizable ferromagnetic coupling between diluted localized spins.\n\nThis article is organized as follows.\nIn Sec.~\\ref{sec:experiments} we present experimental findings of magneto-optical studies on  magnetization dynamics,\nwhich have motivated the theoretical effort.\nThe theoretical model\nthat describes the carrier-induced ferromagnetism in\ndiluted magnetic semiconductor (DMS) quantum wells is introduced\nin Sec.~\\ref{sec:theory}.\nIn Sec.~\\ref{sec:tc} we present MC investigations of\nthe dependence of the critical temperature on the carrier density and spin-independent alloy potential. Section~\\ref{sec:dyn-dom} contains results of our MC studies of spin dynamics and of spin ordering, which serve to interpret the findings of magneto-optical measurements.\nSpecifically,\nsubsection~\\ref{subsec:dyn} is devoted to magnetization dynamics, while\nsubsection~\\ref{subsec:dom} contains a discussion of possible domain sizes.\nOur conclusions are summarized in Sec.~\\ref{sec:conclusion}.\n\n\n\n\\section{Experimental results\n\\label{sec:experiments\nThe experimental results have been obtained for samples grown in Grenoble by molecular beam epitaxy on $\\mathrm{Cd}_{0.88}\\mathrm{Zn}_{0.12}\\mathrm{Te}$ (001) substrates.\\cite{Haury:1997,Kossacki:2004}\nThe modulation-doped $p$-type structures\ncontain a single 8~nm wide  $\\mathrm{Cd}_{1-x}\\mathrm{Mn}_{x}\\mathrm{Te}$ QW embedded by\n$\\mathrm{Cd}_{1-y-z}\\mathrm{Mg}_{y}\\mathrm{Zn}_{z}\\mathrm{Te}$ barriers, in which the Mg\ncontent ($y$ = 0.25-0.28) results in a sizable valence band offset, while\nthe presence of Zn ($z$ = 0.07--0.08) ensures a good lattice match to the substrate.\nThe front barrier is doped by nitrogen acceptors.\nThe distance between the QW and the doping layer is 20~nm, which results in the\nhole density up to $3\\times10^{11} \\mathrm {cm^{-2}}$.\\cite{Kossacki:2004} These holes occupy the ground state subband whose scattering broadening is much smaller than the distance to the next subband, making the system to be truly 2D.\n\nAccording to the well established procedure, magnetic properties of the QW are probed by\nmagnetospectroscopy.\\cite{Haury:1997,Kossacki:2000,Boukari:2002}\nThe photoluminescence (PL) is excited by a cw laser with the photon energies\nbelow the absorption edge of the barrier material.\nA typical PL spectrum of a paramagnetic $p$-type QW consists of a single line\nrelated to the radiative recombination of charged excitons ($\\mathrm{X^{+}}$).\nDepending on its polarization chirality, the emission line exhibits a downward or upward shift in the magnetic field.\nOwing to a strong $sp$-$d$ exchange interaction specific to DMSs, this Zeeman shift is giant (typically over 20~meV in 1~T), and\nits magnitude is proportional to the local magnetization of the Mn spins.\nImportantly, a combination of strain, confinement, and spin-orbit interaction makes that\nthe giant shift vanishes virtually entirely for the in-plane magnetic field.\nBelow a critical temperature, which we identify as the Curie temperature $T_{\\mathrm{C}}$,\na spontaneous splitting of the PL line into two components is observed in the absence of an external magnetic field.\nThe energy distance between these two lines, or the downward shift of the lower line,\nprovides information on an average value of spontaneous magnetization in the regions visited by the holes at given temperature.\n\nSurprisingly, the emitted light at the wavelength corresponding to either of these two lines remains unpolarized,\neven after ramping the field down to zero below $T_{\\mathrm{C}}$.\nA magnetic field of the order of 20~mT, much higher than the demagnetization field for this diluted alloy,\nis necessary to achieve a full circular polarization of the lines at $T \\approx 0.5T_{\\mathrm{C}}$.\\cite{Kossacki:2000}\nThis indicates the absence of a macroscopic spontaneous magnetization at these low temperatures,\nthe conclusion confirmed by a direct magnetization measurements.\\cite{Sawicki:2002}\nMoreover, even focusing detection on the spot with diameter below 1~$\\mu$m does not result in the circularly polarized emission.\nThis could be explained either by a small size of the relevant magnetic domains or by\nfast fluctuations of the magnetization direction.\nAlternatively, a spin-density wave could be considered as the relevant ground state of this reduced dimensionality correlated system.\nHowever, a selective excitation of PL by circularly polarized light leads to circularly polarized PL.\\cite{Kossacki:2002}\nThis indicates that local spin ordering is stable for times longer than the exciton lifetime, estimated to be in the 100~ps range.\n\nSome of us have recently developed a technique for studies of magnetization dynamics by means of time resolved PL measurements\nperformed after a short pulse of the magnetic field.\\cite{Kossacki:2006,Maslana:2006}\nPulses of the magnetic field are produced by a magnetic coil mounted at the surface of the sample.\nThe illumination with a laser beam and the collection of the PL signal are performed\nalong the axis of the coil (Faraday configuration).\nThe coil diameter of 0.5~mm results in a small value of inductance and allows to obtain short rise and fall times of about 10~ns.\nA 2~A current produces a magnetic field of about 40~mT.\nThe time evolution of the PL during and after the pulse is probed with resolution down to 10~ns.\nMagnetization dynamics is studied by analyzing the difference in PL intensities\nfor two circular polarizations after the pulse of the magnetic field.\nResults of measurements shown here have been obtained at different temperatures below and above\n$T_{\\mathrm{C}}$, which is 2.5~K for the studied $\\mathrm{Cd}_{0.96}\\mathrm{Mn}_{0.04}\\mathrm{Te}$ QW.\n\nAbove $T_{\\mathrm{C}}$, in the paramagnetic state, a single PL line is observed at zero field, and\nthe signal induced by the magnetic pulse is related to Zeeman splitting of this line.\nIts relaxation time is found to be shorter than 20~ns.\nSuch a value is just to be expected for (Cd,Mn)Te containing 4\\% of Mn.\\cite{Scalbert:1988, Dietl:1995}\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys1.pdf}\n\\caption{Experimental: Temporal decay of magnetization in the Cd$_{0.96}$Mn$_{0.04}$Te quantum well at various temperatures\ndetermined as a difference in photoluminescence intensities collected for two circular polarizations\nafter a pulse of the magnetic field. The Curie temperature is 2.5~K.}\n\\label{fig:corfun1}\n\\end{figure}\n\nTypical temporal profiles for three different temperatures below $T_{\\mathrm{C}} = 2.5$~K are presented in Fig.~\\ref{fig:corfun1}.\nThe relaxation time becomes much slower when decreasing temperature (see Fig.~\\ref{fig:corfun1}).\nMagnetization persists up to a few microseconds when $T \\approx 0.5T_{\\mathrm{C}}$.\nHowever even for the lowest temperatures, the relaxation time is shorter than 10~$\\mu$s, and\nunder all experimental conditions the remanent magnetization vanishes totally after 20~$\\mu$s.\nThis time scale is much shorter than an acquisition time of standard optical measurements,\nwhich accounts for unmeasurably small values of the coercive field observed experimentally.\n\nIt is interesting to note that the decays are not mono-exponential.\nThe relaxation begins with a rapid decay, faster than 100~ns, but\nit also contains a slower component, in the microsecond range.\nOne could expect that regions of higher hole density have higher local magnetization\nresulting in higher barriers for the magnetization reversal.\nIn order to examine this possibility we carried out relaxation measurements for a constant temperature (below $T_{\\mathrm{C}}$), and\ndifferent detection wavelength which was tuned over the low energy component of the PL line.\nAs mentioned above, this line shows a red shift when decreasing temperature below $T_{\\mathrm{C}}$.\nThis shift is proportional to Mn magnetization, whose magnitude is primarily determined by the hole density.\\cite{Boukari:2002,Kossacki:2004}\nDue to spatial fluctuations in the hole density,\\cite{Maslana:2006} the line is significantly broadened.\nThe low energy side of the line comes from the regions containing higher hole densities,\nwhile the high energy side corresponds to locations visited by fewer holes.\nThis provides an opportunity to trace the relaxation time as a function of the hole concentration.\nThe obtained temporal magnetization profiles are presented in Fig.~\\ref{fig:corfun2}.\nIt is  seen that the contribution of the slow component increases for the regions containing a high hole density.\nThis finding as well as the observed temperature variation of the magnetization decay\ntime will be compared to results of numerical simulations in the next section.\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys2.pdf}\n\\caption{Experimental: Temporal decay of magnetization in the Cd$_{0.96}$Mn$_{0.04}$Te quantum well\ndetermined as a difference in photoluminescence intensities for two circular polarizations\ncollected at various wavelengths after a pulse of the magnetic field. The Curie temperature is 2.5~K.} \\label{fig:corfun2}\n\\end{figure}\n\n\n\n\\section{Theory and simulation procedure\n\\label{sec:theory\nTo understand the experimental results described in Sec.~\\ref{sec:experiments} we carried out Monte Carlo simulations for\nthe subsystem of Mn spins which are at the same time subject to short-range antiferromagnetic and\nlong-range hole-mediated ferromagnetic spin-spin interactions.\nAccording to our results, the presence of these two kinds of competing interactions accounts for\na non-standard character of ferromagnetism in the studied system.\nSince we are mainly interested in modulation-doped $p$-Cd$_{1-x}$Mn$_{x}$Te quantum wells,\nall our results are obtained using the parameter values characteristic for this system, but\nthe methodology we use may as well be applied to other 2D $p$-type II-VI DMS structures.\n\nThe antiferromagnetic interactions originate from  superexchange, the mechanism specific to undoped charge transfer insulators, such as (Cd,Mn)Te.\nThe character and magnitude of these interactions are known from previous extensive experimental studies of magnetic properties of (Cd,Mn)Te.\nIn order to find out how the presence of holes affects the system we compute, in the spirit of the adiabatic approximation,\nthe total energy of the holes at a given configuration of the Mn spins.\nIn this way our approach encompasses automatically the description of carrier-mediated exchange interactions within\neither $p$-$d$ Zener model or  Ruderman-Kittel-Kasuya-Yosida theory.\nThe carrier total energy is evaluated neglecting hole correlations,\nthe approximation that leads to an underestimation of the ferromagnetic coupling.\\cite{Dietl:1997}\nAccordingly, our computed Curie temperatures are expected to be systematically lower than those observed experimentally.\nFurthermore, in accord with the effective mass theory, and owing to a relatively small disorder in our structures,\nwe assume that the in-plane and the perpendicular hole motions can be factorized,\ni.e., that the total envelope function assumes the form\n$\\Psi_{\\sigma}(\\bm{R}) = \\psi_{\\sigma}(\\bm{r})\\varphi(z)$, where $\\bm{R} = (\\bm{r},z)$\nwith in-plane and perpendicular positions described by $\\bm{r} = (x,y)$ and $z$ respectively;\n$\\varphi(z)$ is the wave function of a hole residing in the ground state subband of the QW in question.\nFor the biaxial strain under consideration, the relevant subband is of the heavy hole origin,\nso that the spin-orbit interaction fixes the hole spin in the direction of the growth axis.\nFurthermore, since the heavy holes penetrate only weakly the barriers,\n$\\varphi(z)$ in the form corresponding to an infinite QW is adequate for our computations.\n\nWithin the above model and in the absence of an external magnetic field, the Hamiltonian  $\\mathcal{H}$ of our system takes the form,\n\n\\begin{equation}\\label{htotal}\n\\mathcal{H} = \\frac{p^2}{2m^*} + \\mathcal{H}_{pd} + \\mathcal{H}_{dd}.\n\\end{equation}\n\n\\noindent\nThis Hamiltonian describes noninteracting carriers in the heavy hole parabolic valence subband\nfor which the in-plane effective mass is $m^{*} = 0.25m_{0}$ for the QW in question.\\cite{Fishman:1995}\nSince the coupling between the holes and the Mn ions is weak in (Cd,Mn)Te,\\cite{Dietl:2008}\nwe can consider the $p$-$d$ interaction in a simple contact form,\n\n\\begin{equation}\\label{hpd}\n\\mathcal{H}_{pd} = \\sum_{i} \\frac{1}{3} \\beta j_{z} S_{zi}\n\\delta(\\bm{r}- \\bm{r}_{i}) \\left|\\varphi(z_{i})\\right|^{2},\n\\end{equation}\n\n\\noindent\nwhere the hole spin $j = 3\/2$, $j_{z} = \\pm $ 3\/2,\nis coupled to the Mn ions randomly distributed over the fcc lattice sites $\\bm{r}_{i}$.\nWe treat Mn spins $\\bm{S}_{i}$ as classical vectors with $S = 5\/2$.\nWhile this approximation is certainly qualitatively valid for the large spin in question, quantitatively -- according to the mean-field theory --\nit leads to an underestimation of the $T_{\\mathrm{C}}$ value by $(S+1)\/S$.\nFor the $p$-$d$ exchange integral we take $\\beta =  -5.96 \\times 10^{-2}$~eV~nm$^3$, which corresponds\nto the exchange energy $\\beta N_0 = -0.88$~eV, where $N_0$ is the cation concentration.\\cite{Gaj:1979}\n\nWe have also tested how our results are affected by the spin-independent part of the potentials introduced by Mn ions.\nTo take this alloy disorder into account we add to Eq.~\\ref{hpd} a term\n\n\\begin{equation}\\label{hAD}\n\\mathcal{H}_{\\mathrm{AD}} = \\delta V \\sum_{i} \\delta(\\bm{r}- \\bm{r}_{i})\\left|\\varphi(z_{i})\\right|^{2}\n\\end{equation}\n\n\\noindent\nwith  $\\delta V = 3.93 \\times 10^{-2}$~eV~nm$^3$\nwhich corresponds to the valence band offset $W N_0 = -\\delta V N_0 = -0.58$~eV in (Cd,Mn)Te.\\cite{Gaj:1994,Wojtowicz:1996}\nIt is known that the effect of the Mn isoelectronic impurities on the carrier wave function is governed by the ratio of the total Mn potential $U$ to its critical value $U_c <0$ at which a bound state starts to form,\\cite{Dietl:2008}\n\n\\begin{equation}\\label{uperuc}\nU\/U_c = 6m^*[W - (S + 1)\\beta\/2]\/(\\pi^3\\hbar^2b),\n\\end{equation}\n\n\\noindent\nwhere $b$ is the potential radius. In the case under consideration, the hole spin is fixed along $z$-axis,\nand Mn polarization $\\langle S_z \\rangle$ is typically below 10\\%.\nAccordingly, for $S+1 \\rightarrow \\langle S_z\\rangle$ the total Mn potential is expected to be effectively repulsive for the holes in (Cd,Mn)Te, $U\/U_c <0$. Interestingly, in many important systems, such as oxides and nitrides, the spin-independent part of the magnetic impurity potential is actually attractive.\\cite{Dietl:2008} Accordingly, we have also performed some simulations of magnetization for $\\delta V = - 3.93 \\times 10^{-2}$~eV~nm$^3$\nwhich corresponds to the valence band offset $W N_0 = -\\delta V N_0 = +0.58$~eV.\nHowever, it should be noted that by using the $\\delta$-like potentials, which corresponds to $b\\rightarrow 0$, we actually overestimate $U\/U_c$ and\ndetermine an upper limit of the effect of the alloy potential upon $T_{\\mathrm{C}}$.\nWith this in mind, we have carried out all MC {\\em dynamics} studies presented in Sec.~\\ref{sec:dyn-dom} for $\\delta V = 0$.\n\nThe short-range intrinsic antiferromagnetic (AFM) interaction between the Mn spins is given by,\n\\begin{equation}\\label{hdd}\n\\mathcal{H}_{dd} = -2k_{B} \\sum_{ij}  J_{ij} \\bm{S}_{i} \\cdot \\bm{S}_{j},\n\\end{equation}\n\\noindent\nwith $J_{ij} = -6.3, -1.9, -0.4$~K for the nearest, next nearest, and next next nearest neighbors, respectively.\\cite{Shapira:2002}\nOur MC simulations confirm that these values of $J_{ij}$ reproduce correctly the temperature dependence of the spin susceptibility\nin the absence of holes but we note that other sets of $J_{ij}$ values have also been proposed in the literature.\\cite{Hennion:2002}\n\nThe above approach disregards the presence of long-range dipole-dipole interactions between Mn spins.\nBoth theoretical evaluations and MC simulations we have carried out\nshow that the dipole-diploe interactions can be safely ignored in the case under consideration.\n\nIn order to determine hole eigenfunctions and eigenvalues for a given configuration of Mn spins\nwe assume periodic boundary conditions in the QW atomic layers and diagonalize $\\mathcal{H}$\nin a plane-wave basis with 2D wave vectors truncated\nat a radius $k_c = 5.52 (2\\pi\/L)$ or $k_c = 8.10 (2\\pi\/L)$.\\cite{Schliemann:2001}\nThis part of the 2D $k$-space corresponding to 97 or 213 $k$-states, respectively,\nis sufficient to ensure the convergence for all hole density values considered here.\nThe energy of the holes is determined by summing up the lowest eigenvalues corresponding to a given number of holes $N_\\mathrm{h}$.\nThis procedure assumes that the hole gas is degenerate, i.e., neglects thermal broadening of the hole distribution,\nthe effect considered previously,\\cite{Boukari:2002} and found to be small.\nIn order to keep $k$-space shells filled up,  we have considered $N_\\mathrm{h} = $ 1, 5, 9, 13, 21, 25, 29, 37, 45, 49, and 57,\nwhich correspond to values of hole concentrations $p$ up to $1.11 \\times 10^{11}$~cm$^{-2}$ for the chosen size of the simulation box.\n\nIn our MC simulations we employ the Metropolis algorithm in which the hole eigenfunctions and eigenvalues are updated at each MC sweep\n(in one MC sweep all Mn spins are rotated).\nThis procedure if applied after each single spin rotation would be computationally excessively time consuming and,\ntherefore, we followed the idea of the so-called perturbative Monte Carlo method.\\cite{Troung:1996,Troung:1997,Kennett:2002}\nWe typically keep 2000 initial MC sweeps (to thermalize the system),\nfollowed by $10^{4}$-$10^{5}$ MC sweeps used for a further analysis.\nMost of the numerical results presented in this paper have been obtained for the magnetic field put to zero.\nHowever, our software allows as well simulations in non-zero magnetic fields.\nWe have occasionally used this opportunity to obtain some of the initial configurations.\n\nFor the computations presented here we adopt the simulation box of dimensions $L \\times L \\times L_{W}$, where $L \\leq 350a_o$ and $L_{W} = 8a_o$,\nwith the fcc lattice constant $a_o = 0.647$~nm in our case.\nHence, the quantum well width is $L_W = 5.2$~nm and the area $L^2 = $(226~nm)$^2$.\nWe note that a typical QW width of experimental samples is somewhat grater, $L_W^{exp} = 8$~nm.\nHowever, the adoption of a smaller value of $L_W$ allows us to treat systems with a larger area and,\nat the same time, since within the mean-field theory\\cite{Dietl:1997} $T_{\\mathrm{C}} \\propto 1\/L_W$,\nresults in a partial compensation of a systematic error stemming from treating the spins classically,\n$L_W^{exp}S\/[L_W(S+1)] = 1.1$.\nWe randomly occupy 4\\% of cations sites, which corresponds to 156800 Mn spins.\nIn view of the required accuracy, this large system size makes averaging over the Mn distribution\nand extrapolation to an infinite system size unnecessary.\nThe number of Monte Carlo sweeps needed for the convergence is of the order of $10^4$ to $10^5$.\nHence, we have to use a (pseudo)random number generator with a sufficiently long period.\nAfter testing various generators, the Mersenne Twister developed by Matsumoto and Nishimura has been selected,\nbecause of its huge period and its very high order of dimensional equidistribution.\\cite{matsumoto_nishimura}\n\n\n\\section{Curie temperature\n\\label{sec:tc\nThe critical temperature $T_{\\mathrm{C}}$ has been determined in the standard way:\nby MC simulations we have calculated the temperature dependence of the spin projections\nand the spin susceptibility for both hole and magnetic ion spins.\nTo denote spins we use the notation $\\sigma = s$ (for holes) and $\\sigma = S$ (for Mn ions).\nSimilarly, $N = N_\\mathrm{h}$ (for the number of holes) and $N = N_\\mathrm{Mn}$ (for the number of Mn ions).\nAt each MC sweep, the spin projections per hole or per Mn ion are given by:\n\n\\begin{equation}\\label{sigma-per-particle}\n\\overline{\\sigma_j} = \\frac{\\sum_{i=1}^N \\sigma_{ij}}{N}, \\hspace{0.5cm} j=x,y,z.\n\\end{equation}\n\n\\noindent\nWe symbolize statistical averages of the above quantities by $\\langle ... \\rangle$.\\cite{Newman:1999}\nIn other words, e.g.\n\n\\begin{equation}\\label{sigma-st-av}\n\\langle \\sigma_j \\rangle = \\frac{\\sum_{n=1}^{nsum}\\overline{\\sigma_j}}{nsum}\n\\end{equation}\n\\noindent\nand\n\\begin{equation}\\label{sigma-st-av-abs}\n\\langle |\\sigma_j| \\rangle = \\frac{\\sum_{n=1}^{nsum}\n|\\overline{\\sigma_j}|}{nsum},\n\\end{equation}\n\\noindent\nwhere $n$ denotes successive MC sweeps used for the statistical average and\n$nsum$ denotes the total number of MC sweeps used for the statistical average.\nThe spin susceptibilities per spin are defined in the following manner:\\cite{Newman:1999}\n\n\\begin{equation}\\label{suscept}\n\\chi_{\\sigma_j} = \\frac{1}{T}[\\langle \\sigma_j^2 \\rangle - \\langle \\sigma_j \\rangle^2].\n\\end{equation}\n\\noindent\nWe associate the temperature at which the magnitude of spin susceptibility reaches a maximum\nwith the Curie temperature $T_{\\mathrm{C}}$ of the system.\nUsually, the MC computations have been performed with the step of 0.2~K.\nTherefore, an uncertainty in $T_{\\mathrm{C}}$ values is usually at least $\\pm 0.1$~K.\nSome examples of the temperature dependence of  the spin projections and of the spin susceptibilities\nare presented in Fig.~\\ref{fig:Nh13_29_of_T_+AD_noAD}.\nSpecifically, we have plotted $\\langle |S_z| \\rangle$ and $\\chi_S$ for the Mn ions\n(Figs.~\\ref{fig:Nh13_29_of_T_+AD_noAD}(a) and \\ref{fig:Nh13_29_of_T_+AD_noAD}(c))\nas well as\n$\\langle |s_\\mathrm{h}| \\rangle$ and $\\chi_\\mathrm{h}$ for the holes\n(Fig.~\\ref{fig:Nh13_29_of_T_+AD_noAD}(b) and \\ref{fig:Nh13_29_of_T_+AD_noAD}(d)),\nfor the hole numbers $N_\\mathrm{h} =  13$ and $29$, which correspond to\n$p = 2.54 \\times 10^{10}$ and $5.66 \\times 10^{10}$ cm$^{-2}$, respectively.\nIn Fig.~\\ref{fig:Nh13_29_of_T_+AD_noAD} the case when the alloy disorder is absent (noAD) is compared to the situation of the repulsive (+AD) alloy disorder potential.\nIn Fig.~\\ref{fig:Nh13_45_of_T_noAD_-AD}\nwe show the effect of the attractive (-AD) alloy disorder potential\nby referring to the data obtained without alloy disorder (noAD),\nfor $N_\\mathrm{h} = 13$ and 45, i.e., $p = 2.54 \\times 10^{10}$ and $8.78 \\times 10^{10}$ cm$^{-2}$, respectively.\n\n\\begin{figure*}\n\\includegraphics[scale=0.7]{rys12.pdf}\n\\includegraphics[scale=0.7]{rys13.pdf}\n\\includegraphics[scale=0.7]{rys14.pdf}\n\\includegraphics[scale=0.7]{rys15.pdf}\n\\caption{Monte Carlo simulations:\nthe $z$-component of the absolute spin projections\n$\\langle |S_z| \\rangle$ and $\\langle |s_z| \\rangle$ (open symbols - dotted lines; left scale) and\nthe spin susceptibilities $\\chi_S$ and $\\chi_s$ (full symbols - solid lines; right scale)\nfor the Mn ions (a,c) and for the holes (b,d),\nin the absence of alloy disorder (noAD, $\\delta V = 0$), and for the repulsive (+AD, $\\delta V > 0$) alloy disorder potential.\nA maximum in the temperature dependence of spin susceptibility is identified as the Curie temperature below which the holes induce a long range ferromagnetic order.\nHere the number of holes $N_\\mathrm{h} = 13$ (a,b) and $29$ (c,d), i.e., $p = 2.54 \\times 10^{10}$ and $5.66 \\times 10^{10}$~cm$^{-2}$, respectively.}\n\\label{fig:Nh13_29_of_T_+AD_noAD}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[scale=0.7]{rys16.pdf}\n\\includegraphics[scale=0.7]{rys17.pdf}\n\\includegraphics[scale=0.7]{rys18.pdf}\n\\includegraphics[scale=0.7]{rys19.pdf}\n\\caption{Monte Carlo simulations:\nthe $z$-component of the absolute spin projections\n$\\langle |S_z| \\rangle$ and $\\langle |s_z| \\rangle$ (open symbols - dotted lines; left scale) and\nthe spin susceptibilities $\\chi_S$ and $\\chi_s$ (full symbols - solid lines; right scale)\nfor the Mn ions (a,c) and for the holes (b,d),\nin the absence of alloy disorder (noAD, $\\delta V = 0$), and for the attractive (-AD, $\\delta V < 0$) alloy disorder potential.\nA maximum in the temperature dependence of spin susceptibility is identified as the Curie temperature below which the holes induce a long range ferromagnetic order.\nHere the number of holes $N_\\mathrm{h} = 13$ (a,b) and $45$ (c,d), i.e., $p = 2.54 \\times 10^{10}$ and $ 8.78 \\times 10^{10}$ cm$^{-2}$, respectively.}\n\\label{fig:Nh13_45_of_T_noAD_-AD}\n\\end{figure*}\n\n\nIn Fig.~\\ref{fig:TC_MC_MFA}\nwe summarize our MC results for $T_{\\mathrm{C}}$ as a function of the hole concentration obtained neglecting alloy disorder ($\\delta V = 0$) as well as for $\\delta V = \\pm 3.93 \\times 10^{-2}$~eV~nm$^3$, corresponding to the repulsive and attractive alloy potential introduced by Mn, respectively. We see that the attractive potential washes the ferromagnetism virtually entirely out. A contour plot of hole eigenfunctions demonstrates that holes get localized for the chosen magnitude of $\\delta V$. Our results substantiate therefore the view that delocalized or weakly localized carriers are indispensable to set a long-range order between diluted spins.  Obviously, smaller amplitude of the attractive potential or greater values of hole concentrations will lead to carrier delocalization and the reentrance of a ferromagnetic order.\n\nInterestingly, as seen in Fig.~\\ref{fig:TC_MC_MFA}, the repulsive potential $\\delta V > 0$, the case of (Cd,Mn)Te, also leads to reduced magnitudes of $T_{\\mathrm{C}}$ comparing to the values determined for $\\delta V = 0$. We interpret this finding by noting that in the presence of a repulsive potential, the amplitude of the wave function at Mn ions is diminished comparing to the case $\\delta V = 0$. This effectively reduces the $p$-$d$ coupling and shifts the appearance of the carrier-mediated ferromagnetic order to lower temperatures.\n\nIt is instructive to compare these MC findings to the expectations of the mean-field approximation (MFA),\\cite{Dietl:1997,Haury:1997,Boukari:2002,Kechrakos:2005}\nderived neglecting entirely chemical and thermal fluctuations. We see in Fig.~\\ref{fig:TC_MC_MFA}.\nthat, somewhat fortunately, the MC data for $\\delta V = 0$ and MFA results for classical spins (solid line)\nare in a good quantitative agreement at high hole densities.\nIn particular, both approaches predict that the magnitude of $T_{\\mathrm{C}}$ does not vary with the hole density,\nthe result reflecting the energy independence of the density of states in the 2D case.\nHowever, at low hole densities, $T_{\\mathrm{C}}$ values obtained from the MC simulations tend to decrease,\nthe effect associated with a broadening of the density of states induced by hole scattering on Mn spins\\cite{Dietl:1997,Boukari:2002} and encompassed by our MC simulations. The scattering-induced lowering of $T_{\\mathrm{C}}$ is even more apparent if $\\delta V > 0$.\n\n\\begin{figure}[h!]\n\\includegraphics[scale=0.85]{rys20.pdf}\n\\caption{Curie temperature $T_{\\mathrm{C}}$ as a function of\nthe hole area density $p$.\nSolid symbols show results of Monte Carlo simulations carried out for classical spins, QW width $L_{\\mathrm{W}} \\approx 5.2$~nm, and Mn content 4$\\%$\nneglecting carrier correlations in the absence of alloy disorder ($\\delta V = 0$, squares)\nas well as for repulsive and attractive alloy potentials ($\\delta V  > 0$, triangles and $\\delta V < 0$ circles, respectively).\nResults obtained within the mean-field approximation (i.e., disregarding thermal\nand chemical fluctuations) are shown by the solid line. The dashed line presents the MFA results taking into account carrier correlations with the Landau parameter $A_{\\mathrm{F}} = 2$.}\n\\label{fig:TC_MC_MFA}\n\\end{figure}\n\n\nOwing to mutual compensations between ferromagnetic and antiferromagnetic interactions, the magnitude of $T_{\\mathrm{C}}$ is rather sensitive to the presence of hole correlations. This is shown by dashed line in Fig.~\\ref{fig:TC_MC_MFA}, which has been obtained by incorporating into theory within the MFA  the Fermi liquid Landau parameter $A_{\\mathrm{F}}  = 2$. This procedure increases the ferromagnetic contribution to $T_{\\mathrm{C}}$ by a factor of two but since carrier correlations do not affect the compensating AFM term, the resulting increase of $T_{\\mathrm{C}}$ is actually much greater. Therefore, in order to compare MC results to experimental findings we treat $A_{\\mathrm{F}}$ as a fitting parameter. The theoretical values of  $T_{\\mathrm{C}}$ displayed in Fig.~\\ref{fig:theory_Boukari} has been determined from,\\cite{Dietl:1997,Haury:1997,Boukari:2002,Kechrakos:2005}\n\n\\begin{equation}\n\\chi(T_{\\mathrm{C}}) = \\chi(T_{\\mathrm{C}}^{(\\mathrm{MC})})\/A_F.\n\\end{equation}\nHere $\\chi(T) \\sim T^{-\\alpha}$  is the spin susceptibility of the antiferromagnetically coupled Mn spins in the absence of the holes, where $\\alpha =0.79$ according to our MC simulation, and $T_{\\mathrm{C}}^{(\\mathrm{MC})}$ is the Curie temperature at a given hole concentration, determined from the MC simulations that neglect carrier correlations ($A_F = 1$). We see that for $p = 1 \\times 10^{11}$ cm$^{-2}$\nby  taking $A_F = 1.5$ and $A_F = 3$ for the case $\\delta V = 0$ and $\\delta V = 3.93 \\times 10^{-2}$~eV~nm$^3$, respectively\nwe obtain a good description of the experimental findings.\nIn particular, a decrease of $T_{\\mathrm{C}}$ values with lowering of the hole density observed experimentally\\cite{Boukari:2002} and\nalso visible in our MC simulations,\nresults from scattering broadening of the density of states,\\cite{Dietl:1997} particularly relevant at low Fermi energies. However, the experiment implies that this lowering extents to higher concentrations than the range predicted by the simulations, which indicates that additional scattering mechanisms operate in real samples.\n\n\n\\begin{figure}\n\\includegraphics[scale=0.45]{rys10.pdf}\n\\caption{Curie temperature $T_{\\mathrm{C}}$ as a function of\nthe hole areal density $p$. Solid symbols show results of Monte Carlo simulations carried out for classical spins, QW width $L_{\\mathrm{W}} \\approx 5.2$~nm, and the Mn content 4$\\%$ in the absence of alloy disorder ($\\delta V = 0$, squares) and for repulsive alloy potentials ($\\delta V > 0$, triangles) obtained with the Fermi liquid Landau parameters $A_F = 1.5$ and $A_F = 3$.\nExperimental results\\cite{Boukari:2002} for QW width $L_{\\mathrm{W}} \\approx $ 8~nm and Mn content 3-5$\\%$ for two samples (open circles and triangles).}\n\\label{fig:theory_Boukari}\n\\end{figure}\n\nHaving discussed the ferromagnetic ordering temperature $T_{\\mathrm{C}}$, we turn to the temperature dependence of Mn and hole spin projections, as depicted in Figs.~\\ref{fig:Nh13_29_of_T_+AD_noAD} and \\ref{fig:Nh13_45_of_T_noAD_-AD}. We see that the holes become entirely spin polarized, $\\langle |s_z| \\rangle \\rightarrow 1\/2$, immediately below $T_{\\mathrm{C}}$. On the other hand, the increase of the Mn spin projection $\\langle |S_z| \\rangle$ on lowering temperature is much slower. This is because, the molecular field produced by the spin polarized carriers\nis typically below 1~kOe, much too small, even at 0.1~K, to saturate Mn spins which are coupled by AFM interactions.\n\n\\begin{figure}\n\\includegraphics[scale=0.75]{rys21.pdf}\n\\caption{Monte Carlo simulations:\n$z$-component of the absolute spin projection of Mn spins $\\langle |S_z| \\rangle$ at 0.1~K\nin the absence of alloy disorder ($\\delta V  =  0$, squares) and\nfor repulsive alloy potential ($\\delta V > 0$, triangles). The dashed straight line represents a dependence expected within the MFA.}\n\\label{fig:msiz_of_p_noAD_+AD_T=0.1K}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:msiz_of_p_noAD_+AD_T=0.1K} we compare the magnitudes of the Mn spin projection $\\langle |S_z| \\rangle$\nfrom our MC simulations at 0.1~K with the values expected from the MFA for the saturated carrier spins,\\cite{Dietl:1997,Boukari:2002}\n\\begin{equation}\n\\langle |S_z| \\rangle (T) = \\frac{2^{1\/2}S^2\\beta p\\chi(T)}{6k_{\\mathrm{B}}T\\chi_o(T)L_W},\n\\end{equation}\nwhere $\\chi(T)$ and $\\chi_o(T)$ are the values of Mn spin susceptibilities per Mn ion computed without holes, in the presence and in the absence of the AF interactions, respectively. We see that the MFA quite correctly reproduces the computed magnitudes of $\\langle |S_z| \\rangle$ as a function of the areal hole concentration $p$.\nFurthermore, while  the values of $T_{\\mathrm{C}}$ are systematically smaller in the presence of the repulsive alloy potential comparing to the case $\\delta V N_0 = 0$, the magnitude of $\\langle |S_z| \\rangle$ is seen to be larger in this case. We assign this surprising result to the fact that the repulsive alloy potential delocalizes majority-spin holes,\nwhich can therefore embrace more effectively the magnetic impurities. However, as the hole density $p$ increases, the data with and without alloy potential tend to approach. This is because the dense majority-spin holes can embrace more effectively the magnetic impurities even without help from the repulsive potential.  In the other extreme, when the values of $p$ decreases towards zero, the data with and without alloy potential approach, too. Here, the $T_{\\mathrm{C}}$ magnitude becomes so low that the carriers' spins ceases to be entirely saturated in the presence of the repulsive potential, which lowers the corresponding $\\langle |S_z| \\rangle$ value.\n\n\n\n\n\n\n\n\\section{Interpretation of magneto-optical measurements\n\\label{sec:dyn-dom\nIn this section we present our MC results on Mn magnetization dynamics as a function of\ntemperature and carrier concentration.\nWe also discuss whether it is possible within the proposed model to estimate the typical size\nof magnetic domains.\n\n\\subsection{Magnetization dynamics: autocorrelation function}\n\\label{subsec:dyn}\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys9.pdf}\n\\caption{Monte Carlo simulations: Time evolution of  $S^{z}_{Mn}(t)\/S$  at different temperatures  in a\nCd$_{0.96}$Mn$_{0.04}$Te quantum well. The hole concentration is\n$p = 0.4 \\times 10^{11} \\ cm^{-2}$ ($N_{h} = 21$). Time is\nmeasured in the Monte Carlo steps per Mn site.} \\label{fig:corfun9}\n\\end{figure}\n\nWe have traced time dependence of Mn magnetization in the absence of an external magnetic field for very long runs up to $10^5$ MC sweeps. We have repeated our simulations for different initial configurations including the one with all Mn spins aligned along the $z$-axis. From our findings we conclude that except for the first few MC sweeps the whole time evolution of magnetization does not depend on the initial configuration. The typical time-evolution of magnetization at various temperatures is shown in Fig.~\\ref{fig:corfun9}. For  temperatures well below $T_{\\mathrm{C}}$ the system remains almost unchanged during the whole simulation time. For higher temperatures one can observe occasional global spin inversions: the system flips over between two states of equal energy and spontaneous magnetization that differ only in sign.\n\nThe typical time scale of the observed evolution depends on many factors including temperature and carrier concentration.  To determine this characteristic time scale we calculate the time-autocorrelation $G(\\tau)$ of  magnetization given by\\cite{Newman:1999}\n\n\\begin{equation}\\label{autocor}\n  G(\\tau) = \\frac{\\left\\langle m(t)m(t+\\tau)\\right\\rangle-\\left\\langle m\\right\\rangle^{2}}{\\left\\langle m^{2}\\right\\rangle-\\left\\langle m\\right\\rangle^{2}},\n\\end{equation}\n\nwhere $m(t)$ is the value of  magnetization at time $t$ and\n$\\left\\langle m\\right\\rangle$ is its average value. The autocorrelation function defined by  Eq.~\\ref{autocor} depends only on the time difference $\\tau$. To calculate this autocorrelation function for up to $\\tau=10^{4}$ MCS computation runs as long as at least $10^{5}$ MCS are needed. The total computer time depends on numerical\nparameters of which the most important is the $k$-space cut-off.\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys3.pdf}\n\\caption{Monte Carlo simulations: The magnetization autocorrelation function $G(\\tau)$\nof Mn ions in a Cd$_{0.96}$Mn$_{0.04}$Te quantum well calculated at different temperatures.\nThe concentration of holes $p = 0.4 \\times 10^{11}$~cm$^{-2}$ ($N_{h} = 21$). Time is\nmeasured in the Monte Carlo steps per one Mn site. The thin solid lines describe  exponential\ndecays fitted to the Monte Carlo results} \\label{fig:corfun3}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys4.pdf}\n\\caption{Monte Carlo simulations: The magnetization autocorrelation function $G(\\tau)$\nfor selected values of  hole concentrations  in a\nCd$_{0.96}$Mn$_{0.04}$Te quantum well at $T = 0.7$~K. Time is\nmeasured in the Monte Carlo steps per Mn site.} \\label{fig:corfun4}\n\\end{figure}\n\nFigure~\\ref{fig:corfun3} shows the magnetization autocorrelation functions calculated for the studied system\nat various temperatures below and above  $T_{\\mathrm{C}}$. Above $T_{\\mathrm{C}}$, i.e., in\nthe paramagnetic state, one can observe a very fast decay of the autocorrelation function whereas for temperatures\nbelow $T_{\\mathrm{C}}$ correlations persist for times orders of magnitude longer. Within the present MC approach, it is not possible to translate numbers of MC steps into physical time units in a reliable quantitative way. The qualitative relations between the time scales of the investigated magnetic relaxation processes should nevertheless be reproduced in a correct way. Therefore, we undertake only a qualitative comparison with the experimental data of Sec.~\\ref{sec:experiments}.\n\nIn order to better understand mechanisms that account for magnetization dynamics in a system of competing spin-spin AFM and carrier-mediated ferromagnetic interactions we have also calculated autocorrelation functions for a variety of carrier concentration values keeping the number of Mn ions unchanged. In Fig.~\\ref{fig:corfun4} we display the corresponding results calculated for three different carrier concentrations at the same temperature  $T = 0.7$~K. As one could expect we do observe a significant slowing down of magnetization dynamics with increasing the number of holes in the QW. Such behavior is understandable within the proposed model. Taking into account the magnetic anisotropy -- all hole-spins align along $z$-axis -- and the fact that only the $z$-component of each Mn-spin couples via the $p$-$d$ exchange to the system of holes, one can see that if more holes are present in the system, the value of local magnetization and, hence, the barrier height for magnetization reversal become higher.\nNote that for the carrier concentrations $p = 4.10 \\cdot 10^{10}$~cm$^{-2}$ and $p = 8.78 \\cdot 10^{10}$~cm$^{-2}$ the system is in the ferromagnetic state at $T = 0.7$~K. In contrast,  for $p = 1.75 \\cdot 10^{10}$~cm$^{-2}$ the Curie temperature lies below $T = 0.7$~K (cf. next sections), so that the corresponding points describe the dynamics of magnetization in a paramagnetic QW.\n\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys5.pdf}\n\\caption{Monte Carlo simulations: The magnetization autocorrelation function $G(\\tau)$ of Mn ions in a Cd$_{0.96}$Mn$_{0.04}$Te quantum well at $T = 0.7 T_{\\mathrm{C}}$. The hole density is $p = 0.4 \\times 10^{11}$~cm$^{-2}$ ($N_{h} = 21$). Open circles and full squares:  antiferromagnetic interactions are neglected ($J_{ij} = 0$) and taken into account ($J_{ij} \\neq 0$), respectively. Time is measured in the Monte Carlo steps per Mn site.} \\label{fig:corfun5}\n\\end{figure}\n\nA next important aspect one must understand in order to interpret properly the experimental findings is the way in which the AFM interactions influence the magnetization dynamics. Therefore, we have calculated the\nmagnetization autocorrelation function for the system where all short-range AFM interactions are disregarded.\nIn Fig.~\\ref{fig:corfun5} we compare the obtained magnetic autocorrelation functions for systems with and without AFM interactions calculated at $T=0.7T_{\\mathrm{C}}$ (for each system its own $T_{\\mathrm{C}}$ as previously determined by MC simulations\\cite{Kechrakos:2005} had been used). The results presented in Fig.~\\ref{fig:corfun5} clearly indicate that short-range AFM interactions may strongly accelerate the decay of the ferromagnetic order. In accord with the above observation one can expect that the effect of AFM interactions is\nconsiderably reduced when the layer of Mn ions is thinner than the extend of the hole wave function, in other words, when Mn ions are concentrated close to the maximum (probability) density of holes. To check this we repeated our calculations with all Mn ions occupying only the four central layers of the investigated QW (note that it is eight layers' thick and that for holes the ground state of the infinite well is taken as the \"perpendicular\" part of their wave function, $\\varphi(z)$). The results of our calculations are shown in Fig.~\\ref{fig:corfun11}. One can clearly see that indeed if Mn ions reside only in the central layer, the magnetization dynamics resembles the one for the case when all AFM interactions are switched off. We may as well formulate our conclusion the other way round: more\nMn spins close to the QW edges,  faster  relaxation of magnetization. This finding provides a support for theoretical considerations,\\cite{Boudinet:1993}\nsuggesting that magnetization relaxation of bound magnetic polarons in $p$-type CdTe\/(Cd,Mn)Te QWs occurs\nowing to the AFM coupling to the Mn spins located outside the relevant Bohr radius.\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys11.pdf}\n\\caption{Monte Carlo simulations: The magnetization autocorrelation function $G(\\tau)$ of Mn ions in a Cd$_{0.96}$Mn$_{0.04}$Te quantum well at $T = 0.7 T_{\\mathrm{C}}$. The hole density is $p = 0.4 \\times 10^{11}$~cm$^{-2}$ ($N_{h} = 21$).Open circles and full squares: Mn located in the central layers of the\nQW only and evenly distributed within the whole QW, respectively. Time is measured in the Monte Carlo steps\nper Mn site.} \\label{fig:corfun11}\n\\end{figure}\n\n\\subsection{Measuring the domain size: the two-point spin-spin connected correlation function}\n\\label{subsec:dom}\nLet us now look at the spatial distribution of magnetization and, in particular, whether the presence of competing short-range antiferromagnetic and long-range ferromagnetic interactions results in the formation of magnetic domains.\nOne of the standard ways to estimate the typical domain size is to calculate the spin-spin two-point\nconnected correlation function $G_{c}^{2}(R)$ given by\\cite{Newman:1999}\n\n\\begin{equation}\n  G_{c}^{2}(R) = \\frac{\\left\\langle m(\\textbf{r})m(\\textbf{r}+\\textbf{R})\\right\\rangle-\\left\\langle m\\right\\rangle^{2}}{\\left\\langle m^{2}\\right\\rangle-\\left\\langle m\\right\\rangle^{2}},\n\\end{equation}\n\nWe have calculated spin-spin correlation functions for both: the localized Mn ions and the holes for a number of temperature values and various carrier concentrations. Figure~\\ref{fig:corfun6} shows the correlation functions $G_{c}^{2}(R)$ obtained for the hole subsystem. We observe long-range  correlations (longer than the size of the simulation box) in the ferromagnetic state. For temperatures  below $T_{\\mathrm{C}}$ the hole liquid becomes completely spin-polarized. This has already been predicted in the frame of the mean-field approximation for the 2D hole gas in a QW.\\cite{Dietl:1997} In Fig.~\\ref{fig:corfun7} and \\ref{fig:corfun8} the spin-spin correlation functions for the Mn  subsystem are shown. In Fig.~\\ref{fig:corfun7} the scale has been stretched to make visible how the short-range AFM interactions influence the spin-spin correlation function at small distances. In this way we can clearly see that for the nearest neighbor Mn spins the AFM coupling prevails (the correlation function becomes negative). At long distances, however, below  $T_{\\mathrm{C}}$, the long-range ferromagnetic interactions dominate as may be seen in  Fig.~\\ref{fig:corfun8} that shows the spin-spin correlation functions on a larger scale. As the distance $R$ between the Mn spins becomes large we observe (small) positive long-range correlations for temperatures below $T_{\\mathrm{C}}$. At the same time the correlations decrease to zero when temperature increases towards $T_{\\mathrm{C}}$. The positive correlations observed in the ferromagnetic phase and the lack of correlations in the paramagnetic phase may indicate that magnetic domains do exist, but their typical size is greater than the simulation box. To verify this conjecture further massive simulations would be needed.\n\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys6.pdf}\n\\caption{Monte Carlo simulations: The two-point connected correlation function $G^{(2)}_{c}(R)$ of valence band holes at various temperatures  in a  Cd$_{0.96}$Mn$_{0.04}$Te quantum well. The hole concentration is $p = 0.4 \\times 10^{11}$~cm$^{-2}$ ($N_{h} = 21$).} \\label{fig:corfun6}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys7.pdf}\n\\caption{Monte Carlo simulations: The two-point connected correlation function $G^{(2)}_{c}(R)$ of Mn ions in a Cd$_{0.96}$Mn$_{0.04}$Te quantum well at $T=0.2T_{\\mathrm{C}}$. The hole concentration is $p = 0.4 \\times 10^{11}$~cm$^{-2}$ ($N_{h} = 21$).} \\label{fig:corfun7}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=1.0]{rys8.pdf}\n\\caption{Monte Carlo Simulations: The long-distance parts of two-point connected correlation functions $G^{(2)}_{c}(R)$ of Mn ions at various temperatures  in a Cd$_{0.96}$Mn$_{0.04}$Te quantum well. The hole concentration is $p = 0.4 \\times 10^{11}$~cm$^{-2}$ ($N_{h} = 21$).} \\label{fig:corfun8}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions\n\\label{sec:conclusion\nWe have examined magnetization dynamics in a p-Cd$_{0.96}$Mn$_{0.04}$Te quantum well by magneto-optical studies and by Monte Carlo simulations taking into account the presence of competing short-range antiferromagnetic superexchange and long-range   carrier-mediated ferromagnetic interactions. Single-particle hole energies and eigenfunctions for particular Mn spin configurations have been updated numerically after each Monte Carlo sweep. In addition to the $p$-$d$ exchange interaction, spin-independent alloy disorder has been taken into account and found to be important. In particular, a sufficiently strong attractive potential introduced by the magnetic ion leads to hole localization and to the corresponding disappearance of a long-range order mediated by the carriers. Also a strong repulsive potential, by reducing the magnitude of the carrier wave function at  the magnetic ion, may diminish the magnitude of $T_{\\mathrm{C}}$. This effect is particularly strong in the case of a quantum well, where the direction of the hole spin is fixed rather by the spin-orbit interaction than by the $p$-$d$ coupling, so that the repulsive potential may not be compensated by the $p$-$d$ interaction.\n\nBoth experimental\nand theoretical results demonstrate that the transition from the paramagnetic to ferromagnetic phase\nincreases the magnetization relaxation time. In particular, the relaxation time determined from tracing the time evolution of magnetization after a pulse of the magnetic field is shorter than 20 ns in the paramagnetic state. Lowering temperature below $T_{\\mathrm{C}}$ results in\nan increase of the relaxation time up to only 2 $\\mu$s. Our numerical results  indicate\nthat antiferromagnetic interactions between Mn spins account for this relatively fast magnetization fluctuations and,\nthus, might be responsible for the absence of spontaneous magnetization below $T_{\\mathrm{C}}$, as found in static measurements.\nWhile our Monte Carlo simulations  of the two-point spin-spin connected correlation function show a huge\ninfluence of the short-range antiferromagnetic interactions on the magnetization dynamics, they do not reveal the formation of\ndomains that would be smaller than our simulation cell, $L = 226$~nm.\n\nAt the same time, our Monte Carlo results reveal that magnetization relaxation in the ferromagnetic phase proceeds primarily due to antiferromagnetic couplings to Mn spins residing at the quantum well boundary, as they are weakly polarized by the carriers. Accordingly, magnetization dynamics would be much slow down if the width of the region containing Mn spins were been narrower than the extent of the carrier wave function. We predict, therefore, that spontaneous magnetization and the associated magnetic hysteresis could be observed in quantum wells containing the Mn layer only in its center.\n\n\n\\begin{acknowledgements}\nThe work in Warsaw was supported in part by Technology Agency, by SPINTRA Project of European Science Foundation (ERAS-CT-2003-980409), and by the FunDMS Advanced Grant within the European Research Council \"Ideas\" Programme of EC 7FP.\nPart of the necessary computer time in Athens was provided by\nthe National Grid Infrastructure {\\it HellasGrid}.\nWe thank N.~Papanikolaou for useful discussions.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:Intro}\n\nPhysical systems with a large number of degrees of freedom and complicated\ndynamics can be understood in much simpler terms, if they exhibit a small\nnumber of metastable states. A natural coarse-grained description is then\nfound in terms of residence times in each of the states, joined by\nnearly instantaneous transitions between them.\n\nThis idea has been widely used in the context of thermal equilibrium systems\nthat are characterized by a clear separation of time scales between fast\nintra-state dynamics and slow global relaxation (e.g.~\\cite{haenggi90}), giving\nrise to Markov State Models, see e.g.~\\cite{prinz11}. Here\ntransitions are Poissonian, i.e.~they happen with uniform propensity\n(probability per unit time); equivalently, the waiting time intervals are\nuncorrelated and exponentially distributed. Each transition is described by a\nsingle number, the rate constant.\n\nIn this article we are interested in systems that switch between two metastable\nstates rarely and rapidly, but with a time-dependent propensity.\nSuch time-dependence may be caused by an external force driving the system.\nAlternatively it may also arise due to the presence of internal degrees of\nfreedom that do not fully equilibrate on the macroscopic time scale of\ninterest, giving rise to non-Markovian macroscopic switching dynamics between\nthe metastable states. We consider both equilibrium and non-equilibrium\nsystems, in the sense that their unperturbed microscopic dynamics may or may\nnot be time-reversible.\n\nClearly, in such time-inhomogeneous or non-Markovian systems the\nconcept of a \\emph{rate constant} is inadequate. In this \narticle we investigate in what physical regimes a\ntime-dependent \\emph{rate function} or a history-dependent \\emph{rate kernel}\nmay instead be meaningfully defined, and state the corresponding\nphenomenological rate equations. \nWe then address the question how macroscopic rate\nfunctions or rate kernels can be defined in terms of microscopic\ncorrelation functions; we give microscopic expressions and show how they can be\nused to measure these generalized rates in computer simulations.\n\n\n\\section{Overview}\\label{sec:Metastable}\n\nWe consider systems whose coarse-grained, macroscopic dynamics can be described\nas switching between two macroscopic states $A$ and $B$.   The macroscopic\nstates are assumed to be metastable in the sense that the mean waiting time\nbetween the switching events is much longer than the duration of the switching\nevent itself. This makes it possible to partition the phase space into the\nmetastable regions $A$ and $B$, and a transition region $C$ separating them,\nsuch that\n\\begin{equation} \n\\tau_C \\ll \\tau_{AB},\\tau_{BA},\n\\label{eq:tau_C}\n\\end{equation}\nwhere $\\tau_C$ is a typical duration for a traversal of $C$ (a switching\nevent), and $\\tau_{AB}, \\tau_{BA}$ are the mean waiting times for switching in\nthe forward and backward direction, respectively.  This separation of time\nscales provides the justification for coarse graining the system as a two-state\nsystem, switching rarely but rapidly between the states $A$ and $B$.\n\nWe are interested both in equilibrium systems that are microscopically\nreversible and in non-equilibrium systems with dissipation of energy.\nIf the system is in stationary state, the stability of the states $A$ and $B$\nand the forward and backward probability fluxes, $q_{AB}$ and $q_{BA}$, are\nconstant in time. Nonetheless, if the system has memory, the switching\npropensities will be history-dependent. Furthermore, out of stationary state,\nthe stability of the states and the dynamics of switching between them will\nchange in time. In both of these situations, rate constants $k_{AB}$ and\n$k_{BA}$ do not adequately describe the system, and a time-dependent\ngeneralization of the rate constant concept is called for.\n\n\nFollowing Chandler in his derivation of rate constants in equilibrium systems\n\\cite{chandler78}, we imagine that from experimental observations we know a\ncertain phenomenological expression for the macroscopic switching dynamics to\nbe valid, on times much larger than a certain macroscopic time resolution\n$\\Delta t$.\nThis macroscopic expression is the rate equation. The goal is then to derive a\nmicroscopic expression for the dynamics of the system that is consistent with\nthe rate equation, and allows us to measure the macroscopic rates in a\nmicroscopic simulation. We will thus not derive a macroscopic expression from\nmicroscopic principles; rather, we will \\emph{assume} that the system obeys a\ngiven rate equation, and use this description as a starting point for the\nderivation of microscopic expressions for time-dependent rate functions\n\\cite{chandler78}.\n\nThe propensity of the system to switch from one macroscopic state to another at\ntime $t$ may depend on the current macroscopic state of the system only, in\nwhich case the system is Markovian. More generally, it may depend also upon the\nmacroscopic history, i.e.~the sequence of states visited in the past. For such\na non-Markovian system, the switching propensity could depend on the time that\nhas passed since the last switching event, but it is also conceivable that it\ndepends on the system's dynamics prior to the last switching event.\nIndeed, all information about the history of the macroscopic switching dynamics\nis contained in the sequence of switching times. It is thus natural to capture\nthe history dependence of the switching dynamics by writing the time evolution\nof the system in terms of the times of all previous switching events:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t} P_B(t;t',t'',\\dots)\n&=&k_{AB}(t|t',t'',\\dots)\n  P_A(t;t',t'',\\dots) \\nonumber \\\\\n&& -\n  k_{BA}(t|t',t'',\\dots)\n  P_B(t;t',t'',\\dots), \\nonumber\\\\\n&&\\label{eq:ME}\n\\end{eqnarray}\nIn this Master equation, $P_A(t;t',t'',\\dots)$ denotes the \\emph{joint}\nprobability to be in $A$ at time $t$ and to have switched for the last time\nwithin $(t',t'+\\d t')$ (i.e. from $B$ to $A$), for the second-to-last time in\n$(t'', t''+\\d t'')$ (from $A$ to $B$), and so on. Moreover,\n$k_{AB}(t|t',t'',\\dots)\\d t$ is the conditional probability to leave $A$ in the\ntime interval ($t,t+\\d t$) given that the sequence of switching times was\n$t',t'',\\dots$. An analogous equation holds with $A$ and $B$ interchanged.\nImportantly, this macroscopic phenomenological rate equation \\emph{defines} the\ngeneral rate  kernels $k_{AB}(t|t',t'',\\dots)$ and $k_{BA}(t|t',t'',\\dots)$.\nAn equation for the switching dynamics at time $t$ can\nbe obtained from Eq.~\\ref{eq:ME} by integrating over the switching times prior\nto $t$, \\begin{equation}\\label{eq:MEl}\n\\frac {\\d}{\\d t}P_B(t)= \\int_{t>t'>t''>\\dots}\\frac{\\partial}{\\partial t}\nP_B(t;t',t'',\\dots)\\d t' \\d t''\\dots.\n\\end{equation}\nSince the last, second-to-last, $\\dots$ switches are each unique\nevents, no over-counting occurs here. No approximation has\nbeen made up to this point; note however that \\ref{eq:MEl} is not in \ngeneral a closed equation for $P_{A,B}(t)$. The lower integration limit\ndepends upon the experimental setup and will be discussed in more detail below.\n\nWhile Eqs.~\\ref{eq:ME} and \\ref{eq:MEl} describe the switching dynamics of an\narbitrary two-state system, they cannot be solved in general. They are also too\ndetailed, requiring full information about the macroscopic history. At this\npoint the experiment has to inform us about the most useful and meaningful\nphenomenological model, that is, the minimal model that captures the\nmacroscopic switching dynamics of the system. More specifically, the experiment\nhas to reveal whether the switching propensities do indeed depend on the\ncurrent time $t$ and the previous switching times $t', t'', \\dots$.\nThe decision about the type of macroscopic expression that\ndescribes the experiment best, will then depend both on the physical properties\nof the system under consideration, and on the time resolution and the\nmeasurement uncertainty of the experiment. Below, we will discuss a number of\nstationary and non-stationary situations for which Eqs.~\\ref{eq:ME} and\n\\ref{eq:MEl} can be simplified, and for which microscopic expressions for the\nmacroscopic rate constants can be found.\n\nWe now turn to the microscopic characterization of the system. We consider an\nensemble of trajectories that start at $t=0$ at state space points $x_0$,\nfollowing a specified initial phase-space distribution function $\\rho(x_0)$\n\\footnote{If the microscopic dynamics is not Markovian, the initial condition\nwill have to be supplemented with a prescription of how the system was prepared\nat $t<0$.}.\nIf the dynamics obey detailed balance and microscopic reversibility, and if\n$\\rho(x_0)$ is the canonical distribution, then this is an equilibrium\nensemble. We may instead also consider the relaxation towards thermodynamic\nequilibrium, starting from a non-equilibrium $\\rho(x_0)$. Alternatively, the\nensemble could be that of a non-equilibrium system that does not obey detailed\nbalance and microscopic reversibility. Again, the initial condition could\neither be in stationary state, or out of stationary state, in which case the\nnon-equilibrium system may relax back to a stationary state. Finally,  the\nsystem (time-reversible or not) may never reach a stationary state over the\ntime course $[0,T]$ of the experiment. This can happen if relaxation is slower\nthan $T$, or if the system is driven externally via some protocol $\\phi(t)$,\nfor instance, proteins that are unfolded under the influence of an external\nforce.\n\nTo derive microscopic expressions that are consistent with the macroscopic rate\nequations, Eqs.~\\ref{eq:ME} and \\ref{eq:MEl}, we first need to define functions\n$h_A(x_t)$ and $h_B(x_t)$ that indicate whether the system with configuration\n$x_t$ at time $t$ is in state $A$ or $B$. It is useful to define these\ncharacteristic functions in terms of an order parameter $q(x_t)$ that serves to\nmeasure the progress of the transition between the states $A$ and $B$:\n\\begin{eqnarray}\nh_A(x_t) &=& \\theta[q_A - q(x_t)],\\\\\nh_B(x_t) &=& \\theta[q(x_t) - q_B].\n\\end{eqnarray}\nHere, $\\theta$ is the Heaviside step function. With these definitions, the\nsystem is considered to be in $A$ when $q(x_t) < q_A$, in $B$ when\n$q(x_t)>q_B$, and in the transition region $C$ otherwise. Moreover,\nwe have the relation\n\\begin{equation}\nh_A + h_B\\equiv 1 \\text{, if }q_A=q_B=q^*.\\label{eq:unity} \n\\end{equation}\nThe same relation holds effectively if $q_A<q_B$ and the occupancy of \nthe transition region $C$ is low for all times.\n\n\nThe principal idea is now that the macroscopic rates $k_{AB,BA}(t)$ can be\nderived from the behavior of the microscopic correlation function\n\\begin{eqnarray}\\label{eq:C_t}\nC(t)&=&\\int \\d x_0 \\d x_t\\rho(x_0) p(x_t|x_0)h_B(x_t)\\nonumber\\\\\n&=&\\Avg{h_B(x_t)}\n\\end{eqnarray}\nHere, $p(x_t|x_0)$ is the probability that the system is in state\n$x_t$ at time $t$ given that it started in state $x_0$ at time zero;\n$\\Avg{\\dots}$ denotes an average over the ensemble of trajectories\nthat start in $x_0$, following the phase-space distribution\n$\\rho(x_0)$. The correlation function $C(t)$ gives the probability\nthat the system is in state $B$ at time $t$ given that it\nstarted from the initial distribution $\\rho(x_0)$, which may or may not be\nchosen to fully reside in $A$.  The transition rates are\nderived from the flux into $B$ and out of $B$, and are thus related to\nthe time derivative of $C(t)$:\n\\begin{equation}\n\\dot{C}(t)=\\Avg{\\dot{h}_B(t)},\\label{eq:Cdot_0}\n\\end{equation}\nwhere we use the shorthand notation $h_B(t)=h_B(x_t)$.\nThe task at hand is to rewrite this microscopic expression such that it can\nbe identified with the macroscopic expressions of Eqs.~\\ref{eq:ME} and\n\\ref{eq:MEl}. This is carried out below for a number of different classes\nof systems. In each case, we start with the macroscopic rate equation,\nand then derive a microscopic correlation function from which the macroscopic\nrates can be obtained.\n\n\\section{Markov systems}\\label{sec:StatSys}\n\nIn this section we consider systems that exhibit Markovian macroscopic\nswitching dynamics. By this we mean that it is known from experiment\nthat there exists a macroscopic time resolution $\\Delta t$, on which the\npropensity to switch between the two macroscopic states $A$ and $B$ at time $t$\nis independent of the history of the macroscopic switching dynamics for earlier\ntimes $t'<t$. In this case the rate kernels are independent of the previous\nswitching times, yet may depend on the current time if the system is not\ntime-homogeneous:\n$k_{X{\\bar{X}}}(t|t',t'',\\dots)=k_{X{\\bar{X}}}(t)$, where $X{\\bar{X}}=AB$ or $BA$. We immediately\nconclude that the macroscopic rate equations Eqs.~\\ref{eq:ME}, \\ref{eq:MEl}\nreduce to\n\\begin{equation}\\label{eq:dPBmarkov}\n\\frac{\\d} {\\d t}P_B(t)= k_{AB}(t)P_A(t) -\nk_{BA}(t)P_B(t)\n\\end{equation}\nwith rate functions $k_{AB}(t), k_{BA}(t)$. This equation together with\nits counterpart for $P_A$ forms a closed set of equations for $P_{A,B}$. \nWe remark that the form of Eq.~\\ref{eq:dPBmarkov} can always be\nobtained trivially from Eqs.~\\ref{eq:ME}, \\ref{eq:MEl} by\nmarginalizing. However, the resulting expression will then in general\nonly be valid for a particular experiment. A Markov system is\ncharacterized by the property that Eq.~\\ref{eq:dPBmarkov} with a\n\\emph{fixed} set of rate functions $k_{X{\\bar{X}}}(t)$ correctly describes\nmultiple experiments, differing in their initial conditions and in their\nhistory before $t=0$.\n\nThe fact that memory is lost on the macroscopic time scale $\\Delta t$\nimplies that the system locally equilibrates in the macroscopic states $A$ and\n$B$ on time scales $\\tau_A$ and $\\tau_B$, respectively, that are shorter than\n$\\Delta t$. We can then identify a transient time scale \n\\begin{equation}\n\\tau\\ts{trans}=\\max\\{\\tau_A,\\tau_B,\\tau_C\\}<\\Delta t \\ll\n\\min\\{\\tau_{AB},\\tau_{BA}\\}\n\\label{eq:tau_A_B}\n\\end{equation}\nwhich is the microscopic memory time of the system.\n\nIt is instructive to restrict the attention to trajectories that happen to\nstart in $A$ at $t=0$. To derive microscopic expressions for the macroscopic\nrate constants, we then consider the correlation function\n\\begin{equation}\n\\dot{C}(t)=\\Avg{\\dot{h}_B(t)}_{A_0},\\label{eq:Cdot_0_0}\n\\end{equation}\nwhere the subscript $A_0$ indicates restriction of $\\rho(x_0)$ to this\nsubensemble.\nWe now rewrite this expression  such that its terms can be\nidentified with the macroscopic quantities appearing in\nEq.~\\ref{eq:dPBmarkov}. Taking $q_A=q_B=q^*$ and letting $h_X(t) = h_X(x_t)$,\nwe can insert $h_A(t-\\Delta t) + h_B(t-\\Delta t)=1$ and \n$h_A(t+\\Delta t) + h_B(t+\\Delta t)=1$ into Eq.~\\ref{eq:Cdot_0_0}; the\nimportance of $\\Delta t$ will become clear shortly. This yields\n\\begin{eqnarray}\n\\dot{C}(t)&=&\\Avg{h_A(t-\\Delta t)\\dot{h}_B(t)h_A(t+\\Delta t)}_{A_0}\\nonumber\\\\\n&+& \\Avg{h_A(t-\\Delta t)\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_0} \\nonumber\\\\\n&+& \\Avg{h_B(t-\\Delta t)\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_0} \\nonumber\\\\\n&+& \\Avg{h_B(t-\\Delta t)\\dot{h}_B(t)h_A(t+\\Delta t)}_{A_0}.\\label{eq:Cdot_1}\n\\end{eqnarray}\nWe now condition on\nthe state prior to a transition, by multiplying and dividing the first two\nterms by $\\Avg{h_A(t-\\Delta t)}_{A_0}$ and the last two terms by\n$\\Avg{h_B(t-\\Delta t)}_{A_0}$. This gives\n\\begin{eqnarray}\n  \\dot{C}(t)&=&\\Avg{h_A(t-\\Delta t)}_{A_0}\n  \t\\Avg{\\dot{h}_B(t)h_A(t+\\Delta t)}_{A_0,A_{t-\\Delta t}} \\nonumber\\\\\n  &+&\\Avg{h_A(t-\\Delta t)}_{A_0}\n  \\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_0,A_{t-\\Delta t}} \\nonumber\\\\\n  &+&\\Avg{h_B(t-\\Delta t)}_{A_0}\n  \t\\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_0,B_{t -\\Delta t}} \\nonumber\\\\\n  &+&\\Avg{h_B(t-\\Delta t)}_{A_0}\n  \t\\Avg{\\dot{h}_B(t)h_A(t+\\Delta t)}_{A_0,B_{t-\\Delta t}}.\\label{eq:Cdot_2}\n\\end{eqnarray}\nThe average $\\Avg{\\dots}_{A_0,X_{t-\\Delta t}}$ runs over trajectories that\nstart in $A$ at time $t=0$ and are in $X=A,B$ at time $t-\\Delta t$.\n\nSince the system looses memory on the transient time scale $\\tau\\ts{trans}$,\nthe memory fo the initial state is lost for $(t-\\Delta t)>\\tau\\ts{trans}$,  so\nthat $\\Avg{\\dots}_{X_{t-\\Delta t}, A_0}=\\Avg{\\dots}_{X_{t-\\Delta t}}$.\nIn this regime Eq.~\\ref{eq:Cdot_2} becomes\n\\begin{eqnarray}\n  \\dot{C}(t)&=&\\Avg{h_A(t-\\Delta\n      t)}_{A_0}\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{A_{t-\\Delta t}} \\nonumber\\\\\n  &+&\\Avg{h_A(t-\\Delta\n      t)}_{A_0}\\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta\n      t}} \\nonumber\\\\\n  &+&\\Avg{h_B(t-\\Delta\n      t)}_{A_0}\\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{B_{t\n      -\\Delta t}} \\nonumber\\\\\n  &+&\\Avg{h_B(t-\\Delta\n      t)}_{A_0}\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{B_{t-\\Delta t}}.\\label{eq:Cdot_3}\n\\end{eqnarray}\nWe now rewrite the first and third term in this equation, which describe\ntrajectories that re-cross into their original state $X=A,B$ after a time\n$2\\Delta t$:\n\\begin{eqnarray}\n  \\dot{C}(t)&=&\\frac{\\Avg{h_A(t-\\Delta t)}_{A_0}}{\\Avg{h_A(t-\\Delta t)}}\n  \t\\Avg{h_A(t-\\Delta t) \\dot{h}_B(t)h_A(t+\\Delta t)} \\nonumber\\\\\n  &+&\\Avg{h_A(t-\\Delta t)}_{A_0}\n      \\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta t}} \\nonumber\\\\\n  &+&\\frac{\\Avg{h_B(t-\\Delta t)}_{A_0}}{\\Avg{h_B(t-\\Delta t)}}\n  \t\t\\Avg{h_B(t-\\Delta t)\\dot{h}_B(t)h_B(t+\\Delta t)} \\nonumber\\\\\n  \t\t&+&\\Avg{h_B(t-\\Delta t)}_{A_0}\n  \t\t\\Avg{\\dot{h}_B(t)h_A(t+\\Delta t)}_{B_{t-\\Delta\n  \t\tt}}\\nonumber\\\\\n  &\\equiv&j_{AA}(t;\\Delta t) + j_{AB}(t;\\Delta t)\\nonumber\\\\\n  &+&     j_{BB}(t;\\Delta t) + j_{BA}(t;\\Delta t). \\label{eq:Cdot_4}\n\\end{eqnarray}\nIf the system obeys microscopic reversibility, and if the initial\ncondition $\\rho(x_0)$ coincides with the canonical distribution, then\n$\\Avg{\\dots}$ denotes an equilibrium average. In this case the\nrecrossing terms \n\\begin{equation}\\label{eq:jAA0}\nj_{AA}(t,\\Delta t)=0,\\;j_{BB}(t, \\Delta t)=0\n\\end{equation}\nfor all times $t$ and for all values of $\\Delta t$. This can be seen by noting\nthat the equilibrium average $\\Avg{h_X(x_{t-\\Delta t}) \\dot{h}_B(x_t)\nh_X(x_{t+\\Delta t})}$ changes sign under time reversal and therefore vanishes\nin thermodynamic equilibrium.  Out of equilibrium, the recrossing terms are\nnon-zero in general. However, under the additional condition that over the\ngiven time scale $\\Delta t$ the system is approximately time-homogeneous and\nmemoryless, Eq.~\\ref{eq:jAA0} can be shown to hold nonetheless (Appendix\n\\ref{sec:jAA}). Fig.~\\ref{fig:expl}\ngives a graphical account of these observations.\n\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=6cm]{fig01}\n  \\caption{A visual interpretation of the term\n  \t$j_{AA}=\\Avg{h_A(t-\\Delta t)\\dot{h}_B(t)h_A(t+\\Delta t)}_{A_0}$ of\n  \tEq.~\\ref{eq:Cdot_1}.\n\tIn microscopically reversible systems (a), for each trajectory (blue) \n\tthere exists an identical but time-reflected trajectory (red). \n\tConsequently, $j_{AA}=0$ for all values of $\\Delta t$. In non-equilibrium\n\tsystems that are memoryless and in stationary state (b), for each \n\ttrajectory (blue) there exists a time-shifted trajectory (green) which \n\tcancels the contribution to $j_{AA}$, so that\n\t$j_{AA}=0$ as well, irrespective of time-reversibility. In systems that are\n\tnot time-reversible and remain correlated over $\\Delta t$ (c), time-shifted\n\ttrajectories may fail to contribute to $j_{AA}$, as is the case for the\n\ttrajectory shown in green. Here cancellation cannot occur and \n\t$j_{AA}\\neq 0$.\n\tSee Appendix~\\ref{sec:jAA}.\n\\label{fig:expl}}\n\\end{center}\n\\end{figure}\n\nAccepting Eq.~\\ref{eq:jAA0} for the moment (this will be justified\ncase-by-case below), Eq.~\\ref{eq:Cdot_4} reduces to\n\\begin{eqnarray}\n  \\dot{C}(t)&=&\n  \\Avg{h_A(t-\\Delta\n      t)}_{A_0}\\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta\n      t}} \\nonumber\\\\\n  &+&\\Avg{h_B(t-\\Delta\n      t)}_{A_0}\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{B_{t-\\Delta t}} \\label{eq:Cdot_5}\n\\end{eqnarray}\n\nWe can now relate the microscopic Eq.~\\ref{eq:Cdot_5} to\nthe macroscopic  Eq.~\\ref{eq:dPBmarkov} by identifying\n$\\Avg{h_B(t-\\Delta t)}_{A_0} = P_B(t-\\Delta t)$, \n$\\Avg{h_A(x_{t-\\Delta t})}_{A_0}=P_A(t-\\Delta t)$ and\n$\\dot{C}(t)=\\d P_B(t)\/\\d t$. \nThis yields\n\\begin{eqnarray}\n\\frac{\\d}{\\d t}P_B (t)&=&P_A(t-\\Delta t) \n\t\\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta t}} \\nonumber\\\\\n&+&P_B(t-\\Delta t)\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{B_{t-\\Delta t}},\\nonumber\\\\\n&\\simeq&P_A(t) \\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta t}} \\nonumber\\\\\n&+&P_B(t)\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{B_{t-\\Delta t}}, \\label{eq:dPBdt_2}\n\\end{eqnarray}\nwhere the approximate equality results from the fact that $\\Delta t$ is \nthe time resolution of the macroscopic description.\nComparing this expression with\nthe macroscopic rate equation Eq.~\\ref{eq:dPBmarkov}, we deduce that\n\\begin{eqnarray}\nk_{AB}(t) &=&   \\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta\n      t}},\\nonumber\\\\\nk_{BA}(t)&=&-\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{B_{t-\\Delta t}}. \\label{eq:k_AB_M}\n\\end{eqnarray}\n\nIn the next section, we recapitulate for completeness the classical\nscenario of a Markov system that relaxes towards stationary state\nstarting from a non-stationary distribution \\cite{chandler78}. In the\nsubsequent section we consider Markov systems that are driven via an\nexternal protocol. In both cases, we will first consider the\nphenomenological rate equation, and then demonstrate that the\nmacroscopic rate constants $k_{AB}(t),k_{BA}(t)$ are given by the\nmicroscopic expressions on the right-hand side of Eq.~\\ref{eq:k_AB_M},\nprovided $\\Delta t$ is chosen carefully, as discussed below.\n\n\n\\subsection{Time-homogeneous Markov systems}\n\\label{sec:timehommarkov}\n\nWe imagine that the Markov system has been driven via some protocol into an\narbitrary initial condition $\\rho(x_0)$ at $t=0$, and consider its relaxation\ntowards stationary state. We further imagine that experiments have revealed\nthat the phenomenological rate equation, Eq.~\\ref{eq:dPBmarkov}, can be reduced\nto\n\\begin{equation}\n\\frac{\\d}{\\d t}P_B(t) = k_{AB} P_A(t) - k_{BA} P_B(t),\\label{eq:dPBdt}\n\\end{equation}\nwith rate constants $k_{AB}$ and $k_{BA}$; thus $k_{AB}$ is the\n(constant) propensity that the system switches to $B$, given that it is in $A$.\nThe transitions between the states $A$ and $B$ on the macroscopic time scale\n$\\Delta t$ are described by a time-homogeneous Markov process.\nWe note that even a system with time-homogeneous\nmicroscopic dynamics may fail to obey Eq.~\\ref{eq:dPBdt}, if the system starts\nfrom a highly non-equilibrium initial condition; indeed, the experiment has to\nreveal whether the assumption of constant rates actually holds.\n\nThe solution of Eq.~\\ref{eq:dPBdt} for a system that starts in state $A$\nat time $t=0$ is given by\n\\begin{equation}\nP_B (t) = P_B^{\\infty} (1-e^{-t \/\\tau\\ts{rxn}}), \\label{eq:PB_t}\n\\end{equation}\nwhere $\\tau\\ts{rxn} = (k_{AB}+k_{AB})^{-1}$ is the macroscopic\nrelaxation time of the switch and $P_B^{\\infty}$ is the probability of\nbeing in $B$ in the stationary state, given by $P_B^\\infty =\nk_{AB}\/(k_{AB}+k_{BA}$).\n\nFollowing the discussion above in this special case, we first need to establish\nthat the recrossing fluxes vanish, Eq.~\\ref{eq:jAA0}. This holds whenever the\nsystem is both memoryless and time-homogeneous for time differences larger than\n$\\Delta t$.\nThese conditions are met in the present case, because Eq.~\\ref{eq:dPBdt}, which\nis assumed to describe the experiment, implies that on the time scale $\\Delta\nt$ the system switches between $A$ and $B$ in a  memoryless fashion with\nconstant rates.\nIt then follows that the rates are given by Eq.~\\ref{eq:k_AB_M}:\n\\begin{eqnarray}\nk_{AB} &=&   \\Avg{\\dot{h}_B(t)h_B(t+\\Delta t)}_{A_{t-\\Delta\n      t}},\\nonumber\\\\\nk_{BA}&=&-\\Avg{\\dot{h}_B(t)h_A(t+\\Delta\n      t)}_{B_{t-\\Delta t}}. \\label{eq:k_AB_1}\n\\end{eqnarray}\nWhile $k_{AB}$ and $k_{BA}$ on the left-hand side are independent of\ntime, the expressions on the right-hand side appear to depend on $t$ and\n$\\Delta t$.\n\nWe first discuss the dependence on $t$. In writing down our macroscopic rate\nequation, we have \\emph{assumed} that the system switches between the states\n$A$ and $B$ with constant rates. This presupposes that the system relaxes inside\nthe basins of $A$ and $B$ in between the switching events faster than the\nmacroscopic time $\\Delta t$, leading to Markovian switching dynamics and to\nloss of memory of the initial condition. Importantly, this assumption also\nimplies that during relaxation towards stationary state, the probability of\nbeing in either $A$ or $B$ will change with time, but the state-space\ndistribution \\emph{within} each macroscopic state $A$ and $B$ does not change\nwith time; by deduction, the state-space distribution during relaxation,\nconditioned on being in either $A$ or $B$, must be equal to that of the\nstationary distribution. In other words, while the ensemble brackets\n$\\Avg{\\dots}_{A_0}$ denote an average over trajectories that start from an\narbitrary initial condition $\\rho(x_0)$ in state $A$, this ensemble must,\naccording to our assumption, be effectively equal to the stationary ensemble of trajectories,\nconditioned on starting in $A$.\n\nThis is essentially the content of Onsager's regression hypothesis\n\\cite{onsager31,onsager31a}, which states that the relaxation of an observable\nin a non-equilibrium experiment is proportional to the  relaxation of a\nspontaneous fluctuation of that observable in the equilibrium system. It relies\non the idea that the non-equilibrium initial distribution in a relaxation\nexperiment follows a phase-space distribution that is similar to that of a\nspontaneous fluctuation in the equilibrium system \\cite{chandler87}. The\nregression hypothesis is a well-known theorem for systems close to thermal\nequilibrium \\cite{callen51}, but generalizations also exist for relaxation of\nperturbations towards non-equilibrium stationary state\n\\cite{agarwal72,procaccia79, prost09, seifert10}. For rare switching events in\nequilibrium and stationary non-equilibrium systems, the presence of a dynamical\nbottleneck for switching means that the system can rapidly relax inside the\nbasins of $A$ and $B$, leading to loss of memory of the initial condition. This\nimplies that the regressions theorem also holds, with exponential\ntime-dependence of the relaxation function, at least on a time resolution\n$\\Delta t$ coarser than the transient time $\\tau_{\\rm trans}$.\n\nWe now turn to the dependence on $\\Delta t$. Since the rates are constant, we\ncan time-shift Eq.~\\ref{eq:k_AB_1} to give:\n\\begin{eqnarray}\nk_{AB} &=&   \\Avg{\\dot{h}_B(0)h_B(\\Delta t)}_{A_{-\\Delta\n      t}}\\nonumber,\\\\\nk_{BA}&=&-\\Avg{\\dot{h}_B(0)h_A(\\Delta\n      t)}_{B_{-\\Delta t}}.\\label {eq:k_AB_2}\n\\end{eqnarray}\nThese expressions, which are very similar to those of the reactive\nflux method of Bennett \\cite{bennett77} and Chandler \\cite{chandler78}, hold\nonly for a certain range of $\\Delta t$ values\n\\cite{chandler87}. The macroscopic rate constant $k_{AB}$ is defined as the\n(constant) propensity that the system switches from $A$ to $B$ given that at\nthat moment in time it is in $A$. In contrast, the microscopic expression \n$\\Avg{\\dot{h}_B(0)h_B(\\Delta t)}_{A_{-\\Delta t}}$ is the propensity that the\nsystem switches from $A$ to $B$ minus the propensity that it switches from $B$ \nto $A$ at a certain moment in time, given that at an earlier time $-\\Delta t$\nit was in $A$ and at a later time $\\Delta t$ it is in $B$:\n$\\Avg{\\dot{h}_B(0)h_B(\\Delta t)}_{A_{-\\Delta t}}=\n\\Avg{\\theta[\\dot{h}_B(0)]\\dot{h}_B(0)h_B(\\Delta t)}_{A_{-\\Delta t}}\n-\\Avg{\\theta[-\\dot{h}_B(0)](-\\dot{h}_B(0))h_B(\\Delta t)}_{A_{-\\Delta t}}$. \nIndeed, the microscopic expression takes into account that\nthe system may switch back and forth between the two states a number\nof times, and in fact, for $\\Delta t\\to \\infty$, \n$\\Avg{\\dot{h}_B(0) h_B(\\Delta t)}_{A_{-\\Delta t}}$ becomes equal to \n$\\Avg{\\dot{h}_B(\\infty)}\\Avg{h_B(\\infty)}=0$. Clearly, the macroscopic rate\nconstant can only become equal to the microscopic expression if\n$\\Delta t$ is chosen to be smaller than the typical waiting time,\n$\\Delta t\\ll \\tau_{AB},\\tau_{BA}$. \n\nOn the other hand, $\\Delta t$ cannot be made arbitrarily small. For $\\Delta\nt\\to 0$, Eqs.~\\ref{eq:k_AB_2} reduce to the transition-state approximation\n\\begin{eqnarray}\nk_{AB}^\\mathrm{TST}&=&\\Avg{\\theta(\\dot{q})\\dot{q}\\delta(q-q^*)}\n\\nonumber,\\\\\nk_{BA}^\\mathrm{TST}&=&-\\Avg{\\theta(-\\dot{q})\\dot{q}\\delta(q-q^*)}\n\\label{eq:k_AB_TST}.\n\\end{eqnarray}\nTransition-state theory assumes that every trajectory that crosses the\ndividing surface from $A$ to $B$ will end up in $B$ before it returns\nto $A$ on a time scale $\\tau_{BA}$. However, trajectories may also\nrecross the dividing surface a (large) number of times before they\nsettle in the new state $B$ or the original state $A$.  These correlated\nrecrossings tend to decrease the correlation function\n$\\Avg{\\dot{h}_B(0)h_B(\\Delta t)}_{A_{-\\Delta t}}$ as $\\Delta t$ is\nincreased from zero.  If the recrossings can be resolved\nexperimentally, and if one wishes to characterize them, then\nEq.~\\ref{eq:dPBdt} is not an appropriate model and the rate constants\nhave to be defined differently; we will discuss\nthis scenario in more detail in the next section. Here, we\nhave been assuming that Eq.~\\ref{eq:dPBdt} is an appropriate model. In this\nmodel, the system equilibrates on time scales $\\tau_A<\\Delta t$ and\n$\\tau_B<\\Delta t$ inside the states $A$ and $B$, before\nswitching out of these states on a much longer time scale $\\tau_{AB}$\nand $\\tau_{BA}$, respectively.  This implies that the\ncorrelation functions $\\Avg{\\dot{h}_B(0)h_B(\\Delta t)}_{A_{-\\Delta\n    t}}$ and $\\Avg{\\dot{h}_B(0)h_B(\\Delta t)}_{B_{-\\Delta t}}$ reach a\nplateau for $\\Delta t$ in the range $\\tau\\ts{trans}<\\Delta t\\ll\n\\tau_{AB},\\tau_{BA}$. \n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=7cm]{fig02}\n  \\caption{Microscopic (a) and macroscopic (b) trajectories in a simple\n  model system of diffusion over a flat potential barrier. A particle starts\n  at $q=-2.5$ and then performs a random walk visiting half-integer lattice\n  points, with reflecting boundaries at $q=\\pm 15$. The particle\n  experiences a flat potential barrier given by $U(q)\/k_BT=3\\theta(2-|q|)$;\n  diffusive steps are proposed in either direction and accepted according to a\n  standard Metropolis criterion. The dividing surface defining the states $A$\n  and $B$ is located at $q_A=q_B=q^*=0$.}\n  \\label{fig:trajmarkov}\n\\end{center}\n\\end{figure}\n\n\nFigs.~\\ref{fig:trajmarkov} and~\\ref{fig:Cfig} illustrate these ideas for a\nparticle performing a random walk in a piecewise-flat double-well potential\naccording to Metropolis Monte Carlo dynamics (e.g.~\\cite{frenkel01}). The\nmacroscopic trajectory, shown in Fig.~\\ref{fig:trajmarkov}b, exhibits long\ncommitted episodes, but also short spikes caused by transient recrossings of\nthe dividing surface $q^*=q_A=q_B$.\nWhile the correlation function $C(t)=\\Avg{h_B(t)}_{A_0}$ for this system shown\nin Fig.~\\ref{fig:Cfig}a rises exponentially for macroscopic times, its time\nderivative (Fig.~\\ref{fig:Cfig}b,c) does show a sudden drop for $t \\lesssim\n\\tau\\ts{trans}\\approx 10$; this is due to the rapid correlated re-crossings of\nthe diving surface $q^*$. Importantly, for $t\\ts{trans}<t< \\tau\\ts{rxn}$ the\nderivative of the correlation function exhibits a clear plateau. In this\nregime, the flux of trajectories from $A$ to $B$ is constant.\nIndeed, the rate constant $k_{AB}$ is precisely given by the value of\n$\\dot{C}(t)$ in this regime.\n\n\n\\begin{figure}[tp]\n\\begin{center}\n \\includegraphics[width=7cm]{fig03}\n  \\caption{The correlation function $C(t)$, Eq.~\\ref{eq:C_t} and its time\n  derivative for the model system of Fig.~\\ref{fig:trajmarkov}. Clearly on\n  long times on the order of $\\tau\\ts{rxn}\\simeq 1800$ MC steps the system\n  relaxes exponentially. For very short times $t\\lesssim 10$, the\n  behavior is dominated by correlated transient re-crossings made by\n  trajectories on the barrier top. A plateau emerges between these two times,\n  whose value $\\dot C\\simeq 2.78\\times 10^{-4}$ equals the rate constant\n  $k_{AB}$.}\n  \\label{fig:Cfig}\n\\end{center}\n\\end{figure}\n\nIn summary, we reiterate that we have not derived the macroscopic rate equation\nfrom microscopic laws. By positing the macroscopic rate equation\nEq.~\\ref{eq:dPBdt}, we have made the implicit assumption that the system\nrelaxes insides the basins and looses memory of its dynamics on the\nexperimental time scale $\\Delta t$. The same assumption also entails that the\nnon-stationary ensemble of trajectories is similar to the stationary ensemble\nof trajectories, conditioned on starting in $A$. In other words, if the\nmacroscopic rate equation gives an accurate description of the switching\ndynamics, then this strongly suggests that the Onsager regression theorem\nholds as well.\n\n\n\n\\subsection{Externally driven Markov systems}\n\nWe now consider time-inhomogeneous Markov systems. We imagine that from $t=0$\nthe Markovian system is under the influence of some time-dependent external\nforce $\\phi(t)$ that biases it towards one state or the other. The macroscopic\nrate equation thus has the general form of Eq.~\\ref{eq:dPBmarkov}, and is valid\nfor times longer than a macroscopic time $\\Delta t$ as inferred from\nexperiment. The general solution can be written as\n\\begin{equation}\\label{eq:PABtimeinmarkov}\nP_B(t)=P_B(0) e^{-\\int_0^t \\frac{ \\d t'}{\\tau\\ts{rxn}(t')}}+ \\int_0^t\nk_{{AB}}(t') e^{-\\int_{t'}^t \\frac{\\d t''}{\\tau\\ts{rxn}(t'')} } \\d t',\n\\end{equation}\nwhere we define $\\tau\\ts{rxn}^{-1}(t)=k_{AB}(t)+k_{BA}(t)$.\n\nExamples of externally driven Markov systems include protein unfolding\ndriven by a time-dependent force and (crystal) nucleation at a\ntime-varying temperature. This class also contains systems where one\n(or more) of the degrees of freedom relaxes slowly on the time scale\nof a switching event, such as gene networks where at a given time a\ngene is turned on, which then induces the flipping of a genetic\nswitch. The protein that drives the flipping of the switch could then\nbe viewed as the external force $\\phi(t)$ that acts on the switch. We\nconsider external protocols with finite band-width such that there\nexist a scale of fastest force variation $\\tau_\\phi$ and a scale of\nslowest variation $T_\\phi$, where $T_\\phi$ is at most the duration of the\nexperiment, $T_\\phi\\leq T$.\n\nWhat microscopic scenarios are compatible with the macroscopic rate law,\nEq.~\\ref{eq:dPBmarkov}? The rate law implies that the system looses microscopic\nmemory beyond $\\Delta t$, so that $\\tau\\ts{trans}<\\Delta t$ must hold. As\nbefore, we restrict our attention to systems with a time scale separation\nbetween intrastate relaxation and global relaxation, $\\tau\\ts{trans}\\ll\n\\tau_{AB,BA}$, where $\\tau_{AB}$ and $\\tau_{BA}$ are the mean waiting times for\nswitching  \\footnote{Since the transition rates are now\n  time-dependent, the definition of the waiting times for transitions,\n  $\\tau_{AB}$ and $\\tau_{BA}$ is less clear-cut. A way to obtain these time\n  scales is to take the average `instantaneous waiting times' over the duration\n  of the protocol.\n  E.g.~$\\tau_{AB} = 1\/T\\int_0^T k_{AB}(t)^{-1}\\d t$.}.\n  Under these constraints, there still is a variety of\npossible relations between the time scales of driving and the intrinsic\nmicroscopic time scales of the system.\n\nThe simplest situation is the quasi-static case: While the\nstate-space distribution evolves with $\\phi(t)$ in between the switching\nevents, the system can locally (i.e. within the macroscopic states) adapt to\nthe force, meaning that\n$\\tau\\ts{trans}<\\tau_\\phi$. Importantly, this quasi-static forcing scenario\ndoes not require the force to be slower than the global relaxation;\n$\\tau_\\phi,T_\\phi \\gtrless \\tau_{AB},\\tau_{BA}$. We then expect that it\nshould be straightforward to define time-dependent rate functions in terms of \nmicroscopic correlation functions.\n\nTo derive the microscopic expressions that are consistent with the macroscopic\nEq.~\\ref{eq:dPBdt}, we first note that in this quasi-static case we may choose\n$\\Delta t$ such that $\\tau\\ts{trans} < \\Delta t < \\tau_\\phi,\\tau\\ts{rxn}$. \nOn this time scale $\\Delta t$ the macroscopic\nswitching dynamics is memoryless and approximately time-homogeneous, since the\nsystem adapts instantaneously to the force. Thus Eq.~\\ref{eq:jAA0} applies and\nthe expressions Eqs.~\\ref{eq:Cdot_3}-\\ref{eq:dPBdt_2} hold unchanged.\nWe then indeed arrive at Eq.~\\ref{eq:k_AB_M}. The rates now depend on time $t$,\nbut since the force is quasi-static, the dynamics in each $\\Delta t$ interval\nis governed only by the value of $\\phi$. Time dependence thus enters only via\n$\\phi(t)$: \n\\begin{eqnarray}\nk_{X{\\bar{X}}}(t)&=&k_{X{\\bar{X}}}(\\phi(t))\n\\text{, if }\\tau\\ts{trans}< \\Delta t < \\tau_\\phi.\\label{eq:viaphi}\n\\end{eqnarray}\nFurthermore, when $\\Delta t$ is in the range $\\tau\\ts{trans}<\\Delta t\n<\\tau_\\phi,\\tau\\ts{rxn}$, we can expect that the correlation functions in\nEq.~\\ref{eq:k_AB_M} are constant as a function of $\\Delta t$, in analogy to\nthe time-homogeneous case.\n\n\nFig.~\\ref{fig:jABforced} illustrates these relations in a version of the\nbarrier-diffusion toy model, which is now augmented with an oscillatory force\nwith a single frequency, $f(t) = a k_BT \\sin(2\\pi t\/\\tau_\\phi)$ with\n$\\tau_\\phi=T_\\phi=400$, $a=0.1$. We choose $\\Delta t=20$ so that\n$\\tau\\ts{trans}<\\Delta t\\ll \\tau_\\phi,\\tau\\ts{rxn}$. This separation of time\nscales suggests that we can treat this time-inhomogeneous Markov system as\nquasi-stationary, so that  $j_{AA}\\simeq 0$ (Eq.~\\ref{eq:jAA0}), and time\nenters the rates only via the force. Indeed, we observe that the recrossing\nfluxes approximately vanish, and that at the times when the force crosses zero,\nthe rate constant equals that of the time-homogeneous Markov system of\nFig.~\\ref{fig:Cfig}, supporting this idea.\n\\begin{figure}[htp]\n\\begin{center}\n \\includegraphics[width=7cm]{fig04}\n  \\caption{Time-dependent version of the system of\n    Fig.~\\ref{fig:trajmarkov}, under the influence of an external\n    force. The system is started in stationary state within $A$ and\n    relaxes with a time dependent modulation (a; $\\Avg{h_A}$, blue,\n    $\\Avg{h_B}$, red). Fluxes defined in Eq.~\\ref{eq:Cdot_1} (b;\n    $j_{AB}$, blue, $j_{BA}$, red, $j_{AA}$, black) reflect the state\n    occupancies.  The rate functions, Eq.~\\ref{eq:k_AB_M} (c; colors\n    as in b) are independent of state occupancy. At the\n    zeros of the force (dotted lines) the time dependent rates\n    coincide with the rate constant of the stationary system,\n    $k_{AB}=2.78\\times10^{-4}$ (see Fig.~\\ref{fig:Cfig}), in agreement with \n    a quasi-stationary description on the time scale of the driving.}\n  \\label{fig:jABforced}\n\\end{center}\n\\end{figure}\n\n\nAn alternative simple scenario arises in the opposite case of rapid driving,\nwhere the external force variations are faster than the local relaxation within\nthe macroscopic states, i.e.~$\\tau_\\phi<T_\\phi<\\tau\\ts{trans}$. We can then\nchoose a macroscopic resolution $\\Delta t > \\tau_\\phi,\nT_\\phi,\\tau\\ts{trans}$, so that the macroscopic description effectively\naverages both over microscopic correlations of the system and over the\nvariations in driving. In this case the macroscopic switching dynamics is\ndescribed by the phenomenological rate equation of the time-homogeneous system,\nEq.~\\ref{eq:dPBdt}, although the magnitudes of the rate constants, which are\nindependent of $\\Delta t$, will be renormalized by the time-varying force.\nMoreover, because the system is effectively time-homogeneous and switches in a\nmemoryless fashion between $A$ and $B$ , again Eq.~\\ref{eq:jAA0} holds, so that\nthe rate constants are given by the plateau values of the correlation functions\nin Eqs.~\\ref{eq:k_AB_2} in the regime $\\tau_\\phi,\\tau\\ts{trans}<\\Delta t\\ll\n\\tau_{AB},\\tau_{BA}$.\n\nIn the previous two scenarios we were able to choose $\\Delta t$ to be both\nwell-separated from the driving time scales and within a well-defined plateau\nof the derivative of the microscopic correlation function; this allowed us to\nextract the rate as $k_{AB}=\\dot C(\\Delta t)$, irrespective of the precise\nvalue of $\\Delta t$. We now consider a scenario in which this is not\npossible. Suppose that a certain macroscopic time resolution $\\Delta t$ is\nrequired, which still leads to Markovian switching ($\\tau\\ts{trans}<\\Delta t$)\nbut lies \\emph{within} the frequency band of the driving, $\\tau_\\phi < \\Delta\nt<T_\\phi$. For instance, we may wish to describe a non-stationary experiment\ncarried out with a given time resolution equal to $\\Delta t$. \n\nIn the case that the microscopic relaxation of the system is the fastest\nprocess, $\\tau\\ts{trans} <\\tau_\\phi$ as in the quasi-stationary case studied\nabove, then the correlation function does exhibit a plateau, but in a regime\n$\\tau\\ts{trans} <\\widetilde{\\Delta t} < \\tau_\\phi$ below the macroscopic\nresolution $\\Delta t$. While this does allow us to uniquely define a\nforce-dependent rate constant as $k_{AB}(\\phi)=\\dot{C}(\\widetilde{\\Delta t})$,\nthis rate constant is inaccessible in experiments on the time scale $\\Delta t$.\nThe macroscopic rate constant as measured in the experiment now depends on the\nexperimental time resolution $\\Delta t$:\n\\begin{equation}\nk_{AB}^{\\rm eff}(t)\\simeq (\\Delta t)^{-1}\\int_{t-\\Delta t\/2}^{t+\\Delta t\/2}\nk_{AB}(\\phi(t'))\\d t'.\\label{eq:kABeffmixed1}\n\\end{equation}\n\nAlternatively, when the driving contains components that are faster than\nmicroscopic relaxation, $\\tau_\\phi<\\tau\\ts{trans}$, the system is Markovian on\nthe time scale $\\Delta t$ but retains memory of the driving over times\n$\\tau\\ts{trans}$. In this case ($\\tau_\\phi<\\tau\\ts{trans}<\\Delta t<T_\\phi$) the\ncorrelation function $C(t)$ does not exhibit a well-defined plateau on any time\nscale, and, as a result, a rate constant cannot be defined in a unique,\nprotocol-independent manner. It is of course still possible to define a rate\nfunction in terms of a first passage time density, as a function of the\nexternal protocol; for analytical treatments in model\nsystems with simple driving protocols see for instance \\cite{lindner04} and\nreferences therein.\n\n\n\\section{Non-Markovian systems} \\label{sec:NonStatSys}\n\nFor equilibrium systems, the two observations of rapid switching\n$\\tau_C\\ll\\tau_{AB},\\tau_{BA}$ (Eq.~\\ref{eq:tau_C}) and rapid equilibration\n$\\tau_A\\ll\\tau_{AB};\\tau_B\\ll\\tau_{BA}$ (Eq.~\\ref{eq:tau_A_B}) are often\nintimately connected. Rare events typically arise because of a single large\nfree-energy barrier separating the two states $A$ and $B$. As a consequence,\nthe waiting time tends to be much longer than the switching event\nitself---making a two-state description meaningful---while it also allows the\nsystem to relax inside the basins of $A$ and $B$ in between the switching\nevents---leading to memoryless switching on the macroscopic time scale $\\Delta\nt$. In these equilibrium systems there is only one relevant macroscopic time\nscale $\\tau\\ts{rxn}$, which is associated with the global relaxation of state\noccupancies. However, even for equilibrium systems, inertial effects may lead\nto correlated recrossings \\cite{erp11}, possibly giving rise to non-Markovian\nswitching dynamics on the macroscopic time scale $\\Delta t$; if such correlated\nrecrossings are important, then the macroscopic model of Eq.~\\ref{eq:dPBdt}\nneeds to be refined.\n\nOne can readily find examples of non-equilibrium\nsystems, however, that can be coarse grained as two-state systems, but in\nwhich switching between the two states occurs in a non-Markovian fashion \neven on time scales that are comparable to\nthe typical waiting times $\\tau_{AB}, \\tau_{BA}$. This means that while the\nwaiting time is much longer than the duration of the switching event, there\nexists another time scale in the system that is comparable to the waiting time\nfor switching.\n\nAn example of such a non-Markovian, non-equilibrium system is the bacterial\nflagellar motor.  This rotary motor can run in either clockwise or\ncounterclockwise direction.  The transitions between these two states\nhappen much faster than the typical waiting time for switching,\nsuggesting a two-state description. In experiments, the distribution\nof waiting times was observed to be non-exponential, and the power\nspectrum of the switching dynamics featured a peak around $1 \\mathrm\ns^{-1}$~~\\cite{korobkova06}, which means that there is a characteristic\nfrequency at which the motor switches.\n\nThese data show that the flagellar motor cannot be described as a random\ntelegraph process in which the switching events are independent, and the\nclockwise and counterclockwise intervals are uncorrelated and exponentially\ndistributed; indeed, this system cannot be described by Eq.~\\ref{eq:dPBdt}. The\ndata also imply that motor switching is coupled to a non-equilibrium process\n\\cite{kampen92,tu05,albada09}. It appears that the conformational dynamics of\nthe rotor proteins is coupled to the slow relaxation dynamics of the flagellum,\nwhich in turn is influenced by the rotation of the motor, driven by a proton\nmotive force \\cite{albada09}.\n\nTo describe systems like the bacterial flagellar motor in which memory is\nimportant, we now consider non-Markovian systems that when left unperturbed,\nreach a stationary state (equilibrium or not). As before, we imagine that the\nsystem flips between two macroscopic states $A$ and $B$ that are metastable in\nthe sense that the waiting time between the switching events is much longer\nthan the duration of the switching event itself. These system have memory on\nthe macroscopic time scale $\\Delta t$ which means that the propensity of\nswitching between the two macroscopic states at a given moment in time depends\nupon the history of the switching dynamics.\n\n\nTo derive microscopic expressions for the rate functions in the presence of\nmemory, we start again with the phenomenological description of the system. If\nthe propensity to switch depends upon the macroscopic history even prior to\nthe last switching event, then Eq.~\\ref{eq:ME} cannot be simplified much\nfurther, and remains of little use. However, it is conceivable that the\nswitching propensity depends upon the macroscopic history only since the last\nswitching event, to a good approximation. The bacterial flagellar\nmotor is an example of such a system: when\nthe experimental clockwise and counterclockwise intervals of the flagellar\nmotor were randomly shuffled, the power spectrum was unchanged; moreover, the\n observed power spectrum could be reproduced from the measured\nwaiting-time distributions only \\cite{korobkova06}. This strongly suggests that\nthe different intervals are in fact temporally uncorrelated, so that the\nswitching propensity only depends upon the time that has passed since the last\nswitching event. We refer to this as the \\emph{clock-resetting}\nscenario---after each switching event, the system looses memory; in other\nwords, switching is a renewal process. In what follows we will restrict our\nattention to this case. We do not exclude at this point the presence of\nexternal driving; in that case the system has only a single-switch memory but\nis time-inhomogeneous.\n\n\nIn the clock-resetting scenario, the phenomenological rate equation \nEq.~\\ref{eq:ME} can be simplified:\n\\begin{equation}\n\\frac{\\d}{\\d t}P_B(t)=\\int_{t>t'}k_{A\n  B}(t|t')P_A(t;t') - k_{BA}(t|t')P_B(t;t')\\d t'.\n\\label{eq:dPBdt_NM}\n\\end{equation}\nHere, $P_A(t;t')\\d t'$ is the joint probability that the system is in\nstate $A$ at time $t$ and has switched into this state for the last\ntime within the earlier interval $(t',t'+\\d t')$. It can be decomposed\nas the propensity $q_{BA}(t')$ to enter $A$ at $t'$,  times\nthe survival probability $S_A(t|t')$ to stay in $A$ until at least time $t$:\n$P_A(t;t')=q_{BA}(t')S_A(t|t')$. \n\nThe rate kernel $k_{AB}(t|t')$ is defined as the propensity\nthat the system switches from $A$ to $B$ at time $t$\ngiven that it has switched into $A$ at time $t'<t$ and is\nstill in $A$ at $t$. It is given\nby the propensity $q_{AB}(t|t')\\equiv -\\partial_t S_A(t|t')$ that a trajectory\nthat entered $A$ at time $t'$, switches from $A$ to $B$ at a later time $t$,\ndivided by the probability that it is still in $A$ at time $t$:\n$k_{AB}(t|t')=q_{AB}(t|t')\/S_A(t|t')=-\\partial_t\\ln S_A(t|t')$. If the\nsystem is time-homogeneous, i.e.~without external time dependence,\nthen we have $k_{AB}(t|t')=k_{AB}(t-t')$. If the system is in\nstationary state, then furthermore $P_B(t;t')=P_B(t-t')$. We note that\nthis phenomenological model can be solved analytically\n\\cite{albada09}, see also Appendix \\ref{sec:nonmarkovmacrosol}.\n\n\nBefore we derive microscopic expressions for the rate kernels \nthat are consistent with Eq.~\\ref{eq:dPBdt_NM}, we first discuss the\ntime scales that are relevant in such a system. A two-state\ndescription implies that the duration of the switching event is much\nshorter that the waiting time for switching:\n$\\tau_C\\ll\\tau_{AB},\\tau_{BA}$. Moreover, we expect that inside the\nstates $A$ and $B$ most degrees of freedom relax quickly, within\n$\\tau_A$ and $\\tau_B$ respectively, where $\\tau_A\\ll\\tau_{AB}$ and \n$\\tau_B\\ll\\tau_{BA}$; we can thus identify a fast transient time scale\n$\\tau\\ts{trans} \\simeq \\max\\{\\tau_C,\\tau_A,\\tau_B\\}$. If these were\nthe only time scales in problem, then the macroscopic switching\ndynamics could be described as that of a two-state Poisson process,\nassuming that we choose a macroscopic time scale $\\Delta\nt>\\tau\\ts{trans}$.  However, the observation that the macroscopic\nswitching dynamics exhibits memory on the scale $\\Delta t$, implies\nthat there is another time scale in the problem, $\\tau\\ts{slow}$,\nwhich is at least on the order $\\tau\\ts{slow}\\gtrsim \\Delta t$.\n\nThe bacterial flagellar motor provides a concrete illustration of these ideas:\nwhile the switching dynamics of the motor in the presence of the flagellum is\nnon-Poissonian, the dynamics in the absence of the flagellum is Poissonian,\nwith constant rates, leading to exponential waiting-time distributions\n\\cite{bai10}. The reason for this change in behavior is that the relaxation\ndynamics of the flagellum introduces a slow time scale, which\nbecomes comparable to, and can even set, the typical waiting time for\nswitching. After a switching event, the flagellum first unwinds, but then,\ndriven by the rotation of the motor, winds up in the new direction; this leads\nto an increase in the force on the rotor proteins, which then tends to switch\ntheir conformation, and thereby the rotation direction of the motor\n\\cite{albada09}.\n\nWe extend our diffusive-barrier crossing model to capture the non-equilibrium\nswitching of the bacterial flagellar motor, see Fig.~\\ref{fig:forcetraj}. The\nparticle diffuses in a piecewise constant potential with a single flat barrier\nof height $4k_\\mathrm{B}T$ at $|q|<2$; the state boundaries are at\n$q_A=q_B=q^*=0$. Whenever the particle enters one of the two potential wells\n(say, at $t'$), the clock is reset and a restoring force ramps up over time,\n$f\\ts{clk}=-\\mathrm{sgn}(q) f_0 [1-e^{-(t-t')\/\\tau_\\phi}]$ with time constant\n$\\tau_\\phi=100$ and saturating magnitude $f_0=0.8k_\\mathrm{B}T$.\nThe addition of a restoring force leads to a faster global relaxation, which\nfor the chosen value of $f_0$ is on the order of $\\tau_\\phi$.\nFig.~\\ref{fig:forcetraj} shows that the relaxation of this system is not\nexponential: There is an initial lag time, and relaxation occurs in an\noscillatory fashion, because the restoring force introduces a characteristic\ntime scale for switching, and because particle clocks are taken to\nbe synchronized at $t=0$.\nWe remark that an exponential relaxation observed in a single experiment would\n\\emph{not} by itself imply memoryless switching: For instance, the same system\nwith unsynchronized clocks at $t=0$ relaxes exponentially (not shown).\n\n\n\\begin{figure}[htp]\n\\begin{center}\n \\includegraphics[width=7.5cm]{fig05}\n  \\caption{Particle diffusing with residence-time dependent restoring force.\n  A trajectory (top, blue, left axis) reflects the restoring force (black,\n  right axis). The relaxation of state occupancies when started uniformly in\n  the $B$ state basin (bottom) is oscillatory, and more rapid than without\n  force (cf.~Fig.~\\ref{fig:Cfig}).}\n  \\label{fig:forcetraj}\n\\end{center}\n\\end{figure}\n\nTo derive microscopic expressions for the rate kernels that are\nconsistent with the macroscopic rate equation of Eq.~\\ref{eq:dPBdt_NM}, we need\nto define an indicator function that serves to measure the time since the last\nswitching event. We let $H_X(t,t')\\equiv\\prod_{t\\geq t'' > t'}h_X(t'')$. By\nconstruction, $H_X(t,t')$ takes the value 1 if the given trajectory was in\n$X=A,B$ without interruption from $t'$ to $t$ and 0 otherwise, and $H_X(t,t) =\nh_X(t)$. One also verifies (Appendix \\ref{sec:Hmath}) that\n\\begin{eqnarray}\n\\partial_t H_X(t,t') &=& \\dot{h}_X(t)\n  H_X(t,t') \\text{, and}\\nonumber\\\\\n\\partial_{t'} H_X(t,t')&=&\n  H_X(t,t')\\dot{h}_X(t').\\label{eq:dot_H_X}\n\\end{eqnarray}\nThe temporal derivatives of $H_X(t,t')$ with respect to $t'$ and $t$\n(Eq.~\\ref{eq:dot_H_X}) thus make it possible to count the last entry time and\nthe first exit time out of $X$, respectively. Although the definition of $H$\nmay appear convoluted, it is straightforward to measure in a simulation, by\nrecording switching times along trajectories. Interestingly an\nindicator function similar to $H$ has been employed by Jung \\textit{et al} \n\\cite{jung05} to characterize fluctuations of trajectories in\nglassy systems.\n\nApplying the relations of Eq. \\ref{eq:dot_H_X} and the identity\n$\\dot{h}_A+\\dot{h}_B=0$, we can obtain the following relation:\n\\begin{subequations}\\label{eq:totalhBdot}\n\\begin{eqnarray}\n\\dot{h}_B(t)&=&\\dot{h}_B(t)\\sum_{X=A,B}H_X(t,t)\\nonumber\\label{eq:dot_h_B}\\\\\n&=&\\dot{h}_B(t)\\Bigl[\\sum_{X=A,B} H_X(t,t_0) + \\int_{t_0}^t \\partial_{t'}\nH_X(t,t')\\d t'\\Bigr]\\nonumber\\\\\n&=&\\partial_t H_B(t,t_0) - \\partial_t H_A(t,t_0) \\label{eq:H_X_0}\\\\\n&&+\\int_{t_0}^t\\partial_t\\partial_{t'} H_B(t,t')\n  - \\partial_t\\partial_{t'} H_A(t,t') \\d t'.\\label{eq:H_AB_t}\n\\end{eqnarray}\n\\end{subequations}\nHere the probability flux in and out of $B$ is partitioned into contributions\nfrom trajectories that remained in one state since $t_0$ and then switch at $t$\n(Eq.~\\ref{eq:H_X_0}), and trajectories that had their last switching event at\n$t'>t_0$ before switching at $t$ (Eq.~\\ref{eq:H_AB_t}). The latter terms allow\nus to derive correlation functions which express the rate kernels \ndepending on the last switching event.\n\n\nTo derive a microscopic expression that is consistent with the\n the macroscopic equation \\ref{eq:dPBdt_NM}, we take\nthe ensemble average $\\Avg{\\dots}$ of Eq.~\\ref{eq:totalhBdot} (see also\nEq.~\\ref{eq:Cdot_0}). The ensemble average is defined by the initial\ndistribution $\\rho(x_0)$ as before (although here we do not condition on\nstarting in state $A$), and possibly by a prescription for $t<0$, as discussed\nbelow. We now make the identifications\n\\begin{subequations}\\label{eq:nonMarkovidentify}\n\\begin{eqnarray}\n\\Avg{h_B(t)}&=&P_B(t),\\\\\n\\partial_{t'}\\Avg{H_X(t,t')}\\d t'\n&=&\\Avg{H_X(t,t')\\dot{h}_X(t')}\\d t' \\nonumber\\\\\n&=&P_X(t;t')\\d t',\\\\\n\\frac{\\partial_t\\partial_{t'}\\Avg{H_X(t,t')}}\n{\\partial_{t'}\\Avg{H_X(t,t')}} &=&\n\\frac{\\Avg{\\dot{h}_X(t)H_X(t,t')\\dot{h}_X(t')}}\n{\\Avg{H_X(t,t')\\dot{h}_X(t')}}\\nonumber\\\\\n&=&-k_{X{\\bar{X}}}(t|t'), \\label{eq:kdefstrict}\n\\end{eqnarray}\n\\end{subequations}\nwhere $X=A,B$ and ${\\bar{X}}=B,A$. We can then rewrite the microscopic equation\n\\ref{eq:totalhBdot} in the form of Eq.~\\ref{eq:dPBdt_NM}.\n\\begin{subequations}\\label{eq:dPBdt_NM_BC}\n\\begin{gather}\n\\frac{\\d }{\\d t}P_B(t)=\n\\partial_t\\Avg{H_B(t,t_0)}-\\partial_t\\Avg{H_A(t,t_0)}\n\\label{eq:dPBdt_NM_BC_bdry}\\\\\n+\\int_{t_0}^t k_{AB}(t|t')P_A(t;t')-k_{BA}(t|t')P_B(t;t')\\d t'.\n\\label{eq:dPBdt_NM_BC_int}\n\\end{gather}\n\\end{subequations}\nThe additional boundary terms Eq.~\\ref{eq:dPBdt_NM_BC_bdry} appear here due to\na finite lower integration limit $t_0$.  They can be treated in two ways.  The\nfirst option is taking $t_0\\to-\\infty$. Then the terms \nEq.~\\ref{eq:dPBdt_NM_BC_bdry} vanish, because the system is not expected to\nstay in a given state forever. To extract rate constants from a\nmicroscopic measurement, we then have to be able to determine the integrand in\nEq.~\\ref{eq:dPBdt_NM_BC_int} for negative $t'$. This will be possible for\ninstance if we are dealing with a time-homogeneous system which was prepared\nin a stationary state for negative times, since then $P_X(t;t')=q_{{\\bar{X}} X,\n\\mathrm{ss}}S_X(t-t')$ for all $t'<0$, with a constant, stationary-state\nvalue of the influx $q_{{\\bar{X}} X, \\mathrm{ss}}$; see also Appendix\n\\ref{sec:nonmarkovmacrosol}.\n\nInstead, for non-stationary systems one may wish to explicitly account for the\nswitching events only after $t>0$ without specifying the history before $t<0$.\nThen one would let $t_0\\to0$ and retain the boundary terms,\nEq.~\\ref{eq:dPBdt_NM_BC_bdry}. They give a transient contribution to the\nmacroscopic evolution of the system, which summarizes how the system was\nprepared before $t=0$. We can incorporate this contribution by defining\n\\begin{subequations}\n\\begin{eqnarray}\n\\frac{\\partial_t\n  \\Avg{H_X(t,0)}}{\\Avg{H_X(t,0)}}&=&-k_{X{\\bar{X}}}^{<0}(t),\\\\\n\\Avg{H_X(t,0)}&=&P_X^{<0}(t).\n\\end{eqnarray}\n\\end{subequations}\nIn this case, Eq.~\\ref{eq:dPBdt_NM_BC} becomes\n\\begin{subequations}\\label{eq:dPBmarkovnonmarkov}\n\\begin{gather}\n\\frac{\\d }{\\d t}P_B(t)=k_{AB}^{<0}(t)P_A^{<0}(t)-k_{BA}^{<0}(t)P_B^{<0}(t)\n\\label{eq:dPBmnmm}\\\\\n+\\int_0^tk_{AB}(t|t')P_A(t;t')-k_{BA}(t|t')P_B(t;t')\\d t',\\label{eq:dPBmnmnm}\n\\end{gather}\n\\end{subequations}\nwhich has an effective Markovian part (Eq.~\\ref{eq:dPBmnmm}) and a\nnon-Markovian (Eq.~\\ref{eq:dPBmnmnm}) part.\n\n\nScrutinizing the arguments above more carefully, we should expect that the\nsystem exhibits transient recrossings of the dividing surface $q^*$ that are\nnot persistent on the macroscopic time scale $\\Delta t$. Moreover, these\ntransient recrossings are not expected to lead to loss of memory: only\nmacroscopic switching events, in which the systems fully transitions from one\nbasin of attraction to another, are likely to reset the clock. This means that\nthe identifications made in Eqs.~\\ref{eq:nonMarkovidentify}, like the TST\nexpressions Eqs.~\\ref{eq:k_AB_TST}, are erroneous in that they overcount\ntransient switching events. An additional point concerns the experimental time\nresolution. It is conceivable that the experimental resolution is sufficiently\nhigh that transient crossing events can be detected that do not reset the\nclock. If one wishes to describe these events, then the phenomenological model\nof Eq.~\\ref{eq:dPBdt_NM}, which only describes the crossing events that lead to\nloss of memory, needs to be modified. Here, we consider the case that\nEq.~\\ref{eq:dPBdt_NM} is an appropriate macroscopic model. The task is then to\ncome up with a microscopic description that integrates over the transient\nrecrossings of the dividing surface that do not reset the clock.\n\n\n\nTo integrate over the transient recrossings of the diving surface that\nseparates the macroscopic states $A$ and $B$, we must modify the indicator\nfunction $H_{X{\\bar{X}}}$.  This can be achieved by introducing a  `grace interval'\n$\\Delta t$ over which such short excursions are tolerated. Therefor we define a\nrelaxed version of $h_X$ as\n\\begin{equation}\\label{eq:h_Xtdt}\nh_X(t;\\Delta t) = \\theta\\bigl[\\int_{t-\\Delta t}^th_X(t')\\d t'-\\Delta t\/2\\bigr];\n\\end{equation}\nthis new indicator function takes the value 1 precisely if in the preceding\nshort interval, the system was predominantly in state $X$; the arbitrary\nconvention of using preceding and not e.g.~centered intervals becomes\nunimportant on the macroscopic scale.\nObserve that, as for $h_X$, we have $h_A(t;\\Delta t)+h_B(t;\\Delta t)\\equiv 1$\n(we still take the states to be adjacent, $q_A=q_B=q^*$); the previous\ndefinition $h_X(t)$ is recovered in the limit $\\Delta t\\to 0$.\nUsing this coarse-grained state indicator function, the indicator for\nuninterrupted residence in a state is then defined as before,\n\\begin{equation}\nH_X(t,t';\\Delta t)=\\prod_{t \\geq  t'' > t'}h_{X}(t'';\\Delta t).\n\\end{equation}\nThis expression remains 1 for instance if all excursions from $X$ between\n$t'-\\Delta t$ and $t$ are shorter than $\\Delta t\/2$. By definition,\n$H_X(t,t';\\Delta t)$ is a monotonically decreasing function of $t$.\nFurthermore, $H_X(t,t;\\Delta t)=h_X(t;\\Delta t)$, and $H_X(t,t';\\Delta t) \\to\nH_X(t,t')$ as $\\Delta t\\to 0$.\n\nFig.~\\ref{fig:HXplot} shows the average uninterrupted state occupancies for the\nmodel system introduced above. While the state occupancy $H_A(t,t;\\Delta\nt)=h_A(t;\\Delta t)$ displays the oscillatory global relaxation in this system,\n$H_X(t,t';\\Delta t)$ decays with increasing residence time $t-t'$. This decay\nis slower when $\\Delta t$ is finite due to the fact that brief excursions from\nthe state are tolerated.\n\n\\begin{figure}[htp]\n\\begin{center}\n \\includegraphics[width=7cm]{fig06}\n  \\caption{Uninterrupted state occupancies $\\Avg{H_A(t,t';\\Delta t)}$ for the\n  system from Fig.~\\ref{fig:forcetraj}, for $\\Delta t=0,20$ (solid and dashed\n  lines, respectively). The system starts in state $B$ and \n  accumulates in $A$ with occupancy $\\Avg{H_A(t,t;\\Delta\n  t)}-\\Avg{h_A(t;\\Delta t)}$, shown in gray.\n  Crossings back to $B$ then decrease $\\Avg{H_A(t,t';\\Delta t)}$, shown in\n  colors for fixed values $t'=0,100,200,400$ from left to right.}\n  \\label{fig:HXplot}\n\\end{center}\n\\end{figure}\n\nWith the new definitions for $h$ and $H$, the equalities in\nEq.~\\ref{eq:totalhBdot} still hold. Therefore, also Eq.~\\ref{eq:dPBdt_NM_BC}\nholds unchanged if we make the new identifications\n\\begin{subequations}\\label{eq:nonMarkovidentifydelta}\n\\begin{eqnarray}\n\\Avg{h_B(t, \\Delta t)} &=& P_B(t), \\\\\n\\partial_{t'} \\Avg{H_X(t,t';\\Delta t)}\\d t' &=&P_X(t;t';\\Delta t)\\d t',\\\\\n\\frac{\\partial_t \\partial_{t'}\\Avg{H_X(t,t';\\Delta t)}}\n{\\partial_{t'}\\Avg{H_X(t,t';\\Delta t)}}&=&-k_{X{\\bar{X}}}(t|t';\\Delta t),\n\\label{eq:kdefgrace}\n\\end{eqnarray}\n\\end{subequations}\nreplacing Eqs.~\\ref{eq:nonMarkovidentify}.\n\n\nIf the system exhibits transient recrossings on a time scale $\\tau\\ts{trans}\\ll\n\\tau\\ts{slow}$,  then $k_{X{\\bar{X}}}(t|t';\\Delta t)$ should become independent of\n$\\Delta t$ if we choose $\\Delta t$ to be in the range $\\tau\\ts{trans}<\\Delta\nt<\\tau\\ts{slow}$. Moreover, we can expect that with this choice of $\\Delta t$,\nthe rate constant $k_{X{\\bar{X}}}(t|t';\\Delta t)$ agrees with the experimentally\nmeasured rate constant $k_{X{\\bar{X}}}(t|t')$ when the experimental time\nresolution is indeed of the order $\\Delta t$, such that the transient\nrecrossings are not detected, while the presence of a slow time scale is\ndetected.\n\nThese observations are illustrated in Fig.~\\ref{fig:kPanel}, for the\nclock-resetting model system introduced earlier. We recall that the quantity\n$\\Avg{\\partial_{t'}H_A(t,t';\\Delta t)}=P_A(t;t';\\Delta t)$ measures the\nprobability that the system enters $A$ at time $t'$ and stays in $A$ at least\nuntil $t$; it is thus given by the influx propensity $q_{BA}(t';\\Delta t)$ at\n$t'$ times the survival probability in $A$, $S_A(t|t';\\Delta t)$:\n$P_A(t;t';\\Delta t)=q_{BA}(t';\\Delta t)S_A(t|t';\\Delta t)$. \nFig.~\\ref{fig:kPanel}a shows that $P_A(t,t';\\Delta t)$ decreases with $t-t'$\nbecause $S_A(t|t';\\Delta t)$ decreases with $t-t'$. $P_A(t;t';\\Delta t)$ is\nnon-monotonic as a function of $t'$, reflecting the fact that the flux\n$q_{BA}(t';\\Delta t)$ from $B$ into $A$ shows oscillations\n as the system equilibrates---these arise because the probability that the\nsystem is in $B$ relaxes in an oscillatory fashion, see\nFigs.~\\ref{fig:forcetraj}, \\ref{fig:HXplot}.\n\nThe rate kernel $k_{AB}(t|t';\\Delta t)$ is defined as the flux of trajectories\nthat enter $A$ at time $t'$ and leave $A$ for the first time at time $t$\ndivided by the flux of trajectories that enter $A$ at time $t'$ and remain in\n$A$ till time $t$, see Eqs.~\\ref{eq:kdefstrict}, \\ref{eq:kdefgrace}. This\nconditioning removes the dependence on the start time $t'$ if the system is\ntime-homogeneous. Fig.~\\ref{fig:kPanel}b shows that $k_{AB}(t|t';\\Delta t)$\nindeed depends only on the time difference: $k_{AB}(t|t'; \\Delta\nt)=k_{AB}(t-t';\\Delta t)$, so that we can conclude that our model system is\nindeed time homogeneous. By the same token, we may now average data for the \nsame $t-t'$ but different $t'$, leading to the result shown in\nFig.~\\ref{fig:kPanel}c. It is seen that $k_{AB}(t-t';\\Delta t)$ starts out with a high crossing rate, passes\nthrough a minimum and then rises to saturate at a constant value. The initial\ndrop is due to the rapid correlated recrossings of the diving surface\n$q_A=q_B=q^*$; importantly, the detection of these rapid recrossings is\nstrongly suppressed when the grace interval $\\Delta t$ is chosen to be larger\nthan the microscopic relaxation time $\\tau\\ts{trans}$, $\\Delta\nt\\geq\\tau\\ts{trans}\\approx 10$, as seen in panel c.  The subsequent minimum in\n$k_{AB}(t-t';\\Delta t)$ corresponds to local relaxation of the system inside\nthe basin of the new macroscopic state. The rate kernel remains small as long\nas the force that tends to flip the system back to the other state on a time\nscale $\\tau_\\phi$ is still low. Finally, as the switching force increases with\ntime, the rate $k_{AB}$ increases, to reach a plateau value when the switching\nforce becomes constant.  Fig.~\\ref{fig:kPanel}c also shows that\n$k_{AB}(t-t';\\Delta t)$ becomes independent of $\\Delta t$ when $\\Delta\nt>\\tau\\ts{trans}\\approx 10$ (and $t-t'>\\Delta t)$. This is crucial:\nit allows us to define the macroscopic rate function $k_{AB}(t-t')$, and it\njustifies \\emph{a posteriori} the two-state description with a rate kernel that\ndepends on the time since the last switching event.\n\n\\begin{figure}[htp]\n\\begin{center}\n \\includegraphics[width=7cm]{fig07}\n  \\caption{Time-dependent survival probabilities and rate kernels for\n   the model system from Fig.~\\ref{fig:forcetraj}. (a) The joint\n   influx and survival probability $\\Avg{\\partial_{t'}H_A(t,t';\\Delta\n     t)}=P_A(t;t';\\Delta t)=q_{BA}(t')S_A(t|t')$ as a function of $t-t'$, shown\n   for different values of $t'$ ($t'$=0,100,200,400, as indicated) and\n    for $\\Delta t=0,10$ (solid and dashed lines, respectively).\n   $P_A(t;t';\\Delta t)$ is the probability that the system enters $A$ at time\n   $t'$ and still is in $A$ at $t$. It decreases monotonically\n   with $t-t'$, yet more slowly when the grace interval $\\Delta\n   t=10$ is finite; but is non-monotonic in $t'$, as explained\n   in the text.  (b) The rate kernel $k_{AB}(t|t';\\Delta t)$ as a\n   function of $t-t'$, for different values of $t'$, and for $\\Delta\n   t=5$. It is seen that the rate kernels for the different times $t'$\n   collapse. (c) The rate kernels $k(t-t';\\Delta t)$ averaged over\n   $t'$, for chosen time lags $\\Delta t=0,2,4,10,20$, from top to\n   bottom. It is seen that for $t-t'>\\Delta t$ and $\\Delta\n   t>\\tau\\ts{trans}\\approx 10$, the rate kernels become independent of\n    $\\Delta t$. The macroscopic rate kernel can be identified with this\n    limiting function.}\n  \\label{fig:kPanel}\n\\end{center}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discussion}\n\nIt is instructive to compare the microscopic expression for the rate\nkernel in a non-Markovian system, Eq.~\\ref{eq:nonMarkovidentifydelta},\nwith that for the rate constant in a Markovian system,\nEq.~\\ref{eq:k_AB_2}. Both expressions count only those switching\nevents where the system goes from one macroscopic state to the other\nover a time $2 \\Delta t$.  For a Markov system in stationary state,\nthe recrossing flux $j_{AA}\\propto \\Avg{h_A(t-\\Delta\n  t)\\dot{h}_B(t)h_A(t+\\Delta t)} = 0$, as discussed above (see also\nAppendix \\ref{sec:jAA}). That is, if we chose to count recrossings in\nthe microscopic expression for the rate function instead of using\nEq.~\\ref{eq:k_AB_2}, this would not change the net result except for\nadditional noise. In fact, the Bennett-Chandler expression\n$\\Avg{\\dot{h}_B(t)}_{A_0}=\n\\Avg{\\dot{q}(0)\\delta(q(0)-q^*)\\theta(q(t)-q^*)}\/\\Avg{h_A}$\nalso counts trajectories that come from $B$, cross and recross the\ndividing surface (a number of times), and then go to $B$; however,\ntheir net contribution to the average is indeed zero\n\\cite{chandler78}.\n\nFor a non-Markovian system,  $\\Avg{h_A(t-\\Delta t) \\dot{h}_B(t) h_A(t+\\Delta\nt)}$ may be non-zero even in stationary state, due to the presence of\nmemory over $\\Delta t$. In contrast to the Markov case, here these transient\nrecrossings should be explicitly excluded, for the reasons already mentioned\nabove: If we take the macroscopic rate function to be defined as the propensity\nto switch, given the time that has passed since the last macroscopic switching\nevent, then we should indeed only count the truly macroscopic switching events\nin Eq.~\\ref{eq:nonMarkovidentifydelta}.\n\nWhether a switching event counts as a macroscopic switching event ultimately\ndepends on the chosen macroscopic time resolution $\\Delta t$. We can define a\nmacroscopic switching event in an unambiguous manner if the time scale for\nrecrossing $\\tau\\ts{trans}$, is well separated from the slow memory time scale\n$\\tau\\ts{slow}$ of the system, so that $k(t|t';\\Delta t)$ is independent of\n$\\Delta t$ in the regime $\\tau\\ts{trans}<\\Delta t<\\tau\\ts{slow}$. It may happen\nthat the time scales $\\tau\\ts{trans}$ and $\\tau\\ts{slow}$ are not well\nseparated; then, to match the experimentally measured rate constant, it becomes\ncritical to tune $\\Delta t$ exactly to the experimental time resolution.\nHowever, the rate kernel defined in this way will then depend on the details\nof the experiment at least over some range of residence times $t-t'$, which\nlimits its use as a description of intrinsic properties of the system. Thus in\na non-Markovian system described by Eq.~\\ref{eq:dPBdt_NM}, a time scale\nseparation between rapidly decaying transient memory and a well-defined slow\nmemory time scale is again essential.\n\nIn this context, it is noteworthy that we have used conventional indicator\nfunctions $h_A(t)$ and $h_B(t)$, instead of the indicator functions $h_{\\cal\nA}(t)$ and $h_{\\cal B}(t)$ as introduced by Van Erp and coworkers \\cite{erp03}.\nThe latter are defined such that $h_{\\cal\n  A}(t)$ is 1 if the system was more recently in $A$ than in $B$,\nand vice versa. The advantage of those indicator functions is that by a\njudicious choice of the dividing surfaces $q_A$ and $q_B$, only the macroscopic\nswitching events are counted, i.e.~the crossing events where the system goes\nfrom one basin of attraction to the other \\cite{erp03}. If a clear separation\nexists between the transient time scale and the slow time scale, and\nmacroscopic switching events can thus be defined uniquely (as discussed above),\nthen the indicator functions $h_{\\cal A,B}$ are useful also in the framework\npresented here. For example, these indicator functions would then obviate the\nneed to introduce a grace interval, as we have done for the non-Markovian\nsystem. The advantage of the conventional indicator functions is, however, that\nthe corresponding correlation functions will inform us whether there exists\nsuch a clear separation of time scales: if so, the correlation functions\nexhibit a plateau, from which the macroscopic rate constant can be obtained.\n\n\nFinally, now that we have derived microscopic expressions for\nmacroscopic rate constants for both time-inhomogeneous Markov systems\nand non-Markov systems, the question arises whether these expressions\ncan also be evaluated efficiently in a computer simulation. In the\naccompanying paper \\cite{becker12a} we present a new numerical technique that\nmakes this possible.\\\\\n\n\\subsection*{Acknowledgments} We thank Daan Frenkel, Peter Bolhuis and David\nChandler for many useful discussions. This work is part of the research program\nof the ``Stichting voor Fundamenteel Onderzoek der Materie (FOM)'', which is\nfinancially supported by the ``Nederlandse organisatie voor Wetenschappelijk\nOnderzoek (NWO)''.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOn-chip photonic platforms have allowed for exploration of novel classical and quantum nonlinear phenomena in highly controlled environments. This is due to the ability to precisely tailor the dispersion and nonlinearity for a given interaction \\cite{obrien_photonic_2009,moss_new_2013}. For example, supercontinuum generation (SCG) can be tailored to provide multi-octave spectra \\cite{hickstein_ultrabroadband_2017} and quantum frequency conversion for single photons can be engineered for large spectral translation \\cite{li_efficient_2016}. Due to the availability of robust, inexpensive femtosecond laser sources in the near-infrared (NIR), a current goal for on-chip photonic devices is the conversion of light from the NIR to longer wavelengths in the mid-infrared (MIR). In particular, Er:fiber lasers are desirable pump sources due to their technological maturity and the surrounding telecom infrastructure. In this work, we utilize silicon nitride (Si$_3$N$_4$, henceforth SiN) nanophotonic waveguides to realize wide-band frequency conversion from the NIR to the MIR driven by Er:fiber lasers, and demonstrate tunability via dispersion engineering techniques. Such a compact source of MIR coherent light benefits many multi-disciplinary applications, including frequency comb generation and spectroscopy \\cite{schliesser_mid-infrared_2012}. \n\nNonlinear optical processes in second-order $\\chi^{(2)}$ and third-order $\\chi^{(3)}$ nonlinear media such as difference-frequency generation (DFG) \\cite{erny_mid-infrared_2007,ruehl_widely-tunable_2012,krauth_broadly_2013,cruz_mid-infrared_2015} and SCG \\cite{dudley_supercontinuum_2006,genty_fiber_2007,guo_mid-infrared_2018,grassani_highly_2018} have been explored for MIR generation. Conventionally, materials with strong $\\chi^{(2)}$ nonlinearity have proved difficult to integrate with photonics platforms and often require quasi-phase-matching schemes that add complexity to the fabrication process \\cite{bortz_quasi-phase-matched_1995,mayer_offset-free_2016}. In the case of SCG, the broad spectrum generated comes at a cost of spectral flux in any particular wavelength range. However, controlled frequency conversion that generates light in specific spectral regions is beneficial for many applications including spectroscopy and frequency referencing quantum cascade lasers \\cite{chang_high_2016,cappelli_frequency_2016}.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{ConceptTotalv5.png}\n\\caption{(a) Frequency conversion from the NIR to the MIR via four-wave-mixing: two pump photons at 1.5 \\textmu{}m mix with a 1.0 \\textmu{}m signal photon to generate an idler at 3 \\textmu{}m. (b) Compared to single-mode telecom-grade silica fibers, the dispersion in a nanophotonic waveguide (width $w = 3400$ nm) can be tailored over a broadband region due to the strong influence of the geometric dispersion, and lead to (c) phase-matching for wavelengths far from the pump.}\n\\label{fig:concept}\n\\end{figure}\n\n\nTowards this goal, we utilize optical parametric amplification (OPA) using degenerate-pump FWM in $\\chi^{(3)}$ materials ($\\chi^{(3)}$-OPA) since phase-matching requires careful dispersion engineering and frequency conversion only occurs at desired spectral bands. In such a FWM process, two pump photons are annihilated, providing gain for a signal wavelength, and generate frequency converted light at the idler wavelength, where $2\\omega_{p}=\\omega_{s}+\\omega_{i}$ is the energy conservation relation. The corresponding phase-matching condition is $\\Delta\\beta = 2\\beta_{p}-\\beta_{s}-\\beta_{i}+2\\gamma P = 0$, where $\\beta$ denotes the wave-vector, $\\gamma$ is the nonlinear parameter, and P is the pump peak power. Moreover, numerous materials with large $\\chi^{(3)}$ nonlinearity can be processed with commonplace electron-beam- and photo-lithography techniques. This enables an integrated photonics realization of the $\\chi^{(3)}$-OPA.\n\nIn previous work, such schemes have typically employed single-mode silica-based optical fibers due the favorable anomalous dispersion and low losses around 1.55 \\textmu{}m \\cite{agrawal_nonlinear_2012,radic_parametric_2008,marhic_fiber_2007}. Unfortunately, the tuning bandwidth is limited because the phase-mismatch grows quickly as the detuning from the pump increases (Fig.~\\ref{fig:concept}(b)). To overcome this limitation, higher order dispersion in specialty highly-nonlinear fibers (HNLFs) and photonic-crystal fibers has been used to extend the phase-matching for large-detuning frequency conversion processes \\cite{boggio_730-nm_2008,sharping_octave-spanning_2007}. However, the large absorption in conventional silica fibers beyond 2.4 \\textmu{}m limits the generated wavelengths to the short-wave infrared (1.7--2.4 \\textmu m). On the other hand, strong nonlinear losses at 1.5~\\textmu{}m preclude the usage of Er:fiber technology in silicon- and chalcogenide-based devices and instead necessitate long-wavelength (\\textgreater~2 \\textmu{}m) pump lasers for FWM \\cite{lamont_net-gain_2008,liu_mid-infrared_2010,kuyken_-chip_2011,kuyken_50_2011,lau_continuous-wave_2011,liu_bridging_2012,xing_mid-infrared_2017}, adding complexity to the system. \n\nHere, we exploit the wide transparency region (0.4\\textendash{}4.5 \\textmu{}m) and negligible two-photon absorption of SiN to realize phase-matched $\\chi^{(3)}$-OPA pumped at 1.55 \\textmu{}m to generate MIR light in the first atmospheric window (3--5 \\textmu m) (Fig.~\\ref{fig:concept}(c)). The sub-wavelength cross-sectional areas in the nanophotonic waveguide structures greatly enhance the nonlinearity ($\\gamma \\approx 1000$ W$^{-1}$ km$^{-1}$) and allow broadband dispersion engineering due to the increased influence of the geometric dispersion \\cite{agrawal_nonlinear_2012}. Such dispersion-engineering techniques have allowed for broadband SCG in SiN waveguides \\cite{carlson_self-referenced_2017} and octave-spanning Kerr frequency combs in the micro-resonator geometry \\cite{spencer_optical-frequency_2018}. \n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=\\linewidth]{Fig1v8.png}\n\\caption{(a) Two outputs from an Er:fiber laser are amplified using Er:doped fiber amplifiers (EDFA), and one amplified output is broadened in highly-nonlinear fiber (HNLF) to generate 1 \\textmu{}m light. The amplified 1.5~\\textmu{}m (pump) and 1.0 \\textmu{}m (signal) beams are coupled into a SiN waveguide (inset) to generate MIR light via FWM. (b) When the pulses are temporally overlapped, the phase-matched idler is generated in the 3-\\textmu{}m band (red). Without temporal overlap, the idler is not present (blue). (c) Tunable mid-infrared light is generated from 2.6--3.6 \\textmu{}m for different waveguide widths in the experiment (solid lines) and from theoretical predictions (dashed lines).}\n\\label{fig:MIR}\n\\end{figure}\n\nWe generate tunable MIR light by using a single mode-locked Er:fiber laser (100~MHz repetition rate) to generate the pump pulses at 1.55 \\textmu{}m (degenerate pump) and the signal at 1.07 \\textmu{}m. In this configuration, the laser output is amplified in an erbium-doped fiber amplifier (EDFA) to yield a 50-fs, 1-nJ pump pulse. A second EDFA output is spectrally broadened in HNLF ($\\gamma = 10$ W$^{-1}$ km$^{-1}$) and a bandpass filter around 1.0~\\textmu m isolates the seed pulse (Fig.~\\ref{fig:MIR}a). The two femtosecond-scale pulses are temporally overlapped using a tunable mechanical delay in the 1.5-\\textmu m arm and spatially overlapped at a dichroic mirror before being coupled into the waveguide-chip using an aspheric lens. We note that this setup is analogous to the DFG-based MIR frequency comb generation in $\\chi^{(2)}$ materials such as periodically poled lithium niobate \\cite{cruz_mid-infrared_2015}, but accomplished using $\\chi^{(3)}$ nonlinear media.\n\nThe stoichiometric SiN waveguides are fabricated via low-pressure chemical vapor deposition (LPCVD) with 700-nm thickness on a silicon dioxide substrate. Electron-beam lithography is used to define 1-cm-long ridge waveguides, which are air-clad except at the input and output facets where symmetric oxide cladding helps with coupling by making the waveguide modes symmetric. The waveguide input facet, in addition, has inverse taper couplers for efficient pump coupling (\\textless~2~dB insertion loss) \\cite{cardenas_high_2014}. The waveguide width at the output facet is not tapered, thereby limiting field overlap with the oxide cladding, for which absorption losses in the mid-IR are significant. On-chip pulse energies for the pump and seed are approximately 100 pJ and 50 pJ, respectively. The output is collected using a chalcogenide aspheric lens (0.58 NA) and measured using a free-space Fourier-transform spectrometer (FTS). \n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{Fig3v4.png}\n\\caption{(a) In silicon-nitride waveguides that support phase matching the FWM process (blue), SCG and associated dispersive-wave generation in the mid-infrared is also phase matched (red). This leads to parasitic soliton fission in the waveguide at high pump-pulse energies, and clamps the FWM conversion efficiency.\n(b) Output spectra of a 10-mm long silicon-nitride waveguide with width w = 3400 nm as a function of pump-pulse energy. The signal energy is kept fixed at 10 pJ on chip. The arrows mark the predicted FWM and dispersive-wave phase-matched wavelengths. \n}\n\\label{fig:MIRSim}\n\\end{figure}\n\nThe waveguides with widths 3000--4000 nm have anomalous dispersion at 1.5 \\textmu{}m (Fig.~\\ref{fig:concept}(b)) and support phase matching for the FWM interaction. When the pump and signal pulses are temporally overlapped in the photonic waveguide, MIR light is generated (Fig.~\\ref{fig:MIR}b). The 3.2-\\textmu{}m light corresponds to $\\sim 100$~THz detuning from the pump, made feasible by the higher-order dispersion present in the nanophotonic waveguide geometry (Fig.~\\ref{fig:concept}(b)). While the bandwidth of the MIR light is limited by the group-velocity walkoff between the pump and signal pulses, we note that SiN waveguides can also be engineered to provide group-velocity matching and lead to broadband frequency conversion \\cite{hickstein_self-organized_2018}.\n\nWide tunability from 2.6--3.6 \\textmu{}m is observed as the waveguide widths are changed. The MIR bandwidth and power in each waveguide is approximately constant across the chip. We theoretically model the FWM\/OPA process using the nonlinear Schrodinger equation \\cite{agrawal_nonlinear_2012}. To calculate the dispersion, effective refractive indices are acquired via the WGModes solver \\cite{fallahkhair_vector_2008} using a SiN Sellmeier equation acquired via ellipsometry measurements. The theoretical predictions agree well with the experimental measurements (Fig.~\\ref{fig:MIR}(c)). The slight offset in center frequencies between theory and experiment could arise from thickness variation across the chip and waveguide. The waveguide widths on chip are fabricated in increments of 200 nm, resulting in the discrete tunability shown in Fig.~\\ref{fig:MIR}c. Finer increments have been demonstrated \\cite{carlson_self-referenced_2017}, and would show more continuous tunability. Light in the 2.5--2.9 \\textmu{}m band shows atmospheric water absorption before detection by the FTS. \n\nDue to competing nonlinear processes, increasing the pump energy beyond a certain level does not increase the MIR power. To investigate this effect, we modeled the pulse propagation in a 3400-nm-wide waveguide and plot the output spectra as a function of pump-pulse energy in Fig.~\\ref{fig:MIRSim}. Because the waveguides exhibit anomalous dispersion at the pump, SCG is supported in addition to FWM, leading to additional parasitic effects that limit conversion to the MIR (Fig.~\\ref{fig:MIRSim}(a)).  For example, due to the long-wavelength zero-crossing in the group-velocity dispersion profiles (Fig.~\\ref{fig:concept}(b)), phase-matched dispersive-wave generation in the MIR contaminates the wide-band FWM process. As seen in Fig.~\\ref{fig:MIRSim}(b), at low input pump and signal pulse energies, the MIR idler grows in power linearly. However, pump pulse energies beyond 50 pJ lead to soliton fission and dispersive-wave formation that disturb the large-detuning FWM process and depletes the pump, leading to clamping of the MIR power. To mitigate the effects of SCG on the OPA process, alternative waveguide geometries such as the slot-mode waveguide could be used to provide normal dispersion at the pump while still providing phase-matching for the frequency conversion into the MIR \\cite{moille_phased-locked_2018}. However, in this experiment we simply keep the pump-pulse energy below the threshold for soliton fission.\n\nThe measured MIR power is on the order of 10 \\textmu W and is limited in our present devices by the weak optical confinement of the longer wavelengths and strong absorptive loss in the oxide. This linear loss for the MIR constrains the conversion efficiency in FWM \\cite{agrawal_nonlinear_2012}, though we note that this may be improved by the use of suspended waveguides~\\cite{halir_waveguide_2015}. Accounting for this effect and the $\\sim$5 dB out-coupling loss, we estimate the on-chip power to be $\\sim$100 \\textmu{}W.\n\nWhen strong confinement is available for all the waves, high conversion efficiency and large parametric gain can be expected in the nonlinear process. In the present waveguides, such performance is observed for an all-NIR FWM process. In order to demonstrate these advantages, we employ $\\chi^{(3)}$-OPA using a 1.0-\\textmu{}m (degenerate) pump and a 1.3-\\textmu{}m seed in the 2400-nm-wide waveguide on the same chip. To experimentally realize these conditions, we modify the experimental setup slightly: (i) the input aspheric lens is aligned to favor coupling the 1.0\\textmu m pulse (30 dB higher on-chip pulse energy than the signal) (ii) the 1.5 \\textmu m pulse is spectrally broadened in HNLF and filtered, resulting in a 1.3-\\textmu m pulse (50 fs pulse duration), and (iii) matched input and output aspheric lenses are used. At the pump wavelength (1.07~\\textmu{}m), the group-velocity dispersion is approximately zero (Fig.~\\ref{fig:NIR}(a)), allowing for an exceptionally broadband phase-matching bandwidth (Fig.~\\ref{fig:NIR}(a)). When SCG is suppressed (by restricting the on-chip pump-pulse energy), net parametric gain of up to 25 dB is seen for the 1.3-\\textmu{}m seed, and high-efficiency frequency upconversion into the 800-nm band is also observed (Fig.~\\ref{fig:NIR}(b)). With only 300 pJ of pump-pulse energy, frequency translation exceeding 100 THz is observed along with broadband parametric gain for the seed\\textemdash{}a unique combination made possible by the versatility of the integrated photonics implementation of the experiment.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{FigNIRv3.png}\n\\caption{(a) Dispersion profile (blue) for the w = 2400 nm wide waveguide on the same chip as used in Fig.~\\ref{fig:MIR}, with zero-dispersion crossings at 1.07 \\textmu{}m and 2.3 \\textmu{}m and the corresponding FWM phase-matching curve (red)  for a pump wavelength of 1.07 \\textmu{}m. The FWM phase-mismatch is near-zero for an exceptionally broad bandwidth. (b) Parametric amplification and broadband frequency conversion in the near-infrared. The 1.07 \\textmu{}m pump (red, shaded) \"upconverts\" a 1.3~\\textmu{}m seed (blue, shaded) to 0.85~\\textmu{}m with high-efficiency while providing \\textgreater~20~dB parametric gain (blue, solid). }\n\\label{fig:NIR}\n\\end{figure}\n\nIn conclusion, we demonstrated tunable MIR light from 2.6--3.6 \\textmu{}m in the OPA configuration using SiN nanophotonic waveguides driven by an Er:fiber laser with only 100-pJ-scale pump-pulse energies. The low pulse energy requirements make this system amenable for further photonic integration as well as GHz-repetition rate nonlinear optics \\cite{carlson_ultrafast_2017}. While high absorption loss for the MIR and parasitic phase-matched processes limited the conversion efficiency, alternative waveguide designs can substantially improve MIR transmission and also impede soliton fission by providing normal dispersion at pump. Such structures can yield large, broadband parametric gain for the MIR. In this work, we demonstrated the desired performance in an all-NIR OPA implementation, showing large parametric gain and ultra broadband frequency conversion with only 300~pJ pump-pulse energy. In future work, one could integrate the spectral-broadening stage on the SiN chip and drive the self-seeded OPA process on chip using only a single laser input. This would enable MIR synthesis with a compact foot-print leveraging mature Er:fiber technology, useful for chip-scale sensing and spectroscopy applications, including quantum photonics where the spontaneously driven FWM process can yield entangled photon pairs across the NIR and MIR \\cite{kalashnikov_infrared_2016}. \n\n\\noindent DARPA  SCOUT and DODOS,  AFOSR  (FA9550-16-1-0016),  NRC, NASA, NIST.\\\\\n\\noindent The  authors  thank  Eric Stanton and Daryl Spencer for helpful comments on the manuscript. This work is a contribution of the United States government and is not subject to copyright in the United States of America.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA {\\em Margulis spacetime\\\/} is a geodesically complete flat \nLorentzian $3$-manifold $M^3$ with free fundamental group.\nSuch manifolds are quotients $\\Eto\/\\Gamma$ of \n$3$-dimensional {\\em Minkowski space\\\/} $\\Eto$ by a discrete group $\\Gamma$\nof isometries acting properly on $\\Eto$. \nBy\n\\cite{FriedGoldman,Mess} every complete affinely \nflat $3$-manifold\nhas solvable fundamental group\nor is a  Margulis spacetime.\nIn the latter case the linear holonomy \n$$\n\\pi_1(M) \\cong \\Gamma\\xrightarrow{\\LL} {\\o{SO}(2,1)}\n$$ \nis an embedding of $\\Gamma$ onto a discrete subgroup\n$\\Gamma_0  = \\LL(\\Gamma)$.\nThus, associated to every Margulis spacetime is a complete hyperbolic\nsurface $\\surf = \\hp\/\\Gamma $. \nThis hyperbolic surface has an intrinisic description:\nit consists of parallelism classes of timelike lines \n(particles) in $M$.  \nIn particular $\\Gamma$ is an {\\em affine deformation\\\/} of the\nFuchsian group $\\Gamma_0$, and we say that $M$ is an \n{\\em affine deformation\\\/}\nof the hyperbolic surface $\\surf$. \n\nWe conjecture that every Margulis spacetime $M$ is {\\em tame},\nthat is, admits a polyhedral decomposition by crooked planes.\nA consequence is that $M$ is {\\em topologically tame,\\\/} that is,\nhomeomorphic to an open solid handlebody.\nWe start with the simplest groups of interest,\nthose whose holonomy is a rank two group.  (The cyclic case is\ntrivial.)  We previously established the conjecture when $\\surf$ is a\nthree-holed sphere~\\cite{CDG}.  In this paper, we establish the\nconjecture for its nonorientable counterpart, when $\\surf$ is\nhomeomorphic to a two-holed projective plane.  Every nonorientable\nsurface of negative Euler characteristic contains an embedded\ntwo-holed cross-surface.  For this reason, we expect the two-holed\ncross-surface, like the three-holed sphere, to act as a building block\nfor proving the general conjecture.\n\nJ.\\ H.\\ Conway has introduced the term {\\em cross-surface\\\/} for the\ntopological space underlying the real projective plane $\\R\\mathbf{P}^2$.  Let\n$\\topsurf$ be a {\\em $2$-holed cross-surface,\\\/} that is, the topological\nsurface underlying the complement of two disjoint discs in $\\R\\mathbf{P}^2$.\nAs is common in deformation theory, we fix the topology\nof $\\topsurf$ and consider {\\em marked geometric structures,\\\/}\nthat is, homotopy classes of homeomorphisms $\\topsurf\\rightarrow\\surf$,\nwhere $\\surf$ is a surface with a hyperbolic structure.\n\nThe first part of the paper focuses on the hyperbolic structure.  In\nparticular we describe the space of marked complete hyperbolic\nstructures on $\\topsurf$.  We call this space the {\\em Fricke space}\nof $\\topsurf$ and denote it by $\\mathfrak{F}(\\topsurf)$.  In the second part of the\npaper, we explicitly describe the space of all Margulis spacetimes\narising from $\\topsurf$ as a bundle of convex $4$-sided cones over\n$\\mathfrak{F}(\\topsurf)$.\n\nLike~\\cite{CDG}, the classification involves both the topology of $M$ and the\ngeometry of the deformation space.\nWe show that every affine deformation of $\\surf$ is homeomorphic to a solid\nhandlebody of genus two by constructing an explicit fundamental polyhedron.  \nAlthough the two-holed projective plane is a nonorientable\nsurface, its affine deformations are orientable  $3$-manifolds.\nIn fact, every Margulis spacetime is orientable (Lemma~\\ref{lem:orientable}).\n\nThere are four isotopy classes of \nessential primitive simple closed curves on $\\topsurf$.\nTwo of these curves, denoted $A$ and $B$, \ncorrespond to components of $\\partial\\topsurf$. \nThe other two, denoted $X$ and $Y$, \nreverse orientation and intersect transversely in one point.\n\nIf we fix a hyperbolic structure $\\surf$, \nthe boundary curves $A$ and $B$ correspond either to\nclosed geodesics bounding {\\em funnels\\\/} (complete ends\nof infinite area with cyclic fundamental group) or {\\em cusps\\\/} \n(ends of finite area).\nWhen a curve $K$ bounds a funnel, \n$\\ell_K$ is the (positive) length of the closed geodesic and \nwhen the boundary curve cuts off a cusp $\\ell_K$ is set equal to $0$. \nAn orientation-reversing curve  $K$ always corresponds to a closed geodesic, \nso that $\\ell_K$ represents this length and is always positive.\nFor\n$$\n\\big( \\ell_A, \\ell_B, (\\ell_X,\\ell_Y)\\big) \n\\;\\in\\;\n\\big( [0,\\infty)\\times [0, \\infty) \\big) \\times \\mathbb R_+^2\n$$\nthe lengths satisfy the identity\n\\eqref{eq:LengthIdentity}, and the Fricke space $\\mathfrak{F}(\\topsurf)$\nidentifies with \n$$\n\\big( [0,\\infty)\\times [0, \\infty) \\big) \\times \\mathbb R_+ .\n$$\nThe boundary $\\partial\\mathfrak{F}(\\topsurf)$ has two faces sharing a common edge.\nOne face is defined by $\\ell_A = 0$ and the other by $\\ell_B = 0$.\nThese faces intersect in the line $\\ell_A = \\ell_B = 0$, which\ncorresponds to complete hyperbolic structures with finite area.\n\n\\medskip\n\nFix a linear representation $\\rho_0$\nof $\\pi_1(S)$ in the special orthogonal group ${\\o{SO}(2,1)}$.\nAn {\\em affine deformation\\\/} of $\\rho_0$  \nis a representation $\\rho$ in the group of Lorentzian isometries\nwhose linear part is $\\rho_0$, that is, \n\\begin{equation*}\n\\LL\\circ\\rho = \\rho_0.\n\\end{equation*}\nAn affine deformation $\\rho$ of $\\rho_0$ \nis {\\em proper\\\/} if it defines a proper affine action\nof $\\pi_1(S)$ on $\\Eto$.\nEquivalently $\\rho$ is the holonomy representation of a complete\naffine structure on a $3$-manifold $M^3$ and the isomorphism\n$$\n\\pi_1(S) \\longrightarrow \\pi_1(M^3)\n$$\ninduced by $\\rho_0$ and $\\rho$\narises from a homotopy-equivalence \n$ S \\longrightarrow M^3$.\n\nAffine deformations of the fixed linear representation $\\rho_0$\ncorrespond to {\\em cocycles\\\/} and form a vector space $Z^1(\\Gamma_0,\\rto)$.\nSlightly abusing notation, denote also by $\\mathbb{V}$ the $\\Gamma_0$-module\ngiven by the linear holonomy representation $\\rho_0$ of $\\Gamma_0$ on\nthe vector space $\\mathbb{V}$.  Translational conjugacy classes of affine\ndeformations form the {\\em cohomology group\\\/}\n$H^1\\big(\\Gamma_0,\\mathbb{V}\\big)$.\n\nWhen $\\surf$ is a two-holed cross-surface,\nor, more generally, any surface with $\\pi_1(\\Sigma)$ free\nof rank two,  $H^1(\\Gamma_0,\\rto) \\cong \\mathbb R^3$.\nIn his original work~\\cite{Margulis1,Margulis2}, \nMargulis introduced a \n{\\em marked signed Lorentzian length spectrum invariant,\\\/}\nwhich now bears his name. This $\\mathbb R$-valued class function on $\\pi_1(\\surf)$\ndetects the properness of an affine deformation, and determines an affine\ndeformation up to conjugacy~\\cite{CharetteDrumm_isospectrality,DrummGoldman_isospectrality}.\n\nUnder the correspondence between\naffine deformations and infinitesimal deformations of the hyperbolic structure on\n$\\hp$, Margulis's invariant identifies with the first derivative of the \ngeodesic length function on Fricke space \n\\cite{GoldmanMargulis,Goldman_MargulisInvariant}.\nOriginally defined for hyperbolic\nisometries of $\\Eto$, it has been extended to parabolic isometries \nin \n\\cite{CharetteDrumm}, and to a function on geodesic currents \nin\n\\cite{GLM}. \n\nGiven a curve $K\\subset\\surf$, the Margulis invariant $\\mu_K$ is a linear\nfunctional on $H^1(\\Gamma_0,\\rto)$.  The four functionals associated to the curves\n$A,B,X,Y$ satisfy the linear dependence \\eqref{eq:alphaCDMrelations}.\nLet $\\Dd$ denote the open cone in $H^1(\\Gamma_0,\\rto)$ such that all the quantities $\n\\tilde{\\alpha}_A, \\;\\tilde{\\alpha}_B , \\;\\tilde{\\alpha}_X , \\;\\tilde{\\alpha}_Y.  $ are either all positive or all\nnegative.  Figure~\\ref{fig:DeformationSpace} offers a projective view\nof $\\Dd$; each line corresponds to a zero-set of a $\\mu_K$ for some\ncurve $K$. \n\nProper affine deformations will be investigated via \n{\\em crooked  planes\\\/}, polyhedral surfaces introduced \nin\n\\cite{Drumm_thesis,Drumm_jdg,DrummGoldman_crooked}.  \nA proper affine deformation is tame if it admits a fundamental\npolyhedron bounded by crooked planes.  \nThe {\\em Crooked Plane Conjecture\\\/} \n\\cite{DrummGoldman_crooked,CDG}, which we prove for affine\ndeformations of the two-holed cross-surface, asserts that every\nproper affine deformation is tame.\n\nDenote the subspace of $H^1(\\Gamma_0,\\rto)$ consisting of proper affine\ndeformations by $\\Pr$ and the subset of $\\Pr$ consisting \nof tame affine deformations by $\\Ta$. \nOur main result is that $$\\Ta \\;=\\; \\Pr\\;=\\Dd.$$\n\n\n\\begin{thmmain}\nLet $\\surf$ be a two-holed cross-surface.  Then every proper affine deformation of $\\surf$ is tame and the affine $3$-manifold is homeomorphic to a handlebody\nof genus two.\n\\end{thmmain}\n\nLet us underscore the fact that $\\Pr$ is finite-sided, just like in the\ncase of the three-holed sphere, where proper deformations make up\na three-sided polyhedral cone~\\cite{CDG}. In all other cases, the space of\nproper affine deformations has infinitely many faces. \nIn particular, for the other two surfaces of Euler characteristic\n$ -1$ (or equivalently, surfaces with fundamental group of rank two), \nnamely the one-holed torus and one-holed Klein bottle,\nthe space of proper affine deformations is defined by (necessarily)\ninfinitely many linear conditions~\\cite{Charette_nonproper,CDGp}.\n\n\\subsection*{Acknowledgments}\nWe are grateful to Francis Bonahon, Suhyoung Choi, David Gabai, \nFran\\c cois Labourie, Grisha Margulis and  Yair Minsky, \nSer-Peow Tan for helpful conversations. \nWe also thank the referee for many useful suggestions.\n\n\\section{The two-holed cross-surface}\\label{sec:thx}\n\nIn this first section, we investigate the topology of a two-holed cross-surface.  Then we endow it with a hyperbolic structure.\n\n\\subsection{Curves and the fundamental group of $\\topsurf$}\n\nWe begin by reviewing the topology of the two-holed cross-surface\n$\\topsurf$.  Recall that $X$ and $Y$ are orientation-reversing curves and that $A$ and $B$ bound $\\topsurf$. \n\nRepresent $\\topsurf$ as the nonconvex component\nof the complement of two disjoint discs in $\\R\\mathbf{P}^2$ as follows. \nThe coordinate axes in $\\mathbb R^2$ extend to projective lines in $\\R\\mathbf{P}^2$\nwhich intersect exactly once (at the origin) in $\\mathbb R^2$. Choose two disjoint\nhyperbolas in $\\mathbb R^2$ which intersect the ideal line in pairs of points\nwhich do not separate each other. In $\\R\\mathbf{P}^2$ these hyperbolas extend to\nconics which bound two disjoint convex regions. The complement of these\nconvex regions is a model for $\\surf$.\nIn  Figure~\\ref{fig:hyperbolas}, the complement of $X$ and $Y$  in $\\surf$ is \nfoliated by curves homotopic to the boundary curves $A$ and $B$.\n\nChoosing orientations and arcs from a basepoint  \n$\\basepoint$ to each of these four curves,\ndefine elements of $\\fg$ corresponding to $A, B, X, Y$. \nDenoting these elements also by $A,B,X,Y$, respectively, \nyields the redundant presentation\n\\begin{equation}\\label{eq:redundant}\n\\fg \\;=\\; \\langle A, B, X, Y \\mid X Y = A, \\, \\bar{Y} X = B \\rangle\n\\end{equation}\nNote that any two-element subset of $\\{A,B,X,Y\\}$ \n{\\em except\\\/} $\\{A,B\\}$ freely generates $\\fg$\nand \n\\begin{equation}\nA B = X^2.\n\\end{equation}\nSimple closed curves on $\\surf$ are easily classified. \n(See for example\n\\cite{Bonahon}.)\nIf $\\gamma\\subset S$ is a simple closed curve, then exactly one of the following holds:\n\\begin{itemize}\n\\item $\\gamma$ is contractible and  bounds a disc;\n\\item $\\gamma$ is peripheral and is isotopic to either $A$ or $B$;\n\\item $\\gamma$ is orientation-reversing and is isotopic to either $X$ or $Y$;\n\\item $\\gamma$ is essential, nonperipheral, orientation-preserving and is isotopic to either $X^2$ or $Y^2$.\n\\end{itemize}\n\n\\setcounter{figure}{0}\n\\begin{figure}[b]\n\\includegraphics[scale=1.0\n{hyperbolas4.eps}\n\\caption{The complement of two disjoint discs in \n$\\R\\mathbf{P}^2$ is homeomorphic to a $2$-holed cross-surface.\nThe coordinate axes correspond to the two orientation-reversing \nsimple loops $X$ and $Y$.\nThe two hyperbolas $A$ and $B$ define components of the boundary.}\n\\label{fig:hyperbolas}\n\\end{figure}\n\n\\subsection{Hyperbolic structures on $\\topsurf$}\nConsider a complete hyperbolic surface $S$ and a diffeomorphism (the {\\em  marking\\\/})\n$\\Sigma\\longrightarrow S$. \nChoose a universal covering $\\tS\\to S$, \nand a lift $\\tbasepoint\\in\\tS$ of the basepoint $\\basepoint\\in\\surf$.\nChoose a developing map\n$\\tS\\xrightarrow{\\mathsf{dev}} \\hp$ and holonomy representation \n$$\n\\fg \\xrightarrow{\\rho_0} \\Gamma \\subset \\mathsf{Isom}(\\hp)\\cong\\mathsf{PGL}(2,\\R).\n$$\nThe holonomy representation embeds $\\fg$ as a discrete subgroup\n$\\Gamma_0 = \\rho_0(\\fg)$.\nThe hyperbolic surface $S$ may have finite or infinite area. \n\nWhen $\\surf$ is nonorientable,\nsome elements of $\\Gamma$  reverse orientation.\nWhen $S$ is a two-holed cross-surface as above,\n$\\rho_0(X)$ and $\\rho_0(Y)$ are glide-reflections, \nwhile $\\rho_0(A)$ and $\\rho_0(B)$ are either transvections\nor parabolic isometries of $\\mathsf{H}^2$.\n(Compare Figure~\\ref{fig:FundamentalDomain}.)\n\nHenceforth, when the context is unambiguous,\nwe suppress $\\rho_0$, for example, simply writing $A$ for $\\rho_0(A)$.\n\n\\subsection{Fundamental domains for the two-holed cross-surface}\n\\label{sec:funddomainsHypSurf}\n\nWe now\nconstruct a fundamental domain for the action of the associated group\n$\\Gamma_0$ on the {\\em Nielsen convex region,\\\/} a convex\n$\\Gamma_0$-invariant open subset $\\Omega \\subset\\hp$.  Fundamental\ndomains for incomplete surfaces give rise to fundamental domains\nin $\\Eto$ for complete Margulis space-times.\n\n\nDenote the attracting and repelling fixed points of a hyperbolic\nisometry $W$ of $\\hp$ by $\\xp{W}$ and $\\xm{W}$ respectively.  \nIf $W$ is parabolic \nthen either annotation represents the fixed point on $\\partial\\hp$.\nAssume the fixed points \n\\begin{equation*}\n\\xp{X},\\xp{Y},\\xm{X},\\xm{Y}\n\\end{equation*}\nare in counter-clockwise order. \n(Compare Fig.\\ref{fig:FundamentalDomain}.)\nIn $\\hp$, let $\\mathcal Q$ be the ideal quadrilateral \nwith vertices:\n\\begin{equation*}\n\\xp{B},\\xm{A}, A(\\xp{B}), X(\\xp{B})\n\\end{equation*}\n(Compare Figure~\\ref{fig:FundDomainIdentifiedN1}.)\nThe transvection $A$ and the glide-reflection $X$ identify the four sides\nof $\\mathcal Q$ as follows:\n\\begin{align*}\n\\Big(\\xm{A},\\,\\xp{B}\\Big)     &\\;\\stackrel{A}\\longmapsto\\; \\Big(\\xm{A},\\, A(\\xp{B})\\Big) \\\\\n\\Big(\\xp{B},\\, X(\\xp{B})\\Big) &\\;\\stackrel{X}\\longmapsto\\; \\Big(X(\\xp{B}),\\, A(\\xp{B})\\Big)\n\\end{align*}\nsince $A(\\xp{B}) = X^2(\\xp{B})$.\n(Compare Figure~\\ref{fig:FundDomainIdentifiedN1}.)\nThe diagonal\n$\\Big(\\xm{A},\\, X(\\xp{B})\\Big)$ divides $\\mathcal{Q}$\ninto two ideal triangles. \n(Compare Figure~\\ref{fig:IdealQuadrilateral}.)\nDenote the interior of the complement of a halfplane\n$\\mathfrak{H}$ by $\\mathfrak{H}^c$.\nThe sides of $\\mathcal Q$ bound four disjoint halfplanes which are pairwise\nidentified by $A$ and $X$ to define a Schottky-like system:\n\\begin{itemize}\n\\item\nThe side $\\Big(\\xm{A},\\, A(\\xp{B})\\Big)$ bounds a halfplane $\\mathfrak{H}_A$; \n\\item\nThe transvection $\\bar{A}$ maps \n$\\Big(\\xm{A},\\, A(\\xp{B})\\Big)$ \nto the side $\\Big(\\xm{A},\\,\\xp{B}\\Big)$\nbounding the halfplane $\\bar{A}(\\mathfrak{H}_A^c)$.\n\\item The side \n$\\Big(X(\\xp{B}),\\, A(\\xp{B})\\Big)$ bounds a halfplane $\\mathfrak{H}_X$;\n\\item\nThe glide-reflection $\\bar{X}$ maps \n$\\Big(X(\\xp{B}),\\, A(\\xp{B})\\Big)$ to the side\n$\\Big(\\xp{B},\\, X(\\xp{B})\\Big)$ bounding the halfplane $\\bar{X}(\\mathfrak{H}_A^c)$.\n\\item\nThe four halfplanes \n$$\n\\mathfrak{H}_A, \\bar{A}(\\mathfrak{H}_A^c), \\mathfrak{H}_X, \\bar{X}(\\mathfrak{H}_A^c)\n$$ \nare pairwise disjoint.\n\\end{itemize}\nThe fundamental domain and its corresponding set of identifications for the\nhyperbolic surface $S = \\Omega\/\\Gamma_0$ \nserve as a template for the crooked fundamental domains constructed in\n\\S\\ref{sec:CrookedFD}. The positions of the halfplanes and their bounding\ngeodesics  play a crucial role in ensuring disjointness of the crooked \nplanes.\n\\newpage\n\n\\begin{figure}[t]\n\\psfrag{m}{$X^-$}\n\\psfrag{V}{$X^+$}\n\\psfrag{Z}{$Y^-$}\n\\psfrag{W}{$Y^+$}\n\\psfrag{X}{$X$}\n\\psfrag{Y}{$Y$}\n\\psfrag{A}{$A$}\n\\psfrag{B}{$B$}\n\\psfrag{P}{$A^Y\\,=\\,A^{\\bar{X}}$\n\\psfrag{Q}{\\qquad\\qquad$B^Y\\,=\\,B^X$\n\\psfrag{o}{$\\mathsf{dev}(\\tbasepoint)$}\n\\includegraphics[scale=1.1]\n{c02b1.eps}\n\\caption{\nA fundamental domain for the two-holed cross-surface $\\surf$.\n}\n\\label{fig:FundamentalDomain}\n\\end{figure}\n\n\\begin{figure}[h]\n\\psfrag{X}{$X$}\n\\psfrag{A}{$A$}\n\\psfrag{B}{$\\xp{B}$}\n\\psfrag{C}{$\\xm{A}$}\n\\psfrag{D}{$A(\\xp{B})$}\n\\psfrag{E}{$X(\\xp{B})$}\n\\includegraphics[scale=1.]\n{FundDomainIdentifiedN1.eps}\n\\caption{Identifications of the ideal quadrilateral.\nThe transvection $A$ identifies the bottom two sides,\nwhich share the ideal vertex $\\xm{A}$.\nThe glide-reflection $X$ identifies the top two\nsides, which share the ideal vertex $X(\\xp{B})$.\nThe arrows on the sides indicate how to identify the\nsides.\n}\n\\label{fig:FundDomainIdentifiedN1}\n\\end{figure}\n\n\\begin{figure}[h]\n\\psfrag{A}{$\\bar{X}(\\HX^c)$}\n\\psfrag{B}{$\\xp{B}$}\n\\psfrag{C}{$\\bar{A}(\\HA^c)$}\n\\psfrag{E}{$\\xm{A}$}\n\\psfrag{F}{$\\mathfrak{H}_A$}\n\\psfrag{G}{$A(\\xp{B})$}\n\\psfrag{H}{$\\mathfrak{H}_X$}\n\\psfrag{I}{$X(\\xp{B})$}\n\\includegraphics[scale=1.11]\n{IdealTriShade.eps}\n\\caption{The ideal quadrilateral $\\mathcal Q$ with diagonal,\nand the four halfplanes $\\mathcal Q$ bounds. \n}\n\\label{fig:IdealQuadrilateral}\n\\end{figure}\n\n\n\n\n\n\n\\section{The Fricke space of $\\topsurf$}\nThis section describes the deformation space of\nmarked hyperbolic structures on $\\topsurf$ in\nterms of representations and their characters in $\\SL(2,\\C)$.\nThen we find a simple set of coordinates for $\\mathfrak{F}(\\topsurf)$\nin terms of traces of $2\\times 2$ real matrices.\n\n\n\\subsection{Representing orientation-reversing isometries by matrices}\nFor the following discussion of the Fricke space of the two-holed\ncross-surface, represent $\\hp$ as a totally geodesic hypersurface in\nhyperbolic $3$-space $\\hthree$.\n{\\em Every\\\/} isometry of $\\hp$ extends uniquely to an \n{\\em orientation-preserving\\\/} isometry of $\\hthree$ \npreserving $\\hp\\subset\\hthree.$\n(Compare \\cite{Goldman,Marden}.\nTo use the algebraic machinery of traces in $\\SL(2,\\C)$, \nwe must also lift $\\rho_0$ to a representation $\\trho_0$ in the double\ncovering $\\SL(2,\\C)$ of the identity component \n$$\n\\mathsf{Isom}^+(\\hthree)\\cong\\mathsf{PSL}(2,\\C).\n$$\nSuch a lift is always possible since $\\fg$ is a free group.\nDenote the lifted elements of $\\SL(2,\\C)$ by $A, B, X, Y$ respectively, \nas well.\n\nFollowing  \\cite{Goldman}, represent a lift of an isometry of $\\hp$ as an element\nof $\\SL(2,\\C)$ which is real if the isometry preserves orientation on $\\hp$, \nand purely imaginary it the isometry reverses orientation $\\hp$. \nIn the real case, the matrix lies in $\\SL(2,\\R)$, and is defined up to $\\pm 1$.\nIt may be elliptic, parabolic or hyperbolic. If it is parabolic, then it is conjugate\nto:\n\\begin{equation}\\label{eq:Unipotent}\n\\pm \\bmatrix 1 & 1 \\\\ 0 & 1 \\endbmatrix\n\\end{equation}\nIf it is hyperbolic then it is conjugate to the diagonal matrix:\n\\begin{equation}\\label{eq:Transvection}\n\\pm \\bmatrix e^{\\ell\/2}  & 0 \\\\ 0 & e^{-\\ell\/2} \\endbmatrix\n\\end{equation}\nwith trace $\\pm 2 \\cosh(\\ell\/2)$. \nIt corresponds to a {\\em transvection\\\/} of $\\mathsf{H}^2$.\nIt leaves invariant a geodesic (its {\\em invariant axis),\\\/}\nwhich for the above example corresponds to the imaginary axis in the upper halfplane model, and it displaces points on its axis by length $\\ell$.\n\nIn the purely imaginary case, \nthe corresponding purely imaginary element of $\\SL(2,\\C)$ is $i P$, \nwhere $P\\in\\mathsf{GL}(2,\\R)$ and $\\det(P)\\,= \\, -1$.\nFor example the diagonal matrix\n\\begin{equation}\\label{eq:GlideReflection}\niP = i \\bmatrix e^{\\ell\/2 } & 0 \\\\ 0 & -e^{-\\ell\/2} \\endbmatrix \n\\;\\in\\; \\SL(2,\\C)\n\\end{equation}\nrepresents a glide-reflection of displacement length $\\ell$ along the \ngeodesic in the\nupper halfplane represented by the imaginary axis. \nIts trace equals $2i\\sinh(\\ell\/2)$.\nSince a matrix in $\\SL(2,\\C)$ representing an isometry of hyperbolic space \nis only determined up to multiplication by $\\pm 1$, \nthe matrix in $\\SL(2,\\C)$ representing a glide-reflection of\ndisplacement length $\\ell$ has trace $\\pm 2i \\sinh(\\ell\/2)$.\n\n\\subsection{Trace coordinates}\nSuppose that $\\trho_0$ is a representation in $\\SL(2,\\C)$ \n(preserving ${\\mathsf H}^2\\subset\\hthree$) which covers\na holonomy representation of a marked hyperbolic structure\non $\\Sigma$. \n\nThe character of $\\trho_0$ corresponds to a quadruple\n$(a,b,x,y)\\in\\mathbb R^4$ defined by:\n\\begin{align*}\na &:= \\mathsf{tr}(A) \\\\\nb &:= \\mathsf{tr}(B) \\\\\nx &:= -i\\ \\mathsf{tr}(X)  \\\\\ny &:= -i\\  \\mathsf{tr}(Y) \n\\end{align*}\nsubject to the trace identity\n\\begin{equation}\\label{eq:trace_identity}\na + b + x y \\;=\\; 0,\n\\end{equation}\nwhich arises directly from the ``Basic Trace Identity'' in \\cite{Goldman}:\n\\begin{equation*}\n \\mathsf{tr} (XY) + \\mathsf{tr} (X\\bar{Y}) = \\mathsf{tr} (X) \\mathsf{tr} (Y)\n\\end{equation*}\n\\big(which in turn is just the Cayley-Hamilton theorem for $\\SL(2,\\C)$\\big).\n\nThe {\\em character set\\\/}\n\\begin{equation}\\label{eq:CharacterSet}\n\\mathscr{C} :=\n\\{ (a,b,x,y)\\in\\mathbb R^4 \\mid   a + b = - x y,\\ \\vert a\\vert \\ge 2, \\vert b\\vert \\ge 2 \\}\n\\end{equation}\nidentifies with conjugacy classes of lifts to $\\SL(2,\\C)$ of holonomy representations\nof marked complete hyperbolic structures on $\\Sigma$.\n\n\n\nSince $\\{X,Y\\}$ freely generates $\\fg$, different lifts $\\trho_0$ of $\\rho_0$\ndiffer by {\\em multiplication by a character\\\/}\n$$\n\\fg\\xrightarrow{\\chi}\\{\\pm 1\\}\n$$\nas follows. \nThe group \n$$\n\\mathscr{G}\\ :=\\  \\mathsf{Hom}\\big(\\fg,\\{\\pm 1\\}\\big)\\ \\cong\\  \\mathbb Z\/2 \\oplus \\mathbb Z\/2\n$$ \nacts on $\\mathsf{Hom}(\\fg,\\SL(2,\\C))$ by pointwise multiplication:\n$$\n\\gamma \\stackrel{\\chi\\cdot\\rho}\\longmapsto \\chi(\\gamma)\\rho(\\gamma)\n$$\nwhich is a representation since $\\{\\pm 1\\} \\subset\\SL(2,\\C)$ is central.\n\nOn the quotient of $\\mathsf{Hom}\\big(\\fg,\\SL(2,\\C)\\big)$ \nby $\\mathsf{Inn}\\big(\\SL(2,\\C)\\big)$, \nthe induced $\\mathscr{G}$-action\nis described by the action on traces: \n\\begin{align*}\n(a,b,x,y) &\\mapsto (a,b,-x,-y); \\\\\n(a,b,x,y) &\\mapsto (-a,-b,-x,y); \\\\\n(a,b,x,y) &\\mapsto (-a,-b,x,-y).\n\\end{align*}\nChanging the sign of either $\\mathsf{tr}(X)$ or $\\mathsf{tr}(Y)$ results in changing the sign of $\\mathsf{tr}(A)$ and $\\mathsf{tr}(B)$, \nwhile changing the signs of both $\\mathsf{tr}(X)$ and $\\mathsf{tr}(Y)$ \nleaves the signs of $\\mathsf{tr}(A)$ and $\\mathsf{tr}(B)$ unchanged. \n\nThe Fricke space $\\mathfrak{F}(\\topsurf)$ then identifies with the quotient \n$\\mathscr{C}\/\\mathscr{G}$.\nSince none of  $a,b,x,y$ vanish, the action of $\\mathscr{G}$ is free and $\\mathfrak{F}(\\topsurf)$ also \nidentifies with one of the four connected components of the character set.\n\n\\subsection{Relating traces to lengths}\nThe nonperipheral essential orien\\-ta\\-tion-reversing simple curves $X$ and $Y$ are uniquely\nrepresented by closed geodesics. Similarly the boundary curves $A$ and $B$ are closed\ngeodesics or cusps. Denote the lengths of these geodesics by $\\lX, \\lY,\\lA, \\lB$\nrespectively. Denote the angle of intersection of the geodesic representatives $X$ and $Y$\nby $\\theta$. Explicit matrix representatives are given below:\n\n\\begin{align*}\nX& \\longleftrightarrow \ni\\bmatrix \n\\sinh\\frac{\\lX}{2} + \\cos\\theta \\cosh\\frac{\\lX }{2}  &\n\\sin \\theta  \\cosh\\frac{\\lX}{2}  \\\\\n\\sin\\theta \\cosh\\frac{\\lX}{2} & \n\\sinh\\frac{\\lX}{2} - \\cos\\theta \\cosh\\frac{\\lX}{2} \n\\endbmatrix \\\\\nY & \\longleftrightarrow i\\bmatrix  e^{\\lY\/2 } & 0 \\\\ 0 & -e^{-\\lY\/2 }\n\\endbmatrix.\n\\end{align*}\nTherefore,\n\\begin{align}\\label{eq:trace2length}\nx & =\\; 2 \\sinh\\frac{\\lX}{2} \\\\\ny & =\\;  2 \\sinh\\frac{\\lY}{2} \\notag \\\\\na & =\\; \n-2 \\big(\\sinh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2} \n+ \\cos\\theta  \n\\cosh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2}\\big)\\notag \\\\\nb& =\\;\n-2 \\big(\\sinh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2}\n- \\cos\\theta\n\\cosh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2} \\big). \\notag \n\\end{align}\nThe defining inequalities for these matrices, $x, y > 0$ and $a, b \\le -2$,\ndescribe one component of the set of characters.\nThe \\emph{length identity }\n\\begin{equation}\\label{eq:LengthIdentity}\n\\cosh \\frac{\\lA}{2} \\,+\\, \\cosh\\frac{\\lB}{2}  \\;=\\;  \n2\\ \\sinh\\frac{\\lX}{2}  \\,  \\sinh\\frac{\\lY}{2} .\n\\end{equation}\nresults directly from applying \n\\eqref{eq:trace2length} to \\eqref{eq:trace_identity}.\n\n\n\\begin{thm}\\label{thm:2HXSFrickeSpace}\nThe lengths  of boundary geodesics $\\lA, \\lB \\geq 0$, and the lengths of \norientation-reversing simple closed geodesics  $\\lX,\\lY >0$,\nsubject to \\eqref{eq:LengthIdentity}, \nprovide coordinates for $\\mathfrak{F}(\\topsurf)$. \n\\end{thm}\n\\noindent\nSolve \\eqref{eq:LengthIdentity} for $\\lY$ in terms of the other\nlengths. Thus, $\\mathfrak{F}(\\topsurf)$  identifies with the fibered space whose base\nis the closed first quadrant in $\\mathbb R^2$ ($\\lA, \\lB \\geq 0$) and whose\nfiber $\\mathbb R^+$ ($\\lX >0$) has no boundary.  That is,\n$$\n\\mathfrak{F}(\\topsurf) \\ \\approx \\ \n\\big( [0,\\infty)\\times [0, \\infty) \\big) \\times \\mathbb R_+ .\n$$\n\n\\section{Flat Lorentz $3$-manifolds}\\label{sec:3manifolds} \n\nNow we turn from hyperbolic structures on surfaces\nto flat Lorentz structures on $3$-manifolds.\nAfter a brief review of Lorentzian geometry, \nwe prove that every Margulis spacetime is orientable,\neven when the corresponding hyperbolic surface is nonorientable.\n\n\\subsection{Linear Lorentzian geometry}\nLet $\\mathbb{V}$ denote a Lorentzian $3$-dimensional vector space.\nDenote its group of orientation-preserving linear isometries \n${\\o{SO}(2,1)}$.  In keeping with the notation adopted for $\\mathsf{Isom}(\\hp)$, \nwe use uppercase letters to denote elements of this group.\n\nThe nonzero vectors ${\\mathsf v}\\in\\mathbb{V}$ such that \n\\begin{itemize}\n \\item ${\\mathsf v}\\cdot{\\mathsf v} =0$ are called \\emph{lightlike},\n \\item ${\\mathsf v}\\cdot{\\mathsf v} <0$ are called \\emph{timelike}, and\n \\item ${\\mathsf v}\\cdot{\\mathsf v} >0$ are called \\emph{spacelike}.\n\\end{itemize}\nAny lightlike vector lies on one nappe of the \\emph{lightcone}. \nChoose one of these nappes as the  \\emph{future lightcone}.\nAny vector lying on (lightlike)  or inside (timelike)  the future lightcone is said \nto be \\emph{future-pointing}.\n\nThe correspondence between the hyperbolic plane and the set of\ntimelike lines in $\\mathbb{V}$ induces an identification\n$\\mathsf{Isom}(\\hp)\\cong{\\o{SO}(2,1)}$.  \nTransvections \ncorrespond to {\\em hyperbolic} matrices in ${\\o{SO}(2,1)^0}$ with trace greater\nthan 3. The diagonal matrix \\eqref{eq:Transvection} corresponds to \n$$\n\\bmatrix 1 & 0 & 0 \\\\ 0 &  \\cosh(\\ell) & \\sinh(\\ell) \\\\\n0 & \\sinh(\\ell) & \\cosh(\\ell)  \\endbmatrix\n$$\nwith trace $1 + 2\\cosh(\\ell)$.\nParabolic elements of $\\SL(2,\\R)$ correspond to {\\em parabolic\\\/} matrices in \n${\\o{SO}(2,1)^0}$ with trace equal to 3.  The matrix $iP$ of \\eqref{eq:GlideReflection}\nrepresenting a glide-reflection identifies with the diagonal matrix\n$$\n\\bmatrix 1 & 0 & 0 \\\\ 0 &  -\\cosh(\\ell) & -\\sinh(\\ell) \\\\\n0 & -\\sinh(\\ell) & -\\cosh(\\ell)  \\endbmatrix\n$$\nhaving trace $1 -2\\cosh(\\ell) < -1$.\nThe corresponding isometry of $\\Eto$ preserves orientation but reverses time-orientation.\n\nCall $A\\in{\\o{SO}(2,1)}$ {\\em non-elliptic} if it does not fix a timelike\nline.  Equivalently, $A$ corresponds to a glide-reflection, a\ntransvection or a translation of $\\hp$ as above.\n\nSuppose that $A$ is non-elliptic.  Then the 1-eigenspace for $A$ is a\nline spanned by a spacelike or a lightlike vector.  We choose a\nspecific {\\em neutral vector}, denoted $A^0$, by requiring that:\n\\begin{itemize}\n\\item given a timelike vector ${\\mathsf v}$, $\\big( {\\mathsf v}, A^2({\\mathsf v}), A^0 \\big)$ is a right-handed basis for $\\mathbb{V}$.  \n(This determines a unique direction for $A^0$.\nFurthermore this condition is independent of ${\\mathsf v}$.)\n\\item if $\\mathsf{Fix}(A)$ is spacelike, we choose $A^0$ such that $A^0\\cdot A^0=1$.\n\\end{itemize}\nWhen $A$ is hyperbolic (a transvection or glide-reflection), \nits  eigenvalues are $1$, $\\lambda$, and $\\lambda^{-1}$, \nwhere \n$$\n0< \\lambda^2 <1.\n$$\nIn the above example, $\\lambda = \\pm e^{-\\ell}$.\nChoose $A^-$ to be a future-pointing lightlike \\emph{contracting} eigenvector, \nand $A^+$ to be a future-pointing \\emph{expanding} eigenvector: \n$$\nA^2(A^-) = \\lambda^2 A^-,\\qquad  A^2(A^+) = \\lambda^{-2} A^+.\n$$  \nBoth contracting and expanding eigenvectors are lightlike, and the\nneutral vector  $A^0$ is spacelike and well-defined.   \n(See Figure~\\ref{fig:Frames} and compare\n\\cite{CharetteDrumm}.) \n\\begin{figure}[b]\n\\includegraphics[width=8cm]{Frames2.eps\n\\caption{Defining the direction of $A^0$.}\n\\label{fig:Frames}\n\\end{figure}\n\nWhen $A$ is parabolic, $A^0$ may be either future-pointing or\npast-pointing and its Euclidean length is arbitrary.  By convention,\nwe set $A^+=A^-$ to be a future-pointing vector that is parallel to\n$A^0$.\n\n\\subsection{Affine actions}\nLet $\\Eto$ denote the affine space modeled on $\\mathbb{V}$.  The group of {\\em Lorentzian isometries} of $\\Eto$, denoted $\\mathsf{Isom}(\\Eto)$, consists of affine transformations whose linear part preserve the Lorentzian structure on $\\mathbb{V}$.  Equivalently, $\\g\\in\\mathsf{Isom}(\\Eto)$ if and only if its linear part $\\LL(\\g)\\in\\o{O}(2,1)$.\n\nLet $A\\in{\\o{SO}(2,1)}$ be non-elliptic.  \nSince $A$ is linear and therefore fixes the origin, \nthe cyclic group $\\langle A\\rangle$ it generates does not\nact properly on $\\Eto$.\nHowever, for any vector ${\\mathsf u}\\in\\mathbb{V}$ with nonzero projection on $A^0$,\nthe following affine transformation acts properly and freely on\n$\\Eto$:\n$$\ng: p \\longmapsto o+A(p-o) + {\\mathsf u}\n$$\nwhere $o$ is some choice of origin.  The quotient $\\Eto\/\\langle g\\rangle$ is an open solid torus. \n\nMore generally, if $\\Gamma_0\\subset{\\o{SO}(2,1)}$ is a free group, we obtain an affine action with linear part $\\Gamma_0$ by assigning translation parts to a set of free generators.\n\n\\subsection{Margulis spacetimes}\nRecall that a Margulis spacetime is a quotient $M = \\Eto\/\\Gamma$\nwhose fundamental group $\\Gamma$ is free, and acts properly, freely and discretely\nby Lorentz isometries.\n\n\\begin{lemma}\\label{lem:orientable}\nEvery Margulis spacetime with a nonabelian fundamental group is orientable.\n\\end{lemma}\n\\begin{proof}\nFor any Margulis spacetime $M$, \nthe affine holonomy group $\\Gamma$\nacts freely on $\\Eto$.\nAn affine transformation whose linear\npart does not have $1$ as an eigenvalue\nfixes a point in $\\Eto$.\nTherefore, for every $\\gamma\\in\\Gamma$,\nits linear part \n$\\LL(\\gamma)$ must have $1$ as an eigenvalue.\n\nIf $\\LL(\\gamma)\\in\\o{O}(2,1)\\setminus{\\o{SO}(2,1)}$, then one of its eigenvalues \nequals $-1$.   If $1$ is also an eigenvalue, then $A^2 = I$.\nThus, if $\\LL(\\Gamma)\\not\\subset {\\o{SO}(2,1)}$, \nthen some $\\gamma\\in\\Gamma$ will have \ntrivial linear part, and therefore is\na translation. \nBy \\cite{FriedGoldman}, $\\Gamma$ contains\nno translations.\n\\end{proof}\n\\section{Affine deformations and cocycles}\\label{sec:deformations} \nAffine deformations of a two-holed cross-surface $\\surf$\nare parametrized by Margulis invariants of $X,Y,A,B$.  \nWe start with a general\ndiscussion of the space of affine deformations of the linear holonomy\nof a surface and the Margulis invariant.  At the end of the section,\nwe use the Margulis invariant to provide coordinates for the\ndeformation space for the two-holed cross-surface.\n\n\\subsection{Cocycles}\nLet $\\surf$ be an arbitrary hyperbolic surface with linear holonomy $\\Gamma_0=\\rho_0(\\pi_1(\\surf))\\subset{\\o{SO}(2,1)}$.  \nRecall\nthat an affine deformation of $\\Gamma_0$\nis a lift $\\Gamma_0\\xrightarrow{\\rho}\\mathsf{Isom}(\\Eto)$ such that $\\rho_0 = \\LL\\circ \\rho$.\nLifts correspond to cocycles, that is, maps\n$$\n\\Gamma_0  \\stackrel{{\\mathsf u}}\\longrightarrow \\mathbb{V}\n$$\nsatisfying\n$$\n{\\mathsf u}( XY) \\;=\\; {\\mathsf u}(X) + \\rho_0(X) {\\mathsf u}(Y)\n$$\nHere the affine deformation is defined by:\n$$\np \\stackrel{\\rho(X)}\\longmapsto o+\\rho_0(X)(p-o) + {\\mathsf u}(X).\n$$\nDenote the space of cocycles  $\\Gamma_0\\rightarrow\\mathbb{V}$ by $Z^1(\\Gamma_0,\\rto)$.\n\nWhen $\\Gamma_0$ %\nis free, as is the case for a two-holed cross-surface, a cocycle is determined by its values on a free basis.\nFurthermore, the values on a free basis are completely arbitrary.\n\nTwo cocycles determine translationally conjugate affine deformations\nif and only if they are {\\em cohomologous,\\\/} that is, they differ\nby a coboundary\n$$\n{\\mathsf u}(X) := {\\mathsf v} - \\rho_0(X) {\\mathsf v}\n$$\nwhere ${\\mathsf v}\\in\\mathbb{V}$ is the vector effecting the translation.\nThe resulting set of translational conjugacy classes of affine deformations\n(cohomology classes of cocycles)\ncompose the cohomology group $H^1(\\Gamma_0,\\rto)$. \n\nIn the present case, when $\\Gamma_0$ is free of rank two,\nand $\\dim(\\mathbb{V})=3$, the space \n$Z^1(\\Gamma_0,\\rto)$ of cocycles is $6$-dimensional.\nThe space of coboundaries is $3$-dimensional.\nTherefore the cohomology $H^1(\\Gamma_0,\\rto)$ has dimension $3$.\n \n\\subsection{The Margulis invariant}\n\nContinuing with the notation above, the {\\em Margulis invariant\\\/} of the affine deformation $\\rho(X)$ is the neutral projection of the translational part ${\\mathsf u}$:\n$$\n\\oa{{\\mathsf u}}{X} \\;:=\\; {\\mathsf u} \\cdot X^0\n$$\nThe Margulis invariant \nis everywhere nonzero if and only if the affine deformation\nis free.  Moreover if ${\\mathsf u},{\\mathsf v}$ are cohomologous then $\\oa{{\\mathsf u}}{X}=\\oa{{\\mathsf v}}{X}$.  Furthermore, fixing the cocycle ${\\mathsf u}$, the Margulis invariant is a class function on $\\Gamma_0$.  \n\nThe following basic fact\n(Margulis's Opposite Sign Lemma) \nwas proved in \\cite{Margulis1,Margulis2} \nfor hyperbolic elements and extended in\n\\cite{CharetteDrumm} to groups with parabolic elements. \nThe survey article~\\cite{Abels} containsa lucid description of the ideas in \nMargulis's original  proof. \n\\begin{lemma}\\label{lemma:oppsign}\nLet $g, h\\in\\mathsf{Isom}(\\Eto)$ be non-elliptic.  \nIf the Margulis invariants for $g$ and $h$ have \nopposite signs then $\\langle g , h \\rangle$ does not act properly. \n\\end{lemma}\n\n\n\\subsection{Deformations of hyperbolic structures}\n\nAffine deformations also correspond to \n{\\em infinitesimal deformations\\\/}\nof the marked hyperbolic surface $\\Sigma \\to S$\n\n\\cite{Goldman_MargulisInvariant,GoldmanMargulis}).\nConsider a smooth family $S_t$ of hyperbolic surfaces, with \nsmoothly varying markings \n$$\n\\Sigma \\xrightarrow{m_t} S_t\n$$ \nand holonomy representation \n$\\pi_1(\\Sigma)\\xrightarrow{\\rho_t}\\SL(2,\\C)$.  \nLet ${\\mathsf u}$ be the cocycle tangent to the space of representations\nat $t=0$ .\nBy \\cite{GoldmanMargulis},\nthe Margulis invariant identifies with the derivative of the {\\em geodesic length function $\\ell_\\gamma$,\\\/}\nwhere $\\gamma\\in\\fg$. Specifically, let $\\ell_{\\gamma(t)}\\in\\mathbb R_+$ denote the length of\nthe closed geodesic in $S_t$ in the free homotopy class determined by $(m_t)_*(\\gamma)\\in\\pi_1(S)$.\nThen:\n$$\n\\oa{{\\mathsf u}}{\\rho_0(\\gamma)} \\;=\\;  \\frac{d \\ell_{\\gamma(t)} }{dt} .\n$$\nFurthermore: \n\\begin{equation}\\label{eq:Lengths2Traces}\n\\mathsf{tr}\\big( \\rho_t(\\gamma)\\big) \\;=\\; \\pm 2 \n\\begin{cases}\n\\cosh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~preserves orientation} \\\\\ni \\sinh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~reverses orientation} \n\\end{cases}\n\\end{equation}\nDifferentiating  \\eqref{eq:Lengths2Traces} implies:\n\\begin{equation}\\label{eq:dLengths2Traces}\n\\frac{d}{dt} \\mathsf{tr} \\big( \\rho_t(\\gamma) \\big)   \\;=\\; \\pm\n\\begin{cases}\n\\alpha_\\gamma \\sinh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~preserves orientation} \\\\\ni\\alpha_\\gamma \\cosh\\frac{\\ell_{\\gamma(t)}}{2} &\\text{~if~} \\gamma \n\\text{~reverses orientation} \n\\end{cases}\n\\end{equation}\n\n\n\\subsection{Coordinates for the four-sided deformation space}\n\\label{sec:coordinates}\n\nFixing $A\\in\\Gamma_0$, the Margulis invariant is a\nlinear functional on the space of cocycles $Z^1(\\Gamma_0,\\rto)$, descending to a\nwell-defined functional on $H^1(\\Gamma_0,\\rto)$.  To reflect this, we modify our\nnotation for the Margulis invariant as in~\\cite{CDG}:\n\\begin{align*}\nH^1(\\Gamma_0,\\rto) & \\xrightarrow{\\tilde{\\alpha}_A} \\mathbb R \\\\\n[{\\mathsf u}] & \\longmapsto \\oa{{\\mathsf u}}{A}\n\\end{align*}\n\nNow assume $\\surf$ is a two-holed cross-surface, with elements $A,B,X,Y\\in\\pi_1(\\surf)$ as in \\S\\ref{sec:thx}.  The invariants of elements $A,X,\\bar{X} A=Y$ determine an isomorphism of vector spaces:  \n\\begin{align}\\label{eq:CDM-coordinates}\nH^1(\\Gamma_0,\\rto) &\\xrightarrow{\\tilde{\\alpha}} \\mathbb{R}^3  \\\\\n [u]  & \\longmapsto\n\\bmatrix \\tilde{\\alpha}_A(u) \\\\ \\tilde{\\alpha}_X(u) \\\\\\tilde{\\alpha}_Y(u) \n\\endbmatrix  \\notag\n\\end{align}\n\nConsider a proper affine deformation.  Lemma~\\ref{lemma:oppsign}\nimplies the Margulis invariants of all nontrivial elements have the\nsame sign.  Namely, consider an infinitesimal deformation of the\nhyperbolic structure on $\\surf$ such that every closed geodesic \n{\\em infinitesimally lengthens} or an infinitesimal deformation where\nevery closed geodesic \\emph{infinitesimally shortens}. \n\nDifferentiating \\eqref{eq:trace_identity} yields:\n\\begin{equation}\\label{eq:deriv_trace_identity}\nda \\, +\\, db \\,+\\, y\\ dx \\ \\,+\\, \\ x\\ dy \\;=\\; 0.\n\\end{equation}\nExpress these quantities using \\eqref{eq:trace2length}, \nits derivatives\n\\eqref{eq:dLengths2Traces},  and use the fact that  \nthe Margulis invariants $\\tilde{\\alpha}_A,\\tilde{\\alpha}_B,\\tilde{\\alpha}_X,\\tilde{\\alpha}_Y$\nof $A,B,X,Y$, are\nthe derivatives $d\\lA, d\\lB, d\\lX, d\\lY$ \nrespectively \\cite{GoldmanMargulis,CharetteDrumm}.\nThe differentiated trace identity \n\\eqref{eq:deriv_trace_identity} then implies:\n\n\n\\begin{align}\\label{eq:alphaCDMrelations}\n&\\Big(\\sinh\\frac{\\lA}{2}\\Big)   \\tilde{\\alpha}_A +  \\Big(\\sinh\\frac{\\lB}{2}\\Big) \\tilde{\\alpha}_B = \\\\\n &  \\qquad\\qquad  \\Big(2\\cosh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2}\\Big) \\tilde{\\alpha}_X \\ +\\ \n\\Big(2 \\sinh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2}\\  \\Big) \n\\tilde{\\alpha}_Y. \\notag  \n\\end{align}\nAfter eliminating $\\tilde{\\alpha}_B$ using \\eqref{eq:alphaCDMrelations}, \nthe positivity of $\\tilde{\\alpha}_A$ and\n$\\tilde{\\alpha}_B$ is written as follows:\n\\begin{align}\\label{eq:muAmuXmuY}\n0 \\   & <  \\tilde{\\alpha}_A  \\\\ \n&  < \n  \\bigg( 2\\opn{csch}\\frac{\\lA}{2} \\cosh\\frac{\\lX}{2} \\sinh\\frac{\\lY}{2} \\bigg)\n\\ \\tilde{\\alpha}_X  \\notag \\\\ \n& \\qquad + \n\\bigg( 2\\opn{csch}\\frac{\\lA}{2} \\sinh\\frac{\\lX}{2} \\cosh\\frac{\\lY}{2}\\bigg)\\ \n\\tilde{\\alpha}_Y \\notag\n\\end{align}\nRecall that $\\Dd$ denotes the set of deformations where the\nfunctionals $\\tilde{\\alpha}_X , \\tilde{\\alpha}_Y , \\tilde{\\alpha}_A , \\tilde{\\alpha}_B $ all have the same sign.  Let $\\Dd_+\\subset\\Dd$ denote those whose signs are all positive.  Then the set of lengthening deformations lies inside the\nset $\\Dd_+$\nand is defined in terms of the linear coordinates $\\tilde{\\alpha}_A,\\tilde{\\alpha}_X,\\tilde{\\alpha}_Y$\nby \\eqref{eq:muAmuXmuY} and the conditions $\\tilde{\\alpha}_X > 0$ and $\\tilde{\\alpha}_Y > 0$.\n\n\n$\\Dd_+$ is a four-sided cone, and is thus spanned by four rays.\nEach of these rays can be defined by\nthe linear equation \\eqref{eq:alphaCDMrelations} together\nwith one of the four conditions:\n\\begin{align*}\n\\tilde{\\alpha}_Y, \\tilde{\\alpha}_B > 0,\\; &\\tilde{\\alpha}_X = \\tilde{\\alpha}_A = 0 \\\\\n\\tilde{\\alpha}_X, \\tilde{\\alpha}_B > 0,\\; &\\tilde{\\alpha}_Y = \\tilde{\\alpha}_A = 0 \\\\\n\\tilde{\\alpha}_X, \\tilde{\\alpha}_A > 0,\\; &\\tilde{\\alpha}_Y = \\tilde{\\alpha}_B = 0 \\\\\n\\tilde{\\alpha}_Y, \\tilde{\\alpha}_A > 0,\\; &\\tilde{\\alpha}_X = \\tilde{\\alpha}_B = 0.\n\\end{align*}\n\\noindent \nThe set of shortening deformations lies inside $\\Dd_-$, where \n$$\n\\Dd_- = -\\Dd_+. \n$$ \n\n\\begin{figure}[bh]\n\\includegraphics[scale=2.3]{Quadrilateral.eps}\n\\caption{The four-sided deformation space\nfor a two-holed cross-surface}\n\\label{fig:DeformationSpace}\n\\end{figure}\n\n\\section{Crooked fundamental domains}\n\nIn this final section we prove the Main Theorem stated in the\nIntroduction.  Explicitly, we show that every proper affine\ndeformation admits a crooked fundamental domain bounded by four\ndisjoint crooked planes.  It follows that $\\Eto\/\\Gamma$ is a solid\nhandlebody of genus two.\n\nAs in the case of the three-holed sphere~\\cite{CDG}, \nit suffices\nto consider all affine deformations arising from a single\nconfiguration of crooked planes. This configuration is modeled\non the fundamental domain for $S$ developed in \n\\S\\ref{sec:funddomainsHypSurf}. \nThese crooked plane configurations correspond to \ndecompositions of $S$ into two ideal triangles  bounded by three different geodesics. Some of these crooked\nplane configurations also describe the entire space of proper affine deformations. \nThe more complicated\ncase of the one-holed torus requires the use of multiple configurations\nof crooked planes to cover all the proper affine deformations~\\cite{CDGp}.\nThat is why we obtain a finite-sided\ndeformation space, in contrast with the case of the one-holed torus,\nfor example.\n\nWe provide definitions and some background on crooked planes in the Appendix.\n\n\\subsection{Configuring crooked planes}\\label{sec:CrookedFD}\n\nGiven a spacelike vector ${\\mathsf v}\\in\\mathbb{V}$ and $p\\in\\Eto$, let ${\\mathcal C}({\\mathsf v},p)$\ndenote the crooked plane with direction vector ${\\mathsf v}$ and vertex $p$.\nIts complement in $\\Eto$ is a pair of crooked halfspaces.\n\nAs one-dimensional spacelike subspaces of $\\mathbb{V}$ bijectively\ncorrespond to hyperbolic lines in ${\\mathsf H}^2$, so do parallelism classes of\ncrooked planes.  Thus to any configuration of lines in ${\\mathsf H}^2$ we can\nassociate a corresponding configuration of crooked planes, up to\nchoice of vertices.  In particular, every ideal triangle in $\\hp$\ncorresponds to a triple of crooked planes such that every pair of\ndirection vectors shares a common orthogonal lightlike line.  Moreover\nsuch a triple determines a triple of pairwise disjoint crooked\nhalfspaces, for appropriate choices of vertices.\n\nCrooked planes enjoy the following useful property:\norientation-preserving Lorentzian isometries map crooked planes to\ncrooked planes.  Explicitly, if $g\\in{\\o{SO}(2,1)}$ then:\n$$\ng\\big({\\mathcal C}({\\mathsf v},p)\\big)={\\mathcal C}\\big(\\LL(g)({\\mathsf v}),g(p)\\big).$$ \nMoreover, if these crooked\nplanes are disjoint, then there exists a crooked halfspace $\\mathcal{H}$ in the\ncomplement of ${\\mathcal C}({\\mathsf v},p)$ such that the closures of $\\mathcal{H}$ and $g(\\mathcal{H})$\nare disjoint.  Thus crooked planes are suitable for Klein-Maskit\ncombination arguments, {\\em even for groups containing  glide-reflections}.\n\n\\subsection{Disjointness}\n\nStart with four crooked planes all with the same vertex and\ncorresponding to the ideal quadrilateral $\\mathcal Q$ pictured in\nFigure~\\ref{fig:IdealQuadrilateral}.  We  associate vertices to\neach crooked plane which make them disjoint.  This assignment of\nvertices yields a proper affine deformation of $\\Gamma_0=\\langle\nA,X\\rangle$.\n\nWe introduce \na convention\non \nlightlike vectors.  Recall that glide-reflections in $\\mathsf{Isom}(\\hp)$\nidentify with isometries in ${\\o{SO}(2,1)}$ which interchange the future and\npast null cones.  To facilitate calculations, take all lightlike vectors\nto be future-pointing. If $X$ corresponds to a\nglide-reflection and ${\\mathsf v}$ is a future-pointing vector then write all\nexpressions involving the action of $X$ on ${\\mathsf v}$ in terms of $X(-{\\mathsf v})$\nwhich is again future-pointing.\n\nAs in~\\cite{CDG}, it suffices to consider a triple of crooked planes\ncorresponding to one of the ideal triangles forming $\\mathcal Q$.  Let\n${\\mathsf v}_0$, ${\\mathsf v}_A$, ${\\mathsf v}_X$ be a triple of consistently oriented\nunit-spacelike vectors such that:\n\\begin{align*}\n{\\mathsf v}_0^\\perp & = \\langle\\xm{A},X(-\\xp{B})\\rangle \\\\\n{\\mathsf v}_A^\\perp & = \\langle A(\\xp{B}),\\xm{A}\\rangle \\\\\n{\\mathsf v}_X^\\perp & = \\langle X(-\\xp{B}),A(\\xp{B}) \\rangle.\n\\end{align*}\nThese will be the directing vectors of our triple of crooked planes.  \n\nGiven $(p_0,p_A,p_X)\\in \\Eto \\times \\Eto \\times \\Eto$, \ndefine the cocycle \n${\\mathsf u}_{(p_0,p_A,p_X)} \\;\\in\\; Z^1(\\Gamma_0,\\rto)\n$ \nas follows:\n\n\\begin{align*}\n{\\mathsf u}_{(p_0,p_A,p_X)}(A) &:=  p_A - p_0  \\\\\n{\\mathsf u}_{(p_0,p_A,p_X)}(X) &:=  p_X - p_0.\n\\end{align*}\nSince $A, X$ freely generate $\\fg$, \nthese conditions uniquely determine the cocycle ${\\mathsf u}_{(p_0,p_A,p_X)}$.\n\n\nThis cocycle corresponds to the following triple of crooked planes: \n\\begin{align*}\n{\\mathcal C}_0 & := \\;\n{\\mathcal C}({\\mathsf v}_0 ,p_0) \\\\\n{\\mathcal C}_A & := \\;\n{\\mathcal C}({\\mathsf v}_A , p_A) \\\\\n{\\mathcal C}_X & := \\;\n{\\mathcal C}({\\mathsf v}_X,  p_X).\n\\end{align*}\nIf $\\rho$ is the affine deformation corresponding to\n${\\mathsf u}_{(p_0,p_A,p_X)}$, then ${\\mathcal C}_0$ bounds a crooked halfspace whose\nclosure contains $\\rho(A)^{-1}({\\mathcal C}_A)$ and $\\rho(X)^{-1}({\\mathcal C}_X)$.\nNote that the quadruple\n$$ \n{\\mathcal C}_A,{\\mathcal C}_X,\\rho(A)^{-1}({\\mathcal C}_A),\\rho(X)^{-1}({\\mathcal C}_X) \n$$ \ncorresponds to $\\mathcal Q$.\n\nWe would like these crooked planes to be disjoint.  \nTo this end, consider the three quadrants:\n\\begin{align*}\n\\mathscr Q_0 & :=\\; \\Quad{\\xm{A}, -X(-\\xp{B})}  \\\\\n\\mathscr Q_A & :=\\; \\Quad{A(\\xp{B}), -\\xm{A}} \\\\\n\\mathscr Q_X & :=\\; \\Quad{X(-\\xp{B}),- A(\\xp{B})} \\\\\n\\end{align*}\nBy Lemma~\\ref{lem:disjointasym}, the triple of crooked planes ${\\mathcal C}_0,{\\mathcal C}_A ,{\\mathcal C}_X$ are pairwise disjoint as long as $(p_0,p_A,p_X)\\in \\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X$.\n\n\\begin{thm}\\label{thm:tame}\nLet \n$$\n(p_0,p_A,p_X)\\in \\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X.\n$$\nThen the affine deformation defined by\n$$\n{\\mathsf u}_{(p_0,p_A,p_X)} \\;\\in\\; Z^1(\\Gamma_0,\\rto)\n$$\nis tame.\n\\end{thm}\n\n\\begin{proof}\nApply the Kissing Lemma (Lemma~\\ref{lem:kissing1}) to the crooked planes \nin the halfspace bounded by ${\\mathcal C}({\\mathsf v}_0,p_0)$, to obtain pairwise disjoint crooked planes.  By Theorem~\\ref{thm:disjointCPs}, they bound a fundamental domain.\n\\end{proof}\n\nFigure~\\ref{fig:FourwTwoKiss} shows a quadruple of crooked planes, to which we apply the Kissing Lemma to obtain pairwise disjoint crooked planes. \n\n\\begin{figure}\n\\includegraphics[width=6.3in]{FourwTwoKiss.eps} \n\\caption{Four crooked planes.  The two on the right share a wing.}\n\\label{fig:FourwTwoKiss} \n\\end{figure}\n\n\\subsection{Proper deformations are tame}\nNow we conclude the proof.  Recall that all tame deformations are\nproper.  The Margulis invariants  of the four\nelements $X,Y,A,B$ are all of the same sign for every proper\ndeformation.  That is,\n$$\n\\Ta \\; \\subset \\; \\Pr \\; \\subset \\; \\Dd .\n$$\nWe now show that $\\Dd \\; \\subset \\; \\Ta$ .\n\nAs before, Lemma~\\ref{lemma:oppsign} implies that each of these \nsets has two connected components which are opposites of each other. \nFor proper deformations, \n$$\n\\Pr= \\Pr_+\\cup \\Pr_-\n$$ \nand $\\Pr_-= - \\Pr_+$. \nFor tame deformations, \n$$\n\\Ta= \\Ta_+ \\cup \\Ta_-\n$$ \nand $\\Ta_- = - \\Ta_+$. \nThus \n\\begin{align*}\n\\Ta_+   & \\subset \\Pr_+  \\subset \\Dd_+ \\\\\n\\Ta_-   & \\subset \\Pr_-  \\subset \\Dd_- \n\\end{align*}\nTheorem~\\ref{thm:tame} asserts that the image of the mapping\n\\begin{align*}\n\\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X &\\longrightarrow H^1(\\Gamma_0,\\rto) \\\\\n(p_0,p_A,p_X) &\\longmapsto [{\\mathsf u}_{(p_0,p_A,p_X)}]\n\\end{align*}\nlies in $\\Ta$.\n\n\nIn fact we shall show that this image is exactly\n$\\Dd_+$. The proof for the positive components\nimplies\nthe same statements for the negative components.\n\nWe will in fact consider cohomology classes in the closure of $\\Ta_+$.  Write:\n\\begin{equation}\\label{eq:vertexdef}\n\\begin{array}{rcl}\np_0 &:= &r_0 \\xm{A} - s_0 X(-\\xp{B}) \\\\\np_A &:= &r_A A(\\xp{B}) - s_A \\xm{A}  \\\\\np_X &:=& r_X X(-\\xp{B}) - s_X A(\\xp{B}) \n\\end{array}\n\\end{equation}\nwhere \n$$\nr_0, s_0, r_A, s_A, r_X, s_X \\geq 0.\n$$\nAssigning a zero value to any of these coefficients yields a configuration of \n{\\em kissing\\\/} crooked planes, any two of  which intersect in a point, \na ray or a halfplane.\n\nApplying the definition of a cocycle:\n\\begin{lemma}\\label{lem:yb}\nFor any cocycle ${\\mathsf u}$,\n\\begin{itemize}\n\\item ${\\mathsf u}(Y) \\;=\\; \\bar{X} \\big({\\mathsf u}(A) - {\\mathsf u}(X)\\big)$;\n\\item ${\\mathsf u}(B) \\;=\\; \\bar{Y} \\big({\\mathsf u}(X) - {\\mathsf u}(Y)\\big)$.\n\\end{itemize}\\end{lemma}\n\\noindent\nConsequently:\n\\begin{equation}\\label{eq:udef} \n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  p_A - p_0 \\\\\n{\\mathsf u}(X)\t & = &  p_X - p_0  \\\\\n{\\mathsf u}(Y) \t & = & \\bar{X}(p_A - p_X) \\\\\n{\\mathsf u}(B)\t & = &  \\bar{Y}(p_X - p_0)- \\bar{A}( p_A - p_X )  \n\\end{array}\n\\end{equation}\nTherefore:\n\\begin{equation}\\label{eq:margulisoftriple}\n\\tilde{\\alpha}([{\\mathsf u}_{(p_0,p_A,p_X)}])=\n\\begin{bmatrix}\n(r_A\\xp{B}+s_0X(-\\xp{B}))\\cdot A^0 \\\\\n(-(r_X+s_0)\\xp{B}-s_XA(\\xp{B})-r_0\\xm{A})\\cdot X^0\\\\\n((r_A+r_X+s_X)\\xp{B}-s_AX^{-1}(\\xm{A}))\\cdot Y^0\n\\end{bmatrix}\n\\end{equation}\n\n\\noindent\nThe following lemma is also immediate:\n\\begin{lemma}\\label{lem:alphasXYA}\nLet $(p_0,p_A,p_X)\\in \\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X$ be as above.\n\\begin{itemize}\n\\item \n${\\mathsf u}_{(p_0,0,0)}(Y) \\;=\\; 0$ so \n$\\tilde{\\alpha}_Y({\\mathsf u}_{(p_0,0,0)}) \\;=\\; 0$.\n\\item \n${\\mathsf u}_{(0,p_A,0)}(X) \\;=\\; 0$ so \n$\\tilde{\\alpha}_X({\\mathsf u}_{(0,p_A,0)}) \\;=\\; 0$.\n\\item \n${\\mathsf u}_{(0,0,p_X)}(A) \\;=\\; 0$ so \n$\\tilde{\\alpha}_A({\\mathsf u}_{(0,0,p_X)}) \\;=\\; 0$.\n\\end{itemize}\n\\end{lemma}\n\nThe deformation space $\\Dd_+$ is the (open) convex hull of four rays.\nWe will exhibit a cocycle on each ray and show that its $\\mu$-coordinates are all nonnegative. \n\nFigure~\\ref{fig:FuturePointing} serves as a visual aid in the\ncalculation of $\\mu$-coordinates, by showing where the relevant\nfuture-pointing vectors are located relative to the axes of $X$ and $Y$. \n\n\\begin{figure}[h]\n\\psfrag{m}{$X^-$}\n\\psfrag{V}{$X^+$}\n\\psfrag{Z}{$Y^-$}\n\\psfrag{W}{$Y^+$}\n\\psfrag{X}{$X$}\n\\psfrag{Y}{$Y$}\n\\psfrag{A}{$A$}\n\\psfrag{B}{$B$}\n\\psfrag{P}{$A^Y\\,=\\,A^{\\bar{X}}$\n\\psfrag{Q}{$B^Y\\,=\\,B^X$\n\\psfrag{o}{$\\mathsf{dev}(\\tbasepoint)$}\n\\includegraphics[scale=0.6]\n{TheGeometry.eps}\n\\caption{Computing cocycles on the edge of $\\Dd_+$. The locations of relevant future-pointing vectors are shown.  The figure depicts the case when $A,B$ are hyperbolic; in the case where either isometry is parabolic, the corresponding invariant axis shrinks to a point.}\n\\label{fig:FuturePointing}\n\\end{figure}\n\n\n\\bigskip\n\\noindent{{\\bf First ray:~}$\\tilde{\\alpha}_B^{-1}(0)\\cap\\tilde{\\alpha}_Y^{-1}(0)$}\n\n\\smallskip\n\\noindent\nFor $\\tilde{\\alpha}_B =\\tilde{\\alpha}_Y  =  0$, \nlet $(p_0,p_A,p_X)  =  (-s_0 X(-\\xp{B}), 0, 0)$ with $s_0>0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  s_0 X(-\\xp{B}) \\\\\n{\\mathsf u}(X)\t & = &  s_0 X(-\\xp{B}) \\\\\n{\\mathsf u}(Y) \t & = & \\mathsf{0} \\\\\n{\\mathsf u}(B)\t & = &  \\bar{Y}(s_0 X(-\\xp{B})) = -s_0\\xp{B}\n\\end{array}\n\\end{equation*}\n\n\\noindent\nBecause $\\xp{B}\\cdot \\xo{B} = 0$, we know that\n$\\tilde{\\alpha}_Y({\\mathsf u})=\\tilde{\\alpha}_B({\\mathsf u})=0$. Recall that $X$ is a glide\nreflection, so that the future pointing vector $X(-\\xp{B})$ lies\nabove  \nthe axis of $X$ in\nFigure~\\ref{fig:FuturePointing}. Therefore, $X(-\\xp{B}) \\cdot\n\\xo{X}>0$ and $X(-\\xp{B}) \\cdot \\xo{A}>0 $, so that\n$$\n\\tilde{\\alpha}(A) =s_0 X(-\\xp{B}) \\cdot  \\xo{A} >0\n$$\nand\n$$\n\\tilde{\\alpha}(X) =s_0 X(-\\xp{B}) \\cdot  \\xo{X} >0.\n$$\n\n\\bigskip\n\\noindent{{\\bf Second ray:~}$\\tilde{\\alpha}_B^{-1}(0)\\cap\\tilde{\\alpha}_X^{-1}(0)$}\n\n\\smallskip\n\n\\noindent\nFor $\\tilde{\\alpha}_B = \\tilde{\\alpha}_X  =  0$, let \n$(p_0,p_A,p_X)  =  (0, r_A A(\\xp{B}), 0)$ with $r_A >0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  r_A A(\\xp{B}) \\\\\n{\\mathsf u}(X)\t & = &  \\mathsf{0} \\\\\n{\\mathsf u}(Y) \t & = & \\bar{X}(r_A A(\\xp{B})) =  - r_A Y(-\\xp{B}) \\\\\n{\\mathsf u}(B)\t & = &  -\\bar{A}(r_A A(\\xp{B})) = -r_A \\xp{B}\n\\end{array}\n\\end{equation*}\nClearly  $\\tilde{\\alpha}_X({\\mathsf u})=\\tilde{\\alpha}_B({\\mathsf u})=0$. The future pointing \nvector $Y(-\\xp{B})$ lies to the right of the axis of $Y$ in \nFigure~\\ref{fig:FuturePointing},  so that\n$Y(-\\xp{B}) \\cdot  \\xo{Y}<0$. Because $\\xp{B} \\cdot  \\xo{A}>0$, then $A(\\xp{B}) \\cdot  \\xo{A}>0$.\nThus, \n$$\n\\tilde{\\alpha}(A) =r_A A(\\xp{B}) \\cdot  \\xo{A} >0 \n$$\nand\n$$\n\\tilde{\\alpha}(Y) =- r_A Y(-\\xp{B})  \\cdot  \\xo{Y} >0 .\n$$\n\n\\bigskip\n\\noindent{{\\bf Third ray:~}$\\tilde{\\alpha}_A^{-1}(0)\\cap\\tilde{\\alpha}_X^{-1}(0)$\n\n\\smallskip\n\\noindent\nFor $\\tilde{\\alpha}_A =\\tilde{\\alpha}_X  =  0$, let \n$(p_0,p_A,p_X)  =  (0, -s_A \\xp{A}, 0)$ with $s_A>0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  -s_A \\xp{A}\\\\\n{\\mathsf u}(X)\t & = &  \\mathsf{0} \\\\\n{\\mathsf u}(Y) \t & = & \\bar{X}(-s_A \\xp{A})\\ =\\  s_A \\bar{X}(-\\xp{A}) \\\\\n{\\mathsf u}(B)\t & = &  -\\bar{A}(-s_A \\xp{A})\\ =\\  s_A \\xp{A}\n\\end{array}\n\\end{equation*} \nIt is clear that $\\tilde{\\alpha}_A({\\mathsf u})=\\tilde{\\alpha}_X({\\mathsf u})=0$. Because $\\xp{A}$ is a fixed point\nof $A = XY$, the future pointing vector \n$\\bar{X}(-\\xp{A})=Y(-\\xp{A})$ lies to the left of the axis for $Y$ so that\n$\\bar{X}(-\\xp{A}) \\cdot  \\xo{Y}>0$. It is also evident that $\\xp{A} \\cdot  \\xo{B}>0$.\nThus\n$$\n\\tilde{\\alpha}(Y) =s_A \\bar{X}(-\\xp{A})  \\cdot  \\xo{Y} >0\n$$\nand\n$$\n\\tilde{\\alpha}(B) =- s_A \\xp{A}  \\cdot  \\xo{B} >0 \n$$\n\n\\newpage\n\\noindent{{\\bf Fourth ray:~}$\\tilde{\\alpha}_A^{-1}(0)\\cap\\tilde{\\alpha}_Y^{-1}(0)$}\n\n\\smallskip\n\n\\noindent\nFor $\\tilde{\\alpha}_A\\;=\\;\\tilde{\\alpha}_Y \\; = \\; 0$, let  \n$(p_0,p_A,p_X) \\; = \\; (r_0 \\xm{A}, 0, 0)$ with $r_0>0$.\n\\begin{equation*}\n\\begin{array}{rcl}\n{\\mathsf u}(A)   & = &  -r_0 \\xm{A}\\\\\n{\\mathsf u}(X)\t & = &  -r_0 \\xm{A} \\\\\n{\\mathsf u}(Y) \t & = & \\mathsf{0} \\\\\n{\\mathsf u}(B)\t & = &  \\bar{Y}(-r_0 \\xm{A})\\ =\\  r_0\\bar{Y}(-\\xm{A}) .\n\\end{array}\n\\end{equation*}\nCertainly $\\tilde{\\alpha}_A({\\mathsf u})=\\tilde{\\alpha}_Y({\\mathsf u})=0$. It is also clear that \n$\\xm{A} \\cdot  \\xo{X}<0$.  \n\nIn order to locate $\\bar{Y}(-\\xm{A})$, observe that $\\xm{A}$ is inside the\ncircular arc from $X(-\\xp{B})$ clockwise to  $\\xm{Y}$ \n(see Figure~\\ref{fig:IdealQuadrilateral}). \nBecause $B=\\bar{Y}X$,  $X(-\\xp{B}) = Y(-\\xp{B})$ and $Y$ maps the arc from\n$X(-\\xp{B})$ clockwise to  $\\xm{Y}$  to the arc from $\\xm{Y}$ to $\\xp{B}$. \nIn particular, $\\bar{Y}(-\\xm{A}) \\cdot  \\xo{B}>0$ \nso that: \n$$\n\\tilde{\\alpha}(X) =-r_0 \\xm{A}  \\cdot  \\xo{X} >0\n$$\nand\n$$\n\\tilde{\\alpha}(B) =r_0\\bar{Y}(-\\xm{A}) \\cdot  \\xo{B} >0 \n$$\n\nTherefore, every cohomology class in the interior of the convex hull of these four rays is represented by a cocycle ${\\mathsf u}_{(p_0,p_A,0)}$ such that \n$$\nr_0, s_0, r_A, s_A>0.\n$$  \nThis only yields a configuration of crooked planes where ${\\mathcal C}_0$ and ${\\mathcal C}_A$ are disjoint, but each kisses ${\\mathcal C}_X$.  However, \nEquation~\\eqref{eq:margulisoftriple} implies that by slightly perturbing $s_A$ and $r_0$, we can make $r_X$ and $s_X$ both strictly positive within the same cohomology class.  Finally, since the new values $(p'_0,p'_A,p_X)$ are in $\\mathscr Q_0 \\times \\mathscr Q_A \\times \\mathscr Q_X$,  $[{\\mathsf u}_{(p_0,p_A,0)}]\\in\\Ta_+$ and thus $\\Dd_+\\subset\\Ta_+$ as claimed.\n\\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\input{Introduction}\n\n\\section{System Model}\n\\input{SystemModel}\n\\subsection{Energy Transfer Phase}\n\\input{DownlinkStage}\n\\subsection{Information Transmission Phase}\n\\input{UplinkStage}\n\n\\section{Problem Formulation and Optimal Solution}\n\\label{Section:ProblemFormulationAndOptimalSolution}\n\\input{ProblemFormulationSolution}\n\\subsection{Problem Formulation}\n\\input{ProblemFormulation}\n\\subsection{Single-User WPCNs}\n\\label{Section:SingleUser}\n\\input{SingleUserSystems}\n\n\\subsection{Multi-User WPCNs}\n\\input{MultiUserOptimal}\n\t\\subsubsection{Solution of Problem (\\ref{Eqn:FunctionPsi})}\n\t\\label{Section:OptimalFunctionPsi}\n\t\\input{FunctionSolution}\n\t\\subsubsection{Solution of Problem (\\ref{Eqn:OptimalResourceAlloc})}\n\t\\label{Section:OptimalResourceAllocation}\n\t\\input{ResourceAllocSolution}\n\t\\subsubsection{Solution of Problem (\\ref{Eqn:OptimalTransmitVectors})}\n\t\\label{Section:OptimalVectors}\n\t\\input{VectorsSolution}\n\n\\section{Low-Complexity Design of Multi-User WPCNs}\n\\input{MultiUserSuboptimal}\n\\subsection{Massive MISO WPCNs}\n\\label{Section:MassiveMIMO}\n\\input{MassiveMIMOScheme}\n\\subsection{Suboptimal MRT-based Scheme}\n\\input{ScaledMassiveMIMO}\n\\subsection{Suboptimal SDR-based Scheme}\n\\input{SuboptimalScheme3}\n\n\\section{Numerical Results}\n\\label{Section:SimulationResults}\n\\input{NumericalResults}\n\\subsection{Simulation Setup}\n\\input{SimulationSetup}\n\\subsection{Complexity Analysis}\n\\input{ComplexityAnalysis}\n\\subsection{Performance Analysis}\n\\input{PerformanceAnalysis}\n\n\\section{Conclusions}\n\\input{Conclusions}\n\n\\appendices\n\n\t\\renewcommand{\\thesection}{\\Alph{section}}\n\t\\renewcommand{\\thesubsection}{\\thesection.\\arabic{subsection}}\n\t\\renewcommand{\\thesectiondis}[2]{\\Alph{section}:}\n\t\\renewcommand{\\thesubsectiondis}{\\thesection.\\arabic{subsection}:}\t\n\t\\section{Proof of Proposition \\ref{Theorem:SingleUser}}\n\t\\label{Appendix:PropSU}\n\t\\input{ProofPropSU}\n\t\\section{Proof of Proposition \\ref{Theorem:MuProp1}}\n\t\\label{Appendix:Prop1}\n\t\\input{ProofProp1}\n\t\\section{Proof of Proposition \\ref{Theorem:MuProp2}}\n\t\\label{Appendix:Prop2}\n\t\\input{ProofProp2}\n\t\\section{Proof of Proposition \\ref{Theorem:MuProp3}}\n\t\\label{Appendix:Prop3}\n\t\\input{ProofProp3}\n\t\\section{Proof of Lemma \\ref{Theorem:Lemma}}\n\t\\label{Appendix:LemmaProof}\n\t\\input{ProofLemma}\n\t\\section{Proof of Proposition \\ref{Theorem:MassiveMIMO}}\n\t\\label{Appendix:Prop5}\n\t\\input{ProofProp5}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSystems of coupled bulk--surface partial differential equations arise in many engineering and natural science applications. Examples include multiphase fluid dynamics with soluble or insoluble surfactants~\\cite{GrossReuskenBook}, dynamics of biomembranes~\\cite{bonito2011dynamics}, crystal growth~\\cite{Crystal},  signaling in biological networks~\\cite{Cells}, and transport of solute in fractured porous media \\cite{alboin2002modeling}. In these and other applications, partial differential equations defined in a volume domain are coupled to another PDEs posed on a surface. The surface may be embedded in the bulk or belong to a boundary of the volume domain.\n\nRecently, there has been a growing interest in developing methods for the numerical treatment of bulk--surface coupled PDEs.\nDifferent approaches  can be distinguished depending on how the surface is recovered and equations are treated.\nIf a  tessellation of the volume into tetrahedra is available that fits the surface, then it is natural to introduce finite element spaces in the volume and on the induced triangulation of the surface. The resulting \\textit{fitted} bulk--surface finite element method was studied  for the stationary bulk--surface advection--diffusion equations \\cite{ER2013}, for non-linear reaction--diffusion systems modelling biological pattern formation  \\cite{madzvamuse2016bulk,madzvamuse2015stability}, for the equations of the two--phase flow with surfactants~\\cite{barrett2015stable,barrett2015stable2},  Darcy and transport--diffusion  equations in fractured porous media~\\cite{alboin2002modeling}. \n\n\\textit{Unfitted} finite element methods allow the surface to cut through the background tetrahedral mesh. In the class of finite element methods also known as cutFEM, Nitsche-XFEM or TraceFEM, standard background finite element spaces are employed, while the integration is performed over cut domains and over the embedded surface~ \\cite{cutFEM,TraceFEM}. Additional stabilization terms are often added to ensure the robustness of the method with respect to small cut elements. The advantages of the unfitted approach are the efficiency in handling implicitly defined surfaces, complex geometries,   and  the flexibility in dealing with evolving domains. In the context of bulk--surface coupled problems,  cut finite element methods were recently applied to treat  stationary bulk--surface advection--diffusion equations \\cite{gross2015trace},\ncoupled bulk-surface problems on time-dependent domains \\cite{hansbo2016cut}, coupled elasticity problems \\cite{cenanovic2015cut}. The hybrid  method developed in this paper belongs to the general class of unfitted methods and resembles the TraceFEM in how the surface PDE is treated.\n\nThe methods discussed above treat surfaces and interfaces sharply, i.e. as lower-dimensional manifolds. In the present paper we also  consider sharp interfaces. For the application of phase--field or other diffuse-interface approaches for coupled bulk--surface PDEs see,  for example, \\cite{chen2014conservative,levine2005membrane,teigen2009diffuse}.\n\nIf the finite element method is a discretization of choice for the bulk problem, then it is natural to consider a finite element method for surface PDE as well. However,  depending on application, desired conservation properties, available software or personal experience, other discretizations such as finite volume or finite difference methods can be preferred for the PDE posed in the volume.\nOne possibility to reuse the same mesh for the surface PDE is to consider  a diffuse-interface approach. Alternatively, instead of smearing the interface, one may extend the PDE off the  surface to a narrow band containing the surface\nin such a way that the restriction of the extended PDE solution back to the (sharp) surface solves the original equation on this surface. Further a conventional discretization is built for the resulting volume PDE in the narrow band~ \\cite{BCOS01,olshanskii2016narrow}. The methods based on such extensions, however, increase the number of the active degrees of freedom for the discrete surface problem, may lead to degenerated PDE, need numerical boundary conditions and require smooth surfaces with no  geometrical singularities.\n\nThe present  paper develops a numerical method based on the sharp-interface representation, which uses a finite volume method to discretize the bulk PDE. Our goal is (i) to allow the surface to overlap with the background mesh in an arbitrary way, (ii) to avoid \\rev{building regular surface triangulation}, (iii)  to avoid any extension of the surface PDE to the bulk domain.  To accomplish these goals, we combine the monotone (i.e. satisfying the discrete maximum principle) finite volume method on  general meshes  \\cite{Lipnikov:12,ChernyshenkoFV7:14}   with the trace finite element method on octree meshes from \\cite{chernyshenko2015adaptive}.\nIn the octree TraceFEM one considers the bulk finite element space of piecewise trilinear globally continuous functions\nand further uses the restrictions (traces) of these functions to the surface. These traces are further used in a variational formulation of the surface PDE. Effectively, this results in the integration of the standard polynomial functions over the (reconstructed) surface.  Only degrees of freedom from the cubic cells cut by the\nsurface are active for the surface problem. Surface parametrization is not required, no surface mesh is built,  no PDE extension off the surface is needed.  We shall see that the resulting hybrid FV--FE method is very robust with respect to the position of surfaces against the background mesh and is well suited for handling non-smooth surfaces and surfaces given implicitly.\n\nOne application of interest is the numerical simulation of the contaminant  transport and diffusion in fractured porous media. In this application, transport and diffusion along fractures are often modeled by PDEs posed on a set of piecewise-smooth surfaces, see, e.g.,~\\cite{alboin2002modeling,fumagalli2013reduced,maryvska2005numerical,therrien1996three}; \\rev{see also \\cite{alboin2002modeling,frac1,frac2,frac3} for a similar dimension reduction approach in simulation of flow in fractured porous media}.\nMonotone (satisfying the DMP) finite volume methods on general meshes is the appealing tool for the solution of\nequations for solute concentration in the porous matrix, see, e.g., ~\\cite{ChernyshenkoFV7:14,Droniou:11,GaoWu:13,KapyrinFV7:14,LePotier:08,Lipnikov:12,ShengYuan:11}\n(further references can be found in \\cite{Droniou:14,FVCA7:14}).\n\n However, a straightforward application of this technique to model transport and diffusion along a fracture would require fitting the mesh or triangulating  the surface. For a large and complex net of fractures cutting through the porous matrix \\rev{this is a difficult task \\cite{DPRF}, and} an efficient method avoids mesh fitting and surface triangulations. \\rev{\n Recently, extended finite element method  approximations have been extensively studied in transport and flow problems in fractured porous media, see the review \\cite{flemisch2016review} and references therein.  In XFEM, one also avoids fitting of the background mesh to a fracture, but a separate mesh  is still required to represent the fracture.\n Besides the use of FV for the matrix problem, the approach in the present paper differs from those found in existing XFEM literature in the way the surface problem is discretized.}\n\nWhile the present technique can be applied for tetrahedral or more general polyhedral  tessellations of the bulk domain, we consider octree grid with cubic cells here. This choice is not \\textit{ad hoc}. Indeed, the Cartesian structure and built-in hierarchy of octree grids makes mesh adaptation, reconstruction and data access fast and easy. For these reasons, octree meshes became a common tool in \nin computational mechanics and  several octree-based solvers  are available in the open source scientific computing software, \\cite{DEAL2,Pop03}. However, an octree grid provides \\rev{at most} the first order (staircase) approximation of a general geometry. Allowing the surface to cut through the octree grid in an arbitrary way overcomes this issue, but challenges us with the problem of building efficient bulk--surface discretizations.   This paper demonstrates that the hybrid TraceFEM -- non-linear FV method complements the advantages of using octree grids by delivering \\rev{more accurate treatment of the surface PDE problem}.\n\nThe remainder of the paper is organized as follows. In section~\\ref{s_setup} we recall the system of differential equations, boundary and interface conditions, which models the coupled bulk--interface (or ``matrix--fracture'' in the context of flows in porous media) advection--diffusion problem. Section~\\ref{s_hybrid} gives the details of the hybrid discretization.\nAfter laying out the main ideas behind the method, we discuss the non-linear monotone FV method for the bulk and the TraceFEM for\nthe surface equations, and further we introduce the required coupling.  Section~\\ref{s_numer} presents the results of several numerical experiments with steady analytical solutions on  smooth and piecewise smooth branching surface. We also show the results of numerical simulation of the propagating front of solute concentration through fractured porous media. \n\n\n\n\n\\section{Mathematical model}\\label{s_setup}\nIn this section we recall the mathematical model of the contaminant  diffusion and transport in fractured  porous media.\nAssume the given bulk  domain $\\Omega \\subset \\mathbb{R}^3$ and a piecewise smooth surface $\\Gamma\\subset\\Omega$. The surface $\\Gamma$ may have several connected components. If  $\\Gamma$ has a boundary, we assume that ${\\partial\\Gamma} \\subset{\\partial\\Omega} $.\nThus, we have the subdivision $\\overline{\\Omega}=\\cup_{i=1,\\dots,N}\\overline{\\Omega}_i$ into simply connected subdomains $\\Omega_i$ such that $\\overline{\\Omega}_i\\cap\\overline{\\Omega}_j\\subset\\Gamma$, $i\\neq j$.\n\nIn each  $\\Omega_i$, we assume a given Darcy velocity field of the fluid $\\mathbf w_i(\\mathbf x)$, $\\mathbf x \\in \\Omega_i$.\nBy $\\mathbf w_\\Gamma(\\mathbf x)$, $\\mathbf x \\in \\Omega_\\Gamma$, we denote the velocity field tangential to $\\Gamma$ having the physical meaning of the flow rate through the cross-section of the fracture.  Consider an agent that is soluble in the fluid and transported by the flow in the bulk and along the fractures. The fractures are modeled by the surface $\\Gamma$.\nThe solute \\emph{volume} concentration (i.e., the one in the bulk domain $\\Omega$) is denoted by $u$, $u_i=u|_{\\Omega_i}$. The solute \\emph{surface} concentration along $\\Gamma$ is denoted by $v$.\nChange of the concentration happens due to convection by the  velocity fields $\\mathbf w_i$ and $\\mathbf w_\\Gamma$, diffusive fluxes in $\\Omega_i$, diffusive flux on $\\Gamma$, as well as the fluid exchange and diffusion flux between the fractures and the porous matrix.\nThese coupled processes can be modeled by the following system of equations~\\cite{alboin2002modeling}, in subdomains,\n\\begin{equation} \\label{diffeq1}\n\\left\\{\n\\begin{aligned}\n \\phi_i\\frac{\\partial u_i}{\\partial t} + \\mbox{\\rm div}(\\mathbf w_i  u_i- D_i \\nabla u_i) &= f_i \\quad \\text{in}~ \\Omega_i,\\\\\n u_i&=v \\quad \\text{on}~ {\\partial\\Omega} _i\\cap\\Gamma,\n \\end{aligned}\n\\right.\n \\end{equation}\nand on the surface,\n\\begin{equation} \\label{diffeq2}\n \\phi_\\Gamma\\frac{\\partial v}{\\partial t} + \\mbox{\\rm div}_\\Gamma (\\mathbf w_\\Gamma v - d D_\\Gamma \\nabla_\\Gamma v) =  F_\\Gamma(u)+ f_\\Gamma  \\quad \\text{on}~~\\Gamma,\n \\end{equation}\nwhere we employ the following notations:  $\\nabla_\\Gamma$,  $\\mbox{\\rm div}_\\Gamma$  denote the surface tangential gradient and divergence\n \\begin{wrapfigure}{r}{0.4\\textwidth}\n\\vspace{-20pt}\n  \\begin{center}\n  \n    \\caption{2D illustration of our notation for a domain with triple fraction.\n    }\n    \\label{fig:Schema}\n  \\end{center}\n  \\vspace{-20pt}\n  \\vspace{1pt}\n\\end{wrapfigure}\noperators; $F_\\Gamma(u)$ stands for the  net flux of the solute per surface area due to fluid leakage and hydrodynamic dispersion;   $f_i$ and $f_\\Gamma$ are given source terms in the subdomains and in the fracture;  $D_i$ denotes the diffusion tensor in the porous matrix; the surface diffusion tensor is $D_\\Gamma$.\nBoth $D_i$, $i=1,\\dots,N$, and $D_\\Gamma$ are symmetric and positive definite; $d>0$ is the fracture width coefficient; $\\phi_i>0$ and $\\phi_\\Gamma>0$ are the constant porosity coefficients for the bulk and the fracture.\n\nThe total surface flux $F_\\Gamma(u)$ represents the contribution of the bulk to the solute transport in the fracture. The mass balance at  $\\Gamma$ leads to the equation\n\\begin{equation}\\label{Flux}\n F_\\Gamma(u)= [-D \\mathbf n \\cdot \\nabla u + (\\mathbf n \\cdot\\mathbf w)u]_\\Gamma,\n\\end{equation}\nwhere $\\mathbf n$ is a unit normal vector at $\\Gamma$,\n$[w(\\mathbf x)]_\\Gamma = \\lim\\limits_{\\eps\\to0} w(\\mathbf x-\\eps\\mathbf n)-\\lim\\limits_{\\eps\\to0} w(\\mathbf x+\\eps\\mathbf n)$, $\\mathbf x\\in\\Gamma$,\ndenotes the jump of $w$ across $\\Gamma$ in the direction of $\\mathbf n$.\n\nIf $\\Gamma$ is  \\textit{piecewise} smooth, then we need further conditions on the edges.\nAssume an edge $\\mathcal{E}$ is shared by  $M$  smooth components $\\Gamma_i\\subset\\Gamma$.\nLet $v_{j}=v$ on $\\Gamma_j$, while $\\mathbf w_{\\Gamma,j}=\\mathbf w_{\\Gamma}$, $d_j=d$, $D_{\\Gamma,j}=D_\\Gamma$ on $\\Gamma_j$, and $\\mathbf n_{{\\partial\\Gamma} ,j}$ is the outward  normal vector to ${\\partial\\Gamma} _j$ in the plane tangential to $\\Gamma_j$, cf. Figure~\\ref{fig:Schema}.\nThe conservation of fluid mass yields\n\\begin{equation}\\label{cont_w}\n\\sum_{j=1}^M\\mathbf w_{\\Gamma,j}\\cdot\\mathbf n_{{\\partial\\Gamma} ,j}=0\\quad \\text{on}~~\\mathcal{E}.\n\\end{equation}\nWe assume the continuity of concentration over $\\mathcal{E}$,\n\\begin{equation}\\label{cont_v}\nv_{1}=\\dots=v_{M}\\quad \\text{on}~~\\mathcal{E}.\n\\end{equation}\nWe also assume the conservation of solute flux over the edge. Thanks to \\eqref{cont_w} and \\eqref{cont_v}, this yields the condition:\n\\begin{equation}\\label{cont_F}\n\\sum_{j=1}^M d_j(D_{\\Gamma,j}\\mathbf n_{{\\partial\\Gamma} ,j})\\cdot\\nabla_\\Gamma v_j=0\\quad \\text{on}~~\\mathcal{E}.\n\\end{equation}\n\nFinally, we prescribe Dirichlet's boundary conditions for the concentration $u$ and $v$ on ${\\partial\\Omega} _D$ and ${\\partial\\Gamma} _D$ and homogeneous Neumann's boundary conditions on ${\\partial\\Omega} _N$ and ${\\partial\\Gamma} _N$, respectively, with $\\overline{{\\partial\\Omega} }=\n\\overline{{\\partial\\Omega} _D}\\cup\\overline{{\\partial\\Omega} _N}$ and $\\overline{{\\partial\\Gamma} }=\\overline{{\\partial\\Gamma} _D}\\cup\\overline{{\\partial\\Gamma} _N}$. Initial conditions are given by the known concentration $u_0$ and $v_0$ at $t=0$. We have\n\\begin{equation} \\label{bc}\n\\left\\{\n\\begin{aligned}\n D_i \\mathbf n_{\\partial\\Omega}  \\cdot \\nabla u&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\\\\n u&=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n u|_{t=0}&=u_0 \\quad \\text{in}~ \\Omega,\\\\\n \\end{aligned}\n\\right.\\qquad \\left\\{\n\\begin{aligned}\nD_{\\Gamma}\\mathbf n_{{\\partial\\Gamma} }\\cdot\\nabla_\\Gamma v&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\\\\n v&=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D,\\\\\n v|_{t=0}&=v_0 \\quad \\text{on}~~\\Gamma.\n \\end{aligned}\n\\right.\n \\end{equation}\n\n\\begin{remark}\\rm Bulk--surface coupled systems of advection-diffusion PDEs appear in different applications,\ne.g.  in multiphase fluid dynamics \\cite{GrossReuskenBook} and biological applications \\cite{bonito2011dynamics}. In these and other models, the continuity of the concentration over the embedded surface (second equation in \\eqref{diffeq1}) may \\rev{be replaced by} another suitable   constitutive equation for modeling of the surface adsorption\/desorption.\nFor fluid--fluid interfaces or biological membranes, one often assumes that the surface passively evolves with the flow, and hence there is no contribution of the advective flux to the total flux $F_\\Gamma(u)$ on $\\Gamma$. A standard model for the diffusive flux between the surface and the bulk, cf.~\\cite{Ravera}, is as follows:\n\\begin{equation} \\label{eq4}\n-D_i \\mathbf n \\cdot \\nabla u_i  =  k_{i,a} g_i(v) u_i - k_{i,d} f_i(v),\\quad\\text{on}~\\Gamma,\n\\end{equation}\nwith $k_{i,a}$, $k_{i,d}$ positive adsorption and desorption coefficients that describe the kinetics.  Basic choices for $g$, $f$ are the following:\n\\[\n  g(v)=1, \\quad f(v)=v \\quad \\text{(Henry)}\\quad \\text{or}\\quad g(v)=1- \\frac{v}{v_{\\infty}},\\quad f(v)=v \\quad \\text{(Langmuir)},\n\\]\nwhere $v_\\infty$ is a constant that quantifies the maximal concentration on $\\Gamma$. Further options are given in \\cite{Ravera}.\nOften in literature on the two-pase flows the Robin condition in \\eqref{eq4} is replaced by the ``instantaneous'' adsorption and desorption condition\n\\begin{equation} \\label{eq4a}\nk_{i,a} g_i(v) u_i = k_{i,d} f_i(v),\\quad\\text{on}~\\Gamma,\n\\end{equation}\nThese interface conditions can be also handled through obvious modifications  of our numerical method. \\rev{We include one numerical example with \\eqref{eq4a} and Henry law in Section~\\ref{s_numer}.} At the same time, treating evolving interfaces needs more developments and is not considered here.\n\\end{remark}\n\n\\section{Hybrid finite volume -- finite element method}\\label{s_hybrid}\n\n\\subsection{Summary of the method}\n\nAssume a Cartesian background mesh with cubic cells. We allow local refinement of the mesh by sequential division\nof any cubic cell into 8 cubic subcells. This leads to a grid with an octree hierarchical  structure. This mesh gives the tessellation   $\\mathcal T_h$ of the computational domain $\\Omega$, $\\overline{\\Omega}=\\cup_{T\\in\\mathcal T_h} \\overline{T}$.\nThe surface  $\\Gamma\\subset\\Omega$ cuts through the mesh in an arbitrary way. For the \\textit{purpose of numerical integration},\ninstead of $\\Gamma$ we consider $\\Gamma_h$, a given polygonal approximation of $\\Gamma$. If $\\Gamma$ has a curvature, then $\\Gamma_h$ is reconstructed as a second order approximation of $\\Gamma$. We shall describe the reconstruction algorithm further in the section. We assume that similar to $\\Gamma$, the reconstructed surface $\\Gamma_h$ divides $\\Omega$ into $N$ subdomains $\\Omega_{i,h}$, and ${\\partial\\Gamma} _h\\subset{\\partial\\Omega} $. We do not \\rev{assume} any restrictions on how $\\Gamma_h$ intersects the background mesh.\n\nThe induced tessellation of  $\\Omega_{i,h}$ can be considered as a subdivision of the volume into general polyhedra.\nHence, for the transport and diffusion in the matrix we apply  a non-linear FV method devised on general polyhedral meshes in \\cite{Lipnikov:12,ChernyshenkoFV7:14}, which is monotone and has compact stencil. The trace of the background mesh on $\\Gamma_h$ induces \\rev{a} `triangulation' of the fracture, which is very irregular, and so we do not use it do build a discretization method.\nTo handle transport and diffusion along the fracture, we first consider finite element space of piecewise trilinear functions\nfor the \\textit{volume} octree mesh $\\mathcal T_h$. We further, formally, consider the restrictions (traces) of these background functions on $\\Gamma_h$ and use them in a finite element integral  form over $\\Gamma_h$.\nThus the irregular triangulation of $\\Gamma_h$ is used \\textit{for numerical integration only}, while the trial and test functions are tailored to the background \\textit{regular} mesh. \\rev{Available analysis and numerical experience suggest} that the \\rev{approximation and convergence} properties of this trace finite element method\n\\rev{depend only on the mesh size and refinement strategy for}  the background mesh, and they are independent on how $\\Gamma_h$ intersects $\\mathcal T_h$. The TraceFEM was devised and first analysed in \\cite{ORG09} and extended for the octree meshes in \\cite{chernyshenko2015adaptive}.  A natural way to couple two approaches is to use the\nrestriction of the background FE solution on $\\Gamma_h$ as the boundary data for the FV method and to compute the FV two-side fluxes on $\\Gamma_h$ to provide the source terms for the surface discrete equation. We provide details of each of these steps in sections~\\ref{s_FV}--\\ref{s3} below.\n\n\n\\subsection{Reconstructed surface}\\label{s_rec}\nThe reconstructed surface $\\Gamma_h$ is a $C^{0,1}$ surface that can be partitioned in planar triangular segments:\n\\begin{equation} \\label{defgammah}\n \\Gamma_h=\\bigcup\\limits_{K\\in\\mathcal{F}_h} K,\n\\end{equation}\nwhere $\\mathcal{F}_h$ is the set of all  triangular segments $K$.\nWithout loss of generality we assume that for any $K\\in\\mathcal{F}_h$ there is only \\textit{one}\ncell $T_K\\in\\mathcal{T}_h$ such that $K\\subset T_K$ (if $K$ lies on a face shared by two cells, any of these\ntwo cells can be chosen as $T_K$).\n\nIn practice, we construct $\\Gamma_h$ as follows. For each connected piece of $\\Gamma$ let $\\phi$ be a Lipschitz-continuous level set function, such that $\\phi(\\mathbf x)=0$ on $\\Gamma$. We set $\\phi_h=I(\\phi)$  a nodal interpolant of $\\phi$ by a  piecewise trilinear continuous function with respect\nto the octree grid $\\mathcal{T}_h$.  Consider the zero level set of $\\phi_h$,\n\\[\n\\widetilde{\\Gamma}_h:=\\{\\mathbf x\\in\\Omega\\,:\\, \\phi_h(\\mathbf x)=0 \\}.\n\\]\nIf $\\Gamma$  is smooth, then  $\\widetilde{\\Gamma}_h$ is an approximation to $\\Gamma$ in the following sence:\n\\begin{equation}\\label{eq_dist}\n\\mbox{dist}(\\Gamma,\\rev{\\widetilde{\\Gamma}_h})\\le ch^2_{\\rm loc},\\qquad |\\mathbf n(\\mathbf x)-\\mathbf n_h(\\tilde\\mathbf x)|\\le c h_{\\rm loc},\n\\end{equation}\nwhere $\\mathbf x$ is the closest point on $\\Gamma$ for $\\tilde\\mathbf x\\in\\widetilde{\\Gamma}_h$ and $h_{\\rm loc}$ is the local mesh size.\nWe note that in some applications, $\\phi_h$ is computed from a solution of a discrete indicator function equation (e.g., in the level set or the volume of fluid methods), without any direct knowledge of $\\Gamma$.\n\n\\begin{figure}\n  \\begin{center}a)\n    \\includegraphics[width=0.31\\textwidth]{.\/mesh3d_rotsin.jpg}b)\n    \\includegraphics[width=0.31\\textwidth]{.\/mesh2d_rotsin.jpg} c)\n    \\includegraphics[width=0.29\\textwidth]{.\/mesh2d_rotsin_zoom.jpg}\n    \\caption{a) Example of a  bulk domain with a fracture. In this example, the background mesh is refined near the fracture; b) The reconstructed $\\Gamma_h$; c) The zoom-in of the induced surface triangulation $\\mathcal F_h$.}\n    \\label{fig1}\n  \\end{center}\n\n\\end{figure}\n\\rev{Note that $\\widetilde{\\Gamma}_h$ is still not completely suitable for our purposes, since $\\phi_h$ is \\emph{trilinear}\nand so numerical integration over its zero level is not straightforward. Therefore, we next build a suitable polygonal approximation of $\\widetilde{\\Gamma}_h$ which is our final $\\Gamma_h$.}\nOnce $\\phi_h$ is computed, we recover $\\Gamma_h$ by the cubical marching squares method from \\cite{MCM2} (a variant of the very well-known marching cubes method). The method provides a triangulation of $\\widetilde{\\Gamma}_h$ within each\ncube such that \\rev{$\\Gamma_h$ is  continuous over cubes interfaces}, the number of triangles within each cube is finite and bounded\nby a constant independent of $\\widetilde{\\Gamma}_h$ and a number of refinement levels. Moreover, the vertices of triangles from $\\mathcal{F}_h$ are lying on $\\widetilde{\\Gamma}_h$. This final discrete surface $\\Gamma_h$ is still an approximation of $\\Gamma$ in the sense of \\eqref{eq_dist}. A example of bulk domain with embedded surface and background mesh is illustrated in Figure~\\ref{fig1}.\n\nNote that the resulting triangulation $\\mathcal{F}_h$  is \\emph{not} necessarily  regular, i.e. elements from $T$ may have very small internal angles and the size of neighboring triangles can vary strongly. Thus, $\\Gamma_h$ is not \\rev{ a regular triangulation of $\\Gamma$.} The surface triangulation $\\mathcal{F}_h$ is used only to define quadratures in the finite element method, while approximation properties of the method depend on the volumetric octree mesh.\n\n\n\n\\subsection{Monotone finite volume method} \\label{s_FV}\nFirst we consider a FV method for the advection-diffusion equation \\eqref{diffeq1} in each subdomain $\\Omega_{i,h}$.\nLet $\\mathcal T_{i,h}$ be the tessellation  of $\\Omega_{i,h}$ into non-intersected polyhedra, which is induced by overlapping $\\Omega_{i,h}$ and the background mesh $\\mathcal T_h$.   Since the background mesh is the octree Cartesian, each element $T\\in\\mathcal T_{i,h}$ is either a cube, if it lies in the interior of $\\Omega_{i,h}$, or a cut cube, if ${\\partial\\Omega} _{i,h}$ intersects a background cell from $\\mathcal T_h$. We assume the octree grid is gradely refined, i.e. the sizes of two neighbouring elements of $\\mathcal T_h$ can differ at most by a factor of two.  Such octree grids are also known as balanced. The method applies for unbalanced octrees, but in our experiments we use balanced grids.\nFor the balanced grid,  each interior cell may have from 6 to 24 neighboring cells (cells sharing a face). In the FV method we treat such cells as a polyhedra with up to 24 faces.   Since we reconstruct $\\Gamma$ inside each cell as a triangulated surface without holes, the cut cell from $\\mathcal T_{i,h}$ can be treated as a general polyhedral element as well. By $\\mathcal F_{i,h}$ we denote the set of all faces of polyhedra from $\\mathcal T_{i,h}$.\n\nThe FV discretization below is applied to each subdomain $\\Omega_i$ separately, so we will skip in this section the  redundant index $i$ for the concentration, coefficients and the flow vector field in $\\Omega_i$. Note that ${\\partial\\Omega} _{i,D}$\nincludes the fracture part of the boundary of ${\\partial\\Omega} _i$.\n\nAs the first step,  we assume a time discretization (say, the implicit Euler method) and consider the mixed form of  \\eqref{diffeq1}  and  boundary conditions\n\\begin{equation}\\label{lap}\n\\begin{array}{rclll}\n  \\mathbf q = \\mathbf w u - D \\nabla u, \\quad\n  \\tilde\\phi\\,u+\\mbox{\\rm div } \\mathbf q  &=& f \\quad     & \\mbox{~in} & \\Omega_i,  \\\\\n       u &=& \\tilde{u}_D                     & \\mbox{~on} & {\\partial\\Omega} _{i,D} ,  \\\\\n       -D \\mathbf n_{\\partial\\Omega}  \\cdot \\nabla u &=&0 & \\text{~on}& {\\partial\\Omega} _{i,N},\n\\end{array}\n\\end{equation}\nwhere the right hand side $f$ accounts for the source term and for the values of concentration from the previous time step, $\\tilde\\phi$ is the scaled \\rev{(by the reciprocal of time step)} porosity coefficient\\rev{, and ${\\partial\\Omega} _{i,D}$ includes ${\\partial\\Omega} _i\\cap\\Gamma$, where $\\tilde{u}_D=v$}.\n\nFor a cell $T\\in\\mathcal T_{i,h}$, $\\mathbf x_{T}$ denotes the barycenter of $T$, and $u_T$ denotes the averaged concentration. We formally assign $u_T$ to $\\mathbf x_T$.\nIntegrating the mass balance equation \\eqref{lap} over $T$ and using the divergence theorem, we obtain:\n\\begin{equation}\\label{green2}\n \\tilde\\phi|T|u_T +\\sum\\limits_{\\mathcal F \\in \\partial T} \\mathbf q_\\mathcal F\\cdot \\mathbf n_\\mathcal F  =  \\int_T f \\,{\\rm d} x,\n  \\qquad\n  \\mathbf q_{\\mathcal F} = \\frac{1}{|\\mathcal F|} \\int_{\\mathcal F}  \\mathbf q \\,{\\rm d} s,\n\\end{equation}\nwhere $\\mathbf q_{\\mathcal F} \\cdot \\mathbf n_\\mathcal F$ is the averaged normal flux across face $\\mathcal F$,  and $\\mathbf n_\\mathcal F$  is the normal vector\non $\\mathcal F$ pointing outward for $T$; $|\\mathcal F|$ ($|T|$) denotes the area (volume) of $\\mathcal F$ ($T$).\nThe Dirichlet boundary data on faces $\\mathcal F\\in \\partial\\Omega_D$ will be accounted in the scheme via boundary faces concentration values\n$u_\\mathcal F = \\frac{1}{|\\mathcal F|}\\int_{\\mathcal F}  u_D \\,{\\rm d} s$. We assume that $u_\\mathcal F$ are assigned to barycenters of faces.\n Enforcing homogeneous Neumann boundary conditions on faces from ${\\partial\\Omega} _{i,N}$ is straightforward, for $\\mathcal F\\in  \\partial T\\cap{\\partial\\Omega} _{N}$ the normal flux $\\mathbf q_\\mathcal F\\cdot \\mathbf n_\\mathcal F$ in \\eqref{green2} is  set to 0.\n\nIn the conventional cell-centered FV method, the normal flux $\\mathbf q_\\mathcal F\\cdot \\mathbf n_\\mathcal F$ is replaced by its discrete counterpart $\\mathbf q_{\\mathcal F,h} \\cdot \\mathbf n_\\mathcal F$, which is computed\nfrom cell concentrations $u_T$ and boundary data $u_\\mathcal F$. For simplicity of presentation we shall omit subscript $h$ in notations of the discrete flux.\nThe discrete flux is the combination of the diffusive and convective fluxes and we discretize them separately\nfollowing  \\cite{Lipnikov:10,Lipnikov:12,NikitinVassilevski:10}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=.45\\textwidth]{.\/quadtreecut_MPFA3.pdf}\n\\caption{For the 2D case, the figure illustrates our constructions for the approximation of the directional derivative $\\mathbf l_{\\mathcal F}\\cdot\\nabla u$ on the face (edge in 2D) $\\mathcal F=\\overline{T_+}\\cap\\overline{T_-}$. Bold dots\nshow the barycenters $\\mathbf x_T$ of the cells from $\\mathcal T_{1,h}$ and $\\mathcal T_{2,h}$. The mean values of the concentration are assigned to these barycenters.}\n\\label{fig:dupl}\n\\end{figure}\n\nFor $T\\in\\mathcal T_{i,h}$, we define $\\omega(T):=\\{T'\\in\\mathcal T_{i,h}\\,|\\,\\mbox{area}(\\overline{T}'\\cap \\overline{T})\\neq 0\\}$, the set of all neighboring cells of $T$, and\n $\\omega_{\\partial}(T):=\\{F\\in\\mathcal F_{i,h}\\,|\\,F\\rev{\\subset} \\partial T\\cap \\partial\\Omega_{i,D}\\}$, the set all faces of $T$ with prescribed Dirichlet data.  For $T\\in\\mathcal T_{i,h}$, the set of points $\\mathcal{P}$ collects all barycenters of the elements from $\\omega(T)$ and $\\omega_{\\partial}(T)$.  Furthermore, for each $T\\in\\mathcal T_{i,h}$ we define the bundle of vectors, $\\mathbf v(T):=\\{\\mathbf t\\in\\mathbb{R}^3\\,|\\,\\mathbf t=\\mathbf y-\\mathbf x_T,~\\mathbf y\\in \\mathcal{P}(T)\\}$.\n\nConsider an arbitrary internal face $\\mathcal F$ shared by two cells $T_+, T_-$ from $\\mathcal T_{i,h}$ and assume that $\\mathbf n_\\mathcal F$ points from $T_+$ to $T_-$. We introduce the co-normal vector $\\mathbf l_{\\mathcal F}= D \\mathbf n_\\mathcal F$.\nVector $\\mathbf l_{\\mathcal F}$  can make a nonzero angle with $\\mathbf n_\\mathcal F$ in the case of an anisotropic diffusion tensor.\nTo define the discrete {\\em diffusive} flux on $\\mathcal F$, we first\nconsider three vectors $\\mathbf t_{i}^+\\in \\mathbf v(T_+)$, $i=1,2,3$,\nsuch that for the co-normal vector $\\mathbf l_{\\mathcal F} = D \\mathbf n_f$ we have\n\\begin{equation}\\label{ab}\n  \\mathbf l_{\\mathcal F} =\n  \\alpha_+~ \\mathbf t_{1}^+ + \\beta_+~ \\mathbf t_{2}^+ + \\gamma_+~ \\mathbf t_{3}^+,\n\\end{equation}\nwith non-negative coefficients $\\alpha_+$, $\\beta_+$ and $\\gamma_+$. Such a triplet can be always found,  (in some rare pathological situations, one has to expand $\\mathcal{P}(T_+)$ slightly, cf. \\cite{DanilovVassilevski:09}).\n\nThe normal flux is the directional derivative along the co-normal vector $\\mathbf l_+ := \\mathbf l_{\\mathcal F}$, and hence\nit can also be represented as the linear combination of three derivatives along $\\mathbf t_{i}^+$. The latter  are approximated by central differences (may reduce to one side differences near Dirichlet boundaries). Thus, we get\n$\\mathbf q_\\mathcal F\\cdot\\mathbf n_F\\approx q_+$,\n\\begin{equation}\\label{ab2}\n q_+=\n  \\alpha'_{+}~ (u_+ - u_{+,1}) + \\beta'_{+}~ (u_+ - u_{+,2}) +\n  \\gamma'_{+}~ (u_+ - u_{+,3}),\\qquad u_+=u(\\mathbf x_{T_+}),~ u_{+,i}=u(\\mathbf x_{T_+}+\\mathbf t_{i}^+),\n\\end{equation}\nwhere coefficients $\\alpha'_{+}, \\beta'_{+}, \\gamma'_{+}$ are computed from $\\alpha$, $\\beta$, $\\gamma$ in  \\eqref{ab}\nfor the cell $T_+$, using the simple  scaling with $|\\mathbf t_{i}^+| \/ |\\mathbf l_{\\mathcal F}|$.\nFor the same co-normal vector one has  another decomposition based on $\\mathbf v(T_-)$ vector bundle,\n$\n\\mathbf l_- := -\\mathbf l_{\\mathcal F} =\n  \\alpha_-~ \\mathbf t_{1}^- + \\beta_-~ \\mathbf t_{2}^- + \\gamma_-~ \\mathbf t_{3}^-$, $\\mathbf t_{i}^-\\in \\mathbf v(T_-)$. This decomposition yields another approximation,  $\\mathbf q_\\mathcal F\\cdot\\mathbf n_F\\approx q_-$:\n\\begin{equation}\\label{ab3}\n  q_- =\n  \\alpha'_{-}~ (u_- - u_{-,1}) + \\beta'_{-}~ (u_- - u_{-,2}) +\n  \\gamma'_{-}~ (u_- - u_{-,3}),\\qquad u_-=u(\\mathbf x_{T_-}),~ u_{-,i}=u(\\mathbf x_{T_-}+\\mathbf t_{i}^-),\n\\end{equation}\nwith non-negative coefficients $\\alpha'_-$, $\\beta'_-$ and $\\gamma'_-$. Figure~\\ref{fig:dupl} illustrates the construction in 2D.\n\nNow we can take a linear combination of \\eqref{ab2} and \\eqref{ab3} with non-negative coefficients $\\mu_+$ and $\\mu_-$:\n\\begin{equation}\\label{two-flux}\n  \\mathbf q_{\\mathcal F} \\cdot \\mathbf n_{\\mathcal F} = \\mu_+ q_+ + \\mu_- (- q_-).\n\\end{equation}\nThe discrete flux \\eqref{two-flux}  approximates the differential one if   $\\mu_+$, $\\mu_-$  satisfy\n\\begin{equation}\n\\label{mupm1}\n\\mu_+ + \\mu_- = 1.\n\\end{equation}\nFollowing \\cite{Lipnikov:12}, to construct the {\\it monotone}  FV discretization, we set  both representations of the flux equal:\n\\begin{equation}\n\\label{fvdmp_mus}\n  \\mu_+ q_+ = - \\mu_- q_-.\n\\end{equation}\nIf $| q_+| = |q_-| = 0$, then the solution of \\eqref{mupm1},\\eqref{fvdmp_mus} in not unique. In this case we choose $\\mu_+ = \\mu_- = 1\/2$.\nOtherwise, the solution is given by\n\\begin{equation*}\n\\mu_+ = \\frac{q_-}{q_- - q_+}, \\qquad \\mu_- = \\frac{q_+}{q_+ - q_-}.\n\\end{equation*}\nIf $q_+ q_- > 0$, we avoid   potentially degenerate case by applying the modification from~\\cite{ShengYuan:11} \\rev{(see also formulas in \\cite{Lipnikov:12}, p.~374)}.\nWe note that the resulting multi-point flux approximation is nonlinear and compact, i.e. the stencil includes the values of concentration only from neighboring cells.\n\nTo define the normal component of the discrete \\textit{advective} flux\n$\\mathbf q_{\\mathcal F,a} =\\frac{1}{|\\mathcal F|} \\int_{\\mathcal F}  u\\mathbf w \\,{\\rm d}s$,\nwe adopt the nonlinear  upwind approximation (subscript $h$ is again omitted for the sake of notation):\n\\begin{equation}\\label{vRvR}\n  \\mathbf q_{\\mathcal F,a} \\cdot \\mathbf n_{\\mathcal F} = w_{\\mathcal F}^+  \\mathcal R_{T^+}(\\mathbf x_{\\mathcal F}) + w_{\\mathcal F}^-  \\mathcal R_{T^-}(\\mathbf x_{\\mathcal F}),\n\\end{equation}\nwhere\n$$\nw_{\\mathcal F}^+ =\\frac12 (w_{\\mathcal F}+ |w_{\\mathcal F}|), \\quad w_{\\mathcal F}^- =\\frac12 (w_{\\mathcal F}- |w_{\\mathcal F}|), \\quad w_{\\mathcal F} = \\frac{1}{|\\mathcal F|} \\int_{\\mathcal F}  \\mathbf w\\cdot\\mathbf n_{\\mathcal F} \\,{\\rm d} s,\n$$\n$\\mathcal R_T$ is a linear reconstruction of the concentration\nover cell $T$ which depends on the concentration values from neighboring cells.\n\n\nOn each cell $T$, the linear reconstruction is defined by\n\\begin{equation}\\label{reconst}\n\\mathcal R_T (\\mathbf x) = \\left\\{ \\begin{array}{ll} u_T + \\mathcal L_T \\mathbf g_T \\cdot (\\mathbf x-\\mathbf x_{T}), & \\mathbf x\\in T, \\\\\n                                       0, & \\mathbf x\\notin T, \\end{array} \\right.\n\\end{equation}\nwhere $\\mathbf g_T$ denotes the gradient of the  linear reconstruction of concentration in $\\mathbf x_T$, and $\\mathcal L_T$ is a slope limiting operator. The gradient is recovered from the best affine least-square fit for $u_h$ over a subset of barycenter nodes and, possibly, the boundary data nodes from cells neighboring $T$.\nThe slope limiting operator $\\mathcal L_T$ is introduced to avoid non-physical extrema. It provides the smallest possible changes of the reconstructed least-square slope. Details can be found in  \\cite{Lipnikov:10,Lipnikov:12,NikitinVassilevski:10}.\n\n\nReplacing fluxes in equations \\eqref{green2} by their numerical approximations, we\nobtain a system of nonlinear equations\n\\begin{equation} \\label{sol:system}\n  \\Phi\\mathbf U+ \\mathbf M(\\mathbf U)\\, \\mathbf U = \\mathbf F(\\mathbf U),\n  \\qquad\n  \\mathbf M(\\mathbf U) = \\mathbf M_{dif}(\\mathbf U) + \\mathbf M_{adv}(\\mathbf U),\n\\end{equation}\nwith a diagonal matrix $\\Phi$. For any fixed vector $\\mathbf V$, $\\mathbf M(\\mathbf V)$ is a square sparse matrix,  $\\mathbf F(\\mathbf V)$ is a right-hand side vector.\nMatrix $\\mathbf M_{dif}$ is an M-matrix which has  diagonal dominance in rows.\nThe stencil of this matrix is compact, each row contains non-zero off-diagonal entries corresponding mainly (and in most cases only) to degrees of freedom at the cells sharing a face with the current cell.\n For a  cubic uniform mesh and the Poisson equation, the matrix $\\mathbf M_{dif}$ corresponds to the conventional seven-point stencil.\nAlthough matrix $\\mathbf M_{adv}$ has no diagonal dominance in rows,\nit can be shown, cf.~\\cite{Lipnikov:12}, that the solution to \\eqref{sol:system}\nsatisfies the discrete maximum principle.\n\n\n\\subsection{The trace finite element method} \\label{s_FEM}\n\nConsider now the volumetric finite element space of all piecewise trilinear continuous functions with respect to the bulk octree mesh $\\mathcal{T}_h$:\n\\begin{equation}\n V_h:=\\{v_h\\in C(\\Omega)\\ |\\ v|_{S}\\in Q_1~~ \\forall\\ S \\in\\mathcal{T}_h\\},\\quad\\text{with}~~\n Q_1=\\mbox{span}\\{1,x_1,x_2,x_3,x_1x_2,x_1x_3,x_2x_3,x_1x_2x_3\\}.\n \\label{e:2.6}\n\\end{equation}\nThe surface finite element space is \\textit{the space of traces on $\\Gamma_h$ of all piecewise trilinear continuous functions with respect to the outer triangulation $\\mathcal{T}_h$}  defined as follows\n\\begin{equation}\n V_h^{\\Gamma}:=\\{\\psi_h\\in H^1(\\Gamma_h)\\ |\\ \\exists ~ v_h\\in V_h\\  \\text{such that }\\ \\psi_h=v_h|_{\\Gamma_h}\\}.\n\\label{e:fem-space}\n\\end{equation}\n\nGiven the surface finite element space $V_h^{\\Gamma}$,  the finite element\ndiscretization of \\eqref{diffeq2} is as follows:   Find $v_h\\in V_h^{\\Gamma}$ such that $v_h|_{\\partial\\Gamma_{D,h}}=v_D^h$ and\n\\begin{multline}\n\\int_{\\Gamma_h}\\left(  \\phi_{\\Gamma,h}\\frac{\\partial v_h}{\\partial t}w_h +  d_h D_{\\Gamma,h}\\nabla_{\\Gamma_h} v_h\\cdot\\nabla_{\\Gamma_h} w_h\\, +(\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} v_h) w_h\\right)\\, + (\\mbox{\\rm div}_{\\Gamma_h}\\mathbf w_h) \\, \\rev{w_h}v_h \\mathrm{d}\\mathbf{s}_h\\\\ =\\int_{\\Gamma_h}( F_{\\Gamma,h}(u_h)+ f_{\\Gamma,h}) w_h\\, \\mathrm{d}\\mathbf{s}_h \\label{FEM}\n\\end{multline}\nfor all $w_h\\in V_h^{\\Gamma}$, s.t. $w_h|_{\\partial\\Gamma_{D,h}}=0$ . Here $\\mathbf{w}_h$, $v_D^h$, $d_h$, $D_{\\Gamma,h}$ and $f_{\\Gamma,h}$ are  the problem data lifted from $\\Gamma$ to $\\Gamma_h$, in the case if $\\Gamma\\neq\\Gamma_h$. The bulk domain contributes through the flux $F_{\\Gamma,h}(u_h)$, which is reconstructed from the numerical concentration in the porous matrix.\n\nSimilar to the plain Galerkin finite element for  advection-diffusion equations the method \\eqref{FEM} is prone to instability unless mesh is sufficiently fine such that the mesh Peclet number is less than one.\nFollowing \\cite{ORXimanum}, we consider the SUPG stabilized TraceFEM.  The stabilized formulation reads:\n Find $v_h\\in V_h^{\\Gamma}$ such that\n\\begin{multline}\n \\int_{\\Gamma_h}\\left(  \\phi_{\\Gamma,h}\\frac{\\partial v_h}{\\partial t}w_h +  d_h D_{\\Gamma,h}\\nabla_{\\Gamma_h} v_h\\cdot\\nabla_{\\Gamma_h} w_h\\, +(\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} v_h) w_h\\right)\\, + (\\mbox{\\rm div}_{\\Gamma_h}\\mathbf w_h) \\, \\rev{w_h}v_h \\mathrm{d}\\mathbf{s}_h \\\\\n  +\\sum_{\\rev{T}\\in\\mathcal{F}_h}\\delta_K\\int_{K}(\\phi_{\\Gamma,h}\\frac{\\partial v_h}{\\partial t}-\\rev{d\\mbox{\\rm div}_{\\Gamma_h}D_{\\Gamma,h}\\nabla_{\\Gamma_h}}v_h + \\mathbf w_h\\cdot\\nabla_{\\Gamma_h} v_h + (\\mbox{\\rm div}_{\\Gamma_h}\\mathbf w_h) \\, v_h)\\,\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} w_h\\, \\mathrm{d}\\mathbf{s}_h\\\\ =\\int_{\\Gamma_h}( F_{\\Gamma,h}(u_h)+ f_{\\Gamma,h}) w_h\\, \\mathrm{d}\\mathbf{s}_h + \\sum_{K\\in\\mathcal{F}_h}\\delta_K\\int_{K}( F_{\\Gamma,h}(u_h)+ f_{\\Gamma,h})(\\mathbf w_h\\cdot\\nabla_{\\Gamma_h} w_h)\\, \\mathrm{d}\\mathbf{s}_h\\quad \\forall~ w_h\\in V_h^{\\Gamma}. \\label{FEM_SUPG}\n\\end{multline}\n\\rev{For the definition of $K\\in\\mathcal{F}_h$, $T_K\\in\\mathcal T_h$ we refer to section~\\ref{s_rec}.}\nThe stabilization parameter  $\\delta_K$ depends on $K \\subset T_K$. The side length of the cubic cell $T_K$ is denoted by $h_{T_K}$.  Let $\\displaystyle \\mathsf{Pe}_K:=\\frac{h_{T_K} \\|\\mathbf{w}_h\\|_{L^\\infty(K)}}{2\\eps}$\nbe the cell Peclet number.\nWe take\n\\begin{equation}\n {\\delta_K}=\n\\left\\{\n\\begin{aligned}\n&\\frac{\\delta_0 h_{T_K}}{\\|\\mathbf{w}_h\\|_{L^\\infty(K)}} &&\\quad \\hbox{ if } \\mathsf{Pe}_K> 1,\\\\\n&\\frac{\\delta_1 h^2_{T_K}}{\\eps}  &&\\quad \\hbox{ if } \\mathsf{Pe}_K\\leq 1,\n\\end{aligned}\n\\right. \\label{e:2.10}\n\\end{equation}\nwith some given positive constants  $\\delta_0,\\delta_1\\geq 0$.\n\n\nFor the matrix--vector representation of the TraceFEM one uses the nodal basis of the bulk finite element space $V_h$ rather than tries to construct a basis in $V_h^{\\Gamma}$.  This convenient choice, however, has some consequences. In general, the restrictions to $\\Gamma_h$ of the outer nodal basis functions on $\\Gamma_h$ can be linear dependent or (in most cases) almost linear dependent.\nThis and small cuts of background cells lead to badly conditioned mass and stiffness matrices. In recent\nyears stabilizations have been developed which are easy to implement and result in\nmatrices with acceptable condition numbers, see the overview in \\cite{TraceFEM}.\nIn this paper we use the ``full gradient'' stabilization of the TraceFEM~\\cite{Alg2,Reusken2014}. In this variant of the method, one modifies the surface diffusion part of the  method~\\eqref{FEM} to include the normal part of the gradient:\n\\[\n\\int_{\\Gamma_h}d_h D_{\\Gamma,h}\\nabla_{\\Gamma_h} v_h\\cdot\\nabla_{\\Gamma_h} w_h\\,\\mathrm{d}\\mathbf{s}_h\\quad\\text{yields to}\\quad\n\\int_{\\Gamma_h}d_h D_{\\Gamma,h}\\nabla v_h\\cdot\\nabla w_h\\,\\mathrm{d}\\mathbf{s}_h.\n\\]\nWe note that the method remains consistent on smooth surfaces (up to second order geometric errors), since the true surface solution extended off the surface along normal directions satisfies both variational formulations on $\\Gamma$.   The modification improves algebraic properties of the (diagonally scaled) stiffness matrix of the method~\\cite{Reusken2014}. The full-gradient method  uses the background finite element space $V_h$ instead of the surface finite element space $V_h^\\Gamma$ in \\eqref{FEM}. However, practical implementation of both methods uses the frame of all bulk finite element nodal basis functions $\\phi_i\\in V_h$ such that  $\\mbox{supp}(\\phi_i)\\cap\\Gamma_h\\neq\\emptyset$. Hence the active degrees of freedom  in both methods are the same. The stiffness matrices are, however, different.\n\n\n\\subsection{Coupling between discrete bulk and surface equations}\\label{s3}\n\nThe equations in the bulk and on the surface are coupled through the boundary condition $u_i=v$ on $\\partial\\Omega_{i,h}\\cap\\Gamma_h$ (second equation in \\eqref{diffeq1}) and the net flux $F_{\\Gamma_h}(u)$ on $\\Gamma_h$, which stands as the source term in the surface equation \\eqref{diffeq2}. On $\\Gamma_h$ the solution $v_h$ is defined as a trace of the background finite element piecewise trilinear function. The averaged value\n of $v_h$ is computed on each surface triangle $K\\in\\mathcal F_h$ using a standard quadrature rule. These values assigned to the barycenters of $K$ from $\\mathcal F_h$ serve as the Dirichlet boundary data for the FV method on $\\Gamma_h$.\n The discrete diffusive and convective fluxes are assigned to barycenters of all faces on $\\mathcal T_{i,h}$, $i=1,\\dots,N$.\nSince each triangle $K\\in\\mathcal F_h$ is a face for two cells $T_i\\in\\mathcal T_{i,h}$ and  $T_j\\in\\mathcal T_{j,h}$, $i\\neq j$,\nthe diffusive and convective fluxes are assigned to $K$ from both sides of $\\Gamma_h$.\nThe discrete net flux $F_{\\Gamma_h}(u_h)$ at the barycenter of $K$ is computed as the jump of the fluxes over $K$.\nIn the TraceFEM this value is assigned to all $\\mathbf x\\in K$, and numerical integration is  done over all surface elements $K\\in\\mathcal F_h$ to compute the right-hand side of the algebraic system.\n\nTo satisfy all (discretized) equations and boundary conditions we iterate between the bulk FV and surface FE solvers on each time step.\nWe assume an implicit time stepping method (in experiments we use backward Euler). This results in the following system on each time step.\n\\begin{equation} \\label{system}\n\\left\\{\n\\begin{aligned}\n \\mathcal L u:= \\tilde\\phi u + \\mbox{\\rm div}(\\mathbf w  u- D \\nabla) u &= \\hat{f} \\quad \\text{in}~ \\Omega\\setminus\\Gamma,\\\\\n u&=v \\quad \\text{on}~ \\Gamma,\\\\\n \\rev{D}\\mathbf n_{\\partial\\Omega} \\cdot\\nabla u=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad u&=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n \\mathcal L_\\Gamma v:=\\tilde\\phi_\\Gamma v + \\mbox{\\rm div}_\\Gamma (\\mathbf w_\\Gamma v - d D_\\Gamma \\nabla_\\Gamma v) &=  F_\\Gamma(u)+ \\hat{f}_\\Gamma  \\quad \\text{on}~~\\Gamma,\\\\\n \\rev{D}_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla v=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad v&=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D,\n \\end{aligned}\n\\right.\n\\end{equation}\nthe right hand sides $\\hat{f}$ and $\\hat{f}_\\Gamma$ account for the  solution values at the previous time step. Note that condition \\eqref{cont_v} is satisfied by the construction of trace spaces in the finite element method\nand condition \\eqref{cont_F} is accounted weakly by the TraceFEM variational formulation.\n\n\\rev{\nWe solve the coupled system \\eqref{system} by the fixed point method: Given ${u}^0,{v}^0$, the initial guess,  we iterate for $k=0,1,2,\\dots$ until convergence: \\\\\nStep 1: Solve for ${u}^{k+1}$,\n\\begin{equation} \\label{step1}\n\\left\\{\n\\begin{aligned}\n \\mathcal L {u}^{k+1}&= \\hat{f}~\\text{in}~ \\Omega\\setminus\\Gamma,\\quad {u}^{k+1}={v}^{k} \\quad \\text{on}~ \\Gamma,\\\\\n \\rev{D}\\mathbf n_{\\partial\\Omega} \\cdot\\nabla {u}^{k+1}&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad {u}^{k+1}=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n \\end{aligned}\n\\right.\n\\end{equation}\nStep 2: Solve for ${v}^{\\rm aux}$ and update for ${v}^{k+1}$ with a relaxation parameter $\\omega$,\n\\begin{equation} \\label{step2}\n\\left\\{\n\\begin{aligned}\n \\mathcal L_\\Gamma {v}^{\\rm aux}&=  F_\\Gamma(u^{k+1})+ \\hat{f}_\\Gamma  \\quad \\text{on}~~\\Gamma,\\\\\n \\rev{D}_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla_\\Gamma {v}^{\\rm aux}&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad {v}^{\\rm aux}=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D\\\\\n {v}^{k+1}&=\\omega{v}^{\\rm aux}+(1-\\omega) {v}^{k},\\quad \\omega\\in(0,1],\n \\end{aligned}\n\\right.\n\\end{equation}\n}\n\n\\begin{remark}\\rm \\label{rem1} \\rev{Below we show that the fixed point method is equivalent to a preconditioned Richardson iteration for the discrete Poincar\\'{e}--Steklov operator. Assume that $\\mathcal L$ is linear (this is true for our differential model, but\nthe particular FV discretization applied here is actually non-linear).\nLet's} split $u=u_0+\\hat{u}$, $v=v_0+\\hat{v}$, where $u_0,\\,v_0$ satisfy\n\\[\n\\left\\{\n\\begin{aligned}\n\\mathcal L u_0&= \\hat{f}~\\text{in}~ \\Omega\\setminus\\Gamma,\\quad u_0=0 \\quad \\text{on}~ \\Gamma,\\\\\n D\\mathbf n_{\\partial\\Omega} \\cdot\\nabla u_0&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad u_0=u_D \\quad \\text{on}~ {\\partial\\Omega} _D,\n  \\end{aligned}\n\\right.\\qquad\n\\left\\{\n\\begin{aligned}\n \\mathcal L_\\Gamma {v}_0&=  0  \\quad \\text{on}~~\\Gamma,\\\\\n D_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla {v}_0&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad {v}_0=v_D \\quad \\text{on}~ {\\partial\\Gamma} _D\n \\end{aligned}\n\\right.\n\\]\nNow the iterations \\eqref{step1}--\\eqref{step2} can be written in terms of  $\\hat{u}$ and $\\hat{v}$ parts of the bulk and surface  concentrations:\n\\begin{equation} \\label{step1a}\n\\begin{split}\n\\left\\{\n\\begin{aligned}\n \\mathcal L \\hat{u}^{k+1}&= 0~\\text{in}~ \\Omega\\setminus\\Gamma,\\quad \\hat{u}^{k+1}=\\hat{v}^{k} \\quad \\text{on}~ \\Gamma,\\\\\n \\rev{D}\\mathbf n_{\\partial\\Omega} \\cdot\\nabla \\hat{u}^{k+1}&=0 \\quad \\text{on}~ {\\partial\\Omega} _N,\\quad \\hat{u}^{k+1}=0 \\quad \\text{on}~ {\\partial\\Omega} _D,\\\\\n \\end{aligned}\n\\right. \\\\\n\\left\\{\n\\begin{aligned}\n \\mathcal L_\\Gamma \\hat{v}^{\\rm aux}&=  F_\\Gamma(u^{k+1})+ \\hat{f}_\\Gamma  \\quad \\text{on}~~\\Gamma,\\\\\n \\rev{D}_\\Gamma\\mathbf n_{\\partial\\Gamma} \\cdot\\nabla_\\Gamma \\hat{v}^{\\rm aux}&=0 \\quad \\text{on}~ {\\partial\\Gamma} _N,\\quad \\hat{v}^{\\rm aux}=0 \\quad \\text{on}~ {\\partial\\Gamma} _D\\\\\n \\hat{v}^{k+1}&=\\omega\\hat{v}^{\\rm aux}+(1-\\omega) \\hat{v}^{k},\\quad \\omega\\in(0,1].\n \\end{aligned}\n\\right.\n\\end{split}\n\\end{equation}\nNow we note that $\\hat{u}$ is a (generalized) harmonic extension of $\\hat{v}$  on $\\Omega\\setminus\\Gamma$ and $S_\\Gamma\\,:\\,\\hat{v}\\to F_\\Gamma(\\hat{u})$ is the  Dirichlet to Neumann (discrete) Poincar\\'{e}--Steklov operator. Using this notation,\none can write the surface equation for $\\hat{v}$ in the compact operator form,\n\\begin{equation} \\label{PS}\n(\\mathcal L_{\\Gamma,0}-S_\\Gamma) \\hat{v}= \\hat{F}\\quad \\text{on}~~\\Gamma,\\quad\\text{with}~~ \\hat{F}:=F_\\Gamma(u_0)+ \\hat{f}_\\Gamma.\n\\end{equation}\nWe use zero index in $\\mathcal L_{\\Gamma,0}$ to stress that the operator accounts for homogenous boundary conditions on ${\\partial\\Gamma} $. It is easy to see that \\eqref{step1a} is the Richardson iterative process for the surface\nequation \\eqref{PS}, with the preconditioner $W=\\mathcal L_{\\Gamma,0}^{-1}$ and the relaxation parameter $\\omega$:\n\\begin{equation} \\label{Richards}\n\\hat{v}^{k+1}=\\hat{v}^{k}-\\omega\\,W\\left(\\, (\\mathcal L_\\Gamma-S_\\Gamma) \\hat{v}^k-\\hat{F}\\right),\\quad k=0,1,2,\\dots.\n\\end{equation}\nFrom \\eqref{Richards} we see that a more efficient iterative process based on a different choice\nof preconditioner and employing  Krylov subspaces may be feasible \\rev{(if $\\mathcal L$ is non-linear one may consider Anderson's mixing to accelerate convergence)}.  However, we do not pursue this topic further in this paper.\n\\end{remark}\n\n\n\\section{Numerical results and discussion}\\label{s_numer}\nThis section collects several numerical examples, which demonstrate the accuracy and capability of the hybrid method.\n We perform a series of tests, where we simulate steady and time-dependent solutions  in a bulk domain with an imbedded  fracture. \\rev{We also include an a example with a smooth curved surface (a sphere) embedded in a bulk domain and a given analytical solution for a surface-bulk problem with Henry interface condition.}\n \\rev{To measure the error we shall use $L^2$, $H^1$ and $L^\\infty$ surface and volume norms.\nFor the computed solutions $u_h,v_h$ and true solutions $u,v$, these norms are defined below. In the volume, we set\n\\[\n\\begin{split}\n&\\mbox{err}_{L^\\infty(\\Omega)}:=\\max_{T\\in\\mathcal T_h}|u_h(\\mathbf x_T)-u(\\mathbf x_T)|,\\quad \\mbox{err}_{L^2(\\Omega)}:=\\left(\\sum_{T\\in\\mathcal T_h}\\text{vol}(T)|u_h(\\mathbf x_T)-u(\\mathbf x_T)|^2\\right)^{\\frac12},\\\\ &\\mbox{err}_{H^1(\\Omega)}:=\\left(\\sum_{T\\in\\mathcal T_h}\\text{vol}(T)|\\nabla I(u_h)(\\mathbf x_T)-\\nabla u(\\mathbf x_T)|^2\\right)^{\\frac12},\n\\end{split}\n\\]\nwhere $I(u_h)$ is the $P_1$ least-square interpolant to the values of $u_h$ in barycenters of the cells from $\\omega(\\mathbf x_T)\\cap\\Omega_i$, for $T\\in\\Omega_i$. Over the surface, we set\n\\[\n\\mbox{err}_{L^\\infty(\\Gamma)}:=\\max_{\\Gamma}|v_h-v^e|,\\quad \\mbox{err}_{L^2(\\Gamma)}:=\\|v_h-v^e\\|_{L^2(\\Gamma)},\\quad\n\\mbox{err}_{H^1(\\Gamma)}:=\\|\\nabla_{\\Gamma_h}v_h-\\nabla_{\\Gamma_h}v\\|_{L^2(\\Gamma)},\n\\]\nwhere $v^e$ is the  extension of $v$ from $\\Gamma$ to $\\Gamma_h$ along normal directions to $\\Gamma$.\n}\n\n\n\n\n\\subsection{Steady analytical solution for a  triple fracture problem}\\label{sec_ex2}\n\nOur next experiment deals with the coupled surface--bulk diffusion problem in the domain $\\Omega = [0,1]^3$ with\nan embedded piecewise planar $\\Gamma$. We design $\\Gamma$ to model a branching fracture.  In the basic model,\n$\\Gamma=\\Gamma(0)$ consists of three planar pieces,\n\\[\\Gamma(0) = \\Gamma_{12}\\cup\\Gamma_{13}\\cup\\Gamma_{23},\\quad \\Gamma_{ij}=\\overline{\\Omega_i}\\cap\\overline{\\Omega_j}~~i\\neq j,\n\\]\nsuch that\n\\[\n\\Omega_{1}=\\{\\mathbf x\\in\\Omega\\,|\\, x<\\frac12~\\text{and}~ y>x\\},\\quad \\Omega_{2}=\\{\\mathbf x\\in\\Omega\\,|\\, x>\\frac12~\\text{and}~y>x-1\\},\\quad\n\\Omega_{3}=\\Omega\\setminus(\\overline{\\Omega}_{1}\\cup\\overline{\\Omega}_2).\n\\]\nThis subdivision is illustrated in Figure~\\ref{fig:triple} (left). The pieces $\\Gamma_{ij}$ belong to certain planes of symmetry\nfor the cube, and so the induced triangulation of $\\Gamma(0)$ and the cut cells in the bulk domain are all quite regular.\nTo model a generic situation when $\\Gamma$ cuts through the background mesh in an arbitrary way, we  consider other\ntessellations of $\\Omega = [0,1]^3$ into three subdomains by a surface $\\Gamma(\\alpha)$. The surface $\\Gamma(\\alpha)$ is obtained from $\\Gamma(0)$ by applying the clockwise rotation by the angle $\\alpha$ around the axis $x=z=0.5$. We take $\\alpha=20^o$ and\n$\\alpha=40^o$, the resulting tessellations of $\\Omega$ are illustrated in  Figure~\\ref{fig:triple} (middle and right pictures).\nMore precisely, we define\n\\[\n\\Gamma(\\alpha)=\\{\\mathbf x\\in\\Omega\\,|\\, \\mathbf y\\in\\Gamma(0),~\\mathbf y-\\mathbf x_0=\\mathcal{Q}_\\alpha(\\mathbf x-\\mathbf x_0)\\},~~\\text{with}~\n\\mathcal{Q}_\\alpha=\\begin{bmatrix}\\cos\\alpha&0&-\\sin\\alpha\\\\ 0&1&0\\\\ \\sin\\alpha&0&\\cos\\alpha\\end{bmatrix},~ \\mathbf x_0=({\\footnotesize\\frac12},0,{\\footnotesize\\frac12})^T.\n\\]\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.31\\textwidth]{.\/mesh3d_triple_grey3.jpg} \\includegraphics[width=0.36\\textwidth]{.\/explodeTrip_grey.jpg} \\includegraphics[width=0.31\\textwidth]{.\/meshRot40_explode_grey.jpg}\n\\caption{\\label{fig:triple}\nThe figure illustrates the bulk domain with uniform mesh and the fracture. On the left picture the fracture is set orthogonal to the $xy$-plane, while on the middle and right pictures the fracture is rotated by 20 and 40 degrees.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.32\\textwidth]{.\/mesh2d_grey.jpg}  \\includegraphics[width=0.32\\textwidth]{.\/mesh2d_rot20_2.jpg}\n\\includegraphics[width=0.32\\textwidth]{.\/mesh2d_rot40_grey2.jpg}\n\\caption{\\label{fig:frac}\nThe figure illustrates the induced surface mesh on the fracture, when it cuts through the uniform bulk mesh in different ways.}\n\\end{center}\n\\end{figure}\n\n\nSimilar to the series of numerical experiments with the embedded spherical $\\Gamma$, here we set   the source terms $f_i$ and $f_\\Gamma$ and the boundary conditions such that the solution to the stationary problem \\eqref{diffeq1}--\\eqref{bc} is known.  To define the  solution $\\{v,u\\}$ solving the stationary equations \\eqref{diffeq1}--\\eqref{bc}, we first introduce\n\\begin{equation}\\nonumber\n\\psi_1 = \\left\\{\n\\begin{array}{lr}\n 16(y-\\frac12)^4,& y>\\frac12\\\\\n 0, & y \\le \\frac12\\\\\n\\end{array}\n\\right.,\\quad\\psi_2 = x -y,\\quad \\psi_3 = x + y - 1.\n\\end{equation}\nWe define the solution of the basic model problem ($\\alpha=0$)\n\\begin{equation}\\nonumber\nu(\\mathbf x)=\\left\\{\n \\begin{array}{rl r}\n  & \\sin(2\\pi z)\\cdot\\psi_2(\\mathbf x) \\cdot\\phi_3(\\mathbf x) & \\mathbf x\\in \\Omega_1,\\\\\n  & \\sin(2\\pi z)\\cdot\\psi_1(\\mathbf x) & \\mathbf x\\in \\Omega_2,\\\\\n  & \\sin(2\\pi z)2x\\cdot\\psi_1(\\mathbf x) & \\mathbf x\\in \\Omega_3,\n \\end{array}\n \\right.\\qquad v=u|_{\\Gamma(0)}.\n\\end{equation}\nNote that the constructed exact solution is continuous across $\\Gamma(0)$, but the normal derivatives are discontinuous.\nOther parameters in \\eqref{diffeq1}--\\eqref{diffeq2} are set to be  $\\mathbf{w}=\\mathbf{w}_\\Gamma=0$, $\\phi_1=\\phi_2=\\phi_\\Gamma=0$, $D_1=D_2=D_\\Gamma=I$, and $d=1$. For the problem setup with the rotated fracture, $\\alpha>0$ we set the exact solution\n$v_\\alpha(\\mathbf x)=v(\\mathbf y)$, $u_\\alpha(\\mathbf x)=u(\\mathbf y) $, with $\\mathbf y=\\mathcal{Q}_\\alpha(\\mathbf x-({\\small\\frac12},0,{\\small\\frac12})^T)$.\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{The error in the numerical solution for the steady problem with  triple fracture, $\\alpha = 0$.\n\\label{tab:triple0}}\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&855&\t6.374e-3     &        &4.214e-1&          & 3.920e-2&      \\\\\n&7410&\t 1.698e-3    & 1.84   &1.631e-1&1.36      & 1.276e-2& 1.56 \\\\\n&61620&\t4.235e-4     & 1.97   &6.193e-2&1.39      & 3.506e-3& 1.83 \\\\\n&502440& 1.044e-4    & 2.00   &2.348e-2&1.40      & 1.129e-3& 1.62 \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Gamma$}\n&232& 8.469e-3       &         &2.914e-1 &          &9.280e-3&     \\\\\n&1242&2.003e-3\t       & 1.79  &1.387e-1 & 0.92     &2.779e-3& 1.44\\\\\n&5662&5.588e-4\t       & 1.84  &6.874e-2 & 1.01     &1.217e-3& 1.09\\\\\n&24102&1.791e-4       &  1.64  &3.395e-2 & 1.02     &5.181e-4& 1.18\\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{The error in the numerical solution for the steady problem with  triple fracture, $\\alpha = 20$.\n\\label{tab:triple20}}\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&965&6.319e-3&               &4.208e-1&        &3.754e-2  &    \\\\\n&7872&1.805e-3      & 1.79   &1.661e-1& 1.34   &1.280e-2  &1.55\\\\\n&63592& 5.623e-4    & 1.80   &6.371e-2& 1.38   & 3.411e-3 &1.90\\\\\n&510390&1.602e-4    & 1.81   &2.442e-2& 1.39   & 1.146e-3 &1.57\\\\\\hline\\\\[-2ex]\n\\rev{$\\Gamma$}\n&321& 7.792e-3    &       &2.694e-1&        &2.716e-2&     \\\\\n&1692&2.084e-3    & 1.59  &1.240e-1& 1.12   &5.400e-3& 1.94\\\\\n&7944&7.019e-4    & 1.41  &6.291e-2& 0.98   &2.001e-3& 1.29\\\\\n&{\\color{black}33272}&{\\color{black}2.441e-4}   & {\\color{black}1.52}  &{\\color{black}3.173e-2}& {\\color{black}0.99}   &{\\color{black}7.217e-4}& {\\color{black}1.47}\\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t]\n\\begin{center}\n\\caption{The error in the numerical solution for the steady problem with  triple fracture, $\\alpha = 40$.\n\\label{tab:triple40}}\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&991&\t 5.934e-3   &       & 4.080e-1&      &  3.783e-2 &     \\\\\n&7996&\t 1.700e-3   & 1.80  & 1.621e-1& 1.33 &  1.276e-2 & 1.56\\\\\n&64046&  4.907e-4   & 1.80  & 6.263e-2& 1.37 &  3.515e-3 & 1.86\\\\\n&512258& 1.503e-4   & 1.82  & 2.541e-2& 1.39 &  1.237e-3 & 1.61\\\\  \\hline\\\\[-2ex]\n\\rev{$\\Gamma$}\n&353&     8.167e-3 &       &2.709e-1&     & 2.696e-2 &      \\\\\n&1932&\t  2.146e-3 & 1.66  &1.275e-1& 1.09& 5.566e-3 & 1.85 \\\\\n&8766&    7.115e-4 & 1.59  &6.279e-2& 1.02& 2.063e-3 & 1.31 \\\\\n&{\\color{black}36676}&   {\\color{black}2.538e-4} & {\\color{black}1.49}  &{\\color{black}3.121e-2}& {\\color{black}1.01}& {\\color{black}7.251e-4} & {\\color{black}1.51} \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\label{s_ex2}\n\nThe numerical results for this coupled problem with the triple fracture problem are reported in Tables~\\ref{tab:triple0}--\\ref{tab:triple40}.  We observe stable convergent results for $\\alpha=0$ as well as for more general case of $\\alpha>0$.   An interesting feature of this problem is that the surface $\\Gamma$ is\nonly piecewise smooth.  The bulk grid is not fitted to the internal edge $\\mathcal{E}=\\Gamma_{12}\\cap\\Gamma_{13}\\cap\\Gamma_{23}$,\nand hence the tangential derivatives of $v$ are discontinuous inside certain cubic cells from $\\mathcal T_h^\\Gamma$.\n\\rev{Therefore, a kink in $v$ cannot be represented by the finite element approximation.} This \\rev{may} result in  \\rev{a}  reduction of convergence order. \\rev{Both the performance of the FV method for cut  cells (cut cells inherit a regular structure from the background mesh for $\\alpha=0$, but are very irregular for $\\alpha>0$ ) and the presence of the kink influences the observed convergence rates. }\n\n\\begin{table}[h!]   \\rev{\n\\begin{center}\n\\caption{Iteration numbers in \\eqref{step1}--\\eqref{step2} for the steady problem example in section~\\ref{sec_ex2}.\n\\label{t_conv}}\\smallskip\n\\small\n\\begin{tabular}{r|lll}\\hline\nref. level. & $\\alpha=0$ & $\\alpha=20^o$ & $\\alpha=40^o$ \\\\ \\hline\\\\[-2ex]\n0&\t22   & 74  &24    \\\\\n1&\t29   & 90  &32    \\\\\n2&\t212  & 325 &228    \\\\\n3& 782   & 917 &851   \\\\ \\hline\n\\end{tabular}\n\\end{center}    }\n\\end{table}\n\n\\rev{\nFinally, Table~\\ref{t_conv} shows the performance of the fixed-point iteration~\\eqref{step1}--\\eqref{step2}. We set $\\omega=1$ and take $u^0=0$, $v^0=0$. The stopping criterion was the relative\ndecrease of the Euclidian norm of  both surface and bulk equations residuals by a factor of $10^4$ (a stronger convergence criterion was not found to improve solution accuracy). On each outer iteration, the surface linear subproblem was solved by exact factorization, while a few Picard iterations with exact factorization of linearized problem were done to solve the bulk system in \\eqref{step1}.   The solver does not scale in an optimal way with respect to the mesh size and more research is needed to improve its performance, cf. Remark~\\ref{rem1}. We postpone this topic for the future research. We also note that for time dependent problems studied  below including time-dependent terms and taking initial guess to be the solution from the previous time step improves convergence of \\eqref{step1}--\\eqref{step2} a lot,\nand we typically need 1 or 2 iterations on each time step.}\n\n\n\\subsection{Propagating front in the porous medium with triple fracture}\n\\definecolor{linkcolor}{HTML}{799B03}\n\\definecolor{urlcolor}{HTML}{799B03}\n\n\\hypersetup{pdfstartview=FitH, linkcolor=linkcolor,urlcolor=urlcolor, colorlinks=true}\n\n\\begin{figure}[h!]\n\\begin{center}\na)~~\\includegraphics[width=0.4\\textwidth]{.\/diff3d_0.png}\\qquad b)~~\\includegraphics[width=0.4\\textwidth]{.\/diff3d.png}\\\\\nc)~~\\href{www.math.uh.edu\/~molshan\/OLSH\/pdf_refs\/pdf_ref_video_outx4.html}{\\includegraphics[width=0.4\\textwidth]{.\/conv3d_0.png}}\\qquad d)~~\\includegraphics[width=0.4\\textwidth]{.\/conv3d.png}\\\\\n\\caption{\\label{fig:prop}\nThe figures illustrates the propagating front of the concentration in the fracture and in the bulk: Pictures a), b) show dominating diffusion case, while  c), d) show dominating convection case. Pictures a) and c) \\rev{show} snapshots \\rev{of} the computed solution at time t=0.018, while pictures b) and d) snapshots the computed  solution at time t=\\rev{0.033}. Click on picture~c) to run full animation of the experiment.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\na)~~\\href{www.math.uh.edu\/~molshan\/OLSH\/pdf_refs\/pdf_ref_video_out_new_higher.html}{\\includegraphics[width=0.4\\textwidth]{.\/diff3drot.png}}\\qquad b)~~\\includegraphics[width=0.4\\textwidth]{.\/conv3drot.png}\n\\caption{\\label{fig:prop1}\nThe figures illustrates the propagating front of the concentration in the fracture and in the bulk, with $\\alpha=20^o$: Picture a) shows dominating diffusion case, picture b) shows dominating convection case;  both at time t=0.033. Click on picture~a) to run full animation of the experiment.}\n\\end{center}\n\\end{figure}\n\nIn the last series of experiments we compute the time dependent solution of \\eqref{diffeq1}--\\eqref{bc}. The bulk domain $\\Omega$\nand the fracture $\\Gamma$ are the same as in the previous experiment in section~\\ref{s_ex2}.\nAt  time\n$t_0=0$ we set $u(t_0)=0$ in $\\Omega$ and  $v(t_0)=0$ on $\\Gamma$. On the face $\\{y=1\\}$ of the cube  we prescribe the constant concentration of a contaminant, while on other parts on the boundary the  diffusion flux is  set equal zero. Thus in \\eqref{bc}, we have\n\\[\n\\begin{split}\n\\partial\\Omega_D&=\\partial\\Omega\\cap\\{y=1\\},\\quad\\partial\\Omega_N=\\partial\\Omega\\setminus\\partial\\Omega_D,\\quad \\partial\\Gamma_D=\\partial\\Gamma\\cap\\{y=1\\},\\quad \\partial\\Gamma_N=\\partial\\Gamma\\setminus\\partial\\Omega_D,\\\\  u_D&=1,\\quad v_D=1,\\quad u_0=0,\\quad \\text{and}\\quad v_0=0.\n\\end{split}\n\\]\nThe time independent velocity field transports the contaminant in the bulk and along the fractures.\nWe set\n\\[\n\\begin{split}\n\\mathbf w_i&=2\\kappa(0,-1,0)^T,~~i=1,2,3,\\quad\\text{in}~\\Omega\\\\\n\\mathbf w_\\Gamma&=5\\kappa(0,-1,0)^T~~\\quad\\text{in}~\\Gamma_{12},\\quad\n\\mathbf w_\\Gamma=\\kappa\\mathcal{Q}_\\alpha({\\footnotesize\\frac{1}{\\sqrt{2}}},-{\\footnotesize\\frac{1}{\\sqrt{2}}},0)^T\n~~\\quad\\text{in}~\\Gamma_{23},\\quad\n\\mathbf w_\\Gamma=\\kappa\\mathcal{Q}_\\alpha(-{\\footnotesize\\frac{1}{\\sqrt{2}}},-{\\footnotesize\\frac{1}{\\sqrt{2}}},0)^T~~\\quad\\text{in}~\\Gamma_{13},\n\\end{split}\n\\]\nwhere $\\kappa\\ge0$ is a parameter. One easily verifies the condition \\eqref{cont_w} on the edge $\\mathcal{E}=\\Gamma_{12}\\cap\\Gamma_{13}\\cap\\Gamma_{23}$. Other parameters are set to be\n\\[\nD_1=D_2=0.1\\,I,\\quad d=1,\\quad D_\\Gamma=I, \\quad \\phi_1=\\phi_2=\\phi_\\Gamma=1.\n\\]\n\n\n\n\nThe computed solutions for $\\kappa=1\/8$ (diffusion dominated case) and $\\kappa=8$ (convection plays a significant role)  are illustrated in Figures~\\ref{fig:prop}--\\ref{fig:prop1}. The fracture angle parameter $\\alpha$ was set to $0$ and $20$ degrees, \\rev{respectively}. \n\n\nFor this problem, the exact solution is not known. The computed solution occurs to be physically reasonable. We see no sign of spurious oscillations. \\rev{As expected, the contaminant propagates faster along the fractures.}\n\n\\subsection{Contaminant transport along the fracture.}\n\nIn this test we place  a continuous contaminant source on the upstream boundary of the fracture.\n\\begin{wrapfigure}{r}{0.45\\textwidth}\n\\vspace{-20pt}\n  \\begin{center}\n    \\includegraphics[width=0.4\\textwidth]{.\/mesh16rot.jpg\n    \\caption{The matrix and fracture in the test with contaminant transport along the fracture.}\n    \\label{fig:Front2}\n  \\end{center}\n  \\vspace{-20pt}\n  \\vspace{1pt}\n\\end{wrapfigure}\nThe matrix--fracture configuration for this test is shown in Figure~\\ref{fig:Front2}, $\\Omega=(0,1)^3$, and $\\Gamma=\\{\\mathbf x\\in\\Omega\\,:\\,z+\\frac12x=0.51\\}$. The boundary $x=0$\nis inflow, in the fracture the wind is constant $\\mathbf w_\\Gamma=(w_1,0,w_3)$, $|\\mathbf w_\\Gamma|=1$, and the contaminant source occupies  the part of $\\partial\\Gamma$, \\rev{${\\partial\\Gamma} _D=\\{(0,y,0.51)\\,:\\,y\\in(\\frac14,\\frac34)\\}$, $v_D=1$ on ${\\partial\\Gamma} _D$.}\nWe assume that the  porous matrix  is almost impermeable and so we set $\\mathbf w_i=0$ in $\\Omega_i$ (no flow in the rock) and $D_i=10^{-6}I$, $i=1,2$, \\rev{${\\partial\\Omega} ={\\partial\\Omega} _N$}. In the fracture we assume isotropic diffusion with $D_\\Gamma=10^{-4}I$. Other parameters are the same as in the previous test, \\rev{and $v=0$, $u=0$ at $t=0$}. Therefore, we expect that the contaminant transport happens along the fracture with very small diffusion to the porous matrix. This is a bulk--surface variant of a standard test case of numerical solvers for convection--diffusion problems~\\cite{sun2013mathematical},  and one is typically interested in the ability of a method to capture the right position and the shape of the sharp propagating front and avoid spurious oscillations. For a comparison purpose, one may consider the exact solution for the problems posed\nin a half-plane (or half-space) from~\\cite{leij1990analytical,leij1991analytical}. This solution $C(x,y,0)$ is given in~\\eqref{kapanalsol1}, it solves $C_t-D\\Delta C+{C}_{x}=0$ in  $\\widetilde \\Omega = \\{\\rev{(x,y)}\\in\\mathbb{R}^2\\,:\\, x>0\\}$, with the\nboundary condition\n$\nC(0,y,t)=\\left\\{\n\\begin{array}{c}\nc_0, \\textrm{  when } |y|<a,\\\\\n0,  \\textrm{  when } |y|>a,\n\\end{array}\n\\right.\n\nand  initial conditions: $C(x,y,0)= 0$ in $\\widetilde\\Omega$.\n\\begin{equation}\n C(x,y,t)=\\frac{x c_0}{(16 \\pi D)^{\\frac 12}} \\int \\limits_0^t  \\tau^{-\\frac 32}\n\\left\\{\n\\mathbf{erf} \\left[ \\frac{a+y}{(4 D \\tau)^\\frac 12} \\right]\n+\\mathbf{erf} \\left[ \\frac{a-y}{(4 D \\tau)^\\frac 12} \\right]\n\\right\\} \\cdot \\mathbf{exp} \\left[ - \\left( \\frac{x-\\tau}{(4D \\tau)^{\\frac 12}} \\right) ^2 \\right] d\\tau.%\n\\label{kapanalsol1}\n\\end{equation}\nwhere\n\\begin{equation*\n\\mathbf{erf}(x) = \\frac{2}{\\sqrt{\\pi}} \\int \\limits_0^x e^{-t^2} dt, \\quad \\mathbf{erfc}(x ) = 1-\\mathbf{erf}(x)= \\frac{2}{\\sqrt{\\pi}} \\int \\limits_x^\\infty e^{-t^2} dt.\n\\end{equation*}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/esol_017_full.jpg}  \\\\\n\\includegraphics[width=0.8\\textwidth]{.\/esol_34_full.jpg}  \\\\\n\\includegraphics[width=0.8\\textwidth]{.\/32sol_full.jpg}\n\\caption{\\label{fig:rot20_3d}\nReference 2D solution  (left) and the fracture component of the computed  solution (right) for the contaminant transport along the fracture test\ncase. The solutions is shown for the times $t=0.17,\\,0.34,\\,0.5$.}\n\\end{center}\n\\end{figure}\n\nWe run our simulations with the uniform background mesh,  $h=\\frac1{32}$, $\\Delta t=10^{-2}$. The fracture cuts through the background mesh as illustrated in  Figure~\\ref{fig:Front2} (for better visualization, this figure shows the background mesh for $h=\\frac1{16}$). The computed solution and the `reference' solution is shown in  Figure~\\ref{fig:rot20_3d} at several time instances. We recall that the coupled problem was solved and the contaminant also diffuses into the bulk, but this bulk diffusion was minor. We observe that the\ncomputed solution well approximates the reference one; the computed front has the correct position and \\rev{is not  smeared too much. Moreover,} we do not observe  overshoots or undershoots in $v_h$.\n\n{\\color{black}\n\\subsection{An example with a spherical drop immersed in a bulk}\nWe include one more test case but now with a different interface condition. This is\nthe instantaneous absorbtion--desorption condition \\eqref{eq4a} with the Henry law to define $g_i$ and $f_i$.\nThis condition is common in the literature to model dissolvable surfactant transport in two-phase flows.\nIn this test from~\\cite{gross2015trace} we consider a prototypical configuration for such models consisting of a spherical drop embedded in a cubic domain. We take $\\Gamma$ to be the unit sphere centered at the origin and $\\Omega = [-1.2,1.2]^3$.\nBy $\\Omega_1$ we denote the interior of $\\Gamma$, so $\\Omega_1$ is the unit ball,  \\rev{$\\Omega_2=\\Omega\\setminus\\overline{\\Omega}_1$}. For the velocity field we take a rotating field in the $x$-$z$ plane: $\\mathbf w=\\frac{1}{10}(z,0,-x)$. This $\\mathbf w$ satisfies   $\\mbox{\\rm div}\\mathbf w=0$ in $\\Omega$ and $\\mathbf w \\cdot \\mathbf n=0$ on $\\Gamma$, i.e. the velocity field is everywhere tangential to the boundary and hence the steady interface is consistent with the kinematic condition: $\\mathbf w \\cdot \\mathbf n$ is equal to the normal velocity of $\\Gamma$ for immersible two-pase fluids, e.g.~\\cite{GrossReuskenBook}. We set $\\mathbf w_i=\\mathbf w|_{\\Omega_i}$ and $\\mathbf w_\\Gamma=\\mathbf w|_{\\Omega_\\Gamma}$.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{.\/new_sphere_solmesh.jpg}\\qquad  \\includegraphics[width=0.4\\textwidth]{.\/new_sphere_3Dsolmeshcut.jpg}\n\\caption{\\label{fig:sphere_loc}\nLeft: Induced surface mesh and the surface component of computed solution.\nRight: Cut of the bulk mesh and the volume component of computed solution.}\n\\end{center}\n\\end{figure}\n\nThe material parameters are chosen as $D_1=0.5$, $D_2=1$, $D_\\Gamma=1$ and $k_{1,a}=0.5$, $k_{2,a}=2$, $k_{1,d}=2$, $k_{2,d}=1$, $d=1$.\nThe source terms $f_i\\in L^2(\\Omega)$, $i=1,2$, and $f_\\Gamma\\in L^2(\\Gamma)$   and  data on ${\\partial\\Omega} $ are taken such  that the exact solution of the  \\textit{stationary} equations \\eqref{diffeq1}--\\eqref{diffeq2} is given by\n\\begin{equation} \\label{exsol}\n \n  v(x,y,z) = 3x^2y - y^3, \\quad\n  u_1(x,y,z) = 2 u_2(x,y,z), \\quad\n  u_2(x,y,z) = e^{1-x^2-y^2-z^2} v(x,y,z).\n\\end{equation}\nSince we solve for a steady-state solution, so we  set $\\phi_1=\\phi_2=\\phi_\\Gamma=0$.\n We prescribe Dirichlet boundary conditions on $\\partial\\Omega$, i.e. $\\partial\\Omega_N=\\emptyset$,   $\\partial\\Gamma_N=\\emptyset$, and $\\partial\\Gamma_D=\\emptyset$ in \\eqref{bc}. Conditions \\eqref{cont_w}--\\eqref{cont_F} for this test case are not relevant, since the surface is globally smooth and has no boundary.\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Convergence of numerical solutions in the experiment with a spherical $\\Gamma$ embedded in a cube.\n\\label{tab:sphere_uni}  }\\smallskip\n\\small\n\\begin{tabular}{rr|llllll}\\hline\n&\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}\n&736&\t 3.223e-02 &        & 9.706e-01  &        &  1.072e-01  &      \\\\\n&4920&\t 6.687e-03 &    2.27& 1.555e-01  &    2.64&  3.799e-02  &1.50\\\\\n&36088&  2.005e-03 &    1.74& 5.363e-02  &    1.55&  9.180e-01  &-4.59 \\\\\n&275544& 5.055e-04 &    1.99& 1.825e-02  &    1.56&  2.777e-03  &8.37 \\\\  \\hline\\\\[-2ex]\n\n\\rev{$\\Gamma$}\n&460&      1.670e-02   &    &  2.065e-01&     & 3.863e-02&     \\\\\n&1660&     4.037e-03   &2.05&  9.647e-02& 1.10& 1.060e-02&1.87 \\\\\n&6628&     9.211e-04   &2.13&  4.745e-02& 1.02& 3.881e-03&1.45  \\\\\n&26740&    2.457e-04   &1.91&  2.396e-02& 0.99& 8.875e-04&2.13  \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn this set of experiments we take the sequence of uniform cubic meshes in $\\Omega$, starting with $h=0.3$. The surface $\\Gamma_h$ is reconstructed as described in section~\\ref{s_rec} for $\\phi(\\mathbf x)=1-|\\mathbf x|^2$. The computed solution as well as volume and induced surface meshes are illustrated in Figure~\\ref{fig:sphere_loc}.\nThe computed errors for the bulk and surface concentrations are shown in Table~\\ref{tab:sphere_uni}. For this example, the method\ndemonstrates  optimal convergence: $O(h)$ in the $H^1$ and $O(h^2)$ in the $L^2$ and surface norms. This is consistent with what is known about the convergence of the TraceFEM for linear bulk elements, see e.g. analysis and convergence rates for the same\nexperiment in \\cite{gross2015trace}, where the TraceFEM has been used to discretized equations both on the surface and in the bulk.  For the volume component of the solution, the convergence is close to the second order in the $L^2$ norm and $1.5$ order in the $H^1$ norm.  It is also consistent with the results in \\cite{Lipnikov:12}, where  super-convergence of the method in the $H^1$ norm was observed.  Convergence in the $L^\\infty$-norm is somewhat less regular. We note that the $L^\\infty$ convergence of the TraceFEM and of the non-linear FV method that we used has not been studied before.\n\\medskip\n\nThe aim of the next (final) test is to illustrate the performance of the method for the case of locally refined grids.  The setup is similar to the test with the sphere above, but the coefficients and the known solution are taken different to represent the situation of convection dominated problem with an internal layer. More precisely,  for the velocity field we take  $\\mathbf w=(-y\\sqrt{1-z^2},x\\sqrt{1-z^2},0)$, and set $\\mathbf w_i=\\mathbf w|_{\\Omega_i}$ and $\\mathbf w_\\Gamma=\\mathbf w|_{\\Omega_\\Gamma}$.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{.\/new_sphere_solmesh_cd.jpg}\\qquad  \\includegraphics[width=0.45\\textwidth]{.\/new_sphere_3D_cd.jpg}\n\\caption{\\label{fig:sphere_adap}\nLeft: Induced surface mesh and the surface component of computed solution.\nRight: The bulk mesh and the volume component of computed solution.}\n\\end{center}\n\\end{figure}\n\nThe material parameters are chosen as $D_1=D_2=D_\\Gamma=\\eps$ and $k_{1,a}=0.5$, $k_{2,a}=2$, $k_{1,d}=2$, $k_{2,d}=1$, $d=1$.\nThe source terms $f_i\\in L^2(\\Omega)$, $i=1,2$, and $f_\\Gamma\\in L^2(\\Gamma)$   and  data on ${\\partial\\Omega} $ are taken such  that the exact solution of the  \\textit{stationary} equations \\eqref{diffeq1}--\\eqref{diffeq2} is given by\n\\begin{equation} \\label{exsol}\n \n  v(x,y,z) = {x z}\\mathrm{arctan}\\left(\\frac{2z}{\\sqrt{\\varepsilon}}\\right), \\quad\n  u_1(x,y,z) = 2 u_2(x,y,z), \\quad\n  u_2(x,y,z) = e^{1-x^2-y^2-z^2} v(x,y,z).\n\\end{equation}\nWe take $\\eps=1$ (very smooth solution) and $\\eps=0.01$ (solution has an internal layer along the midplane $z=0$).\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Convergence of numerical solutions in the experiment with a spherical $\\Gamma$ and locally refined mesh.\n\\label{tab:sphere_adap}  }\\smallskip\n\\small\n\\begin{tabular}{r|llllll|llllll}\\hline\n&\\multicolumn{6}{|c|}{$\\varepsilon=1$}&\\multicolumn{6}{|c}{$\\varepsilon=1e-2$}\\\\\n\\#d.o.f. & $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate& $L^2$-norm & rate & $H^1$-norm& rate & $L^\\infty$-norm& rate \\\\ \\hline\\\\[-2ex]\n\\rev{$\\Omega$}&&&&&&&&&&&&\\\\\n5840  &   3.993e-03&       &  8.322e-02&        &  1.706e-02&      &   5.666e-02&       &  3.987e-01&        &  1.945e-01&     \\\\\n43552 &   6.706e-04&2.57   &  2.828e-02&1.56    &  4.926e-03& 1.79 &   2.609e-02&1.12   &  2.440e-01&0.71    &  8.403e-02&1.21 \\\\\n318696&   2.200e-04&1.61   &  1.038e-02&1.45    &  2.158e-02&-2.13 &   1.353e-02&0.95   &  1.708e-01&0.52    &  5.003e-02&0.75 \\\\  \\hline\\\\[-2ex]\n\\rev{$\\Gamma$} &&&&&&&&&&&&\\\\\n1500 &       1.916e-03&    &   3.609e-02&     & 6.850e-03&    &    8.353e-03&    &  4.026e-01&     & 4.538e-02&     \\\\\n6740 &       5.106e-04&1.91&   1.710e-02&1.08 & 1.919e-03&1.84&    1.854e-03&2.17&  1.619e-01&1.31 & 1.335e-02&1.76 \\\\\n25988&       1.400e-04&1.87&   8.924e-03&0.94 & 5.624e-04&1.77&    3.848e-04&2.26&  6.532e-02&1.31 & 3.694e-03&1.85 \\\\  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nWe build a sequence of locally refined meshes as illustrated in~Figure~\\ref{fig:sphere_adap}.\nFor $\\eps=0.01$ the meshes fit the layer and intend  to capture the sharp variation of the solution.\nWe computed numerical solutions on a sequence of 3 meshes, the second mesh is illustrated in ~Figure~\\ref{fig:sphere_adap}. Each mesh has two levels of refinement in the region $|z|<\\frac18$.\nThe convergence of the method is reported in Table~\\ref{tab:sphere_adap}. The optimal order of convergence is attended for the surface component of the solution, but the FV method in the bulk domain shows lower order convergence for the convection dominated case. We conclude that more studies are required to improve the performance of the FV method on such type of meshes.\n\n\\section{Conclusions} \\label{s_concl} The paper proposed a hybrid finite volume -- finite element method for the coupled bulk--surface systems of PDEs. The distinct feature of the method is that the same background mesh is used to solve equations in the bulk and on the surfaces, and that there is no need to fit this mesh to the embedded surfaces. This makes the approach particularly attractive to treat problems with complicated embedded structures of lower dimension like those occurring in the simulations of flow and transport in fractured porous media. We consider the particular monotone non-linear FV method with compact stencil, but we believe that the approach can be carried over and used with other FV methods on polyhedral meshes (e.g. some of those reviewed in \\cite{Droniou:14}) with possibly better performance in terms of convergence rates. In this paper we treated only diffusion and transport of a contaminant assuming that Darcy velocity is given. Extending the method to computing flows in fractured porous media is in our future plans together with the design of better algebraic solvers, doing research on adaptivity, and adding to the method a fracture prorogation model.\n\n}\n\n\\subsection*{Acknowledgments}\nThe work of the first author (the numerical implementation and the numerical experiments) has been supported  by the Russian Scientific Foundation Grant 17-71-10173, the work of the second author has been supported by the NSF grant 1717516, the work of the third author has been supported by the RFBR grant 17-01-00886.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec1}\n\nWave propagation in heterogeneous complex media has been a subject of intensive research interest in recent years. \nAmong various systems, a large part of the conducted studies has been concentrated in families of one-dimensional ($1$D) continuous and discrete models~\\cite{dyson,disorder_14,izrailev,disorder_15}\nby focusing mainly on the localization properties of both the normal modes of finite systems i.e.,~Anderson localization (AL)~\\cite{anderson}, as well as wave propagation in infinite media.\nThe successful extension of AL to many other systems after it was initially formulated for electronic systems, has opened many research frontiers and applications \\cite{bruinsma,schwartz,lahini,billy,roati}. Experimental results on AL (see e.g.,~Refs.~\\cite{schwartz,lahini,billy,roati,jk}) have stimulated further interest in AL for both quantum and classical systems. \n \nRegarding linear disordered $1$D lattices, among different systems, special attention has been given to the tight binding electron model \\cite{harrison},\nthe linear Klein-Gordon (KG) lattice~\\cite{rogers} and the harmonic lattice \\cite{dyson,Ishii,Ishii_2,Kundu}. These models are not only relevant to various physical systems but also represent the linear limits of seminal nonlinear lattices such as the discrete nonlinear Schr\\\"{o}dinger equation (DNLS), the quartic KG, and the Fermi-Pasta-Ulam-Tsingou (FPUT) lattices~\\cite{fpu,kevre,izrailev,ggg}. Within the context of phononic and photonic lattices, these fundamental models have been adopted to describe a variety of physical systems and more recently, they have been used as a testbed for novel wave phenomena~\\cite{deymer2,flachbook}.\n\nA common route to study the wave properties of disordered lattices is through monitoring the time evolution of initially compact wave-packets. For tight binding and linear KG models, the dynamics after the excitation of such an initial condition is characterized by a transient initial phase of spreading, followed by a phase of total confinement to the system's localization length. The width of the wave-packet is of the order of the maximum localization length~\\cite{loc_len}. On the other hand, for the harmonic lattice, along with the localized portion of the energy, there is always a propagating part due to the existence of extended modes at low frequencies. A quantitative description of wave propagation in disordered $1$D systems of one degree of freedom (DoF) per lattice site was formulated in Refs.~\\cite{Ishii, Kundu, Lepri} where wave-packet spreading was quantified using both analytical and numerical methods. Moreover, many variations of these $1$D lattices have been studied extensively in several works including dynamical regimes ranging from the homogeneous linear to the disordered nonlinear~\\cite{Chiaradis,Guebelle,MasonPanos,vassospre2016,Luding2,Sen2017poly,jk,arn_gran,many2}. \n\nAs a natural extension to the above studies, an investigation into the corresponding behavior in disordered lattices with more than one DoF per site seems plausible. Not many studies along such lines have been reported in the literature. The majority of existing works have taken the approach of making generalizations of the tight binding model by assuming a linear coupling between two (or more) $1$D chains~\\cite{ladderPRB,flachlieb}. Such coupling results in changes to the dispersion relation thereby changing the energy transport properties. Our recent work with a linear disordered phononic lattice \\cite{arn_phononic} is indeed an attempt to fill this gap. The wave dynamics of disordered harmonic chains with two DoFs per site appear to be quite interesting, deserving further investigations. Such systems have been useful in modeling macroscopic mechanical devices including granular chains, highly deformable elastic assemblies and origami lattices~\\cite{pichard,florian1,yasuda,yasuda2,deng2,florian2}. This allows for easy  tunability of the system's dispersion due to the geometrical characteristics and material properties, and makes these systems attractive for several applications.\n \nHere we focus on highly deformable architected lattices, characterized by a nonlinear response which enabled the design of new classes of tunable and responsive elastic materials. Several such soft structures have already been reported including bioinspired  soft robots \\cite{shepherd, yang}, self-regulating microfluidics \\cite{Beebe}, reusable energy absorbing systems \\cite{shan,restrepo}, materials with programmable response \\cite{florijn} and information processing via physical soft\nbodies \\cite{nakajima}. Furthermore, soft architected materials present opportunities to control the propagation of elastic waves, since their dispersion properties can be altered by applying a large, nonlinear pre-deformation \\cite{boyce1,boyce2,casadei} or changing the geometry \\cite{deng2}. To date, most of the investigations have predominantly focused on linear stress waves, or soliton solutions of such systems due to the capability of the soft structures to support large-amplitude nonlinear waves. Here, taking a step forward, we study a particular lattice  that supports both translational and rotational waves~\\cite{deng2}. Our main goal is to understand how nonlinear lattice waves propagate in the presence of strong disorder when the DoFs are coupled, as well as the system's chaoticity.\n \nThe rest of this paper is arranged as follows: In Section~\\ref{sec2} we describe the Hamiltonian model of the lattice structure and also formulate the system's equations of motion. The dispersion relation of the system, in addition to its dynamics in the linear limit is also discussed. In Section~\\ref{sec3} we investigate the nonlinear effects on wave propagation under strong disorder, as well as study the system's chaoticity and finally, in Sec.~\\ref{sec4}, we summarize our findings and present our conclusions.\n\\section{The Hamiltonian model and its equations of motion} \\label{sec2}\n\nThe $1$D elastic mechanical lattice studied in this work is assembled from an array of aligned LEGO\\textsuperscript{\\textregistered} crosses connected by flexible links \\cite{celli} as depicted in Fig.~\\ref{fig1}. This system constitutes a highly deformable elastic lattice supporting both translational and rotational waves ($2$ DoFs per site). In Ref.~\\cite{deng2}, the authors describe the general equations of motion for a structure that takes into consideration \nsome of the possible geometrical variations of the lattice. However, for the purposes of this work, we limit ourselves to an aligned, symmetrical structure. The crosses are joined to their neighbors by some flexible hinges which are modeled using a combination of three linear springs. The stretching is modeled by a spring with stiffness $ k_{l}$\nand the shearing is described by a spring $ k_{s}$, whilst the bending is modeled by a torsional spring $ k_{\\theta}$ [see Fig.~\\ref{fig1} (b)].\n\nBy making use of the spatial periodicity $a$, we recast the horizontal deflections $u_n$ at the $n$th lattice site to $U_n = u_n\/a$ and change time units to dimensionless time, $ T = t\\sqrt{ k_{l}\/\\overline{m} } $ and the springs are normalized as $K^{(\\theta)} = 4 k_{ \\theta }\/k_{l} a^2$ and $K^{(s)} = k_{s}\/k_{l}$. The mass of each unit of the cross $m_n$ is normalized such that $M_n = m_n\/\\overline{m}$ where $\\overline{m} $ is the average mass of the crosses. This implies that in the case of the homogeneous chain, $M=M_n  = m_n\/\\overline{m} = 1$ and $\\Gamma=\\Gamma_n  = J_n\/ \\overline{m} a^2$. Herein, $J_n$ denotes the rotational moment of inertia at the $n$th lattice site.\nFor the rest of this work, we assume the parameters of the experimental setup of Ref.~\\cite{deng2} where each cross has mass $m = 4.52$~g ($\\overline{m}=m$), $a = 42$~mm and $J = 605$ g$\\cdot$mm$^2$.\nAlso the spring constants are $k_{l}=71.69$ N$\\cdot$mm$^{-1}$, $ k_{s} = 1.325 $N$\\cdot$mm$^{-1}$ and $k_{\\theta} = 4.85$ N$\\cdot$mm.\n\nThe Hamiltonian $H$, of the top or bottom layer of the system is thus given as (see Ref.~\\cite{deng2} for details)\n\\begin{figure}[t!] \n\\centering\n\\includegraphics[width=12.20cm]{lego_schema.png} \\\\\n\\includegraphics[width=12.20cm]{lego_deflections.png}\n\\caption{\\label{fig1} (a) An architected, highly deformable and elastic mechanical structure which supports translational and rotational waves. (b) Schematic of the cross pairs [by symmetry, the dynamics can be described by either the top row or bottom row of (a)] showing the translational and angular deflections deflections. The connectors (marked in red) model a combination of bending $k_{\\theta}$, shear $k_s$ and stretching springs $k_l$.}\n\\end{figure}\n\\begin{equation}\nH  =\\sum_{n=1}^{N} \\left\\{  \\frac{ M_n \\dot{U}^2_n}{2} +   \\frac{ \\Gamma_n \\dot{\\theta}^2_n }{2} +  \\frac{1}{2}  \\Delta^{n~^{2}}_{_{LH}} + \\frac{K^{(s)}}{2}  \\Delta^{n~^{2}}_{_{SH}}   + \\frac{K^{(\\theta)} }{8 } \\left(  \\delta^{n~^{2}}_{_{\\theta H}}  + \\frac{1}{2}  \\delta^{n~^{2}}_{_{\\theta V}} \\right) \\right\\},\n\\label{hamil} \n\\end{equation}\nwhere the dimensionless deflections are given by, \\\\\n$ \\Delta^{n}_{_{LH}}  = U_{n+1} - U_{n} + \\frac{1}{2} \\left( 2 - \\cos{ \\theta_{n}}  -  \\cos{ \\theta_{n+1}} \\right),$\n$ \\Delta^{n}_{_{SH}}  = \\frac{1}{2} \\left( \\sin{ \\theta_{n+1}}  -  \\sin{ \\theta_{n}} \\right),$\n$  \\delta^{n}_{_{\\theta H}} = \\theta_{n+1} + \\theta_{n}, $ and   $\\delta^{n}_{_{\\theta V}} = 2 \\theta_{n}.$\nIn Eq.~(\\ref{hamil}), $[~\\dot{}~]$ denotes the derivative with respect to time\n\nWe derive the equations of motion from the Hamiltonian Eq.~(\\ref{hamil}) which yields\n\\begin{equation}\n\\begin{aligned}\n M_n  \\ddot{U}_{n}  & =  \\Big[ U_{n+1} - U_{n} + \\frac {1}{2} \\{  2 -  \\cos( \\theta_n)  -  \\cos( \\theta_{n+1}) \\}  \\Big] \\\\\n   - &\\Big[ U_{n} - U_{n-1} + \\frac {1}{2} \\{ 2 -  \\cos(\\theta_n)  -  \\cos( \\theta_{n-1}) \\}  \\Big ],\n\\label{em1}\n\\end{aligned}\n\\end{equation}\n\n \\begin{equation}\n\\begin{aligned}\n \\Gamma_n \\ddot{\\theta}_{n} & =   \\frac{1}{4} K^{(s)} \\cos( \\theta_n)  \\Big[ \\sin( \\theta_{n+1})  -  \\sin( \\theta_{n})   \\Big]  \\\\\n  + &\\frac{1}{4} K^{(s)}  \\cos( \\theta_n)  \\Big[ \\sin( \\theta_{n-1})  -  \\sin( \\theta_{n})   \\Big]  \\\\\n+ &\\frac{1}{4} \\sin( \\theta_n)  \\Big[ 2 (U_{n} - U_{n+1}) + \\cos( \\theta_n)  + \\cos( \\theta_{n+1}) - 2  \\Big] \\\\\n + &\\frac{1}{4} \\sin( \\theta_n)  \\Big[ 2 (U_{n-1} - U_{n}) + \\cos( \\theta_n)  + \\cos( \\theta_{n-1}) - 2  \\Big]  \\\\\n-&\\frac{1}{4} K^{(\\theta)} ( \\theta_{n+1} + 4\\theta_{n} + \\theta_{n-1} ).\n\\label{em2}\n\\end{aligned}\n\\end{equation}\n\n\\subsection{  Homogeneous linear system} \\label{ssec1}\nFirst let us consider the homogeneous system\nand linearize the nonlinear terms (trigonometric terms) in \nEqs.~(\\ref{em1}) and (\\ref{em2}) by assuming small angles, $\\theta_{n+p}$ with $p = \\left\\{-1,0,1\\right\\} $,\nand taking a power series expansion of the appropriate cosine and sine terms to give the first two lowest order terms as\n\n\\begin{equation}\n \\sin \\theta_{n+p} \\approx  \\theta_{n+p} - \\frac{1}{6} \\theta_{n+p} ^3 + \\ldots,\n\\label{eq2}\n\\end{equation}\n\n\\begin{equation}\n \\cos \\theta_{n+p} \\approx  1 - \\frac{1}{2} \\theta_{n+p} ^2 + \\ldots.\n\\label{eq3}\n\\end{equation}\nThe linear parts of Eqs.~(\\ref{eq2}) and (\\ref{eq3}) are plugged appropriately into \nEqs.~(\\ref{em1}) and (\\ref{em2}) to give the linear equations of motion as\n\\begin{equation}\n\\begin{aligned}\n  M_n \\ddot{U}_{n}  & =  U_{n+1} - 2 U_{n} + U_{n-1}  ,\n\\label{em3}\n\\end{aligned}\n\\end{equation}\nand \n \\begin{equation}\n\\begin{aligned}\n \\Gamma_n \\ddot{\\theta}_{n} & =  \\tilde{K} (\\theta_{n+1} -2\\theta_{n} + \\theta_{n-1}) - 6 K^{(\\theta)} \\theta_{n},\n\\label{em4}\n\\end{aligned}\n\\end{equation}\nwhere $\\tilde{K} = K^{(s)} - K^{(\\theta)}$.\nFor the homogeneous case, $M_n = M = 1$ and $\\Gamma_n = \\Gamma$ as already defined. \nA quick glance at Eqs.~(\\ref{em3}) and (\\ref{em4}) reveals that the two sets of DoFs are decoupled in the linear regime. Eq.~(\\ref{em3}) belongs to the linear FPUT class of equations and Eq.~(\\ref{em4}) is a linear KG-type equation.\n\nWe now consider solutions of the form\n\\begin{equation}  \n\\mathbf{X}_n =  \\begin{pmatrix} \n            U_n (t)\\\\\n            \\theta_n (t)\n           \\end{pmatrix} = \\mathbf{X}e ^{i \\omega t - i q n},\n \\label{eq4}          \n\\end{equation}\n where $\\mathbf{X}=[U_0,\\Theta_0]$ is the amplitude vector, $\\omega$ is the cyclic frequency\nand $q$ is the wave number. Inserting Eq.~(\\ref{eq4}) into Eqs.~(\\ref{em3}) and ~(\\ref{em4}), we obtain the following eigenvalue problem for the allowed frequencies \n$\\mathbf{D} \\mathbf{X} = \\Omega^2 \\mathbf{X}$,\nwhere the resultant dynamical matrix is\n$$ \\mathbf{D} = \\begin{pmatrix}\n     2 -2 \\cos q  & 0 \\\\\n      0 &  \\frac{1}{2 \\Gamma}\\left [(K^{(s)} + 2 K^{(\\theta)}) - ( K^{(s)} - K^{(\\theta)}) \\cos q \\right ]  \n         \\end{pmatrix}.\n         $$\nThe corresponding dispersion branch for the transverse DoFs is\n\\begin{equation}\n \\omega^{(U)} = \\sqrt{2 -2 \\cos q},\n\\label{eq5}\n\\end{equation}\nwhilst for the rotational DoFs,\n\\begin{equation}\n \\omega^{(\\theta)} = \\frac{1}{\\sqrt{4 \\Gamma}} \\sqrt{2\\left (K^{(s)} + 2 K^{(\\theta)}) - 2( K^{(s)} - K^{(\\theta)}\\right ) \\cos q  }.\n\\label{eq6}\n\\end{equation}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=12.50cm]{disp.png}\n\\caption{\\label{fig2} The two dispersion branches of the system for translational ($\\omega^{(U)}$ - red curve) and rotational ($ \\omega^{(\\theta)}$ - blue curve) DoFs [See respectively Eqs.~(\\ref{eq5}) and (\\ref{eq6})].}\n\\end{figure}\nThe dispersion relation for the two different DoFs as given by Eqs.~(\\ref{eq5}) and (\\ref{eq6}), are plotted in Fig.~\\ref{fig2}.\nAn important feature for the system is that the rotational mode branch (blue curve in Fig.~\\ref{fig2}) starts at a finite frequency. This means that linear rotational waves are not supported for $\\omega < \\frac{1}{\\sqrt{4 \\Gamma}} \\sqrt{6 K^{(\\theta)}}$ since this frequency domain corresponds to a band gap. In fact, the linear dispersion relation of the rotations can be directly mapped onto the one for the linear KG lattice. \nOn the other hand the transverse displacements follow a typical linear mass-spring dispersion relation.\n\n\\subsection{ Disordered linear system} \\label{ss2b}\nIn order to consider to consider a disordered version of the system, we first note that the disordered linear KG system exhibits AL (See Ref.~\\cite{Kundu} for the corresponding behavior of the linear FPUT). Additionally, even though there are a number of ways to introduce disorder in the system, we model the crosses of the architected lattice assuming disorder in the masses $M_n$, which in turn implies disorder in the rotational moments of inertia $\\Gamma_n$. In practice, disorder in the system can also be achieved by changing the material used to manufacture the LEGO\\textsuperscript{\\textregistered} bricks at each site \\cite{celli} without changing their geometrical dimensions. In dimensionless units, the masses are normalized to unity for a homogeneous chain hence we take this into consideration when choosing the disorder distribution and take $M_n$ from a uniform probability distribution $ f \\big( M_n \\big) $ where\n\\[\n f \\big( M_n \\big)  =\n  \\begin{cases}\n                                   W^{-1}, &-W\/2    < \\text{ $ M_n $} - 1 < W\/2, \\\\\n                                   0 & \\text{otherwise}. \\\\\n    \\end{cases}\n\\]\n$ W$ denotes the distribution width and for this study, we choose $ W = 1.8$ hence $ 0.1 \\leq M_n \\leq 1.9 $. \n\nTo study the dynamical behavior of the system, the equations of motion are integrated using the \\texttt{ABA864} symplectic integrator \\cite{blanes} which has been proved to be very efficient for the accurate integration of large Hamiltonian lattice models \\cite{senyange2018,danieli2019}.\nThis integration scheme allows for energy conservation of the total energy $H$, and keeps the relative energy error $\\Delta H (T)  = \\left|\\dfrac { H (T) - H (0)  }{ H (0) }\\right| <  10^{-5}$ \nwhen the integration time step is set to be $ \\tau = 0.1$. In all our numerical simulations, we employ fixed boundary conditions i.e.,~$ U_{0} = U_{N+1} = 0$, $ \\theta_{0} = \\theta_{N+1} = 0$ ,  $ \\dot{U}_{0} = \\dot{U}_{N+1} = 0$ and $ \\dot{\\theta}_{0} = \\dot{\\theta}_{N+1} = 0$. Furthermore, the considered lattice size is large enough so that the energy does not reach the lattice boundaries. A typical numerical integration of the nonlinear system for $ T = 10^5$ requires a lattice size of at least $2 \\times 10^5$ sites.\n\n\\subsubsection{Translational degrees of freedom}\nWe start our analysis by first exploring the two possible single site initial excitations of velocity and displacement for the translation DoFs i.e.,~\n\n\\begin{equation}\n\\dot{U}_{N\/2}(0)=\\sqrt{2H\/M_{N\/2}}, \\quad \\textrm{or} \\quad U_{N\/2}(0)= \\nu,\n\\label{eq7} \n\\end{equation}\nindependently. In Eq.~(\\ref{eq7}), the scalar $\\nu$ is real and its value is altered to match the desired system energy. We fix the total system energy at $H=H_{0}$ and integrate the system up to $T = 10^5$ time units and observe how the initially localized wave-packet evolves in time. A typical scenario of the dynamics is shown in Figs.~\\ref{fig3}(a)-(b) for $H_0 = 10^{-4}$. For single site velocity initial excitations, the energy distribution displays two main parts, a central localized part, and an expanding peripheral part which is spreading beyond the excitation point [see Fig.~\\ref{fig3}(a)]. Similar spreading characteristics are observed in the dynamics for displacement initial excitations as depicted in Fig.~\\ref{fig3}(b).\n\nFor a more quantitative description, we follow the time evolution of the participation number $P$ and the second moment $m_2$, of the energy distribution to characterize the localization and spreading properties of the wave-packet. The former quantity $P$, is used to quantify the number of highly excited sites in a lattice~\\cite{flachbook} and is computed as\n\\begin{equation} \\label{eqp}\nP = 1\/ \\sum h_n^2,\n\\end{equation}\nwhere $h_n = H_n \/ H$ is the normalization of the site energy $H_n$. In the case of equipartition, for a lattice of size $N$, $ P = N$, while the other extreme gives $ P = 1$, for a wave-packet in which only a single site is highly excited.\nThe second moment $ m_2 $~\\cite{Kundu} of the energy distribution given by\n\\begin{equation} \\label{eqm2}\nm_{2} =  \\sum_n (n-\\overline{n})^2 H_n \/H,\n\\end{equation}\nis a measure of the wave-packet's extent where $\\overline{n} = \\sum_n n H_n\/H$ is the mean position of the energy distribution.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=6.20cm]{Fput_MLin.png}\n\\includegraphics[width=6.20cm]{Fput_DLin.png}\\\\\n\\includegraphics[width=6.20cm]{p_fputlin.png}\n\\includegraphics[width=6.20cm]{b_fputlin.png}\n\\caption{\\label{fig3} Spatiotemporal evolution of the energy distribution for a representative realization with (a) velocity and (b) displacement single site initial excitation in the linear system. The colorbars in (a) and (b) are in log-scale. (c) Time evolution of the average participation number $\\langle P \\rangle$, (d) estimation of the exponent $\\beta$, related to the time evolution of the average second moment through $ \\langle m_2 \\rangle \\propto T^{\\beta} $. For velocity (black curves) and displacement (magenta curves) initial excitations, the mean values $ \\langle \\cdot \\rangle$ are calculated from $100$ disorder realizations and the shaded areas represent the statistical error (one standard deviation). The dashed lines in (d) indicate $\\beta =1.5$ (top) and $\\beta =0.5$ (bottom) and the system energy for all realizations is $ H = 10^{-4} $. }\n\\end{figure}\n\nThe time evolution of $\\langle P \\rangle$ indicates that a  maximum value is reached and remains constant for both velocity and displacement excitations as illustrated in Fig.~\\ref{fig3}(c) by the black and magenta curves respectively. In this work, $ \\langle \\cdot \\rangle$ denotes averages over $100$ disorder realizations. One of the differences between the two cases is that velocity initial excitations yield slightly higher values of $\\langle P \\rangle$ when compared to the $\\langle P \\rangle$ reached for displacement initial excitations. This is due to the fact that more low frequency propagating modes are excited for the case of velocity initial excitation than with displacement initial excitation \\cite{Kundu}. \nThe time evolution of $\\langle m_ 2 \\rangle$ for the two excitations is also different for the same reason.\n \n Regarding the second moment $m_2$, the usual practice is to assume that $\\langle m_2 \\rangle  \\propto T^{\\beta}$. Then the parameter $\\beta$ is numerically estimated by the time local derivative\n\\begin{equation} \\label{eqb}\n\\beta=  \\frac{ d  \\log_{10}\\langle m_2 (T) \\rangle }{ d  \\log_{10} T}.\n\\end{equation}\nThe exponent $\\beta$ is used to quantify the asymptotic behavior of $\\langle m_2 \\rangle$ for sufficiently large times. It is calculated by first smoothing the $m_2 (T)$ values of each disorder realization through a locally weighted difference algorithm~\\cite{cleveland1,cleveland2} and then averaged over all realizations. The computed $\\beta$ for the two cases, saturates to $\\beta = 1.5$ and $\\beta = 0.5$, for respectively velocity and displacement single site excitations as shown in Fig.~\\ref{fig3}(d). This is an expected result because we have already shown the system to be practically a linear disordered FPUT lattice, since the translational DoFs do not couple to the rotational DoFs. \nHere we note that, according to the full system of equations [Eqs.~(\\ref{em1})-(\\ref{em2})] single site translational excitations as given by Eq.~(\\ref{eq7}) will never couple the two DoFs. This is a particularity of the structural geometry under consideration. Thus,  whatever the initial excitation energy, single site initial translations will always lead to the linear behavior summarised in Fig.~\\ref{fig3}. For this reason we shall not consider such initial conditions in Section~\\ref{sec3}. \n\n\\subsubsection{ Rotational degrees of freedom} \\label{sssecR}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=6.20cm]{KG_MLin.png}\n\\includegraphics[width=6.20cm]{KG_DLin.png}\\\\\n\\includegraphics[width=6.20cm]{kg_e4_P.png}\n\\includegraphics[width=6.20cm]{kg_e4_B.png}\n\\caption{\\label{fig4} Similar to Fig.~\\ref{fig3} but for initial angular deflections (magenta curves) and angular deflection time derivatives (black curves). The vertical axes for (a) and (b) are given in $\\log_{10} $. The dashed line in (d) indicates $\\beta=0$. }\n\\end{figure}\n\nWe now turn our attention to the dynamics of single site rotational excitations (angular deflection or the angular deflection time derivative). These excitations are expected to follow the dynamics of a discrete linear KG lattice as already explained earlier in Section.~\\ref{ssec1}. More specifically, we consider independently the initial excitations\n\\begin{equation}\n\\dot{\\theta}_{N\/2}(0)=\\sqrt{2H\/\\Gamma_{N\/2}},\\quad \\textrm{or} \\quad \\theta_{N\/2}(0)= \\mu,\n\\label{eq72} \n\\end{equation}\nwhere $\\mu$ is real and is chosen to match the desired system energy $H=H_0$.\n\nThe energy distribution profiles for both angular deflection time derivative (black curves) and angular deflection (magenta curves) excitations [see Figs.~\\ref{fig4}(a) and (b)] show localized energy distributions for times $T \\gtrsim 10^4$ after an initial phase of wave-packet spreading.\nThe subtle differences between angular deflection time derivative and angular deflection initial excitations in panels (a)-(b) of Fig.~\\ref{fig4} are mainly due to the differences in the initially excited modes of the system for each respective excitation. Thus, the dynamics of the rotations in the linear limit show complete Localization in sharp contrast to the perpetual wave-packet spreading observed for the translational DoFs [Figs.~\\ref{fig3}(a) and (b)]. This is a consequence of the fact that the KG system Eq~(\\ref{em4}) describing rotations, is known to map to the DNLS and experiences localization of all modes \\cite{kivshar,kivshar1993,johansson2006}. To quantify the localization for the initial angular deflections (magenta curves) and angular deflection time derivatives (black curves) we plot the time evolution of $ \\langle P \\rangle $ which reaches constant finite values after $ T \\gtrsim 4 \\times 10^{4} $ [Fig.~\\ref{fig4}(c). A similar behavior is observed for $ \\langle m_2 \\rangle $ (not shown here) for  $ T \\gtrsim 10^{3} $ ] in agreement to what is expected for a linear disordered KG chain. The final saturation of the $ \\langle m_2 \\rangle $ value is clearly depicted in the evolution of the exponent $\\beta$ from the relation $ \\langle m_2 \\rangle \\propto T^{\\beta} $, which eventually becomes $ \\beta =0 $ as clearly seen in Fig.~\\ref{fig4}(d), showing that indeed the system is effectively a $1$D linear disordered KG chain.\n\n\\section{ Disordered nonlinear system} \\label{sec3} \n\nHaving considered the behavior of the system in the linear limit, we further study the fully nonlinear system as described by Eqs.~(\\ref{em1})-(\\ref{em2}). Before proceeding further, we reiterate that single site initial translational velocity ($\\dot{U}_{N\/2}$) or displacement ($U_{N\/2}$) excitations \\textit{do not induce a nonlinear response}. However, initial rotational excitations induce both a nonlinear response on rotations as well as nonlinear coupling between the two sets of DoFs. For the rest of this work we focus exclusively on single site (in the centre of the lattice) initial conditions of angular deflections as well as angular deflection time derivatives. \n\n\\subsection{Weakly nonlinear regime} \n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=14.0cm]{fig5new.png}\n\\end{centering}\n\\caption{\\label{fig5} Weakly nonlinear system,~ (a)-(d) Time evolution of $ \\langle P \\rangle $ (\\ref{eqp}), $\\langle m_2 \\rangle$ (\\ref{eqm2}), $\\beta$ (\\ref{eqb}) and $\\langle \\Lambda \\rangle$ (\\ref{eq41}), respectively. The black (magenta) curves correspond to single site initial angular deflection time derivatives (angular deflection) excitations. The average values $\\langle \\cdot \\rangle$ are computed from $100$ disorder realizations and the lightly shaded areas represent the statistical error (one standard deviation). The dashed blue and green curves which are not always visible due to their overlapping with other curves, respectively show results of the linearized system for the initial conditions given by Eq.~(\\ref{eq72}). All results are for the weakly nonlinear regime with $H = 10^{-8}$. The horizontal line in (c) indicates $\\beta =0$. }\n\\end{figure}\n\nWe start by implementing single site angular deflections and time derivatives of angular deflections as initial excitations for a sufficiently small energy ($ H = 10^{-8}$) so that the system is in the weakly nonlinear regime. With the physical system at hand, this energy corresponds to an initial angle deflections of $\\approx 0.2^{\\circ}$. The time evolution of $\\langle P \\rangle$ is shown in Fig.~\\ref{fig5}(a) where we observe a saturation to a constant value for each type of excitation [angular deflection time derivative (black curve) and angular deflection (magenta curve)]. In fact we also plot, in the same figure, the corresponding linear result (dashed curves) and we observe that the weak nonlinearity does not affect the participation number. Nevertheless, we find that nonlinearity plays a significant role on the evolution of $\\langle m_2 \\rangle$ and $\\beta$. This is illustrated in Figs.~\\ref{fig5}(b)-(c) where both these quantities are found to increase for the last two decades ($T \\gtrsim 10^3$) of the evolution clearly indicating energy spreading  in the system.\n\nTo further explore the spreading it is worthwhile to also investigate the chaoticity of the system using the finite time maximum Lyapunov exponent (ftMLE). It is often found, in systems with multiple degrees of freedom, that energy spreading is due to chaos around the excitation region \\cite{disorder_15,skokos2013,Bob2018}. The ftMLE,\n\\begin{equation} \\label{eq41}\n\\Lambda(T) = \\frac{1}{T} \\ln \\frac { || \\mathbf{w} (T) || }{ || \\mathbf{w} (0) ||},\n\\end{equation}\nis computed using the so-called standard method \\cite{benettin,lec_notes}. $\\mathbf{w} (T)$ is a vector of small perturbations from the phase space trajectory at time $T$ (also called deviation vector) which we denote as \n\n\\begin{equation}\n\\begin{aligned}\n \\mathbf{w} (T)  = & [ \\delta  U_1 (T), \\ldots , \\delta  U_N (T), \\\\ & \\delta  \\theta_1 (T), \\ldots , \\delta  \\theta_N (T), \\\\\n &M_1 \\delta \\dot{U}_1 (T),  \\ldots , M_N \\delta \\dot{U}_N (T), \\\\\n &\\Gamma_1 \\delta  \\dot{\\theta}_1 (T),  \\ldots , \\Gamma_N \\delta \\dot{\\theta}_N (T)], \n \\end{aligned}\n\\end{equation}\nwhere $\\delta  U_n (T)$ and $\\delta  \\theta_n (T)$ indicates small perturbations in positions, while $M_n \\delta \\dot{U}_n (T)$ and $\\Gamma_n \\delta \\dot{\\theta}_n (T)$ indicate small perturbations in momenta for the two DoFs at the $n$th lattice site. The mLE is defined as $ \\lambda = \\lim_{T \\to \\infty} \\Lambda (T) $. In Eq.~(\\ref{eq41}), $|| \\cdot ||$ denotes the usual Euclidean vector norm. For chaotic trajectories, $\\Lambda $ attains a finite positive value, otherwise, $\\Lambda \\propto T^{-1}$ for regular orbits. An efficient and accurate method to follow the evolution of $\\mathbf{w} (T)$ is to numerically integrate the so-called variational equations \\cite{galgani}, which govern the vector's dynamics, together with the Hamilton equations of motion using the tangent map method outlined in Refs.~\\cite{gerlach1,gerlach2,gerlach3}. The mLE can be used to discriminate between regular and chaotic motions since $\\Lambda  = 0$ for regular orbits and $\\Lambda > 0$ for chaotic orbits. The magnitude of the mLE can also be used as a measure of the chaoticity: larger mLE values imply stronger chaotic behaviors.\n\n In Fig.~\\ref{fig5}(d) we show the calculated ftmLE which is found to follow the power law decay $ \\langle \\Lambda \\rangle \\propto T^{-1}$,  and thus the system exhibits regular dynamics. As such we conclude that  the observed energy spreading \\textit{cannot} be attributed to chaoticity as is the case for other lattice models including the single DoF per site KG lattice \\cite{skokos2013,Bob2018}.\n\nIn order to explain the spreading we need to further monitor the energy density in the lattice as a function of time. In Fig.~\\ref{fig6}(a) we show four snapshots of the energy density around the initial excitation point ($n=0$). The spatial distribution of energy is separated into two distinct regions: a large amount of energy localized around  $n=0$  and an extended tail with much lower energy (five orders of magnitude less). In fact, if we consider the propagation of the higher energy central part, especially for $T \\lesssim 10^3$, we clearly observe a leading wave-front (dashed vertical lines) which propagates slower than the main leading wave-front. The latter corresponds to the low energy regions [$H_n \\lesssim 10^{-13}$ in Fig.~\\ref{fig6}(a)] which propagates with a normalized velocity close to one. On the contrary the high energy part around the center [$H_n \\gtrsim 10^{-13}$] is propagating much slower with a normalized velocity of $\\approx 0.1$. These two velocities correspond respectively to the largest group velocities of the translation and rotational branches of the dispersion relations shown in Fig.~\\ref{fig2}. Thus we conjecture that the two distinct parts of the energy distribution, high and low, correspond respectively to the two different types of DoFs i.e., the rotational and translation deflections.\n\nFurthermore, as shown by the snapshots for $T\\gtrsim 10^4$ in Fig.~\\ref{fig6}(a), at later times only the lower (translation) part of the energy continues to spread.\nThis fact is corroborated by calculating the exponent of the second moment for energies lower (larger) than a threshold ($H=10^{-13}$) as shown in Fig.\\ref{fig6}(b). The exponent $\\beta$ for the high energy central part almost vanishes (red curve), indicating no spreading while for the low energy region value of $\\beta$ (black curve) is finite revealing spreading. Thus we conclude this sub-section by explaining the dynamics in the following manner. In the weakly nonlinear regime, the role of the nonlinearity is to induce spreading by stimulating the translation DoFs (which have an FPUT character) through the nonlinear coupling. In this way, the energy spreading in this regime is characterized by a hybrid of KG-like and FPUT-like behaviors. In addition the rate of wave-packet spreading as quantified by $\\beta$ shows no defined asymptotic behavior but rather a dependency on time [Fig. \\ref{fig5}(c)]. This is in accordance with the results obtained for a weakly nonlinear FPUT lattice by Lepri \\textit{et.al.}, \\cite{Lepri}.\n \n\\begin{figure}\n\\centering\n\\includegraphics[width=12.0cm]{fig6newa.png}\\\\\n\\includegraphics[width=12.0cm]{fig6newb.png}\n\\caption{\\label{fig6} (a) Average energy distribution profiles of $100$ disorder realizations for $T= 5\\times 10^2$~(red curve) , $T= 10^3$~(blue curve), $T= 10^4$~(magenta curve) and $T= 5\\times 10^4$~(black curve). The dotted horizontal line marks the energy $ \\langle H_n \\rangle = 10^{-13}$ while the vertical dashed lines indicate the position of the slower wave-front at the indicated times. (b) Time evolution of the exponents $\\beta$ for the central part (solid red curve) and the tails (solid black curve). The lightly shaded regions in (b) indicate the statistical error (one standard deviation). All panels are for angular deflections of system energy $H=10^{-8}$.}\n\\end{figure}\n\n\\subsection{Strongly nonlinear regime} \nLet us move away from the weakly nonlinear regime and increase the nonlinearity of the system by increasing the initial excitation energy to $ H = 5 \\times 10^{-3}$. This energy corresponds to large initial angle deflections of about $30^{\\circ}$. Note that this value of energy leads to strong nonlinear behavior. The system, in this regime, shows a completely different behavior of spreading as is indicated by the increase of $\\langle P \\rangle$ during the time evolution as shown in Fig.~\\ref{fig7}(a). The number of highly excited sites grows in time contrary to what we observed for the weakly nonlinear regime, which shows practically no growth in $\\langle P \\rangle$ at long times. As expected, this increase of participating particles leads to wave-packet spreading and this is confirmed by the time evolution of $\\langle m_2 \\rangle$ which is also increasing as indicated in Fig.~\\ref{fig7}(b). A feature we observe in this strong nonlinear regime is that the dynamics no longer depends on the type of initial condition (angular deflections or time derivatives of angular deflections). This is noticeable from the practically overlapping black (time derivatives of angular deflections) and magenta (angular deflections) curves in Figs.~\\ref{fig7}(b)-(c). To quantify wave-packet spreading, we estimate the exponent of $\\langle m_2 \\rangle \\propto T^{\\beta}$ which is found to acquire values around $\\beta \\approx 2$ as shown in Fig.~\\ref{fig7}(c). This value indicates a very strong spreading corresponding to a near ballistic propagation. We find this result to be quite interesting since the nonlinearity of the flexible architected material under study, is strong enough to bring the system to ballistic behavior, which is not always the case in other systems such as the FPUT and KG lattices~\\cite{pikovsky,flachbook}. In fact, the other example to our knowledge, where near ballistic behavior in a disordered system is observed, is for mechanical lattices featuring non-smooth nonlinearities due to Hertzian forces \\cite{jk}. \n \n Also, in this strongly nonlinear regime the distinct behavior of the two types of DoFs that was observed in Fig.~\\ref{fig6}(a) is now lost. According to Fig.~\\ref{fig7}(d) showing the mean profile of the energy distribution, we identify a large part of the energy being localized around the center and a propagating tail  travelling almost ballistically. This is expected since according to Eqs.~(\\ref{em1})-(\\ref{em2}) the coupling of the two types of DoFs is enhanced at each lattice site when the rotations are of high amplitude. Thus we no longer distinguish between a KG-like and FPUT-like evolution of the energy profiles. We have also considered a range of system energies between $H=10^{-8} - 10^{-3}$ and found that the distinction between KG- and FPUT-like behaviors gradually disappears as the system energy is increased. Some of the results for the intermediate energies  are reported in Ref.~\\cite{ngapasarephd}.\n \nRegarding the chaoticity of the system, for such high initial angles and thus strong nonlinearity, we find the dynamics to be chaotic. This is evident in Fig.~\\ref{fig8}(a) where the mean value of the ftMLE $\\Lambda$ Eq.~(\\ref{eq41}), is shown to be decreasing in a much slower rate compared to regular dynamics (dashed line). This type of chaotic behavior, where the ftMLE does not reach an asymptotic constant value, has recently attracted much attention and appears to be a particular case of chaos spreading \\cite{disorder_15, Bob2018, many2} and it is related to the fact that as the wave-packet spreads, the constant total energy is shared among more sites and consequently the energy per excited site (which plays the role of active nonlinearity strength) decreases. Consequently, the ftMLE which is a global measure of chaos, is also decreasing. \n\nHowever, different to what was found in Refs.~\\cite{disorder_15, Bob2018, many2}, for the architected lattice under study, here the slope of $ \\langle \\Lambda \\rangle$ does not reach a constant value slope even for the \\textit{largest possible } angular deflection  of $45^{\\circ}$.\n %\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12.60cm]{fig1000.png}\n\\end{center}\n\\caption{\\label{fig7} (a)-(c) Similar to Figs.\\ref{fig5}(a)-(c). (d) Average energy density profiles over $100$ disorder realizations at $T=10^2$ (blue curve), $T=5 \\times 10^2$ (red curve) and $T=10^3$ (black curve). Results in all panels are for energy $H= 5 \\times 10^{-3}$.}\n\\end{figure}\n %\nThus, for the sake of completeness and to be able to compare the results regarding the chaos spreading of the soft architected lattice with other models in the literature, we extend our numerical simulations using even larger initial energies. In particular, in Figs.~\\ref{fig8}(b)-(c) we show results using an initial energy of $H=10^{-1}$.\nNote that this value of energy leads to an even stronger nonlinear behavior than the penultimate case above. \n %\n\n %\n %\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12.5cm]{fig8feb.png}\n\\end{center}\n\\caption{\\label{fig8}~(a) Time evolution of the average ftMLE, Eq.~(\\ref{eq41}), $ \\langle \\Lambda \\rangle $ for the strongly nonlinear regime angular deflections and time derivatives of angular deflections as initial excitations for energy $H=5 \\times 10^{-3}$. The red dashed line indicates the power law $\\langle \\Lambda \\rangle \\propto T^{-1}$. (b) Similar to (a) but for energy $H=10^{-1}$. Here the red dashed line indicates the power law $\\langle \\Lambda \\rangle \\propto T^{\\kappa}$, with $\\kappa = -0.28 $. (c) Time evolution of the exponent $ \\kappa $ at energy $H = 10^{-1}$. In all panels the magenta and black curves respectively show results for angular deflections and time derivatives of angular deflections as initial excitations. The magenta horizontal dashed line indicates $\\kappa = -0.28$. }\n\\end{figure}\nWe also observe in Fig.~\\ref{fig8}(b) that the ftMLE reaches a clearly constant slope with which it is decaying. In particular, the slope of the exponent assuming that $\\langle \\Lambda \\rangle \\propto T^{\\kappa}$  is found to be approximately $\\kappa = -0.28$ for both angular deflections and time derivatives of angular deflections. Note that this asymptotic value of the exponent $\\kappa$, is comparable to the corresponding values for the ftMLE in the more studied cases of $1$D lattices namely the disordered DNLS equation and the disordered KG model \\cite{skokos2013, Bob2018}. More precisely, the value  $-0.28$ lies between the values obtained for the so-called weak ($\\kappa = - 0.25$) and strong chaos ($\\kappa = - 0.3$) regimes of these $1$D models having a single DoF per lattice site. \n\n\\section{Summary and conclusions} \\label{sec4} \n\nWe have studied numerically energy spreading and chaos in a nonlinear disordered architected mechanical lattice. The lattice under consideration describes rotating LEGO bricks connected with flexible links which was recently studied experimentally. The in-plane motions of the lattice are described by two DoFs per lattice site i.e., translations and angular deflections. Furthermore, in the linear limit of the aligned structure, the two DoFs per site are completely decoupled. In this state, the lattice shows two distinct behaviors corresponding to the FPUT and KG-like behaviors for translational and rotational DoFs respectively. For both cases, we review results regarding the behavior of energy spreading  under the effect of disorder.\n\nUsing single site angular deflections and time derivatives of angular deflections initial excitations, we studied the system for different strengths of nonlinearity focusing on the weakly nonlinear and strongly nonlinear regimes. For the weakly nonlinear regime we have shown that the total energy density of the lattice is split into two parts: i) a slow spreading part around the excitation point following the KG-like behavior and ii) the fast propagating tails of lower energy which travel with the speed of sound of the corresponding FPUT lattice and are responsible for the evolution of the second moment of the total energy distribution. \nFor sufficiently large initial excitations, the strong nonlinearity of the flexible architected lattice forces the initial wave-packet to spread ballistically and the distinction between a KG- and an FPUT-like behavior is lost. We note here that a ballistic behavior under strong disorder is not easily achieved and here the responsible  physical mechanism is the large geometrical nonlinearity. \n\nAdditionaly, we show that chaos is found to persist during the energy spreading although its strength decreases in time as quantified by the evolution of the system's ftMLE. Here, the power law time evolution of the exponent of the ftMLE is found to acquire a value which lies between the ones obtained for the so-called weak and strong chaos regimes of the well studied nonlinear KG lattice.\n\nOur results show that flexible architected elastic lattices with coupled DoFs per site provide a modern physical platform to study and observe rich wave dynamics which can not otherwise be seen with classical uncoupled fundamental models. Some interesting directions arise from our results like the study and manipulation of energy propagation by tuning the dispersion characteristics of rotations, by changing the shear and bending stiffness's. Furthermore, here we only considered an aligned structure where the two DoFs per site are uncoupled in the linear limit. Extensions to other geometries, where the linear modes are polarised will probably reveal a variety of spreading characteristics and provide a means of controlling energy transport in highly heterogeneous lattices. \n\n\\section*{Acknowledgements}\nA.N. acknowledges funding from the University of Cape Town (University Research Council, URC) postdoctoral Fellowship\ngrant and the Oppenheimer Memorial Trust (OMT) postdoctoral Fellowship grant. Ch. S. thanks the Universit\\'{e} du Mans for its hospitality during his visits when part of this work was carried\nout. We also thank the Centre for High Performance Computing \\cite{chpc} for providing\ncomputational resources for performing significant parts of this paper's computations. We also thank V. Tournat for useful discussions.\n\n\\section*{Author Declarations}\nThe authors have no conflicts to disclose.\n\n\\section*{Data Availability Statement}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec_intro}\n\n\n\nIn four-dimensional General Relativity, a fundamental connection between geometric optics and the algebraic structure of the Weyl tensor is provided by the Goldberg-Sachs (GS) theorem \\cite{GolSac62,NP}, which states that an Einstein spacetime is algebraically special if, and only if, it contains a shearfree geodetic null congruence (cf.~\\cite{Stephanibook,penrosebook2} for related results and generalizations). This theorem plays an important role in the construction of algebraically special exact solutions of the Einstein equations \\cite{Stephanibook}, as the remarkable discovery of the Kerr metric shows \\cite{Kerr63}. \n\nIn recent years, the growing interest in higher dimensional gravity has motivated the study of extensions of the above concepts to  $n>4$ dimensions. An algebraic classification of the Weyl tensor based on the notion of Weyl Aligned Null Directions (WANDs) has been put forward in \\cite{Coleyetal04} (see also \\cite{OrtPraPra12rev} for a recent review). Furthermore, a higher dimensional version of Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) formalisms have been presented in \\cite{Pravdaetal04,Coleyetal04vsi,OrtPraPra07} and \\cite{Durkeeetal10}, respectively. However, simple examples reveal that neither the ``geodesic part'' nor the ``shearfree part'' of the GS theorem extend in an obvious way to higher dimensions (see \\cite{Ortaggioetal12} for a brief summary, along with the original references \\cite{MyePer86,FroSto03,Pravdaetal04,OrtPraPra07,PraPraOrt07,GodRea09,Durkee09}).\n\nThe proper formulation of the geodesic part of the higher dimensional GS theorem has been proven in \\cite{DurRea09} (see also \\cite{PraPraOrt07,Durkee09}): in particular, {\\em an Einstein spacetime admits a multiple WAND if, and only if, it admits a geodesic multiple WAND} -- hence one can restrict to geodesic multiple WANDs {(mWANDs)} with no loss of generality. Very recent work \\cite{Ortaggioetal12} has analyzed the shearfree part in five dimensions by proving necessary conditions on the form of the optical matrix $\\rhob$ (defined below in \\eqref{expmatrix})  following from the existence of an mWAND.\\footnote{See \\cite{Taghavi-Chabert11} for a different formulation of the  ``shearfree'' condition and \\cite{Ortaggioetal12} for a comparison between the two approaches.}  Contrary to the 4d case, the conditions obtained in \\cite{Ortaggioetal12} are, in general, not sufficient. In fact, it seems that in higher dimensions  conditions that are both necessary and sufficient in general do not exist.\n\nIn the present contribution we focus on $n$-dimensional Einstein (including Ricci-flat) spacetimes of type~II (type D being understood as a special subcase thereof) in more than five dimensions and work out the corresponding necessary conditions on $\\rhob$ { {\\em under the assumption that the mWAND is twistfree} (i.e., $\\rhob$ is symmetric, {equivalent to $A_{ij}=0$, cf.~\\eqref{S_A} below)}}.\\footnote{This automatically guarantees that $\\bl$ is geodesic. Except for a few remarks, we assume {that $\\rhob$ is non-zero}, since we are interested in studying its possible non-trivial forms.} \nSince the types N and III were already studied in \\cite{Pravdaetal04}, we will simply connect the results of \\cite{Pravdaetal04} to our analysis when appropriate. As it turns out, in $n>5$ dimensions the constraints on $\\rhob$ are in general not as strong as those for $n=5$ \\cite{Ortaggioetal12}.  Nevertheless, combining our results for a type II spacetime with the results for type III and type N \\cite{Pravdaetal04}  gives \n\n\n\n\\begin{theorem}[Eigenvalue structure of $\\rhob$ for $n\\ge 6$ and $A_{ij}=0$]\n\\label{prop_GSHD}\n\nIn an algebraically special Einstein spacetime of dimension $n\\ge 6$ that is not conformally flat, the $($symmetric$)$ optical matrix {$\\rhob$} of a non-twisting multiple WAND has at least {\\em one double eigenvalue}. In the following special cases stronger conditions hold and the most general permitted eigenvalue structures  are\n\\begin{enumerate}[(i)]\n\t\\item if $\\Phi^A_{ij} \\ne 0$: $\\{a,a,0,\\ldots , 0\\}$ {\\rm\\cite{Ortaggioetal12}}, \n \t\t\\label{PhiA}\n\t\\item if $\\det \\rhob\\not= 0$, $\\Phi_{ij}\\not=0$: $\\{a,a,\\ldots , a\\}$ (Robinson-Trautman, $\\Phi_{ij}\\propto \\delta_{ij}$, type D(bd)), \n\\label{RT}\n \\item if  $\\Phi_{ij}=0$ (type II(abd)): $\\{a,a,b,b,c_1,\\ldots , c_{n-6}\\}$,   \t\t\t\t\t\\label{Phi0}\n\\item\nif the type is N or III: $\\{a,a,0,\\ldots , 0\\}$ {\\rm\\cite{Pravdaetal04}}. \n \\label{NIII}\n \\end{enumerate}\n\\end{theorem}\n\n(The Weyl components $\\Phi^A_{ij}$ and $\\Phi_{ij}$ are defined below in \\eqref{bw0}.) The proof of (\\ref{RT}) and  (\\ref{Phi0}) is a new result and will be given in sections~\\ref{sec_nondeg} and  \\ref{sec_nontwist}, respectively.\nNote that in the above cases (\\ref{PhiA})--(\\ref{NIII}) the matrix $\\rhob$  possesses at least {\\em two} double eigenvalues. In particular, in six dimensions all possible cases can be listed explicitly:\n\n\\begin{corol}[Eigenvalue structure of $\\rhob$ for $n=6$ and $A_{ij}=0$]\n\\label{corol6d}\nIn a 6d algebraically special Einstein spacetime that is not confomally flat, the $($symmetric$)$ optical matrix $\\rhob$ of a non-twisting multiple WAND can have only one of the following eigenvalue structures (where $a,b $ might coincide, or vanish): (i) $\\{a,a,b,b \\}$; (ii) $\\{a,a,b,0 \\}$; (iii) $\\{a,b,0,0 \\}$. If the spacetime is type III or N then the structure is $\\{a,a,0,0\\}$.\n\n\n\n\\end{corol}\n\nNote that we do not claim that all classes compatible with theorem~\\ref{prop_GSHD} or corollary  \\ref{corol6d} are non-empty. \n\nThe structure of the paper is the following. In section~\\ref{sec_genII} we briefly summarize  results for general (possibly twisting) type II spacetimes in arbitrary dimension (already contained in \\cite{Ortaggioetal12}), in particular the so called ``optical constraint''. These will be useful  in  the rest of the paper, where, however, we limit ourselves to the non-twisting case. \nIn section~\\ref{sec_nontw_gen} we set up the study of general non-twisting type II spacetimes. It turns out that it is convenient to study the case of non-degenerate and degenerate $\\rhob$ separately. This is done in sections~\\ref{sec_nondeg} and \\ref{sec_nontwist}, where we derive certain necessary conditions for $\\rhob$ being  compatible with a non-twisting mWAND, \nin particular proving thus theorem~\\ref{prop_GSHD}. Using various explicit examples we show in section~\\ref{sec_counter} that the necessary conditions obtained in sections~\\ref{sec_nondeg} and \\ref{sec_nontwist} are in general not  sufficient. In section~\\ref{sec_totgeod} we elucidate the geometrical implications of theorem~\\ref{prop_GSHD} in terms of existence of integrable null distributions with\ntotally geodesic integral null two-surfaces. In section~\\ref{sec_6D} we study the six-dimensional case (corollary  \\ref{corol6d}) and provide various explicit examples of such metrics.\n\nSome related results are given in the appendices. {{In appendix~\\ref{app_alg_eq} we present new algebraic constraints for general type II Einstein spacetimes. Appendix~\\ref{app_type_N_III} contains a new proof of the canonical form of the optical matrix $\\rhob$ for type III Einstein spacetimes in the case of a non-twisting mWAND $\\bl$ (the original, longer proof was given in \\cite{Pravdaetal04}). In appendix~\\ref{app_shearfree} we summarize various equivalent formulations of the ``geodesic and shearfree'' condition in four dimensions, and we discuss some extensions of those to higher dimensions (including optical structure, optical constraint, integrability of certain null distributions). Appendix~\\ref{app_OS5d} contains the proof (mostly relying on results of \\cite{Ortaggioetal12,Wylleman_priv}) of the existence of an optical structure in all algebraically special spacetimes in five dimensions. In appendix~\\ref{app_shearfreetwist} we present examples of twisting but shearfree mWANDs (in six dimensions). Finally, in appendix~\\ref{app_violating}  examples of Einstein spacetimes of type D with non-degenerate mWANDs violating the optical constraint are given.}}\n\n\n\n\n\\paragraph{Notation}\n\n\n\nThroughout the paper we use higher dimensional GHP formalism developed in \\cite{Durkeeetal10}. We employ a null frame\n\\begin{equation}\n  \\{\\lb \\equiv \\eb_{(0)}=\\eb^{(1)},\n  \\bn \\equiv \\eb_{(1)} = \\eb^{(0)},\n    \\mmb{i}\\equiv\\eb_{(i)} = \\eb^{(i)} \\}\n\\end{equation}\nwith indices $i,j,k,\\ldots$ running from $2$ to $n-1$. The null vector fields  $\\lb$ and $\\bn$  and the orthonormal spacelike vector fields $\\mmb{i}$  obey $\\lb \\cdot \\bn =1$, $\\mmb{i} \\cdot \\mmb{j} = \\delta_{ij}$ and  $\\lb \\cdot \\mmb{i} = 0 = \\bn \\cdot \\mmb{i}.$\nThe optical matrix $\\rhob$ is defined by\n\\be\n \\rho_{ij} = m_{(i)}^a m_{(j)}^b \\nabla_b \\ell_a , \\label{expmatrix}\n\\ee\nand its trace gives the expansion scalar (up to normalization)\n\\be\n \\rho \\equiv \\rho_{ii}.\n\\ee\n{Its (anti-)symmetric parts are, respectively,\n\\be\n S_{ij}\\equiv \\rho_{(ij)} , \\qquad A_{ij}\\equiv \\rho_{[ij]} .\n \\label{S_A}\n\\ee \nWe also define the rank of $\\rhob$ as} \n\\be\n m\\equiv\\mbox{rank}(\\rhob).\n\\ee\nOther Ricci rotation coefficients used in the paper are \\cite{Pravdaetal04,OrtPraPra07,Durkeeetal10} \n\\be\n L_{1i} = n^a m_{(i)}^b \\nabla_b \\ell_a , \\quad \\tau_i\\equiv L_{i1} =  m_{(i)}^a n^b \\nabla_b \\ell_a ,\\quad \\M{i}{j}{0}=m_{(j)}^a\\ell^b \\nabla_b m_{(i)a} , \\quad \\M{i}{j}{k}=m_{(j)}^am_{(k)}^b \\nabla_b m_{(i)a} .\n \\label{L1i_M}\n\\ee\n\n\nBoost weight (b.w.) zero components of the Weyl tensor  \n\\be\n\\Phi_{ijkl} = C_{ijkl}, \\  \\ \\  \\Phi_{ij} = C_{0i1j},\\ \\ \\  2\\Phia_{ij} = C_{01ij}, \\ \\ \\  \\Phi = C_{0101}\n\\label{bw0}\n\\ee\nare subject to\n\\bea\n&& \\Phi_{ijkl} = \\Phi_{[ij][kl]} = \\Phi_{klij}, \\qquad  \\Phi_{i[jkl]}=0, \\label{Phicycl} \\\\\n&& \\Phis_{ij} \\equiv \\Phi_{(ij)}   = -\\pul\\Phi_{ikjk}, \\qquad \\Phia_{ij} \\equiv \\Phi_{[ij]}, \\qquad \\Phi=\\Phi_{ii}, \\label{PhiS} \n\\eea\nand thus they are determined by the $\\frac{1}{12}(n-1)(n-2)^2(n-3)$ components of $\\Phi_{ijkl}$ and the $\\pul (n-2)(n-3)$ components of $\\Phia_{ij}$. In a type II spacetime, at least one of the latter two quantities must be non-vanishing when $\\bl$ is an mWAND (otherwise the spacetime would be type III or more special). \nRecall that the algebraic subtypes II(a), II(b), II(c) and II(d) are defined by the vanishing of $\\Phi$, the symmetric traceless part of $\\Phi_{ij}$, the ``Weyl part'' of $\\Phi_{ijkl}$, and $\\Phia_{ij}$, respectively (see \\cite{OrtPraPra12rev}). \n\n\nThere is no summation in case one or both repeated indices is\/are  in round brackets, unless it is said otherwise. E.g., there is no summation in $\\Phi_{i(j)k(j)}$ and there is summation in $\\Phi_{ijkj}$ and $\\sum_j \\Phi_{i(j)k(j)}$. \n\n\n{In what follows we shall employ the Sachs equation \\cite{Pravdaetal04,OrtPraPra07,Durkeeetal10}\n\\begin{eqnarray}\n  \\tho \\rho_{ij} &=& - \\rho_{ik} \\rho_{kj} \\label{Sachs},\n\\end{eqnarray}\nand the following  Bianchi identities (eqs.~(A10), (A11) of \\cite{Durkeeetal10})    \n \\begin{eqnarray}\n  \\tho \\Phi_{ij}   &=& -(\\Phi_{ik} + 2\\Phia_{ik} + \\Phi \\delta_{ik}) \\rho_{kj}, \\label{A2}\\label{Bi2}\\\\[3mm]\n  -\\tho \\Phi_{ijkl}   &=& 4\\Phia_{ij} \\rho_{[kl]} - 2\\Phi_{[k|i} \\rho_{j|l]} + 2\\Phi_{[k|j} \\rho_{i|l]} \n                       + 2\\Phi_{ij[k|m} \\rho_{m|l]} . \\label{A4}\\label{Bi3}\n\\end{eqnarray}\nIn a parallelly propagated frame,  the GHP directional derivative  $\\tho$ reduces to the NP derivative along the $\\bl$ direction, $D=\\ell^a\\nabla_a$. \nFor an affinely parametrized $\\bl$ with an affine parameter $r$ this is simply $D=\\partial_r$.}\n\n\n\n\n\n\n\nThere are two possible approaches to studying consequences of the Bianchi and Ricci equations. The first approach consists in applying the derivative operator $\\tho$ on certain algebraic equations and in deriving new algebraic constraints, e.g. eq.~\\eqref{thornB8}, using \\eqref{Sachs}--\\eqref{Bi3} (cf. also appendix~\\ref{app_alg_eq} and \\cite{Ortaggioetal12} (section~3.1 and appendix A therein)). In this paper we mainly use the second approach, which consists in solving the Sachs and Bianchi differential equations \\eqref{Sachs}--\\eqref{Bi3} and then in analyzing the compatibility of their solutions with the algebraic equations \\eqref{B4}--\\eqref{thornB8}.\n\n\n\\section{Results for general type II}\n\\label{sec_genII}\n\n\nThis section is devoted to summarizing some useful results that hold for all type II Einstein spacetimes, without assuming that $\\rhob$ is non-twisting. In particular we present the algebraic restrictions on the Weyl components of b.w. zero that follow from the Bianchi identities, to be used in the next sections. More general new results are given in appendix~\\ref{app_alg_eq} for future reference.\nFor genuine type II spacetimes the (unique) mWAND is necessarily geodesic, while for type D spacetimes there always exists a geodesic mWAND \\cite{DurRea09}. In the frame used below we thus take $\\bl$ to be a geodesic mWAND, without loss of generality. \n \n\n\\subsection{Algebraic constraints}\n\n\nFor type II Einstein spacetimes, the Weyl tensor must obey the algebraic constraints (A.12) and\n(B8) of \\cite{Durkeeetal10} (already discussed in \\cite{PraPraOrt07,Durkee09}) and $\\tho$(B8) (derived in \\cite{Ortaggioetal12}), namely \n\\bea\n2\\Phia_{[jk|}\\rho_{i|l]} -2\\Phi_{i[j}\\rho_{kl]} + \\Phi_{im [jk|}\\rho_{m|l]}  = 0, \\label{B4} \\\\\n\\Phi_{kj} \\rho_{ij} - \\Phi_{jk} \\rho_{ij} + \\Phi_{ij} \\rho_{kj} - \\Phi_{ji} \\rho_{jk} \n       + 2 \\Phi_{ij} \\rho_{jk} - \\Phi_{ik} \\rho  + \\Phi \\rho_{ik} + \\Phi_{ijkl} \\rho_{jl} = 0 \\label{B8} , \\\\\n\\left( 2 \\Phi_{kj} - \\Phi_{jk} \\right) \\rho_{il} \\rho_{jl} + \\left( 2 \\Phi_{ij} - \\Phi_{ji} \\right) \\rho_{jl} \\rho_{kl}  - \\Phis_{ik} \\rho_{jl} \\rho_{jl} + \\Phi \\rho_{il} \\rho_{kl} + \\Phi_{ijkl} \\rho_{js} \\rho_{ls} =0. \\label{thornB8}\n\\eea\n\n\nEq. \\eqref{B8} is traceless and its\nsymmetric and antisymmetric parts read, respectively,\n\\bea\n\\left( 2 \\Phi_{kj} - \\Phi_{jk} \\right) S_{ij} + \\left( 2 \\Phi_{ij} - \\Phi_{ji} \\right) S_{jk}  - \\Phis_{ik} \\rho + \\Phi S_{ik} + \\Phi_{ijkl} S_{jl} =0, \\label{B8sym}\\\\\n\\Phi_{jk} A_{ji} + \\Phi_{ji} A_{kj} + \\Phi_{ij} \\rho_{jk} -\\Phi_{kj} \\rho_{ji} +  \\Phia_{k i} \\rho + \\Phi A_{ik} + \\Phi_{ijkl} A_{jl} = 0 .   \\label{B8asym} \n\\eea\n\n\n\nThe reason for investigating algebraic conditions such as \\eqref{B4}--\\eqref{B8asym} (or those of \\cite{Pravdaetal04} in the case of type III\/N) is that they will in general constrain the possible form of $\\rhob$. This way one obtains the standard Goldberg-Sachs theorem in four dimensions and one can arrive at similar conclusions also in higher dimensions, at least with additional assumptions (e.g. on the Weyl type \\cite{Pravdaetal04}, on the number of dimensions \\cite{Ortaggioetal12}, \non the form of the line-element \\cite{OrtPraPra09,MalPra11}, on the asymptotic behaviour \\cite{OrtPraPra09b}, and\/or on optical properties of $\\bl$, as we shall discuss in the next sections).\n\n\n\\subsection{The optical constraint}\n\n\\label{subsec_OC}\n\n\nIt was observed in \\cite{Ortaggioetal12} that the above conditions on the Weyl tensor appear to be less stringent when $\\rhob$ satisfies\nthe {\\em optical constraint} \\cite{OrtPraPra09}\n\\be\n\\rho_{ik} \\rho_{jk} \\propto \\rho_{(ij)} . \\label{OC2}\n\\ee\nIn particular, when this holds eq.~\\eqref{thornB8} is not an extra restriction. \n{This thus suggests that the branch of type II solutions {whose mWAND} obeys \\eqref{OC2} corresponds to the case of a ``generic'' Weyl tensor \\cite{Ortaggioetal12}.}\nIt has been proven that \\eqref{OC2} indeed holds for {the Kerr-Schild vector} of all (generalized) Kerr-Schild spacetimes \\cite{OrtPraPra09,MalPra11} (including Myers-Perry black holes \\cite{MyePer86}), for non-degenerate geodesic double WANDs in asymptotically flat type II spacetimes \\cite{OrtPraPra09b} (see eq.~(14) therein) {and for the unique double WAND of all genuine type II Einstein spacetimes in five dimensions (see footnote~\\ref{foot_typeD} for the type D case) \\cite{Ortaggioetal12,Wylleman_priv}.} \nNevertheless, {double WANDs} violating the optical constraint also exist, as shown by explicit examples constructed below in section~\\ref{subsubsec_examples_distinct} for $n\\ge 6$, in appendix~\\ref{app_violating} for $n\\ge 7$, and in section~6.3 of \\cite{Ortaggioetal12} for $n=5$.\\footnote{We observe that all such examples are of type D. In fact, in five dimensions {\\em all} type D Einstein spacetimes admitting a geodesic mWAND violating the optical constraint are known \\cite{Ortaggioetal12} (and coincide with the class of Einstein spacetimes admitting a non-geodesic mWAND \\cite{DurRea09}). Such ``exceptional'' null directions are necessarily twisting in 5d (but not in higher dimensions). \nHowever, in all such 5d type D spacetimes there always also exists a pair of non-aligned (non-twisting) mWANDs that {\\em do} obey the optical constraint \\cite{Ortaggioetal12}. On the other hand, this is generically not true when $n>5$ (see appendix \\ref{subsubsec_ex2} for a ten-dimensional example admitting exactly two mWANDs, both violating the optical constraint).\\label{foot_typeD}}\nAlthough the above analysis applies only to type D and genuine type II  spacetimes (eqs.~(\\ref{B4})--(\\ref{thornB8}) become trivial identities for more special types), it is worth remarking that the optical constraint is also obeyed by the mWAND of all type N Einstein spacetimes \\cite{Pravdaetal04,Durkeeetal10}. {The mWAND $\\bl$ of type III Einstein spacetimes also obeys the optical constraint provided either \\cite{Pravdaetal04}: (i) the spacetime is five-dimensional; (ii) the Weyl tensor satisfies a genericity condition (see~\\cite{Pravdaetal04}); (iii) $\\bl$ is  {\\it non-twisting} (see also appendix \\ref{app_type_N_III} for a simpler proof in case (iii))}. \n\n \nThe optical constraint implies that $(\\Id- \\frac{2}{\\alpha} \\rhob)$ is an orthogonal matrix \\cite{Ortaggioetal12} and that consequently  $[\\rhob, \\rhob^T]$ vanishes \\cite{OrtPraPra09,Ortaggioetal12}. The optical matrix $\\rhob$ is therefore a {\\em normal} matrix and can thus be put, using spins,  into a convenient block-diagonal form {(see \\cite{OrtPraPra09,OrtPraPra10} for extended related discussions)}, i.e.,\n\\bea\n\\rhob = \\alpha{\\rm diag}\\left(1, \\dots 1, \n\\frac{1}{1+ \\alpha^2 b_1^2}\\left[\\begin {array}{cc} 1 & -\\alpha b_1 \\\\ \n   \\alpha b_1 & 1  \\label{canformL} \\\\\n  \\end {array}\n \\right]\n, \\dots, \n\\frac{1}{1+ \\alpha^2 b_\\nu^2} \\left[\\begin {array}{cc} 1 & -\\alpha b_\\nu \\\\ \n   \\alpha b_\\nu & 1 \\\\ \\end {array}\n \\right] \n , 0, \\dots ,0\n\\right).\n\\eea\nThe block-diagonal form \\eqref{canformL} {is useful for practical purposes because it} allows for determining the $r$-dependence of $\\rhob$ by integrating the Sachs equation \\eqref{Sachs} {(in a parallelly transported frame)} \\cite{OrtPraPra10}. {\\em In the non-twisting case it reduces to a sequence of 1s followed by a sequence of 0s} (up to an overall factor). Note that the symmetric part of each two-block is proportional to a two-dimensional identity matrix, i.e. it is ``shear-free''. In four dimensions the optical constraint implies that either there is a single such block (in which case $\\rho_{ij}$ is shearfree) or that $\\rho_{ij}$ is symmetric with exactly one non-vanishing eigenvalue. However, the Goldberg-Sachs theorem shows that the latter case cannot occur. Therefore in 4d the optical constraint is a necessary condition for $\\bl$ to be a repeated principal null direction but it is not sufficient \\cite{Ortaggioetal12}. \n\n \nOn the other hand, in higher dimensions a vector field $\\bl$ obeying the optical constraint is in general {\\em shearing}, as in the case of Myers-Perry black holes \\cite{MyePer86} (see also \\cite{FroSto03,PraPraOrt07} for a discussion of their optical properties). Together with other results \\cite{Pravdaetal04,PodOrt06,OrtPraPra07}, this has made clear that the shearfree condition is in general ``too restrictive'' in higher dimensions, as opposed to the four-dimensional case. In particular, it was observed in \\cite{OrtPraPra07} that in {\\em odd} dimensions a twisting geodesic mWAND is necessarily shearing. By contrast, twisting geodesic mWANDs with zero shear are permitted in {\\em even} dimensions and they have necessarily $\\det(\\rhob)\\neq 0$ (as can be easily seen in a frame adapted to $A_{ij}$, using the fact that $S_{ij}\\propto\\delta_{ij}$). An explicit example in six dimensions is discussed in appendix \\ref{app_shearfreetwist}.\n\n\n\nIn the non-twisting case, the existence of shearfree spacetimes has been already known for some time in all dimensions -- they are either \\RT \\cite{PodOrt06} or Kundt \\cite{PodZof09} solutions, according to the presence\/absence of expansion.\n \n\n\n\n\n\n\\subsection{Possible generalizations of the geodesic{\\&}shearfree property}\n\n\nIn arbitrary dimensions various geometric conditions can be considered which are different from the standard geodesic{\\&}shearfree condition (considered above) and which, however, become all equivalent (except for the optical constraint, cf.~\\cite{Ortaggioetal12}) in the special case $n=4$.  Further such conditions are discussed in appendix~\\ref{app_shearfree}. \nOne could thus conceive that various formulations of the Goldberg-Sachs theorem are in principle possible when $n>4$. Some of these formulations have been studied in \\cite{Taghavi-Chabert11,Taghavi-Chabert11b,Ortaggioetal12} (see also \\cite{OrtPraPra09}) but none of these gives necessary and sufficient conditions. In the rest of this paper we will discuss necessary conditions determined by the presence of a non-twisting mWAND in an $n$-dimensional Einstein spacetime and we will present several explicit examples. A possible interpretation of these results in terms of the geometric conditions of appendix~\\ref{app_shearfree} will be discussed in sections~\\ref{sec_totgeod} (for $n\\ge6$) and \\ref{subsec_integrability} (for $n=6$).\n\n\n\n\n\n\\section{Non-twisting $\\bl$: general properties}\n\\label{sec_nontw_gen}\n\n\n\nHere we study the optics of a hypersurface orthogonal mWAND $\\bl$ in a type II Einstein spacetime.  This mWAND is automatically geodetic. Thus we have \n\\be\n \\kappa_i=0=A_{ij} ,\n\\ee\n{so that $\\rho_{ij}=S_{ij}$.} Since the algebraic equations \\eqref{B4}--\\eqref{B8asym}  are trivial for  Kundt spacetimes, in what follows we assume $\\rhob\\not= ${\\bf 0}.\n\n\n\n\\subsection{Case $\\Phia_{ij} \\neq 0$}\n\n\\label{sec_PhiA}\n\nThis case has been already analyzed in \\cite{Ortaggioetal12}, {arriving at} point (i) of theorem~\\ref{prop_GSHD} of section~\\ref{sec_intro}. As already observed there, in this case $\\rhob$ satisfies the optical constraint and it is  necessarily degenerate ($m=2$).  Moreover it is shearing, except for $\\rhob=0$ {(which is necessarily the case for $n=4$)}.\n\nIt remains to consider the case when $\\Phi^A_{ij}=0$. \n\n\n\\subsection{Case $\\Phia_{ij}=0$}\n\n\nThis case defines the subtype II(d) {in the notation of \\cite{Coleyetal04}}. For  $\\Phia_{ij}=0$, eq.~\\eqref{B8} reduces to  (recall $A_{ij}=0$ here)\n\\be\n -\\rho \\Phi_{ik} + \\Phi \\rho_{ik} + 2\\Phi_{ij} \\rho_{jk} + \\Phi_{ijkl} \\rho_{jl}=0 ,  \\label{eqn:algI}\n\\ee\nand eq. \\eqref{B8asym}  to \n\\be\n[\\rhob,\\Phib^S]=0.\\label{comm_rho_Phi}\n\\ee\nNote that taking into account  (\\ref{comm_rho_Phi}), eq. (\\ref{eqn:algI}) is symmetric and corresponds to eq.~\\eqref{B8sym}.\n\nSimilarly, eq. \\eqref{thornB8} yields \n\\be\n -\\Phi_{ik} \\rho_{jl} \\rho_{jl} + \\Phi (\\rho^2)_{ik} + 2 \\Phi_{ij} (\\rho^2)_{jk} + \\Phi_{ijkl} (\\rho^2)_{jl} = 0\\label{eqn:rhoC}.\n \\ee\n\n  \n\nThanks to \\eqref{comm_rho_Phi}, one can choose a {basis where both $\\rho_{ij}$ and $\\Phi_{ij}$ are diagonal}, $\\rho_{ij}=\\diag(\\rho_2,\\rho_3,\\dots)$,  \n$\\Phi_{ij}=\\diag(\\Phi_2,\\Phi_3,\\dots)$, { therefore $\\Phi_{ijkj}=0$ for $i\\neq k$}.\nIn this frame, {the off-diagonal components of the algebraic constraint~(\\ref{eqn:algI}) are} \n\\be \n \\sum_j\\rho_{(j)}\\Phi_{i(j)k(j)}=0 \\qquad (\\mbox{for } k\\neq i) .\n \\label{nondiag_constr}\n\\ee\n The diagonal part of~(\\ref{eqn:algI}) can be expressed as  ${\\cal L}_{ij} \\tilde \\rho_j =0$, \nwhere $\\tilde \\rho_{i}$ is the  vector  $\\tilde \\rho_{i}=(\\rho_2, \\rho_3 \\dots)$  \nand the linear operator ${\\cal L}_{ij}$ is given by\n\\[\n  {\\cal L}_{ij} = W_{ij} + \\Phi  \\delta_{ij} + 2 \\diag (\\Phi_{2},\\Phi_{3},\\dots \\Phi_{n}) - \\left[ \n                     \\begin{array}{cccc}\n                       \\Phi_{2}                  & \\Phi_{2}        & \\dots & \\Phi_{2}     \\\\\n                       \\Phi_{3}       & \\Phi_{3}       & \\dots & \\Phi_{3}   \\\\\n                       \\vdots    & \\vdots    & \\dots & \\vdots\\\\\n                       \\Phi_{n} & \\Phi_{n} & \\dots & \\Phi_{n}\n                     \\end{array}\n               \\right]\n\\]\nor, equivalently, ${\\cal L}_{ij} = W_{ij} +( \\Phi + 2\\Phi_{(i)} )  \\delta_{ij} -\\Phi_{(i)} $, where {we have defined the symmetric and traceless matrix}\n\\be\n\tW_{ij} \\equiv C_{(i)(j)(i)(j)} \\qquad {\\mathrm{\\ \\ (no\\ summation)}}.\\label{def_W}\n\\ee\n{Note that $W_{(i)(i)}=0$.} From  the first of \\eqref{PhiS}, it follows  \n\\be\n\\sum_j W_{ij} = - 2 \\Phi_i ,\n \\label{W_phi}\n\\ee\n{which will be used in certain calculations throughout the paper.} (From  \\eqref{W_phi} \nit follows that $W_{ij}=0\\Rightarrow {\\cal L}_{ij}=0$.)  \nThus the sum of the rows of ${\\cal L}_{ij}$ vanishes and so they are  linearly dependent and \n\\be\n\\det {\\cal {\\bm{L}}} =0. \\label{detL}\n\\ee \nTherefore  zero is an eigenvalue of ${\\cal L}$ and  (the diagonal part of) (\\ref{eqn:algI})  admits non-trivial solutions.\n\nNote that (\\ref{eqn:rhoC}) has the form ${\\cal L}_{ij} {\\tilde \\rho}^2_j =0$, where components of  the $(n-2)$ dimensional vector ${\\tilde \\rho}^2$ are squares of components of ${\\tilde \\rho}$.  The characteristic polynomial of ${\\cal L}$ is \n\\be\n \\det({\\cal L}-\\lambda I)=\\lambda^{n-2} + k_{n-3} \\lambda^{n-3} + \\dots + k_1 \\lambda + k_0 ,\n\\ee\nwith $k_0=0$ due to (\\ref{detL}).\nNow zero is a multiple eigenvalue of ${\\cal L}$ iff  $k_1=0$. Let us observe that for a generic form of a type II Weyl tensor $k_1$ is non-vanishing {and thus zero is a single eigenvalue \nof ${\\cal L}$}.\\footnote{For example in five dimensions $k_1 = 12 \\left(\\phi_2 \\phi_4+ \\phi_3 \\phi_4 + \\phi_2 \\phi_3 \\right)$. {If we consider, for instance, five-dimensional} \\RT spacetimes ({which coincide with the Schwarzschild solution plus a possible cosmological constant}), characterized by $\\rho_{ij} \\propto \\delta_{ij}$,  equation \\eqref{eqn:algI} implies $\\Phi_{ij} \\propto \\delta_{ij}$, which clearly leads to $k_1 \\not=0$.} \nIf this is the case then   ${\\tilde \\rho}^2$ is proportional to ${\\tilde \\rho}$ and therefore \n\\be\n\\rho_{ij}= \\alpha \\diag(1,1, \\dots 1,0,\\dots 0).\n\\ee\nRecalling also point (i) of theorem~\\ref{prop_GSHD},\nwe can conclude with \n\n\\begin{prop}\nFor non-twisting type II Einstein spacetimes with $k_1 \\not=0$, all non-vanishing  eigenvalues of the optical matrix $\\rho_{ij}$ coincide. \n\\label{prop_nontwist}\n\\end{prop}\n\nIf $\\Phia_{ij} \\not= 0$, the stronger result of point (i) of theorem~\\ref{prop_GSHD} holds.\n\n\n\nAs we will see in sections~\\ref{subsec_alldistinct} and \\ref{subsec_permitted_6D} (table~\\ref{tab_6D}), \nthere do exist non-twisting type II Einstein spacetimes with distinct non-vanishing eigenvalues of $\\rhob$. \nAccording to Proposition~\\ref{prop_nontwist}, for these spacetimes $k_1$ vanishes. In particular $\\cal L$, and therefore also $k_1$, vanishes for spacetimes with $W_{ij}=0$. We will study special cases with various parts of the Weyl tensor vanishing in the following sections.\n\n\n\\section{Non-twisting $\\bl$: non-degenerate $\\rhob$ ($m=n-2$)}\n\n\n\\label{sec_nondeg}\n\nFirst, let us consider the case of a non-degenerate $\\rhob$ (i.e., $\\det\\rhob\\not= 0$ -- this is relevant, e.g., for asymptotically flat algebraically special spacetimes \\cite{OrtPraPra09b}). When $\\det\\rhob\\not= 0$ one necessarily has (see point (i) of theorem~\\ref{prop_GSHD}) \n\\be\n \\Phia_{ij}=0 ,\n\\ee\nso that the type is II(d). As noticed above we can use a frame in which both $\\rho_{ij}$ and $\\Phi_{ij}$ are diagonal, which is moreover compatible with parallel transport \\cite{PraPra08,OrtPraPra10}. Thus the eigenvalues of the optical matrix  are \\cite{PraPra08}\n\\be\n \\rho_i=\\frac{1}{r-b_i} .\n \\label{rho_i}\n\\ee\n\n{In order to discuss the possible structures of $\\rhob$ it is convenient to discuss separately various cases in which different parts of the Weyl tensor are (non-)zero.}\n\n\n\\subsection{Cases $\\Phi_{ij}\\neq0$ and $\\Phi_{ij}=0\\neq W_{ij}$}\n\n\\label{subsec_nondeg_Phi}\n\nFrom Bianchi equation~\\eqref{Bi2}  one has\n\\beqn\n D\\left(\\frac{\\Phi_{(i)}}{\\rho_{(i)}}\\right)=-\\Phi .\n \\label{nondeg_DPhi_i}\n\\eeqn\nThe case when all  $\\rho_i$ coincide (and are non-zero) is the known \\RT case, for which also all  $\\Phi_i$ must coincide {\\cite{PraPra08,PodOrt06}.}\n{Thus} let us  consider here the case when at least two eigenvalues $\\rho_i$ are different. \n\n\nSince the r.h.s. {of~(\\ref{nondeg_DPhi_i})} is the same for any $i$, we obtain that either all $\\Phi_i=0$, or\n\\be\n \\Phi_{i}=\\rho_i(A+\\Phi^0_{(i)}) ,\n \\label{Phi_i}\n\\ee\nwhere $A=A(r)$ is a function that must satisfy the equation $DA+A\\rho=-\\Phi^0_{j}\\rho_j$ (since $\\Phi=A\\rho+\\Phi^0_{j}\\rho_j$) and, from now on, quantities with a superscript $^0$ do not depend on $r$. Using the intermediate substitution $A(r)=Y(r)\\prod_k\\rho_k$, one arrives at the solution \n\\be\n A=\\left(\\prod_k\\frac{1}{r-b_k}\\right)\\left[-\\sum_j\\Phi^0_{j}\\int\\d r\\prod_{l\\neq j}(r-b_l)+A^0\\right] .\\label{A_nondeg}\n\\ee\nFor subsequent discussions it is useful to rewrite \\eqref{A_nondeg} using partial fraction decomposition, i.e., \n\\be\nA=\\frac{P_{n-2}}{\\prod_k {(r-b_k)}}=a^0+\\frac{P_{n-3}}{\\prod_k {(r-b_k)}}=a^0+\n\\sum_{K=1}^{\\alpha_{max}} \\sum_{\\forall i,  \\alpha_i\\geq K}\\frac{c^{(K)}_{(i)}}{(r-b_i)^K} ,\\label{A_decomposed}\n\\ee\nwhere $P_k$ denotes a polynomial of order $k$ in $r$,  $c^{(K)}_{(i)}$ {do not depend on $r$} and $\\alpha_i$ denotes the multiplicity of $b_i$, with $\\alpha_{max}$ being the maximal multiplicity. {In particular, the term $a^0$ can be determined by looking at} the leading term of \\eqref{A_nondeg} in the limit $r\\to\\infty$, i.e., \n\\be\n A=-\\frac{1}{n-2}\\sum_k\\Phi^0_k+O(r^{-1}) ,\n\\ee\nso that\n\\be\na^0=-\\frac{1}{n-2}\\sum_k\\Phi^0_k.\\label{am}\n\\ee\nIn the case $\\Phi^0_j=0$, $a^0$ vanishes and  one has  simply $A=A^0\/\\prod_k {(r-b_k)}$. \n\n\nNext, the equation for $W_{ij}$ (see \\eqref{Bi3}) can be written as (but recall $W_{(i)(i)}=0$)\n\\beqn\n & & D\\left(\\frac{W_{ij}}{\\rho_{(i)}-\\rho_{(j)}}\\right)=\\frac{\\Phi_{i}\\rho_j+\\Phi_{j}\\rho_i}{\\rho_{(i)}-\\rho_{(j)}} \\qquad (\\rho_{j}\\neq\\rho_{i}) , \\\\\n & & D\\left(\\frac{W_{ij}}{\\rho_{(i)}^2}\\right)=\\frac{\\Phi_{i}+\\Phi_{j}}{\\rho_{(i)}} \\qquad (\\rho_{j}=\\rho_{i}) .\n\\eeqn\nUsing the above expression for $\\Phi_i$, in both cases one can write the solution as\n\\be\n W_{ij}=\\frac{1}{(r-b_{(i)})(r-b_{(j)})}\\left[2\\int\\d r A+(\\Phi^0_{i}+\\Phi^0_{j})r+ W^0_{ij}\\right] .\\label{Wij_nondeg}\n\\ee\n\nNow, imposing $-2\\Phi_i=\\sum_jW_{ij}$ {(recall~(\\ref{W_phi}))} we obtain the constraint\n\\be\n -2(A+\\Phi^0_{i})=\\sum_{j\\neq i}\\frac{1}{r-b_{(j)}}\\left[2\\int\\d r A+(\\Phi^0_{i}+\\Phi^0_{j})r+ W^0_{ij}\\right] .\n \\label{constrW}\n\\ee\n{For $r\\to\\infty$, using \\eqref{A_decomposed} and \\eqref{am} at \nthe leading order (\\ref{constrW}) implies that all  $\\Phi^0_{i}$ coincide, i.e., }\n\\be\n \\Phi^0_{i}=\\frac{1}{n-2}\\sum_k\\Phi^0_k\\equiv{f^0}. \n \\label{Phi_i0}\n\\ee\n$A$ thus becomes\n\\be\n A=-f^0+A^0\\prod_k\\frac{1}{r-b_k} ,\n \\label{A_2}\n\\ee\nwhere we have used $\\sum_j\\prod_{l\\neq j}(r-b_l)=D[\\prod_k(r-b_k)]$, so that\n\\be\n \\Phi_{i}=\\frac{A^0}{r-b_{i}}\\prod_k\\frac{1}{r-b_k} , \\qquad W_{ij}=\\frac{1}{(r-b_{(i)})(r-b_{(j)})}\\left[2A^0r\\prod_k\\frac{1}{r-b_k}+ W^0_{ij}\\right] .\n \\label{nondeg_Weyl_simplif}\n\\ee\n\n\nThe constraint~(\\ref{constrW}) can thus be written as\n\\be\n -2A^0\\prod_k\\frac{1}{r-b_k}=\\sum_{j\\neq i}\\frac{1}{r-b_{(j)}}\\left[2A^0\\int\\d r\\prod_k\\frac{1}{r-b_k}+W^0_{ij}\\right] \\qquad (i=2,\\ldots,n-1) .\n \\label{constrW2}\n\\ee\n\n{It is now useful to discuss separately various subcases with different multiplicity of eigenvalues.} \n\n\\subsubsection{All the $b_i$ coincide: shearfree congruences (Robinson-Trautman spacetimes)}\n\nIn the \\RT case all the $b_i$ {of (\\ref{rho_i})} coincide {(and can be set to zero by shifting $r$, if desired)} and {from~(\\ref{constrW2})} we simply obtain $\\sum_{j}W^0_{ij}=0$. \n\nOn the other hand, let us assume in the following that not all $b_i$ coincide, i.e., at least two of these are distinct, say $b_2\\neq b_3$. \n\n\\subsubsection{Shearing case with all $b_i$ distinct: not permitted}\n\nFirst, if all  $b_i$ are distinct, {i.e., $\\alpha_{max}=1$ in \\eqref{A_decomposed}}, we can compute explicitly the required integral using partial fraction decomposition \n\\be\n \\int\\d r\\prod_k\\frac{1}{r-b_k}=\\sum_k\\frac{\\ln(r-b_k)}{\\prod_{l\\neq k}(b_k-b_l)} \\qquad \\mbox{($b_k$ all distinct)}.\n\\ee\nIt is thus clear that the singularity structure of the l.h.s. and the r.h.s. of (\\ref{constrW2}) (for any $i$) cannot be the same unless $A^0=0$. Therefore, {from~(\\ref{constrW2})} also $\\sum_{j}(r-b_{(j)})^{-1}W^0_{ij}=0$. However, this condition  implies that  $b_i$ cannot be all distinct, so that this case is in fact not permitted (unless $W^0_{ij}=0$, so that $W_{ij}=0$ and therefore also $\\Phi_{ij}=0$, contrary to our assumptions here -- however, we will see in sections~\\ref{subsec_nondeg_Phi3} and \\ref{subsec_nondeg_Phi4}  below that the case with all distinct $b_i$ is ruled out also for $W^0_{ij}=0$). \n\n\n\\subsubsection{Shearing case with at least one $b_i$ repeated}\n\\label{subsec_onerep}\n\nLet us now consider the case when at least one $b_i$ is repeated, say $b_2$, with multiplicity $1<\\alpha_2<n-2$, and there exists $b_3\\neq b_2$ (this is not a restriction since we are excluding here the \\RT case, which corresponds to $\\alpha_2=n-2$). \n\nUsing partial fraction decomposition {similarly as in}~\\eqref{A_decomposed}, the left and right hand sides {of (\\ref{constrW2}) take} the form (for $i=3$)\n\\bea\n\\mbox{lhs}&=&\n-2A^0\\left[\\frac{p_{\\alpha_2}}{(r-b_2)^{\\alpha_2}}+(\\mbox{terms with lower order poles at } b_2 \\mbox{ and with no poles at }b_2)\\right],\\nonumber\\\\\n\\mbox{rhs}&=&\n -2A^0\\frac{p_{\\alpha_2}\\alpha_2}{(\\alpha_2-1)(r-b_2)^{\\alpha_2}} +(\\mbox{terms with lower order poles at } b_2 \\mbox{ and with no poles at }b_2) , \\nonumber\n\\eea\nrespectively, where $p_{\\alpha_2}\\not= 0$. Comparing the highest order terms in $1\/(r-b_2)$ we thus  get\n\\be\n A^0=0, \\qquad \\sum_{j}\\frac{W^0_{ij}}{r-b_{(j)}}=0 .\n\\ee\n\n\nIn particular, it follows from (\\ref{nondeg_Weyl_simplif}) that \n\\be \n \\Phi_i=0 ,\n\\ee\nso that the Weyl type is II(abd), and \\eqref{Wij_nondeg} thus gives \n\\be\n W_{ij}=\\frac{W^0_{ij}}{(r-b_i)(r-b_j)} ,\n \\label{Wij}\n\\ee\nwith $W^0_{ij}=W^0_{ji}$ and $W^0_{(i)(i)}=0$.\n\nNow, because of (\\ref{Wij}) the condition $\\sum_jW_{ij}=0$ constraints the possible multiplicities of  $b_i$ (i.e., of the eigenvalues of $\\rho_{ij}$). First, if $b_i$ are all distinct, we immediately get $W_{ij}^0=0$, {as already discussed above}. Similarly, recalling $W_{ij}^0=W_{ji}^0$ it is easy to see that for $W_{ij}^0\\neq0$ the structures  $\\{a,a,c_1,\\ldots,c_{n-4}\\}$ and  \n $\\{a,a,a,c_1,\\ldots,c_{n-5}\\}$ are also forbidden.\\footnote{To see that \n $\\{a,a,a,c_2,\\ldots,c_{n-5}\\}$ cannot occur, assume $b_2=b_3=b_4$ and all remaining eigenvalues are single. Then $\\sum_jW_{ij}=0$ with (\\ref{Wij}) gives\n$W^0_{i2}+W^0_{i3}+W^0_{i4}=0$ and $W^0_{i\\mu}=0$ ($\\mu\\neq 2,3,4$). Since $W^0_{ij}=W^0_{ji}$, this implies that the only possible non-zero components are $W_{23}$, $W_{24}$ and $W_{34}$, with the conditions $W_{23}^0+W_{24}^0=0$, $W_{32}^0+W_{34}^0=0$ and $W_{42}^0+W_{43}^0=0$. However, this system admits only the solution $W_{23}^0=W_{24}^0=W_{34}^0=0$, so that $W_{ij}=0$.\\label{foot_311}}  \nHowever, the structure  \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ is compatible with such constraints (take, e.g., $b_2=b_3$, $b_4=b_5$ and $W_{34}^0=-W_{35}^0=W_{25}^0=-W_{24}^0\\neq0$, {and the remaining $W_{ij}$ equal to zero}). \n\nTo summarize, we have seen above that in type II Einstein spacetimes with a non-degenerate non-twisting mWAND $\\bl$, one has $\\Phia_{ij}=0$ and\n\\begin{enumerate}\n \\item if $\\Phi_{ij}\\neq0$, then $\\bl$ must be shearfree and the corresponding spacetimes belong to the \\RT class. {This includes, in particular, the result of \\cite{OrtPraPra09b} for asymptotically flat type II vacuum spacetimes (restricted to non-twisting case)}\\footnote{Ref.~\\cite{OrtPraPra09b} used an expansion method. Instead of the condition $\\Phi_{ij}\\neq 0$, a condition on the asymptotic fall-off behaviour of the Weyl tensor was assumed there. In the present notation, that amounts to taking $W_{ij}^0=\\Phi^{\\{3\\}}_{ijkl}=\\Phi^{\\{4\\}}_{ijkl}=0$, which requires that $\\Phi_{ij}\\neq 0$ (otherwise the type would be III). The assumptions of \\cite{OrtPraPra09b} in the non-twisting case were thus stronger than those used here (note indeed that \\RT spacetimes with $W_{ij}^0\\neq 0$ do exist \\cite{PodOrt06}). {(The symbols $\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$ are defined in sections~\\ref{subsec_nondeg_Phi3} and \\ref{subsec_nondeg_Phi4} below.)}} \n\\item if $\\Phi_{ij}=0$ (type II(abd)) the structure of eigenvalues of $\\rho_{ij}$ can be more generic, however for $\\Phi_{ij}=0\\neq W_{ij}$ it must have the multiplicities $\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special). \n\\end{enumerate} \n\n\nLet us now discuss the remaining cases (in which, without loss of generality, we could assume $\\Phi_{ij}=0=W_{ij}$ --  however this will actually not be used to prove the following results).\n\n\n\n\n\\subsection{Case $\\Phi^{\\{3\\}}_{ijkl}\\neq0$} \n\n\\label{subsec_nondeg_Phi3}\n\n\n{Here we assume that a component of $\\Phi_{ijkl}$ with precisely three distinct values of $i,j,k,l$ is non-vanishing,\ni.e., for some $i\\not=k$, we have $\\Phi_{i(j)k(j)} \\not=0$.} This case is possible {only for $n\\ge6$} because we need at least three possible distinct values for $i,j,\\ldots$ and because of the tracefree condition $\\Phi_{ijik}=0$ with $k\\neq j$ (however, {for $n=4,5$ necessarily $\\Phi_{ij}\\neq0$} and from the above discussion one already knows that \\RT is the only possibility).\n\nEq.~\\eqref{B4}  gives\n\\be\n \\Phi_{(i)(j)(i)(k)}(\\rho_{(j)}-\\rho_{(k)})=0 \\qquad (k\\neq j) .\\label{B4offdiag}\n\\ee\nWithout loss of generality, we can assume $\\Phi_{2324}\\neq0$, so that necessarily $b_3=b_4$. \n\nNow, the off-diagonal ($i\\not=k$) components of \\eqref{Bi3} read\n\\be\n-D \\Phi_{i(j)k(j)} = \\Phi_{i(j)k(j)} \\rho_{(j)} + \\Phi_{i(j)k(j)} \\rho_{(k)}.\n\\ee\nTaking into account $\\rho_k = (r-b_k)^{-1}$, we arrive at\n\\be\n\\Phi_{i(j)k(j)} = \\frac{\\Phi^{0}_{i(j)k(j)}}{(r-b_{(j)})(r-b_{(k)})}  \\qquad (i \\not= k). \\label{Phi_ijkj}\n\\ee\n\n\nThen,    \nthe condition $\\Phi_{ijkj}=0$, $k\\not=i$, implies that at least another component $\\Phi_{(i)3(i)4}$ must be non-zero, say $\\Phi_{5354}\\neq0$, and that $b_5=b_2$. The eigenvalue structure must therefore again be  \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special).\n\n\n\\subsection{Case $\\Phi^{\\{4\\}}_{ijkl}\\neq0$} \n\n\\label{subsec_nondeg_Phi4}\n\nHere we assume that there exists a non-vanishing component of $\\Phi_{ijkl}$ with all values of $i, j, k, l$ being distinct. This case is possible only for $n\\ge6$. \n\nSimilarly as in the previous case, eq.~\\eqref{Bi3} implies\n\\be\n \\Phi_{ijkl} = \\frac{\\Phi^{0}_{ijkl}}{(r-b_{(k)})(r-b_{(l)})},  \\qquad  {\\mathrm{\\ (for \\ }} i,j,k,l {\\mathrm{\\ all \\ distinct}}). \n\\ee\nHowever, since $\\Phi_{ijkl}=\\Phi_{klij}$ we obtain $b_i=b_k$ and $b_j=b_l$ (or $b_i=b_l$ and $b_j=b_k$) and the eigenvalue structure is again \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special). \n\n\\subsection{Summary}\n\nNo further cases are possible, since $\\Phi^{\\{4\\}}_{ijkl}=\\Phi^{\\{3\\}}_{ijkl}=W_{ij}=0$ implies the type III. \nTo summarize, {in sections~\\ref{subsec_nondeg_Phi}--\\ref{subsec_nondeg_Phi4}} we have thus shown that\n\\begin{prop}\n\n\n\\label{prop_non_deg2}\n\nFor type II  Einstein spacetimes, {the existence of} a non-twisting, non-degenerate double WAND implies that the algebraic type is necessarily II(d), i.e. $\\Phia_{ij}=0$, or more special, and\n\n\\begin{enumerate}\n\n\\item if $\\Phi_{ij}\\neq 0$ the spacetime is shearfree (Robinson-Trautman), i.e. the eigenvalue structure of $\\rho_{ij}$ is \n $\\{a,\\ldots,a\\}$ {(with $a\\neq0$)}. This is {the only possibility when} $n=4,5$.\\footnote{Since type II(abd) coincides with type III in those dimensions.} \nOne has $\\Phi_{ij}=A^0\\delta_{ij}\/(r-b_0)^{n-1}$, so that the type is D(bd) \n(one can set $b_0=0$ by a shift of the affine parameter $r$).\n\n\n\\item if $\\Phi_{ij}=0$ {(type II(abd))} the structure is  \n$\\{a,a,b,b,c_1,\\ldots,c_{n-6}\\}$ (or more special; $a,b,c_\\alpha\\neq0$). In particular, in six dimensions this means \n$\\{a,a,b,b\\}$ (see also section~\\ref{sec_6D}).\n\n\n\\end{enumerate}\n\\end{prop}\nThis is nothing but a more detailed version of points (\\ref{RT}) and (in the subcase $\\det\\rhob\\neq0$) (\\ref{Phi0}) of theorem~\\ref{prop_GSHD} of section~\\ref{sec_intro}, which are thus proven (the proof of  (\\ref{Phi0}) will be completed in section~\\ref{sec_nontwist} for any value of rank($\\rhob$), cf. remark~\\ref{rem_Phi0}).\nIn particular, (for $n>5$) there are always at least two double eigenvalues. \nIn this sense, we will see that the situation is different in the degenerate case. {Note that spacetimes of point 1. (all explicitly known \\cite{PodOrt06}) obey the optical constraint, whereas those of point 2. do not, in general. Examples of the latter in $n\\ge7$ dimensions will be provided in appendix~\\ref{app_violating}.}\n\n\n\n\n\n\\section{Non-twisting $\\bl$: degenerate $\\rhob$ ($0<m<n-2$) }\n\n\\label{sec_nontwist}\n\nLet $m$ denote the rank of $\\rho_{ij}$. The value $m=n-2$ corresponds to the previously considered non-degenerate case, while $m=0$ defines Kundt. Therefore here we shall restrict to $0<m<n-2$ .\nWe need to define two types of indices to distinguish between non-vanishing and vanishing eigenvalues, namely\n\\beqn\n & & \\rho_p=\\frac{1}{r-b_p} \\qquad (p,q,o,t=2,\\ldots, m+1) , \\\\\n & & \\rho_z=0 \\qquad (z,v,w,y=m+2,\\ldots, n-1) . \\label{rho_z_0}\n\\eeqn\n\nRecall that (point (i) of theorem~\\ref{prop_GSHD}) \nin the non-twisting case with $\\Phi^A_{ij}\\neq0$ one has $m=0,2$ and $\\rho_{ij}=\\mbox{diag}(a ,a ,0,\\ldots,0)$ for any $n>4$, {so that this case does not require further investigation}. \nIn all remaining cases we thus have \n\\be\n \\Phi^A_{ij}=0 \\qquad (m\\neq 0,2).\n\\ee\n{In the following we will give the $r$-dependence of the Weyl components, which is then used to constraint the possible forms of $\\rhob$. In particular, we shall explore under what conditions  all the eigenvalues of $\\rhob$ {can} be distinct (which is not permitted in the non-degenerate case). We shall also discuss some special cases and construct explicit examples.}\n\n\n\\subsection{$\\Phi_{ij}$ and $W_{ij}$ components}\n\n\\label{subsec_Phi_W_deg}\n\nProceeding similarly as in section~\\ref{sec_nondeg} but {(thanks to (\\ref{rho_z_0}))} with the additional equation $D\\Phi_z=0$, one finds\n\\beqn\n & & \\Phi_{p}=\\rho_p(A+\\Phi^0_{(p)}) , \\qquad \\Phi_{z}=\\Phi_{z}^0 , \\\\\n & & W_{pq}=\\frac{1}{(r-b_{(p)})(r-b_{(q)})}\\left[2\\int\\d r A+(\\Phi^0_{p}+\\Phi^0_{q})r+ W^0_{pq}\\right] , \\\\\n & & W_{pz}=\\frac{1}{r-b_{(p)}}(\\Phi^0_zr+W_{pz}^0) , \\qquad W_{zv}=W_{zv}^0 ,\n\n\\eeqn\nwhere \n\\be\n A=\\left(\\prod_o\\frac{1}{r-b_o}\\right)\\left[-\\sum_p\\Phi^0_{p}\\int\\d r\\prod_{q\\neq p}(r-b_q)-\\sum_z\\Phi^0_z\\int\\d r\\prod_{q}(r-b_q)+A^0\\right] .\n \\label{A_degen}\n\\ee\n\n\nNow, imposing $-2\\Phi_z=\\sum_pW_{zp}+\\sum_vW_{zv}$ we obtain the constraints\n\\be\n \\sum_vW_{zv}^0=-(m+2)\\Phi^0_z , \\qquad \\sum_p\\frac{\\Phi^0_zb_p+W^0_{pz}}{r-b_p}=0 .\n \\label{constr_Phi_z}\n\\ee\n\n\nNext, imposing $-2\\Phi_q=\\sum_pW_{qp}+\\sum_zW_{qz}$ gives \n\\be\n -2(A+\\Phi^0_{q})=\\sum_{p\\neq q}\\frac{1}{r-b_{(p)}}\\left[2\\int\\d r A+(\\Phi^0_{q}+\\Phi^0_{p})r+ W^0_{pq}\\right]+\\sum_z(\\Phi^0_zr+W^0_{qz}) .\n \\label{constrW_deg}\n\\ee\nBy comparing the leading terms of the l.h.s. and of the r.h.s. for $r\\to\\infty$ one obtains \n\\be\n m\\Phi^0_{q}+\\frac{1}{m+1}b_q\\sum_z\\Phi^0_z+\\sum_zW^0_{qz}=\\sum_p\\Phi^0_{p}-\\frac{1}{m+1}\\sum_pb_p\\sum_z\\Phi^0_z .\n \\label{constrW_deg_leading}\n\\ee\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$ components} \n\n\\label{subsec_Phi3_4_deg}\n\nTo complete the description of the Weyl tensor, we now give the general form of the $\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$ components (recall that these are non-zero only for $n\\ge6$). In particular, we also discuss constraints following from the assumption that all  non-vanishing eigenvalues of $\\rho_{ij}$ are distinct (useful for later analysis). \n\n\\subsubsection{$\\Phi^{\\{3\\}}_{ijkl}$ components}\n\\label{subsubPhi3deg} \n\nEq.~\\eqref{B4}  gives\n\\be\n \\Phi_{(o)(p)(o)(q)}(\\rho_{(q)}-\\rho_{(p)})=0=\\Phi_{(z)(p)(z)(q)}(\\rho_{(q)}-\\rho_{(p)}) , \\qquad \\Phi_{(q)p(q)z}=0=\\Phi_{(v)p(v)z} , \\quad (p\\neq q, \\ v\\neq z) .\n \\label{A12_deg_Phi3}\n\\ee\nNon-vanishing components must satisfy the tracefree conditions (recall that $\\Phi_{ij}$ is diagonal in the frame we are using)\n\\be\n \\Phi_{poqo}+\\Phi_{pzqz}=0 , \\qquad \\Phi_{zpvp}+\\Phi_{zwvw}=0 \\qquad (p\\neq q, \\ v\\neq z) .\n \\label{trace_deg_Phi3}\n\\ee\n\nThe $r$-dependence, following from~\\eqref{Bi3}, is\n\\beqn\n & & \\Phi_{(p)z(p)v}=\\frac{\\Phi_{(p)z(p)v}^0}{r-b_{(p)}} \\quad (v\\neq z), \\qquad \\Phi_{(z)v(z)w}={\\Phi_{(z)v(z)w}^0} \\quad (v\\neq w) , \\nonumber \\\\\n & & \\Phi_{(p)q(p)o}=\\frac{\\Phi_{(p)q(p)(o)}^0}{(r-b_{(p)})(r-b_{(o)})} ,  \\quad (q\\neq o) \\qquad  \\Phi_{(z)p(z)q}=\\frac{\\Phi_{(z)p(z)(q)}^0}{r-b_{(q)}}  \\quad (p\\neq q) .\n \\label{r_deg_Phi3}\n\\eeqn\nFrom~(\\ref{trace_deg_Phi3}), the quantities in~(\\ref{r_deg_Phi3}) must satisfy the constraints\n\\beqn\n & & \\Phi_{pzqz}^0=0 , \\qquad \\sum_o\\frac{\\Phi^0_{p(o)q(o)}}{r-b_{(o)}}=0 \\qquad (p\\neq q) , \\label{r_deg_Phi3_constr_0}\\\\\n & & \\Phi^0_{zwvw}=0 , \\qquad \\sum_p\\frac{\\Phi_{z(p)v(p)}^0}{r-b_{(p)}}=0 \\qquad (v\\neq z) . \\label{r_deg_Phi3_constr}\n\\eeqn\n\n\nNote that, by the symmetries of the Weyl tensor, {from} the {third} equation of~(\\ref{r_deg_Phi3}) it follows that \n$\\Phi_{(p)q(p)o}\\neq0\\Rightarrow b_q=b_o$ (for $q\\neq o$), in agreement with~(\\ref{A12_deg_Phi3}); using also the {second} equation of~\\eqref{r_deg_Phi3_constr_0} we get\n\\be\n \\Phi^{\\{3\\}}_{(p)q(p)o}\\neq0 \\Rightarrow \\rhob=\\diag(a,a,b,b,c_1\\dots) \\qquad (a,b\\neq0). \n \\label{Phi3_new}\n\\ee\nFurther, {the fourth} equation of~(\\ref{r_deg_Phi3}) gives $\\Phi_{(z)p(z)q}\\neq0\\Rightarrow b_q=b_p$ {(for $q\\neq p$)}, i.e., the structure is $\\{a,a,c_1\\dots\\}$ with $a\\neq0$, in agreement with~(\\ref{A12_deg_Phi3}). For $\\Phi_{z(p)v(p)}\\neq0$, {the second equation of}~\\eqref{r_deg_Phi3_constr} also implies $\\{a,a,c_1\\dots\\}$ with $a\\neq0$. \n\n\nWe are interested, in particular, in determining what are the necessary conditions in order to have all the non-vanishing eigenvalues distinct. From the {above} observations it follows that one necessarily has $\\Phi_{(p)q(p)o}=\\Phi_{(z)p(z)q}=\\Phi_{z(p)v(p)}=0$, therefore the only non-zero $\\Phi^{\\{3\\}}_{ijkl}$ components can be the $\\Phi_{z(w)v(w)}$, however with the constraint $\\Phi^0_{zwvw}=0$ (eq.~(\\ref{r_deg_Phi3_constr})).  \nIt is easy to see that for this to be satisfied in a non-trivial way,  indices $z,v\\ldots$  must run at least over four values, i.e. there must be at least four zero eigenvalues of $\\rho_{ij}$ (excluding Kundt, the spacetime must thus be at least seven-dimensional). In other words:\n\\begin{itemize}\n \\item all non-zero $\\rho_{ij}$ are distinct and $m>n-6$ $\\Rightarrow$ $\\Phi^{\\{3\\}}_{ijkl}=0$. \n\\end{itemize}\n\n\n\n\\subsubsection{$\\Phi^{\\{4\\}}_{ijkl}$ components}\n\\label{subsubPhi4deg} \n\nFrom~\\eqref{Bi3} and the symmetries of the Weyl tensor one finds\n\\beqn\n & & \\Phi_{pqot}=\\frac{\\Phi_{pq(o)(t)}^0}{(r-b_{(o)})(r-b_{(t)})} , \\qquad \\Phi_{pzqw}=\\frac{\\Phi_{pz(q)w}^0}{r-b_{(q)}}  , \\qquad \\Phi_{zvwy}=\\Phi_{zvwy}^0 , \\\\\n & & \\Phi_{pqoz}=0 , \\qquad \\Phi_{pqzw}=0, \\qquad \\Phi_{pzvw}=0 .\n \\label{r_deg_Phi4}\n\\eeqn\nSimilarly as in the case of $\\Phi^{\\{3\\}}_{ijkl}$, we observe that \n{if} $\\Phi_{pqot}\\neq0$ then $b_t=b_q$ and $b_o=b_p$ (or $b_t=b_p$ and $b_o=b_q$), i.e., \n\\be\n \\Phi^{\\{4\\}}_{pqot}\\neq0 \\Rightarrow \\rhob=\\diag(a,a,b,b,c_1\\dots) \\qquad (a,b\\neq0). \n \\label{Phi4_new}  \n\\ee\n\n\n{Additionally,} {if} $\\Phi_{pzqw}\\neq0$ then\n$b_q=b_p$, {so that the structure is} { $\\{a,a,c_1\\dots\\}$ with  $a\\neq0$}. \n\n \nHaving  all the non-vanishing eigenvalues distinct thus requires $\\Phi_{pqot}=0=\\Phi_{pzqw}$ and the only non-zero components can be  $\\Phi_{zvwy}$. Therefore we conclude again that  indices $z,v\\ldots$  must run at least over four values unless $\\Phi^{\\{4\\}}_{ijkl}=0$ (and, again, excluding Kundt, the spacetime must thus be at least seven-dimensional), i.e., \n\\begin{itemize}\n \\item all non-zero $\\rho_{ij}$ are distinct and $m>n-6$ $\\Rightarrow$ $\\Phi^{\\{4\\}}_{ijkl}=0$. \n\\end{itemize}\n\n\n\\subsection{Case $\\Phi_{ij}=0$ (type II(abd))}\n\n\\label{subsec_Phi=0}\n\nThis case (non-trivial only for $n>5$) is obtained by setting\n\\be\n A=-\\Phi^0_p=K^0 , \\qquad \\Phi^0_z=0 \n\\ee\n{in the results obtained in~\\ref{subsec_Phi_W_deg}.}\nFor the $W_{ij}$ components one thus has\n\\be\n W_{pq}=\\frac{W^0_{pq}}{(r-b_{(p)})(r-b_{(q)})} , \\qquad W_{pz}=\\frac{W_{(p)z}^0}{r-b_{(p)}} , \\qquad W_{zv}=W_{zv}^0 ,\\label{Phi=0_Wpq}\n\\ee\nwith\n\\be\n \\sum_vW_{zv}^0=0 , \\qquad \\sum_p\\frac{W^0_{(p)z}}{r-b_{(p)}}=0 , \\qquad \\sum_zW^0_{qz}=0 , \\qquad \\sum_{p}\\frac{W^0_{(p)q}}{r-b_{(p)}}=0 .\n \\label{Phi=0_constr}\n\\ee\n\nIn view of these constraints (and of the properties of $W_{ij}$) we can briefly comment on some special cases:\n\n\\begin{itemize}\n \\item $\\Phi_{ij}=0$, $m=1$ $\\Rightarrow$ $W_{pq}=0=W_{pz}$, \n \\item $\\Phi_{ij}=0$, $m=2$ or $m=3$ $\\Rightarrow$ $W_{pq}=0$, \n \\item $\\Phi_{ij}=0$, $m=n-3$ $\\Rightarrow$ $W_{zv}=0=W_{pz}$, \n \\item $\\Phi_{ij}=0$, $m=n-4$ or $m=n-5$ $\\Rightarrow$ $W_{zv}=0$. \n\\end{itemize}\n\nFor special values of $n$ and $m$ some of these can hold simultaneously, thus leading to $W_{ij}=0$. Recalling the trivial implications $m=0\\Rightarrow W_{pq}=0=W_{pz}$ and $m=n-2\\Rightarrow W_{zv}=0=W_{pz}$ (valid also for $\\Phi_{ij}\\neq 0$), we have in particular \n\\begin{itemize}\n \\item for $n=4$ and $m=0,1,2$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$, \n \\item for $n=5$ and $m=0,1,2,3$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$, \n \\item for $n=6$ and $m=1,3$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$. \n\\end{itemize}\nWhile the first two implications simply reproduce the known result that $\\Phi_{ij}=0\\Leftrightarrow W_{ij}=0$ for $n=4,5$ (for any permitted $m$), the last remark will be useful for later purposes.\n\n\n\n\nUsing \\eqref{Phi=0_Wpq}, \\eqref{Phi=0_constr} and a reasoning similar to that of section \\ref{subsec_onerep} (in the paragraph after eq.~\\eqref{Wij}, including footnote~\\ref{foot_311}) one can also show that\n\\be\n W_{pq}\\neq0 \\Rightarrow \\rhob=\\diag(a,a,b,b,c_1\\dots) \\qquad (a,b\\neq0),  \n \\label{W_new}\n\\ee\nwhich will be useful in the following.\n\n\n\\subsubsection{No repeated non-zero eigenvalues}\n\n\\label{subsubsec_Phi=0_norepeated}\n\nIn case all the $b_p$ are distinct, {equations~(\\ref{Phi=0_constr})} imply\n\\be\n W^0_{pz}=0=W^0_{pq} ,\n\\ee\nso that the only non-zero components of $W_{ij}$ are the $W_{zv}=W_{zv}^0$, with $\\sum_vW_{zv}^0=0$. This implies that, in order to have $W_{ij}\\neq0$, one needs that  indices $z,v,\\ldots$ run at least over  four values, i.e., the cases $m=n-3$, $m=n-4$ and $m=n-5$ imply $W_{ij}=0$. Thus we have proven: $\\Phi_{ij}=0$, all non-zero $\\rho_{ij}$ are distinct and $m>n-6$ $\\Rightarrow$ $W_{ij}=0$. {Together with the results of subsection~\\ref{subsec_Phi3_4_deg}, we see that if such assumptions hold all boost weight zero components must actually vanish, so that we can conclude}\n\n\\begin{prop} \n\\label{prop_IIabd}\nFor type II(abd) ($\\Phi_{ij}=0$) non-twisting Einstein spacetimes with (degenerate) $\\rhob$ of rank $n-6<m<n-2$, the eigenvalue structure of $\\rhob$ is \n$\\{a,a, c_1\\ldots, 0,\\ldots\\}$ (with $a\\neq0$) or more special {(i.e., at least two non-zero eigenvalues of $\\rhob$ coincide)}.\n\\label{prop_Phi=0}\n\\end{prop}\n\n\n\n\nThe condition on $m$ means that either $m=n-3$ or $m=n-4$ or $m=n-5$. We will see {in section~\\ref{subsec_n-3}} below that in the case $m=n-3$ the assumption $\\Phi_{ij}=0$ can in fact be dropped {(since $\\Phi_{ij}\\neq0$ is not possible if all non-zero eigenvalues are distinct).  For $m=n-4$ the eigenvalue structure is thus $\\{a,a,0,0,c_1,\\ldots,c_{n-6}\\}$ and for $m=n-5$ it is $\\{a,a,0,0,0,c_1,\\ldots,c_{n-7}\\}$ (while for any smaller $m$ there are, of course, $n-2-m\\ge 4$ vanishing eigenvalues).} \nRecall also that for $m=n-2$ (non-degenerate case) we had a similar {result, cf.~proposition~\\ref{prop_non_deg2}.}\n\n\n\n\n\n\\subsection{The special case $m=n-3$}\n\n\\label{subsec_n-3}\n\n\n\nFor $n=4$ this case is not possible since it gives $m=1$ (see section~\\ref{subsec_m=1}). \n{In higher dimensions, the simplest spacetimes of this class are obtained by taking a Brinkmann warp {\\cite{Brinkmann25} (see also \\cite{OrtPraPra11})} of a type II Einstein spacetime possessing a non-degenerate mWAND with a single (spacelike) extra-dimension (cf. also \\cite{OrtPraPra11}). They include, e.g., {static} black strings and} this case may thus be of special interest. \n\nSince there is only one vanishing eigenvalue of $\\rho_{ij}$, i.e., $z$ can take only one value, we have $W_{zv}=0$ and, from~(\\ref{constr_Phi_z}),\n\\be\n \\Phi^0_z=0 , \\qquad \\sum_p\\frac{W^0_{pz}}{r-b_p}=0 .\n \\label{constr_m=n-3}\n\\ee\n\n\nTherefore (\\ref{constrW_deg_leading}) reduces to\n\\be\n (n-3)\\Phi^0_{q}+W^0_{qz}=\\sum_p\\Phi^0_{p} .\n\\ee\n\nFrom the results of subsection~\\ref{subsec_Phi3_4_deg}, when $m=n-3$ the only possible non-zero components of $\\Phi^{\\{3\\}}_{ijkl}$ can be $\\Phi_{(p)q(p)o}$, with $q\\neq o$ (in particular, $\\Phi_{pzqz}=\\Phi_{p(z)q(z)}=0$,  see the first of \\eqref{r_deg_Phi3_constr_0}), which satisfy\n\\be\n \\sum_o\\frac{\\Phi^0_{p(o)q(o)}}{r-b_{(o)}}=0 \\qquad (p\\neq q) , \n\\ee\nwhile the only possible non-zero components of $\\Phi^{\\{4\\}}_{ijkl}$ are $\\Phi_{pqot}$ ($\\Phi_{pqoz}=0$, see the first of \\eqref{r_deg_Phi4}) . \nFrom sections \\ref{subsubPhi3deg} and \\ref{subsubPhi4deg} (see \\eqref{Phi3_new}, \\eqref{Phi4_new})     \nit thus follows that if $\\Phi^{\\{3\\}}_{ijkl}\\neq0$ ($\\Leftrightarrow\\Phi^{\\{3\\}}_{(p)q(p)o}\\neq0$ here) or $\\Phi^{\\{4\\}}_{ijkl}\\neq0$ ($\\Leftrightarrow\\Phi^{\\{4\\}}_{pqot}\\neq0$ here)  then the eigenvalue structure is  $\\{a,a,b,b,c_1\\dots\\}$, where  $a,b\\neq0$. Moreover, it is easy to see that when $m=n-3$ both $\\Phi^{\\{3\\}}_{ijkl}$ and $\\Phi^{\\{4\\}}_{ijkl}$  can be non-zero only for $n\\ge7$.  This is of interest only in six dimensions ($\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ identically for $n=4,5$) and can be summarized as \n\\begin{itemize}\n \\item $n=6$, $m=3$  $\\Rightarrow$ $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$. \n\\end{itemize}\n\nIf we additionally assume $\\Phi_{ij}=0$ (then also $W_{zv}=0=W_{pz}$, see comments after \\eqref{Phi=0_constr}), then some of the components $W_{pq}$, $\\Phi^{\\{3\\}}_{p(o)q(o)}$ or $\\Phi^{\\{4\\}}_{pqot}$ have to be non-vanishing (otherwise the spacetime would be of type III). Then using also results of section~\\ref{subsec_Phi=0}  {(eq.~\\eqref{W_new})} it follows that type II spacetimes {with $m=n-3$ and $\\Phi_{ij}=0$} {must have} the eigenvalue structure $\\{a,a,b,b,c_1\\dots c_{n-7},0\\}$. \n\n\n\n\n\\subsubsection{No repeated non-zero eigenvalues}\n\n\\label{subsubsec_m=n-3_no_rep}\n\nIf all $b_p$ are distinct from the second of~(\\ref{constr_m=n-3}) we obtain  \n\\be\n W^0_{pz}=0 ,\n\\ee\nso that $W_{pz}=0$ and all the $\\Phi^0_{q}$ coincide, namely, \n\\be\n \\Phi^0_{q}=\\frac{1}{(n-3)}\\sum_p\\Phi^0_{p}\\equiv f^0 .\n\\ee\nTherefore\n\\be\n A=-f^0+A^0\\prod_p\\frac{1}{r-b_p} ,\n\\ee\nand one can basically proceed as in the non-degenerate case. Comparing the structures of the poles of the l.h.s. and r.h.s. of (\\ref{constrW_deg}) one again arrives at\n\\be\n A^0=0, \\qquad W^0_{pq}=0 .\n\\ee\nCombining this with the previous results we conclude $\\Phi_q=0$, $\\Phi_z=0$, $W_{qz}=0$, $W_{qp}=0$, $W_{zv}=0$, i.e.\n\\be\n \\Phi_{ij}=0=W_{ij} .\n\\ee\nSince here $m=n-3>n-6$, we can use the results of subsection~\\ref{subsec_Phi3_4_deg} to conclude that also $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$, so that all b.w. 0 components of the Weyl tensor vanish. Thus we have proven: {\\it all non-zero $\\rho_{ij}$ are distinct and $m=n-3$ $\\Rightarrow$ all b.w. 0 components vanish.}\nIn other words, {\\it in the case $m=n-3$, the eigenvalue structure of $\\rhob$ must be $\\{a,a,c_1, \\dots ,c_{n-5}, 0\\}$ \n(or more special)}. If, additionally, $\\Phi_{ij}=0$, there is in fact another repeated non-zero eigenvalue \\cite{Wylleman_priv}\\footnote{We thank Lode Wylleman for pointing this out.} {(see the comment just before section \\ref{subsubsec_m=n-3_no_rep}), so that} \n Proposition~\\ref{prop_Phi=0} can accordingly be reformulated as (see also \\cite{Wylleman_priv}) \n \n\\begin{prop} \n\\label{prop_n-3_text}\nIn a type II Einstein spacetime of dimension $n>4$ with a non-twisting multiple WAND of rank $m=n-3$, the eigenvalue structure of $\\rhob$ is $\\{a,a,c_1,\\ldots c_{n-5},0\\}$ where $a,c_\\alpha  \\ne 0$ (and necessarily $\\Phi^A_{ij}=0$ if $n>5$). If, additionally, $\\Phi_{ij}=0$, the structure becomes $\\{a,a,b,b,c_1,\\ldots c_{n-7},0\\}$ where $a,b,c_\\alpha  \\ne 0$.\n\\end{prop}\n\n\n\n\\begin{rem}\n\\label{rem_Phi0}\n{Together with propositions~\\ref{prop_non_deg2} and \\ref{prop_Phi=0}, this shows that, for any possible value of $m$ ($0<m\\le n-2$), the mWAND of type II(abd) (i.e., $\\Phi_{ij}=0$) Einstein spacetimes has  $\\rhob$ of the form $\\{a,a,b,b,c_1,\\ldots c_{n-6}\\}$ (where $a\\ne 0$ if $m=n-5$ or $m=n-4$ and $a,b\\ne 0$ if $m=n-3$ or $m=n-2$ ).} In particular, point~(\\ref{Phi0}) of theorem~\\ref{prop_GSHD} is thus now also proven.\n\\end{rem}\n\n\n\n\n\n\n\n\\begin{rem}\n\\label{rem_5d}\nIn five dimensions (for which always $\\Phi_{ij}\\neq0$), recalling also the results for the non-degenerate case {(proposition~\\ref{prop_non_deg2})}, this enables us to list all the admitted eigenvalues structures, {for $m=3,2,1,0$, respectively}: $\\rho_{ij}=\\mbox{diag}(a,a,a)$,\n$\\rho_{ij}=\\mbox{diag}(a,a,0)$,  \n$\\rho_{ij}=\\mbox{diag}(a,0,0)$,  \nor $\\rho_{ij}=0$ (Kundt). See \\cite{Ortaggioetal12} for more details and examples. \n\\end{rem}\n\n\\begin{rem}\n\\label{rem_6d}\nIn six dimensions, we conclude that the case $m=3$ requires $\\rho_{ij}=\\mbox{diag}(a,a,b,0)$. However, as shown above in this case one has $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ and additionally the implication $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$ holds (see subsection~\\ref{subsec_Phi=0}). Therefore one necessarily has $\\Phi_{ij}\\neq0$ when $n=6$ and $m=3$ (or all Weyl boost zero components would vanish). {See section~\\ref{sec_6D} for more details.}\n\\end{rem}\n\n\n\n\n\n\n\\subsection{The special case $m=1$} \n\n\\label{subsec_m=1}\n\n\n\n\n\nFrom a viewpoint complementary to that of the previous subsection, it is also interesting to analyze the case when only one eigenvalue is non-zero. Let us associate the index $p=2$ to this eigenvalue, while $z,v=3,\\ldots,n-1$. Using the general equations presented above, one obtains (cf. also~\\cite{PraPra08}) \n\\be\n \\Phi_z=\\Phi^0_z , \\qquad \\Phi_2=-\\frac{1}{2}\\sum_z\\Phi^0_z, \\qquad W_{2z}=\\Phi^0_z , \\qquad \\sum_vW^0_{zv}=-3\\Phi^0_z , \n\\ee\nwhile clearly here $W_{pq}=0$ (note that for $n=4$ one gets \n$\\Phi^0_z=0$  \nand $\\Phi_{ij}=0=W_{ij}$, so this case does not occur). Therefore all components of $\\Phi_{ij}$ and $W_{ij}$ are $r$-independent. Note that  $\\Phi_{ij}=0\\Leftrightarrow \\Phi^0_z=0$. When this happens the only non-zero components of $W_{ij}$ can be the $W^0_{zv}$, but with the constraint $\\sum_vW^0_{zv}=0$. Therefore it is easy to see that $W_{ij}$ can be non-zero only if $z,v$ can take at least four values, i.e., $z,v=3,4,5,6,\\ldots$, so that $n\\ge 7$. To summarize \\begin{itemize}\n \\item $n<7$, $m=1$, $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$ .\n\\end{itemize}\nThis can be understood as a special instance of the results of subsection~\\ref{subsec_Phi=0}. \nThis result is of interest in six dimensions, since one always has $\\Phi_{ij}=0\\Rightarrow W_{ij}=0$ in four and five dimensions. Namely, for $n=6$ the case $\\{a ,0,0,0\\}$ is forbidden if $\\Phi_{ij}=0\\neq W_{ij}$. \n\nFrom the results of subsection~\\ref{subsec_Phi3_4_deg}, when $m=1$ the only possible non-zero components of $\\Phi^{\\{3\\}}_{ijkl}$ can be $\\Phi_{(z)v(z)w}$ (with $v\\neq w$), which satisfy\n\\be\n \\Phi^0_{zvzw}=0 \\qquad (v\\neq w) , \n\\ee\nwhile the only possible non-zero components of $\\Phi^{\\{4\\}}_{ijkl}$ are $\\Phi_{zvwy}$. It is easy to see that, since $m=1$,  both sets of components vanish identically unless $n\\ge7$, so that \n\\begin{itemize}\n \\item $n=6$, $m=1$  $\\Rightarrow$ $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$. \n\\end{itemize}\n\n\\begin{rem}\n\\label{rem_6d_m=1}\nSince for $n=6$, $m=1$ one has $\\Phi_{ij}=0$ $\\Rightarrow$ $W_{ij}=0$ (see above), we conclude that in six dimensions spacetimes with $m=1$ require $\\Phi_{ij}\\neq 0$.\n\\end{rem}\n\n\n\\subsubsection{Examples}\n\n\\label{subsubsec_m=1}\n\nFor any $n\\ge5$, simple examples of spacetimes with $m=1$ {(and thus $\\Phi^A_{ij}=0$)} are given by:\n\\begin{enumerate}\n \\item dS$_3\\times$S$_{n-3}$ (or AdS$_3\\times$H$_{n-3}$): $n\\ge5$, $\\Phi_{ij}\\neq 0$, $\\Lambda\\neq 0$, \\label{dSxS_m=1}\n \\item as above but with a Brinkmann warp: $n\\ge6$, $\\Phi_{ij}\\neq 0$, also $\\Lambda=0$ is possible, \\label{Brink_m=1} \n \\item Minkowski$_3\\times$(Ricci-flat)$_{n-3}$: $n\\ge7$, $\\Phi_{ij}=0$, $\\Lambda=0$. \\label{MinkxRc_m=1}\n\\end{enumerate}\n\nIn the first case, if {we restrict to} $n\\ge7$ the second factor space can be in fact {\\em any} Einstein space (with Ricci scalar given by $2(n-3)K$). The metric of the first and third case can thus be given in a unified way as\n\\be\n \\d s^2 =\\left(1+\\frac{K}2ur\\right)^{-2}(2\\d u\\d r+r^2\\d x^2)+\\d\\Sigma^2 ,\n\\ee\nwhere $\\d\\Sigma^2$ is the metric of an $(n-3)$-dimensional Einstein space with Ricci scalar $R_\\Sigma=2(n-3)K$ and $K$ is a constant related to the $n$-dimensional Ricci scalar by $R=2nK$. A geodesic twistfree mWAND $\\bl$ (degenerate but expanding along $x$) is\n\\be\n {\\ell}_a\\d x^a=\\d u .\n\\ee\nNote that here $r$ is an affine parameter along $\\bl$ only in the case $K=0$. {Such product spaces are of type D \\cite{PraPraOrt07} and in fact any null vector field tangent to the three-dimensional Lorentzian factor is an mWAND \\cite{GodRea09,PraPraOrt07}.}\nUsing these results metrics for the second case can also be constructed straightforwardly. {These will be also of type D \\cite{OrtPraPra11}.}\n\n\n\n\\subsection{Case when all non-zero eigenvalues are distinct} \n\n\\label{subsec_alldistinct}\n\n\n\nBy combining the results of subsections~\\ref{subsec_Phi=0} and \\ref{subsec_Phi3_4_deg} we have \n\\begin{itemize}\n \\item all non-zero $\\rho_{ij}$ are distinct, $m=n-4$ or $m=n-5$ and $\\Phi_{ij}=0$ $\\Rightarrow$ all b.w. 0 components vanish,\n \\item all non-zero $\\rho_{ij}$ are distinct and $m=n-3$ $\\Rightarrow$ all b.w. 0 components vanish.\\footnote{By adding the results of \\cite{Pravdaetal04} we can in fact conclude that in both cases the whole Weyl tensor must vanish.} \n\\end{itemize}\nTherefore type II Einstein spacetimes for which all non-zero $\\rho_{ij}$ are distinct require that at least one of the following holds ({obviously $m>0$ or we would simply have Kundt}):\n\\begin{enumerate}\n \\item $1\\le m\\le n-6$ (i.e., at least four eigenvalues of $\\rho_{ij}$ vanish),\n \\item $\\Phi_{ij}\\neq 0$ and $1\\le m\\le n-4$ (i.e., at least two eigenvalues of $\\rho_{ij}$ vanish).\n\\end{enumerate}\nThe first case can occur only for $n\\ge7$: for $n=7$ one can have only $\\rho_{ij}=\\mbox{diag}(a ,0,0,0,0)$, for $n=8$ either $\\rho_{ij}=\\mbox{diag}(a ,0,0,0,0,0)$ or  $\\rho_{ij}=\\mbox{diag}(a ,b ,0,0,0,0)$, etc.. The second case can occur for $n\\ge5$: for $n=5$ one has simply $\\rho_{ij}=\\mbox{diag}(a ,0,0)$, for $n=6$ either $\\rho_{ij}=\\mbox{diag}(a ,0,0,0)$ or $\\rho_{ij}=\\mbox{diag}(a ,b ,0,0)$, for $n=7$ either $\\rho_{ij}=\\mbox{diag}(a ,0,0,0,0)$ or $\\rho_{ij}=\\mbox{diag}(a ,b ,0,0,0)$ or $\\rho_{ij}=\\mbox{diag}(a ,b ,c ,0,0,0)$, etc.. These results are summarized in table~\\ref{tab_distinct_eigenval}. For such solutions {\\em the optical constraint is clearly violated if $m>1$}. The {previously discussed} case $m=1$ has been included for completeness but it is of course ``trivial'' in this context since there is a single non-zero eigenvalue.\n\n\n\\begin{table}[t]\n \\begin{center}\n  \\begin{tabular}{|c|c|l|c|}\n   \\hline Case & $n$ & Possible $m$ & Examples  \\\\\\hline\n    $\\Phi_{ij}\\not=0$ & 5 & 1 & \\\\\n    \t\t\t\t\t   \t\t  & 6 & 1,2 &   \\\\\n     \t\t\t\t\t\t\t\t\t& 7 & 1,2,3 &  \\\\\n     \t\t\t\t\t\t\t \t\t& 8 & 1,2,3,4 &  \\\\\n     \t\t\t\t\t\t\t \t\t& \\vdots &  &  \\\\\n     \t\t\t\t\t\t\t\t \t& $n$ & 1,2,3\\ldots, $n-4$ & (A)dS$_{m+2}\\times$S$_{n-2-m}$(H$_{n-2-m}$) \\\\\\hline\n     $\\Phi_{ij} =0$   & 7 & 1 &  \\\\\n        \t\t\t\t\t\t\t& 8 & 1,2 &  \\\\\n        \t\t\t\t\t\t  & \\vdots &  &  \\\\\n        \t\t\t\t\t\t\t& $n$ & 1,2,3,\\ldots, $n-6$ &  Mink$_{m+2}\\times$(Ricci-flat)$_{n-2-m}$  \\\\\n\\hline\n  \\end{tabular}\n    \\caption{Non-twisting Einstein spacetimes for which there are no repeated non-zero eigenvalues of $\\rho_{ij}$: permitted values of the rank $m$ of $\\rho_{ij}$ in various dimensions {(necessarily $1\\le m \\le n-4$ thanks to propositions~\\ref{prop_non_deg2}  and \\ref{prop_n-3_text})}. {As discussed in the text, different examples can also be generated using Brinkmann's warp, or by replacing the S$_{n-2-m}$(H$_{n-2-m}$) factor by a more general Einstein space. {Note that here solutions with $m>1$ violate the optical constraint.\\label{tab_distinct_eigenval}}\n    }}\n  \\end{center}\n\\end{table}\n\n\n\n\\subsubsection{Examples}\n\n\\label{subsubsec_examples_distinct}\n\nThe case $n=4$ is not possible. Examples with $m=1$ have been already presented {in subsection~\\ref{subsec_m=1} and will not be discussed again here}. For $m\\ge2$ the $n=5$ case is also forbidden {(cf. Remark~\\ref{rem_5d} and \\cite{Ortaggioetal12})}. \nFor $n=6$ the only possibility is $m=2$ (with $\\Phi_{ij}\\neq0$) and one can construct explicit Einstein spacetimes by taking dS$_4\\times$S$^2$ or AdS$_4\\times$H$^2$. These can then be extended to any dimension $n\\ge7$ by a simple Brinkmann warp (and if one starts from dS$_4\\times$S$^2$ the cosmological constant of the resulting spacetime can be arbitrary, cf.~\\cite{OrtPraPra11}), so to have explicit examples of the type $\\rho_{ij}=\\mbox{diag}(a ,b ,0,\\ldots,0)$ (i.e., with $m=2$) for any $n>5$. \n\nHowever, according to the previous comments (see also table~\\ref{tab_distinct_eigenval}), for $n=7$ one can have $m=2,3$, for $n=8$ it is possible $m=2,3,4$ and so on: in general, and for any $n>5$, $2\\le m\\le n-4$ explicit metrics can be constructed similarly as in six dimensions by taking dS$_{m+2}\\times$S$_{n-2-m}$ or AdS$_{m+2}\\times$H$_{n-2-m}$ (where the two factor spaces must have {Ricci scalars} given by $(m+2)(m+1)K$ and $(n-2-m)(m+1)K$, respectively), and appropriate hypersurface orthogonal null congruences living in the (A)dS$_{m+2}$ factor {(explicit examples are given below)}. These have necessarily $\\Phi_{ij}\\neq0$ and $\\Lambda\\neq0$. Using Brinkmann's warp one can also generate Ricci-flat solutions (again with $\\Phi_{ij}\\neq0$) for which, however, the stronger restriction $n>6$, $2\\le m<n-4$ holds (at least if this method is used). Products of Ricci-flat spaces are possible if each factor is at least four-dimensional (therefore there are always at least four vanishing eigenvalues) and the Lorentzian factor is flat and fall into the $\\Phi_{ij}=0$ class, with $\\Lambda=0$ (these metrics belong to the type D \\pp waves mentioned in \\cite{OrtPraPra09}). \n\nSummary:\n\\begin{enumerate}\n \\item dS$_{m+2}\\times$S$_{n-2-m}$ (or AdS$_{m+2}\\times$H$_{n-2-m}$)\\footnote{Note that if $n-2-m\\ge4$ the second factor space can be in fact any Einstein space with Ricci scalar given by $(n-2-m)(m+1)K$.}: $n>5$, $2\\le m\\le n-4$, $\\Phi_{ij}\\neq 0$, $\\Lambda\\neq 0$, \\label{dSxS}\n \\item as above + Brinkmann, also $\\Lambda=0$: $n>6$, $2\\le m\\le n-5$, $\\Phi_{ij}\\neq 0$, \\label{Brink}\n \\item Minkowski$_{m+2}\\times$(Ricci-flat)$_{n-2-m}$: $n>7$, $2\\le m\\le n-6$, $\\Phi_{ij}=0$, $\\Lambda=0$. \\label{MinkxRc}\n\\end{enumerate}\n\nNote that all the above spacetimes {also} belong to the Kundt class, although w.r.t. a null congruence different from the one which is of interest for our discussion (which is expanding and shearing). {In fact they all admit $\\infty^m$ mWANDs:} similarly as in section~\\ref{subsubsec_m=1}, they are of type D (with $\\Phi^A_{ij}=0$) and any null vector field tangent to the Lorentzian factor is an mWAND -- so one can always also find an mWAND with an optical matrix having all non-zero eigenvalues identical.  {However, this mWAND ``degeneracy'' is not the generic situation} (see appendix~\\ref{app_violating} for an example of a class of type D spacetimes which admit (only) two double WANDs, both violating the optical constraint).\n\n\nExplicitly, for cases~\\ref{dSxS} and \\ref{MinkxRc} one can take, e.g., the metric\n\\be\n \\d s^2 =\\Omega^2(2\\d u\\d r+P_q^2\\d x_q^2)+\\d\\Sigma^2 ,\n \\label{ex_distinct}\n\\ee\nwhere $q=2,\\dots,m+1$, $\\d\\Sigma^2$ is the metric of an $(n-2-m)$-dimensional Einstein space with Ricci scalar $R_\\Sigma=(n-2-m)(m+1)K$ and\n\\be\n P_q=r-b_q, \\qquad \\Omega^{-1}=1+\\frac{K}{4}\\left[2r\\left(u-\\frac{1}{2}P_qx_q^2\\right)+P_q^2x_q^2\\right ] ,\n\\ee\n{in which $b_q$ and $K$ are constants.}\nThe first factor space in~(\\ref{ex_distinct}) is a space of constant curvature with Ricci scalar given by $(m+2)(m+1)K$, so that the Ricci scalar of the full spacetime is \n\\be\n R=n(m+1)K .\n\\ee\n\nThe geodesic, twistfree mWAND $\\bl$ and the remaining frame vectors are given by\n\\be\n {\\ell}_a\\d x^a=\\d u , \\qquad n_a\\d x^a=\\Omega^2\\d r, \\qquad m_{(q)a}\\d x^a=\\Omega P_{(q)}\\d x_{(q)} ,\n\\ee\nwhile vectors $m_{(v)a}$ will depend on the specific form of $\\d\\Sigma^2$. The optical matrix $\\rho_{ij}$ is given by\n\\be\n \\rho_{pq}=\\delta_{(p)q}\\frac{(\\Omega P_{(p)})_{,r}}{\\Omega^{3}P_{(p)}} , \\qquad \\rho_{pv}=0=\\rho_{vz} .\n\\ee\nIn the case $K=0$ (i.e., $\\Lambda=0$) one thus has $\\rho_{pq}=\\delta_{(p)q}\/(r-b_{(p)})$. Note, however, that $r$ is {\\em not} an affine parameter when $K\\neq 0$ (one has $\\bl=\\Omega^{-2}\\pa_r$). \n\n\nFor case~\\ref{Brink} one can use the method illustrated in detail in \\cite{OrtPraPra11}. \n\n\n\n\n\\section{Counterexamples}\n\n\\label{sec_counter}\n\n\nIn \\cite{Ortaggioetal12}, a counterexample to the converse of the five-dimensional ``shear-free'' part of the GS theorem was presented, thus demonstrating explicitly that the condition that the optical matrix $\\rhob$ admits a canonical form compatible with a geodetic mWAND is not sufficient for the null geodetic being  an mWAND. Similarly, also our results above give conditions that are (necessary but) not sufficient. In order to demonstrate that, here we present a few ``counterexamples'', i.e., certain Einstein spacetimes that admit a non-twisting null (thus geodesic) vector field $\\bl$ with $\\rhob$ taking one of the permitted ``canonical forms'' (cf. theorem~\\ref{prop_GSHD} and propositions~\\ref{prop_non_deg2}, \\ref{prop_IIabd} and \\ref{prop_n-3_text}) and yet with $\\bl$ not being an mWAND. Note that such counterexamples will necessarily be shearing: a twistfree shearfree null vector field is automatically an mWAND \\cite{PodOrt06,OrtPraPra07}. \n\n\n\n\\subsection{Non-degenerate $\\rhob=\\{a,a,b,b,c_1, \\ldots, c_{n-6} \\}$}\n\nIn the non-degenerate case a ``counterexample'' to the canonical form $\\{a,a,b,b,c_1, \\ldots, c_{n-6} \\}$ (with $a,b,c_\\alpha \\ne 0$ and $a\\neq b$ -- see points (ii) and (iii) of theorem~\\ref{prop_GSHD} and proposition~\\ref{prop_non_deg2}) is given by the following six-dimensional Ricci-flat spacetime, which belongs to a class of metrics considered by Robinson (as described in \\cite{Trautman02b}),\n\\be\n \\d s^2=2\\d u\\d r+2\\cosh^2r\\d w\\d\\bar w+2\\sin^2r\\d\\zeta\\d\\bar\\zeta .\n \\label{robinson}\n\\ee\nHere the hypersuface orthogonal null vector field $\\ell_a\\d x^a=\\d u$ is {\\em not} an mWAND (not even a single one) and yet the corresponding \n$\\rhob$ has the eigenvalue structure $\\{a,a,b,b\\}$ (with $a\\neq0\\neq b$, $a\\neq b$). Associated to $\\bl$ there is also an optical structure \\cite{Trautman02b}.\nNote, however, that spacetime~\\eqref{robinson} is a type N \\pp wave (albeit in {so-called Rosen coordinates, see, e.g., metric~(39) of \\cite{Brinkmann25}}), with a covariantly constant mWAND given by $n_a\\d x^a=\\d r$. \n\n\\subsection{Case $n-6<m<n-2$} \n\nDirect products of~\\eqref{robinson} with flat extra dimensions clearly provide non-WANDs with eigenvalues $\\{a,a,b,b,0,\\ldots,0 \\}$. In particular, for $n=7$ one obtains $\\{a,a,b,b,0\\}$, which is thus a counterexample to the $\\rhob$-form of Proposition~\\ref{prop_n-3_text}. For $n=8,9$ this is also a counterexample to the $\\rhob$-form of Proposition~\\ref{prop_IIabd}. Another counterexample to Proposition~\\ref{prop_n-3_text} can be obtained for $n=6$ by taking the direct product of a vacuum black ring \\cite{EmpRea02prl} with a flat dimension: in the coordinates of \\cite{ElvEmp03}, the hypersurface orthogonal null vector field  (also considered in 5d in \\cite{Taghavi-Chabert11} for different purposes)\n\\be\n \\ell_a\\d x^a = \\frac{\\sqrt{F(y)}}{G(y)}\\d y+\\d\\psi ,\n\\ee\ncorresponds to $\\rhob$ of the form $\\{a,a,b,0\\}$ and is not a WAND. \nAlternatively, the same form $\\{a,a,b,0\\}$ can be obtained by taking the direct product of a 5d vacuum static KK bubble \\cite{GodRea09} with a flat dimension (with a non-WAND $\\ell_a\\d x^a=\\d t+V(r)^{-1\/2}\\d r$ -- this 6d spacetime is of type D but the mWANDs are different from $\\bl$ \\cite{OrtPraPra11}).\n\n\n\\subsection{Case when all non-zero eigenvalues are distinct} \n\nExamples of spacetimes with a non-twisting, non-WAND $\\bl$ having all non-zero eigenvalues distinct can be constructed as a direct product of a 4d \\pp wave with flat space, i.e.,\n\\be\n \\d s^2=2\\d u\\d r+\\cosh^2r\\d x^2+\\sin^2r\\d y^2+\\d z_i\\d z^i .\n \\label{4d_pp}\n\\ee\nHere,  $\\ell_a\\d x^a=\\d u$ has a $\\rhob$ with eigenvalue structure $\\{a,b,0,\\ldots,0\\}$ (with $a\\neq0\\neq b$, $a\\neq b$). This is the form of $\\rhob$ discussed in section~\\ref{subsec_alldistinct} (see also table~\\ref{tab_6D} for the $n=6$ case). Similarly as the spacetime~\\eqref{robinson}, also \\eqref{4d_pp} is a type N \\pp wave with mWAND $n_a\\d x^a=\\d r$. \n\n\\subsection{Case $m=1$}\n\nExamples of non-twisting null congruences with $m=1$ (the case discussed in section~\\ref{subsec_m=1}) that are not multiple WANDs can be obtained from the (Ricci-flat) Newman-Tamburino lift considered in \\cite{Ortaggioetal12}, i.e.,\n\\be\n{\\rm d}s^2 = r^2 {\\rm d} x^2 + x^2 {\\rm d} y^2 - \\frac{4 r}{x } {\\rm d} u {\\rm d} x - 2 {\\rm d} u {\\rm d} r + x^{-2} \\left[c + \\ln (r^2 x^4)\\right] {\\rm d} u^2+\\d z_i\\d z^i , \\label{NTmetric}\n\\ee\nwhere $c$ is a constant. In this case $\\ell_a\\d x^a = {\\rm d} u$ is a {\\em single} WAND  with optical matrix of the form $\\{a,0,\\ldots,0\\}$ (with $a\\neq 0$). \n\n\n\n\n\n\\section{Existence of totally geodesic null two-surfaces}\n\n\\label{sec_totgeod}\n\nTheorem~\\ref{prop_GSHD} comprises the main results proven in the previous sections. In particular, it implies that there always exists at least a pair of repeated (possibly vanishing) eigenvalues and that (recall that $n\\ge 6$) there are in fact at least two such pairs when, e.g.,  $\\det\\rhob\\neq0$ (see also proposition~\\ref{prop_non_deg2}) or $\\Phi_{ij}=0$ (see also propositions~\\ref{prop_IIabd} and \\ref{prop_n-3_text}). The following proposition (see also the definitions in appendix~\\ref{app_shearfree}, in particular eq.~\\eqref{D}) elucidates the geometrical meaning of this fact (and provides a connection with the standard 4d version of the Goldberg-Sachs theorem, cf. appendix~\\ref{app_GS_4D}).\n\\begin{prop} \n In an algebraically special Einstein spacetime of dimension $n\\ge 6$ with a non-twisting {mWAND} $\\bl$, using a parallelly transported eigenframe of $\\rhob$\n \\begin{enumerate} \n \t\\item if the eigenvalue structure of $\\rhob$ is $\\{a,a,c_1,\\ldots, c_{n-5},0 \\}$ with $c_\\alpha\\neq a\\neq 0$ then the totally null distribution ${\\cal D}_{23}$ is integrable with totally geodesic integral surfaces, \n \t\\item if the eigenvalue structure of $\\rhob$ is $\\{0,0,c_1,\\ldots, c_{n-4}\\}$ with $c_\\alpha\\neq 0$ then the totally null distribution ${\\cal D}_{23}$ is integrable with totally geodesic integral surfaces,\n \t\\item if the eigenvalue structure of $\\rhob$ is $\\{a,a,b,b,c_1, \\ldots, c_{n-6} \\}$ with $b\\neq a$ and $a\\neq c_\\alpha\\neq b$ then the totally null distributions ${\\cal D}_{23}$ and ${\\cal D}_{45}$ are integrable with totally geodesic integral surfaces (either $a$ or $b$ can be zero). \n \\end{enumerate}\t\n\\label{prop_integrab}\n\\end{prop}\n\\begin{proof}\nThe proof is similar in all cases and relies on the use of the Ricci identity (11k,\\cite{OrtPraPra07}) (equivalent to the NP equation (A4,\\cite{Durkeeetal10})), which for algebraically special Einstein spacetimes reduces to \n\\be\n \\delta_{[j|} \\rho_{i|k]} =  L_{1[j|} \\rho_{i|k]}+ \\tau_i \\rho_{[jk]}+\\rho_{il}\\M{l}{[j}{k]}\n+ \\rho_{l[j|}\\M{l}{i|}{k]} , \\label{11k}\n\\ee\nwhere $\\delta_i=m_{(i)}^a\\nabla_a$ (recall also the definitions in \\eqref{L1i_M}). Since here $\\bl$ is twistfree we have $\\rho_{[jk]}=0$ in the above equation. Assume the form of $\\rhob$ specified in 1., i.e., $\\rho_{22}=\\rho_{33}=a\\neq 0$ (and all remaining eigenvalues different from $a$) and consider \\eqref{11k} for $i,j,k=23\\hat k\/32\\hat k$ (with $\\hat k\\neq 2,3$). This gives $\\M{2}{\\hat k}{3}=0=\\M{3}{\\hat k}{2}$. Then, \\eqref{11k} with $i,j,k=22\\hat k\/33\\hat k$ gives $\\M{2}{\\hat k}{2}=\\M{3}{\\hat k}{3}$,  which completes the proof of point 1. (see eqs.~\\eqref{2totgeod_real}). The proof of point 2. is identical. The proof of point 3. is analogous, after extending the reasoning to the index pair (45); in this case, however, either $a$ or $b$ can be zero, provided all $c_\\alpha\\neq 0$. \n\\end{proof}\n\\begin{rem}\nIn the case there are further pairs of repeated eigenvalues, the corresponding two-spaces will also define integrable totally geodesic distributions, provided these pairs are not repeated. In particular in even dimensions, if there are $(n-2)\/2$ (non-repeated) pairs of equal eigenvalues there will be $(n-2)\/2$ such totally geodesic two-surfaces.\n\\end{rem}\n\\begin{rem}\nPoint 1. of the above proposition includes, in particular, all non-twisting but expanding type III\/N Einstein spacetimes (for which $c_{\\alpha}=0$ \\cite{Pravdaetal04}) and type II non-twisting non-Kundt spacetimes with $\\Phi^A_{ij} \\ne 0$ (point \\eqref{PhiA} of theorem~\\ref{prop_GSHD}). \nPoint 3. is relevant to the case $\\det\\rhob\\neq0$ with $\\Phi_{ij}=0$ (proposition~\\ref{prop_non_deg2} and point \\eqref{Phi0} of theorem~\\ref{prop_GSHD}), which can only be of type II(abd).\n\\end{rem}\n\\begin{rem}\nThe proposition does not explicitly include the $n=5$ case. However, from Proposition~4 of \\cite{Ortaggioetal12} one can easily obtain the corresponding results (also in the presence of twist). In particular, in the non-twisting case ${\\cal D}_{23}$ is totally geodesic when the eigenvalue structure of $\\rhob$ is either $\\{a,a,0\\}$ or  $\\{0,0,a\\}$ (with $a\\neq 0$ in both cases). The same is true in the case $\\{a,a,a\\}\\neq0$, although this does not follow from (11k,\\cite{OrtPraPra07}) (instead, one can easily adapt the argument of footnote~10 of \\cite{Ortaggioetal12}). \n\\end{rem}\n\\begin{rem}\nRecall that in 4d the Goldberg-Sachs theorem implies that $\\bl$ is geodesic and the eigenvalue structure of $\\rho_{(ij)}$ is $\\{a,a\\}$, which is {\\em equivalent} to ${\\cal D}_{23}$ being integrable (and thus automatically totally geodesic, in 4d, see appendix~\\ref{app_GS_4D}).\n\\end{rem}\n\n\n\n\n\\section{Type II spacetimes with a non-twisting multiple WAND in six dimensions}\n\n\\label{sec_6D}\n\n\n\nAs already mentioned, five-dimensional algebraically special Einstein spacetimes have been studied in detail in \\cite{Ortaggioetal12} and, for $n=5$, the results obtained above are also contained in that reference. The next lower dimension to consider is thus $n=6$ and in this section our general results are specialized to this case. This is of interest also because of a qualitative difference between $n<6$ and $n\\ge 6$ dimensions: for $n<6$ type II spacetime necessarily have $\\Phi_{ij}\\neq 0$ (since type II coincides with type II(c) {for $n=4,5$}), while this is not so in higher dimensions.  \n\n\\subsection{Permitted forms of $\\rhob$}\n\n\n\\label{subsec_permitted_6D}\n\n\n\nOne has to consider the various cases $m=4,3,2,1$ ($m=0$ is Kundt).\n\nFirst, recall that for $m\\neq 2$ point \\eqref{PhiA} of theorem~\\ref{prop_GSHD} implies that $\\Phi^A_{ij}=0$. Let us thus first discuss this case.\n\nNow, for $m=4$ proposition~\\ref{prop_non_deg2} tells us that if $\\Phi_{ij}\\neq 0$ we are in the \\RT class, i.e. $\\rhob=\\mbox{diag}(a,a,a,a)\\neq 0$ (in which case $\\Phi_{ij}\\propto \\delta_{ij}$), otherwise we have $\\rhob=\\mbox{diag}(a,a,b,b)$ (with $a,b\\neq 0$).\n\n\n\nFor $m=3$ we learn from Proposition~\\ref{prop_n-3_text} that $\\rhob=\\mbox{diag}(a,a,b,0)$, while from section~\\ref{subsec_n-3} we know that {$\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ and   $\\Phi_{ij}\\neq 0$ (Remark~\\ref{rem_6d}).} Substituting the form of $\\rhob$ into (\\ref{eqn:algI})--\\eqref{eqn:rhoC} {{and}} (\\ref{eqn:algII}) further reveals that $\\Phi_{ij}=\\diag(\\alpha,-\\alpha,0,0)\\neq 0$.\n\n\nFor $m=1$ one obviously has $\\rhob=\\mbox{diag}(a,0,0,0)\\neq 0$. From section~\\ref{subsec_m=1} we know that { $\\Phi^{\\{3\\}}_{ijkl}=0=\\Phi^{\\{4\\}}_{ijkl}$ and $\\Phi_{ij}\\neq 0$ (Remark~\\ref{rem_6d_m=1}).} \n\n\nFinally, for $m=2$ we can have two possibilities. If $\\Phi^A_{ij}\\neq0$ point \\eqref{RT} of theorem~\\ref{prop_GSHD} \nshows that $\\rhob=\\mbox{diag}(a,a,0,0)\\neq 0$. On the other hand, if $\\Phi^A_{ij}=0=\\Phi^S_{ij}$ we still get $\\rhob=\\mbox{diag}(a,a,0,0)\\neq 0$ from Proposition~\\ref{prop_IIabd}, whereas if $\\Phi^A_{ij}=0\\neq\\Phi^S_{ij}$ the more general form $\\rhob=\\mbox{diag}(a,b,0,0)$ is permitted (with $a\\neq 0\\neq b$).\n\n\nWe have checked in the various permitted cases  that the constraints (\\ref{eqn:algI})--\\eqref{eqn:rhoC} and \\eqref{alg_no_Phi4}--\\eqref{eqn:algII_N}\n allow for nontrivial type II Weyl tensor. {These results are summarized in table~\\ref{tab_6D}, together with a few examples. Examples of spacetimes with distinct non-zero eigenvalues follow from the general discussion of section~\\ref{subsubsec_examples_distinct}, which need not be repeated here.} \n\n\n\\begin{table}[t]\n  \\begin{center}\n  \\begin{tabular}{|c|l|c|c|c|l|c|}\n    \\hline Case & $m$ & Possible form of $\\rhob$ &  O.C. & O.S. & I.T.G. & Examples  \\\\\\hline\n    $\\Phi_{ij} \\not=0$ & 4 & $\\diag(a,a,a,a)$    & $\\surd$ & $\\surd *$ & ${\\cal D}_{23}*$, ${\\cal D}_{45}*$ & RT \\\\\n     $  $ & 3 & $\\diag(a,a,b,0)$     & if $b=a$ & X & ${\\cal D}_{23}$ (if $b\\neq a$) & ?($\\dag$) \\\\\n     $  $ & 2 \\ ($\\Phi^A_{ij}=0$) & $\\diag(a,b,0,0)$    & if $b=a$ &  & ${\\cal D}_{45}$ (${\\cal D}_{23}$ if $b=a$) & {dS$_4\\times$S$^2$} \\\\\n     $  $ & 2 \\ ($\\Phi^A_{ij}\\neq 0$) & $\\diag(a,a,0, 0)$    & $\\surd$ & $\\surd$ & ${\\cal D}_{23}$, ${\\cal D}_{45}$ & ? \\\\\n     $  $ & 1 & $\\diag(a,0,0,0)$    & $\\surd$ & X & & {dS$_3\\times$S$^3$} \\\\\n     $  $ & 0 & $\\diag(0,0,0,0)$   & $\\surd$ & &  & Kundt \\\\\\hline\n     $\\Phi_{ij} =0  $ & 4 & $\\diag(a,a,b,b)$    & if $b=a$ & $\\surd$ &  ${\\cal D}_{23}$, ${\\cal D}_{45}$ (if $b\\neq a$)& ?($\\dag$) \\\\\n     $    $ & 3 & X    & & & & X \\\\\n     $ $  & 2 & $\\diag(a,a,0,0)$  & $\\surd$ &  $\\surd$ & ${\\cal D}_{23}$, ${\\cal D}_{45}$ & ? \\\\\n     $ $  & 1 & X     & & & & X \\\\\n     $ $  & 0 & $\\diag(0,0,0,0)$  & $\\surd$ &  & & Kundt \\\\\n\\hline\n  \\end{tabular}\n    \\caption{Permitted forms of the optical matrix associated with a non-twisting mWAND in a six-dimensional type II spacetime in the two cases $\\Phi_{ij} \\not=0$  and $\\Phi_{ij}=0$ (subtype II(abd)). Recall that $\\Phi^A_{ij}\\neq 0$ is possible only for $m=2$. A question mark indicates subcases that are in principle permitted but for which no examples are known (so we do not claim that these classes are non-empty), while  X corresponds to forbidden subcases. \nAlthough we are not aware of examples with $m=3$ and $\\Phi_{ij} \\not=0(=\\Phi^A_{ij})$ having the most general structure, special solutions with $a=b$ (e.g., a static black string) can be obtained by warping once a type II 5d Einstein metric. {Similarly, generic $m=4$ solutions with $\\Phi_{ij}=0$ are not known, but in the special case $a=b$ they} exist within the \\RT family \\cite{PodOrt06} (the ``$\\mu=0$'' subcase). {These special examples are denoted by a $\\dag$ in the table.} In the O.C. column we have indicated if\/under what conditions $\\rhob$ satisfies the optical constraint. In the O.S. column whether ${\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4}+i\\mbb{5},\\bl\\}$ defines an optical structure and in the I.T.G. column we listed integrable complex two-dimensional distribution that admit totally geodesic integral surfaces (cf. section~\\ref{subsec_integrability} and appendix~\\ref{app_shearfree}). The symbol $*$ means that the corresponding statement is true in a special subcase specified in the text (see section~\\ref{subsec_integrability}). \n\\label{tab_6D}}\n  \\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Additional examples}\n\n\n\nIn addition to the examples mentioned in table~\\ref{tab_6D} and in section~\\ref{subsubsec_examples_distinct}, here we give  two more solutions with $m=1$ in six dimensions (and, necessarily, $\\Phi_{ij}\\neq0$). \n\n\\subsubsection{Example with $\\Lambda\\neq0$}\n\n\n\nWe can use, e.g., AdS$_4\\times$H$^2$ in the form \n\\be\n \\d s^2=\\frac{3\\beta^2}{z^2}(-\\d t^2+\\d\\rho^2+\\rho^2\\d\\phi^2+\\d z^2)+\\frac{\\beta^2}{x^2}(\\d x^2+\\d y^2) .\n\\ee \nFor the null congruence\n\\be\n \\ell_a\\d x^a=\\d t+\\d\\rho ,\n\\ee \nwith the spacelike frame vectors $\\bbm_{(2)}=z\\rho^{-1}\/(\\sqrt{3}\\beta)\\pa_\\phi$, $\\bbm_{(3)}=z\/(\\sqrt{3}\\beta)\\pa_z$, $\\bbm_{(4)}=x\\beta^{-1}\\pa_x$, $\\bbm_{(5)}=x\\beta^{-1}\\pa_y$,  one finds $\\rho_{ij}=\\mbox{diag}(a,0,0,0)$, where $a=z^2\/(3\\beta^2\\rho)$. \n\n\n\n\\subsubsection{Example with $\\Lambda$ arbitrary}\n\nOne can alternatively construct an example with {$\\rho_{ij}=\\mbox{diag}(a,0,0,0)$} by starting from a null geodesic congruence in dS$_3\\times$S$^2$ and adding an extra dimension by Brinkmann's warp. For example, \n\\be\n \\d s^2=f(z)\\left[-V(r)\\d t^2+\\frac{1}{V(r)}\\d r^2+r^2\\d\\chi^2+\\beta^2(\\d\\theta^2+\\sin^2\\theta\\d\\phi^2)\\right]+\\frac{\\d z^2}{f(z)}, \n\\ee \nwith the null congruence\n\\be\n \\ell_a\\d x^a=\\d t+V^{-1}\\d r ,\n\\ee \nwhere  \n\\be \nV(r)=1-\\frac{r^2}{2\\beta^2}, \\qquad f(z)=-\\lambda z^2+2dz+b , \\qquad d^2=\\frac{1}{4\\beta^2}-\\lambda b.\n\\ee\nUsing the spacelike frame vectors $\\bbm_{(2)}=f^{-1\/2}r^{-1}\\pa_\\chi$, $\\bbm_{(3)}=f^{-1\/2}\\beta^{-1}\\pa_\\theta$, $\\bbm_{(4)}=f^{-1\/2}\\beta^{-1}\\sin^{-1}\\theta\\pa_\\phi$, $\\bbm_{(5)}=f^{1\/2}\\pa_z$,  one finds $\\rho_{ij}=\\mbox{diag}(a,0,0,0)$, where $a=\\frac{1}{rf}$. \n\nIt follows from \\cite{PraPraOrt07,OrtPraPra11} that $\\Phi_{ij}\\neq0$, {as it should be}. In general $\\Lambda$ can have any sign but, in particular, we can set $\\Lambda=0$ by choosing the parameter $\\lambda=0$ in $f(z)$. \n\n\n\n\n\\subsection{Optical constraint, optical structures and totally geodesic complex two-dimensional null surfaces}\n\n\\label{subsec_integrability}\n\n\nFirst, in the non-twisting case the optical constraint \\eqref{canformL} is satisfied iff the optical matrix $\\rhob$ possesses only one (possibly repeated) non-zero eigenvalue plus, possibly, some vanishing eigenvalues. This explains column O.C. (``optical constraint'') of table~\\ref{tab_6D}.\n\nNext, when $\\bl$ is non-twisting one can always take an eigenframe of $\\rhob$ to be parallelly transported \\cite{OrtPraPra10,PraPra08} and thus in particular set $\\M{i}{j}{0}=0$. It is then obvious that to each pair of repeated eigenvalues of $\\rhob$ there corresponds an integrable two-dimensional totally null distribution, defined by two unit vectors of the corresponding eigenspace and by $\\bl$ (cf. eq.~\\eqref{D23_6d}). \n\nFurther conditions on the Ricci rotation coefficients can be obtained by considering the Ricci identity (11k,\\cite{OrtPraPra07}), i.e., eq.~\\eqref{11k}, similarly as in section~\\ref{sec_totgeod}.\nConsequences of this equation enable one to prove the statements in columns  O.S. (``optical structure'') and I.T.G. (``integrable totally geodesic'' complex two-dimensional subspace) of table~\\ref{tab_6D}. Since this is rather straightforward, let us just exemplify it to show that when $\\rho_{ij}=\\diag(a,a,b,b)$ (with $b\\neq a$) the maximally totally  null distribution  ${\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4}+i\\mbb{5},\\bl\\}$ defines an optical structure (cf. eqs.~\\eqref{6D_OS}). Namely, from \\eqref{11k} with $i,j,k=224$ and $i,j,k=334$ one gets $\\M{4}{2}{2}=\\M{4}{3}{3}$, while $i,j,k=225$ and $i,j,k=335$ give $\\M{5}{2}{2}=\\M{5}{3}{3}$. Similarly, by taking $i,j,k=442\/552$ and $i,j,k=443\/553$ one obtains, respectively, $\\M{2}{4}{4}=\\M{2}{5}{5}$ and $\\M{3}{4}{4}=\\M{3}{5}{5}$. Next, with $i,j,k=324,234,325,235$ one finds $\\M{3}{4}{2}=\\M{2}{4}{3}=\\M{3}{5}{2}=\\M{2}{5}{3}=0$, while $i,j,k=524,425,534,435$ give $\\M{5}{2}{4}=\\M{4}{2}{5}=\\M{5}{3}{4}=\\M{4}{3}{5}=0$. This implies that eqs.~\\eqref{6D_OS} are satisfied, as we wanted to prove. \n\nLet us finally remark that in the Robinson-Trautman case, corresponding to the first row of table~\\ref{tab_6D}, eq.~\\eqref{11k} reduces to $L_{1i}=0$  \n(cf. also the results of \\cite{PodOrt06,PraPra08}) and thus does not constraint the $\\M{i}{j}{k}$. However, in the special subcase (cf.~\\cite{PodOrt06}) in which the transverse space is of constant curvature one can show by inspection that there exists a natural parallelly transported frame such that ${\\cal D}$ defines an optical structure and ${\\cal D}_{23}$, ${\\cal D}_{45}$ correspond to totally geodesic null two-surfaces.\\footnote{This is a straightforward extension of the discussion in the 5d case, see footnote~10 of \\cite{Ortaggioetal12}.} {{In fact the real $\\mbb{i}$ can be paired arbitrarily in this case, giving rise also to other optical structures and totally geodesic null two-surfaces, and since }} there is a  continuous freedom in  spins there exist {{actually}} infinitely many optical structures. \nThis partly answers a question raised in remark~5.3 of \\cite{Taghavi-Chabert11b}, by showing that also in even dimensions a spacetime may admit more than $2^{[n\/2]}$ optical structures (noting also that in this example replacing $\\bl\\leftrightarrow\\bn$ gives other optical structures). \n\n\n\n\n\\subsection{Type III\/N spacetimes}\n\nBy the results of \\cite{Pravdaetal04}, in this case there are only two admitted structures of $\\rhob$: either $\\rhob=0$ (Kundt spacetimes) or $\\rhob=\\diag(a,a,0,0)\\neq0$. Since eq.~\\eqref{11k} does not contain any Weyl components of zero boost weight, the above discussion still applies and implies that in the case $\\rhob=\\diag(a,a,0,0)\\neq0$ both ${\\cal D}_{23}$ and ${\\cal D}_{45}$ correspond to totally geodesic null two-surfaces and that ${\\cal D}$ defines an optical structure -- explicit examples can be constructed as explained in \\cite{OrtPraPra10}.\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nWe are grateful to Lode Wylleman for useful discussions. The authors acknowledge support from research plan {RVO: 67985840} and research grant no P203\/10\/0749. \n\n\n\n\\renewcommand{\\thesection}{\\Alph{section}}\n\\setcounter{section}{0}\n\n\\renewcommand{\\theequation}{{\\thesection}\\arabic{equation}}\n\n\n\n\\section{Algebraic constraints for type II spacetimes}\n\\setcounter{equation}{0}\n\n\\label{app_alg_eq}\n\n\\subsection{{General mWAND}}\n\nFor completeness and future reference, let us give here some additional algebraic constraints following from the Bianchi and Ricci identities for type II Einstein spacetimes in arbitrary dimensions. \n\nNote that rhs of eq. \\eqref{A4} is not symmetric under $ij\\leftrightarrow kl$ and thus \\eqref{A4}$_{ijkl}-$\\eqref{A4}$_{klij}$ leads to an  algebraic equation\n\\be\n2(\\Phia_{ij} A_{kl}-\\Phia_{kl} A_{ij}) +\\Phi_{k[j}\\rho_{i]l}-\\Phi_{l[j}\\rho_{i]k}-\\Phi_{j[k}\\rho_{l]i}+\\Phi_{i[k}\\rho_{l]j}\n+\\Phi_{ij[k|s}\\rho_{s|l]}-\\Phi_{kl[i|s}\\rho_{s|j]}=0 , \\label{alg_no_Phi4}\n\\ee\nand its trace gives the antisymmetric part of \\eqref{B8}.\nBy multiplying by $\\rho_{lj}$ and using \\eqref{B8}--\\eqref{B8asym} and symmetries of the Weyl tensor, after  lengthy calculations one arrives at an equation with no $\\Phi_{ijkl}$ terms\n\\be\n\\rho \\Phi_{j[i|}\\rho_{j|k]}+\\Phi_{jl}\\rho_{jl}A_{ki}+\\rho_{jl}\\rho_{jl}\\Phia_{ik}\n+\\Phi_{jl}\\left[ 3\\rho_{[j|i}\\rho_{|l]k} +\\rho_{[k|l}\\rho_{j|i]} \\right]\n+2\\rho_{jl}\\left[ \\Phi_{j[i|}A_{l|k]}+2\\Phia_{l[i|}\\rho_{j|k]}\\right]=0.\n \\label{constr_twist}\n\\ee\n\n\\subsection{{Non-twisting mWAND with $\\Phia_{ij}=0$}}\n\nAs already mentioned, by differentiating algebraic constraints and using the Bianchi and Ricci identities one can arrive at further algebraic constraints.\nFor example, in the {\\em non-twisting} case {with $\\Phia_{ij}=0$},\\footnote{{{Recall that in the non-twisting case with $\\Phia_{ij}\\neq0$ the form of $\\rhob$ is fully determined by point \\eqref{PhiA} of theorem~\\ref{prop_GSHD}, cf.~\\cite{Ortaggioetal12}.}}}\nby differentiating~\\eqref{eqn:algI} \nwe arrive at \\eqref{eqn:rhoC} {(equivalent to setting $A_{ij}=0=\\Phia_{ij}$ in \\eqref{thornB8}), which can be rewritten as}\n\\be\n - [\\rhob^2] \\Phi_{ik} + \\Phi (\\rhob^2)_{ik} +  2 \\Phi_{ij} (\\rhob^2)_{jk} + {\\Phi}_{ijkl} {(\\rhob^2)}_{jl}  =0 \\qquad (A_{ij}=0{=\\Phia_{ij}}),\n \\label{eqn:algII_0}\n\\ee\nwhere $[\\rhob^2]$ denotes the trace of {$\\rhob^2$}, etc. The trace of \\eqref{eqn:algII_0} vanishes and multiplying \\eqref{eqn:algII_0} by $\\rho_{ik}$ gives\n\\be\n\t[\\rhob^2][\\Phib \\cdot \\rhob]-[\\rhob][\\Phib \\cdot\\rhob^2]=0 \\qquad {(A_{ij}=0{=\\Phia_{ij}})} .\n\t\\label{trace}\n\\ee\nDifferentiation {of \\eqref{eqn:algII_0}} leads to \n\\bea\n & & -3 \\left[ - [\\rhob^3] \\Phi_{ik} + \\Phi (\\rhob^3)_{ik} +  2 \\Phi_{ij} (\\rhob^3)_{jk} + {\\Phi}_{ijkl} {(\\rhob^3)}_{jl}   \\right] \\nonumber\\\\\n\t\t& & {}+ \\left( \\Phi [\\rhob^2] + [\\Phib \\cdot\\rhob^2]\n\t\t\\right) \\rho_{ik} \n- \\left(\\Phi \\rho + [\\Phib \\cdot \\rhob]\n\\right) (\\rhob^2)_{ik}=0 \\qquad (A_{ij}=0{=\\Phia_{ij}})\n\\label{eqn:algII} ,\n\\eea\nwith trace {of \\eqref{eqn:algII}} vanishing  due to \\eqref{trace}. \n\nFor $\\Phi_{ij}=0$, the $(N-1)$th derivative of  \\eqref{B8}   reads\n\\be\n\t{\\Phi}_{ijkl} {(\\rhob^N)}_{jl} =0, \\quad N =1,2,\\dots \\qquad (A_{ij}=0{=\\Phi_{ij}}).\n\t\\label{eqn:algII_N}\n\\ee\n\n\\section{{Non-twisting type III Einstein spacetimes in arbitrary dimension}}\n\n\n\\setcounter{equation}{0}\n\n\\label{app_type_N_III}\n\n\n\nFor non-twisting Einstein spacetimes of type N, $\\rho_{ij}=\\diag({a},{a},0,\\dots,0)$ \\cite{Pravdaetal04,Durkeeetal10}, where ${a}$ {vanishes} in the case of Kundt spacetimes. Let us present  here a new derivation of the form of the optical matrix for a non-twisting type III Einstein spacetime in arbitrary\ndimension. This is considerably simpler than the original derivation of \\cite{Pravdaetal04}. \n\nFor non-twisting type III, eq. (59) of \\cite{Pravdaetal04} reduces to {(in the notation of \\cite{Durkeeetal10})}\n\\be\n2 \\Psi'_{lij} \\rho_{lk} = \\rho \\Psi'_{kij}.   \\label{typeIIInon-twist}\n\\ee\nIn the Kundt case ($\\rhob=0$) this becomes an identity, while for non-twisting non-Kundt Einstein spacetimes (to which we restrict from now on) necessarily $\\rho\\neq0$ (cf. Proposition~1 of \\cite{OrtPraPra07}) and \\eqref{typeIIInon-twist} gives non-trivial information.\nFollowing \\cite{OrtPraPra10} we choose a parallelly propagated frame with diagonal form of  $\\rho_{ij}=\\diag(\\rho_2,\\rho_3,\\dots)$.\nWithout loss of generality we can assume that $\\Psi'_{2ij} \\not=0$ for some values of $i,\\ j$. Eq.~\\eqref{typeIIInon-twist} then implies\n\\be\n\\rho_2 = \\frac{\\rho}{2}{\\neq0} \\label{typeIIIrho}.\n\\ee\nUsing the $r$-dependence of the non-vanishing eigenvalues of $\\rho_{ij}$ (see eq.~\\eqref{rho_i}), this gives\n\\be\n\\frac{2}{r-b_2} = \\sum_p \\frac{1}{r-b_p},\n\\ee\nwhere $p$ corresponds to all non-vanishing eigenvalues of $\\rho_{ij}$.\nApart from the Kundt case with $\\rho_{ij}=0$, this equation can be satisfied only for $\\rho_{ij}$ of rank 2, with ${a\\equiv}\\rho_2=\\rho_3$.\nThus {\\em the optical matrix for the non-twisting type III Einstein spacetime is either vanishing (Kundt) or \n$\\rho_{ij}=\\diag({a},{a},0,\\dots,0)$}, exactly as in the type N case.\n\n\n\n\n\n\n\\section{On the geodetic and shearfree condition in four and higher dimensions}\n\n\\setcounter{equation}{0}\n\n\\label{app_shearfree}\n\n\n\\subsection{Goldberg-Sachs in four dimensions}\n\n\\label{app_GS_4D}\n\n\nThe four-dimensional Goldberg-Sachs theorem is usually expressed as a statement about a null congruence being geodesic ($\\kappa=0$) and shearfree ($\\sigma=0$). As well-known, this property can be interpreted as different but equivalent geometrical statements, which we now briefly review. This will later be useful for discussing the higher dimensional case.\n\n\n\n\\subsubsection{Integrability of two-spaces}\n\n\\label{subsubsec_integr_d4}\n\nBy considering the commutator $[\\delta,D]$ (see, e.g., p.~77 of \\cite{Stephanibook})\none immediately sees that  $\\bl$ is geodesic and shearfree if and only if the distribution\n\\be\n {\\cal D}=\\mbox{Span}\\{\\bbm,\\bl\\} \n\\ee\nis integrable (cf.~\\cite{penrosebook2}), i.e. $[{\\cal D},{\\cal D}]\\subset{\\cal D}$.\nA similar conclusion holds of course for $\\bar{\\cal D}$. The geodesic{\\&}shearfree property can thus be seen as a statement about the null bivector $\\bl\\wedge\\bbm$ instead of one about the vector field $\\bl$. While the vector field $\\bl$ defines a privileged real null direction lying in ${\\cal D}\\cap\\bar{\\cal D}$, $\\bbm$ is not fixed uniquely (null rotations with $\\bl$ fixed, spins and boost can be performed, under which $\\bl\\wedge\\bbm$ is fixed, up to a rescaling, and $\\kappa=0=\\sigma$ is preserved). The significance of this viewpoint appears clearly when considering complex extensions of the Goldberg-Sachs theorem \\cite{PlebHac75}.\n\n\n\\subsubsection{Conformal structure of the screen space}\n\nGiven a generic null vector field $\\bl$, its flow preserves the conformal structure of the ``screen space'' $L^\\bot\/L$ when \\cite{RobTra83} (see also \\cite{NurTra02,Trautman02a,Trautman02b})\n\\be \n {\\cal L}_\\l g=\\rho g+\\l\\otimes\\xi+\\xi\\otimes\\l  \\quad \\Leftrightarrow  \\quad \\kappa=0=\\sigma .\n \\label{Lie_4}\n\\ee\n\nIn four dimensions only, this is equivalent to requiring that the complex structure of the screen space is preserved by the flow of $\\bl$ \\cite{RobTra83,NurTra02,Trautman02a,MasTag08}.\n\nThis condition can also be derived by imposing Maxwell equations on a null field (two-form) \\cite{robinsonnull} (cf. also, e.g., \\cite{Stephanibook,penrosebook2,NurTra02,Trautman02a,Trautman02b}).\n\n\n\n\\subsection{Conformal structure of the screen space: the standard geodesic{\\&}shearfree condition}\n\n\\label{subsec_screenHD}\n\nRef.~\\cite{RobTra83} (see also \\cite{Trautman02b}) proposed to extend (\\ref{Lie_4}) to higher dimensions as a definition of the geodesic{\\&}shearfree condition. As it turns out,\n\\be \n {\\cal L}_\\l g=\\rho g+\\l\\otimes\\xi+\\xi\\otimes\\l  \\quad \\Leftrightarrow  \\quad \\kappa_{i}=0=S_{ij}-\\frac{S_{kk}}{n-2}\\delta_{ij} ,\n \\label{Lie}\n\\ee\nso that indeed $\\bl$ is geodesic and shearfree in the standard sense. Some comments on algebraically special Einstein spacetimes admitting such a congruence have been already given in section~\\ref{subsec_OC} and appendix~\\ref{app_shearfreetwist}.\n\n\n\nNote that for $n>4$ eq.~(\\ref{Lie}) does {\\em not} follow from Maxwell's equations for a null field (intended again as a two-form), while geodeticity {($\\kappa_{i}=0$)} does, and shear is non-zero for expanding solutions \\cite{Ortaggio07,Durkeeetal10}.\n\n\n\n\n\\subsection{Existence of an optical structure}\n\n\\label{subsubsec_optical}\n\n\n\\subsubsection{Complex notation}\n\n\n\nFor later discussions it will be useful to use a complex basis. Namely, in {\\em even} dimensions one can take the frame $(\\bl,\\bn,\\bmu_A,\\bar\\bmu_A)$ and in {\\em odd} dimensions the frame $(\\bl,\\bn,\\bmu_A,\\bar\\bmu_A,\\bx)$. Except for the unit spacelike vector $\\bx$, all the vectors are null, the complex $\\bmu_A$ are defined by\n\\be\n  \\bmu_2=\\frac{1}{\\sqrt{2}}(\\mbb{2}+i\\mbb{3}) , \\qquad \\bmu_4=\\frac{1}{\\sqrt{2}}(\\mbb{4}+i\\mbb{5}) , \\qquad \\ldots ,\n  \\label{mu}\n\\ee\nand ${\\bar\\bmu_A}$ by their complex conjugates, where $A=2\\mu$, $\\mu=1,\\ldots,(n-2-\\epsilon)\/2$, with $\\epsilon=0,1$ for even and odd dimensions, respectively (in 4d this would be the standard NP frame). The metric becomes\n\\be\n g=\\bl\\otimes \\bn+\\bn\\otimes\\bl+\\bmu_A\\otimes\\bar\\bmu_A+\\bar\\bmu_A\\otimes\\bmu_A+\\epsilon \\bx\\otimes\\bx .\n \\label{complex_g}\n\\ee\n\nThen one can define the complex counterpart of the Ricci rotation coefficients. These are defined as an obvious extension of the real coefficients, e.g.,\n\\be\n \\cL_{AB}=\\mu^a_A\\mu^b_B\\nabla_b l_{a} , \\qquad \\cL_{A\\bar B}=\\mu^a_A\\bar\\mu^b_B \\nabla_b l_{a} , \\qquad \\cM{A}{C}{B}=\\mu^a_C\\mu^b_{{B}}\\nabla_b \\mu_{A a}, \n\\ee\nand other coefficients (and their complex-conjugates) are defined similarly.\n\n\n\n\\subsubsection{Optical structure} \n\n\n\\label{subsubsec_OS}\n\nRef.~\\cite{HugMas88} studied the consequences of Maxwell's equations for a null field defined as an $n\/2$-form in $n$ even dimensions. From these, they arrived at a generalization of the geodesic{\\&}shearfree condition different from the $\\kappa=0=\\sigma$ condition discussed in section~\\ref{subsec_screenHD} (see also \\cite{NurTra02,Trautman02a,Trautman02b,MasTag08}). Recently, this has been extended also to odd dimensions \\cite{Taghavi-Chabert11}. Namely, consider the totally null  $(n-\\epsilon)\/2$-dimensional distribution (recall that $A$ in $\\mu_A$ can be only even)\n\\be\n {\\cal D}=\\mbox{Span}\\{\\bl,\\bmu_2,\\ldots,\\bmu_{(n-2-\\epsilon)}\\} .\n \\label{D_gen}\n\\ee\nIf ${\\cal D}$ and ${\\cal D}^\\bot$ are integrable, i.e.,\n\\be\n [{\\cal D},{\\cal D}]\\subset{\\cal D} , \\qquad [{\\cal D}^\\bot,{\\cal D}^\\bot]\\subset{\\cal D}^\\bot ,\n \\label{Robinson}\n\\ee\n${\\cal D}$ is said to define an ``{\\em optical structure}'' \\cite{Taghavi-Chabert11} (note that ${\\cal D}^\\bot={\\cal D}$ in even dimensions).  In 4d, eq.~\\eqref{Robinson} reduces to the standard conditions $\\kappa=0=\\sigma$ (as discussed in section~\\ref{subsubsec_integr_d4}), which indeed corresponds to the conditions coming from the Mariot-Robinson theorem \\cite{Stephanibook,penrosebook2} for a Maxwell null two-form. In  higher dimensions, using the complex counterpart of the commutators given in \\cite{Coleyetal04vsi}, one finds that (\\ref{Robinson}) is equivalent to \n\\be\n \\cL_{A0}=\\cL_{x0}=\\cL_{AB}=\\cL_{Ax}=\\cL_{xA}=\\cM{A}{B}{0}=\\cM{A}{x}{0}=\\cM{A}{B}{C}=\\cM{A}{B}{x}=\\cM{x}{A}{B}=0 .\n \\label{optical_complex}\n\\ee\n(In fact one first obtains conditions such as $\\cM{A}{B}{C}=\\cM{C}{B}{A}$, but $\\cM{A}{B}{C}=-\\cM{B}{A}{C}$, etc.) The equations above represent the conditions obtained when $n$ is odd, however, if $n$ is even one can simply drop all equations containing $x$ (this will be understood from now on).\n\nNote, in particular, that the vector $\\bl$ is {\\em geodesic} ($\\cL_{A0}=\\cL_{x0}=0$), but generally is not required to be shearfree (except when $n=4$). \nIn even dimensions it has also been observed that (\\ref{Robinson}) means that the complex structure of the screen space is preserved \\cite{NurTra02,Trautman02a,MasTag08}. We have shown in \\cite{Ortaggioetal12} that in 5d a very large class of algebraically special Einstein spacetimes possesses an optical structure. In appendix \\ref{app_OS5d} we extend the results of \\cite{Ortaggioetal12} by showing that, in fact, all algebraically special 5d Einstein spacetimes possess (at least) one optical structure.\n\n\n\n\\paragraph{Optical structure in six dimension} Let us rewrite the conditions \\eqref{optical_complex} in real notation in the special case $n=6$, which is useful for the discussion in the main text (see \\cite{Ortaggioetal12} for the case $n=5$). One readily gets\n\\beqn\n & & \\kappa_{i}=0 \\quad (i=2,\\ldots,5) , \\nonumber \\\\\n & & \\rho_{22}=\\rho_{33}, \\quad \\rho_{23}=-\\rho_{32}, \\quad \\rho_{44}=\\rho_{55}, \\quad \\rho_{45}=-\\rho_{54},  \\nonumber \\\\\n & & \\rho_{24}=\\rho_{35}, \\quad \\rho_{42}=\\rho_{53}, \\quad \\rho_{34}=-\\rho_{25}, \\quad \\rho_{43}=-\\rho_{52}, \\nonumber \\\\\n & & \\M{2}{4}{0}=\\M{3}{5}{0}, \\quad \\M{3}{4}{0}=-\\M{2}{5}{0}, \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad (n=6) \\label{6D_OS} \\\\\n & & \\M{2}{4}{2}-\\M{3}{4}{3}=\\M{2}{5}{3}+\\M{3}{5}{2}, \\quad -\\M{2}{5}{2}+\\M{3}{5}{3}=\\M{2}{4}{3}+\\M{3}{4}{2}, \\nonumber \\\\ \n & & \\M{2}{4}{4}-\\M{2}{5}{5}=\\M{3}{4}{5}+\\M{3}{5}{4}, \\quad \\M{2}{4}{5}+\\M{2}{5}{4}=-\\M{3}{4}{4}+\\M{3}{5}{5} . \\nonumber \n\\eeqn\n\nIn particular, if $\\bl$ is twistfree then $\\rhob$ has two pairs of repeated eigenvalues.\n\n\n\n\n\n\\subsection{Optical constraint}\n\n\n\nBased on results for Kerr-Schild spacetimes, Ref.~\\cite{OrtPraPra09} put forward yet another possible generalization of the shearfree condition for mWANDs in higher dimensions. This is the so called ``optical constraint'', already discussed in section~\\ref{subsec_OC}, which involves only the null direction $\\bl$ (as opposed to the optical structure discussed above). The Lorentz transformation freedom of null rotations preserving $\\bl$, boosts and spins is thus retained in this case (indeed spins can be used to arrive at the canonical form~\\eqref{canformL}), see also \\cite{OrtPraPra10} for related comments. We have shown in \\cite{Ortaggioetal12} that in 5d a very large class of algebraically special Einstein spacetimes admits an mWAND obeying the optical constraint, {{and Ref.~\\cite{Wylleman_priv} extended our result to prove that in fact {\\em all} algebraically special Einstein spacetimes admit such an mWAND} (see also \\cite{OrtPraPra12rev})}.\nNote that in 4d the optical constraint is a necessary condition for $\\bl$ to be a repeated principal null direction but is not {sufficient} \\cite{Ortaggioetal12}.  \n  \n\n\\subsection{Integrability of a (complex) two-dimensional totally null distribution}  \n\n\\subsubsection{${\\cal D}_{23}$ integrable}\n\nAs a further generalization of the geodesic{\\&}shearfree condition one can consider the integrability of the complex two-dimensional totally null distribution\n\\be \n {\\cal D}_{23}=\\mbox{Span}\\{\\bmu_{2},\\bl\\} .\n \\label{D}\n\\ee\nIt is easy to show that ${\\cal D}_{23}$ is integrable if and only if \n\\be\n \\cL_{20}=0, \\qquad \\cL_{B2}=\\cM{2}{B}{0}, \\qquad \\cL_{\\bar B2}=\\cM{2}{\\bar B}{0} \\quad (B\\neq 2) , \\qquad \\cL_{x2}=\\cM{2}{x}{0} .\n \\label{2integr}\n\\ee\nIn 4d this again reduces to the standard $\\kappa=0=\\sigma$ condition.\n\nIn particular, in six dimensions \\eqref{2integr} can be rewritten in real notation as\n\\beqn\n & & \\kappa_{2}=0=\\kappa_{3} , \\qquad  \\rho_{22}=\\rho_{33}, \\quad \\rho_{23}=-\\rho_{32}, \\nonumber \\label{D23_6d} \\\\\n & & \\rho_{42}=\\M{2}{4}{0}, \\quad \\rho_{43}=\\M{3}{4}{0}, \\quad \\rho_{52}=\\M{2}{5}{0}, \\quad \\rho_{53}=\\M{3}{5}{0}  \\qquad\\qquad (n=6) . \n\\eeqn\n\n\\subsubsection{${\\cal D}_{23}$ integrable with totally geodesic integral surfaces}  \n\nWe can strengthen the above conditions by further requiring the integral surfaces of ${\\cal D}_{23}$ to be {\\em totally geodesic}. The corresponding equations read\n\\beqn\n & & \\cL_{B0}=0=\\cL_{x0}, \\qquad \\cL_{B2}=0=\\cL_{x2}, \\qquad \\cL_{\\bar B2}=0 \\quad (B\\neq 2) , \\nonumber \\\\\n & & \\cM{2}{B}{0}=0=\\cM{2}{x}{0}, \\qquad \\cM{2}{B}{2}=0=\\cM{2}{x}{2}, \\qquad \\cM{2}{\\bar B}{0}=0 \\quad (B\\neq 2) , \\qquad \\cM{2}{\\bar B}{2}=0 \\quad (B\\neq 2) .\n \\label{2totgeod}\n\\eeqn\nNow $\\bl$ is necessarily geodesic ($\\cL_{B0}=0=\\cL_{x0}$). In 4d \\eqref{2totgeod} is equivalent to \\eqref{2integr} because $B=2$ is the only possibility (and there are no $x$-components). In real notation \\eqref{2totgeod} can be rewritten as (where we define $\\hat k\\neq 2,3$)\n\\beqn\n & & \\kappa_{i}=0 , \\qquad  \\rho_{22}=\\rho_{33}, \\quad \\rho_{23}=-\\rho_{32}, \\qquad \\rho_{\\hat k2}=0=\\rho_{\\hat k3} \\nonumber \\\\\n & & \\M{2}{\\hat k}{0}=0=\\M{3}{\\hat k}{0} , \\label{2totgeod_real} \\\\\n & & \\M{2}{\\hat k}{2}=\\M{3}{\\hat k}{3}, \\quad  \\M{2}{\\hat k}{3}=-\\M{3}{\\hat k}{2} \\nonumber .\n\\eeqn\n\n\n\n\n\n  \n\n\\section{Optical structures in five dimensions}\n\n\\label{app_OS5d}\n\n\nProposition~4 of \\cite{Ortaggioetal12} gives a set of sufficient conditions for a five-dimensional Einstein spacetime of type II or more special to possess an {\\em optical structure} \\cite{Taghavi-Chabert11}. In particular, it shows that, except possibly for Kundt spacetimes (and for a special subclass ``(iii)'' of genuine type II, later proven not to exist \\cite{Wylleman_priv}), all algebraically special Einstein spacetimes admit an optical structure in five dimensions. It is the purpose of this  appendix to show that this in fact holds also for Kundt spacetimes. (Obviously, if a complex optical structure is integrable its complex conjugate is integrable too and this will be understood in the following.) Combining this with the proof of \\cite{Wylleman_priv} that the special subclass (iii) of genuine type II is empty, we arrive at \n\\begin{prop}\n\\label{prop_integrab_5D}\nIn a five-dimensional Einstein spacetime admitting a multiple WAND~$\\lb$ there always exists an optical structure. In the case of type D spacetimes there exist in fact (at least) two optical structures. \n\\end{prop}\n\n\n\\begin{proof}\nLet us  show that Einstein spacetimes of the Kundt class always possess an optical structure in five dimensions. In other words, we need to show that in such spacetimes there always exists a null frame $\\{\\bl,\\bn,\\mbb{2},\\mbb{3},\\mbb{4}\\}$ such that \nthe totally null distribution\n\\be\n {\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\bl\\} ,\n\\ee\nand its orthogonal complement \n\\be\n {\\cal D}^\\bot=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4},\\bl\\} ,\n\\ee \nare both integrable. This is equivalent to \\cite{Ortaggioetal12} (see also appendix~\\ref{app_shearfree}, and \n{\\eqref{L1i_M}} for the definition of $\\M{a}{b}{c}$)\n\\beqn\n & & \\kappa_{i}=0, \\qquad \\rho_{33}=\\rho_{22}, \\qquad \\rho_{32}=-\\rho_{23},\\qquad \\rho_{24}=0=\\rho_{34}, \\qquad \\rho_{42}=0=\\rho_{43},  \t\t\t  \\label{Dorth_integr_k_rho} \n \\\\  & & \\M{2}{4}{0}=0=\\M{3}{4}{0} , \\qquad \\M{2}{4}{2}=\\M{3}{4}{3}, \\qquad \\M{2}{4}{3}=-\\M{3}{4}{2} .\n \\label{Dorth_integr_M} \n\\eeqn\n\n\nKundt spacetimes admit a metric in the form \\cite{Coleyetal03,ColHerPel06,PodZof09} \n\\be\n \\d s^2 =2\\d u\\left[\\d r+H(u,r,x)\\d u+W_\\alpha(u,r,x)\\d x^\\alpha\\right]+ g_{\\alpha\\beta}(u,x) \\d x^\\alpha\\d x^\\beta , \\label{Kundt_gen}\n\\ee\nwhere $\\alpha,\\beta=2,3,4$ in five dimensions. Here $\\bl=\\partial_r$ is a geodesic, twistfree, shearfree, non-expanding mWAND, so that \\eqref{Dorth_integr_k_rho} is automatically satisfied. Now, define a null frame with $\\ell_a\\d x^a=\\d u$, $n_a\\d x^a=\\d r+H\\d u+W_\\alpha\\d x^\\alpha$ and the spacelike vectors $\\mbb{i} $ living in the three-dimensional transverse (Euclidean) space spanned by the $x^\\alpha$ (their components will be, in particular, independent of $r$). Then one immediately finds\n\\be\n \\M{i}{j}{0}=0, \n\\ee\nso that the first of \\eqref{Dorth_integr_M} is satisfied in this frame. \n\nNext, using the Christoffel symbols given in \\cite{PodZof09} one can check that  \n\\be\n m_{(i)\\a;\\beta}=m_{(i)\\a||\\beta} ,\n\\ee\nwhere the covariant derivatives on the r.h.s. is taken w.r.t. the transverse metric $g_{\\a\\beta}$. \n\nIt follows that\n\\be\n {\\M{i}{j}{k}} ={\\Mt{i}{j}{k}} , \n\\ee\nwhere the connection coefficients on the r.h.s. are those computed w.r.t. the transverse metric $g_{\\a\\beta}$ (and the o.n. frame vectors of the transverse space are simply the ``projections'' $\\tilde m_{(i)\\a}$  of the $m_{(i)\\a}$ vectors). \n\nOne then finds\n\\be\n [\\mbtt{2}+i\\mbtt{3},\\mbtt{4}]=(\\Mt{4}{2}{4}+i\\Mt{4}{3}{4})\\mbtt{4} +[-\\Mt{2}{4}{2}+i(\\Mt{2}{3}{4}-\\Mt{2}{4}{3})]\\mbtt{2}+[\\Mt{3}{2}{4}-\\Mt{3}{4}{2}-i\\Mt{3}{4}{3}]\\mbtt{3} ,\n\\ee\nso that $\\mbox{Span}\\{\\mbtt{2}+i\\mbtt{3},\\mbtt{4}\\}$ is integrable iff\n\\be\n \\Mt{3}{4}{3}=\\Mt{2}{4}{2} , \\qquad \\Mt{3}{4}{2}=-\\Mt{2}{4}{3} .\n \\label{integr_234}\n\\ee \nProving the integrability of ${\\cal D}$ and ${\\cal D}^\\bot$ is thus now reduced to proving the integrability of $\\mbox{Span}\\{\\mbtt{2}+i\\mbtt{3},\\mbtt{4}\\}$. The transverse frame is arbitrary, so we just need to show that there exists at least one frame satisfying this integrability property.\n\nIf we define the complex null vector field\n\\be\n \\bmu=\\frac{1}{\\sqrt{2}}(\\mbtt{2}+i\\mbtt{3}) , \n \\label{def_mu}\n\\ee\nusing \\eqref{integr_234} the required integrability condition $[\\bmu,\\mbtt{4}]=\\alpha\\bmu+\\beta\\mbtt{4}$ reads\n\\be\n \\mu_{a||b}m_{(4)}^a\\mu^b=0 .\n\\ee \nThus by choosing a complex null geodesics $\\bmu$ in the transverse three-space we automatically obtain  the integrability of the corresponding  distributions ${\\cal D}$ and ${\\cal D}^\\bot$ and {(together with the result of \\cite{Wylleman_priv})} our proof is complete.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Shearfree twisting spacetimes (even dimensions)}\n\n\\label{app_shearfreetwist} \n\nTwisting geodesic mWANDs with zero shear are forbidden in odd dimensions \\cite{OrtPraPra07} but they are permitted in {\\em even} dimensions and they have necessarily $\\det(\\rhob)\\neq 0$ (as can be easily seen in a frame adapted to $A_{ij}$, using the fact that $S_{ij}\\propto\\delta_{ij}$). Here we present an explicit example in six dimensions. {To our knowledge, this is the first such example that has been identified.} \n\nFirst, consider the six-dimensional Ricci flat Taub-NUT metric \\cite{ManSte04} \n\\beqn\n \\d s^2= & & -F(r)(\\d t-2n_1\\cos\\theta_1\\d\\phi_1-2n_2\\cos\\theta_2\\d\\phi_2)^2+\\frac{\\d r^2}{F(r)} \\nonumber \\\\\n & & {}+(r^2+n_1^2)(\\d\\theta_1^2+\\sin\\theta_1^2\\d\\phi_1^2)+(r^2+n_2^2)(\\d\\theta_2^2+\\sin\\theta_2^2\\d\\phi_2^2) ,\n \\label{MasSte6D}\n\\eeqn\nwhere\n\\be\n F(r)=\\frac{r^4\/3+(n_1^2+n_2^2)r^2-2mr-n_1^2n_2^2}{(r^2+n_1^2)(r^2+n_2^2)} .\n\\ee\nWe observe that this is a spacetime of type D. A geodetic mWAND is given by\n\\be\n \\ell_a\\d x^a=\\d t+F(r)^{-1}\\d r-2n_1\\cos\\theta_1\\d\\phi_1-2n_2\\cos\\theta_2\\d\\phi_2 ,\n\\ee \nwhile a second one can simply be obtained by reflecting $\\bl$ as $t\\to-t$, $\\phi_1\\to-\\phi_1$, $\\phi_2\\to-\\phi_2$ \\cite{PraPraOrt07}. Using the frame vectors\n\\beqn\n & & m_{(2)a}\\d x^a=\\sqrt{r^2+n_1^2}\\d\\theta_1, \\qquad m_{(3)a}\\d x^a=\\sqrt{r^2+n_1^2}\\sin\\theta_1\\d\\phi_1, \\nonumber \\\\\n & & m_{(4)a}\\d x^a=\\sqrt{r^2+n_2^2}\\d\\theta_2, \\qquad m_{(5)a}\\d x^a=\\sqrt{r^2+n_2^2}\\sin\\theta_2\\d\\phi_2,  \n\\eeqn\none finds\n\\be\n\\rhob = \\left(\n\\begin{array}{cccc} \n      \\displaystyle \\frac{r}{r^2+n_1^2} & \\displaystyle -\\frac{n_1}{r^2+n_1^2} & 0 & 0\\\\ \n\t\t\t\\displaystyle \\frac{n_1}{r^2+n_1^2} & \\displaystyle \\frac{r}{r^2+n_1^2}  & 0 & 0 \\\\ \n\t\t\t0 & 0 & \\displaystyle \\frac{r}{r^2+n_2^2} & \\displaystyle -\\frac{n_2}{r^2+n_2^2}\\\\ \n\t\t\t0 & 0 & \\displaystyle \\frac{n_2}{r^2+n_2^2} & \\displaystyle \\frac{r}{r^2+n_2^2} \n  \\end{array}\n \\right) . \n\\ee\n\n\nOne can easily check that $\\rhob$  obeys the optical constraint~(\\ref{OC2}). {Moreover, we also observe that in the spacetime~\\eqref{MasSte6D} the maximally totally null distribution ${\\cal D}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\mbb{4}+i\\mbb{4},\\bl\\}$\ndefines an {\\em optical structure}  (concept introduced and discussed in \\cite{HugMas88,NurTra02,Trautman02a,Trautman02b,Taghavi-Chabert11,Taghavi-Chabert11b}), and both the totally null distributions ${\\cal D}_{23}=\\mbox{Span}\\{\\mbb{2}+i\\mbb{3},\\bl\\}$ and ${\\cal D}_{45}=\\mbox{Span}\\{\\mbb{4}+i\\mbb{5},\\bl\\}$ are integrable, with totally geodesic integral surfaces (see appendix~\\ref{app_shearfree} for the corresponding definitions and conditions).}\nThe null vector field $\\bl$ is generically shearing ($S_{22}=S_{33}\\neq S_{44}=S_{55}$), however in the special case $n_1=n_2$ (corresponding to the solutions of \\cite{AwaCha02}) it becomes {\\em shearfree} (while still being expanding and twisting). The class of shearfree twisting spacetimes is thus non-empty in higher dimensions. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{Non-degenerate, non-twisting geodesic mWANDs violating the optical constraint ($n\\ge 7$)}}\n\n\\setcounter{equation}{0}\n\n\\label{app_violating}\n\n\nIn the main text we have seen examples of Einstein spacetimes admitting a non-twisting mWAND violating the optical constraint, see e.g. table~\\ref{tab_distinct_eigenval}. However, all of them have a degenerate $\\rhob$ (i.e., $m<n-2$). In this appendix we provide also some examples with $m=n-2$, relevant to theorem~\\ref{prop_GSHD} (point \\eqref{Phi0}) and proposition~\\ref{prop_non_deg2} (point 2.).\n\n\n\\subsection{Metric}\n\nUsing the theory of conformal Einstein spaces \\cite{Brinkmann25} (reviewed, e.g., in \\cite{petrov}), by taking a double Brinkmann warp one can construct the following Einstein space  (satisfying $R_{ab}=2\\Lambda g_{ab}\/(n-2)$):\n\\be\n \\d s^2=\\lambda r^2\\d u^2+2\\d u\\d r+(\\lambda ur-1)^2\\d\\sigma_\\lambda^2+r^2\\d\\Sigma_0^2 , \n \\label{general_violating}\n\\ee\nwhere \n\\be\n \\lambda=2\\frac{\\Lambda}{(n-1)(n-2)} ,\n\\ee\n$\\d\\sigma_\\lambda^2$ is a Riemannian Einstein space of dimension $n_\\sigma$ with Ricci scalar $R_\\sigma=n_\\sigma(n_\\sigma-1)\\lambda$ and $\\d\\Sigma_0^2$ a Ricci-flat Riemannian space of dimension $n_\\Sigma$. In order for $\\d s^2$ to have a non-zero Weyl tensor, the metrics $\\d\\sigma_\\lambda^2$ or $\\d\\Sigma_0^2$ cannot be both of constant curvature, so that either $n_\\sigma\\ge 4$ or $n_\\Sigma\\ge 4$ (or both). The complete spacetime has thus dimension $n=2+n_\\sigma+n_\\Sigma\\ge 7$ (unless it is of constant curvature, which is of no interest to us).\n\nLet us define the vector field $\\bl$ with covariant and contravariant components\n\\be\n \\ell_a\\d x^a=\\d u , \\qquad \\ell^a\\pa_a=\\pa_ r ,\n \\label{l_ex}\n\\ee\nwhich is obviously  null and hypersurface-orthogonal, thus automatically geodesic and twistfree, and $r$ is an affine parameter along it. This can be accompanied by another null vector $\\bn$ (also hypersurface-orthogonal)\n\\be\n n_a\\d x^a=\\d r+\\frac{\\lambda}{2}r^2\\d u ,\n \\label{n_ex}\n\\ee\nwhich satisfies the normalization condition $\\bl\\cdot\\bn=1$.\n\n\n\n\\subsection{Weyl tensor}\n\nFrom the results of \\cite{Brinkmann25} together with those on direct product spacetimes \\cite{PraPraOrt07} it readily follows that\n\\be\n C_{uabc}=0=C_{rabc} .\n \\label{C_ur_ex}\n\\ee \n\nLet us consider a set of $n-2$ spacelike o.n. vectors $\\bbm_{(i)}$ orthogonal to $\\bl$ and $\\bn$, composed of a subset of vectors denoted by $i=A, B, \\ldots$   living in the subspace of $\\d\\sigma_\\lambda^2$ and a subset denoted by $i=I,J, \\ldots$ living in the subspace of $\\d\\Sigma_0^2$. Then the only non-zero Weyl frame components are given by $C_{ABCD}$ and $C_{IJKL}$, and their $r$-dependence is (recall the first definition in \\eqref{bw0}) \n\\be\n \\Phi_{ABCD}=\\frac{1}{(\\lambda ur-1)^2}C^0_{ABCD} , \\qquad \\Phi_{IJKL}=\\frac{1}{r^2}C^0_{IJKL} , \n \\label{C_ABIJ_ex}\n\\ee\nwhere quantities with superscript $^0$ do not depend on $r$ (these are in fact the Weyl components of the respective ``subspaces''). \n\nIn particular, one has\n\\be\n \\WD{ij} =0 .\n\\ee\nNote also that one has generically $W_{ij}\\neq 0$ (recall  definition \\eqref{def_W} and constraint~\\eqref{W_phi}).\n\n\n\n\n\\subsection{Multiple WAND(s) and optics}\n\nFrom the above results it follows that both $\\bl$ and $\\bn$ are {double} WANDs and the spacetime is thus of type~D. In fact a bit more than that, since \n\\be\nC_{abcd}\\ell^d=0=C_{abcd}n^d \\label{Wandcond}\n\\ee\n (cf. \\cite{PraPra05,Ortaggio09} for the meaning of such conditions in terms of the Weyl type)  and the type is D(abd).\\footnote{The construction {of metric \\eqref{general_violating}} is carried out more naturally in a different coordinate system (not adapted to $\\bl$) in which $\\bl$ and $\\bn$ are related by a coordinate transformation (``time-reflection'') that leaves the metric invariant: therefore they share the same geometric properties.}\n\n\nThe optical matrix $\\rhob$ of $\\bl$ is obviously symmetric, it is moreover diagonal and non-degenerate with components\n\\be\n \\rho_{AB}=\\frac{\\lambda u}{\\lambda ur-1}\\delta_{AB} , \\qquad \\rho_{IJ}=\\frac{1}{r}\\delta_{IJ} .\n\\ee\nThere are thus two eigenvalue-blocks of dimension $n_\\sigma$ and $n_\\Sigma$, or $[n_\\sigma,n_\\Sigma]$. For instance, for $n=7$ we can construct explicit solutions with eigenvalues $\\{a,a,a,a,b\\}$, for $n=8$ we can have \n$\\{a,a,a,a,a,b\\}$ and $\\{a,a,a,a,b,b\\}$ and so on (in all cases $a\\neq0\\neq b$, $a\\neq b$). These provide examples for of the shearing spacetimes of  theorem~\\ref{prop_GSHD}  \nand proposition~\\ref{prop_non_deg2}, although an eigenvalue structure {more general than the one of metric \\eqref{general_violating} can in principle exist}. {In particular, the optical constraint is clearly violated (cf. eq.~\\eqref{canformL}).} Note also that we cannot construct a six-dimensional example with this method. By setting the cosmological constant to zero in the above metrics ($\\lambda=0$) one obtains Ricci-flat spacetimes -- however, these are  direct products and $\\rhob$ becomes degenerate, which is of no interest to the present discussion.\n\nBy taking direct products one can trivially generate many similar examples with a degenerate $\\bl$. These will have $\\WDS{ij} \\neq 0$, however still with $\\WDA{ij} =0$ (see \\cite{PraPraOrt07}), in agreement with point (i) of theorem~\\ref{prop_GSHD}. \n\n\n\\subsubsection{On possible additional (m)WANDs}\n\n\\label{subsubsec_additionalW}\n\n\n\nOne might wonder whether spacetimes \\eqref{general_violating} admit other multiple (double) WANDs (different from $\\bl$ and $\\bn$, eqs.~\\eqref{l_ex} and \\eqref{n_ex}) and in particular whether those can obey the optical constraint. Take a generic null vector $\\bk=\\alpha\\pa_u+\\beta\\pa_r+\\gamma^A\\bbm_{(A)}+\\gamma^I\\bbm_{(I)}$ (where the $\\bbm_{(A)}$ [$\\bbm_{(I)}$] have components only in the subspace of $\\d\\sigma_\\lambda^2$ [$\\d\\Sigma_0^2$]), i.e., $\\alpha^2\\lambda r^2+2\\alpha\\beta+\\delta_{AB}\\gamma^A\\gamma^B+\\delta_{IJ}\\gamma^I\\gamma^J=0$. Since for $\\gamma^A=0=\\gamma^I$ $\\bk$ gives the directions of $\\bl$ and $\\bn$ (respectively for $\\alpha=0$ and $\\beta=-\\frac{1}{2}\\alpha\\lambda r^2$), we now assume that $\\gamma^A$ and $\\gamma^I$ are not both zero (so that $\\alpha\\neq0$). Using \\eqref{C_ur_ex}, \\eqref{C_ABIJ_ex} and the Bel-Debever criteria \\cite{Ortaggio09} it follows that: (i) $\\bk$ is a WAND $\\Leftrightarrow C^0_{ABCD}\\gamma^A\\gamma^C=0=C^0_{IJKL}\\gamma^I\\gamma^K$  \n(so that, using also \\eqref{C_ur_ex}, one gets $C_{abcd}k^bk^d=0$); (ii) $\\bk$ is an mWAND $\\Leftrightarrow C^0_{ABCD}\\gamma^A=0=C^0_{IJKL}\\gamma^I$ (in which case, using also \\eqref{C_ur_ex}, one gets $C_{abcd}k^d=0$). \n\nThese conditions on the Weyl tensors of $\\d\\sigma_\\lambda^2$ and $\\d\\Sigma_0^2$ are generically not satisfied (see an example in section~\\ref{subsubsec_ex2}), so in general spacetimes \\eqref{general_violating} do not admit any other WANDs (not even single) apart from $\\bl$ and $\\bn$. However, it is also clear that in special cases additional mWANDs may exist. For example, when either $\\d\\sigma_\\lambda^2$ or $\\d\\Sigma_0^2$ are conformally flat (which is necessarily the case if $n_\\sigma<4$  or $n_\\Sigma<4$) spacetimes \\eqref{general_violating} admit a continuous infinity of mWANDs (generically non-geodesic but some can be geodesic, see an example in section~\\ref{subsubsec_ex1}). \n\n\n\n\n\\subsection{An explicit example with additional mWANDs}\n\n\n\\label{subsubsec_ex1}\n\nFor the sake of definiteness, one can for instance construct an explicit solution for $n=7$ by taking (\\ref{general_violating}) with \n\\beqn\n & & \\d\\sigma_\\lambda^2=V(\\rho)\\d\\tau^2+V^{-1}(\\rho)\\d\\rho^2+\\rho^2(\\d\\theta^2+\\sin\\theta^2\\d\\phi^2) , \\qquad V(\\rho)=1-\\frac{\\mu}{\\rho}-\\lambda \\rho^2 , \\nonumber \\\\\n & & \\d\\Sigma_0^2=\\d z^2 , \\label{NOCmetric}\n\\eeqn\nand a coordinate range such that $V(\\rho)>0$.\n\nTaking $\\bl=\\pa_r$ and the orthonormal vectors\n\\beqn\n & & \\bbm_{(2)}=A^{-1}V^{-1\/2}(\\rho)\\pa_\\tau , \\qquad \\bbm_{(3)}=A^{-1}V^{1\/2}(\\rho)\\pa_\\rho, \\qquad \\bbm_{(4)}=A^{-1}\\rho^{-1}\\pa_\\theta, \\nonumber \\\\ \n & & \\bbm_{(5)}=A^{-1}\\rho^{-1}\\sin\\theta^{-1}\\pa_\\phi, \\qquad \\bbm_{(6)}=r^{-1}\\pa_z , \\qquad A=\\lambda ur-1 ,\n\\eeqn\none finds\n\\be\n \\rho_{22}=\\rho_{33}=\\rho_{44}=\\rho_{55}=\\frac{\\lambda u}{\\lambda ur-1} , \\qquad \\rho_{66}=\\frac{1}{r} .\n\\ee\n\nThe only non-zero Weyl frame components read (recall definition \\eqref{def_W})\n\\be\n W_{23}=W_{45}=\\frac{1}{(\\lambda ur-1)^2}\\frac{\\mu}{\\rho^3} , \\qquad W_{24}=W_{25}=W_{34}=W_{35}=-\\frac{1}{(\\lambda ur-1)^2}\\frac{\\mu}{2\\rho^3} .\n\\ee\n\nHowever, note  that since $\\d\\Sigma_0^2=\\d z^2$ is (conformally) flat, it follows from section~\\ref{subsubsec_additionalW} that metric \\eqref{general_violating} with \\eqref{NOCmetric} admits also other  mWANDs of the form  \n$\\bk=\\frac{1}{r^2}[\\pa_u-\\frac{1}{2}r^2(\\lambda+\\gamma^2)\\pa_r+\\gamma\\pa_z]$, where $\\gamma$ is an arbitrary function.\n\nIt turns out that if $\\gamma_{,r}=\\gamma_{,u}=\\gamma_{,z}=0$ the mWAND $\\bk$ is geodesic, in which case it becomes twistfree iff $\\gamma$ is a constant and it obeys the optical constraint iff $\\gamma^2=|\\lambda|$ (so that it is also twistfree), i.e.,\n\\be\n\t\\bk=\\frac{1}{r^2}\\left[\\partial_u-\\frac{1}{2}(\\lambda+|\\lambda|)r^2\\pa_r\\pm\\sqrt{|\\lambda|}\\pa_z\\right] .\n\\ee\nIn this case the corresponding optical matrix is of the form $\\{a,a,a,a,a\\}$ for $\\lambda>0$ and $\\{a,a,a,a,0\\}$ for $\\lambda<0$ (in both cases $a\\neq0$).\n\n\n\n\n\\subsection{An explicit example without additional mWANDs}\n\n\n\\label{subsubsec_ex2}\n\n{It follows from section~\\ref{subsubsec_additionalW} that in order for metric \\eqref{general_violating} to admit only two double WANDs we need $n_\\sigma\\ge4$ and $n_\\Sigma\\ge4$, i.e., at least 10 spacetime dimensions. If we now take $\\d\\sigma_\\lambda^2$ as in \\eqref{NOCmetric} but} $\\d\\Sigma_0^2$ corresponding to the Riemannian version of 4d Schwarzschild, {by looking at the conditions of section~\\ref{subsubsec_additionalW} it is easy to see that indeed} the {\\em only} mWANDS are the $\\bl$ and $\\bn$ discussed above (eqs.~\\eqref{l_ex} and \\eqref{n_ex}). Such metric thus constitutes an example of an Einstein spacetime with {\\em all double WANDs violating the optical constraint}. Recall also that such mWANDs \\eqref{l_ex} and \\eqref{n_ex} are geodesic and non-degenerate.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nLow-mass starless cores are the earliest observed phase of isolated\nlow-mass star formation.  They are identified via submm\ndust continuum and dense gas molecular lines, they typically contain\na few solar masses, they have sizes of approximately $0.1$ pc,\nand they may form one or a few low-mass \n(M $\\sim 1$ M$_{\\odot}$) stars.\nSeveral hundred starless cores have been\nobserved in the nearest star-forming molecular clouds and isolated Bok\nglobules.  \nRecent large scale\nsurveys of nearby molecular clouds have established a remarkable connection\nbetween the Core Mass Function and the Initial Mass Function of stars\n(e.g., Motte et al. 1998, Johnstone et al. 2000), indicating the importance\nof constraining the evolution of starless cores in order\nto understand the initial conditions of disk and protostar formation.\nTheoretically, the basic core formation and evolution process is still \ndebated between a turbulent-dominated (Mac Low \\& Klessen 2004) \nor ambi-polar diffusion-dominated model (Shu et al. 1987).  \nObservationally, a fundamental challenge is to determine\nthe evolutionary state of a starless core.  Given a set of observations,\ncan we determine how close a starless core is to forming a protostar?\n\nOne step toward understanding the evolutionary state of a starless\ncore is to determine its physical structure ($n(r)$ cm$^{-3}$ and\n$T(r)$ K).  \nRadiative transfer modeling of submillimeter dust continuum emission\nhave successfully fit the density structure with hydrostatic\nconfigurations (Bonnor-Ebert Spheres, BESs), while the calculated \ntemperature structure decreases toward\nthe center of the core due to attenuation of the ISRF (Evans et al. 2001).  \nSince BESs may be parameterized in terms of their central density,\n$n_c$ cm$^{-3}$, it is natural to think that this may be the\nmain evolutionary variable for starless cores.  However, detailed\nmolecular line observations have revealed that the chemical\nstructure ($X(r)$) can strongly vary among cores with similar central\ndensities.  Figure 1 shows the example of L1498 and L1521E, two\nstarless cores in Taurus which have comparable $n_c$ (Tafalla et al. 2006), \nbut very different chemical structures.  L1521E is centrally peaked in C$_3$S\nwith weak NH$_3$ emission while L1498 is centrally peaked in NH$_3$\nand heavily depleted in C$_3$S.  These observations can be explained\nif the cores are evolving at different rates.  \\textbf{In order to\ndetermine the evolutionary state of a starless core, it is\nnecessary to map the chemical structure of the core}.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=9 cm]{Shirley_fig1.ps}\n\\caption{Observations of L1521E (top) and L1498 (bottom) in 850 $\\mu$m, \nC$_3$S,\nand NH$_3$ showing chemical differentiation.  Data are from the ARO-GBT survey\nand Shirley et al. (2005)} \\label{fig}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Chemical Processes in Starless Cores}\n\\label{section1}\n\nThe rate at which molecules are created and destroyed differ for each\nspecies;  therefore, the relative abundance of molecular species may be\nused as a chemical clock.  A prediction that appears to be ubiquitous among\nstarless core chemodynamical models is that there exists\na class of molecules, named ``early-time'' molecules, which\nform rapidly in the cold, moderate density environments typical of nascent\nstarless cores (e.g., CO, C$_2$S, C$_3$S, SO).  The early-time molecule\nabundance peaks typically within a few $10^5$ years and then decreases\ndue to various destruction mechanisms (see below).  Another class of molecules,\nnamed ``late-time'' molecules, build up in abundance slower and\nremain in the gas phase longer at low temperatures and high densities\n(e.g., N$_2$H$^+$, NH$_3$, H$_2$D$^+$).\nIt has been proposed that observations of the abundance ratio\nof species such as [C$_2$S]\/[NH$_3$] may date the chemical maturity of\na core (Suzuki et al. 1992).  Figure 1 shows an example of two cores\nwith very different chemical states in early-time and late-time\nmolecules despite having comparable central densities.  \n\n\n\nThe environments of starless cores are cold ($T < 15$ K) \nand dense ($n_c > 10^4$ cm$^{-3}$), thus many gas phase species\nadsorb onto dust grains.  The best example is the second most abundant\nmolecular species, CO, which freezes out of the gas at a rate of\n$(dn_{\\rm{CO}}\/dt) \\propto n_g T^{1\/2} S n_{\\rm{CO}}$,\nwhere $n_g$ is the dust grain density and\n$S$ is the sticking coefficient (Rawlings et al. 1992).  \nSince the timescale for freezeout depends\non the density and temperature of the core, the amount of CO\ndepletion encodes the history of the physical structure of the\ncore. For instance, a core that evolves slowly (quasi-statically)\nwill have more CO depletion compared to a core that evolves quickly \nto the same $n_c$.  Complicating factors to the simple adsorption\nmodel include competing desorption processes due to direct cosmic ray heating,\ncosmic ray photodesorption, and H$_2$ formation on grains, all of which \nmay be important in starless cores (Roberts et al. 2007).  CO is a destruction agent of many\ngas phase ions (e.g. N$_2$H$^+$ and H$_2$D$^+$), therefore the abundance history\nof these species are directly related to the amount of CO freezout\n(see Figure 2).  The resulting chemical networks in heavily depleted environments\nare very different (see Vastel et al. 2006).  Mapping of starless cores have revealed\na plethora of depleted species toward the dust column density peaks \n(Tafalla et al. 2006).  \n\nDeuterium fractionation is an important chemical diagnostic in low-mass\ncores.  At low temperatures, many chemical reactions involving HD favor\nthe formation of deuterated molecules due to the lower zero-point vibrational\nenergy of deuterated species compared to the hydrogenated species.  The \nclassic deuteration reaction operating in the\nenvironments of starless cores is H$_3$$^+$ + HD $\\rightarrow$\nH$_2$D$^+$ + H$_2$ + $230$ K, where the backreaction is inefficent\nat low temperatures (Vastel et al. 2006).  As the density increases, the temperature\ndecreases, and species such as CO freeze-out, the deuteration of\nhydrogenated species may increases up to four orders of magnitude\nover the cosmic [D]\/[H] $\\sim 10^{-5}$ (Roberts et al. 2004).  Figure 2b illustrates \nthe observed degrees of deuteration of N$_2$H$^+$\nincreases with the amount of CO depletion in the core (Crapsi et al. 2005).  \nSimilarly, since a deuterated species, such as NH$_2$D, \nmay be viewed as an extreme late-time molecule, a comparison between deuterated  \nand early-time molecules from the ARO-GBT survey indicate increasing\nlate-time vs. early-time molecules with increasing central density is\nthe core (Figure 2a).  \n\nThe chemical structure of the core is also important for determining\nwhich species are good probes of the kinematical structure of\nthe core.  Several species (CS, H$_2$CO, HCO$^+$, HCN)\nhave been identified as infall tracers, molecules that are moderately \noptically thick and display asymmetric, self-aborbed line profiles.  \nRecent surveys have attempted to search for evidence of large \nscale infall (e.g. Sohn et al. 2007). \nFurthermore, the linewidths of \nheavy species that lack hyperfine structure, such as C$_3$S (68 amu), \nare dominated by non-thermal motions and can trace turbulent motions\nor large-scale kinematical motions along different lines-of-sight in the core.\n\nA more thorough review of the chemical processes in starless cores\nmay be found in Di Francesco et al. (2005 and references therein).  \nThe comparison of early-time vs. late-time\nspecies, the amount of depletion, the amount of deuteration, and \nthe identification of large scale kinematical motions should be used together to \nelucidate the evolutionary state of a starless core.  Other indicators,\nsuch as the ortho-to-para evolution of symmetric molecules with non-zero spins\nand variations in\nthe structure of ionization fraction should also be explored.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=9 cm]{Shirley_fig2.ps}\n\\caption{Left: The ratio of the ``late-time'' species NH$_2$D to the ``early-time'' \nspecies C$_2$S vs. $n_c$ toward cores where both lines were detected in the \nARO-GBT survey. Right: The deuterium fraction of N$_2$H$^+$ vs. the CO depletion factor\nreported by Crapsi et al. (2005).} \\label{fig}\n\\end{center}\n\\end{figure}\n\n\n\\section{Developing an Evolutionary Sequence}\n\\label{section2} \n\nUltimately, to determine the evolutionary state of a starless core, \nwe should model the molecular line radiaitve transfer\nof each transition along multiple lines-of-sight\nand compare to a grid of chemodynamical models (e.g. Lee et al. 2004).\nThis processes is computationally intensive and the current generation\nof chemodynamical models have not fully explored the parameter space \nof possible conditions in nearby starless cores. \nAn alternative first step is to develop a Boolean evolultionary comparison.  \nA flag of $1$ (more evolved) or $0$ (less evolved) is given to a particular \nobserved property of the core if the chemical criterion is met, and the sum\nof flags represents the observed chemical maturity of the core.\nThis strategy was implemented by Crapsi et al. (2005) for a sample of\n$12$ starless cores and is being extended to the sample of $25$ cores\nwith a larger set of chemical criteria in the ARO-GBT survey (Shirley et al.\n2008, in prep.).   \n\nWhile there is still much work to develop a\ndetailed understanding of the chemical processes within low-mass starless cores, \nit is now possible and necessary to synthesize the chemical evolutionary indicators.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nVisual saliency patterns are the result of a number of factors, including the task being performed, user preferences and domain knowledge. However existing approaches to predict saliency patterns  \\cite{walther2002attentional,hadizadeh2014saliency,wang2015saliency,achanta2009saliency,sharma2012discriminative,yubing2011spatiotemporal} ignore these factors, and instead learn a model specific to a single task while disregarding factors such as user preferences. \\par\n\nBased on empirical results, the human visual system is driven by both bottom-up and top-down factors \\cite{connor2004visual}. The first category (bottom-up) is entirely driven by the visual scene where humans deploy their attention towards salient, informative regions such as bright colours, unique textures or sudden movements. The bottom-up factors are typically exhibited during the free viewing mechanism. In contrast, the top-down attention component, where the observer is performing a task specific search, is modulated by the task at hand \\cite{modellin-search} and the subject's prior knowledge \\cite{deep-fix}. For example, when the observer is searching for people in the scene, they can selectively attend to the scene regions which are most likely to contain the targets \\cite{ einha2008task, zelinsky2008theory}. Furthermore, a subject's prior knowledge, such as the scene layout, scene categories and statistical regularities will influence the search mechanisms and the fixations \\cite{castelhano2007initial, chaumon2008unconscious,neider2006scene}, rendering task specific visual saliency prediction a highly subjective, situational and challenging task \\cite{modellin-search, action-in-the-eye,deep-fix}, which motivates the need for a working memory \\cite{deep-fix, fernando2017tree,Fernando_2017_ICCV}. Even within groups of subjects completing the same task, due to the differences in a subject's behavioural goals, expectations and domain knowledge, unique saliency patterns are generated. Ideally, this user related context information should be captured via a working memory. \\par\n\nFig. \\ref{fig:fig_1} shows the variability of the saliency maps when observers are performing action recognition and context recognition on the same image. In Fig. \\ref{fig:fig_1} (a) the observer is asked to recognise the action performed by the human in the scene; while in Fig. \\ref{fig:fig_1} (b) the saliency map is generated when the observer is searching for cars\/ trees in the scene. It is evident that there exists variability in the resultant saliency patterns, yet accurate modelling of human fixations in the application areas specified above requires task specific models. For example, semantic video search, content aware image resizing, video surveillance, and video\/ scene classification may require a search for pedestrians, for different objects, or recognising human actions depending on the task and video context. \\par\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{.45\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/saliency_predictions_action_rec.pdf} %\n        \\caption{Human action recognition task}\n    \\end{subfigure}\n     \\begin{subfigure}{.45\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/saliency_predictions_context_rec.pdf} %\n        \\caption{Searching for cars\/ trees}\n    \\end{subfigure}\n     \\caption{Variability of the saliency maps when observers are performing different tasks}\n\t\t\t\\vspace{-3mm}\n        \\label{fig:fig_1}\n\\end{figure}\n\nIn recent years, motivated by the success of deep learning techniques \\cite{gammulle2017two,fernando2017soft+}, there have been several attempts to model visual saliency of the human free viewing mechanism with the aid of deep convolutional networks \\cite{deep-fix,deep-ml, vig2014large, kummerer2014deep}. Yet, the usual approach when modelling visual saliency for  task specific viewing is to hand-craft the visual features. For instance, in \\cite{modellin-search} the authors utilise the features from person detectors \\cite{dalal2006human} when estimating the search for humans in the scene; while in \\cite{action-in-the-eye} the authors use the features from HoG descriptors \\cite{dalal2005histograms}  when searching for the objects in the scene. Therefore, these approaches are application dependent and fail to capture the top-down attentional mechanism of humans which is driven by factors such as a subject's prior knowledge and expectation. \\par \n\nIn this work we propose a deep learning architecture for task specific visual saliency estimation. We draw our inspiration from the recent success of Generative Adversarial Networks (GAN) \\cite{goodfellow2014generative, denton2015deep, radford2015unsupervised, salimans2016improved, zhao2016energy} for pixel to pixel translation tasks. We exploit the capability of the conditional GAN framework \\cite{pix2pix}  for automatic learning of task specific features in contrast to hand-crafted features that are tailored for specific applications \\cite{modellin-search, action-in-the-eye}. This results in a unified, simpler architecture enabling direct application to a variety of tasks. The conditional nature of the proposed architecture enables us to learn one network for all the tasks of interest, instead of learning separate networks for each of the tasks. Apart from the advantage of a simpler learning process, this enables the capability of learning semantic correspondences among different tasks and propagating these contextual relationships from one task to another. \\par\nFig. \\ref{fig:networks} (a) shows the conditional GAN architecture where the discriminator $D$ learns to classify between real and synthesised pairs of saliency maps $y$, given the observed image $x$ and task specific class label $c$. The generator $G$ tries to fool the discriminator. It also observes the observed image $x$ and task specific class label $c$. We compare this model to the proposed model given in Fig. \\ref{fig:networks} (b). The differences arise in the utilisation of memory $M$, where we capture subject specific behavioural patterns. This incorporates a subject's prior knowledge, behavioural goals and expectations.\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{.49\\linewidth}\n       \\centering\n        \\includegraphics[width=.99\\linewidth]{fig\/CGANS.pdf} %\n        \\caption{Conditional GAN}\n    \\end{subfigure}\n     \\begin{subfigure}{.49\\linewidth}\n       \\centering\n        \\includegraphics[width=.99\\linewidth]{fig\/proposed.pdf} %\n        \\caption{MC-GAN (proposed model)}\n    \\end{subfigure}\n     \\caption{A comparison of conditional GAN architecture with the proposed model}\n\t\t\t\\vspace{-3mm}\n        \\label{fig:networks}\n\\end{figure}\n\n\\vspace{-2mm}\n\\section{Related Work}\nLiterature related to this work can be broadly categorised into ``Visual Saliency Prediction'' and ``Generative Adversarial Networks'', and these two areas are addressed in Sections 2.1 and 2.2 respectively.\n\n\\subsection{Visual Saliency Prediction}\nSince the first attempts to model human saliency through feature integration \\cite{treisman1980feature}, the area of saliency prediction has been widely explored. Building upon this bottom-up approach, Koch and Ullman \\cite{koch1987shifts} and Itti et al. \\cite{itti1998model} proposed approaches based on extracting image features such as colour, intensity and orientation. These methods generate centre-biased acceptable saliency predictions for free viewing but are highly inaccurate in complex real world scenes \\cite{deep-fix}. Recent studies such as \\cite{valenti2009image,liu2013saliency, lang2012depth, erdem2013visual,porikli2006covariance} have looked into the development of more complex features for saliency estimation.  \\par\nIn contrast, motivated by information theory, authors in \\cite{bruce2006saliency,modellin-search, action-in-the-eye} have taken the top-down approach where task dependent features comes into play. They incorporate local information from regions of interest for the task at hand, such as features from person detectors and HoG descriptors. These models \\cite{bruce2006saliency,modellin-search, action-in-the-eye} are completely task specific, rendering adaptation from one task to another nearly impossible. Furthermore, they neglect the fact that different subjects may exhibit different behavioural patterns when achieving the same goal which generates unique strategies or sub goals that we term user preferences. \\par\n\nIn order to exploit the representative power of deep architectures, more recent studies have been driven towards the utilisation of convolution neural networks. In contrast to the above approaches, which use hand crafted features, deep learning based approaches offer automatic feature learning. In \\cite{kummerer2014deep} the authors propose the usage of feature representations from a pre-trained model that has been trained for object classification. This work was followed by \\cite{deep-fix,deep-ml} where authors train end-to-end saliency prediction models from scratch and their experimental evaluations suggest that deep models trained for saliency prediction itself can outperform off-the-shelf CNN models. \\par\nLiu et al. \\cite{liu2015predicting} proposed a mulitresolution-CNN for predicting saliency, which has been trained on multiple scales of the observed image. The motivation behind this approach is to capture low and high level features. Yet this design has an inherit deficiency due to the use of isolated image patches which fail to capture the global context, composed of the context of the observed image, the task at hand (i.e free viewing, recognising actions, searching for objects) and user preferences. Even though the context of the observed image is well represented in deep single scale architectures such as \\cite{deep-fix,deep-ml} they ignore the rest of the global context, the task description and user behavioural patterns, which are often crucial for saliency estimation. \\par\nThe proposed work bridges the research gap between deep architectures that capture bottom-up saliency features; and top-down methods \\cite{bruce2006saliency,modellin-search, action-in-the-eye} that are purely driven by the hand crafted features. We investigate the plausibility of the complete automatic learning of global context, which has been ill represented in literature thus far,  through a memory augmented conditional generative adversarial model. \n\n\\subsection{Generative Adversarial Networks}\nGenerative adversarial networks (GAN), which belong to the family of generative models, have achieved promising results for pixel-to-pixel synthesis \\cite{arici2016associative}. Several works have looked into numerous architectural augmentations when synthesising natural images. For instance, in \\cite{gregor2015draw} the authors utilise a recurrent network approach where as in \\cite{dosovitskiy2015learning} a de-convolution network approach is used to generate higher quality images. Most recently authors in \\cite{pan2017salgan} have utilised the GAN architecture for visual saliency prediction and proposed a novel loss function which is proven to be effective for both initialising the generator, and stabilising adversarial training. Yet their work fails to delineate the ways of achieving task specific saliency estimation and of incorporating task specific dependencies and the subject behavioural patterns for saliency estimation. \\par\nThe proposed work draws inspiration from conditional GANs \\cite{li2016precomputed,mathieu2015deep,mirza2014conditional,pathak2016context,reed2016generative,yoo2016pixel,zhou2016learning,wang2016generative}. This architecture is extensively applied for image based prediction problems such as image prediction from normal maps \\cite{wang2016generative}, future frame prediction in videos \\cite{mathieu2015deep}, image style transfer \\cite{li2016precomputed}, image manipulation guided by user preferences \\cite{zhou2016learning}, etc. In \\cite{pix2pix} the authors proposed a novel U-Net \\cite{ronneberger2015u} based architecture for conditional GANs. Their evaluations suggested that this network is capable of capturing local semantics with applications to a wide range of problems. We investigate the possibility of merging the discriminative learning power of conditional GANs together with a local memory to fully capture the global context, contributing a novel application area and structural argumentation for conditional GANs. \n\n\\vspace{-2mm}\n\\section{Visual Saliency Model}\n\n\\subsection{Objectives}\n\nGenerative adversarial networks learn a mapping from a random noise vector $z$ to an output image $y, G: z \\rightarrow y$ \\cite{goodfellow2014generative}; where as conditional GANs learn a mapping from an observed image $x$ and random noise vector $z$, to output $y$, given auxiliary information $c$, where $c$ can be class labels or data from other modalities. $G: \\{x,z | c \\} \\rightarrow y$. When we incorporate the notion of time into the system, then the observed image at time instance $t$ will be $x_t$, the respective noise vector $z_t$ and the relevant class label will be $c_t$. Then the objective function of a conditional GAN can be written as, \n\\begin{equation}\n\\begin{split}\nL_{cGAN}(G,D)  =E_{x_t, y_t \\sim  p_{data}(x_t, y_t)}[log(D (x_t, y_t | c_t))] + \\\\\nE_{x_t \\sim p_{data(x_t), z_t \\sim p_z(z_t)}}[log(1-D(x_t, G(x_t,z_t | c_t)))].\n\\end{split}\n\\end{equation}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=.90\\linewidth]{fig\/memory_fig.pdf} %\n        \\caption{Memory architecture: The model receives an input $o_t$ at time instance $t$. A function $f_r^{LSTM}$ is used to embed this input and retrieve the content of the memory $M_{t-1}$. When reading, we use a $softmax$ function to weight the association between each memory slot and the input $o_t$, deriving a weighted retrieval $h_{t}$. The final output $m_t$ is derived using both $\\grave{o}$ and $h_{t}$. Finally we update the memory using memory write function $f_w^{LSTM}$. This generates the memory representation $M_{t}$ at time instance $t+1$, shown to the right.}\n\t\t\t\\vspace{-3mm}\n         \\label{fig:memory_architecture}\n \\end{figure*}\n \nLet $ M \\in \\mathbb{R}^{k*l} $, shown in Fig. \\ref{fig:memory_architecture}, be the working memory with $k$ memory slots and $l$ is the embedding dimension of the generator output,\n\\begin{equation}\no_t= G(x_t,z_t| c_t).\n\\end{equation}\nIf the representation of memory at time instance $t-1$ is given by $M_{t-1}$ and $f_r^{LSTM}$ is a read function, then we can generate a key vector $a_t$, representing the similarity between the current memory content and the current generator embedding via attending over the memory slots such that,\n\\begin{equation}\n\\grave{o}_t=f_r^{LSTM}(o_t),\n\\end{equation}\n\\begin{equation}\na_t= softmax(\\grave{o}_t^T,M_{t-1}),\n\\end{equation}\nand\n\\begin{equation}\nh_t= a_t^TM_{t-1}.\n\\end{equation}\nThen we retrieve the current memory state by, \n\\begin{equation}\nm_t= f^{MLP}(\\grave{o}_t,h_t), \n\\end{equation}\nwhere  $f^{MLP}$ is a neural network composed of multi-layer perceptrons (MPL) trained jointly with other networks. \nThen we generate the vector for the memory update $\\grave{m}_t$ via passing it through a write function $ f_w^{LSTM}$\n\\begin{equation}\n\\grave{m}_t= f_w^{LSTM}(m_t),\n\\end{equation}\nand finally we completely update the memory using,\n\\begin{equation}\nM_{t}=M_{t-1}(1- (a_t \\otimes e_k )^T) + (\\grave{m}_t \\otimes e_l)(a_t \\otimes e_k )^T.\n\\end{equation}\nwhere 1 is a matrix of ones, $e_l \\in \\mathbb{R}^l $ and $ e_k \\in \\mathbb{R}^k$ be vectors of ones and $ \\otimes$ denotes the outer product which duplicates its left vector $l$ or $k$ times to form a matrix.\nNow the  objective of the proposed memory augmented cGAN can be written as, \n\\begin{equation}\n\\begin{split}\nL^*_{cGAN}(G,D)  =E_{x_t, y_t \\sim  p_{data}(x_t, y_t)}[log(D (x_t, y_t | c_t))] + \\\\\nE_{x_t \\sim p_{data(x_t), z_t \\sim p_z(z_t)}}[ log(1-D(x_t, o_t \\otimes tanh(m_t)))].\n\\end{split}\n\\end{equation}\nWe would like to emphasise that we are learning a single network for all the tasks at hand, rendering a simpler but informative framework, which can be directly applied to a variety of tasks without any fine tuning.\n\\subsection{Network Architecture}\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=.4\\linewidth]{fig\/u_new_architecture.pdf} %\n        \\caption{U-Net architecture}\n         \\label{fig:U_Net_architecture}\n\t\t\t\\vspace{-3mm}\n    \\end{figure}\n\nFor the generator we adapt the U-Net architecture of \\cite{pix2pix} (see Fig. \\ref{fig:U_Net_architecture}). Let $Ck$ denote a Convolution-BatchNorm-ReLU layer group with $k$ filters. $CDk$ denotes a Convolution-BatchNorm-Dropout-ReLU layer with a dropout rate of 50\\%. Then the generator architecture can be written as,\nEncoder: C64-C128-C256-C512-C512-C512-C512-C512 followed by a U-Net decoder: CD512-CD1024-CD1024-C1024-C1024-C512-C256-C128 where there are skip connections between each $i^{th}$ layer in the encoder and the ${n-i}^{th}$ layer of the decoder, and there are $n$ total layers in the generator (see \\cite{pix2pix} for details). \nThe discriminator architecture is: C64-C128-C256-C512-C512-C512. \\par\nAll convolutions are 4 x 4 spatial filters applied with stride 2. Convolutions in the encoder and in the discriminator down sample by a factor of 2, whereas in the decoder they up sample by a factor of 2. A description of our memory architecture is as follows. For functions $f_r^{LSTM}$ and $f_w^{LSTM}$ we utilise two, one-layer LSTM networks \\cite{hochreiter1997long} with 100 hidden units and for $f^{MLP}$ we use a neural network with a single hidden layer and 1024 units with ReLU activations. This memory module is fully differentiable, and we learn it jointly with other networks. We trained the proposed model with the Adam \\cite{kingma2014adam} optimiser, with a batch size of 32 and an initial learning rate to 1e-5, for 10 epochs.\n\n\\vspace{-2mm}\n\\section{Experimental Evaluations}\n\n\\subsection{Datasets}\nWe evaluate our proposed approach on 2 publicly available datasets, VOCA-2012 \\cite{action-in-the-eye} and MIT person detection (MIT-PD) \\cite{modellin-search}.  \\par\nThe VOCA-2012 dataset consists of 1,085,381 human eye fixation from 12 volunteers (5 male and 7 female) divided into 2 groups based on the given task. It contains 8 subjects performing action recognition in the given image where as the rest of the subjects are performing context dependent visual search. The subjects in this group are searching for furniture, paintings\/ wallpapers, bodies of water, buildings, cars\/ trucks, mountain\/ hills, road\/ trees in the given scene. The MIT-PD dataset consist of 12,768 fixations from 14 subjects (between 18-40 years), where the subjects search for people in 912 urban scenes. MIT-PD contains only a single task, and we use this dataset to demonstrate the effectiveness of the memory network.\n\n\\subsection{Evaluation metrics}\nLet $N$ denote the number of examples in the testing set, $y$ denotes the ground truth saliency map and $\\hat{y}$ is the predicted saliency map. Following this notation we define the following metrics:\n\\begin{itemize}\n\\item \\textbf{Area Under Curve (AUC): }\nAUC is a widely used metrics for evaluating saliency models. We use the formulation of this metric defined in \\cite{borji2012exploiting}. \n\\item \\textbf{Normalised Scan path Saliency (NSS): }\nNSS \\cite{peters2005components} is calculated by taking the mean scores assigned by the unit normalised saliency map $\\hat{y}^{norm}$ at human eye fixations. \n\\begin{equation}\nNSS=\\frac{1}{N}\\sum_{i=1}^{N}\\hat{y}^{norm}_{i}\n\\end{equation}\n\\item \\textbf{Linear Correlation Coefficient (CC): }\nIn order to measure the linear relationship between the ground truth and predicted map we utilise the linear correlation coefficient, \n\\begin{equation}\nCC=\\frac{cov(y,\\hat{y})}{\\sigma_{y}*\\sigma_{\\hat{y}}},\n\\end{equation}\nwhere $cov(y,\\hat{y})$ is referred to as the covariance between distributions $y$ and $\\hat{y}$, $\\sigma_{y}$ is the standard deviation of distributions $y$ and $\\sigma_{\\hat{y}}$ is the standard deviation of distributions $\\hat{y}$. As the name implies $CC=1$ denotes a perfect linear relationship between distributions $y$ and $\\hat{y}$ where as a value of $0$ implies that there is no linear relationship. \n\\item \\textbf{KL divergence (KL): }\nTo measure the non-symmetric difference between two distributions we utilise the KL divergence measure given by,\n\\begin{equation}\nKL=\\sum_{i=1}^{N}\\hat{y}_{i}log(\\frac{\\hat{y}_i}{y_i}).\n\\end{equation}\nAs ground truth and predicted saliency maps can be seen as 2D distributions, we can use KL divergence to measure the difference between them.\n\\item \\textbf{Similarity metric (SM): }\nComputes the sum of the minimum values at each pixel location between $\\hat{y}^{norm}$ and $y^{norm}$ distributions. \n\\begin{equation}\nSM=\\sum_{i=1}^{P}min(\\hat{y}^{norm}_{i},y^{norm}_{i}), \n\\end{equation}\nwhere\n$\\sum_{i=1}^{P}\\hat{y}^{norm}_{i}=1 $\nand\n$\\sum_{i=1}^{P}y^{norm}_{i}=1 $\nare the normalised probability distributions and $P$ denotes all the pixel location in the 2D maps.\n\\end{itemize}\n\n\\subsection{Results}\nQuantitative evaluations on the VOCA-2012 dataset is presented in Tab. \\ref{tab:tab_1}. In the proposed model, in order to retain the user dependent factors such as user preference in memory, we feed the examples in order such that examples from each specific user go through in sequence. We compare our model with 8 state-of-the-art methods.  The row `human' stands for the human saliency predictor, which computes the saliency map derived from the fixations made by half of the human subjects performing the same task. This predictor is evaluated with respect to the rest of the subjects, as opposed to the entire group \\cite{action-in-the-eye}. \\par\n\nThe evaluations suggest that the bottom-up model of Itti et. al \\cite{itti2000saliency} generates poor results as it does not incorporate task specific information.  Even with high level object detectors, the models of Judd et al. \\cite{judd2009learning} and HOG detector \\cite{action-in-the-eye} fail to render accurate predictions. \\par\nDeep learning models PDP \\cite{jetley2016end} and ML-net \\cite{deep-ml} are able to out perform the techniques stated above but they lack the ability to learn task dependent information. We note the accuracy gain of cGAN model over PDP, ML-net and SalGAN, where the model incorporates a conditional variable to discriminate between the `action recognition' and `context recognition' tasks instead of learning two separate networks or fine-tuning on them individually. Our proposed approach builds upon this by incorporating an augmented memory architecture with conditional learning. We learn different user patterns and retain the dependencies among different tasks, and outperform all baselines considered (MC-GAN (proposed), Tab. \\ref{tab:tab_1}). \\par \nAs further study, in Tab. \\ref{tab:tab_1}, row M-GAN (separate), we show the evaluations for training two separate memory augmented GAN networks for the tasks in the VOCA-2012 test set without using the conditional learning process. The results emphasise the importance of learning a single network for all the tasks, leveraging semantic relationships between different tasks. The accuracy of the networks learned for separate tasks are lower than the combined MC-GAN and cGAN approaches (rows  MC-GAN (proposed) and cGAN, Tab. \\ref{tab:tab_1}), highlighting the importance of learning the different tasks together and allowing the model to discriminate between the tasks and learn the complimentary information, rather than keeping the model completely blind regarding the existence of another task category.\\par\n\\begin{table}[t]\n  \\centering\n  \\begin{adjustbox}{width=.98\\linewidth,center}\n  \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n    & \\multicolumn{4}{|c|}{\\textbf{Task}} \\\\\n        \\cline{2-5}\n     & \\multicolumn{2}{|c|}{\\textbf{Action Rec}}  & \\multicolumn{2}{|c|}{\\textbf{Context Rec}}\\\\\n           \\cline{2-5}\n   \\textbf{Saliency Models} & \\textbf{AUC} & \\textbf{KL} & \\textbf{AUC} & \\textbf{KL} \\\\\n    \\hline\n     \\hline\n   HOG detector \\cite{action-in-the-eye} & 0.736 & 8.54 & 0.646 & 8.10 \\\\\n    \\hline\n   Judd et al. \\cite{judd2009learning} & 0.715 & 11.00  & 0.636 & 9.66  \\\\\n    \\hline\n   Itti et. al \\cite{itti2000saliency} & 0.533 & 16.53  & 0.512 & 15.04 \\\\\n    \\hline\n  central bias \\cite{action-in-the-eye} & 0.780 & 9.59 & 0.685 & 8.82\\\\\n   \\hline\n PDP \\cite{jetley2016end} & 0.875 & 8.23 & 0.690 & 7.98 \\\\\n \\hline\n  ML-net \\cite{deep-ml}& 0.847 & 8.51 & 0.684 & 8.02 \\\\\n   \\hline\n     SalGAN \\cite{pan2017salgan}& 0.848 & 8.47 & 0.679 & 8.00 \\\\\n  \\hline\n   cGAN \\cite{pix2pix} &0.852 &8.24  &0.701 &7.95  \\\\\n    \\hline\n    M-GAN (separate) & 0.848 & 8.54 & 0.704 & 8.00\\\\\n     \\hline\n   \\textbf{MC-GAN (proposed)} &\\textbf{0.901} & \\textbf{8.07 } &\\textbf{0.734} & \\textbf{7.65 } \\\\\n    \\hline\n     \\hline\n  Human \\cite{action-in-the-eye} & 0.922 & 6.14 & 0.813 & 5.90\\\\\n  \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Experimental evaluation on VOCA-2012 test set. We augment the current state-of-the-art GAN method (SalGAN \\cite{pan2017salgan}) by adding a conditional variable (cGAN \\cite{pix2pix}) to mimic the joint learning process instead of learning two separate networks. To capture user and task specific behavioural patterns we add a memory module to cGAN, MC-GAN (proposed), and outperform all the baselines. We also compare training $2$ separate memory augmented GAN networks, M-GAN (separate) without the conditional learning process.}\n\t\t\t\\vspace{-3mm}\n  \\label{tab:tab_1}\n\\end{table}\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\begin{adjustbox}{width=.98\\linewidth,center}\n  \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n    & \\multicolumn{6}{|c|}{\\textbf{Task}} \\\\\n        \\cline{2-7}\n     & \\multicolumn{3}{|c|}{\\textbf{Action Rec}}  & \\multicolumn{3}{|c|}{\\textbf{Context Rec}}\\\\\n           \\cline{2-7}\n   \\textbf{Saliency Models} & \\textbf{NSS} & \\textbf{CC} & \\textbf{SM} & \\textbf{NSS} & \\textbf{CC} & \\textbf{SM}  \\\\\n    \\hline\n     \\hline\n     ML-net \\cite{deep-ml}& 2.05 & 0.71 & 0.51 & 2.03 & 0.64 & 0.42 \\\\\n     \\hline\n     SalGAN \\cite{pan2017salgan}& 2.10 & 0.73 & 0.51 & 2.10 & 0.68 & 0.44 \\\\\n     \\hline\n   cGAN \\cite{pix2pix} & 2.23 & 0.76 & 0.55 & 2.14 & 0.71 & 0.57   \\\\\n    \\hline\n   \\textbf{MC-GAN (proposed)} &\\textbf{2.23} & \\textbf{0.79 } &\\textbf{0.60} & \\textbf{2.20 } &\\textbf{0.77} & \\textbf{0.69 }  \\\\\n    \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Comparison between ML-Net, SalGAN, cGAN and  MC-GAN (proposed) on VOCA-2012}\n\t\t\t\\vspace{-3mm}\n  \\label{tab:tab_3}\n\\end{table}\n\nTo provide qualitative insight, some predicted maps along with ground truth and baseline ML-net \\cite{deep-ml} predictions are given in  Fig. \\ref{fig:fig_action_context_rec}. In the first column we show the input image, and columns ``Action rec GT'' and ``Context rec GT'' depict the ground truth saliency maps for the respective tasks. In columns  ``Our action rec'' and ``Our context rec'' we show the respective predictions from our model, and finally the  column `ml-Net' contains the prediction from the ML-net \\cite{deep-ml} baseline.   Observing columns ``Action rec GT '' and ``Context rec GT'' one can clearly see how the tasks differ based on the different saliency patterns. Yet, the proposed model is able to capture these different semantics within a single network which is trained together for all the tasks. As shown in Fig. \\ref{fig:fig_action_context_rec}, it has efficiently identified the image saliency from low level features as well as task dependent saliency factors from high level cues such as trees, furniture and roads. Furthermore, the single learning process and the incorporation of a memory architecture renders the plausibility of retaining the semantical relationships among different tasks and how users adapt to those. \n\n\\begin{figure*}[!htb]\n   \n   \n   \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006090.jpg} %\n        \n    \\end{subfigure}\n     \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006090_gt_action.pdf} %\n       \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006090_pred_action.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n       \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006090_gt_context.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n         \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006090_pred_context.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006090_mlnet.pdf} %\n        \n    \\end{subfigure}\n   \n    \n    \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006094.jpg} %\n        \n    \\end{subfigure}\n     \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006094_gt_action.pdf} %\n       \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006094_pred_action.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n       \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006094_gt_context.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n         \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006094_pred_context.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006094_mlnet.pdf} %\n        \n    \\end{subfigure}\n\n \n   \n    \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006125.jpg} %\n        \n    \\end{subfigure}\n     \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006125_gt_action.pdf} %\n       \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006125_pred_action.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n       \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006125_gt_context.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n         \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006125_pred_context.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006125_mlnet.pdf} %\n        \n    \\end{subfigure}\n\n   \n     \n    \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006150.jpg} %\n        \n    \\end{subfigure}\n     \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006150_gt_action.pdf} %\n       \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006150_pred_action.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n       \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006150_gt_context.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n         \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006150_pred_context.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006150_mlnet.pdf} %\n        \n    \\end{subfigure}\n\n \n   \n    \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006153.jpg} %\n        \n    \\end{subfigure}\n     \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006153_gt_action.pdf} %\n       \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006153_pred_action.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n       \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006153_gt_context.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n         \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006153_pred_context.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006153_mlnet.pdf} %\n        \n    \\end{subfigure}\n\n   \n        \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006157.jpg} %\n        \\caption{Image}\n    \\end{subfigure}\n     \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006157_gt_action.pdf} %\n        \\caption{Action rec GT}\n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006157_pred_action.pdf} %\n        \\caption{Our action rec}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006157_gt_context.pdf} %\n        \\caption{Context rec GT}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006157_pred_context.pdf} %\n        \\caption{Our context rec}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.160\\textwidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/supplymentry_material\/voc_saliency_maps\/006_2010_006157_mlnet.pdf} %\n        \\caption{ML-net}\n    \\end{subfigure}\n\n    \n     \\caption{Qualitative results for VOCA-2012 dataset and comparisons to the state-of-the-art.}\n\t\t\t\\vspace{-3mm}\n        \\label{fig:fig_action_context_rec}\n\\end{figure*}\n\nTab. \\ref{tab:tab_2} shows the performance of the proposed model with 5 baselines for the MIT-PD test set. The first baseline ``Scene Context'' \\cite{modellin-search} utilises colour and orientation features where as ``Combined'' \\cite{modellin-search} incorporates both scene context features and higher level features from a person detector \\cite{dalal2006human}. Even with such explicit modelling of the task, this baseline fails to generate accurate predictions suggesting the subjective nature of the task specific viewing. With the aid of the associative memory of the proposed model we successfully capture those underlying factors.\n\n\\begin{table}[!h]\n  \\centering\n  \\begin{adjustbox}{width=.98\\linewidth,center}\n  \\begin{tabular}{|c|c|c|}\n \\hline\n    & \\multicolumn{2}{|c|}{\\textbf{Scene Type}} \\\\\n        \\cline{2-3}\n     & \\textbf{Target Present}  & \\textbf{Target absent}\\\\\n           \\cline{2-3}\n   \\textbf{Saliency Models} & \\textbf{AUC}  & \\textbf{AUC}  \\\\\n    \\hline\n     \\hline\n   Scene Context \\cite{modellin-search} & 0.844  & 0.845 \\\\\n    \\hline\n   Combined \\cite{modellin-search} & 0.896 & 0.877 \\\\\n     \\hline\n  ML-net \\cite{deep-ml}& 0.901 & 0.881\\\\\n    \\hline\n    SalGAN \\cite{pan2017salgan}&0.910 & 0.887\\\\\n      \\hline\n    cGAN \\cite{pix2pix} & 0.923 & 0.899  \\\\\n    \\hline\n   \\textbf{MC-GAN (proposed)} &\\textbf{0.942}  & \\textbf{0.903} \\\\\n    \\hline\n    \\hline\n  Human \\cite{modellin-search} & 0.955 & 0.930 \\\\\n \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Experimental evaluation on MIT-PD test set}\n\t\t\t\\vspace{-3mm}\n  \\label{tab:tab_2}\n\\end{table}\n\nIn Tab. \\ref{tab:tab_3} and Tab. \\ref{tab:tab_4} we present the evaluations of NSS, CC and SM metrics. In order to evaluate ML-net, SalGAN and cGAN models we utilise the implementation of the algorithm released by the authors. When comparing the results between the ML-net \\cite{deep-ml}, cGAN \\cite{pix2pix} and SalGAN \\cite{pan2017salgan} models and proposed MC-GAN model a considerable gain in performance is observed in all the metrics considered, which emphasises a greater agreement between predicted and ground truth saliency maps. We were unable to compare other baselines using these metrics due to the unavailability of public implementations.\n\n\\begin{table}[t]\n  \\centering\n  \\begin{adjustbox}{width=.98\\linewidth,center}\n  \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n    & \\multicolumn{6}{|c|}{\\textbf{Task}} \\\\\n        \\cline{2-7}\n     & \\multicolumn{3}{|c|}{\\textbf{Target Present}}  & \\multicolumn{3}{|c|}{\\textbf{Target absent}}\\\\\n           \\cline{2-7}\n   \\textbf{Saliency Models} & \\textbf{NSS} & \\textbf{CC} & \\textbf{SM} & \\textbf{NSS} & \\textbf{CC} & \\textbf{SM}  \\\\\n    \\hline\n     \\hline\n     ML-Net \\cite{deep-ml} & 1.41 & 0.55  &0.41 &1.22 & 0.43 & 0.38 \\\\\n      \\hline\n      SalGAN \\cite{pan2017salgan}& 1.41 & 0.53  &0.44 &1.20 & 0.42 & 0.35 \\\\ \n      \\hline\n   cGAN \\cite{pix2pix} & 1.67 & 0.51  &0.59 & 2.02 & 0.41 & 0.52  \\\\\n    \\hline\n   \\textbf{MC-GAN (proposed)} &\\textbf{2.17} & \\textbf{0.76 } &\\textbf{0.71} & \\textbf{2.34 } &\\textbf{0.75} & \\textbf{0.78 }  \\\\\n    \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Comparison between ML-Net, SalGAN, cGAN and  MC-GAN (proposed) on MIT-PD test set}\n\t\t\t\\vspace{-3mm}\n  \\label{tab:tab_4}\n\\end{table}\n\n\nThe qualitative results obtained from the proposed model along with the ML-net \\cite{deep-ml} network on a few examples from the MIT-PD dataset are shown in Fig. \\ref{fig:fig_context_rec}. We would like to emphasise the usage of a subject's prior knowledge in the task of searching for people in the urban scene. The subjects selectively attend the areas such as high rise buildings (see rows 2, 6) and pedestrian walkways (see rows 1, 3-6), which are more likely to contain humans, which our model has effectively captured. With the lack of capacity to model such user preferences, the baseline ML-Net model generates centre biased saliency without effectively understanding the subject's strategy. \n\\begin{figure}[!htb]\n   \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/Image\/saliency_predictions_31.jpg} %\n        \n    \\end{subfigure}\n     \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/GT\/saliency_predictions_31.pdf} %\n           \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/our\/saliency_predictions_31.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/ml-net\/saliency_predictions_31.pdf} %\n        \n    \\end{subfigure}\n   \n   \n   \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/Image\/saliency_predictions_14.jpg} %\n        \n    \\end{subfigure}\n      \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/GT\/saliency_predictions_14.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/our\/saliency_predictions_14.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/ml-net\/saliency_predictions_14.pdf} %\n        \n    \\end{subfigure}\n   \n \n   \n\\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/Image\/saliency_predictions_18.jpg} %\n        \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/GT\/saliency_predictions_18.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/our\/saliency_predictions_18.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/ml-net\/saliency_predictions_18.pdf} %\n        \n    \\end{subfigure}\n   \n\t\\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/Image\/saliency_predictions_4.jpg} %\n        \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/GT\/saliency_predictions_4.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/our\/saliency_predictions_4.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/ml-net\/saliency_predictions_4.pdf} %\n        \n    \\end{subfigure}    \n   \n     \n   \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/Image\/saliency_predictions_33.jpg} %\n        \n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/GT\/saliency_predictions_33.pdf} %\n    \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/our\/saliency_predictions_33.pdf} %\n        \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/ml-net\/saliency_predictions_33.pdf} %\n        \n    \\end{subfigure}\n \n   \n        \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/Image\/saliency_predictions_5.jpg} %\n        \\caption{Image}\n    \\end{subfigure}\n     \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/GT\/saliency_predictions_5.pdf} %\n        \\caption{GT}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/our\/saliency_predictions_5.pdf} %\n        \\caption{Our}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.24\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/visualisations_2\/ml-net\/saliency_predictions_5.pdf} %\n        \\caption{ML-net}\n    \\end{subfigure}\n\n    \n     \\caption{Qualitative results for MIT-PD dataset and comparisons to the state-of-the-art.}\n        \\label{fig:fig_context_rec}\n\\end{figure}\n\n\n\\subsection{Task Specific Generator Activations}\nIn Fig. \\ref{fig:generator_activations} we visualise the activations from the 2nd (conv-l-2) and 2nd last (conv-l-8) convolution layers of the generator. The task specific learning of the proposed conditional GAN architecture is clearly evident in the activations. For instance, when the task at hand is to recognise actions the generator activations are highly concentrated around the foreground of the image (see (b), (g)), while for context recognition the model has learned that the areas of interest are in the background of the image (see (c), (h)). These task specific salient features are combined and compressed hierarchically and in latter layers (i.e conv-l-8), the networks has learned the most specific areas to focus when generating the output saliency map. \n\n\n\\begin{figure}[htb]\n   \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth, ,height= .82\\linewidth]{fig\/generator_activations\/017_2010_006129.jpg} %\n         \\caption{Input}\n          \\end{subfigure}\n     \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/generator_activations\/layer_4\/017_2010_006129_action_rec.pdf} %\n        \\caption{l-2 AR}  \n         \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/generator_activations\/layer_4\/017_2010_006129_context_rec.pdf} %\n         \\caption{l-2 CR}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth ]{fig\/generator_activations\/layer_12\/017_2010_006129_action_rec.pdf} %\n        \\caption{l-8 AR}\n        \\end{subfigure}\n    \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/generator_activations\/layer_12\/017_2010_006129_context_rec.pdf} %\n         \\caption{l-8 CR}\n          \\end{subfigure}\n    \n   \n      \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth, ,height= .82\\linewidth]{fig\/generator_activations\/017_2010_006217.jpg} %\n        \\caption{Input}\n    \\end{subfigure}\n     \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth]{fig\/generator_activations\/layer_4\/017_2010_006217_action_rec.pdf} %\n         \\caption{l-2 AR}  \n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth, ,height= .82\\linewidth]{fig\/generator_activations\/layer_4\/017_2010_006217_context_rec.pdf} %\n         \\caption{l-2 CR}\n    \\end{subfigure}\n \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth, ,height= .82\\linewidth]{fig\/generator_activations\/layer_12\/017_2010_006217_action_rec.pdf} %\n        \\caption{l-8 AR}\n    \\end{subfigure}\n    \\centering\n    \\begin{subfigure}{.190\\linewidth}\n       \\centering\n        \\includegraphics[width=.95\\linewidth, ,height= .82\\linewidth]{fig\/generator_activations\/layer_12\/017_2010_006217_context_rec.pdf} %\n        \\caption{l-8 CR}\n    \\end{subfigure}\n     \\caption{Visualisation of the generator activations from 2nd (l-2) and 2nd last (l-8) convolution layers for action recognition (AR) and context recognition (CR) tasks. The importance varies from blue to yellow where blue represents the areas of least importance and yellow represents areas of more importance.}\n\t\t\t\\vspace{-3mm}\n\\label{fig:generator_activations}\n    \\end{figure}\n\n\\vspace{-2mm}\n\\section{Conclusion}\n\nThis work introduces a novel human saliency estimation architecture which combines task and user specific information together in a generative adversarial pipeline. We show the importance of fully capturing the context information which incorporates the task information, subject behavioural goals and image context. The resultant frame work offers several advantages compared to task specific handcrafted features, enabling direct transferability among different tasks. Qualitative and quantitative experimental evaluations on two public datasets demonstrates superior performance with respect to the current state-of-the-art. \n\n\\vspace{-2mm}\n\\footnotesize{\n\\subsubsection*{Acknowledgement}\n\\vspace{-2mm}\nThis research was supported by an Australian Research Council's Linkage grant (LP140100221). The authors also thank QUT High Performance Computing (HPC) for providing the computational resources for this research.\n}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \\cite[Lemma IIe]{wiener32}, it states that\n``{\\em If $f(x)$ is a function with an absolutely convergent Fourier series, which nowhere vanishes for real arguments, $1\/f(x)$ has an absolutely convergent Fourier series.}\"\nThe above statement is now known as the  classical Wiener's lemma.\n\nWe say that a Banach space ${\\mathcal A}$  with norm $\\|\\cdot\\|_{\\mathcal A}$\nis  a {\\em  Banach algebra} if \nit has operation of multiplications possessing the usual algebraic properties, and\n\\begin{equation}\\label{banachalgebra.def}\n\\|AB\\|_{\\mathcal A}\\le   K \\|A\\|_{\\mathcal A}\\|B\\|_{\\mathcal A}\\ \\ {\\rm for\\ all} \\ A, B\\in {\\mathcal A},\n\\end{equation}\nwhere $K$ is a positive constant. \nGiven two Banach algebras ${\\mathcal A}$ and ${\\mathcal B}$ such that ${\\mathcal A}$ is a Banach subalgebra of ${\\mathcal B}$,\nwe say that\n ${\\mathcal A}$ is {\\it inverse-closed} in ${\\mathcal B}$ if $A\\in {\\mathcal A}$ and\n$A^{-1} \\in {\\mathcal B}$ implies $A^{-1}\\in {\\mathcal A}$.  Inverse-closedness is also known as  spectral invariance, Wiener pair, local subalgebra, etc \n \\cite{douglasbook, gelfandbook, Naimarkbook, takesaki}.\nLet  ${\\mathcal C}$ be the algebra of all periodic continuous functions  under multiplication, and\n${\\mathcal W}$  be its Banach subalgebra of all periodic functions with  absolutely convergent Fourier series,\n\\begin{equation}\\label{Wieneralgebra.def}\n{\\mathcal W}=\\Big\\{f(x)=\\sum_{n\\in \\ZZ} \\hat f(n) e^{inx},\\ \\\n \\|f\\|_{\\mathcal W}:=\\sum_{n\\in \\ZZ} |\\hat f(n)|<\\infty\\Big\\}.\\end{equation}\nThen the classical Wiener's lemma can be  reformulated as that  ${\\mathcal W}$ is an inverse-closed subalgebra of ${\\mathcal C}$.\n Due to the above interpretation, we  also call the inverse-closed property for a Banach subalgebra ${\\mathcal A}$ as Wiener's lemma for that subalgebra.\n Wiener's lemma for Banach algebras of infinite matrices and integral operators with certain off-diagonal decay\n can be informally interpreted as localization preservation under inversion.\nSuch a localization preservation is of great importance in  applied harmonic analysis, numerical analysis,  optimization\nand many mathematical and engineering fields  \\cite{akramgrochenigsiam, chengsun19, christensen05, grochenigr10,  ksw13, sunsiam06}.\n  The readers may refer to the survey papers \\cite{grochenig10, Krishtal11, shinsun13}, the recent  papers \\cite{fangshinsun20, samei19, shinsun19} and  references therein for  historical remarks and recent advances.\n\n\n\n\nGiven an element $A$ in a Banach algebra ${\\mathcal A}$ with the identity $I$, we define its  {\\em spectral set} $\\sigma_{\\mathcal{A}}(A)$\nand  {\\em spectral radius} $\\rho_{\\mathcal{A}}(A)$  by\n$$\n\\sigma_{\\mathcal{A}}(A):=\\big\\{\\lambda \\in \\mathbb{C} : \\lambda I -A \\text{ is not invertible in }\n\\mathcal{A} \\big\\}\n$$\nand\n$$\n\\rho_\\mathcal{A}(A) := \\max \\big\\{ |\\lambda| :\\lambda  \\in \\sigma_{\\mathcal A}(A)\\big\\}\n$$\nrespectively.\n Let ${\\mathcal A}$  and\n${\\mathcal B}$ be Banach algebras with common identity $I$ and   ${\\mathcal A}$ be a Banach subalgebra of ${\\mathcal B}$.\nThen an equivalent condition for the inverse-closedness of\n${\\mathcal A}$  in ${\\mathcal B}$ is that the spectral set of any\n$A\\in {\\mathcal A}$ in  Banach algebras ${\\mathcal A}$ and ${\\mathcal B}$ are the same, i.e.,\n$$\n\\sigma_{\\mathcal A}(A)=\\sigma_{\\mathcal B}(A).\n$$\nBy the above equivalence, a necessary condition for the inverse-closedness of\n${\\mathcal A}$  in ${\\mathcal B}$ is that\nthe spectral radius of any\n$A\\in {\\mathcal A}$ in the Banach algebras ${\\mathcal A}$ and ${\\mathcal B}$ are the same, i.e.,\n\\begin{equation} \\label{spectralradius}\n\\rho_\\mathcal{A}(A) =\\rho_\\mathcal{B}(A).\n\\end{equation}\nThe above necessary condition is shown  by Hulanicki \\cite{hulanicki} to be sufficient if we further assume that\n$\\mathcal{A}$  and $\\mathcal{B}$ are $*$-algebras with common identity and involution, and that\n $\\mathcal{B}$ is symmetric.\n  Here we say that\n a Banach algebra $\\mathcal B$ is   a $*$-algebra if\nthere is a  continuous  linear {\\em  involution $*$} on  $\\mathcal {B}$\nwith the properties that\n\\begin{equation*}\n(AB)^* = B^* A^*\\  \\text{ and }\\  A^{**} = A\\  \\text{ for all }  A,  B \\in\n{\\mathcal B},\n\\end{equation*}\nand that a $*$-algebra ${\\mathcal B}$ is  {\\em symmetric} if\n$$\\sigma_{\\mathcal {A}} (A^* A) \\subset [0,\\infty )\\ \\ {\\rm for \\ all} \\  A\\in {\\mathcal B}.$$\nThe spectral radii approach \\eqref{spectralradius}, known as  the Hulanicki's spectral method,\n  has been used to establish the inverse-closedness of symmetric $*$-algebras \\cite{branden, gkI, grochenigklotz10,  sunca11,  suntams07, suncasp05},\n  however the above approach does not provide a norm estimate for the inversion, which is crucial for  many mathematical and engineering  applications.\n\n\n\nTo consider norm estimate for the inversion, we recall the concept of norm-controlled inversion   of a Banach subalgebra  ${\\mathcal A}$ of a symmetric  $*$-algebra  ${\\mathcal B}$, which was initiated by Nikolski \\cite{nikolski99} and coined by Gr\\\"ochenig and Klotz \\cite{gkI}. Here\nwe say that\n a Banach subalgebra ${\\mathcal A}$ of ${\\mathcal B}$  admits {\\em\nnorm-controlled inversion} in ${\\mathcal B}$ if there exists a continuous function $h$ from\n$[0, \\infty)\\times [0, \\infty)$ to $[0, \\infty)$\n such that\n\\begin{equation}\\label{normcontrol}\n\\|A^{-1}\\|_{\\mathcal A}\\le h\\big(\\|A\\|_{\\mathcal A}, \\|A^{-1}\\|_{\\mathcal B}\\big)\n\\end{equation}\nfor all $A\\in {\\mathcal A}$ being invertible in ${\\mathcal B}$\n\\cite{gkII, gkI,  samei19, shinsun19}.\n\n\nThe norm-controlled inversion is  a strong version of Wiener's lemma.\nThe classical Banach algebra ${\\mathcal W}$   in \\eqref{Wieneralgebra.def}  is inverse-closed in\nthe algebra\n ${\\mathcal C}$ of  all periodic continuous functions\n  \\cite{wiener32},  however it does not have norm-controlled inversion in\n ${\\mathcal C}$ \\cite{belinskiijfaa97, nikolski99}.\nTo establish Wiener's lemma, there are several methods, including\n the Wiener's localization \\cite{wiener32}, the Gelfand's technique\n\\cite{gelfandbook},   the Brandenburg's trick \\cite{branden},  the Hulanicki's spectral method \\cite{hulanicki}, the Jaffard's  boot-strap argument \\cite{jaffard90},\nthe derivation technique \\cite{grochenigklotz10},\nand the Sj\\\"ostrand's commutator estimates \\cite{shinsun19, sjostrand94}.\n In this  paper, we will use the Brandenburg's trick\n to establish norm-controlled inversion of\n a differential $*$-subalgebra  ${\\mathcal A}$ of a symmetric  $*$-algebra  ${\\mathcal B}$.\n\n\n\n\n\n\n\n\n\nThis introduction article is organized as follows. In Section \\ref{differentialalgebra.section}, we recall the concept of\ndifferential subalgebras and present some differential subalgebras of  infinite matrices with polynomial off-diagonal decay.\nIn Section \\ref{generalizedDS.section}, we introduce  the concept of generalized differential  subalgebras and\npresent some generalized differential subalgebras of integral operators with kernels being  H\\\"older continuous and having polynomial off-diagonal decay.\nIn Section \\ref{normcontrolledinversion.section},  we use the Brandenburg's trick  to establish norm-controlled inversion\n  of   a differential $*$-subalgebra of a symmetric  $*$-algebra, and we conclude the section with two remarks on\n the  norm-controlled inversion with the norm control function bounded by a polynomial and\n  the norm-controlled inversion of nonsymmetric  Banach algebras.\n\n\n\n\n\n\n\n\n\n\\section{Differential Subalgebras}\\label{differentialalgebra.section}\n\n Let ${\\mathcal A}$  and\n${\\mathcal B}$ be Banach algebras such that ${\\mathcal A}$ is a Banach subalgebra of  ${\\mathcal B}$.\nWe say that\n${\\mathcal A}$ is  a {\\em differential subalgebra of order $\\theta\\in (0, 1]$} in ${\\mathcal B}$ \nif there exists a positive constant $D_0:=D_0({\\mathcal A}, {\\mathcal B}, \\theta)$ such that\n\\begin{equation}\\label{differentialnorm.def}\n\\|AB\\|_{\\mathcal A}\\le D_0\\|A\\|_{\\mathcal A} \\|B\\|_{\\mathcal A} \\Big (\\Big(\\frac{\\|A\\|_{\\mathcal B}}{\\|A\\|_{\\mathcal A}}\\Big)^\\theta +\n\\Big(\\frac{\\|B\\|_{\\mathcal B}}{\\|B\\|_{\\mathcal A}}\\Big)^\\theta\n\\Big)\n\\quad {\\rm for \\ all} \\  A, B \\in {\\mathcal A}.\n\\end{equation}\nThe concept of   differential subalgebras of order $\\theta$  was introduced in \\cite{blackadarcuntz91, kissin94, rieffel10} for $\\theta=1$ and\n\\cite{christ88, gkI,     shinsun19} for $\\theta\\in (0, 1)$.\n We also refer the reader\nto \\cite{barnes87, fangshinsun13, gkII, gkI,   grochenigklotz10,   jaffard90, rssun12, samei19, sunca11, sunacha08, suntams07, suncasp05} for various differential subalgebras\nof  infinite  matrices, convolution operators, and  integral operators with certain off-diagonal decay.\n\n\n\n For $\\theta=1$, the  requirement\n\\eqref{differentialnorm.def} can be reformulated as\n\\begin{equation}\n\\|AB\\|_{\\mathcal A}\\le D_0\\|A\\|_{\\mathcal A} \\|B\\|_{\\mathcal B}+ D_0 \\|A\\|_{\\mathcal B} \\|B\\|_{\\mathcal A}\n\\quad {\\rm for \\ all} \\  A, B \\in {\\mathcal A}. \\end{equation}\nSo the  norm $\\|\\cdot\\|_{\\mathcal A}$  satisfying \\eqref{differentialnorm.def}\nis also referred  as a Leibniz norm on ${\\mathcal A}$.\n\n\n\n   Let $C[a, b]$ be the space of  all continuous functions on the interval $[a, b]$  with its  norm defined by\n$$\\|f\\|_{C[a, b]}=\\sup_{t\\in [a, b]} |f(t)|, \\ \\  f\\in C[a, b],$$\n and $C^k[a, b], k\\ge 1$, be the space of all continuously  differentiable functions on the interval $[a, b]$ up to order $k$\nwith its norm defined by\n$$\\|h\\|_{C^k[a, b]}= \\sum_{j=0}^k \\|h^{(j)}\\|_{C[a, b]} \\ {\\rm for} \\  h\\in C^k[a, b].$$\nClearly,  $C[a, b]$  and $C^k[a, b]$ are Banach algebras under function multiplication.\nMoreover \n  \\begin{eqnarray}\\label{Cab.eq}\n\\|h_1h_2\\|_{C^1[a,b]} & = & \\|(h_1h_2)'\\|_{C[a, b]}+ \\|h_1h_2\\|_{C[a, b]}\\nonumber\\\\\n& \\le &\n\\|h_1'\\|_{C[a, b]} \\|h_2\\|_{C[a, b]}+ \\|h_1\\|_{C[a, b]} \\|h_2'\\|_{C[a, b]}\n + \\|h_1\\|_{C[a, b]}\\|h_2\\|_{C[a, b]}\\nonumber\\\\\n& \\le &  \\|h_1\\|_{C^1[a,b]} \\|h_2\\|_{C[a,b]}+\\|h_1\\|_{C[a,b]}\\|h_2\\|_{C^1[a,b]} \\ {\\rm for \\ all} \\ h_1, h_2\\in C^1[a, b],\n\\end{eqnarray}\nwhere the second inequality follows from the Leibniz rule.  Therefore we have\n\n\\begin{thm}\\label{C1ab.thm}\n $C^1[a, b]$ is a differential subalgebra of order one in $C[a, b]$.\n \\end{thm}\n  Due to the above illustrative example of differential subalgebras of order one,\nthe  norm $\\|\\cdot\\|_{\\mathcal A}$  satisfying \\eqref{differentialnorm.def}\nis also used to describe smoothness in abstract Banach algebra  \\cite{blackadarcuntz91}.\n\n\n\nLet\n${\\mathcal W}^1$  be  the Banach algebra of all periodic functions such that\nboth $f$ and its derivative $f'$ belong to  the Wiener algebra  ${\\mathcal W}$, and define the norm on ${\\mathcal W}^1$ by\n\\begin{equation}\\label{differentialwiener.def}\n\\|f\\|_{{\\mathcal W}^1}  =  \\|f\\|_{\\mathcal W}+\\|f'\\|_{\\mathcal W}\n  =  \\sum_{n\\in \\ZZ} (|n|+1) |\\hat f(n)|\n  \\end{equation}\n  for $f(x)=\\sum_{n\\in \\ZZ} \\hat f(n) e^{inx}\\in {\\mathcal W}^1$.\nFollowing the argument used in the proof of Theorem \\ref{C1ab.thm},\nwe have\n\\begin{thm} \\label{wienerdiff.theorem}\n    ${\\mathcal W}^1$ is a differential subalgebra of order one in ${\\mathcal W}$.\n    \\end{thm}\n\nRecall from  the classical Wiener's lemma that  ${\\mathcal W}$ is an inverse-closed subalgebra of\n${\\mathcal C}$, the algebra of all periodic continuous functions under multiplication.\nThis leads to the following natural question:\n\n\\begin{ques}\\label{question1}\n Is ${\\mathcal W}^1$  a differential subalgebra\nof ${\\mathcal C}$?\n\\end{ques}\n\nLet $\\ell^p, 1\\le p\\le \\infty$, be\n the space of all $p$-summable  sequences on $\\ZZ$ with norm denoted by $\\|\\cdot\\|_p$.\n  To answer the above question,\nwe consider  Banach algebras ${\\mathcal C}$, ${\\mathcal W}$ and ${\\mathcal W}^1$  in the ``frequency domain\".\nLet ${\\mathcal B}(\\ell^p)$ be the algebra of all bounded linear operators on  $\\ell^p, 1\\le p\\le \\infty$,\n and let  \n\\begin{equation}\n\\label{tildew.def}\n\\tilde {\\mathcal W}=\\Big\\{A:=(a(i-j))_{i,j\\in \\ZZ},\\  \\| A\\|_{\\tilde W}=\\sum_{k\\in \\ZZ} |a(k)|<\\infty\\Big\\}\\end{equation}\nand\n\\begin{equation}\n\\label{tildew1.def}\n{\\tilde {\\mathcal W}}^1=\\Big\\{A:=(a(i-j))_{i,j\\in \\ZZ}, \\ \\| A\\|_{{\\tilde W}^1}=\\sum_{k\\in \\ZZ} |k| |a(k)|<\\infty\\Big\\}\\end{equation}\nbe  Banach algebras of  Laurent matrices\nwith symbols in ${\\mathcal W}$ and ${\\mathcal W}^1$ respectively.   Then the classical Wiener's lemma can be reformulated\nas that $\\tilde {\\mathcal W}$ is an inverse-closed subalgebra of ${\\mathcal B}(\\ell^2)$,\nand  an equivalent statement of Theorem \\ref{wienerdiff.theorem}\nis that ${\\tilde {\\mathcal W}}^1$ is a differential subalgebra of order one in $\\tilde {\\mathcal W}$.\nDue to the above equivalence,  Question\n\\ref{question1} in the ``frequency domain\" becomes whether  ${\\mathcal W}^1$ is a differential subalgebra of order $\\theta\\in (0, 1]$ in  ${\\mathcal C}$.\nIn  \\cite{suncasp05}, the first example of  differential subalgebra of infinite matrices with order $\\theta\\in (0, 1)$\nwas discovered.\n\n\n\\begin{thm}\\label{W1.thm}\n${\\mathcal W}^1$ is a differential subalgebra of ${\\mathcal C}$ with order $2\/3$.\n\\end{thm}\n\n\n\n\nTo consider differential subalgebras  of infinite matrices in the noncommutative setting, we introduce three noncommutative  Banach algebras of\ninfinite matrices with certain off-diagonal decay.\nGiven $1\\le p\\le \\infty$ and $\\alpha\\ge 0$,\n we define\nthe Gr\\\"ochenig-Schur  family of infinite matrices\nby\n \\begin{equation}\\label{GS.def}\n{\\mathcal A}_{p,\\alpha}=\\Big\\{ A=(a(i,j))_{i,j \\in \\Z}, \\  \\|A\\|_{{\\mathcal A}_{p,\\alpha}}<\\infty\\Big\\}\n\\end{equation}\n\\cite{gltams06, jaffard90, moteesun, schur11,  suntams07, suncasp05},\nthe\nBaskakov-Gohberg-Sj\\\"ostrand family of infinite matrices by\n\\begin{equation}\\label{BGS.def}\n{\\mathcal C}_{p,\\alpha}=\\Big\\{ A= (a(i,j))_{i,j \\in \\Z}, \\  \\|A\\|_{{\\mathcal C}_{p,\\alpha}}<\\infty\\Big\\}\n\\end{equation}\n\\cite{baskakov90, gkwieot89, gltams06, sjostrand94,suntams07}, and\nthe Beurling family of infinite matrices\n\\begin{equation}\\label{Beurling.def}\n{\\mathcal B}_{p,\\alpha}=\\Big\\{ A= (a(i,j))_{i,j \\in \\Z}, \\  \\|B\\|_{{\\mathcal A}_{p,\\alpha}}<\\infty\\Big\\}\n\\end{equation}\n\\cite{beurling49,  shinsun19, sunca11},\n where     $u_\\alpha(i, j)=(1+|i-j|)^\\alpha, \\alpha\\ge 0$, are  polynomial weights on $\\Z^2$,\n \\begin{equation}\\label{GSnorm.def}\n \\|A\\|_{{\\mathcal A}_{p,\\alpha}}\n = \\max \\Big\\{ \\sup_{i \\in \\Z} \\big\\|\\big(a(i,j) u_\\alpha(i, j)\\big)_{j\\in \\Z}\\big\\|_p, \\ \\ \\sup _{j \\in \\Z}\n \\big\\|\\big(a(i,j) u_\\alpha(i, j)\\big)_{i\\in \\Z}\\big\\|_p\n \\Big\\},\n\\end{equation}\n\\begin{equation}\\label{BKSnorm.def}\n\\|A\\|_{{\\mathcal C}_{p,\\alpha}} = \\Big\\| \\Big(\\sup_{i-j=k} |a(i,j)| u_\\alpha(i, j)\\Big)_{k\\in \\Z} \\Big\\|_p,\n\\end{equation}\nand\n\\begin{equation}\\label{Beurlingnorm.def}\n\\|A\\|_{{\\mathcal B}_{p,\\alpha}} = \\Big\\| \\Big(\\sup_{|i-j|\\ge |k| } |a(i,j)| u_\\alpha(i, j)\\Big)_{k\\in \\Z} \\Big\\|_p.\n\\end{equation}\nClearly, we have\n\\begin{equation}\\label{properinclusion}\n{\\mathcal B}_{p,\\alpha} \\subset {\\mathcal C}_{p,\\alpha} \\subset\n{\\mathcal A}_{p,\\alpha}  \\ \\ {\\rm for \\ all}\\ 1\\le p\\le \\infty \\ {\\rm and} \\ \\alpha\\ge 0.\n\\end{equation}\nThe above inclusion  is proper for $1\\le p<\\infty$, while\nthe  above three  families \nof infinite matrices coincide  for $p=\\infty$,\n\\begin{equation}\\label{properinclusioninfinite}\n{\\mathcal B}_{\\infty,\\alpha}={\\mathcal C}_{\\infty,\\alpha}=\n{\\mathcal A}_{\\infty,\\alpha}  \\ \\ {\\rm for \\ all} \\ \\alpha\\ge 0,\n\\end{equation}\nwhich is also known as the Jaffard family of infinite matrices  \\cite{jaffard90},\n \\begin{equation}\\label{Jaffard.def}\n{\\mathcal J}_{\\alpha}=\\Big\\{ A= (a(i,j))_{i,j \\in \\Z}, \\  \\|A\\|_{{\\mathcal J}_{\\alpha}}=\\sup_{i,j\\in \\Z}|a(i,j)| u_\\alpha(i-j)<\\infty\\Big\\}.\n\\end{equation}\n\nObserve that\n$\\|A\\|_{{\\mathcal A}_{p,\\alpha}}=\\|A\\|_{{\\mathcal C}_{p,\\alpha}}$\nfor a Laurent matrix  $A=(a(i-j))_{i,j\\in \\Z}$. Then\nBanach algebras $\\tilde {\\mathcal W}$ and ${\\tilde {\\mathcal W}}^1$\nin \\eqref{tildew.def} and \\eqref{tildew1.def}\nare the commutative subalgebra of the  Gr\\\"ochenig-Schur  algebra ${\\mathcal A}_{1, \\alpha}$\nand the Baskakov-Gohberg-Sj\\\"ostrand algebra ${\\mathcal C}_{1, \\alpha}$ for $\\alpha=0, 1$ respectively,\n\\begin{equation}\\label{wa.re}\n\\tilde {\\mathcal W}= {\\mathcal A}_{1, 0}\\cap {\\mathcal L}={\\mathcal C}_{1, 0}\\cap {\\mathcal L}\n\\end{equation}\nand\n\\begin{equation}\\label{wa1.re}\n{\\tilde {\\mathcal W}}^1= {\\mathcal A}_{1, 1}\\cap {\\mathcal L}={\\mathcal C}_{1, 1}\\cap {\\mathcal L},\n\\end{equation}\nwhere ${\\mathcal L}$ is the set of all Laurent matrices  $A=(a(i-j))_{i,j\\in \\Z}$.\nThe sets ${\\mathcal A}_{p, \\alpha}, {\\mathcal C}_{p,\\alpha}, {\\mathcal B}_{p,\\alpha}$\nwith $p=1$ and $\\alpha=0$ are  noncommutative Banach algebras under matrix multiplication,\nthe Baskakov-Gohberg-Sj\\\"ostrand algebra  ${\\mathcal C}_{1,0}$ and the Beurling algebra ${\\mathcal B}_{1, 0}$ are inverse-closed subalgebras of ${\\mathcal B}(\\ell^2)$ \\cite{baskakov90, bochnerphillips42, gkwieot89,  sjostrand94, sunca11}, however\nthe Schur algebra ${\\mathcal A}_{1,0}$ is not inverse-closed in ${\\mathcal B}(\\ell^2)$  \\cite{tessera10}.\nWe remark that the inverse-closedness of the Baskakov-Gohberg-Sj\\\"ostrand algebra  ${\\mathcal C}_{1,0}$\nin ${\\mathcal B}(\\ell^2)$ can be understood as a noncommutative extension of the classical Wiener's lemma for the\ncommutative subalgebra $\\tilde {\\mathcal W}$ of Laurent matrices in ${\\mathcal B}(\\ell^2)$.\n\n\n\n\n\nFor $1\\le p\\le \\infty$ and $\\alpha>1-1\/p$, one may verify that\nthe Gr\\\"ochenig-Schur family ${\\mathcal A}_{p,\\alpha}$,\nthe Baskakov-Gohberg-Sj\\\"ostrand family  ${\\mathcal C}_{p,\\alpha}$ and the Beurling family ${\\mathcal B}_{p, \\alpha}$\nof infinite matrices form Banach algebras  under  matrix multiplication\nand they are inverse-closed subalgebras of ${\\mathcal B}(\\ell^2)$ \\cite{gltams06, jaffard90, sunca11, suntams07, suncasp05}.\nIn \\cite{sunca11, suntams07, suncasp05},  their differentiability in ${\\mathcal B}(\\ell^2)$ is established.\n\n\\begin{thm} \\label{sundiff.thm}\nLet $1\\le p\\le \\infty$ and $\\alpha>1-1\/p$. Then\n ${\\mathcal A}_{p,\\alpha}$,\n  ${\\mathcal C}_{p,\\alpha}$ and  ${\\mathcal B}_{p, \\alpha}$ are differential subalgebras of  order\n  $\\theta_0= (\\alpha+1\/p-1)\/(\\alpha+1\/p-1\/2)\\in (0, 1)$ in ${\\mathcal B}(\\ell^2)$.\n\\end{thm}\n\n\\begin{proof} The following argument about differential subalgebra property for the Gr\\\"ochenig-Schur algebra ${\\mathcal A}_{p, \\alpha}, 1<p<\\infty$, is adapted from \\cite{suncasp05}.  The reader may refer to \\cite{sunca11, suntams07, suncasp05} for the detailed proof to the\ndifferential subalgebra property for the Baskakov-Gohberg-Sj\\\"ostrand algebra\n  ${\\mathcal C}_{p,\\alpha}$ and  the Beurling algebra ${\\mathcal B}_{p, \\alpha}$.\n Take $A=(a(i,j))_{i,j\\in \\Z}$ and $B=(b(i,j))_{i,j\\in \\Z}\\in {\\mathcal A}_{p, \\alpha}$, and write\n $C=AB=(c(i,j))_{i,j\\in \\Z}$. Then\n \\begin{eqnarray}\\label{sundiff.thm.pf.eq1}\n \\|C\\|_{{\\mathcal A}_{p, \\alpha}} & = &\n \\max \\Big\\{ \\sup_{i \\in \\Z} \\big\\|\\big(c(i,j) u_\\alpha(i, j)\\big)_{j\\in \\Z}\\big\\|_p, \\ \\ \\sup _{j \\in \\Z}\n \\big\\|\\big(c(i, j) u_\\alpha(i, j)\\big)_{i\\in \\Z}\\big\\|_p\\Big\\}\\nonumber\\\\\n &\\le & 2^\\alpha \\max \\Big\\{ \\sup_{i \\in \\Z} \\Big\\|\\Big(\\sum_{k\\in \\Z} |a(i,k)| |b(k,j)| \\big(u_\\alpha(i, k)+u_\\alpha(k, j)\\big) \\Big)_{j\\in \\Z}\\Big\\|_p, \\nonumber\\\\\n  & & \\qquad \\qquad \\ \\  \\sup_{j \\in \\Z} \\Big\\|\\Big(\\sum_{k\\in \\Z} |a(i,k)| |b(k,j)| \\big(u_\\alpha(i, k)+u_\\alpha(k, j)\\big) \\Big)_{i\\in \\Z}\\Big\\|_p\n \\Big\\}\\nonumber\\\\\n & \\le & 2^\\alpha \\|A\\|_{{\\mathcal A}_{p, \\alpha}} \\|B\\|_{{\\mathcal A}_{1, 0}}+ 2^\\alpha \\|A\\|_{{\\mathcal A}_{1, 0}} \\|B\\|_{{\\mathcal A}_{p, \\alpha}},\n \\end{eqnarray}\n where the first inequality follows from the inequality\n $$u_\\alpha(i,j)\\le  2^\\alpha \\big(u_\\alpha(i, k)+ u_\\alpha(k, j)), \\ i, j, k\\in \\ZZ.$$\n\nLet $1\/p'=1-1\/p$,  and define\n   \\begin {equation}\n   \\label{tau0.def}\n   \\tau_0= \\left\\lfloor \\left(\\Bigg(\\frac{\\alpha p'+1}{\\alpha p'-1}\\Bigg)^{1\/p'}\\frac{\\|A\\|_{{\\mathcal A}_{p, \\alpha}}}{\\|A\\|_{{\\mathcal B}(\\ell^2)}}\\right)^{1\/(\\alpha+1\/2-1\/p')}\\right\\rfloor,\\end{equation}\n   where $\\lfloor t\\rfloor$ denotes the integer part of a real number $t$.\n      Then for $i\\in \\Z$, we have\n \\begin{eqnarray}\\label{para1.eq}\n \\sum_{j\\in \\Z} |a(i,j)| & = & \\Big(\\sum_{|j-i|\\le \\tau_0}+\\sum_{|j-i|>\\tau_0}\\Big) |a(i,j)|\\nonumber\\\\\n  & \\le &  \\Big(\\sum_{ |j-i|\\le \\tau_0} |a(i,j)|^2\\Big)^{1\/2} \\Big(\\sum_{|j-i|\\le \\tau_0} 1\\Big)^{1\/2}\\nonumber\\\\\n &  & + \\Big(\\sum_{|j-i|\\ge \\tau_0+1} |a(i,j)|^p (u_\\alpha(i, j))^p\\Big)^{1\/p} \\Big(\\sum_{|j-i|\\ge \\tau_0+1} (u_\\alpha(i, j))^{-p'} \\Big)^{1\/p'}\\nonumber\\\\\n & \\le &  \\|A\\|_{{\\mathcal B}(\\ell^2)} (2\\tau_0+1)^{1\/2}+  2^{1\/p'} (\\alpha p'-1)^{-1\/p'}  \\|A\\|_{{\\mathcal A}_{p, \\alpha}}\n (\\tau_0+1)^{-\\alpha+1\/p'}\\nonumber\\\\\n & \\le &  D\n\n \\|A\\|_{{\\mathcal A}_{p, \\alpha}}^{1-\\theta_0}  \\|A\\|_{{\\mathcal B}(\\ell^2)}^{\\theta_0},\n \n\n   \\end{eqnarray}\n   where $D$ is an absolute constant depending on $p, \\alpha$ only, and\n   the last inequality follows from  \\eqref{tau0.def} and the following estimate\n     $$   \\|A\\|_{{\\mathcal B}(\\ell^2)}\\le \\|A\\|_{{\\mathcal A}_{1, 0}}\\le \\Big(\\sum_{k\\in \\Z} (|k|+1)^{-\\alpha p'}\\Big)^{1\/p'}\n   \\|A\\|_{{\\mathcal A}_{p, \\alpha}}\\le \\Big(\\frac{\\alpha p'+1}{\\alpha p'-1}\\Big)^{1\/p'}\\|A\\|_{{\\mathcal A}_{p, \\alpha}}.$$\n    \n   Similarly we can prove that\n   \\begin{equation}\n   \\label{para2.eq}\n  \\sup_{j\\in \\Z} \\sum_{i\\in \\Z} |a(i,j)| \\le D\n\n \\|A\\|_{{\\mathcal A}_{p, \\alpha}}^{1-\\theta_0}  \\|A\\|_{{\\mathcal B}(\\ell^2)}^{\\theta_0}.\\end{equation}\n Combining \\eqref{para1.eq} and \\eqref{para2.eq} leads to\n   \\begin{equation}\n   \\label{para3.eq}\n  \\|A\\|_{{\\mathcal A}_{1, 0}} \\le D\n\n \\|A\\|_{{\\mathcal A}_{p, \\alpha}}^{1-\\theta_0}  \\|A\\|_{{\\mathcal B}(\\ell^2)}^{\\theta_0}.\\end{equation}\nReplacing the matrix $A$ in \\eqref{para3.eq} by the matrix $B$ gives\n   \\begin{equation}\n   \\label{para4.eq}\n  \\|B\\|_{{\\mathcal A}_{1, 0}} \\le D\n\n \\|B\\|_{{\\mathcal A}_{p, \\alpha}}^{1-\\theta_0}  \\|B\\|_{{\\mathcal B}(\\ell^2)}^{\\theta_0}.\\end{equation}\nTherefore it follows from \\eqref{sundiff.thm.pf.eq1}, \\eqref{para3.eq} and \\eqref{para4.eq} that\n\\begin{equation}\n\\|C\\|_{{\\mathcal A}_{p, \\alpha}}\\le  2^\\alpha  D\n \\|A\\|_{{\\mathcal A}_{p, \\alpha}}\\|B\\|_{{\\mathcal A}_{p, \\alpha}}^{1-\\theta_0}  \\|B\\|_{{\\mathcal B}(\\ell^2)}^{\\theta_0}\n+2^\\alpha  D\n \\|B\\|_{{\\mathcal A}_{p, \\alpha}}\\|A\\|_{{\\mathcal A}_{p, \\alpha}}^{1-\\theta_0}  \\|A\\|_{{\\mathcal B}(\\ell^2)}^{\\theta_0},\n\\end{equation}\nwhich proves the differential subalgebra property for Banach algebras ${\\mathcal A}_{p, \\alpha}$ with $1<p<\\infty$ and $\\alpha>1-1\/p$.\n\\end{proof}\n\nThe argument used in the proof of Theorem \\ref{sundiff.thm} involves a triplet of  Banach algebras\n${\\mathcal A}_{p, \\alpha}$, ${\\mathcal A}_{1, 0}$ and ${\\mathcal B}^2$ satisfying \\eqref{sundiff.thm.pf.eq1} and\n\\eqref{para3.eq}.\nIn the following theorem, we extend the above observation to\n general Banach algebra triplets  $({\\mathcal A}, {\\mathcal M}, {\\mathcal B})$.\n\n\\begin{thm}\\label{triple1.thm}\n Let ${\\mathcal A}, {\\mathcal M}$  and\n${\\mathcal B}$ be Banach algebras such that ${\\mathcal A}$ is a Banach subalgebra of ${\\mathcal M}$\nand ${\\mathcal M}$ is a  Banach subalgebra of ${\\mathcal B}$.\nIf there exist positive exponents $\\theta_0, \\theta_1\\in (0, 1]$ and  absolute constants $D_0, D_1$ such that\n\\begin{equation}\\label{triple1.thm.eq1}\n\\|AB\\|_{\\mathcal A}\\le D_0\\|A\\|_{\\mathcal A} \\|B\\|_{\\mathcal A} \\Big (\\Big(\\frac{\\|A\\|_{\\mathcal M}}{\\|A\\|_{\\mathcal A}}\\Big)^{\\theta_0} +\n\\Big(\\frac{\\|B\\|_{\\mathcal M}}{\\|B\\|_{\\mathcal A}}\\Big)^{\\theta_0}\n\\Big)\n\\quad {\\rm for \\ all} \\  A, B \\in {\\mathcal A},\n\\end{equation}\n and\n\\begin{equation}\\label{triple1.thm.eq2}\n\\|A\\|_{\\mathcal M}\\le D_1 \\|A\\|_{\\mathcal A}^{1-\\theta_1} \\|A\\|_{\\mathcal B}^{\\theta_1} \\ \\ {\\rm for \\ all} \\   \\ A\\in {\\mathcal A},\n\\end{equation}\nthen\n${\\mathcal A}$ is a differential subalgebra of order  $\\theta_0\\theta_1$ in ${\\mathcal B}$.\n\\end{thm}\n\n\\begin{proof}  For any $A, B\\in {\\mathcal A}$, we obtain from \\eqref{triple1.thm.eq1} and\n\\eqref{triple1.thm.eq2} that\n\\begin{eqnarray*}\n\\|AB\\|_{\\mathcal A} & \\le &  D_0\\|A\\|_{\\mathcal A} \\|B\\|_{\\mathcal A} \\Bigg (\\Big(\\frac{D_1 \\|A\\|_{\\mathcal A}^{1-\\theta_1} \\|A\\|_{\\mathcal B}^{\\theta_1}}{\\|A\\|_{\\mathcal A}}\\Big)^{\\theta_0} +\n\\Big(\\frac{D_1 \\|B\\|_{\\mathcal A}^{1-\\theta_1} \\|B\\|_{\\mathcal B}^{\\theta_1}}{\\|B\\|_{\\mathcal A}}\\Big)^{\\theta_0}\n\\Bigg)\\\\\n  & \\le &  D_0  D_1^{\\theta_0}\\|A\\|_{\\mathcal A} \\|B\\|_{\\mathcal A} \\Big (\\Big(\\frac{\\|A\\|_{\\mathcal B}}{\\|A\\|_{\\mathcal A}}\\Big)^{\\theta_0\\theta_1} +\n\\Big(\\frac{\\|B\\|_{\\mathcal B}}{\\|B\\|_{\\mathcal A}}\\Big)^{\\theta_0\\theta_1}\n\\Big),\n\\end{eqnarray*}\nwhich completes the proof.\n\\end{proof}\n\n\nFollowing the argument used in \\eqref{Cab.eq}, we can show that $C^2[a, b]$ is a differential subalgebra of $C^1[a, b]$.\nFor any  distinct $x, y\\in [a, b]$ and $f\\in C^2[a, b]$,  observe that\n$$\n|f'(x)|= \\frac{|f(y)-f(x)-f''(\\xi) (y-x)^2\/2 |}{|y-x|} \\le 2\\|f\\|_{C[a, b]} |y-x|^{-1}+ \\frac{1}{2} \\|f^{\\prime\\prime}\\|_{C[a, b]}\n|y-x|\n$$\nfor some $\\xi\\in [a, b]$, which implies that\n\\begin{equation}\n\\|f'\\|_{C[a, b]}\\le \\max\\big (4 \\|f\\|_{C[a, b]}^{1\/2} \\|f^{\\prime\\prime} \\|_{C[a, b]}^{1\/2},  8 (b-a)^{-1} \\|f\\|_{C[a, b]}\\big).\n\\end{equation}\nTherefore there exists  a positive constant $D$ such that\n\\begin{equation}\n\\|f\\|_{C^1[a,b]}\\le D \\|f\\|_{C^2[a, b]}^{1\/2} \\|f\\|_{C[a, b]}^{1\/2} \\ \\ {\\rm for \\ all} \\   \\ f\\in  C^2[a, b].\n\\end{equation}\nAs an application of Theorem  \\ref{triple1.thm}, we conclude that\n$C^2[a, b]$ is a differential subalgebra of  order $1\/2$ in $C[a, b]$.\n\n\\smallskip\n\nWe finish the section with the proof of Theorem \\ref{W1.thm}.\n\n\\begin{proof}[Proof of Theorem \\ref{W1.thm}]\nThe conclusion follows  from \\eqref{wa1.re} and Theorem \\ref{sundiff.thm} with $p=1$ and $\\alpha=1$.\n\\end{proof}\n\n\n\\section{Generalized differential subalgebras}\\label{generalizedDS.section}\n\n\nBy \\eqref{differentialnorm.def}, a  differential subalgebra ${\\mathcal A}$ satisfies the Brandenburg's  requirement:\n\\begin{equation}\\label{bt.req}\n\\|A^2\\|_{\\mathcal A}\\le 2D_0 \\|A\\|_{\\mathcal A}^{2-\\theta} \\|A\\|_{\\mathcal B}^{\\theta}, \\ A\\in {\\mathcal A}.\n\\end{equation}\nTo consider  the norm-controlled\ninversion of a Banach subalgebra ${\\mathcal A}$ of ${\\mathcal B}$,\nthe above requirement \\eqref{bt.req} could be relaxed to the existence of an integer $m\\ge 2$ such that\nthe $m$-th power of elements in ${\\mathcal A}$ satisfies\n\\begin{equation}\\label{weakpower}\n\\|A^m\\|_{\\mathcal A} \\le D \\|A\\|_{\\mathcal A}^{m-\\theta} \\|A\\|_{\\mathcal B}^{\\theta}, \\ \\  A\\in {\\mathcal A},\n\\end{equation}\nwhere  $\\theta\\in (0, m-1]$ and $D=D({\\mathcal A}, {\\mathcal B}, m, \\theta)$ is an absolute positive constant, see   Theorem \\ref{main-thm1} in the next section.\nFor $h\\in C^1[a, b]$ and $m\\ge 2$, we have\n  \\begin{equation*}\n\\|h^m \\|_{C^1[a,b]}  =  m \\|h^{m-1} h'\\|_{C[a, b]}+ \\|h^m\\|_{C[a, b]}\n\\le m \\|h\\|_{C^1[a, b]} \\|h\\|_{C[a, b]}^{m-1},\n\\end{equation*}\nand hence  the differential subalgebra $C^1[a, b]$ of $C[a, b]$ satisfies\n\\eqref{weakpower} with $\\theta=m-1$.\nIn this section, we introduce some sufficient conditions\nso that\n\\eqref{weakpower} holds for some integer $m\\ge 2$.\n\n\n\n\n\\begin{thm}\\label{triplenew.thm}   Let ${\\mathcal A}, {\\mathcal M}$  and\n${\\mathcal B}$ be Banach algebras such that ${\\mathcal A}$ is a Banach subalgebra of ${\\mathcal M}$\nand ${\\mathcal M}$ is a Banach subalgebra of ${\\mathcal B}$.\nIf there exist an integer $k\\ge 2$, positive exponents $\\theta_0, \\theta_1$, and  absolute constants $E_0, E_1$\nsuch that\n\\begin{equation}\\label{triplenew.eq1}\n\\|A_1A_2\\cdots A_k\\|_{\\mathcal A}  \\le   E_0\n\\Big(\\prod_{i=1}^k\\|A_i\\|_{{\\mathcal A}} \\Big) \\sum_{j=1}^k \\Big(\\frac{\\|A_i\\|_{\\mathcal M}}{\\|A_i\\|_{\\mathcal A}}\\Big)^{\\theta_0}, \\ \\  A_1, \\ldots, A_k \\in {\\mathcal A}\n\\end{equation}\nand\n\\begin{equation}\\label{triplenew.eq2}\n\\|A^2\\|_{\\mathcal M}\\le E_1  \\|A\\|_{\\mathcal A}^{2-\\theta_1} \\|A\\|_{\\mathcal B}^{\\theta_1},\\  A\\in {\\mathcal A},\n\\end{equation}\nthen \\eqref{weakpower} holds for $m=2k$ and  $\\theta=\\theta_0\\theta_1$.\n\\end{thm}\n\n\n\\begin{proof}   By \\eqref{banachalgebra.def},\n\\eqref{triplenew.eq1} and \\eqref{triplenew.eq2}, we have\n\\begin{equation}\n\\|A^{2k}\\|_{\\mathcal A}  \\le     k E_0\n\\|A^2\\|_{{\\mathcal A}}^{k-\\theta_0} \\|A^2\\|_{\\mathcal M}^{\\theta_0}\n\\le k E_0 E_1^{\\theta_0}  K^{k-\\theta_0} \\|A\\|_{{\\mathcal A}}^{2k-\\theta_0\\theta_1} \\|A\\|_{\\mathcal B}^{\\theta_0\\theta_1}, \\ \\  A\\in {\\mathcal A},\n\\end{equation}\nwhich completes the proof.\n\\end{proof}\n\n\nFor a Banach algebra triplet $({\\mathcal A},  {\\mathcal M}, {\\mathcal B})$  in Theorem \\ref{triple1.thm}, we obtain from\n\\eqref{triple1.thm.eq1} and \\eqref{triple1.thm.eq2} that\n\\begin{eqnarray}\\label{triple1.eq3}\n\\|A_1A_2\\cdots A_k\\|_{\\mathcal A} & \\le &  D_0\n\\|A_1\\|_{{\\mathcal A}} \\|A_2\\cdots A_k\\|_{{\\mathcal A}}\n\\Bigg (\\Big(\\frac{\\|A_1\\|_{\\mathcal M}}{\\|A_1\\|_{\\mathcal A}}\\Big)^{\\theta_0} +\n\\Big(\\frac{\\|A_2\\cdots A_k\\|_{\\mathcal M}}{\\|A_2\\cdots A_k\\|_{\\mathcal A}}\\Big)^{\\theta_0}\n\\Bigg)\\nonumber\\\\\n& \\le & \\tilde D_0\n\\Big(\\prod_{i=1}^k\\|A_i\\|_{{\\mathcal A}} \\Big)\n\\sum_{j=1}^k \\Big(\\frac{\\|A_j\\|_{\\mathcal M}}{\\|A_j\\|_{\\mathcal A}}\\Big)^{\\theta_0}, \\ \\ A_1, \\ldots, A_k\\in {\\mathcal A},\n\\end{eqnarray}\nand\n\\begin{equation}\\label{triple1.eq4}\n\\|A^2\\|_{\\mathcal M}\\le \\tilde K \\|A\\|_{\\mathcal M}^2\\le\nD_1^2 \\tilde K \\| A\\|_{\\mathcal A}^{2-2\\theta_1}\\|A\\|_{\\mathcal B}^{2\\theta_1}, \\ \\ A\\in {\\mathcal A},\n\\end{equation}\nwhere $\\tilde D_0$ is an absolute constant and $\\tilde K$ is the constant $K$ in\n\\eqref{banachalgebra.def} for the Banach algebra ${\\mathcal M}$.\nTherefore the assumptions  \\eqref{triplenew.eq1} and \\eqref{triplenew.eq2}\nin Theorem \\ref{triplenew.thm} are satisfied\nfor the Banach algebra triplet $({\\mathcal A},  {\\mathcal M}, {\\mathcal B})$  in Theorem \\ref{triple1.thm}.\n\nFor a differential subalgebra  ${\\mathcal A}$ of order $\\theta_0$ in ${\\mathcal B}$,\nwe observe  that the requirements\n\\eqref{triplenew.eq1} and \\eqref{triplenew.eq2} with  ${\\mathcal M}={\\mathcal B}$,  $k=2$ and $\\theta_1=2$ are met,\n and hence\n \\eqref{weakpower} holds for $m=4$ and  $\\theta=2\\theta_0$.\nRecall that ${\\mathcal B}$ is a trivial differential subalgebra of  ${\\mathcal B}$.\nIn the following corollary,  we can extend the above conclusion to arbitrary  differential subalgebras ${\\mathcal M}$ of ${\\mathcal B}$.\n\n\\begin{cor}\nLet ${\\mathcal A}, {\\mathcal M}$  and\n${\\mathcal B}$ be Banach algebras such that ${\\mathcal A}$ is a differential subalgebra of order $\\theta_0$ in  ${\\mathcal M}$\nand ${\\mathcal M}$ is a differential subalgebra of  order $\\theta_1$ in ${\\mathcal B}$.\nThen \\eqref{weakpower} holds for $m=4$ and  $\\theta=\\theta_0\\theta_1$.\n\\end{cor}\n\n\n\n\nFollowing the argument used in the proof of Theorem \\ref{triplenew.thm}, we can show that\n\\eqref{weakpower} holds for $m=4$ if the requirement \\eqref{triplenew.eq1} with $k=3$ is replaced by the following strong version\n\\begin{equation}\\label{triplenew.eq3}\n\\|ABC\\|_{\\mathcal A}  \\le   E_0\n\\|A\\|_{{\\mathcal A}}  \\|C\\|_{{\\mathcal A}}  \\|B\\|_{{\\mathcal A}}^{1-\\theta_0}\n\\|B\\|_{\\mathcal M}^{\\theta_0}, \\ \\  A, B, C \\in {\\mathcal A}.\n\\end{equation}\n\n\n\n\\begin{thm}\\label{triplenew.thm2}\nLet ${\\mathcal A}, {\\mathcal M}$  and\n${\\mathcal B}$ be Banach algebras such that ${\\mathcal A}$ is a Banach subalgebra of ${\\mathcal M}$\nand ${\\mathcal M}$ is a Banach subalgebra of ${\\mathcal B}$.\nIf there exist positive exponents $\\theta_0, \\theta_1\\in (0, 1]$ and  absolute constants $E_0, E_1$\nsuch that \\eqref{triplenew.eq2} and \\eqref{triplenew.eq3} hold,\nthen \\eqref{weakpower} holds for $m=4$ and  $\\theta=\\theta_0\\theta_1$.\n\\end{thm}\n\nLet $L^p:=L^p(\\R), 1\\le p\\le \\infty$, be the space of all $p$-integrable functions on $\\R$  with standard norm $\\|\\cdot\\|_p$,\nand ${\\mathcal B}(L^p)$ be the algebra of bounded linear operators on\n$L^p$ with the norm $\\|\\cdot \\|_{{\\mathcal B}(L^p)}$.\nFor $1\\le p\\le \\infty, \\alpha\\ge 0$ and $\\gamma\\in [0, 1)$, we define the norm\nof a kernel  $K$ on $\\R\\times \\R$ by\n\\begin{equation}\\label{Ex3-def-norm}\n\\|K\\|_{{\\mathcal W}_{p,\\alpha}^\\gamma}=\n\\left\\{\n\\begin{array}{ll}\n \\max\\Big(\\sup_{x\\in \\R} \\big\\|K(x,\\cdot)u_\\alpha(x,\\cdot)\\big\\|_p,\\\n\\sup_{y\\in \\R} \\big\\|K(\\cdot,y)u_\\alpha(\\cdot,y)\\big\\|_p\\Big)  & {\\rm  if }\\ \\gamma =0\n\\\\\n\\|K\\|_{{\\mathcal W}_{p,\\alpha}^0}+\\sup_{0<\\delta\\le 1} \\delta^{-\\gamma}\n\\|\\omega_\\delta(K)\\|_{{\\mathcal W}_{p,\\alpha}^0} & {\\rm if } \\ 0 < \\gamma <1,\n\\end{array}\n\\right.\n\\end{equation}\nwhere  the modulus of continuity\nof  the  kernel  $K$ is defined by\n\\begin{equation}\\label{Ex3-def-mod}\n\\omega_\\delta(K)(x,y):=\\sup_{|x^\\prime|\\le \\delta, |y^\\prime|\\le \\delta}\n|K(x+x^\\prime, y+y^\\prime)-K(x,y)|, \\ x, y\\in \\RR,\n\\end{equation}\nand  $u_\\alpha(x, y)= (1+|x-y|)^\\alpha,  x, y\\in \\R$ are polynomial weights on $\\R\\times \\R$.\nConsider the set  ${\\mathcal W}_{p,\\alpha}^\\gamma$ of integral operators\n\\begin{equation}\\label{Ex3-def-int-oper}\nTf(x)=\\int_{{\\R}} K_T(x,y) f(y) dy, \\quad f \\in L^p,\n\\end{equation}\nwhose integral kernels $K_T$ satisfy $\\|K_T\\|_{{\\mathcal W}^\\gamma_{p, \\alpha}}<\\infty$, and define\n$$\n\\|T\\|_{{\\mathcal W}_{p,\\alpha}^\\gamma}:=\n\\|K_T \\|_{{\\mathcal W}_{p,\\alpha}^\\gamma}, \\ T\\in {\\mathcal W}_{p,\\alpha}^\\gamma.\n$$\nIntegral operators in ${\\mathcal W}_{p, \\alpha}^\\gamma$ have their kernels being H\\\"older continuous of order $\\gamma$\n and having off-diagonal polynomial decay of order $\\alpha$.\nFor $1\\le p\\le \\infty$ and $\\alpha>1-1\/p$, one may verify that\n ${\\mathcal W}_{p, \\alpha}^\\gamma, 0\\le \\gamma<1$, are Banach subalgebras of\n  ${\\mathcal B}(L^2)$ under operator composition.\n  The Banach algebras  ${\\mathcal W}_{p, \\alpha}^\\gamma, 0<\\gamma<1$, of  integral operators\n  may not form a differential subalgebra of ${\\mathcal B}(L^2)$, however  the triple\n$({\\mathcal W}_{p, \\alpha}^\\gamma,  {\\mathcal W}_{p, \\alpha}^0, {\\mathcal B}(L^2))$ is proved in\n \\cite{sunacha08} to satisfy the following\n\\begin{equation}\\label{Ex3-norm-comp}\n\\|T_0\\|_{\\mathcal B} \\le  D \\|T_0\\|_{{\\mathcal W}_{p,\\alpha}^0} \\le\nD\\|T_0\\|_{{\\mathcal W}_{p,\\alpha}^\\gamma},\n\\end{equation}\n\\begin{equation}\\label{Ex3-norm-2-theta}\n\\|T_0^2 \\|_{{\\mathcal W}_{p,\\alpha}^0} \\le D\n\\|T_0 \\|_{{\\mathcal W}_{p,\\alpha}^\\gamma}^{1+\\theta}  \\|T_0 \\|_{{\\mathcal B}(L^2)}^{1-\\theta}\n\\end{equation}\nand\n\\begin{equation}\\label{Ex3-norm-3-product}\n\\|T_1 T_2 T_3  \\|_{{\\mathcal W}_{p,\\alpha}^\\gamma} \\le D\n\\|T_1 \\|_{{\\mathcal W}_{p,\\alpha}^\\gamma} \\| T_2 \\|_{{\\mathcal W}_{p,\\alpha}^0}\n\\| T_3 \\|_{{\\mathcal W}_{p,\\alpha}^\\gamma}\n\\end{equation}\nholds for all $T_i\\in {\\mathcal W}_{p, \\alpha}^\\gamma, 0\\le i\\le 3$, where $D$ is an absolute constant and\n$$\\theta= \\frac{\\alpha+\\gamma+1\/p}{(1+\\gamma)(\\alpha+1\/p)}.$$\nThen  the requirements  \\eqref{triplenew.eq2} and \\eqref{triplenew.eq3} in Theorem \\ref{triplenew.thm2}\nare met for  the triplet $({\\mathcal W}_{p, \\alpha}^\\gamma,  {\\mathcal W}_{p, \\alpha}^0, {\\mathcal B}(L^2))$,\nand hence\nthe Banach space pair $({\\mathcal W}_{p, \\alpha}^\\gamma,   {\\mathcal B}(L^2))$ satisfies the Brandenburg's  condition \\eqref{weakpower} with $m=4$\n\\cite{fangshinsun13, sunacha08}.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Brandenburg  trick and norm-controlled inversion}\\label{normcontrolledinversion.section}\n\n\nLet\n$\\mathcal{A}$  and $\\mathcal{B}$ are $*$-algebras with common identity and involution, and  let\n $\\mathcal{B}$ be  symmetric. In this section, we show that\n ${\\mathcal A}$ has norm-controlled inversion in ${\\mathcal B}$ if it meets\n the Brandenburg requirement \\eqref{weakpower}.\n\n\n\n\n\n\\begin{thm}\\label{main-thm1}\nLet  ${\\mathcal B}$  be a   symmetric $*$-algebra with its norm\n$\\|\\cdot\\|_{\\mathcal B}$\nbeing normalized in the sense that \\eqref{banachalgebra.def} holds with $K=1$,\n \\begin{equation}\\label{banachalgebra.defnew2}\n\\|\\tilde A\\tilde B\\|_{\\mathcal B}\\le \\|\\tilde A\\|_{\\mathcal B}\\|\\tilde B\\|_{\\mathcal B},\\   \\tilde  A, \\tilde B\\in {\\mathcal B},\n\\end{equation}\nand\n$\\mathcal{A}$   be  a $*$-algebra with its\nnorm\n$\\|\\cdot\\|_{\\mathcal A}$\nbeing normalized too,\n\\begin{equation}\\label{banachalgebra.defnew1}\n\\|AB\\|_{\\mathcal A}\\le \\|A\\|_{\\mathcal A}\\|B\\|_{\\mathcal A}, \\  A, B\\in {\\mathcal A}.\n\\end{equation}\nIf  ${\\mathcal A}$ is a $*$-subalgebra of ${\\mathcal B}$  with  common  identity  $I$ and involution  $*$,  and\nthe  pair $({\\mathcal A}, {\\mathcal B})$ satisfies\nthe Brandenburg requirement \\eqref{weakpower}, then\n${\\mathcal A}$ has norm-controlled inversion in ${\\mathcal B}$. Moreover,\nfor any  $A\\in{\\mathcal A}$  being  invertible in ${\\mathcal B}$ we have\n\\begin{eqnarray}\\label{norm-control1}\n\\|A^{-1}\\|_{\\mathcal A}\n& \\le &\n\\|A^* A\\|_{\\mathcal B}^{-1}  \\|A^*\\|_{\\mathcal A}\\nonumber\\\\\n& & \\times\n\\left\\{\\begin{array}{ll}\n   \\big(2t_0+(1-  2^{\\log_m (1-\\theta\/m)})^{-1} (\\ln a)^{-1}\\big)  a\n \\exp\\Big(\\frac{\\ln m-\\ln (m-\\theta)} {\\ln (m-\\theta)} t_0 \\ln a\\Big)\n  & {\\rm if} \\ \\theta<m-1\\\\\na^2 (\\ln a)^{-1}  (Db)^{m-1} \\Gamma\\Big(\\frac{(m-1) \\ln (Db)}{\\ln m \\ln a}+1\\Big)\n&   {\\rm if} \\ \\theta=m-1,\n\\end{array}\\right.\n\\end{eqnarray}\nwhere $\\Gamma(s)=\\int_0^\\infty t^{s-1} e^{-t} dt$ is the Gamma function, $m\\ge 2$ and $0<\\theta\\le m-1$ are the constants  in  \\eqref{weakpower},\n$\\kappa(A^*A)= \\|A^*A\\|_{\\mathcal B} \\|(A^*A)^{-1} \\|_{\\mathcal B}$,\n$ a= \\big(1- (\\kappa(A^*A))^{-1}\\big)^{-1}> 1$,\n$$b= \\frac{ \\|I\\|_{\\mathcal A}+   \\|A^* A\\|_{\\mathcal B}^{-1}\n \\|A^* A\\|_{\\mathcal A}} {1- (\\kappa(A^*A))^{-1} }\\ge a >1,\n$$\nand\n\\begin{equation}\\label{t0.def0}\nt_0=\\Big( \\frac{ (m-1)(m-\\theta) \\log_m (m-\\theta) \\ln (Db)}{(m-1-\\theta) \\ln a}\\Big)^{\\ln m\/(\\ln m-\\ln (m-\\theta))} \\ {\\rm for}\\  0<\\theta<m-1.\n\\end{equation}\n\n\\end{thm}\n\n\n\\begin{proof}\nObviously it suffices to prove \\eqref{norm-control1}.\n  In this paper, we follow the argument in \\cite{shinsun19} to give a sketch proof.\nLet $A\\in {\\mathcal A}$ so that $A^{-1} \\in {\\mathcal B}$.\nAs $\\mathcal{B}$ is a symmetric $*$-algebra,  the spectrum of $A^*A$ in ${\\mathcal B}$ lies in an interval on the positive real axis,\n\\begin{equation} \\sigma (A^* A)\\subset\n\\big[  \\|(A^* A)^{-1}\\|_{\\mathcal B}^{-1}, \\   \\|A^* A\\|_{\\mathcal B}\\big].\n\\end{equation}\nTherefore\n $B:=I- \\|A^* A\\|_{\\mathcal B}^{-1} A^* A\\in {\\mathcal A}$ satisfies\n\\begin{equation}\\label{B-norm}\n \\|B\\|_{\\mathcal B} \\le 1- (\\kappa(A^*A))^{-1}=a^{-1}<1\n\\end{equation}\nand\n\\begin{equation}\\label{A-norm}\n\\|B\\|_{\\mathcal A} \\le \\|I\\|_{\\mathcal A}+ \\|A^* A\\|_{\\mathcal B}^{-1}\n \\|A^* A\\|_{\\mathcal A}=ba^{-1}.\n\\end{equation}\n\n\nFor a positive integer $n=\\sum_{i=0}^l \\varepsilon_i m^i$, define  $n_0=n$ and\n$n_k, 1\\le k\\le l$, inductively by\n\\begin{equation}\\label{nk.def}\nn_{k}= \\frac{n_{k-1}-\\varepsilon_{k-1}}{m}=\\sum_{i=k}^l \\varepsilon_i m^{i-k}, 1\\le k\\le l,  \\end{equation}\n where $\\varepsilon_i\\in\\{0,1, \\ldots, m-1\\}, 1\\le i\\le l-1$ and $\\varepsilon_l\\in \\{1, \\ldots, m-1\\}$.\nBy \\eqref{weakpower} and \\eqref{banachalgebra.defnew2}, we have\n\\begin{equation}\\label{2-exponents}\n\\|B^{m n_k} \\|_{\\mathcal A} \\le D \\| B^{n_k}\\|_{\\mathcal A}^{m-\\theta} \\| B^{n_k}\\|_{\\mathcal B}^{\\theta}\n\\le D \\|B^{n_k}\\|_{\\mathcal A}^{m-\\theta} \\|B\\|_{\\mathcal B}^{n_k\\theta},\n\\quad k=1,\\cdots, l-1.\n\\end{equation}\n\nBy   \\eqref{banachalgebra.defnew2}, \\eqref{banachalgebra.defnew1}, \\eqref{B-norm}, \\eqref{A-norm}, \\eqref{nk.def} and\n\\eqref{2-exponents}, we obtain\n\\begin{eqnarray}  \\label{tilde-Bn-compu}\n  \\|B^n\\|_{\\mathcal A}  & = &  \\|B^{n_0}\\|_{\\mathcal A}\n\\le \\|B^{m n_1}\\|_{\\mathcal A}\n\\|B\\|_{\\mathcal A}^{\\varepsilon_0}\\le  D \\|B^{ n_1}\\|_{\\mathcal A}^{m-\\theta}\n\\|B\\|_{\\mathcal A}^{\\varepsilon_0}  \\|B\\|_{\\mathcal B}^{n_1\\theta}\n\\nonumber \\\\\n&\n\\le &  D^{1+ (m-\\theta)}\n\\|B^{n_2}\\|_{\\mathcal A}^{(m-\\theta)^2}\n\\| B\\|_{\\mathcal A}^{\\varepsilon_0+\\varepsilon_1(m-\\theta)}\n\\| B\\|_{\\mathcal B}^{ n_1 \\theta+ n_2 \\theta (m-\\theta)}\\nonumber\\\\\n&  \\le &  \\cdots\n \\nonumber \\\\\n&\n\\le & D^{\\sum_{k=0}^{l-1} (m-\\theta)^k}\n\\|B\\|_{\\mathcal A}^{\\sum_{k=0}^l \\varepsilon_k (m-\\theta)^k}\n\\| B\\|_{\\mathcal B}^{\\theta \\sum_{k=1}^l n_k (m-\\theta)^{k-1}}\n\\nonumber \\\\\n&\n=& D^{\\sum_{k=0}^{l-1} (m-\\theta)^k}\n\\|B\\|_{\\mathcal A}^{\\sum_{k=0}^l \\varepsilon_k (m-\\theta)^k}\n\\| B\\|_{\\mathcal B}^{n-\\sum_{k=0}^l \\varepsilon_k (m-\\theta)^k}\n\\nonumber \\\\\n\\qquad &\\le & D^{\\sum_{k=0}^{l-1} (m-\\theta)^k}\nb^{\\sum_{k=0}^l \\varepsilon_k (m-\\theta)^k}\na^{-n}\n\\nonumber \\\\\n\\qquad &\n\\le  &\n\\left\\{\\begin{array}{ll} (Db)^{\\frac{(m-1)(m-\\theta)}{m-1-\\theta} n^{\\log_m(m-\\theta)}}  a^{-n}  & {\\rm if} \\ \\theta<m-1\\\\\n &  \\\\\n(Db)^{(m-1)\\log_m (mn+1)}  a^{-n}  & {\\rm if} \\ \\theta=m-1,\n\\end{array}\\right.\n\\end{eqnarray}\nwhere the last inequality holds since\n\\begin{eqnarray*}\n\\sum_{k=0}^l \\varepsilon_k (m-\\theta)^k\n& \\le &  (m-1) \\sum_{k=0}^l (m-\\theta)^k \\le (m-1) \\left\\{\\begin{array}{ll}\n \\frac{ (m-\\theta)^{l+1}-1}{m-1-\\theta} \\  & {\\rm if} \\ \\theta<m-1\\\\\n l+1 & {\\rm if}  \\ \\theta=m-1\\end{array}\n \\right. \\\\\n & \\le &\n (m-1) \\left\\{\\begin{array}{ll}\n \\frac{ m-\\theta}{m-1-\\theta} n^{\\log_m(m-\\theta)} \\  & {\\rm if} \\ \\theta<m-1\\\\\n\\log_m (mn+1) & {\\rm if}  \\ \\theta=m-1.\\end{array}\\right.\n\\end{eqnarray*}\nObserve that\n $A^*A =  \\|A^* A\\|_{\\mathcal B} (I-B)$. Hence\n$$\n A^{-1}=(A^*A)^{-1}A^* = \\|A^* A\\|_{\\mathcal B}^{-1} \\Bigg( \\sum_{n=0}^\\infty B^n \\Bigg) A^*.\n$$\nThis together with  \\eqref{banachalgebra.defnew1}, \\eqref{tilde-Bn-compu}\nand \\eqref{inverseA.power} implies that\n\\begin{eqnarray}\\label{inverseA.power}\n  \\|A^{-1}\\|_{\\mathcal A}   & \\le &    \\|A^* A\\|_{\\mathcal B}^{-1}  \\|A^*\\|_{\\mathcal A}\n\\sum_{n=0}^\\infty \\|B^n\\|_{\\mathcal A}\n\\nonumber\\\\\n&\\le &  \\|A^* A\\|_{\\mathcal B}^{-1}  \\|A^*\\|_{\\mathcal A}\\times\n\\left\\{\\begin{array}{ll} \\sum_{n=0}^\\infty (Db)^{\\frac{(m-1)(m-\\theta)}{m-1-\\theta} n^{\\log_m(m-\\theta)}}  a^{-n}  &   {\\rm if} \\ \\theta<m-1\\\\\n \\sum_{n=0}^\\infty (Db)^{(m-1)\\log_m (mn+1)}  a^{-n}  &  {\\rm if} \\ \\theta=m-1.\n\\end{array}\\right.\n\\end{eqnarray}\n\nBy direct calculation, we have\n\n\\begin{eqnarray}\\label{m-1.estimate}\n \\sum_{n=0}^\\infty (Db)^{(m-1)\\log_m (mn+1)}  a^{-n} &  \\le &   a \\sum_{n=0}^\\infty \\int_n^{n+1} (Db)^{(m-1) \\log_m (mt+1)} a^{-t} dt\\nonumber\\\\\n& \\le & a^2 (Db)^{m-1} \\int_0^\\infty  (t+1)^{(m-1) \\log_m (Db)} e^{-(t+1) \\ln a} dt\\nonumber\\\\\n& \\le & a^2 (Db)^{m-1} (\\ln a)^{-1} \\Gamma\\Big(\\frac{(m-1) \\ln (Db)}{\\ln m \\ln a}+1\\Big).\n\\end{eqnarray}\nThis together with  \\eqref{inverseA.power} proves \\eqref{norm-control1} for $\\theta=m-1$.\n\n\nFor $0<\\theta<m-1$,  set\n$$s(t)=t- \\frac{ (m-1)(m-\\theta) \\ln (Db)}{(m-1-\\theta) \\ln a} t^{\\log_m(m-\\theta)}.$$\nObserve that\n$$\ns'(t)=1-  \\frac{ (m-1)(m-\\theta) \\ln (Db)}{(m-1-\\theta) \\ln a} \\log_m(m-\\theta) t^{\\log_m (1-\\theta\/m)}.\n$$\nTherefore\n\\begin{equation}\\label{st.eq1}\n\\min_{t\\ge 0} s(t)=s(t_0)=-\\frac{\\ln m-\\ln (m-\\theta)} {\\ln (m-\\theta)} t_0<0\n\\end{equation}\nand\n\\begin{equation}\\label{st.eq2}\n1\\ge s'(t)\\ge  s'(2t_0)=1-  2^{\\log_m (1-\\theta\/m)} \\ \\quad \\ {\\rm for \\ all}\\ \\ t\\ge 2t_0,\n\\end{equation}\nwhere  $t_0$ is given in \\eqref {t0.def0}.\nBy \\eqref{st.eq1} and \\eqref{st.eq2}, we have\n\\begin{eqnarray}\n & & \\sum_{n=0}^\\infty (Db)^{\\frac{(m-1)(m-\\theta)}{m-1-\\theta} n^{\\log_m(m-\\theta)}}  a^{-n}\n  \\le   a \\sum_{n=0}^\\infty \\int_n^{n+1}  (Db)^{\\frac{(m-1)(m-\\theta)}{m-1-\\theta} t^{\\log_m(m-\\theta)}}  a^{-t} dt\\nonumber\\\\\n& = & a\\Big(\\int_0^{2t_0}+\\int_{2t_0}^\\infty\\Big)\n\\exp(- s(t) \\ln a)  dt\\nonumber\\\\\n& \\le &  2at_0\n \\exp(-s(t_0) \\ln a) +    (1-  2^{\\log_m (1-\\theta\/m)})^{-1} a \\int_{s(2t_0)}^\\infty \\exp(-u \\ln a) du \\nonumber\\\\\n & \\le & \\Big  (2t_0+(1-  2^{\\log_m (1-\\theta\/m)})^{-1} (\\ln a)^{-1}\\Big )  a\n \\exp\\Big(\\frac{\\ln m-\\ln (m-\\theta)} {\\ln (m-\\theta)} t_0 \\ln a\\Big) .\n\\end{eqnarray}\nCombining the above estimate with  \\eqref{inverseA.power} proves \\eqref{norm-control1} for $\\theta<m-1$.\n\\end{proof}\n\n\n\nFor $m=2$, the estimate \\eqref{norm-control1} to the inverse $A^{-1}\\in {\\mathcal A}$  is essentially established in \\cite{gkII, gkI} for $\\theta=1$ and \\cite{shinsun19, sunaicm14} for  $\\theta\\in (0, 1)$, and a similar estimate is given  in \\cite{ samei19}. The reader may refer to \\cite{fangshinsun13, grochenigklotz10, sunacha08,  suntams07,\nsuncasp05} for  norm estimation of the inverse of elements in Banach algebras of infinite matrices and integral operators with certain off-diagonal decay.\n\n\n\n\\begin{rem}{\\rm  A good estimate for the norm control function $h$ in\nthe norm-controlled inversion \\eqref{normcontrol} is important for some mathematical and engineering applications.\nFor an element $A\\in {\\mathcal A}$ with $A^{-1}\\in {\\mathcal B}$, we obtain the following estimate from Theorem \\ref{main-thm1}:\n\\begin{equation}\n\\|(A^*A)^{-1}\\|_{\\mathcal A}\n \\le\nC \\|A^* A\\|_{\\mathcal B}^{-1}  a  (\\ln a)^{-1}  \\times \\left\\{\\begin{array}{ll}\n  t_1  \\exp(C t_1)\n  & {\\rm if} \\ \\theta<m-1\\\\\n a  b^{m-1} \\exp\\Big(C \\frac{\\ln b}{\\ln a} \\ln \\big(\\frac{\\ln b}{\\ln a}\\big)\\Big)\n&   {\\rm if} \\ \\theta=m-1,\n\\end{array}\\right.\n\\end{equation}\nwhere  $C$ is an absolute constant independent of $A$ and\n$$t_1= (\\ln b)^{\\ln m\/(\\ln m-\\ln (m-\\theta))} (\\ln a)^{-\\ln (m-\\theta)\/(\\ln m-\\ln (m-\\theta))}.$$\nWe remark that the above norm estimate to the inversion is far away from the optimal estimation for our illustrative differential subalgebra\n$C^1[a, b]$. In fact,  give any $f\\in C^1[a, b]$ being invertible in  $C[a, b]$,\nwe have\n\\begin{equation*}\n\\|1\/f\\|_{C^1[a, b]}\\le\n \\| f'\\|_{C[a, b]} \\|f^{-1} \\|_{C[a, b]}^2+ \\|1\/f \\|_{C[a, b]} \\le  \\|1\/f \\|_{C[a, b]}^2 \\|f\\|_{C^1[a, b]}.\n \\end{equation*}\n Therefore  $C^1[a, b]$ has norm-controlled inversion in $C[a, b]$ with the norm control function\n $h(s, t)$ in \\eqref{normcontrol} being\n $h(s, t)= s t^2$.\nGr\\\"ochenig and Klotz first considered  norm-controlled inversion\nwith the norm control function $h$ having polynomial growth, and they show in\n\\cite{gkII} that the Baskakov-Gohberg-Sj\\\"ostrand  algebra ${\\mathcal C}_{1, \\alpha}, \\alpha>0$\nand  the Jaffard algebra ${\\mathcal J}_\\alpha, \\alpha>1$  have norm-controlled inversion in ${\\mathcal B}(\\ell^2)$ with\n the norm control function\n $h$ bounded by a polynomial. In \\cite{shinsun19}, we \nproved that  the  Beurling algebras  ${\\mathcal B}_{p, \\alpha}$\n with $1\\le p\\le \\infty$ and $\\alpha>1-1\/p$ admit  norm-controlled inversion in ${\\mathcal B}(\\ell^2)$ with\n the norm control function\n bounded by some polynomials.  Following the commutator technique used in \\cite{shinsun19, sjostrand94}, we can establish a similar result for\n the Baskakov-Gohberg-Sj\\\"ostrand  algebras ${\\mathcal C}_{p, \\alpha}$ with $1\\le p\\le \\infty$ and $\\alpha>1-1\/p$.\n\n \\begin{thm}\\label{polynomial.thm}\n Let $1\\le p\\le \\infty$ and $\\alpha>1-1\/p$. Then\n  the Baskakov-Gohberg-Sj\\\"ostrand  algebra ${\\mathcal C}_{p, \\alpha}$\n  and the  Beurling algebra  ${\\mathcal B}_{p, \\alpha}$\n  admit  norm-controlled inversion in ${\\mathcal B}(\\ell^2)$ with\n the norm control function\n bounded by a polynomial.\n   \\end{thm}\n\n It is still unknown whether   Gr\\\"ochenig-Schur algebras ${\\mathcal A}_{p, \\alpha}, 1\\le p<\\infty, \\alpha>1-1\/p$,\n  admit norm-controlled inversion in ${\\mathcal B}(\\ell^q), 1\\le q< \\infty$, with\n the norm control function  bounded by a polynomial.\nIn  \\cite{gkII}, Gr\\\"ochenig and Klotz introduce a differential operator  ${\\mathcal D}$ on a Banach algebra and\nuse it to define a differential $*$-algebra  ${\\mathcal A}$ of a symmetric $*$-algebra  ${\\mathcal B}$, which\n  admits  norm-controlled inversion with\n the norm control function\n bounded by a polynomial.  However, the differential algebra in \\cite{gkII} does not include\n the  Gr\\\"ochenig-Schur algebras ${\\mathcal A}_{p, \\alpha}$,\n the Baskakov-Gohberg-Sj\\\"ostrand  algebra ${\\mathcal C}_{p, \\alpha}$\n  and the  Beurling algebra  ${\\mathcal B}_{p, \\alpha}$ with $1\\le p<\\infty$ and $\\alpha>1-1\/p$.\n   It could be an interesting problem to  extend the conclusions in Theorem \\ref{polynomial.thm}\n   to general Banach algebras such that\n the  norm control functions in the norm-controlled inversion have polynomial growth.\n}\n\\end{rem}\n\n\n\n\n\\begin{rem} {\\rm  A crucial step in the\nproof of Theorem \\ref{main-thm1} is to introduce\n  $B:=I- \\|A^* A\\|_{\\mathcal B}^{-1} A^* A\\in {\\mathcal A}$, whose spectrum is contained in an interval\n   on the positive real axis.\n   The above reduction  depends on the requirements that\n   ${\\mathcal B}$ is symmetric and both ${\\mathcal A}$ and ${\\mathcal B}$ are $*$-algebras with common identity and involution.\nFor the applications to some mathematical and\nengineering fields, the  widely-used algebras ${\\mathcal B}$ of infinite matrices and integral operators\nare  the operator algebras ${\\mathcal B}(\\ell^p)$ and  ${\\mathcal B}(L^p), 1\\le p\\le \\infty$, which are symmetric only when $p=2$.\nIn \\cite{akramjfa09,  fangshinsun13, shinsun19, shincjfa09,  sunacha08, tesserajfa10}, inverse-closedness of localized matrices and integral operators\nin ${\\mathcal B}(\\ell^p)$ and ${\\mathcal B}(L^p), 1\\le p\\le \\infty$, are discussed, and in \\cite{fangshinsun20},\nBeurling algebras  ${\\mathcal B}_{p,\\alpha}$ with $1\\le p<\\infty$ and $\\alpha>d(1-1\/p)$ are shown to admit polynomial norm-controlled\ninversion in nonsymmetric algebras ${\\mathcal B}(\\ell^p), 1\\le p<\\infty$.\nIt is  still widely  open to discuss  Wiener's lemma and\n norm-controlled inversion when  ${\\mathcal B}$ and ${\\mathcal A}$ are not $*$-algebras and\n ${\\mathcal B}$ is not a symmetric  algebra.\n}\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn recent years, many applications related to stochastic differential equations (SDEs) with discontinuous drift coefficient\nhave emerged. These types of equations typically arise in mathematical finance and insurance \\cite{Asmussen.1997,Ichiba.2013,Ichiba.2011,Karatzas.2009},\nengineering applications~\\cite{LaBolle.2000,Olama2009}, economy~\\cite{Shardin.2016,Leobacher2015} or stochastic control problems \\cite{Shardin.2016,Balakrish1980,kushner,touzi}.\n\nThe existence and uniqueness of solutions of SDEs in the standard case, i.e.\\ the case of sufficiently smooth coefficients, is well understood~\\cite{ks}. However, the standard theory on SDEs does not apply anymore in case of a discontinuous drift coefficient, e.g.\\ a piecewise constant drift coefficient, and a special theory is \nneeded to address the question of existence and uniqueness of solutions of such SDEs~\\cite{Kleptsyna.1985,Veretennikov.1981,Zvonkin.1974}. \nThe same is true for the numerical analysis: The convergence behavior of approximation schemes needs to be\nreconsidered and ``research on numerical methods for SDEs with irregular coefficients is highly active.''\\ (\\cite[p.\\ 2]{LSeuler}). In the case of a sufficiently smooth drift and a constant diffusion coefficient, the exact strong rate of convergence is 1 for the Euler scheme, see~\\cite{DGR2003,Kloeden.1999}.\nAt the time, when the main part of the  research presented here was  undertaken,  no comparable result was known in the case of a discontinuous, e.g.\\ piecewise constant, drift coefficient.\nAfter many discussions and investigations, also inspired by a previous version of this manuscript, refined results are now about to be established, see Section \\ref{subsec:equation}.\n\nIn this work, we will focus on numerical approximations of SDEs in the presence of a piecewise constant \ndrift and a constant diffusion coefficient.  We will provide theoretical considerations on the long time behavior of approximated SDE solutions based on results from the theory of ergodic Markov chains. Moreover, we will provide further insight into the numerical behavior of  approximation schemes, in particular the Euler scheme, by analyzing the numerical convergence rates based on a reference solution.\nThe numerical speed of convergence heavily depends on\nthe initial value and properties of the drift coefficient, e.g. drift direction or\njump height. Our tests reveal that for a special class of drift coefficients the numerical convergence rates are higher and independent of\ninitial conditions due to the ergodicity of the Euler scheme and the underlying SDE. We also use the Euler scheme to verify the long time behavior of a rank-based stock market model~\\cite{Banner.2005},\na prominent model in finance to describe the evolution of the capital distribution within the market. \n\nThe remainder of this manuscript is as follows: \nIn Section \\ref{sec:Stability}, we will introduce some theoretical and numerical basics\nand establish the ergodicity of the Euler approximations\nin the case of an  appropriate, piecewise constant drift coefficient.\nIn Section \\ref{sec:SimStudies}, we will discuss numerical convergence properties and further\nfindings of several numerical tests. \nWe will conclude this work in Section \\ref{sec:finance} with the  application from mathematical finance mentioned above, \nwhere SDEs of discontinuous type naturally arise. \n\n\n\\section{Problem Description} \\label{sec:Stability}\n\n\nIn this section, we will introduce our basic setting, i.e.\\ the type of SDE, we are interested in, and some basic terms for the numerical tests. Besides the Euler scheme and its long time properties in our setting, we will also briefly discuss the applicability and performance of some other numerical schemes. \n\n\\subsection{The Equation}  \\label{subsec:equation}\n\n\nIn this manuscript, we will consider time-homogeneous SDEs with piecewise constant drift coefficient and additive noise:\n\\begin{align} \\label{eq:SDE}\ndX_t = \\sum\\limits_{j=1}^{s} \\alpha_j \\cdot \\mathbbmss{1}_{B_j}(X_t)dt + \\sigma dW_t, \\quad t \\geq 0, \\qquad X_0= \\xi.\n\\end{align}\nHere, we have $s \\in \\mathbb{N}$, $\\alpha_j, \\sigma, \\xi \\in \\mathbb{R}$ and disjoint (possibly infinitely many) intervals $B_j \\subset \\mathbb{R}$   for all $1 \\leq j \\leq s$  and $\\left( W_t \\right)_{t \\in [0,T]}$ is a one-dimensional Brownian motion.\n\nThe existence and uniqueness of solutions to this type of SDEs is guaranteed by results of \\cite{Veretennikov.1981,Zvonkin.1974} and \\cite{Kleptsyna.1985}. In \\cite{Veretennikov.1981}, conditions on the drift and diffusion coefficient are derived under which the corresponding SDE has a unique strong solution. As emphasized therein, those conditions are in particular fulfilled for a bounded drift coefficient and a constant diffusion coefficient. Thus, the existence and uniqueness of a strong solution for SDEs of type \\eqref{eq:SDE} is ensured.\n\n\\medskip\n\nFor the numerical analysis of SDEs with discontinuous drift and\/or diffusion coefficient, the situation is more involved.\nIn this manuscript, we will focus on the strong convergence rate of the Euler scheme, which, for a general SDE\n$$ dX_t = f(X_t) dt + g(X_t) dW_t, \\qquad t \\in [0,T],  \\qquad X_0= \\xi,$$ where $f$ and $g$ are such that a unique strong solution exists, is given by\n\\begin{align} \\label{euler_alg}\n\tx_{k+1}^{\\text{expE}} = x_k^{\\text{expE}} + f(x_k^{\\text{expE}}) \\Delta + g(x_k^{\\text{expE}}) (W_{(k+1)\\Delta}-W_{k \\Delta}), \\quad k=0,\\ldots,n-1, \\qquad x_0^{\\text{expE}}= \\xi.\n\\end{align}\nThe underlying time discretization of the time interval $[0,T]$ is $0=t_0<t_1<\\cdots<t_n=T$ with corresponding step size $\\Delta\\mathrel{\\mathop:}= \\frac{T}{n}$, where $n+1$ is the number of grid points.\n\nWhile its behavior is well-known for SDEs with Lipschitz continuous coefficients $f$ and $g$, much less has been known in more general cases, even for SDEs with additive noise and a piecewise constant drift coefficient. The first contribution in this area is -- up to the best of our knowledge -- the work\n \\cite{Gyongy.1998}, where almost sure convergence of the Euler scheme has been established in the case of a one-sided Lipschitz drift coefficient,  a locally Lipschitz diffusion coefficient and the existence of a Lyapunov function for the SDE.\n The results of \\cite{Halidias.2008}  give strong  convergence of the Euler scheme for  SDEs with additive noise in the case of a discontinuous, but monotone drift coefficient, while \\cite{Simonsen2014} establishes\n the almost sure and strong convergence of the Euler scheme for SDEs with additive noise and drift of the form $f(x)=-\\textrm{sign}(x)$.\n Recent contributions with respect to strong approximations of SDEs with discontinuous drift coefficient are a series of articles by Ngo and Taguchi \\cite{NTbest,NTlast,NTmc} and  Leobacher and  Sz\\\"olgyeny \\cite{LSeuler,LSbit,LSaap}, respectively. Very recently M\\\"uller-Gronbach and Yaroslavtseva \\cite{MG-Y} established strong order $1\/2$  for \\eqref{euler_alg} in the case of a scalar equation with piecewise Lipschitz drift and non-additive noise and Neuenkirch et al.~obtained the same convergence order for an adaptive Euler scheme in the multi-dimensional case, see \\cite{NSS}.\nThe weak approximation of SDEs with discontinuous coefficients has been studied in \\cite{kohatsuhiga}, where an Euler-type scheme  based on an SDE with mollified drift coefficient is analyzed.\n\n\\medskip\n\nIn the case of SDE \\eqref{eq:SDE}, the latest result on the strong convergence rate of the Euler scheme\n \\begin{align} \\label{eq:euler}\nx_{k+1}^{\\text{expE}} = x_k^{\\text{expE}} + \\sum\\limits_{j=1}^{s} \\alpha_j \\cdot \\mathbbmss{1}_{B_j}(x_k^{\\text{expE}}) \\Delta + \\sigma (W_{(k+1)\\Delta}-W_{k \\Delta}), \\quad k = 0,\\ldots,n-1, \\quad x_0^{\\text{expE}}= \\xi,\n\\end{align}\nfor the approximation of $X_T$, i.e.\\ the solution at time $T$, is an\n    $L^2$-convergence order $3\/4-\\varepsilon$ in \\cite{SN} for arbitrarily small $\\varepsilon>0$.\n\nFor a better comparison, note that in the standard setting of an SDE with additive noise, where the drift coefficient is sufficiently smooth, the Euler scheme has an exact strong convergence order of $1$, see e.g. \\cite{DGR2003} and \\cite[p.\\ 350f]{Kloeden.1999}.\n\nSo to summarize: The Euler scheme for our non-standard setting of SDE \\eqref{eq:SDE} has at least $L^2$-convergence order $3\/4-\\varepsilon$. However, observing this convergence order numerically will be a different story (see Section \\ref{sec:SimStudies}).\n\n\\subsection{Simulation studies and empirical convergence rates}\nAs already mentioned, we are interested in empirically measuring the strong convergence rate of the Euler scheme. The standard procedure for this is as follows:  The root mean squared  error (RMSE) at time $T$ for the Euler scheme \\eqref{eq:SDE} with step size $\\Delta =T\/n$ is given by\n\\begin{align}\n\te(n) \\mathrel{\\mathop:}=  \\left( \\mathbb{E}\\left \\vert X_T - x_n^{\\text{expE}} \\right\\vert^2 \\right)^{1\/2}.\n\\end{align}\nSince an explicit form  of $X_T$ is unknown in general, one needs to replace $X_T$ in our simulation studies by\na numerical reference solution $X_T^{\\tt num}$, which is computed by the Euler scheme for an extremely small step size $\\Delta=T\/N$ with a very large number of $N+1$ grid points such that this approximation can be considered close enough to the true solution.\nMoreover, also the expectation $ \\mathbb{E} \\vert X_T^{ \\tt num} - x_n^{\\text{expE}}\\vert^2$ is not known explicitly, so we will approximate this expectation by the empirical RMSE\n\\begin{align}\ne_{\\tt emp}(n) = \\sqrt{\\frac{1}{M} \\sum\\limits_{i=1}^{M}\\left\\vert  \\left( X_T^{\\tt num} - x_n^{\\text{expE}}\\right)^{(i)}\\right\\vert^2},\n\\end{align}\nwith a large number $M$ of Monte Carlo repetitions, i.e.\\ $( X_T^{\\tt num} - x_n^{\\text{expE}})^{(i)}$, $i=1, \\ldots, M$, are iid copies of $X_T^{\\tt num} - x_n^{\\text{expE}}$.\nHere, $X_T^{\\tt num}$ and $x_n^{\\text{expE}}$ have the same random input.\nNote that $N$ has to be chosen sufficiently large to generate the numerical reference solution and to avoid oscillations  in $e_{\\tt emp}(n)$, which might occur if $N$ and $n$ are close. The number of repetitions $M$ should also be large enough to have a good approximation of the expectation, i.e.\\ a small Monte Carlo error.\n\n\n\\subsection{Other schemes}\n\n\nA natural idea is of course to consider other schemes than the explicit Euler scheme and to compare them in our simulation studies.\n\n\\subsubsection{The implicit Euler scheme}\n\nImplicit schemes have good stability properties, thus, they are a natural choice to consider. For an SDE with additive noise, where the drift coefficient is sufficiently smooth, the implicit Euler scheme has strong convergence order $1$ (see e.g. \\cite{Alfonsi.2013} and \\cite{Neuenkirch.2014}).\n\nHowever, for SDEs of type \\eqref{eq:SDE}, already the implicit Euler scheme is not well defined. To see this, consider the SDE\n \\begin{align*} dX_t=  \\left( \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (X_t)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (X_t)   \\right) dt +  \\sigma dW_t, \\quad t \\geq 0,  \\qquad X_0=\\xi, \\end{align*} \nwith $\\alpha_1 >0 > \\alpha_2$. The implicit Euler scheme \n$$ x_{k+1}^{\\text{impE}} = x_{k}^{\\text{impE}} +  \\left(\\alpha_1 \\cdot \\mathbbmss{1}_{(-\\infty,0)}(x_{k+1}^{\\text{impE}}) +   \\alpha_2 \\cdot \\mathbbmss{1}_{[0,\\infty)}(x_{k+1}^{\\text{impE}})\\right)\\Delta + \\sigma ( W_{(k+1)\\Delta} - W_{k \\Delta}), \\ k=0,\\ldots, n-1,$$\nrequires to solve, for fixed but arbitrary $z \\in \\mathbb{R}$,  the equation\n$$ y - \\left( \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (y)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (y)   \\right) \\Delta=  z,$$\nwith respect to $y \\in  \\mathbb{R}$. This equation  does not possess a solution if $ z \\in (-\\alpha_1 \\Delta, -\\alpha_2 \\Delta)$, and hence an implicit Euler scheme is not well defined in this setting.\n\n\\subsubsection{The Heun scheme}\n\nThe Heun scheme is another scheme with strong order one for SDEs with additive noise under appropriate smoothness conditions on the drift coefficient. Adapted from \\cite[p.\\ 373]{Kloeden.1999} for SDEs of type \\eqref{eq:SDE}, it is defined by\n\\begin{align*} \n\tx_{k+1}^{\\text{Heun}} &= x_{k}^{\\text{Heun}} + \\frac{1}{2}  \\left(\\sum\\limits_{j=1}^{s} \\alpha_j \\cdot \\mathbbmss{1}_{B_j}(x_{k}^{\\text{Heun}}) + \\sum\\limits_{j=1}^{s} \\alpha_j \\cdot \\mathbbmss{1}_{B_j}(\\Gamma_{k})\\right) \\Delta + \\sigma (W_{(k+1)\\Delta}-W_{k \\Delta}), \\\\\n\t\\Gamma_{k} &= x_{k}^{\\text{Heun}} + \\sum\\limits_{j=1}^{s} \\alpha_j \\cdot \\mathbbmss{1}_{B_j}(x_{k}^{\\text{Heun}}) \\Delta + \\sigma ( W_{(k+1)\\Delta} - W_{k \\Delta}), \\qquad k=0,\\ldots,n-1. \\notag\n\\end{align*}\n\nFor a closer look at the behaviour of this scheme at a discontinuity assume that the drift coefficient is given by  $a(x)=\\pm\\textrm{sign}(x)$. An increment of the  Heun scheme  with $x_{k}^{\\text{Heun}}=x$  is then given by\n\\begin{align*} \n\tx_{k+1}^{\\text{Heun}} - x =   \\frac{1}{2} \\big( a(x)  + a(x + a(x)\\Delta  + \\sigma   ( W_{(k+1)\\Delta} - W_{k \\Delta})) \\big) \\Delta \t\n\t+ \\sigma ( W_{(k+1)\\Delta} - W_{k \\Delta}) .\n\\end{align*}\nSo if no drift change occurs in the Euler step $x + a(x)\\Delta  + \\sigma ( W_{(k+1)\\Delta} - W_{k \\Delta})$,  a Heun step and an Euler step coincide. However, if a drift change occurs in the Euler step, the Heun step\nreads as\n$$ x_{k+1}^{\\text{Heun}} = x +    \\sigma ( W_{(k+1)\\Delta} - W_{k \\Delta}), $$\ni.e., it approximates the drift coefficient by zero and its dynamics are purely diffusion-based in this case. \n\n\\subsubsection{A Wagner-Platen type scheme}\n\nA strong order $1.5$-scheme for SDEs with smooth drift coeffi\\-cient and additive noise is given by a Wagner-Platen type scheme (see e.g. \\cite[p.\\ 383]{Kloeden.1999}), which \nreads  in our setting as\n\\begin{align*}\n\tx_{k+1}^{\\text{Pla}} = x_{k}^{\\text{Pla}} & + a_{k} \\Delta + \\sigma  ( W_{(k+1)\\Delta} - W_{k \\Delta}) \\\\ &\n\t+ \\frac{1}{4}  \\left(a_{k}^{+} - 2a_k + a_{k}^{-}\\right) \\Delta \n\t+ \\frac{1}{2\\sqrt{\\Delta}} \\left(a_{k}^{+} - a_{k}^{-}\\right) \\int_{k \\Delta}^{(k+1)\\Delta} (W_u-W_{k \\Delta}) du,\n\\end{align*}\nwith\n$$ \n\\Gamma_{k}^{\\pm} =x_{k}^{\\text{Pla}} + a_k \\Delta \\pm \\sigma \\sqrt{\\Delta}, \\quad \n\ta_k =a(x_{k}^{\\text{Pla}}), \\quad\n\ta_k^{\\pm} = a(\\Gamma_{k}^{\\pm}), \\quad \\text{with} \\ a(x) = \\sum\\limits_{j=1}^{s} \\alpha_j \\cdot \\mathbbmss{1}_{B_j}(x).$$\n\t\nNow, we look again at the case of a drift coefficient  given by  $a(x)=\\pm\\textrm{sign}(x)$ and stepsize $\\Delta < \\sigma^2$. \nFor a Wagner-Platen step with  $x_{k}^{\\text{Pla}}=x$, it  depends  now on whether\n$$  x + a(x) \\Delta + \\sigma \\sqrt{\\Delta}, \\quad x, \\quad x + a(x) \\Delta - \\sigma \\sqrt{\\Delta} $$\nhave the same sign or not. \nIf this condition is fulfilled, i.e., if $x$ is sufficiently far away from the discontinuity, then, a Wagner-Platen step and an Euler step coincide.\nIf the latter condition is not satisfied, then we have the dynamics\n\\begin{align*}\n\tx_{k+1}^{\\text{Pla}} = x &+  \\frac{1}{2} a(x) \\Delta + \\sigma   ( W_{(k+1)\\Delta} - W_{k \\Delta}) \\notag \\\\\n\t&+ \\frac{1}{2\\sqrt{\\Delta}} \\left(a\\big(x+a(x) \\Delta + \\sigma \\sqrt{\\Delta}\\big)  - a\\big(x+a(x) \\Delta - \\sigma \\sqrt{\\Delta}\\big)  \\right) \\int_{k \\Delta}^{(k+1)\\Delta} (W_u-W_{k \\Delta}) du.\n\\end{align*}\nSo also here, the diffusive dynamic dominates the scheme when taking values close to the discontinuity. \n\n\n\n\\subsection{Ergodicity and stability of the Euler scheme} \\label{sec:ergod}\n\n\nWe will now address the long time properties of the Euler scheme based on results from the theory of ergodic Markov chains. For simplicity, we consider here a special case of SDE \\eqref{eq:SDE}, namely\n\\begin{align*} dX_t=  \\left( \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (X_t)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (X_t)   \\right) dt + dW_t, \\quad t,s \\geq 0,  \\qquad X_0=\\xi, \\end{align*}\nand assume that $$\\alpha_1 >0 > \\alpha_2,$$ i.e.\\ a  drift coefficient, which is pointing towards zero.\nClearly, we have\n\\begin{align} \\lim_{s \\rightarrow 0} \\mathbb{E}(X_{t+s}|X_t=x) = x +  \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (x)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (x), \\qquad t \\geq 0, \\,\\, x \\neq 0, \\label{ce_cont} \\end{align}\ni.e. on average, the solution is moving inwards. Moreover, following e.g.\\ chapter 6 in \\cite{gs}, this SDE admits a unique invariant distribution with Lebesgue density\n$$ \\varphi_{\\infty}(x)= c \\cdot e^{ 2 \\alpha_2 x} \\cdot \\mathbbmss{1}_{[0,\\infty)} (x)  + c \\cdot e^{ 2 \\alpha_1 x}  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (x), \\qquad x \\in \\mathbb{R},$$\nwhere the normalizing constant $c>0$ is such that $\\int_{-\\infty}^{\\infty} \\varphi_{\\infty}(x) dx =1$.\nIn particular, we have\nthat \\begin{align}\\label{conv_cont}\n \\lim_{t \\rightarrow \\infty} {\\mathbb P}(X_t \\leq y) = \\int_{-\\infty}^y \\varphi_{\\infty}(z) dz, \\qquad y \\in \\mathbb{R}, \\end{align}\nand the law of large numbers\n\\begin{align} \\lim_{L \\rightarrow \\infty} \\frac{1}{L} \\int_0^L h(X_t) dt =  \\int_{-\\infty}^{\\infty} h(x) \\varphi_{\\infty}(x) dx  \\qquad \\textrm{a.s.,} \\label{ln}\t\\end{align}\nholds, if $h: \\mathbb{R} \\rightarrow \\mathbb{R}$ is measurable and satisfies $ \\int_{-\\infty}^{\\infty} |h(x)| \\varphi_{\\infty}(x) dx < \\infty$.\n\n\n\\smallskip\n\nIt will turn out that the explicit Euler scheme\n\\begin{align}\\label{e_disc} x_{k+1}^{\\text{expE}, \\xi}= x_k^{\\text{expE}, \\xi} + a(x_k^{\\text{expE}, \\xi}) \\Delta + W_{(k+1)\\Delta}-W_{k \\Delta}, \\quad k=0,1,\\ldots,  \\qquad x_0^{\\text{expE}, \\xi}= \\xi,\\end{align}\nwith\n$$ a(x) = \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (x)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (x), \\qquad x \\in \\mathbb{R},$$ will recover these properties. (Here we also indicate the dependence on the initial value $\\xi$ in our notation.) The Euler scheme \\eqref{e_disc}\ncorresponds to a time homogenous Markov chain with transition kernel\n$$ p_{\\Delta}(x,A) = \\int_A \\frac{1}{\\sqrt{2 \\pi \\Delta}} \\exp\\left( -\\frac{1}{2\\Delta}\\big(y-(x+a(x)\\Delta \\big)^2\\right) dy, \\qquad x \\in \\mathbb{R}, \\quad A \\in \\mathcal{B}(\\mathbb{R}),$$\nand satisfies the discrete counterpart to \\eqref{ce_cont}, i.e.\\\n\\begin{align} \\mathbb{E}( x_{k+1}^{\\text{expE}, \\xi}| x_{k}^{\\text{expE}, \\xi}=x) = x +  a(x)\\Delta, \\qquad k=0,1,\\ldots, \\qquad x \\in \\mathbb{R}. \\label{ce_disc} \\end{align}\n\nNow, we will prove the existence of a unique stationary distribution for the Euler scheme. In particular, due to the discontinuity at zero, the following Proposition  \\ref{ergod_prop} is not covered by the standard references as e.g.\\ \\cite{msh2002} and \\cite{gr1996} for Euler-type discretizations of ergodic SDEs.\nNote that the long time properties of \\eqref{e_disc} have also been heuristically studied in \\cite{Simonsen2013}.\n\nHowever, we can easily verify that $V(x)=e^{ \\tau |x|}$, $x \\in \\mathbb{R}$,  is an appropriate Lyapunov function for the above Markov chain, if $\\tau>0$ is sufficiently small.\nThis is a direct consequence of the well known form of the moment generating function for the folded normal distribution, i.e.\\ \n\\begin{align}\\label{folded} {\\mathbb E} e^{\\tau | \\mu + \\nu W_1|}=e^{\\frac{\\nu^2 \\tau^2}{2}+\\mu \\tau}\\left[1-\\Phi\\left(-\\mu\/\\nu-\\nu \\tau \\right) \\right]+\ne^{\\frac{\\nu^2 \\tau^2}{2}-\\mu \\tau}\\left[1-\\Phi\\left(\\mu\/\\nu-\\nu \\tau \\right) \\right], \\qquad \\tau \\in \\mathbb{R}, \\end{align}\nwhere $\\Phi$ is the distribution function of the standard normal distribution and $\\mu \\in \\mathbb{R}$, $\\nu >0$.\nUsing \\eqref{folded}  with $\\mu=x+a(x)\\Delta$ and $\\nu^2=\\Delta$, we obtain\n\\begin{align*} {\\mathbb{E}}\\big( V (x_{k+1}^{\\text{expE}, \\xi}) | x_k^{\\text{expE}, \\xi} = x \\big) & \\leq e^{ \\Delta \\tau\\left(\\frac{\\tau}{2}  + |\\alpha_2|  \\right)} +  e^{\\Delta \\tau \\left( \\frac{\\tau}{2} - |\\alpha_2| \\right) }  e^{ \\tau x}, \\,\\,\\,\\, \\qquad x \\geq 0, \\\\\n{\\mathbb{E}}\\big( V (x_{k+1}^{\\text{expE}, \\xi}) | x_k^{\\text{expE}, \\xi} = x \\big) & \\leq e^{ \\Delta \\tau\\left(\\frac{\\tau}{2}  + |\\alpha_1| \\right)} +  e^{\\Delta \\tau \\left( \\frac{\\tau}{2} - |\\alpha_1|\\right) }  e^{-\\tau x}, \\qquad x < 0.\n\\end{align*}\nSo, we have\n\\begin{align*} {\\mathbb{E}}\\big( V (x_{k+1}^{\\text{expE}, \\xi}) | x_k^{\\text{expE}, \\xi} = x \\big) & \\leq e^{ \\Delta \\tau\\left(\\frac{\\tau}{2}  + \\max \\{ |\\alpha_1|,  |\\alpha_2|  \\}\\right)} +  e^{\\Delta \\tau \\left( \\frac{\\tau}{2} - \\min\\{ |\\alpha_1|, |\\alpha_2| \\} \\right) }  e^{ \\tau |x|}, \\qquad x \\in \\mathbb{R},\n\\end{align*}\nand choosing $\\tau < 2\\min\\{ |\\alpha_1|, |\\alpha_2| \\}$ gives the desired property \n$$ {\\mathbb{E}}\\big( V (x_{k+1}^{\\text{expE}, \\xi}) | x_k^{\\text{expE}, \\xi} = x \\big)  \\leq C  +  \\gamma  V(x), \\qquad x \\in \\mathbb{R},\n$$\nwith $C>0$, $\\gamma \\in (0,1)$. Since the transition kernel is Gaussian, an application of the quantitative Harris Theorem (see e.g. chapter 15 in \\cite{Tweedie} or Theorem 3.15 (and the following example) in \\cite{eberle}) yields the following geometric ergodicity result:\n\n\n\\begin{proposition}\\label{ergod_prop}\nLet $\\alpha_1>0>\\alpha_2$ and $\\Delta>0$ be fixed. Then, the Euler scheme \\eqref{e_disc} admits a unique stationary distribution $\\mu_{\\Delta}$, which is independent of the initial value $\\xi$. Moreover, \nthere exist $\\beta_{\\Delta} \\in (0,1)$ and constants $\\mathcal{M}_{\\Delta}(\\xi), \\xi \\in \\mathbb{R},$ such that\n$$ \\sup_{A \\in \\mathcal{B}(\\mathbb{R})} \\left| \\mathbb{P}( x_{k}^{\\textrm{expE}, \\xi} \\in A)  - \\mu_{\\Delta}(A) \\right| \\leq \\mathcal{M}_{\\Delta}(\\xi) \\cdot \\beta_{\\Delta}^k, \\qquad k \\geq 1.$$\n\\end{proposition}\n\nChoosing $A=(-\\infty,y]$, we obtain in particular the counterpart to  \\eqref{conv_cont}, i.e.\\\n\\begin{align} \\label{conv_disc}\n \\lim_{k \\rightarrow \\infty} {\\mathbb{P}}(x_k^{\\text{expE}, \\xi} \\leq y) = \\mu_{\\Delta}((-\\infty,y]), \\qquad y \\in \\mathbb{R}. \n \\end{align}\n \n \nNote that the limit distribution is independent of the initial value, as for the underlying SDE. \n\n\\smallskip\n\nFinally, an ergodic Theorem as e.g. Corollary 2.5 in \\cite{eberle} yields also the discrete counterpart  to the law of large numbers \\eqref{ln}: We have\n\\begin{align} \\label{dln}\n\\lim_{L \\rightarrow \\infty}  \\frac{1}{L}\\sum_{k=1}^L h(x_k^{\\text{expE}, \\xi}) = \\int_{-\\infty}^{\\infty} h(x) \\mu_{\\Delta}(dx)\t\\qquad \\textrm{a.s.,}\n\\end{align}\n for all measurable $h: \\mathbb{R} \\rightarrow \\mathbb{R}$ such that $\\int_{-\\infty}^{\\infty} |h(x)|  \\mu_{\\Delta}(dx) < \\infty.$\n\n\n\\bigskip\n\n\n\\section{Simulation Studies} \\label{sec:SimStudies}\n\n\nThis section is concerned with the numerical investigation of SDEs of type \\eqref{eq:SDE}. \nFor the remainder, we will choose $T=1$, $M=10^5$, $N=2^{14}$ and $n=2^{\\tilde{n}}$ with $\\tilde{n} \\in \\{4,\\ldots,10\\}$ (unless otherwise mentioned).\nWe then calculate the corresponding Euler approximation and the empirical RMSE $e_{\\tt emp}(n)$. For simplicity, we omit the upper index of the numerical approximation indicating that the approximation is based on the Euler scheme.\nThe empirical convergence rate is given by the negative slope of the regression line, which we obtain when plotting $\\tilde{n}=\\log_2(n)$ versus $\\log_2\\left(e_{\\tt emp}(n)\\right)$.\nHere, we will focus on two types of drift coefficients: inward and outward pointing drift coefficients. \n\\begin{definition}\nWe will call a drift coefficient $a: \\mathbb{R} \\rightarrow \\mathbb{R}$ {\\it inward pointing}, if there exists $x^* \\in \\mathbb{R}$ such that\n$$ a(x)>0, \\quad x < x^*, \\qquad a(x)<0, \\quad x>x^*,$$\nand {\\it outward pointing}, if there exists $x^* \\in \\mathbb{R}$ such that\n$$ a(x)<0, \\quad x < x^*, \\qquad a(x)>0, \\quad x>x^*.$$\n\\end{definition}\n\nOur numerical investigations are based on several additional key characteristics:\nWe consider the average number of \\textit{drift changes}. As the Euler scheme for SDE \\eqref{eq:SDE} is exact up to the first drift change, another quantity of interest is the \\textit{number of paths with at least one drift change}.\nTo get further insight whether some paths are really far away from the true solution, we measure the \\textit{largest error} that occurs within the considered time interval (not necessarily in the end).\nBesides the error sizes themselves, it is interesting to see what proportion of errors at final time $T$ is large, medium, or small and how this \\textit{distribution of error sizes} depends on the step size.\nFurthermore, we analyze the \\textit{evolution of the error over time} for a fixed step size.\nTo underline the influence of the \\textit{drift direction} towards or away from the discontinuity, we generate plots of several solution sample paths. We will see that the observed empirical\\footnote{We use the expressions \"numerical\" and \"empirical\" rate (respectivley order) of convergence synonymously.} rates of convergence heavily depend on whether the drift coefficient is \\textit{inward or outward pointing}. Whereas for the latter one, there is a dependency on the \\textit{initial value} of the SDE, rates in case of an inward pointing drift coefficient seem to be independent of the initial value, corresponding to Proposition \\ref{ergod_prop}.\nIn addition, we analyze how the \\textit{jump height} (difference in drift values) influences the empirical convergence rate.\n\n\\medskip\n\nAs representatives of the class of SDEs \\eqref{eq:SDE}, we consider here the SDEs given in Table \\ref{tab:overviewSDEs}.\n\n\\begin{table}[htbp]\n\t\\begin{minipage}{\\textwidth}\n\t\t\\centering\n\t\t\\caption{Selection of analyzed SDEs}\n\t\t\t\t\\label{tab:overviewSDEs}\n\t\t\\begin{adjustbox}{max width=\\textwidth}\n\t\t\t\\begin{tabular}{ll}\n\t\t\t\t\\textbf{Drift coefficient} & \\textbf{Corresponding SDE} \\\\\n\t\t\t\t\\hline sign & $dX_t = \\sign(X_t)dt + dW_t$ \\\\ \n\t\t\t\t\\hline minusSign & $dX_t = -\\sign(X_t)dt + dW_t$ \\\\\n\t\t\t\t\\hline 10sign & $dX_t = 10 \\cdot \\sign(X_t)dt + dW_t$ \\\\ \n\t\t\t\t\\hline minus10sign & $dX_t = -10 \\cdot \\sign(X_t)dt + dW_t$ \\\\ \n\t\t\t\t\\hline elementary\\_minus34 & $dX_t = \\left(-3\\cdot \\mathbbmss{1}_{(-\\infty,1.4)}(X_t) + 4\\cdot \\mathbbmss{1}_{[1.4,\\infty)}(X_t)\\right)dt + dW_t$ \\\\\n\t\t\t\t\\hline elementary4minus3 & $dX_t = \\left(4\\cdot \\mathbbmss{1}_{(-\\infty,1.4)}(X_t) - 3\\cdot \\mathbbmss{1}_{[1.4,\\infty)}(X_t)\\right)dt + dW_t$ \\\\\n\t\t\t\t\\hline elementary\\_minus0.6\\_1 & $dX_t = \\left(-0.6\\cdot \\mathbbmss{1}_{(-\\infty,1.4)}(X_t) + \\mathbbmss{1}_{[1.4,\\infty)}(X_t)\\right)dt + dW_t$ \\\\\n\t\t\t\t\\hline elementary1minus0.6 & $dX_t = \\left(\\mathbbmss{1}_{(-\\infty,1.4)}(X_t) - 0.6\\cdot \\mathbbmss{1}_{[1.4,\\infty)}(X_t)\\right)dt + dW_t$ \\\\\n\t\t\t\t\\hline \n\t\t\t\\end{tabular}\n\t\t\n\t\t\\end{adjustbox}\n\t\n\t\n\t\n\t\n\t\\end{minipage}\n\t\\label{tab:OverviewAnalysedSDEs}\n\\end{table}\n\nIn the remainder of this chapter, we will present and discuss some key results of the simulation studies.\n\n\n\\subsection{Key results} \\label{subsec:keyresults}\n\n\nThe  empirical convergence rates obtained by the Euler scheme are given in Table \\ref{tab:EulerMSQRates-step4-finePart} (outward pointing drift coefficients highlighted in light gray, the discontinuity in gray): \n\\begin{table}[htbp]\n\t\\begin{minipage}{\\textwidth}\n\t\\setcounter{mpfootnote}{\\value{footnote}}\n\t\\renewcommand{\\thempfootnote}{\\arabic{mpfootnote}}\n\t\\centering\n\t\\caption{Numerical Euler convergence rates}\n\t\\label{tab:EulerMSQRates-step4-finePart}%\n\t\\begin{tabular}{rcccccc}\n\t\t\\toprule\n\t\t\\textbf{Initial values} & \\textbf{-1} & \\cellcolor{lightgray} \\textbf{0} & \\textbf{1} & \\textbf{2.5} & \\textbf{3} & \\textbf{5} \\\\\n\t\t\\midrule\n\t\t\\rowcolor{verylightgray} sign  & 0.69 & \\cellcolor{lightgray} 0.59 & 0.68 & 0.83 & 1.01 & \\textbf{--} \\\\\n\t\t\\rowcolor{verylightgray} 10sign & \\textbf{--}\\footnotemark[1] & \\cellcolor{lightgray} 0.25  & \\textbf{--} & \\textbf{--} & \\textbf{--} & \\textbf{--} \\\\\n\t\tminusSign & 0.81 & \\cellcolor{lightgray} 0.80 & 0.81 & 0.82 & 0.82 & 0.89 \\\\\n\t\tminus10sign & 0.91  & \\cellcolor{lightgray} 0.91  & 0.91  & 0.91  & 0.91  & 0.91 \\\\\n\t\t\\midrule\n\t\t\\textbf{Initial values} & \\textbf{0} & \\textbf{1} & \\textbf{1.2} & \\textbf{1.25} & \\cellcolor{lightgray} \\textbf{1.4} & \\textbf{2} \\\\\n\t\t\\midrule\n\t\t\\rowcolor{verylightgray} elementary\\_minus34 & 1.17 & 0.37 & 0.38 & 0.40 & \\cellcolor{lightgray} 0.39 & 0.31 \\\\\n\t\t\\rowcolor{verylightgray} elementary\\_minus0.6\\_1 & 0.75  & 0.69  & 0.69  & 0.69  & \\cellcolor{lightgray} 0.71  & 0.70 \\\\\n\t\telementary4minus3 & 0.87 & 0.87 & 0.87 & 0.87 & \\cellcolor{lightgray} 0.87 & 0.87 \\\\\n\t\telementary1minus0.6 & 0.81  & 0.80  & 0.80  & 0.80  & \\cellcolor{lightgray} 0.80  & 0.80 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t\\footnotetext[1]{errors close to machine accuracy; no empirical convergence rate calculated (see also equation \\eqref{exit_prob_2})}\n\t\\setcounter{footnote}{\\value{mpfootnote}}\n\t\\end{minipage}\n\\end{table}%\n\nOur results show that \n\\begin{itemize}\n \\item in general, we loose convergence order one, which the Euler scheme has under standard assumptions for SDEs with additive noise \n \\item and that a crucial factor is whether the drift coefficient is inward or outward pointing: for inward pointing coefficients the guaranteed convergence order $3\/4$ is recovered, which is not always the case for outward pointing coefficients.\n\\end{itemize}\n\\medskip\n\nFurthermore, our numerical tests show that neither using the Heun scheme  nor using the Platen scheme yields a different picture. In particular, convergence rates do not improve significantly, and the schemes do not yield a better resolution of the discontinuity (see Tables \\ref{tab:HeunMSQRates-step4-finePart} and \\ref{tab:PlatenMSQRates-step4-finePart}).\n\n\\begin{table}[htbp]\n\t\\centering\n\t\\caption{Numerical Heun convergence rates, step size $2^{-4}$ onwards}\n\t\\begin{tabular}{rrrrrrr}\n\t\t\\toprule\n\t\t\\textbf{Initial values} & \\textbf{0} & \\textbf{1} & \\textbf{1.2} & \\textbf{1.25} & \\textbf{1.4} & \\textbf{2} \\\\\n\t\t\\midrule\n\t\telementary\\_minus34 & 1.15 & 0.42 & 0.38 & 0.38 & 0.41 & 0.40 \\\\\n\t\telementary4minus3 & 0.77 & 0.77 & 0.77 & 0.77  & 0.77 & 0.77 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t\\label{tab:HeunMSQRates-step4-finePart}%\n\\end{table}%\n\\begin{table}[htbp]\n\t\\centering\n\t\\caption{Numerical Platen convergence rates, step size $2^{-4}$ onwards}\n\t\\begin{tabular}{rcccccccc}\n\t\t\\toprule\n\t\t\\textbf{Initial values} & \\textbf{0} & \\textbf{1} & \\textbf{1.2} & \\textbf{1.25} & \\textbf{1.4} & \\textbf{2} \\\\\n\t\t\\midrule\n\t\telementary\\_minus34 & 1.22 & 0.40 & 0.40 & 0.40 & 0.42 & 0.43 \\\\\n\t\telementary4minus3 & 0.79 & 0.79 & 0.79 & 0.79 & 0.79 & 0.79 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t\\label{tab:PlatenMSQRates-step4-finePart}%\n\\end{table}%\n\n\n\\subsection{Drift direction and initial value} \\label{subsec:InwardOutward}\n\n\nFor an outward pointing drift coefficient, the numerical convergence order even seems to depend on the initial value and the spectrum of orders obtained for different initial values is very broad with values between $0.25$ and $1.17$ (see Table \\ref{tab:EulerMSQRates-step4-finePart}). \n\nOn the other hand, for an inward pointing drift coefficient, the convergence order seems to be independent of the initial value and the spectrum of orders numerically obtained for different initial values and inward pointing drift coefficients is tight with values between $0.80$ and $0.91$ (see Table \\ref{tab:EulerMSQRates-step4-finePart}).\nThe stability of the estimates is due to the ergodicity of the SDE and the Euler scheme in this case, see Subsection \\ref{sec:ergod}.\nThe geometric convergence speed in  Proposition \\ref{ergod_prop}  explains \nwhy the numerical tests for inward pointing drift coefficients yield such stable estimates, independently of the initial value: $X_T^{\\tt num}$ and $x_n$ are, for a sufficiently large number of grid points $n+1$, close to their unique\nstationary distributions, which stabilizes the Monte-Carlo estimates. Also, as pointed out already above, the guaranteed convergence order $3\/4$ is recovered here.\n\nFor the above equations, the structure of the drift coefficient is directly related to the number of drift changes. An inward pointing drift coefficient results in many drift changes, while in the case of an outward pointing drift coefficient, only few drift changes occur. We can further observe that:\n\\begin{itemize}\n \\item[(i)] when starting away from the discontinuity, numerical rates for outward pointing drift coefficients are better than for inward ones;\n \\item[(ii)] when starting close to the discontinuity, outward pointing drift coefficients imply worse numerical convergence rates than inward ones.\n \\end{itemize}\nSo, in the latter case we obtain a positive correlation between the number of drift changes and the numerical convergence rate, which implies that frequent drift changes are not necessarily bad for the quality of the approximation -- quite the contrary seems to apply, which is surprising\nat first glance.\n\n\\medskip\n\nHence, the type of monotonicity of the drift coefficient is of great importance.\nIntuitively, an inward pointing drift coefficient should lead to many drift changes, which suggests that individual drift changes are not of great importance.\nAn outward pointing drift coefficient on the other hand pushes the solution away from the discontinuity implying a low number of drift changes.\n\n\n\\subsection{Jump height}\n\n\nThe intensity of the effects related to inward and outward pointing drift coefficients depends on the jump height, i.e.\\ the distance between assigned drift values. In case of elementary\\_minus34, this distance amounts to 7 whereas it is 1.6 in case of elementary\\_minus0.6\\_1. The empirical convergence rates in Table \\ref{tab:EulerMSQRates-step4-finePart} show: The higher the jump height, the more pronounced are the effects described in Subsection \\ref{subsec:InwardOutward}. Exemplary, there is a difference of $0.8$ in the empirical convergence rates for elementary\\_minus34 for initial values $0$ and $1$ whereas this diffe\\-rence is only $0.06$ for elementary\\_minus0.6\\_1.\nThis phenomenon is related to a scaling property. By enlarging the drift value, the influence of the diffusive part of the SDE is weakened:\nConsider e.g. the SDE\n$$ dX_t = \\alpha \\sign(X_t)dt + dW_t \\label{eq:SDE X_t}$$\nwith $\\alpha \\geq 1$. Using the new variable $Y_t= \\frac{1}{\\alpha} X_t$ we have the dynamics\n$$ dY_t=  \\sign(Y_t)dt + \\frac{1}{\\alpha} dW_t,   $$\nwith a reduced diffusion  coefficient.\n\n\n\n \n\\subsection{Case study of an inward versus outward pointing drift coefficient}\n\n\n\nIn this Subsection, we will analyze the pattern described in \\ref{subsec:InwardOutward} in more detail, exemplary for the drift coefficients elementary4minus3 and elementary\\_minus34.\n\n\\subsubsection{Drift changes}\nFigure \\ref{fig:avNumDriftChanges_ele4minus3ANDele_minus34_start1Komma4} shows the average number of drift changes for both coefficients.  The behavior goes along with the intuitive understanding described above.\nHere  $\\tilde{n}$ is the exponent of the dyadic step size $\\Delta=2^{-\\tilde{n}}$. Note that for step sizes $2^{-4}$ to $2^{-8}$ and elementary\\_minus34 the number of drift changes stays below $2$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{avNumDriftChanges_ele4minus3ANDele_minus34_start1Komma4.pdf}\n\t\\caption{Average number of drift changes for $\\xi=1.4$: elementary4minus3 vs. elementary\\_minus34}\n\t\\label{fig:avNumDriftChanges_ele4minus3ANDele_minus34_start1Komma4}\n\\end{figure}\n\n\\subsubsection{Comparison of solution sample paths}\n\nFigure  \\ref{fig:samplePaths_ele4minus3VSele_minus34_start1Komma4} shows 100 sample paths of the numerical reference solution ($\\Delta=2^{-14}$). The black line represents the discontinuity in the drift coefficient.\n\n\\begin{figure}[h!]\n\t\\subfigure[\\ elementary4minus3, $\\xi=1.4$]{\\includegraphics[width=0.49\\textwidth]{samplePaths_elementary4minus3_start1Komma4_paths1-100_n11.png}}\\hfill\n\t\\subfigure[\\ elementary\\_minus34, $\\xi=1.4$]{\\includegraphics[width=0.49\\textwidth]{samplePaths_elementary_minus34_start1Komma4_paths1-100_n11.png}}\n\n\t\\caption{Comparison of solution paths: elementary4minus3 vs. elementary\\_minus34}\n\t\n\t\\label{fig:samplePaths_ele4minus3VSele_minus34_start1Komma4}\n\\end{figure}\n\n\\smallskip\n\nIn the situation of Figure \\ref{fig:samplePaths_ele4minus3VSele_minus34_start1Komma4}(b), where the solution drifts away from the discontinuity, it is of tremendous importance whether a drift change is captured by the approximation or not: the solution does not stay close to the discontinuity and thus, there are not many chances for a drift correction to take place, see Figure \\ref{fig:importance_captureDriftChange_ele_minus34_1_sample9999}.\nFor the SDE\n\\begin{align} dX_t=  \\left( \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (X_t)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (X_t)   \\right) dt + dW_t, \\quad t \\geq 0,  \\qquad X_0=\\xi, \\label{test_illustrate} \\end{align}\nwith $\\alpha_1<0<\\alpha_2$ and $\\xi >0$ the conditional probability $p(\\xi,\\theta,\\Delta)$ that the exact solution changes its drift over $[0,\\Delta]$ given that the approximation $x_1$ at $t=\\Delta$ has value $\\theta \\geq 0$ (and thus has not changed its drift) satisfies\n\\begin{align} p(\\xi,\\theta,\\Delta):=\\mathbb{P}\\Big{(} \\inf_{t \\in [0,\\Delta]}X_t<0 \\Big{|} X_0=\\xi, x_1 = \\theta \\Big{)}= \\exp \\left( - 2 \\frac{\\xi \\theta}{\\Delta} \\right), \\end{align}\nsee e.g. \\cite{Gobet}, page 169. So the (conditional) probability of missing drift changes is not negligible and even close to one for small $\\xi$  or $\\theta$.\n\n\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{importance_captureDriftChange_ele_minus34_1_sample9999.pdf}\n\t\\caption{Importance of capturing the drift changes for elementary\\_minus34, $\\xi=1$}\n\t\\label{fig:importance_captureDriftChange_ele_minus34_1_sample9999}\n\\end{figure}\n\n\n\\subsubsection{Largest error}\n\nThe latter observation is also reflected in the largest distance for $10^4$ sample paths between the approximation based on step size $2^{-10}$ and the numerical reference solution, see Table \\ref{tab:errorSizes_MSQ_Euler}.\nThe largest distances amount to 1.271 for elementary4minus3 and 4.508 for elementary\\_minus34. \n\\begin{table}[htbp]\n\t\\centering\n\t\\caption{Largest and smallest Euler errors}\n\t\\begin{tabular}{rrrrrrrrrr}\n\t\t\\toprule\n\t\t\\textbf{Initial values} & \\textbf{} & \\textbf{0} & \\textbf{1} & \\textbf{1.2} & \\textbf{1.25} & \\textbf{1.4} & \\textbf{2} \\\\\n\t\t\\midrule\n\t\t\\textbf{elementary\\_minus34} & max   & 0.045 & 1.331 & 2.614 & 3.087 & 4.508 & 0.335 \\\\\n\t\t& min   & 0.0002 & 0.179 & 0.331 & 0.383 & 0.696 & 0.059 \\\\\n\t\t\\textbf{elementary4minus3} & max   & 1.223 & 0.934 & 0.968 & 0.981 & 1.013 & 1.271 \\\\\n\t\t& min   & 0.005 & 0.005 & 0.005 & 0.005 & 0.005 & 0.005 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t\\label{tab:errorSizes_MSQ_Euler}%\n\\end{table}%\n\n\n\n\\subsubsection{Evolution of the error over time}\n\nTo gain even more insight, we compare the empirical RMSE for increasing time $t$ \nof elementary4minus3 and elementary\\_minus34 when starting in the discontinuity $\\xi=1.4$ for step sizes $2^{-4}$, $2^{-8}$ and $2^{-10}$ by plotting the base-2 logarithm of the RMSE against the time (see Figure \\ref{fig:errorOverTime_4_8_10_ele4minus3ANDele_minus34_start1Komma4}).\nWe have added in these figures the following additional information: If the number is not zero, the most frequent times of drift changes corresponding to the chosen step size are indicated. The number of plotted drift change times is based on the average number of drift changes over the simulated sample paths.\n\nFurthermore, if in the corresponding cases drift changes occur, we add the very first drift change (of all simulated paths) of the numerical reference solution and the Euler schemes. They are generated by finding the time at which the first drift change occurs for $10^4$ saved paths and then taking the minimum over all that times. The time is registered as the point of discretization at which a drift change that took place was detected. The very first drift change of the reference solution is marked at a height of zero for a better distinguishability. \nRMSE over time and drift change times are calculated on a basis of $10^{4}$ simulation paths.\n\n\n\\begin{figure}[htbp]\n\t\\subfigure[\\ elementary4minus3, $\\tilde{n}=4$]{\\includegraphics[width=0.45\\textwidth]{rootMSQErrorOverTime_10power5_step4_elementary4minus3_1Komma4.pdf}}\n\t\\subfigure[\\ elementary\\_minus34, $\\tilde{n}=4$]{\\includegraphics[width=0.45\\textwidth]{rootMSQErrorOverTime_10power5_step4_elementary_minus34_1Komma4.pdf}}\n\t\\subfigure[\\ elementary4minus3, $\\tilde{n}=8$]{\\includegraphics[width=0.45\\textwidth]{rootMSQErrorOverTime_10power5_step8_elementary4minus3_1Komma4.pdf}}\n\t\\subfigure[\\ elementary\\_minus34, $\\tilde{n}=8$]{\\includegraphics[width=0.45\\textwidth]{rootMSQErrorOverTime_10power5_step8_elementary_minus34_1Komma4.pdf}}\n\t\\subfigure[\\ elementary4minus3, $\\tilde{n}=10$]{\\includegraphics[width=0.45\\textwidth]{rootMSQErrorOverTime_10power5_step10_elementary4minus3_1Komma4.pdf}}\n\t\\subfigure[\\ elementary\\_minus34, $\\tilde{n}=10$]{\\includegraphics[width=0.45\\textwidth]{rootMSQErrorOverTime_10power5_step10_elementary_minus34_1Komma4.pdf}}\n\t\n\t\\caption{Comparison of the error evolution over time for $\\xi=1.4$ for different step sizes: elementary4minus3 vs. elementary\\_minus34}\n\t\n\t\\label{fig:errorOverTime_4_8_10_ele4minus3ANDele_minus34_start1Komma4}\n\\end{figure}\t\nWe can extract from Figure \\ref{fig:errorOverTime_4_8_10_ele4minus3ANDele_minus34_start1Komma4} at least two features:\n\\begin{itemize}\n \\item[(i)] The error stays constant or even decreases over time for elementary4minus3 -- in contrast to a strong error accumulation over time for elementary\\_minus34. (Note that the ordinate has a base-2 log scale.)\n \\item[(ii)] In the inward pointing drift coefficient case, the error is by several magnitudes smaller than for an outward pointing drift coefficient. \n\\end{itemize}\n\n\nThis illustrates again the  stabilizing effect of an inward pointing drift coefficient and the importance of capturing the first drift changes correctly in case of an outward pointing drift coefficient.\n\n\\subsubsection{Distribution of error sizes}\n\nBesides the empirical RMSE itself, the empirical distribution of the errors in $t=T$ is of interest. The error at final time $T$ is quantified by $\\left\\vert x_N - x_n\\right\\vert$ for step size $\\Delta=T\/n=2^{-\\tilde{n}}$. The histograms in Figure \\ref{fig:hist_10power5_ele4minus3_ele_minus34_n4810_1Komma4}  are based on $M=10^4$ simulations for different step sizes and highlight again the different magnitudes of the  empirical RMSE (abscissa with a base-$2$ logarithm scale).\nAnother feature, which we can extract from the histograms, is a non-negligible part of simulated paths with an error of machine accuracy size for elementary\\_minus34. We will discuss this feature in more detail in the next Subsection. \n\n\\begin{figure}[htbp]\n\t\\subfigure[\\ elementary4minus3]{\\includegraphics[width=0.49\\textwidth]{hist_10power5_elementary4minus3_n4810_1Komma4.pdf}}\\hfill\n\t\\subfigure[\\ elementary\\_minus34]{\\includegraphics[width=0.49\\textwidth]{hist_10power5_elementary_minus34_n4810_1Komma4.pdf}}\n\t\n\t\\caption{Distribution of the error at time $T$ for different step sizes for elementary4minus3 and elementary\\_minus34 with $\\xi=1.4$}\n\t\n\t\\label{fig:hist_10power5_ele4minus3_ele_minus34_n4810_1Komma4}\n\\end{figure}\n\n\\subsection{Rare events and goodness of the regression fit}\n\nIn case of an outward pointing drift coefficient the empirical RMSE and the linear regression estimates become unreliable or at least questionable.\n\n\nFor initial values close to the discontinuity the observed empirical convergence order are in some cases far away from the guarenteed $3\/4$, altough the linear regression typically produces stable results, see Figure \\ref{fig:PlatenEulerRates_elementary_minus34_part1}(b). A possible explanation for this are again the first drift changes. When starting close to the initial value, the first drift changes seem to be very sensitive to the step-size, which results in rather different trajectories of the Euler scheme.\n \nFurthermore, if the initial value is far away from the discontinuity, only very few drift changes occur in the underlying SDE (if at all). Hence, if the step size of the Euler scheme is significantly small, these changes are captured and the error drops drastically. Figure \\ref{fig:PlatenEulerRates_elementary_minus34_part1} illustrates this by comparing the regressions for an initial value $\\xi=0$ away from the discontinuity in $1.4$ and an initial value $\\xi=1$, which is closer to the discontinuity. (The regression has also to deal in Figure \\ref{fig:PlatenEulerRates_elementary_minus34_part1}(a) with two different regimes.)\nNote that the Euler scheme for \\eqref{eq:SDE} is always exact up to the time of the first drift change.\n\\begin{figure}[htbp]\n\t\\subfigure[\\ $\\xi=0$]{\\includegraphics[width=0.49\\textwidth]{RMSQerrorEnd_Euler_10power5_elementary_minus34_0.pdf}}\\hfill\n\t\\subfigure[\\ $\\xi=1$]{\\includegraphics[width=0.49\\textwidth]{RMSQerrorEnd_Euler_10power5_elementary_minus34_1.pdf}}\n\n\t\n\t\\caption{Euler rates of convergence for elementary\\_minus34 for $\\xi \\in \\{0,1\\}$}\n\t\n\t\\label{fig:PlatenEulerRates_elementary_minus34_part1}\n\\end{figure}\n\nMoreover, for an outward pointing drift coefficient, the Euler scheme and the exact solution coincide with high probability, which explains e.g. the errors close to machine accuracy for the drift coefficient $\\textrm{sign}$ and the initial value $\\xi=5$.\nNote that in this setting, the number of paths with at least one drift change is even zero over all saved $10^4$ solution paths.\n\nTo explain this phenomenon, consider again the SDE\n\\begin{align} dX_t=  \\left( \\alpha_1  \\cdot \\mathbbmss{1}_{(-\\infty,0)} (X_t)  + \\alpha_2  \\cdot \\mathbbmss{1}_{[0,\\infty)} (X_t)   \\right) dt + dW_t, \\quad t \\geq 0,  \\qquad X_0=\\xi, \\label{test_illustrate_2} \\end{align}\nwith $\\alpha_1<0<\\alpha_2$. An application of formula (5.13) in chapter 3.5.C in \\cite{ks} gives\n\\begin{align} \\label{exit_prob_2}\n\\mathbb{P} \\left( \\inf_{t \\geq 0} |X_t| >0 \\right) &= 1-e^{2 \\alpha_1 \\xi^{-} -2 \\alpha_2 \\xi^{+}}, \\qquad \\xi \\neq 0.\n\\end{align}\nNote that an initial value $\\xi \\neq 0$ is not a restriction as we analyze the case of an initial value far away from the discontinuity. So, for drift values $-\\alpha_1=\\alpha_2=1$, and an initial value $\\xi=5$, the Euler scheme is exact with a probability of at least $1-e^{-10}$ $\\approx 0.99995460 \\ldots $ \n\nTo summarize: Standard Monte Carlo simulations for testing convergence rates seem to be unreliable in the case of outward pointing coefficients. \nNo stable asymptotic regime seems to be reached by our estimators. Smaller stepsizes or a larger Monte-Carlo sample might be a remedy for this problem, similar to \\cite{arnulf} where moment explosions of the Euler scheme for SDEs with superlinear cofficients are observed in a numerically asymptotic setting. But this is beyond the scope of the present manuscript.\n\n\n\\section{The Euler scheme for the Atlas model}\\label{sec:finance}\n\n\nIn this section, we will use the Euler scheme to simulate the so-called Atlas model, which is a particular first-order market model \\cite{Banner.2005}. In such models, the asset dynamics depend\non the size (measured in terms of market capitalization) of the corresponding firm, which results in an SDE model with discontinuous coefficients.\n\n\\medskip\n\n\\subsection{First-order market models} \\label{subsec:FO-model}\nA \\textit{first-order model} \\cite{Banner.2005} is defined as follows:\nLet $\\gamma, g_1,...,g_d \\in \\mathbb{R}$ and $\\sigma_1,...,\\sigma_d \\in (0, \\infty)$ such that\n\\begin{align*}\n\tg_1<0, \\quad g_1+g_2<0, \\ldots,  \\quad g_1+\\cdots+g_{d-1}<0, \\quad g_1+\\cdots+g_d=0.\n\\end{align*}\nConsider now stocks for which the market capitalizations are given by $X_1,\\ldots,X_d$, where the index  $ i \\in \\{1,2, \\ldots, d \\}$ indicates the name of the firm, and that follow the dynamics\n\\begin{align} \\label{eq:FOmodel}\n\td\\log X_i(t) = \\gamma_i(t) dt + \\sigma_i(t) dW_i(t), \\quad t \\in [0,\\infty),\\qquad i=1,\\ldots,d.\n\\end{align}\nHere, $W_1,...,W_d$ are independent Brownian motions and the {growth rates} $\\gamma_i:[0,\\infty) \\rightarrow \\mathbb{R}$ and {volatilities} $\\sigma_i:[0,\\infty) \\rightarrow (0, \\infty)$ are given by\n\\begin{align} \\label{eq:FOmodelParameters}\n\t\\gamma_i(t) & = \\gamma + \\sum_{k=1}^{d} g_k \\mathbbmss{1}_{\\{r_i(t)=k\\}}, \\qquad \\quad \\sigma_i(t) = \\sum_{k=1}^{d} \\sigma_k \\mathbbmss{1}_{\\{r_i(t)=k\\}}.\n\\end{align}\nThe ranks $r_i(t)$ for the stock $X_i(t)$ at time $t$ arise from the reverse order-statistics:\n\\begin{align} \\label{eq:ranking}\n\t\\max_{1\\leq i\\leq d} X_i(t) \\mathrel={\\mathop:} X_{(1)}(t) \\geq X_{(2)}(t) \\geq \\cdots \\geq X_{(d-1)}(t) \\geq X_{(d)}(t) \\mathrel{\\mathop:}= \\min_{1 \\leq i \\leq d} X_i(t).\n\\end{align}\nTies in the ranking are resolved by giving the firm with a lower  index $i$ the better ranking. So, in such a model the $k$-th largest firm is assigned a growth rate of $\\gamma + g_k$ and a volatility of $\\sigma_k$ over the whole time horizon.\n\n\\smallskip\n\nAccording to \\cite{Banner.2005}, the simplest among the first-order models is the so-called \\textit{Atlas model}, which was introduced in \\cite[Ex.\\ 5.3.3]{Fernholz.2002}.\nWithin the setting of \\eqref{eq:FOmodel} and \\eqref{eq:FOmodelParameters}, choosing\n\\begin{align} \\label{eq:AtlasParameters}\n\t\\gamma = g > 0, \\quad g_k = -g, \\,\\, k=1, \\ldots ,d-1, \\quad g_d = (d-1)g \\quad \\text{and} \\quad \\sigma_i(t) = \\sigma > 0, \\,\\, i=1, \\ldots, d,\n\\end{align}\nleads to the Atlas model. Here, only the smallest stock in the market -- called the {Atlas stock} -- has a nonzero but positive growth rate (for its log-dynamics).\n\n\nBy setting $Y_i(t) \\mathrel{\\mathop:}= \\log X_i(t)$, $i=1,...,d$, as well as plugging in the Atlas parameters \\eqref{eq:AtlasParameters} in our first-order model \\eqref{eq:FOmodel} -- \\eqref{eq:FOmodelParameters}, we obtain the Atlas model in compact form as\n\\begin{align} \\label{eq:AtlasModel}\n\td Y_i(t) &= (d \\cdot g)  \\mathbbmss{1}_{\\{r_i(t)=d\\}} dt + \\sigma dW_i(t), \\quad i=1,...,d.\n\\end{align}\nAs stated in \\cite[Prop.\\ 2.3]{Banner.2005}, the solution of \\eqref{eq:AtlasModel} satisfies the ergodic relation\n\t\\begin{align} \\label{eq:longTermAtlas}\n\t\t \\lim\\limits_{T \\rightarrow \\infty} \\frac{1}{T} \\int_{0}^{T} \\mathbbmss{1}_{\\{r_i(t)=k\\}} dt = \\frac{1}{d} \\quad \\textrm{a.s.}, \\qquad i,k=1, \\ldots, d, \n\t\\end{align}\ni.e., all stocks in the market asymptotically spent at each rank approximately the same amount of time. Similar  ergodic relations also hold  for general first-order market models.\n\n\n\\subsection{Numerical results}\n\n\nFor simulations of the Atlas and general first-order models one has to rely on discretization schemes such as the Euler method.\nIn this subsection, we test whether the Euler scheme is able to recover the long time behavior \\eqref{eq:longTermAtlas}, i.e.,\nwhether the discrete occupation rates \n$$   \\frac{1}{T} \\sum_{\\ell=1}^{T\/\\Delta} \\mathbbmss{1}_{\\{\\widehat{r}_i(\\ell \\Delta)=k\\}}, \\qquad i,k=1, \\ldots, d, $$\nwhere $\\widehat{r}_i$ is the discretized counterpart of \\eqref{eq:ranking} based on the Euler scheme and $T\/\\Delta \\in \\mathbb{N}$, converge to the analytical value.\n\n\nHere, we consider a three-dimensional model with initial log-capitalizations\n${Y(0)=[3.4, 4.1, 5.7]}$ and $\\widetilde{Y}(0)=[1.2, 3.5, 10.8]$,  $\\gamma=0.1$ as market drift and $\\sigma=0.09$ as market volatility\\footnote{The market parameters are inspired by parameters from A.\\ Banner's (INTECH Investement Technologies LLC, Princeton) presentation on ``Equity Market Stability'' given at the WCMF6 conference, Santa Barbara, 2014.}.\nTable  \\ref{tab:LTrankings} presents the discrete occupation rates (averaged over $M=10^3$ repetitions) for $\\Delta=2^{-14}$ and diffe\\-rent values of $T$ as well as the sum of the squared deviations from the analytical asymptotic occupation rate. \nAs hoped, the discrete occupation rates converge to the analytical asymptotic occupation rate of $1\/d=1\/3$ with increasing time horizon.\n\nFurthermore, results suggest that less varying initial capitalizations imply that the numerical values are closer to the analytical result already for shorter time horizons, which coincides with the intuitive understanding.\nWe also simulated the above scenarios with $\\Delta=2^{-10}$ instead of $\\Delta=2^{-14}$: all occupation times where equal with an accuracy of four digits and one third of the $90$ occupation rates differed in the fifth digit. This suggests that -- as soon as the step size is small enough -- a further refinement of the step size is no longer beneficial and the crucial simulation parameter is $T$, the endpoint of the considered time horizon.\n\n \n\\bigskip\n\n\\begin{table}[htbp]\n\t\\centering\n\t\\caption{Discrete occupation rates for the discretized Atlas model}\n\t\\begin{tabular}{|c|c|l|r|r|r|c|c|c|}\n\t\t\\toprule\n\t\t& \\textbf{T} &       & \\multicolumn{1}{l|}{\\textbf{Firm 1}} & \\multicolumn{1}{l|}{\\textbf{Firm 2}} & \\multicolumn{1}{l|}{\\textbf{Firm 3}} & \\multicolumn{3}{c|}{\\textbf{Quadratic deviations}} \\\\\n\t\t\\midrule\n\t\t\\multirow{14}[10]{*}{$Y(0)$} & \\multirow{3}[2]{*}{\\textbf{100}} & \\textbf{Rank 1} & 0.2911 & 0.2895 & 0.4194 & \\multirow{3}[2]{*}{0.0030} & \\multirow{3}[2]{*}{0.0031} & \\multirow{3}[2]{*}{0.0111} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3425 & 0.3662 & 0.2913 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3664 & 0.3443 & 0.2892 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{250}} & \\textbf{Rank 1} & 0.3156 & 0.3161 & 0.3683 & \\multirow{3}[1]{*}{0.0005} & \\multirow{3}[1]{*}{0.0005} & \\multirow{3}[1]{*}{0.0018} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3375 & 0.3463 & 0.3162 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3469 & 0.3376 & 0.3155 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{500}} & \\textbf{Rank 1} & 0.3238 & 0.3246 & 0.3517 & \\multirow{3}[1]{*}{0.0001} & \\multirow{3}[1]{*}{0.0001} & \\multirow{3}[1]{*}{0.0005} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3357 & 0.3400 & 0.3243 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3406 & 0.3354 & 0.3240 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{750}} & \\textbf{Rank 1} & 0.3273 & 0.3273 & 0.3454 & \\multirow{3}[1]{*}{0.0001} & \\multirow{3}[1]{*}{0.0001} & \\multirow{3}[1]{*}{0.0002} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3347 & 0.3379 & 0.3274 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3380 & 0.3348 & 0.3272 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{1000}} & \\textbf{Rank 1} & 0.3288 & 0.3287 & 0.3425 & \\multirow{3}[1]{*}{0.0000} & \\multirow{3}[1]{*}{0.0000} & \\multirow{3}[1]{*}{0.0001} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3344 & 0.3368 & 0.3288 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3368 & 0.3345 & 0.3287 &       &       &  \\\\\n\t\t\\midrule\n\t\t\\multicolumn{1}{|r}{} & \\multicolumn{1}{r}{} & \\multicolumn{1}{r}{} & \\multicolumn{1}{r}{} & \\multicolumn{1}{r}{} & \\multicolumn{1}{r}{} & \\multicolumn{1}{r}{} & \\multicolumn{1}{r}{} &  \\\\\n\t\t\\midrule\n\t\t& \\textbf{T} &       & \\multicolumn{1}{l|}{\\textbf{Firm 1}} & \\multicolumn{1}{l|}{\\textbf{Firm 2}} & \\multicolumn{1}{l|}{\\textbf{Firm 3}} & \\multicolumn{3}{c|}{\\textbf{Quadratic deviations}} \\\\\n\t\t\\midrule\n\t\t\\multirow{19}[10]{*}{$\\widetilde{Y}(0)$} & \\multirow{3}[2]{*}{\\textbf{100}} & \\textbf{Rank 1} & 0.1464 & 0.1447 & 0.7089 & \\multirow{3}[2]{*}{0.0554} & \\multirow{3}[2]{*}{0.0562} & \\multirow{3}[2]{*}{0.2116} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3883 & 0.4654 & 0.1463 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.4653 & 0.3899 & 0.1448 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{250}} & \\textbf{Rank 1} & 0.2581 & 0.2579 & 0.4840 & \\multirow{3}[1]{*}{0.0090} & \\multirow{3}[1]{*}{0.0090} & \\multirow{3}[1]{*}{0.0341} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3555 & 0.3863 & 0.2582 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3864 & 0.3558 & 0.2577 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{500}} & \\textbf{Rank 1} & 0.2949 & 0.2956 & 0.4096 & \\multirow{3}[1]{*}{0.0023} & \\multirow{3}[1]{*}{0.0023} & \\multirow{3}[1]{*}{0.0087} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3448 & 0.3599 & 0.2953 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3603 & 0.3446 & 0.2951 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{750}} & \\textbf{Rank 1} & 0.3082 & 0.3079 & 0.3840 & \\multirow{3}[1]{*}{0.0010} & \\multirow{3}[1]{*}{0.0010} & \\multirow{3}[1]{*}{0.0038} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3407 & 0.3513 & 0.3081 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3511 & 0.3409 & 0.3080 &       &       &  \\\\\n\t\t\\cmidrule{2-9}\n\t\t& \\multirow{3}[1]{*}{\\textbf{1000}} & \\textbf{Rank 1} & 0.3143 & 0.3142 & 0.3715 & \\multirow{3}[1]{*}{0.0006} & \\multirow{3}[1]{*}{0.0006} & \\multirow{3}[1]{*}{0.0022} \\\\\n\t\t&       & \\textbf{Rank 2} & 0.3391 & 0.3467 & 0.3143 &       &       &  \\\\\n\t\t&       & \\textbf{Rank 3} & 0.3467 & 0.3391 & 0.3143 &       &       &  \\\\\n\t\t\\bottomrule\n\t\\end{tabular}%\n\t\\label{tab:LTrankings}%\n\\end{table}%\n\n\\bigskip\n\n\\section{Conclusion and Outlook}\nWe have seen that the numerical approximation of solutions of SDEs with discontinuous drift coefficients is a challenging task, where several particularities arise. We were able to identify two main classes of discontinuous drift coefficients: outward and inward pointing drift coefficients. For the latter class, we analyzed stability properties.\nIt turned out that the main difficulty in measuring the empirical convergence rates is how to appropriately capture drift changes. For inward pointing coefficients, we obtained stable estimates, which are in accordance with the theoretical results. For outward pointing cases, the estimates seem to be unreliable, no stabilizing asymptotic regime seems to be reached for the estimates.\n We tested two higher-order numerical schemes, that are frequently used in a setting where coefficients are sufficiently smooth. However, both schemes did not lead to an improved  behavior.\n\n\\bigskip\n\n\n\n\n\\section*{Acknowledgment}\nThis work was supported by the DFG grant No.\\ GO 1920\/4-1. \nPart of this work was carried out while A. Neuenkirch was visiting \nthe Facultad de Matem\\'aticas de la Universidad de Sevilla; A. Neuenkirch whishes to thank the\nDpto. Ecuaciones Diferenciales y An\\'alisis Num\\'erico \nfor its hospitality and support.\n\nThe publication of this article was funded by the Ministry of Science, Research and the Arts Baden-W\\\"urttemberg and the University of Mannheim.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n   \nIn \\cite{Hardy:1920}, Hardy showed his famous inequality \n\\begin{equation}\\label{EQ:Hardy0}\n\\int_b^\\infty \\left(\\frac{\\int_b^x f(t) dt}{x}\\right)^p dx \\leq \\left(\\frac{p}{p-1}\\right)^p \\int_b^\\infty f(x)^p dx,\n\\end{equation} \nwhere $p>1$, $b>0$, and $f\\geq 0$ is a nonnegative function. The original discrete version of this inequality goes back to \\cite{Hardy1919} making this year the 100${}^{th}$ anniversary of this topic, see \\cite{RS-book} for a historical discussion. Consequently, we do not try to provide a comprehensive historical account here but refer to \\cite{RS-book} for an extensive historical overview of the subject.\n\nIn particular, since then a lot of work has been done on Hardy inequalities in different forms and in different settings. It is clearly impossible to give a complete overview of the literature, so let us only refer to\nbooks and surveys by Opic and Kufner \\cite{OK-book}, Davies \\cite{Davies}, \nKufner, Persson and Samko \\cite{Hardy-weighted-book,KP-Samko},\nEdmunds and Evans \\cite{DE-book}, Mazya \\cite{Mazya-Sobolev-spaces,Mazya-book}, Ghoussoub and Moradifam \\cite{GM-book}, Balinsky, Evans and Lewis \\cite{BEL-book}, and references therein. Hardy inequalities in 1D with weights have been also studied in \\cite{AM,M} in a similar spirit to our approach. Another list of conditions was given in \\cite{DV2}, however, both in more limited settings, as well as, more recently, in \\cite{GKP10}. More precisely, \nin our Theorem \\ref{THM:Hardy1} in the special case $\\mathbb X=\\mathbb R$, the conditions $D_i$ correspond to conditions $A_i(s)$, $s=1,2,3,4$, in \\cite[Theorem 2]{GKP10}. \n\nIn this paper we show that the inequality \\eqref{EQ:Hardy0} actually holds in a much more general setting, also with rather general pairs of weights. However, the weights have to satisfy certain compatibility conditions for such inequalities to hold true, and these conditions are {\\em necessary and sufficient}.\n   \nWe note that the only known cases of our results are essentially only the Euclidean ones. The examples we give on homogeneous groups and hyperbolic spaces are new, but the main result itself is of course more general, completely characterising the weights for the integral Hardy inequality on metric measure spaces. \nThe importance of such results is, in particular, in that they lead to {\\em a variety of hypoelliptic Hardy-Sobolev and other inequalities, once we apply it with the weights associated to Riesz kernels (for hypoelliptic operators}, see \\cite{RY-hypoelliptic}).\n   \n   \n  More specifically, we consider metric spaces $(\\mathbb X,d)$ with a Borel measure $dx$ allowing for the following {\\em polar decomposition} at $a\\in{\\mathbb X}$: we assume that there is a locally integrable function $\\lambda \\in L^1_{loc}$  such that for all $f\\in L^1(\\mathbb X)$ we have\n   \\begin{equation}\\label{EQ:polar}\n   \\int_{\\mathbb X}f(x)dx= \\int_0^{\\infty}\\int_{\\Sigma_r} f(r,\\omega) \\lambda(r,\\omega) d\\omega_r dr,\n   \\end{equation}\n    for the sets $\\Sigma_r=\\{x\\in\\mathbb X: d(x,a)=r\\}$\n    with a measure on it denoted by $d\\omega=d\\omega_r$.\n    The condition \\eqref{EQ:polar} is rather general since we allow the function $\\lambda$ to depend on the whole variable $x=(r,\\omega)$. \n\nThe reason to assume \\eqref{EQ:polar} is that since $\\mathbb X$ does not have to have a differentiable structure, the function $\\lambda(r,\\omega)$ can not be in general obtained as the Jacobian of the polar change of coordinates. However, if such a differentiable structure exists on $\\mathbb X$, the condition \\eqref{EQ:polar} can be obtained as the standard polar decomposition formula. \nIn particular, let us give several examples of $\\mathbb X$ for which the condition \\eqref{EQ:polar} is satisfied with different expressions for $ \\lambda (r,\\omega)$:\n\n\\begin{itemize}\n\\item[(I)] Euclidean space ${\\mathbb R}^n$: $ \\lambda (r,\\omega)= {r}^{n-1}.$\n\\item[(II)] Homogeneous groups: $ \\lambda (r,\\omega)= {r}^{Q-1}$, where $Q$ is the homogeneous dimension of the group. Such groups have been consistently developed by Folland and Stein \\cite{FS-Hardy}, see also an up-to-date exposition in \\cite{FR}.\n\\item[(III)] Hyperbolic spaces $\\mathbb H^n$:  $\\lambda(r,\\omega)=(\\sinh {r})^{n-1}$.\n\\item[(IV)] Cartan-Hadamard manifolds: Let $K_M$ be the sectional curvature on $(M, g).$ A Riemannian manifold $(M, g)$ is called {\\em a Cartan-Hadamard manifold} if it is complete, simply connected and has non-positive sectional curvature, i.e., the sectional curvature $K_M\\le 0$ along each plane section at each point of M. Let us fix a point $a\\in M$ and denote by \n$\\rho(x)=d(x,a)$ the geodesic distance from $x$ to $a$ on $M$. The exponential map ${\\rm exp}_a :T_a M \\to  M$ is a diffeomorphism, see e.g. Helgason \\cite{DV3}.  Let $J(\\rho,\\omega)$ be the density function on $M$, see e.g. \\cite{DV1}. Then we have the following polar decomposition: \n$$\n\\int_M f(x) dx=\\int_0^{\\infty}\\int_{\\mathbb S^{n-1}}f({\\rm exp}_{a}(\\rho \\omega))J(\\rho,\\omega) \\rho^{n-1}d\\rho d\\omega,\n$$\nso that we have \\eqref{EQ:polar} with $\\lambda(\\rho,\\omega)= J(\\rho,\\omega) \\rho^{n-1}.$ \n \\item[(V)] Complete manifolds: Let $M$ be a complete manifold. Let $p\\in M$ and let $C(p)$ denote the cut locus of $p$. Let $D_{p}:=M\\backslash C(p)$ and $S(p;r):=\\{x\\in M_{p}:|x|=r\\}$, where $|\\cdot|$ is the Riemannian length, where  $M_{p}$ stands for the tangent space  at $p$. Then for any $p\\in M$ and any integrable function $f$ on $M$ we have (e.g. see \\cite[Formula III.3.5, P.123]{Cha06}) the polar decompsition\n        \\begin{equation}\\label{pol_decom_manif}\n        \\int_{M}f dV=\\int_{0}^{+\\infty}dr\\int_{r^{-1}S(p;r)\\cap D_{p}}f(\\exp r\\xi)\\sqrt{g}(r;\\xi)d\\mu_{p}(\\xi)\n        \\end{equation}\n        for some function $\\sqrt{g}$ on $D_{p}$, where $r^{-1}S(p,r)\\cap D_{p}$ is the subset of $S_{p}$ obtained by dividing each of the elements of $S(p,r)\\cap D_{p}$ by $r$, and $S_{p}:=S(p;1)$. Here $d\\mu_{p}(\\xi)$ is the Riemannian measure on $S_{p}$ induced by the Euclidean Lebesgue measure on $M_{p}$. We refer to \\cite{Cha06}, \\cite[Chapter 4]{Li12} and \\cite[Chapter 1, Paragraph 12]{CLN06} for more details on this decompsition.\n\\end{itemize} \n\n\n\n Throughout this paper, by $ A\\approx B$ we will always mean that the expressions $A$ and $B$ are equivalent.\n \n\n\\section{Main results}\n\nWe denote by $B(a,r)$ the ball in $\\mathbb X$ with centre $a$ and radius  $r$, i.e \n$$B(a,r):= \\{x\\in\\mathbb X : d(x,a)<r\\},$$ \nwhere $d$ is the metric on $\\mathbb X$. \nOnce and for all we will fix some point $a\\in {\\mathbb X}$, and we will write $${\\vert x \\vert}_a := d(a,x).$$\n\nOur first main result is the following characterisation of weights $u$ and $v$ for the corresponding Hardy inequality to hold on $\\mathbb X$, with the characterisation for the conjugate Hardy inequality given in Theorem \\ref{THM:Hardy2}. The first condition is the Muckenhoupt condition while the other ones are equivalent to it.\n\n\\begin{thm}\\label{THM:Hardy1}\nLet $1<p\\le q <\\infty$ and let $s>0$. Let $\\mathbb X $ be a metric measure space with a polar decomposition \\eqref{EQ:polar} at a. \nLet $u,v> 0$ be measurable functions  positive a.e in $\\mathbb X$  such that $u\\in L^1(\\mathbb X\\backslash \\{a\\})$ and $v^{1-p'}\\in L^1_{loc}(\\mathbb X)$. Denote\n\\begin{align}\nU(x):= { \\int_{\\mathbb X\\backslash{B(a,|x|_a )}} u(y) dy}  \\nonumber\n\\end{align} \nand \n\\begin{align} \nV(x):= \\int_{B(a,|x|_a  )}v^{1-p'}(y)dy\\nonumber. \n\\end{align}\nThen the inequality\n\\begin{equation}\\label{EQ:Hardy1}\n\\bigg(\\int_\\mathbb X\\bigg(\\int_{B(a,\\vert x \\vert_a)}\\vert f(y) \\vert dy\\bigg)^q u(x)dx\\bigg)^\\frac{1}{q}\\le C\\bigg\\{\\int_{\\mathbb X} {\\vert f(x) \\vert}^pv(x)dx\\bigg\\}^{\\frac1p}\n\\end{equation}\nholds for all measurable functions $f:\\mathbb X\\to{\\mathbb C}$ if and only if  any of the following equivalent conditions hold:\n\n\\begin{enumerate}\n\\item $\\mathcal D_{1} :=\\sup_{x\\not=a} \\bigg\\{U^\\frac{1}{q}(x) V^\\frac{1}{p'}(x)\\bigg\\}<\\infty.$\n\\end{enumerate}\n\n\\begin{enumerate}\\setcounter{enumi}{1}\n\\item $\\mathcal D_{2}:=\\sup_{x\\not=a} \\bigg\\{\\int_{\\mathbb X\\backslash{B(a,|x|_a )}}u(y)V^{q(\\frac{1}{p'}-s)}(y)dy\\bigg\\}^\\frac{1}{q}V^s(x)<\\infty.$\n\\item $\\mathcal D_{3}:=\\sup_{x\\not=a}\\bigg\\{\\int_{B(a,|x|_a)}u(y)V^{q(\\frac{1}{p'}+s)}(y)dy\\bigg\\}^{\\frac{1}{q}}V^{-s}(x)<\\infty $, provided that $u,v^{1-p'}\\in L^1(\\mathbb X)$.\n\\end{enumerate}\n\n\\begin{enumerate}\\setcounter{enumi}{3}\n\\item $\\mathcal D_{4}:=\\sup_{x\\not=a}\\bigg\\{\\int_{B(a,\\vert x \\vert_a)}v^{1-p'}(y) U^{p'(\\frac{1}{q}-s)}(y)dy\\bigg\\}^\\frac{1}{p'}U^s(x)<\\infty.$ \n\n\\item $\\mathcal D_{5}:=\\sup_{x\\not=a}\\bigg\\{\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a )}}v^{1-p'}(y)U^{p'(\\frac{1}{q}+s)}(y)dy\\bigg\\}^\\frac{1}{p'}U^{-s}(x)<\\infty$, provided that $u,v^{1-p'}\\in L^1(\\mathbb X)$.\n\\end{enumerate}\n\nMoreover, the constant $C$ for which \\eqref{EQ:Hardy1} holds and quantities $\\mathcal D_{1}-\\mathcal D_{5}$ are related by \n\\begin{equation}\\label{EQ:constants}\n\\mathcal D_{1} \\leq C\\leq \\mathcal D_1(p')^{\\frac{1}{p'}} p^\\frac{1}{q},\n\\end{equation}   \nand \n$$\\mathcal D_1 \\le \\left(\\max(1,{p'}{s})\\right)^\\frac{1}{q}\\mathcal D_2, \\;\\mathcal D_2 \\le (\\max(1,\\frac{1}{p's}))^\\frac{1}{q} \\mathcal D_1,$$ \n$$(\\frac{sp'}{1+p's})^\\frac{1}{q} \\mathcal D_3 \\le \\mathcal D_1\\le(1+sp')^\\frac{1}{q}\\mathcal D_3,$$\n$$\\mathcal D_1 \\le (\\max(1,qs))^\\frac{1}{p'} \\mathcal D_4,\\; \\mathcal D_4 \\le (\\max(1,\\frac{1}{qs}))^\\frac{1}{p'}\n\\mathcal D_1,$$ \n$$(\\frac{sq}{1+qs})^\\frac{1}{p'}\\mathcal D_5 \\le \\mathcal D_1 \\le (1+sq)^\\frac{1}{p'} \\mathcal D_5.$$\n\\end{thm}\n\nIn particular, Theorem \\ref {THM:Hardy1} is an extension of \\eqref{EQ:Hardy0} to the setting of metric measures spaces $\\mathbb X$ with the polar decomposition \\eqref{EQ:polar}: in particular, for $p=q$ and real-valued nonnegative measurable $f\\geq 0$, inequality \\eqref{EQ:Hardy1} becomes \n$$\n\\int_\\mathbb X\\bigg(\\int_{B(a,\\vert x \\vert_a)} f(y) dy\\bigg)^p u(x)dx \\le C \\int_{\\mathbb X} {f(x)}^p v(x)dx,\n$$\nas an extension of \\eqref{EQ:Hardy0}. Indeed, in this case we can take $u(x)=\\frac{1}{x^p}$, $v(x)=1$, $\\mathbb X=[b,\\infty)$, $a=b$, so that Theorem \\ref {THM:Hardy1} implies  \\eqref{EQ:Hardy0}.\n\nFor the results in the case of $\\mathbb X=\\mathbb R$ we can refer to \\cite{Hardy-weighted-book, PS01}, and also to \\cite{PSW07} for inequalities for $q<p$.\nFor $\\mathbb X={\\mathbb R}^n$, the result has been proved in \\cite {Daulti-Verma:Razmadze}, with related inequalities obtained in one dimension in  \\cite{DV2,OK-book}. For related works on hyperbolic spaces we can refer to \\cite{DV4,RY18_manif}, and to \\cite{DV5,RY18_manif} for inequalities on Cartan-Hadamard manifolds, with the background analysis available in   \\cite{DV1,DV3}.\nFor the analysis of Hardy inequalities on homogeneous groups we can refer to\n\\cite{RS-identities,RSY18_Tran}.\n\nLet us also briefly discuss the conjugate Hardy inequality to that in Theorem \\ref{THM:Hardy1}:\n\n\\begin{thm}\\label{THM:Hardy2}\nLet $1<p\\le q <\\infty$ and $s>0$. Let $\\mathbb X $ be a metric measure space with a polar decomposition as in \\eqref{EQ:polar}. Let $u,v >0$ be measurable functions positive a.e. such that\n$u\\in L^1_{loc}(\\mathbb X)$ and  $v^{1-p'}\\in L^1(\\mathbb X\\backslash \\{a\\})$. Let\n\\begin{align}\nU(x)= {\\int_{{B(a,\\vert x \\vert_a )}} u(y) dy}  \\nonumber\n\\end{align} \nand \\begin{align} \nV(x)= \\int_{\\mathbb X\\backslash B(a,\\vert x \\vert_a )}v^{1-p'}(y)dy\\nonumber. \n\\end{align}\nThen the inequality\n\\begin{align}\\label{EQ:Hardycon}\n\\bigg(\\int_\\mathbb X\\bigg(\\int_{\\mathbb X\\backslash B(a,\\vert x \\vert_a)}\\vert f(y) \\vert dy\\bigg)^q u(x)dx\\bigg)^\\frac{1}{q}\\le C\\bigg\\{\\int_{\\mathbb X} {\\vert f(x) \\vert}^p v(x)dx\\bigg\\}^\\frac{1}{p}\n\\end{align}\nholds for all measurable functions f if and only if any of the following equivalent conditions holds:\n\\begin{enumerate}\n\\item $\\mathcal D_{1}^{*} :=\\sup_{x\\not=a} \\bigg\\{U^\\frac{1}{q}(x) V^\\frac{1}{p'}(x)\\bigg\\}<\\infty$.\n\\item $\\mathcal D_{2}^{*}:=\\sup_{x\\not=a} \\bigg\\{\\int_{{B(a,\\vert x \\vert_a )}}u(y)V^{q(\\frac{1}{p'}-s)}(y)dy\\bigg\\}^\\frac{1}{q}V^s(x)<\\infty.$\n\n\\item $\\mathcal D_{3}^{*}:=\\sup_{x\\not=a}\\bigg\\{\\int_{\\mathbb X\\backslash B(a,\\vert x \\vert_a)}u(y)V^{q(\\frac{1}{p'}+s)}(y)dy\\bigg\\}^{\\frac{1}{q}}V^{-s}(x)<\\infty,$\nprovided that $u,v^{1-p'}\\in L^1(\\mathbb X).$\n\\item $\\mathcal D_{4}^{*}:=\\sup_{x\\not=a}\\bigg\\{\\int_{\\mathbb X\\backslash B(a,\\vert x \\vert_a)}v^{1-p'}(y) U^{p'(\\frac{1}{q}-s)}(y)dy\\bigg\\}^\\frac{1}{p'}U^s(x)<\\infty.$\n\\item $\\mathcal D_{5}^{*}:=\\sup_{x\\not=a}\\bigg\\{\\int_{{B(a,\\vert x \\vert_a )}}v^{1-p'}(t)U^{p'(\\frac{1}{q}+s)}(t)dt\\bigg\\}^\\frac{1}{p'}U^{-s}(x)<\\infty,$\nprovided that $u,v^{1-p'}\\in L^1(\\mathbb X).$\n\\end{enumerate}\n\\end{thm}\n\n\n \n\\section{Applications and examples}\n\nIn this section we will give examples of the application of Theorem \\ref {THM:Hardy1} in the settings of homogeneous groups, hyperbolic spaces, and Cartan-Hadamard manifolds.\n\n\\subsection{Homogeneous groups}\n\nLet $\\mathbb G$ be a homogeneous group of homogeneous dimension $Q$, equipped with a quasi-norm $|\\cdot|$.\nFor the general description of the setup of homogeneous groups we refer to \\cite{FS-Hardy} or \\cite{FR}. Particular example of homogeneous groups are the Euclidean space ${\\mathbb R}^n$ (in which case $Q=n$), the Heisenberg group, as well as general stratified groups (homogeneous Carnot groups) and graded groups.\n\nIn relation to the notation of this paper, let us take $a=0$ to be the identity of the group $\\mathbb G$. We can also simplify the notation denoting $\\vert x \\vert_a$ by $\\vert x \\vert$, which is consistent with the notation for the quasi-norm $|\\cdot|.$\n\nIf we take power weights \n$$u(x)= {\\vert x \\vert}^\\alpha \\textrm{ and } v(x)= {\\vert x \\vert }^\\beta,$$\nthen the inequality \\eqref{EQ:Hardy1} holds for $1< p \\le q <\\infty$ if and only if \n$$\n\\mathcal D_1=\\sup_{r>0} \\bigg(\\displaystyle \\sigma\\int_{r}^\\infty {\\rho}^\\alpha {\\rho}^{Q-1}d\\rho \\bigg)^\\frac{1}{q} \\bigg(\\displaystyle \\sigma \\int_{0}^r{\\rho}^{\\beta(1-p')} {\\rho}^{Q-1}d\\rho\\bigg)^\\frac{1}{p'}<\\infty,\n$$\nwhere $\\sigma$ is the area of the unit sphere in $\\mathbb G$ with respect to the quasi-norm $|\\cdot|.$ For this supremum to be well-defined we need to have  $\\alpha+Q<0$\n and $\\beta(1-p')+Q>0.$\nConsequently, we have\n\\begin{multline*}\n\\mathcal D_1=\\sigma^{(\\frac1q+\\frac{1}{p'})}\\sup_{r>0}\n\\bigg(\\displaystyle\\int_{r}^\\infty{\\rho}^{\\alpha+Q-1}d\\rho\\bigg)^\\frac{1}{q}\\bigg(\\displaystyle\\int_0^{r} {\\rho}^{\\beta(1-p')+Q-1}d\\rho \\bigg)^\\frac{1}{p'} \\\\\n= \\sigma^{(\\frac1q+\\frac{1}{p'})}\n\\sup_{r>0} \\frac {{r}^ \\frac{\\alpha+Q}{q}}{{|\\alpha+Q|}^\\frac{1}{q}}\n\\frac{ {r}^\\frac{\\beta(1-p')+Q} {p'}}{{(\\beta(1-p')+Q)}^\\frac{1}{p'}},\n\\end{multline*} \nwhich is finite if and only if the power of $r$ is zero.\nSummarising, we obtain\n\n\\begin{cor}\\label{COR:hom}\nLet $\\mathbb G$ be a homogeneous group of homogeneous dimension $Q$, equipped with a quasi-norm $|\\cdot|$. Let $1<p\\le q <\\infty$ and let $\\alpha,\\beta\\in\\mathbb R$. Then the inequality\n\\begin{equation}\\label{EQ:Hardy1hg}\n\\bigg(\\int_\\mathbb G\\bigg(\\int_{B(0,\\vert x \\vert)}\\vert f(y) \\vert dy\\bigg)^q {\\vert x \\vert}^\\alpha dx\\bigg)^\\frac{1}{q}\\le C\\bigg\\{\\int_{\\mathbb G} {\\vert f(x) \\vert}^p {\\vert x \\vert }^\\beta dx\\bigg\\}^{\\frac1p}\n\\end{equation}\nholds for all measurable functions $f:\\mathbb G\\to{\\mathbb C}$  if and only if \n$\\alpha+Q<0$,  $\\beta(1-p')+Q>0$ and \n$\\frac{\\alpha+Q}{q}+\\frac{\\beta(1-p')+Q} {p'}=0.$\nMoreover, the constant $C$ for \\eqref{EQ:Hardy1hg} satisfies\n\\begin{equation}\\label{EQ:constants-hg}\n\\frac{\\sigma^{\\frac1q+\\frac{1}{p'}}}{|\\alpha+Q|^\\frac{1}{q}(\\beta(1-p')+Q)^\\frac{1}{p'}}\n\\leq C\\leq (p')^{\\frac{1}{p'}} p^\\frac{1}{q}\\frac{\\sigma^{\\frac1q+\\frac{1}{p'}}}{|\\alpha+Q|^\\frac{1}{q}(\\beta(1-p')+Q)^\\frac{1}{p'}},\n\\end{equation}  \nwhere $\\sigma$ is the area of the unit sphere in $\\mathbb G$ with respect to the quasi-norm $|\\cdot|.$\n\\end{cor} \n\n \\subsection{Hyperbolic spaces} \n \n Let $\\mathbb H^n$ be the hyperbolic space of dimension $n$ and let $a\\in \\mathbb H^n$.\n Let us take the weights \n $$u(x)= (\\sinh {\\vert x \\vert_a})^\\alpha \\textrm{ and } v(x)= (\\sinh {\\vert x \\vert_a})^\\beta.$$  \nThen, passing to polar coordinates, $\\mathcal D_1$ is equivalent to\n \n $$\\mathcal D_1\\simeq \\sup_{\\vert x \\vert_a>0}\\bigg(\\displaystyle\\int_{\\vert x \\vert_a}^\\infty(\\sinh{\\rho})^{\\alpha+n-1}d\\rho\\bigg)^\\frac{1}{q}\\bigg(\\displaystyle\\int_0^{\\vert x \\vert_a} (\\sinh{\\rho})^{\\beta(1-p')+n-1}d\\rho \\bigg)^\\frac{1}{p'}.$$\n For the integrability of the first and the second terms we need, respectively,\n$\\alpha+n-1<0$ and  $ \\beta(1-p')+n>0.$\n\nLet us now analyse conditions for this supremum to be finite. \nFor $\\vert x \\vert_a \\gg1 $, it can be written as\n\n\\begin{multline*}\n\\sup_{\\vert x \\vert_a \\gg1} \\bigg(\\displaystyle\\int_{\\vert x \\vert_a}^\\infty (\\exp{\\rho})^{\\alpha+n-1}d\\rho\\bigg)^\\frac{1}{q} \\bigg(\\displaystyle\\int_0^{\\vert x \\vert_a} (\\exp{\\rho})^{\\beta(1-p')+n-1}d\\rho \\bigg)^\\frac{1}{p'}\\\\\n\\simeq \\sup_{\\vert x \\vert_a \\gg1} (\\exp {\\vert x \\vert_a})^{\\bigg(\\frac{\\alpha+n-1}{q}+\\frac{\\beta(1-p')+n-1}{p'} \\bigg)},\n\\end{multline*} \nwhich is finite if and only if $\\frac{\\alpha+n-1}{q}+\\frac{\\beta(1-p')+n-1}{p'}\\le0.$\nFor $\\vert x \\vert_a\\ll 1 $, it can be written as\n\n$\\sup_{\\vert x \\vert_a\\ll 1} \\bigg(\\displaystyle\\int_{\\vert x \\vert_a}^\\infty(\\sinh{\\rho})^{\\alpha+n-1}d\\rho\\bigg)^\\frac{1}{q} \\bigg(\\displaystyle\\int_0^{\\vert x \\vert_a} {\\rho}^{\\beta(1-p')+n-1}d\\rho \\bigg)^\\frac{1}{p'}$\n\n$\\simeq \\sup_{\\vert x \\vert_a\\ll 1} \\bigg(\\displaystyle\\int_{\\vert x \\vert_a}^ R (\\sinh{\\rho})^{\\alpha+n-1}d\\rho +\\displaystyle\\int_{R}^\\infty(\\sinh{\\rho})^{\\alpha+n-1}d\\rho\\bigg)^\\frac{1}{q} \\bigg(\\displaystyle\\int_0^{\\vert x \\vert_a} {\\rho}^{\\beta(1-p')+n-1}d\\rho \\bigg)^\\frac{1}{p'}.$\n\nFor some small $R$ we have ${\\sinh{\\rho} }_{\\vert x \\vert_a\\le \\rho<R} \\approx \\rho$, so that the above supremum is \n\n$\\approx \\sup_{\\vert x \\vert_a\\ll 1}\\bigg( \\vert x \\vert_a^{\\alpha+n}+C_R\\bigg)^\\frac{1}{q} $ ${\\vert x \\vert_a }^\\frac{\\beta(1-p')+n}{p'}.$\n\nNow, for $\\alpha+n\\ge0$, this is \n\n$\\approx \\sup_{\\vert x \\vert_a\\ll 1}{\\vert x \\vert_a}^\\frac{\\beta(1-p')+n}{p'}$, which is finite if and only if $\\frac{\\beta(1-p')+n}{p'} \\ge0$.\nAt the same time, for for $\\alpha+n <0 $ it is\n\n$\\approx \\sup_{\\vert x \\vert_a\\ll 1}{\\vert x \\vert_a}^{\\frac{\\alpha+n}{q}+\\frac{\\beta(1-p')+n}{p'}}$ , which is finite if and only if $\\frac{\\alpha+n}{q}+\\frac{\\beta(1-p')+n}{p'}\\ge0$.\n\nSummarising, we obtain the following\n\n\\begin{cor}\\label{COR:hyp}\nLet $\\mathbb H^n$ be the hyperbolic space, $a\\in \\mathbb H^n$, and let $|x|_a$ denote the hyperbolic distance from $x$ to $a$. Let $1<p\\le q <\\infty$ and let $\\alpha,\\beta\\in\\mathbb R$. Then the inequality\n\\begin{equation*}\\label{EQ:Hardy1hyp}\n\\bigg(\\int_{\\mathbb H^n}\\bigg(\\int_{B(a,\\vert x \\vert_a)}\\vert f(y) \\vert dy\\bigg)^q (\\sinh {\\vert x \\vert_a})^\\alpha dx\\bigg)^\\frac{1}{q}\\le C\\bigg\\{\\int_{\\mathbb H^n} {\\vert f(x) \\vert}^p (\\sinh {\\vert x \\vert_a})^\\beta dx\\bigg\\}^{\\frac1p}\n\\end{equation*}\nholds for all measurable functions $f:\\mathbb H^n\\to{\\mathbb C}$  if and only if \nthe parameters satisfy either of the following conditions\n\\begin{itemize}\n\\item[(A)] for $\\alpha+n \\ge 0 $, if and only if $\\alpha+n<1$, $\\beta(1-p')+n>0$ and $\\frac{\\alpha+n}{q}+\\frac{\\beta(1-p')+n}{p'}\\le \\frac{1}{q}+\\frac{1}{p'}$;\n\\item[(B)] for $\\alpha+n <0 $,  if and only if  $\\beta(1-p')+n>0$  and $0\\leq \\frac{\\alpha+n}{q}+\\frac{\\beta(1-p')+n}{p'}\\le \\frac{1}{q}+\\frac{1}{p'}$.\n\\end{itemize} \n\\end{cor} \n\n\\subsection{Cartan-Hadamard manifolds}\n\nLet $(M,g)$ be a Cartan-Hadamard manifold and assume that the sectional curvature $K_M$ is constant. In this case it is known that $J(t,\\omega)$ is a function of $t$ only. More precisely, if  $K_M=-b$ for $b\\ge0$, then \n$J(t,\\omega)= 1$ if $b=0$, and $J(t,\\omega)=(\\frac{\\sinh \\sqrt{b}t}{\\sqrt{b}t})^{n-1}$ for $b>0$,\nsee e.g. \\cite{BC01-book} or \\cite[p. 166-167]{DV1}.\n \n When $b=0$, then let us take $u(x)=  \\vert x \\vert_a^\\alpha$ and $v(x)=  \\vert x \\vert_a^\\beta$ , then the  inequality \\eqref{EQ:Hardy1} holds for $1<p\\le q<\\infty$ if and only if\\\\\n \n $\\sup_{\\vert x \\vert_a>0} \\bigg(\\displaystyle\\int_{M \\backslash B(a,\\vert x \\vert_a)}\\vert y \\vert_a^\\alpha dy \\bigg)^\\frac{1}{q} \\bigg(\\displaystyle \\int_{B(a,\\vert x \\vert_a)}\\vert y \\vert_a^{\\beta(1-p')} dy\\bigg)^\\frac{1}{p'}<\\infty$.\\\\\n\nAfter changing to the polar coordinates, this is equivalent to            \n\n$\\sup_{\\vert x \\vert_a>0}\\bigg(\\int_{\\vert x \\vert_a }^\\infty  {\\rho}^{\\alpha+n-1} d\\rho \\bigg)^\\frac{1}{q}\\bigg(\\int_0^{\\vert x \\vert_a} {\\rho}^{\\beta(1-p')+n-1}d\\rho \\bigg)^\\frac{1}{p'}$, \\\\\nwhich is finite if and only if conditions of Corollary \\ref{COR:hom} hold with $Q=n$ (which is natural since the curvature is zero).\n\nWhen $b>0$, let us take $u(x)=(\\sinh \\sqrt{b}{\\vert x \\vert_a})^\\alpha$  and $v(x)=(\\sinh \\sqrt{b}{\\vert x \\vert_a})^\\beta$. Then he  inequality \\eqref{EQ:Hardy1} holds for $1<p\\le q<\\infty$ if and only if\\\\\n\n$\\sup_{\\vert x \\vert_a>0} \\bigg(\\displaystyle\\int_{\\mathbb M \\backslash B(a,\\vert x \\vert_a)} (\\sinh \\sqrt{b}{\\vert y \\vert_a})^{\\alpha} dy \\bigg)^\\frac{1}{q} \\bigg(\\displaystyle \\int_{B(a,\\vert x \\vert_a)}(\\sinh \\sqrt{b}{\\vert y \\vert_a})^{\\beta(1-p')}dy \\bigg)^\\frac{1}{p'}<\\infty.$\\\\\n\nAfter changing to the polar coordinates, this supremum is equivalent to \n\n$\\sup_{\\vert x \\vert_a>0}(\\displaystyle\\int_{\\vert x \\vert_a}^\\infty (\\sinh \\sqrt{b}{t})^\\alpha (\\frac{\\sinh \\sqrt{b} t}{\\sqrt{b} t})^{n-1} t^{n-1}dt )^\\frac{1}{q}$\\\\\n$\\quad\\times(\\displaystyle \\int_{0}^ {\\vert x \\vert_a} (\\sinh \\sqrt{b}{t})^{\\beta(1-p')}(\\frac{\\sinh \\sqrt{b} t}{\\sqrt{b} t})^{n-1} t^{n-1} dt )^\\frac{1}{p'}$\\\\\n\n\n$\\simeq\\sup_{\\vert x \\vert_a>0}\\bigg(\\displaystyle\\int_{\\vert x \\vert_a}^\\infty (\\sinh \\sqrt{b}{t})^{\\alpha+n-1}dt \\bigg)^\\frac{1}{q}\\bigg(\\displaystyle\\int_0^{\\vert x \\vert_a} (\\sinh \\sqrt{b} t)^{\\beta(1-p')+n-1}dt \\bigg)^\\frac{1}{p'}, $\\\\\nwhich has the same conditions for finiteness as the case of the hyperbolic space in Corollary \\ref{COR:hyp} (which is also natural since it is the negative constant curvature case).\n\n\n\n\n\n\n \n\n       \n     \\section{Equivalence of weight conditions}\n     \n    In this section we prove that the quantities $\\mathcal D_{1}$--$\\mathcal D_{5}$\ninvolving the weights in Theorem \\ref{THM:Hardy1} are equivalent. However, it will be convenient to formulate it in the following slightly more general form:\n     \n  \\begin{thm}\\label{THM:equivalence}\n    Let $\\alpha , \\beta, s >0$  and let\n    $f  \\in L^{1}(\\mathbb X\\backslash \\{a\\})$, $g \\in L^{1} _{loc} (\\mathbb X)$, be such that $ f,g >0 $ are positive a.e in $\\mathbb X$. Let us denote\n    \\begin{align}\n             F(x):= \\int_{\\mathbb X\\backslash B(a,\\vert x \\vert_a)}f(y)dy\\nonumber,\n    \\end{align}\n    and\n    \\begin{align}\n            G(x):=\\int_{B(a,\\vert x \\vert_a)} g(y)dy\\nonumber.\n    \\end{align}\n    Then the following quantities are equivalent:\n    \\begin{enumerate}\n    \\item$\\mathcal A_1:= \\sup_{x\\not=a}A_1(x;\\alpha,\\beta):= \\sup_{x\\not=a}F^\\alpha(x)G^\\beta(x).$\n    \\item$\\mathcal A_2:=\\sup_{x\\not=a} A_2(x;\\alpha,\\beta,s):=\\sup_{x\\not=a}{\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta-s)\/\\alpha}(y)dy\\bigg)^\\alpha}   G^s(x),$ provided that $G^{(\\beta-s)\/\\alpha}(y)$ makes sense.\n    \\item$\\mathcal A_3:=\\sup_{x\\not=a} A_3(x;\\alpha,\\beta,s):= \\sup_{x\\not=a}\\bigg(\\int_{B(a,\\vert x \\vert_a)}g(y)F^{(\\alpha-s)\/\\beta}(y)dy\\bigg)^\\beta F^s(x),$ provided that $F^{(\\alpha-s)\/\\beta}(y)$ makes sense.     \\item$\\mathcal A_4:=\\sup_{x\\not=a} A_4(x;\\alpha,\\beta,s):= \\sup_{x\\not=a}\\bigg(\\int_{B(a,\\vert x \\vert_a)}f(y)G^{(\\beta+s)\/\\alpha}(y)dy\\bigg)^\\alpha G^{-s}(x),$ provided that $f,g\\in L^1(\\mathbb X)$ and that $G^{-s}(x)$ makes sense.\n    \\item$\\mathcal A_5:=\\sup_{x\\not=a} A_5(x;\\alpha,\\beta,s):=\\sup_{x\\not=a}\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}g(y) F^{(\\alpha+s)\/\\beta}(y) dy\\bigg)^\\beta F^{-s}(x),$\n    provided $f,g\\in L^1(\\mathbb X)$ and that that $F^{-s}(x)$ makes sense.\n    \\end{enumerate}\nMoreover, we have the following relations between the above quantities:\n\\begin{itemize}\n\\item $\\mathcal {A}_1\\le (\\max(1,\\frac{s}{\\beta}))^\\alpha \\mathcal {A}_2 $ and   $\\mathcal {A}_2\\le  (\\max(1,\\frac{\\beta}{s}))^\\alpha \\mathcal {A}_1$;\n\\item  $\\mathcal {A}_1\\le(\\max(1,\\frac{s}{\\alpha}))^\\beta \\mathcal {A}_3 $ and  $\\mathcal {A}_3\\le  (\\max(1,\\frac{\\alpha}{s}))^\\beta \\mathcal {A}_1$;\n\\item  $ (\\frac{s}{\\beta+s})^\\alpha \\mathcal A_4 \\le \\mathcal A_1\\le (1+\\frac{s}{\\beta})^\\alpha \\mathcal A_4 $ and  $ (\\frac{s}{\\alpha+s})^\\beta \\mathcal A_5 \\le \\mathcal A_1\\le (1+\\frac{s}{\\alpha})^\\beta \\mathcal A_5 $.\n\\end{itemize}\n\\end {thm}\n\n   \\begin{proof}[Proof of Theorem \\ref{THM:equivalence}]\n   $\\boxed{{\\mathcal A_1\\approx \\mathcal A_2}}$\n   \n   We will first consider the case $s\\le\\beta$. Then for $\\vert y \\vert_a\\geq \\vert x \\vert_a$ we have \n       $  G^{(\\beta-s)\/\\alpha}(y)\\ge G^{(\\beta-s)\/\\alpha}(x)$. Consequently, we can estimate\n   \\begin{align*}\n   A_2(x;\\alpha,\\beta,s) & ={\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta-s)\/\\alpha}(y)dy\\bigg)^\\alpha} G^s(x)\\nonumber\n \\\\\n &    \\ge{\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)dy\\bigg)^\\alpha G^\\beta(x)}\\nonumber\n\\\\ &\n  =F^{\\alpha}(x)G^{\\beta}(x),\\nonumber\n   \\end{align*}\n  which implies  $\\mathcal A_2 \\ge \\mathcal A_1$.\n    For $s>\\beta$, let us first introduce some notation, using the polar decomposition \\eqref{EQ:polar}. First, we denote\n     \\begin{align*}\n    W(x) & :={\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta-s)\/\\alpha}(y)dy}\\nonumber\\\\\n           & =\\int_{\\vert x \\vert_a}^{\\infty}\\int_{\\Sigma_r}f(r,\\omega) \\lambda(r,\\omega)\\widetilde{G_1}(r)^{(\\beta-s)\/\\alpha} d\\omega_r dr\\nonumber\\\\\n  & = \\int_{\\vert x \\vert_a}^{\\infty}{\\widetilde{W}}(r)dr\\nonumber\\\\\n  & =:\\widetilde{W_1}(\\vert x \\vert_a),\\nonumber\n   \\end{align*}\n   where\n$$\n   \\widetilde{G_1}(r) := \\int_0^r\\int_{\\Sigma_s} g(s,\\sigma)\\lambda(s,\\sigma)d\\sigma_s ds =\\int_0^r\\widetilde{G}(s)ds,\n $$\n with $\\widetilde{G}(s):=\\int_{\\Sigma_s} g(s,\\sigma)\\lambda(s,\\sigma)d\\sigma_s$,\n  and\n$$\n  \\widetilde{W}(r):=\\int_{\\Sigma_r}f(r,\\omega) \\lambda(r,\\omega)\\widetilde{G_1}(r)^{(\\beta-s)\/\\alpha}d\\omega_r.\n$$\nMoreover, we denote \n$$\n   \\widetilde{F_1}(r) : = \\int_r^{\\infty} \\int_{\\Sigma_s} \\lambda(s,\\sigma)f(s,\\sigma)d\\sigma_s ds=\\int_r^{\\infty}\\widetilde{F}(s)ds.\n$$\n\nUsing the function $W$ defined above, we can estimate\n   \\begin{align*}\n   & F^{\\alpha}(x)G^{\\beta}(x) \\\\\n   & = G^{\\beta}(x)\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta-s)\/\\alpha}(y)G^{(s-\\beta)\/\\alpha}(y)W^{(s-\\beta)\/s}(y)W^{(\\beta-s)\/s}(y)dy\\bigg)^\\alpha\\nonumber\n   \\\\\n   &=G^{\\beta}(x)\\bigg(\\int_{\\vert x \\vert_a}^{\\infty}\\int_{\\Sigma_r} \\lambda(r,\\omega) f(r,\\omega)\\widetilde{G_1}^{(\\beta-s)\/\\alpha}(r)\\widetilde{G_1}^{(s-\\beta)\/\\alpha}(r)\\widetilde{W_1}^{(s-\\beta)\/s}(r)\\widetilde{W_1}^{(\\beta-s)\/s}(r)d\\omega_r dr\\bigg)^\\alpha\\nonumber\n   \\\\ & \n   \\le\\bigg(\\sup_{r>\\vert x \\vert_a}\\widetilde{G_1}^{(s-\\beta)}(r)\\widetilde{W_1}^{(s-\\beta)\\alpha\/s}(r)\\bigg)G^{\\beta}(x)\\bigg(\\int_{\\vert x \\vert_a}^{\\infty}\\widetilde{W}(r) {(\\int_r^{\\infty}\\widetilde{W}(s)ds)}^{(\\beta-s)\/s}dr\\bigg)^\\alpha\\nonumber\n \\\\\n&= \\bigg(\\sup_{\\vert y \\vert _a>\\vert x \\vert_a}{G}^{s}(y){W}^{\\alpha}(y)\\bigg)^{(s-\\beta)\/s}\\bigg(\\frac{s}{\\beta}\\bigg)^{\\alpha}\nG^{\\beta}(x) W^{(\\beta\\alpha)\/s}(x)\\nonumber\n\\\\ & \n \\le {\\bigg(\\frac{s}{\\beta}\\bigg)}^{\\alpha}\\bigg(\\sup_{\\vert y \\vert_a >\\vert x \\vert_a}{G}^{s}(y){W}^{\\alpha}(y)\\bigg)^{(1-{\\beta\/s})}\n\\bigg(\\sup_{{\\vert x \\vert_a}>0} G^{s}(x) W^{\\alpha}(x)\\bigg)^{\\beta\/s}\\nonumber\n\\\\ &   \\le {\\bigg(\\frac{s}{\\beta}\\bigg)}^{\\alpha}\\sup_{{\\vert x \\vert_a} >0} A_2(x;\\alpha,\\beta,s).\\nonumber\n   \\end{align*} \n  Therefore, we obtain\n   $$\n \\mathcal A_1 \\le {\\left(\\frac{s}{\\beta}\\right)}^{\\alpha} \\mathcal A_2.\n$$\nHence, we have for every $s>0$ the inequality\n$$\n\\mathcal A_1 \\le {\\left(\\max(1,\\frac{s}{\\beta})\\right)}^{\\alpha} \\mathcal A_2.\n$$\n    Conversely, we have for $s<\\beta$,\n   \\begin{align*}\n & G^s(x)W^\\alpha(x) \\\\\n & =G^s(x)\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta-s)\/\\alpha}(y)F^{(\\beta-s)\/\\beta}(y)F^{(s-\\beta)\/\\beta}(y)dy\\bigg)^\\alpha\\nonumber\\\\\n& =G^s(x)\\bigg(\\int_{\\vert x \\vert_a}^{\\infty}{\\int_{\\Sigma_r} \\lambda(r,\\omega)f(r,\\omega)}\\bigg(\\int_0^r\\int_{\\Sigma_s} \\lambda(s,\\sigma) g(s,\\sigma)ds d\\sigma_s\\bigg)^{(\\beta-s)\/\\alpha}\\nonumber\\\\\n&\\quad\\times\\widetilde{F_1}^{(\\beta-s)\/\\beta}(r)\\widetilde{F_1}^{(s-\\beta)\/\\beta}(r)drd\\omega_r \\bigg)^\\alpha\\nonumber \\\\\n & = G^s(x)\\bigg(\\int_{\\vert x \\vert_a}^{\\infty}{\\int_{\\Sigma_r} \\lambda(r,\\omega)f(r,\\omega)}\\widetilde{G_1}^{(\\beta-s)\/\\alpha}(r)\\widetilde{F_1}^{(\\beta-s)\/\\beta}(r)\\widetilde{F_1}^{(s-\\beta)\/\\beta}(r)d\\omega_r dr\\bigg)^\\alpha.\\nonumber\n\\end{align*}\n  Consequently, we can estimate\n   \\begin{align*}\n & G^s(x)W^\\alpha(x)\n \\\\ &\n \\le\\bigg(\\sup_{r>\\vert x \\vert_a}\\widetilde{G_1}^{(\\beta-s)\/\\alpha}(r)\\widetilde{F_1}^{(\\beta-s)\/\\beta}(r)\\bigg)^\\alpha G^{s}(x) \\bigg(\\int_{\\vert x \\vert_a}^{\\infty}\\widetilde{F}(r)\\left(\\int_r^{\\infty}\\widetilde{F}(s)ds\\right)^{(s-\\beta)\/\\beta} dr\\bigg)^\\alpha\\nonumber  \n \\\\ &=\\bigg(\\sup_{r>\\vert x \\vert_a}\\widetilde{G_1}^{\\beta}(r)\\widetilde{F_1}^{\\alpha}(r)\\bigg)^{(\\beta-s)\/\\beta}  G^{s}(x) {\\bigg(\\frac{\\beta}{s}\\bigg)}^\\alpha \\widetilde{F_1}^{(\\alpha s)\/\\beta}(r)\\nonumber \n \\\\ &\\le\\bigg(\\sup_{\\vert y \\vert_a>\\vert x \\vert_a} G^{\\beta}(y) F^{\\alpha}(y)\\bigg)^{(\\beta-s)\/\\beta}{\\bigg(\\frac{\\beta}{s}\\bigg)}^\\alpha \\bigg(\\sup_{\\vert x \\vert_a>0}G^{\\beta}(x) F^{\\alpha}(x)\\bigg)^{s\/\\beta}\\nonumber\n\\\\   &\\le\\bigg(\\frac{\\beta}{s}\\bigg)^\\alpha \\sup_{\\vert x \\vert_a>0} A_1 (x;\\alpha,\\beta),\\nonumber\n  \\end{align*}\n  which gives   \n      $\n      \\mathcal A_2 \\leq {\\left(\\frac{\\beta}{s}\\right)}^\\alpha \\mathcal A_1.\n     $\n     \n   On the other hand, for  $s\\geq\\beta$, when $\\vert y \\vert_a > \\vert x \\vert_a$  we have\n   $\n   G^{(\\beta-s)\/\\alpha}(y) \\le G^{(\\beta-s)\/\\alpha}(x).\n   $\n   \n   Therefore, we can estimate\n   \\begin{align*}\n   A_{2}(x;\\alpha,\\beta,s) &= G^{s}(x)\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta-s)\/\\alpha}(y)dy\\bigg)^\\alpha\\nonumber\n \\\\ & \n   \\le G^{s}(x)\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)dy\\bigg)^\\alpha G^{(\\beta-s)}(x)\\nonumber\n\\\\\n&=F^\\alpha(x)G^\\beta(x)\\nonumber\n\\end{align*}\n   i.e. $\\mathcal A_2 \\le \\mathcal A_1.$ \nTherefore, we have for $s>0$, the overall estimate\n$$\n\\mathcal A_2 \\le {\\left(\\max(1,\\frac{\\beta}{s})\\right)}^{\\alpha} \\mathcal A_1.\n$$\nHence we have also shown that\n$\n  \\mathcal A_1 \\approx \\mathcal A_2.\n$\n\n  \n  Next we observe that the proof of $\\boxed{\\mathcal A_1 \\approx \\mathcal A_3}$ follows along the same lines as that of $\\mathcal A_1 \\approx \\mathcal A_2$, where we just need to interchange the roles of $F$ and $G$.\n  \n  \\smallskip\n  $\\boxed{\\mathcal A_1 \\approx \\mathcal A_4}$\n \n \\smallskip\n  Let us denote\n  \\begin{align*}\nW_{0}(x) &:= \\int_{B(a,\\vert x \\vert_a)} f(y) G^{(\\beta+s)\/\\alpha}(y)dy\\nonumber\n \\\\    &= \\int_{0}^{\\vert x \\vert_a}\\int_{\\Sigma_r} \\lambda(r,\\omega) f(r,\\omega){G}^{(\\beta+s)\/\\alpha}(r,\\omega)d\\omega_r dr\\nonumber\\\\\n\\\\ \n  &=: \\int_{0}^{\\vert x \\vert_a}\\widetilde{W_0}(r)dr,\\nonumber\n\\end{align*}\n\\\\\n\\\\\nso that we can write\n  $$\n   A_4(x;\\alpha,\\beta,s) = G^{-s}(x)W_0^\\alpha(x).\n  $$   \n  We rewrite\n  $A_1$ as\n   \\begin{align*}\n   A_1(x;\\alpha,\\beta)&=G^{\\beta}(x)\\bigg(\\int_{\\mathbb X\\backslash{B(a,\\vert x \\vert_a)}}f(y)G^{(\\beta+s)\/\\alpha}(y)G^{-{(\\beta+s)\/\\alpha}}(y)dy\\bigg)^\\alpha\\nonumber\n   \\\\\n   &= G^\\beta(x)\\bigg(\\int_{\\vert x \\vert_a}^{\\infty}\\int_{\\Sigma_r} \\lambda(r,\\omega) f(r,\\omega){G}^{(\\beta+s)\/\\alpha}(r,\\omega)G^{-(\\beta+s)\/\\alpha}(r,\\omega) d\\omega_r dr\\bigg)^\\alpha\\nonumber\n \\\\\n   &=G^\\beta(x)\\bigg(\\int_{\\vert x \\vert_a}^{\\infty}\\widetilde{G_1}^{-(\\beta+s)\/\\alpha}(r)\\frac{d}{dr}\\bigg(\\int_0^r\\widetilde{W_0}(s)ds\\bigg)dr\\bigg)^\\alpha.\\nonumber\n   \\end{align*}\n  We can estimate this by  \n   \\begin{align*}\n   & \\le G^{\\beta}(x)\\bigg(\\widetilde{G_1}^{-(\\beta+s)\/\\alpha}(\\infty){W_0}(\\infty) + \\frac{(\\beta+s)}{\\alpha}\\int_{\\vert x \\vert_a}^{\\infty}\\widetilde{G}(r)(\\widetilde{G_1}(r))^{\\frac{-(\\beta+s)}{\\alpha}-1}W_{0}(r)dr\\bigg)^\\alpha\\nonumber\n  \\\\\n&   \\le G^{\\beta}(x)\\bigg(\\sup_{\\vert y \\vert_a >\\vert x \\vert_a} G^{-s}(y) W_{0}^{\\alpha}(y)\\bigg)\\times\n\\\\ & \n\\;\\; \\times \\bigg(\\widetilde{G_{1}}^{-\\beta\/\\alpha}{(\\infty)} + \\frac{(\\beta+s)}{\\alpha}\\int_{\\vert x \\vert_a}^{\\infty}\\widetilde{G_1}^{-(\\beta\/\\alpha)-1}(r) \\frac{d}{dr}\\bigg(\\int_0^r\\widetilde{G}(s)ds\\bigg)dr\\bigg)^\\alpha\\nonumber\n\\\\ &\n   \\le G^{\\beta}(x)\\sup_{\\vert y \\vert_a > 0} {A_{4}}(y;\\alpha,\\beta,s)\\bigg(\\widetilde{G_{1}}^{-\\beta\/\\alpha}{(\\infty)} + \\frac{(\\beta+s)}{\\beta} \\bigg(\\widetilde{G_1}^{-(\\beta\/\\alpha)}(\\vert x \\vert_a) -{\\widetilde{G_1}}^{-\\beta\/\\alpha}(\\infty)\\bigg)\\bigg)^\\alpha\\nonumber\n\\\\\n   &= \\sup_{\\vert y \\vert_a > 0} {A_{4}}(y;\\alpha,\\beta,s) \\bigg[ \\frac{(\\beta+s)}{\\beta} + \\bigg(1-\\frac{(\\beta+s)}{\\beta}\\bigg)\\bigg(\\frac{G(x)}{G(\\infty)}\\bigg)^{\\beta\/\\alpha}\\bigg ]^\\alpha\\nonumber\n \\\\ &\n   \\le \\bigg(1+\\frac{s}{\\beta}\\bigg)^\\alpha \\sup_{\\vert y \\vert_a>0} A_4(y;\\alpha,\\beta,s),\n   \\end{align*}\n  where the expressions like $G(\\infty)$ make sense since $g\\in L^1(\\mathbb X)$.\n   Therefore, we obtain   \n   $$\n   \\mathcal A_1\\le (1+s\/\\beta)^\\alpha \\mathcal A_4.\n   $$\n   To prove the opposite inequality, we assume that\n  $$\n   \\sup_ {\\vert x \\vert_a>0}{ A_1}(x;\\alpha, \\beta)<\\infty.\n  $$\n   Then we have   \n   \\begin{align}\n    A_{4}(x;\\alpha,\\beta,s)&= G^{-s}(x)\\bigg(\\int_{B(a,\\vert x \\vert_a)} G^{(\\beta+s)\/\\alpha}(y)f(y)dy\\bigg)^\\alpha\\nonumber\n   \\end{align}\n\\begin{align}\n&=G^{-s}(x)\\bigg(\\int_0^{\\vert x \\vert_a}{\\widetilde{ G_1}}^{(\\beta+s)\/\\alpha}(r) \\frac{d}{dr}\\bigg(-\\int_r^{\\infty}{\\widetilde{F}(s)}ds\\bigg)dr\\bigg)^\\alpha\\nonumber\n\\end{align}\n\\begin{align}\n&=G^{-s}(x)\\bigg({\\widetilde{ G_1}}^{(\\beta+s)\/\\alpha}(r){\\widetilde{F_1}}(r) \\bigg\\vert_{\\vert x \\vert_a}^0+\\frac{\\beta+s}{\\alpha}\\int_0^{\\vert x \\vert_a}{\\widetilde{F_1}}(r){\\widetilde{ G_1}}^{(\\beta+s)\/\\alpha-1}(r) \\frac{d}{dr}\\bigg(\\int_0^r{\\widetilde{G}(s)}ds\\bigg)dr\\bigg)^\\alpha\\nonumber\n\\end{align}\n\\begin{align}\n\\le G^{-s}(x)\\bigg (\\sup_{0<r<\\vert x \\vert_a}{\\widetilde{ G_1}}^{\\beta}(r){\\widetilde{F_1}}^\\alpha(r)\\bigg)\\bigg(\\frac{\\beta+s}{\\alpha}\\int_0^{\\vert x \\vert_a}{\\widetilde{ G_1}}^{s\/\\alpha-1}(r)\\frac{d}{dr}(\\int_0^r{\\widetilde{G}(s)}ds)dr\\bigg)^\\alpha\\nonumber\n\\end{align}\n\\begin{align}\n\\le {\\left(\\frac{\\beta+s}{\\alpha}\\right)}^\\alpha \\sup _{\\vert y \\vert_a>0}G^\\beta(y) F^\\alpha(y) G^{-s}(x)\\bigg(\\frac{\\alpha}{s} G^{s\/\\alpha}(x)\\bigg)^\\alpha\\nonumber\n\\end{align}\n\\begin{align}\n&= {\\left(\\frac{\\beta+s}{s}\\right)}^\\alpha \\sup_{\\vert x \\vert_a>0} A_1(x;\\alpha,\\beta),\\nonumber\n\\end{align}\nwhere we have used that $f\\in L^1(\\mathbb X)$. Hence we have proved that \n$A_1\\approx A_4.$\n\nThe proof of $\\boxed{A_1\\approx A_5}$ follows\n the same lines as that of the case $A_1\\approx A_4$ if we interchange the roles of $F$ and $G$.\n\\end{proof}\n\n\n\n\n\\section{Equivalent conditions for the Hardy inequality}\n\\label{SEC:proofH}\n\nIn this section we prove Theorem \\ref{THM:Hardy1} and also give some comments concerning Theorem \\ref{THM:Hardy2}.\nWithout loss of generality we can assume that $f\\geq 0$.\nThen we observe that if in Theorem \\ref{THM:equivalence} we take \n$$f(x)=u(x), \\; g(x)=v^{1-p'}(x), \\;\\alpha=\\frac{1}{q}, \\; \\beta=\\frac{1}{p'},$$ \nthen it follows that we have the equivalence of the quantities \n$$\n                          \\mathcal D_{1} \\approx \\mathcal D_{2} \\approx \\mathcal D_{3} \\approx \\mathcal D_{4} \\approx \\mathcal D_{5}.\n$$                   \n   So, we first assume that any one of these equivalent conditions holds true. In particular,  $\\mathcal D_1<\\infty$, and using  polar coordinates \\eqref{EQ:polar}, we have for every $a>0$ that\n \\begin{equation}\\label{EQ:D1-cond}\n \\bigg\\{\\int_a^{\\infty}\\int_{\\Sigma_r}\n \\lambda(r,\\omega)u(r,\\omega)d\\omega_r dr\\bigg\\}^\\frac{1}{q}\\bigg\\{\\int_0^{a}\\int_{\\Sigma_r}\\lambda(r,\\omega)v^{1-p'}(r,\\omega)d\\omega_r dr\\bigg\\}^\\frac{1}{p'}\\le \\mathcal D_1.\n \\end{equation}\nWe denote\n$$\n     h(t):=\\bigg( \\int_0^t \\int_{\\Sigma_s} \\lambda(s,\\sigma) v^{1-p'}(s,\\sigma)ds d\\sigma_s\\bigg)^\\frac{1}{pp'},\n$$\n     \\begin{align}\n     \\widetilde{U}_1(t):= \\int_{\\Sigma_t}  \\lambda(t,\\omega) u(t,\\omega)d\\omega_t\\nonumber,\n     \\end{align}   \n     \\begin{align}\n       F_1(s):= \\int_{\\Sigma_s} \\lambda(s,\\sigma) [f(s,\\sigma) v^{\\frac{1}{p}}(s,\\sigma)h(s)]^p d\\sigma_s\\nonumber,\n     \\end{align}     \n    and                  \n     \\begin{align}\n     H_1(t):=\\int_0^t \\int_{\\Sigma_s} \\lambda(s,\\sigma)[v^\\frac{1}{p}(s,\\sigma)h(s)]^{-p'}d\\sigma_s ds.\\nonumber\n     \\end{align}\nThen using polar coordinates  \\eqref{EQ:polar}, H\\\"older's inequality, and Minkowski's inequality, the left side of \\eqref{EQ:Hardy1} can be estimated as \n   \\begin{align}\\label{EQ:e0}\n  & \\int_\\mathbb X u(x) \\bigg(\\int_{B(a,\\vert x \\vert_a)}f(y)dy\\bigg)^qdx \\nonumber\n  \\\\ &\\leq\\int_0^{\\infty}\\int_{\\Sigma_r}\\lambda(r,\\omega)u(r,\\omega)\n   \\bigg( \\int_0^r \\int_{\\Sigma_s}  \\lambda(s,\\sigma)[f(s, \\sigma)v^{\\frac{1}{p}}(s,\\sigma)h(s)]^pdsd\\sigma_s \\bigg)^\\frac{q}{p}\\nonumber\n   \\\\\n   &\\quad\\times \\bigg(\\int_0^r \\int_{\\Sigma_s} \\lambda(s,\\sigma) [v^{\\frac{1}{p}}(s,\\sigma) h(s)]^{-p'}ds d\\sigma_s \\bigg)^\\frac{q}{p'}d\\omega_r dr\\nonumber\n \\\\\n     &=\\int_0^\\infty \\widetilde{U}_1(r)\\bigg(\\int_0^r F_1(s)ds\\bigg)^\\frac{q}{p}H_1^\\frac{q}{p'}(r)dr\\nonumber\n   \\\\\n  &   \\le \\bigg(\\int_0^\\infty F_1(s) \\bigg(\\int_s^\\infty \\widetilde{U}_1(r) H_1^\\frac{q}{p'}(r)dr\\bigg)^\\frac{p}{q}ds\\bigg)^\\frac{q}{p}.\n    \\end{align}                   \n Denoting \n$$\n     V_1(s):= \\int_{\\Sigma_s}  \\lambda(s,\\sigma) v^{1-p'}(s,\\sigma)d\\sigma_s,\n$$\n     and using inequality \\eqref{EQ:D1-cond} we can estimate\n     \\begin{align}\\label{EQ:e1}\n    & H_1(t) =\\int_0^t \\int_{\\Sigma_r} \\lambda(r,\\sigma)[v^\\frac{1}{p}(r,\\sigma)h(r)]^{-p'}d\\sigma_r dr\\nonumber\n     \\\\\n     &=\\int_0^t \\int_{\\Sigma_r} \\lambda(r,\\sigma) v^{1-p'}(r,\\sigma)\\bigg( \\int_0^r \\int_{\\Sigma_{\\rho}} \\lambda(\\rho,\\omega) v^{1-p'}(\\rho,\\omega)d\\rho d\\omega_{\\rho}\\bigg)^{-\\frac{1}{p}}dr d\\sigma_r\\nonumber\n     \\\\\n     &=\\int_0^t V_1(r)\\bigg(\\int_0^r  V_1(\\rho) d\\rho \\bigg)^{-\\frac{1}{p}} dr\\nonumber\n\\\\\n&=p'\\bigg(\\int_0^t V_1(r)dr \\bigg)^\\frac{1}{p'}\\nonumber\n\\\\\n&=p'\\bigg(\\int_0^t \\int_{\\Sigma_r} \\lambda(r,\\sigma) v^{1-p'}(r,\\sigma)dr d\\sigma_r \\bigg)^\\frac{1}{p'}\\bigg(\\int_t^\\infty \\int_{\\Sigma_r} \\lambda(r,\\sigma) u(r,\\sigma)dr d\\sigma_r \\bigg)^\\frac{1}{q}\\nonumber\n\\\\ \n&\\quad\\times \\bigg(\\int_t^\\infty \\int_{\\Sigma_r} \\lambda(r,\\sigma) u(r,\\sigma)dr d\\sigma_r \\bigg)^{-\\frac{1}{q}}  \\nonumber\n\\\\\n& \\le p' \\mathcal D_1 \\bigg(\\int_t^\\infty\\widetilde{U}_1(s)ds \\bigg)^{-\\frac{1}{q}}.\n     \\end{align}\n     At the same time we can also estimate\n     \\begin{align}\\label{EQ:e2}\n     \\int_s^\\infty \\widetilde{U}_1(t) \\bigg(\\int_t^\\infty \\widetilde{U}_1(\\tau)d\\tau\\bigg)^{-\\frac{1}{p'}} dt\n    &=-p \\int_s^\\infty {\\frac{d}{dt}} \\bigg(\\int_t^\\infty \\widetilde{U}_1(\\tau)d\\tau\\bigg)^ \\frac{1}{p} dt\\nonumber \n     \\\\\n     &=p \\left(\\int_s^\\infty \\widetilde{U}_1(t)dt \\right)^\\frac{1}{p}\\nonumber\n     \\\\\n     &=p \\left(\\int_s^ \\infty \\int_{\\Sigma_t} \\lambda(t,\\omega) u(t,\\omega)dt d\\omega_t\\right)^\\frac{1}{p}\\nonumber\n     \\\\\n     &= p \\left\\{\\left(\\int_s^ \\infty \\int_{\\Sigma_t}\\lambda(t,\\omega) u(t,\\omega)dtd\\omega_t\\right)^\\frac{1}{q} \\right.\\nonumber\n     \\\\\n     &\\quad\\times \\left.\\left(\\int_0^s \\int_{\\Sigma_t} \\lambda(t,\\omega)v^{1-p'}(t,\\omega)dtd\\omega_t\\right)^\\frac{1}{p'} \\right\\} ^\\frac{q}{p}\\nonumber\n     \\\\\n     &\\quad\\times \\left(\\int_0^s\\int_{\\Sigma_t} \\lambda(t,\\omega)v^{1-p'}(t,\\omega)dtd\\omega_t\\right)^{-\\frac{q}{p'p}}\\nonumber\n     \\\\\n     &\\le p \\mathcal D_1 ^\\frac{q}{p} h^{-q}(s), \n     \\end{align}\n     in view of \\eqref{EQ:D1-cond}.\n     Therefore using \\eqref{EQ:e1} and \\eqref{EQ:e2} in \\eqref{EQ:e0}, we have\n     \\begin{align}\n     \\int_\\mathbb X u(x) \\bigg(\\int_{B(a,\\vert x \\vert_a)} f(y) dy \\bigg)^qdx\\leq \\mathcal D_1^q {p'}^\\frac{q}{p'}p \\bigg(\\int_\\mathbb X v(x) {f(x) }^p dx \\bigg)^\\frac{q}{p}.\\nonumber\n     \\end{align}\n     Hence, it follows that \\eqref{EQ:Hardy1} holds with\n     $C\\leq \\mathcal D_1(p')^{\\frac{1}{p'}} p^\\frac{1}{q}$ proving one of the relations in \\eqref{EQ:constants}.\n  \n     Conversely, let us assume that inequality \\eqref{EQ:Hardy1} holds, and consider the function\n     $$f(x)=v^{1-p'}(x) \\chi_{(0,t)} (\\vert x \\vert_a)$$\n     for some $ t>0$, and where $\\chi$ is the cut-off function.\n     With this function, the right hand side of of \\eqref{EQ:Hardy1} takes the form    \n        \\begin{align}\n       &\\bigg(\\int_{\\mathbb X} v(x){ \\vert f(x) \\vert }^pdx\\bigg)^\\frac{1}{p}\n       =  \\bigg(\\int_{\\vert x \\vert_a \\le t} v^{1-p'}(x)dx\\bigg)^\\frac{1}{p}.\\nonumber\n\\end{align}\nAt the same time, the left hand side of of \\eqref{EQ:Hardy1} takes the form\n\\begin{align}\n\\bigg(\\int_\\mathbb X\\bigg(\\int_{B(a,\\vert x \\vert_a)}\\vert f(y) \\vert dy\\bigg)^q u(x)dx\\bigg)^\\frac{1}{q}\n&\\geq \\bigg(\\int_{\\vert x \\vert_a \\geq t} \\bigg(\\int_{B(a,\\vert x \\vert_a)}\\vert f(y) \\vert dy\\bigg)^q u(x)dx\\bigg)^\\frac{1}{q}\\nonumber\n\\\\\n&=\\bigg(\\int_{\\vert x \\vert_a \\geq t} \\bigg(\\int_{\\vert y \\vert_a \\leq t} v^{1-p'}(y) dy\\bigg)^q u(x)dx\\bigg)^\\frac{1}{q}\\nonumber\n\\\\\n&=\\bigg(\\int_{\\vert x \\vert_a \\geq t}  u(x)dx\\bigg)^\\frac{1}{q} \\bigg(\\int_{\\vert y \\vert_a \\leq t} v^{1-p'}(y) dy\\bigg).\\nonumber\n\\end{align}\nAltogether the inequality  \\eqref{EQ:Hardy1} takes the form\n\\begin{align}\n     \\bigg(\\int_{\\vert x \\vert_a \\ge t} u(x)dx\\bigg)^\\frac{1}{q}\\bigg(\\int_{\\vert y \\vert_a \\le t} v^{1-p'}(y)dy \\bigg)\\le C \\bigg(\\int_{\\vert x \\vert_a \\le t} v^{1-p'}(x)dx\\bigg)^\\frac{1}{p},\\nonumber\n     \\end{align}\n     which gives \n     $\\mathcal D_1\\le C$.\n     Hence, we have the equivalence and the second relation in  \\eqref{EQ:constants}.\n\n\\smallskip\n  As for Theorem \\ref{THM:Hardy2}, \nif we take $f(x)= v^{1-p'}$ , $g(x)=u(x)$, $\\alpha= \\frac{1}{p'}$  and  $\\beta= \\frac{1}{q}$ in Theorem \\eqref{THM:equivalence}, we find that \n$$\n\\mathcal D_{1}^{*} \\approx \\mathcal D_{2}^{*} \\approx \\mathcal D_{3}^{*} \\approx \\mathcal D_{4}^{*} \\approx \\mathcal D_{5}^{*}.$$\n\nConsequently, we can show Theorem \\ref{THM:Hardy2} by the argument similar to that in Section \\ref{SEC:proofH} where Theorem \\ref{THM:Hardy1} was proved. \nWe also note that in the case of homogeneous groups, we can actually also derive it from \nTheorem \\ref{THM:Hardy1} by the involutive change of variables $x\\mapsto x^{-1}.$\n\n\\medskip \\noindent\n{\\bf Data accessibility.} No new data was collected or generated during the course of research.\n\n\\medskip \\noindent\n{\\bf Competing interests.} We have no competing interests.\n\n\\medskip \\noindent\n{\\bf Authors' contributions.} The authors contributed equally to this paper.\n\n\\medskip \\noindent\n{\\bf Acknowledgements.}  The authors would like to thank Nurgissa Yessirkegenov for discussions and for checking our calculations. We would also like to thank all 5 referees of this paper for useful comments.\n\n\\medskip \\noindent\n{\\bf  Funding statement.}\nThe first author was supported in parts by the FWO Odysseus Project, EPSRC\ngrant EP\/R003025\/1 and by the Leverhulme Grant RPG-2017-151. \n\n\\medskip \\noindent\n{\\bf Ethics statement.} The work did not involve any collection of human data.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nThank you for downloading this Simon & Schuster eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## Contents\n\nThe Path of the Flood - Map\n\nPennsylvania - Map\n\nAcknowledgments\n\nList of Illustrations\n\nEpigraph\n\nI. The sky was red\n\nII. Sailboats on the mountain\n\nIII. \"There's a man came from the lake.\"\n\nIV. Rush of the torrent\n\nV. \"Run for your lives!\"\n\nVI. A message from Mr. Pitcairn\n\nVII. In the valley of death\n\nVIII. \"No pen can describe . . .\"\n\nIX. \"Our misery is the work of man.\"\n\nPhotographs\n\nAbout David McCullough\n\nList of Victims\n\nBibliography\n\nIndex\nFor Rosalee\n\n## Acknowledgments\n\nThe material for this book was gathered from the files of newspapers, from unpublished reminiscences, from letters and diaries, from Johnstown Flood \"histories\" that were best sellers in their day and from books and pamphlets that were privately printed, from court records, engineering reports, local histories, and rare old maps, from old photographs, and from hours of taped conversations with survivors of the Johnstown Flood.\n\nA bibliography is included at the back of the book, but I want to acknowledge my indebtedness to four works in particular: J. J. McLaurin's The Story of Johnstown, which for all its Victorian embellishments and inaccuracies is the finest by far of the books \"gotten up\" by journalists in 1889; The Reverend David J. Beale's Through the Johnstown Flood, in many ways the best book on the flood and unquestionably the best-written and most reliable of accounts by survivors; A History of Johnstown and the Great Flood of 1889: A Study of Disaster and Rehabilitation, which is a doctoral thesis written by the late Nathan D. Shappee and the only scholarly study of the disaster; and a recently discovered transcription of testimony taken by the Pennsylvania Railroad during the summer of 1889, which has been invaluable. (Most of the dialogue in Chapters 3 and , for example, has been taken directly from this transcription, which, in all, runs to nearly 500 typewritten pages, and no part of which has been previously published.)\n\nI very gratefully acknowledge my debt to the following flood survivors who kindly gave me so much of their time to talk about their experiences, some of whom have since passed on: Mrs. Kate Miltenberger, Mr. and Mrs. Harry Hesselbein, and David Fetterman, all of Johnstown; U. Ed Schwartzentruver of South Fork; Mrs. Gertrude Quinn Slattery of Wilkes-Barre, who also read the manuscript; and Dr. Victor Heiser of New York, who, in addition to vivid descriptions of his own experiences, supplied wonderful insights into the Johnstown of the 1880's, and who read the manuscript.\n\nI am grateful also to the two Johnstown ladies, both survivors, who shared memories of their illustrious family, but who asked that I not mention their names.\n\nI wish to thank too the many others in Johnstown who were helpful, and especially the following: Irving London, who led me to the Pennsylvania Railroad testimony and who made available his extraordinary collection of flood photographs; Harold Strayer; Gustaf Hultman of the National Park Service; Elit Felix; Walter Krebs, president of the Johnstown Tribune-Democrat, who made available the paper's reference library and files; Don Matthews, Jr., also of the Tribune-Democrat, who made numerous helpful suggestions; Frank Dell and Ron Stephenson of Station WJAC; the City Clerk's office; and the staff of the Cambria Public Library.\n\nIn addition, I am much indebted to Edna Lehman, who made available important documentary material at the Cambria County Historical Society at Ebensburg; to the late Robert Heppenstall of Pittsburgh, for information on his father's heroic action; to Mrs. John E. Hannon, Sr., of Detroit, who wrote to me at length about her grandfather, W. Horace Rose; to Mrs. Bernard McGuire of Cresson, who let me borrow a diary kept by her grandfather, Isador Lilly of Ebensburg; to Dr. Philip Bishop, Dr. John White, and Donald Berkebile of the Smithsonian Institution; to Mr. and Mrs. Fred Livengood, Sr., of Somerset; to the late Mrs. O. C. Gaub of Pittsburgh; to Evan Stineman of South Fork; and to Al Danel of St. Michael, Pennsylvania.\n\nAlso I wish to express my gratitude for the facilities offered by the staff of the Pennsylvania Room at the Carnegie Library, Pittsburgh; the Allegheny County Court of Claims, Pittsburgh; the Local History and Genealogy Room at the New York Public Library; the New York Historical Society; the Library of Congress; the University of Pennsylvania; the University of Pittsburgh; the Yale University Library; the White Plains Public Library; and the Boston Public Library.\n\nAnd for their suggestions and encouragement I wish finally to thank Walter McQuade, Walter Lord, Roger Butterfield, David Plowden, Heywood Broun, Jr., David Allison, Frank Fogarty, James Morrison, Royall O'Brien, Charles T. Siebert, Jr.; Audre Proctor, who typed the manuscript; my mother and father; and my wife, Rosalee.\n\nD. McC.\n\n## List of Illustrations\n\nMap of Johnstown at time of Civil War\n\nView of South Fork Fishing and Hunting Club\n\nMoorhead house\n\nJohn G. Parke, Jr.\n\nRemains of South Fork dam\n\nAndrew Carnegie\n\nHenry Clay Frick\n\nPhilander C. Knox\n\nRobert Pitcairn\n\nDaniel J. Morrell\n\nJohn Fulton\n\nTom L. Johnson\n\nCaptain Bill Jones\n\nTwo views of Johnstown\u2014before and after the flood\n\nWreckage near the Methodist church\n\nSchultz house\n\nDebris and \"corpse\"\n\nDevastation in lower Johnstown\n\nPennsylvania Railroad bridge\n\nView of the Point\n\nGertrude Quinn\n\nVictor Heiser\n\nRichard Harding Davis\n\nClara Barton\n\nSurvivors of the flood\n\nRelief rations\n\nAdams Street schoolhouse\u2014temporary morgue\n\nThe broken dam\n\nRelief commissary\n\nSearching among the dead\n\nHappy reunions\n\nLynch law\n\nGrandview Cemetery\n\"We are creatures of the moment; we live from one little space to another; and only one interest at a time fills these.\"\n\n\u2014William Dean Howells\n\nin A Hazard of New Fortunes, 1889.\n\n## I\n\n## The sky was red\n\n### -1-\n\nAgain that morning there had been a bright frost in the hollow below the dam, and the sun was not up long before storm clouds rolled in from the southeast.\n\nBy late afternoon a sharp, gusty wind was blowing down from the mountains, flattening the long grass along the lakeshore and kicking up tiny whitecaps out in the center of the lake. The big oaks and giant hemlocks, the hickories and black birch and sugar maples that crowded the hillside behind the summer colony began tossing back and forth, creaking and groaning. Broken branches and young leaves whipped through the air, and at the immense frame clubhouse that stood at the water's edge, halfway among the cottages, blue wood smoke trailed from great brick chimneys and vanished in fast swirls, almost as though the whole building, like a splendid yellow ark, were under steam, heading into the wind.\n\nThe colony was known as the South Fork Fishing and Hunting Club. It was a private summer resort located on the western shore of a mountain lake in Cambria County, Pennsylvania, about halfway between the crest of the Allegheny range and the city of Johnstown. On the afternoon of Thursday, May 30, Memorial Day, 1889, the club was not quite ten years old, but with its gaily painted buildings, its neat lawns and well-tended flower beds, it looked spanking new and, in the gray, stormy half-light, slightly out of season.\n\nIn three weeks, when the summer season was to start, something like 200 guests were expected. Now the place looked practically deserted. The only people about were a few employees who lived at the clubhouse and some half dozen members who had come up from Pittsburgh for the holiday. D. W. C. Bidwell was there; so were the young Clarke brothers, J. J. Lawrence, and several of the Sheas and Irwins. Every now and then a cottage door slammed, voices called back and forth from the boathouses. Then there would be silence again, except for the sound of the wind.\n\nSometime not long after dark, it may have been about eight thirty, a young man stepped out onto the long front porch at the clubhouse and walked to the railing to take a look at the weather. His name was John G. Parke, Jr. He was clean-shaven, slight of build, and rather aristocratic-looking. He was the nephew and namesake of General John G. Parke, then superintendent of West Point. But young Parke was a rare item in his own right for that part of the country; he was a college man, having finished three years of civil engineering at the University of Pennsylvania. For the present he was employed by the South Fork Fishing and Hunting Club as the so-called \"resident engineer.\" He had been on the job just short of three months, seeing to general repairs, looking after the dam, and supervising a crew of some twenty Italian laborers who had been hired to install a new indoor plumbing system, and who were now camped out of sight, back in the woods.\n\nIn the pitch dark he could hardly see a thing, so he stepped down the porch stairs and went a short distance along the boardwalk that led through the trees to the cottages. The walk, he noticed, was slightly damp. Apparently, a fine rain had fallen sometime while he was inside having his supper. He also noticed that though the wind was still up, the sky overhead was not so dark as before; indeed, it seemed to be clearing off some. This was not what he had expected. Windstorms on the mountain nearly always meant a heavy downpour almost immediately after\u2014\"thunder-gusts\" the local men called them. Parke had been through several already in the time he had been at the lake and knew what to expect.\n\nIt would be as though the whole sky were laying siege to the burly landscape. The rain would drum down like an unyielding river. Lightning would flash blue-white, again and again across the sky, and thunderclaps would boom back and forth down the valley like a cannonade, rattling every window along the lakeshore.\n\nThen, almost as suddenly as it had started, the siege would lift, and silent, milky steam would rise from the surface of the water and the rank smell of the sodden forest floor would hang on in the air for hours.\n\nTonight, however, it appeared there was to be no storm. Parke turned and walked back inside. About nine-thirty he went upstairs, climbed into bed, and went to sleep.\n\nAbout an hour and a half later, very near eleven, the rain began. It came slamming through the blackness in huge wind-driven sheets, beating against the clubhouse, the tossing trees, the lake, and the dark, untamed country that stretched off in every direction for miles and miles.\n\nThe storm had started out of Kansas and Nebraska, two days before, on May 28. The following day there had been hard rains in Kansas, Missouri, Illinois, Michigan, Indiana, Kentucky, and Tennessee. Trains had been delayed, roads washed out. In Kansas, along the Cottonwood River, a dozen farms had been flattened by tornado-force winds and several people had been killed. In northern Michigan and parts of Indiana there had been sudden snow squalls. Warnings had been telegraphed east. On the night of the 29th the U.S. Signal Service issued notices that the Middle Atlantic states were in for severe local storms. On the morning of May 30 all stations in the area reported \"threatening weather.\"\n\nWhen the storm struck western Pennsylvania it was the worst downpour that had ever been recorded for that section of the country. The Signal Service called it the most extensive rainfall of the century for so large an area and estimated that from six to eight inches of rain fell in twenty-four hours over nearly the entire central section. On the mountains there were places where the fall was ten inches.\n\nBut, at the same time, there were astonishing disparities between the amount of rainfall at places within less than a hundred-mile radius. At the South Fork Fishing and Hunting Club, for example, a pail left outside overnight would have five inches of water in it the next morning when the rain was still coming down. The total rainfall at the clubhouse would be somewhere near seven inches. In Pittsburgh, just sixty-five miles to the west as the crow flies, the total rainfall would be only one and a half inches.\n\nBut as the storm beat down on the mountain that night, John G. Parke, Jr., who would turn twenty-three in less than a month, slept on, never hearing a thing.\n\n### -2-\n\nMost of the holiday crowds were back from the cemetery by the time the rain began Thursday afternoon. It had been the customary sort of Memorial Day in Johnstown, despite the weather.\n\nPeople had been gathering along Main Street since noon. With the stores closed until six, with school out, and the men off from the mills, it looked as though the whole town was turning out. Visitors were everywhere, in by special trains from Somerset, Altoona, and other neighboring towns. The Ancient Order of Hibernians, \"a stalwart, vigorous looking body of men,\" as the Johnstown Tribune described them, was stopping over for its annual state convention. Hotels were full and the forty-odd saloons in Johnstown proper were doing a brisk business.\n\nThe Reverend H. L. Chapman, who lived two doors off Main, in the new Methodist parsonage facing the park, later wrote, \"The morning was delightful, the city was in its gayest mood, with flags, banners and flowers everywhere . . . we could see almost everything of interest from our porch. The streets were more crowded than we had ever seen before.\"\n\nThe parade, late starting as always, got under way about two-thirty, marched up Main, past the Morrell place, on by the Presbyterian Church and the park, clear to Bedford Street. There it turned south and headed out along the river to Sandy Vale, where the war dead were buried. The fire department marched, the Morrellville Odd Fellows, the Austrian Music Society, the Hornerstown Drum Corps, the Grand Army Veterans, and the Sons of Veterans, and half a dozen or more other groups of various shapes and sizes, every one of them getting a big cheer, and especially the Grand Army men, several of whom were beginning to look as though the three-mile tramp was a little more than they were up to.\n\nHow much things had changed since they had marched off to save the Union! It had been nearly thirty years since Lincoln had first called for volunteers. Grant and Lee were both dead, and there were strapping steelworkers with thick, black mustaches standing among the crowds along Main Street who had been born since Appomattox.\n\nAt the start of the war Johnstown had been no more than a third the size it was now; and ten years before that, it had been nothing but a sleepy little canal town with elderbushes growing high along Main, and so quiet you could hear the boat horns before the barges cleared the bend below town.\n\nBut ever since the war, with the west opening up, the Cambria Iron Company had had its giant three-ton converters going night and day making steel for rails and barbed wire, plowshares, track bolts, and spring teeth for harrows. The valley was full of smoke, and the city clanked and whistled and rumbled loud enough to be heard from miles off. At night the sky gleamed so red it looked as though the whole valley were on fire. James Quinn, one of Johnstown's most distinguished-looking Grand Army veterans and its leading dry-goods merchant, enjoyed few sights more. \"The sure sign of prosperity,\" he called it.\n\nYears after, Charlie Schwab, the most flamboyant of Carnegie's men, described the view of Johnstown from his boyhood home in the mountain town of Loretto, nearly twenty miles to the northeast.\n\n\"Along toward dusk tongues of flame would shoot up in the pall around Johnstown. When some furnace door was opened the evening turned red. A boy watching from the rim of hills had a vast arena before him, a place of vague forms, great labors, and dancing fires. And the murk always present, the smell of the foundry. It gets into your hair, your clothes, even your blood.\"\n\nMost of the men watching the parade that Memorial Day would have taken a somewhat less romantic view. In the rolling mills they worked under intense heat on slippery iron floors where molten metal went tearing by and one false step or slow reaction could mean horrible accidents. Most of them worked a ten- or even twelve-hour day, six days a week, and many weeks they worked the hated \"long turn,\" which meant all day Sunday and on into Monday. If they got ten dollars for a week's work they were doing well.\n\nA visiting journalist in 1885 described Johnstown as \"new, rough, and busy, with the rush of huge mills and factories and the throb of perpetually passing trains.\" The mills were set just below town in the gap in the mountains where the Conemaugh River flows westward. On the hillside close to the mills the trees had turned an evil-looking black and grew no leaves.\n\nJohnstown of 1889 was not a pretty place. But the land around it was magnificent. From Main Street, a man standing among the holiday crowds could see green hills, small mountains, really, hunching in close on every side, dwarfing the tops of the houses and smokestacks.\n\nThe city was built on a nearly level flood plain at the confluence of two rivers, down at the bottom of an enormous hole in the Alleghenies. A visitor from the Middle West once commented, \"Your sun rises at ten and sets at two,\" and it was not too great an exaggeration.\n\nThe rivers, except in spring, appeared to be of little consequence. The Little Conemaugh and Stony Creek, or the Stony Creek, as everyone in Johnstown has always said (since it is the Stony Creek River), are both more like rocky, oversized mountain streams than rivers. They are about sixty to eighty yards wide. Normally their current is very fast; in spring they run wild. But on toward August, as one writer of the 1880's said, there are places on either river where a good jumper could cross on dry stones.\n\nThe Little Conemaugh, which is much the swifter of the two, rushes in from the east, from the Allegheny Mountain. It begins near the very top of the mountain, about eighteen miles from Johnstown, at a coal town called Lilly. Its sources are Bear Rock Run and Bear Creek, Trout Run, Bens Creek, Laurel Run, South Fork Creek, Clapboard Run and Saltlick Creek. From an elevation of 2,300 feet at Lilly, the Little Conemaugh drops 1,147 feet to Johnstown.\n\nThe Stony Creek flows in from the south. It is a broader, deeper river than the other and is fed by streams with names like Beaver Dam Run, Fallen Timber Run, Shade Creek, and Paint Creek. Its total drainage is considerably more than that of the Little Conemaugh, and until 1889 it had always been thought to be the more dangerous of the two.\n\nWhen they meet at Johnstown, the rivers form the Conemaugh, which, farther west, joins the Loyalhanna to form the Kiskiminetas, which in turn flows into the Allegheny about eighteen miles above Pittsburgh.\n\nAt Johnstown it was as though the bottom had dropped out of the old earth and left it angry and smoldering, while all around, the long, densely forested ridges, \"hogbacks\" they were called, rolled off in every direction like a turbulent green sea. The climb up out of the city took the breath right out of you. But on top it was as though you had entered another world, clean, open, and sweet-smelling.\n\nIn 1889 there were still black bear and wildcats on Laurel Hill to the west of town. Though the loggers had long since stripped the near hills, there were still places within an hour's walk from Main Street where the forest was not much different than it had been a hundred years before.\n\nNow and then an eagle could still be spotted high overhead. There were pheasants, ruffed grouse, geese, loons, and wild turkeys that weighed as much as twenty pounds. Plenty of men marching in the parade could remember the time before the war when there had been panthers in the mountains big enough to carry off a whole sheep. And it had been only a few years earlier when passenger pigeons came across the valley in numbers beyond belief. One January the Tribune wrote: \"On Saturday there were immense flocks of wild pigeons flying over town, but yesterday it seemed as if all the birds of this kind at present in existence throughout the entire country were engaged in gyrating around overhead. One flock was declared to be at least three miles in length by half a mile wide.\"\n\nStill, many days there were in the valley itself when the wind swept away the smoke and the acrid smell of the mills and the air was as good as a man could ask for. Many nights, and especially in winter, were the way mountain nights were meant to be, with millions of big stars hanging overhead in a sky the color of coal.\n\nLooking back, most of the people who would remember Johnstown as it was on that Memorial Day claimed it was not as unpleasant a place as one might imagine. \"People were poor, very poor by later standards,\" one man said, \"but they didn't know it.\" And there was an energy, a vitality to life that they would miss in later years.\n\nMany of the millworkers lived in cheap, pine-board company houses along the riverbanks, where, as the Tribune put it, \"Loud and pestiferous stinks prevail.\" But there were no hideous slums, such as had spread across the Lower East Side of New York or in Chicago and Pittsburgh. The kind of appalling conditions that would be described the next year by Jacob Riis in his How the Other Half Lives did not exist then in Johnstown. No one went hungry, or begging, though there were always tramps about, drifters, who came with the railroad, heading west nearly always, knocking at back doors for something to eat.\n\nThey were part of the landscape and people took them for granted, except when they started coming through in big numbers and there were alarming stories in the papers about crowds of them hanging around the depot.\n\nOne diary, kept by a man who lived outside of town, includes a day-by-day tramp count. \"Wednesday, May 1, 1889, Two Tramps . . . Thursday, May 2, Two Tramps,\" and so on, with nearly a tramp or two every day, week after week.\n\nNew people came to town, found a job or, if not, moved on again, toward Pittsburgh. But for most everyone who decided to stay there was work. Although lately, Johnstown men, too, had been picking up and going west to try their luck at the mills in St. Louis or the mines in Colorado. And lately the jobs they left behind were being filled by \"hunkies\" brought in to \"work cheap.\"\n\nThe idea did not please people much. Nor did it matter whether the contract workers were Italians, Poles, Hungarians, Russians, or Swedes; they were all called Hungarians, \"bohunks\" or \"hunkies.\" But so far, and again unlike the big cities, Johnstown had only a few such men, and most of them lived in Cambria City, just down the river, beyond the new stone bridge that carried the main line of the Pennsylvania across the Conemaugh.\n\nThe vast majority of the people who lined Main Street watching the parade were either Irish, Scotch-Irish, or Cornish (Cousin Jacks, they were called), German or Welsh, with the Germans and the Welsh greatly outnumbering all the rest. There were some Negroes, but not many, and a few of the leading merchants were Jews.\n\nThe Germans and the Welsh had been the first settlers. More of them, plus the Scotch-Irish, had come along soon after to work in the mines and first forges. Quite a few of the big Irishmen in the crowd had come in originally to build the railroad, then stayed on. Johnstown had been an active stop along the Underground Railroad, and a few of the Negroes had come in that way. Others of them came later to work in the tannery.\n\nThere were German and Welsh churches in town, a German newspaper, and several Irish fortunetellers. Welsh and German were spoken everywhere, along with enough other brogues, burrs, and twangs to make a \"plain American\" feel he was in a country of \"feriners,\" or so it often seemed.\n\nThe first white settlers in the valley had been Solomon and Samuel Adams and their sister Rachael, who came over the Allegheny Mountains from Bedford about 1771 and cleared a patch of land near the Stony Creek. Until then the place had been known as Conemack Old Town, after a Delaware Indian village that stood about where the Memorial Day parade had gathered that noon at the foot of Main.\n\nSamuel Adams and an Indian killed each other in a knife fight, and the traditional story is that Rachael was also killed by Indians soon after. Solomon made a fast retreat back to the stockade at Bedford, and it was not for another twenty years or thereabouts that the first permanent settler arrived. In 1794, about the time President Washington was sending an army over the mountains to put down the so-called Whiskey Rebellion in Pittsburgh, Joseph Schantz, or Johns, an Amish farmer from Switzerland, came into the valley with his wife and four children. He cleared off thirty acres between the rivers, raised a cabin, planted an orchard, and laid out a village which he called Conemaugh Old Town\u2014or just Conemaugh\u2014and which he had every hope for becoming the county seat.\n\nWhen the county was established in 1804 and given the old Latin name for Wales\u2014Cambria\u2014Ebensburg, a mountain village fifteen miles to the north, was picked as county seat. Three years later Joseph Johns sold his village and moved on.\n\nThe next proprietor was a long-haired \"York County Dutchman\" (a Pennsylvania German) named Peter Levergood, and from then until the canal came through, the town remained no more than a backwoods trading center. But with the arrival of the canal it became the busiest place in the county. By 1835 Johnstown, as it was by then known, had a drugstore, a newspaper, a Presbyterian church, and a distillery. By 1840 its population, if the nearby settlements were counted, had probably passed 3,000. Then, in the 1850's, the Pennsylvania Railroad came through, the Cambria Iron Company was established, and everything changed.\n\nBy the start of the 1880's Johnstown and its neighboring boroughs had a total population of about 15,000. Within the next nine years the population doubled. On the afternoon of May 30, 1889, there were nearly 30,000 people living in the valley.\n\nProperly speaking Johnstown was only one of several boroughs\u2014East Conemaugh, Woodvale, Conemaugh, Cambria City, Prospect, Millville, Morrellville, Grubbtown, Moxham, Johnstown\u2014which were clustered between the hills, packed in so tight that there was scarcely room to build anything more.\n\nPetty political jealousies and differences over taxes had kept them from uniting. As it was there was no telling where one began or the other ended unless you knew, which, of course, everyone who lived there did. Millville, Prospect, and Cambria City, it was said, lived on the pay roll of the Cambria mills; Conemaugh lived on the Gautier wire works, Woodvale on the woolen mills there, and Johnstown, in turn, lived on all the rest of them. Johnstown was the center of the lot, geographically and in every other way. It was far and away the largest, with a population of its own of perhaps 10,000 by 1889, which was four times greater than even the biggest of the others. The banks were there, the hotels, the jail, and a full-time police force of nine.\n\nThere were five-story office buildings on Main and up-to-date stores. The town had an opera house, a night school, a library, a remarkable number of churches, and several large, handsome houses, most of which were owned by men high up in the Iron Company.\n\nMuch would be written later on how the wealthy men of Johnstown lived on the high ground, while the poor were crowded into the lowlands. The fact was that the most imposing houses in town were all on Main Street, and one of the largest clusters of company houses was up on Prospect Hill.\n\nThe rest of the people lived in two- and three-story frame houses which, often as not, had a small porch in front and a yard with shade trees and a few outbuildings in back. Nearly everyone had a picket fence around his property, and in spite of its frenzied growth, the city still had more than a few signs of its recent village past.\n\nOn the 22nd of May, for example, the town fathers had gathered at the City Council chambers to settle various matters of the moment, the most pressing of which was to amend Section 12 of Chapter XVI of the Codified Ordinance of the Borough of Johnstown. The word \"cow\" was to be inserted after \"goat\" in the third line, so that it would from then on read: \"Section 12. Any person who shall willfully suffer his horse, mare, gelding, mule, hog, goat, cow, or geese to run at large within the Borough shall for each offense forfeit and pay for each of said animals so running at large the sum of one dollar . . .\"\n\nLife was comparatively simple, pleasures few. There were Saturday night band concerts in the park, and lectures at the library. Sundays half the town put on its best and went walking. Families would pick one of the neighboring boroughs and walk out and back, seeing much and, naturally, being seen all along the way.\n\nThere was a new show at the Washington Street Opera House every other night or so. Thus far in 1889 it had been an especially good season, with such favorites as Zozo the Magic Queen (which brought its \"splendid production\" in \"OUR OWN SPECIAL SCENERY CAR\") and Uncle Tom's Cabin appearing in a single week. (Harriet Beecher Stowe's little drama had also changed considerably since the years before the war. The Johnstown performance, for example, featured \"a pack of genuine bloodhounds; two Topsies; Two Marks, Eva and her Pony 'Prince'; African Mandolin Players; 'Tinker' the famous Trick Donkey.\")\n\nThere was also the Unique Rink for roller skating, a fad which seemed to be tapering off some that spring. There was superb fishing along the Conemaugh and the Stony Creek in spring and summer. Downstream from town the river was stained by waste dumped from the mills, but above town the water still ran clear between sun-bleached boulders and was full of catfish, sunfish, mullet, walleyed pike that everyone mistakenly called salmon, trout, eels, and speedy, mud-colored crawfish.\n\nIn spring, too, there was nearly always a good brawl when the circus came to town. In the fall, when the sour gums turned blood-red against the pines, there was wonderful hunting on the mountains, and fresh deer hanging from butcher shop meathooks was one of the signs of the season. In winter there were sleigh rides to Ebensburg, tobogganing parties, and ice skating at the Von Lunen pond across from the new Johnson Street Rail works up the Stony Creek at Moxham.\n\nAnd year round there was a grand total of 123 saloons to choose from in the greater Johnstown area, ranging from California Tom's on Market Street to the foul-smelling holes along the back alleys of Cambria City. California Tom Davis had been a forty-niner. He was one of the colorful characters of Johnstown and the back room of his saloon was the favorite gathering place for those professional men and Cambria Iron officials who liked to take a sociable drink now and then.\n\nBut the average saloon was simply a place where a workingman could stop off at the end of the day to settle the fierce thirst the heat of the furnaces left him with, or to clear the coal dust from his throat. He was always welcome there, without a shave or a change of clothes. It was his club. He had a schooner of beer or a shot; most of the time he spent talking.\n\nLike any steel town Johnstown had a better than average number of hard-line drinking men. On payday Saturdays the bartenders were the busiest people in town. And week after week Monday's paper carried an item or two about a \"disturbance\" Saturday night on Washington Street or in Cambria City, and published the names of two or three citizens who had spent the night in the lockup for behaving in \"frontier fashion.\"\n\nFor those of still earthier appetite there was Lizzie Thompson's place on Frankstown Hill, at the end of Locust Street. It was the best-known of the sporting houses, but there were others too, close by, and on Prospect Hill. And one spring a similar enterprise had flourished for weeks in the woods outside of town, when several itinerant \"soiled doves,\" as the Tribune called them, set up business in an abandoned coal mine.\n\nBut primarily, life in Johnstown meant a great deal of hard work for just about everybody. Not only because that was how life was then, but because people had the feeling they were getting somewhere. The country seemed hell-bent for a glorious new age, and Johnstown, clearly, was right up there booming along with the best of them. Pittsburgh and Chicago were a whole lot bigger, to be sure, and taking a far bigger part of the business. But that was all right. For Johnstown these were the best years ever.\n\nProgress was being made, and it was not just something people were reading about. It was happening all around them, touching their lives.\n\nStreets were bright at night now with sputtering white arc lights. There was a new railroad station with bright-colored awnings. The hospital was new; two new business blocks had been finished on Main. A telephone exchange had begun service that very year, in January, and already there were more than seventy phones in town. Quite a few houses had new bathrooms. The Hulbert House, the new hotel on Clinton Street, had an elevator and steam heat.\n\nThere was a street railway out to Woodvale and another up to Moxham. Almost everyone had electricity or natural gas in his home. There were typewriters in most offices, and several people had already bought one of the new Kodak \"detective\" cameras. \"Anybody can use the Kodak,\" the advertisements said.\n\nInventions and changes were coming along so fast that it was hard to keep up with them all. The town had no debts, taxes were low, and the cost of things was coming down little by little.\n\nOf course, there were some who looked askance at so much change and liked to talk about the old days when they said there had never been so much drinking, no prostitution, and men could still do a day's work without complaining. There was also strong resentment against the company, and quiet talk of trouble to come, though it would have been very hard for most men among those Memorial Day crowds to have imagined there ever being an actual strike in the mills. The miners had tried it, twice, and both times the company had clamped down with such speed and decisiveness that the strikes had been broken in no time.\n\nAnd there were floods and fires and, worst of all, epidemics that hit so swiftly and unexpectedly, terrifying everyone, and killing so many children. The last bad time had been in 1879, when diphtheria killed 132 children within a few months. Death was always near, and there was never any telling when it would strike again.\n\nYear in, year out men were killed in the mills, or maimed for life. Small boys playing around the railroad tracks that cut in and out of the town would jump too late or too soon and lose a leg or an arm, or lie in a coma for weeks with the whole town talking about them until they stopped breathing forever.\n\nBut had not life always been so? Was not hard work the will of the Lord? (\"In the sweat of thy face shalt thou eat bread, till thou return unto the ground . . .\") And yes, death too? (\". . . for dust thou art, and unto dust shalt thou return.\")\n\nAnd besides, was it not a fine thing to be where there was so much going on, so much to keep a man busy and his family eating regularly?\n\nSo far it had been a good year. Except for measles the town seemed pretty healthy. Talk was that it would be a good summer for steel. Prices might well improve, and perhaps wages with them, and there would be no labor trouble to complicate things, as there would probably be in Pittsburgh.\n\nThe Quicksteps, Johnstown's beloved baseball team, had made a rather poor showing so far, losing to Braddock, Greensburg, and McKeesport in a row; but they had beaten Altoona once, and most people felt that about made up for it. The newspapers were full of stories about the World's Fair opening in Paris, and its Eiffel Tower, and about the Oklahoma Territory opening up out west. Towers of structural steel could reach nearly to the heavens, and Americans could turn a dusty prairie into farms and whole new cities overnight. It was some time to be alive.\n\nBut perhaps best of all there seemed such a strong spirit of national unity everywhere. The Constitutional government those Grand Army veterans had fought for had just celebrated its one hundredth birthday that spring and there had been quite a to-do about it in the newspapers and picture magazines. \"A nation in its high hour of imperial power and prosperity looks back a hundred years to its obscure and doubtful beginning . . .\" one article began. That the next one hundred years would be better still, bringing wondrous advantages and rewards to millions of people who had also come from \"obscure and doubtful\" beginnings, seemed about as certain as anything could be; and especially in a place like Johnstown, on a day when flags were flying from one end of town to the other, and the \"Boys in Blue\" were marching again.\n\n\u2022 \u2022 \u2022\n\nWhen the rain started coming down about four o'clock, it was very fine and gentle, little more than a cold mist. Even so, no one welcomed it. There had already been more than a hundred days of rain that year, and the rivers were running high as it was. The first signs of trouble had been a heavy snow in April, which had melted almost as soon as it came down. Then in May there had been eleven days of rain.\n\nThe rivers ran high every spring. That was to be expected. Some springs they ran so high they filled the lower half of town to the top doorstep. A few times the water had been level with first-floor windows along several streets. Floods had become part of the season, like the dogwood blooming on the mountain. Yet, each year, there was the hope that perhaps this time the rivers might behave themselves.\n\nIt had already been such a curious year for weather. A tornado in February had killed seventeen people in Pittsburgh. Not much had happened in Johnstown, but the wind carried off a tin church roof at Loretto. The April snow had been the heaviest of the whole year, with fourteen inches or more in the mountains. And all through May, temperatures had been bouncing every which way, up in the eighties one day, down below freezing two nights later, then back to the eighties again. Now it felt more like March than May.\n\nAbout five the rain stopped and left everything freshly rinsed looking. The Reverend Chapman, back on his front porch after participating at the graveside ceremonies, sat gazing at the park with its big elms and draped chain fence, and thought to himself that he had seldom looked upon a lovelier scene. Or at least so he wrote later on.\n\nThe Reverend had been in Johnstown only a few years, and it had been just the month before that he and his wife, Agnes, had moved into the new parsonage. He had grown up along the canal to the west of Johnstown, at Blairsville, where his father had eked out a living painting decorative scenes and designs on packet boats. His first church had been in Ligonier, on the other side of Laurel Hill. Later there had been churches in New Florence and Bolivar, down the Conemaugh, and half a dozen other places between Johnstown and Pittsburgh. He liked every one of them, he said, but Johnstown was something special. His stone church next door was the largest in town, a landmark, and except for St. Joseph's, the German Catholic church over in Conemaugh borough, no church had a larger membership. His neighbors across the park, Dr. Lowman and John Fulton, were the finest sort of Christian gentlemen, and their homes were as elegantly furnished as any in town. The Dibert bank, Griffith's drugstore, the post office and the Tribune offices on the floor above it, were all but a few steps from the Reverend's front door. The parsonage faced on to Franklin Street, at almost the exact dead center of Johnstown.\n\nNight settled in and the lights came on along Franklin and Main. A few blocks away William Kuhn and Daisy Horner were being married in a small ceremony at the bride's home. At the Opera House Mr. Augustin Daly's New York production of A Night Off, \"The comedy success of two continents,\" was playing to a small house. Daly was the foremost theatrical producer of the day, and A Night Off had been his biggest hit for several years. Like some of his other productions, it was an adaptation from a German comedy, a fact which the Johnstown audience undoubtedly appreciated.\n\nOther than that not much else was going on. Because of the holiday there had been no paper that morning, but according to Wednesday's Tribune, rainstorms were expected late that Thursday; tomorrow, Friday, was to be slightly warmer. The barometric pressure was reported at thirty, temperature from forty-six to sixty-five, humidity at sixty-nine per cent.\n\nAbout nine the rain began again, gentle and quiet as earlier. But an hour or so later it started pouring and there seemed no end to it. \"Sometime in the night,\" according to Chapman, \"my wife asked if it were not raining very hard, and I being very sleepy, barely conscious of the extraordinary downpour simply answered, 'Yes,' and went to sleep, thinking no more of it until morning.\"\n\n### -3-\n\nGeorge Heiser's day did not end until after ten. It was his practice to keep the store open until then. With the saloons along Washington Street doing business on into the night, there were generally people about and he could pick up a little more trade. Either he or his wife Mathilde would be looking after things behind the counter, in among the queen's ware and the barrels of sugar and crackers, the cases of Ewarts tobacco and yellow laundry soap, needles, spools, pins, and Clark's \"O-N-T\" (Our New Thread).\n\nGeorge and Mathilde Heiser, and their sixteen-year-old son, Victor, lived upstairs over the store. They had been at the same location, 224 Washington, for several years now and, at long last, business was looking up. At fifty-two, for the first time in his life, George Heiser was getting on in the world.\n\nWashington Street ran parallel to Main, two blocks to the north. From the Stony Creek over to the Little Conemaugh, the east-west streets\u2014the \"up\" streets they were called\u2014ran Vine, Lincoln, Main, Locust, Washington. Then came the B & O tracks and the B & O depot which was directly across the street from the Heiser store. Beyond the tracks were two more streets, Broad and Pearl, then the Little Conemaugh, and on the other side of that rose Prospect Hill, steep as a roof. With the town growing the way it had been, the Baltimore & Ohio had brought in a spur from Somerset eight years before to try to take away some of the freight business from the Pennsylvania. An old schoolhouse had been converted into a depot, and the steady night and day racket of the trains right by their window had made quite a difference to the Heisers. But with the store doing as well as it was, George Heiser had few complaints.\n\nGeorge had been troubled by bad luck much of his life. During the war, at Fredericksburg, when his unit had been making a rapid withdrawal, his companion in the ranks, a fellow named Pike, got hit and went down grabbing on to George's leg and pleading for George not to leave him. As a result they were both captured and George spent the rest of the war in Libby Prison. Then, years later, at a time when he seemed to be getting nowhere in Johnstown, George had gone off north to Oil City hoping to strike it rich. He wound up running a butcher shop instead and in no time was back in Johnstown, flat broke, wiped out by fire and his own lack of business sense.\n\nGeorge had not marched in the parade that afternoon. His blue uniform seldom ever came out of the big wardrobe upstairs. He was not much for parades and the like. He seldom mixed in politics, never became an enthusiastic church man. He neither smoked nor chewed, though he would take a beer every so often and once a year he liked to make wine down in the cellar. He did enjoy the Grand Army meetings, but that was largely because he enjoyed being with his friends. People liked him. He was a good storyteller, easygoing. He was also about as physically powerful as any man in town, and he was a very soft touch. He could lift a barrel full of sugar, which was considered quite a feat; but turning away a friend whose credit might not be the best seemed more than he was up to.\n\nTwice in his life he had let friends have money when they came to him; twice he had suffered heavily from the loss. Fortunately for the Heisers, Mathilde was a determined and sensible wife. She was the manager of the two, and after George came back from Oil City she took charge. She kept the books, saw that he did not let much go on credit. Nor would she allow him to set any chairs out on the bare wooden floor. Otherwise, she said, his cronies would be sitting about half the day giving the place the wrong sort of appearance. They had at last been able to add a new window to the store front and appearances would be maintained so long as Mathilde had her say.\n\nMathilde's appearance was straightforward and intelligent. She had a fine head of dark-brown hair, a high forehead, and a set to her mouth that suggested she knew where she was going and that chances were good she would get there. She had had a considerable amount of education for a woman of that time and continued to keep up with her reading. Education was the thing, a proper education and hard work. It had been her way of life since childhood in Germany, and she intended to pass it on to the pride of her life, her son.\n\nVictor was her second child. The first, a girl, had died of diphtheria. Victor had been struck down with it too, but he had been stronger. He looked much like his father now, only a ganglier, rawboned version. He was a serious, pink-faced boy, with big feet and blond hair, taller already than most men and far better educated. At sixteen he knew several languages, was well along in advanced mathematics, and had read about as widely as any boy in Johnstown. At his mother's insistence, his life was a steady round of school, homework, and being tutored in one extra course or another. If George Heiser had had his troubles making his way, that was one thing; Victor, Mathilde Heiser was determined, would not just get on, he would excel. There would be no going off to the Cambria works or coal mines for this young man, and no clerking behind a store counter either.\n\nNow and again Victor had his moments away from all that. His father would step in and see that he got some time off. George Heiser had a wonderful way with children. He was forever telling them stories and listening to theirs. He took a great interest in his son and his schemes, one of which was to build a raft and float down the Conemaugh to the Allegheny, then on to the Ohio and Mississippi. Victor had taken some night classes in mechanical drawing at the library and had worked up plans on how the raft should be built. His idea was to catch the Conemaugh when it was high, otherwise he knew he would run aground.\n\nIn summer he would bring home accounts of his long rides out of town to the open country above the valley. Once he and one of his friends had gone all the way to the South Fork dam to take a look at the lake and the summer colony, but they had been sent on their way by the grounds keeper and never got to see much.\n\nOther nights Victor talked about walking to the edge of town to watch the big summer revival meeting. He loved the powerful singing and the whooping and hollering of people \"getting religion.\" On the way home he and his friends would try to imitate what they had seen, laughing and pounding each other on the back as they came along the streets. This was the kind of education George Heiser understood. He had grown up in Johnstown himself. His people had been among the Pennsylvania Germans who first settled the valley.\n\nAfter ten the Heiser store was closed for the day, the lights out downstairs. When the downpour began, George and Mathilde did not think much of it, except that there would almost certainly be high water in the morning. But the thought bothered them very little, except for the inconveniences there might be. They listened to the rain drum on the roof and were glad to be inside.\n\nIf there was such a thing as a typical married couple in Johnstown on the night of May 30, 1889, George and Mathilde Heiser would come about as close as any to qualifying. Together, like Johnstown itself, they combined an Old World will to make good in the New with a sort of earlier-American, cracker-barrel willingness to take life pretty much as it came. Unlike a large number of Johnstown people, they were not directly beholden to the Cambria Iron Company, but their fortunes, like those of the entire valley, depended nonetheless on how red those skies glowed at night.\n\nThey had suffered the death of a child; they had tried their luck elsewhere and had lost. They fought dirt daily, saved every spare nickel, and took tremendous pride in the progress they were making. All things considered, Johnstown seemed a good place to be. It was their home.\n\n## II\n\n## Sailboats on the mountain\n\n### -1-\n\nThe lake had several different names. On old state maps it was the Western Reservoir, the name it had been given more than forty years earlier when the dam was first built. It was also known as the Old Reservoir and Three Mile Dam, which was the most descriptive name of the lot, if somewhat misleading, since the lake was closer to two than three miles long. The Pittsburgh people who had owned it now for ten years, and who had made a number of changes, called it Lake Conemaugh. But in Johnstown, and in the little coal towns and railroad stops along the way to Johnstown, it was generally known as South Fork dam.\n\nSouth Fork was the nearest place to it of any size. Something like 1,500 people lived there in gaunt little frame houses perched on a hill just back from the tracks and the place where South Fork Creek flows into the Little Conemaugh River. Green hills closed in on every side; the air smelled of coal dust and pine trees. It was a town like any one of a half dozen along the main line of the Pennsylvania between Altoona and Johnstown; except for July and August, when things picked up considerably in South Fork.\n\nThe Pittsburgh people were coming and going then, and they were something to see with their troops of beautiful children, their parasols, and servants. Two or three spring wagons and buggies were usually waiting at the depot to take them to the lake. On Saturdays and Sundays the drivers were going back and forth several times a day.\n\nThe ride to the lake was two miles along a dusty country road that ran through the woods beside South Fork Creek, past Lamb's Bridge, then on up the valley almost to the base of the dam.\n\nSeen from down below, the dam looked like a tremendous mound of overgrown rubble, the work of a glacier perhaps. It reared up 72 feet above the valley floor and was more than 900 feet long. Its face was very steep and covered with loose rocks. There were deep crevices between the rocks where, as late as May, you could still find winter ice hiding; but wild grass, bushes, and saplings had long since taken root across nearly all of the face and pushed up vigorously from between the rocks, adding to the overall impression that the whole huge affair somehow actually belonged to the natural landscape. There was hardly any indication that the thing was the work of man and no suggestion at all of what lay on the other side, except over at the far left, at the eastern end of the dam, where a spillway had been cut through the solid rock of the hillside and a wide sheet of water came crashing down over dark boulders. It was a most picturesque spot, and a favorite for picnics. Long shafts of sunlight slanted through a leafy gloom where the mountain laurel grew higher than a man could reach. And at the base of the falls a wooden bridge crossed the loud water and sent the road climbing straight to a clump of trees at the top of the dam, just to the right of the spillway.\n\nThere the road divided, with the left-hand fork crossing another long wooden bridge which went directly over the spillway. But carriages heading for the club took the road to the right, which turned sharply out of the trees into the sunshine and ran straight across the breast of the dam where, about a hundred yards out, the drivers customarily stopped long enough for everyone to take in the view.\n\nTo the right the dam dropped off a great deal more abruptly even than it had looked from below, and South Fork Creek could be seen glittering through the trees as it wound toward Lamb's Bridge.\n\nOn the other side of the road the bank sloped sharply to the water's edge, which was usually no more than six or seven feet below the top of the dam. From there the broad surface of the lake, gleaming in the sunlight, swept off down the valley until it disappeared behind a wooded ridge in the distance.\n\nAlong the eastern shore, to the left, were the hayfields and orchards of the Unger farm, neatly framed with split-rail fences. Beyond that was what was known as Sheep's Head Point, a grassy knoll that jutted out into the lake. Then there were one, two, three ridges, and the water turned in behind them, out of sight, running, so it seemed, clear to the hazy blue horizon off to the south.\n\nAt the western end of the dam the road swung on through the woods, never far from the water, for another mile or so, to the main grounds of the South Fork Fishing and Hunting Club, which, seen from the dam, looked like a colorful string of doll houses against the distant shore line.\n\nFrom the dam to the club, across the water, was about a mile. Except for a few small coves, the narrowest part of the lake was at the dam, but there was one spot, on down past the club, where an east-west line across the water was nearly a mile. A hike the whole way around the shore was five miles.\n\nWhen the water was up in the spring, the lake covered about 450 acres and was close to seventy feet deep in places. The claim, in 1889, was that it was the largest man-made lake in the country, which it was not. But even so, as one man in Johnstown often told his children, it was \"a mighty body of water to be up there on the mountain.\"\n\nThe difference in elevation between the top of the dam and the city of Johnstown at the stone bridge was about 450 feet, and the distance from the dam to that point, by way of the river valley, was just under fifteen miles. Estimates are that the water of Lake Conemaugh weighed about 20 million tons.\n\nThe water came from half a dozen streams and little creeks that rushed down from Blue Knob and Allegheny Mountain, draining some sixty square miles. There was Rorabaugh Creek, Toppers Run, Yellow Run, Bottle Run, Muddy Run, South Fork Creek, and one or two others which seem never to have been named officially. South Fork Creek and Muddy Run were the biggest of them, but South Fork Creek was at least twice the size of the others. Even in midsummer it was a good twenty feet across. Like the others it was shallow, ice-cold, very swift, and just about a perfect place for trout fishing.\n\nIn South Fork there were scores of people who had been out on the dam and had seen the view. There were others who knew even more about the club and the goings on there because they worked on the grounds, tending lawns or waiting on tables at the clubhouse. But for everyone else the place was largely a mystery. It was all private property, and as the club managers had made quite clear on more than one occasion, uninvited guests were definitely not welcome.\n\nThe club had been organized in Pittsburgh in 1879. It owned the dam, the lake, and about 160 acres besides. By 1889 sixteen cottages had been built along the lake, as well as boathouses and stables. The cottages were set out in an orderly line among the trees, not very far apart, and only a short way back from the water. They looked far too substantial really to be called \"cottages.\" Nearly every one of them was three stories tall, with high ceilings, long windows, a deep porch downstairs, and, often as not, another little porch or two upstairs tucked under sharp-peaked roofs. The Lippincott house with its two sweeping front porches, one set on top of the other, and its fancy jigsaw trim, looked like a Mississippi riverboat. The Moorhead house was Queen Anne style, which was \"all the rage\" then; it had seventeen rooms and a round tower at one end with tinted glass windows. And the Philander Knox house, next door, was not much smaller.\n\nBut even the largest of them was dwarfed by the clubhouse. It had enough windows and more than enough porch for ten houses. There were forty-seven rooms inside. During the season most of the club members and their guests stayed there, and the rule was that everyone had to take his meals there in the main dining room, where 150 could sit down at one time.\n\nIn the \"front rooms\" there were huge brick fireplaces for chilly summer nights, billiard tables, and heavy furniture against the walls. In summer, after the midday dinner, the long front porch was crowded with cigar-smoking industrialists taking the air off the water. String hammocks swung under the trees. Young women in long white dresses, their faces shaded under big summer hats, strolled the boardwalks in twos and threes, or on the arms of very proper-looking young men in dark suits and derbies. Cottages were noisy with big families, and on moonlight nights there were boating parties on the lake and the sound of singing and banjos across the black water.\n\nIn all the talk there would be about the lake in the years after it had vanished, the boats, perhaps more than anything else, would keep coming up over and over again. Boats of any kind were a rare sight in the mountains. There were rowboats on the old Suppes ice pond at the edge of Johnstown, and a few men had canoes along the river below the city. But that was about it. Not since the time when Johnstown had been the start of the canal route west had there been boats in any number, and then they had been only ungainly canal barges.\n\nThe club fleet included fifty rowboats and canoes, sailboats, and two little steam yachts that went puttering about flying bright pennants and trailing feathers of smoke from their tall funnels. There was even an electric catamaran, a weird-looking craft with a searchlight mounted up front, which had been built by a young member, Louis Clarke, who liked to put on a blue sailor's outfit for his cruises around the lake.\n\nBut it was the sailboats that made the greatest impression. Sailboats on the mountain! It seemed almost impossible in a country where water was always a tree-crowded creek or stream, wild and dangerous in the spring, not much better than ankle-deep in the hottest months. Yet there they were: white sails moving against the dark forest across a great green mirror of a lake so big that you could see miles and miles of sky in it.\n\nSome of the people in Johnstown who were, as they said, \"privileged\" to visit the club on August Sundays brought home vivid descriptions of young people gliding over the water under full sail. It was a picture of a life so removed from Johnstown that it seemed almost like a fantasy, ever so much farther away than fifteen miles, and wholly untouchable. It was a picture that would live on for a long time after.\n\nThat the Pittsburgh people also took enormous pleasure in the sight seems certain. There was no body of water such as this any where near Pittsburgh. There were, of course, the Monongahela and Allegheny rivers, but they were not exactly clean any longer, and with the mills going full blast, which they had been for some time now, the air around them was getting a little more unpleasant each year. It was a curious paradox; the more the city prospered, the more uncomfortable it became living there. Progress could be downright repressive. But fortunately for the Pittsburgh people, it was very much within their power to create and maintain a place so blessed with all of nature's virtues.\n\nThis water was pure and teeming with fish, and the air tasted like wine after Pittsburgh. The woods were full of songbirds and deer that came down to drink from the mist-hung lake at dawn. There were wild strawberries everywhere, and even on the very hottest days it was comfortable under the big trees.\n\nIn fact, with its bracing air, its lovely lake, and the intense quiet of its cool nights, the South Fork Fishing and Hunting Club must have seemed like paradise after Pittsburgh. Under such a spell even a Presbyterian steel master might wish to unbend a little.\n\nThe summer resort idea was something new for that part of the country. And only the favored few had the time or the money to experiment with it. But so far every indication was that the club was a great success. There had been problems from time to time. Poachers had been a continuing nuisance. The summer of 1888 had been cut short when a scarlet-fever scare sent everyone packing off home to Pittsburgh. Still, everything considered, in 1889 it looked as though the men who had bought the old dam ten years earlier knew what they were doing.\n\n### -2-\n\nThe first member of the South Fork Fishing and Hunting Club to take an interest in the regenerative powers of the Alleghenies was Andrew Carnegie.\n\nCarnegie had been going to what he called \"The Glorious Mountain\" long before the South Fork organization was put together. He had his own modest frame house at Cresson, which was one of the first summer resorts in Pennsylvania and the only one of any consequence in the western part of the state. It was owned by the Pennsylvania Railroad and was located fourteen miles up the line from South Fork at the crest of the Allegheny range.\n\nCresson, or Cresson Springs as it was also known, had been started before the Civil War by a doctor named Robert Montgomery Smith Jackson. The main attractions at Cresson, aside from the mountain air and scenery, were the \"iron springs,\" the best-known of which was the Ignatius Spring, named after \"the venerable huntsman\" Ignatius Adams, who first discovered its life-preserving powers and whose ghost was said still to haunt the place. According to Jackson, \"by drinking this water, dwelling in the woods and eating venison,\" Ignatius had \"lived near the good old age of one hundred years.\" Jackson was against whiskey, slavery, and what he called the \"present tendency to agglomerate in swarms, or accumulate in masses and mobs.\" Those \"gregarious instincts [which] now impel this race to fix its hopes of earthly happiness on city life alone\" would, he was convinced, be the undoing of the race. Life in the country was the answer to practically every one of man's ills, and particularly life on the Allegheny Mountain.\n\nJackson's ambition (\"a mission, solemn as a command from Heaven,\" he called it) was to make Cresson \"the place of restoration for all forms of human suffering.\" He got his friend J. Edgar Thomson, president of the Pennsylvania, interested, and the railroad built a hotel and developed the place, though, as things turned out, along rather different lines. Carnegie, B. F. Jones, and a few other Pittsburgh businessmen, none of whom seems to have been suffering very much, built cottages and the summer trade flourished. Every passenger train bound east or west stopped there. Well-to-do families from Pittsburgh and Philadelphia arrived for summer stays of several weeks.\n\nJackson meanwhile had set down the fundamentals of his philosophy along with a detailed natural history of the Allegheny highlands in a book called The Mountain. He borrowed heavily from Wordsworth and Thoreau, and, in his own way, did about as much as anyone to sum up the wild beauty of the area. Also, in his spare time, he tended bar at the hotel and would be remembered for years after for the two jars he kept prominently displayed on one shelf, flanked on either side by whiskey bottles. In each jar, preserved in alcohol, was a human stomach. One had belonged to a man who had died a natural death, and was, according to all who saw it, an exceedingly unappetizing sight. But it was, nonetheless, an improvement over its companion piece, which, according to its label, had belonged to a man who had died of delirium tremens. When setting out drinks, the doctor seldom failed to call attention to his display. The result was that his bar became the best patronized of any for miles about. Regular customers grew quite attached to the jars; word of them spread far, and along with the iron springs, they appear to have been a major attraction at Cresson for several years.\n\nJackson served in the war; then, in 1865, despite all his good life on the mountain, he died at the age of fifty.\n\nBy 1881, to accommodate the growing trade, the railroad cleared space in a maple grove and built another hotel, The Mountain House, which, with its endless thick-carpeted halls, its many towers and flowing stairways, was easily the grandest piece of architecture in Cambria County.\n\nCarnegie's house, only a short distance from the hotel, was his only real home in Pennsylvania by that time. Though no one held more sway in Pittsburgh, he had not lived there for nearly twenty years. He visited often, and was front-page news when he did, but the rest of the time he was either in New York, Scotland, or at Cresson. He loved Cresson and talked up its charms with great vigor, which is perhaps not surprising for a Scot who had been brought up on Burns and had learned to quote him before he had learned to read. (\"My heart's in the Highlands, my heart is not here;\/ My heart's in the Highlands a-chasing the deer.\") He courted his young wife there in the summer of 1886. (\"A.C. walked home with me in the starlight . . .\" she wrote in her diary at Cresson. \"Such wonderful happiness . . . the happiest day of my life.\") He also very nearly died there of typhoid later that fall; and his mother, who took sick at the same time and lay in the next room down the hall, did die there on November 10. He entertained his distinguished friends at Cresson (Matthew Arnold had stopped over in 1883), and he managed, as was his pleasure, to keep his mind free and above the petty preoccupations of the steel business.\n\nAs one admiring writer of the period explained it: \"All other iron and steel magnates, with the exception of Carnegie, lived in Pittsburgh and were swayed constantly by the local gossip, by the labour troubles, and by the rumours of competition and low prices that floated from office to office. To-day they were elated; tomorrow they were depressed. To-day they bought; to-morrow they sold. Carnegie, on the other hand, deliberately placed himself where these little ups and downs were unnoticed. . . . the news that Coleman had quarrelled with Shinn, or that coke-drawers wanted five cents a day more, was of small consequence. One thing he knew\u2014that civilisation needed steel and was able to pay for it. All else was not worth troubling about.\"\n\nThe place had its hold on him. He went on bird walks; he read; he talked and talked and talked. And if the views and good company were not enough, there were the \"curative powers,\" as the guidebook described them, of the iron springs. And who might more readily endorse such a tonic than the bouncy little ironmaster himself?\n\nBut Cresson had its drawbacks. It was a public place for one, sitting almost on top of the railroad. For another, there was no water. The story goes that there was (and still is) a house at Cresson where the rain water off of one roof eventually ends up in the Atlantic Ocean, while the rain water off the other finds its way to the Mississippi Valley and the Gulf of Mexico. The resort was at the very crest of the Allegheny divide, and though springs were plentiful, every bit of water there was drained off in both directions.\n\nAs a result, except for drives and walks, there was really not much to do at Cresson. Tennis had not yet caught on there. Golf, which was to one day be the great passion of Carnegie and his kind, would not even be attempted in the United States until 1888. But boating was already distinctly fashionable, and fishing had been coming more and more into its own as a gentleman's sport.\n\nPart of the increasing appeal of fishing seemed to be the multitude of trappings it called for. Where once the well-equipped angler needed only the simplest and most inexpensive sort of gear, now in the late 1880's a whole line of elaborate and expensive paraphernalia was said to be necessary. Bait boxes, boots, collapsible nets, cookstoves, silk line, creels, reels and casting rods that cost as much as twenty dollars, even costly books on the subject, were the sign of the true sportsman. All of which seemed to make fishing, and particularly trout and bass fishing, which were generally referred to as a \"science,\" that much more attractive to the man of means. His interest in the sport not only showed his love of the great outdoors, but also that he had both the money and the brains to participate.\n\nOf course, too, few pastimes there were which would take a man so far, in spirit at least, from the rude industrial grind. \"God never did make a more calm, quiet, innocent recreation than angling,\" were the words of Izaak Walton in the newly reissued, and quite costly, two-volume edition of his great work on the subject. One Walton disciple of the 1880's, who as it happens was a Vermonter, wrote about this time, \"I take my rod this fair June morning and go forth to be alone with nature. No business cares, no roar of the city, no recitals of others troubles . . . no doubts, no fears to disturb me as, drinking in the clear, sweet air with blissful anticipation I saunter through the wood path toward the mountain lake.\" Had he been a Pittsburgh steel man he need only have added \"no competitors, no labor agitators.\"\n\nSo all that was needed to improve on Cresson was enough water for boating and fishing. A mountain lake, in short; plus some privacy. It would also be well to be back a way from the railroad, though not too far back, as the railroad was the one and only way to make the trip from Pittsburgh. As it was, the ride took about an hour and fifteen minutes to Johnstown, then another twenty minutes to come on up the mountain. That was about long enough. It would be best, therefore, to be somewhere in the same general area.\n\nThe old reservoir above South Fork certainly must have seemed a perfect solution. It answered every need: It was well back from the railroad; it was only a matter of miles from Cresson; and South Fork Creek was well known as one of the best trout streams in the state. True, the dam needed a great deal of repair work after so many years of neglect, but that could be handled all right, and especially since the property could be had for such a good price. Or so must have run the reasoning of the Pittsburgh men who bought it in 1879. Carnegie was not among them at this particular stage, but no doubt his presence so nearby on the mountain added still another enticing attraction to the scheme.\n\nThe prime promoter behind the move was a onetime railroad tunnel contractor and now-and-then coke salesman and real-estate broker by the name of Benjamin F. Ruff.\n\nRuff bought the dam and the lake from John Reilly, an Altoona Democratic politician and former Pennsylvania Railroad official who was then serving what would be his only term in Congress. Ruff paid $2,000 for the property, which was $500 less than Congressman Reilly had paid for it four years earlier when he had bought it from the Pennsylvania.\n\nRuff then rounded up fifteen other Pittsburgh gentlemen who each, with one exception, bought a single share in the operation for $200. The exception was one of the most interesting young men in Pittsburgh.\n\nIn the spring of 1879, when all this was going on, Henry Clay Frick was only twenty-nine years old, solemn, enigmatic, strikingly handsome, and already worth an even million. The grandson of wealthy, old Abraham Overholt, the Mennonite whiskey maker, Frick had made his own fortune in the coke business, and largely through his dealings with Pittsburgh's number-one coke consumer, Andrew Carnegie.\n\nFrick bought three shares. Ruff kept four for himself. A charter was drawn up stating that the name of the organization was to be the South Fork Fishing and Hunting Club of Pittsburgh, that its \"object\" was to be \"the protection and propagation of game and game fish, and the enforcement of all laws of this state against the unlawful killing or wounding of the same.\" It was also stated that the club's place of business was to be Pittsburgh, in Allegheny County, not in Cambria County where the property was located.\n\nOn November 15, 1879, the charter was approved and signed in the Court of Common Pleas in Allegheny County by Judge Edwin H. Stowe, who for some unknown reason ignored the provision in the law which called for the registration of a charter in the \"office for recording in and for the county where the chief operations are to be carried on.\" Nor did the sportsmen make any effort to conform to the law. Perhaps it seemed a minor point and was overlooked by mistake. In any case, the charter was secured without the knowledge of the authorities in Cambria County, and there would be speculation for years to come as to what might have happened right then and there had they and Judge Stowe gone about their business in strict accordance with the rules.\n\nRuff was to be the president of the new club. The capital stock was to be $10,000, but that was soon increased to $35,000 when it became known how much work was needed to get the dam in shape. For in 1879 the South Fork dam was getting on in years, and the years had been hard on it.\n\nForty-three years earlier, in 1836, the legislature of the state of Pennsylvania had approved funds for the building of a reservoir on the western slope of Allegheny Mountain to supply extra water during dry months for the new canal system from Johnstown to Pittsburgh. The first appropriation was for $30,000, but before the project was finished nearly $240,000 would be put into the dam; and two years after it was finished the whole thing would be obsolete and of no use whatsoever.\n\nThe canal from Johnstown to Pittsburgh was known as the Western Division of the state's \"Main Line\" canal, which had been built to compete with New York's thriving, new Erie Canal. The Western Division officially opened for business in May 1831, when a Johnstown barge pulled into Pittsburgh after traveling 104 miles in less than forty-eight hours.\n\nThe Eastern Division, running from Philadelphia to Hollidaysburg, at the foot of the eastern slope of Allegheny Mountain, was opened the next year. The only thing that then remained to be finished was an ingenious system of railroads and steam-powered hoists designed to get the boats up and over the mountain. But by spring of 1834 that too had been built and was open to traffic. For the first time the wharves of Philadelphia had a direct, nonstop link with the headwaters of the Ohio. It had cost a staggering sum; the state was nearly bankrupt; but that old, formidable barrier to the Pennsylvania route west, Allegheny Mountain, had been bested, and the general course of the country's epic push to the Mississippi and beyond had been set for years to come.\n\nThe system devised for crossing the mountain was widely hailed as one of the engineering wonders of the age. Within a distance of thirty-six miles it overcame an increase in elevation of nearly 1,400 feet, or about twice the elevation the Erie Canal had to overcome along its entire length of 352 miles. Known as the Portage Railroad, it included a series of five inclined planes on each side of the mountain, ten in all, connected with a narrow-gauge railroad. Barges, passengers, freight, everything was hauled up one side and let down the other with hemp ropes thick as a man's leg. It was a thrilling experience for travelers, a goodly number of whom chose to go by way of the Pennsylvania, rather than the Erie Canal, for that very reason.\n\nCharles Dickens, one such traveler, described the ride in his American Notes:\n\nIt was very pretty travelling thus, at a rapid pace along the heights of the mountain in a keen wind, to look down into a valley full of light and softness: catching glimpses, through the tree-tops, of scattered cabins; children running to the doors; dogs bursting out to bark, whom we could see without hearing; terrified pigs scampering homewards; families sitting out in their rude gardens; cows gazing upward with a stupid indifference; men in their shirt-sleeves looking on at their unfinished houses, planning out tomorrow's work; and we riding onward, high above them, like a whirlwind.\n\nThe adventurous journey also included, just to the east of Johnstown, a ride through the first railroad tunnel in the country and, in Pittsburgh, a ride across the first suspension bridge, an aqueduct over the Allegheny River designed by German-born John Augustus Roebling, who would later conjure up that wonder of wonders of the 1880's, the Brooklyn Bridge.\n\nBut from Johnstown west the canal was troubled by water shortages nearly every summer. Operations were interrupted. Business suffered at a time when business had to be especially good to make up for winter, when virtually every moving thing stopped in the mountains for weeks on end, and spring, when the floods came. Particularly troublesome was the canal basin in Johnstown. Despite all the water that rushed into the valley in springtime, along toward mid-July the basin came close to running dry.\n\nThe solution seemed obvious enough. Put a dam in the mountains where it could hold a sufficient supply of water to keep the basin working and the canals open, even during those summers when creeks vanished and only weeds grew.\n\nWork began on the Western Reservoir above South Fork in 1838, after some 400 acres had been cleared of timber. The site had been selected and surveyed by Sylvester Welsh, head engineer for the canal. He proposed an earth dam of 850 feet in length, with a spillway at one or both ends \"of sufficient size to discharge the waste water during freshets, and sluices to regulate the supply for the canal.\" It was also important, he said, that the bed of the spillway be solid rock and that no water be permitted to pass over the top of the dam. The design of the dam was worked out by a young state engineer named William E. Morris, who approved the location because, as he stated in a report made in 1839, it was in an area where there was enough drainage to provide a \"certain\" supply of water. He too proposed an earth dam 850 feet across the top and 62 feet high. He estimated that it would take a year to do the job.\n\nThe contractors chosen were James N. Moorhead of Pittsburgh and Hezekiah Packer of Williamsport. According to lengthy studies made by civil engineering experts years later, they did a competent job. Certainly they went about it with considerable care and patience and despite continuing delays. For, as it turned out, fifteen years passed before the dam was finished.\n\nIn 1842 work was halted because the state's finances were in such bad shape that there was simply no more money to continue the job. For the next four years nothing was done. Then when the work did start again, it was only for another two years. A local cholera epidemic caused \"a general derangement in the business,\" until 1850, when the project again resumed, and for the final time.\n\nThe construction technique was the accepted one for earth dams, and, it should be said, earth dams have been accepted for thousands of years as a perfectly fine way to hold back water. They were in fact the most common kind of dam at the time the South Fork work began and they were the most economical. The basic construction material was readily available at almost any site, it was cheap, and it required a minimum of skilled labor. Virtually any gang of day laborers, and particularly any who had had some experience working on railroad embankments, was suitable. But since the basic raw material, earth, is also highly subject to erosion and scour, it is absolutely essential that a dam built of earth, no matter how thick, be engineered so that the water never goes over the top and so that no internal seepage develops. Otherwise, if properly built and maintained, an earth dam can safely contain tremendous bodies of water.\n\nThe South Fork embankment was built of successive horizontal layers of clay. They were laid up one on top of the other after each layer had been packed down, or \"puddled,\" by allowing it to sit under a skim of water for a period of time, so as to be watertight. It was a slow process. And as the earth wall grew increasingly higher, it was coated, or riprapped, on its outer face with loose rocks, some so huge that it took three teams of horses to move them in place. On the inner face, which had a gentler slope, the same thing was done, only with smaller stones.\n\nThe spillway, as Welsh had stipulated, was not cut through the dam itself, but through the rock of the hillside to which the eastern end of the dam was \"anchored.\" The spillway was about 72 feet wide. The over-all length of the breast was just over 930 feet. The width on top was about 20 feet. The thickness at the base was some 270 feet.\n\nAt about the exact center of the base, there were five cast-iron pipes, each two feet in diameter, set in a stone culvert. They were to release the water down to South Fork, where it would flow on to the Johnstown basin by way of the Little Conemaugh. The pipes were controlled from a wooden tower nearby. On June 10, 1852, the work on the dam was at last completed; the sluice pipes were closed and the lake began to fill in. By the end of August the water was 40 feet deep.\n\nBut about the time the dam was being finished, J. Edgar Thomson, who was then chief engineer for the up-and-coming Pennsylvania Railroad, was making rapid progress with his daring rail route over the mountains, which included what was to become famous as the Horseshoe Curve. The canal was about to be put out of business.\n\nThe Pennsylvania was racing to complete a route west to compete with the New York Central, the Erie, and the B & O, which were each pushing in the same direction. The last part of the run, from Johnstown to Pittsburgh, was ready in late 1852. On December 10, six months after the South Fork dam had been finished, a steam engine made an all-rail run from Philadelphia to Pittsburgh. J. Edgar Thomson became president of the road about the same time, and the company was on its way to becoming within a very few years the biggest and far and above the most powerful single force in the state (and in the Statehouse); the biggest customer for nearly everything, but especially coal, iron, and steel; the biggest employer; and the biggest influence on the way people lived from one end of Pennsylvania to the other. By the end of the '80's it would be the mightiest of the nation's many mighty railroads.\n\nThe effect of the new railroad on the state's troublesome, costly, and beloved canal system was disastrous\u2014almost immediately. Within two years after the railroad opened, the legislature voted to put the \"Main Line\" up for sale for not less than $10 million. Understandably there were no takers. The one likely prospect was the Pennsylvania itself, which could readily use the right of ways. Three years later the sale was made, with the Pennsylvania paying $7.5 million for the system, which included the Main Line, the Portage Railroad, and, as it happened, the South Fork dam.\n\nHaving no use for the dam, the railroad simply let it sit. Nothing whatsoever was done to maintain it. In fact, from 1857, the year the railroad took possession, until 1879, twenty-two years later when the Pittsburgh men took over, the dam remained more or less quietly unattended, moldering away in the woods, visited only once in a while by fishermen or an occasional deer hunter.\n\nAnd it was only five years after the state sold it to the Pennsylvania that the dam broke for the first time.\n\nIn the late spring of 1862, about the time the Union Army under McClellan was sweating its way up the blazing Virginia peninsula, for a first big and unsuccessful drive on Richmond, the mountains of Pennsylvania were hit by heavy thunderstorms. Hundreds of tiny creeks and runs and small rivers went roaring over their banks, and in Johnstown the Tribune ran the first of its musings on what might be the consequences should, by chance, the dam at South Fork happen to let go. Eight days later, on June 10, the dam broke.\n\nThe break was caused by a defect in the foundation near the stone culvert. The accepted theory locally was that various residents had been stealing lead from the pipe joints during the years the dam had been abandoned, that serious leaks had been the result, and that the break had come not long after. Exactly how big the break was is not known, as no records were made and no photographs were taken. The important fact was that though there was much alarm in the valley below the dam, the break caused little damage since the lake was less than half full, the creeks were low, and a watchman at the dam, just before the break, had released much of the pressure by opening the valves. (It was also somewhere along about this time that the wooden tower for controlling the discharge pipes caught fire and burned to the ground.)\n\nFrom then on until the Pittsburgh sportsmen appeared on the scene seventeen years later the lake was no lake at all, but little more than an outsize pond, ten feet deep at its deepest point. At the southern end, grass quickly sprouted across acres and acres of dried-up lake bed and neighboring farmers began grazing their sheep and cattle there.\n\nIn 1875 Congressman Reilly, who had spent most of his working life with the Pennsylvania in nearby Altoona, and who must have thereby known about the dam for some time, bought the property and, like the Pennsylvania, did nothing with it. He just held on to it, apparently on the look for another buyer, which he found four years later in Benjamin Ruff. But before selling at a slight loss to Ruff, he removed the old cast-iron discharge pipes and sold them for scrap.\n\nRuff's idea of what to do about the dam was relatively simple and seemed realistic enough at first. He would rebuild it to a height of only forty feet or so and cut the spillway down some twenty feet deeper to handle the overflow. But when he found that this would cost considerably more than repairing the old break and restoring the dam to somewhere near its original height, he chose the latter course.\n\nThe first indication in Johnstown and thereabouts that a change was in the offing above South Fork was an item in the Tribune on October 14, 1879. \"Rumors\" were reported that a summer resort was to be built by a Western Game and Fish Association. The next day there was a notice calling for fifty men to work, but no name of the organization was given.\n\nFor some reason or other, intentionally or otherwise, the Pittsburgh men kept the correct name of their organization from receiving any kind of public notice. It was a course of action which would later be interpreted as evidence that they had had no desire for anyone to come looking into their business in general, or their charter in particular.\n\nRuff set about repairing the dam by boarding up the stone culvert and dumping in every manner of local rock, mud, brush, hemlock boughs, hay, just about everything at hand. Even horse manure was used in some quantity. The discharge pipes were not replaced, and the \"engineering\" techniques employed made a profound impression on the local bystanders.\n\nThe man immediately in charge of this mammoth face-lifting was one Edward Pearson, about whom little is known except that he seems to have been an employee of a Pittsburgh freight-hauling company that did business with the railroad and that he had no engineering credentials at all.\n\nThe entire rebuilding of the dam ended up costing the club about $17,000, and there was trouble from the start. On Christmas Day, 1879, only a month or so after work had begun, a downpour carried away most of the repairs. Work was discontinued until the following summer. Then, less than a year later, in February of 1881, once again heavy rains caused serious damages.\n\nNo one seems to have been particularly discouraged by all this, however. Along toward the end of March the lake was deep enough for the clubmen to go ahead with their plan to stock it. The first of the small steamboats was being assembled and the clubhouse was close to being readied for the grand opening. In early June the fish arrived by special tank car from Lake Erie, 1,000 black bass, which ended up costing the club about a dollar apiece by the time the last expenses were paid. According to the Tribune, which noted these and all other bits of news it could uncover concerning the club, only three of the fish died, \"one of which was a huge old chap, weighing over three pounds.\"\n\nThe Tribune had also reported earlier that the Pennsylvania was planning to build a narrow-gauge spur from South Fork to the lake and that the clubmen were shopping about for land downriver from Johnstown where they intended to establish a private deer park of 1,500 acres. Neither claim was true, but both seemed perfectly reasonable and fitted in with the picture most Johnstown people had of the club and its members. Would not even the high and mighty Pennsylvania Railroad gladly provide any number of special conveniences for the likes of such men? Was not a deer park a fitting aristocratic touch for their new mountain domain? Certainly money was no problem. Were not the members of the club millionaires to the man?\n\nThe plain truth was that a goodly number of them were; quite a few of them, however, were not, and two or three of them were a great deal better than millionaires. But by the standards of most men, they were, every last one of them, extraordinarily rich and influential. Yet, one of the curious things about the club is that the make-up of its membership, exactly who was who at the lake, was not generally known around Johnstown. If the club appears to have been rather cozy about its name, it was even more so about publicizing who belonged to it. Not until well after the events of May 31, 1889, was a full list of the membership published publicly. And quite a list it was.\n\nThe membership of the South Fork Fishing and Hunting Club, according to its initial plans, was never to exceed one hundred sportsmen and their families. The membership fee was $800. There was to be no shooting on Sundays; and those members who did not have cottages of their own were limited to a two-week stay at the clubhouse. As the summer season of 1889 was approaching, there was a total of sixty-one names on the membership roster.\n\nThe South Fork Fishing and Hunting Club, it should be kept in mind, was a most unostentatious affair by contrast to such watering spots of the time as Newport, Cape May, or the lavish new lakeside resort in New York, Tuxedo Park. There was no opulence. There were no liveried footmen, no Tirolean-hatted gamekeepers such as at Tuxedo, no \"cottage\" architecture to approach the likes of Newport. There was not even a comparison to be made, unless the South Fork group was to be measured by the per capita worth of its members\u2014or the industrial and financial power they wielded\u2014which, everything considered, was often the way such things were measured. On that basis the little resort on Lake Conemaugh was right in the same league.\n\nOne Pittsburgh newspaper called it the \"Bosses Club,\" and aptly so. Carnegie's name by itself on the membership list would have been reason enough. And the same holds for Henry Clay Frick, for much had happened to the young \"Coke King\" since he had first joined with Benjamin Ruff to launch the club.\n\nIn 1881, while in New York, Frick had stopped by the Windsor Hotel on Fifth Avenue to pay a call on Carnegie and his mother and to talk a little business. (Frick happened to be on his honeymoon at the time, but he was not the man to let that, or anything else, stand in the way of progress.) When the meeting was over, he and Carnegie were partners in the coke trade, and from then on it did even better than before. By 1889 the H. C. Frick Coke Company was capitalized at $5 million; it owned or controlled 35,000 acres of coal land and employed some 11,000 men. Moreover, in January of that year, Frick had been made Chairman of Carnegie, Phipps & Company, which meant that he was commander in chief of the whole of the Carnegie iron and steel enterprises, which were by then the biggest in the world.\n\nCarnegie by this time had said something to the effect that he was not much interested in making more money and was spending no more than six months a year at his business. He wanted someone who could manage things. In Frick, whom he did not especially like and whom no one seemed quite able to fathom, he found exactly the right man. By 1889 this humorless, solitary, complex son of a German farmer, who was then still six months from turning forty, was the most important man in Pittsburgh.\n\nAlong with Frick, the club roster included Henry Phipps, Jr., Carnegie's partner since the earliest days of the business, before the Civil War. A pale, painstaking man who stood no more than an inch taller than the five-foot-two Carnegie, Phipps was the financial wizard of Carnegie, Phipps & Company. In the early days he had won certain acclaim for his ability to borrow money for the struggling ironworks and for an old horse he owned, which, according to Pittsburgh legend, was capable of taking him on his rounds of the banks without any guidance whatsoever. By 1889, however, Phipps was one of the three or four top men in the steel business and one of the wealthy men of the country.\n\nBesides the big three\u2014Carnegie, Frick, and Phipps\u2014the Carnegie empire was also represented at South Fork by John G. A. Leishman, the vice-chairman of the firm, and by Philander Chase Knox, a bright little sparrow of a man, who was the company's number-one lawyer as well as the personal counsel for both Carnegie and Frick.\n\nThen there were Robert Pitcairn and Andrew Mellon, two excellent friends to have if you were doing business in Pittsburgh. Pitcairn ran the Pittsburgh Division of the Pennsylvania Railroad, which, as jobs went in those days, was a most lofty position indeed. In terms of pure power and prestige, few men outranked him.\n\nMellon was a shy, frail-looking young man, still in his thirties, and exceedingly quick-witted. Along with his father, old Judge Mellon, he was in the banking business, T. Mellon & Sons, and up to his elbows in the financial doings behind much of Pittsburgh's furious industrial growth. It had been the Mellons who lent Henry Frick the money to buy his first coal land, and who backed him again (with something like $100,000) during the panic of 1873 when he wanted to buy up still more. (One story has it that Judge Mellon, who was a staunch Methodist, made the loan only after Frick implied that he would use his influence to have the Overholt distillery shut down.) Frick and young Andrew were not far apart in age and became fast friends, traveling to Europe together, dealing in business for years, but never calling each other anything but Mister Frick and Mister Mellon.\n\nJames Chambers and H. Sellers McKee ran what they claimed was the largest window-glass works on earth. Durbin Horne and C. B. Shea ran Pittsburgh's leading department store, Joseph Home and Company. D. W. C. Bidwell sold DuPont blasting powder for coal mining. Calvin Wells, A. French, James Lippincott, and John W. Chalfant were in the steel business in one way or other.\n\nAnd so the list went. It included names from the Pittsburgh \"Blue Book\" (Thaw, Laughlin, McClintock, Scaife) and from the lists of directors of several Pittsburgh banks (Schoonmaker, Moorhead, Caldwell). There were among them the founders of the city's new business club, the Duquesne Club, and the new preparatory school for young men, Shady Side Academy.\n\nThere were also among them, it is interesting to note, a future Secretary of the Treasury (Mellon, who would serve under Harding, Coolidge, and Hoover) and a future Secretary of State (Knox, who would be Attorney General under McKinley and Theodore Roosevelt before taking over the State Department under Taft). There was one future Congressman (George F. Huff, who was a banker and coal operator), a future diplomat (John G. A. Leishman, who would be America's first Ambassador to Turkey), and a future President of the Pennsylvania Railroad (Samuel Rea).\n\nFrick and Mellon would not only go on to amass fortunes of spectacular proportions, but would also demonstrate surprisingly good taste in putting together two of the world's finest private art collections. Carnegie, who was already worth many millions, would wind up with more money than any other American except old John D. Rockefeller and would give away well over $300 million of it with no little fanfare.\n\nIn 1889, however, all that was still a good way off. The Carnegie whiskers had not as yet turned their glistening white. No palaces had been built for Carnegie or Frick on New York's Fifth Avenue. No Rembrandt's had been bought, no daughters married off to titled Europeans, no parks donated, no foundations established. Carnegie thus far had built only one of his free libraries, in Braddock, near Pittsburgh and the site of his gigantic Edgar Thomson works, which he had named after his old friend Thomson, who also happened to be his best customer. At the dedication of the Braddock library in March of 1889, Carnegie used the opportunity to say that he would certainly like to build another one in Homestead someday, but that there had been really far too much labor trouble over there. The implication was pretty obvious and suggested that the purpose behind these earliest benefactions at least may not have been altogether altruistic.\n\nGood works, public service, and any ideas about giving away surplus money were all still things of the future.\n\nIt was in fact during the month of May 1889 that Carnegie was finishing up a magazine article to become known as \"The Gospel of Wealth,\" in which he said, and much to the consternation of his Pittsburgh associates, \"The man who dies thus rich dies disgraced.\" The gist of the article was that the rich, like the poor, would always be with us. The present system had its inequities, certainly, and many of them were disgraceful. But the system was a good deal better than any other so far. The thing for the rich man to do was to divide his life into two parts. The first part should be for acquisition, the second for distribution. At this stage the gentlemen of the South Fork Fishing and Hunting Club were attending strictly to the first part. Business was the overriding preoccupation for now, and business in Pittsburgh, either directly or indirectly, meant the steel business, which in 1889 was doing just fine.\n\nTrue, orders for steel rails had tapered off some. Breakneck railroad building in the west had meant palmy days in the accounting offices along the Allegheny and Monongahela. Seventy-three thousand miles of track had been put down in the 1880's, or more than twice as much track as there had been in the whole country when the war ended at Appomattox. Steel production had more than tripled during those ten years. But now there were also orders for all sorts of architectural steel, huge beams and girders for totally new kinds of buildings called \"skyscrapers\" which were going up in Chicago. In 1887 the Government had made its first order for American steel for ship armor. A new kind of Navy was being built, a steel Navy, including, in 1889, a battleship called the Maine, the biggest thing ever built by the Navy, for which Carnegie, Phipps & Company was making the steel plates.\n\nAnd if the rail business was not quite as good as it had been, the United States was, nonetheless, producing about two tons of rails for every one made in England. As a matter of fact, for three years now, the United States had been the leading steel producer in the world. Pittsburgh, which before the war had not had a single mill as big as those in Johnstown, was now throbbing like no other industrial center in the land. The sprawling complex of mills in Chicago had been providing serious competition of late. Prices were not as high as the steel men would have liked to see them (they never were), and there was more talk among them every week about cutting wages. But if the labor leaders could be dealt with (and there was no reason to think they could not be), then there was every reason to believe that Pittsburgh would keep booming for years.\n\nAs far as the gentlemen of the South Fork Fishing and Hunting Club were concerned no better life could be asked for. They were an early-rising, healthy, hard-working, no-nonsense lot, Scotch-Irish most of them, Freemasons, tough, canny, and, without question, extremely fortunate to have been in Pittsburgh at that particular moment in history.\n\nThey were men who put on few airs. They believed in the sanctity of private property and the protective tariff. They voted the straight Republican ticket and had only recently, in the fall of 1888, contributed heavily to reinstate a Republican, the aloof little Harrison, in the White House. They trooped off with their large families regularly Sunday mornings to one of the more fashionable of Pittsburgh's many Presbyterian churches. They saw themselves as God-fearing, steady, solid people, and, for all their new fortunes, most of them were.\n\nQuite a few had come from backgrounds as humble as Carnegie's. Phipps and Pitcairn were Scotch immigrants who had been boyhood pals with Carnegie in what was known as Slabtown, one of the roughest sections of Allegheny, across the river from Pittsburgh. Leishman grew up in an orphanage. Frick, despite the wealth of grandfather Overholt, had started out in business with little more than a burning desire to get rich.\n\nThey and the others were now living in cavernous, marble-floored houses in the new East End section of Pittsburgh. Several made regular trips to Europe, and those who did not always stopped at the finest hotels in New York or wherever else they went. They now considered themselves, each and all, as among the \"best people\" in Pittsburgh. They pretty well ran the city. They were living the good life as they thought the good life ought to be lived. But never for very long did they take their eye off the real business of the human condition as they saw it\u2014which was business. That they should spend some time together in the summer months, away from Pittsburgh, but not too far away, mind you, seemed, no doubt, a perfectly natural extension of the whole process.\n\n\u2022 \u2022 \u2022\n\nThe reaction in Johnstown to their doings on the mountain was mixed. That the Pittsburghers with all their money should think enough of the country around Johnstown to want to summer there was, of course, terribly flattering. As far back as the 1850's there had been some discussion in Johnstown about the area's potential as a summer resort. One Tribune editor decided to set forth Johnstown's charms in no uncertain terms. \"Our scenery is grand beyond description,\" he wrote, \"the atmosphere cool, and invigorating; trout in the neighboring streams large and numerous; drives good; women beautiful and accomplished; men all gentlemen and scholars, hotels as good as the best.\" For all anyone knew, the South Fork venture might lead to other resort developments in the area. The valley might become famous; property values would mount. It was not an unpleasant thought.\n\nThe club had already provided a lot of work in the South Fork area, and much to the irritation of the club's management it had also provided some excellent sport for local anglers. Slipping onto the property in the early morning or toward sundown was no problem for a local man or boy. The well-stocked lake and streams provided any number of suppers in the neighborhood from the time the dam was first restored. The grounds were well posted, but that discouraged nobody especially. If anything, it only added to the fun.\n\nWhen the club responded by putting in fences over the best trout streams, the fences mysteriously disappeared. Relations then deteriorated fast, with the club authorities threatening to shoot any invaders caught on the grounds after dark. During a flood in 1885, one farmer named Leahy, whose property adjoined the lake, decided to rent out fishing space on some of his submerged acreage. The clubmen said he ought not to do that and threatened to take him to court. This had no effect on Leahy, so they tried to buy him out, but he said he was not interested in selling. It was only after an intermediary was brought in, and lengthy negotiations transacted, that the farm was purchased for $4,000 and the fish thereby further protected.\n\nAny love lost locally by such tactics apparently bothered the club management not in the least. A classic undeclared war between poacher and country squire went on for years. It never became a shooting war, despite the threats, but it did leave widespread resentment in the area around the lake that would one day come back to haunt the clubmen.\n\nFar more important, however, was the way people felt about the dam.\n\nEven before the first full season at South Fork got under way in 1881, the dam threw a terrific scare into the people in the valley. On the morning of June 10, during a flash flood, a rumor spread through Johnstown that the dam was about to break. This was the first spring in years that there was a head of water of any size behind the mammoth earth embankment, and it was the first of the many springs from then until 1889 that just such rumors would fly from door to door, across back alleys, up and down Main Street, and all along the line between Johnstown and South Fork.\n\nThis time the Cambria Iron Company sent two of its men to the lake with instructions to make a critical examination. The dam looked perfectly solid to the Johnstown men, and they returned home with their report in time to make the evening edition of the paper. The fact that they had found the water only two feet from the breast of the dam did not seem to disturb them especially, or the editors of the Tribune.\n\nThe paper summed up its story as follows: \"Several of our citizens who have recently examined the dam state it as their opinion that the embankment is perfectly safe to stand all the pressure that can be brought to bear on it, while others are a little dubious in the matter. We do not consider there is much cause for alarm, as even in the event of the dyke breaking there is plenty of room for the water to spread out before reaching here, and no damage of moment would result.\"\n\nThere it was, in one sentence. In the first place, the dam was probably sound, and even if it did fail not a great deal would happen since the dam was so far away. It was a strange piece of reasoning to say the least, but there it was in the evening paper for everyone, including the alarmists, to read and talk about.\n\nStill, that very night, panic swept through the west end of town, which was the lowest end of town and that part which would have been hardest hit by anything coming down the valley of the Little Conemaugh. People were up through the night \"in mortal dread for fear the old Reservoir near South Fork might break,\" the Tribune reported the next day. So apparently the paper could say what it might about \"no damage of moment\"; people were still unsettled, and especially on nights when the dark and the drenching rain blotted out the landscape and imaginations filled with an ancient terror of death raging out of the mountains.\n\nBut nothing happened. Dawn rolled around as usual; the day began. The long shadow of Green Hill slipped back from mid-town as the sun climbed into the sky; life went on. And it looked as though the paper and everyone who thought along the same lines were right after all. There was really no cause to get excited.\n\nFrom then on, practically every time there was high water in Johnstown there would be talk about the dam breaking. One longtime resident was later quoted at length in the newspapers in New York and elsewhere: \"We were afraid of that lake . . . No one could see the immense height to which that artificial dam had been built without fearing the tremendous power of the water behind it . . . I doubt if there is a man or a woman in Johnstown who at sometime or other had not feared and spoken of the terrible disaster that might ensue. People wondered and asked why the dam was not strengthened, as it certainly had become weak; but nothing was done, and by and by they talked less and less about it . . .\" He also evidently had misgivings about the \"tremendous power\" of the men who had owned the dam, for he chose to withhold his name.\n\nOthers came forth at the same time, that is, after May 31, 1889, claiming to have long held doubts about the engineering of the dam, and premonitions of doom for the whole valley of the Conemaugh. Assuredly most of them spoke from deep conviction; but exactly how much widespread, serious, public concern there was, and particularly in the years of the late '80's, is very hard to say.\n\nCertainly there was every reason to have been concerned. The valley from the lake down to Johnstown had sides as steep as a sluice, and there was only one way the water could go if the dam failed. Floods hit the area almost as regularly as spring itself. Johnstown rarely got through a year without water in the streets at least once, and often for several days at a time.\n\nFloods had been a problem from the time of the very earliest settlements in the valley. Lately, for the past ten years or so, they had been getting worse.\n\nThe very first flood anyone had bothered to make a record of in Johnstown destroyed a dam. That was in 1808, and it had been only a small dam across the Stony Creek which had been put in as a millrace for one of the first forges. Then there were the so-called Pumpkin Floods of a dozen years later. They hit in the fall and had swept what looked like every pumpkin in Cambria County down into town. In 1847 another little dam on the Stony Creek broke. During the flood of 1875 the Conemaugh rose two feet in a single hour. In 1880 again another dam broke; it had been built by Cambria Iron as a feeder for the mills and was about sixteen feet high, but it was located below town, so no damage was caused. During the next eight years there were seven floods, including three bad ones in 1885, '87, and '88.\n\nThe reasons were obvious enough to anyone who took the time to think about the problem, which quite a few were doing by 1889. With the valley crowding up the way it was, the need for lumber and land was growing apace. As a result more and more timber was being stripped off the mountains and near hills, and in Johnstown the river channels were being narrowed to make room for new buildings and, in several places, to make it easier to put bridges across.\n\nForests not only retain enormous amounts of water in the soil (about 800 tons per acre), but in mountainous country especially, they hold the soil itself, and in winter they hold snow. Where the forests were destroyed, spring thaws and summer thunderstorms would send torrents racing down the mountainsides; and each year the torrents grew worse as the water itself tore away at the soil and what little ground cover there was left. Then, in the valley, where the water was being dumped ever more suddenly, the size of rivers which had to carry it all was being steadily whittled away at by industry and the growing population. So there was always a little less river to handle more runoff, and flash floods were the inevitable result.\n\nSome men in Johnstown, curiously enough, thought that encroaching on the river channels would simply force the water to dig deeper channels. But this was impossible because the river beds were nearly all rock. When the volume of water increased, the rivers only came up, and often very fast. In the 1885 flood the Stony Creek rose three feet in forty-five minutes.\n\nThe dam was going to break that year, too, and every year, except one or two, up until 1889. At George Heiser's store, people would come in out of the rain to buy something or just to pass the time in a dry, warm place and nearly always someone said, \"Well, this is the day the old dam is going to break.\" It was becoming something of a local joke. Many years later Victor Heiser would recall, \"The townspeople, like those who live in the shadow of Vesuvius, grew calloused to the possibility of danger. 'Sometime,' they thought, 'that dam will give way, but it won't ever happen to us!' \"\n\nWhen there were warnings of trouble up the mountain, very few took them to heart. The dam always held despite the warnings. People got tired of hearing about a disaster that never happened. And after all, was not the dam owned by some of the most awesome men in the country? If there was anything to worry about certainly they would know about it.\n\nThe Tribune continued to imply that there was no cause for alarm. In 1887 the editors again allowed that a break at South Fork would not greatly affect Johnstown, unless it \"occurred in conjunction with a great flood in the Conemaugh Valley which is one of the possibilities not worth worrying about.\" Readers all through town nodded in agreement. On the afternoon of May 31, 1889, shortly before four, one leading citizen was asked how much higher he thought the water would rise in the valley if the dam let go. His answer was \"About two feet.\"\n\n### -3-\n\nIt would appear, in fact, that Johnstown's leading citizens had taken little or no intelligent account of the threat the dam posed, were it not for some highly interesting letters that changed hands during the year 1880.\n\nWhen Benjamin Ruff first began his restoration of the old dam in the fall of '79, the management of the Cambria Iron Company, in the words of its solicitor, Cyrus Elder, became \"extremely exercised\" over the news. The management at that time was a man by the name of Daniel Johnson Morrell.\n\nMorrell was one of the foremost ironmasters of the age, a ruddy-faced Quaker with gray eyes, who wore his whiskers beneath his jowls, so it appeared he was forever sporting a hair scarf. He looked upon the likes of Carnegie as parvenus in the business, brash, unprincipled upstarts who were not real ironmen at all, but harum-scarum drummers who had jumped into something they knew nothing about just to make a quick fortune. Beside the almost elflike Carnegie or Frick, he looked as though he were of another species. He was under six feet tall, but with his massive, thick shoulders and ample girth (he weighed well over 200 pounds), coming along Main Street he looked every bit the most powerful man in town.\n\nBut according to one of his contemporaries, \"With all the responsibilities of his position, with all the care and concern of the great works on his hands, he never seemed worried or out of humor. When he left his desk at the close of the day he seemed to be able to shut off all thought of work; and in the midst of other persons' worry and nervousness in the most distressing times, he would lie down and sleep as contentedly as a child.\"\n\nAside from running the Iron Company, Morrell presided at most town meetings and was President of the Savings Bank and the First National Bank, the water company and the gas company. He had served two terms in Congress and was still a powerful voice in the Republican Party. For many years he was the President of the American Iron and Steel Association, an organization which did as much as any to protect the protective tariff.\n\nHe lived on Main Street in the finest house in Johnstown, a tall brick house with a mansard roof, painted white and set among gardens and shade trees on a lawn that took up a full city block. He had the only greenhouses in town, a full-time gardener, and all his property was enclosed with an ornamental iron fence. Children used to gather by the fence after school, hoping for a chance to look at him. \"Whatever Mr. Morrell wants, well that's it,\" they heard at home. He was the king of Johnstown.\n\nMorrell had been born in Maine, in 1821 (which made him fourteen years senior to Carnegie), but grew up in Philadelphia and started out clerking in a mercantile store. He had moved to Johnstown in the 1850's when the Philadelphia financial backers of the then floundering Cambria Iron Company sent him to see what might be done to keep the works from going bankrupt.\n\nBackwoods iron forges had been in operation in Cambria County for fifty years and more. With plenty of ore, limestone, and coal in the locale, the prospects for turning the Conemaugh Valley into an iron center of some real consequence looked extremely bright. But until Morrell came to town the industry had been beset by repeated failures. Morrell, however, succeeded handsomely. Knowing nothing about the iron business, he reorganized the company, and despite fires and financial panic, he kept his nerve, maintaining to the Philadelphia money men that the works would one day prosper.\n\nBy the start of the Civil War the Cambria Iron Company was the biggest iron-producing center in the country. In addition, Morrell had encouraged some rather primitive and haphazard research into a new pneumatic process for making steel which contributed substantially to dramatic changes in the iron business and, for that matter, in the whole character and growth of the country.\n\nIn 1856 a man named William Kelly, a Pittsburgher by birth, moved from Eddyville, Kentucky, to Johnstown to set up in one corner of the Cambria yard some experimental apparatus which he assembled from scrap-heap parts and pieces. Kelly was in Johnstown off and on for the next three years. He became known among the millworkers as \"The Irish Crank,\" and not without justification. His attempts to \"refine\" molten iron for the rolling mill by blowing air into it had resulted in repeated failures and at least one serious fire, which became known as \"Kelly's Fireworks.\" But later on, in 1862, he came back to try again, this time with an egg-shaped \"converter\" made abroad, and the accepted story is that he had better luck. Kelly would later be credited with having built the converter himself and with developing at Johnstown something very close to what became known as the Bessemer process, a technique for converting iron into steel at far less cost and in considerably larger quantities than had been possible before.\n\nHenry Bessemer, a brilliant English chemist, had devised just such a process at about the time Kelly first arrived at the Cambria works, and, deservedly enough, got nearly all of the credit. The Bessemer converter used a blast of air directed through molten iron to oxidize, or burn off, most of the carbon impurities in the metal to make steel. Previous steelmaking techniques required weeks, even months. The Bessemer process could produce good-quality steel in less than one hour.\n\nIt was one of the important technological innovations of all time, and Morrell was among the first to recognize just what its impact might be. He financed Kelly's erratic pioneering in the technique for close to five years and after the war invested heavily in new Bessemer equipment. In the late '60's and '70's Johnstown was the liveliest steel center in the country, with the most inventive minds in the industry gathering there\u2014the Fritz brothers, George and John, Bill Jones, and the brilliant and energetic Alexander Holley.\n\nMoreover, Morrell had Cambria Iron do something no other steel company experimenting with the Bessemer process dared try, and something that was to prove immensely beneficial to Andrew Carnegie. He used only American workers, training Pennsylvania farm boys to understand and master the new technology, while everyone else in the business was importing English workers already familiar with it. At first there were months of costly setbacks and disappointments in Johnstown, but the results in the long run proved Morrell right.\n\nIn 1867, from ingots made at Steelton, the first Bessemer rails to be rolled on order in the United States came out of the Cambria mill. By 1871 Morrell had one of the first really big Bessemer plants in operation, and for the next five years Cambria would be the largest producer in the country, if not the world.\n\nThe war had brought flush times and dazzling increases in iron production capacities. But now the age of cheap steel was on. By the time of the late 1880's, Cambria Iron had some 7,000 men on the payroll. The works consisted of the Johnstown furnaces Numbers 1, 2, 3, and 4 in one plant, with stacks seventy-five feet high and sixteen feet in diameter at the base. Blast furnaces Numbers 5 and 6 were in a second plant. The hulking Bessemer plant was the main building. Then there was a huge open-hearth building, a rolling mill that was nearly 2,000 feet long, a bolt-and-nut works, and an axle shop.\n\nAdded to that the company owned and operated its own coal mines, coke ovens, and railroads. It was the largest landowner by far in the county, having bought up thousands of acres around Johnstown, coal holdings primarily, which were, in many places, used as tremendous farms where nothing but hay was grown to feed the animals used in the mills and mines. The company also owned some 700 frame houses which it rented to its workers, a big department store, and the Gautier Steel Company, a subsidiary, where Cambria Link Barbed Wire was made.\n\nTo all intents and purposes, Johnstown, in other words, was a company town and an important one at that. And appropriately enough the company ran the place with an iron hand. Labor unions were not to be tolerated, nor were employees who dared even to talk such treason.\n\nFor example, Rule Number 9 of the plant regulations published in 1874 stated: \"Any person or persons known to belong to any secret association or open combination whose aim is to control wages or stop the works or any part thereof shall be promptly and finally discharged. Persons not satisfied with their work or their wages can leave honorably by giving the required notice . . .\"\n\nThe Cambria Iron Company, which meant Mr. Daniel J. Morrell, left no doubts as to where it stood on such matters. So there were no unions in the mill, and inside the high, green fence that surrounded it, work went on around the clock, around the calendar, without any trouble from the help.\n\nIt would be mistaken, however, to imagine Cambria Iron as an entirely overbearing or inhuman organization, grinding down its employees. By the standards of the day, it was quite progressive and looked out for the welfare of its people and the town with uncommon paternalism.\n\nIn his first speech in Congress, Morrell had said, \"The American workingman must live in a house, not a hut; he must wear decent clothes and eat wholesome and nourishing food. He is an integral part of the municipality, the State, and the Nation; subject to no fetters of class or caste; neither pauper, nor peasant, nor serf, but a free American citizen.\" Judged by the standards of his time, he was almost as good as his word.\n\nIn one of its plants the Iron Company maintained the eight-hour day, a practice that had been tried and abandoned by every other steel company, which meant, as one of the trade-union newspapers pointed out, that the only eight-hour mill left in the country was a nonunion mill.\n\nThe town hospital was built by the company and anyone injured on the job received free treatment there. It was the company also which had established the library and a night school where its employees could learn elementary science, mechanical drawing, and engineering. At the company store, Wood, Morrell & Company, which advertised itself as \"The Most Extensive and Best Appointed Establishment in its Class in the United States,\" prices were quite reasonable. At the time the South Fork Fishing and Hunting Club was organized, Cambria Iron had somewhere in the neighborhood of $50 million invested in Johnstown and along the valley. So Morrell, very understandably, had special interest in what the Pittsburgh men were up to. That he held no special good feelings toward some of the clubmen also seems likely.\n\nWhen Carnegie, who had once stoutly proclaimed that \"pioneering don't pay,\" decided the time was right to get into the Bessemer steel business and Carnegie, Phipps & Company built the Edgar Thomson works at Braddock in the early 1870's, he had raided Cambria Iron of its best workers. Among these men, most of whom had been working with pneumatic conversion techniques since Kelly's days in Johnstown, was the tough, gifted little Welshman, Bill Jones.\n\nCaptain Bill Jones, as he was known, had been with Morrell for sixteen years. He had acquired a vast knowledge of the new process and had built a tremendous following among the men at Cambria Iron, largely for his robust, freewheeling willingness not to do things according to the rules if the rules did not suit him. It was the sort of reputation Carnegie took a special interest in. Carnegie never did learn much about steelmaking, but he had a gift for finding men who did, and if they were somewhat unorthodox, so much the better.\n\nWhen the manager of the Cambria works, George Fritz, died and Morrell had to pick a man to replace him, he turned Jones down for the job, giving it instead to another Jones who seemed a steadier sort. At which point Carnegie immediately moved in and offered Bill Jones a two-dollar-a-day job at Braddock. Jones not only accepted but took a number of his \"high-class graduates\" along with him. Within one year, by 1876, the Edgar Thomson mill moved ahead of Cambria in production. Carnegie, on a mountaintop in Italy, literally danced with joy on hearing the news. And as Edgar Thomson continued to break every other production record, it was Bill Jones, not Carnegie or his associates, who got the credit from anyone who knew anything about steelmaking. Jones's papers on production techniques were read before learned societies in Europe. In Pittsburgh he became so important to the Carnegie empire that Carnegie decided to make him a partner, an offer which Jones flatly refused, feeling that he would lose his influence with the men if he ever so openly joined forces with management.\n\n\"Just pay me one hell of a salary,\" Jones said to Carnegie.\n\nTo which Carnegie shot back, \"All right, Captain, the salary of the President of the United States is yours.\"\n\n\"That's the talk,\" said Jones. The salary was $25,000 a year.\n\nSo now, if the Pittsburgh crowd was about to go tampering with the South Fork dam, Daniel J. Morrell wanted to know more about their doings than could be gained from mere hearsay. There was too much at stake to go on their word alone. He had no intention of stopping them, as he made clear later on. He had never been one to stand in the way of progress. He had welcomed innovations throughout his working life, and it seems he never objected to this one on principle. He only wanted to be satisfied that the work was being properly managed. He had seen enough explosions and fires at the mill to have a fair idea of the violent consequences of bungled innovation. He had also had some experience with dams, having personally supervised the installation of several small ones put in near town by the water company.\n\nIn November of 1880 he sent John Fulton to look over the job. Fulton was an engineer by training and profession, but he was also the next in line to succeed Morrell as head of the works. Morrell, in other words, was not just sending any ordinary employee to South Fork.\n\nA lean Ulster County Irishman, Fulton wore his beard close-cropped and had a fix to his mouth like General Grant's. He had wonderfully heavy eyebrows and a resolute gaze that gave him the look of an Old Testament prophet. He was a man to reckon with, one of Johnstown's most ardent temperance leaders and a pillar at the Presbyterian Church, where he taught Sunday school and would be long remembered for closing his Bible classes with the most interminable prayers ever uttered by man.\n\nFulton had made his reputation as a mining engineer and geologist before joining Morrell at Cambria Iron. That there was anyone in Johnstown better qualified to pass judgment on the dam is doubtful. Nor was there any man, save Morrell himself, who was less likely to be dazzled or cowed in any way by the representatives of the club, a factor which Morrell must have taken into account.\n\nFulton was met at South Fork by two club members, Colonel E. J. Unger and C. A. Carpenter, as well as some of the contractors who had worked on the dam. His report was filed in a letter to Morrell dated November 26. The letter began by stating that he had gone as requested to inspect the dam now owned by the \"Sportsmen's Association of Western Pennsylvania.\" (The correct name of the club was evidently still unknown to the Cambria management.) He then said that he did not think the repairs were done in \"a careful and substantial manner, or with the care demanded in a large structure of this kind.\" He stated that he believed the dam's weight was sufficient to hold back the water, but that he had grave misgivings about other aspects of the dam:\n\nThere appear to me two serious elements of danger in this dam. First, the want of a discharge pipe to reduce or take the water out of the dam for needed repairs. Second, the unsubstantial method of repair, leaving a large leak, which appears to be cutting the new embankment.\n\nAs the water cannot be lowered, the difficulty arises of reaching the source of the present destructive leaks. At present there is forty feet of water in the dam, when the full head of 60 feet is reached, it appears to me to be only a question of time until the former cutting is repeated. Should this break be made during a season of flood, it is evident that considerable damage would ensue along the line of the Conemaugh.\n\nIt is impossible to estimate how disastrous this flood would be, as its force would depend on the size of the breach in the dam with proportional rapidity of discharge.\n\nThe stability of the dam can only be assured by a thorough overhauling of the present lining on the upper slopes, and the construction of an ample discharge pipe to reduce or remove the water to make necessary repairs.\n\nMorrell promptly sent the report to Ruff, who responded on December 2.\n\nRuff was not much impressed by Fulton's findings. He pointed out to Morrell that Fulton did not have the correct name of the club, and told Morrell what that name was. He said there was no leak such as Fulton claimed and that Fulton's figures on the comparative weights of the water and the dam were off, since Fulton had overestimated how much water was in the lake. The tone was one of obvious impatience and suggested not very subtly that Morrell would do well to hire himself a more competent man. He ended the letter by saying:\n\nWe consider his conclusions as to our only safe course of no more value than his other assertions. . . . you and your people are in no danger from our enterprise.\n\nVery respectfully,\n\nB. F. Ruff, President\n\nRuff, quite clearly, was not in the least interested in continuing the discussion. The club had managed nicely to keep its affairs private until then, and the idea of any prolonged or possibly complicated negotiations with the Cambria Iron Company had small appeal.\n\nMorrell, however, was unwilling to let it go at that. On December 22 he answered Ruff's letter. After a few opening courtesies, he got to the heart of the issue:\n\n. . . I note your criticism of Mr. Fulton's former report, and judge that in some of his statements he may have been in error, but think that his conclusions in the main were correct.\n\nWe do not wish to put any obstruction in the way of your accomplishing your object in the reconstruction of this dam; but we must protest against the erection of a dam at that place, that will be a perpetual menace to the lives and property of those residing in this upper valley of the Conemaugh, from its insecure construction. In my judgment there should have been provided some means by which the water would be let out of the dam in case of trouble, and I think you will find it necessary to provide an outlet pipe or gate before any engineer could pronounce the job a safe one. If this dam could be securely reconstructed with a safe means of driving off the water in case any weakness manifests itself, I would regard the accomplishment of this work as a very desirable one, and if some arrangement could be made with your association by which the store of water in this reservoir could be used in time of drouth in the mountains, this Company would be willing to cooperate with you in the work, and would contribute liberally toward making the dam absolutely safe.\n\nMorrell, in short, was suggesting exactly what Fulton had urged: give the dam a major overhaul and install a discharge system of some sort. At the same time, he was making it plain that Cambria Iron considered the present job shoddy enough, the situation critical enough, to be willing to help foot the bill to set things right.\n\nThe offer was declined. The matter was dropped\u2014almost.\n\nMorrell felt that just to be on the safe side it might be a good idea to have an inside view of doings at the lake. So he decided to join the South Fork Fishing and Hunting Club, and evidently the Pittsburgh men had no objections. Morrell therefore purchased two memberships in his own name.\n\nIt was not for another nine years that engineers from many parts of the country came to the site of the dam to study what had gone wrong. Fulton's findings appeared to have been correct. But there were four other changes in the dam which Ruff and his men had made which Fulton had not noticed, and these were as crucial to what finally happened as the faults Fulton cited.\n\nTo begin with, in order to provide room for a road across the breast, the height of the dam had been lowered from one to three feet. This would give enough width for two carriages crossing the dam to pass each other comfortably. But it also meant that the capacity of the spillway had been reduced, for now the bottom of the spillway was not ten or eleven feet lower than the crest of the dam, but perhaps only seven or eight feet. This was a very significant change, since it meant that a rising lake would start to go over the top of the dam that much sooner.\n\nThen, too, a screen of iron rods, each about half an inch in diameter, had been put across the spillway to prevent the fish from going over and down into South Fork Creek. The screen was set between the heavy posts which supported the wooden bridge over the spillway. Under normal conditions the combination of posts and screens decreased the spillway capacity only slightly, but they had the potential of decreasing it a great deal should the screens become clogged with debris.\n\nThe third change was probably the most important of all. The dam sagged slightly in the middle, where the old break had been. Exactly how bad the sag was no one was able to say later for certain. It may have been only a foot or two, but according to one study, the crest at the center may have been as much as four feet lower than the ends. The center was where the dam should have been highest and strongest, so in the event that water ever did start over the top, the pressure would be at the ends rather than at the middle. Now the reverse was the case.\n\nTo have seen the sag with a naked eye, and particularly an untrained eye, would have been next to impossible. It is conceivable therefore that it went unnoticed by Ruff and the men who did the reconstruction work. Fulton took no note of it apparently; whether it would have been observed and corrected had experienced engineers been responsible for the reconstruction is a question no one can answer.\n\nWhat it meant in practical terms was that the depth of the spillway was now only about four feet lower than the top of the dam at its center. In other words, if more than four feet of water were going over the spillway, then the lake would start running over the top of the dam at the center where the pressure against it was the greatest.\n\nThe fourth change was unnoticed by Fulton because it had not as yet taken place when he made his inspection. The water then, as he says, was only forty feet deep, which is about the depth it had been kept at during the old days before the first break in 1862. The club, however, brought the level of the lake up to where it was nearly brim full, meaning that the depth ran to sixty-five feet or thereabouts. In spring it sometimes rose even higher. With the lake that full, it was not beyond reason to imagine serious trouble in the event of a severe storm.\n\nBut, as both Fulton and Morrell had made abundantly clear, with the discharge pipes gone, the club was faced with the unfortunate position of not being able to lower the level of the lake, ever, at any time, even if that were its expressed wish.\n\nThe water that high at the dam also meant that the over-all size of the lake was increased. The lake backed up well beyond where it had been in the old days, which lead to the widespread misconception, still current today in and around Johnstown, that the club had actually raised the height of the dam from what it had been.\n\nHow satisfied Morrell was after the business of the letters was over and done with is not known. For when the sun went down behind Laurel Hill on Monday, August 24, 1885, Daniel J. Morrell was in his grave at Sandy Vale. He had been ill for several years, having suffered what appears to have been an advanced case of arteriosclerosis. He had gone into a steady mental decline not long after he took out his membership in the South Fork fishing club. In 1884 he had given up all his various civic responsibilities and retired from business. After that, it seems, senility closed in hard and fast. He was seen almost never, \"lost in mental darkness,\" as one account put it years later. When he died, \"calmly and peacefully\" at eight in the morning on Thursday, August 20, 1885, he was sixty-four years old.\n\nOn Sunday thousands of mourners queued up along the south side of Main Street to go through the iron gates, up the long front walk, and into the big house to view the remains. For three hours the doors were open and a steady procession filed through.\n\nThe next day, from noon until five, the whole town was shut down. The procession that marched out to the cemetery was as fine a display of the town's manhood as anyone had ever seen. Ahead of the hearse tramped men from the Cambria mines and railroads, the rolling mills and blast furnaces, row on row, like an army, followed by the merchants and professional men, the police, the city fathers, men of every sort who worked for or did business with or depended on the Cambria Iron Company, which meant just about everybody. The only sound was the steady beat of their heavy boots and shoes on the cobblestones.\n\nAfter the hearse came the special carriages for the mourners. Bill Kelly and his wife were there; so was Captain Bill Jones, and a Cleveland steel man and family friend named Marcus Alonzo Hanna.\n\nThere never was a bigger or better funeral in Johnstown.\n\nTwo years later, on March 29, 1887, the day a wagonload of fruit trees arrived at his cottage on Lake Conemaugh, Benjamin F. Ruff died suddenly in a hotel in Pittsburgh. The cause of death according to the papers was a carbuncle on the neck.\n\n## III\n\n## \"There's a man came from the lake.\"\n\n### -1-\n\nThe hard, cold rain that had started coming down the night before had eased off considerably by the morning of Friday, May 31. But a thick mist hung in the valley like brushwood smoke and overhead the sky was very dark.\n\nEven before night had ended there had been signs of trouble. At five o'clock a landslide had caved in the stable at Kress's brewery, and anyone who was awake then could hear the rivers. By six everyone who was up and about knew that Johnstown was in for a bad time. The rivers were rising at better than a foot an hour. They were a threatening yellow-brown color and already full of logs and big pieces of lumber that went bounding along as though competing in some sort of frantic race.\n\nWhen the seven o'clock shift arrived at the Cambria mills, the men were soon told to go home and look after their families. By ten there was water in most cellars in the lower part of town. School had been let out, and children were splashing about in the streets with wooden boxes, boards, anything that would make a boat.\n\nOne of the most distinguished residents of the lower part of town was the Honorable W. Horace Rose, Esquire, former cavalry officer, former state legislator, former district attorney of Cambria County, a founder of Johnstown's Literary Society, father of five, respected and successful attorney at law. Rose was a Democrat with a large following among Republicans as well as his own party, and including those Republicans who ran the town and Cambria Iron. He was an expert horseman, slender, erect, and full-bearded, with strong blue eyes and a soft voice which he seldom ever raised.\n\nHe had been born in a log house that had stood at the corner of Vine and Market. At thirteen he had been orphaned when both parents died of cholera within the same hour and had been on his own ever since, first as a bound boy in a tannery, later as a carpenter. When he was nineteen, John Linton, Johnstown's leading lawyer, took him into his office to \"read law.\" Not long after he opened his own office, which he built himself, and got married to Maggie Ramsey of Johnstown. Then came the war, during which he was wounded, captured, and released in time to take part in General Sheridan's Shenandoah Valley campaign.\n\nThe house Horace Rose and his family lived in was downtown, on lower Main Street. He had witnessed nearly every one of Johnstown's floods over the years, and when he heard that the rivers this particular morning were both coming up rapidly at the same time, something they had not done before, he decided to go out after breakfast and see how things were going.\n\nBy the time he and his two youngest sons, Forest and Percy, had finished hitching the team, there was water on the stable floor. Rose took care to use his second-best harness and had one of the boys drive their cow to the hillside, expecting to bring her back down again in a few hours, when the water subsided. Then they climbed into an open wagon and headed down Main, with Rose driving.\n\nHis intention was to pick up anyone toward the river who wished to be evacuated. But by now, which was somewhere near nine, the water that far downtown was too deep to get through safely. So he wheeled around and headed back up Main, going as far as Bedford, where he paused to pass the time with his old friend Charles Zimmerman, the livery-stable owner. For a few moments they watched another cow being led to higher ground.\n\n\"Charlie,\" Rose said, \"you and I have scored fifty years, and this is the first time we ever saw a cow drink Stony Creek river water on Main Street.\"\n\n\"That's so,\" Zimmerman agreed. \"But the water two years ago was higher.\"\n\nAbout then the rain started coming down again as hard as it had during the night, heavy and wind-driven. Rose stopped off long enough to buy his sons rubber raincoats, then proceeded over to the end of Franklin to his office, which was less than a hundred feet from the roaring Stony Creek. For another half-hour he and the boys set about placing his papers and other things well above the flood line of 1887, which was about a foot from the floor. Then they started for home, stopping once more on the way to talk to another old friend, John Dibert, the banker, who was also their next-door neighbor.\n\nThe situation for property holders in the lower part of town was growing serious, Rose and Dibert agreed. This business of flooded cellars every spring had to be corrected. The solution, as they saw it, was to call a meeting to protest the way the Cambria Iron Company had been filling in the riverbanks next to the mills below town. They recalled that town ordinances had fixed the width of the Stony Creek at 175 feet and the Little Conemaugh at 110 feet. This meant that the combined width of the two was 285 feet; but the Conemaugh, which had to carry all the water from both of them, was now less than 200 feet wide near the mills. Obviously, the rivers were bound to back up when flash floods hit, and obviously the Cambria Iron Company would have to restore the riverbed to its original width. With that settled, the two friends parted.\n\nRose went directly home, where he found the water now so deep that he was unable to get near his front door. He sent young Forest with the team to a nearby hillside while he and Percy assembled a makeshift raft and floated over to the back porch. Once inside, like nearly everyone else in town, they busied themselves taking up carpets and furniture. Rose also \"marked with sadness\" that the slowly rising water \"with its muddy freight\" had already ruined his new wallpaper.\n\nThen everyone moved upstairs where the morning took on the air of a family picnic. Forest had been unable to get back to the house after leaving the horses but had signaled from a window across the street that he was high and dry with the Fishers.\n\nHorace Rose called back and forth to Squire Fisher and joked about their troubles while his wife and the others got a fire going in the grate and made some coffee. After a bit Rose got his rifle, went up to the attic, propped himself in a window, and whiled away the time shooting at rats struggling along the wall of a stable in the adjoining lot.\n\nAnd so the morning passed on into afternoon; there was nothing much to do but wait it out and make the best of what, after all, was not such an unpleasant situation.\n\n\u2022 \u2022 \u2022\n\nThere were hundreds of other families, however, who had seen enough. They began moving out, wading through the streets with bundles of clothing and food precariously balanced on rude rafts, or jammed into half-submerged spring wagons. Here and there a lone rowboat pushed up to a front porch or window ledge to make a clumsy, noisy rescue of women and grandfathers, dogs, cats, and children.\n\nSome people were simply heading for higher ground, without any particular place in mind; others were going to the homes of friends or relatives where they hoped there might not be quite so much water, and where, come nightfall, there might be electricity and a dry kitchen.\n\nA few families went over to the big hotels in the center of town, thinking they would be the safest places of all to ride out the storm. Quite a good many, wherever their destination, went a little sheepishly, dreading the looks and the kidding they would get when they came back home again.\n\nThe water by now, from one end of town to the other, was anywhere from two to ten feet deep. It was already higher than the '87 flood, making it, by noon at least, Johnstown's worst flood on record. The Gautier works had closed down at ten, when Fred Krebs, the manager, was reminded by one of the men that the huge barbed-wire plant stood on fill that had been dumped into the old canal basin, and that once upon a time there had been four feet of water right where they were standing. At eleven, or soon after, a log boom burst up the Stony Creek and sent a wild rush of logs stampeding through the valley until they crashed into the stone bridge below town and jammed in among the massive arches.\n\nNot very long after that the Stony Creek ripped out the Poplar Street Bridge; then, within the hour, the Cambria City Bridge went. At St. John's Catholic Church, which stood far uptown at Jackson and Locust, and so, presumably, well beyond reach of spring floods, the water was so deep that the funeral of Mrs. Mary McNally had to be postponed midway through the service and the casket left in the church.\n\nWorst of all, and unlike any other flood in Johnstown's history, there had been a tragic death. A teamster named Joseph Ross, a father of four children, had been drowned when he fell into a flooded excavation while helping evacuate a stranded family.\n\nAlong Main Street, shopkeepers were working feverishly to move their goods out of reach of the water. In his second-floor office overlooking Franklin and Main, George T. Swank, the cantankerous editor and proprietor of the Tribune, began working on what he planned to be a running log of the day's events, with the intention of publishing it in the next edition, whenever that might be.\n\n\"As we write at noon,\" he put down, \"Johnstown is again under water, and all about us the tide is rising. Wagons for hours have been passing along the streets carrying people from submerged points to places of safety . . . From seven o'clock on the water rose. People who were glad they 'didn't live downtown' began to wish they didn't live in town at all. On the water crept, and on, up one street and out the other . . . Eighteen inches an hour the Stony Creek rose for a time, and the Conemaugh about as rapidly.\"\n\nOn the street below his window the current, coming across from the Stony Creek, was rushing by at an estimated six miles an hour.\n\n\u2022 \u2022 \u2022\n\nAcross Main, and three doors down Franklin, the Reverend H. L. Chapman was having a slightly unnerving day.\n\nAfter an early breakfast he had retired to his study to work on his sermon for Sunday. The text he had selected was \"But man dieth, and wasteth away: yea, man giveth up the ghost, and where is he?\" He had barely begun when he was interrupted by the doorbell. Opening the front door, he found his wife's cousin, Mrs. A. D. Brinker, standing on the porch looking terribly frightened.\n\nShe had crossed the park from her home on the other side. She asked Chapman if he had heard about the high water downtown. He said he had not and that he did not think there would be much of a flood.\n\n\"Johnstown is going to be destroyed today,\" she said, and then told him that the reservoir would break and all would be swept away.\n\nChapman was so incredulous he almost laughed in her face, and would have had she not looked so pitifully terror-stricken. It was also not the first time Mrs. Brinker had made just such a forecast.\n\n\"Well, Sister Brinker, you have been fearing this for years,\" Chapman said with patience, \"and it has never yet happened, and I don't think there is much danger.\"\n\nHe invited her to step in and stay with them until after the flood had passed, an invitation she gladly accepted, saying that her husband had insisted on staying home, to \"hold the fort\" as he had put it.\n\nLater, a young student friend named Parker dropped by to see if the Chapmans needed help moving their furniture, but the Reverend told him he expected no trouble, as the new parsonage had a higher foundation than other houses. But the young man stayed on nonetheless.\n\nAbout noon Chapman happened to look out the window long enough to see one of the town's most dignified figures standing in the street in water up to his waist. It was Cyrus Elder, who along with being chief counsel for the Iron Company was now Johnstown's one and only member of the South Fork Fishing and Hunting Club, having acquired the Morrell memberships at the time of the old Quaker's death.\n\nChapman was puzzled by the whole thing and hurried onto the porch to see what he could do. Elder, though a most solid citizen, seems also to have had a sense of humor.\n\n\"Doctor,\" he called, \"have you any fishing tackle?\"\n\nChapman answered that he thought he had.\n\n\"Well,\" said Elder, \"I was in a skiff and it upset and left me here, and I am waiting for a man who has gone after a horse, to take me out, and I might put in my time fishing.\"\n\nWhen the man and horse returned, the hefty Elder, try as he would, was unable to get up on the animal's slippery back. So the rider went off again and returned next with a wagon. This time they had better luck and started off down Main toward Elder's home on Walnut Street but had to turn back and head uptown for Elder's brother's house; all of which was duly noted with amusement from the window of the Tribune by George Swank, who was also Elder's brother-in-law.\n\nElder had arrived back in Johnstown from Chicago just that morning and had been trying for hours to get home to his wife and daughter. From the station platform he had been able to look right across at his house, where, on the front porch, the two women were waving their handkerchiefs at him. They had gestured back and forth about the water and how he might get home, but from then on he had made little progress.\n\nAt midday the Chapmans and their guests sat down to dinner, but Mrs. Brinker was still too unstrung to eat anything. Dinner over, they moved to the study, where they sat quietly chatting beside the gas fire. But the Reverend soon grew impatient with the comings and goings of Lizzie Swing, the Chapmans German servant girl, who kept tramping past the study door on her way from the cellar to the attic with armloads of food.\n\n\"Why does she do that?\" Chapman asked. He was genuinely puzzled by the girl's obvious state of nerves, since she understood almost no English. The Reverend was having trouble maintaining an intelligent sense of domestic calm.\n\n\u2022 \u2022 \u2022\n\nThe train which returned Cyrus Elder to Johnstown had left Chicago the day before at three in the afternoon. It was one of two sections of the Day Express, which had pulled out of Pittsburgh that morning, on time, at 8:10. The two trains arrived at Johnstown about 10:15, and again on time, but were held there for a half-hour or so. The eastbound track on up the valley had washed out. Not until a local mail train came through were the two sections given orders to follow it to East Conemaugh, running east on the westbound track.\n\nDuring the wait at Johnstown, passengers on board watched in fascination the struggles of the flooded city. They waved back and forth to families hanging out of upstairs windows. Some got out for a few minutes and joined the crowds on the station platform and along the near enbankment to watch the railroad crew that was trying to dislodge the logs and drift from the stone bridge. When the trains began moving again, very slowly, around the blind corner of Prospect Hill and on to the East Conemaugh yards two miles ahead, the passengers could see the ugly yellow-brown surge of the Little Conemaugh to their right, now very near their rain-streaked windows. More and more debris swept by and telegraph poles swayed precariously in the strong wind.\n\nThe run to East Conemaugh took about ten minutes, with the mail train in the lead, followed by the first and second Day Express, in that order. At East Conemaugh all trains were stopped and held for further orders. There was trouble up the line at Lilly. Bear Run had risen more than six feet, burst over its banks, and washed out a quarter of a mile of track. Nothing could move east or west. Two earlier eastbound trains, the Chicago Limited and a freight from Derry, had gotten as far as South Fork, where now they too were being held.\n\nThe first section of the Day Express was made up of seven cars\u2014five coaches, a baggage car, and one Pullman. On board were some 90 passengers plus crew. The second section had three sleepers, one baggage and two mail cars, and perhaps 50 passengers and crew. The mail train had only three cars, one for the mail, plus two coaches. Most of its passengers were members of the Night Off company, who were on their way to Altoona for their next performance.\n\nEast Conemaugh was the main marshaling yard for Johnstown. Eastbound trains picked up their \"helpers\" there, the extra engines for the climb over the mountain. There was a huge sixteen-stall roundhouse, water towers, four main tracks, sidings, sheds, repair shops, coal tipples, dozens of locomotives, rolling stock of every description. So that the trains now detained there in the driving rain were only part of a whole concentration of equipment set along a broad flat where the Little Conemaugh makes a sweeping curve between the hills.\n\nFor the people of the little town set just back from the yards, the morning was turning into a fine show. The river was still within its banks but rising fast. There was every sign that the wooden bridge above the station was about to wash out, and rain or no rain, most of the town had gathered along the riverbank to see it happen.\n\nSometime between noon and one o'clock a telegraph message came into the East Conemaugh dispatcher's tower from the next tower up the valley to the east. The message was directed to the yardmaster at East Conemaugh, J. C. Walkinshaw, and to the head of the entire division, Mr. Robert Pitcairn, at Pittsburgh. No one would later recall at exactly what moment the message arrived, and numerous people who should have seen it later claimed that they never did. Nor was there ever agreement on its precise wording, but the consensus was that the message said something close to this:\n\nSOUTH FORK DAM IS LIABLE TO BREAK: NOTIFY THE PEOPLE OF JOHNSTOWN TO PREPARE FOR THE WORST.\n\nIt was signed simply \"Operator.\"\n\nAt Johnstown the message was received at the telegraph office at the depot only a few minutes later. The freight agent, Frank Deckert, was told it had come in; he glanced at it, but he did not stop to read it. As he said later, he knew that \"it was in regard to the dam; that there was some danger of it breaking.\" But it created no alarm in his mind. He had heard such warnings before. When he passed the word along to the few people who happened to be about the station, their response was the same as his, with only one or two exceptions. Two men who were shown the message by Charles Moore, the assistant ticket agent, read it and laughed out loud.\n\nDeckert made no further effort to spread the warning. He did not move his family from their home just down from the station, nor did he bother to send the message over to the central part of town.\n\n### -2-\n\nNo one on the mountain could remember there ever being a night like it.\n\nJohn Lovett, who was seventy-one and had a sawmill on South Fork Creek a quarter of a mile from the head of Lake Conemaugh, said it was the hardest rain he had ever heard. He could not see it, he said, but he could hear it all right, and the creeks got so vicious they carried off logs that had been on his place for forty years. William Hank and Sam Peblin, who had farms farther up the mountain, at the headwaters of South Fork Creek, said much the same thing. Sylvester Reynolds, another farmer, reported that Otto Run, which feeds into Yellow Run, was running four feet deep, compared to its normal depth of two inches. F. N. George, Justice of the Peace at Lilly, said he had never known a cloudburst like it in fifty years. At Wilmore, H. W. Plotner, a druggist who was nearly seventy, said he could recall no worse storm. Dan Sipe, who owned the flour mill on the Little Conemaugh at Summerhill, Sheriff George Stineman, the coal operator at South Fork, and Mrs. Leap, who kept the general store at Bens Creek, all agreed it was the mightiest downpour and the highest water ever in their memories.\n\nThere were also \"weird and unnatural occurrences\" reported. One family by the name of Heidenfelter later described how they had been suddenly awakened and badly frightened by a \"rumbling, roaring sound\" that seemed to come from some indefinable object not far from their house. It was then followed by a terrific downpour, which, according to Mrs. Heidenfelter, sounded as if a gigantic tank had opened at the bottom and all the water dumped out at once.\n\n\"Indeed I thought the last day had come,\" she later told a newspaper reporter. \"I never heard anything like it in my life. I wanted my husband to get up and see what the matter was, but it was dark and he could have done no good. In the morning, as soon as we could see, the fields were covered with water four or five feet deep. . . . People say the noise we heard was a waterspout, but I've never seen one and don't know how they act.\"\n\nApparently the storm did tear big holes in the ground near the Heidenfelter farm, and other families close by the lake reported hearing sounds much like thunder but which they were certain were not thunder.\n\nIn any case it was a wild night on the mountain, and when morning came virtually every farm in the area had swamped cellars and pastures. Freshly plowed fields were sliced through with gullies that carried water as much as three feet deep. Acres of winter wheat and corn planted a few weeks earlier had been washed away. Every backwoods road had turned into a creek; every little mountain spring, run, creek, and stream was on a rampage. The earth could not absorb any more water.\n\n\u2022 \u2022 \u2022\n\nIt was about six thirty that morning when young John Parke awoke in his high-ceilinged room upstairs at the clubhouse on the shore of Lake Conemaugh. He had awakened once before, about an hour earlier, and had heard the rain hammering against the big frame building, but thinking nothing of it, had dropped off to sleep again. Now, outside his window, there was little to be seen but a heavy, white mist that had closed down over the trees and water.\n\nParke dressed quickly, went downstairs, crossed through the main living room, out the porch door into the cold morning, where, for the first time, he heard what he would later describe as a \"terrible roaring as of a cataract\" coming from the head of the lake to the south. He also noticed that during the night the lake had risen what looked to be perhaps two feet. Yesterday the water had been at its usual level, which, he reckoned, was about four to six feet below the crest of the dam. Now, it might be no more than two or three feet from the crest. What he could not tell was how much water was still coming in, and that he knew would be the crucial factor for the next several hours. But the sound from the head of the lake was far from encouraging.\n\nHe went inside again, had breakfast, then, along with a young workman who had been helping on the sewer project, he got hold of a rowboat and started off to have a look at the incoming creeks.\n\n\"I found that the upper one-quarter of the lake was thickly covered with debris, logs, slabs from sawmill, plank, etc.,\" he wrote afterward, \"but this matter was scarcely moving on the lake, and what movement there was, carried it into an arm or eddy in the lake, caused by the force of the two streams flowing in and forming a stream for a long distance out into the lake.\"\n\nAs he and his companion neared the far end, he was astonished to discover that they were rowing over the top of a four-strand barbed-wire fence which stood well back from the normal shore line. Then, rowing against the strong current, they proceeded to cover another hundred yards or more across what was normally a cow pasture. They passed by the place where Muddy Run emptied into the lake and went on to South Fork Creek, which Parke described as \"a perfect torrent, sweeping through the woods in the most direct course, scarcely following its natural bed, and stripping branches and leaves from the trees five and six feet from the ground.\"\n\nThe two of them pulled their boat onto what seemed the driest spot in sight and started up along the creek by foot. For half a mile the woods boiled with water. The trees dripped water, their drenched trunks black against the mist. The very air itself seemed better than half water.\n\nWhen they returned for their boat, they found that the lake had come up enough in that short time to set it slightly adrift. From there they struck out straight for the clubhouse. From what he had seen, Parke knew that the situation at the dam must be growing very serious and that an appreciable letup in the volume of water pouring into the lake was most unlikely.\n\nAs near as he could tell, the lake was rising about an inch every ten minutes. If this were so, it would be only a matter of hours until the water started over the top of the dam, unless something could be done to release more water than the spillway was handling.\n\nAt the clubhouse Parke was told that he was needed at the dam immediately. He went to the stable for his horse and within minutes was galloping off through the cold rain.\n\nThere were close to fifty people at the dam when he came riding out of the woods. There was a clump of bystanders, South Fork men and boys mostly, under the trees over at the far side, next to the spillway. Along the road that crossed over the dam itself, a dozen or so of the Italian sewer diggers were working with picks and shovels, trying, without much success, to throw up a small ridge of earth to heighten the dam. Bill Showers, Colonel Unger's hired man, was also making little progress with a horse and plow. Despite all the rain, the road was so hard packed that thus far they had managed to make only a slight strip of loose earth across the center of the dam hardly more than a foot high.\n\nAt the center of the dam the water level was only two feet or so from the top.\n\nAt the west end another ten or twelve men were trying to cut a new spillway through the tough shale of the hillside but were able to dig down no more than about knee-deep, and the width of their trench was only two feet or so.\n\nAlso among the onlookers were several of the clubmen who had come up from Pittsburgh for the Memorial Day weekend. But the man who was directing things, and deciding what ought to be done as the water advanced steadily toward the crest, was Colonel Elias J. Unger, who had retired from business in Pittsburgh the year before and had only recently been named the club's president and over-all manager. He was living at the lake the year around now, in a modest farmhouse just beyond the spillway.\n\nThe Colonel had started life on another farm in Dauphin County, in the eastern part of the state. His father was a Pennsylvania German, as was his mother, who came from the big and well-known Eisenhower family. At twenty he got a job on the railroad and managed to work himself up from brakeman to conductor to superintendent of the Pennsylvania's hotels, from Pittsburgh to Jersey City, including the one at Cresson, where he was manager for a time and so got to know Carnegie and the others.\n\nAbout the time the South Fork club was being organized, he had gone into the hotel business on his own in Pittsburgh and made even more of a name for himself. By 1888 he was well enough situated to buy the place on Lake Conemaugh and settle down to a quiet retirement in a glorious setting where there was also the added interest of a not very taxing job to keep him occupied, plus, in the summer months at least, the chance to keep up with his Pittsburgh friendships.\n\nUnger had come a long way from Dauphin County. But even so, socially and financially, he was a noticeable cut below the other members of the South Fork fishing and hunting organization. His experience in hotel management, it would appear, had something to do with his position in the club.\n\nThe Colonel had returned home only the night before, after visiting friends in Harrisburg. When he got out of bed that morning at six, it looked to him, he later said, as though the whole valley below was under water, and he was baffled as to what it all meant. He put on his gum coat and boots and walked down the hill in front of the house, crossed the wooden bridge over the spillway and walked out onto the dam, where he began taking measurements of the rising water.\n\nAbout eight thirty Unger's caretaker over at the club grounds, a man by the name of Boyer, came by in a spring wagon with D. W. C. Bidwell, who was on his way to South Fork intending to catch the 9:15 train to Pittsburgh. Bidwell, who evidently had had enough of the soaking weekend at the lake, stopped to ask Unger how things were going.\n\n\"Serious,\" answered Unger, who later that morning was heard to say that if the dam survived the day, he would see that major changes were made to insure that this sort of thing never happened again.\n\nWhen Boyer got back from South Fork, which was sometime near ten, Unger sent him off to bring the Italian work crew down to the dam. He had decided to try digging another spillway at the western end, where he thought the hillside would be solid enough to keep the water from cutting through it too rapidly. There was brief disagreement over the idea, with some of the men protesting that the water would rip through any new wasteway so fast that the dam would quickly fail.\n\n\"It won't matter much,\" Unger said, \"it will be ruined anyhow if I can't get rid of this water.\"\n\nWhen it became clear that even the shallowest sort of ditch could barely be cut through the rocky hillside, Unger then set several of his men to work trying to clear away the debris which by now was clogging the iron fish screens in the main spillway and seriously reducing its capacity.\n\nAmong the bystanders taking all this in was a small fourteen-year-old boy with the big name of U. Ed Schwartzentruver, who, with some of his friends, had been there all morning in the rain watching the excitement. Seventy-six years later, sitting on his porch on Grant Street in South Fork, not quite ninety and nearly blind, he would talk about what he had seen that morning as though it had happened the day before.\n\n\"When this high water come down, there was all kinds of debris, stumps, pieces of logs, and underbrush and it started to jam up those screens under the bridge. The bridge was well constructed of heavy timber. There was a man named Bucannon up there, John Bucannon, who lived in South Fork. Well he kept telling Colonel Unger to tear out that bridge and pull out that big iron screen.\n\n\"But Colonel Unger wouldn't do it. And then when he said he would do it, it was too late. The screens wouldn't budge, they were so jammed in by all that debris.\"\n\nWhen John Parke came up onto the dam on horseback, he did what he could to exhort Unger's men to dig harder and faster, riding back and forth along the breast, shouting orders and moving men from one place to another when he thought it would do some good. But by eleven o'clock it was apparent to everyone that the lake was still advancing as fast as ever before. In fact, by eleven, the water was about level with the top of the dam and had already started to eat into what little had been thrown up by the plow and shovels. On the outer face, near the base of the dam, it looked as though several serious leaks had developed.\n\nAt this point, Colonel Unger decided that perhaps something ought to be done to warn the people in the valley below. The only way was to send a man down. There was a telephone line from the clubhouse to South Fork, but it was used only during the summer season and had not as yet been put in working order.\n\nWith all the rain there had been, the road to South Fork was in very bad shape, but John Parke made the ride in about ten minutes. Parke's relative youth, and the fact that he was not well known in the area, may account for the marginal success of his mission.\n\nFurthermore, the first people to come from the dam to South Fork that morning, Boyer and Bidwell, had already told everyone that there was no danger of the water running over the top. So when Parke came splashing up Railroad Street with his warning, the news was both unexpected and perhaps seemed somewhat questionable. According to testimony made later by Bidwell, Parke stopped in front of George Stineman's supply store, which was across the street from the depot, and where a small crowd had gathered.\n\n\"I saw him come down there,\" Bidwell said, \"and make a statement to the people standing about that the water was then running over the top of the dam, and there was very great danger of it giving way.\" Parke also told two men to go to the railroad's telegraph tower next to the depot and tell the operator to alert Johnstown. But soon after they left, Bidwell, according to one witness, began telling everyone that there was really nothing to get excited about.\n\nThe operator at South Fork that Friday morning was Miss Emma Ehrenfeld. She had come on duty at seven o'clock. It was about noon, she would later estimate, that a man came up into the tower \"very much excited.\"\n\n\"Notify Johnstown right away about the dam,\" he said. \"It's raising very fast and there's danger of the reservoir breaking.\"\n\n\"Who told you all this?\" she asked.\n\n\"There's a man came from the lake,\" he said.\n\nEmma was not quite sure how much to believe of his story. She had seen the man around town and thought his name was Wertzengreist, though she was uncertain about that too. But she said later, \"He is a man that people generally don't have much confidence in, and for that reason, I scarcely knew what to do under the circumstances.\"\n\nShe was also hampered by the fact that her lines west were open only as far as the next tower, four miles down the river at Mineral Point. Beyond Mineral Point there seemed to be a break somewhere and so she had no direct contact with Johnstown.\n\nThe operator at Mineral Point, W. H. Pickerell, was an old hand along the Little Conemaugh, having been there at that same tower for some fifteen years. Emma decided to \"talk\" it over with him on the one good wire she had. She tapped out her problem and waited for an answer. Pickerell told her that the break to the west was caused by the poles falling into the river, and that though he had no way of getting Johnstown, he thought \"it was a thing that there oughtn't to be any risks taken on.\" He said he would take the message and send it on to East Conemaugh by foot if someone should happen along the tracks below his tower.\n\nSo the two of them worked out a message addressed to the yardmaster at Conemaugh and to Robert Pitcairn in Pittsburgh.\n\n\"I wrote the message up,\" Pickerell declared weeks after, \"and repeated it to her and asked her if that would do, and she said that was splendid\u2014to send it that way. I doubled the message and waited and waited.\"\n\nAfter a while a trackman came by. He had been sent from East Conemaugh to flag a landslide at Buttermilk Falls, to the west of Pickerell's tower. Pickerell gave him the folded message and sent him on his way back down the tracks. At Buttermilk Falls, the man, whose name was William Reichard, turned the message over to his boss, the foreman of the division, L. L. Rusher, who set out for East Conemaugh after telling Reichard to go on back up to Mineral Point, in case there should be any more messages.\n\nAs it turned out, Rusher had only to go as far as what was known as \"AO\" tower, which was about a mile and half from Mineral Point and better than a mile upriver from East Conemaugh. From \"AO\" tower west the lines were still clear. Rusher gave the message to operator R. W. Shade, who sent it on immediately. And it was his message which was received by J. C. Walkinshaw at East Conemaugh and by agent Deckert in Johnstown sometime between noon and one o'clock.\n\nIn Pittsburgh, operator Charles Culp, at the Union depot later said he was the one who had received the message there and that he took it right over \"and laid it on Mr. Pitcairn's table in front of him.\" Within an hour Robert Pitcairn, who had a special interest in the South Fork Fishing and Hunting Club, as well as the Pennsylvania Railroad, was sitting in his private railroad car on his way to Johnstown.\n\nBut the telegram drafted by operator Pickerell and Emma Ehrenfeld was only the first of three warnings sent down the valley by way of the Pennsylvania's telegraph system.\n\nAbout twelve thirty, or sometime very shortly after John Parke reined up in front of Stineman's store, another rider was sent from South Fork to check the condition of the dam. His name was Dan Siebert. He worked for J. P. Wilson, who was superintendent of the Argyle Coal Company in South Fork and an old friend of Robert Pitcairn's. Wilson had been asked by Pitcairn some three years earlier to notify him at once if ever he saw any signs of danger at the dam.\n\nSiebert borrowed Wilson's horse and was up and back from the dam inside of twenty-five minutes. He had stayed only long enough to see that near the center of the dam a glassy sheet of water, fifty to sixty feet wide, had started over the top. But Siebert did not seem especially concerned over what he had seen. He was, in fact, according to one witness, \"perfectly cool about it.\"\n\nWilson, however, on hearing Siebert's report, turned to C. P. Dougherty, the Pennsylvania's ticket agent in South Fork, and asked him if he did not think that Mr. Pitcairn should be notified. When Dougherty hesitated, saying there was trouble with the wires downriver, Wilson took it to mean that Dougherty was reluctant to assume the responsibility of such a message and told him to sign his, Wilson's, name.\n\nWhereupon Dougherty went over to the tower, taking along another operator, Elmer Paul, who was more experienced than Miss Ehrenfeld and who Dougherty thought might have better luck getting a circuit. Paul tried the wire for a few minutes but without success.\n\nSo again a message was sent as far as Mineral Point, where it was received at 1:52 by operator Pickerell, who gave it to William Reichard, who walked it down the tracks to \"AO\" tower. From there it was put on the wire to East Conemaugh, Johnstown, and Pittsburgh. The message read:\n\nSOUTH FORK, MAY 31, 1889\n\nR.P. O.D. VIA MP & AO\n\nTHE WATER IS RUNNING OVER THE BREAST OF LAKE DAM, IN CENTER AND WEST SIDE AND IS BECOMING DANGEROUS.\n\nC. P. DOUGHERTY\n\nIt was no more than thirty minutes later that J. P. Wilson came up to the tower himself to have Emma Ehrenfeld send still another warning. The rain was beating down terribly hard by then, and outside they could see the water of the Little Conemaugh and South Fork Creek raging across the flats just below the station. Over near Lake Street, South Fork Creek had flooded the first floors of several houses and the aspens along the banks were whipping about wildly in the wind.\n\nWilson had just heard that a young South Fork boy named John Baker had ridden down from the lake and said that the water had now cut a notch in the center of the dam. Without taking time to write anything down, Wilson dictated a message to Pitcairn which Emma Ehrenfeld put right on the wire.\n\nSOUTH FORK, MAY 31, 1889\n\nR.P.\n\nOD\n\nTHE DAM IS BECOMING DANGEROUS AND MAY POSSIBLY GO.\n\nJ. P. WILSON\n\nWilson waited in the tower long enough to be sure the message had gotten at least as far as Mineral Point; then he warned Miss Ehrenfeld to be on the lookout up South Fork Creek and went out the door\n\nThe time was 2:25. By 2:33 the message had reached East Conemaugh. For some unknown reason, Pickerell this time had been able to get a circuit. Apparently, a wire that had fallen into the river lifted out somehow, and as Pickerell said, \"All at once the wire came all right.\"\n\nThe message was through to East Conemaugh in a matter of minutes, and on to Johnstown and Pittsburgh. Agent Deckert in Johnstown would later state that he had received this particular message sometime near 2:45. He also concluded upon reading it that this time he had best telephone its contents across the way to Hettie Ogle, who ran the central telephone switchboard and Western Union office, just across the river at the corner of Washington and Walnut.\n\nMrs. Hettie Ogle was a Civil War widow who had been with Western Union for some twenty-eight years. At one o'clock the rising water had forced her to move, with her daughter, Minnie, to the second floor of the two-story frame building. Sometime near three she notified her Pittsburgh office of the condition of the dam as reported by Deckert and said that that would be her last message, meaning that the rising water was about to ground her wires. Then she put through a call to the Tribune, where editor Swank was still keeping up his running account of the day's events.\n\n\"At three o'clock,\" he wrote, \"the town sat down with its hands in its pockets to make the best of a very dreary situation. All that had got out of reach of the flood that could, and there was nothing to do but wait; and what impatient waiting it was anyone who has ever been penned in by a flood and has watched the water rising, and night coming on, can imagine. . . .\"\n\nHe described how the Stony Creek carried a live cow down from some point above Moxham and how she struck against a pier of the dislodged Poplar Street Bridge, where she managed to get a foothold for a while but finally, making a misstep, fell into the current and was carried off.\n\n\"At 3:15 the Central Telephone office called the Tribune up to say it had been informed by Agent Deckert, of the Pennsylvania Railroad freight station, that the South Fork Reservoir was getting worse all the time and that the danger of its breaking was increasing momentarily. It is idle to speculate what would be the result if this tremendous body of water\u2014three miles long, a mile wide in places, and sixty feet deep at the breast at its normal stage\u2014should be thrown into the already submerged Valley of the Conemaugh.\"\n\nBut by 3:15 Lake Conemaugh was already on its way to Johnstown.\n\n### -3-\n\nWhen John Parke had arrived back at the dam from his dash to South Fork, he was confident that his warning had been sent on down the valley. Along the way from South Fork he had passed two men struggling through the mud with a sewing machine, and one of them shouted to him, \"We got the sewing machine out, if nothing else,\" which Parke took to be a very good sign. At the dam he found the water had already started sliding over the top, at the center, right above where the old culvert had been. It had taken no time for the water to wash across the little earth ridge that had been thrown up. Now, as he rode his horse out along the breast, the water crossing over the road there was a good six inches deep and getting stronger every minute. Within minutes the sheet of water was a hundred yards wide. But it was all concentrated at the center, clearly illustrating, Parke noted, that the dam dished a little.\n\nIt was now shortly after noon. At the western end the emergency spillway was running about twenty-five feet wide, but only slightly deeper than before. At the main spillway, where the water was roaring through six feet deep or more, the men had started to tear up the floor boards of the bridge and were attempting to remove a V-shaped floating drift made of logs with nails sticking out of them, which had been set out in the water twenty feet or so before the spillway to deter the fish even from venturing toward the screens.\n\nThe men were afraid to go out on the dam now, and so Parke rode across the breast alone, studying the effects of the overflowing water on the face of the dam. He saw that little gullies had already been cut between the riprap, but the damage was not as bad as he thought it would be.\n\nFor a brief moment he gave serious thought to the possibility of cutting another spillway through the dam proper, where there would be no problem digging deep and where the water would do quite a lot of the digging for them. It would have meant the end of the dam, of course; the water, he knew, would bore through any such cut, ripping the dam in two in no time. But by making such a cut near one of the ends, where the pressure was far less than at the center, the water, as fast as it might escape, would still go out a far sight slower than if the whole dam gave way all at once at the middle. It would have meant the certain destruction of the dam, but also far less damage below, he figured.\n\nIt would have been a terribly bold decision, and one which Parke alone would have been in no position to make. It would also be, he concluded, a foolhardy decision. Frightful damage would be caused for certain, and they would be responsible; furthermore, there would then be no way to prove that the dam had been about to break and that they had been left with no other choice. Indeed, Parke decided, there was no certainty at all that the dam was going to break. At which point he hurried to the clubhouse for his dinner.\n\nWhen he returned to the dam, Parke found things had taken a decided turn for the worse.\n\nSeveral big rocks on the outer face had washed away, and the water pouring across the top had cut a hole into the face about ten feet wide and four feet deep. As the water kept pounding down into this hole, he could see it slicing away at the face, a little more every minute, so that the hole began to take the form of a huge step.\n\nThere was absolutely nothing anyone could do now but watch and wait and hope. Parke, Unger, all of them, just stood there looking at the water and the valley stretching away below.\n\nThe rain-drenched crowd gathered on both hillsides had grown considerably since early morning. The news of trouble had spread fast and wide.\n\nSome men had more or less smelled trouble hours before. George Gramling, who had a sawmill on Sandy Run, started off for the dam about eight in the morning, along with Jacob Baumgardner and Sam Helman. The Gramling mill was operated by a small dam which had broken about seven. If a small dam washed out that early, the men reasoned, what might a big dam do later on?\n\nBut mostly people were just curious. The Reverend G. W. Brown, pastor of the South Fork United Brethren Church, for example, like nearly everyone else in the neighborhood, had heard rumors of trouble all day and decided finally to go up to see for himself. When he arrived at the dam it was about ten minutes to three. There was no one actually out on the dam then, just at the ends, and the water was pouring over the breast.\n\nMinutes later he saw the first break. He said it was \"about large enough to admit the passage of a train of cars.\"\n\nJohn Parke said that the break came after the huge \"step\" had been gouged back into the face so near to the water that the pressure caved in the wall.\n\nEd Schwartzentruver called the first break \"a big notch.\"\n\n\"It run over a short spell,\" he said, \"and then about half of the roadway just fell down over the dam.\n\n\"And then it just cut through like a knife.\"\n\nColonel Unger said the water worked its way down \"little by little, until it got a headway, and when it got cut through, it just went like a flash.\"\n\nUnger's man Boyer said, \"It run over the top until it cut a channel, and then it ran out as fast as it could get out. It went out very fast, but it didn't burst out.\"\n\nJohn Parke said, \"It is an erroneous opinion that the dam burst. It simply moved away.\"\n\nAccording to Ed Schwartzentruver, \"The whole dam seemed to push out all at once. No, not a break, just one big push.\"\n\nThe time was ten after three.\n\n## IV\n\n## Rush of the torrent\n\n### -1-\n\nThere were men on the hillsides near the dam who had seen what the force of water could accomplish in mining operations, how a narrow sluice could scour and dig with the strength of a hundred men. Actually anyone who had lived in the area long enough to have seen even the spindliest of the local creeks in April had a fair idea of hydraulics at work. But no one who was on hand that afternoon was prepared for what happened when Lake Conemaugh started for South Fork.\n\n\"Oh, it seemed to me as if all the destructive elements of the Creator had been turned loose at once in that awful current of water,\" Colonel Unger said.\n\nWhen the dam let go, the lake seemed to leap into the valley like a living thing, \"roaring like a mighty battle,\" one eyewitness would say. The water struck the valley treetop high and rushed out through the breach in the dam so fast that, as John Parke noted, \"there was a depression of at least ten feet in the surface of the water flowing out, on a line with the inner face of the breast and sloping back to the level of the lake about 150 feet from the breast.\"\n\nParke estimated that it took forty-five minutes for the entire lake to empty, but others said it took less, more in the neighborhood of thirty-six or thirty-seven minutes. In any case, later studies by civil engineers indicated that the water charged into the valley at a velocity and depth comparable to that of the Niagara River as it reaches Niagara Falls. Or to put it another way, the bursting of the South Fork dam was about like turning Niagara Falls into the valley for thirty-six minutes.\n\nA short distance below the dam stood a farmhouse belonging to George Fisher. Fisher, who had been warned that the dam was about to go and had managed to escape from the house with his family only minutes before, saw everything he owned vanish in an instant.\n\nHuge trees were snapped off or uprooted one after another and went plunging off in the torrent. When the flood had passed and the hollow was still again, the hill opposite the dam had been scraped bare for fifty feet up. Every bush, vine, every tree, every blade of grass, had been torn out. All that remained was bare rock and mud.\n\nThe water advanced like a tremendous wall. Giant chunks of the dam, fence posts, logs, boulders, whole trees, and the wreckage of the Fisher place were swept before it, driven along like an ugly grinder that kept building higher and higher.\n\nAt Lamb's Bridge, the little bridge itself as well as George Lamb's home were destroyed as swiftly as everything else. Lamb had been afraid of the dam but had not fled to higher ground until he heard the roar of the flood bearing down on him. He made a frantic effort to save two pigs but gave it up and got to the hillside with his family in time to see his house climb the face of the water, which, because of the narrowness of the valley at that point, was about sixty feet high. He watched the house roll and toss momentarily; then it was flung against the near hill and smashed to splinters.\n\nFrom where they were the men at the dam could see all this happening as the water raged through the immense gash below them. But just beyond Lamb's Bridge the valley turns sharply to the right and disappears. So now they could only stand there, the rain beating down, and imagine, as much as that was possible, the things taking place beyond that turn.\n\nThe road to South Fork had disappeared, and with most of the dam gone, there was no way back to the clubhouse except the long way, clear around the lake, through flooded woods and fields where the mud would be impossible. So they stayed on, watching the level of the lake sink rapidly down and down, until there was nothing to see but hundreds of acres of dark ooze cut through by a violent yellow stream.\n\nColonel Unger lasted only a short time after the dam failed. He collapsed and had to be carried to his house and put to bed. His work crew, which had been hanging back nearby, waiting for his next orders, then climbed down to where the lake had been and with blankets and baskets and cold bare hands began scooping up the fish that were flopping about in the muck.\n\n### -2-\n\nEmma Ehrenfeld was sitting with her back to the window in the telegraph tower just down from the South Fork depot. She was talking to H. M. Bennett, engineer of the 1165 freight from Derry, and S. W. Keltz, the conductor. The men had left Derry, halfway to Pittsburgh on the main line, the evening before and had been up all night, delayed first at East Conemaugh until five that morning, then held at South Fork since eight.\n\nTired, cold, rain-soaked trainmen had been coming in and out of the tower most of the day, climbing the stairs to ask about news from up the line or just warming themselves by the coal stove on the first floor.\n\nMiss Ehrenfeld had held the Chicago Limited west of the bridge, on the other side of the Little Conemaugh, according to the orders she had received that morning; but with all the talk going around about trouble at the dam, the engineer had grown uneasy about his train standing over there, right where the flood might come. There had been a number of opinions on what to do, and then, after noon, the engineer got up and said he was going to bring the Limited across, orders or no orders.\n\nAfter that there had been more speculating about the bridge. The conductor on the Limited had noticed cracks in one of the piers. The division foreman had been sent for, and when he came down from his house up the tracks and said the piers had looked that way for some time, they cut the Limited's helper loose and ran it across first, very slowly, just to be sure. The Limited followed after and pulled up past the tower and the depot, a half mile or so. By the time that was done with, it was shortly before three.\n\nVery soon after, Emma Ehrenfeld went downstairs to see about the stove. The men had been firing it up so that her little room upstairs was growing uncomfortably warm. The Limited's engineer had come in again and was sitting there having a smoke, trying to dry off some. She passed the time with him for a few minutes, banged shut the door, and went back up the stairs to her desk.\n\nFrom where he was sitting beside her, H. M. Bennett could see the northeastern corner of town neatly framed by the rain-spattered window. In the immediate foreground were the Pennsylvania tracks; just beyond them was Railroad Street, with Stineman's store to the left and the big Stineman house and Pringle's drugstore on either side of it. To the right was the turn where Railroad became Lake Street and headed uptown and out of sight, toward the road to the dam. And way over to the right, on the other side of the coal tipple and the planing mill, he could see South Fork Creek, flooding across the lowlands, through the trees, and reaching among the houses nearest its banks.\n\nSuddenly Bennett noticed distant figures racing toward the hill. He jumped up and rushed to the window.\n\n\"Look at the people running!\" he said. \"I wonder what's wrong?\"\n\nThe other two went immediately to the window and noticed that several people going by in the street below seemed to be shouting something.\n\nAs Miss Ehrenfeld later recalled, Bennett said something about the reservoir and that they ought to get out. Then they saw it coming, spread across the full width of the valley.\n\nSituated as they were, only a few hundred feet from where the creek emptied into the Little Conemaugh, they were in about as good a place as any to see up the valley; but even so, they could not see very far because the abrupt hillside to which the town clung blocked off most of their view. When the water came into sight, it looked very close and enormous.\n\n\"It just seemed like a mountain coming,\" Emma Ehrenfeld said.\n\nConductor Keltz described it as more like a large hill rolling over and over. He judged it was about a hundred feet high.\n\nThe two men turned and dashed down out of the tower. Miss Ehrenfeld was right behind them (\"without waiting to get my hat or anything\"). She raced down the tracks, crossed over to the stairs that led to the coal tipple, ran to the top, and from there followed the crowd running toward the back alleys that led to higher ground.\n\nBennett and Keltz had started for the hill with her, but remembering the fireman and brakeman, who were asleep in the engine of their train on a siding on the other side of the river, they turned and ran for the bridge.\n\nThey made it to the engine, cut it loose, and with the little steam they had, came rolling out of the siding and back across the bridge, heading directly toward the oncoming flood with what looked like no better than an even chance of making safe ground only a few hundred yards away where the tracks swung hard to the left past the station.\n\nContrary to Keltz's estimate, the wall of water closing down on them was probably no more than forty feet high. It was moving straight for the bridge at a rate of perhaps ten to fifteen miles an hour and was driving before it a mass of debris that now included acres of trees, two or three small bridges, numerous mangled houses, dead animals, and rubbish beyond description.\n\nAbout 200 yards from the bridge the water claimed its first human life. Michael Mann, an English coal miner and self-styled preacher who was known in South Fork as \"The Reverend,\" had ignored every warning to leave the shanty he lived in on the banks of South Fork Creek. His body was found a week later, a mile and a half downstream. It was half-buried in mud, stripped of all clothing, and so badly decomposed that it could not be moved. As a result the last remains of the man who would be remembered in the valley as \"The First Victim\" were put into a hole nearby, covered over, and left unmarked.\n\nThe flood crushed right through the planing mill, wrenched the bridge from its piers, bent it as though it had been built with an elbow in the center, and then plowed head on into the mountain on the north side of the Little Conemaugh.\n\nEngineer Bennett's locomotive meanwhile had escaped just about untouched. It had gotten almost to the station when another escaping freight pulled out of a coal siding and blocked the way. The next thing Bennett knew, a huge tree, evidently an advance fragment of the debris, smashed into his locomotive and pitched it halfway off the track. With the water almost on top of them, Bennett, Keltz, and the two others (they were both very much awake by now) jumped to the ground and scrambled onto the other train just as it started pulling away. Seconds later the full brunt of the flood roared past behind them.\n\nBut when it was all over remarkably little damage had been done in South Fork. Stacked on the hillside as it was, the town was almost entirely out of reach of the onslaught. Along with the bridge and the planing mill, some twenty other buildings and houses were destroyed. The bridge, which had been thirty-five feet above the normal water level, was dumped 200 yards up the Little Conemaugh, carried there by the violent backwash created when the water hit the mountainside. There were a few pieces of machinery to be found where the planing mill had been, but that was about all. There was a stone foundation marking where one store had been. A grocery and barbershop went sailing off. J. P. Wilson's stable containing two mules, a horse, and a cow landed behind the depot with the animals unhurt.\n\nStation agent Dougherty's house was tossed into a gully, a total wreck. The depot itself had bobbed up several feet and swirled out over the tracks a ways. Then when the water rushed off downstream, it drifted back again and settled down almost precisely where it had been before, secured by a tangle of telegraph wires and only a little out of plumb.\n\nThe coal tipple was destroyed and so was the telegraph tower. And that was about the size of it, except that there had been three other deaths.\n\nA young man named Howard Shafer had been helping clear the jam of rubbish that had collected under a small bridge on South Fork Creek. When the water came he was unable to climb the steep bank fast enough.\n\nThe other two lost were Thomas Kehoe and Thomas Henderson, another fireman and a brakeman on Bennett's 1165 Derry freight. They had been asleep in the caboose when Bennett and Keltz had cut the engine loose to make their dash over the bridge. The caboose along with four other freight cars was carried away.\n\n\u2022 \u2022 \u2022\n\nPast South Fork the water raged along the valley of the Little Conemaugh, between sharp, wooded bluffs that sent the riverbed swerving back and forth on its way west. A straight line from South Fork to Johnstown would be nine miles, but by the river route the distance was about four miles farther.\n\nFor the first mile or so beyond South Fork the valley runs on a comparatively even line and is nearly 1,000 feet deep. There were no houses, only the railroad, which skirted along the northern banks of the river about forty feet above the normal water level.\n\nThe flood ripped the railroad to shreds, tore out ties, twisted steel rails into incredible shapes, and swallowed up whatever equipment happened to be standing along the way.\n\nA mile down from South Fork the valley narrows abruptly. There the rough hillsides squeezed the great mass of water so that its front wall grew to perhaps seventy to seventy-five feet high. Then, a half mile farther still, the river turns sharply south, traveling nearly two miles out of its way to form an oxbow which is only a matter of yards across. It was here, at the end of the oxbow, that the water smashed into its first major obstacle, a tremendous stone viaduct which had been built more than fifty years earlier to take the old Portage Railroad across the Little Conemaugh and which was still used for the main line of the Pennsylvania.\n\nThe viaduct was one of the landmarks of the country. It stood seventy-five feet high and bridged the river gap with one single eighty-foot arch. Even the biggest locomotives looked tiny by contrast as they chugged across it on their way up the mountain. Faced with a tawny-colored local sandstone, it was, as one engineer said, \"a substantial and imposing piece of masonry,\" which had been built by \"an honest Scotch stonemason\" named John Durno from a design worked out by the same Sylvester Welsh who had picked the site for the reservoir.\n\nThe bridge had been built to save running the railroad clear around the oxbow. A cut had been made across the oxbow, a distance of less than a hundred feet, and the tracks had been run through it to the bridge. At the eastern end of the cut, where the river bends off to the south, the tracks were about twenty feet above the normal water level. But at the western end, where the tracks started over the bridge, they were seventy feet above the river. Thus the river's big two-mile loop to the south accomplished a drop of some fifty feet in elevation, which could have been achieved in less than a hundred feet, if the water were to take the path of the railroad cut.\n\nWhen the flood hit this dividing point, part of the giant wave rushed through the cut at a depth of about twenty feet and plunged down over the top of the viaduct and into the deep river gully below, sweeping with it tons of debris which piled on top of the bridge or wedged between its huge arch.\n\nMeanwhile, the rest of the water, and by far the greater proportion of it, crashed along the longer and more tortuous course of the river bed, heaped up to a height of perhaps seventy feet by the narrow channel and gathering before it the shredded refuse of two miles' worth of heavy timber, rock, and mud. Perhaps six or seven minutes passed before it swung around the last big bend before the bridge. When it struck, it was almost as high as the bridge itself.\n\nThe bridge held momentarily. There was an awful booming crunch as debris piled against stone and virtually sealed off the already clogged arch, and the water surged back and forth, seething with yellow foam, mounting up and up until it was nearly eighty feet high. And then it started spurting over the top of the bridge, gushing between the boulders and mangled railroad cars, the broken planks, ties, and tree stumps that had been dumped there.\n\nNow, for a brief instant (no one knows exactly how long it lasted), Lake Conemaugh formed again some five and a half miles downstream from its original resting place. It gathered itself together, held now by another dam, which however temporary was nonetheless as high as the first one; and when this second dam let go, it did so even more suddenly and with greater violence than the first one. The bridge collapsed all at once, and the water exploded into the valley with its maximum power now concentrated again by the momentary delay.\n\nA mile or so beyond the bridge was the white frame village of Mineral Point, consisting of some thirty houses set in a row along a single street, Front Street, which ran parallel with the river on the north side of the river. It was a pretty little place, quiet, clean, tucked at the foot of the mountainside.\n\nThe river there was quite shallow and filled with rocks. The water was quick and bright, and its steady rushing among the rocks was the first sound people heard when they woke up in the morning and the last they heard as they dropped off to sleep at night. Except for the railroad, which ran along the opposite side of the river well above the roof line of the houses, Mineral Point looked as though it might have been a thousand miles from civilization. The air smelled of the river and of sweet, fresh-cut timber at the sawmill and furniture factory, the town's sole supporting industry, which stood at the far end of Front Street.\n\nThe people who lived in Mineral Point had names like Reighard, Page, Sensebaugh, Gromley, Byers, and Burkhart, and there were perhaps 200 of them, if you counted some of the outlying families that picked up their mail there. The houses all faced the river and had deep lots, running back to where the woods started at the base of the hill. Fruit trees and truck gardens grew wonderfully in the moist soil put down by the river over long geologic ages.\n\nNothing much out of the ordinary had ever happened in Mineral Point. There had been a murder there once. A woman who was new to town and lived off to herself was killed by another stranger, a miner from over in the hard-coal country named Mickey Moore. The accepted story was that he was one of the Molly Maguires and that was the reason behind the killing. He had to carry out some dark oath. But that had been several years back. Mickey Moore had disappeared and life had gone on about as it always had, except that no child liked to stay out in the woods very long after dark. \"Mickey Moore will get you,\" they said to each other, \"sure as anything.\"\n\nBeyond the last house, past the sawmill and around another bend or two in the river, and up on the opposite bank beside the railroad, was Mineral Point tower.\n\n\u2022 \u2022 \u2022\n\n\"I was sitting in the tower, and all at once, I heard a roar,\" W. H. Pickerell testified later. \"I looked up the track, and I seen the trees and water coming. I jumped up and throwed the window up, and climbed out on a tin roof around our office and walked around on it, and I saw the driftwood coming around the curve, and the channel filling up and running over the bank, and I heard voices; I could hear somebody hollowing, but I couldn't see them, and I walked around until the drift came down, and looked out, and perhaps one third of the distance in the river, I saw a man standing on a house roof. He looked over and seen me and recognized me.\n\n\"He says, 'Mineral Point is all swept away, and the people swept away, and my whole family is gone.' I says, 'Is that so?' and I says, 'Do you know anything of my family?', and he says, 'No, I don't; I think they were all drowned.'\n\n\"Christ Montgomery was his name, and I says, 'Cheer up, Christ, don't give up; as long as you're on top, there's hope!'\n\n\"I didn't more than have the words out of my mouth until the drift he was riding made a straight shoot for the shore, and struck one hundred or one hundred and fifty yards west of my office where the river made a short turn, and went all to pieces; shingles flew right up in the air.\n\n\"He got out all right. He grabbed into the bushes just about the time it struck and I didn't see anything of him for a breath, and then he crawled up out of the bushes. After I cheered him up, and told him not to give up, that there was hope for him as long as he was on top; I turned around to walk into my office on this tin roof. I didn't have more than fifteen feet to walk, but I almost fainted when he told me my whole family was drowned. I turned right around to come in the office, and as I climbed toward the window, I looked and saw the house roof striking shore and seen him light, and saw him crawl up on his hands and knees, and saw he was saved, and when I looked above, there was a regular mountain of water coming. He was probably ahead of the main body of water a little.\n\n\"I started without coat or hat, and as it was pouring down raining at the time, I turned around to get my coat and hat, and ran with them in my hand onto the opposite side of the track onto a high bank, and when I looked up the track, the wave wasn't more than a hundred yards off, and I beckoned for this man to get off the track. He wasn't looking for it to come down the track, and he got out on the track ahead of it, and came pretty near getting caught the second time.\"\n\n\u2022 \u2022 \u2022\n\nPickerell did not get caught at all nor, as things turned out, did any of his family, which was true of almost everyone else in Mineral Point.\n\nThe water had been coming up so fast that morning and during the early afternoon that most families had long since pulled out to higher ground by the time the flood fell on Front Street. First floors had been part way under water from about noon on, and there was no seeing the street or the riverbank. Picket fences, chicken coops, and backhouses had been drowned or had floated away as early as dinnertime.\n\nBut when the flood came, the wall of water swept through in such a way that it left almost nothing to suggest that there had ever been such a place as Mineral Point. The town was simply shaved off, right down to the bare rock.\n\nThe number of deaths came to sixteen, and quite a few people, like Christ Montgomery, went racing off on wild downstream rides astride their own rooftops. Christopher Gromley and his son traveled four miles before they were able to leap safely to shore; and three hours later, when they finally made it back to where Mineral Point had been, they found that all the other members of their family, Mrs. Gromley and six more children, were dead.\n\nThe water moved straight on down the valley, picking up a little speed wherever there were fewer turns to eat up its momentum and slowing down wherever the course began twisting again.\n\nEstimates are that, in some places, it may have been moving as much as forty miles an hour. Theoretically, if its weight and the average decline in elevation (thirty-three feet per mile) are taken into account, it had a speed of nearly ninety miles per hour. But the friction created by the rough terrain and the rubbish it pushed before it cut that speed drastically. What is more, its over-all rate of advance was extremely fitful. The wall of debris and water came on not steadily but in an irregular series of thunderous checks and rushes.\n\nAt times, eyewitnesses said later, the debris would even clog the path enough to bring the whole thing to a momentary standstill. All the crushed and tangled sweepings from the dam down would lock clear across the valley, seeming almost more than the millions of tons of pressure from behind could budge.\n\nBut then the whole seething mass would burst apart, with trees and telegraph poles flying into the air, as though blasted by dynamite, and the water would rush forward again, even faster. And as it moved on, the water kept on tossing logs and roots above its surface, as though the whole mass were full of life.\n\nThe friction set up by the terrain and the debris also caused the bottom of the mass of water to move much slower than the top. As a result the top was continually sliding over the bottom and down the front of the advancing wall, like a cake of ice across a slick board. The water, in other words, was rolling over itself all the time it was pressing forward, and this caused a violent downward smashing, like a monstrous surf falling on a beach, that could crush almost anything in its path. A man caught under it had no chance at all. In fact, one of the major problems later on would be finding the bodies that had been pounded deep down into the mud.\n\n\u2022 \u2022 \u2022\n\nWork train Number Two out of East Conemaugh was standing on the track nearest the hillside about a half mile upstream from the Conemaugh yards, at a place called Buttermilk Falls. The engineer sitting inside the rain-soaked cab was a friendly looking man with a round face and a dashing set of muttonchop whiskers. His name was John Hess.\n\nNormally he never worked east of the yards. His division ran west from Conemaugh, as far as Sang Hollow, which was three miles below Johnstown. But today, with trouble almost everywhere along the line, help was being sent wherever it was needed.\n\nHess had gone to his regular engine as usual that morning and had been told to take a work crew down to Cambria City to clear a slide. His conductor was R. C. Liggett, his fireman, J. B. Plummer.\n\nThey had gotten through to Cambria City without any problems and worked there until nearly eleven, when an order came through to go clear up the valley to a landslide at Wilmore, on the far side of South Fork. At Johnstown and East Conemaugh there had been delays of twenty minutes and more, but sometime between noon and one they had started out of the yards, running east along the Little Conemaugh on the track farthest back from the water against the hillside. Less than a mile out they passed a place where a good hundred feet of track on the right had fallen off into the river. Beyond \"AO\" tower they came up on a flagman.\n\n\"I stopped to let him on,\" Hess recalled later, \"and he says, 'You can't go any further.' And I asked him why, and he says, 'The north track is in the river and I don't believe the one you're on is safe,' and I says, 'Whereabouts?' and he says, 'Right through the big cut.' We went through the big cut to where the washout was, and seen it was badly washed, and I says to the conductor, 'I guess we'll have to take it afoot from here, and see where it is safe.' The conductor is an old experienced man, and he looked at the track we were on, and he says, 'It isn't safe, I won't run over that.' It was washed up to the ends of the ties and underneath the track, and undermined it; the ballast was still sticking to the ties; the ties seemed to be holding it up. He says, 'That isn't safe at all,' and we walked on up to Mineral Point, the next tower, and were going to report there but the operator told us he had no communication except with South Fork.\"\n\nThe operator, W. H. Pickerell, also told them about the messages which had been coming through from South Fork concerning the dam.\n\nThe men tramped back down the tracks to \"AO\" tower, where they took time out to eat. When they finished, it was about two o'clock and there was another message from East Conemaugh about a slide at Buttermilk Falls.\n\n\"We came down there,\" Hess said, \"and found the track that we had went up on. The conductor thought at first it was unsafe, and we walked down over it and left the engine above it, and he suggested to cut a couple cars off\u2014we had seven empty flats and the cabin ahead of our engine, and he suggested to cut off a couple cars and run them over to see whether it was safe, and probably we could bring the rest over. So we sent a man with two cars down over this dangerous place, and the bank didn't appear to slip much, and I brought the engine and rest of the train over. That left us on the Conemaugh side of this washout. I went down and the brakeman coupled up those cars that they had sent down ahead, and the conductor took the men with their shovels and went back to the slide about one hundred yards back of where we were laying.\n\n\"I don't suppose we had laid there more than twenty minutes until we heard the flood coming. We didn't see it but we heard the noise of it coming. It was like a hurricane through wooded country.\"\n\nConductor Liggett heard the sound and thought he saw the tops of the trees bend on the flat upstream between the railroad and the river.\n\n\"And I says to the men, 'We'll get away from here,' and I still looked, and then I was satisfied there was something coming. I couldn't see any rubbish or drift, but I saw there was a commotion among the green timber.\"\n\nHe shouted at the men to run. They dropped their tools and started down the tracks looking for a place where they could climb out of the way. But the rocks were too steep. They had to keep running, 200, 300, nearly 400 yards before they found a path.\n\nHess and Plummer still could see nothing, but according to Plummer, Hess said, \"The lake's broke,\" and with that he put on steam, tied down the whistle, and with their gravel cars clattering along in front, they went shrieking toward East Conemaugh and the railroad yards where the two sections of the Day Express stood waiting.\n\nThe Hess ride into Conemaugh would be talked about and described in books and magazine articles for years to come, with Hess in his engine (Number 1124), blazing down the valley, the water practically on top of him, in an incredibly heroic dash to sound the alarm.\n\nHess himself said afterward, \"I didn't know what else to do. I didn't see what else I could do.\"\n\nHe also said that he never did see any water, never waited around that long. Moreover, Plummer estimated that their top speed as they rounded the bend into the yards was no more than twelve miles an hour, which, he said, was the best they could do considering the load they were pushing, the condition of the tracks, and the fact that they had no idea which way the waiting trains on the other side of the blind turn might have been rearranged in their absence.\n\nIt was Hess's intention to keep right on going through the yards, clear to Johnstown, if the track was clear. But it was not. Plummer's guess was that no more than two minutes passed after they had pulled to a stop until the flood came.\n\n\"My brother was up on the bank and saw it coming,\" Plummer said. \"I didn't see it coming at all; he saw it coming though and saw where it was, and he ran down and grabbed hold of me and gouged Hess with his umbrella, and told us to run.\"\n\nWith their whistle still screaming the two men jumped from the cab and started for the hillside.\n\nA locomotive whistle was a matter of some personal importance to a railroad engineer. It was tuned and worked (even \"played\") according to his own particular choosing. The whistle was part of the make-up of the man; he was known for it as much as he was known for the engine he drove. And aside from its utilitarian functions, it could also be an instrument of no little amusement. Many an engineer could get a simple tune out of his whistle, and for those less musical it could be used to aggravate a cranky preacher in the middle of his Sunday sermon or to signal hello through the night to a wife or lady friend. But there was no horseplay about tying down the cord. A locomotive whistle going without letup meant one thing on the railroad, and to everyone who lived near the railroad. It meant there was something very wrong.\n\nThe whistle of John Hess's engine had been going now for maybe five minutes at most. It was not on long, but it was the only warning anyone was to hear, and nearly everyone in East Conemaugh heard it and understood almost instantly what it meant.\n\n### -3-\n\nFor the passengers on board the eastbound sections of the Day Express, the delay in East Conemaugh had been a dreary, monotonous affair. It was going on five hours now since the two trains had pulled to a stop between the river and the little town.\n\nThe first few hours had not been entirely uninteresting. A number of passengers had gone out to look things over. They went walking about in the rain, up and down the tracks, over to the depot or the telegraph tower to see if there was any word on how long they would be held there. Or they picked their way across the tracks to the riverbank where the crowds were gathered and several local men were making great sport of spearing things of interest out of the racing current. And on the other side of the tower, the township bridge looked as though it would go almost any time.\n\nBut when dinnertime had passed and there still seemed no end to the rain and any chance of moving on seemed even less likely, whatever spirit of adventure there had been faded rapidly. The passengers had nearly all returned to the trains. They passed the time as best they could in the dim afternoon light, with the sound of the pelting rain all around them.\n\nElizabeth Bryan of Philadelphia sat looking out the window, while beside her, her friend Jennie Paulson of Pittsburgh read a novel titled Miss Lou. The girls had been to a wedding in Pittsburgh the day before and were on their way to New York, each wearing a small corsage of roses. Another passenger, the Reverend T. H. Robinson, a professor at the Western Theological Seminary in Allegheny, was busy writing a diary of the day's events for his wife.\n\nOthers were doing what they could to amuse their children. Some slept. One elderly gentleman, feeling slightly ill, had had his berth made up and retired for the day. Still others gathered in small clusters along the aisles to talk about the storm and the rising river, service on the Pennsylvania, the dismal prospect of the night ahead, or the possibility of getting a decent meal somewhere.\n\nThere was talk too about the dam farther up the mountain that everyone had been hearing about. But there was not much concern about it.\n\n\"The possibility of the dam giving way had been often discussed by passengers in my presence,\" one man, a bank teller from New Jersey, was later quoted, \"and everybody supposed that the utmost danger it would do when it broke, as everybody believed it sometime would, would be to swell a little higher the current that tore down through Conemaugh Valley. Such a possibility as the carrying away of a train of cars on the great Pennsylvania Railroad was never seriously entertained by anybody.\"\n\nAnother passenger said that though many people may have been uneasy and were keeping \"a pretty good lookout for information,\" the porters comforted them \"with the assurance that the Pennsylvania Railroad Company always took care of its patrons.\"\n\nSo far whoever was directing things in the yard had chosen to move them twice. Twice they had watched the river working in on the tracks where the two trains stood, twice they had been moved forward and toward the hill, to be farther away from the river, and both times they had seen the tracks fall off into the water very soon after.\n\nNow they were on the last sidings next to town, as far from the river as it was possible to be. The second section was on the track beside the depot and closest to town. Then came the first section, on the next track toward the river; and on the other side of it, four tracks over, was the local mail train. The Day Express engines were standing about even with the depot, with Section Two a few cars farther forward. The last cars were nearly on line with the telegraph tower, which stood on the river side of the tracks.\n\nIn the caboose of the mail train, which was nearest of the three to the tower, a fire was going in the stove and the conductors and others of the train crews were sitting about keeping warm between turns at checking in at the tower.\n\nMessages had been coming in and going out of the tower steadily since early morning, and included those from South Fork. One operator, D. M. Montgomery, was later quoted as saying that the South Fork warning was generally well known. \"But of course,\" he added, \"nobody paid any more attention to it than if there hadn't been one at all. I know I didn't for one. It seemed like a rumor and they didn't take any belief in it.\"\n\nCharles Haak, another operator in the tower, and the one who had passed along the first message from South Fork to the yardmaster's office downstairs, said he did not pay much attention to the warnings either.\n\n\"I was a stranger there,\" Haak said. \"I had only been there but eight months, and of course, I listened to other people around there, residents there, and there was talk about the dam breaking, and they said there had been rumors but it never came, and so I thought that was how it would be this time.\"\n\nAs for the decisions on which trains to put where, they were being made by J. C. Walkinshaw, the yardmaster, who had been on duty since six that morning.\n\nWalkinshaw was forty-nine years old, a widower with five children. He had worked for the Pennsylvania since he was seventeen and had been in charge of the East Conemaugh yards for twenty-three years. In a book of short biographical sketches of long-time company employees published later by the railroad, Walkinshaw peers out of a small photograph with wide eyes framed by white hair and whiskers, looking rather astonished and not especially bright. Robert Pitcairn later said that though Walkinshaw suffered from consumption, and so was \"not very efficient\" as yardmaster and \"not very able to stand the physical strain,\" he, Pitcairn, nonetheless considered him amply qualified to look after the company's interest.\n\nWith circumstances as they were, Walkinshaw was left with little choice on what to do with the trains. He could not send them to the east, up the valley, because of the washouts at Buttermilk Falls and farther on. Nor could he send them back down the line toward Johnstown, as there were now reports of washouts in that direction as well.\n\nAbout all he could do was to keep moving them back from the river, which is what he did. But once he had them on the northernmost siding, he concluded that he had taken \"every reasonable precaution\" under the circumstances.\n\nOne other very possible choice, of course, was to move the passengers out of the trains to higher ground. But to ask that many people to go out into the cold wind and rain, into the muddy little town where there might well be problems finding enough shelter for everyone, seemed more than Walkinshaw was willing to do, even though he had full knowledge of the trouble at Lake Conemaugh and was heard by at least one witness to say that if the dam ever broke it would \"sweep the valley.\"\n\nWalkinshaw had been out several times, checking equipment, giving orders, looking at the river. From two o'clock to three there seemed to be no change in the water level. Apparently the worst was over. But then about 3:15 the bridge below the telegraph tower went, causing a great stir among the crowd. Hour by hour the current had eaten away at its foundations, until they let go and the whole thing just dropped down into the water. Sometime shortly after, Walkinshaw went into his office where his son handed him another message about the dam. Then, about a quarter to four, Walkinshaw decided to take a brief rest.\n\n\"I sat down and wasn't in the chair more than a minute until I heard a whistle blow,\" he recalled later.\n\n\"I jumped off my chair, and ran out and hollowed for every person to go away off the road and get on high ground, and I started up the track.\n\n\"Just as I left the office, I saw the rear end of this work train backing around the curve. I started up toward the train, and the minute I saw the train stop, I saw the engineer jump off and run for the hill. Just that minute, I saw a large wave come around the hill.\"\n\nInside the trains there was considerable commotion when the whistle started blasting. People stood up and began asking what the trouble was. Two Negro porters came through, both looking very excited and when asked if this meant that the dam had broken, the first one said he did not know, and the second said he thought it had. Outside, a conductor ran along between the trains shouting, \"Get to the hill! Get to the hill!\"\n\nThe Reverend Robinson said that no one knew what was going on, but that he remembered telling a woman next to him that he thought there was no danger. Then he looked out of the window and saw the wave coming. It appeared to be about 300 yards away, but there was no water to be seen. As one man said, it looked more like a hill of rubbish than anything else. Some people said it looked to be fifty feet high and it was taking everything in front of it. Everyone started for the door.\n\nOn Section One, the train standing between the mail train and Section Two, nearly every passenger got through the doors as fast as possible, but several of them, seeing the mud and rain, turned back. Jennie Paulson and her friend Elizabeth Bryan decided to go back for their overshoes. An old minister from Kalamazoo, Michigan, and his wife saw the flood bearing down and returned to their seats inside. But most people jumped and ran.\n\n\"It was every man for himself and God for us all,\" a New Haven, Connecticut, man named Wilmot said later.\n\nOnce they had clambered out into the rain, the passengers from Section One were faced with an immediate problem. On the next track, directly between them and high ground, stood Section Two.\n\n\"I saw three ways before me,\" the Reverend Robinson wrote afterward, \"climb over section No. 2 or crawl under it, or run down the track with the flood four car lengths and around the train. I instantly chose the latter. No one else followed me so far as I saw, but all attempted the other courses.\"\n\nRobinson made it safely around the train, but between him and the town and the streets which climbed to high ground was a ditch running parallel with the last track. It was about ten feet wide and perhaps five feet deep and rushing with water the color of heavily creamed coffee. Fortunately for him, Robinson arrived at the ditch at a place where a big plank had been laid across it. He was over in seconds and on his way up a steep, mud-slick embankment toward the town.\n\nBut others hesitated at the ditch, or leaped, or fell in and floundered about desperately, panicked. A number of men stopped, then moved back several steps, got a start, and jumped across. George Graham, a doctor from Port Royal, Pennsylvania, made it over this way; then, feeling that he still had time to spare, turned back to see if he could help some of the others.\n\n\"Just to my left, into the ditch, armpit deep, I saw nine women and girls tumble. I instantly grabbed the hand of the first and quickly pulled her out; the meanwhile all the others reached for me at once. I succeeded in saving them all except one old lady.\" Wilmot, the New Haven man, also cleared the ditch, carrying his child in his arms. When he looked back to find out what had become of his wife, he saw her hesitating on the other side, while a man beside her shouted, \"Jump, jump!\" She jumped and made it, and they ran on.\n\nCyrus Schick, a prominent Reading businessman who, with his wife and her sister, had been on his way home from a long health tour in the west, fell headlong into the ditch, as did his sister-in-law, Eliza Stinson. Schick's wife saw him bob up out of the water but then lost sight of him in all the confusion. His body and that of Miss Stinson were not found for ten days.\n\nOn the other side of the ditch the streets were full of running, shouting people. One local girl, a pretty young schoolteacher named Kate Giffen, who lived with her family on Front Street, would later describe racing to her house to pick up a child and seeing the woman who lived next door standing out on her porch screaming. She was the wife of John Hess, and she was screaming that the locomotive whistle still blasting away in the yards below was her husband's.\n\nThe Reverend Robinson found himself all alone, pressing up a back alley.\n\n\"I ran to the second street, and, hoping I might be safe, I turned and looked. The houses were floating away behind me, and the flood was getting round above me. I ran on to the third street and turned again; the water was close behind me, houses were toppling over, and the torrent again pushing round as if to head me off.\"\n\nHe kept on running, and when he turned again, he was high enough to see most of the town and the river valley. He watched a railroad car break loose and bound off in the plunging water, with two men on top trying desperately to keep their balance, moving first to one side then to another, as they headed toward Johnstown. How many passengers there might have been inside he could not tell. Everywhere people were rushing this way and that, some ducking inside doorways, some going for higher ground, stumbling and falling in the muddy streets. As the wave hit Front Street, buildings began falling, one on top of another; some seemed to bounce and roll before they were swept downstream. Locomotives from the roundhouse went swirling about like logs in a millrace.\n\nThe big, brick roundhouse had some nine engines in it when the flood struck. There were also another nineteen or twenty engines elsewhere in the yards, machine shops, a lot of rolling stock, a coal shed, and the three passenger trains. When the wave struck, it was probably about twenty-eight to thirty feet high, though, understandably, it looked a great deal higher to anyone caught in its path. The roundhouse was crushed, as one onlooker said, \"like a toy in the hands of a giant.\" The passenger trains were swamped in an instant. Section One was ripped apart and the baggage car and one coach were flung downstream and its Pullman coach caught fire. Yard engines went spinning off, one after another.\n\nSection Two and the mail train both miraculously survived. Section Two had been standing on an embankment five to six feet high, which certainly had something to do with its good fortune. There were also some freight cars in front of it and a coal shed that toppled across the tracks and helped deflect some of the onrush. But it was the roundhouse which almost certainly did most of the deflecting, and the fact that the valley both curves sharply and broadens out at that particular point along the river undoubtedly contributed to the inconsistent behavior of the oncoming wave.\n\nThe destruction all around the trains was fearful. Forty houses along Front Street were taken away. The Eagle Hotel, the Central Hotel, the post office, the railroad station, several stores, at least half the town was destroyed. The only railroad track left was that under Section Two, the mail train, and a few other pieces of equipment that, for one quirk of fate or another, happened to survive.\n\nThirty locomotives, some weighing as much as eighty tons, were scattered anywhere from a hundred yards to a mile from where they had been standing. One locomotive boiler would be carried all the way to Johnstown. How many lives were lost was never determined exactly. But at least twenty-two passengers from the Day Express sections were killed, including Cyrus Shick and his sister-in-law, Jennie Paulson and Elizabeth Bryan, the minister's wife from Kalamazoo, and F. Phillips, one of the Negro porters.\n\nIn East Conemaugh and Franklin, which was the name of the cluster of houses across the river from the yards, the known death toll came to twenty-eight.\n\nBut when the flood had passed, the engine and tender and six cars of the second section were almost at the exact same place they had been since before noon. They had been shoved along the track some, maybe twenty yards downstream. There was debris jammed in around them; but the sixteen people inside, who out of fear or indecision or dumb luck had stayed on board, were as safe and sound as though virtually nothing had happened. The water had come up over the seats in several cars and the passengers were soaked to the skin and badly shaken by the experience, but the only fatalities from their cars were among those who had tried to make a run for it.\n\nOne such passenger was John Ross of New Jersey, who, it was said later, was about thirty-three years old and a cripple. He had been traveling in one of the sleepers of Section Two, a car in which no one had chosen to hang back. Ross struggled out with the rest and was having a terribly difficult time until one of the train crew, a brakeman named J. G. Miller, came running along, picked him up, and managed to carry Ross some fifty yards or more before he dropped him.\n\n\"I had to drop him,\" Miller said later, \"to save myself. I saw it was either life or death with me, and I dropped him, and went for the hill.\"\n\nThe mail train, which had been standing on an even lower track and within no more than a hundred yards from the river, was also still intact, though it too had been shoved downstream quite a way. Like Section Two, the mail train had been partly sheltered by the roundhouse, but what seems to have saved it was the telegraph tower which fell right onto the engine just as it was being pushed past underneath, and pinned it down there until the water had passed. But unlike any of the other trains, there were no fatalities among its passengers. Everyone got off and onto the hill in time, thanks to the good sense of the crew and, perhaps in part, to the particular nature of the passengers themselves.\n\nLike all the others milling about the yards that morning, the eighteen or so passengers on board the mail train had heard mixed reports about the dam. They were told that if it ever broke it would drown the valley, and they were told that it would raise the level of the river maybe a foot or two. They were told it would take the water one hour to get from the lake to East Conemaugh, and they were told that it would take three hours. But mostly they were told that the dam was an old chestnut and not to think any more of it.\n\nBut their conductor, Charles Warthen, decided to tell them everything he knew, which was not a whole lot more, but he at least made it sound serious. He also told them to get ready to move out at the slightest notice, which was something neither of the conductors of the other trains had chosen to do.\n\nThe trainmen had been sitting in the last car of the mail train, talking about the situation, but for some reason or other, S. E. Bell, who was conductor on Section One, and Levi Easton, conductor on Section Two, made no effort to warn their passengers. The likeliest explanation seems to be that they, like so many others, had no real fear of anything happening.\n\nAll but one or two of Warthen's passengers were from the Night Off company. When they were told what might be expected of them, they quietly went to work rounding up their belongings, and the women began pinning up their skirts.\n\nAbout two o'clock C. J. McGuigan, brakeman on the mail train, had gone to the tower to ask if there was any further news, and the operator (which one it was he did not recall) said, \"Nothing, only another message that the dam is in a very dangerous condition.\" Not knowing anything about the dam, McGuigan asked him what the consequences might be if the dam broke.\n\n\"He kind of smiled,\" McGuigan told the story later, \"and said, 'It would cover this whole valley from hill to hill with water.' I got kind of frightened myself then, and I came right down, and told the passengers the second time to be on the lookout. . . . The ladies got frightened, and one of them wanted to know if they should not better go to the hills now, but the manager of the troupe said 'No, there is no danger yet' . . . The women seemed to be ready for it . . . I think they were very sensible people.\"\n\nMcGuigan then went back to the last car to the other crewmen. When the whistle began blowing, he ran to the passenger coach, shouting that the flood was coming, while conductors Bell and Easton took off for their trains, shouting the same thing.\n\n\"The women were sitting down, and the men were standing up, and they all had their grips and valises in their hands, and the men ran to the upper end of the car, and the ladies to the west end where I was.\n\n\"I assisted them out, and got up and looked through the train, and I couldn't see anybody on the train, and then I ran with two of the ladies, caught hold of their hands, and ran until we came to the ditch . . . and Miss Eberly, she refused to go into the ditch, and I threw her into it, and jumped down and assisted her up on the other side, and ran up the hill.\"\n\nNo one was lost, not even the baggagemaster, J. W. Grove, who decided to jump onto one of the yard engines standing about instead of trying for the hill. Every other loose engine in East Conemaugh was dumped over, driven into the hillside, or swept off with the flood, except the one he picked.\n\nBrakeman McGuigan went about for some time after carrying a picture of Miss Eberly, who was the pretty, young star of the company and actually Mrs. Eberly. She in turn was quoted widely when she returned to New York and described the bravery of the trainmen.\n\nLater the Pennsylvania Railroad, in an effort to establish exactly what had happened at East Conemaugh, conducted its own investigation, which would provide the one full account of the whereabouts of several dozen employees, the official decisions made before the water struck, and the personal decisions made when it was seen rounding the bend behind the Hess train. The study revealed several cases such as that of brakeman McGuigan, but it included many more where the reaction had been a good deal less coolheaded and quite a lot more human.\n\nSamuel S. Miller, for example, was also a brakeman, on the first section of the Day Express, the one on which most of the fatalities occurred. Part of his testimony went as follows:\n\nQ. Where were you when the big wave came?\n\nA. I was partly up on the hill.\n\nQ. What were you doing up there?\n\nA. Well, I was told that it was coming, and I got up on the hill for my own safety. I had gone to the Agent at Conemaugh, he was in the office at Conemaugh station\u2014\n\nQ. Who is he?\n\nA. E. R. Stewart\u2014and I borrowed the key from him for the water closet at the station, and I went in the water closet, and I think I was reading a Commercial-Gazette at the time when I heard the big whistle, and not knowing of any freight moving, I first thought probably it might be a freight engine that was to assist first Day Express up the mountain; I thought maybe they were alarming the passengers to get on the train and wondered why it wasn't a passenger engine whistle. The next thought that came to me was that South Fork dam had broken. I made a hasty exit, and when I got outside, a young fellow came along and said that was what was wrong. He seemed to be in a great hurry, and I asked him if South Fork dam had broken, and he replied, \"Yes, so people say,\" and it seems to me, I told him to run, and I ran too.\n\nQ. You broke for the hill?\n\nA. Yes, sir, I broke for the hill.\n\nQ. You didn't go to your train?\n\nA. No, sir; I got up on the hill probably 110 yards from the station, and looked back, and could see that the water had come. I could see that the water was between the houses at that time. I concluded I wasn't high enough, and I went up onto still higher ground.\n\nQ. You didn't climb a tree?\n\nA. No, sir.\n\nQ. Why didn't you go to your train and help get your passengers out?\n\nA. Well, for my own safety. From the descriptions I had heard, I concluded I had better be on the hill.\n\nQ. You might have gone to your train if you had tried?\n\nA. I could have, but the question was whether I could then have gone to the hill or not.\n\nQ. You believed your life was in danger, did you?\n\nA. Yes, sir.\n\nNow several hundred freight cars, a dozen or more locomotives, passenger cars, nearly a hundred more houses, and quite a few human corpses were part of the tidal wave that surged on down the valley.\n\nBefore it had plowed through East Conemaugh, the water had cut along the valley below Mineral Point, crashing back and forth against the mountainsides as the river channel swung this way and that. A mile or so above East Conemaugh, at the place the railroad men called \"the big cut,\" the Pennsylvania tracks again left the riverbank to take a short cut across another oxbow. Here again the flood had divided briefly, with part of it rushing headlong through the cut, while the rest went with the river on its two-mile loop off to the north. It was a course which sapped much of the wave's potential speed and energy. But from East Conemaugh on to Johnstown the valley opened up considerably and the river headed directly for its meeting with the Stony Creek. Past East Conemaugh the flood was on a straightaway, and there it began to gather speed.\n\nWoodvale got it next. Woodvale was somewhat bigger than East Conemaugh, prosperous, new, and the pride of the Cambria Iron Company. It was a sort of model town, built by the company, and with its clean white houses it looked, as one man said, more like a New England town. It was connected to Johnstown by a streetcar line that ran along its main thoroughfare, Maple Avenue, which was far and away the prettiest street in the valley. Maple Avenue was nearly a mile long and looked like a green tunnel that May. The trees reached over the tracks where the little yellow streetcars rattled by, their horses heading for the stable. When the flood had passed, there would be no trace of Maple Avenue.\n\nAbout 1,000 people lived in Woodvale. There was a woolen mill, built by Cambria Iron, which employed several hundred women. There was the Rosensteel tannery, two schoolhouses, some churches, and no saloons (they evidently were contrary to the Iron Company's idea of a model town).\n\nUnlike East Conemaugh, Woodvale got no warning. It was all over in about five minutes. The only building left standing was the woolen mill, and there was only part of that. At the western end of the town, the end almost touching Johnstown, stood the Gautier works, part of it in Woodvale, part in Conemaugh borough. The huge works sent up a terrific geyser of steam when the water hit its boilers, and then the whole of it seemed simply to lift up and slide off with the water. The tannery went and so did the streetcar shed, along with eighty-nine horses and about thirty tons of hay. When the water had passed, the town was nothing but a mud flat strewn with bits of wreckage. There was only a tiny fringe of houses left along the edges, on the foothills. There was not a tree, not a telegraph pole, not a sign of where the railroad had been. Two hundred and fifty-five houses had been taken off, and there was no way of telling where they had been.\n\nThe official figure for Woodvale's dead would later be set at 314, which means that about one out of every three people in town had been killed.\n\nA number of people had tried to crawl under a freight train that was blocking their way to the hill and had been crushed when the water hit the train and it started moving. Dozens of others had never made it out of their houses. At the woolen mill three men had kept retreating to different rooms and higher floors as the big brick building caved in piece by piece, all around them, until there was only that small part which miraculously withstood everything that was thrown against it.\n\nWhen the wireworks broke up it contributed miles and miles of barbed wire to the mountain of wreckage and water that, once past the wireworks, had only a few hundred yards to go until it struck Johnstown.\n\nIt was now not quite an hour since the dam had given way. The rain was still coming down, but not so hard as before, and the sky overhead was noticeably brighter.\n\nIn Johnstown the water in the streets seemed actually to be going down some. It had been a long, tiresome day in Johnstown, and the prospects for a night without gas or electricity were not especially cheerful, but by the looks of the water and the sky, the worst of it had passed.\n\n## V\n\n## \"Run for your lives!\"\n\n### -1-\n\nMost of the people in Johnstown never saw the water coming; they only heard it; and those who lived to tell about it would for years after try to describe the sound of the thing as it rushed on them.\n\nIt began as a deep, steady rumble, they would say; then it grew louder and louder until it had become an avalanche of sound, \"a roar like thunder\" was how they generally described it. But one man said he thought the sound was more like the rush of an oncoming train, while another said, \"And the sound, I will never forget the sound of that. It sounded to me just like a lot of horses grinding oats.\"\n\nEveryone heard shouting and screaming, the earsplitting crash of buildings going down, glass shattering, and the sides of houses ripping apart. Some people would later swear they heard factory whistles screeching frantically and church bells ringing. Who may have been yanking the bell cords was never discovered, but it was later reported that a freight engineer named Hugh Clifford had raced his train from above the depot across the stone bridge, his whistle going the whole way; and a man named Charles Horner blew the whistle over at Harry Swank's machine shop.\n\nThose who actually saw the wall of water would talk and write of how it \"snapped off trees like pipestems\" or \"crushed houses like eggshells\" or picked up locomotives (and all sorts of other immense objects) \"like so much chaff.\" But what seemed to make the most lasting impression was the cloud of dark spray that hung over the front of the wave.\n\nTribune editor George Swank wrote, \"The first appearance was like that of a great fire, the dust it raised.\" Another survivor described it as \"a blur, an advance guard, as it were a mist, like dust that precedes a cavalry charge.\" One young man said he thought at first that there must have been a terrible explosion up the river, \"for the water coming looked like a cloud of the blackest smoke I ever saw.\"\n\nFor everyone who saw it, there seemed something especially evil about this \"awful mass of spray\" that hovered over \"the black wreck.\" It was talked of as \"the death mist\" and would be remembered always.\n\nThe fact was there had been something close to an explosion up the river, at the Gautier works, when the water rolled over the fires there, which undoubtedly accounted for a good part of what they saw. Horace Rose, who witnessed about as much as anyone, thought so.\n\nAt the first sound of trouble he had rushed to the third floor of his house on lower Main Street and from the front window could see nearly a mile up the valley. Only a few minutes before he had been playfully teasing his neighbors' child, Bessie Fronheiser, from another window downstairs, telling her to come on over for a visit. The distance between the two houses was only about five feet, so he had put some candy on the end of a broom and passed it over to her. That was so successful that he next passed across a tin cup of coffee to Bessie's mother in the same way. She was just raising the cup to her lips when the first crash came.\n\nFrom the third floor Rose could see the long line of the rolling debris, stretching from hill to hill, slicing through the Gautier works, chopping it down and sending up a huge cloud of soot and steam.\n\nThe sight took his breath away. Once clear of the wireworks, the wave kept on coming straight toward him, heading for the very heart of the city. Stores, houses, trees, everything was going down in front of it, and the closer it came, the bigger it seemed to grow. Rose figured that he and his family had, at the most, two or three minutes before they would be crushed to death.\n\nThere would be slight differences of opinion later as to precisely when the wave crossed the line into Johnstown, but the generally accepted time is 4:07.\n\nThe height of the wall was at least thirty-six feet at the center, though eyewitness descriptions suggest that the mass was perhaps ten feet higher there than off to the sides where the water was spreading out as the valley expanded to a width of nearly half a mile.\n\nIt was also noted by dozens of people that the wave appeared to be preceded by a wind which blew down small buildings and set trees to slapping about in the split seconds before the water actually struck them. Several men later described how the wind had whipped against them as they scrambled up the hillsides, grabbing at brush to pull themselves out of the way at the very last instant.\n\nBecause of the speed it had been building as it plunged through Woodvale, the water struck Johnstown harder than anything it had encountered in its fourteen-mile course from the dam. And the part of the city which took the initial impact was the eastern end of Washington Street, which ran almost at right angles to the path of the oncoming wave.\n\nThe drowning and devastation of the city took just about ten minutes.\n\nFor most people they were the most desperate minutes of their lives, snatching at children and struggling through the water, trying to reach the high ground, running upstairs as houses began to quake and split apart, clinging to rafters, window ledges, anything, while the whole world around them seemed to spin faster and faster. But there were hundreds, on the hillsides, on the rooftops of houses out of the direct path, or in the windows of tall buildings downtown, who just stood stone-still and watched in dumb horror.\n\nThey saw the eastern end of Washington Street, the block where the Heiser dry-goods store stood, disappear in an instant. From there the wave seemed to divide into three main thrusts, one striking across the eastern end of town behind the Methodist Church, one driving straight through the center, and the other sticking more or less to the channel of the Little Conemaugh along the northern side of town. Not that there was any clear parting of the wave, but rather that there seemed to be those three major paths of destruction.\n\nEast of the park, Jackson and Clinton streets became rivers of rubbish churning headlong for the Stony Creek. On Main and Locust, big brick buildings like the Hulbert House collapsed like cardboard while smaller wood-frame stores and apartment houses jumped from their foundations and went swirling away downstream, often to be smashed to bits against still other buildings, freight cars, or immense trees caught by the same roaring current.\n\nEvery tree in the park was torn up by the roots and snatched away as the water crossed through the center of town. John Fulton's house caved in, and other big places went down almost immediately after\u2014the Horace Rose house, the John Dibert house, the Cyrus Elder house. The library, the telegraph office, the Opera House, the German Lutheran Church, the fire station, landmarks were vanishing so fast that no one could keep count of them. Then, perhaps no more than four minutes after the water had plunged across Washington Street, it broke past Vine on the far side of town and slammed into the hill which rises almost straight up to nearly 550 feet in back of the Stony Creek.\n\nIt was as though the water had hit an immense and immovable backboard, and the result was much as it had been at South Fork when the wave struck the mountainside there. An immediate and furious backwash occurred. One huge wave veered off to the south, charging up the Stony Creek, destroying miles of the densely populated valley, which, it would seem, had been well out of reach of any trouble from the valley of the Little Conemaugh. Other waves pounded back on Johnstown itself, this time, very often, to batter down buildings which had somehow withstood the first onslaught.\n\nHouses and rooftops, dozens of them with thirty or forty people clinging on top, went spinning off on a second run with the current, some to end up drifting about for hours, but most to pile in to the stone bridge, where a good part of the water headed after striking the hill, and where eventually all the water had to go.\n\nThe bridge crossed the Conemaugh River downstream from the Point where the Stony Creek and the Little Conemaugh come together. Past the bridge, another mile or so west, was the great Conemaugh Gap, the deepest river gorge between the Alleghenies and the Rockies and the flood's only way out of the mountains. But the bridge was never hit by the full force of the water. It had been built far enough down from the Point so that when the wave went grinding over Johnstown, it was shielded by Prospect Hill, and after the wave broke apart against the mountainside, the bridge had to withstand the impact of only a part of the wave.\n\nAs a result the bridge held. Had it been in the direct path and been struck full force, it would have been taken out just like everything else. But as it was, the mountainside took the brunt of the blow, the bridge survived, and the course of events for the next several hours went very differently.\n\nDebris began building rapidly among the massive stone arches. And now it was no longer the relatively small sort of rubbish that had been clogging the bridge most of the day. Now boxcars, factory roofs, trees, telegraph poles, hideous masses of barbed wire, hundreds of houses, many squashed beyond recognition, others still astonishingly intact, dead horses and cows, and hundreds of human beings, dead and alive, were driven against the bridge until a small mountain had formed, higher than the bridge itself and nearly watertight. So once again, for the second time within an hour, Lake Conemaugh gathered in a new setting. Now it was spread all across Johnstown and well beyond.\n\nBut this time the new \"dam\" would hold quite a little longer than the viaduct had and would cause still another kind of murderous nightmare. For when darkness fell, the debris at the bridge caught fire.\n\nNo one knows for sure what caused the fire. The explanation most often given at the time was that oil from a derailed tank car had soaked down through the mass, and that it was set off by coal stoves dumped over inside the kitchens of mangled houses caught in the jam. But there could have been a number of other causes, and in any case, by six o'clock the whole monstrous pile had become a funeral pyre for perhaps as many as eighty people trapped inside.\n\nEditor George Swank, who had been watching everything from his window at the Tribune office, wrote that it burned \"with all the fury of the hell you read about\u2014cremation alive in your own home, perhaps a mile from its foundation; dear ones slowly consumed before your eyes, and the same fate yours a moment later.\"\n\nBy ten o'clock the light from the flames across the lower half of town was bright enough to read a newspaper by.\n\n### -2-\n\nThe water in front of the Heiser store had been knee-deep since early in the afternoon, which was a record for that part of town. In the other floods over the years there had never been any water at all so far up on Washington Street.\n\nPeople had been coming in and out of the store most of the morning joking about the weather, buying this and that to tide them through the day. The floor was slick with mud from their boots, and the close, warm air inside the place smelled of tobacco and wet wool. George Heiser, wearing his usual old sweater, was too busy taking care of customers to pay much attention to what was going on outside.\n\nBut by early afternoon, with the street out front under two feet of water, hardly anyone was about, and the Heiser family was left more or less to itself. A few visitors dropped in, family friends, and an occasional customer. Mrs. Lorentz, from Kernville, sat visiting with Mathilde Heiser upstairs. She had come by alone, without her husband, who was the town's weatherman, and, no doubt, a busy man that day.\n\nSometime near four o'clock George Heiser had sent his son, Victor, out to the barn to see about the horses. The animals had been tied in their stalls, and George, worried that they might strangle if the water should get any higher, wanted them unfastened.\n\nThe barn, like the store front, was a recent addition for the Heisers. It had a bright-red tin roof and looked even bigger than it was, standing, as it did, upon higher ground at the rear of their lot. To get back to it, Victor had left his shoes and socks behind and, with a pair of shorts on, went wading across through the pelting rain. It had taken him only a few minutes to see to the horses and he was on his way out the door when he heard the noise.\n\nTerrified, he froze in the doorway. The roar kept getting louder and louder, and every few seconds he heard tremendous crashes. He looked across at the house and in the second-story window saw his father motioning to him to get back into the barn and up the stairs. Just a few weeks earlier he and his father had cut a trap door through the barn roof, because his father had thought \"it might be a good idea.\"\n\nThe boy was through the door and onto the roof in a matter of seconds. Once there he could see across the top of the house, and on the other side, no more than two blocks away, was the source of all the racket. He could see no water, only an immense wall of rubbish, dark and squirming with rooftops, huge roots, and planks. It was coming at him very fast, ripping through Portage and Center streets. When it hit Washington Street, he saw his home crushed like an orange crate and swallowed up.\n\nIn the same instant the barn was wrenched from its footings and began to roll like a barrel, over and over. Running, stumbling, crawling hand over hand, clawing at tin and wood, Victor somehow managed to keep on top. Then he saw the house of their neighbor, Mrs. Fenn, loom up in front. The barn was being driven straight for it. At the precise moment of impact, he jumped, landing on the roof of the house just as the walls of the house began to give in and the whole roof started plunging downward.\n\nHe clambered up the steep pitch of the roof, fighting to keep his balance. The noise was deafening and still he saw no water. Everything about him was cracking and splitting, and the air was filled with flying boards and broken glass. It was more like being in the middle of an explosion than anything else.\n\nWith the house and roof falling away beneath him, he caught hold of still another house that had jammed in on one side. Grabbing on to the eaves, he hung there, dangling, his feet swinging back and forth, reaching out, trying to get a toe hold. But there was none. All he could do was hang and swing. For years after he would have recurring nightmares in which it was happening to him all over again. If he let go he was finished. But in the end, he knew, he would have to let go. His fingernails dug deep into the water-soaked shingles. Shooting pains ran through his hands and down his wrists.\n\nThen his grip gave out and he fell, backwards, sickeningly, through the wet, filthy air, and slammed down on a big piece of red roof from the new barn. And now, for the first time, he saw water; he was bumping across it, lying on his stomach, hanging on to the roof with every bit of strength left in him, riding with the wave as it smashed across Johnstown.\n\nThe things he heard and saw in the next moments would be remembered later only as a gray, hideous blur, except for one split-second glimpse which would stick in his mind for the rest of his life.\n\nHe saw the whole Mussante family sailing by on what appeared to be a barn floor. Mussante was a fruit dealer on Washington Street, a small, dark Italian with a drooping mustache, who had been in Johnstown now perhaps three years. He had had a pushcart at first, then opened the little place not far from the Heiser store. Victor knew him well, and his wife and two children. Now there they were speeding by with a Saratoga trunk open beside them, and every one of them busy packing things into it. And then a mass of wreckage heaved up out of the water and crushed them.\n\nBut he had no time to think more about them or anything else. He was heading for a mound of wreckage lodged between the Methodist Church and a three-story brick building on the other side of where Locust Street had been. The next thing he knew he was part of the jam. His roof had catapulted in amongst it, and there, as trees and beams shot up on one side or crashed down on the other, he went leaping back and forth, ducking and dodging, trying desperately to keep his footing, while more and more debris kept booming into the jam.\n\nThen, suddenly, a freight car reared up over his head. It looked like the biggest thing he had ever seen in his life. And this time he knew there could be no jumping out of the way.\n\nBut just as it was about to crash on top of him, the brick building beside him broke apart, and his raft, as he would describe it later, \"shot out from beneath the freight car like a bullet from a gun.\"\n\nNow he was out onto comparatively open water, rushing across a clear space which he judged to be approximately where the park had been. He was moving at a rapid clip, but there seemed far less danger, and he took some time to look about.\n\nThere were people struggling and dying everywhere around him. Every so often a familiar face would flash by. There was Mrs. Fenn, fat and awkward, balanced precariously on a tar barrel, well doused with its contents, and trying, pathetically, to stay afloat. Then he saw the young Negro who worked for Dr. Lee, down on his knees praying atop his employer's roof, stark naked, shivering, and beseeching the Lord in a loud voice to have mercy on his soul.\n\nLike the Mussante family, they were suddenly here and gone like faces in nightmares, or some sort of grotesque comedy, as unreal and as unbelievable as everything else that was happening. And there was nothing he could do for them, or anybody else.\n\nHe was heading across town toward the Stony Creek. As near as he could reckon later, he passed right by where Horace Rose's house had stood, then crossed Main and sailed over the Morrell lot, and perhaps directly over where the Morrell greenhouse had been. Almost immediately after that, about the time he was crossing Lincoln Street, he got caught by the backcurrent.\n\nUntil then he had been keeping his eyes on the mountainside, which looked almost close enough to reach out and touch, and on the stone bridge. Both places looked to be possible landings, and either one would do as well as the other.\n\nBut now his course changed sharply, from due west to due south. The current grabbed his raft and sent it racing across the Stony Creek a half mile or so, over into the Kernville section, and it was here that his voyage ended.\n\n\"I passed by a two-and-a-half-story brick dwelling which was still remaining on its foundations. Since my speed as I went up this second valley was about that of a subway train slowing for a stop, I was able to hop to the roof and join a small group of people already stranded there.\"\n\nWhen he had been standing on the roof of his father's barn, looking across the housetops at the avalanche bearing down on Johnstown, he had taken his watch out of his pocket to look at the time. It was a big silver watch with a fancy-etched cover, which had been his fourteenth birthday present from his father. He had snapped it open, because, as he would say later, \"I wanted to see just how long it was going to take for me to get from this world over into the next one.\"\n\nNow, on the rooftop in Kernville, realizing that he had perhaps a very good chance of staying on a little longer in this world, he pulled out the watch a second time.\n\nAmazingly enough, it was still running, and he discovered with astonishment that everything that had happened since he had seen his home vanish had taken place in less than ten minutes.\n\n\u2022 \u2022 \u2022\n\nAgnes Chapman had watched her husband walk to the front door in his bedroom slippers about four o'clock, open it, peer out, and turn around looking, as she told it later on, \"pale and affrighted.\" The Reverend had just seen a boxcar with a man standing on top roll down the pavement in front of the parsonage. As he passed under the tree in the Chapmans' yard, the man had caught hold of a limb and swung himself up onto the roof of the front porch, from which he stepped through the second-story window directly over the Reverend's head.\n\nThe man was the ticket agent from the B & O station, across Washington Street from the Heiser store. Upon hearing the commotion up the valley, he had climbed on top of the car to see what was going on. Then the car had started running with what must have been a small but powerful current preceding the main wave. It swept the car down Franklin, across Locust, too fast for the man to do anything but hang on until he was within reach of the Chapmans' tree.\n\nThe whole scene meant only one thing to the Reverend. The reservoir had broken. He shouted for everyone to run for the attic.\n\nAgnes Chapman, with her seven-year-old granddaughter, Nellie, Mrs. Brinker (their neighbor from across the park), Mr. Parker, and Lizzie, the cook, all made a dash up the front stairs, while Chapman ran to the study to shut off the gas fire. As he turned to go back out to the hall, he saw the front door burst open and a huge wave rush in. He ran for the kitchen and scrambled up the back stairs. A few seconds more and he would have been swept against the ceiling and drowned. The water was up the stairs and into the second floor almost instantly.\n\nBy now the whole family was in the attic, along with the B & O ticket agent and two other young men who had jumped through an open window from a whirling roof.\n\n\"We all stood there in the middle of the floor, waiting our turn to be swept away, and expecting every minute to be drowned.\" Mrs. Chapman said. \"When our porches were torn loose, and the two bookcases fell over, the noise led us to think the house was going to pieces.\"\n\nThe noise everywhere was so awful they had to shout to hear one another. Outside other buildings were scraping and grinding against theirs, or crashing in heaps, and the thunder of the water kept on for what seemed an eternity.\n\n\"We knew . . . that many of our fellow citizens were perishing, and feared that there could be no escape for us,\" the Reverend Chapman wrote later. \"I think none was afraid to meet God, but we all felt willing to put it off until a more propitious time . . .\"\n\nAbout then a man Chapman thought to be \"an Arabian\" came bounding through the window, clad only in underdrawers and a vest. He was drenching wet, shaking with cold and terror, and kept shouting at them, \"Fader, Mudder. Tronk! Tronk! Two, tree hooner tollar, two, tree hooner tollar.\"\n\n\"I think he wanted to tell us he had lost his trunk with two or three hundred dollars he had saved to bring his mother and father over here,\" Chapman later explained.\n\nThe man got right down on his knees and started praying over a string of beads with such frenzy that the Reverend had to quiet him down, as he \"excited and alarmed the ladies.\"\n\nBut despite everything happening outside, the parsonage appeared to be holding on. And when the roar began to die off, Chapman went to the window to take a look. It was, he wrote afterward, \"a scene of utter desolation.\" With darkness closing down on the valley and the rain still falling, his visibility was quite limited. Still, he could make out the tall chimneys and gables of Dr. Lowman's house across the park, poking above what looked to him like a lake spread over the town at a depth of maybe thirty feet. There was not a sign of any of the other houses that had been on the park, but over on the left, where Main Street had been, he could see the dim silhouettes of the bank, Alma Hall, which was the Odd Fellows new building, and the Presbyterian Church sticking up out of the dark water. There were no lights anywhere and no people. \"Everyone is dead,\" Chapman thought to himself.\n\nMrs. Brinker asked him to look to see if her house was still standing. When he said it was not, the others did what they could to console her. The room grew steadily darker, and from outside came more sounds of houses cracking up and going down under the terrible weight of the water.\n\n\u2022 \u2022 \u2022\n\nThe Hulbert House had been the finest hotel in town. It was not so large as the Merchants' Hotel on Main, but it was newer and fitted out \"with all the latest wrinkles\" as one paper of the day put it. Drummers made up most of the trade, and things were arranged to suit them. Breakfast was served early, dinner at noon (a custom most big-city hotels had long since abandoned), and like the other chief hotels in town, each of its rooms had a long extension table where the salesmen could display their wares. \"Through some open door we can always see one piled high with samples of the latest fashions as adulterated for the provincial market,\" wrote a visitor from New York. It was also, for some strange reason, the only hotel in town without a bar.\n\nLocated on Clinton Street, three doors from Main on the east side of the street, it was all brick and four stories tall. Earlier that morning it had looked to quite a number of people like one of the safest places in town.\n\nFor example, Jeremiah Smith, a stonemason who lived in a small frame house over on Stony Creek Street, brought his wife and three children (nine-year-old Florence, seven-year-old Frank, and a four-month-old baby) across town through the rain to the safety of the Hulbert House. How long Smith stayed on with them is not known, but the evidence is he soon went back home again. In any case, he and his house survived the flood. His wife and children were crushed to death when the Hulbert House collapsed almost the instant it was hit by the flood.\n\nIn all there were sixty people inside the building by four o'clock in the afternoon. Only nine of them got out alive.\n\n\"Strange as it may seem, we were discussing the possibility of the dam breaking only a few hours before it really did,\" one of the survivors, a G. B. Hartley of Philadelphia, was later quoted.\n\n\"We were sitting in the office shortly after dinner. Everyone laughed at the idea of the dam giving way. No one had the slightest fear of such a catastrophe.\"\n\nAs the afternoon passed, Hartley moved to the second-floor parlor. He was sitting there talking to a Miss Carrie Richards, Charles Butler of the Cambria Iron Company, and Walter Benford, brother of the proprietor, when they heard shouting in the streets, immediately followed by loud crashes.\n\n\"At first sound,\" Hartley said, \"we all rushed from the room panic-stricken. Why it was I do not know, but we ran for the stairs. Mr. Butler took Miss Richards' hand. She called to me, and I took hold of her other hand. Then we started up the stairs. Mr. Benford did not go with us, but instead ran downstairs where his brother had an office. The scene in the hotel is beyond imagination or description.\n\n\"Chambermaids ran screaming through the halls, beating their hands together and uttering wild cries to heaven for safety. Frightened guests rushed about not knowing what to do nor what was coming. Up the stairs we leapt. Somewhere, I do not know when or how it was, I lost my hold of Miss Richards' hand. I really cannot tell what I did, I was so excited. I still rushed up the stairs and thought Miss Richards and Mr. Benford were just behind and I had reached the top flight of stairs and just between the third and fourth floors, when a terrific crash came. Instantly I was pinned by broken boards and debris . . .\"\n\nHartley then looked up and saw that the building's big mansard roof had been lifted right off and he was looking at nothing more than a sullen sky. In what must have been no more than thirty seconds or so, he managed to scramble out from under the debris and climb onto the roof, which was floating to the side of the crumbling hotel.\n\nF. A. Benford, proprietor of the house, was already on the roof, along with his brother Walter, a traveling salesman from Strawbridge & Clothier named Herbert Galager, and two chambermaids, one of whom had a dislocated shoulder. The roof floated off with the current. The rest of the building just disappeared; the walls fell in and it was gone.\n\n\u2022 \u2022 \u2022\n\nGertrude Quinn was the six-year-old daughter of James Quinn, who, with his brother-in-law, Andrew Foster, ran Geis, Foster and Quinn; Dry Goods and Notions, which stood diagonally across Clinton from the Hulbert House. The two of them, Gertrude would later say, looked like the Smith Brothers on the cough-drop box.\n\nJames Quinn was one of the few prominent men in Johnstown who had been noticeably concerned about the dam since early that morning. He had been to the lake several times over the years and had a clear idea of the volume of water there. If the dam should let go, he had said, not a house in town would be left standing.\n\nThe Quinns lived in one of Johnstown's show places, a three-story, red-brick Queen Anne house newly built at the corner of Jackson and Main. It was surrounded by an iron fence and stood well up off the street, perfectly safe, it was to be assumed, from even the worst spring floods. There were fruit trees and a flower garden in the front yard, a kitchen garden, a barn with one cow and some ducks out back. Inside, everything was the latest\u2014plumbing, icebox, organ, piano, Arab scarves, Brussels carpets, a marble clock from Germany on the mantel.\n\nBesides Gertrude, there were six other children in the family. Vincent, who was sixteen, was the oldest. Helen, Lalia, and Rosemary came next; then Gertrude, Marie, and Tom, who was only a few months old. Rosina Quinn, their mother, was the daughter of old John Geis, who had started the store back in canal days, soon after he arrived from Bavaria. She had worked in the business herself before marrying and was later teased for having five of her seven children in July, which, as everyone knew, was the slow season for dry goods.\n\nThen there was Libby Hipp, the eighteen-year-old German nursegirl, Gertrude's Aunt Abbie (Mrs. Geis), and her infant son, Richard. Aunt Abbie, who was probably no more than twenty-eight years old and a woman of exceptional beauty, had come east for her health from her home in Salina, Kansas. She had had three children in a very short time and needed rest.\n\nJames Quinn was most definitely head of the household. He was a trim, bookish man who had been an officer in the cavalry during the war and still held himself in a like manner. He was President of the Electric Light Company, a member of the school board, and, along with Cyrus Elder, Dr. Lowman, and George Swank, he was one of the trustees of the Johnstown Savings Bank. As a boy he had been taken by his father, a construction worker, to ask for a job in the Cambria mills but had been turned down because he looked too scared\u2014for which he would be forever thankful. For a while before the war he had toyed with the idea of becoming an artist, and one of his early efforts, Rebecca at the Well, done in house paints, hung in the third floor of the new house on Jackson Street. (Later on, his wife would tell him, \"The flood wasn't so bad, when you realize we got rid of Rebecca so gracefully.\")\n\nAt home he was quite exacting about the use of the English language, abhorring slang and insisting on proper diction. He liked cigars. He was quiet, dignified, a strong Republican, and a good Catholic.\n\nThe advertisements he was placing in the Tribune that spring let it be known that Foster and Quinn were offering the finest in Hamburg embroideries, Spanish laces, Marseilles quilts, and \"new French sateens.\" But the store also dealt in carpets, umbrellas, hatpins, hairpins, flannel drawers, striped calico dresses, pearl buttons, black hose, bolsters, and pillowcases.\n\n\"I cannot separate thoughts of parents, brothers, sisters, or home from our store,\" Gertrude would say later. \"When we went there, we became personages . . . the clerks, vying with one another for our attention, were always doing thoughtful little things for us.\"\n\nThe place was big and brightly lighted, with people coming and going, exchanging news and gossip. For the children it was all a grand show, from which they took home strings of stray beads or buttons or some other trinket.\n\nFor Foster and Quinn (father-in-law Geis had long since retired), the place represented an investment of about $60,000 and provided a very good living.\n\nOn the morning of the 31st, James Quinn had gone to the store early to supervise the moving of goods to higher levels. Before leaving home he had told everyone to stay inside. One of his children, Marie, was already sick with measles, and he did not want the others out in the rain catching cold. He did, however, allow young Vincent to come along with him downtown to lend a hand.\n\nAt noon, when he had returned for dinner, the water had been up to his curbstone. He had been restless and worried through the meal, talking about the water rising in the streets and his lack of confidence in the South Fork dam.\n\nA few days before, he and his wife and the infant, Tom, and Lalia had gone to Scottdale for a christening, and Mrs. Quinn and the two children had stayed on to visit with her sister. Now Aunt Abbie and Libby Hipp were more or less running things, and he was doing his best to make sure they understood the seriousness of the situation.\n\n\"James, you are too anxious,\" his sister-in-law said. \"This big house could never go.\"\n\nIn recalling the day years afterward, Gertrude felt sure that her father was so worried that he would have moved them all to the hill that morning, even though he had no special place to take them, if it had not been for Marie. He was afraid of the effect the light might have on her eyes.\n\nAfter dinner he had gone back to the store, and Gertrude slipped out onto the front porch where she began dangling her feet in the water, which, by now, covered the yard just deep enough for the ducks to sport about among the flowers. Everyone who survived the flood would carry some especially vivid mental picture of how things had looked just before the great wave struck; for this child it would be the sight of those ducks, and purple pansies floating face up, like lily pads, in the yellow water.\n\nShortly before four Gertrude's father suddenly appeared in front of her. He took her with one hand, with the other gave her a couple of quick spanks for disobeying his order to stay inside, and hurried her through the door.\n\n\"Then he gave me a lecture on obedience, wet feet, and our perilous position; he said he had come to take us to the hill and that we were delayed because my shoes and stockings had to be changed again. He was smoking a cigar while the nurse was changing my clothes. Then he went to the door to toss off the ashes.\"\n\nIt was then that he saw the dark mist and heard the sound of the wave coming. He rushed back inside, shouting, \"Run for your lives. Follow me straight to the hill.\"\n\nSomeone screamed to him about the baby with the measles. He leaped up the stairs and in no more than a minute was back down with Marie wrapped in a blanket, his face white and terrified-looking.\n\n\"Follow me,\" he said. \"Don't go back for anything. Don't go back for anything.\" Everyone started out the door except Vincent. Just where he was no one knew. Helen and Rosemary ran on either side of their father, holding on to his elbows as he carried the baby. When they got to the street the water was nearly to Rosemary's chin, but she kept going, and kept trying to balance the umbrella she had somehow managed to bring along. The hill was at most only a hundred yards away. All they had to do was get two short blocks to the end of Main and they would be safe.\n\nJames Quinn started running, confident that everyone was with him. But Aunt Abbie, who was carrying her baby, and Libby Hipp, who had Gertrude in her arms, had turned back.\n\nWhen she reached the top of the steps that led from the yard down to the street, Aunt Abbie had had second thoughts.\n\n\"I don't like to put my feet in that dirty water,\" Gertrude would remember her saying. Libby said she would do whatever Aunt Abbie thought best, so they started back into the house.\n\n\"Well, I kicked and scratched and bit her, and gave her a terrible time, because I wanted to be with my father,\" Gertrude said later. How the two women, each with a child, ever got to the third floor as fast as they did was something she was never quite able to figure out. Once there, they went to the front window, opened it, and looked down into the street. Gertrude described the scene as looking \"like the Day of Judgment I had seen as a little girl in Bible histories,\" with crowds of people running, screaming, dragging children, struggling to keep their feet in the water.\n\nHer father meanwhile had reached dry land on the hill, and turning around saw no signs of the rest of his family among the faces pushing past him. He grabbed hold of a big butcher boy named Kurtz, gave him Marie, told him to watch out for the other two girls, and started back to the house.\n\nBut he had gone only a short way when he saw the wave, almost on top of him, demolishing everything, and he knew he could never make it. There was a split second of indecision, then he turned back to the hill, running with all his might as the water surged along the street after him. In the last few seconds, fighting the current around him that kept getting deeper and faster every second, he reached the hillside just as the wave pounded by below.\n\nLooking behind he saw his house rock back and forth, then lunge sideways, topple over, and disappear.\n\nGertrude never saw the wave. The sight of the crowds jamming through the street had so terrified her aunt and Libby Hipp that they had pulled back from the window, horrified, dragging her with them into an open cupboard.\n\n\"Libby, this is the end of the world, we will all die together,\" Aunt Abbie sobbed, and dropped to her knees and began praying hysterically, \"Jesus, Mary, and Joseph, Have mercy on us, oh, God . . .\"\n\nGertrude started screaming and jumping up and down, calling \"Papa, Papa, Papa,\" as fast as she could get it out.\n\nThe cupboard was in what was the dining room of an elaborate playhouse built across the entire front end of the third floor. There was nothing like it anywhere else in town, the whole place having been fitted out and furnished by Quinn's store. There was a long center hall and a beautifully furnished parlor at one end and little bedrooms with doll beds, bureaus, washstands, and ingrain carpets on the floors. The dining room had a painted table, chairs, sideboard with tiny dishes, hand-hemmed tablecloths, napkins, and silverware.\n\nFrom where she crouched in the back of the cupboard, Gertrude could see across the dining room into a miniature kitchen with its own table and chairs, handmade iron stove, and, on one wall, a whole set of iron cooking utensils hanging on little hooks. Libby Hipp was holding her close, crying and trembling.\n\nThen the big house gave a violent shudder. Gertrude saw the tiny pots and pans begin to sway and dance. Suddenly plaster dust came down. The walls began to break up. Then, at her aunt's feet, she saw the floor boards burst open and up gushed a fountain of yellow water.\n\n\"And these boards were jagged . . . and I looked at my aunt, and they didn't say a word then. All the praying stopped, and they gasped, and looked down like this, and were gone, immediately gone.\"\n\nShe felt herself falling and reaching out for something to grab on to and trying as best she could to stay afloat.\n\n\"I kept paddling and grabbing and spitting and spitting and trying to keep the sticks and dirt and this horrible water out of my mouth.\"\n\nSomehow she managed to crawl out of a hole in the roof or wall, she never knew which. All she saw was a glimmer of light, and she scrambled with all her strength to get to it, up what must have been the lath on part of the house underneath one of the gables. She got through the opening, never knowing what had become of her aunt, Libby, or her baby cousin. Within seconds the whole house was gone and everyone in it.\n\nThe next thing she knew, Gertrude was whirling about on top of a muddy mattress that was being buoyed up by debris but that kept tilting back and forth as she struggled to get her balance. She screamed for help. Then a dead horse slammed against her raft, pitching one end of it up into the air and nearly knocking her off. She hung on for dear life, until a tree swung by, snagging the horse in its branches before it plunged off with the current in another direction, the dead animal bobbing up and down, up and down, in and out of the water, like a gigantic, gruesome rocking horse.\n\nWeak and shivering with cold, she lay down on the mattress, realizing for the first time that all her clothes had been torn off except for her underwear. Night was coming on and she was terribly frightened. She started praying in German, which was the only way she had been taught to pray.\n\nA small white house went sailing by, almost running her down. She called out to the one man who was riding on top, straddling the peak of the roof and hugging the chimney with both arms. But he ignored her, or perhaps never heard her, and passed right by.\n\n\"You terrible man,\" she shouted after him. \"I'll never help you.\"\n\nThen a long roof, which may have been what was left of the Arcade Building, came plowing toward her, looking as big as a steamboat and loaded down with perhaps twenty people. She called out to them, begging someone to save her. One man started up, but the others seemed determined to stop him. They held on to him and there was an endless moment of talk back and forth between them as he kept pulling to get free.\n\nThen he pushed loose and jumped into the current. His head bobbed up, then went under again. Several times more he came up and went under. Gertrude kept screaming for him to swim to her. Then he was heaving himself over the side of her raft, and the two of them headed off downstream, Gertrude nearly strangling him as she clung to his neck.\n\nThe big roof in the meantime had gone careening on until it hit what must have been a whirlpool in the current and began spinning round and round. Then, quite suddenly, it struck something and went down, carrying at least half its passengers with it.\n\nGertrude's new companion was a powerful, square-jawed millworker named Maxwell McAchren, who looked like John L. Sullivan. How far she had traveled by the time he climbed aboard the mattress, she was never able to figure out for certain. But later on she would describe seeing many flags at one point along the way, which suggests that she went as far up the Stony Creek as Sandy Vale Cemetery, where the Memorial Day flags could have been visible floating about in the water. Sandy Vale is roughly two miles from where the Quinn house had been, and when Maxwell McAchren joined her, she had come all the way back down again and was drifting with the tide near Bedford Street in the direction of the stone bridge.\n\nOn a hillside, close by to the right, two men were leaning out of the window of a small white building, using long poles to carry on their own rescue operation. They tried to reach out to the raft, but the distance was too great. Then one of them called out, \"Throw that baby over here.\"\n\nMcAchren shouted back, \"Do you think you can catch her?\"\n\n\"We can try,\" they answered.\n\nThe child came flying through the air across about ten to fifteen feet of water and landed in the arms of Mr. Henry Koch, proprietor of Koch House, a small hotel and saloon (mostly saloon) on Bedford Street. The other man in the room with him was George Skinner, a Negro porter, who had been holding Koch by the legs when he made the catch. The men stripped Gertrude of her wet underclothes, wrapped her in a blanket, and put her on a cot. Later she was picked up and carried to the hill, so bundled up in the warm blanket that she could not see out, nor could anyone see in very well.\n\nEvery so often she could hear someone saying, \"What have you got there?\" And the answer came back, \"A little girl we rescued.\" Then she could hear people gathering around and saying, \"Let's have a look.\" Off would come part of the blanket in front of her face and she would look out at big, close-up faces looking in. Heads would shake. \"Don't know her,\" they would say, and again the blanket would come over her face and on they would climb.\n\nGertrude never found out who it was who carried her up the hill, but he eventually deposited her with a family named Metz, who lived in a frame tenement also occupied by five other families. The place looked like paradise to her, but she was still so terrified that she was unable to say a word as the Metz children, neighbors, and people in off the street jammed into the kitchen to look at her as she lay wrapped now in a pair of red-flannel underwear with Mason jars full of hot water packed all around her.\n\nLater, she was put to bed upstairs, but exhausted as she was she was unable to sleep. In the room with her were three other refugees from the disaster, grown women by the name of Bowser, who kept getting up and going to the window, where she could hear them gasping and whispering among themselves. After a while Gertrude slipped quietly out of bed and across the dark room. Outside the window, down below where the city had been, she could now see only firelight reflecting on water. It looked, as she said later, for all the world like ships burning at sea.\n\n\u2022 \u2022 \u2022\n\nThe Reverend Dr. David Beale, pastor of the Presbyterian Church on Main Street, was one of the several hundred people crowded into the cavernous, pitch-dark rooms of the second, third, and fourth floors of Alma Hall. The building was on Main, five doors up from Dr. Beale's church and directly across the street from the park. It was the tallest, largest structure in Johnstown.\n\nDr. Beale had been at home that afternoon, in the Lincoln Street parsonage, which stood directly behind his church, and like his good friend and neighbor the Reverend Chapman, he had been working on his sermon for Sunday. About four he had gone into the parlor to help take up the carpet. Then all at once the house was struck, and in the next few seconds he snatched up the family Bible, his wife turned off the gas, his daughter grabbed the canary cage, and they and several neighbors who had dropped by earlier all dashed up the front stairs. By the time they reached the second floor the water was up to their waists, and a hat rack was driven against Beale's back with such force that it nearly knocked him under. As they reached the third floor a man washed in through the window.\n\n\"Who are you? Where are you from?\" Beale shouted.\n\n\"Woodvale,\" the man gasped. He had been carried on a roof a mile and a quarter.\n\nExpecting at any moment \"to be present with the Lord,\" Beale led the group in a prayer and read aloud from the Bible, his voice straining against the noise of the flood:\n\n\"God is our refuge and strength, a very present help in trouble.\n\n\"Therefore will not we fear, though the earth be removed, and though the mountains be carried into the midst of the sea . . .\"\n\nOutside twisted wreckage, tank cars, freight cars, and what appeared to be every house in sight went tumbling past the window. The Reverend Beale saw J. Q. Benshoff, Johnstown's leading bookseller go by, Mrs. John Fulton and her daughter, and two small children clinging to a roof, both of them nearly naked. For blocks around every building appeared to have been obliterated.\n\nThere were ten people in the Beale attic, counting the newcomer from Woodvale. Soon after, Beale helped save Captain A. N. Hart, his wife, sister, and two small sons by pulling them in through the window. But there was considerable doubt as to how long the frame parsonage could last, and especially when, after a shifting and quieting of the current, the wreckage which had been shoved against the west side of the house began slowly drifting off and the whole building started to jerk and tremble.\n\nA decision was made to try, before nightfall, to walk across the flood\u2014over the debris\u2014to Alma Hall, which was the equivalent of about one block away. Captain Hart went out the window first on the end of a rope, tested a roof that was floating below, found it stable enough, and the other fourteen, plus the Beales' dog, a terrier called \"Guess,\" followed after. The Reverend Beale was the last one out. Then they started off, picking their way over tree trunks, timbers, stepping from one moving house to another, climbing up the sides of roofs sometimes so steep that part of the group on one side would be out of sight from those on the other side, then jumping across sudden spaces of dark water or bridging them with stray planks. At one point one of the girls lost her balance and fell in, vanishing from sight except for her hair which floated on the surface. She was rescued by pulling her back on some long boards, and everyone continued on.\n\nBy now, very near dark, the city was one huge, vile-looking lake anywhere from ten to thirty feet deep, much of it crusted over with a grinding pack of wreckage, across which now other groups of tiny figures, barely visible in the fading light, could be seen groping their way toward the hills or the few buildings still left standing.\n\nAbout the time the Reverend Beale and his party had climbed out of the parsonage, a break had occurred in the railroad embankment to the right of the stone bridge, between the bridge and the depot, and the water began raging through just as though it were a spillway. House after house had plunged through the break like boats running the rapids, many of them loaded with people, dozens to be dashed to pieces when they hit Cambria City below.\n\nBut there were some people who, one way or another, survived the trip to be fished out by rescue teams farther along the river. One of them was Maxwell McAchren, who, after throwing little Gertrude to safety, sailed on toward the bridge in time to be sucked through the break. He wound up riding the mattress straight through Cambria City at the time when a good part of it was being destroyed, past the Cambria works as they were being pounded by the water, and on down the Conemaugh four miles before he was finally pulled to shore by a crowd of men with ropes.\n\nWith the break in the embankment the level of the water over the city began to go down, but only slowly, since the Little Conemaugh and the Stony Creek were still pouring in immense quantities of water, mud, and debris. So as night began, those buildings which had somehow held up against everything so far were still withstanding as much as twenty feet of water, and very often they had had several hundred tons of wreckage dumped against them. For those who would manage to get inside them, the long night ahead would be by far the most agonizing part of the whole ordeal.\n\nWhen the Reverend Beale's group finally reached Alma Hall, there were already close to 200 people inside. At the Dean Canan house there were 60 people in the attic. At least 51 people were in the attic of Dr. Walters' house on Vine Street. The Fred Krebs house had 125 people inside before the night ended. Nearly 200 people were in the upper floors of the Union Street School, more than 100 on top of the Wolfe Building, and at the Morrell house (by then it had been converted into the Morrell Institute, a vocational training school for the Iron Company) there were 175 people. And over at Dr. Swan's tall brick house at the corner of Vine and Stony Creek streets there were close to 90 people, including Horace Rose, who lay stretched out on the floor with a dislocated shoulder, a broken collarbone, several crushed ribs, and half of his face ripped open.\n\nThere had been a few minutes after the flood had fallen on his part of Main Street during which Rose had been at his window, almost hypnotized by the scene outside. He had seen John Dibert's house squashed like a paper bag. Another brick house fell with a crash. A large frame building directly across the street had lifted up and charged right for him. Then there had been a horrible noise, he had felt himself falling, and all was dark.\n\n\"A moment later I felt the press of a heavy shock, a sense of excruciating pain . . . the thought came upon me that I was being crushed to death . . .\"\n\nHis whole right side had been caved in by falling timbers, and he was powerless to free himself. He had heard his youngest son calling for help but had been unable to do anything for him. He had seen his daughter, June, rise up out of the water, then, almost immediately, sink back out of sight. From out of nowhere a small boy had appeared among the chaos and told him his wife had drowned. Then another stranger, this one a young man, seemed, Rose said later, to shoot out of the debris. Rose told him to go help his wife and daughter. The man, Rose learned afterward, was a Pittsburgh dentist named Phillips, and in a few frantic minutes he managed to free Mrs. Rose from the timbers that had fallen on her. Then the majority of the Rose household\u2014Rose, his wife and daughter (she too had been rescued somehow), two of his sons, one maid, \"the strange boy,\" as Rose called him, and an elderly lady who had been pulled off a floating shutter by one of his sons\u2014were all together on a single stout roof which chanced by at the very moment when the last of their house was disappearing in the tide. The roof had been heading toward the stone bridge. But, Rose wrote later, \"Scarcely was the complement of passengers complete, when the current turned, and our ship was driven with terrific velocity directly up the channel of the Stony Creek . . .\"\n\nThen for several more hours they had floated about, sometimes wallowing in dead water, other times rushing rapidly back over a course they had just completed. And through it all Rose lay helpless, in terrible pain, and shaking with severe chills as the cold rain beat down.\n\nThey had seen the spire of St. John's Catholic Church catch fire, which according to most accounts had happened about eight o'clock, and had watched the flames leap clear to the cross on top before the whole thing toppled and fell into the water. At another point they had been becalmed within perhaps a hundred feet of where Rose's office had been on Franklin Street, and listened to the ringing of the ponderous bell in the town clock. The clock was in the steeple of the Lutheran Church, and somehow or other its mechanism was still functioning the same as ever. Through the rest of the night, despite everything, every hour on the hour, it bonged away. The sound had a powerful effect on everyone who heard it.\n\nThen by another sudden change in the surface currents, the roof had been driven off over the main channel of the Stony Creek, where by now the current was again heading downstream. They were carried a hundred yards or more before the roof lodged against the side of the Swan house.\n\nRose was lifted from the roof and through a window. From then until morning he lay listening to buildings breaking up somewhere out in the night and watching the light from the fire at the bridge play across the walls and ceiling.\n\nFor the Reverend Beale and the others inside Alma Hall there had been an immediate fear of fire and what might happen if panic should break out among so many people waiting in the dark. An Alma Hall government had been set up, with Beale and Captain Hart each put in charge of one of the floors. Some whiskey was confiscated, and the use of matches was strictly forbidden because of the likelihood of a natural-gas leak in the basement. A count was made to see how many there were (it came to 264), and the Reverend once again led a prayer.\n\nJames Walters, a lawyer, was named director of the building. Walters had made one of the day's most extraordinary voyages, having been swept from his home on Walnut Street on top of a roof which took him spinning across town until he smashed into the side of Alma Hall, flew headlong through a window, and landed square in the middle of his own office.\n\nThe fourth member of the governing body was the only physician in the building, Dr. William Matthews, who spent the entire night tending to the wounded, without sleep or rest, despite the fact that he had two broken ribs.\n\nIn the Reverend Beale's words, it was a \"night of indescribable horrors.\" The only light was the faint, eerie glow from the fires outside. Up near the long front windows that earlier in the day had looked down into the green treetops of the park, the light was bright enough to recognize a nearby face; but farther back in the deep, high-ceilinged rooms it was nearly pitch-black, and on the stairways between floors there was no light at all.\n\nNearly everyone was wringing wet, filthy, and suffering from the cold. A number of people had most of their clothes torn off. There was no food and no water. There were no blankets, no dry clothes, and no medical supplies. The injured lay shivering in the dark. The rooms were filled with their moaning, with the crying of scared, hungry children, and with a lot of fervent praying.\n\nOutside they could hear the rush of the rain and faint calls for help, a sudden scream, and every now and then the unearthly howling of dogs and other animals, which to many people was the most frightful sound of all.\n\nNor was there any assurance whatsoever that the whole enormous building would not go the way of so many others and crack apart and bury them all under tons of brick and plaster and falling timbers. Everyone was asked to move about as little as possible. According to Beale, \"the expressed opinion of the contractors present\" was that the building would not last the night.\n\nPeople began thinking about whether their own corpses would be recognizable or where they might be buried, if ever their bodies were found. The suspense was unbearable, and it kept on, hour after hour. It seemed morning would never come.\n\n\u2022 \u2022 \u2022\n\nBut Alma Hall stood through the night, as did the Presbyterian Church and its parsonage, Dr. Lowman's house, where a small crowd had gathered in the top floor, and the Methodist parsonage, where the Chapmans and their assorted guests huddled together in the numbing cold praying for morning. The buildings survived because they were on the lee side of the big, stone Methodist Church. Standing as it did, at the corner of Franklin and Locust, on the northeastern corner of the park, the church was one of the first sizable buildings in town to be struck by the wave. Not only had it held, but it had split the wave and so served as a shield for buildings directly in line behind it. (One tale to come out of Alma Hall later on told of a voice in the dark saying, \"We've been saved by the Methodist Church,\" whereupon another voice answered back, \"Only the Catholic Church can save!\")\n\nElsewhere in the night the story was quite different. Buildings caved in or caught fire and burned to the water line. The St. John's fire was the biggest and most spectacular, but there were fires among several houses close by; the Keystone Hotel caught fire and there were one or two small fires over in Kernville.\n\nAnd aside from the many large groups of people gathered inside Alma Hall, Dr. Swan's house, or the other buildings that were still standing, there were any number of smaller groups of four to six, or even one to two, people who spent the night inside their own tiny attics or atop the roofs of little houses that bobbed about with the current. Some were closed in under roof beams, with no windows to look out or escape through; they were still alive, but trapped, and with no way of knowing what might happen next.\n\nAt least one family had jumped into a large bed when the water rushed up their stairs. The bed was borne clear to the ceiling by the water, and the family stayed there, floating inside their own house through the remainder of the night.\n\nAnother family named Williams had their house split in half at the bridge, then went floating up the Stony Creek in what was left of the attic. In the darkness that night Mrs. Williams gave birth to a baby boy; and the family stayed there until morning, soaked, freezing cold, the baby wrapped in a shawl.\n\nScores of others floated on rooftops or freight cars or half-submerged debris, without any protection from the pouring rain. A Mrs. Jacob Malzi hung on to the eaves of a house all night, up to her waist in water. A Miss Minnie Chambers had climbed inside a freight car which had been carried through the cut near the bridge and smashed to pieces against the roof of the Cambria works, where she, miraculously still alive, spent the night holding on to a small pipe that stuck up through the roof. James Shumaker lay half-unconscious across a heap of drifting wreckage all night, his face and arms badly torn and nearly blinded in both eyes by sand and lime.\n\nSeveral people spent the night in trees, hanging on with the water lapping about below, never daring to close their eyes, even for a few moments, for fear they might fall asleep, lose their grip, and drop into the black current. Jacob Horner and his family of eight spent all night in a tree; so did Reuben Bensen and Mrs. Ann Buck, who was eighty years old, and Mrs. John Burket, who had had every bit of her clothing ripped from her back by the flood.\n\nOf the great many people who were lucky enough to get to dry land, there were a number who were in such a state of shock and fear that they just started walking, stopping for nothing, stumbling on blindly through the dripping woods until the first light of morning.\n\nBut by far the worst of the night's horrors was the fire at the bridge. Minnie Chambers, the girl who clung to the roof of the Cambria works, said later that she could hear screaming from the bridge all through the night. William Tice, who owned a drugstore on Portage Street, described what he saw soon after he had been fished out of the water near the bridge.\n\n\"I went up on the embankment and looked across the bridge, which was filled full of debris, and on it were thousands of men, women, and children, who were screaming and yelling for help, as at this time the debris was on fire, and after each crash there was a moment of solemn silence, and those voices would again be heard crying in vain for the help that came not. At each crash hundreds were forced under and slain.\n\n\"I saw hundreds of them as the flames approached throw up their hands and fall backward into the fire, and those who had escaped drowning were reserved for the more horrible fate of being burned to death. At last I could endure it no longer, and had to leave, as I could see no more.\"\n\nFrank McDonald, a railroad conductor who apparently kept on watching, said, \"They reminded me of a lot of flies on flypaper, struggling to get away with no hope and no chance to save them.\"\n\nActually, for anyone to see much of what was going on was extremely difficult, with the rain pouring down, the dark, the smoke, and the wild flames. One after another, houses had been swept against the pileup and quickly took fire. Out of them, crawling on hands and knees, climbing, jumping from place to place, helping one another, small, dark figures had appeared, now silhouetted sharply against the high, wind-whipped flames, now invisible against the black shadow of mangled debris, now emerging again from the smoke and groping their way toward the ends of the bridge. That there looked to be thousands of them and that they seemed insectlike is understandable enough; but evidence is that, at most, perhaps 500 to 600 people were driven into the burning heap, and though exact figures were never settled on, it is likely that all but about 80 of them managed to escape.\n\nA good many escapes were made thanks to the courage of bystanders who rushed in to help. They lifted old people and children from the windows of half-shattered houses. They helped carry the badly injured across the wreckage to the hillsides.\n\nA girl named Rose Clark was trapped near one end of the bridge, half submerged under water, with a broken arm and a broken leg which was pinned down by timbers. A group of men had worked for several hours to free her leg but without success and the fire kept spreading closer. For a short while there was talk among them of cutting her leg off, rather than letting her burn to death, and for a few tense minutes, when the flame was almost on top of them, it looked as though they would have to. But the leg came free at last, and they carried her to safety.\n\nThe fire burned on through the night, and would be still blazing when morning came. In little towns miles away downriver and on the other side of the mountains, people could see a strange, shimmering, blood-red glow in the sky.\n\nBut even for those who had somehow succeeded in getting to the high ground in time, even for those who were uninjured or were lucky enough to have a roof to sleep under, there was the indescribable agony of remembering what they had seen, and not knowing what had become of others. No one really knew for sure the extent of what had happened, but they knew it had been terrible beyond belief, and if the whereabouts of someone was not known, then only the worst could be imagined. All that could be done now was to wait for morning, and hope.\n\n## VI\n\n## A message from Mr. Pitcairn\n\nMr. Robert Pitcairn's private car had been attached to eastbound passenger train Number 18 shortly before noon that morning and rolled out of Pittsburgh's Union Station about an hour later. Mr. Pitcairn was on his way to Lilly to see how serious the storm damage was there, and to look things over at Johnstown and South Fork on the way.\n\nMessages about trouble along the line had been coming in to his office since early morning, including one about the dam. Pitcairn had read it and thought little more of it. First of all, he could not quite understand how Colonel Unger could be sending such warnings, since he knew perfectly well that Colonel Unger had no telegraph wire at the club and that the telephone line was not open yet. And secondly, as he would say later, he simply \"paid little attention to any reports about the South Fork dam, as they had been made perhaps nearly every year.\"\n\nWhen later messages came in from South Fork, from agent Dougherty and Pitcairn's old friend J. P. Wilson, Pitcairn was already on his way east.\n\nPitcairn's knowledge of the dam went back more than thirty years, to the time when the Pennsylvania had first bought it. His old boyhood friend Andy Carnegie had gotten him a job on the railroad, as a ticket agent at Cresson, not long before that. He and Carnegie had been telegraph operators together in Pittsburgh; then they went with the Pennsylvania. Later on, when Andy quit his job as head of the Pittsburgh Division to go into business for himself, Pitcairn had been named to replace him.\n\nBut his first real interest in the dam began when it broke in 1862 and wrecked a lot of railroad property in South Fork. Then nearly twenty years later, when the South Fork club finished its restoration and there was talk in the valley about leaks at the base of the dam, Pitcairn had gone up to see for himself, taking along several of his own people from South Fork. They had given the dam what he felt was a thorough enough going-over. Benjamin Ruff had walked with them, saying that what everyone called leaks were actually springs that came from near the ends of the dam. Ruff also promised that he would strengthen things some, and then they all shook hands and went home.\n\n\"The only point we were afraid of,\" Pitcairn said later, \"was the leaks at the bottom of the dam increasing.\" And he was evidently afraid (or cautious) enough to ask Wilson and others in the area to keep an eye out for him. Whether his subsequent membership in the South Fork Fishing and Hunting Club was, even in part, his way of keeping his own eye out, is not known. Though it seems highly doubtful, since, unlike Daniel J. Morrell, he had every sort of social and business reason for wishing to be there.\n\nNow his train headed out of Pittsburgh along the muddy Monongahela, past the towering black Edgar Thomson works at Braddock, where the British general by the same name had suffered his famous defeat; then on through one little town after another, East McKeesport, Irwin, Jeannette, Greensburg, and out into open hill country. He had been over the route maybe a thousand times during his years with the railroad. He knew every bend, bridge, siding, every water tower, coal tipple, every depot, every barn and farmhouse along the horizon. He could not claim to know all the men, that would be impossible, but he knew the good part of them, and certainly every last one of them knew him. He was their supreme commander. His word was law from Altoona to Pittsburgh, and the portly frame, the bullet head, the pince-nez glasses and walrus mustache were far better-known among them than the rather inconspicuous features of the man who was then President of the United States. And if it were a question as to which one wielded the most authority, there would have been some debate.\n\nMen who had been with the Pennsylvania for a dozen years or more still talked about Pitcairn during the \"Great Strike\" of '77. A lot of them felt that he had been the chief cause for what happened in Pittsburgh. The strike had begun on the B & O in Baltimore and had been spreading fast. Times were hard, and wages had been cut. But Pitcairn had chosen that particular moment to institute new practices on the division which would have meant increased work loads and even more layoffs. \"The men were always complaining about something,\" he would say later. When it was all over, a pitched battle had been fought between militia and a mob of strikers and unemployed, a good part of downtown Pittsburgh had been burned to the ground, and fifty-seven people had been killed. The men had gone back to work, having gained nothing. And Robert Pitcairn's hold on them was stronger than ever.\n\nHe liked to say that the railroads (by which he really meant his railroad) were \"the heart, blood, veins, and arteries of Pittsburgh,\" which, of course, put him in a most important position indeed.\n\nPitcairn was in his mid-fifties. He was an elder of the fashionable new Shady Side Presbyterian Church and a man of considerable financial means. For aside from his earnings on the railroad, he had also managed to put together a good-sized fortune on the side, largely by backing his inventive friend George Westinghouse. Slabtown and the squalid back streets where he and Carnegie and Phipps had grown up were far, far behind him now.\n\nSitting in the upholstered splendor of his private car, he looked appropriately substantial, and quite tired. He had been up late the night before at the telegraph instrument in his home making inquiries about the weather between Pittsburgh and Altoona. Then he had gone off to town earlier than usual that morning, and with the news of the storm growing worse every hour, it had been a difficult day ever since. But now, watching the landscape sweep past his window, he began to realize just how serious things were.\n\nThe rain was coming down in wild, silvery sheets. The whole countryside was awash. Hillsides were mortally veined with angry little creeks. Fields were covered with water that looked to be a foot deep or more. At Latrobe, at the foot of Chestnut Ridge, the Loyalhanna was twice its normal size and well over its banks. Ten miles farther, he saw the Conemaugh for the first time and knew he was up against something unlike anything he had ever experienced. His train was moving very slowly by this time, following the course of the river where it cut through the ridge. On the other side of the ridge, at the village of Bolivar, people were out along the riverbanks watching the torrent rush by. At New Florence the water had spilled through the lowlands, flooding miles of woods and meadows. Pitcairn thought he could actually see the water rising, it was coming up so fast.\n\nAfter New Florence the train pulled through little Nineveh, where men and boys in gum coats, their collars turned up against the biting wind, their hats dripping with rain, stood beside the track watching the cars clack by.\n\nThen the train started into the breath-takingly beautiful Conemaugh Gap, or Packsaddle, the one pass through Laurel Hill to Johnstown. The railroad ran well above the river here, even with the river in its present condition, but above the tracks the mountainside loomed up another 1,500 feet.\n\nSeveral miles farther on in the gorge, at a place called Sang Hollow, about four miles from Johnstown, they stopped. The time by now was about five after four.\n\nPitcairn climbed down from his car and went up to the tower to find out what the trouble was. The operator told him the lines east had gone dead; they tried again several times to get Johnstown, but it was no use. The operator said he could not let them through without clearance, which, according to the rules, was exactly what he was meant to say, even to Pitcairn.\n\n\"I was about making up my mind to proceed cautiously, running carefully, to find the trouble,\" Pitcairn said later, \"when looking east, I saw some debris. The water before this had been muddy, but very little drift. The debris attracted my attention from its singular appearance, being broken up wood entirely, and in very small pieces. In a short time, the telegraph poles commenced to break down, and threatened to take the tower down with it.\"\n\nThen they saw a man coming down the river on some debris, moving very fast. Pitcairn thought the water must have been going by at about fifteen miles an hour. They saw more people coming, hanging on to telegraph poles or what appeared to be parts of buildings or just being swept along and trying desperately and futilely to swim. Pitcairn and the others rushed out to do what they could to save them, but the river carried them off and out of sight.\n\n\"I returned to the telegraph office to see what word I could get, when the people came down by the scores; the water rising very rapidly, and men, women, and children on the drift, and we perfectly helpless.\"\n\nBy this time most of the male passengers on board the train were out on the riverbanks doing everything possible to help. They got hold of long poles and big limbs and held them out over the current as far as they could, hoping maybe the people going by could reach them. They threw ropes, and at one point, one of them actually stripped off his coat and jumped into the water to save a mother and her small child.\n\nHe was a rugged, seventeen-year-old Pittsburgh boy named Bill Heppenstall, who was on his way back to school at Bellefonte, Pennsylvania, after being home ill. A small house had lodged momentarily in some overhanging trees. The men heard a baby crying, but the house was too far out to reach. Heppenstall decided he would go in and get the child. The others tried to talk him out of it, but he got the bell cord out of one of the cars, tied it around him, and swam out to the house. In no time he was back with the child. There were great cheers from the crowd. But he then told them the mother was still back there and started into the water again, this time taking a railroad tie along with him to help hold her up. Just as he got her to shore the house tore loose from the trees and went spinning off downstream.\n\nBy the time it began getting dark, the operator at Sang Hollow had counted 119 people going by, dead and alive. Despite everything they had tried, the men on the riverbank had been able to rescue only seven.\n\nAbout six o'clock Pitcairn ordered the train back down to New Florence. The water was still high, but it did not seem to be getting any higher. He had decided to take the passengers back to Pittsburgh, giving them the option to stop off at New Florence if there were any accommodations to be had.\n\nBut before leaving, Pitcairn got off a message to Pittsburgh. It was directed to the editors of the morning papers, and its exact wording remains unclear. But sometime between five thirty and six the news was out and on the wire. A dam had failed at South Fork and caused a disastrous flood at Johnstown. By then Pitcairn had faced up to the awful realization of what had gone wrong.\n\nHis train rolled ever so slowly back through the gorge, reaching New Florence by perhaps six thirty. The first thing he did there was to write out a still longer and more detailed message, which he then put aside, in the hope that some further word might come in from Johnstown itself. So for the next several hours they sat and waited. The rain hammered down outside; men kept coming in and out talking of more bodies found or the few half-drowned souls they had been able to drag ashore.\n\nThe village sat well back from the river, on high, dry ground. Only a few houses near the river were under water, and few citizens were suffering any serious discomforts. As a result the streets were filled with people going to and from the river, standing in doorways, talking to passengers from Pitcairn's train, or gathered in groups looking at the dull, red glow in the sky to the east.\n\nAbout ten o'clock Pitcairn received word from Johnstown by way of Sang Hollow. One of his men in Johnstown, a W. N. Hays, had managed to get from Johnstown to Sang Hollow on foot. Apparently he had been on the hillside above the west end of the bridge and was able to make his way down the tracks above the rampaging river. Once he reached Sang Hollow, the message was put on the wire to New Florence.\n\nPitcairn was told how things were at Johnstown, and he then sent a second message to Pittsburgh, which would be quoted in the papers there at some length. He reported the number of bodies that had been counted going by at Sang Hollow. He said there was no way clear to Johnstown, but that his information was that the city was \"literally wiped out.\" He said that the debris at the stone bridge was reported to be forty feet high and that it was burning.\n\nThen he said, \"I fear there will be terrible suffering among those saved which should be relieved as soon as possible. In the interest of humanity I think a public meeting should be called early tomorrow to send food, clothing, etc. to those poor people which we will be glad to forward to Johnstown . . . as soon as we can get a clear track there.\"\n\nThis message, like the one before it, went right on the wire. Before midnight the story was across the country:\n\nPittsburgh, Penn. May 31\u2014A rumor, loaded with horror, holds this city in dreadful expectancy tonight. It is said that the bursting of a reservoir, just above Johnstown, a flourishing place in Cambria County, had flooded the town and swept at least 200 of her citizens to death. The news is of a very uncertain character, there being no communication with the district were the flood is reported to have occurred, all the wires being down . . . There is no way to get to the scene of the disaster and full particulars are not expected tonight.\n\nBut the fact was that the rush to Johnstown had already begun hours before. Two trains had been chartered by five Pittsburgh newspapers, and the first of them, the one with the Dispatch and Times men on board, started out from Union Station a few minutes after seven. The second, chartered by the Post, the Commercial-Gazette, and the Chronicle-Telegraph, followed almost immediately after. In New York, Philadelphia, Boston, in Chicago, Cleveland, and St. Louis, reporters picked up their hats and coats and went directly to the nearest depot, taking no time to pack or anything else. Some of them were still in evening clothes after a night at the theater.\n\nThe trains from Pittsburgh got no farther than Bolivar, where the men piled out into the rain and moved among the crowds gathered at the station and along the dark edge of the river. They picked up stories of the bodies and wreckage that had been washing past, about the few rescues that had been made, and the horrid things people had seen happen.\n\nNot long before dark a man and two women had been seen rounding the bend upstream. They were on a raft of some sort, a barn roof most people thought it was, and they were coming on fast, the women down on their knees, the man with his arms around them and looking about for something to grab on to. That had been before the bridges went, and the men on the bridges had been hanging ropes down for the people in the river to get hold of. When the raft shot by under the first bridge, the man reached out for the rope but missed. Then he and the two women were heading for the second bridge, and everyone along the shore line was rooting for them as they watched him telling the women to try for the next rope. As they came under the second bridge, he made a lunge for the rope, got it, and was jerked violently off balance; but seeing that the women had missed, he let go and fell back down on the raft again. The current then swept them toward the bank, where he was able to catch hold of a tree. With an immense effort he managed to pull the two women into the tree with him, but at almost the same instant a large section of the bridge upstream let go with a sudden crash. It came careening down the river, smashed into the tree, carried it away, and drowned the man and the two women.\n\nEveryone in Bolivar had seen the whole thing and they wanted to tell exactly how it happened. Some people in the crowd said they knew the man and said his name was Young. Others said they thought the women looked like mother and daughter, and that they could be heard praying as they went by. The newspapermen wrote it all down, asking questions, taking names.\n\nIt was too dark to see much by the river, but the rush of the water could still be heard plain enough, and tiny, dim specks of light could be seen moving through the trees along the shore where men with lanterns were still watching for possible signs of life.\n\nJohnstown was still twenty miles away. Among the newspapermen there was talk about what to do next. The tracks from Bolivar on were under water and not safe enough to take the train any farther. Most of the men decided to push on in the direction of New Florence, some by foot and some in wagons. The ride up from Pittsburgh had taken quite a long while, with conditions what they were. It was ten thirty when they had pulled into Bolivar. By the time they had slogged through the rain and dark to New Florence, it was getting on toward three in the morning.\n\nMud-spattered, dead-tired, cold, wringing-wet, they moved into whatever dry space there was left in the little town and began interviewing everyone who was willing to talk, which was just about everyone. Several of them got hold of wires to Pittsburgh and started filing their stories.\n\nAt that point about all they could say was that every sign was that \"hundreds if not thousands\" of people had been killed in \"an appalling catastrophe.\" They reported rumors of panic-stricken people fleeing through the woods from the scene of the disaster and of the number of people who had been seen going by in the river at New Florence (counts varied, but eighty-five seems about average). And they sent back what information they could pick up concerning the dam, a good deal of which was inaccurate. Several reporters had the dam 110 feet high and the lake as much as eight miles long and three miles wide. But they did have the name of the South Fork Fishing and Hunting Club by then, reported it accurately, and added that the Pennsylvania Railroad's engineers had inspected the dam once a month, which suggests that Pitcairn was also, by then, doing some talking himself. \"Investigations showed that nothing less than some convulsion of nature would tear the the barrier away and loosen the weapon of death,\" one reporter put on the wire.\n\nAbout four o'clock there was great excitement when a man from Johnstown, a carpenter named McCartney, and his wife came staggering out of the night. He said they had left Johnstown right after the flood struck and had been walking ever since. He told them there was hardly a building left standing in Johnstown, and, in general, substantiated the wildest of the rumors that had been circulating since the night began.\n\nSometime soon after four Pitcairn decided that there was no use staying in New Florence any longer. He boarded his private car, and with the passengers back on board again, his train started for Pittsburgh. It was also about the same time that several of the newspapermen decided that if the McCartneys could make it across the mountain on foot, they could too, and they set off through the woods. With luck, they figured, they could be on the hill above the city by daybreak.\n\n## VII\n\n## In the valley of death\n\nIt was nearly morning when the strange quiet began. Until then there had been almost no letup from the hideous sounds from below. Few people had been able to sleep, and several of the war veterans were saying it was the worst night they had ever been through.\n\nBut in the last chill hour before light, the valley seemed to hang suspended in an unearthly stillness, almost as unnerving in its way as everything else that had happened. And it was then, for the first time, that people began to realize that all those harsh, incessant noises which had been such a part of their lives\u2014mill whistles screeching, wagons clattering over cobblestones, coal trains rumbling past day and night\u2014had stopped, absolutely, every one of them.\n\nAbout five the first dim shapes began emerging from the darkness. But even by six very little stood out in detail. There were no shadows, no clear edges to anything. Some survivors, years later, would swear it had been a bright, warm morning, with a spotless blue sky, which, after the night they had been through, it may well have seemed. But the fact is that though the rain had at last stopped, the weather on that morning of June 1 was nearly as foul as it had been the morning before. The valley looked smothered in a smoky gray film. The hills appeared to be made of some kind of soft, gray-green stuff and were just barely distinguishable from a damp, low sky that was the color of pewter. Odd patches of the valley were completely lost in low-hanging ribbons of mist, and the over-all visibility was reduced to perhaps a mile at best.\n\nStill the view that morning would be etched sharply in the memories of everyone who took it in. Along the Frankstown Road on Green Hill some 3,000 people had gathered. On the rim of Prospect Hill and on the slopes above Kernville, Woodvale, and Cambria City the crowds were nearly as big. Chilled to the bone, hungry, many of them badly injured, hundreds without shoes or only partly clothed against the biting air, they huddled under dripping trees or stood along narrow footpaths ankle-deep in mud, straining their eyes to see and trying hard to understand.\n\nSpread out below them was a vast sea of muck and rubble and filthy water. Nearly all of Johnstown had been destroyed. That it was even the same place was very difficult to comprehend.\n\nThere were still a few buildings standing where they had been. The Methodist Church and the B & O station, the schoolhouse on Adams Street, Alma Hall, and the Union Street School could be seen plain enough, right where they were meant to be. The Iron Company's red-brick offices were still standing, as was Wood, Morrell & Company next door. But everywhere else there seemed nothing but bewildering desolation. The only immediately familiar parts of the landscape were the two rivers churning toward the stone bridge, both still swollen and full of debris.\n\nFrom Woodvale to the bridge ran an unbroken swath of destruction that was a quarter of a mile wide in places and a good two miles long. From Locust Street over to the Little Conemaugh was open space now, an empty tract of mud, rock, and scattered wreckage, where before saloons, stores, hotels, and houses had been as thick as it had been possible to build them. Washington Street was gone except for the B & O station. Along Main, where a cluster of buildings and gutted houses still stood like a small, ravaged island, the wreckage was piled as high as the roofs of the houses.\n\nAt the eastern end of town all that remained between Jackson and Clinton was a piece of St. John's Convent. At the corner of Jackson and Locust the blackened rafters of St. John's Church were still smoking, and where the Quinn house had stood, fifty feet away, there was now only a jumble of rubbish.\n\nAcross the Stony Creek, Kernville had been swept clean for blocks. Virtually everything was gone, as though that whole section had been hosed down to the raw earth. The entire western end of town, near the Point, was now a broad, flooded wasteland. Every bridge was gone except the stone bridge and against it now lay a good part of what had been Johnstown in a gigantic blazing heap.\n\nBelow the stone bridge the ironworks, though still standing, looked all askew, with stacks toppled over and one of the biggest buildings caved in at the end as though it had been tramped on by an immense heel. Cambria City had been ravaged past recognition. At least two-thirds of the houses had been wiped out, and down the entire length of its main street a tremendous pile of mud and rock had been dumped.\n\nWith the light the first small groups of people who had survived the night down in the drowned city could be seen making their way across the debris, most of them heading for Green Hill, where dry ground could be reached without having to cross a river. From Alma Hall and the Union Street School they came in steady little batches, moving up and down over the incredible flotsam. Then, at the same time from Frankstown Road and other near hills, men started moving down into town. And as they came closer, the dim sweep of destruction began to take on a different look. Slowly things came into ever sharper focus.\n\nThe Morrell house could be seen with part of its side sheared off. Dr. Lowman's house stood alone on the park, the only big house still there, but its two-story front porch had been squashed and every window punched in. Colonel Linton's place on lower Main looked as though it had been blasted in two by dynamite, and the black span of an iron bridge was resting where the yard had been. Beyond, houses were dumped every which way, crushed, broken, split clean in half, or lying belly up in the mire, with their floor beams showing like the ribs of butchered animals.\n\nTelephone poles, giant chunks of machinery, trees with all their bark shredded off, dead horses and pieces of dead horses, and countless human corpses were strewn everywhere. \"Hands of the dead stuck out of the ruins. Dead everywhere you went, their arms stretched above their heads almost without exception\u2014the last instinct of expiring humanity grasping at a straw,\" wrote George Gibbs, one of the reporters from the Tribune.\n\nAnd now, too, all the litter of thousands of lives could be seen in sharp detail. Shattered tables and chairs, tools, toys, account books, broken dishes, chamber pots and bicycle wheels, nail kegs, bedquilts, millions of planks and shingles were thrown up in grotesque heaps ten, twenty, thirty feet high, or lay gently shifting back and forth in huge pools of water that covered much of the valley floor like a brown soup.\n\n\"It were vain to undertake to tell the world how or what we felt, when shoeless, hatless, and many of us almost naked, some bruised and broken, we stood there and looked upon that scene of death and desolation,\" David Beale wrote.\n\nThe flood and the night that had followed, for all their terror and destruction and suffering, had had a certain terrible majesty. Many people had thought it was Judgment Day, God's time of anger come at last, the Day of Reckoning. They thought that the whole world was being destroyed and not just Johnstown. It had been the \"horrible tempest,\" with flood and fire \"come as a destruction from the Almighty.\" It had been awful, but it had been God Awful.\n\nThis that lay before them now in the dismal cold was just ugly and sordid and heartbreaking; and already it was beginning to smell.\n\nRescue parties got to work bringing the marooned down from rooftops and went searching among the wreckage for signs of life. Men scrambled over piles of debris to get to the upstairs windows of buildings that looked as though they might fall in at any minute. They crawled across slippery, cockeyed roofs to squeeze through attic windows or groped their way down dripping back hallways where the mud was over their boot tops. It was treacherous work and slow going. Walls were still falling in and fires were breaking out.\n\nAt the stone bridge, gangs of men and boys, many of whom had been there through the night, were still working to free people trapped alive within the burning pile. Young Victor Heiser, who had succeeded in reaching solid ground after his night in a Kernville attic, had made his way down the west bank of the Stony Creek as far as the bridge, where, as he wrote later, \"I joined the rescue squads, and we struggled for hours trying to release them from this funeral pyre, but our efforts were tragically hampered by the lack of axes and other tools. We could not save them all. It was horrible to watch helplessly while people, many of whom I actually knew, were being devoured in the holocaust.\"\n\nAcross the whole of the valley the dead were being found in increasing numbers. And as the morning passed, more and more people came down from the hillsides to look at the bodies, to search for missing husbands and children, or just to get their bearings, if possible. They slogged through the mud, asking after a six-year-old boy \"about so high,\" or a wife or a father. They picked their way through mountains of rubbish, trying to find a recognizable landmark to tell them where their house or store had been, or even a suggestion of the street where they had lived. Or they stood silently staring about, a numb, blank look on their faces. Over and over, later, when the day had passed, people would talk about how expressionless everyone had looked and how there had been so few people crying.\n\nThere was some shouting back and forth among the men. People who had been separated during the night would suddenly find one another. \"What strange meetings there were,\" wrote one man. \"People who had hardly known each other before the flood embraced one another, while those who found relations rushed into each other's arms and cried for very gladness that they were alive. All ordinary rules of decorum and differences of religion, politics and position were forgotten.\"\n\nLone stragglers went poking about looking for only they knew what, many of them strangely clad in whatever odd bits of clothing they had been able to lay hands on. One man, hatless and with a woman's red shawl across his shoulders, came limping along in his stocking feet, using a piece of lath for a cane. He was looking for his wife, Mrs. Brinker, who, as he would soon discover, had survived the night inside the Methodist parsonage and who had long since given him up for dead.\n\nPeople recovered some pathetic belonging or other and carried it carefully back to high ground or began building little personal piles of salvage. There was no order to what went on, no organization, and not much sense. Most people were unable even to look after themselves; they were stunned, confused, trying, as much as anything, to grasp what had happened and what was left of their lives. Where they went from there was something they were not yet ready to think about. Many of them struck off into the country, with no special destination. They just kept walking for hours, looking for food or a dry place to lie down for the night, or, very often, just trying to put as many miles as possible between themselves and the devastated city. They were afraid of the place and wanted no more part of it.\n\nThe problems to be faced immediately were enormous and critical. People were ravenously hungry, most everyone having gone twenty-four hours or more without anything to eat, and now there was virtually no food anywhere. The few provisions uncovered among the ruins were nearly all unfit for eating, and what little else people had was given to the injured and to the children. Moreover, there was no water that anyone felt was safe to drink. Thousands were homeless, hundreds were severely injured. Mrs. John Geis, for example, little Gertrude Quinn's grandmother, had had her scalp torn off from her forehead back to the nape of her neck. Hundreds of others were dazed by lack of sleep or in a state of shock. Dozens of people, as a result of exposure, were already in the early stages of pneumonia. There was almost no dry clothing to be had and no medicines.\n\nPeople had no money, except what change they may have had in their pockets at the time the water struck, and even if they did, there were no stores left at which to buy anything. There was no gas or electric light. Fires were burning in a dozen different places, and no one knew when a gas main might explode. Every telegraph and telephone line to the outside world was down. Bridges were gone, roads impassable. The railroad had been destroyed. And with the dead lying about everywhere, plus hundreds of carcasses of drowned horses, cows, pigs, dogs, cats, birds, rats, the threat of a violent epidemic was very serious indeed.\n\nBut by noon things had begun to happen, if only in a small way. Rafts had been built to cross the rivers and to get over to those buildings still surrounded by water. People on the hillsides whose houses had escaped harm and farmers from miles out in the country began coming into town bringing food, water, and clothing. At the corner of Adams and Main milk was passed out in big tinfuls. Unclaimed children were looked after. A rope bridge had been strung across the Little Conemaugh near the depot, and, most important of all as it would turn out, up at the Haws Cement Works, on the hill at the western end of the stone bridge, several bedraggled-looking newspaper correspondents had established headquarters in a coal shed and were in the process of rigging their own wire down the river to Sang Hollow.\n\nThe men had reached Johnstown about seven in the morning, and like everyone else were cold, dirty, hollow-eyed from no sleep. There remains some question as to which of them arrived first, but William Connelly, who was the Associated Press correspondent in western Pennsylvania, Harry Orr, a telegraph operator for the A.P., and Claude Wetmore, a free-lance reporter working for the New York World, are generally given the credit. Others kept straggling in from New Florence through the rest of the day. But until nightfall the major stories were still being filed out of the little railroad crossing on the other side of Laurel Hill.\n\nNew Florence, Pa. June 1 . . . Seven bodies have been found on the shore near this town, two being on a tree where the tide had carried them. The country people are coming into the news centers in large numbers, telling stories of disaster along the river banks in sequestered places . . . The body of another woman has just been discovered in the river here. Only her foot was above the water. A rope was fastened about it and tied to a tree . . . R. B. Rogers, Justice of the Peace at Nineveh, has wired the Coroner at Greensburg that 100 bodies have been found at that place, and he asks what to do with them.\n\nThat afternoon, at three, a meeting was called in Johnstown to decide what ought to be done there. Every able-bodied man who could be rounded up crowded into the Adams Street schoolhouse. The first step, it was quickly agreed, was to elect a \"dictator.\" John Fulton was the obvious choice, but he was nowhere to be found, so it was assumed he was dead, which he was not. He had left town some days earlier and was at that moment, like hundreds of others, trying desperately to get to Johnstown.\n\nThe second choice was Arthur J. Moxham, a remarkable young Welshman who had moved to Johnstown a few years before to start a new business making steel rails for trolley-car lines. In the short time he had been there Moxham had about convinced everyone that he was the best newcomer to arrive in the valley since D. J. Morrell. His business had prospered rapidly, and it was earlier that spring that he had opened a sprawling new complex of mills up the Stony Creek beside the new town he had developed. He named the business the Johnson Steel Street Rail Company, after his lively young partner, Tom L. Johnson, who, in turn, had named the town Moxham. They paid their men regularly each week, in cash, and did not maintain a company store\u2014all of which had had a marked impact on the town's economic well-being and a good deal to do with their own popularity.\n\nBoth men were energetic, able executives. Both were already wealthy, and both, interestingly enough, were devout followers of the great economic reformer of the time, Henry George, and were equally well known in Johnstown for their impassioned oratory on George's single-tax scheme.\n\nMoxham was a fortunate choice. He took charge immediately and organized citizens' committees to look after the most pressing and obvious problems. Morgues were to be established under the direction of the Reverends Beale and Chapman. Charles Zimmerman and Tom Johnson were put in charge of removing dead animals and wreckage. (That anyone could have even considered cleaning up the mess at that point is extraordinary, but apparently the work began right away, against all odds, against all reason. Trying to bail the rivers dry with buckets would have seemed not much more futile.)\n\nDr. Lowman and Dr. Matthews were responsible for establishing temporary hospitals. Captain Hart was to organize a police force. There was a committee for supplies and one for finance, to which George Swank and Cyrus Elder were assigned.\n\nCaptain Hart deputized some seventy-five men, most of whom were employees of the Johnson Company sent down from Moxham. They cut tin stars from tomato cans found in the wreckage, threw a cordon around the First National and Dibert banks, and, according to a report made days later, recovered some $6,000 in cash from trunks, valises, and bureau drawers lying about.\n\nAs dusk gathered, the search for the living as well as the dead went on in earnest. There seemed to be no one who was not missing some member of his family. James Quinn had already found little Gertrude, but he was still looking for his son Vincent, his sister-in-law and her infant son, and Libby Hipp, the nursegirl, though he had little hope of finding any of them except his son. That Gertrude was alive seemed almost beyond belief.\n\nHe and his other daughters had been luckier than most and had spent the night in a house on Green Hill. At daybreak he had been outside washing his face in a basin when his sister, Barbara Foster, came running up shouting that she had found Gertrude. She had seen her on the porch of the Metz house, still speechless with fright, still unidentified, and almost unrecognizable with her blonde hair tangled and matted with mud, her dark eyes quick with terror. Quinn at first found it impossible to accept what he heard, but started off at a run, the lather still on his face, and the other little girls running behind.\n\n\"When he came near the house,\" Gertrude wrote later, \"I saw him and recognized him at once. I fairly flew down the steps. Just as he put his foot on the first step, I landed on his knee and put both my arms around his neck while he embraced me.\"\n\nQuinn gathered up the child. They both began crying. A small crowd had assembled by now, on the porch and on the street below, and the scene caused several people to break down for perhaps the first time. Then there was a lot of handshaking and Quinn set off with his children to find his son.\n\nVictor Heiser had spent most of the day searching for his mother and father, hoping against hope that somehow they had come through it all alive and in one piece. His own survival seemed such a miracle to him that he could not help feeling there was a chance they might be somewhere in the oncoming darkness looking for him.\n\nAt the bridge late in the afternoon an old man and his daughter were rescued from a house wedged among the burning wreckage. The old man made quite a reputation for himself when, on being helped down into a rowboat, he asked his rescuers, \"Which one of you gentlemen would be good enough to give me a chew of tobacco?\" And on the hillside a few hundred yards away two young ladies who had been stripped naked by the flood were found cowering in the bushes, where they had been hiding through the long day, too ashamed to venture out before dark.\n\nCyrus Elder's wife and daughter were missing. Horace Rose did not learn until late in the afternoon that the two sons, Winter and Percy, from whom he had been separated during the flood were still alive, and that he was the only member of his large family who had even been injured.\n\nHis neighbor John Dibert had already been identified among the dead, as had Mrs. Fronheiser, whom Rose had last seen in her window next door. The bodies of Samuel Eldridge, one of the best-known policemen in town, and Elizabeth Bryan of Philadelphia, who had been on the Day Express, had also been found. But of the other dead found only a small number had as yet been identified for sure.\n\nAt the Adams Street schoolhouse and a saloon in Morrellville, where the first two emergency morgues had been opened, the bodies were piling up faster than they could be properly handled. They came in on planks, doors, anything that would serve as a stretcher, and with no wagons or horses as yet on hand, the work of carrying them through the mud and water was terribly difficult.\n\nEach body was cleaned up as much as was possible, and any valuables found were put aside for safekeeping. Those in charge tried hard to maintain order, but people kept pushing in and out to look, and the confusion was terrific.\n\n\"We had no record books,\" David Beale wrote, \"not even paper, on which to make our records, and had to use with great economy that which we gathered amid the debris or happened to have in our pockets.\"\n\nOne way or other the bodies were numbered and identified, whenever that was possible. Many were in ghastly condition, stripped of their clothes, badly cut, limbs torn off, battered, bloated, some already turning black. Others looked as though they had suffered hardly at all and, except for their wet, filthy clothes, appeared very much at peace.\n\nA Harrisburg newspaperman named J. J. MacLaurin, who had been near Johnstown at the time the flood struck, described a visit to the Adams Street School early Saturday afternoon, where he counted fifty-three bodies stretched on boards along the tops of the desks. \"Next to the entrance lay, in her damp clothing, the waiter-girl who had served my last dinner at the Hulbert House, with another of the dining room girls by her side.\"\n\nHow many dead there were in all no one had any way of knowing, since there was, as yet, almost no communication between various parts of town. But wild estimates were everywhere by nightfall, and with more bodies being discovered wherever the wreckage had been pulled apart, it was generally agreed that the final count would run far into the thousands. Some were saying it would be as much as 10,000 by the time the losses were added up from South Fork to Johnstown, and few people found that at all hard to believe. What may have happened on down the river at Nineveh or New Florence or Bolivar was anyone's guess.\n\nWithin another day the Pennsylvania station and the Presbyterian Church, a soap factory, a house in Kernville, the Millville School, and the Catholic Church in Cambria City would be converted into emergency morgues. But it would be a week before things got down to a system at these places, and not for months would there be a realistic count of the dead. Actually, there never would be an exact, final count, though it is certain that well over 2,000 people were killed, and 2,209 is generally accepted as the official total.\n\nHundreds of people who were lost would never be found. One out of every three bodies that was found would never be identified beyond what was put down in the morgue records. With all the anguish and turmoil of the first few days, such entries were at best a line or two.\n\n. . .11. | Unknown.\n\n---|---\n\n|\n\nA female. \"FL.F.\" on envelope.\n\n. . .17. | Unknown.\n\n|\n\nA man about fifty years of age. Short hair, smooth face.\n\n. . .25. | Unknown.\n\n|\n\nFemale. Light hair. About fifteen years.\n\nLater, more care would be taken to be as explicit as possible.\n\n. . .181. | Unknown.\n\n---|---\n\n|\n\nFemale. Age forty-five. Height 5 feet 6 inches. Weight 100. White. Very long black hair, mixed with grey. White handkerchief with red border. Black striped waist. Black dress. Plain gold ring on third finger of left hand. Red flannel underwear. Black stockings. Five pennies in purse. Bunch of keys.\n\n. . .182. | Unknown.\n\n|\n\nMale. Age five years. Sandy hair. Checkered waist. Ribbed knee pants. Red undershirt. Black stockings darned in both heels.\n\n. . .204. | Unknown.\n\n|\n\nMale. Age fifty. Weight 160. Height 5 feet 9 inches. Sandy hair. Plain ring on third finger of left hand (with initials inside \"C.R. 1869.\") Pair blood stone cuff-buttons. Black alpaca coat. Navy blue vest and pants. Congress gaiters. Red stockings. Pocketbook. Knife and pencil. $13.30 in change. Open-faced silver watch. Heavy plaited chain and locket. Inside of locket a star with S.H., words trade-mark alone a star. Chain trinket with Washington head. Reverse the Lord's prayer. Odd Fellow's badge on pin.\n\nIn all, 663 bodies would be listed as unknown. A few were not identifiable because they had been decapitated. Close to a hundred had been burned beyond recognition, and some so badly that it was impossible even to tell what sex they had been. And many of the bodies found in late June or on into the summer and fall would be so decomposed as to be totally unrecognizable.\n\nPart of the problem, too, was the fact that on the afternoon of May 31 Johnstown had had its usual share of strangers in town, nameless faces even when they had been alive, foreigners who had been living there only a short time, tramps, traveling men new to the territory, passengers on board any one of the several trains stalled along the line, countrypeople who had decided to stay over after Memorial Day. They made up a good part of the unknown dead, and doubtless many of them were among those who were never found at all.\n\nAmong the known dead were such very well-known figures as Dr. John Lee; Theodore Zimmerman, the lawyer; Squire Fisher, the Justice of the Peace, and his entire family; C. T. Schubert, editor of the German newspaper; and Ben Hoffman, the hackman, who, according to one account, \"always got you to the depot in plenty of time\" and whose voice was \"as familiar as train whistle, iron works, or the clock bells.\" (Hoffman had gone upstairs to take a nap shortly before the flood struck and was found with his socks in his pockets.)\n\nThe Reverend Alonzo Diller, the new rector of St. Mark's Episcopal Church, was dead, along with his wife and child. George Wagoner, who was a dentist as well as a part-time preacher, and so one of the best-known men in town, was dead, as were his wife and three daughters. Emil Young, the jeweler, was dead; Sam Lenhart, the harness dealer, was dead; Henry Goldenberg, the clothier, Arthur Benshoff, the bookseller, Christian Kempel, the undertaker, were all dead. Mrs. Hirst, the librarian, lay crushed beneath a heap of bricks, slate, and books that stood where the public library had been.\n\nVincent Quinn was dead, as were Abbie Geis, her child, and Libby Hipp. Mrs. Cyrus Elder and her daughter Nan, Hettie Ogle and her daughter Minnie were dead, and their bodies would never be identified. George and Mathilde Heiser were dead.\n\nNinety-nine whole families had been wiped out. Three hundred and ninety-six children aged ten years or less had been killed. Ninety-eight children lost both parents. One hundred and twenty-four women were left widows; 198 men lost their wives.\n\nOne woman, Mrs. John Fenn, wife of the tinsmith on Locust Street, lost her husband and seven children. Christ Fitzharris, the saloonkeeper, his wife, father, and eight children were all drowned. Charles Murr and six of his children went down with his cigar store on Washington Street; only his wife and one child survived. In a house owned by John Ryan on Washington Street, twenty-one people drowned, including a man named Gottfried Hoffman, his wife and nine children.\n\nAt \"Morgue A,\" the Adams Street schoolhouse, 301 bodies would be recorded in the logbooks. At the Presbyterian Church, which was \"Morgue B,\" there would be 92; at \"Morgue C,\" in the Millville schoolhouse, the total would come to 551 by the time the last entry was made (\"Unknown\") on December 3. And along with the prominent merchants and doctors, the lawyers and preachers, there were hundreds of people with names like Allison, Burns, Evans, Shumaker, Llewellyn, and Hesselbein, Berkebile, Mayhew, McHugh, Miller, Lambreski, Rosensteel, Brown, Smith, and Jones. They made up most of the lists, and in the town directory that was to have been published that June they were entered as schoolteacher, porter, or drayman, clerk, miner, molder, barber, sawyer, dressmaker, or domestic. Dozens of them were listed as steelworker, or simply as laborer, and quite often as widow.\n\nIn that part of the valley through which the flood had passed, the population on the afternoon of the 31st had been approximately 23,000 people, which means that the flood killed just about one person out of every ten. In Johnstown proper, it killed about one out of nine.\n\nBut there were no statistics for anyone to go on that Saturday night. It would be weeks before even a reasonably accurate estimate would be made on the death toll. The business of finding the dead just went very slowly. Young Vincent Quinn's body, for example, was not uncovered until June 7, buried beneath the wreckage in Jacob Zimmerman's yard. Victor Heiser's mother was found about the same time, her clothing still much intact, her body scarcely marked in any way; but the search for George Heiser went on for weeks after, and his body never was identified for certain. Toward the end of June a body was found which Victor was told was his father, but it was by then in such dreadful condition that he was not permitted to look at it.\n\nIn July there would be many days when ten to fifteen corpses would be uncovered. About thirty bodies would be found in August, including that of little Bessie Fronheiser; and so it would go on through the fall. In fact, for years to come bodies would keep turning up in and near the city. Two bodies would be found west of New Florence as late as 1906.\n\nBut by dark that Saturday only a small part of the dead had been accounted for, perhaps no more than 300 or 400, and only a very few had been buried. Most of the living found shelter well back from the city, on Prospect Hill or Green Hill, or on up the Stony Creek, where against the dark mountains tiny windows glowed like strings of orange lanterns. Or they walked to little towns like Brownstown, which was set in a high valley above Cambria City. Victor Heiser spent the next several nights there, along with more than 1,000 other refugees from the flood who were all housed, one way or other, by Brownstown's fifty-three resident families.\n\nHouses, barns, stables, schools, churches, every remaining upright structure for miles around was put into service. Crude tents were fashioned from blankets and bedspreads. Lean-tos were built of planks and doors dragged from the wreckage.\n\nOne man later described smelling the odor of ham frying as he walked along the front street on Prospect Hill, and how he was invited into a small house \"filled with a strangely composed company.\" There were two or three women who had been just recently rescued, and who were \"pitiably pale, and with eyes ghastly at the flood horror.\" There was the hostess, who carried an infant on one hip, \"a divine, a physician, a lawyer, two or three merchants,\" and several others. The dining room was too small to hold everyone, so they ate in shifts, waiting their turn out on the front porch. Below them, almost at their feet it seemed, lay the devastated valley.\n\nThe cold was nearly as cruel as it had been the night before. Pitch-blackness closed down over the mountainsides that crowded so close; but across the valley floor bonfires blazed, torches moved among the dark ruins, and the rivers and big pools of dead water were lighted by the fire that raged on at the stone bridge.\n\nAnd with the deep night, for nearly everyone, came dreadful fear. There was the rational and quite justifiable fear of typhoid fever and of famine. It was entirely possible that a worse catastrophe than the flood itself could sweep the valley in a matter of days if help did not get through.\n\nThere were also rumors of thieves prowling through the night and of gangs of toughs who had come into the valley looking for trouble. Great quantities of whiskey were supposedly being found among the ruins, and drunken brawls were breaking out. People were warned to be on the lookout, that there would be looting and rape before the night was over; and men who had not slept since Thursday night took turns standing guard through the night, watching over their families or what little they may have had left of any earthly value.\n\nPerhaps worst of all, however, was the wholly irrational fear of the very night itself and the nameless horrors it concealed. The valley was full of unburied dead; they were down there among the cold, vile remains of the city, waiting in the dark, and no one could get that idea out of his head for very long. If there were such a thing as ghosts, the night was full of them.\n\nBut despite it all, the hunger, the grief, the despair and fear, people gradually did what they had to; they slept. They put everything else out of their minds, for the moment, because they had to; and they slept.\n\n\u2022 \u2022 \u2022\n\nSunday the weather eased off. It was still cold, but the sky had cleared some, and for the first time in days it looked as though there would be no rain.\n\nSunday they began taking bodies across the Little Conemaugh in skiffs and carrying them to a plot on Prospect Hill where shallow graves were dug in the gravelly soil and the bodies buried without ceremony. (George Spangler, who had been night watchman at the First National Bank, wrote in his diary, \"bisey holing the dead this day I hold 62 to the semitre.\") Sunday a post office was set up at the corner of Adams and Main, and a clearinghouse where everyone who was still alive was meant to come in and register his name and tell what he knew about the rest of his family. Sunday the first patients were cared for in a temporary hospital on Bedford Street. And on Sunday the first relief trains got through.\n\nA train from Somerset came in on the B & O tracks about daybreak. The other train, from Pittsburgh, had arrived at Sang Hollow about ten thirty Saturday night but had been unable to get any closer. From Sang Hollow to Johnstown there was practically nothing left of the old line. There were at least ten miles along the Pennsylvania where it was impossible to tell even where the tracks had been, and several of those ten miles could be accounted for between Sang Hollow and Johnstown.\n\nThe lonely little Sang Hollow depot had become the scene of great activity Saturday night, from about eleven on. Several boxcars had been unloaded and volunteers organized to start moving things upriver by hand. \"The men carried the provisions on their backs,\" one participant wrote, \"over landslides and the trackless roadbeds to points where handcars passed. All night long a procession of lights was moving to and fro from Sang Hollow to the stone bridge.\"\n\nBy morning nearly two carloads of supplies had been deposited at the western end of the bridge and work had begun on a rope bridge to get them over the Conemaugh. But more remarkable still was the fact that early Sunday, perhaps as early as eight in the morning, the Pittsburgh train itself came steaming up the valley clear to the stone bridge. So swiftly had the railroad swung back into action during the night that by dawn enough new track had been put down from Sang Hollow to start the train cautiously on its way. And as it crept through the ruins of Morrellville and Cambria City, men standing in the open doors of the boxcars passed out bundles of bread, cheese, and crackers to the ragged crowds that lined the tracks.\n\nThe supplies had left Pittsburgh about four Saturday afternoon. Pittsburgh had been in a frenzy since early that morning. The Allegheny River had risen sharply during the night, and the riverbanks and bridges were lined with people watching the wreckage sweep past. Already there were rumors that dead bodies had been fished out. \"A sense of intense uneasiness pervaded the air,\" one man wrote.\n\nThere were still precious few facts to go on, but the papers were getting out a new edition every hour, and the news kept growing more and more alarming. Outside the newspaper offices, traffic was snarled by the crowds that pressed in to read the latest bulletins and kept calling for names.\n\nAt one o'clock a mass meeting was held at Pittsburgh's Old City Hall, at which Robert Pitcairn stood up and spoke briefly about what he had seen. \"Gentlemen,\" he said in closing, \"it is not tomorrow you want to act, but today; thousands of lives were lost in a moment, and the living need immediate help.\" Then there was a call for contributions. At the front of the hall two men using both hands took in $48,116.70 in fifty minutes. \"There was no speech making,\" a reporter wrote, \"no oratory but the eloquence of cash.\"\n\nWagons were sent through the city to collect food and clothing. Union Station looked like wartime, swarming with people and with train after train being loaded in the yards. The first train went out with twenty cars full. On board were some eighty volunteers of the \"Pittsburgh Relief Committee,\" a dozen reporters, perhaps thirty police, and, according to one account, Mr. Durbin Home, a member of the South Fork Fishing and Hunting Club, who was on his way to find out what had happened to several friends and relatives who had not been heard from since they left for the lake on Memorial Day.\n\nWhen the rope bridge was finally finished Sunday morning at Johnstown, the men started over with their heavy loads, swaying precariously above the raging river. They came across one at a time and very slowly. And for the next several days, until the stone bridge was open again, they would keep on coming almost without stop.\n\nLater on Sunday several good-sized boats would be hauled up the valley by train and put into service crossing the river, taking men and supplies over and bringing refugees back. On Sunday the boats ferried some 3,000 passengers, coming and going. Monday, they carried 7,000, along with supplies and dead bodies.\n\nWagons loaded down with salt pork, bedding, goods of every kind, rolled down flood-gullied roads from Ebensburg and Loretto, splashing up showers of gummy mud the color of a new baseball glove. Doctors and work crews started off from Altoona, where it was reported 5,000 people were milling about the railroad station. In dozens of little towns along the Pennsylvania toward Pittsburgh, and back along the B & O toward Somerset, church bells were ringing and hundreds of people were coming in from the country with their donations; and all day, one after another, relief trains kept streaming through, many of them with \"For the Johnstown Sufferers\" scrawled in big letters on the boxcars. One train in by the Somerset line carried a whole shipment of tents sent by the governor of Ohio. Another Pittsburgh train, eleven cars long, carried nothing but coffins.\n\nSome of the offerings that were mounting up in Johnstown created more than a little amusement. In their eagerness to help, some people had not bothered to think much about what would be needed. One nicely tied bundle opened Sunday afternoon contained a ball of carpet rags, a paper of tacks, two bags of salt, one baby's shoe, and two darned stockings of different colors. A box of homemade liniment, with \"warm before using\" written on the side, was tossed out of one car. There was a package of worn-out schoolbooks, a Bible with several pages marked, some fancy needlework, even bits of bric-a-brac.\n\nBut almost everything else that came in, however shabby or trivial seeming, was immediately grabbed up and put to good use. A blue dress coat with bright brass buttons that looked every bit of seventy years old was presented to an equally ancient-looking Grubbtown man who wore it away with much pride. Children went shuffling off in shoes several sizes too big for them. Women gladly put on men's coats and hats.\n\nAnd as much as there was coming in, it was nowhere near enough. There were perhaps 27,000 people in the valley who had to be taken care of, who had to be supplied with every kind of basic necessity; and added to them were all those others streaming in to help.\n\nBy nightfall Sunday well over 1,000 people were in from out of town. Something like fifty undertakers had arrived from Pittsburgh. The railroad was bringing work crews in by the trainful. A Pittsburgh fire department had arrived and, remarkably enough, by midnight had just about extinguished the fire at the stone bridge. There was also present a rather stout Republican politician by the name of Daniel Hartman Hastings, the Adjutant General of the state, who, after looking the situation over since morning, had decided it was time the military took over.\n\nA lawyer by profession, the general's only military experience had been at Altoona during the strike of '77. Saturday morning he had hitched up his team and driven nonstop from his home in Bellefonte, seventy miles to the northeast, arriving at Prospect Hill after dark. He had slept that night in the company of several tramps on the floor of the signal tower at the Pennsylvania station and managed to cross over to Johnstown first thing the next morning. He talked to Moxham and his committeemen about calling out the National Guard but was advised strenuously against it. Moxham thought it was important that the people handle their problems themselves; it would do more than anything else, he said, to help them get over their anxieties.\n\nLater in the day, when a company of troops arrived from Pittsburgh, sent by the Pittsburgh Chamber of Commerce, Hastings told them to go back to where they had come from. They had received no proper orders to turn out, he said, and had no business being there. He gave the officer in charge a vigorous dressing down, and back they went.\n\nBut by nightfall another meeting was held with the local officials and it was agreed to draw up a formal request to the governor for troops. For by now it was clear to just about everyone that the job of running things had gone beyond what the Moxham \"dictatorship\" could cope with. In another two days Moxham would resign his authority altogether, and James B. Scott, head of the Pittsburgh Relief Committee, would take over as the civilian head.\n\nRumors of looting and drunken fist fights were now even more exaggerated than they had been the previous night, but now they were not totally unfounded. The Reverend Beale and others later testified to witnessing attempted thefts. On Prospect Hill there seemed to be an inexhaustible supply of whiskey. One husky farm boy who had come down from Ebensburg with a load of provisions stayed long enough to get so drunk that he toppled off the hillside, rolled head over heels down the embankment, and fell into the Little Conemaugh, nearly drowning in minutes. \"God only saved him,\" his father said later, \"and for something better we hope.\"\n\nCaptain Hart's police seemed unable to keep order, and if things were not troublesome enough as they were, one of his lieutenants, a much-respected local lawyer and sportsman named Chal Dick, went riding about on horseback brandishing a Winchester rifle and telling lurid stories about the Hungarians he had seen robbing the dead and how he had already shot a couple of them. The stories spread like wildfire, and with them went more fear and suspicion of any man who spoke with an accent or even looked slightly foreign. People talked of how Paris had been looted by the Germans during the Franco-Prussian War, or harked back to tales of violence and evil doings in the old country at the time of the plagues.\n\nAnd to make matters still worse, it was well known that even more people were on their way. Word was Sunday night that Booth and Flinn, the big Pittsburgh construction company, was sending 1,000 men the next day, and everyone had heard about the kind of riffraff Billy Flinn was known for hiring. He would pack them into freight cars like cattle and then turn them loose into Johnstown. Every last one of them would have to be fed and sheltered, and where was it all to come from? And who was there to police the place?\n\nBut what was not known, even as Hastings sent out his message to the governor, was just how many others were heading for the devastated city. For along with the Flinn crew there were thousands more coming\u2014charity workers, doctors, preachers, men looking for work, smalltime crooks and pickpockets, drifters, farm hands, ladies of the W.C.T.U., former Johnstown people heading back to look for relatives, railroad officials, prostitutes, sight-seers.\n\nFrom Pittsburgh Captain Bill Jones was on his way with three carloads of supplies and a small army of 300 men from the Edgar Thomson works, a number of whom had been with him in the old days at the Cambria mills. In Philadelphia pretty society girls were packing medical supplies and making ready to start off with relief units organized by half a dozen churches. Mr. H. C. Tarr of the Utopia Embalming Fluid Company of Brooklyn had already struck out for Johnstown and would wind up traveling nearly 200 miles by horseback before he got there. In Washington, Miss Clara Barton and her newly organized American Red Cross had boarded a special B & O train.\n\nFor, already, the Johnstown Flood had become the biggest news story since the murder of Abraham Lincoln. On Saturday night, quite late, the reporters camped inside the brickworks had finally gotten a line clear to Pittsburgh, and the words had been pouring out ever since. (\"The awful catastrophe at Johnstown is by all odds the most stupendous fatality ever known in the history of this country. . . .\")\n\nThe news had an effect that is difficult to imagine; by Sunday it was spread across the front page of virtually every paper in the country. On Saturday the papers had hedged on how many had been killed. The New York World had reported 1,500 lives lost; the Times had been even more cautious, saying only that hundreds were dead. But on Sunday the World headlines ran halfway down the page, and though they still had no firsthand facts to go on, the editors had decided to pull out all stops:\n\n10,000 DEAD\n\nJohnstown Blotted Out by the Flood\n\nHALF ITS PEOPLE KILLED\n\nTwo Thousand Burned to Death in the Wreck\n\nALL APPROACHES CUT OFF\n\nIn Pittsburgh the Post-Gazette was selling its editions so fast that it had to reduce its page size temporarily in order not to run out of paper. Everywhere people were talking of little else and wanted to know more, much more; they wanted facts, names, details, pictures. And so along with all the others heading for Johnstown there came more reporters (perhaps a hundred or more), telegraph operators, editors, authors, artists, photographers.\n\nThe great rush to Johnstown, which had begun in Pittsburgh Friday night, was now under way full force. They came, thousands of them, from every station in life and from as far away as California, heading for a place very few of them had ever heard of two days earlier, driven by the most disparate motives imaginable.\n\n## VIII\n\n## \"No pen can describe . . .\"\n\n### -1-\n\nHenry S. Brown of the Philadelphia Press had been sitting at his desk at eleven o'clock Friday night when the news first came in. At 11:25 he was on board a westbound train pulling out of the Broad Street Station, having taken no time to pack or, for that matter, to give much thought to just where it was he was going or what chance he had of getting there. At Harrisburg the train was delayed by floods along the Susquehanna, but Brown stayed on board when the conductor assured him everything would be cleared up in a few hours and that they would be moving on again. At dawn he was told things had changed rather drastically; nothing would be open west for two weeks.\n\nBrown got hold of some maps and decided that if he could get a train to Chambersburg, fifty miles to the southwest, and could hire a team there, he might just be able to drive the rest of the way, which, according to the map, looked to be another hundred miles. He took the Cumberland Valley Railroad out of Harrisburg, but it was not until Sunday afternoon that he reached Chambersburg, located a double team, and started over the Tuscarora Mountain to McConnellsburg, twenty-two miles due west. Halfway over the mountain his wagon broke down, but he managed to borrow another from a farmer. At McConnellsburg he picked up another team and pushed on, along the Pennsylvania Pike (the old Forbes Road), heading for Juniata Crossing. From then on he splattered his way down washed-out roads, forded streams where bridges had been swept away, walked when he had to, crossed Sideling Hill in the dark, changed teams five more times, and never stopped to eat or rest. He reached Bedford about seven Monday morning and, finding no train there as he had expected, went whirling off once again, this time bound for Stoystown behind a pair of snow-white mules. Between Bedford and Stoystown, still traveling the state road, he managed to cross the Allegheny Mountain at a place where the elevation approaches 3,000 feet.\n\nAt Stoystown he would be able to pick up the B & O line from Somerset, but he arrived just in time to miss a relief train there; so rather than wait for another, he hired still one more team and headed on again, following the tortuous route of the raging Stony Creek down to Johnstown.\n\nIt was about seven thirty Monday night when he finally reached Johnstown, after having traveled the hundred miles from Chambersburg in about twenty-eight hours. No more than ten minutes later he was shaking hands with another correspondent by the name of F. Jennings Crute, also of the Press. Crute had left the Philadelphia office at the same time as Brown and had pulled into Johnstown only an hour before, having traveled about seven times as far as Brown. For instead of heading west on the Pennsylvania Friday night, Crute had made the whole trip by rail, first by heading east to New York, then going by way of Buffalo (on the Central) to Cleveland and Pittsburgh.\n\nBrown and Crute went directly to work and, like the other reporters swarming over the place, were soon filing their stories from the brickworks above the stone bridge, which by now had become quite a center of operations. The Pittsburgh papers, the Times, the Press, the Dispatch, the Commercial-Gazette, and the Leader, were all represented. (In an old photograph taken at the end of the stone bridge on Sunday, a group of twenty-one Pittsburgh correspondents pose proudly beneath derby hats, several with cigar in hand, their dark vests crossed by heavy watch chains.) They had taken over two floors of one building, as well as a woodshed. The newcomers squeezed in where best they could, everyone working under tremendous difficulties. Those who had been there for more than twenty-four hours were unshaven, red-eyed, and near collapse from lack of sleep. They were using barrelheads, coffin lids, and shovel bottoms for writing desks, and the words they wrote were put on the wire as fast as was humanly possible.\n\nThe place became known as the \"Lime Kiln Club\" and rapidly gave rise to that special kind of fellowship-through-duress, which so often happens in war. \"The culinary department,\" one of the group wrote later, \"was taken charge of by Tom Keenan of the Press. With an old coffee-pot taken from the debris at the bridge, some canned corned beef, a few boxes of crackers, a few quarts of condensed milk and a bag of unground coffee, he was soon enabled to get up a meal for his starving comrades which was the envy of those in the neighborhood who, while hungry, did not belong to the band of scribes, whom they looked upon as a lot of luxurious revellers.\"\n\nBy late Monday the force of telegraph operators had increased enough to set up night and day shifts. Food became more plentiful, and the presence of the new men did much to boost spirits. The New York Sun reporters had come by the same roundabout route as F. Jennings Crute, while their rivals from the Herald, World, Times, and Tribune had gone more or less the way of Henry Brown. The correspondent of the Chicago Inter-Ocean walked up from Sang Hollow, as did several others, and every one of them was brimful of tales of his experiences.\n\nThe early arrivals at last got some sleep that night, there at the brickworks, while the newcomers found what accommodations they could elsewhere around town. Eight of them, including the Philadelphia men, wound up on the narrow first floor of the signal tower across the river. About midnight they were awakened by a man at the door saying, \"Isn't this terrible. Look at them, human beings, drowned like rats in their hole.\" At which point one of the corpses sat bolt upright and said, \"Get the hell out of here and let us sleep!\"\n\nBut for all the boon companionship and oft-told stories, the hardships endured by \"the gentlemen of the press\" were considerable. Vile-smelling smoke from the still smoldering bridge blew through the windows of the old building where they worked. The floor was shaky and full of holes, and to enter the place in the dark of night was, as one man said, \"to place one's life in jeopardy.\" John Ritenour of the Pittsburgh Post fell twenty feet, wedging between timbers and so severely injuring himself that he had to be sent home. Sam Kerr of the Leader fell off the top of a house lodged in the drift and would have drowned if one of his colleagues had not been on hand to pull him out. Clarence Bixby of the Post fell from the railroad bridge while trying to get across at one in the morning and was badly banged up. And several weeks later, F. Jennings Crute, worn down by lack of sleep and exposure, caught a cold that turned to pneumonia. On December 3 he died.\n\nThe competition between papers was friendly but fierce, with every man scrambling for an advantage. One of them, a William Henry Smith of the Associated Press, had actually been on board a section of the ill-fated Day Express and wrote a long, florid description of the experience. (\"It was a race for life. There was seen the black head of the flood, now the monster Destruction, whose crest was raised high in the air, and with this in view even the weak found wings for their feet.\") But for the rest it was a matter of finding out what was happening amid the chaos around them, and as of Monday night there was plenty happening.\n\nThe city itself was still the most overpowering spectacle. (\"It is a scene that blanches the faces of strong men, and in its multiplying horror is almost beyond description,\" wrote a reporter for the New York Daily Graphic.) The weather had turned dull and cold again, which was unpleasant but welcome news as far as the doctors and sanitation workers were concerned. This way the dead would not decay quite so fast.\n\nBonfires by the hundreds were blazing across the valley where the ungainly and by now putrid carcasses of drowned horses were being cremated. The stench everywhere was terrible, of burned plaster and sodden bedding, of oil-soaked muck, of water thick with every kind of filth, and, worst of all, of still unfound bodies. The correspondents wrote of negotiating the rope bridge over the Conemaugh (\"A slide, a series of frightful tosses from side to side, a run, and you have crossed . . .\") and of the curious things to be found once in town (\"In the midst of the wreck a clothing store dummy, with a hand in the position of beckoning to a person, stands erect and uninjured.\"). They interviewed bystanders (\" 'I have visited Johnstown a dozen times a year for a long time,' said a businessman to-day, 'and I know it thoroughly, but I haven't the least idea now of what part of it this is. I can't even tell the direction the streets used to run.' \"); and they quoted General Hastings as saying that there were 8,000 people dead. (\"Nobody thinks this too small,\" the Sun reporter added. \"Nobody who has been about here an hour would think anything too awful to be possible.\")\n\nSunday night four enormous relief trains had rolled in below the bridge. Monday Billy Flinn brought in 280 teams of horses and 1,300 men. (\"Very few Americans among them,\" wrote one reporter.) Mrs. Lew Wallace, wife of the war hero and novelist, was reported missing from the Day Express. (She had actually taken another train and was safe in Altoona.) John Fulton and Colonel John Linton were both mistakenly reported dead (Fulton was reported \"positively drowned\"), and James McMillan, vice-president of Cambria Iron, was asked when work would start on rebuilding the mills, to which he answered, \"Immediately.\" There was talk of dynamiting the wreckage at the stone bridge, and there was a strong plea from the doctors and the sanitation officials from Pittsburgh to let it burn. The smell of burning flesh among the wreckage was something awful (\"People in New York who remember the smell of the ruins of the Belt Line stables, after their destruction by fire . . . know what the odor is.\"), but fire would cut the odds against a typhus outbreak, and throughout the valley and on downriver, clear to Pittsburgh, typhus had become an overriding concern.\n\nIn Pittsburgh the papers urged everyone to boil his water. From Nineveh, where nearly a hundred bodies had been recovered, Dr. Benjamin Lee, head of the Pennsylvania Board of Health, sent a message to the sheriffs of the four counties between Johnstown and Pittsburgh:\n\nThe State Board of Health hereby directs and empowers you to immediately summon a posse to patrol the Conemaugh river, tear down the drift heaps and remove the dead bodies, both human beings and domestic animals. This is absolutely necessary to protect your county from pestilence.\n\nThe wreckage at the bridge was described in detail, with some saying it covered thirty acres, others claiming it was more like sixty. (It was about halfway in between.) \"I stood on the stone bridge at 6 o'clock,\" wrote a Sun reporter Monday, \"and looked into the seething mass of ruin below me. At one place the blackened body of a babe was seen; in another 14 skulls could be counted . . . At this time the smoke was still rising to the height of 50 feet . . .\" On Wednesday, June 5, a little boy named Eddie Schoefler would be found still alive amid the wreckage. It would be one of the momentous events of the week.\n\nThen, from Sunday on, there had been increased tension over the Hungarians, which was something quite colorful indeed to write about. Thanks to Chal Dick and, by now, many others, tales of \"foul deeds\" perpetrated by the \"fiendish Huns\" were rampant, and only a few reporters bothered to try to check them out. Story after story went on the wire describing how \"ghouls, more like wild beasts\" were slicing off fingers for gold wedding bands, and how angry Johnstown vigilantes were hunting them down. One account described how a woman's body had been decapitated in order to steal her necklace. The Post told how gangs of Hungarians tried to raid unguarded freight cars for food and clothes. Another report said that a Hungarian had been caught in the act of blowing up a safe in the First National Bank. The Daily Graphic described how a crowd cornered a Hungarian at his \"fiendish work\" and strung him up on a lamppost.\n\nThis sample of the over-all tone and content of the reports was written late Sunday:\n\nLast night a party of thirteen Hungarians were noticed stealthily picking their way along the banks of the Conemaugh toward Sang Hollow. Suspicious of their purpose, several farmers armed themselves and started in pursuit. Soon their most horrible fears were realized. The Hungarians were out for plunder. They came upon the dead and mangled body of a woman, lying upon the shore, upon whose person there were a number of trinkets of jewelry and two diamond rings. In their eagerness to secure the plunder, the Hungarians got into a squabble, during which one of the number severed the finger upon which were the rings, and started on a run with his fearful prize. The revolting nature of the deed so wrought upon the pursuing farmers, who by this time were close at hand, that they gave immediate chase. Some of the Hungarians showed fight, but, being outnumbered, were compelled to flee for their lives. Nine of the brutes escaped, but four were literally driven into the surging river and to their death. The thief who took the rings was among the number of the involuntary suicides.\n\nThe \"thugs and thieves in unclean hordes,\" as one writer described them, were nearly always Hungarians, though there was at least one report of two Negroes being shot at by Pittsburgh police when seen robbing a dead body, and there were a few references to \"the worthless Poles.\"\n\nSuch accounts were given a great deal of space by all but a few of the big eastern papers and were featured prominently in the headlines. (\"FIENDS IN HUMAN FORM\" ran the New York Herald headline on Monday. \"DRUNKEN HUNGARIANS, DANCING, SINGING, CURSING AND FIGHTING AMID THE RUINS.\") Lurid illustrations were published, drawn by artists who had only the reporters' stories to go by. One scene showed two bodies dangling from a telephone pole near the riverbank, while in the foreground a \"wild-eyed\" Hungarian, who looks much like a touring company Fagin, is held at bay, knee-deep in water, by a stalwart gentleman with a horse pistol who could very well be Robert E. Lee.\n\nThey were stories which had great appeal to anyone ready to believe in the darker side of humanity and particularly that segment of humanity which spoke with a thick accent, smelled of garlic, and worked cheap. The only trouble was that there was scarcely any truth to the stories, as several correspondents had already begun to suspect. At four Monday afternoon Alfred Reed of the World cabled his editors:\n\nNO LYNCHINGS. I WARNED YOU LAST NIGHT NOT TO PRINT WILD RUMORS, AND AM GLAD TO HEAR YOU HAD ENOUGH CONFIDENCE IN ME TO HOLD OUT SUCH STORIES.\n\nThe next day an angry General Hastings issued a statement that reports of lynchings and rioting were \"utterly devoid of truth,\" sharply criticized the newspapers for publishing them, and suggested that the reporters stick to the facts.\n\nSeveral characters had indeed been caught trying to pilfer the dead and had received some rather rough treatment, including, it appears, enough mock preparations for a lynching to put a terrific scare into one of them; and it is quite possible that a few fingers may have been mutilated by thieves trying to wrench off gold wedding bands. But there were certainly no diamond rings stolen (one survivor doubted that there were more than one or two diamond rings in all Johnstown at that time), no bank safes were blown, and, as David Beale wrote later, no fingers were cut off by human ghouls. Furthermore, the Hungarians themselves apparently had almost nothing to do with what foul doings there were. \"There was little stealing done by the Hungarians,\" Beale wrote, \"and most accounts of outrages attributed to these people were apocryphal; and I am glad to say that all statements of shooting and hanging them were without foundation.\" And to emphasize the validity of this last statement, he said his source was Chal Dick himself.\n\nDick, it seems, had gone slightly out of his head immediately after the disaster and had been suffering from vivid and vicious delusions. His wife and children had been killed and he simply went berserk for about a day or so. By the time he snapped out of it, the damage had been done, and from then on the stories were spread, according to the best evidence, largely by outsiders who had come into the valley.\n\nFor though there may have been relatively little resentment in Johnstown against the Hungarians (or the other Southern European peoples called Hungarians), in Pittsburgh feelings were different. The steel bosses, like Henry Clay Frick, had been bringing them in by the thousands to work in Braddock and Homestead. They were single men mostly, willing to work for the lowest wages, and under the worst conditions, just to save enough to go back home and buy a small farm on the Danube. They got the toughest jobs, worked hard, and were generally hated by the Irish, the German, and American workers. Years later, John Fitch, the historian, interviewed an old Scotch-Irish furnace boss in Pittsburgh about the \"hunkies.\"\n\n\"They don't seem like men to me hardly,\" he said. \"They can't talk United States. You tell them something and they just look and say, 'Me no fustay, me no fustay,' that's all you can get out of 'em.\" When wages were going down, when men were let go at the mills, when the unions suffered setbacks, somehow the Hungarians seemed at the root of things.\n\nThe few Hungarians there were in Johnstown (perhaps 500 of them were living in the valley at the time of the flood) were subjected to days of abuse. Speaking little English, fearful and suspicious even under normal circumstances, they now became so terrified of the angry crowds that hung outside their homes that they dared not go out even to collect their share of the relief provisions. Their children were starving; the men grew desperate. At one point about twenty of them were encouraged to come out to help dig graves in the cemetery above Minersville. After working all day, on their way back home in the dark, they were set upon by a gang armed with clubs and were badly beaten.\n\nBut by midweek the Hungarian scare was over. There were still rumors, but papers like the Philadelphia Press were saying, \"There is not an inch of truth in them,\" and nearly everyone in Johnstown knew that that was so. While in Chicago the Herald wrote that the \"Magyars\" there were \"justly indignant\" over the stories, and tried to resolve the whole unfortunate business by adding, \"The wretches who now prey upon the dead at Johnstown and refuse to aid in the work of rescue, are undoubtedly Bulgarians, Wallachs, Moldavians, and Tartars, classes degraded in all their manners as is the North American Indian. . . .\"\n\n\u2022 \u2022 \u2022\n\nSometime Monday Colonel Unger came into town from South Fork, accompanied by the Shea brothers, John Parke, and one or two other employees of the fishing and hunting club. Understandably, the press was most interested in talking to them.\n\nUnger gave the Pittsburgh Post a brief rundown on what had happened at the dam Friday morning and how he and his men had tried to prevent the disaster. He estimated that the loss to the club was about $150,000 and said that the club members who had been at the lake were all safe and that they had gone off to Altoona.\n\nParke seemed more interested in getting word to his people in Philadelphia that he was alive and in good health, but was quoted by the New York Sun, \"No blame can be attached to anyone for this greatest of horrors. It was a calamity that could not be avoided.\" He said the fault was \"storm after storm\" and that \"by twelve o'clock everybody in the Conemaugh region did know or should have known of their danger.\"\n\nBut an employee by the name of Herbert Webber, who must have been interviewed separately, launched into a long description of the dam the morning before it failed. He told the reporters that at around eleven he had been attending to a camp a mile back from the dam when he noticed that the surface of the lake seemed to be lowering. He could not quite believe what he saw, he said, so he went down and made a mark on the shore, and sure enough he found his suspicions were well founded. For days before, he went on, he had seen water shooting out between the rocks on the front of the dam, so that the face \"resembled a large watering pot.\" The force of the water was so great \"that one of these jets squirted full thirty feet horizontally from the stone wall.\" When he ran up to the dam that morning, he declared, he saw the water of the lake \"welling out from beneath the foundation stones.\"\n\nThe story was preposterous, of course, and had no connection with what actually happened, but the reporters had no way of knowing that. The watering-can image made splendid copy, so out it went, along with everything else.\n\nBut by this time at least one enterprising reporter had already made his way to the club. In Pittsburgh that Monday the headline across the front of the Post read: \"TO THE DAM AT LAST.\" The story had been sent out at nine the night before and said, as Unger had, that the Pittsburgh people were safe and, as Parke had, that warnings had been sent down the valley before the break. In another two days more reporters would show up at South Fork. They would begin looking over the construction of the dam itself and start questioning the local people about the club. South Fork would shortly become the center of a stormy series of events, but for now Johnstown remained the major focus of attention.\n\nOne survivor after another was interviewed and dozens of frightful personal experiences were penciled into reporters' notebooks. The heroism of Bill Heppenstall (Hepenthal several papers spelled it), the adventures of Gertrude Quinn, and John Hess's ride into East Conemaugh were described at length.\n\nNumerous stories were collected of ironic or incredible things that happened. All of Johnstown's three or four blind people had survived the flood. Frank Benford's dun-colored mare was found in an alley next to where the Hulbert House had been, up to her belly in debris, alive, but blinded in both eyes. Old Mrs. Levergood, widow of Jacob Levergood, whose father had owned the town way back in the early days, was found dead, all the way up at Sandy Vale, still seated in her rocking chair.\n\nThen there was the story about the engineer at the Cambria works who early in the afternoon of that fateful Friday had started a letter to an old college friend, \"Thank God, I am through with a day such as I hope never to pass again.\" A \"gay girl\" of the town was said to have jumped from a hotel window during the very worst of the flood in a fatal effort to save a drowning child. There was a Newfoundland dog that supposedly hauled a Woodvale woman to safety and then swam back to save a drowning baby. And another dog, a water spaniel named Romeo, was said to have towed his mistress, Mrs. Charles Kress, to the windows of Alma Hall. And just above the hideous pileup at the stone bridge, on a billboard at the depot, there was a large poster, undamaged by the flood, which several reporters made a point of mentioning. Put there a few days before the flood to announce the arrival of Augustin Daly's A Night Off, its very large headline read, \"Intensely Funny.\"\n\nAmong the best pieces describing the human condition in Johnstown during these days were several by a late-arriving cub reporter for the Philadelphia Press whose name was Richard Harding Davis. The son of two prominent Philadelphia literary figures, he was strikingly handsome, twenty-five, elegant, aloof, and loving every minute of his first real assignment. At Lehigh he had been the most popular figure on campus, despite what one of his classmates described as his \"strict adherence to everything English in the way of dress and manner.\" His first job had been on the Philadelphia Record where he sported a long, yellow ulster, carried a cane, and was rather hard for the old newspapermen to swallow. As it was, he only lasted three months. He was caught one day by the city editor writing up an assignment with his kid gloves on and was promptly fired on the spot.\n\nBut with the Press he had fared better. He had sold a few short stories, interviewed Walt Whitman, and sent some samples of his work off to Robert Louis Stevenson with a request for advice. Stevenson advised writing \"with considerate slowness and on the most ambitious models.\" The slowness Davis would never quite master, nor would he try really; and the only ambitious model he ever seems to have set his sights on was his own very clear picture of himself as the world's most dashing and celebrated foreign correspondent.\n\nAt the time the Johnstown story broke, Davis had been on vacation. It took several days for him to persuade the paper to send him, and when he finally arrived, he got off to a characteristic start. \"A Philadelphia reporter was sent here to finish up the disaster, but the disaster is likely to finish him,\" wrote The New York Times man. On alighting from his train, Davis \"paralysed Newspaper row\" by asking for the nearest restaurant. When it was explained that everyone had to forage on the country, Davis wanted to know where he could hire a horse and buggy, which set off another round of laughter. \"But,\" concluded the Times man, \"he capped the climax by asking where he could buy a white shirt. A boiled shirt here is as rare as mince pie in Africa.\"\n\nBoiled shirt or no, he went to work, concentrating on human-interest stories. He wrote of walking over thousands of spilled cigars and of a pretty, young relief worker named Miss Hinkley of Philadelphia who was\n\n. . . sitting busily writing at a table beside an open window which looked out on the yard of the morgue, and in which forty odd coffins filled with the dead were being examined by the living. Miss Hinkley's hair was as carefully arranged and her tailor-made gown as neat and fresh as if she had stepped that moment from the Quaker City's Rittenhouse Square. Reporters became painfully conscious of clothes that have been slept in for seven nights, and chins that had forgotten razors.\n\nHe wrote about a fist fight down in Cambria City between a local deputy sheriff and a drunken National Guard lieutenant named Jackson, who was put under arrest and sent back to Pittsburgh after several bystanders, including Davis, stepped in to break things up. He described the offers coming in from people all over the country who wished to adopt a Johnstown orphan, and told the story of a man named John McKee, whose body had been found inside a cell of the town jail. McKee had been locked up for twenty-four hours for overcelebrating on Memorial Day.\n\nAnd along with the reporters, working their way among the ruins, came the photographers, lugging their ponderous, fragile equipment. They made pictures of men standing on freight cars in the midst of Main Street, of wagons loaded down with coffins, and of the great barren mud flat where Woodvale had been. The monstrous debris that clogged the city was pictured from virtually every angle, and at least one photographer decided to improve slightly on his composition by having a man lie down and play dead in the foreground. The picture later became one of the most popular stereoptican views of the disaster. But by the time the photographers were about, any body so exposed would have been long since found and removed; moreover, the shirt on the man's back looks a bit too neat and clean and the things around him are a little too nicely arranged.\n\nUpturned houses, gangs of laborers carrying shovels and axes and threading their way through huge dunes of rubbish, like a drab, derby-hatted army moving through the remains of a fallen city, the jerry-built shelters on the hillsides, farm women in poke bonnets working at the commissaries, they all made splendid subjects. But the most popular subject by far was a house owned by John Schultz which had stood on Union Street but was now stranded at the east end of Main. It had been pitched up on its side, and through an upstairs window a gigantic tree had been driven, its roots jutting thirty feet into the air. The building looked as though it had been skewered by some terrible oak-flinging god. One by one men and boys would crawl out on the tree to sit for a portrait, their faces registering no emotion, their feet dangling in light that had little more color than it would have in the final printed photograph. Six people had been in the house when the water struck, and they had all come out alive.\n\nAt one point it was estimated that there were no less than 200 amateur photographers about town, enough in any case that they had become a serious nuisance. So the word went around that if you were an able-bodied man but had no official business in town, then you had to work if you wanted to stay on. It was, as one observer said, a policy which had a \"most salutary effect.\"\n\nHarper's Weekly, Frank Leslie's, and some of the other picture magazines had sent artists to cover the story. There were two or three writers gathering material for quick books. And in New York the World even managed to get Walt Whitman, who had celebrated his seventieth birthday on the day of the flood, to write a poem which was promptly printed on page one.\n\nA voice from Death, solemn and strange, in all his sweep and power,\n\nWith sudden, indescribable blow\u2014towns drown'd\u2014humanity by thousands slain,\n\nThe vaunted work of thrift, goods, dwellings, forge, street, iron bridge,\n\nDash'd pell-mell by the blow\u2014yet usher'd life continuing on, . . .\n\nAnd in the Denver, Boston, and Brooklyn papers, long excerpts were published from a novel called Put Yourself in His Place, which had been written by a well-known English author named Charles Reade nearly twenty years before. Its closing chapters described the bursting of a reservoir and a dreadful flood which were surprisingly similar to what had happened at Johnstown.\n\nReade had based his book on the failure of the Dale Dyke at Sheffield, England, which had taken 238 lives in 1864; but for the millions of Americans who now read the excerpts, the \"Hillsborough\" of his story, with its steel mills and coal mines \"fringed by fair woods,\" and its reservoir in the mountains to the east, seemed so like Johnstown as to be uncanny. The story told of people dreaming of floods, of workmen who had long \"misliked\" the foundation of the dam. When the break came, it was only after a storm had raised the level of the lake so far that it started flowing over the center of the dam. Then down the valley came \"an avalanche of water, whirling great trees up by the roots, and sweeping huge rocks away, and driving them, like corks, for miles.\"\n\nMeanwhile, the headlines blared away, day after day. \"Agony\" . . . \"WOE!\" . . . \"PESTILENCE!\" . . . and by midweek, \"DEATH GROWS\u2014A GIANT! One Pervading Presence Throughout the Conemaugh Valley, FIFTEEN THOUSAND CORPSES, A Tale of Grief That Can Only Be Told in Bitter Tears, Another Day of Utter Despair.\"\n\nThe phrase \"no pen can describe . . .\" kept cropping up again and again, but the pens kept right on describing. The story took up the entire front page of both The New York Times and the World for five straight days. The Boston Post carried little else on its front page for twelve days running. It was called \"The Great Calamity,\" \"The Nation's Greatest Calamity,\" \"The Historic Catastrophe.\" Frank Leslie's said outright, \"It is the most extraordinary calamity of the age.\" Great battles had destroyed more life, said one writer after another, but no battle left such a ghastly trail of horror and devastation. That such a thing had happened in the United States of America in the year 1889 seemed almost more than the editorial writers could accept. Several papers, including Frank Leslie's, allowed that similar slaughter might occur in India or China or other remote lands \"where human life is cheap,\" but how in the world had it ever happened here?\n\nAll over the country newspapers published column after column of names of the dead. An extraordinary amount of space was given over to telling of the \"Slaughter of the Innocents.\" The bodies of women and children found among the wreckage or in the deep, flood-dumped silt downstream were described in grim detail. Every reporter at Johnstown it seems saw at least one dead mother still clutching her dead child, and much was made of the fact that more women died in the disaster than men. (Of the bodies finally recovered 923 were men, 1,219 women.)\n\nVictorian sentimentality had a heyday. The most pathetic-looking \"Johnstown orphans\" imaginable were drawn by New York artists and published beside long accounts of lost children. There were stories published of families tenderly bidding each other a final good-by just as the flood was about to pounce on them and of people tucking farewell notes into bottles before they slipped beneath the water for the last time.\n\nOne publisher, Kurz & Allison of Chicago, eventually got out a color lithograph which became one of the popular works of art of the age. In the upper right-hand corner a dam bursts almost directly on top of a Johnstown where all the women and children are barefoot and many are in their night clothes at four in the afternoon. The women are fainting, falling, down on their knees praying, while the men, most of whom are amply clothed and shod, dash about trying bravely to contend with the rush of fire and water.\n\nThe newspapers, too, went very heavy on the horrors to be seen among the ruins of Johnstown, sometimes stretching the art nearly to the breaking point. This memorable sample, written at Johnstown on Wednesday, June 5, appeared the following day in the Philadelphia Press:\n\n. . . One of the most ghastly and nauseous sights to those unaccustomed to scenes of death is the lunching arrangement for the undertakers. These men are working so hard that they have no time for meals, and huge boilers of steaming coffee, loaves of bread, dried beef and preserves are carried into the charnel house and placed at the disposal of the workers. Along comes one weary toiler, his sleeves rolled up, and apron in front and perspiring profusely despite the cold, damp weather. He has just finished washing a clammy corpse, has daubed it with cold water, manipulated it about on the boards and in the interval before the body of another poor wretch is brought in, gets a cup of coffee and a sandwich. With dripping hands he eats his lunch with relish, setting his cup occasionally beside the hideous face of a decomposing corpse and totally oblivious to his horrible surroundings.\n\nWhatever the reporters may have lacked in the way of facts, they made up for in imagination. Distortions, wild exaggerations, and outright nonsense were published in just about every major paper in the country. One reporter described buzzards (\"incited by their disgusting instinct\") circling over the stone bridge; another claimed the rivers were literally dammed with dead bodies. There were tales of wild dogs ravaging the graves of flood victims and devouring corpses by the dozens. Indianapolis readers were told that \"each blackened beam hides a skull.\"\n\nThere seems little doubt that there was plenty of drinking going on, but one writer for the New York World had men staggering about with whole pailfuls of whiskey. (\"Barrels of the stuff are constantly located among the drifts, and men are scrambling over each other and fighting like wild beasts in their mad search for it.\") A large family was pictured sailing by during the height of the flood singing \"Nearer, My God, to Thee\" in harmony. And of those stories emphasizing the pitiful fate of the innocent, perhaps the most imaginative was one filed from New Florence. A bride from Johnstown (she was supposedly married just before the flood) was quoted: \"Today they took five little children out of the water, who had been playing 'Ring around a rosy.' Their hands were clasped in a clasp which even death did not loosen, and their faces were still smiling.\"\n\nBut the most splendid story of the lot was one about a man named Daniel Peyton, the so-called \"Paul Revere of the Flood,\" who was said to have galloped down the valley on a big bay warning everyone to run for the hills. Peyton (in some versions he is Periton) seems to have evolved out of young John Baker of South Fork and John Parke, Jr. He can also be traced to Charles Reade's Put Yourself in His Place, where a rider sped through the night warning isolated farm families that the water was on the way. In most versions of the Peyton story, including the best-known of several epic poems, The Man Who Rode to Conemaugh, by John Eliot Bowen, which was published first in Harper's Weekly, the hero gallops the length of the valley just ahead of the onrushing wave. (The fact that there was no valley road on which to make such a ride never seemed to bother any of the authors very much.) Pale, his eyes aflame, he cries out \"Run for your lives to the hills!\"\u2014then dashes on.\n\nSpurring his horse, whose reeking side\n\nWas flecked with foam as red as flame.\n\nWither he goes and whence he came\n\nNobody knows. They see his horse\n\nPlunging on his frantic course,\n\nVeins distended and nostrils wide,\n\nFired and frenzied at such a ride.\n\nNobody pays any attention to him. They decide he is a lunatic and jokingly dismiss the whole thing.\n\n\"He thinks he can scare us,\" said one with a laugh,\n\n\"But Conemaugh folks don't swallow no chaff;\n\n'Taint nothing, I'll bet, but the same old leak\n\nIn the dam above the South Fork Creek.\"\n\nIn one version published as a Sunday-school lesson in 1891, the messenger goes into a saloon to spread the warning and winds up getting so drunk he can go no farther. But in either prose or poetry, in the end the flood finally catches up with poor Peyton, and nowhere is the moment more superbly described than in a book titled The Official History of the Johnstown Flood, which was put together that June by two Pittsburgh newspapermen named Frank Connelly and George C. Jenks. Peyton, in their \"official\" account, is about thirty years of age, and though he does not come clear from South Fork, but sticks only to the streets of Johnstown (Connelly and Jenks apparently felt they could not quite get away with the valley ride), his message and manner are essentially the same, and his end comes this way:\n\n. . . At last he completed the circuit of the city and started in search of a place of safety for himself. To the hills he urged his noble steed. Tired out from its awful ride, the animal became slower and slower at every stride, while the water continued to come faster and faster in pursuit. Like an assassin upon the trail of his victim, it gained step by step upon the intrepid rider. But the hills are in sight. No, he is doomed, for at that moment a mighty wave, blacker and angrier than the rest, overtook horse and rider and drew both back into the outstretched arms of death.\n\nAs the Reverend David Beale wrote, \"This fate was very necessary to the story, as it rendered an interview of the hero by another impossible.\"\n\nThough the story appeared in a dozen or more different versions and was accepted outright as fact, it was quickly discredited in Johnstown. For there seemed to be no one who actually saw this Daniel Peyton. Furthermore, as near as anyone knew, and according to every available record, there never was anyone by that name in South Fork, Johnstown, or any other place in the valley. Victor Heiser and some of his friends got so interested in the tale that they spent some time trying to track it down. If there were such a fellow, they wanted to meet him; and if he had been killed as people said he had, then they surely wanted to see him get the credit he had coming. But they never turned up anything, nor did anyone else.\n\nStill, with all the stretching of facts, with all the fabrication and bunk being printed, no one seemed to mind very much. If the horror of what had happened was not described exactly according to facts, people knew that what had happened was still a great deal worse than any words could convey, however accurate. And if a few small fables had been called up for the occasion, well they were really no more extraordinary than a dozen other stories that were \"the God's truth.\"\n\nFor the publishers it was one of the headiest weeks ever. Newspaper circulations broke all records. For days on end, one edition after another sold out almost as soon as it hit the streets. The New York Daily Graphic was selling an unheard of 75,000 copies a day. In Pittsburgh there seemed no letup to the clamor for more news. A new weekly picture newspaper called the Utica Saturday Globe, published in upstate New York but widely circulated, increased its circulation by better than 63,000 with its special addition on the disaster.\n\nSongs were written, including one called \"Her Last Message,\" which was more or less based on Hettie Ogle, and another called \"That Valley of Tears,\" which was about a baby who was swept from the hearthside and drowned in its cradle. The last one, arranged for piano and orchestra, closed with the lines:\n\nAnd there midst all that wreck, with cruel waters laved,\n\nThat babe within its cradle bed tho' dead 'tis saved;\n\nSaved from a life of toil and worldly care,\n\nOh! That we could in thy glorious prospects share.\n\nMagazines such as Harper's Weekly and Frank Leslie's Illustrated Newspaper got out special editions filled with pages of pictures and maps. Books were dashed off in a few weeks and rushed to the printers. Before the year was out, in less than six months, a dozen books would be written and published, most of them little more than an assembly of newspaper accounts, full of repetition, contradictions, and abundant nonsense. Several became best-sellers. One, a rousing period piece called The Johnstown Horror, was on sale in Johnstown itself in less than a month after the tragedy.\n\n### -2-\n\nFor Johnstown the result of the journalists' handling of the story was even more staggering. The enormous sympathy aroused by the newspaper accounts, the pictures, the songs and poems, brought on the greatest outpouring of popular charity the country had ever seen. (And this too, alas, the journalists felt obliged to immortalize: \"As the bow of promise gilded the Oriental sky after the Noachian deluge, so the dark cloud enfolding the Conemaugh Valley had a ray of brightest sun light. A great, grand glorious tide of sympathy for the sufferers swept the land like a conflagration, warming men's hearts to deeds of radiant luster.\")\n\nOn the Saturday following the calamity more than $100,000 had been raised in Pittsburgh. By the time they had finished, the people of Pittsburgh would give $560,000. New York City gave $516,000; Philadelphia, $600,000; Boston, $150,000. Nickels and dimes came in from school children and convicts. Churches sent $25, $50, $100. In Salt Lake City thousands of people turned out for a concert given in the huge Mormon Tabernacle, the proceeds of which were sent to Johnstown.\n\nIn New York's Madison Square Garden, Jake Kilrain, who was to take on John L. Sullivan in another few weeks for the world championship, put on an exhibition fight with Charley Mitchell to raise money for Johnstown. (Sullivan was invited to attend the show but did not, and was soundly hissed when his telegram of apology was read by the manager.) At the Metropolitan Opera House, Edwin Booth played the third act of Othello and raised $2,500. In Washington, John Philip Sousa gave a band concert. In Paris, Buffalo Bill staged a special production of his Wild West Show, which was attended by the Prince of Wales. And in Altoona, on the night of Monday, June 3, there had been a benefit performance presented by the Night Off troupe.\n\nTiffany & Company sent $500. R. H. Macy & Company sent $1,000. Joseph Pulitzer sent $2,000; Jay Gould, $1,000; John Jacob Astor, $2,500. The New York Stock Exchange gave $20,000. An old Confederate soldier sent four $100 Confederate bills, and the citizens of Cupola, Colorado, sent a solid-silver brick.\n\nThere were donations from Nantucket ($1,136.93), Yazoo City, Mississippi ($350), and Tombstone, Arizona ($101). The first check to arrive was supposedly one for $100 from Senator Matthew S. Quay, leader of the Republican forces in Pennsylvania; and Simon Cameron, the state's crusty, old Republican boss, who would be remembered as the man who defined an honest politician as one \"who when he is bought stays bought,\" sat down and wrote out what was said to be one of the last checks he ever signed, for $1,000.\n\nThe Hebrew Benevolent Society of Los Angeles contributed $1,000. The United States Brewers Association sent $10,000. The Pittsburgh Society of Spiritualists collected $100. Money poured in from every state and from fourteen countries overseas. The London Stock Exchange gave $5,000. The total donations from Germany came to $30,000. There was money from the Lord Mayor of Dublin, the Mayor of Belfast, and the Sultan of Turkey. Queen Victoria sent her condolences to President Harrison, and from Washington came more than $30,000, including a check from Harrison for $300. (The President had presided over a mass meeting at the Willard Hotel Tuesday afternoon, looking very small and gray as he sat in a big armchair in the center of the stage. He had made a brief appeal for help, during which, according to one account, his voice trembled, and nearly $10,000 had been raised.)\n\nIn all, the contributions from within the United States would come to $3,601,517.80. The sum from abroad was $141,300.98, making a total of $3,742,818.78. And this does not include the goods of every kind that rolled in by the trainload.\n\nFrom Saturday on the relief trains kept coming without letup. Lumber came in by the carload, and furniture, and barrels of quicklime, embalming fluid, pitch, pine tar, and resin. There was a carload of potatoes from Walla Walla. Minneapolis sent sixteen carloads of flour. Cincinnati sent 20,000 pounds of ham. Prisoners at the Western Penitentiary baked 1,000 loaves of bread. From Arbuckle's in Pittsburgh came 30,000 pounds of coffee, and a New York butcher sent 150 pounds of bologna. Wheeling, West Virginia, sent a whole carload of nails.\n\nBy Friday, June 7, two hundred carloads of provisions had cleared Pittsburgh. At the Pennsylvania depot in Johnstown, and at the B & O depot, the platforms and yards were piled with cans of biscuits, boxes of candles, cheese, lamp chimneys and matches, huge cases of soap and canned goods, bacon by the barrelful, and hundreds of sacks of corn meal. People had donated cots, mattresses, hair combs, pipes, pillows, teakettles, tents, cookstoves, and more than 7,000 pairs of shoes.\n\nBy Thursday, the 6th, the day the Cincinnati hams arrived, any fears there had been of serious food shortages were over. The problem now was distribution. Commissaries had been set up and were run with reasonable efficiency. Several women complained that the prettiest girls were getting more than their share, and often people came climbing back up the hillsides with empty baskets after waiting hours in line. But considering how many more there were to feed, things were exceptionally well organized. The Army had arrived the day before, which helped a lot. Hastings had held off calling in more troops but had decided late Tuesday to bring in the 14th Regiment from Pittsburgh (580 men), and now with their white tents pitched in rows where the park had been and on Prospect, the place began to look like a cross between a military encampment and a wide-open mining town. For along with the Army had come another Booth and Flinn gang of 1,100, and with the railroads bringing in their own crews, with more and more drifters arriving, by the end of the week there would be nearly 7,000 men at work in the valley. They too pitched tents or built rough shanties on the hillsides, upon which they hung names like \"Lively Boys,\" \"Willing Workers,\" and \"The Ladies' Pets.\" They were making $2.00 a day, which was good money; and a man with a team could make $5.00, which was good enough to have attracted men all the way from Ohio and New York. And with the pay went \"room and board.\"\n\nEnormous tents served as so-called dining halls, where several hundred men could be fed at a sitting. Coffee was ladled out of a bucket, bread was passed out in big dishpans, followed by more dishpans, each with a ten-pound slab of butter in it. When meals were ready, \"the rush for the table cast the Oklahoma boom all in the shade,\" wrote the Times.\n\nThe men worked hard and made notable progress. By sundown Thursday more had been accomplished than in all the days since the disaster. The air rang with shouts and the screech of nails yanked out by crowbars. Hundreds of bonfires crackled and sent up columns of black smoke where now barrels of resin were being dumped on to burn up dead animals and the worst of the debris. Teams were hitched to mud-bound machinery and dangerous-looking walls that came down with a crash. There were tools enough now for every man, and by midweek a shipment of dynamite along with an expert to handle it had arrived to break the jam at the stone bridge.\n\nSo far every attempt to pull loose the debris there had failed. Locomotives and a steam winch had been tried, and a gang of lumberjacks from Michigan had done their best but had made hardly a dent. The ugly mass had been driven against the bridge with such savage force, and was so tightly ensnared by miles of barbed wire, that nothing, it seemed, could break it loose. Then for several days the valley echoed with the roar of dynamite as slowly, one by one, great gaps were blasted through the entanglement, which, most people believed, still held many human corpses.\n\nThe expert in charge was a man named Kirk, who was known among the press as \"The Prince of Dynamiters.\" He was short, squat, about fifty years old, with a grizzly beard and a habit, when walking, of waving his hands about as if warning people back from his next explosion.\n\n\"He personally superintends the preparations of all blasts,\" wrote the Times man on the scene, \"and when ready emits a peculiar cry more like a wail than a warning. Then he surveys the atmosphere with the air of a Major General and yells, 'Fire!' The yell often terrifies the spectators more than the explosion itself.\" But on Saturday Kirk managed to scare the daylights out of everybody, and with a good deal more than his yell.\n\nDespite the inroads he was making, he was not at all satisfied with the time the job was taking, so he cut loose with a 450-pound charge, hoping to break the jam with one blow. Nine fifty-pound boxes of dynamite were planted at the base of the mass, each set about thirty feet apart. The whole valley and every building that was still standing trembled with the blast. Horses shied violently. Work stopped all over town. And within no time Kirk had been told in quite explicit language by what seemed like half the surviving population that he was to pull no such stunt again. But a gaping hole had been gouged out of the wreckage and at last daylight could be seen under the arches.\n\nSome men became so exhausted from the work that they had to give up and go home. Everyone by this time had grown more or less accustomed to the grisly look of the place, the smell, and the constant presence of death; but the work itself was grueling, and no matter how much was accomplished, there seemed always an insurmountable lot still left to be done.\n\nThe wreckage spewed across the city was far greater in quantity than all that had stood there before the flood. The water had swept the valley above the city of virtually everything that man had built there over some eighty years and dumped it on Johnstown. Now every last bit of it had to be cleaned up, searched through, burned, or carted away. On top of that, sanitation problems were enormous. One gang of workers did nothing but sprinkle disinfectants over the entire district, and as George Swank later wrote, there was \"a fortune for the man who concocts a disinfectant that won't stink the nose off a person.\" Four thousand barrels of quicklime would be used before the cleanup was finished.\n\nAnother gang was detailed to gather and burn bedding, clothes, and carpets which had been ruined by the water. The houses that had not been swept away were left, as Dr. Matthews said, in the most unsanitary condition imaginable. \"The flood water,\" he wrote, \"was heavily charged with every kind of filth, and whatever this water touched it contaminated. As a result, every house in the flooded district was filled, in most cases to the second floor, with most offensive matter. There was not a place which the flood touched where a man could lay his head with safety.\"\n\nOne of the biggest and most unpleasant jobs was digging out hundreds of cellars. Where houses had been swept from their foundations, it was often close to impossible to tell where they had been, the cellars having been so completely filled in by the flood. But every one of them had to be shoveled clean and their foul contents hauled out of the city.\n\nThe work camps themselves were another part of the problem, with their \"open plumbing, openly arrived at,\" as one doctor described it. Within the first week there were something like ninety-eight doctors at work, who, along with the Army, helped enforce basic sanitary practices. But what helped most of all was the weather. For ten days following the flood, temperatures stayed low, skies clouded, and there were frequent drenching rains. It was, wrote Dr. Lee, head of the sanitation board, a \"great advantage in delaying decomposition.\" It was also miserable weather to work in or to be camping in amid the reeking muck.\n\nOn Friday morning Captain Bill Jones was back home in Pittsburgh, close to a state of collapse after working four straight days with almost no sleep or rest. Jones had paid out of his own pocket for the supplies he had brought to Johnstown and the wages of the men he had taken with him. In an interview with the Pittsburgh Press he gave great credit to the leadership Moxham had shown and added that he had had a hundred or so Hungarians working for him and that they had \"worked like heroes.\"\n\nGeneral Hastings held up handsomely against the strain. He rode about town on a big horse, waving his floppy hat, and, in the main, did a good job. Director James Scott stayed on day after day, working with boundless energy. He wound up growing a full beard during his time in Johnstown and lost thirty pounds.\n\nBut perhaps the most resilient worker of them all, and certainly the one who stirred up the most talk, was a stiff-spined little spinster in a plain black dress and muddy boots who had brought the newly organized American Red Cross in from Washington. Miss Clara Barton and her delegation of fifty doctors and nurses had arrived on the B & O early Wednesday morning.\n\nClara was sixty-seven. She had been through the Civil War, the Franco-Prussian War, and several nervous breakdowns. For a while she had tried running a women's prison in Massachusetts. But since 1881, when, after a long campaign, she had succeeded in establishing an American branch of the International Red Cross, little else had been of real interest to her. And though her position as president was only a part-time job, she had already been down the Ohio by river barge to help during the floods of 1884 and to Texas with food and supplies during the famine of 1887. She had taken her workers to Illinois after a tornado in '88, and later that same year to Florida during a yellow-fever epidemic.\n\nBut these had been minor challenges compared to Johnstown, which she realized the moment she saw the valley from her train window. When the news of the disaster had first reached Washington Friday night, she had postponed doing anything for twenty-four hours; the story seemed too frightful and improbable to be true. But once there she knew that her Red Cross had arrived at its first major disaster. The organization, she had long argued, was meant for just such emergencies, and now she intended to prove it.\n\nShe set up headquarters inside an abandoned railroad car and, using a packing box for a desk, began issuing orders. Hospital tents were to be opened immediately, construction was to start on temporary \"hotels\" for the homeless, and a house-to-house survey was to be conducted to see just how many people needed attention. Hastings, it appears, did not know quite what to make of all this, or of Clara. \"I could not have puzzled General Hastings more if I had addressed him in Chinese,\" she wrote later, adding that \"the gallant soldier could not have been more courteous and kind.\"\n\nIn her stocking feet Clara stood five feet tall. She had a prominent nose, bright, black eyes, and a resolute set to her mouth. She took what little sleep she needed on a hard, narrow cot and had no use for demon rum, bumbling male officials, or for that matter anyone who attempted to tell her how to run her business. \"A keen, steadfast, powerful New England woman,\" she was described as by one writer.\n\nWithin a very short time several large tents were serving as the cleanest, best-organized hospitals in town; six Red Cross hotels, two stories tall, with hot and cold running water, kitchens, and laundries, had been built with some of the fresh lumber on hand; and Clara herself was situated in her own command tent with the Red Cross banner flying overhead. When the survey was completed it was found that a large number of people with serious injuries had been too weak or broken in spirit to do anything to help themselves. Moreover, a surprising number of cases of prolonged shock were discovered, a phenomenon that would also be noted by a correspondent for the Medical News of Philadelphia. \"A profound melancholia,\" he called it, \"associated with an almost absolute disregard of the future\" and evidenced by \"a peculiar intonation of words, the persons speaking mechanically.\"\n\nClara and her people did their best to tend everyone they could. Clara herself worked almost round the clock, directing hundreds of volunteers, distributing nearly half a million dollars' worth of blankets, clothing, food, and cash. She also spoke her mind once or twice to the head of the Philadelphia chapter of the Red Cross, with the result that within a few days neither group would have anything to do with the other.\n\nThere seems little doubt that except for Hastings she was the best-known, among the people of Johnstown, of all the outsiders on hand and certainly the one who would be remembered the longest. She stoutly proclaimed that the Red Cross was there to stay as long as there was work to do. \"We are always the last to leave the field,\" she said. She seemed to be everywhere at once, bouncing through the streets in a buckboard, scrutinizing the way things were being handled, whether she had anything to do with them or not. On one such ride she was accompanied by an Episcopal priest who was afraid she would be jolted to pieces and told her so. \"Oh, this is nothing,\" she shouted back, \"so long as we have no bullets flying around us.\"\n\nClara stayed for five months, never once leaving the scene even for a day. In October, when she did finally pack up and go, it was with all sorts of official blessings and thanks. She was presented with a diamond locket from the people of Johnstown; glowing editorials were written (\". . . too much cannot be said in praise of this lady . . . To her timely and heroic work, more than to that of any other human being, are the people of the Conemaugh Valley indebted . . .\"); and back home in Washington a dinner was given in her honor at the Willard with the President and Mrs. Harrison in attendance. The Red Cross had clearly arrived.\n\nThere were others who would be remembered. A few smalltime crooks slipped in with the crowds. They queued up with the flood victims to collect whatever the Red Cross happened to be handing out at the moment or grabbed what they could from the debris. One or two suspicious-looking characters were nabbed before they had a chance to do much of anything and were swiftly hustled out of town. There were some, too, who simply hung around long enough to educate themselves on the place, then lit off to play on the sympathy of neighboring towns, describing the horrors of the devastation and their own heart-rending experiences in the flood at one back doorstep after another. One of them who went straight to Pittsburgh even managed to take the Relief Committee there for a good deal of cash before he was found out.\n\nA handful of crackpots appeared on the scene, most of whom were of the religious-fanatic sort, and the most memorable of them was a gaunt prophet from Pittsburgh known as \"Lewis, the Light,\" who wore nothing but long, red underwear and passed out handbills that said, among other things:\n\nDeath is man's last and only Enemy, Extinction of Death is his only hope. Your soul, your breath, ends by death. Whew! Whoop! We're all in the soup. Who's all right? Lewis, the Light.\n\nThen there were the \"harpies,\" as some of the newspaper correspondents called them, who apparently came in from Pittsburgh. It was reported that a number of them had been seen at points along the line from Johnstown to Pittsburgh trying to recruit girls for \"their nefarious calling\" among the prettiest flood victims. Several who had tried to board the trains had been put off, but enough of them managed one way or other to reach Johnstown, so that within not too long a time Prospect Hill was making its own lively contribution to the mining-camp spirit of the place. Prospect \"is overridden with bawdy houses and places for the illicit sale of liquor,\" one police official said.\n\nThis in turn added fuel to the cause of the W.C.T.U. ladies who had arrived almost immediately after the disaster and stayed on as the valley filled up with more and more men and as the demand for strong drink grew apace. Through most of June, Hastings kept the lid on liquor sales; the only thing being served was lemonade, which was sold at little makeshift stands through town. When, in July, the liquor ban was lifted, the lemonade concessions went immediately out of business. Twenty-two celebrants were arrested the first day, and George Swank wrote in the Tribune that the lemonade had \"made more sickness than all the beer and whiskey that could be drunk.\"\n\nAnd, of course, there were the sight-seers. They had been coming almost from the first morning after the disaster. They came into South Fork by special trains from Altoona. They arrived in Johnstown on excursion trains that chugged in along the B & O weekend mornings. On Sunday, June 23, several hundred arrived, turned out in holiday attire and carrying picnic baskets. It seemed incredible; but there they were. The wreckage at the stone bridge seemed to fascinate them the most. But they strolled about everywhere, got in the way, set up their lunch parties inside abandoned houses, laughed, took pictures, asked a lot of silly questions, and infuriated nearly everyone except a few enterprising local men who set up booths and began selling official Johnstown Flood relics: broken china, piano keys, beer bottles, horseshoes, buttons, even bits of brick or wood shingles.\n\nHastings and the other officials kept asking the railroads to stop selling tickets to anyone who had no rightful business in Johnstown; before the month was out the railroads had agreed to comply as best they could and the number of visitors dropped off sharply.\n\nActually there would have been many more sight-seers than there were had there not been such widespread fear of disease; and though the cooperation of the railroads helped considerably to diminish the problem, the real turn came when it became known that typhoid fever had actually broken out in the valley.\n\nThe first clear-cut case was identified on Monday, June 10. More cases were found in the next several days, but the news was kept quiet. Within a week, however, the disease had spread swiftly and so had the rumors. By July 25 there would be 215 cases of typhoid within the flooded area and 246 beyond.\n\nThe doctors and sanitation crews, already dog-tired, flew into a frenzy of activity, working night and day to stop the thing and to keep the valley from panicking. Considering what might have happened under the circumstances, they did a spectacular job. But before the summer ended, forty people would die of typhoid; and like those who died of injuries or exposure in the first days after the disaster, they would not be included in the official figure given for the flood's victims.\n\nInterestingly enough, along with the typhoid, there ran a long spell of unusually good health in Johnstown. There were fewer colds, fewer cases of measles, and fewer people complaining of \"spring disorders\" than there would have been under normal conditions. Those who managed to stay healthy through the first frightful days immediately following the disaster, stayed very healthy from then on. The regulations and precautions enforced by the sanitation people undoubtedly had a great deal to do with this, but so, most everyone would later agree, did the fact that there was by the end of the first week something almost like a spirit of exhilaration in the air. There was so much happening all around. Every day there was some kind of excitement. Those who had survived, despite how much they may have suffered, began to discover new energy in themselves. They were alive, and bad as things were that was still a lot better than being dead. And there was now so very much to be done.\n\nThere were exceptions, to be sure. Dr. Swan, for example, was so broken in spirit by what he had been through that he would never be able to practice medicine again. And the only fatality among the polyglot army of people brought in to help was a suicide among General Hasting's troops. Sunday afternoon a moody farm boy with a wife and two children back home became so depressed by what he had seen that he went into his tent and shot himself through the head.\n\nBut for nearly everyone else the almost absurd idea that they were going to pick up and start over again, to rebuild everything, began working like a tonic. They started pulling together what was left of their old lives and got to work on the new.\n\nMost of them had precious little left of the old. Dr. Matthews, who had opened a new practice in Johnstown just a few months before the flood, found not a shred of all that he had owned except for one shaving mug. Of George and Mathilde Heiser's earthly possessions, the only thing recovered was a big wardrobe which young Victor retrieved from the wreckage on Main Street. When he pried open the door, he found his father's old Civil War uniform inside, and in one pocket a single penny. It represented his entire inheritance.\n\nMen like Horace Rose and John Fulton had lost their homes and virtually everything else they owned. James Quinn and his brother-in-law were able to recover scarcely a single board from their dry-goods store. The only possessions recovered from the Quinn home were some books, a photograph album, and some silverware, which were found scattered a mile or more from where the house had stood.\n\nQuinn, like hundreds of others, had sent his children off to Pittsburgh to live with relatives, while he stayed on to do what he could to help. But there were a few days before they left which the children would remember always. For nearly every child who survived, the week after the flood was a time of high adventure. The dynamite blasting at the bridge, the commissaries with their wondrous stacks of goods, the mobs of strangers tramping through the streets, the Iron Company's little wood-burning switching engines with their bell-shaped stacks shunting back and forth moving supplies, the nuns, the soldiers, the Pittsburgh firemen in their long rubber coats, were all something to see.\n\nGangs of small boys made great sport of sneaking past the sentries posted about town, or crawling among the rows of tents. Little prefabricated houses, called \"Oklahomas,\" were being put up everywhere, and often, if he acted sensible enough, a boy could get a chance to help out. And every child, it seemed, took up relic hunting. \"Every yard would yield something if one had the energy to dig,\" Gertrude Quinn wrote later. Gold and jewelry were supposedly being found all over town. One story had it that a crockery jar full of $6,500 worth of gold had been found in the mud where old William Macpherson had had his grocery store. Gertrude Quinn's sister Marie found a two-and-a-half-dollar goldpiece. And it was no trick at all to find half dimes or three-cent pieces or even a shirt stud with what looked like a diamond in it.\n\nOn Sunday, the 9th, the sun broke through for the first time since the flood. The spring green of the hills gleamed in bright morning sunlight, and overhead there were only a few small, soft clouds, and all the rest of the sky was a clean blue. Work went on the same as any other day. The air, already warmer than it had been for weeks, rang with the sound of picks and axes, hundreds of hammers, and, strangely it seemed at first, church bells.\n\nOn the embankment near the depot, just back from where Hastings had his headquarters, the chaplain of the 14th Regiment, H. L. Chapman, and David Beale began conducting the first services since the flood.\n\nThere were no more than thirty people gathered at first, but as time passed the crowd grew. Soldiers hanging about nearby drifted over. People came from the depot and over from the center of town. Beale stood on a packing box and told a story about over-hearing a newcomer in the valley ask a small boy how bad things were in Johnstown, to which the boy was said to have replied, \"If I was the biggest liar on the face of the earth I could not tell you half.\"\n\nThe rest of the service went as might be expected until John Fulton got up and started saying things that soon had the crowd stirring.\n\nHe said the Cambria shops would be rebuilt. \"Amen!\" answered several voices. \"Johnstown is going to be rebuilt,\" he said. \"Thank God!\" someone answered again.\n\nHe said he could not speak for the Gautier works, but he was sure, nonetheless, that they, too, would be rebuilt, and bigger than ever. The Cambria men would be taken care of, he told them, and if you still have your family left, he said, then \"God bless your soul, man, you're rich.\" His sermon was simple: \"Get to work, clean up your department, set your lathes going again. The furnaces are all right, the steel works are all right. Get to work, I say. That's the way to look at this sort of thing . . . Think how much worse it could have been. Give thanks for that great stone bridge that saved hundreds of lives. Give thanks that it did not come in the night. Trust in God. Johnstown had its day of woe and ruin. It will have its day of renewed prosperity. Labor, energy, and capital, by God's grace, shall make the city more thriving than ever in the past.\"\n\n\"Amen!\" again from the crowd.\n\nThat Johnstown should be rebuilt was by now taken as a matter of course. That it should be rebuilt right where it was before was also something everybody took for granted. No matter how dreadfully the valley had been ravaged, it was still their home. In fact, there is no record of anyone ever seriously considering the idea of not rebuilding in that particular place. The only question now was how long was it going to take to get things rolling once more.\n\nThe scene on the hillside would be remembered for years. Fulton, tall and spare, with his iron-gray beard and dark brow, certainly looked and sounded like a man of God; but he was also, as everyone knew, the voice of the Cambria Iron Company speaking, and as sincere as he may have been in asking them to trust in the Lord, his audience had had somewhat more experience putting its trust in the Cambria Iron Company. And either way, any man who could speak for both God and the Iron Company was someone to listen to closely.\n\nThe idea that the stone bridge had actually saved lives was a new one to most people, but the more they thought about it, the more they accepted it. The fire at the bridge seemed to epitomize the worst of the flood's horrors, but the fact was the death toll would have been far greater, perhaps even twice as great, had the bridge collapsed. The whole town would have been plunged down the valley to almost total destruction.\n\nBut it was when Fulton began talking of still another matter that a tense stillness came over the crowd and his every word could be heard over the sounds from the city below.\n\n\"I hold in my possession today,\" he said, \"and I thank God that I do, my own report made years ago, in which I told these people, who, for purposes for which I will not mention, desired to seclude themselves in the mountains, that their dam was dangerous. I told them that the dam would break sometime and cause just such a disaster as this.\"\n\nThen he changed to another subject. But he had said enough. In just two sentences he had hit on something that had been smoldering in people's minds for days. There had been plenty of talk about what had happened at South Fork and about the club. People were bitter, and with their renewed energy had come anger, deep and highly inflammable, and perhaps even contributing to that energy.\n\nIt was not that Fulton had been the first to raise the question of the dam and who was to blame for the flood; what was important was that he, John Fulton, had said what he had. The issue was now right out in the open, full-scale and officially. Moreover, there was no longer any question about which side the Iron Company might be on; and, perhaps most significant of all, if things came to a showdown, which everyone felt sure they would, Fulton, it would appear, held a piece of paper of considerable importance.\n\n## IX\n\n## \"Our misery is the work of man.\"\n\nThe excitement in Pittsburgh continued day after day. Johnstown seemed to be the only thing people were talking about, and the papers carried almost nothing but flood news, with stories running on and on, page after page, and in even greater detail than what was being published elsewhere. The two cities had always had ties, through the steel business and family connections. Now almost everyone in Johnstown, it appeared, had relatives in Pittsburgh.\n\nRefugees from the disaster kept pouring into Union Station by the thousands. The sick and the injured had to be cared for. Children, hundreds of them, lost or orphaned, many wearing tags for identification, had to be fed and looked after. Homes had to be found for them and all the others. On Wednesday, the 5th, four trains full of survivors, most of them women and children, came in.\n\nPeople had dropped everything to help. Ladies' groups were sorting clothes and packing medical supplies in church basements all over town. The Masons, the Republican Club, factory workers were organizing, collecting, donating, and proudly announcing their accomplishments to the papers. The involvement grew so that the local merchants began complaining of a serious drop off in trade. \"From a business point of view\" things were the worst they had been for years, according to one report. Several firms canceled their regular newspaper advertisements in order to express their sympathies for the people of Johnstown, and Young's picture store on Wood Street attracted considerable attention by displaying in its window a painting of South Fork dam done a few years earlier by a local artist.\n\nEverybody, it seemed, had his own latest story from Johnstown. A husband had heard from another man at the mill, a brother had just come back from the railroad depot, a cousin who worked at the Mellon bank had overheard something, two sons from the big Italian family across the alley had actually been there with one of the Flinn gangs, and they all had stories to tell, inside information. The city was alive with the most hair-raising tales and rumors. And nowhere was there more talk, or were things in such turmoil, than at the Pennsylvania depot and yards, where tons of food and supplies were still piling up, and the crowds were so thick, any hour of the day, that you could barely make your way through.\n\nThe railroad itself had never known such times, not even during the worst of the war years. Every schedule had been canceled. All normal business had been stopped. Nothing went east but trains bound for Johnstown, and as it was, the traffic was almost more than could be handled. If there had been only the storm damage to contend with, troubles would have been bad enough; but train after train kept steaming in from across the country, men and supplies had to be kept moving, repair equipment had to be sent forward, and everything that went up had to come back by the same route.\n\nPitcairn, with full authorization from the main office in Philadelphia, did everything possible to speed things up. The Pennsylvania had already donated $5,000 to the relief fund, but that was of small consequence compared to what was accomplished to keep the line open. Pitcairn himself worked almost without letup. All available manpower east and west was rushed into the area, and the cost of everything was assumed by the line.\n\nThis was by far the biggest emergency the Pennsylvania had ever been called on to face, and all its extraordinary power, its almost military-style discipline and organization, its vast resources in men and equipment, were brought to bear on the problem. The results, the swiftness and efficiency with which forces were marshaled, tangles unsnarled, damages repaired, help rushed through, were indeed remarkable and left a lasting impression on everyone involved. For all its highhanded ways, for all the evils people attributed to it, in a crisis the railroad had been worth more than any other organization, including the state, and they would remember that.\n\nStill it would be two full weeks until the line from Harrisburg west would be opened and relief trains could start to Johnstown by way of Altoona. Until then Pittsburgh would remain the one channel through which everything had to pass.\n\nThe Allegheny River, with its endless freight of wreckage, also continued to be an immense fascination. Children were brought from miles away to watch the tawny water slip past the shores, so that one day they might be able to say they had seen something of the Johnstown Flood. The most disreputable-looking souvenirs, an old shoe, the side of a packing box with the lettering on it still visible, were fished out, dripping and slimy, to be carried proudly home.\n\nThere were accounts of the most unexpected finds, including live animals. But the best of them was the story of a blonde baby found at Verona, a tiny river town about ten miles up the Allegheny from Pittsburgh. According to the Pittsburgh Press, the baby was found floating along in its cradle, having traveled almost eighty miles from Johnstown without suffering even a bruise. Also, oddly enough, the baby was found by a John Fletcher who happened to own and operate a combination wax museum, candy stand, and gift shop at Verona.\n\nFletcher announced his amazing discovery and the fact that the baby had a small birthmark near its neck. Then he hired a pretty nineteen-year-old, dressed her in a gleaming white nurse's uniform, and put her and the baby in the front window of his establishment. Within a few days several thousand people had trooped by to look at the Johnstown baby and, it is to be assumed, to make a few small purchases from the smiling Mr. Fletcher. Then, apparently, quite unexpectedly, the baby was no longer available for viewing. The mother, according to Fletcher, had lived through the flood and, having heard the story back in Johnstown, rushed to Verona, identified the birthmark, and went home with her baby.\n\nBut there was another subject that was stirring up far more talk in Pittsburgh. It was not until three or four days after the flood that the rest of the country began growing keenly interested in the South Fork Fishing and Hunting Club, but in Pittsburgh, not surprisingly, the interest had been high since news of the disaster first came through Friday night. And for those who may have forgotten, or who never knew, the nature of the club's membership, the Pittsburgh newspapers were quick to remind them.\n\nIn the beginning there had been some concern over those clubmen who had been at the dam when it failed. But when it became known that they were alive and unharmed, the emphasis immediately shifted to what exactly the other club members might do next.\n\nOn Saturday, at the mass meeting called by Pitcairn, Frick and Phipps had been named to serve on the executive board of the Relief Committee. That night, at the home of another member, Charles Clarke, a number of the clubmen met in private to agree on what their policy should be. At that point, like everyone else in Pittsburgh, they knew very little about what precisely had happened at the dam; but judging from the way things looked, the wisest policy for the moment seemed to be to say nothing, except that no immediate action was planned and that the club would make a donation to the people of Johnstown of 1,000 blankets.\n\nBut, unfortunately for the others, a few members decided to speak their minds all the same. One member, who asked that his name be withheld, told reporters that in the past he had heard questioning about the strength of the dam, but that he had never looked into the matter personally. Then he told a story of riding from the lake down to Johnstown a few years earlier with a driver who had said, \"The time will come when more than you and I will talk about that embankment.\" And he finished up by saying that there were some in Johnstown who used an Episcopal prayer, \"Lord deliver us from mountain floods!\"\n\nAnother member, James McGregor, who gave his name without any hesitation, said he refused to believe that there had been any trouble at South Fork. He was certain the whole thing was a mistake.\n\n\"I am going up there to fish the latter part of this month,\" he said. \"I am a member of the South Fork Fishing Club and I believe it is standing there the same as it ever was.\n\n\"As for the idea of the dam ever being condemned, it is nonsense. We have been putting in from twenty thousand to fifteen thousand dollars a year at South Fork. We have all been shaking hands with ourselves for some years on being pretty clever businessmen, and we should not be likely to drop that much money in a place that we thought unsafe. No sir, the dam is just as safe as it ever was, and any other reports are simply wild notions.\"\n\nHis own notions, which appeared in the papers on the morning of Sunday, June 2, were so wild, and so very tactless in the face of what was by then known of the suffering at Johnstown, that the only possible excuse for making such a statement must have been that he actually believed every word of it.\n\nAnd to make matters worse, he was not alone. Young Louis Clarke next told a correspondent for the New York Herald that there was great doubt \"among the engineers\" who had examined the reservoir whether, after all, it had been that particular dam which broke. Just which engineers he was referring to is unclear, but he was interviewed along with another club member, James Reed, who said that in the past he himself had climbed all over the dam, studying it closely, and that \"in the absence of any positive statement I will continue to doubt, as do many others familiar with the place, that it really let go.\" Perhaps, he then suggested, it had been a dam at Lilly which broke.\n\nReed's comments were of more than passing interest for he was the partner of Philander Knox in the prestigious Pittsburgh law firm of Knox & Reed. If there were to be lawsuits over the disaster, the South Fork Fishing and Hunting Club would almost certainly be represented by Knox & Reed. And already the press was playing up the likelihood that such suits would follow. On June 2 the World published a statement attributed to a prominent lawyer practicing in Allegheny County, who preferred to remain anonymous:\n\n\"I predict there will be legal suits with possible criminal indictments as a result of this catastrophe. I am told that the South Fork Club has been repeatedly warned of the unsafety of its dam, and it comes from good authority . . .\"\n\nOn another page the World published an interview with Jesse H. Lippincott of New York City, who was the son of a club member and who had spent several summers at the lake. The dam, he said, was built almost entirely of solid stone, but if it had indeed broken, the death toll would likely run to several thousand, and \"Pittsburghers will . . . be deprived of their most popular resort.\"\n\nThen, on Monday, the 3rd, reporters from Johnstown reached the dam and started sending a series of dispatches from South Fork which removed once and for all any fantasies about the dam still standing; and out of conversations with people in the neighborhood, they began building a history of the structure which did not bode well for the club members.\n\nFeelings were running very strong against the club at South Fork. Monday after dark an angry crowd of men had gone up to the dam looking for any club members who might have been still hanging about. When they failed to find anyone, they broke into several of the cottages. Windows were smashed and a lot of furniture was destroyed. Then, apparently, they had gone over to the Unger farm to look up the Colonel. The reporters later called it a lynch mob and said they were bent on killing Unger. Whether or not it would have come to that, there is no way of knowing, for Unger by that time was on his way to Pittsburgh. There was a good deal of grumbling among the men as they milled about outside Unger's house; threats were shouted; then the men went straggling off through the night, back down the hollow.\n\nThe clubmen who had been at the lake had gone off on horseback, heading for Altoona, almost immediately after the dam broke Friday afternoon, though one of them, it seems, stuck around long enough to settle his debts with some of the local people. He had no intention of ever coming back again, he told them, which they in turn repeated for the benefit of the newspapermen. They also emphasized that the Pittsburgh people had not made things any better for themselves by pulling out so rapidly at a time when, as anyone could see, there was such a crying need for able-bodied men in the valley. Had they stayed on to help, it was said, then people might have felt somewhat differently toward them. This way there was only contempt.\n\nBut it was when they began describing how the dam had been rebuilt by Ruff and his workers that their real bitterness came through, that all the old, deep-seated resentment against the rich, city men began surfacing. Farmers recalled how they had sold Ruff hay to patch the leaks. A South Fork coal operator who insisted that his name be withheld, but who was almost certainly George Stineman, South Fork's leading citizen, told how, years earlier, he had gone to Johnstown on more than one occasion to complain about the dam's structural weaknesses. Reporters heard that the dam had been \"the bogie of the district\" and how it had been the custom to frighten disobedient children by telling them that the dam would break. The clubmen were described as rude and imperious in their dealings with the citizens of the valley. Reporters were told of the times neighborhood children had been chased from the grounds; and much was made of the hated fish guards across the spillway. Old feuds, personal grudges, memories of insults long forgotten until then, were trotted out one after the other for the benefit of the press.\n\nSomeone even went so far as to claim that several of the Italian workmen employed by the club had been out on the dam at the time it failed and had been swept to their death, thus implying that the Pittsburgh men had heartlessly (or stupidly) ordered them out there while they themselves had hung back on the hillsides.\n\nOne local man by the name of Burnett, who conducted a reporter on an inspection of the dam, told the reporter that if people were to hear that he was from Pittsburgh, they might jump to the conclusion that he was connected with the club and pull him from the carriage and beat him to death. \"That is the feeling that predominates here,\" Burnett said, \"and, we all believe, justly.\"\n\nThe plain fact was that no one who was interviewed had anything good to say about the South Fork Fishing and Hunting Club, its members, or its dam. And when a coroner's jury from Greensburg, in Westmoreland County, showed up soon after the reporters, the local people willingly repeated the same things all over again.\n\nThe jurymen had come to investigate the cause of death of the 121 bodies that had been recovered at Nineveh, which was just across the line in Westmoreland County. They poked about the ruins of the dam, talked to people, made notes, and went home. The formal investigation, with witnesses testifying under oath, was to be held on Wednesday, the 5th.\n\nIn the meantime, Mr. H. W. Brinkerhoff of Engineering and Building Record, a professional journal published in New York, arrived in South Fork to take a look at the dam and was soon joined by A. M. Wellington and F. B. Burt, editors of Engineering News. Most of the reporters remained cautious about passing judgment on the dam, waiting to see what the experts had to say. But on June 5 the headline on the front page of the New York Sun read:\n\nCAUSE OF THE CALAMITY\n\nThe Pittsburgh Fishing Club\n\nChiefly Responsible\n\nThe Waste Gates Closed\n\nWhen the Club Took\n\nPossession\n\nThe indictment which followed, based on a Sun reporter's \"personal investigation,\" could not have been much more bluntly worded.\n\n. . . There was no massive masonry, nor any tremendous exhibition of engineering skill in designing the structure or putting it up. There was no masonry at all in fact, nor any engineering worthy of the name. The dam was simply a gigantic heap of earth dumped across the course of a mountain stream between two low hills. . . .\n\nIn Johnstown on the same day, General Hastings told a World correspondent that in his view, \"It was a piece of carelessness, I might say criminal negligence.\" In Greensburg the Westmoreland coroner's jury began listening to one witness after another testify to the shoddy way the dam had been rebuilt and the fear it had engendered, though two key witnesses had apparently had second thoughts about speaking their minds quite so publicly and refused to appear until forced to do so by the sheriff.\n\nTwo days later, on the 7th, a verdict was issued: \". . . death by violence due to the flood caused by the breaking of the dam of the South Fork Reservoir . . .\" It seemed a comparatively mild statement, considering the talk there had been and coming as it did on the same day as Hastings' pronouncement. But on the preceding day, another cornoner's inquest, this one conducted by Cambria County, had rendered a decision that spelled out the cause of the disaster, and fixed the blame, in no uncertain terms.\n\nThe Cambria jurors had also visited the dam and listened to dozens of witnesses. But their inquest was held to determine the death of just one flood victim, a Mrs. Ellen Hite. Their verdict was \"death by drowning\" and that the drowning was \"caused by the breaking of the South Fork dam.\"\n\nBut then the following statement was added:\n\n\"We further find, from the testimony and what we saw on the ground that there was not sufficient water weir, nor was the dam constructed sufficiently strong nor of the proper material to withstand the overflow; and hence we find the owners of said dam were culpable in not making it as secure as it should have been, especially in view of the fact that a population of many thousands were in the valley below; and we hold that the owners are responsible for the fearful loss of life and property resulting from the breaking of the dam.\"\n\nNow the story broke wide open. \"THE CLUB IS GUILTY\" ran the World's headline on June 7. \"Neglect Caused the Break . . . Shall the Officers of the Fishing Club Answer for the Terrible Results.\"\n\nThe Cincinnati Enquirer said that in Johnstown, as more facts became known, the excitement was reaching a \"fever heat\" and that \"it would not do for any of the club members to visit the Conemaugh Valley just now.\" The Chicago Herald said there was \"no question whatever\" as to the fact that criminal negligence was involved.\n\nAlthough it would be another week before the engineering journals would publish their reports on the dam, the gist of their editors' conclusions had by now leaked to the press. On Sunday, the 9th, The New York Times headline ran:\n\nAn Engineering Crime\n\nThe Dam of Inferior Construction\n\nAccording to the Experts\n\nActually, the engineering journals never worded it quite that way. The full report which appeared in the issue of Engineering News dated June 15 said that the original dam had been \"thoroughly well built,\" but that contrary to a number of previously published descriptions, it had not been constructed with a solid masonry core. (From this some newspapers would conclude that the \"death-dealer\" was nothing but a \"mud-pile.\") The repairs made by Benjamin Ruff, however, had been carried out \"with slight care,\" according to the report. Most important of all, there had been \"no careful ramming in watered layers, as in the first dam.\" But Ruff's work was not the real issue, according to the editors. \"Negligence in the mere execution of the earthwork, however, if it existed, is of minor importance, since there is no doubt that it was not a primary cause of the disaster; at worst, it merely aggravated it.\"\n\nThe primary causes, it was then stated, were the lowering of the crest, the central sag in the crest, the fact that there were no outlet pipes at the base, and the obstruction of the spillway. The details of these matters were carefully described, and it was speculated that the disaster might have been averted that Friday afternoon if the bridge over the spillway and the fish guards had been cut away in time, or if some \"man of great resolution, self-confidence, and self-sacrifice\" had (as John Parke had contemplated) cut the dam at one end, where the original and more firmly built surface would have held up better against the enormous force of erosion.\n\nBut the point the editors of the report seemed most determined to hammer home was that there was no truth to any claims being made that the dam had been rebuilt by qualified engineers.\n\n\"In fact, our information is positive, direct, and unimpeachable that at no time during the process of rebuilding the dam was ANY ENGINEER WHATEVER, young or old, good or bad, known or unknown, engaged or consulted as to the work,\u2014a fact which will be hailed by engineers everywhere with great satisfaction, as relieving them as a body from a heavy burden of suspicion and reproach.\"\n\nMoreover, contrary to some statements made in Pittsburgh since the disaster, they had found no evidence that the dam had ever been \"inspected\" periodically, occasionally, or even once, by anyone \"who, by any stretch of charity, could be regarded as an expert.\"\n\nIn other words, the job had been botched by amateurs. That they had been very rich and powerful amateurs was not considered relevant by the engineering journals, but so far as the newspapers were concerned that was to be the very heart of the matter. It was great wealth which now stood condemned, not technology.\n\nThe club had been condemned by the coroners' juries, General Hastings, and by the engineering experts. The newspapers made no effort to investigate the dam themselves, and only one or two made any effort to present the facts about the dam or to explain even in passing why it had failed. Nor did the editorial writers make an effort to remain even moderately objective until more information became available. The club was guilty, criminally guilty several papers were saying, and that was that. Unlike the Hungarian stories, this one, it seemed, would hold up. It was based on about as solid information as could be hoped for, and in terms of its emotional content, it was perhaps even stronger. Now across the country there arose a great howl of righteous indignation.\n\nFor everyone who had been asking how such a calamity could possibly happen in the United States of America, there now appeared to be an answer, and it struck at the core of something which had been eating at people for some time, something most of them had as yet no name for, but something deeply disturbing.\n\nFor despite the progress being made everywhere, despite the growing prosperity and the prospect of an even more abundant future, there were in 1889 strong feelings that perhaps not all was right with the Republic. And if the poor Hungarians of Johnstown were signs of a time to come when a \"hunky\" could get a job quicker than a \"real American,\" then the gentlemen of the South Fork Fishing and Hunting Club were signs of something else that was perhaps even worse. Was it not the likes of them that were bringing in the hunkies, buying legislatures, cutting wages, and getting a great deal richer than was right or good for any mortal man in a free, democratic country? Old-timers said that with every gain they made people were losing something. If that was so, people were beginning to think a little more about just what it was they might be losing, and to whom. And the more they thought about it, and especially the workingmen, the less they liked it.\n\nIt would be another three years before this kind of feeling would burst out in the terrible violence of the Homestead steel strike in Pittsburgh and Henry Clay Frick would nearly die of a bullet in his neck. And it would be another several years after that before public indignation over the power of the trusts, the giant corporations, and the men who ran them would erupt into public outrage. But the feeling was there in 1889, and it ran a great deal deeper than most people would have supposed. Certainly the language used by the press reflected a level of scorn and bitterness that would have been unthinkable a decade earlier.\n\nThe South Fork Fishing and Hunting Club was now described as \"the most exclusive resort in America,\" and its members were referred to as millionaries, aristocrats, or nabobs. According to the Cincinnati Enquirer not even vast wealth was enough to gain admission, unless it was hereditary. \"Millionaries who did not satisfy every member of the club might cry in vain for admission,\" the Enquirer wrote. \"No amount of money could secure permission to stop overnight at the club's hotel . . .\" The paper said that no one could visit the club without a permit, and called it \"holy ground consecrated to pleasure by capital,\" but added that no one would want to go there now, \"except to gaze a moment at the Desolate Monument to the Selfishness of Man. . . \" J. J. McLaurin, the Harrisburg newspaperman, who was otherwise relatively reliable in his reporting on the disaster, wrote: \"The club was excessively aristocratic, and so exclusive that Tuxedo itself might pronounce the Lorillard ideal a failure. The wealthy members never deigned to recognize the existence of the common clay of the neighborhood, farther than to warn intruders to keep off the premises.\"\n\nLike dozens of others, McLaurin was also infuriated over the idea that the lake had served as a summer resort. He wrote that \"50,000 lives in Pennsylvania were jeopardized for eight years that a club of rich pleasure-seekers might fish and sail and revel in luxurious ease during the heated term.\"\n\nFor an age which by no means looked upon pleasure as something to be expected in life, let alone life's chief objective, the very fact that the lake had been put there solely for pleasure seemed almost more than anyone could take; and in several editorials the writers seemed to imply that if the lake had served some other purpose, some practical purpose, then the tragedy would not have been quite so distressing.\n\n\"It is an aggravation of the calamity to reflect that the reservoir which gave way served no useful purpose, but merely ministered to the amusement of a gentleman's club composed of millionaires,\" wrote a small-town newspaper in New England. \"The dam served no useful end, beyond the pleasure of a few rich men,\" said the Daily Graphic in New York. And the Chicago Herald published a cartoon showing what were supposedly seven clubmen done up in loud-checked coats and diamond stickpins, tossing down champagne on the clubhouse porch, while in the valley below them Johnstown is being wiped out.\n\nLike several other papers, the Herald likened the clubmen to the Romans. \"These wealthy sportsmen, these pleasure-seekers, sat in a secure place, in the amphitheater, like the noble Roman spectators when they gave the signal when the wild beasts were to be admitted into the arena to rend the bodies of the human victims. The Pittsburgh pagans did not give the signal, but they were just as guilty in the fact that they were told that the massacre was about to occur and made no effort to stop it . . .\"\n\nThe effort alluded to here was the failure to remove the fish guards, which, very quickly, had come to symbolize everything repellent about the South Fork Fishing and Hunting Club. \". . . To preserve game for some Pittsburgh swells the lives of fifteen thousand were sacrificed,\" wrote the Herald.\". . . The ghosts of Johnstown are the ghosts of American labor that is dead.\" And a man by the name of Isaac Reed wrote a widely quoted poem which opened with the lines:\n\nMany thousand human lives\u2014\n\nButchered husbands, slaughtered wives,\n\nMangled daughters, bleeding sons,\n\nHosts of martyred little ones,\n\n(Worse than Herod's awful crime)\n\nSent to heaven before their time;\n\nLovers burnt and sweethearts drowned,\n\nDarlings lost but never found!\n\nAll the horrors that hell could wish,\n\nSuch was the price that was paid for\u2014fish!\n\nInterestingly, for all the abuse that was flung at the Pittsburgh people, very few newspapers ever went so far as to mention any specific names of members, and those that did mentioned only a half dozen or so. The Philadelphia Press, for all its superb coverage of what was going on in Johnstown, said hardly a word about the club or its members, and perhaps, as was hinted by another paper, because one club member, Calvin Wells, was a major stockholder in the Press.\n\nAs might be expected, the Pittsburgh papers were extremely cautious about printing anything untoward about the club, or, in some cases, were outright sympathetic toward the renowned members. The Pittsburgh Press, for example, took the position that too much scorn was being heaped on the club, since the dam had been built a long time back and the disaster, therefore, could as easily have happened at some earlier time. The Post-Gazette also felt the clubmen were being unfairly chastised. And Connelly and Jenks, authors of the so-called Official History of the flood, which was being written in Pittsburgh about that time, went out of their way to counteract popular images of opulent splendor at the lake. It was no center of pagan pleasure seeking or vulgar display, they wrote, but a place where the members of the club with their families and friends could \"rough it\" throughout the summer months. It was, they said, a comfortable, homelike place and as different from the \"ordinary fashionable summer resort\" as could be imagined. As for stories of any highhanded ways with the local people, well, \"The place was exclusive only in the sense that a private house or garden is of that character. There was no lofty disregard of other people's rights, nor any desire on the part of the members to set themselves above those around them. The club was a happy family party, and nothing more.\"\n\nForest and Stream, a national fishing and hunting magazine, took strong objection to the \"paragraphs hot with indignation\" that were being published. Such stuff was easy to write, said the magazine's editors, who rose to the defense of the club largely on the grounds that its members were sportsmen who appreciated the beauties of the natural world and so, therefore, were essentially good men. Also, in the opinion of the editors, it was nonsense to condemn the clubmen because their lake was meant for pleasure. \"To maintain a dam to form a lake for pleasure purposes is,\" they argued, \"an enterprise no less legitimate than to build a dam for running a mill wheel.\" If the warnings about the stability of the dam had gone unheeded, perhaps that had been because the members were so preoccupied with the joys of life in the out of doors. And, concluded Forest and Stream there ought to be some compassion for the members, who in their hearts must surely be suffering terribly.\n\nThere were many, too, who looked upon the disaster as a time of the apocalypse. Countless sermons on \"The Meaning of the Johnstown Flood\" were delivered in every part of the land for many Sundays running. One Pittsburgh preacher compared the \"wolf cry\" about the dam breaking to those in his congregation who tired of hearing him on the admonitions of the Lord. Another said that the lesson was to be ever prepared to meet thy Maker.\n\nIn New York the illustrious Reverend T. DeWitt Talmage, using the 93rd Psalm as his text (\"The floods have lifted up, O Lord, the floods have lifted up their voice; . . .\"), told an audience of some 5,000 that what the voice of the flood had to say was that nature was merciless and that any sort of religious attitude toward nature meant emptiness. \"There are those who tell us they want only the religion of sunshine, art, blue sky and beautiful grass,\" said Talmage. \"The book of nature must be their book. Let me ask such persons what they make out of the floods in Pennsylvania.\"\n\nNot a few ministers chose to talk about the spirit of sympathy that was sweeping the country. The New York Witness, a religious newspaper, went so far as to say there was a \"loving purpose of God hidden in the Flood,\" which turned a great many stomachs in Johnstown.\n\nBut the theme that set the most heads nodding in agreement was the old, old theme of punishment from on high. The story of Noah was read from thousands of pulpits. (\"And God looked upon the earth, and, behold, it was corrupt; . . . And God said unto Noah, The end of all flesh is come before me; . . .\") This was The Great American Flood; it had been a sign unto all men, the preachers said, and woe unto the land if it were not heeded. The steel town had been a sin town and so the Lord had destroyed it; for surely only a vile and wicked place would have been visited by so hideous a calamity.\n\nIt was a line of reasoning which many people were quick to accept, for at least it made some sense of the disaster. But it was a line of reasoning which met with much amusement in Johnstown, where, as anyone who knew his way about could readily see, Lizzie Thompson's house and several rival establishments on Green Hill had not only survived the disaster, but were going stronger than ever before. \"If punishment was God's purpose,\" said one survivor, \"He sure had bad aim.\"\n\nThere really was never much mystery in anyone's mind in Johnstown about the cause of the flood. George Swank spoke for just about everyone when he wrote, \"We think we know what struck us, and it was not the hand of Providence. Our misery is the work of man.\"\n\nThe Tribune had started publishing again on the 15th. Swank referred to the Pittsburgh men as \"the dudes\" and said that they wanted \"an exclusive resort where, in all their spotlessness and glory, they might idle away the summer days.\" The people of Johnstown, he said, had never had a chance. \"A rat caught in a trap and placed in a bucket would not be more helpless than we were.\"\n\nDozens of Johnstown people spoke out against the dam, telling the out-of-town newspapermen what an awful menace it had been and describing the dread shadow of fear it had cast over their lives, and nearly every last one of them refused to give his name. The one outstanding exception was Cyrus Elder, Johnstown's only member of the South Fork Fishing and Hunting Club, who said that he had never considered the dam structurally faulty and, contrary to what John Fulton was saying, that he knew of no serious concern about the dam among the Cambria Iron people.\n\nHaving lost his wife and one daughter, his home and just about everything he owned but the clothes on his back, Elder had as much cause as anyone to lash out at the club, and certainly not to do so was to go against the temper of the entire town. But he stuck to his position. He admitted that Johnstown people had long been edgy about the dam and said, \"Therefore, if anybody be to blame I suppose we ourselves are among them, for we have indeed been very careless in this most important matter and most of us have paid the penalty of our neglect.\" It was a brave and most unpopular thing to be saying in Johnstown. The statement was picked up immediately by the newspapers. But his line of reasoning was never given any serious consideration by the popular press, though Engineering and Building Record registered surprise that the men responsible for Johnstown's welfare, not to mention the officials of the Pennsylvania Railroad, with all that they had at stake, had not made sure that the lake over their heads was carefully built in the first place and properly maintained thereafter.\n\nThe railroad, for its part, remained quiet about any involvement it might have had in the dam's past, taking the position, no doubt, that its actions in bringing relief to Johnstown would speak a great deal louder and more favorably than any words\u2014which indeed they did. And once the engineering journals had established that the so-called engineers from the railroad who, according to statements made by Pitcairn, had kept a watch on the dam were in no way qualified to make any sort of intelligent judgment, then there was really very little more that the railroad could say.\n\nBut if the club's guilt had been established as far as the newspapers were concerned, there still remained the matter of paying the penalty, and that such a penalty should be paid seemed self-evident.\n\nOne newspaper after another said that the club should have to make amends for what had happened. Not a little facetiously, The New York Times wrote, \"Justice is inevitable even though the horror is attributable to men of wealth and station, and the majority of the victims the most downtrodden workers in any industry in the country.\"\n\nEven the Boston Post, which except for the Pittsburgh papers was about as conciliatory toward the club as any paper, said that the members had better be prepared to pay up. The Post, quite generously, stressed that the members must have acted as most men would have under the circumstances, \"trusting, perhaps not unjustifiably, to others\" with no thought of imperiling the lives of anyone. \"Even if all that is reported as to the construction of the dam proves true, there is the possibility that personally the owners were not guilty of the reckless parsimony attributed to them.\" Still, added the Post, \"If they were unable or failed to cope with forces of nature which they called into action, the responsibility is theirs, and as they have sown so must they reap, even if the harvest is the whirlwind.\"\n\nAnd behind every editorial was the suggestion of what the Sun said outright: \"If they [the club members] should be held liable in civil suits for damages it is probable that many, if not all of them, will be financially ruined.\"\n\nThe Pittsburgh men had by now given some $6,000 to the relief fund, in addition to the 1,000 blankets, but that did not seem to help their cause much. \"As they are almost all millionaires,\" wrote the New York Daily Graphic, \"the sum is not staggering, but shows that, while they were negligent, they are not heartless. . . . Yet they should do more than they have for the sufferers. It was through their indifference that this great disaster was precipitated upon the residents of the peaceful valley. Remorse, if nothing else, should lead them to alleviate to the fullest extent of their wealth the suffering they have caused.\"\n\nVery shortly thereafter several club members did, in fact, give generously; but, needless to say, it was far from the \"fullest extent of their wealth.\"\n\nHenry Clay Frick, through H. C. Frick Coke Company, gave $5,000. The Mellon family, through T. Mellon & Sons, gave $1,000. The Carnegie Company gave $10,000. There were several gifts of $1,000, $500, and $100. There was also one member who gave $15, and there were about thirty of them who never gave anything.\n\nThe members did suggest that the clubhouse could be used as a home for Johnstown orphans, but the offer was turned down with the excuse that the location was too inconvenient. There was also one member, S. S. Marvin, who actually went to Johnstown to see what he could do to help, and contrary to the many warnings published, he suffered no injuries, or even insults, from the people in the valley. Marvin had been appointed to one of the committees organized by the governor. He was in the baking business in Pittsburgh and had already contributed great quantities of bread. At Johnstown he looked about with absolute dismay and said, \"Johnstown is a funeral,\" an expression the newsmen were quick to pick up.\n\nAs for the other members, they grew increasingly cautious about saying anything. Phipps, Mellon, and Knox said nothing at all. Unger, who was staying with his daughter in Pittsburgh, tried hard to play down the importance of the fish guards, saying that they were only a few feet high. He also reminded the reporters that the dam had been originally built by the state, thus implying that the matter of responsibility, if pursued, might become a very complicated piece of business.\n\nFrick refused to see anyone from the press. Except for Carnegie, Frick was, of course, the best-known and most powerful of the members, and unlike Carnegie, Frick had already had his name published in the papers as one of the members. Moreover, he was, after Ruff, the ranking stockholder in the original organization and one of the few founding members still in the organization. In other words, he was one of the few people who had been involved in the club at the time Ruff made his renovation of the dam. So anything he might have to say would be of great interest, and possibly of great importance to how things might go for the club in the courts.\n\nBut Frick was not talking, and it was probably not so much that he was fearful of saying anything at that particular time as it was that he simply did not talk to the press ever, at any time. It was his standing policy. He was a highly uncommunicative sort anyway and, by nature, abhorred all forms of notoriety. He had no trust in newspapers, no liking for reporters, and talking to them, he was convinced, was bad for business. Only once in his life did he break his rule and speak freely to a reporter, but it was with the understanding that he could edit the copy, which he did, reducing a full column to exactly ten lines.\n\nIn the weeks following the disaster Frick made no public statements, nor did he ever in later years.\n\nCarnegie, on the other hand, had much to say, but never anything to suggest that he had had any connection with the club, and almost no one was ever the wiser, since it would not be for another year or more, when the story had been largely forgotten, that a complete membership list was divulged. Carnegie was in Paris attending the World's Fair at the time the disaster occurred. When a meeting of Americans had been called at the United States Legation, by the American Minister to France, Whitelaw Reid, it had been Carnegie who put forth the resolutions quickly adopted by the assembly. The people of Johnstown were to receive \"profound and heartfelt sympathy\" from their brethren across the Atlantic; they were also to be congratulated for their \"numerous acts of noble heroism\" and especially were they to be admired for the way they had \"preserved order during chaos\" through their own local self-government. How much Carnegie then contributed to the 40,000-odd francs that were pledged is not known.\n\nBut as for the South Fork Fishing and Hunting Club and any thoughts or feelings he may have had concerning its part in what had happened, Carnegie made no mention of that, and there would be none forthcoming. Carnegie wound up his affairs in Paris shortly thereafter, then left for his castle in Scotland, stopping off long enough in London to visit with the American Minister there, Robert Lincoln, the son of Abraham Lincoln.\n\nReporters in Pittsburgh, meanwhile, had been looking into the financial status of the South Fork sportsmen's association and had found, much to their dismay, that, for all the colossal wealth of the men who belonged to it, the club itself was capitalized for a mere $35,000 and there was a $20,000 mortgage still outstanding on the clubhouse. Since any future lawsuits would most likely be brought against the club, and not individual members, the chances for anyone collecting very much appeared to have diminished drastically. And just to be sure that no one missed this particular point, on June 12 James Reed once again granted the press an interview. Reed was a tall, sharp-faced man, quiet-spoken and scholarly looking. His practice included several of Pittsburgh's biggest concerns, as well as the Carnegie interests. His professional prestige was very high. What he had to say, therefore, was carefully taken down and later read with special interest.\n\nThe capital stock of the club would be the extent of the liability, he declared, if, that is, there were any liability, and in his opinion there was not. \"I have tried,\" he said, \"to divest myself of my identity with the South Fork Fishing Club to see if there could possibly be any grounds for a suit against the company or individual stockholders, and I am free to say I have been unable to find any. If a person was to come to me as an attorney and want me to bring suit against the company for damages resulting from the flood, I could not do so, because there are no grounds for such a suit.\"\n\nThen, in conclusion he said, \"As one of the stockholders I most certainly regret the sad occurrence, and I know the rest do; but I cannot see how the organization can be held legally responsible for the breaking of that dam.\"\n\nBut if he could not, there were others who could. At the end of July the first case brought against the South Fork Fishing and Hunting Club was filed at the Allegheny Court House in Pittsburgh, where the club had been originally incorporated. Mrs. Nancy Little and her eight children were suing the club for $50,000 for the loss of her husband, John Little, a woodenware salesman from Sewickley, Pennsylvania, who had been killed at the Hulbert House. The attorneys for the defense were, as had been expected, Knox & Reed, who filed a voluntary plea of not guilty. Then the case was put off for several months.\n\nEarly in August a group of Johnstown businessmen organized to sue the club. They raised some $1,300 to help meet expenses and hired John Linton and Horace Rose to start preparing their case.\n\nLater on, James and Ann Jenkins, backed by some businessmen of Youngstown, Ohio, brought suit for $25,000 for the loss of Mrs. Jenkins' father, mother, and brother, who had been drowned at Johnstown.\n\nThere were also suits against the Pennsylvania Railroad, the most important of which was one filed in September by a Mr. Farney S. Tarbell of Pittsburgh. Tarbell accused the railroad of negligence in the death of his wife and three children, who had been passengers on the Day Express. There were suits for lost luggage, and a Philadelphia company sued for the loss of ten barrels of whiskey, which had been looted from a freight car. This last case was won by the Philadelphia company when a conductor admitted that he had looked the other way when the whiskey was being taken. It was, as things turned out, the only case won by any of those who brought suit against either the club or the railroad.\n\nNot a nickel was ever collected through damage suits from the South Fork Fishing and Hunting Club or from any of its members. The Nancy Little case dragged on for several years, with the clubmen claiming that the disaster had been a \"visitation of providence.\" The jury, it seems, agreed.\n\nThere is no account of how things went in court, as it was not the practice to record the proceedings of damage suits. Nor is there any record of the Jenkins case, though there, too, the clubmen were declared not guilty.\n\nIn the Tarbell case the judge acquitted the railroad, also designating the disaster a \"providential visitation.\" And in Johnstown, after nearly two years of preparation, Colonel Linton and Horace Rose urged their clients to give up their suit, saying that it would almost certainly fail. The club had no assets, they argued, and there was no chance of winning unless individual negligence could be proved and that would be next to impossible since Ruff was dead. So Linton and Rose were paid $1,000 for their services and the suit was dropped.\n\nPerhaps the most frustrating attempt to recover some retribution was carried on by Jacob Strayer, the Johnstown lumber dealer, who set out to sue the club for $80,000. The case sat for years, in one county court after another, as the club kept seeking a change of venue due to local feelings. Then after waiting something like five years without hearing anything, Strayer discovered that his lawyer, unbeknownst to him, had settled out of court (taking $500) and had died shortly after that. Strayer next went bankrupt; the club was long since insolvent; and nothing more happened.\n\nHad the Little case or the Jenkins case been tried in Johnstown instead of Pittsburgh, it is possible that the decisions would have gone the other way, though in Johnstown there would have been small chance of finding twelve men to serve on a jury who would have been able to profess no bias against the club. In the judgment of lawyers who have examined the facts of the disaster in recent years, it also seems likely that had the damage cases been conducted according to today's standards the club and several of its members would have lost. It is even conceivable that some of those immense Pittsburgh fortunes would have been reduced to almost nothing. What the repercussions of that might have been is interesting to speculate. Possibly it would have delayed, perhaps even altered significantly, the nation's industrial growth.\n\nIn trying to evaluate why the cases went as they did, it is, of course, important to keep in mind the tremendous power of the people who were being sued. Their influence and prestige were such that few would have ever dared challenge them on anything. \"It is almost impossible to imagine how those people were feared,\" Victor Heiser would say many years later. They were the ruling class. It was that simple. The papers could rail away to their heart's delight (while seldom ever mentioning any names), but to actually strike out at the likes of the clubmen, even within the confines of the courts, was something else again. Practically speaking, the odds against winning against them were enormous, even had the cases been open and shut, which they were not.\n\nFor to prove that any living member of the club had been personally negligent would have been extremely difficult. And in all fairness, it is quite likely, as the Boston Post suggested, that the clubmen themselves knew no more about the structural character of the dam than did anyone in Johnstown. Like nearly every leading citizen of Johnstown, with the exceptions of Morrell and Fulton, they made the mistake of assuming that the men who had rebuilt the dam had known what they were doing.\n\nThey had been told that the dam was properly engineered and properly maintained, and so, as long as everything went all right, they had no cause to think otherwise.\n\nIn addition, there is no doubt that the storm which brought on the failure of the dam was without precedent; or at least that during the relatively short period of time in which there had been some semblance of civilization in the area (which was less than a hundred years), no one had recorded a heavier downpour. So for such skillful lawyers as Knox and Reed to have argued that the whole dreadful occurrence was an act of God would have been very easy, and judging by the outcome, they made their point with great effect.\n\nCertainly in the eyewitness testimony collected by the Pennsylvania Railroad in preparation for the suits it might have to face, repeated emphasis was placed on proving that no one had ever seen such a storm; and therefore if the \"reasonable precautions\" taken by railroad employees such as yardmaster Walkinshaw had turned out badly, it was only because the storm itself was so very unnatural. (It is also interesting to note that Pitcairn, in defense of Ruff's abilities, agreed openly that Ruff had no engineering training; Ruff was a lot better than any engineer, Pitcairn said.)\n\nStill the heart of the matter remained the dam itself, and judging from occasional comments that appeared in the papers, it seems that the club's defense was based on the proposition that the dam would have broken anyway\u2014even if it had had no structural flaws.\n\nApparently that was a convincing argument, despite the fact that several small dams which had been built near Johnstown to supply the city's drinking water had not failed as a result of the storm; and these, significantly enough, had been built under the personal supervision of Daniel J. Morrell.\n\nThe water in Lake Conemaugh, the attorneys for the defense must have claimed, was coming up so fast on the afternoon of the 31st, and would have continued to come up so fast, even had the dam held past 3:10, that eventually it would have started over the top, and once that happened, sooner or later, the best of earth dams would have failed. Even had there been no sag at the center, even if the spillway had been working to full capacity, the volume of water rushing into the lake was greater than what could get out, and so, they held, the end result would have been the same, except that it would have come later, and perhaps at night when the consequences would have been far more disastrous. It was a specious line of defense, for several reasons.\n\nFirst of all, there is no way of ascertaining for certain whether the inflow of water was such that it would have caused the lake to spill over the breast of the dam for an extended period of time had the dam been higher at the center, instead of lower, and had there been no obstructions in the spillway. There is also no way of telling whether there was a drop off in the volume of water pouring into the lake in the hours following the break. In other words, would there have been enough water rushing off the mountain to keep the lake at a level higher than the breast of the dam (a properly engineered dam, that is) for many hours? It seems unlikely. Moreover, it was clear from the engineering studies made, and from photographs taken of the dam after the break, that it was that part of the dam which had been repaired by Ruff and his crew which went out on the afternoon of the 31st.\n\nBut even if it were assumed, for the sake of argument, that the Ruff repairs were as solid as the original dam, that the spillway obstructions did not greatly diminish its capacity, and that there was no sag at the center to reduce even further the spillway's usefulness, there still remains one very obvious and irrefutable flaw in the dam and in any argument in its defense.\n\nBecause there were no longer discharge pipes at the base of the dam, the owners never at any time had any control over the level of the lake. If the water began to rise over a period of days or weeks to a point where it was becoming dangerously high, there was simply nothing that could be done about it. If, on the other hand, the pipes had still been there, as they were up until they were removed by Congressman Reilly, or if new pipes had been installed by Ruff, then through that abnormally wet spring of 1889 the men in charge of the dam, Unger, John Parke, and others, could have kept the lake at a safe level of say at least ten to twelve feet below the crest of the dam.\n\nSo while there is no question that an \"act of God\" (the storm of the night of May 30-31) brought on the disaster, there is also no question that it was, in the last analysis, mortal man who was truly to blame. And if the men of the South Fork Fishing and Hunting Club, as well as the men of responsibility in Johnstown, had in retrospect looked dispassionately to themselves, and not to their stars, to find the fault, they would have seen that they had been party to two crucial mistakes.\n\nIn the first place, they had tampered drastically with the natural order of things and had done so badly. They had ravaged much of the mountain country's protective timber, which caused dangerous flash runoff following mountain storms; they obstructed and diminished the capacity of the rivers; and they had bungled the repair and maintenance of the dam. Perhaps worst of all they had failed\u2014out of indifference mostly\u2014to comprehend the possible consequences of what they were doing, and particularly what those consequences might be should nature happen to behave in anything but the normal fashion, which, of course, was exactly what was to be expected of nature. As one New England newspaper wrote: \"The lesson of the Conemaugh Valley flood is that the catastrophes of Nature have to be regarded in the structures of man as well as its ordinary laws.\"\n\nThe dam was the most dramatic violation of the natural order, and so as far as a few rather hysterical editorial writers were concerned, the lesson of the flood was that dams in general were bad news. The writers took up the old line that if God had meant for there to be such things as dams, He would have built them Himself.\n\nThe point, of course, was not that dams, or any of man's efforts to alter or improve the world about him, were mistakes in themselves. The point was that if man, for any reason, drastically alters the natural order, setting in motion whole series of chain reactions, then he had better know what he is doing. In the case of the South Fork dam, the men in charge of rebuilding it, those who were supposed to be experts in such matters, had not been expert\u2014either in their understanding of what they did or, equally important, in their understanding of the possible consequences of what they did.\n\nWhat is more, the members of the club and most of Johnstown went along on the assumption that the people who were responsible for their safety were behaving responsibly. And this was the second great mistake.\n\nThe club people took it for granted that the men who rebuilt the dam\u2014the men reputed to be expert in such matters\u2014handled the job properly. They apparently never questioned the professed wisdom of the experts, nor bothered to look critically at what the experts were doing. It was a human enough error, even though anyone with a minimum of horse sense could, if he had taken a moment to think about it, have realized that an earth dam without any means for controlling the level of the water it contained was not a very good idea. The responsibility was in the hands of someone else, in short, and since that someone else appeared to be ever so much better qualified to make the necessary decisions and pass judgment, then why should not things be left to him?\n\nIn Johnstown most men's thoughts ran along the same general line, except that it was the clubmen who were looked upon as the responsible parties. And just as the clubmen were willing to accept on faith the word of those charged with the job of rebuilding the dam, so too were most Johnstown people willing to assume that the clubmen were dutifully looking to their responsibilities. If the dam was in the hands of such men as could build the mightiest industries on earth, who could so successfully and swiftly change the whole character of a city or even a country, then why should any man worry very much? Surely, those great and powerful men there on the mountain knew their business and were in control.\n\nIn the North American Review, in August 1889, in an article titled \"The Lesson of Conemaugh,\" the director of the U. S. Geological Survey, Major John Wesley Powell, wrote that the dam had not been \"properly related to the natural conditions\" and concluded: \"Modern industries are handling the forces of nature on a stupendous scale. . . . Woe to the people who trust these powers to the hands of fools.\"\n\nIt was, however, understandably difficult for the people of Johnstown ever to feel, like Cyrus Elder, that they too had been even partly to blame. Practically everyone felt that he had foreseen the coming catastrophe, and if he had not, like John Fulton, actually put anything down on paper, he, nonetheless, had been equally aware of the troubles with the South Fork dam and every bit as dubious about its future. That the members of the club were never required to pay for their mistakes infuriated nearly everyone in Johnstown and left a feeling of bitter resentment that would last for generations.\n\nAs for the club people, their summers at South Fork were over. The cottages sat high and dry along the vast mud flat which had been Lake Conemaugh and where, here and there, like the remains of some prehistoric age, stood the stumps of great trees that had been taken down more than fifty years earlier just before the dam had been built. By July grass had sprung up along South Fork Creek where it worked its way through the center of the old lake bed, and deer left tracks where they came down to drink.\n\nFor some time several cottages were occupied by Johnstown people. James McMillan, the plumber, and six or seven other men moved their families into the biggest of the houses, apparently with the consent of the owners, and, according to a notice in the Tribune at the end of July, the accommodations were as elegant as ever. But the owners themselves never came back, except for one, Colonel Unger, who not only returned, but lived out the rest of his life on the farm just above the remains of the dam. All the other property was broken up and sold off at a sheriff's auction.\n\nIn Johnstown the Cambria works had started up again by mid-July. It would be a long time before the furnaces were working to capacity, but nearly the full pay roll was being met and slowly things began to return to normal. Estimates were made on the total property damage (about $17 million). The banks opened. The Quicksteps were playing again. By August the Saturday night band concerts had been revived and a piano tuner had come to town. There was ice cream for sale; Haviland china was back on the shelves of those few stores that had been spared. A camera club was started, and the plumbers and steam fitters organized a union.\n\nIn September Mr. and Mrs. Andrew Carnegie came to town to see the flood damage, and before he left Carnegie had agreed to build a new library where the old one had been. In late fall the schools reopened.\n\nNew houses and shops were going up all over town. People who had fled the valley began coming back again. The Quinns were back by October, astonished to find how much had been accomplished and how bad the place still smelled. There was plenty of work still to be done, of course, and plenty of jobs, and it would stay that way for a long time. Just getting the place back to where it had been before would take five years or more.\n\nBut there were many who would leave Johnstown after the flood. For hundreds of people, like Victor Heiser, the disaster had deprived them of every meaningful connection with the place. Suddenly they were alone and there seemed no very strong reason for staying any longer, and particularly if they had ever had an ambition to see something of the world. Had his mother and father survived, Victor Heiser would have remained in Johnstown, probably, he would later speculate, to become a watchmaker. As it was, he left the valley within a year, and after working his way through medical school, spent most of his life as a public-health officer and physician, fighting disease around the world. He would also write a best-selling book on his experiences, An American Doctor's Odyssey, and would be credited with saving perhaps two million lives.\n\nThere were others who could stay no longer because of the memory of what had happened. And David Beale was among those who left out of bitterness over an experience during the frantic days which had followed the disaster. In Beale's case it had been a falling out with some of his congregation over the fact that he had turned the church into a morgue without authorization from the elders. There were plenty who rose to his defense, saying it had been the only intelligent and Christian thing to do under the circumstances and that somebody had to make the decision, but there were enough hard words exchanged to send Beale on his way to another charge in another town.\n\nFor years, too, there would be much speculation on how many of those people listed among the unfound dead were actually very much alive in some faraway place. It seemed reasonable enough to figure that some men, suddenly, in the first dim light of that terrible morning of June 1, had decided that here was an opportune time to quietly slip away to a new and better life. And if one of those names on the unknown list was somebody you had been close to, it was a whole lot pleasanter to think of him living on an apple farm in Oregon or tending bar in a San Francisco saloon than rotting away beneath six feet of river muck somewhere below Bolivar. Furthermore, such speculation seemed well justified when, eleven years later, in the summer of 1900, a man by the name of Leroy Temple showed up in town to confess that he had not died in the flood but had been living quite happily ever since in Beverly, Massachusetts. On the morning of June 1 he had crawled out of the wreckage at the bridge, looked around at what was left of Johnstown, then just turned on his heels and walked right out of the valley.\n\nStories of the flood would live on for years, and in time they would take on more the flavor of legends, passed along from generation to generation. Each family had its tales of where they had been when the wall of water came, where they ran to, who shouted what to whom, who picked up the baby, who went back for the horse, or how they had survived the night. Children who were only four or five years old at the time would live to be old men and women who would describe in the most remarkable detail how they had watched the flood strike the city (from a place where it would have been impossible to have seen the water) or how they had looked at their wrist watch at that exact moment (there were no wrist watches in 1889) and read (at age five!) that it was exactly such-and-such time. There would be stories of how grandfather tried to save an ax handle (\"of all things!\") or how Uncle Otto had thrown away his Bible when he saw what had happened. There would also be a great amount of durable gossip and some rather bad feelings about \"certain people\" who had somehow gotten their hands on more than their share of the relief money and how \"their families are rich to this day because of it.\" And at least one Irish undertaker from Pittsburgh was said to have made \"a positive fortune\" out of the disaster.\n\nThere would also, one day, be signs posted in saloons from one end of the country to the other saying: PLEASE DON'T SPIT ON THE FLOOR, REMEMBER THE JOHNSTOWN FLOOD. At Coney Island and in Atlantic City re-creations of the great disaster would be major attractions for many seasons. And \"Run for the hills, the dam has busted\" would be a standard comedy line the country over for years.\n\nIn Johnstown three babies born on the fateful day would grow up with the names Moses Williams, Flood C. Raymond, and Flood S. Rhodes.\n\nGeneral Hastings would later be elected governor largely because of the name he had made for himself at Johnstown; and when William Flinn later became the Republican boss of Pittsburgh and a state senator, he made it a practice to remind election-year audiences of the job he had done at Johnstown.\n\nTom L. Johnson, who later gave up a lucrative business career to become the highly progressive (some said \"socialistic\") mayor of Cleveland, would use the flood to make a case for his political philosophy. In his autobiography he would write at length about the disaster and its cause and how charity had vitiated local energies (he was still Moxham's man in this regard). The flood, he would conclude, was caused \"by Special Privilege,\" and: \"The need of charity is always the result of the evils produced by man's greed.\"\n\nIn after-dinner speeches at the Duquesne Club, Robert Pitcairn would recall the services rendered by the railroad and ask if a \"heartless corporation\" could have behaved so. Bill Jones never said much about what he did, though he was quoted as saying that perhaps Johnstown ought to rebuild on higher ground. When he returned to Pittsburgh from Johnstown, Jones had only two months to live. At the end of the summer he was killed when a furnace he was working on at the Braddock mill exploded.\n\nThe members of the South Fork Fishing and Hunting Club remained silent. The matter of their part in the flood was simply not mentioned, and as the years passed less and less was heard of it. In another generation it would be just about forgotten in Pittsburgh.\n\nThe more or less agreed-to attitude of Johnstown's business people was also that the flood should be forgotten as soon as possible. There was no sense dwelling on the thing. It was bad for the spirits, and it most certainly was harmful to business.\n\n\"It may be well to consider that the flood, with all its train of horrors, is behind us, and that we have hence forth to do with the future alone,\" said George Swank in the Tribune on the morning of June 1, 1892. It was his conclusion to a long description of the ceremonies held the day before at Grandview Cemetery. The whole city had been shut down and close to 10,000 people had gone up to the new burying grounds.\n\nExcept for the plot for the flood dead, Grandview was still very sparsely occupied. It had been started by Cyrus Elder, John Fulton, and others only a few years before the flood and was laid out a good distance from town up on some of the highest land for miles around. The idea was that here the dead would be safe from spring floods. The view was very grand indeed, stretching off in every direction as far as the eye could carry; but the trees blocked a direct look back down into the great amphitheater among the hills where Johnstown lay, and so the city was wholly concealed, and except for the distant sound from the mills, it was almost as though there was nothing even like a city anywhere near.\n\nOn the afternoon of the 31st, with the new governor present, and with Johnstown's first mayor, Horace Rose, officiating, a large granite monument was dedicated to the \"Unknown Dead Who Perished in the Flood at Johnstown, May 31, 1889.\"\n\nBehind the monument, arranged very precisely row on row, were 777 small, white marble headstones.\n\nThe unknown plot had been purchased by the Relief Commission and the bodies moved there from Nineveh, Prospect Hill, and half a dozen other places during the early fall of 1889. It had taken the time since to raise money for the monument and the nameless headstones. Actually, there were not quite a full 777 bodies buried in the plot; someone had decided to set out a few extra stones just to make an even pattern. But the effect on the immense throng gathered in the warm afternoon sunshine was very great. Against the long sweep of grass and the darker green of the bordering trees, the people stood in their funeral best, clustered in a dark, tight mass, strangely motionless and silent beneath the veiled monument. A few steps beyond, the carriages for the dignitaries were drawn up.\n\nThere were several speeches, the longest and best of which was by the governor, Robert Pattison, who, in offering a lesson to be learned from the disaster, said, \"We who have to do with the concentrated forces of nature, the powers of air, electricity, water, steam, by careful forethought must leave nothing undone for the preservation and protection of the lives of our brother men.\"\n\nThen the choir sang \"God Moves in a Mysterious Way\"; the monument was unveiled, and people started back along the winding road that led down into town.\n\nNEW YORK PUBLIC LIBRARY\n\nJohnstown was still sparsely settled when this map was made about the time of the Civil War. Woodvale (top) and Kernville (lower right) were mostly vacant lots. By 1889 population in the area had tripled.\n\nTHE CAMERA SHOP, JOHNSTOWN\n\nOne of the few surviving photographs of the South Fork Fishing and Hunting Club shows two lake-side boathouses, boardwalks, a rowboat planted with flowers (at lower left) and several cottage fronts.\n\nDAVID G. MCCULLOUGH\n\nThe Moorhead house, once among the finest \"cottages\" on Lake Conemaugh, stands today at the edge of St. Michael, a coal town that, years after the flood, grew up around the abandoned summer colony.\n\nUNIVERSITY OF PENNSYLVANIA\n\nThe heroic \"resident engineer\" at the club, John G. Parke, Jr.\n\nDAVID G. MCCULLOUGH\n\nAt left above, a small railing marks the crest of South Fork dam, the immense ends of which still stand above South Fork Creek. From railing to distant rooftops (the town of St. Michael) was once the northern end of Lake Conemaugh. Large roof is the old clubhouse, only half of which still stands. It is now a miners' bar and hotel.\n\nAndrew Carnegie\n\nFOUR MEMBERS OF THE SOUTH FORK FISHING AND HUNTING CLUB\n\nBROWN BROTHERS\n\nHenry Clay Frick\n\nBROWN BROTHERS\n\nPhilander C. Knox\n\nFROM AUTOBIOGRAPHY OF ANDREW CARNEGLE, HOUGHTON MIFFLIN COMPANY\n\nRobert Pitcairn\n\nDaniel J. Morrell\n\nFOUR OF JOHNSTOWN'S LEADING CITIZENS\n\nJohn Fulton\n\nTom L. Johnson\n\nCaptain Bill Jones\n\nJOHN AND AUGUST NILSON\n\nA wide-angle view of Johnstown taken on the day before the flood gives the valley a broader look than it actually has, makes the hills appear too low-lying, but shows such principal features as the Stony Creek (foreground), the Cambria works (where the smoke rises at far left), Prospect Hill (the rows of white houses to the right of the mills), and the gap (center) through which the wall of water rushed. Kernville is at right. Johnstown proper lies between the gap and the church steeples, and to the left of the river.\n\nThis view was taken from the same point several months after the flood and shows the extent of damage to the lower part of Johnstown (at left) and Kernville. Not shown are Woodvale (which sits in the middle of the gap), where the whole town was wiped out, or Cambria City. Also, though there are still hundreds of houses to be seen in the main part of town, only a relative few were still in one piece, or standing where they belonged. Green Hill, a refuge for thousands of flood survivors, rises on the right of the gap.\n\nTHE CAMERA SHOP, JOHNSTOWN\n\nWith dozens of displaced buildings and tons of debris piled up behind it, the big stone Methodist church (at upper left in the view above) stands unmoved, looking over the desolation of lower Johnstown (in the far distance) and the dim side of Prospect Hill (at upper right).\n\nLIBRARY OF CONGRESS\n\nA favorite subject for the swarms of photographers who rushed to cover the disaster was the house belonging to John Schultz. It had been neatly skewered by a huge tree and then dumped down near the Point. Six people were in it when the wave hit. Miraculously they all came out alive.\n\nLIBRARY OF CONGRESS\n\nThe photograph above is almost certainly a fake. Though hundreds of corpses were strewn among the wreckage, few were found looking quite so neat and clean as this barefoot \"victim,\" and by the time the photographers arrived any body so well exposed would have been long since removed.\n\nTHE CAMERA SHOP, JOHNSTOWN\n\nA view taken less than 24 hours after the flood shows the Cambria Iron offices (the big buildings on the left) and the swamped ruins of the city beyond. At far right is part of the depot. Two Cambria Iron locomotives stand at lower right, inside the high board fence that enclosed company grounds.\n\nWeeks after the flood, workmen tackle the last remains of the debris piled against the Pennsylvania Railroad's massive stone bridge.\n\nTHE CAMERA SHOP, JOHNSTOWN\n\nIn a view from above the Stony Creek, looking toward the Point (the stone bridge and the mills can be seen faintly in the distance), a slim island of surviving houses stands amid acres of mud and ruin.\n\nCOURTESY OF MRS. GERTRUDE QUINN SLATTERY\n\nGertrude Quinn, at the age of 5. This photograph was one of the few Quinn family possessions to survive the flood.\n\nCOURTESY OF DR. VICTOR HREISER\n\nVictor Heiser as he looked at 19, three years after the flood, when he had left Johnstown to begin his college education.\n\nFor Richard Harding Davis, the flood was the first big assignment in a long, colorful career as roving reporter and author.\n\nBROWN BROTHERS\n\nFor Clara Barton, \"Angel of the Battlefield,\" the flood was the first great test of her newly organized American Red Cross.\n\nLIBRARY OF CONGRESS\n\nWith their houses swept away, their money and belongings all gone, many families \"made do\" in rough hillside shelters such as these, built and furnished with the best they could scavenge from the devastated valley.\n\nAt the center of town, where the militia had set up camp, a young sentry stands guard while a survivor signs up for relief rations.\n\nTHE CAMERA SHOP, JOHNSTOWN\n\nThe body of a child is carried into the Adams Street schoolhouse, a temporary morgue where 301 bodies were recorded in the log books.\n\nAn artist's drawing of the broken dam, seen from inside the empty reservoir, shows the spillway (at far right) and the bridge that crossed it. The breach in the dam was about 420 feet across the top.\n\nHARPER'S WEEKLY\n\nAt the Pennsylvania depot survivors crowd one of the several commissaries set up by volunteers from Pittsburgh and other nearby towns.\n\nBROWN BROTHERS\n\nBROWN BROTHERS\n\nScenes such as the one above were repeated many times, as survivors searched among the dead, or suddenly found a lost loved one among the living. But the illustration above, supposedly depicting the demise of three \"Huns\" who had been caught robbing the dead, has no basis in fact. Popular as they were in the nation's press, lurid stories of rampant pillage and bloodshed were soon entirely discredited.\n\nDAVID G. MCCULLOUGH\n\nAt Grandview Cemetery, on a high hill above Johnstown, hundreds of plain marble headstones mark the graves of the flood's unknown dead.\n\nDAVID MCCULLOUGH was born in Pittsburgh, Pennsylvania, and educated there and at Yale. Author of John Adams, Brave Companions, Truman, Mornings on Horseback, The Path Between the Seas, and The Great Bridge, he has received the Pulitzer Prize twice (in 1993, for Truman, and in 2001, for John Adams), the Francis Parkman Prize, and the Los Angeles Times Book Award, and has twice won the National Book Award.\n\nCover design and illustration by Wendell Minor\n\nAuthor photograph by William B. McCullough\n\nVisit us online at www.simonandschuster.com\n\"We have no better social historian.\"\n\n\u2014The New York Times\n\n* * *\n\nREAD THE COMPLETE DAVID MCCULLOUGH COLLECTION\n\n* * *\n\n\"A narrative tour de force . . . expert research and detailed, graceful prose.\" \u2014Publishers Weekly\n\n* * *\n\n\"Combines a novelist's sense of drama with a scholar's meticulous attention to the historical record.\" \u2014The New York Times\n\n* * *\n\n\"Rich in revealing anecdotes and penetrating insights.\" \u2014The Washington Post\n\n* * *\n\n\"A full account of Roosevelt's rise to manhood . . . full of irrepressible vitality.\" \u2014The Denver Post\n\n* * *\n\n\"Full of giant-sized characters and rich in political skullduggery.\" \u2014The New York Times\n\n* * *\n\n\"It will entice a whole new generation of Francophiles.\" \u2014San Francisco Chronicle\n\n* * *\n\n\"That sort of work which brings us to the human center of the past.\" \u2014Los Angeles Times\n\n* * *\n\n\"A first rate example of the documentary method.\" \u2014The New Yorker\n\n* * *\n\n\"If you enjoy good stories well told about interesting people and places, you should read this book.\" \u2014The Plain Dealer (Cleveland)\n\n* * *\n\nDOWNLOAD YOUR COPIES TODAY!\n\n* * *\n\nVisit DavidMcCullough.com\n\n* * *\n\nALSO BY DAVID MCCULLOUGH\n\nThe Great Bridge\n\nThe Path Between the Seas\n\nMornings on Horseback\n\nTruman\n\nBrave Companions\n\nJohn Adams\n\nWe hope you enjoyed reading this Simon & Schuster eBook.\n\n* * *\n\nJoin our mailing list and get updates on new releases, deals, bonus content and other great books from Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## List of Victims\n\nTotal Number Lost, 2,209\n\nThis list, dated July 31, 1890, is the one printed in the Johnstown Tribune fourteen months after the Flood.\n\nGRAND VIEW CEMETERY.\n\n[Buried in private lots in Grand View.]\n\nAlexander, Arailia K., Broad street.\n\nAndrews, John, Sr., 57, John street.\n\nArther, Mrs. Alice, 29, Water street.\n\nBantley, William G., 36, Third Ward.\n\nBantley, Mrs. Ella, 30, Third Ward.\n\nBantley, George L., 6 months, Third Ward.\n\nBarbour, Mrs. Mary, 25, Woodvale.\n\nBarbour, Florence, 4, Woodvale.\n\nBarley, Mrs. Barbara, 56, Woodvale.\n\nBarley, Nancy, 29, Woodvale.\n\nBarley, Viola, 9, Woodvale.\n\nBeam, Dr. Lemon T., 55, Market street.\n\nBeam, Charles C., 4, Market street.\n\nBeam, Dr. W. C., 35, Locust street.\n\nBeam, Mrs. Clara, 32, Locust street.\n\nBeckley, E. E., 23, Main street.\n\nBending, Mrs. Elizabeth, 48, Locust street.\n\nBending, Jessie, 24, Locust street.\n\nBending, Katie, 15, Locust street.\n\nBeneigh, John C., 65, Cambria.\n\nBenford, Mrs. E. E., 63, Hulbert House.\n\nBenford, Maria, 34, Hulbert House.\n\nBenford, May, 26, Hulbert House.\n\nBenford, Louis, 30, Hulbert House.\n\nBenshoff, J. Q. A., 62, Somerset street.\n\nBenshoff, Arthur, 27, Somerset street.\n\nBowman, Nellie, 9, Haynes street.\n\nBowman, Charles H., 7, Haynes street.\n\nBowman, Frank P., 33, Woodvale.\n\nBowman, Emma, 28, Woodvale.\n\nBrinkey, Dr. J. C., 28, Franklin street.\n\nBrinkey, Elmer, 26, Hulbert House.\n\nBuchanan, John S., 69, Locust street.\n\nBuchanan, Mrs. Kate J., 63, Locust street.\n\nBuchanan, Robert L., 20, Locust street.\n\nConnelly, Maud, 6, Franklin.\n\nConstable, Philip E., 60, Broad street.\n\nCope, Mrs. Margaret, 65, Conemaugh.\n\nCope, Ella B., 28, Conemaugh.\n\nCooney, Mrs. Elizabeth.\n\nDavis, Mary Ann, 40, Woodvale.\n\nDavis, Thomas S., 59, Locust street.\n\nDavis, Mrs. Elizabeth.\n\nDavis, Mrs. Susan, 27, Millville.\n\nDavis, Clara, 8, Millville.\n\nDavis, Willie, 3, Millville.\n\nDavis, Eliza M.\n\nDavis, Margaret, E.\n\nDavis, Mrs. Cora B., 25, Water street.\n\nDavis, William L.\n\nDavis, Willard G.\n\nDavis, Mary G.\n\nDelaney, Mrs. Jessie, 29, Vine street.\n\nDelaney, Mrs. Ella A.\n\nDibert, John, 56, Main street.\n\nDibert, Blanche, 9, Main street.\n\nDixon, David, 40, Millville.\n\nDiller, Rev. Alonzo P., Locust street.\n\nDiller, Mrs. Marion, Locust street.\n\nDiller, Isaac, Locust street.\n\nDinant, Lola, Locust street.\n\nDorris, August.\n\nDrew, Mrs. Mark, 62, Millville.\n\nDrew, Mollie, 8, Conemaugh street.\n\nDuncan, Mrs. Sarah A., 23, Woodvale.\n\nDyer, Mrs. Nathan, 64, Somerset street.\n\nEck, Mary Ellen.\n\nEdwards, Mrs. Annie R.\n\nEldridge, Samuel B., Apple Alley.\n\nEldridge, Abram S., 34, Merchants' Hotel.\n\nEtchison, John, 44, Napoleon street.\n\nEvans, Mrs. William F., 63, Union street.\n\nEvans, Maggie, 11, Lewis Alley.\n\nEvans, Kate, 5, Lewis Alley.\n\nEvans, Mrs. Josiah, 36, Vine street.\n\nEvans, Maggie, 16, Vine street.\n\nEvans, Lake, 6, Vine street.\n\nEvans, Ira, 6 months, Vine street.\n\nEvans, Mrs. Maggie, 37, Vine street.\n\nEvans, Mrs. Ann.\n\nEvans, Sadie, 8, Vine street.\n\nEvans, Herbert, 3, Vine street.\n\nEvans, Pearl, 1, Vine street.\n\nEvans, Lizzie.\n\nFails, Francis.\n\nFenn, John, 35, Locust street.\n\nFenn, Genevieve, 9, Locust street.\n\nFenn, Bismarck S., 3, Locust street.\n\nFindlay, Lulu, 16. Woodvale.\n\nFisher, John H., 55, Main street.\n\nFisher, Mary J., 46, Main street.\n\nFisher, Emma K., 23, Main street.\n\nFisher, Ida, 19, Main street.\n\nFisher, Madge, 10, Main street.\n\nFisher Minnie, 21, Main street.\n\nFisher, George, 12, Main street.\n\nFisher, Frank, 9 months.\n\nFleck, Leroy Webster.\n\nFox, Martin, 51, Conemaugh.\n\nFrank, John, Sr., 58, Washington street.\n\nFrank, Mrs. Eliza, 44, Washington street.\n\nFrank, Katie, 19, Washington street.\n\nFrank, Emma, 17, Washington street.\n\nFrank, Laura, 12, Washington street.\n\nFredericks, Mrs. A. G., 45, Millville\n\nFredericks, Mrs. Sarah A.\n\nFrederick, Edmon.\n\nFritz, Maggie, 26, Conemaugh.\n\nFritz, Kate, 22, Conemaugh.\n\nFronheiser, Mrs. Kate, 33, Main street.\n\nFronheiser, Bessie, 8, Main street.\n\nFronheiser, Catherine, 3 months, Main street.\n\nGageby, Mrs. Rebecca, 74, Jackson street.\n\nGageby, Sadie, 27, Jackson street.\n\nGallagher, Prof. C. F., 34, Main street.\n\nGallagher, Lizzie, 29, Main street.\n\nGard, Andrew, Jr., 25, Main street.\n\nGeddes, George, 47, Woodvale.\n\nGeddes, Marion, 17, Woodvale.\n\nGeddes, Paul, 15, Woodvale.\n\nGilmore, Mrs. Margaret, 40, Union street.\n\nGilmore, Anthony, 8, Union street.\n\nGilmore, Llewelyn, 6, Union street.\n\nGilmore, Willy, 4, Union street.\n\nGilmore, Clara, 2, Union street.\n\nGolde, Mrs. Henry, 32, Walnut street.\n\nGriffin, Mary, 47, Walnut street.\n\nHager, Mary E., 33, Washington street.\n\nHager, Mrs. Emma.\n\nHamilton, Jacob, 70, Bedford street.\n\nHamilton, Jessie, 30, Bedford street.\n\nHamilton, Laura, 24, Bedford street.\n\nHamilton, Alex, Jr., 35, Locust street.\n\nHamilton, Mrs. Alex, 30, Locust street.\n\nHamilton, Marion, Locust street.\n\nHamilton, Louther J.\n\nHammer, George K., 19, Moxham.\n\nHarris, Mrs. William T.\n\nHarris, John, 3, Market street.\n\nHarris, Margaret, 47, Market street.\n\nHarris, Wm. L., 23, Market street.\n\nHarris, Winnie, 21, Market street.\n\nHarris, Maggie A., 19, Market street.\n\nHarris, Sarah, 16, Market street.\n\nHarris, Frank, 12, Market street.\n\nHaynes, Walter B., 22, Horner street.\n\nHaynes, Laura C., 20, Horner street.\n\nHennekamp, Rebecca, 24, Franklin street.\n\nHennekamp, Oscar E., 2, Franklin street.\n\nHennekamp, S. E., 27, Lincoln street.\n\nHeidenthal, Harry R.\n\nHeiser, George, 50, Washington street.\n\nHeiser, Mrs. George, 48, Washington street.\n\nHelsel, George, 16, Johns street.\n\nHite, Mrs. Ella, 37, Somerset street.\n\nHochstein, Henry, 30, Conemaugh.\n\nHoffman, Benjamin F., 56, Market street.\n\nHoffman, Mrs. Mary, 43, Market street.\n\nHoffman, Bertha, 19, Market street.\n\nHoffman, Minnie, 16, Market street.\n\nHoffman, Marion, 14, Market street.\n\nHoffman, Florence, 10, Market street.\n\nHoffman, Joseph, 8, Market street.\n\nHoffman, Helen, 4, Market street.\n\nHoffman, Freda, 1, Market street.\n\nHoffman, Mrs. Mary, 41, Washington street.\n\nHohnes, Mrs. Ann, 24, Conemaugh.\n\nHohnes, Mrs. Elizabeth, 80, Lincoln street.\n\nHohnes, Julia, 18, Conemaugh.\n\nHollen, Charles.\n\nHowe, Thomas J., Bedford street.\n\nHowells, William, 59, Union street.\n\nHowells, Maggie, 23, Union street.\n\nHowells, Mrs. Ann.\n\nHughes, Maggie, 22, Sugar Alley.\n\nHughes, Evan, 57, Sugar Alley.\n\nHumm, Geo. C., Merchants' Hotel.\n\nHumphreys, William, 18, Levergood street.\n\nJacobs, Lewis, 41, Cambria City.\n\nJames, Mrs. Ellen M., 42, Main street.\n\nJames, Mollie, 13, Market street.\n\nJones, Mary J.\n\nJones, Reuben, 1, Main street.\n\nJones, James, 32, Conemaugh.\n\nJones, Ann, 9, Conemaugh.\n\nJones, Mrs. W. W.\n\nJones, Edgar R.\n\nJones, Mrs. Mary A., 52, Pearl street.\n\nJones, Eliza, 15, Pearl street.\n\nKarns, Joseph, 50, Locust street.\n\nKeedy, Harry C., 30, Millville.\n\nKeedy, Mrs. Mary, 32, Millville.\n\nKegg, William E., 17, Locust street.\n\nKeiper, Essie J., 24, Franklin.\n\nKeiper, Ralph, 5 months, Franklin.\n\nKennedy, H. D., 32, Stonycreek street.\n\nKeyser, Mrs. John.\n\nKeyser, Ralph.\n\nKeighly, Mary L., 52, Main street.\n\nKidd, Joshua, 65, Walnut street.\n\nKidd, Mrs. Sarah, 60, Walnut street.\n\nKirkbride, Mahlon, 33, Hager Block.\n\nKirkbride, Mrs. Ida, 30, Hager Block.\n\nKirkbride, Luida, 8, Hager Block.\n\nKirlin, Thomas, 40, Conemaugh street.\n\nKirlin, Eddie, 12, Conemaugh street.\n\nKirlin, Frank, 5, Conemaugh street.\n\nKnorr, Mrs. Mary, 45, Jackson street.\n\nKnorr, Emma, 16, Jackson street.\n\nKnorr, Bertha, 14, Jackson street.\n\nKnox, Mrs. Thomas, 45, Somerset street.\n\nKoenstyl, Samuel.\n\nKratzer, Mrs. Mary, Market street.\n\nKuntz, Wade, 21, Morris street\n\nLambreski, Mrs. Barbara, 35, Cambria.\n\nLambreski, Mary, 6, Cambria.\n\nLambreski, John, 4, Cambria.\n\nLayton, Mrs. Elvira.\n\nLayton, William, 58, Broad street.\n\nLayton, Mrs. William, 53, Broad street.\n\nLayton, May, 22, Broad street.\n\nLayton, David, Broad street.\n\nLayton, Ella, Broad street.\n\nLee, Dr. J. K., 48, Main street.\n\nLeitenberger, Mrs. Leah, 68, Vine street.\n\nLeitenberger, Nancy, 48, Vine street.\n\nLeitenberger, Ella, 35, Vine street.\n\nLeitenberger, Eliza, 46, Vine street.\n\nLenhart, Samuel, 58, Clinton street.\n\nLenhart, Mrs. Mary, 56, Clinton street.\n\nLenhart, Annie E., 20, Clinton street.\n\nLenhart, Emma J., 17, Clinton street.\n\nLenhart, Katie M., 13, Clinton street.\n\nLewis, Mrs. Ann.\n\nLewis, Ananias, 41, Millville.\n\nLevergood, Mrs. Jane, 75, Bedford street.\n\nLevergood, Lucy, 45, Bedford street.\n\nLewis, Orrie P., 6, Millville.\n\nLewis, James.\n\nLinton, Minnie, 20, Lincoln street.\n\nLitz, Mrs. John, 74, Morris street.\n\nLlewellyn, Mrs. Margaret, 37, Walnut street.\n\nLlewellyn, Annie, 5, Walnut street\n\nLlewellyn, Sadie, 8, Walnut street\n\nLlewellyn, Herbert, 3, Walnut street.\n\nLlewellyn, Pearl, 1, Walnut street.\n\nLuckhart, Louis, 69, Main street.\n\nLuckhart, Mrs. Adolph, 26, Main street.\n\nLudwig, Charles.\n\nLudwig, Henry G., 34, Bedford street.\n\nLudwig, Mrs. Kate, 35, Bedford street.\n\nMangus, Martha.\n\nMarbourg, Dr. H. W., 56, Market street.\n\nMcDowell, Geo., 29, Pearl street.\n\nMcDowell, Mrs. Agnes, 33, Pearl street.\n\nMcDowell, Lilly, 3, Pearl street.\n\nMcDowell, Georgia.\n\nMcClelland, Mrs. Jennie, 34, Sherman street.\n\nMcConaghy, Mrs. Kate, 68.\n\nMcConaughy, James P., 72, Walnut street.\n\nMcConaughy, Mrs. Caroline M., 65, Walnut street.\n\nMcConaughy, Wallace, 25, Walnut street.\n\nMcConaghy, Robert W.\n\nMcKee, John, 21, Bedford street.\n\nMcKinstry, Mrs. Mary C., 45, Hager Block.\n\nMcKinstry, Annie R., 14, Hager Block.\n\nMcVay, Lizzie, 20, Locust street\n\nMerle, Elmer E.\n\nMoore, Mrs. Charlotte L.\n\nMeyers, Mrs. Elizabeth, 55, Washington street.\n\nMeyers, Mary, 24, Washington street.\n\nMeyers, Mrs. Catherine, 31, Millville.\n\nMeyers, John, 3, Millville.\n\nMiller, Jessie B., 16, Somerset street.\n\nMorgan, Mrs. Charlotte, 49, Millville.\n\nMorgan, Martha, 13, Millville.\n\nMorgan, Minnie, 4, Millville.\n\nMurr, Charles, 41, Washington street.\n\nMurr, Maggie, 14, Washington street.\n\nMusser, Charles, 23, Main street.\n\nNixon, Mrs. Elizabeth, 39, Woodvale.\n\nNixon, Emma R., 16, Woodvale.\n\nNixon, Eddie, 8, Woodvale.\n\nNoro, Kate.\n\nOwens, Gladies, 5 months, Conemaugh street.\n\nOwens, Thomas, 10, Conemaugh street.\n\nOwens, William, 65, Market street.\n\nOwens, Annie.\n\nOwens, Mrs. Mary Ann, 31, Conemaugh street.\n\nOwens, Mary, 8, Conemaugh street.\n\nOyler, Mrs. Mary R., 27, Woodvale.\n\nOyler, John R., 6, Woodvale.\n\nParke, Mrs. Agnes J., 56, Bedford street.\n\nParke, William E., Bedford street.\n\nParsons, Mrs. Eva M., 23, Locust street.\n\nPenrod, William H., 59, Conemaugh.\n\nPeyton, John W., 65, Clinton street.\n\nPeyton, George A., 19, Clinton street.\n\nPeyton, Marcellus K., 16, Clinton street.\n\nPeyton, Julia F., 13, Clinton street.\n\nPhillips, Mrs. Jane M., 68, Market street.\n\nPike, William W., 50, Haynes street.\n\nPike, William W., Jr., 15, Haynes street.\n\nPike, S. Bowen, 10, Haynes street.\n\nPoland, Walter, 5, Market street.\n\nPoland, Frederick, 3, Market street.\n\nPotter, Joseph R., 63, Woodvale.\n\nPotter, Mrs. Sarah, 59, Woodvale.\n\nPotter, Nora G., 17, Woodvale.\n\nPotts, Miss Jane E., 47, Walnut street.\n\nPowell, Richard, 4 weeks, Vine street.\n\nPowell, George, 1\u00bd, Vine street.\n\nPritchard, Henry, 62, Market street.\n\nProsser, Fannie, 22, Market street.\n\nProsser, Bessie, 19, Market street.\n\nProsser, Maria.\n\nPurse, Mary L., Market street.\n\nRaab, George, 44, Clinton street.\n\nRabb, Mrs. George, 88, Clinton street.\n\nRabb, Norma, 16, Clinton street.\n\nRaab, Lizzie, 24, Washington street.\n\nRaab, Emilia, 20, Washington street.\n\nRaab, John C.\n\nRaab, Ella.\n\nRainey, Mrs. Lizzie L., 25, Bedford street.\n\nRainey, Parke, 1\u00bd, Bedford street.\n\nRandolph, George F., 26, Beaver Falls.\n\nReibert, Julius, Washington street.\n\nReese, Sarah, 10, Conemaugh street.\n\nReese, John, 2, Conemaugh street.\n\nReese, Mrs. J. W. Reese, Samuel.\n\nReese, Idris, 3, Vine street.\n\nReese, Gertie.\n\nReese, Mrs. Mary D., 74, Market street.\n\nRepp, Mrs. Catherine, 26, Sherman street.\n\nRhodes, Link, 26, Somerset street.\n\nRhodes, Ellen, 20, Somerset street.\n\nRhodes, Clarence, 10 months, Somerset street.\n\nRipple, Jackson, 34, Apple alley.\n\nRoberts, Howard J., 59, Walnut street.\n\nRoberts, Mrs. Howard J., 50, Walnut street.\n\nRoberts, Otis, 23, Walnut street.\n\nRoberts, Mrs. Lucinda H., 81, Main street.\n\nRobinson, Thomas, 60, Woodvale.\n\nRodgers, Mrs. D. L.\n\nRodgers, Mrs. Rose, 48, Millville.\n\nRoland, Louis, 31, Conemaugh.\n\nRoland, Lizzie, 29, Conemaugh.\n\nRosensteel, James M., 50, Woodvale.\n\nRose, Harry G., 29, Locust street.\n\nRoth, Mrs. Kate, 27, Bedford street.\n\nRoth, John, 38, Potts street.\n\nSchoff, Mrs. E. T., 32, Clinton street.\n\nSchotz, Mrs. Elizabeth, 63, Union street.\n\nSchotz, Annie, 23, Union street.\n\nSchotz, Jennie, 21, Union street.\n\nSchubert, C. T., 39, Stonycreek street.\n\nSeibert, Henry, 58, Woodvale.\n\nShaffer, Jacob, 47, Cambria.\n\nShulteis, Henry, 26, Potts street.\n\nShumaker, John S., 11, Locust street.\n\nShumaker, Edith M., 7, Locust street.\n\nShumaker, Irene G., 5, Locust street.\n\nShumaker, Walter S., 2, Locust street.\n\nSlick, George R., 60, Stonycreek street.\n\nSmith, Mr., 54, Cambria.\n\nSmith, Mrs. Sarah, 72, Walnut street.\n\nStremel, Julius R., 21, Washington street.\n\nStreum, John, 63, Locust street.\n\nStufft, J. Wesley, 37, Woodvale.\n\nStufft, Mrs. J. W., 39, Woodvale.\n\nSuder, Homer, 7. Millville.\n\nSt. John, Dr. C. P., 32, Hulbert House.\n\nStophel, Mrs. Maggie, 21, Baumer street.\n\nStophel, Frank Earl, 4, Baumer street.\n\nStophel, Bertha, Hulbert House.\n\nSwank, Mrs Ella, 29, Main street.\n\nSwank, Jennie, 15, Bedford street.\n\nSwank, Jacob, 61, Bedford street.\n\nSwank, Mrs. Catherine, 57, Bedford street.\n\nSwank, Maud, 11, Napoleon street.\n\nSwank, Fred B., 10, Bedford street.\n\nSwank, Susan, 8, Napoleon street.\n\nSwank, Mrs. Neff, 31, Napoleon street.\n\nSwank, Samuel, 5, Napoleon street.\n\nSwank, Edna, 3, Napoleon street.\n\nStatler, Mrs. Amelia, 51, Park Place.\n\nStatler, May, 23, Park Place.\n\nStatler, Frank E., 17, Park Place.\n\nTeeter, Mrs. Mary, 83, Locust street.\n\nTittle, Cyrus P., 53, Broad street.\n\nTradenick, Edward, 18, Union street.\n\nTurner, May, 15, Main street.\n\nTyler, Jno. T., 29, Stonycreek Township.\n\nThoburn, Thomas, 17, Millville.\n\nThoburn, Jennie, 7, Millville.\n\nThomas, Mrs. Mary A., 39, King street.\n\nThomas, Ida, 7, King street.\n\nUnverzagt, George, Sr., 67, Main street.\n\nUnverzagt, George, Jr., Main street.\n\nUpdegraff, Samuel, 15, Woodvale.\n\nViering, Mrs. Louisa, 38, Conemaugh.\n\nViering, Lizzie, 20, Conemaugh.\n\nViering, Henry, 14, Conemaugh.\n\nViering, Herman, 1, Conemaugh.\n\nVinton, Margaret, 8, Jeannette, Pa.\n\nVon Alt, Mrs. Catherine, 80, Washington street.\n\nWagoner, Dr. George, 63, Market street.\n\nWagoner, Lizzie, 20, Market street.\n\nWagoner, Mrs. Mary L.\n\nWagoner, Frances E., 18, Market street.\n\nWagoner, Cora M., Market street.\n\nWenner, Carl, 32, Locust street.\n\nWenner, Mrs., Locust street.\n\nWenner, Mary, 1, Locust street.\n\nWeaver, Mrs. Sue D., 27, Market street.\n\nWeaver, Martha B., 15 months, Market street.\n\nWeakland, John W., 30, Napoleon street.\n\nWerry, Thomas Albert, 17, Chestnut street.\n\nWhite, Mrs. Mima.\n\nWhite, Mrs. Ella, 34, Union street.\n\nWhite, Mrs. Margaret E.\n\nWhite, Mary P., 20, Market street.\n\nWhite, Maggie, 31, Union street.\n\nWild, Jacob, 72, Main street.\n\nWild, Mrs. Jacob, 58, Main street.\n\nWild, Bertha, 16, Main street.\n\nWilliams, Maggie, 26, Lewis alley.\n\nWilliams, Joseph M., 22, Conemaugh street.\n\nWilliams, William J., Union street.\n\nWilliams, Carrie E., 20, Woodvale.\n\nWorthington, Mrs. Richard, 28, Conemaugh st.\n\nWorthington, Richard, Jr., 1, Conemaugh st.\n\nWorthington, Mamie, 7, Conemaugh street.\n\nWorthington, Annie, 4, Conemaugh street.\n\nYoung, Katie.\n\nYoung, Emil, 48, Levergood street.\n\nYoung, Frank, 16, Levergood street.\n\nYoung, August, 29, Main street.\n\nYoung, Andrew C., 36, Broad street.\n\nZimmerman, Emma, 16, Bedford street.\n\nZimmerman, Theo. F., 34, Locust street.\n\nSANDYVALE CEMETERY.\n\nAbler, August, 28, Conemaugh.\n\nAbler, Mrs. Louisa, 31, Conemaugh.\n\nAbler, George, 11, Conemaugh.\n\nBaldwin, George, 69, Apple alley.\n\nBishop, Charles, 45, Woodvale.\n\nBrindle, Mollie, 25, Conemaugh.\n\nClark, John.\n\nDavis.\n\nDavis.\n\nDavis.\n\nEberle, Joseph, 63, Conemaugh.\n\nForbes, Mrs. Rachael, 38, Pearl street.\n\nForbes, Harry E., 10, Pearl street.\n\nFredericks, Mrs. Annie E., 78, Vine street.\n\nGray, S. Taylor, 37, Woodvale.\n\nGallagher, C. F.\n\nGallagher, Mrs. C. F.\n\nGreenwood.\n\nGreenwalt, Mrs.\n\nGreenwalt, child.\n\nGiven, Jane, Millville.\n\nGiven, Benj. F., Millville.\n\nGreenwood, Jennie, 17, Cambria.\n\nGreenwood, Geo., 55, Cambria.\n\nHammer, Daniel, Railroad street.\n\nHesselbein, Chas., 27, Conemaugh.\n\nHesselbein, Lewis, 23, Conemaugh.\n\nHite, Samuel, 26, Woodvale.\n\nHoffman, Gottfried, 40, Washington street.\n\nHoffman, Harry, 6, Washington street.\n\nHoffman, Daniel.\n\nHoffman, Godfrey, 41, Washington street.\n\nHoffman, Lizzie, 16, Washington street.\n\nHoffman, Mrs. Conrad, 38, Market street.\n\nHoffman, Charles B., 16, Market street.\n\nHoffman, Willie, 15, Market street.\n\nHoffman, Annie, 11, Market street.\n\nHughes, Emma, 26, Potts street.\n\nJones, Thomas, 50, Woodvale.\n\nJones, Mary W., 21, Woodvale.\n\nJones, Richard, 46, Woodvale.\n\nJones, child of Richard, Woodvale.\n\nJones, Clara. 6, Woodvale.\n\nKimpel, Christ., 47, Clinton street.\n\nKnee, Geo. D., 54, Conemaugh.\n\nMeyers, Mrs. Mary, 69, Cambria.\n\nMorgan, Job, 50, Walnut street.\n\nMcClarren, Samuel, 49, Cambria.\n\nMcClarren, Mrs. Jane, 42, Cambria.\n\nMcClarren, Smith, 22, Cambria.\n\nMcClarren, Jno. J., 19, Cambria.\n\nMcClarren, James, 4, Cambria.\n\nPeppler, Wm., 20, Conemaugh.\n\nRaab, Geo. C., 28, Washington street\n\nReese.\n\nReese.\n\nReese.\n\nReese.\n\nRecke, Alex., 35, Washington street.\n\nScheetz, Jacob, 61, Clinton street.\n\nSchnable, Conrad, 38, Baumer street.\n\nSchnable, John, 20, Main street.\n\nStahr, Fred.\n\nStrayer, Mrs. Elizabeth, 47, Market street.\n\nStrayer, Cora, 17, Market street.\n\nStrayer, Bertha, Market street.\n\nTeeter, Mrs. Thomas, Mrs. Edward, Woodvale.\n\nThomas, Edward, 49, Woodvale.\n\nThomas, Lydia, 12, Woodvale.\n\nThomas, Frank, 8, Woodvale.\n\nTross, Mrs. Margaret, 39, Woodvale.\n\nUnverzagt, Lizzie, Washington street.\n\nUnverzagt, Minnie, 27, Washington street.\n\nWill, Casper, 45, Bedford street.\n\nWier, Frank A., 18, Cambria.\n\nWillower, Miss Bella, Somerset street.\n\nWehn, Mrs. Rachel, 57, Main street.\n\nLOWER YODER CATHOLIC CEMETERY.\n\nBlair, Mrs., 50, Woodvale.\n\nBopp, Jacob, 32, Broad street.\n\nBracken, Katie, 21, Woodvale.\n\nBracken, Minnie, 19, Woodvale.\n\nBridges, Chas., 2, Cambria.\n\nBridges, Emma, 18, Cambria.\n\nBrown, Peter, 65, Woodvale.\n\nBrown, Thomas, 24, Woodvale.\n\nBrown, Emma, 20, Woodvale.\n\nBrown, Gertrude, 17, Woodvale.\n\nByrne, John, 32, Hulbert House.\n\nByrne, Ella, 24, Hulbert House.\n\nCarroll, Mrs. Bridget, 70, Conemaugh.\n\nCarroll, Thomas, 30, Conemaugh.\n\nCarroll, Rose, 20, Conemaugh.\n\nClark, Mrs. J. B., 39, Conemaugh.\n\nCronin, Daniel, 50, Vine street.\n\nCullen, James, 55, Cambria.\n\nCullen, Mrs. Ann, 50, Cambria.\n\nCullen, Mrs. Alice, 48, Cambria.\n\nCullen, Annie, 20,| }| Pro'bly dupl'ted  \n---|---|---  \nCullen, Annie, 20,  \n{| Cambria.  \n---|---  \nLocust st.\n\nCush, Mrs. Ann, 55, Cambria.\n\nCush, Daniel, 33, Cambria.\n\nCush, Joseph, 19, Cambria.\n\nCush, Mrs. Tillie, 20, Cambria.\n\nDaily, Mrs. Ann, 60, Locust street.\n\nDaily, Frank, 30, Locust street.\n\nDegnan, Mrs. Mary, 60, Cambria.\n\nDowns, Mrs. Catherine, 55, Millville.\n\nDowns, Mary, 32, Millville.\n\nDowns, Katie, 28, Millville.\n\nDowling, Mrs. Catherine, 42, Market street.\n\nDowling, Mary E., 21, Market street.\n\nDunn, Mary Ann, 25, Prospect.\n\nEarly, Mary, 22, Woodvale.\n\nFitzpatrick, Mrs. Peter, 28, Cambria.\n\nFitzpatrick, Ella, 6, Cambria.\n\nFitzpatrick, Mary, 3, Cambria.\n\nFitzharris, Christ, 42, Franklin street.\n\nFitzharris, Mrs. Margaret, 40, Franklin street.\n\nFitzharris, Christ, Jr., 14, Franklin street.\n\nFitzharris, John, Jr., 12, Franklin street.\n\nFitzharris, Maggie M., 9, Franklin street.\n\nFitzharris, Gertie, 5, Franklin street.\n\nFitzharris, Katie, 7, Franklin street.\n\nGaffney, Catherine, 2, Cambria.\n\nGaffney, John, 4, Cambria.\n\nGallagher, Mrs. Margaret, 32, Washington street.\n\nGallagher, Thomas, 4, Washington street.\n\nGarvey, Bernard, Sr., 62, Cambria.\n\nGrady, Mrs. Abbie, 60, Cambria.\n\nHalleron, May, 5, Washington street.\n\nHayes, Mrs. Jane, 32, Cambria.\n\nHayes, Michael, 12, Cambria.\n\nHayes, Mary, 8, Cambria.\n\nHayes, Rose, 7, Cambria.\n\nHayes, John, 6 Cambria.\n\nHart, Eliza.\n\nHarrigan, Ella, 22, Hulbert House.\n\nHoward, James B., 45, Conemaugh.\n\nHowe, Mrs. Edward, 50, Railroad street.\n\nHowe, Mrs. Bridget, 48, Cambria.\n\nHowe, Maggie, 24, Cambria.\n\nHowe, Lizzie, 22, Cambria.\n\nHowe, Rose, 19, Cambria.\n\nHowe, Gertrude, 13, Railroad street.\n\nKane, John, 20, Cambria.\n\nKane, Mary, 18, Cambria.\n\nKinney, Mrs. Mary, 50, Washington street.\n\nKinney, Mary Ellen, 12, Washington street.\n\nKirby, Wm., 32, Washington street.\n\nKirby, Mrs. Lena, 25, Washington street.\n\nLambert, Johanna.\n\nLavelle, Michael, 22, Broad street.\n\nLavelle, Wm. M. Madden, Kate, 17, Cambria.\n\nMatthews, Thos., 22, Clinton street.\n\nMcAneny, Neal, 50, Cambria.\n\nMcAneny, Mrs. Neal, 45, Cambria.\n\nMcAneny, Rose, 23, Cambria.\n\nMcAneny, Kate, 18, Cambria.\n\nMcAneny, Mary, 13, Cambria.\n\nMcAneny, Wm., 9, Cambria.\n\nMcAneny, Annie, 5, Cambria.\n\nMcAneny, Agnes, 2, Cambria.\n\nMcGee, John, 55, Market street.\n\nMcGinley, James, 34, Conemaugh.\n\nMcVay, Lizzie, 20, Locust street.\n\nMullin, Peter, 50, Conemaugh.\n\nMurphy, Michael J., 34, Brunswick Hotel.\n\nMurphy, Mrs. Mary, 26, Millville.\n\nMurphy, John, 17, Millville.\n\nMurphy, Rose, 14, Millville.\n\nMurphy, Wm., 11, Millville.\n\nMurphy, J. J., 55, Park Place.\n\nMurphy, Lily, 9, Park Place.\n\nNightly, John, 30, Millville.\n\nO'Connel, Capt. Patrick, 70, Washington street.\n\nO'Connel, Margaret, 63, Washington street.\n\nO'Connel, Nora, 60, Washington street.\n\nO'Donnel, Frank, Washington street.\n\nO'Neil, Edward, 3 months, Cambria.\n\nO'Neil, Mrs. Bridget, 28, Cambria.\n\nO'Neil, John.\n\nQuinn, Ellen, Franklin street.\n\nQuinn, John, Franklin street.\n\nRiley, Frances, 15, Cambria.\n\nRiley, Gertrude, 13, Cambria.\n\nRiley, Mary, 18, Cambria.\n\nRogers, Mary, 17, Millville.\n\nRogers, Tatt.\n\nRogers, Mrs. Susan.\n\nRogers, Jane, child.\n\nRyan, John, 55, Washington street.\n\nRyan, Mrs. John, 50, Washington street.\n\nRyan, Maggie, 14, Washington street.\n\nRyan, Mrs. Mary, 73, Washington street.\n\nSagerson, Catherine, 4, Railroad street.\n\nSagerson, Agnes, 2, Railroad street.\n\nSagerson, Thomas, 6 months, Railroad street.\n\nSharkey, Mary, 4, Washington street.\n\nSinniger, Mrs. Mary, Cambria.\n\nSlick, Mrs. Nancy, 55, Fourth Ward.\n\nTakacs, Mrs. Teresa, 31, Cambria.\n\nTakacs, Mrs. John, 21, Cambria.\n\nTokar, Mrs. Dora, 23. Cambria.\n\nTokar, Mary, 4, Cambria.\n\nTokar, Annie, 1, Cambria.\n\nTaylor, Frances.\n\nST. MARY'S CEMETERY.\n\n(Lower Yoder.)\n\nBanyan, Mrs. Rose, 36, Cambria.\n\nBetzler, Mrs. Agnes, 38, Cambria.\n\nBoyle, Charles, Sr., 45, Cambria.\n\nBoyle, Mary, 12, Cambria.\n\nBoyle, Charles, 8, Cambria.\n\nBoyle, Thomas, 7, Cambria.\n\nBrotz, Pancrotz, 55, Cambria.\n\nBrotz, Mrs. Lena, 50, Cambria.\n\nBrady, John, 53, Franklin street.\n\nBrady, Mrs. Julia, 50, Franklin street.\n\nCoby, Elizabeth, Cambria.\n\nCulliton, Mrs. Teresa, 28, Cambria.\n\nDeitrich, Mrs. Amelia, 23, Cambria.\n\nFish, Lena, 17, Cambria.\n\nFisher, Ignatius, 59, Cambria.\n\nFisher, Margaret, 14, Cambria.\n\nFleckenstein, Mrs. Ann, 25, Cambria.\n\nFleckenstein, Regina, 2, Cambria.\n\nGerber, Mrs. Margaret, 41, Cambria.\n\nGerber, John C., 45, Cambria.\n\nGerber, Rose, 8, Cambria.\n\nGerber, Vincent, 6, Cambria.\n\nHanki, Edward, Cambria.\n\nHecker, Mrs. Christ, 58, Cambria.\n\nHeider, Mrs. Ella, 24, Cambria.\n\nHeider, John Leo, 6 months, Cambria.\n\nHessler, Mrs. Fedora, 29, Cambria.\n\nHessler, Mary, 10, Morrellville.\n\nHessler, Joseph, 1\u00bd, Cambria.\n\nHirsch, Eddie, 8, Cambria.\n\nJust, Magdalena, 29, Cambria.\n\nJust, William, 9, Cambria.\n\nJust, Eddie, 4, Cambria.\n\nKintz, Mrs. Mary, 26, Cambria.\n\nKintz, Katie, 19, Cambria.\n\nKintz, Mrs. Mary, 25, Cambria.\n\nKnoblespeice, Maggie.\n\nKoebler, Mrs. George, 60, Cambria.\n\nKropp, Katie, 21, Cambria.\n\nLambert, Johanna, 19, Washington street.\n\nLambreski, Kate, 12, Cambria.\n\nMacheletzky, Stanislaus, 10, Cambria.\n\nMartinades, Mrs. Mary, Cambria.\n\nMiller, Mrs. Annie M., 46, Cambria.\n\nMiller, George, 65, Cambria.\n\nMiller, Eddie, 3, Cambria.\n\nMiller, Annie, 1, Cambria.\n\nNich, Mrs. Margaret, 30, Cambria.\n\nNich, Frank, 6, Cambria.\n\nNich, John, 4, Cambria.\n\nOsterman, Joseph, 38, Cambria.\n\nQuinn, Mrs. Terry, 26, Railroad street.\n\nSchnell, Mrs. Fidel, 68, Cambria.\n\nSchnell, Mrs. Margaret, 60, Cambria.\n\nSchnell, Mrs. F., Cambria.\n\nSchmitt, Mary, 31, Cambria.\n\nSchmitt, George, 4, Cambria.\n\nSchmitt, Sophia, 1\u00bd, Cambria.\n\nShmitt, Fredericka, Cambria.\n\nShmitt, Mrs. Hortena, Cambria.\n\nShmitt, Leo, Cambria.\n\nSinniger, Mrs. Mary, Cambria.\n\nSarlouis, Mrs. Barbara, 48, Cambria.\n\nSarlouis, Mrs. Peter, Cambria.\n\nSnell, Mary, 13, Cambria.\n\nStinely, Mrs. Mary, 35, Cambria.\n\nStinely, Kate, 12, Cambria.\n\nStinely, Joseph, 5, Cambria.\n\nWeber, Mrs. Tresa, 43, Cambria.\n\nWeber, John, 4, Cambria.\n\nWeinzierl, Louis, 41, Cambria.\n\nOLD CATHOLIC GRAVEYARD.\n\n(Conemaugh Borough.)\n\nAkers, Alvar, 54, Upper Yoder.\n\nCoad, Mrs. Mary, 57, Washington street.\n\nCoad, John, 59, Washington street.\n\nConrad, William, 26, Woodvale\n\nHalleran, Mrs. Mary C., 30, Washington street.\n\nHannan, Eugene, 14, Woodvale.\n\nHowe, Abner.\n\nLynch, John, 27, Conemaugh.\n\nLynch, Mary, 16, Conemaugh.\n\nMayhew, Jennie, 18, Woodvale.\n\nMayhew, Joseph, 16, Woodvale.\n\nMayhew, Annie, 12, Woodvale.\n\nMayhew, Earnest, 9, Woodvale.\n\nMayhew, Harry, 6, Woodvale.\n\nMayhew, James, 3, Woodvale.\n\nMcKarley, Mrs. Mary.\n\nNugent, Mrs. Mary Jane, 50, Hager Block.\n\nQuinn, Vincent, 14, Main street.\n\nWehn, Mrs. Laura, 29, Conemaugh.\n\nWehn, Annie, 4, Main street.\n\nWehn, Mary, infant, Conemaugh.\n\nWehn, Joseph, 4, Conemaugh.\n\nWheat, Frank, 28, Clinton street.\n\nGERMAN CATHOLIC CEMETERY.\n\n(Sandy vale.)\n\nBrindle, Mary.\n\nGeis, Mrs. Abbey, 24, Salina, Kansas.\n\nGeis, Richard P., 2, Salina, Kansas.\n\nHable, John, 29, Conemaugh.\n\nHoffgard, Conrad, 18, Clinton street.\n\nHoltzman, Joseph, 35, Woodvale.\n\nHome, William J., 21, Conemaugh.\n\nHome, Emma J., 22, Stormer street.\n\nHornick, John P., 26, Conemaugh.\n\nHornick, Mrs. Amelia, 25, Conemaugh.\n\nHorton, Joseph, Sr., 59, Woodvale.\n\nKeifline, Mrs. Catherine, 56, Conemaugh.\n\nMaloy, Manassas, 45, Clinton street.\n\nMalzi, Jacob, 34, Washington street\n\nMurtha, James, 65, Conemaugh.\n\nMurtha, James, 28, Main street.\n\nMurtha, Mrs. Barbara, 24, Main street.\n\nMurtha, Frank, 6, Main street.\n\nMurtha, Flora May, 4, Main street.\n\nMurtha, Lily, 1, Main street.\n\nOswald, Charles, 44, Third Ward.\n\nOswald, Mary, 19, Third Ward.\n\nQuinn, Vincent D., 16, Main street.\n\nRipple, Maggie B., 27, Merchants' Hotel.\n\nRobine, Christina, 25, Franklin street.\n\nSarlouis, Sophia.\n\nSchnurr, Charles, 40, Conemaugh.\n\nSchnurr, Robert, 27, Smith alley.\n\nSchry, Joseph, Sr., 78, Woodvale.\n\nSchry, Mrs. Joseph, 58, Woodvale.\n\nShellhammer, Lorentz.\n\nShellhammer, Patricius.\n\nSchaller, Joseph, 62, Woodvale.\n\nSchaller, Mrs. Joseph, 62, Woodvale.\n\nSchaller, Annie, 24, Woodvale.\n\nSchaller, Rose, 21, Woodvale.\n\nWerberger, Prof. F. P., 70, Locust street.\n\nVoegtly, Germanus, 62, Conemaugh.\n\n(Geistown.)\n\nRubritz, Peter, 65, Franklin Borough.\n\nRubritz, Mrs. Margaret, 56, Franklin Borough.\n\nRubritz, Maggie, 20, Franklin Borough.\n\nSchiffhauer, John, 62, Washington street.\n\nStenger, John, 12, Main street.\n\nStenger, Leo, 3, Main street.\n\nSteigerwald, William, Conemaugh.\n\nSteigerwald, Mrs. Mary, 38, Conemaugh.\n\nSteigerwald, infant, 1 month, Conemaugh.\n\nPUBLIC PLOT.\n\n[Known to have been found, but bodies never recovered by friends, and buried in Public Plot in Grand View Cemetery.]\n\nArthur, Earl H., 8, Water street.\n\nBaker, son of Andrew.\n\nBohnke, Charles.\n\nBopp, son of Jacob.\n\nBopp, Katie, 9, Broad street.\n\nBricker, Henry.\n\nBurns, John.\n\nBarbour, Harry L., 16, Locust street.\n\nBarker, Mrs. Susan, 28, Woodvale.\n\nBehnke, Charles.\n\nBloch, Louisa, 17, Conemaugh.\n\nBoehler, Mrs. Annie, 39, Conemaugh.\n\nBrawley, George D., 17, Cor. Union and Vine sts.\n\nBrennan, Mrs. Martha, 36, Woodvale.\n\nBrennan, Mary, 16, Woodvale.\n\nBrennan, William, 12, Woodvale.\n\nBrennan, Lewis, 10, Woodvale.\n\nBrennan, Arthur, 7, Woodvale.\n\nBrennan, Frank, 3, Woodvale.\n\nBrown, Sadie, 22, Woodvale.\n\nBruhn, Claus, 58, Conemaugh.\n\nBryan, Wm. A., 45, Mansion House.\n\nCampbell, Peter, 40, Conemaugh.\n\nCasey, William, 48, Cambria.\n\nCornelison, Maggie.\n\nCraig, Thomas A., 32, Market street.\n\nCraig, Mrs. T. A., 30, Market street.\n\nCraig, Christ, 45, Cambria.\n\nCraig, Annie, 13, Walnut street.\n\nCunz, Lydia, 6, Napoleon street.\n\nCunz, Robert, 4 months, Napoleon street.\n\nDillon, James, 35, Napoleon street.\n\nDowney, Mrs. Mary, 55, Pearl street.\n\nDudzik, Andrew, 28, Cambria.\n\nEager, Annie.\n\nEck, Mary Ann, 37, Woodvale.\n\nEck, Lily, 12, Conemaugh street.\n\nEdwards, Mrs. Ann R., 70, Union street.\n\nElsaesser, Andrew, 16, Conemaugh.\n\nEnglish, Joseph, 24, Railroad street.\n\nFagan, Matthew, 40, Millville.\n\nFagan, Mrs. M., 38, Millville.\n\nFagan, Monica, 12, Millville.\n\nFagan, Daniel, 10, Millville.\n\nFagan, Clara, 3, Millville.\n\nFagan, Thomas, 1, Millville.\n\nFaloon, Mrs. Ann E., 63, Pearl street.\n\nFichtner, Mrs. Tillie, 33, Main street.\n\nFiddler, Elmira, Bedford street.\n\nFiddler, Eliza J.\n\nFockler, Herman, 21, Franklin street.\n\nGriffin, Miss Mary.\n\nHamilton, Mary, 33, Bedford street.\n\nHanki, Mrs. Teresa, 40, Cambria.\n\nHause, Mollie.\n\nHellriggle, Chas.\n\nHellriggle, Mrs. Lizzie, 30, Woodvale.\n\nHenry, William, 34, Cumberland, Md.\n\nHocker, Mrs. John, 72, Somerset street.\n\nHop Sing, Franklin street.\n\nHurt, Charles, London, England.\n\nIrwin, Maggie, 22, Hulbert House.\n\nJohnson, David, 45, Conemaugh street.\n\nJones, Mary, 14, Main street.\n\nKast, Clara, 17, Clinton street.\n\nKeene, Katie, 16, Union street.\n\nKeinxstoel, Samuel, 30, Market street.\n\nLarimer, James, 45, Somerset street.\n\nLee Sing, Chinaman, Franklin street.\n\nLucas, Maria, 50, Conemaugh.\n\nMadden, Mrs. Mary, 47, Cambria City.\n\nMack, August.\n\nMcClarren, Cora, 8, Cambria City.\n\nMcCue, Mrs. Melden, Richard.\n\nMaley, Henry.\n\nMosser, Mrs. Mary, 65, Conemaugh street.\n\nMullen, Margaret.\n\nOswald, Mrs.\n\nOwens, Mrs.\n\nOyler, John R.\n\nPhillips, Mrs. Eliza, 48, Union street.\n\nReese, Mrs. Lizzie, 30, Conemaugh street.\n\nReese, Annie, 7, Vine street.\n\nReidel, John C., 60, Conemaugh.\n\nRich, Mrs. Charlotte, 45, Stonycreek street.\n\nRoberts, Mrs. Jennie, 18, Somerset street.\n\nRosenfelt, Solomon, Washington street.\n\nSaylor, Henry.\n\nSchnable, Mrs. Conrad, 35, Baumer street.\n\nSchittenhelm, Anton, Cambria.\n\nSchittenhelm, Anton Jr., Cambria.\n\nShumaker, Mrs. James M., Locust street.\n\nSkiba, Mrs. Stainslous, 32, Cambria.\n\nSkiba, Joseph, 4, Cambria.\n\nSmith, Ralph, 11, Woodvale.\n\nSmouse, Jennie, Hulbert House.\n\nStern, Bella.\n\nStrauss, Moses, 77, Vine street.\n\nStrauch, Henry, 50, Conemaugh.\n\nSmith, Willie, 1, Cambria.\n\nSurany, David.\n\nThomas, John T.\n\nTill, Arthur, 27, Market street.\n\nUnverzagt, Daniel, 66, Washington street.\n\nUnverzagt, Mrs. Daniel, 62, Washington street.\n\nViering, Mrs.\n\nWhite, Mrs. John, 76, Union street.\n\nWagnor, Henry, Cambria.\n\nWarsing, Jane, 24, Coopersdale.\n\nWarkeston, Miss.\n\nWeinzierl, Mrs. Mary, 38, Cambria.\n\nWearn, Willie, 6, King street.\n\nWalford, Frank.\n\nWill, Elizabeth, Conemaugh.\n\nPASSENGERS ON DAY EXPRESS.\n\n[Those marked? bodies never found. Those found lived at the places named, to which places the remains were taken.]\n\nBates, Mrs. Annie, Delavin, Wis.\n\n? Brady, Mrs. J. W., Chicago, Ill.\n\nBryan, Elizabeth M., 20, Philadelphia.\n\nChristman, Mrs. A. C., Dallas, Texas.\n\nDay, John R., 60, Prospect, Md.\n\nDay, Miss, Prospect, Md.\n\nEwing, Andrew, Snow Shoe, Pa.\n\n? Feustermaker, Victor, Egypt, Lehigh County\n\nHarnish, Blanche, Dayton, O.\n\n? Hemingway, Fred, and wife, Kokomo, Ind.\n\nKing, Mrs. J. F.\n\n? Lyon, E., New York.\n\n? McCoy, Mrs.\n\n? McCoy,\u2014\u2014\n\n? McCov,\u2014\u2014\n\nMeisel, Christ, 32, Newark, N. J.\n\nMinich, Kate, Fostoria, Ohio.\n\nPaulson, Jennie, 20, Allegheny City.\n\n? Phillips, Frank (porter), Jersey City.\n\nRainey, Mrs. Sophia, 64, Kalamazoo, Mich.\n\nRoss, John D.\n\nSchrantz, George, Pleasant Gap, Pa.\n\n? Shelly, W., Newark, N. J.\n\nShick, Cyrus, Reading.\n\n? Sible, Mrs. Springtown, Bucks County, Pa.\n\nSmith, Mrs. H. K., 25, Osborn, Ohio.\n\nSmith, R. Wardwell, 3, Osborn, Ohio.\n\nStinson, Eliza, Norristown, Pa.\n\n? Swaney, Mrs. Mary A., 67.\n\nSwineford, Mary A., St. Louis, Mo.\n\nSwineford, Mrs. Ed., St. Louis, Mo.\n\nTarbell, Mrs. Farney, 32, Cleveland, Ohio.\n\n? Tarbell, Grace, 7, Cleveland, Ohio.\n\n? Tarbell, Bertie, 5, Cleveland, Ohio.\n\n? Tarbell, Howard, 2, Cleveland, Ohio.\n\nWeaver, Beneval, Millersburg, Pa.\n\nWoolf, Jennie, Chambersburg, Pa.\n\nMISCELLANEOUS.\n\n[Bodies taken to places named in subheads for burial. The place named in line with name of individual is where they were lost from.]\n\nLOYSBURG, BEDFORD COUNTY, PA.\n\nAaron, Mrs. H. B., 29, Railroad street.\n\nAaron, Flora, 10, Railroad street.\n\nBLAIRS VILLE, PA.\n\nAlexander, John G., 45, Woodvale.\n\nAlexander, Mrs. John G., 45, Woodvale.\n\nBrown, Emma, 20.\n\nMcLaughlin, Mrs. Julia, 60, Cambria.\n\nMiller, Robert, 22, Sixth Ward.\n\nPike, Fanny, 19, Haynes street.\n\nWILMORE, PA.\n\nBeiter, Mathias, 3, Clinton street.\n\nPHILADELPHIA, PA.\n\nButler, Chas. T., Hulbert House.\n\nCarlin, Jonathan, Hulbert House.\n\nCox, James G., Hulbert House.\n\nClark, W. H. L., 50, Hulbert House.\n\nDe Walt, Chas. B., 36, Hulbert House.\n\nDorsey, John D., Hulbert House.\n\nLichtenberg, Rev.\n\nJohn, Locust street.\n\nLichtenberg, Mrs.\n\nMurray, James, 50, Hulbert House.\n\nNathan, Adolph, 40, Main street.\n\nOverbeck, William H., 38, Main street.\n\nSpitz Walter L., Hulbert House.\n\nWoolf, Mrs. M. L., Jackson street.\n\nLOUISVILLE, KY.\n\nMarshall, Chas. A., 34, Hulbert House.\n\nBRADDOCK, PA.\n\nCadogan, Mrs. Mary A., 46, Millville.\n\nCadogan, Ann, 25, Millville.\n\nYoung, Mrs. Kate, 34, Market street.\n\nYoung, Samuel, 13, Market street.\n\nPITTSBURGH, PA.\n\nCreed, David, 60, Washington street.\n\nCreed, Mrs. Eliza, 55, Washington street.\n\nCreed, Maggie, 28, Washington street.\n\nFisher, Moses, 24, Mansion House.\n\nSweeney, Mrs. Ann, 70, Conemaugh.\n\nBENSHOFF'S, CAMBRIA COUNTY, PA.\n\nCuster, William H., 35, Millville.\n\nSTEUBENVILLE, OHIO.\n\nDavis, Frank B., 40, Main street.\n\nDavis, Frank, infant.\n\nSOMERSET, PA.\n\nGaither, Harry, 18, South street.\n\nHouston, Minnie, Hulbert House.\n\nHurst, Nathaniel, 15, Washington street.\n\nSHIPPENSBURG, PA.\n\nDiehl, Carrie, 20, Hulbert House.\n\nWells, Jennie, 22, Hulbert House.\n\nMERCER COUNTY, PA.\n\nDe France, Mrs. H. T., 32, Hulbert House.\n\nNEW YORK, N. Y.\n\nDow, W. F., Hulbert House.\n\nHOLLIDAYSBURG, PA.\n\nFitzharris, John, Sr., 97, Franklin street.\n\nPHILIPSBURG, PA.\n\nEskdale, James, 42, Woodvale.\n\nEskdale, Mrs. James, Woodvale.\n\nBERLIN, PA.\n\nGarman, Grace, 21, Washington street.\n\nPITTSTON, PA.\n\nGroff, Nellie C., 20, Hulbert House.\n\nLEECHBURG, PA.\n\nHill, Ivy, 6, Washington street.\n\nJack, Jennie.\n\nCOVER'S HILL, CAMBRIA COUNTY, PA.\n\nHinchman, Harry, 4, Woodvale.\n\nLong, Samuel.\n\nShaffer, Fred, 21, Conemaugh.\n\nDERRY, PA.\n\nJackson, H. A., 36.\n\nCUMBERLAND, MD.\n\nKatzenstein, Mrs. Ella, Hulbert House.\n\nKatzenstein, Edwin, Hulbert House.\n\nHARRISBURG, PA.\n\nKeis, Charles A., 26, Conemaugh.\n\nWeber, E. Vincent, 26, Woodvale.\n\nWeber, Mrs. Florence, 25, Woodvale.\n\nBUTLER, PA.\n\nBonner, Mrs. Ann, 24.\n\nKenna, Mrs. Alice B.\n\nHUNTINGDON COUNTY, PA.\n\nMcDivitt, Mattie, 32, Water street.\n\nHEADRICK'S, CAMBRIA COUNTY, PA.\n\nAllison, Florence, 12, Texas.\n\nBeck, William J., 30, Woodvale.\n\nBeck, Mrs. Blanche, 29, Woodvale.\n\nWilson, Dr. J. C., 53, Franklin.\n\nWilson, Caroline E., 52, Franklin.\n\nQUAKERTOWN, PA.\n\nSmith, Mrs. J. L., 34, Hulbert House.\n\nSmith, Florence, 9, Hulbert House.\n\nSmith, Frank, 7, Hulbert House.\n\nSmith, infant, 4 months, Hulbert House.\n\nWilson, Charles H., 45, Hulbert House.\n\nARMAGH, PA.\n\nYoung, Sarah C., 66, Court street.\n\nBEAVER FALLS, PA.\n\nLeslie, John S., 30, Levergood street.\n\nYPSILANTI, MICH.\n\nRichards, Carrie, Hulbert House.\n\nRichards, Mollie, Hulbert House.\n\nBALTIMORE, MD.\n\nGoldenberg, Henry, 54, Lincoln street.\n\nHoopes, Walter E., 30, Woodvale.\n\nSmith, Mrs. Alice M., 29, Woodvale.\n\nGREENSBURG, PA.\n\nKilgore, W. Alex., 52, Washington street.\n\nMontgomery, Alex., 55, Stonycreek street.\n\nSEWICKLEY, PA.\n\nLittle, John A., 43, Hulbert House.\n\nBANGOR, PA.\n\nLlewellyn, Mrs. J. J., 27, visiting at J. T. Llewellyn's.\n\nSOUTH FORK, PA.\n\nMullin, James, 24.\n\nJERSEY HEIGHTS, N. J.\n\nMyer, Bernhart.\n\nROME, N. Y.\n\nRichards, John O., 70.\n\nYOUNGSTOWN, OHIO.\n\nWhite, Mrs. Alex., 42.\n\nINDIANA, PA.\n\nZiegler, James B., 24.\n\nREADING, PA.\n\nFediman, W. M., 56, Main street\n\nBLOUGH'S, STONYCREEK TOWNSHIP, PA.\n\nBlough, Samuel, 40, Market street.\n\nBlough, Sophia, 38, Main street.\n\nBlough, child, Main street.\n\nSCALP LEVEL, PA.\n\nOwens, William L., 11, Market street.\n\nOwens, Daisy, 13, Market street.\n\nNICHOLSON, PA.\n\nRosensteel, Mrs. J. M., 35, Woodvale.\n\nRosensteel, Ray Halstead, 18, Woodvale.\n\nNO CEMETERY RECORD.\n\n[Bodies found, but not known where buried.]\n\nAdams, Henry Clay.\n\nAlberter, Anna, 22, Cambria.\n\nAmps, Nicodemus, 42, Cambria.\n\nAmps, Mrs. Teresa, 32, Cambria.\n\nAtkinson, John, 72, East Conemaugh.\n\nBaer, Rosa L., 17, Grubbtown.\n\nBagley, William, Morrellville.\n\nBaird, Charles.\n\nBaker, Mrs. Nelson.\n\nBaker, Mrs. Mary, Woodvale.\n\nBaker, Catherine, 70, Market street.\n\nBaker, Agnes, 68, Market street.\n\nBarley, Myrtle, 11, Woodvale.\n\nBarley, Mamie, 7, Woodvale.\n\nBarley, Effie, 5, Woodvale.\n\nBarley, Laura, 6 months, Woodvale.\n\nBarrett, Jas., 27, Franklin st., St. Charles Hotel.\n\nBerg, Mrs. Marion, 24, Woodvale.\n\nBerkebile, Mahlon, Morrellville.\n\nBlough, Emanuel, 22, Bedford street Blough, infant, School alley.\n\nBowersox, Frank, 22, Market street.\n\nBoyer, Solomon, 62, East Conemaugh.\n\nBradley, Thomas, 42, Conemaugh.\n\nBruhn, Mrs. Anna, 45, Portage street.\n\nBunting, Mrs. Caroline, 45, Woodvale.\n\nBurk, Mrs. Matilda, 38, East Conemaugh.\n\nBurkhard, Mrs. Mollie, 36, Woodvale.\n\nCarr, Alexander, 36, East Conemaugh.\n\nCarr, Sissie, 2, East Conemaugh.\n\nChristie, Andrew C., 50, Woodvale.\n\nClark, Thomas, 42, Union street.\n\nClark, John B., 50, Portage st., and 7 children.\n\nCole, John, Cambria.\n\nConnors, Mrs. Mary, Millville.\n\nCooper, Otto, 8, Kurtz alley.\n\nCooper, Mrs. 38, Kurtz alley.\n\nCouthamer, Mr.\n\nCoy, Mrs. Sarah, 46, East Conemaugh.\n\nCoy, Newton G., 16, East Conemaugh.\n\nCraig, Mrs. Catherine, 40, Walnut street.\n\nCrowthers, infant, 3, Chestnut street.\n\nCummings, Amy, Somerset street.\n\nDavis, Frank, 8, Woodvale.\n\nDavis, Mrs. Philip, 60.\n\nDavis, Mrs. Thomas S., 55, Market street.\n\nDelaney, Mrs. C. W., 59, Conemaugh street.\n\nDimond, Frank, 36, Conemaugh.\n\nDimond, Mrs. Ann, 64, Conemaugh.\n\nDoorocsik, Mrs. Annie, 28, Cambria.\n\nDoorocsik, Miss, 6, Cambria.\n\nDoorocsik, Mary, 4, Cambria.\n\nDorriss, August, 54, Conemaugh.\n\nDoubt, Mrs. William, 63, Cambria.\n\nDougherty, Mary, 16, Cambria.\n\nEberle, Lena, 14, Woodvale.\n\nFails, Dolly F., 15, Union street.\n\nFers, Frank, 23, Millville.\n\nFink, Mary E., 17, Conemaugh street.\n\nFisher, Wolfgang, 33, Main street.\n\nFisher, Noah, East Conemaugh.\n\nFlegle, David G.\n\nFlegle, Miss Annie.\n\nFlinn, Mrs. Mary, Bedford street.\n\nFogarty, Thomas, 50.\n\nForrest, Frank, 12, Locust street.\n\nFoust, Conrad, Woodvale.\n\nGardner, Rose, 20, Prospect.\n\nGill, William, 7, Prospect.\n\nGillen, Laura, Bedford street.\n\nGordon, Susan L., 62, Hager Block.\n\nGreenwood, Mrs. Rose, 33, Conemaugh.\n\nGromley, Lilly, 19, Mineral Point, Pa.\n\nGromley, J. A., 14, Mineral Point, Pa.\n\nHallen, Charles E., 33, Millville.\n\nHarris, Mrs. Mary T., 48, Walnut street.\n\nHartzell, Mr., Market street.\n\nHecker, John, 10, Cambria.\n\nHeckman, Francis, 25, Main street.\n\nHeffley, Edward, 22, Somerset street.\n\nHeine, Henry, 26, Cambria.\n\nHeine, Mrs. Lizzie, 25, Somerset street.\n\nHerman, Edward, Cambria.\n\nHess, William B., 55, Millville.\n\nHipp, Elizabeth P., 20, Main street.\n\nHitchins, Mrs. Cordelia, 35, Market street.\n\nHornick, Agnes, Broad street.\n\nHughes, Mary, 7, Chestnut street.\n\nHughes, Mrs., 64, Union street.\n\nJames, Lena, 26, River avenue.\n\nJames, Maggie, 1, River avenue.\n\nJenkins, Mrs. Susan, 40, Somerset street.\n\nJohill, Joseph, Third Ward.\n\nJohnson, John M., 40, Union street.\n\nJohnson, Mrs. John M., 38, Union street.\n\nJohnson, Mrs. Oliver, 22, Conemaugh street.\n\nJones, Maggie, 29.\n\nKane, John, 45, Union street.\n\nKane, Bridget, 20, Market street.\n\nKeifline, Mary, 4, Conemaugh.\n\nKelly, Charles, Millville.\n\nKunkle, Lizzie, 21, Washington street.\n\nLaban, Mrs. Teresa, 50, Cambria.\n\nLeech, Mrs. Sarah E., 60, Franklin.\n\nLeech, Alice M., 18, Franklin.\n\nLingle, Mrs. Mary J., 44, Pearl street.\n\nLong, Samuel, 60, Vine street.\n\nLotz, Conrad, 64, Sherman street.\n\nLyden, Mary, 20, Merchants' Hotel.\n\nManeval, Clarence, 17, Lincoln street.\n\nMann, Michael, 41, South Fork.\n\nMarczi, Mrs. Mary, 42, Cambria.\n\nMarshall, Wm. H., 23, Clinton street.\n\nMaurer, John, 77, Morris street.\n\nMcAuliff, Laura, 16, Woodvale.\n\nMcDowell, Geo., 8, Pearl street.\n\nMcGuire, Mrs. Mary, 45, Walnut street.\n\nMcHugh, Mrs. D. A., 45, E. Conemaugh.\n\nMcHugh, Gertrude, 16, E. Conemaugh.\n\nMcHugh, Jno. L., 14, E. Conemaugh. McNally.\n\nPatrick, 42, Prospect.\n\nMecke, August, 51, Cambria.\n\nMelczer, Frederick, 28, Cambria.\n\nMiller, Robert, 5, Napoleon street.\n\nMiller, John A., 25, Cambria.\n\nMiller, William, 44, Franklin.\n\nMingle, Sarah.\n\nMonteverde, Mary, 11, Washington street.\n\nMonteverde, Emelia, 7, Washington street.\n\nMorran, James A., 53, Somerset street.\n\nNau, Katie, 20, Bedford street.\n\nNeary, Mrs. Kate, 34, Bedford street.\n\nNeary, Mary Ellen, 11, Bedford street.\n\nNoblespiece, Maggie, 14, Morrellville\n\nNugent, Mrs. Mary Jane, 60, Hager Block.\n\nO'Connell, \u2014, Cambria.\n\nO'Conner, Rose, 20, Locust street.\n\nO'Neal, John, 19, Wood alley.\n\nOswald, Appahmarian, 12, Cambria.\n\nPage, Emma, 11, Mineral Point.\n\nPage, Herman, 6, Mineral Point.\n\nPalmer, Mrs. J. H., 76, Napoleon street.\n\nPartsch, Mrs. Josephine, 59, Woodvale.\n\nPhillips, John, 15, Union street.\n\nRausch, John, 44, Daisytown.\n\nRepp, George, 5 months, Daisytown.\n\nRobine, Eddie, 2, Franklin.\n\nRobine, Willie, 9 months, Franklin.\n\nRodgers, Patrick, 52, Millville.\n\nRodgers, Grace, 5, Millville.\n\nRodgers, Mrs. Mary, 50, Millville.\n\nRoss, Joseph, 30, Conemaugh.\n\nRoth, Annie, 5, Cambria.\n\nRowland, Emma, 32, Market street.\n\nRowland, Ran, 16, Market street.\n\nSamen, Mrs. Annie, 25, Cambria.\n\nSamen, John, 4 Cambria.\n\nSamen, Annie, 3, Cambria.\n\nSamen, Mary, 3 months, Cambria.\n\nSchmidt, Mrs. Frederick, Cambria.\n\nSchmidt, Hortense.\n\nSchmidt. Leo.\n\nSchmitz, Gustave, 33, Clinton street.\n\nSchittenhelm, Max, Cambria.\n\nSnyder, Mrs. Annie, 34, Woodvale.\n\nSpareline, John, 64, Railroad street.\n\nSmith, Mrs. Maggie L., 38, Woodvale.\n\nSmith, Addie, 13, Pearl street.\n\nSmith, Philip, 16, Walnut street.\n\nSmith, Mrs. Amelia, 32, Cambria.\n\nSmith, Mrs. Mary, 52, Conemaugh.\n\nSmith, Philip, 16, Conemaugh.\n\nSlick, Josephine, 20, Woodvale.\n\nSutliff, George, 25, Somerset street.\n\nStern, Bella, 1, Washington street.\n\nStewart, \u2014, Second Ward.\n\nSpicsak, Mrs. Annie, 27, Cambria.\n\nTacey, Peter L., 20, Woodvale.\n\nTrindle, John M., 39, Nineveh.\n\nTrawatha, Mrs. Annie, 60, Conemaugh.\n\nThomas, Mabel, 6, Market street.\n\nThomas, Edward M., 71, Woodvale.\n\nUhl, Mrs. Ludwig, 80, Peter street.\n\nValentine, George M., 42, Market street.\n\nWeisz, Mrs. Martin, 46, Cambria.\n\nWeisz, Jacob, 13, Cambria.\n\nWeisz, Jacob.\n\nWeisz, Isaac, 6, Cambria.\n\nWeisz, Anna, 4, Cambria.\n\nWelsh, Thomas, 60, Cambria.\n\nWeinzierl, Louis, 41, Cambria.\n\nWilliams, Elanor, 7 months, Lewis alley.\n\nWilliams, Mrs. Margaret, 27, Conemaugh street.\n\nWild, Mrs. Margaret, 80, Conemaugh.\n\nWillower, Miss Bella, Somerset street.\n\nWillower, Bertha, Somerset street.\n\nWissinger, Mrs. Catherine, 47, Morris street.\n\nYost, Charlotte, 16, Pine street.\n\nNOT KNOWN TO HAVE BEEN FOUND.\n\nAbele, Katie, 21, Main street.\n\nAbler, Lulu, Woodvale.\n\nAlberter, Teresa, 3, Cambria.\n\nAlexander, Mrs. Martha, Main street.\n\nAllison, Mrs. Jane, 45, Pittsburgh.\n\nAlt, John, 65, Conemaugh.\n\nAlt, Teresa, 20, Conemaugh.\n\nAlt, George, 60, Cambria.\n\nAlt, Mrs. Ann, 75, Cambria.\n\nAmps, Mary, 11, Cambria.\n\nAubrey, Thomas, 45, Conemaugh street.\n\nBacker, George, 27, Conemaugh.\n\nBaker, James, 22, Woodvale.\n\nBaker, Catherine, Market street.\n\nBaker, Lydia, 20, Woodvale.\n\nBaker, Nancy, Market street.\n\nBaker, Richard, 1, Woodvale.\n\nBaker, Mellville, 11, Woodvale.\n\nBaker, Deronda, 5, Woodvale.\n\nBaker, Dolly, Woodvale.\n\nBaker, Clara, 17, Woodvale.\n\nBanyan, John, 7, Cambria.\n\nBanyan, Albert, 4, Cambria.\n\nBanyan, Theodore, 2, Cambria.\n\nBarbour, Howard, 7, Woodvale.\n\nBarbour, John F., 3 months, Woodvale.\n\nBarbour, Mrs. Sarah, 59, Woodvale.\n\nBarker, Edward, 27, Woodvale.\n\nBarker, Clara, 2\u00bd, Woodvale.\n\nBarker, infant, 1 month, Woodvale.\n\nBartosh, Mrs. Hannah, 39, Cambria.\n\nBartosh, Frank, 14, Cambria.\n\nBaumer, Mrs. Eliza, 68, Woodvale.\n\nBeam, Roscoe, 2, Locust street.\n\nBeecher, Mrs. Jane, 44, Woodvale.\n\nBeecher, Mary, 23, Woodvale.\n\nBeck, Alfred, 6. Beck, Roy, 3.\n\nBeckley, Mrs. Mary, 48, Woodvale.\n\nBenson, Mrs. Bessie, 23, Cambria.\n\nBenson, Flora, 3, Cambria.\n\nBare, Mrs. Bare, infant.\n\nBarkley, George.\n\nBarnes, Andrew, Conemaugh.\n\nBarron, Anton.\n\nBarron, Mrs.\n\nBenson, Cora Belle, 1\u00bd, Cambria.\n\nBerkey, Henry S., 45, Clinton street.\n\nBeske, John, 7, Cambria.\n\nBeske, Joseph, 5, Cambria.\n\nBeske, Frank, 3, Cambria.\n\nBeske, Lewis, 1, Cambria.\n\nBetzler, Frank, 9, Cambria.\n\nBetzler, Katie, 7, Cambria.\n\nBishop, Julius, 55, Cambria.\n\nBitner, A. B.\n\nBlair, Alfred, 53, Woodvale.\n\nBlair, Oliver, 25, Woodvale.\n\nBlair, Alfred, Jr., 14, Woodvale.\n\nBlair, Emanuel, 12, Woodvale.\n\nBlair, Rosana, 10, Woodvale.\n\nBloch, Mrs. Rose, 54, Conemaugh.\n\nBloch, Annie, 26, Conemaugh.\n\nBlock, Minnie, 15, Conemaugh.\n\nBloch, Emma, 13, Conemaugh.\n\nBoehler, Barbara, 7, Conemaugh.\n\nBoehler, Annie, 9, Conemaugh.\n\nBogus, William.\n\nBlough, Mrs. First Ward.\n\nBopp, Naomie, 7, Broad street.\n\nBonson, Charles R. Booser, Eddie, 14, Market street.\n\nBowers, George, Woodvale.\n\nBowersox, Mrs. Elia, 16, Market street.\n\nBowersox, Cordelia, 3, Market street.\n\nBowman, Jessie, 4, Woodvale.\n\nBowman, Blanche, 2, Woodvale.\n\nBoyer, Emma, 17, Woodvale.\n\nBoyle, Rose, 6, Cambria.\n\nBoyle, Bridget, 4, Cambria.\n\nBoyle, William, 2, Cambria.\n\nBoyle, Joseph, 8 months, Cambria.\n\nBraden, Patrick, Millville.\n\nBradley, Mrs. Elvira, 39, Conemaugh.\n\nBrawley, Mrs. Maggie, 42, Union street.\n\nBrawley, Robert J., 4, Union street.\n\nBrawley, John.\n\nBrennan, Mrs. Mary Ann, 46, Woodvale.\n\nBrennan, Mary Ann, 23, Woodvale.\n\nBrennan, Ellen, 19, Woodvale.\n\nBrennan, Jane, 16, Woodvale.\n\nBrennan, Agnes, 13, Woodvale.\n\nBridges, Mrs. Jane, 64, Market street.\n\nBrindle. Vincent.\n\nBrindle, Frank.\n\nBrindle, Rose.\n\nBrinker, Henry.\n\nBriney, Matilda, 25. Woodvale.\n\nBrockner, Samuel, 28, Conemaugh.\n\nBrown, Mrs. Magdalena, 58, Cambria.\n\nBrown, Lizzie, 15, Woodvale.\n\nBrown, Mrs., Conemaugh.\n\nBuckhard, Mrs. Elizabeth, 50, Woodvale.\n\nBuckhard, Charles, 19, Woodvale.\n\nBuckhard, Mrs., 63, Woodvale.\n\nBuckley, Mrs. Mary, 48, Woodvale.\n\nBurket, Frank, 14, Washington street.\n\nBurket, Blair, 8.\n\nBurkhard, Howard, 12, Woodvale.\n\nBurkhard, Gussie J., 5, Woodvale.\n\nBurkhard, Charles C., 2, Woodvale.\n\nBurkhard, Mrs. Catherine, 85, Mineral Point.\n\nBurns, Peter, Woodvale.\n\nButler, John, 51, 84 John street.\n\nButler, Robert, 40, Millville.\n\nButler, Mrs., 70, Millville.\n\nButler, Annie, 17, Millville.\n\nButler, Fannie, 14, Millville.\n\nButler, George, 11, Millville.\n\nButler, Mrs. Sarah.\n\nByers, Mrs. Catherine, 46, Mineral Point.\n\nCallahan, Mary, 22, Locust street.\n\nCallahan, Mrs. Frank, Locust street.\n\nCarr, Mrs. Mary, 42, Woodvale.\n\nCarr, William, 7, Woodvale.\n\nCarr, Patrick, 22, Cambria.\n\nCarr, Mrs. Sarah, 20, Cambria.\n\nCartin, Mrs. Thomas, 46, Woodvale.\n\nCartin, Frank, 5, Woodvale.\n\nChristie, Mrs. Lizzie, 46, Woodvale.\n\nChristie, Daisy, 19, Woodvale.\n\nClark, Thomas, Jr., 9, Union street.\n\nClark, Annie, 5, Union street.\n\nClark, Hamilton.\n\nCoad, William, 12, cor. Market and Washington.\n\nCleary, Alice, Cambria.\n\nConrad, John, 21, Woodvale.\n\nConstable, Mrs. Sarah E., 48, Broad street.\n\nConstable, Clara, 16, Broad street.\n\nConstable, George, 39, Franklin.\n\nCope, Ahlum, 70, Conemaugh.\n\nCostlow, Michael, 70, Locust and Union streets.\n\nCostlow, Zita, 6, Woodvale.\n\nCostlow, Juniata, 2\u00bd, Woodvale.\n\nCostlow, Regina, 1, Woodvale.\n\nCraig, William, 8, 314\u00bd Walnut street.\n\nCreed, Kate, 26, 200 Washington street.\n\nCreed, Mary, 16, Washington street.\n\nCrown, Thomas, 51, Conemaugh.\n\nCrowthers, Samuel, 30, Cambria.\n\nCrowthers, Mrs. Verna, 27, Cambria.\n\nCulleton, George F., 1, Chestnut street.\n\nCulleton, John F., 2, Chestnut street.\n\nCummings, Mrs., Somerset street.\n\nCummings, \u2014, Somerset street.\n\nCunz, Mrs. Catherine, 37, Napoleon street.\n\nCunz, Edward, 12, Napoleon street.\n\nCunz, Gussie, 3, Napoleon street.\n\nCurtin, Johanna.\n\nCush, Annie, 17, 112 Railroad street.\n\nCush, Thomas, 1\u00bd, 116 Railroad street.\n\nCuster, Mrs. Emma J., 27, Bedford street.\n\nCurry, Robert.\n\nDarr, George E., 28, Millville.\n\nDavis, Martha, 18, Woodvale.\n\nDavis, Ada, 15, Woodvale.\n\nDavis, Mrs. Ann, 60, Locust street.\n\nDavis, Mrs. Mary, 54, Locust street.\n\nDavis, Delia, 22, Locust street.\n\nDavis, Evan, 16, Locust street.\n\nDavis, Reese, 13, Locust street.\n\nDavis, Mrs. Mary D., 55, Millville.\n\nDeible, Harry, Woodvale.\n\nDeihl, Mrs. Mary, 40, Conemaugh.\n\nDelaney, Charles, 18, 51 Conemaugh street.\n\nDevlin, Melissa, 12, East Conemaugh.\n\nDick, Cornell, 17, Cor. Locust and John streets.\n\nDill, Robert, 26, Woodvale.\n\nDill, Mrs. Robert, 26, Woodvale.\n\nDill, William, 7, Woodvale.\n\nDill, Harry, 3 Woodvale.\n\nDinkel, Adam, 50, Conemaugh.\n\nDishong, Lizzie, 22, Union street.\n\nDluhos, Jacob, 3, Cambria.\n\nDluhos, Mary, 3 months, Cambria.\n\nDolny, Mike, Cambria.\n\nDorillia, Mrs., 30, Cambria.\n\nDorillia, \u2014, Cambria.\n\nDorillia, \u2014, Cambria.\n\nDowns, Willie, Millville.\n\nDudzik, Mike, 21, Cambria.\n\nDudzik, Albert, 21, Cambria.\n\nEarly, Mrs. Ella, 59, Woodvale.\n\nEck, Ellen C., 6, Woodvale.\n\nEck, Edna Marie, 1\u00bd, Woodvale.\n\nEck, John B., 38, Conemaugh street.\n\nEck, Dora, 7, Conemaugh street.\n\nEck, Mabon, 2, Conemaugh street.\n\nEdmonds, Nancy.\n\nEdwards, Roger, 55, Millville.\n\nEdinger, Annie, 19, Millville.\n\nElder, Mrs. Cyrus, 49, Walnut street.\n\nElder, Nannie M., 23, Walnut street.\n\nEldridge, Pennell, 39, Morrellville.\n\nEldridge, Mrs. Sarah T., 71, Woodvale.\n\nEldridge, Mrs. Sallie, 27, Woodvale.\n\nEldridge, Clara, 3, Woodvale.\n\nEldridge, Annie, 1, Woodvale.\n\nElsaesser, Constantine, 44, Railroad street.\n\nElsaesser, Mrs. Frances, 41, Railroad street.\n\nElsaesser, Charles, 13, Railroad street.\n\nElsaesser, Adolph, 11, Railroad street.\n\nElsaesser, Maggie, 10, Railroad street.\n\nElsaesser, Rose, 4, Railroad street.\n\nElsaesser, Mary, 1, Railroad street.\n\nEnglish, John.\n\nEtchison, Samuel, 37, Hulbert House.\n\nEvans, Evan B., 50, Woodvale.\n\nEvans, Susannah, 16, Woodvale.\n\nEvans, Mrs. Mary, 55, Main street.\n\nEvans, Annie, 26, Millville.\n\nEvans, Jennie, 13, Millville.\n\nEvans, Susannah, 9, Millville.\n\nEvans, Idris, 3, Millville.\n\nEvans, Walter, 8, Vine street.\n\nEvans, Albert, 12, Conemaugh.\n\nEvans, Elizabeth.\n\nFairfax, Mrs. Susan, 94, Somerset street.\n\nFairfax, Mrs. G. W., 38, Somerset street.\n\nFedorizen, Miklosz, Cambria.\n\nFenlon, Patrick, 70, Conemaugh.\n\nFendra, E. H.\n\nFenn, John Fulton, 12, Locust street.\n\nFenn, Daisy, 10, Locust street.\n\nFenn, George Washington, 8, Locust street.\n\nFenn, Virginia, 5, Locust street.\n\nFenn, Esther, 1\u00bd, Locust street.\n\nFentiman, Edwin F., 19, Main street.\n\nFees, Frank, 23, Millville.\n\nFichtner, Carrie, Stonycreek street.\n\nFichtner, Annie, 21, Main street.\n\nFiddler, son of Jacob, Cambria.\n\nFindlay, Mrs. Ph\u0153be, 58, Woodvale.\n\nFindlay, Robert B., 17, Conemaugh.\n\nFingerhute, Mary, 18, Conemaugh street.\n\nFingle, Mrs. Mary.\n\nFink, Samuel P., 54, Conemaugh street.\n\nFink, Mrs. Mary, 47, Conemaugh street.\n\nFisher, John, 60, Cambria.\n\nFisher, Johanna, 19, Cambria.\n\nFisher, Kate, 9, Cambria.\n\nFisher, Eddie, 7, Cambria.\n\nFisher, George, 3, Cambria.\n\nFisher, August.\n\nFisher, William.\n\nFitzgerald, Mrs. Catherine. 40, Millville.\n\nFitzpatrick, Eliza, 15 months, Cambria.\n\nFitzharris, Mary J., 16, Franklin street.\n\nFitzharris, Sarah A., 15, Franklin street.\n\nFoling, August, Cambria.\n\nFoster, Mrs. Margaret, 64, Woodvale.\n\nFoster, Maggie, 29, Woodvale.\n\nFrank, August, 26, Washington street.\n\nFrank, Lena, 15, Washington street.\n\nFritz, Mrs. Matilda, 26, Horner street.\n\nFritz, Jane, 2, Horner street.\n\nFritz, Lily, 1, Horner street.\n\nGaffney, Mrs. Ellen, 26, Cambria.\n\nGaither, Willie, 15, South St.\n\nGardner, John.\n\nGeczie, John, 47, Cambria.\n\nGeczie, Veronica, 37, Cambria.\n\nGeczie, Stephen, 8, Cambria.\n\nGeczie, Annie, 4, Cambria.\n\nGeczie, August, 2, Cambria.\n\nGeczie, Belle, 3 months, Cambria.\n\nGeddes, Mrs. George, 40, Woodvale.\n\nGeisel, Julia, 9, Cambria.\n\nGeisel, Rolla, 9, Cambria.\n\nGeraldan, Mrs., 17, Conemaugh street.\n\nGillas, David, 66, Cambria.\n\nGiven, Cora, Millville.\n\nGlass, James, 45, School Alley.\n\nGolde, Harry, 5, Walnut street.\n\nGouchenour, Frank, 31, Conemaugh.\n\nGrant, Mrs. Kate, 24, Cambria.\n\nGrant, Bernard, 5, Cambria.\n\nGrant, John, 3, Cambria.\n\nGray, Mrs. Frances, 36, Woodvale.\n\nGray, Gerald, 5, Woodvale.\n\nGray, Inez, 3, Woodvale.\n\nGreenwood, Mary A., 8, Cambria.\n\nGreitzer, George, 27, Cambria.\n\nGreger, Ann.\n\nGriffith, Mr.\n\nGromley, Mrs. Magdalena, 48, Mineral Point.\n\nGromley, Mary M., 16, Mineral Point.\n\nGromley, Daniel J., 13, Mineral Point.\n\nGromley, Emanuel L., 9, Mineral Point.\n\nGromley, Emma B., Mineral Point.\n\nHager, Mrs. Mary, 62, Washington street.\n\nHagerty, Mrs. Mary J., 36, School alley.\n\nHagerty, Kate, 12, School alley.\n\nHagerty, Stella, 8, School alley.\n\nHaight, Annie, First Ward.\n\nHaltie, Miss.\n\nHaldiman, Hy, Woodvale.\n\nHammers, George.\n\nHammill, Mrs. Catherine, 70, Cambria.\n\nHamilton, Lou.\n\nHannan, Mamie, 22, Woodvale.\n\nHanekamp, Mrs. Louise, 28, Lincoln street.\n\nHanekamp, child.\n\nHarrigan, Mary L., Millville.\n\nHarris.\n\nHart, May, 9, Market street.\n\nHarkey, William G. Hess, Mrs. Haugh, John, Conemaugh.\n\nHayes, Thos., 10, Cambria.\n\nHayes, Annie, 5, Cambria.\n\nHayes, Agnes Gertrude, 3 weeks, Cambria.\n\nHeckman, Miss, 18, Cambria.\n\nHeidenthal, Mrs. Mary, 38, Woodvale.\n\nHeidenthal, Joseph, 14, Woodvale.\n\nHeidenthal, Annie, 12, Woodvale.\n\nHeidenthal, Phoebe, 10, Woodvale.\n\nHeidenthal, Bertha, 6, Woodvale.\n\nHeidenthal, Alfred, 2, Woodvale.\n\nHeingard, Annie, 22, Woodvale.\n\nHeine, Joseph, 1, Cambria.\n\nHeine, Amelia, 8 months, Cambria.\n\nHellenberger, Miss E.\n\nHellreigle, Chas. J., 28, Woodvale.\n\nHenahan, John, 40, Cambria.\n\nHenahan, Mrs. Mary, 24, Cambria.\n\nHenahan, Mary, 7, Cambria.\n\nHenahan, Catherene, 4, Cambria.\n\nHenahan, Frances, 1, Cambria.\n\nHenderson, Thomas, South Fork.\n\nHenderson, Robert, 6 months, Main street.\n\nHenning, John.\n\nHenning, Mary.\n\nHickey, Stephen, 9, Conemaugh.\n\nHicks, Miss Ella, Woodvale.\n\nHimes, Charles C., Conemaugh street.\n\nHimes, Mrs. C. C., Conemaugh street.\n\nHimes,| }| children.  \n---|---|---  \nHimes,  \nHimes,\n\nHinchman, Franklin, 2, Woodvale.\n\nHirsch, Henry, 10, Cambria.\n\nHockenberger, Ann, Napoleon street.\n\nHoffman, Mrs. Mary, 69, Conemaugh.\n\nHoffman, Joseph, 10, Conemaugh.\n\nHoffman, Mary, 8, Conemaugh.\n\nHoffman, Peter, 78, Market street.\n\nHoffman, Frank C., 11, Market street.\n\nHoffman, Sehna, 3, Market street.\n\nHoffman, Lena, 19, Washington street.\n\nHoffman, George, 12, Washington street.\n\nHoffman, Crissie, 9, Washington street.\n\nHoffman, Albert, 4, Washington street.\n\nHoffman, Walter, 2, Washington street.\n\nHoffman, Stella, 6 months, Washington street.\n\nHoffman, Fred W., 42, Conemaugh.\n\nHoffman, Mrs. Jennie, 40, Conemaugh.\n\nHoffman, Lena, 19, Conemaugh.\n\nHoffman, Henry, 65, Conemaugh.\n\nHoffman, Mrs. Mary Ellen, 55, Conemaugh.\n\nHoffman, Stewart, 24, Conemaugh.\n\nHoffman, Mrs., Conemaugh.\n\nHoffman.\n\nHoffman.\n\nHoffman.\n\nHoopes, Mrs. Maria, 25, Woodvale.\n\nHoopes, Ernest, 5, Woodvale.\n\nHoopes, Allen C., 6 months, Woodvale.\n\nHopkins, Hannah, 40, Locust street.\n\nHopkins, Elizabeth, 4, Conemaugh.\n\nHopkins, Geo., 8, Conemaugh.\n\nHopp, Mary E., 7 months, Vine street.\n\nHorner, Miss, Hulbert House.\n\nHorner, Elwood, 15, Levergood street.\n\nHornick, Wm., 23, Conemaugh.\n\nHough, Mrs. Louisa, 48, Conemaugh.\n\nHough, Patrick, 5, Conemaugh.\n\nHoughton, Mrs. Lizzie J., 24, Walnut street.\n\nHowe, Mary E., Washington street.\n\nHowe, Mrs. Nancy, 50, Bedford street.\n\nHowe, Robert G., 8, Bedford street.\n\nHowe, Mrs. W. J.,\n\nHowells, Maggie, 15, Union street.\n\nHowells, John, 25, Union street.\n\nHowells, Wm., 4 days, Union street.\n\nHughes, Lizzie A., 1, Chestnut street.\n\nHurst, Mrs. Minnie, 60, Washington street.\n\nHurst, Emily, 10, Washington street.\n\nHammell, Margaret, 14, Washington street.\n\nIllis, Daniel, Cambria.\n\nJames, John K., 8, Main street.\n\nJames, William, 10, Market street James, Mrs. John.\n\nJames, Benjamin, Third Ward.\n\nJanosky, Mrs. Lena, 27, Market street.\n\nJenkins, John, 20, Upper Yoder.\n\nJenkins, Harvey, 6, Vine street.\n\nJenkins, Thomas, Third Ward.\n\nJenkins, Mrs. Thomas, Third Ward.\n\nJenkins, \u2014.\n\nJohns, Mrs. Josephine, 32, Woodvale.\n\nJohns, Richard, 14, Woodvale.\n\nJohns, Silvie, 11, Woodvale.\n\nJohns, Stephen, 5, Woodvale.\n\nJohnson, Mrs. David, 40, Conemaugh.\n\nJohnson, Geraldine, 17, Conemaugh.\n\nJohnson, George, 17, Union street.\n\nJohnson, William, 15, Union street.\n\nJohnson, Gertrude, 13, Union street.\n\nJohnson, Lottie, 11, Union street.\n\nJohnson, Dollie, 7, Union street.\n\nJohnson, Frederick, 4, Union street.\n\nJohnson, Lulu, 3, Union street.\n\nJohnson, Ellen, Hulbert House.\n\nJones, Mrs. Alice, 65, Millville.\n\nJones, Mrs. Rachael, 41, Main street.\n\nJones, Ella, 11, Main street.\n\nJones, Sarah, 8, Main street.\n\nJones, Abner, 6, Main street.\n\nJones, Ida, 3, Main street.\n\nJones, Thomas, 6, Conemaugh street.\n\nJones, Elmer, 2, Conemaugh street.\n\nJones, Mrs. Jennie, 50, Woodvale.\n\nJones, Williams, 17, Woodvale.\n\nJones, Amanda, 40, Woodvale.\n\nJones, Pearl, 9, Woodvale.\n\nJones, William, 4 months, Woodvale.\n\nJones, James, 19, Pearl street.\n\nJones, Charles, 16, Pearl street.\n\nJones, Emma, Second Ward.\n\nJones, Walter B., 7, Main street.\n\nJones, Mrs. Margaret, 65, Llewellyn street.\n\nJones, Rev. E. W., 56, Vine street.\n\nJones, Mrs. Rev. E. W., 55, Vine street.\n\nKane, Mrs. Lidia, 44, Union street.\n\nKane, Ellsworth, 18, Union street.\n\nKane, Laura, 15, Union street.\n\nKane, Willie, 12, Union street.\n\nKane, Dollie, 10, Union street.\n\nKane, Lester, 2, Union street.\n\nKane, Emma, 21, Prospect.\n\nKane, Mrs. Ann, 60, Cambria City.\n\nKast, Mrs. Charlotte, 43, Clinton street.\n\nKalor, Mrs. Philapena, 67, Conemaugh.\n\nKalor, Jamanes.\n\nKalor, Jane.\n\nKaylor.\n\nKaylor.\n\nKaylor.\n\nKaylor.\n\nKeedy, Clay, 5, Millville.\n\nKeelan, Mrs. Catherine, 55, Cambria.\n\nKeelan, Daphne, 13, Cambria.\n\nKeelan, Edward.\n\nKeelan, Frank.\n\nKeene, Mrs. Elizabeth, 60, Union street.\n\nKeenan, Mrs. Jane, 26, Washington street.\n\nKeiflein, Philamena, Conemaugh.\n\nKeis, Mrs. Caroline, 24, Railroad street.\n\nKeis, infant. Railroad street.\n\nKehoe, Thomas, 24, South Fork.\n\nKelly, Mary M., 30, Millville.\n\nKelly, Mary C., 1\u00bd, Millville.\n\nKelly, Maggie, 17 days, Millville.\n\nKelly, Mrs. Ann, 45, Cambria.\n\nKelly, John W., 24, Cambria.\n\nKidd, Mrs. Jenny, 35, Walnut street.\n\nKidd, Laura, 5, Walnut street.\n\nKilgore, Mrs. W. A., 48, Washington street.\n\nKilgore, Jessie, 15, Washington street.\n\nKilgore, Fred, 12, Washington street.\n\nKilgore, Alex., 9, Washington street.\n\nKimpel, Mrs. Christ, 43, Clinton street.\n\nKinder, Thomas, 40, Moxham.\n\nKing, Mrs. James, 48, Broad street.\n\nKing, Katie M., 24, Broad street.\n\nKing, James, 5, Broad street.\n\nKinney, Mrs. Margaret, 31, Washington street.\n\nKinney,\u2014, 4, Washington street.\n\nKinney, Agnes, 10, Washington street.\n\nKintz, Teresa, 24, Cambria.\n\nKinley, Jane, Bausman alley.\n\nKirkbride, Fannie, 11, Hager Block.\n\nKirkbride, infant, Hager Block.\n\nKirkwood, Finley, 19, Conemaugh.\n\nKirlin, Mrs. Thomas, 32, Conemaugh street.\n\nKirlin, Willie, 2, Conemaugh street.\n\nKnable, John.\n\nKnable, Leonard.\n\nKnox, Thomas, 54, Somerset.\n\nKeohler, Mrs. Philomen, 87, Conemaugh.\n\nKeohler, Wm., 16, Conemaugh.\n\nKraft, Mrs. Maggie, 37, Walnut street.\n\nKraft, Herman, 12, Walnut street.\n\nKraft, Frederick, 10, Walnut street.\n\nKrieger, Katie.\n\nKunkle, Katie, 19, Washington street.\n\nLambreski, Willie, 2, Cambria.\n\nLavelle, Miss Mary, 31, Broad street.\n\nLavelle, Kate, 24, Broad street.\n\nLavelle, Sallie, 18, Broad street.\n\nLavelle, Mrs. Mary, 58, Broad street.\n\nLavelle, John F., 8, Conemaugh street.\n\nLavelle, Edgar R., 4, Conemaugh street.\n\nLavelle, Frances A., 6, Conemaugh street.\n\nLaystrom, Mrs., 30, Union street.\n\nLayton, infant, Broad street.\n\nLewis, Mrs. Lizzie, 28, Lewis alley.\n\nLichtenberg.\n\nLichtenberg.\n\nLichtenberg.\n\nLightner, James, 23, Cambria.\n\nLightner, Mrs. Mary, 21, Cambria.\n\nLightner, Eddie, 1, Cambria.\n\nLohr, Julia, 17, Bedford street.\n\nLonaenstein, Mrs. Ida, 27, Franklin.\n\nLudwig, Charles E., 30, Railroad street.\n\nLuther, Michael, 40, Cambria.\n\nMadden, Willie, 12, Cambria.\n\nMaloy, Mrs. Ann, 35, Millville.\n\nMaloy, Jane, Hulbert House.\n\nMarks, William.\n\nMartin, Edward, 48, River avenue.\n\nMartin, Mrs. Catherine, 40, Millville.\n\nMartin, Mary, 18, Millville.\n\nMartin, Ann, 7, Millville.\n\nMartin, Celia, 7, Millville.\n\nMasters, Margaret.\n\nMasterton, Miss.\n\nMayhew, Annie, 12, Woodvale.\n\nMayhew, Earnest, 9, Woodvale.\n\nMayhew, Harry, 6, Woodvale.\n\nMcAteer, Mrs. Jane, 38, Cambria.\n\nMcAneny, Sarah, 7, Cambria.\n\nMcAley, P.\n\nMcCann, Mrs. John, 30, Railroad street.\n\nMcCann, John, 31, Railroad street.\n\nMcCann, infant, Railroad street.\n\nMcClarren, Mary, 13, Cambria.\n\nMcClarren, Philip, 1\u00bd, Cambria.\n\nMcConaghy, Harry M., 6, Main street.\n\nMcConaghy, Frank A., 2, Main street.\n\nMcCoy, Mr., Railroad street.\n\nMcGrew, Oscar, Conemaugh.\n\nMcGuire, Constantine, 48, Woodvale.\n\nMcGuire, Ann, 19, Woodvale.\n\nMcGuire, Christian, 17, Woodvale.\n\nMcHugh, Kate, 19, Cambria.\n\nMcKeever, Mrs. Mary.\n\nMcKim, Mrs. Polly, 65, East Conemaugh.\n\nMcMeans, William, 33, Conemaugh street.\n\nMcPike, Rosie, 4, Cambria.\n\nMcWilliams, Susie, 13, Pittsburgh.\n\nMelczer, Robert, 35, Cambria.\n\nMelczer, Mrs. Johanna, 30, Cambria.\n\nMelczer, Albert, three weeks, Cambria.\n\nMelczer, Mary, 4, Cambria.\n\nMelczer, John, 2, Cambria.\n\nMerle, Mrs. George, Washington street.\n\nMerle, Mrs. Ida, 29, Washington street.\n\nMerle, Nettie, 5, Washington street.\n\nMerle, Elmer, 2, Washington street.\n\nMeredith, Mr. (probably duplicate).\n\nMeyers, Joseph, 70, Cambria.\n\nMeyers, Lizzie, 11, Millville.\n\nMeyers, Annie, 9, Millville.\n\nMeyers, Stella, 7, Millville.\n\nMeyers, Charlie, 5, Millville.\n\nMeyers, Philip, 1, Millville.\n\nMichalitch, Mrs. Mary, 38, Cambria.\n\nMichalitch, Martin, 6, Cambria.\n\nMichalitch, Mary, 3, Cambria.\n\nMichalitch, John, 1, Cambria.\n\nMiller, Lizzie, 11, Woodvale.\n\nMiller, John, 1, Cambria.\n\nMiller, Mrs. Sophia, 45, Cambria.\n\nMiller, John, 8, Cambria.\n\nMiller, Mary, 12, Horner street.\n\nMonteverde, Mrs. Maria, 40, Washington street.\n\nMonteverde, Joseph, 5, Washington street.\n\nMonteverde, Eleanora, 1\u00bd, Washington street.\n\nMoore, Melda, 20, Main street.\n\nMoreland, Mrs. Margaret, 48, Quarry street.\n\nMorgan, Gertie, 11, Millville.\n\nMorgan, Mrs. Mary R., 66, Conemaugh street.\n\nMorgan, Miss, Conemaugh.\n\nMoser, Heinrich, Cambria.\n\nMoschgat, Amelia, 22, Bedford street.\n\nMullen, Mrs. Mary, 65, Conemaugh street.\n\nMullen, Mrs. Margaret, 47, Prospect.\n\nMumma, Mrs. Eliza, 26, Washington street.\n\nMurphy, Mrs. Kate H., 48, Park Place.\n\nMurphy, Mrs. Maggie, 34, Brunswick Hotel.\n\nMurphy, John, 10, Brunswick Hotel.\n\nMurphy, Clara, 8, Brunswick Hotel.\n\nMurphy, Genevieve, 6, Brunswick Hotel.\n\nMurphy, Martin F., 4, Brunswick Hotel.\n\nMurphy, Maggie, 2, Brunswick Hotel.\n\nMurr, Stella, 16, Washington street.\n\nMurr, Frederick, 11, Washington street.\n\nMurr, Nellie, 6, Washington street.\n\nMurr, Frida, 3 months, Washington street.\n\nNadi, Frank.\n\nNainbaugh, Henry.\n\nNayuska, Mrs. Hannah, 65, Market street.\n\nNew, Frank.\n\nNewell, August.\n\nNewman, Banheim, 68, Washington street.\n\nNich, Peter, 30, Cambria.\n\nNich, William, 2, Cambria.\n\nNich, Lena, 23, Cambria.\n\nNich (infant), Cambria.\n\nNix, Frank, Cambria.\n\nNixon, Fannie, 5, Woodvale.\n\nNeice, Conrad.\n\nO'Brien, Mrs. Sarah, 60, Millville.\n\nO'Brien, Mrs. Ellen, 31, Millville.\n\nO'Brien, Mrs. Catherine, 55, Millville.\n\nO'Callahan, James, 70, Millville.\n\nO'Callahan, Mrs. Bridget, 68, Millville.\n\nO'Callahan, Miss Ella, 25, Millville.\n\nO'Connell, Edward, Cambria.\n\nO'Donnell, Mrs. Julia, 26, Washington street.\n\nO'Donnell, John, 2, Washington street.\n\nOgle, Mrs. Hettie M., 52, Washington street.\n\nOgle, Minnie T., 32, Washington street.\n\nOberlander, Robert, 35, Locust street.\n\nOberlander, Mrs. Robert, 30, Locust street.\n\nOberlander, Mary, 2, Locust street.\n\nO'Lily, Catherine, 20, Cambria.\n\nO'Neill, James, 2, Cambria.\n\nOswald, Eulaliah, 9, Third Ward.\n\nOsterman, Mrs. Victoria, 31, Cambria.\n\nOsterman, Conrad, 4, Cambria.\n\nOsterman, Joseph Jr., 6, Cambria.\n\nOsterman, Mary Ann, 1\u00bd, Cambria.\n\nO'Shea, Mary, Second Ward.\n\nOwens, Mrs. Mary, 62, Market street.\n\nOwens, John, 12, Conemaugh street.\n\nOwens, Amelia, 6, Conemaugh street.\n\nOwens, Willie, 4, Conemaugh street.\n\nOwens, Mrs. Elizabeth, 37.\n\nPfeifer, Charles, 30, Woodvale.\n\nPfeifer, Ella, 21, Woodvale.\n\nPheng, John, Conemaugh.\n\nPhillips, Mary, 16, Union street.\n\nPhillips, Grace, 12, Union street.\n\nPhillips, John J., 14, Market street.\n\nPhillips, David, 12, Market street.\n\nPhillips, Richard, 10, Market street.\n\nPhillips, Mary, 8, Market street.\n\nPhillips, Evan, 6, Market street.\n\nPipple, Mrs., Fourth Ward.\n\nPlummer, Alvin.\n\nPollocks, Louis, 19, Cambria.\n\nPolk, John.\n\nPotts, Mrs. Mary, 29, Market street.\n\nPowell, Mrs. Reese, 74, Main street.\n\nPratt, \u2014, Cambria.\n\nPratt, \u2014, Cambria.\n\nPritchard, Mrs. Henry, 48, Market street.\n\nPritchard, Howell, 9, Market street.\n\nPritchard, Alice, 5, Market street.\n\nPritchard, Rachael, 3, Market street.\n\nPrice, Mrs. Abe, 29, Millville.\n\nProgner, Samuel, 28, Conemaugh.\n\nProsser, Mrs. David, 68, Union street.\n\nPukey, Julius, 23, Cambria.\n\nPukey, Matilda, 1, Cambria.\n\nRaab, Mollie, 18, Clinton street.\n\nRaab, Bertha, 13, Clinton street.\n\nRaab, Katie, 3, Clinton street.\n\nRaab, Mrs. Minnie, 24, Washington street.\n\nRawn, Mrs. Henrietta, 78, Conemaugh.\n\nReam, Mrs. Mary, 34, Woodvale.\n\nReam, Joseph, 10, Woodvale.\n\nReam, Effie May, 6, Woodvale.\n\nReam, Cora, 1, Woodvale.\n\nReam, Frederick E., 23, Third Ward.\n\nReam, Amelia, 20, Third Ward.\n\nReamus, Gussie, 17, Woodvale.\n\nRecke, Mrs. Alex., 29, Washington street.\n\nReed, Charles.\n\nReese, Susie, 14, Millville.\n\nReese, Sarah, Woodvale.\n\nReese, Mrs., 70.\n\nReidel, Mrs. Teresa, 56, Conemaugh.\n\nReilly, Timothy, 27, Millville.\n\nRessler, John.\n\nReynolds, Mrs. Elizabeth, 40, Woodvale.\n\nReynolds, Idella, 14, Woodvale.\n\nReynolds, Columbia, Conemaugh.\n\nRhodes, Frank, 2, Somerset street.\n\nRich, Harry, 16, Stonycreek street.\n\nRichards, Mrs. Margaret, 40, Union street.\n\nRiffle, Mary C., Cambria.\n\nRiley, Mrs. Bridget, 40, Cambria.\n\nRiley, Annie, 8, Cambria.\n\nRiley, Katie, 6, Cambria.\n\nRipple, Emma, 24, Bedford street.\n\nRitter, Katie, 20, Cambria.\n\nRitter, Sophia, 12, Cambria.\n\nRodgers, Mary, Hulbert House.\n\nRodgers, Mary G., 19, Woodvale.\n\nRodgers, Mrs. Mary, Water street.\n\nRoland, Lizzie, 5 months, Conemaugh.\n\nRoose, John, 31, Haynes street.\n\nRosenfelt, \u2014.\n\nRosenfelt, \u2014.\n\nRosenfelt, \u2014.\n\nRosenfelt, \u2014.\n\nRosenfelt, \u2014.\n\nRosensteel, Matilda V., 19, Woodvale.\n\nRoth, Albert, 8, Cambria.\n\nRoth, Mary, 6, Cambria.\n\nRoth, Sebastian, First Ward.\n\nRuth, John.\n\nRowland, Mrs. E. J., 64, Market street.\n\nRyan, Sadie, 16, Washington street.\n\nRyan, Gertie, 3, Washington street.\n\nRyan, Mary, Third Ward.\n\nSagerson, Mrs., 96, Millville.\n\nSalenty, E.\n\nSample, Mrs. Catherine, 63, East Conemaugh.\n\nSarlous, Grace, 16, Cambria.\n\nSavage, Mrs. Bridget, 76, Woodvale.\n\nSaley, Joseph, 50, Millville.\n\nShaffer, Mrs. Mary, 43, Cambria.\n\nShaffer, Carl, 19, Cambria.\n\nSchanvisky, August, 10, Cambria.\n\nSherer, Mrs. Kate, 49, Conemaugh.\n\nSherer, Emma, 24, Conemaugh.\n\nSherer, Mary, 11, Conemaugh.\n\nSchiffhauer, Frances, 19, Washington street.\n\nSchittenhelm, Wilmena.\n\nSchmitt, William J., 7, Cambria.\n\nSchmitt, Mrs. Augustina, 38, Cambria.\n\nSchmitt, August, 8, Cambria.\n\nSchmitt, Anton, 2, Cambria.\n\nSchmitt, Annie, 1, Cambria.\n\nSchmitz, Ferdinand, Cambria.\n\nSchmitz, Gabriel, 50, Conemaugh.\n\nSchmidt, John L., Cambria.\n\nSchonhardt, Victoria, 56, Conemaugh.\n\nSchultz, Mrs. William, Clinton street.\n\nSchultz, Clinton street.\n\nSchultz, Clinton street.\n\nSchultz, Clinton street.\n\nSchultz, Clinton street.\n\nSchultz, Joseph, First Ward.\n\nSchweitzer, William, Conemaugh.\n\nSchweitzer, Catherine E., Conemaugh.\n\nSchurtz, Peter, 38, Conemaugh.\n\nSeibert, Mrs. Elizabeth, 56, Woodvale.\n\nSchaffer, Howard, 21, South Fork.\n\nShea, Mrs. Mary, 30, Locust street.\n\nSheldon, H.\n\nSherman, Mrs. Ann, 35, Market street.\n\nShinkey, Mrs., Second Ward.\n\nShorper, Jacob.\n\nShorper, Jacob, Jr.\n\nSilverman, Moses, Second Ward.\n\nSeigmund, Mrs. Matilda, 52, Woodvale.\n\nSeigmund, Mrs. Carolina, 28, Woodvale.\n\nSeigmund, John, 20, Woodvale.\n\nSinger, Mrs. E. H., Unionport, Ohio.\n\nSiroczki, Mrs. Mary, 30, Cambria.\n\nSiroczki, Mary, 7, Cambria.\n\nSiroczki, Annie, 4, Cambria.\n\nSiroczki, Lizzie, 2, Cambria.\n\nSkiba, Annie, 6, Cambria.\n\nSkiba, Sophia, 1\u00bd, Cambria.\n\nSmith, Harry, 5, Woodvale.\n\nSmith, Hattie, 4, Woodvale.\n\nSmith, infant, 3, Woodvale.\n\nSmith, Alice J., 2, Woodvale.\n\nSmith, Clarence, 6 months, Woodvale.\n\nSmith, George A., 38, Pearl street.\n\nSmith, Mrs. Jennie, 36, Pearl street.\n\nSmith, Charles, 7, Pearl street.\n\nSmith, Alum, 4, Pearl street.\n\nSmith, Effie, 9 months, Pearl street.\n\nSmith, Mrs. Mary, 21, Cambria.\n\nSmith, Mollie, 22, Cambria.\n\nSmith, Mrs. Ann, 55, Cambria.\n\nSmith, Francis, 3, Cambria.\n\nSmith, Charles, 1, Cambria.\n\nSmith, John M., 38, Millville.\n\nSmith, William, 9, Millville.\n\nSmith, Mrs. Mary, Third Ward.\n\nSmith, William, Third Ward.\n\nSmith, Esther, Third Ward.\n\nSmith, Charles, First Ward.\n\nSmith, Richard, First Ward.\n\nSmith, Frank, First Ward.\n\nSnyder, Polly, 14, Woodvale.\n\nSnyder, William, 8, Woodvale.\n\nSnyder, Annie, 6, Woodvale.\n\nSnyder, John, 3, Woodvale.\n\nSnyder, Patrick V., 5 months, Woodvale.\n\nSnyder, Hollis, Woodvale.\n\nSnyder, Mary.\n\nSnyder, Annie.\n\nSnyder, John.\n\nSnyder, Mary E.\n\nSnyder, Harrison V.\n\nSpeers, Mrs. L. E.\n\nSpenger, Mrs. Catherine, 56, Stonycreek street.\n\nSpenger, Edward, 16, Stonycreek street.\n\nSpoller, Mrs.\n\nSpoller, Lee.\n\nStansfield, James C., 30, Woodvale.\n\nStansfield, Mrs. J. C., 25, Woodvale.\n\nStansfield, Ralph, 9 weeks, Woodvale.\n\nSteckman, Fred, 42, Cambria.\n\nStewart, Watson, 60, Pearl street.\n\nStewart, Mrs., 70, Walnut street.\n\nStews, Louis, Walnut street.\n\nStinely, Annie, 4, Cambria.\n\nStinely, infant, 4 months, Cambria.\n\nStork, Casper, 43, Walnut street.\n\nStork, Mary, 38, Walnut street.\n\nStork, John, 20, Walnut street.\n\nStork, Lizzie, 14, Walnut street.\n\nStrauss, Charles S., Conemaugh.\n\nStrayer, Katie, 22, Market street.\n\nStrayer, Bertha, 14, Market street.\n\nStroup, Henry, Conemaugh.\n\nStufft, Vera, 10, Woodvale.\n\nStufft, Earl B., 8, Woodvale.\n\nStufft, Lula B., 6, Woodvale.\n\nStufft, Elda M., 3, Woodvale.\n\nStufft, infant, four months, Woodvale.\n\nSuder, Lizzie, 9, Millville.\n\nSuder, James, 5, Millville.\n\nSullivan, Mrs. Catherine, 55, Millville.\n\nSwank, Leroy, 4, Main street.\n\nSwank, Miss, Morris street.\n\nSweitzer, William, 35, Morrellville.\n\nTemple, Leroy.\n\nThoburn, John, 40, Millville.\n\nThoburn, Mrs. Flora, 36, Millville.\n\nThoburn, John, Jr., 10, Millville.\n\nThoburn, Harry, 1, Millville.\n\nThomas, Tydvil, 19, Millville.\n\nThomas, Mrs. Annie E., 56, Napoleon street.\n\nThomas, Mrs. Ann, 41, Woodvale.\n\nThomas, Albert E., 17, Woodvale.\n\nThomas, Vivian D., 15, Woodvale.\n\nThomas, James Roy.\n\nThomas, Sylvester.\n\nThomasberger, Fannie, 42, Conemaugh.\n\nThomasberger, Nellie, 13, Conemaugh.\n\nThomasberger, Charles, 11, Conemaugh.\n\nThurin, Levi.\n\nTotas, Jacob, Cambria.\n\nTotas, Sophia, Cambria.\n\nTotas, Michael, Cambria.\n\nTotas, Wavreck, Cambria.\n\nTrefts, William S.\n\nTross, W. J. Sr., 43, Woodvale.\n\nTross, Katie, 19, Woodvale.\n\nTross, William, 17, Woodvale.\n\nTross, Conrad, 16, Woodvale.\n\nTross, Charles, 13, Woodvale.\n\nTross, George, 9, Woodvale.\n\nTross, Louis, 7, Woodvale.\n\nTross, Edward, 6, Woodvale.\n\nTucker, Mrs. Margaret N., 45, Woodvale.\n\nTucker, Lillian G., 18, Woodvale.\n\nTucker, Mabel, 6, Woodvale.\n\nTynan, Michael J., 49, Conemaugh.\n\nTynan, Mrs. M. J., 47, Conemaugh.\n\nUnverzagt, Lulu, 23, Washington street.\n\nVallance, David, 55, Conemaugh street.\n\nVallance, Mrs. Sarah, 66, Conemaugh street.\n\nVallance, Annie, 21, Conemaugh street.\n\nValentine, Mrs. Carrie, Market street.\n\nValentine, Alexander L., 14, Market street.\n\nValentine, Annie May, 11, Market street.\n\nValentine, Burt, 7, Market street.\n\nValentine, Howard, 4, Market street.\n\nValentine, Ruth, 1\u00bd Market street.\n\nVarner, Viola, 12, Cambria.\n\nVarner, Sarah, 10, Cambria.\n\nVarner, Ida, 7, Cambria.\n\nVarner, Ella, 5, Cambria.\n\nVarner, infant, six weeks, Cambria.\n\nVeith, Mrs. Carrie, 52, Stonycreek street.\n\nVeith, Emma, 14, Stonycreek street.\n\nVoeghtly, Mrs. Von Alt, Henry, Clinton street.\n\nWagnor, Mrs. Henry, Cambria.\n\nWagnor, Frank, Cambria.\n\nWagnor, John, Cambria.\n\nWalker, Conrad, 27, Clinton street.\n\nWalker, Ida J., 22, Conemaugh.\n\nWalser, Mrs. Ann, Alum Bank, Pa.\n\nWard, Ella, Cambria.\n\nWarren, Edward, 28, Millville.\n\nWaters, Thomas J., 15, Conemaugh.\n\nWatkins, Mary J., 22, Washington street.\n\nWearn, Mrs. Priscilla, 66, Walnut street.\n\nWearn, Richard, 30, King street.\n\nWearn, Mrs. Ella, 27, King street.\n\nWearn, Myrtle, 3, King street.\n\nWeaver, Joseph H., 19, Woodvale.\n\nWeaver, Margaret J., Second Ward.\n\nWebber, Christian, 31, Woodvale.\n\nWehelco, John, Cambria.\n\nWehn, Casper, 80, Clinton street.\n\nWeinzarl, Annie, 13, Cambria.\n\nWeinzarl, Martha, 11, Cambria.\n\nWeinzarl, Sarah, 7, Cambria.\n\nWeinzarl, Mollie, 5, Cambria.\n\nWeinzarl, John, 3, Cambria.\n\nWeinzarl, George, 4 months, Cambria.\n\nWeisc, Rosa, 10, Cambria.\n\nWhite, Annie, 23, Market street.\n\nWhite, Raymond, 4, Youngstown, O.\n\nWickersham, Richard G., 26, Woodvale.\n\nWilson, Mrs. Lavina, 38, East Conemaugh.\n\nWilson, James, 33, Mineral Point.\n\nWilson, Henry, 58, Millville.\n\nWilson, Mr., Cambria.\n\nWiseman, Charles, 26, Conemaugh.\n\nWiseman, Emma, 4, Conemaugh.\n\nWiseman, August, 2, Conemaugh.\n\nWitz, Sarah, Third Ward.\n\nWolf, Anthony, 24, Cambria.\n\nWolf, Albert, 1\u00bd, Cambria.\n\nWolford, Andrew, Conemaugh.\n\nWolford, Conemaugh.\n\nWolford, Conemaugh.\n\nWolford, Conemaugh.\n\nWoren, Richard, Walnut street.\n\nWoren, Mrs. Richard, Walnut street.\n\nWoren, Willie, 6, Walnut street.\n\nWoren, \u2014\u2014\u2014, child, Walnut street.\n\nWoren, \u2014\u2014\u2014, child, Walnut street.\n\nWoren, Mrs. Priscilla, 60, Walnut street.\n\nWoren, Miss, 24, Walnut street.\n\nWoren, Mrs. Thomas, Walnut street.\n\nYocum, Samuel, Third Ward.\n\nYost, Laura, 18, Grant street.\n\nYost, Lottie, Jackson street.\n\nYoung, Mamie, 12, Broad street.\n\nYoung, Katie, 10, Broad street Youst, Mr.\n\nYoust, Eddie.\n\nZellar, Rose.\n\nZern, Miss.\n\nZimmerman, Milton, 19, Locust street.\n\nZimmerman, Morgan, 11, Young's alley.\n\nZimmerman, Owen N.\n\n## Bibliography\n\nAlexander, Edwin P., The Pennsylvania Railroad, A Pictorial History. W. W. Norton & Co., New York, 1947.\n\nAmory, Cleveland, The Last Resorts. Harper & Brothers, New York, 1952.\n\nAndrews, J. Cutler, Pittsburgh's Post-Gazette. Chapman & Grimes, Boston, 1936.\n\nBarton, William E., The Life of Clara Barton, Founder of the American Red Cross. Houghton Mifflin Co., Boston and New York, 1922.\n\nBeale, The Reverend David J., Through the Johnstown Flood. Hubbard Brothers, Philadelphia, Boston, 1890.\n\nBiographical Directory of the American Congress, 1774-1961. U.S. Government Printing Office, Washington, D.C., 1961.\n\nBiographical and Portrait Cyclopedia of Cambria County. Philadelphia, 1896.\n\nBishop, Philip W., The Beginnings of Cheap Steel. Smithsonian Institution, Washington, D.C., 1959.\n\nBridge, James Howard, The Inside History of the Carnegie Steel Company. The Aldine Book Co., New York, 1903.\n\nBruce, Robert V., 1877: Year of Violence. The New Bobbs-Merrill Co., Indianapolis-New York, 1959.\n\nBurgess, George H., and Kennedy, Miles C., Centennial History of the Pennsylvania Railroad Company 1846-1946. Pennsylvania Railroad Co., Philadelphia, 1949.\n\nCarnegie, Andrew, Autobiography. Houghton Mifflin Co., Boston and New York, 1920.\n\nCarson, Herbert N., The Romance of Steel, The Story of a Thousand Millionaires. A. S. Barnes, New York, 1907.\n\nCentury Cyclopedia of History and Biography of Pennsylvania. Century Publishing Co., Chicago, 1904.\n\nChambers, Julius, Lovers Four and Maidens Five, a Story of the Allegheny Mountains. Porter & Coates, Philadelphia, 1886.\n\nChapman, The Reverend H. L., Memoirs of an Itinerant. Privately published, no dates or place of publication listed.\n\nClark, C. B., C. B. Clark's Johnstown Directory and Citizens Register. Altoona, 1889.\n\nClark, Victor S., History of Manufacturers in U.S., Vol. II, 1860-1893. Peter Smith (with permission Carnegie Institute of Washington), New York, 1949.\n\nConnelly, Frank, and Jenks, George C., Official Hisory of the John-town Flood. Journalist Publishing Co., Pittsburgh, 1889.\n\nDiary of Isadore Lilly. Ebensburg, 1889.\n\nDictionary of American Biography. Charles Scribner's Sons, 1937.\n\nDieck, Herman, The Johnstown Flood, Philadelphia, 1889.\n\nEpler, Percy H., The Life of Clara Barton. The Macmillan Co., New York, 1915.\n\nFerris, George T., The Complete History of the Johnstown and Conemaugh Valley Flood. H. S. Goodspeed & Co., New York, 1889.\n\nField, The Reverend C. N., After the Flood. Philadelphia, 1889.\n\nFitch, John A., The Steel Workers. Russell Sage Foundation, New York, 1911.\n\nGeneral Alumni Catalogue of University of Pennsylvania, 1922.\n\nGrand-View Cemetery, An Historical Sketch. Johnstown, 1931.\n\nHall, John W., Fact Sheet of the Flood of 1889. (Compiled for the 75th anniversary of the disaster.)\n\nHarvey, George, Henry Clay Frick. Charles Scribner's Sons, New York, 1928.\n\nHeiser, Victor, M.D., An American Doctor's Odyssey. W. W. Norton & Co., New York, 1936.\n\nHendrick, Burton J., Life of Andrew Carnegie. Doubleday, Doran & Co., New York, 1932.\n\nHendrick, Burton J., and Henderson, Daniel, Louise Whitfield Carnegie; The Life of Mrs. Andrew Carnegie. Hastings House, New York, 1950.\n\nHistorical Sketch and Manual of Shady Side Presbyterian Church. Pittsburgh, 1892.\n\nHolbrook, Stewart H., The Age of the Moguls. Doubleday, New York, 1954.\n\nIllustrated Historical Atlas Combination of Cambria County. Philadelphia, 1890.\n\nInterviews with flood survivors filmed by WJAC-TV, Johnstown, 1964.\n\nJackson, R. M. S., M.D., The Mountain. J. B. Lippincott & Co., 1860.\n\nJohnson, Tom L., My Story. B. W. Huebsch, New York, 1911.\n\nJohnson, Willis Fletcher, History of the Johnstown Flood. Edgewood Publishing Co., Philadelphia, 1889.\n\nJohnstown Borough Council Minute Book, 1889.\n\nJosephson, Matthew, The Robber Barons. Harcourt, Brace and Co., New York, 1934.\n\nLangford, Gerald, The Richard Harding Davis Years. Holt, Rinehart and Winston, New York, 1961.\n\nLorant, Stefan, Pittsburgh, The Story of an American City. Doubleday, New York, 1964.\n\nMcLaurin, J. J., The Story of Johnstown. James M. Place, Harrisburg, 1890.\n\nMatthews, Donald H., Jr., Presbyterians in the Upper Conemaugh Valley. Johnstown, 1957.\n\nMonthly Weather Review. United States Signal Service, May 1889.\n\nMorison, Elting E., Men, Machines, and Modern Times. M.I.T. Press, Cambridge, 1966.\n\nNational Encyclopedia of American Biography.\n\nO'Connor, Harvey, Mellon's Millions. John Day Co., New York, 1933.\n\nO'Connor, Richard, Johnstown the Day the Dam Broke. J. B. Lippincott Co., Philadelphia, 1957.\n\nOperations of the Board of Health in Consequence of the Floods at Johnstown. Harrisburg, 1891.\n\nOrvis, Charles F., Fishing With the Fly. Manchester, Vermont, 1883.\n\nPittsburgh and Allegheny Illustrated Review. Pittsburgh, 1889.\n\nPittsburgh Directory 1887-88.\n\nReade, Charles, Put Yourself in His Place. 1870.\n\nRecollections of flood survivors recorded on questionnaires put out by the Greater Johnstown Chamber of Commerce in commemoration of the 75th anniversary of the disaster; on file at the Cambria County Historical Society, Ebensburg.\n\nRecord Books, Court of Common Pleas. Pittsburgh.\n\nReport of Citizens Relief Committee of Pittsburgh. Pittsburgh, 1890.\n\nReport on the Dedication of the Monument to the Unknown Dead Who Perished in the Flood at Johnstown, May 31, 1889.\n\nReport of the Secretary of the Flood Relief Commission. Harrisburg, 1890.\n\nSesquicentennial of Cambria County. Cambria County Historical Society, Ebensburg, 1954.\n\nShank, William H., The Amazing Pennsylvania Canals. The Historical Society of York County, Pennsylvania, 1965.\n\nShappee, Nathan D., A History of Johnstown and the Great Flood of 1889: A Study of Disaster and Rehabilitation. Thesis filed at the University of Pittsburgh, 1940.\n\nSipes, William B., The Pennsylvania Railroad. Pennsylvania Railroad Co., Philadelphia, 1875.\n\nSlattery, Gertrude Quinn, Johnstown and Its Flood. Wilkes-Barre, 1936.\n\nStorey, Henry Wilson, History of Cambria County, Pennsylvania. Lewis Publishing Co., New York, 1907.\n\nStrayer, Harold H., and London, Irving L., A Photographic Story of the 1889 Johnstown Flood. Johnstown, 1964.\n\nSwank, James M., Cambria County Pioneers. Allen, Lane & Scott, Philadelphia, 1910.\n\n\u2014\u2014\u2014, History of the Manufacture of Iron in All Ages. American Iron and Steel Association, Philadelphia, 1892.\n\nTaft, Pauline Dakin, The Happy Valley (The Elegant Eighties in Upstate New York). Photographer, Leonard Dakin. Syracuse University Press, Syracuse, New York, 1965.\n\nTestimony Taken by the Pennsylvania Railroad Following the Johnstown Flood of 1889. Two copies in possession of Irving London, Johnstown, and the author.\n\nToasts and Responses at the Banquet Given by the Chamber of Commerce of Pittsburgh, May 27, 1892, at the Duquesne Club.\n\nTransactions American Society of Civil Engineers. June 1891.\n\nVan Rensselaer, M. G., In the Heart of the Alleghenies. Allen, Lane & Scott, Philadelphia, 1885.\n\nVeteran Employes Association of the Pittsburgh Division of the Pennsylvania Railroad. Pittsburgh, 1902.\n\nVictims of the Johnstown Flood. Johnstown, 1890.\n\nWalker, James H., The Johnstown Horror. L. P. Miller & Co., Chicago, 1889.\n\nWallace, Paul A. W., Pennsylvania, Seed of a Nation. Harper & Row, New York, 1962.\n\nWatkins, J. Elfreth, History of the Pennsylvania Railroad Company, 1846-96. Withheld from publication, bound proofs.\n\nWeil, The Reverend Carl, The Rider of Johnstown. Pittsburgh, 1892.\n\nWilson, William Bender, History of the Pennsylvania Railroad Company. H. J. Coates & Co., Philadelphia, 1899.\n\nWinkler, J. K., Incredible Carnegie. Vanguard Press, New York, 1931.\n\nWorld's Charity to the Conemaugh Valley Sufferers and Who Received It, The. Harry M. Benshoff, Publisher, Johnstown, 1890.\n\nMAGAZINES\n\nEngineering News\n\nForest and Stream\n\nHarper's New Monthly Magazine, August 1883\n\nHarper's Weekly\n\nIron Age, The\n\nLe Monde Illustr\u00e9, June 29, 1889\n\nLeslie's Weekly\n\nNorth American Review, August 1889\n\nPennsylvania Magazine of History and Biography (which contains, in its July and October issues of 1933, much of a semi-official but incomplete report on the disaster, written by the noted historian, John B. McMaster)\n\nNEWSPAPERS\n\nBoston Daily Globe\n\nBoston Morning Journal\n\nBoston Post\n\nCambria Freeman\n\nChicago Herald\n\nCincinnati Enquirer\n\nGreensburg Tribune and Herald\n\nJohnstown Daily Democrat\n\nJohnstown Daily Tribune\n\nJohnstown Tribune-Democrat\n\nJohnstown Weekly Democrat\n\nLondon Times\n\nMansfield (Mass.) News\n\nNew York Daily Graphic\n\nNew York Herald\n\nNew York Sun\n\nNew York Times\n\nNew York Tribune\n\nNew York World\n\nPhiladelphia Press\n\nPittsburgh Commercial Dispatch\n\nPittsburgh Post\n\nPittsburgh Post-Gazette\n\nPittsburgh Press\n\nRocky Mountain News (Denver)\n\nUtica Saturday Globe\n\n## Index\n\nAdams, Ignatius,\n\nAdams, Solomon, Samuel, and Rachael,\n\nAdams Street schoolhouse (morgue), , ,\n\nAllegheny County, Pennsylvania,\n\nAllegheny Mountain, , , \u201346,\n\nAllegheny Mountains, , , ,\n\nAllegheny River, , , , , ,\n\nAlma Hall (Odd Fellows), , \u2013167, \u201370, , ,\n\nAltoona, Pennsylvania, , , , , , , , , , , , , , ,\n\nAmerican Iron and Steel Association,\n\nAmerican Doctor's Odyssey, An (Heiser),\n\nAmerican Notes (Dickens),\n\n\"AO\" Tower, , ,\n\nArgyle Coal Company,\n\nArnold, Matthew,\n\nAssociated Press, ,\n\nAstor, John Jacob,\n\nAtlantic City, New Jersey,\n\nB&O Railroad, , , , , , , , ,\n\nBaker, John, ,\n\nBarton, Clara, , , \u201331\n\nBaumgardner, Jacob,\n\nBeale, Reverend David, , , , , , , ,\n\nexperience in the flood, \u201367\n\nnamed as emergency morgue head,\n\nBear Creek,\n\nBear Rock Run,\n\nBeaver Dam Run,\n\nBedford, Pennsylvania, ,\n\nBell, S. E.,\n\nBellefonte, Pennsylvania, ,\n\nBenford, F. A., ,\n\nBenford, Walter, \u201357\n\nBennett, H. M., \u2013107\n\nBens Creek, ,\n\nBensen, Reuben,\n\nBenshoff, Arthur,\n\nBessemer, Sir Henry,\n\nBessemer process (as developed at Johnstown), \u201370\n\nBeverly, Massachusetts,\n\nBidwell, D. W. C., , , ,\n\nBixby, Clarence,\n\nBlairsville, Pennsylvania,\n\nBlue Knob,\n\nBolivar, Pennsylvania, , , \u2013181,\n\nBooth, Edwin,\n\nBooth & Flinn,\n\nBoston, Massachusetts,\n\nBoston Post, ,\n\nBottle Run,\n\nBowen, John Eliot,\n\nBoyer, Dan, \u201393,\n\nBraddock, Pennsylvania, , , , , ,\n\nBraddock, General Edward,\n\nBrinker, Mrs. A. D., , , ,\n\nBrinkerhoff, H. W.,\n\nBrooklyn Bridge,\n\nBrown, Reverend G. W.,\n\nBrown, Henry S., \u2013206,\n\nBrownstown, Pennsylvania,\n\nBryan, Elizabeth, , , ,\n\nBucannon, John,\n\nBuck, Mrs. Ann,\n\nBuffalo Bill (William Cody),\n\nBurket, Mrs. John,\n\nBurns, Robert,\n\nBurt, F. B.,\n\nButler, Charles, \u201357\n\nButtermilk Falls, , , ,\n\nCambria City, Pennsylvania, , , , , , , , ,\n\nCambria County, Pennsylvania, , , , , ,\n\nCambria Iron Company, , , , , , , , , , , , , , , ,\n\nleads in pioneering Bessemer steel, \u201371\n\nconcern over South Fork dam, \u201375\n\nplans to rebuild after the flood, \u201337\n\nCape May, New Jersey,\n\nCarnegie, Andrew, , , , \u201361, \u201369, \u201372, , , ,\n\nearly enthusiasm for Alleghenies as resort area, \u201347\n\nreaction to news of the disaster, \u201357\n\nCarnegie, Phipps & Company, , , ,\n\nCarpenter, C. A.,\n\nCentennial of the U.S. Constitution,\n\nChalfant, John W.,\n\nChambers, James,\n\nChambers, Minnie, ,\n\nChambersburg, Pennsylvania, ,\n\nChapman, Reverend H. L., , \u201334, \u201385, , ,\n\nexperience in the flood, \u201356\n\nChestnut Ridge,\n\nChicago, Illinois, , , ,\n\nChicago Herald, , ,\n\nChicago Inter-Ocean,\n\nChicago Limited, \u2013104\n\nCincinnati, Ohio, ,\n\nCincinnati Enquirer, ,\n\nCivil War, , , , , , , , , ,\n\nClapboard Run,\n\nClark, Rose,\n\nClarke, Charles,\n\nClarke, Louis, ,\n\nCleveland, Ohio,\n\nClifford, Hugh,\n\nConemack Old Town,\n\nConemaugh, Pennsylvania,\n\nConemaugh Gap, ,\n\nConemaugh River, , , , , , , , , ,\n\nConey Island, New York,\n\nConnelly, Frank,\n\nConnelly, William,\n\nCoolidge, Calvin,\n\nCresson, Pennsylvania, ,\n\ndevelopment as summer resort, \u201347\n\nCrute, F. Jennings, , ,\n\nCumberland Valley Railroad,\n\nDale Dyke,\n\nDaly, Augustin, ,\n\nDavis, California Tom,\n\nDavis, Richard Harding, , \u201317\n\nDauphin County, Pennsylvania,\n\nDay Express, , , , , , ,\n\nat East Conemaugh, \u201323\n\nDeckert, Frank, ,\n\nDibert, John, , , ,\n\nDick, Chal, ,\n\nDickens, Charles,\n\nDiller, Reverend Alonzo,\n\nDougherty, C. P., \u201396,\n\nDuquesne Club, Pittsburgh, ,\n\nDurno, John,\n\nEast Conemaugh, Pennsylvania, , , , , , , \u201315,\n\ntrains held on morning of flood, \u201387\n\nflood strikes, \u201326\n\nEast McKeesport, Pennsylvania,\n\nEaston, Levi,\n\nEbensburg, Pennsylvania, , , ,\n\nEddyville, Kentucky,\n\nEdgar Thomson works, , , , , ,\n\nEhrenfeld, Emma, \u201397, \u2013105\n\nEiffel Tower,\n\nEisenhower family,\n\nElder, Cyrus, \u201385, , , , , , ,\n\nEldridge, Samuel,\n\nEngineering and Building Record, ,\n\nEngineering News, , \u201348\n\nEpidemics: diphtheria, \u201332\n\ncholera,\n\ntyphus,\n\ntyphoid fever,\n\nErie Canal, ,\n\nFallen Timber Run,\n\nFenn, John, and family,\n\nFire at the stone bridge, \u201350, \u201373,\n\nFisher, Squire, ,\n\nFisher, George,\n\nFishing in the 1880's, \u201348\n\nFitch, John,\n\nFitzharris, Christ and family,\n\nFletcher, John,\n\nFlinn, William, , ,\n\nForest and Stream, \u201352\n\nFoster, Barbara,\n\nFranco-Prussian War, ,\n\nFrank Leslie's Illustrated Newspaper, , ,\n\nFranklin, Pennsylvania,\n\nFrick, Henry Clay, , \u201358, , , , , , , , ,\n\nFrick Coke Company, H. C., \u201358,\n\nFritz, George and John,\n\nFronheiser, Bessie, ,\n\nFulton, John, , \u201377, , , , , , , , ,\n\nstature in Johnstown, \u201373\n\nletter to Morrell concerning the South Fork dam, \u201374\n\ndelivers impassioned speech, \u201337\n\nGalager, Herbert,\n\nGautier Steel Company, , , , , ,\n\nGeis, Mrs. Abbie, \u201362,\n\nGeis, Foster and Quinn, ,\n\nGeis, John, ,\n\nGeis, Mrs. John,\n\nGeorge, F. N.,\n\nGeorge, Henry,\n\nGibbs, George,\n\nGiffen, Kate,\n\nGoldenberg, Henry,\n\n\"Gospel of Wealth,\"\n\nGould, Jay,\n\nGraham, George,\n\nGramling, George,\n\nGrandview Cemetery, , \u201368\n\n\"Great Strike\" of 1877, ,\n\nGreen Hill, , , , , ,\n\nGreensburg, Pennsylvania, , , , ,\n\nGromley, Christopher, and family,\n\nGrubbtown, Pennsylvania, ,\n\nHaak, Charles,\n\nHank, William,\n\nHanna, Marcus Alonzo,\n\nHarding, Warren G.,\n\nHarper's Weekly, , ,\n\nHarrisburg, Pennsylvania, , ,\n\nHarrison, President Benjamin, , ,\n\nHart, Captain A. N., , , ,\n\nHartley, G. B., \u201357\n\nHastings, Daniel Hartman, \u2013202, , , , , , , , , , , ,\n\nHaws Cement Works,\n\nHeidenfelter family,\n\nHeiser, George, \u201338, , , , ,\n\nHeiser, Mathilde, \u201338, , . ,\n\nHeiser, Victor, \u201338, , , , , , , , ,\n\nexperience in the flood, \u201354\n\nHelman, Sam,\n\nHenderson, Thomas,\n\nHeppenstall, Bill, ,\n\n\"Her Last Message,\"\n\nHess, John, \u201315, ,\n\nHipp, Libby, \u201362,\n\nHirst, Mrs.,\n\nHite, Mrs. Ellen,\n\nHoffman, Ben,\n\nHoffman, Gottfried, and family,\n\nHolley, Alexander,\n\nHomestead, Pennsylvania, , ,\n\nHoover, Herbert,\n\nHorne, Durbin, ,\n\nHorner, Charles,\n\nHorner, Daisy,\n\nHorner, Jacob,\n\nHorseshoe Curve,\n\nHow the Other Half Lives (Riis),\n\nHuff, George F.,\n\nHulbert house, , , ,\n\ndestroyed by the flood, \u201357\n\nHungarians, , , , ,\n\nblamed for widespread crime, \u201313\n\nIrwin, Pennsylvania,\n\nJackson, Robert Montgomery Smith, \u201346\n\nJeannette, Pennsylvania,\n\nJenkins, James and Ann (Jenkins case), \u201359\n\nJenks, George C.,\n\nJersey City, New Jersey,\n\nJohnson, Tom L., , ,\n\nJohnson Street Rail Company, ,\n\nJohnstown, Pennsylvania, , \u201335\n\nbefore the flood, \u201338\n\ncanal basin,\n\nhistory of floods, \u201366;\n\ndevelopment of steel industry, \u201371\n\non the morning of May , \u201387\n\nmap,\n\nflood strikes, \u201350\n\non the morning of June 1, \u201389\n\ncitizens organize after the flood, \u201390\n\nstatistics on deaths caused by the flood, \u201396\n\nfirst relief arrives, \u2013204\n\nthe press arrives, \u2013210\n\nHungarian scare, \u201313\n\nas the press described it, \u201321\n\nlegends, \u201324\n\nmassive charity, \u201332\n\nrecovery, \u201338\n\nreaction to trials,\n\nceremonies for the unknown dead, \u201368\n\nJohnstown City Council,\n\nJohnstown Horror, The,\n\nJohnstown Tribune, , , , , , \u201356, \u201364, , , , \u201398, , , , , , ,\n\nJones, B. F.,\n\nJones, Captain Bill, , \u201372, , , ,\n\nJoseph Horne & Company,\n\nJuniata Crossing, Pennsylvania,\n\nKalamazoo, Michigan,\n\nKeenan, Tom,\n\nKehoe, Thomas,\n\nKelly, William, \u201369,\n\nKeltz, S. W., \u2013107\n\nKempel, Christian,\n\nKernville, Pennsylvania, , , , ,\n\nKilrain, Jake,\n\nKiskiminetas River,\n\nKrebs, Fred, ,\n\nKnox, Philander C., , , , , ,\n\nKnox & Reed, ,\n\nKoch, Henry,\n\nKress, Mrs. Charles,\n\nKuhn, William,\n\nKurz & Allison,\n\nLake Conemaugh, , , , , , , , ,\n\nactivities on the morning of the flood, \u201393\n\nlast moments before the dam failed, \u2013100\n\nLamb's Bridge, , ,\n\nLatrobe, Pennsylvania,\n\nLaurel Hill, , , , ,\n\nLaurel Run,\n\nLawrence, J. J.,\n\nLeap, Mrs.,\n\nLee, Dr. Benjamin,\n\nLee, Dr. John,\n\nLeishman, John G. A., , ,\n\nLenhart, Sam,\n\nLevergood, Mrs. Jacob,\n\nLevergood, Peter,\n\n\"Lewis the Light,\"\n\nLibby Prison,\n\nLiggett, R. C., ,\n\nLigonier, Pennsylvania,\n\nLilly, Pennsylvania, ,\n\n\"Lime Kiln Club,\"\n\nLincoln, Abraham, , ,\n\nLincoln, Robert,\n\nLinton, John, , , ,\n\nLippincott, James, ,\n\nLippincott, Jesse H.,\n\nLittle, John,\n\nLittle, Nancy,\n\nLittle case, \u201358,\n\nLittle Conemaugh River, , , , , , , , , , \u2013107, , , , , ,\n\nLord Mayor of Dublin,\n\nLoretto, Pennsylvania, , ,\n\nLovett, John,\n\nLowman, Dr. John, , , , , ,\n\nLoyalhanna Creek, ,\n\nMacy, R. H., & Company,\n\nMadison Square Garden,\n\nMail train, \u201325\n\nMain Line Canal, \u201351,\n\nMaine, U.S.S.,\n\nMalzi, Jacob,\n\nMan Who Rode to Conemaugh, The, \u201322\n\nMann, Michael (\"The First Victim\"), \u2013106\n\nMatthews, Dr. William, , , ,\n\nMcAchren, Maxwell, \u201364,\n\nMcConnellsburg, Pennsylvania,\n\nMcDonald, Frank,\n\nMcGregor, James, \u201342\n\nMcGuigan, C. J., \u201325\n\nMcKee, John,\n\nMcKinley, William,\n\nMcLaurin, J. J., ,\n\nMcMillan, James,\n\nMcMillan, James (plumber),\n\nMcNally, Mrs. Mary,\n\nMedical News,\n\nMellon, Andrew, \u201359,\n\nMellon, T., & Sons, ,\n\nMemorial Day (in Johnstown), \u201323, \u201327, \u201333, ,\n\nMethodist Church, Johnstown, , , , \u201371\n\nMetropolitan Opera House,\n\nMetz family,\n\nMiller, J. G.,\n\nMiller, Samuel S., \u201326\n\nMillville, Pennsylvania,\n\nMineral Point, Pennsylvania, , , \u201311\n\nMinersville, Pennsylvania,\n\nMitchell, Charlie,\n\nMolly Maguires,\n\nMonongahela River, , ,\n\nMontgomery, Christ, ,\n\nMontgomery, D. M.,\n\nMonument to the Unknown Dead,\n\nMoore, Charles,\n\nMoore, Mickey,\n\nMoorhead, James N.,\n\nMorgues and morgue records, \u201396\n\nMorrell, Daniel J., , , ,\n\nleadership in Johnstown, \u201372\n\nconcern about the dam, \u201377\n\ndeath, \u201378\n\nMorrellville, Pennsylvania, ,\n\nMorris, William E.,\n\nMountain, The (Jackson),\n\nMountain House, Cresson,\n\nMoxham, , , ,\n\nMoxham, Arthur J., , \u2013202, ,\n\nMuddy Run, ,\n\nMurr, Charles, and family,\n\nMussante family, ,\n\nNew Florence, Pennsylvania, , , \u201379, \u201382, , ,\n\nNew Haven, Connecticut,\n\nNew York Central Railroad,\n\nNew York, New York, , , , , , , ,\n\nNew York Daily Graphic, , , ,\n\nNew York Herald, , ,\n\nNew York Stock Exchange,\n\nNew York Sun, , , , , ,\n\nNew York Times, , , , , ,\n\nNew York Tribune,\n\nNew York Post,\n\nNew York Witness,\n\nNew York World, , , , , ,\n\nNewport, Rhode Island,\n\nNiagara Falls,\n\nNight Off, A, , \u201325, ,\n\nNineveh, Pennsylvania, , , , ,\n\nNorth American Review,\n\nOfficial History of the Johnstown Flood (Connelly and Jenks), ,\n\nOgle, Hettie, , ,\n\nOhio River, ,\n\nOil City, Pennsylvania, ,\n\nOklahoma Territory,\n\nOrr, Harry,\n\nOtto Run,\n\nOverholt, Abraham, ,\n\nPacker, Hezekiah,\n\nPaint Creek,\n\nPattison, Governor Robert,\n\nParke, General John G.,\n\nParke, John G., Jr., , \u2013100, \u2013102, , , , , ,\n\nat the club on Memorial Day, \u201322\n\ninvestigates Lake Conemaugh on the morning of the flood, \u201390\n\nrides to South Fork with warning,\n\nPaul, Elmer,\n\n\"Paul Revere of the Flood\" legend, \u201323\n\nPaulson, Jennie, , ,\n\nPearson, Edward,\n\nPeblin, Sam,\n\nPennsylvania, state of, , ,\n\nPennsylvania Board of Health,\n\nPennsylvania Canal, see Main Line Canal Pennsylvania Pike,\n\nPennsylvania Railroad, , , , , , , \u201354, , , , , , \u201317, , , \u2013200, \u201340, \u201354,\n\nconducts its own investigation, \u201326\n\nPeyton, Daniel, \u201323\n\nPhiladelphia, Pennsylvania, , , , , , , , , , , , , , ,\n\nPhiladelphia Record,\n\nPhiladelphia Press, , , , , , ,\n\nPhillips, F.,\n\nPhipps, Henry, Jr., , , , ,\n\nPhotographers arrive at Johnstown, \u201318\n\nPickerell, W. H., \u201397, \u201311,\n\nPitcairn, Robert, , , , , , , , , \u201379, , , , , , ,\n\nPittsburgh, Pennsylvania, , , , , , , , , , , , , , \u201353, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,\n\nclub members from, \u201362\n\nreceives first news of the flood,\n\nsends help to Johnstown, \u2013204\n\ncash donations to Johnstown,\n\nclub members interviewed by the press, \u201343\n\nPittsburgh \"Blue Book,\"\n\nPittsburgh Chamber of Commerce,\n\nPittsburgh Chronicle-Telegraph,\n\nPittsburgh Commercial-Gazette, ,\n\nPittsburgh Dispatch, ,\n\nPittsburgh Leader, ,\n\nPittsburgh Post, , ,\n\nPittsburgh Post-Gazette, ,\n\nPittsburgh Press, , , ,\n\nPittsburgh Relief Committee, , , ,\n\nPittsburgh Times, ,\n\nPlotner, H. W.,\n\nPlummer, J. B., , \u201315\n\nPort Royal, Pennsylvania,\n\nPortage Railroad, \u201351, ,\n\nPowell, John Wesley,\n\nPresbyterian Church, Johnstown, , , , , , ,\n\nProspect, Pennsylvania (and Prospect Hill), , , , , , , , , ,\n\nPulitzer, Joseph,\n\nPut Yourself in His Place (Reade), ,\n\nQuicksteps (baseball team),\n\nQuinn, Gertrude, , \u201360, , , ,\n\nexperience in the flood, \u201365\n\nQuinn, James, , \u201361, , , ,\n\nQuinn, Vincent, , ,\n\nRaymond, Flood C.,\n\nRea, Samuel,\n\nReade, Charles, ,\n\nRed Cross, , \u201331\n\nReed, Alfred,\n\nReed, Isaac,\n\nReed, James, , ,\n\nReichard, William, \u201395\n\nReid, Whitelaw,\n\nReilly, Congressman John, \u201349, ,\n\nReynolds, Sylvester,\n\nRhodes, Flood S.,\n\nRichards, Carrie, \u201357\n\nRiis, Jacob,\n\nRitenour, John,\n\nRobinson, Reverend T. H., , \u201321\n\nRockefeller, John D.,\n\nRoebling, John Augustus,\n\nRoosevelt, Theodore,\n\nRorabaugh Creek,\n\nRose, Horace W., \u201382, , , , , ,\n\nexperience in the flood, \u201369\n\nRoss, John, \u201323\n\nRoss, Joseph,\n\nRuff, Benjamin F., , , , , , , , , ,\n\norganizes the South Fork Fishing and Hunting Club, \u201349\n\ncorrespondence with Daniel J. Morrell concerning South Fork dam, \u201376\n\nRusher, L. L., \u201395\n\nRyan, John,\n\nSt. John's Catholic Church, Johnstown,\n\nSt. Michael, Pennsylvania, ,\n\nSalt Lake City, Utah,\n\nSaltlick Creek,\n\nSandy Run,\n\nSandy Vale Cemetery, , , ,\n\nSang Hollow, , , , , , \u201399,\n\nSchantz, Joseph,\n\nSchick, Cyrus, ,\n\nSchoefler, Eddie,\n\nSchubert, C. T.,\n\nSchultz, John, ,\n\nSchwab, Charles,\n\nSchwartzentruver, U. Ed, ,\n\nScott, James B.,\n\nScottdale, Pennsylvania,\n\nSewickley, Pennsylvania,\n\nShade Creek,\n\nShady Side Academy, Pittsburgh,\n\nShady Side Presbyterian Church, Pittsburgh,\n\nShafer, Howard,\n\nShea, C. B.,\n\nShowers, Bill,\n\nSideling Hill,\n\nSiebert, Dan,\n\nSightseers at Johnstown, \u201333\n\nSipe, Dan,\n\nSkinner, George,\n\nSlabtown (in Allegheny, Pennsylvania), ,\n\nSmith, William Henry,\n\nSomerset, Pennsylvania, , , ,\n\nSousa, John Philip,\n\nSouth Fork, Pennsylvania, , , , , , \u201396, , , , , , ,\n\nflood strikes, \u2013107\n\nSouth Fork Creek, , , , , , , , , , , , , , ,\n\nSouth Fork dam, , , , \u201350, \u201364, , , , , \u201364\n\ndescribed, \u201341\n\nconstruction of, \u201353\n\nfirst breaks,\n\nrebuilt, \u201356;\n\nfaults with renovation, \u201377\n\nfirst message from,\n\nsecond message from,\n\nthird message from,\n\nfails,\n\nremains examined by engineering experts, \u201345\n\ncoroner's jury verdicts concerning, \u201346\n\nSouth Fork Fishing and Hunting Club, , , , , , , , , ,\n\non Memorial Day, \u201320\n\ndescribed, \u201344\n\nhistory of organization, \u201350\n\nmembership, \u201362\n\nsubject of publicity, \u201343\n\ndeclared guilty by the press, \u201345, \u201352\n\nlaw suits against, \u201364\n\nSpangler, George,\n\nSteel industry, \u201323, \u201361, \u201370\n\nSteelton, Pennsylvania,\n\nStevenson, Robert Louis,\n\nStineman, Sheriff George, , , ,\n\nStinson, Eliza,\n\nStone bridge, , , , , , \u201373, , , ,\n\nfire breaks out at, \u201350\n\nStone viaduct, \u2013109\n\nStony Creek, , , , , , , , , , , , , , , ,\n\nStorm of May \u201331, , \u201389\n\nStowe, Judge Edwin H.,\n\nStoystown, Pennsylvania,\n\nStrawbridge & Clothier,\n\nStrayer, Jacob,\n\nSullivan, John L.,\n\nSummerhill, Pennsylvania,\n\nSusquehanna River,\n\nSwank, George T., , , , , , , , , , ,\n\nSwank, Harry,\n\nSwing, Lizzie, ,\n\nTalmage, Reverend T. DeWitt,\n\nTarbell, Farney S.,\n\nTarr, H. C.,\n\nTemple, Leroy,\n\n\"That Valley of Tears,\"\n\nThompson, Lizzie, ,\n\nThomson, Edgar, works, see Edgar Thomson works\n\nThomson, J. Edgar, , ,\n\nTice, William,\n\nTiffany & Company,\n\nToppers Run,\n\nTrout Run,\n\nTuscarora Mountain,\n\nTuxedo Park, New York, ,\n\nUncle Tom's Cabin (Stowe),\n\nUnderground Railroad,\n\nUnger, Colonel Elias J., , , , , , , , \u201314, , , ,\n\ntries to stave off disaster at South Fork dam, \u201393\n\nU.S. Geological Survey,\n\nU.S. Signal Service,\n\nUniversity of Pennsylvania,\n\nUnknown dead, ,\n\nUtica Saturday Globe,\n\nVerona, Pennsylvania,\n\nVictoria, Queen,\n\nW.C.T.U., ,\n\nWagoner, George,\n\nWalkinshaw, J. C., , , \u201319\n\nWallace, Mrs. Lew,\n\nWalters, James,\n\nWalton, Izaak,\n\nWarthen, Charles,\n\nWashington, D.C., , , ,\n\nWashington, George,\n\nWebber, Herbert,\n\nWellington, A. M.,\n\nWells, Calvin, ,\n\nWelsh, Sylvester, , ,\n\nWestern Reservoir, , \u201352\n\nWestern Theological Seminary,\n\nWestinghouse, George,\n\nWestmoreland County, Pennsylvania, ,\n\nWetmore, Claude,\n\nWheeling, West Virginia,\n\nWhiskey Rebellion,\n\nWhitman, Walt, ,\n\nWillard Hotel, Washington, ,\n\nWilliams, Moses,\n\nWilmore,\n\nWilson, J. P., \u201397, , ,\n\nWood, Morrell & Company, ,\n\nWoodvale, Pennsylvania, , , , ,\n\nflood strikes, \u201327\n\nWorld's Fair, Paris, ,\n\nYellow Run, ,\n\nYoung, Emil,\n\nYoungstown, Ohio,\n\nZimmerman, Charles, \u201381,\n\nZimmerman, Jacob,\n\nZimmerman, Theodore,\n\nZozo the Magic Queen,\n\nSimon & Schuster Paperbacks\n\nRockefeller Center\n\n1230 Avenue of the Americas\n\nNew York, NY 10020\n\nwww.SimonandSchuster.com\n\nCopyright \u00a9 1968 by David McCullough\n\nAll rights reserved, including the right of reproduction in whole or in part in any form.\n\nSIMON & SCHUSTER PAPERBACKS and colophon are registered trademarks of Simon & Schuster, Inc.\n\nThe Library of Congress has cataloged the Touchstone edition as follows:\n\nMcCullough, David G.\n\nThe Johnstown flood.\n\n(A Touchstone book)\n\nBibliography: p.\n\n1. Floods\u2014Pennsylvania\u2014Johnstown. 2. Johnstown (Pa.)\u2014History. I. Title.\n\n[F159.J7M16 1987] 974.877 86-26056\n\nISBN-13: 978-0-671-20714-4\n\nISBN-10: 0-671-20714-8\n\nISBN-13: 978-1-4165-6122-4 (eBook)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nIntroduction\n\nChapter 1 - The Race for Relevance\n\nChapter 2 - The User Is the Content\n\nChapter 3 - The Adderall Society\n\nChapter 4 - The You Loop\n\nChapter 5 - The Public Is Irrelevant\n\nChapter 6 - Hello, World!\n\nChapter 7 - What You Want, Whether You Want It or Not\n\nChapter 8 - Escape from the City of Ghettos\n\nAcknowledgements\n\nFURTHER READING\n\nNOTES\n\nINDEX\nAdvance Praise for _The Filter Bubble_\n\n\"Internet firms increasingly show us less of the wide world, locating us in the neighborhood of the familiar. The risk, as Eli Pariser shows, is that each of us may unwittingly come to inhabit a ghetto of one.\"\n\n\u2014Clay Shirky, author of _Here Comes Everybody_ and _Cognitive Surplus_\n\n\" 'Personalization' sounds pretty benign, but Eli Pariser skillfully builds a case that its excess on the Internet will unleash an information calamity\u2014unless we heed his warnings. Top-notch journalism and analysis.\"\n\n\u2014Steven Levy, author of _In the Plex: How Google Thinks, Works and Shapes Our Lives_\n\n\"The Internet software that we use is getting smarter, and more tailored to our needs, all the time. The risk, Eli Pariser reveals, is that we increasingly won't see other perspectives. In _The Filter Bubble,_ he shows us how the trend could reinforce partisan and narrow mindsets, and points the way to a greater online diversity of perspective.\"\n\n\u2014Craig Newmark, founder of craigslist\n\n\"Eli Pariser has written a must-read book about one of the central issues in contemporary culture: personalization.\"\n\n\u2014Caterina Fake, cofounder of flickr and Hunch\n\n\"You spend half your life in Internet space, but trust me\u2014you don't understand how it works. Eli Pariser's book is a masterpiece of both investigation and interpretation; he exposes the way we're sent down particular information tunnels, and he explains how we might once again find ourselves in a broad public square of ideas. This couldn't be a more interesting book; it casts an illuminating light on so many of our daily encounters.\"\n\n\u2014Bill McKibben, author of _The End of Nature_ and _Eaarth_ and founder of 350.org\n\n\" _The Filter Bubble_ shows how unintended consequences of well-meaning online designs can impose profound and sudden changes on politics. All agree that the Internet is a potent tool for change, but whether changes are for the better or worse is up to the people who create and use it. If you feel that the Web is your wide open window on the world, you need to read this book to understand what you aren't seeing.\"\n\n\u2014Jaron Lanier, author of _You Are Not a Gadget_\n\n\"For more than a decade, reflective souls have worried about the consequences of perfect personalization. Eli Pariser's is the most powerful and troubling critique yet.\"\n\n\u2014Lawrence Lessig, author of _Code, Free Culture,_ and _Remix_\n\n\"Eli Pariser isn't just the smartest person I know thinking about the relationship of digital technology to participation in the democratic process\u2014he is also the most experienced. _The Filter Bubble_ reveals how the world we encounter is shaped by programs whose very purpose is to narrow what we see and increase the predictability of our responses. Anyone who cares about the future of human agency in a digital landscape should read this book\u2014especially if it is not showing up in your recommended reads on Amazon.\"\n\n\u2014Douglas Rushkoff, author of _Life Inc._ and _Program or Be Programmed_\n\n\"In _The Filter Bubble,_ Eli Pariser reveals the news slogan of the personalized Internet: Only the news that fits you we print.\"\n\n\u2014George Lakoff, author of _Don't Think of an Elephant!_ and _The Political Mind_\n\n\"Eli Pariser is worried. He cares deeply about our common social sphere and sees it in jeopardy. His thorough investigation of Internet trends got me worried, too. He even taught me things \u00e0bout Facebook. It's a must-read.\"\n\n\u2014David Kirkpatrick, author of _The Facebook Effect_\nTHE PENGUIN PRESS\n\nPublished by the Penguin Group Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A. \u2022 Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3 (a division of Pearson Penguin Canada Inc.) \u2022 Penguin Books Ltd, 80 Strand, London WC2R 0RL, England \u2022 Penguin Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd) \u2022 Penguin Books Australia Ltd, 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty Ltd) \u2022 Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi\u2013110 017, India \u2022 Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand (a division of Pearson New Zealand Ltd) \u2022 Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nFirst published in 2011 by The Penguin Press, a member of Penguin Group (USA) Inc.\n\nCopyright \u00a9 Eli Pariser, 2011\n\nAll rights reserved\n\neISBN : 978-1-101-51512-9\n\n__\n\nWithout limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.\n\nThe scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronic piracy of copyrightable materials. Your support of the author's rights is appreciated.\n\nWhile the author has made every effort to provide accurate telephone numbers and Internet addresses at the time of publication, neither the publisher nor the author assumes any responsibility for errors, or for changes that occur after publication. Further, the publisher does not have any control over and does not assume any responsibility for author or third-party Web sites or their content.\n\n<http:\/\/us.penguingroup.com>\nTo my grandfather, Ray Pariser, who taught me that scientific knowledge is best used in the pursuit of a better world. And to my community of family and friends, who fill my bubble with intelligence, humor, and love.\n**INTRODUCTION**\n\nA squirrel dying in front of your house may be more relevant to your interests right now than people dying in Africa.\n\n_\u2014Mark Zuckerberg_ , Facebook founder\n\nWe shape our tools, and thereafter our tools shape us.\n\n_\u2014Marshall McLuhan_ , media theorist\n\n**F** ew people noticed the post that appeared on Google's corporate blog on December 4, 2009. It didn't beg for attention\u2014no sweeping pronouncements, no Silicon Valley hype, just a few paragraphs of text sandwiched between a weekly roundup of top search terms and an update about Google's finance software.\n\nNot everyone missed it. Search engine blogger Danny Sullivan pores over the items on Google's blog looking for clues about where the monolith is headed next, and to him, the post was a big deal. In fact, he wrote later that day, it was \"the biggest change that has ever happened in search engines.\" For Danny, the headline said it all: \"Personalized search for everyone.\"\n\nStarting that morning, Google would use fifty-seven _signals_ \u2014everything from where you were logging in from to what browser you were using to what you had searched for before\u2014to make guesses about who you were and what kinds of sites you'd like. Even if you were logged out, it would customize its results, showing you the pages it predicted you were most likely to click on.\n\nMost of us assume that when we google a term, we all see the same results\u2014the ones that the company's famous Page Rank algorithm suggests are the most authoritative based on other pages' links. But since December 2009, this is no longer true. Now you get the result that Google's algorithm suggests is best for you in particular\u2014and someone else may see something entirely different. In other words, there is no standard Google anymore.\n\nIt's not hard to see this difference in action. In the spring of 2010, while the remains of the Deepwater Horizon oil rig were spewing crude oil into the Gulf of Mexico, I asked two friends to search for the term \"BP.\" They're pretty similar\u2014educated white left-leaning women who live in the Northeast. But the results they saw were quite different. One of my friends saw investment information about BP. The other saw news. For one, the first page of results contained links about the oil spill; for the other, there was nothing about it except for a promotional ad from BP.\n\nEven the number of results returned by Google differed\u2014about 180 million results for one friend and 139 million for the other. If the results were that different for these two progressive East Coast women, imagine how different they would be for my friends and, say, an elderly Republican in Texas (or, for that matter, a businessman in Japan).\n\nWith Google personalized for everyone, the query \"stem cells\" might produce diametrically opposed results for scientists who support stem cell research and activists who oppose it. \"Proof of climate change\" might turn up different results for an environmental activist and an oil company executive. In polls, a huge majority of us assume search engines are unbiased. But that may be just because they're increasingly biased to share our own views. More and more, your computer monitor is a kind of one-way mirror, reflecting your own interests while algorithmic observers watch what you click.\n\nGoogle's announcement marked the turning point of an important but nearly invisible revolution in how we consume information. You could say that on December 4, 2009, the era of personalization began.\n\n**WHEN I WAS** growing up in rural Maine in the 1990s, a new _Wired_ arrived at our farmhouse every month, full of stories about AOL and Apple and how hackers and technologists were changing the world. To my preteen self, it seemed clear that the Internet was going to democratize the world, connecting us with better information and the power to act on it. The California futurists and techno-optimists in those pages spoke with a clear-eyed certainty: an inevitable, irresistible revolution was just around the corner, one that would flatten society, unseat the elites, and usher in a kind of freewheeling global utopia.\n\nDuring college, I taught myself HTML and some rudimentary pieces of the languages PHP and SQL. I dabbled in building Web sites for friends and college projects. And when an e-mail referring people to a Web site I had started went viral after 9\/11, I was suddenly put in touch with half a million people from 192 countries.\n\nTo a twenty-year-old, it was an extraordinary experience\u2014in a matter of days, I had ended up at the center of a small movement. It was also overwhelming. So I joined forces with another small civic-minded startup from Berkeley called MoveOn.org. The cofounders, Wes Boyd and Joan Blades, had built a software company that brought the world the Flying Toasters screen saver. Our lead programmer was a twenty-something libertarian named Patrick Kane; his consulting service, We Also Walk Dogs, was named after a sci-fi story. Carrie Olson, a veteran of the Flying Toaster days, managed operations. We all worked out of our homes.\n\nThe work itself was mostly unglamorous\u2014formatting and sending out e-mails, building Web pages. But it was exciting because we were sure the Internet had the potential to usher in a new era of transparency. The prospect that leaders could directly communicate, for free, with constituents could change everything. And the Internet gave constituents new power to aggregate their efforts and make their voices heard. When we looked at Washington, we saw a system clogged with gatekeepers and bureaucrats; the Internet had the potential to wash all of that away.\n\nWhen I joined MoveOn in 2001, we had about five hundred thousand U.S. members. Today, there are 5 million members\u2014making it one of the largest advocacy groups in America, significantly larger than the NRA. Together, our members have given over $120 million in small donations to support causes we've identified together\u2014health care for everyone, a green economy, and a flourishing democratic process, to name a few.\n\nFor a time, it seemed that the Internet was going to entirely redemocratize society. Bloggers and citizen journalists would single-handedly rebuild the public media. Politicians would be able to run only with a broad base of support from small, everyday donors. Local governments would become more transparent and accountable to their citizens. And yet the era of civic connection I dreamed about hasn't come. Democracy requires citizens to see things from one another's point of view, but instead we're more and more enclosed in our own bubbles. Democracy requires a reliance on shared facts; instead we're being offered parallel but separate universes.\n\nMy sense of unease crystallized when I noticed that my conservative friends had disappeared from my Facebook page. Politically, I lean to the left, but I like to hear what conservatives are thinking, and I've gone out of my way to befriend a few and add them as Facebook connections. I wanted to see what links they'd post, read their comments, and learn a bit from them.\n\nBut their links never turned up in my Top News feed. Facebook was apparently doing the math and noticing that I was still clicking my progressive friends' links more than my conservative friends'\u2014and links to the latest Lady Gaga videos more than either. So no conservative links for me.\n\nI started doing some research, trying to understand how Facebook was deciding what to show me and what to hide. As it turned out, Facebook wasn't alone.\n\n**WITH LITTLE NOTICE** or fanfare, the digital world is fundamentally changing. What was once an anonymous medium where anyone could be anyone\u2014where, in the words of the famous _New Yorker_ cartoon, nobody knows you're a dog\u2014is now a tool for soliciting and analyzing our personal data. According to one _Wall Street Journal_ study, the top fifty Internet sites, from CNN to Yahoo to MSN, install an average of 64 data-laden cookies and personal tracking beacons each. Search for a word like \"depression\" on Dictionary.com, and the site installs up to 223 tracking cookies and beacons on your computer so that other Web sites can target you with antidepressants. Share an article about cooking on ABC News, and you may be chased around the Web by ads for Teflon-coated pots. Open\u2014even for an instant\u2014a page listing signs that your spouse may be cheating and prepare to be haunted with DNA paternity-test ads. The new Internet doesn't just know you're a dog; it knows your breed and wants to sell you a bowl of premium kibble.\n\nThe race to know as much as possible about you has become the central battle of the era for Internet giants like Google, Facebook, Apple, and Microsoft. As Chris Palmer of the Electronic Frontier Foundation explained to me, \"You're getting a free service, and the cost is information about you. And Google and Facebook translate that pretty directly into money.\" While Gmail and Facebook may be helpful, free tools, they are also extremely effective and voracious extraction engines into which we pour the most intimate details of our lives. Your smooth new iPhone knows exactly where you go, whom you call, what you read; with its built-in microphone, gyroscope, and GPS, it can tell whether you're walking or in a car or at a party.\n\nWhile Google has (so far) promised to keep your personal data to itself, other popular Web sites and apps\u2014from the airfare site Kayak.com to the sharing widget AddThis\u2014make no such guarantees. Behind the pages you visit, a massive new market for information about what you do online is growing, driven by low-profile but highly profitable personal data companies like BlueKai and Acxiom. Acxiom alone has accumulated an average of 1,500 pieces of data on each person on its database\u2014which includes 96 percent of Americans\u2014along with data about everything from their credit scores to whether they've bought medication for incontinence. And using lightning-fast protocols, any Web site\u2014not just the Googles and Facebooks of the world\u2014can now participate in the fun. In the view of the \"behavior market\" vendors, every \"click signal\" you create is a commodity, and every move of your mouse can be auctioned off within microseconds to the highest commercial bidder.\n\nAs a business strategy, the Internet giants' formula is simple: The more personally relevant their information offerings are, the more ads they can sell, and the more likely you are to buy the products they're offering. And the formula works. Amazon sells billions of dollars in merchandise by predicting what each customer is interested in and putting it in the front of the virtual store. Up to 60 percent of Netflix's rentals come from the personalized guesses it can make about each customer's movie preferences\u2014and at this point, Netflix can predict how much you'll like a given movie within about half a star. Personalization is a core strategy for the top five sites on the Internet\u2014Yahoo, Google, Facebook, YouTube, and Microsoft Live\u2014as well as countless others.\n\nIn the next three to five years, Facebook COO Sheryl Sandberg told one group, the idea of a Web site that isn't customized to a particular user will seem quaint. Yahoo Vice President Tapan Bhat agrees: \"The future of the web is about personalization ... now the web is about 'me.' It's about weaving the web together in a way that is smart and personalized for the user.\" Google CEO Eric Schmidt enthuses that the \"product I've always wanted to build\" is Google code that will \"guess what I'm trying to type.\" Google Instant, which guesses what you're searching for as you type and was rolled out in the fall of 2010, is just the start\u2014Schmidt believes that what customers want is for Google to \"tell them what they should be doing next.\"\n\nIt would be one thing if all this customization was just about targeted advertising. But personalization isn't just shaping what we buy. For a quickly rising percentage of us, personalized news feeds like Facebook are becoming a primary news source\u201436 percent of Americans under thirty get their news through social networking sites. And Facebook's popularity is skyrocketing worldwide, with nearly a million more people joining each day. As founder Mark Zuckerberg likes to brag, Facebook may be the biggest source of news in the world (at least for some definitions of \"news\").\n\nAnd personalization is shaping how information flows far beyond Facebook, as Web sites from Yahoo News to the _New York Times_ \u2013funded startup News.me cater their headlines to our particular interests and desires. It's influencing what videos we watch on YouTube and a dozen smaller competitors, and what blog posts we see. It's affecting whose e-mails we get, which potential mates we run into on OkCupid, and which restaurants are recommended to us on Yelp\u2014which means that personalization could easily have a hand not only in who goes on a date with whom but in where they go and what they talk about. The algorithms that orchestrate our ads are starting to orchestrate our lives.\n\nThe basic code at the heart of the new Internet is pretty simple. The new generation of Internet filters looks at the things you seem to like\u2014the actual things you've done, or the things people like you like\u2014and tries to extrapolate. They are prediction engines, constantly creating and refining a theory of who you are and what you'll do and want next. Together, these engines create a unique universe of information for each of us\u2014what I've come to call a filter bubble\u2014which fundamentally alters the way we encounter ideas and information.\n\nOf course, to some extent we've always consumed media that appealed to our interests and avocations and ignored much of the rest. But the filter bubble introduces three dynamics we've never dealt with before.\n\nFirst, you're alone in it. A cable channel that caters to a narrow interest (say, golf) has other viewers with whom you share a frame of reference. But you're the only person in your bubble. In an age when shared information is the bedrock of shared experience, the filter bubble is a centrifugal force, pulling us apart.\n\nSecond, the filter bubble is invisible. Most viewers of conservative or liberal news sources know that they're going to a station curated to serve a particular political viewpoint. But Google's agenda is opaque. Google doesn't tell you who it thinks you are or why it's showing you the results you're seeing. You don't know if its assumptions about you are right or wrong\u2014and you might not even know it's making assumptions about you in the first place. My friend who got more investment-oriented information about BP still has no idea why that was the case\u2014she's not a stockbroker. Because you haven't chosen the criteria by which sites filter information in and out, it's easy to imagine that the information that comes through a filter bubble is unbiased, objective, true. But it's not. In fact, from within the bubble, it's nearly impossible to see how biased it is.\n\nFinally, you don't choose to enter the bubble. When you turn on Fox News or read _The Nation_ , you're making a decision about what kind of filter to use to make sense of the world. It's an active process, and like putting on a pair of tinted glasses, you can guess how the editors' leaning shapes your perception. You don't make the same kind of choice with personalized filters. They come to you\u2014and because they drive up profits for the Web sites that use them, they'll become harder and harder to avoid.\n\n**OF COURSE, THERE'S** a good reason why personalized filters have such a powerful allure. We are overwhelmed by a torrent of information: 900,000 blog posts, 50 million tweets, more than 60 million Facebook status updates, and 210 billion e-mails are sent off into the electronic ether every day. Eric Schmidt likes to point out that if you recorded all human communication from the dawn of time to 2003, it'd take up about 5 billion gigabytes of storage space. Now we're creating that much data every two _days_.\n\nEven the pros are struggling to keep up. The National Security Agency, which copies a lot of the Internet traffic that flows through AT&T's main hub in San Francisco, is building two new stadium-size complexes in the Southwest to process all that data. The biggest problem they face is a lack of power: There literally isn't enough electricity on the grid to support that much computing. The NSA is asking Congress for funds to build new power plants. By 2014, they anticipate dealing with so much data they've invented new units of measurement just to describe it.\n\nInevitably, this gives rise to what blogger and media analyst Steve Rubel calls the attention crash. As the cost of communicating over large distances and to large groups of people has plummeted, we're increasingly unable to attend to it all. Our focus flickers from text message to Web clip to e-mail. Scanning the ever-widening torrent for the precious bits that are actually important or even just relevant is itself a full-time job.\n\nSo when personalized filters offer a hand, we're inclined to take it. In theory, anyway, they can help us find the information we need to know and see and hear, the stuff that really matters among the cat pictures and Viagra ads and treadmill-dancing music videos. Netflix helps you find the right movie to watch in its vast catalog of 140,000 flicks. The Genius function of iTunes calls new hits by your favorite band to your attention when they'd otherwise be lost.\n\nUltimately, the proponents of personalization offer a vision of a custom-tailored world, every facet of which fits us perfectly. It's a cozy place, populated by our favorite people and things and ideas. If we never want to hear about reality TV (or a more serious issue like gun violence) again, we don't have to\u2014and if we want to hear about every movement of Reese Witherspoon, we can. If we never click on the articles about cooking, or gadgets, or the world outside our country's borders, they simply fade away. We're never bored. We're never annoyed. Our media is a perfect reflection of our interests and desires.\n\nBy definition, it's an appealing prospect\u2014a return to a Ptolemaic universe in which the sun and everything else revolves around us. But it comes at a cost: Making everything more personal, we may lose some of the traits that made the Internet so appealing to begin with.\n\nWhen I began the research that led to the writing of this book, personalization seemed like a subtle, even inconsequential shift _._ But when I considered what it might mean for a whole society to be adjusted in this way, it started to look more important. Though I follow tech developments pretty closely, I realized there was a lot I didn't know: How did personalization work? What was driving it? Where was it headed? And most important, what will it do to us? How will it change our lives?\n\nIn the process of trying to answer these questions, I've talked to sociologists and salespeople, software engineers and law professors. I interviewed one of the founders of OkCupid, an algorithmically driven dating Web site, and one of the chief visionaries of the U.S. information warfare bureau. I learned more than I ever wanted to know about the mechanics of online ad sales and search engines. I argued with cyberskeptics and cybervisionaries (and a few people who were both).\n\nThroughout my investigation, I was struck by the lengths one has to go to in order to fully see what personalization and filter bubbles do. When I interviewed Jonathan McPhie, Google's point man on search personalization, he suggested that it was nearly impossible to guess how the algorithms would shape the experience of any given user. There were simply too many variables and inputs to track. So while Google can look at overall clicks, it's much harder to say how it's working for any one person.\n\nI was also struck by the degree to which personalization is already upon us\u2014not only on Facebook and Google, but on almost every major site on the Web. \"I don't think the genie goes back in the bottle,\" Danny Sullivan told me. Though concerns about personalized media have been raised for a decade\u2014legal scholar Cass Sunstein wrote a smart and provocative book on the topic in 2000\u2014the theory is now rapidly becoming practice: Personalization is already much more a part of our daily experience than many of us realize. We can now begin to see how the filter bubble is actually working, where it's falling short, and what that means for our daily lives and our society.\n\nEvery technology has an interface, Stanford law professor Ryan Calo told me, a place where you end and the technology begins. And when the technology's job is to show you the world, it ends up sitting between you and reality, like a camera lens. That's a powerful position, Calo says. \"There are lots of ways for it to skew your perception of the world.\" And that's precisely what the filter bubble does.\n\n**THE FILTER BUBBLE'S** costs are both personal and cultural. There are direct consequences for those of us who use personalized filters (and soon enough, most of us will, whether we realize it or not). And there are societal consequences, which emerge when masses of people begin to live a filter-bubbled life.\n\nOne of the best ways to understand how filters shape our individual experience is to think in terms of our information diet. As sociologist danah boyd said in a speech at the 2009 Web 2.0 Expo:\n\nOur bodies are programmed to consume fat and sugars because they're rare in nature.... In the same way, we're biologically programmed to be attentive to things that stimulate: content that is gross, violent, or sexual and that gossip which is humiliating, embarrassing, or offensive. If we're not careful, we're going to develop the psychological equivalent of obesity. We'll find ourselves consuming content that is least beneficial for ourselves or society as a whole.\n\nJust as the factory farming system that produces and delivers our food shapes what we eat, the dynamics of our media shape what information we consume. Now we're quickly shifting toward a regimen chock-full of personally relevant information. And while that can be helpful, too much of a good thing can also cause real problems. Left to their own devices, personalization filters serve up a kind of invisible autopropaganda, indoctrinating us with our own ideas, amplifying our desire for things that are familiar and leaving us oblivious to the dangers lurking in the dark territory of the unknown.\n\nIn the filter bubble, there's less room for the chance encounters that bring insight and learning. Creativity is often sparked by the collision of ideas from different disciplines and cultures. Combine an understanding of cooking and physics and you get the nonstick pan and the induction stovetop. But if Amazon thinks I'm interested in cookbooks, it's not very likely to show me books about metallurgy. It's not just serendipity that's at risk. By definition, a world constructed from the familiar is a world in which there's nothing to learn. If personalization is too acute, it could prevent us from coming into contact with the mind-blowing, preconception-shattering experiences and ideas that change how we think about the world and ourselves.\n\nAnd while the premise of personalization is that it provides you with a service, you're not the only person with a vested interest in your data. Researchers at the University of Minnesota recently discovered that women who are ovulating respond better to pitches for clingy clothes and suggested that marketers \"strategically time\" their online solicitations. With enough data, guessing this timing may be easier than you think.\n\nAt best, if a company knows which articles you read or what mood you're in, it can serve up ads related to your interests. But at worst, it can make decisions on that basis that negatively affect your life. After you visit a page about Third World backpacking, an insurance company with access to your Web history might decide to increase your premium, law professor Jonathan Zittrain suggests. Parents who purchased EchoMetrix's Sentry software to track their kids online were outraged when they found that the company was then selling their kids' data to third-party marketing firms.\n\nPersonalization is based on a bargain. In exchange for the service of filtering, you hand large companies an enormous amount of data about your daily life\u2014much of which you might not trust friends with. These companies are getting better at drawing on this data to make decisions every day. But the trust we place in them to handle it with care is not always warranted, and when decisions are made on the basis of this data that affect you negatively, they're usually not revealed.\n\nUltimately, the filter bubble can affect your ability to choose how you want to live. To be the author of your life, professor Yochai Benkler argues, you have to be aware of a diverse array of options and lifestyles. When you enter a filter bubble, you're letting the companies that construct it choose which options you're aware of. You may think you're the captain of your own destiny, but personalization can lead you down a road to a kind of informational determinism in which what you've clicked on in the past determines what you see next\u2014a Web history you're doomed to repeat. You can get stuck in a static, ever narrowing version of yourself\u2014an endless you-loop.\n\nAnd there are broader consequences. In _Bowling Alone_ , his bestselling book on the decline of civic life in America, Robert Putnam looked at the problem of the major decrease in \"social capital\"\u2014the bonds of trust and allegiance that encourage people to do each other favors, work together to solve common problems, and collaborate. Putnam identified two kinds of social capital: There's the in-group-oriented \"bonding\" capital created when you attend a meeting of your college alumni, and then there's \"bridging\" capital, which is created at an event like a town meeting when people from lots of different backgrounds come together to meet each other. Bridging capital is potent: Build more of it, and you're more likely to be able to find that next job or an investor for your small business, because it allows you to tap into lots of different networks for help.\n\nEverybody expected the Internet to be a huge source of bridging capital. Writing at the height of the dot-com bubble, Tom Friedman declared that the Internet would \"make us all next door neighbors.\" In fact, this idea was the core of his thesis in _The Lexus and the Olive Tree_ : \"The Internet is going to be like a huge vise that takes the globalization system ... and keeps tightening and tightening that system around everyone, in ways that will only make the world smaller and smaller and faster and faster with each passing day.\"\n\nFriedman seemed to have in mind a kind of global village in which kids in Africa and executives in New York would build a community together. But that's not what's happening: Our virtual next-door neighbors look more and more like our real-world neighbors, and our real-world neighbors look more and more like us. We're getting a lot of bonding but very little bridging. And this is important because it's bridging that creates our sense of the \"public\"\u2014the space where we address the problems that transcend our niches and narrow self-interests.\n\nWe are predisposed to respond to a pretty narrow set of stimuli\u2014if a piece of news is about sex, power, gossip, violence, celebrity, or humor, we are likely to read it first. This is the content that most easily makes it into the filter bubble. It's easy to push \"Like\" and increase the visibility of a friend's post about finishing a marathon or an instructional article about how to make onion soup. It's harder to push the \"Like\" button on an article titled, \"Darfur sees bloodiest month in two years.\" In a personalized world, important but complex or unpleasant issues\u2014the rising prison population, for example, or homelessness\u2014are less likely to come to our attention at all.\n\nAs a consumer, it's hard to argue with blotting out the irrelevant and unlikable. But what is good for consumers is not necessarily good for citizens. What I seem to like may not be what I actually want, let alone what I need to know to be an informed member of my community or country. \"It's a civic virtue to be exposed to things that appear to be outside your interest,\" technology journalist Clive Thompson told me. \"In a complex world, almost everything affects you\u2014that closes the loop on pecuniary self-interest.\" Cultural critic Lee Siegel puts it a different way: \"Customers are always right, but people aren't.\"\n\n**THE STRUCTURE OF** our media affects the character of our society. The printed word is conducive to democratic argument in a way that laboriously copied scrolls aren't. Television had a profound effect on political life in the twentieth century\u2014from the Kennedy assassination to 9\/11\u2014and it's probably not a coincidence that a nation whose denizens spend thirty-six hours a week watching TV has less time for civic life.\n\nThe era of personalization is here, and it's upending many of our predictions about what the Internet would do. The creators of the Internet envisioned something bigger and more important than a global system for sharing pictures of pets. The manifesto that helped launch the Electronic Frontier Foundation in the early nineties championed a \"civilization of Mind in cyberspace\"\u2014a kind of worldwide metabrain. But personalized filters sever the synapses in that brain. Without knowing it, we may be giving ourselves a kind of global lobotomy instead.\n\nFrom megacities to nanotech, we're creating a global society whose complexity has passed the limits of individual comprehension. The problems we'll face in the next twenty years\u2014energy shortages, terrorism, climate change, and disease\u2014are enormous in scope. They're problems that we can only solve together.\n\nEarly Internet enthusiasts like Web creator Tim Berners-Lee hoped it would be a new platform for tackling those problems. I believe it still can be\u2014and as you read on, I'll explain how. But first we need to pull back the curtain\u2014to understand the forces that are taking the Internet in its current, personalized direction. We need to lay bare the bugs in the code\u2014and the coders\u2014that brought personalization to us.\n\nIf \"code is law,\" as Larry Lessig famously declared, it's important to understand what the new lawmakers are trying to do. We need to understand what the programmers at Google and Facebook believe in. We need to understand the economic and social forces that are driving personalization, some of which are inevitable and some of which are not. And we need to understand what all this means for our politics, our culture, and our future.\n\nWithout sitting down next to a friend, it's hard to tell how the version of Google or Yahoo News that you're seeing differs from anyone else's. But because the filter bubble distorts our perception of what's important, true, and real, it's critically important to render it visible. That is what this book seeks to do.\n**1**\n\n**The Race for Relevance**\n\nIf you're not paying for something, you're not the customer; you're the product being sold.\n\n\u2014 _Andrew Lewis,_ under the alias Blue_beetle, on the Web site MetaFilter\n\n**I** n the spring of 1994, Nicholas Negroponte sat writing and thinking. At the MIT Media Lab, Negroponte's brainchild, young chip designers and virtual-reality artists and robot-wranglers were furiously at work building the toys and tools of the future. But Negroponte was mulling over a simpler problem, one that millions of people pondered every day: what to watch on TV.\n\nBy the mid-1990s, there were hundreds of channels streaming out live programming twenty-four hours a day, seven days a week. Most of the programming was horrendous and boring: infomercials for new kitchen gadgets, music videos for the latest one-hit-wonder band, cartoons, and celebrity news. For any given viewer, only a tiny percentage of it was likely to be interesting.\n\nAs the number of channels increased, the standard method of surfing through them was getting more and more hopeless. It's one thing to search through five channels. It's another to search through five hundred. And when the number hits five thousand\u2014well, the method's useless.\n\nBut Negroponte wasn't worried. All was not lost: in fact, a solution was just around the corner. \"The key to the future of television,\" he wrote, \"is to stop thinking about television as television,\" and to start thinking about it as a device with embedded intelligence. What consumers needed was a remote control that controls itself, an intelligent automated helper that would learn what each viewer watches and capture the programs relevant to him or her. \"Today's TV set lets you control brightness, volume, and channel,\" Negroponte typed. \"Tomorrow's will allow you to vary sex, violence, and political leaning.\"\n\nAnd why stop there? Negroponte imagined a future swarming with intelligent agents to help with problems like the TV one. Like a personal butler at a door, the agents would let in only your favorite shows and topics. \"Imagine a future,\" Negroponte wrote, \"in which your interface agent can read every newswire and newspaper and catch every TV and radio broadcast on the planet, and then construct a personalized summary. This kind of newspaper is printed in an edition of one.... Call it the Daily Me.\"\n\nThe more he thought about it, the more sense it made. The solution to the information overflow of the digital age was smart, personalized, embedded editors. In fact, these agents didn't have to be limited to television; as he suggested to the editor of the new tech magazine _Wired_ , \"Intelligent agents are the unequivocal future of computing.\"\n\nIn San Francisco, Jaron Lanier responded to this argument with dismay. Lanier was one of the creators of virtual reality; since the eighties, he'd been tinkering with how to bring computers and people together. But the talk of agents struck him as crazy. \"What's got into all of you?\" he wrote in a missive to the \"Wired-style community\" on his Web site. \"The idea of 'intelligent agents' is both wrong and evil.... The agent question looms as a deciding factor in whether [the Net] will be much better than TV, or much worse.\"\n\nLanier was convinced that, because they're not actually people, agents would force actual humans to interact with them in awkward and pixelated ways. \"An agent's model of what you are interested in will be a cartoon model, and you will see a cartoon version of the world through the agent's eyes,\" he wrote.\n\nAnd there was another problem: The perfect agent would presumably screen out most or all advertising. But since online commerce was driven by advertising, it seemed unlikely that these companies would roll out agents who would do such violence to their bottom line. It was more likely, Lanier wrote, that these agents would have double loyalties\u2014bribable agents. \"It's not clear who they're working for.\"\n\nIt was a clear and plangent plea. But though it stirred up some chatter in online newsgroups, it didn't persuade the software giants of this early Internet era. They were convinced by Negroponte's logic: The company that figured out how to sift through the digital haystack for the nuggets of gold would win the future. They could see the attention crash coming, as the information options available to each person rose toward infinity. If you wanted to cash in, you needed to get people to tune in. And in an attention-scarce world, the best way to do that was to provide content that really spoke to each person's idiosyncratic interests, desires, and needs. In the hallways and data centers of Silicon Valley, there was a new watchword: relevance.\n\nEveryone was rushing to roll out an \"intelligent\" product. In Redmond, Microsoft released Bob\u2014a whole operating system based on the agent concept, anchored by a strange cartoonish avatar with an uncanny resemblance to Bill Gates. In Cupertino, almost exactly a decade before the iPhone, Apple introduced the Newton, a \"personal desktop assistant\" whose core selling point was the agent lurking dutifully just under its beige surface.\n\nAs it turned out, the new intelligent products bombed. In chat groups and on e-mail lists, there was practically an industry of snark about Bob. Users couldn't stand it. _PC World_ named it one of the twenty-five worst tech products of all time. And the Apple Newton didn't do much better: Though the company had invested over $100 million in developing the product, it sold poorly in the first six months of its existence. When you interacted with the intelligent agents of the midnineties, the problem quickly became evident: They just weren't that smart.\n\nNow, a decade and change later, intelligent agents are still nowhere to be seen. It looks as though Negroponte's intelligent-agent revolution failed. We don't wake up and brief an e-butler on our plans and desires for the day.\n\nBut that doesn't mean they don't exist. They're just hidden. Personal intelligent agents lie under the surface of every Web site we go to. Every day, they're getting smarter and more powerful, accumulating more information about who we are and what we're interested in. As Lanier predicted, the agents don't work only for us: They also work for software giants like Google, dispatching ads as well as content. Though they may lack Bob's cartoon face, they steer an increasing proportion of our online activity.\n\nIn 1995 the race to provide personal relevance was just beginning. More than perhaps any other factor, it's this quest that has shaped the Internet we know today.\n\n#  **The John Irving Problem**\n\nJeff Bezos, the CEO of Amazon.com, was one of the first people to realize that you could harness the power of relevance to make a few billion dollars. Starting in 1994, his vision was to transport online bookselling \"back to the days of the small bookseller who got to know you very well and would say things like, 'I know you like John Irving, and guess what, here's this new author, I think he's a lot like John Irving,'\" he told a biographer. But how to do that on a mass scale? To Bezos, Amazon needed to be \"a sort of a small Artificial Intelligence company,\" powered by algorithms capable of instantly matching customers and books.\n\nIn 1994, as a young computer scientist working for Wall Street firms, Bezos had been hired by a venture capitalist to come up with business ideas for the burgeoning Web space. He worked methodically, making a list of twenty products the team could theoretically sell online\u2014music, clothing, electronics\u2014and then digging into the dynamics of each industry. Books started at the bottom of his list, but when he drew up his final results, he was surprised to find them at the top.\n\nBooks were ideal for a few reasons. For starters, the book industry was decentralized; the biggest publisher, Random House, controlled only 10 percent of the market. If one publisher wouldn't sell to him, there would be plenty of others who would. And people wouldn't need as much time to get comfortable with buying books online as they might with other products\u2014a majority of book sales already happened outside of traditional bookstores, and unlike clothes, you didn't need to try them on. But the main reason books seemed attractive was simply the fact that there were so many of them\u20143 million active titles in 1994, versus three hundred thousand active CDs. A physical bookstore would never be able to inventory all those books, but an online bookstore could.\n\nWhen he reported this finding to his boss, the investor wasn't interested. Books seemed like a kind of backward industry in an information age. But Bezos couldn't get the idea out of his head. Without a physical limit on the number of books he could stock, he could provide hundreds of thousands more titles than industry giants like Borders or Barnes & Noble, and at the same time, he could create a more intimate and personal experience than the big chains.\n\nAmazon's goal, he decided, would be to enhance the process of discovery: a personalized store that would help readers find books and introduce books to readers. But how?\n\nBezos started thinking about machine learning. It was a tough problem, but a group of engineers and scientists had been attacking it at research institutions like MIT and the University of California at Berkeley since the 1950s. They called their field \"cybernetics\"\u2014a word taken from Plato, who coined it to mean a self-regulating system, like a democracy. For the early cyberneticists, there was nothing more thrilling than building systems that tuned themselves, based on feedback. Over the following decades, they laid the mathematical and theoretical foundations that would guide much of Amazon's growth.\n\nIn 1990, a team of researchers at the Xerox Palo Alto Research Center (PARC) applied cybernetic thinking to a new problem. PARC was known for coming up with ideas that were broadly adopted and commercialized by others\u2014the graphical user interface and the mouse, to mention two. And like many cutting-edge technologists at the time, the PARC researchers were early power users of e-mail\u2014they sent and received hundreds of them. E-mail was great, but the downside was quickly obvious. When it costs nothing to send a message to as many people as you like, you can quickly get buried in a flood of useless information.\n\nTo keep up with the flow, the PARC team started tinkering with a process they called collaborative filtering, which ran in a program called Tapestry. Tapestry tracked how people reacted to the mass e-mails they received\u2014which items they opened, which ones they responded to, and which they deleted\u2014and then used this information to help order the inbox. E-mails that people had engaged with a lot would move to the top of the list; e-mails that were frequently deleted or unopened would go to the bottom. In essence, collaborative filtering was a time saver: Instead of having to sift through the pile of e-mail yourself, you could rely on others to help presift the items you'd received.\n\nAnd of course, you didn't have to use it just for e-mail. Tapestry, its creators wrote, \"is designed to handle any incoming stream of electronic documents. Electronic mail is only one example of such a stream: others are newswire stories and Net-News articles.\"\n\nTapestry had introduced collaborative filtering to the world, but in 1990, the world wasn't very interested. With only a few million users, the Internet was still a small ecosystem, and there just wasn't much information to sort or much bandwidth to download with. So for years collaborative filtering remained the domain of software researchers and bored college students. If you e-mailed ringo@media.mit.edu in 1994 with some albums you liked, the service would send an e-mail back with other music recommendations and the reviews. \"Once an hour,\" according to the Web site, \"the server processes all incoming messages and sends replies as necessary.\" It was an early precursor to Pandora; it was a personalized music service for a prebroadband era.\n\nBut when Amazon launched in 1995, everything changed. From the start, Amazon was a bookstore with personalization built in. By watching which books people bought and using the collaborative filtering methods pioneered at PARC, Amazon could make recommendations on the fly. (\"Oh, you're getting _The Complete Dummy's Guide to Fencing_? How about adding a copy of _Waking Up Blind: Lawsuits over Eye Injury_?\") And by tracking which users bought what over time, Amazon could start to see which users' preferences were similar. (\"Other people who have similar tastes to yours bought this week's new release, _En Garde!_ \") The more people bought books from Amazon, the better the personalization got.\n\nIn 1997, Amazon had sold books to its first million customers. Six months later, it had served 2 million. And in 2001, it reported its first quarterly net profit\u2014one of the first businesses to prove that there was serious money to be made online.\n\nIf Amazon wasn't quite able to create the feeling of a local bookstore, its personalization code nonetheless worked quite well. Amazon executives are tight-lipped about just how much revenue it's brought in, but they often point to the personalization engine as a key part of the company's success.\n\nAt Amazon, the push for more user data is never-ending: When you read books on your Kindle, the data about which phrases you highlight, which pages you turn, and whether you read straight through or skip around are all fed back into Amazon's servers and can be used to indicate what books you might like next. When you log in after a day reading Kindle e-books at the beach, Amazon is able to subtly customize its site to appeal to what you've read: If you've spent a lot of time with the latest James Patterson, but only glanced at that new diet guide, you might see more commercial thrillers and fewer health books.\n\nAmazon users have gotten so used to personalization that the site now uses a reverse trick to make some additional cash. Publishers pay for placement in physical bookstores, but they can't buy the opinions of the clerks. But as Lanier predicted, buying off algorithms is easy: Pay enough to Amazon, and your book can be promoted as if by an \"objective\" recommendation by Amazon's software. For most customers, it's impossible to tell which is which.\n\nAmazon proved that relevance could lead to industry dominance. But it would take two Stanford graduate students to apply the principles of machine learning to the whole world of online information.\n\n#  **Click Signals**\n\nAs Jeff Bezos's new company was getting off the ground, Larry Page and Sergey Brin, the founders of Google, were busy doing their doctoral research at Stanford. They were aware of Amazon's success\u2014in 1997, the dot-com bubble was in full swing, and Amazon, on paper at least, was worth billions. Page and Brin were math whizzes; Page, especially, was obsessed with AI. But they were interested in a different problem. Instead of using algorithms to figure out how to sell products more effectively, what if you could use them to sort through sites on the Web?\n\nPage had come up with a novel approach, and with a geeky predilection for puns, he called it PageRank. Most Web search companies at the time sorted pages using keywords and were very poor at figuring out which page for a given word was the most relevant. In a 1997 paper, Brin and Page dryly pointed out that three of the four major search engines couldn't find themselves. \"We want our notion of 'relevant' to only include the very best documents,\" they wrote, \"since there may be tens of thousands of slightly relevant documents.\"\n\nPage had realized that packed into the linked structure of the Web was a lot more data than most search engines made use of. The fact that a Web page linked to another page could be considered a \"vote\" for that page. At Stanford, Page had seen professors count how many times their papers had been cited as a rough index of how important they were. Like academic papers, he realized, the pages that a lot of other pages cite\u2014say, the front page of Yahoo\u2014could be assumed to be more \"important,\" and the pages that those pages voted for would matter more. The process, Page argued, \"utilized the uniquely democratic structure of the web.\"\n\nIn those early days, Google lived at google.stanford.edu, and Brin and Page were convinced it should be nonprofit and advertising free. \"We expect that advertising funded search engines will be inherently biased towards the advertisers and away from the needs of the consumers,\" they wrote. \"The better the search engine is, the fewer advertisements will be needed for the consumer to find what they want.... We believe the issue of advertising causes enough mixed incentives that it is crucial to have a competitive search engine that is transparent and in the academic realm.\"\n\nBut when they released the beta site into the wild, the traffic chart went vertical. Google worked\u2014out of the box, it was the best search site on the Internet. Soon, the temptation to spin it off as a business was too great for the twenty-something cofounders to bear.\n\nIn the Google mythology, it is PageRank that drove the company to worldwide dominance. I suspect the company likes it that way\u2014it's a simple, clear story that hangs the search giant's success on a single ingenious breakthrough by one of its founders. But from the beginning, PageRank was just a small part of the Google project. What Brin and Page had really figured out was this: The key to relevance, the solution to sorting through the mass of data on the Web was ... more data.\n\nIt wasn't just which pages linked to which that Brin and Page were interested in. The position of a link on the page, the size of the link, the age of the page\u2014all of these factors mattered. Over the years, Google has come to call these clues embedded in the data _signals._\n\nFrom the beginning, Page and Brin realized that some of the most important signals would come from the search engine's users. If someone searches for \"Larry Page,\" say, and clicks on the second link, that's another kind of vote: It suggests that the second link is more relevant to that searcher than the first one. They called this a _click signal._ \"Some of the most interesting research,\" Page and Brin wrote, \"will involve leveraging the vast amount of usage data that is available from modern web systems.... It is very difficult to get this data, mainly because it is considered commercially valuable.\" Soon they'd be sitting on one of the world's largest stores of it.\n\nWhere data was concerned, Google was voracious. Brin and Page were determined to keep everything: every Web page the search engine had ever landed on, every click every user ever made. Soon its servers contained a nearly real-time copy of most of the Web. By sifting through this data, they were certain they'd find more clues, more signals, that could be used to tweak results. The search-quality division at the company acquired a black-ops kind of feel: few visitors and absolute secrecy were the rule.\n\n\"The ultimate search engine,\" Page was fond of saying, \"would understand exactly what you mean and give back exactly what you want.\" Google didn't want to return thousands of pages of links\u2014it wanted to return one, the one you wanted. But the perfect answer for one person isn't perfect for another. When I search for \"panthers,\" what I probably mean are the large wild cats, whereas a football fan searching for the phrase probably means the Carolina team. To provide perfect relevance, you'd need to know what each of us was interested in. You'd need to know that I'm pretty clueless about football; you'd need to know who I was.\n\nThe challenge was getting enough data to figure out what's personally relevant to each user. Understanding what someone means is tricky business\u2014and to do it well, you have to get to know a person's behavior over a sustained period of time.\n\nBut how? In 2004, Google came up with an innovative strategy. It started providing other services, services that required users to log in. Gmail, its hugely popular e-mail service, was one of the first to roll out. The press focused on the ads that ran along Gmail's sidebar, but it's unlikely that those ads were the sole motive for launching the service. By getting people to log in, Google got its hands on an enormous pile of data\u2014the hundreds of millions of e-mails Gmail users send and receive each day. And it could cross-reference each user's e-mail and behavior on the site with the links he or she clicked in the Google search engine. Google Apps\u2014a suite of online word-processing and spreadsheet-creation tools\u2014served double duty: It undercut Microsoft, Google's sworn enemy, and it provided yet another hook for people to stay logged in and continue sending click signals. All this data allowed Google to accelerate the process of building a theory of identity for each user\u2014what topics each user was interested in, what links each person clicked.\n\nBy November 2008, Google had several patents for personalization algorithms\u2014code that could figure out the groups to which an individual belongs and tailor his or her result to suit that group's preference. The categories Google had in mind were pretty narrow: to illustrate its example in the patent, Google used the example of \"all persons interested in collecting ancient shark teeth\" and \"all persons not interested in collecting ancient shark teeth.\" People in the former category who searched for, say, \"Great White incisors\" would get different results from the latter.\n\nToday, Google monitors every signal about us it can get its hands on. The power of this data can't be underestimated: If Google sees that I log on first from New York, then from San Francisco, then from New York again, it knows I'm a bicoastal traveler and can adjust its results accordingly. By looking at what browser I use, it can make some guesses about my age and even perhaps my politics.\n\nHow much time you take between the moment you enter your query and the moment you click on a result sheds light on your personality. And of course, the terms you search for reveal a tremendous amount about your interests.\n\nEven if you're not logged in, Google is personalizing your search. The neighborhood\u2014even the block\u2014that you're logging in from is available to Google, and it says a lot about who you are and what you're interested in. A query for \"Sox\" coming from Wall Street is probably shorthand for the financial legislation \"Sarbanes Oxley,\" while across the Upper Bay in Staten Island it's probably about baseball.\n\n\"People always make the assumption that we're done with search,\" said founder Page in 2009. \"That's very far from the case. We're probably only 5 percent of the way there. We want to create the ultimate search engine that can understand anything.... Some people could call that artificial intelligence.\"\n\nIn 2006, at an event called Google Press Day, CEO Eric Schmidt laid out Google's five-year plan. One day, he said, Google would be able to answer questions such as \"Which college should I go to?\" \"It will be some years before we can at least partially answer those questions. But the eventual outcome is ... that Google can answer a more hypothetical question.\"\n\n#  **Facebook Everywhere**\n\nGoogle's algorithms were unparalleled, but the challenge was to coax users into revealing their tastes and interests. In February 2004, working out of his Harvard dorm room, Mark Zuckerberg came up with an easier approach. Rather than sifting through click signals to figure out what people cared about, the plan behind his creation, Facebook, was to just flat out ask them.\n\nSince he was a college freshman, Zuckerberg had been interested in what he called the \"social graph\"\u2014the set of each person's relationships. Feed a computer that data, and it could start to do some pretty interesting and useful things\u2014telling you what your friends were up to, where they were, and what they were interested in. It also had implications for news: In its earliest incarnation as a Harvard-only site, Facebook automatically annotated people's personal pages with links to the _Crimson_ articles in which they appeared.\n\nFacebook was hardly the first social network: As Zuckerberg was hacking together his creation in the wee hours of the morning, a hairy, music-driven site named MySpace was soaring; before MySpace, Friendster had for a brief moment captured the attention of the technorati. But the Web site Zuckerberg had in mind was different. It wouldn't be a coy dating site, like Friendster. And unlike MySpace, which encouraged people to connect whether they knew each other or not, Facebook was about taking advantage of existing real-world social connections. Compared to its predecessors, Facebook was stripped down: the emphasis was on information, not flashy graphics or a cultural vibe. \"We're a utility,\" Zuckerberg said later. Facebook was less like a nightclub than a phone company, a neutral platform for communication and collaboration.\n\nEven in its first incarnation, the site grew like wildfire. After Facebook expanded to a few select Ivy League campuses, Zuckerberg's inbox was flooded with requests from students on other campuses, begging him to turn on Facebook for them. By May of 2005, the site was up and running at over eight hundred colleges. But it was the development of the News Feed the following September that pushed Facebook into another league.\n\nOn Friendster and MySpace, to find out what your friends were up to, you had to visit their pages. The News Feed algorithm pulled all of these updates out of Facebook's massive database and placed them in one place, up front, right when you logged in. Overnight, Facebook had turned itself from a network of connected Web pages into a personalized newspaper featuring (and created by) your friends. It's hard to imagine a purer source of relevance.\n\nAnd it was a gusher. In 2006, Facebook users posted literally billions of updates\u2014philosophical quotes, tidbits about who they were dating, what was for breakfast. Zuckerberg and his team egged them on: The more data users handed over to the company, the better their experience could be and the more they'd keep coming back. Early on, they'd added the ability to upload photos, and now Facebook had the largest photo collection in the world. They encouraged users to post links from other Web sites, and millions were submitted. By 2007, Zuckerberg bragged, \"We're actually producing more news in a single day for our 19 million users than any other media outlet has in its entire existence.\"\n\nAt first, the News Feed showed nearly everything your friends did on the site. But as the volume of posts and friends increased, the Feed became unreadable and unmanageable. Even if you had only a hundred friends, it was too much to read.\n\nFacebook's solution was EdgeRank, the algorithm that powers the default page on the site, the Top News Feed. EdgeRank ranks every interaction on the site. The math is complicated, but the basic idea is pretty simple, and it rests on three factors. The first is affinity: The friendlier you are with someone\u2014as determined by the amount of time you spend interacting and checking out his or her profile\u2014the more likely it is that Facebook will show you that person's updates. The second is the relative weight of that type of content: Relationship status updates, for example, are weighted very highly; everybody likes to know who's dating whom. (Many outsiders suspect that the weight, too, is personalized: Different people care about different kinds of content.) The third is time: Recently posted items are weighted over older ones.\n\nEdgeRank demonstrates the paradox at the core of the race for relevancy. To provide relevance, personalization algorithms need data. But the more data there is, the more sophisticated the filters must become to organize it. It's a never-ending cycle.\n\nBy 2009, Facebook had hit the 300 million user mark and was growing by 10 million people per month. Zuckerberg, at twenty-five, was a paper billionaire. But the company had bigger ambitions. What the News Feed had done for social information, Zuckerberg wanted to do for all information. Though he never said it, the goal was clear: Leveraging the social graph and the masses of information Facebook's users had provided, Zuckerberg wanted to put Facebook's news algorithm engine at the center of the web.\n\nEven so, it was a surprise when, on April 21, 2010, readers loaded the _Washington Post_ homepage and discovered that their friends were on it. In a prominent box in the upper right corner\u2014the place where any editor will tell you the eye lands first\u2014was a feature titled Network News. Each person who visited saw a different set of links in the box\u2014the _Washington Post_ links their friends had shared on Facebook. The _Post_ was letting Facebook edit its most valuable online asset: its front page. The _New York Times_ soon followed suit.\n\nThe new feature was one piece of a much bigger rollout, which Facebook called \"Facebook Everywhere\" and announced at its annual conference, f8 (\"fate\"). Ever since Steve Jobs sold the Apple by calling it \"insanely great,\" a measure of grandiosity has been part of the Silicon Valley tradition. But when Zuckerberg walked onto the stage on April 21, 2010, his words seemed plausible. \"This is the most transformative thing we've ever done for the web,\" he announced.\n\nThe aim of Facebook Everywhere was simple: make the whole Web \"social\" and bring Facebook-style personalization to millions of sites that currently lack it. Want to know what music your Facebook friends are listening to? Pandora would now tell you. Want to know what restaurants your friends like? Yelp now had the answer. News sites from the _Huffington Post_ to the _Washington Post_ were now personalized.\n\nFacebook made it possible to press the Like button on any item on the Web. In the first twenty-four hours of the new service, there were 1 billion Likes\u2014and all of that data flowed back into Facebook's servers. Bret Taylor, Facebook's platform lead, announced that users were sharing 25 billion items a month. Google, once the undisputed leader in the push for relevance, seemed worried about the rival a few miles down the road.\n\nThe two giants are now in hand-to-hand combat: Facebook poaches key executives from Google; Google's hard at work constructing social software like Facebook. But it's not totally obvious why the two new-media monoliths should be at war. Google, after all, is built around answering questions; Facebook's core mission is to help people connect with their friends.\n\nBut both businesses' bottom lines depend on the same thing: targeted, highly relevant advertising. The contextual advertisements Google places next to search results and on Web pages are its only significant source of profits. And while Facebook's finances are private, insiders have made clear that advertising is at the core of the company's revenue model. Google and Facebook have different starting points and different strategies\u2014one starts with the relationships among pieces of information, while the other starts with the relationships among people\u2014but ultimately, they're competing for the same advertising dollars.\n\nFrom the point of view of the online advertiser, the question is simple. Which company can deliver the most return on a dollar spent? And this is where relevance comes back into the equation. The masses of data Facebook and Google accumulate have two uses. For users, the data provides a key to providing personally relevant news and results. For advertisers, the data is the key to finding likely buyers. The company that has the most data and can put it to the best use gets the advertising dollars.\n\nWhich brings us to lock-in. Lock-in is the point at which users are so invested in their technology that even if competitors might offer better services, it's not worth making the switch. If you're a Facebook member, think about what it'd take to get you to switch to another social networking site\u2014even if the site had vastly greater features. It'd probably take a lot\u2014re-creating your whole profile, uploading all of those pictures, and laboriously entering your friends' names would be extremely tedious. You're pretty locked in. Likewise, Gmail, Gchat, Google Voice, Google Docs, and a host of other products are part of an orchestrated campaign for Google lock-in. The fight between Google and Facebook hinges on which can achieve lock-in for the most users.\n\nThe dynamics of lock-in are described by Metcalfe's law, a principle coined by Bob Metcalfe, the inventor of the Ethernet protocol that wires together computers. The law says that the usefulness of a network increases at an accelerating rate as you add each new person to it. It's not much use to be the only person you know with a fax machine, but if everyone you work with uses one, it's a huge disadvantage not to be in the loop. Lock-in is the dark side of Metcalfe's law: Facebook is useful in large part because everyone's on it. It'd take a lot of mismanagement to overcome that basic fact.\n\nThe more locked in users are, the easier it is to convince them to log in\u2014and when you're constantly logged in, these companies can keep tracking data on you even when you're not visiting their Web sites. If you're logged into Gmail and you visit a Web site that uses Google's Doubleclick ad service, that fact can be attached to your Google account. And with tracking cookies these services place on your computer, Facebook or Google can provide ads based on your personal information on third-party sites. The whole Web can become a platform for Google or Facebook.\n\nBut Google and Facebook are hardly the only options. The daily turf warfare between Google and Facebook occupies scores of business reporters and gigabytes of blog chatter, but there's a stealthy third front opening up in this war. And though most of the companies involved operate under the radar, they may ultimately represent the future of personalization.\n\n#  **The Data Market**\n\nThe manhunt for accomplices of the 9\/11 killers was one of the most extensive in history. In the immediate aftermath of the attacks, the scope of the plot was unclear. Were there more hijackers who hadn't yet been found? How extensive was the network that had pulled off the attacks? For three days, the CIA, FBI, and a host of other acronymed agencies worked around the clock to identify who else was involved. The country's planes were grounded, its airports closed.\n\nWhen help arrived, it came from an unlikely place. On September 14, the bureau had released the names of the hijackers, and it was now asking\u2014pleading\u2014for anyone with information about the perpetrators to come forward. Later that day, the FBI received a call from Mack McLarty, a former White House official who sat on the board of a little-known but hugely profitable company called Acxiom.\n\nAs soon as the hijackers' names had been publicly released, Acxiom had searched its massive data banks, which take up five acres in tiny Conway, Arkansas. And it had found some very interesting data on the perpetrators of the attacks. In fact, it turned out, Acxiom knew more about eleven of the nineteen hijackers than the entire U.S. government did\u2014including their past and current addresses and the names of their housemates.\n\nWe may never know what was in the files Acxiom gave the government (though one of the executives told a reporter that Acxiom's information had led to deportations and indictments). But here's what Acxiom knows about 96 percent of American households and half a billion people worldwide: the names of their family members, their current and past addresses, how often they pay their credit card bills whether they own a dog or a cat (and what breed it is), whether they are righthanded or left-handed, what kinds of medication they use (based on pharmacy records) ... the list of data points is about 1,500 items long.\n\nAcxiom keeps a low profile\u2014it may not be an accident that its name is nearly unpronounceable. But it serves most of the largest companies in America\u2014nine of the ten major credit card companies and consumer brands from Microsoft to Blockbuster. \"Think of [Acxiom] as an automated factory,\" one engineer told a reporter, \"where the product we make is data.\"\n\nTo get a sense of Acxiom's vision for the future, consider a travel search site like Travelocity or Kayak. Ever wondered how they make money? Kayak makes money in two ways. One is pretty simple, a holdover from the era of travel agents: When you buy a flight using a link from Kayak, airlines pay the site a small fee for the referral.\n\nThe other is much less obvious. When you search for the flight, Kayak places a cookie on your computer\u2014a small file that's basically like putting a sticky note on your forehead saying \"Tell me about cheap bicoastal fares.\" Kayak can then sell that piece of data to a company like Acxiom or its rival BlueKai, which auctions it off to the company with the highest bid\u2014in this case, probably a major airline like United. Once it knows what kind of trip you're interested in, United can show you ads for relevant flights\u2014not just on Kayak's site, but on literally almost any Web site you visit across the Internet. This whole process\u2014from the collection of your data to the sale to United\u2014takes under a second.\n\nThe champions of this practice call it \"behavioral retargeting.\" Retailers noticed that 98 percent of visitors to online shopping sites leave without buying anything. Retargeting means businesses no longer have to take \"no\" for an answer.\n\nSay you check out a pair of running sneakers online but leave the site without springing for them. If the shoe site you were looking at uses retargeting, their ads\u2014maybe displaying a picture of the exact sneaker you were just considering\u2014will follow you around the Internet, showing up next to the scores from last night's game or posts on your favorite blog. And if you finally break down and buy the sneakers? Well, the shoe site can sell that piece of information to BlueKai to auction it off to, say, an athletic apparel site. Pretty soon you'll be seeing ads all over the Internet for sweat-wicking socks.\n\nThis kind of persistent, personalized advertising isn't just confined to your computer. Sites like Loopt and Foursquare, which broadcast a user's location from her mobile phone, provide advertisers with opportunities to reach consumers with targeted ads even when they're out and about. Loopt is working on an ad system whereby stores can offer special discounts and promotions to repeat customers on their phones\u2014right as they walk through the door. And if you sit down on a Southwest Airlines flight, the ads on your seat-back TV screen may be different from your neighbors'. Southwest, after all, knows your name and who you are. And by cross-indexing that personal information with a database like Acxiom's, it can know a whole lot more about you. Why not show you your own ads\u2014or, for that matter, a targeted show that makes you more likely to watch them?\n\nTargusInfo, another of the new firms that processes this sort of information, brags that it \"delivers more than 62 billion real-time attributes a year.\" That's 62 billion points of data about who customers are, what they're doing, and what they want. Another ominously named enterprise, the Rubicon Project, claims that its database includes more than half a billion Internet users.\n\nFor now, retargeting is being used by advertisers, but there's no reason to expect that publishers and content providers won't get in on it. After all, if the _Los Angeles Times_ knows that you're a fan of Perez Hilton, it can front-page its interview with him in your edition, which means you'll be more likely to stay on the site and click around.\n\nWhat all of this means is that your behavior is now a commodity, a tiny piece of a market that provides a platform for the personalization of the whole Internet. We're used to thinking of the Web as a series of one-to-one relationships: You manage your relationship with Yahoo separately from your relationship with your favorite blog. But behind the scenes, the Web is becoming increasingly integrated. Businesses are realizing that it's profitable to share data. Thanks to Acxiom and the data market, sites can put the most relevant products up front and whisper to each other behind your back.\n\nThe push for relevance gave rise to today's Internet giants, and it is motivating businesses to accumulate ever more data about us and to invisibly tailor our online experiences on that basis. It's changing the fabric of the Web. But as we'll see, the consequences of personalization for how we consume news, make political decisions, and even how we think will be even more dramatic.\n**2**\n\n**The User Is the Content**\n\nEverything which bars freedom and fullness of communication sets up barriers that divide human beings into sets and cliques, into antagonistic sects and factions, and thereby undermines the democratic way of life.\n\n\u2014 _John Dewey_\n\nThe technology will be so good, it will be very hard for people to watch or consume something that has not in some sense been tailored for them.\n\n\u2014 _Eric Schmidt,_ Google CEO\n\n**M** icrosoft Building 1 in Mountain View, California, is a long, low, gunmetal gray hangar, and if it weren't for the cars buzzing by behind it on Highway 101, you'd almost be able to hear the whine of ultrasonic security. On this Saturday in 2010, the vast expanses of parking lot were empty except for a few dozen BMWs and Volvos. A cluster of scrubby pine trees bent in the gusty wind.\n\nInside, the concrete-floored hallways were crawling with CEOs in jeans and blazers trading business cards over coffee and swapping stories about deals. Most hadn't come far; the startups they represented were based nearby. Hovering over the cheese spread was a group of executives from data firms like Acxiom and Experian who had flown in from Arkansas and New York the night before. With fewer than a hundred people in attendance, the Social Graph Symposium nonetheless included the leaders and luminaries of the targeted-marketing field.\n\nA bell rang, the group filed into breakout rooms, and one of the conversations quickly turned to the battle to \"monetize content.\" The picture, the group agreed, didn't look good for newspapers.\n\nThe contours of the situation were clear to anyone paying attention: The Internet had delivered a number of mortal blows to the newspaper business model, any one of which might be fatal. Craigslist had made classified advertisements free, and $18 billion in revenue went _poof._ Nor was online advertising picking up the slack. An advertising pioneer once famously said, \"Half the money I spend on advertising is wasted\u2014I just don't know which half.\" But the Internet turned that logic on its head\u2014with click-through rates and other metrics, businesses suddenly knew exactly which half of their money went to waste. And when ads didn't work as well as the industry had promised, advertising budgets were cut accordingly. Meanwhile, bloggers and freelance journalists started to package and produce news content for free, which pressured the papers to do the same online.\n\nBut what most interested the crowd in the room was the fact that the entire premise on which the news business had been built was changing, and the publishers weren't even paying attention.\n\nThe _New York Times_ had traditionally been able to command high ad rates because advertisers knew it attracted a premium audience\u2014the wealthy opinion-making elite of New York and beyond. In fact, the publisher had a near monopoly on reaching that group\u2014there were only a few other outlets that provided a direct feed into their homes (and out of their pocketbooks).\n\nNow all that was changing. One executive in the marketing session was especially blunt. \"The publishers are losing,\" he said, \"and they will lose, because they just don't get it.\"\n\nInstead of taking out expensive advertisements in the _New York Times,_ it was now possible to track that elite cosmopolitan readership using data acquired from Acxiom or BlueKai. This was, to say the least, a game changer in the business of news. Advertisers no longer needed to pay the _New York Times_ to reach _Times_ readers: they could target them wherever they went online. The era where you had to develop premium content to get premium audiences, in other words, was coming to a close.\n\nThe numbers said it all. In 2003, publishers of articles and videos online received most of each dollar advertisers spent on their sites. Now, in 2010, they only received $.20. The difference was moving to the people who had the data\u2014many of whom were in attendance at Mountain View. A PowerPoint presentation circulating in the industry called out the significance of this change succinctly, describing how \"premium publishers [were] losing a key advantage\" because advertisers can now target premium audiences in \"other, cheaper places.\" The take-home message was clear: Users, not sites, were now the focus.\n\nUnless newspapers could think of themselves as behavioral data companies with a mission of churning out information about their readers' preferences\u2014unless, in other words, they could adapt themselves to the personalized, filter-bubble world\u2014they were sunk.\n\n**NEWS SHAPES OUR** sense of the world, of what's important, of the scale and color and character of our problems. More important, it provides the foundation of shared experience and shared knowledge on which democracy is built. Unless we understand the big problems our societies face, we can't act together to fix them. Walter Lippmann, the father of modern journalism, put it more eloquently: \"All that the sharpest critics of democracy have alleged is true, if there is no steady supply of trustworthy and relevant news. Incompetence and aimlessness, corruption and disloyalty, panic and ultimate disaster must come to any people which is denied an assured access to the facts.\"\n\nIf news matters, newspapers matter, because their journalists write most of it. Although the majority of Americans get their news from local and national TV broadcasts, most of the actual reporting and story generation happens in newspaper newsrooms. They're the core creators of the news economy. Even in 2010, blogs remain incredibly reliant on them: according to Pew Research Center's Project for Excellence in Journalism, 99 percent of the stories linked to in blog posts come from newspapers and broadcast networks, and the _New York Times_ and _Washington Post_ alone account for nearly 50 percent of all blog links. While rising in importance and influence, net-native media still mostly lack the capacity to shape public life that these papers and a few other outlets like the BBC and CNN have.\n\nBut the shift is coming. The forces unleashed by the Internet are driving a radical transformation in who produces news and how they do it. Whereas once you had to buy the whole paper to get the sports section, now you can go to a sports-only Web site with enough new content each day to fill ten papers. Whereas once only those who could buy ink by the barrel could reach an audience of millions, now anyone with a laptop and a fresh idea can.\n\nIf we look carefully, we can begin to project the outline of the new constellation that's emerging. This much we know:\n\n\u2022 The cost of producing and distributing media of all kinds\u2014words, images, video, and audio streams\u2014will continue to fall closer and closer to zero.\n\n\u2022 As a result, we'll be deluged with choices of what to pay attention to\u2014and we'll continue to suffer from \"attention crash.\" This makes curators all the more important. We'll rely ever more heavily on human and software curators to determine what news we should consume.\n\n\u2022 Professional human editors are expensive, and code is cheap. Increasingly, we'll rely on a mix of nonprofessional editors (our friends and colleagues) and software code to figure out what to watch, read, and see. This code will draw heavily on the power of personalization and displace professional human editors.\n\nMany Internet watchers (myself included) cheered the development of \"people-powered news\"\u2014a more democratic, participatory form of cultural storytelling. But the future may be more machine-powered than people-powered. And many of the breakthrough champions of the people-powered viewpoint tell us more about our current, transitional reality than the news of the future. The story of \"Rathergate\" is a classic example of the problem.\n\nWhen CBS News announced nine weeks before the 2004 election that it had papers proving that President Bush had manipulated his military record, the assertion seemed as though it might be the turning point for the Kerry campaign, which had been running behind in the polls. The viewership for _60 Minutes Wednesday_ was high. \"Tonight, we have new documents and new information on the President's military service and the first-ever interview with the man who says he pulled the strings to get young George W. Bush into the Texas Air National Guard,\" Dan Rather said somberly as he laid out the facts.\n\nThat night, as the _New York Times_ was preparing its headline on the story, a lawyer and conservative activist named Harry MacDougald posted to a right-wing forum called Freerepublic .com. After looking closely at the typeface of the documents, MacDougald was convinced that there was something fishy going on. He didn't beat around the bush: \"I am saying these documents are forgeries, run through a copier for 15 generations to make them look old,\" he wrote. \"This should be pursued aggressively.\"\n\nMacDougald's post quickly attracted attention, and the discussion about the forgeries jumped to two other blog communities, _Powerline_ and _Little Green Footballs_ , where readers quickly discovered other anachronistic quirks. By the next afternoon, the influential _Drudge Report_ had the campaign reporters talking about the validity of the documents. And the following day, September 10, the Associated Press, _New York Times, Washington Post,_ and other outlets all carried the story: CBS's scoop might not be true. By September 20, the president of CBS News had issued a statement on the documents: \"Based on what we now know, CBS News cannot prove that the documents are authentic.... We should not have used them.\" While the full truth of Bush's military record never came to light, Rather, one of the most prominent journalists in the world, retired in disgrace the next year.\n\nRathergate is now an enduring part of the mythology about the way blogs and the Internet have changed the game of journalism. No matter where you stand on the politics involved, it's an inspiring tale: MacDougald, an activist on a home computer, discovered the truth, took down one of the biggest figures in journalism, and changed the course of an election.\n\nBut this version of the story omits a critical point.\n\nIn the twelve days between CBS's airing of the story and its public acknowledgment that the documents were probably fakes, the rest of the broadcast news media turned out reams of reportage. The Associated Press and _USA Today_ hired professional document reviewers who scrutinized every dot and character. Cable news networks issued breathless updates. A striking 65 percent of Americans\u2014and nearly 100 percent of the political and reportorial classes\u2014were paying attention to the story.\n\nIt is only because these news sources reached many of the same people who watch CBS News that CBS could not afford to ignore the story. MacDougald and his allies may have lit the match, but it took print and broadcast media to fan the flames into a career-burning conflagration.\n\nRathergate, in other words, is a good story about how online and broadcast media can interact. But it tells us little or nothing about how news will move once the broadcast era is fully over\u2014and we're moving toward that moment at a breakneck pace. The question we have to ask is, What does news look like in the postbroadcast world? How does it move? And what impact does it have?\n\nIf the power to shape news rests in the hands of bits of code, not professional human editors, is the code up to the task? If the news environment becomes so fragmented that MacDougald's discovery can't reach a broad audience, could Rathergate even happen at all?\n\nBefore we can answer that question, it's worth quickly reviewing where our current news system came from.\n\n#  **The Rise and Fall of the General Audience**\n\nLippmann, in 1920, wrote that \"the crisis in western democracy is a crisis in journalism.\" The two are inextricably linked, and to understand the future of this relationship, we have to understand its past.\n\nIt's hard to imagine that there was a time when \"public opinion\" didn't exist. But as late as the mid-1700s, politics was palace politics. Newspapers confined themselves to commercial and foreign news\u2014a report from a frigate in Brussels and a letter from a nobleman in Vienna set in type and sold to the commercial classes of London. Only when the modern, complex, centralized state emerged\u2014with private individuals rich enough to lend money to the king\u2014did forward-looking officials realize that the views of the people outside the walls had begun to matter.\n\nThe rise of the public realm\u2014and news as its medium\u2014was partly driven by the emergence of new, complex societal problems, from the transport of water to the challenges of empire, that transcended the narrow bounds of individual experience. But technological changes also made an impact. After all, how news is conveyed profoundly shapes what is conveyed.\n\nWhile the spoken word is always directed to a specific audience, the written word\u2014and especially the printing press\u2014changed all that. In a real sense, it made the general audience possible. This ability to address a broad, anonymous group fueled the Enlightenment era, and thanks to the printing press, scientists and scholars could spread complex ideas with perfect precision to an audience spread over large distances. And because everyone was literally on the same page, transnational conversations began that would have been impossibly laborious in the earlier scribe-driven epoch.\n\nIn the American colonies, the printing industry developed at a fierce clip\u2014at the time of the revolution, there was no other place in the world with such a density and variety of newspapers. And while they catered exclusively to the interests of white male landowners, the newspapers nonetheless provided a common language and common arguments for dissent. Thomas Paine's rallying cry, _Common Sense_ , helped give the diverse colonies a sense of mutual interest and solidarity.\n\nEarly newspapers existed to provide business owners with information about market prices and conditions, and newspapers depended on subscription and advertising revenues to survive. It wasn't until the 1830s and the rise of the \"penny press\"\u2014cheap newspapers sold as one-offs on the street\u2014that everyday citizens in the United States became a primary constituency for news. It was at this point that newspapers came to carry what we think of as news today.\n\nThe small, aristocratic public was transforming into a general public. The middle class was growing, and because middle-class people had both a day-to-day stake in the life of the nation and the time and money to spend on entertainment, they were hungry for news and spectacle. Circulation skyrocketed. And as education levels went up, more people came to understand the interconnected nature of modern society. If what happened in Russia could affect prices in New York, it was worth following the news from Russia.\n\nBut though democracy and the newspaper were becoming ever more intertwined, the relationship wasn't an easy one. After World War I, tensions about what role the newspaper should play boiled over, becoming a matter of great debate among two of the leading intellectual lights of the time, Walter Lippmann and John Dewey.\n\nLippmann had watched with disgust as newspapers had effectively joined the propaganda effort for World War I. In _Liberty and the News,_ a book of essays published in 1921, he angrily assailed the industry. He quoted an editor who had written that in the service of the war, \"governments conscripted public opinion.... They goose-stepped it. They taught it to stand at attention and salute.\"\n\nLippmann wrote that so long as newspapers existed and they determined \"by entirely private and unexamined standards, no matter how lofty, what [the average citizen] shall know, and hence what he shall believe, no one will be able to say that the substance of democratic government is secure.\"\n\nOver the next decade, Lippmann advanced his line of thought. Public opinion, Lippmann concluded, was too malleable\u2014people were easily manipulated and led by false information. In 1925, he wrote _The Phantom Public,_ an attempt to dismantle the illusion of a rational, informed populace once and for all. Lippmann argued against the prevailing democratic mythology, in which informed citizens capably made decisions about the major issues of the day. The \"omnicompetent citizens\" that such a system required were nowhere to be found. At best, ordinary citizens could be trusted to vote out the party that was in power if it was doing too poorly; the real work of governance, Lippmann argued, should be entrusted to insider experts who had education and expertise to see what was really going on.\n\nJohn Dewey, one of the great philosophers of democracy, couldn't pass up the opportunity to engage. In The Public and Its Problems, a series of lectures Dewey gave in response to Lippmann's book, he admitted that many of Lippmann's critiques were not wrong. The media were able to easily manipulate what people thought. Citizens were hardly informed enough to properly govern.\n\nHowever, Dewey argued, to accept Lippmann's proposal was to give up on the promise of democracy\u2014an ideal that had not yet fully been realized but might still be. \"To learn to be human,\" Dewey argued, \"is to develop through the give and take of communication an effective sense of being an individually distinctive member of a community.\" The institutions of the 1920s, Dewey said, were closed off\u2014they didn't invite democratic participation. But journalists and newspapers could play a critical role in this process by calling out the citizen in people\u2014reminding them of their stake in the nation's business.\n\nWhile they disagreed on the contours of the solution, Dewey and Lippmann did fundamentally agree that news making was a fundamentally political and ethical enterprise\u2014and that publishers had to handle their immense responsibility with great care. And because the newspapers of the time were making money hand over fist, they could afford to listen. At Lippmann's urging, the more credible papers built a wall between the business portion of their papers and the reporting side. They began to champion objectivity and decry tilted reporting. It's this ethical model\u2014one in which newspapers have a responsibility to both neutrally inform and convene the public\u2014which guided the aspirations of journalistic endeavors for the last half century.\n\nOf course, news agencies have frequently fallen short of these lofty goals\u2014and it's not always clear how hard they even try. Spectacle and profit seeking frequently win out over good journalistic practice; media empires make reporting decisions to placate advertisers; and not every outlet that proclaims itself \"fair and balanced\" actually is.\n\nThanks to critics like Lippmann, the present system has a sense of ethics and public responsibility baked in, however imperfectly. But though it's playing some of the same roles, the filter bubble does not.\n\n#  **A New Middleman**\n\n_New York Times_ critic Jon Pareles calls the 2000s the disintermediation decade. _Disintermediation_ \u2014the elimination of middlemen\u2014is \"the thing that the Internet does to every business, art, and profession that aggregates and repackages,\" wrote protoblogger Dave Winer in 2005. \"The great virtue of the Internet is that it erodes power,\" says the Internet pioneer Esther Dyson. \"It sucks power out of the center, and takes it to the periphery, it erodes the power of institutions over people while giving to individuals the power to run their own lives.\"\n\nThe disintermediation story was repeated hundreds of times, on blogs, in academic papers, and on talk shows. In one familiar version, it goes like this: Once upon a time, newspaper editors woke up in the morning, went to work, and decided what we should think. They could do this because printing presses were expensive, but it became their explicit ethos: As newspapermen, it was their paternalistic duty to feed the citizenry a healthy diet of coverage.\n\nMany of them meant well. But living in New York and Washington, D.C., they were enthralled by the trappings of power. They counted success by the number of insider cocktail parties they were invited to, and the coverage followed suit. The editors and journalists became embedded in the culture they were supposed to cover. And as a result, powerful people got off the hook, and the interests of the media tilted against the interests of everyday folk, who were at their mercy.\n\nThen the Internet came along and disintermediated the news. All of a sudden, you didn't have to rely on the _Washington Post_ 's interpretation of the White House press briefing\u2014you could look up the transcript yourself. The middleman dropped out\u2014not just in news, but in music (no more need for _Rolling Stone_ \u2014you could now hear directly from your favorite band) and commerce (you could follow the Twitter feed of the shop down the street) and nearly everything else. The future, the story says, is one in which we go direct.\n\nIt's a story about efficiency and democracy. Eliminating the evil middleman sitting between us and what we want sounds good. In a way, disintermediation is taking on the idea of media itself. The word _media,_ after all, comes from the Latin for \"middle layer.\" It sits between us and the world; the core bargain is that it will connect us to what's happening but at the price of direct experience. Disintermediation suggests we can have both.\n\nThere's some truth to the description, of course. But while enthrallment to the gatekeepers is a real problem, disintermediation is as much mythology as fact. Its effect is to make the new mediators\u2014the new gatekeepers\u2014invisible. \"It's about the many wresting power from the few,\" _Time_ magazine announced when it made \"you\" the person of the year. But as law professor and _Master Switch_ author Tim Wu says, \"The rise of networking did not eliminate intermediaries, but rather changed who they are.\" And while power moved toward consumers, in the sense that we have exponentially more choice about what media we consume, the power still isn't held by consumers.\n\nMost people who are renting and leasing apartments don't \"go direct\"\u2014they use the intermediary of craigslist. Readers use Amazon.com. Searchers use Google. Friends use Facebook. And these platforms hold an immense amount of power\u2014as much, in many ways, as the newspaper editors and record labels and other intermediaries that preceded them. But while we've raked the editors of the _New York Times_ and the producers of CNN over the coals for the stories they've missed and the interests they've served, we've given very little scrutiny to the interests behind the new curators.\n\nIn July 2010, Google News rolled out a personalized version of its popular service. Sensitive to concerns about shared experience, Google made sure to highlight the \"top stories\" that are of broad, general interest. But look below that top band, and you will see only stories that are locally and personally relevant to you, based on the interests that you've demonstrated through Google and what articles you've clicked on in the past. Google's CEO doesn't beat around the bush when he describes where this is all headed: \"Most people will have personalized news-reading experiences on mobile-type devices that will largely replace their traditional reading of newspapers,\" he tells an interviewer. \"And that that kind of news consumption will be very personal, very targeted. It will remember what you know. It will suggest things that you might want to know. It will have advertising. Right? And it will be as convenient and fun as reading a traditional newspaper or magazine.\"\n\nSince Krishna Bharat created the first prototype of Google News to monitor worldwide coverage after 9\/11, Google News has become one of the top global portals for news. Tens of millions of visitors pull up the site each month\u2014more than visit the BBC. Speaking at the IJ-7 Innovation Journalism conference at Stanford\u2014to a room full of fairly anxious newspaper professionals\u2014Bharat laid out his vision: \"Journalists,\" Bharat explained, \"should worry about creating the content and other people in technology should worry about bringing the content to the right group\u2014given the article, what's the best set of eyeballs for it, and that can be solved by personalization.\"\n\nIn many ways, Google News is still a hybrid model, driven in part by the judgment of a professional editorial class. When a Finnish editor asked Bharat what determines the priority of stories, he emphasized that newspaper editors themselves still have disproportionate control: \"We pay attention,\" he said, \"to the editorial decisions that different editors have made: what your paper chose to cover, when you published it, and where you placed it on your front page.\" _New York Times_ editor Bill Keller, in other words, still has a disproportionate ability to affect a story's prominence on Google News.\n\nIt's a tricky balance: On the one hand, Bharat tells an interviewer, Google should promote what the reader enjoys reading. But at the same time, overpersonalization that, for example, excludes important news from the picture would be a disaster. Bharat doesn't seem to have fully resolved the dilemma, even for himself. \"I think people care about what other people care about, what other people are interested in\u2014most important, their social circle,\" he says.\n\nBharat's vision is to move Google News off Google's site and onto the sites of other content producers. \"Once we get personalization working for news,\" Bharat tells the conference, \"we can take that technology and make it available to publishers, so they can [transform] their website appropriately\" to suit the interests of each visitor.\n\nKrishna Bharat is in the hot seat for a good reason. While he's respectful to the front page editors who pepper him with questions, and his algorithm depends on their expertise, Google News, if it's successful, may ultimately put a lot of front-page editors out of work. Why visit your local paper's Web site, after all, if Google's personalized site has already pulled the best pieces?\n\nThe Internet's impact on news was explosive in more ways than one. It expanded the news space by force, sweeping older enterprises out of its path. It dismantled the trust that news organizations had built. In its wake lies a more fragmented and shattered public space than the one that came before.\n\nIt's no secret that trust in journalists and news providers has plummeted in recent years. But the shape of the curve is mysterious : According to a Pew poll, Americans lost more faith in news agencies between 2007 and 2010 than they did in the prior twelve years. Even the debacle over Iraq's WMDs didn't make much of a dent in the numbers\u2014but whatever happened in 2007 did.\n\nWhile we still don't have conclusive proof, it appears that this, too, is an effect of the Internet. When you're getting news from one source, the source doesn't draw your attention much to its own errors and omissions. Corrections, after all, are buried in tiny type on an inside page. But as masses of news readers went online and began to hear from multiple sources, the differences in coverage were drawn out and amplified. You don't hear about the _New York Times_ 's problems much from the _New York Times_ \u2014but you do hear about them from political blogs, like the _Daily Kos_ or _Little Green Footballs,_ and from groups on both sides of the spectrum, like MoveOn or RightMarch. More voices, in other words, means less trust in any given voice.\n\nAs Internet thinker Clay Shirky has pointed out, the new, low trust levels may not be inappropriate. It may be that the broadcast era kept trust artificially high. But as a consequence, for most of us now, the difference in authority between a blog post and an article in the _New Yorker_ is much smaller than one would think.\n\nEditors at Yahoo News, the biggest news site on the Internet, can see this trend in action. With over 85 million daily visitors, when Yahoo links to articles on other servers\u2014even those of nationally known papers\u2014it has to give technicians advance warning so that they can handle the load. A single link can generate up to 12 million views. But according to an executive in the news department, it doesn't matter much to Yahoo's users where the news is coming from. A spicy headline will win over a more trusted news source any day. \"People don't make much of a distinction between the _New York Times_ and some random blogger,\" the executive told me.\n\nThis is Internet news: Each article ascends the most forwarded lists or dies an ignominious death on its own. In the old days, _Rolling Stone_ readers would get the magazine in the mail and leaf through it; now, the popular stories circulate online independent of the magazine. I read the expos\u00e9 on General Stanley McChrystal but had no idea that the cover story was about Lady Gaga. The attention economy is ripping the binding, and the pages that get read are the pages that are frequently the most topical, scandalous, and viral.\n\nNor is debundling just about print media. While the journalistic hand-wringing has focused mostly on the fate of the newspaper, TV channels face the same dilemma. From Google to Microsoft to Comcast, executives are quite clear that what they call convergence is coming soon. Close to a million Americans are unplugging from cable TV offerings and getting their video online every year\u2014and those numbers will accelerate as more services like Netflix's movie-on-demand and Hulu go online. When TV goes fully digital, channels become little more than brands\u2014and the order of programs, like the order of articles, is determined by the user's interest and attention, not the station manager.\n\nAnd of course, that opens the door for personalization. \"Internet connected TV is going to be a reality. It will dramatically change the ad industry forever. Ads will become interactive and delivered to individual TV sets according to the user,\" Google VP for global media Henrique de Castro has said. We may say good-bye, in other words, to the yearly ritual of the Super Bowl commercial, which won't create the same buzz when everyone is watching different ads.\n\nIf trust in news agencies is falling, it is rising in the new realm of amateur and algorithmic curation. If the newspaper and magazine are being torn apart on one end, the pages are being recompiled on the other\u2014a different way every time. Facebook is an increasingly vital source of news for this reason: Our friends and family are more likely to know what's important and relevant to us than some newspaper editor in Manhattan.\n\nPersonalization proponents often point to social media like Facebook to dispute the notion that we'll end up in a narrow, overfiltered world. Friend your softball buddy on Facebook, the argument goes, and you'll have to listen to his political rants even if you disagree.\n\nSince they have trust, it's true that the people we know can bring some focus to topics outside our immediate purview. But there are two problems with relying on a network of amateur curators. First, by definition, the average person's Facebook friends will be much more like that person than a general interest news source. This is especially true because our physical communities are becoming more homogeneous as well\u2014and we generally know people who live near us. Because your softball buddy lives near you, he's likely to share many of your views. It's ever less likely that we'll come to be close with people very different from us, online or off\u2014and thus it's less likely we'll come into contact with different points of view.\n\nSecond, personalization filters will get better and better at overlaying themselves on individuals' recommendations. Like your friend Sam's posts on football but not his erratic musings on _CSI_? A filter watching and learning which pieces of content you interact with can start to sift one from another\u2014and undermine even the limited leadership that a group of friends and pundits can offer. Google Reader, another product from Google that helps people manage streams of posts from blogs, now has a feature called Sort by Magic, which does precisely this.\n\nThis leads to the final way in which the future of media is likely to be different than we expected. Since its early days, Internet evangelists have argued that it was an inherently active medium. \"We think basically you watch television to turn your brain off, and you work on your computer when you want to turn your brain on,\" Apple founder Steve Jobs told _Macworld_ in 2004.\n\nAmong techies, these two paradigms came to be known as push technology and pull technology. A Web browser is an example of pull technology: You put in an address, and your computer pulls information from that server. Television and the mail, on the other hand, are push technologies: The information shows up on the tube or at your doorstop without any action on your end. Internet enthusiasts were excited about the shift from push to pull for reasons that are now pretty obvious: Rather than wash the masses in waves of watered-down, lowest-common-denominator content, pull media put users in control.\n\nThe problem is that pull is actually a lot of work. It requires you to be constantly on your feet, curating your own media experience. That's way more energy than TV requires during the whopping thirty-six hours a week that Americans watch today.\n\nIn TV network circles, there's a name for the passive way with which Americans make most of those viewing decisions: the theory of least objectionable programming. Researching TV viewers' behavior in the 1970s, pay-per-view innovator Paul Klein noticed that people quit channel surfing far more quickly than one might suspect. During most of those thirty-six hours a week, the theory suggests, we're not looking for a program in particular. We're just looking to be unobjectionably entertained.\n\nThis is part of the reason TV advertising has been such a bonanza for the channel's owners. Because people watch TV passively, they're more likely to keep watching when ads come on. When it comes to persuasion, passive is powerful.\n\nWhile the broadcast TV era may be coming to a close, the era of least objectionable programming probably isn't\u2014and personalization stands to make the experience even more, well, unobjectionable. One of YouTube's top corporate priorities is the development of a product called LeanBack, which strings together videos in a row to provide the benefits of push and pull. It's less like surfing the Web and more like watching TV\u2014a personalized experience that lets the user do less and less. Like the music service Pandora, LeanBack viewers can easily skip videos and give the viewer feedback for picking the next videos\u2014thumbs up for this one, thumbs down for these three. LeanBack would learn. Over time, the vision is for LeanBack to be like your own personal TV channel, stringing together content you're interested in while requiring less and less engagement from you.\n\nSteve Jobs's proclamation that computers are for turning your brain on may have been a bit too optimistic. In reality, as personalized filtering gets better and better, the amount of energy we'll have to devote to choosing what we'd like to see will continue to decrease.\n\nAnd while personalization is changing our experience of news, it's also changing the economics that determine what stories get produced.\n\n#  **The Big Board**\n\nThe offices of Gawker Media, the ascendant blog empire based in SoHo, look little like the newsroom of the _New York Times_ a few miles to the north. But the driving difference between the two is the flat-screen TV that hovers over the room.\n\nThis is the Big Board, and on it are a list of articles and numbers. The numbers represent the number of times each article has been read, and they're big: Gawker's Web sites routinely see hundreds of millions of page views a month. The Big Board captures the top posts across the company's Web sites, which focus on everything from media (Gawker) to gadgets (Gizmodo) to porn (Fleshbot). Write an article that makes it onto the Big Board, and you're liable to get a raise. Stay off it for too long, and you may need to find a different job.\n\nAt the _New York Times,_ reporters and bloggers aren't allowed to see how many people click on their stories. This isn't just a rule, it's a philosophy that the _Times_ lives by: The point of being the newspaper of record is to provide readers with the benefit of excellent, considered editorial judgment. \"We don't let metrics dictate our assignments and play,\" _New York Times_ editor Bill Keller said, \"because we believe readers come to us for our judgment, not the judgment of the crowd. We're not 'American Idol.' \" Readers can vote with their feet by subscribing to another paper if they like, but the _Times_ doesn't pander. Younger _Times_ writers who are concerned about such things have to essentially bribe the paper's system administrators to give them a peek at their stats. (The paper does use aggregate statistics to determine which online features to expand or cut.)\n\nIf the Internet's current structures mostly tend toward fragmentation and local homogeneity, there is one exception: The only thing that's better than providing articles that are relevant to you is providing articles that are relevant to everyone. Traffic watching is a new addiction for bloggers and managers\u2014and as more sites publish their most-popular lists, readers can join in the fun too.\n\nOf course, journalistic traffic chasing isn't exactly a new phenomenon: Since the 1800s, papers have boosted their circulations with sensational reports. Joseph Pulitzer, in honor of whom the eponymous prizes are awarded each year, was a pioneer of using scandal, sex, fearmongering, and innuendo to drive sales.\n\nBut the Internet adds a new level of sophistication and granularity to the pursuit. Now the _Huffington Post_ can put an article on its front page and know within minutes whether it's trending viral; if it is, the editors can kick it by promoting it more heavily. The dashboard that allows editors to watch how stories are doing is considered the crown jewel of the enterprise. Associated Content pays an army of online contributors small amounts to troll search queries and write pages that answer the most common questions; those whose pages see a lot of traffic share in the advertising revenue. Sites like Digg and Reddit attempt to turn the whole Internet into a most-popular list with increasing sophistication, by allowing users to vote submitted articles from throughout the Web onto the site's front page. Reddit's algorithm even has a kind of physics built into it so that articles that don't receive a constant amount of approval will begin to fade, and its front page mixes the articles the group thinks are most important with your personal preferences and behavior\u2014a marriage of the filter bubble and the most-popular list.\n\n_Las \u00daltimas Noticias,_ a major paper in Chile, began basing its content entirely on what readers clicked on in 2004: Stories with lots of clicks got follow-ups, and stories with no clicks got killed. The reporters don't have beats anymore\u2014they just try to gin up stories that will get clicks.\n\nAt Yahoo's popular _Upshot_ news blog, a team of editors mine the data produced by streams of search queries to see what terms people are interested in, in real time. Then they produce articles responsive to those queries: When a lot of people search for \"Obama's birthday,\" _Upshot_ produces an article in response, and soon the searchers are landing on a Yahoo page and seeing Yahoo advertising. \"We feel like the differentiator here, what separates us from a lot of our competitors is our ability to aggregate all this data,\" the vice president of Yahoo Media told the _New York Times._ \"This idea of creating content in response to audience insight and audience needs is one component of the strategy, but it's a big component.\"\n\nAnd what tops the traffic charts? \"If it bleeds, it leads\" is one of the few news maxims that has continued into the new era. Obviously, what's popular differs among audiences: A study of the _Times_ 's most-popular list found that articles that touched on Judaism were often forwarded, presumably due to the _Times_ 's readership. In addition, the study concluded, \"more practically useful, surprising, affect-laden, and positively valenced articles are more likely to be among the newspaper's most e-mailed stories on a given day, as are articles that evoke more awe, anger, and anxiety, and less sadness.\"\n\nElsewhere, the items that top most-popular lists get a bit more crass. The site Buzzfeed recently linked to the \"headline that has everything\" from Britain's _Evening Herald:_ \"Woman in Sumo Wrestler Suit Assaulted Her Ex-girlfriend in Gay Pub After She Waved at a Man Dressed as a Snickers Bar.\" The top story in 2005 for the _Seattle Times_ stayed on the most-read list for weeks; it concerned a man who died after having sex with a horse. The _Los Angeles Times_ 's top story in 2007 was an article about the world's ugliest dog.\n\nResponsiveness to the audience sounds like a good thing\u2014and in a lot of cases, it is. \"If we view the role of cultural products as giving us something to talk about,\" writes a _Wall Street Journal_ reporter who looked into the most-popular phenomenon, \"then the most important thing might be that everyone sees the same thing and not what the thing is.\" Traffic chasing takes media making off its Olympian heights, placing journalists and editors on the same plane with everyone else. The _Washington Post_ ombudsman described journalists' often paternalistic approach to readers: \"In a past era, there was little need to share marketing information with the _Post_ 's newsroom. Profits were high. Circulation was robust. Editors decided what they thought readers needed, not necessarily what they wanted.\"\n\nThe Gawker model is almost the precise opposite. If the _Washington Post_ emulates Dad, these new enterprises are more like fussy, anxious children squalling to be played with and picked up.\n\nWhen I asked him about the prospects for important but unpopular news, the Media Lab's Nicholas Negroponte smiled. On one end of the spectrum, he said, is sycophantic personalization\u2014\"You're so great and wonderful, and I'm going to tell you exactly what you want to hear.\" On the other end is the parental approach: \"I'm going to tell you this whether you want to hear this or not, because you need to know.\" Currently, we're headed in the sycophantic direction. \"There will be a long period of adjustment,\" says Professor Michael Schudson, \"as the separation of church and state is breaking down, so to speak. In moderation, that seems okay, but Gawker's Big Board is a scary extreme, it's surrender.\"\n\n#  **Of Apple and Afghanistan**\n\nGoogle News pays more attention to political news than many of the creators of the filter bubble. After all, it draws in large part on the decisions of professional editors. But even in Google News, stories about Apple trump stories about the war in Afghanistan.\n\nI enjoy my iPhone and iPad, but it's hard to argue that these things are of similar importance to developments in Afghanistan. But this Apple-centric ranking is indicative of what the combination of popular lists and the filter bubble will leave out: Things that are important but complicated. \"If traffic ends up guiding coverage,\" the _Washington Post_ 's ombudsman writes, \"will _The Post_ choose not to pursue some important stories because they're 'dull'?\"\n\nWill an article about, say, child poverty ever seem hugely personally relevant to many of us, beyond the academics studying the field and the people directly affected? Probably not, but it's still important to know about.\n\nCritics on the left frequently argue that the nation's top media underreport the war. But for many of us, myself included, reading about Afghanistan is a chore. The story is convoluted, confusing, complex, and depressing.\n\nIn the editorial judgment of the _Times,_ however, I need to know about it, and because they persist in putting it on the front page despite what must be abominably low traffic rates, I continue to read about it. (This doesn't mean the _Times_ is overruling my own inclinations. It's just supporting one of my inclinations\u2014to be informed about the world\u2014over the more immediate inclination to click on whatever tickles my fancy.) There are places where media that prioritize importance over popularity or personal relevance are useful\u2014even necessary.\n\nClay Shirky points out that newspaper readers always mostly skipped over the political stuff. But to do so, they had to at least glance at the front page\u2014and so, if there was a huge political scandal, enough people would know about it to make an impact at the polls. \"The question,\" Shirky says, \"is how can the average citizen ignore news of the day to the ninety-ninth percentile and periodically be alarmed when there is a crisis? How do you threaten business and civic leaders with the possibility that if things get too corrupt, the alarm can be sounded?\" The front page played that role\u2014but now it's possible to skip it entirely.\n\nWhich brings us back to John Dewey. In Dewey's vision, it is these issues\u2014\"indirect, extensive, enduring and serious consequences of conjoint and interacting behavior\"\u2014that call the public into existence. The important matters that indirectly touch all of our lives but exist out of the sphere of our immediate self-interest are the bedrock and the raison d'\u00eatre of democracy. _American Idol_ may unite a lot of us around the same fireplace, but it doesn't call out the citizen in us. For better or worse\u2014I'd argue for better\u2014the editors of the old media did.\n\nThere's no going back, of course. Nor should there be: the Internet still has the potential to be a better medium for democracy than broadcast, with its one-direction-only information flows, ever could be. As journalist A. J. Liebling pointed out, freedom of the press was for those who owned one. Now we all do.\n\nBut at the moment, we're trading a system with a defined and well-debated sense of its civic responsibilities and roles for one with no sense of ethics. The Big Board is tearing down the wall between editorial decision-making and the business side of the operation. While Google and others are beginning to grapple with the consequences, most personalized filters have no way of prioritizing what really matters but gets fewer clicks. And in the end, \"Give the people what they want\" is a brittle and shallow civic philosophy.\n\nBut the rise of the filter bubble doesn't just affect how we process news. It can also affect how we think.\n**3**\n\n**The Adderall Society**\n\nIt is hardly possible to overrate the value . . . of placing human beings in contact with persons dissimilar to themselves, and with modes of thought and action unlike those with which they are familiar. . . . Such communication has always been, and is peculiarly in the present age, one of the primary sources of progress.\n\n\u2014 _John Stuart Mill_\n\nThe manner in which some of the most important individual discoveries were arrived at reminds one more of a sleepwalker's performance than an electronic brain's.\n\n\u2014 _Arthur Koestler, The Sleepwalkers_\n\n**I** n the spring of 1963, Geneva was swarming with diplomats. Delegations from eighteen countries had arrived for negotiations on the Nuclear Test Ban treaty, and meetings were under way in scores of locations throughout the Swiss capital. After one afternoon of discussions between the American and Russian delegations, a young KGB officer approached a forty-year-old American diplomat named David Mark. \"I'm new on the Soviet delegation, and I'd like to talk to you,\" he whispered to Mark in Russian, \"but I don't want to talk here. I want to have lunch with you.\" After reporting the contact to colleagues at the CIA, Mark agreed, and the two men planned a meeting at a local restaurant the following day.\n\nAt the restaurant, the officer, whose name was Yuri Nosenko, explained that he'd gotten into a bit of a scrape. On his first night in Geneva, Nosenko had drunk too much and brought a prostitute back to his hotel room. When he awoke, to his horror, he found that his emergency stash of $900 in Swiss francs was missing\u2014no small sum in 1963. \"I've got to make it up,\" Nosenko told him. \"I can give you some information that will be very interesting to the CIA, and all I want is my money.\" They set up a second meeting, to which Nosenko arrived in an obviously inebriated state. \"I was snookered,\" Nosenko admitted later\u2014\"very drunk.\"\n\nIn exchange for the money, Nosenko promised to spy for the CIA in Moscow, and in January 1964 he met directly with CIA handlers to discuss his findings. This time, Nosenko had big news: He claimed to have handled the KGB file of Lee Harvey Oswald and said it contained nothing suggesting the Soviet Union had foreknowledge of Kennedy's assassination, potentially ruling out Soviet involvement in the event. He was willing to share more of the file's details with the CIA if he would be allowed to defect and resettle in the United States.\n\nNosenko's offer was quickly transmitted to CIA headquarters in Langley, Virginia. It seemed like a potentially enormous break: Only months after Kennedy had been shot, determining who was behind his assassination was one of the agency's top priorities. But how could they know if Nosenko was telling the truth? James Jesus Angleton, one of the lead agents on Nosenko's case, was skeptical. Nosenko could be a trap\u2014even part of a \"master plot\" to draw the CIA off the trail. After much discussion, the agents agreed to let Nosenko defect: If he was lying, it would indicate that the Soviet Union _did_ know something about Oswald, and if he was telling the truth, he would be useful for counterintelligence.\n\nAs it turned out, they were wrong about both. Nosenko traveled to the United States in 1964, and the CIA collected a massive, detailed dossier on their latest catch. But almost as soon as he started the debriefing process, inconsistencies began to emerge. Nosenko claimed he'd graduated from his officer training program in 1949, but the CIA's documents indicated otherwise. He claimed to have no access to documents that KGB officers of his station ought to have had. And why was this man with a wife and child at home in Russia defecting without them?\n\nAngleton became more and more suspicious, especially after his drinking buddy Kim Philby was revealed to be a Soviet spy. Clearly, Nosenko was a decoy sent to dispute and undermine the intelligence the agency was getting from another Soviet defector. The debriefings became more intense. In 1964, Nosenko was thrown into solitary confinement, where he was subjected for several years to harsh interrogation intended to break him and force him to confess. In one week, he was subjected to polygraph tests for twenty-eight and a half hours. Still, no break was forthcoming.\n\nNot everyone at the CIA thought Nosenko was a plant. And as more details from his biography became clear, it came to seem more and more likely that the man they had imprisoned was no spymaster. Nosenko's father was the minister of shipbuilding and a member of the Communist Party Central Committee who had buildings named after him. When young Yuri had been caught stealing at the Naval Preparatory School and was beaten up by his classmates, his mother had complained directly to Stalin; some of his classmates were sent to the Russian front as punishment. It was looking more and more as though Yuri was just \"the spoiled-brat son of a top leader\" and a bit of a mess. The reason for the discrepancy in graduation dates became clear: Nosenko had been held back a year in school for flunking his exam in Marxism-Leninism, and he was ashamed of it.\n\nBy 1968, the balance of senior CIA agents came to believe that the agency was torturing an innocent man. They gave him $80,000, and set him up in a new identity somewhere in the American South. But the emotional debate over his veracity continued to haunt the CIA for decades, with \"master plan\" theorists sparring with those who believed he was telling the truth. In the end, six separate investigations were made into Nosenko's case. When he passed away in 2008, the news of his death was relayed to the _New York Times_ by a \"senior intelligence official\" who refused to be identified.\n\nOne of the officials most affected by the internal debate was an intelligence analyst by the name of Richards Heuer. Heuer had been recruited to the CIA during the Korean War, but he had always been interested in philosophy, and especially the branch known as epistemology\u2014the study of knowledge. Although Heuer wasn't directly involved in the Nosenko case, he was required to be briefed on it for other work he was doing, and he'd initially fallen for the \"master plot\" hypothesis. Years later, Heuer set out to analyze the analysts\u2014to figure out where the flaws were in the logic that had led to Nosenko's lost years in a CIA prison. The result is a slim volume called _The Psychology of Intelligence Analysis,_ whose preface is full of laudatory comments by Heuer's colleagues and bosses. The book is a kind of Psychology and Epistemology 101 for would-be spooks.\n\nFor Heuer, the core lesson of the Nosenko debacle was clear: \"Intelligence analysts should be self-conscious about their reasoning processes. They should think about how they make judgments and reach conclusions, not just about the judgments and conclusions themselves.\"\n\nDespite evidence to the contrary, Heuer wrote, we have a tendency to believe that the world is as it appears to be. Children eventually learn that a snack removed from view doesn't disappear from the universe, but even as we mature we still tend to conflate seeing with believing. Philosophers call this view na\u00efve realism, and it is as seductive as it is dangerous. We tend to believe we have full command of the facts and that the patterns we see in them are facts as well. (Angleton, the \"master theory\" proponent, was sure that Nosenko's pattern of factual errors indicated that he was hiding something and was breaking under pressure.)\n\nSo what's an intelligence analyst\u2014or anyone who wants to get a good picture of the world, for that matter\u2014to do? First, Heuer suggests, we have to realize that our idea of what's real often comes to us secondhand and in a distorted form\u2014edited, manipulated, and filtered through media, other human beings, and the many distorting elements of the human mind.\n\nNosenko's case was riddled with these distorting factors, and the unreliability of the primary source was only the most obvious one. As voluminous as the set of data that the CIA had compiled on Nosenko was, it was incomplete in certain important ways: The agency knew a lot about his rank and status but had learned very little about his personal background and internal life. This led to a basic unquestioned assumption: \"The KGB would never let a screw-up serve at this high level; therefore, he must be deceiving us.\"\n\n\"To achieve the clearest possible image\" of the world, Heuer writes, \"analysts need more than information.... They also need to understand the lenses through which this information passes.\" Some of these distorting lenses are outside of our heads. Like a biased sample in an experiment, a lopsided selection of data can create the wrong impression: For a number of structural and historical reasons, the CIA record on Nosenko was woefully inadequate when it came to the man's personal history. And some of them are cognitive processes: We tend to convert \"lots of pages of data\" into \"likely to be true,\" for example. When several of them are at work at the same time, it becomes quite difficult to see what's actually going on\u2014a funhouse mirror reflecting a funhouse mirror reflecting reality.\n\nThis distorting effect is one of the challenges posed by personalized filters. Like a lens, the filter bubble invisibly transforms the world we experience by controlling what we see and don't see. It interferes with the interplay between our mental processes and our external environment. In some ways, it can act like a magnifying glass, helpfully expanding our view of a niche area of knowledge. But at the same time, personalized filters limit what we are exposed to and therefore affect the way we think and learn. They can upset the delicate cognitive balance that helps us make good decisions and come up with new ideas. And because creativity is also a result of this interplay between mind and environment, they can get in the way of innovation. If we want to know what the world really looks like, we have to understand how filters shape and skew our view of it.\n\n#  **A Fine Balance**\n\nIt's become a bit in vogue to pick on the human brain. We're \"predictably irrational,\" in the words of behavioral economist Dan Ariely's bestselling book. _Stumbling on Happiness_ author Dan Gilbert presents volumes of data to demonstrate that we're terrible at figuring out what makes us happy. Like audience members at a magic show, we're easily conned, manipulated, and misdirected.\n\nAll of this is true. But as _Being Wrong_ author Kathryn Schulz points out, it's only one part of the story. Human beings may be a walking bundle of miscalculations, contradictions, and irrationalities, but we're built that way for a reason: The same cognitive processes that lead us down the road to error and tragedy are the root of our intelligence and our ability to cope with and survive in a changing world. We pay attention to our mental processes when they fail, but that distracts us from the fact that most of the time, our brains do amazingly well.\n\nThe mechanism for this is a cognitive balancing act. Without our ever thinking about it, our brains tread a tightrope between learning too much from the past and incorporating too much new information from the present. The ability to walk this line\u2014to adjust to the demands of different environments and modalities\u2014is one of human cognition's most astonishing traits. Artificial intelligence has yet to come anywhere close.\n\nIn two important ways, personalized filters can upset this cognitive balance between strengthening our existing ideas and acquiring new ones. First, the filter bubble surrounds us with ideas with which we're already familiar (and already agree), making us overconfident in our mental frameworks. Second, it removes from our environment some of the key prompts that make us want to learn. To understand how, we have to look at what's being balanced in the first place, starting with how we acquire and store information.\n\nFiltering isn't a new phenomenon. It's been around for millions of years\u2014indeed, it was around before humans even existed. Even for animals with rudimentary senses, nearly all of the information coming in through their senses is meaningless, but a tiny sliver is important and sometimes life-preserving. One of the primary functions of the brain is to identify that sliver and decide what to do about it.\n\nIn humans, one of the first steps is to massively compress the data. As Nassim Nicholas Taleb says, \"Information wants to be reduced,\" and every second we reduce a lot of it\u2014compressing most of what our eyes see and ears hear into concepts that capture the gist. Psychologists call these concepts _schemata_ (one of them is a _schema_ ), and they're beginning to be able to identify particular neurons or sets of neurons that correlate with each one\u2014firing, for example, when you recognize a particular object, like a chair. Schemata ensure that we aren't constantly seeing the world anew: Once we've identified something as a chair, we know how to use it.\n\nWe don't do this only with objects; we do it with ideas as well. In a study of how people read the news, researcher Doris Graber found that stories were relatively quickly converted into schemata for the purposes of memorization. \"Details that do not seem essential at the time and much of the context of a story are routinely pared,\" she writes in her book _Processing the News._ \"Such leveling and sharpening involves condensation of all features of a story.\" Viewers of a news segment on a child killed by a stray bullet might remember the child's appearance and tragic background, but not the reportage that overall crime rates are down.\n\nSchemata can actually get in the way of our ability to directly observe what's happening. In 1981, researcher Claudia Cohen instructed subjects to watch a video of a woman celebrating her birthday. Some are told that she's a waitress, while others are told she's a librarian. Later, the groups are asked to reconstruct the scene. The people who are told she's a waitress remember her having a beer; those told she was a librarian remember her wearing glasses and listening to classical music (the video shows her doing all three). The information that didn't jibe with her profession was more often forgotten. In some cases, schemata are so powerful they can even lead to information being fabricated: Doris Graber, the news researcher, found that up to a third of her forty-eight subjects had added details to their memories of twelve television news stories shown to them, based on the schemata those stories activated.\n\nOnce we've acquired schemata, we're predisposed to strengthen them. Psychological researchers call this confirmation bias\u2014a tendency to believe things that reinforce our existing views, to see what we want to see.\n\nOne of the first and best studies of confirmation bias comes from the end of the college football season in 1951\u2014Princeton versus Dartmouth. Princeton hadn't lost a game all season. Its quarterback, Dick Kazmaier, had just been on the cover of _Time_. Things started off pretty rough, but after Kazmaier was sent off the field in the second quarter with a broken nose, the game got really dirty. In the ensuing melee, a Dartmouth player ended up with a broken leg.\n\nPrinceton won, but afterward there were recriminations in both college's papers. Princetonians blamed Dartmouth for starting the low blows; Dartmouth thought Princeton had an ax to grind once their quarterback got hurt. Luckily, there were some psychologists on hand to make sense of the conflicting versions of events.\n\nThey asked groups of students from both schools who hadn't seen the game to watch a film of it and count how many infractions each side made. Princeton students, on average, saw 9.8 infractions by Dartmouth; Dartmouth students thought their team was guilty of only 4.3. One Dartmouth alumnus who received a copy of the film complained that his version must be missing parts\u2014he didn't see any of the roughhousing he'd heard about. Boosters of each school saw what they wanted to see, not what was actually on the film.\n\nPhilip Tetlock, a political scientist, found similar results when he invited a variety of academics and pundits into his office and asked them to make predictions about the future in their areas of expertise. Would the Soviet Union fall in the next ten years? In what year would the U.S. economy start growing again? For ten years, Tetlock kept asking these questions. He asked them not only of experts, but also of folks he'd brought in off the street\u2014plumbers and schoolteachers with no special expertise in politics or history. When he finally compiled the results, even he was surprised. It wasn't just that the normal folks' predictions beat the experts'. The experts' predictions weren't even close.\n\nWhy? Experts have a lot invested in the theories they've developed to explain the world. And after a few years of working on them, they tend to see them everywhere. For example, bullish stock analysts banking on rosy financial scenarios were unable to identify the housing bubble that nearly bankrupted the economy\u2014even though the trends that drove it were pretty clear to anyone looking. It's not just that experts are vulnerable to confirmation bias\u2014it's that they're _especially_ vulnerable to it.\n\nNo schema is an island: Ideas in our heads are connected in networks and hierarchies. _Key_ isn't a useful concept without _lock, door,_ and a slew of other supporting ideas. If we change these concepts too quickly\u2014altering our concept of _door_ without adjusting _lock,_ for example\u2014we could end up removing or altering ideas that other ideas depend on and have the whole system come crashing down. Confirmation bias is a conservative mental force helping to shore up our schemata against erosion.\n\nLearning, then, is a balance. Jean Piaget, one of the major figures in developmental psychology, describes it as a process of assimilation and accommodation. Assimilation happens when children adapt objects to their existing cognitive structures\u2014as when an infant identifies every object placed in the crib as something to suck on. Accommodation happens when we adjust our schemata to new information\u2014\"Ah, this isn't something to suck on, it's something to make a noise with!\" We modify our schemata to fit the world and the world to fit our schemata, and it's in properly balancing the two processes that growth occurs and knowledge is built.\n\nThe filter bubble tends to dramatically amplify confirmation bias\u2014in a way, it's designed to. Consuming information that conforms to our ideas of the world is easy and pleasurable; consuming information that challenges us to think in new ways or question our assumptions is frustrating and difficult. This is why partisans of one political stripe tend not to consume the media of another. As a result, an information environment built on click signals will favor content that supports our existing notions about the world over content that challenges them.\n\nDuring the 2008 presidential campaign, for example, rumors swirled persistently that Barack Obama, a practicing Christian, was a follower of Islam. E-mails circulated to millions, offering \"proof\" of Obama's \"real\" religion and reminding voters that Obama spent time in Indonesia and had the middle name Hussein. The Obama campaign fought back on television and encouraged its supporters to set the facts straight. But even a front-page scandal about his Christian priest, Rev. Jeremiah Wright, was unable to puncture the mythology. Fifteen percent of Americans stubbornly held on to the idea that Obama was a Muslim.\n\nThat's not so surprising\u2014Americans have never been very well informed about our politicians. What's perplexing is that since the election, the percentage of Americans who hold that belief has nearly doubled, and the increase, according to data collected by the Pew Charitable Trusts, has been greatest among people who are college educated. People with some college education were more likely in some cases to believe the story than people with none\u2014a strange state of affairs.\n\nWhy? According to the _New Republic_ 's Jon Chait, the answer lies with the media: \"Partisans are more likely to consume news sources that confirm their ideological beliefs. People with more education are more likely to follow political news. Therefore, people with more education can actually become mis-educated.\" And while this phenomenon has always been true, the filter bubble automates it. In the bubble, the proportion of content that validates what you know goes way up.\n\nWhich brings us to the second way the filter bubble can get in the way of learning: It can block what researcher Travis Proulx calls \"meaning threats,\" the confusing, unsettling occurrences that fuel our desire to understand and acquire new ideas.\n\nResearchers at the University of California at Santa Barbara asked subjects to read two modified versions of \"The Country Doctor,\" a strange, dreamlike short story by Franz Kafka. \"A seriously ill man was waiting for me in a village ten miles distant,\" begins the story. \"A severe snowstorm filled the space between him and me.\" The doctor has no horse, but when he goes to the stable, it's warm and there's a horsey scent. A belligerent groom hauls himself out of the muck and offers to help the doctor. The groom calls two horses and attempts to rape the doctor's maid, while the doctor is whisked to the patient's house in a snowy instant. And that's just the beginning\u2014the weirdness escalates. The story concludes with a series of non sequiturs and a cryptic aphorism: \"Once one responds to a false alarm on the night bell, there's no making it good again\u2014not ever.\"\n\nThe Kafka-inspired version of the story includes meaning threats\u2014incomprehensible events that threaten readers' expectations about the world and shake their confidence in their ability to understand. But the researchers also prepared another version of the story with a much more conventional narrative, complete with a happily-ever-after ending and appropriate, cartoony illustrations. The mysteries and odd occurrences are explained. After reading one version or the other, the study's participants were asked to switch tasks and identify patterns in a set of numbers. The group that read the version adopted from Kafka did nearly twice as well\u2014a dramatic increase in the ability to identify and acquire new patterns. \"The key to our study is that our participants were surprised by the series of unexpected events, and they had no way to make sense of them,\" Proulx wrote. \"Hence, they strived to make sense of something else.\"\n\nFor similar reasons, a filtered environment could have consequences for curiosity. According to psychologist George Lowenstein, curiosity is aroused when we're presented with an \"information gap.\" It's a sensation of deprivation: A present's wrapping deprives us of the knowledge of what's in it, and as a result we become curious about its contents. But to feel curiosity, we have to be conscious that something's being hidden. Because the filter bubble hides things invisibly, we're not as compelled to learn about what we don't know.\n\nAs University of Virginia media studies professor and Google expert Siva Vaidhyanathan writes in \"The Googlization of Everything\": \"Learning is by definition an encounter with what you don't know, what you haven't thought of, what you couldn't conceive, and what you never understood or entertained as possible. It's an encounter with what's other\u2014even with otherness as such. The kind of filter that Google interposes between an Internet searcher and what a search yields shields the searcher from such radical encounters.\" Personalization is about building an environment that consists entirely of the adjacent unknown\u2014the sports trivia or political punctuation marks that don't really shake our schemata but _feel_ like new information. The personalized environment is very good at answering the questions we have but not at suggesting questions or problems that are out of our sight altogether. It brings to mind the famous Pablo Picasso quotation: \"Computers are useless. They can only give you answers.\"\n\nStripped of the surprise of unexpected events and associations, a perfectly filtered world would provoke less learning. And there's another mental balance that personalization can upset: the balance between open-mindedness and focus that makes us creative.\n\n#  **The Adderall Society**\n\nThe drug Adderall is a mixture of amphetamines. Prescribed for attention deficit disorder, it's become a staple for thousands of overscheduled, sleep-deprived students, allowing them to focus for long stretches on a single arcane research paper or complex lab assignment.\n\nFor people without ADD, Adderall also has a remarkable effect. On Erowid, an online forum for recreational drug users and \"mind hackers,\" there's post after post of testimonials to the drug's power to extend focus. \"The part of my brain that makes me curious about whether I have new e-mails in my inbox apparently shut down,\" author Josh Foer wrote in an article on _Slate._ \"Normally, I can only stare at my computer screen for about 20 minutes at a time. On Adderall, I was able to work in hourlong chunks.\"\n\nIn a world of constant interruptions, as work demands only increase, Adderall is a compelling value proposition. Who couldn't use a little cognitive boost? Among the vocal class of neuroenhancement proponents, Adderall and drugs like it may even be the key to our economic future. \"If you're a fifty-five-year-old in Boston, you have to compete with a twenty-six-year-old from Mumbai now, and those kinds of pressures [to use enhancing drugs] are only going to grow,\" Zack Lynch of the neurotech consulting firm NeuroInsights told a _New Yorker_ correspondent.\n\nBut Adderall also has some serious side effects. It's addictive. It dramatically boosts blood pressure. And perhaps most important, it seems to decrease associative creativity. After trying Adderall for a week, Foer was impressed with its powers, cranking out pages and pages of text and reading through dense scholarly articles. But, he wrote, \"it was like I was thinking with blinders on.\" \"With this drug,\" an Erowid experimenter wrote, \"I become calculating and conservative. In the words of one friend, I think 'inside the box.'\" Martha Farah, the director of the University of Pennsylvania's Center for Cognitive Neuroscience, has bigger worries: \"I'm a little concerned that we could be raising a generation of very focused accountants.\"\n\nLike many psychoactive drugs, we still know little about why Adderall has the effects it has\u2014or even entirely what the effects are. But the drug works in part by increasing levels of the neurotransmitter norepinephrine, and norepinephrine has some very particular effects: For one thing, it reduces our sensitivity to new stimuli. ADHD patients call the problem hyperfocus\u2014a trancelike, \"zoned out\" ability to focus on one thing to the exclusion of everything else.\n\nOn the Internet, personalized filters could promote the same kind of intense, narrow focus you get from a drug like Adderall. If you like yoga, you get more information and news about yoga\u2014and less about, say, bird-watching or baseball.\n\nIn fact, the search for perfect relevance and the kind of serendipity that promotes creativity push in opposite directions. \"If you like this, you'll like that\" can be a useful tool, but it's not a source for creative ingenuity. By definition, ingenuity comes from the juxtaposition of ideas that are far apart, and relevance comes from finding ideas that are similar. Personalization, in other words, may be driving us toward an Adderall society, in which hyperfocus displaces general knowledge and synthesis.\n\nPersonalization can get in the way of creativity and innovation in three ways. First, the filter bubble artificially limits the size of our \"solution horizon\"\u2014the mental space in which we search for solutions to problems. Second, the information environment inside the filter bubble will tend to lack some of the key traits that spur creativity. Creativity is a context-dependent trait: We're more likely to come up with new ideas in some environments than in others; the contexts that filtering creates aren't the ones best suited to creative thinking. Finally, the filter bubble encourages a more passive approach to acquiring information, which is at odds with the kind of exploration that leads to discovery. When your doorstep is crowded with salient content, there's little reason to travel any farther.\n\nIn his seminal book _The Act of Creation,_ Arthur Koestler describes creativity as \"bisociation\"\u2014the intersection of two \"matrices\" of thought: \"Discovery is an analogy no one has ever seen before.\" Friedrich Kekule's epiphany about the structure of a benzene molecule after a daydream about a snake eating its tail is an example. So is Larry Page's insight to apply the technique of academic citation to search. \"Discovery often means simply the uncovering of something which has always been there but was hidden from the eye by the blinkers of habit,\" Koestler wrote. Creativity \"uncovers, selects, re-shuffles, combines, synthesizes already existing facts, ideas, faculties, (and) skills.\"\n\nWhile we still have little insight into exactly where different words, ideas, and associations are located physically in the brain, researchers are beginning to be able to map the terrain abstractly. They know that when you feel as though a word is on the tip of your tongue, it usually is. And they can tell that some concepts are much further apart than others, in neural connections if not in actual physical brain space. Researcher Hans Eysenck has found evidence that the individual differences in how people do this mapping\u2014how they connect concepts together\u2014are the key to creative thought.\n\nIn Eysenck's model, creativity is a search for the right set of ideas to combine. At the center of the mental search space are the concepts most directly related to the problem at hand, and as you move outward, you reach ideas that are more tangentially connected. The solution horizon delimits where we stop searching. When we're instructed to \"think outside the box,\" the box represents the solution horizon, the limit of the conceptual area that we're operating in. (Of course, solution horizons that are too wide are a problem, too, because more ideas means exponentially more combinations.)\n\nProgrammers building artificially intelligent chess masters learned the importance of the solution horizon the hard way. The early ones trained the computer to look at every possible combination of moves. This resulted in an explosion of possibilities, which in turn meant that even very powerful computers could only look a limited number of moves ahead. Only when programmers discovered heuristics that allowed the computers to discard some of the moves did they become powerful enough to win against the grand masters of chess. Narrowing the solution horizon, in other words, was key.\n\nIn a way, the filter bubble is a prosthetic solution horizon: It provides you with an information environment that's highly relevant to whatever problem you're working on. Often, this'll be highly useful: When you search for \"restaurant,\" it's likely that you're also interested in near synonyms like \"bistro\" or \"caf\u00e9.\" But when the problem you're solving requires the bisociation of ideas that are indirectly related\u2014as when Page applied the logic of academic citation to the problem of Web search\u2014the filter bubble may narrow your vision too much.\n\nWhat's more, some of the most important creative breakthroughs are spurred by the introduction of the entirely random ideas that filters are designed to rule out.\n\nThe word _serendipity_ originates with the fairy tale \"The Three Princes of Serendip,\" who are continually setting out in search of one thing and finding another. In what researchers call the evolutionary view of innovation, this element of random chance isn't just fortuitous, it's necessary. Innovation requires serendipity.\n\nSince the 1960s, a group of researchers, including Donald Campbell and Dean Simonton, has been pursuing the idea that at a cultural level the process of developing new ideas looks a lot like the process of developing new species. The evolutionary process can be summed up in four words: \"Blind variation, selective retention.\" Blind variation is the process by which mutations and accidents change genetic code, and it's blind because it's chaotic\u2014it's variation that doesn't know where it's going. There's no intent behind it, nowhere in particular that it's headed\u2014it's just the random recombination of genes. Selective retention is the process by which some of the results of blind variation\u2014the offspring\u2014are \"retained\" while others perish. When problems become acute enough for enough people, the argument goes, the random recombination of ideas in millions of heads will tend to produce a solution. In fact, it'll tend to produce the same solution in multiple different heads around the same time.\n\nThe way we selectively combine ideas isn't always blind: As Eysenck's \"solution horizon\" suggests, we don't try to solve our problems by combining every single idea with every other idea in our heads. But when it comes to really new ideas, innovation is in fact often blind. Aharon Kantorovich and Yuval Ne'eman are two historians of science whose work focuses on paradigm shifts, like the move from Newtonian to Einsteinian physics. They argue that \"normal science\"\u2014the day-to-day process of experimentation and prediction\u2014doesn't benefit much from blind variation, because scientists tend to discard random combinations and strange data.\n\nBut in moments of major change, when our whole way of looking at the world shifts and recalibrates, serendipity is often at work. \"Blind discovery is a necessary condition for scientific revolution,\" they write, for a simple reason: The Einsteins and Copernicuses and Pasteurs of the world often have no idea what they're looking for. The biggest breakthroughs are sometimes the ones that we least expect.\n\nThe filter bubble still offers the opportunity for some serendipity, of course. If you're interested in football and local politics, you might still see a story about a play that gives you an idea about how to win the mayoral campaign. But overall, there will tend to be fewer random ideas around\u2014that's part of the point. For a quantified system like a personal filter, it's nearly impossible to sort the usefully serendipitous and randomly provocative from the just plain irrelevant.\n\nThe second way in which the filter bubble can dampen creativity is by removing some of the diversity that prompts us to think in new and innovative ways. In one of the standard creativity tests developed by Karl Duncker in 1945, a researcher hands a subject a box of thumbtacks, a candle, and a book of matches. The subject's job is to attach the candle to the wall so that, when lit, it doesn't drip on the table below (or set the wall on fire). Typically, people try to tack the candle to the wall, or glue it by melting it, or by building complex structures out of the wall with wax and tacks. But the solution (spoiler alert!) is quite simple: Tack the inside of the box to the wall, then place the candle in the box.\n\nDuncker's test gets at one of the key impediments to creativity, what early creativity researcher George Katona described as the reluctance to \"break perceptual set.\" When you're handed a box full of tacks, you'll tend to register the box itself as a container. It takes a conceptual leap to see it as a platform, but even a small change in the test makes that much more likely: If subjects receive the box separately from the tacks, they tend to see the solution much more quickly.\n\nThe process of mapping \"thing with tacks in it\" to the schema \"container\" is called coding; creative candle-platform-builders are those who are able to code objects and ideas in multiple ways. Coding, of course, is very useful: It tells you what you can do with the object; once you've decided that something fits in the \"chair\" schema, you don't have to think twice about sitting on it. But when the coding is too narrow, it impedes creativity.\n\nIn study after study, creative people tend to see things in many different ways and put them in what researcher Arthur Cropley calls \"wide categories.\" The notes from a 1974 study in which participants were told to make groups of similar objects offers an amusing example of the trait in excess: \"Subject 30, a writer, sorted a total of 40 objects.... In response to the candy cigar, he sorted the pipe, matches, cigar, apple, and sugar cubes, explaining that all were related to consumption. In response to the apple, he sorted only the wood block with the nail driven into it, explaining that the apple represented health and vitality (or yin) and that the wood block represented a coffin with a nail in it, or death (or yang). Other sortings were similar.\"\n\nIt's not just artists and writers who use wide categories. As Cropley points out in _Creativity in Education and Learning,_ the physicist Niels Bohr famously demonstrated this type of creative dexterity when he was given a university exam at the University of Copenhagen in 1905. One of the questions asked students to explain how they would use a barometer (an instrument that measures atmospheric pressure) to measure the height of a building. Bohr clearly knew what the instructor was going for: Students were supposed to check the atmospheric pressure at the top and bottom of the building and do some math. Instead, he suggested a more original method: One could tie a string to the barometer, lower it, and measure the string\u2014thinking of the instrument as a \"thing with weight.\"\n\nThe unamused instructor gave him a failing grade\u2014his answer, after all, didn't show much understanding of physics. Bohr appealed, this time offering four solutions: You could throw the barometer off the building and count the seconds until it hit the ground (barometer as mass); you could measure the length of the barometer and of its shadow, then measure the building's shadow and calculate its height (barometer as object with length); you could tie the barometer to a string and swing it at ground level and from the top of the building to determine the difference in gravity (barometer as mass again); or you could use it to calculate air pressure. Bohr finally passed, and one moral of the story is pretty clear: Avoid smartass physicists. But the episode also explains why Bohr was such a brilliant innovator: His ability to see objects and concepts in many different ways made it easier for him to use them to solve problems.\n\nThe kind of categorical openness that supports creativity also correlates with certain kinds of luck. While science has yet to find that there are people whom the universe favors\u2014ask people to guess a random number, and we're all about equally bad at it\u2014there are some traits that people who consider themselves to be lucky share. They're more open to new experiences and new people. They're also more distractable.\n\nRichard Wiseman, a luck researcher at the University of Hertfordshire in England, asked groups of people who considered themselves to be lucky and unlucky to flip through a doctored newspaper and count the number of photographs in it. On the second page, a big headline said, \"Stop counting\u2014there are 43 pictures.\" Another page offered 150 British pounds to readers who noticed it. Wiseman described the results: \"For the most part, the unlucky would just flip past these things. Lucky people would flip through and laugh and say, 'There are 43 photos. That's what it says. Do you want me to bother counting?' We'd say, 'Yeah, carry on.' They'd flip some more and say, 'Do I get my 150 pounds?' Most of the unlucky people didn't notice.\"\n\nAs it turns out, being around people and ideas unlike oneself is one of the best ways to cultivate this sense of open-mindedness and wide categories. Psychologists Charlan Nemeth and Julianne Kwan discovered that bilinguists are more creative than monolinguists\u2014perhaps because they have to get used to the proposition that things can be viewed in several different ways. Even forty-five minutes of exposure to a different culture can boost creativity: When a group of American students was shown a slideshow about China as opposed to one about the United States, their scores on several creativity tests went up. In companies, the people who interface with multiple different units tend to be greater sources of innovation than people who interface only with their own. While nobody knows for certain what causes this effect, it's likely that foreign ideas help us break open our categories.\n\nBut the filter bubble isn't tuned for a diversity of ideas or of people. It's not designed to introduce us to new cultures. As a result, living inside it, we may miss some of the mental flexibility and openness that contact with difference creates.\n\nBut perhaps the biggest problem is that the personalized Web encourages us to spend less time in discovery mode in the first place.\n\n#  **The Age of Discovery**\n\nIn _Where Good Ideas Come From_ , science author Steven Johnson offers a \"natural history of innovation,\" in which he inventories and elegantly illustrates how creativity arises. Creative environments often rely on \"liquid networks\" where different ideas can collide in different configurations. They arrive through serendipity\u2014we set out looking for the answer to one problem and find another\u2014and as a result, ideas emerge frequently in places where random collision is more likely to occur. \"Innovative environments,\" he writes, \"are better at helping their inhabitants explore the adjacent possible\"\u2014the bisociated area in which existing ideas combine to produce new ones\u2014\"because they expose a wide and diverse sample of spare parts\u2014mechanical or conceptual\u2014and they encourage novel ways of recombining those parts.\"\n\nHis book is filled with examples of these environments, from primordial soup to coral reefs and high-tech offices, but Johnson continually returns to two: the city and the Web.\n\n\"For complicated historical reasons,\" he writes, \"they are both environments that are powerfully suited for the creation, diffusion, and adoption of good ideas.\"\n\nThere's no question that Johnson _was_ right: The old, unpersonalized web offered an environment of unparalleled richness and diversity. \"Visit the 'serendipity' article in Wikipedia,\" he writes, and \"you are one click away from entries on LSD, Teflon, Parkinson's disease, Sri Lanka, Isaac Newton, and about two hundred other topics of comparable diversity.\"\n\nBut the filter bubble has dramatically changed the informational physics that determines which ideas we come in contact with. And the new, personalized Web may no longer be as well suited for creative discovery as it once was.\n\nIn the early days of the World Wide Web, when Yahoo was its king, the online terrain felt like an unmapped continent, and its users considered themselves discoverers and explorers. Yahoo was the village tavern where sailors would gather to swap tales about what strange beasts and distant lands they found out at sea. \"The shift from exploration and discovery to the intent-based search of today was inconceivable,\" an early Yahoo editor told search journalist John Battelle. \"Now, we go online expecting everything we want to find will be there. That's a major shift.\"\n\nThis shift from a discovery-oriented Web to a search and retrieval\u2013focused Web mirrors one other piece of the research surrounding creativity. Creativity experts mostly agree that it's a process with at least two key parts: Producing novelty requires a lot of divergent, generative thinking\u2014the reshuffling and recombining that Koestler describes. Then there's a winnowing process\u2014convergent thinking\u2014as we survey the options for one that'll fit the situation. The serendipitous Web attributes that Johnson praises\u2014the way one can hop from article to article on Wikipedia\u2014are friendly to the divergent part of that process.\n\nBut the rise of the filter bubble means that increasingly the convergent, synthetic part of the process is built in. Battelle calls Google a \"database of intentions,\" each query representing something that someone wants to do or know or buy. Google's core mission, in many ways, is to transform those intentions into actions. But the better it gets at that, the worse it'll be at providing serendipity, which, after all, is the process of stumbling across the _un_ intended. Google is great at helping us find what we know we want, but not at finding what we _don't_ know we want.\n\nTo some degree, the sheer volume of information available mitigates this effect. There's far more online content to choose from than there was in even the largest libraries. For an enterprising informational explorer, there's endless terrain to cover. But one of the prices of personalization is that we become a bit more passive in the process. The better it works, the less exploring we have to do.\n\nDavid Gelernter, a Yale professor and early supercomputing visionary, believes that computers will only serve us well when they can incorporate dream logic. \"One of the hardest, most fascinating problems of this cyber-century is how to add 'drift' to the net,\" he writes, \"so that your view sometimes wanders (as your mind wanders when you're tired) into places you hadn't planned to go. Touching the machine brings the original topic back. We need help overcoming rationality sometimes, and allowing our thoughts to wander and metamorphose as they do in sleep.\" To be truly helpful, algorithms may need to work more like the fuzzyminded, nonlinear humans they're supposed to serve.\n\n#  **On California Island**\n\nIn 1510, the Spanish writer Garci Rodriguez de Montalvo published a swashbuckling _Odyssey_ -like novel, _The Exploits of Esplandian,_ which included a description of a vast island called California:\n\nOn the right hand from the Indies exists an island called California very close to a side of the Earthly Paradise; and it was populated by black women, without any man existing there, because they lived in the way of the Amazons. They had beautiful and robust bodies, and were brave and very strong. Their island was the strongest of the World, with its cliffs and rocky shores. Their weapons were golden and so were the harnesses of the wild beasts that they were accustomed to domesticate and ride, because there was no other metal in the island than gold.\n\nRumors of gold propelled the legend of the island of California across Europe, prompting adventurers throughout the continent to set off in search of it. Hern\u00e1n Cort\u00e9s, the Spanish conquistador who led the colonization of the Americas, requested money from Spain's king to lead a worldwide hunt. And when he landed in what we now know as Baja California in 1536, he was certain he'd found the place. It wasn't until one of his navigators, Francisco de Ulloa, traveled up the Gulf of California to the mouth of the Colorado river that it became clear to Cortez that, gold or no, he hadn't found the mythical island.\n\nDespite this discovery, however, the idea that California was an island persisted for several more centuries. Other explorers discovered Puget Sound, near Vancouver, and were certain that it must connect to Baja. Dutch maps from the 1600s routinely show a distended long fragment off the coast of America stretching half the length of the continent. It took Jesuit missionaries literally marching inland and never reaching the other side to fully repudiate the myth.\n\nIt may have persisted for one simple reason: There was no sign on the maps for \"don't know,\" and so the distinction between geographic guesswork and sights that had been witnessed firsthand became blurred. One of history's major cartographic errors, the island of California reminds us that it's not what we don't know that hurts us as much as what we don't know we don't know\u2014what ex\u2013secretary of defense Donald Rumsfeld famously called the unknown unknowns.\n\nThis is one other way that personalized filters can interfere with our ability to properly understand the world: They alter our sense of the map. More unsettling, they often remove its blank spots, transforming known unknowns into unknown ones.\n\nTraditional, unpersonalized media often offer the promise of representativeness. A newspaper editor isn't doing his or her job properly unless to some degree the paper is representative of the news of the day. This is one of the ways one can convert an unknown unknown into a known unknown. If you leaf through the paper, dipping into some articles and skipping over most of them, you at least know there are stories _,_ perhaps whole sections, that you passed over _._ Even if you don't read the article, you notice the headline about a flood in Pakistan\u2014or maybe you're just reminded that, yes, there is a Pakistan.\n\nIn the filter bubble, things look different. You don't see the things that don't interest you at all. You're not even latently aware that there are major events and ideas you're missing. Nor can you take the links you do see and assess how representative they are without an understanding of what the broader environment from which they were selected looks like. As any statistician will tell you, you can't tell how biased the sample is from looking at the sample alone: You need something to compare it to.\n\nAs a last resort, you might look at your selection and ask yourself if it _looks_ like a representative sample. Are there conflicting views? Are there different takes, and different kinds of people reflecting? Even this is a blind alley, however, because with an information set the size of the Internet, you get a kind of fractal diversity: at any level, even within a very narrow information spectrum (atheist goth bowlers, say) there are lots of voices and lots of different takes.\n\nWe're never able to experience the whole world at once. But the best information tools give us a sense of where we stand in it\u2014literally, in the case of a library, and figuratively in the case of a newspaper front page. This was one of the CIA's primary errors with Yuri Nosenko. The agency had collected a specialized subset of information about Nosenko without realizing how specialized it was, and thus despite the many brilliant analysts working for years on the case, it missed what would have been obvious from a whole picture of the man.\n\nBecause personalized filters usually have no Zoom Out function, it's easy to lose your bearings, to believe the world is a narrow island when in fact it's an immense, varied continent.\n**4**\n\n**The You Loop**\n\nI believe this is the quest for what a personal computer really is. It is to capture one's entire life.\n\n\u2014 _Gordon Bell_\n\n**Y** ou have one identity,\" Facebook founder Mark Zuckerberg told journalist David Kirkpatrick for his book _The Facebook Effect_. \"The days of you having a different image for your work friends or coworkers and for the other people you know are probably coming to an end pretty quickly.... Having two identities for yourself is an example of a lack of integrity.\"\n\nA year later, soon after the book had been published, twenty-six-year-old Zuckerberg sat onstage with Kirkpatrick and NPR interviewer Guy Raz at the Computer History Museum in Mountain View, California. \"In David's book,\" Raz said, \"you say that people should have one identity.... But I behave a different way around my family than I do around my colleagues.\"\n\nZuckerberg shrugged. \"No, I think that was just a sentence I said.\"\n\nRaz continued: \"Are you the same person right now as when you're with your friends?\"\n\n\"Uh, yeah,\" Zuckerberg said. \"Same awkward self.\"\n\nIf Mark Zuckerberg were a standard mid-twenty-something, this tangle of views might be par for the course: Most of us don't spend too much time musing philosophically about the nature of identity. But Zuckerberg controls the world's most powerful and widely used technology for managing and expressing who we are. And his views on the matter are central to his vision for the company and for the Internet.\n\nSpeaking at an event during New York's Ad Week, Facebook COO Sheryl Sandberg said she expected the Internet to change quickly. \"People don't want something targeted to the whole world\u2014they want something that reflects what they want to see and know,\" she said, suggesting that in three to five years that would be the norm. Facebook's goal is to be at the center of that process\u2014the singular platform through which every other service and Web site incorporates your personal and social data. You have one identity, it's your Facebook identity, and it colors your experience everywhere you go.\n\nIt's hard to imagine a more dramatic departure from the early days of the Internet, in which _not_ exposing your identity was part of the appeal. In chat rooms and online forums, your gender, race, age, and location were whatever you said they were, and the denizens of these spaces exulted about the way the medium allowed you to shed your skin. Electronic Frontier Foundation (EFF) founder John Perry Barlow dreamed of \"creating a world that all may enter without privilege or prejudice accorded by race, economic power, military force, or station of birth.\" The freedom that this offered anyone who was interested to transgress and explore, to try on different personas for size, felt revolutionary.\n\nAs law and commerce have caught up with technology, however, the space for anonymity online is shrinking. You can't hold an anonymous person responsible for his or her actions: Anonymous customers commit fraud, anonymous commenters start flame wars, and anonymous hackers cause trouble. To establish the trust that community and capitalism are built on, you need to know whom you're dealing with.\n\nAs a result, there are dozens of companies working on deanonymizing the Web. PeekYou, a firm founded by the creator of RateMyProfessors.com, is patenting ways of connecting online activities done under a pseudonym with the real name of the person involved. Another company, Phorm, helps Internet service providers use a method called \"deep packet inspection\" to analyze the traffic that flows through their servers; Phorm aims to build nearly comprehensive profiles of each customer to use for advertising and personalized services. And if ISPs are leery, BlueCava is compiling a database of every computer, smartphone, and online-enabled gadget in the world, which can be tied to the individual people who use them. Even if you're using the highest privacy settings in your Web browser, in other words, your hardware may soon give you away.\n\nThese technological developments pave the way for a more persistent kind of personalization than anything we've experienced to date. It also means that we'll increasingly be forced to trust the companies at the center of this process to properly express and synthesize who we really are. When you meet someone in a bar or a park, you look at how they behave and act and form an impression accordingly. Facebook and the other identity services aim to mediate that process online; if they don't do it right, things can get fuzzy and distorted. To personalize well, you have to have the right idea of what represents a person.\n\nThere's another tension in the interplay of identity and personalization. Most personalized filters are based on a three-step model. First, you figure out who people are and what they like. Then, you provide them with content and services that best fit them. Finally, you tune to get the fit just right. Your identity shapes your media. There's just one flaw in this logic: Media also shape identity. And as a result, these services may end up creating a good fit between you and your media by changing ... you. If a self-fulfilling prophecy is a false definition of the world that through one's actions becomes true, we're now on the verge of self-fulfilling identities, in which the Internet's distorted picture of us becomes who we really are.\n\nPersonalized filtering can even affect your ability to choose your own destiny. In \"Of Sirens and Amish Children,\" a muchcited tract, information law theorist Yochai Benkler describes how more-diverse information sources make us freer. Autonomy, Benkler points out, is a tricky concept: To be free, you have to be able not only to do what you want, but to know what's possible to do. The Amish children in the title are plaintiffs in a famous court case, _Wisconsin v. Yoder,_ whose parents sought to prevent them from attending public school so that they wouldn't be exposed to modern life. Benkler argues that this is a real threat to the children's freedom: Not knowing that it's possible to be an astronaut is just as much a prohibition against becoming one as knowing and being barred from doing so.\n\nOf course, too many options are just as problematic as too few\u2014you can find yourself overwhelmed by the number of options or paralyzed by the paradox of choice. But the basic point remains: The filter bubble doesn't just reflect your identity. It also illustrates what choices you have. Students who go to Ivy League colleges see targeted advertisements for jobs that students at state schools are never even aware of. The personal feeds of professional scientists might feature articles about contests that amateurs never become aware of. By illustrating some possibilities and blocking out others, the filter bubble has a hand in your decisions. And in turn, it shapes who you become.\n\n#  **A Bad Theory of You**\n\nThe way that personalization shapes identity is still becoming clear\u2014especially because most of us still spend more time consuming broadcast media than personalized content streams. But by looking at how the major filterers think about identity, it's becoming possible to predict what these changes might look like. Personalization requires a theory of what makes a person\u2014of what bits of data are most important to determine who someone is\u2014and the major players on the Web have quite different ways of approaching the problem.\n\nGoogle's filtering systems, for example, rely heavily on Web history and what you click on (click signals) to infer what you like and dislike. These clicks often happen in an entirely private context: The assumption is that searches for \"intestinal gas\" and celebrity gossip Web sites are between you and your browser. You might behave differently if you thought other people were going to see your searches. But it's that behavior that determines what content you see in Google News, what ads Google displays\u2014what determines, in other words, Google's theory of you.\n\nThe basis for Facebook's personalization is entirely different. While Facebook undoubtedly tracks clicks, its primary way of thinking about your identity is to look at what you share and with whom you interact. That's a whole different kettle of data from Google's: There are plenty of prurient, vain, and embarrassing things we click on that we'd be reluctant to share with all of our friends in a status update. And the reverse is true, too. I'll cop to sometimes sharing links I've barely read\u2014the long investigative piece on the reconstruction of Haiti, the bold political headline\u2014because I like the way it makes me appear to others. The Google self and the Facebook self, in other words, are pretty different people. There's a big difference between \"you are what you click\" and \"you are what you share.\"\n\nBoth ways of thinking have their benefits and drawbacks. With Google's click-based self, the gay teenager who hasn't come out to his parents can still get a personalized Google News feed with pieces from the broader gay community that affirm that he's not alone. But by the same token, a self built on clicks will tend to draw us even more toward the items we're predisposed to look at already\u2014toward our most Pavlovian selves. Your perusal of an article on TMZ.com is filed away, and the next time you're looking at the news, Brad Pitt's marriage drama is more likely to flash on to the screen. (If Google didn't persistently downplay porn, the problem would presumably be far worse.)\n\nFacebook's share-based self is more aspirational: Facebook takes you more at your word, presenting you as you'd like to be seen by others. Your Facebook self is more of a performance, less of a behaviorist black box, and ultimately it may be more prosocial than the bundle of signals Google tracks. But the Facebook approach has its downsides as well\u2014to the extent that Facebook draws on the more public self, it necessarily has less room for private interests and concerns. The same closeted gay teenager's information environment on Facebook might diverge more from his real self. The Facebook portrait remains incomplete.\n\nBoth are pretty poor representations of who we are, in part because there is no one set of data that describes who we are. \"Information about our property, our professions, our purchases, our finances, and our medical history does not tell the whole story,\" writes privacy expert Daniel Solove. \"We are more than the bits of data we give off as we go about our lives.\"\n\nDigital animators and robotics engineers frequently run into a problem known as the _uncanny valley_. The uncanny valley is the place where something is lifelike but not convincingly alive, and it gives people the creeps. It's part of why digital animation of real people still hasn't hit the big screens: When an image looks almost like a real person, but not quite, it's unsettling on a basic psychological level. We're now in the uncanny valley of personalization. The doppelg\u00e4nger selves reflected in our media are a lot like, but not exactly, ourselves. And as we'll see, there are some important things that are lost in the gap between the data and reality.\n\nTo start with, Zuckerberg's statement that we have \"one identity\" simply isn't true. Psychologists have a name for this fallacy: fundamental attribution error. We tend to attribute peoples' behavior to their inner traits and personality rather than to the situations they're placed in. Even in situations where the context clearly plays a major role, we find it hard to separate how someone behaves from who she is.\n\nAnd to a striking degree, our characteristics are fluid. Someone who's aggressive at work may be a doormat at home. Someone who's gregarious when happy may be introverted when stressed. Even some of our closest-held traits\u2014our disinclination to do harm, for example\u2014can be shaped by context. Groundbreaking psychologist Stanley Milgram demonstrated this when, in an oft-cited experiment at Yale in the 1960s, he got decent ordinary people to apparently electrocute other subjects when given the nod by a man in a lab coat.\n\nThere is a reason that we act this way: The personality traits that serve us well when we're at dinner with our family might get in the way when we're in a dispute with a passenger on the train or trying to finish a report at work. The plasticity of the self allows for social situations that would be impossible or intolerable if we always behaved exactly the same way. Advertisers have understood this phenomenon for a long time. In the jargon, it's called _day-parting_ , and it's the reason that you don't hear many beer ads as you're driving to work in the morning. People have different needs and aspirations at eight A.M. than they do at eight P.M. By the same token, billboards in the nightlife district promote different products than billboards in the residential neighborhoods the same partiers go home to.\n\nOn his own Facebook page, Zuckerberg lists \"transparency\" as one of his top Likes. But there's a downside to perfect transparency: One of the most important uses of privacy is to manage and maintain the separations and distinctions among our different selves. With only one identity, you lose the nuances that make for a good personalized fit.\n\nPersonalization doesn't capture the balance between your work self and your play self, and it can also mess with the tension between your aspirational and your current self. How we behave is a balancing act between our future and present selves. In the future, we want to be fit, but in the present, we want the candy bar. In the future, we want to be a well-rounded, well-informed intellectual virtuoso, but right now we want to watch _Jersey Shore._ Behavioral economists call this present bias\u2014the gap between your preferences for your future self and your preferences in the current moment.\n\nThe phenomenon explains why there are so many movies in your Netflix queue. When researchers at Harvard and the Analyst Institute looked at people's movie-rental patterns, they were able to watch as people's future aspirations played against their current desires. \"Should\" movies like _An Inconvenient Truth_ or _Schindler's List_ were often added to the queue, but there they languished while watchers gobbled up \"want\" movies like _Sleepless in Seattle._ And when they had to choose three movies to watch instantly, they were less likely to choose \"should\" movies at all. Apparently there are some movies we'd always rather watch tomorrow.\n\nAt its best, media help mitigate present bias, mixing \"should\" stories with \"want\" stories and encouraging us to dig into the difficult but rewarding work of understanding complex problems. But the filter bubble tends to do the opposite: Because it's our present self that's doing all the clicking, the set of preferences it reflects is necessarily more \"want\" than \"should.\"\n\nThe one-identity problem isn't a fundamental flaw. It's more of a bug: Because Zuckerberg thinks you have one identity and you don't, Facebook will do a worse job of personalizing your information environment. As John Battelle told me, \"We're so far away from the nuances of what it means to be human, as reflected in the nuances of the technology.\" Given enough data and enough programmers, the context problem is solvable\u2014and according to personalization engineer Jonathan McPhie, Google is working on it. We've seen the pendulum swing from the anonymity of the early Internet to the one-identity view currently in vogue; the future may look like something in between.\n\nBut the one-identity problem illustrates one of the dangers of turning over your most personal details to companies who have a skewed view of what identity is. Maintaining separate identity zones is a ritual that helps us deal with the demands of different roles and communities. And something's lost when, at the end of the day, everything inside your filter bubble looks roughly the same. Your bacchanalian self comes knocking at work; your work anxieties plague you on a night out.\n\nAnd when we're aware that everything we do enters a permanent, pervasive online record, another problem emerges: The knowledge that what we do affects what we see and how companies see us can create a chilling effect. Genetic privacy expert Mark Rothstein describes how lax regulations around genetic data can actually reduce the number of people willing to be tested for certain diseases: If you might be discriminated against or denied insurance for having a gene linked to Parkinson's disease, it's not unreasonable just to skip the test and the \"toxic knowledge\" that might result.\n\nIn the same way, when our online actions are tallied and added to a record that companies use to make decisions, we might decide to be more cautious in our surfing. If we knew (or even suspected, for that matter) that purchasers of _101 Ways to Fix Your Credit Score_ tend to get offered lower-premium credit cards, we'd avoid buying the book. \"If we thought that our every word and deed were public,\" writes law professor Charles Fried, \"fear of disapproval or more tangible retaliation might keep us from doing or saying things which we would do or say could we be sure of keeping them to ourselves.\" As Google expert Siva Vaidhyanathan points out, \"F. Scott Fitzgerald's enigmatic Jay Gatsby could not exist today. The digital ghost of Jay Gatz would follow him everywhere.\"\n\nIn theory, the one-identity, context-blind problem isn't impossible to fix. Personalizers will undoubtedly get better at sensing context. They might even be able to better balance long-term and short-term interests. But when they do\u2014when they are able to accurately gauge the workings of your psyche\u2014things get even weirder.\n\n#  **Targeting Your Weak Spots**\n\nThe logic of the filter bubble today is still fairly rudimentary: People who bought the _Iron Man_ DVD are likely to buy _Iron Man II;_ people who enjoy cookbooks will probably be interested in cookware. But for Dean Eckles, a doctoral student at Stanford and an adviser to Facebook, these simple recommendations are just the beginning. Eckles is interested in means, not ends: He cares less about what types of products you like than which kinds of arguments might cause you to choose one over another.\n\nEckles noticed that when buying products\u2014say, a digital camera\u2014different people respond to different pitches. Some people feel comforted by the fact that an expert or product review site will vouch for the camera. Others prefer to go with the product that's most popular, or a money-saving deal, or a brand that they know and trust. Some people prefer what Eckles calls \"high cognition\" arguments\u2014smart, subtle points that require some thinking to get. Others respond better to being hit over the head with a simple message.\n\nAnd while most of us have preferred styles of argument and validation, there are also types of arguments that really turn us off. Some people rush for a deal; others think that the deal means the merchandise is subpar. Just by eliminating the persuasion styles that rub people the wrong way, Eckles found he could increase the effectiveness of marketing materials by 30 to 40 percent.\n\nWhile it's hard to \"jump categories\" in products\u2014what clothing you prefer is only slightly related to what books you enjoy\u2014\"persuasion profiling\" suggests that the kinds of arguments you respond to are highly transferrable from one domain to another. A person who responds to a \"get 20% off if you buy NOW\" deal for a trip to Bermuda is much more likely than someone who doesn't to respond to a similar deal for, say, a new laptop.\n\nIf Eckles is right\u2014and research so far appears to be validating his theory\u2014your \"persuasion profile\" would have a pretty significant financial value. It's one thing to know how to pitch products to you in a specific domain; it's another to be able to improve the hit rate anywhere you go. And once a company like Amazon has figured out your profile by offering you different kinds of deals over time and seeing which ones you responded to, there's no reason it couldn't then sell that information to other companies. (The field is so new that it's not clear if there's a correlation between persuasion styles and demographic traits, but obviously that could be a shortcut as well.)\n\nThere's plenty of good that could emerge from persuasion profiling, Eckles believes. He points to DirectLife, a wearable coaching device by Philips that figures out which arguments get people eating more healthily and exercising more regularly. But he told me he's troubled by some of the possibilities. Knowing what kinds of appeals specific people respond to gives you power to manipulate them on an individual basis.\n\nWith new methods of \"sentiment analysis, it's now possible to guess what mood someone is in. People use substantially more positive words when they're feeling up; by analyzing enough of your text messages, Facebook posts, and e-mails, it's possible to tell good days from bad ones, sober messages from drunk ones (lots of typos, for a start). At best, this can be used to provide content that's suited to your mood: On an awful day in the near future, Pandora might know to preload _Pretty Hate Machine_ for you when you arrive. But it can also be used to take advantage of your psychology.\n\nConsider the implications, for example, of knowing that particular customers compulsively buy things when stressed or when they're feeling bad about themselves, or even when they're a bit tipsy. If persuasion profiling makes it possible for a coaching device to shout \"you can do it\" to people who like positive reinforcement, in theory it could also enable politicians to make appeals based on each voter's targeted fears and weak spots.\n\nInfomercials aren't shown in the middle of the night only because airtime then is cheap. In the wee hours, most people are especially suggestible. They'll spring for the slicer-dicer that they'd never purchase in the light of day. But the three A.M. rule is a rough one\u2014presumably, there are times in all of our daily lives when we're especially inclined to purchase whatever's put in front of us. The same data that provides personalized content can be used to allow marketers to find and manipulate your personal weak spots. And this isn't a hypothetical possibility: Privacy researcher Pam Dixon discovered that a data company called PK List Management offers a list of customers titled \"Free to Me\u2014Impulse Buyers\"; those listed are described as being highly susceptible to pitches framed as sweepstakes.\n\nIf personalized persuasion works for products, it can also work for ideas. There are undoubtedly times and places and styles of argument that make us more susceptible to believe what we're told. Subliminal messaging is illegal because we recognize there are some ways of making an argument that are essentially cheating; priming people with subconsciously flashed words to sell them things isn't a fair game. But it's not such a stretch to imagine political campaigns targeting voters at times when they can circumvent our more reasonable impulses.\n\nWe intuitively understand the power in revealing our deep motivations and desires and how we work, which is why most of us only do that in day-to-day life with people whom we really trust. There's a symmetry to it: You know your friends about as well as they know you. Persuasion profiling, on the other hand, can be done invisibly\u2014you need not have any knowledge that this data is being collected from you\u2014and therefore it's asymmetrical. And unlike some forms of profiling that take place in plain sight (like Netflix), persuasion profiling is handicapped when it's revealed. It's just not the same to hear an automated coach say \"You're doing a great job! I'm telling you that because you respond well to encouragement!\"\n\nSo you don't necessarily see the persuasion profile being made. You don't see it being used to influence your behavior. And the companies we're turning over this data to have no legal obligation to keep it to themselves. In the wrong hands, persuasion profiling gives companies the ability to circumvent your rational decision making, tap into your psychology, and draw out your compulsions. Understand someone's identity, and you're better equipped to influence what he or she does.\n\n#  **A Deep and Narrow Path**\n\nSomeday soon, Google Vice President Marissa Mayer says, the company hopes to make the search box obsolete. \"The next step of search is doing this automatically,\" Eric Schmidt said in 2010. \"When I walk down the street, I want my smartphone to be doing searches constantly\u2014'did you know?' 'did you know?' 'did you know?' 'did you know?' \" In other words, your phone should figure out what you would like to be searching for before you do.\n\nIn the fast-approaching age of search without search, identity drives media. But the personalizers haven't fully grappled with a parallel fact: Media also shapes identity. Political scientist Shanto Iyengar calls one of primary factors accessibility bias, and in a paper titled \"Experimental Demonstrations of the 'Not-So-Minimal' Consequences of Television News,'\" in 1982, he demonstrated how powerful the bias is. Over six days, Iyengar asked groups of New Haven residents to watch episodes of a TV news program, which he had doctored to include different segments for each group.\n\nAfterward, Iyengar asked subjects to rank how important issues like pollution, inflation, and defense were to them. The shifts from the surveys they'd filled out before the study were dramatic: \"Participants exposed to a steady stream of news about defense or about pollution came to believe that defense or pollution were more consequential problems,\" Iyengar wrote. Among the group that saw the clips on pollution, the issue moved from fifth out of six in priority to second.\n\nDrew Westen, a neuropsychologist whose focus is on political persuasion, demonstrates the strength of this priming effect by asking a group of people to memorize a list of words that include _moon_ and _ocean._ A few minutes later, he changes topics and asks the group which detergent they prefer. Though he hasn't mentioned the word, the group's show of hands indicates a strong preference for Tide.\n\nPriming isn't the only way media shape our identities. We're also more inclined to believe what we've heard before. In a 1977 study by Hasher and Goldstein, participants were asked to read sixty statements and mark whether they were true or false. All of the statements were plausible, but some of them (\"French horn players get cash bonuses to stay in the Army\") were true; others (\"Divorce is only found in technically advanced societies\") weren't. Two weeks later, they returned and rated a second batch of statements in which some of the items from the first list had been repeated. By the third time, two weeks after that, the subjects were far more likely to believe the repeated statements. With information as with food, we are what we consume.\n\nAll of these are basic psychological mechanisms. But combine them with personalized media, and troubling things start to happen. Your identity shapes your media, and your media then shapes what you believe and what you care about. You click on a link, which signals an interest in something, which means you're more likely to see articles about that topic in the future, which in turn prime the topic for you. You become trapped in a you loop, and if your identity is misrepresented, strange patterns begin to emerge, like reverb from an amplifier.\n\nIf you're a Facebook user, you've probably run into this problem. You look up your old college girlfriend Sally, mildly curious to see what she is up to after all these years. Facebook interprets this as a sign that you're interested in Sally, and all of a sudden her life is all over your news feed. You're still mildly curious, so you click through on the new photos she's posted of her kids and husband and pets, confirming Facebook's hunch. From Facebook's perspective, it looks as though you have a relationship with this person, even if you haven't communicated in years. For months afterward, Sally's life is far more prominent than your actual relationship would indicate. She's a \"local maximum\": Though there are people whose posts you're far more interested in, it's her posts that you see.\n\nIn part, this feedback effect is due to what early Facebook employee and venture capitalist Matt Cohler calls the local-maximum problem. Cohler was an early employee at Facebook, and he's widely considered one of Silicon Valley's smartest thinkers on the social Web.\n\nThe local-maximum problem, he explains to me, shows up any time you're trying to optimize something. Say you're trying to write a simple set of instructions to help a blind person who's lost in the Sierra Nevadas find his way to the highest peak. \"Feel around you to see if you're surrounded by downward-sloping land,\" you say. \"If you're not, move in a direction that's higher, and repeat.\"\n\nProgrammers face problems like this all the time. What link is the best result for the search term \"fish\"? Which picture can Facebook show you to increase the likelihood that you'll start a photo-surfing binge? The directions sound pretty obvious\u2014you just tweak and tune in one direction or another until you're in the sweet spot. But there's a problem with these hill-climbing instructions: They're as likely to end you up in the foothills\u2014the local maximum\u2014as they are to guide you to the apex of Mount Whitney.\n\nThis isn't exactly harmful, but in the filter bubble, the same phenomenon can happen with any person or topic. I find it hard not to click on articles about gadgets, though I don't actually think they're that important. Personalized filters play to the most compulsive parts of you, creating \"compulsive media\" to get you to click things more. The technology mostly can't distinguish compulsion from general interest\u2014and if you're generating page views that can be sold to advertisers, it might not care.\n\nThe faster the system learns from you, the more likely it is that you can get trapped in a kind of identity cascade, in which a small initial action\u2014clicking on a link about gardening or anarchy or Ozzy Osbourne\u2014indicates that you're a person who likes those kinds of things. This in turn supplies you with more information on the topic, which you're more inclined to click on because the topic has now been primed for you.\n\nEspecially once the second click has occurred, your brain is in on the act as well. Our brains act to reduce cognitive dissonance in a strange but compelling kind of unlogic\u2014\"Why would I have done _x_ if I weren't a person who does _x_ \u2014therefore I must be a person who does _x._ \"Each click you take in this loop is another action to self-justify\u2014\"Boy, I guess I just really love 'Crazy Train.' \" When you use a recursive process that feeds on itself, Cohler tells me, \"You're going to end up down a deep and narrow path.\" The reverb drowns out the tune. If identity loops aren't counteracted through randomness and serendipity, you could end up stuck in the foothills of your identity, far away from the high peaks in the distance.\n\nAnd that's when these loops are relatively benign. Sometimes they're not.\n\nWe know what happens when teachers think students are dumb: They get dumber. In an experiment done before the advent of ethics boards, teachers were given test results that supposedly indicated the IQ and aptitude of students entering their classes. They weren't told, however, that the results had been randomly redistributed among students. After a year, the students who the teachers had been told were bright made big gains in IQ. The students who the teachers had been told were below average had no such improvement.\n\nSo what happens when the Internet thinks you're dumb? Personalization based on perceived IQ isn't such a far-fetched scenario\u2014Google Docs even offers a helpful tool for automatically checking the grade-level of written text. If your education level isn't already available through a tool like Acxiom, it's easy enough for anyone with access to a few e-mails or Facebook posts to infer. Users whose writing indicates college-level literacy might see more articles from the _New Yorker;_ users with only basic writing skills might see more from the _New York Post._\n\nIn a broadcast world, everyone is expected to read or process information at about the same level. In the filter bubble, there's no need for that expectation. On one hand, this could be great\u2014vast groups of people who have given up on reading because the newspaper goes over their heads may finally connect with written content. But without pressure to improve, it's also possible to get stuck in a grade-three world for a long time.\n\n#  **Incidents and Adventures**\n\nIn some cases, letting algorithms make decisions about what we see and what opportunities we're offered gives us fairer results. A computer can be made blind to race and gender in ways that humans usually can't. But that's only if the relevant algorithms are designed with care and acuteness. Otherwise, they're likely to simply reflect the social mores of the culture they're processing\u2014a regression to the social norm.\n\nIn some cases, algorithmic sorting based on personal data can be even _more_ discriminatory than people would be. For example, software that helps companies sift through r\u00e9sum\u00e9s for talent might \"learn\" by looking at which of its recommended employees are actually hired. If nine white candidates in a row are chosen, it might determine that the company isn't interested in hiring black people and exclude them from future searches. \"In many ways,\" writes NYU sociologist Dalton Conley, \"such network-based categorizations are more insidious than the hackneyed groupings based on race, class, gender, religion, or any other demographic characteristic.\" Among programmers, this kind of error has a name. It's called _overfitting_.\n\nThe online movie rental Web site Netflix is powered by an algorithm called CineMatch. To start, it was pretty simple. If I had rented the first movie in the _Lord of the Rings_ trilogy, let's say, Netflix could look up what other movies _Lord of the Rings_ watchers had rented. If many of them had rented _Star Wars,_ it'd be highly likely that I would want to rent it, too.\n\nThis technique is called kNN (k-nearest-neighbor), and using it CineMatch got pretty good at figuring out what movies people wanted to watch based on what movies they'd rented and how many stars (out of five) they'd given the movies they'd seen. By 2006, CineMatch could predict within one star how much a given user would like any movie from Netflix's vast hundred-thousand-film emporium. Already CineMatch was better at making recommendations than most humans. A human video clerk would never think to suggest _Silence of the Lambs_ to a fan of _The Wizard of Oz,_ but CineMatch knew people who liked one usually liked the other.\n\nBut Reed Hastings, Netflix's CEO, wasn't satisfied. \"Right now, we're driving the Model-T version of what's possible,\" he told a reporter in 2006. On October 2, 2006, an announcement went up on the Netflix Web site: \"We're interested, to the tune of $1 million.\" Netflix had posted an enormous swath of data\u2014reviews, rental records, and other information from its user database, scrubbed of anything that would obviously identify a specific user. And now the company was willing to give $1 million to the person or team who beat CineMatch by more than 10 percent. Like the longitude prize, the Netflix Challenge was open to everyone. \"All you need is a PC and some great insight,\" Hastings declared in the _New York Times._\n\nAfter nine months, about eighteen thousand teams from more than 150 countries were competing, using ideas from machine learning, neural networks, collaborative filtering, and data mining. Usually, contestants in high-stakes contests operate in secret. But Netflix encouraged the competing groups to communicate with one another and built a message board where they could coordinate around common obstacles. Read through the message board, and you get a visceral sense of the challenges that bedeviled the contestants during the three-year quest for a better algorithm. Overfitting comes up again and again.\n\nThere are two challenges in building pattern-finding algorithms. One is finding the patterns that are there in all the noise. The other problem is the opposite: _not_ finding patterns in the data that aren't actually really there. The pattern that describes \"1, 2, 3\" could be \"add one to the previous number\" or \"list positive prime numbers from smallest to biggest.\" You don't know for sure until you get more data. And if you leap to conclusions, you're overfitting.\n\nWhere movies are concerned, the dangers of overfitting are relatively small\u2014many analog movie watchers have been led to believe that because they liked _The Godfather_ and _The Godfather: Part II,_ they'll like _The Godfather: Part III._ But the overfitting problem gets to one of the central, irreducible problems of the filter bubble: Overfitting and stereotyping are synonyms.\n\nThe term _stereotyping_ (which in this sense comes from Walter Lippmann, incidentally) is often used to refer to malicious xenophobic patterns that aren't true\u2014\"people of this skin color are less intelligent\" is a classic example. But stereotypes and the negative consequences that flow from them aren't fair to specific people _even if_ they're generally pretty accurate.\n\nMarketers are already exploring the gray area between what can be predicted and what predictions are fair. According to Charlie Stryker, an old hand in the behavioral targeting industry who spoke at the Social Graph Symposium, the U.S. Army has had terrific success using social-graph data to recruit for the military\u2014after all, if six of your Facebook buddies have enlisted, it's likely that you would consider doing so too. Drawing inferences based on what people like you or people linked to you do is pretty good business. And it's not just the army. Banks are beginning to use social data to decide to whom to offer loans: If your friends don't pay on time, it's likely that you'll be a deadbeat too. \"A decision is going to be made on creditworthiness based on the creditworthiness of your friends,\" Stryker said. \"There are applications of this technology that can be very powerful,\" another social targeting entrepreneur told the _Wall Street Journal._ \"Who knows how far we'd take it?\"\n\nPart of what's troubling about this world is that companies aren't required to explain on what basis they're making these decisions. And as a result, you can get judged without knowing it and without being able to appeal. For example, LinkedIn, the social job-hunting site, offers a career trajectory prediction site; by comparing your r\u00e9sum\u00e9 to other peoples' who are in your field but further along, LinkedIn can forecast where you'll be in five years. Engineers at the company hope that soon it'll be able to pinpoint career choices that lead to better outcomes\u2014\"mid-level IT professionals like you who attended Wharton business school made $25,000\/year more than those who didn't.\" As a service to customers, it's pretty useful. But imagine if LinkedIn provided that data to corporate clients to help them weed out people who are forecast to be losers. Because that could happen entirely without your knowledge, you'd never get the chance to argue, to prove the prediction wrong, to have the benefit of the doubt.\n\nIf it seems unfair for banks to discriminate against you because your high school buddy is bad at paying his bills or because you like something that a lot of loan defaulters also like, well, it is. And it points to a basic problem with induction, the logical method by which algorithms use data to make predictions.\n\nPhilosophers have been wrestling with this problem since long before there were computers to induce with. While you can prove the truth of a mathematical proof by arguing it out from first principles, the philosopher David Hume pointed out in 1772 that reality doesn't work that way. As the investment clich\u00e9 has it, past performance is not indicative of future results.\n\nThis raises some big questions for science, which is at its core a method for using data to predict the future. Karl Popper, one of the preeminent philosophers of science, made it his life's mission to try to sort out the problem of induction, as it came to be known. While the optimistic thinkers of the late 1800s looked at the history of science and saw a journey toward truth, Popper preferred to focus on the wreckage along the side of the road\u2014the abundance of failed theories and ideas that were perfectly consistent with the scientific method and yet horribly wrong. After all, the Ptolemaic universe, with the earth in the center and the sun and planets revolving around it, survived an awful lot of mathematical scrutiny and scientific observation.\n\nPopper posed his problem in a slightly different way: Just because you've only ever seen white swans doesn't mean that all swans are white. What you have to look for is the black swan, the counterexample that proves the theory wrong. \"Falsifiability,\" Popper argued, was the key to the search for truth: The purpose of science, for Popper, was to advance the biggest claims for which one could not find any countervailing examples, any black swans. Underlying Popper's view was a deep humility about scientifically induced knowledge\u2014a sense that we're wrong as often as we're right, and we usually don't know when we are.\n\nIt's this humility that many algorithmic prediction methods fail to build in. Sure, they encounter people or behaviors that don't fit the mold from time to time, but these aberrations don't fundamentally compromise their algorithms. After all, the advertisers whose money drives these systems don't need the models to be perfect. They're most interested in hitting demographics, not complex human beings.\n\nWhen you model the weather and predict there's a 70 percent chance of rain, it doesn't affect the rain clouds. It either rains or it doesn't. But when you predict that because my friends are untrustworthy, there's a 70 percent chance that I'll default on my loan, there are consequences if you get me wrong. You're discriminating.\n\nThe best way to avoid overfitting, as Popper suggests, is to try to prove the model wrong and to build algorithms that give the benefit of the doubt. If Netflix shows me a romantic comedy and I like it, it'll show me another one and begin to think of me as a romantic-comedy lover. But if it wants to get a good picture of who I really am, it should be constantly testing the hypothesis by showing me _Blade Runner_ in an attempt to prove it wrong. Otherwise, I end up caught in a local maximum populated by Hugh Grant and Julia Roberts.\n\nThe statistical models that make up the filter bubble write off the outliers. But in human life it's the outliers who make things interesting and give us inspiration. And it's the outliers who are the first signs of change.\n\nOne of the best critiques of algorithmic prediction comes, remarkably, from the late-nineteenth-century Russian novelist Fyodor Dostoyevsky, whose _Notes from Underground_ was a passionate critique of the utopian scientific rationalism of the day. Dostoyevsky looked at the regimented, ordered human life that science promised and predicted a banal future. \"All human actions,\" the novel's unnamed narrator grumbles, \"will then, of course, be tabulated according to these laws, mathematically, like tables of logarithms up to 108,000, and entered in an index ... in which everything will be so clearly calculated and explained that there will be no more incidents or adventures in the world.\"\n\nThe world often follows predictable rules and falls into predictable patterns: Tides rise and fall, eclipses approach and pass; even the weather is more and more predictable. But when this way of thinking is applied to human behavior, it can be dangerous, for the simple reason that our best moments are often the most unpredictable ones. An entirely predictable life isn't worth living. But algorithmic induction can lead to a kind of information determinism, in which our past clickstreams entirely decide our future. If we don't erase our Web histories, in other words, we may be doomed to repeat them.\n**5**\n\n**The Public Is Irrelevant**\n\nThe presence of others who see what we see and hear what we hear assures us of the reality of the world and ourselves.\n\n\u2014Hannah Arendt\n\nIt is an axiom of political science in the United  \nStates that the only way to neutralize the influence  \nof the newspapers is to multiply their number.\n\n\u2014Alexis de Tocqueville\n\n**O** n the night of May 7, 1999, a B-2 stealth bomber left Whiteman Air Force Base in Missouri. The aircraft flew on an easterly course until it reached the city of Belgrade in Serbia, where a civil war was under way. Around midnight local time, the bomber delivered its cargo: four GPSGUIDED bombs, into which had been programmed an address that CIA documents identified as a possible arms warehouse. In fact, the address was the Yugoslavian Chinese Embassy. The building was demolished, and three Chinese diplomats were killed.\n\nThe United States immediately apologized, calling the event an accident. On Chinese state TV, however, an official statement called the bombing a \"barbaric attack and a gross violation of Chinese sovereignty.\" Though President Bill Clinton tried to reach Chinese President Jiang Zemin, Zemin repeatedly rejected his calls; Clinton's videotaped apology to the Chinese people was barred from Chinese media for four days.\n\nAs anti-U.S. riots began to break out in the streets, China's largest newspaper, the _People's Daily_ , created an online chat forum called the Anti-Bombing Forum. Already, in 1999, chat forums were huge in China\u2014much larger than they've ever been in the United States. As _New York Times_ journalist Tom Downey explained a few years later, \"News sites and individual blogs aren't nearly as influential in China, and social networking hasn't really taken off. What remain most vital are the largely anonymous online forums ... that are much more participatory, dynamic, populist and perhaps even democratic than anything on the English-language Internet.\" Tech writer Clive Thompson quotes Shanthi Kalathil, a researcher at the Carnegie Endowment, who says that the Anti-Bombing Forum helped to legitimize the Chinese government's position that the bombing was deliberate among \"an elite, wired section of the population.\" The forum was a form of crowd-sourced propaganda: Rather than just telling Chinese citizens what to think, it lifted the voices of thousands of patriots aligned with the state.\n\nMost of the Western reporting on Chinese information management focuses on censorship: Google's choice to remove, temporarily, search results for \"Tiananmen Square,\" or Microsoft's decision to ban the word \"democracy\" from Chinese blog posts, or the Great Firewall that sits between China and the outside world and sifts through every packet of information that enters or exits the country. Censorship in China is real: There are plenty of words that have been more or less stricken from the public discourse. When Thompson asks whether the popular Alibaba engine would show results for dissident movements, CEO Jack Ma shook his head. \"No! We are a business!\" he said. \"Shareholders want to make money. Shareholders want us to make the customer happy. Meanwhile we do not have any responsibilities saying we should do this or that political thing.\"\n\nIn practice, the firewall is not so hard to circumvent. Corporate virtual private networks\u2014Internet connections encrypted to prevent espionage\u2014operate with impunity. Proxies and firewall workarounds like Tor connect in-country Chinese dissidents with even the most hard-core antigovernment Web sites. But to focus exclusively on the firewall's inability to perfectly block information is to miss the point. China's objective isn't so much to blot out unsavory information as to alter the physics around it\u2014to create friction for problematic information and to route public attention to progovernment forums. While it can't block all of the people from all of the news all of the time, it doesn't need to.\n\n\"What the government cares about,\" _Atlantic_ journalist James Fallows writes, \"is making the quest for information just enough of a nuisance that people generally won't bother.\" The strategy, says Xiao Qiang of the University of California at Berkeley, is \"about social control, human surveillance, peer pressure, and self-censorship.\" Because there's no official list of blocked keywords or forbidden topics published by the government, businesses and individuals censor themselves to avoid a visit from the police. Which sites are available changes daily. And while some bloggers suggest that the system's unreliability is a result of faulty technology (\"the Internet will override attempts to control it!\"), for the government this is a feature, not a bug. James Mulvenon, the head of the Center for Intelligence Research and Analysis, puts it this way: \"There's a randomness to their enforcement, and that creates a sense that they're looking at everything.\"\n\nLest that sensation be too subtle, the Public Security Bureau in Shenzhen, China, developed a more direct approach: Jingjing and Chacha, the cartoon Internet Police. As the director of the initiative told the _China Digital Times,_ he wanted \"to let all Internet users know that the Internet is not a place beyond law [and that] the Internet Police will maintain order in all online behavior.\" Icons of the male-female pair, complete with jaunty flying epaulets and smart black shoes, were placed on all major Web sites in Shenzhen; they even had instant-message addresses so that six police officers could field questions from the online crowds.\n\n\"People are actually quite free to talk about [democracy],\" Google's China point man, Kai-Fu Lee, told Thompson in 2006. \"I don't think they care that much. Hey, U.S. democracy, that's a good form of government. Chinese government, good and stable, that's a good form of government. Whatever, as long as I get to go to my favorite Web site, see my friends, live happily.\" It may not be a coincidence that the Great Firewall stopped blocking pornography recently. \"Maybe they are thinking that if Internet users have some porn to look at, then they won't pay so much attention to political matters,\" Michael Anti, a Beijing-based analyst, told the AP.\n\nWe usually think about censorship as a process by which governments alter facts and content. When the Internet came along, many hoped it would eliminate censorship altogether\u2014the flow of information would simply be too swift and strong for governments to control. \"There's no question China has been trying to crack down on the Internet,\" Bill Clinton told the audience at a March 2000 speech at Johns Hopkins University. \"Good luck! That's sort of like trying to nail Jell-O to the wall.\"\n\nBut in the age of the Internet, it's still possible for governments to manipulate the truth. The process has just taken a different shape: Rather than simply banning certain words or opinions outright, it'll increasingly revolve around second-order censorship\u2014the manipulation of curation, context, and the flow of information and attention. And because the filter bubble is primarily controlled by a few centralized companies, it's not as difficult to adjust this flow on an individual-by-individual basis as you might think. Rather than decentralizing power, as its early proponents predicted, in some ways the Internet is concentrating it.\n\n#  **Lords of the Cloud**\n\nTo get a sense of how personalization might be used for political ends, I talked to a man named John Rendon.\n\nRendon affably describes himself as an \"information warrior and perception manager.\" From the Rendon Group's headquarters in Washington, D.C.'s, Dupont Circle, he provides those services to dozens of U.S. agencies and foreign governments. When American troops rolled into Kuwait City during the first Iraq war, television cameras captured hundreds of Kuwaitis joyfully waving American flags. \"Did you ever stop to wonder,\" he asked an audience later, \"how the people of Kuwait City, after being held hostage for seven long and painful months, were able to get handheld American flags? And for that matter, the flags of other coalition countries? Well, you now know the answer. That was one of my jobs.\"\n\nMuch of Rendon's work is confidential\u2014he enjoys a level of beyond\u2013Top Secret clearance that even high-level intelligence analysts sometimes fail to get. His role in George W. Bush\u2013era pro-U.S. propaganda in Iraq is unclear: While some sources claim he was a central figure in the effort, Rendon denies any involvement. But his dream is quite clear: Rendon wants to see a world where television \"can drive the policy process,\" where \"border patrols [are] replaced by beaming patrols,\" and where \"you can win without fighting.\"\n\nGiven all that, I was a bit surprised when the first weapon he referred me to was a very quotidian one: a thesaurus. The key to changing public opinion, Rendon said, is finding different ways to say the same thing. He described a matrix, with extreme language or opinion on one side and mild opinion on the other. By using sentiment analysis to figure out how people in a country felt about an event\u2014say, a new arms deal with the United States\u2014and identify the right synonyms to move them toward approval, you could \"gradually nudge a debate.\" \"It's a lot easier to be close to what reality is\" and push it in the right direction, he said, than to make up a new reality entirely.\n\nRendon had seen me talk about personalization at an event we both attended. Filter bubbles, he told me, provided new ways of managing perceptions. \"It begins with getting inside the algorithm. If you could find a way to load your content up so that only your content gets pulled by the stalking algorithm, then you'd have a better chance of shaping belief sets,\" he said. In fact, he suggested, if we looked in the right places, we might be able to see traces of this kind of thing happening now\u2014sentiment being algorithmically shifted over time.\n\nBut if the filter bubble might make shifting perspectives easier in a future Iraq or Panama, Rendon was clearly concerned about the impact of self-sorting and personalized filtering for democracy at home. \"If I'm taking a photo of a tree,\" he said, \"I need to know what season we're in. Every season it looks different. It could be dying, or just losing its leaves in autumn.\" To make good decisions, context is crucial\u2014that's why the military is so focused on what they call \"360-degree situational awareness.\" In the filter bubble, you don't get 360 degrees\u2014and you might not get more than one.\n\nI returned to the question about using algorithms to shift sentiment. \"How does someone game the system when it's all about self-generated, self-reinforcing information flows? I have to think about it more,\" Rendon said, \"But I think I know how I'd do it.\"\n\n\"How?\" I asked.\n\nHe paused, then chuckled: \"Nice try.\" He'd already said too much.\n\nThe campaign of propaganda that Walter Lippmann railed against in World War I was a massive undertaking: To \"goose-step the truth,\" hundreds of newspapers nationwide had to be brought onboard. Now that every blogger is a publisher, the task seems nearly impossible. In 2010, Google chief Eric Schmidt echoed this sentiment, arguing in the journal _Foreign Affairs_ that the Internet eclipses intermediaries and governments and empowers individuals to \"consume, distribute, and create their own content without government control.\"\n\nIt's a convenient view for Google\u2014if intermediaries are losing power, then the company's merely a minor player in a much larger drama. But in practice, a great majority of online content reaches people through a small number of Web sites\u2014Google foremost among them. These big companies represent new loci of power. And while their multinational character makes them resistant to some forms of regulation, they can also offer one-stop shopping for governments seeking to influence information flows.\n\nAs long as a database exists, it's potentially accessible by the state. That's why gun rights activists talk a lot about Alfred Flatow. Flatow was an Olympic gymnast and German Jew who in 1932 registered his gun in accordance with the laws of the waning Weimar Republic. In 1938, German police came to his door. They'd searched through the record, and in preparation for the Holocaust, they were rounding up Jews with handguns. Flatow was killed in a concentration camp in 1942.\n\nFor National Rifle Association members, the story is a powerful cautionary tale about the dangers of a national gun registry. As a result of Flatow's story and thousands like it, the NRA has successfully blocked a national gun registry for decades. If a fascistic anti-Semitic regime came into power in the United States, it'd be hard put to identify gun-holding Jews using its own databases.\n\nBut the NRA's focus may have been too narrow. Fascists aren't known for carefully following the letter of the law regarding extragovernmental databases. And using the data that credit card companies use\u2014or for that matter, building models based on the thousands of data points Acxiom tracks\u2014it'd be a simple matter to predict with significant accuracy who has a gun and who does not.\n\nEven if you aren't a gun advocate, the story is worth paying attention to. The dynamics of personalization shift power into the hands of a few major corporate actors. And this consolidation of huge masses of data offers governments (even democratic ones) more potential power than ever.\n\nRather than housing their Web sites and databases internally, many businesses and start-ups now run on virtual computers in vast server farms managed by other companies. The enormous pool of computing power and storage these networked machines create is known as the _cloud,_ and it allows clients much greater flexibility. If your business runs in the cloud, you don't need to buy more hardware when your processing demands expand: You just rent a greater portion of the cloud. Amazon Web Services, one of the major players in the space, hosts thousands of Web sites and Web servers and undoubtedly stores the personal data of millions. On one hand, the cloud gives every kid in his or her basement access to nearly unlimited computing power to quickly scale up a new online service. On the other, as Clive Thompson pointed out to me, the cloud \"is actually just a handful of companies.\" When Amazon booted the activist Web site WikiLeaks off its servers under political pressure in 2010, the site immediately collapsed\u2014there was nowhere to go.\n\nPersonal data stored in the cloud is also actually much easier for the government to search than information on a home computer. The FBI needs a warrant from a judge to search your laptop. But if you use Yahoo or Gmail or Hotmail for your e-mail, you \"lose your constitutional protections immediately,\" according to a lawyer for the Electronic Freedom Foundation. The FBI can just ask the company for the information\u2014no judicial paperwork needed, no permission required\u2014as long as it can argue later that it's part of an \"emergency.\" \"The cops will love this,\" says privacy advocate Robert Gellman about cloud computing. \"They can go to a single place and get everybody's documents.\"\n\nBecause of the economies of scale in data, the cloud giants are increasingly powerful. And because they're so susceptible to regulation, these companies have a vested interest in keeping government entities happy. When the Justice Department requested billions of search records from AOL, Yahoo, and MSN in 2006, the three companies quickly complied. (Google, to its credit, opted to fight the request.) Stephen Arnold, an IT expert who worked at consulting firm Booz Allen Hamilton, says that Google at one point housed three officers of \"an unnamed intelligence agency\" at its headquarters in Mountain View. And Google and the CIA have invested together in a firm called Recorded Future, which focuses on using data connections to predict future real-world events.\n\nEven if the consolidation of this data-power doesn't result in more governmental control, it's worrisome on its own terms.\n\nOne of the defining traits of the new personal information environment is that it's asymmetrical. As Jonathan Zittrain argues in _The Future of the Internet\u2014and How to Stop It,_ \"nowadays, an individual must increasingly give information about himself to large and relatively faceless institutions, for handling and use by strangers\u2014unknown, unseen, and all too frequently, unresponsive.\"\n\nIn a small town or an apartment building with paper-thin walls, what I know about you is roughly the same as what you know about me. That's a basis for a social contract, in which we'll deliberately ignore some of what we know. The new privacyless world does away with that contract. I can know a lot about you without your knowing I know. \"There's an implicit bargain in our behavior,\" search expert John Battelle told me, \"that we haven't done the math on.\"\n\nIf Sir Francis Bacon is right that \"knowledge is power,\" privacy proponent Viktor Mayer-Schonberger writes that what we're witnessing now is nothing less than a \"redistribution of information power from the powerless to the powerful.\" It'd be one thing if we all knew everything about each other. It's another when centralized entities know a lot more about us than we know about each other\u2014and sometimes, more than we know about ourselves. If knowledge is power, then asymmetries in knowledge are asymmetries in power.\n\nGoogle's famous \"Don't be evil\" motto is presumably intended to allay some of these concerns. I once explained to a Google search engineer that while I didn't think the company was currently evil, it seemed to have at its fingertips everything it needed to do evil if it wished. He smiled broadly. \"Right,\" he said. \"We're not evil. We try really hard not to be evil. But if we wanted to, man, could we ever!\"\n\n#  **Friendly World Syndrome**\n\nMost governments and corporations have used the new power that personal data and personalization offer fairly cautiously so far\u2014China, Iran, and other oppressive regimes being the obvious exceptions. But even putting aside intentional manipulation, the rise of filtering has a number of unintended yet serious consequences for democracies. In the filter bubble, the public sphere\u2014the realm in which common problems are identified and addressed\u2014is just less relevant.\n\nFor one thing, there's the problem of the friendly world. Communications researcher George Gerbner was one of the first theorists to look into how media affect our political beliefs, and in the mid-1970s, he spent a lot of time thinking about shows like _Starsky and Hutch_. It was a pretty silly program, filled with the shared clich\u00e9s of seventies cop TV\u2014the bushy moustaches, the twanging soundtracks, the simplistic goodversus-evil plots. And it was hardly the only one\u2014for every _Charlie's Angels_ or _Hawaii Five-O_ that earned a place in cultural memory, there are dozens of shows, like _The Rockford Files, Get Christie Love,_ and _Adam-12,_ that are unlikely to be resuscitated for ironic twenty-first-century remakes.\n\nBut Gerbner, a World War II veteran\u2013turned\u2013communications theorist who became dean of the Annenberg School of Communication, took these shows seriously. Starting in 1969, he began a systematic study of the way TV programming affects how we think about the world. As it turned out, the _Starsky and Hutch_ effect was significant. When you asked TV watchers to estimate the percentage of the adult workforce that was made up of cops, they vastly overguessed relative to non\u2013TV watchers with the same education and demographic background. Even more troubling, kids who saw a lot of TV violence were much more likely to be worried about real-world violence.\n\nGerbner called this the mean world syndrome: If you grow up in a home where there's more than, say, three hours of television per day, for all practical purposes, you live in a meaner world\u2014and act accordingly\u2014than your next-door neighbor who lives in the same place but watches less television. \"You know, who tells the stories of a culture really governs human behavior,\" Gerbner later said.\n\nGerbner died in 2005, but he lived long enough to see the Internet begin to break that stranglehold. It must have been a relief: Although our online cultural storytellers are still quite consolidated, the Internet at least offers more choice. If you want to get your local news from a blogger rather than a local TV station that trumpets crime rates to get ratings, you can.\n\nBut if the mean world syndrome poses less of a risk these days, there's a new problem on the horizon: We may now face what persuasion-profiling theorist Dean Eckles calls a friendly world syndrome, in which some of the biggest and most important problems fail to reach our view at all.\n\nWhile the mean world on television arises from a cynical \"if it bleeds, it leads\" approach to programming, the friendly world generated by algorithmic filtering may not be as intentional. According to Facebook engineer Andrew Bosworth, the team that developed the Like button originally considered a number of options\u2014from stars to a thumbs up sign (but in Iran and Thailand, it's an obscene gesture). For a month in the summer of 2007, the button was known as the Awesome button. Eventually, however, the Facebook team gravitated toward Like, which is more universal.\n\nThat Facebook chose Like instead of, say, Important is a small design decision with far-reaching consequences: The stories that get the most attention on Facebook are the stories that get the most Likes, and the stories that get the most Likes are, well, more likable.\n\nFacebook is hardly the only filtering service that will tend toward an antiseptically friendly world. As Eckles pointed out to me, even Twitter, which has a reputation for putting filtering in the hands of its users, has this tendency. Twitter users see most of the tweets of the folks they follow, but if my friend is having an exchange with someone I don't follow, it doesn't show up. The intent is entirely innocuous: Twitter is trying not to inundate me with conversations I'm not interested in. But the result is that conversations between my friends (who will tend to be like me) are overrepresented, while conversations that could introduce me to new ideas are obscured.\n\nOf course, _friendly_ doesn't describe all of the stories that pierce the filter bubble and shape our sense of the political world. As a progressive political news junkie, I get plenty of news about Sarah Palin and Glenn Beck. The valence of this news, however, is very predictable: People are posting it to signal their dismay with Beck's and Palin's rhetoric and to build a sense of solidarity with their friends, who presumably feel the same way. It's rare that my assumptions about the world are shaken by what I see in my news feed.\n\nEmotional stories are the ones that generally thrive in the filter bubble. The Wharton School study on the _New York Times_ 's Most Forwarded List, discussed in chapter 2, found that stories that aroused strong feelings\u2014awe, anxiety, anger, happiness\u2014were much more likely to be shared. If television gives us a \"mean world,\" filter bubbles give us an \"emotional world.\"\n\nOne of the troubling side effects of the friendly world syndrome is that some important public problems will disappear. Few people seek out information about homelessness, or share it, for that matter. In general, dry, complex, slow-moving problems\u2014a lot of the truly significant issues\u2014won't make the cut. And while we used to rely on human editors to spotlight these crucial problems, their influence is now waning.\n\nEven advertising isn't necessarily a foolproof way of alerting people to public problems, as the environmental group Oceana found out. In 2004, Oceana was running a campaign urging Royal Caribbean to stop dumping its raw sewage into the sea; as part of the campaign, it took out a Google ad that said \"Help us protect the world's oceans. Join the fight!\" After two days, Google pulled the ads, citing \"language advocating against the cruise line industry\" that was in violation of their general guidelines about taste. Apparently, advertisers that implicated corporations in public issues weren't welcome.\n\nThe filter bubble will often block out the things in our society that are important but complex or unpleasant. It renders them invisible. And it's not just the issues that disappear. Increasingly, it's the whole political process.\n\n#  **The Invisible Campaign**\n\nWhen George W. Bush came out of the 2000 election with far fewer votes than Karl Rove expected, Rove set in motion a series of experiments in microtargeted media in Georgia\u2014looking at a wide range of consumer data (\"Do you prefer beer or wine?\") to try to predict voting behavior and identify who was persuadable and who could be easily motivated to get to the polls. Though the findings are still secret, legend has it that the methods Rove discovered were at the heart of the GOP's successful get-out-the-vote strategy in 2002 and 2004.\n\nOn the left, Catalist, a firm staffed by former Amazon engineers, has built a database of hundreds of millions of voter profiles. For a fee, organizing and activist groups (including MoveOn) query it to help determine which doors to knock on and to whom to run ads. And that's just the start. In a memo for fellow progressives, Mark Steitz, one of the primary Democratic data gurus, recently wrote that \"targeting too often returns to a bombing metaphor\u2014dropping message from planes. Yet the best data tools help build relationships based on observed contacts with people. Someone at the door finds out someone is interested in education; we get back to that person and others like him or her with more information. Amazon's recommendation engine is the direction we need to head.\" The trend is clear: We're moving from swing states to swing people.\n\nConsider this scenario: It's 2016, and the race is on for the presidency of the United States. Or is it?\n\nIt depends on who you are, really. If the data says you vote frequently and that you may have been a swing voter in the past, the race is a maelstrom. You're besieged with ads, calls, and invitations from friends. If you vote intermittently, you get a lot of encouragement to get out to the polls.\n\nBut let's say you're more like an average American. You usually vote for candidates from one party. To the data crunchers from the opposing party, you don't look particularly persuadable. And because you vote in presidential elections pretty regularly, you're also not a target for \"get out the vote\" calls from your own. Though you make it to the polls as a matter of civic duty, you're not that actively interested in politics. You're more interested in, say, soccer and robots and curing cancer and what's going on in the town where you live. Your personalized news feeds reflect those interests, not the news from the latest campaign stop.\n\nIn a filtered world, with candidates microtargeting the few persuadables, would you know that the campaign was happening at all?\n\nEven if you visit a site that aims to cover the race for a general audience, it'll be difficult to tell what's going on. What is the campaign about? There is no general, top-line message, because the candidates aren't appealing to a general public. Instead, there are a series of message fragments designed to penetrate personalized filters.\n\nGoogle is preparing for this future. Even in 2010, it staffed a round-the-clock \"war room\" for political advertising, aiming to be able to quickly sign off on and activate new ads even in the wee hours of October nights. Yahoo is conducting a series of experiments to determine how to match the publicly available list of who voted in each district with the click signals and Web history data it picks up on its site. And data-aggregation firms like Rapleaf in San Francisco are trying to correlate Facebook social graph information with voting behavior\u2014so that they can show you the political ad that best works for you based on the responses of your friends.\n\nThe impulse to talk to voters about the things they're actually interested in isn't a bad one\u2014it'd be great if mere mention of the word _politics_ didn't cause so many eyes to glaze over. And certainly the Internet has unleashed the coordinated energy of a whole new generation of activists\u2014it's easier than ever to find people who share your political passions. But while it's easier than ever to bring a group of people together, as personalization advances it'll become harder for any given group to reach a broad audience. In some ways, personalization poses a threat to public life itself.\n\nBecause the state of the art in political advertising is half a decade behind the state of the art in commercial advertising, most of this change is still to come. But for starters, filter-bubble politics could effectively make even more of us into single issue voters. Like personalized media, personalized advertising is a two-way street: I may see an ad about, say, preserving the environment because I drive a Prius, but seeing the ad also makes me care more about preserving the environment. And if a congressional campaign can determine that this is the issue on which it's most likely to persuade me, why bother filling me in on all of the other issues?\n\nIn theory, market dynamics will continue to encourage campaigns to reach out to nonvoters. But an additional complication is that more and more companies are also allowing users to remove advertisements they don't like. For Facebook and Google, after all, seeing ads for ideas or services you don't like is a failure. Because people tend to dislike ads containing messages they disagree with, this creates even less space for persuasion. \"If a certain number of anti-Mitt Republicans saw an ad for Mitt Romney and clicked 'offensive, etc.,' \" writes Vincent Harris, a Republican political consultant, \"they could block ALL of Mitt Romney's ads from being shown, and kill the entire online advertising campaign regardless of how much money the Romney campaign wanted to spend on Facebook.\" Forcing candidates to come up with more palatable ways to make their points might result in more thoughtful ads\u2014but it also might also drive up the cost of these ads, making it too costly for campaigns to ever engage the other side.\n\nThe most serious political problem posed by filter bubbles is that they make it increasingly difficult to have a public argument. As the number of different segments and messages increases, it becomes harder and harder for the campaigns to track who's saying what to whom. TV is a piece of cake to monitor in comparison\u2014you can just record the opposition's ads in each cable district. But how does a campaign know what its opponent is saying if ads are only targeted to white Jewish men between twenty-eight and thirty-four who have expressed a fondness for U2 on Facebook and who donated to Barack Obama's campaign?\n\nWhen a conservative political group called Americans for Job Security ran ads in 2010 falsely accusing Representative Pete Hoekstra of refusing to sign a no-new-taxes pledge, he was able to show TV stations the signed pledge and have the ads pulled off the air. It's not great to have TV station owners be the sole arbitrators of truth\u2014I've spent a fair amount of time arguing with them myself\u2014but it is better to have some bar for truthfulness than none at all. It's unclear that companies like Google have the resources or the interest to play truthfulness referee on the hundreds of thousands of different ads that will run in election cycles to come.\n\nAs personal political targeting increases, not only will it be more difficult for campaigns to respond to and fact-check each other, it'll be more challenging for journalists as well. We may see an environment where the most important ads aren't easily accessible to journalists and bloggers\u2014it's easy enough for campaigns to exclude them from their targeting and difficult for reporters to fabricate the profile of a genuine swing voter. (One simple solution to this problem would simply be to require campaigns to immediately disclose all of their online advertising materials and to whom each ad is targeted. Right now, the former is spotty and the latter is undisclosed.)\n\nIt's not that political TV ads are so great. For the most part, they're shrill, unpleasant, and unlikable. If we could, most of us would tune them out. But in the broadcast era, they did at least three useful things. They reminded people that there was an election in the first place. They established for everyone what the candidates valued, what their campaigns were about, what their arguments were: the parameters of the debate. And they provided a basis for a common conversation about the political decision we faced\u2014something you could talk about in the line at the supermarket.\n\nFor all of their faults, political campaigns are one of the primary places where we debate our ideas about our nation. Does America condone torture? Are we a nation of social Darwinists or of social welfare? Who are our heroes, and who are our villains? In the broadcast era, campaigns have helped to delineate the answers to those questions. But they may not do so for very much longer.\n\n#  **Fragmentation**\n\nThe aim of modern political marketing, consumer trends expert J. Walker Smith tells Bill Bishop in _The Big Sort,_ is to \"drive customer loyalty\u2014and in marketing terms, drive the average transaction size or improve the likelihood that a registered Republican will get out and vote Republican. That's a business philosophy applied to politics that I think is really dangerous, because it's not about trying to form a consensus, to get people to think about the greater good.\"\n\nIn part, this approach to politics is on the rise for the same reason the filter bubble is: Personalized outreach gives better bang for the political buck. But it's also a natural outcome of a well-documented shift in how people in industrialized countries think about what's important. When people don't have to worry about having their basic needs met, they care a lot more about having products and leaders that represent who they are.\n\nProfessor Ron Inglehart calls this trend postmaterialism, and it's a result of the basic premise, he writes, that \"you place the greatest subjective value on the things in short supply.\" In surveys spanning forty years and eighty countries, people who were raised in affluence\u2014who never had to worry about their physical survival\u2014behaved in ways strikingly different from those of their hungry parents. \"We can even specify,\" Inglehart writes in _Modernization and Postmodernization,_ \"with far better than random success, what issues are likely to be most salient in the politics of the respective types of societies.\"\n\nWhile there are still significant differences from country to country, postmaterialists share some important traits. They're less reverent about authority and traditional institutions\u2014the appeal of authoritarian strongmen appears to be connected to a basic fear for survival. They're more tolerant of difference: One especially striking chart shows a strong correlation between level of life satisfaction and comfort with living next door to someone who's gay. And while earlier generations emphasize financial achievement and order, postmaterialists value self-expression and \"being yourself.\"\n\nSomewhat confusingly, postmaterialism doesn't mean anticonsumption. Actually, the phenomenon is at the bedrock of our current consumer culture: Whereas we once bought things because we needed them to survive, now we mostly buy things as a means of self-expression. And the same dynamics hold for political leadership: Increasingly, voters evaluate candidates on whether they represent an aspirational version of themselves.\n\nThe result is what marketers call brand fragmentation. When brands were primarily about validating the quality of a product\u2014\"Dove soap is pure and made of the best ingredients\"\u2014advertisements focused more on the basic value proposition. But when brands became vehicles for expressing identity, they needed to speak more intimately to different groups of people with divergent identities they wanted express. And as a result, they started to splinter. Which is why what's happened to Pabst Blue Ribbon beer is a good way of understanding the challenges faced by Barack Obama.\n\nIn the early 2000s, Pabst was struggling financially. It had maxed out among the white rural population that formed the core of its customer base, and it was selling less than 1 million barrels of beer a year, down from 20 million in 1970. If Pabst wanted to sell more beer, it had to look elsewhere, and Neal Stewart, a midlevel marketing manager, did. Stewart went to Postland, Oregon, where Pabst numbers were surprisingly strong and an ironic nostalgia for white working-class culture (remember trucker hats?) was widespread. If Pabst couldn't get people to drink its watery brew sincerely, Stewart figured, maybe they could get people to drink it ironically. Pabst began to sponsor hipster events\u2014gallery openings, bike messenger races, snowboarding competitions, and the like. Within a year, sales were way up\u2014which is why, if you walk into a bar in certain Brooklyn neighborhoods, Pabst is more likely to be available than other low-end American beers.\n\nThat's not the only excursion in reinvention that Pabst did. In China, where it is branded a \"world-famous spirit,\" Pabst has made itself into a luxury beverage for the cosmopolitan elite. Advertisements compare it to \"Scotch whisky, French brandy, Bordeaux wine,\" and present it in a fluted champagne glass atop a wooden cask. A bottle runs about $44 in U.S. currency.\n\nWhat's interesting about the Pabst story is that it's not rebranding of the typical sort, in which a product aimed at one group is \"repositioned\" to appeal to another. Plenty of white working-class men still drink Pabst sincerely, an affirmation of down-home culture. Urban hipsters drink it with a wink. And wealthy Chinese yuppies drink it as a champagne substitute and a signifier of conspicuous consumption. The same beverage means very different things to different people.\n\nDriven by the centrifugal pull of different market segments\u2014each of which wants products that represent its identity\u2014political leadership is fragmenting in much the same way as PBR. Much has been made of Barack Obama's chameleonic political style. \"I serve as a blank screen,\" he wrote in _The Audacity of Hope_ in 2006, \"on which people of vastly different political stripes project their own views.\" Part of that is a result of Obama's intrinsic political versatility. But it's also a plus in an age of fragmentation.\n\n(To be sure, the Internet can also facilitate consolidation, as Obama learned when his comment about people \"clinging to guns and religion\" to donors in San Francisco was reported by the _Huffington Post_ and became a top campaign talking point against him. At the same time, Williamsburg hipsters who read the right blogs can learn about Pabst's Chinese marketing scheme. But while this makes fragmentation a more perilous process and cuts against authenticity, it doesn't fundamentally change the calculus. It just makes it more of an imperative to target well.)\n\nThe downside of this fragmentation, as Obama has learned, is that it is harder to lead. Acting different with different political constituencies isn't new\u2014in fact, it's probably about as old as politics itself. But the overlap\u2014content that remains constant between all of those constituencies\u2014is shrinking dramatically. You can stand for lots of different kinds of people or stand for something, but doing both is harder every day.\n\nPersonalization is both a cause and an effect of the brand fragmentation process. The filter bubble wouldn't be so appealing if it didn't play to our postmaterial desire to maximize self-expression. But once we're in it, the process of matching who we are to content streams can lead to the erosion of common experience, and it can stretch political leadership to the breaking point.\n\n#  **Discourse and Democracy**\n\nThe good news about postmaterial politics is that as countries become wealthier, they'll likely become more tolerant, and their citizens will be more self-expressive. But there's a dark side to it too. Ted Nordhaus, a student of Inglehart's who focuses on postmaterialism in the environmental movement, told me that \"the shadow that comes with postmaterialism is profound self-involvement.... We lose all perspective on the collective endeavors that have made the extraordinary lives we live possible.\" In a postmaterial world where your highest task is to express yourself, the public infrastructure that supports this kind of expression falls out of the picture. But while we can lose sight of our shared problems, they don't lose sight of us.\n\nA few times a year when I was growing up, the nine-hundred-person hamlet of Lincolnville, Maine, held a town meeting. This was my first impression of democracy: A few hundred residents crammed into the grade school auditorium or basement to discuss school additions, speed limits, zoning regulations, and hunting ordinances. In the aisle between the rows of gray metal folding chairs was a microphone on a stand, where people would line up to say their piece.\n\nIt was hardly a perfect system: Some speakers droned on; others were shouted down. But it gave all of us a sense of the kinds of people that made up our community that we wouldn't have gotten anywhere else. If the discussion was about encouraging more businesses along the coast, you'd hear from the wealthy summer vacationers who enjoyed their peace and quiet, the back-to-the-land hippies with antidevelopment sentiments, and the families who'd lived in rural poverty for generations and saw the influx as a way up and out. The conversation went back and forth, sometimes closing toward consensus, sometimes fragmenting into debate, but usually resulting in a decision about what to do next.\n\nI always liked how those town meetings worked. But it wasn't until I read _On Dialogue_ that I fully understood what they accomplished.\n\nBorn to Hungarian and Lithuanian Jewish furniture store owners in Wilkes-Barre, Pennsylvania, David Bohm came from humble roots. But when he arrived at the University of California\u2013Berkeley, he quickly fell in with a small group of theoretical physicists, under the direction of Robert Oppenheimer, who were racing to build the atomic bomb. By the time he died at seventy-two in October 1992, many of his colleagues would remember Bohm as one of the great physicists of the twentieth century.\n\nBut if quantum math was his vocation, there was another matter that took up much of Bohm's time. Bohm was preoccupied with the problems created by advanced civilization, especially the possibility of nuclear war. \"Technology keeps on advancing with greater and greater power, either for good or for destruction,\" he wrote. \"What is the source of all this trouble? I'm saying that the source is basically in thought.\" For Bohm, the solution became clear: It was dialogue. In 1992, one of his definitive texts on the subject was published.\n\nTo communicate, Bohm wrote, literally means to make something common. And while sometimes this process of making common involves simply sharing a piece of data with a group, more often it involves the group's coming together to create a new, common meaning. \"In dialogue,\" he writes, \"people are participants in a pool of common meaning.\"\n\nBohm wasn't the first theorist to see the democratic potential of dialogue. Jurgen Habermas, the dean of media theory for much of the twentieth century, had a similar view. For both, dialogue was special because it provided a way for a group of people to democratically create their culture and to calibrate their ideas in the world. In a way, you couldn't have a functioning democracy without it.\n\nBohm saw an additional reason why dialogue was useful: It provided people with a way of getting a sense of the whole shape of a complex system, even the parts that they didn't directly participate in. Our tendency, Bohm says, is to rip apart and fragment ideas and conversations into bits that have no relation to the whole. He used the example of a watch that has been shattered: Unlike the parts that made up the watch previously, the pieces have no relation to the watch as a whole. They're just little bits of glass and metal.\n\nIt's this quality that made the Lincolnville town meetings something special. Even if the group couldn't always agree on where to go, the process helped to develop a shared map for the terrain. The parts understood our relationship to the whole. And that, in turn, makes democratic governance possible.\n\nThe town meetings had another benefit: They equipped us to deal more handily with the problems that did emerge. In the science of social mapping, the definition of a community is a set of nodes that are densely interconnected\u2014my friends form a community if they all don't know just me but also have independent relationships with one another. Communication builds stronger community.\n\nUltimately, democracy works only if we citizens are capable of thinking beyond our narrow self-interest. But to do so, we need a shared view of the world we cohabit. We need to come into contact with other peoples' lives and needs and desires. The filter bubble pushes us in the opposite direction\u2014it creates the impression that our narrow self-interest is all that exists. And while this is great for getting people to shop online, it's not great for getting people to make better decisions together.\n\n\"The prime difficulty\" of democracy, John Dewey wrote, \"is that of discovering the means by which a scattered, mobile, and manifold public may so recognize itself as to define and express its interests.\" In the early days of the Internet, this was one of the medium's great hopes\u2014that it would finally offer a medium whereby whole towns\u2014and indeed countries\u2014could co-create their culture through discourse. Personalization has given us something very different: a public sphere sorted and manipulated by algorithms, fragmented by design, and hostile to dialogue.\n\nWhich begs an important question: Why would the engineers who designed these systems want to build them this way?\n**6**\n\n**Hello, World!**\n\nSOCRATES: Or again, in a ship, if a man having the power to do what he likes, has no intelligence or skill in navigation [\u03b1\u03c1\u03b5\u03c4\u03b7\u03c2 \u03ba\u03c5\u03b2\u03b5\u03c1\u03bd\u03b7\u03c4\u03b9\u03ba\u03b7\u03c2, _aret\u0113s kybern\u0113tik\u0113s_ ], do you see what will happen to him and to his fellow-sailors?\n\n\u2014 _Plato_ , _First Alcibiades_ , the earliest known use of the word _cybernetics_\n\n**I** t's the first fragment of code in the code book, the thing every aspiring programmer learns on day one. In the C++ programming language, it looks like this:\n\nvoid main()   \n{   \ncout << \"Hello, World!\" <<   \nendl;   \n}\n\nAlthough the code differs from language to language, the result is the same: a single line of text against a stark white screen:\n\n _Hello, World!_\n\nA god's greeting to his invention\u2014or perhaps an invention's greeting to its god. The delight you experience is electric\u2014the current of creation, running through your fingers into the keypad, into the machine, and back out into the world. _It's alive!_\n\nThat every programmer's career begins with \"Hello, World!\" is not a coincidence. It's the power to create new universes, which is what often draws people to code in the first place. Type in a few lines, or a few thousand, strike a key, and something seems to come to life on your screen\u2014a new space unfolds, a new engine roars. If you're clever enough, you can make and manipulate anything you can imagine.\n\n\"We are as Gods,\" wrote futurist Stewart Brand on the cover of his _Whole Earth Catalog_ in 1968, \"and we might as well get good at it.\" Brand's catalog, which sprang out of the back-to-the-land movement, was a favorite among California's emerging class of programmers and computer enthusiasts. In Brand's view, tools and technologies turned people, normally at the mercy of their environments, into gods in control of them. And the computer was a tool that could become any tool at all.\n\nBrand's impact on the culture of Silicon Valley and geekdom is hard to overestimate\u2014though he wasn't a programmer himself, his vision shaped the Silicon Valley worldview. As Fred Turner details in the fascinating _From Counterculture to Cyberculture,_ Brand and his cadre of do-it-yourself futurists were disaffected hippies\u2014social revolutionaries who were uncomfortable with the communes sprouting up in Haight-Ashbury. Rather than seeking to build a new world through political change, which required wading through the messiness of compromise and group decision making, they set out to build a world on their own.\n\nIn _Hackers,_ his groundbreaking history of the rise of engineering culture, Steve Levy points out that this ideal spread from the programmers themselves to the users \"each time some user flicked the machine on, and the screen came alive with words, thoughts, pictures, and sometimes elaborate worlds built out of air\u2014those computer programs which could make any man (or woman) a god.\" (In the era described by Levy's book, the term _hacker_ didn't have the transgressive, lawbreaking connotations it acquired later.)\n\nThe God impulse is at the root of many creative professions: Artists conjure up color-flecked landscapes, novelists build whole societies on paper. But it's always clear that these are creations: A painting doesn't talk back. A program can, and the illusion that it's \"real\" is powerful. Eliza, one of the first rudimentary AI programs, was programmed with a battery of therapistlike questions and some basic contextual cues. But students spent hours talking to it about their deepest problems: \"I'm having some troubles with my family,\" a student might write, and Eliza would immediately respond, \"Tell me more about your family.\"\n\nEspecially for people who've been socially ostracized due to quirks or brains or both, there are at least two strong draws to the world-building impulse. When social life is miserable or oppressive, escapism is a reasonable response\u2014it's probably not coincidental that role-playing games, sci-fi and fantasy literature, and programming often go together.\n\nThe infinitely expandable universe of code provides a second benefit: complete power over your domain. \"We all fantasize about living without rules,\" says Siva Vaidyanathan. \"We imagine the Adam Sandler movie where you can move around and take people's clothes off. If you don't think of reciprocity as one of the beautiful and rewarding things about being a human being, you wish for a place or a way of acting without consequence.\" When the rules of high school social life seem arbitrary and oppressive, the allure of making your own rules is pretty powerful.\n\nThis approach works pretty well as long as you're the sole denizen of your creation. But like the God of Genesis, coders quickly get lonely. They build portals into their homespun worlds, allowing others to enter. And that's where things get complicated: On the one hand, the more inhabitants in the world you've built, the more power you have. But on the other hand, the citizens can get uppity. \"The programmer wants to set up some rules, to either a game or a system, and then let it run without interference from anything,\" says Douglas Rush-koff, an early cyberbooster-turned-cyberpragmatist. \"If you have a program that needs a minder to come in and help it run, then it's not a very good program, is it? It's supposed to just run.\"\n\nCoders sometimes harbor God impulses; they sometimes even have aspirations to revolutionize society. But they almost never aspire to be politicians. \"While programming is considered a transparent, neutral, highly controllable realm ... where production results in immediate gratification and something useful,\" writes NYU anthropologist Gabriella Coleman, \"politics tends to be seen by programmers as buggy, mediated, tainted action clouded by ideology that is not productive of much of anything.\" There's some merit to that view, of course. But for programmers to shun politics completely is a problem\u2014because increasingly, given the disputes that inevitably arise when people come together, the most powerful ones will be required to adjudicate and to govern.\n\nBefore we get to how this blind spot affects our lives, though, it's worth looking at how engineers think.\n\n#  **The Empire of Clever**\n\nImagine that you're a smart high school student on the low end of the social totem pole. You're alienated from adult authority, but unlike many teenagers, you're also alienated from the power structures of your peers\u2014an existence that can feel lonely and peripheral. Systems and equations are intuitive, but people aren't\u2014social signals are confusing and messy, difficult to interpret.\n\nThen you discover code. You may be powerless at the lunch table, but code gives you power over an infinitely malleable world and opens the door to a symbolic system that's perfectly clear and ordered. The jostling for position and status fades away. The nagging parental voices disappear. There's just a clean, white page for you to fill, an opportunity to build a better place, a home, from the ground up.\n\nNo wonder you're a geek.\n\nThis isn't to say that geeks and software engineers are friendless or even socially inept. But there's an implicit promise in becoming a coder: Apprentice yourself to symbolic systems, learn to carefully understand the rules that govern them, and you'll gain power to manipulate them. The more powerless you feel, the more appealing this promise becomes. \"Hacking,\" Steven Levy writes, \"gave you not only an understanding of the system but an addictive control as well, along with the illusion that total control was just a few features away.\"\n\nAs anthropologist Coleman points out, beyond the Jocks-and-Nerds stereotypes, there are actually many different geek cultures. There are open-software advocates, most famously embodied by Linux founder Linus Torvalds, who spend untold hours collaboratively building free software tools for the masses, and there are Silicon Valley start-up entrepreneurs. There are antispam zealots, who organize online posses to seek out and shut down Viagra purveyors. And then there's the more antagonistic wing: spammers; \"trolls,\" who spend their time looking for fun ways to leverage technology at others' expense; \"phreaks,\" who are animated by the challenge to break open telecommunications systems; and hackers who break into government systems to prove it can be done.\n\nGeneralizations that span these different niches and communities run the risk of stereotyping and tend to fall short. But at the heart of these subcultures is a shared method for looking at and asserting power in the world, which influences how and why online software is made.\n\nThe through-line is a focus on systematization. Nearly all geek cultures are structured as an empire of clever wherein ingenuity, not charisma, is king. The intrinsic efficiency of a creation is more important than how it looks. Geek cultures are data driven and reality based, valuing substance over style. Humor plays a prominent role\u2014as Coleman points out, jokes demonstrate an ability to manipulate language in the same way that an elegant solution to a tricky programming problem demonstrates mastery over code. (The fact that humor also often serves to unmask the ridiculous pieties of the powerful is undoubtedly also part of its appeal.)\n\nSystematization is especially alluring because it doesn't offer power just in the virtual sphere. It can also provide a way to understand and navigate social situations. I learned this firsthand when, as an awkward seventeen-year-old with all the trappings of geek experience (the fantasy books, the introversion, the obsession with HTML and BBSes), I flew across the country to accept the wrong job.\n\nIn a late-junior-year panic, I'd applied for every internship I could find. One group, a nuclear disarmament organization based in San Francisco, had gotten back to me, and without much further investigation, I'd signed up. It was only when I walked into the office that I realized I'd signed up to be a can vaser. Off the top of my head, I couldn't imagine a worse fit, but because I had no other prospects, I decided to stick out the day of training.\n\nCanvasing, the trainer explained, was a science as much as an art. And the laws were powerful. Make eye contact. Explain why the issue matters to you. And after you ask for money, let your target say the first thing. I was intrigued: Asking people for money was scary, but the briefing hinted at a hidden logic. I committed the rules to memory.\n\nWhen I walked through my first grassy Palo Alto lawn, my heart was in my throat. Here I was at the doorstep of someone I'd never met, asking for $50. The door opened and a harried woman with long gray hair peeped out. I took a deep breath, and launched into my spiel. I asked. I waited. And then she nodded and went to get her checkbook.\n\nThe euphoria I felt wasn't about the $50. It was about something bigger\u2014the promise that the chaos of human social life could be reduced to rules that I could understand, follow, and master. Conversation with strangers had never come naturally to me\u2014I didn't know what to talk about. But the hidden logic of getting someone I'd never met to trust me with $50 had to be the tip of a larger iceberg. By the end of a summer traipsing through the yards of Palo Alto and Marin, I was a master canvaser.\n\nSystematization is a great method for building functional software. And the quantitative, scientific approach to social observation has given us many great insights into human phenomena as well. Dan Ariely researches the \"predictably irrational\" decisions we make on a daily basis; his findings help us make better decisions. The blog at OkCupid.com, a math-driven dating Web site, identifies patterns in the e-mails flying back and forth between people to make them better daters (\"Howdy\" is a better opener than \"Hi\").\n\nBut there are dangers in taking the method too far. As I discussed in chapter 5, the most human acts are often the most unpredictable ones. Because systematizing works much of the time, it's easy to believe that by reducing and brute-forcing an understanding of any system, you can control it. And as a master of a self-created universe, it's easy to start to view people as a means to an end, as variables to be manipulated on a mental spreadsheet, rather than as breathing, thinking beings. It's difficult both to systematize and to appeal to the fullness of human life\u2014its unpredictability, emotionality, and surprising quirks\u2014at the same time.\n\nDavid Gelernter, a Yale computer scientist, barely survived an encounter with an explosive package sent by the Unabomber; his eyesight and right hand are permanently damaged as a result. But Gelernter is hardly the technological utopian Ted Kaczinski believed him to be.\n\n\"When you do something in the public sphere,\" Gelernter told a reporter, \"it behooves you to know something about what the public sphere is like. How did this country get this way? What was the history of the relationship between technology and the public? What's the history of political exchange? The problem is, hackers don't tend to know any of that. And that's why it worries me to have these people in charge of public policy. Not because they're bad, just because they're uneducated.\"\n\nUnderstanding the rules that govern a messy, complex world makes it intelligible and navigable. But systematizing inevitably involves a trade-off\u2014rules give you some control, but you lose nuance and texture, a sense of deeper connection. And when a strict systematizing sensibility entirely shapes social space (as it often does online), the results aren't always pretty.\n\n#  **The New Architects**\n\nThe political power of design has long been obvious to urban planners. If you take the Wantagh State Parkway from Westbury to Jones Beach on Long Island, at intervals you'll pass under several low, vine-covered overpasses. Some of them have as little as nine feet of clearance. Trucks aren't allowed on the parkway\u2014they wouldn't fit. This may seem like a design oversight, but it's not.\n\nThere are about two hundred of these low bridges, part of the grand design for the New York region pioneered by Robert Moses. Moses was a master deal maker, a friend of the great politicians of the time, and an unabashed elitist. According to his biographer, Robert A. Caro, Moses's vision for Jones Beach was as an island getaway for middle-class white families. He included the low bridges to make it harder for low-income (and mostly black) New Yorkers to get to the beach, as public buses\u2014the most common form of transport for inner-city residents\u2014couldn't clear the overpasses.\n\nThe passage in Caro's _The Power Broker_ describing this logic caught the eye of Langdon Winner, a _Rolling Stone_ reporter, musician, professor, and philosopher of technology. In a pivotal 1980 article titled \"Do Artifacts Have Politics?\" Winner considered how Moses's \"monumental structures of concrete and steel embody a systematic social inequality, a way of engineering relationships among people that, after a time, became just another part of the landscape.\"\n\nOn the face of it, a bridge is just a bridge. But often, as Winner points out, architectural and design decisions are underpinned by politics as much as aesthetics. Like goldfish that grow only large enough for the tank they're in, we're contextual beings: how we behave is dictated in part by the shape of our environments. Put a playground in a park, and you encourage one kind of use; build a memorial, and you encourage another.\n\nAs we spend more of our time in cyberspace\u2014and less of our time in what geeks sometimes call meatspace, or the offline world\u2014Moses's bridges are worth keeping in mind. The algorithms of Google and Facebook may not be made of steel and concrete, but they regulate our behavior just as effectively. That's what Larry Lessig, a law professor and one of the early theorists of cyberspace, meant when he famously wrote that \"code is law.\"\n\nIf code is law, software engineers and geeks are the ones who get to write it. And it's a funny kind of law, created without any judicial system or legislators and enforced nearly perfectly and instantly. Even with antivandalism laws on the books, in the physical world you can still throw a rock through the window of a store you don't like. You might even get away with it. But if vandalism isn't part of the design of an online world, it's simply impossible. Try to throw a rock through a virtual storefront, and you just get an error.\n\nBack in 1980, Winner wrote, \"Consciously or unconsciously, deliberately or inadvertently, societies choose structures for technologies that influence how people are going to work, communicate, travel, consume, and so forth over a very long time.\" This isn't to say that today's designers have malevolent impulses, of course\u2014or even that they're always explicitly trying to shape society in certain ways. It's just to say that they can\u2014in fact, they can't help but shape the worlds they build.\n\nTo paraphrase Spider-Man creator Stan Lee, with great power comes great responsibility. But the programmers who brought us the Internet and now the filter bubble aren't always game to take on that responsibility. The Hacker Jargon File, an online repository of geek culture, puts it this way: \"Hackers are far more likely than most non-hackers to either (a) be aggressively apolitical or (b) entertain peculiar or idiosyncratic political ideas.\" Too often, the executives of Facebook, Google, and other socially important companies play it coy: They're social revolutionaries when it suits them and neutral, amoral businessmen when it doesn't. And both approaches fall short in important ways.\n\n#  **Playing It Coy**\n\nWhen I first called Google's PR department, I explained that I wanted to know how Google thought about its enormous curatorial power. What was the code of ethics, I asked, that Google uses to determine what to show to whom? The public affairs manager on the other end of the phone sounded confused. \"You mean privacy?\" No, I said, I wanted to know how Google thought about its editorial power. \"Oh,\" he replied, \"we're just trying to give people the most relevant information.\" Indeed, he seemed to imply, no ethics were involved or required.\n\nI persisted: If a 9\/11 conspiracy theorist searches for \"9\/11,\" was it Google's job to show him the _Popular Mechanics_ article that debunks his theory or the movie that supports it? Which was more relevant? \"I see what you're getting at,\" he said. \"It's an interesting question.\" But I never got a clear answer.\n\nMuch of the time, as the Jargon File entry claims, engineers resist the idea that their work has moral or political consequences at all. Many engineers see themselves as interested in efficiency and design, in building cool stuff rather than messy ideological disputes and inchoate values. And it's true that if political consequences of, say, a somewhat faster video-rendering engine exist, they're pretty obscure.\n\nBut at times, this attitude can verge on a \"Guns don't kill people, people do\" mentality\u2014a willful blindness to how their design decisions affect the daily lives of millions. That Facebook's button is named Like prioritizes some kinds of information over others. That Google has moved from PageRank\u2014which is designed to show the societal consensus result\u2014to a mix of PageRank and personalization represents a shift in how Google understands relevance and meaning.\n\nThis amorality would be par for the corporate course if it didn't coincide with sweeping, world-changing rhetoric from the same people and entities. Google's mission to organize the world's information and make it accessible to everyone carries a clear moral and even political connotation\u2014a democratic redistribution of knowledge from closed-door elites to the people. Apple's devices are marketed with the rhetoric of social change and the promise that they'll revolutionize not only your life but our society as well. (The famous Super Bowl ad announcing the release of the Macintosh computer ends by declaring that \"1984 won't be like _1984._ \")\n\nFacebook describes itself as a \"social utility,\" as if it's a twenty-first-century phone company. But when users protest Facebook's constantly shifting and eroding privacy policy, Zuckerberg often shrugs it off with the caveat emptor posture that if you don't want to use Facebook, you don't have to. It's hard to imagine a major phone company getting away with saying, \"We're going to publish your phone conversations for anyone to hear\u2014and if you don't like it, just don't use the phone.\"\n\nGoogle tends to be more explicitly moral in its public aspirations; its motto is \"Don't be evil,\" while Facebook's unofficial motto is \"Don't be lame.\" Nevertheless, Google's founders also sometimes play a get-out-of-jail-free card. \"Some say Google is God. Others say Google is Satan,\" says Sergey Brin. \"But if they think Google is too powerful, remember that with search engines, unlike other companies, all it takes is a single click to go to another search engine. People come to Google because they choose to. We don't trick them.\"\n\nOf course, Brin has a point: No one is forced to use Google, just as no one is forced to eat at McDonald's. But there's also something troubling about this argument, which minimizes the responsibility he might have to the billions of users who rely on the service Google provides and in turn drive the company's billions in advertising revenue.\n\nTo further muddle the picture, when the social repercussions of their work are troubling, the chief architects of the online world often fall back on the manifest-destiny rhetoric of technodeterminism. Technologists, Siva Vaidyanathan points out, rarely say something \"could\" or \"should\" happen\u2014they say it \"will\" happen. \"The search engines of the future will be personalized,\" says Google Vice President Marissa Mayer, using the passive tense.\n\nJust as some Marxists believed that the economic conditions of a society would inevitably propel it through capitalism and toward a world socialist regime, it's easy to find engineers and technodeterminist pundits who believe that technology is on a set course. Sean Parker, the cofounder of Napster and rogue early president of Facebook, tells _Vanity Fair_ that he's drawn to hacking because it's about \"re-architecting society. It's technology, not business or government, that's the real driving force behind large-scale societal shifts.\"\n\nKevin Kelly, the founding editor of _Wired,_ wrote perhaps the boldest book articulating the technodeterminist view, _What Technology Wants_ , in which he posits that technology is a \"seventh kingdom of life,\" a kind of meta-organism with desires and tendencies of its own. Kelly believes that the technium, as he calls it, is more powerful than any of us mere humans. Ultimately, technology\u2014a force that \"wants\" to eat power and expand choice\u2014will get what it wants whether we want it to or not.\n\nTechnodeterminism is alluring and convenient for newly powerful entrepreneurs because it absolves them of responsibility for what they do. Like priests at the altar, they're mere vessels of a much larger force that it would be futile to resist. They need not concern themselves with the effects of the systems they've created. But technology _doesn't_ solve every problem of its own accord. If it did, we wouldn't have millions of people starving to death in a world with an oversupply of food.\n\nIt shouldn't be surprising that software entrepreneurs are incoherent about their social and political responsibilities. A great deal of this tension undoubtedly comes from the fact that the nature of online business is to scale up as quickly as possible. Once you're on the road to mass success and riches\u2014often as a very young coder\u2014there simply isn't much time to fully think all of this through. And the pressure of the venture capitalists breathing down your neck to \"monetize\" doesn't always offer much space for rumination on social responsibility.\n\n#  **The $50 Billion Sand Castle**\n\nOnce a year, the Y Combinator start-up incubator hosts a daylong conference called Startup School, where successful tech entrepreneurs pass wisdom on to the aspiring audience of bright-eyed Y Combinator investees. The agenda typically includes many of the top CEOs in Silicon Valley, and in 2010, Mark Zuckerberg was at the top of the list.\n\nZuckerberg was in an affable mood, dressed in a black T-shirt and jeans and enjoying what was clearly a friendly crowd. Even so, when Jessica Livingston, his interviewer, asked him about _The Social Network,_ the movie that had made him a household name, a range of emotions crossed his face. \"It's interesting what kind of stuff they focused on getting right,\" Zuckerberg began. \"Like, every single shirt and fleece they had in that movie is actually a shirt or fleece that I own.\"\n\nWhere there was an egregious discrepancy between fiction and reality, Zuckerberg told her, was how his own motivations were painted. \"They frame it as if the whole reason for making Facebook and building something was that I wanted to get girls, or wanted to get into some kind of social institution. And the reality, for people who know me, is that I've been dating the same girl since before I started Facebook. It's such a big disconnect.... They just can't wrap their head around the idea that someone might build something because they like building things.\"\n\nIt's entirely possible that the line was just a clever bit of Facebook PR. And there's no question that the twenty-six-year-old billionaire is motivated by empire building. But the comment struck me as candid: For programmers as for artists and craftsmen, making things is often its own best reward.\n\nFacebook's flaws and its founder's ill-conceived views about identity aren't the result of an antisocial, vindictive mind-set. More likely, they're a natural consequence of the odd situation successful start-ups like Facebook create, in which a twenty-something guy finds himself, in a matter of five years, in a position of great authority over the doings of 500 million human beings. One day you're making sand castles; the next, your sand castle is worth $50 billion and everyone in the world wants a piece of it.\n\nOf course, there are far worse business-world personality types with whom to entrust the fabric of our social lives. With a reverence for rules, geeks tend to be principled\u2014to carefully consider and then follow the rules they set for themselves and to stick to them under social pressure. \"They have a somewhat skeptical view of authority,\" Stanford professor Terry Winograd said of his former students Page and Brin. \"If they see the world going one way and they believe it should be going the other way, they are more like to say 'the rest of the world is wrong' rather than 'maybe we should reconsider.'\"\n\nBut the traits that fuel the best start-ups\u2014aggression, a touch of arrogance, an interest in empire building, and of course brilliant systematizing skills\u2014can become a bit more problematic when you rule the world. Like pop stars who are vaulted onto the global stage, world-building engineers aren't always ready or willing to accept the enormous responsibility they come to hold when their creations start to teem with life. And it's not infrequently the case that engineers who are deeply mistrustful of power in the hands of others see themselves as supreme rationalists impervious to its effects.\n\nIt may be that this is too much power to entrust to any small, homogeneous group of individuals. Media moguls who get their start with a fierce commitment to the truth become the confidants of presidents and lose their edge; businesses begun as social ventures become preoccupied with delivering shareholder value. In any case, one consequence of the current system is that we can end up placing a great deal of power in the hands of people who can have some pretty far-out, not entirely well-developed political ideas. Take Peter Thiel, one of Zuckerberg's early investors and mentors.\n\nThiel has penthouse apartments in San Francisco and New York and a silver gullwing McLaren, the fastest car in the world. He also owns about 5 percent of Facebook. Despite his boyish, handsome features, Thiel often looks as though he's brooding. Or maybe he's just lost in thought. In his teenage years, he was a high-ranking chess player but stopped short of becoming a grand master. \"Taken too far, chess can become an alternate reality in which one loses sight of the real world,\" he told an interviewer for _Fortune._ \"My chess ability was roughly at the limit. Had I become any stronger, there would have been some massive tradeoffs with success in other domains in life.\" In high school, he read Solzhenitsyn's _Gulag Archipelago_ and J. R. R. Tolkien's _Lord of the Rings_ series, visions of corrupt and totalitarian power. At Stanford, he started a libertarian newspaper, the _Stanford,_ to preach the gospel of freedom.\n\nIn 1998, Thiel cofounded the company that would become PayPal, which he sold to eBay for $1.5 billion in 2002. Today Thiel runs a multi-billion-dollar hedge fund, Clarium, and a venture capital firm, Founder's Fund, which invests in software companies throughout Silicon Valley. Thiel has made some legendarily good picks\u2014among them, Facebook, in which he was the first outside investor. (He's also made some bad ones\u2014Clarium has lost billions in the last few years.) But for Thiel, investing is more than a day job. It's an avocation. \"By starting a new Internet business, an entrepreneur may create a new world,\" Thiel says. \"The hope of the Internet is that these new worlds will impact and force change on the existing social and political order.\"\n\nHis comments raise the question of what kind of change Thiel would like to see. While many billionaires are fairly circumspect about their political views, Thiel has been vocal\u2014and it's safe to say that there are few with views as unusual as Thiel's. \"Peter wants to end the inevitability of death and taxes,\" Thiel's sometime collaborator Patri Friedman (grandson of Milton) told _Wired._ \"I mean, talk about aiming high!\"\n\nIn an essay posted on the libertarian Cato Institute's Web site, Thiel describes why he believes that \"freedom and democracy are no longer compatible.\" \"Since 1920,\" he writes, \"the vast increase in welfare beneficiaries and the extension of the franchise to women\u2014two constituencies that are notoriously tough for libertarians\u2014have rendered the notion of 'capitalist democracy' into an oxymoron.\" Then he outlines his hopes for the future: space exploration, \"sea-steading,\" which involves building movable microcountries on the open ocean, and cyberspace. Thiel has poured millions into technologies to sequence genes and prolong life. He's also focused on preparing for the Singularity, the moment a few decades from now when some futurists believe that humans and machines are likely to meld.\n\nIn an interview, he argues that should the Singularity arrive, one would be well advised to be on the side of the computers: \"Certainly we would hope that [an artificially intelligent computer] would be friendly to human beings. At the same time, I don't think you'd want to be known as one of the human beings that is against computers and makes a living being against computers.\"\n\nIf all this sounds a little fantastical, it doesn't worry Thiel. He's focused on the long view. \"Technology is at the center of what will determine the course of the 21st century,\" he says. \"There are aspects of it that are great and aspects that are terrible, and there are some real choices humans have to make about which technologies to foster and which ones we should be more careful about.\"\n\nPeter Thiel is entitled to his idiosyncratic views, of course, but they're worth paying attention to because they increasingly shape the world we all live in. There are only four other people on the Facebook board besides Mark Zuckerberg; Thiel is one of them, and Zuckerberg publicly describes him as a mentor. \"He helped shape the way I think about the business,\" Zuckerberg said in a 2006 Bloomberg News interview. As Thiel says, we have some big decisions to make about technology. And as for how those decisions get made? \"I have little hope,\" he writes, \"that voting will make things better.\"\n\n#  **\"What Game Are You Playing?\"**\n\nOf course, not all engineers and geeks have the views about democracy and freedom that Peter Thiel does\u2014he's surely an outlier. Craig Newmark, the founder of the free Web site craigslist, spends most of his time arguing for \"geek values\" that include service and public-spiritedness. Jimmy Wales and the editors at Wikipedia work to make human knowledge free to everyone. The filtering goliaths make huge contributions here as well: The democratic ideal of an enlightened, capable citizenry is well served by the broader set of relationships Facebook allows me to manage and the mountains of formerly hard-to-access research papers and other public information that Google has freed.\n\nBut the engineering community can do more to strengthen the Internet's civic space. And to get a sense of the path ahead, I talked to Scott Heiferman.\n\nHeiferman, the founder of MeetUp.com, is soft-spoken in a Midwestern sort of way. That's fitting, because he grew up in Homewood, Illinois, a small town on the outskirts of Chicago. \"It was a stretch to call it suburban,\" he says. His parents operated a paint store.\n\nAs a teenager, Heiferman devoured material about Steve Jobs, eating up the story about how Jobs wooed a senior executive from Pepsi by asking him if he wanted to change the world or sell sugar water. \"Throughout my life,\" he told me, \"I've had a love-hate relationship with advertising.\" At the University of Iowa in the early 1990s, Heiferman studied engineering and marketing but at night he ran a radio show called _Advertorial Infotainment_ in which he would remix and cut advertisements together to create a kind of sound art. He put the finished shows online and encouraged people to send in ads to remix, and the attention got him his first job, managing the Web site at Sony .com.\n\nAfter a few years as Sony's \"interactive-marketing frontiersman,\" Heiferman founded i-traffic, one of the major early advertising companies of the Web. Soon i-traffic was the agency of record for clients like Disney and British Airways. But although the company was growing quickly, he was dissatisfied. The back of his business card had a mission statement about connecting people with brands they'd love, but he was increasingly uncertain that was a worthy endeavor\u2014perhaps he was selling sugar water after all. He left the company in 2000.\n\nFor the remainder of the year and into 2001, Heiferman was in a funk. \"I was exhibiting what you could call being depressed,\" he says. When he heard the first word of the World Trade Center attacks on 9\/11, he ran up to his lower-Manhattan rooftop and watched in horror. \"I talked to more strangers in the next three days,\" he says, \"than in the previous five years of living in New York.\"\n\nShortly after the attacks, Heiferman came across the blog post that changed his life. It argued that the attacks, as awful as they were, might bring Americans back together in their civic life, and referenced the bestselling book _Bowling Alone_. Heiferman bought a copy and read it cover to cover. \"I became captivated,\" he says, \"by the question of whether we could use technology to rebuild and strengthen community.\" MeetUp.com, a site that makes it easy for local groups to meet face-to-face, was his answer, and today, MeetUp serves over 79,000 local groups that do that. There's the Martial Arts MeetUp in Orlando and the Urban Spirituality MeetUp in Barcelona and the Black Singles MeetUp in Houston. And Heiferman is a happier man.\n\n\"What I learned being in the ad business,\" he says, \"is that people can just go a long time without asking themselves what they should put their talent towards. You're playing a game, and you know the point of the game is to win. But what game are you playing? What are you optimizing for? If you're playing the game of trying to get the maximum downloads of your app, you'll make the better farting app.\"\n\n\"We don't need more things,\" he says. \"People are more magical than iPads! Your relationships are not media. Your friendships are not media. Love is not media.\" In his low-key way, Heiferman is getting worked up.\n\nEvangelizing this view of technology\u2014that it ought to do something meaningful to make our lives more fulfilling and to solve the big problems we face\u2014isn't as easy as it might seem. In addition to MeetUp more generally, Scott founded the New York Tech MeetUp, a group of ten thousand software engineers who meet every month to preview new Web sites. At a recent meeting, Scott made an impassioned plea for the assembled group to focus on solving the problems that matter\u2014education, health care, the environment. It didn't get a very good reception\u2014in fact, he was just about booed off the stage. \"'We just want to do cool stuff,' was the attitude,\" Scott told me later. \" 'Don't bother me with this politics stuff.' \"\n\nTechnodeterminists like to suggest that technology is inherently good. But despite what Kevin Kelly says, technology is no more benevolent than a wrench or a screwdriver. It's only good when people make it do good things and use it in good ways. Melvin Kranzberg, a professor who studies the history of technology, put it best nearly thirty years ago, and his statement is now known as Kranzberg's first law: \"Technology is neither good or bad, nor is it neutral.\"\n\nFor better or worse, programmers and engineers are in a position of remarkable power to shape the future of our society. They can use this power to help solve the big problems of our age\u2014poverty, education, disease\u2014or they can, as Heifer-man says, make a better farting app. They're entitled to do either, of course. But it's disingenuous to have it both ways\u2014to claim your enterprise is great and good when it suits you and claim you're a mere sugar-water salesman when it doesn't.\n\nActually, building an informed and engaged citizenry\u2014in which people have the tools to help manage not only their own lives but their own communities and societies\u2014is one of the most fascinating and important engineering challenges. Solving it will take a great deal of technical skill mixed with humanistic understanding\u2014a real feat. We need more programmers to go beyond Google's famous slogan, \"Don't be evil.\" We need engineers who will do good.\n\nAnd we need them soon: If personalization remains on its current trajectory, as the next chapter describes, the near future could be stranger and more problematic than many of us would imagine.\n**7**\n\n**What You Want, Whether You Want It or Not**\n\nThere will always be plenty of things to compute in the detailed affairs of millions of people doing complicated things.\n\n\u2014computing pioneer Vannevar Bush, 1945\n\nAll collected data had come to a final end. Nothing was left to be collected. But all collected data had yet to be completely correlated and put together in all possible relationships.\n\n\u2014from Isaac Asimov's short story \"The Last Question\"\n\n**I** recently received a friend invitation on Facebook from someone whose name I didn't recognize, a curvy-figured girl with big eyes and thick lashes. Clicking to figure out who she was (and, I'll admit, to look more closely), I read over her profile. It didn't tell me a lot about her, but it seemed like the profile of someone I might plausibly know. A few of our interests were the same.\n\nI looked again at the eyes. They were a little _too_ big.\n\nIn fact, when I looked more closely, I realized her profile picture wasn't even a photograph\u2014it had been rendered by a 3-D graphics program. There was no such person. My new attractive would-be friend was a figment of software, crawling through friend connections to harvest data from Facebook users. Even the list of movies and books she liked appeared to have been ripped from the lists of her \"friends.\"\n\nFor lack of a better word, let's call her an _advertar_ \u2014a virtual being with a commercial purpose. As the filter bubble's membrane becomes thicker and harder to penetrate, advertars could become a powerful adaptive strategy. If I only get the news from my code and my friends, the easiest way to get my attention might be friends who are code.\n\nThe technologies that support personalization will only get more powerful in the years ahead. Sensors that can pick up new personal signals and data streams will become even more deeply embedded in the surface of everyday life. The server farms that support the Googles and Amazons will grow, while the processors inside them shrink; that computing power will be unleashed to make increasingly precise guesses about our preferences and even our interior lives. Personalized \"augmented reality\" technologies will project an overlay over our experience of the real world, not just the digital one. Even Nicholas Negroponte's intelligent agents may make a comeback. \"Markets are strong forces,\" says Bill Joy, the legendary programmer who cofounded Sun Microsystems. \"They take you somewhere very quickly. And if where they take you is not where you want to go, you've got a problem.\"\n\nIn 2002, the sci-fi movie _Minority Report_ featured personalized holographic advertisements that accosted pedestrians as they walked down the street. In Tokyo, the first _Minority Report_ \u2013style personalized billboard has gone up outside of the NEC corporation's headquarters (minus, for now, the holography). It's powered by the company's PanelDirector software, which scans the faces of passersby and matches them to a database of ten thousand stored photos to make guesses about their age and gender. When a young woman steps in front of the display, it responds instantly by showing her ads tailored to her. IBM's on the case, too; its prototype advertising displays use remotely readable identity cards to greet viewers by name.\n\nIn _Reality Hunger,_ a book-length essay composed entirely of text fragments and reworked quotations, David Shields makes the case for the growing movement of artists who are \"breaking larger and larger chunks of 'reality' into their work.\" Shields's examples are far-ranging, including _The Blair Witch Project, Borat,_ and _Curb Your Enthusiasm;_ karaoke, VH1's _Behind the Music,_ and public access TV; _The Eminem Show_ and _The Daily Show,_ documentary and mockumentary _._ These pieces, he says, are the most vital art of our time, part of a new mode characterized by \"a deliberate unartiness\" and \"a blurring (to the point of invisibility) of any distinction between fiction and nonfiction: the lure and blur of the real.\" Truthiness, in Shields's view, is the future of art.\n\nAs goes art, so goes technology. The future of personalization\u2014and of computing itself\u2014is a strange amalgam of the real and the virtual. It's a future where our cities and our bedrooms and all of the spaces in between exhibit what researchers call \"ambient intelligence.\" It's a future where our environments shift around us to suit our preferences and even our moods. And it's a future where advertisers will develop ever more powerful and reality-bending ways to make sure their products are seen.\n\nThe days when the filter bubble disappears when we step away from our computers, in other words, are numbered.\n\n#  **The Robot with Gaydar**\n\nStanford Law professor Ryan Calo thinks a lot about robots, but he doesn't spend much time musing about a future of cyborgs and androids. He's more interested in Roombas, the little robotic vacuum cleaners currently on the market. Roomba owners name their machines like pets. They delight in watching the little bumbling devices wander around the room. Roombas provoke an emotional response, even a sense of relationship. And in the next few years, they'll be joined by a small army of consumer-electronic brethren.\n\nThe increasing prevalence of humanlike machines in everyday life presents us with new dilemmas in personalization and privacy. The emotions provoked by \"humanness,\" both virtually (advertars) and in reality (humanlike robots) are powerful. And when people begin to relate to machines as we do to humans, we can be convinced to reveal implicit information that we would never directly give away.\n\nFor one thing, the presence of humanoid faces changes behavior, compelling people to behave more like they're in public. The Chinese experiment with Jingjing and Chacha, the cartoon Internet police, is one example of this power. On the one hand, Calo points out, people are much less likely to volunteer private information when being interrogated by a virtual agent than when simply filling out a form. This is part of why the intelligent-agent craze didn't work out the first time around: In many cases, it's easier to get people to share personal information if they feel as though they're privately entering it into an impersonal machine rather than sharing it with people.\n\nOn the other hand, when Harvard researchers Terence Burnham and Brian Hare asked volunteers to play a game in which they could choose to donate money or keep it, a picture of the friendly looking robot Kismet increased donations by 30 percent. Humanlike agents tend to make us clam up on the intimate details of our lives, because they make us feel as if we're actually around other people. For elderly folks living alone or a child recovering in a hospital, a virtual or robotic friend can be a great relief from loneliness and boredom.\n\nThis is all to the good. But humanlike agents also have a great deal of power to shape our behavior. \"Computers programmed to be polite, or to evidence certain personalities,\" Calo writes, \"have profound effects on the politeness, acceptance, and other behavior of test subjects.\" And because they engage with people, they can pull out implicit information that we'd never intend to divulge. A flirty robot, for example, might be able to read subconscious cues\u2014eye contact, body language\u2014to quickly identify personality traits of its interlocutor.\n\nThe challenge, Calo says, is that it's hard to remember that humanlike software and hardware aren't human at all. Advertars or robotic assistants may have access to the whole set of personal data that exists online\u2014they may know more about you, more precisely, than your best friend. And as persuasion and personality profiling get better, they'll develop an increasingly nuanced sense of how to shift your behaviors.\n\nWhich brings us back to the advertar. In an attention-limited world, lifelike, and especially humanlike, signals stand out\u2014we're hardwired to pay attention to them. It's far easier to ignore a billboard than an attractive person calling your name. And as a result, advertisers may well decide to invest in technology that allows them to insert human advertisements into social spaces. The next attractive man or woman who friends you on Facebook could turn out to be an ad for a bag of chips.\n\nAs Calo puts it, \"people are not evolved to twentieth-century technology. The human brain evolved in a world in which only humans exhibited rich social behaviors, and a world in which all perceived objects were real physical objects.\" Now all that's shifting.\n\n#  **The Future Is Already Here**\n\nThe future of personalization is driven by a simple economic calculation. Signals about our personal behavior and the computing power necessary to crunch through them are becoming cheaper than ever to acquire. And as that cost collapses, strange new possibilities come within reach.\n\nTake facial recognition. Using MORIS, a $3,000 iPhone app, the police in Brockton, Massachusetts, can snap a photo of a suspect and check his or her identity and criminal record in seconds. Tag a few pictures with Picasa, Google's photo-management tool, and the software can already pick out who's who in a collection of photos. And according to Eric Schmidt, the same is true of Google's cache of images from the entire Web. \"Give us 14 images of you,\" he told a crowd of technologists at the Techonomy Conference in 2010, \"and we can find other images of you with ninety-five percent accuracy.\"\n\nAs of the end of 2010, however, this feature isn't available in Google Image Search. Face.com, an Israeli start-up, may offer the service before the search giant does. It's not every day that a company develops a highly useful and world-changing technology and then waits for a competitor to launch it first. But Google has good reason to be concerned: The ability to search by face will shatter many of our cultural illusions about privacy and anonymity.\n\nMany of us will be caught in flagrante delicto. It's not just that your friends (and enemies) will be able to easily find pictures other people have taken of you\u2014as if the whole Internet has been tagged on Facebook. They will also be able to find pictures other people took of other people, in which you happen to be walking by or smoking a cigarette in the background.\n\nAfter the data has been crunched, the rest is easy. Want to search for two people\u2014say your boyfriend and that overly friendly intern you suspect him of dallying with, or your employee and that executive who's been trying to woo him away? Easy. Want to build a Facebook-style social graph by looking at who appears most often with whom? A cinch. Want to see which of your coworkers posted profiles on anonymous dating sites\u2014or, for that matter, photos of themselves in various states of undress? Want to see what your new friend used to look like in his drugged out days? Want to find mobsters in the Witness Protection program, or spies in deep cover? The possibilities are nearly limitless.\n\nTo be sure, doing face recognition right takes an immense amount of computing power. The tool in Picasa is slow\u2014on my laptop, it crunches for minutes. So for the time being, it may be too expensive to do it well for the whole Web. But face recognition has Moore's law, one of the most powerful laws in computing, on its side: Every year, as processor speed per dollar doubles, it'll get twice as cheap to do. Sooner or later, mass face recognition\u2014perhaps even in real time, which would allow for recognition on security and video feeds\u2014will roll out.\n\nFacial recognition is especially significant because it'll create a kind of privacy discontinuity. We're used to a public semianonymity\u2014while we know we may be spotted in a club or on the street, it's unlikely that we will be. But as security-camera and camera-phone pictures become searchable by face, that expectation will slip away. Shops with cameras facing the doors\u2014and aisles\u2014will be able to watch precisely where individual customers wander, what they pick up, and how this correlates with the data already collected about them by firms like Acxiom. And this powerful set of data\u2014where you go and what you do, as indicated by where your face shows up in the bitstream\u2014can be used to provide ever more custom-tailored experiences.\n\nIt's not just people that will be easier than ever to track. It's also individual objects\u2014what some researchers are calling the \"Internet of things.\"\n\nAs sci-fi author William Gibson once said, \"The future is already here\u2014it's just not very evenly distributed.\" It shows up in some places before others. And one of the places this particular aspect of the future has shown up first, oddly enough, is the Coca-Cola Village Amusement Park, a holiday village, theme park, and marketing event that opens seasonally in Israel. Sponsored by Facebook and Coke, the teenagers attending the park in the summer of 2010 were given bracelets containing a tiny piece of circuitry that allowed them to Like real-world objects. Wave the bracelet at the entrance to a ride, for example, and a status update posted to your account testifies that you're about to embark. Take a picture of your friends with a special camera and wave the bracelet at it, and the photo's posted with your identity already tagged.\n\nEmbedded in each bracelet is a radio-frequency identification (RFID) chip. RFID chips don't need batteries, and there's only one way to use them: call-and-response. Provide a little wireless electromagnetic power, and the chip chirps out a unique identifying code. Correlate the code with, say, a Facebook account, and you're in business. A single chip can cost as little as $.07, and they'll cost far less in the years to come.\n\nSuddenly it's possible for businesses to track each individual object they make across the globe. Affix a chip to an individual car part, and you can watch as the part travels to the car factory, gets assembled into a car, and makes its way to the show floor and then someone's garage. No more inventory shrinkage, no more having to recall whole models of products because of the errors of one factory.\n\nConversely, RFID provides a framework by which a home could automatically inventory every object inside it\u2014and track which objects are in which rooms. With a powerful enough signal, RFID could be a permanent solution to the lost-keys problem\u2014and bring us face-to-face with what _Forbes_ writer Reihan Salam calls \"the powerful promise of a real world that can be indexed and organized as cleanly and coherently as Google has indexed and organized the Web.\"\n\nThis phenomenon is called ambient intelligence. It's based on a simple observation: The items you own, where you put them, and what you do with them is, after all, a great signal about what kind of person you are and what kind of preferences you have. \"In the near future,\" writes a team of ambient intelligence experts led by David Wright, \"every manufactured product\u2014our clothes, money, appliances, the paint on our walls, the carpets on our floors, our cars, everything\u2014will be embedded with intelligence, networks of tiny sensors and actuators, which some have termed 'smart dust.'\"\n\nAnd there's a third set of powerful signals that is getting cheaper and cheaper. In 1990, it cost about $10 to sequence a single base pair\u2014one \"letter\"\u2014of DNA. By 1999, that number had dropped to $.90. In 2004, it crossed the $.01 threshold, and now, as I write in 2010, it costs one ten-thousandth of $.01. By the time this book comes out, it'll undoubtedly cost exponentially less. By some point mid-decade, we ought to be able to sequence any random whole human genome for less than the cost of a sandwich.\n\nIt seems like something out of _Gattaca,_ but the allure of adding this data to our profiles will be strong. While it's increasingly clear that our DNA doesn't determine everything about us\u2014other cellular information sets, hormones, and our environment play a large role\u2014there are undoubtedly numerous correlations between genetic material and behavior to be made. It's not just that we'll be able to predict and avert upcoming health issues with far greater accuracy\u2014though that alone will be enough to get many of us in the door. By adding together DNA and behavioral data\u2014like the location information from iPhones or the text of Facebook status updates\u2014an enterprising scientist could run statistical regression analysis on an entire society.\n\nIn all this data lie patterns yet undreamed of. Properly harnessed, it will fuel a level of filtering acuity that's hard to imagine\u2014a world in which nearly all of our objective experience is quantified, captured, and used to inform our environments. The biggest challenge, in fact, may be thinking of the right questions to ask of these enormous flows of binary digits. And increasingly, code will learn to ask these questions itself.\n\n#  **The End of Theory**\n\nIn December 2010, researchers at Harvard, Google, _Encyclop\u00e6dia Britannica,_ and the _American Heritage Dictionary_ announced the results of a four-year joint effort. The team had built a database spanning the entire contents of over five hundred years' worth of books\u20145.2 million books in total, in English, French, Chinese, German, and other languages. Now any visitor to Google's \"N-Gram viewer\" page can query it and watch how phrases rise and fall in popularity over time, from neologism to the long fade into obscurity. For the researchers, the tool suggested even grander possibilities\u2014a \"quantitative approach to the humanities,\" in which cultural changes can be scientifically mapped and measured.\n\nThe initial findings suggest how powerful the tool can be. By looking at the references to previous dates, the team found that \"humanity is forgetting its past faster with each passing year.\" And, they argued, the tool could provide \"a powerful tool for automatically identifying censorship and propaganda\" by identifying countries and languages in which there was a statistically abnormal absence of certain ideas or phrases. Leon Trotsky, for example, shows up far less in midcentury Russian books than in English or French books from the same time.\n\nThe project is undoubtedly a great service to researchers and the casually curious public. But serving academia probably wasn't Google's only motive. Remember Larry Page's declaration that he wanted to create a machine \"that can understand anything,\" which some people might call artificial intelligence? In Google's approach to creating intelligence, the key is data, and the 5 million digitized books contain an awful lot of it. To grow your artificial intelligence, you need to keep it well fed.\n\nTo get a sense of how this works, consider Google Translate, which can now do a passable job translating automatically among nearly sixty languages. You might imagine that Translate was built with a really big, really sophisticated set of translating dictionaries, but you'd be wrong. Instead, Google's engineers took a probabilistic approach: They built software that could identify which words tended to appear in connection with which, and then sought out large chunks of data that were available in multiple languages to train the software on. One of the largest chunks was patent and trademark filings, which are useful because they all say the same thing, they're in the public domain, and they have to be filed globally in scores of different languages. Set loose on a hundred thousand patent applications in English and French, Translate could determine that when _word_ showed up in the English document, _mot_ was likely to show up in the corresponding French paper. And as users correct Translate's work over time, it gets better and better.\n\nWhat Translate is doing with foreign languages Google aims to do with just about everything. Cofounder Sergey Brin has expressed his interest in plumbing genetic data. Google Voice captures millions of minutes of human speech, which engineers are hoping they can use to build the next generation of speech recognition software. Google Research has captured most of the scholarly articles in the world. And of course, Google's search users pour billions of queries into the machine every day, which provide another rich vein of cultural information. If you had a secret plan to vacuum up an entire civilization's data and use it to build artificial intelligence, you couldn't do a whole lot better.\n\nAs Google's protobrain increases in sophistication, it'll open up remarkable new possibilities. Researchers in Indonesia can benefit from the latest papers in Stanford (and vice versa) without waiting for translation delays. In a matter of a few years, it may be possible to have an automatically translated voice conversation with someone speaking a different language, opening up whole new channels of cross-cultural communication and understanding.\n\nBut as these systems become increasingly \"intelligent,\" they also become harder to control and understand. It's not quite right to say they take on a life of their own\u2014ultimately, they're still just code. But they reach a level of complexity at which even their programmers can't fully explain any given output.\n\nThis is already true to a degree with Google's search algorithm. Even to its engineers, the workings of the algorithm are somewhat mysterious. \"If they opened up the mechanics,\" says search expert Danny Sullivan, \"you still wouldn't understand it. Google could tell you all two hundred signals it uses and what the code is and you wouldn't know what to do with them.\" The core software engine of Google search is hundreds of thousands of lines of code. According to one Google employee I talked to who had spoken to the search team, \"The team tweaks and tunes, they don't really know what works or why it works, they just look at the result.\"\n\nGoogle promises that it doesn't tilt the deck in favor of its own products. But the more complex and \"intelligent\" the system gets, the harder it'll be to tell. Pinpointing where bias or error exists in a human brain is difficult or impossible\u2014there are just too many neurons and connections to narrow it down to a single malfunctioning chunk of tissue. And as we rely on intelligent systems like Google's more, their opacity could cause real problems\u2014like the still-mysterious machine-driven \"flash crash\" that caused the Dow to drop 600 points in a few minutes on May 6, 2010.\n\nIn a provocative article in _Wired,_ editor-in-chief Chris Anderson argued that huge databases render scientific theory itself obsolete. Why spend time formulating human-language hypotheses, after all, when you can quickly analyze trillions of bits of data and find the clusters and correlations? He quotes Peter Norvig, Google's research director: \"All models are wrong, and increasingly you can succeed without them.\" There's plenty to be said for this approach, but it's worth remembering the downside: Machines may be able to see results without models, but humans can't understand without them. There's value in making the processes that run our lives comprehensible to the humans who, at least in theory, are their beneficiaries.\n\nSupercomputer inventor Danny Hillis once said that the greatest achievement of human technology is tools that allow us to create more than we understand. That's true, but the same trait is also the source of our greatest disasters. The more the code driving personalization comes to resemble the complexity of human cognition, the harder it'll be to understand why or how it's making the decisions it makes. A simple coded rule that bars people from one group or class from certain kinds of access is easy to spot, but when the same action is the result of a swirling mass of correlations in a global supercomputer, it's a trickier problem. And the result is that it's harder to hold these systems and their tenders accountable for their actions.\n\n#  **No Such Thing as a Free Virtual Lunch**\n\nIn January 2009, if you were listening to one of twenty-five radio stations in Mexico, you might have heard the accordion ballad \"El m\u00e1s grande enemigo.\" Though the tune is polka-ish and cheery, the lyrics depict a tragedy: a migrant seeks to illegally cross the border, is betrayed by his handler, and is left in the blistering desert sun to die. Another song from the _Migra corridos_ album tells a different piece of the same sad tale:\n\n _To cross the border_   \n_I got in the back of a trailer_   \n_There I shared my sorrows_   \n_With forty other immigrants_   \n_I was never told_   \n_That this was a trip to hell._\n\nIf the lyrics aren't exactly subtle about the dangers of crossing the border, that's the point. _Migra corridos_ was produced by a contractor working for the U.S. Border Control, as part of a campaign to stem the tide of immigrants along the border. The song is a prime example of a growing trend in what marketers delicately call \"advertiser-funded media,\" or AFM.\n\nProduct placement has been in vogue for decades, and AFM is its natural next step. Advertisers love product placement because in a media environment in which it's harder and harder to get people to pay attention to anything\u2014especially ads\u2014it provides a kind of loophole. You can't fast-forward past product placement. You can't miss it without missing some of the actual content. AFM is just a natural extension of the same logic: Media have always been vehicles for selling products, the argument goes, so why not just cut out the middleman and have product makers produce the content themselves?\n\nIn 2010, Walmart and Procter & Gamble announced a partnership to produce _Secrets of the Mountain_ and _The Jensen Project,_ family movies that will feature characters using the companies' products throughout. Michael Bay, the director of _Transformers,_ has started a new company called the Institute, whose tagline is \"Where Brand Science Meets Great Storytelling.\" _Hansel and Gretel in 3-D,_ its first feature production, will be specially crafted to provide product-placement hooks throughout.\n\nNow that the video-game industry is far more profitable than the movie industry, it provides a huge opportunity for in game advertising and product placement as well. Massive Incorporated, a game advertising platform acquired by Microsoft for $200 million to $400 million, has placed ads on in game billboards and city walls for companies like Cingular and McDonald's, and has the capacity to track which individual users saw which advertisements for how long. Splinter Cell, a game by UBIsoft, works placement for products like Axe deodorant into the architecture of the cityscape that characters travel through.\n\nEven books aren't immune. _Cathy's Book_ , a young-adult title published in September 2006, has its heroine applying \"a killer coat of Lipslicks in 'Daring.' \"That's not a coincidence\u2014 _Cathy's Book_ was published by Procter & Gamble, the corporate owner of Lipslicks.\n\nIf the product placement and advertiser-funded media industries continue to grow, personalization will offer whole new vistas of possibility. Why name-drop Lipslicks when your reader is more likely to buy Cover Girl? Why have a video-game chase scene through Macy's when the guy holding the controller is more of an Old Navy type? When software engineers talk about architecture, they're usually talking metaphorically. But as people spend more of their time in virtual, personalizable places, there's no reason that these worlds can't change to suit users' preferences. Or, for that matter, a corporate sponsor's.\n\n#  **A Shifting World**\n\nThe enriched psychological models and new data flows measuring everything from heart rate to music choices open up new frontiers for online personalization, in which what changes isn't just a choice of products or news clips, but the look and feel of the site on which they're displayed.\n\nWhy _should_ Web sites look the same to every viewer or customer? Different people don't respond only to different products\u2014they respond to different design sensibilities, different colors, even different types of product descriptions. It's easy enough to imagine a Walmart Web site with softened edges and warm pastels for some customers and a hard-edged, minimalist design for others. And once that capacity exists, why stick with just one design per customer? Maybe it's best to show me one side of the Walmart brand when I'm angry and another when I'm happy.\n\nThis kind of approach isn't a futuristic fantasy. A team led by John Hauser at MIT's business school has developed the basic techniques for what they call Web site morphing, in which a shopping site analyzes users' clicks to figure out what kinds of information and styles of presentation are most effective and then adjusts the layout to suit a particular user's cognitive style. Hauser estimates that Web sites that morph can increase \"purchase intentions\" by 21 percent. Industrywide, that's worth billions. And what starts with the sale of consumer products won't end there: News and entertainment sources that morph ought to enjoy an advantage as well.\n\nOn one hand, morphing makes us feel more at home on the Web. Drawing from the data we provide, every Web site can feel like an old friend. But it also opens the door to a strange, dreamlike world, in which our environment is constantly rearranging itself behind our backs. And like a dream, it may be less and less possible to share with people outside of it\u2014that is, everyone else.\n\nThanks to augmented reality, that experience may soon be par for the course offline as well.\n\n\"On the modern battlefield,\" Raytheon Avionics manager Todd Lovell told a reporter _,_ \"there is way more data out there than most people can use. If you are just trying to see it all through your eyes and read it in bits and bites, you're never going to understand it. So the key to the modern technology is to take all that data and turn it into useful information that the pilot can recognize very quickly and act upon.\" What Google does for online information, Lovell's Scorpion project aims to do for the real world.\n\nFitting like a monocle over one of a jet pilot's eyes, the Scorpion display device annotates what a pilot sees in real time. It color-codes potential threats, highlights when and where the aircraft has a missile lock, assists with night vision, and reduces the need for pilots to look at a dashboard in an environment where every microsecond matters. \"It turns the whole world into a display,\" jet pilot Paul Mancini told the Associated Press.\n\nThis is augmented-reality technology, and it's moving rapidly from the cockpits of jet planes to consumer devices that can tune out the noise and turn up the signal of everyday life. Using your iPhone camera and an app developed by Yelp, the restaurant recommendation service, you can see eateries' ratings haphazardly displayed over their real-world storefronts. A new kind of noise-canceling headphone can sense and amplify human voices while tuning other street or airplane noise down to a whisper. The Meadowlands football stadium is spending $100 million on new applications that give fans who attend games in person the ability to slice and dice the game in real time, view key statistics as they happen, and watch the action unfold from a variety of angles\u2014the full high-information TV experience overlaid on a real game.\n\nAt DARPA, the defense research and development agency, technologies are being developed that make Scorpion look positively quaint. Since 2002, DARPA has been pushing forward research in what it calls augmented cognition, or AugCog, which uses cognitive neuroscience and brain imaging to figure out how best to route important information into the brain. AugCog begins with the premise that there are basic limits as to how many tasks a person can juggle at a time, and that \"this capacity itself may fluctuate from moment to moment depending on a host of factors including mental fatigue, novelty, boredom and stress.\"\n\nBy monitoring activity in brain areas associated with memory, decision making, and the like, AugCog devices can figure out how to make sure to highlight the information that most matters. If you're absorbing as much visual input as you can, the system might decide to send an audio alert instead. One trial, according to the _Economist,_ gave users of an AugCog device a 100 percent improvement in recall and a 500 percent increase in working memory. And if it sounds far-fetched, just remember: The folks at DARPA also helped invent the Internet.\n\nAugmented reality is a booming field, and Gary Hayes, a personalization and augmented-reality expert in Australia, sees at least sixteen different ways it could be used to provide services and make money. In his vision, guide companies could offer augmented reality tours, in which information about buildings, museum artifacts, and streets is superimposed on the environs. Shoppers could use phone apps to immediately get readouts on products they're interested in\u2014including what the objects cost elsewhere. (Amazon.com already provides a rudimentary version of this service.) Augmented reality games could layer clues into real-world environments.\n\nAugmented-reality tech provides value, but it also provides an opportunity to reach people with new attention-getting forms of advertising. For a price, digital sportscasts are already capable of layering corporate logos onto football fields. But this new technology offers the opportunity to do that in a personalized way in the real world: You turn on the app to, say, help find a friend in a crowd, and projected onto a nearby building is a giant Coke ad featuring your face and your name.\n\nAnd when you combine the personalized filtering of what we see and hear with, say, face recognition, things get pretty interesting: You begin to be able to filter not just information, but people.\n\nAs the cofounder of OkCupid, one of the Web's most popular dating sites, Chris Coyne has been thinking about filtering for people for a while. Coyne speaks in an energetic, sincere manner, furrowing his brows when he's thinking and waving his hands to illustrate. As a math major, he got interested in how to use algorithms to solve problems for people.\n\n\"There are lots of ways you can use math to do things that turn a profit,\" he told me over a steaming bowl of bibimbap in New York's Koreatown. Many of his classmates went off to high-paid jobs at hedge funds. \"But,\" he said, \"what we were interested in was using it to make people happy.\" And what better way to make people happy than to help them fall in love?\n\nThe more Coyne and his college hallmates Sam Yeager and Max Krohn looked at other dating sites, the more annoyed they got: It was clear that other dating sites were more interested in getting people to pay for credits than to hook up. And once you did pay, you'd often see profiles of people who were no longer on the site or who would never write you back.\n\nCoyne and his team decided to approach the problem with math. The service would be free. Instead of offering a one-sizefits-all solution, they'd use number crunching to develop a personalized matching algorithm for each person on the site. And just as Google optimizes for clicks, they'd do everything they could to maximize the likelihood of real conversations\u2014if you could solve for that, they figured, profits would follow. In essence, they built a modern search engine for mates.\n\nWhen you log on to OkCupid, you're asked a series of questions about yourself. Do you believe in God? Would you ever participate in a threesome? Does smoking disgust you? Would you sleep with someone on the first date? Do you have an STD? (Answer yes, and you get sent to another site.) You also indicate how you'd like a prospective partner to answer the same questions and how important their answers are to you. Using these questions, OkCupid builds a custom-weighted equation to figure out your perfect match. And when you search for people in your area, it uses the same algorithm to rank the likelihood of your getting along. OkCupid's powerful cluster of servers can rank ten thousand people with a two-hundred-question match model and return results in less than a tenth of a second.\n\nThey have to, because OkCupid's traffic is booming. Hundreds of thousands of answers to poll questions flow into their system each night. Thousands of new users sign up each day. And the system is getting better and better.\n\nLooking into the future, Coyne told me, you'll have people walking around with augmented displays. He described a guy on a night out: You walk into a bar, and a camera immediately scans the faces in the room and matches them against OkCupid's databases. \"Your accessories can say, that girl over there is an eighty-eight percent match. That's a dream come true!\"\n\nVladimir Nabokov once commented that \"reality\" is \"one of the few words that mean nothing without quotes.\" Coyne's vision may soon be our \"reality.\" There's tremendous promise in this vision: Surgeons who never miss a suture, soldiers who never imperil civilians, and everywhere a more informed, information-dense world. But there's also danger: Augmented reality represents the end of naive empiricism, of the world as we see it, and the beginning of something far more mutable and weird: a real-world filter bubble that will be increasingly difficult to escape.\n\n#  **Losing Control**\n\nThere's plenty to love about this ubiquitously personalized future.\n\nSmart devices, from vacuum cleaners to lightbulbs to picture frames, offer the promise that our environments will be exactly the way we want them, wherever we are. In the near future, ambient-intelligence expert David Wright suggests, we might even carry our room-lighting preferences with us; when there are multiple people in a room, a consensus could be automatically reached by averaging preferences and weighting for who's the host.\n\nAugCog-enabled devices will help us track the data streams that we consider most important. In some situations\u2014say, medical or fire alerts that find ways to escalate until they capture our attention\u2014they could save lives. And while brainwave-reading AugCog is probably some way off for the masses, consumer variants of the basic concept are already being put into place. Google's Gmail Priority Inbox, which screens e-mails and highlights the ones it assesses as more important, is an early riff on the theme. Meanwhile, augmented-reality filters offer the possibility of an annotated and hyperlinked reality, in which what we see is infused with information that allows us to work better, assimilate information more quickly, and make better decisions.\n\nThat's the good side. But there's always a bargain in personalization: In exchange for convenience, you hand over some privacy and control to the machine.\n\nAs personal data become more and more valuable, the behavioral data market described in chapter 1 is likely to explode. When a clothing company determines that knowing your favorite color produces a $5 increase in sales, it has an economic basis for pricing that data point\u2014and for other Web sites to find reasons to ask you. (While OkCupid is mum about its business model, it likely rests on offering advertisers the ability to target its users based on the hundreds of personal questions they answer.)\n\nWhile many of these data acquisitions will be legitimate, some won't be. Data are uniquely suited to gray-market activities, because they need not carry any trace of where they have come from or where they have been along the way. Wright calls this data laundering, and it's already well under way: Spyware and spam companies sell questionably derived data to middlemen, who then add it to the databases powering the marketing campaigns of major corporations.\n\nMoreover, because the transformations applied to your data are often opaque, it's not always clear exactly what decisions are being made on your behalf, by whom, or to what end. This matters plenty when we're talking about information streams, but it matters even more when this power is infused into our sensory apparatus itself.\n\nIn 2000, Bill Joy, the Sun Microsystems cofounder, wrote a piece for _Wired_ magazine titled \"Why the Future Doesn't Need Us.\" \"As society and the problems that face it become more and more complex and machines become more and more intelligent,\" he wrote, \"people will let machines make more of their decisions for them, simply because machine-made decisions will bring better results than man-made ones.\"\n\nThat may often be the case: Machine-driven systems do provide significant value. The whole promise of these technologies is that they give us more freedom and more control over our world\u2014lights that respond to our whims and moods, screens and overlays that allow us to attend only to the people we want to, so that we don't have to do the busywork of living. The irony is that they offer this freedom and control by taking it away. It's one thing when a remote control's array of buttons elides our ability to do something basic like flip the channels. It's another thing when what the remote controls is our lives.\n\nIt's fair to guess that the technology of the future will work about as well as the technology of the past\u2014which is to say, well enough, but not perfectly. There will be bugs. There will be dislocations and annoyances. There will be breakdowns that cause us to question whether the whole system was worth it in the first place. And we'll live with the threat that systems made to support us will be turned against us\u2014that a clever hacker who cracks the baby monitor now has a surveillance device, that someone who can interfere with what we see can expose us to danger. The more power we have over our own environments, the more power someone who assumes the controls has over us.\n\nThat is why it's worth keeping the basic logic of these systems in mind: You don't get to create your world on your own. You live in an equilibrium between your own desires and what the market will bear. And while in many cases this provides for healthier, happier lives, it also provides for the commercialization of everything\u2014even of our sensory apparatus itself. There are few things uglier to contemplate than AugCog-enabled ads that escalate until they seize control of your attention.\n\nWe're compelled to return to Jaron Lanier's question: For whom do these technologies work? If history is any guide, we may not be the primary customer. And as technology gets better and better at directing our attention, we need to watch closely what it is directing our attention toward.\n**8**\n\n**Escape from the City of Ghettos**\n\nIn order to find his own self, [a person] also needs to live in a milieu where the possibility of many different value systems is explicitly recognized and honored. More specifically, he needs a great variety of choices so that he is not misled about the nature of his own person.\n\n\u2014 _Christopher Alexander_ et al., _A Pattern Language_\n\n**I** n theory, there's never been a structure more capable of allowing all of us to shoulder the responsibility for understanding and managing our world than the Internet. But in practice, the Internet is headed in a different direction. Sir Tim Berners-Lee, the creator of the World Wide Web, captured the gravity of this threat in a recent call to arms in the pages of _Scientific American_ titled \"Long Live the Web.\" \"The Web as we know it,\" he wrote, \"is being threatened.... Some of its most successful inhabitants have begun to chip away at its principles. Large social-networking sites are walling off information posted by their users from the rest of the Web.... Governments\u2014totalitarian and democratic alike\u2014are monitoring people's online habits, endangering important human rights. If we, the Web's users, allow these and other trends to proceed unchecked, the Web could be broken into fragmented islands.\"\n\nIn this book, I've argued that the rise of pervasive, embedded filtering is changing the way we experience the Internet and ultimately the world. At the center of this transformation is the fact that for the first time it's possible for a medium to figure out who you are, what you like, and what you want. Even if the personalizing code isn't always spot-on, it's accurate enough to be profitable, not just by delivering better ads but also by adjusting the substance of what we read, see, and hear.\n\nAs a result, while the Internet offers access to a dazzling array of sources and options, in the filter bubble we'll miss many of them. While the Internet can give us new opportunities to grow and experiment with our identities, the economics of personalization push toward a static conception of personhood. While the Internet has the potential to decentralize knowledge and control, in practice it's concentrating control over what we see and what opportunities we're offered in the hands of fewer people than ever before.\n\nOf course, there are some advantages to the rise of the personalized Internet. I enjoy using Pandora, Netflix, and Facebook as much as the next person. I appreciate Google's shortcuts through the information jungle (and couldn't have written this book without them). But what's troubling about this shift toward personalization is that it's largely invisible to users and, as a result, out of our control. We are not even aware that we're seeing increasingly divergent images of the Internet. The Internet may know who we are, but we don't know who it thinks we are or how it's using that information. Technology designed to give us more control over our lives is actually taking control away.\n\nUltimately, Sun Microsystems cofounder Bill Joy told me, information systems have to be judged on their public outcomes. \"If what the Internet does is spread around a lot of information, fine, but what did that cause to happen?\" he asked. If it's not helping us solve the really big problems, what good is it? \"We really need to address the core issues: climate change, political instability in Asia and the Middle East, demographic problems, and the decline of the middle class. In the context of problems of this magnitude, you'd hope that a new constituency would emerge, but there's a distraction overlay\u2014false issues, entertainment, gaming. If our system, with all the freedom of choice, is not addressing the problems, something's wrong.\"\n\nSomething _is_ wrong with our media. But the Internet isn't doomed, for a simple reason: This new medium is nothing if not plastic. Its great strength, in fact, is its capacity for change. Through a combination of individual action, corporate responsibility, and governmental regulation, it's still possible to shift course.\n\n\"We create the Web,\" Sir Tim Berners-Lee wrote. \"We choose what properties we want it to have and not have. It is by no means finished (and it's certainly not dead).\" It's still possible to build information systems that introduce us to new ideas, that push us in new ways. It's still possible to create media that show us what we don't know, rather than reflecting what we do. It's still possible to erect systems that don't trap us in an endless loop of self-flattery about our own interests or shield us from fields of inquiry that aren't our own.\n\nFirst, however, we need a vision\u2014a sense of what to aim for.\n\n#  **The Mosaic of Subcultures**\n\nIn 1975, architect Christopher Alexander and a team of colleagues began publishing a series of books that would change the face of urban planning, design, and programming. The most famous volume, _A Pattern Language,_ is a guidebook that reads like a religious text. It's filled with quotes and aphorisms and hand-drawn sketches, a bible guiding devotees toward a new way of thinking about the world.\n\nThe question that had consumed Alexander and his team during eight years of research was the question of why some places thrived and \"worked\" while others didn't\u2014why some cities and neighborhoods and houses flourished, while others were grim and desolate. The key, Alexander argued, was that design has to fit its literal and cultural context. And the best way to ensure that, they concluded, was to use a \"pattern language,\" a set of design specifications for human spaces.\n\nEven for nonarchitects, the book is an entrancing read. There's a pattern that describes the ideal nook for kids (the ceiling should be between 2 feet 6 inches and 4 feet high), and another for High Places \"where you can look down and survey your world.\" \"Every society which is alive and whole,\" Alexander wrote, \"will have its own unique and distinct pattern language.\"\n\nSome of the book's most intriguing sections illuminate the patterns that successful cities are built on. Alexander imagines two metropolises\u2014the \"heterogeneous city,\" where people are mixed together irrespective of lifestyle and background, and the \"city of ghettos,\" where people are grouped together tightly by category. The heterogeneous city \"seems rich,\" Alexander writes, but \"actually it dampens all significant variety, and arrests most of the possibilities for differentiation.\" Though there's a diverse mix of peoples and cultures, all of the parts of the city are diverse in the same way. Shaped by the lowest common cultural denominators, the city looks the same everywhere you go.\n\nMeanwhile, in the city of ghettos, some people get trapped in the small world of a single subculture that doesn't really represent who they are. Without connections and overlap between communities, subcultures that make up the city don't evolve. As a result, the ghettos breed stagnation and intolerance.\n\nBut Alexander offers a third possibility: a happy medium between closed ghettos and the undifferentiated mass of the heterogeneous city. He called it the mosaic of subcultures. In order to achieve this kind of city, Alexander explains, designers should encourage neighborhoods with cultural character, \"but though these subcultures must be sharp and distinct and separate, they must not be closed; they must be readily accessible to one another, so that a person can move easily from one to another, and can settle in the one which suits him best.\" Alexander's mosaic is based on two premises about human life: First, a person can only fully become him- or herself in a place where he or she \"receives support for his idiosyncrasies from the people and values which surround him.\" And second, as the quotation at the beginning of this chapter suggests, you have to see lots of ways of living in order to choose the best life for yourself. This is what the best cities do: They cultivate a vibrant array of cultures and allow their citizens to find their way to the neighborhoods and traditions in which they're most at home.\n\nAlexander was writing about cities, but what's beautiful about _A Pattern Language_ is that it can be applied to any space in which humans gather and live\u2014including the Internet. Online communities and niches are important. They're the places where new ideas and styles and themes and even languages get formed and tested. They're the places where we can feel most at home. An Internet built like the heterogeneous city described by Alexander wouldn't be a very pleasant place to be\u2014a whirling chaos of facts and ideas and communications. But by the same token, nobody wants to live in the city of ghettos\u2014and that's where personalization, if it's too acute, will take us. At its worst, the filter bubble confines us to our own information neighborhood, unable to see or explore the rest of the enormous world of possibilities that exist online. We need our online urban planners to strike a balance between relevance and serendipity, between the comfort of seeing friends and the exhilaration of meeting strangers, between cozy niches and wide open spaces.\n\n#  **What Individuals Can Do**\n\nSocial-media researcher danah boyd was right when she warned that we are at risk of the \"psychological equivalent of obesity.\" And while creating a healthy information diet requires action on the part of the companies that supply the food, that doesn't work unless we also change our own habits. Corn syrup vendors aren't likely to change their practices until consumers demonstrate that they're looking for something else.\n\nHere's one place to start: Stop being a mouse.\n\nOn an episode of the radio program _This American Life,_ host Ira Glass investigates how to build a better mousetrap. He talks to Andy Woolworth, the man at the world's largest mousetrap manufacturer who fields ideas for new trap designs. The proposed ideas vary from the impractical (a trap that submerges the mouse in antifreeze, which then needs to be thrown out by the bucket) to the creepy (a design that kills rodents using, yes, gas pellets).\n\nBut the punch line is that they're all unnecessary. Woolworth has an easy job, because the existing traps are very cheap and work within a day 88 percent of the time. Mousetraps work because mice generally establish a food-seeking route within ten feet of where they are, returning to it up to thirty times a day. Place a trap in its vicinity, and chances are very good that you'll catch your mouse.\n\nMost of us are pretty mouselike in our information habits. I admittedly am: There are three or four Web sites that I check frequently each day, and I rarely vary them or add new ones to my repertoire. \"Whether we live in Calcutta or San Francisco,\" Matt Cohler told me, \"we all kinda do the same thing over and over again most of the time. And jumping out of that recursion loop is not easy to do.\" Habits are hard to break. But just as you notice more about the place you live when you take a new route to work, varying your path online dramatically increases your likelihood of encountering new ideas and people.\n\nJust by stretching your interests in new directions, you give the personalizing code more breadth to work with. Someone who shows interest in opera and comic books and South African politics and Tom Cruise is harder to pigeonhole than someone who just shows interest in one of those things. And by constantly moving the flashlight of your attention to the perimeter of your understanding, you enlarge your sense of the world.\n\nGoing off the beaten track is scary at first, but the experiences we have when we come across new ideas, people, and cultures are powerful. They make us feel human. Serendipity is a shortcut to joy.\n\nFor some of the \"identity cascade\" problems discussed in chapter 5, regularly erasing the cookies your Internet browser uses to identify who you are is a partial cure. Most browsers these days make erasing cookies pretty simple\u2014you just select Options or Preferences and then choose Erase cookies. And many personalized ad networks are offering consumers the option to opt out. I'm posting an updated and more detailed list of places to opt out on the Web site for this book, www.thefilterbubble.com.\n\nBut because personalization is more or less unavoidable, opting out entirely isn't a particularly viable route for most of us. You can run all of your online activities in an \"incognito\" window, where less of your personal information is stored, but it'll be increasingly impractical\u2014many services simply won't work the way they're supposed to. (This is why, as I describe below, I don't think the Do Not Track list currently under consideration by the FTC is a viable strategy.) And of course, Google personalizes based on your Internet address, location, and a number of other factors even if you're entirely logged out and on a brand-new laptop.\n\nA better approach is to choose to use sites that give users more control and visibility over how their filters work and how they use your personal information.\n\nFor example, consider the difference between Twitter and Facebook. In many ways, the two sites are very similar. They both offer people the opportunity to share blips of information and links to videos, news, and photographs. They both offer the opportunity to hear from the people you want to hear from and screen out the people you don't.\n\nBut Twitter's universe is based on a few very simple, mostly transparent rules\u2014what one Twitter supporter called \"a thin layer of regulation.\" Unless you go out of your way to lock your account, everything you do is public to everyone. You can subscribe to anyone's feed that you like without their permission, and then you see a time-ordered stream of updates that includes everything everyone you're following says.\n\nIn comparison, the rules that govern Facebook's information universe are maddeningly opaque and seem to change almost daily. If you post a status update, your friends may or may not see it, and you may or may not see theirs. (This is true even in the Most Recent view that many users assume shows all of the updates\u2014it doesn't.) Different types of content are likely to show up at different rates\u2014if you post a video, for example, it's more likely to be seen by your friends than a status update. And the information you share with the site itself is private one day and public the next. There's no excuse, for example, for asking users to declare which Web sites they're \"fans\" of with the promise that it'll be shown only to their friends, and then releasing that information to the world, as Facebook did in 2009.\n\nBecause Twitter operates on the basis of a few simple, easily understandable rules, it's also less susceptible to what venture capitalist Brad Burnham (whose Union Square Ventures was Twitter's primary early investor) calls the tyranny of the default. There's great power in setting the default option when people are given a choice. Dan Ariely, the behavioral economist, illustrates the principle with a chart showing organ donation rates in different European countries. In England, the Netherlands, and Austria, the rates hover around 10 percent to 15 percent, but in France, Germany, and Belgium, donation rates are in the high 90s. Why? In the first set of countries, you have to check a box giving permission for your organs to be donated. In the second, you have to check a box to say you _won't_ give permission.\n\nIf people will let defaults determine the fate of our friends who need lungs and hearts, we'll certainly let them determine how we share information a lot of the time. That's not because we're stupid. It's because we're busy, have limited attention with which to make decisions, and generally trust that if everyone else is doing something, it's OK for us to do it too. But this trust is often misplaced. Facebook has wielded this power with great intentionality\u2014shifting the defaults on privacy settings in order to encourage masses of people to make their posts more public. And because software architects clearly understand the power of the default and use it to make their services more profitable, their claim that users _can_ opt out of giving their personal information seems somewhat disingenuous. With fewer rules and a more transparent system, there are fewer defaults to set.\n\nFacebook's PR department didn't return my e-mails requesting an interview (perhaps because MoveOn's critical view of Facebook's privacy practices is well known). But it would probably argue that it gives its users far more choice and control about how they use the service than Twitter does. And it's true that Facebook's options control panel lists scores of different options for Facebook users.\n\nBut to give people control, you have to make clearly evident what the options are, because options largely exist only to the degree that they're perceived. This is the problem many of us used to face in programming our VCRs: The devices had all sorts of functions, but figuring out how to make them do anything was an afternoon-long exercise in frustration. When it comes to important tasks like protecting your privacy and adjusting your filters online, saying that you can figure it out if you read the manual for long enough isn't a sufficient answer.\n\nIn short, at the time of this writing, Twitter makes it pretty straightforward to manage your filter and understand what's showing up and why, whereas Facebook makes it nearly impossible. All other things being equal, if you're concerned about having control over your filter bubble, better to use services like Twitter than services like Facebook.\n\nWe live in an increasingly algorithmic society, where our public functions, from police databases to energy grids to schools, run on code. We need to recognize that societal values about justice, freedom, and opportunity are embedded in how code is written and what it solves for. Once we understand that, we can begin to figure out which variables we care about and imagine how we might solve for something different.\n\nFor example, advocates looking to solve the problem of political gerrymandering\u2014the backroom process of carving up electoral districts to favor one party or another\u2014have long suggested that we replace the politicians involved with software. It sounds pretty good: Start with some basic principles, input population data, and out pops a new political map. But it doesn't necessarily solve the basic problem, because what the algorithm solves for has political consequences: Whether the software aims to group by cities or ethnic groups or natural boundaries can determine which party keeps its seats in Congress and which doesn't. And if the public doesn't pay close attention to what the algorithm is doing, it could have the opposite of the intended effect\u2014sanctioning a partisan deal with the imprimatur of \"neutral\" code.\n\nIn other words, it's becoming more important to develop a basic level of algorithmic literacy. Increasingly, citizens will have to pass judgment on programmed systems that affect our public and national life. And even if you're not fluent enough to read through thousands of lines of code, the building-block concepts\u2014how to wrangle variables, loops, and memory\u2014can illuminate how these systems work and where they might make errors.\n\nEspecially at the beginning, learning the basics of programming is even more rewarding than learning a foreign language. With a few hours and a basic platform, you can have that \"Hello, World!\" experience and start to see your ideas come alive. And within a few weeks, you can be sharing these ideas with the whole Web. Mastery, as in any profession, takes much longer, but the payoff for a limited investment in coding is fairly large: It doesn't take long to become literate enough to understand what most basic bits of code are doing.\n\nChanging our own behavior is a part of the process of bursting the filter bubble. But it's of limited use unless the companies that are propelling personalization forward change as well.\n\n#  **What Companies Can Do**\n\nIt's understandable that, given their meteoric rises, the Googles and Facebooks of the online world have been slow to realize their responsibilities. But it's critical that they recognize their public responsibility soon. It's no longer sufficient to say that the personalized Internet is just a function of relevance-seeking machines doing their job.\n\nThe new filterers can start by making their filtering systems more transparent to the public, so that it's possible to have a discussion about how they're exercising their responsibilities in the first place.\n\nAs Larry Lessig says, \"A political response is possible only when regulation is transparent.\" And there's more than a little irony in the fact that companies whose public ideologies revolve around openness and transparency are so opaque themselves.\n\nFacebook, Google, and their filtering brethren claim that to reveal anything about their algorithmic processes would be to give away business secrets. But that defense is less convincing than it sounds at first. Both companies' primary advantage lies in the extraordinary number of people who trust them and use their services (remember lock-in?). According to Danny Sullivan's Search Engine Land blog, Bing's search results are \"highly competitive\" with Google's, but it has a fraction of its more powerful rival's users. It's not a matter of math that keeps Google ahead, but the sheer number of people who use it every day. PageRank and the other major pieces of Google's search engine are \"actually one of the world's worst kept secrets,\" says Google fellow Amit Singhal.\n\nGoogle has also argued that it needs to keep its search algorithm under tight wraps because if it was known it'd be easier to game. But open systems are harder to game than closed ones, precisely because everyone shares an interest in closing loopholes. The open-source operating system Linux, for example, is actually more secure and harder to penetrate with a virus than closed ones like Microsoft's Windows or Apple's OS X.\n\nWhether or not it makes the filterers' products more secure or efficient, keeping the code under tight wraps does do one thing: It shields these companies from accountability for the decisions they're making, because the decisions are difficult to see from the outside. But even if full transparency proves impossible, it's possible for these companies to shed more light on how they approach sorting and filtering problems.\n\nFor one thing, Google and Facebook and other new media giants could draw inspiration from the history of newspaper ombudsmen, which became a newsroom topic in the mid-1960s.\n\nPhilip Foisie, an executive at the _Washington Post_ company, wrote one of the most memorable memos arguing for the practice. \"It is not enough to say,\" he suggested, \"that our paper, as it appears each morning, is its own credo, that ultimately we are our own ombudsman. It has not proven to be, possibly cannot be. Even if it were, it would not be viewed as such. It is too much to ask the reader to believe that we are capable of being honest and objective about ourselves.\" The _Post_ found his argument compelling, and hired its first ombudsman in 1970.\n\n\"We know the media is a great dichotomy,\" said the longtime _Sacramento Bee_ ombudsman Arthur Nauman in a speech in 1994. On the one hand, he said, media has to operate as a successful business that provides a return on investment. \"But on the other hand, it is a public trust, a kind of public utility. It is an institution invested with enormous power in the community, the power to affect thoughts and actions by the way it covers the news\u2014the power to hurt or help the common good.\" It is this spirit that the new media would do well to channel. Appointing an independent ombudsman and giving the world more insight into how the powerful filtering algorithms work would be an important first step.\n\nTransparency doesn't mean only that the guts of a system are available for public view. As the Twitter versus Facebook dichotomy demonstrates, it also means that individual users intuitively understand how the system works. And that's a necessary precondition for people to control and use these tools\u2014rather than having the tools control and use us.\n\nTo start with, we ought to be able to get a better sense of who these sites think we are. Google claims to make this possible with a \"dashboard\"\u2014a single place to monitor and manage all of this data. In practice, its confusing and multitiered design makes it almost impossible for an average user to navigate and understand. Facebook, Amazon, and other companies don't allow users to download a complete compilation of their data in the United States, though privacy laws in Europe force them to. It's an entirely reasonable expectation that data that users provide to companies ought to be available to us, and that this expectation is one that, according to the University of California at Berkeley, most Americans share. We ought to be able to say, \"You're wrong. Perhaps I used to be a surfer, or a fan of comics, or a Democrat, but I'm not any more.\"\n\nKnowing what information the personalizers have on us isn't enough. They also need to do a much better job explaining how they use the data\u2014what bits of information are personalized, to what degree, and on what basis. A visitor to a personalized news site could be given the option of seeing how many other visitors were seeing which articles\u2014even perhaps a colorcoded visual map of the areas of commonality and divergence. Of course, this requires admitting to the user that personalization is happening in the first place, and there are strong reasons in some cases for businesses not to do so. But they're mostly commercial reasons, not ethical ones.\n\nThe Interactive Advertising Bureau is already pushing in this direction. An industry trade group for the online advertising community, the IAB has concluded that unless personalized ads disclose to users how they're personalized, consumers will get angry and demand federal regulation. So it's encouraging its members to include a set of icons on every ad to indicate what personal data the ad draws on and how to change or opt out of this feature set. As content providers incorporate the personalization techniques pioneered by direct marketers and advertisers, they should consider incorporating these safeguards as well.\n\nEven then, sunlight doesn't solve the problem unless it's coupled with a focus in these companies on optimizing for different variables: more serendipity, a more humanistic and nuanced sense of identity, and an active promotion of public issues and cultivation of citizenship.\n\nAs long as computers lack consciousness, empathy, and intelligence, much will be lost in the gap between our actual selves and the signals that can be rendered into personalized environments. And as I discussed in chapter 5, personalization algorithms can cause identity loops, in which what the code knows about you constructs your media environment, and your media environment helps to shape your future preferences. This is an avoidable problem, but it requires crafting an algorithm that prioritizes \"falsifiability,\" that is, an algorithm that aims to _dis_ prove its idea of who you are. (If Amazon harbors a hunch that you're a crime novel reader, for example, it could actively present you with choices from other genres to fill out its sense of who you are.)\n\nCompanies that hold great curatorial power also need to do more to cultivate public space and citizenship. To be fair, they're already doing some of this: Visitors to Facebook on November 2, 2010, were greeted by a banner asking them to indicate if they'd voted. Those who had voted shared this news with their friends; because some people vote because of social pressure, it's quite possible that Facebook increased the number of voters. Likewise, Google has been doing strong work to make information about polling locations more open and easily available, and featured its tool on its home page on the same day. Whether or not this is profit-seeking behavior (a \"find your polling place\" feature would presumably be a terrific place for political advertising), both projects drew the attention of users toward political engagement and citizenship.\n\nA number of the engineers and technology journalists I talked to raised their eyebrows when I asked them if personalizing algorithms could do a better job on this front. After all, one said, who's to say what's important? For Google engineers to place a value on some kinds of information over others, another suggested, would be unethical\u2014though of course this is precisely what the engineers themselves do all the time.\n\nTo be clear, I don't yearn to go back to the good old days when a small group of all-powerful editors unilaterally decided what was important. Too many actually important stories (the genocide in Rwanda, for example) fell through the cracks, while too many actually unimportant ones got front-page coverage. But I also don't think we should jettison that approach altogether. Yahoo News suggests there is some possibility for middle ground: The team combines algorithmic personalization with old-school editorial leadership. Some stories are visible to everyone because they're surpassingly important. Others show up for some users and not others. And while the editorial team at Yahoo spends a lot of time interpreting click data and watching which articles do well and which don't, they're not subservient to it. \"Our editors think of the audience as people with interests, as opposed to a flood of directional data,\" a Yahoo News employee told me. \"As much as we love the data, it's being filtered by human beings who are thinking about what the heck it means. Why didn't the article on this topic we think is important for our readers to know about do better? How do we help it find a larger audience?\"\n\nAnd then there are fully algorithmic solutions. For example, why not rely on everyone's idea of what's important? Imagine for a moment that next to each Like button on Facebook was an Important button. You could tag items with one or the other or both. And Facebook could draw on a mix of both signals\u2014what people like, and what they think really matters\u2014to populate and personalize your news feed. You'd have to bet that news about Pakistan would be seen more often\u2014even accounting for everyone's quite subjective definition of what really matters. Collaborative filtering doesn't have to lead to compulsive media: The whole game is in what values the filters seek to pull out. Alternately, Google or Facebook could place a slider bar running from \"only stuff I like\" to \"stuff other people like that I'll probably hate\" at the top of search results and the News Feed, allowing users to set their own balance between tight personalization and a more diverse information flow. This approach would have two benefits: It would make clear that there's personalization going on, and it would place it more firmly in the user's control.\n\nThere's one more thing the engineers of the filter bubble can do. They can solve for serendipity, by designing filtering systems to expose people to topics outside their normal experience. This will often be in tension with pure optimization in the short term, because a personalization system with an element of randomness will (by definition) get fewer clicks. But as the problems of personalization become better known, it may be a good move in the long run\u2014consumers may choose systems that are good at introducing them to new topics. Perhaps what we need is a kind of anti-Netflix Prize\u2014a Serendipity Prize for systems that are the best at holding readers' attention while introducing them to new topics and ideas.\n\nIf this shift toward corporate responsibility seems improbable, it's not without precedent. In the mid-1800s, printing a newspaper was hardly a reputable business. Papers were fiercely partisan and recklessly ideological. They routinely altered facts to suit their owners' vendettas of the day, or just to add color. It was this culture of crass commercialism and manipulation that Walter Lippmann railed against in _Liberty and the News._\n\nBut as newspapers became highly profitable and highly important, they began to change. It became possible, in a few big cities, to run papers that weren't just chasing scandal and sensation\u2014in part, because their owners could afford not to. Courts started to recognize a public interest in journalism and rule accordingly. Consumers started to demand more scrupulous and rigorous editing.\n\nUrged on by Lippmann's writings, an editorial ethic began to take shape. It was never shared universally or followed as well as it could have been. It was always compromised by the business demands of newspapers' owners and shareholders. It failed outright repeatedly\u2014access to power brokers compromised truth telling, and the demands of advertisers overcame the demands of readers. But in the end, it succeeded, somehow, in seeing us through a century of turmoil.\n\nThe torch is now being passed to a new generation of curators, and we need them to pick it up and carry it with pride. We need programmers who will build public life and citizenship into the worlds they create. And we need users who will hold them to it when the pressure of monetization pulls them in a different direction.\n\n#  **What Governments and Citizens Can Do**\n\nThere's plenty that the companies that power the filter bubble can do to mitigate the negative consequences of personalization\u2014the ideas above are just a start. But ultimately, some of these problems are too important to leave in the hands of private actors with profit-seeking motives. That's where governments come in.\n\nUltimately, as Eric Schmidt told Stephen Colbert, Google is just a company. Even if there are ways of addressing these issues that don't hurt the bottom line\u2014which there may well be\u2014doing so simply isn't always going to be a top-level priority. As a result, after we've each done our part to pop the filter bubble, and after companies have done what they're willing to do, there's probably a need for government oversight to ensure that we control our online tools and not the other way around.\n\nIn his book _Republic.com_ , Cass Sunstein suggested a kind of \"fairness doctrine\" for the Internet, in which information aggregators have to expose their audiences to both sides. Though he later changed his mind, the proposal suggests one direction for regulation: Just require curators to behave in a public-oriented way, exposing their readers to diverse lines of argument. I'm skeptical, for some of the same reasons Sunstein abandoned the idea: Curation is a nuanced, dynamic thing, an art as much as a science, and it's hard to imagine how regulating editorial ethics wouldn't inhibit a great deal of experimentation, stylistic diversity, and growth.\n\nAs this book goes to press, the U.S. Federal Trade Commission is proposing a Do Not Track list, modeled after the highly successful Do Not Call list. At first blush, it sounds pretty good: It would set up a single place to opt out of the online tracking that fuels personalization. But Do Not Track would probably offer a binary choice\u2014either you're in or you're out\u2014and services that make money on tracking might simply disable themselves for Do Not Track list members. If most of the Internet goes dark for these people, they'll quickly leave the list. And as a result, the process could backfire\u2014\"proving\" that people don't care about tracking, when in fact what most of us want is more nuanced ways of asserting control.\n\nThe best leverage point, in my view, is in requiring companies to give us real control over our personal information. Ironically, although online personalization is relatively new, the principles that ought to support this leverage have been clear for decades. In 1973, the Department of Housing, Education, and Welfare under Nixon recommended that regulation center on what it called Fair Information Practices:\n\n\u2022 You should know who has your personal data, what data they have, and how it's used.\n\n\u2022 You should be able to prevent information collected about you for one purpose from being used for others.\n\n\u2022 You should be able to correct inaccurate information about you.\n\n\u2022 Your data should be secure.\n\nNearly forty years later, the principles are still basically right, and we're still waiting for them to be enforced. We can't wait much longer: In a society with an increasing number of knowledge workers, our personal data and \"personal brand\" are worth more than they ever have been. Especially if you're a blogger or a writer, if you make funny videos or music, or if you coach or consult for a living, your online data trail is one of your most valuable assets. But while it's illegal to use Brad Pitt's image to sell a watch without his permission, Facebook is free to use your name to sell one to your friends.\n\nIn courts around the world, information brokers are pushing this view\u2014\"everyone's better off if your online life is owned by us.\" They argue that the opportunities and control that consumers get by using their free tools outweigh the value of their personal data. But consumers are entirely unequipped to make this calculation\u2014while the control you gain is obvious, the control you lose (because, say, your personal data is used to deny you an opportunity down the road) is invisible. The asymmetry of understanding is vast.\n\nTo make matters worse, even if you carefully read a company's privacy policy and decide that giving over rights to your personal information is worth it under those conditions, most companies reserve the right to change the rules of the game at any time. Facebook, for example, promised its users that if they made a connection with a Page, that information would only be shared with their friends. But in 2010, it decided that all of that data should be made fully public; a clause in Facebook's privacy policy (as with many corporate privacy policies) allows it to change the rules _retroactively_. In effect, this gives them nearly unlimited power to dispatch personal data as they see fit.\n\nTo enforce Fair Information Practices, we need to start thinking of personal data as a kind of personal property and protecting our rights in it. Personalization is based on an economic transaction in which consumers are at an inherent disadvantage: While Google may know how much your race is worth to Google, you don't. And while the benefits are obvious (free e-mail!), the drawbacks (opportunities and content missed) are invisible. Thinking of personal information as a form of property would help make this a fairer market.\n\nAlthough personal information is property, it's a special kind of property, because you still have a vested interest in your own data long after it's been exposed. You probably wouldn't want consumers to be able to sell all of their personal data, in perpetuity. France's \"moral laws,\" in which artists retain some control over what's done with a piece after it's been sold, might be a better template. (Speaking of France, while European laws are much closer to Fair Information Practices in protecting personal information, by many accounts the enforcement is much worse, partly because it's much harder for individuals to sue for breaches of the laws.)\n\nMarc Rotenberg, executive director of the Electronic Privacy Information Center, says, \"We shouldn't have to accept as a starting point that we can't have free services on the Internet without major privacy violations.\" And this isn't just about privacy. It's also about how our data shapes the content and opportunities we see and don't see. And it's about being able to track and manage this constellation of data that represents our lives with the same ease that companies like Acxiom and Facebook already do.\n\nSilicon Valley technologists sometimes portray this as an unwinnable fight\u2014people have lost control of their personal data, they'll never regain it, and they just have to grow up and live with it. But legal requirements on personal information need not be foolproof in order to work, any more than legal requirements not to steal are useless because people sometimes still steal things and get away with it. The force of law adds friction to the transmission of some kinds of information\u2014and in many cases, a little friction changes a lot.\n\nAnd there are laws that do protect personal information even in this day and age. The Fair Credit Reporting Act, for example, ensures that credit agencies have to disclose their credit reports to consumers and notify consumers when they're discriminated against on the basis of reports. That's not much, but given that previously consumers couldn't even see if their credit report contained errors (and 70 percent do, according to U.S. PIRG), it's a step in the right direction.\n\nA bigger step would be putting in place an agency to oversee the use of personal information. The EU and most other industrial nations have this kind of oversight, but the United States has lingered behind, scattering responsibilities for protecting personal information among the Federal Trade Commission, the Commerce Department, and other agencies. As we enter the second decade of the twenty-first century, it's past time to take this concern seriously.\n\nNone of this is easy: Private data is a moving target, and the process of balancing consumers and citizens' interests against those of these companies will take a lot of fine-tuning. At worst, new laws could be more onerous than the practices they seek to prevent. But that's an argument for doing this right and doing it soon, before the companies who profit from private information have even greater incentives to try to block it from passing.\n\nGiven the money to be made and the power that money holds over the American legislative system, this shift won't be easy. So to rescue our digital environment from itself, we'll ultimately need a new constituency of digital environmentalists\u2014citizens of this new space we're all building who band together to protect what's great about it.\n\nIn the next few years, the rules that will govern the next decade or more of online life will be written. And the big online conglomerates are lining up to help write them. The communications giants who own the Internet's physical infrastructure have plenty of political clout. AT&T outranks oil companies and pharmaceutical companies as one of the top four corporate contributors to American politics. Intermediaries like Google get the importance of political influence, too: Eric Schmidt is a frequent White House visitor, and companies like Microsoft, Google, and Yahoo have spent millions on influence-mongering in Washington, D.C. Given all of the Web 2.0 hype about empowerment, it's ironic that the old adage still applies: In the fight for control of the Internet, everyone's organized but the people.\n\nBut that's only because most of us aren't in the fight. People who use the Internet and are invested in its future outnumber corporate lobbyists by orders of magnitude. There are literally hundreds of millions of us across all demographics\u2014political, ethnic, socioeconomic, and generational\u2014who have a personal stake in the outcome. And there are plenty of smaller online enterprises that have every interest in ensuring a democratic, public-spirited Web. If the great mass of us decide that an open, public-spirited Internet matters and speak up about it\u2014if we join organizations like Free Press (a nonpartisan grassroots lobby for media reform) and make phone calls to Congress and ask questions at town hall meetings and contribute donations to the representatives who are leading the way\u2014the lobbyists don't stand a chance.\n\nAs billions come online in India and Brazil and Africa, the Internet is transforming into a truly global place. Increasingly, it will be the place where we live our lives. But in the end, a small group of American companies may unilaterally dictate how billions of people work, play, communicate, and understand the world. Protecting the early vision of radical connectedness and user control should be an urgent priority for all of us.\n**ACKNOWLEDGMENTS**\n\nWriting may be a lonely profession, but thinking isn't. That's been one of the great gifts of this writing process\u2014the opportunity to think together and learn from some extremely smart and morally thoughtful people. This book wouldn't be the same\u2014and wouldn't be much\u2014without a large team of (sometimes unwitting) collaborators. What follows is my best attempt to credit those who contributed directly. But there's an even larger number whose scholarship or writing or philosophy gave structure to my thoughts or forced me to think in a new way: Larry Lessig, Neil Postman, Cass Sunstein, Marshall McLuhan, Marvin Minsky, and Michael Schudson come to mind as a start. What's good in this book owes a lot to this broad cadre of thinkers. The errors, of course, are all mine.\n\n_The Filter Bubble_ began as a sketched fragment of text jotted down in the first days of 2010. Elyse Cheney, my literary agent, gave me the confidence to see it as a book. Her keen editorial eye, fierce intellect, and refreshingly blunt assessments (\"That part's pretty good. This chapter, not so much.\") dramatically strengthened the final text. I know it's par for the course to thank one's agent. But Elyse was more than an agent for this book\u2014she was its best proponent and critic, constantly pushing it (and me) to be great. Whether or not the final manuscript met that mark, I've learned a lot, and I'm grateful and deeply appreciative. Her team\u2014Sarah Rainone and Hannah Elnan\u2014were also terrific to work with.\n\nAnn Godoff and Laura Stickney, my editors at Penguin Press, are the other two members of the triumvirate that brought this book into existence. Ann's wisdom helped to shape what this book is about and for whom I've written it; Laura's acute eye and gentle questions and provocations helped me see the gaps, leaps, and snags in the text. I'm indebted to both.\n\nThere's another trio that deserves a great deal of credit, not just for getting this book across the finish line in (more or less) one piece, but for inspiring some of the best insights in it. Research assistants Caitlin Petre, Sam Novey, and Julia Kamin scoured the Internet and dug through dusty library books to help me figure out what exactly was going on. Sam, my resident contrarian, constantly pushed me to think more deeply about what I was saying. Julia brought a keen scientific skepticism to the task and protected me from dubious scholarship that I might otherwise have embraced. And Caitlin's great intelligence, hard work, and thoughtful critique were the sources of some of my favorite aha moments. Guys, I couldn't have done it without you. Thank you.\n\nOne of the best parts of the writing process was the opportunity to call up or sit down with extraordinary people and ask them questions. I'm thankful to the following folks for responding to my inquiries and helping inform the text: C. W. Anderson, Ken Auletta, John Battelle, Bill Bishop, Matt Cohler, Gabriella Coleman, Dalton Conley, Chris Coyne, Pam Dixon, Caterina Fake, Matthew Hindman, Bill Joy, Dave Karpf, Jaron Lanier, Steven Levy, Diana Mutz, Nicholas Negroponte, Markus Prior, Robert Putnam, John Rendon, Jay Rosen, Marc Rotenberg, Douglas Rushkoff, Michael Schudson, Daniel Solove, Danny Sullivan, Philip Tetlock, Clive Thompson, and Jonathan Zittrain. Conversations with Ethan Zuckerman, Scott Heiferman, David Kirkpatrick, Clay Shirky, Nicco Mele, Dean Eckles, Jessi Hempel, and Ryan Calo were especially provocative and helpful. Thanks to Nate Tyler and Jonathan McPhie at Google for considering and responding to my inquiries. And strange though it may seem, given my topic, thanks also to my Facebook friends, some real, some virtual, who quickly responded to my queries and were helpful when I was looking for an anecdote or colorful example.\n\nDuring the writing process, I've received invaluable help from a number of institutions and communities. I don't know where I'd be without the summer months I spent researching and writing at the Blue Mountain Center: many thanks to Ben, Harriet, and my fellow fellows for the space to think, advice (especially from Carey McKenzie), and late-night swims. The Roosevelt Institute was gracious enough to offer a place to hang my hat for the last year: thanks to Andy Rich and Bo Cutter for the intellectual stimulation and great conversations. Micah Sifry and Andrew Raseij, two great friends of online democracy, gave me the space to first make this argument at the Personal Democracy Forum. David Fenton has been there to help with every step of this process, from lending his home for writing and thinking to consulting on the title to helping the book find an audience. David, you're a good friend. And Fenton Communications\u2014especially my kind, thoughtful friend Lisa Witter\u2014generously supported the early investigations that set me on the personalization trail.\n\nThere's little I can say to sufficiently thank Team MoveOn, past and present, from whom I've learned so very much about politics, technology, and people. Carrie, Zack, Joan, Patrick, Tom, Nita, Jenn, Ben, Matt, Natalie, Noah, Adam, Roz, Justin, Ilyse, and the whole crew: You're some of the most fiercely thoughtful and inspiring people I've ever met, and I feel lucky to have worked alongside you.\n\nThe manuscript was ready to read only weeks before it was due. Wes Boyd, Matt Ewing, Randall Farmer, Daniel Mintz, my parents, Emanuel Pariser and Dora Lievow, and of course Sam, Caitlin, and Julia were all kind enough to put aside busy lives and plow through it. I shudder to think of what might have gone to print without their notes. Todd Rogers, Anne O'Dwyer, Patrick Kane, David Kirkpatrick, and Jessi Hempel were all kind enough to look at pieces of the book, as well. And I can't say thanks enough to Krista Williams and Amanda Katz, whose brilliant editorial thinking helped nurse some somewhat sickly chapters back to health (Krista, a second thanks for your friendship). Stephanie Hopkins and Mirela Iverac provided invaluable last-minute assistance with the manuscript.\n\nI've saved the greatest and most personal debts for last. I've benefited immeasurably in my life from a string of great teachers: to name a few, Karen Scott, Doug Hamill, and Leslie Simmons at Lincolnville Central School; Jon Potter and Rob Lovell at Camden-Rockport High; and Barbara Resnik and Peter Cocks at Simon's Rock, among others. Whatever perspicuity I have I owe these folks. And I'm lucky enough to have some truly wonderful friends. I can't list all of you here, but you know who you are. I'm especially grateful for the support and love I've received\u2014in good times and bad\u2014from Aram and Lara Kailian, Tate Hausman, Noah T. Winer, Nick Arons, and Ben and Beth Wikler. It's one of my goals in life to be as good a friend as you've been to me.\n\nMy family has also encouraged me and sharpened my thinking every step of the way. Big hugs and even bigger thank-yous to my mother, Dora Lievow, my father and stepmother, Emanuel Pariser and Lea Girardin, and my sister, Ya Jia. Eben Pariser, my brother, not only egged me on but made amazing pizza when I was flagging and helped finalize the manuscript. He's as good a brother as he is a musician (check out his band, Roosevelt Dime, and you'll see how high that compliment is). Bronwen Rice may not be an official family member, but I'll include her here anyway: Bronwen, thanks for keeping me true to myself all of these years.\n\nThere are four final people whose generosity, intelligence, and love I appreciate more than I can fully express:\n\nWes Boyd took a big gamble on a twenty-one-year-old, trusted me more than I trusted myself, and mentored me through eight years at MoveOn. This book draws on many of our conversations over the years\u2014there's no one I enjoy thinking with more. Peter Koechley, my true friend and coconspirator, encouraged me when the going got rough\u2014in the writing process and outside of it. I'm grateful to have a friend who is simultaneously so talented and so decent. Vivien Labaton: I have no sufficient superlatives, so I'll just go with the colloquial. You're the best. And finally, there's Gena Konstantinakos. Gena, you've borne the brunt of this project more than anyone\u2014the months of working weekends and late nights and early mornings at the office, the stress during revisions, and the constantly extended deadlines. You took it in stride and then some, giving me pep talks, helping me sort out chapters on note cards, and cheering me on all the way. I'm still amazed, some days, to wake up with someone so smart, beautiful, talented, principled, and good-spirited in my life. I love you.\n**FURTHER READING**\n\nAlexander, Christopher, Sara Ishikawa, and Murray Silverstein. _A Pattern Language: Towns, Buildings, Construction._ New York: Oxford University Press, 1977.\n\nAnderson, Benedict. _Imagined Communities: Reflections on the Origin and Spread of Nationalism._ New York: Verso, 1991.\n\nBattelle, John. _The Search: How Google and Its Rivals Rewrote the Rules of Business and Transformed Our Culture_. New York: Portfolio, 2005.\n\nBerger, John. _Ways of Seeing._ New York: Penguin, 1973.\n\nBishop, Bill. _The Big Sort: Why the Clustering of Like-Minded America Is Tearing Us Apart._ New York: Houghton Mifflin Company, 2008.\n\nBohm, David. _On Dialogue._ New York: Routledge, 1996.\n\nConley, Dalton. _Elsewhere, U.S.A.: How We Got from the Company Man, Family Dinners, and the Affluent Society to the Home Office, BlackBerry Moms, and Economic Anxiety._ New York: Pantheon Books, 2008.\n\nDewey, John. _Public and Its Problems_. Athens, OH: Swallow Press, 1927.\n\nHeuer, Richards J. _Psychology of Intelligence Analysis._ Washington, D.C.: Central Intelligence Agency, 1999.\n\nInglehart, Ronald. _Modernization and Postmodernization._ Princeton: Princeton University Press, 1997.\n\nKelly, Kevin. _What Technology Wants._ New York: Viking, 2010.\n\nKoestler, Arthur. _The Act of Creation_. New York: Arkana, 1989.\n\nLanier, Jaron. _You Are Not a Gadget: A Manifesto._ New York: Alfred A. Knopf, 2010.\n\nLessig, Lawrence. _Code: And Other Laws of Cyberspace, Version 2.0._ New York: Basic Books, 2006.\n\nLippmann, Walter. _Liberty and the News._ Princeton: Princeton University Press, 1920.\n\nMinsky, Marvin. _A Society of Mind_. New York: Simon and Schuster, 1988.\n\nNorman, Donald A. _The Design of Everyday Things._ New York: Basic Books, 1988.\n\nPostman, Neil. _Amusing Ourselves to Death: Public Discourse in the Age of Show Business._ New York: Penguin Books, 1985.\n\nSchudson, Michael. _Discovering the News: A Social History of American Newspapers._ New York: Basic Books, 1978.\n\nShields, David. _Reality Hunger: A Manifesto_. New York: Alfred A. Knopf, 2010.\n\nShirky, Clay. _Here Comes Everybody: The Power of Organizing Without Organizations._ New York: The Penguin Press, 2008.\n\nSolove, Daniel J. _Understanding Privacy._ Cambridge, MA: Harvard University Press, 2008.\n\nSunstein, Cass R. _Republic.com 2.0._ Princeton: Princeton University Press, 2007.\n\nTurner, Fred. _From Counterculture to Cyberculture: Stewart Brand, the Whole Earth Network, and the Rise of Digital Utopianism._ Chicago: The University of Chicago Press, 2006.\n\nWatts, Duncan J. _Six Degrees: The Science of a Connected Age._ New York: W.W. Norton & Company, 2003.\n\nWu, Tim. _The Master Switch : The Rise and Fall of Information Empires._ New York: Alfred A. Knopf, 2010.\n\nZittrain, Jonathan. _The Future of the Internet\u2014And How to Stop It._ New Haven: Yale University Press, 2008.\n**NOTES**\n\n#  **Introduction**\n\n**1 \"A squirrel dying\":** David Kirkpatrick, _The Facebook Effect: The Inside Story of the Company That Is Connecting the World_ (New York: Simon and Schuster, 2010), 296.\n\n**1 \"thereafter our tools shape us\":** Marshall McLuhan, _Understanding Media: The Extensions of Man_ (Cambridge: MIT Press, 1994).\n\n**1 \"Personalized search for everyone\":** _Google Blog,_ Dec. 4, 2009, accessed Dec. 19, 2010, <http:\/\/googleblog.blogspot.com\/2009\/12\/personalized-search-for-everyone.html>.\n\n**2 Google would use fifty-seven s** _ **ignals**_ **:** Author interview with confidential source.\n\n**6** _**Wall Street Journal**_ **study:** Julia Angwin, \"The Web's New Gold Mine: Your Secrets,\" _Wall Street Journal,_ July 30, 2010, accessed Dec. 19, 2010, <http:\/\/online.wsj.com\/article\/SB10001424052748703940904575395073512989404.html>.\n\n**6 \"Yahoo\":** Although the official trademark is Yahoo!, I've omitted the exclamation point throughout this book for easier reading.\n\n**6 site installs 223 tracking cookies:** Angwin, \"The World's New Gold Mine,\" July 30, 2010.\n\n**6 Teflon-coated pots:** At the time of writing, ABC News used a piece of sharing software called \"AddThis.\" When you use AddThis to share a piece of content on ABC News's site (or anyone else's), AddThis places a tracking cookie on your computer that can be used to target advertising to people who share items from particular sites.\n\n**6 \"the cost is information about you\":** Chris Palmer, phone interview with author **,** Dec 10, 2010.\n\n**7 accumulated an average of 1,500 pieces of data:** Stephanie Clifford, \"Ads Follow Web Users, and Get More Personal,\" _New York Times,_ July 30, 2009, accessed Dec. 19, 2010, www.nytimes.com\/2009\/07\/31\/business\/media\/31privacy.html.\n\n**7 96 percent of Americans:** Richard Behar, \"Never Heard of Acxiom? Chances Are It's Heard of You.\" _Fortune,_ Feb. 23, 2004, accessed Dec. 19, 2010, <http:\/\/money.cnn.com\/magazines\/fortune\/fortune_archive\/2004\/02\/23\/362182\/index.htm>.\n\n**8 Netflix can predict:** Marshall Kirkpatrick, \"They Did It! One Team Reports Success in the $1m Netflix Prize,\" _ReadWriteWeb_ , June 26, 2009, accessed Dec. 19, 2010, www.readwriteweb.com\/archives\/they_did_it_one_team_reports_success_in_the_1m_net.php.\n\n**8 Web site that isn't customized . . . will seem quaint:** Marshall Kirpatrick, \"Facebook Exec: All Media Will Be Personalized in 3 to 5 Years,\" _ReadWriteWeb_ , Sept. 29, 2010, accessed Jan. 30, 2011, www.readwriteweb.com\/archives\/facebook_exec_all_media_will_be_personalized_in_3.php.\n\n**8 \"now the web is about 'me' \":** Josh Catone, \"Yahoo: The Web's Future Is Not in Search,\" _ReadWriteWeb,_ June 4, 2007, accessed Dec. 19, 2010, www.readwriteweb.com\/archives\/yahoo_personalization.php.\n\n**8 \"tell them what they should be doing\":** James Farrar, \"Google to End Serendipity (by Creating It),\" _ZDNet,_ Aug. 17, 2010, accessed Dec. 19, 2010, www.zdnet.com\/blog\/sustainability\/google-to-end-serendipity-by-creating-it\/1304.\n\n**8 are becoming a primary news source:** Pew Research Center, \"Americans Spend More Time Following the News,\" Sept. 12, 2010, accessed Feb 7, 2011, <http:\/\/people-press.org\/report\/?pageid=1793>.\n\n**8 million more people joining each day:** Justin Smith, \"Facebook Now Growing by Over 700,000 Users a Day, and New Engagement Stats,\" July 2, 2009, accessed Feb. 7, 2011, www.insidefacebook.com\/2009\/07\/02\/facebook-now-growing-by-over-700000-users-a-day-updated-engagement-stats\/ .\n\n**8 biggest source of news in the world:** Ellen McGirt, \"Hacker. Drop out. CEO,\" _Fast Company,_ May 1, 2007, accessed Feb. 7, 2011, www.fastcompany.com\/magazine\/115\/open_features-hacker-dropout-ceo.html.\n\n**11 information: 900,000 blog posts, 50 million tweets:** \"Measuring tweets,\" _Twitter_ blog _,_ Feb. 22, 2010, accessed Dec. 19, 2010, <http:\/\/blog.twitter.com\/2010\/02\/measuring-tweets.html>.\n\n**11 60 million Facebook status updates, and 210 billion e-mails:** \"A Day in the Internet,\" Online Education, accessed Dec. 19, 2010, www.onlineeducation.net\/internet.\n\n**11 about 5 billion gigabytes:** M. G. Siegler, \"Eric Schmidt: Every 2 Days We Create as Much Information as We Did up to 2003,\" _TechCrunch_ blog, Aug. 4, 2010, accessed Dec. 19, 2010, <http:\/\/techcrunch.com\/2010\/08\/04\/schmidt-data>.\n\n**11 two new stadium-size complexes:** Paul Foy, \"Gov't Whittles Bidders for NSA's Utah Data Center,\" Associated Press, Apr. 21, 2010, accessed Dec. 19, 2010, http:\/\/abcnews.go.com\/Business\/wireStory?id=10438827&page=2.\n\n**11 new units of measurements:** James Bamford, \"Who's in Big Brother's Database?,\" _The New York Review of Books,_ Nov 5, 2009, accessed Feb. 8, 2011 _,_www.nybooks.com\/articles\/archives\/2009\/nov\/o5\/whos-in-big-brothers-database.\n\n**11 the attention crash:** Steve Rubel, \"Three Ways to Mitigate the Attention Crash, Yet Still Feel Informed,\" _Micro Persuasion_ (Steve Rubel's blog), Apr. 30, 2008, accessed Dec. 19, 2010, www.micropersuasion.com\/2008\/04\/three-ways-to-m.html.\n\n**13 \"back in the bottle\":** Danny Sullivan, phone interview with author, Sept 10, 2010.\n\n**13 part of our daily experience:** Cass Sunstein, _Republic.com 2.0._ (Princeton: Princeton University Press, 2007).\n\n**13\u201314 \"skew your perception of the world\":** Ryan Calo, phone interview with author, Dec. 13, 2010.\n\n**14 \"the psychological equivalent of obesity\":** danah boyd, \"Streams of Content, Limited Attention: The Flow of Information through Social Media,\" speech, Web 2.0 Expo. (New York: 2009), accessed July 19, 2010, www.danah.org\/papers\/talks\/Web2Expo.html.\n\n**15 \"strategically time\" their online solicitations:** \"Ovulation Hormones Make Women 'Choose Clingy Clothes,'\" BBC News, Aug. 5, 2010, accessed Feb: 8, 2011, www.bbc.co.uk\/news\/health-10878750.\n\n**16 third-party marketing firms:** \"Preliminary FTC Staff Privacy Report,\" remarks of Chairman Jon Leibowitz, as prepared for delivery, Dec. 1, 2010, accessed Feb. 8, 2011, www.ftc.gov\/speeches\/leibowitz\/101201privacyreportremarks.pdf.\n\n**16 Yochai Bentler argues:** Yochai Benkler, \"Siren Songs and Amish Children: Autonomy, Information, and Law,\" _New York University Law Review,_ Apr. 2001.\n\n**17 tap into lots of different networks:** Robert Putnam, _Bowling Alone: The Collapse and Revival of American Community_ (New York: Simon and Schuster, 2000).\n\n**17 \"make us all next door neighbors\":** Thomas Friedman, \"It's a Flat World, After All,\" _New York Times,_ Apr. 3, 2005, accessed Dec. 19, 2010, www.nytimes.com\/2005\/04\/03\/magazine\/03DOMINANCE.html?pagewanted=all.\n\n**17 \"smaller and smaller and faster and faster\":** Thomas Friedman, _The Lexus and the Olive Tree_ (New York: Random House, 2000), 141.\n\n**18 \"closes the loop on pecuniary self-interest\":** Clive Thompson, interview with author, Brooklyn, NY, Aug. 13, 2010.\n\n**18 \"Customers are always right, but people aren't\":** Lee Siegel, _Against the Machine: Being Human in the Age of the Electronic Mob_ (New York: Spiegel and Grau, 2008), 161.\n\n**18 thirty-six hours a week watching TV:** \"Americans Using TV and Internet Together 35% More Than A Year Ago,\" Nielsen Wire, Mar. 22, 2010, accessed Dec. 19, 2010, <http:\/\/blog.nielsen.com\/nielsenwire\/online_mobile\/three-screen-report-q409>.\n\n**19 \"civilization of Mind in cyberspace\":** John Perry Barlow, \"A Cyberspace Independence Declaration,\" Feb. 9, 1996, accessed Dec. 19, 2010, <http:\/\/w2.eff.org\/Censorship\/Internet_censorship_bills\/barlow_0296.declaration>.\n\n**19 \"code is law\":** Lawrence Lessig, _Code 2.0_ (New York: Basic Books, 2006), 5.\n\n#  **Chapter One: The Race for Relevance**\n\n**21 \"If you're not paying for something\":** _MetaFilter_ blog, accessed Dec. 10, 2010, www.metafilter.com\/95152\/Userdriven-discontent.\n\n**22 \"vary sex, violence, and political leaning\":** Nicholas Negroponte, _Being Digital_ (New York: Knopf, 1995), 46.\n\n**22 \"the Daily Me\":** Ibid., 151.\n\n**22 \"Intelligent agents are the unequivocal future\":** Negroponte, Mar. 1, 1995, e-mail to the editor, Wired.com, Mar. 3, 1995, www.wired.com\/wired\/archive\/3.03\/negroponte.html.\n\n**23 \"The agent question looms\":** Jaron Lanier, \"Agents of Alienation,\" accessed Jan. 30, 2011, www.jaronlanier.com\/agentalien.html\n\n**24 twenty-five worst tech products:** Dan Tynan, \"The 25 Worst Tech Products of All Time,\" _PC World,_ May 26, 2006, accessed Dec. 10, 2010, www.pcworld.com\/article\/125772-3\/the_25_worst_tech_products_of_all_time.html#bob.\n\n**24 invested over $100 million:** Dawn Kawamoto, \"Newsmaker: Riding the next technology wave,\" CNET News, Oct. 2, 2003, accessed Jan. 30, 2011, <http:\/\/news.cnet.com\/2008-7351-5085423.html>.\n\n**25 \"he's a lot like John Irving\":** Robert Spector, _Get Big Fast_ (New York: HarperBusiness, 2000), 142.\n\n**25 \"small Artificial Intelligence company\":** Ibid., 145.\n\n**26 surprised to find them at the top:** Ibid., 27.\n\n**26 Random House, controlled only 10 percent:** Ibid., 25.\n\n**26 so many of them\u20143 million active titles:** Ibid., 25.\n\n**27 They called their field \"cybernetics\":** Barnabas D. Johnson, \"Cybernetics of Society,\" The Jurlandia Institute, accessed Jan. 30, 2011, www.jurlandia.org\/cybsoc.htm.\n\n**27 PARC was known for:** Michael Singer, \"Google Gobbles Up Outride,\" _InternetNews.com,_ Sept. 21, 2001, accessed Dec. 10, 2010, www .internetnews.com\/bus-news\/article.php\/889381\/Google-Gobbles-Up-Outride.html.\n\n**27 collaborative filtering:** Moya K. Mason, \"Short History of Collaborative Filtering,\" accessed Dec. 10, 2010, www.moyak.com\/papers\/collaborative-filtering.html.\n\n**28 \"handle any incoming stream of electronic documents\":** David Goldberg, David Nichols, Brian M. Oki, and Douglas Terry, \"Using Collaborative Filtering to Weave an Information Tapestry,\" _Communications of the ACM_ 35 (1992), 12:61.\n\n**28 \"sends replies as necessary\":** Upendra Shardanand, \"Social Information Filtering for Music Recommendation\" (graduate diss., Massachusetts Institute of Technology, 1994).\n\n**29 fewer health books:** Martin Kaste, \"Is Your E-Book Reading Up On You?,\" NPR.org, Dec. 15, 2010, accessed Feb. 8, 2010, www.npr.org\/2010\/12\/15\/132058735\/is-your-e-book-reading-up-on-you.\n\n**30 as if by an \"objective\" recommendation:** Aaron Shepard, _Aiming at Amazon: The NEW Business of Self Publishing, Or How to Publish Your Books with Print on Demand and Online Book Marketing_ (Friday Harbor, WA: Shepard Publications, 2006), 127.\n\n**30 \"notion of 'relevant' \":** Sergey Brin and Lawrence Page, \"The Anatomy of a Large-Scale Hypertextual Web Search Engine,\" Section 1.3.1.\n\n**31 \"advertising causes enough mixed incentives\":** Ibid., Section 8, Appendix A.\n\n**32 \"very difficult to get this data\":** Ibid., Section 1.3.2.\n\n**33 black-ops kind of feel:** Saul Hansell, \"Google Keeps Tweaking Its Search Engine,\" _New York Times_ , June 3, 2007, accessed Feb. 7, 2011, www.nytimes.com\/2007\/06\/03\/business\/yourmoney\/03google.html?_r=1.\n\n**33 \"give back exactly what you want\":** David A. Vise and Mark Malseed, _The Google Story_ (New York: Bantam Dell, 2005), 289. **34**\n\n**\"ancient shark teeth\":** Patent full text, accessed Dec. 10, 2010, http:\/\/patft.uspto.gov\/netacgi\/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.htm&r=1&p=1&f=G&l=50&d=PTXT&S1=7,451,130.PN.&OS=pn\/7,451,130&RS=PN\/7,451,13 ,\n\n**35 \"could call that artificial intelligence\":** Lawrence Page, Google Zeitgeist Europe Conference, May 2006.\n\n**35 \"answer a more hypothetical question\":** BBC News, \"Hyper-personal Search 'Possible,' \" June 20, 2007, accessed Dec. 10, 2010, <http:\/\/news.bbc.co.uk\/2\/hi\/technology\/6221256.stm>.\n\n**36 \"We're a utility\":** David Kirkpatrick, \"Facebook Effect,\" _New York Times,_ June 8, 2010, accessed Dec. 10, 2010, www.nytimes.com\/2010\/06\/08\/books\/excerpt-facebook-effect.html?pagewanted=1.\n\n**37 \"more news in a single day\":** Ellen McGirt, \"Hacker. Dropout. CEO,\" _Fast Company,_ May 1, 2007, accessed Feb. 7, 2011, <http:\/\/www.fastcompany.com\/magazine\/115\/open_features-hacker-dropout-ceo.html>.\n\n**37 it rests on three factors:** Jason Kincaid, \"EdgeRank: The Secret Sauce That Makes Facebook's News Feed Tick,\" _TechCrunch_ blog, Apr. 22, 2010, accessed Dec. 10, 2010, <http:\/\/techcrunch.com\/2010\/04\/22\/facebook-edgerank>.\n\n**38 the 300 million user mark:** Mark Zuckerberg, \"300 Million and On,\" _Facebook_ blog, Sept. 15, 2009, accessed Dec. 10, 2010, <http:\/\/blog.facebook.com\/blog.php?post=136782277130>.\n\n**38 the** _**Washington Post**_ **homepage:** Full disclosure: In the spring of 2010, I briefly consulted with the _Post_ about its online communities and Web presence.\n\n**39 \"the most transformative thing\":** Caroline McCarthy, \"Facebook F8: One Graph to Rule Them All,\" CNET News _,_ Apr. 21, 2010, accessed Dec. 10, 2010, <http:\/\/news.cnet.com\/8301-13577_3-20003053-36.html>.\n\n**39 sharing 25 billion items a month:** M. G. Siegler, \"Facebook: We'll Serve 1 Billion Likes on the Web in Just 24 Hours,\" _TechCrunch_ blog, Apr. 21, 2010, accessed Dec. 10, 2010, <http:\/\/techcrunch.com\/2010\/04\/21\/facebook-like-button>.\n\n**42 Acxiom knew more:** Richard Behar, \"Never Heard of Acxiom? Chances Are It's Heard of You,\" _Fortune,_ Feb. 23, 2004, accessed Dec. 10, 2010, <http:\/\/money.cnn.com\/magazines\/fortune\/fortune_archive\/2004\/02\/23\/362182\/index.htm>.\n\n**43 serves most of the largest companies in America:** nternetNews.com Staff, \"Acxiom Hacked, Customer Information Exposed,\" _InternetNews .com,_ Aug. 8, 2003, accessed Dec. 10, 2010, www.esecurityplanet.com\/trends\/article.php\/2246461\/Acxiom-Hacked-Customer-Information-Exposed.htm.\n\n**43 \"product we make is data\":** Behar, \"Never Heard of Acxiom?\"\n\n**44 auctions it off to the company with the highest bid:** Stephanie Clifford, \"Your Online Clicks Have Value, for Someone Who Has Something to Sell,\" _New York Times,_ Mar. 25, 2009, accessed Dec. 10, 2010, www.nytimes.com\/2009\/03\/26\/business\/media\/26adco.html?_r=2.\n\n**44 takes under a second:** The Center for Digital Democracy, U.S. Public Interest Research Group, and the World Privacy Forum's complaint to the Federal Trade Commission, Apr. 8, 2010, accessed Dec. 10, 2010, <http:\/\/democraticmedia.org\/real-time-targeting>.\n\n**44 leave without buying anything:** Press release, FetchBack Inc., Apr. 13, 2010, accessed Dec. 10, 2010, www.fetchback.com\/press_041310.html.\n\n**45 \"62 billion real-time attributes a year\":** Center for Digital Democracy, U.S. PIRG, and the World Privacy Forum's complaint.\n\n**45 the Rubicon Project:** Ibid.\n\n#  **Chapter Two: The User Is the Content**\n\n**47 \"undermines the democratic way of life\":** John Dewey, _Essays, Reviews, and Miscellany, 1939\u20131941, The Later Works of John Dewey, 1925\u20131953_ , vol.14 (Carbondale: Southern Illinois University Press, 1998), 227.\n\n**47 \"been tailored for them\":** Holman W. Jenkins Jr., \"Google and the Search for the Future,\" _Wall Street Journal,_ Aug. 14, 2010, accessed Dec. 11, 2010, <http:\/\/online.wsj.com\/article\/SB10001424052748704901104575423294099527212.html>.\n\n**48 \"don't know which half\":** John Wanamaker, U.S. department store merchant, as quoted in Marilyn Ross and Sue Collier, _The Complete Guide to Self-Publishing_ (Cincinnati: Writer's Digest Books, 2010), 344.\n\n**49 One executive in the marketing session:** I wasn't able to identify him in my notes.\n\n**49 Now, in 2010, they only received:** Interactive Advertising Bureau PowerPoint, report, \"Brand Advertising Online and The Next Wave of M&A,\" Feb. 2010.\n\n**50 target premium audiences in \"other, cheaper places\":** Ibid. **50 \"denied an assured access to the facts\":** Walter Lippmann, _Liberty and the News_ (Princeton: Princeton University Press, 1920), 6.\n\n**50 blogs remain incredibly reliant on them:** Pew Research Center, \"How Blogs and Social Media Agendas Relate and Differ from the Traditional Press,\" May 23, 2010, accessed Dec. 11, 2010, www.journalism.org\/node\/20621.\n\n**52\u201353 \"these documents are forgeries\":** Peter Wallsten, \"'Buckhead,' Who Said CBS Memos Were Forged, Is a GOP-Linked Attorney,\" _Los Angeles Times,_ Sept. 18, 2004, accessed Dec. 11, 2010, <http:\/\/seattletimes.nwsource.com\/html\/nationworld\/2002039080_buckhead18.html>.\n\n**53 \"We should not have used them\":** Associated Press, \"CBS News Admits Bush Documents Can't Be Verified,\" Sept. 21, 2004, accessed Dec. 11, 2010, www.msnbc.msn.com\/id\/6055248\/ns\/politics.\n\n**54 paying attention to the story:** _The Gallup Poll: Public Opinion 2004_ (Lanham, MD: Rowman & Littlefield, 2006), http:\/\/books.google.com\/books?id=uqqp-sDCjo4C&pg=PA392&lpg=PA392&dq=public+opinion+poll+on+dan+rather+controversy&source=bl&ots=CPGu03cpsn&sig=9XT-li8ar2GOXxfVQWCcGNHIxTg&hl=en&ei=uw_7TLK9OMGB8gb3r72ACw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBcQ6AEwAA#v=onepage&q=public%20opinion%20poll%20on%20dan%20rather%20controversy&f=true.\n\n**54 \"a crisis in journalism\":** Lippmann, _Liberty and the News,_ 64.\n\n**56 at this point that newspapers came to carry:** This section was informed by the wonderful Michael Schudson, _Discovering the News: A Social History of American Newspapers_ (New York: Basic Books, 1978).\n\n**57 \"They goose-stepped it\":** Lippmann, _Liberty and the News_ , 4.\n\n**57 \"what [the average citizen] shall know\":** Ibid., 7.\n\n**58 \"distinctive member of a community\":** John Dewey, _Essays, Reviews, and Miscellany, 1939\u20131941, The Later Works of John Dewey, 1925\u20131953,_ vol. 2 (Carbondale: Southern Illinois University Press, 1984), 332.\n\n**59 calls the 2000s the disintermediation decade:** Jon Pareles, \"A World of Megabeats and Megabytes,\" _New York Times,_ Dec. 30, 2009, accessed Dec. 11, 2010, www.nytimes.com\/2010\/01\/03\/arts\/music\/03tech.html.\n\n**59** _**Disintermediation**_ **\u2014the elimination of middlemen:** Dave Winer, Dec. 7, 2005, Dave Winer's blog, _Scripting News_ , accessed Dec. 11, 2010, <http:\/\/scripting.com\/2005\/12\/07.html#>.\n\n**59 \"It sucks power out of the center\":** Esther Dyson, \"Does Google Violate Its 'Don't Be Evil' Motto?,\" _Intelligence Squared US._ Debate between Esther Dyson, Siva Vaidhyanathan, Harry Lewis, Randal C. Picker, Jim Harper, and Jeff Jarvis (New York, NY) Nov. 18, 2008, accessed Feb. 7, 2011, www.npr.org\/templates\/story\/story.php?storyId=97216369.\n\n**60 the Latin for \"middle layer\":** Hat tip to Clay Shirky for introducing me to this fact in his conversation with Jay Rosen. Clay Shirky interview by Jay Rosen, video, chap. 5 \"Why Study Media?\" _NYU Primary Sources_ (New York, NY), 2011, accessed Feb 9, 2011, <http:\/\/nyuprimarysources.org\/video-library\/jay-rosen-and-clay-shirky\/>.\n\n**61 \"many wresting power from the few\":** Lev Grossman, \"Time's Person of the Year: You,\" _Time,_ Dec. 13, 2006, accessed Dec. 11, 2010, www.time.com\/time\/magazine\/article\/0,9171,1569514,00.html.\n\n**61 \"did not eliminate intermediaries\":** Jack Goldsmith and Tim Wu, _Who Controls the Internet? Illusions of a Borderless World_ (New York: Oxford University Press, 2006), 70.\n\n**62 \"It will remember what you know\":** Danny Sullivan, \"Google CEO Eric Schmidt on Newspapers & Journalism,\" Search Engine Land, Oct. 3, 2009, accessed Dec. 11, 2010, <http:\/\/searchengineland.com\/google-ceo-eric-schmidt-on-newspapers-journalism-27172>.\n\n**62 \"bringing the content to the right group\":** \"Krishna Bharat Discusses the Past and Future of Google News,\" _Google News_ blog, June 15, 2010, accessed Dec. 11, 2010, <http:\/\/googlenewsblog.blogspot.com\/2010\/06\/krishna-bharat-discusses-past-and.html>.\n\n**62 \"We pay attention\":** Ibid.\n\n**63 \"most important, their social circle\":** Ibid.\n\n**63 \"make it available to publishers\":** Ibid.\n\n**63 Americans lost more faith in news:** \"Press Accuracy Rating Hits Two Decade Low; Public Evaluations of the News Media: 1985\u20132009,\" Pew Research Center for the People and the Press, Sept. 13, 2009, accessed Dec. 11, 2010, <http:\/\/people-press.org\/report\/543\/>.\n\n**64 \"** _ **New York Times**_ **and some random blogger\":** Author's interview with Yahoo News executive. Sept. 22, 2010. This interview was conducted in confidence.\n\n**65 unplugging from cable TV offerings:** Erick Schonfeld, \"Estimate: 800,000 U.S. Households Abandoned Their TVs for the Web,\" _TechCrunch_ blog, Apr. 13, 2010, accessed Dec. 11, 2010, <http:\/\/techcrunch.com\/2010\/04\/13\/800000-households-abandoned-tvs-web>; \"Cable TV Taking It on the Chin,\" www.freemoneyfinance.com\/2010\/11\/cable-tv-taking-it-on-the-chin.html; and Peter Svensson, \"Cable Subscribers Flee, but Is Internet to Blame?\" <http:\/\/finance.yahoo.com\/news\/Cable-subscribers-flee-but-is-apf-3875814716.html?x=0>.\n\n**65 \"change the ad industry forever\":** \"Google Vice President: Online Video and TV Will Converge,\" June 25, 2010, Appmarket.tv, accessed Dec. 11, 2010, www.appmarket.tv\/news\/160-breaking-news\/440-google-vice-president-online-video-and-tv-will-converge.html.\n\n**66 know people who live near us:** Bill Bishop, _The Big Sort: Why the Clustering of Like-Minded America Is Tearing Us Apart_ (New York: Houghton Mifflin, 2008), 35.\n\n**67 \"watch television to turn your brain off\":** Jason Snell, \"Steve Jobs on the Mac's 20th Anniversary,\" _Macworld,_ Feb. 2, 2004, accessed Dec. 11, 2010, www.macworld.com\/article\/29181\/2004\/02\/themacturns20jobs.html.\n\n**67 thirty-six hours a week:** \"Americans Using TV and Internet Together 35% More Than a Year Ago,\" nielsenwire, Mar. 22, 2010, accessed Dec. 11, 2010, <http:\/\/blog.nielsen.com\/nielsenwire\/online_mobile\/three-screen-report-q409>.\n\n**68 quit channel surfing far more quickly:** Paul Klein, as quoted in Marcus Prior, _Post-Broadcast Democracy_ (New York: Cambridge University Press, 2007), 39.\n\n**68 like your own personal TV channel:** \"YouTube Leanback Offers Effortless Viewing,\" _YouTube_ blog, July 7, 2010, accessed Dec. 11, 2010, <http:\/\/youtube-global.blogspot.com\/2010\/07\/youtube-leanback-offers-effortless.html>.\n\n**69 onto the Big Board, and you're liable to get a raise:** Ben McGrath, \"Search and Destroy: Nick Denton's Blog Empire,\" _New Yorker,_ Oct. 18, 2010, accessed Dec. 11, 2010, www.newyorker.com\/reporting\/2010\/10\/18\/101018fa_fact_mcgrath?currentPage=all.\n\n**70 \"come to us for our judgment\":** Jeremy Peters, \"Some Newspapers, Tracking Readers Online, Shift Coverage,\" _New York Times,_ Sept. 5, 2010, accessed Dec. 11, 2010, www.nytimes.com\/2010\/09\/06\/business\/media\/06track.html.\n\n**71 gin up stories that will get clicks:** Danna Harman, \"In Chile, Instant Web Feedback Creates the Next Day's Paper,\" _Christian Science Monitor,_ Dec. 1, 2004, accessed Dec. 11, 2010, www.csmonitor.com\/2004\/1201\/p01s04-woam.html.\n\n**71 \"creating content in response to audience insight\":** Jeremy Peters, \"At Yahoo, Using Searches to Steer News Coverage,\" _New York Times,_ July 5, 2010, accessed Dec. 11, 2010, www.nytimes.com\/2010\/07\/05\/business\/media\/05yahoo.html.\n\n**72 the newspaper's most e-mailed stories:** Jonah A. Berger and Katherine L. Milkman, \"Social Transmission and Viral Culture,\" Social Science Research Network Working Paper Series (Dec. 25, 2009): 2.\n\n**72 \"Woman in Sumo Wrestler Suit\":** _Huffington Post,_ \"The Craziest Headline Ever,\" June 23, 2010, accessed Dec. 11, 2010, www.huffingtonpost.com\/2010\/06\/23\/craziest-bar-ever-discove_n_623447.html.\n\n**72 sex with a horse:** Danny Westneat, \"Horse Sex Story Was Online Hit,\" _Seattle Times,_ Dec. 30, 2005, accessed Dec. 11, 2010, <http:\/\/seattletimes.nwsource.com\/html\/localnews\/2002711400_danny30.html>.\n\n**72 world's ugliest dog:** Ben Margot, \"Rescued Chihuahua Princess Abby Wins World's Ugliest Dog Contest, Besting Boxer Mix Pabst,\" _Los Angeles Times,_ June 27, 2010, accessed Dec. 11, 2010, <http:\/\/latimesblogs.latimes.com\/unleashed\/2010\/06\/rescued-chihuahua-princess-abby-wins-worlds-ugliest-dog-contest-besting-boxer-mix-pabst.html>.\n\n**72 \"everyone sees the same thing\":** Carl Bialik, \"Look at This Article. It's One of Our Most Popular,\" _Wall Street Journal,_ May 20, 2009.\n\n**73 \"little need to share marketing information\":** Andrew Alexander, \"Making the Online Customer King at The Post,\" _Washington Post,_ July 11, 2010, accessed Dec. 11, 2010, www.washingtonpost.com\/wp-dyn\/content\/article\/2010\/07\/09\/AR2010070903802.html.\n\n**73 \"whether you want to hear this or not\":** Nicholas Negroponte, interview with author, Truckee, CA, Aug. 5, 2010.\n\n**73 \"Gawker's Big Board is a scary extreme\":** Professor Michael Schudson, interview with author, New York, NY, Aug. 13, 2010.\n\n**73 stories about the war in Afghanistan:** Simon Dumenco, \"Google News Cares More About Facebook, Twitter and Apple Than Iraq, Afghanistan,\" _Advertising Age_ , June 23, 2010, accessed Feb. 9, 2011, <http:\/\/adage.com\/mediaworks\/article?article_id=144624>.\n\n**74 \"not to pursue some important stories\":** Alexander, \"Making the Online Customer King.\"\n\n**75 \"periodically be alarmed when there is a crisis?\":** Shirky, interviewed by Jay Rosen.\n\n**75 \"consequences of conjoint and interacting behavior\":** John Dewey, _The Public and Its Problems_ (Athens, OH: Swallow Press, 1927), 126.\n\n#  **Chapter Three: The Adderall Society**\n\n**77 \"contact with persons dissimilar to themselves\":** John Stuart Mill, _The Principles of Political Economy_ (Amherst, MA: Prometheus Books, 2004), 543.\n\n**77 \"reminds one more of a sleepwalker's\":** Arthur Koestler, _The Sleepwalkers: A History of Man's Changing Vision of the Universe_ (New York: Penguin, 1964), 11.\n\n**78 \"but I don't want to talk here\":** Henry Precht, interview with Ambassador David E. Mark, Foreign Affairs Oral History Project, Association for Diplomatic Studies and Training, July 28, 1989, accessed Dec. 14, 2010, <http:\/\/memory.loc.gov\/service\/mss\/mssmisc\/mfdip\/2005%20txt%20files\/2004mar02.txt>.\n\n**78 the two men planned a meeting:** Ibid.\n\n**78 \"all I want is my money\":** Ibid.\n\n**78 \"I was snookered\":** John Limond Hart, _The CIA's Russians_ (Annapolis: Naval Institute Press, 2003), 132.\n\n**78 defect and resettle in the United States:** Ibid., 135.\n\n**79 James Jesus Angleton ... was skeptical:** Ibid., 140.\n\n**79 CIA's documents indicated otherwise:** \"Yuri Ivanovich Nosenko, a Soviet defector, Died on August 23rd, Aged 80,\" _Economist,_ Sept. 4, 2008, accessed Dec. 14, 2010, www.economist.com\/node\/12051491.\n\n**79 subjected to polygraph tests:** Ibid.\n\n**80 sent to the Russian front as punishment:** Richards J. Heuer Jr., \"Nosenko: Five Paths to Judgment,\" _Studies in Intelligence_ 31, no. 3 (Fall 1987).\n\n**80 set him up in a new identity:** David Stout, \"Yuri Nosenko, Soviet Spy Who Defected, Dies at 81,\" _New York Times,_ Aug. 27, 2008, accessed Dec. 14, 2010, www.nytimes.com\/2008\/08\/28\/us\/28nosenko.html?scp=1&sq=nosenko&st=cse.\n\n**80 news of his death was relayed:** Ibid.\n\n**81 full of laudatory comments:** Richards J. Heuer Jr., _Psychology of Intelligence Analysis_ (Alexandria, VA: Central Intelligence Agency, 1999).\n\n**81 \"analysts should be self-conscious\":** Ibid., xiii.\n\n**82 secondhand and in a distorted form:** Ibid., xx\u2013xxi.\n\n**82 \"To achieve the clearest possible image\":** Ibid., xxi\u2013xxii.\n\n**83 \"predictably irrational\":** Dan Ariely, _Predictably Irrational: The Hidden Forces That Shape Our Decisions_ (New York: HarperCollins, 2008)\n\n**83 figuring out what makes us happy:** Dan Gilbert, _Stumbling on Happiness_ (New York: Knopf, 2006).\n\n**83 only one part of the story:** Kathryn Schulz, _Being Wrong: Adventures in the Margin of Error_ (New York: HarperCollins, 2010).\n\n**84 \"Information wants to be reduced\":** Nassim Nicholas Taleb, _The Black Swan: The Impact of the Highly Improbable_ (New York: Random House, 2007), 64.\n\n**85 quickly converted into schemata:** Doris Graber, _Processing the News: How People Tame the Information Tide_ (New York: Longman, 1988).\n\n**85 \"condensation of all features of a story\":** Ibid., 161.\n\n**85 woman celebrating her birthday:** Steven James Breckler, James M. Olson, and Elizabeth Corinne Wiggins, _Social Psychology Alive_ (Belmont, CA: Thomson Wadsworth, 2006), 69.\n\n**86 added details to their memories:** Graber, _Processing the News,_ 170.\n\n**86 Princeton versus Dartmouth:** A. H. Hastorf and H. Cantril, \"They Saw a Game: A Case Study,\" _Journal of Abnormal and Social Psychology_ 49: 129\u201334.\n\n**87 experts' predictions weren't even close:** Philip E. Tetlock, _Expert Political Judgment: How Good Is It? How Can We Know?_ (Princeton: Princeton University Press, 2005).\n\n**88 a process of assimilation and accommodation:** Jean Piaget, _The Psychology of Intelligence_ (New York: Routledge & Kegan Paul, 1950).\n\n**89 the idea that Obama was a Muslim:** Jonathan Chait, \"How Republicans Learn That Obama Is Muslim, _New Republic,_ Aug. 27, 2010, www.tnr.com\/blog\/jonathan-chait\/77260\/how-republicans-learn-obama-muslim.\n\n**89 \"actually become mis-educated\":** Ibid.\n\n**89 two modified versions of \"The Country Doctor\":** Travis Proulx and Steven J. Heine, \"Connections from Kafka: Exposure to Meaning Threats Improves Implicit Learning of an Artificial Grammar,\" _Psychological Science_ 20, no. 9 (2009): 1125\u201331.\n\n**90 \"A severe snowstorm filled the space\":** Franz Kafka, _A Country Doctor_ (Prague: Twisted Spoon Press, 1997).\n\n**90 \"Once one responds to a false alarm\":** Ibid.\n\n**90 \"strived to make sense\":** Proulx and Heine, \"Connections from Kafka.\"\n\n**91 presented with an \"information gap\":** George Loewenstein, \"The Psychology of Curiosity: A Review and Reinterpretation,\" _Psychological Bulletin_ 116, no. 1 (1994): 75\u201398, https:\/\/docs.google.com\/viewer?url=www.andrew.cmu.edu\/user\/gl20\/GeorgeLoewenstein\/Papers_files\/pdf\/PsychofCuriosity.pdf.\n\n**91 \"shields the searcher from such radical encounters\":** Siva Vaidhyanathan, _The Googlization of Everything_ (Berkeley and Los Angeles: University of California Press, 2011), 182.\n\n**91 \"only give you answers\":** Pablo Picasso, as quoted in Gerd Leonhard, Media Futurist Web site, Dec. 8, 2004, accessed Feb. 9, 2011, www.mediafuturist.com\/about.html.\n\n**92 \"On Adderall, I was able to work\":** Joshua Foer, \"The Adderall Me: My Romance with ADHD Meds,\" _Slate,_ May 10, 2005, www.slate.com\/id\/2118315.\n\n**92 \"pressures to use enhancing drugs] are only going to grow\":** Margaret Talbot, \"Brain Gain: The Underground World of 'Neuroenhancing Drugs,'\" _New Yorker,_ Apr. 27, 2009, accessed Dec. 14, 2010, [www.newyorker.com\/reporting\/2009\/04\/27\/090427fa_fact_talbot?currentPage=all.\n\n**93 \"I think 'inside the box' \":** Erowid Experience Vaults, accessed Dec. 14, 2010, www.erowid.org\/experiences\/exp.php?ID=56716.\n\n**93 \"a generation of very focused accountants\":** Talbot, \"Brain Gain.\"\n\n**94 \"an analogy no one has ever seen\":** Arthur Koestler, _Art of Creation_ (New York: Arkana, 1989), 82.\n\n**94 \"uncovers, selects, re-shuffles, combines, synthesizes\":** Ibid., 86.\n\n**95 the key to creative thought:** Hans Eysenck, _Genius: The Natural History of Creativity_ (Cambridge: Cambridge University Press, 1995).\n\n**95 box represents the solution horizon:** Hans Eysenck, \"Creativity and Personality: Suggestions for a Theory,\" _Psychological Inquiry,_ 4, no. 3 (1993): 147\u201378.\n\n**97 no idea what they're looking for:** Aharon Kantorovich and Yuval Ne'eman, \"Serendipity as a Source of Evolutionary Progress in Science,\" _Studies in History and Philosophy of Science, Part A,_ 20, no. 4: 505\u201329.\n\n**98 attach the candle to the wall:** Karl Duncker, \"On Problem Solving,\" _Psychological Monographs_ , 58 (1945).\n\n**98 reluctance to \"break perceptual set\":** George Katona, _Organizing and Memorizing_ (New York: Columbia University Press, 1940).\n\n**99 creative people tend to see things:** Arthur Cropley, _Creativity in Education and Learning_ (New York: Longmans, 1967).\n\n**99 \"sorted a total of 40 objects\":** N. J. C. Andreases and Pauline S. Powers, \"Overinclusive Thinking in Mania and Schizophrenia,\" _British Journal of Psychology_ 125 (1974): 452\u201356.\n\n**99 a \"thing with weight\":** Cropley, _Creativity,_ 39.\n\n**100 \"Stop counting\u2014there are 43 pictures\":** Richard Wiseman, _The Luck Factor_ (New York: Hyperion, 2003), 43\u201344.\n\n**101 bilinguists are more creative than monolinguists:** Charlan Nemeth and Julianne Kwan, \"Minority Influence, Divergent Thinking and Detection of Correct Solutions,\" _Journal of Applied Social Psychology_ , 17, I. 9 (1987): 1, accessed Feb. 7, 2011, <http:\/\/onlinelibrary.wiley.com\/doi\/10.1111\/j.1559-1816.1987.tb00339.x\/abstract>.\n\n**101 foreign ideas help us:** W. M. Maddux, A. K. Leung, C. Chiu, and A. Galinsky, \"Toward a More Complete Understanding of the Link Between Multicultural Experience and Creativity,\" _American Psychologist_ 64 (2009): 156\u201358.\n\n**102 illustrates how creativity arises:** Steven Johnson, _Where Good Ideas Come From: The Natural History of Innovation_ (New York: Penguin, 2010), _ePub Bud_ , accessed Feb 7, 2011, www.epubbud.com\/read.php?g=LN9DVC8S.\n\n**102 \"wide and diverse sample of spare parts\":** Ibid., 6.\n\n**102 \"environments that are powerfully suited\":** Ibid., 3.\n\n**102 \" 'serendipity' article in Wikipedia\":** Ibid., 13.\n\n**103 \"shift from exploration and discovery\":** John Battelle, _The Search: How Google and Its Rivals Rewrote the Rules of Business and Transformed Our Culture_ (New York: Penguin, 2005), 61.\n\n**103 \"database of intentions\":** Ibid.\n\n**104 \"We need help overcoming rationality\":** David Gelernter, _Time to Start Taking the Internet Seriously,_ accessed Dec. 14, 2010, www.edge.org\/3rd_culture\/gelernter10\/gelernter10_index.html.\n\n**105 \"a vast island called California\":** Garci Rodriguez de Montalvo, _The Exploits of Esplandian_ (Madrid: Editorial Castalia, 2003).\n\n#  **Chapter Four: The You Loop**\n\n**109 \"what a personal computer really is\":** Sharon Gaudin, \"Total Recall: Storing Every Life Memory in a Surrogate Brain,\" _ComputerWorld,_ Aug. 2, 2008, accessed Dec. 15, 2010, www.computerworld.com\/s\/article\/9074439\/Total_Recall_Storing_every_life_memory_in_a_surrogate_brain.\n\n**109 \"You have one identity\":** David Kirkpatrick, _The Facebook Effect: The Inside Story of the Company That Is Connecting the World_ (New York: Simon and Schuster, 2010), 199.\n\n**109 \"I behave a different way\":** \"Live-Blog: Zuckerberg and David Kirkpatrick on the Facebook Effect,\" transcript of interview, _Social Beat_ , accessed Dec. 15, 2010, <http:\/\/venturebeat.com\/2010\/07\/21\/live-blog-zuckerberg-and-david-kirkpatrick-on-the-facebook-effect>.\n\n**110 \"Same awkward self\":** Ibid.\n\n**110 that would be the norm:** Marshall Kirkpatrick, \"Facebook Exec: All Media Will Be Personalized in 3 to 5 Years,\" _ReadWriteWeb_ , Sept. 29, 2010, accessed Dec. 15, 2010, www.readwriteweb.com\/archives\/facebook_exec_all_media_will_be_personalized_in_3.php.\n\n**110 \"a world that all may enter\":** John Perry Barlow, A Declaration of the Independence of Cyberspace, Feb. 8, 1996, accessed Dec. 15, 2010, https:\/\/projects.eff.org\/~barlow\/Declaration-Final.html.\n\n**111 pseudonym with the real name:** Julia Angwin and Steve Stecklow, \"'Scrapers' Dig Deep for Data on Web,\" _Wall Street Journal,_ Oct. 12, 2010, accessed Dec. 15, 2010, <http:\/\/online.wsj.com\/article\/SB10001424052748703358504575544381288117888.html>.\n\n**111 tied to the individual people who use them:** Julia Angwin and Jennifer Valentino-Devries, \"Race Is On to 'Fingerprint' Phones, PCs,\" _Wall Street Journal_ , Nov. 30, 2010, accessed Jan. 30, 2011, <http:\/\/online.wsj.com\/article\/SB10001424052748704679204575646704100959546.html?mod=ITP_pageone_0>.\n\n**112 information sources make us freer:** Yochai Benkler, \"Of Sirens and Amish Children: Autonomy, Information, and Law,\" _New York University Law Review_ , 76 no. 23 (April 2001): 110.\n\n**115 \"more than the bits of data\":** Daniel Solove, _The Digital Person: Technology and Privacy in the Information Age_ (New York: New York University Press, 2004), 45.\n\n**116 how someone behaves from who she is:** E. E. Jones and V.A. Harris, \"The Attribution of Attitudes,\" _Journal of Experimental Social Psychology_ 3 (1967): 1\u201324.\n\n**116 electrocute other subjects:** Stanley Milgram, \"Behavioral Study of Obedience,\" _Journal of Abnormal and Social Psychology_ 67 (1963): 371\u201378.\n\n**116 The plasticity of the self:** Paul Bloom, \"First Person Plural,\" _Atlantic_ (Nov. 2008), accessed Dec. 15, 2010, www.theatlantic.com\/magazine\/archive\/2008\/11\/first-person-plural\/7055.\n\n**117 aspirations played against their current desires:** Katherine L. Milkman, Todd Rogers, and Max H. Bazerman, \"Highbrow Films Gather Dust: Time-Inconsistent Preferences and Online DVD Rentals,\" _Management Science_ 55, no. 6 (June 2009): 1047\u201359, accessed Jan. 29, 2011, <http:\/\/opimweb.wharton.upenn.edu\/documents\/research\/Highbrow.pdf>.\n\n**117 \"want\" movies like** _**Sleepless in Seattle**_ **:** Milkman, et al., \"Highbrow Films Gather Dust.\"\n\n**118 \"nuances of what it means to be human\":** John Battelle, phone interview with author, Oct. 12, 2010.\n\n**118 Google is working on it:** Jonathan McPhie, phone interview with author, Oct. 13, 2010.\n\n**119 the \"toxic knowledge\" that might result:** Mark Rothstein, as quoted in Cynthia L. Hackerott, J.D., and Martha Pedrick, J.D., \"Genetic Information Nondiscrimination Act Is a First Step; Won't Solve the Problem,\" Oct. 1, 2007, accessed Feb. 9, www.metrocorpcounsel.com\/current.php?artType=view&artMonth=January&artYear=2011&EntryNo=7293.\n\n**119 \"The digital ghost of Jay Gatz\":** Siva Vaidyanathan, \"Naked in the 'Nonopticon,' \" _Chronicle Review_ 54, no. 23: B7.\n\n**120 \"high cognition\" arguments:** Dean Eckles, phone interview with author, Nov. 9, 2010.\n\n**120 increase the effectiveness of marketing:** Ibid.\n\n**122 pitches framed as sweepstakes:** PK List Marketing, \"Free to Me\u2014Impulse Buyers,\" accessed Jan. 28, 2011, www.pklistmarketing.com\/Data%20Cards\/Opportunity%20Seekers%20&%20Sweepstakes%20Participants\/Cards\/Free%20To%20Me%20-%20Impulse%20Buyers.htm.\n\n**123 \"smartphone to be doing searches constantly\":** Robert Andrews, \"Google's Schmidt: Autonomous, Fast Search Is 'Our New Definition,' \" _paidContent_ , Sept. 7, 2010, accessed Dec. 15, 2010, <http:\/\/paidcontent.co.uk\/article\/419-googles-schmidt-autonomous-fast-search-is-our-new-definition>.\n\n**124 \" 'Not-So-Minimal' Consequences of Television News\":** Shanto Iyengar, Mark D. Peters, and Donald R. Kinder, \"Experimental Demonstrations of the 'Not-So-Minimal' Consequences of Television News Programs,\" _American Political Science Review_ 76, no. 4 (1982): 848\u201358.\n\n**124 \"believe that defense or pollution\":** Ibid.\n\n**124 strength of this priming effect:** Drew Westen, _The Political Brain: The Role of Emotion in Deciding the Fate of the Nation_ (Cambridge, MA: Perseus, 2007).\n\n**125 study by Hasher and Goldstein:** Lynn Hasher and David Goldstein, \"Frequency and the Conference of Referential Validity,\" _Journal of Verbal Learning and Verbal Behaviour_ 16 (1977): 107\u201312.\n\n**126 \"surrounded by downward-sloping land\":** Matt Cohler, phone interview with author, Nov. 23, 2010.\n\n**128 results had been randomly redistributed:** Robert Rosenthal and Lenore Jacobson, \"Teachers' Expectancies: Determinants of Pupils' IQ Gains,\" _Psychological Reports,_ 19 (1966): 115\u201318.\n\n**129 \"network-based categorizations\":** Dalton Conley, _Elsewhere, U.S.A.: How We Got from the Company Man, Family Dinners, and the Affluent Society to the Home Office, BlackBerry Moms, and Economic Anxiety_ (New York: Pantheon, 2008), 164.\n\n**130 \"Model-T version of what's possible\":** Geoff Duncan, \"Netflix Offers $1Mln for Good Movie Picks,\" _Digital Trends,_ Oct. 2, 2006, accessed Dec. 15, 2010, www.digitaltrends.com\/computing\/netflix-offers-1-mln-for-good-movie-picks.\n\n**130 \"a PC and some great insight\":** Katie Hafner, \"And If You Liked the Movie, a Netflix Contest May Reward You Handsomely,\" _New York Times,_ Oct. 2, 2006, accessed Dec. 15, 2010, www.nytimes.com\/2006\/10\/02\/technology\/02netflix.html.\n\n**131 success using social-graph data:** Charlie Stryler **,** Marketing Panel at 2010 Social Graph Symposium, Microsoft Campus, Mountain View, CA, May 21, 2010.\n\n**132 \"the creditworthiness of your friends\":** Julia Angwin, \"Web's New Gold Mine,\" _Wall Street Journal_ , July 30, 2010, accessed on Feb. 7, 2011, <http:\/\/online.wsj.com\/article\/SB10001424052748703940904575395073512989404.html>.\n\n**133 reality doesn't work that way:** David Hume, _An Enquiry Concerning Human Understanding_ , Harvard Classics, volume 37, Section VII, Part I, online edition, (P. F. Collier & Son; 1910), accessed Feb. 7, 2011, <http:\/\/18th.eserver.org\/hume-enquiry.html>.\n\n**133 purpose of science, for Popper:** Karl Popper, _The Logic of Scientific Discovery_ (New York: Routledge, 1992).\n\n**135 \"no more incidents or adventures in the world\":** Fyodor Dostoevsky, _Notes from Underground,_ trans. Richard Pevear and Laura Volokhonsky (New York: Random House, 1994), 24.\n\n#  **Chapter Five: The Public Is Irrelevant**\n\n**137 \"others who see what we see\":** Hannah Arendt, _The Portable Hannah Arendt_ (New York: Penguin, 2000), 199.\n\n**137 \"neutralize the influence of the newspapers\":** Alexis de Tocqueville, _Democracy in America_ (New York: Penguin, 2001).\n\n**138 \"a gross violation of Chinese sovereignty\":** \"NATO Hits Chinese Embassy,\" _BBC News,_ May 8, 1999, accessed Dec. 17, 2010, <http:\/\/news.bbc.co.uk\/2\/hi\/europe\/338424.stm>.\n\n**138 \"most vital are the largely anonymous online forums\":** Tom Downey, \"China's Cyberposse,\" _New York Times,_ Mar. 3, 2010, accessed Dec. 17, 2010, www.nytimes.com\/2010\/03\/07\/magazine\/07Human-t.html?pagewanted=1.\n\n**138 \"an elite, wired section of the population\":** Shanthi Kalathil and Taylor Boas, \"Open Networks, Closed Regimes: The Impact of the Internet on Authoritarian Rule,\" _First Monday_ 8, no. 1\u20136 (2003).\n\n**139 \"Shareholders want to make money\":** Clive Thompson, \"Google's China Problem (and China's Google Problem),\" _New York Times,_ Apr. 23, 2006, accessed Dec. 17, 2010, www.nytimes.com\/2006\/04\/23\/magazine\/23google.html.\n\n**139 \"What the government cares about\":** James Fallows, \"The Connection Has Been Reset,\" _Atlantic,_ Mar. 2008, accessed Dec. 17, 2010, www.theatlantic.com\/magazine\/archive\/2008\/03\/-ldquo-the-connection-has-been-reset-rdquo\/6650.\n\n**139 \"peer pressure, and self-censorship\":** Fallows, \"Connection Has Been Reset.\"\n\n**140 \"sense that they're looking at everything\":** Thompson, \"Google's China Problem.\"\n\n**140 \"Internet Police will maintain order\":** Hong Yan, \"Image of Internet Police: JingJang and Chacha Online,\" _China Digital Times,_ Feb. 8, 2006, accessed Dec. 17, 2010, <http:\/\/chinadigitaltimes.net\/china\/internet-police\/page\/2>.\n\n**140 \"see my friends, live happily\":** Thompson, \"Google's China Problem.\"\n\n**140 \"if Internet users have some porn\":** Associated Press, \"Web Porn Seeps Through China's Great Firewall,\" July 22, 2010, accessed Dec. 17, 2010, www.cbsnews.com\/stories\/2010\/07\/22\/tech\/main6703860.shtml.\n\n**141 \"trying to nail Jell-O to the wall\":** Bill Clinton, \"America's Stake in China,\" _Blueprint_ , June 1, 2000, accessed Dec. 17, 2010, www.dlc.org\/ndol_ci.cfm?kaid=108&subid=128&contentid=963.\n\n**142 \"able to get handheld American flags?\":** Laura Miller and Sheldon Rampton, \"The Pentagon's Information Warrior: Rendon to the Rescue,\" _PR Watch_ 8, no. 4 (2001).\n\n**142 \"border patrols are] replaced by beaming patrols\":** John Rendon, as quoted in Franklin Foer, \"Flacks Americana,\" _New Republic_ , May 20, 2002, accessed Feb. 9, 2011, [www.tnr.com\/article\/politics\/flacks-americana?page=0,2.\n\n**142 thesaurus:** John Rendon, phone interview by author, Nov. 1, 2010.\n\n**143 \"consume, distribute, and create\":** Eric Schmidt and Jared Cohen, \"The Digital Disruption: Connectivity and the Diffusion of Power,\" _Foreign Affairs (_ Nov.\u2013Dec. 2010).\n\n**144 Flatow was an Olympic gymnast:** Stephen P. Halbrook, \"'Arms in the Hands of Jews Are a Danger to Public Safety': Nazism, Firearm Registration, and the Night of the Broken Glass, _St. Thomas Law Review_ 21 (2009): 109\u201341, 110, www.stephenhalbrook.com\/law_review_articles\/Halbrook_macro_final_3_29.pdf.\n\n**145 the cloud \"is actually just a handful of companies\":** Clive Thompson, interview with author, Brooklyn, NY, Aug. 13, 2010.\n\n**145 there was nowhere to go:** Peter Svensson, \"WikiLeaks Down? Cables Go Offline After Site Switches Servers,\" _Huffington Post_ , Dec. 1, 2010, accessed Feb. 9, 2011, www.huffingtonpost.com\/2010\/12\/01\/wikileaks-down-cables-go-_n_790589.html.\n\n**145 \"lose your constitutional protections immediately\":** Christopher Ketcham and Travis Kelly, \"The More You Use Google, the More Google Knows About You,\" _AlterNet,_ Apr. 9, 2010, accessed Dec. 17, 2010, www.alternet.org\/investigations\/146398\/total_information_awareness:_the_more_you_use_google,_the_more_google_knows_about_you _?page=entire.\n\n**146 \"cops will love this\":** \"Does Cloud Computing Mean More Risks to Privacy?,\" _New York Times_ , Feb. 23, 2009, accessed Feb. 8, 2011, <http:\/\/bits.blogs.nytimes.com\/2009\/02\/23\/does-cloud-computing-mean-more-risks-to-privacy>.\n\n**146 the three companies quickly complied:** Antone Gonsalves, \"Yahoo, MSN, AOL Gave Search Data to Bush Administration Lawyers,\" _Information Week_ , Jan. 19, 2006, accessed Feb. 9, 2011, www.informationweek.com\/news\/security\/government\/showArticle.jhtml?articleID=177102061.\n\n**146 predict future real-world events:** Ketcham and Kelly, \"The More You Use Google.\"\n\n**146 \"an individual must increasingly give information\":** Jonathan Zittrain, _The Future of the Internet\u2014and How to Stop It_ (New Haven: Yale University Press, 2008), 201.\n\n**147 \"an implicit bargain in our behavior\":** John Battelle, phone interview with author, Oct. 12, 2010.\n\n**147 \"redistribution of information power\":** Viktor Mayer-Schonberger, _Delete: The Virtue of Forgetting in the Digital Age_ (Princeton: Princeton University Press, 2009), 107.\n\n**148 real-world violence:** George Gerbner, \"TV Is Too Violent Even Without Executions,\" _USA Today_ , June 16, 1994, 12A, accessed Feb. 9, 2011 through LexisNexis.\n\n**149 \"who tells the stories of a culture\": \"** Fighting 'Mean World Syndrome,' \" _GeekMom_ blog, _Wired_ , Jan. 27, 2011, accessed Feb. 9, 2011, www.wired.com\/geekdad\/2011\/01\/fighting-%E2%80%9Cmean-world-syndrome%E2%80%9D\/.\n\n**149 friendly world syndrome:** Dean Eckles, \"The 'Friendly World Syndrome' Induced by Simple Filtering Rules,\" _Ready-to-Hand: Dean Eckles on People, Technology, and Inference_ blog, Nov. 10, 2010, accessed Feb. 9, 2011, www.deaneckles.com\/blog\/386_the-friendly-world-syndrome-induced-by-simple-filtering-rules\/.\n\n**149 gravitated toward Like:** \"What's the History of the Awesome Button (That Eventually Became the Like Button) on Facebook?\" Quora Forum, accessed Dec. 17, 2010, www.quora.com\/Facebook-company\/Whats-the-history-of-the-Awesome-Button-that-eventually-became-the-Like-button-on-Facebook.\n\n**151 \"against the cruise line industry\":** Hollis Thomases, \"Google Drops Anti-Cruise Line Ads from AdWords,\" Web Ad.vantage, Feb. 13, 2004, accessed Dec. 17, 2010, www.webadvantage.net\/webadblog\/google-drops-anti-cruise-line-ads-from-adwords-338.\n\n**151\u201352 identify who was persuadable:** \"How Rove Targeted the Republican Vote,\" _Frontline_ , accessed Feb. 8, 2011, www.pbs.org\/wgbh\/pages\/frontline\/shows\/architect\/rove\/metrics.html.\n\n**152 \"Amazon's recommendation engine is the direction\":** Mark Steitz and Laura Quinn, \"An Introduction to Microtargeting in Politics,\" accessed Dec. 17, 2010, www.docstoc.com\/docs\/43575201\/An-Introduction-to-Microtargeting-in-Politics.\n\n**153 round-the-clock \"war room\":** \"Google's War Room for the Home Stretch of Campaign 2010,\" e.politics, Sept. 24, 2010, accessed Feb. 9, 2011, www.epolitics.com\/2010\/09\/24\/googles-war-room-for-the-home-stretch-of-campaign-2010\/.\n\n**155 \"campaign wanted to spend on Facebook\":** Vincent R. Harris, \"Facebook's Advertising Fluke,\" _TechRepublican_ , Dec. 21, 2010, accessed Feb. 9, 2011, <http:\/\/techrepublican.com\/free-tagging\/vincent-harris>.\n\n**155 have the ads pulled off the air:** Monica Scott, \"Three TV Stations Pull 'Demonstrably False' Ad Attacking Pete Hoekstra,\" _Grand Rapids Press,_ May 28, 2010, accessed Dec. 17, 2010, www.mlive.com\/politics\/index.ssf\/2010\/05\/three_tv_stations_pull_demonst.html.\n\n**157 \"improve the likelihood that a registered Republican\":** Bill Bishop, _The Big Sort: Why the Clustering of Like-Minded America Is Tearing Us Apart_ (New York: Houghton Mifflin, 2008), 195.\n\n**157 \"likely to be most salient in the politics\":** Ronald Inglehart, _Modernization and Postmodernization_ (Princeton: Princeton University Press, 1997), 10.\n\n**159 Pabst began to sponsor hipster events:** Neal Stewart, \"Marketing with a Whisper,\" _Fast Company,_ Jan. 11, 2003, accessed Jan. 30, 2011, www.fastcompany.com\/fast50_04\/winners\/stewart.html.\n\n**159 \"$44 in US currency\":** Max Read, \"Pabst Blue Ribbon Will Run You $44 a Bottle in China,\" _Gawker_ , July 21, 2010, accessed Feb. 9, 2011, <http:\/\/m.gawker.com\/5592399\/pabst-blue-ribbon-will-run-you-44-a-bottle-in-china>.\n\n**160 \"I serve as a blank screen\":** Barack Obama, _The Audacity of Hope: Thoughts on Reclaiming the American Dream_ (New York: Crown, 2006), 11.\n\n**161 \"We lose all perspective\":** Ted Nordhaus, phone interview with author, Aug. 31, 2010.\n\n**162 \"the source is basically in thought\":** David Bohm, _Thought as a System_ (New York: Routledge, 1994) 2.\n\n**163 \"participants in a pool of common meaning\":** David Bohm, _On Dialogue_ (New York: Routledge, 1996), x\u2013xi.\n\n**164 \"define and express its interests\":** John Dewey, _The Public and Its Problems_ (Athens, OH: Swallow Press, 1927), 146.\n\n#  **Chapter Six: Hello, World!**\n\n**165 \"no intelligence or skill in navigation\":** Plato, _First Alcibiades_ , in _The Dialogues of Plato,_ vol. 4, trans. Benjamin Jowett (Oxford, UK: Clarendon Press, 1871), 559.\n\n**166 \"We are as Gods\":** Stewart Brand, _Whole Earth Catalog_ (self-published, 1968), accessed Dec. 16, 2010, <http:\/\/wholeearth.com\/issue\/1010\/article\/195\/we.are.as.gods>.\n\n**167 \"make any man (or woman) a god\":** Steven Levy, _Hackers: Heroes of the Computer Revolution_ (New York: Penguin, 2001), 451.\n\n**167 \"having some troubles with my family\":** \"How Eliza Works,\" accessed Dec. 16, 2010, <http:\/\/chayden.net\/eliza\/instructions.txt>.\n\n**168 \"way of acting without consequence\":** Siva Vaidyanathan, phone interview with author, Aug. 9, 2010.\n\n**168 \"not a very good program\":** Douglas Rushkoff, interview with author, New York, NY, Aug. 25, 2010.\n\n**168 \"politics tends to be seen by programmers\":** Gabriella Coleman, \"The Political Agnosticism of Free and Open Source Software and the Inadvertent Politics of Contrast,\" _Anthropological Quarterly_ , 77, no. 3 (Summer 2004): 507\u201319, Academic Search Premier, EBSCOhost _._\n\n**170 \"addictive control as well\":** Levy, _Hackers,_ 73.\n\n**172 \"Howdy\" is a better opener than \"Hi\":** Christian Rudder, \"Exactly What to Say in a First Message,\" Sept. 14, 2009, accessed Dec. 16, 2010, <http:\/\/blog.okcupid.com\/index.php\/online-dating-advice-exactly-what-to-say-in-a-first-message>.\n\n**173 \"hackers don't tend to know any of that\":** Steven Levy, \"The Unabomber and David Gelernter,\" _New York Times,_ May 21, 1995, accessed Dec. 16, 2010, www.unabombers.com\/News\/95-11-21-NYT.htm.\n\n**174 \"engineering relationships among people\":** Langdon Winner, \"Do Artifacts Have Politics?\" _Daedalus_ 109, no. 1 (Winter 1980): 121\u201336.\n\n**175 \"code is law\":** Lawrence Lessig, _Code_. (New York: Basic Books, 2006).\n\n**175 \"choose structures for technologies\":** Winner, \"Do Artifacts Have Politics.\"\n\n**176 Hacker Jargon File:** The Jargon File, Version 4.4.7, Appendix B. A Portrait of J. Random Hacker, accessed Feb. 9, 2011, <http:\/\/linux.rz.ruhr-uni-bochum.de\/jrgon\/html\/politics.html>.\n\n**177 \"social utility\" as if it's a twenty-first-century phone company:** Mark Zuckerberg executive bio, Facebook press room, accessed on Feb. 8, 2011, <http:\/\/www.facebook.com\/press\/info.php?execbios>.\n\n**178 \"come to Google because they choose to\":** Greg Jarboe, \"A 'Fireside Chat' with Google's Sergey Brin,\" Search Engine Watch, Oct. 16, 2003, accessed Dec. 16,2010, <http:\/\/searchenginewatch.com\/3081081>.\n\n**178 \"the future will be personalized\":** Gord Hotckiss, \"Just Behave: Google's Marissa Mayer on Personalized Search,\" Searchengineland, Feb. 23, 2007, accessed Dec. 16, 2010, <http:\/\/searchengineland.com\/just-behave-googles-marissa-mayer-on-personalized-search-10592>.\n\n**179 \"It's technology, not business or government\":** David Kirpatrick, \"With a Little Help from his Friends,\" _Vanity Fair_ (Oct. 2010), accessed Dec. 16, 2010, www.vanityfair.com\/culture\/features\/2010\/10\/sean-parker-201010.\n\n**179 \"seventh kingdom of life\":** Kevin Kelly, _What Technology Wants_ (New York: Viking, 2010).\n\n**180 \"shirt or fleece that I own\":** Mark Zuckerberg, remarks to Startup School Conference, _XConomy_ , Oct. 18, 2010, accessed Feb. 8, 2010, www.xconomy.com\/san-francisco\/2010\/10\/18\/mark-zuckerberg-goes-to-startup-school-video\/\/.\n\n**181 \" 'the rest of the world is wrong' \":** David A. Wise and Mark Malseed, _The Google Story_ (New York: Random House, 2005), 42.\n\n**182 \"tradeoffs with success in other domains\":** Jeffrey M. O'Brien, \"The PayPal Mafia,\" _Fortune,_ Nov. 14, 2007, accessed Dec. 16, 2010, <http:\/\/money.cnn.com\/2007\/11\/13\/magazines\/fortune\/paypal_mafia.fortune\/index2.htm>.\n\n**183 sold to eBay for $1.5 billion:** Troy Wolverton, \"It's official: eBay Weds PayPal,\" _CNET News,_ Oct. 3, 2002, accessed Dec. 16, 2010, <http:\/\/news.cnet.com\/Its-official-eBay-weds-PayPal\/2100-1017_3-960658.html>.\n\n**183 \"impact and force change\":** Peter Thie, \"Education of a Libertarian,\" _Cato Unbound_ , Apr. 13, 2009, accessed Dec. 16, 2010, www.cato-unbound.org\/2009\/04\/13\/peter-thiel\/the-education-of-a-libertarian.\n\n**183 \"end the inevitability of death and taxes\":** Chris Baker, \"Live Free or Drown: Floating Utopias on the Cheap,\" _Wired,_ Jan. 19, 2009, accessed Dec. 16, 2010, www.wired.com\/techbiz\/startups\/magazine\/17-02\/mf_seasteading?currentPage=all.\n\n**183 \" 'capitalist democracy' into an oxymoron\":** Thiel, \"Education of a Libertarian.\"\n\n**184 \"makes a living being against computers\":** Nicholas Carlson, \"Peter Thiel Says Don't Piss Off the Robots (or Bet on a Recovery),\" _Business Insider,_ Nov. 18, 2009, accessed Dec. 16, 2010, www.businessinsider.com\/peter-thiel-on-obama-ai-and-why-he-rents-his-mansion-2009-11#.\n\n**184 \"which technologies to foster\":** Ronald Bailey, \"Technology Is at the Center,\" Reason.com, May 2008, accessed Dec. 16, 2010, <http:\/\/reason.com\/archives\/2008\/05\/01\/technology-is-at-the-center\/singlepage>.\n\n**184 \"way I think about the business\":** Deepak Gopinath, \"PayPal's Thiel Scores 230 Percent Gain with Soros-Style Fund,\" CanadianHedgeWatch .com _,_ Dec. 4, 2006, accessed Jan. 30, 2011, at www.canadianhedgewatch.com\/content\/news\/general\/?id=1169.\n\n**184 \"that voting will make things better\":** Peter Thiel, \"Your Suffrage Isn't in Danger. Your Other Rights Are,\" _Cato Unbound,_ May 1, 2009, accessed Dec. 16, 2010, www.cato-unbound.org\/2009\/05\/01\/peter-thiel\/your-suffrage-isnt-in-danger-your-other-rights-are.\n\n**185 talked to Scott Heiferman:** Interview with author, New York, NY, Oct. 5, 2010.\n\n**188 \"good or bad, nor is it neutral\":** Melvin Kranzberg, \"Technology and History: 'Kranzberg's Laws,' \" _Technology and Culture_ 27, no. 3 (1986): 544\u201360.\n\n#  **Chapter Seven: What You Want, Whether You Want It or Not**\n\n**189 \"millions of people doing complicated things\":** Noah Wardrip-Fruin and Nick Montfort, _The New Media Reader,_ Vol. 1 (Cambridge: MIT Press, 2003), 8.\n\n**189 \"yet to be completely correlated\":** Isaac Asimov, _The Intelligent Man's Guide to Science_ (New York: Basic Books, 1965),\n\n**190 \"you've got a problem\":** Bill Jay, phone interview with author, Oct. 10, 2010.\n\n**191 ads tailored to her:** Jason Mick, \"Tokyo's 'Minority Report' Ad Boards Scan Viewer's Sex and Age,\" Daily Tech, July 16, 2010, accessed Dec. 17, 2010, www.dailytech.com\/Tokyos+Minority+Report+Ad+Boards+Scan+Viewers+Sex+and+Age\/article19063.htm.\n\n**191 the future of art:** David Shields, _Reality Hunger: A Manifesto_ (New York: Knopf, 2010). Credit to Michiko Kakutani, whose review led me to this book.\n\n**193 interrogated by a virtual agent:** M. Ryan Calo, \"People Can Be So Fake: A New Dimension to Privacy and Technology Scholarship,\" _Penn State Law Review_ 114 , no. 3 (2010): 810\u201355.\n\n**193 Kismet increased donations by 30 percent:** Vanessa Woods, \"Pay Up, You Are Being Watched,\" _New Scientist,_ Mar. 18, 2005, accessed Dec. 17, 2010, www.newscientist.com\/article\/dn7144-pay-up-you-are-being-watched.html.\n\n**193 \"Computers programmed to be polite\":** Calo, \"People Can Be So Fake.\"\n\n**194 \"not evolved to twentieth-century technology\":** Ibid.\n\n**195 identity and criminal record in seconds:** Maureen Boyle, \"Video: Catching Criminals? Brockton Cops Have an App for That,\" _Brockton Patriot Ledger,_ June 15, 2010, accessed Dec. 17, 2010, www.patriotledger.com\/news\/cops_and_courts\/x1602636300\/Catching-criminals-Cops-have-an-app-for-that.\n\n**195 \"other images of you with ninety-five percent accuracy\":** Jerome Taylor, \"Google Chief: My Fears for Generation Facebook,\" _Independent,_ Aug. 18, 2010, accessed Dec. 17, 2010, www.independent.co.uk\/life-style\/gadgets-and-tech\/news\/google-chief-my-fears-for-generation-facebook-2055390.html .\n\n**197 \"The future is already here\":** William Gibson, interview on NPR's _Fresh Air,_ Aug. 31, 1993, accessed Dec. 17, 2010, www.npr.org\/templates\/story\/story.php?storyId=1107153.\n\n**197 your identity already tagged:** \"RFID Bracelet Brings Facebook to the Real World,\" Aug. 20, 2010, accessed Dec. 17, 2010, www.psfk.com\/2010\/08\/rfid-bracelet-brings-facebook-to-the-real-world.html.\n\n**198 \"real world that can be indexed\":** Reihan Salam, \"Why Amazon Will Win the Internet,\" _Forbes,_ July 30, 2010, accessed Dec. 17, 2010, www.forbes.com\/2010\/07\/30\/amazon-kindle-economy-environment-opinions-columnists-reihan-salam.html.\n\n**198 \"some have termed 'smart dust' \":** David Wright, Serge Gutwirth, Michael Friedewald, Yves Punie, and Elena Vildjiounaite, _Safeguards in a World of Ambient Intelligence_ (Berlin\/Dordrecht: Springer Science, 2008): abstract.\n\n**199 four-year joint effort:** Google\/Harvard press release. \"Digitized Book Project Unveils a Quantitative 'Cultural Genome,' \" accessed Feb. 8, 2011, <http:\/\/www.seas.harvard.edu\/news-events\/news-archive\/2010\/digitized-books>.\n\n**200 \"censorship and propaganda\":** Ibid.\n\n**200 nearly sixty languages:** Google Translate Help Page, accessed Feb. 8, 2011, <http:\/\/translate.google.com\/support\/?hl=en>.\n\n**201 better and better:** Nikki Tait, \"Google to translate European patent claims,\" _Financial Times_ , Nov. 29, 2010, accessed Feb. 9, 2010, www.ft.com\/cms\/s\/0\/02f71b76-fbce-11df-b79a-00144feab49a.html.\n\n**202 \"what to do with them\":** Danny Sullivan, phone interview with author, Sept. 10, 2010.\n\n**202 \"flash crash\":** Graham Bowley, \"Stock Swing Still Baffles, with an Ominous Tone,\" _New York Times,_ Aug. 22, 2010, accessed Feb. 8, 2010, www.nytimes.com\/2010\/08\/23\/business\/23flash.html.\n\n**202 provocative article in** _**Wired**_ **:** Chris Anderson, \"The End of Theory: The Data Deluge Makes the Scientific Method Obsolete,\" _Wired_ , June 23, 2008, accessed Feb. 10, 2010, <http:\/\/www.wired.com\/science\/discoveries\/magazine\/16-07\/pb_theory>.\n\n**203 greatest achievement of human technology:** Hillis quoted in Jennifer Riskin, _Genesis Redux: Essays in the History and Philosophy of Artificial Life_ (Chicago: University of Chicago Press, 2007), 200.\n\n**204 \"advertiser-funded media\":** Marisol LeBron, \" 'Migracorridos': Another Failed Anti-immigration Campaign,\" North American Congress of Latin America, Mar. 17, 2009, accessed Dec. 17, 2010, https:\/\/nacla.org\/node\/5625.\n\n**205 characters using the companies' products throughout:** Mary McNamara, \"Television Review: 'The Jensen Project,' \" _Los Angeles Times,_ July 16, 2010, accessed Dec. 17, 2010, <http:\/\/articles.latimes>. com\/2010\/jul\/16\/entertainment\/la-et-jensen-project-20100716.\n\n**205 product-placement hooks throughout:** Jenni Miller, \"Hansel and Gretel in 3D? Yeah, Maybe.\" _Moviefone_ blog _,_ July 19, 2010, accessed Dec. 17, 2010, <http:\/\/blog.moviefone.com\/2010\/07\/19\/hansel-and-gretel-in-3d-yeah-maybe>.\n\n**205 the corporate owner of Lipslicks:** Motoko Rich, \"Product Placement Deals Make Leap from Film to Books,\" _New York Times,_ Nov. 9, 2008, accessed Dec. 17, 2010, www.nytimes.com\/2008\/02\/19\/arts\/19iht-20bookplacement.10177632.html?pagewanted=all.\n\n**207 increase \"purchase intentions\" by 21 percent:** John Hauser and Glen Urban, \"When to Morph,\" Aug. 2010, accessed Dec. 17, 2010, <http:\/\/web.mit.edu\/hauser\/www\/Papers\/Hauser-Urban-Liberali_When_to_Morph_Aug_2010.pdf>.\n\n**207 \"turn it into useful information\":** Jane Wardell, \"Raytheon Unveils Scorpion Helmet Technology,\" Associated Press, July 23, 2010, accessed Dec. 17, 2010 at www.boston.com\/business\/articles\/2010\/07\/23\/raytheon_unveils_scorpion_helmet_technology.\n\n**208 \"turns the whole world into a display\":** Wardell, \"Raytheon Unveils Scorpion Helmet Technology.\"\n\n**208 TV experience overlaid on a real game:** Michael Schmidt, \"To Pack a Stadium, Provide Video Better Than TV,\" _New York Times,_ July 28, 2010, accessed Dec. 17, 2010, www.nytimes.com\/2010\/07\/29\/sports\/football\/29stadium.html?_r=1.\n\n**208 AugCog, which uses cognitive neuroscience:** Augmented Cognition International Society Web site, accessed Dec. 17, 2010, www.augmentedcognition.org.\n\n**209 500 percent increase in working memory:** \"Computers That Read Your Mind,\" _Economist,_ Sept. 21, 2006, accessed Dec. 17, 2010, www.economist.com\/node\/7904258?story_id=7904258.\n\n**209 at least sixteen different ways:** Gary Hayes, \"16 Top Augmented Reality Business Models,\" _Personalize Media_ (Gary Hayes's blog), Sept. 14, 2009, accessed Dec. 17, 2010, www.personalizemedia.com\/16-top-augmented-reality-business-models.\n\n**210 solve problems for people:** Chris Coyne, interview with author, New York, NY, Oct. 6, 2010.\n\n**211 \"reality\" is \"one of the few words\":** Vladimir Nabokov, _Lolita_ (New York: Random House, 1997), 312.\n\n**213 powering the marketing campaigns:** David Wright et al., _Safeguards in a World of Ambient Intelligence_ (London: Springer, 2008), 66, accessed through Google eBooks, Feb. 8, 2011.\n\n**214 \"machines make more of their decisions\":** Bill Joy, \"Why the Future Doesn't Need Us,\" _Wired_ (Apr. 2000) accessed Dec. 17, 2010, www.wired.com\/wired\/archive\/8.04\/joy.html.\n\n#  **Chapter Eight: Escape from the City of Ghettos**\n\n**217 \"the nature of his own person\":** Christopher Alexander et al., _A Pattern Language_ (New York: Oxford University Press, 1977), 8.\n\n**217 \"Long Live the Web\" Sir Tim Berners-Lee, \"Long Live the Web:** A Call for Continued Open Standards and Neutrality,\" _Scientific American_ , Nov. 22, 2010.\n\n**219 \"need to address the core issues\":** Bill Joy, phone interview with author, Oct. 1 2010.\n\n**220 ideal nook for kids:** Alexander et al., _A Pattern Language_ , 445, 928\u201329.\n\n**220 \"distinct pattern language\":** Ibid., xvi.\n\n**220 \"city of ghettos\":** Ibid., 41\u201343.\n\n**221 \"dampens all significant variety\":** Ibid., 43.\n\n**221 \"move easily from one to another\":** Ibid., 48.\n\n**221 \"support for his idiosyncrasies\":** Ibid.\n\n**222 \"psychological equivalent of obesity\":** danah boyd. \"Streams of Content, Limited Attention: The Flow of Information through Social Media,\" _Web2.0 Expo._ New York, NY: Nov. 17, 2007, accessed July 19, 2008, www.danah.org\/papers\/talks\/Web2Expo.html.\n\n**223 how to build a better mousetrap:** \"A Better Mousetrap,\" _This American Life_ no. 366, aired Oct. 10, 2008, www.thisamericanlife.org\/radio-archives\/episode\/366\/a-better-mousetrap-2008.\n\n**223 you'll catch your mouse:** Ibid.\n\n**223 \"jumping out of that recursion loop\":** Matt Cohler, phone interview with author, Nov. 23, 2010.\n\n**226 organ donation rates in different European countries:** Dan Ariely as quoted in Lisa Wade, \"Decision Making and the Options We're Offered,\" _Sociological Images blog,_ Feb. 17, 2010, accessed Dec. 17, 2010, <http:\/\/thesocietypages.org\/socimages\/2010\/02\/17\/decision-making-and-the-options-were-offered\/>.\n\n**229 \"only when regulation is transparent\":** Lawrence Lessig, _Code_ (New York: Basic Books, 2006), 260, http:\/\/books.google.com\/books?id=lmXIMZiU8yQC&pg=PA260&lpg=PA260&dq=lessig+political+response+transparent+code&source=bl&ots=wR0WRuJ61u&sig=iSIiM0pnEaf-o5VPvtGcgXXEeL8&hl=en&ei=1bI0TfykGsH38Ab7-tDJCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBcQ6AEwAA#v=onepage&q&f=false.\n\n**230 \"one of the world's worst kept secrets\":** Amit Singhal, \"Is Google a Monopolist? A Debate,\" Opinion Journal, _Wall Street Journal,_ Sept. 17, 2010, <http:\/\/online.wsj.com\/article\/SB10001424052748703466704575489582364177978.html?mod=googlenews_wsj#U301271935944OEB>.\n\n**231 \"honest and objective about ourselves\":** \"Philip Foisie's memos to the management of the _Washington Post_ ,\" Nov. 10, 1969, accessed Dec. 20, 2010, <http:\/\/newsombudsmen.org\/articles\/origins\/article-1-mcgee>.\n\n**231 \"the common good\":** Arthur Nauman, \"News Ombudsmanship: Its Theory and Rationale,\" Press Regulation: How Far Has it Come? symposium, Seoul, South Korea, June 1994.\n\n**232 that this expectation is one that ... most Americans share:** Jeffrey Rosen, \"The Web Means the End of Forgetting,\" _New York Times Magazine_ , July 21, 2010, www.nytimes.com\/2010\/07\/25\/magazine\/25privacy-t2.html?_r=1 &pagewanted=all.\n\n**235 \"help it find a larger audience\":** Author interview with confidential source.\n\n**237 Google is just a company:** \"Transcript: Stephen Colbert Interviews Google's Eric Schmidt on _The Colbert Report,_ \" Search Engine Land _,_ Sept. 22, 2010, accessed Dec. 20, 2010, <http:\/\/searchengineland.com\/googles-schmidt-colbert-report-51433>.\n\n**237 expose their audiences to both sides:** Cass R. Sunstein, _Republic .com_ (Princeton: Princeton University Press, 2001).\n\n**240 \"we shouldn't have to accept\":** Caitlin Petre phone interview with Marc Rotenberg, Nov. 5, 2010.\n\n**241 and 70 percent do:** \"Mistakes Do Happen: Credit Report Errors Mean Consumers Lose,\" US PIRG, accessed Feb. 8, 2010, <http:\/\/www.uspirg.org\/home\/reports\/report-archives\/financial-privacy\u2013security\/financial-privacy-security\/mistakes-do-happen-credit-report-errors-mean-consumers-lose>.\n**INDEX**\n\n##\n\naccessibility bias\n\n_Act of Creation, The_ (Koestler)\n\nAcxiom\n\nAdderall\n\nadvertars\n\nadvertiser-funded media (AFM)\n\nadvertising\n\naugmented reality and\n\nbrand fragmentation and\n\nday-parting and\n\ndisclosure of personalization in\n\nin social spaces\n\non television\n\nAfghanistan\n\nagents:\n\nhumanlike\n\nintelligent\n\nAlexander, Christopher\n\nalgorithms\n\nCineMatch\n\nEdgeRank\n\nGoogle search\n\nOkCupid\n\nPageRank\n\npolitical districts and\n\nAmazon\n\nKindle\n\nWeb Services\n\nambient intelligence\n\nAmericans for Job Security\n\nAnderson, Chris\n\nAngleton, James Jesus\n\nanonymity\n\nAnti, Michael\n\nApple\n\nNewton\n\narchitecture and design\n\nArendt, Hannah\n\nargument styles\n\nAriely, Dan\n\nArnold, Stephen\n\nart\n\nAsimov, Isaac\n\nAT&T\n\n_Atlantic_\n\nattention crash\n\naugmented cognition (AugCog)\n\naugmented reality\n\n##\n\nBarlow, John Perry\n\nBattelle, John\n\nBay, Michael\n\nbehavioral retargeting\n\nBell, Gordon\n\nBenkler, Yochai\n\nBerners-Lee, Tim\n\nBezos, Jeff\n\nBharat, Krishna\n\nBhat, Tapan\n\nBing\n\nBishop, Bill\n\nBlades, Joan\n\nblogs\n\nBlueCava\n\nBlueKai\n\nBohm, David\n\nBohr, Niels\n\nbooks\n\nadvertising in\n\ndigitized\n\nBosworth, Andrew\n\n_Bowling Alone_ (Putnam)\n\nboyd, danah\n\nBoyd, Wes\n\nBP\n\nbrain\n\nBrand, Stewart\n\nbrand fragmentation\n\nbridges\n\nBrin, Sergey\n\nBurnham, Brad\n\nBurnham, Terence\n\nBush, George W.\n\nBush, Vannevar\n\nBuzzfeed\n\n##\n\nCalifornia\n\nCalo, Ryan\n\nCampbell, Donald\n\nCaro, Robert A.\n\nCatalist\n\ncategories, wide\n\ncensorship\n\nChait, Jon\n\nChina\n\nInternet police in\n\nPabst in\n\nCIA\n\nCineMatch\n\ncities\n\narchitecture and design in\n\nClarium\n\nclick signals\n\nClinton, Bill\n\ncloud\n\nCoca-Cola Village Amusement Park\n\ncode and programmers\n\ncoding, conceptual\n\nCohen, Claudia\n\nCohler, Matt\n\nColeman, Gabriella\n\ncollaborative filtering\n\nconfirmation bias\n\nConley, Dalton\n\ncookies\n\nCort\u00e9s, Hern\u00e1n\n\nCoyne, Chris\n\ncraigslist\n\ncreativity and innovation\n\ncredit reports\n\nCropley, Arthur\n\ncurators, _see_ editors and curators\n\ncuriosity\n\ncybernetics\n\n##\n\nDARPA\n\ndata laundering\n\ndating sites\n\nOkCupid\n\nday-parting\n\nDe Castro, Henrique\n\ndefaults\n\ndemocracy\n\ndialogue and\n\ndesign and architecture\n\nDewey, John\n\ndialogue\n\nDigg\n\nDirectLife\n\ndiscovery\n\ndisintermediation\n\nDixon, Pam\n\nDNA\n\nDo Not Track list\n\nDostoyevsky, Fyodor\n\nDowney, Tom\n\nDuncker, Karl\n\nDyson, Esther\n\n##\n\nEchoMetrix\n\nEckles, Dean\n\nEdgeRank\n\neditors and curators\n\nEliza\n\ne-mail\n\nconstitutional protections and\n\nGmail\n\nengineers\n\nErowid\n\nEurope\n\nevolution\n\nEysenck, Hans\n\n##\n\nFacebook\n\nadvertisements and\n\nEdgeRank and\n\nEverywhere\n\nGoogle and\n\nidentity and\n\nLike button on\n\nlocal-maximum problem and\n\nlock-in and\n\nNews Feeds on\n\npolitical advertising and\n\npolitical involvement and\n\nprivacy policy of\n\nTwitter compared with\n\nfacial recognition\n\nFair Credit Reporting Act\n\nFair Information Practices\n\nFallows, James\n\nFarah, Martha\n\nFBI\n\nFlatow, Alfred\n\nFoer, Josh\n\nFoisie, Philip\n\nFounder's Fund\n\nFoursquare\n\nFrance\n\nFried, Charles\n\nFriedman, Patri\n\nFriedman, Tom\n\nfriendly world syndrome\n\nFriendster\n\n_From Counterculture to Cyberculture_ (Turner)\n\nfundamental attribution error\n\n##\n\nGawker\n\ngeeks\n\nGelernter, David\n\nGellman, Robert\n\ngenetic data\n\nGerbner, George\n\nGibson, William\n\nGilbert, Dan\n\nGlass, Ira\n\nGmail\n\nGoogle\n\nChina and\n\ndashboard of\n\ndigitized books and\n\nDocs\n\n\"Don't be evil\" slogan of\n\nethics and\n\nFacebook and\n\nfacial recognition and\n\nGmail\n\nInstant\n\nlock-in and\n\nNews\n\nOceana and\n\nPageRank\n\nPicasa\n\npolitical advertising and\n\npolitical involvement and\n\nReader\n\nResearch\n\nsearch algorithm of\n\nTranslate\n\nVoice\n\ngovernment\n\nGraber, Doris\n\ngun registration\n\n##\n\nHabermas, Jurgen\n\n_Hackers_ (Levy)\n\nhackers, hacking\n\nHare, Brian\n\nHarris, Vincent\n\nHastings, Reed\n\nHauser, John\n\nHayes, Gary\n\nHeiferman, Scott\n\nHeuer, Richards\n\nHillis, Danny\n\nHoekstra, Pete\n\n_Huffington Post_\n\nhumanlike agents\n\nHume, David\n\n##\n\nIBM\n\nidentity\n\nidentity loops\n\ninduction\n\ninfomercials\n\ninformation gap\n\nInglehart, Ron\n\nInstitute\n\nintelligent agents\n\nInteractive Advertising Bureau (IAB)\n\niPhone\n\nIQ\n\nIraq\n\ni-traffic\n\niTunes\n\nIyengar, Shanto\n\n##\n\njet pilots\n\nJiang Zemin\n\nJobs, Steve\n\nJohnson, Steven\n\nJoy, Bill\n\n##\n\nKaczinski, Ted\n\nKafka, Franz\n\nKalathil, Shanthi\n\nKane, Patrick\n\nKantorovich, Aharon\n\nKatona, George\n\nKayak\n\nKazmaier, Dick\n\nKekule, Friedrich\n\nKeller, Bill\n\nKelly, Kevin\n\nKennedy, John F.\n\nKirkpatrick, David\n\nKlein, Paul\n\nKoestler, Arthur\n\nKranzberg, Melvin\n\nKrohn, Max\n\nKwan, Julianne\n\n##\n\nlanguage translation\n\nLanier, Jaron\n\n_Last Question, The_ (Asimov)\n\nLeanBack\n\nlearning\n\nmachine\n\nleast objectionable programming\n\nLee, Kai-Fu\n\nLessig, Larry\n\nLevy, Steven\n\nLewis, Andrew\n\n_Liberty and the News_ (Lippmann)\n\nLiebling, A. J.\n\nLinkedIn\n\nLinux\n\nLippmann, Walter\n\nLivingston, Jessica\n\nlocal-maximum problem\n\nlock-in\n\n\"Long Live the Web\" (Berners-Lee)\n\nLoopt\n\nLovell, Todd\n\nLowenstein, George\n\nluck\n\nLynch, Zack\n\n##\n\nMa, Jack\n\nMacDougald, Harry\n\nmachine learning\n\nMancini, Paul\n\nMark, David\n\nMayer, Marissa\n\nMayer-Schonberger, Viktor\n\nMcLarty, Mack\n\nMcLuhan, Marshall\n\nMcPhie, Jonathan\n\nMeadowlands\n\nmeaning threats\n\nmean world syndrome\n\nMeetUp\n\nMetcalfe, Bob\n\nMetcalfe's law\n\nMicrosoft\n\nBob\n\nLive\n\nmiddleman, elimination of\n\n_Migra corridos_\n\nMilgram, Stanley\n\nMill, John Stuart\n\n_Minority Report_\n\nMoore's law\n\nMORIS\n\nMoses, Robert\n\nmousetraps\n\nMoveOn\n\nMulvenon, James\n\nMySpace\n\n##\n\nNabokov, Vladimir\n\nNational Rifle Association (NRA)\n\nNational Security Agency (NSA)\n\nNauman, Arthur\n\nNe'eman, Yuval\n\nNegroponte, Nicholas\n\nNemeth, Charlan\n\nNetflix\n\nNetflix Challenge\n\nNewmark, Craig\n\nnews\n\nFacebook feeds\n\nGoogle News\n\npeople-powered\n\nYahoo News\n\nnewspapers\n\neditorial ethics and\n\nombudsmen and\n\n_New York Times_\n\n9\/11\n\nNordhaus, Ted\n\nNorvig, Peter\n\nNosenko, Yuri\n\n##\n\nObama, Barack\n\nOceana\n\n\"Of Sirens and Amish Children\" (Benkler)\n\nOkCupid\n\nOlson, Carrie\n\nOswald, Lee Harvey\n\noverfitting\n\n##\n\nPabst\n\nPage, Larry\n\nPageRank\n\nPalmer, Chris\n\nPalo Alto Research Center (PARC)\n\nPandora\n\nPanelDirector\n\nPareles, Jon\n\nParker, Sean\n\n_Pattern Language, A_ (Alexander et al.)\n\nPayPal\n\nPeekYou\n\npersuasion profiling\n\n_Phantom Public, The_ (Lippmann)\n\nPhilby, Kim\n\nPhorm\n\nPiaget, Jean\n\nPicasa\n\nPicasso, Pablo\n\nPK List Management\n\nPlato\n\npolitics\n\nelectoral districts and\n\npartisans and\n\nprogrammers and\n\nvoting\n\nPopper, Karl\n\npostmaterialism\n\npredictions\n\npresent bias\n\npriming effect\n\nprivacy\n\nFacebook and\n\nfacial recognition and\n\ngenetic\n\nProcter & Gamble\n\nproduct recommendations\n\nProulx, Travis\n\nPulitzer, Joseph\n\npush technology and pull technology\n\nPutnam, Robert\n\n##\n\nQiang, Xiao\n\n##\n\nRapleaf\n\nRather, Dan\n\nRaz, Guy\n\nreality\n\naugmented\n\n_Reality Hunger_ (Shields)\n\nReddit\n\nRendon, John\n\n_Republic.com_ (Sunstein)\n\nretargeting\n\nRFID chips\n\nrobots\n\nRodriguez de Montalvo,\n\nGarci\n\n_Rolling Stone_\n\nRoombas\n\nRotenberg, Marc\n\nRothstein, Mark\n\nRove, Karl\n\nRoyal Caribbean\n\nRubel, Steve\n\nRubicon Project\n\nRumsfeld, Donald\n\nRushkoff, Douglas\n\n##\n\nSalam, Reihan\n\nSandberg, Sheryl\n\nschemata\n\nSchmidt, Eric\n\nSchudson, Michael\n\nSchulz, Kathryn\n\nscience\n\n_Scientific American_\n\nScorpion\n\nsentiment analysis\n\nSentry\n\nserendipity\n\nShields, David\n\nShirky, Clay\n\nSiegel, Lee\n\nsignals\n\nclick\n\nSimonton, Dean\n\nSinghal, Amit\n\n_Sleepwalkers, The_ (Koestler)\n\nsmart devices\n\nSmith, J. 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+{"text":" \nEditor\n\nTheodore Abraham\n\nCase Based EchocardiographyFundamentals And Clinical Practice\n\nEditor\n\nTheodore Abraham\n\nDepts. Neurosurgery & Neurology, John Hopkins Hospital, Baltimore, Maryland, USA\n\nISBN 978-1-84996-150-9e-ISBN 978-1-84996-151-6\n\nDOI 10.1007\/978-1-84996-151-6\n\nSpringer London Dordrecht Heidelberg New York\n\nBritish Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library\n\n\u00a9 Springer-Verlag London Limited 2011\n\nApart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.\n\nThe use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.\n\nProduct liability: The publisher can give no guarantee for information about drug dosage and application thereof contained in this book. In every individual case the respective user must check its accuracy by consulting other pharmaceutical literature.\n\nPrinted on acid-free paper\n\nSpringer is part of Springer Science+Business Media (www.springer.com)\n\nPreface\n\nEchocardiography remains the most commonly used cardiac imaging modality in clinical care despite the emergence of competing technologies such as magnetic resonance and computed tomography. Indeed, it is now considered an extension of the stethoscope and physicians without a primary interest in cardiac imaging, such as interventional cardiologists and internists, often view and interpret echocardiograms. Such an expansion of use has been further driven by the availability of smaller devices including some that are only slightly larger than a smartphone. Coincident with these developments there have been several technology advances in echocardiography such as tissue Doppler\/strain and three-dimensional echocardiography. Furthermore, cardiac imaging is rapidly extending beyond the heart and laboratories are now incorporating various vascular imaging protocols as part of their clinical activity. Lastly, the emerging focus on continuing education and certification in echocardiography puts pressure on physicians to keep abreast of basic and new echocardiography.\n\nThere are several well-written echocardiography textbooks and atlases available. The intent of developing this particular handbook was: (1) to offer a succinct guide to echocardiography incorporating topics ranging from fundamental physical principles to important cardiac pathologies and novel technologies, (2) to provide this knowledge in a novel, condensed format enabling easy and quick reference to topics, (3) to enable a sonographer or echocardiographer to be able to use this handbook as a bedside guide to performance and interpretation of echocardiography and (4) to present content that would assist readers in preparing for competency tests in echocardiography, such as the echocardiography boards.\n\nTo give the reader the benefit of most meaningful and up-to-date content, this book sought to bring together a group of renowned experts in echocardiography with substantial experience in clinical echocardiography and particular expertise in various specific echocardiography technologies. The result is a compact yet highly informative handbook that provides as much if not more knowledge than most textbooks with a significantly less investment of time.\n\nThe concept and motivation to develop this book was born out of my years of interaction with cardiology trainees and sonographers in various echocardiography programs. Unbeknownst to them, I learnt as much from them about echocardiography as they did from me. More importantly, they all helped me develop into a better teacher. Although memory and space preclude me from mentioning all those who have helped in getting me here, there are several individuals who have been particularly helpful with developing this book. This list will unfairly exclude many others who have been equally instrumental in our success and I ask their pardon. My deepest gratitude goes out to Allison Hays, Sourabh Verma, and Erin Michos, our associate editors. The book would not materialize without their efforts. I am also highly appreciative of the kind and generous contributions of all the authors, despite their hectic schedules, that have resulted in the outstanding content in this book. Others who have played a major role in getting us here include Hsin-Yueh Liang, Lea Dimaano, Aurelio Pinheiro, Jacob Abraham, Sue Phillip, Heather Richardson, Ken Cresswell, Vickie Spearman, Joe Wassil, and Nancy Grap. Many thanks to Mary Corretti, Ed Kasper, and Gordon Tomaselli for allowing me the time to complete this book. I would also like to particularly thank Cate Rogers and Grant Weston from Springer who were a delight to work with despite all the delays in the production of this book. Lastly, I am deeply indebted to Roselle and Anya Marithea whose understanding and patience allowed me to do this work, and my parents Marjorie and John, my brothers Jason and Francisco, whose help and encouragement all my life and assistance in the last several months allowed me to bring this book to fruition.\n\nTheodore P. Abraham\n\nContents\n\n1 Physics and Artifacts 1\n\nKenneth D. Horton\n\n2 Basic Ultrasound Views 13\n\nKenneth D. Horton\n\n3 Quantification of Left Ventricular Size and Wall Thickness 23\n\nKhaled Bachour and Luis Afonso\n\n4 Quantification of Left Ventricular Systolic Function 33\n\nYasuhiko Takemoto and Minoru Yoshiyama\n\n5 Noninvasive Hemodynamic Assessment 37\n\nJacob Abraham and Kimberly Chadwell\n\n6 Echocardiographi\u200bc Assessment of Diastolic Dysfunction (DD) and Heart Failure with Normal EF (HFnlEF) 47\n\nAnil Mathew and Luis Afonso\n\n7 Tissue Doppler Imaging and Strain Echocardiography\u200b 63\n\nVeronica Lea J. Dimaano\n\n8 Transesophageal Echocardiography\u200b 85\n\nJulie A. Humphries, Christopher J. Kramer, Partho P. Sengupta and Bijoy K. Khandheria\n\n9 Intraoperative Transesophageal Echocardiography\u200b 103\n\nAlina Nicoara and Madhav Swaminathan\n\n10 Use of Echo Contrast in Routine Practice 117\n\nShizhen Liu and Mani A. Vannan\n\n11 Three-Dimensional Echocardiography\u200b 127\n\nLissa Sugeng, Lynn Weinert and Roberto M. Lang\n\n12 Stress Echocardiography\u200b 143\n\nSebastian Kelle\n\n13 Mechanical Dyssynchrony Assessment 149\n\nGabe B. Bleeker, Nico R. Van de Veire, Theodore P. Abraham, Eduard R. Holman and Jeroen J. Bax\n\n14 Post Cardiac Resynchronizatio\u200bn Therapy (CRT) Optimization Protocol 169\n\nHsin-Yueh Liang\n\n15 Echocardiographi\u200bc Assessment of the Right Ventricle 173\n\nHisham Dokainish\n\n16 Aortic Valve Stenosis and Regurgitation 185\n\nVuyisile T. Nkomo\n\n17 Mitral Prolapse Regurgitation and Mitral Stenosis 195\n\nNuasheen Akhter and Issam A. Mikati\n\n18 Right-Sided Heart Valves Assessment in Disease 219\n\nFarouk Mookadam, Julie A. Humphries, Sherif E. Moustafa and Tahlil A. Warsame\n\n19 The Assessment of Prosthetic Valve Function 235\n\nMengistu Simegn, Anupama Kottam and Luis Afonso\n\n20 The Aorta:\u200b Diseases of the Aorta 251\n\nMary C. Corretti\n\n21 Hypertrophic Cardiomyopathy 271\n\nAllison G. Hays and Aurelio C. Pinheiro\n\n22 Stress Cardiomyopathy 277\n\nJacob Abraham and Ilan S. Wittstein\n\n23 Pericardial Diseases 283\n\nPartho P. Sengupta and Bijoy K. Khandheria\n\n24 Atrial Septal Defect 297\n\nNaser M. Ammash\n\n25 Percutaneous Interventions for Congenital Heart Defects 303\n\nChristian D. Nagy, Richard E. Ringel and W. Reid Thompson\n\n26 Cardiac Tumors and Masses 311\n\nMary C. Corretti\n\n27 Cardiac Transplantation 319\n\nJames Mudd\n\n28 Vascular Imaging 331\n\nElizabeth V. Ratchford\n\nIndex343\n\nContributors\n\nJacob Abraham\n\nDepartment of Medicine, Johns Hopkins Hospital, Baltimore, MD, USA\n\nTheodore P. Abraham\n\nDivision of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nLuis Afonso\n\nDepartment of Cardiology, Detroit Medical Centre,Wayne State University, Detroit, MI, USA\n\nNausheen Akhter\n\nDivision of Cardiology, The Feinberg School of Medicine, Chicago, IL, USA\n\nNaser M. Ammash\n\nDepartment of Internal Medicine, Mayo Clinic, Rochester, MN, USA\n\nKhaled Bachour\n\nInternal Medicine, Division of Cardiology, University of Pittsburgh, Farrell, PA, USA\n\nJeroen J. Bax\n\nDepartment of Cardiology, Leiden University Medical Center, Leiden, The Netherlands\n\nGabe B. Bleeker\n\nDepartment of Cardiology, Leiden University Medical Center, Leiden, The Netherlands\n\nMary C. Corretti\n\nDivision of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nVeronica Lea J. Dimaano\n\nDivision of Cardiology, Johns Hopkins University School of Medicine, Baltimore, MD, USA\n\nHisham Dokainish\n\nDepartment of Cardiology, Baylor College of Medicine, 6620 Main Street-11A.08, 77030 Houston, TX, USA\n\nAllison G. Hays\n\nDivision of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nEduard R. Holman\n\nDepartment of Cardiology, Leiden University Medical Center, Leiden, The Netherlands\n\nKenneth D. Horton\n\nEcho\/Vascular Laboratory, Intermountain Medical Center, Murray, UT, USA\n\nJulie A. Humphries\n\nDepartment of Cardiology, Heart Care Partners, Greenslopes, Queensland, Australia\n\nSebastian Kelle\n\nDepartment of Cardiology, German Heart Institute Berlin, Berlin, Germany\n\nAnupama Kottam\n\nDepartment of Cardiology, Detroit Medical Centre, Wayne State University, Detroit, MI, USA\n\nBijoy K. Khandheria\n\nDivision of Cardiovascular Diseases, Mayo Clinic, Rochester, MN, USA\n\nChristopher J. Kramer\n\nAdvanced Hemodynamic and Cardiovascular Laboratory, Aurora Medical Group, Milwaukee, WI, USA\n\nRoberto M. Lang\n\nDepartment of Medicine, University of Chicago Medical Center, Chicago, IL, USA\n\nHsin-Yueh Liang\n\nDivision of Cardiology, Department of Medicine, China Medical University Hospital, Taichung, Taiwan\n\nShizhen Liu\n\nDepartment of Cardiovascular Medicine, The Ohio State University Medical Center, Columbus, OH, USA\n\nAnil Mathew\n\nDetroit Medical Centre, Wayne State University, Detroit, MI, USA\n\nIssam A. Mikati\n\nDepartment of Medicine, Northwestern University Feinberg School of Medicine, Chicago, IL, USA\n\nFarouk Mookadam\n\nDepartment of Cardiology, Mayo College of Medicine, Scottsdale, AZ, USA\n\nSherif E. Moustafa\n\nDepartment of Cardiovascular Diseases, Mayo Clinic Arizona, Scottsdale, AZ, USA\n\nChristian D. Nagy\n\nPediatric Cardiology\/Adult Cardiology, Johns Hopkins University Medical Center, Baltimore, MD, USA\n\nAlina Nicoara\n\nDepartment of Anesthesiology, Yale University School of Medicine, West Haven, CT, USA\n\nVuyisile T. Nkomo\n\nDivision of Cardiovascular Diseases and Internal Medicine, Mayo Clinic, Rochester, MN, USA\n\nAurelio C. Pinheiro\n\nDepartment of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nElizabeth V. Ratchford\n\nDepartment of Medicine, Division of Cardiology, Johns Hopkins University School of Medicine, Baltimore, MD, USA\n\nRichard E. Ringel\n\nPediatric Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nPartho P. Sengupta\n\nCardiovascular Division, University of California, Irvine, USA\n\nMengistu Simegn\n\nDetroit Medical Centre, Wayne State University, Detroit, MI, USA\n\nLissa Sugeng\n\nSection of Cardiology, Department of Medicine, Non-Invasive Cardiovascular Imaging Laboratory, University of Chicago Medical Center, Chicago, IL, USA\n\nMadhav Swaminathan\n\nDivision of Cardiothoracic, Anesthesiology and Critical Care Medicine, Duke University, Durham, CA, USA\n\nYasuhiko Takemoto\n\nDepartment of Internal Medicine and Cardiology, Osaka City University School of Medicine, Osaka, Japan\n\nW. Reid Thompson\n\nPediatric Cardiology, Johns Hopkins University School of Medicine, Baltimore, MD, USA\n\nNico R. Van de Veire\n\nDepartment of Cardiology, Leiden University Medical Center, Leiden, The Netherlands\n\nMani A. Vannan\n\nDepartment of Cardiovascular Medicine, The Ohio State University Medical Center, Columbus, OH, USA\n\nTahlil A. Warsame\n\nCardiovascular Ultrasound Imaging and Hemodynamic Laboratory, Mayo Clinic Arizona, Phoenix, AZ, USA\n\nLynn Weinert\n\nUniversity of Chicago Medical Center, Chicago, IL, USA\n\nIlan S. Wittstein\n\nDepartment of Medicine, Johns Hopkins University School of Medicine, Baltimore, MD, USA\n\nMinoru Yoshiyama\n\nDepartment of Internal Medicine and Cardiology, Osaka City University School of Medicine, Osaka, Japan\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_1\n\n# 1. Physics and Artifacts\n\nKenneth D. Horton1\n\n(1)\n\nEcho\/Vascular Laboratory, Intermountain Medical Center, Murray, UT, USA\n\nKenneth D. Horton\n\nEmail: kd.horton@comcast.net\n\nFig. 1.1\n\nDefinition of sound and ultrasound. Sound is a mechanical vibration that consists of compressions and rarefactions. Sound waves propagate (travel) through various mediums by interactions between the particles that comprise the medium. Therefore, sound cannot travel through a vacuum. The range of hearing in the human ear is 20\u201320,000 Hz. Sound above 20,000 Hz is called ultrasound\n\nFig. 1.2\n\nWave terminology. The combination of one compression and one rarefaction is called one cycle. The number of cycles completed in 1 s is called the frequency. The frequency is measured in units of Hertz or, for ultrasound, megahertz (one million cycles\/s). The distance occupied by one cycle is called the wavelength and the amount of time occupied by one cycle is the period. The strength of the ultrasound signal is the intensity. The higher the amplitude, the greater the intensity of the ultrasound signal\n\nFig. 1.3\n\nInteractions with tissue. As a sound wave strikes, a difference in the medium (acoustic interface) it is traveling through, some of the sound is reflected back to the transducer (reflection) and some continues to travel through the next medium (refraction). As the sound wave travels through a medium, it loses its strength or intensity. This is called attenuation\n\nFig. 1.4\n\nTransducers. A transducer is a device that changes types of energy. An ultrasound transducer has multiple piezioelectric crystals (elements) that can change electrical energy to mechanical energy \u2013 or vice versa. The crystal is surrounded by a dampening material that prevents \"ringing\" when the crystal is activated. The matching layer has an acoustic impendence between that of the transducer and skin and facilitates the transmission of sound into the body\n\nFig. 1.5\n\nBeam characteristics. In an unfocused transducer, sound leaves the transducer, travels parallel for a period of time and then begins to diverge. The area prior to the divergence is called the near zone and the area after the divergence is called the far zone. Using electronic timing or an acoustic lens, the beam can be focused. The resolution is highest in the area of the focal zone. Ultrasound systems have the ability to move the focal zone along the beam to improve the resolution in areas of interest\n\nFig. 1.6\n\nResolution. Resolution is the ability to see two different objects in the imaging field as two different objects and is measured in unit of distance (millimeters). There are multiple types of resolution in ultrasound. The two most common are (a) longitudinal resolution \u2013 resolution along the axis parallel to the direction of the sound propagation and (b) axial resolution \u2013 resolution on the axis perpendicular to the direction of the sound propagation\n\nFig. 1.7\n\nFrame rate. An ultrasound image is created one scan line at a time. When the scan lines are processed across the field of view (sector) one frame is created. The number of frame that is created in 1 s is called the frame rate and is measured in Hertz. The temporal resolution of a system is determined by the frame rate. The higher the frame rate, the better the temporal resolution. In 2D echocardiography, you should attempt to image at the highest frame rate possible\n\nFig. 1.8\n\nM-mode echocardiography. M-mode (motion-mode) echocardiography is a graphical depiction of the ultrasound signal along a single scan line. The temporal resolution of an M-mode tracing is superior to all other echocardiographic modes because the image is only processing the signal from a single scan line and can be updated thousands of times per second\n\nFig. 1.9\n\n2D Echocardiography. 2D echocardiography is a 2D depiction of the heart. It is usually acquired as a moving picture allowing for assessment of the heart throughout systole and diastole. 2D images are either captured on video tape or as digital loops\n\nFig. 1.10\n\nDoppler echocardiography. As its name implies, Doppler images are created using the Doppler effect. Spectral Doppler tracings can be created using either continuous wave (CW) Doppler or pulsed wave (PW) Doppler. CW and PW Doppler each have distinct advantages and disadvantages that determine when each is used. (a) PW Doppler has the advantage of range resolution, or being able to measure flow of a specific point. Its main disadvantage is there is a limit to how high of a velocity it can measure (Nyquist limit). (b) CW Doppler has an unlimited Nyquist limit and therefore can measure very high velocities. Its disadvantage, however, is it does not have range resolution and measures all flows along the cursor\n\nFig. 1.11\n\nColor flow imaging. Color flow imaging is used to detect the direction and velocity of blood flow. Flow is measured at thousands of points within the color flow sector. By convention, flow toward the transducer is colored red-yellow and flow away from the transducer is colored blue-white. Low flow velocities begin with darker shades and the shade increases as flow velocity increases\n\nFig. 1.12\n\n3D echocardiography. 3D images are obtained using a pyramidal volume of pixels (voxels). Once the image is obtained, it can be rotated and cropped to better visualize any structure within the image. In this example, a mitral annular ring was placed. (a) Assessment of the annular ring from the LA perspective (looking down into the LV) and (b) assessment from the LV perspective (looking up into the atrium). Both images were obtained in the same acquisition\n\nFig. 1.13\n\nTissue Doppler Imaging (TDI). Tissue Doppler imaging is a technology that is used to assess the diastolic and systolic velocities of the heart. (a) TDI measures the direction and velocity of the myocardium (instead of blood flow) and color encodes the signal from the myocardium as red or blue depending on the direction of motion. (b) Using special filters, pulsed wave Doppler is used to measure the velocities during (1) systole, (2) early diastole, and (3) late diastole\n\nFig. 1.14\n\nImaging artifacts \u2013 suboptimal images. The quality of the ultrasound image varies greatly from person to person. Even with today's excellent ultrasound systems, there are patients in whom diagnostic imaging is not possible. Assessment of the heart from multiple windows and image optimization using the ultrasound system controls do not always help. In most cases, the use of commercial contrast media (Definity or Optison) can make an undiagnostic ultrasound study diagnostic\n\nFig. 1.15\n\nImaging artifacts \u2013 shadowing. The amount of ultrasound that is returned to the transducer for processing is proportional to the difference in the media forming an acoustic interface. The stronger the acoustic interface, the more of the signal that returns to the transducer leaving little or none to continue to travel forward. It is the areas beyond that acoustic interface where shadowing occurs. Shadowing is overcome by imaging all areas in multiple windows and views\n\nFig. 1.16\n\nImaging artifacts \u2013 reverberation. Reverberation artifact occurs when an ultrasound signal is bounced back and forth between two acoustic interfaces before returning to the transducer. This results erroneously displayed objects in the imaging sector\n\nFig. 1.17\n\nImaging artifacts \u2013 side lobes. Although the majority of the ultrasound signal is processed from the center of the ultrasound beam, there are small lobes on each side of the signal that also processes returning signals. These are called side lobes. The ultrasound system may receive signals from structures within the side lobes and erroneously display them in the center of the imaging sector\n\nFig. 1.18\n\nImaging artifacts \u2013 range ambiguity. Ultrasound images are processed under the assumption that the sound wave is traveling directly to a structure and then being reflected directly back to the transducer. It is also assumed that sound travels at a constant 1,540 m\/s. If either of these assumptions is violated, structures will be displayed at the incorrect depth within the imaging sector. In the pictured example, the depth of the reflected interface would be too deep because the signal was delayed in returning to the transducer\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_2\n\n# 2. Basic Ultrasound Views\n\nKenneth D. Horton1\n\n(1)\n\nEcho\/Vascular Laboratory, Intermountain Medical Center, Murray, UT, USA\n\nKenneth D. Horton\n\nEmail: kd.horton@comcast.net\n\nFig. 2.1\n\nEchocardiographic windows. When performing an echocardiogram, the transducer is placed in multiple areas of the chest. These areas are commonly referred to as \"windows.\" The most common echo windows are: (a) the left parasternal window, (b) the apical window, (c) the subcostal window, and (d) the suprasternal notch. In certain circumstances, imaging from nonstandard windows is required. Some of the nonstandard windows include the right parasternal window and the mid-clavicular window. In some cases, imaging may need to be performed from any area of the chest from where an image can be obtained (Adapted from Servier Medical Art, www.servier.com, with permission)\n\nFig. 2.2\n\nEchocardiographic views. From each window, the transducer is manipulated to obtain \u00admultiple views of the heart. The different views are obtained by rotating and\/or tilting the \u00adtransducer without actually moving it to a new window\n\nFig. 2.3\n\nThe parasternal window. The parasternal long axis (PLAX) view is obtained by placing the transducer in the three to four left intercostal spaces close to the sternum with the bean oriented toward the patient's right shoulder. This orientation slices through the heart on a long axis from base to apex (Adapted from Servier Medical Art, www.servier.com, with permission)\n\nFig. 2.4\n\nThe parasternal window. The parasternal long axis (PLAX) view is obtained by placing the transducer in the third to fourth left intercostal space close to the sternum with the bean oriented toward the patient's right shoulder. This orientation slices through the heart on a long axis from base to apex\n\nFig. 2.5\n\nParasternal long axis \u2013 color flow image. Color flow Doppler imaging is used to assess the valves for regurgitation. Pictured here is a systolic still frame demonstrating regurgitation of blood flow back through the mitral valve into the left atrium\n\nFig. 2.6\n\nRight ventricular inflow (RVIF) view. The RVIF view is obtained by tilting the transducer toward the left shoulder so the ultrasound beam moves anterior in the chest slicing through the right heart. (a) This view is used to assess the right ventricle, tricuspid valve, and right atrium. (b) Color flow and spectral Doppler are used to assess the valve for tricuspid regurgitation\n\nFig. 2.7\n\nParasternal short axis (PSAX) view. The PSAX view is obtained by remaining in the parasternal window and rotating the transducer clockwise approximately 90\u00b0. Sweeping the transducer from base to apex results in: (a) the basal level view, (b) the mitral level view, (c) the papillary level view, and (d) the apical level view. In the basal and mitral level view, color flow Doppler can be used to assess the valves for regurgitation\n\nFig. 2.8\n\nApical views. The apical window is usually found in the left lateral portion of the chest at the apex of the heart. This can sometimes be located by placing your hand lightly in the area of the apex and feeling for the point of maximal intensity (PMI). The PMI will serve as your starting point; however, small adjustments will need to be made to the transducer location to maximally optimize your image (Adapted from Servier Medical Art, www.servier.com, with permission)\n\nFig. 2.9\n\nApical 4 Chamber (4C) view. All four cardiac chambers are visualized in the 4C view along with the mitral and tricuspid valves. Ventricular and atrial size can be assessed using 2D echo. Color flow and spectral Doppler can be used to assess for valvular regurgitation and stenosis. Left ventricular diastolic function can be assessed using pulsed wave Doppler of the mitral valve and pulmonary veins. In this view, the right ventricular freewall, interventricular septum, and left lateral wall can be assessed for systolic motion\n\nFig. 2.10\n\nApical 5 Chamber (5C) view. (a) The 5C view is obtained by tilting the transducer slightly so the beam moves anterior in the chest slicing through the left ventricular outflow track (LVOT) and aortic valve. (b) Utilizing spectral Doppler, the outflow velocity can be measured. In this example, the high-velocity flow pattern indicates aortic stenosis. The aortic valve area can be calculated by integrating data from the 2D images and Doppler tracings\n\nFig. 2.11\n\nApical 2 Chamber (2C) view. The 2C view is obtained by starting at the 4C view and rotating the transducer clockwise approximately 90\u00b0. In the 2C view, the left ventricle, mitral valve, and left atrium can be seen. The inferior and anterior walls of the left ventricle can be assessed for systolic function. Using color flow and spectral Doppler, the mitral valve can be assessed for regurgitation and stenosis\n\nFig. 2.12\n\nApical 3 Chamber (3C) or long axis view. The apical 3C view is also known as the apical long axis view. Structures seen in the 3C view are the same as the parasternal long axis view. The 3C view is utilized to assess chamber size and function as well as aortic and mitral valve function\n\nFig. 2.13\n\nThe subcostal window. Subcostal views are obtained by positioning the patient flat on their back and placing the transducer just below the xiphoid process. Asking the patient to bend their knees may help relax the stomach muscles. Having the patient take a deep breath often moves the lungs out of the way and results in better images (Adapted from Servier Medical Art, www.servier.com, with permission)\n\nFig. 2.14\n\nSubcostal 4 Chamber view. (a) The subcostal 4C view can be used to assess chamber size and function. Color flow and spectral Doppler imaging can be used to assess valvular function. This is also a good view for assessing for atrial or ventricular septal defects. (b) The subcostal view is a good view for assessing the presence and size of a pericardial effusion\n\nFig. 2.15\n\nInferior Vena Cava (IVC) view. Beginning in the subcostal 4C view and rotating the transducer counterclockwise 90\u00b0 and angling toward the liver, the IVC can be seen in long axis. The IVC can be assessed for diameter and collapsibility during respiration. The size and collapsibility of the IVC are used to estimate right atrial pressure\n\nFig. 2.16\n\nSuprasternal window. The suprasternal window is obtained by placing the transducer in the \"notch\" at the top of the sternum (the manubrium). This window is used for 2D imaging and also for assessment of aortic flows with a dedicated continuous wave Doppler transducer (Pedoff probe) (Adapted from Servier Medical Art, www.servier.com, with permission)\n\nFig. 2.17\n\nSuprasternal window. The suprasternal 2D image is used to assess the ascending, transverse, and descending aorta and its branches. The inominate, left carotid, and left subclavian arteries can be seen branching off the aorta. The right pulmonary artery can also be seen in short axis\n\nFig. 2.18\n\nImage optimization \u2013 overall gain. The gain controls the brightness of your image. Optimizing the gain will result in better detection of structures and endocardial borders. (a) Using excessive gain will result in images that are too bright and may make normal structures appearing calcified. (b) Gains should be set at a point that a balance between image brightness and darkness are reached\n\nFig. 2.19\n\nImage optimization \u2013 compression or dynamic range. Compression or dynamic range are controls that affect the number of shades of gray that are displayed in the image. (a) If these are set too high, the image appears washed out and it may be difficult to visualize endocardial borders. (b) The compression or dynamic range should be set so the blood pool is dark and the tissue is bright. This will result in better endocardial border definition\n\nFig. 2.20\n\nImage optimization \u2013 depth and zoom controls. The image depth and zoom controls can be used to optimize the area of interest. In these examples (a) the depth is set too deep resulting in a small image that will be difficult to assess, (b) the depth is set at the correct level so that all structures can be seen in the apical 4C view, and (c) the use of the \"zoom\" control for better assessment of the mitral valve\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_3\n\n# 3. Quantification of Left Ventricular Size and Wall Thickness\n\nKhaled Bachour and Luis Afonso1\n\n(1)\n\nDepartment of Cardiology, Detroit Medical Centre, Wayne State University, Detroit, MI, USA\n\nLuis Afonso\n\nEmail: lafonso@med.wayne.edu\n\nAbbreviations\n\n  * 2C2 chamber\n\n  * 4C4 chamber\n\n  * LVIDdLeft ventricular inner dimension during diastole\n\n  * LVIDsLeft ventricular inner dimension during systole\n\n  * PWTPosterior wall thickness\n\n  * RWMARegional wall motion abnormalities\n\n  * SWTSeptal wall thickness\n\n## 3.1 LV Wall Thickness and Dimensions\n\n### 3.1.1 Methods\n\n  * Direct 2D\n\n  * 2D-guided M-mode (Table 3.1)\n\nTable 3.1\n\nPros and cons of M-mode and direct 2D methods for assessment of LVIDd, LVIDs, PWT, and SWT | M-Mode | Direct 2D\n\n---|---|---\n\nAdvantages | \u2022?Excellent temporal resolution | \u2022?Preferred with CAD and asymmetrical LV\n\n\u2022?Higher spatial resolution: separates other structures (trabeculae, false tendons) from LV walls | \u2022?Can be used even in oblique parasternal images\n\nDisadvantages | \u2022?Cursor has to be perpendicular to the long axis of the LV | \u2022?Lower spatial resolution: may not be able to separate other structures (trabeculae, false tendons) from LV walls\n\n\u2022?Assume symmetry of LV | \u2022?Lower temporal resolution\n\n### 3.1.2 Technical Points\n\n  * Always from parasternal long axis view (see Fig 3.1)\n\n  * Level: mitral leaflet tips\n\n  * Measurement PWT and SWT, LVIDd and LVIDs is done along the LV minor axis, perpendicular to the long axis of the LV\n\n  * LVIDd, PWT, and SWT are measured during end diastole when both mitral and aortic valves are closed (direct 2D), or at the end of PR segment (M-mode)\n\n  * LVIDs is measured at end systole when both aortic and mitral valves are closed (direct 2D) or at the end of T wave (M-mode)\n\n  * LVID is edged by the blood-endocardium interface edge of the septal and posterior wall, i.e., leading edge of the septum to the trailing edge of the posterior wall\n\n### 3.1.3 Normal Values\n\n  * See Table 3.4\n\n## 3.2 Volumetric Assessment of LV\n\n### 3.2.1 Methods\n\n  * Biplane method of disks (modified Simpson's)\n\n  * Unipolar method of disks\n\n  * Area-length method\n\n  * 3D Volumetric methods: Real-time 3D echocardiography\n\n### 3.2.2 Biplane Method of Disks\n\n  * Using apical 4C and 2C views (Figure 3.2)\n\n  * Long axis of LV is longest dimension extending from the apex to the center of mitral annulas\n\n  * LV cavity is divided into 20 disks perpendicular to the major long axis of the LV\n\n  * Each disk has a height (1\/20 of L), and two dimensions (long and short from the two views)\n\nFig. 3.1\n\n2D guided-linear M-mode dimensions of the left ventricle obtained in end diastole (LVED) and end systole (LVES), along with septal (IVS) and posterior wall (LVPW) thickness, both measured in end-diastole.** ?Bright posterior echoes represent parietal pericardium\n\nFig. 3.2\n\n2D end diastolic and end systolic volumes calculated from the apical 4C (Ap4C) and apical 2C (Ap2C) views in end diastole (LVED) and end systole (LVES), respectively, using the biplane method of disks (modified Simpson's rule). The LV cavity endocardium is traced from the mitral annular plane to the apex of the LV cavity and includes the papillary muscles\n\nFig. 3.3\n\nLV volumes and ejection fraction calculated from a real-time 3D echocardiography full volume acquisition. Regional or segmental function can also be assessed from the time\u2013volume curves (not shown)\n\nFig. 3.4\n\nRelative wall thickness in various forms of LV remodeling (With permission from [1])\n\n  * LV volume is calculated from the summation of volumes of ellipsoid disks along the long axis of LV (Fig. 3.2)\n\n### 3.2.3 Uniplane Method of Disks\n\n  * Variation of biplane method when two orthogonal apical views are not available\n\n  * Utilizes one apical view\n\n  * Assumes circular shape of the disks\n\n### 3.2.4 Length-Area Method\n\n  * Assumes a bullet shaped LV\n\n  * The long axis of the LV is defined as the longest dimension extending from the apex to the center of the mitral annulus in 4C view.\n\n  * The mid-LV cross-sectional area is measured by planimetry at the level of papillary muscles in the parasternal short axis view.\n\n  * Volume ?= ?5 ?\u00d7 ?area ?\u00d7 ?length\/6 (Table 3.2)\n\nTable 3.2\n\nPros and cons of the three methods of evaluation of the LV volumes | Biplane method of disks | Uniplane method of disks | Area-length method\n\n---|---|---|---\n\nAdvantages | \u2022?Recommended whenever possible | \u2022?Require optimal endocardial definition in only one apical view | \u2022?Can be used in the absence of optimal endocardial definition\n\n\u2022?Most accurate in ventricles with RWMA\n\nDisadvantages | \u2022?Require optimal endocardial definition in two orthogonal apical views | \u2022?Assume a regular shape of the LV | \u2022?Assume a regular bullet shape of the LV\n\n|\n\n\u2022?Cannot be used in the presence of RWMA | \u2022?Cannot be used in the presence of RWMA\n\n### 3.2.5 3D Volumetric methods: Real-time 3D echocardiography\n\n  * Semiautomatic border detection of endocardial border (Fig 3.3)\n\n  * Unlike 2D techniques, does not rely on geometric assumptions\n\n  * Immune from plane-positioning errors (eliminates foreshortening)\n\n  * Excellent agreement with cardiac MRI \u2013derived volumes\n\n  * Lower interobserver and intraobserver variability compared to 2D echo techniques\n\n  * Limited by image quality and irregular cardiac rhythms\n\n  * Single beat 3D- volumetric assessment has recently become commercially available\n\n## 3.3 Calculation of LV Mass\n\n### 3.3.1 Methods\n\n  * Linear measurements\n\n  * Volumetric measurements\n\n### 3.3.2 Linear Method\n\n  * LV mass ?= ?0.8 ?\u00d7 ?{1.04 [(LVIDd ?+ ?PTW ?+ ?SWT)3 ?? ?(LVIDd)3]} ?+ ?0.6 g\n\n### 3.3.3 Volumetric Method\n\n  * LV mass ?= ?LV myocardial volume ?\u00d7 ?cardiac tissue density.\n\n  * LV mass ?= ?(epicardial LV volume ?? ?endocardial LV volume) ?\u00d7 ?1.05.\n\n  * Epicardial and endocardial volumes can be measured using the length-area method or the biplane method of disks.\n\n## 3.4 Technical Points\n\n  * Linear method assumes a regular shape of the LV and should not be used in case of geometrical distortions.\n\n  * In the presence of RWMA, biplane method should be used.\n\n## 3.5 Relative Wall Thickness (RWT)\n\n  * RWT ?= ?2 ?\u00d7 ?PWT\/LVIDd\n\n  * Used to differentiate concentric (?0.42) and eccentric (?0.42) remodeling of the LV. See Fig. 3.4 and Tables 3.4. 3.4Pathologic vs. Physiologic (athlete's heart) LVH\n\nTable 3.3\n\nCharacteristics of LVH in athlete's heart vs. pathological left ventricular hypertrophy | Athlete's heart | Pathological hypertrophy\n\n---|---|---\n\nSymmetry | Symmetrical | Symmetrical or asymmetrical\n\nWall thickness | Rarely ?> ?17 mm | Frequently >17 mm\n\nSystolic function | Normal | Normal or abnormal\n\nLV volume | Increased | Increased, decreased or normal\n\nDiastolic function | Normal with E' ?> ?7 cm\/s | Abnormal with E' ?< ?7 cm\/s\n\nStrain | Homogenous and normal | Abnormal and can be paradoxical\n\nE'= medial mitral annular tissue Doppler velocity\n\nTable 3.4\n\nThe reference values and classification of abnormal values of LV dimensions, wall thickness and volumes in men and women based on the guidelines of the American society of echocardiography | Women | Men\n\n---|---|---\n\nNormal | Mildly abnormal | Moderately abnormal | Severely abnormal | Normal | Mildly abnormal | Moderately abnormal | Severely abnormal\n\nDimensions\n\nLVIDd, cm | 3.9\u20135.3 | 5.4\u20135.7 | 5.8\u20136.1 | >6.1 | 4.2\u20135.9 | 6.0\u20136.3 | 6.4\u20136.8 | >6.8\n\nLVIDd\/BSA, cm\/m2 | 2.4\u20133.2 | 3.3\u20133.4 | 3.5\u20133.7 | >3.7 | 2.2\u20133.1 | 3.2\u20133.4 | 3.5\u20133.6 | >3.6\n\nPWT, cm | 0.6\u20130.9 | 1.0\u20131.2 | 1.3\u20131.5 | >1.5 | 0.6\u20131.0 | 1.1\u20131.3 | 1.4\u20131.6 | >1.6\n\nSWT, cm | 0.6\u20130.9 | 1.0\u20131.2 | 1.3\u20131.5 | >1.5 | 0.6\u20131.0 | 1.1\u20131.3 | 1.4\u20131.6 | >1.6\n\nLV volume\n\nDiastolic, mL | 56\u2013104 | 1.5\u2013117 | 118\u2013130 | >130 | 67\u2013155 | 156\u2013178 | 179\u2013201 | >201\n\nDiastolic\/BSA, mL\/m2 | 35\u201375 | 76\u201386 | 87\u201396 | >96 | 35\u201375 | 76\u201386 | 87\u201396 | >96\n\nSystolic, mL | 19\u201349 | 50\u201359 | 60\u201369 | >69 | 22\u201358 | 59\u201370 | 71\u201382 | >82\n\nSystolic\/BSA, mL\/m2 | 12\u201330 | 31\u201336 | 37\u201342 | >42 | 12\u201330 | 31\u201336 | 37\u201342 | >42\n\nLV mass\n\nLinear, g | 67\u2013162 | 163\u2013186 | 187\u2013210 | >210 | 88\u2013224 | 225\u2013258 | 259\u2013292 | >292\n\nLinear\/BSA, g\/m2 | 43\u201395 | 96\u2013108 | 109\u2013121 | >121 | 49\u2013115 | 116\u2013131 | 132\u2013148 | >149\n\nVolumetric, g | 66\u2013150 | 151\u2013171 | 172\u2013182 | >182 | 96\u2013200 | 201\u2013227 | 228\u2013254 | >255\n\nVolumetric\/BSA, g\/m2 | 44\u201388 | 89\u2013100 | 101\u2013112 | >112 | 50\u2013102 | 103\u2013116 | 117\u2013130 | >130\n\nRWT | 0.22\u20130.42 | 0.43\u20130.47 | 0.48\u20130.52 | >0.52 | 0.24\u20130.42 | 0.43\u20130.46 | 0.47\u20130.51 | >0.51\n\nBSA body surface area, LV left ventricle, LVIDd left ventricular inner dimension in diastole, PWT posterior wall thickness, RTW relative wall thickness, SWT septal wall thickness\n\nAdapted from [1]\n\nRecommended Reading\n\nLang RM et al. Recommendations for chamber quantification: a report from the American Society of Echocardiography's Guidelines and Standards Committee and the Chamber Quantification Writing Group, developed in conjunction with the European Association of Echocardiography, a branch of the European Society of Cardiology J Am Soc Echocardiogr. 2005;18(12):1440-1463.PubMedCrossRef\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_4\n\n# 4. Quantification of Left Ventricular Systolic Function\n\nYasuhiko Takemoto1  and Minoru Yoshiyama\n\n(1)\n\nDepartment of Internal Medicine and Cardiology, Osaka City University School of Medicine, Osaka, Japan\n\nYasuhiko Takemoto\n\nEmail: yatakemoto@med.osaka-cu.ac.jp\n\n## 4.1 Introduction\n\nThe assessment of left ventricular (LV) systolic function is an essential part of all echocardiographic examinations and provides crucial and indispensable information for diagnosis, treatment, and prognosis of almost all cardiac conditions.\n\n## 4.2 Global LV Systolic Function\n\nThe most commonly used expression of global LV systolic function is LV ejection fraction (LVEF). LVEF is preferably calculated from two-dimensional (2D) volume measurements using the following formula:\n\nThe biplane Simpson method is most commonly used for measuring 2D LV volumes (LVEDV, LVESV) and LVEF.\n\n## 4.3 Limitations of the Biplane Simpson Method\n\nMyocardial dropout, especially at the apex, is a potential problem. The transducer must be at the true apex and the ultrasonic beam must be through the center of the LV.\n\n## 4.4 Regional LV Systolic Function\n\nRegional LV systolic function is assessed by dividing the LV into segments using a 16-segment model recommended by the American Society of Echocardiography.\n\nA numeric scoring system is adopted based on the contractility of the individual segments.\n\n16-segment model\n\nScoring system\n\n1?\n\nNormal\n\n2?\n\nHypokinesis\n\n3?\n\nAkinesis\n\n4?\n\nDyskinesis\n\n5?\n\nAneurysmal\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_5\n\n# 5. Noninvasive Hemodynamic Assessment\n\nJacob Abraham1  and Kimberly Chadwell\n\n(1)\n\nDepartment of Medicine, Johns Hopkins Hospital, Baltimore, MD, USA\n\nJacob Abraham\n\nEmail: ja@jhmi.edu\n\nHemodynamic assessment is one of the oldest applications of echocardiography (echo). With the increasing prevalence of patients with heart failure, the need to accurately measure hemodynamics is growing. Although cardiac catheterization remains the gold standard, it is invasive and expensive. With standard two-dimensional imaging and Doppler techniques, qualitative and quantitative estimation of cardiac output, ventricular filling pressures, and vascular resistances is possible. Because echo measurements correlate strongly with invasively measured values, echo can be a surrogate for catheterization in a variety of settings. This chapter will review the methods for determining some of the most commonly employed hemodynamic parameters (Fig. 5.1).\n\nFig. 5.1\n\n(a) Calculation of stroke volume (SV) involves careful measurement of the LVOT diameter in systole at the level of the aortic annulus from the parasternal long axis view. An inner-edge to inner-edge method should be employed. The aortic annulus is the most accurate location for stroke volume measurements. (b) Pulse wave Doppler interrogation of LVOT or ascending aorta from the apical five chamber view. The sample volume is placed at the same location as where the diameter was measured. (c) Measurement of the LVOT velocity time integral.![\n$$ {SV }={ }{\\\\mathrm{*}}^{*}\\\\pi *{\\({D\\)}}^{{2}}{}^{*}\\({LVOTVTI\\)}. $$\n](A978-1-84996-151-6_5_Fig1_HTML.gif)Cardiac output equals SV multiplied by heart rate\n\n## 5.1 Cardiac Output (Figs. 5.2\u20135.8)\n\n  * Caveats\n\n  * If using PW Doppler, measure aortic diameter at aortic annulus. If using CW Doppler, measure diameter at sinotubular junction.\n\n  * Aortic valve disease, including bicuspid valve or aortic stenosis\/sclerosis, will render the measurement inaccurate.\n\n  * Velocity and diameter measurements should be made at the same site and under the same heart rate and loading conditions.\n\n  * Best beam alignment is critical to avoid underestimation of stroke volume.\n\nFig. 5.2\n\nThe pulmonary vascular resistance (PVR) is related to the peak velocity of tricuspid regurgitation (V TR) and right ventricular time\u2013velocity integral (RVOT TVI) through the following equation:![\n$$ {PVR}=\\({{V}}_{{TR}}\/{RVOT VTI}\\\\times 10\\)+\\\\mathrm{0.16.} $$\n](A978-1-84996-151-6_5_Fig3_HTML.gif)(a) Measurement of VTR. To avoid underestimation of PVR, tricuspid regurgitation jet should be interrogated from multiple views to obtain the peak velocity. (b) The basal short axis is the best view for determining the RVOT TVI using pulse Doppler. (c) RVOT TVI measurement. Note that RVOT TVI is expressed in units of centimeters, rather than meters. In this example, VTR ?= ?4.2 m\/s and RVOT TVI ?= ?2.47 cm. Therefore, PVR ?= ?(4.2\/2.47 ?\u00d7 ?10) ?+ ?0.16, or 17.2 Woods Units\n\nFig. 5.3\n\nCalculation of the ratio of pulmonary to system blood flow (Qp\/Qs) uses the preceding principles to measure stroke volumes for both right and left ventricles. Although stroke volume calculations can be made at any level, it is generally preferable to use the pulmonic valve and the aortic valve for stroke volume measurement. (a) Measurement of LVOT diameter from parasternal long-axis view. (b) Pulse Doppler of the LVOT from an apical four chamber view. (c) Measurement of RVOT diameter from basal short axis view. (d) Pulse Doppler of the RVOT. In this example,![\n$$ \\\\begin{array}{c}{LVOT D}={2}.{26 cm}\\\\\\\\ {Qs}={1}\/{{4}}^{*}{\\(}^{\\\\pi }{{\\(2}}^{.}{}^{*}{\\(2}.{84}\\)={11}.{{38 cm}}^{{3}}\\\\\\\\ {RVOT D} ={2}.{96 cm}\\\\\\\\ {Qp}={ 1}\/{{4}}^{*}{\\(}^{\\\\pi }{{\\(2}}^{.}{}^{*}{\\(2}.{47}\\)={16}.{{98 cm}}^{{3}}\\\\\\\\ {Qp}\/{Qs}={16}.{98}\/{11}.{38} \\\\\\\\ ={1}.{5}.\\\\end{array} $$\n](A978-1-84996-151-6_5_Fig5_HTML.gif)\n\nFig. 5.4\n\nThe mitral regurgitation velocity (VMR) reflects the pressure gradient between the LA and LV in systole. In the absence of outflow obstruction, the systolic blood pressure (SBP) equals the left ventricular systolic pressure. Application of the Bernoulli equation allows calculation of the left atrial systolic pressure from the mitral regurgitation velocity (VMR). In this patient with severe left ventricular dysfunction, severe MR, and SBP 95:![\n$$ \\\\begin{array}{c}{LA systolic pressure }={ SBP}-{4}{\\(}^{{{V}}_{{MR}}}{}^{ } \\\\\\\\ ={ 95}-{4 \\(4}{\\)}^{{2}} \\\\\\\\ ={ 31 mmHg}.\\\\end{array} $$\n](A978-1-84996-151-6_5_Fig7_HTML.gif)\n\nFig. 5.5\n\nPulse Doppler of aortic valve in patient with severe aortic regurgitation (AR). The jet velocity at end-diastole (VAR ) is 2.97 m\/s. From the Bernoulli equation,![\n$$ {Aortic diastolic pressure}{LV diastolic pressure}=4{\\({{V}}_{{AR}}\\)}^{2} $$\n](A978-1-84996-151-6_5_Fig9_HTML.gif) Therefore, ![\n$$ {LVEDP}={Aortic diastolic pressure}4{\\({{V}}_{{AR}}\\)}^{2} $$\n](A978-1-84996-151-6_5_Fig10_HTML.gif)Diastolic pressure can be used as an estimate of aortic diastolic pressure. In this patient with blood pressure of 110\/65, the estimated LVEDP is![\n$$ \\\\begin{array}{c}{LVEDP }={ 65}-{4\\(2}.{97}{\\)}^{{2} } \\\\\\\\ ={ 3}0{ mmHg}.\\\\end{array} $$\n](A978-1-84996-151-6_5_Fig11_HTML.gif)\n\nFig. 5.6\n\ndP\/dt is estimated from the time interval (dt) required for the mitral regurgitation jet \u00advelocity to increase from 1 to 3 m\/s. The LV pressure rise is derived from the Bernoulli equation (4v12 \u2013 4v22) and assumes that left atrial pressure does not change significantly during systole. The latter assumption makes this equation invalid in the setting of acute mitral regurgitation. In this patient with systolic dysfunction,![\n$$ \\\\begin{array}{l}{dt}={ }0.0{4 s}\\\\\\\\ {dP}\/{dt}={ }\\({{4}}^{*}{{3}}^{{2}}-{{ 4}}^{*}{{1}}^{{2}}\\)\\({mmHg}\\)\/{dt}\\(s\\) \\\\\\\\ ={ 32}\/0.0{4} \\\\\\\\ {} ={ 8}00{ mmHg}\/{s} \\\\\\\\ {Normal dP}\/{dt} > { 1,000 mmHg}\/{s}. \\\\end{array} $$\n](A978-1-84996-151-6_5_Fig13_HTML.gif)\n\nFig. 5.7\n\n(a) M-mode cursor is placed through the mitral valve leaflets in the parasternal long-axis view. (b) Anterior mitral leaflet motion entails early diastolic motion (E) and late diastolic motion (A). In this patient with systolic dysfunction, a third movement, the B hump (B), is seen, indicating elevated LVEDP. The sign has low sensitivity but high specificity for LVEDP ?> ?20 mmHg\n\nFig. 5.8\n\nDemonstration of normal and abnormal relaxation patterns based on mitral inflow velocity and tissue Doppler imaging of the septal mitral annulus (E?). Mitral inflow velocity obtained by placing sample volume at mitral valve leaflet tips. (Left) Normal E\/A ratio and normal E? velocity in a normal subject. Note that E\/E? in patients with normal EF the correlation with wedge pressure is weaker due to the lower correlation of E to filling pressures. (Middle) Reduced E? and indeterminate E\/E? in patient with reduced ejection fraction. (Right) Markedly elevated E and reduced E? in patient with non-systolic heart failure.\n\n  * E\/E? caveats\n\n  * E\/E? 8\u201315 indeterminate for prediction of PCWP\n\n  * Does not predict PCWP if normal relaxation\n\n  * Not reliable in setting of severe mitral regurgitation\n\n  * Falsely low in setting of constrictive pericarditis (annulus paradoxus) (Table 5.1)\n\nTable 5.1\n\nQuantitative hemodynamic measurements\n\nHemodynamic parameter | Echocardiographic correlate | Comment\n\n---|---|---\n\nRight atrial pressure (RAP) | IVC diameter and respirophasic variation | Not valid in positive pressure ventilation except to exclude high RAP\n\nPulmonary artery systolic pressure | 4(VTR)2 ?+ ?RAP\n\n|\n\nPulmonary capillary wedge pressure (PCWP) | E\/E? | E\/E? ?> ?12 predicts PCWP ?> ?15 mmHg in patients with reduced EF\n\nLeft ventricular end-diastolic pressure (LVEDP) | Aortic diastolic pressure \u2013 4(VAR)2\n\n|\n\nLeft atrial pressure (LAP) | SBP \u2013 4(VMR)2 | Exclude patients with acute MR, prosthetic MV, and LVOT obstruction or peripheral arterial disease of the arm\n\nCardiac output (Q) | HR ?\u00d7 ?VTI ?\u00d7 ?Area | Accurate measure of LVOT diameter is critical\n\ndP\/dt | 32\/?t | ?t measured from continuous wave Doppler of MR jet\n\nPulmonary vascular resistance (PVR) | 10 ?\u00d7 ?(VTR\/RVOT VTI) ?+ ?0.16\n\n|\n\n## 5.2 Summary\n\nRecommended Reading\n\nAbbas AE, Fortuin FD, Schiller NB, Appleton CP, Moreno CA, Lester SJ. A simple method for noninvasive estimation of pulmonary vascular resistance. J Am Coll Cardiol. 2003;41(6):1021-1027.PubMedCrossRef02973-X)\n\nAbraham J, Abraham TP. The role of echocardiography in hemodynamic assessment in heart failure. Heart Fail Clin. 2009;5(2):191-208.PubMedCrossRef\n\nLewis JF, Kuo LC, Nelson JG, Limacher MC, Quinones MA. Pulsed Doppler echocardiographic determination of stroke volume and cardiac output: clinical validation of two new methods using the apical window. Circulation. 1984;70:425-431.PubMedCrossRef\n\nNagueh SF, Middleton KJ, Kopelen HA, Zoghbi WA, Quinones MA. Doppler tissue imaging: a noninvasive technique for evaluation of left ventricular relaxation and estimation of filling pressures. J Am Coll Cardiol. 1997;30:1527-1533.PubMedCrossRef00344-6)\n\nXie GY, Berk MR, Smith MD, Gurley JC, DeMaria A. Prognostic value of Doppler transmitral flow patterns in patients with congestive heart failure. J Am Coll Cardiol. 1994;24:132-139.PubMedCrossRef90553-3)\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_6\n\n# 6. Echocardiographic Assessment of Diastolic Dysfunction (DD) and Heart Failure with Normal EF (HFnlEF)\n\nAnil Mathew and Luis Afonso1\n\n(1)\n\nDepartment of Cardiology, Detroit Medical Centre, Wayne State University, Detroit, MI, USA\n\nLuis Afonso\n\nEmail: lafonso@med.wayne.edu\n\n## 6.1 Physiology\n\nA.\n\nAbout half of patients with new diagnoses of heart failure have normal or near normal global ejection fractions (EF). These patients are diagnosed with diastolic heart failure or heart failure with preserved\/normal EF(HFnlEF).\n\nB.\n\nElevated filling pressures are the main physiologic consequence of diastolic dysfunction.\n\nC.\n\nDiastolic function is modulated by myocardial relaxation and myocardial tone but is mainly determined by myocardial stiffness (filling and passive properties of the LV wall).\n\nD.\n\nMyocardial relaxation is determined by:\n\na.\n\nLoad \u2013 increased afterload, late systolic load and\/or elevated preload ?? ?delays myocardial relaxation ?? ?elevated filling pressures\n\nb.\n\nMyocardial inactivation \u2013 in cardiac myocyte, cross bridge detachment caused by calcium exit from cell\n\nc.\n\nNonuniformity (of relaxation in cardiac muscle)\n\nE.\n\nMyocardial stiffness is determined by\n\na.\n\nIntrinsic factors (ex: titin within myocardial cell fibrosis in interstitial, matrix)\n\nb.\n\nExtrinsic factors (ex: pericardial constraint and ventricular interaction)\n\nF.\n\nFour phases of diastole (Fig. 6.1)\n\n  * First Phase \u2013 Isovolumetric Relaxation (IR)\n\nRapid decline in LV pressure due to LV relaxation\n\n  * Second Phase \u2013 Rapid Filling\n\na.\n\nFirst pressure crossover \u2013 end of isovolumic relaxation and mitral valve opening\n\nb.\n\nLA\u2013LV pressure gradient \u2013 acceleration of mitral flow\n\nc.\n\nSecond pressure crossover \u2013 Peak mitral E\n\n(Thereafter, left ventricular pressure exceeds left atrial pressure, decelerating mitral flow)\n\nFig. 6.1\n\nThe Four phases of diastole\n\n  * Third Phase \u2013 Slow Filling\n\nNo pressure differences between LA and LV\n\n  * Fourth Phase \u2013 Atrial Contraction\n\nLA\u2013LV pressure gradient from atrial contraction, resulting in another acceleration of mitral flow\n\n## 6.2 Echocardiographic Parameters of Diastolic Dysfunction\n\n### 6.2.1 Morphologic and Functional Correlates of Diastolic Dysfunction\n\n1.\n\nLeft ventricular hypertrophy\n\n  * Concentric vs. eccentric hypertrophy: Concentric hypertrophy is generally associated with preserved EF and DD. Eccentric hypertrophy is generally associated with reduced EF and DD.\n\n  * LV thickness: measure this to determine if (concentric) LV hypertrophy is present.\n\n2.\n\nLA volume\/area\n\n  * LA volume\/area: reflects the cumulative effects of filling pressure over time and therefore a potential indicator for the presence of diastolic dysfunction.\n\n3.\n\nPulmonary artery pressures\n\n  * Patients with diastolic dysfunction who are symptomatic usually have increased pulmonary artery pressures.\n\n  * In the absence of pulmonary or valvular disease, increased PA pressures indicates the presence of elevated LV filling.\n\n  * Tricuspid regurgitation jets and pulmonic regurgitation jets can be used to estimate PA systolic and diastolic pressures, respectively:\n\n![\n$$\\\\begin{array}{c}PASP=4{\\(}^{T}+RApressure\\\\\\\\ PADP=4{\\(}^{P}+RAPressure\\\\\\\\ \\(RApressureestimatedfromIVC\\)\\\\end{array}$$\n](A978-1-84996-151-6_6_Fig2_HTML.gif)\n\nPA diastolic pressure usually correlates well with invasively measured PCWP (LVEDP) and may be used as its surrogate but this estimation is dependent on quality of PR signal and accuracy of RA pressure estimate.\n\n4.\n\nMitral annular motion\n\n  * Reduced apical motion of the mitral valve during diastole may be a visual clue to reduced e' velocity as seen on tissue Doppler (discussed later).\n\n### 6.2.2 Mitral Inflow Velocities\n\nTechnical tips \u2013 PW Doppler\n\nIn the apical four chamber view, using a 1\u20133 mm sample volume, PW Doppler cursor is placed between the mitral leaflet tips during diastole.\n\nMeasurements: Peak E velocities, Peak A velocities, E\/A ratio\n\nDeceleration time (DT) \u2013 time from peak velocity of E wave to termination of E wave\n\n1.\n\nMitral hemodynamics\n\n  * Mitral E velocity reflects the LA\u2013LV pressure gradient during early diastole and is affected by preload and alterations in LV relaxation.\n\n  * Mitral A velocity reflects the LA\u2013LV pressure gradient during late diastole and is affected by LV compliance and LA contractile function.\n\n  * DT (of E wave) reflects the rate of decline in the early LA\u2013LV pressure gradient and is affected by LV relaxation, LV diastolic pressures post-MV opening and LV compliance.\n\n2.\n\nMitral inflow patterns\n\nFour Mitral Filling Patterns based on E:A ratio and DT (Fig. 6.2):\n\nFig. 6.2\n\nMitral inflow profiles across spectrum of diastolic dysfunction\n\nNormal | E\/A: 1\u20132 | DT: 160\u2013200 ms\n\n---|---|---\n\nImpaired Relaxation | E\/A ?< ?0.8 | DT ?> ?200 ms\n\nPseudonormal | E\/A: 0.8\u20131.5 | DT: 160\u2013200 ms\n\nRestrictive | E\/A ?? ?2 | DT ?< ?160 ms\n\n3.\n\nNormal values (see Table 6.1)\n\n  * Definition of normal depends on age of patient: increasing age ?? ?decrease in mitral E velocity, E\/A ratio, increase in DT and A velocity\n\n  * Variables which affect mitral inflow (other than diastolic function and filling): HR and rhythm, PR interval, cardiac output, mitral annular size, LA function\n\nTable 6.1\n\nNormal values for Doppler-derived diastolic parameters partitioned according to age | Age group (y)\n\n|   \n---|---|---\n\nMeasurement | 16-20 | 21-40 | 41-60 | >60\n\nIVRT (ms) | 50 \u00b1 9 (32-68) | 67 \u00b1 8 (51-83) | 74 \u00b1 7 (60-88) | 67 \u00b1 7 (73-101)\n\nE\/A ratio | 1.88 \u00b1 0.45(0.98-2.78) | 1.53 \u00b1 0.40 (0.73-2.33) | 1.28 \u00b1 0.25 (0.78-1.78) | 0.96 \u00b1 0.18 (0.6-1.32)\n\nDT (ms) | 142 \u00b1 19 (104-180) | 166 \u00b1 14(138-194) | 181 \u00b1 19(143-219) | 200 \u00b1 29 (142-258)\n\nA duration (ms) | 113 \u00b1 17 (79-147) | 127 \u00b1 13 (101-153) | 133 \u00b1 13 (107-159) | 138 \u00b1 19 (100-176]\n\nPV S\/D ratio | 0.82 \u00b1 0.18 (0.46-1.18) | 0.98 \u00b1 0.32 (0.34-1.62) | 1.21 \u00b1 0.2 (0.81-1.61) | 1.39 \u00b1 0.47 (0.45-2.33)\n\nPV Ar (cm\/s) | 16 \u00b1 10 (1-36) | 21 \u00b1 8 (5-37) | 23 \u00b1 3 (17-29) | 25 \u00b1 9 (11-39)\n\nPV Ar duration (ms) | 66 \u00b1 39 (1-144) | 96 \u00b1 33 (30-162) | 112 \u00b1 15 (82-142) | 113 \u00b1 30 (53-173)\n\nSeptal e' (cro\/s) | 14.9 \u00b1 2.4 (10.1-19.7) | 15.5 \u00b1 2.7 (10.1-20.9) | 12.2 \u00b1 2.3 (7.6-16.8) | 10.4 \u00b1 2.1 (6.2-14.6)\n\nSeptal e'\/a' ratio | 2.4' | 1.6 \u00b1 0.5 (0.6-2.6) | 1.1 \u00b1 0.3 (0.5-1.7) | 0.85 \u00b1 0.2 (0.45-1.25)\n\nLateral e' (cm\/s) | 20.6 \u00b1 3.8 (13-28.2) | 19.8 \u00b1 2.9 (14-25.6) | 16.1\u00b1 2.3 (11.5-20.7) | 12.9 \u00b1 3.5 (5.9-19.9)\n\nLateral e'\/a' ratio | 3.1' | 1.9 \u00b1 0.6 (0.7-3.1) | 1.5 = 0.5 (0.5-2.5) | 0.9 \u00b1 0.4 (0.1-1.7)\n\nFrom Nagueh et al Journal of the American Society of Echocardiography Feb 2009\n\n4.\n\nClinical application to patients with depressed and normal EFs\n\n  * Dilated cardiomyopathies \u2013 filling patterns correlate well with filling pressures functional class and prognosis \u2013 better than LV EF\n\n  * Coronary artery disease and hypertrophic cardiomyopathy (EF ?? ?50%) \u2013 filling patterns correlate poorly with hemodynamics\n\n5.\n\nValsalva maneuver and mitral inflow: distinguishing normal from PNF (Fig. 6.3)\n\nFig. 6.3\n\nDiscriminatory utility of the Valsalva maneuver\n\n  * LV preload is reduced during the strain phase (phase II) and changes in mitral inflow can be used to distinguish normal from PNF\n\n  * Valsalva maneuver results in decreased preload during the strain phase. Mitral filling pattern changes to impaired relaxation in patients with PNF post-Valsalva\n\n6.\n\nCaveats\n\n  * LV filling patterns have a \"U-shaped\" relation with LV diastolic function with similar patterns in healthy subjects and patients with Grade II (Pseudonormal) dysfunction.\n\n  * Arrhythmias make filling patterns difficult to interpret sinus tachycardia, first degree AV block: partial or complete fusion of E and A waves, atrial flutter: LV filling heavily influenced by atrial contractions.\n\n### 6.2.3 Pulmonary Vein Inflow Velocities\n\nTechnical tips \u2013 PW Doppler\n\n  * Color Doppler is used to localize area of inflow of right upper pulmonary vein (RUPV) in the apical four chamber view.\n\n  * In this view, using a 2\u20133 mm sample volume, PW Doppler cursor is placed at (RUPV) >0.5 cm into RUPV.\n\n  * Measurements: systolic (S) velocity and components (S1, S2), peak anterograde diastolic (D) velocity, S\/D ratio, atrial reversal (Ar) wave (Ar wave duration, peak Ar velocity in late diastole)\n\n1.\n\nPulmonary vein inflow waveforms\n\nReflect phases of LA function during cardiac cycle(see Fig. 6.4)\n\nReduction in LA compliance and increase in LA filling pressure result in:\n\n    * Decrease in S velocity, increase in D velocity ?? ?decrease in S\/D velocity ratio\n\n    * Increase in Ar velocity, duration and increase in Ar amplitude and duration\n\n2.\n\nNormal values\n\n    * Definition of normal depends on age of patient: Increasing age ?? ?S\/D ratio increases, Ar velocity increases (not greater than 35 cm\/s)\n\n3.\n\nClinical application to patients with depressed and normal EFs\n\nIn patients with depressed EFs, reduced systolic filling fractions (<40%) are related to decreased LA compliance and increased mean LA pressure.\n\n    * Ar duration \u2013 Mitral A duration difference >30 ms indicates elevated LVEDP and can reliably distinguish those with high LVEDP but normal filling pressures in both normal and reduced EFs.\n\n4.\n\nLimitations\n\nDifficulty in obtaining high quality recordings suitable for measurements\n\n    * Tachycardia and arrhythmias make filling patterns difficult to interpret\n\n    * Atrial fibrillation \u2013 loss of atrial contraction and relaxation reduces pulmonary systolic venous flow velocities regardless of filling pressures\n\nFig. 6.4\n\nSchematic illustrating various components of pulmonary venous flow\n\n### 6.2.4 Tissue Doppler Annular Early and Late Diastolic Velocities\n\nTechnical tips\n\nTissue Doppler velocities are a measure of the speed of movement of myocardial tissue in comparison to the inflow velocities, which measure the speed of movement of blood cells. In the apical four chamber view, using a 5\u201310 mm sample volume, tissue Doppler cursor is placed at or 1 cm within septal and\/or lateral insertion sites of mitral valve leaflets (Figs. 6.5 and 6.6)\n\nFig. 6.5\n\nMitral annular tissue Doppler velocity acquisition and waveforms\n\nFig. 6.6\n\nSpectrum of pulmonary flow and mitral annular tissue Doppler velocities in diastolic dysfunction\n\nMeasurements: Early diastolic (e?) and late diastolic (a?) velocities\n\nAdditional value computed from velocities \u2013 E (mitral inflow)\/e? (see below)\n\n1.\n\nHemodynamic determinants\n\n  * e? is determined by LV relaxation, preload (minimally), systolic function and LV minimal pressure\n\n  * a? is determined by LA systolic function and LVEDP\n\n2.\n\nNormal values\n\n  * Definition of normal depends on age of patient: increasing age ?? ?e? decreases, a? and E\/e? increases\n\n3.\n\nClinical application\n\nE\/e? ratio:\n\n  * As diastolic dysfunction progressively worsens, LV filling pressures increase.\n\n  * As diastolic dysfunction progressively worsens, the LA\u2013LV pressure gradient (E) increases and e? velocity declines.\n\n  * Thus, E\/e? ratio is useful as an estimate of LV filling pressures in patients with LV dysfunction.\n\n    * E\/e? ?< ?8 ?= ?normal filling pressures, E\/e? ?> ?15 ?= ?elevated filling pressures\n\n    * If E\/e? ratio ?= ?8\u201314, other echocardiographic indices should be used to estimate LV filling pressures\n\n4.\n\nCaveats\n\n  * In normal subjects, e? is positively related to preload and therefore E\/e? may not always accurately reflect filling pressures.\n\n  * e? velocity is reduced in patients with mitral annular calcification, surgical rings, mitral stenosis and prosthetic mitral valves.\n\n  * e? velocity is increased in patients with moderate to severe primary MR and normal relaxation due to increased flow across the mitral valve.\n\n  * Constrictive pericarditis.\n\n    * Septal e? velocity is increased with constrictive pericarditis due to preserved LV longitudinal expansion compensating for limited lateral and anteroposterior diastolic excursion.\n\n    * Lateral e? velocity may be less than septal e? in constrictive pericarditis\n\n### 6.2.5 Color M-Mode Flow Propagation Velocity\n\nTechnical tips\n\nIn the apical four chamber view, using color flow imaging with a narrow color sector and gain adjustment to avoid noise, the M-mode scan line is placed through the center of the LV inflow blood column from the mitral valve to the apex.\n\nThe color flow baseline is shifted to lower the Nyquist limit so that the central highest velocity is blue. Flow propagation velocity (Vp) is measured as the slope of the first aliasing velocity during early filling, measured from mitral valve plane to 4 cm distally into the LV cavity (Fig. 6.7).\n\nFig. 6.7\n\nColor M-mode obtained from the Ap-4C view. Velocity of flow propagation (Vp or slope) estimated at 49.8 cm\/s (arrow) in this patient with normal diastolic function\n\n1.\n\nHemodynamic determinants\n\nVp is a measure of the early diastolic filling gradient and is determined by pressure gradient between LV base and apex\n\n  * Normal Values: Vp ?>50 cm\/s is considered normal.\n\n2.\n\nClinical application\n\n  * Slowing of Vp is a semiquantitative marker of LV diastolic dysfunction.\n\n  * E\/Vp may be used to predict LV filling pressures \u2013E\/Vp ?? ?2.5 predicts PCWP >15 mmHg.\n\n3.\n\nCaveats\n\n  * E\/Vp may not be reliable if EF is normal.\n\n## 6.3 Estimation of Left Ventricular Filling Pressures and Diastolic Dysfunction in Special Populations (see Fig 6.8)\n\n1.\n\nAtrial fibrillation\n\n  * Variability in cycle length, absence of organized atrial activity and LA enlargement limits the assessment of LV filling pressures in atrial fibrillation if LV EF is depressed, mitral DT (?150 ms) is reasonably accurate as a predictor of increased filling pressures and adverse clinical outcome\n\nFig. 6.8\n\nDiastolic dysfunction assessment in special populations\n\nOther measurements that may be used include deceleration time (DT) and E\/e? and E\/Vp\n\n  * Given variation in cycle length, three nonconsecutive beats with cycle lengths within 10\u201320% of average heart rate or measurements from one cardiac cycle with an RR interval corresponding to HR of 70\u201380 bpm should be used to obtain measurements of diastolic function.\n\n2.\n\nSinus tachycardia\n\n  * If E and A wave are fused, measure the peak velocity of the fused wave as the E wave velocity, do likewise for the e? velocity. E\/e? correlates well with filling pressure\n\n3.\n\nRestrictive cardiomyopathy\n\n  * In restrictive cardiomyopathy, regardless of whether idiopathic or infiltrative, mitral, pulmonary venous and tissue Doppler variables are all good indicators of elevation in filling pressures.\n\n4.\n\nHypertrophic cardiomyopathy\n\n  * E\/A ratio and DT have weak to no correlations with LV filling pressures.\n\n  * E\/e? ratio correlates fairly well w\/LV pre-A pressure.\n\n  * Ar \u2013 (A duration) ?? ?30 ms may be used in this population as in other populations.\n\n5.\n\nMitral stenosis (MS)\n\n  * Mitral stenosis patients will have elevated left atrial pressures due to valvular restriction of LV inflow. However the LV diastolic pressures will be normal or low (unless there is coexisting myocardial disease).\n\n  * Determining LA pressures is more difficult in MS but the following can be used as semiquantitative measures of mean LA pressures:\n\n    * IVRT interval \u2013 shorter interval ?? ?higher early diastolic LA pressure\n\n6.\n\nMitral regurgitation (MR)\n\nDetermining LA pressures is more difficult in MR but the following can be used as semi quantitative measures of mean LA pressures:\n\n  * IVRT interval \u2013 shorter interval ?? ?higher early diastolic LA pressure\n\n  * E\/e? ratio \u2013 is only useful in patients w\/depressed ejection fractions\n\n## 6.4 Other Causes of Heart Failure Symptoms in Patients with Normal EF\n\n### 6.4.1 Differentiating Constrictive Pericarditis from Restrictive Cardiomyopathy (Diastolic Dysfunction)\n\na.\n\nMitral annular velocities\n\n  * Constrictive Pericarditis\n\n    * Pericardium restricts movement of the lateral annulus ?? ?myocardial relaxation preserved BUT lateral annular vertical excursion impaired ?? ?SEPTAL e? normal or increased (?7 cm\/s), lateral e? reduced\n\nb.\n\nSeptal motion\n\n  * Restrictive cardiomyopathy (diastolic dysfunction) \u2013septal motion is normal\n\n  * Constrictive pericarditis \u2013 septal motion shows respiratory shift \u2013 inspiration ?? ?left, expiration ?? ?right\n\nc.\n\nMitral inflow pattern with respiration\n\n  * Restrictive cardiomyopathy (diastolic dysfunction): no variation in mitral inflow pattern\n\n  * Constrictive pericarditis: ?25% increase in mitral inflow E velocity with expiration\n\nd.\n\nHepatic flow vein pattern\n\n  * Restrictive cardiomyopathy (diastolic dysfunction): Diastolic forward flow reversal is augmented during inspiration\n\nDecrease in intrapleural pressures ?? ?increase in diastolic filling\n\n  * Constrictive Pericarditis: Diastolic forward flow reversal is augmented during expiration\n\nVentricular interdependence ?? ?increase in LV filling results in reduction in RV filling\n\n## 6.5 Left Atrium\n\n  * The size and volume of the left atrium (LA) increase with increasing grades of diastolic dysfunction.\n\n  * Mitral inflow parameters and E\/e? reflect instantaneous filling pressure.\n\n  * LA volume, in comparison reflects chronicity of elevated filling pressure.\n\n  * Normal LA volume rules out clinically significant diastolic dysfunction.\n\n  * Normal-appearing mitral inflow likely reflects PNF if the LA volume is increased.\n\nTechnical tips\n\nLA size is measured in end systole (end of T-wave or the frame prior to mitral valve opening). Recommended methodology is the area\u2013length method, using the apical ?4C and 2C views (Fig. 6.9)\n\nFig. 6.9\n\nEstimation of left atrial volume using the area-length method, from the apical-4C (A1) and apical-2C (A2) views, at end-systole. Left atrial volume (ml) = 8\/3? [(A1)(A2)\/ (L*)], where L*= shortest vertical length between Ap-4C and Ap-2C views, measured from back wall to line across hinge points of mitral valve. LA volume may be indexed to body surface area and expressed as LA volume index (ml\/m2)\n\nFig. 6.10\n\nEstimation of LV filling pressure in patients with normal systolic function\n\nFig. 6.11\n\nEstimation of LV filling pressure in patients with depressed systolic function\n\nFig. 6.12\n\nPractical approach to grading diastolic dysfunction\n\nCaveats\n\n  * LA dilation should be interpreted with caution in high flow states such as anemia, pregnancy.\n\n  * Afib and MR can cause atrial enlargement not related to diastolic dysfunction\n\n  * LA dilation in the setting of mitral stenosis occurs due to inflow obstruction\n\n  * Anteroposterior measurement of LA diameter can be misleading and may underestimate\n\n  * LA size as the LA enlarges asymmetrically in the lateral and infero-superior direction\n\nNormal values: LA volume\/BSA (mL\/m2): Normal reference range: 22 ?\u00b1 ?6 mL\/m2\n\nLA area: normal ?< ?20 cm2; 20\u201330 cm2 (mild dilation); 30\u201340 (moderate); >40 cm2 (severely dilation)\n\n## 6.6 Evaluation of Patient with Diastolic Dysfunction (DD) or Heart Failure with Normal EF (HFnlEF)\n\n### 6.6.1 Evaluation, Differential Diagnosis and Subtypes of the DD\/HFnlEF Patient\n\nThe first step in evaluation of a patient with dyspnea due to diastolic dysfunction is to exclude noncardiogenic causes through history, physical and appropriate medical testing.\n\nThe second step is to evaluate for cardiogenic causes of dyspnea by echocardiography.\n\nEvaluating ejection fraction, looking for structural abnormalities and evaluation for the presence of valvular disease.\n\nOnce these have been excluded, in the third step the presence of signs of diastolic dysfunction must be confirmed which are left atrial enlargement or the presence of diastolic function abnormalities.\n\nThe final step in the evaluation of DD\/HFnlEF is to evaluate wall motion, chamber geometry and pulmonary artery pressures to define the subtype of DD\/HFnlEF that is present.\n\n### 6.6.2 Estimation of Elevated Filling Pressures in Normal EF (Refer to algorithm outlined in Fig. 6.10)\n\nThe E\/e? ratio should be used to determine if filling pressures are elevated.\n\n  * E\/e? ratios ?<8 indicate the presence of normal filling pressures.\n\n  * E\/e? ratios ?> ?12 (if lateral e? is measured), 15 (if septal e? is measured) or 13 (if the average of septal and lateral e? are obtained) indicate the presence of elevated filling pressures.\n\n  * E\/e? ratios between 9 and 14 require the use of additional Doppler parameters.\n\nThe most commonly used parameters are LA volume and pulmonary artery systolic \u00adpressures (PAS) as estimated by peak TR velocities.\n\n### 6.6.3 Estimation of Elevated Filling Pressures in Reduced EF (Refer to algorithm outlined in Fig. 6.11)\n\nThe mitral inflow pattern is very useful in determining left atrial filling pressures in this subset of patients.\n\n  * E\/A ratios of <1 and E velocities ?<50 cm\/s indicate the presence of normal filling pressures.\n\n  * E\/A ratios of ?>2 and DT ?<150 ms indicate the presence of restrictive filling and elevated filling pressures.\n\n  * E\/A ratios between 1 and 2 require the use of additional Doppler parameters.\n\nThe most commonly used are E\/e? or pulmonary artery systolic pressures (PAS) as estimated by peak TR velocities.\n\nPAS may be used only in the absence of pulmonary disease.\n\n  * Other measures also may be used to determine normal or elevated filling pressures.\n\n  * LA volume should NOT be used to estimate LV filling pressures in this population because some LA dilatation may occur even when LV filling pressures are still normal.\n\n### 6.6.4 Grading Diastolic Dysfunction (Refer to Fig 6.12)\n\nThe grading scheme is Grade I (mild), Grade II (moderate), or Grade III (severe).\n\nThis scheme was an important predictor of all cause mortality in a large epidemiology study\n\nAside from the parameters mentioned in the figure, age and heart rate should be taken into account in grading diastolic dysfunction\n\n  * Age \u2013 in elderly individuals without cardiovascular disease, grade I diastolic dysfunction is commonly seen and may be considered normal in this population.\n\n  * Heart rate \u2013 increasing heart rate ?? ?reduction in mitral E, E\/A ratio, and annular e?velocities.\n\n  * Grade I Diastolic Dysfunction\n\n    * These patients have reduced diastolic reserve that can be uncovered with stress testing.\n\n    * Volume depleted subjects may have reduced E\/A ratios but normal tissue Doppler velocities.\n\n    * Therefore E\/A ratio ?<0.8 should not alone determine the presence of diastolic dysfunction.\n\n    * LA pressures in these patients are not elevated (exceptions: hypertension, hypertrophic cardiomyopathy).\n\n    * LA volume may be normal\n\n  * Grade II Diastolic Dysfunction\n\n    * LA pressures in these patients are elevated\n\n    * LA volume is increased\n\n  * Grade III Diastolic Dysfunction\n\n    * LV filling may revert to impair relaxation with successful treatment\n\n    * Response to treatment indicates a subtype of grade III with the more favorable prognosis (Grade IIIa) compared with those who do not respond, indicating the subtype of grade III with a high risk of cardiac morbidity and mortality (Grade IIIb\/restrictive filling).\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_7\n\n# 7. Tissue Doppler Imaging and Strain Echocardiography\n\nVeronica Lea J. Dimaano1\n\n(1)\n\nDivison of Cardiology, Johns Hopkins University School of Medicine, Baltimore, MD, USA\n\nVeronica Lea J. Dimaano\n\nEmail: vdimaan1@jhmi.edu\n\nDoppler principle has been routinely used in echocardiography to provide hemodynamic data to the anatomical information offered by two-dimensional (2D) examination. It measures high-frequency, low-amplitude signals from rapidly moving red blood cells enabling quantification of blood flow velocities. Recently, modifications have been applied to the filter settings on pulsed Doppler to allow imaging low-velocity, high-intensity myocardial signal, thereby selectively calculating and displaying on-line myocardial velocity information derived from Doppler shifts created by cardiac motion. This technique referred to as tissue Doppler imaging (TDI) or Doppler myocardial imaging (DMI) provides segmental velocity information utilizing three principal approaches: spectral pulsed-Doppler (PW), color-coded M-mode Doppler display, and 2D color-coded Doppler approach. PW and 2D color-coded TDI are more commonly used in clinical settings. While both techniques provide the same mechanical information, data acquired via each method may differ and may not be interchangeable requiring appropriate labeling to avoid confusion (Figs. 7.1 and 7.2). Pulsed Doppler technique yields peak tissue velocity that is 20\u201330% higher than the mean velocity provided by color-coded Doppler approach; a difference that should be considered when estimating left ventricular filling pressures using the E\/E? ratio.\n\nFig. 7.1\n\nTissue Doppler imaging (TDI) provides segmental velocity information, which can be displayed by (a) pulsed wave or (b) color Doppler technique. While both methods yield the same mechanical information, differences inherent to each technique exist. Velocities obtained via pulsed-wave TDI are higher than that obtained using the color-coded TDI. This difference is of prime importance when estimating left ventricular filling pressures using the E\/E? ratio\n\nFig. 7.2\n\nTissue velocity nomenclature. Relative to an external transducer, a triphasic wave form characterizes the typical tissue velocity curve. Depending on the location of sample volume (mitral annulus vs. myocardium), each wave form is identified by its timing relative to the phase of the cardiac cycle. Following a positive wave representing ventricular systole, two negatively directed waves corresponding to early and late diastole appear as a mirror image of the early (E) late and (A) left ventricular filling velocities. Mitral annular velocity has been frequently interrogated using pulsed wave TDI. Its combination with mitral inflow velocity (E\/E? or E\/Ea) can predict LV filling pressure\n\nTissue velocity depicts myocardial motion at specific locations in the heart and indicates the rate at which a particular point in the myocardium is displaced relative to an external transducer. Integration of velocity over time yields displacement or the absolute distance traversed by that particular point. Longitudinal cardiac motion is such that the base descends toward the generally immobile apex during systole and moves away from it during diastole. A triphasic tissue velocity display represents the magnitude of myocardial motion as the heart contracts and relaxes during systole and diastole, respectively. Because assessment of tissue velocity has its reference on a point outside the heart (transducer), a velocity gradient exists between a more basal and a more apical segment of interest. (Fig. 7.3). As with the classical Doppler technique, tissue velocity assessment is greatly influenced by the angle of insonation and the translational motion of the heart. It is limited by its inability to differentiate passive from active myocardial motion consequent to the \"tethering effect\" wherein a normal more apical segment of the myocardium drags an abnormal basal segment toward the apex.\n\nFig. 7.3\n\nTissue velocity interrogation has been almost exclusively performed in the longitudinal direction in which the base descends toward the generally immobile apex during systole and moves away from it during diastole. Because assessment of tissue velocity has its reference on a point outside the heart (transducer) a velocity gradient (white arrows) exists between a more basal and a more apical segment of the myocardium. Thus, tissue velocity is maximum at the base (V 1), lower in the mid (V 2), and least at the apex (V 3)\n\nDoppler-derived strain rate (SR) and strain measure the speed of myocardial deformation derived from velocity gradient and is expressed in s?1 (Fig. 7.4). SR profile is a triphasic display of the magnitude of myocardial deformation rate as the heart contracts in systole and relaxes in diastole (Fig. 7.5). Integration of SR yields strain. Strain is a measure of myocardial deformation defined as the change in length normalized to the original length and is expressed in percent (Fig. 7.6). Understanding tissue velocity-SR-strain relationship (Fig. 7.7 and 7.8) and the limitations (Fig.7.9-7.12) inherent to the Doppler technique are essential in the analysis and interpretation of these studies.\n\nFig. 7.4\n\nVelocity gradient is used to calculate strain rate. With tissue Doppler, differential velocity of two adjacent segments in myocardium along the ultrasound beam normalized to the distance between these velocities (V 1 and V 2) yields strain rate. Strain rate measures the speed at which myocardial deformation occurs and is expressed as s?1\n\nFig. 7.5\n\nDoppler-derived strain rate is a spatial derivation of the velocity gradient. In contrast to tissue velocity, the points of reference reside within the segment under interrogation. While both entities define the rate at which a particular change occurs (location for tissue velocity and length for strain rate) the triphasic curve depicting the timing of events in one cardiac cycle is displayed on an opposite magnitude. To illustrate, a sample volume is placed at the base of the left ventricular septum using a strain length of 18 mm (distance between points 1 and 2). The speed by which the two reference points (1 and 2) in the myocardium move closer to each other during systole (myocardial contraction) is termed systolic strain rate (SRs). Since the distance between these two reference points becomes progressively shorter, the resulting SRs amplitude is directed negatively. SRs normally peaks at early to mid-systole. The rate by which the reference points in the myocardium move away from each other during relaxation is termed diastolic strain rate. Since the distance between these reference points becomes longer, the resulting early (SRe) and late (SRa) diastolic amplitudes are directed positively\n\nFig. 7.6\n\nStrain denotes myocardial deformation and is defined as the change in length normalized to the initial length. In a one-dimensional object, the only possible deformation of the object is lengthening or shortening. The relative amount of deformation is defined as strain and is expressed in percent. The equation is defined in such a way that lengthening is represented as a positive value for strain while shortening is represented by a negative value (13). Hence, if the initial length is noted to be 5 cm and it shortens to 4 cm, the strain will be ?20%. However, if from 5 cm it lengthens to 6 cm, the strain will be 20%\n\nSpeckle tracking is a relatively angle-independent method of quantifying strain. It enables measurement of myocardial deformation in basically all regions of the myocardium in the longitudinal, radial, and circumferential directions (Fig.7.13-7.15). As it utilizes 2D images, it operates on a lower frame rate and may be inferior to Doppler-derived strain where temporal resolution is concerned. With speckle-tracking, SR is derived from strain, while with Doppler-based technique, it is integrated to yield strain.\n\nMyocardial deformation is the result of the complex interaction of intrinsic contractile force and extrinsic loading conditions applied to a tissue with variable elastic properties. Therefore, changes in preload and afterload, and the changes in myocardial stiffness, are important determinants of the pattern and the magnitude of myocardial deformation. Thus, SR and strain indices are not direct measures of contractility. However, peak systolic strain rate (SRs) was found to correlate best with dP\/dt (an index of contractile function). On the other hand peak systolic strain was found to correlate best with changes in stroke volume and, therefore, was more closely related to changes in global hemodynamics than changes in contractility.\n\nSome of the clinical applications of the TDI and strain imaging include assessment of global and regional left ventricular systolic and diastolic function, differentiating constrictive pericarditis and restrictive cardiomyopathy, ischemia detection and evaluation of viability, early detection of cardiomyopathies, and evaluation of mechanical dyssynchrony (Fig.7.16-7.18). Application of TDI and strain imaging in the evaluation of right ventricular (RV) function is more challenging because of the ventricle's complex shape and its marked load \u00addependence.\n\nFig. 7.7\n\nUsing velocity gradient technique, strain and strain rate are derived from the tissue velocity data. Knowledge of the relationship between these Doppler-derived measures of myocardial function play an important role in the analysis and interpretation of a strain\/strain rate curve\n\nFig. 7.8\n\nTo elucidate on tissue velocity\u2013SR\u2013strain relationship, a narrow sector color Doppler image of interventricular septum using the apical 4-chamber view was acquired from a normal volunteer. Timing of aortic valve opening (AVO) and (AVC) closure (vertical green dashed lines) was measured from the spectral Doppler display of left ventricular outflow tract. Sample volumes were placed at the base and mid-septum. (yellow and green circles, respectively, in the upper left panel of all sets of images). (a) Velocity profile is extracted over one cardiac cycle. Motion during systole, early and late diastole can be identified by the triphasic display. (b) Time-integration of the velocity curve results in the displacement curve. In the longitudinal direction, there is motion toward the transducer during systole and away from the transducer during diastole. The displacement of the more apical segment (green curve) is smaller than that of more basal one (yellow curve). (c) Strain rate curve extracted from the same velocity color-Doppler data in (a). Calculation of the spatial gradient in the myocardial velocities yields an estimate of strain rate. During systole, longitudinal shortening of the ventricle is identified by a negative strain rate value while the two lengthening phases during diastole (passive and active LV filling) are depicted by positive strain rate phases. Presence of multiple peaks may interfere with identification of the true peak diastolic strain rates (yellow curves corresponding to the basal septum tracing). (d) Time integration of strain rate curves yields regional strain curves. It is evident that the wall shortens during systole and lengthens during diastole. In this particular example, the strain curve is characterized by two peaks marked by yellow arrows. Cross-checking with the strain rate curve, a second negatively directed curve appears near the AVC (broken yellow arrow in c). This curve corresponds to the second peak in the strain curve (broken yellow arrow in d) and, therefore, is not the true peak systolic strain\n\nFig. 7.9\n\nLocal coordinates of strain measurement. Using the heart coordinate system (13) myocardial strain can be measured in three directions. The heart shortens and lengthens in the longitudinal direction (a), thickens and thins in radial direction, and shortens and lengthens in circumferential direction (b). (c) When the heart is viewed from the apex, the base (3) and the apex (1) rotate in opposite direction effecting a wringing motion or torsion. (d) During systole (solid yellow arrow), the apex rotates counterclockwise and the base, clockwise (twisting). The heart untwists in diastole (yellow broken arrow)\n\nFig. 7.10\n\n(a) The magnitude of the Doppler-derived strain curve is influenced by the angle of insonation. The line of deformation of a particular myocardial segment should be parallel to the Doppler beam. Thus myocardial wall should be kept parallel to the beam when longitudinal strain is being assessed and perpendicular to it when measuring radial strain. The translational motion of the heart exists in a multi-plane coordinate and myocardial deformation is not unidimensional. In this example of a Doppler-derived strain curve taken from a normal volunteer, sample volumes were placed at the base (yellow circle) and the mid segment (red circle) of the lateral wall in the apical 4-chamber view. The longitudinal strain profile is displayed as two strain curves in opposite magnitudes typical of dyskinesis or dyssynchrony. Lower left panel, inset: A closer look at the position of the strain bars (solid green lines) in relation to the myocardial fiber orientation would reveal that the Doppler beam is approaching a perpendicular relationship to the basal segment and a more parallel relationship to the mid-segment where the respective sample volumes were placed. Hence, during systole, the predominant pattern of myocardial deformation at the base is thickening rather than shortening, resulting in a positively directed strain curve. A schematic diagram is presented in the right panel (b). To address this problem, movement of the sample volume around the basal segment should be performed. During image acquisition, narrow sector images should be taken in addition to proper positioning of the transducer to ensure alignment of wall with the Doppler beam\n\nFig. 7.11\n\nThe amplitude of the Doppler-derived velocity, strain rate, or strain curve is influenced by the insonation angle. When the Doppler beam obliquely impinges on the myocardial wall, the resultant vector (strain) corresponds to a combination of radial, longitudinal, and circumferential strains. When the regions of interest are both parallel to the Doppler beam, as with interventricular septum interrogation, the strain value should normally be approximately the same. It is not uncommon to register a \"numerically\" abnormal value when interrogating left ventricular walls that would normally create an angle with the Doppler beam. In this example from a normal volunteer, sample volumes are placed at the base (yellow) and mid-inferior (aqua) segments in a narrow sector image from apical 2-chamber view. The longitudinal deformation profile displays a peak strain value of 10% (identified by red arrow) in the mid-segment and approximately 16% in the basal segment. Poor alignment of the ultrasound beam (approximated by the direction of the strain length bar in green line) with the inferior wall (longitudinal deformation is marked by white broken arrow) is evident in the images on the left panel with the mid-segment creating a more oblique angle with the ultrasound beam. The low numerical longitudinal strain value in the mid-inferior can be explained by the \"hybrid effect\" of the longitudinal (negative magnitude) and radial (positive magnitude) strain. Thus, care should be taken during acquisition of data sets when interrogating myocardial deformation using this technique. AVO, aortic valve opening; AVC, aortic valve closure\n\nFig. 7.12\n\nTwo-dimensional speckle tracking-derived strain measurement. The interaction of ultrasound with the myocardium produces unique acoustic patterns, or \"speckles.\" These speckles can be tracked overtime and speckle displacement used to calculate tissue velocity and strain. Initial speckle locations (yellow circles) are recorded at end-diastole (left panel) and tracked over time (dotted yellow circles) to their final locations at end-systole (solid yellow circles in the right panel). Integration of displacement yields strain where, in turn, strain rate is derived. This technique is relatively angle-independent because it is not based on the Doppler principle. However, the technique is inferior to Doppler-based imaging in terms of temporal resolution and may not be accurate in timing mechanical events. Speckle-tracking is performed on B-mode images with a much lower frame rate (40\u201390 fps) compared to the Doppler-derived imaging where frame rates are high (100\u2013250 fps)\n\nFig. 7.13\n\nQuad-view of longitudinal strain by speckle-tracking in the LV septum and lateral wall using apical 4-chamber view. Speckle tracking method utilizes automated tracking of the region of interest (ROI). All points in the myocardium are included in the analysis. (a) Automated tracking of septum and lateral wall in the apical 4-chamber view. Each segment of the myocardium is color-coded (dotted line). The color overlay indicates peak systolic strain along the longitudinal direction per color scale on the right upper corner. (b) Longitudinal strain tracings resulting from the analysis of the tracked myocardium in (a). (c) Numerical strain values of the corresponding segments. (d) Colored M-mode display of the strain analysis. Dotted lines (a) and corresponding tracings in (b) yellow, basal septum; aqua, mid-septum; green, apical septum; pink, apical lateral wall; blue, mid-lateral wall; red, basal lateral wall. White dotted line in (b) corresponds to the global strain\n\nFig. 7.14\n\nSpeckle-tracking allows measurement of radial strain in all LV segments, a great contrast to the Doppler-based technique where only the posterior wall can be interrogated due to issues on insonation angle. Radial strain profile is displayed on a positively directed curve that peaks around end-systole heralded by aortic valve closure (AVC). The schematic diagram on the lower left panel indicates the direction of myocardial deformation in systole (solid arrow) and diastole (broken arrows)\n\nFig. 7.15\n\nCircumferential strain profile is displayed on a negatively directed curve that peaks around end-systole heralded by aortic valve closure (AVC). The schematic diagram on the lower left panel indicates the direction of myocardial deformation in systole (solid arrow) and diastole (broken arrows). Measurement of circumferential strain via Doppler-based imaging can only be performed on the lateral wall. Speckle-tracking allows measurement of circumferential strain in all LV segments\n\nFig. 7.16\n\nTDI has been routinely applied in dyssynchrony analysis to identify candidates for cardiac resynchronization therapy (CRT) and is found to be superior to strain rate and strain in predicting response. The intraventricular mechanical delay between the early and late segments predicts response to resynchronization. Two criteria more commonly used in the clinical practice are (1) septal to lateral wall delay >65 ms and (2) standard deviation of the time to peak systolic velocity of 12 segments ?>33 ms. (a) In a normal synchronous heart, regional systolic velocities peak almost simultaneously. Here, the timing of the peak systolic velocity of the septum occurred 20 ms earlier than that of the lateral wall. (b) In failing hearts with left bundle branch block, the lateral\/posterior segment systolic velocity peaks later than septum. Time delay (TD) can be quantified by measuring the difference in the timing of peak systolic velocity between the two opposite walls. In this figure, the septum to lateral wall TD was measured at 110 ms. Alternatively, if one is working on dedicated narrow-sector images (single-wall), TD can be assessed by taking the difference between the time to peak systolic velocities of the basal septum and lateral wall as measured from the onset of electrocardiographic QRS complex\n\nFig. 7.17\n\nWall motion can be quantified by TDI or strain echocardiography. Tissue velocities, strain rates and strain are reduced in ischemia and infarction. However, tissue velocity may not accurately reflect regional function due to tethering. Strain and strain rate identify infarcted segment and correlate with the extent of transmural infarction. Both are less susceptible to cardiac translational motion and tethering. It is encouraged to perform multiple mode of regional analysis to confirm findings that may be confounded by some technical limitations set by the method employed. In this representative TDI curve from a patient with anterior MI, (a) tissue velocity profile may fail to detect hypokinetic\/akinetic segments for reasons explained above. Reduction in the measured Doppler-derived systolic strain rate (b) and strain (c) may be more reflective of regional wall motion abnormalities. Even so, Doppler-derived indices are dependent on the angle of insonation, and this factor may operate on providing a similar reduction in the numerical values of the respective indices. (d) Performing strain analysis by speckle-tracking may eliminate the effect of angle issues and may confirm a true reduction in the myocardial deformation reflected by the Doppler-derived indices\n\nFig. 7.18\n\nSemiautomated measurements of strain by speckle-tracking allows generation of a color-coded image display such as bull's eye plot of the longitudinal segmental strain. The strain color code is depicted in the upper right-hand corner of each image. (a) Normal volunteer with normal strain (shades of red) in all segments. (b) Patient with a myocardial infarction related to proximal left-anterior descending artery lesion demonstrating reduced strain in the anterior septum and anterior and lateral LV free walls. (c) A patient with inferior\u2013posterior myocardial infarction who demonstrated reduced and abnormal (shades of blue) systolic strain in the inferior and posterior walls and preserved strain elsewhere. (d) Patient with nonischemic cardiomyopathy demonstrating reduced strain in all segments and abnormal systolic strains in the anterior septum and anterior LV free wall (shades of blue)\n\nThere exists a significant variability in TDI-derived parameters due to lack of standardized guidelines on data acquisition, partly resulting from the differences in the machine design, and standardized guidelines on post-processing analysis and interpretation. Thus it is difficult to recommend one. It is, however, important to have an insight on the mechanics by which each technique operates and interpret the results in the light of the limitations inherent to each technique.\n\nRecommended Reading\n\nAbraham TP, Dimaano VL, Liang HY. Role of tissue doppler and strain echocardiography in current clinical practice. Circulation. 2007;116(22):2597-2609.PubMedCrossRef\n\nAmundsen BH, Helle-Valle T, Edvardsen T, et al. Noninvasive myocardial strain measurement by speckle tracking echocardiography: validation against sonomicrometry and tagged magnetic resonance imaging. J Am Coll Cardiol. 2006;47(4):789-793.PubMedCrossRef\n\nBax JJ, Bleeker GB, Marwick TH, et al. Left ventricular dyssynchrony predicts response and prognosis after cardiac resynchronization therapy. J Am Coll Cardiol. 2004;44(9):1834-1840.PubMedCrossRef\n\nBountioukos M, Schinkel AF, Bax JJ, et al. Pulsed-wave tissue doppler quantification of systolic and diastolic function of viable and nonviable myocardium in patients with ischemic cardiomyopathy. Am Heart J. 2004;148(6):1079-1084.PubMedCrossRef\n\nDerumeaux G, Loufoua J, Pontier G, et al. Tissue doppler imaging differentiates transmural from nontransmural acute myocardial infarction after reperfusion therapy. Circulation. 2001;103(4):589-596.PubMedCrossRef\n\nDerumeaux G, Ovize M, Loufoua J, et al. Assessment of nonuniformity of transmural myocardial velocities by color-coded tissue doppler imaging: characterization of normal, ischemic, and stunned myocardium. Circulation. 2000;101(12):1390-1395.PubMedCrossRef\n\nDerumeaux G, Ovize M, Loufoua J, et al. Doppler tissue imaging quantitates regional wall motion during myocardial ischemia and reperfusion. Circulation. 1998;97(19):1970-1977.PubMedCrossRef\n\nD'hooge J, Heimdal A, Jamal F, et al. Regional strain and strain rate measurements by cardiac ultrasound: principles, implementation and limitations. Eur J Echocardiogr. 2000;1(3):154-170.PubMedCrossRef\n\nGarcia MJ. Echocardiographic assessment of left ventricular function. J Nucl Cardiol. 2006;13(2):280-293.PubMedCrossRef\n\nGilman G, Khandheria BK, Hagen ME, et al. Strain rate and strain: a step-by-step approach to image and data acquisition. J Am Soc Echocardiogr. 2004;17(9):1011-1020.PubMedCrossRef\n\nGorcsan J 3rd, Strum DP, Mandarino WA, et al. 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New method for evaluating left ventricular wall motion by color-coded tissue doppler imaging: in vitro and in vivo studies. J Am Coll Cardiol. 1995;25(3):717-724.PubMedCrossRef00421-L)\n\nNagueh SF, Middleton KJ, Kopelen HA, et al. Doppler tissue imaging: a noninvasive technique for evaluation of left ventricular relaxation and estimation of filling pressures. J Am Coll Cardiol. 1997;30(6):1527-1533.PubMedCrossRef00344-6)\n\nOmmen SR, Nishimura RA, Appleton CP, et al. Clinical utility of doppler echocardiography and tissue doppler imaging in the estimation of left ventricular filling pressures: a comparative simultaneous doppler-catheterization study. Circulation. 2000;102(15):1788-1794.PubMedCrossRef\n\nPauliks LB, Vogel M, Madler CF, et al. Regional response of myocardial acceleration during isovolumic contraction during dobutamine stress echocardiography: a color tissue doppler study and comparison with angiocardiographic findings. Echocardiography. 2005;22(10):797-808.PubMedCrossRef\n\nPerk G, Tunick PA, Kronzon I. Non-doppler two-dimensional strain imaging by echocardiography\u2013from technical considerations to clinical applications. J Am Soc Echocardiogr. 2007;20(3):234-243.PubMedCrossRef\n\nRushmere RF, Crystal DK, Wagner C. The functional anatomy of ventricular contraction. Circ Res. 1953;1(2):162-170.CrossRef\n\nSutherland GR, Bijnens B, McDicken WN. Tissue doppler echocardiography: historical perspective and technological considerations. Echocardiography. 1999;16(5):445-453.PubMedCrossRef\n\nSutherland GR, Di Salvo G, Claus P, et al. Strain and strain rate imaging: a new clinical approach to quantifying regional myocardial function. J Am Soc Echocardiogr. 2004;17(7):788-802.PubMedCrossRef\n\nUrheim S, Edvardsen T, Torp H, et al. Myocardial strain by Doppler echocardiography. Validation of a new method to quantify regional myocardial function. Circulation. 2000;102(10):1158-1164.PubMedCrossRef\n\nWeidemann F, Wacker C, Rauch A, et al. Sequential changes of myocardial function during acute myocardial infarction, in the early and chronic phase after coronary intervention described by ultrasonic strain rate imaging. J Am Soc Echocardiogr. 2006;19(7):839-847.PubMedCrossRef\n\nWilkenshoff UM, Sovany A, Wigstrom L, et al. Regional mean systolic myocardial velocity estimation by real-time color doppler myocardial imaging: a new technique for quantifying regional systolic function. J Am Soc Echocardiogr. 1998;11(7):683-692.PubMedCrossRef\n\nYu CM, Zhang Q, Chan YS, et al. Tissue doppler velocity is superior to displacement and strain mapping in predicting left ventricular reverse remodelling response after cardiac resynchronisation therapy. Heart. 2006;92(10):1452-1456.PubMedCentralPubMedCrossRef\n\nVan de Veire N, De Sutter J, Bax JJ et al. Technological advances tissue doppler imaging echocardiography. Heart. 2008;Aug;94(8):1065-1074\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_8\n\n# 8. Transesophageal Echocardiography\n\nJulie A. Humphries1  , Christopher J. Kramer, Partho P. Sengupta and Bijoy K. Khandheria\n\n(1)\n\nDepartment of Cardiology, Heart Care Partners, Greenslopes, Queensland, Australia\n\nJulie A. Humphries\n\nEmail: jhumphries@heartcarepartners.com.au\n\nTransesophageal echocardiography (TEE) involves ultrasound imaging of cardiovascular system from the confines of the gastroesophageal track. This helps reduce signal attenuation and permits the use of higher ultrasound frequencies, thereby providing an enhanced spatial resolution. TEE is currently used in approximately 5\u201310% of patients being evaluated in the cardiovascular ultrasound imaging and hemodynamic laboratory.\n\n## 8.1 Protocol for TEE\n\nFasting based on conscious sedation guidelines, an intravenous access, careful history to rule out presence of laryngeal or gastroesophageal diseases, and removal of dentures are prerequisites.\n\n  * Absolute contraindications to TEE include esophageal stricture, diverticulum, tumor, and recent esophageal or gastric surgery.\n\n  * Topical spray, intravenous sedation, a drying agent to minimize oral secretion, and use of appropriate lubrication are helpful.\n\n  * Before the introduction of the scope into the esophagus, the array is rotated to 0\u00b0 to place the transducer into a conventional transverse plane. A bite guard is used always unless the patient is edentulous. The tip of the transducer is advanced into the esophagus gently without force and stopped if any resistance is encountered. After moving the transducer into the desired location, the probe is manipulated to orient the imaging planes for obtaining the desired cross-sectional images.\n\n  * Although each of the views represents echocardiographic image in cross section, moving the probes through the entire extent of a structure permits a rapid three-dimensional evaluation of cardiac structures.\n\nProcedural risks are low in trained hands. However, they need to be explained clearly to the patient. These include transient throat pain, laryngospasm, aspiration, hypotension, hypertension, tachycardia, mucosal bleeding, esophageal rupture, and rare risk of death. Benzocaine topical spary can cause toxic methemoglobinemia. The treatment is administration of methylene blue in addition to supportive measures. Being semi-invasive, appropriate training requirements are needed and have been laid down by both, the American Society of Echocardiography and the British Society of Echocardiography.\n\n## 8.2 Standard TEE views\n\nNormal cardiac structures seen on different transesophageal views (Table 8.1) are shown in Figs. 8.1\u20138.20. Figure 8.21 shows maneuvers utilized for delineation of flow across inter-atrial septum. Transgastric views are shown in Figs. 8.22\u20138.28.\n\nTable 8.1\n\nTransesophageal echocardiography cross sections (Reproduced with permission from the recommendations of American Society of Echocardiography)\n\nWindow (depth from incisors) | Cross section | Multiplane angle range | Structures imaged\n\n---|---|---|---\n\nUpper esophageal (20\u201325 cm) | Aortic arch long axis (s) | 0\u00b0 | Aortic arch, left brachio v\n\n|\n\nAortic arch short axis | 90\u00b0 | Aortic arch, PA, PV, left brachio v\n\nMid-esophageal (30\u201340 cm) | Four-chamber | 0\u00b0\u201320\u00b0 | LV, LA, RV, RA, MV, TV, IAS\n\n|\n\nMitral commissural | 60\u00b0\u201370\u00b0 | MV, LV, LA\n\n|\n\nTwo-chamber | 80\u00b0\u2013100\u00b0 | LV, LA, LAA, MV, CS\n\n|\n\nLong axis | 120\u00b0\u2013160\u00b0 | LV, LA, AV, LVOT, MV, asc aorta\n\n|\n\nRV inflow-outflow | 60\u00b0\u201390\u00b0 | RV, RA, TV, RVOT, PV, PA\n\n|\n\nAV short axis | 30\u00b0\u201360\u00b0 | AV, IAS, coronary ostia, LVOT, PV\n\n|\n\nAV long axis | 120\u00b0\u2013160\u00b0 | AV, LVOT, prox asc aorta, right PA\n\n|\n\nBicaval | 80\u00b0\u2013110\u00b0 | RA, SVC, IVC, IAS, LA\n\n|\n\nAsc aortic short axis | 0\u00b0\u201360\u00b0 | Asc aorta, SVC, PA, right PA\n\n|\n\nAsc aortic long axis | 100\u00b0\u2013150\u00b0 | Asc aorta, right PA\n\n|\n\nDesc aorta short axis | 0\u00b0 | Desc thoracic aorta, left pleural space\n\n|\n\nDesc aorta long axis | 90\u00b0\u2013110\u00b0 | Desc thoracic aorta, left pleural space\n\nTransgastric (40\u201345 cm) | Basal short axis | 0\u00b0\u201320\u00b0 | LV, MV, RV, TV\n\n|\n\nMid short axis | 0\u00b0\u201320\u00b0 | LV, RV, pap mm\n\n|\n\nTwo-chamber | 80\u00b0\u2013100\u00b0 | LV, MV, chordae, pap mm, CS, LA\n\n|\n\nLong axis | 90\u00b0\u2013120\u00b0 | LVOT, AV, MV\n\n|\n\nRV inflow | 100\u00b0\u2013120\u00b0 | RV, TV, RA, TV chordae, pap mm\n\nDeep transgastric (45\u201350 cm) | Long axis | 0\u00b0\u201320\u00b0 (anteflexion) | LVOT, AV, asc aorta, arch\n\nBrachio v Brachiocephalic vein, PA pulmonary artery, PV pulmonic valve, LV left ventricle, LA left atrium, RV right ventricle, RA right atrium, MV mitral valve, TV tricuspid valve, IAS inter atrial septum, LAA left atrial appendage, CS coronary sinus, AV aortic valve, LVOT left ventricular outflow tract, prox proximal, RVOT right ventricular outflow tract, SVC superior vena cava, IVC inferior vena cava, RPA right pulmonary artery, asc ascending, desc descending, pap mm papillary muscles\n\nFig. 8.1\n\nFour chamber mid-esophageal view. LA left atrium, LV left ventricle, RA right atrium, RV right ventricle\n\nFig. 8.2\n\nTwo chamber mid-esophageal view. LA left atrium, LV left ventricle\n\nFig. 8.3\n\nLong axis mid-esophageal view. LA left atrium, LV left ventricle, AO aorta\n\nFig. 8.4\n\n(a) Mitral valve 0\u00b0 view. LA left atrium, LV left ventricle, MV mitral valve. (b) Mitral valve at 0 degrees with color Doppler. (c) Mitral valve \"drop out\" view at 76\u00b0 (usually best seen 60\u201380\u00b0) and shows P3, A2 and P1 segments of the mitral valve leaflets. (d) \"Drop out\" view of mitral valve showing mid A2 segment regurgitation\n\nFig. 8.5\n\n(a) Long axis view of the mitral valve. AL anterior leaflet, PL posterior leaflet. (b) Long axis mitral valve view shows mild mitral regurgitation on color flow Doppler\n\nFig. 8.6\n\n(a) Short axis mid-esophageal view at approximately 60\u00b0. The trileaflet aortic valve (center) and pulmonary valve (arrow) are both seen. MPA main pulmonary artery, RV right ventricle, RA right atrium, LA left atrium. (b) Zoomed up view of the open trileaflet aortic valve. The left main coronary artery (arrow) arises from the left coronary cusp (LCC). RCC right coronary cusp, NCC non-coronary cusp\n\nFig. 8.7\n\nTwo-dimensional biplane view of the open trileaflet aortic valve. Views are at 90\u00b0 orthogonal to each other using the 3D tranesophageal probe\n\nFig. 8.8\n\n(a) Long axis view at 145\u00b0 showing closed aortic valve in long axis. (b) Long axis view at 145\u00b0 showing open aortic valve\n\nFig. 8.9\n\nAortic root and ascending aorta. This view is achieved by withdrawing the probe and reducing the multiplane angle to 110\u2013120\u00b0\n\nFig. 8.10\n\nAortic root dimensions (from left to right): aortic annulus, trans-sinus diameter, sino-tubular junction, proximal ascending aorta\n\nFig. 8.11\n\nStandard lower-esophageal view of the tricuspid valve (at 0\u00b0). There is a Eustachian valve (EV) seen here. The coronary sinus (CS) is also seen\n\nFig. 8.12\n\nTricuspid valve (arrows) seen at 52\u00b0. This is often a good position for color Doppler tricuspid regurgitation signal\n\nFig. 8.13\n\n(a) Mid-esophageal view of right ventricular outflow tract, pulmonary valve (arrow) and main pulmonary artery seen at 74\u00b0. (b) Pulmonary regurgitation color Doppler in the same view\n\nFig. 8.14\n\nBiplane imaging through the pulmonary valve with color Doppler shows the pulmonary valve in short axis (right panel) with central regurgitation\n\nFig. 8.15\n\n(a) Left atrial appendage seen in the mid-esophageal view (36\u00b0). This is an optimal view for Pulse Wave Doppler sample placement. No thrombus seen. The coronary sinus (arrow) is seen coursing between the left atrial appendage and the annulus of the mitral valve. (b) Pulse wave Doppler signal approximately 1 cm into the mouth of the left atrial appendage demonstrating normal velocities. Signals above the line and corresponding with the P wave are ejection velocities, and below the line are filling velocities. Velocities above 0.35m\/s are less likely to develop thrombus. (c) Further left atrial appendage views on multiplane imaging. It is important to image the left atrial appendage in multiple planes as multilobed left atrial appendage can have thrombus in a lobe not seen in other planes. This example is free of thrombus. (d) Left atrial appendage multiplane view at 145\u00b0 shows prominent pectinate muscles which can sometimes be confused with thrombus\n\nFig. 8.16\n\nLeft superior pulmonary vein seen with color flow Doppler. It lies adjacent to the left atrial appendage separated by the \"Q tip\"\n\nFig. 8.17\n\nBiplane color imaging of the left superior pulmonary vein orifice allows for easy identification of the left inferior pulmonary vein (LIPV) seen in the right panel\n\nFig. 8.18\n\nRight superior, middle and inferior pulmonary veins seen on color flow imaging\n\nFig. 8.19\n\n(a) Interatrial septum (arrow). This view demonstrates the retroaortic part of the interatrial septum which represents the most superior aspect. (b) Interatrial septum (zoomed up view from previous image)\n\nFig. 8.20\n\n(a) Interatrial septum seen bowing towards the left atrium in the setting of raised right atrial pressure during inspiration. (b) Large patent foramen ovale (arrow). (c) Left to right color flow seen through large patent foramen ovale. (d) Bubble contrast seen coming through patent foramen ovale during valsalva demonstrating right to left shunting\n\nFig. 8.21\n\n(a) Small secundum ASD with and without color flow Doppler. Color imaging shows small left to right shunt (arrow) (b) Patient with a PFO and an ASD which is very posterior. Separate communications and shunting can be seen on the two images (a) and (b)\n\nFig. 8.22\n\nStandard transesophageal transgastric view showing the left ventricle in short axis at the mid ventricular level\n\nFig. 8.23\n\nTransgastric view of the left ventricle at the level of the mitral valve seen in short axis (valve open)\n\nFig. 8.24\n\nTransgastric view of the left ventricle in the long axis (87\u00b0). This view gives good information about the subvalvular apparatus\n\nFig. 8.25\n\nTransgastric biplane view through the mitral valve which shows the valve in short and long axis simultaneously, with detailed subvalvular apparatus examination\n\nFig. 8.26\n\nTransgastric view of the open tricuspid valve (arrows) in the short axis (left panel) and long axis (right panel)\n\nFig. 8.27\n\nTransgastric view of right ventricular inflow and outflow. RA right atrium, RV right ventricle, AO aorta, MPA main pulmonary artery\n\nFig. 8.28\n\nInferior vena cava (IVC) in long axis seen via the deep transgastric view at 48\u00b0. HV hepatic vein\n\n## 8.3 Application of TEE\n\nIn addition to the usual indications for TEE (suspected endocarditis, source of embolus, and suspected aortic dissection), there are several indications that are unique to critical care patients. These include:\n\n1.\n\nAssessment of unexplained hypotension\n\n2.\n\nSuspected massive pulmonary embolism\n\n3.\n\nUnexplained hypoxemia\n\n4.\n\nComplications of cardiothoracic surgery\n\nLess common indications for TEE in the critical care unit include continuous hemodynamic monitoring, evaluation of potential transplant donors, and guidance of central-line placement. The recent development of transnasal TEE probes may allow for monitoring in the awake patient. The potential benefits of the transnasal probe include less risk for esophageal trauma in patients with varices or coagulopathies and less need for sedation in those with compromised respiratory or hemodynamic status. Figures 8.29\u20138.35 show clinical examples where incremental use of advanced techniques in TEE has been illustrated.\n\nFig. 8.29\n\n52 year old male for routine kidney transplant workup. Incidental finding of mass on mitral valve anterior leaflet. Histological surgical specimen showed non-bacterial thrombotic endocarditis\n\nFig. 8.30\n\nBiplane imaging of the anterior leaflet mass\n\nFig. 8.31\n\nColor imaging showing mitral regurgitation around the mass\n\nFig. 8.32\n\nThree-dimensional imaging of the anterior mitral valve leaflet mass from the left atrial perspective at valve closure\n\nFig. 8.33\n\nThree-dimensional transesophageal imaging of the anterior mitral valve leaflet mass from the left ventricular perspective, showing involvement of the atrial surface on the leaflet\n\nFig. 8.34\n\n(a) Type I Aortic Dissection, Ascending Aorta True Lumen (TL) (Superior), False Lumen (FL) (Inferior). (b) Type I Aortic Dissection, Ascending Aorta with color, Color flowing into false lumen True Lumen (TL) (Inferior), False Lumen (FL) (Superior)\n\nFig. 8.35\n\n(a) Type III Aortic dissection, Short Axis view with color, True lumen (T) (Inferior) False lumen (F) (Superior). (b) Type III Aortic dissection, Short Axis view with color, True lumen (T) (Inferior), False lumen (F) (Superior)\n\n## 8.4 Pitfalls\n\nTEE needs expertise for avoiding potential erroneous diagnosis resulting from misinterpretation of normal and abnormal anatomy.\n\n  * Air within the esophagus and stomach, or an air-filled trachea and bronchi intervening between probe and cardiac structures can consistently produce artifacts or interfere with certain tomographic views (Fig. 8.36).\n\nFig. 8.36\n\n(a) Example of artifact in the mid-esophageal view which interferes with image quality. Artifact can result from poor contact, presence of air in the esophagus, or from intracardiac structures of high density. (b) Example of color flow imaging artifact resulting from esophageal air interfering with probe contact\n\n  * Reverberation signals or ghost shadows are common and result from impedance mismatch resulting in linear artifacts most commonly seen in the upper ascending and mid-descending aorta. Imaging of the upper ascending aorta with the horizontal plane is limited by a blind spot caused by interposed bronchus between the esophagus and upper ascending aorta.\n\n  * Normal anatomy may be interpreted abnormal. These include muscular trabeculations in the atrial appendage mistaken as mass or thrombus, the terminal portion of the partition between the left atrial appendage and left upper pulmonary vein appearing as a globular mass, fat-laden fossa ovalis or lipomatous hypertrophy of the atrial septum interpreted as a mass, and surgical sutures appearing filamentous or pedunculated and interpreted as mass or vegetations.\n\n  * \"Echo-free spaces\": Certain normal structures generate echo-free spaces and may be incorrectly interpreted as cysts or abscesses. These include the transverse and the oblique sinus.\n\n  * Other difficulties that may be encountered during TEE include the adequate visualization and quantification of aortic valve regurgitation, aortic valve and pulmonary valve gradient estimation, and non-foreshortened visualization of the left ventricular apex.\n\n  * Finally, ultrasound transducers generate heat. During prolonged monitoring, the device may be required to be shut down periodically to allow cooling.\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_9\n\n# 9. Intraoperative Transesophageal Echocardiography\n\nAlina Nicoara1  and Madhav Swaminathan\n\n(1)\n\nDepartment of Anesthesiology, Yale University School of Medicine, West Haven, CT, USA\n\nAlina Nicoara\n\nEmail: alina.nicoara@yale.edu\n\nFig. 9.1\n\nAt the mid-esophageal level, at an imaging depth of about 35 cm, several tomographic views can be obtained by adjusting the imaging angle. (a) Mid-esophageal four-chamber view (ME 4-Ch) is obtained at an imaging angle between 0\u00b0 and 20\u00b0 with slight retroflexion of the tip of the probe. This image shows the 4-Ch as well as the mitral and tricuspid valves and it is used to assess the size and function of the heart chambers and the structure and function of the valves. It is the usual starting point for most intraoperative imaging (RA, right atrium; LA, left atrium; RV, right ventricle; LV, left ventricle; MV, mitral valve; TV, tricuspid valve). (b) Mid-esophageal two chamber view (ME 2Ch) is obtained from the same position by advancing the multiplane angle to about 90\u00b0. It visualizes the mitral valve, the anterior and inferior walls of the LV as well as the apical cap, AL, anterior leaflet of the mitral valve; Pl, posterior leaflet of the mitral valve. (c) The mid-esophageal mitral commissural view is obtained at an imaging angle of about 60\u00b0 and is used mainly to assess the structure and function of the mitral valve. The A2 segment of the anterior leaflet is seen in the middle of the image, while the P3 and P1 segments are seen on the left and right sides of the image, respectively. (d) The mid-esophageal long-axis view (ME LAX) at an imaging angle from 120\u00b0 to 130\u00b0 is mostly used to evaluate the motion of the anteroseptal (right side of the image) and inferolateral walls (left side) of the LV; presence of structural abnormalities of the MV such as prolapsed or flail leaflets and LV outflow tract pathology (AoV, aortic valve)\n\nFig. 9.2\n\n(a) The mid-esophageal bicaval view is obtained at an imaging angle of about 90\u00b0 by turning the probe slightly to the right. It is used mainly for the assessment of the integrity of the interatrial septum (IAS) and to guide placement of catheters and cannulae placed intraoperatively (IVC, inferior vena cava; SVC, superior vena cava; LA, left atrium; RA, right atrium). (b) The mid-esophageal aortic valve short-axis view (ME AV SAX) is obtained at the mid-esophageal level by rotating the angle to 30\u00b0\u201360\u00b0, placing the aortic valve in the center of the screen and decreasing the imaging depth to about 10 cm. This image is used to evaluate the aortic valve cusps, measure valve area by planimetry and visualize regurgitant jets using color flow Doppler (AoV, aortic valve; RC, right coronary cusp; LC, left coronary cusp; NC, non-coronary cusp; RV, right ventricle). (c) The mid-esophageal aortic valve long-axis view (ME AV LAX) visualizes the LV outflow tract, aortic valve, and ascending aorta. It is obtained at an imaging angle of about 120\u00b0\u2013160\u00b0. Decreasing the imaging depth to about 10 cm also helps optimize the image. Similar to the SAX view, this view is used to evaluate the aortic valve cusps for the presence of vegetations or other masses and to assess the severity of aortic insufficiency with color flow Doppler (Asc Ao, ascending aorta; MV, mitral valve). (d) The mid-esophageal right ventricular inflow\u2013outflow view is obtained from the ME 4Ch view by advancing the imaging angle to 60\u00b0\u201390\u00b0 and placing the tricuspid valve in the center of the screen. This view is used to visualize the tricuspid valve (TV), the right ventricular outflow tract, the right ventricle free wall, and the pulmonary valve\n\nFig. 9.3\n\nFrom the mid-esophageal level, the probe can be further advanced into the stomach to the transgastric position. (a) Anteflexion of the probe until it makes contact with the gastric wall enables the transgastric mid-papillary short axis view (TG mid SAX). The left ventricle is seen in cross-section as a circular structure at the level of the papillary muscles. This view is mostly used to evaluate the function and size of the left ventricle as well as regional wall motion abnormalities. In addition, pericardial effusions can also be clearly seen. While evaluating wall motion abnormalities, it is important to note that the orientation of left ventricular walls is opposite to that of chest wall imaging. (b) The transgastric basal short-axis view (TG basal SAX) is obtained by further anteflexion and\/or slight withdrawal of the probe. Similar to the TG mid SAX, the basal view is also used to assess the size and function of the LV. (c) The transgastric 2-Ch view is obtained by rotating the imaging angle to 90\u00b0. It visualizes the anterior and inferior left ventricular walls, the mitral valve and the subvalvular apparatus (LV, left ventricle; LA, left atrium; MV, mitral valve). (d) The transgastric right ventricular inflow view (TG RV inflow) is obtained from the TG mid SAX by rotating the imaging angle to about 100\u00b0 and turning the probe slightly to the right. It visualizes the right ventricle in long axis (RV, right ventricle; RA, right atrium; TV, tricuspid valve). (e) The deep transgastric long axis view (deep TG LAX) is obtained by further advancing the probe in the stomach and anteflexing in such a way that the probe is adjacent to the left ventricular apex with the imaging angle at 0\u00b0. This view is used for measuring velocities across the aortic valve and left ventricular outflow tract using spectral Doppler (LVOT, left ventricular outflow tract)\n\nFig. 9.4\n\nTEE can be used to examine the proximal ascending aorta, the distal aortic arch, and the descending aorta. Because of the interposition of the trachea and the left main stem, the distal ascending aorta and the proximal aortic arch cannot always be examined (the echo \"blind spot\"). As these aortic segments are used most frequently for aortic cannulation and cross-clamping, the complete evaluation of these segments requires epiaortic scanning (a) Mid-esophageal ascending aorta short-axis view (ME asc aortic SAX) is obtained from the mid-esophageal level by slightly withdrawing and anteflexing the probe until the ascending aorta is seen in short axis (Asc Ao, ascending aorta; PA, pulmonary artery; right PA, right pulmonary artery; SVC, superior vena cava). (b) From this position, advancing the angle to about 90\u00b0\u2013120\u00b0 will generate the mid-esophageal ascending aorta long axis view (ME asc aortic LAX). Both views may be used to evaluate the ascending aorta for atheroma and dissection flaps, to evaluate the pulmonary artery for the presence of thrombus or for the position of the pulmonary artery catheter, and to assess the dimensions of the ascending aorta. (c) The descending aorta short-axis view is obtained by turning the probe to the left at an imaging angle of 0\u00b0 until the descending aorta comes into view as a circle. The imaging depth should be decreased to about 6\u20138 cm and the aorta should also be examined distally as far as possible. (d) Forward rotation of the imaging angle to 90\u00b0 generates the descending aorta long axis view. Both views of the descending aorta can be used to assess descending aorta for the presence of atheromatous plaque or dissection flaps. (e) By withdrawing the probe in the upper esophagus at an imaging angle of 0\u00b0, the circular shape of the descending aorta changes into a tubular shape of the aortic arch, thus enabling the upper esophageal aortic arch long-axis view (UE aortic arch LAX). (f) From this position, advancing the imaging angle to about 90\u00b0 will generate the upper esophageal aortic arch short axis view (UE aortic arch SAX) (PA, pulmonary artery). Both views can be used to assess the aortic arch for the presence of atheromatous plaque, dissection flaps and other interventions like endovascular aortic stents. The UE aortic arch SAX can also be used to assess velocities across the pulmonary valve, which is oriented well to allow a spectral Doppler beam to be directed along transvalvular flow\n\nFig. 9.5\n\nMitral valve pathology. (a) Mitral valve vegetations appear as irregular, heterogeneous masses originating most often from the atrial side. They are accompanied by valvular regurgitation and are prone to embolization, MV, mitral valve. (b) Posterior leaflet flail as seen in the ME 4-Ch view can be due to chordal elongation or rupture and usually leads to severe mitral regurgitation. (c) Mitral valve regurgitation can be due to degenerative disease of the mitral valve or subvalvular apparatus or due to changes in the geometry of the mitral valve annulus or the left ventricle. It is best assessed and quantified in the mid-esophageal views. (d) One of the methods of quantifying the mitral regurgitation is by measuring the vena contracta, which is the narrowest portion of the regurgitant jet. It is usually measured in the ME LAX view. (e) Measurement of the radius of the proximal isovelocity surface area (PISA) allows calculation of the effective regurgitant orifice area and of the regurgitant volume\n\nFig. 9.6\n\nAortic valve pathology. (a) Aortic valve calcification, fibrosis, and thickening result in decrease in the flexibility of the cusps and fusion of the commissures with limitation of flow and aortic stenosis. The ME AV SAX view allows examination of the cusps for pathology. (b) Turbulent flow through a stenotic aortic valve can be observed with color flow Doppler in the ME AV SAX. (c) Bicuspid aortic valve, seen in ME AV SAX, is the most common congenital malformation of the aortic valve. (d) Aortic valve vegetations originate usually from the ventricular side of the valve are irregular and heterogeneous and lead to valvular regurgitation. TEE is the diagnostic modality of choice for endocarditis. Aortic valve vegetations are seen here in the ME AV LAX view. (e) The same lesions are also seen on the aortic valve cusps in the ME AV SAX view. (f) The ME AV LAX is the preferred view to quantify the severity of aortic valve regurgitation with color flow Doppler. (g) Aortic stenosis and aortic regurgitation can also be quantified using continuous wave Doppler in the deep TG LAX view and analyzing the velocities of the stenotic or the regurgitant lesions. This image depicts concomitant severe aortic stenosis and mild aortic regurgitation\n\nFig. 9.7\n\nProsthetic valves. Transesophageal echocardiography can provide detailed information on prosthetic valves through high-quality two and three-dimensional echo images and Doppler examination. (a) Bileaflet mechanical valve in systole (closed) in the mitral position. (b) Bileaflet mechanical valve in diastole (open) in the mitral position. Prosthetic valves should be evaluated for the movement of the mechanical occluders according to specifications, stability, lack of rocking motion, and absence of attached fibrinous or thrombotic material. (c) An open bileaflet mechanical valve in the aortic position. (d) Bioprosthetic valve in the aortic position. The three supporting struts and the three leaflets are visible in open position. (e) In the ME LAX view, a significant regurgitant jet is visible originating from outside of the annulus of the bileaflet mechanical valve and representing a paravalvular leak\n\nFig. 9.8\n\nMajor vessels. (a) Dissection of the descending aorta with blood flow seen from the true lumen into the false lumen through multiple sites of entry in the descending aorta long axis view (TL, true lumen; FL, false lumen). (b) Large atheromatous plaque seen in the descending aorta in the long axis (left) and short axis (right) views. (c) Thrombus seen in the distal right pulmonary artery in the ME ascending aorta SAX view (Asc Ao, ascending aorta; PA, pulmonary artery)\n\nFig. 9.9\n\nTEE is a valuable tool in diagnosing pericardial effusions as a cause of hemodynamic instability. (a) In the TG mid SAX, the pericardial effusion is seen as an echolucent space posterior to the heart chambers (RV, right ventricle; LV, left ventricle). (b) In the ME 4-Ch view, the pericardial effusion is compressing on the right and left atria distorting the normal anatomy (RA, right atrium; LA, left atrium). (c) ME 4-Ch view in the same patient after drainage of the effusion\n\nFig. 9.10\n\nAssessment of intraoperative diastolic function involves the comprehensive use of spectral Doppler. (a) Transmitral pulsed wave Doppler in the mid-esophageal 4-Ch view (Fig. 9.1a) demonstrating early (E) and late (A) diastolic flow velocities away from baseline. The direction of transmitral flow is in contrast to conventional chest wall imaging. (b) Pulsed wave Doppler of peak velocities in the left upper pulmonary vein. Forward flow in systole (S), diastole (D), and reversed flow (Ar) during atrial contraction can be seen. (c) Tissue velocity of the lateral mitral annulus is shown using tissue Doppler imaging. Two diastolic waves, early (E?) and late (A?) can be seen above baseline and a downward wave in systole (S?) as the base descends during this period. (d) Propagation velocity of early diastolic left ventricular filling assessed using color M-mode. The red arrow illustrates the slope that depicts propagation velocity\n\nFig. 9.11\n\nImaging of the descending aorta for dissections can sometimes be confounded. The bright echo-reflective posterior wall of the aorta can generate a mirror image distal to the actual image of the descending aorta. An intraaortic balloon pump A is seen in the lumen of the aorta, while its mirror image B can be seen in the aortic mirror image. This may be easily mistaken for a dissection\n\nFig. 9.12\n\nTEE is particularly useful in the assessment of inflow and outflow cannulae of ventricular assist devices (VAD). (a) In this mid-esophageal 2-Ch view, a left-sided VAD inflow canula inserted in the left ventricular apex can be seen as an echo-dense object. (b) Color flow Doppler may be used to determine quality of flow across the device for assessment of obstruction. (c) The transgastric mid-short axis view shows the cannula in cross section. (d) While weaning from cardiopulmonary bypass, there may be a significant amount of air that can be seen on TEE as bright echo-reflective bubbles in the left ventricle. It is critical to perform de-airing procedures at this time\n\nFig. 9.13\n\nA common adjuvant procedure during mitral valve surgery is an edge-to-edge repair (Alfieri technique) that is usually performed between the A2 and P2 scallops of the mitral valve for mitral regurgitation, or when there is excessive anterior leaflet tissue. On TEE imaging in the mid-esophageal 4-Ch (a) and transgastric basal short axis (b) views, the effects of the edge-to-edge stitch may be seen on the mitral valve. A double mitral opening can also be appreciated in the short axis views\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_10\n\n# 10. Use of Echo Contrast in Routine Practice\n\nShizhen Liu1  and Mani A. Vannan\n\n(1)\n\nDepartment of Cardiovascular Medicine, The Ohio State University Medical Center, Columbus, OH, USA\n\nShizhen Liu\n\nEmail: shizhen.liu@osumc.edu\n\n## 10.1 Introduction\n\nUltrasound contrast agents, used with contrast-specific imaging techniques, have an established role for diagnostic cardiovascular imaging in the echocardiography laboratory. It has approved indication of Left ventricular opacification (LVO) and enhancement of endocardial border delineation (EBD). Also, a number of investigative reports have confirmed the utility of the off-label use of contrast agents for assessment of myocardial perfusion (MCE).\n\n## 10.2 What Are Ultrasound Contrast Agents?\n\n  * Microbubbles in contrast agent compose of gas and the shell.\n\n  * According to the different gas inside the microbubbles, contrast agents could be divided into two generations.\n\n    * First generation: air\n\n    * Second generation: perfluorocarbon or sulfur fluoride\n\n  * The Second generation agents have better stability and contrast effect and are in use now.\n\n## 10.3 Contrast Agents in Use\n\nAgent | Size, ?m (range) | Gas | Shell composition | Manufacture | Approved in\n\n---|---|---|---|---|---\n\nLevovist | 2\u20133 (2\u20138) | Air | Lipid (palmitic acid) | Bayer Schering Pharma AG (Berlin, Germany) | Canada, Europe, Asia\n\nOptison | 4.7 (1\u201310) | Perfluoropropane | Human albumin | GE Healthcare (Princeton, NJ) | US, Canada\n\nDefinity | 1.5 (1\u201310) | Perfluoropropane | Phospholipid | Lantheus Medical Imaging (North Billerica, MA) | US, Canada, Europe\n\nSonoVue | 2.5 (1\u201310) | Sulfur hexafluoride | Phospholipid | BRACCO Diagnostics (Milan, Italy) | Canada, Europe, Asia\n\n## 10.4 Indication in Clinical Practice \u2013 1 (Fig. 10.1)\n\nLV volumes and LVEF | \u2022Suboptimal image quality: Obesity; chronic obstructive pulmonary disease patients in intensive care settings; mechanically ventilated; chest deformities\n\n---|---\n\n\u2022Optimal image quality: need accurate, precise and repeatable measurements of absolute LV volumes, mass and global systolic performance\n\nFig. 10.1\n\nApical 4 chamber view before (left) and after (right) contrast\n\n## 10.5 Indication in Clinical Practice \u2013 2 (Fig. 10.2)\n\nConfirm or exclude abnormalities in apex | \u2022Apical aneurysm (Fig. 10.2, left)\n\n---|---\n\n\u2022Apical thrombus\n\n\u2022Apical false tendons or trabeculations\n\n\u2022Apical hypertrophic cardiomyopathy\n\n\u2022Endocardial noncompaction\n\nFig. 10.2\n\nLV aneurysm (arrow) is confirmed after contrast (lower panel)\n\n## 10.6 Indication in Clinical Practice \u2013 3\n\nCardiac chamber mass differentiation | \u2022Stasis of blood: swirl within a region of blood stasis\n\n---|---\n\n\u2022Thrombus: complete contrast dropout\n\n\u2022Tumor: reperfuse with contrast after several cardiac cycles\n\nCardiac structure differentiation | \u2022Myocardial rupture\n\n\u2022Pseudoaneurysm\n\n\u2022Giant aortic sinus aneurysm differentiated with RA and\/or cyst\n\nDoppler enhancement | \u2022Tricuspid regurgitation\n\n\u2022Aortic stenosis\n\n\u2022Pulmonary vein flow\n\n## 10.7 Protocol for Routine Contrast Study \u2013 1\n\n  * Laboratory setup\n\n    * Team approach: physicians, sonographers, and nurses\n\n    * Written document: indications; injecting and imaging protocols; personnel responsibilities\n\n    * Reimbursement procedures establishment\n\n    * Contrast agent storage\n\n## 10.8 Protocol for Routine Contrast Study \u2013 2\n\n  * Contrast procedure\n\n    * Patient selection\n\n    * Explain procedure to patient\n\n    * Establish IV access\n\n    * Determine optimal mode of administration (Table 10.1)\n\n    * Optimize equipment settings (Table 10.2)\n\nTable 10.1\n\nAdministration methods of contrast agent\n\nCE administration | Characteristics\n\n---|---\n\nBolus | \u2022Quick and simple to perform with a high level of contrast enhancement.\n\n\u2022Higher incidence of attenuation artifact. (Overcome by slowing the rate of bolus injection or using a dilute bolus.)\n\n\u2022Need a slow flush with enough saline or 5% dextrose and water solution (D5W) to opacify the LV properly.\n\n\u2022Flush is terminated as soon as contrast is seen in the RV and the contrast preset is activated.\n\nContinuous infusion | \u2022Control the injection of microbubbles more efficiently and are essential for quantitative perfusion studies.\n\n\u2022IV setup is usually more complicated and time-consuming.\n\nTable 10.2\n\nEquipment settings for contrast echocardiography\n\nControl | Adjustment\n\n---|---\n\nMechanical Index (MI) | \u20220.4\u20130.6. Select harmonic imaging\n\nOverall gain | \u2022Higher gain does not cause microbubble destruction\n\n\u2022Lower MI requires higher gain\n\nTime gain compensation (TGC) | \u2022Greater TGC gain in greater depths\n\nCompression (dynamic range) | \u2022Medium dynamic range in LVO\n\nTransducer frequency | \u2022Lowest transmit frequency\n\nTransmit focus | \u2022Usually at distal location; move to apex as needed\n\nPersistence | \u2022Low or off\n\nDepth | \u2022Optimize depth to region of interest (e.g., LA is not needed during LVO)\n\n## 10.9 Protocols for Stress Contrast Study\n\n  * Administration\n\n    * In exercise stress echocardiography\n\n      * Baseline and peak stages\n\n    * In pharmacologic stress echocardiography\n\n      * Baseline, low dose, and peak stages\n\n  * For peak images\n\n    * Agent is given 15\u201320 s before termination of the test\n\n    * Followed by a slower bolus of 2\u20133 mL of flush\n\n    * Establish the optimal window in the non-contrast preset\n\n## 10.10 Techniques, Problems, and How to Resolve | Artifact | Causes | Solutions\n\n---|---|---|---\n\nLVO | Attenuation | \u2022High concentration of microbubbles within the LV cavity\n\n\u2022Rib or lung artifact | \u2022Waiting for dissipating over time\n\n\u2022Slowing the speed to inject the flush\n\n\u2022Using a dilute bolus\n\n\u2022Using infusion method\n\n\u2022Repositioning transducer to minimize rib or lung artifact\n\n|\n\nSwirling of the contrast | \u2022MI setting is too high | \u2022Lowering the MI\n\n|  |\n\n\u2022Concentration of contrast is too low | \u2022Increasing the amount of contrast injected\n\n|  |  |\n\n\u2022Moving the focus to the region of the swirling\n\nMCE | False-positive perfusion | \u2022High gain settings | \u2022Using high MI impulse to differentiate: False-positive perfusion shows no reperfusion after high MI flash\n\n|\n\nBlooming | \u2022High gain settings | \u2022Lowering the gain\n\nMotion artifact\n\n|  |\n\n\u2022Always see when using power Doppler imaging methods | \u2022Adjusting pulse repetition frequency (PRF) setting to minimize it\n\n## 10.11 Safety\n\n  * Common adverse effects\n\n    * Infrequent and mild: headache, weakness, fatigue, palpitations, nausea, dizziness, dry mouth, altered sense of smell or taste, dyspnea, urticaria, pruritus, back pain, chest pain, rash\n\n    * Rarely: Allergic and potentially life threatening hypersensitivity reactions\n\n## 10.12 FDA Recommendation for Definity and Optison\n\n  * Contraindication\n\n    * Right-to-left, bidirectional, or transient right-to-left cardiac shunts\n\n    * Hypersensitivity to perflutren\n\n    * Hypersensitivity to blood, blood products, or albumin (applies to Optison only).\n\n    * Intra-arterial injection\n\n  * Patients with pulmonary hypertension or unstable cardiopulmonary conditions\n\n    * Monitoring of vital signs, electrocardiography, and cutaneous oxygen saturation during and for at least 30 min after\n\n## 10.13 Summary\n\nContrast echocardiography is an important and a significant addition to a modern echocardiography laboratory. Successful implementation of contrast echo depends on a team effort, including appropriate dosing and administration of the agent, optimized system settings for contrast-enhanced study, and correct image acquisition and interpretation. Fully understanding the bubble physics and instrument setting, together with the indications, and contraindications for contrast-enhanced studies, will maximize the benefit to patients.\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_11\n\n# 11. Three-Dimensional Echocardiography\n\nLissa Sugeng1  , Lynn Weinert and Roberto M. Lang\n\n(1)\n\nSection of Cardiology, Department of Medicine, Non-Invasive Cardiovascular Imaging Laboratory, University of Chicago Medical Center, Chicago, IL, USA\n\nLissa Sugeng\n\nEmail: lsugeng@medicine.bsd.uchicago.edu\n\n## 11.1 Introduction\n\n  * Two-dimensional echocardiography (2DE) is one of the most widely used and well established noninvasive imaging tools in the cardiology.\n\nFig. 11.1\n\nRT3DE Examination Strategy. Incorporation of a three-dimensional echo examination in routine clinical echocardiographic study may pose challenges due to time constraints. Hence, performing a focused 3D examination is essential. Depending on the patient's pathology, a sonographer may focus on imaging the valve using a zoom mode of imaging to assess valve area or determine a prolapsing segment and acquire a full-volume of the left ventricle to quantitate left ventricular function. If the patient has a cardiomyopathy which requires a functional assessment, then a full-volume acquisition of the left ventricle should be obtained. There are essential steps for 3D imaging, which include: (1) data acquisition, (2) application of cut-planes, and (3) determination of the type of display\n\nFig. 11.2\n\nThere are three modes of acquisition for both transthoracic and transesophageal RT3DE: (1) narrow-angled, (2) zoom, and (3) full-volume. Narrow-angled and zoom mode of acquisition are both truly \"real-time\" since data is obtained in one beat. These two modes of acquisition are a small wedge volume of data and have the advantage of having less respiratory artifact. Adjustments of gain, TGC, compress or as generally referred as \"thresholding\" in 3D terms, are performed on a narrow-angled mode. A zoom mode of acquisition is preferably used for valvular or smaller structures. Ventricular function or atrial chambers need a larger volume scan, so a full-volume mode is chosen. This can be either a four beat acquisition or if higher frame rates are needed a seven beat acquisition which can encompass the entire ventricle\n\nFig. 11.3\n\nDensity settings for transthoracic RT3DE determine the volume resolution and size\n\nFig. 11.4\n\nCut-planes. After 3D volume is acquired, cropping planes are applied to demonstrate certain pathology. There are three cropping tools: (1) auto crop, (2) box crop, and (3) plane crop. Auto crop is useful to automatically demonstrate 50% of a full-volume scan. This enables the operator to ensure that thresholding was sufficient during acquisition. The box crop tool allows six planes to be used to crop the volume acquired in a perpendicular fashion whereas, a plane crop aids the operator to demonstrate abnormalities from any direction\n\nFig. 11.5\n\n3D Display. (1) Volume-rendering, (2) surface-rendering, (3) wire-frame, and (4) 2D tomographic slices\n\nFig. 11.6\n\n3D Orientation is an important aspect of three-dimensional imaging. Imaging from a left atrial and left ventricular perspective, the mitral valve is displayed with the aortic valve at about 12 o'clock similar to a surgical view\n\nFig. 11.7\n\nThese are examples of a normal mitral valve, aortic, and pulmonic valve obtained from a parasternal window. The first image on the left is all narrow-angled acquisitions. A zoom mode acquisition is displayed in the center and right panels\n\nFig. 11.8\n\nExamples of a normal tricuspid valve on top. Imaging of the tricuspid valve is only partially viewed from a parasternal window, so the preferred window to obtain the entire tricuspid valve and annulus is from an apical approach. The first image is a narrow-angled acquisition of part of the right ventricle. A subsequent zoom mode acquisition was obtained to view the tricuspid valve from the right ventricle and the right atrium. The right ventricle is also only partially viewed from a parasternal window; hence similarly to the tricuspid valve the entire right ventricle is obtained from an apical approach. The first image (RV 4CH) is a narrow-angle acquisition of the right ventricle from an apical 4-chamber view. From a full-volume right ventricular data set, several cut planes can be applied to obtain: (1) RV inflow, (2) RV coronal, and (3) RVOT short-axis. RV 4CH, right ventricle 4-chamber; RVOT, right ventricular outflow; a, anterior leaflet; p, posterior leaflet; s, septal leaflet\n\nFig. 11.9\n\nThe interatrial septum (IAS) and interventricular septum (IVS) are both acquired from an apical window. On the left is a narrow-angle acquisition of a left ventricular 4-chamber view. A zoom mode of acquisition is chosen to obtain the IAS and IVS. Both structures can be viewed from the left or right sided chambers\n\nFig. 11.10\n\nThe left ventricle is acquired using a full-volume mode of acquisition. From this pyramidal volume, many cut-planes can be applied to view different walls of the ventricle (top row). Using a box crop, the operator can view the lateral, septal, posterior, and anterior walls. Using the same method, a short-axis plane can be utilized to obtain multiple views of the left ventricle as seen on the bottom panel\n\nFig. 11.11\n\nA fully sampled 3D matrix transesophageal echo (3DMTEE). The image shows routine two-dimensional echo in the top and bottom panels performed on a patient with bileaflet mitral valve prolapse and severe eccentric mitral regurgitation (bottom panel) at 0\u00b0, 60\u00b0, and 120\u00b0. Using a zoom mode, the mitral valve is displayed from a left atrial view in systole (Panel A) and diastole (Panel B). In systole, all three segments of the posterior and corresponding anterior segments are prolapsing. The arrows designate the location of the posterior leaflet indentations demarcating the separation of P1, P2, and P3 which are better seen in diastole\n\nFig. 11.12\n\nIntraoperative evaluation of this same patient in Fig. 11.11 was performed post-repair. Routine 2D imaging was performed at 0\u00b0, 60\u00b0, and 120\u00b0 with and without color. Post-repair there was not any residual mitral regurgitation. Using a zoom mode, the mitral valve is displayed from a left atrial view in systole (Panel A) and diastole (Panel B). In systole, all three segments of the posterior and corresponding anterior segments appear well opposed with a flexible mitral ring. Gore-Tex chords were placed to repair this valve without any resection performed. These chords are not readily seen and in diastole the demarcation of the posterior leaflet indentations can still be appreciated\n\nFig. 11.13\n\nIn the two top panels a long-axis and short-axis of the aortic valve are demonstrated (a and b). There is moderate aortic sclerosis seen on the aortic leaflets as also viewed on a three-dimensional image of the valve from an aortic perspective (bottom panel)\n\nFig. 11.14\n\nExample of a patient with a secundum atrial septal defect demonstrated using simultaneous biplane 2D imaging, which was performed during closure of this defect. The diameter of the ASD measured in 2D was 3.04 and 2.93 cm with sufficient rim to place a device. Three-dimensional echo imaging allows en-face visual assessment of this ASD from a left atrial perspective (bottom). A wire is seen crossing the ASD in 2D as well as 3D\n\n  * Within 50 years of its inception by Edler and Hertz, advancement in transducer and software technology has given birth to fully sampled matrix array transducers, which not only allows traditional two-dimensional echocardiography (2DE) but also has the capability of simultaneous biplane imaging and volumetric three-dimensional imaging.\n\n  * Real-time 3D echocardiography (RT3DE) is truly \"real-time\" when using narrow angled or zoom mode and near \"real-time\" with wide-angled scanning.\n\n  * RT3DE enhances the ability in diagnosis of valvular heart disease by allowing precise imaging planes which can be aligned perpendicular to the valve plane, placed at the tips of the valve leaflets and thus producing a perfect en-face view of the valve orifice.\n\n    * In the assessment of mitral stenosis, three-dimensional echo results in accurate, reliable, and reproducible measurements of mitral valve area compared to conventional 2DE methods.1 \u2013 4\n\n    * Prolapsed or flail segments, valvular dehiscence, and perforation can be identified from a left atrial orientation simulating a \"surgeon's view.\"\n\n    * RT3DE imaging using matrix array probe in patients with good acoustic windows is feasible 83% of the time with adequate visualization of both anterior and posterior mitral leaflets.\n\n    * The anterior mitral valve is best seen from a ventricular perspective regardless of acoustic window, the posterior leaflet from parasternal window, and both commissures well seen from a parasternal or apical window.5\n\n    * RT3DE has also enhanced our understanding of ischemic mitral regurgitation, by imaging papillary muscle displacement.6,7\n\n  * RT3DE is accurate, reliable, and reproducible with less inter- and intra-observer variability in the assessment of left ventricular function and mass, using prior 3DE methods of acquisition and even more recent RT3DE technology.7 - 19 In addition to global left ventricular function and volumes, regional volumes are obtained, which allow quantitation of left ventricular dyssynchrony.20 - 23\n\n  * The development of a fully sampled matrix RT3D transesophageal probe was made possible due to higher density electronics and micro-beam forming architecture within the probe.\n\n  * Instantaneous 3D imaging of the valves and ventricle readily provide information on valvular pathology for surgical planning, and can help assess the results of mitral valve repair.23,24 RT3DE also plays a role in interventional procedures such as balloon mitral valvuloplasty, ASD and left atrial appendage device closure.25 - 27\n\n  * Three-dimensional echocardiography has evolved from a specialized research tool for a highly selected group of researchers that was obscure to the clinician due to cumbersome acquisition, lengthy processing, and off-line analysis. However, RT3DE now is a viable clinical tool mainly attributed to shorter acquisition time and ease of use, which is accommodated by a matrix array transducer and online visualization. It will certainly be a must have tool in most echo labs in the years to come.\n\nReferences\n\n?1.\n\nZamorano J, Perez d I, Sugeng L, et al. Non-invasive assessment of mitral valve area during percutaneous balloon mitral valvuloplasty: role of real-time 3D echocardiography. Eur Heart J. 2004;25(23):2086-2091.PubMedCrossRef\n\n?2.\n\nZamorano J, Cordeiro P, Sugeng L, et al. Real-time three-dimensional echocardiography for rheumatic mitral valve stenosis evaluation: an accurate and novel approach. J Am Coll Cardiol. 2004;43(11):2091-2096.PubMedCrossRef\n\n?3.\n\nSugeng L, Weinert L, Lammertin G, et al. Accuracy of mitral valve area measurements using transthoracic rapid freehand 3-dimensional scanning: comparison with noninvasive and invasive methods. J Am Soc Echocardiogr. 2003;16(12):1292-1300.PubMedCrossRef\n\n?4.\n\nBinder TM, Rosenhek R, Porenta G, Maurer G, Baumgartner H. Improved assessment of mitral valve stenosis by volumetric real-time three-dimensional echocardiography. J Am Coll Cardiol. 2000;36(4):1355-1361.PubMedCrossRef00852-4)\n\n?5.\n\nSugeng L, Coon P, Weinert L, et al. Use of real-time 3-dimensional transthoracic echocardiography in the evaluation of mitral valve disease. J Am Soc Echocardiogr. 2006;19(4):413-421.PubMedCrossRef\n\n?6.\n\nKwan J, Shiota T, Agler DA, et al. Geometric differences of the mitral apparatus between ischemic and dilated cardiomyopathy with significant mitral regurgitation: real-time three-dimensional echocardiography study. Circulation. 2003;107(8):1135-1140.PubMedCrossRef\n\n?7.\n\nWatanabe N, Ogasawara Y, Yamaura Y, et al. Geometric differences of the mitral valve tenting between anterior and inferior myocardial infarction with significant ischemic mitral regurgitation: quantitation by novel software system with transthoracic real-time three-dimensional echocardiography. J Am Soc Echocardiogr. 2006;19(1):71-75.PubMedCrossRef\n\n?8.\n\nGopal AS, Keller AM, Rigling R, King DL Jr, King DL. Left ventricular volume and endocardial surface area by three-dimensional echocardiography: comparison with two-dimensional echocardiography and nuclear magnetic resonance imaging in normal subjects. J Am Coll Cardiol. 1993;22(1):258-270.PubMedCrossRef90842-O)\n\n?9.\n\nGopal AS, Keller AM, Shen Z, et al. Three-dimensional echocardiography: in vitro and in vivo validation of left ventricular mass and comparison with conventional echocardiographic methods. J Am Coll Cardiol. 1994;24(2):504-513.PubMedCrossRef90310-7)\n\n10.\n\nGopal AS, Schnellbaecher MJ, Shen Z, Boxt LM, Katz J, King DL. Freehand three-dimensional echocardiography for determination of left ventricular volume and mass in patients with abnormal ventricles: comparison with magnetic resonance imaging. J Am Soc Echocardiogr. 1997;10(8):853-861.PubMedCrossRef70045-2)\n\n11.\n\nJacobs LD, Salgo IS, Goonewardena S, et al. Rapid online quantification of left ventricular volume from real-time three-dimensional echocardiographic data. Eur Heart J. 2006;27(4):460-468.PubMedCrossRef\n\n12.\n\nJenkins C, Bricknell K, Hanekom L, Marwick TH. Reproducibility and accuracy of echocardiographic measurements of left ventricular parameters using real-time three-dimensional echocardiography. J Am Coll Cardiol. 2004;44(4):878-886.PubMedCrossRef\n\n13.\n\nKing DL, Harrison MR, King DL Jr, Gopal AS, Martin RP, DeMaria AN. Improved reproducibility of left atrial and left ventricular measurements by guided three-dimensional echocardiography. J Am Coll Cardiol. 1992;20(5):1238-1245.PubMedCrossRef90383-X)\n\n14.\n\nKing DL, Gapal AS. Three-dimensional echocardiography: use of additional spatial data for measuring left ventricular mass. Mayo Clin Proc. 1994;69(3):293-295.PubMedCrossRef61073-8)\n\n15.\n\nKuhl HP, Franke A, Merx M, Hoffmann R, Puschmann D, Hanrath P. Rapid quantification of left ventricular function and mass using transoesophageal three-dimensional echocardiography: validation of a method that uses long-axis cutplanes. Eur J Echocardiogr. 2000;1(3):213-221.PubMedCrossRef\n\n16.\n\nKuhl HP, Schreckenberg M, Rulands D, et al. High-resolution transthoracic real-time three-dimensional echocardiography: quantitation of cardiac volumes and function using semi-automatic border detection and comparison with cardiac magnetic resonance imaging. J Am Coll Cardiol. 2004;43(11):2083-2090.PubMedCrossRef\n\n17.\n\nMaehle J, Bjoernstad K, Aakhus S, Torp HG, Angelsen BA. Three-dimensional echocardiography for quantitative left ventricular wall motion analysis: a method for reconstruction of endocardial surface and evaluation of regional dysfunction. Echocardiography. 1994;11(4):397-408.PubMedCrossRef\n\n18.\n\nMor-Avi V, Sugeng L, Weinert L, et al. Fast measurement of left ventricular mass with real-time three-dimensional echocardiography: comparison with magnetic resonance imaging. Circulation. 2004;110(13):1814-1818.PubMedCrossRef\n\n19.\n\nSapin PM, Schroder KM, Gopal AS, Smith MD, DeMaria AN, King DL. Comparison of two- and three-dimensional echocardiography with cineventriculography for measurement of left ventricular volume in patients. J Am Coll Cardiol. 1994;24(4):1054-1063.PubMedCrossRef90869-9)\n\n20.\n\nBacha EA, Zimmerman FJ, Mor-Avi V, et al. Ventricular resynchronization by multisite pacing improves myocardial performance in the postoperative single-ventricle patient. Ann Thorac Surg. 2004;78(5):1678-1683.PubMedCrossRef\n\n21.\n\nKapetanakis S, Cooklin M, Monaghan MJ. Mechanical resynchronisation in biventricular pacing illustrated by real time transthoracic three dimensional echocardiography. Heart. 2004;90(5):482.PubMedCentralPubMedCrossRef\n\n22.\n\nKapetanakis S, Kearney MT, Siva A, Gall N, Cooklin M, Monaghan MJ. Real-time three-dimensional echocardiography: a novel technique to quantify global left ventricular mechanical dyssynchrony. Circulation. 2005;112(7):992-1000.PubMedCrossRef\n\n23.\n\nSugeng L, Shernan SK, Salgo IS, et al. Live 3-dimensional transesophageal echocardiography initial experience using the fully-sampled matrix array probe. J Am Coll Cardiol. 2008;52(6):446-449.PubMedCrossRef\n\n24.\n\nSugeng L, Shernan SK, Weinert L, et al. Real-time three-dimensional transesophageal echocardiography in valve disease: comparison with surgical findings and evaluation of prosthetic valves. J Am Soc Echocardiogr. 2008;21(12):1347-1354.PubMedCrossRef\n\n25.\n\nLodato JA, Cao QL, Weinert L, et al. Feasibility of real-time three-dimensional transesophageal echocardiography for guidance of percutaneous atrial septal defect closure. Eur J Echocardiogr. 2009;10(4):543-548.PubMedCrossRef\n\n26.\n\nShah SJ, Bardo DM, Sugeng L, et al. Real-time three-dimensional transesophageal echocardiography of the left atrial appendage: initial experience in the clinical setting. J Am Soc Echocardiogr. 2008;21(12):1362-1368.PubMedCrossRef\n\n27.\n\nMessika-Zeitoun D, Brochet E, Holmin C, et al. Three-dimensional evaluation of the mitral valve area and commissural opening before and after percutaneous mitral commissurotomy in patients with mitral stenosis. Eur Heart J. 2007;28(1):72-79.PubMedCrossRef\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_12\n\n# 12. Stress Echocardiography\n\nSebastian Kelle1\n\n(1)\n\nDepartment of Cardiology, German Heart Institute Berlin, Berlin, Germany\n\nSebastian Kelle\n\nEmail: kelle@dhzb.de\n\n## 12.1 Purpose\n\n  * Detection of myocardial ischemia and viability\n\n  * Baseline echo to identify regional LV function\n\n  * Wall motion abnormalities (WMA) are one of the earliest signs of myocardial ischemia during stress\n\n  * Exercise or pharmacologic stress\n\n  * Dobutamine is the preferred pharmacological stress agent for the detection of inducible WMA\n\n## 12.2 Indications\n\n  * Inconclusive exercise test or inability to exercise\n\n  * Suspected coronary artery disease\n\n  * Assessment of ischemia in patients with known CAD\n\n  * Patients post revascularization and atypical chest pain\n\n  * Assessing extent of ischemia\n\n  * Guidance for intervention (PCI or CABG)\n\n  * Assessment of viability\n\n  * Preoperative risk assessment\n\n  * Documenting exercise capacity\n\n  * Risk stratification and prognosis\n\n  * Assessment of response to treatment\n\n  * Assessment of low gradient aortic stenosis or mitral valve stenosis\n\n  * Dyspnea of possible cardiac origin\n\n## 12.3 Contraindications\n\n  * Unstable angina pectoris\n\n  * Acute myocardial infarction (within 2 days)\n\n  * Uncontrolled arrhythmias causing symptoms of hemodynamic compromise\n\n  * Symptomatic severe aortic stenosis\n\n  * Uncontrolled symptomatic heart failure\n\n  * Active endocarditis or acute myocarditis or pericarditis\n\n  * Acute aortic dissection or aortic aneurysm\n\n  * Acute pulmonary or systemic embolism\n\n  * Acute noncardiac disorders that may affect exercise performance or may be aggravated by exercise\n\n  * Severe hypertension\n\n  * Hemodynamically significant left ventricular outflow tract obstruction\n\n  * Known HOCM\n\n  * Hypokalemia\n\n## 12.4 Dobutamine\n\n  * Given in incremental doses starting with 5\u201310 ?g\/kg\/min and increasing at 10, 20, 30 and 40 ?g\/kg\/min to simulate exercise\n\n  * Low-dose stages (up to 20 ?g\/kg\/min) viability and ischemia assessment in segments with WMA at rest\n\n  * Half life 2 min\/steady state 10 min\n\n  * Positive inotropy\n\n    * Stimulates LV myocardial function\n\n    * Recruits hibernating myocardium\n\n  * Positive chronotropy\n\n    * Increases heart rate\n\n    * Increases oxygen consumption\n\n    * Induces ischemia\n\n## 12.5 Contraindications for Dobutamine\/Atropine\n\n1.\n\nSevere arterial hypertension (>220\/120 mmHg)\n\n2.\n\nBP ?< ?90 mmHg systolic\n\n3.\n\nUnstable angina pectoris\n\n4.\n\nAcute myocardial infarction\n\n5.\n\nSecond or third degree AV Block or sick-sinus-syndrome (no pacemaker)\n\n6.\n\nSevere aortic stenosis (AVA ?< ?1 cm2)\n\n7.\n\nHOCM\n\n8.\n\nAcute Perimyocarditis or Endocarditis\n\n9.\n\nGlaucoma\n\n## 12.6 Side Effects\n\n  * Palpitations, nausea, headache, chills, urinary urgency, anxiety\n\n  * Angina, hypotension, cardiac arrhythmias\n\n  * Premature atrial or ventricular contractions \u2013 4%\n\n  * Ventricular fibrillation or myocardial infarction: 1\/2,000\n\n## 12.7 Monitoring Requirements\n\n1.\n\nHeart rate and rhythm: continuously\n\n2.\n\nBlood pressure: every 3 min\n\n3.\n\nSymptoms: continuously\n\n4.\n\nWall-motion-abnormalities: every dose increment\n\n5.\n\nImaging ?< ?3 min\n\n6.\n\nCover all segments\n\n7.\n\nST-Segment changes\n\n## 12.8 Visual Assessment of Left Ventricular WMA, the Standard Scoring System is Applied Per Myocardial Segment\n\nScoring system\n\n1, normokinesis\n\n2, hypokinesis\n\n3, akinesis\n\n4, dyskinesis\n\n5, aneurysmal\n\n## 12.9 Performance\n\nPatient instructions:\n\nNo ?-blockers and nitrates 24 h prior to the examination\n\n  * Optimal ECG\n\n  * Select optimal windows (whole window for left ventricle)\n\n  * Use tissue harmonic imaging\n\n  * Highest frame rate\n\n  * Continuous monitoring\n\n  * Cine-loops at baseline, low- and high-dose\n\n  * Record several cycles\n\n  * Use intravenous contrast agents when two or more segments are not well visualized\n\n## 12.10 Termination Criteria\n\n  * Submaximal heart rate reached ([220 \u2013 age] \u00d7 0.85)\n\n  * Maximum dose reached (40 ?g\/kg\/min dobutamine ?+ ?2 mg atropine)\n\n  * Systolic RR decrease ?> ?20 mmHg below the baseline level or decrease ?> ?40 mmHg from a previous level\n\n  * RR increase ?> ?240\/120 mmHg\n\n  * Intolerable symptoms (chest pain, nausea)\n\n  * New or worsening WMA in N ?? ?2 adjacent LV segments\n\n  * ST depression ?> ?3 mm\n\n  * Symptomatic or complex cardiac tachycardia\n\nIschemia is defined as a new WMA or a biphasic response.\n\n## 12.11 Myocardial Viability\n\n1.\n\nMyocardial stunning: result of acute ischemic insult leading to contractile dysfunction despite adequate reperfusion\n\n2.\n\nHibernating myocardium: reversible left ventricular dysfunction due to chronic coronary artery disease that improves after revascularization\n\n## 12.12 Interpretation\n\n  * Report \u2013 description of\n\n    * Hemodynamics\n\n    * ECG changes\n\n    * Clinical symptoms\n\n    * Reached maximum heart rate\n\n    * Stress level\n\n    * Wall motion score at rest and stress\n\n  * Baseline\n\n    * Left ventricular cardiac chamber volumes, ejection fraction, mass and regional wall motion, valves, aortic root\n\n  * Stress\n\n    * New wall motion abnormalities during stress in segments with normal wall motion at rest or worsening or biphasic response in segments with wall motion abnormality at rest\n\n    * Extent of ischemia\n\n    * Assessment of ischemic mitral valve regurgitation\n\n    * Assessment of aortic gradient\n\nFig. 12.1\n\nExercise stress echocardiogram images. Apical four-chamber view showing rest images (top row) and normal left ventricular wall motion. During stress imaging (bottom row) there is an anteroapical wall motion abnormality indicating exercise-induced ischemia in the distribution of the left anterior descending coronary artery.\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_13\n\n# 13. Mechanical Dyssynchrony Assessment\n\nGabe B. Bleeker1  , Nico R. Van de Veire, Theodore P. Abraham, Eduard R. Holman and Jeroen J. Bax\n\n(1)\n\nDepartment of Cardiology, Leiden University Medical Center, Leiden, The Netherlands\n\nGabe B. Bleeker\n\nEmail: g.b.bleeker@lumc.nl\n\nFig. 13.1\n\nCurrently, the resynchronization of left ventricular (LV) dyssynchrony (present before pacemaker implantation) is considered to be the key mechanism of response following cardiac resynchronization therapy (CRT). Identification of LV dyssynchrony is, therefore, needed to select patients with high likelihood of response to CRT. In the current guidelines, a widened QRS complex (>120 ms) is used as a marker of LV dyssynchrony. Recent data, however, have indicated that QRS duration is a poor marker of LV dyssynchrony, with poor predictive value for response to CRT. Since this observation, several echocardiographic techniques, such as color-coded tissue Doppler imaging (TDI), have been evaluated for the ability to directly quantify LV dyssynchrony. (a) Results of a recent study illustrating the lack of correlation between LV dyssynchrony (expressed as the septal-to-lateral delay) and QRS duration in 90 heart failure patients with LV ejection fraction <35% and NYHA class III-IV heart failure (r ?= ?0.26, P ?= ?NS) (Reprinted with permission from Ref.1). (b) Example of a heart failure patient with a narrow QRS complex (110 ms) and a delay of 85 ms between the peak systolic velocity (PSV) of the septum and lateral wall, indicating the presence of LV dyssynchrony (arrows indicate peak systolic velocities, AVO aortic valve opening, AVC aortic valve closure). (c) Example of a patient with a wide QRS complex (160 ms) without a delay in peak systolic velocity between the septum and the lateral wall, indicating the absence of LV dyssynchrony (arrows indicate peak systolic velocities, AVO aortic valve opening, AVC aortic valve closure)\n\nFig. 13.2\n\nThe first step in LV dyssynchrony measurement from TDI is to define the ejection period. Using the \"event timing function,\" the timing of aortic valve opening (AVO) and closure (AVC) can be defined from the pulsed-wave Doppler recording in the LV outflow tract (a). After marking the AVO and AVC in the pulsed-wave Doppler recording, the timing of the ejection phase is automatically displayed in the TDI-analysis window (b)\n\nFig. 13.3\n\n(a) Example of a TDI curve derived from the basal septum in a normal individual in the 4-chamber view. The sample volume is placed in the region of interest (here the basal septum), and a TDI curve will be automatically derived indicating the peak systolic velocity (PSV), which is within the ejection phase between aortic valve opening (AVO) and closure, (AVC), and diastolic parameters (E? and A?). Do not confuse the peak of isovolumic contraction (IVC) with the peak systolic velocity (PSV). (b) Example of a TDI curve derived from the basal lateral wall in a normal individual (4-chamber view, for explanation see Fig. 13.3a). (c) Example of a TDI curve derived from the basal inferior wall in a normal individual (2-chamber view, for explanation see Fig. 13.3a). (d) Example of a TDI curve derived from the basal anterior wall in a normal individual (2-chamber view, for explanation see Fig. 13.3a). (e) LV dyssynchrony can be calculated as the delay in time-to-peak systolic velocity between the basal septum and the lateral wall (referred to as septal-to-lateral delay (2-segmental model, cutoff value 60 ms6). Example of a normal individual without a delay in peak systolic velocity, indicating absence of LV dyssynchrony. (e, f) Another described method to quantify LV dyssynchrony by measuring the time-to-maximum delay between the peak systolic velocities among the four basal LV walls (septum, lateral, inferior, and anterior walls (4-segmental model, derived from Fig. 13.3a\u2013d) (most frequently dyssynchrony is observed between the septum and lateral wall, cutoff value 65 ms7). When the time-to-peak systolic velocity is measured in multiple views, the beginning of the QRS complex can be used as a reference point\n\nFig. 13.4\n\nROC curve analysis to determine the predictive value for response following CRT using the 4-segmental LV dyssynchrony model (maximum delay among four basal segments), demonstrating a sensitivity and specificity of 80% to predict clinical response (defined as improvement in NYHA class ??1 score and improvement ??25% in 6-min walking distance) at a cutoff level of 65 ms for LV dyssynchrony. In addition, this approach yielded a sensitivity and specificity of 92% to predict reverse LV remodeling after CRT (defined as improvement in LV end-systolic volume ??15%) at a cutoff level of 65 ms for LV dyssynchrony. Reprinted with permission from reference8\n\nFig. 13.5\n\nIllustration of the 12-segmental model for LV dyssynchrony measurement (Yu-index8). The time-to-peak systolic velocities during the ejection phase in each view are measured (by using the beginning of the QRS complex as a reference point). The myocardial velocity curves are derived from the basal and mid parts of the following LV segments: septum and lateral wall (4-chamber view, a), inferior and anterior wall (2-chamber view, b) and antero-septal and posterior wall (3-chamber view, c). LV dyssynchrony is defined as the standard deviation of the time-to-peak systolic velocity for all 12 segments (cutoff value 32 ms8)\n\nFig. 13.6\n\nMeasurement of LV dyssynchrony using the 4-segmental model. Example of a patient without a delay in time-to-peak systolic velocity between the septum and the lateral wall (4-chamber view, a) and the inferior and anterior wall (2-chamber view, b), indicating the absence of LV dyssynchrony (Arrows indicate peak systolic velocities, AVO ?aortic valve opening, AVC ?aortic valve closure)\n\nFig. 13.7\n\nMeasurement of LV dyssynchrony using the 4-segmental model. Example of a patient with substantial LV dyssynchrony. In the 4-chamber view, a delay in time-to-peak systolic velocity of 120 ms is observed between the septum and the lateral wall (a) and a delay of 60 ms is present between the inferior and anterior walls (b). The maximum delay among the four segments is well above the cutoff value of 65 ms, indicating the presence of LV dyssynchrony (Arrows indicate peak systolic velocities, AVO aortic valve opening, AVC aortic valve closure)\n\nFig. 13.8\n\nMeasurement of diastolic dyssynchrony within the left ventricle. Example of a patient with a delay of 90 ms between the time to peak E?-wave of the septum and lateral wall. At present, the role of diastolic dyssynchrony in CRT remains to be defined. No cutoff value for diastolic dyssynchrony has been proposed in the literature (Arrows indicate peak E?-wave)\n\nFig. 13.9\n\nMeasurement of interventricular dyssynchrony (dyssynchrony between the left and right ventricles). The first method measures the difference in the pre-ejection times between the pulmonary flow (a, 99 ms) and the aortic flow (b, 137 ms). The difference between both pre-ejection times represents the interventricular dyssynchrony (38 ms in this example, proposed cutoff values range between 40 and 50 ms). Recent studies, however, have now clearly demonstrated that interventricular dyssynchrony is a poor predictor of response after CRT implantation. Another described method for interventricular dyssynchrony calculation is to measure the delay in peak systolic velocity between the LV lateral wall and the right ventricular free wall (c, arrows indicate peak systolic velocities)\n\nTable 13.1\n\nMost commonly used echocardiographic techniques for LV dyssynchrony quantification. All clinically proven parameters of dyssynchrony \u2013 from JACC review\n\nEchocardiographic technique | Parameter | Author\n\n---|---|---\n\nM-mode echocardiography | Difference in systolic inward motion between septum and posterior wall. | Pitzalis et al.2,3\n\nPulsed-wave TDI | Difference in time-to-onset of systolic velocity between septum and lateral wall. | Garrigue et al.4\n\n|\n\n\u2013 Maximum difference in time-to-onset\/peak systolic velocity among 12 basal and mid-LV segments. | Bordachar et al.5\n\n\u2013 Standard deviation in time-to-onset\/peak systolic velocity among 12 basal and mid-LV segments.\n\nColor-coded TDI | Difference in time-to-peak systolic velocity between basal septum and lateral wall (septal-to-lateral delay). | Bax et al.6 (Fig. 13.3)\n\n|\n\nMaximum delay in time-to-peak systolic velocity among four basal segments. | Bax et al.7 (Figs. 13.3 and 13.4)\n\n|\n\nStandard deviation of the time-to-peak systolic velocity among 12 basal and mid-LV segments (Yu-index). | Yu et al.8 (Fig. 13.5)\n\nTissue synchronization imaging | \u2013 Qualitative assessment of LV dyssynchrony based on color maps. | Yu et al.9\n\n\u2013 Standard deviation of the time-to-peak systolic velocity among 12 basal and mid-LV segments.\n\nStrain (rate) imaging (derived from TDI) | Difference in time-to-peak radial strain between septum and posterior wall. | Dohi et al.10\n\n|\n\nDifference in time-to-peak strain between septum and lateral wall. | Popovic et al.11\n\n2D strain (speckle tracking) | Difference in time-to-peak radial strain between septum and posterior wall. | Suffoletto et al.12\n\nReal-time 3D echocardiography | Standard deviation of time-to-minimal regional volume among 16 segments (systolic dyssynchrony index). | Kapetanakis et al.13\n\nTDI tissue Doppler imaging, LV left ventricular\n\nTable based on Ref.14\n\nTable 13.2\n\nAlgorithm for LV dyssynchrony measurement using color-coded TDI. Focus on the criteria you use clinically \u2013 may be 2\u20133 major ones for example, septal to lateral wall delay; maximum delay and Yu index\n\nStep1:\n\n---\n\nDefine the ejection phase by measuring the timing of the aortic valve opening and closure from the pulsed-wave Doppler signal in the LV outflow tract (Fig. 13.2).\n\nStep 2:\n\nUse the TDI curves to measure the time difference between the beginning of the QRS complex and the peak systolic velocity\n\nIn the following LV segments (Fig. 13.3):\n\n\u2013?Basal septum and basal lateral wall (apical 4-chamber view)\n\n\u2013?Basal anterior and basal inferior wall (apical 2-chamber view)\n\nAdditional measurements for the 12-segmental model (Yu-index, Fig. 13.5)\n\n\u2013?Mid-septal and mid-lateral wall (apical 4-chamber view)\n\n\u2013?Mid-anterior and mid-inferior wall (apical 2-chamber view)\n\n\u2013?Basal and mid-antero-septum and basal and mid-posterior LV wall (apical 3-chamber view)\n\nStep 3:\n\nThe measurements from step 2 can be used to calculate the following parameters for LV dyssynchrony:\n\n\u2013?Time-delay between the peak systolic velocity of the basal septum and lateral wall (septal-to-lateral delay, cutoff value 60 ms6).\n\n\u2013?Maximum time delay among four basal segments (septum, lateral, inferior and anterior LV segments, cutoff value 65 ms7).\n\n\u2013?Standard deviation of the time-to-peak systolic velocity among 12 LV segments (Yu-index, cutoff value 32 ms8).\n\nTDI tissue Doppler imaging, LV left ventricular\n\nTable 13.3\n\nPitfalls in LV dyssynchrony measurement using color-coded TDI. Common pitfalls with dyssynchrony analysis and how to overcome them Suggested references: up to five general references where they can go for more details\n\nPitfalls | Solution\n\n---|---\n\nWhich peak should be measured? | The measurement of the peaks should be limited to the ejection phase. Use the event timing function, which marks the opening and closure of the aortic valve (Fig. 13.2).\n\n|\n\nIn the event of multiple positive peaks in the ejection phase, take the peak with the highest systolic velocity.\n\n|\n\nDo not confuse the peak systolic velocity with the peak of isovolumetric contraction.\n\nArrhythmias | Record the TDI-images during at least three consecutive beats in normal sinus rhythm. Avoid measurement of LV dyssynchrony during atrial and ventricular extrasystoles.\n\n|\n\nDuring persistent atrial arrhythmias, LV dyssynchrony should be calculated within the same heartbeat. In addition, the measurements should be repeated\/confirmed within other heartbeats.\n\nNoisy signals | Ensure a frame rate of at least 100 frames per second (the higher the better). A high frame rate can be achieved by narrowing the apical 4-, 2- and 3-chamber TDI recordings down to the left ventricle (i.e., excluding the right ventricle and the atria).\n\nTDI tissue Doppler imaging, LV left ventricular\n\nReferences\n\n?1.\n\nBleeker GB, Schalij MJ, Molhoek SG, et al. Relationship between QRS duration and left ventricular dyssynchrony in patients with end-stage heart failure. J Cardiovasc Electrophysiol. 2004;15:544-549.PubMedCrossRef\n\n?2.\n\nPitzalis MV, Iacoviello, Romito R, et al. Cardiac resynchronization therapy tailored by echocardiographic evaluation of ventricular asynchrony. J Am Coll Cardiol. 2002;40:1615-1622.PubMedCrossRef02337-9)\n\n?3.\n\nPitzalis MV, Iacoviello, Romito R, et al. Ventricular asynchrony predicts a better outcome in patients with chronic heart failure receiving cardiac resynchronization therapy. J Am Coll Cardiol. 2005;45:65-69.PubMedCrossRef\n\n?4.\n\nGarrigue S, Reuter S, Labeque JN, et al. Usefullness of biventricular pacing in patients with congestive heart failure and right bundle branch block. Am J Cardiol. 2001;88:1436-1441.PubMedCrossRef02131-2)\n\n?5.\n\nBordachar P, Lafitte S, Reuter S, et al. Echocardiographic parameters of ventricular dyssynchrony validation in patients with heart failure using sequential biventricular pacing. J Am Coll Cardiol. 2004;44:2154-2165.CrossRef\n\n?6.\n\nBax JJ, Marwick TH, Molhoek SG, et al. Left ventricular dyssynchrony predicts benefit of cardiac resynchronization therapy in patients with end-stage heart failure before pacemaker implantation. Am J Cardiol. 2003;92:1238-1240.PubMedCrossRef\n\n?7.\n\nBax JJ, Bleeker GB, Marwick TH, et al. Left ventricular dyssynchrony predicts response and prognosis after cardiac resynchronization therapy. J Am Coll Cardiol. 2004;44:1834-1840.PubMedCrossRef\n\n?8.\n\nYu CM, Chau E, Sanderson JE, et al. Tissue Doppler echocardiographic evidence of reverse remodeling and improved synchronicity by simultaneously delaying regional contraction after biventricular pacing therapy in heart failure. Circulation. 2002;105:438-445.PubMedCrossRef\n\n?9.\n\nYu CM, Zhang Q, Fung JW, et al. A novel tool to assess systolic asynchrony and identify responders of cardiac resynchronization therapy by tissue synchronization imaging. J Am Coll Cardiol. 2005;45:677-684.PubMedCrossRef\n\n10.\n\nDohi K, Suffoletto MS, Schwartzman D, et al. Utility of Echocardiographic radial strain imaging to quantify left ventricular dyssynchrony and predict acute response to cardiac resynchronization therapy. Am J Cardiol. 2005;96:112-116.PubMedCrossRef\n\n11.\n\nPopovic ZB, Grimm RA, Perlic G, et al. Noninvasive assessment of cardiac resynchronization therapy for congestive heart failure using myocardial strain and left ventricular peak power as parameters of myocardial synchrony and function. J Cardiovasc Electrophysiol. 2002;13:1203-1208.PubMedCrossRef\n\n12.\n\nSuffoletto MS, Dohi K, Cannesson M, et al. Novel speckle-tracking radial strain from routine black-and-white echocardiographic images to quantify dyssynchrony and predict response to cardiac resynchronization therapy. Circulation. 2006;113:960-968.PubMedCrossRef\n\n13.\n\nKapetanakis S, Kearney MT, Siva A, et al. Real-time Three-dimensional echocardiography. Circulation. 2005;112:992-1000.PubMedCrossRef\n\n14.\n\nBax JJ, Abraham T, Barold SS, et al. Cardiac resynchronization therapy. Part 1-Issues before device implantation. J Am Coll Cardiol. 2005;46:2153-2167.PubMedCrossRef\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_14\n\n# 14. Post Cardiac Resynchronization Therapy (CRT) Optimization Protocol\n\nHsin-Yueh Liang1\n\n(1)\n\nDivision of Cardiology, Department of Medicine, China Medical University Hospital, Taichung, Taiwan\n\nHsin-Yueh Liang\n\nEmail: liangsy2@gmail.com\n\n## 14.1 Cardiac Optimization Protocol\n\n### 14.1.1 Scheduling Requirements\n\n1.\n\n2-h slot\n\n2.\n\nSonographer\n\n3.\n\nProgrammer and pacemaker specialist\n\n4.\n\nA physician who is familiar with echo and pacemaker\n\n### 14.1.2 Protocol\n\n1.\n\nInterrogate pacemaker and print out baseline settings, read low rate, hysteresis rate, max tracking rate, AV\/PV delay, and VV offset\n\n2.\n\nDetermine percentage of A sense V sense (ASVS), A sense V pace (ASVP), A pace V sense (APVS), and A pace V pace (APVP) from histogram\n\n3.\n\nSet VV offset as zero (simultaneous bi-ventricular pacing)\n\n#### 14.1.2.1 PV Optimization\n\nSet pacer to the intrinsic rhythm (ASVS), make sure wide QRS from surface ECG on the echo machine and intracardiac electrogram (IEG) on the pacer programmer, usually the intrinsic PV delay is longer than 200 ms\n\nA.\n\nWait for at least 30 s at each setting before start of echo record\n\nB.\n\nPW Doppler on MV at mitral annulus, measure VTI of A wave, and total mitral inflow (Fig. 14.1).\n\nFig. 14.1\n\nVTI of A wave (left) and mitral inflow (right)\n\nC.\n\nPW Doppler at LVOT, measure VTI of LVOT (Fig. 14.2)\n\nFig. 14.2\n\nVTI of LVOT\n\nD.\n\nDecrease PV delay until A sense V pace (ASVP), make sure narrow QRS from the surface ECG and IEG\n\nE.\n\nRepeat this sequence of imagines (point A to C)\n\nF.\n\nDecrease PV delay by 30 ms intervals and repeat this sequence of imagines (point A to C)\n\nG.\n\nRepeat point F until there is mitral A wave truncation (Fig. 14.3).\n\nFig. 14.3\n\nA wave truncation (arrow)\n\nH.\n\nThe optimal PV delay is the setting, which yields maximal LVOT VTI with narrow QRS complex\n\nI.\n\nDetermine the optimal PV delay by VTI of mitral inflow in case maximal LVOT VTI is present at two different settings\n\n#### 14.1.2.2 AV Optimization\n\nSet pacer from APVS (pace at a rate higher than the intrinsic cardiac beats, about 10 beat\/min), make sure wide QRS from the surface ECG and IEG\n\nA.\n\nWait for at least 30 s at each setting before start of echo record\n\nB.\n\nPW Doppler on MV at mitral annulus, measure VTI of A wave and total mitral inflow\n\nC.\n\nPW Doppler at LVOT, measure VTI of LVOT\n\nD.\n\nDecrease AV delay until A pace V pace (APVP), make sure narrow QRS from the surface ECG and IEG\n\nE.\n\nRepeat this sequence of imagines (point A to C)\n\nF.\n\nDecrease AV delay by 30 ms intervals and repeat this sequence of imagines (point A to C)\n\nG.\n\nRepeat point F until there is mitral A wave truncation\n\nH.\n\nThe optimal AV delay is the setting, which yields maximal LVOT VTI with narrow QRS complex\n\nI.\n\nDetermine the optimal AV delay by VTI of mitral inflow in case maximal LVOT VTI is present at two different settings\n\n#### 14.1.2.3 VV Optimization\n\nA.\n\nSet PV delay as the optimal one obtained from the procedure above\n\nB.\n\nChange VV settings as follows and obtain images (point A to C) at each setting\n\ni.\n\nRV pre-excitation by 30 ms\n\nii.\n\nRV pre-excitation by 60 ms\n\niii.\n\nLV pre-excitation by 30 ms\n\niv.\n\nLV pre-excitation by 60 ms\n\nC.\n\nSet AV delay as the optimal one obtained from the procedure above\n\nD.\n\nChange VV settings as follows and obtain images (point A to C) at each setting\n\ni.\n\nRV pre-excitation by 30 ms\n\nii.\n\nRV pre-excitation by 60 ms\n\niii.\n\nLV pre-excitation by 30 ms\n\niv.\n\nLV pre-excitation by 60 ms\n\nE.\n\nThe optimal VV offset (including simultaneous offset) is the setting, which yields maximal LVOT VTI with narrow QRS complex\n\nF.\n\nIf the optimal VV offset at PV delay is different from the one at AV delay, determine the clinically optimal VV offset based on the percentage of ASVP and APVP from the initial interrogation\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_15\n\n# 15. Echocardiographic Assessment of the Right Ventricle\n\nHisham Dokainish1\n\n(1)\n\nDepartment of Cardiology, Baylor College of Medicine, 6620 Main Street-11A.08, 77030 Houston, TX, USA\n\nHisham Dokainish\n\nEmail: hishamd@bcm.tmc.edu\n\n## 15.1 Introduction\n\n1.\n\nEchocardiography is a commonly used noninvasive modality to assess right ventricular (RV) morphology and function.\n\n2.\n\nTwo-dimensional echocardiographic variables include RV size, interventricular septal morphology and motion, and RV function using fractional area change, ejection fraction, and tricuspid annular plane systolic excursion (TAPSE).\n\n3.\n\nConventional Doppler variables include RV stroke volume and cardiac output and pulmonary artery systolic and diastolic pressure using the pulmonic and tricuspid regurgitation signals, respectively.\n\n4.\n\nEmerging echocardiographic techniques to assess RV function include tissue Doppler imaging, speckle-based strain and strain rate imaging, and three-dimensional echocardiography.\n\n5.\n\nA comprehensive echo-Doppler approach to the RV, incorporating many of these variables in concert, can result in an accurate assessment of RV size, function, and hemodynamics.\n\n1.\n\n2D echocardiographic variables in assessing of right ventricular size (Fig. 15.1).\n\nFig. 15.1\n\nIn the top panel, RV diastolic diameter is 3.2 cm and RV systolic diameter is 2.2 cm, consistent with normal RV size In the bottom panel, RV diastolic dimension ?= ?5.4 cm and systolic dimension ?= ?2.6 cm, consistent with moderate to severe RV dilation\n\n  * Most but not all patients with abnormal RV function or significantly elevated pulmonary artery pressures have an enlarged RV.\n\n  * Normal RV diastolic diameter in the apical 4-chamber view is <3.8 cm.\n\n2.\n\nRight ventricular (RV) fractional area change for the assessment of RV function (Fig. 15.2).\n\nFig. 15.2\n\nIn the top panel, RV diastolic area ?= ?27 cm2, RV systolic area ?= ?13 cm2, and FAC ?= ?(27\u201313)\/27 ?= ?52%. In the bottom panel, RV diastolic area ?= ?24 cm2, RV systolic area ?= ?17 cm2, and FAC ?= ?(24\u201317)\/24 ?= ?29%, consistent with mild-to-moderate RV dysfunction. Note should be made that the RV is not dilated in the bottom panel, despite the depressed systolic function\n\n  * Normal values are <32 cm2 for RV diastolic area, <20 cm2 for RV systolic area, and >35% for RV FAC.\n\n3.\n\nRight ventricular (RV) volumes and ejection fraction (EF) (Fig. 15.3).\n\nFig. 15.3\n\nIn the top panel, RV diastolic volume ?= ?130 mL and in the bottom panel, RV systolic volume ?= ?99 mL, giving an RVEF ?= ?(130\u201399)\/130 ?= ?24%, consistent with severe RV dysfunction. RVEF >45% is consistent with normal RV systolic function\n\n  * In patients with severe pulmonary hypertension, echocardiography shows a severely enlarged RV and a small left ventricle (LV) compressed by RV enlargement.\n\n  * The right atrium can be severely enlarged, with the interatrial septum bulging to the left, indicating elevation of RA pressure.\n\n4.\n\nRight ventricular (RV) tricuspid annular plane systolic excursion (TAPSE) (Fig. 15.4).\n\nFig. 15.4\n\nThe top image shows a TAPSE of 2.1 cm in a patient with diabetes but normal RV systolic function. The bottom image shows an abnormal TAPSE of 1.3 cm, consistent with moderate RV dysfunction\n\n  * The image can be viewed by placing an M-mode cursor through the tricuspid annular plane that gives a longitudinal excursion in centimeter \u2013 referred to as TAPSE \u2013 a measure of longitudinal RV systolic function.\n\n  * Normal TAPSE is >1.5 cm, while ?1.5 is associated with RV dysfunction and adverse prognosis since RV disease results in decreased longitudinal RV excursion.\n\n  * The sole use of TAPSE \u2013 or any measure of RV function in isolation \u2013 can result in errors.\n\n  * Many variables of RV performance by echocardiography and Doppler should be used for accurate assessment of RV function.\n\n5.\n\nRight ventricular (RV) eccentricity index (RVEI) (Fig. 15.5).\n\nFig. 15.5\n\nIn the top panel, the longitudinal LV diastolic diameter (D1) ?= ?4.1, while diastolic height ?= ?2.1 cm, and RVEI = 4.1\/2.1 = 1.95. In the bottom panel, longitudinal systolic diameter (D2) = 3.2 cm, systolic height ?= ?1.2 cm, and RVEI =3.2\/1.2 =2.7\n\n  * RV dilates in response to volume overload and pushes the interventricular septum (IVS) toward the left ventricle (LV), compressing the LV.\n\n  * In RV volume overload, the IVS flattens in diastole, and normalizes in systole, as the increase in systolic LV pressure pushes the IVS back toward the RV.\n\n  * The dilation, which is primarily due to RV pressure overload flattens the IVS in both systole and diastole quantified using RVEI, which is the ratio of the LV orthogonal diameters in both diastole and systole.\n\n  * RVEI that remains >1 in both diastole and systole is consistent with pressure overload.\n\n  * High RVEI only confined to diastole, normalizing to ?1 in systole, is consistent with RV volume overload.\n\n6.\n\nComprehensive Doppler assessment of right ventricular function (Fig. 15.6).\n\nFig. 15.6\n\nEchocardiography showed a left ventricular ejection fraction ?= ?34%, and a mildly dilated RV (4.3 cm) which was moderately depressed in function. To calculate RV CO using the RV outflow tract (RVOT) diameter and time velocity integral (TVI), RV cross-sectional area ?= ?(2.6\/2)2 * 3.14 ?= ?5.3 cm2, and RVOT TVI ?= ?8.9 cm, resulting in RV SV ?= ?(5.3 cm2 * 8.9 cm) ?= ?47 mL. Therefore, CO ?= ?47 mL * 95 bpm ?= ?4.47 L\/min, and RV CI = 2.2 L\/min\/m2, indicative of significantly depressed RV function\n\n  * The parameters assessed include Right ventricle (RV) stroke volume (SV), cardiac output (CO), cardiac index (CI), tricuspid regurgitation (TR) velocity, and right atrial pressure (RAP).\n\n  * Normal RV CO is >4.0 L\/min (though this measure is very heart rate-dependent), and normal RV CI is >2.5 L\/min\/m2.\n\n7.\n\nComprehensive Doppler assessment of right ventricular (RV) hemodynamics in a patient with cor pulmonale (Fig. 15.7).\n\nFig. 15.7\n\nComprehensive Doppler assessment of right ventricular (RV) hemodynamics in a patient with cor pulmonale. This 36-year-old woman with primary pulmonary hypertension presented for routine follow-up. On echocardiography, she had a severely enlarged right ventricle (RV) with severely depressed function; the left ventricle (LV) is compressed by the RV. Top left panel shows moderate-to-severe tricuspid regurgitation (TR) due to incomplete leaflet coaptation from RV dilatation. Continuous wave Doppler (top right) shows a dense TR signal with a peak gradient of 62 mmHg. Bottom right panel shows that right atrial pressure is significantly elevated due to a dilated inferior vena cava (>2.0 cm, green arrow), with significant TR causing some reflux into the hepatic veins (white arrow), giving an RAP estimate of 20 mmHg. Therefore, estimated pulmonary artery systolic pressure (PASP) = 62 ?+ ?20 = 82 mmHg. Bottom left panel shows use of the pulmonic regurgitation (PR) signal for estimation of RV end-diastolic pressure (EDP), equivalent to PAEDP in patients without pulmonary outflow tract obstruction. Therefore, estimated PAEDP =24 ?+ ?20 mmHg (RA pressure estimate) ?= ?44 mmHg. Overall, PA pressure can be estimated at 82\/44 mmHg, consistent with severe pulmonary hypertension\n\n8.\n\nDemonstration of right ventricular (RV) systolic dysfunction by tissue Doppler (TD) imaging (Fig. 15.8).\n\nFig. 15.8\n\nDemonstration of right ventricular (RV) systolic dysfunction by tissue Doppler (TD) imaging. This 53-year-old man with atrial fibrillation presented with increasing dyspnea. Echocardiography revealed a severely dilated RV with a diastolic area (top left) of 41 cm2, a systolic area (bottom left) = 24 cm2 and fractional area change ?= ?41%, suggesting normal RV systolic function. However, tissue Doppler (TD) imaging (top right) reveals a systolic annular velocity at the lateral tricuspid annulus = 5.2 cm\/s, consistent with significant RV contractile impairment (Normal TD RV systolic annular velocity is >8 cm\/s using color Doppler-derived TD imaging). Therefore, in the setting of severe TR, although RV FAC is within normal, TD imaging demonstrates evidence of RV systolic dysfunction (see Fig. 15.10)\n\n9.\n\nAdvanced echocardiographic variables in a patient with a normal heart (Fig. 15.9).\n\nFig. 15.9\n\nAdvanced echocardiographic variables in a patient with a normal heart. Tissue Doppler (TD) imaging can be applied to measure the longitudinal velocity of the RV free wall (see Fig. 15.8) to assess function. In this normal subject, the TD systolic annular velocity was 10.1 cm\/s, (normal >8 cm\/s using color-derived TD imaging). Using non-Doppler-based speckle tracking, strain imaging is applied to assess deformation of the RV myocardium. The peak strain in this subject was \u201321% (normal ?< ?\u201316%), while the rate of deformation, strain rate, was \u20131.2 s?1 (normal ?< ?\u20130.9 s?1). These advanced tissue Doppler and speckle-derived strain indices can help identify RV myocardial disease in patients with relatively normal \"conventional\" echocardiographic indices, such as RV fractional area change (see Figs. 15.8 and 15.10)\n\n  * Tissue Doppler (TD) imaging can be applied to measure the longitudinal velocity of the RV free wall to assess function.\n\n  * In normal subjects, the normal TD systolic annular velocity >8 cm\/s using color-derived TD imaging.\n\n  * Using non-Doppler based speckle tracking, strain imaging can be applied to assess deformation of the RV myocardium.\n\n  * The peak strain in normal subject is ?< ?\u201316%, while the rate of deformation, strain rate, is < \u20130.9 s?1.\n\n  * These advanced tissue Doppler and speckle-derived strain indices can help identify RV myocardial disease in patients with relatively normal \"conventional\" echocardiographic indices, such as RV fractional area change\n\n10.\n\nAdvanced right ventricular (RV) echocardiographic variables in a patient with severe tricuspid regurgitation and atrial fibrillation (Fig. 15.10).\n\nFig. 15.10\n\nAdvanced right ventricular (RV) echocardiographic variables in a patient with severe tricuspid regurgitation and atrial fibrillation. This 53-year-old patient (same patient as in Fig. 15.8) underwent further advanced RV quantification. The right ventricle (RV) was moderately to severely dilated (5.3 cm) with RV fractional area change (FAC) = 41% (normal >35%). Tissue Doppler (TD) imaging revealed a systolic annular velocity ?= ?5.2 cm\/s (normal >8 cm\/s using color Doppler-derived TD imaging), indicating depressed systolic longitudinal velocity. Speckle-based strain imaging revealed a peak strain of \u201312% in the RV free wall (normal ?< ?\u201316%), and peak systolic strain rate of ?= ?\u20130.6 s?1 (normal ?< ?\u20130.9%), indicating depressed systolic RV myocardial deformation. Therefore, these TD and speckle-based strain variables point to depressed systolic function in this patient with severe tricuspid regurgitation, despite normal \"conventional\" echocardiographic variables such as FAC\n\n  * Tissue Doppler and speckle-based strain variables can point to depressed systolic function patients with severe tricuspid regurgitation, despite normal \"conventional\" echocardiographic variables such as FAC.\n\n11.\n\nThree-dimensional echocardiography for the assessment of right ventricular (RV) function in a patient with severe primary pulmonary hypertension (Fig. 15.11).\n\nFig. 15.11\n\nThree-dimensional echocardiography for the assessment of right ventricular (RV) function in a patient with severe primary pulmonary hypertension. Full volume acquisition of the heart over six or seven cardiac cycles can produce large volume images of the heart \u2013 which can readily be cropped and rotating in multiple dimensions \u2013 allowing superior morphological and functional diagnosis. The top image depicts the severely dilated RV during diastole, with the tricuspid valve (TV) open (white arrow). The bottom image shows the RV during systole, with the TV closed (yellow arrow), and little change in RV volume compared to diastole, indicative of poor RV systolic function. Note the severely dilated RV and RA compressing the small left ventricle (LV) and left atrium (LA), respectively. The moderator band is also well seen (red arrow)\n\n  * Full volume acquisition of the heart over six or seven cardiac cycles can produce large volume images of the heart, which can readily be cropped, and rotating in multiple dimensions allows superior morphological and functional diagnosis.\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_16\n\n# 16. Aortic Valve Stenosis and Regurgitation\n\nVuyisile T. Nkomo1\n\n(1)\n\nDivision of Cardiovascular Diseases and Internal Medicine, Mayo Clinic, Rochester, MN, USA\n\nVuyisile T. Nkomo\n\nEmail: nkomo.vuyisile@mayo.edu\n\nFig. 16.1a and b\n\nFig. 16.1\n\nMeasurement and calculation of AVA using the continuity equation. (a) Formulas, (AV aortic valve, AVA aortic valve area, LVOT left ventricular outflow tract, A left ventricular outflow tract area, TVI time velocity integral), (b) Example using time velocity integral (TVI)\n\nFig. 16.2a\u2013h\n\nFig. 16.2\n\nComparison of jet profiles from aortic stenosis, mitral regurgitation, and dynamic left ventricular outflow tract obstruction as in hypertrophic obstructive cardiomyopathy. The signal of aortic stenosis (a and b) has a shorter ejection time than that of mitral regurgitation and the mitral regurgitation signal extends beyond ejection and culminates in the mitral inflow signal during onset of diastole (c and d). The duration and peak velocity of aortic stenosis is shorter than that of MR. Dynamic left ventricular outflow tract obstruction may absent at rest (e) and brought on by Valsalva maneuver or amyl nitrite (f and g). Mitral regurgitation is frequently associated with dynamic LVOT obstruction and mitral regurgitation signal may contaminate the signal from LVOT (h). The velocity from mitral regurgitation will always be higher than the velocity from dynamic LVOT obstruction. (a) Aortic stenosis continuous wave signal with guided probe. (b) Aortic stenosis continuous wave signal with non-guided probe. (c) Mitral regurgitation continuous wave signal (arrow) with guided probe. (d) Mitral regurgitation continuous wave signal (arrow) with non-guided probe with mitral inflow signal above baseline (arrowhead) consistent with presence of concomitant mitral stenosis. (e) LVOT continuous signal without dynamic LVOT obstruction at rest. (f) Dynamic LVOT obstruction with Valsalva maneuver. (g) Dynamic LVOT obstruction with amyl nitrite. (h) Dynamic LVOT obstruction signal contaminated with MR signal\n\nFig. 16.3a\u2013c\n\nFig. 16.3\n\n(a\u2013c) Low gradient, low output aortic stenosis and evaluation\n\nTables 16.1\u201316.4\n\nTable 16.1\n\nEtiologies of stenosis and associated lesions\n\nAortic stenosis | \u2022 Valvular\n\n---|---\n\n\u2022 Subvalvular\n\n\u2022 Supravalvular\n\nCauses of aortic valve stenosis | \u2022 Valve calcification\n\n\u2022 Congenital malformation\n\n\u2022 Rheumatic disease\n\n\u2022 Radiation-induced VHD\n\n\u2022 Mucopolysaccharidoses\n\nUnicuspid aortic valve | \u2022 Fatal aortic stenosis in children under 1 year\n\n\u2022 Rare in adults ?0.02%\n\n\u2022 Uni- or acommissural\n\n\u2022 Causes aortic stenosis\n\n\u2022 Associated with aorta dilatation\n\nBicuspid aortic valve | \u2022 1\u20132% of general population\n\n\u2022 From fusion of two cusps\n\n\u2022 Aortic stenosis\/regurgitation\n\n\u2022 Infective endocarditis\n\n\u2022 Sinus of Valsalva aneurysm\n\n\u2022 Ascending aorta dilatation\/aneurysm\n\n\u2022 Aortic coarctation\n\nSubvalvular aortic stenosis | \u2022 Discrete fibrous membrane (90%)\n\n\u2022 Muscular narrowing\n\n\u2022 Prevalence ?6.5% in adults CHD\n\n\u2022 Iatrogenic after VSD patch repair\n\n\u2022 Associated aortic regurgitation\n\nSupravalvular aortic stenosis | \u2022 Focal or diffuse narrowing starting at the sinotubular junction\n\n\u2022 Rare involvement of aortic arch and peripheral vessels\n\n\u2022 Frequently associated with Williams-Beuren syndrome (71%)\n\n\u2022 Aortic valve abnormalities 50% of patients (commonly BAV)\n\nTable 16.2\n\nGrading severity of aortic stenosis (ACC\/AHA guidelines 2006) | AVA (cm2) | Indexed AVA (cm2\/m2) | MG (mmHg) | Velocity (m\/s) | Velocity ratio (no units)\n\n---|---|---|---|---|---\n\nMild | >1.5 | >0.85 | <25 | <3 | >0.50\n\nModerate | 1\u20131.5 | 0.60\u20130.85 | 25\u201340 | 3\u20134 | 0.25\u20130.50\n\nSevere | <1 | <0.60 | >40 | >4 | <0.25\n\nTable 16.3\n\nEtiologies of aortic regurgitation (Circulation. 2006;114:422\u2013429)\n\nValve 46% | Nonvalve 54%\n\n---|---\n\n\u2022 Congenital malformation | \u2022 Aortic dissection\n\nBicuspid | \u2022 Marfan or forme fruste\n\nQuadricuspid | \u2022 Aortitis\n\nTricuspid | \u2022 Cause unclear (maybe associated with hypertension)\n\n\u2022 Endocarditis\n\n\u2022 Rheumatic\n\n\u2022Miscellaneous\n\nTable 16.4\n\nAssessment and grading of aortic regurgitation (American Society of Echocardiography Guidelines 2003) | Mild | Moderate | Severe\n\n---|---|---|---\n\nQualitative and semi-quantitative\n\nJet width in LVOT | <25% | 25\u201364% | ?65%\n\nJet area in LVOT | <5% | 5\u201359% | ?60%\n\nVena contracta width | <3 mm | 3\u20136 mm | >6 mm\n\nColor flow reversal in aorta (thoracic descending aorta) | Brief | Intermediate | Holodiastolic\n\nPressure half-time | ?650 ms | 280\u2013650 ms | <280 ms\n\nQualitative\n\nRegurgitant volume | <30 cc | 30\u201359 cc | ?60 cc\n\nEffective regurgitant orifice area (EROA) | <0.1 cm2 | 0.1\u20130.29 cm2 | ?0.30 cm2\n\nRegurgitant fraction | <30% | 30\u201349% | ?50%\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_17\n\n# 17. Mitral Prolapse Regurgitation and Mitral Stenosis\n\nNuasheen Akhter1  and Issam A. Mikati\n\n(1)\n\nDivision of Cardiology, The Feinberg School of Medicine, Chicago, IL, USA\n\nNuasheen Akhter\n\nEmail: n.akhter@northwestern.edu\n\nThe assessment of mitral regurgitation requires an understanding of the complex anatomy of the mitral valve. The mitral valve complex includes the leaflets, annulus, chordae tendineae, papillary muscles, underlying myocardium, left atrium, and left ventricle.1 Understanding the interaction of these elements is crucial for determining the mechanism of mitral regurgitation. To facilitate communication, the following classification of MR mechanism is widely used. Type I refers to structurally normal valve leaflet mobility, for example, dilated annulus, perforation, or cleft. Type II refers to valves with excessive leaflet mobility as in patients with mitral valve prolapse. Type III refers to restricted leaflet motion. Type IIIa refers to systolic and diastolic restriction of leaflet motion as in cases of rheumatic MR or MR secondary to radiation, connective tissue disorders, or degenerative calcific disease. Type IIIb refers to systolic restriction of leaflet motion as in ischemic MR.2\n\nMitral valve prolapse is said to occur when there is valve leaflet prolapse of 2 mm or more above the mitral annulus in the long-axis parasternal view.3 MVP occurs as a clinical entity with or without thickening (5 mm or greater, measured during diastasis) and with or without MR. Prediction of reparability of the valve is possible with transthoracic echocardiography TTE (97% of cases) and transesophageal echocardiography (TEE).4\n\nIdentification of involved scallops is possible by TEE5 (see Fig. 17.5). Incorporation of jet direction information helps in identification of involved leaflet. The jet is directed in the opposite direction of the prolapsing leaflet. In cases of restriction of leaflet motion the jet moves in the direction of the involved leaflet (which is usually the posterior leaflet). Three dimensional TTE and 2D TEE were more accurate (90% and 87%) than 2D TTE (77%) in identifying involved scallop.6 A thorough exam is essential to identify all the pathologies; in one report 5% of patients had more than one mechanism of MR.7 TEE was able to identify risk factors for post-repair complications such as systolic anterior motion of the anterior leaflet of the mitral valve (tall posterior leaflet > 1.9 cm, posterior leaflet\/anterior leaflet <1, and coaptation distance to septum distance <2.5).8,9\n\nFig. 17.1\n\nNormal MV anatomy.1 The mitral valve MV is composed of several elements. The annulus has a saddle shape in three dimensions. It integrates with the fibrous continuity of the mitral valve anterior leaflet and anterior annulus. The intertrigonal distance is fixed. The anterior leaflet has 60% MV area but subtends only a third of the circumference of the MV. An area of coaptation of 3\u20135 mm is expected normally. Each leaflet is divided into three scallops that are labeled 1\u20133 going from lateral to medial: lateral (A1, P1), middle (A2, P2), and medial (A3, P3) (See Fig. 17.6). Primary chordae extend from papillary muscles to leaflet edge. Secondary chordae extend from papillary muscle to mid undersurface of leaflets. Tertiary chordae extend from posterior LV wall to base of posterior leaflet. Chords from anterolateral papillary muscle supply lateral part of both anterior and posterior leaflets. Similarly chords from posteromedial papillary muscle supply medial part of both leaflets\n\nFig. 17.2\n\nMechanisms of mitral regurgitation.2 A commonly used system for classification of MR was developed by Dr Alain Carpentier. It relates function\/motion of the leaflets to structure. It facilitates communication between cardiologist and surgeon so that feasibility and type of repair can be determined. Three types have been identified. Type I is characterized by normal leaflet length and motion. Examples, annular dilation or leaflet perforation, such as with endocarditis. Type II MR is caused by excessive leaflet motion (most commonly prolapse or flail) usually from myxomatous disease, or by papillary muscle rupture or elongation (See Fig. 17.5) Type III MR is caused by restricted leaflet motion. Type IIIa is classically caused by rheumatic disease with subvalvular involvement (See Fig. 17.12b). Type IIIb is usually caused by ischemic or idiopathic cardiomyopathy with ventricular dilation causing tethering and restricted motion of the leaflets\n\nFig. 17.3\n\nFluid dynamics and measures of mitral regurgitation severity.24 (Adapted from www.\u200bechoincontext.\u200bcom\/\u200bimages\/\u200banat_\u200bpisa003.\u200bjpg). Several concepts of fluid dynamics that apply to passage of fluid through a narrow orifice can be used to assess severity of mitral regurgitation. As fluid approaches a narrow orifice it accelerates in concentric hemispheres or shells. Fluid particles at the same distance from the orifice have the same velocity. This is the underlying principle for the PISA method. As the fluid passes through the narrow orifice it accelerates. The wider the neck of that flow (vena contracta) the more severe is the MR (see Fig. 17.10).Chronic volume overload of the left atrium LA results in LA dilatation and increase in LA pressures. This results in systolic flow reversal in the pulmonary veins (See Fig. 17.11b)\n\nFig. 17.4\n\nIdentification of scallop anatomy of mitral valve by transthoracic echo.4 Four imaging planes to assess the precise localization of prolapsed or flail segments. (a) Inter-commissural plane assessing the continuity of the commissural areas. This is equivalent to vertical midesophageal TEE view with medial angulation (See Fig. 17.5c). (b) Parasternal short-axis view showing the anterior leaflet (A1, A2, and A3) and the three scallops of the posterior leaflet (P1, P2, and P3). This is equivalent to horizontal transgastric TEE view (See Fig. 17.5a). (c) Parasternal long-axis view showing the middle segments of anterior (A2) and posterior (P2) leaflets. This is equivalent to vertical midesophageal TEE view 120\u00b0 with medial angulation (See Fig. 17.5d). (d) Apical four-chamber view showing the anterior para-commissural zone (between P1 and P2). This is equivalent to horizontal midesophageal TEE (See Fig. 17.5c). ANT, anterior; AO, descending aorta; AV, aortic valve; LA, left atrium; LAA, left atrial appendage; LV, left ventricle; POST, posterior; PV, pulmonary vein; RV, right ventricle; TV, tricuspid valve\n\nFig. 17.5\n\nIdentification of scallop anatomy of mitral valve by transesophageal echo.25 (a) Foster et al.25 demonstrated that a systematic approach using TEE is successful in identifying the scallop involved with prolapse\/flail. Left Panel demonstrates the nomenclature. The lateral, middle, and medial scallops are numbered as follows: A1, A2, A3 and P1, P2, P3 for anterior and posterior leaflets, respectively. The cartoon which is modified from Foster25 demonstrates the important anatomic relationships. Left atrial appendage is closest to lateral scallops (A1, P1). The aortic valve is closest to medial scallops (A3 and P3). The middle panel demonstrates a basal transverse transgastric TEE view. This is obtained by withdrawing the probe to the proximal part of the stomach. It is difficult to visualize prolapsing\/flail segments since the ventricular surface of the mitral valve is seen. It does allow identification of elongated\/redundant scallops. Color Doppler in this view allows identification of the origin of the MR jet which also localizes pathology. In right panel color is seen to originate medially in A3, P3 area. (b) Left panel (Modified from Foster25) demonstrates that the lateral scallops of the mitral valve leaflets are cranial to middle and medial scallops. Therefore, when TEE probe is in upper esophagus (green plane) one can visualize A1 and P1. The LVOT and aortic valve are important landmarks that help characterize this location. Pushing the probe in (red plane) LVOT and aortic valve are no longer visualized and A2 and P2 are now seen. Pushing the probe in further (purple plane) A3 and P3 can be examined. An alternative method would be to start at the midesophageal point (red plane) and obtain cranial plane (green) to visualize A1, P1 by anteflexion. Retroflexion of probe provides caudal view (purple plane) and therefore A3, P3. Other planes are needed to confirm findings (see Fig. 17.5c) as individual anatomical variations are common. (c) The mid commissural view (60\u00b0) can be used to confirm findings from other views (Modified from Foster25). Clockwise rotation provides a medial plane (green) cut through anterior leaflet. This allows visualization of A1, A2, and A3 (from left to right of the screen). Slight counterclockwise rotation (red plane) allows visualization of P1, A2, and P3. Further counterclockwise rotation (purple plane) provides a view through a lateral plane demonstrating P1, P2, and P3. The middle panel (green frame) corresponds to a medial vertical cut through mitral valve demonstrating A1 prolapse. The panel on the right (red frame) demonstrates a flail (yellow arrow) medial scallop of posterior leaflet (P3). (d) Other views (30\u00b0, 45\u00b0, and 120\u00b0) can be obtained to confirm findings (See Fig. 17.7)\n\nFig. 17.6\n\nMitral valve anatomy using real time 3D echocardiography. The 3D image is usually displayed in the surgeon's view. This mimics the orientation of cardiac structures in the surgical field as seen by the surgeon. This can be derived from the reference image by rotating the image by 45\u00b0 then flipping it (Fig. 17.6a). The full volume is acquired with 4 consecutive heart beats through the MV at a medium density setting in the 2D 0\u00b0 TEE view, which encompassed the MV annulus and the aortic valve. The next step is image orientation. The MV is displayed from the atrial side with the aortic valve at the 12 o'clock position. In this orientation, the anterior MV leaflet is next to the aortic valve and the posterior MV leaflet is opposite. The scallops are shown from lateral\/left (A1\/P1) to medial\/right (A3\/P3). The final step is to optimize the image by cropping the blue short axis plane down toward the MV and decreasing the gain setting. The resulting image is shown in Fig. 17.6b (Image courtesy of Dr Vera Rigolin)\n\nFig. 17.7\n\nComprehensive assessment of mitral valve prolapse and flail using 2D TTE, TEE, and 3D echocardiography. (Classic P2 Flail) (a) 2D 0\u00b0 TEE view shows a flail P2 scallop with a ruptured chordae tendinae (arrow). (b) 2D 120\u00b0 TEE end-systolic view further demonstrates P2 prolapse. (c) 3D TEE live zoom acquisition in from the atrial side with the aortic valve at the 12 o'clock position displays P2 prolapse (arrow) with a torn chordae tendinae (asterisk). (d) A 3D rendering of the P2 prolapse can be created with Philips Mitral Valve Quantification (MVQ) software. The area coded red indicates part of the mitral valve that is breaking the annular plane and encroaching on the atrial side. This clearly demonstrates that prolapse is occurring in middle posterior scallop (P2). The blue color indicates areas of leaflets that are on ventricular side of annular plane and therefore have restricted motion. This facilitates identification of the mechanisms involved in pathophysiology of MR (still a research tool at this point)\n\nFig. 17.8\n\nMultiple mechanisms of mitral regurgitation examined by multiple modalities. Although multiple mechanisms of MR are not very frequent7 they need to be recognized in advance so that complete repair is done to avoid multiple pump runs and residual MR (A2 Perforation, P2 Prolapse). (a) This is a 2D 0\u00b0 esophageal TEE view, which demonstrates an A2 aneurysm and perforation (arrow) and (b), which shows the P2 prolapse with adjacent flail chordae tendinae. (c) The full volume 3D TEE en face view of the MV from the left atrial perspective displays the A2 perforation (black asterisk), P2 prolapse (white asterisk), and the flail chordae tendinae (arrow). 3D echocardiography allows visualization in one view of multiple pathologies that are physically in different planes. (d) The presence of multiple MR jets is a clue that multiple pathologies exist and should be looked for. This 2D 0\u00b0 esophageal TEE view with color Doppler confirms the two jets (arrows) of mitral regurgitation, anteriorly from the P2 flail and posteriorly from the A2 perforation\n\nFig. 17.9\n\nAssessment of mitral regurgitation by color Doppler jet area. (a) Color Doppler interrogation of severity of MR can be done semiquantitatively by tracing the area of regurgitant signal or, more commonly, qualitatively by visually assessing the size of the jet and comparing it to left atrial area. Relying on this as a sole criterion for assessment of severity of MR is not recommended. Color display is a velocity map so a high velocity jet may generate same area of turbulence as a larger slower moving jet. This method is very sensitive to velocity map, gain, and filter settings. Setting the Nyquist limit too low will exaggerate severity of MR. Proper Nyquist limit is 50\u201360 cm\/s. High gains as is evidenced in Fig. 17.9a by presence of color pixels overlying LA wall will exaggerate severity of MR. Color Doppler will overestimate severity of central jet similar to one seen in Fig. 17.9a. (b) On the other hand eccentric jet (similar to anteromedial jet seen in Fig. 17.9b will be underestimated by color Doppler secondary to entrainment (Coanda effect)\n\nFig. 17.10\n\nVena contracta in grading severity of mitral regurgitation. The MR jet as it passes through the regurgitant orifice narrows down before the jet expands in the LA. The width of that neck correlates with severity of MR. Vena contracta width of 8 mm is consistent with severe MR as in Fig. 17.10. The advantage of this method is that it is simple. Its greatest utility is in identifying patients with MR on either end of the spectrum (mild or severe; it is not very useful in patients with moderate degrees of MR). It works in patient with central as well as eccentric jet but not in patients with multiple jets. Technical recommendations: (1) Parasternal views are preferable. Apical two chamber views are to be avoided. (2) Zoomed views should be used. The zoom box should be as small as possible to maximize magnification of the image and minimize measurement error. This will also image line density and frame rate. (3) The position of the focal zone should be placed at the level of the vena contracta. (4) The color sector should be narrowed again if possible and the pulse repetition frequency (color scale) increased to provide the maximum frame rate. (5) Then the image needs to be frozen. Scrolling back through the cineloop will locate the best representative frame for measurement26\n\nFig. 17.11\n\nQuantitative measures of mitral regurgitation. (a) Mitral regurgitant volume by continuity equation. Advantages of the continuity equation method are that it applies for MR with eccentric jet, late systolic MR jets or even MR with multiple jets. The disadvantages are that numerous measurements are needed. Measurement of annulus is squared resulting in doubling of the relative error. In most cases of severe MR the mitral annulus approaches circular shape. Some practitioners average the four chamber and two chamber diameters. Regurgitant Fraction = (FlowMitral \u2013 FlowLVOT)\/FlowMitral; FlowMitral = (?\/4) * D2 * TVIMitral; FlowAortic = (?\/4) * D2 * TVIAortic; DMitral = 33 mm (4 Chamber) 31 mm (2 Chamber); TVIMitral = 16 cm; DAortic = 20 mm (4 Chamber); TVIAortic = 20 cm; FlowMitral = 0.785 * 10.24 * 16 = 128 mL; FlowAortic = 0.785 * 4 * 20 = 62 mL; Regurgitant volume = FlowMitral \u2013 FlowLVOT = 128\u201362 = 66 mL; Regurgitant Fraction = 66\/128 = 52%. (b) Effective regurgitant orifice area by proximal isovelocity area (PISA) method for grading severity of mitral regurgitation. PISA is based on the continuity principle. When flow converges toward a restrictive orifice, it forms hemispheric isovelocity shells which can be used to calculate an effective regurgitant orifice (ERO) and a regurgitant volume (RV). (A) The aliasing velocity is decreased to approximately 30 cm\/s to create the isovelocity shell. The shell area is 2?(R)2. The (R) is the radius of the shell. The shell area x the aliasing velocity is the flow on the proximal side of the MV. In this example, R = 0.8 cm and the aliasing velocity = 30 cm\/s. The flow is 2 \u00d7 ? \u00d7 (0.815)2(30) = 125 cm3. According to the continuity principle, the flow on the proximal side of the MV = the flow on the distal side of the MV. The distal flow is the area of the effective regurgitant orifice \u00d7 the maximum velocity of the mitral regurgitation obtained by CW Doppler. (B) In this example, the ERO = 125 cm3\/435 cm\/s = 0.29 cm2. The RV is then calculated by the ERO \u00d7 mitral regurgitation VTI. In this example, the RV is 0.29 cm2 \u00d7 146 cm = 43 cm3. This technique is not optimal for eccentric jets, MR with multiple jets, late systolic MR and for patients with deformed valve in whom a proper convergence zone hemisphere cannot be obtained. (C) The PW Doppler of the pulmonary vein demonstrates systolic flow reversal which suggests severe mitral regurgitation\n\nAssessment of the severity of MR should incorporate indirect measures of MR severity (such as LA size, pulmonary venous flow) as well as direct measures for optimal accuracy; reliance on one parameter can often lead to errors. One should question the presence of chronic severe MR if left atrial size is normal or if the mitral inflow shows A-wave dominance. Semiquantitative measures of MR severity (such as jet area by color Doppler, vena contracta, intensity of continuous wave Doppler signal) offer a quick and time-efficient assessment (Table 17.1). Color Doppler techniques assume holosystolic flow; this assumption does not always hold especially in conditions such as mitral valve prolapse and hypertrophic cardiomyopathy in which MR may be a late systolic event. Reliance on color Doppler in these instances may result in overestimation of MR severity.\n\nQuantitative methods improve accuracy and have been shown to predict prognosis even in asymptomatic patients.10 These include regurgitant volume and fraction as determined by continuity equation and effective regurgitant orifice area (EROA) by proximal isovelocity area (PISA) method. Regurgitant volume can also be derived from EROA using the PISA method. A combination of these parameters is used to grade MR severity as mild, moderate, or severe (Table 17.2 and 17.3).11\n\nThe 2D assessment of the stenotic mitral valve helps in diagnosis and management. Most common etiology is still rheumatic heart disease resulting in the typical appearance of hockey stick deformity of the anterior leaflet of the mitral valve (secondary to preferential involvement of leaflet tips) and chordal shortening and fusion (see Fig. 17.12a.)\n\nFig. 17.12\n\nEchocardiographic features of mitral stenosis. (a) Rheumatic mitral stenosis. This figure demonstrates typical findings of rheumatic mitral stenosis (MS). The left panel shows hockey stick deformity of the anterior leaflet of the mitral valve in diastole. This appearance is a manifestation of the preferential involvement of the tips of the leaflets with the inflammatory\/fibrosis process. This leads to relatively preserved mobility of the body with restricted mobility of the tip and the typical appearance. The right panel is a TEE from a different patient. The green arrow shows severe thickening and shortening of the chordae tendinae. Another common feature of rheumatic mitral stenosis is commissural fusion. The presence of commissural fusion may be a good predictor of who will respond to balloon valvuloplasty. (b) Other etiologies of mitral stenosis. Other etiologies of mitral stenosis include radiation and connective tissue disorders. Systemic lupus erythematosus causes diffuse thickening of the body of the valve leaflet (see TEE in Fig. 17.12b) as opposed to the tips in rheumatic heart disease\n\nThe most commonly used score to predict the potential success of balloon valvotomy is the Wilkins score12 (see Table 17.4). Others favor scores based on commissural anatomy.13 No clear superiority of one technique over another has ever been established. Three Dimensional echo is probably superior for visualization of commissures.14\n\nTable 17.1\n\nQualitative and quantitative parameters useful in grading mitral regurgitation severity11 | Mild | Moderate | Severe\n\n---|---|---|---\n\nStructural parameters\n\nLA size | Normala | Normal or dilated | Usually dilatedb\n\nLV size | Normala | Normal or dilated | Usually dilatedb\n\nMitral leaflets or support apparatus | Normal or abnormal | Normal or abnormal | Abnormal\/flail leaflet\/ruptured papillary muscle\n\nDoppler parameters\n\nColor flow jet areac | Small, central jet (usually < 4 cm2 or <20% of LA area) | Variable | Large central jet (usually > 10 cm2 or >40% of LA area) or variable size wall-impinging jet swirling in LA\n\nMitral inflow \u2013 PW | A wave dominantd | Variable | E wave dominantd\n\nJet density \u2013 CW | Incomplete or faint | Dense | Dense\n\nJet contour \u2013 CW | Parabolic | Usually parabolic | Early peaking \u2013 triangular\n\nPulmonary vain flow | Systolic dominancee | Systolic bluntinge | Systolic flow reversalf\n\nCW, continuous wave; LA, left atrium; LV, left ventricle; PW, pulsed wave; RF, regurgitant fraction\n\na Unless there are other reasons for LA or LV dilation. Normal 2D measurements; LV minor axis ? 82 mL\/m2, maximal LA antero-posterior diameter ? 2 cm\/m2, maximal LA volume ? 36 mL\/m2 (2, 33, 35)\n\nb Exception: acute mitral regurgitation\n\nc At a Nyquist limit of 50\u201360 cm\/s\n\nd Usually above 50 years of age or in conditions of impaired relaxation, in the absence of mitral stenosis or other causes of elevated LA pressure\n\ne Systolic dominance or blunting or both\n\nf Pulmonary venous systolic flow reversal is specific but not sensitive for severe MR\n\nTable 17.2\n\nQuantitative measures of grading MR severity11 | Formula | Advantage | Potential Pitfall\n\n---|---|---|---\n\nEROA (cm2) PISA | 6.28 \u00d7 r2 \u00d7 v Nyq\/v CW | Simple, few measurements | Eccentric jets; Non-hemispheric zone of acceleration; Multiple jets\n\nRegurgitant vol. (mL\/beat) PISA method | EROA \u00d7 TVICW | Simple, few measurements | Eccentric jets; multiple steps\n\nContinuity equation method | FlowMitral \u2013 FlowLVOT | Multiple jets, eccentric jets | AF\n\nRegurgitant fraction (%) | (FlowMitral \u2013 Flow LVOT)\/FlowMitral\n\n|  |\n\nEccentric jets; multiple steps, AF\n\nEROA, effective regurgitant orifice area; PISA, proximal isovelocity surface area; V Nyq, velocity of Nyquist limit; v CW , peak velocity of MR continuous wave; DMitral, diameter of mitral annulus; DLVOT, diameter of LVOT annulus; AF, atrial fibrillation;\n\nTable 17.3\n\nQualitative and quantitative parameters useful in grading mitral regurgitation severity11 | Mild | Moderate | Severe\n\n---|---|---|---\n\nQuantitative parametersa\n\n|  |  |\n\nVC width (cm) | <0.3 | 0.3\u20130.69 | ?0.7\n\nR Vol (mL\/beat) | <30 | 30\u201344 | 45\u201359 | ?60\n\nRF (%) | <30 | 30\u201339 | 40\u201349 | ?50\n\nEROA (cm2) | <0.20 | 0.20\u20130.29 | 0.30\u20130.39 | ?0.40\n\nEROA, effective regurgitant orifice area; LV, left ventricle; PW, pulsed wave; RF, regurgitant \u00adfraction; R Vol, regurgitant volume; VC, vena contracta\n\na Quantitative parameters can help subclassify the moderate regurgitant group into mild-to-moderate and moderate-to-severe\n\nTable 17.4\n\nAssessment of mitral valve anatomy according to the Wilkins score13\n\nGrade | Mobility | Thickening | Calcification | Subvalvular thickening\n\n---|---|---|---|---\n\n1 | Highly mobile valve with only leaflet tips restricted | Leaflets near normal in thickness (4\u20135 mm) | A single area of increased echo brightness | Minimal thickening just below the mitral leaflets\n\n2 | Leaflet mid and base portions have normal mobility | Midleaflets normal, considerable thickening of margins (5\u20138 mm) | Scattered areas of brightness confined to leaflet margins | Thickening of chordal structures extending to one-third of the chordal length\n\n3 | Valve continues to move forward in diastole, mainly from the base | Thickening extending through the entire leaflet (5\u20138 mm) | Brightness extending into the mid-portions of the leaflets | Thickening extended to distal third of the chords\n\n4 | No or minimal forward movement of the leavlets in diastole | Considerable thickening of all leaflet tissue (>8\u201310 mm) | Extensive brightness throughout much of the leaflet tissue | Extensive thickening and shortening of all chordal structures extending down to the papillary muscles\n\nThe total score is the sum of the four items and ranges between 4 and 16\n\nTable 17.5\n\nRecommendations for classification of mitral stenosis severity13 | Mild | Moderate | Severe\n\n---|---|---|---\n\nSpecific findings\n\nValve area (cm2) | >1.5 | 1.0\u20131.5 | <1.0\n\nSupportive findings\n\nMean gradient (mmHg)a | <5 | 5\u201310 | >10\n\nPulmonary artery pressure (mmHg) | <30 | 30\u201350 | >50\n\na At heart rates between 60 and 80 bpm and in sinus rhythm\n\nMitral valve area (MVA) measurement by planimetry is considered the reference measurement for assessment of the severity of mitral stenosis.15,16 Optimization of the image is critical (see Fig. 17.13). Three-dimensional echocardiography facilitates this optimization especially for less experienced sonographers.17,18\n\nFig. 17.13\n\nAssessment of mitral stenosis by planimetry. (a) Planimetry by 2D echocardiography. Assessment of severity of mitral stenosis by direct planimetry of mitral valve opening is a Level Recommendation13. It is direct measurement that does not rely on hemodynamic factors like cardiac output. It can be done on transgastric short axis view (Fig. 17.11a) or parasternal short axis view. To optimize results: (1) a zoomed view of MV is done. (2) It is essential to use proper gains. Over gaining image will result in an area measurement that is smaller than the true mitral valve orifice. (3) A sweep from cranial to caudal helps to ensure the smallest area is obtained. This is usually (but not always) at the leaflet tips. (4) Measurement of area should be done at mid diastole. (5) It is essential to make sure that the plane of the orifice being planimetered is perpendicular to the long axis of LV. An oblique cut will overestimate the area and therefore underestimate severity of MS. (b) Planimetry by 3D echocardiography. The main advantage of 3D echocardiography is the ability to achieve a perpendicular en face cut plane of the mitral valve orifice. The measurements are more accurate when performed from the ventricular orientation. Three dimensional derived MVA had better correlation with area calculations derived using the Gorlin formula during cardiac catheterization than 2D and Doppler measurements, such as 2D planimetry, pressure half-time, and flow convergence. 3D measurements have lower intraobserver and interobserver variability. The ease of acquisition and online review of real-time 3D echocardiography facilitates immediate assessment of the mitral valve commissural splitting after percutaneous balloon mitral valvuloplasty in the cardiac catheterization laboratory. This is important since acute changes in compliance make pressure half-time less reliable27\n\nFig. 17.14\n\nDoppler assessment of mitral stenosis by pressure half-time method. (a) The pressure half-time PHT method derives from the principle that the decay of the diastolic pressure gradient between the LA and LV is inversely proportional to MVA and therefore is directly related to MS severity.20 The PHT is the time it takes for the pressure to reach half its original starting value (red line). MVA(cm2) = 220\/PHT (ms). Use of this method for assessment of MS receives a Level Recommendation from the American Society of Echocardiography.13 The advantage of this method is its simplicity. Inadequate recordings may be obtained when there is poor alignment with mitral inflow jet, incorrect gain and filter settings, in presence of aortic regurgitation. Clinical situations in which PHT method does not work well include elderly patients especially in the presence of significant diastolic dysfunction, atrial flutter, and tachycardia and in situations with rapid pressure or compliance changes such as after exercise or balloon valvuloplasty. When the continuous wave signal is not linear or is concave then PHT cannot be measured. (b) In some situations the pressure decay is bimodal. There is an early steeper (red line) and later shallower pressure decay (green line). In these situations the true PHT is reflected by the later slope28\n\nFig. 17.15\n\nDoppler assessment of mitral stenosis by mean gradient. The mean gradient is a Level Recommendation from the American Society of Echocardiography.13 The mean gradient depends on the cardiac output. This leads to underestimation of severity of mitral stenosis in patients with low stroke volume and LV dysfunction. Conversely high output states (like mitral regurgitation, anemia, and hyperthyroidism) will lead to overestimation of the severity of MS. Assessment of mean gradient is done through apical windows. Alignment with MR jet is critical. Color Doppler jet may help in patients in whom valve deformity results in eccentric jet. In patients with atrial fibrillation the mean gradient will vary with cycle length. After a short cycle the gradient is higher (Gradient after short cycle in Fig. 17.15b is 17 mmHg) and after long cycles the gradient is lower (Gradient after long cycle in Fig. 17.15b is 9 mmHg). In patients with atrial fibrillation averaging at least five beats is recommended\n\nFig. 17.16\n\nDoppler assessment of mitral stenosis by proximal isovelocity area method. The PISA method can be used to measure MVA. Unlike the case of MR the convergence does not approximate a hemisphere often because of the angle between mitral valve leaflets. This necessitates the correction factor without which flow and therefore MVA would be overestimated. ![\n$$ \\\\text{MVA}=\\(2\\\\times \\\\Pi \\\\times {\\\\text{r}}^{2}\\\\times {\\\\text{v}}_{\\\\text{Nyquist}}\\\\times \\[\\\\alpha \/180\\]\\)\/{\\\\text{v}}_{\\\\text{Mitral CW}}$$\n](A978-1-84996-151-6_17_Fig18_HTML.gif). For the patient in example 16 r = 1.1 cm; v Nyquist = 0.37 m\/s ;? = 120\u00b0; vMitral CW=1.9 m\/s. ![\n$$ \\\\text{MVA}=\\(2\\\\times 3.14\\\\times {\\(1.1\\)}^{2}\\\\times 0.37\\\\times \\[\\\\alpha 120\/180\\]\\)\/1.9\\\\text{MVA}=1.0{\\\\text{cm}}^{2}$$\n](A978-1-84996-151-6_17_Fig19_HTML.gif). Steps for optimal data collection for the PISA method include: (1) Zoom on mitral valve. (2) Shift baseline up in direction of flow \"toward the LV\" or up in case of TTE. The best Nyquist limit is the one that results in a hemispheric convergence zone. This usually is slightly higher than the Nyquist limit used for MR (35\u201340 cm\/s). (3) Radius is measured from the point of color change to leaflet tips. Leaflet tips may be identified using color suppress function. Measurement is usually made in mid diastole. (4) Angle is then measured. (5) Peak diastolic velocity is then measured\n\nMitral valve area can be obtained using the formula MVA = 220\/Pressure Half Time. This method is limited in patients with abnormal relaxation (severity of stenosis is overestimated19) and in patients with changing compliance (immediately post-valvotomy20,21)\n\nMVA can also be obtained using the continuity equation. This method suffers from the multiple measurements required with potentially compounding error.\n\nThe PISA method can be used to obtain MVA.22 This method is limited by technical demands.\n\nAnother useful measure of mitral stenosis is the mean gradient across the mitral valve (Table 17.5). This has been validated against the gradient obtained invasively in the cath lab using a transeptal approach.23 The heart rate at which the gradient is obtained should be reported consistently.\n\nReferences\n\n1.\n\nOtto CM. Clinical practice. Evaluation and management of chronic mitral regurgitation. N Engl J Med. 2001;345(10):740-746.PubMedCrossRef\n\n2.\n\nCohn LH, Edmunds LH. Cardiac surgery in the adult. 2nd ed. New York: McGraw-Hill; 2003.\n\n3.\n\nFreed LA, Levy D, Levine RA, et al. Prevalence and clinical outcome of mitral-valve \u00adprolapse. N Engl J Med. 1999;341(1):1-7.PubMedCrossRef\n\n4.\n\nMonin JL, Dehant P, Roiron C, et al. Functional assessment of mitral regurgitation by transthoracic echocardiography using standardized imaging planes diagnostic accuracy and \u00adoutcome implications. J Am Coll Cardiol. 2005;46(2):302-309.PubMedCrossRef\n\n5.\n\nEnriquez-Sarano M, Freeman WK, Tribouilloy CM, et al. Functional anatomy of mitral regurgitation: accuracy and outcome implications of transesophageal echocardiography. J Am Coll Cardiol. 1999;34(4):1129-1136.PubMedCrossRef00314-9)\n\n6.\n\nPepi M, Tamborini G, Maltagliati A, et al. Head-to-head comparison of two- and three-dimensional transthoracic and transesophageal echocardiography in the localization of mitral valve prolapse. J Am Coll Cardiol. 2006;48(12):2524.PubMedCrossRef\n\n7.\n\nStewart WJ, Currie PJ, Salcedo EE, et al. Evaluation of mitral leaflet motion by echocardiography and jet direction by Doppler color flow mapping to determine the mechanisms of mitral regurgitation. J Am Coll Cardiol. 1992;20(6):1353-1361.PubMedCrossRef90248-L)\n\n8.\n\nLee KS, Stewart WJ, Lever HM, Underwood PL, Cosgrove DM. Mechanism of outflow tract obstruction causing failed mitral valve repair. Anterior displacement of leaflet coaptation. Circulation. 1993;88(5 Pt 2):II24-II29.PubMed\n\n9.\n\nMaslow AD, Regan MM, Haering JM, Johnson RG, Levine RA. Echocardiographic predictors of left ventricular outflow tract obstruction and systolic anterior motion of the mitral valve after mitral valve reconstruction for myxomatous valve disease. J Am Coll Cardiol. 1999;34(7):2096-2104.PubMedCrossRef00464-7)\n\n10.\n\nEnriquez-Sarano M, Avierinos JF, Messika-Zeitoun D, et al. Quantitative determinants of the outcome of asymptomatic mitral regurgitation. N Engl J Med. 2005;352(9):875-883.PubMedCrossRef\n\n11.\n\nZoghbi WA, Enriquez-Sarano M, Foster E, et al. Recommendations for evaluation of the severity of native valvular regurgitation with two-dimensional and Doppler echocardiography. J Am Soc Echocardiogr. 2003;16(7):777-802.PubMedCrossRef00335-3)\n\n12.\n\nWilkins GT, Weyman AE, Abascal VM, Block PC, Palacios IF. Percutaneous balloon dilatation of the mitral valve: an analysis of echocardiographic variables related to outcome and the mechanism of dilatation. Br Heart J. 1988;60(4):299-308.PubMedCentralPubMedCrossRef\n\n13.\n\nBaumgartner H, Hung J, Bermejo J, et al. Echocardiographic assessment of valve stenosis: EAE\/ASE recommendations for clinical practice. J Am Soc Echocardiogr. 2009;22(1):1-23. quiz 101-102.PubMedCrossRef\n\n14.\n\nMessika-Zeitoun D, Brochet E, Holmin C, et al. Three-dimensional evaluation of the mitral valve area and commissural opening before and after percutaneous mitral commissurotomy in patients with mitral stenosis. Eur Heart J. 2007;28(1):72-79.PubMedCrossRef\n\n15.\n\nBonow RO, Carabello BA, Chatterjee K, et al. ACC\/AHA 2006 guidelines for the management of patients with valvular heart disease: a report of the American College of Cardiology\/American Heart Association Task Force on Practice Guidelines (writing Committee to Revise the 1998 guidelines for the management of patients with valvular heart disease) developed in collaboration with the Society of Cardiovascular Anesthesiologists endorsed by the Society for Cardiovascular Angiography and Interventions and the Society of Thoracic Surgeons. J Am Coll Cardiol. 2006;48(3):e1-e148.PubMedCrossRef\n\n16.\n\nVahanian A, Baumgartner H, Bax J, et al. Guidelines on the management of valvular heart disease: The Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology. Eur Heart J. 2007;28(2):230-268.PubMed\n\n17.\n\nZamorano J, Cordeiro P, Sugeng L, et al. Real-time three-dimensional echocardiography for rheumatic mitral valve stenosis evaluation: an accurate and novel approach. J Am Coll Cardiol. 2004;43(11):2091-2096.PubMedCrossRef\n\n18.\n\nSebag IA, Morgan JG, Handschumacher MD, et al. Usefulness of three-dimensionally guided assessment of mitral stenosis using matrix-array ultrasound. Am J Cardiol. 2005;96(8):1151-1156.PubMedCrossRef\n\n19.\n\nMessika-Zeitoun D, Meizels A, Cachier A, et al. Echocardiographic evaluation of the mitral valve area before and after percutaneous mitral commissurotomy: the pressure half-time method revisited. J Am Soc Echocardiogr. 2005;18(12):1409-1414.PubMedCrossRef\n\n20.\n\nThomas JD, Weyman AE. Doppler mitral pressure half-time: a clinical tool in search of theoretical justification. J Am Coll Cardiol. 1987;10(4):923-929.PubMedCrossRef80290-5)\n\n21.\n\nThomas JD, Weyman AE. Fluid dynamics model of mitral valve flow: description with in vitro validation. J Am Coll Cardiol. 1989;13(1):221-233.PubMedCrossRef90575-5)\n\n22.\n\nMessika-Zeitoun D, Fung Yiu S, Cormier B, et al. Sequential assessment of mitral valve area during diastole using colour M-mode flow convergence analysis: new insights into mitral stenosis physiology. Eur Heart J. 2003;24(13):1244-1253.PubMedCrossRef00208-2)\n\n23.\n\nNishimura RA, Rihal CS, Tajik AJ, Holmes DR Jr. Accurate measurement of the transmitral gradient in patients with mitral stenosis: a simultaneous catheterization and Doppler echocardiographic study. J Am Coll Cardiol. 1994;24(1):152-158.PubMedCrossRef90556-8)\n\n24.\n\nwww.\u200bechoincontext.\u200bcom\/\u200bimages\/\u200banat_\u200bpisa003\n\n25.\n\nFoster GP, Isselbacher EM, Rose GA, Torchiana DF, Akins CW, Picard MH. Accurate localization of mitral regurgitant defects using multiplane transesophageal echocardiography. Ann Thorac Surg. 1998;65(4):1025-1031.PubMedCrossRef00084-8)\n\n26.\n\nRoberts BJ, Grayburn PA. Color flow imaging of the vena contracta in mitral regurgitation: technical considerations. J Am Soc Echocardiogr. 2003;16:1002-1006.PubMedCrossRef00509-1)\n\n27.\n\nZamorano J, Perez de Isla L, Sugeng L, et al. Non-invasive assessment of mitral valve area during percutaneous balloon mitral valvuloplasty: role of real-time 3D echocardiography. Eur Heart J. 2004;25:2086-2091.PubMedCrossRef\n\n28.\n\nGonzalez MA, Child JS, Krivokapich J. Comparison of two-dimensional and Doppler echocardiography and intracardiac hemodynamics for quantification of mitral stenosis. Am J Cardiol. 1987;60:327-332.PubMedCrossRef90236-0)\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_18\n\n# 18. Right-Sided Heart Valves Assessment in Disease\n\nFarouk Mookadam1  , Julie A. Humphries, Sherif E. Moustafa and Tahlil A. Warsame\n\n(1)\n\nDepartment of Cardiology, Mayo College of Medicine, Scottsdale, AZ, USA\n\nFarouk Mookadam\n\nEmail: mookadam.farouk@mayo.edu\n\nTable 18.1\n\nEtiologies of pulmonic stenosis (PS) and associated cardiac abnormalities\n\nClassification | Causes | Associated cardiac anomalies\n\n---|---|---\n\nValvular | Congenital (most common) | Usually isolated\n\nMay also occur as part of more complex congenital lesions such as tetralogy of Fallot (TOF), complete atrioventricular canal, double outlet right ventricle, and univentricular heart\n\nPeripheral pulmonary artery stenosis may coexist with valvular pulmonary stenosis such as in Noonan's syndrome and Williams syndrome\n\n|\n\nAcquired\n\nObstructive tumors\n\nCarcinoid heart disease\n\nRheumatic heart disease | \u2013\n\nSubvalvular (infundibular) | Congenital | Isolated is rare\n\nUsually associated with a large ventricular septal defect (VSD), as seen in TOF\n\n|\n\nAcquired\n\nDouble chamber right ventricle\n\nJet lesion produced by the VSD\n\nSevere right ventricular hypertrophy\n\nIatrogenic: prior surgery or intervention\n\nHypertrophic or infiltrative processes such as hypertrophic obstructive cardiomyopathy or glycogen storage disorders\n\nCompression from a tumor or vascular structure | \u2013\n\nSupravalvular | Congenital | Isolated or in association with other cardiac anomalies\n\nSingle, involving the main pulmonary artery or either of its branches, or multiple\n\nCommon associated defects are pulmonary valve stenosis, VSD, and TOF\n\nPeripheral pulmonary artery stenosis is often seen in association with congenital syndromes, such as congenital rubella syndrome, Williams' syndrome, Noonan's syndrome, Alagille's syndrome, Ehlers\u2013Danlos syndrome, and Silver\u2013Russell syndrome\n\n|\n\nAcquired\n\nIatrogenic: prior surgery or intervention\n\nCompression from a tumor or vascular structure\n\nIsolated pulmonary artery stenosis | \u2013\n\nTable 18.2\n\nDoppler parameters used in grading pulmonary regurgitation [1]\n\nParameter | Mild | Moderate | Severe\n\n---|---|---|---\n\nPulmonic valve | Normal | Normal or abnormal | Abnormal\n\nRV size | Normala | Normal or dilated | Dilated\n\nJet size by color Dopplerc | Thin (usually <10 mm in length) with a narrow origin | Intermediate | Usually large, with a wide origin\n\nMay be brief in duration\n\nJet density and deceleration rate \u2013 CWd | Soft\n\nSlow deceleration | Dense\n\nVariable deceleration | Dense\n\nSteep deceleration, early termination of diastolic flow\n\nPulmonic systolic flow compared to systemic flow \u2013 PWe | Slightly increased | Intermediate | Greatly increased\n\nCW continuous wave Doppler, PR pulmonic regurgitation, PW pulsed wave Doppler, RA right atrium, RF regurgitant fraction, RV right ventricle\n\naUnless there are other reasons for RV enlargement. Normal 2D measurements from the apical four-chamber view; RV mediolateral end-diastolic dimension ?4.3 cm, RV end-diastolic area ?35.5 cm2\n\nbException: acute PR\n\ncAt a Nyquist limit of 50\u201360 cm\/s\n\ndSteep deceleration is not specific for severe PR\n\neCut-off values for regurgitant volume and fraction are not well validated\n\nTable 18.3\n\nEtiologies of tricuspid regurgitation (TR)\n\nClassification | Primary TR | Secondary or functional TR (most common)\n\n---|---|---\n\nCongenital | Cleft valve (usually associated with atrioventricular canal defect)\n\nEbstein's anomaly | _\n\nAcquired | Normal variant\n\nRheumatic valve disease\n\nInfective endocarditis\n\nCarcinoid heart disease\n\nToxic (e.g., Phen-Fen or methysergide)\n\nIatrogenic: pacemaker lead\n\nTricuspid valve irradiation\n\nBlunt or penetrating injuries\n\nTricuspid valve prolapse | Right ventricular dilatation\n\nRight ventricular hypertension (e.g., pulmonary hypertension)\n\nGlobal right ventricular dysfunction secondary to myocarditis, cardiomyopathy, longstanding pulmonary hypertension with fibrosis\n\nSegmental dysfunction secondary to right ventricular ischemia or infarction, arrhythmogenic right ventricular\n\nDysplasia, endomyocardial fibrosis\n\nTable 18.4\n\nEcho parameters of tricuspid regurgitation (TR) severity [1]\n\nParameter | Mild | Moderate | Severe\n\n---|---|---|---\n\nTricuspid valve | Usually normal | Normal or abnormal | Abnormal\/flail leaflet\/poor coaptation\n\nRV\/RA\/IVC size | Normala | Normal or dilated | Usually dilatedb\n\nJet area-central jets (cm2)c | <5 | 5\u201310 | >10\n\nVC width (cm)d | Not defined | Not defined, but <0.7 | >0.7\n\nPISA radius (cm)e | ?0.5 | 0.6\u20130.9 | >0.9\n\nJet density and contour \u2013 CW | Soft and parabolic | Dense, variable contour | Dense, triangular with early peaking\n\nHepatic vein flowf | Systolic dominance | Systolic blunting | Systolic reversal\n\nCW continuous wave Doppler, IVC inferior vena cava, RA right atrium, RV right ventricle, VC vena contracta width\n\naUnless there are other reasons for RA or RV dilation. Normal 2D measurements from the apical four-chamber view: RV mediolateral end-diastolic dimension ?4.3 cm, RV end-diastolic area ?35.5 cm2, maximal RA mediolateral and supero-inferior dimensions ?4.6 and 4.9 cm respectively, maximal RA volume ?33 mL\/m2\n\nbException: acute TR\n\ncAt a Nyquist limit of 50\u201360 cm\/s. Not valid in eccentric jets. Jet area is not recommended as the sole parameter of TR severity due to its dependence on hemodynamic and technical factors\n\ndAt a Nyquist limit of 50\u201360 cm\/s\n\neBaseline shift with Nyquist limit of 28 cm\/s\n\nfOther conditions may cause systolic blunting (e.g., atrial fibrillation, elevated RA pressure)\n\nTable 18.5\n\nEtiologies of tricuspid stenosis (rheumatic, carcinoid, congenital, infective endocarditis, Fabry's, Whipple's disease, giant blood cysts)\n\nClassification | Causes\n\n---|---\n\nCongenital | Congenital tricuspid stenosis\n\nTricuspid atresia\n\nEbstein's anomaly\n\nAcquired | Rheumatic valve disease (most common)\n\nInfective endocarditis\n\nCarcinoid heart disease\n\nToxic (e.g., Phen-Fen or methysergide)\n\nPacemaker endocarditis and pacemaker-induced adhesions, lupus valvulitis, mechanical obstruction by benign or malignant tumors\n\nFabry's disease\n\nWhipple's disease\n\nFig. 18.1\n\nTwo-dimensional echocardiogram. Right ventricular inflow view showing thickened and retracted tricuspid valve leaflets due to carcinoid. RV right ventricle, RA right atrium, STVL septal tricuspid valve leaflet, PTVL posterior tricuspid valve leaflet\n\nFig. 18.2\n\n(a) Two-dimensional echocardiogram. Right ventricular inflow view showing severe tricuspid regurgitation by color flow Doppler in a patient with carcinoid heart disease. (b) Short-axis view showing thickened pulmonic valve leaflets in carcinoid heart disease. RV right ventricle, RA right atrium, TR tricuspid valve, PV pulmonic valve, MPA main pulmonary artery\n\nFig. 18.3\n\nTransesophageal echocardiography showing pulmonary valve thickened and retracted due to carcinoid involvement of the right-sided heart valves. Open pulmonary valve shows doming, LV left ventricle, RV right ventricle, RA right atrium, PV pulmonic valve, MPA main pulmonary artery, AV aortic valve, AC anterior cusp, PC posterior cusp\n\nFig. 18.4\n\nPulmonary regurgitation Doppler signal showing severe pulmonary regurgitation with a short deceleration time and high pulmonary end diastolic pressures. This is in a case of pacemaker induced tricuspid regurgitation. PVEDPV pulmonic valve end diastolic peak velocity, PPM induced TR retraction of septal leaflet\n\nFig. 18.5\n\n(a) Two-dimensional echocardiogram parasternal long axis right ventricular inflow view showing retraction of the septal leaflet by a pacemaker lead, and on the right panel (b) severe tricuspid regurgitation from malcoaptation. (c) Two-dimensional echocardiogram of the RV inflow view showing tricuspid regurgitation jet velocity Doppler signal showing severe tricuspid regurgitation and low pressure gradient. RV right ventricle, RA right atrium, STVL septal tricuspid valve leaflet, ATVL anterior tricuspid valve leaflet, PML pacemaker lead, TR tricuspid valve\n\nFig. 18.6\n\nTwo-dimensional echocardiogram. (a) Right ventricular inflow view showing pacemaker lead obstructing the posterior leaflet of tricuspid valve, RV right ventricle, RA right atrium, STVL septal tricuspid valve leaflet, ATVL anterior tricuspid valve leaflet, PML pacemaker lead, TV tricuspid valve\n\nFig. 18.7\n\nBioprosthetic tricuspid valve replacement. (a) Four-chamber transesophageal view with permanent pacemaker lead externalized to tricuspid ring. RV right ventricle, RA right atrium, LV left ventricle, LA left atrium, PTVL posterior tricuspid valve leaflet, TV tricuspid valve ring\n\nFig. 18.8\n\n(a) Two-dimensional echocardiogram parasternal short axis view at the base of the heart showing tricuspid valve leaflet prolapse. (b) Parasternal short axis at the base showing tricuspid valve regurgitation by color. (c) Tricuspid regurgitation Doppler signal demonstrates normal RVSP. RV right ventricle, LA right atrium, TVP tricuspid valve prolepses, AV aortic valve, MPA main pulmonary artery aneurysm, TR tricuspid regurgitation, RPA right pulmonary artery, LPA left pulmonary artery\n\nFig. 18.9\n\nTransthoracic echocardiogram. Parasternal short axis view at the base showing severe pulmonary regurgitation: Doppler trace demonstrating normal pulmonary artery pressures. PVEDPV Pulmonic valve end diastolic peak velocity\n\nFig. 18.10\n\nCongenital pulmonary stenosis showing main pulmonary artery and left pulmonary artery dilation. (a) Main pulmonary artery measures 3.5 cm and left pulmonary artery measures 3.1 cm, and in comparison, (b) right pulmonary artery measures 1.7 cm. LPA left pulmonary artery, RPA right pulmonary artery\n\nFig. 18.11\n\n(a) Parasternal short axis at the base of the heart showing dilated pulmonary artery and flow acceleration across the stenosed pulmonary valve by color flow Doppler, (b) pulmonary stenosis color flow Doppler, (c) Doppler trace showing high peak and mean gradient across the pulmonary valve, and (d) pulmonary regurgitation showing low pulmonary end diastolic regurgitation velocity indicating mildly raised EDP (8 mmHg + right atrial pressure)\n\nFig. 18.12\n\nTwo-dimensional echocardiogram. Apical four-chamber view showing apical displacement of the septal and anterior tricuspid valve leaflets, leading to atrialization of the right ventricle, moderate size atrialized right ventricle, and severe right atrial enlargement. Crux to TV insertion 20 mm or 1.2 cm\/m > 2. Patient has a BSA of 1.7 m. LA left atrium, LV left ventricle, RA right atrium, RV right ventricle, MV mitral valve, TV tricuspid valve\n\nFig. 18.13\n\nTwo-dimensional echocardiogram. Apical four-chamber view showing apical displacement of the septal and anterior tricuspid valve leaflets. LA left atrium, LV left ventricle, RA right atrium, RV right ventricle, MV mitral valve, STVL septal tricuspid valve leaflet, ATVL anterior tricuspid valve leaflet\n\nFig. 18.14\n\nTwo-dimensional echocardiogram. Right ventricular inflow view showing thickened, large anterior tricuspid leaflet and short, retracted septal leaflet of the tricuspid valve. RV right ventricle, RA right atrium, LV left ventricle, SSTVL short septal tricuspid valve leaflet, LATVL large anterior tricuspid valve leaflet\n\nFig. 18.15\n\nTwo-dimensional echocardiogram. Right ventricular inflow view showing large and mobile anterior tricuspid leaflet and short, retracted septal leaflet of the tricuspid valve. SSTVL short septal tricuspid valve leaflet, LATVL large anterior tricuspid valve leaflet\n\nFig. 18.16\n\nTwo-dimensional echocardiogram. Apical four-chamber view showing severe eccentric tricuspid valve regurgitation. TR tricuspid regurgitation\n\nFig. 18.17\n\nTwo-dimensional echocardiogram. Subcostal four-chamber view showing apical \u00addisplacement of the tricuspid leaflets, leading to atrialization of the right ventricle or Ebstein's anomaly. MV mitral valve, TV tricuspid valve\n\nReference\n\n1.\n\nZoghbi WA, Enriquez-Sarano M, Foster E, et al. Recommendations for evaluation of the severity of native valvular regurgitation with two-dimensional and Doppler echocardiography. J Am Soc Echocardiogr. 2003;16:777-802.PubMedCrossRef00335-3)\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_19\n\n# 19. The Assessment of Prosthetic Valve Function\n\nMengistu Simegn, Anupama Kottam and Luis Afonso1\n\n(1)\n\nDepartment of Cardiology, Detroit Medical Centre, Wayne State University, Detroit, MI, USA\n\nLuis Afonso\n\nEmail: lafonso@med.wayne.edu\n\n## 19.1 Characteristics of Prosthetic Valves: Two General Types (Tables 19.1 and 19.2)\n\na.\n\nMechanical prosthetic valves:\n\ni.\n\nComposition: Titanium or silicon derived with a sewing ring\n\nii.\n\nDurability: 25\u201330 years\n\niii.\n\nRisk of thromboembolism: 0.1\u2013 5.7 patient-years in mitral valve position\n\niv.\n\nObligatory normal regurgitation more common than tissue valves\n\nv.\n\nTypes:\n\n1.\n\nCaged-ball (most thrombogenic)\n\n2.\n\nSingle tilting disc (intermediate thrombogenicity)\n\n3.\n\nBileaflet (least thrombogenic)\n\nb.\n\nBioprosthetic valves:\n\ni.\n\nComposition: Intact valve or made of harvested pericardium. Source: Animal (hetero\/xenograft), human (allo\/homograft), or from patient (autograft)\n\nii.\n\nDurability: 10\u201320% of homografts and 30% of heterografts undergo structural degeneration in 15 years (tears, fibrosis, or calcification)\n\n1.\n\nDeterioration more aggressive in those <40 years\n\n2.\n\nWorse with mitral valves than aortic valves due to high closing pressure\n\niii.\n\nRisk of thromboembolism: equivalent to well anticoagulated mechanical valve (3 months after implantation and endothelialization of sewing ring)\n\niv.\n\nMore resistant than mechanical valves for infection (especially homograft, one reason for their use following endocarditis).\n\nv.\n\nTypes:\n\n1.\n\nStented\n\n2.\n\nStentless\n\n## 19.2 Normal Prosthetic Valve Echocardiograms (Figs. 19.1\u201319.3)\n\n### 19.2.1 2D Echo\n\n1.\n\nApproach to imaging:\n\na.\n\nAssess stability of the sewing ring, motion of the leaflets, disk, or occluder mechanism\n\nb.\n\nIdentify the type of prosthesis\n\nc.\n\nEvaluate for gross structural abnormalities like thrombus, vegetation, etc.\n\n2.\n\nCommon features of prosthetic valves:\n\na.\n\nClicks: opening and closing dense single-line Doppler signals\n\nb.\n\nReverberation artifacts: hall mark of mechanical valves. Single with single tilting disc and double in bileaflet\n\nc.\n\nAcoustic shadowing: Echo lucent as well as color flow (CF) bare area behind the valve that limits visualization of leaflets, thrombus, vegetations; the reason why CF alone should not be used for evaluation of regurgitation (part of or all of regurgitation and perivalvular leak may not be seen)\n\nd.\n\nStruts: in stented valves, struts may be seen protruding into the RV\/LV cavity or aorta.\n\ne.\n\nIn case of prosthetic valve malfunction or endocarditis, TEE is the preferred imaging modality of choice (superior resolution and unobstructed view of atria)\n\nf.\n\nAll prosthetic valves are inherently stenotic. The degree of obstruction is dependent on the type, size, and site of the prosthetic valve.\n\n### 19.2.2 Color Flow Doppler\n\n1.\n\nPattern: in general eccentric\n\n2.\n\nSingle tilting disc: single, eccentric flow\n\n3.\n\nBileaflet: three jets with smaller rectangular central and two larger peripheral\n\n4.\n\nCaged-ball: flow bifurcation around the poppet with \"pair of horns\" appearance\n\n5.\n\nBioprosthetic: single, triangular\n\n6.\n\nProsthetic regurgitation:\n\na.\n\nClosure backflow: early flow reversal to close the occluding mechanism\n\nb.\n\nLeakage backflow: small, continuous retrograde flow as a washing mechanism to prevent thrombus formation on the upstream side\n\n7.\n\nLimitation of color flow Doppler: Flow masking due to acoustic shadowing\n\n### 19.2.3 Spectral Doppler\n\n1.\n\nNormal flow velocity across a valve depends on:\n\na.\n\nType of prosthetic valve: caged-ball mechanical valve and heterograft bioprosthesis have the smallest effective orifice areas and therefore higher gradients, whereas homografts have the largest diameter and hence gradients similar to those of native valves\n\nb.\n\nSize of the prosthetic valve: pressure gradients inversely proportional to valve size\n\nc.\n\nLocation: For mitral and tricuspid valves, gradient increases with increasing heart rate and flow rate, whereas for aortic and pulmonic valves, gradient increases with flow and contractility\n\nd.\n\nCardiac output\n\ne.\n\nHeart rate\n\n2.\n\nIncreased flow velocity may be due to stenosis or regurgitation\n\n3.\n\nAll prosthetic valves are inherently stenotic except for homografts, i.e., flow velocity across prosthetic valve is higher than native valve\n\n4.\n\nProsthetic valves also have inherent regurgitation\n\n5.\n\nAll echos should be compared with prosthetic valve echo fingerprint; an echo done within 3 months of valve replacement\n\n## 19.3 Prosthesis Malfunction\n\nCinefluoroscopy should be recognized as a simple, rapid, inexpensive technique for evaluating prosthetic-valve function, particularly mechanical valves.\n\nTable 19.1\n\nCharacteristics of mechanical prosthetic valves\n\nType | Examples | Description | Hemodynamic profile | Image\n\n---|---|---|---|---\n\nCaged-ball | Star-Edward's | Bulky and more thrombogenic | Significant gradient especially in 19\u201321 mm sizes |\n\n|  |\n\nOldest; rarely used currently | Higher gradients in AV (as high as 20) than MV\n\n|   \n|  |\n\nEffective orifice area (EOA) | Color flow: pair of horns\n\n|   \n|  |\n\nMV = 1.4\u20133.1 sq cm | Not suitable for children\n\n|   \n|  |\n\nAV = 1.2\u20131.6 sq cm\n\n|  |\n\nSingle-tilting disc | Medtronic Hall, Omniscience, Bjork-Shiley | Single disc that swings 0\u201380\u00b0 | Hemodynamics better than cage-ball |\n\n|  |\n\nEffective orifice area (EOA) | PG < 15 in AV, < 5 in MV\n\n|   \n|  |\n\nMV = 1.9\u20133.2 sq cm | Single central orifice\n\n|   \n|  |\n\nAV = 1.5\u20132.1 sq cm | 5\u201310 mL\/beat regurgitation\n\n|\n\nBileaflet | St. Jude, CarboMedics | Less bulky and thrombogenic\n\nPivoted two semicircular discs\n\nAngle changes from 0\u00b0 (closed) to 90\u00b0(open)\n\nThe most commonly implanted\n\nEOA\n\nMV = 2.8\u20133.4 sq cm\n\nAV = 2.4\u20133.2 sq cm | The best hemodynamic profile among mechanical valves |\n\n|  |  |\n\nTwo larger lateral and one smaller central flow pattern\n\n|   \n|  |  |\n\n5\u201310 mL\/beat regurgitation\n\n|   \n|  |  |\n\nPG < 15 in AV and <4.5 in MV\n\n|\n\nTable 19.2\n\nCharacteristics of bioprosthetic valves\n\nType | Examples | Description | Hemodynamic profile | Image\n\n---|---|---|---|---\n\nStented | Heterograft porcine: | Stent maintains 3D relation between the valve leaflets and facilitates implantation | Hemodynamic profile similar to comparable size mechanical valve |\n\n|\n\nCarpertier Edwards\n\n|  |  |   \n|\n\nHancock Porcine\n\n|  |\n\nSingle central flow pattern\n\n|   \n|\n\nMedtronic Intact Porcine\n\n|  |\n\nRegurgitation less than mechanical valve of similar size\n\n|   \n|\n\nHeterograft bovine pericardial Autograft:\n\n|  |  |\n\n|\n\nAutologous Pericardial\n\n|  |  |\n\nStentless | Heterografts: | Less bulky than stented valves | Lower transvalvular gradients and more laminar flow than stented valves |\n\n|\n\nSt. Jude SPV\n\n|  |  |   \n|\n\nMedtronic Freestyle\n\n|  |  |   \n|\n\nHomografts: Harvested from cadavers, could have an aortic conduit | Developed only for AV | For patients with small aortic root |\n\n|\n\nAutografts: Pulmonary autograft (Ross procedure)\n\n|  |  |\n\nTable 19.3\n\nDoppler hemodynamic profiles of 609 normal aortic valve prosthesis\n\nType of prosthesis | Peak velocity (m\/s) | Mean gradient (mmHg) | LVOT-TVI\/AV-TVI\n\n---|---|---|---\n\nHeterograft | 2.4\u00b10.5 | 13.3\u00b16.1 | 0.44\u00b10.21\n\nBall-cage | 3.2\u00b10.6 | 23.0\u00b18.8 | 0.32\u00b10.09\n\nBjork-Shiley | 2.5\u00b10.6 | 13.9\u00b17.0 | 0.40\u00b10.10\n\nJude Medical | 2.5\u00b10.6 | 14.4\u00b17.7 | 0.41\u00b10.12\n\nHomograft | 1.9\u00b10.4 | 7.7\u00b12.7 | 0.56\u00b10.10\n\nMedtronic-Hall | 2.4\u00b10.2 | 13.6\u00b13.3 | 0.39\u00b10.09\n\nTotal | 2.6\u00b10.7 | 15.8\u00b18.3 | 0.40\u00b10.16\n\nAV, aortic valve; LVOT, left ventricular outflow tract; TVI, time velocity integral\n\nFrom Miller et al.1\n\nTable 19.4\n\nApproach to prosthetic aortic valve assessment\n\nCondition | Peak velocity (m\/s) | Mean PG (mmHg) | LVOT\/Ao\n\nTVI ratio | Comment\n\n---|---|---|---|---\n\nNormal | ?3 | <25 | >0.35 | Normal output state\n\nStenosis | >3 | >25 | <0.3 | Pannus or thrombosis, TEE\/fluoroscopy\n\nPatient\u2013Prosthesis mismatch | >3 | >25 | <0.3 | Moderate: EOA: 0.6\u20130.85 cm2\/m2\n\n|  |  |  |\n\nSevere: EOA ?0.6 cm2\/m2\n\nRegurgitation | >3 | >25 | >0.3 | TEE to evaluate\n\nHigh-output state | >3 | >25 | >0.3 | Confirm the absence of regurgitation\n\nTable 19.5\n\nDoppler echocardiographic data for pulmonary valve prosthesis\n\nType of prosthesis | Size (mm) | Peak velocity (m\/s) | Mean gradient (mmHg) | Trivial\/mild prosthetic regurgitation (No.)\n\n---|---|---|---|---\n\nCarpentier-Edwards | 26.5\u00b11.8 | 2.4\u00b10.5 | 12.1\u00b15.3 | 7\n\nPulmonary homograft | 24.2\u00b11.8 | 1.8\u00b10.6a | 8.4\u00b14.8 | 15\n\nAortic homograft | 22.3\u00b11.8 | 2.5\u00b10.4 | 14.4\u00b13.4 | 3\n\nHancock | 26.0\u00b11.8 | 2.4\u00b10.5 | 14.0\u00b15.7 | 1\n\nIonescu-Shiley | 25.0\u00b11.8 | 2.4\u00b10.4 | 12.5\u00b13.5 | 2\n\nSt. Jude Medical | 25 | 2.6 | 12.0 | 1\n\nBjork-Shiley | 25 | 2.0 | 7.0 | 1\n\nCompared with all heterografts combined, P = 0.002\n\nFrom Novaro et al.4\n\nTable 19.6\n\nDoppler hemodynamic profiles of 456 normal mitral valve prosthesis\n\nType of prosthesis | Peak velocity (m\/s) | Mean gradient (mmHg) | Effective area (cm2)\n\n---|---|---|---\n\nHetergraft | 1.6\u00b10.3 | 4.1\u00b11.5 | 2.3\u00b10.7\n\nBall-cage | 1.8\u00b10.3 | 4.9\u00b11.8 | 2.4\u00b10.7\n\nBjork-Shiley | 1.7\u00b10.3 | 4.1\u00b11.6 | 2.6\u00b10.6\n\nSt. Jude Medical | 1.6\u00b10.4 | 4.0\u00b11.8 | 3.0\u00b10.8\n\nFrom Lengyel et al.2\n\nTable 19.7\n\nApproach to prosthetic mital valve assessment\n\nCondition | Peak E velocity | Mean gradient | P1\/2t | ![\n$$ \\\\frac{MV-TVI}{LVOT-TVI} $$\n](A978-1-84996-151-6_19_Fig8_HTML.gif) | Comment\n\n---|---|---|---|---|---\n\nNormal | ?1.9 m\/s | ?5 mmHg | <130 ms | <2.5 | HR 55\u201390, normal output state\n\nObstruction or stenosis | ?2 m\/s | >5 mmHg | >>130 ms | <2.5 | TEE for thrombus pannus formation, and malfunction\n\nPatient-Prosthesis mismatch | Similar to obstruction (above) but is seen at the \"Echo finger-print,\" in the absence of alternative explanation\n\n|\n\nProsthetic valve regurgitation | ?2 m\/s | >5 mmHg | <130 ms | >2.5 | E>1.9 & TVI ratio >2.5 specific for MR\n\n|  |  |  |  |\n\nWhen both are normal; specific for absence of MR\n\n|  |  |  |  |\n\nTEE for the rest\n\nHigh output state | ?2 m\/s | >5 mmHg | <130 | <2.5 | Flow across all valves increased\n\nTable 19.8\n\nDoppler hemodynamic profiles of 82 normal tricuspid valve prosthesis\n\nType of prosthesis | Peak velocity (m\/s) | Mean gradient (mmHg) | Pressure half-time (ms)\n\n---|---|---|---\n\nHeterograft | 1.3\u00b10.2 | 1.3\u00b11.1 | 146\u00b139\n\nBall-cage | 1.3\u00b10.2 | 3.1\u00b10.8 | 144\u00b146\n\nSt. Jude Medical | 1.2\u00b10.3 | 2.7\u00b11.1 | 108\u00b132\n\nBjork-Shiley | 1.3 | 2.2 | 144\n\nTotal | 1.3\u00b10.2 | 3.1\u00b11.0 | 142\u00b142\n\nModified from Connolly et al.3\n\nFig. 19.1\n\nBall-cage prosthesis (Star Edwards) illustrating poppet (*) and characteristic color flow around it\n\nFig. 19.2\n\nTEE images of a bileaflet mitral prosthesis (St. Jude) showing open leaflets and normal regurgitation wash jets upon valve closure (Panel A). Below, an example of a tilting disk prosthesis in mitral position (Panel B) and a single central wash jet (normal finding). Note prominent reverberation artifact below disk\n\nFig. 19.3\n\nMechanical mitral prosthesis showing prominent acoustic shadowing originating from valve ring (enclosed between red bold lines)\n\n### 19.3.1 Prosthetic Valve Stenosis\n\nCauses\n\n  * Outgrowing of prosthetic valve size implanted during childhood with age\n\n  * Prosthesis\u2013patient mismatch: Minimal acceptable effective orifice area for AV (below which prosthesis\u2013patient mismatch is common) = Body surface area (BSA) \u00d7 0.85 cm2.\n\n  * Stenosis in mechanical valves: thrombus, pannus, vegetation (seen typically attached at the base or sewing ring)\n\n  * Stenosis in bioprosthetic valves: cusp degeneration and calcification\n\n### 19.3.2 Prosthetic Valve Regurgitation\n\na.\n\nTypes:\n\n  * Perivalvular: usually associated with dehiscence\n\n  * Valvular\n\nb.\n\nCauses:\n\ni.\n\nMechanical valves: thrombus, pannus, poppet variance, and endocarditis\n\nii.\n\nBioprosthetic valves: cusp degeneration, endocarditis, and torn leaflet (angry bee murmur on Doppler)\n\nc.\n\nGeneral principles:\n\ni.\n\nTTE not adequate for evaluation of mitral, tricuspid, and anterior aortic regurgitation\n\nii.\n\nNormal regurgitation(wash jets):\n\n  * Regurgitant jet area of <2 cm2 for MV and <1cm2 for AV\n\n  * Regurgitant jet length of <2.5 cm for MV and <1.5 cm for AV\n\niii.\n\nVarious flow patterns\n\n### 19.3.3 Evaluation of Prosthetic Aortic Valve (Table 19.3)\n\n1.\n\nAortic valve stenosis (Table 19.4):\n\na.\n\nDetermine maximum velocity and peak gradient\n\nb.\n\nDetermine mean gradient\n\nc.\n\nEstimate velocity or TVI ratio: LVOT\/AV TVI ratio (sewing ring outer diameter may be substituted for LVOT diameter)\n\nd.\n\nCalculate EOA by continuity equation: 0.785 \u00d7 (sewing ring outer diameter)2 \u00d7 LVOT TVI\/AV TVI\n\n2.\n\nProsthetic aortic regurgitation: considered severe if\n\na.\n\nPHT ? 250 ms\n\nb.\n\nMitral inflow pattern (restrictive pattern in acute regurgitation)\n\nc.\n\nHolodiastolic reversals in descending aorta\n\nd.\n\nRegurgitant fraction of ?55%\n\n### 19.3.4 Pulmonic Prosthetic Valve\n\nThe hemodynamic profiles are similar to that of aortic valve (Table 19.5).\n\n### 19.3.5 Evaluation of Prosthetic Mitral Valve (Table 19.6)\n\n1.\n\nProsthetic mitral valve stenosis: (Table 19.7, Fig. 19.4)\n\na.\n\nE velocity: measure 5 cycles in atrial fibrillation\n\nb.\n\nMean gradient by Bernoulli equation using the entire continuous wave inflow spectrum during diastole\n\nc.\n\nEnd-diastolic gradient\n\nd.\n\nPHT: overestimates MVA (not recommended)\n\ne.\n\nMVA by continuity equation: 0.785 \u00d7 LVOT diameter 2 \u00d7 LVOT TVI\/MV TVI\n\nf.\n\nTR velocity\n\ng.\n\nMV\/LVOT TVI ratio\n\n2.\n\nProsthetic mitral regurgitation:\n\na.\n\nMitral inflow peak velocity of >2.5 m\/s\n\nb.\n\nPHT of ? 150 ms\n\nc.\n\nDensity of MR continuous-wave Doppler jet\n\nd.\n\nPISA from TEE\n\ne.\n\nEffective regurgitant orifice area of ? 0.35 cm2\n\nf.\n\nSystolic flow reversal in pulmonary vein Doppler\n\n### 19.3.6 Tricuspid Prosthetic Valve Evaluation (Table 19.8)\n\n1.\n\nFollow similar assessment to that of mitral valve.\n\n2.\n\nCutoff normal parameters are slightly different because of the larger average area; peak E < 1.3 m\/s, mean PG < 4 mmHg, P1\/2t < 140 ms, and TV-TVI to PV-TVI ratio <2.5.\n\n## 19.4 Pannus vs. Thrombus\n\nThe echocardiographic differentiation of pannus and thrombus can be challenging.\n\nFig. 19.4\n\nTEE images of mitral bioprosthesis endocarditis. Note thickened leaflets and echogenic mass consistent with a vegetation (arrow) obstructing mitral inflow. CW Doppler recorded elevated transmitral velocities and gradient (mean gradient: 12 mmHg). Findings indicative of prosthetic stenosis\n\nIn general:\n\n  * Thrombus\n\n    * Sudden dyspnea, new murmur should raise suspicion\n\n    * Presentation acute or subacute\n\n    * Doppler typically reveals elevated transvalvular gradients\n\n    * Cinefluoroscopy very useful for assessment of leaflet mobility\n\n    * Thrombi tend to be mobile, somewhat less echo-dense, and associated with spontaneous echo contrast (SEC or smoke). INR may be subtherapeutic.\n\n  * Pannus\n\n    * Onset generally more insidious; affects mainly mechanical valves\n\n    * Reported as early as 6 months but usually occurs 5\u20136 years post operation\n\n    * In more than 60% of cases variable amounts of thrombus can be found overlying pannus and contribute to the obstruction\n\n    * May encroach into valve orifice and lead to stenosis\/elevated gradients\n\n    * Intermittent valve dysfunction should raise suspicion\n\n    * Ultimately, pannus is a diagnosis of exclusion, and surgical exploration is the gold standard\n\n    * Pannus is highly echogenic, consistent with its fibrous composition, and is usually firmly fixed to the valve apparatus\n\n## 19.5 Other Complications\n\nA.\n\nInfective prosthetic endocarditis\n\nProsthetic-valve infection occurs at some time in 3\u20136% of patients (0.2\u20130.5%\/year all valve types)\n\nEarly endocarditis (occurring less than 60 days after valve replacement) usually results from perioperative bacteremia arising from skin or wound infections or contaminated intravascular devices.\n\nLate endocarditis (occurring more than 60 days after valve replacement) is usually caused by the organisms responsible for native valve endocarditis\n\nVegetations\n\n  * Mechanical valves: sewing rings are commonly affected (unusual on the disc surface).\n\n  * Bioprosthesis: leaflets more than the sewing ring.\n\nB.\n\nPerivalvular leak and valve dehiscence (Fig. 19.5)\n\nFig. 19.5\n\nLong-axis TEE view showing dehiscence of mechanical AV prosthesis from posterior aortic root (arrow) and resulting torrential paraprosthetic aortic regurgitation\n\nDisruption of one or more sutures can affect mechanical and bioprosthetic valves resulting in valve regurgitation\n\nPredisposing factors\n\n  * Endocarditis (should be considered in all patients with valve dehiscence)\n\n  * Marfans syndrome\n\n  * Annular calcification\n\n  * Atherosclerotic roots\n\nC.\n\nHemolysis:\n\nTypically suggests paravalvular leakage due to partial dehiscence of the valve or infection. Patients with a caged-ball valve or with multiple prosthetic valves have an increased incidence and more severe degree of hemolysis than those with tissue valves. It is more common in men and in patients with heart failure. Beta blockers attenuate degree of hemolysis (by reducing the dp\/dt).\n\nLab findings:\n\n  * ? LDH\n\n  * ? Haptoglobin\n\n  * Schistocytosis\n\n  * Reticulocytosis\n\n  * Hemosiderinuria\/hemoglobinuriasis\n\nD.\n\nValve bed abnormalities (pseudoaneurysm, ring abscess)\n\nTEE is superior to TTE for the detection of endocarditis and invasive complications such as paravalvular abscesses, myocardial abscess, pseudoaneurysms, valve dehiscence, or leaflet perforations (Fig. 19.6).\n\nFig. 19.6\n\nShort-axis TEE views at the level of AV prosthesis showing a large loculated abscess with multiple septate cavities (asterisks)\n\n## 19.6 Miscellaneous\/Caveats\n\nA.\n\nMicrocavitation: Not an uncommon finding and needs to be recognized (Fig. 19.7).\n\nFig. 19.7\n\nParasternal long-axis view in a patient with tilting-disk aortic valve prosthesis illustrating microcavitation\/microbubble (Mcav) formation in the LV outflow tract and aortic root (arrows) during valve closure\n\n  * As the local pressure drops below the vapor pressure of the liquid, vapor-filled microbubbles are formed.\n\n  * This phenomenon has been observed only on the tilting disc type and bileaflet valves.\n\n  * No evidence of bubble formation on bioprosthetic valves.\n\n  * Has no clinical implications(benign finding).\n\nB.\n\nPressure Recovery (Fig. 19.8)\n\nFig. 19.8\n\nSchematic explaining the pressure recovery phenomenon. Pressure recovery distal to the aortic valve orifice leads to lower cath-derived gradient (P1\u2013P3) relative to gradient obtained with spectral Doppler (P1\u2013P2)\n\nPressure recovery is the variable increase in lateral pressure downstream from a stenotic orifice. Downstream from the orifice, flow expands and decelerates with a corresponding decrease in kinetic and increase in potential energy, a phenomenon called \"pressure recovery.\"\n\nClinical implications:\n\n  * Aortic pressure measured by cath distal to the orifice (mid ascending aorta) is higher than at the orifice (pressure recovery)\n\n  * Aortic valve gradient measured by catheterization is LOWER than that measured by Doppler (which measures \"unrecovered pressure\")\n\n  * Aortic valve area calculated by catheterization may be \"larger\"\n\n  * Phenomenon more relevant in patients with small or normal (not dilated) roots\n\n  * In prosthetic valves, it is a peculiar feature of bileaflet mechanical valve in the aortic position primarily because of measurement of velocity across the small central orifice that over estimates the pressure gradient.\n\nC.\n\nPatient\u2013Prosthesis Mismatch (PPM)\n\nPPM describes a state in which the effective orifice area (EOA) of a normally functioning heart valve prosthesis is too small in relation to the patient's body size; this situation results in high transvalvular pressure gradients in the setting of a structurally normal valve.\n\nTypically PPM is observed with aortic replacements but may also occur with MV prosthesis.\n\nFor example: A patient with a BSA of 1.3 m2 should get a valve with minimum EOA of 1.1 cm2\n\nMinimal acceptable effective orifice area for AV (where prosthesis\u2013patient mismatch is common) = BSA \u00d7 0.85 cm2 (corresponding value for mitral valve is BSA \u00d7 1.2 cm2)\n\n  * Severe mismatch = EOA of ? 0.6 cm2\/m2\n\n  * Moderate mismatch= > 0.6 ? 0.85 cm2\/m2\n\n  * Mild mismatch = >0.85 cm2\/m2\n\nClinical implications: Patients with severe PPM have\n\n  * ? Functional and exercise capacity\n\n  * ? Regression of LVH\n\n  * ? Recovery of coronary flow reserve\n\n  * Impaired blood coagulation status\n\n  * ? Late mortality\n\n  * ? Adverse cardiac events after AVR\n\nStentless valves provide larger EOA and more favorable hemodynamic profiles compared to their stented counterparts.\n\nReferences\n\n1.\n\nMiller F Jr, Callahan J, Taylor C, et al. Normal aortic prosthesis hemodynamics; 609 prospective Doppler examinations. Circulation. 1989;80(Suppl 2):II-169.\n\n2.\n\nLengyel M, Miller F Jr, Taylor C, et al. Doppler hemodynamic profile in 456 clinically and echo normal mitral prosthesis. Circulation. 1990;82(Suppl 3):III-43.\n\n3.\n\nConnolly HM, Miller FA Jr, Taylor CL, Naessens JM, Seward JB, Tajik AJ. Doppler hemodynamic profiles of 82 clinically and echocardiographically normal tricuspid valve prostheses. Circulation. 1993;88:2722-2727.PubMedCrossRef\n\n4.\n\nNovaro GM et al. Doppler hemodynamics of 51 clinically and echocardiographically normal pulmonary valve prostheses. Mayo Clin Proc. 2001;76:155.PubMedCrossRef63122-4)\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_20\n\n# 20. The Aorta: Diseases of the Aorta\n\nMary C. Corretti1\n\n(1)\n\nDivision of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nMary C. Corretti\n\nEmail: mcorre+1@yhmi.edu\n\n## 20.1 Aorta\n\n  * Largest vascular structure in the body\n\n  * Vessels are multicellular, and serve numerous dynamic biologic, physiologic, and anatomic functions\n\n  * Numerous conditions affect the aorta and branch vessels\n\n  * Numerous consequences from various disease processes and environmental exposures.\n\n## 20.2 Anatomy of the Thoracic Aorta\n\n## 20.3 Conditions Affecting the Aorta\n\n  * Atherosclerosis\n\n  * Hypertension\n\n  * Inflammatory diseases\n\n  * Infection\n\n  * Trauma\n\n  * Aortic valve disease\n\n  * Genetic collagen vascular conditions: Marfan's, Ehler's Danlos, Loeys-Deitz, Sinus of Valsalva aneurysm\n\n  * Iatrogenic injury\n\n## 20.4 Aorta-Atherosclerosis\n\n  * Atherosclerotic disease can manifest in the aorta in a variety of ways:\n\n    * Aneurysmal dilitation\n\n    * Atherothrombic plaque\n\n    * Dissection\n\n    * Intramural hematoma\n\n    * Penetrating ulcer in the aortic plaque\/or vessel wall\n\n    * Rupture-transection of the aorta\n\n    * Pseudoaneurysm\n\n## 20.5 Imaging of the Aorta\n\n  * Transthoracic echo exam should include full view of the aortic root, mid ascending aorta, and arch view displaying the upper ascending and proximal descending aorta\n\n  * TEE also provides similar views with exception of the blind spot due to tracheal interference at the level of the aortic arch\n\n  * CT and MRI are gold standards to evaluate the full extent of the aorta and disease processes.\n\nAorta\n\nTransthoracic Suptrasternal Notch View\n\nNormal\n\n  |\n\n---|---\n\nAortic Arch with upper ascending aorta in box view, and branch vessels | Aortic arch with upper descending aorta and branch vessels off the arch\n\nBicuspid Valve and Mid Ascending Aortic Dilitation due to Associated Aortopathy\n\nTransthoracic Parasternal Long Axis View\n\nMarfan's Syndrome\n\nDilated Sinuses of Valsalva-ST Junction\n\nNormal AV\n\nTransthoracic Parasternal Long Axis View\n\n## 20.6 Aortic Plaque\n\n### 20.6.1 General Classification as Noted on TEE\n\n  * Mild: plaque or intimal thickening: ?1 mm\n\n  * Moderate: 2\u20133.9 mm\n\n  * Severe: >4 mm\n\n  * Severe\/complex: ? ?4mm with ulceration and\/or superimposed thrombi\n\nAortic Arch with Severe Atherosclerotic Plaque\n\nTransesophageal Echo View of the Aortic Arch\n\n## 20.7 Classification of Aortic Dissection\n\n  * DeBakey classification\n\n    * Type I \u2013 proximal and descending aorta\n\n    * Type II \u2013 proximal only\n\n    * Type IIIa \u2013 proximal descending\n\n    * Type IIIb \u2013 complete descending and abdominal aorta\n\n  * Stanford classification\n\n    * Type A \u2013 proximal and descending aorta\n\n    * Type B \u2013 descending aorta\n\nAortic Dissection\n\nType A Proximal Aortic Dissection.\n\nCircumferential dissection flap beginning at the ST junction.\n\nDilitation (aneurysm formation) of the mid ascending aorta.\n\nTransesophageal view\n\nAortic Dissection\n\nType B Cross-section of a Descending Aortic Dissection\n\nAortic Dissection\n\nTransesophageal echo view of the arch with color flow Doppler.\n\nAortic Dissection with Thrombus in the False Lumen\n\nTransesophageal Echo View of a section of the descending aorta.\n\nThe aorta is aneurysmal with discrete dissection flap and organized thrombus in the false lumen.\n\n## 20.8 Functional Assessment of Aorta and Other Vascular Structures\n\n  * Numerous imaging modalities to assess vascular structures\n\n  * Various invasive and noninvasive testing modalities offer some degree or aspect of function\n\n  * Vascular function is complex and requires assessment beyond intraluminal occlusion assessment in the over-all evaluation of structure and function in health and disease states.\n\n## 20.9 Vascular Function\n\n  * Ultrasound-evaluations structure and flow assessment\n\n  * Artery tonometry \u2013 noninvasive acquisition of arterial waveforms to measure parameters of arterial stiffness\n\n## 20.10 Pulse Waveform Velocity Analysis\n\n  * Each pulsation of the heart generates a velocity of the pressure wave that is transmitted throughout the peripheral vascular system. The waveform biomechanical properties of the arterial system includes arterial wall stiffness.\n\n  * Ascending aortic pressure waveform can be measured from the carotid artery, femoral artery or radial artery using noninvasive techniques such as applanation tonometry and Doppler ultrasound.\n\n  * The peak systolic pressure is represented by Ps. Pd is the minimum diastolic pressure. An inflection point Pi in the waveform identifies the merging point of the beginning upstroke of the reflected pressure wave. AIx, augmentation index; AP, augmented pressure; Pi, inflection point; PP, pulse pressure.\n\nAortic Pulse Velocity Waveform\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_21\n\n# 21. Hypertrophic Cardiomyopathy\n\nAllison G. Hays1  and Aurelio C. Pinheiro\n\n(1)\n\nDivision of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nAllison G. Hays\n\nEmail: ahays2@jhmi.edu\n\n## 21.1 Introduction\n\n  * Hypertrophic cardiomyopathy (HCM): relatively common genetically heterogeneous disease characterized by myocardial hypertrophy with myocyte disarray and impaired LV performance (Fig. 21.1).\n\nFig. 21.1\n\nHypertrophic obstructive cardiomyopathy. The ventricular septum is markedly thickened\n\n  * HCM is characterized by asymmetric left ventricular hypertrophy typically involving the septum, but may involve other segments.\n\n  * Diagnostic criteria for asymmetric septal hypertrophy on echocardiography: septal thickness >15 mm, and septal to posterior wall ratio >1.3.\n\n  * Asymmetric septal hypertrophy is not pathognomonic for HCM and may be seen in other conditions such as glycogen storage disease.\n\n## 21.2 Patterns of Hypertrophy\n\n  * Anterior portion of LV septum (96%)\n\n  * Basal septum (15%)\n\n  * Concentric variant (1\u20135%)\n\n  * Apical hypertrophic variant (1\u20133%)\n\n  * Midventricular septal variant (1%)\n\n## 21.3 Obstructive Gradients\/Provocative Maneuvers\n\n  * Resting LV Outflow Tract (LVOT) obstruction defined as a peak gradient >30 mmHg.\n\n  * Resting LVOT obstruction in HCM patients has important prognostic value and predicts death and advanced heart failure.\n\n  * Many HCM patients do not have resting LVOT obstruction, and provocative maneuvers should be performed such as amyl nitrite, Valsalva or upright exercise.\n\n  * Mid cavitary or apical cavitary obstruction is not uncommon in rare HCM variants.\n\n## 21.4 LVOT Obstruction: Echocardiographic Features\n\n  * Mid systolic notching\n\n  * Early aortic valve closure\n\n  * Course systolic fluttering of the aortic valve\n\n  * Fibrotic septal changes at the level of leaflet-septal contact\n\n  * Doppler signal across the LVOT: characteristic signal with a late-peaking dagger-shaped appearance\n\n  * Severe obstruction: mid-systolic drop in LV velocity ? characteristic \"lobster claw\" Doppler tracing abnormality\n\n## 21.5 Systolic Anterior Motion (SAM)\n\n  * SAM (Figs. 21.2\u201321.4) of the anterior mitral valve leaflet with or without LVOT gradient is indicative of HCM and with a high specificity (Fig. 21.5).\n\nFig. 21.2\n\nHypertrophic cardiomyopathy. (a) Systolic anterior motion (SAM) of the anterior mitral valve leaflet is shown, contributing to the obstruction of the left ventricular outflow tract. (b) Color flow imaging showing posteriorly- directed mitral regurgitation secondary to SAM\n\nFig. 21.3\n\nApical hypertrophic cardiomyopathy. Apical four-chamber view during systole showing markedly increased apical wall thickness\n\nFig. 21.4\n\nContinuous wave Doppler from the left ventricular outflow tract illustrating a late peaking \"dagger-shaped\" signal indicating a dynamic outflow obstruction\n\nFig. 21.5\n\nM-mode echocardiography of the mitral valve (MV) in the parasternal short axis view. Example of systolic anterior motion of the MV leaflet\n\n  * Characterized by mid-late systolic posterolaterally directed mitral regurgitation.\n\n  * Dominant mechanism may be due to hydrodynamic \"drag\" or the pushing force of flow, rather than due do a \"Venturi\" effect.\n\n  * Leaflet elongation and anterior\/inward displacement of papillary muscles also contribute to SAM.\n\n## 21.6 Ejection Dynamics\n\n  * Systolic function typically normal or hyperdynamic.\n\n  * Ejection fraction usually preserved despite significant decrease of longitudinal contractile function as assessed by strain and strain rate.\n\n  * Diastolic dysfunction often present with impaired relaxation.\n\n  * A reduction in chamber compliance, increased stiffness, and heterogeneity of relaxation contribute to diastolic dysfunction.\n\n  * In late stages of the disease, progressive myocardial fibrosis results in impaired systolic function, myocardial thinning and cavity dilation.\n\n## 21.7 Conditions Mimicking HCM\n\n  * Hypertensive heart disease and athletic heart may mimic HCM.\n\n  * Long term exercise training causes LV remodeling and physiologic LVH, making the diagnosis of HCM difficult.\n\n  * The presence of wall thickness ?>12 mm (males) and >11 mm (females) in a trained athlete suggests HCM.\n\n  * In athletic LVH, hypertrophy is symmetric and occurs without diastolic dysfunction.\n\n  * Typically, cessation of exercise (for 6\u20138 weeks) causes 2\u20135 mm regression in wall thickness in athletic LVH but not in HCM.\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_22\n\n# 22. Stress Cardiomyopathy\n\nJacob Abraham1  and Ilan S. Wittstein\n\n(1)\n\nDepartment of Medicine, Johns Hopkins Hospital, Baltimore, MD, USA\n\nJacob Abraham\n\nEmail: ja@jhmi.edu\n\n## 22.1 Stress Cardiomyopathy: Definition\n\n  * Also called Takotsubo cardiomyopathy, transient apical ballooning syndrome, and broken heart syndrome\n\nFig. 22.1\n\nExample of SCM, Apical variant with basal sparing during systole. Diastole (left panel) and systole (right panel) from apical four-chamber view. Note mid-apical akinesis with normal basal wall motion (arrows)\n\nFig. 22.2\n\nExample of SCM with mid-ventricular sparing during systole. Diastole (left panel) and systole (right panel) from apical four-chamber view. Note focal hypokinesis of the mid-ventricle (arrow)\n\nFig. 22.3\n\nExample of SCM, basal variant with basal ballooning during systole. (Diastole (left panel) and systole (right panel) from parasternal long-axis view. Note akinesis of mid and basal segments (arrows))\n\nFig. 22.4\n\nExample of RV involvement during SCM with RV apical dyskinesis. Diastole (left panel) and systole (right panel) from right ventricular inflow view. Note mid-apical hypokinesis of the right ventricle\n\nFig. 22.5\n\nExample of Apical variant SCM with LV mural thrombus (arrow). (Well-circumscribed thrombus (arrow) present at the apex in apical ballooning variant.)\n\n  * Acute heart failure precipitated by sudden, intense emotional or physical stress characterized by:\n\n    * Three distinctive patterns of LV wall motion abnormalities (\"ballooning patterns\") that extend beyond a single coronary artery distribution\n\n    * Recovery in days to weeks\n\n    * Mild troponin elevation despite severe wall motion abnormalities\n\n    * Evolving ECG abnormalities including ST elevation, T-wave inversion, and QT-prolongation\n\n  * Diagnosis is based on clinical criteria and may be difficult to distinguish from myocardial infarction without coronary angiography\n\n## 22.2 Echocardiographic Evaluation of SCM\n\n  * Left ventricular variants\n\n    * Apical\n\n    * Mid-ventricular\n\n    * Basal\n\n  * RV involvement\n\n  * Apical thrombus\n\n## 22.3 Apical Variant (Fig. 22.1)\n\n## 22.4 Mid-Ventricular Variant (Fig. 22.2)\n\n## 22.5 Basal Variant (Fig. 22.3)\n\n## 22.6 RV Involvement (Fig. 22.4)\n\n  * RV involvement in SCM has been associated with greater hemodynamic compromise, longer length of stay, and increased mortality\n\n## 22.7 Apical Thrombus (Fig. 22.5)\n\n  * May be seen with apical variant\n\n  * Recovery of apical wall motion confers high embolic risk\n\n  * Anti-coagulation is recommended as prophylaxis until wall motion recovers\n\nSuggested Reading\n\nElesber AA, Prasad D, Bybee KA, et al. Transient cardiac apical ballooning syndrome: prevalence and clinical implications of right ventricular involvement. J Am Coll Cardiol. 2006;47(5):1082-1083.PubMedCrossRef\n\nHahn JY, Gwon HC, Park SW, et al. The clinical features of transient left ventricular nonapical ballooning syndrome: comparison with apical ballooning syndrome. Am Heart J. 2007;154(6):1166-1173.PubMedCrossRef\n\nWittstein IS, Thiemann DR, Lima JA, et al. Neurohumoral features of myocardial stunning due to sudden emotional stress. N Engl J Med. 2005;352(6):539-548.PubMedCrossRef\n\nWittstein IS. Acute stress cardiomyopathy. Curr Heart Fail Rep. 2008;5(2):61-68.PubMedCrossRef\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_23\n\n# 23. Pericardial Diseases\n\nPartho P. Sengupta1  and Bijoy K. Khandheria\n\n(1)\n\nCardiovascular Division, University of California, Irvine, USA\n\nPartho P. Sengupta\n\nEmail: partho@uci.edu\n\n  * The diagnosis of pericardial effusion was one of the first clinical applications of echocardiography.\n\n  * Transthoracic echocardiography is generally sufficient in the evaluation of patients with pericardial diseases.\n\n  * Transesophageal echocardiography is useful in measuring pericardial thickness, in the evaluation of diastolic function, and in detecting loculated pericardial effusion and other structural abnormalities of the pericardium.\n\nFig. 23.1\n\nCongenital absence of pericardium. 2D echocardiogram of a 53-year-old male patient with an unusual echo window in the absence of thoracic wall deformity. Note the unusual appearance of the heart on standard parasternal window (a) and the need to reach the anterior axillary line for obtaining the apical four-chamber view of the cardiac chambers (b). Diagnosis of congenital absence of pericardium is suspected on X-ray which may reveal prominent pulmonary artery contour and flattened elongated contour of LV apex (snoopy sign) (c) and confirmed by 2D echo and MRI (d). Note the absence of parietal and visceral pericardial coverings and interposed lung tissue between inferior wall of the left ventricle and the diaphragm (d)\n\n## 23.1 Pericardial Effusion\n\n  * Differentiating pleural effusion from pericardial effusion.\n\n  * The potential spaces around the heart when filled with fluid or blood is detected as an echo-free space.\n\n  * Pleural effusion can be distinguished from pericardial effusion by identifying the descending aorta. The descending aorta is anterior to pleural effusion and posterior to pericardial effusion.\n\n  * The reflection of the pericardium around the pulmonary veins tends to prevent collection of pericardial fluid behind the left atrium. Hence an echo-free space behind the left atrium is more likely to be pleural than pericardial effusion.\n\nFig. 23.2\n\nPericardial effusion. The space around the heart when filled with fluid or blood can be seen as an echo-free space (arrows). Note the differences in location of left sided pleural effusion (arrow 1, a\u2013c) and pericardial effusion (arrow 2, d\u2013f)\n\n## 23.2 Loculated Pericardial Effusions\n\n  * Pericardial effusions can be loculated and eccentric, a situation commonly encountered after cardiac surgery.\n\n  * The presence of an anterior echo-free space in the absence of a posterior echo-free space should be interpreted with caution since this may result from a large epicardial fat pad.\n\n  * The conditions resulting in an anterior echo-free space are listed in Table 23.1.\n\nTable 23.1\n\nConditions producing an anterior echo-free space\n\nEpicardial fat pad\n\n---\n\nHernias of foramen of Morgagni\n\nPericardial cyst\n\nPericardial tumors\n\nLeft atrial enlargement\n\nThrombus\n\nMassive ascites\n\nTable 23.2\n\nEchocardiographic artifacts seen in large pericardial effusions\n\nMitral valve prolapse\n\n---\n\nTricuspid valve prolapse\n\nSystolic anterior motion of the mitral valve\n\nEarly systolic closure of the aortic valve\n\nMidsystolic notching of the pulmonary valve\n\nParadoxical septal motion\n\nAttenuated motion of the posterior wall of the aortic root\n\nTable 23.3\n\nPitfalls in the diagnosis of pericardial effusion\n\n1. Excess gain settings may mask a pericardial effusion by \"filling-in\" the echo-free area.\n\n---\n\n2. The presence of left pleural effusion can make the demonstration of a pericardial effusion difficult. A left pleural effusion may be best demonstrated echocardiographically by placing the transducer in the left axillary line.\n\n3. An anterior echo-free space can be produced by an epicardial fat pad or by a homogenous, localized thrombus or by a pericardial cyst. In general, it is unlikely that an anterior clear space results from pericardial effusion in the absence of a posterior effusion.\n\n4. The presence of a loculated effusion can also cause a false-negative diagnosis, unless the pericardial space is examined carefully in different views.\n\nTable 23.4\n\nHemodynamic and echocardiographic features of constrictive pericarditis compared with restrictive cardiomyopathy\n\nFeature | Constrictive pericarditis | Restrictive cardiomyopathy\n\n---|---|---\n\nParadoxical pulse | Present in one-third of cases | Absent\n\nPericardial knock | Present | Absent\n\nProminent Y descent in JVP | Present | Variable\n\nRight and left sided filling pressures | Equalized within 5 mmHg | Left sided pressures at least 3\u20135 mmHg more than right\n\nFilling pressures more than 25 mmHg | Rare | Common\n\nRVSP ?> ?50 mmHg | No | Common\n\n\"Square root\" sign | Present | Variable\n\nRVEDP\/RVSP | ?0.33 | <0.3\n\nDiscordant respiratory variation of ventricular peak systolic pressures | Right and left ventricular peak systolic pressure variations are out of phase | Right and left ventricular peak systolic pressure variations are in phase\n\nPericardial thickness | Usually increased\n\n|\n\nAtrial size | Mild enlargement, usually of the left atrium | Biatrial enlargement, usually severe\n\nVentricular wall thickness\n\n|  |\n\nUsually increased\n\nSeptal bounce | Present | Absent\n\nMitral or tricuspid regurgitation | Usually absent or mild | Often present\n\nRespiratory variation in left\u2013right pressures or flow | Exaggerated\n\n|\n\nMitral inflow | Inspiratory E less than expiratory E (?25% change)\n\nDT usually ?160 ms | No respiratory variation of E velocity\n\nIncreased E\/A ratio ?2.0\n\nDT <160 ms\n\nTricuspid inflow | Inspiratory E greater than expiratory E (?40% change) | Mild respiratory variation in E velocity (?15%)\n\nPulsed wave Doppler of hepatic vein | Decreased diastolic forward flow with expiration\n\nMarked diastolic flow reversal, which increases with expiration compared to inspiration | Systolic forward flow less than diastolic forward flow\n\nDiastolic flow reversal in the hepatic vein is more prominent with inspiration\n\nRVSP right ventricular systolic pressure, RVEDP right ventricular end-diastolic pressure, E early rapid filling wave, A filling wave due to atrial systole, DT deceleration time\n\nFig. 23.3\n\nLoculated pericardial effusions image shows the appearance of anterior (a) and posterior (b) loculated pericardial effusions\n\n## 23.3 Pericardial Deposits\n\n  * Echocardiogram can give clues as to the cause of effusion. Metastatic deposits can be noted on visceral pericardium as echogenic masses in patients with neoplastic pericardial effusion. The presence of fibrinous strands in the pericardial fluid suggest an infective pathology.\n\nFig. 23.4\n\nPericardial deposits. Image showing metastatic deposits on visceral pericardium as echogenic masses (arrows, a and b)\n\n## 23.4 Cardiac Tamponade: Echocardiographic Features of Cardiac Tamponade\n\n  * The presence of an effusion even when large does not indicate its hemodynamic significance.\n\n  * The presence of an effusion must be correlated with the clinical features and other echocardiographic parameters of cardiac filling and transvalvular flow.\n\n  * The earliest echocardiographic sign of hemodynamic compromise is right cardiac chamber diastolic compression, which may precede other signs like pulsus paradoxus.\n\n  * The thinner and more compliant right cardiac chambers manifest the first signs of hemodynamically significant effusion as evidenced by compression of these chambers in diastole when their filling volume is lowest and therefore the effects of the external pericardial constraint are greatest. The signs appear first as right ventricular free wall collapse early in diastole.\n\n  * The right ventricular collapse may be absent despite cardiac tamponade in patients with preexisting elevated right ventricular pressures because of coexisting \u00adpathology. Echocardiographic artifacts and pitfalls seen with large pericardial \u00adeffusions which may confound the interpretation of features of cardiac tamponade are shown in Tables 23.2 and 23.4.\n\nFig. 23.5\n\nCardiac tamponade. Image showing the presence of an effusion (a). The right cardiac chambers manifest the first signs of hemodynamically significant effusion evidenced by compression in diastole when filling volume is lowest and effect of pericardial constraint is maximum appearing first as right ventricular free wall collapse early in diastole (b, arrows)\n\n## 23.5 Transvalvular Flow in Cardiac Tamponade\n\n  * The presence of exaggerated respiratory variation in the transvalvular flow is an important indicator of a hemodynamically significant effusion.\n\n  * In cardiac tamponade the intrapericardial pressure falls substantially less than intrathoracic pressure.\n\n  * The gradient between the pulmonary veins and the left sided cardiac chambers decreases with inspiration, thus reducing early transmitral flow, which on pulsed wave Doppler is seen as decreased early diastolic transmitral velocity. This also results in increased filling during late diastole, with greater dependence on atrial systolic contribution.\n\n  * The corresponding increase in tricuspid valve flow and augmented right heart filling during inspiration further compromises left heart filling through diastolic interactions mediated by the septum. These flow variations correlate with pulsus paradoxus and may precede chamber collapse.\n\nFig. 23.6\n\nIn cardiac tamponade, the gradient between the pulmonary veins and the left sided cardiac chambers decreases with inspiration, thus reducing early transmitral flow seen on pulsed wave Doppler as decreased early diastolic transmitral velocity (a). The corresponding increase in tricuspid valve flow and augmented right heart filling during inspiration (b) is seen\n\n## 23.6 Pericardiocentesis\n\n  * Pericardiocentesis is life saving, but a blind percutaneous pericardiocentesis can result in several complications like pneumothorax, puncture of the cardiac wall, and death.\n\n  * Monitoring pericardiocentesis echocardiographically helps in localizing the fluid, optimal site of puncture, depth of the pericardial effusion, and hence the distance from puncture site to the effusion, thus increasing the patient's safety.\n\n  * Following initial aspiration of pericardial fluid, injection of saline helps creating a contrast echocardiogram, and if the contrast echoes located in the pericardial space are not intracardiac the needle-tip location is confirmed in the pericardial space.\n\nFig. 23.7\n\nPericardiocentesis. Echo helps in localizing the fluid, optimal site of puncture, depth of the pericardial effusion (a). The shaft of the needle is not difficult to see (b), but it is difficult to be certain that the entire needle is being visualized. Following initial aspiration of pericardial fluid (c), injection of saline helps creating a contrast echocardiogram (d)\n\n## 23.7 Pericarditis\n\n  * The echocardiogram may be normal or show thickened parietal and visceral pericardium with either absent or minimal effusion unless pericardial effusion is associated with pericarditis.\n\n  * Transesophageal echocardiogram has higher diagnostic yield than transthoracic echocardiogram in assessing the thickness of pericardial layers with the transgastric view reported to give high-quality images of the pericardium useful in diagnosing chronic pericarditis.\n\nFig. 23.8\n\nPericarditis. Transesophageal echocardiogram is very useful in assessing the thickness of pericardial layers (arrows)\n\n## 23.8 Constrictive Pericarditis\n\n  * M-mode signs reflecting rapid early diastolic filling in constrictive pericarditis.\n\n  * In presence of thickened pericardium, abrupt anterior or posterior motion of the septum in early diastole is seen in most patients with constrictive pericarditis and reflects a rapid change in transseptal pressure gradient during early diastole caused by unusually vigorous early ventricular filling.\n\n  * The left ventricular posterior wall demonstrates a rapid early relaxation with posterior movement during early diastole followed by an abrupt cessation of such movement during mid and late diastole.This flat motion of the left ventricular posterior wall during mid and late diastole corresponds to abrupt transition of rapid ventricular filling in patients with constrictive pericarditis.\n\n  * The net diastolic left ventricular posterior wall endocardial movement posteriorly is less than 1 mm in constrictive pericarditis compared to normal where the posterior wall endocardial posterior movement ranges from 1.5 to 4 mm. The abrupt motion of the septum and posterior wall can be discerned on tissue Doppler imaging in the form of high velocities. The hemodynamic and echocardiographic features of constrictive pericarditis compared with restrictive cardiomyopathy are shown in fig. 23.4.\n\nFig. 23.9\n\nConstrictive Pericarditis. The unusually vigorous early ventricular filling (a, arrow 1), in presence of thickened pericardium and abrupt anterior or posterior motion of the septum in early diastole is seen in most patients with constrictive pericarditis and reflects a rapid change in transseptal pressure gradient during early diastole. The left ventricular posterior wall demonstrates a rapid early relaxation with posterior movement during early diastole followed by an abrupt cessation of such movement during mid and late diastole (b, arrow 2)\n\nFig. 23.10\n\nTissue Doppler and pulsed wave Doppler in constrictive pericarditis. Note the extent of pericardial thickening on cardiac computed tomography (a). An early diastolic mitral annular velocities (E' >8 cm\/s) at lateral or septal mitral annular corner can distinguish constrictive pericarditis from restrictive cardiomyopathy (b). On pulse wave flow Doppler imaging, early diastolic mitral flow is reduced in constriction with the onset of inspiration (c), whereas an increase in mitral inflow early diastolic velocity greater than or equal to 25% occurs with expiration. The pulsed Doppler recording of hepatic venous flow mirrors the right atrial pressure tracing. Pulsed wave Doppler recordings from hepatic vein in constriction (d) show marked diastolic flow reversal, which increases with expiration compared to inspiration, although it is not unusual to see significant diastolic flow reversals during both inspiration and expiration in patients with advanced constriction\n\nFig. 23.11\n\nPericardial masses. Two-dimensional echocardiography can help in identifying pericardial masses. (a and b) This example shows a solid mass in pericardium (arrow) arising from metastatic spread of bronchogenic adenocarcinoma\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_24\n\n# 24. Atrial Septal Defect\n\nNaser M. Ammash1\n\n(1)\n\nDepartment of Internal Medicine, Mayo Clinic, Rochester, MN, USA\n\nNaser M. Ammash\n\nEmail: ammash.naser@mayo.edu\n\nThe population of adults with congenital heart disease (CHD) is rapidly growing as a result of improvement in diagnostic techniques, medical and surgical expertise. It is estimated that 85% of infants born with CHD survive into adulthood, some of whom have never had any intervention or surgery. This patient population represents a heterogeneous group that includes simple defects such as atrial septal defects, ventricular septal defects, pulmonary stenosis, coarctation of aorta, and complex defects such as Tetralogy of Fallot, Ebstein's anomaly, pulmonary atresia, transposition of the great artery, and univentricular heart. Adults CHD survivors have special needs and require expertise care even after surgical repair since there is usually no cure after repair of CHD and many residua and sequelae are being recognized. The challenges of caring for these adult survivors of CHD can be diagnostic, medical, electrophysiologic, surgical, and psychological. Many of these patients require reoperation that should be performed by surgeons with expertise in that field. The improved outlook for adults with CHD could be best sustained by having adult congenital specialists working in close collaboration with electrophysiologists, cardiac surgeons, cardiac imaging including echocardiography (ECHO), magnetic resonance imaging, computer tomography, as well as obstetricians, psychiatrists, intensivists, transplant service, and medical subspecialists. This multidisciplinary approach toward the care of adults with CHD is the best we can offer for these survivors. This chapter will illustrate the most important echocardiographic features of the common CHD seen before and after repair.\n\n## 24.1 Atrial Septal Defect\n\nAtrial septal defects (ASD) are common CHD seen in adulthood causing left-to-right shunt across the atrial septum with secondary volume overload of the atria and the right ventricle. Both transthoracic (TTE) and transesophageal (TEE) Echos play a very important role in the assessment of these patients before and after surgical or percutaneous closure. In addition to the detection of ASD, Echo assesses the degree of shunting, severity of volume overload, and the presence and severity of pulmonary hypertension, tricuspid regurgitation, and associated CHD most commonly partial anomalous pulmonary venous connection (PAPVC) (Figs. 24.1\u201324.4).\n\nFig. 24.1\n\nDiagrammatic illustration of the different ASD demonstrating the most common septum secundum ASD (1). Please note the location of the defect in middle of the septum as compared to the septum primum ASD (2) in close proximity to the tricuspid valve, sinus venosus ASD in close proximity to the superior vena cava, and the least common coronary sinus ASD in proximity to the coronary sinus and inferior vena cava\n\nFig. 24.2\n\nTEE view (a) obtained at the base of the heart demonstrating a secundum ASD with left-to-right shunt between the left atrium (LA) and the right atrium (RA). In (b), a three-dimensional view of secundum ASD delineating clearly the margins of the defect as compared to the two-dimensional TEE images\n\nFig. 24.3\n\nApical 4 chamber view demonstrating a primum ASD (arrow) in close proximity to the mitral and tricuspid valves (a). On parasternal short-axis view (b and c), a cleft in the anterior mitral valve leaflet is seen associated with mitral regurgitation. Typically, these patients have leaflet anterior fascicular block on electrocardiogram. LA, left atrium; RA, right atrium; LV, left ventricle; RV, right ventricle\n\nFig. 24.4\n\nSinus venosus ASD as noted by TTE (a) and TEE (b). On both studies, note the defect which is superior to the fatty limbus of the atrial septum (AS) and in close proximity to the superior vena cava (SVC). RA, right atrium; LA, left atrium; RAA, right atrial appendage\n\n### Summary\n\n  * ASD are common defects of the atrial septum, leading to left-to-right shunt\n\n  * ASD cause enlargement of the atria and right ventricle and can lead to pulmonary hypertension\n\n  * Imaging of the atrial septum is best done with TEE, as compared to TTE especially in the adult patient\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_25\n\n# 25. Percutaneous Interventions for Congenital Heart Defects\n\nChristian D. Nagy1  , Richard E. Ringel and W. Reid Thompson\n\n(1)\n\nPediatric Cardiology\/Adult Cardiology, Johns Hopkins University Medical Center, Baltimore, MD, USA\n\nChristian D. Nagy\n\nEmail: cnagy3@jhmi.edu\n\n## 25.1 Atrial Septal Defect Closure\n\n  * Atrial septal defect (ASD) closure leads to improved functional status, reduces the risk of progressive right-sided failure, and prevents the development of severe pulmonary hypertension or paradoxical embolism.\n\n  * Percutaneous device closure using a number of different devices has become the treatment of choice for secundum ASDs, whereas other types of ASDs are treated surgically.\n\n  * Device closure of ASD is guided by a combination of fluoroscopic and echocardiographic imaging. Intracardiac echocardiography (ICE) is now being used more commonly for adults and transesophageal echocardiography (TEE) employed in special situations and small children.\n\n  * Indications for ASD closure are right heart dilation by echocardiography, MRI, or CT (Qp:Qs >1.5:1) in the absence of advanced pulmonary arterial hypertension.\n\n  * Patients with an ASD too large for device closure, inadequate atrial septal rims to permit stable device deployment, or those with proximity of the defect to the AV valves, the coronary sinus, or the vena cavae are referred for surgical repair.\n\n  * Device closure is a safe and effective procedure in experienced hands, with major complications such as cardiac perforation or device embolization occurring in less than 1% of patients. Successful closure is achieved in up to 95% of patients. Although small residual shunts are often seen on echocardiography at the end of the procedure, these are hemodynamically insignificant, and most will close spontaneously within 1 year.\n\nFig. 25.1\n\nTEE apical 4-chamber view demonstrating a large secundum ASD (a). In Fig. 25.1b the defect has been closed using an Amplatzer septal occluder. Figure 25.1c and d show angiographic correlates. Under TEE guidance, the ASD device is positioned across the interatrial septum and then released\n\nFig. 25.2\n\nTEE view demonstrating a large secundum ASD. In addition to determining the maximal diameter of the defect and total atrial septal length, measurement of the surrounding rims to the aorta, right upper pulmonary vein, superior vena cava, inferior vena cava, mitral valve, and coronary sinus, is warranted in order to assess candidacy for percutaneous closure\n\nFig. 25.3\n\nICE demonstrating ASD closure. In Fig. 25.3a both the left and right atrial disks have been deployed; however the device is still attached to the delivery cable. In Fig. 25.3b the device has been released. Note the septal occluder in stable position, anchored between the aortic rim and the posterior rim\n\n## 25.2 Ventricular Septal Defect Closure\n\n  * In adults, congenital ventricular septal defect (VSD) device closure in the catheterization lab is relatively uncommon, reserved primarily for defects in the apical muscular septum, which are difficult to close surgically.\n\n  * Acquired VSD is a rare, but serious, complication of myocardial infarction occurring in <1% of cases, and mortality is high if untreated.\n\n    * Early surgical cure is difficult, and in selected patients percutaneous closure may be an alternative treatment option.\n\n    * Individuals most likely to benefit from transcatheter VSD occlusion are those with persistent hemodynamic compromise who are medically difficult to manage.\n\nFig. 25.4\n\nApical 4-chamber TTE view (a) demonstrating device closure of a post-myocardial infarction muscular VSD (in this case using an Amplatzer septal occluder) in the apical interventricular septum. In Fig. 25.4b the VSD is visible angiographically and in Fig. 25.4c the defect is closed with the VSD device\n\n## 25.3 Patent Ductus Arteriosus Closure\n\n  * Patent ductus arteriosus (PDA) is a congenital cardiac disorder that can be first identified at almost any age.\n\n  * The clinical implications vary and can include development of heart failure and pulmonary hypertension.\n\n  * Transcatheter techniques have replaced surgical therapy in most patients with PDA and recent advances in echocardiography have resulted in better detection and characterization of PDA and avoiding complications.\n\nFig. 25.5\n\nFig. 25.5a shows the 3-D echo image demonstrating an Amplatzer duct occluder closing a PDA. The larger end faces the aorta and the smaller end is positioned toward the pulmonary artery. Figure 25.5b shows the angiographic correlate. A catheter is positioned in the aorta. The closure device is deployed in the ductus arteriosus and still attached to its delivery cable and the device is released after confirming the position\n\nTable 25.1\n\nStructural heart conditions amenable to percutaneous cath lab interventions\n\nSecundum atrial septal defect\n\n---\n\nPatent foramen ovale\n\nMuscular ventricular septal defect\n\nPatent ductus arteriosus\n\nFenestrated Fontan\n\nAtrial baffle leaks after Mustard\/Senning operation for transposition of the great arteries\n\nAortic to pulmonary artery collaterals\n\nPulmonary arteriovenous malformations\n\nVenous collaterals\n\nCoarctation of the aorta\n\nBranch pulmonary artery stenosis\n\nStenosis of the systemic venous baffle after Mustard operation\n\nBalloon atrial septostomy\n\nPercutaneous balloon valvotomy for pulmonary stenosis\n\nAortic balloon valvotomy for aortic stenosis\n\nMitral balloon valvotomy for mitral stenosis\n\nPercutaneous pulmonary valve implantation\n\nPercutaneous aortic valve replacement\n\nTable 25.2\n\nComparison of different echocardiographic techniques | Transthoracic echocardiogram | Transesophageal echocardiogram | Intracardiac echocardiogram\n\n---|---|---|---\n\nIndications | Cardiac biopsy | ASD\/PFO closure | ASD\/PFO closure\n\n|\n\nSeptostomy | VSD closure\n\n|   \n|\n\nPericardiocentesis | Fenestrated Fontan closure\n\n|\n\nAdvantages | Noninvasive | High sensitivity | No sedation required\n\n|\n\nNo sedation required | Posterior structures well imaged | Inferior atrial septum well imaged\n\n|\n\nAnterior structures well imaged\n\n|  |\n\nDisadvantages | Inconvenient in cath lab | Deep sedation required | Additional venous access required\n\n|\n\nLower resolution of small structures | May compromise airway | Limited visualization\n\n|  |\n\nAnterior structures difficult to assess\n\n|\n\n## 25.4 Summary Box\n\nASD, VSD, and PDA are congenital defects amenable to percutaneous device therapy. Imaging studies such as echocardiography with its various modalities including TTE, TEE, 3-D echo, and ICE play a crucial role in the diagnosis and management of congenital heart disease.\n\nRecommended Reading\n\nMinette MS, Sahn DJ. Ventricular septal defects. Circulation. 2006;114:2190-2197.PubMedCrossRef\n\nMullen MJ, Dias BF, Walker F, et al. Intracardiac echocardiography guided device closure of atrial septal defects. J Am Coll Cardiol. 2003;41:285-292.PubMedCrossRef02616-5)\n\nSchneider DJ, Moore JW. Patent ductus arteriosus. Circulation. 2006;114:1873-1882.PubMedCrossRef\n\nWarnes CA, Williams RG, Bashore TM, et al. ACC\/AHA 2008 guidelines for the management of adults with congenital heart disease. J Am Coll Cardiol. 2008;52(23):e1-121.CrossRef\n\nWebb G, Gatzoulis MA. Atrial septal defects in the adult: recent progress and overview. Circulation. 2006;114:1645-1653.PubMedCrossRef\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_26\n\n# 26. Cardiac Tumors and Masses\n\nMary C. Corretti1\n\n(1)\n\nDivision of Cardiology, Johns Hopkins University, Baltimore, MD, USA\n\nMary C. Corretti\n\nEmail: mcorret1@jhmi.edu\n\n## 26.1 Categories of Tumors\/Masses\n\n  * Primary benign tumors\n\n  * Primary malignant tumors\n\n  * Intracardiac thrombus\n\n  * Intracardiac vegetations\/infections\n\n  * Degenerative valvular changes creating mass effect\n\n  * Pericardial masses or cysts\n\n  * Normal variants\n\n## 26.2 Cardiac Masses: Benign\n\n  * Myxoma-most common\n\n  * Lipoma\n\n  * Fibroma\n\n  * Valvular \u2013papillary fibroelastoma\n\n  * Hemangioma\n\n## 26.3 Cardiac Masses: Malignant\n\n  * Angiosarcoma \u2013 most common in adults\n\n  * Rhabdomyosarcoma \u2013 more common in children\n\n  * Mesothelioma\n\n  * Fibrosarcoma\n\n  * Lymphoma\n\n  * Leiomyosarcoma\n\n  * Thymoma\n\n## 26.4 Myxoma\n\n  * Comprise the majority of benign intracardiac tumors\n\n  * Usually on a stalk attached to the fossa ovalis (mid portion of the interatrial septum) protruding into the left atrium, or right atrium. They can develop in the ventricules as single masses, or as a multiple presence in the Carney syndrome-genetic defect in the long arm of chromosome 17.\n\n  * Tumor is of mesenchymal origin, appears amorphous, gelatinous and usually mobile on a stalk. Sessile myxomas can also occur.\n\n  * Potential for embolization is high thus requiring surgical removal upon diagnosis.\n\n  * Diagnosis is made often with transthoracic echo and transesophageal echo, if transthoracic echo image quality is limited.\n\n## 26.5 Atrial Myxoma (Fig. 26.1)\n\n## 26.6 Papillary Fibroelastoma: Benign Valvular Tumor\n\n  * Second most common primary benign tumor in the adult\n\n  * Small in size with characteristic stippled or frond-like edge that appears to vibrate in motion at the tumor-blood interface. The tumor emanates from a stalked central core which often arises from a valvular structure (more commonly on the aortic and mitral valves, less so on the tricuspid and pulmonary valves.\n\n  * Potential for embolization is present\n\n### Papillary Fibroelastoma (Fig. 26.2)\n\n### Valvular Papillary Fibroelastoma (Fig. 26.3)\n\n## 26.7 Fibroma (Fig. 26.4)\n\n## 26.8 Sarcoma\n\n  * Most common malignant cardiac tumor in adults, and second to myxomas in frequency though still rare.\n\n  * Derived from mesenchymal cells causing wide array of different histologic types: angiosarcoma, rhabdomyosarcoma, fibrosarcoma, leiomyosarcoma, osteosarcoma.\n\n  * These can be seen as bulky tumors in the chambers, and more so infiltrating the myocardium and pericardium.\n\n## 26.9 Malignant Tumor: Burkitt B Cell Lymphoma (Fig. 26.5)\n\n## 26.10 Mimickers of Cardiac Tumors (Figs. 26.6 and 26.7)\n\n  * Eustachian ridge and Chiari network: Normal variant of an embryologic remnant of the Eustachian valve from the IVC. This is a long mobile string like structure that can attach from the Eustachian ridge (IVC\/RA junction) to the interatrial septum.\n\n## 26.11 Lipomatous Hypertrophy TEE bicaval view\n\n(Fig. 26.8)\n\nFig. 26.1\n\n(a) TEE-midesophageal short axis aortic valve view. Tumor attached to the left atrial side fossa. Ovalis portion of the interatrial septum. (b) TEE-midesophageal bicaval view. Tumor attached to the left atrial side. Interatrial septum\n\nFig. 26.2\n\nPapillary fibroelastoma. Transthoracic short axis view of aortic valve. Round echodense fibroelastoma noted on the right coronary cusp (X)\n\nFig. 26.3\n\nValvular papillary fibroelastoma. Transthoracic parasternal long axis view. Round tumor on a stalk on the aortic side of the aortic valve\n\nFig. 26.4\n\nTransthoracic four chamber view in diastole (left image) and mid systole (right image). Fleshy tumor mass on the proximal ventricular septum\n\nFig. 26.5\n\nMalignant tumor: Burkitt B cell lymphoma. Transthoracic subcostal view. Tumor mass in the right atrium, with infiltration of tumor in the myocardium. Pericardial effusion present\n\nFig. 26.6\n\nMimickers of cardiac tumors. Transthoracic parasternal short axis view. Chiari network \u2013 X\n\nFig. 26.7\n\nIntracardiac thrombus in transit. Transthoracic parasternal short axis view\n\nFig. 26.8\n\nEchogenic fatty tissue along the interatrial septum and sparing the fossa ovalis of the interatrial septum. Dumb-bell shape appearance and echogenicity of the tissue is characteristic of this anatomic variant\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_27\n\n# 27. Cardiac Transplantation\n\nJames Mudd1\n\n(1)\n\nXXXX, yyyy\n\n## 27.1 Ventricular Assist Device\n\n  * Ventricular assist devices (VADs) are placed in the right, and most commonly, the left ventricle to support patients with end stage heart failure.\n\n  * Echocardiography is used to evaluate the following issues with VADs:\n\n    * Apical cannula position\n\n    * Assessment of ventricular unloading\n\n      * Frequency of aortic valve opening\n\n      * Position of the septum:\n\n        * Leftward shift may indicate an under filled left ventricle from excessive unloading or abnormalities with preload.\n\n        * Rightward shift may indicate poor LV unloading due to low pump speed or problems with pump filling.\n\n    * Right ventricular size and function with left sided VAD\n\n## 27.2 VAD: Parasternal Long Axis\n\n  * This view should attempt to show the VAD cannula protruding through the apex (arrow). The purpose of this view is to determine the orientation of the cannula and identify obstruction to blood inflow (clot or tissue).\n\n## 27.3 VAD: Apical Short Axis\n\n  * Much like the parasternal view, this view helps to determine if there is an abnormally aligned cannula or clot, or other debris preventing blood inflow through the cannula.\n\n## 27.4 VAD: Aortic Valve Function\n\n  * Careful examination of the aortic valve with and without color flow Doppler should be performed capturing at least 10 beat in all views.\n\n  * Frequent opening of the aortic valve signals potential insufficient ventricular unloading either from low pump speeds or problems with pump filling.\n\n  * Significant aortic regurgitation may also develop.\n\n  * Continuous aortic regurgitation throughout the cardiac cycle is sometimes seen with an LVAD.\n\n## 27.5 Heart Transplantation\n\n  * Important differences include larger atria and abnormal septal motion as seen after other cardiac surgeries.\n\n  * Most transplanted hearts are anastamosed with a bicaval technique resulting in a slightly larger left atrium and normal sized right atrium.\n\n  * Previous techniques resulted in biatrial enlargment as seen in this image.\n\n## 27.6 Echocardiography in Heart Transplantation\n\n  * Routine echocardiography is used to evaluate for changes in ejection fraction and\/or regional wall motion abnormalities caused by:\n\n    * Cellular or humoral immunologic rejection\n\n    * Transplant associated coronary artery disease\n\n  * Transplant patients may also develop right ventricular dysfunction if there was preexisting pulmonary hypertension.\n\n  * Echocardiography can also be used to guide endomyocardial biopsy to evaluate for rejection.\n\n## 27.7 Heart Transplant Rejection\n\nEvidence of new hypertrophy in a heart transplant patient may be a sign of cardiac rejection as seen in this patient with suspected humoral rejection. The ejection fraction may be preserved, demonstrate regional wall motion abnormalities or diffuse hypokinesis in rejection\n\n## 27.8 Dilated Cardiomyopathy\n\n  * Dilated cardiomyopathy refers to enlargement of the left ventricle and a depressed \u00adejection fraction.\n\n  * Causes:\n\n    * Coronary artery disease\n\n    * Familial\n\n    * Peripartum\n\n    * Viral\n\n    * Idiopathic\n\n    * Sarcoidosis\n\n    * Hemachromatosis\n\n    * Primary valve disease\n\n  * Echocardiography is used to evaluate for:\n\n    * Regional wall motion abnormalities\n\n    * Severity of mitral regurgitation\n\n      * Mitral regurgitation is common in dilated cardiomyopathy typically resulting from stretching of the valve annulus.\n\n    * Presence of apical thrombus\n\n    * Pulmonary hypertension\n\n    * Aneurysm or pseudoaneurysm formation after myocardial infarction\n\n## 27.9 Dilated Cardiomyopathy Mitral Regurgitation\n\nApical four chamber view demonstrating eccentric mitral regurgitation.\n\nDoppler signal demonstrates significant mitral regurgitant velocity\n\n## 27.10 Dilated Cardiomyopathy\n\n### IVC Diameter\n\nThe IVC is often dilated in patients with heart failure and can be used as a metric for volume overload and\/or right heart dysfunction. Lack of respiratory variation is also a sign of venous congestion.\n\n### Ventricular Thrombus\n\nRV thrombus.\n\nPedunculated LV trhombus\n\n## 27.11 Amyloid Cardiomyopathy\n\nAmyloid cardiomyopathy demonstrates significant hypertrophy, depressed left ventricular function. A distinguishing feature is a diminished mitral annular tissue Doppler signal.\n\n## 27.12 Bicaval Orthotopic Heart Transplant\n\n  * The recipient SVC and IVC are anastomosed to the donor heart without altering the size of the right atrium.\n\n  * The recipient pulmonary veins and some surrounding tissue are retained to anastomose to the donor left atrium.\n\n\u00a9 Springer London 2010\n\nTheodore Abraham (ed.)Case Based Echocardiography10.1007\/978-1-84996-151-6_28\n\n# 28. Vascular Imaging\n\nElizabeth V. Ratchford1\n\n(1)\n\nDepartment of Medicine, Division of Cardiology, Johns Hopkins University School of Medicine, Baltimore, MD, USA\n\nElizabeth V. Ratchford\n\nEmail: eratchf1@jhmi.edu\n\n## 28.1 Introduction\n\nVascular imaging has now become routine in many echocardiography laboratories. With increasing recognition of the systemic nature of atherosclerosis, cardiologists are becoming more involved in imaging outside the heart both for clinical and research purposes.\n\n## 28.2 Outline\n\n1. ?\n\nVascular ultrasound principles\n\n2. ?\n\nCarotid duplex\n\n3. ?\n\nAbdominal aorta ultrasound\n\n4. ?\n\nResearch techniques in vascular imaging\n\n  * Carotid intima-media thickness\n\n  * Brachial reactivity\n\n  * Augmentation index\n\n## 28.3 Vascular Ultrasound Principles\n\n### Vascular Ultrasound: Conventions\n\n  * Several principles and conventions are unique to vascular ultrasound:\n\n  * The Doppler angle is set at 60\u00b0.\n\n  * The head is on the left-hand side of the screen and the feet are to the right.\n\n  * With color flow, red is for arteries and blue is for veins.\n\n  * Flow towards the transducer is positive; flow away from the transducer is negative.\n\n### Vascular Ultrasound: Techniques\n\n  * All sampling should be done at an angle of 60\u00b0 or less (between the beam and the flow direction or vessel wall).\n\n  * The sample volume should be at the center of the vessel and should be set as small as possible.\n\n  * The cursor should be parallel to the vessel wall.\n\n### Vascular Ultrasound: Diagnostic Criteria\n\n  * In any given artery, a doubling of the peak systolic velocity (PSV) on spectral Doppler generally indicates more than 50% stenosis.\n\n  * Percent stenosis refers to diameter reduction.\n\n    * For example, 50% diameter reduction is equivalent to a 75% area reduction.\n\n  * The velocity in the area of stenosis (or just distal to the stenosis) is typically compared to a normal, more proximal portion of the artery.\n\n  * In general, diagnosis is based on velocity criteria and the presence or absence of plaque.\n\n  * While there are published criteria, each vascular laboratory is must establish and then internally validate its own velocity criteria.\n\n## 28.4 Carotid Duplex\n\n### Indications for Carotid Duplex (Fig. 28.1)\n\n  * Carotid bruit\n\n  * Amaurosis fugax\n\n  * TIA\/Stroke\n\n  * Follow-up of known stenosis or post-stent or post-CEA\n\n  * Syncope\n\n  * pre-CABG (controversial)\n\n### Carotid Duplex Protocol\n\n  * Look for presence or absence of plaque\n\n  * Note morphology of the plaque\n\n    * Calcified (echogenic) or echolucent or heterogeneous\n\n  * Degree of stenosis\n\n    * Based on spectral Doppler (angle-corrected)\n\n      * A quantitative measurement of hemodynamic changes\n\n    * Parameters include: peak systolic velocity (most important), end-diastolic velocity, and ICA\/CCA ratio\n\n      * Spectral broadening \u2013 evidence of turbulence\n\n      * ICA\/CCA ratio: peak systolic velocity in the ICA divided by the velocity in the distal CCA measured 2 cm proximal to the bifurcation.\n\n### Sample Carotid Diagnostic Criteria\n\nDiameter reduction | Peak systolic velocity (cm\/s) | ICA\/CCA ratio\n\n|   \n---|---|---|---\n\nNormal | <105 | <1.8 | No plaque\n\n1\u201339% | <105 | <1.8 | Plaque imaged\n\n40\u201359% | 105\u2013159 | <1.8\n\n60\u201379% | ?160 | 1.8\u20133.7\n\n80\u201399% | ?240 (EDV ? 135) | >3.7\n\nEDV end-diastolic velocity, ICA internal carotid artery, CCA common carotid artery\n\n### B (\"Brightness\" Mode): Useful for Plaque Analysis (Fig. 28.2)\n\nFig. 28.1\n\nColor Doppler image of the carotid bifurcation\n\nFig. 28.2\n\nAn 8\u201311 MHz linear transducer is typically used to image the carotid artery. This image illustrates plaque in the bifurcation\n\n### Color Doppler (Fig. 28.3)\n\nFig. 28.3\n\nLeft; Color Doppler image of the stenosis at the bifurcation. Right; Color Doppler image of the stenosis in the proximal ICA\n\nColor Doppler is useful for detecting areas of turbulence to help localize stenosis.\n\n### Waveform Morphology (Figs. 28.4 and 28.5)\n\nFig. 28.4\n\nThe spectral Doppler waveform in the internal carotid artery (left) is typical of a \"low resistance\" vessel with more diastolic flow. The waveform in the external carotid artery (right) is \"high resistance\" with minimal diastolic flow\n\nFig. 28.5\n\nThe spectral Doppler waveform in the common carotid artery (left) shares features of both the internal and external carotid arteries which are downstream. The waveform in the vertebral artery (right) is \"low resistance\" with more diastolic flow similar to the internal carotid artery\n\n### ECA Versus ICA (Fig. 28.6)\n\nFig. 28.6\n\nIn addition to the characteristic waveform morphology, several features help to distinguish the external carotid artery (ECA) from the internal carotid artery (ICA): (1) the ECA is typically smaller; (2) the ECA has branches (left); (3) oscillations can be seen in the ECA when the temporal artery is tapped, referred to as the \"temporal tap\" (right)\n\n### Vertebral Artery Waveform in Subclavian Steal (Fig. 28.7)\n\nFig. 28.7\n\nSubclavian stenosis is suspected in a patient (most commonly asymptomatic) with a differential in brachial pressures of more than 12\u201315 mmHg. Retrograde flow may be seen in the vertebral artery (as shown here) in the presence of subclavian stenosis which is termed \"subclavian steal.\" If accompanied by symptoms such as syncope or vertigo, it is referred to as \"subclavian steal syndrome\"\n\n### Carotid Stenosis (Fig. 28.8)\n\nFig. 28.8\n\nThis image illustrates 80\u201399% stenosis of the proximal internal carotid artery with a peak systolic velocity of 286.9 cm\/s; the ICA\/CCA ratio was 4.4. Turbulence is seen on color Doppler and spectral broadening is seen in the spectral Doppler waveform\n\n## 28.5 Abdominal Aorta Ultrasound\n\n### Abdominal Aortic Aneurysm (AAA)\n\n  * The abdominal aorta is easily imaged in the echo lab and can be seen with the cardiac transducer.\n\n  * A curvilinear convex (2\u20134 MHz) transducer is preferred.\n\n  * The patient must fast for 6\u20138 h prior to the exam to limit the interference of bowel gas.\n\n### AAA\n\n  * In general, an aneurysm is defined as an increase in the diameter of any artery that is 1.5 times normal.\n\n  * For example, the abdominal aorta is normally 2 cm in diameter, and AAA is defined as an aortic diameter greater than 3 cm.\n\n### Aortic Ultrasound (Fig. 28.9)\n\nFig. 28.9\n\nAortic measurements are obtained in both transverse (left; 5.9 cm aneurysm) and longitudinal (right; normal aorta) views. It is common to see plaque or thrombus within an aneurysm (left)\n\n### Aorta Protocol\n\n  * Transverse AP (anterior-posterior) and longitudinal AP measurements as well as spectral Doppler waveforms should be obtained:\n\n    * In the proximal aorta near the superior mesenteric artery\n\n    * In the mid aorta near the renal arteries\n\n    * In the distal aorta near the bifurcation\n\n    * In the bilateral common iliac arteries\n\n  * Normal aortic velocities are <100 cm\/s\n\n## 28.6 Research Techniques in Vascular Imaging\n\n  * Carotid intima-media thickness\n\n  * Brachial reactivity\n\n  * Augmentation index\n\n### Carotid Intima-Media Thickness\n\n  * Carotid Intima-Media Thickness (CIMT) measurement is a non-invasive ultrasound technique used most commonly in the assessment of subclinical atherosclerosis.\n\n  * Increased CIMT correlates well with cardiovascular risk.\n\n  * Sensitive and reproducible; protocols vary.\n\n### CIMT\n\n(Fig. 28.10)\n\n  * Once the CIMT measurement is obtained, it must be compared to an appropriate reference population depending on the protocol.\n\n  * Specialized \"edge-detection\" software and three-lead EKG monitoring for gated images are recommended.\n\n  * For further reference on CIMT, see the Consensus Statement of the American Society of Echocardiography (Stein et al., J Am Soc Echocardiogr. 2008).\n\nFig. 28.10\n\nThe best method is to obtain R-wave gated images of the far wall of the distal common carotid artery. An example of CIMT measurement is shown here\n\n### Brachial Reactivity\n\n  * Brachial reactivity testing is used to measure endothelial function, a response referred to as \"flow-mediated dilatation\" (FMD).\n\n  * A blood pressure cuff is inflated to approximately 250 mmHg for 5 min and then rapidly released to induce reactive hyperemia; dilatation of the brachial artery occurs in response to shear stress.\n\n  * High-frequency ultrasound images of the brachial artery are gated to the R wave of the QRS complex.\n\n  * Reduced flow-mediated dilatation (FMD) is thought to be a sign of endothelial dysfunction and correlates with cardiovascular risk.\n\n### Augmentation Index\n\n  * Augmentation Index (AI) is a surrogate measure of systemic arterial stiffness.\n\n  * AI tends to be negative in young people, zero at approximately 35 years of age, and positive thereafter.\n\n  * Applanation tonometry may be employed at the radial artery to derive an ascending aortic pressure waveform.\n\n  * The AI is calculated as the difference between the first and second systolic peaks of the ascending aortic pressure waveform.\n\n  * The central pulse pressure is the systolic pressure minus the diastolic pressure.\n\n  * The AI is expressed as a percentage of the central pulse pressure.\n\nIndex\n\nA\n\nAbdominal aortic aneurysm (AAA)\n\nAmerican Society of Echocardiography Guidelines 2003\n\nAmyloid cardiomyopathy\n\nAorta\n\nabdominal\n\natherosclerosis disease\n\nconditions affecting\n\nfunctional assessment\n\nimaging\n\nthoracic\n\nvascular function\n\nAortic dissection\n\nclassification\n\nthrombus, false lumen\n\ntransesophageal echo view\n\ntype A proximal\n\ntype B cross-section\n\nAortic plaque\n\nAortic pulse velocity waveform\n\nAortic regurgitation\n\netiologies of\n\ngrading and assessment\n\nAortic stenosis\n\netiologies of\n\ngrading severity\n\nlow gradient and low output\n\nAortic valve\n\npathology\n\nprosthesis\n\npulse Doppler measurement\n\nAortic valve area (AVA)\n\nAsymmetric septal hypertrophy\n\nAtherosclerotic disease\n\nAthlete's heart, left ventricular hypertrophy\n\nAtrial myxoma\n\nAtrial septal defect (ASD) closure\n\nICE\n\nlarge secundum ASD, TEE view\n\nAtropine\n\nAttenuation\n\nAugmentation index (AI)\n\nAV optimization protocol\n\nAxial resolution\n\nB\n\nBall-cage prosthesis\n\nBeam characteristics\n\nBicaval orthotopic heart transplant\n\nBicuspid aortic valve\n\nBileaflet mitral prosthesis\n\nBioprosthetic valves\n\ncharacteristics\n\ntricuspid valve replacement\n\nBiplane method of disks\n\nBiplane Simpson method\n\nBurkitt B cell lymphoma\n\nC\n\nCardiac masses and tumors\n\nbenign\n\nBurkitt B cell lymphoma\n\ncategories\n\nfibroma\n\nlipomatous hypertrophy TEE bicaval view\n\nmalignant\n\nmimickers\n\nmyxoma\n\npapillary fibroelastoma\n\nsarcoma\n\nvalvular papillary fibroelastoma\n\nCardiac optimization\n\nAV optimization\n\nPV optimization\n\nscheduling requirements\n\nVV optimization\n\nCardiac tamponade\n\nechocardiographic features\n\ntransvalvular flow\n\nCardiomyopathy\n\namyloid\n\nconstrictive vs. restrictive\n\ndilated\n\ncauses\n\nechocardiography\n\nIVC diameter\n\nmitral regurgitation\n\nventricular thrombus\n\nhypertrophic cardiomyopathy ( see also Hypertrophic cardiomyopathy)\n\nasymmetric septal hypertrophy\n\ncharacteristics\n\nconditions mimicking\n\ndiagnostic criteria\n\nejection dynamics\n\nLVOT obstruction\n\npatterns\n\nprovocative maneuvers\n\nstress cardiomyopathy\n\napical thrombus\n\napical variant\n\nbasal variant\n\ncharacteristics\n\ndiagnosis\n\nechocardiographic evaluation\n\nmid-ventricular variant\n\nRV involvement\n\nCarotid Duplex\n\ncolor doppler\n\nECA vs. ICA\n\nindications\n\nplaque analysis\n\nprotocol\n\nsample diagnostic criteria\n\nstenosis\n\nsubclavian stenosis\n\nwaveform morphology\n\nCarotid intima-media thickness (CIMT)\n\nCarotid stenosis\n\nCinefluoroscopy\n\nCircumferential strain profile\n\nClicks\n\nColor-coded tissue Doppler (TDI) imaging\n\nColor flow Doppler imaging\n\nColor flow imaging\n\nColor M-mode flow propagation velocity\n\nComprehensive Doppler assessment\n\nright ventricle echocardiography\n\nright ventricle hemodynamics\n\nCongenital heart defects (CHD)\n\natrial septal defect closure\n\npatent ductus arteriosus closure\n\nventricular septal defect closure\n\nConstrictive pericarditis\n\nvs. restrictive cardiomyopathy\n\ntissue and pulsed wave Doppler\n\nventricular filling\n\nContinuous wave (CW) Doppler\n\nContrast agents\n\nadministration methods\n\ndefinition\n\nFDA recommendations\n\nindications, clinical practice\n\nroutine study\n\nsafety\n\nstress\n\ntechniques, problems, and solutions\n\nin use\n\nContrast echocardiography\n\nD\n\n2DE. See Two-dimensional echocardiography\n\n3DE. See Three-dimensional echocardiography\n\nDeBakey aortic dissection classification\n\nDefinity\n\nDiastole phases\n\nDiastolic dysfunction\n\nechocardiographic parameters\n\ncolor M-mode flow propagation velocity\n\nmitral inflow velocities\n\nmorphologic and functional \u00adcorrelations\n\npulmonary vein inflow velocities\n\ntissue Doppler velocities\n\nleft atrium (LA)\n\nand left ventricular filling pressures\n\npatient evaluation\n\ndifferential diagnosis and subtypes\n\nelevated filling pressures\n\ngrading\n\nphysiology\n\nDiastolic dyssynchrony measurement\n\nDiastolic function\n\nDilated cardiomyopathy\n\ncauses\n\nechocardiography\n\nIVC diameter\n\nmitral regurgitation\n\nventricular thrombus\n\nDobutamine\n\nDoppler-derived strain rate\n\namplitude\n\nmagnitude\n\nvelocity gradient\n\nDoppler echocardiography\n\n2D strain speckle tracking\n\nDynamic left ventricular outflow tract obstruction\n\nE\n\nEccentricity index, right ventricle assessment\n\n2D echocardiography\n\n3D echocardiography\n\nExternal carotid artery (ECA)\n\nF\n\nFibroma\n\nFlow-mediated dilatation (FMD)\n\nFrame rate\n\nFrequency\n\nG\n\nGlobal LV systolic function\n\nGrading diastolic dysfunction\n\nH\n\nHeart failure with normal ejection fraction (HFnlEF)\n\ncauses\n\npatient evaluation\n\ndifferential diagnosis and subtypes\n\nelevated filling pressures\n\ngrading\n\nHeart transplantation\n\nechocardiography\n\ntransplant rejection\n\nHemolysis\n\nHypertrophic cardiomyopathy (HCM)\n\nasymmetric septal hypertrophy\n\ncharacteristics\n\nconditions mimicking\n\ndiagnostic criteria\n\nejection dynamics\n\nLVOT obstruction\n\npatterns\n\nprovocative maneuvers\n\nsystolic anterior motion\n\napical hypertrophic cardiomyopathy\n\nleft ventricular outflow tract\n\nmitral regurgitation\n\nmitral valve leaflet\n\nI\n\nImaging artifacts\n\nrange ambigity\n\nreverberation\n\nshadowing\n\nside lobes\n\nsuboptimal images\n\nInfective prosthetic endocarditis\n\nInternal carotid artery (ICA)\n\nInterventricular dyssynchrony\n\nIntraoperative transesophageal echocardiography, 107\n\nadjuvant procedure during mitral valve surgery\n\naortic valve pathology\n\nintraoperative diastolic function assessment\n\nmajor vessels\n\nmid-esophageal bicaval view\n\nmid-esophageal level\n\nmitral valve pathology\n\nprosthetic valves\n\nL\n\nLeft atrial pressure (LAP)\n\nLeft atrium (LA)\n\nLeft ventricular dyssynchrony\n\nalgorithm\n\nechocardiographic techniques\n\nmeasuremnet\n\npitfalls\n\nresynchronization\n\nLeft ventricular ejection fraction (LVEF)\n\nLeft ventricular end diastolic pressure (LVEDP)\n\nLeft ventricular opacification (LVO)\n\nLeft ventricular (LV) quantification\n\ndirect 2D methods\n\nmass calculation\n\nM-mode dimensions\n\npathologic vs. physiologicl LVH\n\nsystolic function\n\nvolumetric assessment\n\nwall thickness\n\nLeft ventricular systolic function\n\nglobal\n\nregional\n\nLength-area volumetric method\n\nLevovist\n\nLoculated pericardial effusion\n\nLongitudinal resolution\n\nLVEF. See Left ventricular ejection fraction\n\nM\n\nMechanical dyssynchrony assessment\n\ndiastolic dyssynchrony\n\ninterventricular dyssynchrony\n\nleft ventricular\n\nmeasurement\n\nresynchronization\n\nROC curve analysis\n\n4-segmental model\n\n12-segmental model\n\nTDI curve\n\nMechanical prosthetic valves\n\nacoustic shadowing\n\ncharacteristics\n\nMicrocavitation\n\nMid-esophageal view, TEE\n\nartifact\n\nfour chamber\n\nleft atrial appendage\n\nlong axis\n\npulmonary artery\n\npulmonary valve\n\nright ventricular outflow tract\n\nshort axis\n\ntwo chamber\n\nMitral inflow velocities\n\nMitral prolapse regurgitation\n\ncolor Doppler jet area assessment\n\nfluid dynamics and measurement\n\nmechanisms of\n\nmultiple mechanisms of\n\nnormal anatomy\n\nqualitative and quantitative parameters\n\nquantitative measures\n\nvena contracta\n\nMitral stenosis\n\nDoppler assessment\n\nechocardiographic features\n\nplanimetry\n\nrecommendations for classification\n\nMitral valve\n\nanatomy\n\nassessment\n\nnormal\n\nreal time 3D echocardiography.\n\nscallop\n\npathology\n\nprolapse\n\nprosthesis\n\nMitral valve area (MVA)\n\nM-mode echocardiography\n\nModified Simpson biplane method\n\nMotion-mode echocardiography\n\nMyocardial deformation\n\nMyocardial perfusion (MCE)\n\nMyocardial relaxation\n\nMyocardial stiffness\n\nMyxoma\n\nN\n\nNoninvasive hemodynamic assessment\n\nmitral regurgitation velocity\n\nM-mode cursor\n\nnormal and abnormal relaxation patterns\n\npulmonary to system blood flow ratio\n\npulmonary vascular resistance\n\npulse Doppler, aortic valve\n\nstroke volume calculation\n\nNyquist limit\n\nO\n\nOne cycle\n\nOptison\n\nP\n\nPannus vs. thrombus\n\nPapillary fibroelastoma\n\nPatent ductus arteriosus (PDA) closure\n\nPathological left ventricular hypertrophy\n\nPatient-prosthesis mismatch (PPM)\n\nPeak systolic pressure\n\nPericardial effusion\n\ndiagnosis, pitfalls\n\nechocardiographic artifacts\n\nleft atrium\n\nloculated\n\nmetastatic deposits\n\npericardium, congenital absence\n\nvs. pleural effusion\n\ntransesophageal echocardiography\n\nPericardiocentesis\n\nPericarditis\n\nconstrictive\n\nvs. restrictive cardiomyopathy\n\ntissue and pulsed wave Doppler\n\nventricular filling, 291\n\ntransesophageal echocardiogram\n\nPerivalvular leak and valve dehiscence\n\nPlanimetry\n\nmitral stenosis assessment\n\nmitral valve area measurement\n\nPoint of maximal intensity (PMI)\n\nPressure half-time method\n\nPressure recovery\n\nProsthesis malfunction\n\nprosthetic aortic valve evaluation\n\nprosthetic mitral valve evaluation\n\nprosthetic valve regurgitation\n\nprosthetic valve stenosis\n\npulmonic prosthetic valve\n\ntricuspid prosthetic valve evaluation\n\nProsthetic valve echocardiograms\n\ncolor flow Doppler\n\n2D echo\n\nspectral Doppler\n\nProsthetic valves\n\naortic\n\nbio\n\ncharacteristics\n\nmechanical\n\nmitral\n\npulmonary\n\ntricuspid\n\nProximal isovelocity area (PISA) method\n\nPulmonary artery systolic pressure\n\nPulmonary capillary wedge pressure (PCWP)\n\nPulmonary valve prosthesis\n\nPulmonary vascular resistance (PVR)\n\nPulmonary vein inflow velocities\n\nPulmonic regurgitation\n\nDoppler parameters\n\nDoppler signal\n\nPulmonic stenosis (PS)\n\ncongenital\n\netiologies of\n\nPulsed wave (PW) Doppler\n\nPulsed wave tissue Doppler imaging\n\nPulse waveform velocity analysis\n\nPV optimization protocol\n\nQ\n\nQuantitative hemodynamic measurements\n\nR\n\nReal-time 3D echocardiography (RT3DE)\n\nexamination strategy\n\nmitral valve anatomy\n\ntransthoracic and transesophageal\n\nvolumetric assessment of left ventricular\n\nRegional LV systolic function\n\nscoring system\n\n16-segment model\n\nRelative wall thickness (RWT)\n\nResolution\n\nReverberation artifacts\n\nRheumatic mitral stenosis\n\nRight atrial pressure (RAP)\n\nRight ventricle echocardiography\n\nadvanced echocardiographic variables\n\ncomprehensive Doppler assessment\n\n2D echocardiographic variables\n\neccentricity index\n\nfractional area change\n\nsystolic dysfunction\n\nthree-dimensional echocardiography\n\ntricuspid annular plane systolic excursion (TAPSE)\n\nvolumes and ejection fraction\n\nS\n\nSarcoma\n\n4-Segmental model\n\n12-Segmental model\n\nShadowing artifact\n\nSide lobes artifact\n\nSonoVue\n\nSound\n\ndefinition\n\ninteractions with tissue\n\nSpeckle tracking, 64\u201365\n\nSpectral Doppler\n\nStanford aortic dissection classification\n\nStenotic mitral valve\n\nStrain imaging\n\nStrain rate profile\n\nStress cardiomyopathy (SCM)\n\napical thrombus\n\napical variant\n\nbasal variant\n\ncharacteristics\n\ndiagnosis\n\nechocardiographic evaluation\n\nmid-ventricular variant\n\nRV involvement\n\nStress echocardiography\n\ncontraindications\n\ndobutamine\n\nindications\n\ninterpretation\n\nmonitoring requirements\n\nmyocardial viability\n\nperformance\n\npurpose\n\nside effects\n\ntermination criteria\n\nvisual assessment\n\nStruts\n\nSubclavian stenosis\n\nSubvalvular aortic stenosis\n\nSupravalvular aortic stenosis\n\nT\n\nTakotsubo cardiomyopathy. See Stress \u00adcardiomyopathy\n\nTAPSE. See Tricuspid annular plane systolic excursion\n\nTDI. See Tissue Doppler imaging\n\nThoracic aorta\n\nThree-dimensional echocardiography (3DE)\n\napical window\n\n3D display\n\n3D orientation\n\nparasternal window\n\nright ventricle assessment\n\nTissue Doppler imaging (TDI)\n\ncolor-coded\n\ndyssynchrony analysis\n\npulsed wave\n\nwall motion\n\nTissue Doppler velocities\n\nTissue synchronization imaging\n\nTissue velocity\n\ninterrogation\n\nnomenclature\n\nTransducer\n\ndefinition\n\nflow in\n\nTransesophageal echocardiography (TEE)\n\naortic root\n\naortic valve\n\nascending aorta\n\nindications\n\nanterior leaflet mass\n\nanterior mitral valve leaflet mass\n\nmitral regurgitation\n\nmitral valve anterior leaflet\n\ntype I aortic dissection\n\ntype III aortic dissection\n\ninteratrial septum\n\nintraoperative\n\nadjuvant procedure during mitral valve surgery\n\naortic valve pathology\n\nintraoperative diastolic function assessment\n\nmajor vessels\n\nmid-esophageal bicaval view\n\nmid-esophageal level, 104\n\nmitral valve pathology\n\nprosthetic valves\n\nleft superior pulmonary vein\n\nmid-esophageal view\n\nartifact\n\nfour chamber\n\nleft atrial appendage\n\nlong axis\n\npulmonary artery\n\npulmonary valve\n\nright ventricular outflow tract\n\nshort axis\n\ntwo chamber\n\nmitral valve\n\npitfalls\n\npulmonary valve, biplane imaging\n\nsmall secundum ASD\n\ntransgastric views\n\nbiplane view\n\ninferior vena cava\n\nleft ventricle\n\nopen tricuspid valve\n\nright ventricular inflow and outflow\n\ntricuspid valve\n\nTransthoracic echocardiogram\n\nTransthoracic real-time 3D echocardiography\n\ndensity settings\n\nmodes of acquisition\n\nTricuspid annular plane systolic excursion (TAPSE)\n\nTricuspid regurgitation (TR)\n\necho parameters\n\netiologies of\n\nTricuspid stenosis\n\nTricuspid valve prosthesis\n\nTwo-dimensional echocardiography (2DE)\n\napical four-chamber view\n\nparasternal long axis right ventricular inflow view\n\nparasternal short axis view\n\nright ventricular inflow view, 222\u2013223, 226\n\nsubcostal four-chamber view\n\nTwo-dimensional speckle tracking-derived strain measurement\n\nU\n\nUltrasound\n\napical views\n\ncontrast agents\n\nadministration methods\n\ndefinition\n\nFDA recommendations\n\nindications, clinical practice\n\nroutine study\n\nsafety\n\nstress\n\ntechniques, problems, and solutions\n\ndefinition\n\nechocardiographic windows\n\nimage optimization\n\ninferior vena cava view\n\nparasternal short axis view\n\nparasternal windows\n\nright ventricular inflow view\n\nsubcostal views\n\nsuprasternal window\n\ntransducer\n\ntypes of resolution\n\nvascular imaging\n\nabdominal aorta, 8\u20139, 338\n\ncarotid Duplex ( see Carotid Duplex)\n\nconventions\n\ndiagnostic criteria\n\ntechniques\n\nUnicuspid aortic valve\n\nUniplane method of disks\n\nV\n\nValve bed abnormalities\n\nValvular papillary fibroelastoma\n\nVascular imaging\n\naugmentation index\n\nbrachial reactivity\n\ncarotid intima-media thickness\n\nultrasound\n\nabdominal aorta, 8\u20139, 338\n\ncarotid Duplex ( see Carotid Duplex)\n\nconventions\n\ndiagnostic criteria\n\ntechniques\n\nVelocity gradient\n\nDoppler-derived strain rate\n\nstrain rate measurements\n\nVena contracta, mitral prolapse regurgitation\n\nVentricular assist devices (VADs)\n\naortic valve function\n\napical short axis\n\nechocardiography\n\nparasternal long axis\n\nVentricular septal defect (VSD) closure\n\nVolumetric assessment of left ventricular\n\nbiplane method of disks\n\nlength-area method\n\nreal-time 3D echocardiography\n\nuniplane method of disks\n\nVV optimization protocol\n\nW\n\nWavelength\n\nWave truncation\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nRiptide Publishing\n\nPO Box 1537\n\nBurnsville, NC 28714\n\nwww.riptidepublishing.com\n\nThis is a work of fiction. Names, characters, places, and incidents are either the product of the author's imagination or are used fictitiously. Any resemblance to actual persons living or dead, business establishments, events, or locales is entirely coincidental. All person(s) depicted on the cover are model(s) used for illustrative purposes only.\n\nA Summer Soundtrack for Falling in Love\n\nCopyright \u00a9 2018 by Arden Powell\n\nCover art: Shayne Leighton, parliamentbookdesign.wordpress.com\n\nEditor: Carole-ann Galloway\n\nLayout: L.C. Chase, lcchase.com\/design.htm\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without the written permission of the publisher, and where permitted by law. Reviewers may quote brief passages in a review. To request permission and all other inquiries, contact Riptide Publishing at the mailing address above, at Riptidepublishing.com, or at marketing@riptidepublishing.com.\n\nISBN: 978-1-62649-852-5\n\nFirst edition\n\nOctober, 2018\n\nAlso available in paperback:\n\nISBN: 978-1-62649-853-2\n\nABOUT THE EBOOK YOU HAVE PURCHASED:\n\nWe thank you kindly for purchasing this title. Your nonrefundable purchase legally allows you to replicate this file for your own personal reading only, on your own personal computer or device. Unlike paperback books, sharing ebooks is the same as stealing them. Please do not violate the author's copyright and harm their livelihood by sharing or distributing this book, in part or whole, for a fee or free, without the prior written permission of both the publisher and the copyright owner. We love that you love to share the things you love, but sharing ebooks\u2014whether with joyous or malicious intent\u2014steals royalties from authors' pockets and makes it difficult, if not impossible, for them to be able to afford to keep writing the stories you love. Piracy has sent more than one beloved series the way of the dodo. We appreciate your honesty and support.\n\nWhen Kris Golding leaves his dusty Kansas hometown for a fresh start in New York, he thinks an apartment and a job are waiting for him. But when he finds neither, rather than admit defeat, he takes his chances busking\u2014and meets Rayne Bakshi of international rock band The Chokecherries. Rayne needs a new guitarist, and gives Kris his first break since leaving home.\n\nRayne wears makeup and glitter and thinks nothing of kissing Kris in front of twenty thousand screaming fans for the attention. Instantly infatuated, Kris begins to question whether he might have a crush on Rayne\u2014could he be bisexual? But since Kris originally claimed to be straight, Rayne's wary of getting involved offstage.\n\nAs their tour gains momentum, Kris's sexuality becomes the least of his troubles. Between his conservative brother hell-bent on \"rescuing\" him from his life of debauchery, a peacock that may or may not be the avatar of a cult god, and a publicity stunt that threatens to upend the band, Kris is definitely not in Kansas anymore.\nFor all the bands I've ever loved.\n\nAbout A Summer Soundtrack for Falling in Love\n\nTour Schedule\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nDear Reader\n\nAcknowledgments\n\nAlso by Arden Powell\n\nAbout the Author\n\nMore like this\n\nJune 12: Hershey, PA\u2014Hersheypark Stadium\n\nJune 13: Camden, NJ\u2014BB&T Pavilion\n\nJune 14: Newark, NJ\u2014Prudential Center\n\nJune 15: Charlottesville, VA\u2014John Paul Jones Arena\n\nJune 17: Grand Rapids, MI\u2014Van Andel Arena\n\nJune 18: Indianapolis, IN\u2014Bankers Life Fieldhouse\n\nJune 20: Orlando, FL\u2014Amway Center\n\nJune 21: Sunrise, FL\u2014BB&T Center\n\nJune 22: Tampa, FL\u2014Amalie Arena\n\nJune 23: Birmingham, AL\u2014Legacy Arena\n\nJune 24: New Orleans, LA\u2014Smoothie King Center\n\nJune 25: Dallas, TX\u2014American Airlines Center\n\nJune 26: The Woodlands, TX\u2014Cynthia Woods Mitchell Pavilion\n\nJune 27: Austin, TX\u2014Austin360 Amphitheater\n\nJune 29: Phoenix, AZ\u2014Talking Stick Resort Arena\n\nJune 30: Salt Lake City, UT\u2014Vivint Smart Home Arena\n\nJuly 1-6: Purple Sage Music Fest, NV\n\nKris Golding wasn't the kind of person to fly across the country without a plan. His was foolproof, or at least fool resistant\u2014but, like most plans, was quickly crushed by circumstances outside his control. When his plane took off from Kansas, he was content in the knowledge that he had a job and a couch, if not an apartment, waiting for him. By the time he touched down in New York, everything had gone terribly wrong.\n\n\"What do you mean she kicked you out?\" he asked his cousin as patiently as he could. The cell connection was choppy as he stepped off the bus in Manhattan after two hours on public transport.\n\n\"Well it's her place, right?\" Marty said, his voice wheedling over the line. \"It's her name on the lease. So when she found out about me and Maria, she called it quits. Threw all my stuff out in a garbage bag on the street.\"\n\n\"Wait, who's Maria?\"\n\n\"Somebody I work with at the club.\"\n\n\"The club where you said you had a job lined up for me,\" Kris clarified.\n\n\"Yeah, man, but I got fired today.\"\n\n\"What? Why?\"\n\n\"Cuz Maria, she's the owner's girl\u2014\"\n\nKris held the phone away from his ear to keep from throwing it into traffic. He took a deep breath before returning it to hearing range.\n\n\"So I don't have a job,\" he said, cutting off whatever his cousin had been saying. \"And your boss won't introduce me to his music-industry guys.\"\n\n\"I'm sorry, man, the timing, it's just\u2014\"\n\n\"And I don't have a place to stay.\"\n\nMarty made a helpless noise.\n\n\"Great. That's amazing. Thanks.\"\n\n\"Hey, I'm in the same boat, okay? You could be a little more sympathetic.\"\n\nKris ended the call and shoved his phone back into his pocket.\n\n\"Well fuck,\" he said, to no one in particular. He slung his duffel bag of worldly goods over his shoulder, his guitar case secure on his back, and set off into the streets of New York to figure out what to do next.\n\nThe misfortune of his situation didn't detract from New York's beauty, even as his nerves started to gnaw at him. The whole city looked glittering and pulsing with possibility\u2014the place where dreams came true. The buildings loomed high, scraping the clouds in endless panels of polished glass. Sure, everything was huge and impressively daunting, but Kris hadn't come all the way out here to admit defeat, no matter how intimidating the city proved.\n\nHe had two hundred dollars in his bank account and a pocketful of loose change. It was barely enough to buy a plane ticket home, but his parents would spring him one if he asked. He would work in his dad's garage to pay them back, and he'd have to keep his tail between his legs for the rest of his life because the one time he'd tried to get out and make something of himself, he'd come crawling back barely six hours later. His parents could be as supportive as they liked, but it wouldn't change the fact that he'd tried and failed. And then there was his brother, Brad, who had told him New York was a bad idea.\n\n\"What's wrong with Kansas?\" Brad had demanded. \"You can make an honest living here. All those big-shot music-industry guys you look up to, they've never done a real day's work in their lives. There's nothing but freaks in that line of work, Kris. You're better off steering clear.\"\n\nKris couldn't bear Brad's smug face if he went back defeated.\n\nIf he had to be homeless, at least the weather was good. The June air was warm, the bits of sky he could see between the buildings were blue, and the sun was aggressively bright, like it had something to prove. There were a few hours left before sunset, and even if it cooled down overnight, he had a coat in his bag. He could weather it. He had enough money to stay at a hostel, but only for a night, and he was loath to spend more than he had to. Two hundred dollars and change wouldn't stretch far. No, he would sleep on a bench, and scour the streets for any hiring signs. He could work retail or fast food; he could find a shelter or a friendly couch to crash on until he'd saved enough for a place of his own. He wasn't going to wind up freezing to death on a street corner, gnawed on by rats, as soon as the season changed. He was going to get a job, and he would be fine.\n\nIt wasn't a great plan, but it was something.\n\nHe started walking.\n\nOne problem in applying for jobs was that they wanted an address on the application form. The bigger stores did background checks Kris couldn't pass without a permanent residence, and if they didn't, they weren't hiring anyway. The smaller shops didn't offer more than a few hours a week, and unless he wanted to juggle three conflicting schedules, he'd never make enough to feed himself, let alone get an apartment. His desperation mounted as store after store turned him away, leaving his stomach a knot of anxiety and his palms itching with sweat.\n\nHe ended up on a bench at the edge of Central Park watching the pigeons. They were fearless in the way rural birds never were, flashing their colors and strutting back and forth. They quickly determined that he had nothing of value to offer, and ignored him in favor of accosting passing pedestrians for crumbs. Rejected by one of the lowest of the city's inhabitants\u2014though a step up from the rats\u2014Kris again weighed his options.\n\nHe still couldn't go home.\n\nInstead, he texted his parents to tell them he'd landed safely, and he'd call them in a few days once he got settled. He was safe, after all; he might be close to panic, but they didn't have to be. He was an adult and he could look after himself. In theory, anyway. He had a bottle of water from the plane, an apple, and a granola bar. His nerves were like a guitar string wound up too tight and liable to snap at any second, and that put a damper on his appetite; what he had would last him till morning.\n\nHe settled in for a night on the bench. It wasn't the most comfortable place he'd ever been, but he couldn't say it was the worst, either. He didn't want to lie down in case someone recognized him for a vagrant and called the cops. If he looked like he was just resting a moment, or waiting for someone, he should be fine. He was dressed well enough. No one could tell that he didn't have anywhere else to go.\n\nHe crunched through his apple, more for something to do than out of hunger. It tasted sour, but it kept him awake. The sky slipped into pink as the sun sank behind the buildings, lighting them up in a warm glow as it passed. Kris couldn't hate the city. He had chosen New York partly because his cousin was here\u2014working for a guy who had so many connections in the music industry that Marty had sworn he could set Kris up with a gig in no time\u2014but mostly because it was as far from rural Kansas as he could imagine. He loved his parents and he loved his town, but it had been stifling. He didn't want to work in his dad's garage until he was sixty. He might not know exactly what he did want to do with his life, but he was determined to find out.\n\nSo far he was finding out that park benches were uncomfortable, and city pigeons were more intimidating than any bird had the right to be.\n\nAs the sun went down and the crowds thinned, Kris wrapped himself in his coat and prepared for a long night. The smaller shops locked up for the night as the streets finally emptied. The moon blinked out between the buildings but the stars stayed hidden behind the haze of clouds and the solid blocks of skyscrapers. He paced for a bit, exploring the park with a measured gait, his hands in his pockets as he tried to keep his blood moving. When he couldn't stop his feet from dragging or his eyes from closing, he found another bench and huddled down to roost again.\n\nIt didn't get cold, exactly, but it got cool, and the hours dragged into eternity and back. He drifted off once or twice, but the park was never fully deserted and he didn't trust that he'd wake up to find his bags still on him. He wouldn't particularly miss his clothes if the duffel bag got stolen, but his guitar\u2014he couldn't bear the thought of losing that, even if Marty had lost him his chance at a career with it.\n\nHe watched the sun rise from the wrong side of the morning for the first time since he was a teenager. The sky lightened in strips: silver, then pale yellow, before making room for blue. The pigeons cooed and rustled their wings, waking one by one and then all at once, and the subways rumbled underground, a constant churning noise. Straightening, he stretched his back until it gave a satisfying crack, and shuffled to his feet. The coffee he bought from a street vendor was black and bitter, and he was ready to shoot the caffeine straight into his veins if he had to. He was tired in a way that made everything feel thin and slightly unreal, like the world was hidden behind a film he could see through, but couldn't quite part.\n\nAfter his first swallow, he crossed out of the park to the nearest caf\u00e9, busy enough in the morning rush that he could slip into the washroom unnoticed. As he splashed cold water over his face, he hoped he didn't look too haggard. He could stretch it another day before needing to shave; his stubble always took a while to show through, and he was fair enough that it never gave him much of a five-o'clock shadow anyway. Coffee in hand, he returned to his bench, sipping the drink and trying to make it last.\n\nToday he would find somewhere to work. Twenty hours a week in retail might not be his dream of making a living off his music, but it would be better than nothing. He wasn't going to sleep on the bench again.\n\nThe pigeons looked skeptical.\n\nHe flipped them off and resumed drinking his coffee. The caffeine hit him like a ton of bricks, slamming into him all at once and setting his heart kicking behind his ribs. He took a deep breath and flexed his fingers. It was a drastic way to start the morning, but he was definitely awake now. The film separating him from reality peeled back and left him blinking into the sun like a newborn fawn, all wobbly-legged and uncertain.\n\nFind a job, he told himself. Take the first one you can. This isn't rock-bottom. This isn't even close.\n\nAll he had to do was wait for the stores to come back to life. Most would probably open around nine\u2014it was just past dawn. He had a few hours to kill. The coffee place across the road wasn't hiring, but maybe he could find a record shop or a music store that needed help.\n\nHe took out his guitar.\n\nHe couldn't say why he did it. There was no conscious thought behind the action. It was like the hand of some divine entity had reached down from the clouds and prodded him, right between the eyes, and said, Hey, you. Play me something. Music had always been like that for him, talking to him the way gods spoke to prophets: in the pure, undiluted language of the universe.\n\nHis guitar was a big acoustic thing, the wood polished until it glowed warm and gold. He'd bought it from a pawnshop for fifty bucks when he was thirteen; it had taken him all year to save up his allowances for it. In high school, he'd used to dream about his band making it big and playing in Madison Square Garden. It had been a couple of guys from his year and his little sister on drums. It fell apart before graduation and he'd put that particular dream aside, but he'd thought he could try his hand as a session musician, if nothing else. All he needed was one lucky break to get his foot in the door. He wouldn't call himself a prodigy, but he was good, and even if he'd never left Kansas to seek his fortune in the industry, he would have kept playing for himself.\n\nAs he tuned his guitar by ear, he ghosted his fingers over the frets, and gave it a strum. The chords rang out clear and true, and he drew in a deep breath as his nerves started to knit themselves together again.\n\nBack home he had played in front of live audiences, albeit small ones\u2014barbeques or house parties or just sitting out on the porch with his family in the summer evenings, strumming out chords overtop of the crickets. He had never busked before, though; his hometown was too small for it, and he'd never been as desperate for quick money as he was now. Easing into a bluegrass riff, half-remembered and half-invented, he kept his eyes down as he played. The song petered out after a few minutes, finding its natural ending like bluegrass always did, and he let the city eke back into his consciousness.\n\nSomeone tossed a crumpled bill into the case at his feet, and he smiled reflexively at the young woman who vanished back into the crowded street. He took a swig from his coffee and adjusted the guitar in his lap. Maybe one more song wouldn't hurt. One more song, and then he would start looking for music shops in the area.\n\nBy 9 a.m. he had thirty dollars in his case and a steady stream of attention from the park's passersby. By ten he had as much as he would have made from a day's work at any minimum-wage gig. His fingers ached and he was so hungry his stomach nearly drowned out the music with its complaints, but no one had yelled at him for loitering, and the pigeons hadn't tried to make away with his money. He couldn't stop smiling.\n\nHe laid his guitar down in the nest of bills and loose coins and closed his case. The little gathering he'd attracted gradually wandered off, one or two people pressing a last offering into his hand as they left. It was only when the path was clear that he noticed the one man who had yet to depart, leaning against the broad trunk of a tree, his arms crossed as he watched Kris unabashedly.\n\nKris tucked his last few bills in his pocket, raised his brows, and met the man's gaze head-on. In response, the man peeled himself from the tree and approached. He was tall, dark, and handsome, almost ridiculously so, with legs that went on for miles, and a wild mane of deep brown hair that stopped just above his shoulders. He was dressed all in black, chic rather than goth, his shirt open at the collar to reveal a hint of tattoos against the brown skin of his chest.\n\n\"Hi there,\" the man said. He slung his hands in his pockets and smiled, bright and easy. \"I was watching you play\u2014you're amazing.\"\n\n\"Thanks,\" Kris said. \"It wasn't a planned performance, but these things happen.\"\n\n\"Do you mind if I sit? I'm Rayne. Rayne Bakshi.\" Rayne offered a hand. He wore a ring on nearly every finger, and when Kris took it, his grip was warm and firm. He had a little mercury tattoo on the base of his left thumb.\n\n\"I was about to go find some food, but sure,\" Kris said. His stomach whined but he ignored it. There was something about the man that demanded his full attention, and a tingly feeling swept through Kris's veins, suggesting he was perched on the edge of something momentous. He didn't get that feeling very often; it felt careless to disregard it, especially in so strange a time as this. \"I'm Kris.\"\n\nRayne grinned and slid onto the bench beside Kris. \"I'll be quick,\" he promised. \"I wanted to ask you about your music, if you have a second.\"\n\n\"Okay,\" Kris said. Up close, he could make out a smattering of freckles across Rayne's nose.\n\n\"You've been out here for hours. Are you a busker?\"\n\n\"I'm between jobs,\" Kris said. \"Technically.\"\n\n\"Are you a professional musician?\"\n\n\"Not exactly. I was a mechanic. The stars didn't align when I tried to change tracks, so now I'm . . . whatever will have me, I guess. It's kind of a long story.\"\n\n\"But you'd like to be? A professional musician, I mean. I happen to need a guitarist, and here you are, so.\"\n\nKris stared at him. Rayne stared back. His eyes were the color of sea glass, a perfectly clear, pale green, startlingly light against the rest of his features. His eyelashes were unreasonably long. His face held no trace of insincerity; Kris must have misheard him.\n\n\"Sorry?\"\n\n\"Also a long story. Do you like burgers? There's this great place a few blocks over\u2014it's a bit early, but let me buy you brunch, and we'll talk.\"\n\n\"Sure,\" Kris said slowly.\n\n\"No pressure,\" Rayne assured him. \"But the food is really good.\"\n\nKris's stomach howled like a coyote, right on cue. \"Burgers sound great.\"\n\nHe wasn't one to turn down a free meal, especially not now with his money situation being what it was, but more than that, he was intrigued. Hope sparked in his chest; maybe he hadn't lost his chance after all.\n\nKris let Rayne carry his duffel bag, counting on him not to make a run for it. Rayne kept up a steady stream of chatter as they walked, which Kris, sleep-deprived and overcaffeinated, mostly tuned out. It didn't seem to matter to Rayne whether Kris joined in; he was perfectly capable of carrying the conversation by himself, which Kris appreciated. As the streets grew busier, Rayne slipped through the crowds like a fish, with Kris floundering in his wake.\n\nWhen Rayne had said \"burger place,\" Kris had not pictured anything like the restaurant Rayne took him to. It was long, with ambient lighting and private booths lining the walls. The entire back wall was an aquarium; the fish inside shimmered and flickered back and forth, lit up in purple and green. It was classier than any burger joint Kris had ever seen, and he faltered as he walked in. The rest of his body was preoccupied with getting fed, and didn't care about his insecurities. Rayne seemed oblivious on all counts, and led him to a secluded booth without letting up his one-sided conversation. Kris slunk in after him, trying to avoid eye contact with the staff. At the table, he stuttered as he read over the menu before settling on an innocuous-sounding cheeseburger and fries, and tried not to look at the cost. Unless they hand-reared the cows themselves, there was no reason for the prices to be so high. Rayne ordered a veggie burger with a salad, and returned the menus to the waitress with a smile that had the girl blushing and scurrying to the kitchen.\n\n\"So I have this band,\" Rayne said, twirling his straw between his fingers. \"And due to unfortunate circumstances involving a lot of heroin and some even worse life choices, our guitarist is out of commission.\"\n\n\"I'm sorry,\" Kris said.\n\n\"Don't be. His mom had to call us to explain that he's in rehab\u2014a week before we leave for tour. Don't get me wrong,\" Rayne added, \"I'm glad he's getting help, but his timing leaves a lot to be desired. Anyway.\" He took a sip of his water. \"Don't do heroin.\"\n\n\"Right,\" Kris said. \"No, that wasn't high on my list. When you say band\u2014\"\n\n\"The Chokecherries,\" Rayne said, as if that clarified anything.\n\nIt sounded vaguely familiar, but not enough to bet on.\n\n\"We've been auditioning people for the past six days, but no one's worked out yet. The label's threatening to send a session musician on tour with us, which would be fine, but only as a last resort. We're a band; we should have chemistry, you know?\"\n\n\"Right,\" Kris agreed. \"Of course. How popular exactly is your band?\"\n\nThe waitress arrived with their plates, blushing furiously again and unable to meet Rayne's eye. Rayne smiled like he was used to it, thanking her before she fled the scene. Rayne was attractive enough to cause that kind of reaction anyway, but Kris suspected he might actually be famous. Kris filed that away to deal with after eating. His cheeseburger was huge, nearly the size of the plate, and his mouth flooded with drool the second he smelled it. He chomped down a fry before returning his attention to Rayne, who looked amused.\n\n\"We're international,\" Rayne said. \"Where are you from?\"\n\nKris's breath caught for a second before he swallowed. International was a lot bigger than Marty's club owner could have offered. \"Kansas. I don't think I know The Chokecherries.\"\n\nRayne took a bite of his burger. He didn't seem in the least offended. \"Do you want to?\"\n\n\"I'll be honest with you: I don't have a lot else going on right now.\"\n\nRayne flicked through his phone for a second before handing it to Kris, who reluctantly set his burger aside. There was a video on screen, waiting for him to press Play.\n\n\"We have two albums out now,\" Rayne said. \"Our second went platinum; we're heading out on tour the day after tomorrow. The shows are already sold out. All we need is a guitarist.\"\n\nKris frowned and pressed Play. He could tell the music was good right away: the production quality was high, the guitar slick, and the bass throbbing. The band was dressed in leather jackets, torn jeans, and bright T-shirts, caught halfway between punk and impossible glamour. Their instruments caught the studio lights and flashed them around; their hair gleamed; and their makeup demanded attention. But it was the vocals that stood out above everything else, twining through the music, perfectly on key. Kris couldn't catch any hint of auto-tune on the track, but no one had that kind of range anymore, not since Mercury. The Rayne in the video closed his kohl-lined eyes and purred into the mike, his fingers wrapping around the stand as he sang about a satisfaction just out of reach.\n\n\"That drumbeat is really familiar,\" Kris said. \"I think my sister listens to you.\"\n\n\"Then she has good taste. So? Are you interested?\"\n\nAs the video ended and the screen went dark, Kris put the phone back on the table. The Chokecherries were a far cry from his old high school band, in aesthetic and musical talent both. They looked like a pantheon of young gods, and Kris was wearing a flannel shirt and worn-out converse sneakers. They also sounded different than any contemporary band he'd heard in a long time, seductive and aggressive all at once. Rayne sat across the table, watching him with an impossibly hopeful expression.\n\n\"I feel like this is a weird lucid dream or something,\" Kris said. \"Maybe I fell asleep out there after all.\"\n\n\"What can I say to convince you you're awake?\" Rayne asked. \"Actually\u2014what can I say to convince you to play guitar for us, too?\"\n\n\"For real?\"\n\n\"I watched you play for an hour this morning. You're miles beyond anyone we've auditioned so far.\" Rayne leaned forward, his hands folded before him on the table. He looked so earnest that Kris squirmed under the attention, and then immediately pretended he hadn't. \"You've got talent, Kris. Real, raw talent, the kind most people practice lifetimes to get close to. Whatever you want, I can give you\u2014just audition for my band.\"\n\n\"Whatever I want,\" Kris echoed dumbly. His day couldn't be more surreal if one of the fish leaped out from the tank and offered him three wishes.\n\n\"Fame? You got it. Money? I promise, we will get you a really nice contract. You want to travel? Meet your favorite band? Whoever it is, we can swing it, I guarantee.\" Rayne drummed his fingers against the table. His rings glinted in the low light of the restaurant. His nails were painted black, matte and immaculate. \"Whatever it takes to get you to drop everything else, I will make sure you get it.\"\n\nKris's head spun and he held up one hand. Rayne's mouth snapped shut instantly, and Rayne waited, all ears, for Kris's bargain.\n\n\"You pretty much had me at the burger,\" Kris said. \"Even if I had something else to drop, this is\u2014 You're serious? You want me in your band?\"\n\nRayne broke into a broad grin. \"I can keep buying you burgers, if that's all it takes. I can buy you a burger every single day. Ultimately, it's up to our manager to accept new members, so this isn't a legally binding arrangement just yet, but I think you could be the one.\"\n\n\"You might have to switch it up with pizza like, once a week or so. I wouldn't want to get sick of them.\" Kris ate another fry to give his heart a chance to return to normal speed. \"You don't even know me, though. You just saw me playing on a bench. I could be anybody.\"\n\n\"I can tell you're not a junkie; I've gotten pretty good at reading those signs, unfortunately. Am I wrong?\"\n\n\"No, I'm not into drugs.\"\n\n\"Are you an ax murderer?\"\n\n\"No.\"\n\n\"Cultist?\"\n\n\"No.\"\n\n\"Republican?\"\n\n\"God, no.\"\n\n\"Then I'm not going to worry too much,\" Rayne said. \"Tell you what\u2014we've got a private show tonight, before we hit the road. If you're interested, come meet the rest of the band, audition and charm our manager, play with us, then sleep on it. Okay?\"\n\n\"About that,\" Kris said, carefully studying the sesame seeds on his burger bun. \"Sleeping on it.\"\n\n\"I know it's short notice, but it's just a little show. We're only playing covers, so you don't even have to learn our songs.\"\n\n\"No, I mean, I'm kind of technically homeless? At the moment? So I don't really have anywhere . . . to sleep. As it were.\" Kris bit his lip and glanced at Rayne.\n\nRayne looked at him for a long moment, his expression intrigued. \"You did say it was a long story,\" he said eventually.\n\n\"On the plus side, it doesn't involve heroin?\" Kris offered. He tried to drink his water too quickly and nearly choked.\n\n\"How about we finish eating and you tell me all about it,\" Rayne suggested.\n\nKris shoved the rest of the burger in his mouth and nodded. \"So I have this cousin . . .\"\n\n\"I've never auditioned for a band before,\" Kris said. \"I played with some guys in high school, but we were just messing around. You guys are legit.\"\n\n\"Yeah, but unless you try to murder our manager or start spouting conspiracy theories about the music industry to the people who sign our checks, you'll be fine.\" They were sitting side by side in the back of a sleek black cab that matched Rayne's outfit as if it had been hand-picked for that reason. He smiled reassuringly at Kris from across the seat. \"Just play like you were doing back in the park, okay? You've got this.\"\n\nKris nodded. His stomach twisted itself into knots and set itself on fire. He fidgeted, lacing his fingers together and undoing them again. Rayne reached over and patted his knee. The touch sent sparks zinging up and down Kris's leg, and while it did nothing to loosen the Gordian knot inside him, he tried to leach a little comfort from it anyway.\n\nHe hadn't been this nervous when he'd planned to audition for that club owner, but there was something distinctly different between that imagined scenario and going to meet an actual international band. Kris didn't know The Chokecherries; he wasn't suddenly terrified of meeting his idols. This was just a bunch of musicians getting together to play some music\u2014and if they didn't like him, he'd go back to sleeping on the streets and begging for loose change to keep from starving to death. Or, somewhat less drastically, he'd go back home to live with his parents again. It was a totally no-pressure deal. Rayne had only met him an hour ago, and he believed in him. Kris could do this.\n\n\"You look like you expect them to eat you alive,\" Rayne said. \"They probably won't. Being a cannibal is on that same list as ax murderer and Republican, you know. Breathe.\"\n\nKris nodded and curled around Rayne's borrowed phone. Tucking his earbuds in, he skimmed through the music collection. He passed playlists of foreign music, their titles long and with more syllables than he was used to, as well as lists of more familiar punk and indie bands, before he found The Chokecherries. As they headed out of Manhattan to the studio in Queens, he focused on memorizing their singles. Kris watched every video, running through the playlist from start to finish, studying the faces until he could recognize all the band members at a glance. Rayne stood out first and foremost, with dark makeup on his eyes and glitter on his cheekbones, his nails black as he played with the mike cord. There was something dangerous about his stage persona that Kris couldn't see in real life\u2014something dark and predatory that made Kris's stomach flip, half-excited and half-terrified. Rayne could be an entirely different species, he was so far removed from anyone back home. Beside him, Rayne drummed his fingers against the door in a soundless beat.\n\nThe rest of the band was just as captivating. A Japanese girl with ashy hair played keyboards, her eyes and lips and nails all painted gold, her expression never wavering as her hands flew over the keys. On drums, a black dreadlocked man sat shirtless, tattoos dancing over his body as he moved like they were alive. Bass was taken by a slight, dark-haired figure nearly vibrating with energy, wielding the instrument like a weapon. And last\u2014there: the former guitarist who had landed Kris the opportunity in the first place. He was tall and lean, almost gaunt. Kris couldn't tell he was a junkie just from looking at him, but makeup could cover a multitude of sins. The man kept his eyes down, either rapt in the music or a million miles away.\n\nKris's gaze flickered back and forth between the Rayne on screen and the one beside him, trying to reconcile the two. It was disorienting, but no more so than the rest of Kris's time in New York so far. He could roll with it.\n\nThe songs slipped from sultry to sexual to aggressively upbeat and back again, and no matter what Rayne did with his voice, Kris wanted more. He didn't try to focus on the guitar just yet. Instead, he let the music wash over and through him as he tried to figure out what the band was all about.\n\nThey weren't going to eat him alive. They weren't. Rayne wouldn't let them.\n\nKris took a deep breath, closed his eyes, and let himself imagine being someone famous. He rekindled his old fantasy and pictured a sold-out crowd at Madison Square Garden, the heat from the stage lights, a roar of adulation\u2014nothing between him and the oblivion of a million screaming fans but his guitar.\n\nHe opened his eyes and tugged one earbud out to dangle around his neck. \"Rayne? I don't have an electric guitar.\"\n\nRayne glanced over. \"You can play one though, right?\"\n\n\"Sure, but I can't afford one unless your record label wants to give me an advance on signing.\"\n\n\"Oh, don't worry about that. I'll get you one this afternoon.\"\n\nKris's mouth went dry. \"Business expenses?\"\n\n\"It's all budgeted in,\" Rayne assured him. \"Don't worry about a thing. I got you.\"\n\nThe studio was a little building with a burly security guy at the door, who seemed to be doing his best to come across as a more intimidating feature of the brickwork. He nodded to Rayne and looked Kris up and down with an impressively impassive expression. Kris waved.\n\n\"This is Butch,\" Rayne said. \"He's our guard dog.\"\n\n\"Hi, Butch,\" Kris said.\n\nButch sighed. \"Get inside, Bakshi. They're waiting on this big surprise you said you had.\"\n\n\"Please don't say that's me,\" Kris whispered as Butch held the door for them.\n\n\"That's you! Hey guys, look what I've got!\"\n\nThe Chokecherries turned their heads as one.\n\n\"Is it a guitarist?\" the bassist asked.\n\n\"Got it in one! Everybody, this is Kris. Kris, this is everybody.\"\n\nThe band let out a chorus of hellos as they crowded near, inquisitive and smiling. Rayne caught Kris's hand and tugged him over to meet them halfway.\n\n\"This is Maki, Stef, Lenny, and Brian,\" Rayne introduced. \"Maki on keyboards, Stef on bass, and Len on drums. Brian's our manager. Guys, Kris is going to audition for guitar. I picked him up off the street and promised him none of you are cannibals.\"\n\n\"Hey,\" Kris greeted. He offered them an awkward wave, his other hand still trapped in Rayne's, which no one was commenting on. \"For the record, I never actually thought anyone was a cannibal, but it's nice to know for sure.\"\n\nThe band was less intimidating in real life than they were in the videos. They were dressed casually, hair down, without the flashy stage makeup. Stef wore a pair of thick-framed hipster glasses and a beanie. Everyone was in jeans and T-shirts. Only Brian, the manager, looked skeptical, his arms crossed as he eyed Kris over the tops of his glasses.\n\n\"You picked him up off the street,\" Brian said. \"Rayne . . .\"\n\n\"He just arrived in the city! Listen to him play before you judge me for adopting strays.\"\n\nBrian sighed and dropped his arms. \"It's not strays I'm worried about, it's getting involved with another Fink. Kris? You want to play us something?\"\n\n\"Fink?\" Kris asked.\n\n\"The last guy,\" Rayne supplied.\n\n\"Right.\"\n\nKris slipped away from Rayne, missing his warmth immediately\u2014there was comfort in having his hand held, but he needed to focus\u2014and took his guitar out. He slung the strap over his shoulder, fiddled with the tuning one more time just to have something to do, and shifted on his feet.\n\n\"So what do you want to hear?\"\n\n\"Whatever you want,\" Rayne said.\n\n\"Anything but 'Wonderwall,'\" Stef said. \"If you play 'Wonderwall' I'm vetoing Rayne's vote to bring you in.\"\n\n\"You don't have veto power,\" Rayne said, \"and we're only pretending this is a democracy.\"\n\n\"Ignore them,\" Brian cut in. He had a patient, long-suffering air about him that must have come from wrangling rock stars all day. \"Play anything.\"\n\nKris nodded, blew out his breath, and played.\n\nAs he played, everything else stopped. He didn't worry about money or employment or how his future hung on this one audition: there was nothing in his head but the music. His hands flew over the frets, the strings biting into his fingertips with every touch, and the music swelled to fill the studio with a rich, heavy sound.\n\n\"That's ours!\" Rayne exclaimed as Kris moved into the chorus. \"Did you learn that in the car ride over?\"\n\n\"It's not perfect,\" Kris said, letting the notes peter out. \"I didn't have time to memorize it; there are bits I had to make up as I went along.\"\n\n\"But you got all that by ear in forty minutes,\" Rayne said.\n\nKris shrugged. \"I guess?\" he offered sheepishly.\n\nHe transitioned to a seventies medley of Bowie, Queen, and T. Rex, and Rayne flung his arm around Brian's shoulders, beaming ear to ear. \"We can keep him, right?\"\n\n\"He's not high and he's not jonesing,\" Brian said reluctantly. \"I can tell that much just from looking. Kid? You got any bad habits we should know about? Anything likely to interfere with the band?\"\n\nKris shook his head, hugging his guitar like a shield.\n\n\"You know I wouldn't take that kind of risk this close to the tour,\" Rayne said. \"We're desperate, not suicidal. I've got a good feeling about him, Brian.\"\n\nBrian grunted. \"Kris? You want in?\"\n\n\"Please,\" Kris said. \"Not to guilt-trip you or anything, but the alternative is moving back in with my parents and embracing my Midwest heritage. Or living under a bridge somewhere. Just so you know.\"\n\nBrian winced. \"Let's try to avoid that. Here's what I'll do.\" Everyone else quieted. \"We'll take you on for a trial period. We've got sixteen shows between New York and the Nevada festival. If you make it through all that without a hitch, we'll talk about signing you on for good. But you make one wrong move, and I've got a session musician on speed dial who already knows all the tracks, and is just waiting to jump in and cover for us. All right?\"\n\n\"Okay,\" Kris said quickly. \"I'll be on my best behavior, one hundred percent. No wrong moves.\"\n\nBrian seemed skeptical, but he nodded. \"I'll get a contract drawn up tying you to the tour as far as Nevada.\"\n\nKris's heart skipped giddily and he tried not to let it show.\n\n\"Everybody on board?\" Rayne asked.\n\n\"The kid can play,\" Lenny acknowledged.\n\n\"Of course we're on board,\" Maki said.\n\n\"You'd pitch a fit if we weren't,\" Stef concluded.\n\n\"They're all thrilled to have you here,\" Rayne said, fixing the band with a stern look. \"Right?\"\n\n\"Yeah, we're just fucking with you,\" Stef said. \"You sound great. Not that Fink was a tough act to follow, but whatever.\"\n\nMaki shook her head. \"Fink was always a mess. I'm surprised he lasted this long.\"\n\n\"He could play guitar but he was no prodigy,\" Stef agreed. \"And all that fucking heroin was a pain.\"\n\n\"You shouldn't speak ill of the dead,\" Rayne said mildly.\n\n\"He's not dead; he's in rehab.\"\n\n\"Yeah, but he's dead to me. So, Kris\u2014no drugs.\"\n\n\"No, of course not,\" Kris promised. He glanced at Brian. \"Uh. Weed?\"\n\n\"Doesn't count,\" they all said in unison.\n\n\"No hard drugs,\" Rayne amended. \"Weed and party drugs are okay. Partake responsibly.\"\n\n\"I've never done anything besides weed.\"\n\n\"He's cute,\" Maki said decisively, and just like that, Kris was accepted into the fold.\n\n\"I'll pay you back for it as soon as I can,\" Kris said.\n\nThey stood on the sidewalk in front of the music store. The neon letters in the window read OPEN, and the display was full of bright, shiny new instruments Kris didn't have a chance in hell of affording anytime soon.\n\n\"Shut up and let me buy you a guitar,\" Rayne said pleasantly.\n\nInside, the store was low-lit but spacious. Aisles of cords, cables, speakers, and amps filled the front half of the room, giving way to keyboards as they moved farther in. Rayne steered him to the left side near the back where, tucked away in a little nook, the electric guitars and basses were covering every inch of the three walls. They were glossy and bright and came in every size, shape, and color, and Kris's eyes glazed over with want just looking at them.\n\nRayne nudged his shoulder. \"Find one you like.\"\n\n\"Any one?\"\n\nThey both glanced at the double-necked beast on the back wall. It had to weigh more than Kris.\n\n\"Maybe not that one,\" Rayne amended. \"Anything else.\"\n\nKris stepped into the nook with the sensation of stepping into a whole new world. He reached out to touch the nearest guitar. Rayne waited in the main body of the store, letting him explore on his own, for which Kris was grateful. The one-eighty degree spin his life had taken since landing in New York was still catching up to him, as dizzying as a hurricane and keeping him off-balance. Right now, looking at the wall of guitars and knowing he was going to take one of them home\u2014or out of the store, anyway, wherever \"home\" was\u2014he was nearly in tears.\n\nHe checked the nearest price tag and swallowed. His acoustic had cost him fifty bucks. These were significantly more.\n\n\"Pick one up, try it out,\" Rayne said. \"Play me something.\"\n\nKris forced his attention off the numbers and onto the guitars. He landed on a Squier Mini Strat in the back corner, and had lifted it from its hook before he was aware of his actions. The body was a dark cherry red, wood grains showing through black near the base. Rayne hummed in approval. The Fender was smaller than the others\u2014Kris was short and needed a guitar with a shorter neck to match. He knelt down, balancing it across his lap, and gave it a strum. Unplugged, the sound was faint, but the notes were true. He closed his eyes and plucked out a scale, skipping up and down the frets.\n\n\"You like it?\" Rayne asked. \"Plug her in and play me a song.\"\n\nThere was an amp waiting in the corner beside a stool and a pair of headphones. Kris plugged in and spun out a riff to match the Green Day song on the overhead speakers, and Rayne grinned in return. The guitar felt warm and alive under his hands. He liked the look and the feel of it\u2014it was just the money niggling at him.\n\n\"You want to try any others?\" Rayne asked.\n\n\"This one's talking to me.\"\n\nThe longer Kris sat with it, the more attached he got. He played until Rayne wandered off to look at something, and as soon as his back was turned, Kris snuck a peek at the price. It wasn't the most expensive guitar on the wall; maybe it was even the cheapest. That didn't make it affordable.\n\nThe cherry-red finish glinted up at him.\n\n\"You sure you don't want to get me a used one somewhere?\" Kris asked, calling Rayne back.\n\n\"You looked at the price, didn't you?\"\n\n\"Maybe.\"\n\n\"You make it hard for people to buy you stuff, you know that?\" Rayne said. \"You like it, right? Best out of all the ones they've got?\"\n\n\"Yeah, but\u2014\"\n\n\"No buts.\" Rayne nudged him until he stood. \"You're in the band; you need a guitar. I promise, the label can afford a decent instrument or two.\"\n\nKris shifted, cradling the guitar in his arms like a baby, and relented.\n\nRayne grinned. \"You'll get used to the whole rock-star thing. Wait till we hit LA. You're allowed to be a diva there. Hell, you're expected to be.\"\n\n\"I'm from Kansas. We don't have divas or rock stars there. We have wheat. And cows.\"\n\n\"Yeah, you look like a cowboy kind of guy.\" Rayne held out his arm, beckoning Kris in. Kris stepped up and Rayne caught him around the shoulders, steering him out of the nook. \"Come on. We've got a show tonight.\"\n\nThey ended up at the register with the guitar, a hard case, a strap, and a pack of extra strings\u2014Rayne said that Fink had left behind his old amplifier, and Kris was more than welcome to inherit it.\n\n\"Hey, Rayne,\" the cashier greeted them. \"Find everything okay?\"\n\n\"This is baby's first electric guitar,\" Rayne said, ruffling Kris's hair.\n\n\"Your boy's got good taste,\" she said, ringing up the instrument. \"You guys have fun?\"\n\n\"I think I'm in shock,\" Kris said. \"Should I wait outside while you pay for this?\"\n\n\"Don't be dumb,\" Rayne replied as the girl punched the numbers in. \"After this I'm buying you coffee\u2014one of those fancy ones with the ridiculous names and ten hundred toppings.\"\n\n\"Thanks,\" Kris said helplessly.\n\n\"Get used to it, babe. You're living in the fast lane now.\"\n\nThe total flashed up on the register, and Kris broke out in a sweat and had to look away. Rayne paid with his business account without batting an eye, and Kris was only allowed to carry the stuff to their car after fighting Rayne for it first. Butch, who'd been waiting patiently behind the wheel of the '66 Mustang, put the top down and drove them around the block to the nearest coffee shop. Kris bolted from the car, vaulting over the door before Rayne had undone his seat belt, and flung himself into the shop, adamant that he would get in line and pay before Rayne could.\n\nThe barista spelled both their names wrong, and Rayne laughed at him the entire drive back to the studio. Somehow, that didn't stop Kris from enjoying his drink, though he suspected the caffeine was going to fry his nerves. They could get fried\u2014the important thing was staying awake to play the show.\n\n\"You good?\" Rayne asked as they pulled back into the studio lot, twisting in his seat to look back at Kris. Butch killed the engine and disembarked, warning them not to linger too long.\n\nKris rolled his cup between his hands and breathed. \"Yeah, I'm okay. I'm just trying to wrap my head around everything. I feel like I left Kansas in the real world, and I landed in some parallel dimension where everything is just . . .\" He met Rayne's gaze. \"Wild. Unbelievable. And good? But like.\" He let the condensation from the cup drip over his fingers. Rayne smiled encouragingly. \"I have no idea what I'm doing,\" Kris finished.\n\n\"I didn't give you a lot of time to think about it\u2014\" Rayne began, but Kris cut him off with a shake of his head.\n\n\"I want this,\" he said firmly. \"This could be the best thing that's ever happened to me. You met me a few hours ago, and you've already bought me a guitar, man! This isn't how real life works. I should be working twenty hours a week as a cashier making minimum wage in some crappy convenience store, not\u2014\" He gestured up and down to Rayne.\n\n\"The guitar suits you,\" Rayne said. \"You look good with it.\"\n\n\"It outclasses me by a mile.\"\n\n\"You wouldn't have picked it if you thought it wouldn't work for you, and you wouldn't have blown that audition out of the water if you weren't meant for this.\"\n\n\"I want it to suit me,\" Kris said. \"I think it could, but . . . maybe a version of me that doesn't exist yet.\"\n\n\"It will,\" Rayne promised. \"Maybe you think I'm nuts for offering you all this when I barely know you, but listen\u2014I have a good instinct for people. When I saw you in the park, even before I heard you play, I knew you were worth taking a chance on. You're pure potential right now, and we're going to shape you into something incredible. Besides, Brian won't let you fuck up too badly.\"\n\nKris thought about the little red guitar in the trunk of the Mustang and tried to imagine himself onstage with The Chokecherries, in leather and makeup and too much glitter, playing for a crowd of thousands\u2014his old dream, glimmering on the brink of coming true. His head spun at the thought of it.\n\n\"That,\" Rayne said. \"Whatever you were thinking right there. That's what we're going to make you into.\"\n\n\"I'm in.\" Kris's mouth was dry and his heart beating too fast. \"Just tell me what to do.\"\n\nRayne described the show as \"intimate,\" but the venue looked plenty big to Kris. The set list had twelve tracks, all covers the band had chosen once it became apparent Fink wasn't coming back and they couldn't train a new guitarist on their original material in time. Kris knew all the songs and could play most of them; those he couldn't, he was confident he could fake. Bowie, the Rolling Stones, Nirvana\u2014they were so deeply ingrained in Kris's mind he could play them in his sleep. He'd cut his teeth on The White Stripes' song, learning to play back in middle school.\n\n\"You're going straight from Britney Spears to Nine Inch Nails?\" he asked.\n\n\"Have to keep them guessing,\" Rayne said. \"They'll love it.\"\n\nIt was already 5 p.m. The show was slated to start at eight. Kris was running on caffeine and not much else, and he hadn't played with a real live band in ten years. Brian walked him through all the equipment as the others warmed up. The tech was familiar, albeit fancier and more expensive than Kris was used to, but muscle memory kicked in and he managed well enough, even if he felt like a kid pretending to be a grown-up.\n\n\"You're fine,\" Brian determined. \"We'll get you in better shape by the time we hit the road, but you'll do for now.\"\n\n\"Right.\" Kris took a deep breath. \"I've got this.\"\n\n\"Ready?\" Rayne asked.\n\nKris shifted his guitar around until it felt like part of his body and nodded. \"Okay. Go.\"\n\nHe kept to himself at first, concentrating on the music and the equipment and learning how the different players fit together. The Chokecherries were so familiar with one another they could communicate without words, and Kris felt left behind\u2014like he was stuck on the ground trying to understand the beauty of a flock of birds. Maki and Lenny were the easiest to pin down; they stayed by their stations, Lenny emanating quiet ease while Maki seemed effortlessly cool, keeping to herself until she needed to trade sarcastic asides with Stef. Stef, meanwhile, took up every inch of space they could manage, strutting back and forth, their hair falling in their face as they thrashed their bass around with a palpable aggression.\n\nAnd then there was Rayne.\n\nHe stayed near his mike stand, giving directions and suggestions between tracks, sometimes pacing, but with nowhere near the energy Kris had seen in the videos. Rather than making the rehearsal feel dull in comparison to them, the air was thick with anticipation, like the room was waiting for the tiger to burst from its cage.\n\nBy the time they had run through every track, it was one hour till showtime. They tucked themselves away backstage in a single dressing room, waiting for the audience to pile in.\n\n\"It's a going-away show,\" Rayne said. \"Mostly friends and family. It's really small; a few hundred people, tops. We're not even dressing up for it.\"\n\nKris nodded and pretended he wasn't being eaten alive by nerves. The others were all comfortably settled in their preshow rituals, and he didn't want to interrupt to demand reassurances. He knew the songs, could use the equipment, and the rehearsal had gone fine. Rayne believed in him. Kris swallowed his anxiety, wiped his palms on his jeans, and smiled. It felt shaky, but no one questioned it. He spent the last hour as a ball of nervous energy, pacing backstage, tuning his guitar, trying to sit, drinking too much water and then running to the bathroom every five minutes. He could hear the venue starting to fill, a low thrum of voices and bodies behind the curtain, waiting for the band.\n\nHe left his flannel shirt folded over the back of the dressing room couch and hoped he didn't look too out of place. The Chokecherries were exotic even in plain clothes, and Kris wondered if he might have preferred taking the stage for the first time behind a mask of makeup and high fashion.\n\n\"You good?\" Rayne asked brightly.\n\n\"Yeah,\" Kris croaked, holding his guitar in a death grip.\n\n\"It's going to be great.\" Rayne slung an arm around Kris's neck and toppled him off-balance into a hug. \"They're going to love you.\"\n\n\"Right. No, totally. I'm not even nervous.\"\n\nRayne flashed him one last smile before the voices reached a crescendo and he lunged forward to take the stage. The sudden well of screams\u2014a few hundred, but god, the noise\u2014was the last thing Kris heard before he followed Rayne into the blinding lights.\n\nKris couldn't remember a single detail about the show once it was over. As soon as Rayne was done with the introductions\u2014\"The Chokecherries\" left a rich taste in Kris's mouth: sweet as honey, and just a little bit dirty\u2014and they launched into their first song, his mind went blank and all that was left was the music. It passed like a fever dream: a rush of sound and color that was over as quickly as it had begun. He remembered the feel of the strings under his fingers, and the way the floor vibrated with the heartbeat of the drums, but little else. He couldn't have answered to his own name, but he knew Rayne's. The crowd chanted it like a prayer.\n\nHe returned to himself when he staggered backstage, disoriented and high on adrenaline. Setting his guitar on the couch, he dropped down beside it like his strings had been cut. His ears were ringing and his knees were weak. He'd never felt more alive in his life.\n\nComing in after him, Rayne was flushed and bright-eyed, his hair damp with sweat. He looked ecstatic.\n\n\"Did I do good?\" Kris felt slightly drunk, like all the stress had whooshed out of him during the show, and champagne bubbles had sprung up to take its place.\n\n\"Baby, you did great,\" Rayne said, hauling Kris to his feet to pass him around to the rest of the band like a party favor.\n\nThey all clapped him on the back and ruffled his hair, obviously exhausted and pleased with themselves. Lenny handed him a beer from the minifridge, and Kris cracked it open as they cheered him on. The audience had seemed to love the set and he hadn't messed anything up too badly\u2014not that he could remember, anyway\u2014but it was the band's approval he needed. He caught Rayne's eye amid the group, and Rayne smiled like a cat. Kris let out the breath he'd been holding all evening. He wasn't going to starve to death on the streets. He was going to play festivals and tour the world with a platinum-selling glam-punk band. It wasn't his Madison Square Garden fantasy, but it was better, because it was real.\n\nThey went straight from the show to a party at Rayne's penthouse in SoHo. Butch drove the band in a long black limo, slipping in and out of traffic as they left the venue behind. Sandwiched between Rayne and Stef, Kris wondered whether the never-ending somersault in his stomach was going to level out soon. It didn't seem inclined to, but then, maybe it was his body's way of telling him to stop drinking coffee and start considering solid food again sometime soon.\n\nInstead he was going to give it more alcohol.\n\nRayne's penthouse was bigger than any house in Kris's hometown, with floor-to-ceiling windows and a balcony overlooking the city. Stairs led to an open-concept loft where the bedrooms were housed, and framed art pieces colored the walls. An Indian Buddha statue sat on a table in one corner, gleaming gold and surrounded by leafy houseplants.\n\nA black girl with a huge halo of hair greeted them at the door. She had a bright smile and, like Rayne, was dressed to the nines, jewelry glinting at her throat and fingers.\n\n\"Angel, this is Kris, our new guitarist,\" Rayne said, putting his arm around her and kissing her cheek as they crossed the threshold en masse. \"Kris, this is Angel. She's my best girl, and she's coming on tour with us.\"\n\n\"Nice to meet you,\" Kris said.\n\nAngel offered her hand for a shake. Her nails were long and painted with glitter, and she smelled like vanilla and sugar cookies.\n\n\"He's a little bit drunk,\" Rayne said.\n\n\"No, I'm not.\" He was, a little. He'd made peace with his lightweight tendencies years ago.\n\n\"There's food and drinks in the kitchen,\" Rayne directed. \"More people are going to show up within the hour, so if you want to eat, get in there.\"\n\nHe shepherded Kris through to the kitchen with one hand on the small of his back, and Kris, pleasantly buzzed, let himself be steered. There was something reassuring about knowing he would never get lost in New York, because Rayne would never let him out of arm's reach.\n\nThe kitchen island was piled high with trays of nachos, pizza, and fruit and vegetable platters. Bowls of chips and dips lined the counters, with bottles of soda and booze and empty cups standing sentinel behind them. Kris grabbed a paper plate and piled it high. He hoped getting some food in him would take the edge off the tipsiness, but suspected it was half excitement.\n\nHe spent the next three hours in a daze of company, food, and booze. Midnight ticked past and the party grew from the band to friends of the band, and friends of friends, and relatives Kris couldn't keep track of. The girls were all flawlessly made up, charming and dazzlingly sharp, and the men all looked like models, nearly as made up as the girls. Rayne's social circle wasn't only model-beautiful, but beautifully androgynous too. Everyone wore designer clothes; even the punks looked groomed. Kris, used to faded denim and sneakers so old the soles were worn through, drank more and talked faster to keep up.\n\nHe could play the hell out of a guitar, but surrounded by all these glittering socialites, all stars or stars in the making, he felt like a moth in the company of butterflies. His hair was a nondescript brown, his eyes much the same. His mom called them hazel, but she was being generous. He was small, skinny\u2014\"pocket-sized,\" one of his old girlfriends had joked. Definitely nothing that would stand out in a crowd.\n\nJust after 1 a.m., he escaped to the balcony, desperate for some fresh air to clear his head, and ensconced himself on the porch swing nestled in amid a throng of plants.\n\n\"Hey,\" said Angel, approaching with a drink in her hand. \"How you doing?\"\n\n\"Hi.\" Whatever her perfume was, it was intoxicating. \"I'm good. I'm fine. I haven't slept in forty-two hours, but that's cool.\"\n\n\"Rayne said he found you on a street corner this morning. This your first time at a dig like this?\"\n\n\"This is my first time for a lot of things,\" Kris said. \"I'm breaking personal records for how many firsts I've had today. Do you want to sit?\"\n\nAngel slid onto the swing beside him and balanced her drink in her lap.\n\n\"Rayne said you're going on tour with them?\" Kris didn't want to jinx it and say us without signing a contract first.\n\n\"I do makeup and wardrobe. Rayne says it's my job to make everybody pretty.\"\n\n\"Everybody looks pretty pretty already, from where I'm standing.\"\n\nKris could tell when a guy was attractive, though he could never say that kind of thing back home. There, they took pride in their ruggedness, like moisturizer was an affront to their masculinity, and they'd rather die than get called pretty. Here the men preened like peacocks, and Rayne out-peacocked them all, with his earrings, heeled boots, and glittering makeup: the kind of beautiful Kris had never seen before but wanted to see a lot more of.\n\nAngel hummed knowingly and took a drink.\n\n\"You going to make me all fancy for the shows too?\"\n\n\"Yep,\" Angel said.\n\n\"Clothes and makeup and glitter and everything,\" he clarified.\n\n\"The whole shebang. You good with that?\"\n\nHe thought about it. He didn't know how he'd look, dolled up like the rest of the band was in their videos. Would he even recognize himself in the mirror?\n\nMaybe it was time for a change.\n\nThen again, he was a few drinks past tipsy.\n\n\"Sure.\" He shrugged easily. \"Sounds fun; why not?\"\n\n\"You'll be fine,\" she agreed.\n\n\"He didn't find me on a street corner, though. That sounds so trashy. I was on a park bench.\"\n\n\"Ah, you're right. That's much more respectable.\"\n\nShe bumped their shoulders together with a teasing smile, and Kris was almost drunk enough to think kissing her would be a good idea. The one sober part of his brain holding the fort cleared its metaphorical throat and suggested that if they were all going to be touring together, making a move on Rayne's self-described best girl wouldn't be the smartest move.\n\n\"You're really pretty,\" his mouth said, with zero input from his brain.\n\n\"Thanks.\" She was still smiling. \"You're pretty cute yourself.\"\n\n\"I told you so.\" Rayne came out through the back door, his boots clicking against the balcony floor. \"People told me you guys had hidden away out here. I came to make sure you were still having a good time.\"\n\n\"I needed some air,\" Kris said, smiling up at him. Rayne was as pretty as Angel, and just as kissable. Kris bet he knew how to moisturize.\n\n\"He was trying to decide whether to kiss me,\" Angel explained.\n\n\"Whoops. Did I interrupt?\" Rayne asked with a teasing grin.\n\n\"I wasn't,\" Kris said, and tamped down the urge to admit he would consider kissing Rayne too. \"I definitely was not. Also, I'm drunk.\"\n\n\"You are,\" Rayne said. \"You want to come back inside? Maki's making these mixed drinks that are, like, ninety percent sugar\u2014they're amazing. You have to try one.\"\n\nThe sober part of his brain, rapidly drowning in alcohol, shook its head. \"I should probably call it quits, actually. Try to get some sleep before morning.\" Kris froze. \"Um.\"\n\n\"Shit, I'm sorry, I never thought about where to put you tonight,\" Rayne said. \"I should have got you a hotel or something. I've got rooms upstairs if you want to crash here.\"\n\n\"Is that okay? I didn't even think\u2014 I'm still all jet-lagged and weird.\"\n\n\"Don't worry about it.\" He coaxed Kris to his feet, collecting his empty beer can from him as he went. \"Go get some sleep.\"\n\n\"Thanks.\" Standing brought Kris a fresh wave of dizziness, and with it, the realization that he was drunker than he thought. \"Bye, Angel. Night, Rayne.\"\n\n\"You can think about kissing me next time,\" Angel said with a wink.\n\n\"Nope,\" Kris said. \"Drunk brain. No kissing anybody.\"\n\n\"Good night, Kris,\" Rayne said, laughing. \"Don't hurt yourself.\"\n\nKris gave them a sloppy salute before heading back into the warm buzz of the party. He waved to everyone he recognized as he passed them and then headed up the stairs to the loft, one hand on the rail for balance. Upstairs was quieter, unspokenly off-limits to the partygoers, and Kris shimmied into the first bedroom he saw, flopping face-first onto the mattress without bothering to remove his shoes, which dangled over the end of the bed. As long as he left them hovering in the air like that, he wouldn't feel rude.\n\nIt was a good party, and he liked Rayne's friends, even if they outpaced and outclassed him. Maybe he could sleep for an hour and then rejoin them, or at least help clean up after. But then, maybe Rayne was rich enough to have a maid. Maybe Butch did cleaning on top of security and chauffeuring.\n\nPreoccupied by thoughts of rock stars and limousines, Kris tipped from drowsiness into sleep between one breath and the next.\n\nHe blinked awake, groggy and disoriented, when someone entered the room. The clock on the bedside table glowed soft and red, the numbers informing him it was 4:13 a.m. His mouth felt like something had curled up and died in it.\n\n\"Ughh?\" he said.\n\n\"Sorry. I didn't think\u2014\"\n\n\"Rayne?\" Kris lifted his head and tried to find a face in the shadows. Everything looked blue and fuzzy. \"Party over? Meant to come back downstairs. Sorry.\"\n\n\"It's okay. Go back to sleep, Kris.\"\n\n\"Am I in your bed?\"\n\n\"I'll take one of the guest rooms.\"\n\nKris frowned. That wasn't right. But before he could protest, Rayne was gone again, the door pulled shut behind him. Kris dropped back to the pillow and back into sleep, unconscious before Rayne's footsteps had faded.\n\nHe woke with the dawn, feeling less zombie-ish than he had at four, but not by much. He shuffled down the hall in search of the bathroom, mentally noting which room was Rayne's so he wouldn't make the same mistake twice. After, he headed downstairs, where he found the remnants of the party scattered over tables and counters, though the guests themselves must have left sometime in the night. He shrugged and found a garbage bag to shift greasy paper plates and food scraps into, moving from room to room as the sun rose. He helped himself to leftover pizza as he went, the cheese congealed but to his slightly hungover self, perfect. The remaining booze he capped and set aside, not looking at it straight on in case his stomach decided to object.\n\nKeeping busy stopped his brain from overthinking things, like how he'd wanted to kiss Rayne the night before. Sober, he had more important things to dwell on. They were leaving for the tour the next day, and Kris still needed to sign a contract and learn the songs. Could he afford an apartment in New York after the tour was over? Would he even want to stay in the city?\n\nHe found a broom tucked away beside the fridge and started sweeping the kitchen, fanning out to the living room and then the balcony, just in case. That was where Rayne found him\u2014Rayne, sporting an incredible show of bedhead and, judging by his bright eyes and easy smile, apparently unaffected by the previous night's events.\n\n\"Hey, Kris,\" Rayne said around a yawn. \"Are you sweeping my balcony?\"\n\n\"Maybe.\"\n\n\"Okay, cool. You want breakfast? Coffee? You can help yourself to anything in the fridge.\"\n\n\"I had pizza, but coffee would be amazing.\"\n\nRayne's nose scrunched up. \"The pizza that was sitting out all night?\"\n\nKris shrugged. It wouldn't kill him.\n\nThey returned to the kitchen, and Rayne got the coffee started\u2014the machine was huge and chrome and insanely complicated\u2014and Kris leaned against the counter as the smell filled the kitchen and made his mouth water.\n\n\"Sleep well?\" Rayne asked.\n\n\"Like a rock. Sorry I took your bed. I wasn't really on top of my game last night.\"\n\n\"I noticed,\" Rayne said, smiling easily. \"It's fine. Did you have fun at the party?\"\n\n\"It was great, but I won't remember half those people's names.\"\n\n\"That's okay. Barely any are coming on tour with us.\"\n\n\"About that . . .\"\n\nRayne stilled. \"You're not changing your mind, are you? Because that's not allowed.\"\n\n\"No! Not at all. I just\u2014 The contract?\"\n\nRayne relaxed, though he looked faintly alarmed around the eyes. \"It doesn't tie you to anything further than the festival, but you do have to promise you won't leave us hanging in the middle of nowhere without a guitarist. And Brian's reserving the right to kick you out at the first sign of trouble, just in case.\"\n\n\"Right,\" Kris said. \"It's just, I've never signed a contract before. Should I get a lawyer to look it over?\"\n\n\"Do you have a lawyer?\"\n\n\"No?\"\n\n\"The label's lawyers will explain everything,\" Rayne promised. \"No tricks, no mind games. You play guitar, and they give you money. Speaking of!\" He dug into his jeans' pocket\u2014 They were plastered on so tight that Kris didn't think they could fit anything in there at all, but Rayne proved him wrong and pulled out a wad of bills wrapped in an elastic band. \"Here's your cut from the show last night.\" He handed it to Kris with a grin. \"I wasn't sure about your banking situation what with the whole, you know, sleeping on the street, so I thought cash would be safer.\"\n\n\"I have a bank account,\" Kris said faintly. \"This is, like, two hundred dollars.\"\n\n\"You should probably deposit it, then. We can stop and do that on the way.\"\n\n\"The way to what?\"\n\n\"Business first, and then the fun stuff. Drink up and get changed, babe. We're going to make you famous.\"\n\nThe trick to surviving a day of shopping with a friend, Kris had learned years ago, was to let that friend do whatever they wanted. Back home it had been his girlfriend, and all he'd had to do was stand around, tell her she looked good, and hold her bags.\n\nShopping with Rayne and Angel promised to be considerably more interactive. They walked from Rayne's apartment into the heart of SoHo, where block after block of shops crowded the streets, all connected by roads thronged with pedestrians who had no apparent regard for the cars or bikes trying to eke their way through. Butch promised to come by later with the car to collect their bags, and Kris mentally prepared himself for a day of walking, shopping, and spending more money than he could imagine on clothes.\n\n\"We'll start you with jeans and a tee,\" Rayne said, leading him into their first store, a tiny boutique with a green storefront, ironically called The Emporium.\n\n\"I have jeans and tees,\" Kris said.\n\n\"You do.\"\n\n\"Is there something wrong with them?\"\n\n\"No! Of course not.\"\n\n\"You can tell me if there is. I know I'm not up to your fancy rock star standards yet. That's why we're here, right?\"\n\n\"We're shopping for clothes that are a little more fitted,\" Angel said, in an admirable attempt at diplomacy. \"And black.\"\n\nKris glanced down at the jeans he was wearing. They were a basic straight-leg denim wash. \"You want a pair that'll make my ass look good?\" he guessed. \"Because I hate to break it to you, but I don't really have one.\"\n\n\"We'll work something out,\" Rayne assured him.\n\nAfter trying on five different pairs of jeans, Kris was feeling weirdly smug about it.\n\n\"Okay,\" Rayne said. \"You really don't have an ass.\"\n\nKris turned around like he was trying to see himself better in the mirror, but he was actually just flaunting his complete lack of assets. \"I don't know; you want to keep trying? I heard that girls have these padded butts they wear under their pants to make themselves bigger. You want to get me one of those?\"\n\n\"You're the worst,\" Rayne said. \"You're like a couple of toothpicks propping up a pipe cleaner. You're tiny.\"\n\n\"I'm pocket-sized.\"\n\nKris did like the jeans, though. They clung to his legs and made him seem even smaller, somehow; the back pockets were embellished with sequins and stones in little looping patterns. The stitches up the side seams were lighter than the fabric of the pants.\n\n\"Hey,\" he said. \"Are these girls' jeans?\"\n\nRayne and Angel glanced at each other.\n\n\"They fit better, don't they?\" Angel asked.\n\nKris looked back in the mirror. They really did. He actually had thighs and calves in them instead of the skinny, shapeless sticks his men's-style jeans gave him.\n\n\"Huh,\" he said. \"Okay. What else they got in the girls' section?\"\n\nThey left The Emporium with him wearing one outfit and carrying two additional pairs of jeans and an armful of tops: tees, tanks, and what Kris was resolutely refusing to think of as blouses. They were silky and kind of see-through, sure, but he had to draw the line somewhere.\n\nWhen he said it out loud, Angel flatly replied, \"They're blouses. The sooner you make peace with that, the sooner we can move on.\"\n\nKris wrinkled his nose.\n\n\"You said you liked them,\" Rayne pointed out. \"You want to take them back? We can do it right now. We're not even at the car yet.\"\n\n\"I do like them. They're flattering.\"\n\nThey were, but half of that was Rayne telling him they looked good on him. Kris had found himself preening under the attention, and though the trying on and taking off of outfits had been exhausting, not to mention stiflingly hot, there was something about being dressed up to fit Rayne's whims that had given him a little thrill. Rayne hadn't been shy about expressing his appreciation, either, raking his gaze over Kris or stepping into Kris's space to adjust some part of the outfit to his liking. Angel had been quieter, but Kris had figured out how to read her easily enough. She might not say as much out loud, but her opinions resided in her eyebrows and in the shape of her mouth, and she'd seemed pleased with the results so far.\n\nStill.\n\n\"I feel like a kid playing dress-up,\" he said, scrutinizing his reflection in the car window. From the neck down, he could slip into any one of The Chokecherries' videos and feel at home. From the neck up, he was still a nobody from his dusty little hometown. If he came out of this shopping excursion looking half as put together as either Rayne or Angel, he'd be surprised.\n\n\"We'll get you there,\" Angel said. \"Now, about your hair . . .\" It was just dark enough to escape being called mousy, but too light to make a statement. \"You mind if I change it up?\" she asked. \"Dye it, cut it?\"\n\n\"There's not a lot here to cut.\"\n\n\"I'll make it work.\"\n\nHe shrugged. \"Knock yourself out, then. I'm in your hands.\"\n\nThey tossed the bags in the trunk and headed for their next stop. Rayne detoured and got smoothies from a trendy little caf\u00e9 where everything was vegan, organic, and obscenely expensive, and they drank them as they walked.\n\n\"Now you need your statement pieces,\" Rayne explained around his straw. \"At least one really good jacket, some boots, some belts. Something with bling. Do you like scarves?\"\n\nKris had never worn a scarf except in winter, but that probably wasn't what Rayne meant.\n\n\"I'm open to scarves.\" Kris was open to anything. He liked the attention, which he had expected, but he also liked being dressed up like a mannequin, which he hadn't. When Rayne and Angel led him into the second shop and took him through the women's section, it felt illicit; his stomach flipped with nerves, and he kept glancing around to see if the shop assistants were looking at him. They weren't. He clearly wasn't even a blip on their radar.\n\n\"You're enjoying this more than I thought you would,\" Rayne said.\n\n\"A guy can't shop?\"\n\nRayne held his hands up in surrender. \"I'm just saying!\"\n\nKris picked up something sparkly and gold, covered in tassels and sequins. He was pretty sure it was a scarf.\n\n\"So, jackets,\" he said. \"If they're not denim or leather I'm kind of lost.\"\n\n\"We can do leather! Come over here.\"\n\nRayne dragged him into another room in the store, where rack after rack of jackets and coats filled the floor.\n\n\"You don't want anything too heavy or it'll look like it's eating you.\"\n\nKris was barely five six; he knew all too well the limits of his wardrobe options. He wandered through the racks, letting his attention drift until a little black piece with too many gold zippers and buckles to make any sense caught his eye. The leather was impossibly soft, the metal impossibly bright. He shrugged it over his shoulders and let it settle, searching out his reflection in one of the mirrors along the walls.\n\n\"Picture that, but with blond hair and eyeliner,\" Angel said.\n\nKris could see it. His heart thudded harder and he tugged at the cuffs. They came down to his knuckles, the zipper tags jingling.\n\n\"This one,\" he said.\n\n\"You don't want to try anything else?\" Rayne asked.\n\n\"Nope. This is it.\"\n\nHe handed it to Rayne, who glanced at the tag before adding it to the pile. \"Vegan leather, nice. Points for you. Lunch break and then go find you some shoes?\"\n\n\"I've got shoes.\"\n\n\"You could have new ones.\"\n\n\"These are comfy.\"\n\nThey were so worn in they had no choice but to be comfortable. He wasn't opposed to getting new shoes, but he had never seen the point if his current ones were still functional. Although these were practically slippers now.\n\n\"I will buy you the exact same pair if you really want to stick with Converse,\" Rayne said, \"but the ones you're wearing won't survive the tour.\"\n\nKris wiggled his toes in them. \"Fine. But they have to be high-tops.\"\n\nRayne hauled their loot to the counter, and Kris watched the assistant ring it up. There was the jacket, a collection of scarves, one covered in skulls, and a sequined top that Kris honestly wasn't sure how to wear, but which Rayne had assured him would look amazing.\n\n\"You're getting better at not panicking about the money,\" Rayne observed as the assistant wrapped the items in tissue paper.\n\n\"I've figured out that panicking won't stop you from spending it, so I've decided not to bother.\"\n\n\"Smart move.\"\n\nThey loaded the bags into the car, the trunk getting full, and went in search of a place for lunch. The sun was warm on Kris's face, the breeze ruffled his hair, and Rayne and Angel walked on either side of him, jostling into him occasionally. Sometimes Rayne's hand brushed his, and it sent electric tingles up and down Kris's arm.\n\nPhysical intimacy wasn't done in Kris's hometown. It was reserved as an expression between couples, and anything outside of courtship was put down as an accident. Guys didn't hug; there was no such thing as a casual touch outside of a romantic relationship. So Kris had filled that void with his family and the girls he dated, or wanted to date, until he hadn't realized there was a void at all. Rayne didn't work like that. He simultaneously dragged that void out into the open and set about filling it before Kris could say culture shock. Kris could get used to it. It had only been two days, and he would already miss it if it stopped.\n\nRayne took them to an Indian caf\u00e9 and laughed at Kris when he tried to make sense of the menu. He didn't recognize half the words, never mind the dishes. He finally gave up, ordered an iced tea, and let Rayne recommend him something. He never found out what it was, but it was colorful and tasted good, so he couldn't complain. Rayne told him about the cities he was most excited to play in, their opening band, and the next tattoo he was going to get. Angel talked about the band's fashion, the club she owned, and where they were playing in her hometown of New Orleans. Kris watched them, ate his food, and basked in the warmth and the company.\n\n\"One night ago I was sleeping on a bench,\" he said suddenly, sliding into a gap in the conversation.\n\nThey both looked at him.\n\n\"Sorry,\" he said. \"I wasn't going anywhere with that. It just hit me.\"\n\n\"The world works in mysterious ways,\" Rayne said.\n\nThe second realization hit him like a brick to the face. \"I need to learn two albums' worth of songs before our first show.\"\n\n\"Total faith in you,\" Rayne assured him, though he seemed slightly worried. \"You've already got a couple of them down. As long as you know the tune you can make the rest up as you go, right?\"\n\n\"You want me to fake the riffs? Won't people notice?\"\n\n\"Not fake them so much as improvise. It's a tour\u2014the fans will expect it to sound a little different from the albums.\"\n\n\"Yeah. No, I can\u2014I can totally do that.\" Kris nodded and tried to laugh. It wasn't very convincing.\n\n\"Let's get him some shoes and head back,\" Angel said, clearing her plate. \"Before the boy has a nervous breakdown.\"\n\n\"It's not a breakdown,\" Kris said. \"I just need a minute.\" He held up his fingers to measure the smidgeon of a minute he was taking. \"Just to get my bearings.\"\n\n\"You're good though, right?\" Rayne checked. \"You still want this?\"\n\n\"I'm good,\" Kris promised. He could absolutely learn two albums in twenty-four hours.\n\nBleach, he learned, stung like a bitch. He bit his lip to keep from squirming as Angel wrapped the top of his head in plastic to let the stuff set. It burned his scalp, digging in like fire ants, and he hated it.\n\n\"You'll get used to it,\" she said, seemingly unconcerned about how his entire head was aflame. \"After the first few times you won't even feel it anymore.\"\n\n\"Uh-huh,\" he said, not believing a word of her lies. Her hair was all natural. She was clearly much smarter than he was. He turned the box over in his hands, rereading the warnings for the tenth time. \"You sure this won't blind me or anything?\"\n\n\"As long as you don't rub it in your eyes.\"\n\n\"I wasn't planning to.\"\n\nThey were in Rayne's penthouse, crowded into the bathroom-cum-salon on the lower level. Angel had apparently converted it herself: bright bare bulbs shone in stripes around the mirror, which took up nearly an entire wall; another wall was packed with neat little shelves housing every kind and color of makeup, dye, and polish under the sun. It was glamorous, hyperfeminine, streamlined to the point of peak efficiency, and reminded Kris of some weird science fiction spaceship. It was incredible, and he was scared to touch any of it.\n\n\"Are you bringing all this on tour with you?\" he asked.\n\n\"Only the basics,\" she said. \"Everyone's got their own style figured out except for you, so I can narrow it down to the bare essentials.\"\n\n\"How do I figure out my style?\"\n\n\"We'll start with where you're comfortable and go from there. Lenny just wears a bit of eyeliner; Maki spends up to an hour in the chair before a show. I'll make sure you look good, but you don't have to wear anything you don't want.\"\n\nKris caught Rayne's eye in the mirror. \"No, I'm down to, like, experiment.\"\n\n\"I'm guessing you never tried this back in Kansas,\" Angel said.\n\n\"I never tried a lot of things in Kansas.\"\n\nIn the mirror, Rayne's lips curved in a slow smile. Kris grinned to himself and dropped his gaze. \"In for a penny, right? I want you to make me pretty.\"\n\nHe stayed at Rayne's place again that night. The next morning they would leave for the tour, heading across the country toward Nevada for the Purple Sage Music Fest, a six-day desert festival, playing sold-out arenas along the way. Rayne brushed aside Kris's offer of finding a hotel, saying it would be more convenient for everyone if he just stayed where he was. One less trip for Butch to make in the morning when he collected everyone. Kris caved without more than a token protest. A hotel room sounded lonely after two days of constant company.\n\nThat evening, as the sun started to slip below the horizon, he spread his belongings over the guest room bed and looked at the trappings of a life that wasn't yet his. The little red Fender lay on top, gleaming up at him. It was even brighter and more beautiful by itself, without the competition from the hundred other guitars in the shop, and Kris loved it with his entire heart. He was bringing his acoustic too; he couldn't bear leaving it behind after all these years, even if he wouldn't play it onstage. Folded neatly in a borrowed suitcase were his new stage clothes. He got a buzz just from looking at them, like they had the power to turn him into someone new, and he couldn't wait to find out who that person was.\n\nHis phone buzzed with an incoming text from his mom. Kris, have just heard from your cousin. PLEASE CALL US.\n\nHe winced, moved the guitar aside to perch on the edge of the bed, and video-called home. It connected immediately and his parents and younger sister, Cass, stuttered into frame. He was relieved to see that Brad, his older brother, was absent\u2014while his parents were remarkably open-minded for small-town folks, Brad had leaned increasingly to the right ever since falling in with the wrong crowd at community college, and there was no way he'd be pleased about Kris wearing girls' jeans or joining a glam band, no matter how punk it was.\n\n\"Hi, everybody,\" he said. \"So, I've got news.\"\n\n\"Kris,\" his mom said. \"Kris, your aunt called and told us about Marty. What happened? Why didn't you come home?\"\n\n\"I didn't want to give up that quickly, but it worked out! I was just going to call you and tell you the news.\"\n\n\"Good news?\" his dad asked.\n\n\"Really good news.\" He felt shaky, balancing on the precipice of saying it out loud for the first time to someone else. \"I've got a job and a place to stay, at least for now. The money's good, and I'm excited for it.\"\n\n\"What job?\" his dad pressed.\n\n\"Um, I joined a band . . .\"\n\nCassie gave an excited squeak.\n\n\"A real band,\" he added. \"We're leaving for tour tomorrow\u2014I signed a contract and everything. It's totally professional.\"\n\n\"What band?\" Cass asked.\n\n\"The Chokecherries?\"\n\nShe let out an inarticulate yell and leaped from the couch, pacing with barely contained energy. \"Like, as a roadie?\" she demanded.\n\n\"As their guitarist.\"\n\nShe made a noise like a pterodactyl.\n\n\"Do you know them, sweetie?\" his mom asked.\n\n\"They're huge,\" Cass gushed. \"Kris, you met them? You met Rayne Bakshi?\"\n\n\"Yeah,\" Kris said. \"Rayne's really nice. I'm staying at his penthouse. And I like their music. This is basically the best thing that could've happened to me.\"\n\n\"You're staying at Rayne Bakshi's penthouse.\"\n\n\"He's in the next room; don't make this weird.\"\n\n\"I have a poster of him in my bedroom,\" Cassie said. \"I learned drums playing along to their songs. Oh my god.\"\n\n\"Hang on,\" his dad said. \"You signed a contract?\"\n\n\"It's totally legit,\" Kris assured him. \"They had lawyers walk me through it and everything. They were really professional. There was a limo.\"\n\n\"And you're getting paid?\" his mom checked.\n\nKris could still see the amount dancing behind his eyelids every time he blinked. There were a lot of zeroes. \"I'm getting paid. It's good money, Mom. I'm going to be okay. I'll send you the tour schedule when I figure it out\u2014we're going west to this Nevada festival first with a few stops on the way, and then heading back east again.\"\n\n\"You're going to be a rock star,\" Cassie stressed.\n\n\"I came here hoping to land a session gig or something, do a few anonymous backing tracks, but this is so much better than anything Marty's boss could have set me up with. It fell into my lap right when I needed it most, and it would be crazy stupid to turn it down, you know?\"\n\n\"Rayne Bakshi fell into your lap,\" Cassie said. \"Oh my god, Kris.\"\n\n\"We're happy for you,\" his mom promised, \"but you'll be careful, won't you? You hear stories about rock bands, and life on the road . . .\"\n\n\"Heroin,\" Cassie coughed.\n\n\"I'll be fine, Mom. We're all adults.\"\n\n\"I know you are! I just worry.\"\n\n\"You don't have to anymore. I've got a contract and a full-blown benefactor and a job I think I'm going to love. If you need to worry, worry about my idiot cousin who got himself kicked out and fired in a single move.\"\n\nHis parents sighed in unison.\n\n\"Marty was never a bright kid,\" his dad said. \"I'm glad you didn't end up staying with him.\"\n\n\"Same. Rayne has a really nice place, you know. There's a balcony.\"\n\n\"Of course there's a balcony. Send me pictures,\" Cassie said.\n\n\"I'm not sending you pictures of his apartment, you little stalker.\"\n\n\"Okay, that was a bit stalkerish,\" she conceded. \"Send me a picture of him instead? Take a selfie! Is he still in the next room?\"\n\n\"He's probably busy; leave him alone. Anyway, I'm sure you'll see videos from the tour.\"\n\n\"I will,\" Cassie said. \"I'll show them to you guys,\" she added to their parents. \"Now that Kris is with The Chokecherries, you have to see everything they've ever done.\"\n\nTheir parents visibly steeled themselves, but nodded.\n\n\"Listen, I'm going to go to bed,\" Kris said. \"We're heading out first thing, and I've had a really long couple of days. I'll text lots, okay?\"\n\n\"Skype when you can,\" his mom said. \"It's nice to see your face. Did you do something with your hair?\"\n\n\"Maybe? Love you! Bye!\"\n\nHe ended the call and stretched out on the bed on top of all his new clothes, his phone held against his heart, and he stared up at the ceiling as the day's last light chased itself over the walls and down below the horizon. After finishing community college, he had worked in his dad's garage, and the years had passed without him noticing. It wasn't until Cassie had decided she was going away to get a bachelor's degree out of state\u2014the first of all the Goldings, to their parents' pride and joy\u2014that Kris had realized he could leave too. Now Cassie was almost done college and Kris was twenty-five, and he felt like he had a whole decade of misspent youth to catch up on. Joining a band and crossing the country on a sold-out tour sounded like a solid way to start. He couldn't stop smiling.\n\nThe Chokecherries shared a tour bus while a second bus housing their opening act, Passionfruit, would join them at their first show in Pennsylvania. The bus was a tight fit, but overall roomier than Kris had expected, with a bathroom at the back and a fridge at the front. There were eight bunks and a couch crammed in, leaving little leg space, but Kris didn't take up much room.\n\n\"The international leg of the tour is all flying,\" Rayne said, \"so I really fought for the buses while we stayed in the States. It's a classic, you know? I wanted that experience.\"\n\n\"I get it,\" Kris said. \"It makes it more about the journey than the destination.\"\n\nRayne beamed. \"Exactly!\"\n\nHersheypark Stadium was only a few hours away, and though the whole band and crew were buzzing with energy, Kris toppled straight into nerves. Despite going to bed early the night before, he'd stayed awake for hours fantasizing about the tour, and now he was ready to crash. He sat on the couch, his back pressed to the window, and listened to The Chokecherries' albums on repeat, struggling to keep up on his guitar. The others gave him space, careful not to interrupt, though he must have been driving them nuts. After the third hour, Rayne finally dropped into the seat beside him, and Kris tugged his earbuds out.\n\n\"How's it going?\" Rayne asked.\n\n\"I don't want to fuck up in front of twenty thousand people.\" The words rushed out in a single panicked breath.\n\n\"You played our New York show fine. Pretend it's like that again.\"\n\n\"Yeah, no, I was scared shitless then too,\" Kris admitted. \"I just didn't say anything. And that was only playing covers, but now it's your music, and I might be freaking out a tiny bit.\"\n\n\"Do you think you're going to fuck up?\"\n\nKris shrugged and gnawed his lip.\n\n\"I probably will,\" Rayne offered. \"I mess up lyrics all the time. If anyone asks, I say I did it on purpose, but I don't actually notice. Stef fucked up a bass solo on our first tour.\"\n\n\"Epically fucked,\" Stef agreed from the other side of the bus. \"I was so high I couldn't remember what song we were doing, so I did a Flea cover instead. It was awesome.\"\n\n\"I just want to make a good impression,\" Kris said. \"I thought the nerves would go away now that I've done it once, but I think they've gotten worse.\"\n\nRayne put his arm around Kris's shoulders, and Kris leaned into him, careful not to stick him with the neck of the guitar. Rayne was warm and solid and his hair smelled good.\n\n\"Believe me when I say you're going to kick ass onstage tonight. You're a natural performer, Kris. You might not realize it yet, but you will.\"\n\nRayne said it with a conviction Kris didn't know how to fight, and besides, he wanted to believe it. He turned, leaned his back against Rayne's shoulder, and popped his earbuds back in, returning to track one of their set list as he hefted his guitar back into place. He could do this. Talking about his nerves hadn't made them go away, but Rayne's confidence was hard to resist.\n\nBy the time they rolled into the venue, the borrowed confidence wasn't good enough anymore, and the butterflies in Kris's stomach were ready to eat him alive. His legs shook a little as he disembarked onto solid ground. Rayne intervened as the second bus pulled up behind them, dragging Kris over to meet Passionfruit before he could get any more worked up.\n\n\"Angel introduced us during last year's festival circuit,\" he explained, \"and we just knew we had to do something together. These guys have incredible energy onstage; you have to see them play. They're insane in all the best ways.\"\n\nPassionfruit was a four-man band, and they spilled out of their bus to greet The Chokecherries like long-lost friends, the one at the forefront pulling Angel into a hug before even saying hello.\n\n\"New guitarist, Kris,\" Rayne said. \"Kris, Billie\u2014and Jay, Hatchwork, and Knocks. Vocals, guitar, bass, and drums, in that order.\"\n\n\"Hey,\" the band chorused.\n\n\"Hi,\" Billie said, detaching from Angel's side.\n\nHe was ghostly pale, with messy black hair and smudges of maybe makeup, or maybe signs of exhaustion under his eyes, and he had the softest voice Kris had ever heard. Jay was M\u00e9tis and sporting a devil's lock and a mischievous spark in his eye; Hatchwork wore an impressively waxed auburn moustache; and Knocks was skinny, dark, and had a head of curls that would give Rayne a run for his money. The others disbanded almost immediately after introductions, wandering off in different directions to explore the venue, until only Billie remained.\n\n\"You ready for this?\" he asked.\n\n\"Haven't decided yet,\" Kris said. \"Might sit and panic about it a bit longer.\"\n\n\"That's a valid option,\" Billie agreed.\n\n\"No one's panicking,\" Rayne said. \"We're going straight into rehearsal, and by the time we're done, he'll be raring to go.\"\n\n\"Or that,\" Kris said.\n\n\"Cool,\" Billie said. \"I should go find Jay before he sets something on fire. See you around!\"\n\nKris stared after him. \"Fire?\"\n\n\"Honestly, it's best not to ask,\" Rayne said. \"Come on, come get your stuff set up.\"\n\nRehearsal went as well as Kris could hope. He fucked up a few times, but so did everyone else. Playing a Chokecherries' set felt different from playing covers: the energy was tighter, almost crackling in the air, so sharp Kris could taste it. The butterflies in his stomach spiraled faster in synchronized loops, threatening to choke him, but he swallowed them down and played like his life depended on it. When Rayne finally called a break for lunch and the others dispersed for snacks and water, Kris stayed where he was, his fingers clenched around the neck of his guitar.\n\n\"You're doing great,\" Rayne said. \"You want food?\"\n\n\"Actually great, or you're just saying that so I'll stop freaking out?\"\n\n\"I can't mean both at once?\" Rayne prodded him until Kris relented and lifted the guitar from around his shoulders. \"Eat,\" Rayne ordered. \"Or drink some water, at least. The doors open at seven; we take the stage at nine.\"\n\nIt was 1 p.m. Kris tried to remember the last time he'd had a full night's sleep: before the flight, for sure. Maybe he should get another coffee.\n\n\"Jay's got energy drinks,\" Rayne said, as if reading his mind, \"but they might kill you. Imbibe with caution.\"\n\n\"Caution. Roger that.\"\n\n\"I want to run through this one more time during sound check at five,\" Rayne said, \"and Angel wants you in her chair by six.\"\n\nKris swallowed and nodded.\n\n\"Don't play your fingers to the bone before then, okay?\"\n\nRayne put his hand on Kris's shoulder and ran it down Kris's arm in a long stroke, leaving tingles trailing in his wake. Kris leaned into it, desperate for any reassurance he could get, and Rayne didn't hesitate before pulling him into a hug.\n\n\"I get it,\" Kris said into Rayne's shoulder. \"Why you guys are so touchy-feely. It's because you're all on the verge of a mental breakdown at any second, isn't it? I thought you were all, like, in touch with your emotions and stuff, but you're actually just hella stressed all the time.\"\n\nRayne huffed out a laugh and rubbed circles between Kris's shoulder blades, and Kris went boneless against him, practically purring.\n\n\"You'll get used to it,\" Rayne said. \"The preshow nerves never go away completely, but you'll find ways to manage them. Or you'll get a therapist and a nice antianxiety prescription. Either way, you'll play the shows and you'll be fine.\"\n\n\"Can I keep getting hugs in the meantime?\" Kris asked.\n\nRayne squeezed him tighter. \"For sure. Anytime you need one, you come find me.\"\n\n\"Your hugs are really great.\" Kris dug his chin into Rayne's shoulder\u2014he had to stand on his toes to reach it\u2014and from there he nuzzled into Rayne's mane of hair. Rayne's shampoo was citrusy, but his cologne smelled like sandalwood and peppermint. Kris breathed in deep to carry it with him through the show. \"Macho guys never hug other guys. They're missing out.\"\n\n\"Yeah, they are,\" Rayne said, and ruffled his fingers through Kris's hair. \"You're tiny and pretty and you play guitar like a fucking prodigy. You want to fling yourself into my arms, you do it. That's a sacrifice I can make.\"\n\n\"Take one for the team,\" Kris mumbled.\n\nRayne pushed him back far enough to look him in the eye. \"By the end of the night, you're going to be a rock star, and you're going to love it.\"\n\n\"Promise?\"\n\n\"Cross my heart, babe. You're going to have it all.\"\n\nKris went to find Angel later in the afternoon. She was holding court in the Passionfruit dressing room, engrossed in a conversation about eye shadow brands with Billie. Both seemed to be taking it very seriously. Kris knocked on the doorframe and leaned in. \"Am I interrupting?\"\n\n\"Billie's a makeup aficionado too,\" Angel said. \"This is Kris's first time getting done up,\" she added to Billie. \"He's never tried before.\"\n\n\"Oh cool,\" Billie said. \"You'll love it. I started doing mine in high school\u2014emo phase, you know.\" He shrugged self-deprecatingly. \"One of those things you never really shake.\"\n\n\"Did you need something?\" Angel asked.\n\n\"No, I'm just having a tiny breakdown about the show tonight, so I thought I'd come hang out here.\" Kris scuffed his toe across the floor. \"I mean, if that's okay. I can go somewhere else if you're busy. I'm looking for a distraction.\"\n\n\"I can go get Jay,\" Billie offered. \"He's, like, a walking distraction.\"\n\n\"Setting stuff on fire?\" Kris asked.\n\nBillie frowned. \"Not on purpose. He's accident prone, and the accidents don't always follow the laws of physics, so I've found it's best to prepare for the worst possible outcome and err on the side of caution.\" He paused. \"Actually, I should go find him.\" He slid from his chair, passed Kris, and departed with a backward wave.\n\n\"Billie and I were roommates in art school,\" Angel said. \"He's a doll. That's where I learned that the best way for me to deal with stress is to do makeup, and since you came to me, that's my suggestion. Get your makeup done. You'd have to do it in another hour or two anyway, so you might as well.\"\n\n\"Sure, why not. Beats pacing for another hour, right? And if I practice any more, my fingers are going to fall off.\"\n\n\"That would suck,\" she agreed. She gestured for him to follow her to The Chokecherries' room, where Kris hopped up into the chair and folded his hands in his lap.\n\n\"Close your eyes and let me work some magic on you. You'll feel better in no time.\"\n\nKris obeyed. The dressing room setup was wildly different from her salon at Rayne's: still brightly lit with a huge mirror, but not nearly as streamlined or personalized. It didn't seem to make any difference to Angel; she could probably work in the dark and it would come out great. Her hands were steady against his face, wielding brushes and pencils and other, more exotic instruments Kris couldn't identify. He kept his eyes shut the whole time, and as she worked, she chatted\u2014about her club in New Orleans, how she met Billie and Jay in art school, and how Billie was the first person she'd ever come out as trans to, and about her plans for after the tour. Her words washed over Kris like a stream, and he gradually relaxed, muscle by muscle. He didn't have to talk back\u2014he tried a few times, and she told him to stop moving his face\u2014but he hummed here and there, just to show he was still paying attention.\n\nAfter maybe half an hour, her talking lulled for a second and he twitched.\n\n\"Okay,\" she said. \"Open your eyes.\"\n\nKris blinked and stared into the mirror reflecting a face he couldn't recognize. It looked bewildered but very, very pretty.\n\nOver his shoulder, Angel smiled. \"What do you think?\"\n\nKris opened his mouth and all that came out was \"Wow.\"\n\n\"Good wow?\"\n\nHis lips were dark red, like oxblood; his eyes were done up with smoky eye shadow in jewel tones, and rimmed with thick black kohl. The mascara made his lashes feel like spider legs. He blinked, his reflection blinked back, and the fact that it was really him slammed into him. The makeup didn't have the exaggerated contours of drag, and he didn't look like a girl, but he didn't look much like a boy anymore either. He was caught somewhere between the binaries in a space he hadn't realized existed, and it thrilled him to his core.\n\n\"It's really drastic in the light,\" Angel said, \"but onstage it'll be amazing. Just wait till the others see you.\"\n\n\"Wait, don't bring anyone in yet,\" Kris said. \"I want a minute to admire myself first.\"\n\nHe looked like some hedonist's wet dream from the seventies, androgynous and fey.\n\n\"You do that,\" she said. \"I'm going to do your hair.\"\n\nHe still wasn't used to the blond. It was so pale it was nearly white, and next to the makeup it made him seem ghostly. Angel had buzzed the sides short back in New York, and now she fluffed it up along the top, teasing it with hairspray before easing wax along the roots and the bangs. It wasn't long enough to really style, but Angel promised she could work magic with anything as long as it sat still, so Kris waited, making faces in the mirror until he could recognize himself again. Angel fixed his hair up like a cockatoo's ruff and stood back with her hands on her hips.\n\n\"Rayne is going to drop dead when he sees you.\"\n\n\"In a good way?\"\n\n\"In the best way,\" she promised.\n\nKris grinned. A pleased little blush spread over his cheeks, visible even through the makeup, but he didn't care. Angel had been right\u2014the routine and the attention had calmed him, and now that he had a mask in place, he felt ready to take on anything. He changed into his stage clothes, feeling heady and invincible, and from there went backstage for sound check and one last rehearsal.\n\nIt didn't take long for Rayne to notice him. Kris kept his head down and his attention trained on his guitar. He didn't stop when he heard Rayne's boots clicking over the floor behind him, and he didn't stop when Rayne circled around to stand in front of him and stare, either.\n\n\"It's good, right?\" Kris asked, aiming for confidence and falling just short.\n\n\"Fuck, Kris.\"\n\nHe glanced up.\n\nRayne was studying him in open appreciation, from the outfit to the makeup and hair and back again. Kris licked his lips self-consciously and came away with the bitter taste of lipstick on his tongue.\n\n\"Gorgeous,\" Rayne proclaimed, though he seemed like he wanted to say something more.\n\nKris fidgeted with his guitar and ducked his head, infinitely pleased. His face was hot.\n\n\"You're going to inspire a thousand crises of teenage sexuality,\" Rayne said, slinging his arm around Kris's shoulders.\n\n\"Just what I always wanted,\" Kris deadpanned. \"My mom will be so proud.\"\n\n\"She should be. You ready?\"\n\n\"I don't know.\"\n\n\"Close enough. I still get nervous before a show. Like I've got a live wire running through me. It's good.\" Rayne nodded, seemingly more to himself than to Kris. Kris nodded along anyway.\n\nPassionfruit, Kris learned, was not just an opening act. They weren't worried about converting The Chokecherries' fans to their music; they brought their own fans who screamed for them and knew all their lyrics, and Passionfruit whipped them into a frenzy. Kris watched from the side, burning with secondhand adrenaline. They were half-feral and completely unrecognizable from when Kris had first met them. Billie was unstoppable. Whether Jay was slamming into him like a battering ram or the crowd was surging up against security to mount the barrier, he never missed a note, singing his throat raw.\n\n\"The crowd looks ready to riot,\" Kris said.\n\nRayne laughed. \"You didn't get many punk bands out your way, huh?\"\n\n\"We were more of a country town. Banjos and shit.\"\n\n\"That's terrifying,\" Rayne said mildly. \"You ready?\"\n\nOnstage, Passionfruit didn't wind down so much as drop abruptly out of their last song. Billie said something about love and respect\u2014the crowd screamed\u2014the drum kit crashed\u2014and the band came staggering backstage as the lights went dark. They were all soaked in sweat but their smiles were incandescent; The Chokecherries hugged or patted them as they passed.\n\n\"Twenty minutes,\" Rayne said in Kris's ear.\n\nThe crew nipped out to turn the stage over for them, clearing away things the fans had thrown\u2014pamphlets and flowers and the occasional item of intimate clothing\u2014and wiping up the spilled water. Kris vibrated with restrained energy as the crowd chanted in a low rumble of anticipation, waiting to overflow. His blood felt like mercury in his veins.\n\nSomething must have shown in his expression because Rayne said, \"Come here,\" and drew him into an embrace, tucking him in close so Kris couldn't protest.\n\nKris pressed his face against the ridges of Rayne's jacket buckles, careful not to disturb his makeup, and breathed in. Rayne smelled warm, like spices, and his jacket was clean and leathery.\n\n\"You've got this,\" Rayne said. \"They're going to love you.\"\n\n\"Promise?\"\n\n\"Promise. I do, so they have to, because I'm not getting rid of you anytime soon.\"\n\nOne by one, the rest of the band slipped onto the stage until it was only Rayne and Kris left behind. The crowd sounded insatiable. Lenny kicked up the drums to a roar of screams, and Stef opened with a slinky bass line minutes later. By the time Maki set the keyboard in motion the roar was deafening\u2014twenty-seven thousand voices all demanding Rayne.\n\n\"Your turn,\" Rayne said.\n\nHis eyes were dark in the dim lights, and Kris could feel his heart skipping through their clothes. Kris gave him one last squeeze for luck before ducking through the curtain and taking the stage.\n\nIt was blinding. The noise was unreal, even through his earpieces; the stage pulsed under his feet with every beat. He could barely see the crowd through the stage lights\u2014they were a sea of dark shapes, no faces, only voices. He took his position on stage left, planted his feet, and tried to breathe. The music helped. He matched his breaths to the rhythm of the bass and struck up the chords to the intro as the screams reached a crescendo, and he knew Rayne was coming.\n\nThey charged into the intro song as Rayne took the stage, the crowd falling back as soon as he started to sing. By the time they reached the chorus, the pace was blistering, the crowd manic, and Kris's nerves were seared away under the onslaught. He felt bare\u2014no insecurities, no fears, no stress. It was like he belonged on the stage, like he'd been born there, and there was nowhere else in the universe he should be instead.\n\nThe intro finished with a flourish and the crowd roared as Rayne grinned and took the mike from its stand.\n\n\"Hi, guys,\" he said, then waited for the cries to die down again. \"It's great to be back. A lot's changed in the past two years\u2014maybe you heard we got a new guitarist? Well, maybe not. We've been keeping him a secret.\"\n\nKris twanged out a few chords in response and Rayne laughed. A ripple ran through the crowd.\n\n\"I like him,\" Rayne continued. \"I hope you guys do too. But first, let me introduce the more familiar faces. On bass, we have Stef Morganstern.\"\n\nStef thumbed out something dark and sexual as Rayne swayed to the beat, center stage.\n\n\"On drums, Lenny Lawson.\"\n\nEvery beat brought Kris closer to his own introduction, and his heart lodged in his throat, nervous and tight. The butterflies in his stomach were so intense he thought their hurricane might carry him away.\n\n\"On keyboards, the lovely Maki Ito.\"\n\nMaki played a little scale, so sharp it had teeth.\n\n\"And finally, on guitar.\" Rayne turned to him. The lights lit him up from behind like he was haloed. \"Our brand-new member, Kris Golding!\"\n\nKris launched into his solo like he was throwing himself off a cliff. His hand stayed steady on the strings, though he couldn't remember a single note they'd practiced together. He flew into an improvisation, barely knowing what the next note might be, and above it all, the crowd screamed their greeting.\n\n\"Now you know,\" Rayne said, \"this is his first time touring. In fact, this is the very first show over a hundred people that he's ever played. So I want you to give him a lot of love, okay? Give him as much love as you can.\" He let them scream themselves hoarse a minute longer before continuing. \"And obviously I'm Rayne Bakshi, and we are The Chokecherries.\"\n\nIf Kris hadn't taped the set list to the floor in front of him, he would have been lost. As it was, he swam through the next few songs back-to-back with barely a pause to catch his breath. How Rayne did it, he didn't know. Kris only got lost once, stumbling for a second at the start of the third track, but if anyone noticed they didn't say. He invented a melody when he had to; it was too early to rely on muscle memory. But even when he couldn't remember the name of the song they were playing, the words to the next verse, or exactly how his solo was meant to go, he breathed in the thick air, crackling with energy, and reveled in it.\n\nThere was nothing better than sweating through his shirt under the volcanic glow of the lights, basking in the attention of twenty-seven thousand fans, all screaming Rayne's words back to him. He lost himself in the rhythm, concentrating on playing. Rayne was on fire, and the crowd was wild, swelling and crashing like a wave against the barrier. Kris looked up in time to see Rayne prowling over and a reckless feeling surged through him, his fingers never slowing against the strings. When Rayne was close enough to touch, Kris wondered what he was doing, but that wonder stuttered out in a split second as Rayne closed the gap, curling his hand behind Kris's neck to draw their mouths together.\n\nKris shut his eyes and saw red, a flood of heat that started in his lips and surged through him like a bolt of lightning. Every atom in his body sang, and when he inhaled, all he could smell was Rayne: peppermint and citrus and sandalwood and sweat. His knees buckled and he keened, leaning into Rayne for balance, his guitar the only thing stopping him from melting into the embrace.\n\nHe opened his eyes when Rayne let go, dazed and dizzy, and Rayne smiled like a promise before turning back to the crowd. Kris fumbled a note before regaining his wits. His legs were still weak; he locked his knees and braced himself to keep from falling. Across the stage he caught Stef's gaze. Stef winked and blew him a kiss.\n\nKris dropped his eyes back to his guitar, staring at his own hands like he was twelve and just learning to play. The rest of the show passed in a blur of colored lights and heat and a beat so throbbing that Kris could barely tell whether it was the bass, the drums, or his own pulse. As they crashed into their final track, Kris felt drunk from success. The last note from Stef's bass ripped through the air, and he staggered offstage to untangle himself from his guitar and dump a bottle of water over his head. Rayne collided with him like a freight train, wrapping his arms around Kris's chest from behind and pressing his face into Kris's neck. Kris sagged back against him like he'd been waiting for it all his life.\n\n\"Encore in five,\" Rayne announced, his hands trailing over Kris's shoulders as he withdrew.\n\nKris paced backstage, running his fingers through his hair. The water had done nothing to the spray; whatever Angel used, it was impervious to the elements and kept Kris's hair in place like a crown. When Rayne finally called them back to the stage, Kris felt more jittery than before, but stepping under those lights and hearing the roar of the crowd let him breathe again.\n\n\"We were perfect,\" Rayne said.\n\nThey shed their instruments and most of their clothes backstage, downing water like they were dying. Kris peeled his shirt off, not caring who was watching, and all the while Rayne circled them like a shark, unable to keep his hands off anyone for too long.\n\n\"You like your first show?\" Stef asked.\n\n\"Perfect,\" Kris echoed, panting through his grin.\n\nRayne grinned back, catching his hand and tangling their fingers together, and for a second Kris thought Rayne was going to kiss him again. His enthusiasm was infectious, and so was his skin hunger. Kris launched himself at him, and Rayne wrapped his arms around Kris's middle, lifting him off the floor. Burying his face in Rayne's neck, Kris laughed.\n\n\"I'm in a band,\" he said, breathless as soon as his feet touched the ground. \"I'm touring in a band. I'm going to be famous.\"\n\n\"As soon as those videos go online, you will be,\" Stef said. \"You're Rayne Bakshi's new boy toy.\"\n\n\"There are worse things to be,\" Rayne said.\n\nHe still hadn't let go, though Kris was standing on his own two feet again. They kept their arms wrapped around each other, leaning in. Kris was trapped in orbit and wildly, excruciatingly happy. He felt so high coming offstage he thought his heart might burst straight from his chest\u2014he understood why so many bands got caught up in drugs and booze and parties trying to sustain the sensation. He shivered at the memory of the kiss, his lips burning as he raised his hand to brush his fingers over them. Rayne just pulled him closer, one hand mussing through Kris's hair as the other planted itself warmly over his heart.\n\n\"Yeah, there are worse things,\" Kris agreed.\n\nThe band eventually separated, wandering off to find food or drinks or a bathroom, and Kris headed for the bus, needing to lie down before he fell over. His phone buzzed with a text before he even left the venue.\n\nCass: OMG YOU DOG.\n\nIt took him a solid minute to realize that of course his sister would have been scouring the internet for concert footage, and of course there would be a lot to find. He had kissed Rayne in front of twenty-seven thousand people\u2014or rather, Rayne had kissed him, and he'd kissed back.\n\nHis parents were going to see that footage.\n\nOops? he replied.\n\nShe called him less than a second later.\n\n\"Hi, Cassie.\"\n\n\"That show looked so good!\" she yelled. He held the phone away from his ear. \"You looked amazing\u2014holy shit, I couldn't recognize you at first! If I hadn't known it was you, I never would have guessed.\"\n\n\"Thanks?\" he hazarded.\n\n\"So good,\" she stressed. \"Amazing. Was it good? Did you like it?\"\n\n\"The show?\"\n\n\"No, dumbass\u2014yes, the show! I'm not actually going to grill you about the kiss,\" she added. \"If it was anyone else up there, I'd say it was hot, but as my brother, you made it weird.\"\n\nHe decided not to dig into that. \"The show was mind-blowing,\" he said instead. \"It was everything I ever wanted. You have no idea, Cass, it was . . . everything.\"\n\n\"I bet,\" she said wistfully. \"Shit man, I'm so happy for you. Mom and Dad watched the video, by the way. You should've seen their faces.\"\n\n\"Uh, maybe not.\"\n\n\"No, you should have! They're going to call you next, you know, so you should call them first. They seem cool with it. I told them it was a performance thing. Fan service, you know? Lots of guys get freaky onstage for attention. Anyway, the video was pretty crappy; it was too low-res for them to get really scandalized.\"\n\n\"That's good?\" Kris offered, hesitantly.\n\n\"Was it scandalous, though?\"\n\nKris thought about the hot press of Rayne's lips on his, the brush of his hair against Kris's face. \"I don't know. It happened so quickly.\"\n\n\"Did he slip you the tongue?\"\n\n\"I thought you weren't grilling me about this.\"\n\nShe crowed in triumph.\n\n\"No!\" he said. \"No, he didn't! Or\u2014I don't think he did. Or I did.\"\n\nThey both paused.\n\n\"I told you he was hot,\" Cass said.\n\nKris sighed. \"Bye, Cass. I'll text you later, 'kay?\"\n\n\"Bye, Kris! Give my love to Rayne!\" She ended the call with a wet smacking noise and, from what Kris could tell, zero shame.\n\nHe texted his parents rather than call them, a quick note that he was heading out with the band and he'd call in the morning. Then, like a complete coward, he turned his phone off and hoped his brother wouldn't find the footage along with the rest of his family.\n\nAs far as Kris's hometown was concerned, he was straight. He'd never kissed a man before Rayne, and he'd definitely never done more than that. He might have thought about it with different people over the years, but since he liked girls and girls liked him, he'd seen no reason to push himself into uncharted territory that would have gotten his teeth kicked in in certain parts of town. That, and everyone knew some distant relative or friend of a friend who'd been disowned for getting involved with the wrong sex. Though Kris couldn't imagine his parents doing anything like that, he couldn't help but heed the warnings.\n\nSo he knew bisexuality was a thing, and that it almost definitely applied to him, but he'd never had the chance to take it for a test run, as it were, back in Kansas.\n\nBeing on tour with The Chokecherries was providing ample opportunity.\n\nAnyone could see Rayne was attractive. Girls wanted him and men wanted to be him, and maybe there was a little crossover between the two. Kris wasn't entirely sure on which side of the line he fell\u2014though he had suspected it ran to the former, even before they'd kissed\u2014but he had the chance to find out, and that was more than most people could say.\n\nHe shook his head. His post-show nap would have to wait; he needed to see Rayne.\n\nKris found him lounging on the couch in their dressing room, sharing a smoke with Angel. Rayne was still dressed for the stage, black from head to toe, with rhinestones on his boots and glitter in his hair.\n\n\"Want a hit before Butch drags us back to the bus?\" Rayne asked, offering the pipe. It was glass, handblown, with a multitude of colors swirling around from stem to bowl.\n\nKris accepted it and took a drag. The weed helped bolster his courage. \"Thanks. Can I talk to you for a sec?\"\n\n\"Yeah, of course.\" Rayne blinked, then visibly pulled himself together. \"Oh\u2014are we cool? I should have asked earlier. I get carried away sometimes, and boundaries, you know, blur.\" He raked his hand through his hair, looking sheepish. \"Do I need to apologize?\"\n\n\"For offending my delicate sensibilities?\"\n\n\"Or something,\" Rayne agreed.\n\nKris's lips tingled in memory of the kiss, and his gaze dropped to Rayne's for a second of its own accord. Angel glanced back and forth between them, cleared her throat, and stood. \"I'm going to go find Billie and the guys,\" she said. \"You guys just . . . have a chat.\" She patted Kris on the chest as she left.\n\nKris took another hit as he considered how to best approach this. However interested Kris might be, Rayne was the one keeping him in the band and off the street, and getting involved with the boss seemed like a monumentally bad idea\u2014especially with Brian threatening to watch for the slightest reason to kick him out of the band. Anyway, it wasn't as if Rayne had kissed him and Kris had fallen instantly in love. He needed to draw a line, make sure they were both clear about where it lay, and then be careful not to cross it.\n\n\"The thing is, I like girls,\" Kris said. \"I've only ever been with them.\n\n\"Okay,\" Rayne said slowly. \"Are you trying to let me down gently or something?\"\n\n\"I don't want any, like, misunderstandings,\" Kris said, sinking into the couch, his knee knocking into Rayne's as he handed the pipe back. \"I'm not complaining, like, at all. I like kissing. I just want to make sure we're on the same page.\"\n\n\"That I'm not harboring any latent, pent-up desires for you?\" Rayne asked, lifting the pipe to his lips again.\n\n\"Right. Exactly.\" Kris's insides burned at the thought. He doused them with the threat of being kicked out of the band if things turned sour.\n\n\"I don't know. Are you offering?\" Rayne jostled Kris's shoulder as he grinned. \"No. It was a lot of adrenaline, and it was our first show together, and I got a bit overexcited. But I make a point not to get involved with straight guys, so it won't happen again.\"\n\n\"Why would a straight guy get involved with another guy at all?\" Kris asked blankly.\n\nRayne shrugged. \"They want to prove how open-minded they are, or they want to experiment, or whatever. Doesn't matter. I won't do it. That way lies heartbreak and misery, believe me.\"\n\nKris opened his mouth with no idea what he was about to say. Before he could find out, Butch stuck his head around the corner, eyed them, and waved them over. \"Bus time, guys. Up and at 'em.\"\n\n\"Hold that thought,\" Kris said to Rayne as they got up, each taking one more long drag on the pipe. He needed a minute to get his brain straightened out. That line he'd drawn looked a little shaky.\n\nKris thought about the kiss for the next hour, tucked away in his bunk as the bus trundled down the highway to New Jersey for their Camden and Newark shows. He lay on his back, his guitar across his ribs, strumming up and down the frets with no real intent. When, after sixty minutes of practicing riffs, he couldn't shake the memory of the kiss from his lips, he set his guitar aside and went to find Rayne. The weed was still curling through his thoughts, heavy and sweet, urging him to erase lines and push boundaries.\n\nRayne was stretched out along the couch in as private an alcove as there was on the bus, his boots propped up on one arm with his head on the other, looking like a giant cat, relaxed and languid.\n\n\"Hey,\" Kris said, testing the limits of his faculties, which were rapidly expanding in all directions. \"So I was thinking about that kiss.\"\n\nRayne rolled his head to the side to glance at him. \"What about it?\"\n\nKris took a deep breath and let his words out in a rush. \"You want to do that again?\"\n\nRayne perked up, lifting his head.\n\n\"Onstage,\" Kris clarified. \"For the show.\"\n\n\"As a regular thing?\"\n\n\"Yeah, like, to amp up the tension and stuff. They'll eat that shit up. Or that's what my sister says, anyway. You said no straight guys, but I'm\u2014\" He cut himself off before he could out himself, and changed tracks. \"I mean, if it's just for the show, that doesn't count, right?\"\n\n\"Kissing,\" Rayne said.\n\n\"Or more. I'm not asking you to maul me out there, but I thought. I wouldn't mind?\"\n\nRayne's eyebrows were up near his hairline. \"Straight.\"\n\nKris shrugged. He didn't want to publicly commit to a label until he was a hundred percent sure of it, and if Rayne thought he was straight\u2014ish\u2014in the meantime, that kept things simple. That, and his Midwest upbringing left him cautious of announcing he was any variety of queer at all, even when he was surrounded by the most welcoming people he could imagine. He just needed the chance to figure out what he liked without the risk of anyone getting hurt, and if Rayne was into it . . .\n\n\"You'd let me do . . . anything?\" Rayne asked.\n\n\"I'm pretty sure there are public decency laws and stuff, but anything legal, sure.\"\n\n\"Kris Golding,\" Rayne said, like he was savoring the taste. Kris liked how his name sounded in Rayne's mouth. \"You're a bit of a wild card, aren't you?\"\n\n\"Is that a yes?\"\n\nRayne sat up, dropping his feet to the floor, and stood, tall and suddenly looming. Kris tilted his head back and waited, his breath caught in his throat, as Rayne moved his hand to Kris's hair. He went slowly enough that Kris could duck away if he wanted, but Kris didn't move. Rayne threaded his fingers through Kris's crown.\n\n\"You've never been the center of attention before,\" Rayne said in a low voice. \"You sure you want that?\"\n\n\"As your boy toy?\" Kris asked, repeating Stef from earlier. \"You worried people might think we're fucking?\"\n\nRayne's grip tightened for a second. \"No. Aren't you?\"\n\nKris butted his head against Rayne's hand. \"Nah. It's just kissing. We'll tell everyone it's for the show; it's not like I joined a band and suddenly turned gay. The fans will love it, my family will understand, and you can handle the press.\"\n\nRayne's eyes went dark, the green almost eclipsed by pupil.\n\n\"We don't have to.\" Kris bit his tongue, needling the tip with his teeth as he waited.\n\n\"Honestly? I'm wondering how far you'll let me push it.\"\n\nKris prodded him in the chest, under the collarbone where his chains and pendants started to tangle. \"Probably pretty far. Try it and find out.\"\n\n\"Kissing,\" Rayne said.\n\nKris's lips buzzed with a phantom touch.\n\n\"Hair pulling?\" Rayne tugged on a lock to demonstrate. Kris leaned with it, sparks zinging through his scalp.\n\n\"I'm going to grow it out. By the end of tour it'll be long enough to yank around for real.\"\n\nRayne's lips parted for a second. \"Can I touch you?\"\n\n\"You're touching me right now.\"\n\n\"For the show, brat.\"\n\n\"Sure,\" Kris said, and purposefully didn't ask what kind of touch Rayne meant. When Rayne touched him skin to skin, it burned like a furnace and sent him shivering all over in its wake. \"You don't need to write up a contract, you know. You can just do whatever feels right. Whatever you think will work.\"\n\n\"Carte blanche.\" Rayne looked like he wanted to eat Kris alive.\n\n\"Yeah. That.\" Kris's mouth was desert-dry. He ran his tongue over his lips and Rayne's eyes went unfocused for a second before he seemed to snap back into himself and put space between them.\n\n\"God, you're something else. And I thought I might have upset you with that kiss.\"\n\n\"I'm pretty easygoing.\"\n\n\"I'm so lucky I found you.\"\n\n\"I'm the lucky one,\" Kris countered. \"But we're cool?\"\n\n\"We'll mess around onstage and put on a show,\" Rayne said. \"No strings, no complications, no messy feelings.\"\n\nThat sounded perfect. Kris bumped their fists together. \"Sounds good, man.\"\n\nRayne sat back down on the couch, smiling like this was an everyday transaction. Kris thought about lying down and trying to sleep as his skin buzzed from the memory of Rayne's lips on his with twenty-seven thousand people screaming for more. His thoughts were slow and rambling, the high still curling around his brain and making it hard to focus on anything but that kiss.\n\n\"You should find out where Angel gets her stuff,\" he said absently. \"This is good shit.\"\n\n\"She knows what's up,\" Rayne agreed, tipping his head back and letting his eyes fall closed.\n\nThe days turned into a week as they went from New Jersey to Virginia, then Michigan, and Indiana; down to Florida for three shows, to Alabama for one, until they were thirteen days in and playing in New Orleans to twelve thousand people, and Kris had barely had time to catch his breath, let alone work out the details of his attraction to Rayne. In between shows, Rayne did press, giving interviews and sound bites for the flashing cameras, coy when they asked about his love life and honest when they wanted to talk music. Kris didn't know how he had the patience for it, but Rayne took the business side of his music seriously, and gave the press just enough intrigue to keep The Chokecherries in the public eye.\n\nThe band threw themselves into their shows without hesitation. They never played a crowd under ten thousand, and as soon as the people knew to expect Rayne and Kris's midshow kiss\u2014they always had at least one, but as the shows went on the number climbed\u2014they screamed and cheered for more. Kris had worried that with the booze and drugs out of their systems, he and Rayne would reconsider the discussion they'd had after Hershey. It was the most pointless worry Kris had ever had in his life. Rayne's daring only grew as the tour went on, kissing Kris harder, for longer, and more often, running his hands through Kris's hair, tugging on his shirt, playing with his belt while Kris tried not to miss a chord.\n\nThe fan sites swarmed with videos and theories. Cassie sent Kris links to fanfiction featuring his and Rayne's imagined backstage love life, which he quickly learned to avoid without opening, though the first one stuck with him. There was something unsettling about strangers speculating on the details of his nonexistent sex life, especially in such graphic detail\u2014though he could admit it was hot, from under his blanket of embarrassment.\n\nThe more people reacted, the more Rayne pushed, until Kris was looking forward to Rayne's kisses as much as he looked forward to his solos, and their energy burned and crackled as they ramped it up with every show they played. After two weeks of near daily making out, and hugs and casual touches in between, Kris was so acclimatized to Rayne that he thought he might go into withdrawal if he went a day without him.\n\nLuckily, that didn't seem likely\u2014the fans and the press ate up the shows, and the rest of the band seemed delighted by the reaction. Kris was in heaven. By the time they played New Orleans, he felt like he could do this for the rest of his life.\n\nAs soon as they finished their set, Angel was waiting to whisk them offstage and back to the bus, cramming both bands in one, to see her club.\n\n\"Is it like a strip club?\" Kris hazarded, glancing sideways at her to gauge her reaction to his guess.\n\n\"The White Rabbit's a burlesque club,\" she corrected, as Butch turned the engine over and pulled out onto the road. \"We do strip shows, but not like you're thinking.\"\n\nKris's hometown had a bar where the girls dressed provocatively, but technically there was no stripping. Whatever went on in the back room was done under the table, and Kris had never asked about it. He'd only been once, and that at Brad's behest\u2014it had taken him a week to shake the sleazy feeling and look his girlfriend in the eye again. Angel didn't seem capable of running a joint like that.\n\n\"So what's burlesque like?\" he asked.\n\nAngel just smiled, sphinxlike, and Rayne laughed. It didn't answer a thing, but Kris nestled in between them and watched the city stream by, all glowing neon and car taillights, and decided he didn't mind finding out firsthand.\n\n\"My godfather sold me the building for a fraction of its worth,\" Angel explained, \"and I turned it into something completely new.\"\n\n\"It's a beautiful club,\" Rayne added. \"Billie helped design it, right?\"\n\n\"Yeah, you bought the place like a year after graduation?\" Billie said. \"And you asked me to help with bits. It was fun, like a giant installation project.\"\n\n\"That was before I met Rayne,\" Angel said.\n\n\"And she didn't introduce me to Billie and Passionfruit until last year, when we\u2014Passionfruit and The Chokecherries\u2014were both playing the same festival together, and Angel was touring with us as our makeup artist,\" Rayne said.\n\n\"Huh,\" Kris said. \"Small world.\"\n\n\"Not too small, or we'd get bored,\" Angel said. \"Here we are.\"\n\nThe bus slowed and pulled over to the side of the street. The White Rabbit was identified by a neon rabbit on a door that was otherwise unmarked, squeezed in between the brick walls of two other clubs. The rabbit glowed fat and white, neither inviting nor off-putting as Angel led them through the door, down the narrow stairs, and onward to greet the bouncer at the bottom. They slipped through without a cover charge, and the guard held the second door open for them as they entered the club beyond.\n\nThe White Rabbit was sumptuous, decadent, and sybaritic, like it had rolled out from an erotic Victorian fantasy. Leather couches lined the walls; the bar glittered with a hundred different kinds of alcohol in glass bottles and crystal decanters. The walls were a deep, rich red like bloodshed or rose petals, the ceiling was covered in tiny mirrors that winked in the changing light, and the place was full of the most beautiful people Kris had ever seen, dressed in leather or lace or lingerie or, in some cases, very little at all. On the stage at the back, someone hidden by huge ostrich plumes like a flapper pinup posed under a spotlight, nothing but long legs, pale skin, and high heels.\n\n\"They're expecting me to take the stage,\" Angel said. \"Grab a seat and enjoy the show.\" She pressed a quick kiss to Rayne's jaw before darting away behind the stage to change.\n\n\"Drinks,\" Rayne said. \"She'll say they're on the house, but we're not doing that. I've got this round.\"\n\nKris ordered something stiffer than his usual; it seemed like he was going to need it. They crowded around a couple of tables that had been saved for them near the stage, and Kris tried to figure out how to look at the other people in the room without being rude. Some of them wore masks and little else, making it impossible to meet their eyes. Kris settled on staring at their shoulders and trying not to blush. Rayne laughed at him, and Kris hid his face behind his drink.\n\nWhen Angel took the stage, the music swelled and the conversation quieted. The beat was sultry and seductive, and she walked out in an enormous fur wrap that covered her from throat to thigh, her heels so high she was almost walking en pointe. She was like a showgirl from the Moulin Rouge, her mouth dark and pouty, her eyes glittering in the low lights of the club. Kris wondered again how he had stumbled out of his universe and into this one\u2014it must have been here all along, just waiting for him to pull back the surface and take a peek.\n\nAngel danced like every eye in the world was on her, and she knew how to give them exactly what they wanted. The fur came away inch by inch revealing a smooth expanse of dark-brown skin, interrupted by the crisscrossing straps of her lingerie, refusing to give away the whole picture at once. She caught Kris's eye, and his mouth went so dry he nearly choked on his own tongue. Rayne put his arm around Kris's shoulders and spoke in his ear, close enough to be heard over the music.\n\n\"Better than you expected?\"\n\n\"Different,\" Kris agreed.\n\nThe music deepened, and Angel stepped from the stage, leaving her fur in a heap behind her. She wore a bra and garter set designed to look like flower petals, and Kris had never seen a more beautiful girl before in his life. She approached Jay, who sat nearest the stage, and curled her fingers under his chin as everyone jeered and wolf-whistled.\n\n\"I'd do this for Billie,\" she said, \"but I know he's shy when he's not onstage.\" She sidled in closer until she was right over his lap, and Jay dropped his head back, laughing helplessly under her attention.\n\n\"I'm filming this and sending it to your fianc\u00e9e,\" Billie cackled.\n\n\"She'll love it,\" Jay countered. \"She'll just be mad she's not here in person.\"\n\nAngel rolled her eyes without pausing in her dance. She swayed her hips, careful not to touch him, her arms crossed above her head as Jay clearly tried to work out where to put his hands without making it awkward. He eventually settled on tucking a folded bill into the top of her stocking. She kissed him on the cheek, leaving a rose-petal of lipstick behind, and stepped back, still moving to the beat.\n\n\"Who else wants to dance with me?\"\n\nRayne looked at Kris, who squeaked.\n\n\"Oh, honey,\" Angel said, stepping closer on those impossible heels until she was near enough to smooth her hand through his hair. \"You don't have to have a dance if you don't want one.\" She leaned in until her lips ghosted over his cheek, and his eyes fluttered shut of their own accord. \"But if you change your mind, you let me know.\"\n\nShe left him and sauntered back to the stage, where the music swelled again and the lights flared to welcome her back. Kris's heart was beating a mile a minute now, and Rayne's arm stayed heavy around his shoulders.\n\n\"You know when you're a kid at a restaurant and someone tells the waiter it's your birthday?\" Kris said.\n\n\"You think being offered a lap dance is as bad as having a roomful of strangers sing at you?\"\n\n\"How red is my face right now? I feel like I'm burning up.\"\n\n\"It makes you look healthy,\" Rayne said.\n\nAngel's number ended and despite Kris's embarrassment, he kept his eyes on her until she disappeared backstage, and the club exploded in applause.\n\n\"Let's get you another drink,\" Rayne decided.\n\nKris couldn't agree more.\n\nHe was buzzed but not drunk when Angel returned, wearing a silk robe over the lingerie, and shoes that, while still aggressively fashionable, seemed considerably less dangerous to walk in. With all the alcohol in his system, Kris had a harder time keeping his eyes from wandering; Angel caught on straightaway and smiled.\n\n\"Like what you see?\" she asked. \"You're allowed to look as long as we're both in the club. That's the whole point of me being onstage.\"\n\n\"You look . . . really nice,\" Kris stuttered, trying his hardest to stay polite.\n\nShe laughed and put her hand on his arm. \"You're the sweetest thing.\"\n\n\"You'd never know I found him on the roadside like a kitten in a cardboard box,\" Rayne agreed. \"Such a gentleman.\"\n\n\"Every time you tell this story, it gets worse,\" Kris complained, without meaning it. \"I was busking!\"\n\n\"You didn't have a permit.\"\n\n\"You need a permit to busk? That might be the first illegal thing I've ever done in my life. Besides the weed, I guess.\"\n\n\"Weed doesn't count.\"\n\n\"I'm sure it won't be the last thing, anyway,\" Angel said. \"Next round is on the house, and, Rayne, don't even try to sneak the bartender your money. I see you. Let's get turned up.\"\n\nKris wasn't sure how he ended up dancing in between Rayne and Angel half an hour later. He wasn't drunk enough to excuse it, but between the post-show high, the exhaustion preceding it, and the giddiness that had followed him ever since meeting Rayne, he might as well be ten shots in. Rayne's hands were on Kris's hips as they moved in tandem to the music, a slow, rhythmic beat slinking out from the speakers. There was a bare inch of space between their bodies; little enough that he could feel Rayne's body heat through his clothes, but they didn't touch. Angel caught his eye and nudged him back to close the gap. Kris's shoulders met Rayne's chest as Rayne caught and held him there, lightly, so Kris could squirm away if he wanted\u2014but he didn't move. He leaned back and let Rayne turn him around so they were face to face, and looked up to meet Rayne's eye.\n\n\"Hi,\" Rayne said.\n\nIn close quarters, surrounded by people, it would be second nature to reach up and kiss him. Kris fought the urge and slipped from Rayne's arms before he could do anything impulsive. No strings, no complications, and no causing trouble. \"I'm going to get another drink.\"\n\nTaking a deep breath, he headed for the bar, signaling for a refill of whatever he'd had last time. He slid onto a stool beside a stranger: tall and blond, not attached to either band, but clearly not a White Rabbit performer, either. The man was wholesomely handsome in a way that made him stand out among the club's sultrier inhabitants, and Kris gave him a smile before reaching for his wallet to pay for his drink.\n\n\"Hey, let me get that,\" the man said, smiling back and offering a few bills to the bartender. \"I'm Tom.\"\n\n\"Kris. Thanks for the drink.\"\n\n\"Are you performing tonight?\"\n\n\"Oh, I'm not\u2014 I'm in a band,\" Kris explained, stupidly proud of being able to say that. \"We're just hanging out.\"\n\nKris was still wearing his makeup from the show, but doubted it had survived without smudging. Angel and the other burlesque performers were flawless, not a line or hair out of place. Maybe Tom was drunker than Kris.\n\n\"I've been coming here for the past few months,\" Tom was saying. \"I haven't seen you before but I assumed\u2014I thought you must be one of the dancers. You're very pretty.\"\n\n\"Thanks,\" Kris said again, preening a little. He liked being called pretty; no one ever had before he met Rayne. \"So what do you do, Tom?\"\n\n\"I'm in seminary school,\" Tom said brightly. \"I want to be a youth pastor.\"\n\nKris blinked. \"And you're hanging out in a burlesque club?\"\n\nTom blushed and the freckles on his nose stood out. \"I just wanted to see what it was like. I haven't taken any vows yet.\"\n\n\"Huh.\"\n\n\"There's beauty in everything,\" Tom said, clearly warming to his subject now that Kris hadn't run away screaming. \"I don't want to preach about sin and shame; I want to appreciate God's work in everyone and everything.\" He looked into the depths of his glass. \"I might be drunk.\"\n\n\"I'm not judging you, man,\" Kris said. \"I think it's great that you've got an open mind.\"\n\n\"I don't want to condemn people,\" Tom said, \"not when they haven't hurt anyone. Look at you\u2014a nice young lady\u2014just because we met in a place like this, doesn't mean\u2014\"\n\n\"Hang on,\" Kris said.\n\nTom blinked at him, his expression open and painfully sincere.\n\n\"Never mind,\" Kris said. The alcohol made it seem like too much effort to correct him, and besides, he was strangely flattered by the attention, however misguided. \"Doesn't mean what?\"\n\n\"That you're not blessed,\" Tom said. \"Everyone in here\u2014we're all blessed. Can I get you another drink?\"\n\n\"Why not.\"\n\nKris was tucking into his second, something pink and sugary and with a significantly higher alcohol content than his usual, because girly drinks didn't fuck around, when Rayne joined them, flagging the bartender down.\n\n\"Making friends?\" he asked Kris.\n\n\"This is Tom,\" Kris said. \"He thinks I'm the prettiest girl in the whole club.\" Rayne raised one eyebrow and Kris shrugged. \"What's up?\"\n\n\"Brian,\" Rayne said, with uncharacteristic sourness. Kris made a concerned noise and Rayne waved him off. \"Business stuff. Don't worry about it. I'm just not drunk enough for the conversation he wants to have. I'll tell you about it later.\" The bartender slid him his drink and Rayne took a gulp as he put his money down. He swallowed and smiled. \"Seriously, don't worry. You two kids have fun getting acquainted. Nice to meet you, Tom.\" Rayne saluted them with his drink and slipped back onto the dance floor.\n\nKris watched him go, wondering if he should chase him down and press him for details, but Tom spoke before he could decide.\n\n\"Are you on tour?\"\n\nKris shook Rayne's moodiness off. If it was due to business, he'd find out the cause sooner or later. \"We're heading west to Nevada for the Purple Sage Music Fest. It's my first festival. It's my first lots of things.\" Two weeks into the tour didn't feel any realer than it had on the first day, and he smiled around his straw, bright and tipsily enthusiastic. \"Have you ever been?\"\n\n\"No, I've never done much either,\" Tom said.\n\n\"Well, if you have a pre-priesthood bucket list, add it. A lot of people have, like, spiritual awakenings at music events. Maybe you'll get something out of it.\"\n\n\"Maybe,\" Tom agreed, hesitantly. \"I'd like to try as many spiritual experiences as I can.\"\n\n\"Isn't that blasphemous?\"\n\n\"I don't think so. God is everywhere, after all.\" Tom examined his drink again. \"I think He is, anyway.\"\n\n\"You're the one with the calling,\" Kris said with a shrug. \"You must know better than most.\" Tom seemed reassured, and Kris swallowed the last of his drink and clapped him on the shoulder. \"Come see us if you decide to head that way. I'm going to go find my friends.\"\n\nHe found Angel sitting on one of the long, low couches, her legs crossed and a drink in her hand as she looked around her club with the air of a queen surveying her castle. Rayne was beside her on the arm of the couch, and there was something uncharacteristically tense in the way he was holding himself that made Kris pause in his approach, and then hang back amid a throng of dancers.\n\n\"I haven't decided anything yet,\" Rayne was saying, \"but I don't want Kris to hear it from anybody but me. Okay?\"\n\nAngel raised her hands. \"I'm steering clear of all of this.\"\n\nRayne glanced around but apparently didn't catch sight of Kris, then slid down onto the couch cushions, put his arm around Angel's shoulders, and whispered directly into her ear. Intrigued, Kris snuck in nearer, having drunk too much to feel properly guilty about eavesdropping. Something was bothering Rayne, and if Kris was even tangentially involved, he wanted to know what it was. He took up residence behind a giant potted fern at the far end of the couch and tried not to feel overly ridiculous. He missed the first half of Rayne's whispered sentence, but caught \"publicity stunt\" near the end.\n\n\"Me and some guy at the festival,\" Rayne finished.\n\n\"What kind of stunt?\" Angel whispered back.\n\nKris didn't catch what followed, but Angel gasped, and flashed a wicked smile that she quickly hid behind her hand. Rayne heaved a sigh. \"I shouldn't be talking about this at all.\"\n\n\"Security breach.\" Angel nodded sagely.\n\n\"I should go.\"\n\n\"You do that. Get some sleep, and when you've got your head on straight, then you decide what to do about it. No decision-making when you're drunk. You got that?\"\n\n\"Don't tell Kris.\"\n\nRayne swam back into the sea of dancers, and Angel shook her head at his retreating back. Kris, crouched on the floor in the shadows of the plant, tried to make sense of what he'd heard, but all he got for his troubles was a twist in his stomach that had more to do with Rayne keeping secrets than from the alcohol. It served him right for eavesdropping in the first place. He stood, careful not to disturb the leaves and give himself away, and circled around to approach Angel as if he hadn't been hiding amid her d\u00e9cor for the past two minutes. The whole thing left him feeling a bit like a scumbag.\n\nAngel straightened when she caught sight of him and smiled, extending her free hand to beckon him in.\n\n\"Hi,\" Kris said. \"Am I interrupting?\"\n\n\"Of course not; I was just admiring the view. Come sit.\"\n\nKris sat gratefully, tipping sideways into Angel's shoulder. He stole a sip of her drink and she laughed. \"Enjoying yourself?\"\n\n\"I'm good. I like it here. In the club, with the band and everybody.\" He forced the million questions on the tip of his tongue to the back of his mind. He didn't want to think about secrets, not even business-related ones, not when he was so close to having everything he'd ever dreamed of. Shuffling down, he rested his head in her lap. \"This okay?\"\n\n\"Mm. You're an affectionate drunk, aren't you?\"\n\n\"You're just really nice. And comfortable. What's owning a club like?\"\n\n\"A lot of work. I love it, but some days it feels like this place is determined to fall apart around me.\"\n\n\"You can't tell from looking.\" Kris couldn't, anyway; the place was beautiful, and Angel had clearly poured every drop of love she had into it.\n\n\"Boy, this whole place is smoke and mirrors.\" She petted his hair, and he closed his eyes, drifting contentedly on a current of drunkenness. \"I need a live-in repairman to keep on top of things. I wouldn't trade it for the world, though. Not for anything. What I should do is start managing it full-time again, get everything back on track.\"\n\n\"It must be nice to have somewhere to come home to. Somewhere that's all yours.\"\n\n\"Kansas isn't home for you?\" she asked.\n\n\"Nah. Not my hometown, anyway. It's a good place to grow up, or a good place to retire, but there's nothing to do in between. I had to get out.\"\n\n\"Lucky thing Rayne found you.\"\n\n\"Lucky I met him. Lucky I met you too. Need you to keep making me pretty.\"\n\n\"Rayne seems to appreciate it.\"\n\nKris snorted and flicked her knee. \"I appreciate it. I never got to play around with any of this stuff back home. I didn't have the guts. Now you're dressing me up in girls' clothes and makeup and I'm\u2014 Onstage, with the\u2014 It's fun. I wish I could have started years ago.\"\n\n\"Well, you're too old now,\" Angel said matter-of-factly. \"Can't make up for lost time.\"\n\n\"I'm baby-faced.\"\n\n\"Over the hill. Ancient.\" She fluffed his hair. With all the spray in it, it stayed in whatever position she put it. Kris was counting down the days until it would be long enough for her to have some real fun styling. \"Soon you'll go bald, and then what? Career, over.\"\n\n\"I'll get a wig. Rayne would get me a wig, right?\"\n\n\"Honey, I think Rayne would get you anything you asked for.\"\n\nKris hummed, pleased. \"Good.\" He poked her knee again. \"He's a good kisser, you know.\"\n\n\"I figured. A couple hundred thousand people figured.\"\n\nKris grinned and turned to look up at the ceiling, where the mirrors glittered and winked above them. He remembered first meeting Angel; he'd been drunk then too, and had spent the night wondering if he should kiss her, or maybe kiss Rayne. In the end Rayne had made the choice for him, and he didn't regret how it had played out. He didn't regret anything about that night, or anything about the tour after. The constant travel and playing had left him exhausted, but the drinks had taken the edge off and left him boneless and content. Angel was warm beneath him, her fingers twining through his hair, and he was suddenly, intensely grateful for the turn his life had taken.\n\n\"You're a good friend, you know that?\" he said. \"Rayne's lucky to have you. You're just\u2014you're a really a good person, and I'm glad I met you.\"\n\nShe laughed, and he felt it all the way through his body. \"How much have you had to drink, hun?\"\n\n\"Some,\" he admitted, \"but it's still true.\"\n\n\"Well, thank you.\"\n\nTheir conversation paused for a second, and the questions Kris had tried to bury came tumbling back. He chewed on the inside of his lip before finally asking, \"Do you know what's bugging Rayne? He was annoyed with Brian, but he wouldn't tell me what about. Did he talk to you about it?\"\n\n\"It's business stuff, but it might turn out to be nothing. If it is something, he wanted to tell you himself. Nothing to worry your pretty little head over, in any case.\"\n\nEven drunk, Kris was skeptical. \"Promise?\"\n\n\"You haven't been around long enough to see the business side of things,\" she said. \"I promise: if it's really important, Rayne or Brian will call a meeting. This is just . . .\" She shrugged. \"Rayne likes to make a fuss once in a while. He's a diva at heart, you know. Then he'll get over it and move on.\"\n\nKris couldn't imagine Angel lying, so he took her at her word and let it slide. Rayne would tell him eventually, and in the meantime, he would pretend he'd never heard anything at all.\n\nThey lapsed back into silence, watching the dancers move over the floor like fish shimmering in a pond. They were beautiful, all leather and lace, and Kris wondered again how he had ended up in such a place, and why it had taken him so long to realize it was where he belonged.\n\n\"Angel?\"\n\nShe blinked and looked down at him. \"Hm?\"\n\n\"When did you know you were trans?\"\n\nShe paused. \"That's a big question,\" she said eventually. \"I didn't start calling myself trans till I was almost through art school, but I did drag for a while before that. I took baby steps to figure it out. Some people know and are out from the minute they're born, and others take a more scenic route, like me. There's no wrong way to do it.\"\n\n\"So you didn't just, like\u2014\" Kris wet his lips. He could still taste the sugar from his last drink on them. \"You didn't just wake up one morning and start dressing in girls' clothes, and like . . .\"\n\n\"Turn queer?\"\n\n\"Yeah.\"\n\n\"I don't think it works like that.\" She smoothed his hair down again. \"Why? You got something on your mind?\"\n\n\"The clothes, and the makeup, and stuff. I like it. I like wearing it.\" He bit his lip and glanced up at her. \"I can still be a boy if I do that, right?\"\n\n\"The gender police aren't going to come arrest you because you like wearing skinny jeans, hun. If you want to be a boy, you're a boy.\" She tapped his nose. \"Just like kissing Rayne onstage doesn't make you gay.\"\n\nHis ears burned, suddenly too hot under the club lights. \"I still like girls,\" he blurted, trying to force his blush back before it gave him away.\n\n\"Sure you do,\" she replied, obviously fighting a smile. \"You can still like girls and like kissing Rayne at the same time. They've even got a word for folks who like both. Hell, they've got a couple of words.\"\n\nHe poked her. \"Stop making fun.\"\n\n\"I'm not, sweetie. I'm just saying.\"\n\n\"I don't know if I'm, you know. Ready. For those words yet.\"\n\nHe felt her hum more than he heard it. \"I get that. Anything in particular holding you back?\"\n\nHe shrugged ineffectually. \"Not really? Where I grew up\u2014Kansas isn't a great place for stuff like that. Experimenting, or coming out, or . . . any of it. I guess I'm still wrapping my head around things.\" He chewed on his lip for a minute before tipping his head back to look at her properly. \"You won't tell anybody, right?\"\n\n\"Course not. You can tell whoever you want when you're ready.\" She paused. \"Rayne would be excited to hear it, though. Everybody would, but especially him. Just so you know.\"\n\n\"He's a good guy,\" Kris agreed. \"I know he'd be supportive of whatever.\" He caught a glimpse of Angel rolling her eyes as he settled more comfortably. \"What?\"\n\n\"Nothing, hun.\"\n\nHe huffed and crossed his arms over his chest, still lying on his back with his head on her lap. He knew he looked ridiculous, but he didn't mind. \"He's right, you know,\" he said, apropos of nothing. \"The not-a-priest-yet over there.\" He waved to the bar.\n\n\"Tom? He's been coming here a few months now. He's a good regular. What's he say?\"\n\nKris smiled. He could feel his blush lingering over his nose; it felt like he was glowing. \"That we're blessed. That we're all blessed.\"\n\nThey rolled into Texas at ten the next morning, a solid ten hours before their Dallas show. While the majority of Passionfruit and The Chokecherries seemed happy to bury themselves in their bunks and take the time to sleep in, Rayne was out the door as soon as the bus stopped moving. Kris followed him, blinking in the sudden sun. Whatever had been bothering Rayne at the club seemed to have dissipated overnight, which Kris hoped meant Angel had been right about it being an overreaction. In the meantime Kris, though still curious, was happy to let it go.\n\n\"My tattoo artist from LA has a guest spot at one of the parlors here this week, and I've got an appointment,\" Rayne said, fairly buzzing with energy.\n\nKris was aware of a certain internet faction's obsession with Rayne's tattoos. He'd just never seen more than two in person in any detail before. He knew the mercury sign on Rayne's left thumb and the burst of rose blooms on the side of his neck, but the others remained hidden, only caught in stolen glimpses when Rayne changed backstage. The pictures on the internet didn't do them justice, and anyway, Kris was trying to feel as little like a stalker as possible.\n\nCassie, who emailed him scanned magazine spreads, wasn't helping.\n\n\"What are you getting?\" Kris asked.\n\n\"A snake. She sent me the drawing a month ago, and I've been freaking out ever since. It's going to be amazing. Do you want to come?\"\n\n\"Yes. Can I? I've never been to a tattoo parlor before.\"\n\n\"It's going to take the whole day,\" Rayne warned. \"It's a long session.\"\n\n\"If I get bored I'll go walk around, or I'll have a nap in the corner,\" Kris said. \"No big deal. I want to see.\"\n\nThey walked to the tattoo parlor, Rayne insisting it was just far enough to stretch their legs, while Kris was happy to tag along and soak up the sun. The only thing to identify the place was a koi fish painted on the door; there was no sign or window or anything else to indicate the nature of the building. Rayne pushed through the door and led Kris up a narrow staircase wallpapered in art, over the landing, and into the shop above. A long black couch sat in front of the reception desk and framed art hung on the walls. The buzz of tattoo guns filled the air as a handful of artists worked at their stations, scattered around the room.\n\n\"Rayne!\"\n\nThe woman who greeted them was maybe forty, with spiky black hair and inked art covering every inch of exposed skin below her jaw.\n\n\"Hey, Jiao,\" Rayne said with a wave. \"This is Kris; he's just here for company. Kris, this is Jiao Fang, my artist.\"\n\n\"Nice to meet you, Kris,\" Jiao said. \"Come over here, take a look at the final design.\"\n\nShe led them to her station in the back corner. The two walls were covered floor to ceiling in art: pencil sketches, charcoal, watercolor, ink\u2014Kris's mouth dropped open at the sheer skill of it. There were dragons and phoenixes and flowers, unidentifiable monsters, fish, knights, and portraits drawn with such delicate attention to detail that he couldn't fathom the time they must have taken. He could understand Rayne's devotion to Jiao; if he were going to have a drawing tattooed on his body forever, he would want it from someone as skilled as her.\n\nJiao fished a drawing from her desk drawer and handed it to Rayne. It was marked with a faint grid, but on top of the lines was a thick, coiled serpent, its scales immaculately rendered, flashing its fangs as it reached back for its own tail.\n\n\"This is what I sent you earlier,\" Jiao said. \"If you're happy with it, we're all set.\"\n\n\"It's perfect,\" Rayne breathed. \"I love it.\" He beckoned Kris over. \"Isn't it perfect? Jiao's done every tattoo I have. She's the best there is.\"\n\n\"I believe it,\" Kris said.\n\nJiao smiled and retrieved the drawing. \"I'll transfer it to the tattoo paper and we'll be ready in a minute.\" She headed off in the direction of the printer to do just that, leaving Kris and Rayne alone.\n\n\"Where are you putting it?\" Kris asked.\n\n\"My arm. It'll start here\"\u2014Rayne pointed to his shoulder, right where it met his chest\u2014\"and wrap around down to my elbow.\"\n\n\"That's huge. Won't it hurt?\"\n\n\"Yeah, but in a good way.\"\n\nKris's eyebrows lifted of their own accord.\n\n\"What? You don't think there's different kinds of hurt? This is worth it because you get to walk away with something amazing at the end. It's not like you're suffering for no reason.\"\n\n\"Uh-huh, sure.\"\n\nRayne waved him off. \"Get one yourself and you'll see what I mean.\"\n\nKris looked back at the wall of art. He'd never really considered it before, but then, he'd never considered a lot of things before The Chokecherries. Not seriously considered, anyway. His eyes were smudged from yesterday's makeup, his nails were painted black, and he had left home for the big city and joined a rock band he'd never heard of before. It occurred to him that he was the perfect candidate for a spontaneous tattoo.\n\n\"Huh,\" was all he said aloud.\n\nBefore Kris could elaborate, Rayne took his shirt off and Kris lost his train of thought entirely. He could see Rayne's tattoos.\n\nRayne had two birds perched on his chest, one near either shoulder. The left wore a crown, and the right had an arrow through its heart. Under his throat, running parallel to his collarbone, was an old-fashioned key. His right shoulder was capped with flowers, the same roses that crawled up the side of his neck to sit under his jaw, always half hidden by his hair, opposite where the snake's tail would start. The tattoos were all blackwork, and all recognizably Jiao's. Kris wanted Rayne to turn around to see if he had any more on his back. Maybe there were others below the waist of his jeans.\n\nHe swallowed.\n\n\"Like them?\" Rayne asked, his tongue poking out between his teeth.\n\n\"They're beautiful,\" Kris replied, and Rayne's eyes softened and lost their teasing edge.\n\n\"Thanks. I've got these ones too.\"\n\nHe turned; between his shoulder blades was a mandala like the sun, and when he lifted his hair there was a second mandala hidden along his hairline, peeking out down the back of his neck, like the fan of a peacock's tail. Kris inched closer to get a better look. The ink was older there, but it was still Jiao's work.\n\n\"The one on my neck was my first,\" Rayne said, dropping his hair and turning back. \"I had an undercut then; it's a full mandala, but I grew my hair out and you can't see the top half anymore. It's supposed to be\u2014\"\n\n\"A peacock,\" Kris finished. \"I can tell. It suits you,\" he added. \"They all do.\"\n\nRayne preened but whatever he was about to say was cut short by Jiao's return. The snake was printed on transfer paper, the design in reverse, and she smoothed it over Rayne's arm bit by bit, adjusting the angles as she went, starting at his chest and working her way down. When she removed the last of the paper, the snake coiled blue around Rayne's arm with not a scale out of place. Rayne twisted around to see it from all angles as Jiao held up a mirror, waiting patiently as he examined every inch of it.\n\n\"Okay, I'm set,\" he finally announced, sliding onto the chair. Jiao took up her position at his side, snapping her gloves into place, and her gun buzzed to life. Kris pulled a nearby stool closer and settled in to watch.\n\nWhatever Rayne said about the pain, it didn't show on his face. His hair fanned out over the chair's headrest, and he chatted back and forth with Jiao and Kris like he was out for coffee, rather than getting a tiny needle dragged repeatedly over his skin. Kris winced for him, but couldn't look away. The fingers of Jiao's gloves turned black with ink as she wiped away the excess, moving slowly and methodically over her design. The snake came to life under her hands one scale at a time, and Kris stared, transfixed. Rayne didn't watch. Occasionally something else leaked over his skin besides ink, and Kris leaned in closer, intrigued.\n\n\"Is that blood?\" he asked.\n\n\"Yep.\" Jiao swiped it away with her cloth. \"Inner bicep has thin skin. He'll bleed more here.\"\n\n\"Sounds fun.\"\n\n\"It's not that bad right now,\" Rayne said, \"but I guarantee I'll be bitching about it tomorrow.\"\n\n\"Adrenaline,\" Jiao said wisely. \"He likes it though. I gave him his first piece six years ago, when I opened my shop in LA. He's been coming back ever since.\"\n\nThe black ink was smoky and rich against the brown of Rayne's skin, and Kris wanted to touch the healed ones, just to see if they were as smooth and indistinguishable as they seemed.\n\nMaybe later he could ask.\n\n\"Benji is free this afternoon,\" Jiao said, \"if you're interested in a small piece. Her appointment canceled, so she left the slot open for walk-ins.\" She didn't look away from her work.\n\n\"No pressure,\" Rayne said. \"Your first tattoo, you should know for sure.\"\n\nKris wet his lips. \"Isn't your first tattoo supposed to be something meaningful? Because I'm feeling pretty impulsive here.\"\n\n\"Tattoos can be whatever you want,\" Jiao said. \"Get one if you want one. You should be sure you want it, but it's only ink.\" She shrugged. \"Not the end of the world.\"\n\nKris was saved from answering when his phone buzzed with an incoming call from his brother. \"I'm going to take this.\" He patted Rayne's knee. \"You keep at it, champ.\"\n\nHe crossed to the emptier side of the parlor and sank onto the couch. \"Hey, Bradley.\"\n\n\"Hey, Kris. Sorry I didn't call earlier. I heard you joined a band.\"\n\n\"Yeah, I did. It's going great. How's work?\"\n\n\"Good, good,\" Brad said absently. \"Listen\u2014I've seen some of the concert footage.\"\n\nKris picked at his nail polish and waited.\n\n\"You doing okay?\" Brad asked. \"Mom and Dad told me how plans fell through with Marty; you must have been really stressed trying to find something. I get if you just grabbed the first thing that came your way, but there are always other options, you know? You can always come home.\"\n\n\"I like the band, Brad.\"\n\n\"I know you like playing guitar, but you're a country kid, Kris. This\u2014what you're doing onstage\u2014you know that's not you, right?\"\n\n\"Pretty sure it is,\" Kris said, a thread of annoyance winding through him. \"And I never played country. I played bluegrass.\"\n\n\"I'm worried about you.\"\n\n\"Well, don't be,\" Kris shot back. \"I'm happy. I'm having a great time.\"\n\n\"They've got you dressing up and wearing makeup like some kind of a\u2014\"\n\n\"Bradley,\" Kris warned.\n\nBrad let out his breath. It came across as a crackle of static over the line. \"You can do better than this. Don't change into something you're not just to fit in.\"\n\nKris ended the call, his fingers shaking, and shoved his phone back into his pocket with more force than was necessary. A girl with short rainbow hair was sitting at an empty station, flipping through her phone and pretending not to hear to his conversation.\n\nKris raked his hands through his hair before approaching her. \"Hey, are you Benji?\"\n\n\"That's me.\"\n\nHe smiled, still shaky. \"Can I make one of those walk-in appointments Jiao was talking about? I think I need a tattoo after all.\"\n\nBenji walked him through the paperwork, and he scribbled his signature swearing he wasn't drunk and wouldn't blame them for any complications or change of heart later on, and she led him to her chair, sat him down, and handed him a thick portfolio of flash designs.\n\n\"They've all got their prices marked, or if you want something not in the book, we can make a deal,\" she said. \"Did you have anything particular in mind?\"\n\nKris flipped through the pages of artwork. There were death's-head moths, skeletons, hearts, and daggers\u2014all the classics, all with their original spin. The art was flawless. He chewed on his lip as he thought it over, his heart racing. \"Can I just get a star?\" he finally asked.\n\n\"Like an outline?\"\n\n\"Yeah, just a five point star, right here.\" He pointed to the inside of his left wrist, just below the joint.\n\n\"Sure,\" Benji said. \"Our base charge is eighty bucks, and a star should only take a couple of minutes. I won't charge you more than that.\"\n\nHe handed her the portfolio back and tried to calm his heartbeat. She pulled a marker out, took his wrist, and drew a perfect freehand star on his skin, the lines straight and the points sharp. \"Like this?\"\n\nHe examined it. His pulse still felt like it was going to burst out of his skin, but he nodded and tried not to grin like an idiot. \"Yes, please. Do that.\"\n\nShe donned her gloves, fitted a new needle to her gun, and he gave her his wrist. The first bite of the needle didn't hurt as much as he'd expected, but he still went tense from it. It felt like a hard, biting drag against his skin, but Benji's hands were steady, and he was more excited than uncomfortable. She paused to wipe away the ink and he watched a bright-red drop of blood well up in its wake.\n\n\"So it's like scarring, right? That's why it bleeds?\"\n\n\"It needs to go deep enough for the ink to set permanently,\" Benji said. \"It's technically an open wound until it scabs over; that's why aftercare is so important.\"\n\n\"Gross.\"\n\n\"Yeah,\" she agreed, and wiped the last of the ink away with a smile. \"But not gross enough to stop people. You're all done.\"\n\nHis wrist now sported a perfect five-point star. The lines looked puffy and tender.\n\n\"I'll wrap it up for you, and then you can pay Alicia at the front desk. All good?\"\n\n\"Perfect. So perfect, thank you so much.\"\n\nShe covered the tattoo in a plastic medical film that stuck to his skin, explaining what she was doing as she worked. He nodded along, unable to take his eyes off his wrist.\n\n\"Alicia will give you a pamphlet with everything written down,\" Benji added at the end, \"but it's pretty easy. Leave this on for twenty-four hours, and then replace it whenever it gets dirty over the next week.\" She handed him a few sheets of the medical product, already cut to fit his tattoo, the plastic waiting to be peeled from its paper backing. \"This isn't plastic wrap from the kitchen,\" she warned. \"They're not interchangeable. This stuff is breathable and antibacterial, and using regular plastic wrap will leave you open to infection. Okay?\"\n\nKris nodded emphatically. \"Special medical wrap, no kitchen supplies. Got it.\"\n\n\"After a week, you can leave it uncovered. Keep it clean and moisturized, and you're good to go. A little one like this should be healed in no time.\"\n\nAfter paying at the desk, which was decorated with sugar skulls and tiny statues of samurai warriors, Kris wandered back to Rayne and Jiao. The snake didn't have a head yet, but it was coming along.\n\n\"You got one!\" Rayne said, clearly torn between delight and accusation. \"I wanted to watch.\"\n\nKris thrust his wrist out, bursting with pride. The star was blurry under the ink and the blood, which collected in the wrap's creases, but he didn't care how gory it looked. It was perfect.\n\n\"Nice,\" Rayne said. \"You decided, just like that?\"\n\n\"It's my body,\" Kris said. \"I get to do whatever I want with it, right?\"\n\n\"Course,\" Rayne agreed, but he caught Kris's hand and held it. \"You good?\" He glanced at Kris's phone, which was sticking out of his shallow pocket.\n\n\"I'm fine,\" Kris promised. \"It was dumb family stuff. Your snake is awesome though\u2014you still bleeding?\"\n\n\"He'll keep bleeding till I'm done,\" Jiao said.\n\nRayne gave a dismissive wave. \"Who cares about that. Did yours hurt? Your first tattoo on the inside of your wrist\u2014that's brave.\"\n\n\"Nah, it's only little. And this plastic stuff is really cool. I thought I was going to get all wrapped in gauze or something, but now I can keep an eye on it the entire time.\"\n\n\"It's the same wrapping they use on burn victims,\" Jiao said. \"It's less work for you than cleaning and moisturizing it every day as it heals. You leave it on for a few days, and when you peel it off, you're all done.\"\n\n\"Burn victims,\" Kris said. \"Don't they use baby foreskins in that stuff?\"\n\nRayne froze. They looked at each other.\n\n\"They have amazing medical properties,\" Jiao said with a shrug. \"Try not to think about it.\"\n\nKris glanced down at the plastic film. Underneath, the excess ink blotted and welled. \"Yeah, you know what? Pretend I never asked.\" He patted Rayne's knee reassuringly. \"Just pretend I never said anything at all.\"\n\n\"Right,\" Rayne said faintly.\n\n\"I've been using that stuff on you for years,\" Jiao said. \"Don't be childish.\"\n\nKris sniggered and Rayne smacked him. He deserved it.\n\nIt took another hour for Jiao to finish the snake. Rayne talked less as time passed, the discomfort finally catching up to him, but Kris stayed by his side until it was done. His own wrist started to sting as the adrenaline wore off, but Rayne was right: it was the good kind of pain. It felt like commitment. No matter what happened on tour, or where he ended up after, he had a piece of it under his skin now, and even if he never wore makeup or girls' jeans or dyed his hair again, his star would stay with him. He rubbed his thumb along the perimeter of the wrap and smiled to himself as Jiao's gun buzzed away, etching the snake into Rayne's arm. Soon the snake and the star would both be healed, smooth and indistinguishable to the touch, and when they were hidden under shirts and jackets, no one else would know they were there.\n\nThey had three back-to-back shows in Texas, and each show ratcheted the sexual tension between them up another notch. The night after their tattoos, Rayne set himself on Kris like a dog on a rabbit, and twelve thousand people screamed their approval. Dizzy from the taste of fame, Kris couldn't tell whether it was the roar of the crowd or Rayne's lips on his that had his blood pumping so hard, but he didn't question it. It was good\u2014the lurch of his stomach, the weakness in his knees, the way he saw stars when he closed his eyes, listening to Rayne's voice soar above the music. Every show, Rayne pulled his hair and petted his chest and stalked him, intent and predatory, radiating want. The crowd's screams always reached their crescendo when they kissed, almost as loud as when they played their encore, and Kris, breathless and drunk on the music and the smell of Rayne's cologne, lapped it up like he was starving.\n\nHe knew how they looked together. Cassie sent him the videos afterward: the same kiss from a hundred different angles, the reverent gasps and shrieks from a thousand different mouths. Backstage after the shows, Kris's whole body tingled with phantom touches. He could remember every brush of Rayne's skin against his own\u2014hand to hand, or at his throat, against his scalp, inside his collar and down his chest, a slow, dragging tease\u2014\n\nHe was more of an exhibitionist than he had thought, but then, the stage was the stage. If he wasn't up there for the attention, what was the point?\n\n\"Just don't get arrested,\" Brian said, when it became evident that their public petting was only going to get heavier. \"Keep all your private bits private. If this leads to public indecency charges, that's on your head, Kris. You're still on a trial run here. And Rayne? You know what happened last time, and none of us want a repeat of that mess.\"\n\n\"That's not happening again,\" Rayne said firmly. \"I have this under control.\"\n\nBrian seemed skeptical. \"And what we talked about in New Orleans\u2014\"\n\n\"I know,\" Rayne said. \"I said I'd tell you when I decided.\"\n\nKris looked between the two of them, not wanting to get on anyone's bad side by asking for clarification.\n\nBrian sighed and waved them away. \"Fine, okay. Other than that, go nuts.\"\n\nPassionfruit joined in on their antics immediately following Brian's reluctant blessing.\n\n\"Don't you guys have fianc\u00e9es?\" Kris asked before the first of their newly sexually charged shows.\n\n\"Yeah, and they've both given their very enthusiastic permission.\" Jay held up his phone; on the screen was a text message that consisted of nothing but exclamation marks, followed by a second one that read GOD YES SEND PICS.\n\n\"Huh,\" Kris said. \"Well, whatever works, right?\"\n\n\"Just a couple of straight guys being dudes,\" Billie agreed, fixing his makeup with a handheld mirror. \"Do you think red eye shadow is too much?\"\n\n\"What is this, 2005?\" Jay asked.\n\n\"Too late, I'm doing it anyway.\"\n\nPassionfruit approached their new stage play with the same aggression they poured into their music: raw and desperate, as compared to Rayne's simmering sensuality. Still, it seemed to work for them. Jay and Billie dragged each other around the stage, hands fisted in one another's hair, and Billie dropped to his knees to scream his lyrics out to Jay's guitar from below.\n\nKris watched from the side. He couldn't deny it was hot, in a feral way. Whatever chemistry Jay and Billie had before\u2014and he was pretty sure Billie had been joking when he'd said they were both straight\u2014it came boiling over now, frothing at the mouth and leaving the crowd hoarse with lust. The band came backstage invariably soaked with sweat and brimming with energy. Jay thumped Kris appreciatively on the chest as he passed.\n\n\"Great idea, man. The best.\"\n\n\"It really adds something to the performance,\" Billie added, watching Jay strip off his sodden shirt and upend an entire bottle of water over his head.\n\n\"Happy to help,\" Kris said. \"Hope the girlfriends like it.\"\n\n\"They'll be so jealous they can't see it in person,\" Jay said.\n\nKris nodded and elected not to pry any further.\n\nThe thing about life on the road, Kris decided sixteen days in, was that it was fucking exhausting. When he stole naps during the day the bus rocked him to sleep like he was a toddler, and at night he passed out cold, impervious to every outside stimulus, from snoring to traffic to blown tires, but he never seemed any better rested. The bunk wreaked havoc on his back, and he longed for a single night without his mattress moving under him. When Brian announced after their Austin show\u2014two stops away from Purple Sage\u2014that they were staying in a hotel, Kris nearly cried from sheer relief. A real bed, in a real room\u2014he was going to sleep for twelve hours straight and there wasn't a force in the entire universe that could stop him.\n\n\"So, there's been a slight mix-up,\" Brian said apologetically that night.\n\n\"Please tell me we have a room.\"\n\nEveryone else had already shuffled off to their respective suites for the night; it was only Kris and Rayne left in the hotel lobby with Brian.\n\n\"You do,\" Brian said. \"But you're sharing.\"\n\n\"Not a problem,\" Kris said quickly. \"No problem at all.\"\n\n\"There's only one bed.\"\n\nKris glanced at Rayne.\n\n\"They can bring up a cot though, can't they?\" Rayne said.\n\n\"I'm not sleeping on a cot,\" Kris said. \"I might as well stay on the bus.\"\n\nRayne, the diva, gave him a look that said he was most certainly not sleeping on any cot either.\n\n\"Forget it. I don't care. We can share a bed, right, Rayne?\"\n\n\"It's queen-sized,\" Brian said, seeming like he wanted to crawl straight into his own bed, and damn their sleeping arrangements.\n\nKris poked Rayne in the arm, careful that it wasn't the tattooed one. \"Your call, big shot. Are we sharing or not?\"\n\n\"We can share,\" Rayne said. \"As long as I can lay down within the next ten minutes, I really don't care.\"\n\n\"Make it sound like such a chore,\" Kris grumbled. Rayne reeled him into a sideways hug, and Kris dropped the act, ducking his head with a grin and hugging him back.\n\n\"Good. Perfect. Here's your key; we're checking out at eight. Don't be late.\" Brian dropped the key in Rayne's palm and departed without a backward glance. Kris couldn't blame him. He and Butch had been taking turns driving, and if Kris was tired, they must be nearly comatose.\n\n\"Dibs on the shower,\" Rayne said, pushing Kris toward the elevator.\n\nReaching their floor a minute later, Kris stumbled along, dragging his bag down the corridor. As soon as they arrived at the room, he face-planted into the pillow. He melted into the mattress as Rayne started the shower; the pipes rattled and the suite filled with the familiar hum of running water. Kris rolled onto his back and stared at the ceiling, absently listening to Rayne move in the bathroom, the spray shifting as he stepped into the stream. Kris had never lived this closely with anyone before, not even his own family. Touring was like living on top of one another for every minute of the day, a constant crush of forced intimacy, yet it felt good: there was always someone in touching distance, always someone to talk to, no matter the hour. Kris had taken to it like he'd been waiting all his life.\n\nHe understood how it could be overwhelming. Maki disappeared on a regular basis, slipping away to rebuild her personal space without a dozen eyes on her. Stef wore noise-canceling headphones nearly constantly unless they were onstage. Sometimes Kris needed room to breathe too, but he found reassurance in never being alone. He was so comfortable with his bandmates already, and Rayne in particular, like he'd known him a million years.\n\nWhen Rayne came out from the bathroom, Kris was flicking through his phone, looking at conversations he'd had with people months ago, old acquaintances he'd never deleted from his contacts, even after leaving Kansas and assuming he'd never see them again. He felt like a different person now. Would any of them recognize him anymore?\n\n\"Your turn,\" Rayne said.\n\nHe wore nothing but a towel slung low around his hips, inviting Kris's attention. Rayne didn't seem to know how to be shy, not about his body or anything else. Kris took the bait and whistled as leeringly as he could while pretending to be unaffected. Rayne laughed.\n\nIt was hard not to look, though, and Kris didn't fight the urge for long. Rayne was broad-shouldered and narrow-hipped, with the faintest suggestion of abs. His skin was warm and rich in the low lamps of the hotel suite, like the darkest shades of amber: gold where he caught the light, and brown in the shadows. He moved like a panther, elegant and self-assured, tiny rivulets of water running down his body over his tattoos as he crossed the room. Kris wet his lips and glanced away.\n\n\"You're dripping everywhere.\"\n\n\"I'll leave a big housekeeping tip.\" Rayne flopped onto his back on the other side of the bed.\n\nKris headed for the bathroom before he could see how far the towel shifted.\n\nIn the shower, he lathered up in complimentary body wash and shampoo. The water was hot, the pressure was perfect, and Kris was not reliving the last kiss they'd shared onstage.\n\nIn fact, he wasn't fantasizing about Rayne at all. This preoccupation was just a matter of bad timing. He hadn't gotten laid in over four months since he and his girlfriend had broken up before he left for New York, and they'd been drifting apart for longer before that. They had parted on mutual terms, and there was no bad blood between them, but still\u2014he was getting antsy. And since Rayne's attention onstage was the only action he was getting, it was only natural for him to get a little distracted. It didn't mean he wanted to do anything about it.\n\nRayne was the kind of tall, dark, and handsome that could make even the most heterosexual man grudgingly admit he was good-looking, and Kris was far from straight. Some might say Rayne was too pretty, but Kris liked that. He liked how Rayne could switch from being a total dork offstage, overenthusiastic, with a braying laugh, to the sultry, slinky predator who stalked around in front of the crowd, and he liked how Rayne was equally sincere in both roles. But they never kissed except during the show. Sex was for the stage, and the stage alone. Passionfruit was equally close, but then, they were odd too. Kris didn't have any kind of baseline for normality anymore. His parents didn't comment on it, his sister jeered and congratulated him, and he hadn't talked to Brad since that phone call in the tattoo parlor.\n\nWould it be easier if he'd been a Chokecherries' fan prior to meeting them? No, it was better this way. Less awkward. He shouldn't overthink things. If he were smart, he'd leave what they did onstage onstage like they'd agreed, and that would be that. He'd compartmentalize.\n\nHe sighed, turned the water all the way to the cold side, and stood under the spray until he was shivering. Getting involved with a bandmate was way too risky when his position in the band was so tenuous. Besides which, he still didn't feel ready to come out. It was ridiculous; he was as safe as he would ever be, and the fans and the press all assumed he was some kind of queer anyway. But his heart stuttered when he thought about making that final leap and saying it out loud, so he bit his tongue and kept it under wraps.\n\nBack in the room, Rayne, now dressed in pajama bottoms, was watching something bright and loud on the television.\n\n\"I thought hotel nights were for hookers and booze,\" Kris said. \"You're watching a musical.\"\n\n\"If you want hookers, you can get your own room,\" Rayne said. \"There's booze in the minibar, though.\"\n\nKris fluffed his pillow and joined him, settling in to get comfortable. \"Nah, I'm good. What is this?\"\n\n\"An Indian movie. I used to watch them with my mom all the time, but I haven't been keeping up the last couple of years.\"\n\n\"Like Bollywood? Aren't those movies all singing and dancing?\"\n\nOn screen, a man in a chariot slashed through an enemy army.\n\n\"A lot of them,\" Rayne agreed, \"but not this one.\"\n\n\"Holy shit, that was epic.\"\n\n\"Big budget.\" Rayne nodded.\n\nIntrigued, Kris kept his eyes glued to the screen as the battle raged on, men getting torn apart in slow motion. \"So movies were a thing for you and your mom?\"\n\n\"She was an actress in India before she moved to the States with my father. He's Persian. I got more of his looks, but I got her flair for the dramatic.\"\n\n\"That's so cool. They must be proud of you, following in her footsteps, huh?\"\n\n\"My dad died a few years before I started the band, and my mom moved back to India not long after, to be with her family.\" Rayne seemed wistful, but not sad. \"I hope he'd be proud of me. I know she is.\"\n\nKris shuffled sideways to fit himself under Rayne's arm, nestling in against his chest. \"I bet they both are. Look at you, man. You're living the dream.\"\n\nRayne hummed.\n\n\"Do you miss them?\"\n\n\"Sometimes,\" Rayne said, after a moment. \"I still visit my mom, and I'll always miss my dad a little bit, but I'm okay.\"\n\nKris gave him a comforting squeeze. \"Is India really like this?\" he asked, nodding to the movie.\n\n\"Kind of? I mean, this is fantasy, but it's recognizable. You could come see it in person, if you wanted.\"\n\n\"Is the tour going through India?\"\n\n\"No, but we could go on our own. The rest of the band's already been, but the two of us could visit. I could show you around.\"\n\nKris twisted around to look up at him. \"Seriously? I've never been outside the States before. I'd love that.\"\n\n\"I'll make it happen,\" Rayne promised.\n\nKris smiled and settled in to watch the movie. He'd already missed the beginning and his vision was starting to blur, making the subtitles a challenge, but the cinematography was beautiful. On screen, an elephant reared up, balancing on its hind legs. \"You should get an elephant,\" Kris said sleepily. \"Make it part of the show. Or, like, put it in your next video or something. That would be awesome.\"\n\nRayne seemed to give that thought way too much consideration. \"I don't think we could use it in the live show. A video would be good though. I should ask Brian.\" He dug out his phone and started texting.\n\nTwo minutes later Brian called. \"Absolutely not.\"\n\nKris giggled into his pillow.\n\n\"No wild animals. Never mind the liabilities\u2014do you know how hard it is to wrangle a bus full of musicians? And you want to add an entire elephant to the mix? No. Stop watching TV and go the fuck to sleep, Rayne. And Kris? I know you're listening, and I know this was your idea. Sleep. Now.\"\n\nHe hung up.\n\n\"I'll work on him,\" Rayne decided.\n\n\"Maybe we can start with like, a horse, and work our way up.\"\n\n\"Or a snake,\" Rayne said around a yawn. \"A big ball python or something. They're exotic. They'd look great in a video.\"\n\n\"For sure,\" Kris agreed. The camera panned over an ancient cityscape as the soundtrack wailed. \"We can get a snake. A snake must be easier to wrangle than a horse or an elephant.\" He tried to think of everything he knew about India, which wasn't much. \"You want a peacock? They're Indian, right? They've got those at the zoo just roaming around, mingling with the visitors and stuff. I bet we could steal one.\"\n\n\"You want to steal me a peacock? That might be the most romantic thing anyone's ever said to me.\"\n\nA delightful shiver shot through Kris at the word romantic, but he crushed it down. \"Fuck yeah, I'll steal you a peacock. Also, you should probably raise your standards, but that's your business. But I will absolutely steal you a giant bird to further your rock star aesthetic if that's what you want.\"\n\nRayne tugged him sideways so Kris sprawled the rest of the way across his chest. \"You're the best. Don't get arrested though.\"\n\n\"Why not? That's totally rock and roll. How's anybody going to take me seriously if I've never been arrested?\"\n\n\"Brian would have a heart attack and cancel your contract,\" Rayne pointed out. \"And then I'd have to leave you in jail to stay on his good side.\"\n\n\"That's cold, man.\"\n\n\"That's show business.\"\n\nA jungle stretched across the TV, calmer now that the battle had passed, and Kris's eyes kept drifting closed of their own accord. Listening to the background music, he could pick out similar themes in The Chokecherries' songs: an undercurrent of Indian influence he'd never noticed before.\n\n\"Early night?\" Rayne asked.\n\n\"So rock and roll.\" Kris snuggled deeper into the pillow, Rayne's heat a constant burn against his side. \"Going to leach your body heat a bit longer, then I'll move back to my side.\"\n\n\"I knew you were just using me.\"\n\nRayne ruffled Kris's hair until Kris batted at him. On screen, a chariot horse blinked at the camera like it was judging them. Kris didn't care.\n\nSomething nudged him. \"Hey. You awake?\"\n\nKris blinked groggily, willing his eyes to focus. The clock on the bedside table read 3:03. He had been awake, more or less, drifting in that heavy in-between place and flirting with consciousness. \"Mm?\" Rayne was a warm presence against his back, and Kris rolled over to face him. Their knees bumped under the covers as Kris propped himself up on one elbow and yawned. \"What's up?\"\n\n\"Want to talk to you.\" Rayne looked delicious in the dark, his hair unmanageably mussed from the pillow and his eyes soft from sleep.\n\n\"At three in the morning?\" Kris asked around another yawn. \"Okay, sure. Talk about what?\"\n\n\"There's this band going to Purple Sage\u2014Dead Generation. They're signed to the same label as us. They're new, but they're good.\"\n\nKris woke up properly. This must be about Rayne's secret from the club. \"You want to add them to our opening act?\"\n\nRayne shook his head. His hair fell in his eyes, and Kris was momentarily distracted.\n\n\"Their front man, Calloway, got outed in a gossip rag the other week, and he and the label have decided to roll with it. They've asked me\u2014the label has, I mean\u2014to step in for a publicity stunt. Something to put the narrative back in their control.\"\n\nKris blinked and tried to make any kind of sense out of Rayne's words. He must be more asleep than he'd thought. \"Sorry, what? You're going to have to spell it out for me, man. I'm not versed in all this cryptic industry talk yet.\"\n\n\"Me and Calloway,\" Rayne said. \"They want us to go on a few dates, let the paparazzi get a few pictures, spread a few rumors. Just for the length of the festival.\"\n\n\"They want you to . . . pretend to date some guy. Oh my god, they're pimping you out!\"\n\n\"No, that's\u2014 Well, sort of. It's an image thing. Now that he's been outed, the label has to decide how to reinvent his brand. They chose me. Everybody loves me, and I've been out since before I signed. And the papers have been dying to dig into my personal life for ages now. Two birds, one stone.\"\n\n\"This is what Brian was talking to you about in the White Rabbit?\" Kris asked incredulously.\n\n\"You knew about that?\"\n\n\"I overheard you talking to Angel about something that night. You had said it was business stuff.\"\n\n\"Right, well, Brian's not impressed with the idea either, but I said I'd consider it.\"\n\n\"Okay,\" Kris said, slightly dazed. \"Fake dating. That makes sense. It's so obvious; why didn't I think of it?\"\n\nRayne rolled his eyes fondly and prodded Kris in the shoulder. \"I'm not pretending the music industry is remotely sane.\"\n\n\"So are you going to do it?\"\n\nRayne was still for a moment. Rayne being still meant he was deep in his thoughts, and that was never a good thing. He was the kind of guy who made split-second decisions and never looked back, like lightning in a bottle. He wasn't careless, but he wasn't given to overthinking. Kris leaned in and tapped him on the forehead, right between the eyes.\n\n\"I don't know,\" Rayne said finally. \"That's why I'm asking you.\"\n\n\"What? Why? What have I got to do with your fake love life?\"\n\n\"Right now, you are my fake love life,\" Rayne pointed out, and Kris's heart gave a thrilling thud. \"A good chunk of people\u2014fans, tabloids\u2014think we're a thing, and they'll keep thinking that as long as we keep doing what we do onstage, no matter what we say. If Calloway and I do this stunt, it'll throw all that for a loop, so I thought I'd better ask before deciding.\"\n\nNot six hours ago Kris had been fantasizing about Rayne in the shower. Now, lying nose to nose in bed together, his thoughts weren't much further ahead, and that wasn't helpful when Rayne was trying to talk about something important. Kris forced himself to concentrate and consider Rayne's proposal. \"Would you having a fake boyfriend affect our shows?\"\n\n\"Brian's advising us to drop the make-outs while I go out with Calloway, but he says that's up to us. People will talk about it either way.\" Rayne smiled wryly. \"The press loves a scandal.\"\n\nKris took a deep breath and closed his eyes for a second. His brain was still fuzzy, not used to functioning at this hour, let alone making publicity decisions. Rayne was a solid presence in the dark beside him.\n\nKris didn't have any right to monopolize Rayne's attention\u2014or his affection, his sleepy brain added traitorously. Or his touches, or his kisses, onstage or off. Rayne was just being considerate, giving Kris a heads-up like this.\n\n\"If the label thinks it's a good idea . . .\"\n\n\"Brian's skeptical,\" Rayne said, \"but he always is. He stressed it was entirely my choice. They're not going to make me do anything I don't want to.\"\n\n\"They're just going to pointedly encourage you?\"\n\n\"Right.\" Rayne sighed.\n\n\"Do you know this Calloway?\"\n\n\"Never met him. Here, take a look.\"\n\nRayne fumbled for his phone before handing it to Kris, and the screen lit up with a picture of the singer in question. Calloway was heavily freckled under his sun-kissed glow, with coiffed ginger hair and a blinding smile. He was attractive, but anyone could look like a model with enough airbrushing.\n\n\"Irish?\" Kris guessed.\n\n\"So they tell me. I've listened to his band, and they're good. They're not big yet, but they could be, given the chance.\"\n\n\"Okay. So he's pretty, and he's got talent. That sounds like your type, right?\"\n\n\"Evidently. So? Are you going to be my voice of reason and tell me it's a terrible plan?\"\n\nKris thought about it. He and Rayne weren't dating. They weren't even friends with benefits. In fact, they weren't anything at all besides bandmates trying to put on a good show, and that was exactly how they were going to stay, because Kris was a goddamn professional and he wasn't going to ruin his shot at making it big with the band. If Rayne were involved with somebody else, even for a single week, even just for the tabloids, that would keep him strictly off-limits from Kris and his increasingly overactive imagination. A week would be more than enough time for Kris to pull himself together and get over whatever this was. It was perfect.\n\n\"Actually,\" Kris said, \"I think it's a great idea.\"\n\n\"Really?\"\n\n\"Yeah. Like you said, two birds, one stone. And you could help jump-start this guy's career, by the sound of it. That's got to be good karma, right?\"\n\n\"Right,\" Rayne agreed. Was it Kris, or did he look almost disappointed? No, it was just the way the shadows played over his face.\n\nKris plastered on a smile and socked Rayne on the shoulder. \"It's cool, man. Go sow your fake wild oats.\"\n\n\"We'll have to change our shows a bit,\" Rayne reminded him.\n\nJust the thought of their shows left Kris burning up from remembered kisses. Cold water, he thought. Freezing, ice-cold water. The least-sexy feeling imaginable. \"We'll figure something out.\" He paused, then asked, \"Hey, what was that other thing Brian was talking about earlier? Something about not wanting a repeat of what happened last time?\"\n\nFor a moment it didn't seem Rayne was going to answer, and then he blew out his breath with a rueful smile. \"He was worried about you and I getting involved onstage. He knows my history with straight boys, and what happened before\u2014it got messy and ended badly. As these things do.\"\n\n\"Heartbreak and misery?\" Kris guessed.\n\n\"Something like that.\" Rayne slumped lower, propped up on one elbow. \"It was with Fink,\" he finally admitted. \"Before he got into the hard drugs. We used to fool around\u2014not during the shows, but on tour\u2014and I wasn't great at keeping my feelings in check then. When I told him I'd fallen for him, he just laughed.\"\n\n\"What a dick,\" Kris said, disbelievingly.\n\n\"Yeah.\" Rayne shrugged. \"He said he was still straight, and what we were doing was just for fun. 'A good time,' he called it. I was sitting there pining while he was fucking around the whole time with any groupie who looked his way, acting like nothing had changed between us. I couldn't do the same. After that his drug use got too bad to ignore, and you know how that turned out.\"\n\n\"I'm sorry.\"\n\n\"It was only a matter of time before he left, even if he'd stayed clean. I would have kicked him out myself, I think, heroin or not. Or Brian would have. I could barely be in the same room as him, near the end.\"\n\n\"You can't blame yourself for him getting hooked on heroin, though. Or for bailing on the band like that.\"\n\n\"No, of course not, but I don't think I helped. Brian's right to warn me off you.\" Rayne smiled and shook his hair back. \"It's fine, though. We're good, right? Brian has nothing to worry about.\"\n\n\"We're good,\" Kris promised, catching Rayne's hand to hold it. \"And hey, good riddance, right? You're better off without the guy.\"\n\nRayne cracked a genuine smile and squeezed his fingers. \"You're a marked improvement over him,\" he agreed softly.\n\nKris's insides went warm and melty at that, though his brain was too fuzzy to process everything Rayne had shared. He smiled back and rubbed his thumb reassuringly over Rayne's knuckles. \"You got anything else you need to talk about? Any deep, dark secrets you're harboring? Lip-synching onstage? The long lost prince of Genovia?\"\n\nRayne hesitated for a split second before shaking his head, his curls catching the only light in the room. \"No, I'm done. No secrets, no royal bloodlines.\"\n\n\"Too bad. Guess you'll have to make do being a millionaire rock star.\" Kris's head was dropping of its own accord back toward his pillow as sleep rushed in to meet him. \"Fancy rock star with a fake boyfriend,\" he mumbled, reaching over to swat at Rayne one last time. \"You're like a walking fantasy, man. All those fics Cassie talks about, they've got nothing on the real you.\" The last thing he saw before his eyes slipped shut was Rayne's expression, caught somewhere between startled and wondrous.\n\nHe was surrounded by warmth. Sunlight filtered through the curtains, covering everything in pools of gold. The bedsheets were rumpled, and the body under him arched as Kris's veins flooded with heat. It was the lazy, unquestionable kind of enjoyment that came from familiarity, when he knew every inch of his partner's body and they knew his. Kris didn't even need to open his eyes; they were connected spirit to spirit, pleasure crashing through them with no regard for where one body ended and the next began. Kris dropped his head to his partner's shoulder and breathed in deep. They smelled like oranges and lemon peel, so fresh his mouth flooded with water. He tasted the salt of their skin as he moved his hips in lazy circles.\n\n\"Oh god, fuck,\" he panted, chasing his climax. It felt like burning coals in the pit of his stomach, spreading out through every limb.\n\n\"You feel so good,\" his partner breathed, nipping at his ear. Kris shuddered. \"You feel so good . . . come on, come on\u2014\"\n\n\"Wait,\" Kris gasped. \"Wait, I want to kiss you.\" He leaned down to find his partner's lips, and when their mouths met, his eyes flew open for the first time. He knew that kiss. Looking back at him, sun-dappled in the bed, lay Rayne, his hair tousled and his eyes bright and flooded with want.\n\n\"Oh, fuck,\" Kris said, and woke up.\n\nIt was morning, and Rayne was spooning him from behind, his face buried in Kris's shoulder. Kris was painfully, achingly hard. So, he noticed, was Rayne. Rayne's breathing was slow and steady, fanning over the back of Kris's neck. It felt damp and too hot. Kris shifted, trying to escape it, but only succeeded in pushing himself farther back into Rayne's embrace.\n\nKris paused and considered his options. Rayne was asleep, no matter what his libido was thinking. Kris could extricate himself\u2014very carefully\u2014and slip away to the bathroom, have an excruciatingly cold shower, and pretend nothing had happened. Rayne would probably be awake by the time he came back, and neither of them would have to talk about it. Or Kris could pretend to go back to sleep until Rayne woke and left for the bathroom, and again, neither would have to mention anything. Or least advised: he could stay exactly where he was and wake Rayne up to call him on his accidental intimacy. It was a terrible idea, especially considering their last conversation concerning Calloway, but Kris could still see his dream-Rayne glimmering behind his eyelids every time he blinked. That made it hard to remember why, exactly, he should be keeping his distance.\n\nKris held his breath and waited to see if his erection wanted to subside on its own.\n\nIt didn't.\n\nHe took the third option, shifted onto his stomach, and elbowed Rayne in the ribs.\n\n\"Dude,\" Kris said.\n\nRayne whuffled in his sleep and held him tighter.\n\n\"You are getting way up in my business here, man.\" Kris twisted around just enough to see Rayne blink awake, looking baffled and adorably out of place. He stayed pressed against Kris's back from chest to knee.\n\n\"Huh,\" said Rayne, his voice rough and low from sleep. \"You don't seem to mind.\"\n\n\"You have to make an honest guy out of me now.\" Kris kept his tone deliberately light. \"Roses and fancy dinners and a big fat ring. No fake boyfriend for you after all.\"\n\n\"You're so high maintenance.\" Rayne yawned. \"Fine, we'll detour through Vegas and get married by some Elvis impersonator, how about that?\"\n\n\"Perfect.\"\n\nKris wriggled onto his side, pretending to search for a way out from under Rayne's arm, but actually pressing his ass back against Rayne's hips, testing how far he could push things. Rayne groaned and buried his face in the pillow.\n\n\"You're a menace. Stop moving like that.\"\n\n\"Like what?\" Kris asked, all innocence.\n\n\"You're doing this on purpose.\"\n\n\"So go jerk off in the shower, rock star.\"\n\n\"Aren't you supposed to be straight? Straight boys don't do this, unless the world's been lying to me this whole time.\"\n\nKris bit his lip and hid his face so Rayne couldn't see. \"Maybe I'm just getting you back for rubbing up on me all morning.\"\n\nHe couldn't admit he was still chasing that high from his dream, trying to get close to Rayne in any way he could before they reached the festival and whatever complications it was going to bring. It wasn't smart, whatever the excuse. Rayne narrowed his eyes, and Kris had a second to brace himself before Rayne flattened him onto his stomach, swung a leg over Kris's thighs, and pinned his wrists to the pillow. Kris noted Rayne was keeping their hips conspicuously apart.\n\n\"What are you going to do?\" Kris asked around the pillow.\n\n\"You are such a brat.\"\n\nKris shrugged as best he could, which was minimal. \"You like it.\" He struggled against Rayne's hold, but only for show. Rayne was taller and heavier than him; he didn't stand a chance if Rayne really wanted to keep him pinned. He was pretty sure Rayne was bluffing, though.\n\n\"You going to stare at the back of my head all day, or you going to make good on that threat?\" Kris bucked his hips up, and Rayne jolted up to avoid contact. Kris grinned. \"Didn't think so.\"\n\nHe could feel Rayne's glare through his skull. Rayne held him there for a moment longer as if considering him, before dropping his full weight on Kris's back. Kris wheezed as he deflated, the air fleeing his lungs.\n\n\"Brat,\" Rayne repeated, more fondly now that Kris had been unilaterally defeated.\n\nKris pulled one hand free to smack Rayne in the shoulder. \"Off,\" he wheezed. \"Dying. You win.\"\n\n\"You sure?\" Rayne asked. \"You sure this wasn't what you wanted?\"\n\n\"You're still poking me.\"\n\nRayne shifted around and jabbed his elbows into Kris's ribs.\n\n\"Okay!\" Kris managed. \"Okay, I give up. You win. Off.\"\n\nRayne didn't move, and Kris wondered if he was really going to be crushed to death under an annoyed, blatantly aroused rock star first thing in the morning. He'd just started drafting his final will and testament when Rayne grinned and pressed a smacking kiss to Kris's shoulder and dismounted, hopping off the bed.\n\n\"First shower,\" Rayne announced.\n\n\"Roses and dinner,\" Kris repeated. \"The most expensive shit you can find.\"\n\n\"I can do coffee and a muffin. Maybe weed, if Angel's generous.\"\n\n\"Deal.\"\n\nRayne disappeared into the bathroom, and Kris rolled onto his back, flinging one arm over his face as he palmed himself through his shorts with the other. It was a fluke, he repeated to himself. It had been a while, and Rayne was the only person he'd touched in months. He wasn't in love\u2014he wasn't even in lust, whatever his subconscious was saying\u2014and he was not going to rub one out over his boss-slash-best-friend, no matter what kind of thrill had shot through him when Rayne had pressed him into the mattress like that. Rayne was going to go meet his publicity-stunt-fake-boyfriend, and Kris wasn't going to get between them, because he and Rayne were just friends. Friends and bandmates. Brian would kill him if he pulled a Fink and fucked up the tour, and Kris would kill himself if he pulled a Fink and fucked up Rayne's heart.\n\n\"Fuck me,\" Kris complained to the empty room.\n\n\"Marriage first!\" Rayne hollered from the bathroom.\n\nKris pulled the pillow over his face and groaned into it. He was such a mess.\n\nModern spirituality is a strange thing, often made difficult by the unrelenting fast pace of the twenty-first century, but people find ways to make ends meet. Some attend church; some read Buddhist teachings. Some surround themselves in nature and breathe deeply until they feel at peace.\n\nLeif had started a cult.\n\nHe hadn't meant to, at the time. But it had gotten away from him, as those things tend to do, and before he knew it people had been coming to him for guidance and spiritual well-being, and he could hardly have turned them away with nothing to believe in. He might have fucked up, but he was damn well going to take some responsibility for once in his life and see it through.\n\nSo: cult.\n\nSeven Years Earlier:\n\nThe exact moment it began, he was sitting in the rickety foldout chair in front of his rickety little trailer, parked in a lot of other semipermanent ramshackle homes in the Mojave Desert. Leif did his best with what he had, but he'd always felt there was something missing. The outside of his trailer was rough and weather-beaten, and inside wasn't much better. He might as well be living in a cave somewhere, for all the home touches he lacked.\n\nHis soul was as neglected as his physical surroundings, and like every other hole in his life, he filled his missing sense of purpose with drugs. When his life changed, he was extraordinarily high on a combination of weed, ecstasy, and LSD, to the point where he was barely tethered to reality. His head swam with shapes and colors; his mind was so far open that the universe was pouring in through his third eye and his soul was pouring out and connecting to every other living being in the cosmos, and he understood everything. It was beautiful, but when he came down, it wouldn't have been marked as truly spiritual.\n\nIt was the peacock that tipped the experience into life-altering territory.\n\nLeif had never seen a peacock in real life before, and at first he wasn't convinced the bird was real at all. It was too big, too colorful, too . . . much, to be a real animal existing in three-dimensional space. He flew down to Leif from the sky, surrounded by lights of a billion different colors, colors Leif had never seen before, beyond the spectrum of human visibility. The peacock's wings beat in slow motion as he descended, and Leif's mouth fell open in slack-jawed awe as they fanned his face. The breeze was warm and, it seemed to him, originating from another plane of reality. When the peacock landed on the ground before him, folding his wings back and shaking out his tail in a series of short ruffled waves, Leif was convinced he wasn't looking on any mortal bird, but some divine creature sent from heaven to bring enlightenment to his life.\n\nHe was incredibly high, and he knew he was incredibly high, but what really cemented the idea of the peacock being divinity was that he felt the same once the drugs wore off. Hours later, when he was settled back in his own skin and his mind worked slowly and linearly again, the peacock still gave him that same sense of otherworldly reverence. The peacock regarded Leif with bright, shiny eyes that seemed too intelligent to belong to a bird, and his feathers were so glossily iridescent that Leif thought there must be LSD still left in his system.\n\n\"You're something else, aren't you?\" he whispered, his voice hoarse and his mouth dry.\n\nThe peacock screamed, an unearthly wail that sent goose bumps running up and down his arms, and the hairs on the back of his neck raising in alarm.\n\nHe called his friend Red.\n\nRed was a uniquely belligerent man whose only source of inner peace came from the drugs he imbibed, and he made sure to carry a wide variety on him at all times, a pill for every occasion. He was the type of man more likely to stare at a wall for six hours when he was high than talk about the wonders of the universe, but he and Leif had been crossing paths for years, and found it easier to call themselves friends than anything else. Red was as logical and down to earth as anyone Leif had ever met, if prone to brawling. If anyone could offer some perspective on the peacock, it was him.\n\n\"What do you mean you saw God?\" Red grunted over the phone.\n\n\"Not God-god, a different god,\" Leif said. The peacock watched him, seemingly judging his every word. \"The god of . . . truth, or beauty or\u2014 I don't know. Enlightenment.\"\n\n\"You know you sound nuts, right?\"\n\n\"Yeah, I know. But listen, get over here and judge for yourself, okay? I'm telling you, there's something about this thing. He's not a normal bird.\"\n\nRed grunted again and hung up, but he came. He pulled in on his big hulking motorcycle, the machine choking out fumes, and parked beside Leif's trailer. By then the peacock was perched on the edge of the aluminum roof, his tail cascading down over the window, and he looked back over his shoulder to watch Red's approach. Leif rose from his chair to greet him.\n\n\"That's a hell of a bird,\" Red said eventually. \"Bigger than I thought. Don't know about it being any god, though. You sure all that shit you smoke hasn't fried your brain?\"\n\n\"I haven't tried anything you haven't,\" Leif retorted, but it was the peacock that demanded the last word. He shrieked and launched himself from the roof to assault Red in a flurry of feathers and talons, and Red shrieked in return and fell to the ground, his arms up to protect his face.\n\n\"Okay, all right!\" he hollered, curling up while trying to fend the bird off. \"I believe you! It's a god!\"\n\nThe peacock hopped back a pace and cocked his head to one side, watching him. Red slowly got to his knees, his arms and face covered in scratches. He and Leif watched the bird, wary in case he launched a second attack. Instead, the peacock took another step back and fanned out his tail, keeping his gaze fixed on them all the while.\n\nNow, Leif wasn't given to superstition. He hadn't been raised in a barn. He'd gone to college and studied poetry and philosophy, looking for anything to give his life a little meaning, though the student loans had piled up too quick and he'd dropped out before finishing his degree. Then he'd gone off grid for a while to avoid the debt collectors, and things had spiraled from there. The point was, he'd had an education, and he wasn't the sort to unquestioningly follow the first thing that showed any promise of spiritual fulfillment. But when the peacock spread his tail, Leif dropped to his knees alongside Red, his hands pressed to the dusty ground as he leaned forward until his forehead touched the dirt. Submitting himself like that brought the strangest sense of serenity, like all his worldly troubles were drifting away, out of reach and suddenly meaningless in the face of this beautiful, radiant creature.\n\nAnd sure, maybe the drugs had fried his brain. It was possible. But the thing was, once they started showing the peacock to other people, they all agreed that there was something about the bird as well. The peacock made them feel things about life and beauty and the universe that they'd never felt before, and if that wasn't the sign of some kind of godhood, Leif didn't know what was. That, and the fact that as soon as the peacock arrived, his luck had started to change. Things started to go right again for the first time in years. He'd found a twenty-dollar bill on the ground that very day, and that was only the beginning.\n\nFrom there, the cult seemed like a natural progression of events. He and Red were the only members at first, and they didn't call it a cult then. But the more time they spent communing with the bird, the more attention they attracted from like-minded individuals. Boar came next, a great hulking mountain of a man, like a Viking berserker from centuries past, looking for a community and a purpose. By that time Leif had determined his own purpose in life, which was to devote himself entirely to the peacock in return for his feathery blessing, and he was happy to bring others into the fold. Boar took to worship like it was a lifeline saving him from drowning, and everything escalated quickly after that.\n\nThey named their god to better worship it: Incandescent and All-Seeing. They shaved their heads like monks, forsaking all earthly beauty, and tattooed themselves with the images and messages they saw in their visions. Leif sold his trailer and bought a motorcycle, and traveled to hold court in music festivals and hippie camps, preaching the gospel that flowed from the universe straight into his brain. He had a gift for talking; all those years of reading poetry had left him with a deep, lilting intonation and an unusually developed vocabulary that caused his audience to sit up and pay attention, even the skeptics. Leif didn't actively recruit people to the order, but more did join, here and there. Some stayed for a few days, some for a few years. It always hurt when they left, for whatever reason, but the peacock never seemed inclined to smite them for their abandonment, so Leif didn't either.\n\nThey rarely strayed out of Nevada. The peacock had come to them in the desert, and in the desert they would stay, haunting the highways on their motorcycles as their god watched over them from above. They took money where they found it, whether from hustling pool at roadside dives, or jacking cars for parts when out-of-state drivers broke down in the desert and had to pull over and call for help. It wasn't always a legal living, but Leif thought it was an honest one. Spiritually honest: that was the important thing. Otherwise, the cops should have caught them by now. Bald and tattooed, they were hardly inconspicuous: dressed all in dusty motorcycle leathers, roaming the desert like great earthbound vultures. They had to be on the right path. Otherwise, the peacock would have abandoned them, and they would have gone to jail.\n\nNow:\n\n\"Where to next?\" Red asked.\n\nThere were four of them now, and they had set up camp off the side of the highway, their bikes parked in the shade of the towering cacti. The desert stretched out vast and orange in all directions as the sun began its descent.\n\n\"Purple Sage,\" Leif said, looking up at the clouds. One of their old members would be there\u2014Calloway, who had turned his back on them, seduced by the call of worldly fame and fortune. Leif didn't have any intention of trying to win him back to the fold, but they had parted on poor terms, and he wanted closure. The new kid who had replaced him just wasn't the same. And the peacock had liked Calloway, before he'd left. Had barely screamed at him at all, and never tried to peck at him the way He did Red and Boar.\n\nLeif sighed. Nothing was the same anymore. He wondered sometimes whether he should have founded the order at all. A selfish part of him wished he'd never called Red that day, all those years ago, that he had kept the peacock a secret between him and the universe. Would that have been such a terrible life?\n\nThe peacock screamed and rustled His wings warningly, and Leif abruptly cut off that train of thought. \"Purple Sage,\" he repeated, more firmly this time. He'd made his choice, and he wasn't going to back down now. He couldn't. \"We'll find new people there.\"\n\nThey were three hours out from the festival grounds, and Kris could already feel the change. It wasn't just that it was drier, the desert soaking up the sun and chasing every last shred of moisture from the air: everything looked sharper in the desert, more vivid, like reality was suddenly realer. The atmosphere crackled with expectation, and Kris couldn't keep still.\n\n\"I've never been to a music festival before,\" he said to Cassie on the phone. He was curled up in his bunk as they trundled down the highway, trying to block out the band's chatter. \"I don't know what I'm doing.\"\n\n\"You'd never been to a gig anywhere bigger than a dive bar, either,\" she said reasonably. \"You'll figure it out. We're coming to see you, by the way.\"\n\n\"Wait, what? Who's we?\"\n\n\"All of us,\" Cass said. \"Me and Brad and Mom and Dad. We're on the road now. We should be there tomorrow. We wanted to see you play live!\"\n\nKris tensed up instinctively. \"Why is Brad coming? This isn't his scene.\"\n\n\"He said it had been a while since you two hung out.\" He imagined her offering a careless shrug. \"He said we should have gone to a closer show, but Mom and Dad thought it'd be cool to make a whole trip out of it, so here we are.\"\n\n\"Well, shit. Okay. We'll be here.\"\n\n\"And you'll be awesome. I have to go, but stop freaking out! Go find Rayne and make him give you a hug.\"\n\nKris sighed. \"Say hi to everybody for me. Love you. Bye.\"\n\n\"Bye!\"\n\nHe ended the call and climbed out of his bunk. While he hadn't been avoiding Rayne since the hotel, he hadn't been actively seeking him out, either. He found him on the couch chatting with Lenny, and sat down beside him and buried his face in Rayne's shoulder, slinging his arms around Rayne's waist. Rayne put his arm around Kris's back without missing a beat in his conversation.\n\n\"Hey,\" Kris said during a pause, mostly to Rayne's hair. \"My family's coming to see us play tomorrow.\"\n\n\"That's great!\"\n\nKris groaned.\n\n\"Is it not great?\"\n\n\"I'm worried my parents are going to give me some kind of talk about life choices because of the makeup and stuff, and the last time I talked to my brother we yelled at each other, and I need you to not kick me out of the band after you meet my sister.\" He chanced a glance up. Rayne's eyebrows were raised, and Lenny looked amused. Stef and Maki came over to join them, likely drawn by the promise of drama.\n\n\"That's a lot,\" Rayne said eventually. \"Is there a reason I should kick you out of the band?\"\n\nKris shrugged and nestled closer. \"She's a fan. She has posters of you in her bedroom. It's weird.\"\n\n\"As long as she doesn't have a shrine and an altar, it should be fine. And if your parents or your brother start giving you shit, text me, and I'll say it's an emergency and I need you onstage right this second, no excuses.\"\n\n\"Thanks.\"\n\nRayne's hand drifted up to Kris's hair, and Kris purred and leaned into it as Rayne scratched his scalp.\n\n\"It's going to be fine though.\"\n\n\"Course it is,\" Kris agreed. \"I'm not even panicking. It's totally cool. What's the festival going to be like?\"\n\n\"Festivals have a different energy than playing a show in a stadium for a few hours,\" Rayne said. \"More of a marathon than a sprint.\"\n\n\"It's hard to describe,\" Lenny said thoughtfully. \"You have to let it happen and decide for yourself.\"\n\n\"You take a desert, right,\" Stef chimed in, \"and you fill it with drugs and music and kids looking for meaning, or love, or just a week of stories to tell after the fact, and mostly everyone's there for the experience and it's all good, but I've never played a festival where things didn't get weird.\"\n\n\"Weird,\" Kris repeated.\n\n\"Not in a bad way,\" Maki said, \"but Stef's right. Festivals are . . . different.\"\n\n\"None of this is filling me with confidence, guys.\"\n\nRayne squeezed him tighter. \"It's like if everywhere else is the real world, then festivals are just slightly off-kilter from it. Everything gets turned up to eleven.\"\n\nKris glanced out the window in time to see a biker gang pass by in a rumble of tattoos and black leather.\n\n\"I don't know if that sounds like a place I want my parents to show up,\" he said.\n\n\"They probably remember Woodstock,\" Stef said with a shrug. \"Bet they know more about the scene than you think.\"\n\nKris decided he'd rather not think about that too hard. \"There's still my brother.\"\n\n\"What's the worst that could happen?\" Rayne asked optimistically.\n\n\"He's a Republican.\"\n\n\"Oh.\"\n\nKris buried his face in Rayne's shoulder while Rayne patted his arm.\n\n\"Maybe it won't be that bad?\" Rayne offered. \"You'll be pretty busy; you can avoid him if you have to. We'll help run interference.\"\n\n\"We totally will,\" Stef agreed. \"Like secret agent bodyguards.\"\n\n\"At least Cass should have fun,\" Kris mumbled into Rayne's shirt.\n\n\"You will too. We'll get set up and play our show and meet Calloway and Dead Generation later on, and everything will be great.\"\n\nKris nodded and clung tighter for a second, savoring the contact, before straightening. Once Rayne and Calloway were involved, there'd be no more of that for a while. Kris and Rayne could technically keep sneaking cuddles in private where the press couldn't catch them, but it seemed safer to stop entirely while Calloway was in the picture, so Kris wanted to store up what he could while he had the chance. \"No, you're right. Everything will be fine.\"\n\nThe last of the stages were still being built when they rolled in. The festival sprawled like an oasis in the desert, a tiny city of tents, trailers, and scaffolding all glittering hot as a mirage under the sun. Kris watched it unfold around him, his nose pressed to the window, as the bus made its way through to their campsite. They parked in a clearing, and it dawned on Kris that they were going to be living out of the bus for the next week until the festival wrapped up, outdoor toilets and showers and all.\n\nIt was still better than being homeless on the streets of New York.\n\nBy the time they started rehearsing, Kris's anxiety had settled into a nervous thrum of excitement. The songs came easily now, and he barely had to improvise at all anymore. He had their set list memorized, and he could anticipate Rayne's vocals as easily as he could his own solos. And he knew Rayne's habits onstage too\u2014how he looked when he was wound too tight, his expression and the set of his shoulders when he was about to prowl over and kiss Kris senseless. Kris kept one eye on him during rehearsal, watching for the signs, but they didn't come. They rarely did, except during the live shows, like Rayne wanted to bottle it all up until the pressure was too much and he couldn't keep it down any longer.\n\nKris liked it that way. It kept things clear between them, and clarity was what he needed, no matter what his traitorous body suggested to the contrary. A solid line between what they did onstage and how they were off it.\n\nHe still thought about that dream when he was trying to fall asleep at night. He tried not to dwell on it, but it crept in through the cracks in his resolve, and before he knew it, he'd be half-hard from a memory that had never even happened.\n\nIt was getting ridiculous.\n\nThis stunt with Calloway was the best thing for it. And while Rayne and Calloway were busy, Kris would throw himself into the festival and play so hard his mind didn't have time to wander. He and Rayne would cut down on the fan service onstage, and offstage, Rayne and Calloway would be boyfriends, at least while their pictures were being taken. It would all be good. And then later, when his head was clear\u2014after the tour, when Rayne wasn't influencing his decisions anymore\u2014then he would come out as bi.\n\nThe first night of the festival\u2014when the shadows grew long in the setting summer sun, the last of the stages had finally been erected, and the lanes were flooding with crowds of thousands at a time\u2014they took to the stage and Kris realized with a heavy drop of horror that his plan was not going to work.\n\nRayne kissed him like he wanted to eat him alive. He tangled one hand in Kris's hair, wrapped the other around his throat, and reeled him in like a fish on a hook and held him there, helpless, in front of their screaming fans. Kris's knees buckled and he nearly fell, but Rayne held him fast, licking into every inch of his mouth like he had something to prove. Kris moaned, uncaring if Rayne's mike picked it up, and licked back. His hands kept time on his guitar, moving mindlessly over the chords as his whole body lit up with want, his brain blank except for a chorus of Rayne, Rayne, Rayne.\n\nAfter the show, he stumbled backstage, dry-mouthed with a pounding heart and no plan except to douse himself in the coldest water he could find. It was better than nothing, though he doubted it would help. He had made it this far without rubbing one out because of Rayne, and he wasn't about to start now. His guitar stayed on him like armor, even as the rest of the band pulled him into their customary post-show embrace, sweating and panting and bright-eyed all around. He returned it, still riding the high himself, but this time trying to hide his blatant arousal from anyone else.\n\nHe didn't know why he bothered. Rayne got worked up onstage all the time; it was a combination of adrenaline and elation and Kris didn't want to flatter himself but he imagined he was no small part of it, either. There were entire websites dedicated to pictures of Rayne getting overly excited onstage, and Rayne never bothered denying them, so why should Kris?\n\nHe adjusted the guitar across his lap and skittered back to their dressing room, a separate trailer set up behind the stage, to get changed and clean the makeup from his face. Rayne's kiss had destroyed his lipstick, smudging it and making him look more wanton than usual. Rayne hadn't escaped unscathed either, finishing the show with his own lips darker than usual, but it had only made his smile all the more enticing. The crowd had screamed and wailed, begging them to continue.\n\n\"You good?\" Rayne asked from the doorway. The lipstick was smeared around his mouth, berry-dark. \"You bailed pretty fast.\"\n\n\"Needed water.\" Kris managed a smile. \"I'm good. It's all cool.\"\n\nRayne sidled in, ignoring Kris's attempted brush-off. \"Still worried about your family coming?\"\n\nKris's family had been the last thing on his mind. \"Yeah, that.\"\n\n\"Calloway's arriving with Dead Generation tomorrow morning; we'll tone our shows down once he gets here. Your parents don't have to see you and I doing all that in person. I still haven't committed to anything, though. Not for sure.\" Rayne's expression was a mixture of tentative hope and nervousness, and Kris didn't like seeing it on his face.\n\n\"But you're going to, right?\" Kris asked, pushing aside the way his stomach flipped at the thought. \"I think you should. If the label wants you to, then it must be a good career move. And who knows, maybe you and Calloway will hit it off for real.\"\n\nRayne's face shuttered for a split second before he smiled. \"Yeah, maybe we will.\"\n\nKris smiled back and adjusted his guitar strap, resolutely ignoring the way he felt faintly sick at the thought of Rayne getting involved, offstage and for real, with a guy who wasn't him. \"Cool,\" he said aloud. Everything was going to be fine.\n\nKris's family arrived the next morning, hours before The Chokecherries were due to take the stage again. The festival was a hive of constant activity, with additional bands and attendees arriving every hour, the tents and stages bursting with music and partygoers day and night. Dead Generation arrived around ten, but Rayne said they were busy setting up, and he was going to find them later. The energy was unlike anything Kris had ever felt\u2014maybe it was just because the air was hazy with weed and everyone seemed to be high on one thing or another, but it felt brimming with humanity at its best and most expressive, and Kris thought he could get high off that alone.\n\nThere was really a lot of weed.\n\nBut the combination of performing and connecting with fans he never thought he'd have left him feeling more at home in his skin than ever before, so when his family arrived on site he was dressed half in his stage clothes and half casually, with his hair ruffed up in a faux-hawk and eyeliner smudged around his lids. He wasn't fully done up and wouldn't be until closer to showtime, but he still got a thrill from seeing himself in the mirror with makeup on, so it was creeping further and further into his offstage life. He couldn't apply a convincing smoky eye yet, but if all he needed was a bit of kohl, he was set.\n\nThe downside to his increase in comfort was that he forgot he was wearing the stuff when he met his family.\n\nTo his parents' credit, they didn't comment. Cassie did, but nonverbally\u2014she pointed at him from behind their parents' backs, exaggeratedly miming at his face while grinning and giving him the thumbs-up. Brad stiffened and said nothing. Kris opened his mouth to apologize preemptively, then changed his mind and smiled instead.\n\n\"Thanks for coming all the way out here. You really didn't have to.\"\n\n\"Of course we did,\" his dad said. \"If you're going to go off touring the world, the least we could do was come see you before you left the country.\"\n\n\"I appreciate it. I, uh, I guess Cassie's been keeping you up-to-date on all the shows?\"\n\n\"You do look like you're having fun up there,\" his mom commented mildly, and Kris nearly choked on his tongue. \"That Rayne knows how to put on a show, doesn't he?\"\n\n\"He's a natural all right,\" his dad agreed.\n\nCass elbowed Kris in the ribs, still grinning.\n\n\"Lunch?\" Kris blurted. \"Let's get lunch. They've got decent food here, unless you want to drive out to an actual restaurant.\"\n\n\"Here's good,\" his mom said. \"We're here for the experience, after all.\"\n\nKris led them through the picnic area to the food trucks, where vendors of every possible cuisine had set up shop, from the vegan elites to the place where you went when you were drunk at 3 a.m. and the fridge was empty. They ordered their food and sat at a picnic table. Kris tried not to fidget. He was in his element, and his family was happy for him.\n\nExcept Bradley. Brad didn't seem happy.\n\nKris resolved to ignore him until he couldn't anymore.\n\n\"You're going to introduce us to the band, right?\" Cass asked, shoveling rice into her mouth. \"And Passionfruit too? They look fun. Do you think anyone would let me try their drum kit?\"\n\n\"You can ask about drumming. And I already warned Rayne you were coming, so sure, you can meet everybody,\" Kris said. \"That won't be weird for me at all.\"\n\n\"We met your last band,\" his dad pointed out.\n\n\"In high school,\" Kris replied. \"This is a bit different.\"\n\n\"Your last one didn't dress you up like a girl and grope you in front of a million people,\" Brad said.\n\nEveryone stilled and Kris sighed internally. Of course he wouldn't be allowed to ignore Brad.\n\n\"It's for the show, Brad,\" he said, repeating their press line with as much patience as he could muster.\n\n\"Hell of a show,\" Brad replied.\n\n\"I think it's great,\" Cass cut in. \"People are loving it. Don't crush his entertainment dreams, Brad.\"\n\nBrad held up his hands. \"I'm not here to crush anything. I just want you to realize how it looks to other people.\"\n\n\"It looks fucking punk,\" Cass insisted.\n\n\"Okay,\" his dad said, \"let's everybody cool down. Kris, why don't you tell us about the rest of the tour? What are your plans after the festival?\"\n\nKris dived in gratefully, talking about the band's plans and where Rayne wanted to go, and where the international tour would take them\u2014not that Kris was officially signed on for that part\u2014but he kept stealing glances at Brad out of the corner of his eye. Brad didn't interrupt again, but he didn't relax, either. Kris finally texted Rayne under the table, demanding that he show up and provide a distraction.\n\nRayne came over a few minutes later; Kris knew his arrival by the sharp intake of Cassie's breath.\n\n\"Sorry to interrupt,\" Rayne said, and Kris let out a sigh of relief. He loved his family, he really did, but negotiating Bradley's idiot ideals of masculinity was the last thing he wanted to deal with. \"I was going to borrow Kris for a minute, but if he's busy . . .?\"\n\nKris stood, grabbed Rayne by the wrist and made a gesture like Vanna White. \"So hey, this is Rayne! Rayne Bakshi. Rayne, this is my family.\"\n\nHis parents smiled and Cassie gazed at Kris imploringly, stars in her eyes, pleading for him to stay. He relented, grudgingly abandoning his plan of escape. \"You guys want to walk us back to the . . . stage? Bus? Where are we going, Rayne?\"\n\n\"The bus,\" Rayne said. \"There's been a slight incident.\" He held out his hand to Frank, smiling widely. \"And you're Kris's dad! I've heard about you. All of you,\" he amended. \"Did you have a good drive?\"\n\nKris's dad shook Rayne's hand and smiled back, clearly relieved to find Rayne capable of small talk. \"A good drive, yes, thank you. Beautiful landscape, beautiful country. And it looks like you've got decent weather for the festival too.\"\n\n\"Couldn't ask for better,\" Rayne agreed, offering his hand to Kris's mom next, who seemed charmed. Rayne had that effect on people.\n\n\"And my sister, Cassie,\" Kris said.\n\nCass squeaked, grinned, and thrust her hand out at lightning speed. \"Hi. Rayne. Rayne Bakshi. Hey.\"\n\n\"Hey,\" Rayne said, his eyes sparkling. \"I heard you keep my picture in your room.\"\n\n\"Kris!\"\n\nRayne laughed and patted her shoulder. \"No, I'm flattered. He promised there wasn't a shrine or anything, so it's fine. You want to meet the rest of the band?\"\n\n\"I get why Kris likes you,\" Cass said, allowing his hand to stay. \"You're awful. I love it.\"\n\nThey set out together back to the buses, and if anyone noticed that Brad never got introduced or shook Rayne's hand, no one commented on it.\n\nKris was surprised to find that there really had been an incident with the bands: Knocks, Passionfruit's drummer, had broken his foot and been whisked away to the nearest hospital, and the remaining band members were huddled outside The Chokecherries' bus, nursing various alcoholic beverages and looking morose.\n\n\"What happened?\" Kris asked.\n\n\"Jay tried to kill our drummer,\" Billie said.\n\n\"It was an accident,\" Jay said.\n\n\"You tackled him into the middle of his drum kit. You're lucky you didn't break his neck.\"\n\n\"I didn't know he was going to get tangled up in the pedals like that.\"\n\nBillie sighed. \"And now we're trying to borrow Lenny for our show tonight instead.\"\n\nLenny raised his drink in a salute.\n\n\"You're going to play two shows back-to-back every night for the rest of the festival?\" Kris asked.\n\n\"That's what I said,\" Rayne cut in, glowering at Passionfruit.\n\n\"We need a drummer,\" Billie returned, \"and Lenny knows all the songs. Mostly. Kind of.\"\n\nRayne cleared his throat. \"Everyone, this is Kris's family. They're here to see the show.\"\n\nCassie waved. Kris's parents smiled and fidgeted, keeping a polite distance while Brad stood stony-faced, arms crossed and feet planted firmly on the ground.\n\n\"Cass wanted to meet the bands, and the timing kind of sucks, but hey?\" Kris tried.\n\n\"The timing's fine,\" Rayne said. \"It's a festival. There's always something going on.\"\n\nBillie sighed again, more heavily this time.\n\n\"You know I didn't do it on purpose,\" Jay muttered, sotto voce.\n\n\"We're talking about this later,\" Billie said.\n\nKris winced and turned back to his family. \"So, this is it. Passionfruit; The Chokecherries. Check out the glamorous life of bands on tour.\"\n\n\"Accidents happen,\" his mom said diplomatically. \"They're rarely the end of the world.\"\n\n\"But obviously you're all very busy, and we should get out of your hair,\" his dad said.\n\n\"No, it's fine!\" Billie said. \"It'll be fine. Do you want a drink? We've got beer.\" He glanced at Cassie. \"Or water, or Coke?\"\n\n\"I'll have a Coke,\" Cassie said, happily undermining her parents' attempt at retreat. \"Is your drummer going to be okay?\"\n\n\"He'll be fine,\" Jay stressed, pulling a can from the cooler for her. \"Grab a seat, hang out with us.\"\n\nCassie sat down on top of the cooler and looked at the rest of her family expectantly.\n\n\"We really don't want to get in anyone's way,\" their mom began.\n\nAs much as he'd like to avoid Brad, Kris felt bad giving his parents the same treatment. \"Stay until we go for makeup in another hour,\" he said. \"It'll be fun.\"\n\nRayne nodded. Cassie was already making herself at home as if she were among old friends, smiling and chatting away. Brad scowled\u2014almost imperceptibly; Kris didn't think anyone else would notice\u2014but took a seat with the others. Hatchwork, the mustachioed Passionfruit bassist, drew Kris's parents into a conversation about their hometown, which they were only too glad to have. Lenny was engrossed in a quiet argument between Rayne and Billie.\n\n\"You've met Rayne now,\" Kris said to Cass. \"This is Maki and Stef\u2014and that's Lenny, over there.\"\n\n\"Hey,\" Stef said with a smile.\n\n\"Hi.\" Cassie smiled back. \"I've seen your videos. Love your work.\"\n\nMaki caught Kris's eye and gave him a knowing smile. Kris rubbed his hand over his face, not caring about smudging his makeup anymore.\n\n\"I can play drums, you know,\" Cassie said.\n\nEveryone paused and looked at her.\n\nShe shrugged. \"I'm pretty good. Kris can vouch for me. Just putting that out there.\"\n\n\"Cassie,\" Brad said.\n\n\"What? I am.\"\n\n\"How fast can you learn a new song?\" Billie asked, leaning in.\n\n\"I know most of your set already,\" she said, taking a swig from her drink. \"I checked out your albums when Kris said you were opening for Rayne, and I've been watching all your concerts since you started the tour. I bet I could play most of it right now. See?\" She offered her phone to show off a recording of her on her kit in the garage. She played as steady as a metronome, and she smashed out her solos with the energy of a hurricane.\n\n\"Damn,\" Billie said. \"That's actually amazing.\" Cassie preened as Billie glanced around at the rest of his band. \"What do we think, guys?\"\n\n\"Hell with it,\" Jay said. \"Let's audition her. What have we got to lose?\"\n\n\"You can't be serious,\" Brad said, but though their parents exchanged a few whispered words, they didn't voice any objections.\n\n\"Don't worry,\" Rayne said, ignoring Brad entirely in favor of addressing Kris's parents. \"There's a billion people around to keep an eye on her. She won't get into any trouble.\"\n\nCassie beamed and finished her drink. \"No trouble at all,\" she promised, and then she was sauntering off with Passionfruit like that had been her plan all along.\n\nBrad turned to their parents. \"You're really going to let her go off like that without\u2014\"\n\n\"Oh, give her a break,\" Kris cut in. \"It's a festival, she's an adult. Let her enjoy herself for a minute. Jesus.\"\n\n\"I'm sure it's fine,\" their mom said. \"Like Mr. Bakshi said, there are plenty of people around to watch out for her.\"\n\nKris flashed Brad a spiteful smile. He wasn't sure what to make of Cassie auditioning for Passionfruit, but if it annoyed Brad, he was all for it.\n\nLeif was at home in music festivals, and Purple Sage was one of his personal favorites. He sat with his disciples in a clearing between a ring of tents, their bikes parked nearby and the peacock wandering the grounds, as was His wont. The bird never strayed too far, and it wasn't as if He could get lost, so Leif let Him be. He talked about his spirituality in a low, calm voice as he etched a fresh tattoo on Boar's leg. They had attracted a few onlookers, including a twitchy, strung-out youth named Travis who watched wide-eyed as the needle dug into Boar's skin and the ink set.\n\n\"Will you do me next?\" he asked, but before Leif could answer, a black girl with a huge halo of curly hair stumbled into their circle as she rounded the corner of the nearest tent.\n\n\"Sorry,\" she said. \"Didn't mean to barge in on you like that.\"\n\nLeif glanced up from his work to see if she was interested in hearing about their god, and she stared back, apparently not expecting to run into such a gathering. Leif knew how they appeared to casual passersby: pale as bone and covered in strange scribbled tattoos, their heads shaved, and dressed head to toe in scuffed-up bike leathers. Anyone who wanted to hear about the universe, their god, and their coming salvation had to see past their rough exteriors to the truth.\n\n\"It's no problem,\" Rikki said to the young woman.\n\nRikki didn't fit in with the others yet. He was the order's newest recruit: the youngest of them, and the slightest, too, with big pale eyes and a face too earnest to make a living the way they did, though he believed in their god as fervently as Leif could wish. They'd work on him.\n\nThe woman gave Rikki a short nod and sidestepped out of their midst. \"Enjoy the festival,\" she said, and left quickly, without looking back.\n\nLeif returned his attention to Boar's tattoo, but he didn't miss the way Rikki watched the woman as she disappeared into the mess of tents.\n\nIt was easy for Kris to forget that his family was in the audience by the time The Chokecherries took the stage. Passionfruit killed it, playing even more aggressively than usual, as if to make up for losing a member, but Cassie took to it like breathing. Apparently she had nailed the audition, and while Passionfruit had still wanted to test her in front of a live crowd, they had been vibrating with enthusiasm about her skills, telling everyone who passed that she was a genius and they needed to steal her for the rest of their tour. She broke a stick halfway through the set, and Passionfruit looked pleased as anything about it. When they traipsed offstage, she smacked Kris on the chest as she passed, sweaty and wrung out, grinning from ear to ear.\n\n\"This is great!\" she yelled. \"Way better than our band in high school. I get why you do it!\"\n\n\"Nice playing, kid,\" Stef said. \"You're a natural.\"\n\n\"Hell yeah, I am. Go get it!\"\n\nWhen their set began, Kris played as he always did, keyed up with tension the second until the opening riff started and Rayne took the stage. By then he wasn't thinking about his parents or Calloway or the future at all. Rayne commanded the crowd's attention with a crook of his finger and shake of his head, and he stole Kris's just as easily. The Chokecherries played with the same passion they always did, and Rayne sang with the same raw energy, but this time, when Kris waited for their midshow kiss, it didn't come.\n\nKris frowned and tried to catch Rayne's eye, but Rayne wasn't looking.\n\nThere was nothing wrong with changing it up. They could always do the kiss later.\n\nLater came and went, and Kris grew increasingly twitchy behind his guitar. He had thought they would keep their shows as they were until Calloway entered the picture, for the fans' sake. Had Rayne already met with Calloway and begun the stunt without telling Kris?\n\nWhen they only had three tracks left, Kris finally tore a page from Passionfruit's book and launched an attack, barreling into Rayne from the side and jabbing his shoulder into Rayne's chest, all without missing a chord. Rayne took a fistful of Kris's hair, whether to hold him in place or push him away, Kris didn't know. It didn't matter. All he needed was Rayne's hands on him, somewhere, anywhere, especially if it was going to be the last time. He butted into Rayne's space, demanding more, but Rayne still didn't kiss him. He came close, pulling Kris's back in against his chest so Kris could feel their heartbeats thundering in tandem, even above the bass and the kick of the drums, and his lips ghosted over Kris's temple, but they didn't touch.\n\nKris leaned back, eyes closed as he coaxed the riff from his guitar, and dared Rayne to touch him for real. He let the music swell around him and Rayne's heat soak through his shirt as his mouth parted, throat bared to the sky, waiting.\n\nWhen Rayne shuddered, Kris felt it in his whole body. They moved into Stef's solo, and Rayne bent forward, his hand splayed against Kris's chest, and he mouthed, \"You little brat,\" against the shell of Kris's ear.\n\n\"Why did you skip our thing?\" Kris mouthed back. \"Is it Calloway?\"\n\n\"I haven't even met him yet.\"\n\n\"Then stop fucking around.\"\n\n\"Your parents are watching!\"\n\n\"I'm not asking you to fuck me onstage!\"\n\nRayne dug his nails into Kris's skin, and Kris rocked against him. \"Just kiss me before we go backstage again, and give the people their money's worth.\"\n\n\"You're incorrigible.\"\n\nKris twisted around in Rayne's grasp, flinging one arm around Rayne's neck, the other holding his guitar between them, to meet Rayne's gaze. They stared at each other for a second, Kris daring him to do something, anything\u2014and Rayne finally smiled in exasperation and leaned down to press their lips together. It was more chaste than usual, but the crowd screamed right on cue all the same, and Kris shut his eyes and basked in it. Rayne's grip in his hair tightened as he laughed into Kris's mouth, smudging lipstick everywhere.\n\n\"You're the worst,\" Rayne said when they parted.\n\n\"You started it,\" Kris said, his heart pounding at the thought that that might have been their last kiss until the festival was over. \"Now you have to follow through.\"\n\nRayne shook his head, his hair falling into his eyes, and stepped up to the mike stand again. Kris let him go this time as Stef pounded out the last of their solo, and Rayne opened his mouth for the final chorus. Kris returned to his spot, hot all over. He was definitely going to have withdrawal symptoms if he couldn't do that again the next day.\n\nBrad found him backstage after the show. By that point Passionfruit was long gone and the rest of The Chokecherries had dispersed as Kris took an extra minute to pull himself together before rejoining Rayne. Kris glanced up as Brad approached with a guarded expression and leaned against a set of scaffolding, watching Kris for a moment. Kris set his jaw and wiped his face with his sleeve, clearing up the excess sweat near his hairline as he waited for Brad to speak.\n\n\"I get that you're doing it for the show,\" Brad finally said, \"and you don't want to let your fans down. I can respect that. But can you seriously not see how he's using you? He's a predator, Kris!\"\n\n\"Oh, yeah, he's really taking advantage of me up there.\" Kris rolled his eyes and put his guitar away. \"Just let it go, man. I'm having a good time.\"\n\n\"Listen to me. You're my little brother, and I worry about you. And this? What you're doing? I know you think this is the only chance you've got of fulfilling your dream or whatever, but have you stopped to ask yourself if it's actually worth it?\"\n\nKris paused and narrowed his eyes. \"Are you worried somebody's forcing me to do all this shit, or are you worried I'm actually into it?\"\n\nBrad's mouth twisted.\n\n\"No, tell me,\" Kris said. \"What's worse\u2014your little brother being bullied into wearing girly clothes and getting groped onstage, or it being all his idea?\"\n\n\"Kris\u2014\"\n\n\"What's worse?\" Kris demanded. \"That I'm a victim or a pansy?\"\n\n\"I didn't come here to fight,\" Brad snapped. \"If you don't want my support, fine!\" He turned and stalked off the way he'd come, away from the stage and back in the direction of the family car.\n\n\"Fuck you too!\" Kris yelled after him, before he disappeared into the crowds.\n\nCass poked her head around the corner. \"Bad timing?\"\n\n\"Fucking Bradley, sanctimonious asshole,\" Kris muttered.\n\n\"Yeah,\" she agreed. \"Just wait till he tries to warn Mom and Dad about the dangers of me running away with a punk band.\"\n\n\"The Passionfruit guys are sweethearts,\" Kris said automatically. \"Wait, are you really joining the band?\"\n\n\"I passed the audition,\" she said with a shrug, \"and they say I did good on the live show. They need a drummer, at least until Knocks is back on his feet. It's almost as cool as running away to join the circus, and I've wanted to do that since I was a kid.\"\n\n\"Cool,\" Kris said faintly.\n\n\"Cool,\" Cassie confirmed. \"Have you seen Stef? I want to talk to them about something.\"\n\n\"Right, yeah, I'm sure. They were heading back to the buses, last I heard.\"\n\nCassie grinned. \"Thanks, bro. See you later.\"\n\n\"Cassie? No hooking up on the buses! Tour rule!\"\n\nShe waved over her shoulder. \"Sure, sure. Whatever you say.\"\n\nKris flopped onto the baggy couch and pressed a water bottle against his face. Why Brad had even wanted to come to Nevada, he didn't know. The idea that Brad suddenly wanted to spend time with Kris after years of minimal contact was laughable. They hadn't been close since before high school, and by the time they both finished community college, they'd had little in common outside their family tree. If Brad had come along on the Golding road trip out of some misguided attempt to show Kris the error of his ways and guide him back to the conservative, heteronormative light, he was going to have to try harder than that. Even if Kris left the band after they finished the tour and never wore girls' clothes or makeup again, he wasn't going to forget how incredible it had felt to kiss Rayne, or that dream in the hotel, or how he wanted to\u2014\n\nHe rubbed his fingers over his eyes and shoved his thoughts back into order. He could barely remember why he had wanted to keep Rayne off-limits in the first place. Closing his eyes, he let a fantasy run wild for a second. He would come out to Rayne; they'd do a press release and kiss onstage. Then maybe they'd kiss offstage too. He'd talk to Brian and assure him that what had happened with Fink would never happen with Kris, and his family would come around quickly enough\u2014apart from Brad. Maybe he should come out just to spite his brother.\n\n\"Stop it,\" he said aloud, sternly. Fantasy aside, there was too much at risk in propositioning Rayne\u2014like his career, and his friendship, and his dignity, if Rayne rejected him. Rayne had never shown any interest in being more than friends, and Kris didn't need to invite awkwardness into their lives, or joblessness into his.\n\nHis fantasy ran in full Technicolor, suggesting a few other things he could invite instead.\n\nHe groaned. \"That's so unhelpful,\" he muttered to his brain.\n\nThe fact that he wanted to sleep with Rayne\u2014not hypothetically, not in a dream, but in a definite, real, messily undeniable way\u2014didn't get the due diligence it deserved. It seemed like the kind of revelation that should come down like a ton of bricks, but mostly Kris was just annoyed.\n\nThe timing sucked, but Kris was willing to take the blame for that. He should have figured out his attraction earlier; he should have admitted it the second that dream had come crashing through his subconscious. But just because Rayne had dragged his bisexuality over the threshold from a probability to a mouth-watering surety\u2014right at the moment Calloway was due to enter the picture\u2014that wasn't cause for a crisis. It was cause for a plan.\n\nHe had more than a crush, and it was only getting stronger as the days went on; that was fine. His plan could work with that. Rayne would do his stunt with Calloway while Kris worked up the nerve to come out, and if by the time the stunt was over Kris still wanted to sleep with Rayne? Then he'd tell him that too.\n\nKris was itching to meet Calloway in person and get things underway. Neither Rayne nor Brian had announced anything about the stunt to the rest of The Chokecherries yet, but they clearly knew something was winding Kris up. Kris wasn't about to admit to having feelings for Rayne and to depending on a publicity stunt with a stranger to get him over them, so they all circled around each other pretending everything was normal. It was exhausting, and he was strung so tight he might snap, but when he was a second away from flinging himself from the bus to pace the festival, Rayne beckoned him over. Dead Generation was ready.\n\nKris headed across the grounds with Rayne and Angel, his nerves twisting around like it was his career on the line and no one else's. Technically, he didn't need to meet Calloway at all. Rayne was perfectly capable of making a decision about the stunt on his own. But Kris wasn't going to turn down any chance to hang out, no matter how stressful, so he accepted Rayne's invitation to tag along. Rayne looked brightly optimistic and Angel seemed amused by the whole thing, and Kris let them walk ahead while he pulled himself together.\n\n\"You're not slick, you know,\" Angel said eventually, her voice pitched softly enough that Kris assumed she was speaking to Rayne alone. \"You can't stand publicity stunts. You've always said if somebody's music can't stand on its own, it's not worth listening to.\"\n\n\"I have no idea what you're talking about,\" Rayne replied loftily.\n\n\"You talk to . . .?\"\n\n\"He told me I should go for it.\"\n\nAngel glanced back at Kris, who wondered if he should hang farther back to let them talk in peace. He waved, and Angel smiled at him, then turned to Rayne and murmured something too low for Kris to catch.\n\n\"If I were secretly pining for him, sure,\" Rayne said, his voice light, and Angel didn't seem inclined to push the matter further.\n\nThe Dead Generation stage was easy to find, their banner a huge black-and-white thing with skulls and crossbones. Beside them, an androgynous creature with pink hair was rigging a banner that read Neurts and the Synthetic Skunks between two sets of scaffolding for the opening act. Calloway sat on the edge of the stage, kicking his sneakers against the side as he took a swig from his water bottle. When he saw them approach, he lifted his hand in an easy greeting and hopped down to meet them. He was even more attractive in real life: taller than Rayne, with strong shoulders and a vitality that infused his every movement like he was living entirely in the present, and keen to make the most of it. He was less of a twink than Kris\u2014and Kris suspected twink was basically Rayne's type\u2014but that wasn't a fair comparison. Kris was like, the Platonic Ideal of twinkdom. A few tattoos scrawled around Calloway's arms and peeked out from under the neck of his shirt. Kris tried to imagine him and Rayne in glossy tabloid photos together, arms around each other's necks and knowing smiles on their lips. They would look flashy like rock royalty, and the press would eat it up.\n\n\"Hi,\" Calloway said, offering his hand to whoever wanted to take it first. \"I'm Cal. Nice to meet you.\" He had an easy smile and his words lilted with a rough Irish accent.\n\nRayne took his hand with an answering smile, turning on the same charm Kris recognized from his press interviews. \"Rayne. Nice to meet you too. This is my guitarist Kris, and Angel, my makeup artist and . . . bodyguard?\"\n\n\"I'm a lot of things,\" Angel said, taking his hand and flashing a smile. \"Moral support and common sense, at the moment.\"\n\n\"Yeah, we could use some of that,\" Calloway agreed, rubbing the back of his neck with a sheepish air. When Kris shook his hand next, his grip was warm and sturdy. Every inch of exposed skin was covered in freckles, peeking out from in between tattoos, and the hairs that dusted his forearms were gold from the sun. \"I've heard about you, of course. You've been making quite a splash in the tabloids with your recent shows.\"\n\n\"I don't read the tabloids,\" Kris said.\n\n\"Neither did I, till quite recently.\" Cal looked back to Rayne. \"So, how do you want to do this? Your manager said you were still thinking it over, and I understand your not wanting to jump straight in. Or at all.\"\n\n\"Well, I thought I'd come and meet you, and we'd have a chat,\" Rayne said easily, \"maybe get to know each other a bit. See if we get along well enough to fake a relationship for a week.\"\n\n\"Right, right, that won't be awkward at all.\" Calloway smiled again, and Kris could see why the label wanted to keep him around. A pretty face went a long way in the industry, if Rayne's stories were anything to go by.\n\n\"Let's head to the picnic tables and grab some food,\" Rayne suggested. \"Talk things out. Angel? Kris? You want to join us?\"\n\n\"I wouldn't miss it,\" Angel said, and Kris nodded along agreeably.\n\nThey grabbed some food and took a seat in the picnic area, Rayne sitting between Angel and Kris with Calloway across from them, as if they were in an interview. Rayne still had his press face on, the one that made him look perfectly at ease, but which Kris knew to be a carefully constructed mask. He thought about the first time he met Rayne and they'd gone to that burger place\u2014that had been an interview too, of sorts, but he was sure Rayne had been nothing but genuine the entire time. The thought helped melt some of the nerves away.\n\nCalloway isn't replacing me. It's a stunt to boost album sales, and it'll be over as soon as we leave Purple Sage.\n\n\"A lot of Dead Generation fans love The Chokecherries,\" Calloway said. \"If I have to pull a stunt like this, I'm glad it's with you, at least. Though I'm surprised you're considering it at all\u2014I thought you and Kris had a thing going on.\"\n\n\"I haven't agreed to anything yet,\" Rayne pointed out before Kris could say anything, \"and no. Kris is straight.\"\n\n\"Are you?\" Calloway asked him. \"My god, that's dedication. I'd never have guessed.\"\n\nKris swallowed the panicked lump in his throat\u2014this was his perfect chance to say otherwise\u2014but by the time he opened his mouth, Cal was talking again.\n\n\"I assumed the label was pushing you into this to cover up you and Kris, actually, though I couldn't figure out why. But this is cleaner. And hey, if things go well, who knows what could happen?\" He winked.\n\nKris glanced at Rayne and tried to detect any glimmer of interest behind the perfectly professional veneer. Rayne just looked amused.\n\n\"We'll see how it goes,\" Rayne agreed, and Cal grinned wider in response. \"So how did you get outed? Not to pry, but if the press comes up asking me about a sex tape or something, I'd like to be prepared.\"\n\n\"Now that you mention it\u2014 No, they caught me at a gay club with a friend. The label said if I got a girlfriend and laughed it off, it would blow over fast enough, but I couldn't bear the thought of lying, only to come out on my own years later.\"\n\nA rush of guilt burned through Kris, and he laced his fingers together on the tabletop until his knuckles went white. Angel glanced at him behind Rayne's back, her brows knitted in concern, but he shook his head. If Rayne or Cal noticed anything amiss, they pretended not to, for which Kris was grateful.\n\n\"So they set this up instead,\" Cal continued. \"It really wasn't very scandalous at all, I'm afraid. I'd at least have liked a good story out of it.\"\n\nCalloway was charming enough; he wasn't obnoxious like some front men could be, and he had an easy air to him, like he wasn't inclined to take things personally. If Rayne turned him down, Kris guessed they could all still be friends. Beside him, Rayne relaxed, likely reading those same signs.\n\n\"Tell you what,\" Rayne said, and Calloway brightened and sat up straighter. \"I like what I've heard of your band, and you don't seem like a serial killer. I'll do it. Give me one of your Dead Generation shirts to wear for my next show\u2014that'll be enough to start the rumor mill on its own.\"\n\nCalloway grinned and leaned over the table to take Rayne's hand again. \"This'll be fun,\" he promised. Up close, his eyes were a dark, sparkling blue, and when he turned his smile on Kris, Kris forced himself to smile back. His inner turmoil wasn't Cal's fault, after all.\n\n\"You should come meet the rest of the band,\" Angel said. \"And you three should have a talk about what this means for The Chokecherries' shows, with the kissing and all that.\"\n\nRayne elbowed her and she elbowed him back harder, still smiling brightly at Cal.\n\n\"Kris and I agreed to tone things down while you and I are together,\" Rayne said. \"No more kissing, but we want to keep a little teasing, if that's cool with you. The crowds are into it, you know?\"\n\n\"For sure,\" Calloway agreed, grinning between the two of them. \"It's all for the fans. Definitely nothing else going on.\"\n\nKris bit his tongue to hide his wince and managed to keep his smile in place.\n\nRayne sighed. \"You're going to be trouble, I can tell. Come on, come meet the others. Let's get this rolling.\"\n\nCalloway detoured back to his stage to grab a shirt for Rayne, and Angel looped her arm through Rayne's as they waited. Kris stood, leaning his hip against the table as he soaked up the sun.\n\n\"So?\" Angel asked. \"What do you guys think? You like him?\"\n\n\"I agreed to go out with him, didn't I? Or pretend to, anyway,\" Rayne said.\n\n\"You think he's cute?\" she pressed. \"Maybe hit that thing for real? He clearly wants to.\"\n\n\"Stop it, you.\"\n\n\"It could do you some good,\" she said with a shrug, then glanced at Kris, who kept his face carefully neutral. \"This is your perfect chance to unwind and have some fun, and you don't even have to hide it from the press. Hell, you get to flaunt it!\"\n\n\"You're a terrible influence.\"\n\n\"What do you think, Kris?\" Angel asked pointedly. \"If they're both into each other, can you think of one good reason why they shouldn't give it a shot?\"\n\nKris remembered their inebriated conversation in the White Rabbit\u2014how he'd told her he liked kissing Rayne, and how he'd never declined her offer to find him a label other than \"straight\" to encompass that. She would never out him without his say, but he wished she would, just to get it over with, since he was clearly having trouble doing it himself. Before he could form a semicoherent reply, Calloway came loping back, waving a black T-shirt with the Dead Generation logo sprawled across the chest. He tossed it to Rayne, who shrugged out of his old shirt in a single smooth movement. As Rayne changed, Kris's and Calloway's gazes met and Calloway grinned knowingly before turning away.\n\nKris had seen Rayne shirtless a thousand times by now\u2014privacy was a foreign concept on tour\u2014but the sight was affecting him more and more. He'd have to be less obvious about it if Calloway could see through him so easily.\n\nFully clothed again, Rayne put his arm around Cal's shoulders and pulled him along as the four of them walked back to The Chokecherries' stage. Cal fell in line easily, matching his longer strides to the group's.\n\n\"So what did you do before your band?\" Angel asked.\n\n\"Oh, this and that. Dropped out of university and traveled around. Bit of a vagrant, really. Had a brief fling with a cult; that was exciting, I can tell you.\"\n\nThey all made an inquisitive noise.\n\n\"Yeah, not my wisest life choice, but it served as a wake-up call. I cleaned up my act, and now I'm here.\"\n\n\"Wait, you're not joking?\" Kris demanded.\n\n\"About the cult? Nah. It'll make a good chapter in my memoir, if I ever get big enough to write one.\"\n\nAngel darted in, her eyes fixed on the arm Calloway had slung around Rayne's waist. \"Those tattoos . . .\"\n\n\"Ah, yeah, I'm covering most of the older ones up. I'm a work in progress.\"\n\nShe frowned and took his hand, bumping Rayne aside with her hip to swap places and walk next to Calloway instead.\n\n\"I've seen designs like this before,\" she said, turning his hand over and examining his arm in full. The tattoos were unintelligible, a million miles from what Jiao Fang and her ilk could do. There were bits of writing in languages and alphabets Kris didn't know, scrawled pictures of flowers or strange animals up and down his arms. There was the outline of an eye on the back of his left hand.\n\n\"Yeah,\" Cal said, \"I don't know what most of them are either.\"\n\n\"Where did you get them?\" Rayne asked.\n\n\"A guy I knew had his own kit. I'm lucky I never got an infection, but I was an idiot kid. He explained what they all were at the time, but it never made a lot of sense. I can barely remember, now. That was years ago. It all feels like a dream, honestly.\"\n\nAngel dropped his hand and stared at him. \"Did you have your head shaved back then?\"\n\nHe blinked. \"Yeah, actually. We all did.\"\n\n\"What?\" Rayne asked, glancing between the two of them.\n\n\"That's where I saw these tattoos before, on those skinhead guys I ran into earlier,\" Angel said. \"They all had them. You're not joking about the cult. They're here.\"\n\n\"Are they?\" Calloway looked around like he expected them to leap out from behind the nearest tent. \"I'm not surprised. I met them at a music festival in the first place. They like these sorts of places. They're not so bad,\" he added. \"They're not like some ritual death cult or anything. They just go around on their bikes and talk about the universe, really. Like big leathery hippies.\"\n\n\"I'm . . . a bit lost,\" Rayne admitted. \"Angel? You met a cult?\"\n\n\"I missed that part too,\" Kris said.\n\n\"Yeah, I ran into them earlier. I didn't know they were a cult, but I got a weird feeling about them.\" She eyed Calloway. \"You sure they're all right?\"\n\nCalloway shifted uncomfortably. \"They were never dangerous when I was with them,\" he offered. \"I don't think they know I'm here. It's not like we've kept in touch. My manager knows about them, though not the details. I never even mentioned it to the label.\"\n\nThe Chokecherries' stage came into view and they all slowed.\n\n\"Okay,\" Rayne said. \"Just . . . try not to mention it to the rest of the band? Definitely don't mention it to Brian. That's the kind of thing that'll give him an aneurysm, and we like Brian. We need to keep him around.\"\n\nCalloway nodded sheepishly. \"Right. Sorry. Pretend I never said anything.\"\n\nThey headed backstage to where the rest of The Chokecherries were waiting, and Kris took a deep breath. They were committed to the stunt. As long as he didn't interfere, it would go off without a hitch. Brian and the label would be happy, Rayne and Cal would be happy, and he would have enough time to settle his nerves before coming out. And hopefully his feelings for Rayne would dissipate too. He had this covered.\n\n\"Hey, guys,\" Rayne said, tugging Calloway forward to make introductions. \"This is Cal, from Dead Generation, and we're . . . seeing each other?\" The Chokecherries immediately broke into catcalls, and Rayne rolled his eyes at them. \"It's not too late to call everything off, you know.\"\n\n\"Nah, you've got this,\" Kris said. \"It'll be great.\"\n\n\"Course it will,\" Cal agreed. \"So, since the paparazzi have been so keen on making up stories about your stage kisses, would you prefer rumors of cheating, or a threesome now that I've butted in?\"\n\nThe catcalls went up again, led mainly by Stef.\n\nRayne glanced at Kris with an exasperated smile. Angel scowled at all three of them like she wanted to smack some sense into their heads, a line of aggravation between her brows.\n\n\"We don't need either,\" Kris said firmly. \"We'll quit messing around, like Brian suggested, and we'll remind the press that the show stuff was just for the fans, like we've been saying all along. All three of us can tell them at once, if they want. No cheating, and no threesomes.\"\n\n\"A little speculation is unavoidable, but I'm not worried,\" Rayne said. \"The rest of you,\" he added, addressing Stef, Len, and Maki, \"no stirring the pot. It's only while the festival is running; you can give me shit for it later.\"\n\n\"We will,\" Stef assured him.\n\n\"Nothing personal,\" Maki told Calloway. \"He needs to be kept in his place.\"\n\n\"You're all terrible,\" Rayne said. \"Cal? Second thoughts?\"\n\n\"No, no. If everyone's happy to go along with it, I think it should be fun.\" Calloway caught Kris's gaze. \"Assuming everyone is happy?\"\n\nKris put on a smile. \"It's cool.\" He almost believed it, too. Calloway and Rayne looked good together, and that was the most important thing as far as the paparazzi were concerned. And Calloway seemed amiable: maybe a year or two older than Kris, sure of himself without being arrogant, and quick to smile. \"I'm happy for you guys. I hope you have a great fake relationship.\"\n\nCalloway didn't stay long, citing the need to go back and prep his band for the incoming press. Rayne went with him, flashing one last smile back at The Chokecherries before he and Calloway disappeared into the festival, their arms around each other with the same casual intimacy Rayne showed everyone in his chosen group.\n\nKris sat on the edge of the stage and watched them go. The rest of the band dispersed, flitting out between the tents to see different attractions, and Kris leaned back as the clouds drifted through the bright-blue sky. Angel joined him a moment later, dropping down at his side and mirroring his position.\n\n\"Hey, you,\" she said. \"What do you think?\"\n\n\"The press will love it.\"\n\n\"Sure, but what do you think?\" She punctuated the you with a poke to his shoulder.\n\n\"I think the same thing I thought when I told Rayne it was a good idea. And Cal seems like a decent guy. They could pull it off.\"\n\n\"Hm.\"\n\nHe turned to face her. \"What, hm?\"\n\n\"Level with me a minute. All the kissing you two do onstage, and all the hand-holding and cuddling up you do off it, you really don't have a thing for him?\"\n\nKris froze, and in that split second of hesitation he knew he'd waited too long. Angel's expression softened. \"No,\" he said weakly. \"No, it's not . . . We're not like that.\"\n\n\"Kris, honey. I love you, but lying's not your forte.\"\n\n\"It's just a crush. It doesn't mean anything.\" He bit his tongue, hoping his lie didn't sound as blatant as it felt. \"And now that Calloway's here and they're doing this whole thing, I'll get over it.\"\n\n\"You really don't want to tell him?\"\n\n\"I don't want to fuck up and make things awkward. Brian's still got me on a trial run, and I don't want to do anything that could get me kicked out of the band.\" He reached over and squeezed her fingers. \"Honestly. It's just a crush because we're around each other all the time, and he's a great kisser. And he's . . . Rayne. But it's fine. I'm not in love with him, and he's definitely not in love with me. I've got this totally under control.\"\n\n\"Why don't you want him to know you're not straight?\" she asked bluntly. \"You said you weren't ready, and I respect that, but I can see you pining for him. I could see you pining from the moon.\"\n\n\"I'm going to tell him,\" Kris promised. \"But I don't want to complicate things, especially not while he's doing this with Cal. As soon as their stunt's over, we'll talk. And I'm not pining.\"\n\nShe didn't look like she believed him, but he steeled himself and nodded like it was the absolute truth. She finally sighed and patted his knee.\n\n\"Okay, hun. Whatever you say.\"\n\nKris took a deep breath to calm his pounding heart. Angel would keep his secret. As far as she was concerned, it was just a crush and it was under control.\n\nNothing could be farther from the truth.\n\nControl was only ever an illusion. Kris knew that, on some deep level, but superficially he still had hope. He lost the last of it\u2014hope and control both\u2014after Angel brought out her costume trunk.\n\nIt wasn't her fault\u2014 Kris didn't want to blame her for anything that happened. She was simply the catalyst that sent everything else whirring into motion like a terrifyingly unstoppable doomsday device. It started with a pair of gold lam\u00e9 booty shorts.\n\n\"We're doing a Rocky Horror tribute,\" Billie said.\n\n\"I have costumes,\" Angel added.\n\n\"Okay,\" Kris said. \"Rocky Horror. Why not.\"\n\n\"I'm Frankie, obviously,\" Rayne said. \"Stef dibs'd Riff Raff, as if anyone else would want him. Maki's Columbia, Billie's Meatloaf\u2014Eddie\u2014and everyone else is a Transylvanian. We're throwing everybody in corsets and glitter.\"\n\n\"What about me?\" Kris asked, feeling the answer in his gut already.\n\n\"You're Rocky, of course,\" Angel said with a wink. \"Not much of a bodybuilder, but I never did like a man with too many muscles.\"\n\n\"I would've thought I was more of a Janet,\" Kris said, \"but hey, whatever floats your boat.\"\n\nAngel pressed the shorts into his hand, and Kris had to turn them over three times to figure out what he was holding. They seemed a lot smaller in person than he remembered from the movie, and he remembered them being tiny. \"Where's the rest of them?\"\n\n\"Oh hun,\" Angel said. \"The rest of it's all you.\"\n\nThe important thing Kris needed to remember about Purple Sage was that it wasn't part of the real world. It might seem like it was, being in the middle of a desert that could be found on any map, and the tickets were paid for with very real money, but the festival itself\u2014the part where the music happened\u2014that was somewhere else entirely. That part was just left of reality, where the laws of nature and common sense and outside society didn't apply. That was the reason Kris didn't question the Rocky Horror getups, or the way Angel carried a veritable cornucopia of recreational drugs, or his own tumultuous, headfirst tumble into heart-throbbing infatuation with Rayne, which he could no longer deny, though he was trying. Questioning was for the real world, as were consequences. He put the shorts on.\n\n\"I think I need a bit more to wear,\" he said, examining his reflection in the mirror in the private dressing-room-trailer he and Rayne had appropriated.\n\nHe didn't mind his body. Sure, he was skinny, with no real muscle definition, and maybe his limbs were a little too coltish to belong on a man of twenty-five, but there was nothing offensive about it. It left his gender ambiguous, almost, at the right angle and in the right light. He'd never really appreciated that before the tour. Less ambiguous in nothing but shorts that barely covered the tops of his thighs, but that was probably the point.\n\n\"Can I get one of those corsets too?\"\n\n\"Sure,\" Rayne said absently. \"You have to feel comfortable performing.\"\n\nKris turned from the mirror to find Rayne rolling his fishnets up with agonizing care, sitting on one chair with his foot propped up on another. His corset was unlaced, hanging around his middle with no thought for modesty, and he was barely covered by a pair of black shorts under his garter belt.\n\nKris swallowed.\n\n\"I don't recommend wearing stockings, whatever else you're thinking,\" Rayne said. \"This isn't the first time I've worn them, but I hope it's the fucking last. And fishnets aren't even that bad compared to hose.\" He reached his thigh and clipped the first garter in place, shaking his hair from his face to glance up at Kris. \"Don't do it,\" he repeated, but he was smiling.\n\n\"Noted,\" Kris said, his mouth dry. He coughed. \"No stockings. I need something, though. I can't go out there like this.\"\n\n\"It's good, though. You sure you need to cover all that up?\"\n\n\"When I'm wearing my guitar I'm going to look naked,\" Kris pointed out. \"I'll look like a slapstick sketch. You have to give me more to work with here.\"\n\n\"Fine, fine. Crush all my dreams at once.\" Rayne rooted through a bag of feather boas, gloves, and costume jewelry to pull a corset from its depths. Like the shorts, it was gold. Unlike the shorts, it covered more than a few square inches of skin.\n\n\"Awesome,\" Kris said, holding it up against his torso as he returned to the mirror. All the gold brought the browns in his eyes to life and made his hair seem even paler. Once Angel did his makeup, he would look like nothing he'd ever seen before. Not even his first time in makeup could compare to this, with so many glittering sequins and so much bare skin.\n\nHe'd never worn a corset before. Or, unlike Rayne, any kind of hosiery\u2014or high heels, or\u2014\n\nThe list was endless. There were so many things he had never worn or done or thought about before The Chokecherries, and now he wanted to try all of them at once. He glanced at Rayne in the reflection; he was busy fastening a rope of pearls around his throat.\n\n\"Hey, rock star, you want to help me into this?\" Kris asked.\n\nRayne looked up, his expression inscrutable in the tarnished mirror.\n\nKris hefted the corset. \"I can't reach the laces by myself, unless you want me to wear it backwards.\"\n\n\"No, I can help you do it the right way around.\" Rayne stood slowly. His legs were so long out of his usual jeans. He padded quiet as a panther over to Kris and stopped just behind him, not quite touching, but near enough for Kris to feel his heat. \"Here, hold it around you like this.\"\n\nKris obeyed, offering the edges to Rayne without taking his eyes from their reflections. The mirror was an old one, its surface going silvery and dim, but looking at it, secreted away in the tiny trailer that was all half-lights and dark corners, Kris felt hidden from the rest of the universe, like there was no one in the entire world but him and Rayne. Within this room, no one else existed: not the fans, not the band, not Calloway or the press. It was a dangerous, intoxicating feeling that curled around his ribs as surely as the corset did, and settled in his heart like a bad idea.\n\nIt would be so easy to kiss Rayne here, where no one else could see them.\n\nHe couldn't do it.\n\n\"How tight do you want me to go?\" Rayne asked.\n\n\"Tight enough to look good.\"\n\n\"Such an exhibitionist.\"\n\n\"Always,\" Kris said.\n\nHe sank his teeth into his lip at the first bite of the laces; Rayne gave no quarter as he worked his way up, but Kris liked the pressure. It was firm in the way the best hugs are firm, and if it left him a little breathless, that only added to the experience. He watched his body reshape in front of him as the corset forced his waist in, giving the illusion of hips\u2014not much of one, as he had always been flat as a plank in all directions, but enough to stop and take notice.\n\n\"Okay?\" Rayne checked in.\n\nKris pressed his hand against his belly and felt nothing but firm, unyielding fabric. His stomach fluttered and he smoothed his hand down. \"Perfect.\"\n\nRayne tied the laces off and then stopped to admire his work. His eyes were dark in the mirror, and Kris held his breath as he awaited Rayne's verdict. The fans would be delighted, but it was only Rayne he wanted to impress. That was easier to admit when it was just the two of them, even if he couldn't say it aloud.\n\n\"You look good enough to eat,\" Rayne said, his voice a pitch too low to be teasing.\n\n\"Yeah?\" Kris caught Rayne's gaze and held it, his heart beating fast as a hummingbird. \"What you going to do about it?\"\n\nRayne's grip tightened on Kris's shoulder for a second, his fingers digging into his bare skin as surely as the corset laces dug into his back, and then he let go and stepped away, shaking his head with a rueful smile.\n\n\"Let's get you into Angel's chair and try not to break any hearts on the way.\"\n\nKris let Rayne guide him from the trailer, infinitely disappointed in his own lack of courage for not pushing further.\n\nThings only got worse once the makeup was on.\n\nPassionfruit and The Chokecherries crammed into the space backstage, all of them centered around Angel and her makeshift makeup station like she was a queen holding court. Passionfruit did their own makeup, for the most part, though Billie and Angel were collaborating more often as Billie's tastes grew increasingly outlandish. Jay stayed punk and gleefully mocked him, which Billie ignored, and this instance was no exception. Cassie flitted around between the two bands, exploring Angel's makeup collection, unable to hold still, until Angel promised to do her up in turn, if she would just keep her hands to herself for five minutes. Cass compromised by sitting on Stef's lap. Stef, busy applying dark shadows under their eyes, didn't seem to mind.\n\nRayne's makeup was heavier than usual for his Frank-N-Furter look\u2014Angel dusted him with black and purple from his lashes to his brows, painting his eyes with kohl and his lips with a wine-dark red that made his mouth obscenely shiny and wet. Kris, she coated in the gold Maki favored, so his eyelids glittered and caught the light with every blink. She gave him his usual red lipstick, and then painted his nails black, the same black Rayne most often wore.\n\n\"Now sit tight and let that dry,\" she instructed, moving on to Maki's Columbia makeup next. \"If you smear it, I'm putting you in time-out.\"\n\n\"Yes, ma'am.\" Kris placed his hands palm-down over his knees and tried not to fidget.\n\nCassie, though making good on her promise to sit still, lacked any incentive to keep quiet. \"So, Angel, are you a professional makeup artist? How'd you meet Rayne?\"\n\n\"It wasn't long after I opened my club, and he was just starting the band,\" Angel said.\n\n\"We didn't have a record deal yet\u2014we didn't even have Brian yet\u2014but I was determined to play every club and dive bar in the country that would have me,\" Rayne continued. \"So I piled everybody into a van and we headed out, usually booking one or two shows ahead as we drove.\"\n\n\"It was insane,\" Stef said.\n\n\"We all nearly quit at one point or another,\" Maki added. \"Our original drummer did. We met Lenny en route.\"\n\nLenny tipped an imaginary hat.\n\n\"So we rolled into New Orleans with no real plan, but I'd heard rumors of this new club that sounded queer-friendly, and\u2014\" Rayne shrugged expansively \"\u2014the rest is history.\"\n\n\"Just like that?\" Cassie asked.\n\n\"Just like that,\" Angel agreed. \"I didn't start touring with Rayne until after he got his record deal, but we kept in touch the whole time. I visited him in LA a few times and nabbed myself a studio space in his house.\"\n\n\"Not New York?\" Kris asked.\n\n\"New York's my second home,\" Rayne said. \"My LA place is bigger; it came first. As soon as I had enough pull with the label, I got Angel signed as our official fashion director. Now we don't do anything without her.\"\n\nAngel shrugged. \"I can't complain.\"\n\n\"I can't believe I didn't think to steal you first,\" Billie groused.\n\n\"To be fair, honey, Passionfruit was pretty grungy when you started,\" she said. \"I love you, but I wouldn't have had much to work with.\"\n\nBillie coaxed his hair into a giant bouffant. \"How about now?\"\n\n\"Baby steps,\" Angel said.\n\n\"Why burlesque?\" Cassie asked. Stef finished their makeup and slung their arms around Cassie's waist, holding her in place.\n\n\"I started with drag shows back in art school,\" Angel replied, taking over the design of Billie's hair, \"but I wanted something more . . . sexual. I liked the camp, but it felt like I was wearing my own skin and pretending it was a costume. Burlesque seemed like the next natural step.\"\n\n\"More sexual,\" Kris repeated.\n\nShe shrugged. \"It's fun. And it's safe, playing onstage like that. There's no room for misunderstandings when you're on a stage and everyone else is paying to watch.\"\n\n\"I get that. I'd never have tried this if it weren't for the stage.\" He drummed his fingers across his knees, waiting for the polish to dry. \"I don't think I'd actually strip, though.\"\n\n\"It's not for everyone,\" Angel agreed. \"Billie would never do it. He's body-shy.\"\n\nBillie rolled his eyes but didn't stop her from fixing his hair.\n\n\"I found it really helped with my body image, personally,\" she continued. \"And now that I have the body I want, I like showing it off. I worked too hard on it for it to go unappreciated.\"\n\n\"And unpaid for?\" Kris guessed.\n\n\"It's worth every penny. You said no to a dance last time; what about now?\" She laughed at the expression on his face. \"No? What about from your man Rayne? He can dance, you know.\"\n\n\"I can,\" Rayne assured them, though he threw Angel a look Kris couldn't interpret. \"I'm a great dancer.\"\n\n\"Can you strip?\" Cassie asked.\n\n\"Can't you go back to being all shy and flustered around him?\" Kris asked. \"He's supposed to be your idol.\"\n\n\"Nah, I'm over it. Can you, though?\"\n\n\"Not wearing a corset,\" Rayne said, but when he met Kris's gaze he was positively predatory. Whatever bait Angel had thrown him, he'd taken it. \"But I can do other stuff.\"\n\nJay whooped and finished drawing on the last of his eyeliner. \"Give Kris a dance and I'll give Billie one.\" Kris couldn't tell whether it was a dare or a threat. Either way, with his nails still wet, he couldn't defend himself.\n\nRayne stood in a long, languid motion. \"How about it, Kris?\" He rested one boot on the edge of Kris's chair, in between his thighs. \"Say no.\"\n\nKris should say no for the sake of his sanity, but Rayne looked delicious made up like that, and Kris was only human. He raised his hands in the universal symbol of surrender\u2014ostensibly to keep his nails from smearing, but mostly to keep himself from trying to touch. \"Go ahead. See who gets more embarrassed.\"\n\nRayne laughed.\n\n\"Bad move, honey,\" Angel said.\n\n\"You know me better than that, Kris,\" Rayne said. \"I don't even know what shame is.\"\n\nRayne could move: he had the coordination, the rhythm, and the sex appeal to do it. He danced onstage, or in clubs, or on the bus; something as simple as rocking his hips to the beat had driven Kris to distraction more than once. Kris had just never seen it from so close an angle before.\n\nRayne returned both feet to the floor and shimmied up, his stance wide, until he stood over Kris's lap, and rested his arms across Kris's shoulders, bending at the waist to reach. He moved slowly, either teasing or giving Kris time to back out, Kris didn't know. He didn't care. His breath hitched in his chest and he blushed, hard. Swallowing, he tried to brush it off and look nonchalant.\n\nIt didn't work. His face was burning up; he could feel the fever-bright heat on his cheeks and in his eyes, and the way his lips fell open of their own accord as Rayne leaned in, his hair brushing Kris's face, to whisper in his ear.\n\n\"You can still say no,\" he breathed, for no one but Kris.\n\n\"Do it, you giant diva. Show me what you got.\"\n\nRayne laughed again, lower this time, and tossed his hair back. \"Keep your hands up, baby. No touching.\"\n\nJay whistled. \"Look at him, all professional. Does it cost extra to touch?\"\n\n\"Kris isn't touching anything,\" Angel said. \"He tries it, I'll skin him. I'm not redoing his nails before the show.\"\n\n\"No touching,\" Kris promised, not taking his gaze from Rayne.\n\nRayne's tattoos showed above the line of the corset, the matching birds dark against his skin. The corset was barely high enough to cover his nipples, and if he shifted in just the right way, they peeked above the border, dusky brown and getting hard from the friction. Rayne either didn't notice or didn't mind as he started to move, striking up a rhythm as serpentine as the tattoo that wound around his arm. Kris still hadn't touched it. The wrapping was off, but after only a week it had to still be tender. Rayne looked like a walking, breathing piece of art, and Kris wanted to fall to his knees in worship, one way or another. He bit his tongue and kept his hands away.\n\nLike when they kissed onstage, Kris was aware of the cacophony of shouts and appreciative jeers around them. Like onstage, he tuned them out and drank Rayne in like it was only the two of them in the whole world. But their stage antics never got close to this kind of concentrated attention. While Kris knew Rayne was playing up the sex appeal for the others, it was still so much closer, so much hotter, so much more of everything that he felt it in every single atom of his being. It had never been like this offstage before, like Rayne was trying to turn Kris on for real\u2014and it was working.\n\nRayne danced like his body was made for it. He kept his arms wound around Kris's neck, twisting his body down toward Kris's lap but always stopping just shy of contact, his hips working in a mesmerizing pattern that left Kris's mouth dry and his blood flooding south.\n\nKris wanted to unlace Rayne's corset hole by hole and watch the muscles shift under his skin as Rayne moved; he wanted to put his hands on him and feel them to be sure this was real. He wanted Rayne to sit down in his lap, and he wanted to push his hands through Rayne's hair and drag him down until their lips met, makeup be damned, and kiss him until Rayne was as breathless and shaky as he always left Kris.\n\nRayne was supposed to be doing this with Calloway, not Kris, but there were no paparazzi backstage. No one else had to know.\n\nWhether it was the dream, the dance, or just the inevitable deepening of friendship to love, something in Kris flipped. Rayne made him want to fling himself into a jungle of unmapped sexuality, and to hell with ever finding his way out again. Kris wanted to taste him and touch him and be touched by him, and with every passing second the thought of announcing his sexuality to the world seemed less intimidating and more exciting. His nerves gave way to curiosity, which gave way to want that squirmed in his belly, taking up residence with his flock of butterflies, which fluttered wildly every time Rayne moved.\n\nKris didn't touch him. Not even when Rayne pressed his lips to Kris's cheek in a lingering kiss clearly designed to leave a dark-red mark against his skin, nor when Rayne straightened and stepped back as the bands cheered and wolf-whistled. Completely wrecked and so turned on he was light-headed, Kris returned his hands to his knees and tried to pull himself together.\n\nAngel fanned herself with a flyer, affecting a wide-eyed, overwhelmed expression. \"Damn, boy. You ever want out of music, come back to the White Rabbit with me.\"\n\n\"I think I got him a little worked up,\" Rayne said in fake apology.\n\nKris groaned and covered his face with the crook of his arm, careful not to smudge anything. His heart was beating in triple time and all he could think of was Calloway, fucking Calloway\u2014he wasn't serving as the deterrent Kris needed him to be. If they were dating for real, Kris would never interfere, but it was fake, and his whole body was burning for Rayne.\n\n\"You better take care of that before we hit the stage,\" Rayne said.\n\n\"No touching,\" Angel said, though she couldn't keep a straight face this time. \"Sorry, Kris.\"\n\n\"It's okay,\" Rayne decided. \"It's a compliment. I'm flattered.\"\n\n\"Cassie, I know you filmed that whole thing,\" Kris said, not removing his arm from his face. \"For the love of god, don't show our parents. Don't show anyone, ever.\"\n\n\"I'm conflicted,\" Cassie admitted, still comfortably settled on Stef's lap. \"On the one hand, you're my brother, and there's no way anything you're involved with can be hot. On the other hand\u2014oh my god, Rayne. Oh my god.\"\n\nRayne nudged at Kris until he dropped his arm to glare at him. Rayne grinned, mercifully not looking at Kris's shorts.\n\n\"I don't know what you were trying to prove,\" Kris said. \"I can't remember. But you did it: Congratulations. You proved the thing. I hate you.\"\n\n\"No, you don't,\" Rayne said fondly. \"Come on, Rocky. Up you get. If you play a good show, you might even get a repeat performance sometime.\"\n\nThere seemed to be something wistful in his tone, lingering just below the surface and unnoticeable unless you were specifically listening for it. A second later Kris thought he must have imagined it, because Rayne had him by the wrist and was pulling him to his feet, laughing with the others. Kris accepted his help in getting up, but was privately sure that a repeat performance would actually kill him, especially if he still couldn't touch. He needed to tell Rayne how he felt; it didn't matter if it was unrequited. Not telling was going to be the actual literal death of him if Rayne kept up like this.\n\nCalloway dropped by before the show, talking animatedly about how the press was already paying ten times more attention to Dead Generation on the basis of a few rumors about Rayne wearing their shirt. Rayne got caught up in his high spirits with admirable ease, and rather than watch the two of them play boyfriends, Kris joined Angel on her quest to the food trucks in search of an apple pretzel.\n\n\"Not that I don't appreciate the company,\" she said, \"but you seem like you're avoiding something.\"\n\n\"Nope,\" Kris said determinedly. \"I'm just giving them space to do their thing, because I'm a supportive friend.\"\n\n\"Uh-huh.\"\n\n\"And I'm trying to get used to moving around in this thing.\" It was weird to walk in the corset; it pulled his spine so straight he felt like a toy soldier, though he looked like anything but. The sequins caught the sun and flashed it around with every step, and his limbs seemed much paler and skinnier than he remembered them being. \"And I like hanging out with you.\" He nudged her with his elbow. \"If you want me to get lost, you can say so.\"\n\n\"No, you can stay. Honestly, I'm glad you're here. I'm not used to wearing so much glitter in public, and having this many eyes on me outside of the club makes me nervous.\"\n\n\"I'm probably not helping you blend in though, am I?\"\n\n\"Not especially, no.\"\n\nAngel's afro always ensured that she would stand out in a crowd, but with the added sequins and sparkles, Kris didn't doubt that every eye in the festival was on her. She might feel self-conscious, but she looked stunning. He grabbed her hand and squeezed it, and she shot him a grateful smile.\n\nAs soon as they set foot in the picnic area, a wolf whistle pealed out. Kris cast around for the source, but Angel just squared her shoulders and marched on, making a beeline for the pretzel wagon. Kris hovered by her side as she made her purchase, but the moment they turned to head back to the stage, they collided with a very tall, very wide man dressed in dusty leather, who sent them stumbling back a pace.\n\n\"Watch where you're going,\" the man grunted.\n\n\"You the one who cat-called me?\" Angel asked. The man's lip curled and Angel rolled her eyes. \"Fuck off.\"\n\nShe went to step around him. Kris stayed glued to her side, though he had no idea what he'd do if he had to intervene, but the man blocked their way. An expression of dumb belligerence sat on his face. \"You don't talk to me like that.\"\n\n\"Yeah, I do. Now move, or I'll get you thrown out,\" Angel said. Kris tried to look intimidating, though he barely came up to the man's chest, and was dressed only in a corset, his sneakers, and a pair of uncomfortably formfitting shorts.\n\nWhatever the man was about to say next\u2014and it was doubtlessly going to be asinine\u2014was interrupted by his friends. One of them gave a sharp whistle and the big one's gaze snapped over.\n\n\"Boar! Leave that. Come on back,\" the other man ordered.\n\n\"Yeah, go on, dog,\" Angel said.\n\nBoar glared and stepped forward\u2014Kris and Angel both darted away\u2014when a second one at the table shouted, \"She disrespects us!\" and vaulted the picnic bench, hurtling toward them like a burly red missile.\n\nAngel smashed her pretzel into Boar's face, its apple filling still hot from the oven, grabbed Kris by the wrist, and booked it. Shouts followed them, from all four of the men now, but Kris didn't dare look back. They lost them after a few turns, the tents too closely packed for a longer chase. Once they were sure the men were gone, Kris and Angel slowed their pace to a halt.\n\n\"Fuck,\" Angel breathed, and finally dropped Kris's arm.\n\n\"Fuck,\" Kris agreed. \"And you lost your pretzel too.\"\n\n\"Forget the pretzel\u2014those were the same guys I saw earlier, the ones Calloway said were a cult.\"\n\nKris craned his neck to glance back the way they'd come, but there was no sign of the men. \"No shit. That was the cult? Cal said they weren't bad guys!\"\n\n\"Maybe Cal didn't know them so well.\"\n\nThey adjusted their course for The Chokecherries' stage, slower now that they weren't being pursued. Kris took the time to catch his breath as the adrenaline faded, his heart thumping back to a normal pace.\n\nPassionfruit was just starting to set up when they arrived, Billie running through a mike check as festival goers milled around, drawn by the movement onstage. Kris relaxed, confident they weren't going to get jumped by a cult of skinheads in broad daylight with so many witnesses around. Tom the priest-to-be was there too, and waved when he saw them, bright and blond in the sun. Kris waved back and tried on a smile, but stilled as the youngest of the cultists appeared by the far side of the stage, making his way toward them. Onstage, Billie paused to watch, and Tom hesitated in his approach.\n\n\"Hey,\" said the cultist. He was tall and lanky and impossibly pale, no older than Kris, and didn't seem like he'd eaten a decent meal anytime in the past year. Kris looked to Angel for direction, and Angel crossed her arms and leveled a stoic glare at the interloper.\n\n\"Sorry about your pretzel,\" the kid said. \"And, um, about Boar. And Red. They get riled easily when they're on the booze, and they forget their manners.\"\n\n\"What do you want?\" Angel asked.\n\n\"I brought you a replacement?\" He held out a little package, butter seeping through the paper. \"To make up for it, if you want.\"\n\nOnstage, Billie cleared his throat conspicuously, and Kris glanced up. The singer appeared ready to jump down and fight the kid if he made one wrong move toward Angel, though Kris doubted that even if he and Billie teamed up they'd be much good in an altercation.\n\n\"What's your name?\" Angel asked.\n\nThe kid broke into a smile that lit up his eyes. They were big and pale and silvery, but they didn't seem malicious. He felt Angel relax fractionally beside him. \"I'm Rikki. Do you want it?\"\n\nAngel accepted the replacement pretzel carefully, like it might blow up in her hand. \"I'm Angel. This is Kris. Your friends are assholes, you know that?\"\n\nHis smile slipped but he nodded. \"Yeah. Sorry. I should get back before they miss me.\"\n\n\"Thanks for the pretzel, Rikki.\"\n\nHis smile bounced back into place before he loped off to find his cult. The pretzel dripped butter over Angel's fingers as she and Kris watched him go.\n\n\"You're aware that guy is in a cult, right?\" Billie said from the stage.\n\nTom, perhaps sensing that not all was well, ducked back into the crowd with a little nod of goodbye and a promise to come back later to see the show. Angel tore a chunk off the pretzel with her teeth.\n\n\"So, cults, huh?\" Kris said. \"When everybody was telling me how weird festivals could get, nobody mentioned the bit about cults.\"\n\n\"We should probably tell Rayne and Cal about this,\" Angel said, heaving a sigh and casting her gaze heavenward.\n\n\"And Brian?\" Kris asked.\n\nThey glanced at each other, wincing in unison. Angel shook her head. Maybe not Brian, at least not yet.\n\nKris didn't get the chance to talk to Rayne or Calloway before the show. While Passionfruit played their set, the two of them were busy flirting with the paparazzi and each other, wrapped up to the extent that Kris was reluctant to interrupt. They really did make a good-looking couple: their arms slung around each other's shoulders, sharing earbuds as they leaned in close to whisper and laugh together, seemingly oblivious to the outside world. Of course it was all an act\u2014Kris saw the way Rayne kept the flashing cameras in his peripherals at all times, making sure he and Cal were angled so the paparazzi saw exactly what he wanted them to see, nothing more or less.\n\nHe could always tell them about the cult after the show, though he was no longer sure he had anything to tell. Neither he nor Angel had caught the cult doing anything particularly dangerous; they were just assholes, and that wasn't cause for a group meeting. Lots of people were assholes, especially when you were wearing a corset, a lot of body glitter, and not much else.\n\nInstead, Kris took the time before the Rocky Horror show to think things over. Passionfruit had taken the stage almost immediately after Kris and Angel got back, leaving Kris a very small window to gather his thoughts. He pushed the cult aside temporarily and returned to the issue of coming out to Rayne, sexuality and feelings both. It was love, he could finally admit: full-fledged, cavity-inducing, spine-tingling love. It was hard to think backstage with Passionfruit playing a few feet away, even more raucous and impulsive than usual, as Billie belted out a Meatloaf tune to the roar of the crowd. Kris's corset didn't help either; he was hyperconscious of it, and the constant pressure made his stomach flip in the most distracting way, like the butterflies in there were planning a revolt. Yet despite the butterflies, the corset, and Passionfruit's aggressive brand of music-making, Kris couldn't stop thinking about the curve of Rayne's mouth, half hidden in the shadows of his mane of messy hair, as he slowly danced over Kris's lap.\n\nRayne, apparently oblivious to Kris's turmoil, wasn't helping. His flair for the dramatic combined with the energy implicit in a Rocky Horror tribute show left Kris feeling delightfully debauched. He was grateful for being able to hide behind his guitar, because his shorts hid nothing, and Rayne knew it. He seemed to take perverse pleasure in winding Kris up\u2014he spent the show draping himself over Kris, rubbing up against him, curling his fingers through Kris's hair or around his throat. Kris didn't fight a second of it\u2014but they didn't kiss. They touched, and teased, and Kris leaned in when Rayne came up behind him and wrapped one arm around his chest, tugging him close to sing in Kris's ear, but their lips never met, and this time, Kris didn't push it. People needed to believe Rayne and Calloway were a thing, and Kris wasn't going to interfere with that, no matter how he felt. But he wasn't the only one getting excited.\n\nThe crowd kept yelling for more, hurling themselves against the barricade, singing along until their voices were wrecked, hands in the air, jumping in time. Calloway was up near the front, his ginger hair burnt gold in the stage lights, arms thrown up over his head as he sang along with Rayne. The cult was there too, all pale tattooed skin and black leather off to the side. They could have passed for punks, but there was something separate about them\u2014they didn't throw themselves into the music like the rest of the crowd. Even among the misfits who made up a music festival, they didn't look like they belonged. Tom didn't look like he belonged either, but from what Kris could see from the stage, he seemed to be enjoying himself.\n\nThe band played their set and Kris stayed at Rayne's side the whole time. He'd thought the corset would get more comfortable the longer he wore it, but it never loosened and he couldn't stop thinking about how the laces dug into his skin and squeezed him into a new shape. Rayne's wasn't as tight as his\u2014he needed the room to sing.\n\nAs they reached the end of their set, their lipstick messed beyond hope, Stef started up the bass line to \"Sweet Transvestite.\" The crowd dropped to silence for a single second and then surged back louder and shriller than before. Rayne caught Kris's eye across the stage and grinned, fluffed up his feather boa, and strutted to the mike.\n\nHe'd said he couldn't strip in a corset, but he managed fine. He had laced his backwards, the ties all up the front, to make for easier removal. The boa went first\u2014he leaned over the edge of the stage and wrapped it around Calloway's neck like a gift, dropping a fleeting kiss on Cal's lips before retreating\u2014and then the laces, one hole at a time. Kris didn't pretend to look anywhere else but at him. His heart skipped a beat and lodged in his throat, trembling in anticipation as the corset dropped. The crowd screamed, and Rayne winked and hit the chorus. When he next stalked over to Kris, one hand coming up to trace the lines of Kris's corset, Kris dropped his gaze to his guitar and closed his eyes, dry-mouthed and desperate for the attention. Rayne was more skin than clothes, sleek and fit, his muscles smooth and lithe and barely showing. He seemed to glow under the stage lights, his tattoos coming to life as he moved, and Kris still couldn't touch him. Kris played until his fingers ached, but that hurt less than having all that in front of him, unable to reach out and feel it.\n\nHe couldn't even tell if Rayne was doing it on purpose. What Rayne knew, or suspected, or thought he knew about Kris's feelings, Kris had no idea\u2014other than that he was embarrassingly easy to turn on, but everyone knew that, now. That had been very publicly proven. It didn't mean Rayne guessed anything. He was just messing around, high on the adrenaline from the show, and since Calloway wasn't onstage\u2014\n\nKris wished they'd never agreed to Brian's suggestion of cutting the midshow make-outs. Rayne still lavished him with attention, but it was Cal he was kissing, not Kris, and it was killing him. He'd never been prone to jealousy before, but this was enough to tie him up in knots.\n\nThe song ended and Rayne returned to center stage to say their goodbyes, when a scuffle broke out beyond the barricade. Kris couldn't see what happened, but a girl shouted, and Rikki the cultist punched one of his fellows, and then all hell broke loose. Security swarmed over to tear them apart as Rayne turned back to the stage, frowning, and reached for the mike.\n\n\"There's no fighting at a Chokecherries' concert,\" he said sternly, his voice reverberating without the music to back him. \"We're here for peace and love, guys. Get your shit together.\"\n\n\"He tried to grab me!\" a girl yelled from the front.\n\nRayne's frown deepened. \"Okay, PSA time, kids. Now, I know my fans are better than that. You'd never grab anybody without permission. So listen\u2014you see somebody doing that? Touching, grabbing, trying to throw somebody into the pit who doesn't want to be there? You punch them in the face.\"\n\n\"Peace and love,\" Kris muttered from the side.\n\n\"Peace and love take a back seat when people are getting harassed. We're all here to have a good time, right? So look out for each other.\"\n\nSecurity finally succeeded in hauling the cultists off one another. Rikki glowered at his cohorts, who spat at him. They finally separated, security letting them off with a warning, and skulked away into the twilight. Kris saw Calloway disappear into the crowd, away from the site of the brawl.\n\n\"Look out for each other, be kind, and take no shit. We'll see you tomorrow afternoon. Go forth with love!\"\n\nHow he could deliver a halfway rousing speech dressed only in fishnets, boots, shorts, and a garter belt, Kris had no idea, but Rayne managed it. They traipsed backstage, trailing costume parts in a parade of glitter and sequins.\n\n\"I heard something going down, but I couldn't see from side stage,\" Angel said as she helped Kris unlace his corset. Rayne had disappeared to find Calloway right after the show, and Kris tried not to feel abandoned, missing his post-show dog pile of sweaty hugs and bright smiles. The rest of The Chokecherries were still there, but it wasn't the same without Rayne. Angel, probably sensing that, had stepped in to neatly fill the void. \"What happened?\"\n\n\"Cal's cult friends started shit, and Rayne yelled at them. Well, not yelled, but you heard him. Security dragged them out; I didn't see what happened after that.\"\n\nAngel dropped the corset and turned her back as Kris shimmied out of the shorts and pulled his real clothes on.\n\n\"Well, I'll be happy if we don't see them again for the rest of the festival,\" Angel said. \"And I doubt it'll do Cal any good to run into them, either.\"\n\n\"No, well, Rayne's keeping him pretty busy.\" He bit his tongue, aware that had come out more bitterly than he'd meant. \"I'm dressed; you can turn back around,\" he added as he put his guitar away and avoided meeting Angel's eyes.\n\n\"Come on,\" she finally said. \"Let's go get food and distract you from all that pining.\"\n\n\"I'm not pining for anybody,\" Kris retorted as they headed into the sea of tents again. \"I just have to get used to Rayne spending all his time with somebody else. It's no big deal.\"\n\nAngel mm-hmm'd skeptically, but as they turned the corner of a large but empty tent, they stumbled smack into the middle of a fistfight that stopped whatever Angel had been about to say.\n\nIt wasn't a fight so much as a beating: the three older cultists were surrounding Rikki, Boar and Red laying into him with boots and fists as he curled up on the ground, trying to protect his face and underbelly, while a third man stood back a pace, his arms folded as he looked on impassively. It was with him whom they collided when they turned the corner, and they all tripped back as the fight paused.\n\n\"Sorry,\" Kris said automatically, and then frowned. \"Hang on, what the fuck.\"\n\n\"Walk away,\" the man warned.\n\n\"Get off him,\" Angel countered. Rikki curled up tighter, like a hedgehog.\n\n\"Mind your business,\" Red growled, pulling himself up tall to loom toward her.\n\nKris bristled and tensed his hands into fists, knowing full well that Red could crush him like a bug but willing to throw a punch if he had to. But before he got the chance, Angel swung her handbag and hit Red full in the face. The buckles caught his cheek with just enough force to cut, and he staggered back with his hand pressed to his face in shock, like he'd never seen his own blood before.\n\n\"Get lost before I set security on you,\" she said, and hefted her bag up again when they didn't immediately leave.\n\nThey fled.\n\n\"Shit,\" Kris said. \"That was amazing. Do you carry a brick in there in case of emergencies?\"\n\nShe handed him her bag and he peeked inside. It was mostly makeup, and very hefty. Angel toed at Rikki, still on the ground, with her boot. \"Hey, pretzel-boy. You alive?\"\n\nHe uncurled inch by inch, but didn't try to get up. She knelt down and put her hand on his shoulder. Her touch was gentle, but he flinched away all the same. Kris held Angel's bag in both hands and shifted his weight from one foot to the other.\n\n\"You want me to call the paramedics?\" he asked.\n\n\"No,\" Rikki said, finally lifting his face. \"I'm okay. I can't afford the hospital.\"\n\n\"No broken bones?\" Angel asked. \"Internal bleeding?\"\n\n\"I don't think so.\"\n\nAngel kept her hand on him as he righted himself, pushing up on his elbows and checking himself for injuries as he went. His nose was bleeding, his lip split, and at least one eye was going to go black, but he was all in one piece, even if that piece looked the worse for wear.\n\n\"You got somewhere safe to stay?\" Angel asked. \"Somewhere away from them?\"\n\nRikki sat, pulling his knees to his chest. His eyes were shiny with unshed tears and he seemed impossibly young. He shook his head. \"We're family,\" he explained. \"We came to the festival together. They said they were going to trash my bike.\"\n\n\"How come they turned on you?\" Kris didn't trust the other cultists as far as he could throw them, but it was hard to look at Rikki and see any kind of threat.\n\n\"Boar took offense when I called him out for getting handsy with that girl,\" Rikki said, eyes downcast. \"Leif told him off, but he still said I was causing more trouble than I was worth today.\"\n\n\"Listen,\" Angel said. \"I don't want to leave you by yourself out here. You might have a concussion, or those guys might come back for another round. You sure we can't call the paramedics?\"\n\n\"I'm fine,\" Rikki repeated. \"I've been kicked around enough to know when to worry.\" He offered a wan smile, ruined by the blood in his teeth. \"I've done some kicking of my own before, anyway. Not like I don't deserve it.\"\n\nKris glanced at Angel. Her gaze was fixed on Rikki, who hunched his shoulders against her scrutiny.\n\n\"Right,\" she said decisively. \"Up you get. Come back to the bus. Let me keep an eye on you through the night to make sure your brain's not going to fall out.\"\n\n\"Can I do anything?\" Kris asked her in an undertone. \"Should I tell somebody?\"\n\nShe hesitated. \"Hang out with us for a minute, just to make sure there's no trouble.\"\n\nHe nodded and pulled his phone from his pocket, ready to call for help if things went south. Rikki let Angel help him to his feet, and he swayed there a second before steadying himself against her. He was taller than Kris by almost a head, but he was all gangly limbs and coltish proportions, no matter how the leather jacket tried to bulk him up. They set off at a careful pace and he went along meekly, letting Angel keep her hand on his arm. Kris suspected it was the first time in a long time he'd been touched gently by anyone.\n\nWhen they got to the buses Rikki shuffled obligingly onto the bus couch, and stayed quiet while Angel checked his pupils.\n\n\"You're probably right about being fine,\" she mused as she held his jaw, shining her phone's flashlight into one eye and then the other, \"but I ran into a beating or two back in the day, and I know exactly how much it sucks.\"\n\nKris fetched Rikki a water bottle from the minifridge and perched in the driver's seat while Angel finished her examination. As soon as she proclaimed him unlikely to keel over and die, Kris tugged her aside to whisper, low enough that Rikki couldn't hear, \"You can't just adopt a stray skinhead. Rikki might be okay, but he's still in a cult, and those guys are nuts.\"\n\n\"We couldn't leave him there to get beaten to death,\" she objected. \"And he seems all right. He gave me a pretzel, remember?\"\n\n\"Cult,\" Kris stressed.\n\nThey both glanced down the bus to Rikki, on the couch. Rikki smiled hopefully.\n\n\"You said you guys were a family,\" Angel said, at a normal volume. \"You mean related?\"\n\n\"Oh, no. I don't have a real family. A blood family, I mean. I don't think any of us do.\"\n\n\"What do you guys do? You said you have bikes.\"\n\n\"Yeah, we've got our motorcycles. We drive around, we do . . . well, I guess most of it's illegal.\" He rubbed his hand over his head sheepishly. \"Leif\u2014he's our leader\u2014he hustles people at poker. Carjacking too, stuff like that. Just to get enough money to stay afloat and keep the Avatar safe.\"\n\nKris and Angel exchanged looks before coming to a mutual agreement that they didn't want to dig into the avatar part yet.\n\n\"Have you ever been arrested?\" Kris asked.\n\n\"No.\" Rikki blinked up at them with earnest eyes. \"We're protected. We have the favor of the gods.\"\n\n\"Right,\" Angel said slowly. \"Of course you do.\"\n\n\"Cult,\" Kris mouthed to her.\n\n\"You know a guy called Calloway?\" she asked Rikki. \"Redhead, Irish? About your height?\"\n\n\"I know the name,\" he offered. \"The others talk about him sometimes, but I've never met him. I think he used to be in the order before I came.\"\n\nKris resolved to press Calloway for details as soon as he saw him again. \"What do you guys do besides the hustling and carjacking?\"\n\n\"We serve the Avatar. We spread His message and we worship Him as He deserves.\"\n\nKris could actually hear the capital letters, and he did not like the sound of that one bit.\n\n\"Okay,\" Angel said. \"What are we talking about, exactly? Should we be looking out for sacrificial altars? Black magic rituals?\"\n\n\"Oh, nothing like that,\" he said earnestly. \"No, it's all . . . love. Love, and beauty, and unraveling the secrets of the universe. Leif preaches it much better than I do. And we come to places like this to listen to the music, too. Leif knows almost every band here. You're with The Chokecherries, right?\"\n\n\"We are.\"\n\nRikki nodded. \"He's a fan. Anyway, I'm just grateful they took me in.\"\n\n\"Yeah, I can see that. Hang on one sec.\" She flashed him a smile before turning away to tap out a text to Rayne, which Kris read over her shoulder. Apparently Cal's old cult has a god. Also, they're violent. Steer clear of them.\n\nOn the couch, Rikki rested his head against the arm, a bruise already forming around his eye. He didn't look dangerous so much as lost, and if Kris knew anything about The Chokecherries, it was that they weren't good at leaving people to fend for themselves.\n\nRikki stayed on the bus to rest, promising not to wander off or get in trouble, while Kris and Angel returned to the stage. Evening was setting in, the sunset sweeping over the festival grounds in a great pink and gold wash like the desert was on fire, and people were lighting up their glow sticks like fireflies as the music and the dancing continued.\n\n\"You think Rayne and Calloway are meeting the press again tonight?\" Kris asked. \"They got off to a good start, so I guess they'll want to keep pushing it, right?\"\n\n\"Yeah, Rayne said he was going out with Cal again this evening,\" Angel said.\n\n\"Cool, cool. Get some pictures, give some interviews . . .\" Kris let the sentence hang. He didn't know where he was going with it, and it was too late to reel it back in now. \"What should we do about the cult?\" he asked instead.\n\n\"We'll tell security to keep an eye out, and try not to run into them again,\" Angel said firmly. \"Come on. Let's go find you a distraction.\"\n\n\"From the cult, or from Calloway?\"\n\nShe grabbed his hand and dragged him backstage with a wink. \"Both.\"\n\nAngel set her purse on the table where she normally did the band's makeup, digging through its many hidden pockets before triumphantly pulling out a packet of semitranslucent, crystalline drugs. Kris had never done anything besides weed or booze, not because he was leaning toward straight edge, but because back home those were the safest options. Everything else was liable to be cut with so much crap that it would as soon kill you as get you high, and Kris had never been curious enough to take the risk.\n\nAngel guaranteed that she had good drugs.\n\n\"I get my weed from a friend in LA who runs a dispensary, and I get everything else from another friend in Louisiana,\" she explained, holding the packet up between her thumb and her forefinger. \"I don't fuck with the hard stuff, but as far as recreational drugs? I know them, I've got them, and I only carry the best.\"\n\nFootsteps sounded from the stage, and she and Kris looked up as Rayne came around the corner.\n\n\"I was just introducing Kris to my stash,\" Angel said by way of greeting. \"You joining us?\"\n\n\"Yes, please.\" Rayne ambled over to sling his arm around Kris's shoulders, leaning into him in a brief hug before straightening again. \"What have we got lined up?\"\n\n\"I know it's not weed, but that's as far as I can guess,\" Kris admitted. \"It's not cocaine, is it? Because I'm not sure I want that.\"\n\n\"Nope, no coke,\" Angel said. \"That's a hard drug and I'm not about it. This is pure MDMA.\" The little plastic bag was full of pearly shards of crystal that sparkled in the light. \"Or as pure as I've ever found, anyway. You want to try?\"\n\nKris wet his lips. \"Should I be worried?\"\n\n\"If you are, you won't be for long. But no pressure, hun.\"\n\nShe eased the crystals from their bag onto the table, and Rayne took first choice, licking his finger and picking up a single shard on the tip. It glittered there for a second like fairy dust before he wrapped it inside a bit of tissue, put it in its mouth, and let it disappear. Angel followed suit.\n\n\"Can you have a bad trip on MDMA?\" Kris asked, unable to tear his gaze from the crystals. \"Like, if it's my first time, will anything like that happen?\"\n\nRayne and Angel glanced at each other.\n\n\"In my experience?\" Angel said. \"You should be pretty chill. Worst-case, you might get panicky wondering how hard it's going to hit, but if you only take a little, you should be coasting on good vibes all night long.\"\n\n\"If you start feeling weird, find security or one of the paramedics,\" Rayne advised. \"They'll take care of you. Or we will.\" He ruffled Kris's hair. \"Whatever you need, we've got you.\"\n\n\"You want, or should I pack it up?\" Angel asked.\n\n\"Hell with it,\" Kris said. \"No, I want to try.\" He dabbed a shard on his fingertip and stared at it for a second. \"You guys will watch out for me, right?\"\n\n\"Course we will,\" Rayne promised. \"Here, wrap it up like this. You don't want to taste that shit.\"\n\n\"Drink lots of water and don't get overheated,\" Angel said. \"It should hit you in about half an hour.\"\n\nKris popped his finger in his mouth and swallowed. \"Half an hour.\"\n\nRayne caught his hand and tugged him toward the door. \"Come on, baby. Let's dance.\"\n\n\"Wait. You and Calloway have to\u2014\"\n\n\"He's meeting us in a bit. I'm not waiting by myself until he shows up. Come dance with me.\"\n\nThey wound up in a tent on the far side of the festival grounds, where electro music blared and neon lights that hung in the air like jellyfish lit the place up as if it were underwater. Kris danced with Rayne at his back, rubbing up against him in teasing brushes, simultaneously reveling in it and shot through with tension, waiting for Calloway to arrive and put a stop to things. Angel danced with anyone who looked at her, her smile flashing white in the darkness.\n\nThe drug hit him at the half-hour mark like it had been on a countdown. Between one breath and the next, the entire universe rushed into him. He blinked, and when he opened his eyes, the neon lights were brighter, the music sweeter, his body cleaner from the inside out, like every bad thought had been scrubbed away. His heart leaped in his chest from the sheer wonder of it all, and he turned to Rayne for confirmation that it was real. Rayne put his hands on either side of Kris's face and drew him in, and the touch was electrifying: it ran through Kris's whole body, top to bottom, and he shivered with the pleasure of skin against his skin. He couldn't understand why he'd waited so long before saying anything, or why he'd thought burying his feelings was a good idea. He had love in him, and he needed to let it out, into the world where it belonged.\n\n\"This is perfect,\" Kris said, and it was.\n\nHe felt sparkling, like he had never done a bad thing in his life and nothing bad could ever happen again\u2014he was nothing but love, and love surrounded all of them\u2014it was the only thing that mattered in the world. When he looked in Rayne's eyes, he knew Rayne understood it too. He put his hands on Rayne's waist and petted him there, just to feel the fabric under his hands.\n\n\"Dance,\" Rayne told him.\n\nKris turned and danced. He danced like nothing else mattered, and maybe nothing else did\u2014only his sweat and his heartbeat and the feeling of Rayne pressed against his back, his hands drifting over Kris's body, closer than they got onstage. The lights glowed and made patterns in the air; he could see the shape of the music as it pounded from the speakers in a rhythmic thump thump thump that demanded his body move to meet it. He danced until the music filled him up and he breathed in colors and his head flooded with elation, and he had to stagger back from Rayne just to breathe.\n\n\"You okay?\" Rayne asked, following after him.\n\n\"I'm perfect,\" Kris said. \"I'm perfect\u2014I never knew I could feel like this.\"\n\n\"Like what?\"\n\nKris didn't know how to describe how he felt made of love\u2014the same particles that made up the whole universe, but not the universe he had lived in up till now: a different one, a purer one, like it had been at the time of the Big Bang, or even earlier, before anything else got in the way.\n\n\"I love you,\" Kris said instead. It tumbled out of his mouth like he couldn't contain it a second longer, alive and spoken in the same instant it occurred to him. He loved Rayne, completely and absolutely, and now he'd finally said it aloud. He kept talking, too engrossed in cataloging every feeling to panic. \"I know you think this is the drugs talking, but I need you to know\u2014I love you, more than I've ever loved anything. You're everything to me. It's not a crush, or a\u2014 I love you.\"\n\nRayne touched Kris's hair, his cheeks, his mouth. \"I love you too. You're beautiful.\"\n\nKris felt beautiful, and if Rayne said it, it must be true. Rayne breathed beauty; he bled and sang and exuded it with every minute he spent on Earth. He was beauty incarnate. Kris reached out with one trembling hand to rest it on Rayne's shoulder, where the wild roses twined under his shirt. Though Kris couldn't feel them through the fabric, he imagined them pulsing with a life of their own. He imagined laying Rayne down and stripping him bare and tracing every one of his tattoos line by line, until he had them ingrained in his mind's eye, never to fade. Opening his mouth, his thoughts raced a thousand miles ahead of his body, intending to say these fantasies aloud, safe in the dark glowing crush of the crowd. Rayne looked at him with soft wondrous eyes, his lips parted as if waiting with baited breath for Kris to speak.\n\n\"Rayne . . .\"\n\nThey both moved in at once, as if in slow motion. Kris's heart thudded in his chest like it was underwater, his blood rushing through his veins and tingling in his lips in anticipation of the\u2014\n\n\"Sorry, am I interrupting?\"\n\nKris froze as his heart skipped faster and his blood spread through his face in a wave of heat. He turned to find Calloway standing a few feet off, watching them with a smile, even as a faint line furrowed his brow.\n\n\"Cal.\" Rayne smiled and held out his hand. \"You made it.\"\n\n\"You sure I shouldn't come back later?\"\n\nKris wished Cal would. He had forgotten, in that rush of love and understanding, that Calloway was coming, that that had been the plan all along. He was so close to telling Rayne everything\u2014they were on the same level, he and Rayne, bonded by the music and the neon lights. He'd been going to tell Rayne with more than words\u2014they had been so close!\u2014but he'd needed to make sure Rayne knew what he was saying first. But now, with Calloway here, his certainty stuttered. He didn't know what to do.\n\n\"No, stay,\" Rayne said, taking Calloway by both hands and drawing him closer. \"We were waiting for you.\"\n\n\"You're both very high.\"\n\n\"You're not?\"\n\nCalloway shook his head. \"I stopped all that when I quit the cult. I don't even drink anymore.\"\n\n\"You can dance, though.\"\n\n\"Yeah, I can dance. Rayne, listen.\" He glanced at Kris, but when Kris didn't say anything, he continued. \"I know where a few guys from the press are hanging out. If we go over there now, by tomorrow everyone will be talking.\"\n\n\"Okay,\" Rayne said, tracing the lines of one of Calloway's tattoos up his arm until it disappeared under his sleeve. Kris squirmed, longing to do the same to Rayne, somewhere quieter and more private.\n\n\"Rayne.\" Calloway was laughing, his touch gentle as he caught Rayne's hand. \"Are you listening?\"\n\n\"Yeah, I am. We're going to find the press. Kris? Will you be okay staying with Angel?\"\n\nKris had lost Angel somewhere in the crowd. He pictured her dancing with a hundred strangers under those incandescent jelly lights, swimming through the sea of bodies like a minnow dancing under the waves.\n\n\"Kris,\" Rayne repeated.\n\nHe and Calloway both looked concerned, clearly waiting for Kris's response. Kris could see their auras, shimmering around their silhouettes like heat waves. Calloway was attracted to Rayne, he realized, even beyond the basic friendly attraction that had prompted him to agree to the publicity stunt in the first place. They were still holding hands, Cal's thumb brushing Rayne's almost unconsciously. Kris watched as tiny sparks of light like fireflies jumped from the point of contact.\n\nHe had missed his chance. No\u2014he had given his chance away when he'd told Rayne to agree to the whole scheme. It was supposed to have made things easier, but it had made everything impossibly worse. His thoughts whirled uselessly. He was too high to solve this problem, too high to . . .\n\nHe made himself smile and relax and meet their questioning gaze. \"I'm good. I'm okay. You guys go . . . go do your thing.\"\n\nRayne glanced at Calloway before letting go of his hands and stepping in close to Kris again. Kris leaned in instinctively to meet him, their bodies hot and too complicated for him to consider with his mind like this. He touched Rayne's arm, trailing up to his shoulder and his neck. The wild roses again. He leaned in\u2014\n\nCalloway gave a polite cough from over Rayne's shoulder.\n\n\"Kris?\" Rayne said, softly, like he didn't understand at all.\n\n\"I need water,\" Kris said abruptly. Rayne and Calloway needed to convince the world they were dating, and Kris had no place in that. Cal's aura shimmered like gossamer, all wound up in Rayne's. Kris pulled his hand back, missing the touch before it was even gone, and retreated a step, forcing distance between them until his head cleared. \"I'm going to walk outside,\" he said. \"I want to see . . . everything.\"\n\nAngel reemerged from the crowd, flushed with sweat from dancing.\n\n\"Will you be okay by yourself?\" Rayne asked. \"It's still your first time on molly.\"\n\n\"I'm good,\" Kris said firmly. \"I'll meet you at the bus later, okay?\"\n\n\"You sure?\"\n\n\"I'm sure. Keep dancing. Go find your paparazzi.\"\n\nRayne smiled, his hesitation melting away like it had never been, and Angel pressed a bottle of water into Kris's hands. \"Remember, if you feel weird, just find festival security,\" she told him. \"They'll get you back to us.\"\n\n\"I'll be okay,\" he promised. Angel kissed his cheek, and when he touched it after he found that her lipstick had left a little mark on his skin.\n\nCalloway took Rayne by the shoulders, and leaned in close to whisper something in his ear, and Rayne smiled and laughed, low and dark and full of promise. Kris slipped from the tent before he had to see them kiss, leaving behind the jellyfish swimming through the air above the dancers' heads, and into the night air. It was crisp and fresh and it filled his lungs with the sweetest taste he could imagine, the kind he hadn't known since childhood\u2014like bonfires, with pinprick stars in the void, fireflies, and frogs chirping, forever out of sight. It was a relief to replace Rayne, and all the feelings Rayne evoked\u2014love like he couldn't even articulate\u2014with the air and the stars and the insects, if only for a moment.\n\nHe walked in no particular direction at all. Half the time he spent staring up at the sky, watching the constellations shimmer in and out of existence, and the other half winding around tents and dancers and makeshift campsites under the open sky. He felt connected to every single person he passed. They were all children, all finding themselves on the same planet, all spinning through space and time and life together. They were all love. He hummed to himself as he walked, skipping between Rayne's lyrics and songs he remembered his mother singing to him when he was a child. He basked in the beauty of the world and the fate that had led him to this particular moment, this moment out of a million, billion possibilities, and then he saw the peacock.\n\nHe had wandered to the edge of the festival grounds where tents were scarce and people were scarcer, but it was lit up with pot lights that bathed the desert in a golden glow. Looking around, he cast glances over his shoulder, but there was no one nearby to confirm that the bird was real.\n\nThe bird looked at Kris with bright eyes and bobbed his head. He was the most strikingly regal creature Kris had ever seen, but what a peacock was doing in the middle of the Mojave Desert, Kris couldn't guess. The peacock didn't seem lost, in as far as any peacock could seem lost or not, but he obviously didn't belong. Unfortunately, Kris and the peacock were the only living beings in sight.\n\n\"Hey there,\" he said.\n\nThe peacock strutted closer, his head cocked as he regarded Kris with frank curiosity.\n\n\"I'm Kris. What are you doing out here?\"\n\nHe didn't answer, for which Kris was grateful. Instead, as if pleased to have found an appreciative audience, the peacock spread his tail feathers in a magnificent fan, and Kris's knees buckled in the face of such overwhelming splendor, and he sank to the ground to stare.\n\nHe needed to find Rayne.\n\nThe thought came to him unbidden but, once in his head, was undeniable. Kris didn't want to experience this alone, and Rayne was the only person Kris could imagine appreciating it as he did. Rayne and the peacock were kindred spirits, after all, bound by the mandala on the back of Rayne's head. Kris dug his phone from his pocket and texted him, his fingers skittering over the letters.\n\nYou have to see this, he wrote. Come find me by the lights.\n\nThe peacock strutted closer, apparently annoyed by Kris's lapse in attention, but preened, appeased, when Kris raised his phone to snap a picture. It came out blurry and unidentifiable, but he sent it anyway.\n\nRayne texted back a series of question marks.\n\n\"Okay,\" Kris said, staring at his phone screen. \"This isn't going to work.\" He pushed himself to his feet, swaying unsteadily as he found his balance, and held out his hand to the peacock. \"I hope you're a cooperative kind of bird, because it's really important that Rayne meets you.\"\n\nThe peacock rustled his fan, blinked, and seemed to shrug. Kris took that as consent and proceeded with his plan. The peacock was amicable enough about letting him approach, and though Kris was nervous about putting his hand in range of the beak, the bird didn't startle or snap or appear remotely concerned about the strange, drug-addled human trying to touch him. In fact, he seemed perfectly at ease with the idea.\n\n\"Good boy,\" Kris said, like the peacock was a strange, colorful dog instead of a misplaced exotic bird. The bird's feathers were soft, and so bright they seemed to glow when he touched them. Kris shuffled nearer, and the peacock obligingly folded his tail down, and allowed Kris to pick him up and tuck him under his arm, his tail trailing to the ground behind him as he settled in against Kris's side.\n\nWarmth unfurled in Kris's chest at his success. Rayne was going to love it.\n\nKris didn't remember passing out, but the next thing he knew, he was waking up again. It happened slowly. He drifted out of his dreams bit by bit, until his brain was awake but his body wasn't, and another indistinguishable amount of time passed before he could convince his eyes to open. He would have been happy to float in the ether awhile longer, but Rayne approached his bunk\u2014Kris recognized his footsteps even half-asleep\u2014paused, and said, \"What the hell is that.\"\n\nKris blinked. His mouth was so dry his tongue was glued to his palette, and he had to unstick it an inch at a time before he could talk. It felt like he'd been clenching his jaw in his sleep. The trade-off for a night of life-altering revelations, he supposed.\n\n\"What?\" he croaked. He rolled over\u2014a wave of dizziness followed him, with nausea on its tail\u2014and found himself face to face with a very large, very glossy blue bird. \"Oh.\"\n\n\"Kris, why do you have a peacock in your bunk?\"\n\nThe bird cocked his head at Rayne and bobbed closer; Kris put his arm around the peacock to keep him from escaping.\n\n\"I have an explanation,\" Kris said, \"but it involves a lot of drugs.\"\n\n\"You had one hit of MDMA, the same as the rest of us, and no one else appropriated a wild bird.\"\n\n\"I don't think it's actually wild,\" Kris said, fishing around his bunk for his water bottle. The bird nestled in under his arm and regarded Rayne with an appraising eye. \"It seems pretty friendly.\" He took a swig, wincing as the water burned its way down his parched throat.\n\n\"Should I ask where you got it, or should I keep my plausible deniability?\"\n\n\"I didn't steal it,\" Kris protested. He sat up, fighting how his head spun at the movement, and ducked out of the bunk to join Rayne on the bus floor. The bird perched on the edge of the mattress and ruffled his wings. \"He was just wandering the grounds, all by himself. I wanted to show you.\"\n\n\"That's what you were texting me about?\"\n\n\"It was too dark to take a good picture.\"\n\n\"So you brought it back to the bus, but then you fell asleep before you could show me,\" Rayne guessed. \"Did you use it as a pillow all night?\"\n\n\"He didn't mind,\" Kris said awkwardly. The peacock screamed pleasantly. \"Anyway, it's a present. I said I'd get you a peacock, and I did.\"\n\nRayne stared at the bird. The bird stared back.\n\n\"I was having some kind of revelation last night about life, the universe, and everything, and the peacock seemed really important at the time. You have to admit it's perfect for your look. Like, it's not an elephant, but it's pretty good.\"\n\n\"It is,\" Rayne agreed, his reluctance plainly slipping fast as the bird stared him down.\n\n\"And I couldn't just leave him wandering the desert. Peacocks aren't supposed to be all the way out here. The coyotes would get him.\"\n\n\"They would.\"\n\n\"So?\"\n\nThe bird fluttered down to the bus floor, measured how much room he had, and spread his fan, shaking it in Rayne's face until he made the appropriate appreciative noises. Then, despite the extremely tight fit, the peacock began strutting up and down the length of the bus, screaming, until everyone else poked their bleary faces from their bunks to see what the fuck all the commotion was.\n\n\"Meet our new mascot,\" Rayne said helplessly.\n\nThe peacock leaped up to land on Rayne's shoulder, where he surveyed his new kingdom with a proud and godly air, his tail feathers trailing down Rayne's back like a cape.\n\n\"You do look good,\" Stef said, before retreating back behind their curtain. \"We should bring it onstage next time. Now for fuck's sake, let me go back to sleep.\"\n\nThe bird pecked at a strand of Rayne's hair and seemed perfectly content to stay on his shoulder.\n\n\"Okay then,\" Rayne said. \"I guess we're keeping it.\"\n\nKris wasn't infused with light and love and an understanding of the universe anymore, but he still got a buzz of satisfaction from knowing his instinct to share the bird with Rayne had been the right one.\n\nKris found Angel sitting outside with Rikki, both drinking coffee from Styrofoam cups. They sat cross-legged facing each other, Rikki gazing at Angel with unabashed attention, which Angel seemed to accept as her due.\n\n\"Morning,\" Kris said. His dizziness was mostly gone, though his stomach was still churning uneasily. \"You two do anything fun last night?\"\n\n\"We got high; we talked. We slept outside on the ground, for some reason.\" Angel grimaced. \"My back is killing me.\"\n\n\"You said you needed to connect to nature,\" Rikki provided. \"The stars were all sparking in the sky; we watched them for hours. It seemed important to you, so I didn't try to talk you out of it.\"\n\n\"Yeah, no, that was a bad idea.\" She turned to glance up at Kris. \"He told me more about his order's spirituality, their god and all that. That was something.\"\n\nKris's eyebrows went up. \"Oh yeah? Is he like you guys, that whole shaved-head, biker-gang style?\"\n\n\"Oh, no. He doesn't look anything like us,\" Rikki said. \"He's the most beautiful being in all creation. He embodies it. We look like this so He won't think we're trying to compete.\"\n\n\"He doesn't sound very compassionate,\" Kris said. \"Maybe you should upgrade.\"\n\n\"Reject the patriarchy,\" Angel said, raising her fist while bringing her coffee up to her mouth again with her other hand.\n\nRikki blinked. \"I wouldn't know how to find another god. Or a goddess. Or . . . any other kind. I wouldn't know where to start.\"\n\n\"You think they're going to let you back in the cult?\" Kris asked. \"I mean the gang. Um. The family?\"\n\n\"I don't know,\" Rikki confessed. \"I don't know what to do if they won't take me back.\"\n\nAngel nudged him with her boot. \"You can stay with us till you figure it out. We'll watch out for you.\"\n\n\"I don't want to be abandoned by my god.\"\n\n\"I'll find you a new one,\" she promised. \"Don't worry, hun. You'll find the family you deserve.\"\n\nRikki seemed doubtful, but he asked, \"Can I meet yours? It's okay to say no,\" he added quickly. \"Most people would rather not meet me, so I get it.\" He bit his lip with a hopeful expression, but not much of one.\n\nAngel looked at Kris. Kris looked at Rikki. He didn't have any opposition to introducing the kid to the others, though he wasn't sure what Calloway's reaction would be. Cal seemed confident in having left the cult behind, but Kris didn't want to put that to the test with the publicity stunt at stake. Though Rayne could walk away from the stunt with his career intact, the press would still have a field day trying to connect him to a cult, and Kris couldn't imagine Brian being too pleased about that.\n\n\"We'll avoid Cal for now,\" Angel said, evidently following Kris's line of thought without having to ask. \"And try not to mention the whole god thing just yet, okay?\"\n\nRikki nodded, entirely eager to please.\n\n\"All right, come on, then.\" Angel stood up in increments, stretching her back as she rose.\n\n\"Do you know where Rayne and Cal ended up last night?\" Kris asked, aiming for casual and missing it by a mile.\n\nAngel regarded him knowingly. \"I wasn't with them, but I've heard some things.\"\n\nKris swallowed. \"Is that good or bad?\"\n\n\"Well, it's good for Calloway.\" She took Rikki by the arm. \"Let's go find breakfast. Kris? I'll see you around?\"\n\n\"For sure,\" Kris said faintly. Angel waved goodbye over her shoulder and left him there, wondering what exactly \"good for Calloway\" meant, and if that meant it was bad for Kris. He needed more information before going to see Rayne again, and if he needed information about paparazzi rumors, he needed to find Cassie.\n\nShe was sitting behind the drum kit on their otherwise empty stage, tapping out a beat as she nodded along to the rhythm. Kris watched her for a minute, wondering what might have happened if they'd stuck together in their old high school band after all, before he stepped up to get her attention.\n\nShe set her sticks down and grinned from ear to ear, which was a strong indicator that something had indeed happened last night, and she knew exactly what it was.\n\n\"Guess what!\"\n\nHe really didn't want to. \"What?\"\n\n\"Calloway propositioned Rayne last night. It was filthy. If I didn't know better, I'd think they were the real thing.\"\n\nKris's throat went dry. \"What did Rayne say?\"\n\n\"Why? You jealous?\"\n\n\"Cassie.\"\n\nShe rolled her eyes. \"They didn't start screwing in the middle of the festival, obviously. Here, check this out. The pics from last night are already online. The label must be ecstatic\u2014look at them! Don't they look good together?\"\n\nShe handed him her phone, the screen lit up with tiny pictures. They were too small for Kris to see many details, but didn't leave much to the imagination. Either Rayne and Cal were that dedicated to their performance, or there was something more going on between them. Kris bit his lip and scrolled down.\n\n\"This is a video. That's, uh. Daring.\"\n\n\"It's not a sex tape, genius, it's an interview.\"\n\nSuccumbing to curiosity against his better judgment, Kris pressed Play. Rayne and Calloway swam into focus, Rayne's arm slung casually over Cal's shoulders, with Cal's hand at his waist. They appeared intimately at ease together, and though Kris knew Rayne had been high, he didn't look it.\n\n\"So are you two serious?\" the interviewer asked.\n\n\"It's too soon to say,\" Rayne said easily. \"Maybe it'll turn into something and maybe it won't; we're just enjoying the chance to hang out at the festival.\"\n\nCal nodded along. He obviously wasn't as practiced as Rayne at acting for the press, but Kris was sure he'd get there. Rayne was the best teacher he could hope for, after all.\n\n\"So where does your burgeoning romance leave your guitarist, Kris Golding? You're familiar with the rumors that you two were an item on the down-low, of course?\"\n\n\"He's not into guys,\" Rayne said immediately, and Kris winced. \"We always made it very clear to everyone\u2014press and fans alike\u2014that what we did was just a performance, and I stand by that.\"\n\n\"Kris is a solid guy,\" Cal added, leaning into Rayne's side and entangling their fingers. \"He's a good friend and a great bandmate, but there's really nothing else there. I'm not a home-wrecker.\"\n\nThe interviewer laughed obligingly. \"Okay, so Golding isn't interested. What about you, Rayne? You never had any feelings on your end? I hate to push this,\" she added, \"but my viewers would be out for blood if I let it slide.\"\n\nKris held his breath as he waited for Rayne's reply.\n\n\"No, none of that, I'm afraid,\" Rayne said, and he winced apologetically at the camera as if in sympathy for his disappointed fans. \"I don't get involved with straight boys anymore, no matter how enthusiastic they are. I learned that lesson ages back, and it's not an experience I'm keen to repeat.\"\n\n\"Fair enough. Are your shows going to change, if you and Calloway do get serious?\"\n\nRayne and Cal shared a glance. They were sitting so close they were practically in each other's laps, having apparently decided subtlety was the enemy.\n\n\"We'll have to discuss that when it comes up,\" Rayne finally said. \"For now, Kris and I are toning down some of the more risqu\u00e9 performances, just while Cal and I get our bearings.\"\n\n\"The fans are very invested in your perceived relationship with Golding,\" the interviewer pressed.\n\nCal intervened before Rayne had to.\n\n\"They put on a hell of a show and I'm not trying to mess with that,\" he said. \"I'm sure we can work something out to everyone's satisfaction. The fans don't have to worry about a thing.\" He smiled broadly and pressed a kiss to Rayne's jaw, lingering there until Rayne laughed and obligingly turned to meet his lips.\n\nThe video clip cut out, and Kris stared at the end screen for a second before handing the phone back to Cassie.\n\n\"Cool,\" he croaked. \"It looks like it's going well. I'm happy for them.\"\n\nShe narrowed her eyes. \"No, you're not. You're being weird.\"\n\n\"No, really. If they want to get together for real, that's . . . awesome. I'm just going to go . . . elsewhere. For a minute. And, uh.\"\n\nCassie watched him critically. \"You want to tell me what's going on, or am I supposed to guess? You don't want me to guess.\"\n\n\"You don't have to guess anything. There's nothing going on.\"\n\nCassie's expression intensified.\n\n\"Shut up,\" Kris said. \"It's true.\"\n\nHe nodded definitively, turned on his heel, and left the stage. He heard Cassie pick up her sticks again, a snappy little beat following him as he went. He hadn't lied; he would absolutely congratulate Rayne and Cal if they decided to give it a shot. But he really fucking hoped they didn't, because Kris wasn't just going to tell Rayne he was bi: he was going to tell him he was in love with him too, sober and for real, and he wasn't going to risk waiting till after they left Purple Sage to do it.\n\nLeif rarely raised his voice or lost his temper, but when neither he nor Red nor Boar could locate the peacock when they got up that morning, he came close. He shut his eyes and forced his temper down, his hands in fists at his sides.\n\n\"We'll find Him,\" he said, his voice admirably calm. \"And find Rikki.\" The boy had been causing trouble the day before; he might have retaliated by stealing the Avatar away. Leif concentrated on his breathing meditation, and gradually his temper cooled to something more controllable. Maybe Leif had been too harsh on the boy\u2014letting Red and Boar give him that beating had been one thing, but threatening to strip his bike down and sell it for scrap metal might have pushed Rikki over the edge. \"Fan out; search the grounds. Find them both.\"\n\nHe set off through the heart of the festival toward the picnic area. Eventually, the boy would need to eat. And when he did, Leif would be waiting.\n\n\". . . the last time you ate a real meal?\" a familiar voice said.\n\nLeif strained his ears, recognizing it but not remembering from where until the speaker stepped into the clearing where the tables were set up. It was the young black woman he had encountered twice already\u2014she'd hit Boar in the face with that pretzel, and hit Red later with her handbag when she interrupted their fight. And at her side was Rikki, the insolent little whelp\u2014Leif was right that the boy had defected. The Avatar, however, was nowhere to be seen.\n\n\"You're skin and bones,\" the woman continued.\n\n\"I've always been like that,\" Rikki replied.\n\nHe was all limbs, like a scarecrow that had learned to walk. His cuts were healing up though, even if he was still too pale and the skin around his eyes too dark to look healthy. But he would live; Red and Boar hadn't done him any lasting damage. Leif was sure they would have if he'd let them; their inherent violence was getting increasingly difficult for him to rein in. He couldn't say what would happen when he finally lost control.\n\n\"We'll get you some food, then you can meet the bands. Though we should wait before introducing you to Calloway, just in case. Kris is right; that sounds like a mess waiting to happen.\"\n\nLeif blinked at the mention of Calloway. It was too great a coincidence to be anyone but his former order member, the one who had abandoned them not so long ago. He waited until they were near enough before he stepped out from his hiding place to intercept them.\n\n\"You,\" he glowered, \"have stolen the Avatar.\"\n\nRikki stopped dead in his tracks, one arm flung in front of his companion like he could protect her. Leif stiffened and puffed himself up, though he kept one eye on the woman in case she decided to launch another offensive.\n\n\"Who stole it?\" Rikki demanded.\n\nLeif paused. \"You didn't?\"\n\n\"No! I never touched it.\" Rikki folded his arms over his chest and glared. Leif, broader if not taller, glared back more formidably.\n\n\"Rikki was with me all night,\" the woman cut in. \"He hasn't been anywhere near your avatar, whatever it is.\"\n\nLeif studied her long and hard, and she bore his scrutiny with cool eyes, her chin held high. Finally, he determined she was telling the truth, and turned back to Rikki. \"If it wasn't you, then it was someone else, and you're going to help us find it.\"\n\nRikki looked at the woman, visibly hesitating. Leif ground his teeth together, his patience fraying. This was the longest the peacock had been away from him since His arrival, and His absence grated at him, worse than road rash. Did his god's protection extend that far? Would his luck finally run dry without the bird? He didn't want to test it.\n\n\"Of course I'll help you,\" Rikki finally said, when the woman did nothing to intervene. \"We're family, right?\"\n\n\"The Avatar is more important than that,\" Leif replied. Rikki wouldn't be welcomed back into the fold, but he could serve his purpose before they cut him loose. \"Come. We should talk.\"\n\nRikki nodded and glanced back at the woman one last time.\n\n\"You know where to find me,\" she said.\n\nLeif took Rikki by the arm and dragged him away. The boy kept looking back over his shoulder as they walked, his expression lost, but unafraid. The last thing Leif heard before exiting the picnic grounds was the woman uttering a heartfelt \"Fuck.\"\n\nKris needed to work up the courage to talk to Rayne. He was sure that once his feelings were out in the open, even if they were unrequited, he would feel better than he did bottling them up and pretending there was nothing there. Once Rayne had all the facts, the ball would be in his court, and he could do what he wanted. He was pretending to date Calloway\u2014so what? Unless they made it real, it didn't have to get in the way. The publicity stunt had always been a weak excuse to keep Kris's feelings from evolving.\n\nBut then, Cal and Rayne were getting along well, and Kris couldn't have imagined Cal's attraction to Rayne the night before. The only difference between Kris and Cal was that Calloway was openly gay, and it was his job to make people believe he and Rayne were an item. That gave Cal the upper hand, but Kris wasn't going to give up. Rayne could choose Kris, or Cal, or neither, but the weight would be off Kris's chest. He would reassure Brian that what had happened with Fink would never happen with Kris, because Kris wasn't straight, and more than that, he was in love.\n\nNo more secrets. He was going to let it all out.\n\nBut first, he got drunk.\n\nAnd then he got drunker.\n\nIt was all Jay's fault. After the first few days of touring, the bands hadn't usually imbibed too heavily. They'd partied hard for a short time, and then the novelty had worn off and the reality of surviving bus life with a hangover sunk in, and they'd stuck to mostly social drinking. But once in a while, someone wanted to turn up, and insisted on dragging everyone else along for the ride.\n\nWhen their set ended that afternoon, Kris staggered backstage and Jay greeted him by pushing a bottle of whiskey into his hands.\n\n\"Tonight we drink!\" Jay crowed. A ragtag cheer, more confused than enthusiastic, went up around him. The peacock, which Rayne had christened Freddie Mercury, shrieked. Brian had read them the riot act about what would happen if the bird got hurt onstage, or hurt one of them, or turned on the audience, as if he were a stampeding elephant after all, and not just an overgrown and unusually vain bird. However, it turned out to be too much work trying to keep him offstage and safe. Like his namesake, Freddie had a thirst for fame, and escaped his handlers to insinuate himself in the center of attention at every opportunity, to Brian's despair. Of course Rayne also enjoyed the extra attention, and Kris was pleased that his gift\u2014however drug-induced\u2014had gone over so well.\n\nFreddie seemed equally pleased with his newfound stardom.\n\nOnly Calloway was uneasy about it, asking repeatedly where Freddie had come from and appearing worried when Kris told him that the bird had been wandering the festival grounds, then downright shaken when Kris offered up his drug-addled spiritual experience on finding him.\n\n\"Maybe you shouldn't share that story too widely,\" Calloway suggested. \"People might come looking for him.\"\n\n\"What, like animal-control guys? Do you need a license to tour with a bird like this?\"\n\n\"Just be careful, that's all I'm saying. Maybe he escaped from a zoo or somebody's private collection or something. It might cause trouble for you.\"\n\n\"I'm sure Brian's taking care of any legal stuff.\" But Calloway's concern was palpable. They might be in competition for Rayne's attention, but it was hard to dislike the guy, especially when he was so clearly distressed. Cruelty didn't come naturally to Kris in any case, so he smiled and gave Calloway a reassuring thump on the arm. \"Don't worry about it. It's not like we adopted a stray tiger to take onstage.\"\n\nKris got drunker than he had since their going-away party, maybe more so. He wound up on the bus couch with his head in Angel's lap and his feet in Jay's, watching Rayne tell a story about his first show with Passionfruit, and trying not to look at Cass and Stef sitting on top of each other in a near acrobatic feat of intimacy that he really didn't need to see, like, ever. Calloway stood near the bunks, leaning against them as Rayne talked. Kris's attention kept drifting from Rayne to Cal and back again, the alcohol pushing him to search for clues. Were they together for real? Should he ask? No, Rayne would have told him.\n\n\"And then the drum kit caught fire,\" Rayne said with a flourish, nearly spilling his drink.\n\n\"Knocks, you have the worst luck,\" Cassie said.\n\n\"My luck's just fine,\" Knocks said, tapping his rocker boot and pointing to Jay. \"It's this little asshole who keeps fucking me over.\" Knocks was more or less ambulatory since getting back from the hospital, but unable to play drums in the cast. So while Cass stayed behind the kit, he'd taken up rhythm guitar rather than sit out entirely.\n\n\"How was I supposed to know it was flammable?\" Jay asked. \"It was a total accident, and I already apologized about your foot today. Anyway, you're having a great time on guitar.\"\n\nKnocks glared in a way that said he knew Jay was right, but he refused to give him the satisfaction of agreeing.\n\n\"Anyway,\" Cass said loudly. \"We're cool. Right?\"\n\n\"We're fine,\" Billie said. \"Jay's a live wire; we all know to give him a wide berth onstage.\"\n\n\"As if,\" Cassie snorted. \"You guys can't keep your hands off each other for five seconds.\"\n\n\"Stealing all our ideas,\" Rayne said mournfully. \"You used to be so much more innovative, Billie. What happened?\"\n\n\"We're not\u2014 Jay, stop it,\" Billie said, as Jay dropped Kris's feet to the cushions and rose, advancing on Billie. \"It's just for the show!\"\n\n\"Giving me a wide berth,\" Jay said. \"I'll show you a wide berth\u2014\"\n\n\"That doesn't even make sense!\" Billie howled, before Jay pounced and took him from his chair to the floor.\n\nCassie whipped out her phone without moving from Stef's lap. \"Sending this to both your girlfriends,\" she sang. Jay flipped her the finger without getting off Billie, who thrashed weakly, making strangled cries of exaggerated anguish as he tried to shove Jay off. Kris made himself comfortable, stretching out along the couch.\n\n\"I feel like I should intervene,\" Rayne said, \"but the view's too good. I can't bring myself to do it.\"\n\n\"Fuck you,\" Billie said from the floor. \"I would've helped you, you traitor.\"\n\n\"I don't know if I'd be complaining,\" Calloway said.\n\n\"Well, I might if it was Jay,\" Rayne mused.\n\n\"Fuck you!\" Jay laughed. \"I'm a fucking catch and you know it.\"\n\n\"I don't know, you're not quite my type . . . I mean, if it were Kris or Cal, that's a different story.\"\n\n\"If I were what?\" Kris asked.\n\n\"Engaging Rayne in an impromptu wrestling match,\" Angel said.\n\n\"Oh. Do you want me to?\" Kris asked Rayne, looking for somewhere to put his drink. \"I can totally do that, hang on.\" He handed his drink to Angel and got to his feet. The change in elevation brought the drunkenness on suddenly, and he had to take a moment to collect himself.\n\n\"Don't think you're going to be wrestling anybody, sweetie,\" Angel said, her hand on his hip to steady him.\n\n\"Nope, I got this. Bit drunker than I thought, but I got this.\" He leveled one finger at Rayne. \"You stand there and hold still, okay?\"\n\n\"I'm not sure this is a great idea,\" Rayne said, but he didn't move. Calloway backed up a pace, out of the line of fire.\n\n\"Get him!\" Jay yelled from the floor.\n\nKris made a valiant attempt, but the floor tipped sideways under him and Rayne met him halfway, sweeping him up in an embrace that took Kris off his feet, laughing helplessly as Rayne swung him around before setting him down again.\n\n\"That's not wrestling,\" Kris said. \"This is just, like, a hug.\"\n\n\"I like hugging better.\"\n\n\"Get a room,\" Cassie called.\n\nIt was the best idea Kris had ever heard, but he swallowed it with the rest of his whiskey and smiled and tried not to let his heart show too obviously. He had no idea whether it worked, but Rayne laughed and poured him another drink. Calloway watched them without saying a word.\n\nThey carried on until well after midnight. The stars peeked out one by one until the sky was a dazzling array of rhinestones against the black, and the desert air grew cool and sharp. Somewhere beyond the festival perimeter, a coyote howled intermittently. Gradually the bus quieted until everyone had retired save for Kris, Rayne, and Calloway. Cal didn't drink, and outstripped them in terms of sobriety by miles, but seemed happy to keep them company. Finally, when Passionfruit had returned to their bus and the rest of The Chokecherries had lost the battle with consciousness, he stretched and got to his feet, gesturing to the door. Rayne nodded and drew them both outside, leaving the others to sleep.\n\n\"I should head back to my bus,\" Cal said, his voice soft in the night.\n\n\"We'll walk you.\" Rayne put his arm around Cal's waist and leaned into him, looking back expectantly at Kris. \"Coming?\"\n\n\"Don't want to get in your way,\" Kris said.\n\n\"I don't think there's anyone around worth putting on a show for,\" Calloway said. \"Come on, walk with us.\"\n\nKris walked a pace behind them, cataloging how their arms looped around each other's bodies and how Rayne leaned against Cal, his boots scuffing through the dust as he walked, remarkably sure-footed, through the maze of tents and stages. Kris was pleasantly drunk and getting sleepy, though the sharpness of the air canceled out some of the haze. Rayne and Cal complement each other nicely, he thought blurrily. Rayne's slender, darker figure bent flower-like toward Cal's broader, sun-kissed frame. When they reached the Dead Generation bus, they paused, three points under the vast night sky.\n\n\"I'll see you tomorrow,\" Calloway said. \"Take care of yourselves, all right?\"\n\nRayne darted in to press a kiss to his jaw, his curls brushing Cal's face for a second before he pulled back. \"We'll be fine. It's just us and the stars out tonight.\"\n\nThey let go of each other and Calloway nodded to Kris. \"Good night.\"\n\n\"Good night,\" Kris echoed as Cal disappeared into his bus.\n\nRayne swayed back to him and dropped an arm over his shoulders. \"Okay?\"\n\n\"Do you like him?\" Kris asked, his heart in his mouth as he waited for Rayne's reply.\n\n\"Sure I like him. Don't you?\"\n\n\"No, I mean\u2014Cassie said he propositioned you last night.\"\n\nRayne broke into a smile. \"Yeah, he did. I have to say, he made a very tempting offer.\" The night cast his face in shadows of blue and violet and his expression turned thoughtful. \"The possibility's there. He asked if I wanted to do more, since we were already\u2014for the press, you know? I didn't say yes, but I didn't turn him down, either.\"\n\n\"Do you want to?\" Kris asked, biting the tip of his tongue. If Rayne said yes, Kris was done. If he said no\u2014\n\n\"A lot of things seem like a good idea on MDMA,\" Rayne said vaguely. \"I might . . . I could like him. Or more than like. But I don't, not yet.\"\n\n\"Is he a good kisser?\"\n\nRayne grinned and pulled Kris tighter, so Kris tripped over his own sneakers and stumbled against Rayne's side. They both laughed in hushed voices as they righted themselves.\n\n\"Yeah, he is. We had fun.\"\n\n\"A better kisser than me?\" Kris pushed, not entirely joking.\n\n\"Course not. You're my favorite, baby. Always will be.\"\n\nRayne pressed a kiss to Kris's temple, sloppy and a bit off the mark as he laughed, but Kris closed his eyes and hummed, warm all the way through. He knew Rayne was only teasing, but that didn't mean he was lying. Kris tilted his head back to take in the stars. The sky looked infinite, and he found a weird comfort in his own insignificance. Gazing up at the stars like that, it didn't matter what he did or didn't do; the universe didn't care.\n\n\"Hey, Rayne. Remember what I told you last night? When I was high?\"\n\n\"About the peacock?\"\n\n\"No, dumbass, about you.\"\n\nRayne bit his lip. Kris looked away. He had the whiskey bottle by the neck, dangling loose from his fingers, its glass catching the scarce light left to them.\n\n\"You want a nightcap?\" Kris asked. \"Come on, come back to the stage.\"\n\nThe festival never slept, even if individuals had to. Kris and Rayne wound through music tents and raves and drum circles until they reached their stage, which was dark and empty for the night, and crept through the back to their tiny makeshift dressing room, where Kris set the whiskey bottle down on the table and took a minute to consider how drunk he was. Drunker than Rayne; they were evenly matched for consumption, but Kris's size had always made him a lightweight. Drunk enough to do something reckless, like tell Rayne how he felt for real. Maybe not drunk enough to regret it in the morning.\n\nHe'd have to wait and find out.\n\n\"When I told you I loved you,\" he said, \"and you said you loved me too. You said I was beautiful.\"\n\n\"Yeah?\"\n\n\"I meant it.\"\n\n\"You were so high,\" Rayne giggled.\n\n\"And now I'm so drunk,\" Kris agreed. \"Tell me I'm pretty?\"\n\n\"God, you're the prettiest.\" Rayne reached out to pet Kris's hair and Kris leaned into it, his eyes fluttering closed for a second at the feeling. \"Even without the makeup, your face\u2014 I love that you wear it, though. I love that you don't care what anybody thinks.\"\n\n\"I care.\"\n\n\"Not enough to stop.\" Rayne carded his fingers through the tufts and down the back of Kris's neck to hold him there, warm and steady. Kris blinked up at him.\n\n\"Fuck, Kris,\" Rayne breathed. \"You don't know what you do to me.\"\n\n\"You and Calloway\u2014\"\n\n\"We're just friends, I don't know if we'll ever be anything but friends, I don't even know if I want to be more\u2014fuck, it's not like you and me, you know that\u2014\"\n\nWith more coordination than the drink should allow, Kris pulled Rayne in for a kiss. He closed his eyes, expecting the roar of the crowd; his heart skipped a beat when it never came. The only sounds were their own hurried breathing, the wet sound of their mouths meeting, and the rustle of their clothes as they pressed together. Kris threw himself into the kiss like he was dying for it, and Rayne returned with equal fervor. They only broke apart when they had to pant for breath, their foreheads touching as they held each other's faces, eyes wild and chests heaving.\n\n\"I want\u2014\" Kris began, but he didn't know how to finish. The booze swam in his head, dizzying and pushing him off-balance. Rayne hadn't let go: he stood with his eyes closed, his hair falling in his face. His lips were dark and swollen from the kiss, and Kris darted in to nip at them, pulling a desperate moan from Rayne's throat.\n\n\"Kiss me,\" Kris said against Rayne's skin. \"Just kiss me\u2014\"\n\nRayne kissed him. He started at Kris's jaw and bit and sucked his way down Kris's throat until Kris was panting breathless syllables into the dark, one hand knotted in Rayne's hair. By the time Rayne reached his collarbone, mouthing at the bend where his neck met his shoulder, Kris was aching for it. He wound his arms around Rayne's neck and buried his face in Rayne's hair, breathing in the smell of his shampoo, needing to feel and touch and taste nothing in the world but him.\n\nHe didn't know which of them moved first; maybe they moved at the same time, an inevitable collision of bodies and wants. The first touch brought the best kind of friction, relieving and desperate both at once, and Kris bucked his hips forward, instinctively seeking more.\n\nWhen their hips met, they both froze.\n\n\"Okay?\" Kris asked, not raising his face from Rayne's hair.\n\nRayne stroked his hands up and down Kris's sides. \"Fuck,\" he whispered. His voice was rough. Kris shuddered and twitched forward again, a fraction of an inch.\n\n\"We're really drunk,\" Rayne said.\n\n\"Is that a no?\"\n\n\"No, god, no. Please don't stop.\"\n\nAs far as sex went, it was the least sophisticated Kris had had since his adolescence, but he didn't care. Burning up in the wake of Rayne's touches, he was too drunk for anything more complex. He finally had Rayne and Rayne wanted him back. He couldn't believe he'd wasted so much time.\n\nThey made out like teenagers, rubbing up against each other fully clothed, stealing kisses with every breath. Kris went into it harder and faster than he'd meant, crashing up against the edge of orgasm without any finesse, wanting to draw it out longer but too desperate for Rayne's touch to manage. He came without even undoing his jeans, gasping out a curse in the dark and clinging to Rayne like a lifeline. Rayne pressed his open mouth to Kris's neck, muttering a steady litany of fuck, fuck, fuck against his skin, and followed a second after.\n\nAfterward, neither of them spoke at first. On the other side of the room, the mirror stood like a silent witness. Rayne's pupils were huge, blown out with lingering lust and a look so fond that Kris's heart leaped giddily in his chest. He was less drunk than he'd been a minute before, but drunk enough to dare to run his finger over Rayne's face, from his brow down his nose to trace the curve of his lips. Rayne smiled against his fingertip and Kris smiled back, helpless and glowing.\n\n\"Good?\" Kris asked, before he could stop himself.\n\n\"You're perfect,\" Rayne said, catching Kris's hand. \"How are you even real?\"\n\n\"Magic.\"\n\nRayne kissed his knuckles, a wet press of his lips, and led him down the steps and past the stage, back into the night. The air caressed Kris's skin like a lover. He felt on top of the world, and when they returned to the bus and he crawled into his bunk, his eyes drifting closed as the last of the whiskey chased itself through his blood, he could still taste Rayne on his tongue, like cinnamon hearts and sea salt. He fell asleep promising himself that it wouldn't be the last time.\n\nKris woke with a hangover so bad he thought he might die. He hadn't drunk enough to black out so there was no reason he should feel so close to death, but maybe there was something in the whiskey that did it; he didn't know. He got up as slowly as he could, his head throbbing, and resolved to never accept a drink from Jay ever again. Playing that afternoon was going to be a nightmare. He'd be lucky not to puke halfway through the set. He settled one hand protectively over his stomach while the other groped blindly along the bus as he staggered outside into the sun.\n\nHe instantly wished he'd stayed in bed. The sun was blinding and his head protested with a scream so shrill he actually flinched; he forced his eyes open and found Freddie staring at him judgmentally. He flipped the bird off and stumbled his way to the picnic area, dreading solid food but hoping a Gatorade might fix him. He hadn't had enough water the night before, and felt like he'd been keelhauled fifty miles down the desert highway.\n\nDespite feeling like death, there was an undeniable spring in his step, and he gave a bright wave to Cassie and Stef breakfasting together as he passed. He'd finally gotten with Rayne and it had been amazing, and no amount of alcohol-induced misery could take that away from him.\n\nRayne was sitting at a picnic table beside Calloway, nursing a coffee and a quiet conversation. He looked bleary and barely able to keep himself upright, while Cal appeared healthier, but vaguely apologetic about something. Kris couldn't hear their words, but they seemed intimate. Cal straightened and nudged Rayne in the side at Kris's approach, and Rayne let whatever he'd been saying trail off.\n\n\"Morning,\" Calloway greeted. \"Coffee?\"\n\n\"Morning,\" Kris said, \"and yeah, I need it.\"\n\n\"I'll get you one,\" Cal offered, his hand on Rayne's shoulder. \"Anything in it?\"\n\n\"Just black, thanks. Extra-large. Extra-triple-large. Or bigger.\"\n\nCal nodded and headed off to the vendor, glancing over his shoulder as Kris leaned one hip against the table.\n\n\"Morning, sunshine,\" Kris said to Rayne. \"How's your head?\"\n\nRayne grunted and tipped nose-first into his drink. \"Probably not as bad as yours. About last night . . .\"\n\nKris broke into a stupid grin. \"Yeah?\"\n\nRayne patted the table. \"Sit.\"\n\nKris slid onto the bench on the opposite side. Calloway returned a second later with the coffee, and reclaimed his place beside Rayne. That was where Kris wanted to sit, pressed up close beside him to leach his warmth in the morning sun, but they were in public and Cal and Rayne had appearances to keep up.\n\n\"You look miserable,\" Cal said to Rayne. \"There's a guy with a camera just over there\u2014give us a smile?\"\n\n\"I am miserable,\" Rayne complained. \"I'm hungover, among other things.\" He smiled anyway, and Kris doubted anyone else would notice how it seemed strained around the edges. He wondered what the \"other things\" were.\n\n\"Listen,\" Cal said. \"Why don't you two go someplace private for a chat? I'll come find you later and we'll do lunch.\"\n\nRayne nodded and beckoned for Kris to stand.\n\n\"Okay, sure,\" Kris said, glancing between the two of them in confusion. They had obviously been talking before his arrival\u2014about the night before? Rayne wasn't meeting his gaze, but Calloway carried an air of sympathy. \"Thanks for the coffee,\" Kris said, at a loss for any other words.\n\nRayne led him back to the dressing room, closing the door behind them and leaning against it. He did look miserable, the tension etching lines between his shoulders like he was expecting a blow. Kris couldn't imagine from whom.\n\n\"About last night?\" Kris prompted.\n\n\"We were really drunk.\"\n\n\"Yeah.\" Kris frowned. \"You don't think it was a mistake, do you?\"\n\n\"No,\" Rayne said quickly.\n\n\"Because I don't regret it. I regret the timing, and I regret drinking that much, but I needed some liquid courage to\u2014you know. It might get messy with us both being in the band, and I never want to compromise that, or your stunt with Calloway, but I meant what I said, Rayne.\"\n\n\"What you said when you were high on molly.\" Rayne raised his hands before Kris could protest. \"I believe you. I just\u2014 We had a good thing going on, onstage, didn't we? It was all for the show. And now I don't know what you want from me.\"\n\nKris floundered. It had never occurred to him last night that they weren't communicating on the same level. \"I want . . . more. Of what we did. I liked it. I thought you did too?\"\n\nRayne raked his fingers through his hair, tousling it up as he looked away. \"Kris, when you joined the band, you told me you liked girls. You said you were straight.\"\n\n\"I do like girls. Back home, I always said I was straight, but that was different.\"\n\n\"And when we were messing around, we said that was for the fans. No strings, no complications.\"\n\n\"Last night wasn't for the fans,\" Kris pointed out.\n\n\"Last night's why we're having this conversation. Kris . . .\"\n\n\"I said that I'd only ever been with girls, which was true. I never got a chance to explore anything in Kansas, and I didn't want to accidentally lead you on, or let you lead me on, because we're in a band together and I didn't want to fuck that up. And I still don't, but I do want . . . more. Of everything. From you.\"\n\n\"You've never even been with a man.\"\n\n\"I'm trying to be with one right now,\" Kris said.\n\nRayne sighed.\n\nKris's heart dropped like a rock. He folded his arms like he could keep it from dropping any farther, and gnawed on the inside of his lip. \"This isn't how I imagined this going,\" he admitted. \"I know you're into me. I know you were into me last night.\"\n\n\"I've tried it on with straight guys before,\" Rayne reminded him, \"and it's not something I want to do again. If you're bi, or pan, or . . . whatever else, that's great, and I'm happy for you. But I don't want to be the person you experiment with to figure it out.\" He raked his hair back from his forehead and finally met Kris's eyes. Kris couldn't make out his expression in the shadows, but it seemed like it might be one of regret.\n\nKris cast around for any way to drag the conversation back on track. \"We had fun though, didn't we?\" he tried. \"It was good\u2014\"\n\nRayne's expression shuttered and Kris knew he'd fucked up.\n\n\"Wait, that's not\u2014\"\n\n\"I'm sorry, Kris. I can't go through another scenario like Fink where I get dragged through the mud for someone else's entertainment, especially if they don't even feel\u2014\" Rayne took a deep breath. \"It's not fair to me\u2014it's demeaning, and I can't\u2014I won't do it.\"\n\n\"That's not what I meant,\" Kris finished in a small voice.\n\n\"We were drunk,\" Rayne said. \"And I've got a chance with Cal. I like him and he likes me, and he's openly gay, Kris. He knows what he's about, and he told me so right from the start.\"\n\n\"You and Cal are just a publicity stunt.\"\n\n\"So are you and me,\" Rayne said gently.\n\nKris swallowed. \"Fine. I'll see you onstage.\"\n\nRayne nodded and slipped from the room. As soon as he was gone, Kris leaned back against the wall, his head throbbing from a hangover and from grief. He couldn't figure out where he'd gone so wrong.\n\n\"What did you do?\" Cassie asked bluntly.\n\nThey were waiting for their parents to come by and take them for brunch; the sun was sweltering and the highway asphalt was hot enough to melt rubber.\n\nKris shuffled around and didn't answer. Cassie rolled her eyes.\n\n\"I know something happened,\" she said. \"You got up this morning like you were on top of the world, and now you're all mopey. What's up?\"\n\n\"Me and Rayne,\" Kris finally said.\n\nCass whooped.\n\n\"No, no whooping. I fucked up and now it's all weird.\"\n\nShe quieted, but it took visible effort. \"But you two did get it on,\" she confirmed. \"Like, offstage, in private, just the two of you? How far did you get? Third base? Homerun?\"\n\n\"I hate sports metaphors. We, uh.\" He made a gesture. Cassie screeched. He sighed and covered his face with his hands. \"We were so drunk, Cass. It was such a mistake.\"\n\n\"Is that what he said, or are you having an identity crisis because you got off with a dude?\"\n\n\"Why does everyone keep assuming I don't know what I want?\" Kris demanded. \"I didn't wake up one morning and think, 'Oh, I wonder what it's like batting for the other team.' Like, I'm into guys. I always have been. Just because I haven't had a ton of\u2014any\u2014experience, that doesn't mean I don't know what I like.\"\n\n\"You've never talked about it before,\" Cassie pointed out.\n\n\"Neither have you,\" he countered.\n\n\"Point. So you're\u2014what? Bi?\"\n\n\"Bi,\" he said. \"I'm bisexual.\" It was the first time he'd said it aloud, and it felt scary and thrilling and relieving all at once, like going out in makeup for the first time, or lacing up a corset.\n\n\"Okay, well, does Rayne know that?\" Cassie asked. \"Or does he still think you're straight? And why would you even let him think that if it's not true? These guys shit rainbows, dude. You're not going to get a better opportunity than this to come out.\"\n\n\"He thinks I want to use him to experiment.\" Kris's insides went leaden at the reminder, and he curled in on himself, wishing he could disappear.\n\n\"Oh my god, are you in love with him?\"\n\n\"I\u2014\" He couldn't be, not now that Rayne had so thoroughly rejected him, no matter what Kris had thought he'd seen in Rayne's face in that dressing room the night before\u2014\n\n\"You are,\" Cassie breathed. \"Oh, shit. That sucks, man.\"\n\n\"Well, he doesn't want that,\" Kris said, with heavy resignation. \"He was talking about maybe trying to get somewhere with Cal. So it doesn't matter.\"\n\n\"You should really talk to him.\" She poked him in the arm. \"You guys could still work it out.\"\n\n\"Maybe?\"\n\n\"Talk to him, Kris. You're both adults\u2014physically, if not mentally some days\u2014and this might just be a huge misunderstanding.\"\n\nKris wanted to hole up somewhere dark and lick his wounds, but not talking was what had led to this mess in the first place. And regardless of whether he and Rayne could work things out, Cassie would be after him like a dog with a bone until he made an attempt. He sighed. \"Fine. We'll talk.\"\n\nShe settled back. \"You better. See me missing my chance to have Rayne Bakshi for a brother-in-law\u2014I don't think so.\"\n\nTheir parents' car pulled up and they got to their feet.\n\n\"As soon as we get back,\" Kris promised. \"I'll find him and we'll talk it out.\"\n\nTheir parents took them outside the festival to a little diner on the side of the highway, a quaint, family-owned place that proudly advertised its famous pies. Cassie hijacked the conversation, steering it clear of Kris with an ease born of years of jostling for attention. Kris didn't object; he only pitched in occasionally to corroborate some of her less believable Passionfruit anecdotes, all while Brad sat quietly, humming here and there to contribute, but mostly leaving well enough alone. Kris was glad for the respite\u2014he didn't want to pick a fight in the middle of the restaurant, not with his parents trying to treat him, but he didn't think he could resist if Brad started in on him.\n\n\"The festival's nearly over,\" his dad said. \"Are you looking forward to a change of scenery?\"\n\n\"It'll be good to be back on the road,\" Kris agreed. Purple Sage had left a sour taste in his mouth now, and he didn't want to set foot in that dressing room ever again. The fact that he and Rayne would still be sharing a bus\u2014he didn't know what to do about that, but quitting the band was out of the question.\n\nI'm going to talk to Rayne as soon as I get back, he told himself firmly. Cassie was right: he had probably blown this whole thing out of proportion. He should have taken Rayne's history with Fink into account and been more sensitive; he would apologize for that, and they could salvage things. Even if Rayne didn't want anything more than friendship, surely they could still have that.\n\n\"I think I'm still a bit hungover,\" he apologized.\n\nCassie slid his plate into her spot without a word and tucked in to the remains of his meal. \"Lightweight,\" she said cheerfully, around a mouthful of pancake. \"I think it'll be great to hit the road! I've never been on a road trip before, unless you count driving out here.\"\n\n\"With your Passionfruit band?\" their mom asked.\n\n\"And The Chokecherries,\" Cassie said, moving onto the hash browns. \"At least until Knocks can play again, but I'm having a great time, and the guys all like me, so.\" She shrugged. \"We'll see how it plays out. I'm happy to stay as long as I can.\"\n\n\"You'll be careful though,\" their dad said. \"They seem like a rowdy lot.\"\n\n\"I'll be careful, Dad. Jay's and Billie's girlfriends are joining us like two stops over anyway.\"\n\nTheir parents looked mollified at the thought of chaperones.\n\n\"And what about Rayne?\" their mom asked Kris. \"I've heard rumors flying about some young man he's been spotted with a few times now.\"\n\nBrad clenched his fingers around his fork.\n\n\"I don't know that it's anything serious,\" Kris said, keeping his voice even. \"I think it's hard to date on tour. I wouldn't want to try it.\"\n\n\"Well, there'll be plenty of time for that after,\" she said, \"what with you being so successful now. I've seen how the girls scream in the audience, you know, throwing all sorts of things at the stage. Young women will always love a man in a rock band.\"\n\n\"Seen some of the boys screaming too,\" their dad added.\n\nKris coughed and quickly took a drink of water. \"I'm just happy they like the music,\" he choked out.\n\nCassie thumped him on the back without looking up from her plate. \"I'm dating Stef, by the way,\" she said. \"The bassist? You guys met the first day we got here.\"\n\nKris sank deeper into his seat, grateful the attention was off him again. What he needed was to collect himself before his parents asked for any more details about Rayne, or Cal, or anything else.\n\nTheir parents glanced at each other.\n\n\"We'll have to talk to\u2014er,\" their dad began.\n\n\"Them,\" Cassie supplied.\n\n\"It would be nice to get to know them better,\" their mom finished. \"We'll take you both out to dinner before we leave.\"\n\nCassie brightened before glancing at Brad, who thinned his lips but only said, \"Congratulations.\"\n\n\"That would be nice, Mom. I'm sure Stef will like that.\"\n\n\"They definitely won't turn down a free dinner,\" Kris said. \"I didn't know you were actually dating.\"\n\n\"This is a family restaurant, Kris,\" Cassie said. \"We're calling it dating.\"\n\nTheir dad cleared his throat as the waitress returned with fresh coffee. \"Anyone want pie? I'm getting pie.\"\n\nHe and their mom both ordered; Cassie, having already eaten her waffles and most of Kris's pancakes, begged off.\n\nKris jumped at the chance for a minute alone. \"I'm going to go for a walk, see if that'll help my head. I'll meet you back here when you're ready to go, okay? Enjoy the pie.\"\n\nHe headed out to do a few laps around the parking lot, hoping the sun would bake the angst from his brain.\n\nBrad joined him a minute later.\n\n\"Hey,\" he said. \"We got off on the wrong foot the other day. Can we talk?\"\n\nKris squinted at him, then shrugged. It couldn't go worse than their last talk; it definitely couldn't go worse than his talk with Rayne.\n\n\"Jump in the car,\" Brad offered. \"Let's go for a drive.\"\n\nBrad slid into the driver's seat and waited for Kris to buckle up next to him before pulling out onto the highway. It stretched endlessly in either direction, cutting through the rust-colored desert rocks. Brad steered away from the festival, drumming his thumbs against the wheel. Neither of them spoke for the first few minutes. Something with a banjo twanged on the radio.\n\n\"Are you really happy?\" Brad finally asked.\n\nKris didn't know where to begin, but he wasn't going to prove Brad right by saying so. \"Sure I am. You still hung up on how I look onstage?\"\n\n\"You're wearing makeup right now.\"\n\nKris stole a glance in the side mirror. Whatever he'd been wearing last night had smudged around his eyes, making him look more goth than glam.\n\n\"It's hardly drag. So?\"\n\nThe highway lines passed under them in streaks.\n\n\"You and Rayne. Tell me he's never tried anything with you offstage.\"\n\nKris laughed. It sounded unpleasant, even to his ears. \"You trying to protect my virtue, Brad? That's a losing fucking battle, man. I've done more in the past week than I've done in my entire life, and if me and Rayne did do something, what's it to you? We're both adults.\"\n\nBrad's mouth twisted. \"You're still my little brother. I don't like seeing you like this.\"\n\n\"Successful?\" Kris offered. \"Getting paid to make music and hang out with some really cool people I respect and admire?\"\n\n\"It's degrading.\"\n\n\"Sorry, what?\"\n\n\"I'm not saying anyone's forcing you to do anything,\" Brad said, \"but someone's clearly influencing you. Makeup, girls' clothes, letting some gay rub up on you like that\u2014you would never have agreed to any of that before you moved away.\"\n\n\"No shit. You giving Cassie this talk too, or am I just special?\"\n\n\"She's twenty. Every college girl goes through a phase like this; it'll pass. You\u2014you're old enough to know better.\"\n\n\"Wow. Say that to her face\u2014she'll have your balls, man. I wish I'd had half her guts at twenty to be myself.\"\n\n\"Mom and Dad think you can take care of yourself,\" Brad continued, as if Kris hadn't spoken. \"Cassie will always side with you to spite me. But I didn't come all the way out here just to fight with you, Kris.\"\n\n\"No? Why did you come, Brad? Because I don't believe for a second that you missed me. We've barely talked since you went all right wing, and honestly, I was good with that. That was working for me. So what changed? Why are you suddenly pulling this concerned-older-brother crap that we both know is bullshit?\"\n\nBrad's mouth was set in a thin, hard line. \"Did you ever think how your little stage antics would reflect on the rest of us? Dressing up like this, the makeup, the kissing, like you're some kind of sex toy for him to play with up there, while he's fucking some other guy the rest of the time\u2014did you stop and think how that would make the rest of us look?\"\n\n\"Mom and Dad don't care\u2014\" Kris began, and then he stopped. \"I'm embarrassing you.\"\n\nBrad's knuckles whited out around the steering wheel.\n\n\"All your straitlaced conservative friends found out about me, and you freaked.\" Kris barked out a laugh. \"This is pathetic, man! You came all the way out to Nevada to bully me into quitting the band so you could save face with those assholes? Like that was ever going to work. Just get better friends.\"\n\n\"You don't understand anything,\" Brad said coldly, in a tone that meant Kris was exactly right. \"But I want you to know, I'm doing this for your own good.\"\n\nA thread of fear slipped through Kris. They were fifteen minutes out from the diner, and another fifteen after that from the festival. They hadn't passed a single car on the entire drive. \"What are you talking about?\" he asked, fighting to keep his voice level.\n\nBrad pulled onto the shoulder and let the car roll to a halt. \"Consider this an intervention. Give me your phone.\"\n\n\"Like hell!\"\n\n\"Now!\" Brad snapped.\n\n\"Fuck you. I'm calling Mom.\"\n\nKris made it to the second ring before Brad grabbed his wrist and smacked the phone away, sending it clattering to the car floor.\n\n\"Get out of the car,\" Brad said, his hand still wrapped bruisingly hard around Kris's wrist.\n\nKris tried to yank his arm free to no avail. \"You're kidding. You're fucking nuts, Brad. Do you really think this is going to get you anywhere?\"\n\n\"Everything's going to be fine,\" Brad said through gritted teeth. \"Now get out of the goddamn car before I drag you out.\"\n\n\"Fuck you.\"\n\n\"You think you can play guitar with a broken wrist?\" Brad twisted his hand, and Kris's bones ground up against one another until he swore and hit the car door with his free hand.\n\n\"Fine! Fucking\u2014fine. Let go of me.\"\n\nBrad let him go incrementally. His fingers left white marks like bands, which throbbed hotly around Kris's otherwise red wrist. Kris rubbed it as he braced himself against the door and squared his shoulders. \"Asshole.\"\n\nBrad growled and before Kris could defend himself, lunged over like a juggernaut to force the passenger side door open, undo the seat belt, and shove Kris out to land on his ass on the road.\n\n\"What the fuck!\" Kris scrambled to his feet, but Brad had already pulled the door closed again and locked it.\n\n\"I'm telling Rayne you want out,\" Brad said through the window, retrieving Kris's phone from under the seat.\n\n\"He'll never believe you.\"\n\n\"I'm texting it from your phone. I'm not an idiot; I know you two fought earlier. He'll believe it if you're the one who says so. And then you can move on with your life, and forget this whole dumb stunt ever happened. Go back to acting like a man again.\" He rolled the window down far enough to toss a water bottle into the dust at Kris's feet. \"Somebody will pick you up before you starve to death. You'll thank me for this, one day.\"\n\n\"Fuck you,\" Kris said. \"Fuck you with a giant fucking chainsaw, Brad, I swear to god, if you drive away\u2014\"\n\nBrad drove off before Kris could finish the threat, a spurt of dust exploding from the back tires and, to add insult to injury, sending Kris into a coughing fit that only made his lingering headache worse.\n\nThe car disappeared from sight, heading toward the diner where the rest of his family waited. He wondered what Brad would tell them.\n\n\"Lying bastard asshole.\"\n\nNo one replied.\n\nOnce Leif put his plan to recover the Avatar in motion, it was easy. According to the grapevine, it was The Chokecherries who had stolen it, so all he had to do was go to them to get it back. He knew the band; he had been to enough festivals to have encountered them prior to Purple Sage, and Rayne Bakshi stood out even among the throngs of music artists and crowds of partiers. Finding the band wouldn't be an issue. Convincing them to return the Avatar without getting security involved would be a greater challenge, but Leif had a plan for that as well.\n\n\"Travis, would you like to do a favor for the All-Seeing God?\"\n\nTravis nodded fervently. He wasn't ideal as far as new order members went\u2014he was too skittish, too twitchy, and too strung out for Leif's liking, but Leif had worked with worse before. Travis seemed curious about the order, and most importantly, eager to please.\n\n\"I want you to go to the north edge of the grounds and create a diversion. As loud and as riotous as you can manage.\"\n\nTravis's eyes lit up and he nodded again, even more frantically this time. \"I can do that. What kind of diversion do you need?\"\n\n\"Something big enough to distract the festival security. All of them. Can you do that?\"\n\n\"Hell yeah. I know a guy with fireworks\u2014give me ten minutes and it's done.\" He hesitated, seemingly on the brink of running off to find the guy in question. \"And after, you'll introduce me to your god, right? You'll let me talk to Him?\"\n\n\"If you do this well, you'll be one of us,\" Leif promised.\n\nTravis's face split in a grin, and he bounded away to wreak havoc on the north side, leaving Leif and the rest of the order to find Calloway.\n\nIt wasn't difficult. All Leif had to do was walk and let the All-Seeing God guide his steps. Fifteen minutes later he found Calloway at his stage, his hair shining like a golden beacon in the sun where he sat under the great sweeping banner that called people to worship him and his band. Leif had been a little disappointed when he'd learned of Dead Generation; Cal had always been a good disciple, and to find he'd thrown the order aside for a taste of fame still sat uncomfortably.\n\nRed, Boar, and Rikki followed Leif like great loping shadows, awaiting instruction. Rikki had been reluctant to rejoin them, and he'd soon leave the order one way or another. Leif was willing enough to let him go peacefully, especially now that Travis was primed to take his place, but Red's temper was shortening by the day, and Boar had never liked the boy. It was doubtful he'd be allowed to go without some bloodshed.\n\nHe put it from his mind and stepped into Calloway's line of sight. The man froze like a rabbit, and they stared at each other for a long moment. A cluster of security personnel at the far edge of the stage kept an eye on them but didn't move to intervene.\n\n\"Leif,\" Cal finally croaked. \"What are you\u2014\"\n\n\"Don't bother calling for help,\" Leif warned. \"Security will be busy elsewhere in a minute. This will be easier if you cooperate.\"\n\nOn cue, the nearest security officer's radio screeched to life, and he frowned, raising it to his ear as the person on the other end said something about \"a disruption\" between sharp bursts of static. The officer nodded to his fellows and they took off at a run, heading north.\n\nCalloway glanced around furtively before edging closer and dropping his voice. \"Why are you here?\"\n\n\"The Avatar is missing and you know where it is.\"\n\nCal looked panicked. \"No, I don't.\"\n\n\"The Chokecherries have been bringing a peacock to their shows. You know Rayne Bakshi. You're going to take me to him to help negotiate its return.\"\n\n\"I don't know what you're talking about,\" Cal insisted, but his eyes gave him away.\n\nLeif stepped forward and Cal scrambled back, but not quickly enough\u2014catching him by the forearm, Leif dragged him in close.\n\n\"You're coming with me,\" he growled, \"and you're going to ensure this negotiation goes smoothly for everyone involved. Do you understand?\"\n\n\"Why can't you talk to him yourself?\"\n\n\"Because I don't want to escalate things if I don't have to.\" They both briefly eyed Red and Boar. \"He trusts you. He's more likely to return the Avatar if you ask than if I do, and then no one has to get hurt.\"\n\nCal hesitated before finally nodding, dropping off the stage to take his place at Leif's side. Leif kept his grip firm around Cal's arm and set off through the festival without looking back.\n\n\"Slow down,\" Cal hissed, stumbling in his efforts to keep up. \"If you're trying not to look suspicious\u2014\"\n\n\"I don't care why they've stolen it,\" Leif said as he walked, ignoring Calloway's protests entirely. \"I don't care what they were thinking. We need the Avatar back; I don't have to tell you its importance. Once we have it, we'll let His Serene Majesty decide their fate.\"\n\n\"I'm sure this is all a huge misunderstanding,\" Cal said, trying in vain to pull his arm from Leif's grasp. Leif tightened his grip until Calloway's bones shifted under his fingers, and Cal stopped struggling. \"Maybe the bird just wandered off on its own.\"\n\nRed and Boar scoffed from behind them, and Leif paused long enough to level a truly fearsome glower in Cal's direction. Cal shrank back and shut his mouth.\n\n\"We're going to The Chokecherries. If you try any tricks, I'll make you regret them.\"\n\nCal nodded, eyes downcast, and pointed in the direction to go. Leif nodded stonily. They walked in silence, and whenever Calloway looked like he might try to lead Leif astray, Leif tightened his grip until Cal got back in line. Calloway might have left the order, but he wasn't stupid; he knew the All-Seeing God's power, and he knew Leif wasn't so easily fooled.\n\n\"What are you going to do to them?\" Calloway asked in muted tones.\n\n\"Whatever His Serene Majesty dictates, as always.\"\n\nTruth be told, Leif had no interest in punishing those who had stolen the Avatar any more than would mollify Boar and Red. He was growing tired\u2014not of his god, nor of caring for the Avatar, but of the never-ending parade of festivals and acolytes, spreading the word day after day to countless unbelievers. It was exhausting, and after so many years, it was beginning to take its toll. He wanted to retire, and care for the Avatar in peace. Maybe he would rent another little trailer, somewhere he could sleep every night and call home, eating ramen and drinking beer and letting someone else preach about the wonders of the universe.\n\nBut Boar and Red were still fervent in their faith, and they would demand retribution for this slight. All Leif wanted was to keep them content so they would leave him be.\n\nUnfortunately for The Chokecherries, what kept them most content was violence.\n\nCalloway led Leif and his ragtag order to The Chokecherries' bus, and there he stopped, digging his heels into the earth. \"I don't want any part of this. I never wished you any ill will when I left, Leif.\"\n\nLeif inclined his head. That was true, despite the hurt he'd felt at the time.\n\n\"I have a career now,\" Cal continued, his voice pleading. \"I have the chance to really make something of myself. I can't\u2014 I'll do anything to keep that alive. Please, just\u2014\"\n\n\"You'll be fine,\" Leif interrupted, \"as long as you do as I say. You'll find a lot has changed since you left; I have far less patience than I did. But you were a good one; I'd hate for anything to happen to you as a result of this . . . unfortunate incident.\" He gestured to the bus. \"Get your friends.\"\n\nCalloway tugged his arm free, and this time Leif allowed it. Cal crept to the bus door, standing open in the desert heat, and paused to peer up the steps while keeping out of sight from the bus's occupants.\n\n\"I don't see the peacock,\" he whispered.\n\n\"Find out where He is,\" Leif returned. \"Maybe they've hidden Him in one of the bunks.\"\n\nCalloway put one finger to his lips and turned all his attention to the conversation within the bus. Leif followed suit with his arms crossed, waiting one pace back.\n\nRayne Bakshi sat on the bus couch like he'd been shot, staring at his phone with an expression somewhere between shock and heartbroken bafflement. The last time Leif had felt either of those emotions was when he'd woken to find the peacock gone, and it was doubtful the rock star had ever lost anything half as important. He watched as Angel scooted in beside the man and leaned into his shoulder to look at the phone.\n\n\"It's fucking Fink all over again,\" Rayne said.\n\nAngel frowned and took the phone. She read: \"'I've finally had the chance to think things over, and I don't want to do this anymore. I'm sorry I have to do it this way, but I feel it's my only chance to get out. I never should have let you do the things you did, or let things get as far as you took them. Sorry I couldn't give you more notice, but this is best for everyone. Please don't try to get in touch.'\"\n\n\"He's gone,\" Rayne said.\n\n\"You have a show in four hours. He wouldn't bail on that, no matter what happened.\" Angel twisted around to sit cross-legged on the couch and face him. \"But something did happen between you two. You finally hooked up in private, didn't you? He was practically glowing when I saw him this morning.\"\n\n\"We hooked up,\" Rayne admitted. \"We were both so, so drunk. It should never have happened. God, why didn't I listen to Brian? He's going to have a fit when he finds out.\"\n\nCalloway winced in apparent sympathy, and Leif dropped his hand on Cal's shoulder, wordlessly ordering him to focus.\n\n\"Kris didn't look like he regretted it. He looked pretty damn happy about it, from what I could see,\" Angel said.\n\n\"That was before we talked.\"\n\nShe fixed Rayne with a stare so unimpressed that he flinched.\n\n\"I told him I didn't want to be his experimentation project,\" he said. \"Which, if I'd been halfway sober the night before, I would have said in the first place and we could have avoided all this.\"\n\n\"But you didn't.\"\n\n\"Because I got distracted by him kissing me when he didn't have to just for the show,\" Rayne said miserably. \"I was so fucking happy he wanted to, I didn't stop to think it through.\"\n\n\"I've seen the way that boy looks at you,\" Angel said. \"If you think he's using you to experiment, you're fucking blind, honey.\"\n\nRayne seemed unconvinced.\n\n\"Call him if you don't believe me,\" she said.\n\n\"His phone's turned off.\" He pushed his hands through his hair, bleeding distress.\n\n\"Then talk to Cassie. She must know where he went.\"\n\n\"I don't have time for this,\" Leif told Cal in an undertone. \"They haven't mentioned the Avatar once. If you're just stalling\u2014\"\n\n\"I'm not!\" Cal insisted, his voice pitching up.\n\nRayne and Angel froze, looking first at each other and then to the door.\n\n\"I've had enough,\" Red growled, shoving forward to take his place at Leif's side behind Calloway. \"This is a waste of time. We can make them tell us where the Avatar is.\"\n\n\"Don't\u2014\" Cal started, but before Leif could intervene, Red drew his hunting knife from the sheath at his hip.\n\n\"Red\u2014\" Leif began warningly, but it was too late: Cal broke away from the others and bounded up the bus steps to slam into the driver's seat.\n\n\"I'm so sorry,\" he gasped. \"I didn't want to bring them here. You have to get away\u2014\"\n\nRed's shadow stretched up the steps and into the bus ahead of him, the knife's blade long and glinting dully in his fist. \"Rayne Bakshi,\" he said, the sun reflecting off the knife and his shaved skull like a twin spotlight. \"You've stolen something important from us, and we're going to take it back.\"\n\nLeif rubbed his hand over his face and took a fortifying breath. Things were escalating very quickly.\n\n\"I'm so sorry,\" Calloway repeated. He had Rayne by the arm, guiding him quickly through the grounds as Leif and the rest of the order followed behind. Rikki held Angel in a similar way, and Leif watched her in case she chose to launch another handbag-based assault, and Rikki in case his fraying loyalty finally broke and he abandoned the order once and for all. Leif saw the way he looked at the woman, like she had hung the stars and the sun. It was how he should be looking at the Avatar.\n\nBut Angel didn't seem to like her chances against the entire order at once, not now that they had their knives out, and let Rikki guide her, however begrudgingly.\n\n\"They found me,\" Cal was saying in a strained whisper Leif chose to ignore. \"They knew you had the bird, and they said\u2014 I couldn't risk it. I'm not with them anymore, I swear I'm not, and I never told them anything, but they knew.\"\n\n\"What are you even talking about?\" Rayne asked.\n\n\"We're getting the rest of your band,\" Red cut in. \"We tried this your way,\" he added, turning to Leif, \"using Calloway as a negotiator, and he betrayed us. So now we do it like this.\"\n\n\"I didn't betray anyone!\" Cal snapped. \"You pulled a knife! Everything would have been fine\u2014\"\n\n\"Am I being held hostage?\" Rayne demanded.\n\n\"No!\" Cal said. \"Well, technically, yes, a bit. Please cooperate. I don't like this any more than you do.\"\n\n\"This is insane,\" Rayne said, but no one paid him any attention.\n\nFestival security had all but disappeared, presumably converging on the north side of the grounds where Travis was providing his distraction, leaving Leif, the order, and their hostages to move through the festival unimpeded. When they encountered Rayne's personal security guard, Rayne asked him through gritted teeth to fetch something from the far side of the festival grounds. Though Rayne was clearly banking on him to see it for the wild-goose chase that it was, the guard ambled off agreeably. Leif took it as a sign that luck was still on his side, despite Red's mutiny, and smiled grimly to himself. It was the kind of smile that made other people back away nervously and avoid eye contact.\n\nAfter that it was a simple matter of rounding up the remaining Chokecherries and their opening act. The order steered them at knifepoint with unerring finality into an empty tent at the southernmost border of the festival, as far from Travis and the security personnel as possible. Red had Cal by the scruff of his shirt, his knife held up warningly the entire time, while Boar took up the rear, looming mountainously as he shepherded the captives along. Rikki kept his hand on Angel's arm, and no matter how she glared at him, he ducked his head and refused to meet her eye.\n\nThe order deposited them in the tent and herded them into the center, where a large pole was erected like something out of an old Western, and tied their hands behind their backs, tethering them to the pole in prisoners-of-war style. Calloway was tied up alongside them; he would never rejoin the order, especially not after this debacle, and Leif had no further use for him. By the time the order was done securing everyone, both bands were accounted for, with the exception of Kris Golding, Stef Morganstern, and the new girl with Passionfruit.\n\n\"I feel like we're missing some really important context here,\" Billie said, his voice slightly strangled despite his calm words.\n\n\"We are the Worshippers of His Serene Majesty, the Incandescent and All-Seeing God,\" Leif said, drawing on every ounce of his poetic theatricality as he wrestled control of the situation back from Red, \"and you have stolen His Avatar.\"\n\n\"Um,\" said Billie.\n\n\"What the fuck,\" said Jay.\n\n\"Rayne,\" said one of the captives, in a deliberately calm voice, like the sea before the storm, \"can you explain?\" He was the only one Leif didn't recognize: middle-aged, beleaguered, and obviously not a performer. He must be Bakshi's manager, Brian.\n\n\"Oh my god,\" said Rayne. \"It's the fucking peacock.\"\n\nRed and Boar visibly bristled, and Leif raised his hand to hush them.\n\n\"You don't speak of His Serene Majesty in those tones,\" Red hissed, ignoring him.\n\n\"The bird,\" Brian repeated. \"The one Kris found the other night.\"\n\n\"That's the one,\" Rayne confirmed.\n\n\"Well, give it back so I can return to civilization and press kidnapping and unlawful confinement charges.\"\n\n\"I don't have it! Do I look like I have a peacock on me right now? He's probably wandering around outside somewhere, like he was when Kris found him.\"\n\n\"You should try keeping your Serene God on a leash next time,\" Jay suggested to the order. \"Since this seems to be an ongoing problem.\"\n\nBoar glowered and Red stepped forward, brandishing his knife as if he intended to gut his captive then and there. Jay flinched back even as he curled his lip in a snarl, and the rest of the hostages went rigid.\n\n\"You shut your mouth, boy,\" Red warned, the blade dangerously close to Jay's throat. \"Or I'll shut it for you.\"\n\n\"Jay, shut the fuck up,\" Billie whispered frantically.\n\nLeif wondered, not for the first time, whether he could actually hold Red and Boar back if they decided to kill their hostages. They had never killed anyone before\u2014not in the time Leif had known them, at least\u2014but then, no one had ever stolen their Avatar before, either. The Avatar gave their lives meaning, and Leif realized, watching them loom over the hostages, just how far they were willing to go to protect that meaning. He sighed and wished again that he had never told anyone about the peacock at all.\n\n\"Rayne, I swear to god, if I die in here, I am holding you personally responsible,\" Angel hissed to her companion.\n\n\"Nobody's dying!\" Brian said loudly.\n\n\"How is any of this my fault?\" Rayne demanded.\n\n\"Kris brings you a peacock and you don't question where he got it?\"\n\n\"Are you saying I should have assumed it was attached to a cult? And you,\" he added, twisting in his ropes to face Calloway. \"You knew! Why didn't you warn us?\"\n\n\"There are lots of peacocks in the world,\" Calloway said under his breath. \"I couldn't be sure it was the same one. I told Kris to be careful\u2014\"\n\n\"Where is Kris, anyway?\" Brian interrupted.\n\nLeif and the order frowned.\n\n\"He left,\" said Rayne.\n\n\"For brunch,\" Brian said slowly.\n\n\"The band.\"\n\nBrian's face went white, then red, and a vein in his forehead throbbed like it was going to burst. For an instant Leif was less worried about the hostages meeting their end by Red and Boar's hands, and more worried about the health of the band's manager.\n\n\"He has a contract,\" Brian said, still speaking with deliberate slowness, as if he could put the universe back in order through sheer force of will alone.\n\n\"I'm sorry,\" Rayne said, his misery evident in his voice. \"I know you warned me\u2014\"\n\n\"We'll talk later,\" Brian said.\n\n\"This Kris is the one who stole the Avatar,\" Leif said, hauling the conversation back under control.\n\n\"He didn't know it was the avatar,\" Calloway said quickly. \"He just thought it was a fancy bird. He didn't mean anything by it.\"\n\nLeif silenced him with a look. \"Then why did he steal it?\"\n\n\"He wanted to show me,\" Rayne said. \"He thought I'd like it. And he was high as balls, but, uh.\" He glanced at Brian. \"I don't think that's a legal defense.\"\n\nBrian shook his head.\n\n\"This is insulting,\" Boar said. \"His Serene Majesty the All-Seeing God should smite you for this.\"\n\n\"Let him fucking try it,\" Jay said. \"You'd think he would've done it already, though. Since we stole his avatar and everything.\"\n\n\"Please stop antagonizing the cult,\" Billie said.\n\n\"They ain't shit.\"\n\nRed stepped in and dealt Jay a vicious backhanded blow across the face, snapping his head to the side as the rest of the hostages burst out in yells, struggling against their ropes. Jay glared and jutted his chin out like he was actually going to fight the order from the floor with no hands.\n\nLeif rubbed his temples and wished he were high enough to deal with this calmly. When they got out of this, he was going to try switching Red and Boar from psychedelics to sedatives for a while.\n\nAs Jay and the others continued to bait Red and Boar, shouting increasingly derogatory things about their god, Brian cleared his throat and Leif circled around to him to begin negotiations. After a few minutes it became apparent that the man had no concept of the peacock's importance to either the order or the All-Seeing God, so Leif dismissed him to turn his attention back to Rayne.\n\n\"Ah,\" said the rock star. \"Yeah, you missed a bit.\"\n\nLeif stared. There was an empty space beside him where Angel had previously been sitting, and she and Rikki were nowhere to be seen. The ropes had been hacked away\u2014Rikki's switchblade, Leif realized belatedly. No one had thought to disarm him, despite his obviously wavering allegiance. A long slice in the side of the tent's canvas confirmed their escape.\n\n\"Yeah, you dumb fucks,\" Jay crowed from the other side of the pole. \"That was worth a black eye, right?\"\n\nRed aimed a kick at him but Jay just laughed. He sounded slightly manic.\n\n\"Oh, for fuck's sake,\" said Leif.\n\nKris kept up a steady litany of curses under his breath as he trudged back toward the festival, his shoulders set and his jaw squared. The sun was deathly hot and the highway stretched bare, disappearing over the horizon into the eternity of rocky orange desert on either side. It would be pretty if he weren't so fucking furious. He tried to calculate the distance in his head\u2014over half an hour by car, on the highway\u2014and when he might reach the festival grounds. Whatever fuckery Brad was causing, The Chokecherries had a show at four, and it was already half past one.\n\nKris didn't like his chances.\n\nAlso, his math was telling him it was going to take eight hours to walk back, and that didn't sound right, but he couldn't check the numbers because Brad had stolen his fucking phone.\n\nHe was going to throttle Brad with his bare hands when he finally caught up to him.\n\nA car engine rumbled in the distance, and he flung himself into the road in his haste to flag it down. It slowed to a crawl as it drew up alongside him, pulling over to the shoulder and rolling the window down.\n\n\"Hey, Kris. You need a ride?\"\n\nKris squinted and pushed his hair back to make sure he wasn't looking at a mirage. Tom smiled at him from inside the car, leaning over to wave as Kris panted in the sun.\n\n\"Fuck, please. It's an emergency.\"\n\nKris stumbled into the car and slammed the door shut behind him, gulping down his now-lukewarm water while Tom watched in consternation.\n\n\"You okay?\"\n\nKris wiped his mouth, shook his head, then nodded. \"I need to get back to the festival. My brother's sabotaging my career and Rayne thinks I hate him, which I don't, but he might hate me. I don't know. I need to fix everything. Can you drive me back?\"\n\nTom looked taken aback, but he adjusted the rearview mirror and set his hands back on the wheel. \"Sure, of course.\" He pulled onto the road and Kris sank into his seat. Tom glanced at him before cranking the air-conditioning. \"Tough day, huh?\"\n\n\"Understatement. But I'll make it work. I'll talk to Rayne\u2014everything will be fine. Or, at least\u2014almost fine. Manageable, anyway.\"\n\n\"That's a good start,\" Tom said encouragingly. \"Is there anything I can do?\"\n\n\"Honestly, just being in the right place at the right time like that was the most I could ask for. If you hadn't picked me up, I don't know where I'd be.\"\n\n\"It's dangerous to hitchhike. I always want to believe in the best in everyone, but you hear stories about girls\u2014 Well. I'm sure you've heard them.\"\n\n\"Lucky me it was you, then.\"\n\nThey drove in silence for another mile.\n\n\"So is Rayne your boyfriend?\" Tom asked eventually. \"I only ask because what you said earlier\u2014 Well, it sounds like relationship troubles. I don't mean to pry.\"\n\n\"Seems Rayne's love life is all anybody wants to talk about.\"\n\n\"Sorry. Forget I asked.\"\n\n\"No, it's okay.\" Kris took a deep breath, scrubbed his hand over his face, and confessed everything in a single exhalation. \"You've seen our shows. You know what we do onstage. Up until last night, that was all we did\u2014and then I pushed for more because I got tangled up in feelings for him, but I didn't have the balls to say it without a drink first, so he didn't take me seriously. He's fake-dating somebody else for the publicity but now he's talking about doing it for real, and if he does, I've lost my chance to convince him I'm not just messing around. Everything's fallen apart, and I don't know how to fix it. I wouldn't know how to fix it even without my brother interfering. God knows what he said to Rayne.\" He sucked his next breath in with a gasp of relief. It was easier to talk to a stranger than a friend, and anyway, wasn't that the whole point of a priest? \"And now he thinks I've pulled the same shit on him that his heroin-addict ex-guitarist did earlier\u2014fucking around and then quitting the band\u2014which is the furthest thing from the truth!\u2014but I need to tell him that so he believes me.\"\n\n\"I'm sure it's not as bad as you imagine,\" Tom offered. \"Things rarely are.\"\n\n\"I'm in love with him, you know.\"\n\n\"I guessed,\" Tom said. \"I'm not blind. I hoped you might\u2014 But I guessed.\"\n\n\"I should have told him I was bi right from the start, but it's so stupid\u2014I was scared, you know? For no reason. It's not like anything would have happened to me. Even my parents . . .\" His parents would be fine with it. They'd always been decent about that stuff, if not necessarily vocal in their support. \"When you get ordained and you start preaching and stuff, promise you'll never say anything to make people scared of coming out? Not even accidentally. Because it gets right inside your head, man, so that even when you know nothing bad's going to happen, you're still nervous to do it, years later on the other side of the country. And it sucks.\"\n\n\"I promise,\" Tom said, and crossed himself with a painfully earnest expression. \"No, I would never. No one deserves that.\"\n\nKris stared out the window. The desert stretched on and on, like a green screen on a loop in a cheap movie. \"I don't know about God, but somebody's been playing a hell of a joke on me since I touched down in New York.\"\n\n\"The Lord works in mysterious ways. What else can I say? I feel like I should have more answers if I'm going to be a priest and offer guidance, but I don't. I'm not sure what that says about my calling.\" Tom shrugged helplessly. \"All we can do is our best.\"\n\nKris tapped his fingers against the door. His nail polish was chipping; he should fix that when he got back to the bus. There was something soothing about dragging the brush over each nail, one by one. Angel had been right about the calm a makeup ritual could bring.\n\n\"I'm not a girl,\" Kris said.\n\nTom blinked.\n\n\"Since we're confessing shit,\" Kris added. \"I thought you should know.\"\n\n\"Okay,\" Tom said, after a minute. \"You just said you were bi, and I didn't realize\u2014now I feel a bit silly, but okay.\"\n\n\"You seriously thought\u2014\"\n\n\"I met you in the White Rabbit. I assumed you were one of those, you know, punky androgynous girls. I was tipsy. You never corrected me.\"\n\n\"Sorry. I found it kind of flattering at the time. I was figuring some stuff out. Are you mad?\"\n\n\"I wish you'd said something sooner, but no. Getting mad never makes anyone feel better about anything.\"\n\n\"I like that,\" Kris said thoughtfully. \"That's a good philosophy. Sorry if I, like, embarrassed you.\"\n\n\"Oh, I've had worse. I can absolve you though, if you like,\" Tom offered. \"Not officially. I'm not sworn in yet. But I think, as far as misdeeds go, your brother's outstrips yours by a long shot. There's no sin in wanting to feel loved.\"\n\n\"I hope not,\" Kris said. The highway in front of them shimmered in the heat, and he held his breath, waiting for the festival grounds to roll into sight.\n\nKris threw himself from the car before Tom had it in park, and made a beeline for the buses. He'd make it up to Tom later\u2014a bottle of wine, the really nice stuff, or expensive chocolates or\u2014something. A house, maybe. He'd figure it out later.\n\nThe buses were empty.\n\nHe tried the stage next, anxious but not alarmed yet. When the stage was empty too, his anxiety started to solidify into a rock in the bottom of his stomach, and he had to tell himself in no uncertain terms that Brad hadn't convinced both bands to pack up and leave altogether. He poked around the back of the stage, peering into the shadows and skirting the edges of the dressing room, when he heard Angel from outside, her voice unusually raised, shout, \"You kidnapped us!\" Kris wheeled around to follow the sound.\n\n\"You held us hostage like a bunch of thugs!\"\n\n\"I'm sorry.\" A second voice, softer\u2014Rikki. \"They said\u2014 I didn't know how to stand up to them. They just want the Avatar back.\"\n\n\"It's a bird,\" Angel said flatly. \"Get another one.\"\n\n\"That's not\u2014\"\n\n\"Nope. You telling me one bird is worth all our lives?\"\n\nKris rounded the next tent, trying to figure out what the hell they were talking about.\n\n\"I'm getting security if I have to drag them here kicking and screaming, and this is all going to be over,\" Angel said. \"No birds, no hostages\u2014\"\n\nKris caught sight of them by a tower of scaffolding between two empty stages a few yards off and veered toward them.\n\n\"You can't call the cops!\" Rikki blurted. \"We'll all get arrested.\"\n\n\"Yeah?\"\n\n\"Please don't send me to jail.\" Rikki was still, his eyes downcast except to sneak glances up every now and then, shining and hopeful and full of regret. Angel opened her mouth to reply, but Kris interrupted before she got the chance.\n\n\"Angel!\"\n\n\"You!\" She smacked his arm as soon as he came into reach, then grabbed his hand and gave him a relieved squeeze.\n\n\"Me. Where is everybody? And did Rayne get a weird text from me earlier saying I quit the band?\"\n\nAngel narrowed her eyes. \"He did. You've got some explaining to do, but it'll have to wait. The skinhead cult is holding the bands hostage in a tent and security's tied up dealing with some nutcase who started a fire on the far side. Apparently he's barricaded himself on a stage with a bunch of fireworks and is trying to incite a riot. Where's that damn peacock you stole the other day? The cult wants it back.\"\n\n\"Sorry, what part of that requires less explanation than a text?\"\n\nShe took him by the arm and dragged him along through the tents, intent in her direction. \"The peacock. They want it. Badly. They call themselves The Worshippers of His Serene Majesty, the Incandescent and All-Seeing God. That peacock? That's their god. And you went and stole the thing because you thought Rayne would think it was pretty. Which he did, and it is, but oh my god. And Rikki defected and helped me escape, which is the only reason I'm not throwing him to the wolves right now.\"\n\n\"Okay,\" Kris said. \"Awesome. Hi, Rikki. And I didn't steal the peacock\u2014I just picked him up and carried him around for a while. It's not like he was locked up on private property.\"\n\n\"He likes to wander,\" Rikki said. \"Leif said it would be degrading to keep Him in a cage, since He's a god and everything.\"\n\n\"Uh,\" said Kris. \"Do you believe he's a god?\"\n\nRikki shuffled uncomfortably.\n\n\"He's still thinking about it,\" Angel supplied.\n\n\"I mean, I know He's not literally a god,\" Rikki said. \"But Leif and the others were always really sure about it . . .\"\n\n\"Crisis of faith later,\" Kris decided. \"Where are they? It's three against three now, right? We can bust them out.\"\n\nAngel looked skeptical, but nodded anyway.\n\n\"They'll have noticed we're missing by now,\" Rikki said, \"but they can't do anything about it if they want to keep guarding the others. Do you have any weapons?\"\n\n\"What? No, I don't have any weapons. Just\u2014 Take me to the tent, and we'll try to bluff them into giving up. Unless you think we can fight them,\" he added. He didn't know how to fight, but he could try.\n\n\"Let's try to avoid that, if possible,\" Angel said. \"They've got knives.\"\n\n\"They have knives? This is insane. This is a terrible plan.\" Kris's voice pitched higher, and he tried to tamp down on it before it evolved into full-fledged hysterics. \"Rikki, I'm sorry, but I'd rather you get arrested than anybody get stabbed.\"\n\n\"That's okay,\" Rikki said. \"I have a knife too, if that helps.\"\n\n\"Rikki's right, though,\" Angel admitted grudgingly. \"Cops aren't the best idea. They'll want to do things by the book: bring in backup and a hostage negotiator, the whole nine yards. That gives Leif and his friends too much time to get stabby if they panic. Besides which, the cops will take too damn long to get here.\"\n\nKris pushed both hands through his hair, aware he was exuding stress all the way into outer space. We're not going to die at the hands of a bunch of stab-happy bird enthusiasts, he told himself sternly. Who even needs cops? Or backup? Fuck. He took a deep breath. \"Okay. Let's go. Aim for minimal bloodshed.\"\n\nAngel led them through the festival to the tent in question. The cult's motorcycles were parked outside, hulking like shiny black beasts awaiting their masters' return. Boar stood outside the entrance, apparently watching for any passersby who strayed too near. Angel, Rikki, and Kris crept up from the side, ducking behind whatever cover they could find.\n\n\"We got out there,\" Rikki said, pointing to the slice in the side of the tent. \"We can get back in that way, unless you want to fight Boar.\"\n\n\"His name is Boar?\" Kris asked. \"Jesus Christ. No, I'm not fighting a skinhead named Boar. Let's sneak in.\"\n\nThey slunk up to the side of the tent and Rikki crawled through first, his muscles bunching as he fought to keep quiet. Angel slipped through after him, with Kris following on her heels, wondering how the hell his life had come to this.\n\nRed was waiting for them, arms crossed and a scowl on his face.\n\n\"Ah, shit,\" Kris said.\n\n\"Kris?\" Rayne said. Kris couldn't see him around Red, but Passionfruit, The Chokecherries, and Cal were huddled around the center pole, sitting on the floor, tied up and looking so ridiculous that Kris did a double take.\n\n\"Hey, Rayne,\" he said. \"Bad timing, but we have to talk later.\"\n\n\"No talking,\" Red growled.\n\nRikki pulled his knife.\n\n\"No!\" Angel shouted. \"Nope. No knives. Rikki, put it down.\"\n\n\"You little punk,\" Red said. \"You think you can pull a knife on me? I'll take you apart with my bare hands.\"\n\nAngel jerked Rikki's arm down. \"Nope. Nobody's pulling any knives or fighting anybody. Rikki, stop it.\"\n\n\"But we have to\u2014\"\n\n\"No,\" she stressed.\n\nRed smacked the knife out of Rikki's hand like a bear swatting a fly. \"Traitor. You can join the others.\"\n\n\"Wait!\" Kris shouted. Everyone stopped. \"Wait. This is about the peacock, right? I'm the one who stole him; it's me you want. Let everyone else go and I'll get him back for you.\"\n\n\"That's not good enough,\" Red said. Boar lingered in the tent entrance, near enough to hear the conversation while still keeping a lookout. Leif watched them, weary looking and exasperated. \"If you'd returned Him after an hour, maybe we could let this affront go, but it's been days. The All-Seeing God deserves a sacrifice to right this insult.\" Red glanced at Boar and Leif. Boar nodded enthusiastically; Leif very deliberately shut his eyes and pinched the bridge of his nose.\n\n\"What?\" Cal cut in. \"We've never sacrificed people!\"\n\n\"We do now.\" Boar frowned. \"The All-Seeing God needs to know these people are sorry.\"\n\n\"We're sorry!\" Kris yelled. \"Jesus, don't\u2014 It's just a fucking bird!\"\n\nThat was, of course, the wrong thing to say, Kris reflected as Red dragged him to the pole and knotted his hands behind him. Diplomacy had never been his strongest suit, especially not when people were threatening to sacrifice him and all his friends to a glorified feather duster. Maybe that was something he should work on, now that he was living in the public eye.\n\n\"Hey, Rayne,\" he said glumly. He was tied up in between Rayne and Cal. It was sweltering in the tent, pressed up against so many other bodies, all of them sweating from stress and making the enclosed space more humid than the desert had any right to be.\n\n\"Hey, Kris,\" Rayne returned, his tone carefully unemotional. \"You came back.\"\n\n\"I never left. My brother\u2014\"\n\n\"Stop talking,\" Leif ordered.\n\nKris obeyed, but leaned his shoulder against Rayne's in a silent bid for understanding. After a second, Rayne leaned back.\n\nIt was a start.\n\nThere were few things worse, Kris decided, than listening to three giant skinheads discuss whether or not they were going to kill you, when you were tied up with no hope of escape. The prospect of dying without knowing how Rayne felt about him was one of those few things.\n\n\"We need to talk,\" Kris repeated under his breath, while the cultists argued back and forth about whether or not to kill them.\n\n\"I'm listening,\" Rayne said. \"It's not like I have anywhere to be, after all.\"\n\n\"Whatever that text said, it wasn't me.\"\n\nRayne tilted his head. They were trying not to look at each other in case their conversation became obvious, but the tension between them was palpable.\n\n\"It said you regretted everything, you should have stopped me before we went so far, and you were leaving the band,\" Rayne said.\n\n\"That was Brad.\"\n\n\"Why was Brad texting me from your phone?\" Rayne asked, his tone suspicious but his face bewildered.\n\n\"He stole it and kicked me out of the car. The point is, it's not true. Did you seriously think I was just going to run out on you guys like that?\"\n\n\"No? But the things he said\u2014\"\n\n\"He made them up, because he's a lying bastard who lies!\" Kris said, louder than he meant to. The cult paused in their conference and stared at them. \"And you guys,\" Kris continued. \"I thought my biggest challenge today was going to be stopping Brad from imploding my career, but you really blew it out of the park, didn't you?\"\n\n\"Don't antagonize them,\" Cal muttered.\n\n\"Fuck antagonizing them! I've had a shitty day since I got up. I didn't need to be kidnapped on top of that. Fucking peacock cults, what the fuck? Why didn't you warn us when you saw I brought the bird back? You were all vague and unhelpful\u2014how was I supposed to guess the peacock was from a goddamn cult? You never said a thing!\"\n\n\"I didn't think they were going to do this!\" Cal protested. \"I couldn't know for sure about the peacock, and if it was a different one, I didn't want to cause trouble for no reason, not with the label already taking a chance on me\u2014\"\n\n\"Cults are more important than your music career! Jesus. I'm out. If they want a human sacrifice, fucking kill me already, man. Put me out of misery. I'm ready. I'm done.\"\n\nLeif shifted uncomfortably. \"It's not human sacrifice. It's\u2014\"\n\n\"Justice,\" Red cut in. \"The All-Seeing God needs justice, so it's only right\u2014\"\n\n\"That we give Him an offering worthy of His Serene Majesty,\" Boar explained.\n\n\"And you're not,\" Red added.\n\nKris felt irrationally offended. \"Then who is?\"\n\nThe cult turned to Rayne.\n\nThe tent broke out in a cacophony of shouts from the hostages, all at once.\n\n\"You can't sacrifice him!\" Kris shouted above the din, grappling desperately for any excuse, no matter how outlandish. \"He's the chosen one!\"\n\nThe tent fell silent like someone had pulled the power cord.\n\n\"That's why I took the bird,\" Kris said, talking slowly as he tried to invent a story out of thin air. \"Because your god chose Rayne as the next . . . prophet.\"\n\nCal and Rikki stared at him with huge eyes.\n\n\"You're lying,\" Leif said.\n\n\"No, I'm not! Rayne, show them\u2014show them the peacock tattoo!\"\n\nRayne looked at him wildly. \"Right. That's right, I have a peacock tattoo.\" He bent forward as far as the ropes would allow, not that anyone could see anything though his mane of hair.\n\nLeif stepped forward, a frown marring his brow, to grab Rayne by the back of the neck and push his hair aside. The mandala was plain to see against his skin, even crisscrossed by the necklaces. Red and Boar crowded in to see, and all three of them stood there for a minute while Rayne, still bent double, waited, looking increasingly nervous.\n\n\"See?\" Kris said. \"That's a sign! You can't deny that's a sign.\"\n\n\"That doesn't mean\u2014\" Boar began.\n\n\"Yes, it does!\" Kris shouted.\n\nOn cue, an eerie scream went up from outside the tent, and the cult froze like they'd been electrocuted.\n\n\"See?\" Kris insisted.\n\nLeif let go of Rayne's head and Rayne reared up just in time for Freddie Mercury to come flying into the tent like divine retribution, screaming the whole time and pushing the cultists aside to land in Rayne's lap in a whirlwind of feathers and talons.\n\n\"Look,\" Angel cut in. \"Your god is beauty incarnate, right? So who's a better earthly incarnation of beauty than Rayne Bakshi?\"\n\n\"He's literally perfect,\" Kris added. \"The peacock knows it. Do you want to risk the wrath of the All-Seeing God by flouting his will like this?\"\n\nLeif, still frowning, stepped forward, flicking his knife out. Everyone on the floor flinched. The peacock flared his wings warningly, but Leif only knelt, reached around, and cut Rayne's rope. He drew Rayne to his feet\u2014Rayne was shaky, but drew himself up tall\u2014and studied him up and down as if examining a prime cut at the butcher's.\n\nRayne kicked Kris surreptitiously in the ankle in an obvious bid to know the plan. Kris winced and then shrugged. Freddie, devoid of lap in which to sit, began screaming again.\n\nLike flipping a switch, Rayne cleared his throat and donned his stage persona like a robe.\n\n\"Your god is moving up in the world,\" he said, his voice low and smooth. It wasn't the voice of a hostage bargaining for his life: it was the voice of someone who held all the cards, and knew how to play them. \"He wants to be admired by more than just four people. He wants to be adored by millions. That's why he chose me\u2014I'm already loved the world over.\"\n\n\"But we've done everything He ever wanted,\" Red said plaintively. \"Aren't we good enough anymore?\"\n\n\"You did wonderfully,\" Rayne soothed, his voice like honey, \"and you'll be rewarded for it. He's not dismissing you. This is just the next step of his plans.\"\n\n\"But He should have told us His plans,\" Boar protested. \"We would have carried them out for Him.\"\n\nRayne straightened up another inch and glared. The cult took an involuntary step back.\n\n\"He should have?\" Rayne demanded. \"Maybe he thought you were getting too presumptuous; maybe that's why he left you. He's in me, now. I'm All-Seeing, and if you claim to worship me, prove it. Otherwise, I'll leave you here in the desert and find worthier acolytes.\" He glanced down at Kris, who nodded encouragingly. \"I have the tattoo.\"\n\nThe cult nodded, slowly.\n\n\"You saw how Freddie\u2014how the avatar came to me.\"\n\nThey murmured among themselves in quiet agreement.\n\n\"Have you ever seen anyone more beautiful than me?\"\n\nA frown and a ripple of debate.\n\n\"Have you?\" Rayne demanded, steelier this time.\n\nKris was biased, but still, he couldn't see how they could deny it. From Rayne's dark complexion and the way his skin held the warmth of the sun like a jewel, to the sea-glass color of his eyes, to his curls\u2014thick and silky, never a split end or a strand out of place\u2014legs for miles and narrow hips\u2014\n\nKris swallowed, distracted. Of course Rayne was the most beautiful man they'd ever seen. To think otherwise was blasphemy.\n\n\"I'll worship you,\" he said, over the cult's muttering.\n\nRayne and Cal both looked at him, startled.\n\n\"I will too,\" Angel said.\n\n\"And me,\" said Rikki.\n\n\"Fuck, sure,\" Jay agreed. \"Why not.\"\n\n\"Good speech,\" Billie added.\n\n\"Stef will be furious they missed this,\" Maki said, \"but I'll worship you.\"\n\n\"And me,\" Lenny sighed, regret plain in his words.\n\nHatchwork and Knocks chimed in, clearly lost, but amenable as long as it got them untied.\n\n\"Jesus Christ,\" Brian sighed. \"Rayne\u2014\"\n\nRayne cleared his throat pointedly and tapped his foot.\n\n\"Your Serene Majesty,\" Brian corrected himself.\n\n\"And me,\" Cal said. \"I'll worship you too.\" He glared at the cult before turning his gaze back to Rayne, who nodded to him.\n\n\"See?\" Rayne said to the cult. \"New acolytes.\"\n\nThe cult jostled one another for a second longer before reaching their decision and prostrating themselves, foreheads touching the floor. Rikki twitched but, being bound, couldn't join them. Cal closed his eyes for an instant with a wistful sigh. Rayne took a step back and looked around, shooting Kris a helpless glance before resuming his haughty stage character. Freddie walked between their prone forms, inspecting each one as they held quaveringly still, and everyone tied to the pole held their breath, waiting.\n\n\"Attention!\" a voice blared from outside, crackling through a megaphone. Everyone jumped. The three cultists twitched and leaped to their feet, hands going to their knives. \"We have you surrounded,\" the voice continued. \"Come out with your hands up.\"\n\nThe cult glanced at one another. The megaphone screeched with static.\n\n\"You have until the count of three,\" the voice said. \"On three, we're coming in, and we're not afraid to use force. One\u2014\"\n\nFreddie shrieked and took wing, flying directly for Leif's face. Flinching, Leif ducked back with a shout as the bird drove him toward the tent entrance with beating wings and a tail that seemed far too heavy to fly. He fell through the opening, a scuffle sounded, and the megaphone blared again.\n\n\"Your compatriot has surrendered. I repeat, come out with your hands up.\"\n\nRed and Boar exchanged a look and silently obeyed. As soon as they were out, the tent entrance flared open and the sun came streaming in, revealing Cassie, wearing a security guard's vest, and a very pleased Stef, who was wielding a megaphone. Butch headed an assortment of sooty-looking security personnel as cops and medics swarmed in the background, forcing the cultists into cuffs. The press, confused and overexcited, stood by with their cameras flashing.\n\n\"What's up?\" Cassie asked. \"You guys look like you had a fun afternoon.\"\n\n\"Please untie us,\" Kris said. \"Now, please. Immediately.\"\n\n\"Sure, sure,\" she said easily, pulling a knife from god knew where. \"So a cult, huh? An entire hostage scenario! And I missed the whole thing. I'm glad you're not dead, though; Mom and Dad would have had a fit. What happened with you, anyway? Brad said you weren't feeling good so he drove you back to the festival early\u2014did you get your talk with Rayne?\"\n\n\"I'm standing right here,\" Rayne said.\n\n\"Brad,\" Kris said. \"Is he here? I'm going to kill him.\"\n\n\"Yeah, he'll probably be here in a sec,\" she said. \"Why are you killing him?\"\n\nShe finished sawing through his ropes and he rubbed the feeling back into his wrists before trying to stand. Rayne took his elbow and helped balance him, while fishing his phone from his pocket and handing it to Cass. \"Read the last texts from Kris.\"\n\nAs Cass did, her eyebrows climbed higher and higher until they threatened to disappear into her hairline. \"Brad sent these? What a dick. I missed a lot.\"\n\n\"What we get for sneaking off to make out,\" Stef said, cutting Passionfruit free. \"Who wants a quickie when you can get kidnapped instead?\"\n\n\"Seriously,\" Cass agreed. \"And then there was that fire on that north-side stage! Most of security's still dealing with that. They caught the guy, though. See?\"\n\nShe pointed to a couple of personnel who weren't manhandling the cultists: instead, they had in custody a skinny, twitchy-looking man who seemed to have been caught in an explosion. He was covered in ash from head to toe and littered with scratches, his hair standing on end and his eyebrows missing, but beaming manically. He shouted, squirming around in the security guard's grip as he tried to wave at Leif, who stoically ignored him.\n\n\"Huh,\" said Kris. \"He's connected to the cult too?\"\n\n\"Wild stuff.\" Cassie nodded. \"You okay though, Kris?\"\n\n\"Mentally?\" Kris asked. Rayne still hadn't let go of his arm. His hand was warm, but he stood farther off than he normally did, like he wasn't sure he was allowed to touch. \"We still need to talk,\" Kris said. \"That's what I was coming to do before the fucking peacock cult got involved.\"\n\n\"Go,\" Cassie shooed. \"Not far\u2014the cops will want to talk to you. But go talk!\"\n\nKris led Rayne through the side slit in the tent, ducking around the congregation of bemused but warily entertained law enforcement at the front, to stand a few yards back in an illusion of privacy, hidden by the tent's shadow.\n\n\"So,\" Kris said.\n\n\"I'm sorry,\" Rayne said immediately. \"I never should have believed you would drop everything and run like that. I know you wouldn't, not for any reason, no matter how it looked. You're not Fink. You're nothing like him.\"\n\n\"I'm in love with you.\"\n\nRayne blinked, lips parted in shock. Kris doubled down and rolled with it.\n\n\"I've been in love with you this entire time.\" He forged on, speaking over the crazy pounding of his heart and the way his palms broke out in sweat. \"Since way before we messed around. Maybe since you bought me that guitar and offered me a contract. And it wasn't messing around,\" he corrected himself. He needed to be completely transparent this time. \"I needed to get drunk before I could make a move on you, but I didn't regret it, and I wasn't experimenting. I like guys too, Rayne. I've always liked guys, even when I was scared to say so back home. I'm bi, always have been, and I didn't need to get my hands on your dick to figure that out. But mostly I just like you. And I'm pretty sure you like me too. So, uh, anytime you want to say something, feel free to jump in. Please.\"\n\nRayne wet his lips. He looked lost, and like the sun coming through the clouds, radiant and hesitant and overwhelmed all at once. \"Okay,\" he said. \"I like you too. A lot. And I'm sorry for what I said this morning\u2014I was scared of getting hurt, and I thought getting involved with you was going to hurt a lot. It usually does,\" he added, running his hand through his hair, \"when straight boys are involved, and after Fink, I never wanted to take that chance again. I didn't want to risk the band, and I didn't think the sex would be worth the fallout.\" He dropped his gaze. \"I didn't want to risk my heart again, either. Fink trampled it pretty ruthlessly, even if I was over him by the time he quit the band for good.\"\n\n\"Fink's a dick,\" Kris said bluntly, \"and I'm sorry for everything I did that made you think I was like him. I should have told you I was bi sooner, but I'd never said it aloud before. I didn't want to complicate things, especially once Calloway showed up, but I still should have said something.\"\n\nHis heartbeat had calmed now that Rayne had heard him out, though as the adrenaline faded it left him weak-kneed and in need of a hug. Rayne made an aborted movement like he wanted to reach for him, but Kris shook his head. \"Obviously I managed to complicate everything anyway. I wasn't planning on falling for you, if that helps.\"\n\n\"Complicate everything by encouraging me to do this stunt with Calloway, even though you liked me the entire time, for example?\"\n\nKris's heart leaped at Rayne's cautiously teasing tone. \"I was hoping that if you were off-limits, it would help me get my head on straight and I'd get over you,\" he admitted. \"It kind of backfired.\"\n\n\"That night in the hotel, when I asked you whether I should do that stunt, I was hoping you'd tell me it was a dumb idea,\" Rayne said. \"I was looking for some sign that you wanted me too.\"\n\nKris stared at him for a second before he started to laugh. \"Oh man, did we ever fuck up. So, you and Cal . . . still just fake?\"\n\n\"Yeah. We talked this morning, after you and I . . . after you left, but before Brad sent those texts. Cal said he wasn't ready for anything serious, but he didn't feel right about messing around when I was clearly hung up on someone else. And I'm not really into the whole casual-sex thing anyway.\" Rayne paused. \"Of course, now he might be under arrest with the rest of the cult. I should go see about that.\"\n\n\"I'm sure you can convince the cops it wasn't his fault. Though you might have to fight Brian first. He's pissed.\"\n\n\"I can't blame him.\"\n\nKris shifted from one foot to the other. \"So, are we okay?\"\n\n\"Yeah. I should have seen this coming, though. I knew you'd be trouble the minute I saw you.\"\n\n\"The good kind of trouble?\" Kris asked, daring to look Rayne in the eye.\n\n\"I think so. I hope so.\"\n\nRayne caught his wrist and tugged him near. Kris let Rayne reel him in, inch by inch, until they were standing a breath apart. Tipping his head back, he met Rayne's gaze, simultaneously shy and daring. His stomach flipped like it did when he'd first seen his own reflection after Angel had made him up, or when he'd walked out of the change room wearing girl's clothes for the first time, or when Rayne had come prowling up to him onstage and the crowd's roar swelled to a crescendo. He was going to kiss Rayne Bakshi\u2014Rayne Bakshi was going to kiss him\u2014in broad daylight, when they were both sober, and there was no audience to impress. He couldn't stop smiling as he leaned in. Rayne's hands hovered just above his shoulders, like he couldn't believe Kris was real, and\u2014\n\n\"You!\" a voice thundered.\n\nThey jumped apart.\n\nBrad Golding stopped a yard away, glowering, one finger pointed accusingly, at which of them, Kris couldn't tell.\n\n\"You never learn, do you?\" Brad demanded. \"I thought I took care of this, but obviously\u2014\"\n\n\"Shut up,\" Kris interrupted. \"What is even wrong with you? This is none of your business, man. Let it go.\"\n\n\"No. You\u2014\" Brad jabbed his finger at Rayne. \"You're exploiting him for your sick gay exhibitionist fantasies, and he's going along with it because he thinks he needs the job. You messed with his head, and now he's confused, and he's dragging the rest of us down with him. I'm bringing him home to work in Dad's garage until he figures his shit out again, and we can go back to being a normal family!\"\n\nKris punched him.\n\nIt wasn't a very good punch, as Kris had never had much practice, but it made a satisfying thwack when it connected, and the pain that shot up Kris's arm from his knuckles spoke of a job well done. Brad reeled away clutching his jaw.\n\n\"Oh, shit,\" Rayne said.\n\nKris shook out his hand. \"I'm going to need an ice pack. You think the medics have one?\"\n\n\"Yeah, probably.\"\n\nBrad was bent double now, holding his face in obvious pain.\n\n\"Should we just . . . leave him?\" Rayne asked.\n\n\"You just wait until Mom and Dad hear about this,\" Brad swore.\n\n\"What are you, five?\" Kris said. \"Get over yourself or I'll hit you again.\"\n\n\"I feel like I should be more offended about the exploitation comment,\" Rayne said, \"but he's clearly nuts.\"\n\n\"Republican,\" Kris agreed. \"If you want to fight him to defend your honor, I'll totally back you, though.\"\n\nBrad, seething, walked up and shoved Rayne hard in the chest. Rayne staggered back a step, then shoved him in return. Shoving quickly turned to grappling, completely ineffectual on both sides, while Kris tried to figure out whether he should pull them apart or join in. Rayne was the taller of the two, and more fit, but Brad was clearly running on unadulterated anger, while Rayne had just escaped a hostage scenario and looked faintly baffled by the whole thing. When Brad resorted to hair-pulling, Kris intervened, wading into the fray to grab hold of Brad's ear\u2014the only handhold he could manage\u2014and yanking. Brad howled and pulled harder on Rayne's hair; Rayne yelped, and the police and medics, who were gathered at the front of the tent presumably straightening out the story behind the cult, finally took notice and poked their heads around.\n\n\"We're being attacked!\" Kris yelled, pointing at Brad.\n\nBrad finally let go, looked around wild-eyed, and booked it. He ran straight past the cops to the nearest motorcycle, which he leaped onto as if he had any idea how to ride the thing. He kicked the engine to life and made a bid for freedom. For a minute it seemed like he might succeed\u2014Kris wasn't about to chase him down, and they'd run into each other at the next family gathering anyway\u2014but the bike let out a shuddering cough and lurched to one side. At that precise moment Freddie Mercury, apparently seeing his own reflection in the shiny chrome of the motorcycle's body and deciding it was a threat, charged the thing head-on and latched on to Brad's scalp with an unearthly wail, louder and shriller than the engine. Brad yelled and drove straight into the nearest tent, which collapsed around him in a wave of canvas as the bike made a horrible noise and literally fell apart underneath him. The engine choked, sputtered, and went dead from under the wreck. Freddie fluttered out to land a few feet away, rearranging his wings and looking pleased with himself.\n\nKris and Rayne watched the tent settle. There was a conspicuous broken-motorcycle-shaped heap in the center of it, and it wasn't moving.\n\n\"Do you think he's all right?\" Rayne asked eventually.\n\nKris couldn't muster any convincing amount of concern. \"Let the medics figure him out,\" he said. \"Let's go see how everybody else is doing.\"\n\nRayne slung his arm around Kris's shoulders like he'd done the day they first met, and Kris leaned into him, comfortable and warm, as the butterflies in his stomach finally settled into something hopeful rather than anxious, and together they walked back around to the front to rejoin the rest of their band.\n\nIn the end, the fallout went like this:\n\nBrad ended up with a broken leg, some cracked ribs, and a colorful assortment of scrapes and bruises. He might have escaped unscathed had he not made the mistake of grabbing Rikki's half-dismantled bike instead of one of the other cultists' more functional ones, but Rayne said that was karma in action. Kris wondered if he should feel bad about not feeling bad, but if anyone deserved a hospital trip, it was Brad.\n\nThe cult was arrested on twelve counts of kidnapping and unlawful confinement\u2014Travis, the fireworks arsonist, was likewise arrested and charged\u2014though Rikki narrowly escaped thanks to having been tied up at the time of their rescue, and Angel's fierce insistence on vouching for him.\n\nCalloway likewise avoided charges, though that wasn't to do with Angel, but with the label stepping in to smooth things over. Kris didn't hold a grudge, though Angel said she did, a little. Rayne seemed sympathetic, but not enough to resume their publicity stunt for the remainder of the festival\u2014not that Brian would have let him anywhere near Cal following that whole debacle. Cassie was insufferably smug about the whole scenario, though Kris probably would have been too if he'd mounted a rescue mission like that while twelve people had let themselves be captured and tied up by three or four pseudo-religious skinheads.\n\nButch confessed that he spent the afternoon alternating between kicking himself for leaving them unguarded in their moment of need, and quietly laughing at them for being taken down so easily. He swore he would never take anything Rayne said at face value again, in case he was actually being coerced into sending his security away. Rayne apologized at great length and promised to negotiate a raise, which Butch said he would use to invest in a GPS tracker implanted under Rayne's skin so he couldn't get himself kidnapped again. Brian seemed uncomfortably keen on that idea, not that Kris could blame him.\n\nNeither Passionfruit nor The Chokecherries were any worse for wear following their adventure, though Brian was verging on apoplectic as he conferred with the cops and made long-distance calls to lawyers, trying to cement the charges and get everyone's stories straight. Kris did his best to stay out of Brian's path, hoping that as long as Brian was preoccupied with those legal matters, he wouldn't have time to kick Kris out of the band for bringing the peacock to Rayne and starting the whole mess in the first place.\n\nAfter giving their statements to the cops, the bands only had a minute to spare before stumbling onto the stage for their afternoon show. Passionfruit played a good set, their adrenaline giving way to giggles as they bounced off one another, reveling in their freedom. Kris watched the show from side stage, pressed up against Rayne, their hands entangled between them. They hadn't tried to kiss again. Kris knew they would during their show, and he'd love it as he always did, but after\u2014when they left the stage, and the crowd finally went quiet\u2014they were going to tuck themselves away somewhere dark and private, away from prying eyes. He could feel the promise in Rayne's heartbeat and the heat from his body, and the light that glinted in his eyes, and in the curl of his hair.\n\n\"Are you going to tell them you've been promoted to godhood?\" Kris asked, standing on tiptoe to reach Rayne's ear and be heard above Passionfruit's din.\n\n\"I do like it,\" Rayne admitted. \"I think I'd make a good god, don't you?\"\n\n\"Like your ego needs any more stroking,\" Kris scoffed, but his heart still skipped a beat at the thought of worshipping Rayne like that, him laid out on an altar while Kris knelt for him, mouth watering.\n\nRayne squeezed Kris's hand tighter.\n\nKris didn't remember much of the show. The lights burned and the crowd screamed and Rayne was on fire, prowling to and fro like he couldn't keep his hands off Kris, and singing like he needed the whole world to sing with him. Kris sang too, away from any mike\u2014he couldn't hear his own voice over the rush of blood and the screams from the audience and the pounding music, but he sang until his throat was raw and his head was spinning and he thought, Yeah, I can see where the cult is coming from. There was nothing like losing yourself in the oblivion of something beautiful, and Rayne\u2014he was perfect. He demanded adoration, and the crowd was only too happy to give it to him.\n\nTheir stolen kisses came in rushes, hot and wet and all too brief, snuck in between verses. They were smiling too widely to do it properly, but Kris didn't care\u2014he didn't care about a thousand people watching, either, because this time he knew it was for real. The kisses didn't taste any sweeter for it, not yet. They were still a shot of sweat and cologne, bumped noses and teeth, but with every one Rayne reinforced the promise that soon\u2014as soon as they got offstage\u2014he was going to take Kris apart atom by atom, and Kris was going to love every second of it.\n\nKris finished the show in a daze, drunk on possibility. They didn't stay for an encore\u2014\"Tomorrow,\" Rayne said, \"for the last show, but not now\"\u2014and Kris handed his guitar to Cassie without a word, not even stopping to get changed, as Rayne grabbed his hand and dragged him back to their bus, while the rest of the bands whooped and wolf-whistled behind them.\n\n\"What happened to the no-fucking-on-the-bus rule?\" Kris asked.\n\nRayne paused, one foot on the step. \"Is that what we're doing right now?\"\n\nKris wanted to do everything and then some. In lieu of answering, he climbed the last step of the bus and reached up to wrap his arms around Rayne's neck and kissed him on the mouth.\n\nIt was better than kissing drunk or kissing onstage. Kris waited for the screams, or fireworks, or for the earth to tilt off its axis. None of it came. All he felt was warmth, starting in his lips like the burn from eating too many peppermints, and traveling down his body to burst around his heart and pool in his stomach, low and burning, until every inch of his body was transfused with it. He felt like he was glowing, but he didn't want to open his eyes to look.\n\nRayne was solid against him, his hands running up Kris's shoulders to tangle in his hair, teeth tugging at Kris's bottom lip like he wanted to eat him alive. Kris would have let him.\n\n\"Can we\u2014\" Rayne began.\n\n\"Yes.\"\n\nRayne laughed and backed Kris farther into the bus, toward the bunks. They stopped only when the backs of Kris's knees hit the edge of his mattress. Rayne met his gaze, a question in his eyes, before pushing Kris down to sit. \"I thought about this,\" he confessed, \"a lot more than I meant to. I imagined a hotel where I could spread you out on a real bed and take my time with you. Show you how good it can be.\"\n\n\"Yeah?\"\n\n\"Really draw it out,\" Rayne promised. \"Make it last for days.\"\n\nKris blushed hard and his mouth went dry. He could practically feel his pupils blowing out, and he straightened, reaching for Rayne. There was little room in the bus, and as soon as they both tried to cram into Kris's bunk there would be even less, but he was up for the challenge. He didn't need to take up much space.\n\nRayne laughed. \"God, I can't believe I ever thought\u2014 'Historically straight,' as if you were ever\u2014\"\n\n\"Stop talking,\" Kris ordered, grabbing Rayne's wrists. \"Either kiss me or take your shirt off, I don't care which.\"\n\nRayne pulled his wrists free and unbuttoned his shirt one hole at a time, teasingly slow. The silk shifted, revealing his skin in glimpses, and Kris's heart beat faster with every inch Rayne exposed.\n\n\"Tell me what you want,\" Rayne said. \"Tell me what you thought about.\" He finally let his shirt fall open, and dropped it behind him on the bus floor. Kris shuffled forward until he was sitting on the very edge of the mattress, his feet on the floor, with Rayne standing between his knees, the opposite wall of bunks within touching distance of Rayne's back. Kris put his hands on Rayne's hips, his thumbs brushing the skin just above his belt.\n\n\"Get down here.\"\n\nRayne ducked and crawled into the bunk beside Kris, all warm skin and still too many clothes.\n\n\"I need to memorize every single one of your tattoos now,\" Kris told him.\n\n\"Okay,\" Rayne said, breathlessly. \"Sounds good.\"\n\nKris started at Rayne's throat, on the side of his neck where the wild roses were. He kissed his way down to their leaves on Rayne's shoulder, leaving tiny lipstick marks like petals as he went. His lipstick was purple this time, so dark it was nearly black. He couldn't tell the difference between the ink and the plain skin by taste alone\u2014it was all sweet and salty\u2014but he was diligent in tracing every line, either with his tongue or his fingers. He kissed his way across the key under Rayne's collarbone. He kissed the two birds on his chest. His breath ghosted over the snake that curled down Rayne's arm; the last of the scab had since fallen away, but it was still so fresh in Kris's memory that he was scared to touch it. He ran his fingers over the mandala on the back of Rayne's neck and the sun between his shoulder blades, slow and deliberate, while Rayne panted and shivered under him. Kris was in no rush. Their last time had been so hurried\u2014he didn't regret it, not for a second\u2014but now, he wanted to take his time.\n\nHe nipped Rayne's skin, and Rayne tightened his grip in Kris's hair, warningly. Kris laughed, his breath huffing over Rayne's chest. \"You can tell me to stop,\" he pointed out, \"if you have a better idea.\"\n\n\"I might,\" Rayne said.\n\n\"Does it involve taking off your pants?\" Kris dipped lower, down past Rayne's ribs to follow the faint line of his abs, then over to bite at the jut of his hip bones. He traced the veins in Rayne's stomach down to where they disappeared under his belt. \"You hiding any more ink down there?\"\n\n\"No.\"\n\nKris sat up. \"Guess I'm done, then.\"\n\n\"You're the actual worst,\" Rayne informed him, and pulled him into a kiss. Whatever was left of Kris's lipstick smeared between them, and he moaned into Rayne's mouth. When they pulled apart, Rayne looked fucked and all they'd done was make out; Kris couldn't imagine how he appeared, with his makeup trashed and his hair mussed. Debauched might be the word for it.\n\nHe could always get more debauched.\n\n\"You, sit,\" he said, and climbed over Rayne and out of the bunk to kneel on the bus floor, manhandling Rayne into the position he wanted: upright, with his feet on the floor and knees spread wide. Rayne went obediently, and soon Kris was sitting between his thighs, at eye level with his belt buckle. Kris wet his lips, his insides turning somersaults.\n\n\"I don't really know how to do this, so you don't get to complain about technique, okay?\"\n\n\"Baby, you could bite it off and I'd probably thank you,\" Rayne said. \"I mean, try not to, but the bar is low.\"\n\n\"Personally, I'm just going to try not to choke.\" Kris took a deep breath. He was shaky, but the expression on Rayne's face\u2014like Kris was awe-inspiring, down on his knees like this\u2014was exhilarating. He reached for Rayne's belt. \"Okay, rock star. Let's see what you've got.\"\n\nRayne threaded his fingers through Kris's hair and tugged to get Kris to look up again. \"Hey,\" Rayne said. \"Love you.\"\n\nKris's butterflies flipped over and he grinned. \"Love you too.\" He slid Rayne's belt from its buckle and smoothed his hands over Rayne's hips. \"Okay. Here we go.\"\n\nAfter, they lay side by side, crammed in the narrow bunk, sweaty and glowing.\n\n\"I still owe you dinner and roses,\" Rayne commented, dragging his fingers up and down Kris's side like he was petting a giant cat.\n\nKris hummed and stretched, tangling their legs together in the sheets. \"We're pretty close to Vegas. We could do a stopover before we head to LA. Hit the strip, find a little neon chapel, and get hitched on the down-low . . .\"\n\nRayne flicked him in the ear, and Kris laughed and squirmed away.\n\n\"Can you imagine what Brian would say?\" Rayne asked. \"He wouldn't let us out of his sight for the rest of our contracts. Maybe never.\"\n\n\"Okay, no shotgun wedding.\" Kris shuffled over to lie on his side, propped up on one elbow to look Rayne in the eye. Rayne gazed up at him, blissed out and adoring. \"Roses and dinner for sure, though. No cop-outs this time.\"\n\nRayne caught Kris's hand and brought it to his mouth, pressing kisses over Kris's knuckles. \"No cop-outs this time,\" he promised. \"Like I said in New York\u2014whatever you want, I'll get you. Anything in the whole world, babe, it's yours.\"\n\n\"Well, I want to stay in the band,\" Kris said. \"After my contract's up, I want to sign on for good, if Brian's not out for my head.\"\n\nRayne nodded. \"He likes you, really. He'll let you stay.\"\n\nKris smiled, relieved. \"And I want to do this again. Or something like this.\"\n\n\"I wasn't joking about the hotel,\" Rayne said immediately. \"Or if you don't want a hotel\u2014anywhere. Anytime. Whatever you want, that's what I want.\"\n\n\"Will you still take me to India and show me around? With or without the band, I'd really like that.\"\n\n\"Yes, absolutely. I'll take you everywhere and show you everything. What else?\"\n\nKris bit his lip. \"Mostly I just want you, any way I can get you.\"\n\nRayne kissed him softly. \"I'm sorry I didn't believe you the first time you told me. I panicked. I thought you were too good to be true.\"\n\nKris rested his head on Rayne's chest and let Rayne play with his hair. \"I'm pretty sure I've fucked up my fair share in this. Let's call it even.\" He dropped a kiss to the nearest bird tattoo.\n\nRayne laughed and tugged his hair. \"So we're good?\"\n\n\"We're good.\" Kris kissed the underside of Rayne's jaw and flung an arm around his ribs, nestling in close. \"We're perfect.\"\n\nThe next morning, Kris, Angel, and Rikki sat side by side in the shade of the bus, their legs stretched out in the sand, watching the clouds drift by. They were white and fluffy like cotton candy, and Kris wanted to remember the image forever: the bright-blue sky and the fierce orange desert sand, cacti on the horizon, and the endless highway winding through it all. Angel's thigh was warm where it pressed against his, casual and unobtrusive. Rikki sat on her far side, and Kris shut his eyes, breathing in the hot, dry air, and took a moment to enjoy the company. They were a million miles from Rayne's penthouse where he and Angel had first met, and Kris couldn't imagine life without her now\u2014her or Rayne.\n\n\"You and Rayne are a sure thing now, for real?\" she asked.\n\n\"For sure for real,\" he confirmed.\n\n\"Finally. It was driving me crazy trying to get you two on the same page, you know. Like you were both determined to make things as hard as possible.\"\n\n\"Sorry. Next time I'll listen to you from the start.\"\n\n\"Smarten up and there won't need to be a next time,\" she suggested.\n\n\"That's fair.\" He couldn't speak for Rayne, but now that he had him, Kris had no intention of letting this relationship slip through his fingers. He poked Angel in the shoulder. \"Thanks for not giving up on us.\"\n\nShe rolled her eyes and swatted him away, but she was smiling.\n\n\"What will you do after the festival?\" Rikki asked Angel.\n\n\"I'll finish the tour, then head back home to my club for a while. What about you? You've got no gang anymore.\"\n\nRikki watched the sky for a minute, his expression contemplative. \"I think that's good, though. I don't know what I'll do, but I'll figure something out. Like a fresh start.\" He dropped his gaze and turned his hands over in his lap, knotting his fingers together before asking softly, like it had been bothering him for a while: \"How come you took a chance on me?\" He kept his head down like he was afraid of what he might see in their faces.\n\n\"Did you not want us to?\" Angel asked.\n\nHe shrugged.\n\nShe touched his chin and raised his head. His gaze flickered before settling on hers, his eyes wide and cautious. \"I'm a black trans girl living in America: I have to believe in the good in people. If I weren't an optimist, I might as well be dead. I'll help you figure things out, if you want.\"\n\nWhen Rikki leaned in to kiss her, he did it slowly, like he expected her to move away or shove him back, but she did neither. Kris cleared his throat and got to his feet.\n\n\"I'll leave you to it,\" he said. They ignored him, so he ambled out of the shadow, scuffing the dust with his sneakers as he walked. It was a beautiful day, and cults notwithstanding, he'd be sorry to leave the desert. Still, he expected LA would be just as breathtaking, if in a different way. Hopefully with less kidnapping.\n\nHe returned ten or fifteen minutes later and dropped back down beside them like he'd never left. They both seemed bonelessly content, their fingers tangled together in the dust between their legs, shy smiles on their faces, eyes downcast. Kris thought back to his night on MDMA, when every touch sparked like electricity. Angel and Rikki were practically glowing; was that how Kris had seemed when he looked at Rayne? He elected not to comment apart from jostling Angel with his shoulder as he got comfortable.\n\n\"So?\" he asked.\n\n\"He said he'd come back to the White Rabbit with me. Help out with repairs, that kind of stuff.\"\n\nRikki nodded, something reverent in his gaze. \"Anywhere. Anything, as long as it's with her.\"\n\nKris decided to trust Angel's judge of character. \"Well, good. I'm happy for you.\"\n\nThe clouds had long since drifted past, leaving the sky bright blue and blue alone. Rikki curled up against Angel, leaning his head to rest on her shoulder, and traced shapes on the back of her hand with his thumb.\n\n\"You see that blond guy over there?\" Kris asked, nodding to Tom, who was wandering the grounds, seemingly elated by his surroundings, and possibly high. \"He's going to be a priest someday. If you still have questions about finding a new god, you should talk to him sometime.\"\n\n\"I don't know if I want a new god,\" Rikki said.\n\nThe way he looked at Angel, he might have already found one.\n\nTom paused in his walk and glanced over. He brightened and waved. Kris and Angel waved back.\n\n\"I'll introduce you,\" Angel said. \"Never too early for him to start practicing sermons, right?\"\n\n\"Oh, I gave him a head start on that already,\" Kris said.\n\n\"Okay,\" Rikki said, turning so he was nestled more securely against Angel's shoulder, her hair brushing his face. \"I'd like that. Anything you want.\"\n\nAngel hummed and held him close.\n\n\"So, back to the White Rabbit, huh?\" Kris said. \"You're done with the music industry?\"\n\n\"Not until this tour is over,\" Angel said. \"I'll get Rayne set up with a new makeup artist long before I call it quits. The Chokecherries will get by fine without me. I miss my club; it deserves some undivided attention for a time.\"\n\n\"Course it does. It's your baby.\"\n\n\"And we'll still see each other,\" she added. \"Hell, Rayne will probably book you to play there before the year is up. We'll be just fine.\"\n\nKris wasn't worried. He felt on top of the world, like he had everything he'd ever wanted. It was hard to worry in that state.\n\n\"Tell me about your club?\" Rikki asked.\n\nAngel smiled and tipped her head back to look up at the sky. \"It's a burlesque club. You know anything about burlesque?\"\n\nTheir parents came to say goodbye that afternoon, whisking Kris and Cassie out for lunch one last time before they parted ways. Brad had been left behind at the hotel, apparently sulking like a toddler but unwilling to start another fight with his leg in a cast and his ribs bruised to hell and back, in almost as bad shape as his dignity\u2014that, and the fact that Freddie Mercury was still roaming the grounds as part of Rayne's ensemble.\n\n\"The doctors say he'll recover just fine,\" their mom said as they sat around the diner table, \"though he'll be sore awhile.\"\n\n\"As soon as he's come down from his painkillers, we'll be giving him a talking to, you believe me,\" their dad added. \"It's not how we raised him\u2014not how we raised any of you\u2014and we won't stand for it any longer, not as long as he calls himself a Golding.\"\n\n\"I'm sure that'll go over great,\" Kris said. \"Rayne's offering to help pay the medical bills, by the way.\"\n\n\"Oh, dear\u2014no, we couldn't possibly accept that,\" their mom said, visibly flustered. \"Tell him thank you, but we'll get by.\"\n\nKris shrugged. He had protested the idea too at first, but Rayne was as stubborn as he was loaded, and the battle had been short-lived.\n\n\"I'm not a saint,\" Rayne had said. \"I'm not offering because I think he deserves a second chance. I just want to see his face when he realizes he owes me his life, and I won't let him pay me back to settle the score. Anyway, your parents shouldn't have to shoulder his bills all by themselves.\"\n\nKris was confident Rayne could win his parents over as well.\n\n\"You and Rayne, though,\" his mom said.\n\n\"Us,\" Kris agreed.\n\n\"The young man he was seeing earlier wasn't serious?\"\n\n\"That's a funny story, but no. They're not together. They never were.\"\n\n\"But you two are? Should we expect him for Christmas?\"\n\n\"I think so, and . . . I don't know? But it's good. I'm happy.\" He let his grin burst out. \"I'm really happy.\"\n\nHis parents beamed back at him as Cassie leaned over for a fist bump.\n\n\"So, Knocks is getting his cast off in a couple of weeks,\" Cassie said, smoothly changing the subject as she chased her waffles through a sea of syrup, \"but the guys have been talking.\"\n\n\"Knocks has been playing rhythm guitar in the meantime,\" Kris supplied for their parents.\n\n\"He likes it,\" Cassie said. \"A lot. So much, actually, that they asked if I wanted to stick around for the rest of the tour.\"\n\n\"Is that something you want?\" their mom asked.\n\n\"Hell yes, absolutely. This is the best thing I've ever done. I was thinking . . .\"\n\nTheir parents looked at each other in trepidation.\n\n\"About what happens after the tour. Assuming I stay on as their drummer, of course.\" She speared a waffle piece with a strawberry on it and popped it in her mouth. \"Because they said they like playing with me, and if Knocks wants to keep playing guitar, well, after the tour they're going to start work on their next album.\"\n\n\"You want to join the band for real,\" Kris said.\n\n\"Obviously.\"\n\n\"You've got your last year of college starting in September,\" their dad pointed out, in a careful, nonjudgmental voice.\n\n\"That's the part I was thinking about,\" Cassie said. \"I can put school on hold and go back next year if this doesn't work out, but I won't get another chance to join a band like this. Not without starting from scratch and forming my own, you know?\" She looked back and forth between them.\n\n\"This isn't something you should jump into,\" their dad began.\n\n\"Like I jumped into drumming for them?\"\n\nHe held up his hands. \"It's your future, Cassie, and your education on the line. You've only known them a week.\"\n\n\"We're not saying don't do it,\" their mom added. \"We're saying sleep on it a little longer.\"\n\n\"These are all hypotheticals,\" Cassie said. \"We have to finish the tour first. Hell, we have to get Knocks's cast off first. I just . . . I think it would be really cool. I'm happy here. I feel like I could do it forever.\"\n\nTheir parents softened.\n\n\"Both my babies growing up and joining bands,\" their mom said. \"Touring the world. What a turn.\"\n\n\"I'm twenty-five,\" Kris said, without much rancor. \"I've been grown-up for a while.\"\n\n\"You look like you're going through a second puberty, though,\" Cassie pointed out cheerfully. \"You're wearing more makeup than I did in high school.\"\n\nKris kicked her under the table and she kicked back, grinning.\n\n\"As long as you're happy,\" their mom said again.\n\n\"We are,\" they chorused together, and Kris had never felt anything truer in his life.\n\nWhen they returned to the festival, he called Brad. He was still angry; Brad getting thrown through the wringer hadn't changed that.\n\n\"Hey, listen: we're heading to LA in a day and then we're going for the international leg of the tour, so we won't see each other again for a while. I wanted to make sure you knew something, before I left.\"\n\n\"What?\" Brad croaked. He sounded terrible, but if the doctors said he was going to be fine, Kris wasn't going to worry about it.\n\n\"We hooked up, me and Rayne. I sucked him off and he told me I was pretty, and we had a really good time. And in another day or two, after we're back on the road, we're going to book a hotel and I'm going to ask him to fuck me, for real.\"\n\nBrad sputtered incredulously over the line.\n\n\"And I'm going to fucking love it, Brad, and there's nothing you can do about it.\"\n\nHe ended the call before Brad could respond and turned off his phone with a smile. Though his heart was racing, he felt insurmountable.\n\n\"Are you going to be okay?\" Kris asked.\n\nHe and Calloway stood by The Chokecherries' bus. Passionfruit was wrapping up their final show, and he was due to take the stage in a few minutes. The last show of Purple Sage.\n\n\"I feel like I owe everyone an apology, though I'm not sure what to say,\" Cal said. \"I'm grateful Rayne's not pressing charges.\"\n\n\"He's good like that.\"\n\n\"I should have spoken up as soon as I saw you with the bird. But I was scared the label was going to drop me if I caused the slightest problem, and I hoped . . . you know. I hoped I was wrong about the cult and everything. I hoped the whole thing would blow over before it could come to a head.\" Cal rubbed the back of his neck. \"I fucked up on that one, and I'm sorry for it.\"\n\n\"No lasting damage done,\" Kris said amiably, \"but yeah, an earlier heads-up would have been nice.\"\n\nCal nodded. \"I feel like I should apologize on behalf of the label, as well, for that whole publicity stunt. I suppose I got in the way a bit, didn't I?\"\n\n\"The stunt definitely wasn't your fault. I fucked that part up on my own. We all made our fair share of mistakes the past week or two.\"\n\n\"I suppose we did. Granted, some were bigger than others.\" Cal shook his head with a wry smile. \"But you and Rayne are an item now, I hear? That's good. He was fucking pining for you, man, as sure as you were for him. Absolutely painful to watch, I'm telling you.\" He tentatively touched Kris's shoulder. \"I'm glad you got it sorted. Sorry I wasn't much help.\"\n\n\"Sorry your stunt got cut short,\" Kris said sincerely. \"I guess you never got enough press coverage to really boost your career like you wanted, huh? Now all people are talking about is the hostage-cult thing.\"\n\n\"Ah, it's for the best, I think. I'm not sure I'm ready for the sort of fame Rayne has. I'll take a few years to grow into it.\" Cal tipped his head back to squint at the sun. \"Actually, I thought I might take a break once this bit of touring is over, see a counselor, maybe. Talk about the whole cult scenario, get it off my chest.\"\n\n\"Yeah?\"\n\nButch hollered at them from the stage, beckoning Kris over.\n\nKris clapped Cal on the arm. \"Listen, it was cool meeting you, despite everything. Good luck with the band and the counseling, if you go that route. I think it'd be good for you.\"\n\nCal smiled widely, the same smile from that picture Kris had seen way back before he'd met him. \"Good luck to you too. We should keep in touch\u2014Rayne Bakshi's fake love interests. Though of course you're the real thing now.\"\n\n\"We should. We will,\" Kris promised as Butch yelled again, this time with greater insistence.\n\n\"Go on,\" Cal said, laughing. \"Give him one last kiss from me, will you?\"\n\nKris grinned and shivered\u2014he could kiss Rayne now, he was allowed, and there was no one to stop him. Waving goodbye to Calloway, he broke into a run, heading to the stage for their last show as his heart tripped with excitement in the best way possible.\n\nHe arrived backstage just as Lenny drummed out his cue. Slinging his guitar strap over his shoulder, he ran to take his place in front of the crowd. Rayne caught him before he went out, pressing close to drop a kiss to Kris's cheek. Kris twisted around to kiss him back\u2014mostly teeth, he couldn't stop smiling\u2014before slipping out onto the stage.\n\nTheir intro song flared up around him as he shouted a hello to the crowd, which roared back at him in adulation. Standing there, soaking up the noise and the heat and the love, he felt like he was flying. He got to kiss Rayne again and mean it, and keep meaning it, for as long as they were together. He couldn't think of anything better.\n\nWhen Rayne took the stage, Kris could feel the crowd's excitement all the way down to his bones. He kept pace with the rest of The Chokecherries as they careened through their intro song, and then everything dropped off, music and screaming cheers alike, as Rayne slunk up to the mike.\n\n\"This is our last show of the festival,\" Rayne said. The crowd cheered. \"But these kinds of things stay wild right to the end. Did everybody have a good time?\" Screams. \"Yeah? It got crazy there for a minute, let me tell you. But as long as it turns out all right in the end, that's what matters. You have to remember that.\" He turned to look at Kris. \"Now, this show\u2014I want to dedicate it to all the lovers out there. I don't care if you're fifteen or fifty or ninety-three, if it's your first love or if you met an hour ago or if you've been married for thirty years. All the lovers, and everybody who's still looking: this show is for you.\" He paused for Lenny to tap out a beat. \"So let's get started. You know how it goes. You going to show me you love me?\"\n\nThe crowd screamed and stomped and flung their hands up in worship, and Kris closed his eyes and let it all wash over him. When he opened them, Rayne was watching the crowd, leaning on his mike stand with both hands as he basked in their attention, the stage lights casting a halo down around his curls. He was larger than life, his skin shimmering with glitter, and Kris had never seen anything more beautiful in his life.\n\n\"Yeah, just like that,\" Rayne said. \"Okay. Let's roll.\"\n\nDear Reader,\n\nThank you for reading Arden Powell's A Summer Soundtrack for Falling in Love!\n\nWe know your time is precious and you have many, many entertainment options, so it means a lot that you've chosen to spend your time reading. We really hope you enjoyed it.\n\nWe'd be honored if you'd consider posting a review\u2014good or bad\u2014on sites like Amazon, Barnes & Noble, Kobo, Goodreads, Tumblr, Twitter, Facebook, and your blog or website. We'd also be honored if you told your friends and family about this book. Word of mouth is a book's lifeblood!\n\nFor more information on upcoming releases, author interviews, blog tours, contests, giveaways, and more, please sign up for our weekly, spam-free newsletter and visit us around the web:\n\nNewsletter: riptidepublishing.com\/newsletter\n\nTwitter: twitter.com\/RiptideBooks\n\nFacebook: facebook.com\/RiptidePublishing\n\nGoodreads: tinyurl.com\/RiptideOnGoodreads\n\nTumblr: riptidepublishing.tumblr.com\n\nThank you so much for Reading the Rainbow!\n\nRiptidePublishing.com\n\nThank you first and foremost to my beta readers: Lin, for reading my very first draft and telling me about New York; Neurtsy, for assuring me it was funny and fact-checking my drug scenes; and K.D., for general encouragement and positivity, even when I'm sure I was driving her mad.\n\nThanks to my parents, Earla and Jamie, for listening to me ramble on about writing and rewriting for months on end, and promising to buy a copy (or ten) when it was done.\n\nThank you to my entire publishing team at Riptide, who made my book beautiful, but most specifically my editor: for pulling me from the purgatory of the slush pile, for excising my draft of errant semicolons, and for helping shape this book into something much better than I could have done on my own.\n\nReset to Zero\n\nThe Botanist's Apprentice\n\nArden graduated from St. Francis Xavier University with an Honours degree in English literature and the realization that essay writing is just another form of making up stories. They also came away with an overriding and all-abiding love of semicolons, to the general dismay of their editors. Arden lives in Ontario with a dog, a fellow human, and an unnecessary number of houseplants.\n\nWebsite: ardenpowell.wordpress.com\n\nTwitter: twitter.com\/ArdenPowell\n\nGoodreads: goodreads.com\/author\/show\/8053059.Arden_Powell\nEnjoy more stories like A Summer Soundtrack for Falling in Love at RiptidePublishing.com!\n\nRunning with Scissors\n\nKeeping secrets on a tour bus isn't easy.\n\nriptidepublishing.com\/products\/running-with-scissors\n\nRock N Soul\n\nHe's a ghost. I'm a bellboy. What could possibly go wrong?\n\nriptidepublishing.com\/products\/rock-n-soul\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nS T O N E W A L L\n\nD A V I D\n\nC A R T E R\n\nS T. MARTIN'S G R I F F I N   N E W Y O R K\nSTONEWALL.  \n _Copyright \u00a9 2004 by David Carter.  \nAll rights reserved. Printed in the United States of America.  \nNo part of this book may be used or reproduced in any manner whatsoever  \nwithout written permission except in the case of brief quotations  \nembodied in critical articles or reviews. For information, address  \nSt. Martin's Press, 175 Fifth Avenue,  \nNew York, N.Y.  \n10010_.\n\n_www.stmartins.com_\n\n_Library of Congress Cataloging-in-Publication Data_\n\n_Carter, David_.\n\nStonewall: the riots that sparked the gay revolution \/ by David Carter.\n\n_p. cm._\n\n_Includes bibliographical references (p. 311) and index (p. 323).  \nISBN 0-312-20025-0 (hc)  \nISBN 0-312-34269-1 (pbk)  \nEAN 978-0312-34269-2  \n1. Homosexuality. 2. Lesbianism. 3. Gay liberation movement\u2014United States. I. Title_.\n\n_HQ76.C3155 2004  \n306.76'6'0973\u2014dc22_\n\n2004040226\n\n_First St. Martin's Griffin Edition: June 2005_\n\n_10 9 8 7 6 5 4 3 2 1_\n_To the gay street youth  \nwho fought and bled at Stonewall_\n_I have no doubt we shall win,  \nbut the road is long, and red with monstrous martyrdoms_.\n\n\u2014OSCAR WILDE,  \nLETTER TO GEORGE IVES\n\n# C O N T E N T S\n\n_Prologue_\n\nSETTING THE STAGE\n\n1. Greenwich Village, USA\n\n2. Oppression, Resistance, and Everyday Life\n\n3. On the Street\n\n4. The Stonewall Inn\n\n5. The Skull\n\n6. Dawn Is Just Breaking\n\nTHE STONEWALL RIOTS\n\n7. A Friday Night Out\n\n8. \"We're Taking the Place!\"\n\n9. Lancing the Festering Wound of Anger\n\n10. \"Christopher Street Belongs to the Queens!\"\n\n11. \"They've Lost That Wounded Look\"\n\nGAY LIBERATION\n\n12. Seizing the Moment\n\n13. \"We're the Gay Liberation Front!\"\n\n14. The Heroic Age\n\nConclusions\n\n_Author's Note_\n\n_Acknowledgments_\n\n_Notes_\n\n_Bibliography_\n\n_Oral Histories_\n\n_Credits for the Photographs_\n\n_Index_\n\n# Prologue\n\nThe Stonewall Riots were a series of violent protests and street demonstrations that began in the early morning hours of June 28, 1969, and centered around a gay bar in the Greenwich Village section of New York City. These riots are widely credited with being the motivating force in the transformation of the gay political movement. Everything else about these riots\u2014how they started and who was responsible; who were the patrons of the Stonewall Inn; who were the owners; what happened at the riots; who was there and who was not; and pretty much everything else revolving around these riots\u2014has been a bone of contention between various individuals and interest groups within and without the gay world. This book attempts to bring together everything that is known about the Stonewall Inn, the riots themselves, and the life and times of the people involved (gay men, lesbians, transvestites, and others) to present the clearest possible picture of what happened and why.\n\nIt was only a few decades ago\u2014a very short time in historical terms\u2014that the situation of gay men and lesbians was radically different from what it is today. At the end of the 1960s, homosexual sex was illegal in every state but Illinois. Not one law\u2014federal, state, or local\u2014protected gay men or women from being fired or denied housing. There were no openly gay politicians. No television show had any identifiably gay characters. When Hollywood made a film with a major homosexual character, the character was either killed or killed himself. There were no openly gay policemen, public school teachers, doctors, or lawyers. And no political party had a gay caucus.\n\nIt is common today to trace the tremendous gains made for lesbian and gay rights since the early 1970s back to the Stonewall Riots of 1969, when gay men, transvestites, and lesbians fought the police during a routine raid on a popular gay club in Greenwich Village. It is also commonly asserted that the riots, which continued on and off for six days, marked the beginning of the \"gay rights movement.\" Gay people had founded a political movement for the rights of gay people prior to Stonewall, although of modest means, and it was the Stonewall Riots that resulted in the birth of the Gay Liberation Front (GLF) and later of the Gay Activists Alliance (GAA). These exemplars of a new kind of gay organization, imbued with the militant spirit of the riots that engendered them, soon inspired thousands of gay men and lesbians across the country\u2014and ultimately around the world\u2014to join the movement for gay civil and human rights.\n\nThe immense changes precipitated by the Stonewall Riots make us want to know more about this event and to understand its causes. Yet the riots have seemed to a large degree inexplicable. Why did a sustained resistance occur when the police raided this particular club? Why did gay men take a stand at that time and at that place?\n\nResearch conducted for this history revealed that the riots occurred for a number of reasons having to do with timing, social history, cultural changes, and local history and geography, as well as political events. Moreover, many of these causes were as subtle as they were real and thus have long eluded notice. The factor of time further complicates determining the event's causes: not only did these causes and contributing factors not occur simultaneously but they were spread out over wide vistas of time, with some of the underlying causes occurring decades prior to the event while others were as fresh as that week's headlines. How these various strands eventually came together to create a turning point for the gay rights movement is the subject of this history. Given the varied nature of the riots' causes, their origins at different points in time, and their multiplicity, the way in which these factors converged to create the Stonewall Riots is an intricate story. Yet almost all the causes lay where the riots took place, in America's bohemia, Greenwich Village.\n\n# S E T T I N G\n\n# T H E\n\n# S T A G E\n\n#\n\n# Greenwich Village, USA\n\n## G R E E N W I C H V I L L A G E  \nB E F O R E T H E S T O N E W A L L R I O T S:  \nH I S T O R Y A N D T R A D I T I O N S\n\nIn the late 1960s, Tony Lauria, known to his friends and associates as Fat Tony, the son of an important Mafioso named Ernie, decided to open a gay bar in Greenwich Village. He did so despite the unhappiness it caused his father, a man so conservative that it was said that meeting with him was almost like having an audience with the pope. Ernie had made his fortune in traditional Mafia operations such as the carting business and felt that running a \"fag bar\" was for people on the lower echelons of the Mafia hierarchy. The father had high ambitions for Tony and sent him to Xavier, a Catholic preparatory school. Despite the quality education Tony had received\u2014as shown by his good diction\u2014he preferred to hang out on the street with other neighborhood boys whose thick Italian accents made them sound like actors playing mobsters. The father's success, his lofty aspirations for his son, and his displeasure at his son's barroom venture suggest that an archetypal father-son conflict may have been behind Fat Tony's decision.\n\nFat Tony's father owned the apartment building on the southwest corner of Waverly Place and Sixth Avenue, in which he and Fat Tony lived. An impressive structure, the proud building towers over its neighbors and displays knights carved in stone on its facade. At seventeen stories it is high enough that its upper floors command a fine view of the neighborhood.\n\nLooking east, Washington Square Park, for many the very epitome of the Village, is just a block away. The popular park has a history hardly suspected by the many New Yorkers who think of it only as a pleasant place to walk, little knowing, for example, that the golden leaves they enjoy strolling through in the fall have grown out of the bodies of other New Yorkers. In the city's early days, when most of New York's population still inhabited Manhattan's southern tip, a marsh covered the park. During the 1798 cholera epidemic the city desperately needed an out-of-town paupers' graveyard and drained the marsh to meet this exigency. On the northwest corner of Washington Square an extremely tall English elm tree stands only about ten feet from the park's edge. A straight line drawn west from this tree would practically hit the front of Tony's father's building. Some say the tree on the corner is the oldest tree in the entire five boroughs. Oldest or not, the Hanging Elm earned its name when Manhattan established a public gallows and chose it as the site for executions. Perhaps the city chose the tree because of its proximity to the paupers' graves, which allowed the city authorities to dispatch its least valued citizens without bothering to haul the bodies away. When the graveyard had consumed ten thousand bodies, the poor were not even shown the minimal respect ordinarily granted a final resting place, for the graveyard was converted into a military parade ground. Eventually, the beggars and the criminals had their revenge. When the army brought in their heavy artillery to show it off, the weight proved too much for the decaying bodies to support and the weapons collapsed into the unmarked graves of the poor. After that unexpected defeat, the city turned the site into a park.\n\nTo the west of Ernie's building lay another vista with a compelling past. The street that runs from the Hanging Tree to the front of the mob-owned apartment building intersects Christopher Street just where Fat Tony's new business was situated\u2014but not before passing the Northern Dispensary, the city's oldest clinic, where Edgar Allan Poe had been a patient. (Poe would no doubt have appreciated the irony that immediately adjacent to the building where sick people had gone seeking to escape the grave there was once a sausage factory; in recent years, as if inadvertently betraying its ancestry, the ground floor of the same building housed a leather clothing store.)\n\nFat Tony could hardly have found a street with a more colorful history than Christopher Street for his new business. The oldest and longest street in Greenwich Village, Christopher Street at one time extended beyond its current length to about the middle of what is now West 8th Street between Fifth and Sixth avenues. Where Christopher Street used to begin was the location of the Eighth Street Bookshop, the most influential bookstore in Manhattan for Beat literature. In 1964, when Allen Ginsberg\u2014better known to some for being openly homosexual than for his poetry\u2014returned from his long stay in India, he stayed in a room above the store while looking for a place of his own. It was at the Eighth Street Bookshop that Al Aronowitz, the _New York Post_ reporter who had written some of the first articles on the Beats, dropped by one day with a young folk singer he wanted Ginsberg to meet, namely, Bob Dylan.\n\nGiven Christopher Street's length and history, it is not surprising that a walk down even a couple of its blocks can provide a sampling of the long Village tradition of bohemian life and its influence across time. In walking from where the Eighth Street Bookshop stood to the Northern Dispensary on Christopher Street, the short physical distance traveled suggests a parallel journey of ideas over a vast expanse of time. Allen Ginsberg found inspiration in the work of Edgar Allan Poe, calling him \"the first psychedelic poet,\" making this short tour all the more suggestive, for while 8th Street had the Eighth Street Bookshop as a cutting-edge literary presence in the 1950s, in the late 1960s the street became one of the main purveyors of psychedelic posters and clothes.\n\nWhere Christopher Street begins today, at its intersection with Greenwich Avenue, we find the former site of Luke Connor's, a popular gathering place for the actors and writers of the Provincetown Players, an association of some of the twentieth century's most important talents in the theater, such as Eugene O'Neill. A few doors in from Greenwich Avenue is 11 Christopher Street, where the influential poet e. e. cummings once lived. Farther down the block, just two doors west of where the Stonewall Inn would open, was the Lion's Head, a pub popular with writers. This bar offered refuge for creative spirits from playwright Lanford Wilson and composer David Amram to writers James Baldwin, Norman Mailer, and Frank McCourt. At the end of this block of Christopher Street, at the corner of Seventh Avenue South, _The Village Voice_ 's office was in a building whose jutting triangular shape resembled a ship's prow, suggesting the forward-looking aspirations of the innovative writers and artists who worked for _The Voice;_ among them, photographer Berenice Abbott, underground filmmaker Jonas Mekas, and poet Frank O'Hara. Through its encouragement of Off Off Broadway theater, _The Voice_ helped to expand the very concept of theater.\n\nIn a subtle way, another element of the avant-garde history of these few blocks ties into this history. Just as the Village's population rose and fell over the decades and centuries, so did its reputation as a bohemian quarter. (It was the Village's reputation for unconventional lifestyles that first attracted gay people to the area around the turn of the century, as they sensed that a place known for wide tolerance might even accept sexual nonconformists.) When both Seventh Avenue and the Seventh Avenue subway line were extended south into the Village, the new easy access led to a rediscovery of the area as a bohemian enclave. This in turn led to a burgeoning of the real bohemian scene and the birth of a tourist-trap imitation one. It was in part because of the propinquity of the new subway station that Sheridan Square, close by Christopher Street, became the epicenter for both kinds of venues. These were composed mainly of clubs and another Village institution, tearooms, which were very modest restaurants that often catered to a particular clientele.\n\nWith the onset of Prohibition, artists, intellectuals, and gay men and lesbians began to socialize more and more in tearooms, since bars could no longer serve alcohol. Among the rare early American books to depict lesbian love is the autobiography of \"Mary Casal,\" _The Stone Wall_ , published in 1930. That same year, two former stables at 51 and 53 Christopher Street were merged into one building at the ground-floor level and became Bonnie's Stone Wall, which soon gained a reputation as one \"of the more notorious tearooms\" in the Village, a reputation not easily earned in a time when tearooms were routinely raided by the police. It seems reasonable therefore to assume that in naming her \"notorious\" business Bonnie's Stone Wall, the owner, presumably Bonnie, was alluding to the new memoir to send a coded message to lesbians that they would be welcomed there. The tearoom's notoriety does not seem to have harmed business, for Bonnie's Stone Wall was one of the rare cases of a tearoom that not only survived but also evolved into a full restaurant. Decades later, Bonnie's Stone Wall had lost its rebellious edge and become a popular place to hold wedding receptions and banquets and had even become a particular favorite of policemen. By the 1940s its name had already changed to the more bucolic-sounding Bonnie's Stonewall Inn, and by the 1960s it had been changed again to the Stonewall Inn Restaurant. It was the former Stonewall Inn Restaurant that, in 1967, having sat vacant for some time after a fire gutted it, metamorphosed into the gay club the Stonewall Inn. While the staid restaurant's uproarious origins were quite forgotten by the second half of the twentieth century, it seems as if fate had marked the place from its very beginnings as a site of homosexual rebellion.\n\nChristopher Street's origins go back to the time when the area that became Greenwich Village seemed remote from Manhattan's southern end, where the Dutch founded the city. When the Dutch government wanted to reward Wouter Van Twiller, the third director of New Amsterdam, with a farm, he was given two hundred acres of land within the present-day Village, near the Indian settlement known as Sapponckanican. With the passage of time, the original Dutch farmland was subdivided and resubdivided as the population of farmers slowly grew. The tempo of populating this portion of Manhattan sped up dramatically when a series of four epidemics of yellow fever and cholera struck lower Manhattan between 1791 and 1805. Early New Yorkers fled to what was then seen as an outpost so distant that they could not imagine a plague following them that far.\n\nThese flights from plague eventually transformed the Village from a rural country hamlet into an area so populous that it became necessary to lay down roads. From their earliest days, Villagers have shown a certain appreciation of their own traditions and a willingness to defend them. Perhaps the earliest manifestation of this trait occurred when the first Village roads were planned by the residents who were careful to see that the roads followed the footpaths left by the original Indians as well as those added by the early settlers. A number of the Village's original streets were therefore laid out because Indians and farmers, with their close ties to the earth, had followed paths that seemed natural to them, so that the Village's streets were created for the convenience of human feet and not for wheeled vehicles. When an attempt was made in 1817 to impose a grid plan on all of Manhattan's streets, the citizens of Greenwich Village successfully resisted the plan and the Village became the only part of Manhattan north of the Wall Street area where the new street plan was not implemented. This resistance shows that the Villagers' sense of their community as a unique place and their resistance to conformity have deep roots. To understand Villager psychology, this ingredient of feistiness must be factored in. Villagers have long been willing to fight for what they want as well as to applaud those who have the courage to stand up for their beliefs. When the antecedent to the subway trains, elevated trains (the \"el\"), were built too close to Village residences, housewives angered by the trains' loud racket are said to have stacked bricks in their kitchens to throw at the passing trains. Villagers were so proud of a leader of a riot that they named a street for him: Gay Street was named for attorney Sidney Howard Gay, the editor of the _Anti-Slavery Standard_ , to honor him for his role in the 1834 abolition riots.\n\nThe New York Public Library has a collection of 54,000 photographs of old New York, taken between the 1870s and the 1940s. The first appearance in this collection of the two buildings that would one day become the Stonewall Inn bar is a photograph taken in 1899 that shows two white horses drawing a trolley as they approach 51 and 53 Christopher Street. It is appropriate that a pair of horses are featured in the earliest image of these buildings, since they were both built to serve as stables in an era that relied heavily on animal muscle for transportation. The horses bespeak a period when Americans lived closer to the land, a slower time when people were not so alienated from their own natures or from their fellow beings. Even in bustling Manhattan, businesses took the time to pay attention to the amenities: of these two Christopher Street stables, one was home to the all-black horses that delivered goods for Saks Fifth Avenue, and one of the stable keeper's duties was to paint the horses' hooves black to match their coats.\n\nHowever, 1899 was also the year that Henry Ford started the Detroit Automobile Company and that New York City got its first fleet of taxicabs. The next image in the library's collection of these buildings is taken in 1928, and 53 Christopher Street is a French bakery. Number 51 still has \"The Jefferson Livery Stable\" emblazoned on its facade, but these words are obscured by a larger poster nailed on top of it proclaiming that the building is about to be altered into \"most desirable STUDIO APARTMENTS.\" A clear view of 51 and 53 Christopher Street is blocked by a car and a delivery truck.\n\nWhile Greenwich Village grew by fits and starts\u2014and had occasional declines\u2014it maintained a certain level of isolation until well into the twentieth century because its irregular street plan impeded a direct flow of traffic into the Village. The increasing popularity of both the automobile and the recently introduced subway system added to the public pressure to extend Seventh Avenue. At the close of World War I, Seventh Avenue, which used to end at 11th Street, was extended south, with its new section named Seventh Avenue South. _The Greenwich Village Historic District Designation Report_ , which documents 1969 Greenwich Village in detail, states that due to the extension south of both Seventh Avenue and the West Side subway line, \"the physical isolation of Greenwich Village from the main traffic routes of the City was lost forever.\" As horses and stables became rarer and rarer, gas stations and automobile supply stores filled up the small triangular plots the cut-through had made along Seventh Avenue South. The triumph of impersonal mechanical speed over a gentler and more natural mode of transportation meant that the last horses at 51 and 53 Christopher Street were evicted, and in 1930 the two buildings became one.\n\nAt the west end of the block on which the Stonewall Inn club would later open, a number of streets\u2014Christopher, Seventh Avenue South, West 4th, Sheridan Square, and Grove\u2014crisscross and converge. The resulting effect of a traffic hub in an entertainment district with busy pedestrian traffic has struck more than one observer as a scaled-down version of Times Square. Running underneath the surface of Christopher Street, a PATH commuter train connecting the Village to New Jersey runs at almost a right angle to the Seventh Avenue subway line, dropping off passengers both on lower Christopher Street, close to the Hudson River, and on 9th Street near Sixth Avenue, only about 150 feet from where Christopher Street begins. The Sixth Avenue subway was extended south at the end of the 1920s, and an entrance to the West 4th Street station, with its seven subway lines, is only a little farther away from the Stonewall Inn's door than the Christopher Street station. All in all, the Stonewall Inn was only a block away from eight subway lines, only about two short blocks away from a PATH train station, and between three major avenues: Sixth Avenue, Greenwich Avenue, and Seventh Avenue South. More significantly, the club was only a short city block and a half from Greenwich Avenue, the premier cruising ground for gay men in New York City in the 1960s. Moreover, the new bar was located the same distance from what these men called The Corner, the intersection of Greenwich Avenue and Christopher Street, the most popular meeting place for gay men on all of Greenwich Avenue. The new bar could not possibly have been more centrally located.\n\nIn addition to the highly centralized location of the Stonewall Inn, its immediate geography also had some unusual features for New York City. First, by Manhattan standards there is an unusual amount of empty space around the Stonewall Inn. Directly in front of the club lies Christopher Park and, just a little to the east of the park, the area around the triangular Northern Dispensary is fairly open, being the intersection of three streets (Grove, Christopher, and Waverly Place). Indeed, _The Greenwich Village Historic District Designation Report_ noted the way the geography of Waverly Place by the Northern Dispensary gives \"the feeling of openness, so rare in our streets today.\"\n\nThat feeling of openness sprang from the Village tradition of citizens standing up to those in power to protect their interests, for Christopher Park had formerly been the site of an awful tenement. When it burned to the ground in a fire that killed more than forty persons, Villagers protested the idea of another residential building being built there and insisted that the land be turned into a park.\n\nOther events in the history of the immediate area surrounding the small park demonstrate the Village's long history of open-spiritedness. Near the base of the triangle formed by Christopher Park is 59 Grove Street, where Thomas Paine died. Paine's pamphlets played such a vital role in starting the American Revolution that Thomas Jefferson praised him as the only American who could write better than himself, and William Blake put him in a class with Jesus as a worker of miracles for overthrowing \"all the armies of Europe with a small pamphlet.\" In spite of these accolades, Paine was so unconventional as a freethinker that he often ended up in trouble. After he went to France to support the French Revolution, he was imprisoned. When New York State gave him a free home in New Rochelle to honor him for his role in the American Revolution, his unconventional beliefs so outraged his conservative neighbors that they tried to kill him several times. At the end of his life when he needed a refuge, the Village provided it.\n\nOn the south side of Christopher Park, directly across from the buildings that became the Stonewall Inn, was the home of Phillip Stokes. Stokes had a great interest in Eastern mysticism, and so, in 1932, he invited the Indian spiritual master Meher Baba to stay in his home. It was there that Meher Baba first met the American public. He had been observing silence since 1925, communicating by means of a small board he carried on which the letters of the alphabet were painted. At the gathering at the Stokes home, Meher Baba pointed to the letters on the board to spell out his message, saying that he was observing silence only so that he could one day break it, which he said would bring about a spiritual upheaval: \"America has tremendous energy, but most of this energy is misdirected. I intend to divert it into spiritual and creative channels.\" Asked how his speaking would help such current social problems as those of politics, economics, and sex, he answered that when he spoke the results would be gradual and would affect all aspects of life. \"New values and significance will be attached to matters which appear to baffle solution at the present moment,\" he asserted. He later explained that constructive and creative forces were being released that, although working silently, would \"bring about the transformation of man.\" Harbingers of radical change\u2014artistic, political, and spiritual\u2014surrounded the diminutive triangular park.\n\nAt the west end of the block where Stonewall was located, there is a confusion of street names: Seventh Avenue South is generally called Seventh Avenue, and Sheridan Square\u2014shaped like a triangle in spite of its name\u2014is commonly referred to as West 4th Street, while everyone calls the park across the street from the Stonewall Inn \"Sheridan Square,\" even though its official designation is Christopher Park. That a statue of General Philip Sheridan looks out over Christopher Park\u2014and not Sheridan Square\u2014may have caused the confusion, whereas the statue of a Civil War general opposite an establishment named Stonewall caused at least one person to assume that the statue represented Stonewall Jackson.\n\nThe several one-way streets intersecting around Christopher Park confuse even Village residents. Someone sitting in Christopher Park, facing west and surveying the corner lampposts on the park's western end, will see that some of the lampposts have two signs, each saying ONE WAY. Because of the oblique angles at which streets intersect there, the one-way signs seem to point in opposite directions. The way streets come together into a hub here results directly from the extension of Seventh Avenue farther south. When the extension was created, it cut through the existing crazy-quilt pattern, created a century earlier when the Villagers fended off the imposition of the grid plan. With the \"quilt\" being cut again, its pattern became even crazier around the sheared-off edges, and nowhere more so than around Christopher Street, where a person standing today can see the almost surrealistic shape of some buildings sliced apart by what Villagers referred to with dread as the cut-through.\n\nThe east end of Christopher Park can be just as confusing. There Waverly Place achieves the distinction that makes it unique among all the streets of New York City: the only street in the city to wrap around itself, it does so where it crosses Christopher Street, throwing itself around the three-sided Northern Dispensary (as a New Yorker once put it, \"Where Waverly intersects Waverly intersecting Waverly\"). From this point, as if having succeeded in confusing itself, Waverly takes a sudden sharp bend like the irregular path of a drunk and continues east to Sixth Avenue. But not before running into Gay Street on its northern side, a surprisingly short street that in its own diminutive existence of one block cannot keep itself straight, veering back upon itself at an odd angle, only to turn back into Christopher Street.\n\nChristopher Street became so gay in the 1970s that some gay men wrongly assumed that Gay Street had been named in their honor. Still, from a turn-of-the-century perspective, this confused knot of streets could certainly be termed, however anachronistically, a queer geography. It was in the midst of this tangle of irregular streets and triangular open spaces that the largest gay club of the 1960s opened.\n\nTo comprehend the club's significance, however, it is first necessary to understand the various social and political forces that were shaping the specific features of the homosexual world that the Stonewall Inn would inhabit. Some of these forces were national, and many of them were specific to New York City. These factors provide the context that explains why the Stonewall Inn became a special place for gay men in general and for particular subsets of the gay population in particular. A survey of the legal situation of homosexual men and women in the United States and particularly New York City in the 1950s and 1960s will provide a framework to understand the dynamics at work in both the local gay population as well as the history of gay political organizations in New York City. The successes and failures of gay political activism did much to influence the reception and fate of the Stonewall Inn\u2014and even influenced the kind of club it would be.\n\n## T H E L A W\n\nThe homosexual is an inveterate seducer of the youth of both sexes, and... is not content with being degenerate himself; he must have degenerate companions, and is ever seeking younger victims.... Some male sex deviants do not stop with infecting their often-innocent partners [with homosexuality]: they descend through perversions to other forms of depravity, such as drug addiction, burglary, sadism, and even murder. Once a man assumes the role of homosexual, he throws off all moral restraints.\n\n\u2014 _Coronet_ , 1950\n\nThis history is one in which almost all the action takes place in the 1960s. The phrase _the sixties_ inevitably brings to mind images of freedom and rapid social and political change. The irony is that for almost the entirety of that decade homosexual men and women, far from experiencing a great burst of freedom, found themselves in the worst legal position they had been in since the republic's birth.\n\nBecause of a Puritan heritage, America's laws had traditionally oppressed those who engaged in same-sex lovemaking. With the increasing shrillness of the far right after World War II, exemplified by both a rabid anticommunism and the demand for total conformity that characterized the 1950s, laws aimed at homosexuals became so harsh that at times they were draconian.\n\nThe Defense Department hardened its policy of excluding homosexual servicemen and -women, tripling the World War II discharge rate and, in a reversal of prior practice, generally giving less-than-honorable \"blue discharges.\" These punitive discharges stripped thousands of veterans of the benefits that had been promised them in the GI Bill of Rights. After Lt. Roy Blick of the Washington, D.C., vice squad testified before the Senate in 1950 that 5,000 homosexuals worked for the government (a figure he had invented), the Senate authorized a full investigation into the matter by a subcommittee chaired by North Carolina senator Clyde Hoey. The Hoey subcommittee's report said that \"those who engage in overt acts of perversion lack the emotional stability of normal persons.\" Having concluded that \"one homosexual can pollute an entire office,\" the subcommittee urged that the military's recent purge of homosexuals be the model for civilian agencies.\n\nThe Civil Service Commission and the FBI complied by initiating an intense campaign to ferret out homosexuals by, for instance, correlating morals arrests across the United States with lists of government employees and checking fingerprints of job applicants against the FBI's fingerprint files.\n\nAfter Eisenhower became president, he signed Executive Order 10450, in April 1953, which added \"sexual perversion\" as a ground for government investigation and dismissal. The government shared police and military records with private employers, resulting in the dismissal of hundreds.\n\nWhile McCarthyism encouraged the toughening of laws toward homosexuals because they were believed to be security risks, America's Puritan tradition was producing hysteria over child molestation. Homosexuals were believed to be the main culprits. As the right-wing demonization of homosexuals proceeded apace, the negative qualities attributed to them overlapped until it became a common assumption that any homosexual man or woman was so beyond the pale that he or she must also partake of the most forbidden ideological fruit of all: communism. Homosexuals thus became handy scapegoats for both of these postwar obsessions. Antihomosexual laws were correspondingly made more severe.\n\nStates passed new laws that either stiffened the penalties for homosexual sex or created new categories to criminalize. For example, California governor Earl Warren thought the sex offender problem so serious that in 1949 he convened a special session of the state legislature to deal with the issue. That session passed laws that increased the penalties for sodomy and invented a new crime: loitering in a public toilet. The name of anyone convicted of spending too much time in a toilet was registered with the state. Twenty-nine states enacted new sexual psychopath laws and\/or revised existing ones, and homosexuals were commonly the laws' primary targets. In almost all states, professional licenses could be revoked or denied on the basis of homosexuality, so that professionals could lose their livelihoods.\n\nBy 1961 the laws in America were harsher on homosexuals than those in Cuba, Russia, or East Germany, countries that the United States criticized for their despotic ways. An adult convicted of the crime of having sex with another consenting adult in the privacy of his or her home could get anywhere from a light fine to five, ten, or twenty years\u2014or even life\u2014in prison. In 1971 twenty states had \"sex psychopath\" laws that permitted the detaining of homosexuals for that reason alone. In Pennsylvania and California sex offenders could be locked in a mental institution for life, and in seven states they could be castrated. At California's Atascadero State Hospital, known soon after its opening as \"Dachau for Queers,\" men convicted of consensual sodomy were, as authorized by a 1941 law, given electrical and pharmacological shock therapy, castrated, and lobotomized. _Gay Law_ author William N. Eskridge Jr. summed up the legal status of homosexuals at the beginning of the 1960s: \"The homosexual... was smothered by law.\"\n\nNor were transvestites spared. In New York State an old antilabor statute, passed in the nineteenth century to suppress tenant farmers who donned disguises to demonstrate against their landlords, was dusted off to use against men and women who dressed in the clothes of the opposite sex. In practice, New York police used the guideline that any person wearing fewer than three articles of clothing appropriate to their sex was, according to subsection 4 of section 240.35 of the New York Penal Code, \"masked... by unusual or unnatural attire or facial alteration\" and thus subject to arrest.\n\nIt has often been pointed out that no specific statute outlawed being homosexual, that only homosexual acts were illegal. While this is technically true, the effect of the entire body of laws and policies that the state employed to police the conduct of homosexual men and women was to make being gay de facto a crime. The very harshness of the laws made judges generally unwilling to sentence homosexual men, lesbians, and transvestites to such inhumane punishment. Instead, judges tended to hand out light fines or to place those convicted on probation. But the random or selective use of far harsher penalties and the potential threat of their use, combined with other sanctions and harassments, major and minor, official and nonofficial, more than sufficed to keep the vast majority of homosexual men and women well within the lines that society had drawn for them. Thus lesbians and homosexual men lived in an uneasy state of fear and spiritual suffocation.\n\nOnce all manner of sanctions had been created to make it difficult for homosexual men and women to meet their own kind, the police aggressively patrolled the few places where homosexuals could mingle: bars and bathhouses (both private and public) and outdoor cruising places such as streets, parks, and beaches. Agents planted microphones in park benches and used peepholes and two-way mirrors to spy on homosexuals in public rest rooms.\n\nWhile the law classified homosexuals as criminals and the scientific establishment used psychology to medicalize homosexuality into an illness, gay men and lesbians found almost universal moral condemnation from religions, whether mainstream or obscure. Thrice condemned\u2014as criminals, as mentally ill, and as sinners\u2014homosexuals faced a social reality in post\u2013World War II America that was bleak, if not grim.\n\nDoric Wilson had come to New York in 1958 from Washington State to pursue a career as a playwright and often came to the Village to meet other gay men. One Sunday he circled a block on Greenwich Avenue. He had circled the block twice already and hadn't met anyone. In the early sixties it could be hard to meet someone, the encounter was so ritualized. The shops were closed on Sundays, but he circled the block looking at the shop windows; then he walked by without looking. Then he went around the block again, passed an attractive man, and stood and feigned a studious interest in the goods displayed in the window. Would the good-looking fellow stop and also stare? Maybe the other man would signal his interest by tapping his foot as they both window-shopped. Then Doric could ask the other man if he knew the time or had a light for a cigarette? They started the long, drawn-out procedure of approach. Did he live here? What did he think of the Village? And so forth. Suddenly Doric felt a sharp pain in his back. The blow was accompanied by a command: \"Move on, faggots! Move on!\" Doric didn't bother to argue with the police officer. He just moved on.\n\nOne reason Edgar Allan Poe went to the Northern Dispensary was because it was practically next to the small row house where he had found a very inexpensive apartment. The house where Poe lived happens to be across the street from the building Fat Tony's father owned. Barry Miles wrote that \"American bohemianism is said to have started with Edgar Allan Poe,\" and the incongruous juxtaposition of the mobster's large apartment building with the bohemian's small row house in some ways parallels the paradox in which gay New Yorkers in the 1960s found themselves trapped. As word increasingly got out nationwide that there were large numbers of gay people in Greenwich Village, more and more gay men and lesbians were drawn to New York City. Eventually New York had the largest gay population in the United States, and the Village increasingly served as a center for the growing homosexual subculture. Sociologists Martin Weinberg and Colin Williams, who studied New York's homosexual population in the late 1960s, wrote that they found twenty-six bars, twelve nightclub\/restaurants, four hotels, and two private clubs that catered to homosexuals, as well as all five of the city's lesbian bars, in the Greenwich Village area. They added: \"Cruising areas abound.\"\n\nParadoxically, New York was also the city that most aggressively and systematically targeted gay men as criminals. George Chauncey's _Gay New York_ traces the rise of a visible gay population in the city from the 1890s to 1940, as well as the rise of powerful antivice societies to combat prostitution, alcohol consumption, homosexuality, and other behavior viewed as immoral. In response to the great influence that these societies exerted over politicians, police vice squads\u2014which New York City was the first to create\u2014attempted to control homosexuals by observing locales where they congregated, using decoys to entice them, and raiding gay bars and baths.\n\nWhen Prohibition ended in 1933, the state passed many laws to regulate the sale and consumption of alcoholic beverages. As the word _Authority_ in its name suggests, the laws that created the State Liquor Authority (SLA) made it undemocratic, giving it practically total leeway in administering and enforcing these laws. The SLA interpreted the laws so that even the presence of homosexuals\u2014categorized as people who were \"lewd and dissolute\"\u2014in a bar made that place disorderly and subject to closure. The result made New York City the \"most vigorous\" investigator of homosexuals before World War II. Responding to right-wing pressure after the war, New York City modernized its stakeout, decoy, and police raid operations, and continued to haul in thousands of homosexuals, sometimes just for socializing at a private party. More commonly, the police arrested gay men at bars and in cruising areas. By 1966 over one hundred men were arrested each week for \"homosexual solicitation\" as a result of police entrapment.\n\nMaking it illegal for bars to serve homosexuals created a situation that could only attract organized crime. The Mafia entered into the vacuum to run gay bars, which in turn set up a scenario for police corruption and the exploitation of the bars' customers. These victims were not likely to complain, because they had nowhere else to go and because they feared the mob. Moreover, the involvement of the Mafia in gay clubs further increased the legal vulnerability of gay men and lesbians.\n\nIn the mid-1960s\u2014the very time when a wave of freedom, openness, and demand for change was cresting\u2014New York City increased its enforcement of antihomosexual laws to such an extent that it amounted to an attempt to impose police-state conditions onto a homosexual ghetto.\n\n## R E S I S T A N C E\n\nThroughout much of American history lesbians, homosexual men, and transvestites at times offered resistance\u2014however weak, fleeting, or ineffectual\u2014to the social and legal institutions that oppressed them. The first sustainable political resistance began in 1950 in Los Angeles when Harry Hay founded the Mattachine Society. This marked the birth of the homophile movement. The founders of the Mattachine Society used the word _homophile_ because they believed that this new term, with its incorporation of the Greek word for love, could help counter the stereotype of homosexuals as obsessed with sex. The Mattachine Society took its name from medieval societies in which those playing the role of fool could speak truth to those in power.\n\nHay and his colleagues articulated in a statement of purpose a plan of action in which homosexuals who lived \"isolated from their own kind\" were, first, unified and then \"educated\" in order to develop an \"ethical homosexual culture\" similar to that of other minorities. After progress had been made toward the first two goals, the intent was \"to push forward into the realm of political action.\" The Mattachine Society began taking political action sooner than planned when a police officer entrapped one of its founders. The new organization resolved to fight the charge, a radical idea at the time. Astonishingly, Mattachine won, making their name renowned among California homosexuals and spurring a membership surge.\n\nNot long afterward, however, word began to circulate of the radical politics of some of the Society's founders, most notably Hay who had been a member of the Communist Party. When it became clear that the organization would falter if Hay did not withdraw from it, he did so at the Mattachine Society's 1953 national convention. The organization then incorporated as an educational and research group, espousing the belief that by providing accurate information about homosexuality to the public it could \"eliminate discrimination, derision, prejudice and bigotry.\" The Mattachine's new leaders, including those in New York City, reasoned that since they had no validity in society's eyes, they could borrow from those who did, namely, psychologists and psychiatrists, the supposed experts on homosexuality. The vision of political activism was lost.\n\nIn Chicago, sixteen-year-old Craig Rodwell, a high school junior in 1957, left his mother's apartment each weekday to walk to school. Although he had the money to pay the bus fare, he walked the entire distance. Craig walked to and from school on days when it was hot as well as on days when the stiff winter winds that howled in off Lake Michigan blasted the city, making pedestrians scurry to escape the misery outdoors. Money was hard to come by for Craig, whose mother's salary as a secretary was stretched thin to support him and his older brother. But Craig was a very determined young man, and he was especially eager to save all the money he could. The more he saved, the sooner he could realize his dream. Besides saving money by walking to school, Craig started asking for money for his birthday and for Christmas.\n\nOn many days after arriving at school, he walked to the library, went straight to the unabridged dictionary resting on its stand, and began to turn the pages. He continued looking until he came to the word. The word he had come to the library to see over and over again. _Homosexual_. He stared at the word as if to make sure that it was still in the same place and on the same page where he had so often seen it. To Craig, it proved that he existed.\n\nNeeding to stare at a word in the dictionary to validate his very being showed just how radically Rodwell's experience of his sexuality had changed. Only a few years before, he had easily expressed his sexuality with other boys, for he had spent much of his childhood and early adolescence in an all-boys institution run by the Christian Science church. \"We lived for our free time when we could go out in the woods. We liked the kissing and the holding hands and other things too. It was all very natural,\" he recalled. When a housemother occasionally saw Craig being intimate with another boy, she would tell them that they could not do what they were doing. Her injunctions always angered Craig.\n\nAs for Craig's being at the institution, there, too, money had played a role: his mother had sent him there after her husband had deserted her. With no income, she had to spend all her time working to support herself and her sons. And so Craig was sent away to the Christian Science School until he was fourteen, when he returned to Chicago to live with his mother. By this time his mother had remarried, and the family economic situation had improved enough that Craig could take ballet lessons.\n\nSome months after returning home, Craig was walking one night to the el with Frank, a gentle Italian dishwasher, when suddenly two police cars, their lights whirling, converged on the two. Policemen got out of the cars and separated and questioned Craig and Frank. The police refused to believe that it was the fourteen-year-old Rodwell who had picked up the older man. The district attorney put heavy pressure on Craig to lie by saying that the older man had approached him and given him money. Craig refused. Frank was sentenced to five years for having sex with a minor, even though Craig explained that he had claimed to be older. No doubt it was Craig's noncooperation with the authorities that caused them to recommend he be sent to a reformatory.\n\nThis was more than Craig's mother could bear after she had already been separated from him for so much of his life. She broke down crying and screaming, got down on her knees, and begged the juvenile hearing officer to have mercy. The officer relented and said that if Craig's mother sent him to a private psychiatrist, she would place him on probation instead of sending him away. Despite the arrest and barely escaping reform school, Craig's confidence in the rightness of his behavior was not shaken. On the contrary, he now became, in his own words, \"an angry queer.\"\n\nAbout two years later Craig met a man named Harry at Chicago's gay beach. Harry told him about the Mattachine Society and showed him some of their publications. Craig was immediately fascinated with the idea that gay people were actually organizing to do something about the way they were treated. Craig decided to go to the Chicago chapter of the Mattachine Society right away. Once at the address listed in the publication, his heart sank as he scanned the building directory in vain for the words _Mattachine Society_. He wandered the halls looking for a door with the word _Mattachine_ on it. Finally he realized that what he had imagined as a large, busy headquarters was just a mailing address. Craig felt the bottom fall out of his life.\n\nNot long afterward he heard that all queers lived in Greenwich Village. He immediately decided to move there as soon as possible, which meant after graduating from high school. He started walking to school.\n\nIn the nation's capital, on some evenings in 1959 Franklin Kameny lay awake in his bed unable to fall asleep. Malnutrition had caused him to lose so much weight that his bones hurt where they touched the mattress. Less than two years before, he had had a good job, as befitted someone with a Ph.D. in astronomy from Harvard. The job had come from the U.S. government, which had hired Kameny to make more accurate maps for the military.\n\nHe had once served in the military himself, having left college in 1943 to fight in Europe on World War II's front lines. Frank was intellectually precocious, so that when he started studying astronomy after the war, he made steady progress in his field. In the 1950s he went to a Tucson observatory to do the research for his Ph.D. thesis. After he had completed his research, he decided to travel around the country by bus on his way back to Cambridge. On a stop in San Francisco a stranger had approached him as he walked through the bus terminal. When the man walked up to Frank and groped him, plainclothes police officers emerged from their hiding place and arrested Frank and the stranger. Unfamiliar with legal proceedings, Frank did not fight the charge, since he was told that after being on probation for three years, his arrest record would be expunged. Because of the promise of expungement, Frank had not worried about the old arrest record when he applied for a job with the military.\n\nAt the end of 1957, while working on a military project in Hawaii, Kameny received a phone call summoning him to Washington immediately. Once there, he met with a representative of the Civil Service Commission.\n\n\"We have information that leads us to believe you are homosexual. Do you have any comment?\"\n\n\"What's the information?\"\n\n\"We can't tell you.\"\n\n\"Well, I can't give you an intelligent answer if you can't tell me, and in any case, I don't think it's any of the government's business.\"\n\nSoon after that terse exchange, Kameny received a standard dismissal letter. Frank decided to fight back. In the meantime, he could not find a job, because almost all the positions for observational astronomers such as himself were with the U.S. government. Unfortunately, Frank did not know anything about working at jobs other than the kind for which he had been trained. As Frank pursued his solitary fight, his funds dwindled, and he lived on food such as Jell-O, hot dogs, and mashed potatoes. On the best days, he had a pat of butter to put on his mashed potatoes. On the worst days, he had nothing to eat.\n\nHe followed the administrative appeal process all the way through to its end without success. After twenty months, Kameny finally found a science and engineering job that did not require a security clearance. For almost anyone else, that would have been the end of the fight. It was perfectly clear to Frank, however, that while he had done nothing wrong, the government had. Frank Kameny decided that the United States would have to change its policies.\n\nRandy Wicker was so determined to unravel the mysteries surrounding homosexuality in order to understand himself that he majored in psychology and sociology. He was disappointed in his studies, however, for in the academic literature he found only case studies and the standard theories of the day, such as the Freudian orthodoxy that male homosexuality was caused by a son having a weak father and a dominant mother. Yet all Randy's reading about homosexuality ultimately paid off in a different manner than he had expected when he read about a very large place where homosexuals went to meet one another: Greenwich Village. The following summer, he traveled to New York to explore the Village. Hanging out in his first gay bar, Lenny's Hideaway, Randy heard of a mysterious and very powerful organization called the Mattachine.\n\nThomas Lanigan-Schmidt grew up in the sister towns of Linden and Elizabeth, New Jersey, towns that, while having their own governments and boundaries, blend into and overlap each other on the ground. These are gritty industrial towns, whose chief industries in the 1950s and '60s were represented by companies like Singer Sewing Machine, General Motors, and Gordon's Gin. Growing up there in a time before there were any laws against air pollution, Tommy had trouble seeing how to drive his paper route through the pollution that mingled with the morning fog. The gray air that engulfed Tommy penetrated his nostrils with a sulfurous stench that reminded him of rotten eggs.\n\nPerhaps Tommy drew with bright colors to make a contrast with his gray surroundings. He loved to make pictures of parrots and peacocks. When he depicted tropical fish, he favored Siamese fighting fish because of their rich, vibrant colors.\n\nTommy attended a Catholic grammar school. There the nuns noticed Tommy's talent for drawing and put him in charge of decorating the bulletin boards. Unlike the nuns, who encouraged Tommy's artistic temperament, Tommy's father feared his son's interests indicated a lack of masculinity. In an attempt to change Tommy, his father had persuaded him to work the paper route. His father had explained that this was the path to independence and promised Tommy that he could buy anything he wanted with the money he earned.\n\nTommy set his heart on an object in the local Woolworth's that he considered an item of rare beauty: a little opalescent pink-and-gold teacup made of very thin china, with a delicate stem on its bottom and an elaborate gold handle. He dutifully saved the small amounts of money from his paper route for its purchase.\n\nWhen the great day finally came, Tommy proudly carried the cup home. Rather than being pleased at seeing Tommy use the money to buy something he wanted, his father was enraged that his son had bought a lady's teacup. The next morning Tommy found the teacup in the middle of the kitchen table, lying on its side, its small pink bottom snapped off. Tommy carried the broken teacup up to his room, carefully placed it in an Easter egg box, and hid the box.\n\nThe agitated twenty-four-year-old with a trim build, blue eyes, and brown hair was practically yelling at Dick Elman, WBAI's public affairs director. \"Why do you have these people on that don't know a damn thing about homosexuality?\" he demanded. The furious young man's piercing voice, the result of a deviated septum, certainly was irritating. Randy Wicker continued, \"They don't live it and breathe it the way I do.... I spend my whole life in gay society.\" Wicker was infuriated because the previous night New York's progressive radio station had broadcast a program on homosexuality in which the only persons talking had been straight psychologists and psychiatrists. Predictably, they had given the psychological establishment's party line on the subject. Elman replied, \"Get a group together, and I'll come up and do an interview.\"\n\nThese were words Wicker had been waiting to hear. For years Randy had tried everything he could think of to get out accurate information about homosexuals and homosexuality. His first brainstorm had come in 1958 when he had persuaded the Mattachine Society of New York (Mattachine\u2013New York) to publicize its lectures. Randy had printed and distributed hundreds of flyers, which had drawn an overflow audience. He also captured the attention of the police, who then pressured the Mattachine Society's landlord into throwing the organization out on the street. The loss of the brand-new quarters was especially painful because members of the society had just spent months renovating them. Half of the organization's fifteen members thought more firebrands like Randy were just what the organization needed and pointed to all the people who had come to the lecture. The other half pointed to the lost lease as evidence that rabblerousers like Wicker would doom the organization.\n\nWicker's next venture took place about three years later when he learned of a public forum called a speak-out that was held on the corner of MacDougal and West 3rd streets. A speaker could stand on a soapbox and talk about issues considered radical or offbeat. Randy trembled as he mounted the box, fearful that the crowd might stone him. He decided to talk about a gay bar that operated just down the street. He explained that the bar paid off the cops and that this was not in the interest of the cops, for it corrupted them. In turn, the cops corrupted the city government. It therefore served everyone's interest for homosexuals to have the right to gather in their own places. To Randy's amazement, his peroration was greeted with a round of applause. After another positive encounter at a speak-out, Randy concluded that whatever the shortcomings of their government, the American people were accepting and wanted to be informed.\n\nRandy became most angry when he read the misinformation printed whenever the media discussed homosexuality. They carefully fed the public images of homosexuals as communist spies, child molesters, or men who played a stereotypical \"queen\" role by plucking their eyebrows and speaking in a high voice.\n\nIn his determination to break the media silence on homosexuality, Wicker tried everything from having matchbooks printed with a one-line homosexual rights message to advertising unsold homophile publications for a nominal price as a \"sample packet of homosexual publications,\" ads that were rejected by most newspapers and magazines, including _The Village Voice_.\n\nWhen Randy had buttons made that said \"Equality for Homosexuals\" and wore one to a gay party, he sold a button to everyone there. Usually, however, he had little luck in promoting the cause in the homophile community. He also found the machinery of group decision-making inordinately slow for his taste and concluded that a considerable number of the Mattachine's members were either too frightened or brainwashed to fight for their rights. The group that exemplified this overly cautious mentality for Randy was the West Side Discussion Group, a gay social group sponsored by the Mattachine Society that tended to be home to the most conservative members and affiliates of the homophile movement. The liberal attitude toward homosexuality in the late fifties and early sixties was that homosexuals were not criminals but mentally ill people who might be cured if given understanding and a lot of expensive therapy. The West Side Discussion Group was so conservative, however, that some of its members were not even able to digest the liberal theory. One of them said to Wicker, \"How can we expect the police to allow us to congregate? Let's face it; we're criminals. You can't allow criminals to congregate.\" Randy blew up at the man, saying, \"It disgusts me. Why do I have to sit here and listen to idiots like you say things like that? You've let society brainwash you.\"\n\nIn the spring of 1962 Randy finally got so fed up that he formed his own one-man operation, the Homosexual League of New York. He had business cards printed and presented himself to WBAI as the public relations director of the \"organization.\"\n\nOn a humid spring night several weeks after Wicker had confronted Elman, the WBAI public affairs director arrived at the Upper West Side apartment of a friend of Randy's. With the windows open, the eight men Wicker had assembled sat on the floor with Elman to tape a candid discussion of their lives. Realizing that it was breaking new ground, WBAI sent out a press release about the broadcast, bringing the event to the attention of Jack O'Brian, a columnist with the _New York Journal-American_. O'Brian was deeply offended by the very idea of such a program. Labeling Wicker an \"arrogant, card-carrying swish,\" O'Brian suggested the station change its call letters to WSICK. The media-savvy Wicker realized that he had just been handed an opportunity and, with characteristic energy, carried copies of O'Brian's article to newspaper and magazine offices until it had reached the desk of every news editor in the city.\n\n_The New York Times_ did two stories on the broadcast. In a review headlined \"Radio: Taboo Is Broken,\" _The Times_ pronounced that the program \"succeeded... in encouraging a wider understanding of the homosexuals' attitudes and problems.\" _Newsweek_ gave the event a full page. _The Realist_ used several of its issues to print an entire transcript of the ninety-six-minute program. And _Escapade_ , a _Playboy_ imitator, printed a condensed version of the transcript. The _New York Herald Tribune_ and _Variety_ carried favorable articles as well.\n\nThe immediate payoff had been golden, but the resourceful Wicker leveraged it into platinum. He assembled copies of the transcripts and the extensive press coverage and sent them to writers whom he continued to buttonhole, offering \"field trips\" into homosexual New York. As historian John D'Emilio noted, the coverage of homosexuality in the national media from 1962 to 1965 increased markedly as a result of Wicker's campaign. Articles on the subject in the _Reader's Guide to Periodical Literature_ increased sixfold between 1962 and 1963 alone. A crack had been made in the wall of media silence.\n\nArthur Evans took his belt off, went to the closet in his small dormitory room, and opened the door. After scrutinizing the clothes bar in the closet, he decided his idea would work. He could attach his belt to the bar, wrap the belt around his neck, and end his life.\n\nMost people would have figured that a student in his junior year of college at an Ivy League school who was bright enough to be there on a full scholarship would have everything in life to look forward to. All the more so considering that he had come from a poor, working-class family in a dismal neighborhood of York, Pennsylvania.\n\nArthur was tortured by his overpowering feelings of attraction toward other boys. He had experienced these feelings since age eleven, when puberty hit early and hard. He knew that his longings were dangerous and was careful not to act on them, believing that if other boys found out how he felt he'd be beaten up or killed. When he tried to read about homosexuality in the school library, he found only Freudian theory; in the Bible, condemnation.\n\nArthur felt he had to escape York. He found the means to leave when he won a scholarship from the local paper company to study chemistry at Brown University. Though Brown at that time enrolled only males, Arthur at first found no sign there of other men who felt as he did.\n\nArthur was beginning to be interested in political activism. He started the Free Thinkers Society of Brown with his roommate, Michael. While Brown was a secular institution, it had a weekly chapel service at which attendance was compulsory. Considering themselves militant atheists, Arthur and Michael hatched a plan to protest the service. They demonstrated in front of the chapel, handing out leaflets and urging those who agreed with them not to stand but to sit through the chaplain's prayer. Arthur and Michael got five students to join them. This surprised Arthur, for this was before campus protests became common.\n\nThe protest made the front page of one of Arthur's two hometown newspapers. The Glatfelter Paper Company, which had given Evans his scholarship, was run at that time by Christian fundamentalists. After the article on the protest appeared, Glatfelter informed Arthur that it was revoking his scholarship. Arthur then got in touch with Joseph Lewis, the millionaire head of the National Free Thinkers Society, and explained the threat Glatfelter had made. Lewis wrote to Glatfelter saying that if they terminated Evans's scholarship, he would sue them and drag their name through the mud in the media. Glatfelter then backed down. This victory over the paper manufacturer became a shaping event for Arthur's future.\n\nEvans's triumph convinced him to change his major to political science. Arthur concluded that the confrontation with Glatfelter offered several lessons for successful political activism: take a principled stand, be smart about it, get a lot of media attention, and have some friends who can pull strings for you.\n\nMeaningful as it was, Evans's victory did not solve the problems of loneliness and sexual frustration. Instead, the feelings intensified. Even worse, other students were coming on to him. Evans couldn't believe they were like him, so they could only be coming on to him because they wanted to get him to give himself away. This was more than he could bear.\n\nEvans decided to act. He went to the closet and had just started the preparations for ending his life when a wave of deep dread came over him. An image of standing on the edge of a precipice above a pit so deep it had no bottom welled up before him. A sudden wave of nausea swept over Evans, and he stepped back from the edge.\n\nSoon afterward, in the spring of 1963, Arthur picked up a national magazine and read in it that a lot of homosexuals lived in Greenwich Village. That he was not alone after all was a complete and sudden revelation. Evans decided to give up his scholarship and move to New York City.\n\nDick Leitsch was born in Kentucky, which he called \"the land of horse-shit, tobacco and the mint julep.\" His Kentucky childhood was in many ways typical. For example, on Saturdays he went with his friends to the movies. But while his companions fantasized about being cowboys, westerns failed to engage Dick's imagination. He was, however, mesmerized by the great screen actresses of the forties and fifties. The lives they portrayed in urban settings made Kentucky seem decidedly lackluster. Dick began to dream of \"going to New York, drinking cocktails, smoking cigarettes, and having stupid love affairs, just like Bette Davis.\"\n\nWhile Leitsch may have found Kentucky lacking in cultural sophistication, there was some sophistication about homosexuality, both in the larger society and in Dick's Catholic family. Dick's mother had gay male and lesbian friends who were couples and came over to the house and visited. His parents were totally accepting of his homosexuality when the first man with whom Leitsch had sex indiscreetly sent him flowers the following morning with a love note attached.\n\nHis mother asked, \"Is it true? Is this your boyfriend?\"\n\n\"I had sex with him last night.\"\n\n\"Well, Father Valentine used to be a chaplain in a prison and said that you would probably grow up to be a homosexual. We've been watching and thought that you probably were. I guess he was right.\"\n\nDick's parents continued to be supportive, urging him to introduce them to his friends. Meanwhile, Dick's childhood fantasies about New York as the epitome of sophistication had continued to grow as he listened at night to live radio broadcasts from the Stork Club and the Copacabana.\n\nAfter moving to Cincinnati in 1958, he and Derrick, a man who had lived in New York but had not liked the city, became lovers. Leitsch ultimately persuaded Derrick to give it another try. Derrick went back first, got reestablished, and wrote to Dick to come join him. Arriving at Pennsylvania Station in early 1959, Leitsch was greeted by Derrick. They got into a taxi and, as the cab passed through Times Square, Dick saw the crossroads of the world and felt intensely happy. It looked just as he had imagined it, except that in color it looked, if possible, even better.\n\nOn Sundays Tommy Lanigan-Schmidt often went with his friend \"Mark\" Vag to the Greek Catholic church Mark attended. Tommy had easily embraced the Catholic church in which he was raised, but when the nuns sent him to the grammar school library to do his bulletin board artwork, he discovered the highly mystic traditions of the Russian and Greek Orthodox churches. The school library had Greek Orthodox volumes because there was a local population of Greek Catholics, who belonged to a branch of Catholicism that blended Roman Catholic and Greek Orthodox teachings.\n\nTommy had been shocked as a Cub Scout when his den mother, in spite of Mark being a member of the den, had warned them to stay away from the local Greek Catholic church because the priest there had a wife. He also felt sorry for Mark because the local toughs would mock his name, walking up to him and saying loudly, \"Hey, Mark FAG!\"\n\nGood male friends were not plentiful in Tommy's young life. The tougher kids in school picked on him. Between classes they would walk up to him and hit him hard in the stomach.\n\nTommy had befriended Mark, and Mark in turn invited Tommy to attend his church. The rich fabric used by the priests, the highly ritualized ceremonies, and the abundant use of incense immediately appealed to Tommy's spiritual and aesthetic sensibilities. The local Greek Catholic church had held an air of mystery for young Tommy, especially when he saw its doors standing open on Easter mornings.\n\nNow that he was in high school, he was old enough to stay up till midnight, the symbolic hour of deepest darkness, when the Greek Catholic Easter service begins. As Tommy entered the church, something seemed odd. Gradually, he realized that no candles were lit. A life-size portrait of Jesus on a piece of cloth was arrayed over a large coffin-shaped box. A priest emerged and told everyone to leave the church. As the worshipers left, they intoned slowly and repeatedly, \"Christ is risen from the dead,\" in a droning dirge that was deep and sad.\n\nAs they left the church some worshipers stumbled in the darkness. By late night in a factory town like Linden, the air smells heavy because the factories release much of their effluvium in the evenings, and by this time of night the polluted air had settled down to street level. The foul odors intensified the plainsong's sadness.\n\nThe chant continued mantralike until all the congregants were outside. Then the singing stopped and the words were intoned. The pace of the chant quickened until it was almost frenetic. Suddenly a priest holding a large gold Bible stepped forward. He was wearing the brightest vestments Tommy had ever seen: gold embroidery spangled with silver threads shimmered in the candles' light. The priest read the story of Jesus' disciples going to the tomb and seeing the angel who said, \"He is risen.\" The crowd repeated, \"Christ is risen.\" The priest approached the church's double doors and knocked loudly on them with his crook: _Bang! Bang! Bang!_ And the doors sprang open.\n\nTheir candles lit, there was a rush of fire as the churchgoers hastened inside, chanting, \"And on those in the tombs bestowing life!\" a moment in the ritual that represents being brought back from death's realm into life. Inside, Tommy found the church transformed: every chandelier was turned on, all of the church's bells began pealing, the choir started to sing, and the church, previously bare, was filled with flowers. As he stepped back outside the church into Linden's foul air, Tommy had an inexplicable feeling of hope. This moment was to return to him unexpectedly years later.\n\nSoon after his arrival, Dick Leitsch was introduced to New York City's gay scene, including the neighborhood he initially settled in, the Upper West Side. Curiously, many of the places that were pointed out to Dick had recently been closed down, from Milano's, a rough trade bar, to the All-State Caf\u00e9, a dance bar, to Omega, a hustler bar, as well as many others. This shuttering of gay businesses was the result of a recent and massive campaign of police harassment under Mayor Robert Wagner. During the crackdown the licenses of gay businesses had been revoked. Many people had been arrested, and many more had been entrapped and harassed. The mayor's campaign against gay men was the main topic of conversation in the gay male world. Although Leitsch found a much bigger homosexual scene in Manhattan than he had known in Louisville, he noticed that it was more underground. \"I just couldn't believe that was happening in New York, which I thought was the capital of civilization. It had never occurred to me that it was going to be more conservative than my hometown.\"\n\nAs Arthur Evans settled into his new life in New York City, he often went to Washington Square Park to sit and read. Politics remained very much on his mind, and as he contemplated the park's arch, built to honor George Washington, he noticed a quotation by the revolutionary inscribed on the arch's top: \"Let us raise a standard to which the wise and the honest may repair.\" Whenever he returned to the park, he read the inscription, which seemed to speak to him.\n\n#\n\n# Oppression, Resistance, and Everyday Life\n\nDick Leitsch went to Greenwich Village from time to time, but he found gay men there a bit cliquish and self-consciously bohemian. He found a real communal social life with other gay men in the area along Central Park West, where homosexual men went to relax by socializing at evening time.\n\n\"Every night after dinner we'd go walk on Central Park West and we'd run into all of our friends, the people we worked with, and half the West Side gay people\u2014which were thousands\u2014on Central Park West between 59th and 86th. In those days, there was one long bench from corner to corner, solid with gay men. Hundreds and thousands of them walked back and forth singularly, in couples, and in groups.\"\n\nThe huge chattering crowds made a considerable din, which led to complaints from nearby residents, and the police would make from three to five sweeps along the park most nights. A sweep might consist of a single car or as many as six at a time, each with large bright lights mounted on its top. While driving by and shining the lights on the gay men, the police used megaphones to order, \"Move on. Keep moving, faggots. Keep moving.\" Dick recalls that the gay men would all get off the benches and walk up the street very slowly, as though they were going home. Once the police cars disappeared from view, the gay men would scramble back to the benches, wait for the police to circle again, and repeat the same charade.\n\nLate in the winter of 1962 a suave, tall, and handsome thirty-two-year-old Wall Street banker was cruising Central Park West. When he reached 88th Street, he noticed an attractive man and approached him. The banker's name was Harvey Milk and the man he had cruised was Craig Rodwell.\n\nCraig Rodwell and Harvey Milk had little in common. Milk was Jewish and a native New Yorker, whereas the midwestern Rodwell described himself as a Christian Scientist. The banker held the right-wing political values common in his profession, while the young ballet student was politically progressive. Whereas Rodwell was eager to take on the whole world's prejudice against homosexuality, Milk was very secretive about his sex life.\n\nWhen Craig had arrived in New York in the late summer of 1959, he was only eighteen, yet he felt it had taken him forever to get to a city with an active Mattachine chapter. On arriving at the Mattachine\u2013New York office, Rodwell was told that he could not join because he was too young: he would have to wait until he was twenty-one. Craig could not wait that long. On his own he made up flyers urging people to join the Mattachine Society and put them in the mailboxes of Village residences with two men's names on them, figuring that the recipients were probably a gay couple.\n\nWhile Rodwell was studying at the American Ballet School thanks to a scholarship, he had little money left over for living expenses. He got a low-paying job in a flower factory and did not make enough money to go to gay bars. One night while he was cruising in Washington Square Park a police officer gave Craig the usual order to move on. Rodwell retorted, \"This is harassment of homosexuals.\" The officer immediately arrested him and took him to jail. Craig's resistance was unprecedented. That night in jail one policeman after another came by Rodwell's cell to see the upstart homosexual. \"What'sa matter, lose your purse?\" baited one. Craig shot back, \"What'sa matter, never seen a faggot before?\"\n\nRodwell continued to haunt Manhattan's cruising grounds until he met Milk, with whom he would have his first serious romance. Each morning, Milk, a born romantic, gave Craig a wake-up call invariably accompanied by a joke. On Christmas morning, Harvey, the son of Orthodox Jews, arrived early, laden with presents that showed thoughtfulness and a romantic touch. Craig was soon swept off his feet.\n\nBut the pair's divergent politics tested the romance. While in his early teens Rodwell had idolized Adlai Stevenson, Milk had championed Barry Goldwater. When Rodwell told Milk about another of his evening forays putting flyers into mailboxes, Milk yelled at him, \"You shouldn't do that to people! Getting those in mailboxes will make people paranoid that everyone knows about them being gay.\"\n\n\"You're just thinking about how you would react if it showed up in your mailbox,\" Rodwell retorted.\n\nThere were other problems beyond politics. When Milk discovered that he had gonorrhea, he concluded Craig had been unfaithful, for Milk had not slept with anyone else after the two had started dating. When Milk confronted Rodwell, Craig admitted to having sex with other men, even though he was in love with Harvey.\n\nIt was not only homosexuals from out-of-state who heard about the Village and found their way there. Jerry Hoose recalls his first trip there as an eighteen-year-old: \"Around the end of '63 or the beginning of '64, I heard about Greenwich Village. I was living in Brooklyn and kept dreaming about finding other gay people. One day I just said, 'I got to find it!'\n\n\"And I got on the train and got off at 8th Street. I got to Sixth Avenue and 8th Street, at that time the Mecca: it was Greenwich Avenue from Sixth Avenue to Mama's Chick'N'Rib on Charles Street. I crossed Sixth Avenue and I saw all these very gay-looking people, and I said, 'Well, I've found my world.'\"\n\nTree is another native New Yorker who discovered the Village as a gay place in his teen years. His favorite place was not a bar but the restaurant many gay men from the era still remember affectionately, Mama's Chick'N'Rib. With perhaps a touch of hyperbole, Tree remarks, \"It was considered the gay hangout of the world.\" Tree was at Mama's so often he ended up working there in the late 1950s.\n\n\"This place was more crowded than any bar. Of course the bars in those times were all owned by the Mafia\u2014they charged you to get in, watered down drinks, a little roughing up if you got too drunk. But Mama's Chick'N'Rib was a home. We had a waiter there named Jack. They called him Joan Crawford. And there was myself, Jerry, Garth, and a few others.\n\n\"Now, certain nights, when it was all friends in the place, we wouldn't let anybody in. We would lock the door because we were packed. And we'd fog up the windows by turning the heat and air-conditioning on, so there was a little dancing, a little kissing here and there. Everybody would eat and drink for free all night long and then in the morning when everybody knew Mama was coming, Jack would pass around two boxes, one for the register and one for the tips. Everybody would throw money in both of them. And at the end of the night Mama looked in the cash register and she was so happy. She made five, six hundred dollars and Jack and Garth and everybody that worked there was happy because they made like fifty, sixty dollars for their tips, which is a lot of money in the fifties. Mama never knew the difference.\n\n\"She was a little old Greek lady that you had to love her or hate her....In all the years she had Mama's she only wore two dresses. One was blue and one was black. She had a favorite strand of pearls and wore them every day. She looked on all the customers in her place as her children. It was 'my boy' this, and 'my boy' that, and as she talked to you she always played with her pearls. She could hold a handful of coins for a minute and then tell you to the penny how much was in her hand.... She counted french fries, sliced tomatoes that you could read a newspaper through. If you dropped food on the floor, it went back on the plate. It was that kind of place, but movie stars were seen in there.\"\n\nEven though Mama's was not run by the Mafia, as the place catered to gay people, Mama still had to pay off the police. \"Brown bag Friday it was called. Cops would come around to bars and restaurants that catered to gay people, and they would have a little brown bag like a container of coffee with no coffee in it. It was all full of money for the neighborhood precincts\u2014uptown, downtown, midtown. 'You want your windows broken?' 'No.' 'Well, you let all these fags hang out in there.' So, we had to give a little envelope every once in a while.\"\n\nOn Labor Day weekend of 1962, Craig Rodwell headed for the beach like millions of his fellow Americans. The beach Craig chose, however, was the one at Riis Park, where thousands of gay men and lesbians congregated. After he had enjoyed the beach for a while, he headed for the bathhouse, where hot dogs and soda were sold. Craig had heard the talk on the beach that since this was a homosexual beach, on Memorial Day and Labor Day the police enforced a blue law written early in the century that outlawed a man wearing a bathing suit that did not cover his navel and reach at least the midway point between his knees and his groin.\n\nCraig left the beach without a towel nonetheless. When he reached the boardwalk, the policeman stationed there started writing him a ticket. Craig protested that this was harassment of homosexuals and that they all knew what was going on, it was a quota system, and...\n\nA crowd began to collect.\n\nThe policeman quickly grabbed Craig and hustled him into a small room maintained by the police inside the bathhouse. As soon as the policeman had Craig behind the closed door, he slammed him down on the hard floor. \"Faggot!\"\n\nCraig Rodwell had always been told to tell the truth as part of his Christian Science upbringing. He had learned that following that precept could be dangerous for himself as well as for others. When Frank had been arrested in Chicago, he had lied, saying he was Craig's uncle. Craig had told the truth and the dishwasher had been sent to prison for years. The district attorney had told Craig to lie and he had not, which was why his mother had had to get down on her knees, crying, to keep her son from also being sent away.\n\nAfter being arrested at Riis Park, Craig appeared before a judge who gave him a court date. The kindly judge who had taken him into his chambers said the fine would be routine and to be sure and bring the amount to pay it, around twelve dollars. Two weeks later Craig went to court carrying the cash. He found, however, that a charge of inciting to riot had been added to the charge of violating the dress regulation. Asked by the judge what had occurred, Craig said that the charge of inciting to riot made no sense, that he'd been immediately hauled into a small room away from the public. The judge dismissed the charge. Asked about the other violation, Craig turned to look into the face of the judge, who was sitting right next to him. Craig explained how the law worked, that it was an old law currently used only to harass homosexuals.\n\nCraig saw the immediate change in the judge's eyes on hearing the word _homosexuals_. His face turned beet red as he banged his hammer down on the gavel, looked away from Craig, and gave him the maximum penalty, twenty-five dollars. Since Craig did not have that much money on him, he had to go to jail for three days.\n\nRodwell remembered that after his arrest in the park, the police had come by his cell for hours to taunt him. He knew they had been looking for an excuse to work him over, so he had held his tongue except for the one mild rejoinder when he had asked the guard if he was the first \"faggot\" he had seen. He also remembered that when he'd been asked his name at Riis Beach, after he had answered, the officer had taunted him by asking him to give it over and over again.\n\nAsked his name at the prison, he refused to give it. Within a short time, a huge guard walked into the holding area where Craig was.\n\n\"All right, where's the guy who won't talk?\"\n\nSomeone pointed to Craig. The guard slammed Craig's head against the wall, took his wallet from his pocket, and held it upside down so that its contents fell out.\n\n\"Pick it up!\"\n\nHis head bleeding, Craig got down on his knees to gather up the pieces of paper. Through his tears, he could make out the photograph of his mother.\n\nSeveral days had passed since Craig had not returned home, and Collin, Rodwell's roommate, had no idea what had happened to him. The phone rang and Collin answered. Harvey Milk again. No, Collin still didn't know where Craig was. Milk had slowed down on calling Craig after he had gotten gonorrhea from him, but with Craig gone and his roommate having no idea why, he was very concerned.\n\nWhen Craig was released from prison, he got together with Harvey. Craig noticed Milk's strange expression on seeing Rodwell's shaved head. When the wake-up calls came even less frequently, Craig understood that in the eyes of the conservative Milk, he was now not only unfaithful but also a marked man who might endanger Milk's very successful career.\n\nShattered over the end of the relationship, Craig decided to take his life. He got rid of his cats, gave notice at his job, and dropped out of school. He bought a bottle of the depressant Tuinal. When Collin went out to a double-feature movie one night, Craig started taking the pills. He'd already written a note asking Collin to call Craig's aunt on finding the body. Above all, Craig didn't want his mother to get the news from a policeman.\n\nWhen Craig Rodwell opened his eyes in Harlem Hospital, he saw a police officer. Although Craig was tied down, the very sight of a cop so infuriated him that he bolted upright with enough energy to break the restraints. His roommate, Collin, had been bored at the movies and come home early and found Craig unconscious. Chance had caused his attempted suicide to fail, but he wasn't sure he wanted to be alive.\n\nA long difficult period followed. After a month in the Bellevue psychiatric ward he was transfered to St. Luke's psych ward. Then a wandering period in which Craig moved to Pleasantville, New York, Chicago, and on to Hollywood. As soon as he got off the bus in Los Angeles, Craig realized he'd made a mistake. He stayed for about a month, during which time John Kennedy was assassinated. By January 1964 Craig was back in New York.\n\nStriving to give meaning to his life, Craig decided to work for the gay movement and volunteered to work full-time at the Mattachine Society. At the Mattachine office he typed the newsletter, edited it, stuffed envelopes, and did anything else that needed doing. He found using all his free time in this way \"just thrilling.\" Craig Rodwell had found his life's calling.\n\nThe marginal quality of homosexual existence was rendered literal in the Village along the waterfront. While the shores of oceans, lakes, and rivers are still used in many cities by gay men as places to meet because of the privacy they afford, in the postwar Village the banks of the Hudson River became a meeting place because of the trucking industry. Commercial trucks that hauled produce and other cargo into the city usually unloaded their goods either at warehouses or at the waterfront loading docks. While the trucks were waiting for a load to take back out of the city, the empty trucks were parked unattended, with the backs unlocked.\n\nMarle Becker recalls the orgies that took place in the trucks: \"There was no fence. The trucks would back right up to the river and it was pitch-dark in there. Every now and then you could see somebody with a cigarette in back of a truck. There would sometimes be two or three hundred people in them.\" While Becker enjoyed the sex in the trucks, he adds: \"If we had our druthers we would have been happier checking into a hotel like any other couple, but that wasn't always an option for us.\" As for the criticism aimed at homosexual men by heterosexuals for having sex along the riverbank, Becker remembers how on his first visit to New York City he and his lover had been refused a hotel room because they were gay: \"There wasn't any place for us to go. If you didn't have an apartment or if you had roommates or what have you [the only option] was to have sex in Central Park or the trucks or some out-of-the-way place where hopefully you didn't get caught and arrested.\" Even an invitation by another person to his place could be problematic: would the other person turn violent or turn out to be a plainclothes policeman? \"It was like, should I trust this person, shouldn't I trust this person? You didn't know from one minute to the next what was going to happen. So it was safer to be with two hundred people in the back of a truck.\"\n\nAdding to the Village waterfront's appeal as a gay outpost, there were at various times in the postwar era a few waterfront bars that catered to homosexuals. The best-known of these, Keller's, operated in the sixties as a seaman's bar during the day and a gay bar at night.\n\nIn the postwar era, even in the Village, most social life for gay men took place in bars. Under increasing police pressure, the gay male bar and cruising scene migrated from West 8th Street to Greenwich Avenue between Sixth Avenue and Seventh Avenue South, until eventually in the early 1960s Greenwich Avenue became gayer than 8th Street.\n\nAlthough police pressure had been very severe in 1960, it probably reached its apogee beginning in 1963 under the administration of Mayor Robert Wagner as the city began a concerted drive to make homosexuals invisible in time for the 1964\u201365 World's Fair in New York City.\n\nPhilip Bockman arrived in New York City in 1963 after falling victim to a campaign of harassment at the University of Michigan. Called into the dean's office the day after he had refused to give the names of other gay students to a police officer who had roughed him up, he was told that unless he \"changed his ways\" he \"really should not be registered at the school.\" He recalls, \"I did come to New York, only to find that Mayor Wagner had this crackdown on the gay bars. Everything was closed up when I got here. The only gay bar open was the Cherry Lane. There were a few others that operated very surreptitiously. In 1964 for the World's Fair they started entrapping people as much as they could to clean up the homeless, the prostitutes, and the homosexuals. A lot of people were caught in that dragnet. I remember going to bars and some of the undercover policemen got to be known because they would always be in the bars and some people got to know who they were. I remember a couple of instances where we would form a circle around them and we'd spread the word to everybody in the bar. Sometimes we'd even follow them from bar to bar.\"\n\nEdmund White, who came to New York in 1962, recalls: \"During the World's Fair time, the mayor didn't want there to be a gay image to the city so he closed virtually all the bars. It felt to me, like whereas gays used to cruise in a rather furtive way on Greenwich Avenue, they were now coming down Christopher Street and moving farther and farther down toward the water.\"\n\nIt seems likely that gay men who cruised Greenwich Avenue and did not find a sexual partner had established the practice of circling the block again, and Christopher Street was one of the most convenient places to turn back for such a maneuver. The sex scene at the trucks and one or two waterfront bars also began to draw men there, and the most direct route from Greenwich Avenue was down Christopher Street.\n\nOne winter night in 1964 Dick Leitsch was walking down Greenwich Avenue and spotted a boy who struck him as very pretty standing in the doorway of a bakery at the corner of Greenwich Avenue and Christopher Street. It had been almost two years since Leitsch's love affair with Derrick had ended when Derrick decided for the second time that he did not like New York City and had decamped for Florida. The young man Leitsch had eyed started walking down Christopher Street, stopped, and leaned up against a building. The maneuver struck Leitsch as odd, for gay men cruised on Greenwich Avenue and on 8th Street, not on Christopher Street. Nonetheless, Leitsch followed him. The two men struck up a conversation, and Dick accepted Craig Rodwell's invitation to his apartment.\n\nCraig and Dick started seeing each other, but when Rodwell told Leitsch that he was in the Mattachine Society, Leitsch laughed at him. Dick had been to a Mattachine\u2013New York meeting in 1962 and heard Albert Ellis, a psychologist on the board, give a talk on homosexuality as illness and was appalled when Ellis got a standing ovation from the membership. Leitsch may have been bemused by Rodwell's involvement in Mattachine, but he soon learned that it was total. \"Every time I called Craig and said, 'You want to go to the movies,' he said, 'I have to go to Mattachine.' Finally, just to be with him, I started hanging out at Mattachine.\"\n\nConcerning the relationship between Leitsch and Rodwell, there is little that the two men agreed on except that it did not last long and ended badly. Rodwell maintained that he never wanted more than a sexual relationship and that Leitsch was obsessed with him to the point of being violent; Leitsch states the opposite, that it was Rodwell who \"decided that he was going to be my lover.... He seizes on you and won't let go.\" Whatever the truth of the matter, the two agree that it was because of their sexual involvement that Leitsch got involved in Mattachine\u2013New York.\n\nAlthough Rodwell had not been able to join formally because of his age, he had been active in Mattachine prior to 1964. He and Randy Wicker strongly advocated that the organization abandon the educational and research approach and take a militant stance. Frank Kameny had, with Jack Nichols, founded the Mattachine Society of Washington (Mattachine-Washington) in 1961. After Kameny and Nichols met Rodwell and Wicker, the four of them worked together to share strategies and support one another in their quest for a militant homophile movement. They were further supported by and worked with a lesbian couple, Barbara Gittings and Kay Tobin.\n\nLesbians had started their own organization, the Daughters of Bilitis (DOB), not long after the Mattachine Society was founded. If lesbians were less likely to draw the attentions of the police than homosexual men, that was in part because they had fewer institutions, whether social, political, or cultural. The needs of American lesbians were so basic in the 1950s that DOB turned inward to try to meet those needs by providing discussion groups and get-togethers. Barbara Gittings founded the New York chapter of DOB in 1958 and served as its president until 1961, the same year she met Kay Tobin. When Gittings met Frank Kameny in 1963, he greatly influenced her thinking and attitudes. \"If you take the position that Frank has (that homosexuality is fully on par with heterosexuality) then you get a very clear view of what you have to do. He really crystallized my thinking.\" In 1962, Barbara had become editor of DOB's publication, _The Ladder_. Influenced by Kameny's and Wicker's militancy, she used her editorship to try to edge DOB toward an activist stance.\n\nAs a dialogue got under way between militants Rodwell, Wicker, and Kameny and members of the Mattachine\u2013New York board, the board began to listen more seriously to what the militants had to say. One board member in particular, Julian Hodges, a rising star within Mattachine\u2013New York, seemed more open than the others. When he became the organization's president-elect (the equivalent of vice president) he invited Kameny to give the organization's 100th public lecture.\n\nIn July, Kameny came up to New York from Washington, proclaiming like a fiery Old Testament prophet the message that a militant civil rights and social action approach was the way for gay men and lesbians to achieve their rights. Kameny argued that the homophile movement should be modeled on the black civil rights movement as formulated by nonviolent militant leaders such as Rev. Martin Luther King Jr. Kameny warned of the dangers inherent in the position that homosexuality was a mental illness. Having used his scientific background to study the psychological literature, Kameny stated that \"until and unless valid, positive evidence shows otherwise, homosexuality, per se, is neither a sickness,... a neurosis, a psychosis, nor a malfunction of any sort.\"\n\nKameny courageously carried his logic to its conclusion, saying, \"Not only is homosexuality... not immoral, but... homosexual acts engaged in by consenting adults are moral,... right, good, and desirable, both for the individual participants and for... society.\"\n\nThe effect of Kameny's passionate and eloquently argued speech was electric. Dick Leitsch, who read Kameny's speech over a dozen times, credited it with making him political for the first time in his life.\n\nLeitsch started putting in so many hours volunteering at the Mattachine office that Hodges asked him to become his assistant. Leitsch became editor of the organization's newsletter as well and used it to further the militants' agenda by emphasizing coverage of legal, political, and governmental events that bore on homosexuality and by writing in support of the black civil rights movement.\n\nHodges and his supporters decided to run a slate of militant officers in the 1965 Mattachine\u2013New York elections. Hodges's slate included Kameny, the progressive psychologist Dr. Hendrick Ruitenbeek, and Dick Leitsch. Their ally Randy Wicker, ever the individualist, ran for the office of secretary as an independent. Leitsch was urged to run for president-elect and accepted the nomination in spite of not wanting the position. (Leitsch had urged Rodwell to run for president-elect, but Rodwell refused.) In his campaign statement, written by Rodwell, Leitsch promised to work to end police entrapment, discrimination by government and private agencies, and discrimination and prejudice in general against homosexuals. The militants won the May election in a sweep, making Julian Hodges the president of Mattachine\u2013New York.\n\nThose were heady times for the insurgent activists. Having elected a slate of militants to run New York's Mattachine Society, they pressed forward with what were probably the two most controversial (and for the militants exciting) ideas in the homophile movement: public demonstrations in support of homosexual rights and an attack on the psychological establishment's position that homosexuality was a mental illness.\n\nPicketing proved to be the most exciting experience for the militants. The first picketing was to protest against Castro's plans to put homosexuals in concentration camps. These protests\u2014one in Washington, D.C., at the White House and one the next day in New York at the UN\u2014went so well that the activists decided to mount a series of picket lines at strategic sites in Washington. The participants found the experience to be so liberating that they felt elated.\n\nCraig Rodwell considered the picketing at the White House to be \"the most wonderful day in my life.\" Saddened that the series of picket demonstrations was ending, he had a sudden brainstorm: the activists should picket once a year on July 4 in Philadelphia, where the United States had been born, to remind the American people that there was one minority group that still lacked basic rights. Rodwell's idea was accepted, and in 1965 activists traveled to Philadelphia to march in what came to be called the Annual Reminder.\n\nIt was during the time when Leitsch was just getting to know Julian Hodges that, out on a walk one day, they came across a politician who wanted to talk to them. The politician was campaigning to be district leader of the Village Independent Democrats (VID), then a new organization. He told Leitsch and Hodges about how Illinois had decriminalized sodomy and why New York should also do the same. The politician went on to say that adultery and divorce should be legalized as well. Although both men agreed with his position, Leitsch had walked away feeling that there was something very obnoxious about the politician, whom he had not taken very seriously. The politician lost the election, and his attempt to win elected office was derisively labeled \"the SAD Campaign\" because of Ed Koch's emphasis on legalizing sodomy, adultery, and divorce.\n\nDoric Wilson, feeling miserable as well as confused, entered the Cherry Lane Bar. His play _And He Made a Her_ , an Off Off Broadway hit, had just been moved to the Cherry Lane Theater as part of that theater's Monday evening playwrights series. Doric had considered this move to the well-respected Cherry Lane as proof of the professional theater world's stamp of approval. But the version of the play he had just seen did not at all resemble the one he had previously seen performed at the Caffe Cino. The new director had killed all the play's humor. Doric felt total despair, for he now saw his future endangered instead of assured. A drink in the Cherry Lane Bar, where gay men often gathered, would give him a chance to collect himself.\n\nA stranger came up to him and offered to buy him a drink.\n\n\"No, leave me alone. I'm in a rotten mood,\" was Doric's terse reply as he exited the bar.\n\nWilson walked to another bar. The same man who had approached him in the Cherry Lane had followed him and again walked up to Wilson. Doric said, \"No, get away from me!\" He left that bar and found a third one.\n\nBy the time he entered the third bar, he was beginning to calm down. The man from the Cherry Lane Bar showed up again. Feeling better, Doric yielded to the persistent fellow and accepted the drink, though it seemed only fair to warn him that he was not open to any further advances.\n\nDoric faced his would-be suitor: \"Look, I'm not interested in you. I've had one of the worst days of my life, but if you're going to follow me around, yes, you can buy me a drink, but I'm not going to be good company.\"\n\nThe man took out a badge and announced that Doric was under arrest.\n\nOne day in October of 1964 Leitsch picked up _The New York Times_ and read the headline about Greenwich Village. He learned that Ed Koch had made a 180-degree turn: he and Carol Greitzer, Democratic leaders of the First Assembly District, were demanding that something be done about \"the problem of... homosexuals who have been congregating on Village Square at Eighth Street and the Avenue of the Americas.\" Two days later _The Times_ reported that Koch was \"head[ing] a drive to rid the area\" of homosexuals \"and other undesirables.\" The article explained that after Koch had met with Police Commissioner Michael J. Murphy, Murphy had announced \"plans to increase surveillance of Greenwich Village to curtail loitering and solicitation by homosexuals.\" In the article Koch boasted \"that Commissioner Murphy had assured him that more 'effective' measures would be taken to curb activity of homosexuals.\"\n\nThe drive Koch and Greitzer launched led to the closing of a number of gay bars and the entrapment of hundreds of homosexuals, leaving many of them unable to get licenses or well-paying jobs for years into the following decade. In April of 1965, Koch asked Wagner for another \"cleanup\" of the Village and got it, during which more gay bars were closed and thousands of gay people were harassed on the Village's sidewalks. Three months later, Koch got yet another drive started, this one targeting Christopher Street.\n\nEntrapment was an issue much on Dick Leitsch's mind as he continued to volunteer at the Mattachine headquarters. At times in the midsixties the phone would ring in the Mattachine office all evening with calls for help from desperate men trying to save their families, their livelihoods, and their reputations. As Leitsch talked to one victim after another, he noticed certain patterns.\n\nLeitsch realized that the police explanation for arresting gay men\u2014that they were locking homosexuals up because they were prowling the city making unsolicited advances\u2014was patently false. The entrapped men had been arrested by police officers in plain clothes who had gone into gay bars and cruising areas dressing and acting as though they were interested in sex. When the gay men had responded in kind, the cops arrested them. Gay men also noticed that the policemen who did this kind of work tended to be very attractive men who dressed provocatively for the occasion. The policemen who went into public bathrooms were not supposed to display themselves, but Leitsch heard over and over from those arrested that the men they met in a toilet often not only were displaying their penises but also had erections and were masturbating to look as inviting as possible.\n\nOther patterns emerged as well as Leitsch continued to answer the calls for help. For example, it was not uncommon for the police to suggest the names of certain attorneys who might be able to defend the men arrested for solicitation and\/or loitering. When those charged consulted the attorneys, they were asked to pay exorbitant fees, far beyond the normal charge for such an infraction. Leitsch also noticed that after the charges had been dropped, the police officers, previously so serious and gruff, suddenly became quite amiable. Entrapped gay men commonly concluded that one reason for the high fees charged by the lawyers the police steered them to was to grease the wheels of \"justice.\"\n\nLeitsch had run for president-elect of Mattachine\u2013New York on a platform of ending entrapment. This issue was arguably the most pressing in the homosexual community, and it also seemed to Leitsch that it should be the easiest to address since it was so obviously unpleasant and went against everything that most Americans stood for. Once Leitsch had been elected, the question was how to go about ending entrapment.\n\nGay men in New York had learned that besides major events such as a world's fair, elections were also likely to trigger an increase in police entrapment. In May of 1965 John Vliet Lindsay, the six-foot-three lean, good-looking Republican congressman from New York's Silk Stocking District, announced that he would run against the incumbent mayor, Democrat Robert Wagner.\n\nAs the election campaign grew heated in the summer of 1965, Leitsch was in the Mattachine\u2013New York office one afternoon when a reporter from the _New York Post_ (then a paper with progressive politics) named Joseph Kahn came by to discuss a proposed change in the sodomy law that was under consideration in the state legislature. As Leitsch and Kahn tried to talk, they were constantly interrupted by the phone ringing with calls from entrapped men. When Leitsch explained to Kahn both how entrapment worked and how common it had become within the male homosexual community, Kahn reacted skeptically. Leitsch made the reporter an offer: would he like to talk to some entrapment victims?\n\nTommy Lanigan-Schmidt graduated high school in 1965 and went to New York City to apply to the Pratt Institute to study art. At the end of the application process he was told that they would let him enroll because he drew so well but that he wasn't going to graduate because he lacked self-discipline.\n\nWhen Tommy enrolled at Pratt he found the instructors uninspiring and the way they taught very limiting. Moreover, the art movement held up as a model in the classes Tommy took was Bauhaus, the antithesis of the medieval art he loved. With money a problem, he also did not see how he could complete a degree. Tommy made a conscious decision: to keep his dignity, he would stay in Pratt and learn the material taught, but he would make sure he failed.\n\nIn June of 1966 he received an official letter of dismissal. On that day, he returned home to Linden and showed the letter to his father, who was exasperated by the news.\n\nTommy's father determined to take matters into his own hands. Within hours, he took Tommy to downtown Linden and had him sign up on a ditch-digging crew. It was summer work that through the right connections might eventually lead to a union job for his son. Maybe it wasn't too late to make a man out of Tommy after all, his father thought.\n\nOn signing up to join the crew, Tommy saw that the other crew members were some of the tougher kids he had gone to high school with. Tommy knew it would have given them all a laugh to bury him alive. On leaving the office, he walked outside, took the dismissal letter out of his pocket, and looked at it. He already considered the letter his own personal declaration of independence. A moment of great clarity came to him, and he knew what he had to do.\n\nTommy went home. His father asked him, \"You want to go get the newspaper?\" Tommy said yes, and his dad handed him seven cents. He had sixty-five cents in his pocket. Seventy-two cents wouldn't get him to New York from Linden, but it would from Elizabeth. Tommy walked to the train station in Elizabeth, bought a ticket, and climbed aboard. Arriving at Penn Station, he headed down to the Village. He'd make it as an artist on his own somehow.\n\nChange seemed to be everywhere in the air in the fall of 1965 as John Lindsay was elected mayor. An aura of excitement pervaded the city as the new mayor prepared to take office, his youthful good looks inspiring comparisons with John Kennedy, whose election had likewise inaugurated a season of hope. New York's homosexuals also felt cautiously hopeful about the new mayor. They had fared worse under the Wagner administration than under any other in the city's history. Maybe conditions would improve for them with Lindsay.\n\nAlthough Dick Leitsch was also curious about how the new mayor would treat the gay community, a major change had already taken place in his life. As he remembered, one day Julian Hodges had come to him and said that they had to talk. Hodges confessed to Leitsch that he had been misappropriating Mattachine\u2013New York funds. Leitsch replied that Hodges must inform the organization's treasurer of this wrongdoing. Mattachine\u2013New York's treasurer, the head of a major Wall Street accounting firm, agreed to make up the shortfall but told Hodges that he would have to leave town. Mattachine\u2013New York accepted Hodges's resignation, which made Dick Leitsch the organization's president just in time for the Lindsay administration.\n\nOn a March night in 1966 the top brass of the new Lindsay administration were having a tough time keeping order during a raucous meeting at Judson Memorial Church in the Village. Police Commissioner Howard Leary, Chief Inspector Sanford Garelick, and Lindsay's human rights commissioner, William Booth, had all come there to pour oil on troubled waters after a cleanup campaign aimed at drug use, noise, congestion, and homosexuals in the Village had made some very public pratfalls. Two weeks prior, Garelick, New York's highest-ranking uniformed officer, had personally supervised an attempt to seal off an entire section of the Village around MacDougal Street, where young people drawn to the city went looking for a good time. The chief of police had ordered fourteen blocks around MacDougal Street closed to traffic on a Friday night, a peak time for the rowdy youths who came to enjoy the nightlife. Soon 1,500 youths were sitting in the middle of the empty streets, clapping and chanting, \"Up with the Village and down with the police.\" A half hour later cars were allowed inside the area. The press had a field day mocking the police, and residents were up in arms.\n\nCrowded into the church, Villagers had their boisterously independent spirit on full display as various political factions vied to control the agenda and the evening's tone.\n\nGarelick was getting discouraged by the chaos when suddenly a nasal voice sounded from the room's rear. \"I'm from a minority group in the community that is rarely heard from. Why aren't private nonracketeer businessmen allowed the legal right to run restaurants for homosexuals just as they run restaurants for heterosexuals?\" asked Randy Wicker.\n\nSanford Garelick, speaking in his best professional tone, replied, \"We have to enforce the law on licensed premises. You say repression; I say enforcement...\"\n\nWicker was having none of it. \"Are sexual deviates supposed not to eat? And what about those plainclothesmen, whose psychology even the homosexual doesn't understand, who come into places dressed in tight pants to lure people into illicit acts?\"\n\n\"Entrapment,\" intoned Garelick, \"is a violation of our rules and a violation of our procedure.\"\n\nWhen Wicker pointed out that many solicitations in gay bars were made by plainclothes policemen, Garelick denied it. This prompted Aryeh Neier, the NYCLU's executive director, to walk up to the podium and say that Garelick showed \"a certain naivet\u00e9\" in denying that entrapment occurs, before adding, \"It's alarming to think that the Chief Inspector doesn't know that a large number of police spend their duty hours dressed in tight pants, sneakers, and polo sweaters' to bring about solicitations.\" The audience erupted into a loud ovation and shouts of \"Bravo!\"\n\nPressed on how frequently police officers tried to entrap homosexuals, Garelick demurred but stated, \"I'm very severe in my condemnation of entrapment.\" Looking obviously uncomfortable, he added that he hoped that if incidents of entrapment did occur, the public would report them to the police \"as they would crimes.\"\n\nDick Leitsch had to allow himself a smile. His plan of placing Mattachine members in the audience to ask the right questions had worked beautifully, as had the entire campaign he and other Mattachine\u2013New York members had relentlessly pursued since Lindsay had taken office. The timing of the Kahn series on the vice police in the _New York Post_ had been perfect. The Wagner administration had ignored the articles and all the questions they raised, but Kahn had confronted Police Commissioner Walsh on the entrapment of homosexuals in particular and Walsh had said that entrapment was not police department policy. The _Post_ had then run an editorial calling for an end to entrapment. Leitsch and other Mattachine members had called the _Post_ to urge them not to let the issue fade, and the newspaper had promised to keep the heat on. They did, exposing both the seamy nature of such police operations as well as how wasteful they were of city resources and police manpower.\n\nWhen Garelick left Judson Memorial Church that evening, he no doubt felt relieved that the meeting was over. Unfortunately for him, his troubles about homosexual entrapment were not. When Dick Leitsch arrived home late that evening, his phone rang. He picked it up and heard the voice of John Lassoe, an Episcopal priest and Mattachine\u2013New York supporter. The information Lassoe had was perfect: a vice plainclothesman had just entrapped a heterosexual priest at Julius', only a few blocks away from where Garelick had been speaking. Leitsch hung up the phone and started calling reporters to make sure they had the story in time for the next day's papers.\n\n_The New York Times_ put the story of Garelick's request that the public report instances of entrapment to the police next to a story about the entrapment occurring simultaneously in the same neighborhood. Shortly afterward the _New York Post_ ran a story detailing a typical instance of bathhouse entrapment: a policeman, standing by his locker, dressed in his underwear, clutched his groin as he moaned. When a forty-year-old Brooklyn tailor asked him if he was ill or needed help, the tailor was arrested. Columnist Pete Hamill spoke with the man's devastated wife, who asked as she sobbed what would happen to her husband. In spite of her crying, she managed to speak softly so that her children would not hear the conversation.\n\nAfter he had read the press reports about the entrapment at Julius' coinciding with Garelick's denunciation of entrapment, Leitsch rolled a sheet of paper into the typewriter and started a letter to Sanford Garelick. He had, after all, asked to be informed of any instances of entrapment.\n\nLeitsch's cleverness, while understandable, had not been needed. By now the new administration had clearly gotten the message, which was confirmed when Lindsay invited Leitsch to a meeting. The meeting brought together various factions: officials of the Lindsay administration, Village politicians, and representatives of Village groups that had been harassed by the recent police drives. The meeting took place at the Village Vanguard and got off to a humorous start when Allen Ginsberg voiced concern about poets and avant-garde theater people caught in the cross fire of the Mafia, the police, and the city's cabaret laws. Ginsberg told Lindsay there had to be someone in the License Department who could handle culture.\n\n\"He's right behind you,\" Lindsay answered, indicating the deputy commissioner from the License Department.\n\n\"But is he culturally hip?\" Ginsberg pressed.\n\n\"I can assure you,\" Lindsay deadpanned, \"he is not.\"\n\nWhen the subject of entrapment arose, the mayor said that \"he had told the police department that the practice would stop immediately and would not be resumed\" so long as he was mayor. Ed Koch grew red-faced, jumped up, and started yelling in protest. Leitsch smiled broadly as he sat next to the apoplectic Koch. He and Mattachine\u2013New York had won the war on entrapment.\n\n## S L A P O L I C Y\n\nEven before the success of the campaign to end entrapment, Leitsch had already started an effort to end police harassment of gay bars, which he saw as the second-most pressing problem facing the local gay population.\n\nAccording to both Dick Leitsch and Randy Wicker, in the late 1950s there were a considerable number of gay bars that operated openly in New York City. Leitsch puts the number of gay bars at \"more than forty\" in 1959 and says that they were businesses that had been around for a long time and were controlled by \"private individuals.\" But a great wave of bar closings began in 1960, which Leitsch felt was the fallout of a large investigation into corruption within the State Liquor Authority (SLA). When the New York City Police began a systematic campaign in 1960 to close all gay bars, they all lost their licenses, with one exception, the bar of the Cherry Lane Theater (though even that bar finally lost its license during Koch's 1966 antigay crusade).\n\nThe result of the 1960 crackdown, according to Leitsch, was that \"the organized crime syndicate saw an opening and rushed in, opening bars all over town.\" Leitsch claims that \"during one period\" Mafia gay bars were able to remain open by paying off \"strategic people\" but that \"when that became more difficult after the scandals in the State Liquor Authority, the bars began to operate only until they were raided, at which time they would pick up the entire operation, bar, employees, clientele and all, and move to another place in the same neighborhood, with the licenses in another name.\"\n\nLeitsch later commented in a lecture that whenever a crackdown occurred, \"homosexuals knew the 'Mafia' would find some way to supply us with a place to meet and socialize. The sad philosophy of the gay world was that expressed by Brecht's Mother Courage: 'Our only hope lies in corruption.' \" Leitsch saw the problem as the SLA having \"surround[ed] the liquor business with so many rules and petty restrictions that honest men cannot survive,\" so that the business \"reverts to the hands of those crooked enough to have the knowledge and lack of scruples to ferret out and work with crooked liquor agents.\"\n\nWith so many powerful interests having so much at stake, how could Mattachine\u2013New York go about making gay bars legal? To try to determine how to approach the problem, Mattachine\u2013New York paid an attorney to do a detailed study of New York's Alcoholic Beverage Control (ABC) Law. The memorandum that was prepared found: \"Contrary to the contention of many bar operators, there is no provision in New York which flatly prohibits homosexuals from gathering in bars and there is no provision which flatly prohibits bars from serving homosexuals.\" However, the memorandum's author immediately added, Section 106(6) of the law, which prohibited a licensee's premises from becoming \"disorderly,\" had been interpreted as not allowing licensed premises to serve homosexuals or for homosexuals to gather in drinking places as \"a reflection of the attitude of the State Liquor Authority and... most of our courts.\" As Leitsch and his colleagues studied the memorandum, it seemed to them that before they could get very far in trying to make gay bars legal, what was initially needed was some clarity about the New York State Liquor Law. The liquor law's vagueness with regards to serving alcohol to homosexuals rendered the issue murky.\n\nDanny Garvin ordered a beer and settled against the bar to relax with his navy buddies. He felt a tap on his shoulder and was told to turn around and face the bar. \"Why?\" Danny asked. The bartender explained that this was the bar's rule to prevent being closed on charges of solicitation. Danny did not understand how he was supposed to cruise if he was facing the bar. Wasn't that the point of coming to this bar, to meet other men?\n\nDanny remembered the first time he had walked into Julius'. Some of his navy mess mates had brought him here, perhaps to show off their worldliness. They had said that though he was only seventeen he would not have any trouble getting a drink. The bar had seemed so ordinary that Danny had no inkling that it was a homosexual bar until a rather stereotypically feminine-looking older man, seeing Danny and his buddies' haircuts, had spoken to them: \"I bet you three are servicemen. I can tell fortunes. Let me see your palms.\" Taking one of each of their palms, he had pretended to study them before announcing, \"Well, I'm going to tell you, one of you is homosexual, but I'm not going to tell you which one. I'm going to give all three of you my phone number, and the one that is will know to call me!\"\n\nThat was how it always seemed to work out for Danny: the only men approaching him were older men whose sole interest in him was sexual. That was just fine, he told himself. He was mainly interested in getting his rocks off, and men seemed more sexually available than women.\n\nOne reason the all-male crowd at Julius' hadn't struck Danny as unusual was that it seemed like any of the neighborhood bars where men went to socialize in Inwood, the middle-class immigrant neighborhood he had grown up in. An all-male gathering in a corner bar was common enough in Inwood, perhaps in part because of the strong Irish presence there. So much of Danny's identity seemed wrapped up with his Irish heritage. His grandfather James, freshly arrived in America, had helped to assemble the Statue of Liberty, and Danny's father, Michael, had smuggled messages for the IRA in Ireland until he was caught. It was only because Michael had been born in America that the British had released him, but only on the condition that he return to the States.\n\nAnother part of Danny's Irish heritage was the guilt that seemed to come with growing up in a devout Catholic family. When he reached puberty his older sister had told him that if he ever started to have an erection or experienced a wet dream, he should get out of bed, fall to his knees, and pray to the blessed virgin Mary. No, he wasn't really gay: he was just getting his rocks off. At least he hoped so, because judging from all he knew of it\u2014the older men who had come on to him in public toilets or theaters\u2014it seemed a very seamy life. It was such a confusing scene, too, with all the lingo he didn't understand, like men calling each other she and Mary Duggan. And there were strange terms like _flame queen_ and _scare drag_. Sometimes he couldn't even follow it.\n\n_\"The closet door was so tight back then.\"_ That's how he would try to explain it years later.\n\nThe stratagem Leitsch and his colleagues devised to seek clarity on the state's liquor laws revealed the extent to which they modeled their activism on the black civil rights movement. In a move that was both creative and ingeniously simple, they determined that several members of the Society would enter a bar, announce themselves as homosexuals, and ask to be served. If they were refused service, as they expected to be, they would make a complaint against both the bar and the SLA for violating their constitutional rights to free assembly and equal accommodation. To make sure that they made their point, they invited the press along.\n\nThey sent telegrams to the press announcing that three Mattachine members would show up at noon on April 21, 1966, at the Ukrainian-American Village Restaurant on the Lower East Side to ask for drinks. That restaurant had been targeted because it displayed a sign saying: \"If You Are Gay, Please Go Away.\"\n\nWhile the conception of the challenge to the SLA may have been ingenious, it seems to have been ill-fated from the start. By this time Craig Rodwell had quit the Mattachine Society because Mattachine\u2013New York's board would not agree to some of Rodwell's proposals to do more public outreach. However, Leitsch had planned to mount his challenge with John Timmons, and Timmons had insisted on inviting Craig along, to which Leitsch reluctantly agreed. Craig, in turn, told Randy Wicker about the event and Wicker, ever eager for publicity, showed up uninvited. Wicker's presence annoyed Leitsch, for although it was Mattachine\u2013New York that had coordinated the gay challenge at the Judson Church and Wicker was the organization's secretary, he had identified himself that night as speaking for his own one-man operation, the Homosexual League.\n\nThe four men arrived at the Ukrainian-American Village Restaurant later than planned and found it closed. The press had informed the manager of the planned action and asked how he would respond. The manager had thwarted their plans by simply closing shop. The activists then walked over to a Howard Johnson's on Sixth Avenue and 8th Street, where Dick Leitsch read a statement identifying the group as homosexuals and asking for service. The manager, Emile Varela, doubled over in laughter and ordered a waiter to bring the men bourbons, saying that he knew of no regulation against serving homosexuals. Having failed twice, the men improvised a third target, the Waikiki, another Sixth Avenue spot popular with gay men. There the men were again served. After huddling yet again, the men walked two blocks west to Julius', where the bartender was also willing to serve them. Leitsch talked to the manager, who explained that as they were already facing a license suspension he did not want any more trouble. When Leitsch explained how valuable his refusing to serve them could be and promised that Mattachine\u2013New York would help get them legal assistance with their current case, the manager agreed to play along. Refusing to serve the men, he told the press that since the men had said they were homosexual, he would not serve them, saying, \"I think it's the law.\" Dick Leitsch then told the press that he would file a complaint in Mattachine\u2013New York's name with the SLA.\n\nMattachine\u2013New York got extensive media coverage for what _The New York Times_ dubbed a Sip-in, but the SLA refused to play their part in the game the organization had set up. The day of the Sip-in, the media asked the SLA's chief executive officer, Thomas Ring, about his agency's policies concerning homosexuality. After years of suspending and revoking licenses when homosexuals had been entrapped in bars, Ring said that the SLA \"do[es] not discriminate against homosexuals\" and claimed that their regulations \"leave service to the discretion of the management.\" Four days after the Sip-in, the SLA told the press that it \"would take no action against bartenders or liquor licensees who refuse to serve drinks to homosexuals.\" The SLA chairman, Donald Hostetter, asked about the Mattachine\u2013New York complaint, answered, \"We would take no action on such a complaint,\" but added, \"This might be a matter for the [New York City] Commission on Human Rights.\" The SLA had decided to pretend that it did not discriminate and had no oversight over such discrimination, while shuffling the problem off to another entity. Commissioner Booth was eager to cooperate and, based on the city's law that outlawed discrimination on the basis of sex, declared that \"[d]enial of service to a homosexual... would come within those bounds.\"\n\nSubsequent events and explanations would eventually clarify that the SLA's new line was that homosexuals could be in bars and be served liquor as long as they were not \"disorderly,\" by which the post-Sip-in SLA meant that homosexuals who behaved largely as heterosexuals were allowed to\u2014kissing, touching each other intimately, or dancing in an overtly sexual manner\u2014were a disorderly presence and hence could endanger a bar's license. In the meantime, Booth had been reluctantly forced to conclude that the New York City Commission on Human Rights was limited by law to investigating sex discrimination only in the field of employment and so was powerless to take action against a bar that refused to serve homosexuals.\n\nThe publicity gained by the Sip-in did have the effect, however, of inspiring a number of bars outside New York City, especially in New Jersey, to get in touch with Mattachine\u2013New York and request its assistance in fighting their own state liquor authorities. Over the coming years Mattachine\u2013New York would continue to whittle away at the tools the police and the region's state liquor authorities used to harass bars and restaurants that served a lesbian and\/or gay male clientele. Most significantly for New York City, the police department could no longer use entrapment against gay men as a tool to deprive a business of its liquor license.\n\nFor all that had been achieved by Mattachine\u2013New York under Dick Leitsch's leadership, there were a lot of strains and divisions among the leading East Coast homophile activists. Leitsch had been furious when Mattachine-Washington did not pay what he saw as their portion of the cost of producing both a 1965 regional conference and the _Eastern Mattachine Magazine_ , a joint publication venture of the two organizations. After a falling-out over this matter, Leitsch felt alienated from Kameny.\n\nRandy Wicker had made so much money selling his gay buttons that he was busy preparing to launch a shop that would specialize in political buttons, and he began to withdraw his energies from Mattachine\u2013New York.\n\nBy the spring of 1966 Craig Rodwell had become alienated from the Mattachine Society because of a disagreement over a project he had proposed. Rodwell had wanted Mattachine\u2013New York to open a storefront operation to make the organization more visible and accessible. His idea was that from such an office the organization could make available pamphlets and magazines published by the homophile movement as well as books on the subject of homosexuality. When the board rejected the idea, he quit Mattachine\u2013New York in protest and began to spend his summers working on Fire Island cleaning rooms at a hotel popular with gay men. He planned to save his money and open his own bookstore.\n\nAfter two seasons of working at Fire Island, Rodwell had enough money to open a bookstore that would stock only gay and lesbian reading material. Rodwell named the store the Oscar Wilde Memorial Bookshop. He intended the store to function as a center for the gay and lesbian community at a time when there was no such community center in New York. After quitting Mattachine\u2013New York, Rodwell took a leaf from Randy Wicker and started his own organization, which he named HYMN, for Homophile Youth Movement in Neighborhoods. While the organization never had more than a few members, it gave Craig a public platform. Rodwell emphasized the idea of identity and activism by setting up the organization so that \"memberships are not sold... you 'join' by being gay\u2014because this is a _movement_ ,\" as he explained to a reporter. A few months after opening his bookstore, Craig started publishing a newsletter that he named the _Hymnal_ , which he intended to use to inspire gay men and lesbians to become politically active. As Barbara Gittings had done with _The Ladder_ , Rodwell highlighted news of militant actions and major successes, especially from Europe, where the gay legal and social situation was much better than in America. The _Hymnal_ 's first issue informed readers that Britain's House of Commons had given initial approval to a law that would legalize homosexual sex.\n\n\"Why don't you just tell them you're queer?\"\n\nDanny Garvin, only 17 in early 1967, just shook his head once from left to right. He couldn't believe what he was hearing. First Chuck says that Danny has to leave because he's AWOL and under eighteen and Chuck could get arrested, and now Chuck's telling him to tell the navy he's homosexual!\n\nIt seemed that everything that could go wrong in several hours had. After getting drunk the previous night he'd overslept and not reported back to base on time. Then he discovered that he'd left his military ID in the liquor store where he'd picked up the booze for the impromptu party last night. When he had called his sister, she'd said the FBI and the CIA were looking for him. That was when he had really panicked. Now he'd run to Chuck, the one gay person his own age he'd gotten to know, and he's telling him to get lost. Danny left Chuck's, went to a drugstore, and bought some razor blades.\n\nThe Mattachine Society of New York had had a partial success in legalizing gay bars through the so-called Sip-in, beginning a legal challenge to the practice of denying licenses to the bars that served homosexuals. The incomplete victory created ambiguous results. While the SLA had retreated into a strategy of obfuscation instead of joining the battle, the success of Mattachine\u2013New York in ending entrapment and changing the SLA policy so that businesses with liquor licenses could serve homosexuals resulted in legitimate businessmen beginning to invest in non-Mafia clubs. In March of 1967 Mattachine won a very significant ruling from New York's highest court: the SLA could not revoke a license on the basis of homosexual solicitation. The SLA's ability to prevent bars from serving gay men and lesbians was being eroded.\n\nThe Mafia apparently could read the writing on the wall just as well as the SLA. Running gay bars would become increasingly like running any other bar, and so the profits would go from exorbitant to ordinary. The Mafia was not interested in ordinary profits.\n\nA year after these Mattachine successes, _The New York Times_ , always eager to be the first to spot a new trend, reported on its front page that because of the declining profits in the gay bar business due to the encroaching legalization, the Mafia had a new strategy: operating gay bars as private clubs. Since they were private, the paper explained, these clubs enjoyed \"the same legal sanctuary as an individual's home against police inspection without a court warrant.\" _The Times_ reported that some of these new clubs charged high annual dues, extra fees for admission to special rooms, and high prices for setups or illegal liquor.\n\nThis strategy created new challenges for the police. Courts were refusing to issue search warrants \"without specific complaints and evidence,\" and patrons of these businesses were generally unwilling to give the police the kinds of detailed information needed for warrants. When plainclothes police tried to enter the clubs to gather evidence, they had \"been ejected bodily when they failed to pass a recognition test.\" These new kinds of operations, _The Times_ reported, \"have tight security and member identification.\"\n\nDanny Garvin looked at the piece of paper the U.S. Navy wanted him to sign. It seemed a reasonable exchange: he wanted an honorable discharge; the navy didn't want to pay any benefits. By this time he'd had enough of both the navy and psychiatric wards. After he had cut himself with two razor blades and only succeeded in scratching his wrist up, he knew that he couldn't bring himself to commit suicide. He had then gone to Bellevue on the advice of a psychiatrist and asked to be admitted. The navy had come and picked him up from Bellevue and transferred him to St. Alban's Naval Hospital.\n\nThere had been plenty of strange moments during the three months he had spent at St. Alban's. Volunteering as a movie projectionist, he had shown the film _Gigi_ seven or eight times a week to soldiers just back from Vietnam who were missing legs or whose faces had been burned away. Only a few weeks ago when he'd obliged a nurse by helping to take a body to the morgue, workers there had grabbed his hand and slapped it on the ice-cold chest of a corpse that had been stored there for God knows how long. But by far the most absurd aspect of Danny's hospitalization was his inability to discuss what troubled him the most, his homosexuality, because he wanted the honorable discharge.\n\nDanny picked up a pen and signed the general release. Hours later he walked out of St. Alban's Naval Hospital, where he had turned eighteen. He was the same age his father had been when the British had given him his freedom, also in exchange for getting something they had wanted.\n\nIt seemed appropriate therefore that it was also St. Patrick's Day. Danny had touched death, but he was alive.\n\nHe would go to Julius' that night to celebrate.\n\n#\n\n# On the Street\n\nWhile gay men and women in the 1960s were pressured into keeping their sexual lives very closeted, they succeeded in creating a diverse subculture. Then as now, the lesbian and gay male community had its own divisions and subcultures based on everything from sex, to sexual proclivities, to class and race. While it is not within the scope of this history to survey all or even most of these groups then existing in New York City, one group that demands our attention was a band of very young gay men whose social life existed mainly on the street. Although these youths stood in contrast with the activists because of their lack of respectability, the two groups shared two common traits beyond their sexual orientation: the social life of both was centered on the Village, and the members of these two groups were among the very few homosexuals of the era who chose to be visible.\n\nWhen Tommy Lanigan-Schmidt arrived in New York City, he had no plan about how he was going to survive, let alone realize his dreams of becoming an artist. He did not know anyone who could help him get a job, and he initially looked to other homeless gay youths to help him survive. With no money and no job, Tommy turned to panhandling.\n\nOne cold night a man walked up to him and asked, \"Are you all right, Mary?\"\n\nTommy, confused, protested, \"I'm not Mary!\"\n\n\"Mary, Grace, Allison\u2014what difference does it make? After all, we're all sisters. Aren't we?\"\n\n\"But I'm a man!?\"\n\n\"Listen, Jesus would not want someone freezing in the street. If you need a place to stay, I have one.\"\n\nThe street queen who had so kindly offered Tommy lodging was alternately known as Opera Jean, because of his love of opera, and Sister Tooey, as he was infamous for going through people's medicine cabinets searching for the downer Tuinal.\n\nTommy soon found a very inexpensive apartment, thirty-five dollars a month, on the Lower East Side. It was in a terrible building, with around six of its apartments burned out. He had almost no money at all after paying for his rent and food, and so he lived very close to the street and found his friends there among the drag queens, street queens, and runaways.\n\nThere was Stanley, a queen who was always in drag and prone to bizarre pronouncements. When Stanley told Tommy that he had gone to a famous school, Tommy innocently asked him which one. Stanley answered exasperatedly, \"You're crazy! 'A Famous School' was carved above the door! We operated on lions and alligators in biology class, not frogs!\" Congo Woman, a \"very nasty\" black queen, used to throw bricks through store windows to get whatever she wanted, usually some drag item such as a dress or a wig. She also carried a stone in her purse just to have handy to throw at people. Irish Sylvia used drugs fairly regularly, and so when he fell to his death from the roof of the St. George Hotel many assumed that he had fallen because he was high, but there were rumors that he had been pushed. Tommy recalls that \"that was the first queen funeral that I went to.\" Bambi, who got his name because of his large eyes, was never seen without a bottle and a bag. He had a habit of standing in the middle of the street to stop traffic and then banging on drivers' car windows as he demanded change. Tommy emphasizes Bambi's unpredictability by saying that he was \"a creature of the moment.\" Nelly, a Latino youth also known as Betsy Mae Kulo (as in Spanish for \"kiss my ass\"), was so young-looking that when he went hustling in drag no one would have thought him a man. Orphan Annie was so called because his white skin, red Afro, and unique eyes immediately called to mind the hollow-eyed comic strip character. He claimed to have an extremely wealthy grandmother who sent him money. Another youth who wore a Beatles haircut and had good teeth almost never said anything but always wore a suit or a coat and usually a tie. Sometimes the shirt would get very dirty, but Tommy never saw this young man without a coat and figured this was his way of trying to hold on to some dignity.\n\nWhereas the youth with the Beatles haircut looked quite conventionally male, a few of the youths were extremely feminine. Tommy recalls, for example, one boy who, although he always wore men's clothes, had such feminine mannerisms that the overall effect\u2014his body language in combination with his fine features, long hair, and plucked eyebrows\u2014was decidedly feminine.\n\nAlmost all the boys, some as young as fourteen, were runaways. A few actually lived with their families and went home at night but made the streets their second home. Most of them had come from abusive families. They were predominantly white, with a few black and Latino youths.\n\nWE WERE STREET RATS. PUERTO RICAN, BLACK, NORTHERN AND SOUTHERN WHITES. \"DEBBY THE DYKE\" AND A CHINESE QUEEN NAMED \"JADE EAST.\" THE SONS AND DAUGHTERS OF POSTAL WORKERS, WELFARE MOTHERS, CAB DRIVERS, MECHANICS AND NURSES AIDES (JUST TO NAME A FEW). UNTIL PROPERLY INTRODUCED IT WAS DE RIGUEUR ARGOT TO CALL EVERYBODY \"MISS THING,\" (AFTER THIS, IT WAS DISCRETIONARY USAGE).*\n\nBob Kohler is a native New Yorker who has lived in the Village almost all his adult life. After being seriously wounded while serving in the navy in World War II, Kohler worked as a talent agent. At forty, Bob had decided to take some time off from work to consider where his life was going. Kohler lived in the Village on Charles Street and had a schnauzer named Magoo that had to be walked several times daily. It so happened that Magoo's route went by Christopher Park. During his furlough, Bob went by the park so often he noticed the gay street youth who often hung out there and began to talk to them.\n\n\"They loved my dog. That was the big thing. They would ask to walk the dog, and they became even younger with the dog. That became like a link, being able to walk the dog. I was old enough in those days to be their father. I think in the beginning they were very surprised that I wasn't looking for anything at all, which is one of the reasons they trusted me. I think that was basically the bond from them to me, the dog and the old person.\"\n\nBit by bit Kohler became extremely interested in the youths and their stories. Sometimes he would circle the park with Magoo and one or two of the youths. As they walked, one of the youths would betray his tender age by swinging around a parking meter that had been hit by a vehicle so that, loose in the sidewalk, it leaned at an angle. The more Bob listened to the young men talk, the more he sensed the pain they felt over having lost their connections with their fathers: \"I always felt that what the kids saw in me was a father. At least for most of them that was their biggest problem.\"\n\nBob often gave them small amounts of money to buy food at Smilers Deli, a neighborhood convenience store. \"You'd give them a quarter for potato salad and another quarter for macaroni salad, and you had to give them another quarter for Ex-Lax because they were so constipated from all the starches. But that's what they lived on, back and forth to Smilers. You could almost see the dent in the road.\"\n\nThe stories of the street youth's behavior are often shocking. As Bob Kohler explained, \"They were rotten kids. Of course they had been made rotten.\" One young man Tommy met on the street had an enormous burn scar covering his face and most of the rest of his body. His mother had decided that she didn't want men to be \"tempted\" anymore by her son's good looks and had held his face in the flames herself. He was not the only one among the street youth who had been burned. Bob Kohler recalled one boy among the street youth who bore the impression of a clothes iron on one of his buttocks, also made by one of his parents. In other cases parents or other relatives had thrown boiling water on them.\n\nWE SAT ON THE CURB-GUTTER AROUND THE CORNER FROM A DANCE-BAR CALLED THE STONEWALL. HE HAD WOUNDS SUTURED UP AND DOWN HIS ARMS. THE ARMY HAD REJECTED HIM FOR BEING \"A QUEER.\" HIS FATHER HAD THROWN HIM OUT OF THE HOUSE THROUGH A GLASS DOOR.\n\nOnce a youth urgently said to Bob, \"If I could only have my breakdown, if I could only have my breakdown, then I know they would help me.\" Bob asked him what he was talking about. The boy explained, \"They would always help my mother when my mother had her breakdowns. They would take her away and then she would come back and she'd be fine.\"\n\nKohler recalls, \"Death was a very, very common thing. You didn't miss a beat when somebody said... 'She died,' or 'She got shot.'... There were always bodies being fished out from the river.... I know of two that I knew. I remember one queen waking up on a very high rooftop in the blazing sun, stoned out of her mind from the night before and had slept through the whole day, burned to a crisp, and just wandered around the roof, just fell off. One called the wrong person a nigger and was murdered.\"\n\nExposure to the elements worsened their chances of survival. \"They slept in doorways in the rain. They slept on the benches when it was nice. So it was a very harsh life that they lived, and in the winter especially.\" Tommy says simply that \"most of the people I knew from back then were already dead within four years after I met them.\"\n\nAccording to Kohler, the kids operated on a constant cycle from searching for drugs, to getting high, to crashing, back to seeking more. Many of them would take anything they could lay their hands on. Still, they had their preferences even as they bartered, stole, and begged for drugs. \"The down of the day was Darvon. Then Black Beauties, Christmas Trees. The downs were to take away the pain that was always there. The ups were to keep them going, so that they could go out on tears all night. They never slept. It was just collapse, hoping they made it to a bench before they couldn't go another step.\"\n\nThey would shoplift from the local A & P grocery, or sometimes they would steal from Smilers. Several would enter together and two would pretend to fight to distract the staff so that a third could steal food. Another way of stealing a meal was known as \"eat and tip,\" as in \"tip out the door.\" This ploy was mainly used at the local Howard Johnson's when the youth wanted what for them was a really good meal. Located on Sixth Avenue at 8th Street, the Howard Johnson's was cavernous and had a very long counter with a zigzag shape. The youths would situate themselves far from the exit the waitresses used to get from behind the counter and after eating, at a signal, would all get up at the same time and run out.\n\nAs much as Bob cared for the youths, he did not idealize them. Even when they walked Magoo, he did not completely let down his guard: \"I used to watch, because I thought they're going to either eat him or sell him if somebody offered them five bucks.\n\n\"I became many things. I became a father confessor. A stasher of stolen goods. I became a patsy at times, too, because these kids lived to get over, and I was no exception. Just because I was as much of a friend as they could have, I still was somebody that they wanted to get over on. So I never turned my back. But they would tell me all their troubles.\"\n\nBilly, a blond youth also known as Miss Billy, was one of the boys Bob was closest to: \"He was a cute, cute kid. A real tall kid and very young. He was gay, gay, gay, but he was not a drag queen. I sort of became a protector of Billy. I tried to help him more than the others. Billy leaned on me more. I bailed him out of jail a couple of times.\" Billy was one of the youths who lived at home but spent as much time as he could on the streets.\n\nJust because some of the youths had homes did not necessarily mean that they had easy lives. Robert Rivera, more usually known as Birdie, for example, came from a Puerto Rican family in the Bronx. Although his Jewish father had a male lover, Birdie recalls that \"my parents didn't want me anywhere near the house.\" He became the lover at age eleven of Joe, a police officer who became his legal guardian, but Joe, who insisted that Rivera wear dresses, also beat him.\n\nAccording to Birdie, by 1965, while he was still in high school, he and other gay youth had formed their own gang, the Commando Queens. Members had to be over eleven and under eighteen. Their code of ethics was: \"You had to be kind to someone every day; you had to make sure your makeup was okay; couldn't be dirty; had to protect somebody who was getting beat up, someone who was queer; couldn't wear bras or girdles, no leather.\"\n\nBirdie became a very tough street youth whose philosophy was to hit before being hit. He and other street youth friends of his staked a claim to Riker's, a restaurant at the intersection of Christopher Street and Seventh Avenue South, and were intimidating enough that they \"took it over from the winos.\" They ran wild in the subways and didn't necessarily spare other gay people, such as the old men and cabdrivers they hustled: \"We used to terrorize those queens\u2014would tease them about who they had sex with and for what. We called them 'dog food.' \" Birdie's closest friends were Martin Boyce and Tommy Lanigan-Schmidt, both of whom recall Birdie's fearlessness.\n\nWhen the street youth were able to get a little money they would sometimes find transient lodging in cheap hotels like the Albert, the very popular Broadway Central (until it collapsed), the Earl, the Marlton on 8th Street, the Haliburt on University Place, and the Keller.\n\nWhen they were able to get into a hotel, they almost always ended up getting thrown out. Bob Kohler went to the Essex to visit Orphan Annie. After he and some of Annie's friends had been sitting and having a normal conversation for some minutes, Annie suddenly leaped up, picked up the radio, threw it out the window, and sat back down. No one acted as if anything unusual had happened, so Bob also kept quiet. Minutes later, Annie, for no apparent reason, picked up a lamp and threw it out the window.\n\nCross-eyed Sylvia instructed Tommy in how to make what the street queens called a drape dress, a resourceful technique for a queen who had no female attire and needed to find some in a hurry: \"Check in a place like the Hotel Albert and be sure to have a shopping bag with you. Since the curtains have hems on their tops and bottoms and a lining on the back, you simply cut the seam off the top and the bottom, slip in between the cloth and the lining, put on a belt, and attach a broach. After you've put your male attire in the shopping bag, you're ready to leave the hotel in your new dress.\" The Broadway Central's large windows made it particularly popular with New York's street queens.\n\nTommy explains, however, that usually the street youth were thrown out of hotels not because of vandalism but either from generosity or their own survival needs: they either tended to let their homeless comrades stay the night or brought johns to their rooms.\n\nThe cabaret laws, laws against homosexuality, and the SLA laws complicated life at the few businesses that might have had some special appeal to these youths as gay young adults. For example, the Tenth of Always was an ice-cream parlor run by the Mafia and targeted at gay teenagers. The prohibition on selling alcohol to persons under eighteen years of age was compensated for by selling hamburgers and sodas at hugely inflated prices. The Tenth allowed dancing, an activity that attracted teenagers, notwithstanding the small size of its dance floor. But Tommy recalls how the staff would yell at customers if they even sat close to each other. Since the Tenth did not have the required license to allow dancing, when the police made one of their frequent stops by the place the Mafia staff would turn on a chandelier as a signal for everyone to stop dancing.\n\nWhy had these youths been so totally abandoned? Apparently the reason is that most of them were much more feminine in behavior than the average homosexual man of the time. They were a band of youths from New York City and around the country who were generally not wanted by their families or schools or hometowns because they were so obviously queer. Hearing that the Village was the best place for them, they found their way there. At least in the Village they could find others like themselves\u2014or so they thought\u2014but other gay men shunned them. With the exception of Marsha Johnson, drag queens also shunned them. People in the apartment buildings around Christopher Park threw things at them from their windows. They were a world unto themselves, cut off by their age and by being so out of control. The interest Bob Kohler took in these youths and the kindness he showed them were unique.\n\nUnfortunately, the youths directed the rage they felt at being so completely abandoned against one another. Kohler recalls, \"They were constantly at odds. In a given week every one of them would stop speaking to every other one of them, and then they'd be back as sisters again. Most of them carried either a pair of scissors or a nail file, the preferred weapons because if they got arrested they would claim either that they were going to beauty school or that they were cutting people's hair and doing nail filing, assertions that the police could not disprove.\n\n\"It was nothing to be sitting there and it would be fine, and all of a sudden you'd hear a bottle break and that was always the weapon, the broken bottle, and they'd go for the face. If it was a drag queen, they'd go for the tits to deflate them. They would fight over pills. They would fight over whose bench that was. They would stake out benches.\n\n\"The fight would be about nothing. It would be over in a finger snap. Getting hurt was not important. Getting dead was not important. They'd rip their toes open wearing stupid open-toe sandals, and their feet sometimes would be so swollen up. There was no value placed on these kids by the gay community, by the medical community, by anybody, but mostly by themselves.\"\n\nThe boys' sense of humor helped dissipate some of the anger. Tommy recalls \"When gay people talk to each other there can be this really competitive performance. Street queens have that in machine-gun motion. It's so rapid-fire in their competition because they like to get each other to laugh and they like to get each other laughing at how clever and smart and interesting they are. So usually they would rather get each other laughing than hitting or knifing each other. There was a respect for someone that could really turn a phrase in a crazy way.\"\n\nSometimes, too, camp humor could be used to express group solidarity while trying to shock heterosexuals. In a spirit of youthful fun, at times when the street youth would be cruising on Greenwich Avenue, they would spontaneously throw their arms around each other and sing to the tune of \"It's Howdy Doody Time\" their own gay-affirming lyrics:\n\n_We are the Village Girls!_\n\n_We wear our hair in curls_.\n\n_We wear no underwear:_\n\n_We show our pubic hairs!_\n\nHumor was also the one weapon the street youth could use against the police.\n\nTHE COPS (SINGULAR AND PLURAL) WERE GENERICALLY KNOWN AS \"LILY LAW,\" \"BETTY BADGE,\" \"PATTY PIG\" OR \"THE DEVIL WITH THE BLUE DRESS ON.\"\n\nTommy recalls, \"The bubble-gum machine meant the cop car, because back then they only had one light in the middle. So someone might say, 'Lilly and the bubble-gum machine' or just 'Here comes Lil.' Because it was the sixties, sometimes she was called Patty Pig, which was more easy for the cops to interpret, so it wasn't said that much. Lilly Law they mostly didn't know. Especially when we said 'Lil': 'Lil, there she is.' Some queens called her Alice Bluegown. And she rode in the bubble-gum machine, so if someone said there were a lot of bubble-gum machines out, we knew there were a lot of cops there.\"\n\nProstitution (or hustling, as it is called in the gay world) was not only a way to get cash by trading sex but also gave the youths an opportunity to steal from johns. The youths would ask Bob, \"Can you hold this?\" and his apartment came to be filled with stolen items. When they would go down to the piers, they would also ask him to hold their money\u2014if they had any\u2014because they themselves were often robbed while hustling.\n\nAccording to Kohler, \"Very, very few, if any, were drag queens. They lived by panhandling, stealing, and hustling down at the piers at night. This is the only time that they ever went in drag, and it was pathetic drag. I mean, they had no money. They sometimes were able to steal wigs from 14th Street.\" Being very young, with their features not fully formed, many of them succeeded in impersonating women by using their own natural looks with just a little makeup and a few additions to their own clothing. With long hair being the style for young men in the mid-1960s, many of them could even dispense with wigs or falls. In the summer they usually wore shorts and at other times flowing bell-bottom pants, which was the fashion for young men at the time Bob met the street youths: \"The street kids had bag-woman drag. They wore sandals and tied their blouses or shirts in a big knot midriff, and that was basically it. The drag queens that were down at the piers couldn't have fooled a blind man. It was just ragged, hausfrau drag, whatever they could get together.\" Some simple jewelry like earrings might be added and the bell-bottoms would sometimes be in a lightweight, sheer material, so they looked like a long gown.\n\nAn example of a young man who hustled at the waterfront, though not in drag, was Jackie Hormona. He enhanced his natural good looks by dying his hair blond, putting on a little facial powder to make his skin appear smoother, and applying a touch of eyeliner. Although his overall appearance was masculine, like most of the street boys he was a bit more feminine in comportment than most gay men of the time. While Kohler explains that Jackie was known as Hormona because he wanted to take hormones, Jerry Hoose thinks otherwise, attributing the name to camp humor: \"Even his name, Hormona, was sort of like a put-on, really. Because he really wasn't that outrageous, like a drag type. The 'Hormona' was an exaggeration, sort of a goof. A lot of us back then took on exaggerated kinds of names and things just to be outrageous. But I don't think he had any plans on changing his sex or anything.\" While Hoose recalls the subtle use of makeup, he never saw Jackie wear a dress or use lipstick.\n\nOne of the more enigmatic of the street youth, Jackie seems to have maintained a certain emotional distance between himself and others. Tommy knew him so little at the time that he didn't even know his street name: \"She was always walking by and waving and then kept walking.\" Jerry Hoose agrees, saying that although he always got along with Jackie, \"Not really many people were very intimate with Hormona. She was sort of a loner, more than the rest of them.\" Hoose remembers havng the impression that Jackie was from the South, for there seemed to be a slight southern twang in his speech. Hoose also recalls that Jackie \"always gave the impression like he was from money. Whether that was true or not, who knows? But that's the impression he gave a lot of us, that he's living on the streets and doing all this shit because he wanted to, but not because he had to.\"\n\nHoose knew Jackie from cruising areas: \"Jackie was a regular on Greenwich Avenue\u2014and on Christopher Street, sitting on the stoops, getting drunk. I saw Jackie drunk many, many, many times. You know, bottles in a brown paper bag, stuff like that.\"\n\nHoose links the constant drinking to another of Jackie's characteristics: \"He would just do anything to make a splash. He really had no fear. He was extremely wild, one of those I-have-nothing-to-lose kind of characters.\"\n\nJust how wild Jackie could be Hoose remembers from the place he saw Hormona most often, the trucks: \"The trucks used to be raided like two or three times a night. Out of nowhere police cars would come from all directions, with the police smashing their nightsticks against the trucks. Everybody would fly out. They'd beat us\u2014try to grab people, smash people in the head with the nightsticks. People would run in all directions, but there'd be a small crowd of people here and there like me that screamed back, but I remember he would like go a little further than me\u2014throw things at the police. He was pretty fearless. Jackie wouldn't just sit around and take it.\"\n\nBeyond Jackie's nothing-to-lose attitude, an additional explanation for his resistance to police oppression may lie in an attribute that Tommy recalls in Jackie, that of simple fairness: \"Jackie had good moral leverage, being someone who wouldn't get into fights or wouldn't encourage stupid fights. He was the kind of person that if two people are fighting would say something like 'You know, it's not worth fighting about' or 'It's your friends, so why are you acting like this?' He was one of those people that had the nerve to tell people they should treat each other better. Most people were more expedient than fair. There was a very typical kind of opportunism among street queens, who were robbing each other a lot. So if someone could step in and get people to be a little more levelheaded, I always respected that. And it was unusual, because that person had to have a certain presence and confidence to not get murdered doing that. I would just be quiet and fade into the background because I didn't know who was going to go off, but Jackie would say, 'Don't do that to people.'\"\n\nAnother of the street youth who hustled and did so in drag at times was a black male known as Nova or, more fully, Zazu Nova, Queen of Sex. A good-looking man, Nova had a strikingly soft complexion of rich color, with a long nose and a distinctly African silhouette. A bit aloof, he carried himself almost as if he were royalty. Jerry Hoose thought he looked good as a man but made a rather masculine woman, for Nova was a very tall, muscular man with large hands. Tommy heard that Nova came from upstate New York. A staunch Unitarian, he took great pride in his religious upbringing. By all accounts, Nova had a fierce temper, even to the point of having a violent nature. The word on the street was that Nova not only had been in prison several times; there were also rumors that he had served time for murder.\n\nOne of Tommy's best friends from the era is Martin Boyce, who lived at home but socialized on the streets. Martin knew Nova was a force to reckon with, for he had seen her in action. Boyce was exiting Grand Central Station at three o'clock one morning when he ran into Nova on 42nd Street. As Martin greeted Nova, he grabbed her breast because it looked real and he wanted to see what it was made of. He didn't realize that five men had surrounded them. When one of them said, \"Look at that. He's touching that tit and doesn't get nothing out of it,\" Martin realized that he and Nova were about to be jumped. Fortunately, Nova was prepared: \"She had in her purse a chain that could tie up a truck. She swung that fucking thing\u2014it really looked like an anchor was attached to it. She whipped that fucking thing out and they fled.\"\n\nWhile Marsha P. Johnson was not one of the street youth per se, she was very much of the street. Born in Elizabeth, New Jersey, the city Tommy Lanigan-Schmidt grew up in, Malcolm Michaels Jr. moved to New York the same year Tommy left home for good, 1966. The two youths also shared a deep religious sensibility, but Marsha Johnson was deeply disturbed, so that while her religious pronouncements could have the oracular tone of direct experience, they also had the unsettlingly weird quality of statements made by persons suffering from psychosis: \"Sometimes I have visions. In one of them, there were ten suns shining in the sky, gorgeous and freaked out, like the end of the world. I love my saints, darling, but sometimes the visions can be scary.\" Her off-center nature is also shown by the way she used materials she found on the street for her wardrobe: costume jewelry earrings became ornaments not for her ears but her hair, sometimes she wore red plastic high heels, and she might deck herself out in flowers, either real ones discarded by flower merchants or plastic flowers or artificial fruit from a five-and-dime. But this describes Marsha years after she arrived in the Village. In the early days she tended to go out mainly in semidrag and called herself Black Marsha. (When she later dropped the _Black_ and started calling herself Marsha P. Johnson, she explained that the _P_. stood for \"Pay it no mind.\")\n\nThat energy and craziness were essential characteristics of Marsha's life is seen in her acknowledgment that by 1979 she had had several attempts made on her life by johns, eight nervous breakdowns, and innumerable arrests\u2014after one hundred she had stopped counting. But the characteristic most often cited by those who knew Marsha best was her essential goodness. Tommy remembers seeing Marsha stealing a loaf of bread early one morning from a delivery left behind a shopkeeper's gate and then passing it to a homeless person. Randy Wicker flatly stated that Marsha \"was the most generous person I ever knew.\" And a volunteer at a thrift shop where Marsha bought her outfits, recalling \"the fella that used to buy all the gowns,\" described Marsha as a \"wonderful, sweet person.\"\n\nBut Robert Heide recalls another side to the childlike Marsha, who could be both helpless and charming. He sometimes saw a demon emerge, especially when Marsha was in his male persona as Malcolm. \"I think we all have that to some degree, but apparently in Malcolm\/Marsha's case there was this real duality and it would take hold. There was a schizophrenic personality at work, for Malcolm Michaels could be a very nasty, vicious man, looking for fights. You could say hello to him and if he was Malcolm that day, he might not recognize you or you might be in trouble or a fight might even ensue.\" Heide's experience of this side of Malcolm is borne out by Randy Wicker, who took Marsha in as a roommate decades later. While Marsha was generally an ideal living companion, on one occasion she wrecked Randy's residence. A 1979 _Voice_ article also reported that Marsha's \"plumed saintliness\" was \"volatile\" and quoted a Christopher Street shop manager who described her as a \"bully under that soft sweet manner\" and listed a roster of gay bars from which she had been banned.\n\nDespite her toughness, the incessant police harassment stung. Marsha was as much an habitu\u00e9 of 42nd Street as she was of the Village waterfront, and she spent much time in the cheap \"hot-spring\" hotels that prostitutes favored for their liaisons. Sitting with her fellow prostitutes in the small hotel rooms, she had a recurring fantasy, that one day they would be able to walk around and not get busted by the police: \"That was a dream we all had, sitting in those hotel rooms or in the queens' tanks of the jails.\"\n\n*The extract in full capitalization is from _Mother Stonewall and the Golden Rats_ , a work of art in the form of a handbill by Thomas Lanigan-Schmidt.\n\n#\n\n# The Stonewall Inn\n\n_There were some lesbians, hustlers, married people, single people, some transvestites, but not too many. It was the heart and soul of the Village because it had every kind of person there_.\n\n_\u2014Philip Eagles_\n\nOpening the Stonewall Inn was a venture that took little money and even less imagination. When the Stonewall Inn Restaurant closed after a fire in the midsixties, the building sat empty for some time, until Fat Tony converted the space into a gay club with a minimum of effort. Harry Beard, a waiter at the Stonewall, recalls \"one of the managers remarking, 'Yeah, paint the joint black and give it a little class.' So, consequently, the walls and the ceiling and anything else that happened to get in the way were painted black.\" Apparently the color was chosen not only because so many gay bars at the time were painted black but also because one coat of black would cover the burned wood and save the expense of replacing it. An extensive contemporary description of the club noted: \"There's a certain hastiness about the look of the place. It seems to have only recently been converted from a garage into a cabaret in about eight hours and at a cost of under fifty dollars.\"\n\nThis early reviewer was correct in his estimation that only a small amount of money had been used to open the bar. Chuck Shaheen, who helped Fat Tony open the Stonewall Inn, estimates the total spent at less than a thousand dollars. However much money may have been expended on remodeling, Shaheen is precise about how much cash was used to start the club: \"Tony invested $2,000 and three guys invested $500 each, and that's what started the Stonewall. I remember distinctly the $3,500 being the exact amount... because Tony always said, 'We did this on $3,500.' \" For his partners Tony chose Zookie Zarfas, who was in the firecracker business, \"Tony the Sniff,\" and \"Joey,\" apparently because all three were childhood friends. But there was another more significant partner who did not have to put up any money: Matty Ianello, known as Matty the Horse, had been given control of that section of the West Village by the Genovese family and so, according to Mafia protocol, automatically got a cut from mob businesses in that territory. Without investing a dime, he was considered \"the real boss, the real big boss.\" The modest investment would soon make the four partners who did put up money a tidy fortune.\n\nThe mobsters did have the white Greek Revival columns that had graced the entrance to the restaurant demolished, perhaps so they could more easily install the heavy doors they wanted to provide security for their new bar. Around this same time, a fresh coat of stucco was applied to the building's exterior, hiding the brick that had become exposed as the building had deteriorated during the Stonewall Inn Restaurant's decline.\n\nNot even much imagination was used in naming the new \"club\": all that Fat Tony did was lop _Restaurant_ off the name of the former enterprise, which allowed him to keep the enormous rusting sign hanging just above the club's entryway. Fat Tony did not even bother to have the word _Restaurant_ painted over.\n\nIn spite of recent modifications forced upon the SLA, because Fat Tony intended to serve a clientele that he could not legally sell liquor to he needed a ruse. The ploy that Fat Tony used was a common Mafia ploy for running a gay bar, the fiction that the operation was a bottle club, a private club serving its members. To that end a charter that had been issued to the Red Swan Social Club in 1929 was obtained for the new club. In a true bottle club, members could bring their bottles and leave them at the club with their names on them. Waiters at such a club could pour a drink from the member's private stock and the club members would tip the waiter for doing so. But the staff of the Stonewall Inn just made up names, wrote them on strips of paper, and attached the strips to bottles to fool the police. In practice, the club served anyone the doormen admitted.\n\nIf little imagination was used to create the Stonewall Inn, the club did break new ground by being a large gay club in a rather open area and on a main thoroughfare, instead of on a side street, as was more typical in New York City, which is what Tommy Lanigan-Schmidt remembers noticing about the Stonewall Inn the first time he saw it.\n\nA customer entering the Stonewall Inn would see that the entrance was in the middle of a brick edifice, directly under the hulking metal sign and between two large horizontal windows that, as was true of the windows of almost all street-level gay bars in New York City at the time, were blacked out so that the police and the public could not see inside, providing the clientele privacy and security. For extra protection, the windows were reinforced from the inside with plywood, which was further reinforced with two-by-fours to prevent the police from being able to simply break through the windows and rush inside.\n\nThe thick oak doors were rendered more secure by steel doors inside them and several inside locks intended to slow down the police in a raid. Each of the front doors had two small vertical openings cut into it at eye level. One of the four openings served as a peephole to screen potential customers, giving the establishment somewhat the feel of a speakeasy. Dawn Hampton, a black heterosexual woman and former torch singer who had worked in many gay clubs, remembers that \"at that time there was a lot of entrapment going on, police coming to the door and pretending they were of that life.\" Chris Babick, a frequent customer, recalls that \"the door of the Stonewall had wrought iron bars across this little peephole, a little wooden thing that slid open. And the man inside would look at you and, if you looked like you belonged there, would let you in.\"\n\nAnother technique used to screen out undercover police or heterosexuals who might not blend in well was to ask would-be customers to describe the inside of the Stonewall Inn as proof they had been there before. The most common test of entry, however, was simple recognition by the doorman. Blond Frankie, who had worked at a number of gay clubs before working at the Stonewall, had a nearly photographic memory for faces, which earned him a spot at the door. Harry Beard remembers that \"a favorite ploy of the vice cops was to pass themselves off as a typical patron, and the Stonewall... would always say, 'This is a private club, for members only.' I assume that the standard response in those days was, 'Well, I've been here before.' And Frankie would just look at their face and go, 'Well, no, you haven't.'\" The entrance's final defense was a switch that the doorman could throw to turn on the bright white lights inside as a signal that the bar was being raided.\n\nWhen a customer had passed inspection, the door was opened and the bouncers would be found sitting at a table to collect $1.00 for admission on weekdays and $3.00 on weekends. On the weekends after a patron plunked his money down, one of the doormen would tear two tickets off a roll and hand them to him. Each ticket could be traded for a drink. (Rolls of tickets of different colors were rotated to prevent customers from cheating on readmission and\/or holding unused tickets for a later weekend.) A special lamp was also used for door control: customers wanting to step outside and return later could tell the doorman they were coming back, and he would stamp their hands with an invisible ink that would show up under the lamp's blue light.\n\nCustomers were asked to sign a book as part of the Stonewall Inn's fiction of being a private club. The book-signing ritual served in part to prevent straight people from entering. One regular club goer remembers that in the summertime the front doors to the Stonewall Inn would sometimes be open. \"They would stop straight people who were coming in: 'No, this is a private club.' That's the reason for the book to sign in.\" Theoretically, the book could also have been introduced in court as evidence of the Stonewall Inn being a bottle club. Perhaps it functioned too as a kind of prop to mock or try to confuse the police. Certainly some customers were in on the joke, for they scrawled campy names such as Elizabeth Taylor and Donald Duck in the book. But there was also a measure of self-protection in using the false names: if the police did seize the book, it could not implicate anyone as a \"deviate.\"\n\nUpon entering, those who wanted to check their coats turned to the left, where the checkroom was located on the left side of a dead-end corridor, directly behind one of the large windows. Hampton remembers that homosexual men dressed more formally then than in the years after the gay liberation movement launched a more casual and identifiably gay style of dress: \"Most of the men were wearing outer jackets. They weren't wearing so much leather back then or so much denim at that time. There was a whole different mind-set: there were some men who wore hats in those days.\"\n\nBeing a heterosexual woman did not stop Hampton from enjoying a warm relationship with the club's clientele: \"I wasn't hitting on anybody. I wasn't trying to straighten anybody out or anything like that, so I was very accepted. A lot of the kids called me Mommie. It used to be a fine joke for me when someone would say something to me about having children and I said, 'I have a ton of them.'\"\n\nTo get into the main bar\u2014located on the east side of the Stonewall Inn\u2014those with nothing to check simply turned to the right at the end of the lobby, passed through an inner door that was usually open, and stepped down one step.\n\nA customer who had gotten this far was now into the Stonewall Inn's largest room, whose most prominent feature was a long bar, with many round stools. While the clientele was generally in their upper teens and lower twenties, those at the front bar tended to be white men in their upper twenties and lower thirties, with men over the age of thirty-five being unusual, though by no means unheard of. On the wall in back of the customers seated at the long bar was a narrow ledge on which customers could rest their drinks.\n\nAt the end of the bar was an area reserved for dancing, with the music provided by a jukebox stocked with singles, which could be played for ten cents a song or three for a quarter. The music made it possible for customers to dance, which was the Stonewall Inn's main attraction from the day it opened its doors.\n\nOn St. Patrick's Day of 1967 Danny Garvin headed downtown, figuring that he would enjoy an evening at Julius' before heading to Chicago to stay with his sister. He would go to Illinois to pull his life back together, but being Irish, he would never have considered not celebrating the holiday. An older man walked up to the fresh-faced youth and demanded, \"What are you doing in here?\"\n\n\"Well, I'm having drinks.\"\n\n\"You should be around the corner with all the chicken.\"\n\nMore language confusion. Danny gave the older man a quizzical look.\n\n\"That new bar that opened up around the corner tonight, Stonewall. They're all over there.\"\n\nDanny walked one block over to the Stonewall Inn and entered. The place was \"fairly well packed\" with young people, but what really struck Danny was that people were \"dancing all over the place.\" Shocked anew, he thought, _Men don't dance with other men! This will never last!_\n\nChris Babick was also impressed the first time he went to the Stonewall Inn by the sight of men dancing, but for him the reaction was positive: \"And there were men dancing with men. And when I [saw] the two men\u2014there were several couples dancing together\u2014I had such a thrill in my stomach. It was... like an electric shock. And it was so fucking exciting.\"\n\nDawn Hampton recalls, \"I was there when they originally opened it as a dance palace. People came there because they loved to dance.\" _The Homosexual Handbook_ , published in 1968, said: \"The younger, more agile and more sensationally demi-dressed, jerk and bump on the rather large dance area at the end of the [front] room.... Spotlights are pointed directly down and they light the dancing youths dramatically.\" All contemporary accounts of the Stonewall Inn are unanimous in their agreement on the centrality of dancing to the club. Dawn Hampton told an interviewer, \"It's not like the places that had back rooms in the last twenty years. It was just a place where they loved to dance. They loved to be together and wanted to be with their own kind.\"\n\nAn habitu\u00e9 of the club concurs: \"There was a sense of community feeling in the Stonewall. You would meet your friends there. It was a fun time, a good time. There was a warmer feeling in the bars, a nicer feeling, I'd say, than there is now [1989]. Maybe because bars are advertised now.... You didn't know then, so someone had to point the Stonewall out to you, or let you know, or somehow you just happened to see a bunch of men walking in there. Because you wouldn't see two guys on the street holding hands then, so it was more of an in place. Gay people were different back then. Guys would ask you to dance.\"\n\nOf course dancing was not the only activity that went on in the front room. Vito Russo remembered the front room as a place for cruising and camping. For people who wanted to sit, there were some tables set up in the club, usually for two, with little candles on them.\n\nDrinks were priced at a dollar each, a high price for the era. There was pressure from the Stonewall staff to order a drink very soon after entering: \"You got a ticket, but they wanted you to buy your drink _zip_ when you went in and that was the main bar,\" recalls Chuck Shaheen. On weekend nights, after customers had gotten two drinks by trading in their tickets, the staff would then pressure them to buy more. Even though drinks were priced high, the mob was only passing out watered-down drinks. Even so, the staff was pouring liquor that had been stolen or obtained from bootleggers. Shaheen remembers that it all \"came from Matty the Horse's company,\" explaining that \"none of the liquor was brand-name liquor. We would go in the back at the beginning of our shift and take Dewar's bottles and pour whatever swill we could get into it. The same with vodka. It wouldn't be Smirnoff. Nothing would be what it said it was.\" The club kept one bottle each of a couple of call brands for favored customers. According to Shaheen, everything that was sold in the Stonewall Inn came only from Mafia suppliers, the liquor, the cigarettes, and even the music on the jukeboxes.\n\nThe lighting was very dim throughout. The first room had a low ceiling that combined with the darkness to create a cavern-like feel. Smoky air filled the bar, for the club's only ventilation came from several small, inadequate air conditioners. However, the men liked to wear perfumes, such as Tabu and Ambush, both marketed to women, whose aromas gave the place a rich, saturated atmosphere.\n\nIn the wall dividing the front from the back room was a doorway with swinging black doors through which one passed, stepping up one step, to the back room. There was one other route between the two main rooms\u2014through the bathrooms at the club's rear. The rest rooms had doors on both ends, which allowed them to be used as passageways. The women's room was distinguished by having a red lightbulb, which irritated some of the scare queens (men who affected a feminine presentation without trying to pass as women) and transvestite club goers, for the dim light made it difficult for them to adjust their makeup.\n\nThe men's room was distinguished by having its own attendant. Beard remembers him as \"an old black guy named John. He used to dispense colognes and underarm deodorants, and razors and soap and towels and toilet paper and such.\" Michael Fader remembers John as heterosexual. He sat in the corner, at the ready to give a club goer a shot of cologne on his neck for a tip. A rest room attendant in a place as down-at-the-heels as the Stonewall Inn sounds improbable; however, he may have been there to discourage men from having sex in the rest room and making the Stonewall even more legally vulnerable.\n\nWhether reached via the main doorway or by cutting through one of the club's rest rooms, a customer who reached the Stonewall Inn's second room found himself in a setting different from the front room. While the first room was dominated by its long bar, the second room was much smaller and had only a small bar and even dimmer lighting.\n\nHarry Beard recalls that the back room had \"a service bar with a few stools around it.\" The back room had some cheap wood paneling, a ledge to set drinks on, and a few tables, with a bench against the wall providing additional seating. While usually only one floor waiter circulated in the front room, a number of floor waiters worked the back room, and they pushed drinks very aggressively.\n\nTommy preferred the music on the jukebox in the back room. There the records were releases by artists who put a lot of feeling into their songs, such as Otis Redding and Stax recording artist Carla Thomas (called the \"Queen of Soul\" by Redding). In the front room, the jukebox offered more mainstream performers such as the Beach Boys. The back room, with its more soulful music, made up in spirit whatever it lacked in size. Its lively feeling derived from its being the favored place of the homeless youth, as well as of young blacks and Puerto Ricans. Tommy Lanigan-Schmidt recollects how some people called the front room \"the white room,\" because of its racial makeup and its music. For similar reasons, patrons dubbed the back room the \"black\" or \"Puerto Rican room.\" Always crowded, the younger and the \"in\" crowd claimed the back room as their domain. The back room's flagstone floor gave it an advantage for energetic dancers.\n\nJust a few feet inside the doorway one found the Stonewall Inn's most unexpected feature, a full-sized wishing well constructed of stone and cement. Surrounded by benches, the five- or six-foot-tall well had a roof over it. In the days when the club operated as a restaurant, the wishing well had often graced wedding receptions, and many pictures of brides and grooms had been carefully posed with the well as a prop. Now the well stood incongruously in the dim light amid the jukebox's constant blare, its potential for romance and charm not even noticed by the Stonewall staff, who used its interior as a makeshift storage space. Tommy recalls how he never knew what he might see inside it: bags of ice, empty boxes, or boxes full of beer temporarily parked there on the way to being consumed.\n\nThe back room had its own social geography, with the street end being where the most marginal of the Stonewall's customers congregated, including Tommy: \"It was where the most down-and-out queens hung out, because that was the furthest out place.\"\n\nThe question of different parts of the Stonewall being favored by different clientele raises the long-debated issue of what kind of club it was after all. Who went there and who did not? Did some group or groups predominate so that it can be categorized as a particular kind of gay bar?\n\nWhile the clientele was mainly male and young, Hampton says that the bar had \"some of everything... a lot of them were businessmen and didn't want their families to know, naturally.\" She states that the mixture included various ethnic groups: \"They were mixed Spanish, whites, and blacks, but there were more whites than the others.\" Tommy Lanigan-Schmidt agreed that \"somehow, it covered the range... the whole mixture of everyone gay at the time.\" Robert Bryan said the mixed clientele reminded him of how a gay bar in a small city would draw all kinds of gay people \"and you get the women, and the leather, and the collegiate, and the drag queens,\" and he felt the similarity was explained by a corresponding cause, that the Stonewall Inn \"was _the_ gay bar in the city.\"\n\nFader says that \"eighty percent [of the crowd] you wouldn't know anybody there from [somewhere] else,\" before making the critical point for understanding the Stonewall Inn: that \"anyone who was in the margins of gay society would be free to go there, because they were totally accepted.\" Bryan says: \"The street queens were the predominant force in there, although there were all types of people,\" with the result that \"it did have an overall kind of a trashy reputation, and certainly, that was the predominant ambience: trashy, low, and tawdry.\" Vito Russo similarly recalled: \"It was a bar for the people who were too young, too poor or just too much to get in anywhere else. The Stonewall was a street queen hangout in the heart of the ghetto.\" Perhaps Allen Young characterized it best when he said the club was the \"favorite hang-out of the freest of the gay people\u2014those most likely to be labeled 'fag' and 'drag queen.'\" The nature of the Stonewall's crowd made Young so uncomfortable that he \"preferred the more up-tight and sedate (read, 'masculine') crowd at Danny's, a few blocks closer to the waterfront.\"\n\nOne of the more disputed points about the Stonewall clientele is to what extent women went there and how many of those were lesbians. Hampton says that \"all gay men went there. Very few, if any, gay women at all. Usually I would practically be the only woman around.\" Shaheen said, \"It was 98 percent men.\"\n\nCertainly some of the women were heterosexual. One Stonewall regular recalls, \"Some women who would go to the Stonewall were hippie straight women, or some woman who would look like a drag queen.\" One gay man who went to the Stonewall Inn with a female\u2014although a very young one\u2014was Chris Babick. When Chris started going there in 1968 he was a seventeen-year-old high school senior but looked so much younger that he relied on his fifteen-year-old friend Peggy to buy him drinks. Shaheen adds that \"the floormen [waiters who circulated selling drinks] always loved girls to come in because [the floormen] were straight.\"\n\nEarly Stonewall researcher Tina Crosby concluded in 1974: \"The screening process at the door effectively excluded women; each person I talked with remembered the Stonewall as an exclusively male bar, with the only exception being an occasional tough lesbian or female friend of one of the male patrons.\" Jennifer \"Hardy\" (not her real last name) remembers seeing a lesbian of this very type on her first visit to the Stonewall Inn as a seventeen-year-old runaway: \"There was this woman in there, and she was big.... That's not what I was looking for. I didn't want somebody that would control me and beat me up and that's kind of the impression I got from her, that she was strong and that she was mean. I don't know if she was a bouncer or what she was, but I steered clear of her.\" The presence of this kind of lesbian at the Stonewall reflects the predominance in the lesbian social scene at that time of the butch-femme model.\n\nThe one dissenting voice about the kind of lesbians who frequented the Stonewall is Shaheen, who remembers beautiful hairdressers, which suggests that \"femme\" lesbians patronized the Inn and raising the intriguing possibility that they might have been mistaken for heterosexual women or even for \"fag hags,\" since they brought along their male homosexual counterparts: \"There were lesbians and they were not of the very butch type of lesbians in those days. Remember that its biggest clientele were hairdressers.... They were big tippers, and some of them were women and some of them were gay.\"\n\nThe last hotly contested issue about the clientele is the extent to which transvestites came to the Stonewall. The general consensus confirms the usual presence of a few. Certainly all credible witnesses agree that the Stonewall Inn never became a \"drag queen bar.\" But what determined if a drag queen was admitted? Shaheen said simply that \"it all had to do with the doorman knowing you.\" Some of the drag queens Shaheen recalls going to the Stonewall Inn are Tiffany, Desiree, Spanola Jerry (a hairdresser), and Tammy Novak, but he quickly added, speaking of the first three, \"They were spenders. That's why they got in,\" making it plain that not any transvestite could walk up to the door at any time and be admitted. As for Tammy Novak, she clearly belonged to a special category, having lived with Fat Tony and Chuck Shaheen.\n\nThe presence of drag queens at the Stonewall Inn has been much exaggerated over the years for a number of reasons. One of the first is a terminology problem. The word _queen_ was more widely used in the late sixties to indicate any gay man who was not conventionally masculine, whereas today the word usually occurs in the phrase _drag queen_ or indicates a very feminine gay male. Thus when a contemporary person reads about \"a whole bunch of queens,\" the picture that may come to mind is one of transvestites, whereas the 1960s usage probably simply indicated a group of gay men, with the understanding that none of them were totally straight-acting.\n\nComplicating the picture is the existence in the late 1960s of gay men known as _scare queens_ or _flame queens_. One of the club's regular customers explains the meaning of these terms that describe a kind of gay male who became practically nonexistent not long after 1969: \"What you had back then was the flame queens, which were very similar to the character Emory in _The Boys in the Band:_ they were supereffeminate, hair would be teased, they would wear eye makeup, Tom Jones\u2013type shirts, maybe hiphuggers, bright colors. A good example of this was Brandy Alexander, who used to perform at the 82 Club and who was an extra in the movie _Midnight Cowboy_. I remember Brandy saying to me she would have to go up there and do the streets, and even then, even though Brandy was a performer in drag\u2014in the movie [when] they would show you the drag queens on 42nd Street, they're flame queens. You couldn't do the whole drag thing [in the film], because that's not how people dressed: you would be arrested. So as far as the Stonewall being all these drag queens, no, there were flame queens. Even Barbara Eden [a male, Barbara Eden being a camp name] who worked there in the coat check didn't do drag except once in a blue moon.\"\n\nThis distinction helps resolve some of the seemingly contradictory accounts about how many transvestite and transsexual customers patronized the Stonewall. For example, Chris Babick spoke to Michael Scherker as if the transvestite population at the Stonewall was quite sizable. However, when Babick describes what he wore, it fits the description given earlier of \"scare\" or \"flame\" queens: \"I had bleached hair and I started wearing makeup [because] I had this acne problem. My favorite outfit was these black pair of hiphuggers, bell-bottoms, and my straight girlfriend embroidered daisies on the bottom of them, this trim.... And I had this favorite cotton satin-type shirt.... And I'd wear that with this big belt with a big buckle and my boots.\" It also seems clear that while there was high drag at the Stonewall, there was also low drag, which practically fits into the previously mentioned category of \"flame queen,\" as is the case with the following person Edmund White saw there: \"There was one [drag queen] that was always there, this very tall black one who I called the whirling dervish because he would get out by himself on the dance floor and turn around with his leg just shooting through the air. He was under thirty and would wear pedal pushers and a midriff shirt and maybe a wig, but the kind of thing that he could take the wig off and put it in his pocket and look kind of like a boy, because I think a lot of times these boys had to go back to Harlem at night. They could make the transition pretty quick\u2014maybe they carried shopping bags and checked those. I'm not quite sure, but I have a feeling they did a quick change on the subway as they were going up to Harlem late at night.\" From White's description, one can imagine that someone who saw the whirling dervish with his wig on might describe him as a drag queen (with the understanding that he was not trying to pass as a woman), but someone who saw him without the wig would undoubtedly consider him a \"flame queen.\" Perhaps the most accurate way to characterize the Stonewall Inn in terms of what we currently call transgender identity is that while there was a lot of gender transgression going on there, this was largely a reflection of the male homosexual world of the time, that most men who went there were conventionally masculine, but that there was a considerable minority of men there who ran the gamut from men effeminate in their mannerisms, to scare or flame queens, to a few transvestites and some transsexuals.\n\nFinally, the most important point about the clientele at the Stonewall Inn is that all segments of the gay and lesbian community, including a strong representation of the more marginal elements, defined the Stonewall Inn as a special place in the homosexual world of greater New York, giving it a unique status at that time. Tommy Lanigan-Schmidt was so struck by this feature on his first visit that he described the Stonewall as a kind of temple of homosexuality: \"If the Tenth of Always was like a little parish church, the Stonewall would be like St. Peter's in Rome.... It was big in its scope.... Somehow, it covered the whole mixture of everyone gay at the time.\"\n\nWith two dance floors, the Stonewall Inn became wildly popular. Harry Beard recalls, \"We had a real healthy segment of people with cash, a lot of people from the bar business, since at that time we were the largest gay club anywhere in the country, probably the largest club period.\"\n\nAnd so it was that when the Stonewall Inn opened its doors for business on the night of March 18, 1967, immense profits began to pour in. Chuck Shaheen recalls that \"we used to take out of there between $5,000 and $6,000 on a Friday night and [again] on Saturday.\" With the initial investment being $3,500 and the monthly rent only $300 a month, the Stonewall Inn made back all of its investment and then some the first night, so that after a few hours of operation it was essentially pure profit every day for nearly two and a half years. Shaheen explains that to prevent the loss of such large sums of money, part of the bouncers' job was to \"go around and pick up our banks,\" an errand on which they carried baseball bats. The money was kept in cigar boxes, which had several advantages over cash registers. The first time the police raided the club they seized the cash registers as evidence of the illegal sale of alcohol. After that, the club used cigar boxes, not only because the expensive machines would be lost in future raids, but they could also be used as evidence against the owners, who, as operators of a purported bottle club, were not supposed to be selling anything. After removing the money from the cigar boxes, the bouncers wrapped rubber bands around the rolls of cash and casually tossed them into the Stonewall Inn's office. Chuck Shaheen eventually moved up from being a bartender and lived with Fat Tony as a kind of man Friday, and one of his duties was to run the money over to \"the family bank,\" the Bank of Commerce on Delancey Street, where \"most of the money was put through.\"\n\nOf course it was impossible to make so much money in such a public setting without being noticed. The huge profits certainly caught the eye of Craig Rodwell. Since practically the only social outlet gay people had in New York City was bars, it infuriated Rodwell that these places were controlled by the Mafia and not by the gay community. \"Bars have always been our only place, our haven in a sense. I was always furious that the mob controls so much of our social life.\" Having worked in gay bars, Rodwell had learned how much the management loathed their gay customers: \"They like our money and hated our guts. And that was one of the major issues around the Stonewall. There was that collusion between the cops and the mob, and we were like caught in the middle all the time.\"\n\nCharacteristically, Rodwell took an activist approach to the bar situation. Having his own publication and the opportunity to distribute it in his bookstore, he used the _Hymnal_ to critique the implications of the Mafia's control of the gay bar scene. About the Stonewall's finances Rodwell wrote that the bar was \"one of the... more financially lucrative of the Mafia's gay bars in Manhattan.\"\n\nSuch heady profits also interested the police, who felt they had a right to a piece of the pie. It was common knowledge when the Stonewall Inn opened that it was run by the Mafia and that the police were paid off by the club, the standard practice for lesbian and homosexual clubs in New York City. One reason that the payoffs were common knowledge is that, as Kevin Brew noted, \"It's the kind of thing you don't see in actuality. You don't see the bartender paying the police, but you knew something was going on because they left the bar alone (after the payoff).\" Craig Rodwell noted the same thing in _Hymnal_ in his list of characteristics that explained how to tell if a club was Mafia run: \"Policemen will make periodic and mysterious appearances to talk with the goons at the door.\" Shaheen confirms this method of payment: \"The police were very much on the payroll of the Stonewall.... The Sixth Precinct. I can only account for the Sixth Precinct.\" Asked how often the police came by to collect, Shaheen replied, \"Once a week. There were two [envelopes]. Because in those days there was an officer and that was his beat, and then someone in the precinct, usually the captain or the desk sergeant, would get an envelope.\" As for the amount paid, Shaheen said he did not know the exact amount but was positive it was hundreds of dollars a week.\n\nThe Stonewall was in Manhattan's Sixth Precinct, and from all available evidence, there is no reason to doubt that the Sixth Precinct was paid off by gay bars and clubs. For example, Tim Callahan, who worked at the bar Danny's for around six months during the time the Stonewall was open, said that he saw the precinct captain come by several times. Each time the same thing happened: the captain would have a drink with the bar's owner during which time they seemed to have a congenial conversation, after which they would both walk out to the captain's car. While Tim never actually saw money change hands, he had no doubt that this was taking place. During his time at Danny's, these friendly visits from the captain were the only times Tim ever saw a policeman enter the bar.\n\nOther rare contemporary evidence of the police receiving bribes comes from a letter Dick Leitsch wrote:\n\nThose two cops who came to visit us... have informed me that they discovered one of the cops whose name I gave them was taking payoffs from that club [the Stonewall Inn], and he's been dismissed from the force. The other one is still under investigation and surveillance, though they think he's clean. They're not going to take any action against the Stonewall.\n\nThe amount the Stonewall Inn paid to the Sixth Precinct was $1,200 per month, according to a _Pageant_ magazine article on police corruption. The article was published less than five months after the Stonewall Inn opened. While the article did not name the bar, it sounds like the Stonewall Inn: \"a rather shady bar serving homosexuals.\" The article added that in terms of payoffs, the bar was the largest contributor to the police department. Proof that the bar mentioned was the Stonewall Inn comes from the $1,200 figure being given three times afterward as the amount of the Stonewall's monthly payoff (twice in 1978 and once in 1988) by Ed Murphy, who was associated with the bar.\n\nThat the Stonewall Inn was a Mafia club had many ramifications. While homosexual men were provided with a place to socialize, the downside could be considerable, for the Mafia had little or no concern for their clientele's welfare. But as one person on New York's gay scene in the fifties and sixties explained, since clubs were closed down so often by the police, gay people were desperate for places to meet and usually had to settle for dirty and dangerous environments.\n\nThe Stonewall had both. Chuck Shaheen explains the Stonewall Inn's main affront to hygiene: \"We had no running water behind the main bar. We were unable to clean the glasses. We had two sinks. We filled them up at the beginning of the shift with water. Today you need to have all kinds of hot water and detergents. We didn't use that. Because of the volume, we would just take anybody's glass and run it through the water and refill it and serve it to anybody else.\" This practice apparently caused an outbreak of hepatitis among the Stonewall's customers.\n\nThe lead article in the _Hymnal_ 's first issue attacked Mafia control of gay bars, with the Stonewall Inn serving as the epitome of all that was wrong with such clubs. The article included a report on a hepatitis outbreak: \"New York HYMNAL received a report... that the Stone Wall was going to be closed by the Health Department because it was alleged that a number of cases of Hepatitis (which has reached epidemic proportions among the homosexual community) had been traced to the Stone Wall's bar.\" As the lesbian publication _The Ladder_ pointed out, such risk to customers came as a natural consequence of the SLA's ban on licensing businesses to sell alcohol to homosexuals: \"Since the SLA refuses to issue licenses to gay bars, these bars are generally run... under unsanitary conditions.\"\n\nThe Mafia's refusal to invest in basic amenities literally spilled over into other inconveniences for customers. The front bar had a regular built-in sink and for a second sink made use of a rubber tub. At the beginning of the shift, each was filled with clean water. With hundreds of people passing through the bar on a weekend night, the water eventually got dingy enough that even the Mafia felt compelled to freshen it. \"Mike\" the porter remembers that one of his jobs was to empty the rubber tub. Since the club lacked the proper facilities for doing so, he would take the tub to the men's room and empty the contents in a toilet. Eventually, with the Stonewall's many customers flushing the toilets before and after the water from the rubber tub had been poured in, the total volume would exceed the capacity of the club's pipes and the toilets would overflow. The result was that the bathroom floors, as Lanigan-Schmidt recalls, seemed constantly wet \"with an unpleasant-looking water.\" Rodwell's _Hymnal_ lamented the \"filthy john\" and called the Stonewall Inn \"the tackiest joint in town.\" Robert Heide and his friends referred to it as \"The Cesspool.\"\n\nHowever, the Stonewall Inn posed an even greater physical threat to its customers: the building had no fire exits. Given the crowds that gathered there on the weekends, that the club had no rear exit, that it had only one doorway between its front and back rooms, and that customers fleeing a fire would have had to pass through a narrow lobby before exiting the building, a fire in the Stonewall could have had terrible consequences.\n\nNot satisfied with the huge profits made from diluted booze, the greedy Mafia owners also dealt in drugs. Rodwell explained that \"of course in any of those [Mafia bars] there's going to be a huge drug crowd. The drugs are pushed in there like crazy. That's one reason the syndicate keeps their control of gay bars. Just selling liquor is one of the minor things they do in those places.\" Vito Russo agreed, writing: \"It was possible to buy any known substance available in capsule form.\" Shaheen recalls that \"all the hairdressers were very into Desbutols, which was actually Desoxyn, but it was mixed with Nembutol,\" a combination intended to \"smooth you out as it helps you up.\" One club goer who helped supply some of the drugs denied that hard drugs were readily available: \"It would be the place you would go to cop. If you knew the right person in there, that's where you would get your acid... marijuana or whatever. Nothing like heroin or anything like that.\"\n\nThe chief purveyor of drugs at the Stonewall was the legendary Maggie Jiggs. (Though everyone called him Maggie, he did not work or live in drag. Tish, a female impersonator from the era, explained, however, that \"you knew she was gay when you saw her.\") Tish had gotten friendly with Jiggs in Providence, when Tish was living in Connecticut and commuting to Providence for school. One night as Tish took the commuter bus home, he was surprised to see Jiggs. \"I said to her, 'Where are you going?!' and she said, 'I'm going to New York.' I say, 'Really?' She says, 'Oh, yeah. I just robbed a john and I got $100, and I'm going to go to New York.'\"\n\nDespite being chubby, having a receding hairline and a clubfoot, and standing only about five and a half feet tall, Jiggs had an outgoing personality that made him hugely popular. When Tish moved to New York, he was surprised to find Jiggs a ubiquitous presence in Manhattan's gay bars.\n\n\"I start going to the clubs, and I see she's behind the bars. I would go to some new bar and I would [say], 'Well, Jesus Christ, Jiggs, what are you doing here!?' 'Oh, well, they sent me over here.' And every time I'd go to another club, Jiggs was there! Then we'd leave that club, and we'd go to an after-hours club: Jiggs would be behind the bar! She had a following. If a new bar opened up, the boys\u2014as I like to call them\u2014would say, 'Send that Jiggs there; he's got a good crowd.' You know, we all liked Jiggs.\"\n\nJiggs had a prop that he always used in his work, a tall gold-colored chalice set with stones, which normally sat on the bar. Tish used to tease Jiggs about it. \"I'd say, 'A little offering! Drop it in.' And I'd say, 'Don't forget: Jiggs is going to go to Europe, a little offering for Jiggs.'\" After circulating with the chalice, Tish would then turn to Jiggs and say, \"Well, so far, Jiggs, you've got just enough to get to the airport. You've got to work a little harder.\"\n\n\"She'd slip you the drinks and you'd give her the money and after a time she wouldn't ring it up; she'd put it in her cup, you see. So we all said, 'Oh, she's such a conniver!' But we all liked her and the bosses always knew, if you're working in a gay bar, you're going to take a little off the top.\"\n\nJiggs was even better known for the cigar box he always carried, which served among other purposes as a container for the drugs he peddled. (Jiggs was so well known for the cigar box he used that when he died, his ashes were buried in one.)\n\nStealing from the Mafia was dangerous, even for a popular bartender. Tree recalls that \"the person who collected the entrance fee, or the doorman, or the guy in the shiny suit... walked around watching to see if the bartenders were ringing up the drinks or if the waiters were reselling the entrance tickets.... Arms and legs were broken on a few bartenders and waiters when they got caught.\" In spite of her well-known light fingers, Jiggs survived unharmed because, Tish explains, \"Jiggs knew just about how much to take.\"\n\nEven with police payoffs, illegal bars were raided on an average of once a month but more frequently during an election campaign when politicians, eager to impress the electorate with how \"clean\" they were keeping the city, pressured the police to be more vigilant. Why did the police raid after being paid off? Because they were ordered to, because it was their job, and because of complaints from the Stonewall Inn's neighbors, never very happy with the gay men the Inn drew to the neighborhood.\n\nThe Sixth Precinct did at least warn the Stonewall that a raid was imminent. Chuck Shaheen explains: \"We would be alerted that we were going to be raided before we opened, so all kinds of things could be done. There was always less money on the premises. There was always less liquor... because what they would do when a bar was raided in those days, if you had cash registers, they would impound the cash register... [and] the liquor, and... let all of the customers leave. They would try and identify the people who worked there. It was never, and I repeat never, any of the real management or owners. The bosses would disappear very quickly. That's why they always had a gay person on the door, because that's the person who had to take the heat and actually get arrested.\n\n\"The white lights on the dance floor would go on and... we would have time to jump from behind the bar and mingle. And people would get rid of their drugs and all that stuff. But they were never really interested in looking for drugs and things like that, unless you still had it on your person or something, if you were one of the unlucky people to be chosen\u2014supposedly at random from the crowd by the police\u2014as an employee. We [employees] would immediately jump from behind the bar and try and take the money out of the box [and] put it in our pocket. We'd jump over the bar because we knew they had to break through the front door, which was a very strong door. [The Mafia owners] didn't want them to get the money. They would hold it as long as they could.\"\n\nThere was such close cooperation between the police and the gay bars that the police would time the raids to minimize the disruption to the bars' business. Shaheen explained, \"A normal raid would be before midnight, which would enable us [to prepare for it]... we really never got busy until after midnight on the weekend. Or they would raid it on a Tuesday night. Whoopee. If they raided it on the weekend, it was always that we would know it was going to be before midnight, before we even officially set up.\"\n\nAlthough the Stonewall Inn employees (as opposed to the owners) were occasionally taken away by the police, arrest was not viewed with any alarm. \"We were never frightened of being arrested. The police would just randomly pick people off and say, 'You, you, you, and you.' If we were arrested and taken into the Sixth Precinct, which we were many times, we automatically knew we would be out in a matter of hours. Many times it would happen that people who did not work there were arrested. But the lawyers had them out so quickly, they never even left the precinct house.\" The lawyer the Stonewall Inn used was a \"family lawyer,\" Enid Gerling.\n\nWhile there was dancing in both of the Stonewall's rooms, dancing had more of a communal character and was more expressive in the back room. Tommy Lanigan-Schmidt feels that the real power of the Stonewall for its admirers had its genesis in the music played there. From his perspective as one of the gay street youth, he saw three main ingredients at play: being gay; being unconstrained, because these youths were living from day to day; and being adolescent, they had tremendous sexual energy. Tommy explains, \"The music is carrying the articulation of this emotional need that can't be articulated on the outside publicly. You picture a bunch of kids and the towns they came from: they hear the songs on the radio, Diana Ross or the Shangri-Las, and the way they relate to it is they identify it as someone of their own sex, but they're not telling anyone. But the song is articulating that feeling both through the music and the words. At the Stonewall, all these kids who had to hold this inside in high school can now articulate this completely. At the Tenth of Always I was very conscious of their stopping people from dancing: you were never allowed the full flow of that. The Mafia people there were very gruff about telling you not to dance; they'd really treat you like queers, like you were disgusting. At the Tenth of Always it was always a little tiny dance floor and you'd hear the Shangri-Las' song\u2014something like 'walk across the room and give him a great big kiss'\u2014and you could look across the room and see sexy Vinnie over there and you could get into all the longing, but you couldn't go over and ask him to dance.\n\n\"At the Stonewall Inn the articulation becomes much more complex because not only could you ask him to dance, but a lot of other people could ask him to, so then you could get into _all_ of the feelings, like feeling sorry for yourself in a really beautiful way like a teenager is supposed to. A teenager loves to have that frustration articulated: 'Oh, I wanted to ask him to dance, and he doesn't even know I exist! And there he is dancing with that awful queen!' 'Cause I would get into this thing: 'That queen doesn't know what's good for that man, and I know what's good for him.' And then I'd go and articulate other things that I heard in a movie like, 'Oh, you have hands like Michelangelo's _David_ ,' and the guy would look at me and say, 'So what?' But the usual articulation coming in there was not from Michelangelo; it was from the Supremes and Martha and the Vandellas. So there's this really tremendous articulation seeping into people's subconscious and deepest needs, plus the place becoming a refuge and a strength.\"\n\nThree months after Danny Garvin left New York City to stay with his family in Illinois, he returned.\n\n\"I was at the point that I would have sex with men, but I wouldn't kiss; I was still in the early stages of coming out. Then I ran into this guy one night who picked me up in a Bickford's [a chain restaurant], and he was with a few friends of his. His name was Charlie, and I stayed overnight. The following night we went to the Stonewall and Charlie asked me to dance. And I said, 'No, no, no. I don't dance with men; I don't do that.' So he said, 'Why?' I said, 'I don't want anyone to see me.' He said, 'But you know, everybody in here is gay.' And it was just the fear, you know?\n\n\"So as the night went on we had a few more drinks; he asked me if I found any of the men who were with him attractive; I said 'Yeah, Frank.' And I started crying, 'cause nobody had ever asked me.\n\n\"The following night we went back to the Stonewall, and then Frank came over to ask me to dance. And I realized if I didn't dance with this guy, we probably wouldn't get it on together. It was the first man I was attracted to; it was the first time I danced. I felt embarrassed and nervous, and I realized that I liked this guy. You know, it was my first love. So we dance. It was nice. We slow danced to 'Let It Be Me,' which became our song. That was the first time I allowed myself to dance, to be part of gay life. To go ahead and kiss a man in public.\"\n\nLanigan-Schmidt feels that the appeal of the Stonewall Inn came not only from dancing to music but also from the songs' lyrics. He cites the lyrics of several songs performed by Martha Reeves and the Vandellas as examples of lyrics that could have a particular resonance for those who went to the Stonewall Inn. One example is \"Third Finger, Left Hand.\" In this song the singer exults that although \"friends said it couldn't be done,\" she has succeeded in marrying the man of her dreams.\n\nTommy found magic in the lyrics, applying the idea of the attainment of something said to be impossible to his own situation as a gay man. \"It translates the same way it would to a heterosexual, but it also has other translations, because the whole thing is, it says: 'Friends said it couldn't be done.' I really would think things like, _Here we are dancing!_ because I was very aware of looking and saying, 'There's men dancing with each other and dancing slow dances with each other.' That entered very consciously in a way that impressed me. So a song like this would be like a big, celebratory thing, every word.\"\n\nTommy analyzes \"Forget Me Not\" as a poignant example of how pop music could speak to gay people. The song is written from the point of view of a woman telling her boyfriend good-bye as he prepares to sail for Vietnam. The singer tells her lover that he should remember her when he is away so that when he feels lonely or in despair they will still be united.\n\nTommy points out that not only did people generally never think of gay men as having soldiers for lovers and being separated from them by the war but that \"someone could go out and have a one-night thing with a soldier and hear that song and that song could be the way they hold on to that forever. Or they could actually have a regular steady boyfriend in the army. To have that affair or quickie sex and then come to a place and be able to dance that into a song was a major innovation of the Stonewall, because it wasn't happening anyplace else.\n\n\"Another thing in gay life that happens since you're a little kid is the ability for self-ridicule that we developed to survive, which is about laughing at ourselves with a certain poignancy. You've got all these queens in the street, and they can be pretty mean to each other, but we all get laughing. And these songs can play into that, too. All teenagers make up their own crazy words for what is not in the song. The street queens would do it, too. That song 'Trains and boats and planes are passing by to Paris and Rome for someone else but not for me,' by Dionne Warwick or someone. And these street queens would go, 'Faggots and dykes and queers are passing by; it means a trip to Paris and Rome for someone else but not for me.' Now, you could analyze it as hopes and dreams to go to Paris or Rome, or someone else is getting it, but I'm not, but it's not so serious-sounding that you couldn't laugh at it.\n\n\"At the Stonewall people would invent a line like that and then it would spread like wildfire, where a bunch of people could scream out that line while they're dancing and get into it. And it might fade away in a couple of days or it might be something they'd say forever. I think the best metaphors for Stonewall are words like _home_ , because when you're not living anywhere regularly, home becomes where you make it. A bunch of queens singing that gives each other a mutual strength, and that familial sense of the street queens, it's a kind of tribal loyalty.\"\n\nDick Leitsch also noted the special place the Stonewall Inn held for the gay community's most rejected members:\n\nThe \"drags\" and the \"queens,\" two groups which would find a chilly reception or a barred door at most of the other gay bars and clubs, formed the \"regulars\" at the Stonewall. To a large extent, the club was for them.... Apart from the Goldbug and the One-Two-Three, \"drags\" and \"queens\" had no place but the Stonewall....\n\nAnother group was even more dependent on the Stonewall: the very young homosexuals and those with no other homes. You've got to be 18 to buy a drink in a bar, and gay life revolved around bars. Where do you go if you are 17 or 16 and gay? The \"legitimate\" bars won't let you in the place, and gay restaurants and the streets aren't very sociable.\n\nThen too, there are hundreds of young homosexuals in New York who literally have no home....\n\nJobless and without skills\u2014without decent clothes to wear to a job interview\u2014they live in the streets, panhandling or shoplifting for the price of admission to the Stonewall. That was the one advantage to the place\u2014for $3.00 admission, one could stay inside, out of the winter's cold or the summer heat, all night long. It also saved the kids... from getting arrested as vagrants.\n\nThe tribal energy that Lanigan-Schmidt noticed might express itself in any of a number of ways while dancing. For example, Tommy remembers how the pin spotlights in the front room were playfully used: \"Queens used to use this expression, 'Give me vogue.' They would do these poses that they figured were like models. They'd suck in their cheeks, but they'd try to get just where the pin lights near the jukebox came in. The queens from the back room would go into the front room to show off because in the front room they'd be noticed more.\"\n\nDanny Garvin also remembers how music at the Stonewall Inn fostered communalism, especially in the back room: \"In the back room\u2014because the room was large enough\u2014is where you'd find most of the people would do their line dances. I can remember learning a dance there called 'The Spider.' And it would be a camp, because they'd get ten or fifteen of us all lined up and we'd all go across and somebody would scream, 'Hit it, girl!' and everybody would go down and touch the floor and get back up, and move on.\"\n\nAnd the back room had all the right ingredients to make it the more sexually charged of the two rooms: it had dimmer lighting, was the favored hangout of the street youths, and had better music and dancing. Harry Beard recalls, \"The back bar was for the socializers and talkers and people that wanted to play touchy-feely under the tables.\"\n\nCharles Burch found the dancing in the back room so sexual as to be inspiring. An avid reader of Walt Whitman, he believed in the poet's vision of love in its physical aspects as so free that it was universal. To him the Stonewall seemed \"terribly wonderful\" because it was a place to sexually connect with strangers, the realization of his Whitmanesque fantasies. \"To me it's nice when they play a slow tune that you could use for the kind of dancing where you get up close. 'Cause then I could do my sex dancing with some utter stranger. And that was what I loved.\"\n\nAnother testimony to the power of dancing at the Stonewall comes from Danny Garvin. Asked when it was that he first became aware of being different and how he felt about it, he responded, \"It would have been in the Stonewall that night when I first danced with that guy. I knew I wasn't drinking. I knew I wanted to go to bed with him. I was so scared to [dance with him]... And there was that fear of 'suppose somebody from my neighborhood sees me.' You know, the closet door was so tight.... There were no positive role models. Who'd know what the Mattachine Society was at seventeen?... So, yeah, the Stonewall's the first place where I started to accept myself being gay.\"\n\nWhile other clubs and settings may have offered places for socializing and making romantic and sexual liaisons and while there were a number of clubs where people could dance, the Stonewall was the only sizable place where gay men could express their sexuality freely and openly for sustained periods of time. Bathhouses offered places where gay men could actually have sex\u2014that was the bathhouses' sole purpose\u2014but socializing there was largely incidental to sexual activity. In other words, the Stonewall Inn was the only place where gay men could express all sides of their personalities. Not only could gay men dance freely, but there also could be a certain expansiveness about their dancing, given the physical size of the club and the tight security provided by the Mafia.\n\nTommy Lanigan-Schmidt succinctly sums up the meaning of the Stonewall Inn in this regard when he says, \"It was as real as the street, and also\u2014what was the biggest step in there\u2014at the Tenth of Always, when you'd dance with someone, you always knew that any minute the chandelier would flick on and you'd have to go sit down. At the Stonewall, everyone just kept dancing. It was open a few years before those riots. It sunk its roots deep because it had longevity and because you were never told not to dance.\"\n\nWith its owners interested only in exploiting gay people, the Stonewall was not an ideal place for gay men. Yet it offered its patrons three crucial things: space, security, and freedom. Added to these were longevity and the continuity that longevity made possible. Through the power of music and dance, the club fused these elements to create among most of its regular customers a sense of gay community and identity and thus a loyalty to the Stonewall Inn.\n\n#\n\n# _The Skull_\n\nWhile Chuck Shaheen clearly stated in his interview with historian Martin Duberman how Tony Lauria and his three partners became the prime movers behind opening the Stonewall Inn, when he discussed another of the Stonewall's underworld managers, he never explained how Ed Murphy, a shadowy figure known as The Skull, came to be a manager at the Stonewall Inn. This is fitting for Murphy, who seems to have been so ubiquitous in the gay bar scene in Manhattan that his presence was often taken for granted. Danny Garvin remembers seeing him \"just hanging out\" at the Stonewall Inn and at Danny's. Tommy Lanigan-Schmidt describes him in a similar vein at the Tenth of Always: \"He was always there like a Buddha. You know, he had that presence. I never saw him do anything but sit where he was. Wherever he sat, he sat and that was Murphy.\" It was as if Murphy, an obese man with a gray beard and gray suit, had a knack for insinuating himself into the gay bar scene.\n\nWhile the Stonewall Inn had its negative aspects, for most of its customers these were not usually major issues, especially when weighed against what the club had to offer on the positive side or, for that matter, when compared to most of the other New York City gay bars then in operation. But as it turns out, some of the Stonewall's patrons had more to fear from the club's operators than high-priced weak drinks and occasional police raids.\n\nOne warm evening as Tommy Lanigan-Schmidt headed up Greenwich Avenue with several of his friends, he was delighted to see one of his heartthrobs, Tano, a masculine, good-looking Puerto Rican youth. In spite of his attraction to Tano, the reason for Tommy's glee in spotting him was not because he hoped it would lead to anything romantic or even sexual. Gay men on the street passed the word to each other that Ed Murphy fancied the youth as a favorite boyfriend. Tommy had heard many different rumors on the street about Murphy, suggestions that he was powerful and connected and owned lots of gay bars. To Tommy it all boiled down to \"vague rumors of bad things,\" and so he had kept a certain distance from Tano in spite of his strong attraction. Tano's only response, however, to Tommy's circumscribed friendliness was complete aloofness. Frustrated, Tommy decided to play sour grapes by calling Tano Miss Polka Dot, for Tano often wore polka-dot shirts, like the one he had on this evening. Tommy took a perverse pleasure in baiting him, for it upset Tano enough that Tommy could see the very masculine Latino fighting down an urge to attack him. When Tommy noticed Tano that night, his first thought was, _Oh, good! I'm going to get to call him Miss Polka Dot again and see his angry, sexy face_. But on approaching the youth, Tommy heard the screech of an automobile's brakes and noticed a dark car from which several men emerged. They pulled Tano into the vehicle before speeding off. It was the last time Tommy saw Tano.\n\nIn the coming weeks two versions of a story emerged on the street to explain the youth's disappearance, both of which connected his kidnapping to Ed Murphy. In one version, Tano had stolen something from Murphy. The second rumor was that he had enraged Murphy by becoming involved with someone else. The kidnapping of a man right in front of his eyes had startled Tommy, but when he remembered that he had heard Murphy could have people rubbed out the event no longer seemed so surprising. The suspicion that Murphy was involved in the murders of youths goes back at least to the early sixties. Stephen van Cline recalls, for example, that Murphy had been involved with the early 1960s waterfront gay bar called Dirty Dick's, where, he says, a number of young men were seen for the last time.\n\nEdward Francis P. Murphy had become a hellion at a tender age. Repeatedly thrown out of Catholic schools, he came to the attention of the police at age nine when he attacked the owner of a neighborhood banana store and trashed his fruit stand. After several other brawls, Murphy was sent to a special problem school on nearby Hudson Street. The time he spent there with other juvenile delinquents did not improve his temperament, for he later hit a police officer over the head with a milk bottle in a dispute over Murphy's shoeshine box. For this infraction he was packed off to a reform school in Dobbs Ferry.\n\nWhen Murphy was released from reform school in 1943, he entered the army after a short stint in the gay bar business at the Pink Elephant, an establishment run by the Jewish Mafia. After combat in France, he left the army in 1946 and worked as a bouncer at the Moss Bar on Eighth Avenue.\n\nApparently it was his work as a bruiser for hire that led the ambitious and physical Murphy into a wrestling career. Murphy realized that he had a \"tough top\" and decided to make his head do double duty. He shaved it, anointed himself Skull, and put his cranium to work as a battering ram. The tactic worked well enough to get him into professional wrestling, where spectators came to know him for his special death hold. Murphy's aggressive style seemed to make him a natural for wrestling. A television viewer from the era recalled seeing him rip the microphone out of the announcer's hands, stomp on his opponent's head when he was unconscious, and throw chairs at old women who booed him.\n\nBy the late 1940s Murphy had teamed up with a gay friend to rob dentist offices, targeting them for their shipments of gold from dental laboratories. After robbing seventy-three dental offices, Murphy was finally caught in 1947 and served ten years in prison, the maximum time, spending much of it in isolation for assaults.\n\nWhen he left prison, he worked as a bouncer in gay bars such as the Cork Club, the Bali, Mais Oui, Sans Souci, the 415, the Terrace, and Artie's. Murphy seems to have had a penchant for moonlighting, for he also began to work as a hotel house detective.\n\nWhile the preceding record of Murphy's early career is taken from his own accounts, it was in his work in the hotel business that he first came into the public record in a way that has a direct bearing on gay history. In August of 1965 the police arrested Murphy while he was working for the New York Hilton as a house detective. He was charged with being the head of a ring that had preyed on hotel guests, extorting $100,000 from \"rich playboys and executives.\" While the newspaper report made it clear that sex had been used in the extortion scheme, only those who read between the lines would understand that the blackmailers had targeted homosexual men.\n\nSix months later Murphy's name hit the New York media in a much bigger story that was explicitly connected to homosexual victimization. This time Murphy was accused of being involved in a ring of blackmailers that preyed primarily on homosexual men. The ring had been centered in New York and Chicago, although its reach was nationwide. While the scale of the extortion ring was known to be large from the beginning, by the time the investigations were completed its scope and size were staggering: having operated for almost ten years, the ring had victimized close to a thousand men and taken in $2 million. Equally impressive was how many of these men were highly successful. Among those listed at the time or in more recent newspaper reports were the head of the American Medical Association, two army generals, Admiral William Church, a Republican member of Congress from New Jersey, a Princeton professor, \"a leading motion-picture actor,\" \"a musician who has made numerous appearances on television,\" heads of business firms, \"a much-admired television personality,\" and \"a British producer.\" None of the individuals named here were identified in the contemporary press coverage of the blackmail ring, a credit to both the men prosecuting the cases, Andrew Maloney and Robert Morgenthau, as well as the newspapers that covered the story.\n\nThe investigation began with a fairly routine arrest, when a man impersonating a detective in Grand Central Station was apprehended by a real detective, James McDonnell.\n\nAlthough the ring used several scams, by far the most common one was to send attractive young men into hotels where wealthy homosexual men came to meet such youths. The decoy would lure the man to a hotel room, attack him, and run out of the room with his wallet or other valuables and identification. Then, weeks later, after having found out which men had the most to lose, whether in terms of their families, professions, or reputations, other members of the ring, usually a pair, would call on those deemed vulnerable. The pair would produce police badges from New York or another jurisdiction and an arrest warrant. The \"policemen\" would then threaten to arrest the victim, exposing him as a homosexual. They would soon offer to \"forget about it\" if the victim would give them anywhere from a few hundred to several thousand dollars.\n\nThe nerve the blackmailers displayed was breathtaking. Tracking their victims down in their hometowns and calling on them in their homes was routine for the ring. They called on the New Jersey congressman on Capitol Hill and marched him out of his own office door so he could fly home to pick up $50,000. A surgeon was pulled out of an operating room. Members of the ring went to the West Coast and got inside a nuclear plant. The extortionists were discussing the price of their silence with a nuclear scientist when the scientist's superior walked into the office. The fast-thinking victim introduced the blackmailers as two detective friends of his who were visiting from New York. The scientist, thinking he was dealing with real detectives, suggested that they might want to see the plant. His superior approved and the extortionists were taken throughout a plant that was normally under the heaviest security strictures. When the scientist was invited to New York to testify against those who had victimized him, he at first refused to believe that they were not detectives, saying he was disgusted with the New York police. After all, as the Mattachine Society pointed out at the time, this operation did resemble the New York Police Department's use of police decoys to entrap gay men.\n\nNot surprisingly, these stories sometimes ended tragically. When Admiral William Church, the head of the New York Naval Yards, was approached by Detective McDonnell in Washington, D.C., he refused to accompany McDonnell back to New York to testify before a grand jury. Church told McDonnell that he would drive up the following day. Instead, he drove his car to a Maryland motel and put a bullet in his head.\n\nA particularly noteworthy feature about the contemporary news accounts of this story is that the initial newspaper stories listed Edward Murphy as one of the arrested extortionists, but afterward his name disappeared from the newspaper coverage. For example, in nine _New York Times_ articles on the case published between February 18, 1966, and July 12, 1967, Murphy was never mentioned again after the initial February 18, 1966 story. Yet the initial news story made it clear that Murphy was one of three ringleaders, if not the head of the entire operation. In March of 1968 the _New York Mattachine_ _Newsletter_ asked why Murphy had not been sentenced for his role in the blackmail ring. The newsletter stated that not only had Murphy served several prior prison terms, but he also had recently pled guilty under a federal indictment to extortion charges and was under \"a number of indictments\" at the state level. After pleading guilty to the federal extortion charge, Murphy had merely been put on probation for five years. The newsletter reported that Mattachine\u2013New York \"had been informed that Murphy's sentence has been so often postponed because he had made a 'deal' to turn state's evidence, and the delays are to work out another 'deal' to lighten his sentence.\"\n\nMurphy apparently did give evidence against the two other main figures in the ring: John J. Pyne, a Chicago police officer, and Sherman Chadwick Kaminsky, a Bronx native who also went by the name of Paul Vargo.\n\nHowever, that Murphy, after having been to prison several times before, could get off with serving only part of a five-year sentence merely for giving evidence against his co-conspirators, despite shaking down approximately one thousand men over almost ten years\u2014including a number of men of very eminent social rank as detailed earlier\u2014is astonishing. Moreover, while several names emerged in the newspaper accounts as key players in the scheme, word on the street said that the gang had one ringleader. The _New York Mattachine Newsletter_ of March 1968 named Ed Murphy as that person. The theory of a single person at the head of the enterprise is bolstered by a letter from Richard Inman, a homophile activist battling police extortion of homosexuals in South Florida, written to Mattachine-Washington cofounder Jack Nichols in 1965. In the letter Inman stated that he knew via a friend inside the FBI that there was one \"boss man of the syndicate's homo shakedown detail for the whole U.S.\" That Inman wrote the letter before the police uncovered the national ring adds to its credibility.\n\nThe law of averages dictates that the more prominent and wealthy (and therefore correspondingly more vulnerable) the men Murphy targeted, the more likely he was, on average at least, to get the blackmailer's ultimate prize: men who had the most to lose by any publicity while also having the most money to spend on silence. But Murphy did even better in the waters he trawled: he landed the biggest fish of all, one whose value exceeded even that of money, because this one had the ability to keep law enforcement off his back. Murphy's operation landed none other than the nation's top law enforcement officer, J. Edgar Hoover. The same net that hauled in Hoover brought along a bonus prize: J. Edgar Hoover's \"longtime companion,\" Clyde Tolson, who was, moreover, an associate director of the FBI.\n\nAllen Ginsberg, a man who loved both to gather and pass on gossip, had known since the late forties that J. Edgar Hoover was homosexual. A friend of Ginsberg's happened to be in Washington, D.C., when he ran into the FBI chief. Hoover grabbed the man through his pants, inspiring Allen to write about being \"groped by the FBI in the halls of Congress.\"\n\nAs he recalled homosexual life in the era before Stonewall, Ginsberg explained, \"There was one very tall young guy, rather portly, who later became an insurance executive, who went down to Washington and was wandering through the upper corridors of some very good hotel right near the White House and was accosted, for erotic purposes, by J. Edgar Hoover no less, and told me about that in 1947. So I always had Hoover's number, although it was very difficult to prove. Like Cardinal Spellman, another gent who was supposed to be a closet queen and supposedly had young men accompany him on his trips. That was, of course, never breathed in the newspaper, although both of those guys were fervent anti-Communists and heavy moralists and all in favor of all sorts of censorship. Nonetheless, among the underground or elegant gay world, there were many rumors about them, and even, as I say, one or two encounters, particularly with J. Edgar Hoover.\"\n\nThat Hoover was homosexual and Clyde Tolson his lover is currently generally accepted. The history of _The Homosexual Handbook_ , published in 1968, shows, however, that by the late 1960s, not only was Hoover's homosexuality whispered rather widely in the homosexual world but also that Hoover was, understandably, extremely sensitive about any public suggestion of this information. The book's last chapter, titled \"Uncle Fudge's List of Practical Homosexuals Past and Present with Very Short Biographical Notes\u2014A Hearsay Reference Work,\" includes Hoover's name:\n\nJ. Edgar Hoover: _Celibataire_ , the director of the Federal Bureau of Intelligence [sic], he has for several decades remained the _eminence froide_ of our national great society.\n\nAfter the book appeared, pressure from the FBI caused it to be withdrawn. The publisher soon reissued the book, but with Uncle Fudge's list one name shorter.\n\nKnowing that J. Edgar Hoover was homosexual, Ginsberg made an intuitive leap to explain Hoover's odd and consistent denial of the existence of organized crime. Why would the head of law enforcement for the entire United States claim to know less than what the average citizen knew from reading the newspaper, that organized crime certainly did exist, and abundantly? In 1987, well before a witness came forward to talk about Hoover appearing at parties dressed as a woman, Ginsberg explained his surmise to Obie Benz: \"[Hoover] insisted there was no organized crime. In fact, in those years [the late 1940s] I had the fantasy that the Mafia might have secret movies of J. Edgar Hoover in the basement with some big, hairy Mafia Lothario and were blackmailing him so he'd lay off organized crime, because he insisted there was no organized crime.\"\n\nFrom information published in the 1993 book _Official and Confidential: The Secret Life of J. Edgar Hoover_ , by Anthony Summers, Ginsberg's intuition has been proved correct. Not only did Summers discover that the Mafia had photographic evidence implicating Hoover in homosexual activity, but it also came to light that Hoover at times dressed in female attire. Research conducted for this book strongly suggests that Ed Murphy had one or more of these photographs, which allowed him to avoid serving time in prison for leading an extensive national blackmail ring.\n\nJohn Paul Ranieri, a former prostitute interviewed for this history, provided critical testimony for corroborating and better understanding the larger implications of Murphy's criminal enterprises for gay history. Ranieri said that as a youth from Westchester County he had been forced by blackmail and Mafia-supplied drugs into a prostitution ring in which he remained active for three years before he escaped the mob's control. He claimed that a number of youths in the ring had disappeared after they got careless with talk, for while most of the customers were more or less average homosexual men with money, the regular clientele, according to Ranieri, also included famous men such as Malcolm Forbes, Cardinal Spellman, Liberace, U.S. senators, a vice president of the United States, one of the most famous rock musicians, and J. Edgar Hoover. The mob's order, according to Ranieri, was strictly \"Keep your zipper open and your mouth shut.\"\n\nRanieri said that he met J. Edgar Hoover at private parties at the Plaza Hotel and that Hoover's name was never mentioned. Hoover was always in drag, and Ranieri said he could tell that the FBI director was sure that no one recognized him. Ranieri said that he had ensured his own survival by having in his possession a photograph of himself with Hoover, given to him by the photographer.\n\nHow does the preceding information link Ed Murphy with J. Edgar Hoover? The connection is made evident in a news story written shortly after Hoover's homosexuality and transvestism became public. When Summer's book was published, a newspaper story about the 1960s national homosexual blackmail ring suddenly appeared after a quarter of a century of silence on the subject. Without mentioning Murphy's name, it quoted law enforcement sources who had worked on the case as saying that their investigation into the nationwide blackmail ring had turned up a photograph of Hoover \"posing amiably\" with the racket's ringleader and had uncovered information that Clyde Tolson, Hoover's lover, had himself \"fallen victim to the extortion ring.\" After federal agents joined the investigation, both the photograph of Hoover and the documents about Tolson disappeared.\n\nInformation uncovered in researching this history suggests that having the goods on Hoover, Ed Murphy continued to blackmail homosexual men, using the Stonewall Inn as a prime locale for this new extortion operation. There he targeted professional gay men, especially those working on Wall Street.\n\nAccording to the same 1968 _New York Mattachine Newsletter_ that asked why Ed Murphy had not been sentenced for his role in the national blackmail ring, Murphy had an interest in several gay bars in New York, including the Stonewall Inn\u2014the only club identified in the article\u2014and these clubs' membership lists had been used for blackmail:\n\nMSNY has also been informed that Murphy has an interest in the Stone Wall, a club on Christopher Street, and several other gay clubs in New York. Our source claims that the membership lists of some of these clubs are used to further extortion and shake-down schemes.\n\nStronger evidence of Murphy using the Stonewall Inn for blackmail comes from a 1969 publication that quoted the Mattachine Society of New York. In the 1960s it was not easy to find gay bars and so, as a service, the Mattachine Society of New York began compiling lists of these bars. Eventually a local guide was published by a small business venture as the _Gay Scene Guide_ , and the publishers, on friendly terms with Mattachine\u2013New York, quoted material from that organization's newsletter. Still, the bar lists remained just that, lists of gay bars, with but one exception: In the 1969 _Gay Scene Guide_ , after the entry for the Stonewall Inn, there was a long quote from the _New York Mattachine Newsletter_ inserted in the midst of the brief descriptions of gay bars:\n\nThe following news item was reported in the March 1968 _Mattachine (N.Y.) Newsletter_ , and is presented here in condensed form... the Mattachine Society Inc., of New York, was instrumental in aiding D.A. Frank Hogan's office with information that led to the arrests of a number of blackmailers:\u2014\"Edward F. P. Murphy, an ex-convict who is alleged to have been the head of the national ring which recently was active in extorting money from homosexuals... has served prison terms for larceny and for carrying deadly weapons, and was arrested for impersonating an officer, and for extortion... under Federal indictment on extortion charges... permitted to plead guilty and received a five-year probation. On a number of indictments in the state courts, Murphy pleaded guilty on May 16, 1966... sentencing has been postponed six times... he could get up to 15 years in prison as a second offender, on the robbery charge alone. MSNY has also been informed that Murphy has an interest in the Stone Wall, a club on Christopher Street, and several other gay clubs in New York.\" We _caution our readers NEVER_ to use your real name when cruising, NEVER to give your address to a questionable bar or club, and remember, that trick or hustler you've just picked up may be \"working\" for the management! We urge you, if you've been intimidated or blackmailed in the past, to report it to the D.A.'s office, or to M.S.N.Y.\n\nThat a local guidebook took the unusual step of inserting a warning against giving out information to employees of the Stonewall Inn in the middle of a bar listing is more than suggestive. Indeed, several years later Dick Leitsch was quite forceful about Murphy's role at the Stonewall Inn, writing that he \"seemed to be the manager of the place.\"\n\n_The Homosexual Handbook_ reported that the \"burly\" at the door of the Stonewall Inn \"keeps boxes that hold, or are rumored to hold, thousands of cards upon which are printed the particulars of the many thousands of customers,\" which echoes some of the ideas found in the _New York Mattachine Newsletter_ article and may explain part of how the blackmail routine operated: did some customers naively believe that they were really being screened for membership in a private club and that this screening would protect them by both excluding police officers and helping to establish the identity of the place they patronized as a legitimate private club?\n\nBeyond Murphy's involvement in the Stonewall Inn and in blackmailing gay men, he was deeply involved in male prostitution. Chuck Shaheen, who had a very high regard for Murphy, told Martin Duberman, \"I knew Eddie Murphy for a long time.... He was into young boys. Most definitely. And was very, very involved with procurement of young boys.\" Danny Garvin recalls how he would \"always see these hustlers hanging out with [Murphy]. He had connections, and these hustler kids would hang out with him.\" Tommy explains why the Mafia would operate the Tenth of Always as an ice-cream parlor in terms of Murphy's predilections: \"The Tenth of Always had a kind of particular feeling, that you knew you were there because Murphy liked chicken. In there I felt like I was in some surreal Catholic Youth Organization dance, because everybody was like my age or younger, and the drag queens just looked like regular high-school girls, and the hustlers looked like regular high-school boys. And then it really looked crazy because everyone was sitting, sipping these sodas, and it was like\u2014there's no word to describe\u2014it wasn't a brothel, a bawdyhouse, or whatever. It was like the pickings of johns: that's what it was set up for.\" Bob Kohler, who hated Murphy passionately, cited as evidence of Murphy's loathsomeness that he paid the youths he pimped with counterfeit money.\n\nResearch on this book uncovered a couple of elusive references to a prostitution ring that was run on the second floor above the Stonewall Inn. When one looks at all the available evidence today, there is little doubt that such a ring operated out of the floor above the Stonewall Inn, although few knew it existed. But by Murphy's own account, \"upstairs the Mafia retained a room.\"\n\nCraig Rodwell's _Hymnal_ provides rare contemporary evidence that the Mafia recruited youths specifically at the Stonewall Inn for their criminal activities. Continuing its campaign to warn gay men against Mafia exploitation, the _Hymnal_ reported that \"some of the 'queenie-boppers' on Greenwich Ave. and at the Stone Wall have been approached to be [heroin] delivery boys.\"\n\nOne day, as John Paul Ranieri strolled around the Village during an interview for this history, he pointed out various places that he remembered from his life there in the late 1960s. As we wandered the area around Christopher Street, walking east on West 10th Street, we passed a parking garage on the northern side of the block where the Stonewall Inn used to be. Ranieri then pointed to a door in the reddish-brown wall a few steps above the sidewalk and said that door was the one he always passed through, entering via West 10th Street, to go to the second floor above and to the rear of the Stonewall Inn to pick up his orders. The office above the Stonewall Inn was, he said, one of two locations to which he reported for instructions on his assignations.\n\nAccording to several articles written by the gay bartender Tree, who worked as a page boy on the floor of the New York Stock Exchange in the early sixties, it was \"the world's biggest closet.\" \"Between the dirty ol' men brokers, married and single, and the employees of the NEW YORK STOCK EXCHANGE there were more gay men than in any gay bar at a single time.\" And if the stock exchange itself was the world's biggest closet, it also boasted, according to Tree, in the men's room on its third floor, \"one of the best T-ROOMS [public rest rooms used for sex] around... spoken about with affection all over the world.\" As if in evidence of the T-room's international pull, two 1990s articles that Tree wrote specifically about homosexuality in the Wall Street area in the 1960s include photographs of Konrad, a beautiful blond youth. The articles explain that Konrad had come on vacation from Durban, South Africa, when Tree met him in the crowded T-room. \"We fooled around a little but the traffic in the men's room was too much.\" Apparently the stock exchange T-room was so popular that it could become practically impossible even to get into at times. As Tree explained, \"Word passes on from one to another.\"\n\nBarry Perrin, a Stonewall Inn habitu\u00e9, remembered the warning he had been given about blackmail after he came out: \"Of course, there was a lot of blackmail. You heard about that sort of thing going on if somebody found out you were gay. I didn't know it then [in the late 1960s]. I found out since [in the early 1970s] that even the bartenders at Stonewall did that. That I didn't know, but they didn't bother with me, I suppose, because I was nineteen and they didn't expect me to have much of a career at that point, or money to give them.\n\n\"The Stonewall staff apparently\u2014and this is not from my memory but from what I've heard since\u2014used to talk to guys that probably looked like they were a little more successful, probably a little bit older. If they found out that you worked for a law firm or a stock brokerage firm or anything like that, you were blackmailed... especially in the stock market areas\u2014it's not that large\u2014you could be blackballed from a whole industry, and that was absolutely so.\n\n\"The good-looking bartenders in Stonewall, who were probably connected to the Mafia\u2014they themselves were probably not actually Mafia, but those were their bosses and they were told to do this, and they did it\u2014the good-looking waiters would get friendly with guys and then find out where they were. It's really so insidious, when you're talking to somebody that you find nice\u2014he's being nice, pleasant finally. Eventually you tell him where you work, and then all of a sudden this happens. It's just so awful.\"\n\nAccording to Perrin, blackmail at the Stonewall worked in a manner parallel to the national ring: those who later became blackmail victims were initially victims of a robbery that included taking documents. The ring then researched which men might be wealthy enough to be vulnerable to extortion. At the Stonewall the waiters did the research.\n\nBut beyond the Stonewall being linked to blackmailing Wall Street employees and blackmailer Ed Murphy being connected to the Stonewall Inn, is there any piece of evidence that directly connects Murphy himself to blackmailing men on Wall Street? There is: In the late 1970s, a friend of Murphy's told Morty Manford that Ed Murphy said that he had been informing on the mob to the FBI. Murphy's friend went on to explain that because the mob had discovered that The Skull was an informer, Murphy had decided to come out, stop working with the mob, and quit informing because he \"wants to become a good guy.\" Manford reacted skeptically to Murphy's claim of being an FBI informer by saying, \"I've heard... that he was involved... in a ring that was blackmailing homosexual men down at Wall Street.\"\n\nDeputy Inspector Seymour Pine had not been moved to Manhattan's First Division of the Public Morals police for long before being summoned to a meeting with his commanding officer. Detective Charles Smythe, who shared joint responsibility for Morals First Division, accompanied him. Pine was glad to have Smythe as his partner, as they knew each other quite well, having fought side by side in World War II. The men enjoyed the comfortable trust old army buddies who have been through combat together naturally feel toward each other.\n\nThe war had considerably disrupted Pine's life, but being Jewish, he had not minded: he was eager to make his contribution to the fight against Hitler. In fact, it was Pine's religion that had determined his career as a police officer, for while he had wanted to join the FBI, his father had discouraged him, knowing that it was difficult for Jews to become FBI agents. Encouraged by his father to become a police officer instead, Pine had taken courses in government and public administration at Brooklyn College before graduating from there in 1941. He entered the police department right away and also enrolled in St. John's Law School but was drafted before finishing his probationary period in the police department.\n\nThe young draftee proved to be quite a catch for the army. Pine had been captain of his college's wrestling team, played football, been president of the school's athletic association, and studied judo in the police department. With such a diverse athletic background and his training in fighting, it is understandable that during his basic training at Fort McClellan in Alabama he readily spotted deficiencies in the army's hand-to-hand fighting techniques. After he convinced his superiors that there were better ways to train soldiers in hand-to-hand fighting, he was sent as an instructor to infantry school. There he wrote a manual on hand-to-hand combat for the infantry soldier that became the army's official manual on the subject. Pine's accomplishment in writing this manual was considered proof at the time that someone who had not gone to West Point could make important contributions to the military.\n\nFrom infantry school Pine was sent to officer candidate school, graduating from the Provost Marshal General School as a second lieutenant and then being immediately assigned to teach hand-to-hand combat there. But Pine longed to be in Europe fighting, and so when he was ordered overseas he turned down an offer by the school to remain there as an instructor. Shipped to Europe at the end of June 1943, Pine was assigned to the Allied Military Government of Occupied Territories (AMGOT) both because he had been a police officer and because he had a knowledge of Italian. He served in AMGOT for a year in Sicily and Italy before being reassigned to the American School Center in England, where he instructed servicemen who were going to be running a military government in the event of victory. He was next transferred to General Patton's staff in military government but requested service in an infantry unit. His request was granted, and he was assigned to a Massachusetts outfit, the 26th Infantry Unit.\n\nServing in AMGOT, he had seen his share of hard combat. \"We would try to capture records so that we could preserve the towns. We would have to be up near the fighting when they were ready to go in, and if the shelling would start very often we would get into a foxhole with somebody else, and they were very happy that we would get in with them, when the ground was shaking and the shells were dropping\u2014you felt as if you were going to explode. You would feel the walls of the foxhole actually shake against you, reverberate, 'cause these shells were falling all around you.\" Pine was tough enough that not only could he keep his cool under such intense fire, but he also was able to calm other soldiers down. \"In between [the shelling] you'd go around and talk to your guys and break that tension, this tremendous fright, where everybody is crapping in his pants.\"\n\nIn Europe, working as a liaison with the underground forces, he was injured in a mine explosion, after which he was hospitalized for approximately thirteen months. Released from the hospital, he retired from the army as a captain and resumed his civilian job as a police officer on limited duty, as he needed hospital treatment three times a week for the following two years.\n\nAs his years in the police force passed, Pine rose through the ranks, eventually attaining the rank of deputy inspector. Before Pine's last promotion he was a captain in Brooklyn's Tenth Division. He impressed Chief Inspector Sanford Garelick by how he handled himself in a tricky situation in a Coney Island race riot. Garelick offered Pine a promotion, but Pine refused it, explaining to Garelick that he had organized a program in which he and all his officers volunteered to work with brain-injured children in patterning: manipulating the arms and legs of the children in an attempt to help them learn how to crawl. As the program was strictly voluntary and Pine had organized it, he felt that if he left the area the program would come to an end. In 1967 Garelick promoted Pine against his wishes but kept him in Brooklyn's Tenth Division. This meant moving him into Public Morals, the only opening to which Garelick could promote Pine and keep him in the Tenth Division.\n\nPine had great success in fighting the mob in Brooklyn and was known not to tolerate any corruption by his officers, and so in the spring of 1969 Garelick transferred him to a troubled unit, Manhattan's First Division Morals. The First Division included all of Manhattan from 35th Street down to Battery Park at the island's southern tip. The types of crime he had to deal with in Manhattan were generally similar to those he had investigated in Brooklyn, except that now he became involved in cases of pornography, censorship, and gay bars. He was also told that it was his responsibility to meet with representatives of the Mattachine Society, which he did from time to time.\n\nAt this particular meeting with their First Division commanding officer, Pine and Smythe learned that financial irregularities discovered in Europe reached back to the deputy inspector's bailiwick. Interpol had noticed that an unusual number of negotiable bonds were surfacing in foreign countries and had requested that the New York Police Department look into the matter: were these legal or counterfeit? The NYPD investigation had found evidence of a collusion between the Mafia and the employees of depositories to steal large numbers of bonds. Moreover, it appeared the Mafia was able to put the squeeze on certain people with access to the bonds because the Mafia had learned that some of them were homosexual. From studying the police reports on various gay clubs the commanding officer had concluded that the center of this activity lay in the general area of the city in which the Stonewall Inn was located. The commanding officer singled out the Stonewall for particular attention: police reports stated that a lot of big cars stopped in front of the club to drop off wealthy straight people who went into the Stonewall. A source of outside money?\n\nThe commanding officer made his wishes clear: He wanted those clubs put out of business. Particularly the Stonewall Inn.\n\n#\n\n# _Dawn Is Just Breaking_\n\nAs Craig Rodwell sat down to write an article for the _Hymnal_ 's fourth issue, due out in May of 1968, he felt the homophile movement was stalled locally. It was natural then that he turned to events on the West Coast, which he found inspiring. As he warmed up to what he felt could be accomplished in New York, he extolled the accomplishments of the Society for Individual Rights (SIR) in San Francisco: \"In three years, it has opened a community center with varied social activities, a theater, a storefront, and has become a force to be reckoned with in San Francisco. Candidates for public office seek their support.\" After mentioning that SIR had 3,500 active members\u2014Mattachine\u2013New York then had about 550\u2014Rodwell wrote: \"The same thing can and must be done in New York City.\"\n\nBy the mid-1960s San Francisco already had a rich history of homosexual activism. In 1961, in response to San Francisco gay bars being shaken down by the police and alcohol authorities, an openly gay drag queen, Jos\u00e9 Sarria, ran for the position of city supervisor by circulating a petition among the city's gay population, helping to set off \"developments that fed a steadily growing stream of gay political activity in San Francisco,\" as historian John D'Emilio has written. One of the new endeavors sparked by Sarria's campaign was the formation of a group known as the League for Civil Education (LCE), which began printing the _LCE News_ , \"the first sustained attempt to bring the movement into the world of the gay bar... its circulation in San Francisco alone [after one year] exceeding nationwide figures of _One, The Ladder_ , and the _Mattachine Review_.\" In spite of the _LCE News_ 's militant tone, by 1963 three mayoral candidates had purchased ads in it.\n\nIn 1962 bar owners and employees had formed the Tavern Guild, which retained a lawyer and bail bondsman for anyone arrested in or near a gay bar and coordinated the fight against California's Alcohol and Beverage Control Department (ABC). Soon SIR formed with a more open and democratic approach than that used by the LCE and California's Mattachine organization \"to create a community feeling that will bring a 'Homophile Movement' into being.\" Recognizing the social needs of gay men, SIR held not only dances and parties but also meditation groups and art classes. It ran a major education campaign on venereal disease with the support of the city's public health department and held voter registration drives. The Guild also published an attractive magazine, _Vector_ , which was sold on newsstands throughout the city. In 1965 the Guild began holding candidates nights each fall where political office seekers faced a gay audience that endorsed pro-gay candidates, including those who have since become leading figures in San Francisco and California politics: Willie Brown, John Burton, and Dianne Feinstein. So successful was SIR that in 1966 it opened the country's first gay community center. By 1967 it was the country's largest homophile organization.\n\nCritical to the success of organizations such as the LCE, SIR, and the Tavern Guild, however, was the Council on Religion and the Homosexual, or CRH, a coalition of homophile leaders and clergy, which, through providing the \"cloak of the cloth,\" as San Francisco historian Paul Gabriel has phrased it, lent an aura of credibility to these organizations.\n\nCRH had incorporated in 1964 and is especially known for the events that unfolded at one of its first undertakings, a New Year's Eve ball at California Hall to raise money for the homosexual community. When the ball was held, the San Francisco Police Department showed up in force and staked out California Hall with paddy wagons, huge klieg lights, movie cameras, and photographers. As guests entered the dance, the police filmed and photographed every one of the hundreds of attendees. Finally, the police, who had promised the clergy not to interfere, found a pretext to shut the dance down and arrested several persons who had helped organize the event. This only galvanized the ministers who had put on the affair, having learned firsthand that what they had been told about homosexuals being harassed and oppressed was not an exaggeration.\n\nWhile the story of California Hall is sometimes celebrated as \"San Francisco's Stonewall,\" the story of Vanguard and Compton's Cafeteria is a similarly compelling one and also has its roots in both CRH and the Glide Memorial Foundation. The Foundation had been started by the wealthy Lizzie Glide to fund a church in the Tenderloin, her vision being of a church that would serve sailors and itinerant workmen.\n\nBy the 1960s, San Francisco's Tenderloin had turned into a district very much on the down-and-out. Its cheap hotels and restaurants made it the section of town where runaway youths gravitated. There homeless gay youths sold themselves, as did transvestites and female prostitutes. Drugs were widely available, and it was a center for police corruption, as its very name suggested: The term _tenderloin_ was imported from New York, where it denoted a district so rich in graft that a police officer who worked there could afford to buy his family a choice cut of meat. As more and more families left the area around Glide for the suburbs, Glide Memorial Church, with its very large endowment, was left with a small and aging congregation. In the 1960s young people were attending church less often, and so churches began to reexamine what forms their ministries should take. In this progressive era, many churches responded to the drop in attendance by emphasizing social services. A number of denominations started inner city ministries to reach out to youths, helping poor youths deal with such problems as drugs, job training, and housing, almost always among black populations. Many of the ministers who did such work became interested in rethinking theology and in getting involved in protest movements.\n\nLouis Durham was sent from Nashville, the national education and training headquarters for the Methodist Church, to lead the effort to revive the dying Glide Memorial Church. After performing an evaluation, he hired three young ministers, the first being Ted McIlvenna, considered a rising star. He had had a personal vision, giving him the belief that in the twentieth century, and especially since the Second World War, a huge biblical revelation, like that found in the Old Testament books of prophets was happening. The vision that had been revealed to McElvena was of a coming into consciousness of the divinity of human sexuality. Ted saw his life's work as opening the world to this revelation. Constantly on the go, he met with everyone on the forefront of sexuality issues, from Kinsey to the leaders of the Mattachine Society.\n\nIn 1963, Ted began to work with young homosexuals in the Tenderloin, where he met a youth who had been castrated by an adult. McIlvenna was appalled, but seeing the disfigured adolescent made his commitment to gay youth total. As McIlvenna began to investigate the youths' situation, he realized that there were no services available for them: no overnight shelters or counseling or vocational services.\n\nTed McIlvenna then brought ministers Don Kuhn and Cecil Williams, the latter a black minister fresh from battling segregation in Kansas, to Glide in 1964. The work these men were doing in the Tenderloin was soon supplemented each year by a church intern who also did youth outreach work. Especially key was Ed Hanson, the youth intern from 1965 to 1966.\n\nAfter years of work, the Glide Foundation ministers were able to bring together members of the homophile movement and Tenderloin neighborhood activists to form the Central City Citizens Council (CCCC). The CCCC published the _Tenderloin Report_ , which documented the lack of education among those living in the Tenderloin, as well as the high poverty and the lack of social services and housing. The CCCC also fought for and won a portion of federal poverty program funds that had been designated for San Francisco.\n\nThe funds were used to create the Central City Poverty Program. And so the interns and the minister set about helping the street youth, the runaways, and the hustlers to organize themselves. Although the ministers hired by Glide had worked primarily with black and other ethnic minority populations, they believed that the approach used in those communities should work with a population of inner city youths who were mainly homosexual. The ministers had found that it was much more effective to hire someone from a disadvantaged group to organize that community to help itself rather than bringing in an outsider to do the same work.\n\nNeil Secor, the first intern to work with the Tenderloin youths, began by inviting them to meet weekly in his living room. The first sessions were almost like consciousness-raising sessions, meetings that served for heterosexuals and homosexuals to get to know, trust, and understand each other. Once they were able to work together, Secor got the youths to organize, and then they began to use Glide's offices. When Secor left and Hanson arrived, the youths were ready to formalize their organization as Vanguard.\n\nThe group of mostly gay youths chose the name of Vanguard by July of 1966 and elected one of their own, Jean-Paul Marat (a pseudonym), as Vanguard's president. An issue of _The Berkeley Barb_ , an underground newspaper, gives a thumbnail portrait of Marat as thin and pale, with wavy black hair, reddish cheeks, and thick, babyish lips. Vanguard soon started a magazine, called _V_ , which, like many underground newspapers of the time, featured original and compelling artwork, all done by Marat. The only requirement for membership in the group was that a person be \"a kid off the street.\" Vanguard embraced the belief of self-empowerment advocated by the Glide Foundation's young, idealistic ministers. The first known news article about the group quotes a member as saying, \"We believe we can take care of a large portion of our problems without the interference of the federal government, head shrinkers or older people, most of whom do not at all understand the problems of the kids.\"\n\nA flyer put out by the group describes their understanding of the problems facing them and how to solve them: \"We protest police harassment of youth in the area when the big time speculators seem to work openly and receive _no attention_.... We protest being called 'queer,' 'pillhead,' and being placed in the position of being outlaws and parasites when we are offered no alternatives to this existence.... We demand justice and immediate corrections of the fact that most of the money made in the area is made by the exploitation of youth by so-called normal adults who make a fast buck off situations everyone calls degenerate, perverted and sick.\"\n\nIt is remarkable that Vanguard had from its inception the militantly unapologetic tone that would characterize the gay liberation movement, which would not be born for another three years. _V_ is also remarkable for its common use of the more affirming _gay_ as opposed to _homosexual_ or _homophile_. Similar to the later Gay Liberation Front's (GLF's) emphasis on self-criticism, the youths showed a willingness to address not only society's shortcomings but their own as well.\n\nAn editorial on prejudice by Marat talked about the gay youths' difficulties in accepting one another: \"Day after day I hear complaints about the prejudice that the straight society has against the gay society. Let's look at our own prejudices. We ostracize people because they do this or that or the other in bed. We make snide comments about the drag queen who isn't quite convincing enough. The 'leather boys' are the butt of many jokes and much ridicule.... If we want society to accept us as we are, we are going to have to start accepting ourselves and others like us.\"\n\nBesides self-help articles, listings of organizations to go to for assistance, and articles on politics, _V_ also featured many articles written by the youths about their own experiences. One of the most poignant pieces is a poem titled \"The Hustler\" that was sent in anonymously and published under a photograph of an attractive young man with a resigned and sad expression:\n\nI sell my love for dollars,  \nIf you can spare the time,  \nA hug is but a nickel,  \nA kiss is but a dime.\n\nI'll go to bed for twenty,  \nAll night for just ten more.  \nNow don't get the idea  \nThat I am just a whore.\n\nFor if I didn't sell my love,  \nWhere else would it go?  \nI have no one to give it to;  \nNo one who'd care to know.\n\nSo open up your wallet  \nAnd show me what you've got.  \nAnd whether I should love you  \nA little or a lot.\n\nOne of the few hangouts available to the street youths in the Tenderloin was a branch of a local chain restaurant called Compton's, especially popular with gay youths, hustlers, and transvestites. According to Tamara Ching, \"It was just a place that we would come and hang out. If you didn't manage up a trick that night, it's where you would hang out with your friends. We'd stay there and then eat breakfast and go home and sleep until it was time to get up and go out on the prowl again.\" Crema Ritz even compared Compton's to \"a community center.\"\n\nHistorian Susan Stryker has found that the gay youth were tolerated at Compton's for years because the evening manager was an older effeminate homosexual. When he died suddenly in the spring of 1966 and new management came in, a decision was made to discourage the patronage of these poor customers who sat for long periods of time and ordered little food. The diner hired Pinkerton security guards to harass them, and invited the police inside for the same reason. The security guards insulted the gay and transvestite customers and manhandled those who did not drink their coffee fast enough. On July 18, the newly formed Vanguard organized a picket line at Compton's to protest the harassment and discrimination. Around twenty-five persons picketed in the evening, from ten o'clock until midnight.\n\nOn a hot night the following month, when a policeman grabbed the arm of a transvestite, she threw a cup of coffee in his face. As if on signal, other gay customers began throwing cups, saucers, and trays at the police and security guards. Compton's immediately closed, \"and with that, the gays began breaking out every window in the place.\" As other gay men ran outside to escape the breaking glass, the police tried to catch them and put them in patrol wagons, according to the only known written account of the disturbance. Those leaving Compton's fought hard, with gay men hitting the police in their groins and \"drag-queens smashing them in the face with their extremely heavy purses.\" One police car had all of its windows broken and a newspaper stand close to the restaurant was burned down as \"general havoc [was] raised that night in the Tenderloin.\" The following night, when the restaurant turned away transvestites, another picket line of \"drag-queens, hair fairies [West Coast equivalent of 'scare drag'], conservative Gays, and hustlers\" picketed the cafeteria. The conflict ended when the restaurant's newly installed windows were smashed a second time.\n\nGiven that the riot at Compton's occurred when the gay street youth of the Tenderloin were receiving significant support from Glide Memorial Church ministers, it is reasonable to conclude that the support these youths, who were regularly abused, received from Glide emboldened them to resist oppression by the police and security guards. No record exists of any resistance by gay or transgendered men on a similar scale prior to the event at Compton's. Susan Stryker, who has studied the history of services available to transgendered persons in San Francisco, offers an additional reason for the Compton's incident. Shortly before the disturbance, a doctor in the area became the first in the U.S. to offer sex-change reassignment surgery. Since his office was only five blocks from Compton's and since transgendered persons took a leading role in the disturbance, Stryker concludes that this new service may have played a role in triggering the riots: transgendered men, a highly despised minority within a minority, were being given hope for the first time ever and thus felt assertive enough to resist oppression. *\n\nCraig Rodwell circled around in the 100-degree heat for at least the twentieth time. There was nothing he could do about it being summer or the lack of shade. He had, after all, chosen this time and this place to protest. Having come all the way to Philadelphia for the Annual Reminder, he wasn't going to miss picketing. He wished, though, that he didn't have to wear a coat and tie. But Craig lost this argument every year. He and some others thought that the idea of a strict dress code for a picket demonstration was absurd. Still, others, particularly Frank Kameny, argued that since they were picketing to establish their right to employment, they must look employable. Maybe next year Rodwell could persuade his colleagues to loosen the rules controlling every aspect of their behavior as they picketed. Meanwhile, he just kept circling.\n\nWhen Arthur Evans had moved to Greenwich Village he began to read a lot of poetry, which changed his understanding of life: \"William Blake had a big impact on me, and I got very excited about Blake. His vision of the recovery of sex is related to the struggle against imperialism and industrialism and the machine: the reaching for the organic, getting down to sinewy nerves and muscles that go way back into history before the mind. And they're more than a machine.... It's the living pulse of the planet coming out.\"\n\nWalt Whitman inspired Evans with his celebration of the body electric and his vision of democracy: \"Whitman speaks of democracy as spiritual practice, something that you have to live and practice the way you would practice a musical instrument, with the same degree of passion and commitment. That sensibility, I think, is in Walt Whitman and it involves sex as well.\"\n\nWhen Evans chanced upon Allen Ginsberg's work, he recognized a kindred spirit to his beloved Whitman: \"Ginsberg has influenced my basic perceptions of things more so than any other living thinker, but it's hard to put it into conceptual words because it's more than that. It affects feelings as well. I was electrified by his poetry. First of all because I thought it was very powerful poetry, and secondly because I thought he was drawing on things that were very important to me. The Vietnam War was not just a conflict. It was saying something significant about American history and it was saying something important about human spirituality. Where did the Vietnam War come from? It came from America's spiritual history and from its sexual history. These things\u2014sex, politics, history, personal life\u2014all these things are interconnected, and Allen Ginsberg, as I saw it, was articulating this vision, expressing these connections between things.\"\n\nFor Evans, as for so many others of his generation, the national conflict over the Vietnam War did much to define him. In 1967 he had enrolled in Columbia University's doctoral program in philosophy. Evans loved philosophy and no doubt in quieter times would have completed his Ph.D. in several years, but Columbia University was, like many others, roiled by the decade's political upheavals. Columbia had claimed it was not supporting the war in Vietnam, but the Students for a Democratic Society (SDS) did research that proved that Columbia was lying, which was one of the reasons for the uprisings there.\n\nDuring the student occupation of the campus in May of 1968, Grayson Kirk, the university's president, chose the middle of the night to call in the police, who beat hundreds of students and faculty members. Although Evans did not happen to be present when the police violence occurred, the next day he saw lots of professors and students going around campus with bandages on their heads, a sight that changed Evans: \"That was a radicalizing experience. At that point I realized the administration of this university is the enemy. They're part of the military-industrial complex. These people are not supporters of learning. They are not my friends.\"\n\nIn August Arthur went to Chicago to protest the war in Vietnam at the Democratic Convention. Many thousands of protesters were heading to the Windy City, for it seemed certain that Lyndon Johnson's vice president, Hubert Humphrey, would get the Democratic Party's nomination to be president although antiwar candidate Senator Eugene McCarthy had won more votes in the primaries than Humphrey had. Mayor Richard Daley, the quintessential machine party politician, was determined to prevent the protesters from demonstrating.\n\nIn the weeks before the convention Allen Ginsberg had had an increasingly bad feeling about how things would turn out in Chicago. He had been one of those who had suggested holding the Festival of Life in Chicago as a counterexample to Democratic Party machine politics, and still felt that he should attend in hopes of minimizing the violence that seemed likely. He and longtime colleague Ed Sanders held exercises in Chicago to train protesters in breathing and chanting techniques that if\u2014and only if, Ginsberg well knew\u2014practiced by a large portion of a crowd could help maintain calm when violence and confusion threatened to erupt. Ginsberg also paid attention to more mundane matters, such as meeting with city officials to try to get a permit for a sound system so that those trying to maintain calm could communicate with those protesting.\n\nAs Evans took part in a protest in one of Chicago's parks and was attacked by the police, he noted the event's surreal quality. Simultaneous with the ground attack, he saw tear gas canisters flying through the air with gas beginning to emerge from them even as he heard Ginsberg chanting the Hindu mantra \"Om.\"\n\nEvans found that Ginsberg's chanting \"gave it a sense of significance, that this wasn't just a bunch of people running around. Very important values were at stake here, about meaning and life and our proper role in history and how to deal with violence. All these issues were hanging in the air, and he sort of put that into context by that very simple gesture. These demonstrations were a meditative act, and we were crossing the boundary between politics and spirituality.\"\n\nIn 1968 at the Stonewall Inn one night, Danny Garvin met a man named Craig who invited him to visit him in the gay commune where he lived at the corner of Bleecker Street and Sixth Avenue. This commune, having no grand overarching vision or structure, was rather nonchalant, mainly a place where young gay men and their friends lived and hung out while enjoying hippie culture.\n\nYet the commune was not simply a gay male crash pad. Its members discussed and debated the great issues of the day such as the war in Vietnam, women's rights, consciousness-raising, and the legalization of marijuana. Their standard reading material was the local alternative publications, _The Village Voice_ and the _East Village Other_. Members and their friends listened to albums by the favorite musicians of the day, such as the Beatles, Buffy Sainte-Marie, and the Jefferson Airplane. At other times, a friend might drop by with a guitar and everyone would sit around singing popular songs as they shared bottles of wine.\n\nDanny remembers the end of the sixties as a time of rapid changes: \"All of a sudden men were growing longer sideburns. It was happening throughout society, but it was happening in the gay community a little bit quicker. Gay men all of a sudden would wear bell-bottoms. Clothes became unisex. Men wore hip-huggers. Jewelry became somewhat unisexual, too, because you had men and women wearing love beads and chains and bells.\" When Danny went wearing a bell to meet Frank, the man he had first danced with, Frank began to lose interest. \"You're not becoming one of those, are you?\" he asked.\n\nRemarkably, although the members of the commune had debated the war in Vietnam and the oppression of women they had never discussed the meaning of being gay, even though they lived in a gay commune. Danny recalls that even at that time he figured that he would eventually \"settle down with a woman.\" Garvin attributes the lack of such political awareness among his gay hippie friends to just how oppressive the situation was for homosexuals. Garvin recalls a practice, common at the time, called dropping dimes, in which one gay man betrays another by calling his parents and telling them that their son is gay. \"I knew gay men, usually in their later twenties or early thirties, who eventually got married and would just all of a sudden step out of gay life.\" The generation gap also affected gay men: the older gay men in suits and ties presented an image rejected by young gay hippies in the Village.\n\nGay men suffered from self-hatred as well. While pop and rock music celebrated (hetero)sexuality with joy and abandon, gay men still gathered in bars to listen to \"The Ballad of the Sad Young Men,\" a song that portrayed gay life as morbidly as its title suggests. They also threw acidic lines from Edward Albee's _Who's Afraid of Virginia Woolf?_ at one another and often quoted the depressing line \"Show me a happy homosexual and I'll show you a gay corpse\" from an Off Broadway play of the period, _The Boys in the Band_. Danny recalls that \"gay life didn't seem a very positive thing to grow old into.\"\n\nThe Stonewall Inn had modified its decor in response to the changing times, though only in the front room. Black lights were added to the dance floor in the first room and a light show was installed behind the bar: lights underneath lit up dark and light squares in a checkerboard pattern. But the main innovation was go-go boys who danced on weekends in gilded cages at each end of the bar. As they danced in paisley or silver or gold lam\u00e9 swimsuits or bikini underwear, light projections swam over their nude bodies. Their cages were perched on top of rickety platforms behind the bar where customers could see but not reach them.\n\nAccording to a number of witnesses, though not all, the clientele at the Stonewall Inn had changed with the passage of time as well. Edmund White felt that the patrons included more blacks and Latinos and that drugs had become much more popular at the club. Two persons interviewed by Tina Crosby in the early seventies told her that \"by 1969 it was getting a 'burnt-out feeling' and was past the height of its popularity.\" However, most persons from the era recall few, if any, changes in clientele over the years. The _1969 New York City Gay Scene Guide_ , published by the Mattachine Society, wrote: \"It continues operating amid persistent rumors of closing. Observers note that 'go-go boys'... have failed to attract the dwindling crowds.\" But maybe this was wishful thinking on the Mattachine's part. After all, Rodwell had written practically the same thing a year before in the June\u2013July 1968 issue of the _Hymnal_ : \"The Stone Wall... is still in operation, unfortunately. Hoping to save their declining business, the Mafia management instituted 'go-go boys' on platforms.\"\n\nAs the decade approached its end\u2014and especially in 1969\u2014the subject of homosexuality was more and more in the air, whether in the context of the arts or political events or merely as a topic of conversation. To give but several examples, in mid-July of 1968 _The Wall Street Journal_ published a lead front-page article that noted the growing militancy among homosexuals who were fighting for \"a piece of the action\" in America. The response to the article was so strong that a later issue devoted an entire \"Letters to the Editor\" column to correspondence about the article. Craig Rodwell highlighted the article in the _Hymnal_ , noting: \"It is said that _The Wall Street Journal_ is six months ahead of society in indicating trends.\" In mid-1968, Criswell, the television psychic, appearing as a guest on the _Tonight_ show, predicted that within several years exclusively homosexual communities would spring up throughout the country. Whereas in 1962 Randy Wicker had made a splash in the media by going on the radio as an openly gay man, before 1968 ended, a group of gay men in New York City that included Bill Weaver and Charles Pitts began broadcasting on WBAI a weekly program on homosexuality called _The New Symposium_ , \"with the aim of inspiring a sense of social identification within our subculture.\" Every member of the program staff appeared under his own name.\n\nIn January 1969, _Time_ magazine published the first cover story by a national magazine on homosexuality, and _Hair_ , whose lyrics contained several positive homosexual references, became the bestselling record album of the year.\n\nGay folk even took the number of the year as a hopeful symbol, sixty-nine long having been a code word for homosexuality, based on reading the numbers as a depiction of a gay couple engaging in mutual oral sex.\n\nEven _Hair_ 's most popular song, which proclaimed \"the dawning of the age of Aquarius\" as an age of \"harmony and understanding,\" has as its source, no matter how few were aware of it, a myth that celebrated homosexual love. The age of Aquarius is named for the Greek server at the banquet of the gods who bears liquid refreshment to the deities, whence his popular astrological designation as the Water Bearer. But the story of Aquarius is that of Ganymede, who, in Greek mythology, was the most beautiful youth on earth. His beauty was so enchanting that even Zeus himself could not resist his charms, and the supreme god of the Greeks, taking the form of an eagle, flew down to earth, seized him, and transported him back to Olympus. After this \"rape,\" Ganymede served the Greek deities when they dined, while Hera, Zeus's consort, grew jealous as Zeus paid more attention to the beautiful youth than to her. But being a mortal, Ganymede did eventually die. Zeus's love for the youth was so great, however, that rather than let him go the way of all flesh, he bestowed immortality upon Ganymede by transforming him into the constellation known as Aquarius.\n\nEarl Galvin recalled how when the calendar turned to 1969, gay men began to say to one another, \"Maybe this will be our year.\" The _New York Mattachine Newletter_ 's gossip columnist, D.D., noted that according to Chinese astrology, in which each year is associated with an animal, 1969 was the year of the rooster, which pleased D.D., who saltily commented that a \"year of the cock\" sounded appealing. If the world was on the threshold of entering an age named to celebrate same-sex love, the coinciding of a rooster year could be portentous, for according to the Chinese astrological system, a rooster year is one in which \"politically and domestically, the downtrodden make their voices heard, the aggrieved or tyrannized will stand up for themselves, and bullies will now get their due comeuppance.\"\n\nIn New York City, apart from steady progress on court rulings that continually ate away at the SLA's ability to revoke or withhold liquor licenses from gay bars, there were few indications of progress. For example, although the New York Police Department had ceased entrapping homosexual men, the Transit Police, who were not controlled by the mayor's office, continued the practice. Then an off-duty Transit cop, Colin Kelly, killed a gay man, John Allison, and another gay man who was with him at the trucks near the waterfront. The circumstances of the killings and subsequent police conduct caused the Mattachine Society to suspect that the men had been murdered. The February _New York Mattachine Newsletter_ reported that two Transit cops, after having arrested three men in a washroom, emptied out their wallets. When one of the men protested, the cop asked him how he would like to be carried out of the washroom with a \"shiv\" in his body. (A shiv is the crude but deadly handmade knife prisoners fashion.) The same newsletter recounted how another Mattachine member had stopped at 1:15 in the morning to talk with some friends near the Stonewall Inn. When two police officers came along and told everyone to leave, he refused to comply, pointing out that the group was neither loud nor obstructing traffic, whereupon he was handcuffed and taken to jail.\n\nJust as the Transit cop who had shot and killed two gay men at the trucks seemed about to go free, there was violence and extortion at the waterfront. Trucking firms and pier owners had hired private policemen to protect their property and vehicles from being damaged by the gay men who used them for sex, but some of the private detectives shook down the gay men. Mattachine\u2013New York tried to alert others to the dangers of pickpockets, muggers, and blackmailers: \"The area has become a mecca for up-tight hoodlums looking for a 'queer' to beat up. One of their favorite games is to shove a homosexual into the cesspool known as the Hudson River. [...] At least four people have drowned in the filth after hitting their heads on pier footings.\" Another man who had performed sex on \"an evidently willing partner\" was beaten so badly afterward that he lost an eye. Urging gay men to stay away from the trucks to avoid physical danger as well as the chance of arrest, the Mattachine Society suggested the baths as an alternative for those compelled to seek out anonymous sex.\n\nIronically, after Mattachine\u2013New York had urged gay men who felt a need for anonymity to seek it in the baths rather than on the waterfront, the next issue of the organization's newsletter carried the lead headline \"Bathhouse Raided.\" The Continental Baths, where Bette Midler would be discovered a few years later, was already becoming a fixture of New York City's gay scene. On February 20 it was raided by the police and twenty-two men were arrested. Before the police left, they wrecked the bathhouse. The bathhouse owners said that the raid had happened because they had refused to pay off the police. The newsletter also reported that Colin Kelly had been absolved of any guilt in the killing of the gay men on the waterfront.\n\nOn the West Coast, militancy was increasing far beyond that displayed in previous years by SIR and the Tavern Guild. Leo Laurence, a thirty-six-year-old reporter for San Francisco's KGO radio, had covered the protests at the 1968 Democratic convention. Witnessing young antiwar protesters putting their bodies on the line had made him a radical. Laurence began writing for both _The Berkeley Barb_ , an underground newspaper, and SIR's _Vector_. Early in 1969 Laurence was elected by SIR's membership to edit _Vector_ , and he wrote at the time that \"this is the beginning of a new revolution in San Francisco, the Homosexual Revolution of 1969. When the black man became proud, he became more militant. That same power is starting to hit the homosexual movement in the Bay Area.\" Laurence called for gay people to form coalitions with other militant and radical groups such as the Black Panthers and the antiwar movement.\n\nBecause Laurence believed it was important to be honest about one's sexuality, he had a picture of himself and his lover, Gale Whittington, with the latter shirtless and Laurence embracing him, published in the _Barb_. Gale, who worked as an accounting clerk at the States Steamship Line, was immediately fired from his job. Laurence soon lost his position as editor of _Vector_ as well after he characterized West Coast homosexual organization leaders as \"timid... middle-aged up-tight conservatives.\" The article was titled \"Gay Revolution\" and Laurence complained that such leaders were \"hurting almost every major homosexual organization on the West Coast and probably throughout the nation.\" While they loved to say, \"Gay is good,\" at gay meetings, Laurence complained, they were afraid to say it in public, which to him only made them so many hypocrites. \"About the only people with that kind of courage are the new breed of young gay kids. And that's just why organizations like SIR keep them out. The old-timers are scared that these kids will come in and really create a gay revolution.\"\n\nFired from his editorship position and thrown out of SIR, Laurence started a new organization called the Committee for Homosexual Freedom (CHF). Far from being intimidated by what he had experienced, Laurence kept up the attack. \"We are organizing a campaign that will show the State Lines, or any other employer, that they don't indiscriminately fire homosexuals who are good employees and get away with it.... The social revolution that is sweeping the country has given new pride to the Blacks and is now giving fire to the homosexuals,\" Laurence predicted in the _Barb_ 's pages. After demanding that State Lines rehire Gale, he said: \"If they don't, militant homosexuals will show that company what Gay Power really means.\" He finished with a prediction: \"The public has a big surprise coming this year if they think they can push homosexuals around and get away with it.\" Laurence appealed for help: CHF needed people to make signs and posters as well as to demonstrate.\n\nSoon dozens of homosexuals under twenty-five were showing up to help, as well as heterosexuals. It was decided to mount a picket line in front of the State Lines' offices in San Francisco's financial district. The picket demonstration would be held every weekday starting at noon for one hour. On the first day of the picket line over fifty people demonstrated.\n\nState Lines refused even to meet with CHF to discuss their demands. The picket line grew and soon a young minister from Los Angeles by the name of Troy Perry was trying to organize support from that city. Only six months prior Perry had held the first service for a church he was forming in Los Angeles for homosexuals. Spirits remained high on the picket line, as freedom songs rang out, more office workers came to join in, and blacks driving by raised their fists in solidarity. One man organized a phone-in, urging supporters to call States Lines every day and ask for a different department. Michael Cooke, a twenty-nine-year-old who had been thrown out of the University of Texas in 1961 for making a speech about homosexuality, said, \"Fear and intimidation have ruled the gay world for two thousand years. The only legacy this has brought me is the feeling I have precious little to lose. The time is ripe for some militancy.\" Just how much the social and political upheavals of the previous few years had prompted the protest at State Lines was shown not only by Laurence's own experiences in Chicago but also by the presence on the picket lines of gay people who had participated in antidraft groups, the Stanford Sit-In, and a strike at San Francisco State College, as well as other veterans of the bloody events of 1968 Chicago.\n\nAs the State Lines picket demonstration gained momentum, a gay man died after he was shot in the head by a plainclothes police officer who had tried to entrap him in a Berkeley park. When the district attorney indicated that no inquest was necessary, Larry Littlejohn, SIR's president, said, \"It looks like an official coverup.\" A mock funeral motorcade was organized from Glide Methodist Church to the Berkeley Hall of Justice to protest the killing, as the CHF picket line grew grim... and spread to Los Angeles, where Troy Perry led fifteen marchers (mostly young gay men and a number of heterosexual theology students) who attracted a crowd of two hundred onlookers. Reverend Perry observed that with so many onlookers, \"amazed at the courage of our people,\" the picket demonstration \"was the biggest thing in downtown L.A.\"\n\nWhen a Tower Records store in San Francisco fired a clerk, Frank Denaro, on the mere suspicion that he was homosexual, CHF launched a weekend picket at the store specifically geared to stop customers from going inside. The tactic proved effective, as a large number of customers stopped at the picket line, then climbed back inside their cars and drove away.\n\nIn May, a young leftist in San Francisco named Carl Wittman took note of all the militant actions occurring in the city and sat down to write about the events' implications. Before a year had passed, the essay, eventually titled \"A Gay Manifesto,\" would become one of the defining documents of the gay liberation movement. In it Wittman wrote:\n\nSan Francisco is a refugee camp of homosexuals. We have fled here from every part of the nation, and like refugees elsewhere, we came not because it is so great here, but because it was so bad there....\n\nAnd we have formed a ghetto, out of self-protection. It is a ghetto rather than a free territory because it is still theirs. Straight cops patrol us, straight legislators govern us, straight employers keep us in line, straight money exploits us. We have pretended that everything is OK, because we haven't been able to see how to change it\u2014we've been afraid.\n\nIn the past year there has been an awakening of gay liberation ideas and energy. How it began we don't know; maybe we were inspired by black people and their freedom movement; we learned how to stop pretending from the hip revolution....\n\nWhere once there was frustration, alienation, and cynicism, there are new characteristics among us. We are full of love for each other and are showing it; we are full of anger at what has been done to us. And as we recall all the self-censorship and repression for so many years, a reservoir of tears pours out of our eyes. And we are euphoric, high, with the initial flourish of a movement.\n\nIn New York City, the month of May found Mayor Lindsay facing a tough reelection battle. Running against Lindsay for the Republican Party nomination was state legislator John J. Marchi, whom Mattachine\u2013New York described as \"one of the bitterest opponents of homosexual law reform.\" The Democratic candidates were not generally progressive, and some of the better-known candidates included Robert Wagner, whose dismal record regarding homosexuals was only too well known, and Mario Procaccino, who was such a \"law and order\" candidate that even his fellow Democrats considered him repressive.\n\nOn June 17, Lindsay lost his own party's primary to Marchi. The race was thrown into a confused frenzy as Lindsay realigned himself and sought to save his political life by running a fusion campaign, with his name appearing on both the Liberal and Independent parties' lines. Mattachine\u2013New York reported that Marchi's victory speech was \"a panegyric to 'old values' of cleanliness and godliness,\" with much of it sounding like \"a veiled warning to the sexually unorthodox, the minority groups, the people who don't abide by the 'old morality,' and others, to run for the hills.\" Procaccino then tried \"to out-reactionary Marchi\" and succeeded to such an extent that \"he alarmed even former Mayor Wagner.\" In this context, as a growing number of bar raids took place, New York's gay population wondered if it was not seeing a return to the bad old days they had always lived through at election time.\n\nWhat came to be called the counterculture had many manifestations, from communes, to alternative foods, to underground newspapers. One of its characteristic innovations was the alternative schools that sprang up all across America. These organizations were usually free or low-priced and very modest in scope. One of the best-known in New York City was Alternate U., a free school and organizing center founded around 1966 by Tom Wodetski. Located on the Village's northern edge at the corner of 14th Street and Sixth Avenue, its several classrooms and one office inhabited a former dance studio on the corner building's second floor. The school emphasized action, not academics, and encouraged its students to take the skills and ideas they had learned\u2014whether in a class in Marxist theory taught by Stanley Aronowitz or in community organizing by Flo Kennedy\u2014out into the world in order to change it. Although freewheeling, Alternate U. did have a board.\n\nIn early 1969, John O'Brien volunteered at Alternate U. and was soon asked to join the board. He accepted the invitation in spite of being preoccupied with sorting out his feelings about being gay. His sexuality predominated his thoughts, for he had recently been thrown out of the Young Socialists Alliance (YSA), the youth group of the Marxist Socialist Workers Party (SWP), because he was gay.\n\nHe remembered how on his birthday in January 1969 he had been brought in for questioning by the organizers of the New York chapters of the YSA and the SWP. He could tell that they were reluctant to believe that he was homosexual: not only was he a hard-working party member, but his very masculine demeanor\u2014including a very muscular physique\u2014did not fit the gay stereotype. He also knew he could deny the accusation and they would believe him and the matter would be forgotten. But for O'Brien this was the moment of truth. He decided not to compromise his integrity and was summarily thrown out.\n\nThis expulsion upset him, for he had devoted his life since his early teen years to progressive and revolutionary causes. No one in the New Left could doubt his dedication: he had been demonstrating and fighting the police in the streets since his youth, when he had sneaked away from home to demonstrate for black civil rights in Alabama and returned with his hand scarred by a police dog. During the student occupation of Columbia University the SDS had put him in charge of security for one of the occupied buildings. O'Brien often pulled these sorts of dangerous assignments not only because of his muscles, but also because growing up in Spanish Harlem had made him tough.\n\nO'Brien had not served on Alternate U.'s board long when he met a young man by the name of Bill Katzenberg. Bill also belonged to the SDS and had been referred to John by Alternate U. after showing up there saying he wanted to start a gay radical group. But starting a gay anything was about the last thing O'Brien wanted to consider given his previous experiences with gay groups: He had tried to join the picket line at one of the Annual Reminders and had not been allowed to because his clothing did not fit the dress code. A gay student group at Columbia University, the nation's first, had picketed outside one of the buildings there during the student occupation. O'Brien had joyously joined them, but the students had ignored him. Perhaps they had suspected him of being a spy or a provocateur because he didn't look gay. Sex was the only gay thing that seemed to work for O'Brien.\n\nKatzenberg struck O'Brien as being both highly motivated and sincere. He told O'Brien that he wanted to organize radical gay people into a group and that he wanted the group to do more than just complain: it should be a group geared toward action. O'Brien was tempted but decided not to join. He would lend his name to help sponsor the group at Alternate U., and he would help them find a place to meet. Other than that, Katzenberg would have to sink or swim on his own.\n\nBut Katzenberg continued to come to Alternate U. to talk to O'Brien and finally managed to convince him to join the new group. A turning point came when Katzenberg introduced O'Brien to Jerry Hoose, who was very much a part of the sex scene at the trucks and the cruising scene on Christopher Street. Hoose, O'Brien recalls, seemed to know everyone who hung out on Christopher Street. According to O'Brien, in the spring or summer of 1969 Hoose supported the idea of a new group and introduced him and Katzenberg to other people on the street with whom they talked about gay oppression.\n\nOne of the men Katzenberg conversed with was a lithe and handsome young carpenter from Brooklyn by the name of Marty Robinson, who had no shyness about being gay. When his prominent parents had offered him a trip to Europe if he would renounce his homosexuality, he had not thought twice before turning them down. His \"natural aggressiveness\" had caused Katzenberg to approach him about creating a \"new butch image for male homosexuals.\" But Marty had already experimented with the extremes of acting \"ultrabutch\" as well as \"a little screaming and camping\" before going back to just being himself. He turned down Katzenberg's proposal to start a group called the Pink Panthers, because \"[t]he whole idea was a big turn-off to me. There's nothing worse than to try to be a 1950's butch!\"\n\nO'Brien recalls, \"We had a number of discussions, most of them actually standing on street corners on Christopher Street, mostly on the northwest corner of Bleeker. We'd stand there, sometimes on the stoops a little further down, sometimes in the Silver Dollar Restaurant, which was a big hangout for us. We met there. We started, the three of us, then were able to recruit a couple more people, becoming like five people, and we agreed on putting an ad in the paper and finally having a meeting.\" When _The Village Voice_ refused the ad, O'Brien and Katzenberg went to the SDS-affiliated newspaper _Rat_ , which accepted the small classified ad, which announced: \"A group of young radical homosexuals will meet... to develop a critique of heterosexual supremacy, both in society and within the Movement.\"\n\nMattachine\u2013New York's June newsletter brought plenty of bad news for its readers. The body of a man, killed by strangulation, had been pulled out of the Hudson, appearing to be yet another \"dock scene\" victim. In the previous month and a half, three men, apparently gay, had been robbed and murdered on Manhattan's East Side. In May a new gay private club had opened in the East Fifties without any licenses. When the police raided it and arrested the management, the police were well within the law, but they then illegally arrested every customer and charged them with \"disorderly conduct.\" The police also issued a warning: in the future they intended to arrest any person they found in an unlicensed club. Even D.D.'s gossip column had bad news: the police were unusually active in the public parks and the Hilton Hotel was \"hot with private dicks.\"\n\nIn light of so much bad news, the newsletter's lead story\u2014that the New York City's Civil Service Commission (CSC) had grudgingly, after fighting for two years in court, agreed that homosexuality was not an \"absolute disqualification\" for holding city jobs\u2014offered little reason for celebrating. All the more since although this ruling only affected hiring by New York City itself, there were still a number of exceptions to the new policy. The Welfare Department was exempt from this ruling, and the CSC itself quickly pointed out that \"it would probably refuse to hire a homosexual as a policeman or fireman.\" The CSC spokesman then added, \"An admitted homosexual, when the acts are frequent and recent, would probably not be qualified to be a guard in city penitentiaries, a children's counselor, or a playground attendant.\"\n\nAs the newsletter reported that Canada was in the process of legalizing homosexual sex and that West Germany's new laws making gay sex legal would probably take effect by Labor Day, it informed New Yorkers that both _The Village Voice_ and _The New York Times_ had refused to accept ads for _The Homosexual Handbook_ , which had just sold out its second printing of 50,000 copies. While New York's homosexuals read of how U.S. allies, including Canada, were legalizing homosexual sex, they themselves were being fired, blackmailed, beaten, knifed, shot, strangled, thrown into the Hudson, and killed. Meanwhile, the most liberal city government in a decade couldn't bring itself, even after a two-year court battle, to think homosexuals fit to fight fires or work in penitentiaries, and two of the city's most liberal publications would not even run an ad for a book about homosexuality.\n\nToward the end of June word spread through New York's gay community of vigilantism in Queens. A number of homosexual men had found a convenient trysting spot in Kew Gardens, a public park. When men in the neighborhood took umbrage and decided they needed \"to protect their wives and children\" (even though none were ever in the park in the middle of the night), they formed a vigilante committee to harass the men who went there. Using walkie-talkies to coordinate their efforts, a group of as many as forty men would patrol the park, where they surrounded and confronted gay men, shining bright lights in their faces and ordering them to leave. But gay men were not so easily cowed by the decade's end and some refused, asserting their legal right to be on public land. Neighborhood youths also took to going to the park and beating and robbing the homosexuals they found there. When gay men still showed up, the vigilantes got saws and axes and, a night or two after Lindsay lost the primary, chopped down the park's trees. When a concerned citizen saw the mayhem going on, he admonished one of the vigilantes who swung an ax at him. When the man reported the attempted violence to the police, they did nothing to stop the cutting. Other citizens reported the trees' destruction, only to see police cars drive up, and an officer get out and chat with the vigilantes, then leave without taking any action.\n\nIn Greenwich Village, there was no lack of police action at the end of June, as five gay bars were raided over three weeks. Three of the five bars raided were among the most popular gay spots in town, and several bars\u2014the Checkerboard, the Tel-Star, and the Sewer\u2014were closed for good.\n\nIn April _The Advocate_ had announced the appearance of a new gay publication called _Queen's Quarterly_. Started by friends of Craig Rodwell's, the magazine had a philosophy of gay pride and empowerment that practically jumped off the magazine's pages at the reader, starting with its attractive art direction and continuing on through its editorials to articles about gay sexuality that were both frank and positive. The magazine's tenor even prompted heterosexual _Village Voice_ reporter Howard Smith to refer to its readers as \"healthy homosexuals.\"\n\nThe magazine's third issue, which came out in the summer, featured a black-and-white photograph of a beautiful youth made by the famed early-twentieth-century homosexual photographer, the Baron von Gloeden. The photograph of the Sicilian youth was surrounded by yellow and purple pansies as if to accentuate the youth's beauty while subversively standing on its head the use of the flower's name to denigrate gay men. The summer issue featured a fashion spread on sunglasses, a discussion of wine, an article on sadomasochism, an interview with transvestite actress Mario Montez, vacation tips, and an article on how to make a gay marriage last, as well as an article on how to defend oneself if attacked. The article on self-defense included photographs of a nude man with the most vulnerable parts of the anatomy labeled, from the testicles, to the Adam's apple, to the bridge of the nose, and advised that if the reader had \"the time, inclination, and dough, by all means\u2014learn Karate and Judo.\"\n\nThe lead editorial, titled \"Dawn Is Just Breaking..., \" announced:\n\nQQ wants to... present a total picture of gay life\u2014but we're happy being ourselves.... [W]e cannot be something we are not.hellip; We must be ourselves.... Our message is simple: Stop apologizing for yourself... face the facts and accept yourself as you are... there is a real place for us in this world... be proud and excited about being gay\u2014we have earned our place in society and it must now learn its lesson that we are here to stay and that our voice is loud and strong. Dawn is just breaking for gay guys... this is the age of the gay renaissance.\n\nBy the end of June 1969, Pine's and Smythe's campaign to close down Mafia gay bars in the Village had been in full swing for some time. On Tuesday, June 24, Seymour Pine led a raid on the Stonewall Inn, arresting bar personnel and confiscating the bar's liquor. There was a lot of resentment about this raid on the part of gay men in the Village who complained about it over the following nights. Ronnie Di Brienza, a twenty-six-year-old musician, who apparently was inside the Stonewall Inn during the Tuesday night raid, described his feelings and thoughts about the event:\n\nI have had a lot of shit thrown my way, but... I was basically a pacifist. However... how many times can one turn the other cheek?...\n\nBasically, I am not gay, but I am not straight either...\n\nThe establishment and their elite Gestapo, the pigs, have been running things too long. First you had the Negro riots a few years back, which woke up white cats like myself to the fact that, though I am white, I am just as much considered a nigger as the black man is. From those early battles came the more intense militant organizations who, like myself, are sick and tired of being niggers, and want to become real and human. We have reached the bottom of the oppressed minority barrel... gay people... too, have turned the other cheek once too often.\n\nDi Brienza summed up the mood in the Village after the Tuesday raid by saying: \"Predominantly the theme was 'this shit has got to stop!'\"\n\nAs Pine prepared to leave the Stonewall Inn at the evening's end one of the bar's owners sneered, \"If you want to make a bust, that's your business. We'll be open again tomorrow.\"\n\nThe words stung Pine, in part because he knew they were true. But they also sounded to him like a challenge. He would be back.\n\n*This account of the history of Vanguard, the Council on Religion and the Homosexual, and the Compton's revolt is based upon unpublished research by Paul Gabriel and Susan Stryker.\n\n# T H E\n\n# S T O N E W A L L\n\n# R I O T S\n\n#\n\n# _A Friday Night Out_\n\n\"Steve\" Ritter felt on top of the world as he began the day on Friday, June 27. Not only was it his birthday, but he turned eighteen that day, so he knew when he woke up that legally he had reached adulthood: he still couldn't vote, but he could be drafted, and he could drink. Other major social and personal milestones marked this legal coming-of-age: he would graduate from high school in three days; he had found a lover, a muscle builder who wanted Steve to move in with him so they could live as \"man and wife\"; and later that same night Steve was starting a job as a waiter at the Tenth of Always. It didn't bother Steve much that he'd be working in an after-hours place, for he was about to begin his first full-time job, and one at which he could dress in the women's clothes he felt at home in. Besides, the muscle builder had even promised to pay for the expensive process of going through sex reassignment surgery, which meant that Steve would no longer have to hide who he was.\n\nBut tonight he was going to have the big blowout he'd been planning for the entire past week with Kiki, his cross-dressing hairstylist friend: they would both go out \"dressed\" that night to the Stonewall Inn to celebrate Steve's coming of age. Steve had been to the Stonewall and had always gotten in using phony ID, but he had registered for the draft weeks before as required by law so that on the day he turned eighteen he could begin to carry his draft card as required. Tonight he would impress the doormen at the Stonewall by showing them some real ID.\n\nSteve spent the better part of the day getting everything he would need to dress at Kiki's ready. He went to his mother's closet and surveyed her outfits. Luckily for Steve, his mother was a full-figured woman, making it possible for him to fit into her clothes. He selected a really nice empire-waisted black-and-white cocktail dress. Almost sleeveless, the high-waisted dress had a V-shaped back that came down to the middle of his spine and featured a more modest V-shape in the front, while the garment's hem stopped several inches above the knee. All in all, the outfit combined a sophisticated look with sexiness, making it the perfect choice for the evening.\n\nSteve took the dress to his bedroom, where he already had a pair of shoes borrowed from a friend: strapless black shoes with four-inch heels. In addition to the shoes and dress, Steve had assembled a pair of black stockings, some Cover Girl makeup that he had bought for himself, and other makeup he had stolen from his mother. Steve had longish hair, which would help it blend in with the fall he had purchased. He also had a large pocketbook, so that he could roll up a pair of \"boy's clothes\" and hide them there in case he needed to change in a park on the way home.\n\nSteve arrived at Kiki's house in Brooklyn Heights in the late afternoon. They had decided to meet and dress here, since Kiki's neighborhood was a bit more tolerant than Steve's.\n\nAfter exchanging greetings, the two friends began the pleasurable task of slowly transforming themselves into the women they felt themselves to be. For Steve the evening at Kiki's was one of simple fun and innocent self-indulgence: \"I just spent time making up. It would take us hours, and at that time we painted for the gods: it would take us three or four hours to make up.\" However, the preparations were not lengthy simply because of the infinite care the boys took in applying their makeup: \"We spent a whole lot of time kiki'ing around\u2014fooling around. You'd get your hair set. We'd sit around slinging a few cocktails. I was drinking gin and tonic, then vodka and orange juice. We were slamming them back as we were getting dressed.\"\n\nHours later, having completed the transformation that left her male identity as Steve behind, \"Maria\" picked up her pocketbook and walked out the door with Kiki when the car service they had ordered arrived. Maria remembers, \"I was nervous. People would say I was a drag queen, but I always felt like a woman, so this was more natural dress than I'd usually be in.\" But nervousness was not the dominant emotion Maria was feeling that night as the car sped toward Manhattan. \"I felt like I had finally come of age and that I was growing up. I was going to be able to finally live the life I wanted to live. There was an exhilaration because I was growing up and finally getting some freedom.\"\n\nAs Friday evening came to a close, Deputy Inspector Seymour Pine was also feeling pretty good. He had seen to it that the evening's raid on the Stonewall Inn packed some extra punch. He was tired of busting these Mafia clubs only to see them reopen the following day. And now the Mafia managers had thrown a challenge in his face, one that still stung. Well, he would bust them good this time. It may not be a stake through the heart, but he'd bet it would take the Stonewall more than twenty-four hours to reopen with its bars sliced into pieces and hauled out the door. He already had the search warrant that Kenneth Convoy, the district attorney assigned to Public Morals, had gotten Judge Schawn to issue the day before, authorizing Pine to search the premises, seize alcohol, and have the bars cut up and removed along with the Stonewall's vending equipment.\n\nPine had taken the unusual step of getting a warrant to prevent one of the Mafia's lawyers from attempting to use the lack of one as a technicality to have charges against the Stonewall Inn dismissed. He had also asked the city to send an inspector from the Department of Consumer Affairs, which they had promised to do. Having a city inspector along should strengthen the case, for while Pine was capable of citing the Stonewall for infractions of any city laws, it looked better in court if it was done by the city's own experts in overcrowding and the cabaret laws, the regulations that governed nightclubs. As usual, he had requested a federal agent from the Bureau of Alcohol, Tobacco and Firearms. While the Bureau did not always have an agent to spare, they did this night, for Pine's office had received a message stating that one of their agents would be there, which was welcome news. He had no doubt the Stonewall Inn was watering down its liquor, which was a violation of federal law. Federal charges were more serious than state violations, and if the local courts did not take these charges seriously, federal judges might. And while the state courts were subject to local political pressure, the federal courts were more likely to be beyond local suasion. Besides, the more charges, the better it looked in court. Maybe if he hit the Stonewall Inn often enough and hard enough, he just might succeed in shutting it down for good.\n\nWhile Pine needed the policewomen he had requested for this raid, he particularly hated using women undercover, for it always seemed to him that they faced even greater risk than his men. The women went inside in advance, in part so that they could say in court exactly who did what: who mixed and poured the drinks, who served them, which of the men inside the club was giving orders, and so forth. While Pine's undercover men could do that, it was easier to get the women inside the clubs. But he especially needed the women to assist in the arrest of any transvestites. Any men the agents snared in women's clothes would be examined to determine if they were simply wearing women's clothes or were transsexuals who had undergone a sex change: if they had had the operation they would not be arrested. As part of their job, the women were expected to mingle with the crowd and get friendly. And if a Mafia owner leaned on one of them to be more than friendly, duty called for her to play along. Pine cringed at such awful work, and he wondered how those who were married stood it.\n\nTo make sure that all went smoothly and to minimize the risk to the policewomen, he had arranged a meeting for later that Friday evening around midnight in the First Division headquarters on East 21st Street. While such a review before a raid was routine, Pine sometimes skipped it. Tonight the review would be held: Pine wanted to be sure that nothing went awry.\n\nWhen Jennifer Hardy's gay roommate headed out that evening, she decided to tag along. First they stopped to get a bite to eat and ran into some of her roommate's friends. Eventually they ended up at the Stonewall Inn as they usually did. Jennifer, only seventeen, had arrived in New York from California as a runaway a couple of weeks earlier. This was her third visit to the Stonewall. She stayed and ordered a number of drinks even though she did not really care for the Inn, which seemed oppressive to her with its extreme darkness and poor ventilation. Besides, the whole place seemed like the kind of bar one had to slink into. Jennifer also felt very uncomfortable in such an all-male environment. She wanted to find some women. On her first visit to the bar she had noticed one lesbian there, but the woman was so much older than Jennifer that she had not been interested. She wanted to find someone around her own age, someone full of life.\n\nAround ten o'clock the car carrying Maria and Kiki pulled up by the United Cigars Store on Seventh Avenue South where the two young \"women\" paid the driver and headed for the Stonewall Inn. As they approached the bar they ran into a couple of Kiki's friends, gay men older than Maria. One of the men appeared to be in stage makeup, with plucked eyebrows, and a small amount of eyeliner. His appearance made Maria wonder if he was a professional actor or someone who did drag.\n\nAs Maria approached the Stonewall Inn's doors, she had her draft card out, ready for her moment of triumph. \"The last couple of times that I had been there, it was like sneaking in with the grown-ups.\"\n\nAs she entered the Stonewall, Maria proudly displayed her draft card to the doorman.\n\n\"Look, I got proof.\"\n\n\"You've been here before, haven't you?\"\n\n\"Yes.\"\n\n\"Well, you little bastard! We could have really got in trouble for that. Don't you know that we could get arrested?\" The doorman would not stop his embarrassing harangue. \"We could be closed down for letting you in without proof.\"\n\nThe verbal slap in the face was in sharp contrast to the congratulations Maria had anticipated. Crestfallen, she followed Kiki inside.\n\nLeaving the Bleecker Street commune, Danny Garvin headed to Danny's, his current bar of choice. He walked over to Christopher Street, waited in line to get inside what was now the most popular gay men's bar in the Village, and purchased his two drink tickets. On entering, he traded in one of the tickets for a soda and then stationed himself at the short service bar in the club's rear.\n\nAfter hanging out for about a half hour, he was surprised to see none other than Keith Murdoch approaching. Danny had gotten involved with Keith after he had split up with Frank, who had disliked Danny's hippie attire. While Danny had not seen Keith since he had left for college the previous summer, he had continued to long for him, in part because he found he had so much in common with Keith. Moreover, they had hit it off sexually. Keith's stunning good looks, dirty blond hair, blue eyes, and cleft chin had not hurt, either.\n\n\"Wow! What are you doing here?\"\n\n\"Wow, how's it going?\"\n\n\"You back in the city now?\"\n\n\"Yeah. You look good.\"\n\n\" _You_ look good.\"\n\nThe two friends started catching up on what had been going on in each other's lives. Danny told Keith that he had moved into an all-gay commune. They went on to talk about all the incredible social and political changes rocking the country, especially the war in Vietnam. Keith learned that Danny had attended a Be-in in Grand Central Station that had turned violent when the police had in effect rioted, charging into the crowd. Eventually they got around to comparing notes on their experiences smoking marijuana.\n\nKeith asked, \"Do you know where I can get some?\"\n\n\"Well, I have some back at the commune,\" Danny answered. \"Why don't we go back there?\"\n\nKeith agreed, and after they finished using up their drink tickets, the two set off for the commune. Their conversation turned to the subject of music, which like everything else seemed to be in a radical state of flux. Each eagerly asked the other if he had heard the latest albums by their favorite artists, including the Beatles and Judy Collins.\n\nWhen they reached the commune, they smoked some marijuana and then headed for Danny's bed.\n\nDick Leitsch threw his suitcase on his bed and started filling it. He and his lover, Bob Amsel, were to leave for Europe on Sunday. It'd been forever since Leitsch had had any rest, so this was a trip he was really looking forward to. He certainly could not have afforded a European vacation on the meager salary he got from Mattachine\u2013New York, but luckily Bob had a well-paying job. As Dick began packing, he turned on the radio.\n\nAt the midnight meeting, Seymour Pine reviewed the plans and his instructions for the bust of the Stonewall Inn with all those who would conduct the raid: the two women he had borrowed for the evening from Chinatown's Fifth Precinct, the five Public Morals officers besides himself, and Inspector Adam Tatem from New York City's Department of Consumer Affairs. It did not matter that the federal agent was not at this meeting, for it was up to him to get into the club on his own. After they had busted the Stonewall, the federal agent would simply identify himself to the police and then wait around while they gathered up the liquor so that he could take a sample for one of the Bureau's laboratories to analyze.\n\nThe review of the plans finished, the four undercover police officers\u2014the two male members of the Public Morals squad and the two female police officers\u2014and Tatem left Pine's office first as they needed time to get inside the Stonewall Inn and observe its operations.\n\nAs Kiki and Maria settled in at the front room bar for an evening of fun and celebration, Kiki ordered a round of drinks to toast Maria's eighteenth birthday. Later, with the music of the Supremes playing in the background, Maria bought a drink or two for herself while waiting for friends who had promised to come help her celebrate.\n\nBut the friends never arrived.\n\nJohn O'Brien never had to think about what he was going to do on a Friday night. He was so passionate about politics that every Friday night he looked forward to going to his favorite place to discuss political issues: the east side of Sixth Avenue below 8th Street. There he'd find activists hawking pamphlets as they worked the crowds who came to buy incense and beads from hippies who sold these goods on the sidewalk. It was not only a great place to hang out and talk, but there also were a good number of gay men among the political debaters and shoppers, so it was an excellent place to cruise as well.\n\nAbout half an hour after the first undercover officers had departed from the First Division office, Seymour Pine, Charles Smythe, and the two other plainclothes police officers left the 21st Street office. None of the four used police department vehicles, and Pine and Smythe traveled together in Pine's automobile. They drove to Greenwich Village, parked separately within a few blocks of 51 Christopher Street, and walked to their designated rendezvous, the interior of Christopher Park.\n\nThere the four men stood and watched the Stonewall Inn's entrance, waiting for the two undercover men to exit, which they did after a while. Everything was going according to plan, so Pine confidently waited for the female undercover agents to leave the bar, which would be his cue to raid the place.\n\nTommy Lanigan-Schmidt decided to visit his favorite bar late in the evening. When he got to the Stonewall, he saw that Johnny Shades was at the door. Tommy had no idea if Johnny Shades slept with men, but he found him very sexy. Johnny, however, far from even noticing Tommy, was dismissive of him. On this night, Johnny refused even to let Tommy inside. Disappointed, Tommy took off down Christopher Street.\n\nSeymour Pine continued to stare at the entrance to the Stonewall, but the policewomen still had not come out. Had something gone wrong? Pine was aware that some members of the Mafia knew police procedure even better than some police officers, and the two female officers were carrying guns in their purses. Perhaps the women had been discovered. Were they in danger? Should he go in now or wait? As Pine nervously weighed his options, he noticed that the place seemed unusually busy and mentioned this to Smythe, who seconded the observation.\n\nJennifer Hardy was getting rather drunk. _Shit_ , she thought. _I want to get out of here_. She walked outside, crossed the street, and entered Christopher Park to smoke a cigarette. \"I was kind of milling around in the park and trying to clear my head a little bit, because I was getting a real funny feeling in my head. I felt alone that night. I felt really lonely, really alone.\"\n\nCraig Rodwell and his lover, Fred Sargeant, left their friends' apartment near New York University where they had just finished a game of bridge. As they headed back to their Bleecker Street apartment, Craig noticed that it was unusually hot, even for late June.\n\nPine continued to wait. Still there were no policewomen. He discussed the situation with Smythe and his two men. _What could have happened?_ he wondered. He decided to wait awhile longer before going inside.\n\nDisappointed at her friends' standing her up, Maria tried to make the best of the evening by talking to some of the club goers, still eager to share her excitement about her coming-of-age and the freedom she anticipated. \"I knew in my heart that I wasn't gay, but I felt that these people were more accepting than any other people I had met. I felt happy to be with them, yet sorry for myself and for them that we had to hide\u2014hide who we were, hide who we loved, hide who we wanted to be.\"\n\nA club patron walked up to one of the Stonewall Inn's jukeboxes, dropped in a coin, surveyed the offerings, and selected a song by the Rolling Stones. The androgynous Mick Jagger's voice cut urgently through the dark, perfume-scented air inside the Stonewall Inn, singing, \"I can't get no satisfaction.\"\n\n#\n\n# _\"We're Taking the Place!\"_\n\nSeymour Pine finally decided that he could wait no longer, for it now seemed a real possibility that the undercover policewomen inside the Stonewall Inn could be in danger. Turning to his men, he said, \"Let's go, fellas.\"\n\nAnd so it happened that around one-twenty on the morning of Saturday, June 28, four plainclothes police officers from the First Division morals squad wearing dark three-piece suits and ties, two patrolmen, one carrying a radiotelephone, and Detective Charles Smythe, all led by Seymour Pine, walked through Christopher Park's north gate and crossed Christopher Street to raid the Stonewall Inn. Pine stopped at the Stonewall Inn's thick wooden double doors and announced, \"Police! We're taking the place!\" As always, they were delayed at the door for several seconds. By the time Pine, Smythe, and one of the other two officers who had been waiting in Christopher Park\u2014the officer carrying the radiotelephone initially stayed just outside the Stonewall Inn because the bulkiness of the unit would make him too vulnerable in case something went wrong\u2014were stepping across the Stonewall Inn's threshold, they were joined by the two undercover men who had been inside earlier. Once the raiding party (except for the officer in charge of the radiotelephone) was inside, they used a pay phone to notify the Sixth Precinct of their action and ask for backup.\n\nOn entering the front room, Pine caught a glimpse of Inspector Tatem sitting at the bar talking animatedly with a transvestite. Obviously Tatem was trying to put the make on her, but the officers, not wanting to give him away, merely walked by him without saying anything.\n\nMorty Manford was dancing when the music suddenly stopped and the lights were turned up. Abruptly stopping in midbeat, he noticed \"some very vicious men\" moving through the bewildered crowd.\n\nMichael Fader, a twenty-six-year-old insurance salesman, was confused by the white lights that he associated with closing time. He had been in his favorite club such a short while that he didn't even feel settled in yet. The Rochester native knew from experience that the Rochester police did not harass homosexuals. If someone threw a rock through a gay bar window in his hometown, the police arrested the rock thrower, not the homosexuals, so naturally Michael did not equate the arrival of the police with harassment. \"It crossed my mind that it was a fire drill or something. I didn't know what to think. I learned very quickly that the police were raiding the place. It immediately flashed through my mind, _What for?_ \"\n\nPhilip Eagles was standing close to the dance floor by the bar in the front room, having a drink and watching the go-go boys dance in their gold-lam\u00e9 bikinis, when \"there was a commotion, and I saw policemen filing into the bar, and I thought, _Here we go again!_ \"\n\nJoel S. remembers the night as a regular one until the lights came on. \"People were just kind of standing there talking to each other, 'What's going on?'\u2014waiting to see some kind of direction. 'Are the lights going to go off, are we going to be able to start dancing again, or...?' \"\n\nAs soon as Maria Ritter noticed the police, however, she did not need any elucidation. \"The cops had gone into the back room and started pushing people out. I realized it was something when I saw a cop. I headed for the bathroom, hoping there was a window. If there had been I would've gone out it. I have this strange recollection of wanting to get the hell out of there.\" Maria went through the back room and entered the women's rest room. No sooner had she closed the door than a police officer ordered her out. When she emerged, he grabbed her by the arm, and said, \"Over there!\" directing her to the east side of the club's front room.\n\nAs whispers went around the club that the place was being raided, customers rushed to locate friends. \"Are we going to be arrested?\" one shocked young man asked. Another terror-stricken man moaned, \"I'll lose my job. What will happen to me? My family! Oh no, no, no!\" Then the police sealed the doors of the bar.\n\nMaria remembers that \"people were trying to get out, but to no avail. It was havoc; it was chaotic\u2014it seemed things happened Bam! Bam! Bam!\" Michael Fader's recollection is similar: \"Things happened so fast you kind of got caught not knowing. All of a sudden there were police there and we were told to all get in lines and to have our identification ready to be led out of the bar. And I noticed while waiting in line to leave, the policemen were running around the facilities and the room.\"\n\nAnother witness described how the customers felt: \"I was anxious. Everybody was, not knowing whether we were going to be arrested or what was happening next. I wouldn't say that I was afraid. It was a nervous mood that set over the place.\"\n\nThe sudden arrival of the cops and the blare of the lights had transformed the scene from one of festivity to sadness. The jukeboxes fell silent, and the shimmering go-go boys left their cages to put on their street clothes.\n\nPine called the female police officers to the side and got information from them on whom to arrest. He first wanted to know why they had not come outside. The women explained that soon after their arrival the bartenders changed shift and, wanting to identify the workers in both shifts, they had been waiting for the change of staff to finish.\n\nAs the police officers moved around the club, the four undercover agents who had been inside pointed out the managers and workers, noting which ones had poured drinks and who had served them. Other policemen gathered up the alcohol. Pine noted that many of the bottles on the bars bore labels, although none in the storage room did. As the Public Morals police moved through the club, uniformed officers from the Sixth Precinct who had been on foot patrol began to show up outside. Having checked in with headquarters on their call boxes they had been told, \"Get your ass over to the Stonewall; they're going to make a bust there.\"\n\nMaria noticed a number of other \"women\" in the area where she had been directed to wait, as the police moved quickly to separate the Stonewall customers into groups according to their official interests. Adam Tatem was shocked to see the \"woman\" he had been putting the make on pulled to the side as a transvestite.\n\nTwo decades later Pine recalled the raid's beginning: \"We immediately\u2014I don't remember who it was\u2014'Everybody get his identity cards out,' and that kind of thing. And we began moving everybody out. In the meantime, the people who were actually working there, in the club, were isolated and put in the back [west] room, and we began taking information from them, names and so forth.\"\n\nBut even this early in the evening nervousness was not the only emotion customers in the bar were feeling. Michael Fader remembers that, standing in line waiting to exit, \"I thought, _I don't want to leave!_ That crossed my mind very strongly. I just got here and I'm supposed to leave? And stand in line to get checked for my ID? And I felt myself boiling up inside, getting more and more angry. And then I noticed the police were doing some damage to the room. The side [west] room had benches along the wall that you could sit on. They had a hollow cavity underneath\u2014and the police were ripping apart these benches, tearing them apart. I thought, _What are they doing that for?_ \" Suddenly he remembered the Democratic National Convention the previous August, when he had watched on television as the Chicago police had rioted, savagely beating the crowd of nonviolent protesters gathered in front of the Hilton Hotel.\n\nAlmost immediately after entering the Stonewall Inn, the police encountered resistance. Philip Eagles witnessed customers \"giving the cops lip\" by saying things such as \"We're not taking this\" and \"I'm not showing you my ID.\" Philip asserts that he and some other customers in the front room also initially resisted showing the police their ID or only did so with \"a lot of attitude.\" One man refused to leave the club, and Pine demanded to see his identification. When the man complied, Pine made a note of the name to follow up on later. As the customers were made to line up, word passed down the line that the police were not letting people without identification out. Those in line began to conspire together to share their identification, figuring that if a person had an extra form of ID on him and it did not have a photograph, that ID could be lent to a person who had none. Michael Olenick remembers that someone in his group of four friends had an extra Bloomingdale's credit card that he lent to a friend of theirs without ID.\n\nWhatever grumbling there may have been from the gay men, the police soon ran into more significant resistance from other patrons in each of the two rooms.\n\nAccording to Pine, \"We had a couple of the transvestites who gave us a lot of flak. We'd have policemen standing at the door and most of the transvestites that frequented these places were known to us, so you sort of weeded them out and said, 'Okay, you stand over here, everybody else out.' The transvestites were picked up near where the bathrooms were, because we used to at least walk them over there, because that's where they were going to be checked. And the policewoman would take them by the elbow and say, 'Okay, let's go in and check you out,' and usually, that was it. They would say, 'All right, honey, I'm a man.' I don't think anybody really checked them. I think it was almost like we were satisfied in embarrassing them, that 'we know who you are and get out!' So you took one collar just to add to the evidence that this place was illegal. Whoever was the unlucky one was the one who gave you lip. We did have the right to check for transvestites, and as transvestites, most of them were prostitutes as well, and we were concerned about that.\" Still, Pine insists, \"if the place didn't turn into a rhubarb, we probably would have thrown everybody out, including those we verified as transvestites, except the management. I mean, there was no plan to take these people.\"\n\nHowever, this night, according to Pine, the transvestites resisted by refusing to go into the Stonewall's bathrooms to be \"examined.\" \"We separated the few transvestites that we had, and they were very noisy that night. Usually they would just sit there and not say a word, but now they're acting up: 'Get your hands off me!' 'Don't touch me!' They wouldn't go in, so it was a question of pushing them in, fighting them.\"\n\nTo Seymour Pine, it was the transvestite resistance that made the raid take on a broader scope than its intended original targets, the Stonewall Inn's owners and employees: \"So, we then decided that we would take everybody in. We collected all the liquor, and those transvestites that had given us some trouble, we decided they would stay, and everybody else out.\" Pine put the transvestites and some of the bar personnel into the back room to hold them, but that allowed the transvestites' insurgent attitude to spread to the Stonewall personnel under arrest.\n\nWhile the police had their hands full with the transvestites in the second room, they had been getting more than lip in the front room as well. According to Philip Eagles, there were some lesbians standing against the back wall who, when approached by the police, said, \"We have a right to be here,\" and, \"What are you doing?\" Philips states that the cops were \"feeling some of them up inappropriately or frisking them,\" so it seems likely that the lesbians' challenging the cops and the police frisking of them are related, although it is impossible to say which occurred first. Either way, Philip says that the lesbians \"were being pushed around and bullied\" and this plus the frisking of them made \"everybody generally very uncomfortable.\"\n\nAccording to all other accounts, however, most patrons in the front room only seemed aware of a long and nervous wait, which might be because they were not close enough to the lesbians to witness their plight. Manford recalled that \"confusion and uncertainty reigned... until they were ready to move us out.\" Manford estimated that it was \"ten or fifteen minutes\" after being lined up that the club goers started to leave the place.\n\nBut for many others inside the bar, whatever the actual passage of time may have been, it felt much longer than fifteen minutes. Raymond Castro, a twenty-seven-year-old Puerto Rican baker known to his friends as Ray, had been in the back room standing next to the jukebox watching a couple of drag queens dancing by themselves to a Stevie Wonder tune just prior to the raid. He remembers that the police \"kept us there for so long, it was almost like a hostage situation. The police wouldn't even let you get near them. They wouldn't answer any questions. You couldn't ask them anything. They wouldn't tell you anything. The only ones you could actually still speak to were a couple of the [Mafia] bosses, and all they would tell you is, 'Calm down. Everything's going to be all right, nothing to worry about.' \"\n\nAround this time patrol cars from the Sixth Precinct began to roll up and park directly in front of the Stonewall Inn, to reinforce the two foot patrolmen already present.\n\nWhile most customers were waiting in line in the front room, Inspector Smythe continued to move around the club, supervising the seizing of all the alcohol from the two bars and the club's storage room, he or the other officers making notes as they went along, for each bottle would need to have a tag indicating exactly where the bottle had been found, so that when the seized liquor reached the Sixth Precinct each bottle could be cataloged into the evidence book. Pine was periodically checking on Smythe, one time asking him, \"You're almost finished?\" and a bit later, \"We need another patrol wagon right now?\" By the time Smythe had finished rounding up all the alcohol, he found he had twenty-eight cases of beer and nineteen bottles of liquor on his hands.\n\nThe question of the seized alcohol in fact weighed directly on the need for more patrol wagons: Pine knew that alcohol transported in patrol cars often did not reach the precinct house, so he preferred that it be transported in patrol wagons. Moreover, because the Stonewall Inn had a lot of waiters and Pine had not counted on arresting several transvestites, he would definitely need more than one patrol wagon.\n\nPine next had to figure out how to remove the bars and the jukeboxes. After inspecting the equipment, he notified Emergency Service that he wanted both the bars inside the Stonewall Inn cut up and removed along with the jukeboxes.\n\nMaria Ritter felt worse than nervous as she waited to see what her fate would be. By now not only had her combined coming-of-age birthday and high school graduation celebration been spoiled, but also arrest with all its dire consequences for a pre-op transsexual seemed inescapable. \"My biggest fear was that I would get arrested. My second biggest fear is that my picture would be in a newspaper or on a television report in my mother's dress!\" During the wait, thoughts of her father weighed heavily on her mind. Although he was a research scientist who worked for a pharmaceutical company, her father struck Maria as sounding like Archie Bunker's twin. How could she possibly explain to a man who always called her Butch how she came to be arrested in his wife's dress in a Greenwich Village gay bar on her eighteenth birthday? As awful as the prospect of facing her father seemed, there were still worse fears: \"I had already heard what happens to queers in jail, so... take it from there.\"\n\nFinally the line of detained club goers did begin to move, for Pine, having separated those he wanted to arrest from those he did not, wanted the latter out of the way so that he and his officers could finish inventorying the evidence and taking the names and addresses of those they were arresting without any interference. The process of getting the club goers who were not arrested to exit the Stonewall did not go quickly. As Joel S. noted, every one of the approximately two hundred customers held by the police had to stop and produce identification at the front desk, where the police checked \"every single one\" of the IDs produced. \"It was a tedious chore just getting out of there.\" Those who lacked identification were herded into the coatroom to await questioning by the police.\n\nAs the patrons began to exit the bar in a single file during the early morning hours of Saturday, June 28, the police officers expected them to disappear silently and gratefully into the night as had always happened after raids on homosexual bars. But this night was to be different: after the patrons left the raided club, they stood in the street, watching and waiting, their attention focused on the entrance to the Stonewall Inn.\n\nWhy did the Stonewall Inn's patrons stand in the street and watch that night? Part of the answer is probably simple: the police had chosen to raid the bar at a peak time\u20141:00 A.M. on a summer Saturday morning\u2014so that with a large number of customers in the club when it was raided, there were enough people present to form a crowd. Also, the larger the crowd, the more patrons there were who might be there with friends, hence a larger number who might be inclined to wait and see if their friends were arrested\u2014or beaten. However, a rarely noted contemporary account attributes the eruption of anger that night to the previous raid on the Stonewall Inn, noting that customers had already complained about the earlier raid:\n\nOn Tuesday night... the Stonewall Inn on Christopher Street was raided.... The Stonewall... has survived... for the past three years.... However, the pigs decided to start playing political games... because when did you ever see a fag fight back?... Now, times are a-changin'. Tuesday night was the last night for bullshit.\n\nOn Wednesday and Thursday nights grumbling could be heard among the limp wristed set. Predominantly, the theme [w]as, \"this shit has got to stop!\"\n\nNot only was the previous raid on the Stonewall Inn discussed, but so were the trees that had been cut down in Kew Gardens. The gay men who stood outside the Stonewall talked about the destruction of their lovers' lane as well as about how many of their other clubs had been raided in recent weeks: the Snake Pit, the Checkerboard, and the Sewer. Now not only had the Stonewall Inn been hit twice in one week, but also tonight's raid had come on a Friday night and at the evening's peak. Standing in the crowd, Morty Manford sighed. _Damn_ , he wondered. _Why do we have to put up with this shit?_ 29\n\nJoel S. was among the first few dozen or so patrons to be released. He remembers that \"as we filed out the crowd started to gather in front of the place. I guess more and more people just started to mill around the front. People were kind of staying with friends, looking and hanging out. I mean the street wasn't packed, but there were a lot of people there, and they were spread all around in front of the bar, towards Waverly Place. I remember seeing some of the more ostentatious drags walking in twos or threes down Christopher towards the bar and hearing loud, shrieking little sentences. I don't really know what they were saying, just kind of drawing attention to themselves. They were marching along, trying to show a little anger or annoyance, trying to make some noise.\" Joel noticed the temperature: \"It was a hot, seething night. A real New York summer night.\n\n\"I guess everybody was wondering, _What's going on_?\u2014a little annoyed. _Is something going to happen? Why is this taking so long?_ In fact, some of the drag queens were kind of chanting and skipping along. It was entertaining.\"\n\nFrom his second-floor perch in a storage room The _Village Voice_ rented on the west side of Seventh Avenue South, Howard Smith looked up from his typewriter and noticed a number of police cars on Christopher Street below. Smith kept a pair of binoculars on his desk to inspect people who asked for him at the _Village Voice_ office across the street. Intrigued, he reached for the binoculars and peered at the scene. While he could not tell just what was happening, it seemed out of the ordinary. As he moved around New York City Smith had made a habit of following police cars and fire trucks as well as investigating any other interesting scene he happened upon. _Wow!_ he thought. _Right here! I wonder what it is?_ He also noticed the size of the crowd. \"It was growing very quickly. Every time I'd blink, there were more people.\" His curiosity piqued, he picked up his blue plastic press pass, hung it around his neck, stuffed some notepads into his pockets, ran downstairs, and \"walked up to the cops and tried to figure out what was happening. I saw that it was the Stonewall.\"\n\nThe commotion in the streets had also drawn Lucian Truscott from the Lion's Head to see what was causing all the racket. Truscott, an army lieutenant on leave, was also an aspiring writer and was using his time off from the military to do some occasional writing for _The Village Voice_. Observing the scene and deciding to watch for a while, he soon ran into his _Voice_ colleague. Truscott and Smith talked briefly about what was going on, and when Truscott realized that he did not have a press pass Smith let him into the _Voice_ office to pick one up so that he, too, could go behind police lines. The two men parted company, not to see each other again that night.\n\nThe crowd on Christopher Street continued to grow as the club's ejected patrons reached the pavement where they also were joined by a considerable number of tourists who, having come to the Village on a Friday night looking for excitement, had found it for free on the street.\n\nOne of those now in the crowd was Tommy Lanigan-Schmidt, having decided that enough time had passed since that mean Johnny Shades had refused to let him inside that he could again try his luck at getting past the doorman. Although Tommy had been to the club more times than he could recall, it just so happened that he had never been there during a raid, so that coming across a raid of the club he practically considered home took him aback: \"It surprised me because it was such a secure place that I thought, _Well, how could this happen?_ I immediately felt a kind of indignation because this was like sacred.\"\n\nAround this time Danny Garvin and Keith Murdoch were approaching the downtown side of Christopher Park as they headed north on Seventh Avenue South. After they had finished making love, they had decided to end the evening by dancing at the Stonewall. On the walk over they had talked about what it was going to be like when the revolution they and so many others anticipated happened: even the Beatles were singing about revolution. \"We all figured that the Black Panthers were going to start the revolution. All of a sudden, you see people coming out of the Stonewall, so you knew it was raided.\" High on pot, Garvin's first reaction was: \"The revolution had started!\" Garvin estimates the size of the crowd when he arrived at around 100 or 150 people.\n\nMost patrons exited quietly, but the small crowd cheered some of the club's favorites, many of whom camped it up. As one young man swished by the detective posted at the door, he tossed the classic come-on line at him: \"Hello there, fella!\" Others of the departing men took bows and were hailed by their friends. As customers reacted to the crowd's applause, they became bolder and more confident in their outrageousness. _Voice_ reporter Smith described the crowd as \"prancing high and jubilant,\" while Truscott observed: \"Wrists were limp, hair was primped, and reactions to the applause were classic.\" Said one excited patron, \"I gave them the gay power bit, and they loved it, girls.\" Another ad-libbed, \"Have you seen Maxine? Where _is_ my wife\u2014I told her not to go far!\"\n\nThe scene was not all hilarity, however. Smith, who managed to form a bond with Seymour Pine on the spot, kept near him and observed the police at close hand. He noticed that the police handled the exiting patrons roughly, now hurrying one out quicker than he could comfortably move, now giving another a parting kick.\n\nAs Garvin watched, he saw that \"it started getting ugly. You had attitude: 'Don't touch me!,' which then would ignite the crowd: 'Go get 'em!' So then it became a show. Who's coming out, who's exiting? No one knew that it was going to turn into a riot. In everybody's mind all it was was a bar raid. People were being kicked out of the bar, so there was going to be a little campiness, there's going to be stars coming out onto the street.\" Garvin recalls that some of the exiting men threw their arms up and out in a V shape as if they were performers making a grand entrance on a stage.\n\nAround this time Tom, a young man who worked for the East Village underground newspaper _Rat_ , out with a friend for a beer, stumbled across the scene in front of the Stonewall and asked some of the men in the crowd what had happened.\n\n\"They raided the joint, the fucking bastards.\"\n\n\"Why?\"\n\n\"Operating without a liquor license.\"\n\nA black man yelled, \"Shit, man, they's out like always to chase us down and give us a good fuck. They ain't got nothin' else to do during the summer.\"\n\nAnother member of the crowd approached Tom and asked him if he wanted to buy some speed.\n\nNoticing the crowd's \"skittish hilarity,\" Howard Smith paused in his note taking to peer up at the moon. It was full.\n\nA paddy wagon appeared, seemingly from nowhere. It pulled up part of the way on the sidewalk in front of the Stonewall Inn, just a little east of the entrance, and parked against the flow of traffic, facing east. Joel S. recalls that at this time \"the area was kind of loosely crowded, and the noisy drags, walking along, attracted other viewers. The crowd started getting bigger and bigger.\"\n\nCraig Rodwell and Fred Sargeant were crossing Sheridan Square on their way home after a game of bridge when they saw the crowd gathered in front of the Stonewall Inn. Craig and Fred approached to get a closer look and saw the patrol wagon parked in front of the bar, telling them that they had stumbled upon a raid. While there was nothing new about that, Craig noticed \"a feeling in the air that something was going to happen.\" Craig immediately decided that he was not about to miss out on this, and he and Fred climbed the seven steps to the top of the tallest stoop on Christopher Street, the one just west of the Stonewall Inn, to get a bird's-eye view of whatever was about to unfold. _This_ , Craig thought, _is different_.\n\nThe arrival of the paddy wagon marked the first time that evening that the seriousness of the raid came home to the men and women gathered on Christopher Street. From his vantage point above the street Craig noticed how very quiet the crowd had become.\n\nSeymour Pine exited the Stonewall Inn, prepared to begin loading his haul of prisoners and evidence to the Sixth Precinct station house on Charles Street, but found that he had only one patrol wagon. As Pine recalls, \"This presumably should have been the end of the situation, because the raid was already over. Now, all we had to do was put them in the patrol wagon.\" But Pine saw that he had a problem: \"The crowd had grown to ten times the size: it was really frightening.\" This surprised and confused Pine: \"So many showed up immediately, it was as if a signal were given. And that was the unusual thing because usually, when we went to work, everybody disappeared. They were glad to get away. But this night was different. Instead of the homosexuals slinking off, they remained there, and their friends came, and it was a real meeting of homosexuals.\" Still, Pine was not too concerned about the problem, for certainly neither he nor his men had ever encountered hostility from a crowd of homosexuals. Pine learned that only one patrol wagon had arrived, so he tried to order a second one, but when the officer with the radiotelephone made the call, a mysterious message came over the line: \"Disregard that call.\" This struck Pine as very odd, for the calls went via CB to a central unit. True, anyone in the Sixth Precinct would automatically hear the announcement as well, but why would anyone want to countermand it? Pine would try again later. Having cleared all of the people he did not want to arrest outside the Stonewall Inn, Pine could now load his prisoners.\n\nThe first prisoners to be loaded inside the wagon were members of the Mafia, who were brought out of the club one by one. As the mob members exited the Stonewall and were shoved inside the patrol wagon, the crowd reacted variously, sending up Bronx cheers for the hated Mafiosi, jeering the police, and clapping. Meanwhile, Craig Rodwell, watching the familiar ritual of another gay club being busted, felt the anger build up inside of him until he could stand it no longer and suddenly let loose a yell that pierced the night: \"Gay Power!\" Fred immediately nudged Craig, saying, \"Shut up!\" But everyone heard the cry that reverberated through the still night air and a few took it up and repeated it. But this idea seemed too unreal, too radical, to be taken seriously, and the newly heard slogan soon dissolved into giggles.\n\nThe non-Mafia employees of the club were the next ones loaded into the paddy wagon: bartenders, hatcheck boys, and even John, the black men's room attendant. As Michael Fader watched those arrested being loaded into the paddy wagon, the only one he recognized was John, who had \"always been a gentleman and helpful.\" As he saw this meek middle-aged heterosexual black man climb into the paddy wagon, the whole thing struck Michael as being so absurd and unnecessary. \"It was just kind of a strange cross section of people.\" His sense of social justice offended, Fader began to feel even angrier. The behavior of the police also galled the young salesman: \"They had no sensitivity whatsoever. They weren't brutal in terms of clubbing people, but there was condescension and a contempt.\"\n\nA few cheers went up as the bar employees continued to be loaded into police vehicles, and then someone began to sing one of the most familiar protest songs from that era of protest, \"We Shall Overcome.\" A few in the crowd started singing along, but after a few verses this, too, seemed too dignified to be taken seriously by a bunch of homosexuals and the crowd began to camp on the solemn lyrics. By almost all accounts, until somewhere around this time most of the crowd on the street were largely maintaining a sense of good humor about everything that was happening.\n\nThe next to enter the paddy wagon were three men in full drag, described by Truscott as being among the \"more blatant queens.\" Bob Kohler had arrived on the scene after the patrol wagon and spotted the street kids he knew from Christopher Park standing close to the park's exit onto Christopher Street. He walked up to them with his dog, Magoo, and watched as the well-dressed drag queens were put into the paddy wagon. Kohler recalled how \"the queens... were waving, and the kids were saying, 'Have a good rest,' and, 'Oh, I'm glad they're taking her. She needs a rest,' and, 'Oh, Lily Law's got you, girl!'\"\n\nThe buoyant humor displayed during the riots is noteworthy, especially on the first night, as the crowd deployed its keen sense of camp. Fader remembers, \"A couple of times the police would come out with someone and take him to the paddy wagon, [the crowd] would make quips to the police. There were some people with really good senses of humor, and the crowd would all crack up and laugh. That was intermingled with growing and intensive hostility.\"\n\nThe first hostile act outside the club occurred when a police officer shoved one of the transvestites, who turned and smacked the officer over the head with her purse. The cop clubbed her, and a wave of anger passed through the crowd, which immediately showered the police with boos and catcalls, followed by a cry to turn the paddy wagon over. Edmund White, having chanced upon the scene, noted that \"the cops, used to the cringing and disorganization of gay crowds, snort off. But the crowd doesn't disperse.\" Tom witnessed the same police\u2013crowd interplay: \"People began beating the wagon, booing, trying to see who was being hauled out and off. Several pigs were on guard and periodically threatened the crowd unless they moved back. Impossible to do. 'Nobody's going to fuck around with me. I ain't going to take this shit,' a guy in a dark red tee-shirt shouted, dancing in and out of the crowd.\" White described the temper of the crowd: \"Everyone's restless, angry and high-spirited. No one has a slogan, no one even has an attitude, but something's brewing.\"\n\nGay men began to go to the many pay phones around the Sheridan Square area and call up friends, telling them to rush down to the Stonewall. Others ran throughout the neighborhood shouting that the Stonewall was being busted, and word of the raid passed through the night like a fever.\n\nFader saw a policeman come out of the Stonewall who, though obviously \"trying to disperse the crowd,\" had no idea about how to proceed: \"He didn't know who was from the bar and who wasn't, because by then a few people had come from other areas. I was on the end of the crowd, probably closer to the Lion's Head, but close to the door. I called him a pig.\" Brandishing his nightstick, the policeman grabbed Fader and said, \"Just say that again!\" Fader's experience in many other demonstrations served him well: \"I was very cool at that moment. I had some knowledge, and I thought, _I'm not going to walk into that trap. He's standing there with a revolver and a club. I'm a loser on this one_. I said, 'That would be rather silly, since you have a gun and a billy club, and I don't. I don't see any point in that!' So I remember him taking the billy club and shoving me back.\"\n\nTom noted that with every passing minute, the crowd grew angrier. \"Pennies ricocheted off the van, a beer can hit the door.\" One reason that the crowd was becoming so angry was that they were aware that the police had other people inside the Stonewall Inn and they believed that these people were being beaten by the police. Fred Sargeant voiced this opinion soon after Friday night in a radio interview he gave to WBAI's _New Symposium II:_ \"The kids felt that some of the other kids were being kept inside and being beaten up by the police. I don't know whether it really happened that way or not, but the rumor spread.\" This belief no doubt explains Tom's reporting that \"the crowd... began shouting for different people that they knew were being held. 'We want Tommy, the blond drag queen.' Shouts went up... Suddenly Tommy appeared in blond wig, etc., and walked coolly out the door. Shouts and screaming. 'We want Tommy!' Tommy, not held by the pigs, smiled and suddenly took off into the crowd to the left.\"\n\nRodwell, conscious as always of the Mafia exploitation of gay men, yelled, \"Get the Mafia out of the bars!\"\n\nOne of the drag queens about to enter the patrol wagon was Maria Ritter. \"The cop who put me in was kind of teasing and flirting with me when he helped me get in. He said, 'Jesus, I can't even believe you're a boy!' I said, 'I'm not a boy! You don't understand, I'm a girl, but it's real hard for me to explain to you.' I guess he was just going by the voice and kind of like spooking me. He said, 'Come on, let's go.' I was put in a paddy wagon with a whole group of older drag queens, and I'm thinking to myself, _I'm dead_. They opened the door to get a few more who were dressed in drag into the police bus, and I very gingerly stepped down and kind of shimmied out, saying, 'Excuse me.' As I was walking away the cop who put me in saw me. He said, 'Hey, you! Come over here!' He was a young guy who looked as if he was maybe interested, and I begged him, 'Please! It's my birthday; I'm eighteen. And my mother's going to kill me!' And I was crying, and my makeup was going. He almost looked the other way, kind of gave me the hand motion, and I kept walking. If he comes after me and grabs me by the scruff of the neck, he's going to do it, but if he's not, he doesn't. And he didn't.\"\n\nThe next bar patron to be taken from the Stonewall was a lesbian, and she was decidedly not in a good mood. The existence of this lesbian and her supposed role in the Stonewall Riots have always been among the most controversial aspects of the riots, with some prominent commentators displaying skepticism about her. It is therefore important to look closely at all the available evidence concerning her. Tall and stout, with a short, mannish haircut, she was wearing pants and what one witness described as \"fancy, go-to-bar drag for a butch dyke.\" She clearly fit the role known at the time as a butch lesbian. Jennifer Hardy, who had been watching the raid from Christopher Park, remembers her as a husky \"bull dyke,\" of \"bigger size\" and with a \"nasty temper.\" Steve Yates, who, like Maria Ritter, had had his birthday celebration inside the Stonewall abruptly ended by the raid, went to Christopher Park and stood outside the fence, where his friends joined him. He remembers the lesbian as a \"rather beefy, good-sized woman\" and a \"typical New York butch.\" Another woman who witnessed the scene and wrote about it to a friend a few days later described her similarly: \"a dyke\u2014stone butch.\"\n\nAccording to Harry Beard, a former Stonewall employee, the lesbian's fight with the police had begun inside the bar. She had been visiting a bar employee who was a friend, Beard relates, when the raid occurred. Arrested for not wearing the three pieces of clothing correct for her gender according to New York law, she was handcuffed and, while in the hallway and just a few short steps away from the entrance, was \"yanked\" by a policeman. She told the officer, \"Don't be so rough.\" According to Beard, the policeman's response was to hit her in the head with a billy club. (In two other versions, one given by Beard in 1980 and another one given in 1989 by Beard and two friends, Gene Huss and Don Knapp, it was a request to the police officer to loosen her cuffs that resulted in a blow to her head. Yates's account might corroborate Beard's in part, for he remembers that \"they were manhandling her _out the door_ [emphasis added] to try to push her into a squad car.\" Yates also described her as \"one rather beefy, good-sized woman who had probably given them a ration of shit back,\" which also seems to fit in with Beard's assertion that she had complained to the police while still inside the Stonewall Inn. Like Beard, Yates remembered handcuffs: \"They had her pushed down with her hands cuffed behind her.\"\n\nThere is no doubt that, furious for whatever reason, she put up a fight. Yates says, \"She was giving them their money's worth,\" and remembers that there were three or four policemen on her. She fought them all the way from the Stonewall Inn's entrance to the back door of a waiting police car. Once inside the car, she slid back out and battled the police all the way to the Stonewall Inn's entrance. An unknown woman who recorded the scene in a letter emphasized the lesbian's fury: \"Everything went along fairly peacefully until... a dyke... lost her mind in the streets of the West Village\u2014kicking, cursing, screaming, and fighting.\" But after she reached the Stonewall the police pulled her back to the police car and again placed her inside it. She got out again and tried to walk away. This time an officer picked her up and heaved her inside. Yates estimates that the struggle between the police and the lesbian lasted between five and ten minutes. According to yet another account, at around this time a woman\u2014possibly this same lesbian\u2014urged the gay men watching her struggle to help her: \"Why don't you guys do something!\"\n\nBob Kohler remembers that at about this time \"a couple of the kids threw some change over. I got mad and said, 'Stop throwing your money! I probably gave you that money. Stop throwing it!' The cops closed the paddy wagon, got rid of the paddy wagon, because they obviously felt something was going to happen.\"\n\nAs the heroic fight by the lesbian who had twice escaped the car neared its end, the crowd erupted. The anonymous author of the letter wrote that the woman's fighting \"set the whole crowd wild\u2014berserk!\" Both the _Voice_ reporters are agreed that it was the lesbian's struggle with the police that ignited the riot. Truscott wrote: \"It was at that moment that the scene became explosive.\" Smith's account pinpoints the policeman bodily throwing her inside the car on the third and final attempt to put her in the vehicle as the moment \"the turning point came.\"\n\nThe _Berkeley Barb_ account seems to agree with Smith's version. Smith wrote that after the lesbian was thrown into the car, \"The crowd shrieked, 'Police brutality!' 'Pigs!' A few coins sailed through the air. I covered my face. Pine ordered the three cars and paddy wagon to leave with the prisoners before the crowd became more of a mob. 'Hurry back,' he added, realizing he and his force... would be easily overwhelmed if the temper broke. 'Just drop them at the Sixth Precinct and hurry back.' \"\n\nLeo E. Laurence wrote, \"Pigs were loading her into the wagon when she shouted to a big crowd of bystanders: 'Why don't you guys do something!' That did it. The crowd rushed the police wagon as someone yelled: 'Let's turn it over.' The pig driver drove off escaping the angry crowd.\"\n\nIndeed, it seems possible that it was this extremely masculine lesbian struggling with the police that _Rat_ reporter Tom took for a man when he wrote that \"the pigs started hauling off in a squad car a guy they had dragged outside. The crowd protested wildly, booing, shrieking, 'Up against the wall, faggots!' 'Beat it off, pigs!' A few tried jumping out to disrupt the no-man's-land between the door and the squad car, but most as yet were reticent about provoking any pig violence. Someone tried to sell me some speed, and another asked me for a knife. 'I'm gonna slice up those motherfuckers' tires.' \" That the masculine-appearing lesbian was the person Tom observed seems all the more probable since he mentions that an onlooker immediately asked for a knife to \"slice... those motherfuckers' tires,\" for according to Harry Beard, all four tires of the police car with the lesbian were slashed.\n\nSteve Yates and Jennifer Hardy likewise remember the lesbian's resistance as triggering the crowd's violence. Yates recalled the reaction of his friend Gino, an occasional sexual partner whom he knew only by his first name and describes as an intelligent working-class Puerto Rican \"rabble-rouser\": \"My friend Gino had had considerably too much to drink at this point, and he started yelling at the police. They'd started dragging a woman out in handcuffs, and he started yelling at them to 'let her go! Leave her alone!' and people started yelling for them to 'let her alone,' 'Leave her alone,' 'Let her go!' I never knew her name or knew anything about her. I just remember that was the thing that Gino saw that he just could not [bear to] see. He saw it as a great injustice. There was a loose cobblestone in the pit [surrounding] a tree.\" Gino, who sometimes labored as a construction worker, became so enraged by the woman's mistreatment that he wrenched the heavy cobblestone out of the ground and threw it all the way across Christopher Street. It landed on the trunk of a police car with a terrible screech, \"scaring the shit\" out of a policeman who was standing next to the car, says Yates. \"That's when the police backed in and barricaded themselves in the Stonewall.\"\n\nHardy's recollection agrees with Yates's in that they both depict a gradual buildup of the crowd's anger that suddenly let loose in a violent outburst: \"It just seemed like the catalyst for the whole thing to break loose was her getting out of the car and screaming and getting shoved back or thrown back into the car and then getting out again, because she was getting real pissed off. And the crowd started getting real ugly and bigger. The fun mood disappeared real fast and people started shouting. And then all of a sudden somebody said, 'Let's get 'em!' And cops just vanished.\"\n\nIndeed, while there seems no doubt that the lesbian who twice escaped the police car played a leading role in increasing the crowd's anger that Saturday morning in June, it seems very probable that more than one lesbian was seen resisting the police that night outside the Stonewall Inn\u2014or was being mistreated by the police\u2014and also contributed to the crowd's anger. Kevin Dunn, who had seen a fair percentage of the gay men leaving the Stonewall Inn camping and posing, noticed that \"the lesbians who came out were not in a good humor to do a little pose. They were resistant about being busted. And they had a more serious tone, being arrested.... The lesbians had been thrown in... and sealed [inside the patrol wagon] and not let go of, [and] the crowd seemed to get pretty ornery about that. It was like 'Don't do this to these ladies!' It was sort of chauvinistic in a way, but it was very interesting that they got angrier, it seemed, when the lesbians were thrown in.\" And just as Philip Eagles had seen a small clutch of lesbians inside the club, he also remembers there being more than one outside. He and his friends had gone across the street from the Stonewall Inn and stood on the sidewalk next to Christopher Park to watch what would happen. \"The paddy wagon was eventually driven up in front of the bar and parked on the park side of Christopher rather than on the bar side and drag queens were pushed into the paddy wagon and the cops were roughing up some of the lesbians. I believe one of them was hurt or was bloodied to some extent. Of course, the butch lesbians were among the first to start fighting back as I remember... and so they were getting beaten and hit.\"\n\nThere is no doubt, however, that at this stage of the evening many things were happening simultaneously in the angry crowd. One of the events that further stoked the crowd's fury revolved around Ray Castro. After he had been allowed to leave the club, Ray had walked around a bit and then returned to the front of the Stonewall Inn. When one of the doors opened he spotted a friend much younger than himself inside who gestured to him in a manner that said to Ray, _Do something!_ Ray surmised that his friend was fearful because he did not have ID on him. Castro decided that he would try to find someone in the crowd with some phony ID so that he could go back inside and give it to his worried friend. But before Ray could do anything at all he suddenly found himself being pushed by plainclothes police.\n\n\"At that point I started pushing back and wound up with two plainclothes police pushing me. The next thing I know, there's two plainclothes cops and two uniformed police in the melee. I was knocked to the ground by one of their billy clubs, [which,] put between my legs, tripped me. At that point the handcuffs got put on me, and they had a paddy wagon right in front of the entrance to the Stonewall. When I got shoved up to the door of the paddy wagon, I had two policemen on each side of me. I didn't quite go willingly into the paddy wagon. I didn't want to be arrested. Even though I was handcuffed, I jumped up and [put] one foot on the right of the door, and one foot on the left of the door. I sprung up like a jumping jack and pushed backwards, knocking the police down to the ground, almost against the wall of the Stonewall. Well, they finally dragged me into the paddy wagon.\"\n\nRay's fierce struggle with the police was recorded by Tom: \"A couple more were thrown into the van. We joined in with some who wanted to storm the van, free those inside, then turn over the van. But nobody was yet prepared for that kind of action. Then a scuffle at the door. One guy refused to be put into the van. Five or six cops guarding the van tried to subdue him with little success. Several guys tried to help free him. Unguarded, three or four of those in the van appeared then quickly disappeared into the crowd. This was all anyone needed.\"\n\nFader recalls seeing some of those he had seen placed in the patrol wagon earlier escape. \"The police were undermanned; they had big crowds forming. They thought everybody would just quietly go away and they'd be all matter-of-fact, but no one was in the mood to sit and be good or do whatever the cops told them to do.\" Fader saw the police \"leave the van unattended\u2014the doors were open, so they left.\" His account concurs with Tom's that the prisoners' escape excited the crowd. \"That raised the emotional level, the excitement of them getting away.\"\n\nManford also witnessed prisoners being left unattended but felt that the officers had done so on purpose. \"After everybody who was going to be released was, the prisoners were herded into a paddy wagon parked right on the sidewalk in front of the bar. They were left unguarded by the local police, and they simply walked out and left the paddy wagon to the cheer of the throng. There's no doubt in my mind that those people were deliberately left unguarded.\"\n\nIt seems certain that one of the escaped prisoners was blackmailer and Stonewall manager Ed Murphy. As Murphy told the story, he was handcuffed to \"Frankie\" (very likely Blond Frankie) and the two escaped into the crowd and then took a taxi down to Keller's, where an S and M queen was able to free them from the handcuffs. Manford's belief that the police allowed prisoners to escape seems plausible when one considers that the Sixth Precinct was paid off handsomely by the Stonewall Inn and some of the police present must have known Murphy's role in paying them off. Furthermore, letting the prisoners escape may have been retaliation for Pine's not informing them of a raid in their precinct in advance.\n\nWhile at this point in the night a full-scale riot was probably inevitable, Tom credited Castro's struggle with assuring that violence did erupt.\n\n\"Several others tried [assisting] the guy held by the cops,\" Tom wrote, \"but the latter escaped into the Stonewall. Soon the van pulled out leaving the street unguarded. A few pigs outside had to flee for their lives inside and barricade themselves in. It was too good to be true. The crowd took the offensive.\"\n\nDanny Garvin recalls that it was the police pushing back at the crowd, some of whom were throwing money and other projectiles at the officers, that caused the gay throng to make a useful discovery: a large stack of new bricks at a construction site on Seventh Avenue South. \"All of a sudden a cop would get hit, so there'd be an attitude. Then anger would start on each side. The cops then started to get a little bit physical. They would come at us with nightsticks, and we would have to disperse onto Seventh Avenue, where the people were able to see the bricks.\"\n\nAs the paddy wagon left, it was accompanied by three police cars with their sirens screaming. The crowd, however, was beyond being intimidated by mere sirens and the caravan had to push slowly through the furious protesters, who, enraged, pounded on the police vehicles. Danny Garvin recalls the noise as \"people would run over, grab the paddy wagon, and start shaking\u2014ba-boom! ba-boom!!\" Truscott heard the \"cry [that] went up to push the paddywagon over, but it drove away before anything could happen. With its exit, the action waned momentarily.\"\n\nAt least one\u2014and perhaps more\u2014of these vehicles had some of its tires slashed, slowing them down further. Pine, noticing that the crowd was getting out of control, urgently told one of the departing drivers, \"Hurry back! Just drop them at the Sixth Precinct and hurry back!\" After the police vehicles left for the station house, Pine had eight plainclothes detectives with him, including the two undercover policewomen who had been sent in ahead of the raid and Gil Weissman, a police officer from the Sixth Precinct. Weissman stayed behind with Pine and the initial raiding party, making him the only uniformed police officer left on the scene.\n\nDave Van Ronk, a prominent folksinger, was yet a third person out celebrating his birthday on that discordant evening. Two female friends had taken him out for dinner and drinks at the Lion's Head, Van Ronk's favorite watering hole, just two doors down from the Stonewall Inn. Sitting in the dining room of the basement establishment, he heard the wail of the departing police cars but, being a New Yorker, thought nothing of it. But then he began to hear \"a lot of yelling and screaming\" and saw people running in the street, though from his limited view out of the basement window he could see only feet and legs from the knees down. His curiosity piqued, he decided to step outside. His companions tried to discourage him, but he assured them, \"I'll be right back.\"\n\nOnce outside he asked a man in the crowd what was happening and learned that the Stonewall had been raided. \"Somebody was yelling out that they didn't pay off the cops.\" But then \"somebody [else] said, 'Let's pay them off!' and started to throw change at them.\" The rain of coins began with pennies, which made pinging sounds as they hit the pavement and the Stonewall Inn's windows, accompanied by jeers and the shout of \"Dirty Copper!\" While the heterosexual Van Ronk had never thought much about gay rights, he did not side with the police. \"I had been involved in antiwar demonstrations where the police descended on us like armed locusts. What I saw was yet another example of police arrogance and corruption. As far as I was concerned, anybody who'd stand against the cops was all right with me, and that's why I stayed. The cops had made themselves fairly unpopular over the years with tear gas, with dogs. Every time you turned around the cops were pulling some outrage or another. I reached in my pocket and tossed a quarter or just some pennies and around that time the heavy artillery cut in. I assume that some of the street people in the park had decided to join the fray because beer cans started flying over our heads.\"\n\nShouts of \"Pigs!\" and \"Faggot cops!\" filled the night air.\n\nAs pennies and dimes pelted him, Howard Smith backed up against the Stonewall Inn's door. He noticed that by now the detectives held \"at most a 10-foot clearing.\"\n\nNickels were the next thing to be thrown.\n\nFollowed by quarters.\n\nA glass bottle was lobbed.\n\nThen another one came flying through the dark air. And another.\n\nA flying object clipped Gil Weissman under the right eye. \"He hollers, and his hand comes away scarlet.... They (the police) are all suddenly furious,\" Smith wrote. Three of the officers ran into the crowd to try to scare the rioters away from the door. A beer can glanced off Deputy Inspector Smythe's head.\n\nVan Ronk decided that \"around that time, I guess I'd seen all I needed to, and I thought I'd go back and have a drink and tell my friends what was going on.\" But at that very moment Seymour Pine suddenly leaped into the middle of the crowd and grabbed him around the waist. The musician fell backward, pulling Pine down to the ground with him. Van Ronk decided to put up a fight, rather than be arrested. With the help of two police officers, Pine succeeded in separating Van Ronk from the crowd and hauled him inside the Stonewall Inn.\n\nVan Ronk pointed out that at six-feet-five, \"I'm pretty conspicuous\u2014I was one of the first ones they grabbed.\" While Van Ronk did not precisely remember the others who were seized by the police, he felt certain that the officers did arrest several other protesters around the same time they arrested him.\n\nPine's version of why he arrested Van Ronk is quite different: \"They were coming up and flipping coins into the faces of the policemen. I saw this one fellow come up and do that, and he hit the patrolman in the eye. He reached for his eye, and when he took it away there was blood. And when this fellow saw it, he got pretty nervous, and he immediately ran back into the crowd. And I went right after him.\"\n\nThe cops hauled Van Ronk, his head banging against the ground, into the Stonewall Inn's front room, where they placed a handcuff on his left hand and attached the other end of the handcuff to the flange of a radiator, close to the floor. Van Ronk said he was then kicked several times. He ruefully noted the irony that, handcuffed and lying on the floor, \"from there on all I could get was a worm's-eye view. The episode ended as it began for me, looking at legs.\"\n\nHaving secured Van Ronk inside the Stonewall Inn, Pine exited the club to survey the situation. As all kinds of objects continued to crash around the police, Pine decided that his only recourse was to take refuge inside the Stonewall Inn: it was either that or risk being assaulted by a wild mob. The crowd had begun to sense the policemen's fear. Tom wrote: \"A few plainclothesmen were surveying the crowd, obviously panicked,\" while Ronnie Di Brienza wrote in the _East Village Other_ that \"during the height of the action, you could see the fear and disbelief on the faces of the pigs.\"\n\nPine recalls that immediately after he hauled Van Ronk out of the crowd, \"he [Van Ronk] was rushed inside the place. In the meantime we [the police] were right near the windows, near the doors... And then it got so bad I ordered them back into the Stonewall.\" Pine told his officers, \"Let's get inside. Lock ourselves inside; it's safer.\" Before retreating inside the Stonewall, Pine turned to Howard Smith and, with a paternal tone, asked him, \"You want to come in? You're probably safer.\"\n\nSmith felt confused by Pine's offer but knew that he had little time to choose between two clear alternatives. Smith debated in his mind, \"If they go in and I stay out, will the mob know that the blue plastic thing hanging from my shirt is a press card, or by now will they assume I'm a cop too? On the other hand, it might be interesting to be locked in with a few cops, just rapping and reviewing how they work. In goes me.\" When Smith made his choice, he thought reinforcements would arrive in a few minutes, so the risk seemed negligible. Besides, Smith had always liked seeing things closer than other people. That's why he had gotten police plates on his car, so that he could pursue anything that looked interesting.\n\n\"Oh, I'll go inside,\" Smith told Pine.\n\n\"Fine,\" Pine answered. \"Come on. Right now.\"\n\nSmith hastened to join Pine, Smythe, Weissman, the two undercover policewomen, the four other plainclothes morals officers, and Consumer Affairs inspector Adam Tatem as they quickly retreated inside the Stonewall Inn and shut its doors.\n\n#\n\n# _Lancing the Festering Wound of Anger_\n\nHoward Smith had never been inside the Stonewall Inn and was immediately struck by how dark and dank it was and how strongly it smelled of beer. As soon as Pine and his officers had closed the club's doors, they used the tables inside to barricade the doors. Tables and other objects were also piled up behind the wooden structures inside the Stonewall Inn's windows to reinforce them: the police were now using the very plywood forms reinforced with two-by-fours to try to keep gay people on the outside that the Mafia had built to keep the police out.\n\nBut the objects used as barricades proved poor insulation against the crowd's rage, for at this time, in Pine's own words, \"all hell broke loose.\" He called again for assistance over his portable radio and again got no response.\n\nWhile Smith had taken refuge with the police inside the Stonewall Inn, Truscott had climbed on top of a residential garbage can next to the stoop Craig Rodwell and Fred Sargeant had mounted earlier to get a better view of the action. There Truscott witnessed the immediate reaction to the police retreat: \"The exit left no cops on the street, and almost by signal the crowd erupted into cobblestone and bottle heaving.\" Indeed, Truscott almost fell to the sidewalk when two men whisked the garbage can he was standing on from under his feet and threw it through the Stonewall Inn's west window, where it sat, supported by the window ledge.\n\n\"Some of those guys on the streets [had been] pushing the sides of the paddy wagon and rocking it back and forth and stuff. [When] they pulled the paddy wagon away, they threw the garbage cans through the window. When they threw the garbage cans through the window, that was it. Until then there was just a bunch of people in the street yelling and raising hell.\"\n\nA general assault now began on the Stonewall Inn using anything and everything the crowd outside could get its hands on: garbage, garbage cans, pieces of glass, fire, bricks, cobblestones, and an improvised battering ram were all used to attack the police holed up inside the Stonewall Inn. This was the high point of the violence on the part of the crowd of what would become known as the Stonewall Riots, a fury aimed at the police inside the Stonewall Inn, as if all those in the crowd assembled that night on Christopher Street had decided to personally revolt against the police to express a collective _cri de coeur_.\n\nMichael Fader remembers it this way: \"We all had a collective feeling like we'd had enough of this kind of shit. It wasn't anything tangible anybody said to anyone else, it was just kind of like everything over the years had come to a head on that one particular night in the one particular place, and it was not an organized demonstration. It was spontaneous. That was the part that was wonderful.\n\n\"Everyone in the crowd felt that we were never going to go back. It was like the last straw. It was time to reclaim something that had always been taken from us. It was something that just happened. All kinds of people, all different reasons, but mostly it was total outrage, anger, sorrow, everything combined, and everything just kind of ran its course. It was the police who were doing most of the destruction. We were really trying to get back in and break it free. And we felt that we had freedom at last, or freedom to at least show that we demanded freedom. We didn't really have the freedom totally, but we weren't going to be walking meekly in the night and letting them shove us around\u2014it's like standing your ground for the first time and in a really strong way, and that's what caught the police by surprise. There was something in the air, freedom a long time overdue, and we're going to fight for it. It took different forms, but the bottom line was, we weren't going to go away. And we didn't.\"\n\nThe police, who had blithely assumed that since they were just dealing with a bunch of fairies they would be unchallenged, found that it was as if the fey beings had suddenly and inexplicably metamorphosed into raging tigers. Pine, author of the U.S. Army's manual for hand-to-hand combat in World War II and victim of a mine explosion in the bloody Battle of the Bulge, said simply, \"There was never any time that I felt more scared than I felt that night.\" Pine's partner, Charles Smythe, who had fought alongside him in Africa and Sicily in World War II, would recall \"I was still shaking an hour later. Believe me, I've never seen anything like it.\"\n\nWhile Truscott saw\u2014indeed, felt\u2014the garbage can being thrown, according to _Rat_ reporter Tom, it was the man in the dark red tee-shirt Tom had earlier seen dancing in and out of the crowd who had the honor of throwing the opening volley after the police retreated inside the Stonewall Inn: \"The cat in the tee-shirt began by hurling a container of something at the door. Then a can or stone cracked a window. Soon pandemonium broke loose.\" Morty Manford recalled watching one of the second-floor windows above the Stonewall Inn break: \"With the shattering of glass the crowd [went] 'Ooooh!' \" For Manford, the breaking of the glass was \"a dramatic gesture of defiance.... We had just been kicked and punched around symbolically by the police. They weren't doing this at heterosexual bars. And it's not my fault that the local bar is run by organized crime and is taking payoffs and doesn't have a liquor license.\" When Manford saw the window break, \"there was a slight lancing of the festering wound of anger at this kind of unfair harassment and prejudice.\"\n\nWhether the anonymous man in the dark red tee-shirt was the first to throw something as the riot reached its climactic stage on Friday night or not, people closer to the street played a key role in unleashing the crowd's fury. Kevin Dunn, a nineteen-year-old gay man, dressed in hippie attire and a true believer in the peace movement, recalls how he stood thinking to himself, \"'I'm sick of being told I'm sick' and went to grab something\u2014I don't know if it was a halfway-filled milk carton\u2014it was some kind of a carton\u2014and I was just about ready to throw it, but I stopped and said, 'But you're not supposed to be violent, you're against violence.' \" But as Kevin hesitated, \"a big, hunky, nice-looking Puerto Rican guy\u2014but big mouth\u2014yelling out (at the police) next to me... took that thing out of my hand and threw it! And it was one of the first things that got thrown at the Stonewall. And I just thought after he did that, 'Ah, what the hell! Yeah!' And then I started scrambling to pick up whatever I could find.\" Indeed, the street element seems to have been critical from the riot's inception. According to Bruce Voeller, shortly after the police let the Stonewall patrons not under arrest leave the club, \"a crowd gathered and some of the watchers jeered the police. After a few interchanges, a young Puerto Rican taunted the gays, asking why they put up with being shoved around by cops.\" Voeller goes on to write that according to some accounts, it was this same Puerto Rican who unleashed the barrage of objects thrown at the police by hurling a beer can, adding that it is possible that the young man was not gay.\n\nBob Kohler agrees that the street element was important but credits the gay street kids he knew from Christopher Park: \"Billy was standing next to me, and I remember the first thing, Billy started to run and I grabbed him, because Billy had a court case pending. I said, 'Stay out of this, Billy! Don't get involved!' because it just looked like it was going to be a little skirmish kind of thing. You had suddenly the taunts. It wasn't, 'Oh, there's the girls in blue.' It was nasty, and suddenly all kinds of things were being thrown. Garbage was being thrown, cans. People were fighting. People were yelling. I just remember suddenly the mood getting very ugly. I remember being taken off guard and being scared, because people were sort of running towards the cops, and it was obvious that there was going to be trouble. That I saw. A lot of the street kids started to run towards the Stonewall.\" Before they took off, many of them handed him their stolen wallets, credit cards, and \"so-called diamond rings.\"\n\nThat at least three of these street kids were literally on the front line of this action is documented by a photograph taken that night by freelance photographer Joseph Ambrosini. The caption underneath the photograph\u2014published in the Sunday edition of the New York _Daily News_ just a little more than twenty-four hours after the riot began\u2014describes the photo as showing the crowd attempting to \"impede police arrests outside the Stonewall Inn.\" On the left of the photograph is Jackie Hormona, face-to-face with a police officer, his left hand reaching toward the officer. To his right one sees the feminine young man Tommy knew from the street with the plucked or shaved eyebrows. To the right of that youth stands the young man Tommy always saw wearing a coat or suit and usually a tie. The face of this young man, who sports a Beatles haircut, is partly obscured by another youth with his back to the camera. In addition to this photographic evidence regarding the role played by the street kids on Friday night, there is the memory of Jerry Hoose, who was summoned there by a phone call from his close friend John Goodman. Goodman told Hoose on his arrival at the scene \"that Hormona had kicked a cop, maybe, or punched a cop and then threw something through the window, and then everybody got going. But he was there and he attributed it to Jackie, and I believe that because she was a lunatic. And all the other queens like Zazu Nova Queen of Sex and Marsha P. Johnson had got involved. It wasn't just the drag queens; it was the street people outside of the Stonewall.\"\n\nMany other credible witnesses offer similar testimony concerning the gay street youth. Lanigan-Schmidt says, \"What I know definitely from my own experience is that the people who did the most fighting were the drag queens and the hustlers. [They] fought with the same ferocity they would fight with when any situation of survival put their sense of dignity on the line, very much like Bob Dylan's 'When you ain't got nothing, you got nothing to lose.' \"\n\nIt is as if on the morning of June 28, 1969, America symbolically got back the anger she had created by her neglect of her most despised children: the fairies, queens, and nelly boys she had so utterly abandoned, saying she did not want them.\n\nThe question of transvestite participation in the riots is complicated by differences in the gay male community that have developed since the Stonewall Riots. Scare and flame queens hardly exist anymore, and Garvin insists that it was flame queens who contributed most to the fighting: \"When people say, you know, it was the drag queens that started it, it wasn't the drag queens, it was the flame queens. The ones who were getting angriest and giving attitude were the flame queens.\" While Fader is a credible witness who says that \"no one group did any more bravery than any other group that I saw,\" the preponderance of witnesses who are both credible and who witnessed significant amounts of the action agree that the most marginal groups of the gay community fought the hardest\u2014and therefore risked the most\u2014on this and the following nights. (It is worth noting in this context that the lesbian who set the crowd afire with her physical courage was, from all available evidence, non-gender-conforming and, very possibly, transgendered, to use today's parlance.)\n\nStill, it is important to note that it was not only homeless or street youths who were fighting the police. Men like Robert Bryan, a middle-class college graduate, joined the fray. According to him, just before the police barricaded themselves inside the Stonewall, \"every so often, they would sort of reach out and grab somebody from the crowd and pull them in, and they were beating them in the doorway with clubs. They had a circle around; they'd grab somebody at random and beat them in the doorway. I was so appalled at what was going on that I came running up behind one of these policemen who was bent over with a club and, with all my force, kicked him in the seat and knocked him over. Well, at this point, one of the policemen charged out of the crowd after me, and I took off, running. He chased me four or five blocks, almost all the way home to Thompson Street, and I finally escaped from him.\"\n\nJohn O'Brien says that three groups were \"going after the police\": \"You had an underclass of kids who were either hustlers or homeless and a few people like me who were political agitators and troublemakers. Fortunately that inspired some others who were students, which made up the other element of people.\"\n\nWhile it seems beyond dispute that most of the acts of violence during the riots were committed by the more marginalized members of the gay and lesbian community, it is evident that when the police retreated into the Stonewall Inn everybody outside was angry. As Craig Rodwell recalled: \"[After the police drove off in the paddy wagon] there was just like a flash of group\u2014of mass anger.... And this may be five or six hundred people.\"\n\nTruscott wrote that \"the crowd erupted into cobblestone and bottle heaving.\"\n\nManford recalled: \"And it escalated. A few more rocks went and then somebody from inside the bar opened the door and stuck a gun out. Their arm was reaching out with a gun telling people to stay back. And then withdrew the gun, closed the door and went back inside.\"\n\nInside the Stonewall Inn, Howard Smith heard \"the shattering of windows, followed by what we imagine to be bricks pounding on the door, voices yelling. The floor shudders at each blow.\" As the crowd of hundreds shouted \"Gay power!\" and \"We want freedom!\" garbage cans, bottles, beer cans, and any and everything else that could be found was hurled at the Stonewall Inn. Pine recalls how \"now they really in earnest started to come after us. We covered everything [but] whatever we could find to put up against the windows and the doors didn't last very long. They began to batter this down and made some holes.\"\n\nSmith and the police \"found little holes in the wood [behind the Stonewall's windows], little peepholes, so we could look out into the street to see how big the crowd was without getting hurt because they wouldn't know we'd be looking through these little pinholes. And I took my turn and looked out. And the crowd was enormous by then.\" To Smith, it seemed like thousands of people.\n\nSmith asked Pine, \"Where are the reinforcements?\"\n\n\"I don't know. There must be some mix-up,\" Pine answered.\n\nSmith had the impression that Pine did not even want to talk about it.\n\nAs the caravan of police vehicles departed the Stonewall Inn with their prisoners, John O'Brien, out for his usual Friday night recreation of debating politics while cruising, heard the sirens but, like Dave Van Ronk, thought nothing of it. However, when O'Brien saw some young men running down Waverly Place, his interest was piqued enough to ask them what was going on. When they explained to him that the police were raiding the bars and that they were escaping, he decided to walk up Christopher Street and take a look for himself. When he arrived he walked over to the sidewalk on the north side of Christopher Park and saw an enraged crowd yelling and throwing a barrage of objects at the Stonewall Inn. When it was explained to him that the police had raided the club and were now trapped inside, O'Brien needed no further encouragement to join in.\n\nOne of the first things O'Brien saw after he arrived at the scene was men stuffing pieces of paper into cracks at the bottom of the plywood inside the Stonewall's window and holding cigarette lighters to them. Seasoned street fighter that he was, O'Brien could hardly believe his eyes as he watched the youths deploy their lighters. \"Some people went up to it, they'd light it, they'd run away.\" But at this point in the evening O'Brien says that very few people stood directly in front of the Stonewall for long, because they feared the police might fire on the crowd. \"There was a little bit of unsureness about what's next, and everybody just played on the spontaneity of what happened right before that.\" O'Brien thinks at this point people in the crowd were \"just egg[ing] each other on. We wanted to outdo the last person.... Nobody knew how far this was going to go.\" Just as Garvin recalls bricks being used for work on a building, O'Brien remembers that work on the street was being done on Seventh Avenue South close to the Christopher Street subway station and this made it easier to uproot cobblestones, which \"played an important part... [when] people went looking for [something to escalate with].\"\n\nO'Brien noticed a group of people on the sidewalk who were trying to uproot the parking meter that the street kids used to swing on, first trying to dig it out of the ground and then rocking it back and forth to free it from the sidewalk. The men wrestling with the meter looked like street people to John. \"One was a dirty blondish-brownish biker type, and a couple of other guys like that. One muscular guy who was with a couple of other guys working together got it out of the ground.\"\n\nSeveral gay men now used the parking meter as a battering ram on the Stonewall Inn's doors. Danny Garvin remembers that the use of the converted parking meter got an enthusiastic response from the crowd: \"It was like, 'Okay! Boom! Boom!' \" It took more than one attack, however, on the Stonewall Inn's doors to force them open. As the men continued to pound the heavy doors with the parking meter, others in the crowd grabbed whatever they could find to throw. The attack on the police created a cacophony as the sounds of glass shattering up and down the street mixed with the pounding of the parking meter on the doors, while cries of \"Liberate the bar!\" filled the air. All this was punctuated by breaking bottles thrown at the demonstrators from apartment dwellers along Grove Street who wanted to get some sleep. White witnessed the continuing assault on the Stonewall's doors: \"The ramming continues; the boys back up to the park fence, take a flying start, collide with the door as the crowd cheers wildly. Cries of 'We're the pink panthers!' A mad Negro queen whirls like a dervish with a twisted piece of metal in her hand and breaks the remaining windows. The doors begin to give.\"\n\nSeveral observers insist that the crowd only wanted to get back inside the Stonewall Inn to continue partying, implying that the riots had no political or historical meaning. Such a reading ignores the deeper side of this instinctual reaction of the crowd, who sensed that something valuable to them was being taken away and reacted accordingly. (Seymour Pine's intention was to shut the Stonewall down for good.) Fader explains this with considerable insight: \"The crowd was really wanting to bust in and go back in; it was that strong a feeling. We felt like we'd been booted out of our home for no reason.... And the police were in there, and they weren't coming out. We wanted to get back at them somehow. And of course there was no way to really do that with guys who have guns and sticks, and the best we could do, and there was not too much to deal with\u2014pavement stones and people throwing [things]\u2014they didn't have the equipment, so that's why they used a parking meter as a battering ram. We wanted really as [much as] one could without equipment to go after the police and free the Stonewall or whatever you want to call it. And of course you couldn't. People were throwing rocks and different kinds of things. They were symbolic gestures, and they were enough. It was a spontaneous thing. It was a totally emotional\u2014everyone having to live this life where we had to sneak in and all of that stuff, and then the place that everybody likes, to have it taken away in such a brutal way, we wanted to try and rectify this somehow to make it right and get back.\"\n\nInside the Stonewall Inn, the police and Howard Smith grew more anxious when they heard the windows being shattered, and then a loud banging made by what Smith assumed to be \"bricks pounding on the door,\" all mixed in with yelling voices. Smith felt the floor shudder at each blow the doors received.\n\n\"Aren't you guys scared?\" he asked.\n\nThe police answered that no, they weren't, but Smith noticed that they looked \"at least uneasy.\"\n\nThe night reverberated again with the boom of the parking meter on the Stonewall's doors.\n\nWhile most of the crowd were expressing their anger at the police, Tommy Lanigan-Schmidt stood in wonder as he watched the scene unfolding in front of him. To Tommy, who thought naturally in terms of biblical allegories and art criticism, the big double doors of the Stonewall Inn seemed like church doors, so that when he saw them drawn shut by the policemen who had retreated inside the club, he had begun to wait. He wasn't clear exactly what it was he was waiting for but at the same time felt he was waiting \"for something that you know is going to happen.\" Dim echoes of memories from a night years past had suddenly begun to resound inside him.\n\n_In the middle of the night: death, and a crowd that has been made to leave a building goes outside and, reluctant to leave, mills around, waiting. Words are chanted over and over, faster and faster. The yellow flames of lit candles. At the chanting's climax the priest bangs loudly on the double doors, which suddenly spring open. It is the Resurrection_.\n\nWith a final loud blast, the doors of the Stonewall Inn swung wide open.\n\nSuddenly Tommy felt that it was Easter again: a new beginning in the middle of the night.\n\nMeanwhile, Bob Kohler, still standing next to the park fence, noticed several people in the park quietly and methodically pouring a liquid into several empty Coke bottles, a couple of which were distributed by a man he knew by sight, who kept the rest of the bottles for himself.\n\nAs the mayhem continued, the police inside stayed busy as they tried to keep the Stonewall's doors shut, for the mob attacked the doors again each time the police closed them. Each time the doors opened, the crowd threw all kinds of objects inside at the police. Smith wrote that \"bottles and beer cans fly inside. Pine and his men immediately rush to shut it.\" On at least one occasion, \"[a] customer in the crowd picked up a shard of glass from the smashed windows and lobbed it inside. It bounced off the inner wall.\"\n\nOutside the Stonewall Inn, pandemonium reigned as the protesters gave full vent to their fury. Martin Boyce, who spent much of this evening with his friends Tommy Lanigan-Schmidt and Birdie Rivera, recalls how \"all of a sudden the whole street now had had it, and windows started cracking, and people attacked cars and moved cars back and forth, but let the straight people out, who were in terror, really, but nobody hurt them, ever. It's amazing how controlled the rage was, even though it was so hard. This black bus driver, I remember, was caught in the middle. He was laughing, but nervously. I don't think they'd ever seen anything like that, and we hadn't either.\n\n\"Birdie was a natural-born fighter. If she could grab something, no matter what, Birdie was going to get in on it. And Birdie grabbed some oranges and spread them across this car's windshield. I was saying, 'All right, girl, go! That's my sister! Now, that's the sister I love!' But the guy inside opened the door [saying], 'You motherfuckin' bitch!' \" Birdie and Boyce had assumed that because the driver was conventionally masculine-looking he was straight. They both suddenly realized that Birdie had picked on a fellow gay man in error. At that point, they both fled the enraged driver.\n\nThe riot had its surreal moments. The gay street youth kept running up to Bob Kohler now and then, loading him down with wallets and various other stolen goods. Trying to hold on to his armfuls of contraband, keep his dog, Magoo, from escaping, while restraining Billy from joining in the mayhem, Kohler at times resembled a circus clown who was pulling off an especially tricky balancing act.\n\nSomewhere around this time, Craig Rodwell and Fred Sargeant called the _New York Post_ , the _Daily News_ , and _The New York Times_ \"because we wanted to make sure this got in the papers.\" Craig explained, \"I immediately knew that this was the spark we had been waiting for for years.\"\n\nThe police's problem became even more complicated when the rioters started battering the Stonewall's west window's wooden subwall with the parking meter. \"One of the big plywood windows gives, and it seems inevitable that the mob will pour in. A kind of tribal adrenaline rush bolsters all of us; they all take out and check pistols. I see both policewomen busy doing the same, and the danger becomes even more real,\" Smith wrote. Inside the Stonewall Inn, the wooden wall having almost given in so terrified Smith that, having visions of being trampled by the mob, he went searching for a weapon behind the Stonewall's long bar. There he managed to find a fire ax next to a fire hose on the wall. He quickly stuck it under his belt \"like a scimitar because I wanted to keep writing notes, and Pine found that very funny.\"\n\nPine's mood changed quickly when he realized how vulnerable he and his men were. \"We shut the doors and tried to barricade them with tables that were in the place, and we tried to pile things up against the plywood that was backing up the windows. They broke all the windows and crashed open doors regardless. And it was at that point that I realized that my men were very nervous. And I was very apprehensive.\"\n\nOutside, Kohler saw a bottle go sailing through the air and land in the Stonewall Inn, followed by a small trail of smoke emerging from the club.\n\nThe \"Molotov cocktails\" that the crowd began to throw into the Stonewall made Pine and his officers even more anxious: \"There were bottles that came in that exploded with some kind of flame, and we were able to put those out with the fire hose that we had. We were very worried because we didn't know how long we could put these Molotov cocktails out, because they were gasoline and all we had was water. They didn't have the kind of fire extinguishers that would put out a fuel fire.\"\n\nDick Leitsch, who had cut short his packing and rushed downtown as soon as he heard about the rioting on the radio, attests to the kind of fuel being used on the Stonewall Inn: \"I saw [a rioter] take a can of lighter fluid, a Ronson or something, out of his pocket and squirt it all over this plywood, over the front window of Stonewall, and put a match to it. So there were flames. They were blue and had little yellow tips, and lighter fluid is very combustible. It burns, and it's gone.\"\n\nPine had no way of knowing what substance was contained in the bottles landing inside the Stonewall Inn. Given all the tension that Pine and his officers were already under, it is easy to see how they might have feared that they were in danger of dying in a conflagration.\n\nThis fear for their lives combined with the inability to communicate with the outside world put the police and Howard Smith in a frantic frame of mind: \"Every time we tried to use the [portable] radio to call for assistance, a message came back, 'Disregard that call.' Somebody else apparently had our frequency in the crowd, and so we couldn't get a message through. The phone lines apparently were cut, because we couldn't use the phone.\"\n\nPine's men found the fire hose on the wall and a fire extinguisher nearby. They unrolled the hose and used it and the extinguisher to put out the fires caused by the bottles that were lobbed inside at them. However, the fire extinguisher was soon empty, leaving the police inside the Stonewall with only water to fight what Pine and his officers believed to be gasoline bombs. Spontaneously all the law enforcement agents\u2014including the women\u2014took their pistols out. Pine surveyed the officers under his command: \"Everybody was lined up and they had their guns out, and that's when I became frightened that they would shoot, and I ordered them not to shoot, that I would be the first. You're so tense and it's the easiest thing in the world for a shot to go off at that time, and one shot going off will start everybody else off. That's what I feared, because the people were so close to us that there would have been people killed over nothing. The crime that we were arresting anybody for that night didn't warrant anybody losing his life for, and I was afraid that that's what was going to happen. I don't think there was anybody there who wasn't really perspiring\u2014I mean _really_ sweating\u2014because it was touch and go.\"\n\nThe fear of a fiery death, the great din of voices from outside, and the incessant reverberations from metal and rock pounding against the walls and doors of the club were about to overwhelm the police officers when an old war memory suddenly reawakened inside Pine: \"It came back to me as if it was yesterday... this tremendous fright, where everybody is crapping in his pants. And this was a situation that was equivalent to that.\"\n\nPine now knew what he had to do. \"I went down the line with the policemen, and I started to call each man by name: 'How do you feel, Joe?' \" As he addressed each officer, Pine placed his hand on him. \"And they all answered. It's the only way you break the tension, once there was physical contact between them and they have to answer you, 'cause you're holding your breath.\n\n\"Then I got to the reporter, who was standing there with a fire ax in his hand, and I asked him how he was, and he said, 'I'm okay, but I'd feel a lot better if you had the ax and I had the gun.' \"\n\nEven with Pine's cool head and leadership, it was still hard for the police to withhold their fire, for in such a tense situation time seems to pass much more slowly than in ordinary life. At one point, one of the officers stationed at the Stonewall's entrance momentarily lost his composure and threw his revolver at the crowd. It hit the door frame and landed on the ground. Pine remembers why the situation was so difficult: \"I knew that at any moment it could break into a real killing. It was very hard to keep holding while they were attacking and you were doing nothing.\"\n\nBut Pine also knew the dangers of not \"doing nothing\": \"I'd been in war situations where somebody panics and, instead of holding fire, shoots; and once that first shot is off, everybody lets go.\" Even so, Pine felt that if they had fired, they still would have been killed: \"If they would have broken through, no matter what we would've fired they would've continued going. It's the old riot situation: the people from the back don't know what's going on in front and they keep going and we would've been killed.\" Smith exactly shared Pine's estimate of the situation: \"It looked like they [the police] were gonna be killed. Because what if the crowd did break through and he yelled: 'Fire!' I don't think that would have mattered. There were so many people in the mob, only the people in the front would be hit and then we'd be overrun. I don't think firing would have saved anyone's life. It looked that enraged, the mob.\" Smith expressed this in his 1969 account by writing: \"By now the mind's eye has forgotten the character of the mob; the sound filtering in doesn't suggest dancing faggots any more. It sounds like a powerful rage bent on vendetta.\"\n\nHoward Smith agrees that Pine's presence of mind was critical in preventing a bloodbath:\n\nI was sure we were gonna be killed, but he was very, very good. He went from guy to guy. He made sure their guns were ready and all that. But he said, \"Anybody who fires their gun without me saying 'Fire!' is gonna be in big, big trouble. You'll be walking the loneliest beat on Staten Island for the rest of your career.\" He was very threatening and very in control of the guys, even though it was very apparent he was scared also. But he didn't want anybody to start firing because I think he knew one cop fires, they all start, and people could really be killed. And he seemed to have a very good sense that you don't do that.\n\nPine was very sharp. He was definitely in command of his men. Some he specifically said: \"You stand over there and don't leave that spot. You watch the door. You watch the corridor. You, I want in the back part of the bar in case there is some way in that we don't know about. You stand back there.\" He definitely positioned everybody. He always had at least two [police in the corridor] facing the door with guns drawn. It was the first time in my life\u2014and ever since then when I read this in articles about any police riot anywhere\u2014where they talk about good supervision and how important it is, how there always has to be a sergeant or lieutenant in any important cop event\u2014I had never understood that until I saw him working in that bar. And I was very impressed. It wasn't like, \"Oh, well, we're in here. They come in, we shoot them.\" Absolutely not. He was being very careful, but he had very little to work with.\n\nIndeed, Pine had less to work with than Smith realized. In 1969 automatic firing handguns were rare. The officers would have only been able to fire as rapidly as they could pull the trigger, but also most of the guns the police had on them that evening would have had to be reloaded after firing five or six shots. As angry as the crowd was, Pine did not feel that he and his men could have stopped it with only several shots each. Moreover, Pine's own gun, a .38mm Smith & Wesson with a wooden handle, had a very short barrel, only about one inch long, reducing its range of accuracy to only about eight to ten feet.\n\nWhile a couple of officers remained posted at the Stonewall's door, Smythe and some other officers and even Smith searched for an escape. Among the prisoners held inside was a bartender who had the keys to the place, and Pine intermittently barked questions at him, demanding to know what various things were and peppering him with queries about the club's layout.\n\nOutside, Garvin recalls, \"it was like being in a war. People were crying. People were cut up. I mean, people would throw bricks, but you didn't always hit a cop. Sometimes you'd hit another queen. So you didn't know when you saw someone cut were they cut because the cop hit the guy or were they cut because of running and falling or what? It was great, but you didn't want to get hit by nightsticks. Yet I had to see what was happening. I had to see! This was unbelievable. My God! It was like\u2014these are the guys at the Stonewall who were\u2014my God, look at... They busted open the doors. I can't believe it.\"\n\nFinally the police succeeded in finding a vent in the back up near the roof, and they struggled to get the smaller of the two policewomen outside through this opening. Pine instructed the woman to go across the roof and to climb down\u2014but not on Christopher Street\u2014report the fire at the Stonewall to the firehouse on the adjacent block, and use a telephone to send an emergency signal for assistance.\n\nThe crowd outside again focused its fury on the Stonewall's western window. Tom watched as \"a sort of wooden wall blocking out the front plate glass windows was forced down.\" Smith wrote: \"One of the big plywood windows gives, and it seems inevitable that the mob will pour in.\" But just then the police inside turned the Stonewall Inn's fire hose on the crowd, hoping to stop the rioters. From the inside, Howard Smith reported: \"The detectives locate a fire hose, [but] can't see where to aim it, wedging the hose in a crack in the door. It sends out a weak stream. We all start to slip on water and Pine says to stop.\" Outside, Truscott saw the youths \"cavort in the spray\" in \"momentary glee.\" Tom wrote scornfully: \"The pigs carried futility to the extreme and turned the fire hose on the mob through the door. Jeers, derision. One of the kids shouted 'Grab it! Grab his cock!' \" While the police did momentarily stop the siege, they did so only because they had provided some entertainment\u2014at their own expense\u2014for their tormentors, adding yet another humiliation to the already considerable roster of such embarrassments the raiders had already suffered that evening.\n\nThe ineffectual fire hose could divert the crowd for only so long, and, as Truscott observed, the rioters soon \"were able to regroup forces and come up with another assault,\" apparently a simultaneous one on both the door and the west window. Edmund White wrote: \"The door is broken down, and the kids, as though working to a prior plan, systematically dump refuse from waste cans into the Wall, squirt it with lighter fluid, and ignite it. Huge flashes of flame and billows of smoke.\"\n\nFinally, it seemed to Pine that he and his officers might have to open fire on the crowd to have any chance of saving their lives: \"We're inside and the fires are coming in and we're putting them out\u2014all the time we're dodging the bricks that they were throwing in\u2014and then they crashed through with this parking meter. And then I was sure we were gonna have to fire, but I was very reluctant to give the order, and I still kept saying, 'Nobody fire! Nobody fire! Let's back up if we have to. Help's going to be coming.' \"\n\nThe crowd outside repeatedly attempted to set a fire through the now-gaping hole where the west window had been. \"Some then lit a trash can full of paper afire and stuffed it through the window. Flames leaped up.\" Morty Manford witnessed the same event: \"People took a garbage can, one of those wire mesh cans, and set it on fire and threw the burning garbage into the premises. The area that was set afire is where the coatroom was.\" That night the closet was set on fire both symbolically and literally.\n\nWhile watching the riot, Truscott had heard more than one person say, \"Let's get some gas,\" yet he still found it a \"shock\" when \"the blaze of flame... appeared in the window of the Stonewall.\" Di Brienza watched as \"some small, scrawny, hoody-looking cat threw a can of lighter fluid through the broken window, and lit it up.\" On the other side of the Stonewall Inn's brick front, Smith saw an \"arm at the window. It squirts a liquid into the room, and a flaring match follows. Pine is not more than 10 feet away. He aims his gun at the figures.\"\n\n\"Then,\" Pine recalled, \"as luck would have it, we heard the sirens.\"\n\nThe sirens that Pine, his officers, and Howard Smith heard inside the club did not belong to the police but to two fire trucks that circled the area, no doubt because they could not get through the packed streets. White wrote: \"Two fire engines pull up,\" accompanied by the patrol wagon Pine had urged to \"hurry back.\" Manford saw the \"fire engine [start] coming down the block,\" evidence that the policewoman who had managed to escape the Stonewall had informed the fire department that the Stonewall was burning.\n\nHoward Smith recalls, \"When I heard the sirens, I was pretty damn happy.\" Pine remembers all the tension disappearing \"because we knew the Tactical Patrol Force would come in very large numbers.\" The Tactical Patrol Force, then known as the TPF or, more commonly, the riot police, was an elite group of police held in reserve in various parts of the city where trouble might be expected to break out. They had special riot equipment such as tear gas and shields and instead of hats wore helmets with plastic visors.\n\nThose trapped inside the Stonewall Inn felt relieved on hearing the sirens, but they had been so unnerved by the fury of the crowd that they were reluctant to venture outside. Smith recalls: \"We made sure there were really a lot of cops there, and then we went out.\" The first reaction of the police and Smith was to check each other to see that each of them was all right. After the head count, they giggled for several minutes to release all the pent-up tension. As the laughter died down, Smith began catching up on his note taking and jotted down: \"The people around me change back to cops. They began examining the place.\"\n\nThe Sixth Precinct police began to move the crowds away from the Stonewall's doors, and Pine, his officers, and Smith walked outside the club into the welcoming night air. Smith, still shaken, stuck close to Pine. \"I didn't want trouble from the newly arrived cops, and I didn't want trouble from the crowd. I was very afraid. I felt I had to find a safe way to re-go through the crowd and back to my office so people wouldn't think 'There goes a cop!' \"\n\nSeveral local radio patrol cars pulled up with their red lights dancing, and one of them soon sent out a mobilization signal, either a 10-13, which means \"policeman in trouble\u2014respond,\" or a 10-41, which would bring the TPF. Whether in response to the signal just sent by the patrol car or to the earlier telephone call by the policewoman who crawled through the vent, police cars from the Fourth, Fifth, and Tenth precincts soon arrived bearing seventeen officers\u2014one sergeant and four or five patrolmen from each of the precincts\u2014all of them predesignated to respond to emergency calls. The seventeen joined the handful of officers who had already shown up from the Sixth Precinct. Pine next used the radio cars to call for an ambulance to take Gil Weissman to nearby St. Vincent's Hospital to have the wound to his eye taken care of.\n\nAlthough the TPF had not yet arrived, Pine now had the patrol wagon he had so sorely been lacking and decided to go ahead and load the alcohol and the remaining prisoners in it. He had the prisoners marched out of the Stonewall Inn to the patrol wagon, parked on Christopher Street near the corner of Seventh Avenue South, presumably because enough of the crowd had not yet been dispersed for the patrol wagon to park right in front of the Stonewall Inn.\n\nAmazingly, Pine still had not learned from the fiery experience he had just been through: \"Even though we had just had all this trouble from these docile homosexuals\u2014who weren't docile anymore\u2014we still didn't think that anybody wouldn't go peacefully.\" He let the prisoners leave the Stonewall with only Van Ronk handcuffed.\n\nVan Ronk vividly remembered leaving the Stonewall Inn: \"I heard sirens. The door opened. Some cops came in and they rousted us, 'Up, out!': that was it. The cops made us [into] sort of a flying wedge to get through [the crowd] to the paddy wagon. There were more people out there when I came out than when I went in. Things were still flying through the air, cacophony\u2014I mean just screaming and yelling, sirens, strobe lights, the whole spaghetti. From what I saw, that mob was not cowed. It would have taken something to get them to disperse. They were loaded for bear.\"\n\nPine recalls, \"Fights erupted with the transvestites, who wouldn't go into the patrol wagon. Some [transvestites] who hadn't even been in the Stonewall came over and started a fight with our guys.\" Marle Becker witnessed the transvestites fighting from across the street: \"All I could see about who was fighting was that it was transvestites and they were fighting furiously.\" Becker's report of the fury with which the transvestites fought is believable, as Pine says that \"some of our policemen [were] roughed up by the transvestites. The only fighting we did with anybody was putting them into the patrol wagons, and that was only the transvestites. The other guys went peacefully.\" The new transvestites were added to the haul Pine had already made.\n\nThere is another possible interpretation of the preceding account: Pine says that no transvestites went on the first patrol wagon trip, but reliable accounts recorded in 1969 have transvestites being loaded on the patrol wagon before Pine and his officers were barricaded inside the Stonewall Inn. Also, Van Ronk said he does not recall any transvestites being in the patrol wagon on his trip over to the police station. It is possible therefore that Pine's memory is inaccurate and that the incident he remembers is actually the combined accounts of the two times the patrol wagon was loaded that evening. In this case, Becker's witnessing of the transvestites fighting the police would be not of the second but of the first loading of the paddy wagon. The problem with this solution of the puzzle, however, is that Becker's positioning of the patrol wagon matches Van Ronk's recollection of where the patrol wagon was when he entered it: up close to Seventh Avenue South rather than directly in front of the Stonewall Inn, where it seems the first patrol wagon was loaded. Either way\u2014whether the transvestites resisted being loaded into the patrol wagon on the first or second trip (or, more improbably, on both trips)\u2014it is probably more significant to note that the transvestites fought as they were put inside the patrol wagon and that the crowd was still not cowed when other police officers rescued Pine and his officers from inside the Stonewall Inn.\n\nBefore the patrol wagon left Christopher Street, the TPF arrived in two buses. They came down Christopher Street from the west\u2014against the flow of traffic\u2014and parked between Seventh Avenue South and the Stonewall, close to the corner. Bob Kohler watched as the TPF got off the buses. \"I had been in enough riots to know that the fun was over. These guys had helmets and lived to break heads.\" Moreover, they had special reason to be angry that evening: \"The cops were totally humiliated. This never, ever happened. They were angrier than I guess they had ever been, because everybody else had rioted. Everybody in America who had a beef had already rioted, but the fairies were not supposed to riot. And nobody else had ever won. The cops realized that just by having to call in reinforcements, just by barricading\u2014no other group had ever forced cops to retreat before, so the anger was just enormous. I mean, they wanted to kill.\"\n\nConcerning the TPF, Pine himself recalls, \"They were not gentle,\" but offers another rationale for their anger: \"They saw the damage that was done to the place, and we had had this policewoman tell them that they were throwing fire bombs in there, and so they were prepared for real action. They scattered the crowds, and then they [the crowd] regathered. They were swinging their nightsticks at the crowd, breaking up these groups of people, and that's when we took off. But that was where my job ended.\" As he left the scene Pine still does not seem to have grasped the meaning of what had happened as the result of the raid he had just led: \"There was no inkling, when the TPF came, that this was going to be a protracted operation that was going to last not only through the night but the following day and night.\" Pine and Smythe left for the Sixth Precinct station house to log in the evidence they had gathered and process the prisoners they had arrested.\n\nHoward Smith also left, returning to his desk overlooking Christopher Park. \"I could see it all from my window when I went back upstairs as it continued for a while, and I tried to write it right away.\"\n\nCraig Rodwell was part of the crowd that the TPF was deployed on Christopher Street to clear. \"They appeared in a V wedge, coming up Christopher Street towards Seventh Avenue because cars couldn't get through or nothing. And we slowly backed up\u2014the same speed as they came forward. And they got to the Square. But meanwhile all of us had just gone around the block and were behind them the other way. And they didn't know that at first. They had just assumed, _Well, we'll just show all of our billy clubs and all of our men's equipment and then the faggots will go home and that would be it_. But it wasn't it. And it's like a tug-of-war that went on for a few hours that night. They would chase us down the street and we'd just go around the block and come back and chant things and throw bottles.\"\n\nO'Brien remembers that \"when they tried to clear the streets is when people resented it, 'cause it came down to 'Whose streets are these? They are our streets. And you cops are not from this area; this is our area. It's gay people's streets.' And I think that was the attitude expressed in different formats.\"\n\nO'Brien feels certain that some of the crowd, both gay and nongay, including those on the streets and those in cars, were purposefully not moving, pretending to merely be curious or helplessly stuck in the crowd. \"People just came by to see what's happening and they couldn't [move] because there were rows of people behind them and the rows behind deliberately [would not move]. A lot of them knew that they were interfering with the cops, and [that] was that passive kind of resistance by a lot of gays. They just stood, but it was their way of showing resistance, and some of the people the cops went after just for standing there, because they want the streets cleared, because then they can bring more police cars in.\" O'Brien adds that it \"was brave to do even that, and it allowed people like me [who wanted to engage the police more directly] to take advantage of the police having to respond to the crowds to go after the police. Because every time their attention was turned towards trying to clear the streets and their backs to us we'd be throwing things at the cops or in front of the cops, going after us.\"\n\nThe police objective was to clear the streets, and given the crowds and the narrowness of the one-way street in front of the Stonewall Inn, the police did not have an easy task, whereas all the crowd had to do was block the section of the street in front of the Inn and traffic was brought to a standstill. To this end a car was overturned, approximately in front of the stoop next to the Stonewall. As O'Brien explained, \"You can wail your siren all day, you're still not getting through if those vehicles aren't moving, and they're not moving if one's overturned and there's a crowd there.\"\n\nAs the TPF and the crowd faced off, some of the youths called upon their street repertoire, probably inspired by the sight of the TPF advancing on them in a line formation. Suddenly the gay street youths linked their arms around one another and kicked Rockette style as they sang their old reprise, but this time changing the word _Village_ to _Stonewall_ :\n\nWe are the Stonewall Girls,  \nWe wear our hair in curls.  \nWe wear no underwear:  \nWe show our pubic hairs.\n\nPerhaps the street youths' humor was more effective than they hoped, for now the fire trucks were used not for firefighting but to help disperse the crowd. Their hoses were turned on the crowd, scattering them.\n\nBeing hosed down was only a temporary setback for the protesters. Over and over that night they re-formed, playing a game of cat and mouse with the TPF, for as Garvin accurately states, \"because of the location, there was no way to contain us.\" The confused geography of the Sheridan Square area of the Village worked very effectively against the TPF. As Bob Kohler watched, the street kids \"were constantly getting over on the TPF. The TPF would chase somebody this way, then the kids would start something behind them so that attention would be taken, and then the TPF would come [the other] way, and then more kids would start something behind them. So the TPF were constantly off guard. It was keeping them on the run constantly.\"\n\nThe kick-line routine was used two or more times that night. Kohler, as well as others who watched, were amazed at the courage of the street kids who dared to mock the TPF to their faces: \"The TPF were down at one end of the street, and the kids lined up in their little Rockette line, and they would kick their legs up, and the TPF would run after them. Suddenly these kids were coming from the other end, and there were maybe four [of them]. They had their arms linked and they were doing Rockette kicks and going towards the TPF. They were taunting them, calling them 'the girls in blue' again and 'Lily Law,' and they would get about as close to them as they felt they should. Then they'd start to run. Some of them, of course, didn't run fast enough.\"\n\nTwo who did not run fast enough were Martin Boyce and Birdie. Boyce recalls, \"We started taunting the cops, 'We are the Village girls, we wear our hair in curls, we wear our dungarees above our nelly knees, and with the constables, we simply hypnotize,' and we're going on, and the police rushed us, and that's when I realized this is not a good thing to do, because they got me in the back with a [night] stick. And I said, 'No, that's not for me.' Miss Birdie was more emboldened by all that, but I wasn't. I split from Birdie at that point.\"\n\nBut others in the crowd did not get off so easily. According to Kohler, \"People were beaten who had no part in it. I remember a very good-looking kid who said, 'What's going on? What's going on?' I said, 'Just stay out of the way.' He said, 'What's it all about?' 'It's a little too complicated.' About twenty minutes later, I saw him being dragged into a police car with blood all over him. He had just come down to find out what's going on, but it was indiscriminate. They lashed out at anybody, at anything.\"\n\nDanny Garvin also witnessed police brutality. So fascinated was he by the evening's events that at one point he went into an apartment building at 98 Grove Street that overlooked the most open area near the Stonewall Inn, the area where the triangular tip of Christopher Park narrows down to a daggerlike point and where Waverly Place juts west around the Northern Dispensary. A friend of his lived in the building, and Garvin thought that from there he could get a better view of all that was happening.\n\n\"I saw a bunch of guys on one side and the cops over there, and the cops with their feet spread apart and holding their billy clubs straight out. And these queens all of a sudden rolled up their pants legs into knickers, and they stood right in front of the cops. There must have been about ten cops one way and about twenty queens on the other side. They all put their arms around one another and started forming a kick line, and the cops just charged with the [nightsticks] and started smacking them in the heads, hitting people, pulling them into the cars. I just can't ever get that one sight out of my mind. The cops with the [nightsticks] and the kick line on the other side. It was the most amazing thing. What was more amazing was when the cops charged. That's when I think anger started. And the cops were used to us calling [them] Lily Law, so the cops were used to some kind of camp coming from us. And all of a sudden that kick line, which I guess was a spoof on their machismo, making fun of their authority. Yeah, I think that's when I felt rage. Because... people were getting smashed with bats. And for what? A kick line.\"\n\nJohn O'Brien was one of those the police were running after. He was in a group of about fifteen or twenty people who banded together and continually taunted the police into chasing them. O'Brien remembers that initially there were two different bands taunting the police, with O'Brien's group repeatedly running up and down Christopher Street from in front of the Stonewall Inn down to Waverly Place and then turning left on Waverly. O'Brien saw that among those who made up the band he had joined up with were some of the youths he and Bill Katzenberg had talked with about the need for gay people to organize. He quickly shared his knowledge of street-fighting tactics by shouting as he ran, \"No, not that way! Don't go up that street, because the cops can follow us. Go down this street and go around. Go opposite of the cars and go through the cars!\" O'Brien explains that \"as someone who had fought the police for a long time I knew that you do not help the police by running in the same direction as the traffic: you go the opposite so they can't get to you in the cars, so the police had to go on foot.\"\n\nThe protests continued into the night. Angry gay men set fires in trash cans and broke store windows and then desperately scoured for more objects to throw. Bob Kohler saw Craig Rodwell go by screaming, \"Gay power! Gay power!\" Occasionally Kohler and some of the street youths would take refuge in Christopher Park, because, for whatever reason, the police did not pursue people into the park that night.\n\nTommy Lanigan-Schmidt's experience must have been typical of many who did not fight but ran along with the crowds through the streets: \"I myself was more part of like a mob that was waving in and out like the ocean. I was part of a mob that had a kind of deep identity and was acting as one force.\" Tommy tended to stay with his friends Martin Boyce and Birdie. Tommy noticed that evening that time took on a special quality. \"[A riot] creates its own time, like weddings, funerals, wakes. A wedding or funeral has something that isn't the time of a clock ticking. It's a time of its own system. It also probably has a time like something of a dream, too.\" Interestingly, Michael Fader noticed the very same phenomenon: \"There was no sense of time; it was all ongoing.\"\n\nAlthough Danny Garvin had lost track of Keith Murdoch \"as soon as the bricks started flying,\" Garvin had continued to participate in the evening's action: \"People kept wanting to come back around to see what was happening. They would chase us up on the end [to] Seventh Avenue; we'd come back around on Tenth Street. They would push us down Christopher Street; people would just sneak around Gay Street and shoot back up onto Christopher.\"\n\nWhile some of the people in the crowd of protesters were having fun and others were frightened, some, such as Ed White's friend Charles Burch, found a certain joie de vivre and exaltation in fighting the TPF: \"They represented an invincible force, which was really fun, because I am enough of a masochist and a romantic that I love the idea of going up against people that you could never look bad losing to, because they were invincible. So you could just try to hurt them and not get hurt yourself if possible. And I developed, in that first encounter, a sense of street-fighting tactics, of how to harass and get away with it; of how to taunt and provoke a response and somehow try to not get hurt. And just years and years of all the resentments and humiliations and things that can come down on the head of a gay person were really\u2014I was really experiencing liberation and radicalization and everything, _bang!_ , right then and there.\"\n\nWhen Jerry Hoose arrived on Christopher Street that night, in response to the phone call from John Goodman, his first response was: \"Thank God!\" Hoose explains that \"I was a very angry person. I'd been waiting for this to happen. I knew it was going to happen. I said, 'Great!' I was the happiest person on the face of the earth; I'm sure I had tears of joy. And I was willing to do anything. I wanted to get into it. Everybody was angry. We were angry people, and we had a lot of reason to be angry. Lot of reason.\n\n\"It was just, to me, more of a feeling of joy, and I think it was to a lot of people. It was like, 'It's about time.' We had finally done something. I was a little sad because I didn't think that we were going in the right direction. I wanted to get the crowd to move down to City Hall and all that, but at least we had done something. And the thing I was most worried about was [that] this would be it. I said, 'This can't be it! This can't be it!' because we were mad. Everybody I knew was mad.\"\n\nO'Brien agrees, \"What was exciting was other people wanted to fight, too. What excited me was that I finally was not alone, that I had found my gay brothers. Until then I had been isolated and alone.\"\n\nEventually, the riot began to peter out, mainly because the rioters were tired or bored. But even as it ended, there was a palpable excitement in the air. William Wynkoop, who lived at 146 Waverly Place at the foot of Gay Street, had been awakened around two o'clock in the morning by the sound of people running. Hearing intermittent shouting as well, he had decided to go outside and investigate. After he had looked around awhile and stood in front of the Stonewall Inn talking with a witness about the evening's events, he noticed that the crowd was beginning to disperse: \"All through Sheridan Square there were people who had been in the crowd and watching what was going on in the bar, [and they] had moved away, I gather, and were standing in groups talking on Christopher Street, all around Sheridan Square, on Seventh Avenue, and over on Grove Street on the other side of Christopher Park, just standing around in groups talking. And here it was, the middle of the night!...They were... moving away because the excitement was over, but they were still so fascinated by it that they were stopping in groups and talking about it. I finally went to bed, probably about four. And when I left, there were still people standing in groups, talking.\"\n\nWhen Kevin Dunn had heard all the sirens, he moved away from the immediate scene, but remained in the area: \"I went back after when things had calmed. And the one thing I remember is that there were little groups of people, maybe like a handful or two on different corners, talking about what had happened and there was this very quiet kind of excitement and buzz in the air. You know, about [whispering], 'Boy, did you catch what happened?' \"\n\nIndeed, the feeling was so real that it could be felt by someone happening across the scene, even if that person was not gay. John Fisk, a heterosexual man who lived in the neighborhood and who had a job in the arts that ended late at night, had a ritual of walking to Smilers for a sandwich after work. As he walked to Smilers that evening, although Fisk did not see any fighting nor any police around, he did notice more than the usual number of people sitting on the stoops and noted an electricity in the air that he felt to be positive in some sense. When he went into Smilers, the counterman made him the largest sandwich he'd ever seen and handed it to him without a word but with a huge smile.\n\nDick Leitsch remembers that \"after a while, everybody thought, _Well, this is boring. All we're doing is running around the block, here. We've done it ten times now and it's dull. Let's do something else_. So we sort of vanished. I remember Christopher Street being so empty, except for the cops, and seeing very few civilians anywhere around. Then I remember the sky being very dark, and there being a terrific moon, and the Village being eerily quiet.\"\n\nNear dawn a number of the individuals who had witnessed the riot lingered on. Bob Kohler recalls, \"We were sitting across the street at the park, and you would see smoldering [garbage] baskets and the street was broken glass. The Stonewall window was smashed, and there were cops all standing around like storm troopers. You'd look a block away and you could see trash cans still smoldering.\n\n\"It was very surreal; it really was. This didn't look like the Village. It didn't look like Christopher Street. We were sitting there, and a couple of kids were bleeding. Nobody really knew what happened. It was like, 'Oh, wow. What was that all about?' It was that kind of thing. The kids had some makeshift bandages. None of the kids were hurt that badly. Other people did get hurt very badly. But that was the end.\"\n\nMartin Boyce's recollections of the end of the night portray a dawning sense of pride: \"Morning came on Christopher, and those broken windows and pieces of cloth inside and diamondlike glass all over. It was a riot, no doubt about it, and there were just exhausted survivors looking dazed. We knew what happened. We all did it. It was like, 'Oh.' Because, you know, the low skyline. There was a certain beauty to the aftermath of the riot. It was a very extraordinary kind of beauty, something to make art out of later. Not directly, but Tommy Schmidt, I'm sure, could see the beauty of shattered glass and certain kind of fag decorations being blown in the wind, by the window. It was obvious, at least to me, that a lot of people really were gay and, you know, this was our street.\"\n\n#\n\n# _\"Christopher Street Belongs to the Queens!\"_\n\n## S A T U R D A Y N I G H T\n\nChris Babick had gone out Friday night with Bob, the twenty-seven-year-old antique dealer he was dating, for Chinese food and had decided he didn't want to deal with the hassles he often ran into in bars as a seventeen-year-old who appeared even younger. Instead they had decided to just go home. Bob woke up first on Saturday and turned on the radio. Soon he was yelling at his still-asleep boyfriend, \"Chris! Chris, you're not going to believe what happened!\"\n\nChris tried hard to focus. \"What? What?\"\n\n\"They rioted. The police raided the Stonewall and they were rioting!\"\n\nChris could not believe his ears. \"And I\u2014it was incredible. Incredible. The fucking queens had rioted.\" Chris later admitted that he did not know what he would have done had he been at the Stonewall the first night, \"But I know what I did the rest of the weekend: Saturday and Sunday nights I was there. Of course Bob and I called our friends.\"\n\nThroughout the day, as the streets were swept and the smashed windows of the Stonewall Inn were boarded up and painted black, the unbelievable news was spreading rapidly through New York's gay grapevine. Many who heard about the insurrection were so excited that they headed down to the Stonewall Inn to see for themselves what had happened. Others sensed that there would have to be some kind of follow-up that night and wanted to be part of it.\n\nAs gay men and women trickled down to the Village, numerous slogans began to appear on the Stonewall Inn's boarded-up windows and the bar's brick facade (some of which were obviously written by the Stonewall Inn personnel), including: \"We are Open,\" \"There is all college boys and girls in here,\" \"Support Gay Power\u2014C'mon in, girls,\" \"Insp. Smyth looted our: money, jukebox, cigarette mach, telephones, safe, cash register, and the boys tips,\" \"THEY WANT US TO FIGHT FOR OUR COUNTRY [BUT] THEY INVADED OUR RIGHTS,\" \"GAY PROHIBITION CORUPT$ COP$ FEED$ MAFIA,\" \"How Can Inspector Smythe Drive a $15,000 Car on HIS Salary?\", \"Support Gay Power,\" and \"Legalize Gay bars and lick the problem.\" Also attached to the front of the club were two copies of the _Daily News_ story about Friday night's events, which struck _Voice_ reporter Lucian Truscott as odd, given the highly negative reporting.\n\nBy around nine o'clock in the evening, enough gay men and lesbians had shown up around the Stonewall Inn to form small groups on the sidewalks. As their numbers steadily grew, police officers in the area would not let them stand still but kept them moving, hoping to prevent a repeat of the previous night's events. But within two to three hours, those numbers had grown into at least a couple of thousand. One lesbian reported that so many people came on Saturday that \"all over the Village... hippies joined the queers & straight places turned gay for the week-end\u2014it was complete madness in NYC.\"\n\nThe Stonewall Inn was the initial focal point for the gathering of mostly gay men along with a few lesbians and a sprinkling of straight supporters. The crowd reached its greatest size late in the evening, probably between ten o'clock and midnight. After the protesters had gathered, they began to sing\u2014including the \"We are the Stonewall Girls\" tune used the night before\u2014and chant, facing off with police officers in front of the Stonewall Inn. The initial chants were slogans such as \"Gay power,\" \"We want freedom now,\" and \"Equality for homosexuals,\" but before long the rallying cries became more militant as the demonstrators shouted: \"Christopher Street belongs to the queens!\" and \"Liberate Christopher Street!\" Some protesters denounced the police for harassing homosexuals. As members of Mattachine\u2013New York and Craig Rodwell handed out literature and people talked excitedly about the meaning of the previous night's events, the area around Christopher Park took on the air somewhat of a political fair\u2014or even a political free-for-all. Edmund White wrote the best and fullest contemporary account of the scene:\n\nA mad left-wing group of straight kids called the Crazies is trying to organize the kids, pointing out that Lindsay is to blame (the Crazies want us to vote for Prococino [ _sic_ ], or \"Prosciutto\" as we call him). A Crazy girl launches into a tirade against Governor Rockefeller \"Whose Empire must be Destroyed.\" Straight Negro boys put their arms around me and say we're comrades (it's okay with me\u2014in fact great, the first camaraderie I've felt with blacks in years).\n\nTruscott described the crowd as being made up of \"onlookers, Eastsiders, and rough street people who saw a chance for a little action.\" Leitsch saw them as \"gay people from all over town, tourists, Villagers, and the idly curious,\" adding that while the crowd had been entirely gay in the beginning, as tourists came along they would inquire about what was going on and, when told that the protest was against the shutting down of a gay club, would become very supportive and either stay to watch or join. Leitsch saw one middle-aged woman, there with her husband, who, after telling a policeman that he should be ashamed of himself, shouted at him, \"Don't you know that these people have no place to go, and need places like that bar?\"\n\nTruscott recorded the raucous scene of speeches and chanting spiced by a lot of camp humor and openly gay behavior:\n\nHand-holding, kissing, and posing accented each of the cheers with a homosexual liberation that had appeared only fleetingly on the street before. One-liners were as practiced as if they had been used for years. \"I just want you all to know,\" quipped a platinum blonde with obvious glee, \"that sometimes being homosexual is a big pain in the ass.\" Another allowed as to how he had become a \"left-deviationist.\" And on and on.\n\nLittle wonder then that a sizable number of the somewhat older gay men who watched, including some who had returned from Fire Island just to see what was going on, \"had strained looks on their faces and talked in concerned whispers as they watched the up-and-coming generation take being gay and flaunt it before the masses.\" Not all those who took a dim view of Saturday night's events were conservative or older: the young radical Ronnie Di Brienza wrote that \"Saturday night was very poor. Too many people showed up looking for a carnival rather than a sincere protest. Queens were posing for pictures, slogans were being spouted out, but nothing really sincere happened in the way of protest.\"\n\nCraig Rodwell disagreed: \"Generally [on Saturday] it was an angry mood, a lot of chanting, a lot of hand-holding, a lot of assertion of being gay... it was a way of saying, 'We're tired of hiding, tired of leading two lives, tired of denying our basic identity.' A general assertion [by] gay people [of] newfound collective pride.\"\n\nChris Babick had a similar reaction: \"It was an absolutely exhilarating experience to know that... they had defied authority. And it was... like the beginning of a lesbian and gay value system. From going to places where you had to knock on a door and speak to someone through a peephole in order to get in. We were just out. We were in the streets. I mean, can you imagine? And when you went to the Stonewall\u2014people did hang out in [Christopher Park]... but you didn't make a lot of noise because the cops would harass you, beat you, whatever. So it was just a matter of two or three people at a time would walk up the street, ring the bell or knock on the door... and... disappear into this cavern, this place called the Stonewall in which there was lively, lively activity. But all of a sudden we were out on the streets. We were there. And I was there.... Here's the homosexual standing on the streets. And it was incredible. And... gay men and lesbians came from all over.\"\n\nWhile the protests continued outside the Stonewall Inn, the Mafia owners were doing their best to attract customers back inside. The owners seem to have realized that the entire context of running their particular business had changed, for not only had there been a riot the previous night, but also the allegations of their illegal activities were in the media and there was a police presence right outside their door. No wonder that when they reopened on Saturday they served only soft drinks, which they gave away.\n\nOutside the Stonewall Inn the chants were repeated louder and louder and more and more frequently. In the midst of this rising emotion, the club owners stepped outside and addressed the gay crowd: \"C'mon in and see what da pigs done to us. We're honest businessmen here. We're American-born boys. We run a legitimate joint here. There ain't nuttin bein' done wrong in dis place.\" In spite of the new sign on the door stating: \"This is a private club. Members only,\" the owners invited \"Everybody\" to \"come and see.\" But the crowd was growing restless.\n\nIt had also grown to a multitude. Craig Rodwell recalled that \"thousands of our people came down... all around 10th Street and all the way up Christopher and around the Square [and] Seventh Avenue.\" Dick Leitsch saw an old woman trying to make her way through the throngs who, while refusing the help of many who tried to assist her, was apparently bewildered and frightened by the unimaginable sight of masses of homosexuals openly gathering in the streets, for she trembled with obvious fear as she walked, all the while muttering to herself, \"It must be the full moon, it must be the full moon.\" The crowds came in spite of the terrific heat. While the previous day had been hot and humid, Saturday was worse: it would be the hottest June 28 in New York City history.\n\nWhen the sidewalks could no longer contain the crowds, people started spilling over into the streets. It was at this point, Rodwell recalls, that \"we decided to block off Christopher Street.\"\n\nRodwell and hundreds of others chose the beginning of Christopher Street at Greenwich Avenue as the place to stop traffic simply because that was where most vehicles entered Christopher Street. Both Rodwell and his lover, Fred Sargeant, say the decision was made to turn Christopher Street into a gay street by saying that no straight people were allowed on the street: taxis and buses would not be admitted unless they were carrying gay people, a turning of the tables intended to make heterosexual people think twice about the nature of discrimination based on sexual orientation and a militant statement that gay people were entitled to something of their own: if America was going to force lesbians and gay men into ghettos, then the ghettos' inhabitants would insist on running them.\n\nApparently the blockade began not so much as an attempt to stop traffic as to harass the occupants of vehicles who were not gay or who would not indicate support for the demonstrations. Rodwell recalled that \"cars that came through and buses, we just rocked them back and forth.\" Leitsch witnessed a showdown between the crowd and a city bus:\n\nA bus driver blew his horn at the meeting, and someone shouted, \"Stop the Bus!\" The crowd surged out into the street and blocked the progress of the bus. As the driver inched ahead, someone ripped off an advertising card and blocked the windshield with it. The crowd beat on the sides of the bus and shouted \"Christopher Street belongs to the queens!\" and \"Liberate the street.\"\n\nLeitsch reports that while the police intervened and were able to persuade the crowd to let the bus pass, the demonstrators immediately came up with a new tactic to slow down the traffic: a human chain was formed across the street and cars were only let through one at a time.\n\nWhile the demonstrators were serious about their intent, an incident with a taxi driver was a warning that they needed to exercise some caution. Unaware of what was happening, a taxicab turned onto Christopher Street from Greenwich Avenue and immediately was caught up in the crowd, the demonstrators apparently not realizing that the challenge to them was unintentional. As the crowd started rocking the taxi back and forth, Rodwell looked in the taxi and saw that both the passengers in the backseat and the cabdriver were terrified, \"scared out of their skulls.\" It seemed to Rodwell and others that the driver was having a heart attack, so a number of demonstrators joined arms to protect the taxi, and it backed up and out onto Greenwich Avenue. Bob Kohler and Craig Rodwell would later hear that the taxi driver died, apparently the only fatality from the riots.\n\nIn an apparently separate incident with a taxicab witnessed by Jamake Highwater, when the cabdriver turned into the crowd, demonstrators banged in the sides of the taxi, jumped on the cab's hood, and paraded on its top. When the driver got out and attacked some members of the crowd, \"about fifteen jumped him. Meanwhile, about 15 others were trying to let the passengers get to freedom.\"\n\nAround this same time, a little farther west, Rodwell saw a barrage of bottles being thrown, apparently by crowds gathered on the Grove Street side of Christopher Park at the police on the other side. One Christopher Street merchant stood in her shop's doorway and berated the police, telling them to behave themselves. Whenever the police headed toward her, she ran into her shop and locked the door. As some of the policemen began acting nasty toward members of the crowd, one youth found garbage can lids and threw them discus style so that they bounced \"neatly off\" the cops' helmets.\n\nRodwell saw this period of time on Saturday night as similar to the time the previous evening between the lesbian resisting arrest and the crowd trying to break back into the Stonewall Inn: so much was being done simultaneously by so many different people that there is no way any one person could have taken it all in. Although the chronology of events is more straightforward on Saturday night than it was on Friday night, there were many more participants spread over a larger area on the second night. If on the previous night gay people had been angry in reaction to the police raid, tonight they were on the offensive. In claiming Christopher Street for their own, they were determined to remove the police and any other hostile parties from it... or at least make it very uncomfortable for them to hold on to it, like territory that an army wins but can only occupy with difficulty against a hostile populace. Rodwell understood it this way: \"[Saturday night] was the first time in history that there was a general assertion of anger by gay people. A public assertion of real anger that was just electric. And even the old queens just came down to walk around the block to see what was going on. They were heroes in a sense, too. Just the fact that they came down. Being there, just standing there watching, they added to the sheer magnitude of the numbers of people there, which was very, very impressive.\"\n\nOne middle-aged gay man who came to see the demonstrators was Dr. Howard Brown, the former chief health officer of New York City, who had resigned from the Lindsay administration in 1968. Typical of many successful gay men of his generation, he was very concerned about passing for straight. He recalled his reaction to stereotypical gay men when, in his capacity as night mayor, a new office created by Lindsay to make the city more responsive to citizens, he had been given a tour of New York City's prison, the Tombs. There he had been shown the section reserved for homosexual prisoners: \"Almost all the men in the crowded cells were demonstrably effeminate. I could not identify with them, and I froze, too terrified to ask any questions about their treatment.\" Over the weekend the uproar from the riots was loud enough that, even though his apartment was a four-minute walk away, with his windows open he could hear the shouting. Intrigued, he left his apartment and headed for the Stonewall Inn. He found that the demonstrators:\n\nwere like the homosexuals I had seen in the Tombs\u2014most of them obviously poor, most of them the sort of limp-wristed, shabby, or gaudy gays that send a shiver of dread down the spines of homosexuals who hope to pass as straight. I could not have felt more remote from them. And yet, at the same time, the scene brought to mind every civil rights struggle I had ever witnessed or participated in.\n\nBy now, fires were again being set in trash cans as they had been the night before. Witnesses report seeing blazes set up and down the street as if in ancient times when bonfires, prepared in advance on mountaintops, would be lit to relay important news over great distances.\n\nFire from another source was also being used to send signals that night. As Doric Wilson stood on the street Saturday night, he looked up toward The Corner and saw red sparks falling from on high, through the night air, as in a gentle rainfall. The sparks were coming from New York City's women's prison, the House of Detention\u2014usually referred to as the House of D\u2014located at the intersection of Greenwich Avenue and Christopher Street. It was a prison with a lot of black and Latina prisoners, many of whom were lesbians. The prisoners were setting toilet paper on fire and dropping it from their cell windows to show their support for the rioters. Chris Babick recalls \"that whole week the women were screaming, cheering us on.... The whole jail, it seemed like, was alive with people, with activity, because the streets were alive with activity. Everything vibrated.\"\n\nIndeed, the crowd was so whipped up with anger that the police became targets. Craig Rodwell was only six feet from an empty squad car with three policemen standing nearby when he saw Marsha Johnson climb to the top of a lamppost and drop a bag containing a heavy object on the car's windshield, shattering it. After the windshield broke, the police immediately jumped into the car but then reached out and grabbed the nearest person they could\u2014a man who had nothing to do with the assault on the car\u2014pulled him into the squad car, and drove off, beating their hapless victim. As Rodwell lamented, \"It was just the nearest faggot. You know, 'We'll teach you,' kind of thing.\" Leitsch witnessed a similar incident in which a police car came up Waverly Place and stopped at the intersection of Christopher Street. The car's occupants sat and stared angrily at the crowd until a concrete block landed loudly on the car's hood. The crowd pulled away and \"then, as one person... surged forward and surrounded the car, beating on it with fists and dancing atop it.\" Soon after the police inside radioed for help, the crowd permitted the vehicle to leave.\n\nAt one point in the evening, Leitsch saw someone attack an officer in a police car:\n\nAnother car, bearing a fat, gouty-looking cop with many pounds of gilt braid, chauffeured by a cute young cop came through. The fat cop looked for all the world like a slave-owner surveying the plantation, and someone tossed a sack of wet garbage through the car window and right on his face. The bag broke and soggy coffee grounds dripped down the lined face, which never lost its \"screw you\" look.\n\nIn a showdown with a bus, possibly the same one described earlier, the police persuaded the crowd to pull back but only to have themselves become the next victims of the enraged citizenry: \"They descended onto the prowl car. They first knocked off the flashing red light. Then they started shaking the squad car sideways as if to tip it over.\"\n\nAccording to _The New York Times_ , the Sixth Precinct had been unable to control the situation and so the TPF were called in for the second night in a row, but according to Dick Leitsch, the Fourth, Fifth, Sixth, and Ninth precincts had already brought in \"a hundred or so cops who had no hope of controlling the crowd of nearly two thousand people in the streets\" and it was only then that the TPF was called in.\n\nIt was approximately 2:15 A.M. when the TPF units assigned to the East Village arrived in the West Village. About 150 TPF officers arrived, with around one hundred of them getting out at The Corner while another fifty were dropped off at Seventh Avenue South and Christopher Street.\n\nAs the TPF met with a number of high-ranking police officers at Greenwich Avenue and Christopher Street, the crowd sporadically tossed beer cans at their vans and cars. Without any warning, two police officers rushed into the crowd, plucked a youth from it at random\u2014one who, according to Leitsch, \"had done absolutely nothing\"\u2014and carried him off to a patrol wagon. As they did so, while the two arresting officers still held the young man, four other police officers began to pound the boy's face, stomach, and genitals with nightsticks. A \"high shrill voice\" yelled, \"Save our sister!\" and there was a heavy pause, during which, Leitsch recounts, \"the 'butch' looking 'numbers' looked distracted.\" Then \"momentarily, fifty or more homosexuals who would have to be described as 'nelly,' rushed the cops and took the boy back into the crowd.\" Like the Red Sea, which parted to let the Hebrews escape Pharaoh's army only to then close itself against that same army, the queens \"formed a solid front and refused to let the cops into the crowd to regain their prisoner, letting the cops hit them with their sticks, rather than let them through.\"\n\nThe police top brass, having finished their strategic consultations, had the TPF form themselves into flying wedges as they had done the night before. The TPF then marched up and down Greenwich Avenue, forcing parts of the crowd onto 10th Street and others onto Sixth Avenue, although from Sixth Avenue the crowds simply came back south onto Christopher Street. The TPF next marched up and down Greenwich Avenue but this time linked their arms, and as they came to a side street those on the ends of the lines broke off and \"chased demonstrators down the side streets and away from the center of the action.\"\n\nHaving cleared Greenwich Avenue, the TPF focused on opening up Christopher Street between Greenwich Avenue and Seventh Avenue South. It so happened that a number of the young men who had been hanging out in front of the Stonewall Inn had finally grown bored and decided to head to The Corner to see what was happening there when they ran right into the TPF heading toward them. As the TPF advanced, they were a formidable sight. Almost three decades later, Dick Leitsch could still recall the TPF's appearance as they deployed that night on Christopher Street: \"They formed a phalanx in the Greek style. They lined up, shoulder to shoulder, across Christopher Street, and they had big, plastic shields that were half-circles about four feet tall. They covered from the crotch to the top of the head, so they would protect their groin and head from bullets or bricks thrown at them. The shields were semicircular and had handles, and they were inside this half a tube. They also had riot helmets and billy sticks. So they formed this shoulder-to-shoulder phalanx across Christopher Street and then just marched, very slowly\u2014very slowly, not rushing or anything\u2014down the street. It was a solid wall.\" Charles Burch recalls the impressive sight the TPF made when deployed in line formation: \"There was this huge phalanx of blue-helmeted cops with shields, and the light reflecting from their blue plastic-ness [made] an aura of silvery sheen over them.\"\n\nThe TPF succeeded in sweeping the crowd down Christopher Street as far as Waverly Place, where the riot police stopped, with the kids who had just before been hanging out by the Stonewall Inn at the head of the crowd. As they had the previous night, the youths again formed a chorus line and started singing and dancing right in front of the TPF.\n\nAs Robert Bryan watched the face-off between the men in the kick line and the TPF, he was amazed by the gay youths' courage. \"The queens\u2014they were extremely effeminate young men\u2014formed this kicking line all across Christopher Street, and started to do a Rockettes kick. And singing, 'We're the Stonewall Girls, we wear our hair in curls, we don't wear underwear..., ' as it went. And the police started moving ahead, moving towards them.\" The TPF advanced at a slow and steady pace, inching forward. \"And the queens did not move; they just continued to kick and to sing as the police just moved closer and closer and closer; and you just wondered how long are they going to keep this up before they break and run? The police got closer and closer to them with their clubs and their helmets and their riot gear and the whole thing; and I thought it was just very inspiring, their bravery, like Bunker Hill or 'Don't fire until you see the whites of their eyes.' They were waiting until the very last minute, and it wasn't until the police were eight feet away from them that the crowd finally broke and ran.\"\n\nIronically, the TPF's psychology of using machismo to try to intimidate protesters whom society had branded as deficient in masculinity\u2014and hence courage\u2014ultimately raised the question of who, indeed, was braver: the TPF hiding behind their shields and helmets, equipped with guns and billy clubs, with all the force of the law and the approval of society behind them, or the gay men\u2014with most of those in the kick line being effeminate to some degree\u2014the objects of society's scorn and ridicule, offering their vulnerable bodies as targets and armed with nothing more than their intelligence and humor?\n\nWhen the protesters finally did run, they merely repeated their tactic from the night before: as the TPF advanced up along Christopher Park toward Seventh Avenue South with the goal of clearing all of Christopher Street, the protesters turned southeast on Waverly, went up Gay Street, and again came out behind the TPF on Christopher Street forming a chorus line, all the while screaming and taunting the TPF.\n\nThe guerrilla tactic of using the Village's many side streets to circumvent the police tactics for clearing the streets with sweeps of the TPF was used by the protesters over and over again, as was the kick-line routine. And just as had happened when the \"nelly\" men rescued a gay man from the clutches of the TPF, Leitsch noted that most of the remarkable examples of physical courage he saw during the riots were done by the more effeminate men, the real \"queens\":\n\nIt was an interesting sidelight on the demonstrations that those usually put down as \"sissies\" or \"swishes\" showed the most courage and sense during the action.... The most striking feature of the rioting was that it was led, and featured as participants, \"queens,\" not \"homosexuals.\" \"Homosexuals\" have been sitting back and taking whatever the Establishment handed out; the \"queens\" were having none of that. The \"butch\" numbers who were around the area and who participated peripherally in the action remained for the most part in the background. It was the \"queens\" who scored the points and proved that they were not going to tolerate any more harassment or abuse.... Their bravery and daring saved many people from being hurt, and their sense of humor and \"camp\" helped keep the crowds from getting nasty or too violent.\n\nThis assertion by Leitsch is corroborated by a memory of Bob Kohler's: a \"plea scrawled on the sidewalk in front of the Stonewall: BUTCHES, WHERE ARE YOU NOW THAT WE NEED YOU?\"\n\nConventionally masculine men were involved in the fighting, but in light of Leitsch's, Kohler's, and other testimony, it seems irrefutable that a highly disproportionate amount of the physical courage displayed during the riots came from the more effeminate men in the crowd.\n\nThe one portion of the Village that the police could close without too much difficulty was the very focus of the protest: the block the Stonewall Inn was on. Before the night was over, the police closed off the block, allowing no one to enter Christopher Park, Christopher Street between Waverly Place and Seventh Avenue South, or Grove Street from either end of the south side of Christopher Park. Shutting down what New Yorkers refer to as Sheridan Square was a victory for the police, but only a partial one, for several reasons. First, there were still plenty of gay people out on the streets challenging the police, who had seized only one small piece of the contested area, enlarged significantly since the previous night. Second, the closing came at an economic price to a number of nongay businesses. Truscott noted the loss of Saturday night business \"even at the straight Lion's Head and 55 [the bar next to the Lion's Head at 55 Christopher Street].\" Another _Voice_ columnist, Walter Troy Spencer, bemoaned that one bar owner on Christopher Street (no doubt one of these same two bars) lost $500 in one evening \"of the indirect embargo.\" Keeping the ghetto's occupants under even partial control for one evening on one block could be expensive. Indeed, things were so out of hand that on the other side of Seventh Avenue South, the United Cigar Store, Riker's restaurant, and Smilers Deli all closed down of their own volition.\n\nThe TPF continued to fight to regain control of other streets beyond the Christopher Park area as the same cat-and-mouse game between police and protesters of the night before was reenacted, but this time with more police officers, more rioters, and more anger on both sides. Not surprisingly, the TPF was brutal. Leitsch simply wrote: \"[T]he TPF again lived up to its reputation for violence and brutality.\" A letter to _The Village Voice_ written by Kevin Liscoe complained that he had \"witnessed many senseless brutalities which there could be no justification for. One guy walking on Sheridan Square across from the disturbance was with his chick (obviously not into it) when a TPF came from behind and just split his head open with his club.\" Truscott reported witnessing one person getting a busted head at or near Sheridan Square and noted that \"the cops amused themselves by arbitrarily breaking up small groups of people up and down the avenue.\"\n\nIn spite of the TPF's numbers and superior equipment, the crowds of protesters were not cowed and made full use of their wit, anger, and resourcefulness in fighting the TPF. As the TPF moved through the streets, they swung their nightsticks with abandon, using them, Leitsch noted, like swords. But when one police officer grabbed \"a wild Puerto Rican queen\" and raised his arm to club him, the queen suddenly asked the cop, \"How'd you like a big Spanish dick up your little Irish ass?\" The officer was so startled he stopped his nightstick in midswing and the man escaped. On another occasion two police officers were chasing a crowd of a hundred or more demonstrators down Waverly Place when one of the demonstrators suddenly realized that they outnumbered the police. He immediately shouted that they should catch the police, rip their clothes off, and screw them on the spot. As the crowd turned on them, the police quickly reversed themselves and ran for blocks with the angry crowd in hot pursuit, yelling, \"Catch them! Fuck them!\"\n\nAmid all these sorties and skirmishes between protesters and the police, Leitsch again spotted the middle-aged woman he had seen several hours earlier telling a police officer that he should be ashamed of himself. This time she and her husband and two other heterosexual couples were in the midst of a large group of homosexuals trying to escape the TPF as they chased them with nightsticks.\n\nStill, with between two and three hundred police on the scene, most of them heavily armed, the streets were reopened and most of the crowds dispersed. It had taken a major deployment of New York City's Finest to wrest control of a small section of the city back from its homosexual citizenry. For approximately two hours Christopher Street had indeed belonged to the \"queens.\"\n\nYet this still was not the end of the evening. While the TPF had restored order by 2:30 A.M., when the bars closed at 3:00 A.M. customers from the Village's many gay bars came by to see what was going on. According to _The New York Times_ , the abundance of helmeted police made the crowd even angrier than the previous night's raid on the Stonewall Inn had, and the new arrivals were quickly organized into a second attempt to \"liberate Christopher Street.\" Accounts of this attempt are extremely sketchy, apparently because with such a heavy police presence the attempt was short-lived. Still, even then, a group of the new protesters broke off and attempted to take over the IND subway line at the corner of Sixth Avenue and Waverly Place. There was a bit of a confusion during which the police tried to figure out if they could enter an area controlled by the Transit Police. After the bureaucratic and political points had been discussed, it was decided that they could, and they entered the station and chased the demonstrators out.\n\nIt was not until around 3:30 A.M. that the police finally succeeded in dispersing all of the crowds. The TPF remained on the scene for some time later. It is even debatable the extent to which the clearing of the streets meant a victory for the police. According to Leitsch, many men had taken advantage of the heavy police presence in the heart of the Village to go to the docks for some recreational sex. Leitsch writes: \"The docks were packed tight with homosexuals having the times of their lives. After all, everything was perfectly 'safe'\u2014all the cops were on 'The Corner'!\"\n\n#\n\n# \"They've Lost That Wounded Look\"\n\n## S U N D A Y\u2013W E D N E S D A Y\n\nAs New Yorkers walked through the Christopher Park area of Greenwich Village on Sunday, teams of gay people\u2014each consisting of one man and one woman\u2014passed out a flyer headlined \"Get the Mafia and the Cops Out of Gay Bars.\" Issued in the name of the Homophile Youth Movement, the typed document proclaimed that \"the nights of Friday, June 27, 1969, and Saturday, June 28, 1969, will go down in history as the first time that thousands of Homosexual men and women went out into the streets to protest the intolerable situation which has existed in New York City for many years\u2014namely, the Mafia (or syndicate) control of this city's Gay bars in collusion with certain elements in the Police Dept. Of the City of New York.\" Written by Craig Rodwell, the flyer reasoned that since January 1968, when Judge Keating had ruled that even close dancing between homosexuals was legal, there was \" _nothing illegal, per se, about a Gay bar_.\" The flyer therefore urged gay businessmen to open legal bars that would not overcharge and that would have \"a healthy social atmosphere.\" It also suggested that homosexual men and women \"boycott places like the Stonewall,\" in order to \"get criminal elements out of\" gay bars, and write Mayor Lindsay to demand \"a thorough investigation and effective action to correct this intolerable situation.\"\n\nBy Sunday it indeed seemed justifiable to hail the two previous nights as historic. Rodwell recalls that even after the first night, \"all kinds of people [were] getting together to discuss what are we going to do, because everybody sensed that nothing's going to be the same after this. We just knew. I'm sure there were some people there that were saying, 'Oh, I don't want to get involved in this,' and ran away, but for those of us that were there\u2014and there were many thousands of us\u2014we sensed that it was a moment in history. There was a lot of very animated talks all over the place. 'What's going to happen now? What are we going to do?' \" Rodwell was not alone in sensing the importance of events: his estranged lover, Dick Leitsch, writing on behalf of Mattachine\u2013New York titled his account of the uprising: \"The Hairpin Drop Heard Around the World,\" _dropping a hairpin_ being homosexual argot from the era for dropping a hint that one is gay.\n\nSunday, Monday, and Tuesday nights remained relatively calm due to the police having learned several lessons from the first two nights of the riots: that in the aftermath of Friday night's raid it would take many more police officers than were initially deployed on Saturday to discourage gay men and women and their allies from attempting to seize the area around the Stonewall, which the homosexual citizens saw as their turf; that the police needed to arrive early in large numbers to preempt the gay citizenry from seizing the initiative; and that such a maneuver must be executed in a low-key manner or their mere presence might precipitate the outbreaks they were hoping to end. (At least the police used this approach on Sunday night and it worked.) The second factor working in the police's favor was that the weekend had passed. Friday and Saturday nights' events had begun late and lasted until the early morning, something most people could not do on an evening before a workday, especially if they had already done so on the previous night or two. Thirdly, some of the demonstrators felt that they had already made their point and nothing more was to be gained by again repeating what was becoming a predictable scenario. Was it worth risking more injuries and arrests, as well as damage to the neighborhood that gay people were fighting to claim as their own?\n\nStill, Rodwell and other activists felt that the protests must continue as long as possible to try to ensure that lesbians and gay men would be heard at last, that the events of the last two days would have some kind of lasting result and not be seen as a fluke.\n\nMattachine\u2013New York, however, after talking to the mayor's office and the police, joined police efforts to stop the protests. A prominent sign went up on Sunday afternoon on the front of the Stonewall Inn's east window:\n\nWE HOMOSEXUALS PLEAD WITH  \nOUR PEOPLE TO PLEASE HELP  \nMAINTAIN PEACEFUL AND QUIET  \nCONDUCT ON THE STREETS OF  \nTHE VILLAGE\u2014MATTACHINE\n\nMattachine officials came to the Stonewall Inn and talked to people who showed up in an effort to discourage them from protesting.\n\nThese attempts to stop the protesters did not work. On Sunday night gay men again turned out in significant numbers on Christopher Street, including a sizable leather contingent. Truscott condescendingly wrote that \"Sunday night was a time for watching and rapping. Gone were the 'gay power' chants of Saturday, but not the new and open brand of exhibitionism. Steps, curbs, and the park provided props for what amounted to the Sunday fag follies as returning stars from the previous night's performances stopped by to close the show for the weekend.\" There were fewer protesters, however, than there had been on the previous two nights, and the police presence was very high. Dick Leitsch wrote that \"there were never enough people to outnumber the large squads of cops milling about\" and that \"all the cops in town seemed to be near The Corner again.\" Truscott felt that \"it was slow going.\" _The Berkeley Barb_ reported that on Sunday \"the word passed for another gay demonstration. Help was expected from the SDS, but they never showed up.\"\n\nNonetheless, there were small acts of resistance by gay people\u2014beyond demonstrating against the police publicly for the third day in a row, which was remarkable in itself: several gay youths took daring advantage of the heavy police presence in the Christopher Street area to make a guerrilla raid on the Sixth Precinct's headquarters. They went to the police station and slapped Day-Glo blue and fuschia bumper stickers reading \"Equality for Homosexuals\" on the police cars, a patrol wagon, and the personal cars of cops who had left their vehicles parked while on duty. Whether any police officers who drove such subverted cars home unawares were able to laugh about it is unknown, but there is evidence that at least some of the police force appreciated the use of humor by New York's gay populace during the Stonewall Riots. A member of Mattachine\u2013New York who was detained by the police reported overhearing the following exchange between two police officers:\n\nOne [cop] said he'd enjoyed the fracas. \"Them queers have a good sense of humor and really had a good time,\" he said. His \"buddy\" protested: \"Aw, they're sick. I like nigger riots better because there's more action, but you can't beat up a fairy. They ain't mean like blacks, they're sick. But you can't hit a sick man.\"\n\nIndeed, there is good evidence that after two days the police had finally learned something about queer community relations: With the Stonewall still open as a \"free store,\" the police were \"begging\" homosexuals to go inside the club, a rich irony that Dick Leitsch noted with glee. And when the TPF made a sweep of the area to clear it, they did so in a nonconfrontational way, not wearing their helmets and in a \"controlled and very cool\" manner.\n\nLucian Truscott was standing in front of the United Cigar Store on Seventh Avenue South watching things wind down when he ran into Taylor Mead, an avant-garde film and literary figure. They stood on the corner and had been talking things over for a while when up walked Allen Ginsberg. Ginsberg had heard about the \"Stonewall battle\" the previous day and had decided to drop by to see what was going on and \"to show the colors.\"\n\nSome gay activists happened by and filled Mead and Ginsberg in on the previous two nights' events, to which Ginsberg reacted enthusiastically: \"Gay power! Isn't that great! We're one of the largest minorities in the country\u2014ten percent, you know. It's about time we did something to assert ourselves.\" A sudden realization seemed to come over Ginsberg, who told Truscott, \"You know, I've never been in there,\" and decided on the spot to visit the Stonewall Inn. As Ginsberg proceeded toward the Stonewall, he practiced his lifelong belief in peace, shaped by his pilgrimage to India, where he had studied Hindu meditation and chanting techniques for calming and quieting the mind. As he walked over he held up the first two fingers of his hand in the V shape used to signify peace in the late 1960s and greeted the members of the TPF not with the epithet \"pig\" but by saying \"hello\" to them. Truscott noted that \"it was a relief and a kind of joy to see him on the street. He lent an extra umbrella of serenity to the scene with his laughter and quiet commentary on consciousness, 'gay power' as a new movement, and the various implications of what had happened.\"\n\nWhen Truscott entered the Stonewall Inn with Ginsberg, the poet immediately began to bounce and dance \"wherever he moved.\" This was also the first time Truscott had been inside the club, and he noted that rock music was being played from speakers placed all around the room, suggesting that a sound system had been brought in, necessitated because the police had destroyed the jukeboxes. Truscott thought that the club looked like \"a Hollywood set of a gay bar.\"\n\nTruscott recounts that Ginsberg \"was real excited by it all. I mean, he was just bouncing up and down with glee. He just thought it was great that all these kids had stood up to the police and that there was a revolution happening. I mean, that's the way he was behaving. And when we went into the Stonewall, he went and danced with a whole bunch of kids.\"\n\nAfter about an hour, Ginsberg and Truscott left the Stonewall Inn and Ginsberg headed toward Manhattan's Lower East Side. As Truscott walked with him, Ginsberg explained what homosexual experience used to be like, drawing the contrast with the poet's experience that evening, and concluded, \"You know, the guys there were so beautiful\u2014they've lost that wounded look that fags all had ten years ago.\"\n\nThe two men parted company at Cooper Square. Ginsberg waved to Truscott and yelled, \"Defend the fairies!\" as Truscott watched him \"bounce\" across the square.\n\nGinsberg's characterization of the change the Stonewall Riots had brought about was so perceptive that when the early gay activist Allen Young interviewed Ginsberg for the literary magazine _Gay Sunshine_ , the only question that Young asked him about Stonewall concerned the circumstances behind Allen's statement. Ginsberg replied:\n\nI wasn't there at the riot. I heard about it, and I went down the next night to the Stonewall to show the colors. A crowd was there, and the place was open. So I said, the best thing I can do is to go in; the worst that can happen is I'll calm the scene. They're not going to attack them when I'm there. I'll just start a big Om.\n\nI didn't relate to the violent part. The trashing part I thought was bitchy, unnecessary, hysterical. But, on the other hand, there was this image that everybody wanted to make that they could beat up the police, which apparently they managed to do. It was so funny as an image that it was hard to disapprove of, even though it involved a little violence.\n\nAlthough Ginsberg's reply to Young is the lengthiest commentary on the riots he ever provided to an interviewer, it did not speak directly to the question about the loss of the \"wounded look.\" Truscott, the person to whom Ginsberg made the comment, thinks he understands what inspired the poet. Pointing out that Allen did not generally hang out in gay bars and using the example of the jubilant looks on the faces of the street youths in the famous Fred McDarrah photographs of the rioters, Truscott explains that \"I think his idea of what a gay bar was at that time was kind of like Julius': a bunch of middle-aged men, standing around in crew-necked sweaters having a drink. What happened was he got to see this real mixed population of queens and drag queens and these young street kids, and some of them were doubtlessly hustlers. All of them were young and had long hair. I mean, this doesn't look like a bunch of middle-aged, unhappy gay men at Julius'. And that's what he was responding to: all these sort of happy-go-lucky young guys, having a good time being gay, and not unhappy about it and not unproud about it and everything. That's what he was responding to, and that's why he said they've lost that wounded look. And that was the truth. Look at the pictures. You can see it on everybody's faces.\"\n\nGinsberg later said about the Stonewall Riots: \"All of a sudden at the height of the antiwar movement, at the height of the black liberation movement, after the triumph of liberation of the word [the end of print censorship], all of a sudden the cops were in there again trying to bust some guys... right in the center of Sheridan Square, the most bohemian traditional place in Greenwich Village!\"\n\nIf Sunday night was \"slow,\" then Monday and Tuesday nights were practically dead. In fact, there is only one contemporary account of those nights, that written by Dick Leitsch for the _New York Mattachine Newsletter_. According to him, while both nights were calm because few gay people were on the streets, by this time both the police and the protesters had become short-tempered. One police officer tried to pick a fight with passing gay men by repeatedly challenging them, saying, \"Start something, faggot; just start something. I'd like to break your ass wide open.\" When one man finally turned and said, \"What a Freudian comment, Officer!\" the cop attacked the man and arrested him, placing him in a patrol wagon to be taken to jail.\n\nWhile Leitsch says that some of the police did maintain \"enormous 'cool,' \" it is clear that others tried to provoke violence, apparently because of their recent humiliations. Leitsch saw two officers in a patrol car repeatedly drive around the area, shouting obscenities at passersby.\n\nHowever, if some police sought to cause trouble, they were not the only ones: so did some gay men. A cop, stationed at the corner of Waverly Place and Christopher Street, made a show of swinging his nightstick while tossing insults at pedestrians. He made too inviting a target for one subversive to resist. A man Leitsch describes as \"[a] wildly 'fem' queen,\" firecracker at the ready, snuck up behind the unsuspecting pillar of society, lit the explosive, and dropped it between the officer's feet. Leitsch describes the result: \"It exploded and he jumped into the air in a leap that Villella would have envied, landing on a part of his anatomy that one queen called a 'money-maker.' He got up screaming like a peasant woman and swinging his stick.\" When the same man tossed another firecracker at the policeman, a general fight broke out in which members of the crowd managed to steal the badge off the policeman's shirt.\n\nSome gay men were so angry that militancy at times overlapped into aggressiveness. Journalist Tom Burke witnessed an exchange at Waverly Place between a youth and a Village resident who walked up and down the street trying to calm the gay crowd, reassuring them that the heterosexual residents sympathized with \"the oppressed homosexual.\" The portly older man in a white shirt tried to reason with a youth with long red hair wearing an acid-green tank top. The youth was having none of it.\n\n\"The hell you say! You don't impress me,\" the youth shouted. \"You are straight and you are my enemy! Don't give me that phoney liberal bull. _You_ made the laws. Nixon's silent army!\" As onlookers laughed, the young man leaned in close to the older one, the youth's face now deep red. \"Now we are gonna get _you!_ \"\n\nThe older man's face turned ashen, and Burke could tell he wanted to flee but was too frightened of what the young man might do.\n\n\"I was in Vietnam, man, how does that grab you? Huh? Huh? And, man, I'll screw your daughter. _But I'll screw your son first!_ \"\n\nOn Wednesday night the riots came back full force. The reason for the sudden resurgence was twofold. The more obvious cause was the appearance of _The Village Voice_ on Wednesday evening, which gave very prominent play to Truscott's and Smith's accounts of the uprising. Dominating the front page, the two stories were run next to each other along with two photographs by Fred McDarrah, one showing some of the rioters and the other the graffiti written on the Stonewall Inn's boarded-up windows. While the stories may have enticed some curious citizens who had not yet heard of the riots to the Christopher Park area to see what was transpiring, the main effect of these accounts was to inflame the gay populace. While Smith and Truscott had done a good job of reporting the rebellion, the tone of Truscott's article\u2014while often sympathetic and objective\u2014was at times one of derision and insensitivity, with phrases like \"Limp wrists,\" \"the Sunday fag follies,\" and \"gay cheerleaders.\" That Smith's article was less offensive than Truscott's, with only two derogatory expressions (the phrase \"dancing faggots\" and the word \"dyke\"), was lost on the readers who, understandably, lumped the two articles together as one. While there was a lot of purposefully effeminate behavior on the part of some in the crowd, the two articles not only focused on this behavior but also ignored the courage shown by conventionally masculine gay men. (The only woman mentioned in the two articles is the lesbian resisting arrest, characterized by both men with the same single word _dyke_ ). Edmund White attributed the writers' tone to their wanting to assure \"readers that the authors are straight.\" Not only did _Voice_ reportage enrage the gay population; it also came quite close to consuming _The Voice_ itself in flames, when a group of gay people considered setting fire to _The Voice_ 's office that night. The anger that led to this discussion of whether to burn down the _Voice_ office was apparently caused by not only the reporting on the riots but also _The Voice_ 's various past slights toward gay men and lesbians. Leitsch wrote: \"That paper's editorial policy has long infuriated most homosexuals, as the paper pretends to be 'liberal' and avant-garde, but actually is conservative and uptight about homosexuality.\" Leitsch let it be known that he held _The Voice_ responsible to a large degree for the flaring up of the rioting that he and others in the Mattachine Society were trying to quell: \"They published two long 'put down' articles about the Christopher Street incidents which contributed heavily to the anger that incited the Wednesday rioting after two relatively peaceful days.\"\n\nThe second reason that rioting resumed in full force on Wednesday is because various radical Left groups came to protest. Leitsch, a man whose politics were eclectic, combining views that ranged from very progressive to those more typically held by the Right, saw the radical Left groups as \"exploiters [who] had moved in and were using the gay power movement for their own ends.\"\n\nWhile Kohler, a leftist, felt less negative about the influx of nongay Left groups, his reading of the crowd's composition and its significance is similar to Leitsch's:\n\n\"A lot of people kept coming down, asking what happened, asking the kids, me, anybody that was around [for] information. You had a lot of curiosity seekers. Because something had happened. Everybody was aware that there was a riot, so a lot of people had questions.\n\n\"You had, for lack of a better word, _provocateurs_ who were seeing an incident that could be used for the good of their movement or a different movement or a coalition of movements. This was one more. It was the last. There was nobody else left to riot, so people with some foresight, more foresight than I [had] at the time, could see that this could again strengthen the radical movement, that people they hadn't really thought about were now up there in the front.\n\n\"The big thing on everybody's mind, especially the police and movement people, was that these were the only rioters that had gotten the best of the police. So that gave them a special strangeness. I mean, people just [wondered], _How could that happen?_ Because there'd been riots with Yippies and there'd been riots with the SDS and Abbie Hoffman, but nobody had ever gotten the best of the police before. The police were never put on the run, and suddenly they were put on the run by the fairies, so those people were very curious: did this really happen?\n\n\"There was no carnival atmosphere; it was just people wanting to know what happened. A lot of grumbling and the kids were laughing and talking it over, getting ready for the night, a couple of people leafleting.... But it was not an exceptional day of any kind. It was just a rehashing.\n\n\"The riot no longer belonged to the kids. That was very obvious. So, yes, there was a difference in the makeup. There were more people rioting that could not be easily categorized, and a lot of that had to do with people that came over from other areas. The straight movement moved in heavily that night as a support. I'm assuming it was as a support. They were, of course, using it for their own ends, too, but you were aware the riot did not belong to the kids [Wednesday] night.\"\n\nThe action on Wednesday night began around 10:00 P.M. According to one report, at that time \"a motorcade of police drove down Christopher Street looking for trouble.\" There was a crowd of at least \"several hundred youths\" gathered near the Stonewall Inn described by the then-sympathetic _New York Post_ as \"shoving and bottle-hurling.\" _The New York Times_ reported it as a chanting crowd and estimated its size at five hundred strong. Leitsch characterized the crowd as composed of \"Black Panthers, Yippies, Crazies and young toughs from street gangs all over the city and some from New Jersey.\"\n\nAround 10:30 P.M. \"queens\" lit some trash at the corner of Waverly Place and Christopher Street, apparently resulting in more fires being set on Christopher Street, to which both the TPF and the fire department responded. While some members of the Mattachine Society were trying to stop such acts, the Mattachine Society leadership was trying to use the riots to advance the cause by distributing handouts stating that \"the disorders showed that 'gay people are reaching the end of their patience.' \"\n\nIndeed, it appears that by Wednesday night patience was in exceedingly short supply on all sides. While the _East Village Other_ reported that \"everything became more than serious,\" _The New York Times_ described the crowd as \"hostile,\" and Leitsch wrote that:\n\nthe street people were no longer half-serious, half-camping. The cops... had taken the offensive and massive retaliation was their goal. Some seemed quite ready to depopulate Christopher Street the moment anyone would give them permission to unholster their guns. Failing that, some of them, particularly some of the TPF men, tried to achieve the same objective with their nightsticks.\n\nRonnie Di Brienza described the scene he witnessed:\n\nOne really fat Bircher-type pig grabbed a friend of mine, who was promptly beaten in front of two hundred people by three other pigs, and then carted off to a waiting patrol car. This was it. From no where the crowd swelled to an estimated thousand, and the battle was on. One head, standing on the corner of Waverly, was unfortunate enough to yell out \"pig\" just when the man was behind him. Well, in front of 1,000 witnesses, he was pummeled, dragged, kicked and lifted down the length of Christopher Street to a waiting squad car on Seventh Avenue....\n\nMy buddy received seven stitches over his left eye for his participation in a freedom of assembly rally.\n\nBetween the anger of the demonstrators and that of the police\u2014with the police of course being better equipped and trained\u2014it is not surprising that many demonstrators were injured. Leitsch describes the bodies of the wounded that littered Seventh Avenue South between Christopher and West 10th streets: \"Young people, many of them queens, were lying on the sidewalk, bleeding from the head, face, mouth, and even the eyes. Others were nursing bruised and often bleeding arms, legs, backs and necks.\"\n\nNor were demonstrators the only ones hurt. At least one police officer, Richard Adkins, a patrolman, was injured seriously enough by a bottle that hit him on the left side of his face that he was taken to St. Vincent's Hospital.\n\nThere is consensus that on this night, as on the first night of the Uprising, it was the effeminate men who did most of the fighting. Di Brienza wrote: \"I have never seen anything worse than an infuriated queen with a bottle, or long nails. Believe me, get their ire up, and you face the wrath of all the Gods that ever lived.... Revolution is being heard on Christopher Street, only instead of guttural MC-5 voices, we hear it coming from sopranos, and altos.\" Indeed, Leitsch takes this view even further, saying that not only were the \"queens\" braver than the masculine gay men, they were also braver than the Black Panthers, Yippies, and other leftists who were there on Wednesday night: \"The exploiters had moved in... blacks and students who want a revolution, any kind of revolution... swelled the crowd... but 'graciously' let the queens take all the bruises and suffer all the arrests. (If they have no more courage than they displayed on Christopher Street, their revolution is a long way off.)\"\n\nFive persons were arrested on Wednesday night. The charges against four of them was harassment; the charges against the fifth person are unknown.\n\nOn Wednesday night Village shops were broken into, apparently by the nongay protesters, for as Dick Leitsch pointed out, \"all the most unlikely places were looted\": shops whose owners were sympathetic to the homophile movement and to the street queens were broken into and the \"fag shops\" that exploited the gay community by selling overpriced wares were not hit.\n\nWhile the police and protesters fought their hardest on Wednesday night, the strife finished quickly. As Ronnie Di Brienza wrote at the end of his account of what he called, with some understatement, the Stonewall Incident, \"This all ended within an hour, and peace was restored.\" He concluded his report with a pithy summation of the meaning of the events he had witnessed from beginning to end: \"But the word is out. Christopher Street shall be liberated. The fags have had it with oppression.\"\n\n# G A Y\n\n# L I B E R A T I O N\n\n#\n\n# Seizing the Moment\n\nThe Stonewall Riots had barely ended when Bill Katzenberg called Charles Pitts, the coproducer of the WBAI series _The New Symposium_. The 1968 radio program had been so successful that the program's producers had mounted a new series, _The New Symposium II_. Katzenberg asked Pitts if he would be interested in helping to start a gay leftist group. Pitts agreed to meet with Katzenberg and invited Pete Wilson, Randy Wicker's lover, who had often appeared on _The New Symposium II_ , to come along to the meeting. Wilson had not only appeared as an openly gay man on the 1969 radio series; he had also marched in most of the early homophile demonstrations and joined the League for Sexual Freedom, an organization Wicker had cofounded that advocated sexual freedom for all along libertarian lines. When Pitts and Katzenberg met, Pitts was so immediately inspired that even before Wilson arrived, he crafted the memorable lines that captured the new defiant spirit of gay New Yorkers. Soon scores of gay New Yorkers would find a piece of paper thrust in their hands that asked: \"Do You Think Homosexuals Are Revolting?\" The same flyer rejoined: \"You Bet Your Sweet Ass We Are,\" before promising: \"We're going to make a place for ourselves in the revolutionary movement.\" The flyer urged all who wanted to help in this new endeavor to show up at the July 24 Alternate U. meeting, the same date that O'Brien and Katzenberg had reserved at Alternate U. before the Stonewall uprising.\n\nUnbeknownst to Katzenberg, O'Brien, and Pitts, a young man named Michael Brown who had witnessed the riots sensed an opening to join newly politicized gay men with blacks, antiwar activists, and other forces on the Left. Although Brown worked in an interior decorating firm on Wall Street and had been on the staff of Hubert Humphrey's 1968 presidential campaign, he considered himself both a socialist and countercultural. He had been thrilled during the riots to see so many gay men be public about their sexuality, for he felt disgusted by the secrecy of gay life and the sordid tone that sometimes resulted from leading such a shadow life. Brown had thrown himself into the effort to support the Stonewall Riots, primarily by helping Dick Leitsch mimeograph and distribute Leitsch's account of the riots, \"The Hairpin Drop Heard Around the World.\" The mimeographed \"Hairpin\" hailed the event as \"the first gay riots in history\" and \"the opening shot\" of what could be \"a long, hot summer\" unless reforms were made.\n\nWhen Brown explained his ideas and feelings to Dick Leitsch before the riots were even over, Leitsch told him to be realistic and underlined the importance of Mattachine\u2013New York not jeopardizing its relationship with those in authority. But Brown insisted that it was an absolute necessity for Mattachine to respond to the riots proactively.\n\nDavid Scott was a twenty-one-year-old who had gotten caught up in supporting the riots when he met Michael Brown handing out mimeographed copies of \"The Hairpin Drop\" near Christopher Park. Scott was so eager that nearly every night after work he went up to volunteer at the Mattachine offices, helping to mimeograph flyers and then distribute them in the area around Christopher Park. While he was at the Mattachine offices he met Marty Robinson, John O'Brien, Dick Leitsch, and Madolin Cervantes, a heterosexual Mattachine member and one of its most enthusiastic supporters. He recalls a major discussion that he heard about a week after the riots among Madolin, Marty, Michael, Dick, and possibly John. \"I remember Madolin saying, 'Oh, we should be nice. Gay people are known as being nice, sweet people.' And like Marty and [others] of them said, 'No! This has all got to change. It's time to get radical. The civil rights movement didn't get where they are by being nice and quiet. The antiwar movement didn't get where it is. We have to be militant, and we have to confront the authorities.' \"\n\nLeitsch yielded to Brown's entreaties and agreed to form an Action Committee and to schedule a meeting where all concerned could express their feelings and reactions to the riots. The meeting was scheduled for July 9 at Freedom House in midtown. Headed \"GAY POWER,\" the flyer advertising the meeting was circulated during the riots over July 1 and 2. The flyer declared that the riots had shown \"that gay people are reaching the end of their patience. At the bottom of the calls for GAY POWER and HOMOSEXUAL EQUALITY is a realization that we can influence our existence\u2014if we can only come together.\" By July 5 a new flyer appeared. This one, titled \"Homosexual Liberation Meeting,\" recognized that the riots presented a rare opportunity for movement forward that must not be lost: \"Many of us in the community have been heartened by the appearance of a new spirit this past two weeks. Now is the time to take a stand on our own behalf. We cannot let the homosexual community fall back into a period of indifference and inaction because we have seen that this leads to persecution and explosive bitterness.\" It concluded with a tenet of the Left: \"[N]o one is free until everyone is free!\"\n\nHow ready the times were for such a new direction is demonstrated not only by the riots themselves but also by how individuals were affected by the uprising in terms of political awareness. When Marty Robinson witnessed one of the kick lines squaring off with the TPF on the first night of the riots he had had a kind of epiphany, discovering in that moment that gay men's sexuality was \"something precious, something worth fighting for.\" That night, he had lain awake in bed, unable to sleep as he thought about \"the responsibility of being in the movement and the importance of doing things that are good for people.\" He concluded: \"If I go out and politicize and repeat and demonstrate this liberation over and over again we will change the way homosexuals live.\" In Marty's mind, that night sealed his commitment to the gay political movement.\n\nMartha Shelley, who had only recently resigned as one of the officers of New York's small branch of the Daughters of Bilitis, had seen the riots on their second night as she was showing the Village to some DOB members visiting from Boston. She had not been aware that she was witnessing a homosexual riot, so that when her visitors had asked her what was going on she had played the New York sophisticate, calmly saying, \"Oh, just a riot. We have them here all the time,\" and had continued her walking tour. But on reading the Sunday _New York Times_ the following afternoon and learning the nature of the riot, she received a jolt. She, too, had barely slept, and not only because she had missed the last bus home to New Jersey after showing her guests around. \"I was slightly feverish from lack of sleep, so I lay on my couch tossing and turning, so excited by it, thinking, 'We have to do something. We have to have a protest march.' I thought, _What am I going to do?_ I thought, _I can make a speech or something_. I said, 'Yes, stand up in public and get shot at.' [But] that part of me that says, _You'd rather die than be called a coward_ , came to the fore and said I have to do it.\" Martha went to the phone and called Jeannie and Eleanor, a lesbian couple who kept the local DOB running. After explaining what was going on, Martha said, \"Look, we've got to do something. Let's have a march.\" Jeannie and Eleanor suggested that Martha go to the Mattachine Society. \"If they agree to it, we'll jointly sponsor it.\"\n\nMichael Brown, Martha Shelley, and Marty Robinson were not the only ones to sense the change in the air. According to historian Toby Marotta, several homosexuals with New Left or countercultural values went to Mattachine\u2013New York, Daughters of Bilitis, and Craig Rodwell to urge radical new action. Those who called on Mattachine\u2013New York and DOB found that \"their budding ideas about gay political activity were very different from the homophile outlooks of the leaders in charge.\" Those who went to Rodwell, who was trying to get people to join HYMN, agreed with him that his \"need to keep his bookstore legal and self-sustaining\" limited the kind of activities that HYMN, as an extension of the bookstore, could undertake.\n\nAs Brown hustled to find recruits for his Action Committee, one of the first he got to join was Bill Weaver, a gay man he had worked with to organize the large New York City antiwar march down Fifth Avenue. Soon Earl Galvin, Marty Robinson, and Martha Shelley were members as well.\n\nHow word had spread through connections at Alternate U. showed how ripe the time was for a new kind of movement. Susan Silverman, the director of programming at Alternate U., had helped set up the July 24 meeting. Silverman had started to think about women's issues in 1967 when only seventeen and a member of the Queens College SDS chapter. There she had been encouraged by feminist Robin Morgan to join New York Radical Women, New York City's first such feminist group. Through New York Radical Women she had participated in some of the key events in the new women's liberation movement, such as the 1968 feminist protest at the Miss America Pageant. Although these demonstrations had given an outlet to Silverman's passionate interest in feminist issues, they had done nothing to help her integrate her lesbian feelings, which she kept secret. Feeling isolated within WITCH, the radical feminist guerrilla theater collective, she quit the organization and began to work at Alternate U. She also ran a feminist workshop at Alternate U. in which she met Lois Hart, a follower of the recently deceased silent Eastern master, Meher Baba. Hart was the first woman Silverman had ever met who was willing to talk about being a lesbian, and both would attend the July 24 meeting.\n\nBut even before the first community meeting was held at Freedom House, another event took place that showed how ambiguously some in the gay community viewed the recent riots, as well as the extent to which the incipient movement would have its work cut out.\n\nIronically, Randy Wicker, the homophile movement's first East Coast militant, had been horrified by the riots. For years he had worked hard to counter stereotypical portrayals of homosexual men, and the reports he heard of the riots sounded like his worst nightmare come true: effeminate boys were in the streets camping it up while drag queens joined gay men in a chorus line to kick their heels at cops and others set trash cans on fire. And they were getting more press than Randy had received even on his best day. Wicker feared that a crazed queen might set one of the Village's old residences on fire and a grandmother or a child might be incinerated alive, obliterating overnight all the progress homophile activists had so painstakingly achieved over the past decade. Wicker had also gone up to the Mattachine\u2013New York office to beseech Leitsch, but to calm things down rather than rev them up. Wicker proposed diverting energy into a positive and peaceful channel, such as the effort the organization already had under way to raise money to replace the trees the vigilantes had cut down in Queens. Somehow the management of the Electric Circus, one of Manhattan's biggest and hippest nightclubs, which was just beginning to see business slacken, heard of the riots and had the idea of opening its doors to gay men and women. Mattachine accepted the offer and the nightclub put out a flyer titled _Oh Boy!_ , extolling itself as offering the city's best recorded sound and light show and saying: \"We don't think it's necessary for gay people to be quizzed at the door, packed into over-crowded, over-heated, over-priced, Mafia-controlled sewers. If you all come, and if the experiment works\u2014it could be beautiful... beautiful enough to do every week.\" The flyer noted that \"we'll be open to the general public as usual, but we're especially encouraging gay people to come\u2014and we really hope that everyone will dance together and dig one another.\" The Electric Circus invited gay people to come and dance on the night of July 6, a Sunday, promising that all the proceeds would go to the Mattachine fund to buy new \"Trees for Queens.\"\n\nWhen July 6 did come around, everything began smoothly. An acid rock band played a medley of popular songs, and some straight couples even joined gay ones on the dance floor. When the music stopped shortly after midnight, Wicker, wearing an American flag shirt (similar to the one that Abbie Hoffman had stolen from him and had been famously seen wearing on television), addressed the crowd and began talking about \"gay power,\" the phrase made famous by the riots. But Wicker also criticized the Stonewall Riots, saying: \"Rocks through windows don't open doors.\" Apparently the Circus had not bothered to inform all of their staff about their guests that Sunday evening, for after Randy had spoken only a few words, one of their employees suddenly realized he might be surrounded by homosexuals. He asked a blond youth standing next to him if he was \"with them\" and when the man replied affirmatively began to pummel him. Jack Nichols, who witnessed the scene, wrote that after the staff member was quelled, he was \"led from the premises screaming and shouting like a madman.\" The violence ended the evening, and Wicker offered to give the bloodied youth a ride home. In the car, Randy learned that the young man had fought at the Stonewall Riots. When Randy asked him what he thought about it all, the youth answered, \"All I know is that I've been in this movement three days, and I've been beaten up three times.\"\n\nWhen the July 9 meeting called by the Mattachine was held, close to a hundred people attended. One of those present was Bob Kohler, who stood up and tried to get attention for the street youths by asking for donations of money or clothing for them: \"We're on the brink of something here. Organizations are forming, and we're hearing 'gay liberation,' but these kids are sitting in the park. I'm doing what I can.\" But those at the meeting were not in a charitable mood and told him to \"sit down and shut up.\" Most of those at the meeting wanted to hold a demonstration to protest police harassment. A vote was taken and the idea of the march won overwhelmingly. The people who wanted to have a march were told to hold a planning session in a room to the rear, to which Martha Shelley and others repaired. It was a hot day, and the newly formed march committee opened cans of beer to quench their thirst as they began to work. The activists decided that the committee should have a name. As various names were proposed, someone put forward \"the Gay Liberation Front,\" a name modeled on the Communist Vietnamese National Liberation Front. Martha Shelley recalls her spontaneous reaction when she heard the phrase: \"I started pounding on the table, saying, 'That's it! That's it! We're the Gay Liberation Front!' \" In her enthusiasm, Shelley hit her hand on her beer's pop top over and over until she began to bleed, as she continued to scream, \"Oh, all right!\"\n\nThe group raised such a ruckus in their joy that Dick Leitsch came in demanding, \"What's going on here? What is this Gay Liberation Front?\" Shelley recalls: \"He was really upset. He thought that we were going to have another organization. There were seven gay organizations in New York, some consisting only of two people and a newsletter. He wanted there to be one gay organization, with him at the head of it. So we hastened to reassure him, 'Oh, no, we're not starting an organization. We're just a march committee.' \"\n\nBut Brown wanted something still more militant in tone and pressed for a second meeting, urging that it be held in the Village, since it was \"the heart of the gay community.\"\n\nThe next meeting was therefore planned to take place at St. John's Episcopal Church on Waverly Place on July 16. In the meantime, it was decided that the action should be both a protest and a commemoration of the Stonewall Riots. There would be a rally with speeches in Washington Square Park, followed by a march to the Stonewall Inn on the one-month anniversary of the riots. An ad was placed in _The Village Voice_ that announced the July 27 demonstration as co-sponsored by the Daughters of Bilitis and Mattachine\u2013New York. The display ad urged the reader to \"SUPPORT GAY POWER\" by attending the demonstration and wearing a lavender armband.\n\nBetween the meetings of July 9 and 16 Brown recruited Earl Galvin, Bill Weaver, and Martha Shelley to join the Mattachine Action Committee, or MAC. As members of MAC were leafleting for the July 16 meeting, they saw flyers advertising a special picket to be held on Bastille Day at the House of Detention for Women in support of the inmates, especially Black Panthers Afeni Shakur (mother of the future rapper Tupac Shakur) and Joan Bird. To show that gay people were now ready to join other oppressed minorities, MAC members joined the July 14 demonstration, which included John O'Brien. For the first time the two streams that had been independently moving toward starting a militant gay activist organization converged. They also got their first press coverage as militant and radical homosexuals in _Rat:_ \"[T]he demonstrators, chanting and singing, marched around the Women's House led by black people who had come to visit their friends inside and who had joined the solidarity demonstration. Then gay people, who now call themselves 'Pink Panthers,'... led the parade from the Women's House to the Stonewall Inn\u2014scene of the Christopher Street riots.\"\n\nThe flyer put out by Mattachine\u2013New York on July 10 called for gay people to end their isolation and apathy by attending the second gay liberation meeting, claiming that the \"positive response\" to the first gay power meeting had shown that \"homosexuals are no longer going to sit back and be apathetic pawns for every politician who comes along.\"\n\nAs it turned out, apathy would be the least of Mattachine's problems at the July 16 meeting. On that night, around two hundred people gathered in a large room with a low ceiling. Things got off to a bad start when Dick Leitsch arrived late. From there it went downhill\u2014or uphill, depending upon one's point of view. Tom Burke recorded what happened:\n\nDick Leitsch, in a staid brown suit, strides to the front. With professional aplomb, he reopens the meeting. Police brutality and heterosexual indifference must be protested, he asserts; at the same time, the gay world must retain the favor of the Establishment, especially those who make and change the laws. Homosexual acceptance will come slowly, by educating the straight community, with grace and good humor and...\n\nA tense boy with leonine hair is suddenly on his feet... \"We don't want acceptance, goddamn it! We want respect! Demand it! We're through hiding in dark bars behind Mafia doormen. We're going to go where straights go and do anything with each other they do and if they don't like it, well, _fuck them!_ [...]\n\n\"Well, now, _I_ think,\" says [Mattachine's secretary] Mrs. Cervantes, \"that what we ought to have is a gay vigil, in a park. Carry candles, perhaps. A peaceful vigil. I think we should be firm, but just as amicable and sweet as...\"\n\n\"Sweet!\" The new speaker resembles Billy the Kid. He is James Fouratt, New Left celebrity [...]\n\n\"Sweet! Bullshit! There's the stereotype homosexual again, man! Soft, weak, sensitive! Bullshit! That's the role society has been forcing these queens to play, and they just sit and accept it. We have got to radicalize, man! Be proud of what you are, man! And if it takes riots or even guns to show them what we are, well, that's the only language that the pigs understand!\"\n\nWild applause.[...]\n\nDick Leitsch tries to reply, but Fouratt shouts him down.\n\n\"All the oppressed have got to unite! The system keeps us all weak by keeping us separate.[...]\"\n\nAgain and again, Dick Leitsch tugs frantically at his clean white tie, shouting for the floor, screaming for order. He is firmly ignored.\n\nThe old wineskin was not able to contain the new wine. Nor was the Mattachine meeting on July 16 the first indication that the gay youth inspired by Stonewall would not follow the forms that an earlier generation had. As Craig Rodwell had ridden to Philadelphia on July 4 on the bus he had chartered for the Annual Reminder, he noticed that the people on the bus, most of whom he had personally recruited, were very animated. Much of the talk on the bus was about the Stonewall Riots, which had ended only a little more than twenty-four hours before the bus had departed New York.\n\nWhen Lilli Vincenz arrived from Washington, D.C., for the Annual Reminder, she immediately noticed a change. Instead of a small and sedate group in conservative dress, she found a boisterous crowd double the usual size, wearing jeans and T-shirts and brimming over with excitement about the riots. \"It was clear that things were changing. People who had felt oppressed now felt empowered,\" she recalls.\n\nWhen the demonstration started, its organizers told the demonstrators to follow the stringent rules always used at the Annual Reminder: pickets had to walk in a circle, single file and without chanting. For half an hour Craig marched in silence in the terrible heat, with no shade, clad in a suit and tie. Inwardly, he and others were seething to cut loose, to \"really make a statement.\" Then he noticed that two young women right in front of him were simultaneously breaking two of the picket demonstration's rules: they were not marching single file and they were holding hands. Craig was just thinking, _Oh-h-h, isn't that wonderful!_ when Frank Kameny, his face red as a beet, moved in between the two women and Craig and with a karate-chop blow broke their hands apart, saying, \"None of that! None of that!\"\n\nFurious, Craig immediately convinced about ten couples he had brought from New York to march holding hands. Bill Weaver scratched out the bland slogan on his picket sign and scrawled: \"SMASH SEXUAL FASCISM!\" on it. Rodwell noticed that the media were \"going crazy\" at the sight of queers actually holding hands in public as they marched in front of Independence Hall on July Fourth. He then spotted Kameny talking to a journalist. By prior agreement, the event's organizers had designated Kameny as the official spokesman, the one person authorized to talk to the media. Rodwell strode up to Kameny and barged in. \"Did you hear about what's going on in New York and the riots last week?\" Craig demanded. \"We're tired of not being able to hold hands in public, and the leadership of our demonstration has to change.\"\n\nThe bus ride back to New York City was just as animated as the one going down had been, but this time instead of the talk being about the Stonewall Riots, it was about the Annual Reminder and dress codes. As he listened, Craig realized that he had just attended the last of the Annual Reminders. Never again would gay people agree to such a controlled demonstration.\n\nThe July 24 meeting at Alternate U. drew about forty people, including Katzenberg, O'Brien, Pitts, Wilson, Hart, Silverman, Brown, Weaver, Galvin, Shelley, Fouratt, and Hoose. The crowd of mostly radical homosexuals was astonished at their own numbers. They decided to have each attendee introduce him- or herself and talk a little about his or her background. After these introductions, the group agreed to meet again in one week. By this time, Brown, Weaver, and Galvin had quit MAC, whose leadership was assumed by Marty Robinson and Martha Shelley.\n\nRobinson and Shelley were the main speakers at the Gay Power vigil and march to the Stonewall Inn six days after the first Alternate U. meeting. By 2:00 P.M. five hundred gay men and lesbians had gathered around the fountain in Washington Square Park. A new logo for the nascent revolution was displayed that day: a lavender banner with two female symbols interlinked on the left and two male symbols interlinked on the right was unfurled. Lavender armbands and sashes were distributed to the crowd. Martha Shelley stood on the fountain's rim to address the throng. She praised those present for their courage in attending such an open meeting: \"The time has come for us to walk in the sunshine. We don't have to ask permission to do it. Here we are!\" As she developed her theme, Shelley's tone became more militant: \"Brothers and sisters, welcome to this city's first gay-power vigil. We're tired of being harassed and persecuted. If a straight couple can hold hands in Washington Square, why can't we?\" The crowd applauded and went into a delirium of screams, giggles, and shouts of \"Sock it to 'em!\" and \"Long live the queen!\" Shelley continued, \"We will no longer be victimized by straight people who are guilt-ridden about sex. We're tired of flashlights and peeping-tom vigilantes. Tired of marriage laws that punish you for lifting your head off the pillow.\" After Shelley finished by naming some famous homosexuals, Marty Robinson addressed the crowd.\n\nThe passionate young carpenter declared that \"Gay power is here! Gay power is no laugh. There are one million homosexuals in New York City. If we wanted to, we could boycott Bloomingdale's, and that store would be closed in two weeks.\" He asked everyone present to join with groups like Mattachine\u2013New York and Daughters of Bililis to press the fight for equality. \"We will not permit another reign of terror. Let me tell you, homosexuals, we've got to get organized. We've got to stand up. This is our chance!\"\n\nThe crowd then marched down 4th Street to Sheridan Square in a four-by-four column, clapping in cadence and shouting \"Gay power\" and other slogans. As traffic was halted for them to cross Sixth Avenue, the protesters gained confidence. This was the first openly gay march not only in New York City but on the East Coast and many could not believe what they were seeing. As _The Village Voice_ reported, \"Even 4th Street's resident gypsies unearthed an Instamatic from the bowels of their trailer and snapped away.... Maybe it wasn't just a joke. Maybe there really was a gay power.\"\n\nAs the crowd assembled in and around the park across from the Stonewall Inn, Marty Robinson appealed for money, as well as for the creation of a gay newspaper. At this point there was tension among some of the protesters, for according to John O'Brien, Robinson had made an agreement with him that after reaching Sheridan Square they would march on the Sixth Precinct station. When someone shouted out that they should march on to the police station, Robinson ignored it and the crowd sang \"a curiously moving 'We Shall Overcome' \" before Robinson and Shelley both urged the crowd to disperse, apparently afraid of potential violence if the march continued to the police station. O'Brien felt betrayed and decided that from then on he would have nothing to do with the Mattachine Society. To the _Village Voice_ reporter it seemed that \"[g]ay power had surfaced.... A mild protest to be sure, but apparently only the beginning.\"\n\nOn July 31 at the second gay militant meeting at Alternate U., the assembly voted to call the new organization the Gay Liberation Front. They chose the name in part as a tribute to the National Liberation Front in its war with the South Vietnamese and U.S. governments. They also selected the name in hope that the new political entity would indeed be a \"front,\" that is, not simply a new organization but a unified alliance with all other gay and lesbian groups. But the vote on the new name would be one of the few noncontroversial votes in the Gay Liberation Front's short and highly fractious history.\n\nIronically or fittingly, the big fight at the first official meeting of the Gay Liberation Front (or GLF, as it soon became known), as Charles Pitts, one of the founders, explained, was over whether the new group \"should be [about] self-enlightenment (as a kind of consciousness-raising type thing) or integration immediately with other revolutionary or militant movements.\" The meeting voted and a slight majority, around 52 percent, favored concentrating on homosexual issues for the immediate future. The outcome caused a lot of shouting and disruptions as the more radical members, intent now on forming their own group, walked out. At this point many people who had attended, including Marty Robinson, were so disgusted they walked out of what now appeared to be a divided movement, leaving the floor to the more revolutionary types. In the second room a more organized meeting proceeded, so that many who had voted against joining the SDS-type radicals joined those in the second room. Jerry Hoose, who was not a leftist, stayed. He explained in retrospect that \"the victory of the radicals was sealed when the responsibility for preparing a statement for _Rat_ announcing GLF's birth was delegated to Michael Brown, Lois Hart, and Ron Ballard,\" the last two members in particular being among the most radical of all of GLF's members.\n\nThe article for _Rat_ in which the Gay Liberation Front made its debut was done in the form of an interview, with the supposed interviewer unidentified. To the first question, \"What is the Gay Liberation Front?,\" the answer given was direct, if grounded in theory:\n\nWe are a revolutionary homosexual group of men and women formed with the realization that complete sexual liberation for all people cannot come about unless existing social institutions are abolished. We reject society's attempt to impose sexual roles and definitions of our nature. We are stepping outside these roles and simplistic myths. We are going to be who we are. At the same time, we are creating new social forms and relations, that is, relations based upon brotherhood, cooperation, human love, and uninhibited sexuality. Babylon has forced us to commit ourselves to one thing... revolution.\n\nThe answer to the second question indicated the strong Marxist strain that would dominate the organization. This ideology influenced both GLF's analysis of the plight of lesbians and gay men as well as the position held by a majority of its members, that true homosexual revolutionaries should, indeed must, ally themselves with other groups oppressed by capitalism:\n\n_What makes you revolutionaries?_\n\nWe formed after the recent pig bust of the Stonewall, a well known gay bar in Greenwich Village. We've come to realize that all our frustrations and feelings of oppression are real. The society has fucked with us... within our families, on our jobs, in our education, in the streets, in our bedrooms; in short, it has shit all over us. We, like everyone else, are treated as commodities. We're told what to feel, what to think [....] We identify ourselves with all the oppressed: the Vietnamese struggle, the third world, the blacks, the workers... all those oppressed by this rotten, dirty, vile, fucked-up capitalist conspiracy.\n\nAsked how capitalism oppressed homosexuals in particular, GLF pointed to the closet, which it saw as the result of a social \"system of taboos and institutionalized repressions\" to control \"sexual expression.\" GLF's analysis of the gay political situation appropriately brought in both Freud and feminism, for marriage was named \"one of the most insidious and basic sustainers of the system\" of sexual repression because, through it, \"a male worker is given the illusion of participating in the power of the ruling class through economic control of his children and through the relation he has with his wife as sexual object and household slave.\"\n\nTo end gay oppression, GLF said that it would \"[relate] the militancy generated by the bar bust and by increasing pig harassment to a program that allows homosexuals and sexually liberated persons to confront themselves and society.\" GLF said it would achieve this through creating encounter groups, holding demonstrations, putting on dances, starting a newspaper, \"and just by being ourselves on the street,\" emphasizing the value of being \"out of the closet\" that would, in contrast to the homophile movement, be one of the hallmarks of the gay liberation movement. GLF already had \"specific plans\" that included opening a coffeehouse and starting both \"a working commune, and experimental living communes.\"\n\nWhile the analysis and the language used by GLF in its coming-out announcement were both revolutionary and highly militant, in practice, as historian Donn Teal recognized, \"GLF did not become SDS-type revolutionary. Its public appearances ranged, as did its membership, from support of Movement-sponsored protests to confrontations over American anti-homosexuality, personal, political, and organizational.\" Toby Marotta concurred: \"Although Brown, Hart, and Ballard took steps [in the _Rat_ article] to identify GLF with all the major concerns of the Movement, they made concessions to win the support of activists eager to deal first with gay issues.\"\n\nIt is also true that because of some of the positions GLF adopted at its inception, particularly that of trying to form alliances with other radical causes, less than four months after its birth GLF would fatally split. In the meantime, the single most important thing that needed to happen did: enough people had recognized the unique opportunity created by the Stonewall Riots to form a new, radically militant homosexual organization. The energy that the riots had released was not allowed to dissipate.\n\n#\n\n# \"We're the Gay Liberation Front!\"\n\nOn a Friday night Martha Shelley spun around joyously in the dance hall to the pounding beat of the rock music that filled the room. It mattered little that this room enlivened by happy faces and animated bodies usually functioned as an Alternate U. meeting room or that the blaring music came from a makeshift stereo and tape deck system. What did matter to Martha and the hundreds of other gay men and lesbians crowded into the room was that they had come together to create this celebratory event where they felt free. Intense bursts of color generated by a light show obscured the political posters that normally provided the plain hall's only decoration. While some men and women taking a break between dances sat in the wooden folding chairs lined up along the sides of the room, others sat in the windows overlooking the street, their feet dangling three floors above the sidewalk. As everyone gathered on the floor to participate in the line dance, the primal tribal feeling grew stronger. The men and women joined together and snaked around the room hooting and shouting, then spiraled into a tight mass before quickly unwinding with such terrific force that Nikos Diaman's glasses snapped in two as he was thrown into another reveler's body.\n\nMartha loved the joy and freedom she found in these communal rituals. Such celebratory dances felt entirely different from her experiences of lesbian bars before the summer of 1969. The watering holes she had frequented were dimly lit and small. As she had danced in those small bars she had felt uptight and uncomfortable. At the GLF dances the coat check was free and a beer cost a quarter. At these dances, friendly GLF hosts wearing name tags constantly circulated looking for anyone who seemed alone, and when they found a wallflower they talked to the man or woman and encouraged him or her to dance. Here the lights were low enough to be romantic, but not so low as to suggest you were in hiding. Off this main room couples found smaller, dimly lit rooms furnished with mats and chairs, a setting that invited quiet conversation and romantic encounters. These elements created an atmosphere of sexiness and camaraderie, as shown by the way the dancers smiled, hugged, kissed, and even pinched each other.\n\nSoon there would be dances exclusively for GLF women, which made them even more ecstatic: \"I remember when we had the lesbian-only dances, we could get undressed. We danced, you know, naked to the waist. Some people even took off all their clothes. I never went that far, but any chance to take off my shirt\u2014Ooh, all right! You know, hot night, summer in the city. And we danced in circles that were like an expression of community, not just coupling off. I'm sure we did too [many] drugs, though I never had that much of a problem with it. People slept around. Sometimes they stayed in couples for long times. But there was this feeling of ecstasy and freedom. It was great.\"\n\nJim Owles was equally enthusiastic about the GLF dances: \"The first time I danced with a man was at a GLF dance, and I really loved it! It was terrific!\" A born fighter and individualist, he had never had sex with another man until his twenty-second year, after he had been discharged from the air force, primarily for protesting the Vietnam War. Jim was still twenty-two when he arrived in New York City just in time to read news accounts about the Stonewall Riots. When he saw a notice about a meeting of \"militant homosexuals\" he decided to attend. At first he felt unsure about whether he should join the more radical group that was forming or stay with MAC. He discussed his reservations with Marty Robinson, who, like Jim, wondered if the radicals' only real intention was to provoke violence to bring about a revolution. Eventually Jim and Marty became lovers and both ended up joining GLF. Although out of the closet for less than a year, Jim wholeheartedly threw himself into the most militant gay organization ever. The young radical even became GLF's treasurer.\n\nThe GLF dances were so popular that people came not just from Long Island or New Jersey but also from as far away as Amherst and Baltimore. While other groups that used Alternate U. sometimes had dances, they attracted only small groups of people. For these gay dances, the line stretched down the block.\n\nThe reasons for the dances' success were clear to Jerry Hoose and Bob Kohler, who organized them. Kohler recalls, \"For the first time\u2014hallelujah!\u2014gay dances would be non-Mafia-run. It would be the first time that it would be gay dances by gay people for gay people, with the money that was handed in at the door going to gay issues and gay causes. This was a fabulous first. I get chills when I think about it. I mean, what had we had up till that point? Our entire existence revolved around oppressive, Mafia-run gay establishments where they hated faggots, where you risked your life.\" For Hoose, who disliked gay bars, the contrast could not have been more marked. He hated everything about the Stonewall Inn except its dance floor and had witnessed Ed Murphy brutalizing a customer at one of the other establishments at which Murphy worked: \"If you got drunk or did anything wrong [in a Mafia gay bar] you not only could get thrown out, you could get murdered. I mean, no one cared.\" Not only did GLF offer gay men and women a free and supportive environment, but also the dances were reasonably priced and those who could not afford to pay were let in for free. Hoose recalls that \"it was such a good feeling at the door because lots of people didn't have a penny and they were treated just as nicely as people who were handing us ten dollars instead of five dollars or three dollars.\" The dances were so appreciated that those who gave more than the suggested donation outnumbered those who asked to be excused from contributing.\n\nNor were gay and lesbian bars neglected as GLF began to exercise its philosophy of street-level activism there. John O'Brien recalls, \"One of the things we had to win from these bars was the right to post flyers. We demanded, and won, the right to have bulletin boards. A lot of gay people today take that for granted, that the cigarette machine has newspapers on top of it or that there are bulletin boards in these places. That was a demand won. That was something that was given with reluctance by these bar owners. And the bigger the mob connection, the less likely they wanted to have bulletin boards or material distributed. They wanted their customers as ignorant and as controlled as possible, with the bouncers at the door serving watered-down liquor. And if you didn't like something, they'd push you around. It was okay for the police to come and raid, because they paid off the police anyway and occasionally there would be a raid. So what? They don't care what happens to you, and they certainly didn't care if you were beaten up on the street outside of a gay bar.\"\n\nWhen two lesbians dancing in a Mafia lesbian bar refused to let a businessman cut in on them, the man punched one of them in the face while a Mafia hood watched and did nothing. When this was reported to GLF, a group of GLF men and women organized a takeover of the bar. As the group arrived, they started playing the jukebox and dancing while making a point of not buying any drinks. The owners did not understand what was happening, and Martha Shelley was selected to go talk to them. \"We were a little afraid because, you know, Mafia, right? You don't know if they're going to bring out the submachine guns and start blowing you away. So I go up to talk to this guy, and I'm standing there with my knees shaking, and I tell him why we're there, to protest that our people aren't getting taken care of, et cetera, et cetera. He says, 'Do you know who I am?' I said, 'I don't know, and I don't care, but we're the Gay Liberation Front!' \"\n\nWhile such moments were scary, they were also empowering. In those early, heady days of gay liberation, the moments of exuberance certainly outnumbered those of fear. GLF reveled in puncturing the smug superiority most heterosexuals of the era felt about their own sexual orientation. Shelley recalls such an instance from GLF's first winter when she took to the streets to sell the organization's newspaper. As she stood in the snow on a Village street corner in a pair of sneakers and a torn leather jacket, yelling \"Get your copy of _Come Out!_ , newspaper of the Gay Liberation Front,\" a well-dressed middle-class couple passed by, pushing a stroller and looking at Shelley with horror. Just as they passed her, Shelley said, \"Get your copy of _Come Out!_ Read what your kid's going to be like when he grows up.\" Shelley recalls, \"And they jumped, and I just loved it. It was a way of giving the world the finger for what they were doing to me.\"\n\nThere was also a special joy in the simple fact of being openly gay twenty-four hours a day in the public square, as practiced, for example, in demonstrations. Shelley described participation in a typical GLF demonstration as \"marching down the streets, arm-in-arm, shouting, 'Power to the people. Give me a _G;_ give me an _A;_ give me a _Y_.' Shouting at the top of our lungs. Dressed in whatever wild and freaky costumes we felt like. Forget the nylons, unless you were a drag queen, I guess. It was let it all hang out. Unzip your fly, and it's all out there, or unzip your mind.\" As John Lauritsen, who had been involved with two homophile groups in Boston before joining GLF, wrote, \"The Gay Liberation Front was a quantum leap forward. No more special pleading. No more apologies. Here was a radical organization, wild, woolly, and wonderful, ready to fight militantly for homosexual freedom.\"\n\nFlush with money from the success of its dances, GLF had moved quickly to fund a series of radical projects: a bail fund for its members, an underground-style newspaper, and a program of free lunches for the poor.\n\nIt was the launch of _Come Out!_ that precipitated GLF's first major demonstration and its first victory. Immediately after its official formation on July 31, and true to its word in _Rat_ , GLF set to work on its own publication. GLF composed an ad for the August 7 _Village Voice_ with the provocative lead-in \"Gay Power to Gay People.\" The ad's purpose was to solicit articles, photographs, and artwork for the new gay publication. After accepting the ad and payment for it, _The Voice_ ran it without the lead-in. Furious GLFers expressed \"general outrage\" at their regular Sunday meeting and discussed an action against _The Voice_ but decided they did not have enough evidence of bad faith to justify it. The group decided to submit another ad using the word _gay_ for _The Voice's_ next issue. That ad for a GLF dance carried the lead-in \"Gay Community Dance.\" After the ad was again accepted with no comment, the person who had placed the ad got a phone call from _The Voice_ the following day, explaining that it was the newspaper's policy \"to refrain from printing obscene words in classified ads.\" Asked why anyone would consider the word _gay_ obscene, the person representing _The Voice_ said that \"the staff had decided 'Gay' was equatable with 'fuck' and other four-letter words.\" Either the ad would have to be changed or it would not run. Since GLF wanted the ad to run and the word _homosexual_ was not an acceptable substitute, GLF accepted _homophile_ to get the ad run, with Leo Martello remarking that the then-outmoded word sounded like a nail file for homosexuals. _The Voice_ promised that it would provide GLF with a written explanation of their opposition to the words _gay_ and _homosexual_.\n\nGLF had asked for the written explanation because they planned to use it as part of the basis for a lawsuit against _The Voice_. The promised explanation was never delivered, and GLF initiated its suit, which necessitated GLF's serving a copy of a letter stating their proposed action to _The Voice's_ publisher, Ed Fancher. When Fancher continually refused to meet anyone from GLF at his office, GLF was forced to serve him at his apartment. As the letter was delivered, GLF asked to speak with Fancher about _The Voice's_ classified policy. He refused to do so, mumbling that GLF \"should not have done such an outrageous thing as to come to his place of residence\" before firmly closing the door in the GLF representatives' faces.\n\nFeeling that Fancher had literally and figuratively closed the door on dialogue, at its next general meeting GLF decided to hold an action and a demonstration aimed at _The Voice_. On the morning of September 12, at 9:00 A.M., GLF threw a picket line up around the _Voice_ offices, mere feet from the Stonewall Inn. GLF protesters recalled Smith's and Truscott's casual use of terms such as _faggot_ and _dyke_ in their coverage of the Stonewall Riots and Walter Troy Spencer's column in which he had referred to the riots as the \"Great Faggot Rebellion.\" In his column, instead of worrying about basic issues of justice and decent treatment of the city's lesbian and gay male population, Spencer had bemoaned that \"one Christopher Street bar operator estimates that a single night of indirect embargo cost him $500 business\" and that the riots may have caused friction among the Sixth Precinct, the TPF, and the Public Morals Division of the police department. Was this the newspaper that Fancher himself had envisioned, and therefore named, as the _voice_ of \"the displaced, disaffected, dissatisfied, and the unhappy\"?\n\nGLF set up a table to give out coffee and collect signatures on a petition condemning _The Voice_ 's policy. They also brought along 5,000 copies of a flyer explaining the GLF action. When Fancher arrived at 10:00 A.M., he was presented with a proclamation of GLF's grievances before he quickly disappeared inside _The Voice_ 's offices. All through the day the picketing and petitioning continued, accompanied by chanting. By late afternoon not many of the flyers remained and many signatures had been gathered on the petition. At 4:30 the GLF submitted a classified ad saying: \"The Gay Liberation Front sends love to all Gay men and women in the homosexual community.\" Soon after, Howard Smith emerged from the building and requested three GLF representatives to meet with Fancher.\n\nOnce inside, the representatives were greeted with a cry of outrage: how could the GLF have chosen the so-liberal _Voice_ as a target? But negotiations were soon undertaken. While Fancher explained that it was _Voice_ policy not to censor the content of the copy its writers submitted (which GLF accepted with the proviso that they reserved the right to oppose anything _The Voice_ did print), he soon relented on the advertising policy, saying that henceforth the newspaper would not alter ads after payment and would print the words _gay_ and _homosexual_ in its classified ads. One of the GLF emissaries leaned out the window and flashed a V-for-Victory sign to the crowd below. The _Voice_ triumph came just in time for GLF's first issue of its own publication, _Come Out!_ On its cover a banner line proclaimed: \" _Village Voice_ Goes Down.\" A victory dance was held in celebration.\n\nArthur Evans and his lover, Arthur Bell, had read about the Stonewall Riots, but the event had not made a big impression on them. Evans, then a student at Columbia University, confesses that although he was a political activist, he had never made a connection between his gayness and the political struggles that were roiling the country. Regarding Stonewall, he says, \"I didn't realize the implications of it. It just went over my head.\"\n\nBut two months after the riot, Evans and Bell were walking in the Village on a balmy, foggy night when a youth, apparently Orphan Annie, handed them a leaflet announcing the existence of GLF. They both decided to attend a meeting to see what it was like. As Evans was to say over three decades later, \"It changed our lives, both of us, forever.\"\n\nEvans was immediately captivated: \"It was exciting because we met people there who were very proud of being lesbian and gay and had a developed sense of politics and political struggle, and who were concerned about the great issues of the day, which were imperialism, racism, and peace. So it was wonderfully exciting to see all this great consciousness bursting out in a gay and lesbian context. The two lobes in my brain came together, making me realize, _Oh, well, my political life is related to my sexual life. There's a connection here_. I saw that connection by just interacting with people there. It wasn't a preached connection; it was a lived connection. They were real people, who embodied it in who they were. That's the most powerful example of all. It was an electrifying, wonderful experience.\"\n\nAround the time that GLF was confronting _The Village Voice_ , an important evolutionary change occurred that would have great implications for the organization. Members of the organization were encouraged to form cells within GLF, structures that would generally function autonomously. When the cell structure began to be implemented, Evans helped start one called the Radical Study Group, most of whose members would later become published authors. One of the first books the group read was Engels's _The Origin of the Family, Private Property, and the State_ , which had a great influence on Evans's thinking. For Evans, participating in the Radical Study Group was the most meaningful learning experience in his life in terms of reading, a marked contrast with his experience at Columbia, where he was still enrolled. \"The graduate students were walking around, sort of in a daze, trying to get through their exams. The professors were still pretending that the 1960s hadn't happened. The idea that a gay and lesbian sensibility could be relevant to philosophy, they would scoff at that, or scoff at the idea of gay and lesbian history. It was just totally beyond the faculty's comprehension. So I would be in that academic environment, which was so backwards and rigid and foolish, and then get together with my friends, where we were discussing these texts, and it was so lively and invigorating and wonderful, because we were bringing our personal experience to it.\"\n\nEven before Jim Owles and Marty Robinson had joined GLF, as members of MAC, they had discussed how to mobilize gay people into an effective political force. Numerous actions that were taken just before and after the formation of GLF had been conducted under Marty Robinson's scrutiny as experiments in militant organizing: the march on the one-month anniversary of the Stonewall Riots' outbreak, the \"Hang Out\" that was held on Christopher Street stoops, and the protest held where the trees in Kew Gardens had been cut down. While Marty did not complete his formal education, he had an extraordinary native intelligence, as Arthur Evans attests: \"Marty Robinson is the smartest person I've ever known. He wasn't the most learned, but I've never known anybody as smart as Marty was: a brilliant thinker, theatrical, and funny. Marty was a true genius.\" After analyzing his experiences with the July and August actions, Marty, with Jim's assistance, was ready to take their past tactics a creative step further in September as New York's mayoral race moved into its final phase.\n\nOn a warm autumn day, Mario Procaccino, the Democratic nominee for mayor, was campaigning in Queens, with his entourage not letting most citizens get too close. When his guards let a short youth with a Beatle haircut through to shake the candidate's hand, Procaccino pumped the young man's hand and beamed at him. The youth suddenly asked him, \"Mr. Procaccino, what are you going to do about the oppression of the homosexual?\" The smile disappeared and a look of concern came over his face as the candidate began to pat Jim Owles's hand as if in condolence: \"Young man, I can see that you're very interested in this problem. That is one of the many problems that we must face in New York. It is sick rather than criminal, and we must show understanding and compassion for them.\"\n\nAt the Gotham Young Republican Club, State Sen. John Marchi, the far-right-wing Republican nominee, had just finished a speech on urban crises when a member of GLF stood up and asked, \"Senator Marchi, are you aware of the emerging militancy within the homosexual community, and how does this relate to your views on law and order? Will homosexuals become targets or will you be responsive to their needs?\" A stunned silence hit the room and, a reporter noted, \"For the first time that evening the Senator lost his cool, elegant articulate style.\" Marchi tried to avoid answering by saying that he did not feel it necessary to speak on the matter, for \"it was being considered by some committee and was a topic for the state legislature.\" The questioner persisted, \"Senator, it's not just for the legislature. As mayor you have control of the police force. How will this affect the lives of New York's eight hundred thousand homosexuals?\" Marchi answered in a staccato voice, \"I will enforce the laws and prevailing mores of society.\" He would be questioned two more times about the issue before he left the room.\n\nAs a candidates' forum organized by the League of Women Voters on October 1 at the Temple Torah in Queens began, thirteen GLF members were scattered in the audience of two thousand. In advance, they had all submitted questions about homosexuality. After the forum had droned on for an hour and a half, it was clear that, as they had suspected, none of their questions would be addressed by the candidates. A decision had been made prior to the meeting to target the most reactionary candidates, Marchi and Procaccino. Suddenly Marty Robinson stood up in the middle of the audience and said, \"It's 1776, Mr. Procaccino! The homosexual revolution has begun!\" As Marty and then other GLF members peppered Procaccino and Marchi with questions, the audience, many of its members young, began to take their side, hooting at the candidates. When the police tried to remove Marty and Jim Owles, members of the League of Women Voters surrounded them to protect them. When the GLF protesters went outside, much of the audience followed them to talk with them, finding the rest of the proceedings inside boring. Gay liberation politics made the news, with the October 1 confrontation covered by NBC-TV's one o'clock news and the _New York Post_. It would not be the last time.\n\nAfter returning from the Fifth Annual Reminder, Craig Rodwell developed his own idea about how to propagate the energy released by the Stonewall Riots: create a gay holiday by moving the date of the Reminder from July 4 to the riots' anniversary and by changing the place of observance to New York City, the site of the riots. At first he shared his idea only with his lover, Fred Sargeant, and two lesbian friends of theirs who were members of the New York University Lesbian and Gay Student Union, Linda Rhodes and Ellen Broidy. They all liked the idea and, in October at another meeting of the foursome in Rodwell's Bleecker Street apartment, they outlined plans for the event.\n\nFittingly, the move to make the celebration official came in Philadelphia at the November 1969 Eastern Regional Conference of Homophile Organizations (ERCHO). Since Rodwell had offended protocol with his outburst at the Fifth Annual Reminder, he realized that if he was too publicly identified with a resolution to create the holiday it might fail. He therefore let Ellen Broidy take the lead in introducing the resolution. It read:\n\nThat the Annual Reminder, in order to be more relevant, reach a greater number of people, and encompass the ideas and ideals of the larger struggle in which we are engaged\u2014that of our fundamental human rights\u2014be moved both in time and location.\n\nWe propose that a demonstration be held annually on the last Saturday in June in New York City to commemorate the 1969 spontaneous demonstrations on Christopher Street and this demonstration be called CHRISTOPHER STREET LIBERATION DAY. No dress or age regulations shall be made for this demonstration.\n\nWe also propose that we contact Homophile organizations throughout the country and suggest that they hold parallel demonstrations on that day. We propose a nationwide show of support.\n\nThe resolution passed with the support of all the organizations except for Mattachine\u2013New York, which abstained.\n\nFor all of its creative and passionate radicalism, GLF also had a dark side. Over time, events convinced Robinson and Owles that the revolutionaries in GLF were, as Marotta has pointed out, opposed to \"securing reforms that would only make homosexuals complacent about the system\" the revolutionaries wanted to overthrow. For example, to make the gay population more politically aware, Marty Robinson had encouraged Ralph Hall to write a column in _Gay Power_ , a gay underground-style newspaper that had sprung up in the wake of the Stonewall Riots, taking its name from the riots slogan that seemed to best express the event's meaning. Some GLF members then wrote an article for _Come Out!_ that dismissed _Gay Power_ as \"an enterprise designed to make money.\" When an open meeting was held to review material being considered for the GLF newspaper, a consensus was reached to not print the article critical of _Gay Power_. The _Come Out!_ cell then went ahead and printed the article anyway. When a GLF assembly voted not to print a second issue of _Come Out!_ but to use the money instead to start a community center, the _Come Out!_ cell declared its independence. Lois Hart, who belonged to the _Come Out!_ collective, described the secession as a natural \"evolution,\" to which Marty Robinson and others responded that this was \"nothing less than an attempt to steal GLF's mouthpiece, which belonged to all because it had been financed with money raised by all.\"\n\nThere was no shortage of self-righteousness and shrill pronouncements (not to mention extraordinary rudeness) to be heard at GLF meetings and found in the pages of _Come Out!_ When Barbara Gittings and Kay Tobin came to meetings, they were sharply questioned because of their age and more conservative styles of dress about why they were attending. Kay was called a fascist for wearing a LINDSAY FOR MAYOR button. Madolin Cervantes, who had worked hard for homosexual equality for years, was rudely challenged about her right to participate in a conference of homophile organizations because she was heterosexual. Consciousness-raising sessions often amounted to little more than character assassinations. As Martha Shelley put it, \"The downside is that we had so much anger that we also turned it against each other.... It was like if you disagreed, you were a threat.\" Even Bill Katzenberg, the organization's prime mover, quit the organization before it was three months old \"because he thought the radicals spent a ridiculous amount of time criticizing him because he was not open about his homosexuality when working in SDS.\" It was not long before the other main founder, Michael Brown, also quit GLF in disgust.\n\nMost infuriating for many of GLF's members was the way in which the organization through striving for the purest form of democracy became both inefficient and undemocratic. The examples given earlier illustrate how democracy was subverted in the name of autonomy. But it was also the case that any matter that had once been decided could be reopened at any future meeting, parliamentary procedures being considered too restrictive by GLF. Given the freewheeling style of GLF meetings, it could take a very long time to reach consensus on any matter. Understandably, it tried the patience of many to see matters that had been painstakingly discussed and decided reconsidered over and over at meeting after meeting, a practice that drove Arthur Evans in particular to distraction.\n\nBut of all the contentious issues that wracked this most contentious organization (which claimed not to be an organization at all but a loose collection of individuals and small groups of individuals), the most divisive of all was the issue that had bedeviled it from its first meeting, whether it should be a one-issue organization or whether it should ally itself with all other progressive causes.\n\nIn mid-November 1969, that issue came to a head over whether GLF should make a large contribution of money to the Black Panthers. John O'Brien had introduced a motion to donate $500 to the Panthers because he wanted to achieve solidarity with them. After ferocious debate, the motion was defeated. Those who opposed the motion made many arguments against it. Among the most compelling points was that the Panthers often employed language even more virulently homophobic than anything used by the reviled Marchi or Procaccino. In a move to reopen debate on an issue that had already been decided, the following week there was a call for a recount, and the motion passed.\n\nFollowing the vote, Jim Owles resigned as treasurer. Marty Robinson had already quit a couple of weeks prior. That night many members walked out, something that had never happened before. These resignations marked the beginning of the end for the fledgling gay liberation organization, but the birth of another one.\n\n#\n\n# The Heroic Age\n\nArthur Evans had become friendly with Marty Robinson and Jim Owles not long after joining GLF because he felt a strong affinity with their approach to gay politics. It was natural, therefore, that when he began to have thoughts about the need for an alternative to GLF, he turned to one of these men. On a rainy Sunday early in November, Arthur invited Jim to join him and Arthur Bell for brunch. At the end of the meal, over coffee, Evans broached the subject of starting a new group.\n\nJim told the two Arthurs that he and Marty had been thinking along the same lines. In fact, they had already discussed the idea with some of their closer friends. Jim expanded on his analysis of what he saw as one of GLF's main weaknesses, that in trying to align itself with all other oppressed groups, \"homosexual needs were being sluffed over for the big picture.\" He cited how a recent weekly Sunday night meeting had been disrupted by someone arriving with a report that women were being discriminated against at the Electric Circus, whereupon half the people attending left to join in a protest at the Circus. A month prior, in early October, Jim and Marty had considered forming a group dedicated exclusively to the cause of gay liberation, with a constitution, preamble, goals, and purposes. Everyone in the group who wanted to support other causes would be free to do so on their own. All this sounded good to Evans and Bell, and so Jim invited them to meet with him and Marty at Jim's apartment the following night. And so it was that the three men\u2014Marty Robinson, Jim Owles, and Arthur Evans\u2014who would be the most important figures in forming the new gay liberation group, as well as the three persons most prominently identified with it, began work to make it a reality in early November of 1969.\n\nThe small group invited only trusted friends to form a core group of thirteen who would meet during November and December to debate just what form the new group should take. Specifically, they worked on language for a preamble and constitution. Those thirteen were Marty Robinson, Jim Owles, Arthur Evans, Arthur Bell, Kay Tobin, Donn Teal, Tom Doerr, Fred Orlansky, Fred Cabellero, Robin Souza (\"Gary Dutton\"), Leo Martello, Richard Flynn, and Steve Adams. The first meeting of all who wanted to participate in the \"new group,\" as they initially called themselves, took place on November 24. (Arthur Evans missed the meeting to attend a reading by Allen Ginsberg.) After everyone agreed to screen new potential members to keep spies from GLF out, the new group's purpose was discussed. The activists agreed that the group would be radical and activist and \"would take responsible actions only, carefully planned within the framework of our society... without violence.\" Various possible undertakings were discussed, as well as the issue of what they should call themselves. Brainstorming for a name produced nothing satisfactory, and it was at a later meeting that the group settled on the straightforward Gay Activists Alliance. They also agreed that the group should first draw up \"a strong preamble\" and then use that text as a basis to further structure the group. The task of drafting the preamble was assigned to Arthur Evans.\n\nAfter taking suggestions, Evans sat down to work. He recalled: \"What I was concerned about in writing the preamble for the Gay Activists Alliance and the work on the constitution was to incorporate personal lived experience. I had become very suspicious of abstract political rhetoric, even when I agreed with it. I've been heavily influenced by Marxist and anarchist thinking, but I felt that in our movement we had to start with experiential confrontations first: the light would come out of the events themselves. Let's set up the stage so that the right events happen and the light will come out of that.\n\n\"That's what I experienced when I was in Chicago at the demonstration there... that the experience of the activity transcends any ideology. The ideology is important; I'm not saying it's not: I'm a philosopher. I'm interested in ideas and ideology. But... the lived experience... always transcends it....\n\n\"And that's what I saw GAA as doing and saying. You want to talk about revolution, the really revolutionary thing is... the situation of people on the street acting, in that dramatic context.... That releases an energy that goes on forever and keeps transforming people's lives, and that just can't be captured in any ideological formula.... That was one of my objections to GLF. I kept feeling they were trying to capture revolution in an ideological formula, and my feeling was that all formulas are lies.\"\n\nAt the next meeting, Evans distributed his draft of the preamble, and later took critical feedback over the telephone. As the constitution evolved, it contained two critical provisions: that GAA would not endorse any candidates for public office and that meetings would follow _Robert's Rules of Order_ in order to keep the organization democratic.\n\nAfter one more meeting and some revisions, the final versions of the preamble and constitution were refined and adopted at a meeting attended by nineteen people in Arthur Bell's apartment on December 21, which happened to be the winter solstice. The group of activists held an election and selected Jim Owles as their president and Marty Robinson delegate at large. To celebrate the birth of their new organization, they drank champagne.\n\nThe preamble as adopted read:\n\nWe as liberated homosexual activists demand the freedom for expression of our dignity and value as human beings through confrontation with and disarmament of all mechanisms which unjustly inhibit us: economic, social, and political. Before the public conscience, we demand an immediate end to all oppression of homosexuals and the immediate unconditional recognition of these basic rights:\n\nThe right to our own feelings. This is the right to feel attracted to the beauty of members of our own sex and to embrace those feelings as truly our own, free from any question or challenge whatsoever by any other person, institution, or moral authority.\n\nThe right to love. This is the right to express our feelings in action, the right to make love...\n\nThe right to our own bodies. This is the right to treat and express our bodies as we will, to nurture them, to display them, to embellish them...\n\nThe right to be persons. This is the right freely to express our own individuality under the governance of laws justly made and executed...\n\nTo secure these rights, we hereby institute the Gay Activists Alliance, which shall be completely and solely dedicated to their implementation and maintenance,... disdaining all ideologies, whether political or social, and forbearing alliance with any other organization except for those whose concrete actions are likewise so specifically dedicated.\n\nIt is, finally, to the imagination of oppressed homosexuals themselves that we commend the consideration of these rights, upon whose actions alone depends all hope for the prospect of their lasting procurement.\n\nAccording to Arthur Evans, he used the word _activist_ in the first paragraph to emphasize that the members of the organization were not just thinkers but people who acted to secure gay rights. Evans modified the Declaration of Independence's phrase \"let facts be submitted to a candid world\" to \"before the public conscience\" to place GAA's call to arms firmly in the American revolutionary tradition.\n\nEvans considered the right to feelings particularly important, especially as it was mentioned nowhere in the U.S. Constitution. As Evans believed that rights begin with feelings, he thought it necessary to acknowledge the right to feelings and not to only thought and behavior as traditional law did. That is, \"It doesn't matter what your morality is or what your laws are or what your thoughts are: We are entitled to our feelings, and we're especially entitled to feelings of love and beauty. The right to one's own body was written partly in affirmation of people who today are called transgendered people.\"\n\nIn the penultimate paragraph Evans intended to draw the major distinction between GLF and GAA, as he felt the imperative task was to seize the opportunity to create a vehicle that could create an uncompromised sense of gay and lesbian identity. To Evans this identity seemed the necessary bedrock at that historic juncture.\n\nThe final paragraph was intended as an appeal to all living in the closet. As Evans saw it, \"I'm appealing to myself as that lonely child, saying, 'This is the avenue through which you can grow and express your power and become a person.' \"\n\nGAA had forged a revolutionary document. Remarkably, GAA often succeeded in living up to its lofty ideals.\n\nAn advertisement was placed in _The Village Voice_ for the first public meeting of GAA. The event was held at what would become the organization's regular meeting place, the Church of the Holy Apostles on 28th Street and Ninth Avenue. Marty Robinson reported that GAA's first action would be a petition to present to none other than Carol Greitzer, the councilwoman who had often called on the police to arrest gay men. The GAA petition demanded that Greitzer introduce a bill outlawing discrimination in employment on the basis of homosexuality and that she work to end all discriminatory restrictions on gay businesses and remove state laws that criminalized homosexual conduct. Blank petitions were handed out for circulation. This, the activists reasoned, would provide them with a means to meet the public while introducing many citizens to the idea of gayness as a political cause.\n\nFurther progress was soon made in organizational structure. Committees were set up for publicity and political action, as well as a street committee to distribute petitions and pamphlets and to make signs for demonstrations, and a combination pleasure and fund-raising committee. When Marty Robinson ran into _Gay Power_ 's editor, he offered to run a regular GAA column if someone from the group would write it. Arthur Bell volunteered for the job.\n\nAs the eager activists busied themselves with these and other projects, they also fired off letters to newspapers that had made denigrating comments about homosexuals; met with a senior columnist at the _New York Post_ to complain about the same matter; and paid surprise visits to VID and VID's Human Rights Committee to begin to soften up Greitzer as a target.\n\nThe media-savvy group realized that it could be highly beneficial to have a logo or symbol, which was provided by Marty Robinson's new lover, Tom Doerr, a graphic design artist. (Marty and Jim had broken up as a couple in the fall around the same time Marty got involved with the shy and attractive Doerr.) The symbol Doerr suggested was that of the lambda, the eleventh letter of the Greek lowercase alphabet, which resembles an inverted letter _Y_. Tom designed the symbol as chrome yellow on a blue background. The lambda was chosen because in chemistry and physics it symbolizes a complete exchange of energy, \"that moment that's witness to absolute activity.\"\n\nBy the end of February GAA had already grown from fifteen to forty members and thus had adequate mass to plan a large action. On the morning of March 5, GAA members traveled to city hall, planning to enter the building and demand that Mayor Lindsay\u2014newly reelected in large part thanks to gay voters who remembered his actions to end entrapment\u2014take a public stance to end both job discrimination and police harassment. Apparently someone had tipped off both the press and city hall, for GAA members getting off the subway carrying posters were greeted by photographers and reporters. As they approached city hall they saw that police were present in overwhelming numbers, with many on horseback. When the GAA members tried to enter city hall, the police stopped them. They asked the police why other citizens were allowed to enter city hall and they were not. The police answered that \"only private citizens\" could enter, not groups. The activists were warned to \"keep out\" and \"get back,\" but the slight Jim Owles, whom Marty Robinson would later affectionately call \"the scrappiest little faggot in New York City,\" tried to push past the police. After the police bodily evicted Jim, they put up barricades around the city hall promenade. A minor city functionary offered the members of GAA a meeting with a slightly less minor official. GAA stood their ground, insisting on a meeting with the mayor. They were told no dice, that Lindsay was in Buffalo. The activists were not so easily dissuaded, however. They threw up a picket line and stayed for hours, chanting and holding their signs high when not talking to the press. Men kissed men as cameras recorded the scene, and a guitar player with long hair began to improvise songs about Lindsay, sodomy, love, and \"the little piggies that protect city hall.\" The lunch crowd generally reacted favorably, seeming to enjoy the demonstrators. Arthur Bell described the scene:\n\nThe sight... they saw and the answers [the press] got quickly negated any concept of namby-pamby homosexuality. Here, for all Wall Street, City Hall, and the press of America, were honest-to-God flesh-and-blood gorgeous, gorgeous gays. Stereotypes? Gone with the wind.\n\nFinally Michael Dontzin, the mayor's counsel and adviser, walked up to the protesters and agreed to meet with Jim Owles, a photographer from _Gay Power_ , and Arthur Bell, armed with his tape recorder. A general discussion ensued about harassment of gay folk, including specifics about bars, bathhouses, and entrapment by the Transit Police. Whereas Dontzin insisted that the mayor was supportive and sensitive to the issue and cited the progress of recent years, Owles insisted that the days of backroom promises were over: what they wanted was a public stand from the mayor. Moreover, Lindsay and other political figures from now on were going to have to listen to gay gripes, talk to the gay community in public forums, and \"help us when help was needed.\" Dontzin assured them he would present the mayor with their demands when he returned from Buffalo and, as the protesters left, promised to get back to them shortly. Owles recognized Dontzin's talk as political softshoe, however, and did not take it seriously, though Bell thought Dontzin sincere. All the same, as Bell left City Hall that Thursday, he reasoned that it was best to remain prepared to attack whoever attacked the community next.\n\nHe did not have to wait long. The following night Seymour Pine raided the gay bars the Zodiac and 17 Barrow Street. On Saturday night he raided the Snake Pit, an after-hours illegal but non-Mafia bar that, in spite of its name, had a reputation for being one of the friendlier bars in town.\n\nPine claimed that he had had a great many complaints against the Snake Pit for noise and that he was closing down illegal bars, gay or not. Certainly the bar was patently illegal, having neither a liquor license nor a certificate of occupancy, and like the Stonewall Inn was an overcrowded firetrap with only one exit. Around two hundred men were crowded into an area with only one exit that probably would have had an occupancy limit of sixty customers if it had been inspected and licensed by the city.\n\nWhen Pine raided the Snake Pit he had trouble identifying the management, who had slipped into the crowd of customers. But more seriously, Pine was once again faced with a crowd of customers who milled about outside a raided gay bar and did not disperse. The scene seemed explosive to Pine, and the last thing he wanted was a repeat of last summer's rioting. He surveyed the situation and made up his mind: at the Stonewall raid he had let the customers go. Now he had another crowd of gay club customers on his hands as he had at the Stonewall. To prevent a recurrence of that dreaded night of last June, he decided to handle it differently this time: he would arrest all the customers still inside and sort out the bar's management from them at the station house.\n\nIt took a long time for the police to transport over 160 customers in small groups via patrol wagons from the bar to the Sixth Precinct station house. One of the customers who was held inside the Snake Pit the longest was Alfredo Diego Vinales, who had recently started frequenting the bar. Vinales was a twenty-three-year-old Argentinean with an expired visa living in East Orange, New Jersey. Before he was put into a patrol wagon, one of Vinales's friends noticed that he seemed very nervous and extremely frightened. Lawrence (not his real name), one of the club's employees, asked one of the police, \"What rights do we have?\" The cop simply answered, \"Shut your fucking mouth.\"\n\nVinales and the 162 other men taken to the Charles Street station were herded up to the station's second floor, where the scene was chaotic. The police were also verbally abusive of the arrested men, repeatedly calling them faggots and making other derogatory remarks and threats. As summons were about to be issued to all the men taken from the Snake Pit, Pine explained that their identification would not be checked and that no one would have to pay bail.\n\nThe Spanish-speaking Vinales either did not hear or did not grasp the implications of Pine's explanation. Terrified of being deported for being homosexual, Vinales suddenly bolted up a flight of stairs. As one person present in the station house would later recall, \"Suddenly I heard a sound, like something falling, then screams.\" The screams he heard were those of Vinales, who had tried to escape by jumping from a window to the roof of the next building. He had missed and fallen onto a fence below, where he had been impaled on six of its fourteen-inch-long ice pick\u2013sharp iron spikes. When it became obvious that the police could not simply lift Diego off of the fence, Rescue Company No. 1 of the city fire department was called in to use an electric saw to cut off a portion of the fence so that they could transport Vinales to the hospital.\n\nAfter the firemen arrived, a police officer said to one of them, \"You don't have to hurry, he's dead, and if he's not, he's not going to live long.\"\n\nWhile the police may have initially intimidated those they had arrested, these were men who had witnessed the birth of a new militancy within the gay community over the last nine months. As word spread through the station of what had happened to Vinales, the men inside first talked to one another to keep their spirits up. Then, as the crowd had done at the beginning of the Stonewall Riots, they began to sing. First \"America the Beautiful.\" Then the more political \"We Shall Overcome.\" Then a chant first heard during the previous summer's riots began to resound inside the Sixth Precinct station: \"Gay power! Gay power! Gay power!\"\n\nAs the police became preoccupied with Vinales, some of the arrested men wandered into what apparently was the captain's office and took over a pair of phones. They called _The New York Times_ , the _New York Post_ , and the _Daily News_ to apprise them of what had happened. _The Times_ and the _Post_ were not responsive, but the _Daily News_ seemed keenly interested and dispatched a photographer to the Charles Street station. One man called a lawyer. Another man called Marty Robinson and Jim Owles. To even up the score some more, they took record books from the office and dumped them down a shaft.\n\nAt ten o'clock on Sunday morning, Arthur Bell felt groggy as he wandered into his bathroom and turned on the radio. As he listened to the news, word came of a police raid on an after-hours bar in Greenwich Village in which two hundred were arrested and an unidentified man had been seriously hurt. The injured man had been taken to St. Vincent's Hospital. Bell jolted awake as he immediately realized that a gay bar had been raided once again. Bell roused Evans, signaling to him to listen to the news as Bell went to the phone and started making calls to learn what had happened.\n\nNews of the tragic incident had spread throughout the gay community, interspersed with rumors that Vinales had died or was dying, so it did not take Bell long to grasp what had happened. When he could not reach Jim or Marty, he called Bob Kohler and suggested that GLF and GAA and all the rest of the homosexual community join in a protest action that evening. Kohler did not think a protest could be pulled together on such short notice. At noon, Bell finally reached Jim and Marty, who had already gotten the news via the phone call from the police station.\n\nIt was Bell who suggested that all GAA members be called to an emergency meeting. Only two hours later, thirty GAA members were gathered at Jim and Marty's apartment. Everyone agreed that it was essential to act that day to signal the community's outrage. A protest march was decided on, to be followed by a vigil at St. Vincent's Hospital, where Vinales, who had not obliged the police by dying, lay in a coma.\n\nA pamphlet was written by Marty Robinson about the incident, which he began with the resounding lines: \"Any way you look at it\u2014that boy was PUSHED!! We are ALL being pushed.\" The march would be on the Sixth Precinct station and would start from Christopher Park, the site of the riots. GAA swung into high gear as Robinson managed to find a mimeograph machine on a cold Sunday. GAA members divided into teams to deliver the flyers to different parts of the gay world. Bell worked the phone, talking to the news media and encouraging them to cover the evening's protest. Bell found that a number of the reporters with whom he spoke remembered him from the City Hall protest a few days earlier.\n\nLater in the day, Bell noticed that Marty \"was like a tiger on the prowl, pacing back and forth excited and jumpy\": GLF had invited a number of nongay movement groups, so a very large turnout was expected.\n\nBy 9:00 P.M. five hundred demonstrators had assembled at Christopher Park, the site of the Stonewall Riots. The crowd included men and women from GAA, GLF, Homosexuals Intransigent (another recently formed gay organization), and HYMN, many gay people who were not members of organizations, heterosexuals, Yippies, members of a women's liberation group, as well as the Reverend Robert O. Weeks, the very supportive pastor of the Church of the Holy Apostles. From the park the angry demonstrators marched to the Sixth Precinct, chanting \"Say it loud: gay is proud\" and \"Stop the killings,\" as well as the emblematic refrain from the Stonewall Riots: \"Gay power!\" It had been decided the tone was to be angry but orderly, and marshals from GAA and GLF supervised the march to keep it from spilling over into the streets. As the GLF banner accompanied the marchers, Villagers gay and straight spontaneously left their apartments to join the march, while others watched astonished from their windows.\n\nWhen the crowd arrived at the police station they found that busloads of blue-helmeted TPF officers had been brought to greet them, as well as hundreds of uniformed police officers and \"countless\" plainclothesmen who encircled the area, guarding the station.\n\nBut the protesters were not cowed by this show of police force, massive as it was. Instead, the crowd called for Salvatore Salmieri, the captain, yelling, \"We want Salmieri! We want Salmieri!\" When Salmieri did not appear, the crowd chanted, \"Who gets the pay-off? The police get the pay-off!\" and \"There's the Mafia in blue!\"\n\nOwles and the Reverend Weeks approached the police, and Jim suggested that a few representatives from the crowd be allowed to enter the station and talk to the commanding officer to lower the risk of rioting by the crowd. His request was turned down.\n\nOwles turned from the police to the agitated crowd. He shouted, \"Our brother lies near death at St. Vincent's Hospital! We've made our point to the police by our numbers! Now we will march in solemn procession to the hospital.\" The crowd calmed down somewhat and walked over to the hospital, with hundreds carrying lighted candles. There the crowd held a silent vigil march across the street from the hospital. They then returned to Christopher Park, but only after retracing the footsteps of the rioters from the previous summer by spontaneously detouring onto Gay Street as they headed up Christopher Street. To have been able to turn out a crowd half a thousand strong in cold weather in a few hours to march on a police station on behalf of gay rights was a measure of how far the new gay liberation movement had progressed in the seven months since its birth.\n\nThe raid on the Snake Pit and Diego Vinales's tragic accident received fairly intense media coverage in New York City, the most attention the issue of homosexuality had received since the Stonewall Riots. The substantial media coverage that the Snake Pit raid received was a direct result of the way the Stonewall Riots had politicized the gay world: there were presently not only two organized groups of highly militant, activist gay men and lesbians but also three gay newspapers where before there had been none: in addition to _Gay Power_ and _Come Out!_ , there was _Gay_ , which had been started by Jack Nichols and Lige Clarke. The erotic and wildly successful newspaper _Screw_ had given Lige and Jack a column to write on homosexuality in 1969, called \"The Homosexual Citizen.\" The column had proven so popular that Lige and Jack had soon realized that a gay newspaper might be financially sustainable. Also, after Stonewall the couple realized that the homosexual community was entering a historic phase, and they wanted to create a newspaper of record for the era. _Screw_ 's publisher, Al Goldstein, agreed to finance the new paper, and Jack and Lige hired some of their friends from the old homophile movement, such as Kay Tobin, Dick Leitsch, and Randy Wicker, to help record the unfolding history of gay liberation.\n\nWhile both _Gay_ and _Gay Power_ had around twenty-five thousand readers each, the new movement was also having some success with the nongay press. The protest at City Hall was covered by the _New York Post, Women's Wear Daily, Long Island Press_ , WCBS, WINS, WBAI, and WABC-TV, as well as making the front page of the Spanish-language newspaper _El Diario_. _The Village Voice_ noted that the number of gay bars in New York City seemed to have doubled since the Stonewall Riots.\n\nFor these and other reasons, the Snake Pit incident got so much attention that it galvanized the just-born gay liberation movement. Not only did the raid pump more energy into a movement that was already highly energized, but it also inspired GAA to take their previous strategy of creative, direct confrontations a critical step forward to create a new form of political protest, combining militancy, guerrilla theater, and gay sensibility in the form of camp. It was this innovative tactic of bold, face-to-face encounters, created by Marty Robinson, that eventually won the fledgling movement significant media coverage and finally put the issue of lesbian and gay rights permanently on the political map.\n\nArthur Evans recalled how the Snake Pit incident (and GAA's political analysis of the root cause) inspired the invention of what GAA came to call a zap: \"John Lindsay was the mayor of New York who had gotten elected by a plurality of black and gay votes, a Republican liberal with a nice plastic smile, very gracious on television but really unwilling to put any teeth in his rhetoric. As a result, there was a step-up of police raids on gay bars.\n\n\"The Snake Pit incident truly outraged us, and we put out a leaflet saying that, in effect, regardless of how you looked at it, Diego Vinales was pushed out the window and we were determined to stop it. We decided that the politics that were going on that allowed this to happen [was something that] we felt in our personal lives: it wasn't just an abstract political issue with us. People were being impaled. People were terrified, being arrested, careers broken. There was no division for us between the political and personal. We were never given the option to make that division. We lived it.\n\n\"So we decided that people on the other side of the power structure were going to have the same thing happen to them. The wall that they had built protecting themselves from the personal consequences of their political decisions was going to be torn down and politics was going to become personal for them. That meant, in effect, that we were going to disrupt Mayor Lindsay's personal life and the personal life of his family as a result of the political consequences of his administration. So we decided that every time he appeared in public or every time we could get to him, we would make life as personally uncomfortable for him as we could and remind him of the reason why.\"\n\nAs April began, it seemed timely to call on the mayor anyhow, as Jim Owles's estimate of Dontzin's sincerity had been accurate: in the month since their meeting Dontzin had neither taken any phone calls nor answered any letters from GAA.\n\nMonday, April 13, 1970, dawned a perfect spring day, which seemed appropriate for the ceremony planned for that day, the one hundredth anniversary of New York City's most prominent cultural institution, the Metropolitan Museum of Art. At 9:55 in the morning Mayor Lindsay ascended the front steps of the city's great temple of culture, where he was greeted by the museum's director, Thomas Hoving. A band played the national anthem, and then Mr. Hoving wittily introduced the mayor as \"our landlord,\" bringing a grin from Lindsay.\n\nLindsay began to speak, thanking museum patrons for caring \"deeply enough about the city and art to recognize their mutual dependency.\" As Lindsay spoke, no one seemed to pay any attention to a young man with dark hair, wearing a baseball jacket, who was slowly walking up the museum's steps. Lindsay was saying that for an entire century the city and the museum had \"nourished each other\" as the young man got closer to the podium. The mayor had just begun to praise the museum's new plaza for \"extending the museum's beauty out into the street,\" when Marty Robinson spoke into the microphone, his voice suddenly interrupting Lindsay's bland address with the question, \"When are you going to speak out on homosexual rights, Mr. Mayor?\" Lindsay smiled but did not say a word as two helmeted policemen rushed Marty off the scene.\n\nGAA members were scattered throughout the crowd, armed with leaflets, and ready to get in a line that was to form to shake the mayor's hand. As GAA members shook Lindsay's hand, they asked him questions about why he was not supporting gay rights and attempted to give him their leaflets. Some held on to the mayor's hand so long that the police had to pull them apart. While Lindsay toured the museum, one activist stuck a leaflet in his hand. Jim Owles then walked up to Lindsay and said, \"You have our leaflet. Now when the hell are you going to speak to homosexuals?\" Lindsay never quit smiling, nor did he ever say anything in response.\n\nSix days later Lindsay was blindsided again. The mayor had a weekly half-hour television program on WNEW titled _With Mayor Lindsay_ that was taped three hours before it aired. GAA had decided weeks in advance to target the April 19 show, so its members had been writing in for tickets for weeks. By mid-April they had around forty tickets for the April 19 date, enough to make up a third of the audience. That Lindsay still had not responded to the organization gave them a legitimate reason to \"zap\" him a second time.\n\nAt four o'clock thirty-seven men and at least one woman, Kay Tobin, gathered in Bell's apartment. There Arthur Evans coached them on when and how to chant, applaud, and stamp their feet in order to make the experience as unnerving as possible for Lindsay, hoping either to get him to loosen up or for some of them to be arrested. Two hours later they left Bell's apartment, which was near the television studio, departing in small groups so as not to attract attention. They had taken care to dress in a number of different styles so as to blend in with the rest of the studio audience.\n\nAt the studio Arthur Bell ran into Lindsay's counsel, Dontzin.\n\n\"I see you have some of your people here.\"\n\n\"Why haven't we heard from you?\"\n\n\"I didn't know I was to get back to you. Let's get together next week and talk.\"\n\nThat must have struck Bell as more than odd, as he had left many messages with Dontzin's secretary. When Bell pointed this out to him, Dontzin apologized, saying that he \"was over his head with work\" and really was eager to meet with Bell and Jim Owles. In fact, he would tell his secretary to set up an appointment with them for the following week.\n\nDontzin then tried to home in on what GAA had planned. How many GAA members were in the audience? Bell played dumb.\n\nWhen Lindsay appeared on the set, it was obvious that he and his staff knew that the deck had been stacked, for Lindsay was visibly nervous. His hands crammed into his trouser pockets, he crossed and uncrossed his legs repeatedly.\n\nSoon the music that announced the show's opening was played, which, providentially, was Leonard Bernstein's \"Something's Coming.\" The guest that day was Arthur Godfrey, who was appearing as an advocate for the new ecological movement.\n\nThe show started without incident as Lindsay read a Passover greeting off a TelePrompTer, but when Godfrey and the mayor began discussing ecology, the mayor's hands twiddled and he crossed his legs yet again. Marty Robinson's political innovation was already succeeding in tearing down the invisible but real wall that insulates the powerful from the negative emotional impact of their policies.\n\nWhen Godfrey observed that \"soot is what the housewife sees the most of,\" Arthur Evans rushed up to the mayor shouting, \"Homosexuals want an end to job discrimination!\" as another GAA member followed him, saying, \"Let that man speak!\" Security guards grabbed the disrupters and pulled them through an exit door. There was a huge noise created by dozens of stamping feet, accompanied by shouts of \"Answer the question! Answer the question!\" A member of the audience yelled out, \"Are you in favor of repeal of the sodomy laws?\"\n\nThe television cameras stopped rolling, and Lindsay quietly spoke: \"My counsel, Michael Dontzin, will meet with those who want to see him outside.\" The mayor's good-cop act was followed by a member of his staff who darkly but incoherently threatened, \"You cannot disrupt a public meeting under threat of arrest. You either leave in peace or are under arrest.\" As Bell would later write in his _Gay Power_ column, \"Nobody pays him [any] mind.\"\n\nBoth Lindsay and Godfrey, Bell wrote, looked \"terribly uncomfortable.\" As a cameraman said, \"Let's pick it up from...\" a GAA member heard a straight person behind him say, \"Lindsay is phony as hell. He has to read off tapes. He can't answer questions unless he has prepared answers. Now he's threatening arrest.\"\n\nWhen the taping resumed, GAA members in the audience proceeded according to plan, taking statements made by Lindsay or Godfrey, reinterpreting them in gay terms, and firing them back. When Godfrey mentioned abandoned cars, Phil Raia shouted, \"And what about abandoned homosexuals?\" The camera stopped and Raia was escorted out of the studio. When Lindsay made a comment about one-way bottles, Jim Owles shouted, \"What about a one-way mayor\u2014nonreturnable?\" Owles gave a V-for-Victory sign as he was led outside. The funniest improvisation came when Lindsay's observation \"If you're stuck in a traffic jam, it's illegal to blow your horn,\" elicited the response: \"It's illegal in New York to blow anything.\" At that point, according to Bell, the looks on the faces of Godfrey, Lindsay, and the program directors said, _What next_?\n\nTen days later, Marty Robinson, Jim Owles, and Arthur Evans were ushered into Michael Dontzin's City Hall office, accompanied by Arthur Bell and Kay Tobin to cover the meeting for their respective gay newspapers, _Gay Power_ and _Gay_. Present from the city besides Dontzin were Deputy Mayor Aurelio and Harry Taylor, chief of patrol for the New York Police Department, who was there to act as Police Commissioner Leary's representative.\n\nAurelio started the meeting by saying that he did not like GAA's public confrontations with the mayor, which he hoped would cease. Bell retorted that if Lindsay would speak publicly on gay issues, there would be no need for the confrontations.\n\nMarty Robinson then opened a notebook that contained a list of demands. First on the list was a \"moratorium on police raids and harassment to give time to the authorities to work on solutions to the underlying problems of the State Liquor Authority and Police Department corruption.\" Bell noticed that Chief Taylor winced as the indictment of his police force was read, inspiring Bell to say, \"Since the raids on the Stonewall and the Snake Pit and the resulting riots, homosexuals will no longer sit back and take shit from the police. One of the reasons we're here is to forewarn you that spontaneous riots might break out again this summer if police harassment continues.\" Bell's statement in turn inspired Taylor to chime in that Seymour Pine had been transferred back to Brooklyn. Aurelio added, \"It's not the policy of the police department to harass homosexuals per se.\"\n\nArthur Evans spoke up next, observing that there were two kinds of harassment that homosexuals experienced: harassment that has a legal basis and harassment the police did while knowing that there was no legal justification, making the arrests only for the purpose of intimidation. Evans was insistent: \"This last type of harassment must stop immediately. We will not tolerate it. Often police direct verbal abuses at homosexuals that are disgusting. We demand that a directive go out that this police practice stop.\"\n\nAfter more demands by the GAA representatives and assurances by the public officials that they understood GAA's concerns and would work with GAA to address them, the meeting ended. This time it was Bell who left the room doubting the city officials' sincerity about a summer moratorium on raids.\n\nGAA was growing steadily by leaps and bounds, yet Craig Rodwell was having difficulty getting GLF or GAA to regularly send representatives to the meetings he was holding to organize the first Christopher Street Liberation Day (CSLD). After the resolution endorsing the creation of CSLD had passed at ERCHO the previous November, all groups that were ERCHO affiliates were invited to send representatives to the meetings to plan the first observance of the \"gay holiday.\" But few did. This was in part because of distance, for most ERCHO organizations were outside New York City.\n\nCraig and a small number of workers labored on to plan the observance, with Fred Sargeant, Michael Brown, and GLF treasurer Marty Nixon doing most of the work, assisted by Mattachine member Foster Gunnison.\n\nAs was so often the case with lesbian and gay organizing, ideology tended to derail the work at hand. Craig had to ride herd constantly over members of the committee to keep them focused on making the CSLD march a success. As each meeting began, Rodwell tried to forestall debates by reminding the other committee members that although all of them came from different political, sexual, and racial backgrounds, their job was \"to put together a mass community march\" that would bring all the elements of the community together. When discussion did veer off into the theoretical, Craig would let the discussion go on for a while before raising a practical point with a remark such as, \"Who's going to take these leaflets up to the Bronx? 'Cause this woman is offering to put leaflets around on telephone poles. Somebody has to take them up to her.\"\n\nOnce when Craig made a motion to put a notice in _The Village Voice_ to advertise the march, a self-styled radical went off on a tirade, asking, \"How can we spend a hundred and twenty dollars on ads when there are people starving to death in this city?\" Craig then made a motion to scrap the advertising budget for the march and spend the money instead on buying hot dogs to pass out in the Bowery. Taken aback, the man opposed to buying the ad suddenly withdrew his objections.\n\nBy May of 1970 GAA had collected six thousand signatures on their petition addressed to Carol Greitzer, but the councilwoman had refused to accept the petitions. On May 11, Jim Owles met with Greitzer at City Hall and got a frigid reception, with Owles reporting that she \"didn't want anything to do with [GAA].\" She would neither sponsor nor cosponsor an employment discrimination bill, nor would she testify at the City Human Rights Commission.\n\nFrom an article in _Gay Power:_\n\nTwo evenings later... thirty-five members of Gay Activists Alliance confronted Mrs. Greitzer at an open meeting of the Village Independent Democratic Club.\n\n... Before entering the big meeting hall, she... made small talk with a man she called Marty. But then she entered, and suddenly, bedlam. A roar she will never forget.\n\nPoor Carol fidgeted.... She told Marty she was going to leave. She was persuaded to hear it out.... And she and all of those V.I.D.'ers... heard Arthur Evans say, \"Carol Greitzer refused to accept the petitions. She said she would not sponsor a job discrimination bill.... If she doesn't relate to the homosexual cause, the Village Independent Democratic Club doesn't relate and we are prepared to sit in.\"\n\nMrs. Greitzer turned to platform chairman Robert Egan, and said, under her breath, \"I don't want to make a statement.... Tell them I have a terrible cold. I didn't refuse those petitions.... I had too many things to carry....\"\n\n... Carol reneges. Her dark eyes flash.... \"The Attorney General is the person who has done most this past year with civil rights legislation.... I can't get it done.\"\n\nSomeone says, \"You refuse to represent us in City Council?\"\n\nMrs. Greitzer says, \"Is there a specific piece of legislation you're talking about?\"\n\nArthur Evans says, yes, and elaborates the points in the petition.\n\nGreitzer retorts, \"My stand is there is no way of getting this through... not even with bombs.\"\n\nJim Owles says, \"The very least we expect is a commitment, Mrs. Greitzer.... You've never issued a positive statement about homosexuals. You are guilty of a crime of silence.\"\n\nAll join in, [chanting] \"Guilty of a crime of silence.\"...\n\n\"Will you back us up?\" asks someone from G.A.A.\n\n\"Yes,\" says Mrs. Greitzer...\n\n\"Will you co-sponsor a bill?\"\n\n\"Yes,\" says Mrs. Greitzer in a tone that can only be described as exasperated defeat.\n\n\"Do you accept the petitions?\"\n\n\"Yes,\" says Mrs. Greitzer, who fortunately has one free arm for lugging that night.\n\nAs a week of celebrations to commemorate the Stonewall Riots was in full swing, members of GAA planned their own special celebration by sitting in at the Republican State Committee headquarters. Arthur Evans, Marty Robinson, Tom Doerr, Jim Owles, Phil Raia, Cary Yurman, and Arthur Bell and a few other GAA members arrived at the headquarters a little after noon on June 24.\n\nThey were greeted by an emissary who said, \"If you have legitimate grievances, I will see to it that they are forwarded to the right party.\"\n\nArthur Evans replied, \"We want Rockefeller to come out and fight for homosexual rights. Rockefeller is guilty of a crime of silence, and we are not leaving until we get a satisfactory answer to our demands.\"\n\nThe employee with whom they were speaking complained, \"You did not call for an appointment! You have not made a legitimate request!\"\n\nArthur Bell described GAA's quick response to the bureaucratic run-around the employee was trying to pull on them: \"We just unwound and made ourselves comfortable on the floor of the reception room until they got the picture.\"\n\nMeanwhile, downstairs, in midtown Manhattan, a GAA picket line had been thrown up, with demonstrators chanting, \"Two-four-six-eight, gay is just as good as straight! Three-five-seven-nine lesbians are mighty fine!\" and, \"Say it loud: Gay is proud!\" As the day wore on, the pickets began to hold hands, kiss, and hug. Every half hour a member of the sit-in contingent would send a report downstairs that nothing was happening. TV and other news media went inside to photograph and interview the GAA members as they sat in.\n\nAt 5:00 P.M., with the protesters still there, the office could not be closed. At 6:30 a phone call came from the chair of the New York Republican Party agreeing to meet with a representative of GAA. Jim Owles said he would do it provided a member of the press could come along. The chairman would not agree to that, and Owles would not give in on his demand. The police soon showed up, and those sitting in huddled and agreed that Evans, Owles, Raia, Robinson, and Doerr would remain to be arrested\u2014the first people ever arrested for a gay sit-in in New York.\n\nThe five arrested men, soon dubbed the Rockefeller Five, were then taken to be arraigned. Activists worked the phones and by 8:00 had forty people at the Criminal Court. As the five entered, the forty GAA members stood up and held hands, causing a commotion in the courtroom.\n\nAll five were booked on a charge of criminal trespass and let free on their own recognizance. A trial was set for August 5. Outside the courtroom, Marty Robinson made a statement to a _New York Post_ reporter: \"We are trying to use political power to achieve changes that will benefit homosexuals in the state. We want homosexuals to know who has been responsible for inaction regarding their civil rights, and we also wish to charge the state with corruption, such as the State Liquor Authority's nonissuance of licenses to gay bars.\"\n\nRecalling the quotation he had often read on the arch in Washington Square, \"Let us raise a standard to which the wise and the honest may repair,\" Arthur Evans says, \"I think that that's what GAA was doing. We were setting an example for people who were ready to come to that standard and join the struggle and make an example of their lives. That was extraordinary because it was a concept of democracy that almost doesn't exist in this society, a Walt Whitman type of democracy. Walt Whitman speaks of democracy and adhesiveness and the dear love of comrades. All that was in GAA. Democracy for us wasn't voting once every four years in an election which the Supreme Court would then throw out. It wasn't that kind of democracy.\n\n\"It was a direct, immediate democracy. We decided how we were going to govern ourselves and we governed ourselves that way. We decided who our leaders were. We elected them; we knew them all, face-to-face. We went into battle with them. We put ourselves at risk with them. We evaluated in person what we did. When we made a mistake, we evaluated it; we corrected it.\n\n\"So what was wonderful about it, I think, was that we discovered what democracy is. We learned that what we call democracy in this society is really a shadow of democracy. We learned to create a democratic life of work and struggle and sex and love and commitment, and we controlled it all, in person, face-to-face.\n\n\"So we were inspired to take that seed of lived democracy and make it spread into the body politic, to make it spread into the world. And that, I think, was one of GAA's greatest achievements. GAA was a school for democracy.\"\n\nIn the future lay many more zaps that would be planned and executed with brilliant panache and inspiring courage. It came to GAA's attention that an agency named Fidelifacts routinely collected information on individuals who were homosexual and sold that information to prospective employers. GAA inquired about the practice only to be told that \"if it walks like a duck and quacks like a duck, we call it a duck.\" Members of GAA organized a picket demonstration against the company and Marty Robinson came dressed as a duck. When _Harper's_ magazine printed a highly insulting article on homosexuality and then refused to print an apology or a rebuttal by GAA, members of GAA showed up at their offices unannounced bringing cookies and coffee and while some approached staff members saying, \"I'm a homosexual; have a doughnut,\" others occupied the office of the magazine's editor. When a member of the city's marriage bureau made gratuitous insults about homosexuals wanting to marry, that office was taken over by GAA members who answered the phones and informed callers that \"today we're only issuing marriage licenses to homosexuals. Are you a homosexual?\" And when John Lindsay continued to withhold his full backing to passage of a gay rights law for New York City, GAA members sabotaged the announcement of his bid for the U.S. presidency by handcuffing themselves to the railings of the top balconies of Radio City Music Hall the moment Lindsay strode to the podium. As Lindsay tried to speak, the activists released hundreds of flyers stating their demands on the audience, all the while shouting for Lindsay to address their concerns.\n\nThese brilliant tactics and much more\u2014the constant organizational work done by GAA and GLF, the dances they held, the publications they put out, the centers they started\u2014succeeded in making the gay liberation movement flourish. As Frank Kameny stated, \"By the time of Stonewall, we had fifty to sixty gay groups in the country. A year later there was at least fifteen hundred. By two years later, to the extent that a count could be made, it was twenty-five hundred.\"\n\nLegal gay bars also proliferated in New York City in the early seventies. While Mattachine\u2013New York had done much of the work to make this possible through its legal efforts, Dick Leitsch feels that the main reason for their spread was Diego Vinales's impalement as a result of the raid on the Snake Pit: \"The photos of that sickened the public and outraged even the cops. The public attitude was that the cops (and liquor agents) were out of control [and] being unnecessarily brutal.\"\n\nAs for the Stonewall Inn, the club that played such a pivotal role in bringing about this revolution, it survived the riots by about three months: passersby noticed a STORE FOR RENT sign appeared in its window sometime in the month of October 1969. There are several reasons that could explain its quick demise. First, as an expression of anger at Mafia exploitation, Craig Rodwell urged the gay populace to boycott the club. Second, the bar tried to do business as a juice bar, no longer serving alcohol. This is linked to the third reason, which is probably the true reason the club failed: Chuck Shaheen said that after the riots, the club was too notorious to remain open. Shaheen attributed this in part to timing, for the media was beginning to focus on the issue of police corruption. Shaheen explains, \"We were told, This is it. Forget this. This has gotten too much notoriety, too much everything. We can never let this place open again.\" As for Fat Tony, Chuck Shaheen merely stated that he was \"eventually\" killed without providing any further details. In another part of the interview Shaheen said that Fat Tony, who was a heavy user of drugs, including crystal methamphetamine, talked freely when he was on drugs and that Fat Tony had become an \"embarrassment\" to other mobsters. This is one reason that Shaheen knew as much as he did about the Stonewall's business operations, including things Shaheen said he should not have known. If what Shaheen was told about his former boss's demise is true, perhaps Fat Tony died because he was not careful enough about observing the Mafia code of silence.\n\nThe Mafia-owned bars were initially opposed to the annual celebrations of the Stonewall Riots. But Ed Murphy, seeing that the marches doubled in size between the first and second marches, founded the Christopher Street Festival Committee in 1972 and by 1974 had succeeded in reversing the march's direction. Whereas the first marches began in the Village at the site of the riots and went up to Central Park, Murphy manipulated matters so that the march started uptown and went down to the Village, so that he and his cronies could once again make money off the gay populace as they drank in the bars and ate food at the festival. As the Stonewall Riots became more famous, he took to riding in the annual marches in a car, calling himself \"the first Stonewaller\" to try to take credit for making the riots happen in the first place.\n\nMurphy officially \"came out\" in 1978, claiming to have quit both his careers as a criminal freelancer and an FBI informer because he wanted to work for gay liberation. He then began to put in appearances at political events to talk about gay rights. He told one prominent gay publication that \"he is the only person from Stonewall who is still interested in gay rights; the others 'are too interested in making money in bars.' \" The last place Murphy worked at before his death from AIDS in 1989 was Trix\u2014a hustler bar. In 1992, Randy Wicker led a fight to wrest back control of the festival from Murphy's associates.\n\nAs for the gay street youth, the main instigators of the Stonewall Riots, Bob Kohler dragged some of them to the first meetings of GLF and tried to get them involved. But the youths lacked the political understanding, experience, and emotional maturity to participate in a meaningful way. While they sometimes took part as foot soldiers in the early gay liberation movement\u2014handing out flyers or attending demonstrations\u2014they did not play a role in the politicization of the gay world.\n\nOn Sunday, June 28, 1970, Craig Rodwell went to Sheridan Square to prepare for the first Christopher Street Liberation Day march, not quite sure what to expect. Crowds were to assemble on two nearby streets, Washington Place and Waverly Place between Sixth and Seventh avenues. The first signs were not encouraging. People were throwing eggs from buildings in the assembly areas at those preparing to march, and probably fewer than a thousand people had gathered. Still, Craig told himself, a thousand would be the largest gay demonstration ever. Even his friends on the committee who had planned the event with him had predicted that they would never get more than a thousand people to march all the way from the Village to Central Park, where there was to be an assembly, or Gay-in, at the Sheep Meadow.\n\nThe police did not want to let Craig delay starting the march, despite his desire to do so. He felt certain that more people would eventually show up. Before the march began, those in the crowd were warned not to wear glasses or loose jewelry around their necks in case someone attacked them. A number of local heterosexuals had shown up to watch. How would they react? The crowd seemed nervous. Sam Agostino and his friend had brought their dogs along so that if the marchers were attacked, they could just leave the march and claim that they were out walking their dogs. Half of the gay crowd that had gathered hung back on the sidewalks, trying to decide if they should join in and step out into the street.\n\nWhen the march started, participants walked so quickly because of their fear of violence that later the event was jokingly referred to by some as the \"first run\" rather than the \"first march.\"\n\nBut step off it did, led by a simple banner at the head of the march that read: \"Christopher Street Gay Liberation Day 1970.\" Alongside the banner a blond youth carried an American flag, followed by the Gay Activists Alliance contingent, two hundred strong. Lesbians and gay men had come from Philadelphia, Washington, and Baltimore to participate. Representatives of the Mattachine Society of New York marched, as did members of the Daughters of Bilitis. A new radical lesbian group, the Lavender Menace, joined in, along with representatives from Yale, Rutgers, and New York University. Marching with the Gay Liberation Front contingent were some of the homeless street youth. All in all, approximately twenty different groups were represented.\n\nRight away those marching could see that there were a lot of gay people on the sidewalks, trying to decide if it was safe to join in. Bit by bit they did. The march grew larger and larger, expanding beyond what had seemed probable or even possible. As the numbers increased, the crowd became more and more excited and spirited, inspiring still more gay people on the sidelines to join. Foster Gunnison had come up with his own scheme for calculating the crowd's size, and as the march progressed, he dashed back and forth tracking numbers and performing computations. Finally he ran up to Craig and excitedly announced that at least two thousand people were marching.\n\nFortunately, there was no violence, which Craig attributed in part to bewilderment on the part of heterosexual onlookers: \"I think people were so shocked to see thousands of queers. At first they just can't believe it. 'Queers' was that one person they saw walking their French poodle with the rhinestone collar about ten years ago.... They just couldn't deal with it. They would stand with their mouths open, a blank face.\"\n\n_The Village Voice_ recorded the astonished looks: \"No one could quite believe it, eyes rolled back in heads, Sunday tourists traded incredulous looks, wondrous faces poked out of air-conditioned cars.\"\n\nAs various high points along the route were reached, marchers could look back and get a sense of the march's size. Many could not believe its length. Some marchers in the middle were even more amazed when they noticed that looking in both directions they could see neither the head nor the end of the march.\n\nThe GAA contingent started to chant, \"Out of the closets and into the streets!\" over and over.\n\nAnother group chanted, \"Give me a _G_!\"\n\n_\"G!\"_\n\n\"Give me an _A_!\"\n\n_\"A!\"_\n\n\"Give me a _Y_!\"\n\n_\"Y!\"_\n\nAnd on until, \"What do you have?\" was answered with full-throated yells of \"Gay power! Gay power! Gay power!\"\n\nThe march was fifteen blocks long by the time it hit 22nd Street, where a woman leaned out of an office window to throw streamers of film to celebrate the marchers, who in turn applauded her. As the sun shone brightly, more and more men removed their shirts, lending an erotic cast to the march. Signs included: \"I am a lesbian and I am beautiful,\" \"Hi, Mom!,\" \"Smash Sexism,\" \"Me too\" (on a dachshund), and \"Homosexual is not a four-letter word.\"\n\nThe cathartic moment for most marchers was yet to come, however. As the marchers reached first Central Park and then the Sheep Meadow, they came to a high point by a granite outcropping that gave an almost panoramic view of the march. There, at that high point, many stopped and cried tears of joy in a moment they would never forget, as they looked out at the vast numbers of gay men and lesbians who had turned out to support each other by marching proudly in the open.\n\nFranklin Kameny could hardly believe the crowd of thousands he saw in the Sheep Meadow. It had been only five years and three months since he and Jack Nichols had organized the first picket demonstration by a gay organization. On that day the ten people who participated had been worried that rocks would be thrown at them and so they had not alerted the press about their plans.\n\nIt had been Kameny who had coined and propagated the phrase \"Gay is good,\" and in his 1968 essay with that title he had written: \"I say that it is time to open the closet door and let in the fresh air and the sunshine; it is time to doff and discard the secrecy... to live your homosexuality fully, joyously, openly, and proudly, assured that morally, socially, physically, psychologically, emotionally, and in every other way: _Gay is good_. It is.\" At that first Stonewall Riots commemoration in 1970, it must have seemed to him that his 1968 vision was already coming true.\n\n# Conclusions\n\nThe received wisdom about the Stonewall Riots is that such an event was inevitable. This proposition is untenable, however, for there were many raids on gay bars before and after the raid at the end of June 1969 on the Stonewall Inn, and none of them resulted in any kind of sustained uprising.\n\nIt is equally untenable to maintain that the riots' occurrence around the particular club, the Stonewall Inn, was simply fortuitous, that the events that took place there could have happened at any number of gay bars. On the contrary, the Stonewall Inn both as a social institution and as a geographic site had a number of unique or special features that, taken together, explain why the riots erupted there: the club was located in a large homosexual ghetto, so thousands of gay men and women could learn of the event quickly and become involved; within that ghetto, the club was strategically located, being only about two hundred feet from the ghetto's epicenter, The Corner; the club was centrally situated among several major transportation systems and nodes, which made it easy for those who wanted to become involved to get there quickly; the layout of the streets around the club was a hub design, which gave the rioters the advantage of controlling the streets around the Stonewall, for this pattern made it easy for them to enter and exit the area and correspondingly difficult for the police to seal it off; the combination of being a transportation nexus, having a hublike design, and having a lot of foot traffic meant that there were many public telephones within feet of the club that those supporting the riots could use to notify the press and friends; the club was the city's largest gay club; the club had a significant amount of open space in front of it, so that there was room for a sizable crowd to collect; most of the streets in the Stonewall's immediate vicinity were narrow, one-way streets that gave the advantage to pedestrians and the disadvantage to police vehicles; the street pattern around the club is highly irregular, which made it confusing for the riot police, many of whom lived in other boroughs, whereas many of the demonstrators lived in or frequented the Village and were familiar with the area; unlike most gay bars of the era, which were short-lived, the Stonewall had been in existence for years, which contributed to the sense of identification with the club felt by a significant number of its patrons; the club was popular with all segments of the community, so that when it was attacked, the entire community felt under attack; its broad appeal notwithstanding, the Stonewall Inn was particularly popular with the most marginal members of the gay community, gay homeless youths, so that they felt a special loyalty to the club; because of their anger, their age, and their alienation, these gay homeless youths were ideal candidates to fight in a riot; the club was across the street from one of the main places where these marginalized persons congregated; and the club offered the largest venue for dancing for gay men and lesbians and was the place where they could dance most freely, endowing the club with a special meaning as a site for full and free self-expression. These factors came together with others to create the Stonewall Riots.\n\nSeveral conditions that helped precipitate the event relate to timing: the raid occurred during an election campaign (which had traditionally been the time for the worst harassment operations), making gay people assume that the raid's only purpose was to oppress them; it came late on a Friday night in an entertainment district in the week that schools were letting out, which meant that the club was very crowded and that there were many persons out on the streets; that the raid, atypically, took place late in the evening also meant that the crowd inside the bar had had adequate time to consume alcohol, lowering their inhibitions; it was the first hot weekend of the summer, and riots often coincide with high temperatures, apparently because heat increases irritability; the raid came shortly after a campaign of harassment by vigilantes in Queens, and some gay people present at the Stonewall that night were aware of those incidents; it was the last in a series of raids on gay bars, all in the same neighborhood; and it was the second raid that week on the Stonewall Inn.\n\nThat the Stonewall Inn was located in New York City further contributed to the riots' success: only in a very large city were there gay activists with the specialized skills to take on leadership roles to help shape and direct the event so that it could realize its potential. Consider Craig Rodwell and John O'Brien. The first was a gay activist who knew all the other gay activists in the city, and the second a radical who knew street-fighting tactics. That two such activists happened upon the riots within minutes of each other is not something that could have occurred in many other places. New York's size is relevant to another contributing factor: the riots took place not just in the world's media capital but on a block with two significant media sites: _The Village Voice_ , the country's leading alternative newspaper at the time, was only about fifty feet away, and the Lion's Den, the Village's main social center for journalists, was even closer. That, of all the reporters who covered the riots, Howard Smith, Lucian Truscott, and the unknown _Rat_ reporter\u2014the three who wrote the most detailed accounts of the riots' outbreak\u2014simply stumbled across the event is a striking coincidence. It demonstrates how the power of certain features of the Stonewall's physical and social geography, key in creating the riots, was exponentially enhanced by being in New York City: had the Stonewall's immediate geography been the same, for example, but located in another city that was not a media center, would the event have been covered by the press at all and, if it had been, would such local coverage have mattered?\n\nThere are other factors behind the riots that are significant on the macro level. First, Greenwich Village in the 1960s was both the best-known gay community in America and the place where homosexuality was most aggressively policed. This demonstrates that the basic preconditions were in place for a possible revolt by the members of that ghetto. Second, historians have observed that revolutions tend to happen after periods of liberalization. From all I have learned about the history of the homosexual community in New York City during the 1960s, I feel that this principle is an underlying cause of the Stonewall Riots. Indeed, one sees the same phenomenon occuring in another gay and transgendered community in the 1960s that almost exactly parallels the situation in New York City. Compton's Cafeteria in San Francisco was the principal refuge for that gay community's most marginalized members, transgendered persons and gay street youth. After both of these communities were given hope\u2014by transsexual surgery becoming obtainable and by ministry provided by the Glide Memorial Church and the Council on Religion and the Homosexual\u2014these communities rioted when their sole place of refuge was endangered. Like the Stonewall Riots, this smaller but also violent uprising occurred during a period when harassment was easing in San Francisco because of successes by a local homophile movement. Given the same results in parallel situations, to ignore this principle noted by historians would be to willfully ignore the evidence. It would also make the riots' occurrence during the Lindsay administration difficult to explain: that is, why did the riots occur during the most enlightened administration on matters homosexual New Yorkers had ever seen, particularly as it followed the most repressive administration gay New Yorkers had ever endured, that of Mayor Wagner?\n\nNew York Mattachine's at-times ambivalent response to the riots aside, given this historic tendency it seems probable that without the successful campaign by the Mattachine Society of New York to end entrapment and its partial success in legalizing gay bars, the riots might not have occurred. Indeed, this view was espoused by none other than Craig Rodwell. When Michael Scherker, a historian of the riots, asked him whether \"in the month or weeks before the Stonewall Riots... was there anything different... that... was a preparation?\" Rodwell answered, \"I think the four or five years before the Stonewall Riots were preparation.... Mattachine finally got going.... We finally started having a large membership and... we started to get some media coverage of our issues.... I think largely through the actions of the gay organizations, SIR in San Francisco, Mattachine here in New York in particular, and Mattachine in Washington.\"\n\nFinally, that the Stonewall Riots occurred during a period of great social change and unrest\u2014the civil rights and the antiwar movements in particular\u2014has to be added to the list of factors that caused the riots. The sixties was a time so open that even society's basic beliefs about sex, gender, and sexuality were being questioned on multiple fronts: by the reborn movement for women's rights, the sexual revolution, and androgynous styles in fashion. Challenging bedrock beliefs about sex and gender led to questioning assumptions about sexual orientation.\n\nThis intersection of many varied factors\u2014geographical, social, political, and cultural\u2014at the Stonewall Inn on Christopher Street shows that when Craig Rodwell, pressed by Michael Scherker to explain what had caused the riots, answered, \"There are certain... events in history where... everything comes together at one particular moment,\" he was closer to the truth than he suspected. Certainly it was an event of a kind rare in history, one where without any planning of any kind all the necessary elements came together in just the right way to start a revolutionary change in human consciousness that is profoundly for the better. Rather than being an inevitable event that could have happened almost anywhere, the riots could have occurred only at the Stonewall Inn.\n\nFinally, that the preconditions existed for such an event to not only occur but to bear such positive fruit is demonstrated by the beginnings of a new gay militancy first on the West Coast, as evidenced by, for example, Leo Laurence's activism and Carl Wittman's \"Gay Manifesto,\" and later on the East Coast with Bill Katzenberg trying to start a gay Left group in the months before the riots and _Queen's Quarterly_ urging gay men to learn judo and karate. These attitudes and actions show the influence of the New Left, another critical element of the event's underpinning.\n\nHaving considered what did cause the riots, it is worth considering another event that is often said to have caused them, the death of Judy Garland. The bibliography\u2014that humble and uniquely useful research tool\u2014gives us four essential pieces of information relevant to the theory that Garland's death caused the riots: (1) No eyewitness account of the riots written at the time by an identifiably gay person mentions Judy Garland. (2) The only account written in 1969 that suggests that Garland's death contributed to the riots is by a heterosexual who sarcastically proposes the idea to ridicule gay people and the riots. (3) \"D.D.'s New York,\" the gossip and news column of the _New York Mattachine Newsletter_ , discussed the riots once; that column begins with a description of how the local gay world has been changed since the riots and about halfway through the two-page column there is a lament over Garland's death, with other news items interposed. What makes this last piece of writing so striking is that the author is enamored of celebrities, much of the regular fare of the column being social events, movies, plays, and entertainers. The other purpose the author used the column for was to boost gay politics. Having a strong interest in both gay politics and entertainers, he was precisely the kind of person who, had there been a causal connection between the two events, would have noticed and touted it. On the contrary, what is conspicuous here is that the gossip maven makes the fallout from Stonewall the leading story, places six news items between Garland's death and the Stonewall Riots, and never connects the two events. Rather than being linked in the _only_ piece of indisputably gay writing at the time of the riots where Garland's death and the riots are both mentioned, the two events are not related. (4) _Esquire_ magazine carried an article at the end of 1969 about the emergence of the \"New Homosexual\" in the riots' wake that briefly discusses Garland but does not credit her death with inspiring the riots. The sexual orientation of the author, Tom Burke, is not clear from the article, but it is clear that he is conversant with and appreciative of gay society and culture. Although the author notes that Garland's passing coincided with the riots, he sees this as symbolic, because he sees Garland as emblematic of the \"Old Homosexual,\" who, already out of touch, became hopelessly pass\u00e9 with the changes the riots precipitated. Rather than inspiring the riots, Garland is seen as symbolically allied with the old order the riots ended and hence as a contrasting or oppositional symbol.\n\nIf the preceding is not sufficient to demonstrate that this great artist's tragic and untimely death could not have caused the riots, consider that the main fighters in the riots were street youths. These young men were not of the generation that listened to Garland, their music being rock, soul, or both. Finally, author and activist Vito Russo, a great fan of Garland's, went to pay his respects at her casket on the day the riots would begin and then witnessed the riots that night. He wrote that \"it wasn't good old Judy's death or even the full moon that caused all the trouble that night.\" Like the author of the _Esquire_ article, Russo saw Garland's funeral taking place on the day the riots broke out as symbolic of a sea change, not as its cause. He told historian Eric Marcus, \"[Her burial that day] historically marks the end of the old gay world and the beginning of a new one.\"\n\nThe question of who gets credit for starting the riots is one that deserves consideration. The question, however, contains a premise: that an individual or group of individuals can be singled out as the prime mover in a complex process that many persons collectively created. This is important for two reasons. First, as John O'Brien pointed out, there was a continuum of resistance ranging from silent persons who ignored the police orders to move to those who threw objects at the police. O'Brien maintains that it was because of those persons standing around and blocking the streets and sidewalks and keeping the police from being able to operate efficiently that he and others were able to engage in their tactics as effectively as they did: if there had been only about fifteen youths lobbing objects at the police the young men would have been quickly caught or chased away. Second, I wrote the account of the first night to reflect my understanding of what happened, namely, that until the definitive outbreak of rioting when the police retreated inside the Stonewall Inn, there was throughout the evening both a _gradual_ buildup of anger and, correspondingly, a gradual escalation in the release of that anger. In the course of that buildup there were numerous turning points, some more critical than others. With these qualifications noted, I think it is clear that special credit must be given to gay homeless youths, to transgendered men, and to the lesbian who fought the police. Among these, we can name three individuals known to have been in the vanguard: Jackie Hormona, Marsha Johnson, and Zazu Nova.\n\nA common theme links those who resisted first and fought the hardest, and that is gender transgression. While we do not know how the lesbian who fought the police saw herself, we do know that her clothing was masculine, in keeping with her general demeanor. We know from Pine's testimony that the first significant resistance that he encountered inside the bar came from transvestites, and Joel S. places them among the first outside the bar to resist. Marsha Johnson and Zazu Nova were both transvestites, and, as the reader has seen, the street youth were, generally speaking, effeminate men. All available evidence leads us to conclude that the Stonewall Riots were instigated and led by the most despised and marginal elements of the lesbian, gay, bisexual, and transgendered community. My research for this history demonstrates that if we wish to name the group most responsible for the success of the riots, it is the young, homeless homosexuals, and, contrary to the usual characterizations of those on the rebellion's front lines, most were Caucasian; few were Latino; almost none were transvestites or transsexuals; most were effeminate; and a fair number came from middle-class families.\n\nWhat can one make of Seymour Pine's claims that he was ordered to put the Stonewall Inn out of business because it was being used as a site to blackmail gay men who worked in the Wall Street area? Having researched the matter as far as possible, I have concluded that Ed Murphy was using the Stonewall Inn for just that purpose. Most of the information on Murphy and blackmail that I found I came upon incidentally as I gathered other material. That I happened upon most of this information piecemeal and inadvertently increases my confidence in this evidence.\n\nThere is no definitive evidence to say conclusively whether Pine is telling the truth on this point. Because his role was so critical and a full report of his account has not been available, I have given his version of the events concerning the Stonewall Riots and have checked it where possible. Having compared his statements about the events with others, most of which were unpublished and thus unknown to him, I can say that the other accounts have on some matters corroborated his, including, for example, statements made by Ed Murphy. For example, Pine said that one of the undercover policewomen who were sent inside the club was from Chinatown's Fifth Precinct; that the two women were chosen based on their sizes (one petite and one large-framed) to look like a butch-femme couple; and that he suspected that one reason they had not exited on time was that they had been indulging in some partying. Murphy said that \"the two cops were drunk.... Even the policewoman was half-crocked. She was a Polynesian broad. And she's been coming there as a dyke.\" Finally, no account I know of disproves any statements by Pine.\n\nProfessionals who have studied the reliability of witness testimony have established two hallmarks of veracity: a willingness to acknowledge what one does not know and a consistency over time in one's recollections. The following factors weigh strongly in favor of Pine's account of the cause of the raid. When I interviewed Pine, he did not hesitate to say when he did not know, could not remember, or was not positive about information. On one occasion, he made an error, noted it, and later drew it to my attention. He has also been consistent over the years in what he has said on the subject. For example, in Morty Manford's papers I found a January 1972 term paper that he wrote while at Columbia University in which he analyzed gay milieux. In the section on gay bars he writes: \"The Gay bar scene has improved considerably in the past few years (since the _Stonewall_ Rebellion). Gays are not hiding as much (opportunities to blackmail Gay clients don't work as well for the syndicates as they used to), and Gay Liberation has flexed its muscles on a number of occasions, keeping individual bar managements in line.\" A footnote appended to the parenthetical statement about blackmail states that \"New York City Police Lieutenant Seymore [ _sic_ ] Pine reported to me that the syndicate's main source of income from Gay bars was not on the liquor, but rather on the securities information they were able to blackmail out of the bar's Wall Street clients.\"\n\nPine's consistency as demonstrated by this quotation is all the more remarkable in that it dates back to, at most, thirty months after the riots, long before there was the intense historical examination of them that began, approximately, with the approach of their twentieth anniversary. Pine gave the same explanation to David Isay in 1989 that he gave to me and also informally to tourists in Israel when he was living there in the mid-1990s.\n\nFurther corroboration of Pine's version of events was found in Dick Leitsch's papers. The following statement is from testimony Leitsch gave before the New York State Assembly at its \"Hearings on Homosexuality\" on January 7, 1971:\n\nLast year, following a wave of thefts from Wall Street brokerage houses, the State issued an order that all employees in the financial industry be fingerprinted. Because of this, many old and trusted employees had to be let go, because bonding companies will not insure known homosexuals and the fingerprint checks turned up evidence of old arrests. Banking and many other fields requiring bonds are off-limits to homosexuals because of this policy of bonding companies.\n\nThis testimony corroborates Pine on several points: first, that there had been an unusual quantity of thefts (\"a wave\") from brokerage houses; second, that \"many\" of those who were in a position to steal such bonds were, in fact, homosexual; third, the time given in Leitsch's testimony fits the time line given by Pine, who says he got the information about the thefts around mid-1969. Leitsch's testimony, given in January 1971, refers to the fingerprint order being issued \"last year,\" or in 1970, _following_ the wave of thefts. If the connection between the stolen bonds and Mafia extortion of Wall Street workers only came to light in late 1968 or early 1969, then a new policy put into effect roughly a year later, sometime in 1970, corroborates Pine's chronology. Further, if Pine was trying to fabricate a cover story based on these historical events to make himself look better in the eyes of history, it seems probable that he would have pointed this material out to me or to someone else or suggested that such material be searched for. Not only did he not point directly or indirectly to the policy changes Leitsch described, but I have found no indication anywhere that he was aware of this information. Moreover, on two occasions where Pine's versions of events at the riots differed substantially from accounts given by others whose presence there is indisputable, I have determined to my own satisfaction that Pine's statements were the accurate ones.\n\nFinally, it is notable that Pine's narrative has its own internal logic when looked at in the context of other new information contained in this history. If Murphy had the goods on J. Edgar Hoover, as I believe he did, then the investigation's origin overseas in Interpol makes perfect sense. Hoover did not control Interpol, and the results of the New York Police Department investigation, as far as Pine says he was told it went, did not implicate Hoover. Therefore, no alarms would have been sounded that would have alerted Hoover to try to quash the investigation.\n\nIf Pine is telling the truth about the reason he was told to shut down the Stonewall Inn, which seems highly probable, and if Murphy was using the Stonewall Inn to blackmail homosexual men on Wall Street, which seems certain, does this change the meaning of the Stonewall Riots? First, common sense and a historical sense would lead us to conclude that the police department's fervor for closing down the Stonewall Inn would have been inspired more by a concern for protecting the wealth of powerful persons and institutions and only secondarily, if at all, by a concern for protecting gay men. Second, while such a cause would give the riots a new context, I do not feel that it changes their meaning. The portrait of the Stonewall Inn as an institution that emerges from the historical record is a very mixed one: while it was a site of refuge and safety for some so that it functioned almost as a sort of community center, for others it was a site of exploitation and degradation. Does this mean that the Stonewall Riots should not serve as a symbol for lesbian, gay, bisexual, and transgendered people today?\n\nI feel that the riots can and should still serve as a potent and inspiring story. The reasons I feel this way are twofold. First, by way of analogy, let us look at the history of race riots in the United States. The first race riot in American history in which African-American people were not the victims of violence but its instigators took place in New York City in 1935. The specific incident that sparked that riot was a belief that the police had beaten or even killed a youth of color. After the riot was over, it turned out that the alleged event never took place. For me, this mistake does not at all diminish the rightness of those who rioted. The affected community was routinely exploited in many ways, including by white merchants who refused to hire black workers. In other words, those who rioted may have been incorrect on that occasion about that specific injustice, but they were certainly right in the aggregate that their community was being ruthlessly exploited by merchants. The 1935 riot was an attempt by the powerless to end that injustice. Just so, several persons I interviewed told me that they felt that the terrible anger that erupted on the first night of rioting at Stonewall was not simply anger at the injustices of that night or of recent weeks but anger that went back to the harsh mistreatment of the community by Mayor Wagner. While I do not support the use of violence as a method of social and political change, I think there are exceptions to this principle. I agree with prominent author, essayist, and editor of gay and lesbian books, Michael Denneny that in this case, given that the community had no reason to believe anything other than that it was being oppressed, the violence was not only justified, it was also necessary: gay men had always been stigmatized in our culture for their perceived lack of masculinity, and their violent resistance against the police proved that these gay men possessed both moral and physical courage.\n\nSecond, Craig Rodwell's response is relevant, for the Stonewall Inn had no harsher critic than Rodwell. Yet when he saw the bar being raided he reacted with anger. As the reader has seen, he became the riots' primary supporter, doing his utmost to keep the riots going and to publicize them. Moreover, it was Rodwell who had the idea of celebrating the riots annually. His example shows that it is possible to recognize the ambivalence of the Stonewall Inn while celebrating the lesbian and gay resistance to the oppression that occurred there. In celebrating the event, the emphasis has always been placed on gay militancy and not on the club: thus the gay community was urged to boycott the Stonewall Inn after the riots, and the celebrations were originally called Christopher Street Liberation Day to underline the idea that gay people were claiming their own territory.\n\nIn conclusion, let me quote from the article \"Where It Was,\" published in a gay periodical on the riots' fifth anniversary:\n\nThe gay Stonewall was actually an illegal after hours club, operated against the laws... the hoods tending the door and roaming the floors were only too obvious. The place was packed, overcrowded and a fire hazard to boot. But \"they\" just kept piling them in.... Two and sometimes three bars dispensed drinks until sun up.... There were dark hidden corners where anything went and where some of the \"customers\" dispensed other things.... Before its arrival... that area of Christopher Street was relatively quiet.... Then came the throngs hanging around on the streets in front of the Stonewall. They sat on cars... pushed [drugs] in Sheridan Square.... Finally the neighborhood was up in arms, and rightly so.\n\nMorty Manford, then president of GAA, wrote in reply:\n\n\"Where It Was\"... bemoans the appropriateness of the Stonewall Inn as a rallying point for the Gay Liberation Movement. There is no question the bar was illegal, dirty, dingy and exploitative: in 1969 it was virtually all we had.... Crumby as it was, it was ours.... The raid was like a number of other raids I'd been in with the important exception that we fought back. A spirit of pride encapsulated in anger was articulated for the first time in modern history by Homosexuals as a group. That spirit said both explicitly and symbolically: \"We are sick of being pushed around; we are sick of being denied our Constitutional rights; and we will persevere until injustice, exploitation, harassment and discrimination end.\" There is no getting away from the fact the Stonewall Inn is a symbol of the past. It was a departure point for Gays like the Bastille was for the French people.\n\nManford concludes his letter by reminding us of some fundamental truths: \"We have undertaken a struggle of tremendous proportions that will understandably be tough, face its set-backs and take a long period of time. It is our struggle. We owe it to ourselves to discern our own best interests and support united efforts with other gays to achieve our liberation.\"\n\nManford is correct: the true legacy of the Stonewall Riots is the ongoing struggle for lesbian, gay, bisexual, and transgender equality. While this fight is far from over, it is now a worldwide movement that has won many significant victories, most of them flowing from those six days in the summer of 1969 when gay people found the courage to stand up for themselves on the streets of Greenwich Village.\n\n# A U T H O R ' S N O T E\n\nThe Stonewall Riots are the critical turning point in the movement for the rights of gay men and lesbians as well as for bisexual and transgendered people. This six-day struggle by gay people with the police for control of a gay ghetto constitutes an important event in American and world history, for it ultimately led to the inclusion of sexual orientation as a protected category in the civil and human rights movements. This was a significant broadening of these important historic movements and the beginning of the reversal of millennia of oppression.\n\nAs is true of so many gay men and lesbians, I have long been fascinated with this historic turning point. One of the reasons the event has been so captivating is that it seemed to both erupt out of nowhere and effect radical change with precipitate speed: less than six months after the Stonewall Riots _Esquire_ magazine announced the birth of the \"new homosexual\"\u2014and that was before the Gay Activists Alliance (GAA) was born. When I began work on this book, it was my intent to discover everything that I could about the riots in the hopes of creating a much fuller account of this event than had previously existed. I also hoped that I would be able to offer new insights into the event itself as well as to discover explanations for some questions that had long resisted resolution. Having completed this history, I feel I owe it to the reader to give some brief account of how I conducted the research for this volume.\n\nFor the most part I relied on standard methodology: a careful survey of the bibliography, thorough research in archives, and extensive interviewing of witnesses and participants. I interviewed not only persons who were at the riots, but people who participated in the movement for homosexual equality before and after Stonewall, as well as a few others who were simply participants in gay life in the Stonewall era. Those not at the riots were interviewed because they had information that would help explain the context in which the riots occurred and the immediate political changes they precipitated. To understand how radical political changes grew out of the riots and what these changes were is important because the political sea change that occurred after Stonewall is the reason the riots are important. In other words, the Stonewall Riots are important because of the political change they brought about _through_ the birth of the gay liberation movement, a wave of new gay and lesbian political organizations characterized by New Left values, including activism and militancy. Journalists and other writers, however, tend not to connect the events, instead explaining the significance of Stonewall as if one day there was a riot in New York City by gay men and then, in some mysterious way, as a result gay people across the nation formed political organizations. The truth of the matter is that Stonewall intimately links two interdependent phenomena. Bob Kohler made this salient historical point with admirable pithiness when we met one day on Greenwich Avenue. He said, \"The Stonewall Riots would be totally insignificant if the Gay Liberation Front hadn't been formed after the riots.\" After a pause, he added, \"Of course, I guess it took something like Stonewall to make the Gay Liberation Front possible.\"\n\nI began my research with a careful survey of the known bibliography, concentrating especially on the accounts written at the time of the riots. I then searched archives and discovered a number of articles published in 1969 that had not been previously noted. An unpublished letter, handwritten by an eyewitness in 1969, was discovered by historian James Sears as I was researching this book. The final addition to previously unknown 1969 reports came from a person I interviewed who had recorded a radio interview with an eyewitness either during the riots or a few days afterward. Access to interviews conducted by previous researchers with witnesses who were either dead or not locatable provided critical information, as did earlier interviews by other researchers with persons I was able to further interview. When it is not indicated in the body of this work whether someone is speaking to myself or to another interviewer, that information is contained in the notes or in the list of oral histories.\n\nWhile many persons have claimed that the 1969 accounts of the riots either conflict with each other or are not credible, the more I studied them, the more I felt that they were both highly reliable and did not conflict with one another. I discovered that if I simply reversed the order of two events in one account, then there were no major discrepancies among the 1969 accounts.* I then collated them all into one document, arranged chronologically and organized by subject matter. I found not only almost no contradictions among these accounts, but that collectively they provided a fairly detailed history of this event, and especially of the riots' outbreak.\n\nI was able to supplement these contemporary accounts with a wealth of information from oral histories. While interviewing witnesses and participants, I used maps of the area around the Stonewall Inn and asked interviewees to mark on the maps the places where they saw events occur and the position of persons and vehicles as well as where they themselves were situated. This technique sometimes provided corroboration of the testimony of other witnesses. The issue of corroboration leads to what I consider the most important aspect of the interview process for this history. Because the events at the Stonewall Riots have been so disputed, I was careful not to tell any interviewee prior to the interview what other persons had told me nor of the conclusions I had reached based upon my research. This is a critical point, for the reader will see that I was able to achieve anywhere from a moderate to a high degree of corroboration of many of this history's new findings based solely on the many interviews conducted. Further corroboration for new information was also generally obtained from archival and bibliographic materials. I did not give information about the accounts of other interviewees or share my conclusions not only so as not to influence the testimony each person might give me, but also\u2014and primarily\u2014because there has been a small number of persons whose spurious accounts, while having entered the bibliography, do not withstand careful scrutiny. It is in part because of this mere handful of persons that the contemporary accounts of the Stonewall Riots have come to be viewed with a degree of mistrust. This false testimony has naturally also made the informed public quite skeptical of most if not all accounts of the riots, and therefore I have not referred to these inaccurate accounts.\n\nBy a stroke of good fortune, my residence is only a block and a half from the former Stonewall Inn. It is also the collective good luck of those who care about gay history that the Greenwich Village Historic District, which encompasses the area where the Stonewall Riots occurred, was created the very month the insurrection occurred. While the creation of this historic district did not freeze the Village as if in amber, it did preserve the vast majority of the buildings around the locality where the riots occurred and has kept the layout of the streets essentially the same as it was in the summer of 1969. Thus I have been able to walk by the site of the riots countless times during the decade it took me to research and write this book. Being able to contemplate the setting frequently has given me a number of insights about the riots and the factors that caused them.\n\nWhile I have made every effort to quote accurately from the oral histories used in this volume, I have made the normal kinds of editorial changes, such as editing out false starts; deleting sentence fragments that, leading nowhere, would tend to confuse the reader; and supplying words (indicated by brackets) to clarify the speaker's meaning. On several occasions I have combined two or more quotations by the same person from different interviews (or, very rarely, combined an interview with a written record they left, such as a published article) into one quotation for the sake of clarity. Whenever this has been done, the sources of the original quotations are given in the notes.\n\nNames that are given in quotation marks when they first appear indicate use of a pseudonym created for this book. There are three instances in this history where accounts refer to persons without even a first name being known. As noted in each case, I created names for those persons for narrative effect: Tano, Tom, and Lawrence. I chose these names to honor the memory of deceased friends and a family member, all of whom died of AIDS. Tom Casteel and Tano Delgado, lovers in the mid-1980s, were the first new friends I made when I moved to New York. Lawrence Goudreau Jr. was the brother of my sister-in-law, Lynn.\n\nIt is part of gay culture that some gay men refer to other men with feminine pronouns, often quickly switching the gender references about one person back and forth. I have sometimes used this device when it was employed by the persons whose interviews I am quoting, not to confuse the reader but to convey the feeling and flavor of the milieu and personalities.\n\nI have generally tried to use the terminology of the 1960s in writing this history except when doing so might cause confusion or the effect might be jarring. I have also used more current terms when it seemed preferable for the sake of clarity. I have used the current term _transgender_ sparingly for two reasons: first, because the word is of recent mintage, and second, because it is an umbrella term, encompassing both transvestites (crossdressers) and transsexuals, and it usually seemed worthwhile to preserve the specificity of these more precise terms.\n\nFinally, this history is one in which the major actors were male: the Mattachine Society of New York, whose leadership was practically all male, was the only gay political organization in New York City before the riots with any political clout; the Stonewall Inn's clientele was overwhelmingly male; men probably predominated in the riots even more so than they did inside the Stonewall Inn; and most of the founders and leaders of the Gay Liberation Front (GLF) and the Gay Activists Alliance (GAA) were also male.* Given the need to keep the main thread of the story moving forward, it has not been possible to cover lesbian life in a full or representative way, any more than it has been to give a full picture of transgender or bisexual life at the time. I have tried to limit myself to describing events, persons, and communities only to the extent necessary to give the reader an adequate understanding of what led up to the Stonewall Riots and how the riots in turn led to the creation of the gay liberation movement. The same logic applies to the emphasis on New York in this history: since the Stonewall Riots took place in New York and GLF and GAA were born there, activities elsewhere have been brought in only when they seemed directly relevant to the story at hand.\n\nAn archivist who assisted me patiently for years as I worked on this book worried aloud that if too much information was definitely known about the Stonewall Riots, the event would lose its mythological power, the power to inspire. I can only say that for myself, the new material is as complex and ambiguous as life itself, making the story richer. On a more personal note, I would like to add that while working on this book, I met some of the most intelligent, generous, and interesting persons I have ever had the pleasure of knowing. Together they taught me much about life and about human nature, so that I will always be grateful for all that Stonewall has taught me.\n\nDAVID CARTER\n\n_Greenwich Village_\n\n_1994\u20132004_\n\n*I am referring to the Howard Smith account published in the _Village Voice:_ If one reverses the order of the arrest of Dave Van Ronk and the police retreating into the Stonewall Inn bar (and key witnesses disagreed with Smith's chronology on this one matter, saying that Van Ronk was arrested before the police retreated into the bar, not afterward) then there is no real disagreement about the events portrayed in the 1969 accounts, other than some disagreements about the meaning of these events.\n\n**I deeply regret that Lois Hart, a key figure in GLF, appears to have left no account of her role in that organization.\n\n# A C K N O W L E D G M E N T S\n\nThe first persons I must thank are those who granted me interviews, entrusting me with their personal histories. I was often touched by the openness with which so many people spoke about their lives. I also am deeply appreciative of how patiently the persons I interviewed answered every question I had, asking them to recall details about events that took place decades ago. While I offer my sincere thanks to all those I interviewed, I would feel negligent if I did not name certain persons who showed extraordinary patience and generosity in answering my every question over numerous interviews: Thomas Lanigan-Schmidt, Danny Garvin, John O'Brien, Bob Kohler, Seymour Pine, and Jack Nichols. Tommy Lanigan-Schmidt also went to extraordinary lengths to help me understand several aspects of gay life critical to this history: without him I never would have understood either the gay street youth or what the Stonewall Inn meant to them, and thus would have missed the most critical elements of this history.\n\nDick Leitsch not only gave me many interviews and faithfully provided detailed answers to follow-up queries, but gave me unfettered access to his personal papers, which proved to be an extremely valuable source. Other pioneering figures who shared their personal papers include Randy Wicker, Barbara Gittings, Kay Tobin Lahusen, Jack Nichols, and Bob Kohler. A special thanks to Joe Kennedy and Robin Souza for assistance in finding GAA sources and documentation, and to historian Donn Teal for help with various leads.\n\nMy work has necessarily built upon the work of previous researchers, whether they have been historians who have made a particular study of the Stonewall Riots or writers who included Stonewall as one of their many interests. Of special usefulness has been the work done by early researcher Tina Crosby, the pioneering work done by writer Michael Scherker, the first real historian of the Stonewall Riots, the interviews conducted by David Isay for his 1989 radio documentary, the research of historian Martin Duberman for _Stonewall_ , his history of the Stonewall Riots, and interviews conducted by Eric Marcus for his book, _Making History_. David Isay, Martin Duberman, Eric Marcus, Nikos Diaman, Paul Cain, and the family of Michael Scherker all generously made the material relevant to the Stonewall era in their possession available to me. Historian John Loughery was generous in sharing documents with me and providing leads. Ken Lustbader provided me with early drafts of his master's thesis on homosexuality in Greenwich Village and also provided me with floor plans and other documentation on the Stonewall Inn. The late Joel Honig kindly provided me with copies of documents and generously shared his unbridled enthusiasm for gay history. It is an honor to thank historians Jonathan Ned Katz and Joan Nestle for their encouragement and support. Frank Toscano, retired from the New York Police Department, assisted me with several questions about the NYPD.\n\nThe number, quality, and accessibility of lesbian and gay archival collections has greatly increased over the last decade and a half, and this book would not have been possible without them. Much basic work remains to be done in collecting and preserving gay history, and I urge the readers of this book to donate materials and funds to your local gay and lesbian archives. Among the many wonderful amateur and professional archivists and librarians I met during my research, I must single out Rich Wandel, founder and archivist of the Lesbian and Gay Community Services Center National Archive of Lesbian and Gay History in New York City, for special thanks. Rich is one of the many unsung heroes of the gay community who volunteers countless hours selflessly, does high-quality work, and seeks no recognition for himself. I must also give a special nod to Henny Brandhorst, Vincent van der Kaap, and Jack van der Wel, the director of Homodok, at Homodok, the Gay\/Lesbian Archives and Information Center in the Netherlands, which did amazing bibliographic research for this history. I also wish to thank the librarians and staff of the following archives and libraries: the New York Public Library's Manuscripts and Archives Division, where I made extensive use of various collections, but particularly the International Gay Information Center Archives; the map division of the New York Public Library; the Irma and Paul Milstein Division of United States History, Local History and Genealogy Division of the New York Public Library; the Lesbian Herstory Archives of the Lesbian Herstory Educational Foundation, Inc., in New York City; Columbia University's Oral History Research Office, with special thanks to Mary Marshall Clark; the Rare Book and Manuscript Library of Columbia University's Butler Library; the Tamiment Institute Library of New York University; the Pat Parker\/Vito Russo Center Library in New York City; the Gay, Lesbian, Bisexual & Transgender Library\/Archives of Philadelphia, with special thanks to Steven Capsuto for transferring interviews taped in the early 1970s to MP3 files so that they could be studied; the Human Sexuality Collection in the Division of Rare and Manuscript Collections of Cornell University, in Ithaca, New York, with thanks to collection curator Brenda Marston; the Museum of the City of New York; Jim Lomax, head of the Heritage Committee and archival programs of the SAGE Oral History Project in New York; the Gay, Lesbian, Bisexual, Transgender Historical Society of Northern California in San Francisco, with special thanks to Executive Director Susan Stryker, Paul Gabriel, and Willie Walker; the June L. Mazer Lesbian Archives of Los Angeles, California; the Dodd Research Center of the University of Connecticut Libraries in Storrs, Connecticut, with thanks to Rutherford Witthus, Curator, Literary and Natural History Collections, for help with the Foster Gunnison papers; the One Institute and Archives in Los Angeles; the James C. Hormel Gay & Lesbian Center of the San Francisco Public Library, with thanks to Jim Van Buskirk, program manager; the Gerber\/Hart Library of Chicago, with thanks to Karen C. Sendziak, Gerber\/Hart historian and curator, archives; and the Special Collections Library of Duke University in Durham, North Carolina, with thanks to Elizabeth Dunn and Virginia Daley; and Mike Lanza at New York's criminal court records division.\n\nFor very generous assistance in fund-raising, I would like to thank Bob Rosenthal and the Committee on Poetry, as well as Phillip Ward, Larry Maas, and David Tsang. David Tsang is a jewel of a friend and helped support me in many different ways during the years I worked on this book.\n\nI was honored to receive a grant from the Greenwich Village Society for Historic Preservation to write a nomination to place the site of the Stonewall Riots on the National Register of Historic Places. The nomination, which also created the Stonewall Historic District, later resulted in the district being declared a National Historic Landmark. The nomination could not have succeeded without the three preservationists who were coauthors of the nomination: Andrew Dolkart, Gale Harris, and Jay Shockley.\n\nDeborah Lattimore did an outstanding job of producing a great number of accurate interview transcripts under tight deadlines. Amy Endler also did wonderful work as a transcriptionist. Thanks to Peter Hale for transferring MP3 files of interviews to CD-ROM format.\n\nThanks to all the periodicals that helped me find persons to interview, and a special thanks to Rex Wockner for helping me reach the gay media with requests for witnesses to interview.\n\nThanks to Tony Carini for his plumbing expertise.\n\nMany friends and family members have been very generous in their support during the decade I have researched and written this book. Without trying to name them all, and in no particular order, I would like to thank Mark Christianson and John Kretch; Nina Mankin; Hiram Perez and the late Ray Wenner for providing wonderful hospitality in Miami; Jack Nichols for help with housing arrangements in Cocoa Beach; Henry Santiago; J.; Craig Smith; Stephen van Cline; John Jeseren; Charles Haynes and Christopher Wilson; Charles Lombardo; Steve Wolf; Jane Greenlaw; Pauline Park; Jean-Pierre Boch\u00eane and Olivette Halton; Tad Crawford; E. E. Krieckhaus; Allan Vogt; Robert Locke; Scott Barton; Steve Silberman; Tracy Turner; and Gregg Firth.\n\nPerhaps my greatest debt of all is to Michael Denneny, who first encouraged me to write this book and who gave me unstinting support both as an editor and as a friend. Conversation with a man of true culture like Michael was always a pleasure, and I learned much from him. Because of his seminal role in gay publishing, I consider myself honored to have worked with the man I think of as the dean of gay letters.\n\nWhen Michael Denneny left St. Martin's, I was fortunate that Keith Kahla was the editor who picked up where Michael had left off. Keith's ability to identify the weak spots in a manuscript is fortunately equaled by his ability to suggest practical solutions for fixing them. Keith's assistant, Steve Eichinger, also provided valuable assistance in completing this volume. Robert Cloud expertly shepherded the book through the final stages of revision and correction. My thanks also go to Doric Wilson, not only for granting me an interview but also for his careful preparation of this book's index.\n\nMy brother, William C. Carter, and his wife, Lynn, were very supportive of me as I researched and wrote. Bill was very kind to review the manuscript with the utmost care, and his many valuable suggestions greatly improved this history. My niece, Josephine Monmaney, and her husband, Terry, provided wonderful hospitality during my stay in Los Angeles.\n\nArlo McKinnon not only carefully read various drafts of this book and gave me invaluable, sensitive, and intelligent feedback, he was a mainstay of support as I worked to bring the book to completion. Helping in many different ways without stint, he was the soul of generosity: friends do not come any truer.\n\nFinally, a special thanks to my parents who, while not alive to see this work published, made it possible by inculcating in me an interest in and a respect for history.\n\n# N O T E S\n\nMore than a hundred people were interviewed during the course of researching this book. In addition, several other historians made available to me their independent interviews. Some of these interviews were used only as background information. Others are quoted or summarized within the text. When not otherwise cited in the notes below, a quotation will be from these interviews.\n\nIn these notes, Mattachine\u2013New York is abbreviated as MSNY. The full titles of the _New York Mattachine Newsletter (NYMN)_ and some other sources are also abbreviated, and the shortened form is noted the first time that the source is cited.\n\n## P R O L O G U E\n\n1. In June 1969, the month the Stonewall Riots began, Connecticut became the second state to legalize homosexual sex.\n\n## 1 : G R E E N W I C H V I L L A G E, USA\n\n1. The last names of Fat Tony and his father, Ernie, are not given here. (Lauria is not the family name.) Information in this chapter on Fat Tony, his father, and other Mafia owners of the Stonewall Inn is taken from the Shaheen-Duberman interview.\n\n2. Bill Morgan, _A Walking Tour of Jack Kerouac's City_ (San Francisco: City Lights Books, 1997).\n\n3. The history of how the Village was colonized by gay men and lesbians is recounted in George Chauncey's _Gay New York: Gender, Urban Culture, and the Making of the Gay Male World, 1890\u20131940_ (New York: Basic Books, 1994).\n\n4. Nancy Adair and Casey Adair, _Word Is Out_ (San Francisco: New Glide Publications, 1978), p. 335, and Jonathan Katz, _Gay\/Lesbian Almanac: A New Documentary_ (New York: Harper and Row, 1983), p. 304, and _Gay American History_ (New York: Thomas Y. Crowell, 1976), pp. 548\u2013556.\n\n5. Extreme notoriety of Bonnie's Stone Wall: Rick Beard and Leslie Cohen Berlowitz, eds., _Greenwich Village: Culture and Counterculture_ (New Brunswick: Rutgers University Press, 1993), hereafter _GV_ , p. 49. That Bonnie's Stonewall Inn was located at 51 and 53 Christopher Street is established by a 1940s menu, displayed in January 2003 on www.villagewaterfront.org, the Web site of the Federation to Preserve the Greenwich Village Waterfront & Great Port.\n\n6. On Village tearooms, see Terry Miller, _Greenwich Village and How It Got That Way_ (New York: Crown, 1990), pp. 38\u201339 and 223\u2013224; Luc Sante, _Low Life_ (New York: Farrar Straus Giroux, 1991); and Caroline F. Ware, _Greenwich Village, 1920\u20131930_ (Berkeley: University of California Press, 1994), especially pp. 52\u201354 and 96\u201397.\n\n7. _GV_ , p. 36.\n\n8. _GV_ , p. 38, and Kenneth T. Jackson, ed., _The Encyclopedia of the City of New York_ (New York: New-York Historical Society, 1995), hereafter _Encyclopedia_ , p. 506.\n\n9. Miller, _Greenwich Village_ , p. 2. The author is grateful to preservationist Ken Lustbader for making available his very valuable master's thesis, \"Landscape of Liberation: Preserving Gay and Lesbian History in Greenwich Village\" (Columbia University, 1993), as well as construction plans and diagrams relating to the original Stonewall Inn Restaurant and the later Stonewall Inn club.\n\n10. _GV_ , p. 42.\n\n11. Nos. 51 and 53 built as stables: _The Greenwich Village Historic District Designation Report_ , Vol. 1, p. 117 (New York: New York City Landmarks Preservation Commission, 1969), hereafter _Report_. Introduction of taxicab fleet: _Encyclopedia_ , p. 68. Information on Saks Fifth Avenue horses supplied by Matt McGhee, who heard it from the stable employee's grandson.\n\n12. New York Public Library, Irma and Paul Milstein Division of United States History, Local History and Genealogy, Views of New York, Microfiche 795, D4, D6, D7, and E1.\n\n13. Lustbader.\n\n14. _Report_ , Vol. 1, p. 13.\n\n15. Lustbader.\n\n16. \"The Hairpin Drop Heard Around the World,\" _New York Mattachine Newsletter_ , hereafter _NYMN_ , July 1969.\n\n17. _Report_ , p. 153. The space around the Stonewall Inn was much more open in the 1960s than it is today. A traffic divider was added at the east end of Christopher Park in the 1990s and a viewing garden was installed in Sheridan Square, which was only covered by asphalt in the 1960s. Unfortunately, the small amount of open space in Village streets diminishes with each passing decade.\n\n18. _GV_ , pp. 291\u201392.\n\n19. Kitty Davy, \"Baba's First World Tour, 1932,\" Part II, _Awakener_ 12: 3, Summer 1968, p. 3, and Meher Baba House, October 16, 1994, letter re 88 and 90 Grove Street; _84 Questions and Answers on Avatar Meher Baba_ (compiled and published by A. C. S. Chari, India: 1969), p. 49.\n\n20. This discussion of Greenwich Village history is generally drawn, except where otherwise noted, from _GV_ (especially the chapter \"Straight Down Christopher Street\"), _Report_ , and _Encyclopedia_.\n\n21. _Coronet_ , September 1950, pp. 101\u20138, quoted in Neil Schlager, ed., _Gay\/Lesbian Almanac_ (NY: St. James Press, 1998), p. 63.\n\n22. Information on Blick and Hoey is taken from David K. Johnson's \"Homosexual Citizens,\" _Washington History_ , Fall\/Winter 1994\u201395, pp. 44\u201363, and William N. Eskridge Jr., _Gay Law: Challenging the Apartheid of the Closet_ (Cambridge: Harvard University Press, 1999), pp. 68\u201372.\n\n23. Arno Karlen, _Sexuality and Homosexuality_ (New York: W. W. Norton, 1971), p. 609.\n\n24. Eskridge, _Gay Law_ , p. 98.\n\n25. New York Penal Code; \" 'Drag' and the Laws\u2014and a Drag Ball,\" _NYMN_ , January\u2013February 1967; and \"Ku Klux Klan Says It Will Seek Permission for a Masked Rally in City,\" _New York Times_ , hereafter _NYT_ , May 2, 2000.\n\n26. Karlen, _Sexuality and Homosexuality_ , p. 610, and Peter Fisher, _The Gay Mystique_ (New York: Stein and Day, 1972), pp. 133\u201334.\n\n27. Barry Miles, \"The Beat Generation in the Village,\" in _GV_ , p. 165.\n\n28. Martin Weinberg and Colin J. Williams, _Male Homosexuals: Their Problems and Adaptations_ (New York: Oxford University Press, 1974), pp. 31, 44\u201345. The authors did their fieldwork between 1966 and 1968, supplemented by two weeks in 1970.\n\n29. Chauncey, _Gay New York_.\n\n30. Eskridge, _Gay Law_ , p. 44. Eskridge points out that although the vice societies were originally organized to combat prostitution, during and after World War I homosexuality became an important focus of their activities.\n\n31. Chauncey, _Gay New York_ , pp. 338\u201342, cited in Eskridge, _Gay Law_ , p. 46.\n\n32. Eskridge, _Gay Law_ , p. 46.\n\n33. Karlen, _Sexuality and Homosexuality_ , p. 610.\n\n34. Stuart Timmons, _The Trouble with Harry Hay_ (Boston: Alyson, 1990).\n\n35. Molly McGarry and Fred Wasserman, _Becoming Visible: An Illustrated History of Twentieth-Century Gay Life in America_ (New York: New York Public Library, 1998), p. 43.\n\n36. Unless stated otherwise, the information on Frank Kameny used in this book comes from Kay Tobin and Randy Wicker, _The Gay Crusaders_ (New York: Paperback Library, 1972), Eric Marcus's _Making History_ (New York: HarperCollins, 1992), and Paul D. Cain's interview with Kameny.\n\n37. _The Voice_ eventually relented after Wicker kept returning with the same ad copy over and over.\n\n38. Dan Wakefield, \"The Gay Crusader,\" _Nugget_ , June 1963, pp. 51\u201352, 71\u201372.\n\n39. \"Jack O'Brian Says,\" _New York Journal-American_ , July 9, 1962. Details about the taping come from \"Minority Listening,\" _Newsweek_ , July 30, 1962, p. 48.\n\n40. _NYT_ , July 16, 1962, p. 47; _Realist_ , August, September, October 1962; _Variety_ and _Herald Tribune_ cited in Edward Alwood, _Straight News_ (New York: Columbia University Press, 1996), p. 47; and _Escapade_ , February 1963, cited in John D'Emilio, _Sexual Politics, Sexual Communities:_ _The Making of a Homosexual Minority in the United States_ (Chicago: University of Chicago Press, 1983), p. 159.\n\n41. Toby Marotta, _The Politics of Homosexuality_ (Boston: Houghton-Mifflin, 1981), p. 27.\n\n42. John D'Emilio, _Sexual Politics_ , p. 160.\n\n43. Leitsch letter to Bob Amsel, September 7, 1967. The author is indebted to Dick Leitsch for unfettered access to his papers, hereafter Leitsch papers.\n\n44. Mark is not the real name of Tommy's friend.\n\n## 2 : O P P R E S S I O N, R E S I S T A N C E, A N D E V E R Y D A Y L I F E\n\n1. Kay Tobin and Randy Wicker, _The Gay Crusaders_ , pp. 67\u201368.\n\n2. Edited together from Tree-Carter interview and Tree, \"Beating Around the Bush,\" _Private Lives_ , September 1991.\n\n3. As Jerry Hoose has explained to the author, there were places other than directly on the waterfront where numerous empty trucks were parked that were used by gay men for sex in this era, but those directly on the waterfront were the best known. All of the locations that Hoose knew about were on the western end of the Village and thus proximate to the waterfront.\n\n4. 1960 crackdown: Leitsch letter (unsent?) to Stephen M. Goldfarb, n.d., Leitsch papers; on the effect of the World's Fair crackdown, see Steven A. Rosen, \"Police Harassment of Homosexual Women and Men in New York City 1960\u20131980,\" _Columbia Human Rights Law Review_ 12, No. 2, Fall\u2013Winter 1980\u201381, pp. 167\u201368, and Lustbader.\n\n5. Lustbader.\n\n6. Leitsch letter to Amsel, September 20, 1967, and Leitsch-Marcus interview.\n\n7. On Leitsch's reaction to Rodwell's turning onto Christopher Street, conversation with Dick Leitsch and Jonathan Ned Katz at the Candle Bar.\n\n8. Leitsch-Marcus interview and Leitsch letter to Amsel, September 22, 1967.\n\n9. Leitsch interview with Paul D. Cain, August 16, 1995, quoted here. Leitsch's tone toward Craig in interviews changed markedly after the publication of Martin Duberman's _Stonewall_ (New York: Dutton, 1993), which only gives Craig's version of events, Duberman having never met or interviewed Leitsch. In the interviews before the publication of _Stonewall_ , Leitsch's tone is positive or neutral about Rodwell; afterward his tone is entirely negative. Leitsch told me that he always wondered why he was overlooked by historians writing about the era. After reading Rodwell's account of the era's history in _Stonewall_ , Leitsch says he realized that Rodwell had been \"poisoning the well of history\" for decades by giving erroneous accounts of events, especially with regards to Leitsch. To sort out all such claims and counterclaims between Rodwell and Leitsch and their respective supporters and detractors would take an entire volume in itself. I have tried to limit myself concerning Leitsch and Rodwell to reporting what both seems to be immediately relevant to the story at hand and can be reasonably established. Whatever future research may reveal, there is no doubt that both men played vital roles in recent gay history and that each of them also made very significant contributions to the movement for homosexual equality.\n\n10. D'Emilio, _Sexual Politics_ , p. 166, and Marotta, _Politics of Homosexuality_ , p. 30.\n\n11. Leitsch-Marcus interview.\n\n12. Leitsch-Carter interview. \"Police Are Added in Washington Sq.,\" _NYT_ , August 5, 1964; \" 'Village' Assured of Added Police,\" _NYT_ , August 10, 1964; Randy Wicker, \"The Wicker Report\u2014It's Koch vs. DeSapio Again,\" _Eastern Mattachine Magazine_ , July 1965, pp. 9\u201310; and \"Politicians Make Strange Bedfellows!\" _Gay_ , May 25, 1970.\n\n13. In later years a member of the Mattachine Society told Leitsch that he was himself a vice police officer and that a number of the officers who volunteered for this kind of work were homosexual, suggesting that some of the men on the vice squad who had sexually interacted with other gay men at least acknowledged their homosexuality to themselves: Leitsch-Carter interview. Indirect corroboration of the possibility that a good number of such officers could have been homosexual themselves comes from an interaction Leitsch had during a chance encounter with a judge who recognized him as \"the guy from that homosexual organization.\" Complaining about a Transit Police officer who regularly brought men he had arrested in bathrooms before him, the judge told Leitsch that \"every defendant\" the officer brought in \"said it was the cop who went down on them, then arrested them because they wouldn't pay him.... I told him the next time he showed up, I'd give him three months in prison for sucking cock in public, and I wrote a letter to his boss, telling him the same thing.\" Leitsch letter to Amsel, n.d., Leitsch papers.\n\n14. \"The World of the Homosexual,\" _Star Chronicle_ , third in a series of articles that began on September 27, 1965.\n\n15. Martin Duberman gives a different version of the events surrounding Hodges's resignation. He did not, however, attempt to interview Dick Leitsch for his recollections of these events.\n\n16. This account of the meeting at Judson Church is taken from \"Public Meeting on MacD.: Pirandello Writes Script,\" _Village Voice_ , April 7, 1966; eight-page undated, untitled typescript in Leitsch papers; \"Meeting on Village Cleanup,\" _NYMN_ 11, no. 3; \"Garelick Urges Public to Report Police Trapping of Homosexuals,\" _NYT_ , April 2, 1966.\n\n17. An eight-page undated typescript and an eleven-page undated, untitled typescript, apparently notes for the 1966 annual report by Leitsch to MSNY, the latter hereafter referred to as 1966 Annual Report, Leitsch papers. See also \"Lindsay Placates Coffeehouse Set,\" _NYT_ , May 3, 1966. Leitsch-Carter interview (on Koch jumping up).\n\n18. Ten-page untitled speech on the Stonewall Riots, hereafter Stonewall Speech, and undated letter from Leitsch to Stephen M. Goldfarb, Leitsch papers; Leitsch-Carter interview; and Randy Wicker letter to _NYT_ , October 23, 1963, Wicker papers.\n\n19. Stonewall Speech. On SLA corruption, see, for example, \"Ex\u2013F.B.I Agent Links State Liquor Aide to Mafia,\" _NYT_ , November 30, 1967, p. 50. Mentioned in this article is Matthew Ianello, the Genovese crime family member who controlled mob activity in the Village and was thus considered the \"real\" owner of all the Mafia's gay bars in the Village.\n\n20. Frank Patton Jr. of Ellis, Stringfellow & Patton, representing Mattachine\u2013New York, \"Memorandum [of Law]: The Homosexual and the New York Alcoholic Beverage Control Law,\" Leitsch papers.\n\n21. Leitsch's 1966 Annual Report, Leitsch papers.\n\n22. Telegram sent to twelve members of the press, April 20, 1966, Leitsch papers.\n\n23. Leitsch-Carter interview and \"3 Deviates Invite Exclusion by Bars,\" _NYT_ , April 22, 1966.\n\n24. \"S.L.A. Won't Act Against Bars Refusing Service to Deviates,\" _NYT_ , April 26, 1966.\n\n25. Booth letter to Leitsch, June 16, 1966. Leitsch papers.\n\n26. \"Gay Is Good,\" Craig Rodwell, _Queen's Quarterly_ , Summer 1969, p. 39.\n\n27. \"Signs of the Times,\" _Hymnal_ 1, No. 1, February 1968.\n\n28. Charles Grutzner, \"Mafia Buys Clubs for Homosexuals,\" _NYT_ , November 30, 1967.\n\n## 3 : O N T H E S T R E E T\n\n1. Material for this chapter on the street youths is taken primarily from interviews of Thomas Lanigan-Schmidt, Bob Kohler, Martin Boyce, and Jerry Hoose by David Carter. An interview of Lanigan-Schmidt by Martin Duberman was also used, as was the article \"From Night of Rage, Seeds of Liberation,\" by Rick Bragg, _NYT_ , June 23, 1994. The primary source for material about Birdie Rivera is the interview of him by Michael Scherker.\n\n2. When I interviewed Bob, he asserted that the street youth always hung out in Christopher Park. Tommy Lanigan-Schmidt, however, is emphatic that the park was just one of many places the youths frequented. He recalls sitting more often on stoops on Christopher Street than in Christopher Park. It makes sense that Bob would overemphasize the park, since that is both where he first met the youths and where he continued to meet them. The material about Bob Kohler is inserted here even though Bob did not meet the youths until the spring of 1969, for this is where the material fits logically.\n\n3. Overpriced hamburgers: Barbara Judith Marie, \"Stonewall Remembered Book 1,\" www.inch.com\/~kdka\/stonewall.htm.\n\n4. Because of Hormona's fair features and youthful good looks, he probably could have easily passed as a woman had he wanted to, adding plausibility to Hoose's contention that Jackie did not have a transgender identity and did not want to undergo sexual reassignment surgery. Female hormones were not that difficult to obtain at the time, and had this been Jackie's wish, one would think that had he gone in drag he could have made a fair amount of money as a cross-dressing prostitute because of his good looks and then been able to purchase hormones, which he did not do. We know this to be so, for in the early 1980s Jackie Hormona moved in next door to Hoose. Hoose noted then that the mildly effeminate side of Jackie had totally disappeared so that his presentation was quite conventionally masculine. Note, too, that Hoose is not sure what kind of makeup Jackie applied to smooth over facial imperfections but is sure it was something of the nature of foundation or facial powder. I have chosen facial powder as Hoose stressed the subtlety of the effect.\n\nHoose believes that it was around 1968 that he first saw Jackie Hormona.\n\n5. Physical appearance: O'Brien-Carter interview.\n\n6. Lanigan-Schmidt\u2013Carter telephone conversation, June 31, 1998.\n\n7. Prison, murder, muscular: Hoose-Carter interview.\n\n8. Boyce-Carter interview.\n\n9. Sources used for Marsha Johnson are Steven Watson, \"The Drag of Politics,\" _Village Voice_ , June 25, 1979; Andy Newman, \"Death of a Drag Queen,\" _Hoboken Reporter_ , July 19, 1992; Boyce-Carter interview; and Heide-Carter interview. I am grateful to Steven Watson for making available to me a copy of his _Voice_ article as well as a copy of his original interview with Marsha. Randy Wicker kindly provided me with a copy of the _Hoboken Reporter_ article.\n\n## 4 : T H E S T O N E W A L L I N N\n\n1. Painted black after fire: Leitsch-Carter interview; contemporary description: Angelo D'Emilio, _The Homosexual Handbook_ (New York: Ophelia Press, 1968).\n\n2. Shaheen-Duberman interview. Duberman states that the three partners grew up on Mulberry Street in Little Italy and cites as his source his interview with Chuck Shaheen, but this information is not in the interview. Joey is a made-up name: Shaheen could never remember this person's name in the interview. An Anthony Verra was arrested during the Stonewall Riots and charged with unlawful sale, consumption, and storage of liquor according to the _Sunday [New York Daily] News_ (Dennis Eskow, \"3 Cops Hurt As Bar Raid Riles Crowd,\" June 29, 1969, hereafter Eskow), but there is no further information in the article that would allow one to determine whether this person was one of the bar owners.\n\n3. Fred W. and Timothy S. McDarrah's _Gay Pride: Photographs from Stonewall to Today_ (New York: A Cappella Book, 1994), p. 2, has a photograph of the Stonewall Inn Restaurant taken in mid-1966 with Greek Revival columns and much of the second floor's brick wall exposed by the deterioration of the old stucco. I am grateful to Stephen van Cline for the detail about the columns.\n\n4. License: Beard-Scherker interview; names on bottles: Shaheen-Duberman interview.\n\n5. Beard-Scherker interview.\n\n6. Oak and steel doors: Ed Murphy, \"1969 at Stonewall,\" _Equal Times_ , June 1, 1989; locks: Beard-Scherker interview. The _Equal Times_ article, after describing \"two big oak doors,\" says: \"Inside that were steel doors,\" which seems to describe two separate sets of doors but could be read to mean steel encased in oak. Openings in door: See photo by Diana Davies in illustration section of this volume.\n\n7. Describe the inside: unpublished paper by David C. Scott. Blond Frankie's last name may have been Esselourne, according to an article supposedly written by the doorman that appeared in _Gay Community News_ (Frank Esselourne, \"Doorman Remembers,\" June 23, 1979, p. 10). Worked at many gay clubs before Stonewall: Hampton-Marcus interview.\n\n8. Rotated ticket colors: Perrin-Carter and Shaheen-Duberman interviews; lamp with blue light: Shaheen-Duberman interview.\n\n9. Garvin-Scherker interview.\n\n10. Book used to discourage straight customers and signing false names as protection against police: Garvin-Scherker and Garvin-Carter interviews; book could be used in court: Shaheen-Duberman interview; campy names signed in book: Sheldon Ramsdell, letter to Michael Scherker, June 17, 1988.\n\n11. Step down: Garvin-Scherker and Beard-Scherker interview; \"through an inner door that was usually open\": Beard-Scherker interview.\n\n12. \"Many round stools\": Lanigan-Schmidt\u2013Carter interview.\n\n13. Conclusions on age and race based on numerous interviews.\n\n14. Garvin-Carter interview.\n\n15. Lanigan-Schmidt\u2013Carter and Perrin-Carter interviews.\n\n16. See, e.g., _Gay Activist_ , June 1971.\n\n17. Garvin-Scherker interview.\n\n18. Vito Russo, \"Still Outlaws,\" _Gay News_ , No. 170, p. 16.\n\n19. Garvin-Carter interview.\n\n20. Manford in Tina Crosby, \"The Stonewall Riot Remembered,\" January 1974, Lesbian Herstory Archives: \"The atmosphere was cavern-like; it was very dark\"; Arcangelo, _The Homosexual Handbook_ , describes \"the inky atmosphere.\"\n\n21. Perry Brass in Crosby, \"Stonewall Riot\": \"there was no ventilation\"; _Gay Activist_ , June 1971: \"the fetid, incredibly smoky air.\"\n\n22. Lanigan-Schmidt\u2013Carter interview.\n\n23. Lanigan-Schmidt\u2013Carter interview.\n\n24. Garvin-Scherker interview.\n\n25. Joel S.-Carter interview.\n\n26. Perrin-Carter interview.\n\n27. Joel S.-Carter interview.\n\n28. This room was referred to as the back room because it was the last room one entered: one could not enter it directly from the front door. It was not a back room in the sense of a place for having sex. Lights were dimmer in back room: Joel S.\u2013Carter and Beard-Scherker interviews.\n\n29. Tables: Beard interview; benches: Beard-Scherker and Fader interviews; ledge: Garvin interview; paneling is visible in McDarrah photograph (see McDarrah and McDarrah, _Gay Pride_ , p. 4). Number of waiters: Garvin; their aggressiveness: Lanigan-Schmidt.\n\n30. Lanigan-Schmidt\u2013Carter interview.\n\n31. Lanigan-Schmidt\u2013Carter interview; Shaheen-Duberman interview.\n\n32. Joel S.-Carter interview.\n\n33. Beard-Scherker interview.\n\n34. Garvin-Scherker interview.\n\n35. Lanigan-Schmidt\u2013Carter interview; drawings by Garvin and Michael Konnon; Shaheen-Duberman interview.\n\n36. May 10, 1998, conversation with Larry, employee at the Waverly Theatre, who told author he attended the wedding reception for his sister at the Stonewall Inn Restaurant in 1958.\n\n37. Lanigan-Schmidt\u2013Carter interview.\n\n38. Lanigan-Schmidt\u2013Duberman interview.\n\n39. Russo, \"Still Outlaws.\"\n\n40. Karla Jay and Allen Young, eds., _Out of the Closets: Voices of Gay Liberation_ (New York: Douglas Book Corp., 1972), pp. 6\u20137.\n\n41. Garvin-Scherker interview.\n\n42. Crosby, \"Stonewall Riot.\"\n\n43. \"Hardy\"-Carter interview. Hardy is a family name created by the author at Jennifer's request.\n\n44. It should be noted that the terms _drag queen, crossdresser_ , and _transvestite_ are being used here because these are the predominant terms from the era and we do not necessarily know how these people identified themselves in that era.\n\n45. Garvin-Carter interview. Tommy Lanigan-Schmidt points out that today people tend to equate the scare drag of the sixties with the gender-fuck (men purposefully donning female attire without trying to pass as women by, for example, simultaneously wearing dresses and beards) movement that emerged in the seventies in San Francisco. Tommy emphasizes that much of the scare-drag look (e.g., the shirt tied at midriff that could be quickly untied) was about being able to quickly change from a male to a female persona to avoid being arrested or attacked.\n\n46. Note that even by contemporary New York City standards, the Stonewall Inn would still be considered a moderate-sized bar.\n\n47. I use this date because Danny Garvin is able to date the club's opening since he had been discharged from the navy that very day. He also remembers the event because it opened on the weekend of St. Patrick's Day and he had just turned eighteen.\n\nFurther corroboration for this date is found in an interview of Kevin Dunn by Michael Scherker on June 5, 1989, in San Francisco, in which Dunn states that he first went to the Stonewall Inn in March of 1967. When Scherker asked him, \"How did you end up at the Stonewall?\" Dunn replied, \"Some guy told me about this hot _new_ bar\" (emphasis added).\n\n48. \"Mike\"-Carter interview. (Mike is a fictitious name for the former porter of the Stonewall Inn.)\n\n49. Rodwell-Scherker interview.\n\n50. Craig Rodwell, \"Mafia on the Spot,\" _Hymnal_ 1, No. 1, February 1968, pp. 1\u20132.\n\n51. Brew quoted in Crosby, \"Stonewall Riot.\"\n\n52. Rodwell, \"On the Spot.\"\n\n53. Leitsch letter to Amsel, September 18 [1967], Leitsch papers. While Leitsch's letter does not state that the police were in uniform or from the Sixth Precinct, that seems a reasonable assumption, as it was normally the vice squad's job to investigate and catch corrupt police officers.\n\n54. Murphy gave the $1,200 figure to _The Village Voice_ , to Emerald City Television, and to the organization Seniors in A Gay Environment (SAGE). The SAGE account was used in a SAGE theatrical presentation in 1988, and excerpts from the play were published in _Equal Times_ in 1989. See Bob Dolan, \"When the Law Reaches into Your Pocket,\" _Pageant_ 23, No. 2, August 1967, pp. 40\u201346; Arthur Bell, \"Skull Murphy: The Gay Double Agent,\" _Village Voice_ 23, No. 19, May 8, 1978; Murphy, \"1969 at Stonewall\"; and Murphy interview with Gene Stavis of Emerald City Television. Note that the $1,200 a month figure agrees with Shaheen's estimate of several hundred dollars a week.\n\n55. Tree-Carter interview.\n\n56. _Ladder_ , \"Gay Power in New York City,\" October\u2013November 1969, p. 40. Rodwell related in \"On the Spot\" how he tried to report the unsanitary conditions to various city agencies and how unresponsive the agencies were.\n\n57. \"Mike\"-Carter interview.\n\n58. I am indebted to plumbing consultant Tony Carini for explaining to me how a substantial volume of water poured into a toilet could cause a delayed overflow.\n\n59. Carl Lee, \"It's What's Happening,\" _Hymnal_ 1, No. 5, June\u2013July 1968.\n\n60. Seating capacity information on floor plans of the Stonewall Inn filed with the City of New York gives the total capacity of the club at 190: \"185 persons and 5 employees.\" However, Seymour Pine said that on the evening he raided the club it had 200\u2013300 persons inside.\n\n61. A number of people pointed out the club's dangers as a fire trap after the bar closed, e.g., Perry Brass in Crosby, \"Stonewall Riot.\" Other comments on the lack of fire exits are: \"The Stonewall couldn't get a license because it lacked a rear fire exit.\" (\"Stonewall 'Historic'?\" _Advocate_ , August 15, 1973, p. 16), and \"there were no fire exits\" (Leitsch, _NYMN_ , August 1969). Note, too, that the windows were carefully boarded up.\n\n62. Rodwell in Crosby, \"Stonewall Riot.\"\n\n63. Russo, \"Outlaws,\" p. 16.\n\n64. Garvin-Scherker interview.\n\n65. Shaheen-Duberman interview.\n\n66. Tree telephone conversation, ca. June 1999.\n\n67. \"Beating Around the Bush,\" _Private Lives_ , January 1993, p. 100.\n\n68. Enid Girling is also known from the period for being one of the few attorneys available for helping gay men arrested as a result of entrapment, but is controversial for charging large fees for doing so.\n\n69. Dick Leitsch, \"The Stonewall Riots[:] The Gay View,\" _NYMN_ , August 1969, pp. 13\u201314.\n\n## 5 : T H E S K U L L\n\n1. Lanigan-Schmidt\u2013Duberman and Lanigan-Schmidt\u2013Carter interviews. Note that Ed Murphy was known to be especially attracted to Latino and black youths, which could lend credibility to the rumor that the kidnapped Puerto Rican was a boyfriend of Murphy's. Tano is a made-up name for the man Tommy and his friends called Miss Polka Dot, as his real name is unknown.\n\n2. Stephen van Cline\u2013Carter telephone conversation, April 21, 1996.\n\n3. This account of Murphy's life is based on interviews he gave. The sources used are Arthur Bell, \"Skull Murphy: The Gay Double Agent,\" _Village Voice_ , May 1978, pp. 1, 17\u201319; John Hoglund, _New York Native_ obituary, March 27, 1989, \"Eddie Murphy\u2014from Hellion to Hero,\" _Private Lives_ 2, Issue 6, December 1987; \"SAGE Stories,\" _Equal Times_ , June 1, 1989. Some of the details in the preceding sources are corroborated in a general way by Glenn Person, \"CS People,\" _Christopher Street_ , Issue 66, July 1982; in an interview by Gene Stavis on Emerald City Television; and in an interview given to Chris Davis, n.d. (apparently 1983 or 1984, since Murphy refers to the gay rights bill having been introduced before the city council of New York City for \"thirteen years\"). Regarding the violent incidents in his early life, Murphy consistently portrays himself as a victim standing up for himself against his oppressors, a line that interviewers Bell and Hoglund apparently bought, e.g., (Bell): \"One day when shining shoes for quarters, a nasty Irish cop laced into him [Murphy] and broke his shoeshine box. The natural thing to do was to hit the sonovabitch over the head with a milk bottle.\" Hoglund, after characterizing Murphy as \"above all else, a fighter for [...] human dignity,\" quotes Murphy as saying that \"the teacher hit me\u2014and I hit back!!!\"\n\n4. \"Detective at Hotel Is Held in Extortion,\" _NYT_ , August 5, 1965, p. 19.\n\n5. All of the information in this chapter about the blackmail ring quoted as coming from contemporary news accounts is from the following newspaper and magazine articles: \"Gay Bar Closed,\" _NYMN_ , March 1968, p. 5; \"Nine Seized Here in Extortion Ring,\" _NYT_ , February 18, 1966; \"17 Indicted in Hush-Money Shakedown,\" _Herald Tribune_ , 1966; \"Nationwide Ring Preying on Prominent Deviates,\" _NYT_ , February 3, 1966; \"3 Indicted Here as Sex Extorters,\" _NYT_ , June 1, 1966; \"Grab City Cop as Sex-Blackmail Kingpin,\" _Chicago Sun-Times_ , June 25, 1966; \"Detective Accused as a Top Extorter,\" _NYT_ , July 1, 1966; \"Blackmailer of Deviates Gets 5 Years,\" _New York Post_ , August 16, 1966; \"Gets 5 Years in Extortion of Homosexuals,\" _Daily News_ , 1966; \"Blackmailer Gets Five-Year Sentence in Homosexual Case,\" _NYT_ , August 17, 1966; \"Gets 5 Yrs. in Extortion of Homos,\" _Daily News_ , August 17, 1966, p. 9; \"Lands at JFK, FBI Is There,\" _Daily News_ , September 18, 1966; \"More Indictments Due in Blackmail Case,\" _New York Post_ , September 28, 1966, p. 62; \"Blackmail Paid by Congressman\"; _NYT_ , August 17, 1967; \"Member of 70-Man Ring Preying on Homosexuals Given 5 Years,\" _NYT_ , July 12, 1967; and \"2 Found Guilty in Chicago in Extortion of Homosexual,\" _NYT_ , December 9, 1967.\n\nAlthough most of the victims, including those men who were married and had children, were undoubtedly homosexual or bisexual, it seems possible from contemporary accounts that some small percentage of the victims may have been strictly heterosexual.\n\n6. William McGowan, \"Before Stonewall,\" _Wall Street Journal_ , June 16, 2000, p. W13.\n\n7. These nine articles are from Dick Leitsch's personal papers.\n\n8. Richard Inman, letter to Jack Nichols (Warren Adkins, pseud.), September 19, 1965, quoted in James T. Sears's _Lonely Hunters: An Oral History of Lesbian and Gay Southern Life, 1948\u20131968_ (Boulder, Colo.: Westview Press, 1997), p. 244.\n\n9. Ginsberg\u2013Benz interview.\n\n10. _Straight News_ , p. 269, and Donn Teal, _The Gay Militants_ , p. 65. Description of Hoover is on p. 267 of what appears to be either the first edition of Arcangelo, _The Homosexual Handbook_ , or a pirated copy of it.\n\n11. Ginsberg\u2013Benz interview.\n\n12. John Paul Ranieri had a lifetime of mental health issues, and I did not always find his accounts of events to be reliable. For these reasons, I only used information from him that was corroborated by other sources.\n\n13. Murray Weiss, \"J. Edgar's Slip Was Showing,\" _New York Post_ , February 11, 1993. I am grateful to Randy Wicker for providing me with a copy of this article. Very suggestive in this context is that Murphy would publicly say in 1978\u2014before it became public information, as it did in the 1990s, that the Mafia had photographs of Hoover involved in sex acts\u2014that he knew that J. Edgar Hoover \"was one of my sisters\" (see Bell, \"Skull Murphy\").\n\n14. It is very suggestive of Murphy's powers to silence opposition that while the aforementioned article ends with the statement: \"We also have other information on Mr. Murphy that will be printed here next month,\" the following month's newsletter did not so much as mention Ed Murphy's name.\n\n15. _New York City Gay Scene Guide_ , Vol. 2, p. 11 (1969).\n\n16. Stonewall Speech. The relevant passage says: \"A man who seemed to be the manager of the place had been arrested several times for shaking down homosexuals and had been recently named as a member of a national blackmail ring preying on homosexuals.\" It is interesting to note that Leitsch originally typed \"blackmailing\" before changing the wording to \"shaking down.\" Leitsch also wrote: \"A man named 'Scumback' Murphy was always around the Stonewall, and was said to be one of the operators of the place.\" Dick Leitsch, \"The Snake Pit Raid: Some Afterthoughts,\" _Gay_ 1, No. 10, April 13, 1970.\n\n17. Arcangelo, _Homosexual Handbook_ , p. 130.\n\n18. Shaheen-Duberman interview.\n\n19. Garvin-Carter interview.\n\n20. Lanigan-Schmidt\u2013Duberman interview.\n\n21. Kohler-Carter telephone conversation, January 11, 1999. In the _Emerald City_ interview, Murphy talks about being involved in a counterfeit ring but claims that he was acting in the ring as a government agent to break the ring.\n\n22. The only published references to this the author is aware of are Miller, p. 42, which also speaks of rumors and Robert Heide and John Gilman, _Greenwich Village: A Primo Guide to Shopping, Eating, and Making Merry in True Bohemia_ (New York: St. Martin's Griffin, 1995) p. 138.\n\n23. _Equal Times_ , June 1, 1989, p. 19.\n\n24. Carl Lee, \"It's What's Happening,\" _Hymnal_ 1, No. 6, August\u2013September 1968, p. 2.\n\n25. In our talks, as with all the other persons interviewed for this book, no indication had been given to Ranieri of the author's knowledge of the Stonewall Inn's operations nor any other information collected nor conclusions reached, so as not to prejudice testimony. Thus Ranieri had never been told by the author of this history of the rumors that the second floor of the Stonewall Inn was used for prostitution.\n\n26. Ranieri-Carter conversation, June 27, 1998. My handwritten notes on the entrance say: \"Gambino headquarters [were at the] back of 2nd floor\u2014[one] entered by garage. [Ranieri] Thinks front of 2nd floor [was] junk and storage. There may've been an emergency connection from back of 2nd floor to front of 2nd floor, [Ranieri is] not sure.\"\n\n27. \"Beating Around the Bush,\" _Private Lives_ , February 1992, August 1993, and December 1993.\n\n28. Barry Perrin shared his memories of what he had been told about blackmail at the Stonewall Inn spontaneously, without any prompting from the author.\n\nThe question naturally arises about the likelihood that a well-known person would risk giving information about himself to a membership club. A letter by Leitsch to his lover Bob Amsel documents that Cardinal Spellman himself signed the guestbook of the gay bar the Mystique: \"[John Lassoe and I] mapped out a tentative campaign for revising the sodomy and solicitation laws.... We're gambling that we can get enough grass-roots support... to counter-balance the inevitable Roman Catholic opposition. John thinks it's useless, but I'm going to begin another campaign to talk to Cardinal Spellman and see if I can persuade him to remain neutral, if he won't support reform. His name is in the guest book of the Mystique, and he's been seen other places, and I'm not above a little blackmail if it keeps him in line!\" Leitsch letter to Amsel, n.d., Leitsch papers.\n\n29. McGowan, \"Before Stonewall.\"\n\n30. New York Public Library Manuscripts and Archives Division, International Gay Information Center tape 02851, dated February 5, 1978: Morty Manford with Bob Battenberg and Arthur Bell re Ed Murphy. Arthur Bell also reacted with skepticism to Murphy's claims of being an informant: \"I don't believe it. Four or five years ago... the guy in charge of the Joint Strike Force Against Crime... showed me trees [that diagrammed] how everyone [in the mob] was related to everybody else, and he told me Ed Murphy was a dangerous guy and someone to watch out for.\"\n\n31. Information in this chapter on Seymour Pine's career and the order to close the Stonewall Inn comes from Pine-Carter interview.\n\n32. Pine-Isay interview.\n\n33. When Pine was interviewed by Carter he had trouble remembering when he was transferred to Manhattan but thought it was in 1969. He told Isay in 1989 that he had come into Manhattan in 1968. _The Village Voice_ , which interviewed him in 1970 (Jonathan Black, \"The Boys in the Snake Pit: Games 'Straights' Play,\" March 19, 1970), says he arrived in May 1969, so I have used that date. Note also that many press accounts of the Stonewall Riots talk about a \"new\" commanding officer, although some say new at the Sixth Precinct. Asked if there had there been any recent change of commanding officers at the Sixth Precinct, Pine said that there had not been.\n\n## 6 : D A W N I S J U S T B R E A K I N G\n\n1. \"McCarthy Wins Poll,\" _Hymnal_ 1, No. 4, May 1968.\n\n2. The summation given here of gay organizing in San Francisco in the 1960s is taken from D'Emilio's _Sexual Politics_ , pp. 182\u201391.\n\n3. Date of July 1966 is from \"Young Rejects Form Own Organization,\" _Cruise News & World Report_ 2, No. 7, July 1966, which announces the start of the new organization.\n\n4. \"Exiles of Sin, Incorporated,\" _Berkeley Barb_ , November 11, 1966.\n\n5. _We Protest_ , flyer, Donald Lucas papers, Gay and Lesbian Historical Society of Northern California. I am indebted to Susan Stryker for calling this flyer and all the other materials quoted here on Vanguard to my attention.\n\n6. Jean-Paul Marat, \"On Prejudice,\" _V_ 1, No. 2, October 1966, p. 1.\n\n7. Stryker-Carter interview.\n\n8. \"Young Homos Picket Compton's Restaurant,\" _Cruise News & World Report_ 2, No. 8, August 1966, and \"History of Christopher Street West,\" (San Francisco) _Gay Pride_ program, June 25, 1972.\n\n9. Evans\u2013Carter interviews. Although the perception is still widespread that Whitman was the major influence on Ginsberg's poetic development, Ginsberg was clear that Blake was his main inspiration. It is notable that Evans was moved by both Blake and Whitman, both important to Ginsberg, and then was heavily influenced by Ginsberg as well. See, e.g., Allen Ginsberg _Spontaneous Mind: Selected Interviews 1958\u20131996_ , ed. David Carter (New York: HarperCollins, 2001).\n\n10. Garvin-Carter interview and Garvin-Scherker interview.\n\n11. Garvin-Scherker, Garvin-Carter, Shaheen-Duberman, and Beard-Scherker interviews; \"The Night They Raided Stonewall,\" _Gay Activist_ , June 1971; Russo, \"Still Outlaws\"; _1969 New York City Gay Scene Guide_ , Vol. 2; and Arcangelo, _The Homosexual Handbook_.\n\n12. White-Carter interview and Crosby, \"Stonewall Riot.\"\n\n13. \"Gay Power Gains,\" _Hymnal_ 1, No. 6, August\u2013September 1968. The date of the _Wall Street Journal_ article was July 17, 1968; the letters to the editor column containing responses to the article was published in August.\n\n14. \"It's What's Happening\" and \"On Stage,\" _Hymnal_ 1, No. 5, June\u2013July 1968.\n\n15. Lori Reid, _The Complete Book of Chinese Horoscopes_ (Shaftesbury, Dorset; Rockport, Mass.: Barnes & Noble Books, 1997), p. 123.\n\n16. \"Transit Cops Shake Down Wrong Man\" and \"More Police Harassment,\" _NYMN_ , February 1969.\n\n17. \"Cop Kills Two\u2014May Go Free\" and \"Docks, Darkness and Danger,\" _NYMN_ , March 1969.\n\n18. Bathhouse wrecked and no bribes paid: _Los Angeles Advocate_ , April 1969, p. 22. \"Bathhouse Raided,\" \"Dock Killings,\" and \"Your Friendly SLA,\" _NYMN_ , April 1969.\n\n19. D'Emilio, _Sexual Politics_ , p. 230, and _Vector_ , April 1969.\n\n20. \"Gay Rebel Gets Shafted by Uptight Boss,\" _Berkeley Barb_ , April 11, 1969.\n\n21. \"Pink Panthers Gay Revolution Toughening Up,\" _Berkeley Barb_ , April 18\u201324, 1969.\n\n22. \"Killer Cops at Large\" and \"Gay Strike Turns Grim,\" _Berkeley Barb_ , April 25\u2013May 1, 1969, and \"Group Will Act\" and \"Gay Strike Hits Southern Front,\" _Berkeley Barb_ , May 2\u20138, 1969.\n\n23. \"Gays Get Tougher,\" _Berkeley Barb_ , May 23\u201329, 1969.\n\n24. Carl Wittman, \"A Gay Manifesto,\" in Karla Jay and Allen Young, eds., _Out of the Closets: Voices of Gay Liberation_ (New York: Douglas Book Corp., 1972), p. 330.\n\n25. \"Coming Mayoralty Contest,\" _NYMN_ , May 1969.\n\n26. Alternate U. history from telephone conversation with Mike Bradley, October 26, 2003.\n\n27. Tobin and Wicker, _Gay Crusaders_ , p. 173. I conclude that this approach was made by Katzenberg because he was the main person pushing the idea for such an organization and Hoose and O'Brien have no recollection of making the pitch to Marty Robinson. Further evidence that the approach to Robinson came from Katzenberg or someone in his group is that the first press mention of the group that would very soon become known as the Gay Liberation Front is of a demonstration that John O'Brien was present at and it says that the group called itself the Pink Panthers. \"From Outside the Pen,\" _Rat_ 2, No. 15, late July 1969, p. 4.\n\nHoose says he has no memory of meeting O'Brien before the riots and thus does not count himself as a founder of the group that would become the Gay Liberation Front, but when I first interviewed O'Brien, he told me that he had spoken to Jerry Hoose about this and that Hoose did not recall these conversations and meetings. If O'Brien were fabricating this, it seems highly unlikely that he would include Jerry Hoose in the roster of those trying to start a militant gay group before Stonewall when he knew that Hoose would deny any memory of these events. Moreover, in an article in the program of San Francisco's first gay pride march ( _Gay Pride_ , June 25, 1972), an article about how gay pride observations began (\"History of Christopher Street: How It All Began!\") lists Hoose as one of four founders of the Gay Liberation Front. (The other three it credits are Michael Brown, Martha Shelley, and Jim Owles.) Incidentally, Jerry Hoose told the author in an October 26, 2003, telephone conversation that it was he who had coined the phrase _gay pride_ at one of the meetings planning the first celebration of the Stonewall Riots.\n\n28. _Rat_ , July 24, 1969.\n\n29. _NYMN_ , June 1969.\n\n30. David Bird, \"Trees in a Queens Park Cut Down As Vigilantes Harass Homosexuals,\" _NYT_ , July 1, 1969; Don Jackson, \"Reflections on the N.Y. Riots,\" _Los Angeles Advocate_ , October 1969, p. 11.\n\n31. Letter to the editor signed by Kevin Liscoe, July 10, 1969, _Village Voice_ , p. 4; \"Gay Power in New York City,\" p. 40; and \"Reflections.\"\n\n32. \"Dawn Is Just Breaking...,\" _Queen's Quarterly_ 1, No. 3, Summer 1969, p. 5. Most ellipses are in the original editorial.\n\n33. Ronnie Di Brienza, \"Stonewall Incident,\" _East Village Other_ 4, No. 32, July 9, 1969, p. 2.\n\n34. Di Brienza, \"Stonewall Incident,\" and Pine-Carter interview.\n\n## 7 : A F R I D A Y N I G H T O U T\n\n1. The experiences of persons recounted in this chapter are based upon interviews and the following: \"Steve\" Ritter's talk at \"The Forgotten Stonewall,\" a Stonewall commemoration, June 16, 1994, and Garvin-Carter telephone conversation, December 12, 1999. In this chapter, for dramatic reasons, most of the thoughts and feelings of the persons reported in interviews have not been given as direct quotations.\n\n2. That the search warrant, number 578, dated June 26, 1969, was over twenty-four hours old when the police initiated the raid is significant, for there have been several variations of a rumor that the raid on the Stonewall Inn the night of June 27 began because of a dispute between the police and the bar earlier that day. Information on the search warrant is from the police log for June 28, 1969, obtained by Michael Scherker under the Freedom of Information Act.\n\n3. \"Satisfaction\" played prior to police raid: Barry Cunningham, _Gay Power: The Homosexual Revolt_ , (New York: Tower Publications, 1971.)\n\n## 8 : \"W E'R E T A K I N G T H E P L A C E!\"\n\nWhile the account in this book of the Stonewall Riots is based on many sources, it makes extensive use of the invaluable accounts published at the time, especially Lucian Truscott's \"Gay Power Comes to Sheridan Square\" and Howard Smith's \"Full Moon over the Stonewall\" (both in _Village Voice_ , July 3, 1969), Dick Leitsch's \"The Hairpin Drop Heard Around the World\" ( _NYMN_ , August 1969), Ronnie Di Brienza's \"Stonewall Incident\" ( _East Village Other_ 4, No. 32, July 9, 1969), and the anonymous \"Queen Power: Fags Against Pigs in Stonewall Bust\" ( _Rat_ 2, no. 14, July 9\u201323, 1969).\n\n1. According to the Sixth Precinct's log for June 28, 1969, obtained by Michael Scherker under the Freedom of Information Act, the raid began at 1:20 A.M. This agrees with the account given in the underground newspaper _Rat_ , which gives the time of the raid as \"[a]bout 1:15.\" By all evidence, the _Rat_ account of the riots is possibly the most precise reporting of the events of Friday night. That the official police account coincides with an unofficial account within five minutes is very strong corroboration.\n\n2. Established from photograph by Joseph Ambrosini in Eskow.\n\n3. Morty Manford, May 1978, student paper, \"The New York Gay Political Movement, 1978: An Organizational and Strategy Critique,\" hereafter Manford 1978, and a 3-page incomplete typescript, \"Stonewall Remembrances,\" hereafter Manford \"Remembrances.\" All Manford manuscripts cited are from the Morty Manford Papers, Manuscripts and Archives Division, the New York Public Library, Astor, Lenox and Tilden Foundations. These accounts of the riots were eventually published as part of an article, \"Why We Should Postpone New York's Gay Rights Bill: A New Look at Gay Political Strategy,\" in _Gaysweek_ , October 16, 1978, pp. 10\u201316.\n\n4. Pine-Carter interview.\n\n5. According to Pine's 1989 interview with David Isay, they called for backup just before entering the Stonewall Inn, but since that is Pine's recollection twenty years after the event, I have chosen to go with what Howard Smith was told by the police, presumably Pine, in 1969: they phoned after all the officers were inside the Inn: Smith, \"Full Moon.\" Also, I have chosen to go with Pine's later statement to Isay that they \"requested a couple of police officers and patrol wagons,\" since a patrol wagon did indeed come, rather than accept the _Voice_ version that he requested \"a patrol car and some officers.\"\n\n6. Note, however, that according to one patron who claimed to be inside the bar when the raid began, the police initially \"kept the music and dancing going so's not to get everyone up.\": _Rat_ , July 9, 1969. Perhaps this seeming inconsistency could be explained by the presence of the two undercover female police in the bar before the raid.\n\n7. Manford later described the raiding officers as a combination of federal Treasury officials and Sixth Precinct policemen. Morty Manford interview with Eric Marcus, hereafter Manford 1989, and Manford \"Remembrances.\" Manford identified some of the raiding officers as federal Treasury officials in not only the 1978 and 1989 accounts but also a January 1972 paper, \"A Comparative Study of Gay Liberationists and Columbia Homosexuals,\" hereafter Manford 1972. (See also Manford, \"Why We Should Postpone.\") This assertion suggests Pine told Manford this, as the Bureau of Alcohol, Tobacco and Firearms is part of the U.S. Department of Treasury and Manford 1972 has a footnote citing Pine.\n\nPhotograph accompanying Eskow article shows that the suits plainclothes officers were wearing that night were dark.\n\n8. Manford 1978 and Manford 1989.\n\n9. Manford 1989.\n\n10. Eagles-Carter interview.\n\n11. Pine-Carter interview. Pine added that \"I guess they had a few drinks, too. You know, that was their job, to be part of that crowd.\"\n\n12. Pine-Carter interview.\n\n13. Pine-Carter interview.\n\n14. Pine-Carter interview. Pine demanded to see identification and noted the man's name.\n\n15. Eagles-Carter interview. Possible corroboration of this account of lesbian resistance inside the club soon after the raid began comes from Bruce Shenton, who told the author that he heard about a week after the riots that the genesis of it all was a lesbian confrontation inside the bar. Shenton-Carter interview.\n\n16. \"Confusion and uncertainty\": Manford 1978, and \"ten or fifteen minutes\": Manford 1989.\n\n17. Pine-Carter interview.\n\n18. Alcohol found in storage room: Pine-Carter interview.\n\n19. Paper attached to each container and evidence book: Pine-Carter interview.\n\n20. Quantities of alcohol seized: Eskow. That it was Smythe who cataloged the alcohol: Pine-Isay and Pine-Carter interviews. Cataloging their names and addresses: Pine-Carter interview.\n\n21. Pine estimates the length of time it took between the beginning of the raid and getting all the customers he chose not to arrest outside as from fifteen to twenty minutes. Pine-Carter interview.\n\n22. _Rat_ , July 9, 1969.\n\n23. Manford 1989. Manford saw symbolism in their being held in this area, as if inside a closet. \"Little did the police know the ironic symbolism of that. But they found out fast.\"\n\n24. While this account never explicitly states that the complaining about the raids on the Stonewall took place inside the Stonewall Inn itself, it all but says so, leaving one with that tacit understanding, Di Brienza, \"Stonewall Incident.\"\n\n25. Jackson, \"Reflections.\"\n\n26. Jackson, \"Reflections.\"\n\n27. Walter Troy Spencer, \"Too Much, My Dear,\" \"Last Call\" column, _Village Voice_ , July 10, 1969, p. 36.\n\n28. \"Gay Power in New York City,\" p. 40.\n\n29. Manford 1978.\n\n30. Lanigan-Schmidt\u2013Duberman interview.\n\n31. Truscott, \"Gay Power.\"\n\n32. Manford 1989.\n\n33. Smith, \"Full Moon.\"\n\n34. Truscott, \"Gay Power,\" and Smith, \"Full Moon,\" p. 1. Tom is a made-up name for the anonymous _Rat_ reporter.\n\n35. Joel S.\u2013Carter. The direction of the patrol wagon is corroborated by other witnesses, including Maria Ritter.\n\n36. This is Joel S.'s description of the scene at the time the first paddy wagon arrived.\n\n37. Rodwell-Ardery interview. \"The raid was just starting at that time. And we noticed the crowds. We went over there,\" and Rodwell-Duberman: \"When we first arrived there... it was a small crowd... it had just started to gather.\" Game of bridge: Randy Shilts, _The Mayor of Castro Street_ (New York: St. Martin's Press, 1982), p. 42.\n\n38. Rodwell-Scherker interview. While in 1986 Craig Rodwell told Scherker that when he arrived there were two police cars in front of the Stonewall Inn, when he gave an account of the riots to Breck Ardery in 1970 he said that when he arrived \"there was a paddy wagon pulled up.\" I have decided to use the Ardery account concerning the police vehicles present, not only because it is so close in time to the event itself but also because according to Ed White's letter to Alfred Corn, it was while Mafia owners were being loaded into the van that \"[s]omeone shouted 'Gay Power,' \" for White's letter has the ring of someone noting the cry for the first time. He even says \"others took up the cry,\" which sounds similar to Rodwell telling Scherker, \"I can remember yelling 'Gay power' at one point, and people cheering, like this was the first time they'd ever heard something like that, you know. And then people throwing pennies or something.\" Note, too, that most accounts do not have money being thrown until a patrol wagon is on the scene and being loaded with prisoners.\n\n39. Rodwell-Scherker interview.\n\n40. Rodwell-Ardery interview.\n\n41. Pine-Isay and Pine-Carter interviews. Although Pine describes activities in the back room as happening mainly while the patrons were leaving, it is reasonable to assume that they began these activities while the patrons were lined up and waiting. Otherwise, what would the eight police officers have been doing during that time? Note, too, that in the Eskow account the police \"began clearing the bar [...] _[a]fter_ 28 cases of beer and 19 bottles of liquor were confiscated\" (emphasis added). In the Isay interview, Pine stated that the real motivation for removing the bars was to put pressure on the club's owners.\n\nPine never did learn who was giving the orders to disregard his calls for more patrol wagons and reinforcements. As the Sixth Precinct was well paid by the Stonewall Inn and as Pine had violated protocol by not informing them of the raid in advance and the Sixth Precinct was able to countermand the orders, it seems probable that these cancellations did originate from the Sixth Precinct in retaliation.\n\n42. Pine was quite clear in his interviews with the author that he made sure everyone he did not want to arrest was outside the club before he attempted to load any prisoners into vehicles.\n\n43. White-Corn letter. Both White and Truscott (\"Gay Power\") report negative and positive reactions from the crowd as the bar owners and employees are arrested. The accounts are ambiguous and could suggest that they were being either booed because of their own exploitation of gay people or cheered because of their connection with a bar that was a gay institution. Similarly, one could read the accounts as suggesting that the police were being booed (or cheered) for arresting them. White: \"As the Mafia owners were dragged out one by one and shoved into the wagon, the crowd would let up Bronx cheers and jeers and clapping. Someone shouted 'Gay Power,' others took up the cry\u2014and then it dissolved in giggles. A few more prisoners\u2014bartenders, hatcheck boys\u2014a few more cheers.\" Truscott: \"Suddenly the paddywagon arrived and the mood of the crowd changed. Three of the more blatant queens\u2014in full drag\u2014were loaded inside, along with the bartender and doorman, to a chorus of catcalls and boos from the crowd.\" While it is quite possible that different persons in the crowd reacted differently (some seeing the police as the enemy for arresting the mobsters, others applauding these arrests, and still others feeling that both were the enemy), I have interpreted the boos and catcalls at this point at being aimed primarily at the Mafia members and associates, per the account given by Vito Russo to Eric Marcus: \"People were... screaming at who they perceived to be the enemy,... the owners of the bar\u2014people who worked there.\" Even though Russo told Marcus he arrived later, after the patrol wagons had left, if some members of the crowd were screaming at the Mafiosi then, it seems probable that they would have done so earlier as well.\n\n44. Joel S.\u2013Carter interview: \"Sound carried very well that night.\"\n\n45. White-Corn letter.\n\n46. Fader-Carter interview. Harry Beard also saw John in the paddy wagon. Scherker-Beard interview.\n\n47. White-Corn letter.\n\n48. It is worth noting, however, that humor can be a way of sublimating anger, and various writers have noted over the years that as an oppressed minority, homosexuals found\u2014and continue to find\u2014a number of creative ways to express that anger, with camp humor being one of them.\n\n49. White-Corn letter.\n\n50. Truscott, \"Gay Power,\" p. 1.\n\n51. White-Corn letter.\n\n52. Based on numerous reports.\n\n53. _The New Symposium II_ , a WBAI Sunday gay radio program, broadcast transcript. The precise date of the program is unknown. The broadcast begins with an introductory statement by Charles Pitts saying, \"On _The New Symposium_ tonight we will hear an interview with a gay bar owner. But first we want to play a tape we recorded yesterday when we talked to a young man who was in the vicinity of the Stonewall bar during the riots there Friday and Saturday nights of last week.\" If Pitts considered Sunday to mark the beginning of a new week, then it is possible the program was taped on June 29, but it seems more likely from the description of the riots as taking place \"last week\" (instead of \"over the last two days,\" for example) that the interview with Sargeant was taped on Saturday, July 6, 1969.\n\n54. There can be practically no doubt that this was Tammy Novak that the _Rat_ reporter saw exiting. Not only does the description of Tommy fit Tammy Novak, but also Tree explained in a telephone conversation during the summer of 1999 that Tammy Novak's name as a boy was Tommy.\n\n55. When Ritter says \"older\" we need to note that she had just turned eighteen, so she just means older than herself.\n\n56. See Duberman, _Stonewall_ , p. 197, Rodwell-Duberman interview, and Maida Tilchen, \"Mythologizing Stonewall,\" _Gay Community News_ , June 23, 1979, p. 16.\n\n57. Note that the descriptions of her clothing given by several witnesses agree to a considerable degree, especially when one considers that the _Voice_ articles give no description of her clothing or any other details about her. Indeed, a careful reading of the two _Voice_ articles shows that only _one_ word, _dyke_ , is employed in the entirety of the _Voice_ 's reporting to describe this woman. It is therefore remarkable that witnesses Yates's, Hardy's, and Beard's accounts describing her appearance and dress agree in such detail, since the _earliest_ of these was given twenty years after the event. Also, Scherker's interview with Beard has never been published (although it is quoted in _Stonewall_ ) and when I interviewed Yates he had no particular interest in the lesbian, his interest centering more on his friend Gino's reaction to her mistreatment, for Yates thought it was perhaps Gino's cobblestone toss that had triggered the riots. In Mike Long's _San Francisco Sentinel_ article (\"Stonewall: The Night the Girls Said No!\" June 22, 1989, pp. 3\u20134) she is described as wearing \"a man's black leather suit.\" Beard described her to Scherker as being \"fully dressed in male clothing.\" It should be noted that while Duberman cites inconsistencies between Beard's accounts as given to Scherker and to the _San Francisco Sentinel_ , this is not necessarily the case, for while Beard's is the only name quotations are attributed to in the article, some quotes in the article are _not_ attributed, and the article begins by stating that \"three men [are] determined to set the record straight.\" (The other two men, shown in a photo with Beard, are Gene Huss and Don Knapp.) Also, whereas Duberman says (p. 299, n. 39) that Beard \"describes the cop as 'hitting her over the head with his billy club,' \" in the article this is not a direct quotation but a statement by the reporter. Duberman assumes that this information came from Beard, but it could just as well have come from Huss or Knapp. Still, I consider Beard a source to be used with some care, seeming at times given to exaggeration, as when in his interview with Scherker he said that twenty-five drag queens lined up in a chorus line and, under Beard's leadership, kicked the front door of the Stonewall Inn open.\n\nDescriptions of the lesbian's clothing are as follows: \"fully dressed in men's clothing\" (Beard-Scherker interview); \"a man's black leather suit\" (Beard and\/or Knapp and\/or Huss in Duberman, _Stonewall_ ); \"Dark suit. Sport coat, white shirt, tie,\" which Yates summed up as \"fancy, go-to-bar drag for a butch dyke\"; and \"I think she was in pants\" (Hardy, when asked about her clothes).\n\nThe similarities in the independent descriptions of the woman's body are also striking, again, especially after so many years. Yates describes her as \"beefy, good-sized,... probably in her late thirties, dark, very short... mannish haircut, typical New York butch, not Anglo, probably Italian or Jewish: dark hair, dark features.\" Hardy described her as \"bigger, huskier, not a femme, not a real feminine person. Not even a feminine person. Bigger size, nasty temper. She was at least as tall as I am; I'm five-seven. If I remember right, she had darker hair.\" Asked her impression of the lesbian's age, Hardy emphasized that she was only seventeen in June 1969 and has never been a good judge of age and therefore could only guess \"late twenties, early thirties.\"\n\nThe descriptions of this woman as a butch or bull-dyke type further increase the likelihood of these accounts being accurate because of another important piece of unpublished evidence. In the Lesbian Herstory Archives among the records of the film _Before Stonewall_ is the paper written by Tina Crosby on the Stonewall Riots as a piece of folk historical research, \"The Stonewall Riot Remembered.\" Crosby interviewed eight people, seven of whom either went to the Stonewall Inn or had participated in the riots. Crosby wrote that \"each person I talked with remembered the Stonewall as an exclusively male bar, with the only exception being an occasional _tough_ lesbian [emphasis added] or female friend of one of the male patrons.\" This paper is dated January 1974. The early date of Crosby's interviews, combined with the paper's unpublished status, make it valuable corroboration on this point.\n\n58. Hardy, who at five feet, seven inches, is tall for a woman, says that the arrested lesbian was at least as tall as she.\n\n59. Yates-Carter interview.\n\n60. This letter (referred to as the \"Rhoda letter\") was discovered by James T. Sears, who published an excerpt from it in his book _Lonely Hunters_. The letter is found in the Atlanta Lesbian Feminist Alliance Papers, Special Collections Library, Duke University. Unfortunately, not all pages of this letter have been preserved. Also, the manuscript at Duke is a photocopy of the original letter and not the original itself. Whether the original manuscript survives is unknown.\n\n61. Beard-Scherker interview.\n\n62. Long, \"Stonewall,\" June 22, 1989. Note that the two possibilities are not mutually exclusive: the lesbian could have both requested that the cuffs be loosened and protested the rough treatment.\n\n63. Beard's versions of the events surrounding the lesbian seem all the more plausible given the role in them of the handcuffs bothering the lesbian, for the role of handcuffs was also very much on the mind of Dave Van Ronk. Van Ronk explained that the handcuffs the police used on him were Spanish cuffs, metal handcuffs specifically made so that if one struggles to escape, they automatically tighten. However, the cuffs can also tighten simply from general movement. Van Ronk remembers this well because he made his living playing guitar: his cuffs had tightened and by the time he was released from the handcuffs he had lost some feeling in one of his hands. Note also that Steve Yates remembers the lesbian having her hands cuffed _behind_ her back, which is the way Van Ronk remembers his hands being cuffed, another indirect corroboration: it makes sense that the police would cuff all or most of the prisoners arrested in the same place at the same time in the same manner and (presumably) with the same kinds of handcuffs.\n\n64. Truscott, \"Gay Power,\" p. 1, and Smith, \"Full Moon,\" p. 25.\n\n65. Leo E. Laurence, \"Gays Hit N.Y. Cops,\" _Berkeley Barb_ , July 4\u201310, 1969, p. 5. Laurence wrote: \"Ironically it was a chick who gave the rallying cry to fight.\" The _Magpie_ , a Los Angeles gay publication, in an August 1969 article by Bobbie Huff, \"N.Y. Po-lice Scream at Queens\" (p. 14), appears to quote the _Barb_ account but instead of calling the woman a \"chick\" identifies her as a \"female customer.\"\n\n66. Laurence, \"Gays Hit N.Y. Cops.\"\n\n67. However, chronologically the event in _Rat_ seems too early in the evening and may very possibly have involved a male prisoner. Harry Beard, \"I Was There... the Stonewall Riot,\" _New York Gay Press_ 1, No. 1, July 1980, p. 5. \"The squad car tried to escape with its prisoner. Too late... the cops weren't going anywhere on four slashed tires.\" That police vehicles' tires were slashed is corroborated by four separate mentions in three independent accounts: the _Rat_ account, Beard's account, and two quotations by Morty Manford: a single short phrase in Manford 1978, \"Slashed tires of the paddy-wagon,\" and Manford 1989: \"I can't claim credit for the small acts of violence that took place. I didn't break any windows. I wasn't the one who had a knife and cut the tires on the paddy wagon. I didn't hit a cop and I didn't get hit by a cop.\"\n\n68. Yates-Carter interview. Note that according to Yates, Gino told everyone he was Italian, although Yates said he and his friends \"knew\" Gino was Puerto Rican.\n\n69. Hardy-Carter interview.\n\n70. According to \"I Was There,\" by Harry Beard, the patrol car he saw the lesbian put in had all of its tires slashed, which suggests the lesbian originally in the police car might have been subsequently placed inside the patrol wagon to make driving the police car easier.\n\n71. Dunn-Scherker interview.\n\n72. That Eagles thinks that the injured lesbian he saw had blood on her head suggests that this may be the same lesbian Beard says he saw get hit on the head. If this is the case, it could bolster the scenario laid out in note 70 whereby the lesbian in the police car could have been moved to the patrol wagon, which would also mean that the same lesbian could have been seen entering two vehicles, and thereby magnifying the impression some people had of the number of lesbians present. However, I accept the testimony of Beard, Eagles, and Dunn that more than one lesbian was present, although I think there were probably only two or three lesbians at most inside the Stonewall Inn on this occasion. Another difficulty with Eagles's account about the lesbians is that the arrested Raymond Castro was placed inside a patrol wagon with a lesbian and it is very clear that she was not injured.\n\n73. There can be little doubt that Castro was the person Tom saw fighting the cops, for note both that Tom puts the man \"at the door\" of the Stonewall where Castro was by his own account and Tom writes that \"5 or 6 cops... tried to subdue him.\" Castro told the author that when he got to the police station an officer of higher rank suggested that he was an animal because it had taken five policemen to subdue him. It was very probably Ray whom Fader also had in mind when he said, \"I think there was a fellow\u2014he was a strong guy, and I think he got in kind of a fight with them. They put him in the wagon.\" Fader-Carter interview. Castro's arrest Friday night is also corroborated via Carter's interviews with both Joel S. and Perrin.\n\n74. Manford-Marcus interview. Manford attributed this act by the local police to their being conscripted into the raid by \"treasury police,\" but a more reasonable interpretation is that the Sixth Precinct was angry about not being notified by the Public Morals Division of the raid until the last moment, especially if this would endanger their payoffs from the Stonewall Inn. One also has to wonder whether from where Manford was standing he could see the police struggling to subdue Ray Castro. If not, Manford may not have known why the police were leaving the rear of the van.\n\n75. Murphy-Davis interview.\n\n76. The original account has a typographical error here: \"tried the resting the guy\"; _assisting_ is the only word that would seem to both fit the context and be capable of being misread as \"the resting.\"\n\n77. Smith, \"Full Moon,\" p. 25.\n\n78. Truscott, \"Gay Power.\"\n\n79. The slashed tires may also help explain why it took the patrol wagon so long to return: perhaps the tires were changed at the Sixth Precinct or another patrol wagon had to be found.\n\n80. Smith, \"Full Moon.\" That the officer Pine spoke to was one of the drivers of the vehicle: Pine-Isay interview. In that interview Pine does not say which driver of which vehicle he spoke to.\n\n81. Smith, \"Full Moon.\"\n\n82. Eskow identifies the officer as Gil Weissman.\n\n83. Pine-Isay interview; Smith, \"Full Moon\": \"the only uniformed cop among them.\" He is identified as Gil Weissman in Eskow; see note 82.\n\n84. Van Ronk's birthday was not until the thirtieth of the month, so he was celebrating early, presumably because, as was the case with Steve Yates, it was the weekend closest to his actual birthday.\n\n85. _NYT_ , June 29, 1969, and Leitsch, _NYMN_ , July 1969.\n\n86. \"Pennies were flung at the door of the bar and the mass of evicted homosexuals tittered in defiance.\" Manford 1978.\n\n87. \"Dirty Copper!\": Leitsch, _NYMN_ , August 1969.\n\n88. Smith, \"Full Moon.\"\n\n89. Under the eye: Smith, \"Full Moon.\" Right eye: Police log, June 28, 1969. Pine in his interview with the author gave a very different account of the incident: Pine is very clear that he saw Van Ronk purposefully approach Weissman and flip a coin directly into his face, causing the injury. Pine further asserts that far from picking someone at random, he knew who had injured Weissman and followed him into the crowd.\n\n90. Smith, \"Full Moon.\"\n\n91. Smith, \"Full Moon.\"\n\n92. Pine-Isay interview.\n\n93. Smith, \"Full Moon.\"\n\n94. Pine-Isay interview.\n\n95. Van Ronk explicitly rejects Howard Smith's account of the police beating him, even going so far as to say that the scene inside the Stonewall Inn was so confused that he is not sure whether the kicks he received were purposeful or inadvertent. Van Ronk\u2013Carter interview.\n\n96. Smith, \"Full Moon.\" Note that according to Pine, it was Smith who approached him: \"At that point, a reporter from the _Village Voice_ came up and asked if he could stay on our side to see what we were doing, and he would cover it from that angle, rather than from out front. He showed me his press card; that was okay with me.\" Pine-Carter interview.\n\n97. Smith, \"Full Moon.\"\n\n98. Smith-Carter interview.\n\n## 9 : L A N C I N G T H E F E S T E R I N G W O U N D O F A N G E R\n\n1. Pine-Isay interview.\n\n2. Truscott, \"Gay Power.\"\n\n3. Truscott-Carter interview.\n\n4. Truscott-Carter interview. Note how closely this agrees with the account in _Rat_ of the meaning of what happened at that time: \"A few pigs outside had to flee for their lives... and barricade themselves in. It was too good to be true. The crowd took the offensive.\"\n\n5. Fader-Carter interview. Note the similarity to a quote published a year later on Stonewall's first anniversary: \"It was our Verdun: Thou shalt not pass and all that.\" Hans Knight, \" 'Other Society' Moves into the Open,\" _Sunday Bulletin_ , July 19, 1970.\n\n6. David Isay, _Remembering Stonewall_ , 1989 Pacifica Radio Archive.\n\n7. _Newsweek_ , October 27, 1969.\n\n8. Manford 1989. Photographic corroboration of Manford's account can be seen in a photograph by Diana Davies taken of the front of the Stonewall Inn shortly after the riot, that shows the lower left pane of the double window immediately above the Stonewall Inn's entrance broken.\n\n9. Manford 1989.\n\n10. Bruce Voeller, \"Stonewall Anniversary: Assessing the Activist Years,\" _Advocate_ , July 12, 1979, p. 30. Voeller continues: \"Stories about the following events differ. In some, a transvestite picked up a rock and threw it at the police; in others, the young Puerto Rican, who may have been nongay, threw a beer can. Whoever acted first started an avalanche.\" That it is Voeller who wrote this is all the more interesting, given Voeller's strong interest in the Stonewall Riots. Unfortunately, Voeller does not say if he was at the riots, whether he witnessed any of this, and, if not, what his sources are.\n\n11. The photograph by Ambrosini is reprinted on the cover of this book.\n\nHoose-Carter interview. The quote is what Hoose remembers his friend the deceased John Goodman telling him immediately after Hoose arrived. Robert Bryan corroborates John Goodman's presence at the riots on Friday. Besides the testimony of Goodman about the leading role of Zazu Nova, we have John O'Brien's testimony that he saw Nova there the first night and in the company of Marsha Johnson, although O'Brien did not see Nova fighting but getting her photo taken on a corner with Marsha. Still, O'Brien arrived after the initial outburst that forced the police to retreat inside the Stonewall Inn, so he could not have seen the first instigators of violence.\n\nTestimony about the primacy of Marsha Johnson's role is widespread but generally second- or even thirdhand. For instance, both Robert Heide and John O'Brien saw Marsha in the crowd outside the club on the first night and both independently described her as being in semidrag, not full drag. Heide says he saw her \"just in the middle of the whole thing, screaming and yelling and throwing rocks and almost like Molly Pitcher in the Revolution or something. I mean, a loud yelling and screaming, 'You can't do that. Who the hell, who the fuck!' \" He is also clear, however, that he cannot say that she was the first to throw an object: \"I didn't see this, but I did speak with someone who claims that Marsha was the first one to throw a stone at a policeman.\" Heide-Carter interview.\n\nAs is true here with Heide, several others interviewed for this history said that soon after the first night of the riots they heard that she was the person who really started it. There is also an intriguing account that Robin Souza told the author that he heard first from Marty Robinson in the fall of 1969 and from Morty Manford in the mid-1970s: that after the police entered the bar Marsha threw a shot glass into a mirror, saying, \"I got my civil rights!\" and that Morty said that was what started the riots. Souza said that in the Gay Activists Alliance that was called \"the shot glass that was heard around the world.\" While I believe Souza is telling the truth when he says that Morty told him this story, there are several problems with it. First is that Morty left several accounts of the riots before his death, including several written by him, and did not once include any mention of Marsha. (This raises the possibility that Morty may have censored his own recorded accounts, fearing that to publicly credit a severely mentally ill transvestite prostitute with starting the Stonewall Riots, and thus indirectly the gay liberation movement, could have been used effectively by the movement's opponents.) Second, Seymour Pine said that there were no black transgendered persons inside the bar.\n\nIt seems clear that whether it was Jackie Hormona who was the first person to be violent (or to perform the first act of violence that precipitated an outpouring of violence), it seems reasonable to conclude that Marsha Johnson was almost indubitably among the first to be violent that night and may possibly even have been the first. Marsha's inability to speak with any coherence and focus is no doubt the reason we do not have a clear account of her own recollections.\n\nTo this date there are no known interviews or accounts by Zazu Nova. This may have something to do with her apolitical nature. According to John O'Brien, who knew Nova from the trucks, \"Basically Nova was very much into her looks. I mean, she was very narcissistic... very much so... was worried about her looks and her face and everything.... She was an attitude queen, but not a political queen.\" O'Brien-Carter interview.\n\n12. Truscott, \"Gay Power.\"\n\n13. Manford 1989.\n\n14. \"No place for Gaiety,\" _New York Post_ , June 28, 1969.\n\n15. O'Brien noted that the main concentrations of people doing most of the action were at the west end of the Stonewall Inn, close to the stoop that Craig Rodwell and Fred Sargeant had climbed to witness the action and directly across the street from the stoop. Truscott, who the reader will remember had perched himself on top of a garbage can on the east side of the stoop to better witness the action, also remembers the way the crowd focused its anger on the Stonewall's west window.\n\n16. O'Brien, like Kohler, was aware of the meter being loose.\n\n17. O'Brien's description of one of the men who uprooted the meter is corroborated by Fader: \"I remember one guy, kind of a muscular, heavy guy, blond guy\u2014some people had started to pull a parking meter.\" (Fader-Carter interview.) O'Brien's description of the men uprooting the parking meter being masculine in appearance but also with a street feel to them also matches Edmund White's description of the same scene: \"Some adorable butch hustler boys pull up a _parking meter_ [emphasis in original], mind you, out of the pavement and use it as a battering ram.\" White-Corn letter.\n\n18. Eskow agrees that \"windows were smashed.\"\n\n19. White-Corn letter.\n\n20. White-Corn letter.\n\n21. Smith, \"Full Moon.\"\n\n22. _Rat_ , July 9, 1969, and Di Brienza, \"Stonewall Incident.\"\n\n23. Kohler-Carter interview.\n\n24. Smith, \"Full Moon.\"\n\n25. Beard, \"I Was There,\" p. 5. That a piece of glass was thrown in this manner at the police through the open door is corroborated fully by Sargeant's account given to WBAI and partially by _Rat_ 's statement: \"the cops inside were scared shitless, dodging projectiles and flying glass.\"\n\n26. Kohler-Carter interview.\n\n27. Rodwell-Scherker interview October 9, 1986, and Rodwell-Duberman interview.\n\n28. While Smith wrote that it was a \"large wrench,\" Pine is adamant that it was the Stonewall Inn's fire ax that Smith took to protect himself with and not a wrench: Pine-Carter interview.\n\n29. Leitsch-Carter interview.\n\n30. Used hose to put fire out: Eskow and Leitsch, \"Hairpin.\"\n\n31. _Rat_ , July 9, 1969.\n\n32. Smith, \"Full Moon.\"\n\n33. Pine cannot recall whether it was he or Smythe who discovered a way to get the policewoman outside the bar. Since Pine states that his main focus was very much on keeping his men from firing, I have written this account as if it was Smythe who got the policewoman out, which Pine told an interviewer he finds a logical assumption.\n\n34. Smith-Carter interview.\n\n35. \"Wooden wall... was forced down\": _Rat_ , July 9, 1969.\n\n36. \"[T]he mob will pour in\": Smith, \"Full Moon.\"\n\n37. Smith, \"Full Moon.\"\n\n38. Truscott, \"Gay Power.\"\n\n39. \"[C]ome up with another assault\": Truscott, \"Gay Power.\"\n\n40. \"[T]he door is broken down\": White-Corn letter. The account of lighting a fire inside the Stonewall is corroborated by Dave Van Ronk, who told the author that he recalled walking past the Stonewall Inn shortly after the riots and seeing that the bottom part of the entrance door had been burned.\n\n41. \"[L]it a trash can full of paper afire and stuffed it\": _Rat_ , July 9, 1969.\n\n42. Manford 1989.\n\n43. The symbolism of the coatroom being burned was noted by Eric Marcus in his interview with Morty Manford.\n\n44. Di Brienza, \"Stonewall Incident.\" Craig Rodwell similarly recounted, \"I remember [a] person coming up and squirting lighter fluid into the broken windows and then throwing matches in,\" though, Rodwell added, \"it never caught.\" Rodwell-Scherker interview.\n\n45. Smith, \"Full Moon.\"\n\n46. In the last paragraph of page 1 of Manford \"Remembrances,\" Manford makes a kind of impressionistic and dramatic collage of the evening's high points and mentions \"fire trucks circling; the hosing-down of the crowd.\"\n\n47. White-Corn letter. Note that Manford's use of the plural in note 46 is a corroboration of White on there being more than one fire truck at the scene: \"fire trucks\" and \"two fire engines.\"\n\n48. Manford 1989. Original has \"started\" rather than \"start.\"\n\n49. TPF: Pine-Isay and Pine-Carter interviews.\n\n50. Smith, \"Full Moon.\"\n\n51. Police clearing the crowds away from the Stonewall Inn's entrance: Pine-Carter interview.\n\n52. Garvin-Carter interview.\n\n53. Sixth Precinct police log for June 28, 1969.\n\n54. Police predesignated: Pine-Carter interview.\n\n55. Calling ambulance: Pine-Carter interview.\n\n56. St. Vincent's Hospital: Sixth Precinct police log for June 28, 1969.\n\n57. Van Ronk\u2013Carter interview. Location of patrol wagon corroborated by Marle Becker.\n\n58. Becker-Carter interview. Asked how he could be sure they were transvestites from such a distance, Becker responded that they were all wearing dresses and that the only people he regularly saw at the Stonewall Inn in dresses were men.\n\n59. Pine-Carter interview.\n\n60. Pine-Carter interview.\n\n61. Apparently Rodwell means Christopher Park. In the manner of most New Yorkers, he seems to be referring to the park as Sheridan Square.\n\n62. Rhoda letter. Note that the evidence that it was the fire truck hoses that were used on the rioters is inferential, for the Rhoda letter uses the plural, _fire hoses_ , and by all accounts there was only one fire hose in the Stonewall Inn. That the fire trucks' hoses were used on the crowd is confirmed by a brief mention in the collage-style paragraph written by Morty Manford ca. 1978: \"the hosing-down of the crowd,\" which immediately follows the phrase \"fire trucks circling.\" (See note 46 above.) One suspects that for propaganda purposes most witnesses have not wanted to recall the hosing down of the crowd, for it would be the one definite instance during the riot when the protesters were humiliated.\n\n63. Rodwell-Scherker interview.\n\n64. Telephone conversation with John Fisk, October 21, 1997.\n\n## 10 : \"C H R I S T O P H E R S T R E E T B E L O N G S T O T H E Q U E E N S!\"\n\n1. Babick-Scherker interview.\n\n2. Truscott, \"Gay Power.\"\n\n3. I am grateful to Danny Garvin for providing the fuller text of this slogan, the beginning of which is not visible in the McDarrah photographs, making it clear that the slogan is referring to the Vietnam War.\n\n4. Fred McDarrah, the _Village Voice_ photographer, took photographs of some of the rioters on Saturday night. The first slogan quoted here is visible in one of those photographs. The photograph of the second slogan was made the same weekend, if not on Saturday. For a copy of the photographs with these slogans, see McDarrah and McDarrah, _Gay Pride_ , pp. xxxvi and 3.\n\n5. Dick Leitsch, \"The Stonewall Riots[:] The Police Story,\" _NYMN_ , August 1969, pp. 5\u20136.\n\n6. _NYT_ , June 30, 1969.\n\n7. Rodwell, from leaflet titled _Get the Mafia and the Cops Out of Gay Bars_. Rodwell issued the flyer in the name of HYMN.\n\n8. Truscott, \"Gay Power.\"\n\n9. Rodwell-Ardery interview.\n\n10. Rodwell-Duberman interview.\n\n11. Within two to three hours: Rodwell-Ardery interview.\n\n12. While Rodwell told Duberman that \"people started to come down here in the early evening, eight o'clock probably or something like that,\" most contemporary accounts have the crowds assembling late in the evening and early morning. Fred Sargeant, Craig's lover, told WBAI Radio within days of the riots that the crowd assembled around 12:00 A.M. (This latter is from my notes made after hearing the WBAI interview, so it is based on my memory and not a transcript.) While it is possible that people began to filter into the neighborhood around 8:00 P.M., it does not sound as if the crowds were truly sizable and identifiable as crowds until around midnight.\n\n13. Rhoda letter.\n\n14. _NYT_ , June 29 and 30, 1969; Leitsch, _NYMN;_ and Truscott, \"Gay Power.\"\n\n15. _NYT_ , June 30, 1969; Leitsch, _NYMN_ , July 1969; and Laurence, \"Gays Hit N.Y. Cops.\" Craig Rodwell told Martin Duberman that people started \"to come down here in the early evening, eight o'clock probably... and gradually it got to be 8:30, 9:00 o'clock.... Just thousands of people. Spilling over into the street. And that's when we decided to block off the street.\" However, Rodwell told this to Duberman over twenty years after the events. I have noticed in my many interviews with Stonewall Riots witnesses that the element they consistently are the least reliable on is that of time, and they also tend to consistently place times of events earlier than they in fact happened. I base my attribution of error to them because both news and police reports recorded at the time of these events tend to generally agree on the timing of events. Since contemporary news and police reports agree that the crowd started to act up probably around 12:45 A.M. at the earliest, I give these contemporaneous records more weight than Rodwell's much later interview with Duberman.\n\n16. Truscott, \"Gay Power.\"\n\n17. Leitsch, _NYMN_ , July 1969.\n\n18. Leitsch, _NYMN_ , August 1969.\n\n19. _NYT_ , June 30, 1969.\n\n20. Unfortunately, the accounts that mention the handing out of literature do not make it clear what pieces were handed out except that Edmund White's letter to Alfred Corn states that the Mattachine Society was handing out the pamphlet _What to Do If You Are Arrested_. It would appear very likely that what was handed out by Rodwell was an earlier draft of the leaflet he gave out on Sunday, _Get the Mafia and Cops Out of Gay Bars_ , and that similarly the Mattachine Society of New York was also handing out an early version of Leitsch's \"The Hairpin Drop Heard Around the World.\"\n\n21. White-Corn letter. Note that White's account corroborates O'Brien's assertion that he invited members of the Crazies to the demonstration and that they came.\n\n22. Truscott, \"Gay Power.\"\n\n23. Leitsch, \"Hairpin.\"\n\n24. Leitsch, \"Hairpin.\"\n\n25. Truscott, \"Gay Power.\"\n\n26. Di Brienza, \"Stonewall Incident.\"\n\n27. Babick-Scherker interview.\n\n28. Leitsch, _NYMN_ , July 1969.\n\n29. Sign on door: _NYT_ , June 29, 1969.\n\n30. Owners trying to coax customers into the Stonewall Inn as chants grew louder and more frequent: Truscott, \"Gay Power.\"\n\n31. Crowd growing restless: Truscott, \"Gay Power.\"\n\n32. Laurence, \"Gays Hit N.Y. Cops\"; Rodwell-Duberman interview.\n\n33. Old lady trying to get through crowd: Leitsch, _NYMN_ , July 1969.\n\n34. As of 2003, June 28, 1969, was still the hottest June 28 in New York City history: <http:\/\/www.weather.com\/weather\/climatology\/USNY0996?climoMonth=6>.\n\n35. Rodwell-Duberman interview. Leitsch's account corroborates Rodwell on the crowd's overflowing from the sidewalks into the street: \"As the mob grew, it spilled off the sidewalk, overflowed Sheridan Square Park, and began to fill the roadway.\" Leitsch, _NYMN_ , July 1969.\n\n36. When they decided to block off Christopher Street: Rodwell-Duberman interview. According to Leitsch (\"Hairpin\"), as the crowd began to spill over into the streets \"[o]ne of the six cops who were there to keep order began to get smart and cause hostility.\" Leitsch's very next sentence begins: \"A bus driver blew his horn at the meeting, and someone shouted, 'Stop the Bus!' \" which seems to suggest that the decision to occupy Christopher Street may have been instigated by hostility from unsympathetic heterosexuals, but both Rodwell's and Sargeant's accounts carry no such suggestion. The reader needs to remember that Leitsch was writing as the head of Mattachine\u2013New York and, while his account is both an admirable job of reporting and an important historical record, one should expect it to carry somewhat of a bias in favor of the protesters by emphasizing negative behavior on the side of the police and downplaying any such behavior by the demonstrators.\n\n37. Sargeant on WBAI interview late June or early July 1969 and Rodwell-Ardery interview. It is significant that this assertion is made by both individuals independently in two interviews very close to the time of the riots, yet this aspect of the riots has never been commented upon in any of the literature on the Stonewall Riots.\n\n38. In 1990s terminology, this would be the right to gay (or queer) space. This interpretation of the meaning of excluding nongay traffic is the author's own and is not based on interviews with any Stonewall Riots participants.\n\n39. A witness indicates that vehicles of people who showed support for gay people were allowed through.\n\n40. Leitsch, \"Hairpin.\"\n\n41. Rodwell-Duberman interview.\n\n42. Dick Leitsch wrote to the author in an October 30, 2003, e-mail that he had never heard of this purported death and believed it to be urban legend, but Kohler wrote about it in 1970 and in an e-mail to the author, also on October 30, 2003, confirmed that Rodwell had criticized him \"for being insensitive to the cab driver's death by using it in an article,\" confirming that Rodwell also believed the man to have died. The article \"Where Have All the Flowers Gone\" ( _Come Out!_ 1, No. 2, January 10, 1970), about the street youths, reminisced about the riots. In it Kohler wrote: \"I remembered when... Christopher Street was a battleground... and the Cab Driver who had a heart attack when his cab was overturned (in time of peace and war\u2014the man said\u2014many sparrows fall!).\" Like Rodwell, Kohler also recalls the incident as taking place on the night of Saturday, June 28, 1969. Kohler told the author in a phone conversation on the same day that he sent the e-mail that he remembered that the man was very old.\n\n43. In 1969 author Jamake Highwater went by the name J. Marks. He is quoted extensively on Saturday night's events by Leo Laurence in \"Gays Hit N.Y. Cops.\"\n\n44. Unfortunately, Rodwell did not describe the situation in enough detail to make the meaning of this incident clear. Again, since most New Yorkers refer to Christopher Park as Sheridan Square and as a group of policemen had been stationed in front of the Stonewall Inn and in the account just before this Craig is positioned on Christopher Street close to Greenwich Avenue, there can be no doubt that he is talking about Christopher Park, as Sheridan Square is not visible from Christopher Street close to Greenwich Avenue. According to Rodwell, the barrage injured people (at least some of them police, presumably) and made the police very angry.\n\n45. Youth bouncing garbage can lids and shopkeeper scolding the police: Leitsch, _NYMN_ , July 1969.\n\n46. Howard Brown, M.D., _Familiar Faces, Hidden Lives: The Story of Homosexual Men in America Today_ (New York: Harcourt Brace Jovanovich, 1976), pp. 12, 19\u201320.\n\n47. Wilson to Carter in conversation, September 4, 2003; Babick-Scherker interview.\n\n48. Leitsch, \"Hairpin.\"\n\n49. Leitsch, \"Hairpin.\"\n\n50. Highwater (J. Marks) quoted in Laurence, \"Gays Hit N.Y. Cops.\"\n\n51. _NYT_ , June 30, 1969.\n\n52. Leitsch, \"Hairpin.\"\n\n53. _NYT_ , June 30, 1969.\n\n54. Leitsch, \"Hairpin.\"\n\n55. Leitsch, \"Hairpin.\"\n\n56. Leitsch, \"Hairpin.\"\n\n57. Truscott, \"Gay Power.\"\n\n58. Truscott, \"Gay Power,\" and Highwater (J. Marks) in Laurence, \"Gays Hit N.Y. Cops.\"\n\n59. White-Corn letter.\n\n60. Edited together from two _NYMN_ articles. The first and last sentences of this extract are from the August 1969 issue of _NYMN;_ the rest is from the July 1969 issue.\n\n61. Kohler, \"Where Have All the Flowers Gone.\"\n\n62. Note also in this regard that in Leitsch's statement he qualifies his own words by saying \"for the most part\" and \"participated peripherally.\" He is not saying, of course, that all the physical courage and bravery was on the part of the more effeminate members of the crowd but that as a group, they were the most courageous.\n\n63. Although none of the accounts state explicitly that Grove Street, running on the south side of Christopher Park, was shut down between Waverly Place and West 4th Street, it is only logical to assume that it was, given the apparent police objective of controlling the area around the Stonewall Inn, which was the focal point of protests on both Friday and Saturday nights.\n\n64. Truscott, \"Gay Power.\"\n\n65. References to Spencer are from Walter Troy Spencer's \"Last Call\" column in the July 10, 1969, _Village Voice_ , titled \"Too Much My Dear,\" a highly insulting piece of reportage, p. 36.\n\n66. White-Corn letter.\n\n67. Brutality of TPF: _NYMN_ , August 1969.\n\n68. Kevin Liscoe letter to the editor, _Village Voice_ , July 10, 1969, p. 4.\n\n69. Truscott, \"Gay Power.\"\n\n70. Demonstrators on Waverly Place realizing they outnumbered their pursuers: Leitsch, _NYMN_ , July 1969.\n\n71. Jack Nichols and Lige Clarke, \"Homosexual Citizen\" column, _Screw_ , July 25, 1969, p. 16.\n\n72. Three heterosexual couples fleeing TPF with homosexuals: Leitsch, _NYMN_ , July 1969.\n\n73. Number of police officers between two and three hundred: Leitsch, _NYMN_ , July 1969.\n\n74. Bars closing at 3:00 A.M. and patrons coming to Sheridan Square area: Leitsch, _NYMN_ , July 1969.\n\n75. _NYT_ , June 30, 1969.\n\n76. Truscott, \"Gay Power.\"\n\n77. According to the June 30 _NYT_ , the TPF was withdrawn at 4:00 A.M.; Leitsch puts the time of withdrawal at 5:30 A.M. _NYMN_ , July 1969.\n\n78. Leitsch, _NYMN_ , July 1969.\n\n## 11 : \"T H E Y'V E L O S T T H A T W O U N D E D L O O K\"\n\n1. Rodwell-Scherker interview.\n\n2. See chapter 8, introductory unnumbered note.\n\n3. The sign is visible in a photograph taken by Fred McDarrah on Sunday, June 29, 1969. Leitsch told the author that the message was written by the Stonewall's owners, who asked him to okay it, and that he barely glanced at it, telling them that he didn't care if they put it up or not. Such a claim does not seem likely when one compares the literacy and neat lettering of this sign with the crudeness and semiliteracy of the signs put up by the Mafia. Even if the sign were the Mafia's idea, okaying the sign (as Leitsch says he did, however offhandedly), especially with the use of the name of the organization he headed attached to it, Leitsch became obligated to take a large amount of the responsibility for the sign.\n\n4. Crowd turnout Sunday, leather contingent: Leitsch, \"Hairpin.\"\n\n5. Truscott, \"Gay Power.\"\n\n6. All the cops at The Corner: Leitsch, \"Hairpin.\"\n\n7. Truscott, \"Gay Power.\"\n\n8. Laurence, \"Gays Hit N.Y. Cops.\"\n\n9. Leitsch, \"Hairpin.\" Colors of bumper stickers: Tom Burke, \"The New Homosexuality,\" _Esquire_ , December 1969, pp. 178, 304\u201318.\n\n10. Police conversation reportedly overheard by Mattachine\u2013New York member: Leitsch, \"Hairpin.\"\n\n11. Police begging homosexuals to go to the Stonewall Inn: Leitsch, \"Hairpin.\"\n\n12. Nonhelmeted TPF sweep: Truscott, \"Gay Power.\"\n\n13. Mead and Ginsberg in front of United Cigar Store: Truscott-Carter interview.\n\n14. Alison Colbert, \"A Talk with Allen Ginsberg,\" _Partisan Review_ 38, No. 3, 1971, pp. 289\u2013309, reprinted in Allen Ginsberg, _Spontaneous Mind: Selected Interviews 1958\u20131996_ , David Carter, ed. (New York: HarperCollins, 2001).\n\n15. \"To show the colors\": Ginsberg interview by Allen Young, first published in _Gay Sunshine_ , No. 16, January\u2013February 1973, pp. 1, 4\u201310, reprinted several times, including in Ginsberg, _Spontaneous Mind_ (see note 14).\n\n16. Ginsberg walking to Stonewall and dancing: Truscott, \"Gay Power.\"\n\n17. Appearance inside Stonewall: Truscott, \"Gay Power.\"\n\n18. \"After about an hour\": Truscott-Carter.\n\n19. Truscott walking Ginsberg home: Truscott, \"Gay Power.\"\n\n20. Young, _Gay Sunshine_.\n\n21. Ginsberg interview for film _Before Stonewall_ transcript, Lesbian Herstory Archives.\n\n22. Cop at Waverly Place as victim of firecrackers and having his badge stolen: Leitsch, \"Hairpin.\" (The badge was found the following day by the police stuck in a string of pickled pig's feet hanging from a tree in Washington Square Park.) Edward Villella was principal dancer with the New York City Ballet in the 1960s.\n\n23. Burke, \"New Homosexuality.\" The article does not give the date Burke witnessed this encounter, but it is clearly several days into the riots.\n\n24. The date of the _Voice_ pieces can easily lead to confusion about chronology. The edition with Truscott's and Smith's articles is dated July 3, Thursday, but although _The Voice_ was dated Thursday it came out late on Wednesday night (another useful corroboration that events on Wednesday night happened late in the evening, incidentally).\n\n25. I do not think it fair to characterize Smith's phrase \"the gay customers freshly ejected from their hangout, prancing high and jubilant in the street,\" as derogatory. As the reader has seen from the accounts in this volume, there was a great deal of very flamboyant behavior at the time of the riots. This was especially true on the first night at the very time Smith is reporting on with the preceding phrase, i.e., the time when the customers were leaving the Stonewall Inn before the protest began to turn serious. A more objective assessment would be that in this instance Smith probably was quite accurate in his recording of the prevalent tone of the camp protest at that particular moment.\n\n26. Smith, \"Full Moon,\" and Truscott, \"Gay Pride.\"\n\n27. White-Corn letter.\n\n28. Leitsch, _NYMN_ , August 1969.\n\n29. Leitsch, _NYMN_ , August 1969.\n\n30. John Thomas, letter to the editor, _Village Voice_ , July 10, 1969, p. 4.\n\n31. _New York Post_ , July 3, 1969.\n\n32. _NYT_ , July 3, 1969.\n\n33. Leitsch, _NYMN_ , August 1969.\n\n34. Protesters' setting trash on fire at Waverly and Christopher: Di Brienza, \"Stonewall Incident\"; fires set along Christopher Street: _NYT_ , July 3, 1969.\n\n35. The TPF and fire department responding: Di Brienza, \"Stonewall Incident.\" Besides the TPF, the only other police presence reported in the media for Wednesday night is that of the Sixth Precinct. _NYT_ , July 3, 1969.\n\n36. _New York Post_ , July 3, 1969.\n\n37. Di Brienza, \"Stonewall Incident.\"\n\n38. _NYT_ , July 3, 1969.\n\n39. Leitsch, _NYMN_ , August 1969.\n\n40. Di Brienza, \"Stonewall Incident.\"\n\n41. Leitsch, _NYMN_ , August 1969.\n\n42. \"Cop Injured 5 Seized in Village,\" _New York Post_ , July 3, 1969. _NYT_ (July 3, 1969) similarly noted that \"members of the Tactical Patrol Force and police of the Charles Street station [...] were the targets occasionally of bottles and beer cans.\"\n\n43. Di Brienza, \"Stonewall Incident.\"\n\n44. Leitsch, _NYMN_ , August 1969.\n\n45. While the July 3 _NYT_ reported that \"at least four\" were arrested, according to both the letter from Thomas published in the _Voice_ and the July 3 _New York Post_ , five persons were arrested.\n\n46. Leitsch, _NYMN_ , August 1969.\n\n47. Di Brienza, \"Stonewall Incident.\" The July 3, 1969, _New York Post_ account, the only other published account that addresses the duration of Wednesday's rioting, corroborates Di Brienza's statement that Wednesday night's events lasted approximately one hour.\n\n## 12 : S E I Z I N G T H E M O M E N T\n\n1. The account in this chapter of the series of meetings and actions that took place during the month between the Stonewall Riots and the founding of the Gay Liberation Front, unless otherwise noted, is based primarily on Toby Marotta's _The Politics of Homosexuality_ and secondarily on Donn Teal's _The Gay Militants_.\n\n2. While Katzenberg had started to organize a gay Left organization before the riots, the radicalizing effect the street actions had is demonstrated by the difference in tone between the two ads for the same meeting that appeared in _Rat_. The ad placed by Katzenberg and O'Brien before the riots stated: A group of young radical homosexuals will meet this week to develop a critique of heterosexual supremacy, both in society and within the movement. The second ad used Pitts's language: Do you THINK HOMOSEXUALS ARE REVOLTING? YOU BET YOUR SWEET ASS WE ARE. We're going to make a place for ourselves in the revolutionary movement. We challenge the myths that are screwing up this society.\n\n3. _Gay Power_ flyer with date notated, \"7\/1\/69\u20137\/2\/69,\" Leitsch papers.\n\n4. _Homosexual Liberation Meeting_ flyer with date notated \"7\/5\/69\u20137\/6\/69,\" Leitsch papers.\n\n5. _Gay Crusaders_ , pp. 173\u201374; Raul Ramirez, \"I Remember Stonewall,\" _San Francisco Examiner_ , June 4, 1989.\n\n6. Shelley-Scherker interview.\n\n7. Marotta, _Politics of Homosexuality_ , p. 71.\n\n8. _Oh Boy!_ flyer, Leitsch papers; Leitsch-Carter interview; Wicker-Marcus interview; [Lige] Clark and [Jack] Nichols, \"Remember the Stonewall,\" _Gay_ , June 29, 1970; McDarrah and McDarrah, _Gay Pride_ , p. xxxv. While the claim of three days doesn't quite seem to fit with the chronology of the riots and the event at the Electric Circus, these are the youth's words as Wicker remembers them.\n\n9. Shelley-Scherker interview. Bob Kohler heard Shelley proclaiming the name Gay Liberation Front and so has always incorrectly credited her with inventing the name.\n\n10. _Village Voice_ , July 24, 1969, Leitsch papers.\n\n11. \"From Outside the Pen,\" _Rat_ 2, No. 15, [late July] 1969, p. 4; O'Brien-Carter interview.\n\n12. Mattachine\u2013New York flyer, Leitsch papers.\n\n13. Burke, \"New Homosexuality.\"\n\n14. Kimberly Scott, \"Stonewall's Aftermath Spurred D.C. Successes,\" _Washington Blade_ , May 25, 1994.\n\n15. Kameny disputes Rodwell's contention that he used a blow to separate the two women's hands, saying that he simply walked up to the women and passed his hand between them, thereby separating them.\n\n16. Jonathan Black, \"Gay Power Hits Back,\" _Village Voice_ , August 1, 1969, pp. 1, 3; \"Gay Power in New York City,\" quoted in Gay Community News 20, Nos. 1\u20132, June 1994, p.3; O'Brien-Carter interview.\n\n17. Ellipses are in the original article.\n\n18. \"Gay Revolution Comes Out,\" _Rat_ , August 12\u201316, 1969.\n\n## 13 : \"W E'R E T H E G A Y L I B E R A T I O N F R O N T!\"\n\n1. Jonathan Black, \"The Boys in the Snake Pit,\" _Village Voice_ , March 19, 1970, and N. A. Diaman's _Making Gay History_ , an unpublished manuscript. I am indebted to Diaman for letting me quote from _Making Gay History_ for the description of a GLF dance and the GLF meetings where votes were taken over donating money to the Black Panthers.\n\n2. Tobin and Wicker, _Gay Crusaders_ , pp. 29\u201345.\n\n3. Diaman, _Making Gay History_.\n\n4. O'Brien-Carter interview.\n\n5. John Lauritsen, \"The First Gay Liberation Front Demonstration,\" _New York Native_ , June 26, 1989, p. 20.\n\n6. Mike Brown, Michael Tallman, and Leo Louis Martello, \"The Summer of Gay Power and the _Village Voice_ Exposed,\" _Come Out!_ 1, No. 1 November 11, 1969, pp. 10\u201311.\n\n7. On Robinson using the early actions as experiments, see Marotta, _Politics of Homosexuality_ , pp. 135\u201339, and Evans-Carter interview.\n\n8. Teal, _Gay Militants_ , pp. 48\u201350; Tobin and Wicker, _Gay Crusaders_ , p. 168; and _Come Out!_ , November 11, 1969.\n\n9. Rodwell told Duberman in the 1990s that he had the idea for the event on the bus ride back from the Fifth Annual Reminder, but he had told Michael Scherker in 1986 that he had the idea some weeks after his return to New York City. However, on the record \"June 28, 1970: Gay and Proud,\" made for the occasion of the first Christopher Street Liberation Day, Rodwell told interviewer Breck Ardery in 1970 that he and his three friends had brainstormed the entire concept in October of 1969, without saying if or when he himself first had the idea.\n\n10. [Craig Rodwell], \"Gay Holiday,\" _Hymnal_ 2, No. 1, January 1970, p. 8; Rodwell-Scherker interview; Rodwell-Duberman interview; Marotta, _Politics of Homosexuality_ , pp. 164\u201365; and Teal, _Gay Militants_ , p. 300.\n\n11. Marotta, _Politics of Homosexuality_ , p. 90.\n\n12. O'Brien-Carter interview.\n\n13. Teal, _Gay Militants_ , pp. 89\u201390, and Arthur Bell, _Dancing the Gay Lib Blues_ , p. 16.\n\n## 14 : T H E H E R O I C A G E\n\n1. Unless otherwise indicated by a note, this account of the history of the formation of the Gay Activists Alliance and the first six months of its existence is drawn from Bell, _Dancing_ , and Teal, _Gay Militants_.\n\n2. Date of early October is from _GAA Reunion_ 1, No. 6, December 3, 1994, hereafter _Reunion_. Jim and Marty conceived of the organization at 29 Jones Street.\n\n3. GAA 1971 flyer, quoted in _Reunion_.\n\n4. Bell, _Dancing_ , pp. 21\u201323. The text of the preamble is given here exactly as printed in Bell.\n\n5. Bell, _Dancing_ , p. 31, and Souza-Carter interview.\n\n6. \"Lawrence\" is a made-up name for the Snake Pit employee who anonymously gave Arthur Bell an account of the raid and subsequent events at the Sixth Precinct police station in _Gay Power_ , No. 13.\n\n7. This is an inference drawn from a report of the men inside the station calling GAA and another report that Owles and Robinson received a phone call at around that same time notifying them of what was happening, though there are other ways that the news could have reached Robinson and Owles.\n\n8. This account of the Snake Pit raid and the accident Vinales suffered is based on Bell, _Dancing_ ; Teal, _Gay Militants;_ Disk Leitsch, \"The Snake Pit Raid: Some After-thoughts,\" \"The Snake Pit Raid:\" \"500 Angry Homosexuals Protest Raid,\" \"Patrons Tell of Raid from Inside,\" and \"Responsible Cop Cops Out,\" all in _Gay_ 1, No. 10, April 13, 1970; Leo Louis Martello, \"Raid Victim Impaled on Fence,\" _Los Angeles Advocate_ , April 29\u2013May 12, 1970; Arthur Irving (pseud. for Arthur Bell), \"Gay Activists Alliance News and Other Events\" and \"The Morning of the Snake Pit: An Interview,\" _Gay Power_ , No. 13; \"Jumper's Condition Still Grave,\" _New York Post_ , March 10, 1970; and Philip McCarthy, \"Impaled, Saved by Fire 'Docs,' \" _Daily News_ , March 9, 1970. Salmieri's name is misspelled in the article \"500 Angry Homosexuals.\" Thanks to Frank Toscano, formerly of the Sixth Precinct, for verifying the spelling of the captain's name.\n\nFortunately, Vinales lived, despite the seriousness of his injuries.\n\n9. Martello, \"Raid Victim.\"\n\n10. Black, \"Boys in the Snake Pit.\"\n\n11. Scherker-Evans interview.\n\n12. Teal, _Gay Militants;_ Bell, _Dancing;_ \"Gay Activists Alliance News,\" _Gay Power_ , No. 16; and \"GAA Confronts Lindsay at Channel 5,\" _Gay_ 1 No. 14, May 11, 1970.\n\n13. Marotta, _Politics of Homosexuality_ , pp. 166\u201367.\n\n14. Rodwell-Scherker interview.\n\n15. \"Zapping with Carol: Hello Bella,\" _Gay Power_ 1, No. 18. International Gay Information Center Archives, Manuscripts and Archives Division, The New York Public Library, Astor, Lenox and Tilden Foundations.\n\n16. _Gay_ , \"5 Gay Activists Arrested in Sit-in,\" 1, No. 23, July 13, 1970.\n\n17. Franklin Kameny interview for _Stonewall: Myth, Magic and Mobilization_ , Public Radio International, 1994. While Kameny may exaggerate slightly here, consider that John Paul Hudson's _The Gay Insider, USA_ (New York: Stonehill Pub, 1972), published in 1972 (and very likely therefore reflecting the gay world of 1971), takes 627 pages to discuss gay places and groups in America. It lists, for example, merely under the category of gay \"Religious Organizations,\" eighty of them. It also lists 108 gay publications and 386 \"gay liberation\" groups. See also Marotta, _Politics of Homosexuality_ , pp. 322, 324\u201325; Teal, _Gay Militants_ , pp. 196, 205, 310; and D'Emilio, _Sexual Politics_ , p. 238.\n\n18. E-mail from Dick Leitsch to author, October 29, 2003.\n\n19. On the date of the Stonewall's closing, see Carl Lee, \"It's What's Happening,\" p. 2 (this closing date is corroborated by the \"Bar Beat\" column in _Gay Power_ 1, No. 3); Shaheen-Duberman interview.\n\n20. On the march's doubling in size each year, see Craig Rodwell's \"Gay and Free,\" _QQ Magazine_ , December 1971; on gay bars opposing the commemorations of the Stonewall Riots, see Craig Rodwell quoted in Arthur Bell, \"Hostility Comes Out of the Closet,\" _Village Voice_ , June 28, 1973, pp. 1, 16, 18; on Murphy's founding of the Christopher Street Festival Committee, see February 12, 1993, open letter from Randy Wicker to members of Community Board 2; on Murphy's funeral and last place of employment, see \"A Call About Ed Murphy,\" _New York Native_ , February 20, 1989; for Murphy quoted in a prominent gay publication see Glenn Person, \"CS People,\" _Christopher Street_ , Issue 66, pp. 10\u201311; on Wicker's attempt to displace Murphy's associates from the festival, see J. Pederzane, \"Charges of Skimming Roil '92 Christopher Street Festival,\" _NYT_ , June 26, 1992.\n\n21. Rodwell-Scherker interview.\n\n22. Jonathan Black, \"A Happy Birthday for Gay Liberation,\" _Village Voice_ , July 2, 1970, pp. 1, 58.\n\n23. Tobin, _Gay Crusaders_ , p. 134.\n\n24. Bell, _Dancing;_ Teal, _Gay Militants;_ Duberman, _Stonewall;_ Lacey Fosburgh, \"Thousands of Homosexuals Hold a Protest Rally in Central Park,\" _NYT_ , June 29, 1970; Black, \"Happy Birthday\"; Rodwell-Scherker interview; Agostino-Carter interview; and Leo Skir, \"Notes from the Underground\u2014The Road That Is Known,\" _Evergreen Review_ , September 1970, pp. 16, 20, 74\u201377.\n\n## C O N C L U S I O N S\n\n1. I am indebted to John O'Brien for making this point to me.\n\n2. For the effect of high temperatures on behavior, see Emma Brockes and Oliver Burkeman, \"Blame It on the Sunshine,\" _The Guardian_ , May 30, 2001, as well as the report of the National Advisory Commission on Civil Disorders. Note that the fighting at Compton's Cafeteria also occurred during hot weather. Rodwell also cited the weather as a factor: \"I immediately knew that this was the spark we had been waiting for for years. It's hard to explain. It has to do with the weather and the time of the day and the week and who's around and circumstances, whatever.\" Rodwell-Duberman interview.\n\n3. It seems possible that at least one of the other journalists who initially covered the event may have also come from the Lion's Head, for the _Daily News_ account features a photograph by Joseph Ambrosini, a freelance photographer. The picture was taken at the very beginning of the outbreak from _behind_ police lines at a time when the police were trying to keep people away from the Stonewall Inn. Was Ambrosini also in the Lion's Den and did he call the newspaper to alert them to the story?\n\n4. See, e.g., D'Emilio, _Sexual Politics_ , and Marcus, _Making History_. Given the several parallels, it is worth asking whether the interest Bob Kohler took in the homeless gay youths a couple of months before the riots played any part in their willingness to fight.\n\n5. Scherker-Rodwell interview.\n\n6. Spencer, \"Too much my dear.\"\n\n7. \"D.D.'s New York,\" _NYMN_ , August 1969.\n\n8. Burke, \"New Homosexuality.\"\n\n9. Russo, \"I Remember Stonewall.\"\n\n10. Charles Kaiser suggested to the author that Storm\u00e9 DeLarverie (see _The Gay Metropolis: 1940\u20131996_ [Boston: Houghton Mifflin, 1997], p. 198) was this woman, but she could not have been. To cite only a few of the problems with this thesis, DeLarverie's story is one of escaping the police, not of being taken into custody by them, and she has claimed that on that night she was outside the bar, \"quiet, I didn't say a word to anybody, I was just trying to see what was happening,\" when a policeman, without provocation, hit her in the eye (\"Stonewall 1969: A Symposium,\" June 20, 1997, New York City). DeLarverie is also an African-American woman, and all the witnesses interviewed by the author describe the woman as Caucasian. Finally, there has been much speculation about who this woman could have been if a lesbian did play a key role. Storm\u00e9 was well known in the local lesbian community at the time of the riots and has remained so ever since, and it is highly improbable that this woman who was seen by hundreds of people could have been a person of note in the community, else she would have been identified at the time or shortly thereafter.\n\n11. It is remarkable\u2014and no doubt inevitable given human psychology\u2014that in the popular imagination the number of transvestites at the riots is always exaggerated. Readers will note that in the McDarrah photos of the riots there is one transgendered person and none of the persons I interviewed, some of whom knew her, ever saw her actively involved in the riots. (Note that the McDarrah photographs, which do feature the street youths, were taken late on Saturday night during one of the lulls in rioting, when nothing in particular was happening. Truscott-Carter interview.) The Ambrosini photo does not show a single transvestite. Craig Rodwell told researcher Michael Scherker that \"one of the myths about Stonewall is it was all drag queens. I mean, drag queens are part of what went on. Certainly one of the most courageous, but there were maybe twelve drag queens... in thousands of people.\" Scherker-Roadwell interview.\n\n12. Stavis-Murphy interview.\n\n13. Manford 1972.\n\n14. David Isay, _Holding On_ (New York: W. W. Norton, 1996); letter from Nancy LaMarr, _Transgender Tapestry_ , issue 80; Fall 1997, pp. 9\u201310.\n\n15. \"Testimony of Dick Leitsch\/Executive Director Mattachine Society Inc. of New York\/ Hearings on Homosexuality\/New York State Assembly\/January 7, 1971,\" p. 8, Leitsch papers. The same news was noted in two gay publications in March 1970: \"Wall Street Purge Causes Job Losses,\" _Gay_ 1, No. 8, March 15, 1970, p. 3, and \"Wall Street Purge,\" _NYMN_ , March 1970, pp. 7\u20138. The latter article states that a \"rash of thefts of negotiated stocks and bonds from banks and securities brokers\" led Attorney General Louis Lefkowitz to decree that \"all employees in the financial district must be fingerprinted. This was because of a reputed Mafia connection with the thefts.\" The mention of a \"Mafia connection\" is further corroboration of Pine. See also \"361 Arrest Records Uncovered by Wall Street Fingerprinting,\" _NYT_ , February 5, 1970.\n\n16. This history is given in the PBS series _New York City: A Documentary History_.\n\n17. Gene Heil, \"Where It Was,\" _Where It's At_ , July 15, 1974, pp. 30\u201331. International Gay Information Center Archives, Manuscripts and Archives Division, The New York Public Library, Astor, Lenox and Tilden Foundations.\n\n18. Morty Manford, letter to the editor, _Where It's At_ , July 21, 1974, Morty Manford Papers, Manuscripts and Archives Division, The New York Public Library, Astor, Lenox and Tilden Foundations.\n\n# B I B L I O G R A P H Y\n\nAdair, Nancy, and Casey Adair, _Word Is Out: Stories of Some of Our Lives_. San Francisco: New Glide Publications, 1978.\n\n_Advocate_. \"Stonewall 'Historic'?\" (August 15, 1973): 16.\n\nAlwood, Edward _. Straight News: Gays, Lesbians, and the News Media_. New York: Columbia University Press, 1996.\n\nBeard, Harry. \"I Was There... the Stonewall Riot.\" _New York Gay Press_ (1:1; July 1980): 5.\n\nBeard, Rick, and Leslie Cohen Berlowitz, editors, _Greenwich Village: Culture and Counterculture_. New Brunswick: Rutgers University Press, 1993.\n\nBell, Arthur. _Dancing the Gay Lib Blues: A Year in the Homosexual Liberation Movement_. New York: Simon and Schuster, 1971.\n\n___ [pseudonym Arthur Irving]. \"Gay Activists Alliance News and Other Events.\" _Gay Power_ (No. 13): 4.\n\n___ [pseudonym Arthur Irving]. \"The Morning of the Snake Pit: An Interview.\" _Gay Power_ (No. 13): 5.\n\n___ [pseudonym Arthur Irving]. \"Gay Activists Alliance News.\" _Gay Power_ (No. 16).\n\n___ [pseudonym Arthur Irving]. \"Zapping with Carol: Hello, Bella.\" _Gay Power_ (No. 18): 4.\n\n___. \"Skull Murphy: The Gay Double Agent.\" _Village Voice_ (May 8, 1978): 1, 17\u201319.\n\n_Berkeley Barb_. \"Gay Rebel Gets Shafted by Uptight Boss\" (April 11, 1969): 11.\n\n___. \"Pink Panthers Gay Revolution Toughening Up\" (April 18\u201324, 1969): 11.\n\n___. \"Gay Strike Hits Southern Front\" (May 2\u20138, 1969): 11.\n\n___. \"Gay Strike Turns Grim\" (April 25\u2013May 1, 1969): 7.\n\n___. \"Group Will Act\" (May 2\u20138, 1969): 11.\n\nBird, David. \"Trees in a Queens Park Cut Down As Vigilantes Harass Homosexuals.\" _New York Times_ (July 1, 1969).\n\nBlack, Jonathan. \"Gay Power Hits Back.\" _Village Voice_ (July 31, 1969): 1, 3.\n\n___. \"The Boys in the Snake Pit: Games 'Straights' Play.\" _Village Voice_ (March 19, 1970): 1, 61\u201364.\n\n___. \"A Happy Birthday for Gay Liberation.\" _Village Voice_ (July 2, 1970): 1, 58.\n\nBragg, Rick. \"From Night of Rage, Seeds of Liberation.\" _New York Times_ (June 23, 1994).\n\nBrockes, Emma, and Oliver Burkeman. \"Blame It on the Sunshine.\" _Guardian_ (May 30, 2001).\n\nBrown, Howard, M.D. _Familiar Faces, Hidden Lives: The Story of Homosexual Men in America Today_. New York: Harcourt Brace Jovanovich, 1976.\n\nBrown, Mike, Michael Tallman, and Leo Louis Martello. \"The Summer of Gay Power and the _Village Voice_ Exposed!\" _Come Out!_ (1:1; November 14, 1969): 10\u201311.\n\nBurke, Tom. \"The New Homosexuality.\" _Esquire_ (December 1969): 178, 304\u201318.\n\nChari, A. C. S., compiler and publisher. _84 Questions and Answers on Avatar Meher Baba_. India: 1969.\n\nChauncey, George. _Gay New York: Gender, Urban Culture, and the Making of the Gay Male World, 1890\u20131940_. New York: Basic Books, 1994.\n\n_Chicago Sun-Times_. \"Grab City Cop as Sex-Blackmail Kingpin\" (June 25, 1966).\n\n[Clark,] Lige, and Jack [Nichols]. \"New York Notes.\" _Los Angeles Advocate_ (April 1969): 22.\n\n___. \"Pampered Perverts.\" _Screw_ (July 25, 1969): 16. (The title of the article, \"Pampered Perverts,\" was chosen by Al Goldstein, the publisher of _Screw_. This article was printed in the Homosexual Citizen column of _Screw_ , a forum created by Goldstein. While Nichols and Clark were the sole authors of the articles for this column, they had no control over the title Goldstein assigned to each issue's column.)\n\n___. \"Remember the Stonewall!\" _Gay_ (No. 21; June 29, 1970): 5.\n\nColbert, Alison. \"A Talk with Allen Ginsberg.\" _Partisan_ _Review_ (38:3; 1971): 289\u2013309; reprinted in Allen Ginsberg, _Spontaneous Mind: Selected Interviews 1958\u20131996_ , edited by David Carter.\n\nCrosby, Tina. \"The Stonewall Riot Remembered.\" Lesbian Herstory Archives of the Lesbian Herstory Educational Foundation, Inc.\n\n_Cruise News and World Report_. \"Young Rejects Form Own Organization\" (2:7; July 1966): 1.\n\n___. \"Young Homos Picket Compton's Restaurant _\"_ (2:8; August 1966): 1 and continued.\n\nCunningham, Barry. _Gay Power: The Homosexual Revolt_. New York: Tower Publications, 1971.\n\n_Daily News_ \"Gets 5 Years in Extortion of Homos\" (August 17, 1966).\n\n___. \"Lands at JFK, FBI Is There\" (September 18, 1966).\n\n___ (Sunday Edition). \"3 Cops Hurt As Bar Raid Riles Crowd\" (June 29, 1969).\n\nD'Arcangelo, Angelo (Josef Busch). _The Homosexual Handbook_. New York: Ophelia Press, 1968.\n\nDavy, Kitty. \"Baba's First World Tour, 1932, Part II.\" _Awakener_ (12:3; Summer 1968): 1\u201321.\n\nD'Emilio, John. _Sexual Politics, Sexual Communities: The Making of a Homosexual Minority in the United States_. Chicago: University of Chicago Press, 1983.\n\nDi Brienza, Ronnie. \"Stonewall Incident.\" _East Village Other_ (4:32; July 9, 1969): 2.\n\nDolan, Bob. \"When the Law Reaches into Your Pocket.\" _Pageant_ 23:2; (August 1967): 40-46.\n\nDuberman, Martin. _Stonewall_. New York: Dutton, 1993.\n\nEskow, Dennis. \"3 Cops Hurt As Bar Raid Riles Crowd.\" _New York Daily News_ (June 29, 1969).\n\nEskridge, William N., Jr. _Gay Law: Challenging the Apartheid of the Closet_. Cambridge: Harvard University Press, 1999.\n\nEsselourne, Frank. \"Doorman Remembers.\" _Gay Community News (_ June 23, 1979): 10.\n\nFederation to Preserve the Greenwich Village Waterfront and Great Port. www.villagewaterfront.org.\n\nFisher, Peter _. The Gay Mystique: The Myth and Reality of Male Homosexuality_. New York: Stein and Day, 1972.\n\nFosburgh, Lacey. \"Thousands of Homosexuals Hold a Protest Rally in Central Park.\" _New York Times_ (June 29, 1970).\n\n_GAA Reunion_ (1:6; December 3, 1994): 4.\n\n_Gay_. \"Wall Street Purge Causes Job Losses\" (No. 8; March 15, 1970): 3.\n\n___. \"The Snake Pit Raid\" (1:10; April 13, 1970): 2.\n\n___. \"500 Angry Homosexuals Protest Raid\" (1:10; April 13, 1970): 3.\n\n___. \"Patrons Tell of Raid from Inside\" (1:10; April 13, 1970): 3.\n\n___. \"Responsible Cop Cops Out\" (1:10; April 13, 1970): 10.\n\n___. \"GAA Confronts Lindsay at Channel 5\" (No. 14; May 11, 1970): 3.\n\n___. \"5 Gay Activists Arrested in Sit-In\" (No. 23; July 13, 1970): 3.\n\n_Gay Activist_. \"The Night They Raided Stonewall\" (1:3; June 1971): 4.\n\n_Gay Power_. \"Bar Beat\" (1:3): 7, 18.\n\nGinsberg, Allen. Interview by Allen Young. _Gay Sunshine_ (No. 16; January\u2013February 1973): 1, 4\u201310.\n\n___. _Spontaneous Mind: Selected Interviews 1958\u20131996_ , edited by David Carter. New York: HarperCollins, 2001.\n\n_The Greenwich Village Historic District Designation Report_ , 2 volumes. New York: New York City Landmarks Preservation Commission, 1969.\n\nGrutzner, Charles. \"Mafia Buys Clubs for Homosexuals.\" _New York Times_ (November 30, 1967).\n\nHaines, Steve. \"Killer Cops at Large.\" _Berkeley Barb_ (April 25\u2013May 1, 1969): 7.\n\nHarrington, Stephanie. \"Public Meeting on MacD.: Pirandello Writes Script,\" _Village Voice_ (April 7, 1966): 1, 24\u201325, 28.\n\nHeide, Robert, and John Gilman. _Greenwich Village: A Primo Guide to Shopping, Eating, and Making Merry in True Bohemia_. New York: St. Martin's Griffin, 1995.\n\nHeil, Gene. \"Where It Was.\" _Where It's At_ (July 15, 1974): 30\u201331.\n\n_Herald Tribune_. \"17 Indicted in Hush-Money Shakedown\" (1966: undated clipping): n.p.\n\n\"History of Christopher Street: How It All Began!\" [San Francisco] _Gay Pride_ [program] (June 25, 1972).\n\nHoglund, John. \"Eddie Murphy\u2014From Hellion to Hero.\" Part 1. _Private Lives_ (2:6; December 1987): 30, 32.\n\n___. \"Obituaries: Edward F. Murphy.\" _New York Native_ (March 27, 1989): 9.\n\nHudson, John Paul. _The Gay Insider, USA_. New York: Stonehill Pub., 1972.\n\nHuff, Bobbie. \"N.Y. Po-Lice Scream at Queens.\" _Magpie_ (August 1969): 14.\n\nIsay, David. _Holding On_. New York: W. W. Norton, 1996.\n\nJackson, Don. \"Reflections on the N.Y. Riots.\" _Los Angeles Advocate_ (October 1969): 11.\n\nJackson, Kenneth T., editor. _The Encyclopedia of New York City_. New Haven: Yale University Press; New York: New-York Historical Society, 1995.\n\nJay, Karla, and Allen Young, editors. _Out of the Closets: Voices of Gay Liberation_. New York: Douglas Book Corp., 1972.\n\nJohnson, David K. \"Homosexual Citizens: Washington's Gay Community Confronts the Civil Service.\" _Washington History_ (Fall\/Winter 1994\u201395): 44\u201363.\n\nKaiser, Charles. _The Gay Metropolis: 1940\u20131996_. Boston: Houghton Mifflin, 1997.\n\nKarlen, Arno. _Sexuality and Homosexuality: A New View_. New York: W. W. Norton, 1971.\n\nKatz, Jonathan. _Gay American History: Lesbians and Gay Men in the U.S.A.: A Documentary_. New York: Thomas Y. Crowell, 1976.\n\n___. _Gay\/Lesbian Almanac: A New Documentary_. New York: Harper and Row, 1983.\n\nKatz, Leonard, and Norman Poirier. \"Blackmailer of Deviates Gets 5 Years.\" _New York Post_ (August 16, 1966).\n\nKepner, Jim. \"Our Movement Before Stonewall\" (booklet printed by Kepner). October 1989; revised 1994.\n\nKnight, Hans. \"'Other Society' Moves into the Open.\" [Philadelphia] _Sunday Bulletin_ (July 19, 1970).\n\nKohler, Bob. \"Where Have All the Flowers Gone.\" _Come Out!_ (1:2; January 10, 1970): 14.\n\n_The Ladder_. \"Gay Power in New York City.\" (October\u2013November 1969): 40.\n\nLaMarr, Nancy. \"Transcriptions.\" _Transgender Tapestry_ (Issue 80; Fall 1997): 9\u201310.\n\nLaurence, Leo E. \"Gay Revolution.\" _Vector_ (April 1969): 11, 25.\n\n___. \"Gays Get Tougher.\" _Berkeley Barb_ (May 23\u201329, 1969): 31.\n\n___. \"Gays Hit N.Y. Cops.\" _Berkeley Barb_ (July 4\u201310, 1969): 5.\n\nLauritsen, John. \"The First Gay Liberation Front Demonstration.\" _New York Native_ (June 26, 1989): 20.\n\nLee, Carl. \"It's What's Happening.\" _New York Hymnal_ (1:5; June\u2013July 1968): 2.\n\n___. \"It's What's Happening.\" _New York Hymnal_ (1:6; August\u2013September 1968): 2.\n\n___. \"It's What's Happening.\" _New York Hymnal_ (2:1; January 1970): 2.\n\nLeitsch, Dick. \"Cop Kills Two\u2014May Go Free.\" _New York Mattachine Newsletter_ (March 1969): 1\u20132.\n\n___. \"Coming Mayoralty Contest.\" _New York_ _Mattachine Newsletter_ (May 1969): 1\u20132. (Not signed.)\n\n___. \"The Hairpin Drop Heard Around the World.\" _New York Mattachine Newsletter_ (July 1969): 21\u201323.\n\n___. \"Gay Riots in the Village.\" _New York Mattachine Newsletter_ (August 1969): 1\u20133.\n\n___. \"The Stonewall Riots[:] The Gay View.\" _New York Mattachine Newsletter_ (August 1969): 13\u201314.\n\n___. \"The Stonewall Riots[:] The Police Story.\" _New York Mattachine Newsletter_ (August 1969): 5\u20136.\n\n___. \"Police Raid on N.Y. Club Sets Off First Gay Riot.\" _Los Angeles Advocate_ (September 1969): 3, 11.\n\n___. \"The Snake Pit Raid: Some Afterthoughts.\" _Gay_ (1:10; April 13, 1970): 13.\n\n___. \"Politicians Make Strange Bedfellows!\" _Gay (_ May 25, 1970).\n\nLiscoe, Kevin. \"Letters to the Editor.\" _Village Voice_ (July 10, 1969): 4.\n\nLong, Mike. \"Stonewall: The Night the Girls Said No!\" _San Francisco Sentinel_ (June 22, 1989): 3\u20134.\n\nLustbader, Ken. \"Landscape of Liberation: Preserving Gay and Lesbian History in Greenwich Village.\" Master's thesis, Columbia University, 1993.\n\nManford, Morty. \"A Comparative Study of Gay Liberationists and Columbia Homosexuals.\" Student paper, Columbia University, 1972.\n\n___. \"The New York Gay Political Movement, 1978: An Organizational and Strategy Critique.\" Student paper, Columbia University, 1978.\n\n___. \"Why We Should Postpone New York's Gay Rights Bill: A New Look at Gay Political Strategy.\" _Gaysweek_ (October 16, 1978): 10\u201316.\n\n\"Marat, Jean-Paul.\" \"On Prejudice.\" _V_ (1:2; October 1966): 1.\n\nMarcus, Eric. _Making History: The Struggle for Gay and Lesbian Equal Rights, 1945_ \u2013 _1990: An Oral History_. New York: HarperCollins, 1992.\n\nMarie, Barbara Judith. \"Stonewall Remembered Book 1,\" www.inch.com\/~kdka\/stonewall.htm.\n\nMarotta, Toby. _The Politics of Homosexuality_. Boston: Houghton-Mifflin, 1981.\n\nMartello, Leo Louis. \"Raid Victim Impaled on Fence.\" _Los Angeles Advocate_ (4:5, April 29\u2013May 12, 1970): 1, 6.\n\nMcCarthy, Philip. \"Impaled, Saved by Fire 'Docs'.\" _Daily font_ (March 9, 1970).\n\nMcDarrah, Fred W., and Timothy S. McDarrah. _Gay Pride: Photographs from Stonewall to Today_. New York: A Cappella Books, 1994.\n\nMcGarry, Molly, and Fred Wasserman. _Becoming Visible: An Illustrated History of Twentieth-Century Gay Life in America_. New York: New York Public Library, 1998.\n\nMcGowan, William. \"Before Stonewall.\" _Wall Street Journal_ (June 16, 2000).\n\nMiles, Barry. \"The Beat Generation in the Village.\" In Beard and Berlowitz, _Greenwich Village: Culture and Counterculture_. New Brunswick: Rutgers University Press, 1993.\n\nMiller, Terry. _Greenwich Village and How It Got That Way_. New York: Crown, 1990.\n\nMorgan, Bill. _The Beat Generation in New York: A Walking Tour of Jack Kerouac's City_. San Francisco: City Lights, 1997.\n\nMorty Manford Papers, Manuscripts and Archives Division, The New York Public Library, Astor, Lenox, and Tilden Foundation.\n\nMurphy, Ed. \"1969 at Stonewall.\" _Equal Times_ , 1989 Gay Pride Commemorative Special Edition (1:10; June 1, 1989): 19.\n\n_National_ _Star Chronicle_. \"The World of the Homosexual.\" (undated clipping: third in a series of articles that began with 5:1; September 27, 1965).\n\nNewman, Andy. \"Death of a Drag Queen.\" _The Hoboken Reporter_ (July 19, 1992).\n\n_Newsweek_. \"Minority Listening\" (July 30, 1962): 48.\n\n___. \"Policing the Third Sex\" (October 27, 1969): 81.\n\n_New York City Gay Scene Guide_ , vol. 1, no. 2. New York: Apollo Book Company, 1969.\n\n_New York Hymnal_. \"Signs of the Times\" (1:1; February 1968): 5.\n\n___. \"McCarthy Wins Poll\" (1:4; May 1968): 1,3.\n\n___. \"On Stage\" (1:5; June\u2013July 1968): 5\u20136.\n\n___. \"Gay Power Gains\" (1:6; August\u2013September 1968): 1, 9.\n\n_New York Journal-American_. \"Jack O'Brian Says\" (July 9, 1962).\n\n_New York Mattachine Newsletter_. \"Meeting on Village Cleanup\" (11:3; April 1966): 1\u20133.\n\n___. \" 'Drag' and the Laws\u2014and a Drag Ball\" (January\u2013February 1967): 5\u20137.\n\n___. \"Gay Bar Closed\" (March 1968): 5.\n\n___. \"More Police Harassment\" (February 1969): 14.\n\n___. \"Transit Cops Shake Down Wrong Man\" (February 1969): 14.\n\n___. \"Docks, Darkness, and Danger\" (March 1969): 3\u20134.\n\n___. \"Bathhouse Raided\" (April 1969): 1\u20132.\n\n___. \"Dock Killings\" (April 1969): 5.\n\n___. \"Your Friendly SLA\" (April 1969): 16.\n\n___. \"D.D.'s New York\" (August 1969): 18\u201319.\n\n_New York Native_. \"A Call About Ed Murphy\" (February 20, 1989): 10.\n\n_New York Post_. \"Cop Injured 5 Seized in Village\" (July 3, 1969).\n\n___. \"The Gay Anger Behind the Riots\" (July 8, 1969).\n\n___. \"Jumper's Condition Still Grave\" (March 10, 1970).\n\n___. \"No Place for Gaiety\" (June 28, 1969).\n\n_New York Times_. \"Police Are Added in Washington Sq.\" (October 5, 1964).\n\n___. \"'Village' Assured of Added Police\" (October 7, 1964).\n\n___. \"Detective at Hotel Is Held in Extortion\" (August 5, 1965).\n\n___. \"Garelick Urges Public to Report Police Trapping of Homosexuals\" (April 2, 1966).\n\n___. \"3 Deviates Invite Exclusion by Bars\" (April 22, 1966).\n\n___. \"S.L.A. Won't Act Against Bars Refusing Service to Deviates\" (April 26, 1966).\n\n___. \"Lindsay Placates Coffeehouse Set\" (May 3, 1966).\n\n___. \"3 Indicted Here as Sex Extorters\" (June 1, 1966).\n\n___. \"Detective Accused as a Top Extorter\" (July 1, 1966).\n\n___. \"Blackmailer Gets Five-Year Sentence in Homosexual Case\" (August 17, 1966).\n\n___. \"Blackmail Paid by Congressman\" (May 17, 1967).\n\n___. \"Ex-F.B.I. Agent Links State Liquor Aide to Mafia\" (November 30, 1967).\n\n___. \"2 Found Guilty in Chicago in Extortion of Homosexual\" (December 9, 1967).\n\n___. \"4 Policemen Hurt in 'Village' Raid\" (June 29, 1969).\n\n___. \"Police Again Rout 'Village' Youths\" (June 30, 1969).\n\n___. \"361 Arrest Records Uncovered by Wall Street Fingerprinting\" (February 5, 1970).\n\n___. \"Ku Klux Klan Says It Will Seek Permission for a Masked Rally in City\" (May 2, 2000).\n\nPederzane, J. \"Charges of Skimming Roil '92 Christopher Street Festival.\" _New York Times_ (June 26, 1992).\n\nPerson, Glenn. \"CS People: Ed Murphy.\" _Christopher Street_ (Issue 66; July 1982): 10.\n\n_Queen's Quarterly_. \"Dawn Is Just Breaking...\" (1:3; Summer 1969): 5.\n\nRamirez, Raul. \"I Remember Stonewall.\" _San Francisco Examiner_ (June 4, 1989).\n\nRanzal, Edward. \"Member of 70-Man Ring Preying on Homosexuals Given 5 Years.\" _New York Times (_ July 12, 1967).\n\n_Rat_. \"Queen Power: Fags Against Pigs in Stonewall Bust\" (2:14; July 9\u201323, 1969): 6.\n\n___. \"From Outside the Pen\" (2:15; \"late July 1969\"): 4.\n\n___. \"Gay Revolution Comes Out\" (August 12\u201316, 1969).\n\n_The Realist_. August 1962, September 1962, October 1962.\n\nReid, Lori. _The Complete Book of Chinese Horoscopes_. Shaftesbury, Dorset; Rockport, Mass.: Barnes and Noble Books, 1997.\n\nRodwell, Craig. \"Mafia on the Spot.\" _New York Hymnal_ (1:1; February 1968): 1\u20132. (Not signed.)\n\n___. \"Gay Power Gains.\" _New York Hymnal_ (1:6; August\u2013September 1968): 1, 9.\n\n___. \"Gay Is Good.\" _Queen's Quarterly_ (Summer 1969): 39.\n\n___. \"Gay Holiday.\" _New York Hymnal_ (2:1, January 1970): 8. (Not signed.)\n\n___. \"Gay and Free.\" _QQ Magazine_ (3:6; December 1971): 22\u201327.\n\nRosen, Steven A. \"Police Harassment of Homosexual Women and Men in New York City 1960\u20131980.\" _Columbia Human Rights Law Review_ (12; 1980\u20131981): 159\u2013190.\n\nRoth, Jack. \"Nine Seized Here in Extortion Ring.\" _New York Times_ (February 18, 1966).\n\n___. \"Nationwide Ring Preying on Prominent Deviates.\" _New York Times_ (March 3, 1966).\n\nRusso, Vito. \"I Remember Stonewall.\" _Soho Weekly News_ (6:38; June 21\u201327, 1979): 12\u201313, 22. (The same article was published, slightly edited, as \"Still Outlaws.\" _Gay News_ [No. 170]: 16\u201317.)\n\nSante, Luc. _Low Life: Lures and Snares of Old New York_. New York: Farrar Straus Giroux, 1991.\n\nSchlager, Neil, editor. _Gay and Lesbian Almanac_. New York: St. James Press, 1998.\n\nScott, David C. \"My Autobiography.\" Student paper, New College of California, 1983.\n\nScott, Kimberly. \"Stonewall's Aftermath Spurred D.C. Successes.\" _Washington Blade_ (May 25, 1994).\n\nSears, James T. _Lonely Hunters: An Oral History of Lesbian and Gay Southern Life, 1948_ \u2013 _1968_. Boulder, Colo.: Westview Press, 1997.\n\nShilts, Randy. _The Mayor of Castro Street: The Life and Times of Harvey Milk_. New York: St. Martin's Press, 1982.\n\nSkir, Leo. \"Notes from the Underground\u2014The Road That Is Known.\" _Evergreen Review_ (September 1970): 16, 20, 74\u201377.\n\nSmilon, Marvin. \"More Indictments Due in Blackmail Case.\" _New York Post_ (September 28, 1966).\n\nSmith, Howard. \"Full Moon over the Stonewall.\" _Village Voice_ (July 3, 1969): 1, 25.\n\nSpencer, Walter Troy. \"Too Much, My Dear.\" _Village Voice_ (July 10, 1969): 36.\n\nTate, Laurence. \"Exiles of Sin, Incorporated.\" _Berkeley Barb_ (November 18, 1966): 5.\n\nTeal, Donn. _The_ _Gay Militants_. New York: Stein and Day, 1971.\n\nThomas, John. \"Letters to the Editor.\" _Village Voice_ (July 10, 1969): 4.\n\nThompson, Mark, editor. _Long Road to Freedom: The_ Advocate _History of the Gay and Lesbian Movement_. New York: St. Martin's Press, 1994.\n\nTilchen, Maida. \"Mythologizing Stonewall.\" _Gay Community News_ (June 23, 1979): 16.\n\nTimmons, Stuart. _The Trouble with Harry Hay_. Boston: Alyson, 1990.\n\nTobin, Kay, and Randy Wicker. _The Gay Crusaders_. New York: Paperback Library, 1972.\n\nTree. \"Beating Around the Bush.\" _Private Lives_ (September 1991): 86\u201388.\n\n___. \"Beating Around the Bush.\" _Private Lives_ (February 1992): 74\u201378.\n\n___. \"Beating Around the Bush.\" _Private Lives_ (January 1993): 100.\n\n___. \"Beating Around the Bush.\" _Private Lives_ (August 1993): 86\u201388.\n\n___. \"Beating Around the Bush.\" _Private Lives_ (December 1993): 84\u201385.\n\nTruscott, Lucian. \"Gay Power Comes to Sheridan Square.\" _Village Voice_ (July 3, 1969): 1, 18.\n\nVoeller, Bruce. \"Stonewall Anniversary: Assessing the Activist Years.\" _Advocate_ (July 12, 1979): 30.\n\nWakefield, Dan. \"The Gay Crusader.\" _Nugget_ (June 1963): 51\u201352, 71\u201372.\n\nWare, Caroline F. _Greenwich Village, 1920\u20131930: A Comment on American Civilization in the Post-War Years_. Boston: Houghton-Mifflin, 1935; reprint Berkeley: University of California Press, 1994.\n\nWatson, Steven. \"The Drag of Politics.\" _Village Voice_ (June 25, 1979): 72\u201373.\n\nWeinberg, Martin S., and Colin J. Williams. _Male Homosexuals: Their Problems and Adaptations_. New York: Oxford University Press, 1974.\n\nWeiss, Murray. \"J. Edgar's Slip Was Showing _.\" New York Post_ (February 11, 1993).\n\nWhite, Edmund. \"Letter to Ann and Alfred Corn.\" In David Bergman, editor, _The Violet Quill Reader: The Emergence of Gay Writing After Stonewall_ , 1\u20134. New York: St. Martin's Press, 1994.\n\nWicker, Randy. \"The Wicker Report\u2014It's Koch vs. DeSapio Again.\" _Eastern Mattachine Magazine_ (July 1965): 9\u201310.\n\nWilde, Oscar. _The Complete Letters of Oscar Wilde_ , edited by Merlin Holland and Rupert Hart-Davis. New York: Henry Holt, 2000.\n\nWittman, Carl. \"A Gay Manifesto.\" In Jay and Young, _Out of the Closets: Voices of Gay Liberation_ , 330\u2013342.\n\n# O R A L H I S T O R I E S\n\nThe following interviews were conducted by the author, unless a second name appears, indicating a different interviewer. The use of quotation marks around a name indicates a pseudonym.\n\nAgostino, Sam: Feb. 26, 1998.\n\nBabick, Christopher\u2014by Michael Scherker: Dec. 10, 1988.\n\nBeard, Harry\u2014by Michael Scherker: Aug. 3, 1988.\n\nBecker, Marle: Jul. 30, 1999.\n\nBockman, Philip: Aug. 14, 1999; July 13, 2001.\n\nBoyce, Martin: Sept. 17, 1998.\n\n\"Bruce\": Sept. 16, 1997.\n\nBryan, Robert: Sept. 3, 1999.\n\nBurch, Charles: Apr. 10, 1998.\n\nCallahan, Tim: Apr. 5, 2001.\n\nCastro, Raymond: Oct. 10, 1999.\n\nDunn, Kevin\u2014by Michael Scherker: June 5, 1989.\n\nEagles, Philip: Oct. 1, 1998; Aug. 21, 1999.\n\nEvans, Arthur: Dec. 16, 17, 2000; Oct. 13, 2002. Also interviewed by Michael Scherker: June 8, 1989.\n\nFader, Michael: Apr. 8, 1998.\n\nGabriel, Paul: Dec. 17, 2000.\n\nGalvin, Earl\u2014by Nikos Diaman: Dec. 7, 1992.\n\nGarvin, Danny: Feb. 12, Mar. 14, May 27, 1998; Sept. 29, Nov. 30, 1999. Also interviewed by Michael Scherker: Dec. 15, 1988.\n\nGinsberg, Allen\u2014by Obie Benz: Oct. 17, 1987.\n\nGittings, Barbara: Dec. 15, 2000. Also interviewed by Paul D. Cain: n.d.\n\nHampton, Dawn\u2014by Eric Marcus: Jan. 1994.\n\n\"Hardy,\" Jennifer: Sept. 1, 1997.\n\nHeide, Robert: Aug. 19, Sept. 21, 1999.\n\nHoose, Jerry: Mar. 5, 17, 1998; Aug. 23, 2001; Oct. 16, 17, 25, 2003.\n\nJoel S.: Jan. 5, 1998.\n\nKohler, Bob: Oct. 7, 17, 1997; Mar. 12, 24, 28, 1998; Apr. 5, 2003.\n\nKonnon, Michael: May 5, 1998.\n\nLanigan-Schmidt, Thomas: Mar. 13, 21, Dec. 12, 1998; Sept. 30, 2000; May 6, 2001; Jul. 21, 22, 2002; Aug. 4, 9, Oct. 18, 2003. Also interviewed by Martin Duberman: May 2, 1992.\n\nLeitsch, Dick: Dec. 9, 1997; Mar. 2, 1999; Jul. 21, 2001. Also interviewed by Eric Marcus: Jan. 23, 1989; \u2014by Eric Marcus: Jan. 1994; and by Paul Cain: Aug. 16, 1995.\n\nManford, Morty: Dec. 9, 1989\u2014by Eric Marcus.\n\n\"Mike,\" Stonewall Inn's porter: Mar. 2, 21, 1998.\n\nMurphy, Ed\u2014by Chris Davis: circa 1983; and by Gene Stavis: May 22, 1978.\n\nNichols, Jack: Oct. 4, 7, 8, 1996.\n\nO'Brien, John: Mar. 21, 1997; June 23, 30, Jul. 4, 1998.\n\nOlenick, Michael: Oct. 30, 1999.\n\nPerrin, Barry: Jul. 14, 1999.\n\nPine, Seymour: Sept. 18, Oct. 26, 28, Nov. 14, 1999; Aug. 12, 2000. Also interviewed by David Isay: ca. spring 1989.\n\nRanieri, John Paul: Mar. 17, 24, 30, 1998.\n\nRitter, \"Steve\"\/\"Maria\": Nov. 9, 1997; Apr. 19, 2000.\n\nRivera, Robert\u2014by Michael Scherker: June 10, 1989.\n\nRodwell, Craig\u2014by Breck Ardery: ca. spring 1970; by Michael Scherker: Oct. 9, Nov. 20, 1986; and by Martin Duberman: 1990, 1990, Nov. 14, 1990; and Jan. 25, 1991.\n\nRusso, Vito\u2014by Eric Marcus: Dec. 21, 1988.\n\nScott, David C.: Apr. 24, 1998; Jan. 26, 1999. Also interviewed by Michael Scherker: June 10, 1989.\n\nShaheen, Chuck\u2014by Martin Duberman: Nov. 20, 1991.\n\nShelley, Martha: Dec. 14, 2000. Also interviewed by Michael Scherker: June 5, 1989.\n\nShenton, Bruce: Aug. 31, 1999.\n\nSmith, Howard: Feb. 2, 1998.\n\nStryker, Susan: Dec. 15, 2000.\n\nTish: Jan. 15, Mar. 10, 1998.\n\nTree: Sept. 16, 1997.\n\nTruscott, Lucian K.: Mar. 17, 1997.\n\nvan Cline, Stephen: Apr. 21, 1996; Apr. 18, 1997.\n\nVan Ronk, Dave: Sept. 8, 1997.\n\nWhite, Edmund: Sept. 10, 1997.\n\nWicker, Randy: Nov. 13, 1997; Jul. 6, 2001. Also interviewed by Michael Scherker: Aug. 8, 1988.\n\nWilson, Doric: Aug. 22, Sept. 4, 1999.\n\nWynkoop, William: Nov. 29, 1997.\n\nYates, Steve: Oct. 8, 1996.\n\n# C R E D I T S F O R T H E  \nP H O T O G R A P H S\n\nThe author gratefully acknowledges the following sources for the material reprinted in the photo section:\n\nPhoto Section pages:\n\n2 (top) reprinted courtesy of Milstein Division of United States History, Local History & Genealogy, The New York Public Library, Astor, Lenox and Tilden Foundations.\n\n2 (bottom) reprinted courtesy of Tree.\n\n3 (top), 5 (middle) reprinted courtesy of Fred W. McDarrah.\n\n3 (bottom), 7 (bottom) reprinted courtesy of Kay Tobin Lahusen.\n\n4, 5 (bottom) reprinted courtesy of Diana Davies Collection, Manuscripts and Archives Division, The New York Public Library, Astor, Lenox and Tilden Foundations.\n\n5 (top) reprinted courtesy of Thomas Lanigan-Schmidt.\n\n6 (top), 7 (top), 8 (middle) reprinted courtesy of Fred N. Orlansky.\n\n6 (bottom) reprinted courtesy of New York Daily News, L.P. Reprinted with permission.\n\n8 (top) taken from \"Gay Freedom 1970,\" reproduced by permission of Queen's Quarterly Publishing Co., Inc.\n\n# I N D E X\n\nThe index that appeared in the print version of this title was intentionally removed from the eBook. Please use the search function on your eReading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.\n\nAbbott, Berenice\n\n_Advocate_ magazine\n\nage of Aquarius\n\nAgostino, Sam\n\nAlbee, Edward\n\nAlbert (hotel)\n\nAlcohol and Beverage Control Department (California)\n\naction against\n\nAlcoholic Beverage Control (ABC) Law\n\nAlexander, Brandy\n\nAll-State Caf\u00e9 (dance bar)\n\nAllison, John\n\nAlternate U.\n\nalternative schools\n\nAmbrosini, Joseph\n\nAmram, David\n\nAmsel, Bob\n\n_And He Made a Her_ (Wilson)\n\nAnnual Reminder (Independence Hall\/Philadelphia); dress and behavior code\n\n_Anti-Slavery Standard_\n\nAronowitz, Al\n\nAronowitz, Stanley\n\nArtie's (bar)\n\nAurelio, Richard\n\nBabick, Chris\n\nand the Stonewall Riots\n\nBaldwin, James\n\nBali (bar)\n\n\"Ballad of the Sad Young Men, The,\"\n\nBallard, Ron\n\nBambi (street queen)\n\nbar culture (East Coast)\n\nextortion and blackmail\n\ngay bars, legalizing\n\nhomophile movement and\n\nkidnapping and murder, suspicion of\n\nMafia involvement with\n\nnumber of bars in the 1950s\n\nand payoffs\n\npolice and bar cooperation\n\nraids and closings\n\nbar culture (West Coast)\n\nand payoffs\n\nTavern Guild (San Francisco)\n\nbars (East Coast)\n\nAll-State Caf\u00e9\n\nArtie's\n\nBali\n\nCheckerboard\n\nbars (East Coast) _(continued)_\n\nCherry Lane\n\nCork Club\n\nDanny's\n\nDirty Dick's\n\n415 (club)\n\nJulius'\n\nKeller's\n\nLenny's Hideaway\n\nMais Oui\n\nMilano's\n\nMoss Bar\n\nMystique\n\nOmega\n\nPink Elephant\n\nSans Souci\n\n17 Barrow Street\n\nSewer\n\nSnake Pit, The\n\nTel-Star\n\nTerrace\n\nZodiac\n\n_See also_ State Liquor Authority; Stonewall Inn\n\nbathhouses\n\nBeard, Harry\n\nBeat subculture\n\nBecker, Marle\n\nBell, Arthur\n\nand GAA\n\n_Gay Power_ column\n\nand GLF\n\n_Berkeley Barb_\n\nBilly (a.k.a. Miss Billy) (street queen)\n\nBird, Joan\n\nBlack Panthers, homophobia of\n\nblackmail. _See_ extortion\n\nBlake, William\n\nBlick, Roy\n\nBlond Frankie\n\nBob (Chris Babick's boyfriend)\n\nBockman, Philip\n\nBonnie's Stone Wall (Bonnie's Stonewall Inn)\n\n_See also_ Stonewall Inn\n\nBooth, William\n\nBoyce, Martin\n\n_Boys in the Band, The_ , (Crowley)\n\nBrew, Kevin\n\nBroadway Central (hotel)\n\nBrown, Howard\n\nBrown, Michael\n\nand GLF\n\nBrown, Willie\n\nBryan, Robert\n\nBurch, Charles\n\nBureau of Alcohol, Tobacco and Firearms\n\nand the Stonewall Inn raid\n\nBurke, Tom\n\nBurton, John\n\nCalifornia\n\nhomosexuals, laws restricting\n\nCalifornia Hall\n\nCRH New Year's Eve ball, police raid of\n\nCallahan, Tim\n\ncamp and street slang\n\nCanada\n\nhomosexual acts, legalizing of\n\nCasteel, Tom\n\ncastration\n\nas punishment for homosexuality\n\nCastro, Raymond\n\narrest of\n\nCentral City Citizens Council (CCCC)\n\nCentral City Poverty Program\n\nCervantes, Madolin\n\nCharlie (Danny Garvin's friend)\n\nChauncey, George\n\nCheckerboard (bar)\n\nCherry Lane bar\n\nChing, Tamara\n\nChristopher Park (Sheridan Square)\n\nstreet kids and\n\nChristopher Street\n\nThe Corner (Greenwich Ave. and Christopher St.)\n\nhistory of\n\nPATH station\n\nStonewall Riots, blocking the day after\n\nChristopher Street Festival Committee\n\nChristopher Street Liberation Day (CSLD)\n\nfirst march\n\ngay movement support of\n\nGay-In (Sheep Meadow, Central Park)\n\nChurch, William\n\nentrapment and suicide of\n\nChurch of the Holy Apostles\n\nCivil Service Commission (CSC)\n\nhomophobia of\n\nand homosexual employment\n\nClarke, Lige\n\nCollin ((Craig Rodwell's roommate)\n\nColumbia University uprisings\n\n_Come Out!_ (GLF newspaper)\n\nCommando Queens (gang of street kids)\n\nCommittee for Homosexual Freedom (CHF)\n\nCompton Cafeteria (San Francisco) riot\n\nCongo Woman (street queen)\n\nconsciousness-raising\n\nContinental Baths\n\nConvoy, Kenneth\n\nCooke, Michael\n\nCork Club (bar)\n\nCorner, The (Greenwich Ave. & Christopher St.)\n\n_Coronet_ magazine\n\nCouncil on Religion and the Homosexual (CRH)\n\nNew Year's Eve ball police raid\n\ncounterculture\n\nalternative schools\n\nCrazies\n\ngay communes\n\nCraig (gay commune member)\n\nCriswell\n\nCrosby, Tina\n\nCross-eyed Sylvia (street queen)\n\ncrossdresser as a term\n\nCuba (and homosexuals)\n\nconcentration camps for\n\nlaws restricting\n\ncummings, e. e.\n\nD'Emilio, John\n\nDaley, Richard\n\nDanny's (bar)\n\nDaughters of Bilitis (DOB)\n\nDefense Department\n\nhomosexuals, exclusion of\n\nDeLarverie, Storm\u00e9\n\nDelgado, Tano\n\nDemocratic Convention (Chicago)\n\nprotests at\n\nDenaro, Frank\n\nDepartment of Consumer Affairs\n\nand the Stonewall Inn raid\n\nDerrick (Dick Leitsch's lover)\n\nDesiree (drag queen)\n\nDi Brienza, Ronnie\n\nStonewall Inn, Tuesday raid\n\nStonewall Riots\n\nDiaman, Nikos\n\nDirty Dick's (bar)\n\nDoerr, Tom\n\nDontzin, Michael\n\ndrag queen as a term\n\ndropping dimes\n\ndrugs\n\nat the Stonewall Inn\n\nand street kids\n\nDuberman, Martin\n\nDunn, Kevin\n\nDurham, Louis\n\nDylan, Bob\n\nEagles, Philip\n\nand the Stonewall raid and riot\n\nEarl (hotel)\n\nEast Germany and homosexuals\n\n_East Village Other_\n\n_Eastern Mattachine Magazine_\n\nEastern Regional Conference of Homophile Organizations (ERCHO)\n\nEden, Barbara (drag queen)\n\neffeminate men\n\ncourage of during Stonewall Riots\n\nEgan, Robert\n\nEighth Street Bookshop\n\n82 Club (nightclub)\n\nEisenhower, Dwight David\n\nElectric Circus\n\nQueens tree replacement benefit\n\nelectrical shock therapy\n\nas punishment for homosexuality\n\n11 Christopher Street\n\nElman, Dick\n\nentrapment\n\nand death\n\nJudson Church meeting (1966)\n\nJohn Lindsay and\n\npolitical campaigns and\n\nTransit Police and\n\n_Escapade_ magazine\n\nEskridge, William N., Jr.\n\n_Esquire_ magazine\n\nEssex (hotel)\n\nEvans, Arthur\n\ncoming out\n\nand Allen Ginsberg\n\nand GAA\n\nand GLF\n\npolitical action of\n\non Marty Robinson\n\nand the Snake Pit raid\n\nExecutive Order 10,450\n\nextortion (and blackmail)\n\nWall Street employees\n\nFader, Michael\n\nand the Stonewall raid and riot\n\nFancher, Ed\n\nFBI\n\nhomophobia of\n\n_See also_ Hoover, J. Edgar\n\nFeinstein, Dianne\n\nFidelifacts, GAA zap of\n\n55 bar\n\n59 Grove Street\n\nFisk, John\n\nflame queens\n\n_See also_ street kids; transvestites\n\nForbes, Malcolm\n\n\"Forget Me Not,\"\n\n415 (bar)\n\nFouratt, James\n\nFrank (Danny Garvin's friend)\n\nFrank (Craig Rodwell's friend)\n\nFreedom House\n\nGabriel, Pauln\n\nGalvin, Earl\n\nGarelick, Sanford\n\nGarland, Judy\n\nas cause of the Stonewall Riots\n\nGarvin, Danny\n\nand the gay commune\n\nand Ed Murphy\n\nNavy, discharge from\n\nand the year 1969\n\nand the Stonewall Inn\n\nand the Stonewall raid and riot\n\nsuicide attempt\n\n_Gay_ (newspaper)\n\n\"gay\" as a term\n\nGay, Sidney Howard\n\nGay Activists Alliance (GAA)\n\nChristopher Street Liberation Day\n\nFidelifacts zap\n\nfocus of\n\nformation of\n\nCarol Greitzer, petition presented to\n\n_Harper's_ magazine zap\n\nlambda symbol\n\nJohn Lindsay, zaps confronting\n\nmarriage bureau zap\n\nand the media\n\nmeetings, structure of\n\nmembership\n\npolitical actions of. _See also below_ zaps\n\npreamble and constitution of\n\nRepublican State Committee headquarters sit-in (Rockefeller Five)\n\nSnake Pit raid protest march\n\nzaps\n\ngay communes\n\n_Gay Law_ (Eskridge, Jr.)\n\nGay Liberation Front (GLF)\n\nbars, demonstrations against\n\ncell structure\n\nchanting\n\nChristopher Street Liberation Day\n\n_Come Out!_\n\nconsciousness-raising sessions\n\nfocus of\n\nformation of\n\nand \"gay\" as a term\n\nGLF dances\n\ninfighting and internal disputes\n\nMarxist ideology and\n\npolitical candidates, confronting\n\nRadical Study Group\n\nSnake Pit raid protest march\n\n_Village Voice_ , demonstrations against\n\ngay lifestyle (pre-Stonewall)\n\nbathhouses\n\ncamp and street slang\n\ngay communes\n\n_New York City Gay Scene Guide_ (Mattachine\u2013NY)\n\nself-hate\n\nstreet cruising\n\ntrucks, sex at\n\nthe year 1969, omens concerning\n\n_See also_ bar culture\n\n\"Gay Manifesto, A\" (Wittman)\n\ngay men\n\neffeminate men\n\nlegal status\n\nmedia depiction. _See also_ media\n\npre-Stonewall situation of\n\ngay movement (post-Stonewall) (East Coast)\n\nAlternate U. meetings\n\nChristopher Street Liberation Day\n\nGay Activists Alliance (GAA)\n\nGay Liberation Front (GLF)\n\nGay Power vigil and march\n\nHomosexuals Intransigent\n\nMattachine Action Committee (MAC)\n\nriots as origin of\n\n_See also_ homophile organizations\n\ngay movement (pre-Stonewall) (East Coast)\n\nbar protests\n\n\"Equality for Homosexuals\" buttons\n\ngay communes\n\nHomosexual League of New York\n\nHYMN (Homophile Youth Movement in Neighborhoods)\n\nJudson Church meeting (1966)\n\nOscar Wilde Memorial Bookshop\n\nPink Panthers\n\nState Liquor Authority anti-gay regulations, challenges to\n\nWBAI broadcast and\n\n_Gay New York_ (Chauncey)\n\n_Gay Power_ newspaper\n\nGay Power vigil and march\n\n\"gay pride\" as a term\n\nGay Street\n\n_Gay Sunshine_\n\ngender-fuck\n\nGerling, Enid\n\nGino (friend of Steve Yates)\n\nGinsberg, Allen\n\nFestival of Life (Chicago Democratic Convention)\n\non the sexuality of J. Edgar Hoover\n\nand the Stonewall Riot\n\nGittings, Barbara\n\nGlide, Lizzie\n\nGlide Memorial Church\n\nGlide Memorial Foundation\n\nGloeden, Baron von\n\ngo-go boys\n\nGodfrey, Arthur\n\nGoldstein, Al\n\nGoldwater, Barry\n\nGoodman, John\n\nGotham Young Republican Club\n\nGoudreau, Lawrence, Jr.\n\nGreenwich Village\n\nbohemian presence in\n\nChristopher Park (Sheridan Square)\n\nChristopher Street\n\nThe Corner (Greenwich Avenue and Christopher Street)\n\ngay population of\n\nhistory of\n\nNorthern Dispensary\n\nstreet layout\n\nsubway and PATH access to\n\nWashington Square Park\n\n_Greenwich Village Historic District Designation Report, The_\n\nGreitzer, Carol\n\nGAA petition presented to\n\nGrove Street\n\nGunnison, Foster\n\n_Hair_ (Ragni\/Rado\/MacDermot)\n\n\"Hairpin Drop Heard Around the World, The\" (Leitsch)\n\nHaliburt (hotel)\n\nHamill, Pete\n\nHampton, Dawn\n\nHanson, Ed\n\nHardy, Jennifer\n\nand the Stonewall raid and riot\n\n_Harper's_ magazine, GAA zap of\n\nHart, Lois\n\nand GLF\n\nHay, Harry\n\nCommunist party affiliation\n\nHeide, Robert\n\nHighwater, Jamake\n\nHodges, Julian\n\nHoey, Clyde\n\nHoffman, Abbie\n\nhomophile organizations (post-Stonewall) (East Coast). See gay movement (East Coast)\n\nhomophile organizations (pre-Stonewall) (East Coast)\n\nDaughters of Bilitis (DOB)\n\ndivisions among leaders\n\nEastern Regional Conference of Homophile Organizations (ERCHO)\n\nMattachine Society of New York (Mattachine\u2013New York)\n\nMattachine Society of Washington (Mattachine\u2013Washington)\n\npublications. _See also_ media (homophile and gay movement)\n\nWest Side Discussion Group\n\nhomophile movement (pre-Stonewall) (West Coast)\n\nCommittee for Homosexual Freedom (CHF)\n\nCouncil on Religion and the Homosexual (CRH)\n\nLeague for Civil Education (LCE)\n\nMattachine Society of Los Angeles\n\nSociety for Individual Rights (SIR)\n\nTavern Guild (San Francisco)\n\nVanguard\n\nhomophobia\n\nEd Koch and\n\nMcCarthyism and\n\nin the media\n\nQueens, anti-gay vigilantism in\n\n_Village Voice_ and\n\n\"Homosexual Citizen, The\" (Nichols\/ Lige)\n\n_Homosexual Handbook_\n\nHomosexual League of New York\n\n\"Homosexual Liberation Meeting\" (flyer)\n\nHomosexuals Intransigent\n\nHoose, Jerry\n\nand GLF\n\nand Mafia-run bars\n\nand the term \"gay pride,\"\n\nHoover, J. Edgar\n\nHormona, Jackie\n\nHostetter, Donald\n\nHouse of Detention for Women\n\nBastille Day picketing\n\nStonewall riot, inmates and\n\nHoving, Thomas\n\nHoward Johnson's (Greenwich Village)\n\npolitical action (sip-in) at\n\nstreet kids and\n\nHumphrey, Hubert\n\nHuss, Gene\n\n\"Hustler, The,\"\n\nhustling (male prostitution)\n\nHYMN (Homophile Youth Movement in Neighborhoods)\n\n_Hymnal_ newsletter\n\nIanello, Matty (the Horse)\n\nIndependence Hall (Philadelphia)\n\nAnnual Reminder\n\nInman, Richard\n\nInterpol bond investigation\n\nIrish Sylvia (street queen)\n\nJack (waiter at Mama's Chick'N'Rib)\n\nJefferson, Thomas\n\nJewish Mafia\n\nJiggs, Maggie\n\nJoel S.\n\n\"Joey\" (partner in the Stonewall Inn)\n\nJohn (men's room attendant)\n\narrest of\n\nJohnson, Lyndon\n\nJohnson, Marsha P. (Malcolm Michaels, Jr.)\n\ngenerosity of\n\nand the Stonewall Riots\n\nJulius' (bar)\n\nKahn, Joseph\n\nKameny, Frank\n\nand the Annual Reminder (Philadelphia)\n\nand government clearance\n\nand Mattachine-Washington\n\nKaminsky, Sherman Chadwick. _See_ Vargo, Paul\n\nKatzenberg, Bill\n\nKeating, Judge\n\nKeller (hotel)\n\nKeller's (bar)\n\nKelly, Colin\n\nKennedy, Flo\n\nkidnapping, Mafia and\n\nKiki (cross-dressing hairstylist)\n\nKirk, Grayson\n\nKnapp, Don\n\nKoch, Ed\n\nhomophobia of\n\nKohler, Bob\n\nand the GLF dances\n\nand Ed Murphy\n\nand the Stonewall Riots\n\nKonrad (South African youth)\n\nKuhn, Don\n\n_Ladder_ (DOB publication)\n\nLahusen, Kay Tobin. _See_ Tobin, Kay\n\nlambda symbol\n\nLanigan-Schmidt, Tommy\n\nand Ed Murphy\n\nand the Stonewall Inn\n\nand the Stonewall raid and riot\n\non the streets\n\nLassoe, John\n\nLaurence, Leo E.\n\nand the Stonewall Riot\n\nLauria, Ernie\n\nLauria, Fat Tony\n\nLavender Menace (lesbian group)\n\n_LCE News_\n\nLeague for Civil Education (LCE)\n\nLeague for Sexual Freedom\n\nLeary, Howard\n\nlegal status (of gay men and lesbians)\n\nExecutive Order 10,450\n\nhomosexual acts, legalizing\n\npublic toilet, loitering in\n\npre-Stonewall\n\nsodomy laws\n\nLeitsch, Dick\n\ncoming out\n\nJudson Church meeting (1966)\n\nand Mattachine\u2013New York\n\nand Ed Murphy\n\nNew York City, life in\n\non payoffs\n\nand police entrapment\n\npolitical activism (post-Stonewall)\n\nand Craig Rodwell\n\nand the State Liquor Authority\n\nand the Stonewall Inn\n\non _The Village Voice_\n\nLeitsch, Dick (Stonewall raid and riot)\n\nFriday\n\nSaturday\n\nStonewall Inn's window, sign in\n\nSunday-Wednesday\n\nLenny's Hideaway (bar)\n\nlesbians\n\nlegal status\n\nliterature\n\nmedia depiction\n\npre-Stonewall situation of\n\nStonewall raid, police manhandling\n\nStonewall raid, reaction to\n\nWomen's House of Detention inmates, solidarity with Stonewall rioters\n\nLewis, Joseph\n\nLiberace\n\nLindsay, John V.\n\nGAA zaps confronting\n\nLion's Head pub\n\nLiscoe, Kevin\n\nLittlejohn, Larry\n\nlobotomies\n\nas punishment for homosexuality\n\nLuke Connor's (hangout)\n\nMcCarthy, Eugene\n\nMcCourt, Frank\n\nMcDarrah, Fred\n\nMcDonnell, James\n\nMcIlvenna, Ted\n\nMafia\n\nChristopher Street Liberation Day, opposition to\n\ndrug dealing\n\nand gay bars\n\nand hustling (male prostitution)\n\nJewish Mafia\n\nand the Stonewall Inn\n\nWall Street employees, blackmail of\n\nMagoo (Bob Kohler's dog)\n\nMailer, Norman\n\nMais Oui (bar)\n\nMaloney, Andrew\n\nMama's Chick'N'Rib\n\nManford, Morty\n\nand the Stonewall raid and riot\n\nMarat, Jean-Paul (pseudonym)\n\nMarchi, John J.\n\nMarcus, Eric\n\nmarijuana, legalization of\n\nMarlton (hotel)\n\nMarotta, Toby\n\nmarriage bureau, GAA zap of\n\nMartello, Leo\n\nMartha Reeves and the Vandellas\n\nMarxist Socialist Workers Party (SWP)\n\nMattachine Action Committee (MAC)\n\nMattachine Society of Los Angeles\n\nas an educational and research organization\n\nand political action\n\nMattachine Society of New York (Mattachine\u2013New York)\n\n_Eastern Mattachine Magazine_\n\ngay bars, legalizing\n\n_New York City Gay Scene Guide_ (Mattachine\u2013New York)\n\nJulian Hodges as president\n\nDick Leitsch as president\n\nand political action\n\nStonewall Inn's window, sign in\n\nStonewall Riots, ambivalent response to\n\n_See also New York Mattachine Newsletter_\n\nMattachine Society of Washington (Mattachine\u2013Washington)\n\nMead, Taylor\n\nmedia\n\ndemonstrations against\n\nand the early gay movement\n\nand GAA\n\nhomophobia of\n\non Ed Murphy's extortion career\n\nand the post-Stonewall movement\n\n\"sip ins\" (political action in bars)\n\nSnake Pit raid\n\nand the Stonewall Riots\n\nmedia (homophile and gay movement)\n\n_Advocate_ magazine\n\n_Come Out!_ (GLF newspaper)\n\n_Eastern Mattachine Magazine_\n\n_Gay_\n\n_Gay Power_ newspaper\n\n_Gay Sunshine_\n\n_Hymnal_ newsletter\n\n_Ladder_ (DOB publication)\n\n_LCE News_\n\n_New York City Gay Scene Guide_ (Mattachine\u2013New York)\n\n_New York Mattachine Newsletter_\n\n_Queen's Quarterly_\n\n_V_ magazine (Vanguard group)\n\n_Vector_ magazine (Tavern Guild)\n\nMeher Baba\n\nMekas, Jonas\n\nMichael (roommate of Arthur Evans)\n\nMidler, Bette\n\nMilano's (bar)\n\nMiles, Barry\n\nMilk, Harvey\n\nMorgan, Robin\n\nMorgenthau, Robert\n\nMoss Bar\n\n_Mother Stonewall and the Golden Rats_ (Lanigan-Schmidt)\n\nmurder, Mafia and\n\nMurdoch, Keith\n\nMurphy, Ed (the Skull)\n\nChristopher Street Liberation Day, appropriation of\n\ncriminal career\n\nextortion and blackmail\n\nFBI, as alleged informant for\n\nas \"the first Stonewaller,\"\n\nJ. Edgar Hoover, alleged blackmail of\n\nkidnapping and murder, suspicion of\n\nat the Stonewall Inn\n\nStonewall Riots, arrest and escape\n\nWall Street employees, blackmail of\n\nMurphy, Michael J.\n\nMystique (bar)\n\nNBC-TV\n\nNeier, Aryeh\n\nNelly (a.k.a. Betsy Mae Kulo) (street queen)\n\n\"New Homosexual,\"\n\n_New Symposium, The_ (WBAI program)\n\n_New Symposium II, The_\n\nNew York City (and gays)\n\nlegal oppression of\n\npopulation of\n\n_See also_ Greenwich Village\n\n_New York City Gay Scene Guide_\n\nNew York _Daily News_\n\n_New York Herald Tribune_\n\nNew York Hilton, blackmail ring at\n\n_New York Journal-American_\n\n_New York Mattachine Newsletter_\n\n\"D.D.'s New York,\"\n\non Ed Murphy\n\n_New York Post_\n\nNew York Radical Women\n\nNew York Stock Exchange\n\ntoilet action\n\nWall Street employees, blackmail of\n\n_New York Times_\n\n_Newsweek_\n\nNichols, Jack\n\nand _Gay_\n\nand Mattachine-Washington\n\n1964 World's Fair\n\nand harassment of homosexuals\n\n1969 (the year)\n\nomens and predictions concerning\n\nNixon, Marty\n\nNorthern Dispensary\n\nNova (Zazu Nova, Queen of Sex);\n\nNovak, Tammy (drag queen)\n\nO'Brien, John\n\nand GLF\n\npolitical activities (post-Stonewall)\n\nand the Stonewall raid and riot\n\nO'Hara, Frank\n\nO'Neill, Eugene\n\n_Official and Confidential: The Secret Life of J. Edgar Hoover_ (Summers)\n\nOlenick, Michael\n\nOmega (bar)\n\nOpera Jean (a.k.a. Sister Tooey) (street queen)\n\nOrphan Annie (street queen)\n\nOscar Wilde Memorial Bookshop\n\nOwles, Jim\n\nand GAA\n\nand GLF\n\nSnake Pit raid\n\n_Pageant_ magazine\n\nPaine, Thomas\n\nPeggy (Stonewall Inn customer)\n\nPennsylvania\n\nhomosexuals, laws restricting\n\nPerrin, Barry\n\nPerry, Troy\n\npharmacological shock therapy\n\nas punishment for homosexuality\n\nPhiladelphia\n\nAnnual Reminder at Independence Hall\n\nPine, Seymour\n\nFirst Division of the Public Morals, appointment to\n\nand the Interpol bond investigation\n\nmeets with Mattachine Society of New York\n\nmilitary career\n\npost-Stonewall transfer\n\nPine, Seymour (Stonewall raid and riot)\n\nearlier raid (June 24)\n\npolice using guns, option of\n\nand policewomen\n\npreparation for\n\nreason for raid\n\nsearch warrant\n\nStonewall Inn, trapped inside\n\nDave Van Ronk's arrest\n\nPink Elephant (bar)\n\nPink Panthers\n\nPitts, Charles\n\nPoe, Edgar Allan\n\npolice, street slang for\n\npolice brutality\n\nPolice Department\n\nbar raids and closings\n\nentrapment\n\nFirst Division of the Public Morals\n\nharassment\n\npayoffs and corruption\n\nundercover cops, homosexuals as\n\n_See also_ Stonewall Riots; Sixth Precinct; Tactical Patrol Force\n\nProcaccino, Mario\n\nProvincetown Players\n\npsychological establishment and homosexuality\n\npublic toilets, loitering in\n\nPyne, John J.\n\n_Queen's Quarterly_\n\nQueens (New York City borough), anti-gay vigilantism in\n\nRaia, Phil\n\nRanieri, John Paul\n\n_Rat_ newspaper\n\n_Realist_ magazine\n\nRed Swan Social Club. _See_ Stonewall Inn\n\nRepublican State Committee headquarters\n\nGAA sit-in (Rockefeller Five)\n\nRiker's restaurant\n\nRing, Thomas\n\nRitter, Maria (Steve)\n\nand the Stonewall raid and riot\n\nRitter, Steve. _See_ Ritter, Maria\n\nRitz, Crema\n\nRivera, Birdie (Robert) (street queen),\n\n59\n\nRobinson, Marty\n\nand GAA\n\nand GLF\n\npolitical activism (post-Stonewall)\n\nSnake Pit raid\n\nRockefeller Five\n\nRodwell, Craig\n\nand the Annual Reminder (Philadelphia)\n\nChristopher Street Liberation Day\n\ncoming out\n\nand Dick Leitsch\n\nand the Mafia's control of bars\n\nand Mattachine\u2013New York\n\nand the media\n\nand Harvey Milk\n\nHYMN (Homophile Youth Movement in Neighborhoods)\n\nOscar Wilde Memorial Bookshop\n\npolice, problems with\n\npolitical activism\n\npolitical activism (post-Stonewall)\n\nand the Stonewall Inn\n\nsuicide attempt\n\nRodwell, Craig (Stonewall raid and riot)\n\non the cause of\n\nChristopher Street, blocking\n\nFriday night\n\nthe future, considering\n\nSaturday\n\nSunday\n\nRuitenbeek, Hendrick\n\nRussia and homosexuals\n\nRusso, Vito\n\nSt. John's Episcopal Church\n\nSalmieri, Salvatore\n\nSan Francisco\n\nhomophile movement in\n\nTenderloin district\n\nSan Francisco State College strike\n\nSans Souci (bar)\n\nSapponckanican (Indian settlement)\n\nSargeant, Fred\n\nSarria, Jos\u00e9\n\nscare drag\n\nscare queens\n\n_See also_ street kids; transvestites\n\nSchwan, Judge\n\nScherker, Michael\n\nScott, David\n\n_Screw_ (newspaper)\n\nSecor, Neil\n\n17 Barrow Street (bar)\n\nSeventh Avenue\n\nSeventh Avenue South\n\nSeventh Avenue subway (IRT)\n\nChristopher Street\/Sheridan Square station\n\nSewer (bar)\n\nShades, Johnny\n\nShaheen, Chuck\n\nShakur, Afeni\n\nShelley, Martha\n\nand GLF\n\npolitical activism (post-Stonewall)\n\nSheridan, Philip\n\nSheridan Square (Christopher Park)\n\nshock therapy\n\nas punishment for homosexuality\n\nSilver Dollar Restaurant\n\nSilverman, Susan\n\nSixth Avenue subway (IND)\n\nWest 4th Street station\n\nSixth Precinct\n\nSnake Pit raid protest march\n\npayoffs to\n\nwarning of raids\n\n_See also_ Stonewall Riots\n\nSmilers Deli\n\nSmith, Howard\n\nStonewall Inn, trapped inside\n\nStonewall raid and riot, observing\n\n_Village Voice_ article\n\nSmythe, Charles\n\nSnake Pit, The (bar)\n\ncustomers chanting and singing\n\ngay rights movement, effect on\n\nmedia and\n\nSeymour Pine's raid on\n\nAlfredo Vinales, impalement of\n\nSociety for Individual Rights (SIR)\n\nsodomy laws\n\nSpanola Jerry (drag queen)\n\nSpellman, Cardinal Francis\n\nSpencer, Walter Troy\n\n\"Spider, The\" (dance)\n\nStanford Sit-In\n\nStanley (street queen)\n\nState Liquor Authority (SLA)\n\ncorruption in\n\nhomosexuals, regulation of\n\nStates Steamship Line, picketing\n\nStevenson, Adlai\n\nStokes, Phillip\n\n_Stone Wall, The_ (\"Mary Casal\")\n\nStonewall Inn (bar)\n\nalcohol, watering\n\nclientele\n\nclosure\n\ncommunity feeling, sense of\n\ndecor and layout of\n\ndrug dealing and use\n\nearly history of\n\nfinancial success of\n\nfire exits, lack of\n\ngo-go boys\n\nJohn (men's room attendant)\n\nlesbians at\n\nlocation of\n\nMafia involvement with\n\nmusic and dancing\n\npayoffs\n\nas a private club\n\nprostitution at\n\nracial mix\n\nraids, police warning of\n\nsanitation\n\nas a street queen hangout\n\ntransvestites and transsexuals\n\nWall Street employees, Mafia blackmail of\n\nwindows and doors, reinforcement of\n\nwomen at\n\nStonewall Inn raid\n\narrests\n\nBureau of Alcohol, Tobacco and Firearms\n\nDepartment of Consumer Affairs\n\nearlier raid (June 24)\n\nlesbian reaction to\n\nlesbians, police manhandling\n\nSeymour Pine's preparation for\n\npolicewomen and\n\nresistance. _See also_ Stonewall Riots\n\nSixth Precinct and\n\ntransvestites and\n\nStonewall Inn Restaurant\n\nStonewall Riots\n\ncasualties\n\ncause of and credit for starting\n\nchanting and protest songs\n\ncrowd gathering\n\neffeminate men and\n\nfull moon and temperature as cause\n\ngay rights movement, effect on\n\nhumor, camp, and kick lines\n\nmedia reaction to\n\nneighborhood reaction to\n\npolice brutality\n\npolice surprised by\n\nradical Left provocateurs and\n\nstreet kids and\n\ntransvestites and\n\n\"We are the Stonewall Girls,\"\n\nthe year 1969, omens concerning\n\nStonewall Riots (Friday night)\n\narrest that initiated\n\narrests, reaction to\n\nRay Castro's arrest\n\ncobblestone, bottle, and garbage can throwing\n\ncoin tossing\n\ncrowd, fire hoses turned on\n\ncrowd, street kids robbing\n\ncrowd, Tactical Patrol Force (TPF) attempting to control\n\nfire trucks arrive\n\nfires and Molotov cocktails\n\nlesbian who resisted arrest\n\npaddy wagon escapes\n\nparking meter as battering ram\n\npolice using guns, option of\n\npolice vehicles, tires slashed on\n\nStonewall Inn, police escape from\n\nStonewall Inn, police trapped inside\n\nwindows breaking\n\nStonewall Riots (Saturday)\n\nboarded-up windows, slogans on\n\nChristopher Street, blocking\n\nCrazies exploiting\n\nfires and projectiles\n\nJudy Garland's death and\n\nthe gathering crowd\n\nneighborhood bar business, effect on\n\nTactical Patrol Force (TPF) and\n\ntaxi driver, death of\n\ntaxicab overturned\n\ntemperature\n\nStonewall Riots (Sunday\u2013Wednesday)\n\narrests\n\ncasualties\n\npolice, interaction with\n\npolice badge, theft of\n\nradical Left involvement\n\nshops broken into\n\nStonewall Inn's window, sign in\n\n\"Stonewall Riot Remembered, The\" (Crosby)\n\nstreet cruising\n\nstreet kids (street queens)\n\nbehavior\n\nbravery of\n\nand Christopher Park\n\nflame queens\/scare queens\n\ndrugs and\n\nand GLF\n\nhustling\n\nand the Stonewall Inn\n\nand the Stonewall Riots\n\n_See also_ transvestites\n\nStryker, Susann\n\nStudents for a Democratic Society (SDS)\n\nSummers, Anthony\n\nTactical Patrol Force (TPF) (Stonewall riots)\n\nbrutality\n\nFriday night\n\nSaturday\n\nand Snake Pit raid protest march\n\n\"Tano\" (Puerto Rican youth)\n\nTatem, Adam\n\nTavern Guild (San Francisco)\n\nTaylor, Harry\n\nTeal, Donn\n\nTel-Star (bar)\n\n_Tenderloin Report_ (CCCC)\n\nTenth of Always (juice bar)\n\nTerrace (bar)\n\n\"Third Finger, Left Hand,\"\n\nTiffany (drag queen)\n\n_Time_ magazine\n\nTimmons, John\n\nTish (female impersonator)\n\nTobin, Kay\n\nTolson, Clyde\n\n\"Tom\" ( _Rat_ reporter)\n\nand the Stonewall raid and riot\n\nTommy (street kid)\n\n_Tonight_ show\n\n\"Tony the Sniff\" (partner in the Stonewall Inn)\n\nTower Records (San Francisco), CHF picketing\n\nTransit Police, and entrapment\n\ntransvestite as a term\n\ntransvestites (and transsexuals)\n\nflame queens\/scare queens\n\nlaws concerning\n\nat the Stonewall Inn\n\nand the Stonewall raid and riot\n\n_See also_ street kids\n\nTree (bartender)\n\ntrucks, sex at\n\nTruscott, Lucian (and the Stonewall raid and riot)\n\nTruscott, Lucian _(continued)_\n\nFriday night\n\nSaturday\n\nSunday\n\n_Village Voice_ article\n\nUkrainian-American Village Restaurant\n\npolitical action (sip-in) at\n\nUnited Cigar Store\n\nUnited Nations, picketing\n\n_V_ magazine (Vanguard)\n\nVag, \"Mark,\"\n\nvan Cline, Stephen\n\nVan Ronk, Dave\n\narrest of\n\nand the Stonewall Riots\n\nVan Twiller, Wouter\n\nVanguard\n\nVarela, Emile\n\nVargo, Paul (Sherman Chadwick Kaminsky)\n\n_Variety_ magazine\n\n_Vector_ magazine (Tavern Guild)\n\nVietnam War, opposition to\n\nVillage Independent Democrats (VID)\n\n_Village Voice_\n\nand censorship\n\nGLF demonstrations against\n\nhomophobia of\n\nand Off Off Broadway\n\nand the post-Stonewall movement\n\nStonewall Riots coverage\n\nVinales, Alfredo Diego\n\nVincenz, Lilli\n\nVoeller, Bruce\n\nWagner, Robert\n\nWaikiki (bar)\n\npolitical action (sip-in) at\n\nWall Street employees\n\nMafia blackmail of\n\n_Wall Street Journal_\n\nWashington Square Park\n\nas cruising area\n\nWaverly Place\n\nWBAI\n\n_The New Symposium_\n\n_The New Symposium II_\n\nand the Stonewall Riots\n\nRandy Wicker broadcast\n\n\"We are the Stonewall Girls,\"\n\nWeaver, Bill\n\nWeeks, Father James\n\nWeinberg, Martin\n\nWeissman, Gil\n\nWest 4th Street\n\nWest Germany\n\nhomosexual acts, legalizing\n\nWest Side Discussion Group\n\n\"Where It Was\" (article)\n\nWhite, Edmund\n\nand the Stonewall raid and riot\n\nand the Stonewall Riots aftermath\n\non _Village Voice_ Stonewall coverage\n\nWhite House, picketing\n\nWhitman, Walt\n\nWhittington, Gale\n\n_Who's Afraid of Virginia Woolf?_ (Albee)\n\nWicker, Randy\n\nHomosexual League of New York\n\nJudson Church meeting (1966)\n\nLeague for Sexual Freedom\n\nand Marsha P. Johnson\n\nand Mattachine\u2013New York\n\nand the media\n\nStonewall Riots, reaction to\n\nWBAI broadcast\n\nWilde, Oscar, vii\n\nWilliams, Colin\n\nWilliams, Cecil\n\nWilson, Doric\n\nentrapment of\n\nWilson, Pete\n\nWilson, Lanford\n\nWittman, Carl\n\nWodetski, Tom\n\nwomen's rights\n\nWynkoop, William\n\nYates, Steve\n\nYoung, Allen\n\nYoung Socialists Alliance (YSA)\n\nzaps\n\nZarfas, Zookie\n\nZodiac (bar)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \n#  LADIES OF THE CANYONS \n\n#  LADIES OF THE CANYONS\n\nA League of Extraordinary Women and Their Adventures in the American Southwest\n\nLESLEY POLING-KEMPES\n\nThe University of Arizona Press   \nwww.uapress.arizona.edu\n\n\u00a9 2015 The Arizona Board of Regents  \nAll rights reserved. Published 2015\n\nPrinted in the United States of America  \n20 19 18 17 16 15 6 5 4 3 2 1\n\nISBN-13: 978-0-8165-2494-5 (paper)\n\nCover designed by Leigh McDonald   \nCover photos: ( _top_ ) Carol and Jack at Canyon de Chelly, Carol Stanley collection; ( _bottom left_ ) Natalie Curtis Burlin, Santa Fe, 1917, Natalie Curtis Burlin Archives; ( _bottom right_ ) Mary Cabot Wheelwright, Collection of the Wheelwright Museum of the American Indian.\n\nLibrary of Congress Cataloging-in-Publication Data   \nPoling-Kempes, Lesley, author.   \nLadies of the canyons : a league of extraordinary women and their adventures in the American Southwest \/ Lesley Poling-Kempes.   \npages cm   \nIncludes bibliographical references and index.   \nISBN 978-0-8165-2494-5 (pbk. : alk. paper)  \n1. Southwest, New\u2014History\u201419th century. 2. Southwest, New\u2014Social conditions\u201419th century. 3. Women\u2014Southwest, New\u2014History. 4. Women\u2014Southwest, New\u2014Biography. I. Title.   \nF786.P85 2015   \n979'.03109252\u2014dc23   \n2015001524\n\n  This paper meets the requirements of ANSI\/NISO Z39.48-1992 (Permanence of Paper). \nThis book is for Jim and his inexhaustible joy and contagious enthusiasm for every adventure. \nWomen, when they are old enough to have done with the business of being women, and can let loose their strength, must be the most powerful creatures in the world.\n\n_\u2014Isak Dinesen_\n\n#  CONTENTS\n\nPrologue: Ladies of the Canyons\n\nChapter 1. Imperfectly Independent\n\nChapter 2. Land of Ineffable Light and Sudden Shadow\n\nChapter 3. Capturing the Wind\n\nChapter 4. _The Indians' Book_\n\nChapter 5. \"Time and the World\"\n\nChapter 6. Secrets and Dualities\n\nChapter 7. Creating a Brave New World\n\nChapter 8. Desert Euphoria\n\nChapter 9. Walpi Rendezvous\n\nChapter 10. \"A Fine Woman to Be Out With\"\n\nChapter 11. The City Different\n\nChapter 12. Remarkable Time, Remarkable Place\n\nChapter 13. A Rare Thing: Becoming Oneself\n\nChapter 14. \"A Country of Things in Light\"\n\nChapter 15. Death in Paris\n\nChapter 16. City of Ladies\n\nChapter 17. New Women of the Rio Grande\n\nChapter 18. \"Roughneck and Low-Brow\"\n\nChapter 19. Everything and Nothing\n\nAfterword: Something of the Spirit\n\nAcknowledgments\n\nAbbreviations\n\nNotes\n\nSelected Bibliography\n\nIllustration Credits\n\nIndex\n\n#  LADIES OF THE CANYONS \n#  PROLOGUE\n\n_Ladies of the Canyons_\n\nI BEGAN THE QUEST for the ladies of the canyons more than a decade ago. At the time, I was looking for a woman who had gone missing in history. Her name was Carol Bishop Stanley, and although she was the founder of Ghost Ranch, the place that would achieve celebrity status as the faraway nearby home of the American artist Georgia O'Keeffe, there was little to nothing written about Carol in the Ghost Ranch story. From oral histories and a handful of written accounts and letters, I learned that in 1931 Carol, a middle-aged divorcee from Boston, moved all that she owned to el Rancho de los Brujos, the Ranch of the Witches, the place she called Ghost Ranch, and with limited financial resources set about creating a home on the high desert of northern New Mexico. But how and why this woman had come to live in this austere, remote, and wildly beautiful landscape was lost to living memory.\n\nO'Keeffe never acknowledged Carol's role in the creation of Ghost Ranch. But that was in character, since O'Keeffe rarely acknowledged that anyone, male or female, held a prior claim to Ghost Ranch, although she herself never legally owned more than ten of the ranch's nearly thirty thousand acres. Arthur Pack, the millionaire who bought the ranch from Carol in 1935, did mention Carol Pf\u00e4ffle (her married name) in his memoir _We Called It Ghost Ranch_ , and in doing so provided some of the only recorded glimpses of her life: \"Close-in under the sheltering protection of magnificent buttes and sheer sandstone cliffs huddled a single low adobe building whose every door and window staggered crookedly. From it appeared a woman who spoke in cultured tones unmistakably Bostonian. . . . This homestead, which her ex-husband was said to have won in a poker game, was all that Carol Pf\u00e4ffle had left in the world except for a beautiful grand piano.\"\n\nPack likely knew more about Carol's life than he shared in his memoir\u2014like how a cultured woman from Boston came to be living alone (with a grand piano) in a decrepit homestead on the edge of the Colorado Plateau in the first years of the Depression. But Pack, like most chroniclers of his era, did not consider the immigration story of an ordinary woman worthy of more than a passing reference that gave context to his own story. \"When you go looking for missing persons, you may not find them,\" the historian Virginia Scharff wrote, \"but you are bound to find out a lot of other things.\"\n\nI did find Carol Stanley. I also found a lot of other people and things I did not even know to go looking for. While searching archives and public records for Ghost Ranch's founding mother, I discovered a narrative that connected Carol to an informal alliance of remarkable women. Women who, like Carol, had left the security and comfort of conventional society in the first decades of the twentieth century and journeyed to Indian Country in search of a wider, deeper view of themselves and the world. Natalie Curtis Burlin, Alice Ellen Klauber, Mary Cabot Wheelwright, and Carol Bishop Stanley were members of an intrepid group of ladies whose lives became entwined with and altered by the people and landscape of the Southwest outback. Their ranks included Louisa Wade Wetherill, Alice Corbin Henderson, Elsie Clews Parsons, the sisters Amelia Elizabeth and Martha Root White, Mabel Dodge Luhan, Margretta Dietrich, Mary Austin, Elsie Shepley Sergeant, and Willa Cather.\n\nUnearthing Carol Stanley's half-century-long journey from her grandfather's beach house on Nahant Island, Massachusetts, to the ramshackle outpost built in Yeso Canyon, New Mexico, by infamous cattle rustlers meant following a road through the Southwest blazed by plucky and resourceful women. Within this landscape they imagined and created a new home territory, a new society, and a new identity for themselves and for all the women who would follow them.\n\nThe saga that emerged was far greater than Carol Stanley and Ghost Ranch. The story of the lady of the canyon became the stories of the ladies of the canyons and included Boston Brahmins and the Greenwich Village avant-garde, New York's Armory Show and the birth of modern art, a president and Indian music, the Snake Dance at Walpi and the Ute battle at Cottonwood Gulch, San Diego's Balboa Park and Santa Fe's Museum of Art, Florence before World War I and Paris in the Roaring Twenties. It was the story of New Women stepping bravely into the New World, of Anglo America waking up to Native America, of inconspicuous success and ambitious failure. \"My thoughts now are the thoughts of a stone,\" Alice Corbin wrote after her move to New Mexico in 1916. \"My substance now is the substance of life itself.\"\n\nHerein is the story of the ladies of the canyons, nearly buried by negligence and the inevitable avalanche of the decades that quietly claim the details and narratives of even remarkable lives. The ladies' trail remains true and shows the way into the vast territory of stone and sky and untold history. \n\n#  _1_\n\n##  IMPERFECTLY INDEPENDENT\n\nNo one can predict at the outset where the life stream will lead, but those moments of fissure, rupture, diversion, and frustration require choice and can even become springboards to opportunity.\n\n##  \u2014 _Virginia Scharff_\n\nON A SPRING MORNING IN 1903, Natalie and George Curtis walked together down a dusty street in the border town of Yuma, Arizona, as far south and west as one could wander in the continental United States. Dressed in expedition khaki and wide-brimmed hats to shade their pale faces, their leather boots colored by the fine sand of the Sonoran Desert, Natalie and George talked and laughed with the comfortable ease of good friends, or sister and brother. They were both.\n\nGeorge DeClyver Curtis was tall and lanky, with an amiable, welcoming manner. His younger sister, Natalie, was blonde and petite, with intense blue eyes, and an easy, contagious smile. The Curtis siblings were newcomers to Yuma, gringos off the train. Their appearance certainly gave them away, as did their accents\u2014upper-class Eastern Seaboard with the diction of the Greenwich Village\u2013Washington Square blue blood laced with the soft _r_ acquired in Harvard Yard and the parlors of the Boston Brahmins. How had these two aristocratic Anglo-Americans come to be renting rooms in a flea-bitten, cactus-and-snake-infested railway settlement on the farthest uncivilized edge of the great American desert?\n\nAlthough the Colorado River irrigated a narrow corridor along the town's edge, Yuma was derided as the hottest, driest hellhole in the desert Southwest. It was not a particularly attractive or remarkable landscape\u2014not like the iconic Indian Country of chiseled spires and carved canyons to the north\u2014and there was no culture or industry here beyond the railroad yard and the fenced fortress of the state penitentiary. The community's population comprised Spanish Americans who lived on the Arizona side of the Colorado River and the thousand or so Quechan Indians who lived on their reservation on the California side.\n\nNatalie Curtis was twenty-seven in the spring of 1903, on her inaugural journey into the vast, wild territory of the American Southwest. Her brother, George DeClyver Curtis, was thirty-two. George had come alone to Arizona in the winter of 1900 looking for reprieve from debilitating asthma. George found his good health and a lifestyle that suited him on an Arizona ranch, and was a renewed man when he returned for a visit to the Curtis home in Greenwich Village a year later. Natalie, herself in poor physical and emotional condition, saw the remarkable changes in her older brother and decided she, too, needed to experience the healing air and light of southern Arizona.\n\nNatalie, George, and their siblings Constance, Bridgham, and Marian (an older sister, Julia, died as a child) enjoyed every advantage available to those born into Victorian New York City's upper class. In the family brownstone at 27 Washington Place, the Curtis children were surrounded by New York City's cultural and intellectual elite. Their father, Edward Curtis, called Bogey by Natalie, was a distinguished doctor on the faculty of the College of Physicians and Surgeons and had served on the staff of the army surgeon general during the Civil War. Mother Augusta, known as Mimsey, oversaw a household where everyone in the family circle excelled at _something_. Overachievers were the norm at the informal and animated family suppers at the Washington Place dining table. Boldness and self-confidence were encouraged in the Curtis household, even among the women, and success in a chosen endeavor a given, at least among the men.\n\nGeorge and Bridgham were Harvard educated, an option not available to Natalie, her sisters Connie and Marian, or any other woman in Victorian America. Until she climbed aboard the train that would take her to the desert in 1902, Natalie had lived most of her life within the warm, comfortable, but confining rooms of the Washington Place house where she was born in 1875. She did travel abroad with her family, and attended the Brearley School, a private day school for privileged girls. Music was considered a female-appropriate educational pursuit, and was also an important social grace for upper-class Victorian women. Natalie's education included piano lessons when she was a very young child. \"There was always music in the home,\" Natalie's brother Bridgham remarked years later. Music and music teachers. Soon music was all Natalie cared and talked about. A visit with little Natalie Curtis meant an afternoon of piano duets and song.\n\nNatalie's young voice was trained in Anton Seidl's festival chorus, which accompanied the New York Philharmonic, and she was a featured soloist with the National Conservatory Chorus. She took piano classes with the great Anton\u00edn Dvo\u0159\u00e1k at the Conservatory of Music, and studied privately with the German composer Arthur Friedham and the Italian composer Ferruccio Busoni, both of whom had studied in Europe with Franz Liszt. By her teen years, Dr. Curtis's daughter Natalie was recognized as one of the most talented and serious young pianists, male or female, in New York City.\n\nAlthough her father was not the Wagnerian enthusiast that Natalie was, Bogey and Natalie were in the audience for every one of Wagner's operas performed at the Metropolitan Opera House. Bogey often regretted taking Natalie to the Met, however, because she became so emotionally involved and musically stimulated. At the end of one Wagner opera, when the audience broke into applause Natalie zealously pounded the opera house floor with a crutch (she had a lame leg at the time) until Bogey reached over and snatched the wood stick from her hands. \"Papa says that music is a bad habit with me,\" she wrote a girlfriend, \"and that I am nothing more nor less than a musical drunkard.\"\n\nWhile her contemporaries had crushes on stage actors, Natalie's infatuations involved opera singers, pianists, and conductors. At seventeen, she was smitten by Walter Damrosch, a handsome, enigmatic, married, and very popular-among-the-women Wagnerian conductor. Damrosch, who Natalie nicknamed \"It\" and described as Adonis-like in her letters to her friend Bessie, was ten years Natalie's senior. Even so, Natalie managed to attract the celebrity conductor's eye, and in the early 1890s the two began a friendship that would last several decades.\n\nBy her eighteenth birthday, Natalie was a musical tour de force whose entire life was organized around music. Four to six hours of piano practice was the norm, with choral classes, voice, and harmony lessons fitted in around schoolwork several times each week. A rigorous daily practice schedule, devised and enforced by Natalie, so wore down the patience and frayed the nerves of the Curtis family that a toneless practice clavier was placed on Natalie's keyboard to mute the sound, and eventually she was given a techniphone keyboard that made only soft clicks.\n\nNatalie's punishing schedule never stopped, even during holidays at the Curtis summer home, Wave Crest, in Far Rockaway, Long Island. While George and her other siblings went swimming and boating, rode horses, and sunned on the beach, Natalie commuted into the humid, hot city for her piano lessons. She practiced six to seven hours daily at Wave Crest, and when she wasn't practicing, Natalie was consuming the works of Goethe, Tennyson, Keats, and Schopenhauer.\n\nNatalie's zealous commitment to her music, and her inability to moderate that commitment, would be responsible for her greatest achievements and most devastating disappointments. \"Halfhearted\" was not in her psyche or her vocabulary. The relentless routine she maintained at Wave Crest threatened her health, and exhaustion spawned depression and doubt. Still, she persisted with her routine, even using brandy and strong coffee to get through her lessons. Finally, Mimsey and Bogey said enough, and Natalie was sent to the Berkshire Hills in Massachusetts for rest and recovery.\n\nAt eighteen, Natalie was the age when women were expected to marry, not begin careers as concert pianists. Natalie's parents understood that their daughter's talent was worthy of the professional stage. They understood, too, that a woman, a proper woman, did not choose the stage over marriage and a family. But the world was about to undergo a cultural revolution, and educated, talented, ambitious young women like Natalie Curtis would find themselves among the vanguard who would, almost in spite of themselves, untie the stays and dismantle the walls that defined Victorian American women. Natalie, just sixteen, wrote a friend: \"I don't know myself, but I want to have a kind of profession just as a man would, so that I could be perfectly independent. Don't you think that it would be awful to feel as some girls do that they _must_ marry? Just as though they were so many cows!\"\n\nBut for all her bravado and progressivism, Natalie _was_ conflicted about the implications of a career: if she were a concert pianist she would have to leave home. And unless she was married, a woman, even a gifted one, did not leave home to pursue a profession in the 1890s. \"I feel very well that my place is at home, in my family,\" Natalie wrote Bessie in 1893. \"I know that Papa would feel my absence tremendously and I owe him _so much_ that I can never pay him that the least I can do is to stay with him if my presence give him any pleasure.\" Leaving Mimsey and Bogey to go out into the world, Natalie confided to her girlfriend, would make her feel like an \"ungrateful hog.\"\n\nGeorge graduated from Harvard that year and, not burdened by his younger sister's sense of debt or duty to Mimsey and Bogey, began a career as a librarian at the Astor Library in New Haven, Connecticut.\n\nThe Curtis home \"breathed of liberal reform, practical idealism, moral courage and love of the fine arts.\" Located two blocks from Washington Square, Natalie's world was the invigorating, upscale, fashionable, semibohemian community of Greenwich Village. Henry James was born a few doors down at 21 Washington Place, and the Vanderbilts lived in the next block at number 10. The Curtis brownstone had been the family home since Natalie's grandfather bought it in 1839. Natalie's father and his four brothers spent their childhoods in the quiet, gracious neighborhood. The older Curtis boys married or went off to the Civil War, and Bogey, the youngest, inherited the house on Washington Place, where he settled with his wife, Augusta, and raised Natalie and her siblings.\n\nThe impetus for Natalie and George's unconventional journey into the rugged outback of the American West could be at least partly attributed to the example set by their father's half-brother, George William Curtis. The influence of Uncle George on the character and imagination of Natalie in particular cannot be overstated. George Curtis was a remarkable individual, and his life would serve as a template for his niece, who shared his spirited sense of adventure, sharp intellect, welcoming manner, and tactful but resolute temperament.\n\nGeorge Curtis, twenty years Bogey's senior, was more of a grandfather than an uncle to the Curtis children. Although he never attended college, by the 1880s George Curtis held honorary degrees from Harvard, Brown, and Columbia, and was the chancellor of the University of the State of New York. He was a self-taught scholar, an insatiable seeker, and a prolific writer who for twenty years was a popular and influential political editor at _Harper's Weekly_.\n\nAlthough George Curtis had homes near Boston and on Staten Island, he was an experienced outdoorsman who encouraged Natalie and George to learn about the simple joys found in a life lived close to the land. As a young man, he was introduced to the gifts of nature at Brook Farm, where he was schooled in transcendentalism by Ralph Waldo Emerson, Henry David Thoreau, and their league of New England poet-farmers and philosophers.\n\nGeorge Curtis's friends came to include the literary elite and politically powerful of his time. Fluent in four languages (he taught himself Italian, French, and German), he traveled the world in his twenties before returning to the United States and settling into a career as a political writer. Although George Curtis was always comfortable in the company of writers\u2014his closest acquaintances were Irving, Bryant, Lowell, Thackeray, and Longfellow\u2014by the Civil War he had become a respected and high-profile American statesman, and had twice served as a delegate at Republican national conventions. In 1869 he was offered (but turned down) the nomination to be secretary of state under President Grant.\n\nA fervent supporter and promoter of Abraham Lincoln, George Curtis was devastated by the president's assassination. Both George Curtis and Natalie's father gazed upon the slain body of President Lincoln\u2014George Curtis as a colleague and devotee at the memorial service, Bogey as a member of the surgeon general's team at the autopsy.\n\nLike many of their neighbors in Greenwich Village, the Curtises lost several family members in the Civil War, including Bogey's brother and George Curtis's stepbrother Joe, a lieutenant colonel who fell in the Battle of Fredericksburg. George Curtis's brother-in-law, Robert Shaw, was just twenty-seven when he died at the head of his regiment, the famous Massachusetts 54th, the first all-Negro regiment in the Civil War. Colonel Shaw and his troops stormed Fort Wagner in South Carolina on July 18, 1863. One-fourth of the regiment fell that day. Shaw was among the first killed, and the victorious Confederates stripped, robbed, and buried his body in a mass grave with his Negro comrades. When Uncle George's wife, Anna Shaw, and her family learned of Robert's dishonorable burial in South Carolina, the army offered to disinter his body and return it to Boston for a proper burial with full honors. Shaw's father, Frank, responded: \"We can imagine no holier place than that in which he lies, among his brave and devoted followers, nor wish for him better company.\"\n\nA confirmed Independent by the 1880s, George Curtis's weekly political column in _Harper's_ reached tens of thousands of devoted readers. With the spoken and written word his platform of choice, the gentle, even-tempered George Curtis had a \"big manner,\" Walt Whitman wrote of his friend, \"and without being offish his personality had a large swing, as if it had plenty of time and space in which to live.\"\n\nFireside chats in the Curtis parlor in Greenwich Village were attended by the progressive individuals who were reshaping American principles, politics, and aesthetics in the second half of the nineteenth century. Natalie's civic and cultural education, and her moral conscience, were supplemented and influenced by the conversations and activities of her Uncle George and his famous peers, who debated political theory and social and educational reform around the Curtis dinner table.\n\nFor the precocious and gifted Natalie, George William Curtis exemplified a life devoted to intellectual exploration, creative expression, social reform, and the unrestrained pursuit of one's passion. Natalie adored him, and as she grew into maturity as a musician and a woman, and struggled to find her place\u2014and to leave home to find that place\u2014in the new world of the twentieth century, Uncle George's spoken and written wisdom guided her choices and bolstered her resolve.\n\nMainstream Protestantism was undergoing challenge from unconventional and esoteric religious movements in the 1890s. Natalie and Mimsey attended meetings of the Theosophical Society that introduced Buddhism, Sufism, and Hinduism to New York's upper class. Theosophy encouraged the study of comparative religion, philosophy, and science, and also endorsed the concepts of karma and reincarnation. Life was not haphazard, but part of a grand scheme. Theosophy affirmed teenage Natalie's intuition that her art and her life were one, and that music was her divine destiny.\n\nNatalie was more than a coming-of-age feminist; she was an emerging humanist. Where traditional, Anglocentric Americans saw the Other in race, religion, and culture, Natalie recognized common ground. It was a perspective that would cause her to challenge conventional ideology, and push her to pursue a destiny far from Washington Square in the exotic and \"uncivilized\" territory of Native America.\n\nIN THE SUMMER OF 1896, Natalie went to Europe with her brother George. She was twenty-one, and it was her second journey across the Atlantic. Natalie and George traveled to London, through the Rhine region of Germany, and across Italy. By late fall, the siblings had settled into rooms in Paris, where George enrolled at the \u00c9cole des Beaux-Arts and Natalie was accepted into voice classes at the revered Paris Conservatory.\n\nIt was to be a winter devoted to study and creative stimulation, but it all came to an end when Natalie suffered an acute attack of appendicitis. Mimsey boarded the first ship to France and took Natalie by train to Switzerland for medical treatment. After Natalie recovered her health, Natalie and Mimsey journeyed to Bayreuth, Wagner's home and now the mecca for all things Wagnerian, and Natalie enrolled to study the operas she had loved since childhood. Natalie's reputation as an accomplished young American pianist opened doors to Bayreuth's inner circle of musicians and socialites, and Natalie and her mother were introduced to Wagner's widow, Cosima, with whom they spent a memorable New Year's Eve 1896.\n\nBack in residence at Washington Place that spring, Natalie received an invitation from her former teacher Anton Seidl to perform a piano concerto with the Philharmonic Orchestra. Natalie, now twenty-two, was straddling two worlds: the world of the professional, paid musician, and the world of a proper single woman who lived at home with her parents. Seidl recognized an enormous talent in Natalie Curtis and urged her to take her place on the professional stage. The invitation flattered even as it terrified Natalie. She understood that although she would be equal to and respected by New York's top musicians, her professional career would be limited by her gender. She also understood that she would be expected to step down from the professional stage when she married. And if she did not marry, she would be sentenced to the life at the periphery of conventional society experienced by all single women who dared to pursue their creative passion.\n\nSeidl's invitation to join him on the professional stage pushed Natalie's conflicts to the surface. Even in a progressive household like the Curtises', it was understood that daughters would marry and raise a family. Even an accomplished musician like Natalie was biding her time until she was betrothed. As a married woman with children, she could teach music and continue to play publicly, but not for pay. Music would become her hobby, not her vocation. Natalie's passion for music, indeed for anything, would have no place in a Victorian woman's home.\n\nNatalie's fate as a concert pianist was not decided by intellectual debate within the Curtis family, or in private conversations with her mentors and friends. Instead, it was Natalie's obsessive demand for musical excellence that ended her dream of becoming a professional musician. A debilitating daily practice routine, coupled with the sobering reality that her artistic dreams were laden with uncertainty, broke Natalie's health, and she collapsed in the fall of 1897.\n\nGeorge claimed his sister's concert career ended with \"a strained hand, due to over-zealous piano practice.\" Bridgham's account of Natalie's collapse was less specific about the cause, but he, too, acknowledged the magnitude of that moment in Natalie's life. While preparing for the most important concert of her young life, Natalie suffered nothing less than \"a severe physical breakdown that made all thought of a career as a pianist out of the question.\"\n\nIt was a line of demarcation in the life of Natalie Curtis. She was as emotionally devastated as she was physically broken. Hindsight suggests that Natalie understood the confines of the cultural prison that she, even as the daughter of progressive parents, was sentenced to accept. Natalie's physical and emotional withdrawal from the performance with the Philharmonic avoided a head-on collision between the reality that was a proper woman's place in upper-class New York society and the artistic path she had dreamed of and pursued since childhood.\n\nNatalie had no helpful mentors and no answers, personally or professionally. What she needed was to retreat to a cabin in the woods near a place like Walden Pond where she could find solace and glean wisdom from the elders. But the elder who could have given Natalie an astute and hopeful perspective on life and art, on the individual's path and the grand scheme, her Uncle George, had been dead for five years.\n\nNatalie's convalescence was slow. She remained in the brownstone on Washington Place, and although she was physically unable to play the piano, began to compose pieces for voice, piano, and violin. Gradually, her strength returned, and she played a little piano and sang her new songs for family gatherings. But her grand and passionate dreams of a career under the lights of the world's great concert halls were snuffed. \"Whenever she played her touch was sensitive and lovely,\" George wrote, \"and her voice, whether in singing or in lecturing, had a sweet and moving quality of its own.\"\n\nNatalie was an adept composer, and several of her first compositions were published by G. Schirmer: \"Dearest Where Thy Shadow Falls,\" published in 1898, and \"Song from Pippa Passes,\" in 1899. One long composition based on the poems of Robert Louis Stevenson, \"Songs from a Child's Garden of Verses,\" was published in 1902.\n\nNatalie seemed resigned to the life of an unambitious introvert. Considering the world-renowned maestros she had trained with and the prestigious concert halls she had performed in, playing a little piano and singing a few songs for an audience of close friends and family must have been as satisfying to Natalie Curtis as a trot among the stable ponies would be for a derby-trained thoroughbred. Natalie was a finely honed artist of \"unquestioned brilliant promise\" who seemed to have lost her confidence and gumption. At twenty-five years of age, she was grappling with the dark night of the soul.\n\nGeorge returned from Europe and his studies at the Beaux-Arts and settled into life in New Haven. His job in the stacks of the Astor Library suited him, but the damp climate of New England aggravated his asthma, and in 1900, worn down by frequent and incapacitating attacks, George did what ailing men of means could choose to do: he quit his job, packed his bags, and boarded a train for the arid, pristine air of the American West.\n\nThe Curtis family had friends and relatives in the Los Angeles area, but George went to rural Arizona and looked for work in the small towns along the Santa Fe Railway. Although his equestrian skills were limited and his knowledge of cows nil, George was hired by one of the large cattle outfits. Life in the saddle under the big sky of the West\u2014a life of long days on horseback, short nights on the hard ground, with meals cooked on a campfire\u2014rehabilitated his body and stimulated his mind in much the same manner as the life on Brook Farm had restored and inspired his Uncle George.\n\nTo Bogey and Mimsey Curtis and their friends on the Eastern Seaboard, the American Southwest was an exotic territory at its best and an uncivilized, even savage frontier at its worst. The public relations department of the Santa Fe Railway was working overtime at the turn of the century to change Anglo-America's perception of the southwestern United States. In pamphlets and newspaper articles the American public was told to \"See America First\" and that the landscape and indigenous cultures of the western half of the continent were as exotic, dynamic, and worthy of exploration as anything found in Europe or the Middle East.\n\nThroughout 1900 and 1901, George sent letters and picture postcards of his travels through California and Arizona home to Natalie and the family in New York. His words and images sparked Natalie's dulled imagination and diminished spirit. And although a call to adventure was not a conventional choice for a twenty-seven-year-old woman, especially if she were single and of dubious health, Natalie announced in 1902 that she wanted to join George out west.\n\nBogey and Mimsey, painfully aware that Natalie was not recovering her stamina and her famously energetic joie de vivre sequestered within the rooms of the Washington Place brownstone, consented to her request. George came east for a family visit in the fall, and when he boarded the train west again, Natalie was at his side.\n\nThe siblings went to Southern California and stayed with family friends near Los Angeles, where the winter weather was balmy and bright. On New Year's Day 1903, Natalie and George went to Pasadena for the outdoor celebrations and events associated with the Tournament of Roses. There, a new friend, Charles Lummis, invited them to an event showcasing Navajo singers from Arizona.\n\nThe Navajos had been invited to Los Angeles by the controversial author, editor, and outdoorsman Charles Fletcher Lummis and were camped on land near his home, El Alisal, in Pasadena. It is possible that Lummis, a political activist and journalist, had known Natalie and George's uncle George William Curtis years before in Boston. Lummis was born and raised in Massachusetts and had attended Harvard, where his classmates included Theodore Roosevelt. By 1903, and Natalie's arrival in Los Angeles, Lummis's _The Land of Poco Tiempo_ had been in print for a decade. The book was a popular success and practically required reading for Yankee tourists and everyone else who ventured to \"See America First\" (the phrase coined by Lummis) and make the bold journey into the Southwest.\n\nLummis's episodic narrative about the land of \"pretty soon\"\u2014the Territory of New Mexico\u2014introduced Americans like Natalie to the people, geography, history, and essence of the Southwest. Lummis was a disciple of the open air, close-to-the-earth lifestyle championed by Thoreau, Emerson, and John Burroughs, and advocated by Uncle George, and had walked the length of New England as an undergrad. In 1884, at the age of twenty-five, Lummis left his desk job at the _Scioto (OH) Gazette_ and began a trek\u2014he called it a \"tramp\"\u2014across the American continent. Following railroad lines and wagon roads where they could be found, Lummis walked 3,507 miles in 143 days. His destination was Los Angeles, and his six million footsteps wove through natural and cultural wonders in Colorado, New Mexico, Arizona, and finally Southern California. Lummis's tramp included stopovers at the pueblos of Tesuque and Isleta in New Mexico, and his time living among the natives of these ancient villages had redirected the course of his life and work.\n\nAt dinner parties and social events at El Alisal (Place of the Sycamores) in the hills east of Los Angeles, Natalie and George were introduced to the Los Angeles artists, intellectuals, and social reformers who congregated at Lummis's home. It was a lively introduction to the rich, complicated, and many-layered cultural and political society of the twentieth-century Southwest. El Alisal was a cozy, rustic, enchanting adobe and stone home hand built by Lummis and his native friends in the oak, cactus, and sagebrush Arroyo Seco near Pasadena (later called Highland Park). The house was striking in style and character and would be recognized as among the first to exhibit the no-frills, clean, clutter-free, close-to-nature design, construction, and interior decor aesthetic of the California arts and crafts movement.\n\nNatalie was plunged into a new visual, cultural, and intellectual environment at El Alisal. Lummis, like Natalie, was driven by a \"hunger in the head,\" a passionate need to explore the whys and wherefores of the world. Editor of _Out West_ magazine, Lummis was the national spokesman for the paradise he had found in California. His clean, straightforward prose about the idyllic life and pristine landscape found out west was nearly as responsible as the Santa Fe Railway's promotion department for the mass immigration of artists, writers, health seekers, and adventurers into the Southwest in the first decades of the twentieth century.\n\n**Figure 1.** Interior view of El Alisal, the home of Charles F. Lummis, ca. 1920.\n\nLummis told stories about roaming the Pajarito (Little Bird) Plateau in New Mexico with the great Swiss archaeologist Adolph F. Bandelier. On foot, on horseback, and on burro, Lummis and Bandelier explored the canyon and mountain country west of the old city of Santa Fe, the region known as El Rito de Los Frijoles (Creek of the Beans) that would one day become Bandelier National Monument. \"Thousands of miles of wilderness and desert we trudged side by side,\" Lummis wrote of his days with Bandelier, \"camped, starved, shivered, learned and were glad together.\"\n\nNatalie had enjoyed the out-of-doors in New England and on the Atlantic coast, and she had twice crossed the Atlantic and stayed in Paris, the Swiss Alps, and the lush German forests near Bayreuth. But Lummis described a landscape the likes of which she had never seen or even imagined, a place that was \"a picture, a romance, a dream . . . a land of ineffable lights and sudden shadows . . . where the rattlesnake is a demigod, and the cigarette a means of grace . . . the heart of Africa beating against the ribs of the Rockies.\"\n\nAt their home in the Los Angeles foothills, Lummis and his wife, Eva, held dinner parties and \"noises\" (salons) to which artists, writers, politicians, and activists regularly flocked. Newcomers like Natalie and George mingled with \"the great, the near-great, the merely colorful, and assorted hangars-on\" who on any given night could include John Muir, John Burroughs, and a California labor and Indian rights activist, suffragette, and writer named Mary Austin.\n\nLummis and his friends were involved in the \"roiling caldron of Indian rights\" and with their Sequoyah League, founded in 1902, were active participants in the heated national political debate surrounding Indian reform policies. Lummis used the pages of _Out West_ magazine to criticize Washington's Indian education programs, and took his grievances about federal Indian programs, especially the Indian boarding schools, to his college pal President Roosevelt. Theodore Roosevelt was frequently peeved and even outraged by Lummis and the Sequoya League's methods of activism, but to his Washington staff's chagrin, the president welcomed the provocative California writer into the White House.\n\nThrough Lummis, Natalie was introduced into the society, issues, and people of Southern California. California was a foreign country with its own traditions, history, and lifestyle. Even the lively salons in Arroyo Seco had an unmistakably western flair. Lummis's wardrobe of choice (even for visits to the White House) was a Spanish bolero jacket, corduroy trousers and matching sport coat, and leather moccasins, with a wide sash around his waist, and a colored kerchief knotted about his head or neck. Khaki skirts and riding slacks, cowboy boots, Levi's, broad-brimmed hats, and colorful ties and belts were the norm for California women and men. The staid attire and formal social graces deemed proper and acceptable back in New York and Boston had no place in Southern California. Natalie had stepped into a new world, and her body, mind, and spirit responded with vigor and enthusiasm to the myth and the reality of the American West.\n\nLummis was not a musician, but he had been collecting and recording the folk music of the Spanish and the indigenous Indians of the Southwest since the mid-1880s. Natalie certainly asked Lummis about the region's musical traditions, and studied his notes and listened to his recordings. And she was certainly Lummis's most educated guest at the dance performance given by the Navajos at the Tournament of Roses celebration in Pasadena on New Year's Day 1903.\n\nAfter several months in Southern California, Natalie was strong enough to go with George to Arizona. They left Los Angeles on the train bound for the southwestern-most corner of the United States\u2014Yuma, Arizona. The seven-hour journey took them through snow-covered mountains and across a flat cactus-studded desert near the Salton Sea. Exactly why they chose Yuma, in the very middle of the continent's largest, driest, hottest environment, the Sonoran Desert, is not known. Yuma was, in George's words, a \"humid hellhole . . . a town filled with the oft-scourings of Arizona\" where the best and only lodging was found at the Hotel Gandolfo, where guests dined on rice and cabbage, and paid a dollar a night for rooms that were \"not good.\"\n\nLummis had traveled through the region and would have warned Natalie and George about spring on the desert. \"High noon in southern Arizona. The sun is a flood of infinite fire, wherein earth and sky are saturated. The heavens are an arch of burnished brass. The blinding landscape seems the thin crust over a sea of boiling lava. . . . Touch a palm to that sand, and you shall acquire blisters as from a red-hot stove.\"\n\nThe Colorado River and the green bosque that clung to the moisture found along its banks were the town's only inviting and attractive features. All travel around dusty, sunburned Yuma was by wagon, on horseback, or on foot. Natalie visited the state penitentiary and took notes for an article, but her interest from the start was not the so-called civilized side of the river, but the Native American community living on the reservation on the other side.\n\nThe reservation of the Yumas (Quechan, or Cochan), a Yumanspeaking tribe of desert farmers on the south side of the Colorado River, was the home to six hundred Indians. Natalie was impressed by the Yumas' peaceful, self-contained community, and that not one of their members had ever been incarcerated in the state prison. \"In spite of their proximity to the town,\" she wrote in what was among the first of dozens of articles about the Southwest, \"these Indians are comparatively free from many of the vices unconsciously acquired by a primitive people who are ignorant of new temptations.\"\n\nNatalie was stepping into a world rarely glimpsed by Anglo-American women\u2014the homeland of Native America. A few women had broken ground before her: anthropologist-ethnologist Matilda Coxe Stevenson had worked with her husband in the 1880s at Zuni and other New Mexico pueblos, and Alice C. Fletcher of the Peabody Museum had begun work in the 1880s among the Sioux. But as a single white woman of education and means, traveling at her leisure across Indian Country seeking to further her understanding of the Southwest's indigenous cultures, Natalie was a pioneer. \n\n#  _2_\n\n##  LAND OF INEFFABLE LIGHT AND SUDDEN SHADOW\n\nThe moonlight flooded that great, silent land. . . . The senses were too feeble to take it in, and every time one looked up to the sky, one felt unequal to it, as if one were sitting deaf under the waves of a great river of melody.\n\n##  \u2014 _Willa Cather_\n\nWHEN THE CURTIS SIBLINGS came to Yuma in the spring of 1903, their gear included a \"talking machine,\" an Edison phonograph and recorder that may have been given to Natalie in Los Angeles by Lummis. The dance-songs performed with \"lusty vigor\" by the Navajos at the New Year's Day performance in Pasadena, coupled with Lummis's stories of the indigenous people he had met and recorded during his treks in the outback, had sparked Natalie's interest in native song, and she intended to go into the desert in search of music.\n\nJust as she had boldly introduced herself to the great maestros at the Met, and gained admittance into the inner sanctum of Wagner's family home in Bayreuth, Natalie walked across the bridge that spanned the Colorado River and introduced herself to the Yuma Indians living on the other side. Who precisely served as her guide is not known, but when she crossed the river that day, Natalie began an exploration into the heart, mind, and music of Indian America that would become her passion and her vocation for the rest of her life.\n\nNatalie's initiation into the \"primitive\" world of Native America began with conversations with Indian elders. Chiparopai (also spelled Hiparopai in Natalie's writings) was an elderly Quechan woman who spoke English and Spanish. Like many Yumas, Chiparopai lived miles from the river and the white railroad town of Yuma, preferring the open desert where she could \"feel the air wide around me.\"\n\n**Figure 2.** George Curtis and Natalie Curtis, playing a hand game with an Indian friend, ca. 1904.\n\nNatalie met Chiparopai seated beneath a plum tree during one of the old woman's visits to relatives who lived \"by the wells.\" Natalie told Chiparopai that she was interested in understanding and knowing the Yuma people, and asked Chiparopai if she could take notes about their conversation. Chiparopai agreed, and answered Natalie's questions with a blunt and poignant honesty. \"Never mind if we are poor,\" the elder told Natalie in the shade of a thatched shed. \"What good is life if we are not happy? . . . White people think that money is everything; we Indians think that happiness is more.\"\n\nNatalie commented on Chiparopai's persistent racking cough and asked why so many Quechan were ill. \"It seems as though you white folk bring poison to us Indians,\" the elderly woman answered. \"Sickness comes with you, and hundreds of us die.\"\n\nThe irony was not lost on Natalie: she and her brother George had come to the desert to be healed, to live.\n\nIn her conversations with Chiparopai, Natalie learned how Indian Americans felt about Washington's assimilationist policies that banned their language, religion, and dress and forcibly placed their children in government boarding schools. Especially brutalizing to Indian children was the Bureau of Indian Affairs' (BIA) strictly enforced practice of cutting off their long hair. Chiparopai also admonished white America's preference for stiff clothes and hard collars. Natalie assured Chiparopai that not all white people agreed with federal policies regarding Indian education.\n\nNatalie recorded Chiparopai singing \"Song of the Mocking-Bird,\" and asked her about her tribe's spiritual beliefs. \"How would we know how to live if we did not believe in something greater than ourselves?\" the old woman asked Natalie. \"And when we die?\" Natalie asked. \"Then it is all happiness,\" Chiparopai answered. \"When we _die_ we are all alike.\"\n\nNatalie published an article in the _Craftsman_ about her conversations with Chiparopai. She prefaced the story with the disclaimer that she, as the recorder, sought only to convey an individual Indian's opinions and thoughts about the changes brought to their world by the whites. She was not interested in conducting an academic study of Indian society or religious beliefs, but rather in learning who the people of Native America were and how they thought about their world and culture. She was also very interested in their music. \"You want to know my people?\" Chiparopai asked Natalie. \"Live among them. See them in their fields, planting and harvesting; see them with their children; hear them sing!\"\n\nNatalie took Chiparopai's counsel and, with George, traveled deeper into Indian Country to find and hear the music of the native people. They boarded the train with the bulky Edison recorder and wax cylinders and several bags of gear and headed east across Arizona. George was comfortable traveling by horse and wagon, but the Arizona wilderness had few maintained roads in 1903, and the Curtis siblings traveled by train whenever possible.\n\nIn his first years in the Southwest, George had lived and worked on a cattle ranch near Houck, Arizona. The cowboy way and a life spent close to the land and far from people fit the former librarian's temperament. As his uncle George William Curtis had done in his formative years near Walden Pond, George DeClyver Curtis, nearly thirty when he came to Arizona, began to write. Unlike his uncle, George had no ambitions to become a career journalist and had little interest in achieving a name for himself. (In 1914 George would publish his only novel, _The Wooing of a Recluse_ , under the pseudonym Gregory Marword.) But George was a competent and thoughtful writer, and even when living in a tent in the outback, he regularly recorded his adventures and experiences in his diary.\n\nIn 1903 guest ranches set up to accommodate eastern dudes looking to play cowboy and to explore Indian Country did not yet exist. Natalie's first experience among cowboys and cattle was at a working ranch, probably near Houck, where life was a \"hand to hand tussle with an elemental world\" shaped by \"scorching sun, wind, sudden cloudbursts, thirst, exhaustion, violent danger, loneliness, [and] isolation from the ties of home-women.\"\n\nUnless Natalie and George were in a community like Yuma along one of the railroads that cut across the territory, or were staying at one of the Protestant missions or government schools on the reservations, Natalie's living quarters that spring and summer were exactly the same as George's.\n\n> I have ridden with cowboys, sung with them, seen round-ups and bronco-busting, spent months amid a thousand head of cattle on one of the loneliest ranges in Arizona, devoid of all intercourse with the outside world save for visits of wayfaring Indians or stray sheep-herders seeking water at our springs. . . . When the men left to \"ride the range\" and I was alone in the cabin with a Colt revolver for companion; when I heard the plaintive sob of the wood-pigeon in the cedars\u2014the prelude to the choir of coyotes who would howl at the rising moon; when I realized that should any strange man try to enter (in which case I had been instructed to \"shoot\") I might call forever and though the sandstone cliffs might send back my voice no human soul could hear or come\u2014then I understood the note of utter loneliness that sounds in many a cowboy song.\n\nNatalie could never have undertaken such adventures on the southwest frontier without her brother George. Just how much of their adventures were shared in letters sent to Bogey and Mimsey is unknown, but the details of her solitary nights in an isolated cabin with a Colt revolver to keep her company were certainly omitted from the siblings' missives to their family on Washington Place.\n\nFrom the cattle ranch near Houck, George and Natalie went by horse and wagon north and west toward what the early Spanish explorers and now the American anthropologists called Tusayan, or Hopiland. Their destination, the ancient mesa-top village of Oraibi, was a several-day journey across the Navajo Reservation through some of the most rugged, beautiful, harsh, and remote country in North America. Weather, the availability of water, and the physical condition and psychological stamina of brother and sister determined the speed and distance of their wagon's daily progress.\n\nAfter two days of slow progress across \"that Arizona wilderness of beauty known as the 'Painted Desert,'\" Hopi Third Mesa rose into view on the northwest horizon. The stone and mud pueblo of Oraibi appeared carved from the stone island of Third Mesa, which soared six hundred feet into the searing blue sky. Surrounded by an ocean of sand that fell away like waves into the four directions, the home of the Hopi (or Moqui) was as exotic, primal, and spectacular as anything Natalie Curtis had ever seen.\n\nOraibi had a trading post as early as 1897, and by 1903 the U.S. government had built several buildings below Third Mesa near the springs and wells that provided water to the natives. By midsummer there would be small patches of green cultivated fields in the flat land below the village mesa, but in spring the sand land boasted only small flecks of color that were the fragile desert wild flowers. \"The scarcity of water can hardly be conceived by those who have not been there,\" Natalie wrote after her first visit to Hopiland. \"Every drop used in Oraibi has to be brought for a distance of two miles, one-half of which is up the steep trail, and carried all that long way in heavy earthen jars on the backs of toiling women.\"\n\nFor nearly a thousand years, the Hopis of Third Mesa, like their tribal neighbors on the cliff-top villages on First and Second Mesas, had hauled all of their water, food, and provisions on their backs up the trails carved into the stone to their homes on top of the mesa. The Hopi were a historically peaceful tribe (their name means \"quiet people\" or \"people of peace\"), and for several hundred years they had managed to maintain a distance from European explorers and then American missionaries and other cultural invaders. But by the early twentieth century and Natalie's first visit to Oraibi, the policies of the U.S. government had begun to erode Hopi independence, and the people of Tusayan regarded the white world as wicked, and white men, and most white women, as deceitful.\n\nBefore Natalie and George departed Los Angeles in late January, Lummis may have suggested that their desert itinerary include a visit to the Hopi village of Oraibi. At that time, Lummis's Sequoyah League was launching a rigorous and covert campaign meant to undermine and expose the BIA's assimilationist policies sanctioned by Roosevelt's secretary of the interior, Ethan Allen Hitchcock. The league's efforts focused on the superintendent of Indian affairs, Charles E. Burton, at Keam's Canyon on the Hopi Reservation. Stories about overzealous BIA agents kidnapping Indian children from their homes and dragging them off to the government school had Lummis and his friends up in arms.\n\nAll of the solutions to the \"Indian problem\" in the United States after the Civil War advocated the same approach: the management and altering of \"savages\" into \"civilized\" Christian farmers. To implement the assimilation of Indians into the white world, the BIA opened boarding and day schools throughout Indian America. By stripping children of all outward signs of Indianness, and introducing educational programs meant to undermine and weaken their cultural identity, the U.S. government meant to bring all Native Americans into the \"civilized\" fold of white America.\n\n\"Kill the Indian, and save the man.\" The now infamous statement of Colonel Richard Henry Pratt, director of the Carlisle Indian School in Pennsylvania, pretty much summed up the federal government's agenda for Indians and Indian policy in the United States.\n\nTo young Indians sent to one of the federal Indian schools, a white man's education meant learning to speak English and never _ever_ speaking their native language while at school. Hair was cut short or shorn off completely, and traditional garments were burned and replaced by \"citizens'\" clothes\u2014white people's trousers, shirts, and shoes. Every Indian student slept in a bed, prayed to Jesus, used a toilet, and ate only white food with a knife and fork.\n\nThe remote, peaceable people of Hopiland were not spared the insensitive programs of the BIA meant to eradicate their language, culture, and religion. In 1902 Indian commissioner William A. Jones had sent out a letter to Indian agents and school superintendents, including Burton, which identified the native customs to be modified or eliminated. Long hair and traditional clothing were added to the government's formal list of forbidden behavior, which already banned use of native language or song in Indian schools. When the commissioner's rigorous instructions for assimilation of the Indians reached Burton at his office at Third Mesa, he instructed his agents to energetically, even brutally, enforce the reforms.\n\nIn April 1903, as Natalie and George made their way in their wagon across the Arizona outback toward the country of the Hopis, Charles Lummis and _Out West_ magazine launched a series of articles titled \"Bullying the Moqui\" that specifically targeted the alleged atrocities perpetuated by Burton at his schools on the Hopi Reservation, especially at Keam's Canyon.\n\nLummis's inflammatory articles about Burton drew the attention of the White House. In May 1903 Lummis was granted time with Roosevelt on his presidential train as it traveled from the Grand Canyon to Los Angeles. Although Lummis had access to the president for almost two full days, and presented in detail his grievances against Burton, Roosevelt was not ready to fire his superintendent of Indian affairs.\n\nLummis's _Out West_ continued to publish stories about the mistreatment of Indian students at Burton's boarding schools. Most of Lummis's allegations of abuse were based on secret investigations by a woman named Gertrude L. Gates, and on the testimony of an Oraibi schoolteacher, Belle Axtell Kolp. Earlier in the spring of 1903, Lummis had dispatched Gates, of Pasadena, to Hopiland to collect information on Burton's activities. Gates posed as an invalid who had come to the high desert for her health, but Burton's agents became suspicious and confronted her. Gates confessed that she was in Hopiland to spy on Burton, and she was unceremoniously escorted off the reservation.\n\nBurton's office may have been leery of Gates because she appeared on the Hopi Reservation just after the departure of Kolp, who had publically renounced Burton's education policies and resigned her teaching post in Oraibi in mid-February. Although Kolp, also from Pasadena, was not one of Lummis's spies, her eyewitness accounts of the mistreatment of Indian children in government schools helped launch the Sequoyah League's campaign against Burton. (Kolp was well versed in frontier politics and politicians\u2014her uncle was Samuel B. Axtell, one-time governor of New Mexico and later the territory's chief justice.) Upon her return to Pasadena, Kolp wrote a lengthy deposition outlining government offences against the Hopis at Oraibi. This deposition and her experiences as an Indian schoolteacher became the basis for a series of articles Lummis wrote and published in _Out West_ magazine throughout the spring and summer of 1903.\n\nThe BIA did investigate Burton's activities, but cleared the superintendent of most charges. And although the situation between the federal government and Native America, especially at Oraibi, remained a tinderbox, and federal Indian policy continued to be harshly criticized by reformers all over the United States, Roosevelt insisted Lummis issue a formal apology to Burton.\n\nNatalie and George knew about Lummis's many grievances against federal Indian policy. But in May 1903, when they pulled their horse and wagon up before the government buildings at the foot of Third Mesa, Natalie most likely did not know how she might be mistaken for yet another of Lummis's female subversives. That Natalie and George managed to move into a room in the BIA housing at New Oraibi is a testament to Natalie's compelling personality and George's gentle manner. However, the atmosphere in Hopiland was charged with suspicion and distrust, and their activities were closely scrutinized.\n\nIf Natalie had access to copies of _Out West_ and Lummis's articles about Burton's \"reign of terror,\" and if she knew about Gertrude Gates and her eviction from Indian Country, the antagonistic environment did not dampen Natalie's enthusiasm to visit the Hopis. However, it likely dampened her acknowledgment of the purpose of her journey through Indian Country\u2014to hear and record Indian music. The singing of native songs and chants was strictly forbidden. Had a federal official asked to inspect Natalie and George's gear, they would have known that the diminutive, engaging, and educated musician named Natalie Curtis was most certainly crossing the high desert to break the law, because the wagon was loaded with crates packed with a bulky Edison phonograph.\n\nAfter Natalie and George settled into rooms among the other white outsiders in the government buildings at the foot of Third Mesa, Natalie, with George and a translator\u2014one of the English-speaking Hopi boys who had attended a government school\u2014hiked up the steep, exposed, cliff-hugging trail to old Oraibi. With her field notebook in hand, and dressed in what would become her desert uniform\u2014khaki skirt, white long-sleeved shirt, lace-up leather boots, and a broad-brimmed hat\u2014Natalie introduced herself to the villagers.\n\nOraibi, like each of the Hopi villages of First, Second, and Third Mesas, was a cluster of several-storied mud and stone dwellings built around small plazas perched on the narrow top of an island mesa hundreds of feet above the shimmering floor of the desert. The first Spanish priests in the Southwest built a chapel and mission near Oraibi in the early years of the seventeenth century, but the Hopis peaceably and tenaciously resisted Catholicism and maintained their own spiritual traditions. After the Spanish were thrown out of Arizona and New Mexico in the uprising of 1680, the Franciscans never returned to Hopiland. The Hopis adopted the Spaniards' sheep, burros, and horses, and when Natalie arrived in 1903, the desert landscape below Third Mesa boasted orchards of peach and apricot trees.\n\nThe Hopis were a predominantly vegetarian culture, and the daily meal included maize (corn), beans, squash, watermelons, and sunflowers. Corn was the primary food source, but among the Hopi, as among all desert dwellers, water was the supreme life source. A rainstorm was a celebrated event, and every adult and child participated in the capture, collection, and storage of the precious commodity. Hopi was and is a matrilineal, monogamous society, and Natalie was introduced to women who owned and occupied their own dwellings in the villages. Women also owned cultivated land, including some of the orchards and horses.\n\nThe village of Oraibi and the people of Hopiland surpassed Natalie's expectations. Every act of life was ceremonial, poetic, symbolic. \"I was not prepared to find a people with such definite art-forms, such crystallized traditions, beliefs, and customs.\" The music of the Hopi \"astounded\" her. \"I had not been in Oraibi twenty-four hours before my mental picture of the scope of music faded like a desert mirage, revealing a new and far-off horizon stretching boundless as the desert itself\u2014a new world of art.\"\n\nNatalie made friends easily, and her residence below Third Mesa was soon visited by her Hopi neighbors from dawn until dusk.\n\n**Figure 3.** Hopi village of Oraibi, ca. 1907.\n\n> My workshop, as I called it, was one of the \"government houses\" at the foot of the mesa. . . . In one of these houses I had my phonograph, and here the Indians collected daily with true Hopi curiosity to peek in at the windows, to stare at the \"Pahana\" [white or American woman], and to join in the fun and excitement of sin in into the machine \n> \n> Indeed, I had but to be seen issuing from the house where I lived, to be followed by a stream of Hopis\u2014women with babies on their backs, men on their way to the fields, children just off to school. The Indians opened my door and entered my little house as unceremoniously as they did each other's. I always welcomed them, and offered them the customary empty boxes to sit on. There were never enough to go around, and half my visitors sat on the floor. But this was Hopi style; boxes are an innovation of civilization.\n\nIn the month and a half that Natalie was among the Hopis, she cultivated relationships with everyone who would talk with her. With native guides, she traveled by wagon into the vast and often labyrinthine landscape of deep unnamed canyons, sheer cliffs, wide washes, and treacherous rock chasms to reach the desert camps and outposts where Hopi singers and storytellers were found. Natalie spent entire days on foot or in the saddle, and slept in a bedroll on the ground under the stars or in a cabin or tent near a trading post if one was nearby.\n\nSix months after leaving New York, Bogey and Mimsey would not have recognized their daughter, the confident desert traveler. Natalie's physical and emotional strength increased with every week spent under the sky of Indian Country. Her body was healed by the landscape, and her heart had reconnected with her passion in the music of the people who called the desert home.\n\nDuring one desert trek, a sandstorm engulfed Natalie and her Hopi guide and literally blew away the road. They walked into the night, and as the sky cleared and became luminous with stars, Natalie's guide began to sing. The wild beauty of the music, and the soulful, uninhibited quality of the guide's voice, astonished Natalie. Even as her classically educated ear struggled to understand the pitch and cadence of his song, Natalie attempted to sing along. \"I joined my voice to that of my guide. Not a soul to hear, not one to comment or criticize\u2014only the desert stretching broad and far, only the craggy buttes and cliffs, only the heavens and those great bright stars. What freedom to sing one's very soul out into such a wide vast night! We lifted our faces to the sky and sang.\"\n\nEvery act of Hopi life was accompanied by song. Planting fields, grinding corn, dancing, putting children to sleep, watching the approach of the \"yellow light\" (dawn), the setting of the sun, rain\u2014every season, every daily chore, was honored by song. Among the Hopis, Natalie was surrounded by music. For Natalie, whose entire life had been the pursuit of music, life _was_ music. In the remote desert world of the Hopis, Natalie was among people who understood what she had always known: music was _life_.\n\nBefore attempting to talk with any of the Hopis in Oraibi, Natalie went to meet the village leaders. The most important meeting was her introduction to the elderly _kikmongwi_ (village chief) Lololomai, his daughter, Noyasoya, and his sister, Ponianomsi. Lololomai (Very-Good) was about eighty years old in 1903, and was one of only a few Hopi who had traveled out of Hopiland. In the mid-1880s Lololomai had gone with the trader Thomas Keams to Washington to ask President Chester Arthur to stop Navajo encroachment onto Hopi lands. Impressed by Euro-American civilization, Lololomai asked the federal government to open a school at Keam's trading post. By 1903, however, Lololomai's support of federal Indian schools, especially those under Superintendent Burton, had ended.\n\nNatalie sought the elderly chief 's approval of her intention to write down and also record on the Edison phonograph the songs of his people. She was taken by her interpreter, \"a Hopi lad, who, although blind, led with sure foot the way up the steep, rocky trail to the village,\" to meet Lololomai, whom they found \"seated on his housetop, spinning, for in Hopi-land it is the men who spin and weave.\"\n\nAfter Natalie explained that she had come to talk about \"something that concerns your people,\" Lololomai invited her to climb the ladder and join him on the rooftop. From the flat-topped roof with its 360-degree view of Third Mesa and Hopiland, Natalie told Lololomai, through her interpreter, that if Hopi songs were not written down, they would be forgotten. Lololomai understood the value of recording his people's songs, but he was fearful of repercussions against his village if Superintendent Burton learned about their collaboration. Natalie assured the chief that she would not reveal the names of collaborators, nor speak of their sessions to the whites associated with the federal government. The chief gave Natalie's work his blessing, and in late spring of 1903 Natalie began to collect and record the songs of the Hopi.\n\nAlthough Natalie and George were offered food and shelter in the government-issue headquarters at Third Mesa, and Natalie made friends at Oraibi and the other Hopi villages, her presence, like that of all whites, was the source of heated discussion among the Indian people. It was a volatile time in Tusayan, and at Oraibi Natalie was involved with individuals who were at the center of the political firestorm that would soon be known as the Oraibi Split.\n\nBeginning in the late 1800s, there was disagreement among the Hopis of Oraibi as to the tribe's cooperation and interaction with white America. By 1900 the disagreement had become a rift that divided the village into two factions, the Friendlies and the Hostiles. The Friendlies, who included Chief Lololomai and his family, although ambivalent about interaction with the white world that sought to diminish their traditional way of life, believed that at least a _little_ cooperation with the whites was necessary for peaceful survival. The Hostiles wanted nothing to do with the Two Hearts, the white-skinned people Hopi prophesy had predicted would come with guns and poison. (Smallpox had arrived in 1899 and was epidemic by 1901.) And when Burton and the BIA began to forcibly remove Hopi children to boarding schools, even Chief Lololomai began to question his friendly attitude toward the whites.\n\nIf Natalie knew how fractured and volatile the native community on and below Third Mesa was in 1903, she did not let it stop her recruitment of prospective collaborators. And in spite of her membership in the white race, the people of Hopiland opened their homes and shared their music and stories. \n\n#  _3_\n\n##  CAPTURING THE WIND\n\nSingle women making their way to individual destinies\u2014who in the home circle understands them? If they try to share what they have found in their further reach, who wants it?\n\n##  \u2014 _Elizabeth Shepley Sergeant_\n\nBY THE FIRST OF JUNE, news reached Natalie in Oraibi that a close friend of the family had died in New York. George was working at a cattle ranch when he received the same news, and he traveled to Third Mesa, packed up Natalie and her gear, and went with her to Winslow, where they caught the Santa Fe Railway's eastbound train.\n\nNatalie had been summoned home by Mimsey and Bogey to sing at the memorial service of the family's deceased friend. Natalie returned, the dutiful daughter, but it was immediately obvious to the family that nine months in the West had changed her. She was not the professionally ambivalent, emotionally fragile woman who had left Washington Place last fall on the arm of her older brother. Months spent in primitive quarters in frontier towns, in austere rooms at government agencies, and in cowboy camps and outposts had transformed and matured Natalie.\n\nAt the memorial service, a suntanned, buoyant Natalie surprised her family and friends by sharing several of the songs she had learned from the Hopis. She spoke briefly about the people and cultures \"so rapidly vanishing\" in the American West, and how she believed it was important to record and preserve Indian music. That the Indians of America had music was a revolutionary concept to most New Yorkers, even among the Curtises' progressive circle of friends. \"Cultured people in the east, except the few who were actively interested in the work of Indian welfare . . . knew little or nothing of the Indian,\" Natalie's younger brother, Bridgham, wrote. \"To them he probably appeared as the romantic painted 'savage' of Cooper's tradition, or as the taciturn and treacherous enemy of the border warfare days, or as the debauched and grotesque outcast of the frontier town; and to them I think Natalie's relation of what she had seen and heard was a revelation.\"\n\nNatalie didn't so much settle into life on Washington Place as she unsettled the life of the Curtis family with her tenacious pursuit of a new mission. On arrival in the city, Natalie drafted her first article about the Southwest and handed it to one of Uncle George's colleagues, who published the piece in _Harper's Weekly_. Boldly titled \"An American-Indian Composer,\" it was to be the first of scores of magazine articles Natalie would write about the people, music, and landscape of the American Southwest in the next two decades.\n\nBogey and Mimsey knew from experience with Natalie's previous self-appointed tasks that her new calling would likely not be a casual hobby, but a rigorous immersion. For Natalie to undertake her new mission, she had to overcome two substantial obstacles: she needed funding to cover the cost of expeditions into Indian Country, and she needed to find a way around the federal law that essentially undermined her ability to find and work with native collaborators during her expeditions.\n\nWithin a few weeks of returning to New York, Natalie contacted the one person who could quickly and absolutely grant her safe passage through Indian Country: President Theodore Roosevelt. By mid-July she had secured a personal audience with the president at the Roosevelt's summer home, Sagamore Hill on Oyster Bay, across Long Island from Natalie's family home in Far Rockaway. She was able to gain access to the president so easily and quickly because of Roosevelt's long association with Uncle George Curtis. George William Curtis was considered a founding member of the Republican Party, and he was also among those who, in league with Roosevelt and Henry Cabot Lodge, famously revolted against the \"old-line\" Republican Party at the stormy party convention in 1884 and created the Independent Republicans, aka the \"Mugwumps.\"\n\nGeorge Curtis did not live to see his friend become governor of New York or president of the United States. But in his niece, Roosevelt surely saw much of his friend's charisma, creative passion, and intelligence. Roosevelt likely recognized a bit of himself as well in the diminutive person of Natalie Curtis, a woman who shared his own \"tremendous capacity for work and a driving need to be productive.\"\n\nIn Roosevelt's Sagamore Hill study, Natalie candidly outlined what she believed were the numerous injustices perpetrated by the federal government on the land and culture of the Indians of the American West. Natalie pitched her cause with unflinching conviction. The president knew that every word she uttered, every opinion she offered, was the truth as she understood it. Her Uncle George had done precisely the same in lectures and speeches and in his political columns in _Harper's_.\n\nNatalie would look back at her meeting with the president of the United States when she \"was but little more than a girl\" and recall Roosevelt's capacity to consider a \"radically opposite view-point.\" Roosevelt patiently listened to Natalie's comprehensive litany about \"the benighted policies of the Indian Bureau in Washington\" and its \"ruthless campaign of destruction of all things pertaining to the indigenous culture of the red race.\" Of course, he was well versed in these complaints. Even as Natalie sat in his study at Sagamore Hill, Roosevelt's college pal Charles Lummis was publishing inflammatory articles about Superintendent Burton and about Roosevelt's Bureau of Indian Affairs in _Out West_ magazine. It was a prickly subject, but something about Natalie Curtis's plea and her \"straight-from-the-shoulder truthfulness\"\u2014and the exotic, strangely beautiful songs she carried from the faraway desert village of Oraibi\u2014captured Roosevelt's imagination and his admiration.\n\nNatalie had impeccable credentials. She was a classically trained musician, the daughter of a respected New York doctor, and the niece of a friend and political ally. But perhaps of more importance to the president in 1903, Natalie Curtis was an astute eyewitness of current events in Indian Country. She had traveled into Native America and undertaken what she believed was important work in a difficult, even hostile environment. She had also managed to make friends with the elusive Hopis. And she meant to return.\n\n\"Roosevelt respected anyone, regardless of color or class, who could produce under pressure.\" When the twenty-eight-year-old Natalie Curtis came to his summer home to talk about Indian policy, Roosevelt recognized a person of perspicacity committed to a serious mission.\n\nNatalie told the president she planned to continue with her recording of Indian music. In defense of her law-breaking task, she insisted that any attempt to educate the native people by the white people ought to be framed and informed by a thorough study of Indian culture. \"They can teach us as well as we them,\" she lectured the president.\n\nRoosevelt's recognition of the value of native culture was, in Natalie's words, instantaneous:\n\n> \"I am thoroughly in sympathy with the idea of preserving and, if possible, of developing the art, the music, the poetry of the Indian,\" [Roosevelt] said, with that emphasis of conviction that made his decision like powerful strokes of a forging hammer beating out results. \"It fits in with my policies of conservation, and I consider this question\u2014the conservation of Indian art in our education of Indians\u2014important enough to include in my next Message to Congress.\"\n\nBefore Natalie departed Sagamore Hill, Roosevelt assured her that although he did not know anything about Indian song-poems or native poetry, her effort to \"keep this song literature alive\" had his \"immediate and hearty support.\"\n\nRemarkably, within hours of Natalie's departure from the Roosevelt home, the president penned and sent off several official letters, including one addressed to federal Indian agents for Natalie to carry into Indian Country. \"I desire that every opportunity be given Miss Natalie Curtis to travel through all Indian reservations, and to fully investigate and report on, to me or any official under me, all matters pertaining to the wellbeing, the education, the artistic development and industries of the Indians.\"\n\nThat same day, July 22, 1903, Roosevelt sent a letter to Ethan Allen Hitchcock instructing his secretary of the interior to consider Natalie's proposals regarding the preservation of Indian art and music. \"Her desire is to see the exceptionally interesting artistic side of the life of these Indians preserved. . . . She feels\u2014and I think she is entirely right\u2014that the one side in which American life is weak is the artistic, and that we ought not to throw away anything which will give us a chance to develop artistically.\"\n\nHitchcock and his staff, one of whom described Natalie Curtis as \"an intense little person,\" were most unhappy about the president's instructions. Hitchcock was already disgruntled with Roosevelt's willingness to take advice and counsel from ordinary folks. Now the president was considering bringing to Congress proposals made by a person\u2014and a woman!\u2014Hitchcock considered to be nothing more than a do-gooder.\n\nCopies of Roosevelt's letters dispatched on her behalf reached Natalie at Wave Crest several days later. She wrote Roosevelt a thank-you note on July 29. \"Though you say that your power is not unlimited, the incisive force, clarity of purpose and promptness of action with which you have met each issue in your career make us feel that any movement receives an enormous impetus by your championship.\"\n\nNatalie took the opportunity to once more urge Roosevelt to make the finding of solutions to the Indian question a primary focus of his administration. She added that she would be investigating the Indian schools in the East as well as those in the American West. \"I am full of plans,\" she wrote to Roosevelt, \"but shall not take your time further.\"\n\nBut Natalie would take a good deal more of Roosevelt's time that summer, and in many summers to come.\n\nIn spite of high-level naysayers and politically powerful disapprovers, Natalie's self-proclaimed mission had overcome a major obstacle\u2014the federal government. Her passion was morphing into a formal vocation, but her vision still needed to secure sponsorship. Bogey and Mimsey Curtis were moderately wealthy, but they could not, or would not, finance the comprehensive project now evolving in Natalie's mind: the recording of not just the music of the Hopis and the Navajos, but of every Indian tribe in the United States.\n\nNatalie spent the summer and fall in New York, transcribing the songs she had secretly recorded in Hopiland. She also took her presidentially sanctioned plan for a research trip to gather more Indian music and ceremonial material to wealthy family friends and colleagues from whom she hoped to garner financial support. Natalie explained to potential patrons that her mission was to preserve Indian music as close to the original form, sound, and translation as possible. Natalie's classical training and her finely tuned ear, combined with her socially gregarious and charismatic nature, made her well qualified for the task.\n\nAlthough Natalie accepted the prevalent attitude of white America that the Native American world was in its twilight years, their traditions and culture all destined for extinction in the tidal wave that was the modern, industrial world, Natalie believed in the Indian world's inherent value. She also believed that white Americans, beginning with herself, had much to learn from Indian America.\n\nLATER THAT SUMMER, on one of those \"stifling days in late August when all New York panted and closed blinds,\" Natalie left Wave Crest and the cooling breezes found along the Long Island shoreline and went into the city to visit Charlotte Osgood Mason. Natalie and her mother had met Mason, the friendly but overbearing wife of a wealthy New York physician, in the 1890s at the meetings and lectures given by the Theosophists. Mason was a generous philanthropist and likely at the top of Natalie's list of possible benefactors for her Indian music project. When Mason asked Natalie to come to her Park Avenue apartment \"on a mysterious errand,\" Natalie obliged.\n\nIt was only after Natalie reached her apartment that the reason for Mason's summons was revealed. \"We looked from the window at a crowd that had assembled without. Standing before the front door, in the midst of the crowd, we saw an old friend\u2014Frank Mead\u2014fresh from the West, with the breath of the Arizona desert still upon him. Behind him in the crowd was a darker face looking out above an enormous black-and-white Apache basket urn.\"\n\nFrank Mead, a Philadelphia architect Natalie had met with George in Arizona, and his basket-toting friend, Yavapai (known as Mojave-Apache at that time) chief Pelia, were to be Mason's guests. Pelia was wide-eyed and awed, and moved about the luxurious apartment \"as if in a dream.\" Once everyone settled down\u2014Pelia was reluctant to release the basket and held it in both hands for most of Natalie's visit\u2014Mead informed Natalie of what Mason already knew: that he and Pelia had come east so that the Yavapai chief might ask President Roosevelt to grant the Yavapais their own reservation at an old Arizona military post, Camp McDowell. White settlers had encroached on the Yavapai's lands in the Verde Valley, and numerous requests over several years to officials in Washington for resolution of this problem had fallen on deaf ears. The Yavapai were homeless, hungry, and desperate.\n\nIt was obvious why Mason had beckoned Natalie into the city on such a humid, hot August day. Mason had surely heard all about Natalie's informal and very successful meeting with Roosevelt at Sagamore Hill, and she wanted Natalie's help in obtaining a meeting for Mead and Chief Pelia.\n\nHad the request been for someone other than Frank Mead, and for a cause other than Indian rights, she might have been annoyed with Mason. Mead, a New Jersey native, had traveled and lived in North Africa and the Middle East, where he was influenced by architecture that reflected cultural style and indigenous design and also blended with the landscape. In 1902 he returned to the United States, gave up his position at an architecture firm in Philadelphia, and headed into the American Southwest. The dress, traditions, and architecture of Native America resonated with the iconoclastic Mead, and when Natalie met him in Arizona, he had become involved with the Indian rights movement.\n\nSeveral months before the August meeting in Mason's apartment, Natalie had penned a letter that introduced her new friend Frank Mead to her new friend Charles Lummis. Mead wanted to meet Lummis and others in Southern California who shared his architectural passion for the aesthetic that would soon emerge into the Arts and Crafts movement. Mead also wanted to connect with Lummis's circle of Indian rights advocates. Natalie told Lummis that the thirty-eight-year-old Mead shared his artistic and idealistic concerns, and that he was a rare person who exemplified Polonius's dictum \"to thine own self be true.\"\n\nNatalie agreed to help Mead and Chief Pelia meet the president, and the evening after she returned to Long Island, she sent a note to Roosevelt about Pelia. The next morning, Natalie had a response: the president would see Mead and the chief within the week.\n\nNatalie accompanied Mead and a \"well soaped and combed\" Pelia and his oversized Yavapai basket to the president's home on Oyster Bay. Pelia presented Roosevelt with the basket, a gift from his people, and then pleaded for a reservation for his homeless tribe. Mead followed Pelia's plea with a practical plan for obtaining the land the Yavapai wanted to live upon and the establishment of a reservation at Fort McDowell, northeast of Phoenix.\n\nAfter listening to Mead and Pelia's impassioned pitch for a reservation for the Yavapai, Roosevelt agreed without \"five minutes' hesitation\" to throw his support behind their proposal. The president must have been able to look past Mead's social eccentricities, because he soon appointed Mead special agent in charge of securing the Yavapai's lands and settling disputes with the white squatters.\n\nAfter Mead and Pelia's late-August visit, Natalie wrote the president a lengthy letter outlining why Mead ought to be considered for the job as Roosevelt's supervisor of all pueblo Indians and allowed to set up a \"model agency\" in Arizona where Indian arts would be fostered. Roosevelt did not adopt nor implement all of Natalie's ideas, but he did appoint Mead his \"Special Supervisor\" to oversee the BIA agents and superintendents in four states and two territories in the Southwest.\n\nReflecting on Roosevelt's ability to act decisively and quickly on matters he believed warranted immediate relief, Natalie wondered if the \"red tape\" that so frequently tied the hands of decision-makers in Washington was actually a regional affliction: \"It is a fact that life in the West does produce 'real men.' Theodore Roosevelt owed his power of action and his driving force to the stimulus of those early days on a cattle ranch. . . . If Washington had been perched within the neighborhood of Pike's Peak instead of having been sunk in a sluggish, swampish flat, who knows but that red tape might unwind as steadily and swiftly as a stock ticker?\"\n\nShortly after Mead and Pelia's meeting with the president, Charlotte Mason offered her patronage to Natalie's Indian music project. Natalie did not leave for the Southwest immediately, but remained on the East Coast, lecturing about Indian music and the urgent need to record and preserve it. Her plea attracted another important sponsor and advocate: the New York\u2013based businessman and philanthropist George Foster Peabody.\n\nGeorge Foster Peabody was a progressive thinker, social reformer, and a self-made millionaire; he is remembered by the prestigious\n\nPeabody Awards in broadcasting, which are named for him. Peabody was a partner in the Brooklyn-based Spencer Trask and Company that during the 1880s and '90s went into business in several very lucrative fields, including electrical infrastructure and the financing and construction of railroads. Peabody was an active member of the Episcopal Church, which advocated the education of minorities, especially in the South. Beginning in the 1880s, Peabody served as a trustee and board member of several schools and colleges, including the Hampton Institute in Hampton, Virginia, a school that offered African and Native American students training in a wide variety of trades and creative skills.\n\nPeabody may have learned of Natalie's Indian project in conversations with his colleague Charlotte Mason (who also sat on the Hampton Institute's board of trustees), or Natalie may have taken the initiative herself and told Peabody she was raising money to collect and preserve Indian music. Peabody liked the project and by the autumn of 1903 gave Natalie one thousand dollars for her next expedition into Indian Country.\n\nWith money in her pocket, Roosevelt's letter of safe passage in her hand, and brother George again by her side, Natalie returned to Hopiland in the fall. No longer afraid of Burton and his BIA police, Natalie spoke openly about her mission, and the federal agents who had made her work nearly impossible six months before now \"strewed my path with courtesies and called on me before breakfast with offers of help.\"\n\nNatalie headed out on horseback with \"only a pencil and notebook, a helpful brother, and at times some young Indian who had been to school and learned English\" to find the eldest Hopi chiefs and singers. The bulky Edison and its wax cylinders were left in her room below Third Mesa. Handwritten notations were the only practical recording method in the desert outback. To accurately record an Indian song or chant on paper was a task of almost unimaginable difficulty, even for a professional musician. George was awed by his sister's gift. \"Natalie's ear for music was so marvelous that she could set down, without the help of a phonograph, the strange melodies and rhythms that are so different from the white man's music; though to be sure the Indian singer might have to repeat several times before [Natalie] knew that she had the song correctly.\"\n\nChief Lololomai was still living with his sister, Ponianomsi, and a nephew in their home in Oraibi. Natalie asked the chief if he would let her record his songs so that they would not be lost to future generations. \"Your song is heard, then dies like the wind that sweeps the cornfields and is gone, none knows whither,\" Natalie told Lololomai. But if his songs were written down, his people \"even to the children of their children, could know your song as if yourself were singing.\"\n\nLololomai was in his eighties, and he agreed with Natalie's assessment that his songs were destined to be lost \"like a wind-blown trail.\" But he worried that there would be repercussions if they proceeded with the recording. Natalie shared her letter from the Great Chief in Washington, and the elder Hopi agreed to work with her.\n\nWith a translator beside her, her pencil and notebook, and a wide hat to shade her face from the bright October sun, Natalie sat with the chief on his Oraibi rooftop and recorded a thank-you song sung after the village corn harvest. It was a ponderously slow chore, recording on paper Hopi words, tone, notes, and rhythm.\n\n\"In rhythmic monotone the old man crooned beside me,\" Natalie later wrote of that sky-touching session on top of Third Mesa. \"Long and diligently I worked at the recording, with the glare of the hot sun on my paper. It was no light task to fix the chant in musical notation.\" Even Lololomai commented on Natalie's laborious task to make \"those black marks on the paper.\" Natalie told the chief, \"Your song is a wild blackbird to me, and it may be that the sun will move far along the sky before I have captured it.\"\n\nAfter several hours on the rooftop, Natalie was satisfied with her transcriptions. Lololomai knelt in prayer as dusk enveloped the ancient pueblo.\n\n> The shadows on the village street grew long. The sun was sinking. Here and there a lone Hopi was returning from below with laden burro. Soon the trail would be dotted with home-coming Indians. We sat long in silence, Lololomai, the blind boy, and I. I watched the glow enfold the desert with the mystery of dying day. . . . \n> \n> Silent still, the blind boy and I took our downward way upon the rocky trail. To my companion, in his night, the deepening shadow bore no import, but a twilight sadness lay upon my spirit. I thought of the garnered Hopi corn. Will there be many more plantings of poetry and harvesting of song? Darkness closed in. But off beyond in silver glory rose the moon.\n\nThroughout the fall of 1903, Natalie recorded dozens of native singers. Her task was challenging because of the remoteness and inaccessibility of the Hopi villages, the complexity of their language, and the tonal variations in their music. She likened capturing Hopi music on paper to capturing the wind as it swept down the sides of a rock mesa. \"To seize on paper the spirit of Hopi music is a task as impossible as to put on canvas the shimmer and glare of the desert. Hopi music is born of its environment. . . . There, in that wide land, under the blaze of the Arizona sun, amid the shifting color of the tinted sands and the purple-blue of the sharp-shadowed rocks must the songs be heard to be heard truly.\"\n\nWith George, Natalie traversed Indian Country on horseback, in a wagon, and where necessary, on foot. To reach a singer in a sheep camp or in another village meant traveling across the enormous, empty, high desert of the Colorado Plateau, navigating unmarked routes that crossed ledges and canyons, streambeds, and rock shelves and circled buttes and mountains. Their path was always directed by water and the location of a spring or water cache known by the native guides.\n\nIn 1903 and 1904, and during her subsequent journey through the Southwest in 1905, nearly sixty Hopi songs were recorded in Natalie's notebook and on the phonograph. Letters home enabled the Curtis family back east to follow Natalie and George's progress across Indian Country.\n\n\"It was a long and exhausting work, physically and mentally,\" Bridgham wrote of his sister's recording expeditions. \"As a rule her party lived in the guest rooms at the Indian Government schools, but frequently the quarters were primitive. There were long rides and drives over rough country, irregular meals\u2014often none at all. Her health, frail at best, was seriously taxed by the hardships. . . . At times it seemed almost impossible to finish the task, but her courage and resolution never faltered.\"\n\nThe task was daunting and the conditions difficult. Even so, Natalie was always reluctant to leave the \"spiritual beauty\" and the \"song-poets\" of the Hopi, especially Third Mesa and her friends at Oraibi. \"It was sunset when for the last time I climbed the steep trail to the village of Oraibi,\" she wrote the evening before her departure for Williams, Arizona, and the train for home. \"The level desert seemed a lilac sea, and the outlines of the craggy table-lands were sharp against the flaming sky. Many weeks had passed since I had left the railroad to take the long two-days' drive across the 'Painted Desert' to the Hopi villages. . . . To-morrow I must leave the desert and its freedom for my distant Eastern home.\"\n\nNatalie's sense of home was shifting with each visit to the Southwest. On one of her early visits, the Hopi gave her a name that pleased her immensely: Tawimana, \"Woman Who Sings like a Mockingbird\" or Song Maid. Natalie signed her Hopi name alongside her American name on Christmas cards and personal correspondence the rest of her life.\n\nNatalie and George departed Hopiland by way of the village of Hano on First Mesa, where they met the celebrated potter Nampayo and her family. From First Mesa they headed their wagon south through the vast country of the Hopis' historic enemy, the nomadic Navajos. Natalie found the Navajos to be \"industrious, independent, and fearless\" with \"the strength of the warrior and the simplicity of the shepherd . . . the Navajo is the true son of the silence, the awe, the grandeur of the desert.\" Natalie made friends among the Navajos and over the next two years would record more than ten of their ceremonial chants and songs.\n\nAfter her return to New York, the president summoned Natalie to \"come at once\" to a luncheon at the White House. It was a simple affair by Washington standards, but the just-off-the-desert Natalie, still adjusting to the noise and pace of modern life, was intimidated by the event and the guests seated around Roosevelt's table:\n\n> I remember the entire simplicity of that luncheon\u2014lamp chops, and for dessert baked apples\u2014although the party was twelve or fourteen in number. The directness with which Roosevelt introduced the subject of the luncheon\u2014the American Indian\u2014was most embarrassing to me, who felt myself nothing but a little \"snip\" in the presence of gray-haired Senators and Cabinet members; for as we found our places at the table he turned his glasses full upon me and exclaimed, with terrifying emphasis, \"Miss Curtis will now sing for us an Indian grace.\"\n\nNatalie obliged her host and sang an Indian prayer for the senators and cabinet members, all men, seated at the long table. As the luncheon progressed, it became apparent to Natalie that Roosevelt had assembled this group specifically \"to talk about Indians.\" He told the guests \"in detail\" about Natalie's translations of \"Indian song-poems\" and her recordings of Indian music that she had brought to his attention the year before at Sagamore Hill.\n\nAfter lunch, Roosevelt asked Natalie to perform songs learned during her recent visit to Indian Country. The president led everyone into \"the big ballroom, where stood the grand piano, which was quite superfluous for a rendition of Indian songs.\" Although unprepared, Natalie again obliged the president and performed several musical poems of the desert; she also gave an impromptu talk about her \"harvestings of Indian songs\" to the captive audience from Capitol Hill.\n\nNatalie was beholden to Roosevelt, but her response to the president was more than gratitude. Natalie considered Roosevelt one of the great heroes of her time. She may, too, have seen a bit of herself in the president, if she allowed such a comparison. \"Roosevelt never lost one moment's time in the accomplishment of a purpose,\" she wrote, \" . . . he never preambled or beat about the bush; he snapped at a subject with sharp surety, and the near-sighted eyes behind the thick glasses seemed always intently focused.\"\n\nRoosevelt told Natalie, \"I can never afford to be in arrears with anything that I have to do; I must always be abreast or ahead of my tasks.\" Natalie could not have said it better.\n\n#  _4_\n\n##  THE INDIANS' BOOK\n\nThere is another sort of beauty playing always about the Pueblo country, beauty of cloud and rain and split sunlight. . . . Everywhere peace, impenetrable timelessness of peace, as though the pueblo and all it contains were shut in a glassy fourth dimension, near and at the same time inaccessibly remote.\n\n##  \u2014 _Mary Austin_\n\nNATALIE SPENT THE SUMMER ON the East Coast, working on her recordings and transcriptions at the relentless pace her family was accustomed to, if not always comfortable with. She went to Maine and recorded love, war, medicine, and dance songs, as well as the hunting and creation stories of the surviving tribes of the once-great Wabanaki nation of the Penobscots, Passamaquoddies, Micmacs, and Maliseets. She also secured more funds, including a thousand dollar gift from George Foster Peabody that would enable her to expand the scope of her project to include the music of virtually every Indian tribe in the continental United States.\n\nIn late fall, Natalie and George donned their travel clothes, packed their khakis, boots, and broad-brimmed hats into their expedition bags, and boarded a train for the Dakotas. It was November, and the lovely, cool days of fall had turned into the cold, wet days of the coming winter. It was not an ideal time of year to be traveling into the rural Midwest, but Natalie's mission had evolved into the writing of a major book about Indian music, and her task would not be deterred by bad weather.\n\nBefore they headed for the Dakotas, Natalie and George made a short stopover in St. Louis to see the Louisiana Purchase Exposition celebrating Jefferson's 1804 land deal with the French and the opening of the Trans-Mississippi West. Fairgoers saw the first electric ovens and range tops, and could stand alongside the first automobiles. The Lincoln Museum, built around the log cabin where Abraham Lincoln was born, and Festival Hall, with its gold-leafed dome larger than the one on top of St. Peter's Basilica in Rome, were the exposition's showstoppers.\n\nWhile George took voluminous notes, Natalie interviewed and collected songs, stories, and folklore from individuals from the Navajo, Crow, Pawnee, and Chiricahua Apache tribes who were participants in the fair's Indian exhibits.\n\nIn particular, Natalie wanted to meet Chief Geronimo of the Apaches, who was featured in William Cody's, aka Buffalo Bill's, Wild West Show on The Pike (the fair's entertainment boulevard). Natalie and George found \"the most famous of Apache war-leaders\" making and selling bows and arrows in the Indian Department. Except for newspaper articles, very little had been written about Geronimo. The white world knew about the chief and the Chiricahua Apache people only through the dramatic and often inaccurate stories and legend-worthy accounts given by U.S. military personnel or BIA officials. Geronimo did, however, have a relationship with Roosevelt (the Apache chief would be among the luminaries to ride in the president's 1905 inaugural parade), and Roosevelt may have suggested Natalie seek out the elderly chief at the fair.\n\nNatalie and Geronimo had no common language, but with a translator they conversed at length in the Southwest Building. Between seventy and eighty years of age, the chief, whose eyes flickered with the \"tiger-flash\" of a warrior, agreed to let Natalie record on paper several of his songs. The difficulties of recording Geronimo in the exposition hall were nearly as great as recording Chief Lololomai on the rooftop in Hopiland. But Natalie persevered and, with an interpreter, recorded the words and music of at least two of Geronimo's songs, one of which was \"so old that none knows who made it.\"\n\nWith notebooks bulging with new material and a lengthening list of collaborators, Natalie and George left St. Louis on November 6 on the 1 p.m. train for Chicago. Two days later, they disembarked from the warmth and comfort of the Pullman car in Sioux City, Iowa, where they did not remain even one night. They transferred their gear directly into a wagon and headed through a cold, drizzling rain to austere quarters at the Winnebago Indian agency in Nebraska.\n\nIn the next few days the rain turned to snow on the Winnebago Reservation. After several days of recording sessions with local musicians, Natalie and George left the Winnebago school and headed their horse and wagon across the frozen winter landscape to the Sioux agency headquarters at Pine Ridge and Rosebud, South Dakota. Natalie set up her workshop in a hotel and, with the safe passage granted by Roosevelt's letter, went with George in search of Sioux musicians and storytellers in town and out on the reservation.\n\nFor nearly eight continuous weeks Natalie's workshop was set up and taken down in numerous hotel rooms and government headquarters, hauled about on foot and on horseback, and packed into wagons and moved by train cars across Nebraska, the Dakotas, Wisconsin, and Oklahoma. Natalie and George witnessed ceremonies and dances rarely seen by non-Indians: the hand games and the mescal rites of the Arapahos and Cheyennes, the Antelope Ceremony and wind songs of the Kiowas, and the clandestine Spirit-Dance of the Ghost Dance religion (considered to have been \"the trouble\" that caused U.S. troops to massacre more than three hundred Sioux men, women, and children at Wounded Knee), still performed by the Dakotas and Pawnees.\n\nThe Curtises boarded a train for Oklahoma City. After meeting an Indian policeman, Hiamovi, a half-Sioux, half-Cheyenne who was a high chief of both the Dakotas and Cheyennes, they altered their itinerary and spent the next month among his people in Oklahoma. During the winter on the Great Plains, Natalie recorded over forty-five collaborators from six tribes, each of them named and their contributions meticulously catalogued in her notebooks.\n\nJust before Christmas, Natalie and George departed the frigid country of the Plains Indians and headed into the country of the Puebloans, the bright, high-altitude desert landscape of New Mexico and Arizona. They spent the holidays with Charlotte Mason near Phoenix, where Frank Mead was living among the Yavapai on their new reservation at Fort McDowell.\n\n\"It was Christmas Day when Mrs. Mason and I drove in a canvas-topped cart through the Arizona desert to Camp McDowell,\" Natalie wrote. \"Pelia [the basket-carrying chief Natalie had met in New York City fourteen months before] and his wife had driven in to the railway at Phoenix to meet us. Mr. Mead was personally conducting us to the house of the missionary agent on the reservation, and our visit to the Indians had the warmth of a home-coming.\"\n\nOn Christmas Day, Pelia and his tribe gathered for a night dance. Natalie, George, Mason, and Mead were the honored guests: \"A huge bonfire burned on the open desert; its flames seemed to lick the deep blue-purple of the sky,\" Natalie wrote of that Christmas night. \"Brilliant moonlight\u2014such moonlight as Easterners have never seen\u2014touched cactus and mesquite and lit the horizon line of hills.\"\n\nThe New Year of 1905 found Natalie and George at the Indian school in Phoenix, where Natalie recorded students from southwestern tribes. From Phoenix, they traveled east by wagon to New Mexico, where for the next two months Natalie and George lived among the people of Zuni, Acoma, and Laguna Pueblos. By March, Natalie was out of money, and the glorious, exhausting four-month expedition through the Indian Country of the Midwest and Southwest came to an end.\n\nNatalie returned to New York City, but she did not move back into the Washington Place home of her parents. The Curtis family also maintained a residence at 33 West 69th Street, and it was into this apartment, where her brother George and possibly Bridgham resided, that Natalie unpacked her voluminous notebooks and her weather-worn garments from her four-month expedition.\n\nHer research and collection of Indian music were more or less completed, and Natalie's task shifted to her next goal: the assembling and editing of a book about Native American song, dance, ceremonies, and stories.\n\nWithin a month of her return to the East Coast, Natalie was writing and speaking about her experiences. In a small pamphlet published by G. Schirmer, she reproduced three of the corn-grinding songs recorded just a few months earlier at Laguna Pueblo. As she prepared the material for her book, Natalie penned articles for popular audiences about what she had seen and learned. Although she hoped her research about Indian music would earn the respect of her peers, her primary audience was not academia. Natalie wanted to preserve Indian music, but she also wanted to educate average Americans about the federal assimilationist policies that she believed would bring extinction to Native American culture. She wrote articles for the broadly based, predominantly Anglo-American audience that regularly read _Harper's_ , _Outlook_ , _Craftsman_ , and _Southern Workman_ magazines, and newspapers like the _New York Times_ and _New York Evening Post_.\n\nBy conventional turn-of-the-century standards, Natalie Curtis was a confirmed spinster when she turned thirty in April 1905. How she thought about herself as a single woman on her thirtieth birthday can only be guessed at. She had lived most of the last two years far from home in the hinterlands of the American West where an unmarried woman was able to move about and work relatively free of public scrutiny or societal disapproval. If Natalie had a romantic interest or liaison during her late twenties, in New York or out west, neither she nor her family mentioned it in their letters and diaries.\n\nThe West was a frontier for both the body and soul of those Anglo-Americans, men and women, who ventured into its vast, uncivilized landscape. In contrast, the upper-class world of New York City was still a stronghold of proper Victorian society, and a single woman like Natalie, who insisted on pursuing a vocation, even an artistic one, was treading a thin line between appropriate and inappropriate behavior.\n\nIn May 1905 Charlotte Mason came to Natalie's social and creative rescue when she offered her the use of her apartment in Washington, DC. Natalie needed a place where she could work on her Indian music book without interruption from friends and family, and she accepted Mason's offer and moved alone to Washington in late spring.\n\nBut there were numerous distractions in Washington. Natalie socialized with officials from Roosevelt's Indian affairs department and accepted invitations to society functions where she could promote Indian policy reform. Although there was a continent between them, Charles Lummis remained in frequent contact with Natalie. In 1905 he was especially insistent that she become an active member of the Southwest Society of the Archaeological Institute of America. Considering that Natalie lacked affiliation with the academic community, and that her credentials for her work among the Indians consisted of her passionate belief that native music and stories needed to be preserved, the invitation to join Lummis's circle of scientists, anthropologists, and ethnologists was precisely the sort of endorsement Natalie wanted for her work. But she declined the AIA position, telling Lummis that \"every moment of my time is so absolutely taken up.\"\n\nBut Lummis _really_ wanted Natalie on the membership roster and told her so in several more letters. By 1905 Lummis had fallen from favor with Roosevelt, and he knew that her ability to get the ear of \"the most powerful American\" would be a valuable asset for both the AIA's and his own agendas. Natalie was likely privy to the disparaging comments and opinions of Lummis circulating drawing rooms on both the East and West Coasts. There was gossip about Lummis's indiscretions with women (he reportedly kept a log of his sexual exploits), and one California-based editor warned Roosevelt that \"Lummis is awful sharp, but a consummate crank of the first water and will get you or anybody else in trouble.\"\n\nNever one to be swayed by gossip or to choose her friends for their social standing or political influence, Natalie was able to remain friends with Lummis and to also stay out of the trouble around him for decades. But it was not an easy friendship. Lummis continued his campaign to enlist Natalie into the AIA until December 1905, when he learned that she had joined the New York branch. Lummis wrote Natalie that he would prefer she be affiliated with the AIA's southwest chapter but added, \"So long as you are in the Institute I suppose we must not quarrel.\"\n\nMembership in the AIA did bring validation and recognition to Natalie and her work among the Indians. The Archaeological Institute of America was twenty-five years old and had backed field expeditions and surveys by America's most eminent anthropologists and ethnologists, including Lummis's friend the Swiss anthropologist Adolph Bandelier. The same year that Natalie joined the AIA, the New Mexico\u2013based anthropologist and AIA member Edgar Lee Hewett was in Washington lobbying for the passage of the federal Antiquities Act. Natalie may have met Hewett in Washington in 1905. If not, she would meet him in the near future. Hewett was touring the country on behalf of the AIA, encouraging cities to open local chapters and to financially and politically support the relatively new field of American anthropology.\n\nThe AIA was very good company for Natalie to keep and offered her introductions to the southwestern anthropology and ethnology community and to prominent leaders of the Indian policy reform movement. But Natalie could not stop work on her book to become a political activist. Throughout 1905, in spite of accelerated national activity around Indian rights and a burgeoning public interest in southwestern anthropology and archaeology, Natalie kept her head in her transcriptions and field notes and her hands on her typewriter.\n\nHer book, entitled _The Indians' Book_ , was completed by early 1906 and soon after accepted for publication by Harper Brothers. Natalie sent a copy of the manuscript to Roosevelt and asked him for an endorsement. Roosevelt's handwritten letter was placed at the front of _The Indians' Book_. \"These songs cast a wholly new light on the depth and dignity of Indian thought, the simple beauty and strange charm\u2014the charm of a vanished elder world\u2014of Indian poetry.\"\n\nRoosevelt's words echoed the dominant sentiment among Anglo-Americans that Native America was in its last years, and that the \"childlike\" red race would soon vanish. This limited and racist view of Indian America would be slowly replaced by the modern understanding of the intrinsic worth and enduring vitality of Native America. Although _The Indians' Book_ reflected the prevalent Euro-centric view of Indian America, it was a major accomplishment and successfully gave the American public the first comprehensive collection of native songs, stories, and ceremonies. Natalie's summaries of each tribe also introduced the general reader to the complex and numerous indigenous cultures that had existed and thrived on American soil for hundreds of years.\n\n_The Indians' Book: An Offering by the American Indians of Indian Lore, Musical and Narrative, to Form a Record of the Songs and Legends of Their Race_ , recorded and edited by Natalie Curtis, was published by Harper Brothers in 1907. The first edition contained colored title pages and artwork. The book was illustrated by Hinook-Mahiwi-Kilinaka, the Winnebago artist Angel De Cora. Several years older than Natalie, De Cora had already achieved considerable attention as an artist when she agreed to illustrate _The Indians' Book_. A graduate of Smith College, De Cora was a member of the American Academy of Design by the age of thirty-one, a rare honor at that time for a woman of any age or race.\n\nHarper Brothers send out hundreds of review copies of _The Indians' Book_ to scholars, editors, and politicians. By the end of 1907, Natalie's position as an authority on American Indian music was well established. Her personal and professional life would never be the same.\n\nAware that her audience would be reading about Indian America for the first time, Natalie assembled the material in _The Indians' Book_ in an accessible and reader-friendly format. Arranged by geographical location and tribal affiliation (Eastern, Plains, Lake, Northwestern, Southwestern, and Pueblo Indians), the chapter for each of the eighteen tribes began with a historical overview and summary of the tribe's religious beliefs, symbols, language, food, family, and village customs. Natalie also included contextual narratives about landscape and environment, and vignettes of her experiences among the American Indians.\n\nThe book's transcriptions of Indian songs and musical scores were rendered in Natalie's own hand and included her notes about tempo and nuance. The words of the songs were written both in their native tongue and in English translation. With the help of one of her editors, Kurt Schindler, Natalie designed a transcription system that arranged musical phrases like lines of verse and made \"the song flash before the eye like the form of a stanza in poetry.\"\n\nThere was no previous publication on Indian music or folklore to which _The Indians' Book_ could be compared. The Indians were the authors of the book, Natalie stated in the introduction, and every singer, poet, dancer, medicine man, and storyteller was acknowledged by name and tribe. Natalie was simply the recorder, her task to collect, edit, and arrange the Indians' contributions.\n\nWith the release of _The Indians' Book,_ Natalie's life was dominated by public appearances and speaking engagements. The book was praised by her colleagues in the upper echelon of the professional music world, including Walter Damrosch, the conductor with whom she had been smitten as a teenager. It was a heady time in Natalie's life, an intoxicating year of personal and professional adulation and affirmation. Almost a decade had passed since her physical and psychological collapse and subsequent withdrawal from the professional stage. Finally, her peers understood exactly what the gifted Miss Curtis had been doing out west all these years.\n\nBy the New Year of 1908, Natalie's head and heart longed for the uncluttered spaces and open skies of Indian Country, and she announced her plans to return to the West. George was ready to leave New York, too, and together they packed their trunks with desert khakis and riding boots, said farewells to the family at Washington Place, and journeyed by train into the land of the sun. Natalie and George arrived in Escondido, California, on January 6 and moved into rented rooms in the town of Riverside. George wanted to find work on a local ranch or fruit farm, and the next morning, after meeting a Mr. Bailey, he left Natalie in Riverside and took a four-hour ride on the mule-drawn cart of the local mail carrier to a ranch owned by Bailey's friends the Mendenhalls.\n\nNatalie had come to Riverside specifically to talk with Tawakwaptiwa, Chief Lololomai's nephew whom she had met at Oraibi several years before. Known around Riverside by his white name, Wilson Fredericks, Tawakwaptiwa was living at the Sherman Institute with his wife and children. Natalie wanted to give Tawakwaptiwa a copy of _The Indians' Book_ , in which he was a named and much-honored contributor. She also wanted to find out how her Hopi friend and his family were surviving forced relocation at the government Indian school in Riverside.\n\nTawakwaptiwa and forty of his fellow villagers from Third Mesa had been exiled from Hopiland in 1906 by Roosevelt's commissioner of Indian affairs, Francis Leupp. The event that precipitated this exile was among the most significant altercations in Hopi history. Following Chief Lololomai's death in 1906, Tawakwaptiwa had become chief and leader of Third Mesa, the largest and most powerful of the Hopi communities. Tawakwaptiwa, like his uncle Chief Lololomai, was a Friendly and believed that compromise with Washington could be maintained without the Hopi people losing their cultural integrity. And like his uncle, Tawakwaptiwa hoped to avoid confrontation between the Friendlies and the Hostiles on Third Mesa, since he believed a split would bring the disintegration of historic Oraibi.\n\nIn the autumn of 1906, shortly after Lololomai's death, the decades-old conflict between the Friendlies and the Hostiles erupted into confrontation when the Hostiles declared that coexistence with the Friendlies was no longer possible on Third Mesa. The Hostiles, led by Youkeoma, wanted only one faction to live at Oraibi.\n\n**Figure 4.** Natalie and Tawakwaptiwa in Riverside, California, 1907.\n\nThe Hopi, true to their People of Peace reputation, commenced the settlement of their conflict not with violence and weapons (although members of both groups were carrying knives, clubs, and even bows and arrows) but with an athletic contest. Late in the day on September 8, the Friendlies and the Hostiles gathered on a level stretch of hard ground northwest of the village of Oraibi. A line was drawn in the sand from east to west. Youkeoma took a position on the Oraibi side of the line and challenged Tawakwaptiwa's men to force him back over the line and away from the village. If the Friendlies succeeded, Youkeoma and his followers would leave Oraibi.\n\nEach faction lined up behind its strongest man, a signal was given, and the great push began. After a long, hard war between the two lines of men, Youkeoma acknowledged defeat. The Hostiles packed up their families and belongings, and 298 villagers departed Oraibi. Youkeoma moved his people seven miles to the east, where they settled and built the village of Hotevilla (or Hotevila).\n\nAlthough the Hopi had peaceably settled their civil war, Youkeoma was arrested by federal agents and imprisoned at Fort Wingate near Gallup. In the weeks before what historians would call the Oraibi Split, Tawakwaptiwa, nervous about violence in the village, had sought the counsel of government officials, including Commissioner Leupp. Even so, after the Hostiles had departed Third Mesa, Leupp deemed Tawakwaptiwa a threat to government control of the Hopis and temporarily stripped him of his chieftainship. To avert any additional challenges to federal policies, Leupp ordered that Tawakwaptiwa and forty of his followers be sent to the Sherman Institute for four years. Leupp then authorized federal troops, augmented by armed Navajo policemen, to seize eighty-two children from Oraibi and take them to the boarding school at Keam's Canyon.\n\nThe reaction by Leupp to the Oraibi Split seemed harsh, heavy-handed, and insensitive to people like Natalie who hoped the White House was actively reforming federal Indian policies. If Natalie broached the subject or registered a complaint about events at Third Mesa with Roosevelt, it was not recorded. Roosevelt trusted Leupp, but also understood that his commissioner \"though an exceedingly honest and fearless man . . . is very prejudiced when he makes up his mind.\"\n\nWhen Natalie met with Tawakwaptiwa at the Sherman Indian School in January 1908, he and his family had been living in California for two years. His hair was short, he spoke nearly fluent English, and he was delighted to see Natalie and to receive a copy of the book in which his songs were recorded. But Tawakwaptiwa was no longer the free-spirited musician of whom Natalie had written, \"Of all the Hopi poets, none sings a gladder song than Tawakwaptiwa (SunDown-Shining) . . . a Hopi untouched by foreign influence, the child of natural environment, spontaneous, alert, full of life and laughter.\"\n\nTawakwaptiwa was now a deeply disillusioned man, acutely antagonized by the white environment he had been thrust into at the Sherman Institute. The betrayal by the U.S. government that he had worked so conscientiously to accommodate had made him bitter and far less tolerant of the non-Indian world. In many ways, Leupp's effort to diffuse Tawakwaptiwa's power and Hopi tribal identity and independence backfired. In spite of the school's best efforts, the Hopis returned home determined to maintain their Hopi traditions. \"I could talk like a gentleman, read, write, and cipher,\" Oraibi native Don C. Talayesva wrote of his education at Sherman, \"quote a hundred verses of Scripture . . . debate, shout football yells, swing my partners in square dances. . . . But I wanted to become a real Hopi again, to sing the good old Katcina [ _sic_ ] songs, and to feel free to make love without fear of sin or a rawhide.\"\n\nNatalie was able to remain friends with Tawakwaptiwa, and she would see him again in Hopiland (he would be released from the Sherman Institute, return to Oraibi, and reassume his position as chief in 1910). But the harmonious independence, historic self-government, and tribal unity so valued by Tawakwaptiwa and the Hopis, and so revered by Natalie, had been weakened by the interfering and insensitive hands of white men. \n\n#  _5_\n\n##  \"TIME AND THE WORLD\"\n\nPut aside for a while all that mesh of complex activities that is entangling your health, find some placid travelling-companion and come West. I shall meet you at the railroad with ponies and a packhorse . . .\n\n##  \u2014 _Gregory Marwood, aka George DeClyver Curtis_\n\nWHEN NATALIE AND GEORGE RETURNED to California in 1908, they found Frank Mead enjoying life in San Diego and working in partnership with the architect Irving J. Gill. Southern California's Mediterranean climate and outdoorsy, close-to-the-earth culture fit Mead's nonconformist lifestyle, and his eclectic Moorish designs resonated with the residents of San Diego. Among Mead and Gill's first important clients was Wheeler J. Bailey, for whom they designed and built Hilero, a home destined to become a landmark on the rocky bluffs north of San Diego along a wild stretch of beach called La Jolla. Bailey was the same man with whom George had struck up a conversation on his first morning in Riverside. Bailey and the Mendenhalls, whose ranch George may have worked on, and their associates and friends would become the foundation of George's social circle as he relocated his life to Southern California.\n\nWithin this circle of wealthy and creative Californians was an energetic, educated, and artistic woman named Alice Ellen Klauber, whose friendship quickly became an important anchor in Natalie's life. Alice was a native Californian, born in 1871 to a prominent merchant's family in San Diego. Like Natalie, Alice was petite and small boned, and made up for her size with a quick wit, a generous nature, and a bold, inquisitive intellect. Serene, self-assured, and dignified, Alice made friends easily.\n\nWhen she met Natalie, Alice was thirty-seven years old and unmarried. She had plenty of family money and owned her own apartment in downtown San Diego where, surrounded by her sisters and her friends, she lived a happy and independent life as a single woman. Alice had traveled throughout the United States, twice been to Europe, and in January 1908 had just returned from seven months abroad, including an extended stay in Italy where she studied painting with the American master William Merritt Chase.\n\nAlice's father, Abraham Klauber, had immigrated as a young man to California from Czechoslovakia and had built successful wholesale grocery businesses in San Francisco and San Diego. The nine Klauber children were raised in a home that valued and encouraged art, music, literature, and the finer things of American culture. Since childhood, Alice had loved to draw and paint in much the same obsessive manner that Natalie had loved to play the piano.\n\nAlthough her early years were spent in the frontier town of San Diego, Alice attended high school and came of age in the \"Paris of the West,\" San Francisco, in the progressive and active Jewish community of Oakland. Her formal training as a painter began in 1884 at the first art school on the West Coast, the Art Students League (the San Francisco Art Association), where Alice attended classes and socialized with the West's emerging artists and writers.\n\nAlthough a serious student and a talented painter, a career as a painter was hardly among Alice's or any woman's options in the late 1800s. The curriculum at the Art League included a program for aspiring art teachers, but Alice had neither the inclination nor the need to become a schoolteacher. She just wanted to paint. Her parents indulged her art classes, but expected Alice to marry and raise a family. Exactly how she would combine the socially sanctioned role as wife and mother with the unsanctioned, indeed unheard-of, life of a woman painter, Alice did not know.\n\nAfter completing high school in the 1890s, Alice returned to San Diego. The little town on the Pacific had boomed while the Klaubers were in San Francisco. With the completion of the Santa Fe Railway tracks into downtown and the community's connection to the rest of the United States via the comfort of the train, San Diego's population surged from 2,600 in 1880 to nearly 40,000 in 1887.\n\n**Figure 5.** Alice Klauber.\n\nAlice came close to marriage after meeting the intense young artist Arthur Putnam in the summer of 1894. Putnam was a sculptor who, like Alice, had studied at the Art Students League in San Francisco. When Alice met Putnam he was living with his mother on a lemon ranch near San Diego and working as a land surveyor with his brother George. Putnam looked like a cowboy-artist and was charming and straightforward. According to a biographer and friend, he was \"extremely handsome, lean as a wolf, with strong, clear-cut features . . . and large brilliant dark eyes.\"\n\nThe intensity of Alice and Putnam's romantic relationship was not recorded by Alice, but they did spend one summer together sharing their favorite pastime, exploring and drawing the flora and fauna found in the canyons and hills near San Diego. They fell in love, but they would not and could not marry. Alice had money and Putnam did not. In the successful, self-made merchant family of the Klaubers, a gifted but financially insecure artist like Arthur Putnam was not considered an acceptable suitor for their daughter. The relationship ended when Putnam departed San Diego for Chicago and an apprenticeship with the famed animal sculptor Edward Kemeys. Although Alice and Putnam exchanged many letters, and in later years Alice corresponded with Putnam's wife, Grace Storey, Alice never saw Putnam again.\n\nAt the turn of the century, the Klauber family bought nearly a thousand acres of rolling hills and canyons in the country outside San Diego. Alice called the family retreat Encanto, Enchantment, and it became her haven where she spent languid days alone painting, reading, and walking and sketching the countryside. At the age of thirty, Alice was drawn to the solitary life of a painter, but as the spinster daughter of one of San Diego's wealthiest and most influential families, she was duty bound to participate in the social and civic life of the community.\n\nIn contrast to cities on the Eastern Seaboard of the United States, in San Diego a single woman was an accepted member of Southern California's society of independent, educated, progressive, and forward-looking folks involved in building a twentieth-century cultural hub on the shores of the Pacific. Two of Alice's sisters, Ella and Leda, were artists and shared Alice's need for creative space and time. Another sister, Stella, was a photographer. Ella eventually married, as did a fifth sister, Laura.\n\nAlice did her civic duty and participated in the opening of the city's first art gallery by the Wednesday Club, a women's group, in 1905. But community art events and informal painting classes by visiting artists did not and could not nurture Alice's muse, and her development as an artist was in isolation. San Diego was a cultural backwater on the California frontier, and Alice had almost no peers beyond her sisters and the small handful of women and men painters who visited Southern California.\n\nShe did have, however, what an unmarried woman needed to undertake serious creative work: money, her own shelter and studio space, and time to herself. Alice also lived in the West, where even single women enjoyed more freedom of movement physically and socially than women in the conservative industrial centers of the East Coast. By the turn of the century, women in the American West had access to better educational opportunities and enjoyed positions of leadership in their communities. And women in western states won the right to vote decades before their counterparts back east\u2014Wyoming in 1869, Utah in 1870, and Colorado in 1893. California would give Alice and her sisters the vote in 1911.\n\nAlice had been born and raised in a region that in the first decade of the twentieth century was becoming very attractive to women like Natalie Curtis who sought to emancipate themselves from conventional notions of femininity. Whereas the East Coast of America was tame, familiar, domesticated, and ordered, the West Coast was the opposite. Alice offered Natalie an introduction to the refreshingly nonconformist society found in Southern California. And Natalie's arrival in San Diego gave Alice an educated and sophisticated friend who understood and encouraged her need to experience the world of art and culture beyond Southern California.\n\nWith several of her siblings, Alice toured Europe in 1906. But it was just that\u2014a tour. Alice wanted to study the work of the grand masters with contemporary masters. She wanted to experience Europe not as a woman of leisure but as an artist. The next year, as a mature woman of thirty-six, she was accepted as a student in the European Summer School of William Merritt Chase. On June 8, 1907, Alice boarded the SS _Romanic_ at the Charleston Docks in Boston, bound for Punta Delgada and Naples. She traveled with a female cousin, O. Epstein, whom Alice rarely mentioned in the accounts she wrote almost daily in a small red notebook.\n\nWilliam Merritt Chase was sophisticated, cosmopolitan, and flamboyant, and with his friend and rival Robert Henri was considered to be among the most important teachers of American artists in the first decade of the twentieth century. Chase's students included Charles Demuth, Marsden Hartley, Georgia O'Keeffe, John Marin, Rockwell Kent, and Edward Charles Volkert. In 1896 Chase had begun the Chase School of Art, which became the New York School of Art (later Parsons School of Design), with Chase and Henri the principal instructors.\n\nChase was a portrait painter of the first rank and was among the first American artists to paint impressionist landscapes. He knew everyone who counted in American art, and by the summer of Alice's class in Italy, he was \"the reigning force among the loose constellation of painters called American Impressionists.\" Chase's students were taught his eclectic painting style, which employed rapid brush strokes and a strong color palette and mirrored the energy and expansion found in New York City at the turn of the century. Among Chase's students in New York in 1907 was the young Georgia O'Keeffe, who, like all of Chase's students, was in awe of him: \"When he entered the building a rustle seemed to flow from the ground floor to the top that 'Chase has arrived!'\"\n\nAlice was in Italy with a modern master, and also with a teacher who distinguished himself among his female students by his pronounced lack of male chauvinism. Chase encouraged women of talent to become professional artists, and almost a decade earlier had chaired the committee that made Cecilia Beaux the first woman to receive the prestigious Gold Medal in art from the Carnegie Institute.\n\nAfter twelve days at sea on an ocean that was \"smooth as glass,\" Alice and her classmates disembarked in Naples, where they were greeted by Chase's assistant, a young painter named Walter Pach. Pach, just twenty-four years old that summer, was already on the faculty at the Teacher's College at Columbia University. Pach was a gifted artist\u2014he had met Chase while studying at the New York School of Art\u2014and after six summer tours as Chase's assistant, he was well on his way to becoming a specialist in the new world of modern art emerging in Europe.\n\nDuring the first few weeks of class, Alice was intimidated by Chase and his harsh criticisms of student work. But she persevered, and by late July was among the select group that went with Chase and Pach on private museum tours or out to dinner. There was an immediate connection between the young New Yorker Walter Pach and the thirty-something Californian Alice Klauber. They gravitated to one another on class field trips in the Italian countryside and met at sidewalk caf\u00e9s for long talks about art and travel, painting and painters, and life in the modern world of the twentieth century.\n\nEverything about that summer in Italy\u2014the instruction, the setting, the social life, her friendship with Pach\u2014was life-changing for Alice. With Chase at the helm, Alice and her classmates went on study and painting trips to Assisi, Bellagio, Padua, Perugia, Pisa, Milan, Venice, and Verona. She also ventured out into the countryside on her own, particularly to rendezvous with two California friends, Leo and Gertrude Stein, who were spending the summer at a small villa near Florence. Leo and Gertrude, and their older brother, Michael, were childhood friends of Alice's in Oakland. Michael and his wife, Sally, and their son had followed Leo and Gertrude to Paris, where by 1907 Leo and Gertrude's apartment at 27 rue de Fleurus, with its collection of canvases by unknown painters named Picasso, Matisse, and C\u00e9zanne, was the recognized epicenter of the Parisian avant-garde.\n\n**Figure 6.** Walter Pach.\n\nGertrude and Leo Stein spent the summer of 1907 at the Casa Ricci in Fiesole. On several afternoons in July, Alice left the noise and heat of Florence and joined the Steins at their villa high above the Arno valley. Their back patio was a verdant oasis of potted plants and lush flowers reminiscent of Alice's home on the shores of the Pacific Ocean. Like Alice, Gertrude and Leo loved to walk, and they often hiked together across the Tuscan countryside through poppy fields and groves of fragrant olive trees. One afternoon, the Steins walked with Alice to the home of the Renaissance scholar Bernard Berenson at Settignano. Alice \"enjoyed the house very much indeed but especially his books in three rooms and all over the hall.\"\n\n**Figure 7.** Leo, Gertrude, and Michael Stein.\n\nWe can imagine that on one of these excursions to the Stein's villa above Florence, Alice took along her new friend Walter Pach and introduced him to her childhood friends. Alice does not record such an event in her diary, but by late summer of 1907 Pach and the Steins were acquainted, and Pach had an invitation to visit their apartment and salon in Paris. Alice could not have known the importance of the introduction at the time, since Pach was just stepping into the swirling current of modern art, a movement for which he would soon be recognized as a major voice and authority.\n\nGertrude Stein would become famous for her word portraits of friends. Of Walter Pach, she wrote: \"Some things are to him beautiful things, some things are to him desirable things . . . learning anything is to him a natural thing, succeeding in living is to him a natural thing, teaching any one is to him a necessary thing, teaching everything is to him a necessary thing.\"\n\nWith Chase, Pach, and their class, Alice visited the studio and summer home of Henri Matisse near Florence. Pach told Alice he was ambivalent about Matisse's paintings, but throughout the summer, he and Alice dined frequently with Matisse and his wife, and by late summer they were friends. Alice enjoyed Matisse as a person and a painter, and after she left Italy, she regularly corresponded with the painter.\n\nIn the summer of 1907, Pach was beginning what would become a stellar career as an art critic and historian and was submitting articles and art reviews to American magazines. In August, Alice was with Pach in a bookstore in Rome when he found a copy of the latest _Scribner's_ , which included one of his articles. Alice and the storeowner watched a euphoric Pach \"skip around the shop\" with the magazine. The article would be the first of scores of published essays and reviews by Pach that would introduce European modern art to America.\n\nIn Italy, Alice was among artists who were the vanguard of the modern movement. When Chase's summer school came to a close on September 1, Alice, now fully emancipated as an artist and as a woman, did not take the boat home to America but remained on the continent. Her diary entries never acknowledged her feelings for Pach, but she confessed that the dissolution of the class and the end of the summer caused her great emotional and physical suffering.\n\nWith her cousin \"O,\" Alice remained in Europe for another three months. Walter Pach remained on the continent as well, and in October, after Alice toured Germany, Belgium, and the Netherlands, she rendezvoused with him in Paris. Gertrude and Leo Stein were back in residence at 27 rue de Fleurus, and throughout her four weeks in Paris, Alice, like Pach, was frequently a guest at their salon. The Steins took Alice to museums and galleries and may have been alongside her in the Place de la Concorde crowd that watched \"the flying machine\" in the sky over Paris on October 26.\n\nAlice and her cousin sailed to America from England on November 18. After a short visit with Chase at his studio at 303 Fifth Avenue in New York, Alice boarded the train for California and was home by Christmas. Alice had been away from San Diego and her family for seven months. If she pined for the stimulation and freedom found in Europe, or for the companionship of Pach, she kept it to herself.\n\nPach wrote to Alice even before she climbed on the boat for America. His first letter spoke of his frustration that he had not seen more of her before she departed for England. Pach, fluent in four languages, signed the letter \"Suyo, ato y afmo\" (Yours, sincerely and affectionately), Walter Pach.\n\nThroughout the winter and spring, Pach's letters to Alice chronicled his life in Paris among the painters and writers who were breaking bold new ground in art and literature. He told her he was still struggling to like the work of their friend Matisse, and that he had discussed his style with Claude Monet, who was also not a fan of Matisse's canvases. The Steins were almost always mentioned in Pach's letters. \"The Steins send you their love,\" he wrote Alice from the Pension Innocenti, Florence, in August 1908, \"which I need not tell you is no small thing, even though they do extend its warmth to many people.\"\n\nA year after they parted in France, Pach wrote Alice, \"It is just a year since I missed you on my return from Amiens. Gee, I wish there were another opportunity to see you. I wouldn't let any chances get mixed with it.\"\n\nIn San Diego, Alice was disconnected physically and intellectually from the stimulating society of artists and intellectuals she had moved among in Europe. Unwilling or unable to relocate to Paris, Alice did what she could to bring the energy of the modern art world to Southern California, and made plans to open a gallery to exhibit and sell the work of contemporary American and European artists. Pach offered his assistance, although he could not come and join her in California. Needing money, he had moved back to New York in the fall of 1908 and gone to work full-time as a commercial photographer with his father. Pach was intensely unhappy with his job and wrote Alice that he found little to like in New York City.\n\nThe content, tenor, and extent of Alice's letters to Pach can only be guessed at. Although throughout his life he collected and carefully archived all of his correspondence, nothing from Alice was found in his voluminous collection of letters. It is believed that Pach destroyed her letters when he married in February 1914.\n\nWOMEN WHO PURSUED any creative work in the first decades of the twentieth century had to be independent, ambitious, stubborn, unconventional, and tenacious. They also had to endure the loneliness and isolation experienced by all New Women of the modern age. Having a friend to share the paradoxes of the disorderly and undomesticated New World was rare, and the arrival in San Diego of the worldly, accomplished, and kindred spirit Natalie Curtis must have seemed a godsend to Alice Klauber in the lonely winter of 1908.\n\nGeorge Curtis made California his home country when he bought land in the foothills of the Laguna Mountains near Ramona, twenty miles southeast of Escondido. Over several years, George constructed a house, a horse barn, and chicken coop, planted a pecan grove and a kitchen garden, plowed and planted fields of alfalfa, and became a beekeeper.\n\nNatalie stayed with George in his rural retreat as often as she stayed in San Diego with Alice and her friends. Natalie might have been considering permanent residence in Southern California, but in 1909 her father became gravely ill, and she hurried cross-country to Washington Place. Bogey's illness was terminal, and he would be bedridden with paralysis the last years of his life. Natalie put aside her work and her dreams of a life in the West, and for nearly four years \"except to fulfill the most pressing engagements near at hand . . . never left [Bogey's] side.\"\n\nNatalie did manage at least one visit to George and her friends (she saw Lummis, among others, in Los Angeles) and the wide-open spaces of the West in May 1910. But even as she moved into the life of an emancipated woman, she understood that in a time of family crisis, her place as an unmarried daughter was beside her mother and her father. During the years of her father's illness, Natalie was forced to make peace with her divided self, which loved and needed her nuclear family but yearned for the freedoms found among people and places far away from their Greenwich Village world. \"I hunger and thirst so for a bed on the ground that, once freed, I don't believe I can ever go into New York chains again,\" she wrote Ale\u0161 Hrdli\u010dka. \"What a problem\u2014to have a primeval soul and one's home in New York!\"\n\nWhile she was living in New York City, Natalie's sponsors Charlotte Mason, George Peabody, and Hollis Frissell, the principal of the Hampton School in Virginia, urged her to do for Negro music what she had done for Indian music. Natalie had recorded Negro folk songs and spirituals as early as 1904, but in 1910 she began a comprehensive study of African American music with the staff and students of the Hampton School.\n\nImportant to Natalie's survival during the years of Bogey's illness were her friendships with musicians living in New York who shared her advocacy of musical education for America's poor. Kurt Schindler, a twenty-eight-year-old Berlin-born musician and a staff conductor at the Metropolitan Opera House, had become a friend when he worked at G. Schirmer, and she sought his advice while writing _The Indians' Book_. In 1909 Schindler founded New York's MacDowell Chorus (renamed the Schola Cantorum in 1912). After Natalie returned to New York, she and Schindler joined forces to promote music education for New York City's immigrant and minority children.\n\nMimsey Curtis was already involved with the kindergarten reform movement that developed in tandem with the settlement house movement famously established at Jane Addams's Hull House on Chicago's West Side in 1889. Gertrude House, a settlement house associated with the Chicago Kindergarten Institute, had incorporated music in their curriculum in the early 1890s. Natalie wanted to bring music to children at the settlement houses in New York and took her idea to Mimsey and the board (on which Mimsey served as vice president) of the Mary Walton Free Kindergarten for Colored Children on West 63rd Street.\n\nWith the backing of Mimsey's organization, Natalie assembled her own board of influential individuals, which included Schindler, the violinist David Mannes, the brother musicians Frank and Walter Damrosch, the patron of the arts George Foster Peabody, and the attorney and family friend Elbridge Adams, and founded the Music School Settlement for Colored People, which opened in New York City in November 1911.\n\nThe men and women involved with early twentieth-century philanthropic and reform organizations were people of enormous verve, compassion, and talent. Natalie's work at the music settlement houses introduced her to several individuals with whom she would form lasting and important friendships. Among these was Charles Winfred Douglas, an Episcopalian priest and a professionally trained musician and linguist who may have served on Natalie's music school's board of directors. Douglas, a canon in the Diocese of Fond du Lac, Wisconsin, was a talented musicologist who had studied Gregorian chants in Europe among the Benedictine monks, and was among the first Americans to translate the ancient musical form called plainchant or plainsong. But Douglas's most remarkable characteristic, at least to Natalie, was his interest in and knowledge of the American Southwest.\n\nIn 1911 Douglas was the music director of the Community of St. Mary, a boarding school for girls in Peekskill, a town on the Hudson River just north of New York City. Douglas kept an apartment in Manhattan and frequently came into the city to see his colleagues Frank and Walter Damrosch and his good friend Kurt Schindler. Born in Oswego, New York, in 1867, Douglas was tall and big boned, imposing, confident, brusque, and good-humored, with an orator's voice and a preacher's shepherding bossiness. Like Natalie, Douglas was an overachieving doer and seeker. When he was in full health, Douglas was a man of tremendous energy. But also like Natalie, Douglas's passion for his work, and his relentless creative drive, frequently pushed him to exhaustion.\n\nMuch like Natalie's inaugural trip to Arizona, Douglas's first journey to the American West was precipitated by a physical collapse brought on by an obsessive work schedule. In 1893, after simultaneously obtaining degrees from Syracuse University (in music) and St. Andrew's Divinity School (where he was ordained), Douglas came to New York City to work as music curate at the Church of the Redeemer and to teach music at St. John's parochial school. In his first winter in New York, twenty-seven-year-old Douglas, overworked, exhausted, and ill, collapsed while practicing the organ one night after choir rehearsal. He was taken to Presbyterian Hospital where he lay near death with the complications of double pneumonia for nine weeks.\n\nAfter several months of convalescing in New York, Douglas was sent out west where it was hoped his lungs would heal in the high, dry air. Douglas boarded a train for Denver, where he met and married Mary Josepha Williams, a graduate of Gross Medical School and the joint director of a small sanatorium on Pearl Street in Denver. Dr. Jo, as she was called, was the daughter of a wealthy Detroit family with major landholdings in Colorado and Denver's first woman doctor. She owned several hundred acres of land near the Evergreen stage stop, and with Douglas built a spectacular summer home on the property they named Camp Neosho.\n\nIn the early 1900s Douglas began lecturing at Northwestern's School of Music and helped to develop the music programs at Gertrude House in Chicago. Freed of financial concerns by his wife's wealth, Douglas traveled between an apartment in New York City and his home in Evergreen, where his only son, Frederic (Eric) Huntington Douglas, was born in 1897. Douglas sailed for England in May 1901 and spent the next year in England, Scotland, Germany, and France listening to and studying Gregorian chants and ancient plainchant. When he returned to the United States, Douglas was determined to bring the heavenly chanting music of plainsong into mainstream American liturgical music.\n\nIn the fall of 1902, Douglas and his wife and son moved into an apartment in New York City on West 84th Street. Douglas resumed his duties as curate at the Church of the Redeemer, and joined Damrosch's Musical Art Society. By the winter of 1903, the canon's health broke again. After months at various convalescent centers and retreats in New York, Pennsylvania, and Massachusetts, Douglas headed out west.\n\nFor six months, Douglas lived in a tent and slept in a bedroll in the wilderness of the Rio Grande of New Mexico, on the high desert of the Kaibab Plateau, and in the deep canyons of the Colorado River in Arizona. He lived alone and with the Indians he met on his journey, including the Hopis, who adopted Douglas and named him \"Tall Pine Tree.\" Douglas submersed himself in the culture and art of the native tribes and began to collect pueblo pottery and baskets and Navajo rugs and blankets.\n\nSix months on the high desert healed Douglas, and in 1904 he donned his city clothes and pastoral collar and returned to work. Now a vocal advocate of Indian art and culture, and a student of Indian music, Douglas lectured to audiences in Chicago and New York about Native America. When exactly Douglas became cognizant of the work of Natalie Curtis is not known, but by 1907, and the publication of _The Indians' Book_ , he had surely met her. Schindler likely introduced them. Douglas was a board member of Schindler's Schola Cantorum from its inception, and Schindler had been friends with Natalie since editing her _Indians' Book_. By the fall of 1912, Douglas's appointment book noted frequent meetings with Natalie.\n\nBogey died just after the Thanksgiving holiday in late November 1912. George came from California and joined his mother and siblings to bury Edward Curtis in the family cemetery in Providence, Rhode Island. Augusta was seventy-two years of age. Natalie was thirty-seven. Augusta may have hoped her spinster daughter would settle permanently in the house on Washington Place. But she knew, too, that Natalie had unfinished business\u2014an unfinished life\u2014on the other side of the continent. And now that George was healthy and happy in his own home in California, even Bridgham agreed that only the American West could restore Natalie's health and her spirit.\n\nNatalie and George thought it was time their mother saw for herself the country that had claimed the hearts of two of her children, and in April 1913 Natalie boarded the train west with Mimsey. For Augusta Curtis it was a journey into an exotic, foreign land. The West Coast of the United States, and the desert and mountains that served as a line of demarcation between civilized and frontier America, was still very much another country.\n\nMimsey had heard in detail about Natalie's adventures in the Southwest, and she may have read all or parts of George's novel _The Wooing of a Recluse_ , which would be published the next winter by the Devin-Adair Company. The novel was a grand romantic monologue by an unnamed cowboy written in letters to his lover, Pandora, back in New York. The narrator's transition from a life in upper-class New York City to a cattle ranch in Arizona very much paralleled the life of George DeClyver Curtis. Although George's cowboy protagonist went east to reunite with his love, George the cowboy-writer never lived in the East again, but set down deep roots in his land near Ramona. If George Curtis ever had a gal back in New York or Boston, she lost him to his new love, Southern California.\n\nGeorge wrote under a pseudonym, Gregory Marwood, but his Harvard classmates, friends, and family in New York and Boston understood that the romantic novel of letters from an Arizona cowboy to his society girl back east offered glimpses into the author's own life. And throughout the novel there were passages that seemed written just for his sister Natalie.\n\n> Put aside for a while all that mesh of complex activities that is entangling your health, find some placid travelling-companion and come West. I shall meet you at the railroad with ponies and a packhorse, and you may roam this unseeing land with as little regard for the conventionalities as when you once slide down the roof of our icehouse. . . . I will be your guide, cook and horse-wrangler; you need only ride and be glad . . . for time and the world will be all yours if you so choose, and in the freshness of a natural life you will forget that you were once in a fair way to be an invalid.\n\nBack in California again, Natalie immersed herself in the natural life where she could, in George's words, live with little regard for conventionalities. George met Natalie and Mimsey at the San Diego railroad station, possibly with Alice Klauber, who had a car and was a good friend of both George and Natalie. Alice's extended family in San Diego included half a dozen siblings, and the Curtis women had invitations to stay in the homes of several prominent families, including Alice's mother, Theresa Klauber, who had been a widow for less than two years. Mimsey stayed a month in California. With Natalie and Alice as her companions, she explored the little city of San Diego and went out for overnight retreats in La Jolla. They also stayed with George at his ranch in the rolling hills thirty miles inland.\n\nIn late April, Natalie, George, and Mimsey motored north of San Diego with Alice to Hilero, Wheeler Bailey's weekend retreat on the La Jolla bluffs above the Pacific Ocean. Bailey, called Uncle Wheeler by Alice and Natalie, invited Alice and the Curtises to spend a few nights at the fabulous house designed by Irving J. Gill and Frank Mead.\n\nThe auto journey out to Hilero's breathtaking perch above the ocean was through a paradisiacal and pastoral landscape of green fields and wild flowers where sheep and cattle grazed. Hilero was only five years old, but it had already achieved significant national attention from builders and designers for its Moorish architectural elements, and for its museum quality collection of Native American arts and crafts. Although Natalie appreciated the collection of Indian art that decorated the walls and windowsills of the house, Hilero's greatest treasure for a musician was the red Steinway piano covered by a Navajo rug on the large first-floor portico.\n\nNatalie wrote about their days at Hilero as \"a sort of California idyl, a song of the Pacific coast.\" Their meals were served at long tables on the covered porches and out in the gardens. If they ate in the dining room, the enormous doors were wide open to the sounds of the surf and the sea birds. Natalie, George, and Alice, perhaps Mimsey too, chose to leave the comfort of their beds and private rooms and slept on the large second-floor balcony under the stars.\n\nNatalie, George, Mimsey, Bailey's relative Wheeler North, and Alice signed Hilero's guestbook before departing on April 28. George wrote in French and English about \"M\u00e9lodie \u00e9cossaise, sung at Hiliro [ _sic_ ], where even Auld lang syne [ _sic_ ] has an original touch.\" Natalie filled an entire page of the guest book with the musical notes and the Navajo (with English translation) words of a Navajo peace chant, adding \"Indian thoughts inspired by a wind-swept day, and a night on the porch of Hiliro [ _sic_ ]\u2014the 'Blanco Hogan' of the Pacific slope!\"\n\nMimsey departed California for New York in the first week of May. Natalie remained on the West Coast and moved between George's ranch and her rented rooms in downtown San Diego. By 1914 she was living in the Palomar Apartments, a building designed by Frank Mead and partners at the corner of 6th and Maple, within walking distance of Alice's home in downtown San Diego. From her rooms at the Palomar, Natalie could watch the construction of the buildings and gardens of the San Diego Panama-California Exposition that would open in 1915. Begun in 1912, the 1,400-acre exposition grounds of Balboa Park would define modern San Diego. Although the exposition was to celebrate the opening of the Panama Canal, the museums and exhibits of Balboa Park would catapult San Diego into the twentieth century, and bring Southern California's art, history, and natural resources national attention.\n\nSouthern California was fiercely hot and dry in June and July 1913. Natalie frequently stayed for weeks at a time on the \"unseeing land\" of George's ranch. He was digging a well, installing a pump, and trenching a pipeline that would bring water to the house. Natalie was strong again and worked alongside George doing ranch chores until noon. After lunch, they napped through the heat of the afternoon, and in the evenings they read aloud to one another from the newspapers and magazines Mimsey sent from New York. \"Cleared poison oak under Merlin's oak,\" George wrote in his lined pocket diary on July 18, \"and read a little Dante and snoozed there in the afternoon. N[atalie] and I walked up to the spring when the sun went, and brought back a pail of water.\"\n\nWith Alice, Natalie motored up and down the coast, and out to La Jolla for overnights with Alice's friends. She also went north to Pasadena for several days and visited Lummis. They listened to Lummis's recordings of Indian songs and discussed Washington's latest political appointments and their Indian policies.\n\nLife in California was not all play. Natalie worked on several pieces for publication including two articles about Indian music for the _Outlook_. But her benefactors, like George Foster Peabody, although pleased to know she was in good health, believed she ought to return to the East and get back to the real work that awaited her attention. Her comprehensive piece about African American musicians, \"The Negro's Contribution to the Music of America,\" had come out in the _Outlook_ in March, and Natalie needed to complete the project at the Hampton School.\n\nIf Natalie heard Peabody's call for her return to New York, she did not heed it. Instead, in the summer of 1913 she pushed deeper into Indian Country and the wild frontier. \n\n#  _6_\n\n##  SECRETS AND DUALITIES\n\nWomen are not supposed to take up much room, or to go very far from home, or to stay away for long. They are not supposed to be by themselves. They are supposed to hear and obey, to come back when they are called.\n\n##  \u2014 _Virginia Scharff_\n\nWHILE NATALIE WAS KEEPING VIGIL beside her dying father in his room near Washington Square, a similar family drama was transpiring in the J. Bishop Johnson house on the tiny Massachusetts island of Nahant. On a clear day, the gold dome of the Massachusetts State House was visible from the front porch of the Johnson home on Dorothy Cove. Even so, urban Boston and the professional music world in which J. Bishop's granddaughter Carol had lived and worked for more than a decade belonged to a universe that existed far beyond the shores of the fishing community on the small island.\n\nIn 1912, when Caroline Bishop Stanley was summoned home, she was employed as a piano teacher at the prestigious Hannah More Girls' Academy in Maryland. As the only child and single daughter of a widowed, ailing mother, Carol was bound by duty and principle to leave her position as a music instructor and return to Nahant. Carol's mother, Fannie, lived in the Bishop family house on Dorothy Cove with Carol's grandmother, and Nahant's oldest resident, eighty-four-year-old Almira Choate Johnson. Coming home to care for her mother on Nahant meant Carol had to resign from one of the few professional positions available to a woman with a bachelor's degree in music in 1912. It also meant Carol, at thirty-two, was completely dependent on her family.\n\nCarol Stanley had lived and breathed music since she was a young child. Unlike her future friend Natalie Curtis, Carol's education and departure from the bosom of the family on the island of Nahant was a hard-won accomplishment, nearly every step taken in defiance of the communal and familial tide of expectation. Carol's father, Edwin Stanley, had died on Nahant in the 1880s when Carol was a small child, and Carol and her mother had gone to live with her grandparents in their house on the beach. Grandmother Almira was widowed in 1896, so by the turn of the century, the J. Bishop Johnson house had become the domain of two widows and a serious-minded young girl who wanted only one thing in life: to play the piano.\n\nThe history of Nahant island was the story of the extended Johnson family. There were two distinct and separate Johnson lineages, and Carol's mother's family was related to the latecomers, the Maine-based Johnsons, who had come to the island in the mid-1800s. The first Johnsons predated Carol's family by two centuries, and were listed among the first three families to settle the island in the 1600s. In spite of their lack of residential longevity, Carol's family was considered island royalty, especially since her grandfather's first wife, Caroline, was a Johnson of the first pedigree.\n\nCarol's maternal grandfather, known around the island as J. Bishop, came to Nahant from Maine with several of his brothers. By the mid-1800s the Johnson brothers had a thriving fish business, and J. Bishop built a family home at the corner of Willow and Valley Roads beside the Johnson fish market, a short walk from the waters of Massachusetts Bay.\n\nCarol was born on December 16, 1879. By that time, there were Johnsons belonging to one of the two Johnson families in every classroom, business, and neighborhood in the village of Nahant and out along the island's shoreline. The dual lineage of the Johnson name befits the island, as Nahant is a place of dualities and twins, of mirrors and magnification, and the island's history is laced with stories of the seen and unseen, above and below land and water. The name Nahant was derived from the Indian word _nahanteau_ (twins), and the island is actually two islands, Little and Big Nahant, that are connected by tombolos, isthmuses, that flood at high tide. Before the raised cart road was built in the mid-1800s, islanders could only reach the mainland and the neighboring town of Lynn on horseback or in buggies when the tide was out. This limited and rhythmic accessibility created a community where business revolved around the tide tables\n\n**Figure 8.** Forty Step Beach, Nahant, Massachusetts, 1906.\n\nIn the earliest accounts of fishermen and sailors there was frequent mention of a peculiar optical illusion, called doubling, seen on the sea near Nahant island. When certain weather and cloud conditions existed, ships and landmarks appeared to be mirrored in the cloud ceiling. This doubling was so endemic to Nahant that early historians included illustrations of the phenomenon, and poets wrote of the mirrored and magical world above and below Nahant.\n\nNahant was a place of doubles and also of secrets. Along the coastline where the land has been battered and shaped by the waves of the Atlantic there are numerous caves and secret passages: Spouting Horn was so named because of the sound the sea makes at it crashes through this formation on the rocky cliff on the island's eastern tip. Swallows' Cave is a deep channel seventy-two feet long and eight feet high that opens into the sea and is accessible only by climbing down rock cliffs to the cave's entrance at low tide. Nearer the center of the island, on the high land near the village proper, is the Witches' Cave, where several women eluded capture and prosecution during the Salem witch trials in 1692.\n\nCaroline Bishop Stanley was in body and in spirit a _Nahanteur_. She had a twin brother, Edward, who lived only two months. Caroline had an older brother who died during childbirth, so after her twin died, Caroline was raised an only child. In true mirroring style, Carol was named for her grandfather J. Bishop Johnson's first wife, Caroline Johnson, who had died in childbirth thirty years before.\n\nAlthough the public death record confirms that Carol's twin died in February 1880, there was never again an acknowledgment in private or public accounts that Carol's mother had given birth to twins. We assume that Carol was told about her twin brother, but the keeping of such a secret would not have been out of character among the Johnsons or any other family with deep roots on the island of Nahant.\n\nCarol's childhood on Nahant in the 1880s and '90s coincided with the island's heyday as the resort and second-home community of Boston's elite. As the northeast seaboard was transformed by American industry and Boston became crowded and dirty, city folk looked east across the water to rural Nahant, a pastoral island of peace and quiet untouched by industrial America. The island off the shore of the bustling town of Lynn was a quiet backwater during the New England winter. But each summer, Nahant's population more than doubled with day-trippers from Boston who came in droves on the daily ferries, and seasonal residents who moved into their grand cottages for the summer.\n\nIn 1821 Colonel Thomas H. Perkins and William Paine bought East Point, the \"Ram Pasture,\" and built a fine hotel. By 1903 the island boasted fourteen hotels, several of which were grand affairs that catered to the ultrawealthy. On Nahant the Brahmins could escape the heat and pollution of the city, and often a stay on the island prompted visitors to build second-home \"cottages\" along the beaches and up in the village. Although Boston's Brahmins brought to the island the culture and money of Boston high society, they shed the posturing and formalities inherent to urban life. The family breadwinner could continue to commute to his work on Beacon Hill or near the state house and commons while his wife and children could soak up the healthy sea air and sun while swimming and boating, or riding horseback at the stables.\n\nThe island's part-time residents included Boston's most prominent families. Samuel A. Eliot, father of Charles W. Eliot, the future president of Harvard, kept a residence on the island, and the family of George H. Mifflin of the publishing house spent summers in their home on Nahant. Natalie's uncle George's close friend Henry Wadsworth Longfellow, the iconic American poet, was perhaps Nahant's most famous resident. Longfellow lived in Cambridge and taught at Harvard, but beginning in 1850, he came each summer with his family and lived in their home directly across the cove from Carol's grandfather's fish market. Another friend of George Curtis's, the statesman Henry Cabot Lodge, was a child on Nahant. Lodge's father, John E. Lodge, had come to the island in the 1830s, and after he married Anna Cabot in 1842 and moved to Beacon Hill to raise their two children, he returned to build a grand summer house on Nahant. As a U.S. senator, Henry Cabot Lodge retreated to Nahant for solitude and respite from politics. Although busy with the matters of Washington and the world, he served as Nahant town moderator for twenty years and remained a trustee for the Nahant public library for over four decades.\n\nLodge was not a progressive Republican, like Theodore Roosevelt, but aligned himself with the conservatives who dominated the Senate when Roosevelt was president. Although often at political odds, Lodge was a welcomed friend at Roosevelt's home, Sagamore Hill, and Roosevelt came to Nahant to visit Lodge at his family retreat.\n\nAROUND NAHANT, Caroline Stanley was the local girl who excelled on the piano. In 1894, when Carol (called Carrie as a child) was fifteen, she played \"Smiling Springtime\" at the Nahant graduation exercises. There were only three graduates that year (all of the children on Nahant attended one school), but the whole town turned out for the event, which included orchestral performances, several violin and piano solos, including Carol's, six student recitations, and an address titled \"The Nation's Progress\" by graduating senior Charles Cabot Johnson.\n\nCarol Stanley graduated Nahant School in a similar ceremony in 1899 and entered Mount Holyoke College the next fall. But the Massachusetts women's college didn't offer the level of music instruction Carrie wanted, and in the autumn of 1900 she transferred to the New England Conservatory of Music in Boston.\n\n**Figure 9.** Pleasant Street School, Nahant, 1880s. Carol Bishop Stanley is in the front row, second from right.\n\nThe New England Conservatory of Music was undergoing dramatic growth in the first years of the twentieth century. In Carol's third year, the conservatory moved out of cramped quarters in the South End of Boston and into a new building near the Boston Symphony Orchestra's headquarters. The school began an affiliation with Harvard, and in June 1903, when the conservatory's new Jordon Hall and its modern, elegant, and acoustically perfect stage and auditorium opened, the New England Conservatory boasted, and rightly so, that it could compete with any music school in the world.\n\nIndeed, by 1906 even the _New York Observer_ touted the conservatory's formidable offerings, telling New Yorkers that \"Boston is the recognized musical center of the country\" where is found \"the New England Conservatory of Music, the best equipped institution of its kind, and in quality of work rivaling kindred institutions in foreign countries \"\n\nCarol and her classmates were the first to occupy the conservatory's new residential housing on Hemenway Street, which was an easy block's walking distance from the school's offices, class and rehearsal rooms, and library and bookstore on the corner of Huntington Avenue and Gainsborough Street. The conservatory was in the heart of Boston's music world, and the school's population boomed to almost two thousand students. Carol studied piano with George W. Proctor and music theory with the young American composer Harry N. Red-man. She joined the staff of _The Neume_ , the conservatory's yearbook, and in the fall of 1903 was the seventy-eighth member inducted into the Zeta chapter of Alpha Chi Omega, a national sorority dedicated to attaining \"the highest musical culture and to cultivat[ing] those principles that embody true womanhood.\"\n\n**Figure 10.** New England Conservatory of Music on Franklin Square, Boston.\n\nConservatory student musicians performed solo recitals and participated in the school's orchestra and chorus in public performances throughout the school year. Jordan Hall hosted the Boston Symphony Quartet's six public concerts a year, and the Apollo and McDowell Clubs and the Choral Art Society also held their annual concert series at the conservatory's signature facility.\n\n**Figure 11.** New England Conservatory of Music graduates, 1905. Carol Stanley is in the third row, second from right.\n\nNahant island native Carol Stanley was being educated at one of the finest conservatories in America. She was also learning the realities of modern urban life. And every day from 1900 until her graduation on June 20, 1905, Caroline Bishop Stanley was being indoctrinated into the ranks of the twentieth century's New Women.\n\nThe Johnson family was comfortably middle-class, but Carol's mother, Fannie, was not wealthy, and at least a portion of Carol's tuition and board was paid for by her grandmother Almira. Carol likely received additional tuition monies from Almira's Brahmin family or from the Edward C. Johnson family, with whom Carol lived in the summers during her college years. Carol earned money teaching piano lessons on Nahant, and also worked as the nanny for the Edward Johnson family when they came to stay at their summer house on the island. The E. C. Johnsons, relatives of Carol's grandfather's first wife, Caroline, were the wealthy owners of the Boston dry goods house C. F. Hovey & Co.\n\nThe E. C. Johnsons moved in Boston's most exclusive social circles. Their daughter, Carol's step-cousin Charlotte Howe Johnson, married Curtis Guild Jr., and when he became the forty-fifth governor of Massachusetts in 1905, the Edward Johnsons attended every gala event and social function at the governor's mansion. But even among the proper Brahmins, daughters could wander from the straight and narrow and fall into the wilder side of society. When Carol was a single woman living and working in Maryland, she learned that Edward Johnson's daughter Mary Frothingham Johnson, her cousin and former piano student, had eloped with a count named Vittorio Oriandini Guidi. The Boston gossip columnist had a field day with the humiliating announcement, saying the count \"was cutting quite a wide swath in Boston and Brookline society\" and although he was the \"social lion of the hour\" was in fact a sewing machine salesman.\n\nWhen young Mary Johnson ran off with and married the Italian sewing machine salesman at the Providence, Rhode Island, Episcopalian church, she personified the improper, independent, headstrong, unchaperoned New Woman the Victorian Brahmins of 1907 feared and shunned. The news of her elopement served as a reminder to all families with single daughters: keep the girls at home until they marry, and supervise their interactions with all potential suitors, even titled ones.\n\nCarol was twenty-six and a self-supporting music teacher at the Hannah More Academy near Baltimore when the gossip about the new Countess Guidi reached her apartment on the strictly guarded grounds of the oldest Episcopalian girls' school in America. If Carol had a love interest in Reisterstown, she knew better than to share the details of her social life with the Stanley-Johnson clan back in Boston or out on Nahant island. Such was the paradox of the educated, single woman at the turn of the century: to keep the family peace, she kept the extent of her ambitions and independence to herself, and although she was frequently crossing the boundary between the Victorian and the modern world, she still longed for family approval and support. Dualities and secrets were second nature to Carol, and she kept her explorations into the frontier of the New Woman to herself, and spoke of her professional and personal concerns only with a small circle of colleagues and confidants.\n\nNatalie Curtis and Canon Winfred Douglas were among the friends with whom Carol could talk of her life as a single woman and of her love of music. With no record of their early friendship, it is impossible to determined whom Carol met first, Douglas or Natalie. In November 1906 Cardinal Gibbons of Baltimore held a high mass to confirm ninety-one novitiates at St. John's Church in Frederick, Maryland. It was a gala event that included an orchestra and chorus. Carol B. Stanley was a member of the professional choir that performed at the mass, which was attended by dozens of ministers from various denominations and by politicians, military officers, and college presidents from the Baltimore region. Winfred Douglas had just accepted the post as music director at the Community of St. Mary and moved with his family to Peekskill, New York. Was Father Douglas in Baltimore for the gala high mass in November 1906, and if so, was he introduced to the young music teacher, Carol Stanley of Nahant?\n\n**Figure 12.** Canon Charles Winfred Douglas.\n\nThe next fall, in October 1907, the _Baltimore Sun_ announced that the Hannah More Academy had opened the scholastic year with more than a hundred women students. The school's new teachers included a member of the music department, a Miss Carol B. Stanley. Living and teaching at the Hannah More Academy was as good as it got for a young female music teacher, and Carol remained on the resident staff for five years. She might have stayed another ten, but in the late fall of 1912 Carol learned that her mother, Fannie, was dying of carcinoma of the uterus. Carol resigned her position at the academy and returned to the Johnson House on Nahant.\n\nThrough the next winter and spring, Carol cared for her bedridden mother and kept house for her elderly grandmother Almira. In early June, when Nahant gardens began to bloom and the islanders anticipated the return of the summer tourists and seasonal residents, Fannie fell into a coma.\n\nOn Sunday, June 8, 1913, when Carol was tending her comatose mother, unusually ominous clouds began to gather in the sky over Nahant island stretching to the coastal town of Marblehead on the mainland. By early afternoon a great storm moved into Massachusetts Bay. As the day turned dark and the winds surged, Carol, like all her neighbors on Dorothy Cove, tied down everything around the house and yard.\n\nThe daily steamer, the _General Lincoln_ , was approaching Nahant's Bass Point wharf with a full load of Bostonians preparing to disembark for a Sunday afternoon on the pier or on one of the island's lovely beaches. When the steamer was ten minutes out, the storm broke. The captain pulled the _General Lincoln_ away from shore and, in gale-force winds, turned the steamer stern first into the storm and then let it be carried down the coast.\n\nFrom her mother's bedroom window, Carol may have watched the _General Lincoln_ reverse its course as the flags marking the channel across the harbor were blown from their moorings. The Johnson house shutters slammed and banged, and anything not tethered and secured was thrown into the air. Carts on the road to the village were overturned by the wind, and dust clouds reduced visibility on the island to a few yards.\n\nWith the very first clap of thunder\u2014a violent, concussive sky-splitting crack\u2014a bolt of lightning struck the Johnson house, traveled one of the house's vent pipes down three stories, and exploded in the first-floor lavatory. The bolt instantly reversed and shot back up the staircase, ripping the woodwork as it ascended to the roof, where it splintered shingles and tore off the saddle board.\n\nNeighbors who had seen the lightning strike fought the winds and torrential rain to reach the Johnson house to see if anyone was hurt. Carol and the eighty-five-year-old Almira were shocked and traumatized by the ferocity and scope of the lightning strike but physically unscathed. Fannie remained unconscious throughout the violent storm. Several hours later, however, Fannie Johnson Stanley passed away.\n\nAs reported in a local newspaper, Dr. Cusick, the family physician, was of the opinion that the \"shock of the bolt\" and the ferocity of the storm had not hastened Fannie's death.\n\nFannie was buried in the tiny cemetery on Nahant. Carol, an unemployed woman of thirty-four, remained on the island for the next year, living with and caring for her grandmother, teaching piano lessons, and wondering what she would or could do with the rest of her life. Her mother's brother, Otis, and his wife and two daughters lived in town, and Carol knew virtually everyone who lived in the village or in the grand summer homes out along the shoreline. Even so, Nahant island was no longer Carol's home of choice.\n\nCarol's grandfather had been a man of determination and vision, and his hard work as a fisherman and market owner had brought him wealth. Her grandmother, Almira Choate Estes Johnson, also had her own family money. She was the granddaughter of Ebenizer Choate, a relative of the New York lawyer and statesman Joseph Hodges Choate. The Choates had been part of the seasonal community on Nahant island since the first grand hotel was built on East Point by Colonel Thomas Perkins in the 1820s. As the granddaughter of one of the wealthiest and most respected dignitaries on the island, the middle-aged spinster Carol Stanley had no real financial or social worries _if_ she settled down within the Johnson family domain on Nahant.\n\nBut Carol could not settle down on Nahant. Her heart remained focused on Boston, Baltimore, and the world of professional music she had been part of for a brief, wonderful decade. She continued to interact with musicians at the music settlement houses, specifically Boston's South End Music School, about which Natalie wrote in 1911. Carol practiced and sometimes performed with local orchestras, and volunteered and played piano at the Seaman's Bethel of the Boston Port and Seamen's Aid society in North Boston. But on most days during the year after her mother's death, Carol was isolated as a woman and a professional musician in the family house on the quiet beach of Dorothy Cove.\n\nIf Carol did indeed know Natalie Curtis and Winfred Douglas by 1913, her desire to return to the world beyond the island was likely intensified by their stories of the people and places of Arizona, New Mexico, Colorado, and California. However, Carol kept her thoughts and yearnings to herself and remained on Nahant, the obedient granddaughter, until Grandmother Almira's death the following year. After that, the second half of Carol's life would begin with an epic journey into the deepest, most remote and unforgiving wilderness yet found in the American Southwest. \n\n#  _7_\n\n##  CREATING A BRAVE NEW WORLD\n\nYou either have to be utterly common place or else do the thing people don't want, because it has not yet been invented. No really new and original thing is wanted: people have to learn to like new things.\n\n##  \u2014 _Willa Cather_\n\nIN 1913 NEW YORK CITY was fertile ground for radical discourse and social experimentation, a place where American intellectuals interested in new ideas about politics, society, the arts, and women found energy and voice. The hub of this radical New World\u2014the Greenwich Village salon of Mabel Ganson Dodge\u2014was located near Washington Square just a few blocks from Natalie's family home.\n\nMabel Dodge was born in Buffalo in 1879, the daughter of a wealthy industrialist. In the first decade of the twentieth century, Dodge and Alice Klauber had several friends in common in Italy and France, although neither woman makes mention of ever meeting the other in Europe or America. Dodge lived in Florence for several years beginning in 1905, and she dined and socialized, as Alice did, with the Bernard Berensons at their villa in the Tuscan countryside. Dodge became friends with the Steins in Italy, and her visits to their famous Paris salon in 1911 completely transformed her ideas about men, women, art, and life.\n\nWhen Dodge returned to America in November 1912, she moved into an apartment on Fifth Avenue near Washington Square and within the year established her own salon for weekly gatherings of Greenwich Village bohemians and radicals. Dodge was a feminist, or on her way to becoming one, and as a New Woman was determined she would not be bound by traditional roles and ideas. Like the New Art and the New Psychology, the New Woman was creating an innovative paradigm that denounced the Victorian social barriers and gender limitations that had repressed their parents' generation. Like Stein in Paris, Dodge and her Greenwich Village crowd of \"capitalists, anarchists, artists, writers, and actresses\" debated sex and society, marriage, monogamy, the work of men and women, and child care and were imagining and creating a modern society that encouraged personal, professional, and artistic freedom.\n\nSome historians claim it was \"the restlessness of women\" that caused \"the development called Greenwich Village, which existed not only in New York but all over the country.\" The New Woman was to be independent, self-defined, and physically capable, and not to be at the mercy of her biology. She was to determine the direction of her own life, with or without a man, and could assume positions of power in politics, education, and reform movements.\n\nAs Natalie, Alice, Carol, and Mabel Dodge had learned by 1913, the New Woman was not an easy persona to integrate into their daily lives. In spite of their best efforts to be independent and self-directed, women in the early twentieth century, even those with education and money, were conflicted and even hindered by the deeply ingrained need for a husband, for security, and for acceptance in their family and a place in society.\n\nIt is not known whether Natalie knew of her neighbor Mabel Dodge and her Greenwich Village salon, but several years later, in 1917, the two women's paths were destined to cross two thousand miles away in an adobe casita near the plaza of old Santa Fe.\n\nBEFORE NATALIE AND MIMSEY departed New York for California in 1913, they, like most everyone in New York City, witnessed the most important event in the history of American art: the opening of the International Exhibition of Modern Art, known as the Armory Show. Held in the cavernous halls of the 69th Regiment Armory in New York City, the exhibition introduced European and American modern painting and sculpture to an artistically na\u00efve and largely ignorant American public.\n\nThe Armory Show's major organizers were the American painters Walt Kuhn and Arthur B. Davies, and Alice's friend Walter Pach, a painter and an emerging American authority on modern art. Pach was living again in Paris, where he was often found at the home of Leo and Gertrude Stein, and also at the residence of their older brother, Michael. The Steins were involved with the Armory Show from its inception. Leo, \"at the pinnacle of his reputation as the first prophet of the new art\" (many considered Leo to be the discoverer of Picasso), assisted Pach in the securing of loans from Parisian art dealers.\n\nPach was the Armory Show's chief European liaison, loan, and sales agent. Throughout 1912 he approached and secured loans from the best European moderns, many of whom were his close personal friends, and he was directly responsible for securing the exhibition's most scandalous pieces: Marcel Duchamp's _Nude Descending a Staircase, No. 2_ , Matisse's _Blue Nude_ , and Brancusi's _Sleeping Muse_.\n\nThe show brought to the United States the first major exhibition of modern art, and the 1,250 works by three hundred American and European artists included the most innovative of the Europeans, including C\u00e9zanne, Duchamp, Matisse, and Picasso. Four thousand people flocked to the Armory Show's opening night on February 17, 1913, to view the finest examples of Symbolism, Impressionism, Post-impressionism, Neo-Impressionism, and Cubism. But even urban and progressive New Yorkers were not ready for modern art. The exhibit's month-long run in New York City, and the subsequent exhibitions at the Art Institute of Chicago and at the Copley Society of Art in Boston, shocked the American public, unleashed vociferous outrage from the press, and ultimately altered the course of art history in America.\n\nThe New American Art movement emerged from the same soil and at the same time as the New American Woman. And like the New Woman, the New Art that invaded America via the Armory Show would be the object of public distain and ridicule for many years before it gained widespread acceptance.\n\nThe year before the momentous opening of the Armory Show in New York City, Alice Klauber once again left the peaceful but artistically unchallenging environment of San Diego for Europe. In June 1912 she sailed for Spain with the iconic American artist and teacher Robert Henri. Henri was a member of the \"Ashcan\" group of painters, New York realists who rebelled against both \"the dead hand of tradition and the foreign hand of Paris Modernism.\" Henri would become one of the first American painters associated with the moderns, and with his cohorts advocated a New Art with a new ideal of beauty\u2014the common and nonheroic people and activities of everyday life.\n\n**Figure 13.** Robert Henri ( _standing center_ ) and class, summer 1912, Spain. Alice Klauber is third from right.\n\nHenri was a member of The Eight, a group of painters whose New York exhibition in 1908 caused a sensation in American contemporary art. In 1910 Henri and The Eight organized the larger Exhibition of Independent Artists that included Rockwell Kent, George Bellows, and several other contemporary art renegades who, like Henri, challenged the academic traditions defined and defended by Alice's first teacher, William Merritt Chase.\n\nIn Henri's class, and to some extent in Chase's classroom in Italy five years earlier, Alice was an observer of the aesthetic power struggle unfolding between the dominant artists and teachers of the American modern art movement. Chase was ideologically opposed to all the modernist movements, American and European, and did not care for the narrative subject style favored by Henri's Ashcan group. Although a leader and a force in contemporary art, Henri was not himself a fan of the American avant-garde movement that was influenced and ultimately eclipsed by the European moderns Pablo Picasso, Albert Gleizes, Constantin Brancusi, Jacques Villon, Henri Matisse, and others.\n\nIn 1911 Henri and the members of The Eight formed the Association of American Painters and Sculptors. Henri, the organization's first director, and the association initiated the conversation in 1912 that gave birth to the Armory Show. Henri might have been the grand master of the International Exhibition, but in 1913 he was jostled from power and directorship by a colleague, the painter Arthur B. Davies.\n\nON JUNE 7, 1912, Alice, Henri and his wife, Marjorie Organ (called Margie \"O\" or simply \"O\"), Henri's assistants Randall Davey and Wayman Adams, the painters Meta Gehring and Esther Stevens Barney, and the dozen other members of Henri's summer class boarded the _Carpathia_ in New York, bound for Gibraltar. Just six weeks earlier, the _Carpathia_ had become the most famous rescue ship in the world when it was the first to reach the _Titanic_ 's survivors on the night of April 14. Alice and her traveling companions were on the _Carpathia_ 's first transatlantic crossing since that fateful night when the ship had reversed it course to Europe and returned to New York with its decks filled with passengers rescued from the Atlantic.\n\nAlice's crossing ended uneventfully at Gibraltar on June 14. During the next month, Alice and the rest of the group followed Henri across a sweltering hot Spain, stopping to paint in Granada, Seville, and Madrid. In Cordova in late June, after sketching the famous mosque, Alice lunched with Gertrude Stein and Alice B. Toklas on the old square. Stein and Toklas had been in Spain for almost two months, visiting the same cities, art museums, and architectural landmarks that Alice and Henri's class were touring. The relentless Spanish summer heat was inhibiting the class's progress and rearranging itineraries. On a particularly steamy noon in Cordova, Alice met Stein and Toklas at a sidewalk caf\u00e9. The upcoming Armory Show dominated art news on both sides of the Atlantic, and with Walter Pach orchestrating the selection of artwork to be sent from Europe to New York, we can imagine Stein shared with Alice details of Pach's life in Paris.\n\n**Figure 14.** Robert Henri on board the _Carpathia_ , June 1912.\n\nThe next day, the group went on to Madrid, Segovia, and Toledo, where the class painted for the next several weeks. Stein and Toklas went to Seville, where Stein suffered a severe attack of colitis that prompted the two women's quick departure for Gibraltar and the cooler breezes found along the Mediterranean.\n\nThroughout the summer, Henri lectured on art, literature, theater, and philosophy and instructed his students to get away from their canvas and experience the world. His lessons continued after class around dining tables and in outdoor caf\u00e9s. One of Henri's favorite themes was the challenge of the solitary life necessary for the serious artist. \"Few have the courage and stamina to see it through. You have to make up your mind to be alone in many ways.\"\n\nAlice, forty-one that summer, knew all about the solitary life of an artist, and had experienced the numerous ways in which a single woman of a certain age felt alone. Like Natalie, Alice knew that immersion in creative work was the only antidote to her ennui. \"And dont [ _sic_ ] get discouraged,\" Henri told Alice. \"Don't let discouragement stop you. You can't afford that. Work . . . work all the time.\"\n\nThroughout their months in Spain, Alice talked art and life with Henri during long train rides, at tables in sidewalk caf\u00e9s, while walking through cathedrals and museums, and at her easel in the studio. Henri preached that \"life and art cannot be disassociated\" and encouraged his students to be \"all wrapped up in life\" because an artist's education took place as much out of the studio as in the studio. He advocated a new freedom of expression and sense of adventure, and even as his students studied the grand masters, Henri reminded them to make their own art. \"[The masters] made their language,\" Henri told his painting students. \"You make yours.\"\n\nAlice took copious and meticulous notes that recorded verbatim Henri's wit, philosophy, and wisdom. She understood the historic importance of her conversations with Henri, and after she returned to the United States, she transcribed his informal talks, lectures, and her personal journal entries about her time with the great teacher into a single notebook labeled \"Robert Henri lectures.\"\n\nAs she had in 1907, Alice remained in Europe after the class disbanded in early fall. During her month-long stay in Paris, Alice saw the Steins many times. If she met with Walter Pach, in public or private, she did not record the meeting in her travel journal. Henri and Margie \"O\" also stayed in Paris and did see Pach that October. Pach was still single but within the year would become engaged to Magdalene (Magda) Frohberg, a young German woman he became acquainted with the same summer he met Alice in Florence.\n\nAlice returned to San Diego proselytizing Henri's contemporary art ideology. She also returned home inspired and eager to begin new work in her La Jolla studio. She continued to be the driving force behind a small gallery downtown that promoted local artists. But after her trip to Spain, Alice was ready to bring contemporary, even modern, art and artists, including Henri, to Southern California. Alice's instincts were aligned with the times, since San Diego and Southern California were precisely poised to begin the transformation from a frontier backwater to a nationally recognized center of contemporary art and culture. And Alice was perfectly positioned to assist in that transformation.\n\nThe construction of the exhibit halls, grounds, and gardens of San Diego's Panama-California Exposition at Balboa Park dominated Southern California's civic agenda for several years. The November 11, 1911, ground breaking on Balboa Park's 1,400 acres of treeless hills and barren canyons set in motion three years of construction and beautification that would profoundly alter the persona of San Diego and environs. The grounds of the park underwent extensive landscaping, with the natural sagebrush and chaparral hillsides transformed into a lush landscape with the planting of several thousand trees, including hundreds of fragrant eucalyptus. San Diego harbor was rebuilt and improved, the trolley system was enlarged, the streets of downtown were paved, and the Santa Fe Railroad built an attractive new Spanish Colonial depot.\n\nThe designing of the Panama-California Exposition at Balboa Park instigated an interest in and rebirth of Southern California's Spanish architectural traditions, and also brought to San Diego Mediterranean and Moorish designs like those first employed by Frank Mead and Irving J. Gill at Bailey's Hilero house in La Jolla. The towers, archways, gardens, domes, reflecting pools, and fountains of the new park brought a romance and refinement to the scruffy frontier of San Diego. The California Building with its Spanish-Moorish tower and Cabrillo Bridge, designed by Gill, which spanned deep Cabrillo Canyon and connected the town to the park, became iconic landmarks on the twentieth-century San Diego skyline.\n\nBy the spring of 1913 and Natalie's return to the West, Alice was involved with two challenging projects. The first was a little book of Henri's teachings based on her notes and journal entries from her summer in Spain. The second was the creation of Balboa Park's fine art exhibition, which was scheduled to open to the public in early 1915.\n\nIn April, Alice sent off to Henri a typed and edited manuscript of the lectures he had given in Spain. Alice asked him for his opinion of the transcriptions, and of her twenty-two-page introduction to the text. She also asked Henri if he would grant her permission to publish the manuscript.\n\nHenri, in Ireland with a class for the summer, was evasive about publication. After months of correspondence, he admitted to Alice that Margery A. Ryerson was just then working on a book that promised to be the complete collection of his lectures and criticism. Henri did not want Alice's manuscript published ahead of Ryerson's, but more importantly, he had instructed Ryerson to withhold publication of her book until after his death. Disappointed, Alice scrapped her book project and tucked her notes and manuscript into a drawer.\n\nAS ONE OF SAN DIEGO'S most accomplished painters with personal connections to important art dealers and prominent artists in America and Europe, Alice was named chairman of the Fine Arts Department of the Panama-California Exposition. This role thrust Alice into the forefront of the most important event in Southern California cultural history to date. Alice edited the official exhibition catalogue and also participated in the selection of art and artists for the exposition. She met and worked with Edgar Hewett and the anthropologists, archaeologists, and artists, many of whom, like Hewett, were from Santa Fe, brought to San Diego to create the buildings and exhibitions in Balboa Park.\n\nCharles Lummis may have provided the first link in the historic chain that connected the art and archaeology community of San Diego with its counterpart in Santa Fe and the soon-to-be-state of New Mexico. In a conversation in 1911 with Colonel D. Charles Collier, the director-general of the San Diego Exposition, Lummis had learned that Collier was looking for a director of the Balboa Park exhibitions. Lummis suggested Collier consider Lummis's friend and camping buddy Edgar Lee Hewett for the job. Hewett was the director of both the Museum of New Mexico and the School of American Archaeology housed in the old Palace of the Governors on the plaza in Santa Fe.\n\nCollier took Lummis's advice and hired Hewett as director of exhibitions. Hewett, widely known as \"Doctor Archaeology\" and called \"El Toro\" (the bull) by friends and foes alike, was given the daunting task of overseeing the vast ethnology and art exhibits that would highlight the culture, art, and lifestyle of the Mayan and Aztec Indians of Mesoamerica and the Native Americans of the southwest United States. The theme of the San Diego exposition was \"the Progress of Man and his achievements in the completion of the Panama Canal.\" With his second wife, Donizetta (his first wife, Cora, died in 1905), Hewett began to commute between Santa Fe and San Diego as he directed the planning and building of the ambitious Science of Man exhibitions. With an initial budget of $100,000 granted by the Smithsonian for the anthropology exhibits, Hewett planned five grand rooms that would tell the story of man's evolution and development.\n\nHewett brought in Al\u011bs Hrdli\u010dka, the director of physical anthropology at the National Museum of Natural History in Washington, to develop the exhibit on human evolution. Hrdli\u010dka was a Bohemian-born physician and a physical anthropologist who had become friends with Natalie in Washington, DC, in 1912. Hrdli\u010dka liked the gutsy ethnomusicologist immediately, and their friendship was anchored in their like-mindedness concerning the reformation of Washington's Indian policies.\n\nHrdli\u010dka was one of the first of dozens of high-profile experts in American archaeology, ethnology, and art Hewett drafted into service to create the exhibitions that became the fair's headliners. Anthropology received top billing at the Panama-California Exposition and deservedly so. More than five thousand specimens of ancient pottery, apparel, and tools were brought from around the world into the Museum of Man. Hewett was also responsible for the Science of Man exhibits, for which Hrdli\u010dka spent three years literally searching the world for artifacts and fossils to display in the Primal Man, Java Man, Neanderthal Man, and Cro-Magnon Man exhibits.\n\nAs director of exhibits, Hewett supervised the Indian Arts Building, the California Building's Mayan exhibits, and the Painted Desert, an open-air, mini\u2013theme park complete with replicas of Taos Pueblo and a cliff dwelling that would be \"staffed\" by natives of Hopi, Zuni, the Rio Grande Pueblos, and from the Navajo and Apache reservations. (The Painted Desert was underwritten by the Santa Fe Railway, which was vigorously promoting the Southwest, especially New Mexico, \"The Land of Enchantment,\" in national publicity campaigns.)\n\nIn twentieth-century New Mexico, art and anthropology coexisted as professional and creative communities, and the activities and goals of the artists and the scientists frequently overlapped in mutually beneficial civic projects and public events. Hewett solicited members from both disciplines, and Balboa Park profited from inspiration from New Mexico's visual artists and expertise from the state's anthropologists and archaeologists. Hewett imported Santa Fe\u2013based painters Carlos Vierra and Gerald Cassidy to create the permanent murals for Balboa Park's Central American and Science of Man exhibits. The ethnologist John Peabody Harrington was hired to create models of Indian villages and homes in the Indian Arts building, and the New Mexico\u2013based artist Kenneth Chapman, considered to be the leading authority on pueblo pottery, partnered with Jesse Nusbaum, Hewett's six-foot two-inch, twenty-six-year-old photographer and all-around assistant at the Mesa Verde excavations, to create the life-sized villages and dwellings of the Painted Desert outdoor exhibit. When the fair opened in 1915, Hewett's, Chapman's, and Nusbaum's New Mexico friends, the husband and wife potters Maria and Julian Martinez of San Ildefonso Pueblo, were in residence and gave daily demonstrations of native ceramic art at the Painted Desert village.\n\nFROM HER RENTED ROOMS at the Palomar Apartments, Natalie could view the progress of Hewett's exhibitions, and walk across the boulevard to Cabrillo Bridge and stroll through the new pedestrian-friendly grounds of Balboa Park. With so many colleagues and friends involved daily with the exposition\u2014Alice, Hewett, Hrdli\u010dka, Lummis, plus Maria and Julian Martinez, and the very elderly Geronimo, who participated in the Painted Desert exhibitions\u2014Natalie frequently visited the park and offered her expert advice to whoever would listen.\n\nIn Natalie's opinion, the \"live\" demonstrations of Indian arts and crafts at the Painted Desert villages did not do native art justice, and she suggested privately to Alice that a showcase be created where the fair-going public could view exemplary examples of American Indian art. With Hrdli\u010dka, Natalie designed an Indian Art Industries exhibit that would explain and display Indian ceramics, particularly the work of her Hopi-Tewa friend Nampeyo, Navajo blankets and rugs on loan from the collection of Canon Winfred Douglas, plus a selection from Frank Mead's collection of rare Navajo silver. Natalie also proposed an exhibit devoted to the handcrafts and artwork of Angel De Cora's students at the Carlisle Indian School.\n\nIt was not a practical proposal and Natalie knew it. She admitted to Alice that even if the exhibit appealed to Hewett, she and Hrdli\u010dka had estimated their Indian Art Industries display would cost a thousand dollars. With the war in Europe draining funds from American philanthropists, Natalie confessed she did \"not feel competent to raise so much money,\" and the idea was scrapped.\n\nIn spite of the national publicity blitz by the Santa Fe Railway and the Fred Harvey Company promoting the beauty and cultural treasures endemic to the American Southwest, in 1913 the capital \"city\" of Santa Fe was considered by most Americans too primitive and foreign to hazard a visit. It was the town's exotic, nonconformist qualities, however, that made it a haven for artists, archaeologists, and ethnologists who used Santa Fe as a base camp for their explorations of the vast country and multicultural people of New Mexico.\n\nIf they had not yet heard talk about the new \"expat\" colony of artists and intellectuals taking root in old Santa Fe, Natalie and Alice certainly learned of it from Hewett and his team of creative and scientific overachievers who came from New Mexico to build Balboa Park. Hewett became friends with George Curtis, likely through their common pal Alice, who was a member of the San Diego chapter of the School of American Archaeology. Hewett is mentioned in George's diary as early as May 1913. Natalie socialized with Hewett when he worked in San Diego, but it is possible she had met El Toro several years earlier, perhaps through Lummis at an AIA meeting, or as early as 1906 when Natalie and Hewett were both temporary residents of Washington, DC.\n\nWith Hewett and a dozen other Santa Fe luminaries actively involved with creation of the Panama-California Exposition, it was fitting that the fair organizers invited the new state of New Mexico to submit a plan for their own building and exhibitions in Balboa Park. The New Mexico state legislature responded enthusiastically to San Diego's call for \"something different\" and enlisted Ralph Emerson Twitchell to oversee the state's displays, and chose the architects and brothers Isaac H. Rapp and William M. Rapp to design the New Mexico Building.\n\nFor inspiration, the Rapps viewed Santa Fe artist Carlos Vierra's paintings of mission churches. In particular, they studied the design of the two-hundred-year-old Franciscan monastery and church of San Esteban del Rey perched on the island mesa of Acoma Pueblo. The two-story adobe chapel flanked by two bell towers, the only Franciscan mission to survive the Pueblo Revolt of 1680, became the Rapps' model for the New Mexico Building in Balboa Park.\n\n**Figure 15.** New Mexico Building, Balboa Park, 1916.\n\nJesse Nusbaum oversaw construction of the New Mexico Building, and Twitchell designed and oversaw the creation of the building's exhibits depicting the long history of New Mexico and its people. The completed structure was a stunning and historically accurate copy of the Franciscan mission church at Acoma. Hewett, the Rapps, Twitchell, Vierra, Nusbaum, and all of the New Mexicans who assisted in the design and creation of the Mission Revival masterpiece in Balboa Park knew they had participated in something special.\n\nIn 1915 Hewett proposed to the New Mexico legislature that a building of nearly identical design be re-created on the Santa Fe plaza. The proposal was approved, and in 1917 a second reconstruction of the Acoma church (incorporating designs from six of New Mexico's Franciscan mission churches) would open its doors as the Museum of Fine Arts.\n\nAlice threw her prodigious energy into organizing the Fine Arts Department and also into assembling her energetic and influential women friends into a committee, the Women's Board, whose purpose was to bring to Southern California the first major show of modern American art. With the financial support of the San Diego community, Hewett wanted the art show to have the status boost of a high-profile general director. Exactly when Hewett learned of Alice's access to American contemporary art star Robert Henri can only be guessed, but by 1914 Alice and Hewett were negotiating with Henri about overseeing the fine art exhibit at Balboa Park.\n\nAlice had been exchanging newsy, chatty, and informal letters with Henri since the end of their European trip in 1912. She repeatedly urged Henri and \"O\" to come for an extended stay in San Diego, and after a particularly severe winter in New York, the Henris finalized their plans to visit Alice's home on the Pacific Ocean in the summer of 1914. \"Looking forward to California as a place where the sun will warm me up to the right heat of production,\" Henri wrote Alice. \"Where I can luxuriate in work, sunshine, fruit, flowers, good food, not have to dress, not entertain or be entertained, nothing but work and sun and the afore said.\"\n\nIn the same March letter, Henri told Alice that in February Walter Pach had married Magdalene \"Magda\" Frohberg in a small, private ceremony in New York City.\n\nThe Henris planned to remain on \"the other edge\"\u2014the West Coast\u2014for seven months. Alice arranged for Henri, Margie \"O,\" and \"O's\" sister, Violet, to lease a house near her own studio in the rural beach village of La Jolla. (The Henri's Richmond Court cottage was not far from Bailey's cliff-hugging Hilero and was designed by Frank Mead's former partner and Hilero's co-builder, Irving J. Gill.) While living in La Jolla, Henri used Alice's studio for public demonstrations and also to work on his own paintings. Henri did ten portraits in Alice's coastal studio, one of which, an oil \"sketch\" of his wife, titled _Marjorie \"O,\"_ Henri gave to Alice. He signed the drawing: \"dedication to Miss Klauber, a souvenir of the summer of 1914.\"\n\nDuring that summer Hewett met Henri in San Diego and asked him to cochair, with Alice, the California Building's fine arts exhibit. Still smarting from his loss of influence and power with the Armory Show the year before, Henri welcomed the personal and professional reception he received in Southern California, and agreed to curate and manage the fine arts exhibition.\n\nFor Balboa Park's fine art exhibit Henri assembled forty-nine works representative of American modern art. Seven of the invited artists were from the original eight \"Ashcan\" painters who had shown together in 1908: George Bellows, Maurice Prendergast, William Glackens, George Luks, Guy Pene DuBois, John Sloan, Carl Sprinchorn, and Henri. (Everett Shinn did not exhibit at Balboa Park.) All of Henri's invitations were to artists whose work embodied what he considered to be _American_ modern art\u2014Joseph Henry Sharp, Childe Hassam, and Ernest Lawson, among others\u2014not the European-style Henri disliked and that was favored by the Stieglitz group, John Marin, Max Weber, Marsden Hartley, and Alfred Maurer.\n\nWith Henri's urging, Hewett sent an invitation to Arthur B. Davies, the director of the Armory Show who had elbowed Henri out of power a few years before. Davies declined to submit a canvas to the Balboa Park art show. Henri was not surprised and told Hewett that Davies \"was not essential to the success of the exhibition.\"\n\nAs chair of the Fine Arts Committee, and a founding member of the San Diego Archaeological Society, which sponsored the fine art show, Alice assumed charge of the gallery and supervised installation of the paintings, all of which followed very precise color and design instructions given to her by Henri. It was a small but noteworthy exhibition, especially because it gave major American painters the opportunity to, in George Bellows's words, \"show on the other edge\" for the first time.\n\nAt midnight on New Year's Eve 1914, President Woodrow Wilson, on the other edge of the continent in Washington, pressed a button that lit the buildings, plazas, avenues, pavilions, fountains, and gardens on the fairgrounds at Balboa Park and officially opened the Panama-California Exposition. Among the dignitaries who witnessed Balboa Park's grand opening was Theodore Roosevelt, who was a passenger in the first automobile to cross Cabrillo Bridge.\n\nThe Balboa Park art exhibition precipitated the birth of Southern California's art colony. Both can be attributed to Henri's visit to San Diego and the efforts of Alice and other women artists who were versed in and moved within the American contemporary art movement. Henri's influence on these women was substantial. Alice's classmate in Spain Meta Gehring married another of Henri's students, Bert Cressey, and moved to Southern California in 1914. The painters Henrietta Shore and Esther Stevens, also Henri's prot\u00e9g\u00e9es and friends of Alice's, settled in Los Angeles and San Diego. \"Their activities encouraged the birth of the region's culture as we know it today.\"\n\n**Figure 16.** California Tower, Balboa Park, 1915. Oil painting by Alice Klauber.\n\nShore would help found the Los Angeles Modern Art Society in 1916, and after the Balboa Park exposition, Alice would be a founding member of the La Jolla Art Association and the San Diego Art Guild and be instrumental in the opening of the Fine Arts Gallery in San Diego. \n\n#  _8_\n\n##  DESERT EUPHORIA\n\nThe desert will take care of you. At first it's all big and beautiful, but you are afraid of it. Then you begin to see its dangers, and you hate it. Then you learn how to overcome its dangers. And then the desert is home.\n\n##  \u2014 _John Wetherill_\n\nIN THE SPRING OF 1913, after several months of easy living in the California sunshine at George's ranch and at her apartment in San Diego, Natalie was ready to don her desert boots, wide-brimmed hat, and khaki riding skirt and return to the desert outback. Discussion began among Natalie's friends on both coasts about itineraries and possible expedition routes in the Arizona canyonlands. Winfred Douglas was part of this discussion from the start. Douglas was living in Peekskill, New York, where he continued to serve as director of music at the Episcopalian boarding school for girls, St. Gabriel, at the Community of St. Mary. Located at the top of a small mountain overlooking the Hudson, the cloistered Community of St. Mary was a haven for Douglas's overworked colleagues, such as Natalie's friend Kurt Schindler, who used a cottage at St. Mary's for weekend retreats away from the bustle of New York City.\n\nDouglas and Schindler shared an uncommon expertise in and enthusiasm for ancient and obscure musical traditions. Douglas, a member of Schindler's Schola Cantorum board of directors, was collaborating with Schindler on the editing of sixteenth-century Spanish motets. Like Schindler, Douglas had spent much of his professional life researching ancient liturgical songs and folk music, and after his studies among the Benedictine monks in Europe and England, he had become an American authority on Gregorian chants and plainsong. Schindler advised Douglas on his ongoing effort to translate and adapt the English texts called \"plain-chant\" and was helping Douglas locate obscure sources for songs of the Crusades (only available in French) and English madrigals.\n\nIn the winter of 1913 Schindler was already working extraordinarily long hours when he accepted the position as choral director for Temple Emmanuel in Manhattan. With the demands on his time and energy that accompanied the phenomenal success of his Cantorum (formerly the MacDowell Chorus), Schindler was courting physical collapse. In the first week of March, just before Easter, Schindler's hectic rehearsal and performance schedule, and the daily management of his chorus of 250 singers, put the thirty-one-year-old in bed with exhaustion. Douglas, famously guilty of wrecking his own health with an unsustainable work ethic, told Schindler he needed to get away from the city, hop the train to Peekskill, and recuperate in the peace and quiet of Douglas's residence at St. Mary's.\n\nDouglas was in poor health himself that spring, overwhelmed by the weekly commute between his home in Peekskill and his apartment at 127 West 87th Street in New York. By March, the canon was looking ahead to the summer break and the Douglas family's annual return to their palatial, two-story, seventeen-room log home at Camp Neosho (named for Dr. Jo's mother, Mary Neosho Williams) near Evergreen, Colorado. But even at Camp Neosho, Douglas had a full work schedule, since he was the founder and the director of the Evergreen summer music conference.\n\nAs he had in the past when his health deteriorated, in the spring of 1913 Douglas returned to Camp Neosho. While his friend Schindler recuperated in New York, Douglas spent a month in the Rocky Mountains. He sent letters to his young friend (whom he addressed as \"My Dear Boy\"), urging Schindler to consider making the trip west for a long stay in the Douglas's home in the Rocky Mountains. In March, Douglas began to plan the Arizona sojourn that would, he hoped, include Natalie, and he invited Schindler to \"join some of us in as much as you will of the Desert Journey.\"\n\nThe \"some of us\" on the roster for the desert journey that would cross Navajoland to Canyon de Chelly and culminate at the Hopi pueblo of Walpi were Douglas's teenage son, Eric, a friend and guide named Ted Whitaker, Natalie, Natalie's San Diego friend Alice Klauber, and Kurt Schindler. By late spring Douglas, an experienced outdoorsman and a natural, eager, if sometimes overbearing camp leader, became the self-appointed expedition point person and general organizer of the itinerary, outfitters, guides, and supplies.\n\nGeorge Curtis did not sign up for this excursion with his sister. He had animals, gardens, and a ranch to tend, and it was hellishly hot in Southern California that year, the temperature most days soaring to over one hundred degrees. George and his neighbors were on a twenty-four-hour fire watch from midsummer until late September. \"The country is like tinder,\" George wrote in his diary, \"a fire coming up the valley or starting at the roadside, would take the house.\" George could not leave his ranch and risk the safety of his home and livestock.\n\nThe departure date was set for late July. Natalie advised Alice on expedition necessities and preparations. Her own camping gear had been packed away for years in one of George's sheds on the ranch, and with his help, Natalie aired out blankets and sleeping bag, ordered supplies, mended garments, and readied it all for train and eventually horse travel.\n\nDouglas wrote an apprehensive Schindler in New York that their expedition would commence the last week of July or in very early August. He assured Schindler, the definitive city slicker tenderfoot around horses and everything to do with the desert and camping, that all the necessary camping gear would be provided by outfitters. All Schindler had to do was climb on the westbound train and make his way to Colorado.\n\nAlthough a native Californian and avid outdoorswoman, Alice, like Schindler, had never ridden horseback across the high desert of Arizona or camped there. Alice trusted Natalie and her knowledge of the outback, but she was ambivalent about following a stranger, even an ordained one, into the wilderness. George may have assured Alice that she would be in capable hands. Although he could not vouch for Douglas's wilderness skills, George could and did vouch for Natalie's.\n\nWith Douglas, Natalie planned their three-week sojourn so as to conclude their expedition at Third Mesa in Hopiland where Natalie planned to rendezvous with Theodore Roosevelt at the Snake Dance on August 21. With Roosevelt's public announcement of his and his sons' plans to attend the Snake Dance, and his promise to write three articles for the _Outlook_ about his summer adventures in Arizona, the remote village of Walpi anticipated \"such crowds of white people as no Hopi mesa had probably ever before beheld.\" A historic number of tourists would likely make the rigorous trek across the nearly roadless, amenity-barren Arizona desert just to watch the ex-president watch the Hopi Snake Dance in late August. Even so, Natalie welcomed the chance to introduce her friend to Hopiland and, with Douglas, coordinated their expedition route so as to be in camp below Walpi during the third week of August.\n\nThe Roosevelt family expedition into the Arizona outback began on July 14, several weeks ahead of Natalie and Douglas's. With his sons, Archie, nineteen, and Quentin, fifteen, and their cousin, Harvard undergrad Nicholas, Roosevelt began the six-week expedition at the South Rim of the Grand Canyon. Their itinerary would take them down into the canyon, across the Colorado River, and north to Kayenta, where, with guide John Wetherill, they planned to cross the dangerous and challenging Rainbow Trail to reach the legendary Rainbow Bridge, one of the most remote landscapes in North America. From here, the Roosevelts would traverse the desert southwest to Walpi on top of First Mesa.\n\nIt was to be an epic expedition, but it was exactly the sort of physically demanding journey Roosevelt yearned for in 1913. Eight months before, he had lost his bid for a third term in the White House in a humiliating defeat to Woodrow Wilson. Although Roosevelt had enjoyed the public's veneration after he survived an assassination attempt, his former party members and legions of political allies were furious with him because he ran as a Progressive and spoiled the chances of the Republican candidate, incumbent William Howard Taft, at beating Wilson.\n\nRoosevelt had spent the winter of 1913 hunkered down at Sagamore Hill, isolated, lonely, and depressed. He kept himself in the public eye, at least on paper, by writing a column for _Outlook_ magazine, for which he was a staff member. When the Armory Show opened in February, Roosevelt decided to go see for himself what all the fuss was about. He chose March 4, the day of Wilson's inauguration, to investigate the \"bedlam of aesthetic debate\" that was the International Exhibition of Modern Art in New York City.\n\nIn the two weeks since it opened, the Armory Show had attracted record-breaking crowds, caused traffic jams throughout midtown Manhattan, and received the kind of newspaper copy that was every publicist's dream. Widely ridiculed and satirized by the press, hailed as a \"Chamber of Horrors,\" the show's headliners, Matisse and the Parisian painters, were dubbed _enfants terribles_. The single most notorious work in the show, Marcel Duchamp's _Nude Descending a Staircase_ , brought to America by Walter Pach, was likened to an \"explosion in a shingle factory.\"\n\nNot one scared off by controversy, Roosevelt entered the vast hallways of the 69th Street Armory determined to come to his own conclusions about modern art, and to write down his thoughts in a review for the _Outlook_. Roosevelt's guides through the cavernous galleries were Henri's rival and nemesis Arthur Davies, the painter Walt Kuhn, and the American muralist Robert W. Chanler. Chanler's brother, William Astor Chanler, had lived in Wyoming (where his friends included Butch Cassidy and the Hole in the Wall Gang) and had soldiered with Roosevelt's Rough Riders. Robert Chanler painted the American West, and during his tour of the Armory Show, Roosevelt stopped and gazed a long while into Chanler's mural of a ceremony called the Snake Dance in a faraway Indian village called Walpi.\n\nRoosevelt must have felt a familiarity with the exotic, primitive scene and people depicted in Chanler's painting. Natalie had told him stories about the Hopis and their cliff-hugging villages, and Roosevelt had read her accounts of sunset meetings with old chief Lololomai on a flat roof under the open sky of Oraibi. Perhaps all that modern and even unsettling futurist art surrounding Roosevelt in the Armory made it apparent that what he needed to reconnect with his sense of purpose was time alone under the wide sky found in the primitive outback of the American Southwest.\n\nAt fifty-four, Roosevelt was looking for redemption and distraction. Throughout his life he had turned to physical tests of endurance in the great outdoors to overcome disappointments and setbacks. In the spring of 1913, Roosevelt accepted an invitation to visit Brazil and explore the Amazon River, a larger-than-life trip that would commence in the fall. He also announced to his sons Archie and Quentin that in July they were going with him on an expedition deep into the Arizona desert.\n\nThe Roosevelt family pack trip began with a predawn descent on horseback into the Grand Canyon in the light of the nearly full moon on July 15. Their route through Navajoland traversed the canyon and ascended the North Rim into the Kaibab National Forest, crossed the Colorado at Lee's Ferry, and finally headed northeast across the harsh, bald, and breathlessly hot sandland plateau toward Kayenta. By Roosevelt's account, they passed herds of buffalo and Navajo sheep, confronted venomous rattlesnakes, and cooked supper over campfires beneath sunsets that turned the heavens ruby red before fading to a pale turquoise.\n\nThis was what a world-weary, politically wounded, middle-aged Roosevelt had come to the Southwest in search of: an encounter with the primordial universe.\n\nIn late July, Kurt Schindler shed his starched white dress shirt and elegant black tails, put away his sheet music and conductor's baton, and with a duffel bag carrying new khakis and oiled-leather lace-up boots, climbed on the train in New York and commenced the journey to his own encounter with the Southwest. Schindler arrived in Morrison, Colorado, on July 26 and, following Douglas's instructions, took Skerrett's automobile service from the depot up into the Rocky Mountains to Camp Neosho in Evergreen. Douglas was away on business, so Schindler spent a week walking the mountain trails and enjoying the fragrant pine forest sanctuary that was Camp Neosho before catching the train to Colorado Springs, where he connected with Douglas and his fifteen-year-old son, Eric.\n\nThe three took the Santa Fe train south to Albuquerque and west to Williams, Arizona, where they met up with Douglas's friend Ted Whitaker and at least one guide who went with the men on the stage up to the Grand Canyon. After several days on the South Rim, Douglas and his son left Schindler, Whitaker, and their Navajo guide at the canyon and hopped the stage to Holbrook, where Natalie and Alice would disembark their train at the Santa Fe depot on Friday, August 8.\n\nClimbing on the train bound for Arizona with her camping gear was, for Natalie, the final leg of a long-anticipated homecoming to Indian Country. For Alice, however, departing the genteel and civilized city of Los Angeles with bags that carried only those clothes and personal items deemed necessary for survival in the desert was unsettling.\n\nAlice may have kept her ambivalence about the expedition from Natalie, but she admitted in her diary the imbecility of undertaking such a rigorous trip with strangers. She had never met the expedition's leader, Canon Douglas, although she had heard all about the good and able musician-priest from Natalie and George. Alice had also heard about New York conductor sensation Kurt Schindler from Natalie and read about him, too, as Schindler's chorus was receiving rave reviews and national attention. Even so, except for Natalie, Alice was about to depart civilization with complete strangers.\n\nAlice loved to be alone in nature and sought solitude on La Jolla's rocky shoreline and in the hills and scrub desert outside of San Diego. She had thrice sailed across the Atlantic for extended, even rigorous travels through Europe and the Mediterranean. But the physically demanding journey by horseback she was about to undertake with Natalie into the remote, uninhabited, dangerous, and still very much uncivilized southwest wilderness was an extreme adventure the likes of which the forty-two-year-old Alice had never attempted.\n\nThat first night, Natalie and Alice left California and rode the Santa Fe train across a bright, moon-infused desert to Arizona. Alice was unable to sleep in their shared Pullman car, her discomfort as much psychological as material. \"I felt sick enough to ponder the imbecility of undertaking the trip with strangers,\" she scribbled in her small journal. The next afternoon Alice's spirits were lifted by thundershowers that cooled the smoldering desert, and her well-being and optimism returned when the train stopped for supper at the comfortable, immaculate, and professionally staffed Harvey House in the tiny outpost of Ashfork, Arizona. After dinner and a tour of the Harvey House Indian Curios room, Natalie and Alice reboarded the train at dusk and continued east to their rendezvous in Holbrook with Douglas.\n\nAs their train came into Holbrook, the moonlight on the desert, and the absolute stillness of the night, cast a surreal quality onto their arrival. Alice and Natalie's midnight train was met by the good Canon Douglas, who alone (Eric was asleep) carried their bags to their room in a hotel. The hotel was not on a par with a Harvey House, and Holbrook was nothing more than the Santa Fe depot by the single track and a dirt street with a few stores, a bank, and a post office. If Alice had concerns about the quality of their lodgings, and was still harboring doubts about the wisdom of following the priest into the outback, Douglas assuaged her unease and discomfort when he brought the women a late supper in their room.\n\nA night of sleep in a lumpy bed in a third-rate hotel completely refreshed Alice. With Natalie, she breakfasted with Douglas and Eric. After a day of errands and preparations that included a visit to the bank, and a last laundry and bath, Natalie and Alice went with the Douglas men to meet Schindler's train.\n\nIt was a happy and much-anticipated reunion for Natalie and Schindler, who had not seen one another since Natalie left New York with Mimsey last winter. Alice had heard all about the brilliant and eccentric musician Kurt Schindler. Even so, his appearance and demeanor astonished her. After meeting him at the Holbrook depot that evening, Alice wrote: \"With all N.C.'s hints and suggestions [I] was not prepared in fullness\u2014curious, sensitive, attenuated being\u2014looks like a Navajo (Zuni) Indian who had just crawled out of a sunless cavern completely bleached out + with a much quickened interest in his fellow man which a Navajo would succeed much better in hiding.\"\n\nThe Curtis-Douglas entourage enjoyed one last supper in civilization at a small Holbrook diner called The Cottage. And then, with two teams and a wagon carrying their camping gear, a wrangler friend of Douglas's named Ted Whitaker, and several more saddle horses and pack animals, the two women and four men left Holbrook for the open desert. At eight in the evening, Saturday, August 9, their journey began much like Roosevelt's three weeks before under the surreal light of the quarter moon, with Mercury, the evening star, sparking in the west for an hour after sundown.\n\nThey struck north for Ganado and the settlement at Hubbell's trading post, where they would camp a few nights among the folks who attended the annual late-summer Navajo fair. It was a trek that would take the horses and wagons four days following the tracks of previous teams that had made this journey before them. Even with the fine quarter moon lighting their way, Douglas and Whitaker lost the trail several hours out, and they had to retrace their tracks for several miles. They made camp where they found the trail again, and even though it was impromptu, and they'd lost several hours looking for the route north, everyone slept well. The dawn was a fine one, and before breakfast Alice slipped away with her sketchbook to make a drawing of the sun lighting the sky over the empty, silent Arizona sandland.\n\n**Figure 17.** Desert Expedition, July\u2013August 1913. _Left to right:_ Eric Douglas, Kurt Schindler, Alice Klauber, Natalie Curtis, Winfred Douglas.\n\nDouglas, Natalie, and Whitaker had traveled between Holbrook and Ganado and were familiar with the route to Cottonwood Wash and a campsite Douglas had used two years before. To Alice's eyes, though, the landmarks that marked the trail across the desert were often imperceptible, and she, like Schindler, had to abandon all notions of self-reliance and place her fate in the hands of the expedition leader.\n\nEarly in the afternoon on their first day out, Alice and Natalie shed their travel skirts for more appropriate desert riding apparel\u2014split skirts of heavy cotton\u2014and left the wagon and rode horseback alongside Douglas, Eric, and Schindler. Schindler was stoically managing blisters, sore muscles, sunburn, and chafed surfaces after a long, hot day in the unfamiliar embrace of a western saddle. On previous trail rides, Natalie had come to terms with the nongenteel aspects of riding astride a horse. But Alice had never ridden a western saddle and had to quickly adjust her sense of propriety. \"Don't like it astride a bit,\" she wrote in her journal, \"and couldn't ride side-saddle in my skirt if I could get a side saddle.\"\n\nThe horses and wagon followed a route that was never more than tracks in sand or ruts through arroyos, and that took the riders up steep mesas, across slick rock, and around jagged stone outcroppings to Twin Buttes. They were in the fantastic country of the Desierto Pintado, the Painted Desert, the region the Navajo called Halch\u00ed\u00edtah (Among the Red Areas). At six thousand feet above sea level, the huge swath of eccentrically eroded and multicolored sandstone, clay, and volcanic rock was brilliantly striated. The hues and forms of the desert changed hourly with the movement of the sun and the shadows of the clouds. It was fine country to be swallowed up in, and a visual feast for a painter. Alice likened the landscape to the Holy Land and remarked how the colors of the desert belonged in a Jules Gu\u00e9rin painting.\n\nEach day they moved farther into the wild. Natalie welcomed the transformation wrought by desert life. She, like Douglas, knew the Colorado Plateau outback would work its magic on Alice, Schindler, and young Eric, even as it had on her and George a decade ago. The physical and emotional challenges endemic to wilderness travel on horseback, and the confidence and resiliency earned as those challenges were successfully met and overcome, were life altering. Natalie wrote, \"For us the sordid bargaining of our narrowed city lives, the fret and whirl of petty currents that bear us so far from our inner goal\u2014all were forgotten in that enchanted land.\"\n\nDawn on August 12 came with a soft, pastel-colored sky and a herd of Navajo goats wandering into their camp near Indian Wells. Alice was the first to dress and leave the tent she shared with Natalie, and she hiked a sand hill and completed a sketch before Douglas even made the breakfast fire.\n\nDesert life settled into a comfortable routine. Expedition members became accustomed to their assigned tasks and chores, and confident in their riding and camping abilities. Douglas and Whitaker pushed the horses and riders north, their destination Hubbell's trading post, a two days' ride in the wide riverbed of the Pueblo Colorado Wash that cut through the hard desert of the Din\u00e9 to Ganado.\n\nSchindler took to wearing a bandana tied about his head Navajo-style. Douglas rode with his shirt sleeves rolled to his elbows, but Natalie and Alice kept their sleeves buttoned down around their wrists, their shirt collars up, and their necks wrapped with bandanas. Sunup till sundown, everyone lived beneath a broad-brimmed hat. Even with so much precaution, the sun exposure at this altitude was extreme and even dangerous, and three days into the desert, everyone was sunburned somewhere on their body.\n\nA drenching thunderstorm late on the afternoon of August 12 sent the expedition up and away from the Pueblo Colorado Wash, which became a raging river with flash flood waters. They found shelter in the cedar trees on the side of a mesa, and after several hours the storm cleared and left the red desert fragrant beneath a perfectly formed double rainbow. Douglas and Whitaker led them in the approaching dusk to a camp on a hill near a dramatic volcanic mountain that the men climbed at sundown. Alice and Natalie remained by the campfire, listening to the haunting songs of Navajo shepherds camped nearby. After dark, when Douglas and the men had returned from their hike up the volcanic cone, one of the Navajo singers came to their camp and exchanged songs with Natalie.\n\nTraders and ranchers, anthropologists, adventurers, and adventurous tourists frequented the route Douglas and company were following between Holbrook and Ganado. The Din\u00e9, the Navajos, who lived in family groups scattered across their 25,000-square-mile reservation, were becoming accustomed to _Bilagaana_ (non-Navajo) horse and wagon expeditions crossing Navajoland. Even so, encounters on such an enormous frontier were infrequent and remarkable. The usually reserved Din\u00e9 often approached Natalie and Douglas's entourage. White women were rare in this remote country, and Natalie and Alice were a curiosity to the natives they encountered on the trail.\n\nOn August 13 they rode into L\u00f3k'aah niteel (Wide Reeds) and Hubbell's trading post at Ganado. After four days on horseback in the glittering monotony of sun, rock, sky, and horizon, in a land where mesas floated like islands in a sand sea, coming into the Ganado valley was like riding into modern civilization. Of course, Ganado was not _modern_ : the homestead of John Lorenzo Hubbell, known by everyone as Don Lorenzo, was one of very few parcels privately owned within the boundaries of the gigantic Navajo Reservation. It had amenities like water, fresh food, and buildings that boasted wood floors and glass windows, but the post was extremely isolated, fifty miles west of Gallup and sixty-five miles northeast of Holbrook.\n\n**Figure 18.** Hubbell's trading post, 1890s. John Hubbell is holding the rug.\n\nThe red stone and log trading post served as the stage stop, post office, supply store, farm and ranch outfitter, wool warehouse, stable and blacksmith, and trader of wool, goat, and sheep skins and the finest Navajo blankets, rugs, and silver. Ganado also provided modern medical services at the first and soon largest Indian mission and hospital, which had been built by the Presbyterians in 1901.\n\nThe waters of the Pueblo Colorado River filled a reservoir that irrigated fields of alfalfa, wheat, oats, and barley, a vineyard and an orchard of fruit trees, and the Hubbell family's kitchen garden. The trading post also kept chickens and hogs, sheep, goats, cattle, saddle horses, and a small flock of peafowl to keep the rattlesnakes away.\n\nFor horse and wagon travelers in 1913, riding in off the desert to Hubbell's in mid-August was akin to finding an oasis in the Sahara. Don Lorenzo was a friend of both Douglas's and Natalie's (Natalie and George had stayed with the Hubbells on previous trips), and he arranged for their group to camp in a prime location in the trees near the trading post, and to take several meals a day in his family's house. This gave Natalie and company the best of both worlds\u2014the privacy and space of their own tents at a distance from the busy trading post, and the cool and comfortable porches, dining and sitting rooms of the Hubbell family home.\n\nThe thick-walled stone and mud dwelling housed a private museum of native crafts and was decorated floor to ceiling, and across the vigas of the ceiling itself, with Indian baskets, carvings, ceramics, and textiles. The paintings of contemporary artists who had stayed with the Hubbells were displayed throughout the home, and the shelves of the library in the grand salon held signed editions of America's best authors, including Mark Twain and Zane Grey. And every floor in the house boasted the finest examples of Navajo rugs found anywhere in the world.\n\nLike Louisa and John Wetherill's trading post and home more than a hundred miles to the northwest in Kayenta, Hubbell's was a destination that outback travelers planned their itineraries around. Don Lorenzo had lived nearly all his life among the Navajos and spoke their language and understood the subtleties and complexities of their culture. Like the Wetherills' near Monument Valley, the Hubbell home was a gathering place for archaeologists and anthropologists, writers and artists exploring the country of the mysterious Din\u00e9.\n\nThe Hubbell dinner table was usually a hub of energetic conversation, and Natalie and Douglas looked forward to evenings with Don Lorenzo. However, in 1913, following the death of Don Lorenzo's wife, Lena Rubi, the atmosphere around the trading post was somber, and Natalie's group hardly socialized with Hubbell.\n\nThey camped in the cedar trees of Ganado for three nights. The ground was flat and hard, and pockmarked by petrified wood and cactus, but there were the luxuries of ample shade and plenty of water for washing and bathing. Although Natalie, Alice, Douglas, and Eric enjoyed a daily dip in the Pueblo Colorado River, Schindler had become a devotee of the cowboy way and continued to take his daily bath in the camp's Dutch oven.\n\nSix hundred Navajos and Hopis came to Ganado on August 13 and 14 for the annual Indian fair. There were two days of races and contests on horseback, and in the evenings there was storytelling around campfires. When word got out around the trading post and camp that Natalie was the woman who collected Indian songs, more than twenty Din\u00e9 came to her and Alice's tent wanting to share chants and stories.\n\nOn Monday, August 15, Natalie, Douglas, and friends struck their comfortable camp in the cedar trees and said good-bye to their horses, which were left in Don Lorenzo's barn. They were to continue north to Chin Lee (Chinle) in Don Lorenzo's \"Ford machines.\" It was thirty-five miles of very slow going on a primitive road north through the wild and lovely Nazlini Canyon, home to a community of small black bears sacred to the Navajos. To their west, across a wide ocean of buff-colored sand and low featureless plateaus, was Hopiland. Below the horizon to their east were the cool pine-covered slopes of the Chuska Mountains that marked the Arizona\u2013New Mexico border.\n\nRecent rains had packed down the sand, but the wheels of the two Model Ts, considered good desert vehicles because of their four-cylindered, twenty-horse-powered engines, were no match for the thick clay and mud. The drivers had to navigate the road's high centers, deep arroyos, and scrabbled inclines, and Alice and Natalie commented that horses and wagons were a better choice than the lurching, swerving Ford machines. Navajos came and went on horseback, and travelers in wagons passed by the noisy automobiles, both groups destined to arrive hours ahead of Natalie and Alice's company.\n\nThey were driving due north to the great Canyon de Chelly, the labyrinthine canyon where sixty years before the Navajos hid from Kit Carson and the U.S. military in the ancient, inaccessible cliff villages abandoned long ago by the Anasazi, the ancestors of the Hopis. Douglas had explored Canyon de Chelly before, but it would be an inaugural visit for Natalie.\n\nThe two Fords and their eight passengers arrived at John Kirk's trading post near Canyon de Chelly in the late afternoon. Kirk's log and stone store was tucked into cottonwood trees against a small bluff a half mile from the entrance to the great canyon. Even so close to the massive stone labyrinth, the pastoral landscape surrounding the trading post gave travelers no clue about their proximity to the legendary canyon.\n\nDouglas had arranged for new drivers and guides\u2014Ken Shelly, a Navajo from Fort Defiance, and a second guide named Brady\u2014to accompany their group into the canyon. After an hour in the trading post, where Natalie, Schindler, and Alice talked with the locals who came in to trade and to have a cup of Arbuckle's coffee, Douglas shepherded everyone into the automobiles, and with new supplies of food and a load of dry firewood, they drove into the canyon.\n\n**Figure 19.** Spider Rock, Canyon de Chelly, 1873.\n\nThere would have been nothing that Douglas or Kirk or their guides could have told Alice, Natalie, and Schindler that would have prepared them for the hauntingly exquisite beauty that was Canyon de Chelly. It was dusk when they passed through the stone-walled gateway into the heart of the canyon. Sheer, smooth, red and gold stone rose hundreds of feet into the air on either side of the cottonwood-lined river. Until they departed the canyon two days later, all Natalie and Alice would see of the Arizona sky was the slice of blue held between the stone rims of Canyon de Chelly's vertical walls.\n\nDarkness fell and a million stars crowded the sky above the canyon. The automobiles drove through a tunnel of darkness that followed the river, the skilled Navajo drivers careful to keep the automobiles' tires from the canyon's famous yellow quicksand that was known to suck vehicles into muck so deep they vanished completely.\n\nThe full moon finally reached the sky beyond the stone rim and flooded the canyon with a luminous, otherworldly light. They set up camp on a high ledge of fine dry sand alongside a great bluff that gleamed in the moonlight. On the flat surface of smooth stone, Natalie, Alice, Douglas, and the men ate a late supper of bread and mackerel, olives and grape juice, with a nightcap of hot chocolate. The night was too beautiful and clear for tents, and no one bothered to even unpack the tarps, but slept in bedrolls on the open ground in the shade made by an enormous stone monolith. Alice lay awake and stared into the radiant nightscape that surrounded her, listening to the sound of horses hobbled somewhere in the canyon.\n\nDawn came in a soft blush of color to the canyon rims. An hour later, the hot rays of the sun reached down the southwestern walls. The floor of the canyon was smooth white sand, and the river, lined by lush groves of cottonwood and fruit trees, wound through a chasm edged by sheer stone walls, rock towers, and polished buttes that stood more than eight hundred feet above the canyon floor.\n\nThey saddled up the horses brought in by the guides and rode deeper into the canyon past summer hogans, cornfields, orchards of peach trees, and small corrals of horses and goats that belonged to the Navajos who lived in the canyon. Their destination was the White House ruin tucked on a ledge under an immense stone overhang. After scrambling about the old pueblo, they rode up the tributary of Ca\u00f1on del los Muertos and from a stone ledge in the shade pondered the gruesome fate of Navajos who were slaughtered by the Spanish in Mummy Cave several hundred feet above them.\n\nTwo days later, Natalie, Alice, Schindler, Douglas, Eric, and Whitaker reemerged from Canyon de Chelly. At Kirk's trading post they purchased supplies, repacked and organized their gear into the Model Ts, and headed west for Keam's Canyon, Hopiland. It was Sunday, August 18, but after ten days in the outback, the calendar of the \"civilized\" world ceased to have meaning. Alice's journal entries, like her sense of time, had become untethered from dates and days of the week. Alice and Schindler now understood what Natalie and Douglas knew: when you traveled into the deep outback of Indian Country, conventional time became irrelevant. Hours and days were measured by the movements of the sun and the moon, and time was of significance only in relation to the distance one needed to cross between sunup and sundown.\n\nNatalie flourished in the primordial landscape, and delighted in the days and weeks spent \"wandering, roofless and free, over the sun-scorched deserts . . . for in them we quite lost the trail of the twentieth century and of our materialistic and commercial civilization. . . . Even the wrinkles and crow's feet that lined our white faces . . . were burned away. In the silence of the desert and the sweep of pure winds our souls were washed clean, till we too seemed true children of the Earth-Mother and the Sky-Father.\"\n\n#  _9_\n\n##  WALPI RENDEZVOUS\n\nThe spell of the desert comes back to me, as it always will come. I see the veils, like purple smoke, in the ca\u00f1ons, and I feel the silence. And it seems that again I must try to pierce both and to get at the strange wild life of the last American wilderness\u2014wild still, almost, as it ever was.\n\n##  \u2014 _Zane Grey_\n\nTHE ROAD TO KEAM'S CANYON angled across the high desert to the south and west from Chinle, directly into the ocean of buff-colored sand plateaus. The same day they left Kirk's at Canyon de Chelly, Natalie and Douglas's expedition left the mixed blessings of the Ford machines and transferred onto horses, with a team and wagon once again carrying their camping gear. The weather was cooler and they were able to ride fifteen miles to Salina Springs, where they set up camp in a thicket of cedar trees above the spring. The next morning, August 18, they rode off the high plateau east of Low Mountain and followed Jadito Wash toward Peach Orchard Springs, Keam's Canyon. By noon, they rode down rock and sand bluffs into a herd of goats grazing in the peach trees, and stopped before the trading post owned by Don Lorenzo and run by his son Lorenzo Hubbell Jr.\n\nAn ancient village site of the Hopis, by 1913 Keam's Canyon was as anglicized as any community found in Indian Country. The BIA offices were within sight of the notorious Keam's Canyon boarding school that Natalie, Lummis, and scores of other Indian rights activists had condemned for years.\n\nSuperintendent Leo Crane was not a fan of Natalie Curtis's work among the Indians and would not have welcomed her arrival to Keam's Canyon. However, in August 1913 Crane had been briefed by Washington officials that Natalie and her friends were slated to rendezvous with Theodore Roosevelt at the Snake Dance at Walpi. Crane and his agents were instructed to accommodate Roosevelt and Curtis and to provide an office at Third Mesa where the ex-president and the ethnomusicologist could meet.\n\nAgent Crane had served as superintendent of the Hopis for only two years, yet his tyrannical style of management (he continued to send Navajo policemen into the Hopi villages to round up schoolchildren) and his palpable animosity toward all Indians, who, in his estimation, lived \"on a moral plane little above their livestock,\" made him deeply disliked in Tusayan. Crane was suspicious of traders like Hubbell and the \"open-mouthed\" tourists, curio hunters, actresses, authors, and moving-picture fans who revered all things Indian. And Crane absolutely _detested_ the annual heathen affair called the Snake Dance that brought hordes of Indian-loving outsiders into whatever Hopi village was sponsoring the ceremony. Crane wanted outsiders kept outside, and the \"immoral\" Hopis kept under his firm, civilizing hand.\n\nA persistent and bothersome thorn in Agent Crane's side was Natalie's collaborator and friend Tawakwaptiwa, now the _kikmongwi_ of Old Oraibi. The Oraibi Split of 1906 had precipitated social and political realignments among the Hopis, and Chief Tawakwaptiwa was seen as both a hero and a scoundrel among his own people. Crane detested Tawakwaptiwa and was particularly perturbed by the adulation and recognition Miss Natalie Curtis had given Lololomai's nephew in _The Indians' Book_.\n\nEven if Crane had no patience for her work in preserving Indian music and culture, he understood the place of privilege Tawimana (Song Woman) Natalie Curtis held in Hopiland. He would be outwardly courteous and helpful to Natalie and her traveling companions, but he would be relieved when the spectacle of the Snake Dance was over and all meddlesome outsiders, famous and not, departed Indian Country.\n\nAfter Natalie's friends enjoyed a leisurely stopover at the Hubbell Keam's Canyon store, they remounted and rode east eleven miles across a well-used but very rocky road to Polacca, the outpost below the imposing rock promontory of First Mesa. The sand drifted in deep dunes across the road that wound up and down treacherous ledges and bald rock outcroppings. It was obvious to Alice why the Ford machines were left at Canyon de Chelly: a dozen autos a week were stranded on this road, defeated by the steep ascents or swallowed up by sand that reached to their hubcaps.\n\nNatalie, Alice, Douglas, and the others reached First Mesa in the afternoon. In the shade of a large peach orchard they set up tents, tarps, and their outdoor kitchen within walking distance of the stone trading post, yet another establishment owned by Don Lorenzo Hubbell. They would remain at this camp beneath First Mesa for at least the next week.\n\nIt had been eight years since Natalie had trod the trail up to the Hopi villages of Walpi and Sichomovi and the Tewa-Hopi village of Hano on the slender island of rock that was First Mesa. She was eager to see Nampayo and before dusk led everyone but Alice (who confessed to her diary that she was cross, tired, and of no help anyway) up the narrow path to Hano, perched on the mesa six hundred feet above the desert floor. The sinking sun cast purple and rose hues onto the desert below the trail, and the air was fragrant with the scent of cedar fires and burnt sage. As they ascended the mesa, the dusk silence was laced with the calls of ravens, the bells of goats, and the shouts of the village children.\n\nNatalie took Douglas, Schindler, Eric, and Whitaker directly to Nampayo's home, where they were warmly welcomed by Nampayo and her family. Nampayo was nearly blind now, but continued to make pottery with her daughter Nellie. Nampayo's pottery, like that of the Tewa artist Maria Martinez at San Ildefonso Pueblo in New Mexico, had become the gold standard by which Native American ceramics were evaluated. With the publicity boost brought to her pottery by the Indian Department of the Fred Harvey Company and the Santa Fe Railway, Nampayo's ceramics were collected by museums and by connoisseurs of fine Indian arts and crafts.\n\nAfter a quick tour of Hano in the soft light of the August dusk, Natalie and Douglas led Eric, Schindler, and Whitaker back down the mesa. Later that night, from the warmth of her bedroll on the open ground below First Mesa, Natalie pondered the fragility of the traditional life of her Hopi friends. Nampayo's pottery and the craftwork of other emerging native artists were bringing collectors directly into the villages and communities of Native America. Staring up at the stars twinkling through the branches of the peach trees, Natalie wondered if it was possible to stop or even slow \"the tide of Anglo-Saxon iconoclasm that would sweep the Hopi Indian villages from their ancient sites \"\n\n**Figure 20.** Walpi, First Mesa, Arizona, ca. 1907.\n\nNatalie's desert idyll ended the next morning. Roosevelt and his expedition were coming in on horseback from the north, and the news of the former president's arrival moved through the native and nonnative communities on and below First Mesa like an electric current. After weeks without even a passing thought about the condition of her appearance, Natalie was suddenly conscious of her disheveled clothes and tangled hair.\n\n\"Weeks in the Arizona desert do indeed restore one to an atavistic, primitive state in outward appearance as in inner peace of mind,\" Natalie reflected. \"I was travel-stained from head to foot, and on the front of my khaki riding-skirt flared with placid disregard of all conventions a large round circle of black axel-grease, where the camp wagon had branded me for my affront of trying to climb in over the front wheel.\"\n\nAlice, Douglas, and Schindler understood how important Natalie's appointment with Roosevelt was to her. But with each of them looking equally rumpled and unkempt\u2014skin sun dried and burned, hair matted under hats, lips chapped, boots scuffed, shirts discolored by sweat and horsehair\u2014Natalie's sudden need to be presentable, indeed _professional_ , became the subject of much camp humor. Douglas and Schindler thought the grease stain decorating the front and center of Natalie's riding skirt made a fine Indian sun symbol, and ought to be left alone. But if she insisted on removing the grease stain, the men told Natalie nothing short of a bath in gasoline would do the job. Burning the skirt seemed a more prudent solution.\n\nNatalie left her unsympathetic compadres beside the breakfast fire and marched off alone in search of gasoline. In the BIA buildings at Polacca, agency workers were mopping floors and scrubbing surfaces \"for Roosevelt.\" Next door at the schoolhouse, the teacher was cleaning up the students and the classroom for the president, and had no time to help Natalie with her stain. She did, however, offer Natalie the use of any cup she could find. After locating a handle-less vessel in the school's kitchen, Natalie went outside and found a man standing beside his parked automobile.\n\n\"I need gasoline,\" Natalie told him, gesturing at her skirt. After telling Natalie she was welcome to as much gasoline as she could extract from his auto's tank, the man quickly left to join the crowd gathering on the far side of the mesa to greet Roosevelt's expedition as they rode in from the desert.\n\nNatalie had no idea where the vehicle's gas tank was located and was accepting defeat when a horse and rider appeared.\n\n\"Presently a handsome and deep-tanned young cowboy in blue overalls passed me, leading some horses,\" Natalie later wrote of that August morning. \"I wonder if you could get me some gasoline from this car? . . . I am allowed to have it. You see, we are all trying to clean up in honor of Colonel Roosevelt\u2014he is expected today some time.\"\n\nThe young man smiled, dismounted, and within moments \"lay in utter chivalry under the car, milking the gasoline into my cup.\" He soon slid out from under the auto, handed Natalie the handle-less cup filled with gasoline, gathered up the horses he was leading, remounted his horse, and with a tip of his hat trotted away.\n\nCarefully balancing the cup with the precious cleaning agent, Natalie walked back toward the camp in the orchard. As she crested a low rise, she stopped: \"A stalwart figure on a cow pony was riding up the hill, alone and unattended; without any flourish of trumpets, Colonel Theodore Roosevelt had arrived! His sunburned face was partly shaded by a big Stetson hat, a red bandanna fluttered at his throat, and he too looked as though gasoline might improve his khakis. I waved my sombrero and cried, 'Hail to the chief!'\"\n\nRoosevelt had arrived at First Mesa. Undetected by his great herd of fans, only the tiny, tanned, smiling Natalie Curtis welcomed the president to Hopiland. Theodore Roosevelt enjoyed a Big Entrance before a large, boisterous, adoring crowd\u2014Henry James described him as \"the mere monstrous embodiment of unprecedented and resounding Noise\"\u2014but on this particular August morning under the hot sun and brilliant cerulean sky of Indian Country, Theodore Roosevelt rode into the outpost at Polacca the iconic western loner. Even his doting and protective sons were not with him.\n\nRoosevelt recognized his friend immediately and leaned down from his saddle to grasp Natalie's outstretched hand. It was so like Roosevelt, Natalie would later say, to appear where no one expected him \"ahead of the rest of the world.\"\n\nThe night before, while Natalie had contemplated the universe from her bedroll under the peach trees, Roosevelt had done much the same on the hard ground of a mud flat in the desert north of First Mesa. With his sons Archie and Quentin and his nineteen-year-old cousin Nicholas, and with John Wetherill and two Navajo guides, Roosevelt had ridden three days through the mountains of Black Mesa and down across the roadless country of low cliffs and blood-red buttes between Kayenta and Walpi. After six weeks on horseback, camping on rims, in pine forests, and under the great Rainbow Bridge in canyons so untouched by humans as to be unnamed and unmapped, Roosevelt had become immersed in what Natalie fondly called the desert dream. \"Only those who live and sleep in the open,\" Roosevelt wrote in his notebook, \"fully realize the beauty of dawn and moonlight and starlight.\"\n\nAfter several minutes of casual chat, Roosevelt told Natalie she needed to meet his son and turned in his saddle and called out for Archie. A few moments later, the handsome cowboy-knight in overalls who had filled Natalie's cup with siphoned petrol rode up alongside his father. \"We have met to-day before,\" a smiling Archie said to a speechless Natalie.\n\n**Figure 21.** Theodore Roosevelt, Kayenta, August 1913.\n\nThere was almost no gasoline in Natalie's cup, as it had quickly evaporated into the dry air. But Natalie didn't care anymore about cleaning up her skirt. No explanations were necessary. She was among kindred spirits.\n\nWinfred Douglas, wondering what had become of Natalie and her quest for petrol, wandered over to the schoolhouse and found her with the Roosevelts. After introductions, it was agreed that Natalie and Douglas would meet with Roosevelt the next day in the Indian agency near the school where Agent Crane had set aside spacious quarters for Roosevelt. Douglas had seen the Snake Dance in 1911, and Roosevelt wanted context before witnessing the ceremony. He also asked Natalie if she would accompany him on his tour of Walpi. Roosevelt had only a few days to absorb Hopi culture and draft his _Outlook_ article about the people of Hopiland. Natalie agreed to help.\n\n**Figure 22.** Theodore Roosevelt at Walpi, August 20, 1913.\n\nRoosevelt actually had plenty of expert consultants with him at First Mesa. He was traveling with John Wetherill, surely the most knowledgeable and revered guide in Navajoland. Don Lorenzo Hubbell had come to Polacca from Ganado to assist Roosevelt during the Snake Dance. Hubbell was a personal friend of the Roosevelts, and had visited the family in February during a trip to Washington for a meeting at the BIA. John, as Roosevelt called Hubbell, had also served as a driver and guide for various legs of the Arizona expedition.\n\nLater that day, Natalie took Alice and Schindler to see Second and Third Mesas. They rode twelve miles up and down a steep and difficult road to Second Mesa, where they visited friends of Natalie's at the village of Shongopovi. As the afternoon passed into the evening, Natalie, Alice, and Schindler pressed on to Third Mesa, where Natalie hoped to reconnect with Chief Tawakwaptiwa at Old Oraibi.\n\nThey spent the night at New Oraibi, a community of federal-issue dwellings and offices at the base of Third Mesa. Natalie was distressed by New Oraibi's lack of indigenous architectural characteristics, and by the U.S. government's continued insistence on funding and constructing buildings artistically and practically out of synch with Hopiland. The traditional thick-walled, flat-roofed dwellings stacked several stories high, with shaded overhangs and terraces, had for centuries successfully served the Hopis as they worked, played, socialized, and celebrated in every season of the year. At New Oraibi, the traditional Hopi village was replaced by rows of impractical pitched tin-roofed dwellings with too many windows that were frying-pan hot in the summer and ice cold in the winter. Natalie had hoped to find improvements in the federal government's management of Indian Country. Instead, she found the feds had continued their campaign to weaken the traditional cultures and lifeways of Native America.\n\nAfter a night in one of the government buildings of New Oraibi, Natalie woke Alice and Schindler at dawn and led them up the steep foot trail to see Tawakwaptiwa at his home in Old Oraibi. To Natalie's great sorrow, she found Tawakwaptiwa was the chief of a diminished pueblo. The once-vital village where Natalie and Chief Lololomai had sat and watched sundown from a rooftop was disintegrating back into the dust of Third Mesa. Natalie blamed the community's ruination on the U.S. government's relentless insistence that Hopi children be forcibly taken from their homes and placed in boarding schools whose goal was to obliterate the cultural heritage of Indian America.\n\n**Figure 23.** Alice Klauber and Chief Tawakwaptiwa, August 1913.\n\nIn spite of the community's deterioration, the Hopi of Third Mesa continued to perform the Flute Ceremony. Natalie, Alice, and Schindler joined the mostly native audience to watch one of the most poetic of the traditional Hopi dances. The drama was an invocation of water for the springs below the pueblo, and dancers were dressed as symbols of the sun, the rainbow, and the blossoming earth. The Flute Dance was of keen interest to Kurt Schindler, who incorporated folk music and indigenous rhythms into his own compositions.\n\nWhile Natalie was at the Flute Ceremony at Third Mesa, Douglas, who had lived among the Hopis a decade before, assumed the role of cultural tour guide for Roosevelt's entourage. With his son, Eric, Douglas retrieved Roosevelt from his rooms in the well-scrubbed schoolhouse and walked the colonel, his two sons, and cousin Nick up the trail to the villages on top of Third Mesa. Roosevelt, who had several days before successfully crossed the infamously difficult and nerve-testing Rainbow Trail to reach Rainbow Bridge, remarked on the precipitousness of the very narrow and steep cliff-hugging path.\n\nDouglas took the Roosevelt's to meet Nampayo and her family. Nampayo's husband was an old friend of Douglas's, someone, he explained to Roosevelt, to whom he owed much. Winfred Douglas was a man who inspired great loyalty among his friends, and equally great animosity among those who disliked him. What Roosevelt's personal opinion of Canon Douglas was can only be guessed. His _Outlook_ article shared only professional comments about Douglas. Cousin Nick, however, _loathed_ Father Winfred Douglas, whom he had met on this very mesa in 1911.\n\nDouglas was a gregarious extrovert, confident and outspoken about his professional and intellectual accomplishments, and ever ready to offer pastoral counsel, comfort, and assistance. Nick Roosevelt had encountered Douglas at the 1911 Snake Dance at Walpi when he was in high school in Phoenix, and he thought the canon was an \"insufferable idiot\" who \"held forth in stentorian tones\" as he explained the Snake Ceremony to a woman friend who was \"the most remarkable bird\" the teenager had ever seen. \"She was attired in less than knee length skirts, was middle agish, stout, bespectacled and cock-like, had her hair piled mountain high over her forehead, wearing upon it a minute green Alpine hat, on the top of which sat in an attitude of possible premeditated flight, an enormous bird.\"\n\nIn 1911 Douglas had apparently been in Hopiland with one of his more eccentric friends, likely a European. Whoever she may have been was of no concern to Nick: he did not like the canon in 1911, and his opinion of Douglas would not improve in August 1913. Nick, like Roosevelt's sons Archie and Quentin, guarded the former president's privacy, and thought Douglas didn't deserve so much access to Roosevelt. As Nick saw it, Douglas was able to get close and \"get his claws\" into Roosevelt only because he was traveling with Roosevelt's friend Natalie Curtis, \"apparently a real person, and a woman of great good sense.\"\n\nNatalie, Alice, and Schindler returned to the peach orchard at the base of First Mesa and spent the afternoon resting in camp. While they napped in the shade, John Hubbell took Roosevelt up First Mesa. Roosevelt had been invited to a private meeting with the Snake priests in their kiva, the ancient heart of Walpi.\n\nThe meeting between the Snake priests and the man they called the Great White Father from Washington lasted only a quarter of an hour that hot afternoon on August 20. Roosevelt's descent by ladder into the kiva was the climax of an extraordinary personal journey through Indian Country. In the sacred subterranean chamber, the Snake priests had been preparing for the Snake Ceremony for nine days. Roosevelt knew there would be snakes\u2014 _lots_ of snakes\u2014but he was unprepared for the large number of _poisonous_ snakes slithering untethered and uncontained about the kiva floor throughout his audience with the priests.\n\nThe smoky, dimly lit kiva was decorated with eagle feathers, animal skins, lightning sticks, and paintings of snakes. Roosevelt sat on a blanket in the middle of the chamber with his back just eight feet from \"thirty-odd rattlesnakes, most of them in a twined heap in one corner, but a dozen by themselves scattered along the wall.\" There was also a ceramic pot \"containing several striped ribbon-snakes, too lively to be left at large.\" One man spoke English and translated the priests' greetings and Roosevelt's questions, but the conference in the kiva that afternoon was mostly silent. Roosevelt did his best to mimic the priests' nonchalance about the venomous participants, which were mostly kept to one side of the kiva by a snake watcher who used a fan made of eagle feathers to redirect those reptiles which wandered too close to Roosevelt. \"Every move was made without hurry and with quiet concern,\" Roosevelt wrote, noting that the religious life of the Hopi contained relics of \"an almost inconceivably remote and savage past,\" but also \"a mystic symbolism which has in it elements that are ennobling.\"\n\nArchie, Quentin, and Nick greeted Roosevelt when he climbed out through the top of the kiva into the blindingly bright pueblo plaza \"burning with enthusiasm.\"\n\nThe remainder of that afternoon and long into the evening, the schoolhouse below First Mesa functioned as Roosevelt's impromptu office. Nick served as clerk and timekeeper for the droves of dignitaries, politicians, and celebrities who appeared to pay their respects to the ex-president. Governor George W. P. Hunt of Arizona came to Polacca from Phoenix with a press entourage of photographers, motion picture people, and a writer from _Collier's_ magazine. Even though he was the wounded Republican-turned-Progressive spoiler, a photo op with Roosevelt was highly valued by politicians.\n\nNatalie's scheduled appointment with Roosevelt commenced in the evening, after everyone (Natalie, Alice, Douglas, Schindler, and Whitaker, plus the Roosevelt clan) trekked up First Mesa at dusk to observe the Antelope Dance, a ceremonial precursor to the Snake Dance. Douglas joined Natalie and Roosevelt at the schoolhouse for a discussion that lasted long into the night. Roosevelt wanted Natalie's and Douglas's input on the _Outlook_ article he was drafting about the Hopis. But Natalie had her own agenda, and as Roosevelt well knew from a dozen meetings at Sagamore Hill and in the White House, she was not easily deterred.\n\nNatalie bluntly asked Roosevelt what right the U.S. government had \"to stretch forth an autocratic hand arbitrarily to change the village life of this ancient and peaceful folk?\" She then proceeded to lecture him on the function and design of the historic Hopi villages that for centuries had existed in harmony with their natural surroundings. Natalie complained about buildings like the schoolhouse they sat in, designed with no eye to either beauty or usefulness, that the BIA forced upon the Hopis. And, she asked Roosevelt, just when _would_ the feds study the Indian culture they sought to change? \"Until some great foundation like the Carnegie . . . or Rockefeller shall endow funds for a study of this kind,\" Natalie concluded, \"and until the whole matter of white-man control of Indians can be lifted out of politics, our efforts at Indian education are bound to be like the rain-shedding roofs in the desert\u2014misfits in spirit as in fact.\"\n\nRoosevelt had heard these grievances from Natalie before. But that August night he was hearing them while sitting in a BIA schoolroom at the base of First Mesa, deep in Indian Country. In _The Indians' Book_ Natalie wrote that one would only understand Hopi music when it was heard in the environment from which it was born, in the wide land, amid the shifting color of the tinted sands and the purple blue of the sharp-shadowed rocks\u2014the very landscape Roosevelt had been immersed within for the last six weeks. That evening, Roosevelt had stood on the lip of the mesa and watched dusk descend to the haunting rhythm of the Antelope Dance, and that afternoon he had shared with dozens of snakes the packed mud floor of a kiva that echoed with the chants of the Snake priests. Roosevelt had stepped into the heart of Native America and he had been changed by it. Roosevelt assured Natalie he would use the power of his pen to educate the American public about Indian culture, \"for somehow people do listen to me.\"\n\nRoosevelt had drafted \"Across the Navajo Desert\" about his trip from the Grand Canyon to Kayenta and Rainbow Bridge. He read the article aloud to Natalie and Douglas and asked for corrections or suggestions. Natalie offered a few changes to strengthen the text. \"I was amazed at the electric snap with which the Colonel would grasp a suggestion,\" Natalie later wrote, \"and without a moment's hesitation put it into a written form far clearer and more succinct than the suggester could have supplied.\"\n\nThe article, penned in longhand, would be mailed to the _Outlook_ offices in New York from Hubbell's at Ganado the next evening. Natalie was impressed by Roosevelt's productivity while traveling on horseback through the outback, and asked him when and how he had found the time and circumstances in which to write. Roosevelt smiled and commented that he liked \"the strenuous life.\"\n\nBefore dawn on August 21, with the steady light of two morning stars, Mercury and Venus, glimmering over the desert, and with the glow of the night fires of Walpi flickering on the rim of the mesa above them, Natalie, Alice, Douglas, Eric, Schindler, and Whitaker left the warmth of their bedrolls and made a pot of coffee over the campfire. Before leaving the schoolhouse the night before, Natalie promised Roosevelt she would come for him before sunrise, and together they would walk up the steep, dark trail to Walpi. (Archie, Quentin, and Nick were not housed with Roosevelt, but camped nearby with John Wetherill.) In the half-dark, Natalie dressed, filled a thermos with coffee, and went to find Roosevelt. But Roosevelt's room was empty, and as she walked back to the camp in the peach orchard, Natalie reminded herself that one must indeed get up _very_ early to keep up with the colonel.\n\nNatalie and her campmates skipped breakfast and scrambled up to the summit of First Mesa. They found Roosevelt high on a stone ledge silhouetted against the \"yellow line\" of the dawn. With his two sons and cousin, Roosevelt, led by John Wetherill, had ridden on horseback up the First Mesa trail. Roosevelt brushed off Natalie's apology for being so tardy, but accepted her thermos of hot coffee. As the sun rose above the horizon, the mostly Hopi crowd waiting on the edge of First Mesa cheered the return of ceremonial runners as they completed a six-mile race.\n\nWith sunrise came the first trickle of what would become the great flood of visitors to First Mesa. The Snake Dance would begin around 5 p.m., and throughout the day hundreds of white tourists and dignitaries ascended the vertigo-inducing trail to Walpi. Nick Roosevelt estimated some eight hundred people made the climb to the old village and that the audience was four times as large as that which attended the Snake Dance in 1911.\n\nAlice and Schindler were introduced to Roosevelt that morning. Alice liked Roosevelt immensely, and she thanked him for his good work for the Indians and for America. Alice had carried her copy of _The Indians' Book_ in her saddlebag for eleven days, and she asked Roosevelt to inscribe it. Inside the front page he penned: \"In a Hopi Indian pueblo . . . To Alice Klauber with all good wishes, from Theodore Roosevelt, Walpi August 21, 1913.\"\n\nNick Roosevelt ignored Douglas when they stood face-to-face on Walpi, but he was impressed by and eager to be introduced to Kurt Schindler, \"the terrific genius so much spoken of at the Opera House in New York's musical circles.\" Schindler knew Nick's sister, Lorraine Roosevelt Warner of Boston (whom Nick thought Natalie resembled in size and demeanor). Like everyone else standing around Roosevelt, Schindler had spent weeks traveling by horseback across the Arizona desert, and the usually pressed, shaven, and showered Schindler scarcely resembled the conductor toasted by New York's effete music circles. Schindler, introduced to the Roosevelts \"as being no tenderfoot,\" was as brown as the natives, and although he \"wore the costume of a musician\" (his hair hung below his ears and was tied down with a bandana), the morning Nick met him, Schindler \"had apparently forgotten to put on a shirt.\"\n\nAfter lunch, Natalie, Alice, Douglas et al. left the shade of the orchard and walked again up to Walpi. By mid-afternoon, all the dirt roads leading to First Mesa were crowded with people coming for the Snake Dance. \"They came on foot, in wagons, and on horseback, and from the mesatop we watched the automobiles crawling over the sand like an eruption of ugly and alien beetles. Such a heterogeneous crowd as flocked from all directions to watch the Colonel watch the Indians!\"\n\nNatalie had been assigned a seat beside Roosevelt, his sons, and Governor Hunt, front and center to the ceremony that would be performed on the plaza. Douglas, Alice, Schindler, and their campmates, like John Wetherill and Don Lorenzo Hubbell, had to find a place in the great mob. The crowd around the plaza became so large that people climbed up onto the rooftops and lined the village's rock walls. The pueblo's small plaza ended abruptly in a sheer drop to the desert floor hundreds of feet below, but dozens of spectators, anxious for a view of the dance and the former president watching the dance, took positions along the mesa's dangerous precipice.\n\nThe whole scene on Walpi reminded Nick of the Republican Convention. Reporters, photographers, and \"moving picture men\" joined the \"mob of the idle curious and the near great.\" Before the Roosevelts reached their assigned seats at the base of a wall to one side of the plaza, the colonel was stopped several dozen times by men and women asking for an autograph, or reaching out to shake his famous hand. Roosevelt obliged everyone.\n\nNatalie walked into Walpi alongside Roosevelt. In the last two weeks of camping, she had somehow managed to keep one outfit free of desert grime, dust, mud, horsehair, and axle grease, and she floated into the Snake Dance wearing a pristine white cotton shirt and skirt, her face shaded by an enormous white hat. The clicking sound of rapid-fire Kodak cameras followed Roosevelt's entourage as it moved through the narrow walkways of the Hopi village. By Nicholas's estimate, they were photographed exactly sixteen million times.\n\nBefore the dance commenced, the crowd, now stacked three stories high on ledges and rooftops surrounding the plaza, was given a preliminary speech about the sacredness of the ceremony and instructed that there was to be silence throughout its duration. The speaker was an Indian service official. Douglas and Natalie actually instigated the preperformance cultural lecture, although they had suggested Roosevelt stand and address the crowd before the ceremony. This idea was quickly nixed as inappropriate.\n\n**Figure 24.** Snake Dance, Walpi, August 21, 1913. Theodore Roosevelt and Natalie Curtis (in white hat) are in the center of the second row.\n\n**Figure 25.** Natalie, Theodore Roosevelt, Quentin and Archie Roosevelt, and Governor George W. P. Hunt, Snake Dance, Walpi, August 21, 1913.\n\nThe ceremony commenced when the snakes\u2014the same serpents seen by Roosevelt on the floor of the kiva the day before\u2014were brought into the plaza in bags by the Snake priests and placed in the _kisi_ , a shallow pit covered with a board that represented the underworld. The snakes' arrival in the crowded plaza was followed by the solemn line of chanting Antelope and Snake priests. Two Antelope priests accompanied each individual Snake priest as he was given one of the snakes and proceeded trance-like around the plaza. The rattlesnakes were first carried in the priests' hands. When the snakes were transferred to the priests' mouths, the assisting Antelope priests helped the dancer support the weight of the snake and also stroked and calmed the rattlesnakes with feathers. (Roosevelt saw several dancers bitten by snakes, but the priests did not seem to suffer ill consequences from the poisonous strikes.)\n\nAs the mass of onlookers pressed in within an arm's length of the dancers and their venomous partners, the procession of Snake and Antelope priests chanted and stomped four times around the plaza. At the end of the spirited, dramatic ceremony, the snakes were tossed down into a circle. With their bare hands, each of the Antelope priests retrieved half a dozen snakes each and dashed out of the plaza with them and down the steep sides of First Mesa to the desert floor, where they released the sacred reptiles into the four directions.\n\nAs the mob of spectators climbed down from their perches on Walpi's dwellings and walls, Natalie said her farewells to Roosevelt. She hoped to see him back east after his trip to South America later in the fall. Natalie wove her way out of the throng pressing in to see Roosevelt and found Alice and Douglas. After regrouping with their campmates, they walked across the mesa to Hano to watch the sunset from Nampayo's home.\n\nThe legions of white people headed back to their autos and civilization, and Hopi First Mesa returned to normal. Roosevelt and his family packed up their expedition gear, said farewell to their horses, Navajo guides, and their very good friend John Wetherill, and piled into John Hubbell's two Ford machines to drive into the night to Ganado.\n\nNatalie and Douglas's expedition remained in Hopiland another week. Natalie set up a workshop at their campsite near Polacca, and many Hopi friends came to share stories and songs with her in the shade of the peach trees. One of the Snake priests who had sat with Roosevelt in the kiva, named Harry, was an old friend of Natalie's, and he spent an afternoon telling stories and singing with Tawimana and her white friends. Natalie sang almost as many songs for the Hopis as they sang for her. Schindler scribbled notes throughout Natalie's sessions with native singers. When he returned to New York, he planned to incorporate the native melodies and folk rhythms of Hopiland into his own compositions.\n\nOn the morning of August 24, Canon Douglas held an outdoor mass in the orchard near their campsite. After an Episcopalian service under the wide Arizona sky, Natalie, Alice, and the men packed up camp and walked the trail to Hano one more time to say good-bye to Nampayo.\n\nOver the next few days they camped in the open country near Second and Third Mesas. Natalie wanted to connect with several Hopi singers, including Chief Tewaquaptewa, who came to see her when she returned with her group and set up camp below Old Oraibi. Alice carried _The Indians' Book_ in her saddlebag, and on the same page signed by Natalie and Roosevelt, the chief wrote _Kwaptiwe_ , \"Hopi Chief\/Orababi\/August 24, 1913.\"\n\n**Figure 26.** Ted Whitaker, Alice Klauber, Kurt Schindler, Natalie Curtis, Winfred Douglas, and Hopi chief, First Mesa, August 1913.\n\nDuring one of these private sessions with the Hopi chiefs and priests, Alice received her Hopi name\u2014Koiyahoinim. Ted Whitaker was also given a Hopi name, but no one's diary recorded what that name was.\n\nAfter several days in the village of Hotevilla, the community founded by the Hostiles who departed Oraibi in 1906, Natalie and friends rode east to the Toreva Spring and camped near Second Mesa. At the ancient Hopi village of Shongopovi, they were invited to observe the Flute Ceremony. The ceremony beside the spring at Shongopovi was rarely if ever viewed by outsiders. Douglas later wrote about the procession and dance, and he became credited as perhaps the first white man to see this particular Hopi ceremony.\n\nNews and letters from the outside world awaited Natalie, Alice, and the others on their return to the BIA offices in Polacca. After a final stop for supplies at Hubbell's trading post, they rode southwest out of Tusayan into the Painted Desert of the Navajos. It was very hot. The monsoons had passed, and the dry landscape shimmered with ghostly mirages as they rode southeast for Indian Wells. After a final night under the open sky of the high desert, they packed up the horses one more time, and their great journey across Indian Country came to an end the next afternoon when they rode into Holbrook.\n\nAfter a fitful night in hotel rooms that felt confining and airless, they carried their gear and bags to the Santa Fe Railway depot and boarded the eastbound train. Two hundred miles east across the desert, the train stopped at Laguna Pueblo, New Mexico. Everyone disembarked except for Schindler, who was to remain on the train bound for Chicago and New York. After twenty-two days of fellowship on the desert, Natalie, Alice, Douglas, Eric, Schindler, and Whitaker had developed the bonds known by all who share the trail through a beautiful and dangerous land. Saying farewell to Schindler was difficult.\n\nNatalie intended to spend a few weeks at Laguna visiting friends and working on her field notes, and she set up her tent near the old section house, where she was a guest of the Marmon family. Alice, who wanted to explore the pueblo and sketch and paint the people and landscape, remained in Laguna with Natalie. Douglas had one more destination on his itinerary, and with his son and Whitaker, he left Laguna and rode eighteen miles across the New Mexican desert to Acoma, the Sky City Pueblo, whose mission church was the architectural template for the New Mexico Building under construction at Balboa Park. The men spent several days exploring the massive rock islands of Acoma and its neighbor, Enchanted Mesa. During this expedition, Douglas found and successfully climbed a legendary trail that led nearly straight up the side and onto the top of Enchanted Mesa. Douglas would be hailed as the first white man to locate and climb the ancient \"lost\" trail of the Acoma Indians. Whether the trail was actually lost, and whether Douglas was, in fact, the first white man to find and ascend that trail, remain open to discussion. However, Douglas's membership in the elite Explorers Club was in recognition of two \"firsts,\" both of which occurred on his expedition with Natalie in 1913: his witnessing of the sacred Hopi Flute Ceremony and his ascension of the difficult and unmarked trail to Enchanted Mesa. From its inception in 1905 until 1981, the Explorers Club recognized the feats and adventures of men only. That Natalie and Alice may well have been the first white women invited to attend the Hopi Flute Ceremony, and that they witnessed the ceremony twice, on August 23 at Third Mesa and on August 26 at Second Mesa was never honored or officially recorded.\n\n**Figure 27.** Southwest point of Mesa Encantada (Enchanted Mesa) near Acoma, New Mexico, ca. 1900.\n\nOn Thursday, September 4, their grand desert dream came to an end. Natalie and Alice said good-byes to Douglas, Eric, and Whitaker at the Laguna station and watched them depart on the 4:30 train for Denver. Their farewell was blessed with a good omen from the Snake gods: thunder rumbled from clouds over the western horizon, and the promise of rain scented the wind ahead of the storm. Later that evening, Alice went by wagon with the Marmon boys through pouring rain to the depot and boarded the 11 p.m. train for Los Angeles. Natalie remained in New Mexico. Alice hated to leave her alone at Laguna, but knew Natalie wanted and needed to \"get a bit of real work done here when there's no interruptions.\"\n\nThe heat that dogged Natalie, Alice, and company across the Arizona desert had been bearable because of the relief brought by the seasonal monsoon thunderstorms. Southern California was suffering the same intense heat that summer, but was not blessed with the rain that fell on the neighboring state of Arizona. Alice returned home to San Diego through a parched landscape. The country surrounding George Curtis's ranch was a tinderbox by late summer, and on September 16 a terrific wildfire swept across the top of the valley near George's house. George fought a fierce wind and cleared dead brush from around his house and beehives, and his little ranch was spared.\n\nIt was still hellishly hot and dry in late September, but a drizzle of rain finally dampened the fires, and George was able to work on his ranch \"without the demonic image of smoke on the horizon.\" On the night of the full harvest moon, October 15, Natalie returned to California on the train. George retrieved her and brought her to the ranch. Over the next several days they unpacked Natalie's bags, aired out her gear, and then repacked and stowed the camping equipment in his shed.\n\nLater that week, Natalie and George together read Roosevelt's article \"The Hopi Snake Dance,\" which was published in the October 18 issue of _Outlook_. Roosevelt praised Natalie's work collecting Indian music and reiterated her plea to the federal government to create an educational system that would perpetuate Indian poetry and music. Roosevelt also gave American readers a detailed and colorful recounting of his experiences among the Hopis, especially about his time in the \"grey twilight\" of the kiva of the Snake priests where he encountered the \"symbolism and dark savagery\" of their \"elder world.\"\n\nPerhaps because Roosevelt was still assessing his feelings about the modern art seen at the Armory Show, he commented on the importance of Nampayo's ceramics and the Native American aesthetic. \"A great art must be living,\" he wrote, \"must spring from the soul of the people; if it represents merely a copying of an imitation, and if it is confined to a small caste, it cannot be great.\"\n\nIt must have made Natalie smile when Roosevelt suggested in his article that the federal government needed to design and construct dwellings appropriate to place and culture, and challenged the decision to force the Hopi off their mesa-top villages and into white man's\u2013style communities built \"precisely like some ten million other cheap houses.\"\n\nEven Washington outsider Frank Mead, Natalie and George's eccentric friend who had found an appreciative audience for his North African and indigenous American building designs, received a nod from Roosevelt. \"Mr. Frank Mead . . . has done admirable work of the kind by adapting Indian architectural ideas in some of his California houses.\"\n\nRoosevelt's six weeks of submersion in Indian Country, particularly his three days and two nights at Walpi and his conversations with Natalie at the schoolhouse at the foot of First Mesa, had transformed his thinking. \"Give [a Hopi] a chance to lead his own life as he ought to,\" Roosevelt wrote, \"and realize that he has something to teach us as well as to learn from us.\"\n\n#  _10_\n\n##  \"A FINE WOMAN TO BE OUT WITH\"\n\nThe personality of which she was so tired seemed to let go of her. The high, sparkling air drank it up like blotting-paper. It was lost in the thrilling blue of the new sky and the song of the thin wind in the pi\u00f1ons. The old, fretted lines which marked one off, which defined her . . . were all erased.\n\n##  \u2014 _Willa Cather_\n\nALMIR A CHOATE JOHNSON DIED in her bed of heart complications in March 1914. She left her spinster granddaughter, Carol Stanley, the house built by J. Bishop Johnson across Dorothy Cove from Longfellow's former residence. Almira also left her unmarried granddaughter a sizable bank account. With a home and money of her own, her grandmother gave Carol what she as a Choate and as a married woman always had: security.\n\nAt thirty-five, Carol had earned the means and the right to live out her days as a respected woman in the community on Nahant island. But Carol was not interested in living out her days as the island's spinster music teacher. The staid life of nursemaid and helper that had defined the last two years had left her tired, dissatisfied, and restless. She held on to an image of life beyond the woman's sphere of home and community service, and yearned to return to the stimulation and challenge she had known in Boston and Baltimore. The house on Dorothy Cove and the security of her own bank account were gifts of incalculable value to a single woman, but Carol had experienced too much of the world beyond her family. She still wanted to do something with her music and with her life. And whatever that something was, it was not found on Nahant.\n\nThe Lynn and Nahant Street Railway connected the island to the mainland trains into Boston, and Carol had continued to socialize with professional colleagues and friends during her mother's illness. She had also worked as a volunteer at two organizations in Boston\u2014the South End Music School, where she interacted with Natalie and Canon Douglas, and the Boston Port and Seaman's Aid Society. As the daughter and granddaughter of fishermen, Carol understood the travails of seamen and their families, and gave piano concerts for boarders and visitors at the Seaman's Bethel in Boston's North Square. The Seaman's Bethel was a Methodist and Episcopalian charity organization founded by the \"Sailor Preacher\" Father Edward Thompson Taylor in the 1820s. Like the South End Music School, the bethel was staffed by mostly female volunteers from Boston's wealthier families who worked to improve the moral, religious, and living conditions of Boston's working-class folks.\n\nNatalie wrote about Boston's South End Music settlement school in the _Craftsman_ in 1911, and may have met Carol during one of her tours of Boston settlement houses. Natalie and Carol were probably not aware that their ancestors may have known one another years before: Natalie's uncle George Curtis entertained Joseph Choate, a relative of Carol's great-grandmother Elizabeth Choate, at his home in Ashfield, Massachusetts, decades before Natalie and Carol were born.\n\nBoston was populated with friends from Carol's school days at the New England Conservatory of Music. Carol also had Brahmin relatives and Nahant friends living on Beacon Hill: her father's first wife's relatives, the Edward C. Johnson family, several Cabots, Lodges, and a few Lowells whom Carol had known since childhood when they lived or summered on Nahant. But Carol was not a Brahmin, and her social circle, although educated and worldly, was made up of musicians and intellectuals, not heiresses and business barons.\n\nCarol continued to play piano on the professional stage, although her opportunities were limited while she lived out on Nahant. Even though she was in her thirties, and past the conventional age when women married, by the fall of 1914 Carol was dating a professional musician. Carol was no beauty, but she was witty and smart, and had pleasant, attractive features and a good fashion sense. She also had enough money to be considered independently wealthy.\n\nCarol's family on Nahant, specifically her mother's brother, Otis Johnson, feared the musician courting his niece had his eyes, not on Carol, but on her very attractive inheritance. Details of the liaison are scant, but Carol evidently had a serious affair with a violinist with whom she shared the Boston concert stage. By Carol's own account, the affair ended scandalously (the violinist may have eloped with a wealthy society patroness), and Carol's friends, and possibly her family too, suggested she needed to leave town.\n\nWinfred Douglas may have been among those friends who urged Carol to sever ties with Boston, and he may have helped orchestrate her departure from New England by recommending that she join the staff at the Kindergarten Institute in Chicago. By late fall of 1914, Carol closed up her Nahant home on Dorothy Cove, packed up her belongings, and departed Boston for Chicago.\n\nFrom Douglas and Natalie, Carol had heard dozens of stories about desert expeditions and Indian ceremonies, about cultures and landscapes as foreign to Nahant island as the civilizations found along the Nile. Carol never recorded when the suggestion was first made that she go with Douglas on a journey into Indian Country, but in the spring of 1915, after a winter in the Midwest, Carol traveled to Evergreen, Colorado, where she embarked on a journey to Arizona with Douglas. She couldn't have known it at the time, but her introductory journey with Douglas into the American Southwest would change the course of her life and, in doing so, would significantly alter and influence the lives of dozens of other people, some of whom would leave bold and transformative marks on Southwest history.\n\nSometime in late May 1915, Carol and Douglas took the train south and west from Denver to Durango and eventually Mancos, Colorado. From the narrow-gauge station in Mancos, they journeyed across the country below Mesa Verde and the cliff dwellings famously \"discovered\" by John's brother Richard Wetherill and his brother-in-law Charles Mason in the winter of 1888. From Cortez, they headed into McElmo Canyon and reached the San Juan River near Bluff. They continued downriver to the bridge at Mexican Hat and the last leg of their journey south and west across the Utah-Arizona desert to Kayenta and their destination, the home of Louisa and John Wetherill.\n\n**Figure 28.** The Wetherill Trading Post and home, Kayenta, Arizona, ca. 1912.\n\nDouglas had become friends with the couple on previous expeditions in the Four Corners country. In 1915 Douglas was a defense witness for an upcoming trial in Denver, and on this trip with Carol he had come to see Louisa Wetherill about locating other witnesses living on the Ute reservation. Douglas stayed a few days at the Wetherills' discussing the trial with Louisa (John Wetherill was not in Kayenta but on an expedition with a group in the outback), and then he left Kayenta for Utah.\n\nCarol, a guest at the Wetherills' throughout Douglas's meetings with Louisa, must have impressed Louisa because Carol did not leave with the canon, but remained her houseguest for the next few months. Living at the Wetherills' trading post home provided Carol with an extraordinary introduction to Indian Country. As Roosevelt and his family had learned in the summer of 1913, Louisa's home in Kayenta was a desert palace in the heart of the most spectacular landscape in all of Indian Country. Louisa, like her husband, had achieved legendary status among the Navajos and Utes who lived on this high desert, and among the white people who wanted to experience one of the last great American frontiers.\n\nLouisa Wade Wetherill was known as Asthon Sosi, Slim Woman, among the Navajos. She had lived in Kayenta with her husband, a trader, backcountry guide, and internationally known amateur archaeologist, since 1910. Louisa's stone house on the red desert twenty-four miles south of Monument Valley hosted adventurers, traders, writers, archaeologists, and former presidents. All expeditions to the great Rainbow Bridge began and ended at the Wetherills', and nearly all of these were led by John.\n\nOn August 14, 1909, John Wetherill, and his companions Dr. Byron Cummings and William B. Douglass, entered the historic record as the first white men to reach Rainbow Bridge. Outback insiders knew that it was Louisa Wetherill who enabled their \"discovery,\" since in November 1908 she had convinced the Navajo known as One-Eyed Man to guide her and John to the legendary stone rainbow the Navajo called Nonnezoshe. Louisa's securing of a Navajo guide was nothing short of extraordinary. Nonnezoshe was sacred to the Navajo, but the Din\u00e9 had forgotten the appropriate prayer for the great bridge, and normally no Navajo would venture near it.\n\nLouisa Wetherill was a respected _Bilagaana_ among the native people of the Four Corners region. Born in 1877 in Nevada, Louisa grew up in Mancos, Colorado, near the ranch of her future husband's family. She spoke nearly fluent Navajo and was known and trusted by the Navajos, Utes, and Paiutes she had lived among since childhood. After witnessing Louisa's interactions with the Indians who came to trade at the Wetherills' post during his stay in Kayenta, Roosevelt commented that she was not simply versed \"in archaeological lore concerning ruins and like, she was also versed in the yet stranger and more interesting archaeology of the Indian's own mind and soul. . . . [Louisa] not only knows [the Navajos'] language; she knows their minds. . . . They trust her so fully that they will speak to her without reserve about those intimate things of the soul which they will never even hint at if they suspect want of sympathy or fear ridicule.\"\n\nThe Wetherill outpost at Kayenta was a desert oasis. Louisa diligently watered a garden that included Virginia creeper and rosebushes shaded by a large stand of box elders and a front yard that boasted a grass lawn. For eastern dudes who might forget they were in the middle of a very hot and dry desert that received only a few inches of rain a year, Louisa posted a \"Keep Off the Grass\" sign alongside her fragile green lawn.\n\n**Figure 29.** Louisa Wade Wetherill and Sitsosie, a Navajo guide and friend of the Wetherills, Olijato, 1909.\n\nAt Kayenta, Carol had a room in a dwelling unlike any she had seen in New England. The floors of the Wetherill home were packed dirt softened by Navajo rugs. The walls were decorated with more Navajo tapestries and replicas of Navajo sand paintings, Indian baskets and artifacts, and dozens of original paintings given to Louisa and John by the artists who had stayed with them. The roof was constructed of poles packed with earth, and to keep debris from sifting down onto food and drink, lengths of unbleached muslin was tacked to the ceiling over the dining table, which frequently accommodated ten or more guests. Like their counterparts at the Hubbells' in Ganado, the Wetherills' bookshelves were crammed with books on Southwest history and archaeology.\n\nThe home of Louisa Wetherill was, in Roosevelt's words, a delight to the senses, \"clean, comfortable, with its bath and running water, its rugs and books, its desks, cupboards, couches, and chairs, and the excellent taste of its Navajo ornamentation.\"\n\nKayenta was the gateway to the most iconic landscape of the American Southwest: Monument Valley and its otherworldly landscape of sculpted red buttes, stone needles and towers, obelisks, and colossal spires began a few miles to the north. Comb Ridge and El Capitan, Church Rock, carved the horizon to the northeast, and on the horizon to the south was the great limestone rampart of Black Mesa, whose rock fingers reached fifty miles across the desert to the Hopi mesas. Fifteen miles to the west of Kayenta, a primitive road crossed Marsh Pass into verdant Tsegi Canyon (Navajo National Monument), where John Wetherill had stumbled upon the cliff ruins of Kiet Seel in the 1890s and located the Anasazi cliff dwellings of Betatakin in 1909.\n\nThe small cluster of stone and mud-chinked cedar post buildings that was the Wetherills' home was an island outpost surrounded by a wilderness of sand and rock. Perhaps this was why Carol felt at home in Kayenta: she was well accustomed to island living, and to windows that framed views of infinity.\n\nCarol was living in a land of legendary places and people. Zane Grey's novel _The Rainbow Trail_ was published in 1915, the same summer that Carol came with Douglas to the Southwest. Grey's story, set in Hopi and Navajo land, introduced urban readers to the rugged canyonlands of Arizona. Two years before Carol moved in with the Wetherills, Grey had stayed in those same rooms and had gone with John Wetherill to Nonnezoshe. Grey scratched his name into a stone near the base of the bridge, and three months later his signature was found by Roosevelt and his sons when they followed Wetherill to the sacred monument.\n\nNatalie's and Douglas's descriptions of Arizona's space, color, and hot light could only partially prepare Carol for her experiences at the Wetherills'. Carol learned what Louisa had come to know as a young woman: from this hard country came beauty.\n\nThe Wetherills' home became the headquarters and supplier for the surveyors and archaeological teams studying and excavating the cliff dwellings and Anasazi sites in Tsegi Canyon and in the canyonlands along the San Juan River near Bluff, Utah. In 1909, before beginning work at Balboa Park, Edgar Hewett and Jesse Nusbaum had stayed here and traveled with John Wetherill into the vast unmapped country Hewett was surveying for the Archaeological Institute of America.\n\nWhen Carol arrived for her several months' residency, the Wetherills' outpost was the base camp for a young Harvard PhD, Alfred V. \"Ted\" Kidder, who with Samuel James Guernsey was completing a second season of fieldwork in the caves and ruins near Marsh Pass. Ted Kidder was an affable, good-humored, bearded fellow who had been on Hewett's field teams in Bluff and McElmo canyons, and at Mesa Verde. Kidder had also worked at Hewett's archaeological field school (frequently visited by Hewett's pal Charles Lummis) in the cliff dwellings of the Rito de los Frijoles on the Pajarito Plateau in northern New Mexico.\n\nIf Carol's residency overlapped with Kidder's at the Wetherills' that spring, they surely talked about goings-on in and around Boston, where Kidder lived with his wife. They may have also talked about the land of enchantment, New Mexico, where Kidder had spent several summers in the canyon, mesa, and pine-forested mountain country west of Santa Fe near a town called Espa\u00f1ola. Kidder loved all things New Mexican, and he may have sparked Carol's interest in the ancient pueblo country of the Rio Grande Valley. Within the year, Carol would be in Santa Fe, and like Kidder, Lummis, Hewett et al., she would be smitten by the enchanting country of New Mexico.\n\nThe first automobiles that used the rough wagon roads broken by John Wetherill and his partner, Clyde Colville, reached the Wetherill trading post and the farthest-flung post office in the United States in 1914. With the motorcars came a different type of visitor into the country of Monument Valley: auto tourists, who, Louisa said, were \"impatient of delay, frightened of hardship, desiring the comfort of settled places.\" The Wetherills, like the Hubbells, had been negotiating the desert in Model Ts a few years before the first tourists. But as Natalie, Alice, Douglas, and their co-travelers learned in 1913, even with the best of drivers behind the wheel, auto travel across sand dunes and rock ledges on roads that were nothing more than trails was laborious and slow going. Most people came to Kayenta on horseback or, as Douglas and Carol did in the spring of 1915, by horse and buggy.\n\nDouglas and Carol journeyed to Kayenta because Douglas was a subpoenaed witness in the trial of Tse-ne-gat, also known as Everett Hatch, the young son of Chief Polk (Billy Hatch) of the Utes. The year before, in March 1914, Tse-ne-gat was charged with the robbery and murder of Juan Chacon, a Mexican shepherd, near Navajo Springs, Colorado. After a warrant was issued for his arrest, Tse-ne-gat had gone directly to Louisa for advice. After hearing his story, Louisa was certain the young Ute had been set up by the real murderers, and she told him to go directly to the Ute Mountain Indian agent in Colorado and tell his story. While en route from Kayenta to the Ute agency, Tse-ne-gat was told by an agency policeman that no one would believe he was innocent. Afraid that he would be jailed when he reached the Ute agency, Tse-ne-gat went into hiding in the vast Indian Territory of the Four Corners region.\n\nThrough the fall and winter of 1914, a federal posse unsuccessfully searched for Tse-ne-gat on the Ute, Paiute, and Navajo reservations. In February 1915 violence erupted between a frustrated posse composed of three Colorado sheriffs and twenty-six armed cowboys and Chief Polk's tribe living in teepees in Cottonwood Gulch near Bluff, Utah. The sound of gunfire brought over fifty more armed white men from Bluff and Blanding to the posse's aid, and several dozen Ute warriors camped on the San Juan River with Chief Posey rushed to help Chief Polk. The Battle at Cottonwood Gulch lasted only twenty-four hours, but several members of the posse were killed, and dozens of Utes and white men were wounded. One Ute warrior and two Ute women were killed, and a young Ute child was wounded.\n\nChiefs Polk and Posey and their warriors escaped into the desert, and the Utes and their allies, the Paiutes, declared a state of war with the U.S. government, warning that any whites who came near their camps would be shot on sight.\n\nIn the winter of 1915, while Carol was planning her first trip to the American Southwest from the civilized comfort of a Chicago apartment, a posse of fifty Navajo policemen was assembling to pursue Polk and his band of Utes. But even the Navajos could not locate and apprehend the renegade Indians. Frustrated and fearful, the local BIA agents requested help from Washington, and by spring 1915 the Four Corners Indian uprising was handed over to Brigadier General Hugh L. Scott of Virginia.\n\nBecause Louisa was trusted by the Utes, Paiutes, and Navajos, the Indian agents working with General Scott suggested she travel into the now very hostile territory of the Utes, find the camp of Chief Polk and his family, and convince them to bring Tse-ne-gat in for what the feds promised would be a fair trial. General Scott even promised to protect him from both the white posse and the Navajo police.\n\nLouisa agreed to intervene and put out the word among her Navajo and Ute friends that Scott could be trusted. Chief Polk and his family came into Kayenta and, after consulting with Louisa, took Tse-ne-gat to Mexican Hat, an outpost on the San Juan River, and surrendered him to General Scott.\n\n**Figure 30.** Louisa Wetherill and Utes, Kayenta, June 1915. The woman in the doorway may be Carol Stanley.\n\nNewspaper writers across the United States recounted in colorful detail the Bluff War and the Battle of Cottonwood Gulch, and the hunt for Tse-ne-gat. To readers in Boston and New York, even in Chicago, the West was yet a wild place, and the region called the Four Corners a frontier where untamed and hostile Indians roamed free. In June 1915, when Carol traveled by buggy with Winfred Douglas from the train depot in Mancos, Colorado, across the mountains and canyonlands to Kayenta, Arizona, she was moving into a region whose residents, both Indian and white, were still reeling from the violence at Cottonwood Gulch.\n\nIt was a nervous and even dangerous time in Indian Country, and tourism was not encouraged. Exactly how it came to pass that Douglas left Carol alone with the Wetherills\u2014whether this was an impromptu arrangement made after Carol met Louisa, or was the plan from the beginning of their journey\u2014in a time of such hostility, it speaks volumes about the Wetherills' reputation in the Four Corners. It also reveals a good deal about the character of Carol Stanley.\n\nIn June 1915, in the pages of the only diary ever kept by Louisa Wetherill, Carol emerges into the written record of the American Southwest. Louisa's diary chronicles her quest to locate the Indian witnesses and interpreters Tse-ne-gat's attorney needed brought to Denver. The trial would begin on July 6 and time was running out, so after Douglas left Kayenta, Louisa decided to go into Ute country herself, locate the witnesses and interpreter, and get everyone including herself onto the train for Denver in time for the opening arguments.\n\nLouisa departed Kayenta at supper time on Saturday, June 26, in a buggy packed with enough supplies and provisions for a four-day, fifty-mile journey across the desert to the Ute agency at Navajo Springs. This was no journey for a tenderfoot. Louisa would be roughing it, following leads to Ute camps where there would be no amenities like those found at trading posts. In spite of the challenges, Louisa invited Carol to accompany her and Emma Charley, a young English-speaking Navajo, and John Chief, a friend of the Wetherills.\n\nLike Natalie's and Roosevelt's camping expeditions, Carol's outback adventure began in the long shadows before sunset. Louisa's buggy with Carol and Emma Charley, followed by John Chief and several horses, headed north past El Capitan and into the dusk. Before dark engulfed the desert, the light of the full moon illuminated the landscape. Carol's first camp in Indian Country was on a sand hill at the edge of Monument Valley. They did not pitch tents, and from her bedroll under the open sky, Carol gazed at a dreamlike moonscape of stone spires. She was ecstatic, and even Louisa remarked in her diary about the beauty of the June moon on the landscape.\n\nThe next day the weather was hot, and deep sand made for slow progress. They were following a road\u2014faint tracks obvious to Louisa but barely perceptible to Carol\u2014that wound around the monolithic buttes, stone arches, and rock islands. After sundown cooled the desert floor, they continued to push on, and the moon rose and lit their way to Spencer's store in Goodridge. Louisa's son, Ben, and her husband, John, met their buggy and helped them set up camp. Since leaving Kayenta two nights before, Louisa's party had crossed more than forty miles of desert.\n\nLouisa was surprised and delighted to find John and Ben at Spencer's, since they had been gone a month, guiding a Yale professor through the canyons of the Colorado River. Carol likely met John Wetherill for the first time that night, although she would have known of him from reading the popular writings and stories of Zane Grey and Theodore Roosevelt.\n\nThe next morning, Monday, June 28, Ben Wetherill and John Chief rode south to Kayenta. John Wetherill joined Louisa, Carol, and Emma Charley bound northeast across the desert for Bluff. The heat was worse than the day before, and when they found grass and water for the horses at Lime Creek, they stopped for a long lunch break. There was no shade to be found for miles, so Carol and Louisa stretched a tarpaulin from the buggy to a sage bush, and everyone crawled under and ate lunch under the tarp that protected them from the blistering sun.\n\nLouisa wasn't feeling well that afternoon, and when they reached Bluff at supper time, she was put to bed by their hosts, the Perkinses, friends who lived near the San Juan River. During the twenty-mile journey that day Carol had moved from the seat of the buggy onto the back of one of John Wetherill's horses. It was her first time in a western saddle, and very possibly her first time on a horse. Although Nahant had an equestrian community with stables that served the island's summer residents, Carol's family did not keep horses. With John and Louisa's calm encouragement, Carol rode cowboy style\u2014astride\u2014across a sand, cactus, sage, and stone plateau six thousand feet above her sea-level home on Dorothy Cove. \"Miss Stanley is a fine woman to be out with,\" Louisa wrote in her diary that evening in Bluff. \"She has been riding horse back a good deal today so is pretty tired as she has never ridden before. She done fine though.\"\n\n**Figure 31.** Carol Stanley in the desert.\n\n_A fine woman to be out with_. Although she did not know it, the sunburned, muscle-bruised, and saddle-sore Yankee music teacher had received the supreme compliment from Louisa Wetherill.\n\nWinfred Douglas was also at the Perkinses' home in Bluff that night. The next morning after breakfast, Douglas left in an automobile with Frank Hyde and a Mr. Cornell of Bluff, both of whom, like Douglas, had been subpoenaed by the defense to appear as witnesses at the trial of Tse-ne-gat. Douglas would accompany them to Durango, where they were all to catch the train for Denver.\n\nLouisa still had to locate several Ute witnesses and also an interpreter before she could follow Douglas to Denver. After a messenger told Louisa that Chief Polk was camped above McElmo Canyon and that he was very anxious to talk with her, Louisa, Carol, Emma, and John Wetherill headed out in the buggy and on horseback. The road to McElmo Canyon was terrible, and it took four hours for them to cross ten miles. Exhausted, they slept that night on the open ground beside the buggy. In the morning, they were found by Polk's men, who led them to the Ute camp on a mesa high above McElmo Canyon.\n\nFor the next several nights, Carol and the Wetherills stayed in the camp of Chief Polk and his family, and went with them to Navajo Springs below Ute Mountain where the Sun Dance ceremony was to be held. Hundreds of Utes and Paiutes came to see this dance, and at sunset on the last day of June, Carol stood with Louisa and gazed out at the teepees of the Indian village where dancers in beaded costumes were gathered for the ceremonies. \"It was a beautiful picture to see them riding here and there,\" Louisa wrote, \"and the beautiful red and gold clouds over the blue mountain, and to hear the drum beating, and the whistles blowing, and the Indian singing the Sun Dance Song.\"\n\nCarol woke before dawn and walked with Louisa and John into the Ute village to watch the Rising Sun Ceremony. The dancers had been dancing for two days and a night without water or food, and one by one they went mad and fainted to the ground.\n\nLouisa's search for witnesses led them from camp to camp across the Ute reservation. Carol was a natural horsewoman, and by July she preferred the saddle to the buggy seat. Louisa asked Carol several times to witness her conversations with agency personnel and telegraph operators, and to listen in on and witness telephone calls to officials. After several more days of riding and camping, Louisa located the witnesses she needed, including Chief Polk's wife, who was subpoenaed to appear in Denver, and also secured a trustworthy interpreter for those witnesses.\n\nOn the Fourth of July, Carol was in the tiny frontier town of Mancos, Colorado, where Louisa and John had grown up and met. The community of ranchers and traders celebrated Independence Day with contests and games, pistols and firecrackers. Businesses and adobe houses were decorated with American flags and red, white, and blue bunting, but the Fourth of July in this small Colorado town bore no resemblance to the annual Yankee extravaganza Carol had known back in Boston and out on Nahant island.\n\nCarol said good-bye to Louisa at Mancos and, with John Wetherill and Emma Charley, headed back across the desert to Kayenta. Louisa and Tse-ne-gat's mother took a Pullman car from Durango to Denver. Louisa was bored and lonely in the city, although Winfred Douglas, his wife, Dr. Jo, and other friends of the Wetherills came to visit her at the Kaiserhoff Hotel. The trial commenced on July 6, and after three days of testimony and seven hours of deliberation by the jury, Everett Hatch, aka Tse-ne-gat, was found innocent of murder.\n\nThe verdict cited a lack of evidence against the young Ute, and later accounts by those in the courtroom said the jury was particularly swayed by the strength of the testimonies given by the defense witnesses, specifically the statements given by a pair of Mormons from Bluff who insisted the boy was innocent. These two Mormons may well have been Frank Hyde and Mr. Cornell, the subpoenaed witnesses who traveled with Winfred Douglas from Bluff to Denver.\n\nCarol left Kayenta by midsummer. She eventually went back to Chicago and worked at least temporarily at the Chicago Kindergarten Institute. For Christmas that year she may have visited Nahant. She may have seen Natalie in New York and told her of her life-changing experiences with the Wetherills in Indian Country. Carol left no diary, so we can only speculate about the details of her life in the months after she left Kayenta. However, from the moment she left the Southwest, Carol was planning her return. And when she does return nine months later with Natalie, Carol, like Natalie, is determined to make the Southwest her home country. \n\n#  _11_\n\n##  THE CITY DIFFERENT\n\nWith a burro to ride and a burro to drive, There is hardly a man so rich alive.\n\n##  \u2014 _Alice Corbin_\n\nIN 1916 SANTA FE, ALTHOUGH now the capital of the brand-new state of New Mexico, was still situated in a region of the United States that most Americans deemed exotic at best and primitive and downright dangerous at worst. In March of that year, \"civilized\" Americans in Boston and New York were reading about the jaw-dropping exploits of the Mexican _bandito_ Francisco \"Pancho\" Villa and the raid by his guerilla band of armed revolutionaries on the town of Columbus in southern New Mexico. Seventeen Americans were killed by the Villistas, and a small army under General John J. Pershing went into Mexico in pursuit of the guerillas. New Mexico's National Guard was serving as border patrol, but Americans were discouraged from traveling in southern New Mexico.\n\nNew Mexicans themselves were appalled by the exaggerated news accounts and blatant misinformation making headlines back east. \"Supposedly intelligent persons\" mistakenly reported that Villa's guerillas were encamped in canyons near Santa Fe, and that his Zapata bandits were terrorizing Albuquerque's streets. Although hundreds of miles of mountains and desert lay between the Mexican border and Santa Fe, to a New Yorker or Bostonian reading about Pancho Villa in the newspapers, all of New Mexico was under siege.\n\nH. H. Brook and C. B. Ruggles, the owners of Cliff Dwelling Pack Outfitters, which guided archaeologists and tourists across the mountains and into the canyons of the Pajarito Plateau west of Santa Fe, worried that their fledgling business would be ruined by inflammatory headlines in eastern newspapers, rubber-stamped their outgoing mail with their own headline: \"This region [Frijoles Canyon, Los Alamos Ranch, and the Ramon Vigil Ranch] is 400 miles from the Mexican Border and the war cloud causes less interest here than in New York City. Not the least uneasiness need be felt in visiting this wonderful country.\"\n\nRegardless of the headlines and the belief among the gentry back east that New Mexico was the home of desperados and banditos, on April 10, 1916, Miss Natalie Curtis of New York City and Miss Caroline B. Stanley of Boston arrived in Santa Fe for what they planned to be an extended stay. The ladies' appearance in the capital city, and their registration at the Hotel De Vargas, was noted in the \"Personals\" column of the _Santa Fe New Mexican_. Throughout the spring and summer, as news of the Villistas' guerilla raids, arrests, and hangings dominated the front pages of the _New Mexican_ , the comings and goings of Carol and Natalie\u2014numerous camping excursions in Indian Country, day visits to the pueblos along the Rio Grande\u2014were documented in the newspaper's gossip and social columns.\n\nNatalie and Carol visited the popular tourist sights in the old capital city and explored the quaint streets and small shops near the historic plaza, but they were not in Santa Fe for a vacation. They had come to New Mexico to make permanent homes and to begin new lives in this country.\n\nAfter Carol departed Kayenta in the summer of 1915, she returned to Chicago and worked on the teaching staff at the Chicago Kindergarten Institute, where Winfred Douglas lectured and also was a member of the board of directors. In partnership with Northwestern University, the institute had begun a postgraduate program in music education, and Carol may have been considering a long-term teaching position there. In the fall of 1915, she went to the meetings of the Musical Education League of Chicago and became an active member of Chicago's suffragettes.\n\nIn Chicago, Carol had opportunities for permanent employment in a welcoming community of like-minded professional women. But she could not forget what she had seen and felt in Kayenta and in Indian Country and, with Natalie, made a plan to return.\n\n**Figure 32.** Hotel De Vargas, Santa Fe, ca. 1920.\n\nIn 1914 Natalie heeded the call of duty and left George's ranch in California and returned to New York City to continue her work on Negro folk songs. The project, underwritten by George Foster Peabody and the Hampton Institute, pleased many of her friends and colleagues, including Charlotte Mason. Everyone hoped Natalie would do for American Negro music what she had done for American Indian music with the publication of _The Indians' Book_. Throughout 1914 and 1915, and until her departure for Santa Fe in April 1916, Natalie traveled throughout the American South collecting folk songs, spirituals, and poems. This collection would eventually fill the four volumes titled _Negro Folk Songs_ , published by G. Schirmer in 1918 and 1919.\n\nNatalie's brother Bridgham and her mother, Mimsey, still lived in New York, and Natalie had a comfortable apartment on West 76th Street in the same neighborhood as Winfred Douglas. New York City was a stimulating environment in 1915, and the epicenter of American cultural change. Natalie had many friends living in the city, including her outback expedition pals Douglas and Schindler. Schindler's compositions and choral arrangements were now laced with the peculiar rhythms and atonal chants and songs heard during his 1913 journey through Indian Country, and the concert programs of his Schola Cantorum juxtaposed variations of Hopi and Navajo music with Bach, Mozart, and Schubert.\n\nYears before, while working on _The Indians' Book_ , Natalie had shared several Indian melodies with her former teacher Ferruccio Busoni. In February 1915 Natalie traveled to the Academy of Music in Philadelphia to hear Busoni's use of those melodies in his composition _Indian Fantasy_. With her friend Percy Grainger, the Australian composer and folklorist, she listened to the Philadelphia Orchestra rehearse _Indian Fantasy_ under the direction of Leopold Stokowski. Busoni himself played the piano for what was an emotionally charged performance for Natalie. \"With the first bars of the orchestral introduction the walls melted away, and I was in the West, filled once again with that awing sense of vastness, of solitude, of immensity.\"\n\nOne year later, she made her return to the immense vastness with her friend Carol Stanley, who, like Natalie, was determined to create a home in the Southwest, specifically in Santa Fe. The capital of New Mexico was on the cusp of historic change. Statehood in 1912 had prompted the Anglo-driven Chamber of Commerce and Women's Board of Trade to launch formal campaigns to attract new residents and more tourists to Santa Fe.\n\nNatalie likely suggested to Carol that they go to Santa Fe because she had many colleagues and friends here, with Hewett the energetic hub of Anglo society. Hewett's establishment of the School of American Archaeology and the Museum of New Mexico in Santa Fe in 1909 had turned the ancient city into home base for legions of anthropologists, ethnologists, geologists, and linguists migrating into the state. And Hewett's promotion of New Mexico artists, architecture, and cultural riches at the San Diego exposition in 1915 had prompted cross-country train travelers to add stopovers in Albuquerque and Santa Fe.\n\nBy 1916 Santa Fe was touting itself as the \"Granada of America\" and, in league with the Santa Fe Railway and the Fred Harvey Company, was aggressively promoting New Mexico as a destination with natural and cultural sites that rivaled those found in Europe and the Mediterranean. The exhibitions at Balboa Park had developed an appreciation and understanding of Southwestern arts and crafts among the American public, and the Harvey Company's Indian Buildings along the Santa Fe Railway offered collectors and tourists easy access to Native American wares. Instead of a primitive, dangerous backwater, New Mexico was now hailed as America's pristine backyard with stunning and spectacular scenery, charming villages and pueblos, and indigenous cultures still connected to the land, a retreat in the American West that offered peace and renewal to those weary of modern, industrial, urban life.\n\nCharles Lummis and Edgar Hewett went so far as to predict that Santa Fe, the headquarters for the Archaeological Institute of America's School of American Archaeology, would soon take its rightful place alongside the world's great cultural centers of Athens, Rome, and Jerusalem.\n\nWhile working on the Balboa Park exhibits, Hewett had also managed to supervise the renovation of the old Palace of the Governors on the Santa Fe plaza which had commenced in 1909. In his absence, Hewett left the renovation in the able hands of Jesse Nusbaum, Sylvanus Morley, and Carl Lotave, and by 1913 the renovated Palace of the Governors gave the Santa Fe plaza a historic landmark worthy of national attention.\n\nAt the Hotel De Vargas, Carol and Natalie had rooms just a block from the heart of old Santa Fe, the Palace of the Governors, and the plaza that dated to the arrival of the Spanish in 1610. Narrow dirt roads and foot trails led from the plaza into neighborhoods of adobe dwellings with flat roofs of packed dirt. Seen from a distance, the buildings of Santa Fe seemed carved from the earth. The village was watered by the _Acequia Madre_ \u2014the Mother Ditch\u2014and there were lush vegetable and flower gardens, orchards of mature fruit trees, and graceful cottonwood and elm trees shading the neighborhoods and _caminos_ nestled in the foothills and canyons of old Santa Fe.\n\nThe Sangre de Cristo (Blood of Christ) Mountains rose twelve thousand feet into the sky above Santa Fe. The town itself sat at nearly seven thousand feet above sea level, and was blessed with a clear, clean, dry, high-altitude climate. Never too hot, never too cold, the salubrious air of old Santa Fe was scented with mountain pine and aspen, and canyon sage and cedar. Even at midsummer, the nights were cool, and in winter the noon sun was bright and warm, and lunch could be taken out of doors.\n\nIt was the climate that attracted Santa Fe's first colony of expats. Tuberculars and others with chronic respiratory complaints left the East Coast and Midwest and checked into Santa Fe's Sunmount or St. Vincent's sanatoriums. Patients often arrived on stretchers, gasping for a last breath. In New Mexico, lungers found their breath and recovered their health, and in Santa Fe's thin, dry air and pristine atmosphere, most didn't just survive, they thrived.\n\nSt. Vincent's Sanatorium was built in 1911 near St. Vincent's Hospital located alongside St. Francis Cathedral a few blocks from the plaza. Sunmount was under the tutelage of Dr. Frank E. Mera, who with his brother, Dr. Harry P. Mera Jr., bought the tent-city sanatorium in 1902. Perched on a hill just north of the old plaza, by 1916 Sunmount was more health resort than hospital. Patients could be housed in spacious cottages with their families for months at a time. The main house had modern medical facilities, but also had ample and comfortable space for the frequent and numerous social events sponsored by Sunmount's residents. These included poetry readings, evening lecture programs with local archaeologists and artists (many of whom were former patients), and performances by the Sunmount thespian group.\n\nSunmount's patient population included numerous artists and writers who, after experiencing the light and life of the Indian-Spanish community nestled up against the foothills of the Sangre de Cristos, remained in Santa Fe and became permanent residents. When Natalie and Carol arrived in the spring of 1916, the Sunmount's roster of convalescing residents included the nationally known poet Alice Corbin. On April 1, the _New Mexican_ noted the recent arrival in Santa Fe of Mr. and Mrs. William Penhallow Henderson of Chicago. William Henderson was a painter of considerable success and national renown. Alice Corbin, his wife, was a respected poet and a founder and associate editor of _Poetry_ magazine, one of the most influential and important literary magazines of the time.\n\nCorbin was thirty-five when she was diagnosed with advanced TB. With her family, she left Chicago, \"the warm center of her world,\" for New Mexico, \"the ragged edge of the universe.\" \"I had been thrown out into the desert to die,\" she wrote of her March 1916 exile to Santa Fe, \"like a piece of old scrap-iron, or a rusty Ford.\"\n\n**Figure 33.** Alice Corbin and William Penhallow Henderson.\n\nWith their nine-year-old daughter, Little Alice, Alice and Will Henderson moved into a cottage at Sunmount where it was believed Big Alice might die. But with the care of the staff and the healing attributes of the clear air of Santa Fe, Alice Corbin recovered from tuberculosis. Instead of ending her life in \"desolate exile,\" Corbin \"found a new world of beauty.\"\n\nLike a dozen other creative artists and writers who convalesced at Sunmount, Alice and Will Henderson were enamored of the people and lifestyle of the quaint town of Santa Fe, and invigorated by the variety and the vitality of the indigenous cultures thriving in New Mexico. The Hendersons moved out of Sunmount and into a primitive adobe dwelling on the mostly uninhabited Telephone Road (so named for the telephone line that used its right of way), and in the next few years built a comfortable home where they raised their daughter, pursued their art and craft, and became the nucleus of Santa Fe's art colony.\n\nNatalie had suffered respiratory ailments since she was a child, and at forty-one, her health continued to concern her family and friends. She spent the winter of 1916 in California, and was in good health when she arrived in Santa Fe in April. However, in the months that followed, Natalie and Carol embarked on several strenuous expeditions, and by midsummer Natalie would take a room at St. Vincent's Sanatorium for a few days of rest and recuperation.\n\nNatalie and Carol were introduced to the Hendersons that summer, perhaps by their common friend, Edgar Hewett. At the Hendersons' adobe home on the renamed Camino del Monte Sol (Road to the Sun Mountain, the name used by the locals) there could always be found a friendly and energetic group of artists and archaeologists sharing coffee, conversation, or cocktails. As Alice Corbin regained her health, she promoted Santa Fe as a writer's colony and, imagining a dynamic new literary arts center in the West, encouraged her many literary friends to come out to New Mexico and visit her splendid home.\n\nThe American writer Willa Cather and her companion, Edith Lewis, came to Santa Fe in the summer of 1916. Cather and Lewis had visited Mancos and Mesa Verde the summer before, and had traveled on the same Denver and Rio Grande narrow-gauge railroad from Denver, up and over La Veta Pass, and across the valley to Durango which had brought Carol and Douglas to Mancos a few months ahead of them. In Mancos, Cather heard the story of John Wetherill's brother's discovery of Mesa Verde while looking for lost cattle in the canyons below the cliff dwellings. With Lewis, Cather spent a week climbing about the unexcavated cliff dwellings in the rough country of Mesa Verde's Cliff and Soda Canyons.\n\nDuring her stopover in Santa Fe, Cather visited the villages and pueblos of the Espa\u00f1ola Valley and explored Taos and the small villages found in the mountains north along the Rio Grande. Cather was working on the first draft of _My \u00c1ntonia_ , but the landscape and people of the Southwest had cast their spell on her, and she had begun to gather the material that would become _The Professor's House_.\n\nSanta Fe's population and infrastructure were undergoing a historic transformation the spring that Carol and Natalie unpacked their bags in the Hotel De Vargas. The new Museum of Art, designed by the same Rapp brothers who had designed the \"Cathedral in the Desert,\" the New Mexico Building at Balboa Park, was under construction on the northeast corner of the plaza. The new art museum, as it was colloquially called by the locals, was under the direction of Edgar Hewett, who was also the director of the School of American Archaeology (which in 1917 changed its name to the School of American Research). In Santa Fe, artists and archaeologists shared the same professional and social circles.\n\nHewett and the chamber of commerce were energetically promoting Santa Fe as \"The City Different.\" Hewett's New Mexico exhibits at Balboa Park had aroused national attention in the art and culture of New Mexico, and capitalizing on the popularity of all things New Mexican, he invited nationally known artists to spend a summer in Santa Fe. The chamber of commerce helped visiting painters find inexpensive housing, of which there was a plethora, and Hewett offered artists work and exhibit space free of charge at the Palace of the Governors. Artists and writers began to arrive in droves. Santa Fe, like Paris, promised artists and intellectuals an inexpensive Bohemian life unrestrained by traditional social rules and mores. \"No paved streets, no automobiles, one sewer line. . . . A passenger could ride all over town in a horse-drawn taxi for a quarter.\"\n\nThe word was getting out: In New Mexico, a new society based on art and creativity was flourishing within and alongside some of the oldest and most picturesque indigenous communities found in North America.\n\nAlthough the new art museum would give the community its first real exhibition space, the residential artists valued Santa Fe's primitive qualities as much as its exquisite sunsets and sparkling vistas. In 1916 New Mexico, there were few automobiles and only a handful of roads that could accommodate them. Transportation of people and goods was by horseback, in buggies and wagons, and on the backs of burros. Electricity, indoor plumbing, telephones, and central heating were rare in private homes and most businesses. Spanish was spoken more than English, and although the tourists' hotels, restaurants, and upscale shops and businesses located around the plaza had the amenities and services found in modern America, the persona of Santa Fe and environs was still that of the frontier outpost at the end of the Santa Fe Trail.\n\nHewett's tactics to attract a colony of artists to Santa Fe were very successful. Several years before the new museum opened in November 1917, the studios in the Palace of the Governors were home to Balboa Park artists Gerald Cassidy, Donald Beauregard (who was with Wetherill, Cummings, and Douglas when they reached Rainbow Bridge), and Carlos Vierra (who with his wife remained in Santa Fe after recovering his health at Sunmount), and to Santa Fe part-time and full-time residents William Penhallow Henderson, Paul Burlin, and George Bellows, among others. And in the summer of 1916, after much effort, Hewett brought American art headliner Robert Henri to Santa Fe.\n\nHewett initially wanted Henri's advice about the contemporary art exhibit that would open the art museum in 1917. But by the fall of 1915, Hewett wanted to convince Henri to come to Santa Fe and set up his painting studio in one of the spaces at the renovated Palace of the Governors. Having Henri in residence would bring the Santa Fe art colony and the formation of the new art museum national attention. After months of correspondence, the Henris accepted Hewett's invitation and made plans to arrive in New Mexico by midsummer of 1916, for a several months' stay in old Santa Fe.\n\nAFTER ALL OF THREE DAYS in the comfortable and civilized accommodations of the Hotel De Vargas, Carol and Natalie packed up their gear and struck out for the Pajarito Plateau west of Santa Fe. Their destination was the Pajarito Club at the Ramon Vigil Ranch, a few miles and several canyons from Edgar Hewett's cliff-hugging field school in El Rito de los Frijoles.\n\nThe Pajarito Club, a small, rustic guest ranch tucked into Pajarito Canyon, had been owned since 1913 by four Michigan-based partners: Roy D. Chapin and Henry Joy, executives of the Hudson Motor Company and the Packard Motor Company, respectively; and the brothers Paul and David Gray, Detroit bankers. The Pajarito Club was built primarily as a vacation retreat for its wealthy owners, their families, and friends.\n\nThe Ramon Vigil Ranch and the Pajarito Club were managed by a fifth partner, Ashley Pond, who lived with his wife and three children at the club. Pond was a former member of Roosevelt's Rough Riders, who, after coming to New Mexico to recover his health, had become an avid outdoorsman. Pond's dream was to start a boys' school in New Mexico where young men from back east would be educated in the great outdoors.\n\nIn April 1916, when Natalie and Carol made their first visit to the Ramon Vigil Ranch, Pond and his Detroit partners were very much at odds, financially and otherwise, about the purpose and future of the Pajarito Club. Was it to remain a rustic outpost for private expeditions, or ought the club become a luxurious dude ranch that catered to paying guests? This country was the same plateau that Charles Lummis had hiked and explored with Adolph Bandelier before 1900, and within the 32,000 acres of the Ramon Vigil Ranch were dozens of unexcavated archaeological sites, including prehistoric cliff houses. Pond wanted the partners to invest in improvements to the existing lodge and cabins so the club could accommodate more guests. But the Detroit partners wanted the ranch they rarely used\u2014the Gray brothers had yet to lay eyes on the place\u2014to cost them no money until they figured out what they wanted to do with the remote property. At the very least, Pond thought the owners ought to visit the ranch and appreciate the truly spectacular piece of New Mexico real estate they owned.\n\nExactly what Natalie and Carol knew about the Pajarito Club's status during their first stay at the ranch is unknown. They likely chose to stay at the guest lodge because of its proximity to Hewett's field school in nearby Frijoles Canyon. Although archaeological teams rented the rustic cabins of the Pajarito Club during their fieldwork at sites on the Pajarito Plateau, ranch guests were more often wealthy visitors from back east looking to explore the New Mexican wilderness for a week or two.\n\nThe canyon of El Rito de los Frijoles, in which Hewett and his students and colleagues had spent years excavating cliff ruins and ancient pueblo villages, was a six-hundred-foot-deep cut in the massive Pajarito Plateau on the southeastern flank of the Jemez Mountains. Named Pajarito (Little Bird) by Hewett, the region was considered to be one of the richest locales of archaeological sites in the West, and perhaps in the world.\n\nLocated in the pi\u00f1on and ponderosa pine country west of the Rio Grande at the base of the Jemez Mountains, the plateau had been formed by two massive volcanic eruptions more than a million years ago. The crater from those eruptions was an enormous grassy bowl known as the Valles Caldera, or Valle Grande. Over millions of years the landscape surrounding the caldera was eroded and carved into a rugged plateau with steep, narrow canyons wedged between elongated mesas of porous rock that extended like fingers away from the crater.\n\nTo reach the Pajarito Plateau, Natalie and Carol likely hired one of the vehicles owned by E. J. Ward's Rocky Mountain Camp Company. Ward's outfit had been taking visitors into the outback on horseback since 1910 and in automobiles since 1914, and his drivers were considered the most experienced with New Mexico's famously dreadful roads. If the weather held, it took three hours to drive the twenty-seven miles from the Hotel De Vargas to the Pajarito Plateau. If it rained or snowed, or an axle broke, it could mean spending the night camped alongside the road.\n\nThe Denver and Rio Grande narrow-gauge railroad\u2014affectionately known as the Chile Line\u2014did serve Santa Fe at Guadalupe Station, and travelers for the Pajarito could take the little train north and west to Otowi Station on the Rio Grande, connecting with a wagon or an automobile to cross the river at Buckman and continue into the steep canyons of the Pajarito. But for most travelers, it was more convenient to hire a car and a guide and be driven from Santa Fe to the plateau.\n\nThe automobile and wagon road for the Pajarito Plateau followed the Pojoaque Wash to the pueblo of San Ildefonso on the Rio Grande, the same village where Natalie's friends, the famed potters Maria and Julian Martinez, lived. From here the road angled south along the river until it reached the \"town\" of Buckman, which was just a cattle corral, lumber-loading platform, and small depot where the Chile Line, which began in Utah, reached the end of the line in the deep bottom of White Rock Canyon. The river crossing there, the Buckman Bridge, a rock and timber assemblage no wider than a single vehicle, had served Hewett, Lummis, Kidder, Chapman, Nusbaum, and legions of Pajarito-bound explorers and archaeologists for nearly a decade.\n\n**Figure 34.** Julian Martinez, Ceremonial Cave, Rito de los Frijoles, Bandelier National Monument, 1910.\n\nFrom Buckman Bridge, wagons and autos began the steep ascent to the Pajarito Plateau on the old Buckman sawmill road, a narrow, rutted, rocky trail that climbed nearly straight up the canyon on switchbacks across stone shelves. Ward's drivers were accustomed to the ledge-hugging road that clung precariously to rims hundreds of feet above canyons, but the journey to the Pajarito Plateau and the Pajarito Club was often an exhausting and nerve-wracking affair for their passengers.\n\nIn the spring of 1916, one of the Rocky Mountain Camp Company drivers was a former Jemez Mountains forest ranger named Richard LeRoy Pf\u00e4ffle. Roy Pf\u00e4ffle owned a livery stable in Espa\u00f1ola and guided dudes on horseback into the outback of northern New Mexico and Arizona. He also drove visitors from Santa Fe to the Ramon Vigil Ranch, and Pf\u00e4ffle could have been the driver behind the wheel of the Rocky Mountain Camp Company's Oberlin touring car that carried Natalie and Carol to the Pajarito Plateau in the third week of April 1916.\n\nPf\u00e4ffle was compact and athletic, a handsome, talkative bachelor cowboy in his early thirties. Years later, Carol's friends would remember that she and Pf\u00e4ffle first met on the platform of the Denver and Rio Grande Railway station in the small town of Espa\u00f1ola, New Mexico. Canon Douglas and his son, Eric, were in Santa Fe several times that summer, and engaged Roy Pf\u00e4ffle's services on expeditions into Indian Country and to the Grand Canyon. Carol could have met Pf\u00e4ffle through Douglas, or vice versa. However they first met, the music teacher from Boston and the cowboy guide from Iowa liked one another in a way that changed everything for each of them by early summer.\n\nNatalie and Carol liked the rustic quality of the Pajarito Club and returned for extended stays several times that spring. They rode over to Hewett's excavation sites in Frijoles Canyon, and went with guides, likely Pf\u00e4ffle, on horse camping trips in the Jemez Mountains. Anticipating a summer in the outback, Natalie wanted her own equipment, and she wrote George to send her bedroll and camping gear stored in his ranch shed.\n\nThe Detroit owners of the Pajarito Club and the Ramon Vigil Ranch were becoming more and more frustrated with Pond's management style, which they believed was costing them money. Pond may have shared details of the struggles with his Michigan partners with Natalie and Carol, or they may have heard rumors about the pending dissolution of the partnership from Pf\u00e4ffle or others who worked in the Santa Fe tourist business. Natalie and Carol were looking for a place in New Mexico where they could put down roots, and as the gossip about Pond's potential departure from the Pajarito Club escalated, the two friends began to talk about what they would do if they were the managers of the guest ranch on the Pajarito.\n\nBecause the Ramon Vigil Ranch was wedged between canyons containing some of the most extensive archaeological sites found in the Southwest, and the Pajarito Club was only a few miles from Frijoles Canyon, there was already discussion about making the guest ranch into a cultural center that would offer visitors historic context to the cliff dwellings and ancient villages of the Pajarito. In 1914 Pajarito Club owner Roy Chapin had vigorously pushed for the passage of House Bill No. 14739, which would have established the National Park of the Cliff Cities on the forest service lands surrounding the Ramon Vigil Ranch. The bill did not pass, but by 1916 Chapin, the only one of Pond's partners who had made the trek to New Mexico to see the ranch, was still interested in how the Pajarito Club could be a destination for tourists visiting Frijoles Canyon and other ruins under excavation by Hewett and company.\n\nHow Natalie and Carol came to know about Chapin's interest in developing the Ramon Vigil into a cultural center is not known. But by May, Natalie was eager to share their ideas and plans for the ranch with her brother George and told him in letters that he needed to come to New Mexico. Immediately.\n\nOn May 19, Alice Klauber and her friend Mrs. Heinricks drove out from San Diego to visit George at his ranch. During her visit, Alice learned that George did not want to go to New Mexico. She would have sympathized with his reluctance to leave his ranch: he was still digging wells and wrestling with the ranch water system, was building an apiary, and was at war with gophers. But Alice had stopped over in Santa Fe at least once, and she knew all about the special aura of New Mexico; she understood Natalie's enthusiasm for the life and people of Santa Fe. In addition, she, like George, had made friends with New Mexico artists and archaeologists when they worked on the exhibitions for the Balboa Park fair. Everyone who went to New Mexico became enamored of the place. And in 1916, something special was happening in Santa Fe.\n\nOn May 20, Natalie sent two telegrams to George insisting he get on the train for New Mexico. In his journal George swore about Natalie's summons, but, bound by duty and love to his sister, and perhaps a bit curious about what she was so eager to talk with him about, he packed his bags, left the ranch animals, garden, orchard, and beehives under the care of a neighbor, and jumped on the stage north to Los Angeles, where on Monday, May 22, he caught the 9 p.m. train for Lamy, New Mexico.\n\nAlice may have been tempted to go with George on this impromptu excursion, but she remained in San Diego. Since Henri would be in Santa Fe by mid-July, Alice was making plans to be in the City Different by late summer when she could rendezvous with her former teacher and friend.\n\nThe Santa Fe train was running four hours late, and George's rail excursion across the Arizona\u2013New Mexico desert was hot and tedious. Two days later, he disembarked at Lamy, and the heat and discomfort of the train car dissolved into the cool, clear New Mexico morning. A bus took the train passengers on a road that wove across the sage- and pi\u00f1on-studded foothills into Santa Fe. Natalie met George on the plaza, and after unpacking his bags at the hotel, they met up with Carol (whom George called Caroline) and walked to the Rocky Mountain Camp Company stables, where they saddled up horses and went out riding in the hills above the old town.\n\nEven the reluctant tourist George found the landscape and people of Santa Fe irresistible, and on the first evening he was enchanted by the setting sun that turned the landscape into \"a fairy land of purple mountains.\" The next morning, he was up early for a stroll around the plaza. With Carol and Natalie, he found Hewett at his office in the Palace of the Governors, and they went for a tour of the new art museum under construction across the road.\n\nDuring the next week there were supper parties under the stars, morning horseback rides into the foothills, and leisurely afternoons spent wandering adobe neighborhoods with Carol and Natalie. In spite of himself, George fell under the spell of the people and the mystique of old Santa Fe. \"Starlit air or some sweet influence, soothed me earlier to sleep,\" George wrote in his diary. \"The sacred mountain at dawn was still wrapped in clouds.\"\n\nNatalie had also summoned the Curtis family's financial adviser, Elbridge Adams, to Santa Fe, and he arrived from New York on May 29. Natalie wanted Adams's advice about the ambitious proposal she and Carol were preparing for Roy Chapin and the Pajarito Club owners in Detroit. George and Carol remained in Santa Fe while Natalie took Adams on the Chile Line to Buckman, where they were met by a car that took them up the mesa to the Ramon Vigil Ranch.\n\nNatalie and Carol were about to propose to Chapin that the Pajarito Club be developed into a major Indian arts, crafts, and cultural center for the Pajarito Plateau. Frank Mead was part of the professional team Natalie was assembling. Adams would serve as legal counsel, and as such, Natalie and Carol wanted him to carry their proposal to Chapin in Detroit and negotiate on their behalf.\n\nThe idea to build a cultural center to showcase Indian art and history had been a topic of conversation between Natalie and her friends long before she came to New Mexico in 1916. \"Before we ever saw Pajarito,\" Natalie wrote Chapin, \"we had in mind the creation of a centre in the Southwest for people of culture, who, nevertheless, know little of the interest (historical, artistic and ethnological) nor of the beauty and value of this Spanish-Indian part of the United States.\"\n\nEarly in 1916, Natalie, Frank Mead, George, and likely Alice had discussed the creation of an Indian art and crafts ranch in Southern California. Although Balboa Park had generated significant interest in Indian art and culture, Southern California lacked indigenous attributes important to their concept, and by spring the consensus was to find a location in the Rio Grande district of northern New Mexico where the landscape, the density of archaeological sites, and the vital life of the puebloan people provided the perfect combination of scenic and cultural attractions.\n\nIt was an ambitious undertaking, but after touring the Ramon Vigil Ranch and reviewing their formal plan, Adams believed that the women's proposal was sound and agreed to take it to Chapin. Adams left Santa Fe in late June\u2014George had departed New Mexico several weeks earlier\u2014and presented Natalie and Carol's ideas to Chapin in his office at the Hudson Motor Company.\n\nRoy Dikeman Chapin was a Michigan native who at the age of twenty-eight had directed the consortium of engineers and merchants that founded the Hudson Motor Car Company in 1908. Chapin was an enormously powerful individual in the American auto industry, but he was also approachable and \"friendly to an exceptional degree\" with a \"gift of participating in another's outlook as though it were his own.\" After meeting with Adams, Chapin was genuinely interested in Natalie and Carol's proposal for his remote New Mexican ranch. After Adams's visit, Chapin penned a letter to Natalie telling her that he liked the women's ideas about the development of the Pajarito, and that he wanted to meet with Frank Mead as soon as possible.\n\nNatalie and Carol received news of Chapin's interest in their proposal in a letter from Adams carried to them in the saddlebag of a cowboy who found the women camping somewhere in the outback. Natalie hastily penciled a short response to Chapin telling him that she and Carol were on a camping expedition and that she would contact Mead as soon as they returned to \"quasi civilization\" in mid-July. Natalie apologized to Chapin for the condition of her letter that \"goes to Santa Fe in a guide's pocket.\"\n\nUpon their return to Santa Fe, Carol remained in residence at the Hotel De Vargas, but Natalie, who was not feeling well, stayed in bed in rented rooms on Buena Vista Street. Both women corresponded with Chapin, and by July he invited Natalie and Carol to come to Detroit.\n\nWinfred Douglas and Eric were in Santa Fe several times that summer, and they hired Pf\u00e4ffle to be their outfitter and guide on at least two expeditions into Indian Country. Douglas was an informal consultant for the Pajarito project, and his name was mentioned in at least one letter to Chapin because he was an old friend of the Gray brothers, the Detroit bankers who had yet to visit New Mexico.\n\nAshley Pond was not included in the flurry of letters passed back and forth between the women in Santa Fe and the ranch partners in Detroit. Chapin's insistence that Pond not be privy to their discussion about the ranch's future bothered Natalie and Carol, who had become friends with the \"kindly and good-hearted\" Pond and his family. Although they did not want their plans to offend him or damage their relationship, negotiations between Natalie, Carol, and the Detroit partners proceeded without Pond's knowledge until August, when Chapin finally wrote to Pond of the women's plans.\n\nThe ladies' plan would transform the Pajarito Club's lodge and cabins into the headquarters and outfitter for the ethnologists, archaeologists, writers, artists, and tourists who wanted to be immersed in the great canyons, mesas, and mountains of the Pajarito Plateau. To maintain the pristine beauty of the country, all expeditions would be by foot or horse only. The mission of the new Ramon Vigil guest ranch would be to celebrate and promote the art and culture of the Indian Southwest. The finest Native American arts and crafts would be displayed in a museum and sold in the museum shop. All new construction or renovation at the Pajarito Club would employ the architectural style championed by Frank Mead in Southern California and applauded by Roosevelt in his 1913 _Outlook_ article. Natalie sent Chapin photographs of Mead's Spanish-Indian buildings in the event that he was unfamiliar with southwestern style.\n\nChapin told Natalie and Carol that their plans \"fit in very closely with what all of us, as owners, have in mind.\" Chapin's only reservation had to do with finances, since his partners weren't looking to spend more money on their rustic hunting camp. Even so, Chapin thought a meeting with Mead would advance a plan that would be mutually satisfactory.\n\nProvidentially, by late summer Ashley Pond informed his Ramon Vigil partners that he and his family were going to leave the Pajarito Club in the fall. Pond had long held the dream of starting a boys' school in the New Mexico outback, and in late summer of 1916 he had struck a deal with the owner of the adjacent Brooks Ranch to create an outdoor school that would promote \"a sound mind and a healthy body.\" By fall, Pond planned to formally open the Los Alamos Ranch School on a high mesa just a few miles across the Pajarito Plateau from the Pajarito Club.\n\nAfter several lengthy camping expeditions with Carol and friends, Natalie's health was failing again, and by August it was decided that Carol would journey alone to Detroit to meet Roy Chapin. Armed with detailed plans for a cultural center and Indian heritage museum on the Ramon Vigil Ranch, Carol climbed onto the train from Lamy to Chicago and Detroit on August 16.\n\nUnfortunately, recent telegrams from Nahant had brought Carol the devastating news that her inheritance had been squandered by poor management. Carol still owned the Johnson house on Dorothy Cove, but with her bank account now greatly diminished, she needed a secure situation. Carol's new twist of misfortune added urgency to negotiations with Chapin.\n\nAdams had introduced their plan and opened the door for Natalie and Carol to negotiate with the Detroit men. Now it was up to Carol to convince these captains of industry that their proposal and credentials were solid, and that she and Natalie, with the advice and input of friends and advisers that included Mead, Douglas, Hewett, Adams, Lummis, and Alice Klauber, could develop the Pajarito Club into a cultural center.\n\nCarol evidently gave Chapin a professional and convincing presentation, because he told his partners that he was \"tremendously impressed\" and found Carol Stanley to be very attractive and to have a charming personality, \"quite the type that we all desire to have in charge of the property.\" After Carol departed Detroit for Santa Fe, Chapin wrote Henry B. Joy and the Gray brothers\u2014all of whom were vacationing on Cape Cod\u2014that they needed to seriously consider underwriting the cost of the proposed heritage center. \"While you plutocrats are enjoying sea breezes, I am sitting here sweltering in the heat, trying to put our Pajarito Ranch on the map. . . . Something has transpired that looks as though it might be particularly interesting.\"\n\nCarol socialized with Chapin's wife and, by the end of her visit, was comfortable enough with Chapin to tell him that her once-sizable inheritance had been lost to mishandling and negligence by the estate executor. \"Neither Miss Curtis or Miss Stanley have any income of any great amount. Miss Stanley had hoped to finance the scheme such as Miss Curtis has outlined, from her inheritance, but she discovered very recently that the Executor of her Mother's Estate had dissipated practically all of the funds and she is now in a position where her income is not enough to keep going and as she puts it very plainly,\u2014she must go to work.\"\n\nIt was a grand scheme for a unique piece of real estate in the New Mexico wilderness, a scheme that needed considerable funding from a small group of investors who lived in the modern world of automobile production and high finance. Chapin seemed confident enough in Natalie and Carol's management abilities, but his partners were reluctant to finance such a grandiose project on a remote mesa on the ragged edge of the New Mexican frontier with two single women at the helm.\n\n#  _12_\n\n##  REMARKABLE TIME, REMARKABLE PLACE\n\nLooking back, it is as if these men and women knew of their place in history even as they sat around the evening campfires at Puye and Rito. . . . Magic itself informed the early years of [Hewett's] school.\n\n##  \u2014 _Marjorie Lambert_\n\nTHROUGHOUT THE SUMMER and fall of 1916, Santa Feans observed the daily progress of the massive pueblo-mission building that was the new art museum emerging from the sandy dirt on the northwest corner of the plaza. The museum's Saint Francis Auditorium was a jaw-dropping feat of brick construction that lifted over two stories above West Palace Avenue. The auditorium's ceiling was held up by massive timber beams and carved corbels that were hoisted into position by cables. When the museum opened in November 1917, it promised to change the skyline of the old city and become the cultural, artistic, and social hub of twentieth-century Santa Fe.\n\nJust a short walk across Lincoln Avenue from the new museum, the colony of artists in residence at the Palace of the Governors was industriously preparing for a late-August group show of New Mexico landscapes and portraits. To Hewett's delight, Henri was in residence and working in the old Palace, and had also agreed to hang the late-summer exhibition, which would include his Balboa Park compatriots Carlos Vierra, Gerald Cassidy, and Kenneth Chapman, and additional artists new to Santa Fe like the New York modernist Paul Burlin, who had been one of the youngest artists to exhibit at the 1913 Armory Show.\n\nHewett's art exhibitions at the Palace of the Governors, and soon at the Museum of Fine Arts, implemented Henri's \"open-door\" policy that allowed any artist working in New Mexico to participate in public shows. Modeled on the exhibition practices favored by Paris artists, which removed the critic as gatekeeper and encouraged a nonacademic, nonjuried environment, this policy let the artists themselves choose what to exhibit. Henri and Hewett's \"open door\" at the old Palace and a year later at the new art museum instigated a considerable migration of prominent painters to New Mexico, and the formation of the Santa Fe modernists' movement by 1920.\n\nAlice joined her expat pals in Santa Fe in August, arriving at Guadalupe Station at high noon on Saturday, August 12. Even with so many friends and colleagues living within a few blocks of the narrow-gauge railroad depot, no one met her train. Alice made her way to the Hotel De Vargas, checked in, enjoyed a light lunch in the hotel dining room, and, still unable to connect with Natalie or Henri, set out to see the little town.\n\nAlice walked across the plaza to the Palace of the Governors and found the front doors locked. She was told that everyone\u2014the Hewetts, the Henris, Natalie, and her friends\u2014had left town and driven out to see the dances at Santa Clara Pueblo. Alice wandered the charming alleyways and dirt roads of Santa Fe and perused the small shops around La Fonda and the plaza. She was just beginning to feel lonely when Natalie telephoned her at the hotel. Natalie and Carol Stanley, whom Alice had not yet met, were recovering from a strenuous expedition to Acoma and Zuni Pueblos during which they both came down with \"a queer dysentery\" that had placed them in beds in a Gallup hotel for five days. Carol apparently was on the mend, but Natalie had been bedridden and staying in rented rooms on Buena Vista Street for most of the past three weeks.\n\nAlice went in search of Henri and found him taking in the pleasant evening from a chair on the front porch of his rented abode on De Vargas Street. Henri was sharing the house with his wife, Margie \"O,\" his sister-in-law, Violet, their friend Lucy, and several dogs. Henri told Alice that Hewett took care of everything, and that everyone in the Henri household was quite satisfied with life in the little capital. Alice ate supper with the Henri clan, and then walked with them to the plaza to people watch. A rustling in the trees heralded a drenching downpour that sent them scurrying to the Hotel De Vargas. Soaked to the skin, the Henris called for a buggy and said good-night to an equally sodden Alice. Alice retired to her room, changed out of her drenched clothing, and wrote in her diary what a truly glorious first day it had been in old Santa Fe.\n\nThe next morning, Carol, who kept her Santa Fe residence in the Hotel De Vargas, knocked on Alice's door and introduced herself. Carol was in the company of a woman named Dorothy Kent, and they invited Alice to go with them to Tesuque Pueblo northwest of Santa Fe.\n\nDorothy Kent's home was in Tarrytown, New York, a community on the Hudson River seventeen miles south of Peekskill, where Winfred Douglas lived. Dorothy was a twenty-six-year-old painter and a classically trained violinist. She was the younger sister of the up-and-coming artist Rockwell Kent, who, coincidentally, had been a friend and classmate of Walter Pach's at the New York School of Art. Rockwell Kent was among those who, with Pach, had joined Henri and the other members of the Ashcan school of \"realist renegades\" in the 1910 Exhibition of Independent Artists.\n\nDorothy Kent had studied painting with Homer Boss in New York. Through her association with Boss, and with her brother, Rockwell, with whom she had a famously contentious relationship, Dorothy was tangentially involved with the birth of the Society of Independent Artists in 1916. At the same time that Henri and Hewett were advocating an \"open-door\" exhibition policy in Santa Fe, Pach, Boss, Kent, and their circle of New York modern artists, writers, and musicians were drafting the society's \"no-prize, no-jury\" mission policy. Dorothy, like her brother, would participate in the society's first exhibition at Grand Central Palace, New York City, in 1917.\n\nOn her third day in Santa Fe, Alice met her \"Santa Fe self which is a new woman.\" Santa Fe Alice rose early and stayed out with friends until close to midnight. She sketched in her notebook and painted watercolors of the streets and people of old Santa Fe. She dined with the Henris and the Hewetts, and the new museum's construction foreman, Jesse Nusbaum, whom Alice had become friends with in San Diego. She went to the movies with the Henri clan, and with Natalie, Carol, and Dorothy Kent. Alice was reading Lummis's _Land of Poco Tiempo_ and took her book to a bench among the hollyhocks in the Palace of the Governors courtyard near Henri's studio, where he was working on a portrait of a puebloan, Willie Begay.\n\nOn Wednesday, August 16, the same day that Carol climbed on the Santa Fe train bound for Chicago and Detroit and her meeting with Chapin, Alice and a friend, Miss L. W. Wilson, took the Chile Line northwest from Santa Fe on the first leg of a long day's trip to reach Hewett's field camp at Puye near Santa Clara Pueblo. The summer monsoons had filled the Rio Grande to flood level, and after Alice climbed off the train and crossed the river at the Buckman Bridge, she was met by Crescensio Martinez, who would take her by wagon to Hewett's camp.\n\nCrescensio, the brother of Julian Martinez, was a gifted self-taught watercolorist. Alice had met Crescensio several years before at Balboa Park. He was now working in a studio near Henri's in the Palace of the Governors and, with Awa Tsireh (also known as Alfonso Roybal), also of San Ildefonso Pueblo, Fred Kabotie of Hopi, and Ma-Pe-Wi of Zia, was part of the first wave of pueblo artists whose work would bring national recognition to contemporary Native American art.\n\nFrom Buckman Bridge, Crescensio and his horse and wagon took Alice twelve miles north along the Great River via the lands of San Ildefonso and Santa Clara Pueblos, and then west into Santa Clara Canyon and the cliff dwellings under excavation by Hewett at Puye. As dusk fell, Crescensio pulled the wagon up to a camp at the base of massive cliffs. There were two snapping bonfires, and some fifteen people\u2014all relatives of Crescensio's\u2014moving between a dozen tents. Alice joined them and shared her picnic supper in exchange for hot coffee. It was a congenial and lively camp at the foot of the cliffs, but it was not Alice's final destination. After supper, still in her traveling skirt and laced boots, with duffel bag in hand, Alice followed Crescensio on foot up the dark cliffs. They climbed a ladder straight up the stone walls, and followed the deep grooves of ancient trails across narrow ledges of volcanic rock to the ridgetop camp of Dr. Archaeology, Edgar Hewett.\n\nAlice was welcomed into a camp community of more than twenty students and professional archaeologists. After an hour beside their campfire under the sparkling star-filled sky of the Pajarito Plateau, Alice was taken to her tent. She climbed into her bedroll and within minutes collapsed into sleep.\n\nThe men and women sitting around the campfire that night were participating in a summer field school that would be remembered as a charmed chapter in the story of archaeology in the American Southwest. On the high ledges of Puye and in the canyon of El Rito de los Frijoles, which she would visit several days later, Alice was among the world's most rarefied assembly of archaeologists. It was a remarkable time in a remarkable place. \"Most of those who came to study, work on, and contribute to Hewett's programs at Puye . . . and the Rito de los Frijoles . . . were to become giants in the field of American anthropology and also legends in their own lifetime.\"\n\nHewett's field schools and their sponsoring organization, the School of American Archaeology, were \"peopled with giants.\" Adolph Bandelier, Alice Fletcher, and Charles Lummis all worked with Hewett's school in some capacity. Jesse Nusbaum, Ted Kidder, Kenneth Chapman, Sylvanus Morley, John P. Harrington, Donald Beauregard, F. W. Hodge, Neil Judd, Maude Woy, Barbara Friere-Marreco, even San Ildefonso potters Maria and Julian Martinez, worked alongside El Toro, the enigmatic, domineering, forceful bull who was Edgar Hewett.\n\nThe field school's day commenced at dawn with breakfast served at a long plank table under a large canvas tarp. On her first morning at Puye, Alice went with Hewett's team to an excavation site where she watched the unearthing of a large ceramic vessel. After a picnic lunch, Alice climbed a narrow trail down the cliff side to a cave where two young Santa Clara potters, Romana and Maximiana, were teaching Hewett's students the fundamentals of traditional pueblo ceramics. At five o'clock, Alice, Hewett, and his crew ascended the cliff back to camp. Alice had tea in Donizetta Hewett's tent and participated in the hour-long Spanish class before supper. Evening was spent, once again, around the campfire under the wide, star-studded sky.\n\nEach day for the next week, Alice's routine was the same. She often accompanied Hewett on his treks to various excavation sites, including one six-mile hike across several canyons and up a cliff ladder to a site on a neighboring mesa. Hewett was excavating a large kiva, and he asked Alice to sketch petroglyphs and artifacts as they were found.\n\nAlice departed Hewett and his field school at Puye late in the afternoon of August 23. She went by a wagon to San Ildefonso Pueblo, where she spent the night on a hillside beside Maria and Julian Martinez's home. Maria gave Alice breakfast and a tour of the old pueblo before she went on foot to the depot at Buckman, where she met the narrow-gauge train that took her up the Rio Grande Gorge to Taos.\n\nAfter several days of hiking into the mountains and sketching the Spanish villages near Taos, and touring and sketching Taos Pueblo, Alice climbed back on the little train and traveled back down the gorge to Espa\u00f1ola and Santa Fe. It was Saturday, August 26, and Carol Stanley returned from her trip to Detroit on the same afternoon.\n\nAlice, Carol, and Dorothy Kent joined Hewett, Henri, and most of Santa Fe's residents at the opening of the exhibit of New Mexican paintings by Henri and the other artists in residence at the Palace of the Governors. Natalie, however, remained in bed, as she was still not feeling well, and her circle of close friends began to quietly discuss moving her into one of the local sanatoriums.\n\nAlso absent from the festivities surrounding the art show opening were Winfred Douglas and Roy Pf\u00e4ffle, who were making their way back from weeks in the outback. In early August, Douglas and son, Eric, had gone with Roy on a trail-scouting expedition through New Mexico and Arizona that ended at the Wetherills' in Kayenta. From Kayenta, John Wetherill had taken Douglas, Eric, and Roy across the infamously brutal Rainbow Trail. Roosevelt and his sons, the party of Zane Grey, and a few dozen lesser-known but no less intrepid individuals had similarly followed John Wetherill into the notorious canyon country to Nonnezoshe, the great Rainbow Bridge.\n\n**Figure 35.** Alice Klauber at the home of Julian and Maria Martinez, San Ildefonso Pueblo, August 1916.\n\nAfter several days of hard riding, walking, and scrambling across the hot, steep, slick rock ledges of the high plateau, the party stood beneath the bridge. On August 24, Pf\u00e4ffle and the Douglases added their names to the fabled registry tucked in the rocks. It was likely Pf\u00e4ffle's and Douglas's first visit to the sacred arch, but it would not be their last.\n\nThe August pack trip Roy Pf\u00e4ffle took with the Douglas men was historic in the annals of modern Southwest expeditions, since they were the first to travel by horseback from Santa Fe across Navajo and Hopi country to Rainbow Bridge. Pf\u00e4ffle and other guides routinely took horses and riders on the train from Lamy to Gallup, where they disembarked, packed up, and headed by horseback into Indian Country. Pf\u00e4ffle's historic outing with Douglas and his son began on horseback in Santa Fe and continued north and west across New Mexico into Navajoland, and west and north across Arizona to Kayenta. It was an epic achievement\u2014they rode nine hundred roadless miles in twenty-eight days, and successfully blazed a trail through Ca\u00f1on del Muerto at the east end of Canyon de Chelly. Pf\u00e4ffle and Douglas mapped a route from Santa Fe to Canyon de Chelly that would become the standard for horse expeditions in the next decade.\n\nIn late August, Natalie was still bedridden with a lung disorder, and with the lingering effects of the nasty germ picked up on her trip with Carol into the outback in July. Although Natalie remained involved in the discussions about the Pajarito Club's future, Carol realized that if Chapin and his partners agreed to their proposal, Natalie could not possibly live on the plateau and manage the ranch. Carol, alone, would have to go oversee the Pajarito Club for the winter.\n\nCarol, Alice, Douglas, and all of Natalie's Santa Fe friends were anxious about her ill health. On the last day of August, Alice and Carol went to St. Vincent's and made arrangements for Natalie to move into one of hospital's private rooms. The next day, they moved her into the sanatorium.\n\nSt. Vincent's was near the old plaza just off the corner of Palace Avenue and Cathedral Place, and in 1916, with seventy-five beds, it was Santa Fe's largest hospital. Private rooms were furnished with home-style beds, rocking chairs, and wide windows with a view of Palace Avenue or the cathedral gardens. Visitors were welcome, and Natalie had many during her convalescence. One person in particular, Paul Burlin, remained by her bedside.\n\nOver the spring and summer of 1916, Natalie and Paul Burlin had fallen in love. Carol knew the extent of their relationship because she was with Natalie when the courtship began. Burlin had gone on camping trips with Carol and Natalie's friends, and when Natalie moved out of the Hotel De Vargas, she had rented rooms in a little house in the same neighborhood as, if not alongside, Burlin's casita on Buena Vista Street. But Alice had only just met Burlin a few weeks before, at the Palace of the Governors where he had a studio near Henri's, and she did not grasp his significance to Natalie until she walked into Natalie's hospital room early one morning and found the artist sitting beside her bed.\n\nIt was obvious to Alice that Burlin was devoted to Natalie. In the next few days, he was never gone long from her room at St. Vincent's, and Natalie was delighted by his presence and his attention. Alice slept one night on the sofa in Natalie's room at the hospital and visited her several times each day, but throughout Natalie's four-day internment Burlin was the first to arrive at her room in the mornings and the last to leave at night.\n\nAt first glance, they seemed an unlikely couple\u2014she was forty-one, he was just thirty that summer\u2014but Natalie and Burlin had much creative and intellectual common ground. Burlin was a nationally known artist who had studied in New York and Europe before leaving the world of modern art to embark on a solo adventure in the American West. The son of an English father and a Polish mother, Isadore \"Paul\" Burlin was raised in New York City and London, and studied at the National Academy of Art and the Art Students League. In his twenties, Burlin traveled through Europe, where his work was influenced by the European avant-garde, especially by C\u00e9zanne's composition and Matisse's use of color.\n\n**Figure 36.** Paul Burlin.\n\nBut it was during his first journey to the American Southwest in 1910 that Burlin, just twenty-four, encountered what would be the most significant influence on his art: the landscape, culture, and spirituality of indigenous Native America. Burlin came to the Southwest looking for a pristine landscape and primitive culture. He found both in New Mexico. Armed with a gun given to him by New York friends, Burlin traveled by horse and wagon through the Southwest outback and \"learned to sing out of doors just to keep the demons away.\" Although nervous about the remoteness and isolation, and intimidated by the Indians and Hispanic Americans he encountered in New Mexico, Burlin fell in love with the unspoiled frontier.\n\nBurlin kept a residence on 51 West 10th Street in New York, but after 1913 he spent a substantial part of each year in what he referred to as the simple, unpretentious town of Santa Fe. He always rented the same tiny adobe dwelling on the corner of Buena Vista and College Streets (later Old Santa Fe Trail), and he was among the first of the New York painters and expats to move into the studio space provided by Hewett in the renovated Palace of the Governors.\n\nBy 1916 when Burlin met Natalie, he was a knowledgeable and competent outdoorsman, and he became part of the regular gang of good friends with whom Natalie and Carol went into the outback. Roy Pf\u00e4ffle was frequently the guide on their informal jaunts in the deep, vast network of canyons on the Pajarito and to the high country of ponderosa and aspen forests in the Sangre de Cristo and Jemez Mountains.\n\nDuring the days of Natalie's convalescence, Alice and Burlin became acquainted in conversations in the hospital, and over lunches and informal suppers at eateries on the plaza. Alice became interested in Burlin's very modern, often dark and somber art and went to view his new canvases at his studio at the Palace of the Governors.\n\nOn Monday, September 4, while Natalie recuperated in her room at the sanatorium, her friends gathered for an end-of-summer luncheon at a caf\u00e9 on the plaza. Carol, Dorothy Kent, the Henris, Alice, Paul Burlin, and the newly returned Roy Pf\u00e4ffle and Canon and Eric Douglas all recounted their summer travels and discussed departure itineraries and autumn plans. On Tuesday, Alice was at the Palace of the Governors and watched Burlin, Carlos Vierra, and Will Henderson hang \"the Taos Group\" in one of the galleries. By Thursday, Natalie moved out of St. Vincent's and, with Alice and Paul Burlin, went by automobile out to the Pajarito, where Ashley Pond helped move Natalie into one of the cottages in the canyon. In the cool quiet of the ponderosa forest, she planned to read, write letters, walk, and sleep until she was fully recovered. Pond knew of Natalie and Carol's negotiations with Chapin and his partners, but he was looking forward to the opening of his boys' ranch school, and Pond and Natalie were able to maintain their amicable relationship.\n\nAfter a visit with the Henris and the Hewetts in Santa Fe, Alice caught the Santa Fe train from Lamy to the Grand Canyon, where she spent a few days hiking and sketching. Then she headed for San Diego and her home on the Pacific.\n\nNatalie remained at the Pajarito Club until late September. During her weeks on the high plateau, she wrote Chapin several letters reaffirming her and Carol's interest in building the cultural center at the Ramon Vigil Ranch. Burlin may have stayed at the Pajarito Club for all or part of Natalie's convalescence and journeyed with her back to Santa Fe on September 27. Natalie's return to the capital was mentioned in the \"Personals\" of the _Santa Fe New Mexican_ , where it was also noted that she planned to join an expedition to the Grand Canyon in two days. Natalie would not return to New Mexico for the winter, but would continue on to San Diego, where she would spend the winter.\n\nNatalie wrote Chapin that she was planning to take the train and meet Carol and other friends at the Grand Canyon in mid-October. Unbeknownst to Natalie, Carol had left Santa Fe six days earlier with Roy Pf\u00e4ffle.\n\nRoy Pf\u00e4ffle and several other guides from the Rocky Mountain Camp Company had left Santa Fe on horseback on September 13 for another trail-blazing expedition through Indian Country. Roy's July journey with Douglas had created a buzz among the outfitters and guides who worked around Santa Fe, and Roy was happy to share what he and Douglas had learned. Roy also wanted to scout additional routes between Santa Fe and Canyon de Chelly via Chaco Canyon and Washington Pass (Narbona Pass) to Luckachukai. Carol and Dorothy Kent must have proven themselves adept campers and proficient horsewomen, because they were invited to ride with Roy and the other professional guides on an arduous three-week-long adventure in the outback.\n\nTheir whereabouts for three weeks were not recorded in anyone's journal or letters, and it is not until October 2, in Gallup, New Mexico, that the expedition reemerged into recorded history. On that day, at two o'clock in the afternoon, in the small railroad town on the edge of the Navajo Reservation, Caroline Bishop Stanley of Boston, Massachusetts, married Richard Leroy Pf\u00e4ffle of Santa Fe, New Mexico. The Reverend David A. Sanford of Gallup's Episcopalian Church of the Holy Spirit officiated. The reverend's wife, Mary Sanford, and the bride's good friend, Miss Dorothy Kent, witnessed the ceremony. The McKinley county clerk entered the bride's date of birth as December 16, 1884, her age thirty-four. Carol was actually born in 1879 and was thirty-six, but no one at the impromptu ceremony, including the math-challenged clerk, seemed concerned about the age of the suntanned, saddle-and-trail-toned bride.\n\nRoy Pf\u00e4ffle was recorded as being thirty-one years of age, although his stated year of birth was 1882, making him thirty-three. The age of the bride and groom was surely of less interest to the witnesses gathered in Gallup that October afternoon than the fact that the new Mr. and Mrs. Pf\u00e4ffle were both born on December 16. What Roy knew of Carol's life story is unknown, but at some moment during their six-month courtship, she must have mentioned to him that he shared a birthday with her and her long-lost twin, Edward.\n\nTwo days later, the newlyweds and the other members of the expedition returned on the train to Santa Fe. The success of their trip by horseback from Santa Fe to Chinle received a glowing write-up in the _New Mexican_ on October 5. However, the newspaper story did not include the Gallup marriage. That announcement would wait until Carol sent off personal telegrams and letters about her marriage to friends and family back in Boston and Nahant.\n\nThree weeks after the fact, Carol and Roy's secret marriage was formally announced in the _New Mexican_. The new Mrs. Pf\u00e4ffle, \"well-known in musical circles in the east,\" was introduced to Santa Feans as a member of the Choate family of Massachusetts. Roy Pf\u00e4ffle was already well known in northern New Mexico, since he had worked for several years with the forest service in the Jemez Mountains above the Pajarito, and then as a guide with the Rocky Mountain Camp Company.\n\nBy late October, Roy had severed his ties with the Rocky Mountain Camp Company, and the Pf\u00e4ffles moved to Espa\u00f1ola, where Roy had a livery stable and the couple hoped to open their own business. Chapin had sent Carol a warm letter of congratulations after he learned of her marriage to Pf\u00e4ffle, but he had also told her that because his partners were ambivalent about the cultural center, the project was shelved. With Roy's reputation as one of the region's premier guides, and with Carol's social connections back east, it seemed reasonable that the Pf\u00e4ffles could attract sufficient clientele to build a viable outfitting business. Even out in the tiny railroad town of Espa\u00f1ola.\n\n**Figure 37.** On the trail to Rainbow Bridge, ca. 1924.\n\nBefore joining up with the Rocky Mountain Camp Company, Roy had operated a small outfitting business out of the O\u00f1ate Hotel in Espa\u00f1ola. Roy also had family in the town: his widowed mother, Margaret, known as Gaga, and two younger sisters, Marie and Lois, had been living at the O\u00f1ate for several years. Carol had met Roy's family and was aware that his mother and sisters depended on him financially and personally. But exactly how all their lives would mesh after Carol moved into the O\u00f1ate with the other Pf\u00e4ffle women, no one could say.\n\nIn early November, Carol and Roy packed up their horses and rode west into the Arizona outback for a honeymoon in Indian Country. Their destination was the Wetherills at Kayenta. The couple retraced the route they had taken in September, and also scouted trails and campsites north and west from Chinle across Hopiland for their new outfitting business. Several weeks later, the Pf\u00e4ffles rode into Kayenta. Just eighteen months had passed since Carol had learned to ride a horse with Louisa and John Wetherill, but that New England music teacher was long gone. The new woman known as Carol Pf\u00e4ffle was very much at home on the bright, hot desert of Navajoland, and with John Wetherill and her new cowboy husband, she rode across the Rainbow Trail to the legendary bridge on the far side of the world.\n\n**Figure 38.** Rainbow Bridge, ca. 1920. \n\n#  _13_\n\n##  A RARE THING\n\nBecoming Oneself\n\nOn the other side of the Sangre de Cristo Mountains there is a great welter of steel and flame. . . . I know nothing of it here.\n\n##  \u2014 _Alice Corbin_\n\nTHE UNITED STATES FORMALLY entered World War I on April 6, 1917. The American effort to mobilize what would by 1918 total over four million military personnel immediately affected all U.S. industry and business. Detroit auto companies, including Pajarito Club owners Roy D. Chapin and Henry B. Joy, cut their automobile production in half and reconfigured their factories to serve the war industry.\n\nIn late 1916 Hewett proposed that his field school become a private corporation under the State of New Mexico. This new entity would continue to conduct field schools and respond to the research mandates of the Archaeological Institute of America, but the change in status would allow Hewett to go after much-needed financial support. In December 1916 the institute unanimously approved Hewett's proposal, and the School of America Research (SAR), with headquarters in the Palace of the Governors, was incorporated in early 1917. However, America's entry into the European conflict brought the Pajarito field schools and all excavations in the Southwest under Hewett's auspices and SAR to a halt, since \"the services of the staff and all the facilities of the School and Museum were promptly placed at the disposal of the Governor and the State Council of Defense.\"\n\nSanta Fe's new art museum was pushing toward completion and its scheduled opening in late November, although the museum staff and space were redirected to serve the activities of the local Red Cross and the Women's Naval Auxiliary. Patriotic rallies were held on the plaza in front of the Palace of the Governors and the new museum, and the Santa Fe art community channeled its prodigious creative energy into the war effort. Alice Corbin became director of publicity for the Santa Fe Women's War Effort, and her husband, Will, with Sheldon Parsons and Paul Burlin, learned from Taos artist Ernest L. Blumenschein how to use their skills with paint and brush to create \"range-finder paintings\"\u2014detailed landscapes of the French countryside to be used by army machine gunners to develop target, distance, and range assessment skills.\n\nIn January 1917 Carol and Roy Pf\u00e4ffle opened their outfitting business at the O\u00f1ate Hotel in Espa\u00f1ola. Advertised in newspapers and tourist brochures as the most experienced guide service for major expeditions to Chaco Canyon, the Natural Bridges of southern Utah, and Mesa Verde in Colorado, the Pf\u00e4ffles' business, like all tourist-based endeavors in the Southwest, received a substantial boost when the United States entered the war. Wealthy Americans canceled their trips to Europe and the Mediterranean and instead booked vacations to the American West, many for the first time. It took a war overseas, but the cultural and natural wonders of the Southwest were finally getting the attention that Lummis had been clamoring for since he first uttered \"See America First\" in the previous century.\n\nEspa\u00f1ola was a quiet, pastoral village on the Rio Grande near its confluence with the Chama River, and was flanked by the tribal lands of the Tewa pueblos of Santa Clara to the south and San Juan to the north. The only horse, wagon, or automobile road between Santa Fe and Taos passed through the center of town, and Espa\u00f1ola was linked to Colorado by the narrow-gauge tracks of the Denver and Rio Grande Railroad. The oldest foot and wagon routes to the Four Corners and Indian Country followed the Chama River valley north and west from Espa\u00f1ola to Abiqui\u00fa, and the wonders of the great Pajarito Plateau and the Jemez Mountains were accessible by horse through the canyons and mesa country to the south and west of town.\n\n**Figure 39.** Carol Stanley Pf\u00e4ffle, 1920.\n\nHeadquartered at the O\u00f1ate Hotel, the Pf\u00e4ffles' outfitting company was located at the gateway to Indian Country. Clients arrived by the D&RG railway and stayed at the hotel before and after their expeditions with Roy into the outback. Although Carol often went on these expeditions into the country she loved, her primary job was business manager. The Pf\u00e4ffles needed to expand their clientele, so beginning in 1917 Carol went each winter to Chicago, New York, and Boston to give lectures about her experiences in the American West to friends and former colleagues, hoping to sign them up for their own adventure in New Mexico.\n\nCarol supervised the daily operations of the hotel where Carol, Roy, and the extended Pf\u00e4ffle family also made their home. Carol's mother-in-law, Gaga, not only lived with Carol and Roy, she worked for them, too. For Carol, and for her in-laws, it was an uncomfortable and complicated family arrangement from the start.\n\nRoy Pf\u00e4ffle had taken care of his mother and younger sisters, Marie and Lois, since the violent death of his father, Frederick Harley Pf\u00e4ffle, in a train wreck near Avoca, Iowa, in 1907. Fred Pf\u00e4ffle was a fireman for the Rock Island Railroad, and after his death, Roy's mother and sisters left Council Bluffs and went to live with Roy on his homestead in South Dakota. Times were tough, and in 1910 twenty-six-year-old Roy gave up his job clerking at a store, boarded up his homestead, and moved with his mother and sisters to the home of relatives in the frontier ranching community of Sapello, New Mexico. Roy found work as a cowboy on ranches near Las Vegas, and eventually became a forest ranger in the mountain country near Santa Fe and Espa\u00f1ola.\n\nWhen Carol Bishop Stanley of Boston met Richard LeRoy Pf\u00e4ffle of Council Bluffs at the Espa\u00f1ola railroad depot, they were the embodiment of the western story romanticized in a dozen dime novels. She was the educated and genteel lady of the East who had left everything to find something of her own out west. He was the penniless son of German immigrants with little formal education and no experience of the modern, urban world. The match shocked Carol's and Roy's families, although among their friends in Santa Fe the marriage was really not all that remarkable. Carol was in middle age, her inheritance had been reduced to pennies, and she was alone. She could not or would not return to Nahant. Carol needed a partner in New Mexico if she was going to make a home there. Roy Pf\u00e4ffle was in his early thirties, handsome and well spoken, industrious, self-reliant, as popular with the men as with the ladies, and skilled as a guide and horse wrangler. Although he had a wild streak\u2014he loved to play poker\u2014he was liked and respected in social and business circles around Santa Fe and northern New Mexico. Everyone knew Roy had taken good care of his widowed mother and sisters for a decade, and by 1916 he was ready to have a partner in his life.\n\nRoy's elopement with the educated, middle-aged spinster with the odd, _r_ -less Boston accent brought significant challenges to his mother's household in Espa\u00f1ola. Overnight, the home and business of Roy Pf\u00e4ffle was run by his new wife, and Gaga's diaries reveal now and again a mother-in-law's repressed disgruntlement with the haughty, strong-willed, aristocratic eastern lady who stole her son's heart and moved into her family's home.\n\nIn the winter and spring of 1917, while Carol was figuring out married life with her new husband and his family in New Mexico, Natalie was in San Diego preparing to marry Paul Burlin. Burlin, himself suffering from tuberculosis, spent most of the winter of 1916\u201317 in the warm climate of Southern California. By January, Natalie and Burlin had discussed their intention to marry with George, who was given the task of telling Mimsey of Natalie's engagement plans. George had met Burlin the summer before in Santa Fe and heartily endorsed his sister's engagement to the young painter. But George was a Californian now and lived by the relaxed status quo found in the new society of the \"other edge,\" the West Coast. He knew Mimsey and the Curtis family back east would not necessarily celebrate Natalie's engagement to Paul Burlin.\n\nEarly in 1917 Natalie and Burlin traveled cross-country together to attend the opening of an exhibition in New York City of Paul's new work alongside that of Man Ray. Although the New York reviewers liked Burlin's Santa Fe work, Natalie's family, especially her mother, was ambivalent about his modern style. Mimsey was also ambivalent about her daughter's engagement to Burlin.\n\nNatalie announced that she and Paul wanted to be married in New York City. Burlin had colleagues and friends from his art school days living in the city, and some of his immediate family (with whom he was not close) still resided in New York; and almost all of Natalie's friends and professional colleagues lived in or near Manhattan. New York seemed the natural choice for their wedding.\n\nNatalie's plan to marry in New York City garnered a tepid response from the Curtis family. Although Mimsey and her siblings _liked_ Burlin, there were several matters that inhibited their support of him as Natalie's choice for a husband. He was not wealthy, and although highly respected in the modern art world, he hardly made enough income to support himself, let alone a wife. Burlin was also ten years Natalie's junior, and he was an immigrant. A Jewish immigrant.\n\nThe Curtises, for all their social-reformist agendas and liberal, progressive political alliances, were members of New York's elite old-money crowd. Mimsey made it known to Natalie that her marriage to Burlin would be an awkward affair in her hometown, and the discussion about a Curtis-Burlin wedding ceremony in New York ceased. Even their engagement was kept a secret. Mimsey later admitted to George that she had been greatly relieved that Natalie quickly understood the potential awkwardness of the event and did not pursue a New York City wedding where \"the Montagues and Capulets would not blend.\"\n\nIf Natalie was disappointed in her family's understated but potent upper-class snobbery, or was surprised by the undercurrent of anti-Semitism flowing just beneath the polite surface of New York's intellectual elite, she never said so publicly. With Burlin, Natalie returned to California and the welcoming environment of George's rural ranch and Alice's home in San Diego. By late March, Natalie and Burlin were happily residing in bohemian Santa Fe, where their unlikely romance didn't seem so unlikely.\n\nCarol was delighted to have Natalie back in New Mexico, and left her mother-in-law in charge of the O\u00f1ate to spend nights with Natalie in Santa Fe. Espa\u00f1ola and the O\u00f1ate Hotel did not offer Carol much opportunity for intellectual or creative stimulation, and she sometimes went alone to Santa Fe for evenings with Natalie and their friends.\n\nNatalie attempted to explain to family and friends her love for Burlin and the life they shared in faraway New Mexico. She told her Aunt Nat that with Paul Burlin she had at last found the \"Great Happiness\" in a relationship that was born in the out-of-doors where, in \"the enforced and inevitable intimacies of camp-life, people come to know each other through and through.\" Her husband-to-be took such good care of her, Natalie said, that she believed she would never again be ill. She and Paul shared an \"affection based on sound friendship, mutual aims and absolute intellectual and spiritual rapport,\" and Natalie assured her Aunt Nat they would have an enduring happiness \"even without riches.\" Natalie's letter to her aunt was written during a visit to Carol's new home in Espa\u00f1ola. The two friends found comfort in their similar situations, and shared thoughts and feelings about their family's reactions to their decisions to marry years \"after first youth.\" Perhaps hoping to give her forthcoming New Mexico marriage context, Natalie told her Aunt Nat that she was staying with \"a dear friend of mine, a Boston Girl who is married and living out here.\"\n\nCarol well understood the camaraderie and intimacies developed during outback expeditions, having herself fallen in love with and married her cowboy guide after knowing him for just six months. And as Carol may have done several months before, Natalie defended her choice for a husband to friends and family back east. \"Of course my family would have liked me to marry a _rich banker_! or some version of the well-to-do and 'tried [tired?] businessman' but I could never have been really happy with anyone but an artist of some kind, because my whole make-up demands that kind of understanding.\"\n\nNatalie penned similar letters to friends, including her patrons, Peabody, Mason, and Frissell, each of whom was surprised but ultimately supportive. But they were worried, too. Worried that Natalie would not return and complete her work on Negro folk songs, although she assured everyone that, with Paul by her side, she would return to the South and resume her study.\n\nTHE ROBERT HENRIS returned to Santa Fe in June 1917 and moved into the Safford house at 601 Palace Avenue. Planning to remain in Santa Fe until the art museum opened in late November, Henri set up a studio in the adobe dwelling. He wrote Alice that Santa Fe was \"more beautiful and delightful than ever.\"\n\nDuring his second summer in Santa Fe, Henri immersed himself in the domestic and artistic life of the small town at the foot of the Sangre de Cristos. His home was a pleasant walk from the old plaza and the Palace of the Governors. Henri shared the house and yard with his wife and her sister, their friend Lucy, also a painter, Hilda the cook, Flossy the feline, Patsey [ _sic_ ] the canine, four burros \"whose names I do not know,\" plus \"a chorus of bees, wasps, bumble bees, lizards, children of the Earth [Jerusalem crickets], horned toads and etc. etc.\"\n\nThe new art museum was close to completion, and the anticipation and planning for its grand opening gave Santa Feans a happy distraction from the distressing news of the war. But like other towns across America, Santa Fe's social agenda was dictated by the organizations and activities involved with the war effort. Henri described to Alice the Cowboy Ball, a benefit for \"the Navy boys\" held at the Armory, for which \"we, the 'art colony'\" made posters that were awarded to the best costumes. \"It was a success,\" Henri wrote, \"and the costumes make a good spectacle notwithstanding my own opinion that having a cowboy costume party in cowboy country is like taking coals to Colgate.\"\n\nHenri's studio on Palace Avenue was within walking distance (as was most everything in Santa Fe) of the rustic, three-room adobe Paul Burlin rented on the hillside corner of Buena Vista and Old Santa Fe Trail. The Henris, like most people in town, knew Burlin and Natalie Curtis were engaged (by 1917 Natalie and Burlin may have been cohabitating in the little house on Buena Vista). There was no date set yet for the wedding, but Natalie and Burlin's relationship was the talk and toast of the town. Henri wrote Alice in mid-June, \"Yes, there are nuptials in the air at Santa Fe\u2014It's no secret and both bride and Burlin are radiantly transformed. May they be blessed with much happiness and not a large family!\"\n\nThe marriage of Natalie Curtis and Paul Burlin took place at high noon in the federal courthouse a few blocks north of the Santa Fe plaza on July 25, 1917. It was a quiet and simple affair conducted by Supreme Court Justice Clarence J. Roberts. Carol and Roy Pf\u00e4ffle and a friend from San Diego, Alice E. Chaffee, witnessed the brief ceremony. Although the marriage license had been obtained five days before, Alice Klauber did not or could not make it to Santa Fe for the wedding. After the ceremony, the wedding party joined friends at one of the restaurants on the plaza for a celebration that lasted into the evening and included nearly everyone in Santa Fe's expat community of artists, archaeologists, tour guides, politicians, physicians, and shopkeepers.\n\nHenri wrote Alice a comprehensive description of Natalie's wedding, and in his missive about the happy day, he took the opportunity to express his feelings about desert khaki: \"Miss Curtis (Mrs. Burlin) looked fine. I hardly knew her out of kaki (however that is spelled) and I should advise all women never to wear kaki except when roughing it very roughly, and even then I see no reason to have to wear such a colorless color. They looked happy and I hope they will make a great go of it.\"\n\nHenri told Alice that the Burlins' honeymoon was \"in the Penitentes country,\" a nickname for Rio Arriba County and the region of northern New Mexico where the mysterious and once-forbidden Catholic folk religion Los Penitentes (the penitent ones) had thrived in rural communities since the late 1700s. In 1917 the Hispanic villages of the Rio Grande near Espa\u00f1ola, north along the Chama River, and throughout the mountains near Taos still had active albeit clandestine Penitente organizations that met regularly in their local _moradas_ (meeting places). During his years in the forest service, Roy Pf\u00e4ffle had worked with and earned the trust of dozens of local ranchers, many of whom were Penitente _hermanos_ (brothers). With friends active in the village moradas, Roy knew more than most Anglos about the Penitentes. The Pf\u00e4ffles' O\u00f1ate Hotel was likely where the Burlins spent at least part of their \"Penitentes country\" honeymoon. They may have gone with Roy and Carol by auto (if Paul was hoping to paint) or by horseback to visit the moradas in Alcalde on the Rio Grande or in villages like Abiqui\u00fa on the Chama River. Burlin's painting titled _Penitentes' Santuario_ dated \"before 1918\" may have been painted during the August honeymoon.\n\nA Southwestern-style wedding announcement designed by Natalie and Paul celebrating the iconic mountains, mesas, and pueblos that were the beloved landscape of their new life was sent out to friends and colleagues on both coasts by early August. Charles Lummis, although working as an adviser on the new museum and frequently seen in Santa Fe and New Mexico, a place he referred to as \"my own country,\" was not in town for Natalie's wedding. Nor was he aware of her romance with Burlin. Upon receiving the wedding announcement Lummis wrote: \"Dear Natalie Curtis Plus: As the Spanish say: 'When least you expect it up jumps the jack-rabbit!' . . . I had no prevision of any such news as I have just received. But I am surely glad now, as always, of any good fortune or happiness that comes to you.\" Lummis hoped Natalie and Paul's celebrating would \"get over as far into the wilderness\" as his home near Los Angeles so that \"we shall have the pleasure of having you under our roof.\"\n\nIn late summer, Mimsey sent out her own announcement of Natalie's Santa Fe wedding. Her mailing list included more than a thousand names, and it included Paul Burlin's friends and acquaintances. Mimsey wrote George that \"Paul's list included many Goldberg's, Frybergs . . .\u2014not in the Social Register.\"\n\nAs the Great War raged on overseas, the tourist business in the Southwest continued to thrive. The Pf\u00e4ffles' O\u00f1ate Hotel was filled to capacity, and Roy took a succession of expeditions into Indian Country throughout the summer. In late August, Carol went with him on horseback to the Hopi Snake Dance at Walpi. The conflict in Europe was a vague and distant concern for people on the reservations, although the war had brought a noticeable surplus of cash to Indian Country. The U.S. government needed wool for uniforms and blankets, and by late 1917 Navajo sheepherders had more money in their pockets than ever before. However, the Navajos and Hopis were losing their loved ones in a war with another sort of enemy. In the year since Carol and Roy had crossed Navajo and Hopi land on their honeymoon trip, a smallpox epidemic had decimated entire Hopi and Navajo families, and their losses and grief hung like a dark cloud over Indian Country.\n\nOn the road across Hopiland, Carol and Roy crossed trails with a Santa Fean named Nathan B. Stern. Stern was a land developer, and in the late summer of 1917 he was coincidentally the manager of Roy Chapin's Ramon Vigil Ranch on the Pajarito. During a trailside conversation under the hot sun of the high desert, the Pf\u00e4ffles learned that Stern was on his way to join his regiment in California and was bound for France within a month. Stern needed\u2014actually was legally bound to find\u2014someone to move onto and manage the Pajarito Club.\n\nStern realized that the Pf\u00e4ffles had precisely the skills and experience needed to manage the remote guest ranch, and then and there he asked Carol and Roy if they would relocate their outfitting business to the Pajarito Plateau and assume the management of the Ramon Vigil Ranch.\n\nCarol and Roy were understandably nervous about accepting the position. Their business was doing very well out of the O\u00f1ate in Espa\u00f1ola. The Pajarito Club was a magnificent guest ranch, but it was located in a very remote canyon on the high plateau of the mountains. Roy had worked as a forest ranger in the Jemez Mountains and well knew how the Buckman road was terrible to dangerous in any season.\n\nBut how could they say no? Just one year ago, Carol and Natalie had prayed for the opportunity to manage and live at the Pajarito Club. The Pf\u00e4ffles could hardly pass up the chance to run an outfitting business from the heart of the greatest archaeological region in North America.\n\nStern had to know their decision that day, and before they climbed back into their saddles and rode into their respective horizons, the Pf\u00e4ffles and Nat Stern brokered a deal and sealed it with a handshake under the big sky of Hopiland.\n\nBy early November, Carol and Roy moved to the Pajarito with mother Gaga and Roy's sister Lois. During their first two weeks on the plateau, the Pf\u00e4ffles had a full house with thirty-six guests in the main lodge and cottages of the Pajarito Club. It was an auspicious beginning, and seemed to indicate that neither the war nor the colder weather diminished America's fascination with the ancient, untouched-by-modern-time world of the Pajarito Plateau.\n\nThe Museum of Art opened with great fanfare and grandiloquence on Saturday evening, November 24, 1917. The old Algodones Bell in the Acoma Tower of the museum announced to Santa Feans and to the several hundred visitors to the City Different that the day of celebration had arrived. The dedication exercises would take place over several days, but an enormous crowd of more than two thousand people came to the opening evening gala. Hewett, master of ceremonies, spoke briefly, and then the crowd inside St. Francis Auditorium and outside on the road sang \"The Battle Hymn of the Republic.\" And the new art museum's doors were thrown open.\n\nBefore the first speaker (Dr. David R. Boyd, president of the University of New Mexico) took the podium in the auditorium, the crowded hallways and reception rooms of the museum echoed with an emotionally charged rendition of \"America\" sung by the several thousand visitors filling the halls and courtyard\u2014a moment of spontaneous gratitude for the young men from New Mexico and all over the United States fighting overseas.\n\nAlice came to Santa Fe and attended the museum's opening night with the Henris, the Burlins, and the archaeologists and artists she had met in San Diego when they worked on Hewett's exhibitions at Balboa Park\u2014Jesse Nusbaum, Ken Chapman, Carlos Vierra, George Bellows, and a dozen others. Hewett, now director of both New Mexico's Museum of Art and San Diego's Museum of Man, had sent out hundreds of invitations. Roosevelt was personally invited to attend the opening ceremonies, but declined. He wrote Hewett that he had promised his family he would not travel over the holidays (it was Thanksgiving weekend).\n\n**Figure 40.** Burros and wagon loaded with firewood in front of the Museum of Art, Santa Fe, New Mexico, ca. 1918.\n\nAlthough she missed Natalie's Santa Fe wedding, Alice was very much involved with the San Diego wedding of George DeClyver Curtis to Josephine (Lora) D. Jones on September 8, 1917. Alice provided the flowers for the bride, who was twenty-seven years old and staying at the Palomar Apartments. Alice may have also arranged the couple's honeymoon lodgings at her friend Miss Chase's bungalow in La Jolla. If Natalie and Paul traveled to San Diego for George and Lora's wedding, George, who was forty-six when he married, did not mention it in his diary.\n\nIn the months before the museum's opening, Henri, anticipating the surge of artists and archaeologists who would come for the celebrations, asked Alice to ship him some liquor because Santa Fe was now a dry town. Alice, daughter of a prominent California merchant, always brought Henri gifts of cigars and cigarettes. But in the fall of 1917, he sent Alice a very specific gift list, and before she left San Diego for New Mexico, she shipped Henri eight bottles of American gin, four bottles of Italian vermouth, and five boxes of Melachrino cork-tip cigarettes.\n\nThroughout the fall the Henris hosted spirited preopening dinner parties at their casa at 601 Palace Avenue for the painters and archaeologists working in and around the Palace of the Governors and the new art museum. Henri's association with the new museum brought many nationally recognized painters to Santa Fe. Prominent contemporary artists George Bellows and Leon Kroll came in late summer to tour and paint northern New Mexico, and also to help Henri assemble the art exhibit at the museum. Bellows, a former student of Henri's at the New York School of Art, set up a studio in the old Palace. Leon Kroll came to Santa Fe from Colorado Springs to rendezvous with Bellows and Henri, and managed to paint several canvases in between the socializing and festivities.\n\nKroll's canvas _Santa Fe Hills_ , Bellows's _Chimayo_ , and several of Henri's painted during the summer and fall of 1917 were included in the art museum's opening exhibition. Alice had at least one painting in the opening exhibit, whose roster of artists was a virtual who's who in American art: Henry Balink, Donald Beauregard, George Bellows, Oscar E. Berninghaus, Ernest L. Blumenschein, Paul Burlin, Kenneth Chapman, Edgar S. Cameron, Gerald Cassidy, E. Irving Couse, Katherine Dudley, W. Herbert Dunton, William Penhallow Henderson, E. Martin Hennings, Robert Henri, Victor Higgins, Alice Klauber, Leon Kroll, Ralph Myers, Arthur F. Musgrave, Sheldon Parsons, Bert G. Phillips, Grave Ravelin, Julius Rolshoven, Doris Rosenthal, Joseph H. Sharp, Walter Ufer, Carlos Vierra, and Theodore Van Soelen. After the show many of the artists donated their paintings to the art museum, thus beginning the museum's superb collection of twentieth-century art.\n\nHenri's letters to colleagues back east left no doubt as to his passion and enthusiasm for the new museum and the emerging art colony of Santa Fe. \"Santa Fe may do the rare thing and become itself,\" he wrote to the painter and new San Diego resident Henry Lovins. \"The painters are all happy. The climate seems to suit well both temperaments\u2014to work or not to work\u2014and here painters are treated with that welcome and appreciation that is supposed to exist only in certain places in Europe. Being of the 'to work' temperament myself, I am having a fine time.\"\n\nWalter Pach did not leave his desk in New York City and travel to Santa Fe to review and critique the three hundred canvases by forty contemporary artists, nearly all of whom were well known to him. Pach may have passed up the opportunity to chronicle the opening of a major art museum in the American West (still a frontier on the maps of most New Yorkers), and probably peeved a few of his art world colleagues like Henri and Bellows, but the editors of the _New Mexican_ did their best to make up for the lack of national press coverage. The newspaper's columnists effusively expressed the local sentiment that Santa Fe's museum was worthy of national if not international attention and that the opening exhibition \"ranks with the Carnegie gallery in Pittsburg, the Chicago Art institute, the Corcoran Art gallery in Washington and others.\" Although the Museum of Art in Santa Fe invited such comparison, the editor added, it was so artistically and architecturally unique, so completely in a class by itself, as to be an institution \"without parallel in the country.\"\n\nNatalie spoke to a full house in St. Francis Auditorium (which seated more than 600) on Tuesday morning, November 27. She gave a lecture about Indian music and ceremonial dance, and introduced San Ildefonso pueblo dancers, who performed the Eagle Dance on the auditorium stage.\n\nA photograph of Natalie Curtis Burlin taken at that time captured the confident forty-two-year-old newlywed astride her horse on the street near the new museum. Dressed in formal riding clothes, laced boots, and a broad-brimmed hat (but no khaki), Natalie, like her hometown of Santa Fe, seemed to have done the rare thing and become herself.\n\n**Figure 41.** Natalie Curtis Burlin, Santa Fe, 1917.\n\nThe photograph became Natalie's 1917 Christmas card and was mailed to friends and family on both coasts. The copy she sent to Lummis was inscribed \"To Mr. Lummis, Sage of the Southwest the thought of whom Dwells in the 'New Museum' at Santa Fe and in the hearts of many friends.\"\n\n#  _14_\n\n##  \"A COUNTRY OF THINGS IN LIGHT\"\n\nI found out that the sunshine in New Mexico could do almost anything with one: make one well if one felt ill, or change a dark mood and lighten it. It entered into one's deepest places and melted the thick, slow densities. It made one feel _good_. That is, alive.\n\n##  \u2014 _Mabel Dodge Luhan_\n\nON A BRIGHT AFTERNOON IN mid-December 1917, Natalie and Paul Burlin were visited by the New York socialite Mabel Dodge Sterne, who had come to Santa Fe by train the night before. She came with her husband, the painter Maurice Sterne, who was a friend of Paul Burlin's from New York City. Sterne was also a former student of Robert Henri's, and he had arrived in New Mexico in October and set up a studio in a small adobe just across the road from the Burlins' home at the corner of Buena Vista and Old Santa Fe Trail.\n\nWhen Mabel and Maurice walked into the Burlins' front room, Alice Corbin was on the Taos daybed knitting a khaki sweater for the Red Cross. Alice Corbin and Natalie warmly greeted Mabel Sterne, but their welcome fell on deaf ears. Mabel had been in the capital less than twenty-four hours, and in spite of her husband's best efforts, had already decided she disliked the little city. \"Santa Fe was the strangest American town I had ever seen. In that bright December sunshine the plaza was a queer mixture of oddities. There were wood-carts drawn by burros, with the short lengths of pi\u00f1on wood made into square piles and set in sort of high cages, and these were led by dark Mexicans. . . . I saw only two or three automobiles. People rode horseback or drove two-seated buggies. . . . The only Americans I saw who looked like people I was accustomed to, were the shopkeepers in the few stores I went into.\"\n\nMabel especially loathed the obligatory social call to her husband's friends the Burlins, whom he praised as \"pioneers.\" For two months, Maurice Sterne had been enjoying the people and environment of Santa Fe, and had encouraged his wife to join him in New Mexico and begin a new life in the town at the foot of the Sangre de Cristos. But Santa Fe's expat community did not resonate with the thirty-eight-year-old Mabel.\n\n\"Going across to tea at the Burlins . . . was the first really tiresome thing that happened,\" Dodge wrote of her first day in New Mexico. \"There was a good deal of bright paint about\u2014yellow and blue\u2014and Paul Burlin's modernist paintings here and there on the walls. Natalie Curtis, his wife, was a little old doll that had been left out in the sun and the rain. She had faded yellow hair, cut in a Buster Brown bob, and faded blue eyes. Paul was much younger and looked fresher. He had curly red hair above a round forehead, and the absent, speculative thoughtful look of an intelligent Jew.\"\n\nMabel, who had yet to experience firsthand the effects of New Mexico's high altitude sun and year-round aridity on the skin, hair, health, and overall disposition of those who lived in the Southwest, was stiff-necked, ill humored, and unsmiling throughout the visit. She was aggravated and bored by the tea-time talk of modern writers and _Poetry_ magazine (Alice and her co-editors, Harriet Monroe and Carl Sandburg, had recently produced a special issue of _Poetry_ devoted to Indian chant and song and its influence on modern American poetry), the wartime efforts of the Santa Fe Red Cross, and the Burlins' and Hendersons' plans to spend Christmas Eve and Day at Santa Domingo Pueblo.\n\nBefore Mabel and Maurice Sterne departed the Burlins', Alice Corbin suggested they come tomorrow for tea at the Hendersons' home, just a few steps away on the Camino del Monte Sol. Mabel did not respond to the invitation but instead announced, to her husband's surprise, that she planned to motor up to \"a place called Taos\" the next day. And so she did. Thus began Mabel Dodge Sterne's (later Mabel Dodge Luhan) very high-profile and famous life in Taos, New Mexico.\n\n**Figure 42.** Mabel Dodge Luhan.\n\nIn spite of her unfriendly behavior and comments, Mabel Dodge Sterne knew there was a good deal more to Natalie Curtis Burlin than faded hair and dry skin. Mabel was enormously interested in primitive cultures and was certainly aware of Natalie's extensive travels into and writings about Indian America. Mabel's home at 23 Fifth Avenue in Greenwich Village was only a few blocks from Natalie's family home on Washington Place. They may have not met face to face until that December day in Santa Fe, but Mabel and Natalie surely knew something of each other's activities.\n\nMabel also had a home in the town of Croton-on-Hudson, New York, just a few miles from Winfred Douglas's home at St. Mary's in Peekskill. Douglas certainly spoke of Natalie's book and adventures in the Southwest to Mabel, as he was at least partially responsible for Mabel's interest in and inaugural journey to New Mexico. Years later, she mentioned Douglas when she described her first trip to the \"little known neighborhood\" of the Southwest: \"I armed myself with letters of introduction to several individuals who stood out in that unknown and unexplored land. Among them, one was to Ford Harvey [relative of Fred Harvey] and another to Lorenzo Hubbell. These letters were both given me, after some effort to find people who had been to New Mexico, by a priest named Father Douglas who had lived near a tribe named Hopi. . . . He had told me strange, wonderful stories about those people.\"\n\nMabel later claimed that she went so quickly to Taos because Santa Fe was too tame and conventional. But Santa Fe in 1917 could hardly be accused of either of these characteristics. For Mabel, the real problem with Santa Fe was its population, however small, of erudite, creative, and audacious New Women who were already making their mark on the place. Mabel needed her own space to make her mark, and the day after the insufferable tea party at the Burlins', Mabel Dodge Sterne motored up to Taos with her soon-to-be-ex-husband, Maurice, and her son, John Evans, and began her own new life.\n\nThe Burlins' home on the hill, at the corner of a narrow dusty lane called Buena Vista and the rutted dirt road that would later be formally named Old Santa Fe Trail, was in the beating heart of the Santa Fe art colony that flourished almost overnight after the Great War ended in November 1918. Within a stone's throw of Natalie and Paul's front door lived the British painter Arthur F. Musgrave, who, like Alice Corbin, had come in 1916 for a stay at Sunmount, recovered his health, and settled into life in Santa Fe. The painter Louise Crow lived next door to the Burlins, and Sheldon Parsons, the Hendersons, Will Shuster, and Carlos and Ada Vierra all made their homes in adobe houses close by.\n\nMarsden Hartley came for a stay at Mabel's new Taos salon in the spring of 1918. Paul Burlin knew Hartley in New York, and he went with Mabel to meet Hartley's train at the Lamy station on June 10. Alice Klauber's friend, and Gertrude's brother, Leo Stein had recently begun his own infatuation with New Mexico. No longer on speaking terms with his famous sister, Leo was living full-time in New York City. After his pal Mabel moved to New Mexico, it was Leo who encouraged Hartley to go out west and visit her utopian world in Taos. Stein joined Hartley at Mabel's that summer and loved everything about Taos. But Hartley tired of the town and their hostess, and in late summer moved down to Santa Fe and into the casita next to the Burlins in the vibrant Buena Vista neighborhood of artists, writers, and musicians.\n\nWith Natalie and Paul Burlin, Hartley toured the pueblos of the Rio Grande and met Natalie's many native friends. Hartley was a member of Alfred Stieglitz's \"soil-and-spirit\" group of artists who believed that the spiritual essence of the new American modernism would be found in the American soil. Natalie's long-held belief that Indian music, song, and ritual were a pure and original American art form paralleled Hartley's philosophy of an \"essential reality\" found in the landscape and people of the Southwest. With Natalie and her husband as sympathetic and educated guides, Hartley traveled into the remote Indian and Spanish American country of northern New Mexico. Hartley stayed only two years in Santa Fe, but the landscape, like the indigenous people, forever changed him as an artist and a person. \"New Mexico is essentially a major sensation, for my eye, at least,\" he wrote. \"The sense of form in New Mexico is for me one of the profoundest, most original, and most beautiful I have personally experienced. It must be 'learned.' . . . It is not a country of light on things. It is a country of things in light.\"\n\n**Figure 43.** Marsden Hartley, Randall Davey, and John Sloan on the patio of the Palace of the Governors, 1919.\n\nSanta Fe was as welcoming to the women artists, writers, and archaeologists (although there were few female archaeologists before 1920) as it was to the men who flocked there during and after the war. The author and playwright Mary Hunter Austin arrived in Santa Fe in early November 1918. Austin was in town to investigate land laws and customs of the Native Americans and Hispanic Americans of New Mexico, and Natalie and Alice Corbin were among her first consultants. Within a few weeks of her arrival, however, Austin's sociopolitical energies were rechanneled into more civic artistic endeavors when she offered to help Santa Fe establish a community theater like the one she had begun in Carmel, California.\n\nWith Austin's assistance and direction, the same artists, writers, and archaeologists whose skills and energies built the new art museum created the first Santa Fe community theater. The opening season featured productions written and produced by locals, including the crowd-pleasing one-act spoof entitled _Tranquilina, An Archaeological Absurdity in One Fragment, Discovered, Excavated and Decoded by Waldo C. Twitchell_. The boundaries between the New Mexico art and archaeology communities had always been blurred, and by 1918, on- and offstage, the merging of Santa Fe's scientific and artistic colonies was a fait accompli.\n\nThat same year, Austin wrote the introduction for _The Path on the Rainbow_ , an anthology of Indian songs and chants. Contributors included Natalie, Alice Corbin, Matilda Coxe Stevenson, John Peabody Harrington, and J. Walter Fewkes. Austin's introduction noted the \"instinctive movement of the American people is for a deeper footing in their native soil\" and praised Natalie, Alice Fletcher, Frank Cushing, and Washington Matthews for \"the clarity and sincerity of their literal translations\" of Indian poetry, music, and dance. \"I know of no task so salutary to the poet who would, first of all, put himself in touch with the resident genius of his own land.\"\n\nThe art and anthropology communities merged in the husband- and-wife team of Natalie and Paul Burlin when they created a dance-drama called _The Deer Dance_. Natalie wrote the musical score based on adaptations of San Ildefonso Pueblo deer, buffalo, and eagle dance ceremonials, and Paul designed the sets and the costumes. Although it was performed locally, the Burlins' dream to bring the production to New York and Boston never materialized.\n\nALTHOUGH THE PAJARITO PLATEAU was less than thirty miles from Santa Fe, the newly renamed guest ranch, the Rancho Ramon Vigil, where Carol and Roy now lived was so removed from the modern world as to be in another country. The winter of 1918 was warm with almost no measurable snowfall, and the road from Pajarito Canyon to Buckman, where Carol retrieved mail, supplies, and arriving guests, of which there were several dozen, remained dry and passable. But even dry, the journey to and from the ranch was hazardous for horses, wagons, or automobiles. And even with a bridge over the Rio Grande at the Buckman depot, there was no guarantee that a vehicle or horse could cross the river.\n\nThe Buckman Bridge was famous for disappearing. If the river was high, the bridge was often washed out, carried downstream by the waters of the Rio Grande. Without the bridge, the only way a vehicle or wagon could cross the river was on the open trestle of the D&RG Railroad. To access the trestle \"bridge\" the driver of a car would back up so as to align the tires with the railroad track and \"bump its way across the ties with the river swirling ominously below.\"\n\nEven when the Buckman Bridge was intact and above water it was an unstable, worrisome structure, especially for first-time visitors to the Pajarito. Once across the Rio Grande, the road to Carol and Roy's guest ranch cut up the very steep and very high mesa on the other side. For the Pf\u00e4ffles' guests, just getting to the Rancho Ramon Vigil was a test of nerves. \"The original road went in from Buckman, across wobbly old Buckman Bridge, then slantwise in what looked deceptively like one straight gash up the formidable escarpment of the Pajarito Plateau in a two mile stretch of ruts and rock called Buckman Hill.\"\n\nIn spite of the challenges involved with reaching the guest ranch, throughout the winter and spring of 1918 the five guest rooms in the two-story main house and the thirteen rooms in the eight cabins were filled with lodgers who paid twenty-one dollars a week for the privilege of a bed and meals on the remote plateau. Many of these folks were friends, or friends of friends. Charles Lummis was among Carol's first guests and came twice in December 1917, his second visit coinciding with Carol and Roy's birthday party on December 16. Mother Gaga's birthday was on December 17, so Carol and Roy's birthday cake included candles for Gaga, and there was plenty of sparkling burgundy for Lummis and the ranch guests.\n\nIn January, Hewett's friend, and patron of the new art museum, Frank Springer came for several days, and Natalie and Carol's attorney and business advocate, Elbridge Adams, made his way up the plateau with several Rothchilds and one of the Rockefellers. With Roy and Carol as guides, the Adams entourage spent the week touring Frijoles Canyon and other archaeological sites on the Pajarito. Natalie's horse, Buck, was stabled with Carol and Roy's horses on the plateau, and the Burlins and the Hendersons came to the plateau for horse excursions and overnights. Charles Lummis returned in March during Easter and went \"Penitente hunting\" with Carol and Roy and a crowd of unnamed friends that may have included the Hendersons and the Burlins.\n\nExcept for expected and routine car and truck troubles\u2014overheated engines, blowouts, punctured tires, and burst radiators\u2014it was an uneventful winter. Gaga and a friend did get stranded in their car on Buckman Hill one afternoon and had to walk home, and several guests became temporarily lost while hiking about the canyons. But Roy and Carol managed to find everyone and return them safely to ranch headquarters before dark.\n\nIn April and May, a Boston acquaintance of Carol's and Natalie's from the South End Music Settlement School in Boston, Mary Cabot Wheelwright, came with a friend to stay at the ranch on the Pajarito. Carol's personal staff now included Alice Pring, an Englishwoman who joined the household that winter as Carol's maid. Gaga, Carol, and Alice worked dawn to dusk on laundry, dishes, meals, and guest entertainment and transportation needs. Gaga's diary records that a young man from Denver named \"Erick\" worked as Roy's ranch hand throughout that winter and spring. The diary does not reveal the man's full name, but he may be Eric, Canon Douglas's son. As managers of a guest ranch, Roy and Carol's daily routine was a relentless repetition of mundane but necessary chores. Carol fled the plateau for overnights in Santa Fe with Natalie, and for ten days in February she went alone to Denver to stay with the Douglases in Evergreen.\n\n**Figure 44.** Carol Stanley Pf\u00e4ffle, Jack Lambert ( _far right_ ), and Eric Douglas ( _second to left_ ) at Rancho Ramon Vigil, 1918.\n\n**Figure 45.** Carol Stanley Pf\u00e4ffle and expedition at Los Alamos Ranch School, ca. 1925.\n\nThe lack of snow and the January heat wave (Gaga recorded the temperature on the house porch at ninety degrees on January 2) meant a _very_ dry spring. Roy, having worked as a forest ranger in this country, well understood the implications of a snowless winter. The streams of the Pajarito canyons would be low flow at best, bone dry at worst. And with drought came the threat of forest fires.\n\nThroughout the winter and spring, Carol kept Roy Chapin and his partners, who were looking for a buyer for the Rancho Ramon Vigil, apprised of the ranch's activities. Chapin was living in Washington, DC, serving on the War Department's Highways Transport Committee. Henry Joy was a captain with the U.S. Signal Corps and was also living in DC. The Gray brothers remained in Detroit.\n\nCarol hated bothering Chapin with trivial issues, but when the drought lowered the water table on the plateau and the ranch wells began to run dry, she was forced to share the dire news with him. \"We are, frankly, terribly nervous about the water,\" Carol wrote Chapin after she returned to the Pajarito from Denver in late February. \"All there is in the whole canyon is a tiny bit in the well near the tennis court. We serve on the table and for bathing, weird dark brown fluid which naturally scares most of our guests, and has lost us some. Boiling fails to improve its color, and our immaculate Mother Pf\u00e4ffle has been reduced almost to tears because one new comer asked if her pitcher might be 'cleaned occasionally.'\"\n\nIt seemed like divine intervention when, in late February 1918, Roy was asked to accompany a wealthy Denver mining magnate, James R. Thorpe, on a property tour in the Tesuque hills near Santa Fe where Thorpe wanted to build an upscale resort. Thorpe liked what he saw and on April 18, 1918, bought 152.8 acres of land from the family of William Scoville and Edith Pulitzer Moore. This property in the pi\u00f1on-juniper foothills of the Sangre de Cristo Mountains four miles from the plaza of Santa Fe, named Bishop's Ranch, included the chapel and retreat once used by the French Jesuit archbishop Jean-Baptiste Lamy.\n\nIt was a spectacular location for a guest ranch. Unlike the rustic and isolated Rancho Ramon Vigil, Bishop's Ranch had the luxuries of electric lights and heating systems, indoor plumbing, landscaped gardens, orchards, and _water_. (The estate had two lakes on the property.) Before Thorpe even placed an offer on the Tesuque property he asked the Pf\u00e4ffles if they would like to oversee the creation and management of his new Bishop's Ranch, soon renamed Bishop's Lodge.\n\nCarol immediately wrote Chapin that she and Roy were considering Thorpe's offer. They still owned the small outfitting business in Espa\u00f1ola, and Carol told Chapin that Roy thought they could feasibly run a \"circuit business\" and manage all three operations.\n\nThe Rancho Ramon Vigil became an untenable situation even for the resolute Pf\u00e4ffles when the plateau wells went dry in early March. \"Our sword of Damocles has fallen,\" Carol wrote Chapin. \"The well has gone dry, and now, with a house full of people, we are hauling water from nearby canyons.\"\n\nAn Espa\u00f1ola sheep rancher and businessman, Frank Bond, and his brothers purchased the Ramon Vigil from Chapin and company by late April, and Carol, Roy, Gaga, Roy's sister Lois, and most of the Pf\u00e4ffles' employees moved down to the new Bishop's Ranch by May.\n\nIn the spring of 1918, while talking to a local bear hunter and outfitter named C. B. Ruggles in front of the Capital Pharmacy near the Santa Fe plaza, Roy was introduced to a young cowboy named Jack Lambert. Ruggles knew Roy and Carol were setting up the new Bishop's Ranch and suggested Roy consider Lambert as a ranch hand. When Roy learned Lambert was a seasoned cowboy and outdoorsman, he invited the Santa Fe newcomer to join him for lunch at a local diner.\n\nEverett Vey \"Jack\" Lambert was all of twenty when he met Roy Pf\u00e4ffle in Santa Fe. Born in Okarche, Oklahoma, in 1898, Lambert had spent several years living on his horse. As a teenager he worked as a cowpuncher, pack guide, camp cook, taxi driver, even a bridge builder (on the Green River) as he made his way across the desert and mountains of the Southwest. The ruggedly handsome Lambert was tall and thin, even tempered, courteous, and thoughtful. When in the company of ladies, he was a perfect gentleman, and he was as comfortable on a horse in the deep outback as he was in a corral filled with high-maintenance, wide-eyed dudes just off the train from the East.\n\n**Figure 46.** Jack Lambert and Roy Pf\u00e4ffle.\n\nJack Lambert, like Roy Pf\u00e4ffle, was a self-educated, self-made, self-reliant man born and raised in the hardscrabble country of the Great Plains. Roy and Jack liked each other immediately, and over lunch at the diner near the plaza, Roy offered Lambert a job at the new Bishop's Ranch. Lambert accepted, and thus began a twelve-year partnership between Lambert and Roy and the extended Pf\u00e4ffle family.\n\nRoy and Jack were still pulling boards off the windows and renovating guest cabins when the first guests arrived at Bishop's Ranch. Carol oversaw the daily management of the ranch while Roy and Lambert organized and packed up expeditions, often from Roy's livery stable in Espa\u00f1ola, and headed out into the Four Corners Indian Country. Bishop's Ranch guests went on auto and horse trips to Chaco Canyon, Canyon de Chelly, the Natural Bridges of Utah, and through the Mesa Verde region. That first year Roy led at least two groups on epic pack trips to Kayenta and the Rainbow Bridge between May and August.\n\nCharles Lummis came to see Bishop's Ranch in July and, with Carol and Roy and their Overland touring car, went up to the Pajarito. The Pf\u00e4ffles continued to maintain the lodge at the Rancho Ramon Vigil for the Bond brothers, and with Lummis, they hiked Frijoles Canyon during the day and stayed nights at the former Pajarito Club.\n\nRoy Chapin came with his wife and three-year-old son, Roy Jr., in October for a two-week retreat at Bishop's Ranch. Carol had become friends with Chapin's wife, Inez, during her trip to Detroit, and the Pf\u00e4ffles accompanied the Chapin family on an auto tour of northern New Mexico that included an overnight at the Pajarito Club. The rumors of an armistice dominated conversations that October, and during dinners with guests and family at the Bishop's Ranch, Chapin shared an insider's view of the war, and also read aloud from \"some very interesting speeches\" brought from his office in Washington.\n\nMr. Hubbell, either Don Lorenzo or one of his sons (Gaga's diary entry does not specify), came from Ganado to visit the Pf\u00e4ffles at the new Bishop's Ranch several times that fall and winter. Hubbell's first visit coincided with Carol and Roy's anniversary party, and Hubbell joined the Chapins and the other Bishop's Ranch guests in dining and dancing with the Pf\u00e4ffles and the ranch staff late into the night. Hubbell returned again before Christmas and was a guest at Carol and Roy's birthday party on December 16.\n\nRoy's anniversary present to Carol that year was a grand piano\u2014the first piano she had had in New Mexico\u2014which was placed in the Bishop's Ranch main house. Carol played an informal concert on October 16 for the guests, who included Judge and Mrs. Abbot, who had a ranch in Frijoles Canyon; the Chapins, who were departing for the East in the morning; and Hubbell. A perfectly sublime fall evening of music and friends came to a sudden end when Carol fainted. Although Gaga commented in her diary that \"we had a time bringing her to,\" she provided no further details about Carol's collapse. Carol fainted several more times in the next weeks and was bedridden with influenza after Christmas.\n\nNow that Carol had a piano in the house, she offered music lessons to neighbor children and friends. She also gave her teenage sister-in-law, Lois, voice and piano lessons. Lois, however, was an unenthusiastic student and preferred horse or auto excursions in the mountains to afternoons inside the lodge at a piano with her older brother's wife. Like her mother, Lois resented Carol's executive position in Roy's businesses (Carol kept the books and dictated the house budget) and was critical of her shopping holidays to Denver and Chicago.\n\nA good bit of Lois's disgruntlement with her sister-in-law was really a teenager's declaration of independence. Lois frequently left the ranch in Tesuque to horseback ride with friends or go into the movies in Santa Fe. Lois especially liked to do anything in the company of Jack Lambert, who, although only three years her senior, was not considered a proper suitor, at least by Gaga, Roy, and even Carol.\n\nThe Great War came to an end on November 11, 1918. Santa Feans, like folks in towns and cities across America, awoke at dawn to loud whistles that announced the signing of the armistice and the cessation of hostilities. The bells of the capital churches and the new museum joined the shrill whistles from city hall and the Santa Fe Water and Light Company.\n\nBy midmorning, there were boisterous demonstrations around the plaza and \"every motor car in town was in motion, and every one of them decorated with bunting.\" Children on \"wobbling cycles\" joined the throngs of people who carried the flags of the United States, Italy, and France through downtown Santa Fe, past the art museum and around the plaza, and Carol and Roy drove into Santa Fe by midday to confirm the news and to celebrate with friends.\n\nThe end of the war marked the end of an era in Santa Fe and the American Southwest. The sense of a frontier landscape separated geographically and temperamentally from the rest of the United States and the world would rapidly diminish within the next year, and a great surge of visitors and new residents would make their way to the Land of Enchantment.\n\nIN 1920 NATALIE RECEIVED a small inheritance and with it purchased the three-room adobe casita near the corner of East Buena Vista Street and Old Santa Fe Trail that she and Paul had been living in for several years. After adding a small kitchen and purchasing a few handcrafted and locally made benches, chairs, and tables, the Burlins settled very comfortably into married life.\n\nThe little house and its \"hand's breath of land\" sat on a hill and had fine views to the north and east of the roads to the plaza and of the foothills of the Sangre de Cristos. Although considered modern by New Mexico standards\u2014the house had running water and indoor plumbing, a telephone and electric lights, and its three quaint rooms \"fashioned with a wand\" in true Santa Fe style\u2014the Burlins' adobe was not, by a Greenwich Village New Yorker's criteria, much to write home about. Indeed, had Natalie's Washington Place family come to Santa Fe and stepped through the rough wood doorway into her mud and timber, dirt-roofed abode, they would have been appalled.\n\nNatalie's adobe in old Santa Fe would not have shocked George, though. Like Natalie, George Curtis had slept in the mud, stone, and wood dwellings that were home to the Southwest's native peoples, and he had lived in rustic cabins and tents in far-flung cow camps. Natalie suggested that George come and stay with her and Paul in Santa Fe, although her only guest room was the breezeway between rooms in which she and Paul often slept in the summer. But George couldn't get to Santa Fe. He and his wife, Lora, had their hands full with their ranch.\n\nAlthough Henri did not return to Santa Fe until 1922, he continued to send his friends and colleagues to New Mexico. In the summer of 1919, after listening to several years of Henri's incessant, effusive praise about the light and character of New Mexico, the artist and teacher John Sloan decided to undertake the arduous cross-country auto trip \"to investigate the situation in Santa Fe.\" Sloan convinced a friend, Randall Davey, to join him on this adventure, and with their unenthusiastic wives, the two painters drove a 1912 ninety-horsepower Simplex chain-drive touring car from New York to Santa Fe in 1919.\n\n\"It was a great day for John Sloan when he discovered New Mexico,\" Walter Pach would later write about Sloan and company's arrival in Santa Fe. \"But it was also a great day for New Mexico.\"\n\nBy 1919 Sloan was nearly as prominent in American art as Henri. He was one of the Ashcan Eight painters, was a founder and president of the Society of Independent Artists, and had exhibited alongside Henri in the Independent Exhibition in 1910 and the Armory Show in 1913. Sloan was also a close acquaintance of Walter Pach, who had been his student in 1906 when Sloan and Henri were instructors at the New York School of Art.\n\nSloan and Davey liked the light, the landscape, and the people of Santa Fe. They also liked Hewett's and the art museum's open-door exhibition policy, Santa Fe's nonacademic creative environment, and the exceptional community of artists who welcomed them with open arms. And once they climbed out of the noisy, cramped, dusty touring car, their wives took a liking to Santa Fe, too.\n\nWithin a week, Hewett set Sloan and Davey up in studios at the Palace of the Governors and helped them find comfortable quarters to rent near the plaza. Dolly and John Sloan loved their summer in Santa Fe and became seasonal residents, returning every summer for the next thirty years. But Randall Davey and his wife, Isabel, liked Santa Fe so much they wanted a permanent place of their own, and in 1920 they bought the historic house and orchard of Candelario Martinez at the top of the Santa Fe canyon east of town.\n\nAlice Klauber had met Randall Davey in the summer of 1912 when the twenty-five-year-old Davey was Henri's affable and capable assistant in Italy. When he arrived in Santa Fe in 1919, Davey was a nationally known and award-winning painter enjoying much success on the American art scene. He had exhibited in the Armory Show in 1913, and in 1915 won the Julius Hallgarten Prize of the National Academy of Design and honorable mention at Balboa Park's Art Exhibition. Davey had studied architecture at Cornell, and he personally directed the enlargement and restoration of the structure that had served as Santa Fe's first sawmill in 1847. Located near the entrance to the Santa Fe National Forest on what became Upper Canyon Road, the Davey home became a favorite gathering place for Santa Fe's burgeoning colony of artists, writers, musicians, and archaeologists. Although Davey continued to teach semesters at the Chicago Art Institute, he and his wife and son made Santa Fe their primary residence.\n\nDavey became one of Santa Fe's most devoted residents, and surely spoke for the Burlins, Hendersons, Pf\u00e4ffles, and a few dozen other New Mexico expats when he wrote: \"I wouldn't trade my life here where I can hunt, shoot, ride, for all that committee going and boot-licking you've got to do in a city for anything. An artist might starve for food here, but he'll starve spiritually in a place like New York.\"\n\nAlthough Natalie had finally found a devoted partner with whom she could share her love of the land and people of the West, and now had a home and place of her own in the affirming and stimulating environment of Santa Fe, she kept her promise to Frissell,  Peabody, and Mason and returned to the East and resumed her study of African American music. In 1918\u201319 Natalie's complete study was published in four volumes titled _Negro Folk-Songs_. As she had in _The Indians' Book_ , Natalie included cultural context and background information on the words, tunes, and symbolism of African American music, music she believed was an essential and vital part of modern America's life and culture.\n\nKurt Schindler added Natalie's arrangements of Negro spirituals to the Schola Cantorum program in New York in July 1918. Eighteen months later, Frank Damrosch and the Musical Art Society of New York performed two of Natalie's concert arrangements of African American Christmas songs in their annual Christmas program.\n\nIn 1919, while Natalie was back in New York City working on her books of Negro music, Carol contacted Winfred Douglas and, with his assistance, enrolled Roy's sister Lois in the private school run by the good sisters at St. Mary's in Peekskill where Douglas was music director. Although Carol was friends with the wranglers and cowboys that worked for her, and had herself dramatically eloped with a cowboy just a few years before, she hoped Lois would not marry a cowboy. At least not for a few years. Everyone around Bishop's Ranch knew that Jack Lambert was sweet on seventeen-year-old Lois. Only physical separation would avert premature matrimony.\n\nGaga and Carol did not agree on much, but they agreed that Lois needed to be sent east to finish high school, and by the fall of 1919 Lois Pf\u00e4ffle was sent to live in Peekskill under the watchful tutelage of Canon Winfred and Dr. Jo Douglas.\n\nJack Lambert drove Lois to the station at Lamy to catch the train for Chicago. He did not, however, stay to watch Lois depart: \"Poor Jack,\" Lois wrote her mother, \"he didn't stay until the train left. He just went down and then left.\"\n\nDuring her first months in Peekskill, Lois was introduced to the art and culture of New York City by the Douglases and by Dorothy Kent, who was at home in nearby Tarrytown. Lois stayed with the Douglas family over Christmas, and also remained with them for the summer holiday. She became an adept urban traveler and went weekly by train or motorcar into New York for lunch, shopping, and evenings at the movies. She went to numerous Carnegie Hall events with the Douglases and with Dorothy Kent and was frequently in the audience for Schindler's choral concerts. During her first winter at St. Mary's, Lois met Natalie for the first time at Dorothy's home, where Natalie often came to ice skate on the Kents' pond.\n\n**Figure 47.** St. Mary's, Peekskill, New York, where Winfred Douglas lived and where Lois Pf\u00e4ffle was sent to finish high school.\n\nIn March 1920 Lois went into New York with Kent and the Douglases to visit Natalie at her studio. They braved a cold and pouring rain to have tea with Jesse Nusbaum and Frederick Webb Hodge, friends of Natalie's and Douglas's whom Lois may have already met in New Mexico. Nusbaum lived in Santa Fe and was a frequent camper on the Pajarito Plateau. Hodge had worked at Frijoles Canyon in 1910 alongside Hewett and Nusbaum, and returned to the Southwest each summer for extended fieldwork. A prominent anthropologist and writer, and the former director of the Bureau of American Ethnology, Hodge edited the massive _Handbook of North American Indians_. When Lois met him at Natalie's in 1920, Hodge was the director of the New York Museum of the American Indian and with Nusbaum was working during the summer months on the excavation of Hawikuh at Zuni Pueblo in New Mexico.\n\nAlthough Carol and Roy orchestrated Lois's separation from Jack Lambert, within a year of his arrival Lambert had become indispensable to the Pf\u00e4ffles and was essentially a full partner in their outfitting business. He had also become Carol and Roy's most trusted friend. As such, in the next decade Lambert, like Carol, would witness and suffer the personal and professional consequences of Roy Pf\u00e4ffle's escalating gambling and alcohol addiction.\n\nRoy, like most men who came of age as a cowboy or homesteader, learned to play cards at a young age. He was good at poker, but he lost as much as he won, and as his gambling became habitual, so did his drinking. In late 1919 James Thorpe, owner of Bishop's Ranch, fired Roy Pf\u00e4ffle as his ranch manager, claiming Roy was mismanaging the new guest ranch. In fact, Bishop's Ranch was doing very well at the time. Thorpe may have heard about Roy's poker escapades and hoped that by firing him he was distancing Bishop's Ranch from the seedier side of northern New Mexico.\n\nHowever, it is also possible that Roy's dismissal was a defensive move by Thorpe, because at that precise time, November 1919, the Pf\u00e4ffles, in partnership with Lambert, were assembling the funds with which to purchase property in Alcalde, where they planned to build their own guest ranch. The Pf\u00e4ffles' attorney, Francis O. Wilson of Santa Fe, was negotiating a deal with the family of Elias Clark for several historic and decrepit adobe buildings and 34,000 acres of land bordered by San Juan Pueblo and the eastern bank of the Rio Grande.\n\nIt was a spectacular location for a guest ranch. Alcalde, a tiny Penitente village four miles north of Espa\u00f1ola, was located on the road that connected Santa Fe to Taos through the tectonic chasm called the Rio Grande Gorge. Although more than a thirty-mile drive over rough roads to Santa Fe, Alcalde was located near the confluence of the Chama and Rio Grande Rivers, and the Clark property had unimpeded access to the best horse and auto routes north and west to Colorado and Arizona Indian Country. The Sangre de Cristo Mountains dominated the length of the eastern horizon, and the Jemez Mountains and the Pajarito Plateau, a half day's ride across the lands of three Tewa pueblos (San Juan, Santa Clara, and San Ildefonso), rose above the river valley to the south and west.\n\nThe purchase of the Alcalde property, at least partially underwritten by loans from friends and private investors, was completed by February 1920. With Lambert, the Pf\u00e4ffles began the colossal task of rebuilding the decrepit dwellings, outbuildings, and corrals. Carol planted flower and vegetable gardens, furnished and decorated the adobe casitas in the style and manner she had admired at Louisa Wetherill's home in Kayenta, and renovated the historic hacienda into a rustic but charming ranch headquarters.\n\nCarol and Roy named their guest ranch San Gabriel. The name paid homage to San Gabriel del Yungue, the first Spanish settlement in New Mexico, established four miles downriver from the Pf\u00e4ffles' ranch in 1598. But Carol was also honoring her own migration story. The Episcopalian school of St. Mary's in Peekskill, New York, was built on Mount San Gabriel, high above the Hudson River. It is possible that during a visit to Winfred Douglas's home at the school on Mount San Gabriel, Carol had begun to imagine her journey to the Wetherills and a new life in the Southwest.\n\nBy spring, Carol had produced a brochure about San Gabriel Ranch that became part of the Santa Fe Railway's postwar campaign promoting tourism in the Southwest. The ranch letterhead was graced with the simple \"Little Bird\" petroglyph found in abundance in the ancient villages on the Pajarito Plateau. During their three years at the Rancho Ramon Vigil and Bishop's Ranch, Carol had assembled a large mailing list, and she sent out announcements of the new guest ranch to former and prospective clients and to friends in Chicago, Boston, New York, and Southern California. By summer of 1920 the dude ranch called San Gabriel was on the map.\n\nSan Gabriel's resident staff included the most experienced cow punchers, dude wranglers, wilderness guides, and automobile drivers and mechanics found in New Mexico: horse wranglers Jack McKinley, Alfred \"Slim\" Jarmon, and Sandy MacLean; driver, auto mechanic, and guide and historian Orville Cox; and two cowpunchers remembered only by their nicknames, Shorty and Weary. Roy had worked alongside most of these men in the forest service and the Rocky Mountain Camp Company, and with Jack Lambert as ranch manager, and Carol as general manager and bookkeeper, San Gabriel had an excellent and respected foundation. Carol's English maid, Alice Pring, and Roy's mother, Gaga, also moved to the new ranch.\n\nFive years had passed since Carol's arrival at the Wetherills' trading post. Since that time, she had lived in hotels and rented rooms in Santa Fe, and at ranches and outposts in some of the most remote and beautiful settings found in the Southwest. But until San Gabriel on the Rio Grande, Carol had not lived in a place she could call her own.\n\n**Figure 48.** Roy and Carol Pf\u00e4ffle, San Gabriel, ca. 1922.\n\nLois graduated from St. Mary's in the spring of 1921. After saying her good-byes to Dr. Jo and Canon Douglas, who she would see out west within a month or so, Lois boarded the train for home. Her mother had come to visit Lois in New York, but Lois had not been back to New Mexico in two years. Jack Lambert was waiting exactly where she had left him\u2014on the train station platform in Lamy, New Mexico. Jack still had eyes only for Lois. Lois was nearly twenty-one and, after seeing a bit of the modern world, knew what she liked and whom she loved, and several weeks later, on June 23, Lois married her favorite cowboy, E. V. \"Jack\" Lambert. \n\n#  _15_\n\n##  DEATH IN PARIS\n\nOne will never remember A greater thing when one dies Than sunlight falling aslant long rows of corn, Or rainy days heavy with grey sullen skies.\n\n##  \u2014 _Alice Corbin_\n\nTHE BURLINS SPENT EACH SUMMER in their simple adobe home in old Santa Fe, but Natalie's speaking and lecture engagements, and Paul's need to participate in the contemporary art world, necessitated long absences from New Mexico. If Natalie and Paul Burlin came for a retreat at Carol's dream ranch on the Rio Grande, it was never recorded.\n\nUnfortunately for Paul Burlin and his contemporaries in the modern art movement, after the Great War the American public retreated to the intellectual and artistic conservatism popular in the years before the war. Even back east, the progressivism initiated by the Armory Show fell from favor as Americans sought to return to prewar normalcy. Like the paintings of his avant-garde colleagues, Burlin's ultramodern, somber canvases became the target of violent verbal attack by critics in New York and Philadelphia. Burlin had weathered harsh criticism before, but by 1920 he had lost all patience with American conservatism, and possibly with its not so subtle anti-Semitism as well. Bohemian Santa Fe and the clean, uncluttered faraway atmosphere of New Mexico had given his artistic spirit space and refuge in the years before the war. But in late 1920 even Santa Fe was feeling too American for Burlin, and he told Natalie he wanted to move back to Europe where modern art was celebrated instead of chastised.\n\nBurlin departed for Paris in the winter of 1921. Natalie remained alone in New York to promote her new book, _Songs and Tales from the Dark Continent_. In May, she sailed for Europe and joined her husband in London. Burlin was reenergized by Great Britain, but Natalie was unable to engage her artistic passion in soggy England. Before leaving America, she had told her family that she wanted to expand and revise _The Indians' Book_ , but in London, Indian music and the physical and spiritual landscape from which it came, like Natalie's three-room adobe house on the hill in old Santa Fe, seemed to belong to a former life.\n\nMary Austin was in London promoting _The Trail Book_ , and during a conversation with her Santa Fe neighbor, Natalie realized how her mind and heart atrophied when she was not engrossed in creative work. \"I know I have been rather spineless here,\" Natalie wrote Mimsey, \"but I can't seem to get up steam.\"\n\nNatalie ached for her work, her home, and her sense of purpose, all of which were deeply rooted in the soil and people of the American Southwest. But she soldiered on, the supportive wife, and in July went with Paul to Paris to set up house in the City of Lights.\n\nAfter the war, Paris quickly established itself as the cultural capital of the world, and the Burlins joined the migration of English-speaking expatriate artists and intellectuals drawn to the city's beauty and history, and to the respect accorded the arts by the French. In Paris, Paul Burlin was welcomed into a community of painters who were publicly celebrated for their innovation and experimentation. And Natalie found an enthusiastic audience for her work with American Indian and Negro music.\n\nLife in postwar Paris was a perfect fit for the Burlins. Natalie wrote to George Peabody of the city's pulsating atmosphere, \"full of artists, musicians, all bustling with ideas and new expression\" that she found \"encouragingly purposeful.\" The Burlins moved into a comfortably appointed pension on the rue Jeanne d'Arc and settled into a stimulating and satisfying community of like-minded artists and intellectuals. The Burlins were introduced to the movers and shakers of bohemian Paris by a friend of Paul's, the French painter Albert Gleizes, and his wife, Juliette. Gleizes was the recognized founder of Cubism (he cowrote with Jean Metzinger the movement's treatise, _Du \"Cubisme,\"_ in 1912), and was among those modern French painters brought to America and the Armory Show by Walter Pach. Gleizes had joined Pach, John Sloan, George Bellows, Rockwell Kent, Homer Boss, Maurice Sterne, and other friends of Natalie and Paul's to organize, in 1916, the American version of the French Society of Independent Artists.\n\n**Figure 49.** Natalie Curtis Burlin, ca. 1921.\n\nIn the spring and summer of 1921, Albert Gleizes guided the Burlins through Paris. Natalie, who spoke French with adequate fluency, also went with Paul into the French countryside. Although its population and landscape was war-ravaged and scarred, she was encouraged by the spirit of the French people, and by the palpable intellectual excitement surrounding the artists and writers working in Paris. \"Everybody in the art-world is here,\" Natalie wrote to Peabody and to her family back in the States, \"and it is a great source of inspiration and development to us both.\"\n\nBy midsummer, Natalie's creative ennui was replaced by the effervescent fire that always presaged a new chapter in her work. In the canvases of Picasso, Braque, and the French moderns Natalie saw striking similarities to the abstract art and design elements of Indian America. She wrote Mimsey about these French painters who had \"broken away from representation _completely_ and _entirely_ ,\" and used nothing in their paintings that recalled a visual object \"any more than designs on Indian pottery look like visual rain-storms, or real corn-fields.\"\n\nParis was also experiencing a modern movement in music. Futurist instruments used in a concert at the Th\u00e9\u00e2tre des Champs-Elys\u00e9es were to Natalie the musical equivalent of modern painting. Natalie penned an article for the _Freeman_ titled \"A Futurist Experiment\" about industrial society's need to nurture and embrace the spiritual and emotional complexities of the individual. She understood the need of contemporary musicians to throw off the shackles of the past and make \"music unfettered by the present orchestral limitations.\" But she concurrently recognized how humans living in the postwar world were overwhelmed and numbed by the sterility of modern machines. Music continued to be a source of healing. \"There is a cosmic music . . . all about us,\" she wrote, \"full of beauty to which we are only half awake.\"\n\nGleizes personally presented Natalie's books about African American and Indian music to French publishers. She was invited to lecture to Paris audiences about her life and work among the Indians of America, and by midsummer European editors expressed interest in translating and publishing new editions of her books.\n\nIn early September, Alice joined Natalie in Paris and moved into a small hotel on the Left Bank for a several months' stay. Alice was in Paris to attend with Natalie the International Congress of Art History held at the Sorbonne in September and October. Alice was the appointed secretary for the American delegation, which counted only four delegates (tiny Belgium had seventy-eight representatives): Harvard music professor Edward Burlingame Hill; respected but conservative painter Cecilia Beaux, who was in her seventies; secretary Alice Klauber; and the American delegation chairman, Robert W. de Forest, president of the Metropolitan Museum of Art. (For reasons never understood by Alice, Chairman de Forest did not make it to Paris for the meeting.) Natalie had been invited to give a lecture at the Sorbonne about her work in American folk music, but she was not an official delegate to the conference.\n\nAlice's California friends Michael and Sally Stein, like Michael's sister, Gertrude, were still living in Paris. Leo Stein remained in America, exploring the bohemian colonies of Taos and Santa Fe. After his first visit in 1918, Leo, smitten by New Mexico, bought his own adobe casita in Taos and began to work in \"the only landscape . . . where Nature is as aesthetic as Art; the only landscape that can compete with the great painters.\"\n\nIn Paris, Walter Pach was a valued and popular member of the artistic and literary coterie that passed in and out of Gertrude Stein's salon at 27 rue de Fleurus. After the war and throughout the 1920s, the home of Stein and Alice B. Toklas was the favored territory for the American expats Stein called \"the lost generation\" (Ernest Hemingway, F. Scott Fitzgerald, Juan Dos Passos, Sherwood Anderson, et al.) whose work would become synonymous with the great literature of the twentieth century.\n\nWithin several days of her arrival in Paris, Alice received a telephone call from Magda Pach, Walter's wife, asking her to come to their apartment before they left on an autumn trip to Germany. Magda and Walter Pach had a six-year-old son and a stable and satisfying life in Europe. Alice spent just one hour with the Pach family, but the visit was enough to renew their friendship. Alice found Pach to be a happier and \"better rounded off' man than the person she had known years before. Now thirty-eight, he was among the world's most authoritative and influential writers about modern art. Following their brief reunion, Alice and Walter Pach began a sporadic, tender, and respectful correspondence about art and life that continued into the 1940s.\n\nIn October, Alice and Natalie went \"by bus, car, + taxi\" to see Michael and Sally Stein at their spacious and airy apartment on the rue Madame. Like Leo and Gertrude, the Michael Steins had been collecting modern art since the early years of the twentieth century, and the walls of their Paris apartment showcased the finest modern painters. (In 1906 the art collection in their San Francisco home included the first three works by Matisse brought to the United States.) Although Alice remained a close friend of Michael and Sally Stein, her 1921 diary does not mention a visit to Gertrude and Alice Toklas, whom she had last seen when they lunched in Italy in 1912. Natalie and Paul Burlin saw Leo Stein several times in New Mexico, but if Natalie ever met Leo's famous sister Gertrude in 1921, it was not recorded.\n\nThroughout the fall, Paul Burlin left Paris for extended painting and sketching outings in the French countryside. Alice and Natalie shared the glittering days of a very warm autumn in Paris with long walks along the Seine, leisurely lunches at sidewalk caf\u00e9s, and evenings at concerts and the opera. They took cabs, buses, and trains in, around, and out of the city to visit the grand historic sites, famous gardens, and museums of Paris and its environs. They were joined by their friend Alice Chaffee, who was in Paris for several months (and who with Carol and Roy Pf\u00e4ffle had witnessed Natalie and Paul's vows before the justice of the peace in Santa Fe four years earlier).\n\nThe meetings of the art history congress began the third week of September, and for the next several weeks, the two Alices and Natalie attended dozens of talks and lectures at the Sorbonne. On Monday morning, October 3, Natalie was scheduled to deliver her much-anticipated lecture about American folk music. In the weeks before the event, she was uncharacteristically nervous about her presentation. Unlike most of the English-speaking presenters, Natalie had decided to give her paper in French. Juliette Gleizes assisted her with the translation and coached her pronunciation. Even with Juliette's expert help, Natalie remained anxious about the event, and confessed in a letter to Peabody that it was \"a petrifying job to give a talk in a language not your own.\"\n\nBut Natalie wasn't just nervous about speaking to an international audience in French. She was uncomfortable about the order of the program, which placed her lecture directly after Professor Hill's. Edward Burlingame Hill was a composer and assistant professor of music at Harvard, where he had graduated in 1894, one year after Natalie's brother George. Hill's students would include several who would achieve considerable fame, notably Leonard Bernstein. Natalie respected Hill, although he, like fellow American delegate Cecilia Beaux, was an academic, a traditionalist, and a confirmed conservative on artistic matters. Hill's and Beaux's opinions on American art were diametrically opposite Natalie's, and their speeches to the Sorbonne congress would venture that America was too young to have its own music or art traditions.\n\nBeaux, an accomplished portrait painter and the first woman elected to the National Institute of Arts and Letters, made no secret of her disgust for the French moderns. She especially disparaged Gleizes and the Cubists, whose work she found \"irritating, violent, exaggerated, and produced by mad idiots.\" Once while standing before a canvas in a Paris gallery, Beaux remarked to the woman standing next to her that the avant-garde painting \"looked like nothing but a map with odd splashes of color stuck about.\" The woman was Gertrude Stein, and their conversation ended with Stein's curt retort \"This is realism, a still life group.\"\n\nNatalie knew that Hill's lecture would assert that American composers (including him) were incapable of creating music that could be characteristically called American because America had nothing resembling its own music. In a private conversation, Hill told Natalie that \"the reason we haven't any great music in America is because we have no folk music. I suppose you'll say we have?\"\n\nNatalie told the professor that America certainly did have its own folk music and that her presentation would dispute the premise of his lecture. But later that night, Natalie realized she could not in good conscience get up before the audience at the Sorbonne and bluntly contradict and possibly embarrass Professor Hill. Hill was the formal American designate in music. Natalie was just an American in Paris who had been invited to talk about her self-directed work among the American Indians and Negroes. Natalie went to the presiding officer and asked to be removed from the program. He refused. American folk music, he told her, was of particular interest to this year's assembly, at least among the non-Americans.\n\nNatalie and Alice sat together in the audience and listened to Hill deliver his address. Hill had made one modification to his thesis: instead of his rigid declaration that there was no folk music in America, he explained that while there was folk music _made_ in America, it belonged to the Indians and the Negroes, and thus was not _American_ folk music.\n\nHill's insulting, elitist, and racist doctrine provided Natalie with the fuel she needed to deliver the lecture of her life. Natalie took the podium and gave the speech she had so carefully rehearsed in French. But she added an impromptu response, in French, to Hill's claim that Indians and Negroes were not American. Natalie's twenty years in the field collecting and absorbing the stories, songs, myths, art, and life of Native and African Americans came with her onto the Sorbonne stage where she, by the accounts of those present, spoke eloquently and passionately about American art and music.\n\nNatalie dismantled Hill's assertion that Negroes and Indians were not Americans, pointing out that American Negroes, \"good enough 'Americans' to die for American ideals in our wars,\" had a long tradition of folk music and songs \"that are the very voice of our South.\"\n\nNatalie had long harbored resentment at \"the notion that only New England with Harvard College [not forgetting it was her Uncle George and both brothers' alma mater] at its 'hub'\" was considered American. \"All America is not New England\u2014it is an agglomeration of races that has given us a folk-lore almost as rich and diverse as that other agglomeration of races that we call Russia.\" The music of America, she told the multinational audience, was not found in universities and schools \"but out in the great expanse of territory that stretches from the Atlantic to the Pacific Oceans, and from Canada to Mexico.\"\n\nHalfway through her speech, the French and Belgian delegates were enthusiastically nodding their heads with approval. \"I didn't mean to get into any kind of controversy,\" Natalie would later tell friends, \"but when I sang those songs about the American maize, about the big, hot American sun that rides his turquoise horse across our Rocky mountains; those chants that have come out of _America_ itself\u2014the audience was literally electrified.\"\n\nProfessor Hill's myopic assertions about what exactly could be called American music were dismantled by a diminutive, spirited, self-taught, self-directed scholar whose credentials as an ethnomusicologist had been earned over two decades in the farthest reaches of the American outback. Natalie was sorry that her speech had \"tread all over Mr. Hill's toes,\" but the audience's response to her performance\u2014she sang several Indian songs and demonstrated the use of Indian rattles\u2014and the groundswell of support immediately following her speech alleviated her misgivings. Her speech may have offended Hill and his conservative colleague Cecilia Beaux, but the international delegates showered Natalie with compliments and praise. \"We not only know you\u2014we felt you,\" a Greek woman told Natalie. \"You should go all over this world with this voice from the America that we do not know.\"\n\nAlthough Professor Hill's theories had been summarily dismissed by Natalie's fervent and informed remarks, Natalie's gentle charm, sincerity, and great, grand, genuine love of America and Americans won Hill's respect. \"We parted very good friends,\" Natalie wrote in her missive about the event, \"even though [Hill] accused me of saying the 'schools and universities had never done anything for music in America'!\"\n\nNatalie was resolute in her belief in the value of the people, landscape, and music of indigenous America. \"I feel so proud of the wonder and force of the American background,\" Natalie summarized in her letter home, \"the big hinter-land of our Country that is so striking in character, so forceful, so dynamic, so red-blooded. It's not what the European-educated composers do that counts, but the thing that's of the land itself, the thing that's in the soil.\"\n\nAlice returned to her hotel (she had moved into the Hotel de L'Univers on the rue Croix-des-Petits-Champs) and that night, alone in her room, wrote in her diary how Natalie's presentation \"went off with her usual finesse.\" For the next three weeks, Natalie basked in the afterglow of her performance at the Sorbonne. There was renewed interest from editors and publishers for a French version of _The Indians' Book_ , and Natalie talked about returning to her work on American folk music. After the conference ended, Alice and Natalie, often with the Gleizes and the Steins, lunched at caf\u00e9s (their favorite was the Medici Grill), browsed bookshops, listened to music, and strolled the boulevards. Paul Burlin, who had set up a studio in Marseilles, spent most of that gloriously sunny October painting in the French countryside.\n\nOn Sunday, October 23, the unseasonably balmy weather overnight turned cold and rainy. Paris was famous for such a quick change of seasons. Young Ernest Hemingway, experiencing autumn in Paris for the first time that October, wrote how the bad weather \"would come in one day when the fall was over. You would have to shut the windows in the night against the rain and the cold wind would strip the leaves from the trees.\"\n\nAlice went out that first cold, wet evening with friends to see a performance of _The Three Musketeers_. When they emerged from the theater, Paris was deluged with rain, and after a long, chilly wait by the curb, Alice found a taxi and gratefully reached the warmth of her tiny hotel room. The next morning, she slept in and wrote letters to her sisters in California. Just about the time she thought she ought to call Natalie and ask her to join her for lunch, Alice was summoned to the phone in the hotel lobby. She picked up the phone expecting Natalie to be the caller. But it was Natalie's \"hotel woman,\" and in French and broken English, the woman told Alice that Natalie had been hurt. Worried and confused, Alice returned to her room, dressed, and went by taxi to Natalie's hotel. It was only when she reached Natalie's rooms that Alice understood that Natalie had been hospitalized the night before with very serious injuries.\n\nThe previous day, while Alice was at the theater, Natalie had spent the afternoon with friends and then taken a bus across Paris back to her hotel. At 5:45 p.m., at the corner of the boulevard du Mountparnasse and the rue Campagne-Premi\u00e8re, Natalie got off the bus. It was dusk and pouring rain, and as she walked into the street, a motor car struck her down. The driver, a Dr. Poujand speeding down the boulevard to a medical emergency, never saw the petite woman in the rain near the bus stop.\n\nFrom Natalie's hotel, Alice telegraphed Paul Burlin in Marseille. The authorities had tried to reach him the night before, but had sent the message to the wrong address. Alice telephoned the Gleizes and with them went to the hospital, where they all expected to find an injured but stable Natalie resting in a bed. Instead, they learned that Natalie had never regained consciousness and had died two hours after the accident.\n\nAlbert and Juliette Gleizes went to meet Paul's train. Numb and dazed, Alice, alone in the hospital room with Natalie's body, wondered why, the day before, she had had no presentiment of the accident. \"It is unbelievable that Natalie should have been snuffed out like that and I dining within two blocks and no premonition of any sort . . . I was walking . . . feeling happy!\"\n\nIn the terrible hours and days that followed, Juliette Gleizes calmly and compassionately guided Alice and a grief-stricken, nearly dysfunctional Paul Burlin through the French police reports and official documents, and the planning of Natalie's cremation and memorial service. Even as Alice suffered and grieved Natalie's passing, she understood that Natalie's quick death was precisely the way Natalie would have wanted it. \"Natalie died without knowing anything of the pain and distress of it all,\" Alice wrote to Mimsey Curtis. \"She looked in death serene and beautiful\u2014her slight smile, the great tranquility of the room in which she rested, the beauty of the flowers banked about her would have meant much to you as they did to me. I went often, and I stayed long alone with her for the very peace it brought me and I am sure that the veil that separates us is a very thin one. She never feared death and said that she hoped the end would come quickly when it did come. The dear blessed girl has gone without pain or suffering.\"\n\nNatalie's memorial service was held on All Saint's Day, October 31, at the P\u00e8re Lachaise Cemetery in Paris. Albert Gleizes read selections from Mercereau, \"Come raggio di sol\" was performed beside the coffin, and the hymn \"Abide with Me\" was sung twice by the mourners. After the service, Alice, Alice Chaffee, Michael and Sally Stein, and Albert and Juliette Gleizes walked alongside Paul Burlin behind the coffin the several miles to the crematorium. Some fifteen thousand people came out to honor Natalie that cold day.\n\nIn the weeks after Natalie's death, Paul Burlin stayed in the Gleizeses' home. Alice was cared for by Michael and Sally Stein.\n\nNews of the tragedy reached George in a telegram from his sister Connie on October 28, five days after Natalie's death. At the time, George and Lora were on holiday in La Jolla, dining and taking in the sea air with several of Alice's friends, the Hillyers and the Nivens, and a Miss Mary Bickel. The day after the telegram with the horrible news reached him, George received a last letter from Natalie written in Paris on October 7. And several days later, George and all of the Curtis family received a final missive from Natalie, written on October 16, which shared the details of her phenomenal success at the Sorbonne.\n\nGeorge was \"knocked all of a heap\" by the news of the brutal accident and wrote in his diary that he \"felt dull and stupid\" as he went about the chores and duties of the ranch. The random senselessness of the accident, and the sudden loss of his cherished sister and best friend, devastated George Curtis. He wrote his younger brother, Bridgham, \"What a brilliant mind, what character, what driving force, with the tender sympathy and understanding that found the best in everyone, and gave her power to help people to help themselves. The world seems to me dingy and mean, without her. What a blind, brutally stupid waste there is in her taking-off. . . . It's enough to make the human race hold an indignation meeting and pass resolutions of censure.\"\n\nThe Curtis family found some consolation in recalling Natalie's calm acceptance of life and death, and her unshakable belief in what her Indian friends called the Great Spirit. \"She never had the slightest feeling about death,\" Mimsey told George in a letter, \"and would be perfectly willing to go at any moment\u2014except to leave Paul. I never met a person so sure of the circumstance of the soul.\"\n\nPaul Burlin confided to Mimsey he had never known happiness in any abundance until he met and married Natalie. To Paul, Natalie was a saint and \"self abnegation was the key note of that sweet gentle woman\u2014my wife.\"\n\nWith Natalie's younger sister Connie, Mimsey climbed on a boat to France and helped a bereft Paul pack up Natalie's possessions. Alice assisted however she could and then, on November 17, said farewells to Mimsey and Connie, who accompanied Paul, carrying Natalie's ashes, on the ocean voyage back to America.\n\nWinfred Douglas was called upon to conduct the private graveside service at the family cemetery in Providence, Rhode Island. George and Lora did not come east, but her brother Bridgham, her sisters, Connie and Marian, and childhood friend Bessie Day stood graveside with Mimsey and Paul Burlin. Canon Douglas made a few remarks about the life and person of Natalie Curtis Burlin, and Paul placed Natalie's ashes into her father, Bogey's, grave. And it was done.\n\nIn the next months and years there were a dozen concerts and events given in Natalie's honor on both coasts and in Santa Fe. At the memorial service held at the Hampton Institute, Elbridge Adams spoke about Natalie's unique character, and read a letter from Kurt Schindler announcing the establishment of a Hampton scholarship in Natalie's name.\n\nTheodore Roosevelt had died two years before Natalie. At sixty years of age, his death was considered premature for such an energetic and vital man who brought so much to his world. When Natalie learned of Roosevelt's death, she remembered how he had presaged his death in 1913. \"I have only about six more years of the strenuous life before me,\" Roosevelt told Natalie during a conversation in the shade of the peach trees below First Mesa. At that time, Natalie found Roosevelt's \"unconscious measuring of his days\" fatalistic. Six years later, she had a new understanding of Roosevelt's words. \"Though his going was untimely and tore a gap in American life that no other personality can fill, it seemed characteristic. Decrepitude and age were not for him, whose mind and body so loved hard work and hard play.\"\n\nIn the months following Natalie's death in Paris, her final musings about Roosevelt's life and death offered her friends and family a comforting perspective as they grappled with her sudden and untimely passing at the age of forty-six, the zenith of her life as a woman and a creative artist. \"And so [she] will remain in our minds a powerful figure vibrant with life to the end, even as we saw [her] on the Hopi buttes at dawn\u2014up and on the mountaintops before the world was half awake.\"\n\n#  _16_\n\n##  CITY OF LADIES\n\nThey do as they please, they say what they think, and nobody cares, for everyone is busy doing likewise.\n\n##  \u2014 _Mabel Dodge Luhan_\n\nEXACTLY WHEN AND HOW Carol learned of Natalie's death is not known. Winfred Douglas may have wired the Pf\u00e4ffles with the awful news. Just two months before, the canon and his son had stayed at San Gabriel following a month-long outing with Roy in the \"savage desert\" of drought-parched Arizona. Roy and Douglas were once again trailblazing (this time across Hopiland to Kayenta and the Rainbow Bridge, south across the Kaibab plateau and into the Grand Canyon, a route that included a nerve-wracking human and pack animal crossing of the new suspension bridge over the Colorado River). Carol and Roy's sister Lois, who had lived with the Douglases in Peekskill, were practically kinfolk to Douglas and his wife, Dr. Jo. And Roy, who was \"always patient\" with Douglas \"around the horses,\" and with whom Douglas had now twice crossed the character-testing Rainbow Trail, had a special place in the canon's life. \"I had a whole month of comradeship with Richard,\" Douglas wrote to his wife at the end of the 1921 expedition, \"who is to me the perfect man friend\u2014one in ten million.\"\n\nIn late autumn of 1921, the _Santa Fe New Mexican_ and _El Palacio_ , the publication of the Museum of New Mexico, published lengthy obituaries of Natalie, and a local memorial was organized by Carol Pf\u00e4ffle, Alice Corbin, and their friend and part-time Santa Fean Elizabeth Shepley Sergeant. Bridgham Curtis's farewell to his sister, given several weeks before at her memorial service in Providence, was read by Alice Corbin at the service in Santa Fe, attended by a multitude of Natalie's many friends and colleagues from the Santa Fe Society of the Archaeological Institute, Hewett's School of American Research, the Fine Art Museum, the pueblos of the Rio Grande, and the extended community of Santa Fe artists and archaeologists.\n\nPaul Burlin did not attend the Santa Fe memorial, and he would never return to Santa Fe and the house near the sandy corner of Buena Vista and Old Santa Fe Trail. After traveling back to the United States with Mimsey and Connie Curtis, and interring Natalie's ashes in the family plot in the North Burial Ground in Providence, Paul went back to Europe and set up a studio in Germany, where he could live and work \"where most of the 'modern' painters sell.\" The people and place of New Mexico, once the source of creative and personal inspiration and affirmation, became synonymous with everything Burlin had loved and lost when Natalie died.\n\nBy June 1922, Burlin rented Natalie's first and last home to Witter Bynner, a Harvard-educated poet and scholar who had worked at _McClure's_ magazine, where he was a colleague of Willa Cather's. Although Bynner was acquainted with Robert Henri and John Sloan in New York, it was Alice Corbin who convinced him to visit Santa Fe. During his early spring stopover in the capital, Bynner fell ill with influenza. Alice urged him to move into Sunmount Sanatorium, where, she assured him, he would be safer than in a New York trolley car. After six weeks at Sunmount, Bynner wanted to remain in old Santa Fe, where he found a community of resourceful people \"washed clean of the war. . . . I was writing to friends who lived on another planet.\" Bynner learned that the Burlins' former home in the Hendersons' neighborhood was for rent, and he moved into the house by June.\n\nBynner loved Natalie's \"little shack\" and its \"three quaint rooms\" on sight and wanted to make the adobe his own, but he couldn't afford Paul Burlin's asking price of $2,500. While waiting for Burlin to lower the price, the house was bought out from under Bynner by Margretta Stewart Dietrich.\n\nDietrich, like Bynner, was new to Santa Fe, and had come with her sister, Dorothy Stewart. Dietrich was a Bryn Mawr graduate and classmate of another New Mexico newcomer, Elizabeth Shepley Sergeant. She was also the wife of the former governor of Nebraska, the current president of the Nebraska Women's Suffrage Association, and had a sizable income with which she was buying and renovating Santa Fe properties.\n\nLike Bynner, Dietrich fell in love with Natalie's \"sweet little old adobe house\" and its undulating walls plastered with \"pinkish mud from La Cienega.\" For the next three years, Dietrich rented the dwelling to an indignant Bynner, who believed the house was meant to be his. Bynner relentlessly pestered Dietrich to sell him the adobe, and in 1925 she relented and sold it to him.\n\nBynner lived happily for the rest of his life in the house in the midst of the \"mumble jumble of chumminess\" that was Santa Fe's art and archaeological community. Bynner also purchased the adjacent property and enlarged the Burlins' small hillside lot to a full acre. He eventually transformed Natalie's three-room casita into a palatial compound of more than twenty-five rooms in several multistoried buildings. Over the next four decades, Bynner and his partner, the writer Willard \"Spud\" Johnson, hosted frequent and legend-worthy parties in the house. The once-humble casita of newlyweds Natalie Curtis and Paul Burlin was destined to become a Santa Fe landmark listed on the National Register of Historic Places.\n\nIn September 1922, when Bynner was still just a renter, Mabel Dodge Sterne brought D. H. and Frieda Lawrence to visit. The Lawrences were just off the train from the east, and because it was a long and challenging drive to Taos through the Rio Grande Canyon, Mabel asked Bynner to accommodate them all for the night. Bynner welcomed Mabel and her friends, and Frieda and D. H. Lawrence spent their first night in New Mexico in Natalie's adobe a few blocks from the Santa Fe plaza. Although Bynner and the Lawrences were destined for a famous falling-out with Mabel, everyone had a lovely time that evening. Bynner and the Lawrences became friends, and within the year, much to Mabel's ire, Bynner would travel with them to Mexico.\n\nLeo Stein, now a seasonal resident of Taos, had encouraged his friend Lawrence to come to New Mexico and immerse himself in what Stein considered a transcendent landscape. Lawrence was not disappointed and was captivated by Santa Fe and Taos. \"New Mexico was the greatest experience from the outside world that I have ever had. It certainly changed me forever. . . . The moment I saw the brilliant, proud morning shine high up over the deserts of Santa Fe, something stood still in my soul, and I started to attend.\"\n\nIn May 1922, the Museum of Art opened its first annual exhibition of western artists. The seventy-eight artists included local favorites John Sloan, William Henderson, J. H. Sharp, Sheldon Parsons, Will Shuster, Willard Nash, and Californian Alice Klauber. Alice did not return to Santa Fe for the gala opening, but remained in self-imposed exile in Europe.\n\nAfter Natalie's death, Alice remained in damp, dreary Paris until mid-December. She was not sleeping well, and hoping to escape the depression and listlessness that had dogged her since Natalie's accident, she departed Paris for Lyons and Marseilles, and eventually Florence, a city where she had been happy years before. But Alice was restless and unfocused, and soon left Italy for Germany. Her melancholy would persist for two years, and she would spend those years a nomad in Italy, Spain, Germany, Switzerland, northern Europe, England, and France. \"So may this shadow on my soul,\" Alice wrote in a poem, \"Lift away when I am whole.\"\n\nIn the fall of 1923, while traveling through Belgium and France, Alice was reading _The Ladybird_ , D. H. Lawrence's newest collection of novellas (published in the United States under the title _The Captain's Doll_ ). In her diary she quoted Lawrence several times. \"A man can only be happy following his own inmost need.\" \"Exactly, I will lay down the law for nobody, not even myself. The thought of death and the afterlife saves me from doing any more. . . . As the thought of Eternity helps me.\"\n\nNeither contemporary literature nor the great art and architecture of Europe consoled Alice. She was in her mid-fifties, and the sudden death of Natalie initiated soul-searching reflection on her past, present, and future. Alice began to write poetry about love and loss, nature and death that would be privately published in 1928.\n\nThree years after Natalie's death, at the end of 1924, a much older, well-traveled, and certainly transformed Alice moved back to America and her home in San Diego. Like everyone who had known and loved Natalie, Alice would never be completely at peace with Natalie's abrupt passing. But as her poetry revealed, Alice did come to an appreciation of death as the eternal mystery, and life\u2014and love\u2014as precious and fleeting. Although she kept the details and names to herself, and her letters to male friends like Pach were destroyed or lost, Alice left intimations in her poetry about her love life and discreetly alludes to romances that even in middle age continued to frame the landscape of her heart. In \"Relapse\" she wrote:\n\n> I make each day a weary task,  \n>  To close my heart to him, as then!  \n>  And now, as punishment but ask  \n>  It break, and let him in again.\n\nAnd in her poem \"Tryst\":\n\n> I feel the sweet unclosing leaves  \n>  Refreshing in the limpid dew.  \n>  I hear the lily sigh. She grieves  \n>  That I should wait the night for you.\n\nIn the month after Natalie's death, Carol Pf\u00e4ffle went into the desert to grieve. After the guest season of 1921 ended, Carol, possibly with Roy, left San Gabriel Ranch and went to Hopiland and the Grand Canyon. One November morning, in the Harvey Curio room at Hopi House on the South Rim of the canyon, Carol had an extraordinary encounter with several Hopi singers. Carol shared this chance meeting with Elbridge Adams in a letter written on her forty-third (and Roy's thirty-sixth) birthday. \"Just after news came of Natalie's death I met a Hopi whose uncle was Natalie's teacher for some time. I told him of her death. After a long silence he said 'But she cannot die. She is singing now\u2014somewhere with her Hopi friends.'\"\n\nIn spite of the room being filled with tourists, the Hopi elder and his friends began to softly and reverently chant in the Harvey gift shop. Carol recognized the songs\u2014Natalie had taught them to Carol years before. When the singers were silent again, the elder Hopi told Carol, \"[Natalie] sang like Indian. Have to have spirit of Indian for white woman to sing that way.\"\n\nAdams shared Carol's letter with the story of her serendipitous and affirming encounter with the Hopi singers with Mimsey Curtis, who sent it on to George. Carol wrote the letter as winter descended on the Rio Grande Valley. At the time, she was struggling to maintain a hopeful attitude about San Gabriel Ranch and its large mortgage, about her husband and his late-night poker games, and about her mother-in-law, Gaga, who, she confided to Adams, \"while the dearest mother that ever lived, is a born pessimist.\"\n\nFriends back in New England were concerned about Carol's decision to live in rural New Mexico. \"My good friends write with a kind of worry and suggestions that my life is not as rich as they might wish it to be,\" she wrote Adams. Carol wanted to pack up her friends and \"put them down in this blessed valley where they may see for themselves how wonderful my life is.\"\n\nSan Gabriel would be quiet until the start of the tourist season in late March. The New Mexico winter and the absence of guests offered a welcomed respite for Carol and Roy, Gaga, and newlyweds Jack and Lois Lambert. Even so, empty beds and idle saddle horses meant there was no income, and if the spring and summer did not bring a full house of dudes and dudettes, it could mean foreclosure on the Pf\u00e4ffles' 34,000-acre dream ranch on the banks of the Rio Grande.\n\n**Figure 50.** Carol Stanley Pf\u00e4ffle and Jack Lambert, Canyon de Chelly, ca. 1922.\n\nIn spite of everything, Carol was very much in love with Roy and his \"Coeur de Lion,\" and told Adams she would not let their life be ruled by petty concerns and financial worries.\n\nAt San Gabriel, the only apparent remnant of Carol's former life in Boston was the Steinway grand piano in the hacienda's front room. Carol played music for the ranch guests and taught piano to the Spanish American children at San Juan Pueblo and at the local mission school. Carol and Roy traveled to Chicago and Boston at least once each year. To her friends and former colleagues back east, Carol had chosen an unconventional husband and a tenuous, even risky lifestyle. But it suited Carol.\n\n\"Think of us having music,\" Carol wrote at the close of her birthday letter to Adams, \"going quail and duck shooting, riding over the foothills and getting a good long breath to last us over what we pray may be a grand rush from early spring to late fall.\"\n\nPerhaps it was Carol's unwavering optimism in the face of so much uncertainty that prompted her pueblo neighbors to christen her with the Tewa name T'apokwj, Sun Lake.\n\nIN SANTA FE AND ALL of northern New Mexico, the 1920s saw a tide of tourists and new residents, and each year Carol and Roy's prayers were answered and a grand rush of guests did indeed descend upon San Gabriel Ranch from early spring until late fall. The Pf\u00e4ffles' ranch on the Rio Grande was featured in virtually every national article about New Mexico tourism, and San Gabriel Ranch received top billing in the travel brochures produced by the Santa Fe Railway. Carol commissioned Gustave Baumann, a German-born painter and printmaker who had joined the Santa Fe art colony in 1918, and whose work had won a gold medal at the Balboa Park exhibition, to design woodcuts for the San Gabriel Ranch pamphlet. Each December, the Pf\u00e4ffles commissioned New Mexico artists to create custom-made Christmas cards (including two by Alice Corbin's husband, William Henderson, that featured Roy and his horse) to send to friends, family, and former and prospective clients.\n\nWith Natalie gone, Carol no longer had a room in which to stay overnight in Santa Fe. Because ranch guests often arrived on the late train and needed to remain in Santa Fe before driving on to Alcalde \u2014a several-hour trip in one of the ranch Packards\u2014and because Carol often wanted time away from the guest ranch, in 1926 the Pf\u00e4ffles leased a four-room apartment from the Baca family at 537 Canyon Road. The Pf\u00e4ffles' pied-\u00e0-terre was just over the garden wall from the Prada property under renovation by Margretta Dietrich, and was within walking distance to just about everything and everyone in Santa Fe.\n\n**Figure 51.** San Gabriel Ranch Christmas card by William Penhallow Henderson, ca. 1925.\n\nThe Santa Fe art colony was flourishing. Artists and writers were no longer just summer residents with temporary studios in the palace or rented rooms near the plaza, but were buying plots of land and constructing or renovating their own permanent dwellings. Building with adobe was inexpensive. Mud and adobe bricks, rough-cut lumber, and ponderosa beams for ceiling vigas made up a home builder's materials list. The rustic, handmade artisan quality that was revered as Santa Fe style was doable by even the novice adobe mason. By the 1920s Carol had become proficient with adobe construction, having worked alongside the local crews renovating Bishop's Lodge and San Gabriel Ranch, and was comfortable with every step in the process from adobe brick making to hand-plastering exterior and interior walls with mud made with the colored sands of local arroyos.\n\nThe town of Santa Fe was expanding into the foothills, especially along the Camino del Monte Sol where the Hendersons lived. In the early 1920s Santa Feans watched with amusement the building efforts of five newcomers, _Los Cinco Pintores_ (The Five Painters), and their efforts to create studio homes on the Camino. The five\u2014Fremont Ellis, Willard Nash, Wladyslaw (Walter) Mruk, Jozef Bakos, and William Howard Shuster\u2014came to Santa Fe after hearing of the colony's camaraderie and spirit from their mentors Robert Henri, Randall Davey, and John Sloan. Each of the five was less than thirty years of age when they arrived, and all of them became essential members of the art community.\n\nWith little money or construction experience, but plenty of energy, time, and determination, each of Los Cinco Pintores set out to build a home of his own in the foothills near town. Frequent construction mishaps\u2014toppling of wobbly and unanchored adobe walls, dubious design and infrastructure decisions (the five installed their own electrical and plumbing systems)\u2014were chronicled by their neighbors, who affectionately referred to them as those \"five little nuts in five adobe huts.\" Even the _New Mexican_ frequently commented on the progress of the five's \"interesting studio homes\" that gave the Camino \"the appearance of an enormous anthill.\"\n\nLos Cinco Pintores persevered, and each managed to complete a habitable studio-home. During the summers, the five moved into tents and rented out their handmade abodes to tourists who paid good money to stay in the charmingly rustic casitas. Ellis, Nash, Mruk, Bakos, and Shuster all became nationally respected painters. Nearly a century later, the handmade adobe dwellings built by Los Cinco Pintores remain standing and habitable\u2014and worth a great deal of money\u2014on the Camino del Monte Sol.\n\nSanta Fe and New Mexico in the 1920s also became the destination for educated and emancipated women of means from New York City, Boston, Philadelphia, and Chicago. Natalie Curtis Burlin, Mary Austin, Alice Henderson, Mabel Dodge Luhan (Mabel married Taos native Tony Luhan in 1923), and others discovered New Mexico before 1920, and their high-profile lives attracted scores more women who, like them, were seeking a place where they could work and live free of the gender-based restrictions and expectations endemic to their class and culture back home.\n\nThe modern women who made homes in Santa Fe and its environs became a force to be reckoned with when they joined together and championed cultural and political causes, before and after they won the right to vote. Their influence and sway over Indian land protection issues and the preservation of indigenous arts and crafts, and the ease with which they assumed and maintained power in Santa Fe in the 1920s, gave the City Different the additional designation of the City of Ladies.\n\nElizabeth \"Elsie\" Shepley Sergeant, with whom Carol Pf\u00e4ffle and Alice Corbin planned Natalie's Santa Fe memorial service, had come to see New Mexico after hearing about the charms of the Great Southwest from her good friend Willa Cather. Sergeant was close in age to Carol\u2014thirty-nine when she came to Santa Fe in 1920\u2014and was also born near Boston, in Brookline, Massachusetts. Like Carol and Natalie, Elsie Sergeant had little family income, and in spite of her connections to a well-known and respected New England family (her sister, Katherine, married E. B. White and was the fiction editor for the _New Yorker_ from 1925 until 1960), had to support herself to maintain an independent life. Educated at Bryn Mawr, Sergeant was an early proponent of the social reform and settlement house movement, and worked as a journalist in New York City, where she wrote expos\u00e9s about the sweatshops, slum tenements, and exploitation of the city's immigrant population.\n\nSergeant was a confidant and close friend of Willa Cather, whom she met in 1910 at _McClure's_ magazine. Cather told Sergeant details about her adventures in the Southwest and in 1914 took her to the Museum of Natural History to see the exhibition of artifacts and ceramics that Dr. Edgar Hewett and friends had excavated in canyons in a faraway place called New Mexico. Sergeant was enchanted by Cather's descriptions of the Southwest and wanted to see Hewett's field camps for herself, but she would not get to New Mexico until after the war.\n\nIn 1917 Sergeant was sent to France to write about the Great War for the _New Republic_. She was seriously wounded in the leg by a land mine and had a long recovery in Paris before returning to New York City. Cather was even more insistent that Sergeant needed to go to the Southwest, and in 1920 a wounded and weary Sergeant took Cather's advice and set out to see the landscape of the American West. Sergeant's physical and psychological recovery began on the train journey with a fellow Bryn Mawrter, Gertrude Ely. \"The luminous light that burns on the Arizona desert, out of long miles of untouched sage and sand. . . . Yes, that's where I want to be, on an observation car traveling swiftly into the Southwest. Losing myself in a shimmer of fine dust.\"\n\nSergeant and Ely made their way to Santa Fe and stayed in Tesuque at Bishop's Ranch, now called Bishop's Lodge. The Pf\u00e4ffles continued to organize the expeditions and pack trips for guests at Bishop's Lodge, and \"lazy old yellow Buck,\" Natalie's horse, remained in the Tesuque stable after the Pf\u00e4ffles moved out to Alcalde. Sergeant was introduced to Carol and perhaps met Natalie, too, in the summer of 1920.\n\nSergeant and Ely rented Natalie's gelding and other horses for overnight trips, and Sergeant finally saw with her own eyes Hewett's excavations on the Pajarito Plateau and toured the Tewa pueblos of the Rio Grande she had read about in the Museum of Natural History back in New York. The freedom Willa Cather, Natalie, Carol, and other women had found in the Southwest was indeed a reality, even for a middle-aged, wounded spinster from New York. \"But the moral of the tale is that the real way to see New Mexico is simply to go off on a horse, and then on and on, rejoicing that nobody knows or cares whether you ever come back, taking adventure and beauty and night's lodging as they come.\"\n\nDuring their first summer in Tesuque, Sergeant and Ely decided to become at least seasonal residents and together purchased an adobe ruin down the road from Bishop's Lodge. With building plans drawn up by Alice Corbin's husband, Will, and a crew of local workers, they commenced renovation of the crumbling mud house. By July 1921 Sergeant returned and moved into the partially renovated adobe. Although the dwelling was still barely habitable\u2014she shared the house with a gopher snake that moved freely through the rooms\u2014Sergeant loved the rustic life in the Tesuque hills, and joined the creative community she found a few miles up the road in Santa Fe at the homes of Randall Davey and Alice Corbin.\n\nWilla Cather had told Sergeant that she would find a new identity in the American West. And so she did. \"The truth is, it is enough to _live_ in this country. Just to live. Work isn't necessary for the salvation of the soul.\"\n\nBut Sergeant did work, as she needed to generate income to sustain even a frugal life, and she wrote about her experiences on the exotic frontier of New Mexico in a series of articles for _Harper's Magazine_ titled \"The Journal of a Mud House.\" She also joined in the civic and cultural activities championed by Santa Fe's growing league of vigorously active women. With Alice Corbin, Mary Austin, Carol Pf\u00e4ffle, and the newly arrived (and fellow Bryn Mawr graduates) sisters Elizabeth (Amelia) and Martha White, Sergeant joined the simmering political battle surrounding Indian land rights that erupted with the Bursum Bill.\n\nIn 1921 New Mexico senator Holm O. Bursum introduced legislation that would allow non-Indian settlers living on Indian lands to take legal title to those lands if they could prove ten years of continuous possession. Hewett and the American archaeological community, joined by the Santa Fe art and literary colony, launched a loud and forceful protest to stop the bill. The national campaign against the Bursum Bill was organized by John Collier, the research and publicity agent for the Indian welfare department of the National Federation of Women's Clubs. With the publicity and press brought to the issue by such internationally known writers as D. H. Lawrence and Carl Sandburg, and with the tireless and continuous efforts by numerous regional and local women's organizations, the bill was defeated in 1923.\n\nWith the defeat of the Bursum Bill, New Mexico's artists, writers, archaeologists, and activists realized how their united voices could mobilize and influence public opinion on a national scale. Soon after the Bursum Bill fight, Collier and company formed the American Indian Defense Fund, and in 1925 the Santa Fe art and archaeology community, focusing their energy on the preservation of Indian art, created the Indian Arts Fund.\n\nThe Indian Arts Fund was actually the result of an accident that had occurred in 1922 during a dinner party at Elsie Sergeant's pink-walled adobe hut on the Tesuque River. Sunmount cofounder Dr. Harry Mera (who was serving as the Santa Fe County health officer) and his wife, Reba, Amelia and Martha White, and at least one other guest were seated about Sergeant's rustic supper table when Sergeant's maid dropped an old Zuni _olla_ (jar). The apologetic maid told Sergeant she would clean up the mess and throw the broken jar into the Tesuque River. Harry Mera, a respected amateur archaeologist whose specialty was dating and classifying ceramics, took the pieces of the Zuni jar from the maid and said he would take them to Ken Chapman for reconstruction. The evening's conversation at Sergeant's dinner table became a discussion about the need to protect historic Indian ceramics, and within a few weeks Sergeant, Mera, and friends started the Pueblo Pottery Fund.\n\nThe preservation effort initiated by the Pueblo Pottery Fund was expanded to include the collection and protection of all Indian arts and crafts and was incorporated as the Indian Arts Fund in 1925. The founding members included Chapman and Mera; artists Frank Applegate, Andrew Dasburg, and J. O. Nordfeldt; writers Alice Corbin, Mary Austin, and Elsie Sergeant; archaeologists Samuel J. Guernsey, Frederick Webb Hodge, Alfred V. Kidder, Sylvanus Morley, and Jesse Nusbaum; and Santa Fe attorney Francis Wilson, land manager Nate Stern, Amelia White, Carol Pf\u00e4ffle, and Carol's new Alcalde neighbor, Mary Cabot Wheelwright.\n\n#  _17_\n\n##  NEW WOMEN OF THE RIO GRANDE\n\nThe _new woman_ means the woman not yet classified, perhaps not classifiable, the woman _new_ not only to men, but to herself.\n\n##  \u2014 _Elsie Clews Parsons_\n\nTHE PF\u00c4FFLES' SUCCESSFUL and popular San Gabriel Ranch brought a steady influx of tourists and dudes to the small village of Alcalde from late March until late October. The region from Espa\u00f1ola north along the Rio Grande to Alcalde also became a desirable location for seasonal and permanent homes for Carol's friends. In 1923, after a brief marriage that produced a son, newly divorced Dorothy Kent came for a stay at San Gabriel. Kent loved what Carol had created, and deciding it was time for her, too, to have a home of her own in New Mexico, she purchased a property adjacent to San Juan Pueblo on the Rio Grande a few miles southwest of the Alcalde village plaza. With help from Carol and a crew of local workers, Kent renovated a primitive two-room adobe into a peaceful ranchito with a painting studio, barns for horses, rambling flower and vegetable gardens, and several shaded courtyards. Kent would live out her life at the ranch she named Chinguague (Tewa for \"the wild place\") and would die here in her own bed in 1981.\n\nMary Cabot Wheelwright returned to New Mexico for an extended stay at San Gabriel Ranch in 1920. Mary and Carol, although friends who had met at Boston's South End Music School, which Mary founded and funded, were from very different worlds. Mary was the definitive blue-blood Brahmin. Her mother was a fifth-generation Cabot, and she was raised in the heart and soul of good old Boston, where \"the Lowells talk to the Cabots, and the Cabots talk only to God.\" Mary's and Natalie's relatives sipped tea in the same New England drawing rooms, and Mary's parents' home on Mt. Vernon Street was frequented by many of the same contemporary luminaries found at Natalie's Uncle George Curtis's home, notably Ralph Waldo Emerson.\n\nIn New Mexico, however, the class boundaries that defined the society of old Boston blurred and even dissolved. Mary still had the financial resources and freedom from economic concerns that Carol could only dream about, but Carol had the experience, contacts, and equipment that Mary needed to negotiate the frontier of the Southwest.\n\nMary's new life in New Mexico began when she was forty, precipitated by the death of her widowed mother, Sarah Perkins Cabot. Born a year before Carol, in 1878, Mary did not attend college but in keeping with the custom of New England's upper classes was educated at home by tutors and governesses. As a young woman, tall, angular, aristocratic Mary was awkward and opinionated, intelligent, curious, and a nonconformist. The decidedly unfeminine Mary, for all her wealth and position, never attracted suitors or married. By middle age and her journey to the Southwest, she was a sheltered but powerful spinster. \"She was an entity,\" one of her relatives remembered. \"You knew when you were talking with Mary that she was somebody.\"\n\nBy the time of her mother's death in 1917, Mary was ready to experience life beyond her parents' Beacon Hill palace. Once released from her life sentence as the dutiful, unmarried daughter who cared for her parents, Mary \"woke up\" and began to travel the world in search of intellectual stimulation, spiritual inspiration, and personal renewal. She was blessed with an abundance of family money, but she also inherited the Cabot's practical, fiercely independent character, and a \"strong interest in nature and a need to get away from civilization.\"\n\nBy 1920 middle-aged Mary was increasingly preoccupied with a personal study of the myths and symbols of the world's great religions. After exploring the mystical and spiritual traditions of Europe, Greece, and Egypt, she turned her attention to the world of Native America found in Arizona and New Mexico.\n\n**Figure 52.** Mary Cabot Wheelwright.\n\nMary and a traveling companion, Katie Scott, had come to New Mexico just after Mary's mother died in 1918, and stayed on the Pajarito with Carol at the Rancho Ramon Vigil. Returning after her travels overseas, she immersed herself in the emancipated society of women and men gathering in and around Santa Fe. Mary especially appreciated the bohemian culture of Santa Fe where material wealth was not the measure of a person's worth. \"It was exciting to come to a region where individuals counted so much, coming as I had from a community where family customs and possessions made a great deal of difference.\"\n\nAt San Gabriel Ranch, Mary unpacked her trunks and set up her headquarters from which to explore the Southwest. San Gabriel and its owners gave Mary \"a wonderful introduction to the country,\" with Roy giving the guests \"an atmosphere of casual adventure\" and his \"Choate from Massachusetts\" wife, the \"musical, imaginative and adventuresome\" Carol \"bringing the appreciation of a cultivated woman to the imaginative quality of the Indian and Spanish life\" of the region.\n\nThe village of Alcalde was, in Mary's opinion, a \"very primitive place\" where the Spanish-speaking residents still adhered to the traditions, customs, and superstitions of their forefathers, and where daily conversations frequently included stories of _brujas_ (witches) and supernatural goings-on. The villagers were devout Catholics, and the Penitente morada on the village plaza, literally over the wall from San Gabriel Ranch's main office, was close enough that guests could hear and observe the clandestine activities of the hermanos. The Pf\u00e4ffles had a good relationship with their Spanish neighbors, and in exchange for the Pf\u00e4ffles' silence about the activities of the Alcalde hermanos, the villagers allowed Carol, Roy, and their wide-eyed San Gabriel guests, including Mary, to huddle in the shadows and observe the procession at Lent of the primitive Penitente death cart.\n\nMary was fascinated by the traditions and Old World culture of Spanish New Mexico, but it was the world of the American Indians that especially captured her imagination. The mysticism and ceremonies of the Navajo were of particular personal interest to Mary and her quest to understand her place in the world. In Indian rituals, Mary, like one of her favorite writers, D. H. Lawrence, hoped to find \"freedom from two of the curses of civilization: loneliness and boredom.\"\n\nIn the symbols and myths of the American Indians, Mary saw commonalities with other world religions she had studied in her travels, especially those found in the East Indies. \"I remember quite well when this idea came to me\u2014of religion as a great tide which moved men all around the world. It gave me a feeling of peace and took away much of the feeling of loneliness from my life.\"\n\nDuring her stay at San Gabriel in 1920, Mary heard the details of Carol's arrival in the Southwest five years before and of her several months' stay with the Wetherills at Kayenta, \"at that time one of the wildest and most lawless parts of the West.\" Carol's stories of her expeditions through Indian Country stirred Mary's desire to experience for herself \"the untouched wilderness, where [Carol] had met untamed Indians\" and witnessed private Ute, Hopi, and Navajo ceremonies.\n\n\"I began to hear from Mrs. Pfaffle and others about the Navajo religion, the ceremonies they had,\" Mary wrote, \". . . finally Mrs. Pfaffle and Miss [Evelyn] Sears and I went on a trip to try and get in touch with some of this life.\"\n\nWith San Gabriel's premier auto wrangler and guide, Orville Cox, Mary went to Gallup for the annual Navajo Ceremonial. During this event, where hundreds of Navajos came to dance and trade, and hundreds of tourists came to buy Navajo rugs and jewelry, Mary purchased a remarkable, one-of-a-kind ceremonial tapestry woven by the legendary Navajo medicine man Hostiin (Hosteen) Klah. Klah was perhaps the most influential cacique on the Navajo Reservation, the first Navajo man to become a master weaver, and the very first Navajo, male or female, to weave sacred Yeibichai sandpainting designs into a tapestry.\n\nFollowing the devastation brought to the Navajos by the flu epidemic of 1919\u201320, the fifty-three-year-old Klah worried about the preservation of Din\u00e9 ceremonials and chants. Among the Navajo, nothing was ever recorded or written down, and the details of all songs, chants, and ceremonies were orally passed from medicine man to apprentice. Although he had a young apprentice, Beaal Beyal, Klah realized that recording his sacred sandpaintings\u2014traditionally destroyed after use\u2014in woven tapestries would tangibly preserve valuable knowledge of the Navajo Way for future generations.\n\nEven though Klah's intention was to save the spiritual traditions of his people, his first Yeibichai weaving, the Whirling Log painting, caused so much distress among the Navajos (who feared the rendition of a sacred sandpainting in permanent form would bring disaster to the tribe) that Klah's gigantic twelve-by-twelve-foot outdoor loom had to be guarded at night.\n\nThe Hail Chant tapestry that Mary bought in late summer of 1920 and its medicine man creator were Mary's gateway into the world of Navajo symbolism and mysticism. Mary returned from Gallup to San Gabriel and asked Carol to arrange an excursion into the outback so that she could witness authentic Navajo ceremonials and, perhaps, locate and talk with Hostiin Klah.\n\n**Figure 53.** Hostiin Klah.\n\nCarol had met Klah during her travels through Navajoland with Louisa Wetherill, and she knew the best way to locate the famous cacique and the ceremonies taking place on the reservation was to contact Arthur and Franc Newcomb, traders at Nava south of Shiprock. After making arrangements to stay in one of the Newcombs' guest cottages, and with confirmation of at least one Navajo ceremony being held near Canyon de Chelly, Carol and her driver, Orville Cox, went into Indian Country with Mary and her friend, Evelyn Sears.\n\nLike Carol, Mary was raised near the sea in a family of sailors, and the immense, unpeopled expanses of the desert were strangely similar to those found on the open ocean. Mary was an experienced sailor, and Indian Country's nearly featureless landscape edged by the wavering line of an unreachable horizon was an environment in which she was comfortable. \"My particular release and joy in the East was sailing and cruising on the sea in the summer, and when I came to the desert I found it gave me a similar feeling of vastness and escape, and I came to love that, too.\"\n\nThe auto route from Alcalde to Chinle and Canyon de Chelly eventually became nothing more than rutted tracks. Carol had made arrangements for them to stay overnight at Cozy McSparron's trading post, but when they arrived at Canyon de Chelly they learned there was a _Tlegi_ or Night Chant ceremony taking place that very night at another location in the desert. Although they had traveled all day and it was already dark, Orville gassed up the Lincoln and with the three women headed out into the night to find the Navajo camp.\n\nCarol and Orville managed to find the ceremony as much by the chanting and singing, which Mary thought sounded very much like coyotes, as by the fires and smoke of the Indian camp. Mary, Carol, Evelyn Sears, and Orville stood and watched the dancers through \"a pall of smoke so thick it was like a tent.\" With their eyes smarting from acrid smoke and tears running down their cheeks, they climbed back into the Lincoln and drove over the Chuska Mountains. It began to snow, and by dawn when they reached the trading post at Crystal, New Mexico, they were negotiating a blizzard. The guest quarters at the post were filled with storm-stranded travelers, and the three women slept on the floor of the unheated lumber room.\n\nFrom Crystal they drove twenty miles northeast on a road that was \"nothing but a track going up and down and across all the arroyos\" to Nava and the Newcombs' trading post. The weather had turned bitter cold and it continued to snow. While the car bumped and slipped across the rough terrain at the foot of the Chuska Mountains, Mary \"saw a thing I have never understood to this day, a solitary man on horseback, pursued by an eagle.\" Like the Hopis, the Navajos captured eagles and used their feathers in sacred ceremonies. The man Mary observed had likely just raided the eagle's nest.\n\nAt Nava, the Newcombs told the San Gabriel travelers that Hostiin Klah was performing a Night Chant ceremony at a trading post seventy miles to the east. After a fitful night of sleep in rooms that had formerly housed chickens, the women climbed into Orville's trusty auto and headed back into the desert. Their destination was Kimpeto (Kimbeto), and their mostly roadless route took them east again and skirted the northern end of Chaco Canyon.\n\nThe four rooms of the Kimpeto trading post were packed with people sleeping wherever they could find a space. Mary slept under a bed already occupied by a nurse. At dawn, everyone trekked through heavy snow to find the ceremony. It was everything Mary had hoped for: \"We managed to reach the ceremony nearby where the dancers were to be seen through the whirling snow, while the fires blew out sideways. Out of this turmoil appeared Klah, calm and benign. He had been conducting the ceremony in the Hogan. He joined us for a while and I got a very strong impression of power from him.\"\n\nThe next morning, the weather cleared, and while Mary and her San Gabriel compatriots were making their breakfast over a cook stove shared with several dozen other campers, Klah walked into the trading post. He sought out Mary and asked her if she was the one who had bought his sandpainting blanket at Gallup. During their first conversation, Mary asked Klah the meaning of the dance she had just seen. When he asked her why she wanted to know, she told him that she was interested in religion. Klah studied Mary for a moment and then asked, \"How deep is the sea?\"\n\nKlah's question startled Mary, since her primary fear and fascination since childhood had been the ocean. In the trading post, Klah began to tell Mary children's stories and Navajo morality tales that gave no significant insight into Navajo ceremonies and religious beliefs. Mary, known for her impatience and lack of social graces, stopped Klah mid-tale and told him she wanted to hear \"the great stories.\"\n\nThus began the relationship between the Yankee spinster Mary Cabot Wheelwright and the Navajo medicine man Hostiin Klah. Mary returned to San Gabriel Ranch but stayed only long enough to confirm the location of Klah's next ceremony, organize and pack up her gear, and, with Orville as her driver, head back into Indian Country. Mary would repeat this journey deep into the land of the Navajos by automobile and eventually on horseback dozens of times in the next few years.\n\nHostiin Klah's role in the social, religious, and daily life of the Din\u00e9 was hardly summed up by his title of medicine man. As a Navajo shaman, Klah was priest, physician, historian, artist, and teacher, virtually a \"centralized source of Indian culture, a human library of primitive lore.\" Recognized as one of the preeminent ceremonial practitioners of his time, Klah's esoteric knowledge of complex Navajo rituals was acquired through decades of study with an elder cacique. In his fifties when he met Mary, Klah had memorized and mastered four complete ceremonies (Nightway, Hailway, Chiricahua Windway, Mountainway), each of which incorporated complex chants, sandpaintings, prayers, and rituals that took days to perform. Klah also carried in his head the tribe's oral histories, myths, and symbols used in many more ceremonies.\n\nThere was no elder held in higher esteem by the Din\u00e9 than Hostiin Klah. His mother, Slim Woman, had sur vived the Long Walk to Bosque Redondo, and her father was the great Navajo chief Narbona. In his lifetime Klah had witnessed the escalation of U.S. government intervention in the life and culture of the Navajos, and seen the incursions of meddling missionaries who sought to eradicate his people's \"heathen\" religious practices. Klah feared the Navajo Way was facing extinction.\n\nBy 1920 and his first meeting with the wealthy Bostonian Mary Wheelwright, Klah had been told by the Great Spirit to document and preserve Navajo stories, rituals, and religious objects. Klah continued to pass along his knowledge to his understudy, but he also employed his remarkable skills as a weaver to permanently document the sacred sandpaintings and designs carried in his head.\n\nKlah was a _nadle_ , a transformed \"man-woman\" deserving of special status and prestige among the Navajos. There is disagreement among his biographers as to whether Klah was born a hermaphrodite or emasculated as a child by the Utes. Whatever his biological and sexual orientation, Klah's career as a venerated medicine man and a master weaver enabled the preservation of valuable and sacred Navajo religious customs.\n\nThere was a remarkable and nearly instantaneous connection between Klah and Mary, and their friendship seemed preordained in both their lives. Klah recognized that Mary's quest to understand Navajo religion was sincere, personal, and without ulterior motives. He also understood that she had the resources and the sensitivity to gather and preserve his people's stories and ceremonies. And not unlike Natalie's self-appointed task to record the music of Native America begun twenty years before, Mary's collaboration with Klah to preserve the stories and rituals of the Din\u00e9 became a consuming vocation that directed and engaged her intellect and passion for the rest of her life.\n\nMary followed Klah to the farthest camps and gatherings in Navajoland. Because note taking was forbidden in a ceremonial hogan, Mary's recording of Klah's rites and chants were done from memory after the ceremony. Franc Newcomb often accompanied Mary and would memorize the actual sandpainting associated with a ritual while Mary would memorize the myth (translated by Klah's apprentice, Beyal) told in a ceremony.\n\nOver the next decade, using a recording machine when she and Klah worked privately and relying on her memory when she attended a ceremony, Mary amassed a comprehensive collection of Navajo chants, myths, and ceremonies. Klah, Beyal, and Mary met at the homes of traders, at schools, even in the underground sample room of the Gallup Harvey House. Klah and Beyal also came to San Gabriel Ranch to work with Mary. In nonceremonial recording sessions, Klah and Mary worked two three-hour sessions a day, or until Klah became weary. Mary always wrote her notes in longhand because a typewriter interrupted Klah's flow, and she never asked questions during a session because Klah's stories were \"told in a kind of rhythm which should not be broken any more than necessary.\"\n\nIn the 1920s the only published materials about Navajo religion and ceremonies were by an army surgeon, Washington Matthews, who had lived at Fort Wingate in the 1890s. Matthews had gained the confidence of the Navajos under his care and had been invited to observe sacred dances and ceremonies. The surgeon kept diaries and records and published his experiences in articles and academic monographs about Navajo myths, legends, rituals, silverwork, sandpaintings, and weavings before his death in 1905. Matthews's published studies became Mary's \"Bible on the subject\" of Navajo religion.\n\nSan Gabriel Ranch was comfortable and private and provided Mary with everything she needed for long stays in New Mexico, including the wranglers and guides necessary for travel into the outback. Mary liked the San Gabriel cowboys and was stimulated by their \"relationship to my kind of woman\" even as she \"found that being from Boston was a distinct handicap, and I was determined that one of my missions was to convince cowboys that it was possible for a person to be a good sport and also drink tea.\"\n\nIn the Southwestern outback, women were expected to shed the prissy helplessness acceptable back east and pull their own weight, whether on top of a horse or alongside it, through dust storms, thunder, lightning, pouring rain, pounding hail, burning heat, or howling snowstorms at high noon or in the dead dark of night. Mary enjoyed being treated \"as a person who would be brave in an emergency\" and the cowboy notion that she, like all women on the trail, was able \"to live up to a difficult situation.\"\n\nJust as she had overcome her fear of the sea and become a proficient sailor and navigator, Mary became a capable horsewoman and competent traveler of the high desert of Indian Country. As Carol, Alice, and Natalie before her, Mary had to learn to ride astride a western saddle. By her third season in Alcalde, she was ready for a grand expedition through the outback. With her friend Evelyn Sears, and with Jack Lambert\u2014a father now, his daughter, Louise, a little more than a year old\u2014as their guide, Mary departed Alcalde in late April 1923 for a pack trip that would cover 550 miles in twenty-nine days, and take her across the magnificent and brutal Rainbow Trail.\n\nLambert had a natural, friendly way with city slickers like Mary, but he never pampered a dude or a dudette, no matter their age or experience. Even for someone used to a western saddle, the route called for long days on the trail with stops few and far between. With Lambert's encouragement, dudes came to be secure on a horse and proficient around camp. Lambert taught his charges how to make biscuits in the Dutch oven, create a bed on the ground, and pass an evening staring contentedly into the fire or up at the stars.\n\n**Figure 54.** Mary Cabot Wheelwright and Jack Lambert, ca. 1925.\n\nKayenta was still the gateway to the last real frontier in the continental United States. In 1923 the half dozen stone, wood, and mud buildings of John and Louisa Wetherill's trading post and home remained a virtually inaccessible outpost on the vast red and orange desert. There were still no roads north and west from Kayenta, and the trail from the Wetherills' to Nonnezoshe was no easier when Mary attempted it in 1923 than it had been for the expeditions of Roosevelt and Zane Grey a decade before.\n\nA journey to Rainbow Bridge was a lesson in improvisation and endurance. Whatever was useless or frivolous was left at the Wetherills'. If Mary had read the harrowing accounts of previous travelers across the raw, scalded desert near Navajo Mountain and the harsh landscape of the Rainbow Trail, she did not let it sway her resolve to see the legendary stone bridge.\n\n\"No one who has not seen it can imagine the difficulties of the trail,\" Mary wrote, \"where it was necessary for the horses to climb up the slick rocks, or petrified sand dunes and where we went afoot and the horses follow Ben Wetherill's horse as best they could. At times I felt I could not possibly get through. At one point the trail was so steep that I could only get up by holding onto the tail of my horse.\"\n\nThe better-bred horses, Mary observed, like the better-blooded humans, were the ones that balked and lost their nerve in the most difficult stretches of the trail. Mary, however, defied her breeding and with her guides Jack Lambert, Everett Cheetham, and Ben Wetherill, and with every one of their horses and pack mules, crossed the Rainbow Trail and reached Nonnezoshe. She slept on the sand under the great stone arch in full moonlight, and like pilgrims before her, woke in the night and gazed into the canyon. \"The other sleepers lay calm and white in the starlight,\" Zane Grey wrote from his desert bedroll, \"There was something nameless in that canyon, and whether or not it was what the Indian embodied in the great Nonnezoshe . . . the truth was that there was a spirit.\"\n\nOn the morning of May 8, 1923, Mary, Evelyn Sears, Jack Lambert, and their companions signed their names in the registry kept in a metal box under the bridge. Mary was pilgrim number 168.\n\nMary did manage to convince the San Gabriel horse wranglers and auto guides that a person who drank tea could also be a good sport. Jack Lambert was perhaps Mary's first convert, and he, like all the San Gabriel staff, learned that although Mary may have looked the pampered maiden, she was in fact from a line of women who had traveled to the rough edges of the world and \"none of them minded much being uncomfortable.\"\n\nDorothy Kent went with Roy, Lambert, and Carol to Rainbow Bridge in the mid-1920s and insisted on taking along her violin. It was an awkward and certainly nonessential item, but Lambert secured Kent's instrument case to a horse, and the violin went with their expedition over the ridiculously difficult Rainbow Trail. If Carol and Kent recorded their names in the bridge log, the pages from their visit were lost. But Kent's was likely the first violin to reach the sacred arch, and she was surely the first graduate of the Juilliard School of Music to play classical music in the light of the full moon at Rainbow Bridge.\n\nIn the autumn of 1923, while out riding with Carol through the golden-leafed cottonwoods of the Rio Grande bosque, Mary saw the abandoned, two-story adobe mansion with sagging wraparound porches called Los Luceros. Carol and Roy had purchased the dilapidated house and the surrounding 138 acres of fields, orchards, and outbuildings on the east banks of the Rio Grande in January of that year. The Pf\u00e4ffles were using the property to grow alfalfa and to pasture San Gabriel's horses, but they had no plans to rehabilitate the old hacienda.\n\nMary wanted to see the interior and toured the house with Carol. Cattle had broken into the first floor and had wreaked havoc on doorways and floors, and portions of the forty-two-inch-thick adobe walls had been damaged by Rio Grande floodwaters. The once-manicured vegetable and flower gardens and the massive apple orchard were unrecognizable and overgrown. The property was in ruins, but Mary fell in love with Los Luceros on sight.\n\nSoon after that autumn ride, Carol and Roy sold Mary the circa 1840 two-story adobe dwelling and six acres of land along the banks of the Great River. Before Mary left New Mexico for the East, she placed Carol in charge of the renovation of the once-elegant Luceros mansion. A local builder, Ted Peabody of Espa\u00f1ola, who had worked with the Pf\u00e4ffles in the renovation of San Gabriel, became Carol's foreman and head carpenter for Los Luceros.\n\nLos Luceros was a classic Territorial-style adobe (although two-story dwellings of any kind were rare in northern New Mexico), but most of the renovations done in 1923\u201324 reflected the popular Santa Fe style and used Spanish-Pueblo Revival design elements. Mary told Carol she wanted New England touches like bay windows, built-in cabinets, bookshelves, and closets. Outside, Mary wanted flagstone walkways (unheard of in rural New Mexico), a duck pond, and a lawn of bluegrass and clover. The hacienda's exterior walls were whitewashed instead of stuccoed, and the window trim and doors were all painted a dark Yankee green.\n\n**Figure 55.** Los Luceros, 1923.\n\nAfter more than a year of renovation, Los Luceros was a ranch worthy of a Boston Brahmin, and Mary called New Mexico home for at least part of each year. She was not the only woman creating a place of her own along the Rio Grande in the early 1920s. The Pf\u00e4ffles sold off more of their property to the millionaire writer Marie Tudor Garland, who built the H & M Ranch with her husband, Henwar Rodakiewicz (renamed the White Swan Ranch after they divorced) alongside the Alcalde plaza. By the mid-1920s the dirt road that went north from Espa\u00f1ola past San Juan Pueblo into the Rio Grande Gorge and up to Taos and Mabel's turf wound past the ranches of four New Women\u2014Dorothy Kent, Marie Garland, Carol Pf\u00e4ffle, and Mary Wheelwright.\n\nIn the fall of 1923 Elsie Clews Parsons joined the neighborhood of New Women along the Rio Grande when she moved into one of Carol's guest cottages for the first of many prolonged stays at San Gabriel. Parsons was a notable addition to the valley community, since she was among the world's most unconventional, prolific, and outspoken sociologists, anthropologists, and feminists. She was born in Grosvenor House at 10 Fifth Avenue in 1874, one year before her Greenwich Village neighbor Natalie Curtis. Her father, Henry Clews, founded the New York banking firm that bore his name, and her mother, Lucy Madison Worthington, was a descendent of President James Madison. As a teenager, Elsie Clews rebelliously opted out of the debutante social life expected of the daughters of the very wealthy and instead pursued an undergraduate degree at Barnard College, and eventually a masters and PhD in sociology at Columbia.\n\nEven after marrying Herbert Parsons, a New York lawyer and Republican congressman who was a close friend and political ally of Theodore Roosevelt, and after giving birth to six children, four of whom lived, the restless, inquisitive, and brilliant Mrs. Parsons insisted on pursuing a life and career \"characterized by a strenuous revolt against convention.\" In the first decade of the twentieth century, she focused her independent research on the problems of the modern family, and on how, in her opinion, those problems were the result of the subordinate status of women. Parsons's book _The Family_ , published in 1906, scandalously recommended the concept of \"trial marriage\" and proposed the popular feminist manifesto that women would only become fit wives and mothers when they were given the same opportunities for personal and professional development as men. _The Family_ raised aristocratic eyebrows and ruffled the feathers of conformists and conservatives, and following its publication Mrs. Parsons was dropped from the New York Social Register.\n\nParsons's personal and professional life was shaped by her passionate feminism, her dazzling intellect, and her fearless temperament. \"Women try hard to live down to what is expected of them,\" she declared in 1913. In New York City, Parsons, like her neighbor Mabel Dodge, was a member of the Greenwich Village feminist discussion group Heterodoxy, begun in 1912. She was also an erudite and vociferous participant in the lively discussions about sex, marriage, and the emancipation of women held weekly at Mabel Dodge's apartment near Washington Square.\n\n**Figure 56.** Elsie Clews Parsons at San Gabriel Ranch.\n\nBefore she came to the Southwest and began her studies of the cultures and people of Native America, Parsons published more than seventy-five academic papers and magazine articles and four books that boldly challenged traditional American ideals regarding women's reproductive rights, the rights of children and adults in marriage, the burdens placed on women who married, and the domination of women by men. Discouraged by the slow progress of American cultural reform movements, especially the feminist movement, in 1916 Parsons turned her full attention to the Indian cultures of the American Southwest.\n\nIt was a turning point in her life and in her career. In the fall of 1913 Parsons visited the New Mexico pueblos of Acoma and Laguna, but it was her journey to Zuni Pueblo in 1915 that initiated her extended ethnological study of the pueblo people. After the death of an infant, and the discovery that her husband, Herbert, was having an extramarital affair, Parsons sought solace in New Mexico. The solitude of the land and the satisfaction she found in fieldwork gave the forty-one-year-old Parsons a personal and professional remedy for a midlife crisis.\n\nBetween 1915 and 1917 Parsons escaped the social and physical confines of upper-class New York City in the wide-open spaces of New Mexico and Arizona and conducted her research while living in a tent near a pueblo, or in a cave if she was working on the Pajarito Plateau. \"This cave room faced south,\" Parsons wrote of her cliff-hugging quarters at Puye, the same pueblo ruin where Alice camped with Hewett in 1916, \"and that night I looked out from its frame on the moonlit talus below and the pines beyond and thought that whether Indian or White one was fortunate indeed to live for a time in a world of such beauty.\"\n\nElsie Clews Parsons's father died in 1923, and with a hefty inheritance, the already liberated Parsons became an independently wealthy woman who could travel and live precisely as she liked. When she moved into the adobe casita at San Gabriel Ranch in November 1923, she had lived at Laguna (where she worked alongside Natalie's friend Franz Boas), Acoma, Zuni, and Jemez Pueblos. She had stayed with Mabel Dodge in Taos, and had also lived at Hopi First Mesa and was an honorary member of one of the native families.\n\nOften the sole woman on anthropological research trips into the outback, Parsons was breaking down the taboo that had long prevented academic camaraderie between men and women and pushing back the social barrier that kept women barred from all fieldwork. She herself underwrote the research trips of her male and female colleagues, and in 1918 was instrumental in the formation of the Southwest Society, an organization that raised money and funded anthropological fieldwork.\n\nWhen she moved into San Gabriel, Parsons was forty-nine and a new grandmother. And although she was still the wife of Herbert Parsons, Elsie Clews came to San Gabriel Ranch with her lover of five years, the novelist Robert Herrick.\n\nFrom her home base in Alcalde, Parsons intended to study the culture and collect the folk stories of the Tewa people living along the Rio Grande at San Juan, Santa Clara, and San Ildefonso Pueblos. Parsons may have met Carol and Roy Pf\u00e4ffle during previous visits to New Mexico, since their ranch was a popular lunch stop for automobile travelers on the road between Santa Fe and Taos. At San Gabriel Ranch, Parsons had access to the neighboring Tewa pueblos, but she also had access to the Pf\u00e4ffles, who by the 1920s had developed an extensive network of friends and confidants in the Tewa pueblos and Hispanic villages of the Rio Grande and Chama River valleys. Although the Tewa pueblos allowed formal visits by tourist groups and permitted limited access to their ceremonial dances, they were guarded and tight-lipped about their myths, social structure, and kinship patterns. Parsons needed Carol and Roy to personally open doors to their friends in northern New Mexico. \"Imitating the secretiveness observed in all the Rio Grande pueblos, I settled in Alcalde, the Mexican town two or three miles north of San Juan,\" Parsons wrote in the opening pages of her book about the Tewa, \"and here, thanks to my helpful and understanding hosts of San Gabriel ranch, I secured informants from San Juan, Santa Clara, and San Ildefonso.\"\n\nNovember weather in New Mexico is pleasant and moderate, and Carol was pleased to have their ranch filled with guests almost until Christmas of 1923. Parsons and Herrick stayed from mid-November until mid-December. In letters to her husband, Parsons wrote of San Gabriel's proximity to \"the seven mountains & blue sky, mostly.\" Although the food was delicious and her room was warm and quiet, Parsons pined for the utter solitude she had known in cliff-hugging caves and mesa-top campsites and complained about having to share the ranch with other guests. Nonetheless, San Gabriel suited her. \"A dude ranch indeed, and I stick to it. My informants are brought out from the pueblo.\" Parsons would return to the adobe casita at San Gabriel until 1926 when she bought her own ranch several miles downriver near the pueblo of Santa Clara.\n\nThe residents of Santa Fe and the villagers of northern New Mexico were becoming accustomed to unconventional and unescorted white women moving through and often into their communities. But even among the expats, the tall, thin Parsons, with her chiseled face and intense, confident gaze, was remarkable. Like Natalie and Alice, once she set foot on New Mexico's soil, Parsons happily shed all semblance of her fashionable high society upbringing and donned the standard outback uniform of leather boots, wide-brimmed hat, and khaki. At a time when women were forbidden to smoke in public, Parsons defiantly wore her cigarette holder and lighter on a chain around her neck, and declared that her greatest enjoyment was to have \" _at the same time_ a cigarette, a cup of coffee, and an open fire. . . . It is very hard to get all three together. It is easier among Indians than among ourselves.\"\n\nParsons's reentry to life in New York City was reminiscent of Natalie's, and was disconcerting for her and for her family. Parsons's mother found her just-back-from-the-desert-daughter's appearance to be \"perfectly dreadful,\" even \"scandalous,\" in that she \"wore khaki clothes\" and \"an old felt hat and she was touching up her hair and of course it hadn't been touched up while she'd been away, so she wore a bandanna tied around it, and then these saddle bags full of manuscripts.\"\n\nIn New Mexico, Parsons and her dusty skirts, saddlebags stuffed with stenographer notebooks, and unkempt hair blended into the emerging community of like-minded women who embodied in greater and lesser proportions the evolving characteristics of twentieth-century feminism. In Santa Fe and in the Spanish villages along the Rio Grande, Parsons joined Carol Stanley, Mary Wheelwright, Dorothy Kent, Marie Garland, Mabel Dodge Luhan, Alice Corbin, Elsie Shepley Sergeant, Margretta Dietrich and her sister Dorothy Stewart, and the sisters Amelia and Martha White in the first wave of what would become a historic groundswell of single and married women who re-created their lives in northern New Mexico.\n\nIn December 1925, following the accidental death of her husband (killed while demonstrating a motor-drive bicycle just given to his son Mac), Elsie Clews came to San Gabriel with her two youngest children, Mac, fourteen, and Herbert Jr., sixteen. Elsie was still involved with Herrick, with whom she would journey to Egypt and the Sudan in April, but he did not accompany Elsie Clews and her grieving children to New Mexico in 1925.\n\nThe Christmas vacation gave Elsie a rare long, uninterrupted holiday with her two sons. Although she did some fieldwork at the nearby pueblos, her primary focus was the simple joys found in New Mexico during the holiday season and reconnecting with her own sense of purpose. \"I ride horseback daily, and it has done me good,\" she wrote her daughter Lissa, who was married with a newborn baby. \"Most of all though the interest in my old job has been restorative. . . . The beauty of the country I have never succeeded in getting across to you or anyone who has not seen it for himself.\"\n\nGaga Pf\u00e4ffle's diary records the comings and goings of \"Dr. Parsons and her boys\" throughout the month they lived in the casita at San Gabriel. Mary Wheelwright was at Los Luceros for Christmas that year and came almost daily to San Gabriel to dine or play cards (seven-up was a favorite), ride horses, or go for walks with the Pf \u00e4ffles' guests and extended family. \"Mother Lambert\" was at the ranch visiting her son Jack, his wife, Lois, and their three-year-old daughter, Louise. Roy and Lois's sister, Marie Wolfe, and her husband also came for Christmas that year. Marie Garland, George King, Alice and Bill Henderson, Mabel Dodge and Tony \"her colored friend\" (as Gaga referenced Mabel's husband, Tony Luhan) came with Spud Johnson (now living with Witter Bynner in Natalie's former home in Santa Fe) for lunch and leisurely conversation around the San Gabriel dining room table just before New Year's.\n\n**Figure 57.** Margaret \"Gaga\" Pf\u00e4ffle at her San Gabriel Ranch casita, ca. 1924.\n\nRoy and Orville Cox drove the Parsons family to Frijoles Canyon for day hikes, up the road above the Rio Grande to Taos, and northwest along the Chama River to Abiqui\u00fa. On Christmas Eve, Mary Wheelwright went with the Parsons family, Gaga, Roy, Mother Lambert, and the extended Lambert-Pf\u00e4ffle family to watch the all-night dances at the nearby pueblos.\n\nOnly Carol was missing from the holiday festivities. From early December until mid-January, she was at the Mayo Clinic in Minnesota. She spent her forty-sixth, and Roy's fortieth, birthday in a bed in the Rochester clinic, and underwent surgery on December 22 for a condition that Gaga did not specify or elaborate upon in her diary. \n\n#  _18_\n\n##  \"ROUGHNECK AND LOW-BROW\"\n\nIrony drives every good story about women's lives in the West. The best stories run like the braided streams of the region's great river system, liberally strewn with ironies and ambiguities.\n\n##  \u2014 _Virginia Scharff_\n\nSAN GABRIEL WAS RIDING THE economic high brought to the Southwest in the mid-1920s by the surge in American train and automobile travel. Dude ranches in northern New Mexico were suddenly the rage among savvy easterners, and dudes reserved rooms at San Gabriel and other guest ranches in the West months before the season began in late March.\n\nAfter six years of hard work, the Pf\u00e4ffles' guest ranch and outfitting business was operating at capacity and making real money. San Gabriel Ranch was the featured centerpiece of the Santa Fe Railway's promotion \"America's Most Famous Fifty Mile Square.\" The ranch was expensive and exclusive, and before a potential guest was granted a reservation Carol's assistant, Minnesotan Margaret Phillips (who would marry a San Gabriel wrangler, Sandy MacLean), looked up their credit report in Dun & Bradstreet. If someone was not favorably listed, they were not given a reservation at San Gabriel.\n\nCarol's guest roster included members of America's most influential families. Jack Lambert guided the family of the Santa Fe Railway executive J. Sanford Otis into the Four Corners, and Nicholas Roosevelt, now a seasoned writer with the _New York Times_ , stayed at San Gabriel and went on at least one extended expedition into the outback with Carol's wranglers and equipment. Winfred Douglas stayed at San Gabriel every summer, and if one of his visits happened to coincide with Nick Roosevelt's, perhaps age and maturity had mellowed Roosevelt's adolescent dislike of the Episcopalian canon he had met at Walpi.\n\nFrom early May until late October, Roy, Jack, and the San Gabriel cowboys, Sandy MacLean, brothers Jack and Lester McKinley, and Pete Coleman, packed up and guided groups of between two and twelve individuals, male and female, adults and children, to Chaco Canyon, Canyon de Chelly, Mesa Verde, Hopiland, the Grand Canyon, Monument Valley, Bryce and Zion Canyons, and the Natural Bridges of Utah. Orville Cox offered guests excursions through Indian Country in one of the ranch's touring cars. And at least once each season, Roy or Jack led a few intrepid guests to Kayenta and, with John Wetherill or Albert Smith, crossed the legendary trail to Rainbow Bridge.\n\nIn late June 1925, Willa Cather and her companion, Edith Lewis, left the comforts of the Hotel La Fonda and the pleasant camaraderie of artists and writers found in old Santa Fe for the rural village experience of San Gabriel Ranch. Cather wrote Elsie Sergeant, who was not in New Mexico that summer, how Alcalde was \"awfully hot.\" In spite of the June heat, Cather found the Pf\u00e4ffles' ranch on the banks of the Great River to be exactly what she needed. Since her first journey to the Southwest more than a decade before, Cather had loved hiking and riding in the desert and mountain outback, and sleeping in rustic, close-to-the-earth quarters. \"I have a regular Zane Grey mind,\" Cather wrote, \"roughneck and low-brow is [the] name for me.\"\n\nCather was reading the proofs of her new novel, _The Professor's House_ , and within the San Gabriel Ranch community, she was in the company of individuals\u2014Carol and Roy, Mary Wheelwright, Dorothy Kent, Jack Lambert, Orville Cox\u2014who had personal experience with the place (the canyon and cliff country of Mesa Verde) and the story (based in part on the Wetherills' discovery and excavation of the Mesa Verde) central to _The Professor's House_. In Carol and her Alcalde crowd were individuals whose lives had parallels to that of Cather's character, Tom Outland: Carol, Dorothy Kent, and Mary Wheelwright, among many others, had rejected the disillusionment, rigid structure, and materialism of modern life for the psychological space, spiritual freedom, and unconventional society of New Mexico.\n\nIn late June, Tony and Mabel Dodge Luhan drove the forty miles from Taos to Alcalde to retrieve Cather and Lewis, who spent the next two weeks at Los Gallos with Mabel and her other illustrious houseguests, including D. H. and Frieda Lawrence. Cather returned to Santa Fe in late July, and during that stay in the old capital found and read the book, _The Life of the Right Reverend Joseph P. Machebeuf_ , that launched her research and writing of _Death Comes for the Archbishop_. The Bishop's Ranch (Bishop's Lodge) in Tesuque where Carol, Roy, and family had lived in 1918 and 1919 was the home of the first archbishop of Santa Fe, Jean-Baptiste Lamy, who had attended seminary in France with Machebeuf and built Santa Fe's Cathedral Basilica of St. Francis of Assisi. Cather based her character Jean Marie Latour on Archbishop Lamy.\n\nThe prosperity the Pf\u00e4ffles had worked and hoped for had become a reality. In 1925 San Gabriel Ranch expanded to include a mountain retreat, El Rito Canyon Ranch, a homestead located in the high mountains between the villages of El Rito and Canjilon, New Mexico. By 1926 the renamed Canjilon Camp was a well-appointed haven with a grand main lodge and headquarters, half a dozen private log cabins, corrals of saddle and pack horses, a resident kitchen and housekeeping staff, and wranglers. Guests seeking the cool air and silence found at eight thousand feet above sea level could stay comfortably on the mountain for weeks at a time.\n\nSan Gabriel's success enabled the Pf\u00e4ffles to maintain the four-room apartment in Santa Fe on Canyon Road, and Carol and Roy enjoyed overnights in town. Carol also continued to come in alone for Indian Arts Fund board meetings, and for solitary retreats from the guests, staff, and family of San Gabriel Ranch. On July 19, Carol and Roy were among the guests at the gala bathing party that celebrated the opening of the White sisters'\u2014and Santa Fe's first\u2014swimming pool. Martha and Amelia Elizabeth White's home, El Delirio (The Madness), at 660 Garcia Street had been transformed from a humble adobe casita surrounded by sage and cactus hills in 1923 into a sprawling estate designed by Alice Corbin's husband, William P.\n\n**Figure 58.** Jack, Lois, and Louise Lambert at Canjilon Camp, 1923.\n\nHenderson. Amelia White was a founding member of the Indian Arts Fund, for which Carol served a six-year term as a trustee and executive committee member. Amelia, like her Bryn Mawr classmates Elsie Sergeant and Margretta Dietrich, had been an active participant in the defeat of the Bursum Bill in 1922, and also joined Mary Austin and the Santa Fe art colony's vehement opposition to and defeat of the Southwest Federation of Women's Clubs' attempt (supported by Hewett) to build a Chautauqua retreat in Santa Fe.\n\nThe swimming party in July 1926 included nearly every member in good standing (Hewett was not invited) of Santa Fe's famous, forceful, and illustrious art, literary, and anthropology colony. As with most of the White sisters' parties at El Delirio, there was a theme to the bathing party, and Carol and Roy, like all the guests, were expected to show up in costume. The theme for the July bash around the new swimming pool was a Mayan ceremony, and guests dressed in beaded and fringed skirts, moccasins, stick and feather headdresses, sandals, silver belts, bathing suits, and very little else. The poet Witter Bynner wrote the script for the evening's main event, the \"Ceremonial Sacrifice at the Sacred Pool,\" and Alice Corbin directed the final act, the sacrificial pool dunking of the virgins, the title roles played by the two White sisters.\n\nIt was a time of gaiety, exuberance, and innocence, and Santa Feans enjoyed \"the ferment of the times without the lost feeling of the Paris group of American expatriates.\"\n\nThe Roaring Twenties brought years of prosperity and happiness to the Pf\u00e4ffles and their friends in northern New Mexico. But there was a shadow side to Carol's life that the San Gabriel staff and her closest friends were acutely aware of: Roy's gambling and drinking. By the late 1920s, the tension and discord created by Roy's addictions began to affect the people around him, and Carol's life and the community of San Gabriel Ranch began to unravel.\n\nThe New Year of 1927 began ominously with an automobile accident. Roy was driving Carol down the steep, snow-packed mountain road from Canjilon Camp when the Lincoln overturned, injuring Carol's leg and sending her to the hospital. In the first week of July, Mother Gaga, out on a picnic with Carol's secretary and a guest, nearly drowned in the swollen waters of the Rio Grande when the road gave way and their Chevrolet plunged sixty feet into the river. Gaga and her friends were \"saved almost by a miracle\" when two men from San Ildefonso Pueblo heard their cries and, after a herculean effort, pulled each of the three from the rising waters with ropes and a team of horses. The car was sucked into quicksand and never seen again. A traumatized and hypothermic Gaga was taken to the hospital in Santa Fe where she was treated for a broken nose and numerous bruises.\n\nCarol was not sleeping well and suffered migraine headaches throughout the summer. By late fall Roy's gambling debts had dug so deeply into San Gabriel's funds that the Pf\u00e4ffles were forced to enter into mortgage agreements with family and friends. Mary Wheelwright alone loaned San Gabriel twenty thousand dollars.\n\nEven with a wife and small child at the ranch, Jack Lambert preferred to spend the dude season on horse expeditions of four to six weeks in Indian Country. But by 1927 he could no longer ignore his brother-in-law's alcoholism. After a particularly challenging horse and auto journey to the Grand Canyon with a doctor whose wife became violently ill after eating poisonous mushrooms, Lambert returned to Alcalde, found Roy \"was drinking,\" and decided it was time to severe his ties with San Gabriel Ranch.\n\nBy season's end, Jack asked the Pf\u00e4ffles for a letter of recommendation. Three days before Carol and Roy's birthday in December, Roy reluctantly penned a statement To Whom It May Concern recommending Jack as a pack and motor trip manager, adding that he regretted Mr. Lambert was leaving his employ.\n\nThe end of their ten-year partnership was a bitter and painful ordeal for both Roy Pf\u00e4ffle and Jack Lambert. For Lambert, who considered the ranch and the community on the Rio Grande his home, moving his wife and daughter out of the adobe Jack himself had built at San Gabriel was among the lowest moments of his life. \"How I loved that place!\" Lambert said years later. \"Losing it almost broke my heart.\"\n\nWith wife, Lois, and daughter, Louise, Jack moved into Santa Fe, where the Lamberts managed a little restaurant opened by the White sisters in Sena Plaza. But the Lambert marriage was in trouble, and by 1929 Jack and Lois were divorced. Jack went looking for ranch work in Arizona. Lois and seven-year-old Louise went east and lived in Tarrytown near or with Dorothy Kent and Canon Douglas and his wife, Dr. Jo, while Louise attended the junior St. Mary's School in Peekskill.\n\nJack Lambert returned to New Mexico and worked as the manager of the White sisters' El Delirio. Although he was married five times, Lambert had only one child, Louise. His fifth and final wife was Margery Ferguson, a Hewett-trained archaeologist and the curator at the Museum of New Mexico for over thirty years. Lambert and Ferguson married in 1950, and they lived the next forty-one years on Garcia Street in Santa Fe until Lambert's death in 1991.\n\nThe winds of unrest swirled like dust devils through the once-peaceful gardens and grounds of San Gabriel Ranch. During an autumn visit in the late 1920s, Hostiin Klah pulled ranch wrangler Sandy MacLean aside and said, \"Tell my friend [Carol] that her man is hurt.\" Around San Gabriel and Alcalde, Klah's psychic abilities were widely respected, and Gaga, Carol, and the ranch staff waited nervously for Roy to return from Santa Fe. Within a few hours of Klah's ominous statement, Roy had not returned, and Carol learned that his horse had lost its footing in the rodeo parade and fallen, injuring Roy.\n\nCarol kept meticulous account books for San Gabriel and knew there was absolutely no disposable income. But Roy continued to gamble, and his nights at the poker tables of northern New Mexico fueled Carol's migraines and escalating sense of doom. However, in 1928 Roy participated in a card game that would change Carol's life and offer her, paradoxically, one last chance to make a permanent home in the Southwest.\n\nThe actual where and when of the poker game is the stuff of myth and legend. Somewhere in Rio Arriba County, perhaps in a village bar, a ranch barn, or at a kitchen table, or maybe out under the stars by a campfire at some cow camp, Roy Pf\u00e4ffle and at least one member of the Alfredo Salazar family played a high stakes game of cards. What Roy Pf\u00e4ffle wagered in that game can only be speculated upon: as of late December 1927, San Gabriel Ranch was incorporated with shareholders and was heavily mortgaged. The Pf\u00e4ffles still owned Canjilon Camp and approximately six hundred acres in the mountains above El Rito, but it is unlikely Roy could have staked the land without Carol's consent and cosignature. Roy may have held title to one or more of the San Gabriel Lincolns, and to its horses, saddles, and expedition equipment. But in the winter of 1928, Roy Pf\u00e4ffle had little to nothing to lose in a poker game.\n\nThe Salazars were sheep ranchers who had lived for generations on the high desert of the Piedra Lumbre (Shining Stone) basin below Cerro Pedernal ten miles north of Abiqui\u00fa. The family had extensive landholdings in Rio Arriba County that included an abandoned and reputedly haunted homestead at the mouth of Yeso (Gypsum) Canyon under the spectacular cliffs on the northern edge of the Piedra Lumbre basin. Called El Rancho de los Brujos (Ranch of the Witches) by the locals, the Rito del Yeso homestead consisted of one dilapidated _jacal_ (cedar post and mud) dwelling, a few corrals and sheds, and the homestead's most valuable asset, a perennial source of water.\n\nRoy had visited the haunted homestead under the great cliffs north of Abiqui\u00fa many times. He watered horses at the stream and used the corrals for San Gabriel expeditions en route to Navajoland to the northwest and Mesa Verde to the north. There was also access to the mountains and to Canjilon Camp via a narrow, rocky, horse and mule trail that climbed up and out of a box canyon where the waters of the Yeso stream literally oozed from sheer rock walls.\n\nBy 1928 no one had lived in the spirit-infested landscape of the Rito del Yeso homestead for nearly thirty years. The Salazar family may have been looking to rid themselves of the place and purposefully lost the deed to the gringo Roy Pf\u00e4ffle in the fabled poker game. Ghosts and witches were not taken lightly in northern New Mexico, and local residents had kept clear of the Ranch of the Witches since the late 1800s when the homestead had been the hideout for a mean bunch of cattle rustlers led by the notorious Archuleta brothers. From the 1880s until the mid-1890s, cattle stolen by the Archuleta gang was held in Yeso Canyon. Any rancher unlucky enough to stumble upon the Archuleta's operation was never seen again. An undetermined number of bodies were believed buried in the red and gold sands under the cliffs, but the saddles, firearms, and horses of the murdered could be counted among the possessions of the Archuletas.\n\nThe Archuletas' reign of terror ended in the 1890s when one brother murdered the other during an argument concerning a gold payment. The remaining Archuleta was either hung by a sheriff's posse or run out of town by the local ranchers. By 190O the ghosts of those murdered by the cattle rustlers created such an incessant racket in the canyons and sand hills of the northern edge of the Piedra Lumbre that not even seasoned shepherds and cowboys from the nearby communities of Abiqui\u00fa, Coyote, Ca\u00f1ones, or Youngsville would stay a night at the place. Only a gringo outsider would be foolish enough to want El Rancho de los Brujos.\n\nRoy won the deed to HES 127, the 157-acre homestead at the Ranch of the Witches, in the early spring of 1928. In her account ledger, Carol recorded that the place she called Ghost Ranch, valued at six thousand dollars, was _acquired_ that year. It took the Salazar family ten or twelve months to legally sign over the deed, but in February 1929 the place of Ghost Ranch was recorded in the Rio Arriba County records to be the property of one Caroline S. Pf\u00e4ffle. Roy was not mentioned on the deed.\n\nEven before the stock market crash in October 1929, San Gabriel Ranch was in a death spiral. Mary Wheelwright, Marie Garland, and Florence Dibell Bartlett had loaned the Pf\u00e4ffles substantial amounts of money, hoping to keep the ranch, and the local economy it supported, afloat. The walls of the guest ranch flanked the Alcalde village church and plaza, and San Gabriel generated jobs and brought cash into the rural community. Because it was on the main road to Taos from Santa Fe and all points south, San Gabriel's dining room had become a popular meal stop for auto travelers. San Gabriel Ranch had essentially put Alcalde on the map.\n\nBut after Lambert departed as manager, and Roy's addictions became impossible to ignore, the ranch's neighborhood support faltered. Bartlett purchased the land that her house El Mirador was built on (she had leased the property throughout the 1920s), and Mary Wheelwright told her lawyer, Francis Wilson (also the Pf\u00e4ffles' attorney), that there would be no more loans to bail out the Pf\u00e4ffles.\n\nThe crash of 1929 brought the New Mexico tourism industry to an abrupt end. For Carol, the last months of 1929 marked the collapse of San Gabriel Ranch and her partnership with Roy. Divorce was surely on Carol's mind, but she waited another eighteen months before formalizing the end of her marriage.\n\nIronically, it was the haunted, decrepit homestead at Ghost Ranch that offered Carol her last best chance for redemption. The three-room mud and wood ruin was little more than a leaky flat roof and crooked walls, but it boasted a spectacular view of red and gold badlands, carved buttes, faraway mountains, and the distinctive flat-topped mountain called Pedernal. The Piedra Lumbre basin, 6,500 feet above sea level, was located on the easternmost lip of the Colorado Plateau and geologically and spatially resembled the high desert country Carol had first known at Kayenta. The Chama River cut from north to south across the basin, but excepting the narrow verdant canyon through which flowed the Rito del Yeso, this was a parched and uninhabitable landscape.\n\nThe Ghost Ranch homestead Roy won in the poker game bordered the historic Piedra Lumbre land grant, a fifty-thousand-acre parcel of community grazing land granted to a prominent military family, the Mart\u00edn Serranos, by the king of Spain in 1766. By the 1920s the grant was no longer community grazing land, but had become the possession of A. B. Renehan, one of the lawyers of the notorious Santa Fe Ring that bought or \"legally\" seized land held by native New Mexicans. After Roy won the deed to the Rito del Yeso, the Pf\u00e4ffles evidently asked Francis Wilson to make inquiries into the ownership of the Piedra Lumbre lands adjacent to the homestead, because in late 1928 Carol and Roy were negotiating with the owners to buy the northern third of the land grant.\n\nFor the last sixteen years, Carol had rented out her house on Nahant island. The home built by J. Bishop Johnson was all that was left of her inheritance and represented her only physical connection to her past life. If Carol had any sentimental feelings for the home of her grandparents, she set them aside when her life with Roy began to unravel. If the Nahant house were sold, Carol would have the money to underwrite a new beginning on the Piedra Lumbre.\n\nDuring Christmas of 1928 Carol and Roy traveled east to initiate the sale of the Johnson home on Dorothy Cove to the man who had leased the house for the last decade. The building that was once the Johnson fish market had burned to the ground several years before, so the sale was for the plot of land near the beach and the three-story house that had been struck by lightning the same June night that Carol's mother had died in her bed.\n\nIt took a year for the sale to become final, a small miracle considering that the year included the collapse of the American banking system and the economic ruination of most businesses and individuals. In March 1930 Francis Wilson, Esq., of Santa Fe and Healey & Healey, attorneys at law, Lynn, Massachusetts, exchanged the final papers, and Carol relinquished all legal ties to the island of Nahant.\n\nThe sum was undisclosed, but letters suggest Carol received thirty thousand dollars for the Johnson house. The sale enabled the Pf\u00e4ffles\u2014specifically Carol\u2014to buy the northern third of the Piedra Lumbre land grant (16,582 acres) and also an adjacent section of land called the Arroyo Seco, from the widow of A. B. Renehan in June 1930. In August, Roy signed the jointly held deed for the Piedra Lumbre land over to Carol. Two months later, Carol initiated divorce proceedings, and all business correspondence was placed on the letterhead of Caroline B. Stanley, president of San Gabriel Ranch, Canjilon Camp, and a place called Ghost Ranch.\n\nCarol's decision to divorce Roy, with whom she remained friends, and go it alone at fifty years of age may have found reinforcement within the circle of independent, educated, \"roughneck and lowbrow\" women who came and went at San Gabriel Ranch throughout the 1920s: Elsie Clews Parsons, Mary Wheelwright, Willa Cather and Edith Lewis, Dorothy Kent, Mabel Dodge. All of these women lived lives that subtly or overtly sanctioned the code of the New Woman. Marriage was a choice, not a dictum, and in New Mexico even middle-aged divorced women were given a place in society.\n\nThe shareholders, who included Mary Wheelwright and several members of Roy Pf\u00e4ffle's immediate family, foreclosed on San Gabriel Ranch in July 1931. Mary took possession of 132 acres of Los Luceros land still held by the Pf\u00e4ffles. Bartlett claimed 75 acres of the San Gabriel land that surrounded her two-story home, El Mirador. Orville Cox and San Gabriel's fleet of touring cars\u2014a 1928 and a 1930 Lincoln seven passenger, and one 1929 five passenger (the enormous twelve-passenger Lincoln remained Carol's property)\u2014joined Hunter Clarkson's \"Couriercars\" associated with the Fred Harvey Company's tour business. Gaga gave up the only home she had ever owned (Roy and Carol had given her one of San Gabriel's casitas) and moved in with her daughter Marie and her husband and family in Santa Fe.\n\nIn mid-June 1931, before Carol walked away from her home on the Rio Grande, she drove her former husband to a sanatorium in Pueblo, Colorado. Roy was left in the care of professionals (likely recommended by the Pf\u00e4ffles' good friend Canon Douglas) in a residency program where, with time and treatment, it was hoped that Roy's alcohol- and tobacco-ravaged body would find the trail back to healthy and productive living. Carol returned alone to New Mexico to begin a new life on the high desert beyond Abiqui\u00fa.\n\nIn a letter to Elsie Sergeant, Willa Cather remarked how trekking about the Southwest outback had taught her the meaning of Balzac's statement \"In the desert there is everything and nothing\u2014God without mankind.\" If Carol had not come to an understanding of Balzac's words before the summer of 1931, she surely appreciated their meaning when she moved all that she owned into the everything and nothing that was the faraway place called Ghost Ranch.\n\n#  _19_\n\n##  EVERYTHING AND NOTHING\n\nI want to go right back into that canyon and be mauled about by its big brutality, though all my bruises are not gone yet. It's a country that drives you crazy with delight, and that's all there is to it. I can't say anything more intelligible about it.\n\n##  \u2014 _Willa Cather_\n\nFEW OF CAROL'S FRIENDS would have been surprised if, after losing San Gabriel Ranch and divorcing Roy, she had left New Mexico and returned to live out her life on the East Coast. But in the years since 1915 and her first expedition into Indian Country with Louisa Wetherill, Caroline Bishop Stanley had become an adopted daughter of the desert, for better and for worse. Carol knew, too, that New England would hardly welcome her back as one of its own. Not only was she a divorcee, her ex-husband was a cowboy and a gambler drying out in a hospital in Pueblo, Colorado. And following the bankruptcy of her ranch, Carol had chosen to sell off her ancestral home on Nahant to finance yet another venture in the middle of Nowhere, New Mexico. At fifty-two, she had secured the black sheep status in the Johnson-Stanley clan of Massachusetts.\n\nBy late summer of 1931, Carol moved all that she could lay claim to\u2014clothing and personal items, the piano Roy had given her as an anniversary present in 1918, a few good horses, tack and saddles, some furniture, several Indian rugs, and dishes and cookware\u2014twenty miles up the Chama River to the haunted homestead on the northern edge of the wide bright Piedra Lumbre basin. Alice Pring, Carol's English maid and friend, and several of her tried and true San Gabriel cowboys went with her to help her build a new ranch.\n\nTo reach the homestead, now called Ghost Ranch, at the base of the red and gold cliffs, Carol departed the main road along the Chama River and followed a rutted horse and wagon track due east into the badlands. A cow, ox, or horse skull (customarily removed and replaced with whatever animal's skull was readily available) was kept propped on the entrance gate over a primitive wooden sign that said \"Ghost Ranch\" with an arrow pointing into the sandlands beneath the stone cliffs.\n\nThe Painted Desert landscape of Ghost Ranch bore no resemblance to the lush riverside setting of San Gabriel Ranch. San Gabriel was situated in a village on one of the main roads through northern New Mexico. Ghost Ranch was an isolated outpost with a watering hole, fifteen miles away from the tiny village of Abiqui\u00fa and the nearest telephone and post office. Santa Fe could be reached by automobile in a half day in good weather, but all journeys south meant navigating the narrow, precipitous road through the five-hundred-foot deep Chama River canyon. Carol was comfortable driving dangerous roads that clung to narrow ledges, but most of her city slicker clients were not.\n\nThis region of northern New Mexico was still considered the frontier in the 1930s, even by New Mexicans. Spanish was the first language, and the traditional Old World Roman Catholic ranchers, farmers, and villagers of Rio Arriba County maintained a lifestyle independent of modern Anglo America. The community of Abiqui\u00fa was a unique blend of Indian and Spanish Americans. The historic village began as a pueblo built around a Franciscan mission given to _gen\u00edzaros_ (detribalized Native Americans) by the Spanish in the 1700s. Abiqui\u00fa, the Piedra Lumbre basin, and the lands of Ghost Ranch were the homelands of the prehistoric Tewa used in historic times by the Navajos, Utes, and Apaches. The Spanish called the Piedra Lumbre _la Tierra de Guerra_ , the Land of War, and considered the region to the north and west too dangerous to settle; the basin remained an unpeopled frontier used only by a handful of intrepid ranchers until well after U.S. occupation in the mid-@@@1800s.\n\n**Figure 59.** The old homestead at Ghost Ranch that became Carol Stanley's home in 1931.\n\nIn 1931, when Carol moved into the _jacal_ dwelling built by the infamous Archuleta brothers, she became the first Anglo woman ever to live at the Ranch of the Witches. She was also the first full-time resident, male or female, to move into the homestead in thirty years. Carol's only neighbor on the northern side of the hundred-square-mile basin was an elderly Navajo, Juan de Dios Gallegos, who had a house on the main road beside the Chama River at the bridge called Vadito de Chicos (Crossing of the Willows).\n\nGhost Ranch and the Piedra Lumbre basin were nearly as remote as the Wetherills' home on the edge of Monument Valley. The landscape was nearly as dramatic too, although on a smaller scale. Red and gold stone mesas rose like islands from the multicolored sands of a painted desert, and narrow spires and pinnacles carved by wind and weather stood out against the skyline. The vertical wall of shining stone rose seven hundred feet from the badlands and was banded in dusty orange, red, yellow, lavender, grey, white, and buff that shimmered and gleamed at sundown.\n\nTo the southwest the horizon followed the lift and fall of the Jemez Mountains defined by the peaks of Tschicoma and Polvadera. The volcanic country of Frijoles Canyon and the Pajarito Plateau was just thirty miles as the crow flies south across the sierras. And standing like a blue sentinel 9,800 feet into the sky above the basin was the knife-like mountain, Cerro Pedernal. Known as Tsiping (Flint-topped Mountain) to Carol's friends at San Juan and the other Tewa-speaking pueblos of the Rio Grande, the _cerro_ was sacred to the Navajo and Apache. Carol had used Pedernal's distinctive, angular profile to orient horse expeditions into Indian Country for nearly two decades. Now the _cerro_ was the dominant feature on her western horizon and marked her home country.\n\nTo Carol, Ghost Ranch and the Piedra Lumbre were a familiar and soothing landscape of magnificent light and infinite space. Where others had seen only an evil, haunted hideout, she recognized one of the great, good-spirited places where the human heart beats up against the soul of the earth. And so in spite of the place having no modern conveniences, no rail connection, terrible roads, and virtually no infrastructure, Carol envisioned a home and guest ranch under the cliffs.\n\nWith the moneys left from the sale of her Nahant house, Carol began construction of ranch headquarters, a dining room, several guest casitas, and a bunkhouse and machine shed. Ted Peabody, Carol's principal foreman for the Los Luceros renovation and several building projects at San Gabriel, was able to sketch out her ideas and make rough plans from which to build the first casitas and facilities for the new guest ranch. A labor crew was assembled from local villages and moved into a camp on the ranch. With thousands of adobes made in the field before La Patrona's (the Spanish workmen's name for Carol) three-room casita under the cottonwoods, the first guest cottages were built.\n\n**Figure 60.** Pedernal Mountain seen from Carol Stanley's home at Ghost Ranch.\n\n**Figure 61.** Ghost Ranch headquarters, ca. 1935.\n\nThe cowboys from San Gabriel lived in tents pitched in the sage field near headquarters, and had meals over the campfire or on Carol's patio when Alice Pring cooked supper in the small kitchen in the old homestead. In the evenings, the workers sat out under the stars and listened to Carol playing her piano by candlelight\u2014there would be no electricity for at least the next year\u2014and classical pieces long ago rehearsed and performed at the New England Conservatory of Music echoed out into the thin, cool air of the high desert canyon. By 1933 the Ranch of the Witches had begun its transformation to the Faraway Nearby, to the Good Country, the magic place known as Ghost Ranch.\n\nNearly all of the Ghost Ranch staff was made up of the cowboys and wranglers who had worked with Carol and Roy in Alcalde, and even earlier at Bishop's Ranch: Pete Coleman, Alfred \"Slim\" Jarmon, shy Archie McKellor, and the McKinley brothers Jack, Marvin, and Lester. A friend of Roy's, Pete Dozier, a half-French, half-Indian guide from Santa Clara Pueblo, left a position at Bishop's Lodge and came to live and work for Carol at Ghost Ranch. And San Gabriel's widely respected and experienced driver and auto mechanic Orville Cox rejoined Carol and her new outfit at Ghost Ranch by 1933.\n\nWithout Roy Pf\u00e4ffle or Jack Lambert as partners, Carol had no manager or foreman. She made herself the Ghost Ranch manager and named Lloyd Miller, a Texas cowboy who had lived and worked at San Gabriel for several years, ranch foreman. Miller was a compact man with an easygoing nature who had made his living in the saddle as a cowboy and a horse jockey since he was a teenager in the Texas Panhandle. Miller had moved into the community at San Gabriel in the mid-1920s, and by 1930 was a valuable member of the ranch staff. A year before the Pf\u00e4ffles divorced, there was talk around Alcalde and in Santa Fe that even though Lloyd Miller was ten years Carol's junior, and she was married, he was sweet on her. And Carol was doing nothing to discourage Miller's attention.\n\nThe guest ranch under the cliffs at the mouth of Yeso Canyon was becoming a comfortable oasis on the high desert. It was surrounded, however, by country where bandits and ruffians still roamed. The most notorious bandits of the modern era, the Night Riders of Rio Arriba, had been apprehended by 1931, but residents in remote villages and ranches from Espa\u00f1ola to Tierra Amarilla were still nervous, and most everyone carried a gun. Carol carried a pistol at all times, and told her visitors to explore and enjoy the beautiful country of her new home but always in the company of one of the Ghost Ranch cowboys.\n\n**Figure 62.** Lloyd Miller, Ghost Ranch foreman and Carol Stanley's second husband.\n\nCarol went into Santa Fe infrequently after moving out to Ghost Ranch. It was a hard day's journey by car, and Carol could no longer lease the apartment on Canyon Road for leisurely overnights in the capital. Orville Cox went into Espa\u00f1ola and Santa Fe to retrieve guests and to run errands for the ranch. Once or twice a month, Carol dined at La Fonda with friends and had her hair done at the De Vargas Beauty Shop. But the days of shopping sprees to Denver and Chicago for dresses, boots, and jewelry were over. Carol paid her ten-dollar annual dues and remained an active member in the Archaeological Institute of America. But as the 1930s progressed, Carol Stanley's name disappeared from the Santa Fe society pages and from the member rosters of art and cultural boards.\n\nBy 1933 Ghost Ranch could comfortably accommodate twenty-five guests in four adobe casitas. There was a bunkhouse for the cowboys, and a ranch headquarters with an office, lounge, and dining room. The corral held a string of good saddle horses and expedition equipment filled the sheds. Carol had at least two touring cars: an indestructible Lincoln named the Clay Car because it survived a dunking in the Chama River, and an older touring car that Orville dubbed Bad News.\n\nEvery cottage had a fine view to the west of Pedernal and the Piedra Lumbre sandlands and basin. Interiors were furnished with sturdy wood tables, chairs, and _trasteros_ (cupboards) made by Ted Peabody and Pete Dozier. Carol's Navajo rugs\u2014many of which had been purchased during her first trips to Canyon de Chelly and Ganado\u2014were hung on the walls, and her collection of Indian baskets and pottery decorated shelves and window ledges. Like the Wetherills' trading post home at Kayenta, Carol's Ghost Ranch was an oasis of shade and comfort on the hot, bright desert. The ranch headquarters and casitas were shaded by young trees planted by Carol's gardener, and the water of the Yeso stream fed an orchard, a large vegetable garden, and flower beds of hollyhocks and Carol's favorite flower, blue morning glories.\n\nGhost Ranch had the potential of becoming a fine guest ranch. But the ranch would survive only if Carol could immediately fill its rooms with paying guests. And in the 1930s in New Mexico, paying guests were as scarce as rain.\n\nThe paralysis brought to the American tourist industry by the Great Depression was intensified in the Southwest by the suffocating heat and drought of the Dust Bowl. Only the ultrawealthy continued to come to the Southwest for vacation and retreat. Ghost Ranch might have folded a year after it opened had it not been for the timely and seemingly destined arrival of three very wealthy men and their families. The first was Arthur Newton Pack, cofounder and editor of _Nature Magazine_ , who had come with his staff and stayed at San Gabriel in the late 1920s. Pack's father, Charles Lathrop Pack, after making his fortune in the lumber industry, realized the need to conserve and protect forests. Pack helped found the American Forestry Association and raised his son to appreciate nature. After Arthur graduated from Harvard, he funneled his family's money into conservation causes. By the time Pack came to see Carol at her new ranch under the cliffs north of Abiqui\u00fa, he was one of America's most important writers about and crusaders for environmental protection and education.\n\nThe Packs made their home in Princeton. In 1933 Arthur and his wife, Eleanor (Brownie), were told by a doctor that they had to move their ailing daughter, Peggy, to a drier climate. Pack learned that Carol and Roy Pf\u00e4ffle had divorced and that Carol had begun a new ranch on a homestead north of Abiqui\u00fa. Pack had ridden across the Piedra Lumbre and well remembered the \"gorgeous cliffs there in Navajo country. They had been like a magnet drawing me back . . . and now all at once here was a legitimate excuse. . . . We made our plans to go.\"\n\nThe Packs wrote Carol and asked if they and their three children could come to Ghost Ranch in early spring and stay the entire summer. Ghost Ranch was not yet open for business, but Carol could hardly turn the Pack family away, since they promised to bring her new dude ranch its first guests and income. Pack wrote about his family's arrival to Ghost Ranch in 1933: \"The main road of poorly graded clay had been rough enough, but after a dozen miles we turned off on a mere track marked only by a cow's skull propped against a rock, slid down an incredibly steep hill, crossed a creek on a narrow log bridge and wound up on the other side beneath a spectacular array of cliffs seen years before.\"\n\nFor their first night at Ghost Ranch, Arthur, Brownie, and their children placed bedrolls out under the sky bright with the moon and listened to Carol play the Steinway piano in the tiny, candlelit dwelling that was now her home. The very next day, Pack asked Carol to sell him land on which to build a house. Carol took Pack and Brownie out to the Painted Desert under the cliffs where they chose a building site with a grand and glorious view of Pedernal. With Carol's help, Pack and his wife designed a simple U-shaped adobe dwelling. Ted Peabody supervised construction, and by Christmas the Pack family moved out of Princeton and into the house on the Piedra Lumbre they called Rancho de los Burros, Ranch of the Burros.\n\nPack brought Carol's new venture the economic stability it desperately needed. He became Carol's partner in the Piedra Lumbre Cattle Company, and within the first year after his family's move to Ghost Ranch, he also held the mortgage for most of Carol's property. But Pack did more than just inject a sizable cash flow into Ghost Ranch: he brought his wealthy friends to the ranch.\n\nThe Pack home in Princeton shared a garden wall with the Robert Wood Johnson family. Bob Johnson and his brother, Seward, of the Johnson & Johnson medical empire, were good friends with Arthur and Brownie Pack, and Bob Johnson traveled with Arthur on several expeditions into the Canadian outback. After the Packs secured their piece of paradise on the Piedra Lumbre and moved into Rancho de los Burros, they began to share their love of Ghost Ranch with close friends. In 1934 Bob and Maggie Johnson came to see for themselves the place that had so enchanted their former neighbors. The Johnsons liked what they found at Ghost Ranch, and leased land near the ranch headquarters from Carol; soon they built a two-story adobe summerhouse for their extended family.\n\nIn spite of the distance from Taos and Santa Fe, and the rigorous and demanding road from Espa\u00f1ola to Abiqui\u00fa and north to the Piedra Lumbre, Ghost Ranch became a destination for the rich and famous living in or visiting New Mexico in the 1930s. Mabel and Tony Luhan brought carloads from Taos to Carol's new ranch, and Mary Wheelwright came out from Alcalde with nieces, cousins, and friends. Members of the Fred Harvey family stopped in for lunch with Carol, and the conductor Leopold Stokowski came several times each summer for overnights with his wife, Evangeline Johnson, the sister of Seward and Robert Wood.\n\nIn July 1933 Carol was visited by the retired Chicago architect Edward H. Bennett and his son, Ted. The Bennetts had grown bored with the \"polo-playing crowd\" found in Tesuque near Bishop's Lodge, and had come out to Abiqui\u00fa to see if Ghost Ranch was indeed a different sort of dude ranch. Carol had met Bennett before (he may have been a Chicago friend of Winfred Douglas). Bennett and his circle in the upper-class Chicago suburbs of Winnetka, Highland Park, and Oak Park\u2014including Archibald MacLeish, the family of George Hemingway (uncle of Ernest), and the Santa Fe Railway executives J. Sanford and Stuart Otis\u2014were already in Carol's client book in the 1920s.\n\nBennett, one of the primary architects of the Chicago Plan and the designer of Buckingham Fountain, the centerpiece of the city's Grant Park, had recently retired and was looking for a summer retreat far from the tourist scene in Santa Fe. Carol showed father and son one of the newly completed and available guest cottages near ranch headquarters. The Bennetts liked what they saw, and they returned the next day with the whole family, who spent the entire summer at Ghost Ranch. Before they left in the fall, Carol sold Edward Bennett 212 acres of land directly under the Puerto del Cielo, Gate of Heaven, a chimney-like formation in the cliffs west of Arthur Pack's home site. Within the year, the Bennetts' Casa Monte Rojo, Red Hill House, was completed.\n\nBy the summer of 1934, the Pack and the Bennett families were living under the cliffs, and the adobe cottages clustered around the old homestead were filled with paying guests\u2014rich folks who were, in the words of Ted Bennett Jr., \"waiting out the depression.\" Robert Wood Johnson's house was under construction and would be completed by winter, and at the mouth of the canyon east of headquarters there was a new swimming pool filled with the cold water of the Yeso stream. Auto and horse expeditions departed weekly for the Four Corners and Indian Country, a homesteader named Dave Burnham called squares at the weekly square dance in the ranch headquarters, and every Friday night Carol's cowboys held a rodeo and invited local riders, bronco busters, and ropers to show off their skills for the ranch dudes. Carol loved to throw a good party for friends and visitors up at headquarters or on her front patio, but after witnessing the havoc wreaked by Roy's alcoholism, Carol enforced a no-liquor law at Ghost Ranch for both staff and guests.\n\nThe dude ranch under the gleaming cliffs beyond old Abiqui\u00fa had become a topic of conversation in social circles back east. Pack, delighted with Ghost Ranch, raved about his new home in New Mexico to friends in Princeton and New York City. One of these friends was David McAlpin, the grandnephew of John D. Rockefeller (whose family had stayed at Carol's Canjilon Camp for a month in 1930). After hearing Pack's descriptions of the magical high desert place called Ghost Ranch, McAlpin decided to go to New Mexico and see Pack's home for himself.\n\nSoon afterward, at a party in New York City in the winter of 1934, McAlpin told a friend, the painter Georgia O'Keeffe, about Arthur Pack's adobe home on the high desert of New Mexico at a ranch that was possibly \"the best place in the world.\" O'Keeffe had spent time in New Mexico, including a four-month stay at Mabel's in Taos in 1929, and several weeks in Alcalde in a cottage at Marie Garland and Henwar Rodakiewicz's ranch. The H & M (Henwar and Marie) Ranch was built on land sold to Garland by the Pf\u00e4ffles, and the property, like San Gabriel, flanked the Alcalde plaza. Following Jack and Lois Lambert's departure from San Gabriel in 1928, Garland had purchased their adobe home and additional San Gabriel acreage.\n\nO'Keeffe returned to Alcalde and the H & M ranch in the summer of 1934. She had learned to drive and went on solo painting excursions up the Chama River valley toward Abiqui\u00fa and the country she called the Faraway. O'Keeffe loved the little hills and landscape near the old village, and she also liked the space, light, and the utter lack of people. O'Keeffe had become weary of Taos and Santa Fe's frenetic social scenes. Even Marie Garland's little ranch in rural Alcalde hosted all-night parties attended by large mobs of artists and writers from Santa Fe and Taos. Now that O'Keeffe knew there was a guest ranch out on the high, bright desert of the Piedra Lumbre, she set out to find \"someplace where the people do not run me crazy\u2014,\" the place called Ghost Ranch.\n\nIn midsummer, in the company of young Charles Collier, son of John Collier, who was now the commissioner of Indian affairs, O'Keeffe headed out to find the most beautiful landscape in the world. It was a frustrating day: although they successfully navigated the road up through the Chama canyon and reached the Painted Desert below the Cliffs of Shining Stone, Collier and O'Keeffe could not find the road across the badlands to Ghost Ranch. Several days later at San Juan Pueblo, O'Keeffe saw a pickup truck with \"GR\" painted on its door parked out front of the Kramer Mercantile. She waited by the truck, and when the driver returned, she asked him for _very_ specific instructions about the location of the road to Ghost Ranch. The driver explained that there was no sign marking the ranch gate, only a skull. With the directions firmly in hand, O'Keeffe drove her Model A back out onto the Piedra Lumbre and alone found the road to Ghost Ranch.\n\nIt was August and high summer, and the sky over Pedernal was intensely blue. The road led O'Keeffe across the hot, dry landscape of red-gold sandlands and deeply carved arroyos to a cluster of adobe buildings beneath the sheer walls of the luminous cliffs. \"Perfectly mad looking country\u2014hills and cliffs and washes too crazy to imagine all thrown up into the air by god and let tumble where they would. It was certainly as spectacular as anything I've ever seen\u2014and that was pretty good.\"\n\nFrom the front steps of the Ghost Ranch headquarters, the world that O'Keeffe was looking to find, to paint, to immerse herself in, unfolded to the horizon. \"It was a new world,\" O'Keeffe said years later. \"And I thought it was my world.\"\n\nThus began the half-century love affair between the painter Georgia O'Keeffe and the place of Ghost Ranch. O'Keeffe would eventually claim ten acres of the ranch as her own, but in August 1934 the Faraway was owned by Carol Bishop Stanley. And it was to Carol's crooked little house under the cottonwood trees that O'Keeffe went to inquire about a renting a room.\n\nO'Keeffe had been introduced to Carol when she stayed at Marie Garland's in 1929, but in O'Keeffe's accounts of her first years at Ghost Ranch she never acknowledges Carol Stanley by name. Perhaps Carol's ownership of Ghost Ranch, and her remarkable, even accidental role as the creator of modern Ghost Ranch, were not easily conceded by O'Keeffe. From the moment O'Keeffe connected with Ghost Ranch, she claimed it as her own. Legal ownership was of little interest to her. Whatever O'Keeffe formed an affection for\u2014the Faraway Nearby, Ghost Ranch\u2014belonged to her. She was especially fond of Pedernal. \"It's my private mountain,\" she declared decades after she first came to the ranch. \"It belongs to me. God told me if I painted it enough, I could have it.\"\n\nWhen O'Keeffe arrived at Carol's doorstep that August day in 1934, the ranch's guest cottages were all spoken for. Carol did, however, have one room unoccupied for just one night. One night at Ghost Ranch was better than nothing, and O'Keeffe drove all the way back to Alcalde\u2014forty miles over very poor roads\u2014packed up a few necessities, and returned to the ranch by evening. During the night, the child of one of the families staying at the ranch became very ill with appendicitis. By morning, the family had departed for the hospital in Santa Fe, and their casita was available for the remainder of the summer. O'Keeffe gleefully claimed the cottage. \"I went immediately to Ghost Ranch, and I never left.\"\n\n**Figure 63.** Georgia O'Keeffe at Ghost Ranch, ca. 1935.\n\nBY THE LATE 1920S, Mary Wheelwright's collaboration with Klah had amassed a significant and rare collection of tapestries, rugs, and recordings of Navajo ceremonies and chants. After the Night Riders' robbery at Los Luceros of Navajo rugs and jewelry (the items were recovered in late summer of 1931), Mary knew she had to find a secure and permanent home for the collection. In 1927 she offered the recordings, sacred items, rugs, and transcriptions of chants and stories\u2014and the money necessary to build a museum devoted to Navajo ceremonialism and religion\u2014to the new Laboratory of Anthropology in Santa Fe.\n\nThe Laboratory of Anthropology, recently funded by John D. Rockefeller, was to be built on fifty acres of land donated by the White sisters in the hills southeast of the Santa Fe plaza. The architect John Gaw Meem had been selected to design a Pueblo Revival\u2013style complex for the new facility that would house, among other things, the substantial and valuable collection amassed by the Indian Arts Fund begun by Harry Mera and friends at Elsie Sergeant's Tesuque dinner party in 1922.\n\nWhen Mary approached the lab's new board of directors\u2014a virtual who's who in American anthropology, including Hewett, Kenneth Chapman, Jesse Nusbaum (who was the lab's first director), Natalie's friends and colleagues Franz Boas and F. W. Hodge, Alfred \"Ted\" Kidder (the young archaeologist at the Wetherills' the month Carol arrived in 1915), and Elsie Clews Parsons, among others\u2014they were in the early phases of planning the institution's future. The directors, although delighted to consider the acquisition of Mary and Klah's collection, were immediately averse to Mary's one condition: that the museum to be built (with her money) for her collection be an octagonal structure patterned after a traditional Navajo hogan (hooghan).\n\nSeveral members of the new board claimed the hogan design was incompatible with Meem's Santa Fe\u2013style structure that would house the laboratory. But it was not just Mary's architectural preference that the conservative men on the board found objectionable. Several members argued that Mary and Klah's collaboration had no place in the collections of a scientific organization. Mary's plans \"represent an artistic interpretation and are not the real thing at all,\" Francis I. Proctor, a Boston eye doctor, reasoned, because \"in a scientific institution everything should be founded on the truth.\"\n\nMary's vision of a museum celebrating Navajo ceremonialism and her institutionally unsanctioned and amateur research partnership with Hostiin Klah had run into the stone wall of the academic patriarchy. Mary did not hold an advanced degree, or any other degree for that matter. She was also a wealthy, entitled, determined, progressive, unmarried woman, and the accomplished men of the Laboratory of Anthropology were not about to let an independent female dictate policy or suggest aesthetic directives.\n\nAlice Corbin understood what Mary was up against and told Amelia Elizabeth White what she thought of the lab's board of directors: \"The Committee is, I think, frankly _scared_ , because the Hogan is unique and different. Though this is just what would make it an asset. But they have no architectural imagination and no courage.\"\n\nTrue to form, Mary was not intimidated by the esteemed members of the board, nor was she in awe of the academic community. She was thick-skinned to criticism about her supposed lack of scientific rigor, and she shrugged off comments by conservative scholars about her eccentric and unconventional relationship with Navajo cacique Hostiin Klah.\n\nUndaunted by the board's initial opposition to her proposal, Mary persevered with her intention to build a museum linked to the Laboratory of Anthropology that was devoted to Navajo religion. For the next five years, she wrote letters, made calls, submitted and resubmitted ideas and designs, negotiated, cajoled, socialized, and haggled about the project with the Laboratory of Anthropology board members.\n\nAnd then she gave up.\n\nMary sensed that time was running out for Hostiin Klah. In 1935 she withdrew her offer to the lab of her collection and went forward with her museum with help from her friend Amelia Elizabeth White. In a move that surely incensed several of those staunchly scientific board members, White donated the land adjacent to their Laboratory of Anthropology to Mary and Klah's museum of Navajo religion.\n\nIf the boys of the Santa Fe art and anthropology colony had forgotten, they were quickly reminded: in the City of Ladies, the ladies were a mighty\u2014and often resolutely united\u2014force to be reckoned with.\n\nWith her remarkable and straightforward gift, Amelia White empowered Mary Wheelwright to build the Museum of Navajo Ceremonial Art. Mary chose Alice Corbin's husband, William Henderson, as the architect for the project. With Klah as his consultant, Henderson designed a larger-than-life hogan complete with interlocking cribbed-log ceiling and an east-facing door that opened directly onto the pi\u00f1on-studded foothills of the Sangre de Cristos. The museum's primary directive was to provide a sanctuary for Klah's recordings and sacred paraphernalia. Klah wanted the collection to assist future generations of Navajo singers, and he gave Mary explicit instructions that his ceremonial items be available for study in perpetuity.\n\nKlah conducted the traditional House Blessing ceremony at the groundbreaking, but he did not live to see the museum completed. In the early winter of 1937 influenza raged across the Navajo Reservation, and Klah, after caring for dozens of sick neighbors, succumbed to pneumonia. He died in the Rehoboth Hospital near Gallup in late February. Per Klah's instructions, his body was placed in a \"white\" casket and buried in the cemetery in Tohatchi, New Mexico, where his male relatives kept vigil for four days and nights and chanted the traditional Navajo prayers that accompanied his departing spirit along the Rainbow Trail.\n\nWith the passing of Hostiin Klah \"went many a precious secret of the tribe's religious life.\" Mary was in Florida when Klah died, and when she returned to New Mexico, she called upon Arthur and Franc Newcomb to help her get permission from Klah's relatives to disinter his casket and bring his body to Santa Fe for burial near the new museum. After assuring Klah's family that the removal of the casket would be discreet and that the exact time would be known only by the funeral director and his workers (the family did not want to know when his body was above ground), Mary sent a hearse from Santa Fe to retrieve Klah's body. He was reburied on the hill beside the nearly completed museum. Mary had wild plants and herbs important to Klah's ceremonies\u2014desert goldenrod, water grass, grass daisy, and dwarf sage\u2014brought from Navajoland planted in the earth surrounding his grave.\n\nNine months after Klah's death, the House of Navajo Religion opened in Santa Fe in November 1937. Klah's nephew, Big Man's Son, oversaw the House Blessing ceremony. Twenty-one Navajos, including fifteen of Klah's nieces and nephews, came to Santa Fe in a private bus arranged by Mary. Attendance at the museum dedication was by invitation only and was conducted in the Navajo way, with Klah's Navajo family seated on the south side of the massive hogan, and the few white people\u2014Mary, Alice Corbin (named the museum's first curator) and William Henderson, the Newcombs, and the workers who built the museum\u2014seated on the north side of the octagon. Before they departed for their homes in Navajoland, Klah's family sang the \"Traveler's Song,\" a prayer for safe passage for Mary, who was about to embark on a trip to China.\n\nThe museum's hogan design that had garnered such vociferous disapproval from the Laboratory of Anthropology architectural team received national attention and was awarded Honorable Mention for \"Works of Major Importance in Architecture\" from the Architectural League of New York in 1938.\n\nIn the next decades Mary and Klah's renamed Museum of Navajo Ceremonial Art (later called the Wheelwright Museum of the American Indian) became a repository for sound recordings, paintings, manuscripts, sandpainting tapestries, and valuable Navajo ceremonial materials. Washington Matthews's papers that had been Mary's \"bible\" in the first years of her work with Klah were transferred to the Santa Fe museum from the University of California, Berkeley, and dozens of valuable ethnographic films, an archive of reproductions of more than five hundred sandpaintings, Navajo song texts and recordings by Mary and others working in the field, and manuscripts from other major studies of Navajo religion were placed in the museum's permanent collection.\n\nCAROL LOVED GHOST RANCH as much as any home she had ever known. But love didn't pay the massive bills associated with the startup of a guest ranch. Even with the Packs, Bennetts, Johnsons, and O'Keeffe in residence, and with every bed taken by dudes throughout the summers of 1933 and '34, Carol could not bring in enough income to cover Ghost Ranch's operating expenses. Her personal funds from the sale of the Nahant house had been exhausted by the construction of the ranch headquarters, guest cottages, bunkhouse, animal sheds, corrals, and auto garage. The historic drought had dried up the meager grasslands of the Piedra Lumbre, and by early 1935 Carol and Arthur Pack's Piedra Lumbre Cattle Company was defunct.\n\nGhost Ranch operated in the red during its first years, and no one but a millionaire could absorb those sorts of losses. Pack was a millionaire who loved the country and the ranch life, and he offered to buy out Carol in March 1935. \"It became evident that Carol couldn't keep the place,\" Ted Bennett Jr. recalled years later. \"For all her charm, there wasn't enough money coming in. Arthur Pack wanted to buy it from her, and this turned out to be a blessing in disguise. For a naturalist, he was very wealthy, and the considerable cost of the Piedra Lumbre grant was no burden to him.\"\n\nPack bought Ghost Ranch from Carol for seventy-five thousand dollars. Fifty thousand dollars were deposited into Carol's Santa Fe bank account; the remaining twenty-five thousand paid off her mortgage to Pack. The deal included sixteen thousand acres of land; the ranch's buildings, sheds, and barns, wagons, generators; all of the furniture in all of the casitas, headquarters, and bunkhouse; and Carol's fleet of pickup trucks and her four twelve-passenger Lincoln touring cars. Pack became the owner of the ranch's saddle horses, pack burros, herd of cows, one bull and one sow, and Carol's 150 chickens, 25 ducks, and 86 turkeys. Pack also took possession of some but not all of Carol's Indian art collection\u2014Carol kept several of the rugs purchased years ago at Canyon de Chelly and Ganado. The Steinway piano remained with Carol.\n\nPack asked Carol to stay in residence on the ranch until late fall and the end of the 1935 guest season. All of the cowboys, guides, and drivers employed by Carol were kept on, and Lloyd Miller was retained as the head wrangler, although Pack knew Miller would remain at Ghost Ranch only as long as Carol was living there. Carol had fallen in love with Miller, and he with her. By 1935 the relationship that had begun quietly during the last tumultuous years of Carol and Roy's marriage was now a widely accepted liaison. In September, Carol journeyed with Harriet Ropes Cabot (Mary Wheelwright's relative), O'Keeffe, and Mabel and Tony Luhan to the Jicarilla Apache fiesta at Stone Lake. In spite of losing yet another home and ranch, and having just said good-bye to her maid, Alice Pring, who returned to England after working in Carol's household for fifteen years, Carol was optimistic and happy, and told her companions that she would marry Lloyd before Christmas.\n\nCarol's friends were nervous about her decision to marry again. Lloyd Miller, although respected as one of the nicest, kindest, most dependable cowboys in northern New Mexico, had spent his whole life around the racetracks of Texas and New Mexico. Lloyd loved to train racehorses, and he loved to bet on racehorses. And although Lloyd did not drink, he had won and lost more money at the track than anyone cared to remember or to mention to Carol.\n\nIn October, after most of the guests had departed Ghost Ranch, Carol went to Kayenta to visit Louisa and John Wetherill. Exactly twenty years had come and gone since Carol had learned to ride astride a horse and journeyed through Indian Country with Louisa. Carol was fifty-five years old. She had built, managed, and lost two remarkable and significant ranches, had married and divorced, and with Lloyd Miller, was about to embark on what would likely be the last chapter of her life.\n\nCarol took a Boston friend, Helen Appleton, with her to the Wetherills'. Mid-October was a grand time of year to see Monument Valley, and Carol and Helen moved into agreeable but crowded quarters in the Wetherill home. A Mr. and Mrs. C. B. Green of Williamstown, Massachusetts, Pamela Parsons and Jack Stacy of Santa Fe, and Bill Brimhall of Gallup were also staying at the Wetherills'. The guesthouse near the trading post was being used by a group from Flagstaff that included the painter Mary-Russell F. Colton and her husband, Harold S. Colton. On October 14, Carol signed Louisa's guest book as Caroline B. Stanley of Los Luceros, New Mexico, and gave as her mailing address the post office in Chamita, New Mexico.\n\nThe friendly advice and counsel Louisa Wetherill may have given Carol during her October visit was never recorded. In her Louisa, Carol surely found support for her decision to sell her home in Nahant and remain in New Mexico following the loss of San Gabriel. But Louisa's opinion of her decision to remarry must remain unknown.\n\nOne month later, on November 14, Caroline Bishop Stanley of Nahant, Massachusetts, married Lloyd Miller of Content, Texas, at the United Brethren Church in Espa\u00f1ola, New Mexico. The marriage certificate revealed the curious fact that Lloyd Miller, like Roy, shared Carol's birthday of December 16. Born in 1890, the groom, eleven years younger than the bride, may not have known the precise difference in their ages, since the marriage license erroneously claimed Carol was born in 1886. Lloyd also may not have known that he was the second man Carol had married who shared her and her deceased twin brother's birthday.\n\nWith the fifty grand left from the sale of Ghost Ranch, the Millers bought a small ranch on the Rio Grande near Los Luceros. Cottonwood Ranch was developed to accommodate Lloyd's quarter horse breeding and training business. It was an expensive operation to get up and running, and the Depression had yet to loosen its grip on the economy. There was no money coming in, and Lloyd, who was hanging out around the local racetracks, lost a lot of money gambling. By 1940 Carol's bank account was drained and the Millers faced foreclosure.\n\n\"We all saw it coming,\" Dorthy Fredericks remembered. \"Carol turned her money over to Lloyd just like she had to Pf\u00e4ffle before him. Carol began to avoid her old friends\u2014never went to lunches at La Fonda or any social event where she might meet people who knew what had happened. Carol couldn't take the humiliation of being broke twice by men.\"\n\nCottonwood Ranch was sold and with it everything Carol owned, including her Steinway piano. Carol and Lloyd departed New Mexico with one good quarter horse stallion and a borrowed saddle and moved to Arboles, Colorado, a small town just over the New Mexico border at equal distance from Durango and Pagosa Springs. Surrounded by the Navajo and Southern Ute Reservations at the northernmost end of the geologic formation that reached south to Chaco Canyon, Carol was still in the country where her affair with the American Southwest had begun. She had ridden through this landscape dozens of times, and camped alongside the streams and in the mountain meadows en route to Mesa Verde. Carol Stanley Miller had lost everything she owned except the country she loved.\n\nThe Millers moved into a two-room adobe outside the small town. After the war began in 1942, Lloyd opened a feed store in Arboles, and Carol served as the town postmistress. There was no money for Carol's annual trip to Boston, and she would never again visit the Douglases in Denver or any of her friends in Chicago. Carol never returned to Ghost Ranch, or to Santa Fe or Alcalde. She simply slipped out of sight.\n\nA few of Carol's friends, mostly the cowboys, stayed in touch with her after she left New Mexico. In 1946 Dave Burnham and Ghost Ranch cowboy Slim Jarmon went to Arboles and dropped in on Carol and Lloyd. Carol, now in her mid-sixties, wore patched jeans and a wrinkled plaid shirt, her former wardrobe of stylish dresses and fashionable riding clothes a distant memory. To Burnham and Jarmon's surprise, Carol cooked supper (neither man had ever seen Carol prepare a meal except over a campfire in the outback), and with Lloyd, they spent a pleasant evening around the kitchen table talking about old times. Lloyd took Burnham and Jarmon out to the corral to see his two quarter horses and spoke optimistically about their chances at the track. But Carol's health was not good, and it was obvious to their old friends that the Millers had hit rock bottom.\n\nCaroline Bishop Stanley Miller died in that two-room adobe house outside Arboles on the afternoon of December 4, 1948. The cause of death was thrombosis of the coronary artery. She was sixty-eight years of age, and her death certificate, signed by Lloyd, correctly gave her birthdate as December 16, 1879. Carol was buried in the small cemetery near Pagosa Springs on December 7, Pearl Harbor Day.\n\nThere was no announcement of Carol's death in the _Santa Fe New Mexican_ , nor in any newspaper in Boston or Nahant. The cowboys who had worked with Carol at the Ramon Vigil, Bishop's Ranch, San Gabriel, and Ghost Ranch learned of her passing the old-fashioned way, person to person, and according to Dave Burnham's daughter, Dorthy, \"all of the cowboys showed up for her funeral. _All_ of them.\"\n\nLloyd Miller left Colorado after Carol's death, and died several years later back in New Mexico. Carol's first husband, Roy Pf\u00e4ffle, outlived her by seven years. After treatment in the hospital in Pueblo, Roy returned to New Mexico in the mid-1930s and took jobs wrangling horses and dudes where he could find them. Roy eventually married a teacher named Helen Fowler Nolan, who taught school in the coal-mining town of Gramerco on the Navajo Reservation a few miles north of Gallup. They shared a good and simple life, and Roy remained employed and sober until Helen died after World War 11.\n\nRoy moved into Gallup in the 1950s, and tried to make a living as a cattle wrangler at the rail yards. But the work he did so effortlessly as a younger man was too much for his aged body. In his last letter, dated November 7, 1955, Roy confessed to his sister Lois and her husband, Tom, that he had almost not survived the last two weeks of thirteen-hour shifts as a night watchman in the cattle yards. After being knocked down by a steer, and spending several weeks laid up in bed in a poorly heated Gallup boarding house, Roy decided that \"the old boy isn't what he used to be.\" Roy was hoping to get enough money to leave Gallup and join his sisters and their families in Santa Fe. \"Things are pretty rough here now,\" he told Lois at the close of his letter.\n\nRichard Leroy Pf\u00e4ffle died in Gallup six weeks later, on December 21, 1955. He was buried on Christmas Eve in the Fairview Cemetery in Santa Fe. Roy's mother, Margaret \"Gaga\" Pf\u00e4ffle, had been laid to rest in Santa Fe just eleven months before.\n\n#  AFTERWORD\n\nSomething of the Spirit\n\nHere is the desert of silence, Blinking and blind in the sun\u2014An old, old woman who mumbles her beads And crumbles to stone.\n\n##  \u2014 _Alice Corbin_\n\nMARY WHEELWRIGHT AND ALICE KLAUBER both lived to see their eightieth birthday. Mary's primary residence remained in the Northeast among the Brahmins, but even as she aged, she continued to make an annual pilgrimage to Los Luceros on the Rio Grande. Her last visit was in May 1958, just two months before she died.\n\nAlice, however, did not go back to New Mexico again after Natalie's death in Paris in October 1921. Following three years of a self-imposed European exile, she returned to San Diego in late 1924. She was in her mid-fifties, and after years of living out of steamer trunks in hotels and rented rooms in foreign places, Alice was ready to settle down in the ocean-side community that had been her home since birth.\n\nBefore she embarked for Paris and the meeting of the International Congress of Art History at the Sorbonne, Alice and a small group of San Diego artists had founded Friends of Art, a society of like-minded creative people who shared the idea that \"the real status of the community is measured not alone by its commercial achievements, laudable as they may be, but by the things which make for culture.\" By the time of Alice's return to California from Europe in the mid-1920s, discussions about the building of an art museum in San Diego had become formalized, and Friends of Art had merged with the Art Guild of San Diego. Alice and other members of the renamed Fine Arts Society became the directors of San Diego's first fine art museum. While the new museum was under construction, the directors' meetings were held in the New Mexico Building at Balboa Park, a place whose rooms held the echoes and memories of a dozen of Alice's closest friends, including Natalie.\n\nAlice was not just a founding member of the Fine Arts Gallery of San Diego (later renamed the Museum of Art), she was a major donor, and also served as the museum's first curator. Her personal art collection given to the museum over several decades reflected her remarkable life among some of the most influential and important artists of her time. After visiting Japan in the 1930s, Alice also became an avid collector of and authority on Asian art. In 1935, with her friend Elsie Kimberley, she created San Diego's Asian Arts Committee, and in the ensuing years her collection of Japanese woodblock prints, Persian miniature paintings, and rare Chinese porcelains became the nucleus of the San Diego Museum's Oriental art collection.\n\nIn her sixties Alice was teaching art to children and adults, and comfortably assumed the role of San Diego's first lady of the arts. But beneath the civic dynamo and celebrated matron of the local arts, the skilled and serious painter persisted and continued to seek the nourishment found in her private studio among her paints and canvases. Alice's paintings appeared regularly in exhibitions at the San Diego Museum of Art and were exhibited at the California Pacific International Exposition in 1935.\n\nBy the time of her death in her bed in Lemon Grove, California, in early July 1951, Alice was an important and respected member of the Southern California fine art community. But she was unknown in the American art world. Although her teachers and advocates included many of the most prestigious artists of her time, Alice never sought fame or expected recognition as a painter. Hardly surprising, as she was often apologetic about her passion to paint, and rarely if ever used the term \"artist\" to describe herself. Indeed, the time and society into which Alice was born hardly encouraged a woman to make such a bold and self-affirming declaration, or to entertain the pursuit of such an identity. \"Locally her social activities seemed more newsworthy than her achievements as an artist,\" one art historian wrote of Alice, \"if a paucity of newspaper accounts is considered any criteria.\"\n\nIn spite of the prevailing climate of expectation that urged women to stay home and concern themselves with domestic matters, Alice traveled to exotic and often challenging destinations in pursuit of her muse. However, her travel journals, diaries, and surviving letters offer only cryptic and abbreviated references to her own work and rarely shed light on her private thoughts or opinions about her irrepressibly independent spirit. Natalie was the rare friend with whom Alice discussed the challenges and conflicts endured by women artists, and Natalie's death left her with few women friends who personally understood the dilemmas of the creative modern woman.\n\nLike Natalie, Carol, Mary, and their spirited female contemporaries in the early twentieth century, Alice was a New Woman not so much by sociopolitical choice as by disposition and circumstance. Alice's poems leave a few veiled clues to her feelings about life as a single woman and an artist. The poem aptly titled \"Untitled\" perhaps sums up her acceptance of her place in the world.\n\n> The least of His reporters I   \n> Who take some small note from His sky and  \n>  let the Universe pass by.\n\nMary Wheelwright died in her own bed in her summerhouse, the White Hen, on Sutton Island, near Mount Desert, Maine, in July 1958. She was eighty years of age and, like Alice, was vigorous and sharp-minded until the end of her life. Mary knew her death was near\u2014she'd been told by her doctor that she had serious heart trouble. On her last day of life, she spent the morning in her garden. The White Hen faced west and had a fine view of the sea. After a good lunch, Mary retired to her room to read. With her book in her hand, she dozed off into a sleep from which she never awoke.\n\nTwo of Mary's cousins, sisters Aimee and Rosamond Lamb of Boston, were visiting Sutton Island that day, and they watched in silence as the undertaker carried Mary's body on a stretcher out of the summerhouse and down to a small rowboat tied to the dock. The sisters placed flowers on the canvas-wrapped body and said their farewells by the water.\n\nSutton Island was famous for its fog, and that afternoon the Lamb sisters stood with Mary's bereft house staff and watched as the little boat carrying her body was rowed out into the mist. \"It was really just what she would have liked,\" Rosamond said years later. \"Could anything be more appropriate for Mary than just to go out into the void like that, into the deep fog, just by herself in a boat.\"\n\nMary's body was buried among the Brahmins in the family cemetery in Boston, but if her Indian friends and their beliefs held sway over her spirit, Mary's soul was far away on the Rainbow Trail.\n\nA memorial service was held for Mary Cabot Wheelwright in Santa Fe at the Museum of Navajo Ceremonial Art. Oliver La Farge, whose novel about the Navajo people, _Laughing Boy_ , won the Pulitzer Prize in 1930, recalled the obstacles that Mary and Klah had overcome to make the museum a reality. Mary's unscientific methods, La Farge told the audience, were ultimately responsible for the preservation of extraordinary and irreplaceable materials. La Farge praised Mary and her ability to study and respond to \"a very remarkable primitive religion as a whole instead of approaching it only from the rather timid point of view of the anthropologist seeking his purely so called objective data.\"\n\nThe memorial concluded with a written statement from the Navajo tribal chairman, Paul Jones. \"It is certainly most unusual and difficult for a museum to achieve the purpose of keeping alive something of the spirit of the religion of the people who never had a written language. . . . The Navajo people will be forever grateful to [Mary] for . . . building the thing of the spirit into visible and physical form in the Museum of Navajo Ceremonial Art.\"\n\nTHE MUSEUM OF NAVAJO CEREMONIAL ART (Wheelwright Museum of the American Indian) gave visible and physical form to the remarkable, unconventional life and work of Mary Cabot Wheelwright. It is more difficult to find tangible reminders or memorials for the other women whose journeys to their individual destinies are recounted in this book. Their lives and their contributions to the story of their times most often remain nebulous, something of the spirit.\n\nNatalie Curtis Burlin's work among the Hopis and dozens of other Native American tribes, and her articles, books, recordings, and transcriptions are valued and recognized by specialists in the field of ethnomusicology. Alice Klauber is revered among those who work at and frequent the museum in San Diego she helped to build, but Alice's name, like her own paintings and her friendships with Henri, the Steins, and Pach, garner little public interest. Carol Bishop Stanley's intrepid and risky move onto the faraway desert beyond Abiqui\u00fa resulted in the creation of one of the most iconic places in American art history, yet her story remains only tenuously tethered to Ghost Ranch and the world-famous relationship, between O'Keeffe and the ranch, that her movements and choices inadvertently precipitated.\n\nOf course, a life is not necessarily more valuable because it is broadly remembered or publicly applauded. Even celebrated worldly success may ultimately amount to a handful of sand scattered in the winds of time, and stupendous failure may spark a fire that burns brightly for decades. When all is said and done, a life well lived leaves only silence and beauty behind on the trail, and footprints and dreams merge seamlessly into the landscape.\n\nThe last words spoken at Mary Wheelwright's memorial service in Santa Fe were from an ancient Navajo song-prayer. We can imagine the prayer intended for Mary resonated among her kindred spirits Carol, Natalie, and Alice and all their fellow travelers across the wide, bright sky of their beloved Indian Country.\n\n> Before her it is beautiful.  \n>  It shows the way  \n>  Behind her it is beautiful  \n>  It shows the way  \n>  This that is beautiful  \n>  It shows the way  \n>  May she ever walk in beauty.\n\n#  ACKNOWLEDGMENTS\n\nIT MAY TAKE A VILLAGE TO raise a child, but it takes a continent of friends and colleagues to write a book. I was able to undertake the grand journey to find the ladies of the canyons because I was supported every step of the way by family, friends, scholars, and editors. Kristen Buckles at the University of Arizona Press was enthusiastic about the project from the get-go and agreed to administer a research fund that afforded me the luxury of two years devoted almost entirely to this book. This fund was initiated by major gifts from Dona Bolding and Roger Hamilton, and from Diane and Tom Arenberg and family. I am humbled by their generosity and sense of adventure. The project received additional funding from Mary Ann Bumgarner, Joyce and Rex Davidson, Rebecca DeLair, Isabel and Sam Jewell, Carol and William Padden, Willie Picaro, Ann Sherman, and Suzanne and Robert Snow. You are all patrons of the arts in the truest sense. I am blessed as a writer to have such a splendid circle of history-loving friends and benefactors.\n\nThe stories told in _Ladies of the Canyons_ were waiting to be discovered in archives, libraries, and private collections located in the Southwest, Midwest, and on both coasts. My search was facilitated by the expertise and energy of professional librarians and researchers, by relatives of the women themselves, and by professional and amateur historians who shared precious clues and valuable keys to these women's lives and the world they were part of.\n\nLea Armstrong of the Wheelwright Museum of the American Indian gave advice and expert counsel as I sought to understand the connections between Mary Cabot Wheelwright and Carol Bishop Stanley. Cheri Falkenstien-Doyle, also of the Wheelwright Museum, guided me through the photo and manuscript collections held in the Wheelwright archives.\n\nAlfred Bredenberg's articles and publications about Natalie Curtis Burlin brought important attention to Natalie's life and work, and his Natalie Curtis Burlin Archive is an invaluable source of information and context for Natalie's life and work. Michelle W. Patterson, author of _Natalie Curtis: A Life in Native and African American Music_ , kindly shared her extensive research and professional insights and gave coherence to Natalie's professional life.\n\nAuthor historians Carolyn Brucken, Chief Curator, Autry National Center, and Virginia Scharff, Professor of History and Director of the Center for the Southwest at the University of New Mexico, offered valuable and thoughtful insights and suggestions during the first months of research. Historian and author Don D. Fowler was a source of expert and friendly advice about Southwest archaeology.\n\nJames Grebl, Manager, Library & Archives, The San Diego Museum of Art, graciously assisted and guided me through the Klauber archive. Nada Borsa was an invaluable source of information about the Klauber family and San Diego's cultural history.\n\nLaura Holt, Librarian, School for Advanced Research, Santa Fe, facilitated my many research visits to the SAR library and cheerfully endured my many requests to view files, books, and papers held in the archives.\n\nPatricia Hibbard, Jack Lambert's granddaughter and Roy Pf\u00e4ffle's grandniece, graciously copied and identified family photos, sent me the diaries of Margaret \"Gaga\" and Lois Pf\u00e4ffle, and helped me comprehend her family's story. Peggy Terrell shared scrapbooks, diaries, and newspaper clippings, and her own memories of the Pf\u00e4ffle family saga.\n\nHarvey Leake, grandson of Louisa Wade and John Wetherill, brought to my attention Louisa Wade Wetherill's diary and its astonishing revelations about Carol Stanley, provided me with historic photographs, and identified people and locations. Waldo M. Wedel alerted me to the extraordinary anecdotes about Natalie, Douglas, and Theodore Roosevelt at Walpi in the 1913 diary of Nicholas Roosevelt.\n\nBeverley Spears, FAIA, architect in charge of the historic Los Luceros restoration, shared her substantial knowledge of Los Luceros's past and present and gave me an insider's tour of the magnificent property she helped to preserve. Robert Archuleta of El Mirador opened the grounds and historic buildings of the Pf\u00e4ffles' San Gabriel Ranch on the Rio Grande. Isabel Ziegler shared memories of Dorothy Kent, Carol Stanley Pf\u00e4ffle, and Mary Cabot Wheelwright. Philip Blood graciously showed me Dorothy Kent's former home in Alcalde, New Mexico. John Steinle, Administrator, Hiwan Homestead Museum, Evergreen, Colorado, assisted my research into the life and family of Canon C. Winfred Douglas.\n\nCalantha Sears, Nahant Historical Society volunteer curator extraordinaire, shared her over ninety-year knowledge of Nahant during a five-star tour of her magical home island. Dona Bolding was my savvy guide through the streets of old Boston and the cliff-hugging roads of Nahant island. Joyce and Rex Davidson explored with me many a road less traveled, and with Jim helped me locate Carol Stanley's high mountain treasure, Canjilon Camp. Christine Mather showed me Santa Fe's back roads and historic alleys and answered all my questions about Santa Fe style. My cousin Reid Meloy, with his wife MJ and daughter Olivia, gave me a room with a view to the La Jolla world that Alice Klauber loved and painted. Cristina McCandless was my companion during two days of intense excavation in the archives of the San Diego History Center and the San Diego Museum of Art in Balboa Park. The University of Arizona Press loaned me the services of two student interns, Sarah Parker and Corrine Ng, for which I am most grateful. Thank you all!\n\nMy partner and husband, Jim Kempes, explored the desert outback near Kayenta and Chinle, Arizona, with me, and was my scout through astonishing Hopiland. My son, Chris Kempes, was my navigator through greater Los Angeles and shepherded me through the research collections at the UCLA, Huntington, and Braun libraries. My daughter, Mari, and my daughter-in-law, Suzanne, provided unwavering encouragement and optimism, especially during the homestretch, and my parents, Ann Reid and David Poling, were, as always, tireless cheerleaders throughout the journey to write this book.\n\nI am also indebted to the following individuals and institutions: Joan Bacharach, Lisa Riedel, and Rosemary Sucec of the National Park Service, Rainbow Bridge Registry; Brandon E. Barton, Duplication Services Coordinator, UCLA Library Special Collections; Eric Berkemeyer, Curatorial Assistant, Native Arts, Denver Art Museum; Marjorie Bixler; Ed Chamberlin, Museum Curator, Hubbell Trading Post National Historic Site; Heather Cole, Assistant Curator of Modern Books & Manuscripts and Curator of the Theodore Roosevelt\n\nCollection, Houghton Library, Harvard University; Sarah Dana, Research Archivist, The Archives of the Episcopal Church; Lesley Daniel, Library Assistant, Royal Academy of Music, London; Amanda Donohue, Alumnae Information Services, and Patricia J. Albright, Archivist, Mount Holyoke College; Bonnie Ayers D'Orlando, Assistant Curator, and all of the volunteers and staff at the Nahant Historical Society; Cecilia Esposito, Director, and Charline Gannon, Plattsburgh State Art Museum; Arcadia Falcone, Public Services Intern, Harry Ransom Center; Scott Ferris; Mary French, Curator of Archives, The Explorers Club; Claudia Gallardo de Campbell, New Mexico State Monuments, Department of Cultural Affairs; Gina Giang, Department of Manuscripts, Sue Hodson, Curator of Literary Manuscripts, and Natalie Russell, Library Assistant, Literary Manuscripts, The Huntington Library; Tomas Jaehn, DCA, The Fray Ang\u00e9lico Ch\u00e1vez History Library; Jane Kenealy, Archivist, and the staff of the San Diego History Center, Balboa Park; George and Mary Jeanette Kennedy, and Judith and Harold W. Lavender; Daniel Kosharek, Photo Curator, Palace of the Governors; Lisa Kulyk-Bourque, Reference Librarian, Lynn Public Library; Frederick Lewis; Mike Martin; Robin Martin; Mary Morganti, Director of Library and Archives, California Historical Society; Liza Posas, Head Librarian, Braun Research Library, Archivist, Autry National Center; Maryalice Perrin-Mohr, the New England Conservatory of Music; Earle E. Spamer, Ann Reinhardt, Valerie-Anne Lutz, and Michael Miller of the American Philosophical Society; Hope Shannon, Executive Director, The South End Historical Society, Boston; Amy S. Wong, Library Special Collections, Charles E. Young Research Library, UCLA; Martha Yates, Ghost Ranch Education and Retreat Center. \n\n#  ABBREVIATIONS\n\nACH\/HRC  |  Alice Corbin Henderson papers, Harry Ransom Center, University of Texas Austin   \n---|---  \nCWD\/DMA  |  Charles Winfred Douglas papers, Denver Museum of Art   \nGRA  |  Ghost Ranch Archives, Abiqui\u00fa, New Mexico   \nKS\/NYPL  |  Kurt Schindler papers, New York Public Library   \nLAHM\/PPC  |  Los Alamos Historical Museum, Peggy Pond Church collection   \nLC\/BRL  |  Charles Lummis Collection, Braun Research Library, Southwest Museum, Los Angeles   \nLOC  |  Library of Congress, Washington, DC   \nMCW\/WM  |  Mary Cabot Wheelwright papers, Wheelwright Museum of the American Indian   \nNCBA  |  Natalie Curtis Burlin Archive, Raleigh, North Carolina   \nNHS  |  Nahant Historical Society   \nRK\/AAA  |  Rockwell Kent papers, Archives of American Art, Smithsonian Institution   \nSDHC  |  San Diego History Center   \nSDMA  |  San Diego Museum of Art Library and Archives   \nSRA  |  New Mexico State Records and Archives   \nUCLA  |  Special Collections, UCLA   \nWACC\/NPS  |  Western Archeological and Conservation Center, National Park Service, Tucson, Arizona \n\n#  NOTES\n\n###  Epigraph\n\nIsak Dinesen, \"The Monkey,\" _Seven Gothic Tales_ , 109.\n\n###  Prologue\n\n1. Pack, _We Called It Ghost Ranch_ , 23.\n\n2. Scharff, _Twenty Thousand Roads_ , 12.\n\n3. Corbin, \"From the Stone Age,\" in _Red Earth_ , 37.\n\n###  Chapter 1\n\nVirginia Scharff, \"Introduction: Women Envision the West,\" I.\n\n1. Bridgham Curtis, \"Memorial.\"\n\n2. George DeClyver Curtis, \"In Memory of Natalie.\"\n\n3. Natalie Curtis to Bessie Day, November 30, 1891, NCBA.\n\n4. Patterson, _Natalie Curtis Burlin_ , 42.\n\n5. Ibid., 47.\n\n6. Natalie Curtis to Bessie Day, June 29, 1890, NCBA.\n\n7. Natalie Curtis to Bessie Day, May 1893, NCBA.\n\n8. Bridgham Curtis, \"Memorial.\"\n\n9. Frank Shaw to Lincoln Stone, in Foote, _Seeking the One Great Remedy_ , 120. Robert Gould Shaw and the Massachusetts 54th Infantry are the subject of the 1989 film _Glory_. A memorial to Shaw and his men is on Beacon Street in Boston, across from the Massachusetts State House.\n\n10. Walt Whitman quoted in Milne, _George William Curtis_ , 232.\n\n11. Bridgham Curtis, \"Memorial.\"\n\n12. George DeClyver Curtis, \"Natalie Curtis Burlin,\" unpublished biography; Bridgham Curtis, \"Memorial.\"\n\n13. George DeClyver Curtis, \"Natalie Curtis Burlin.\"\n\n14. Clements, _Native American Verbal Art_ , 47.\n\n15. Lummis, _The Land of Poco Tiempo_ , x.\n\n16. Ibid., ix.\n\n17. Ibid., 2\u20133.\n\n18. Starr, _Inventing the Dream_ , 84.\n\n19. Edwin R. Bingham, _Charles F. Lummis, Editor of the Southwest_ , quoted in Fowler, _Laboratory for Anthropology_ , 251.\n\n20. Diary of George DeClyver Curtis, January 11, 1908.\n\n21. Lummis, _The Land of Poco Tiempo_ , 140.\n\n22. Natalie Curtis, \"The Words of Hiparopai.\"\n\n###  Chapter 2\n\nWilla Cather, \"The Bohemian Girl,\" 440.\n\n1. Natalie Curtis, _The Indians' Book_ , xxviii.\n\n2. All quotations from Chiparopai from Natalie Curtis, \"The Words of Hiparopai,\" 294, 297.\n\n3. Natalie Curtis, \"A Western Reverie,\" 591.\n\n4. Natalie Curtis, \"An American Indian Composer,\" 626. In 1849 the BIA was transferred to the newly created U.S. Department of the Interior. For years thereafter, the bureau was known variously as the Indian office, the Indian bureau, the Indian department, and the Indian Service. The Interior Department formally adopted the name Bureau of Indian Affairs for the agency in September 1947.\n\n5. Clements, _Native American Verbal Art_ , 157.\n\n6. Talayesva, _Sun Chief_ , ix.\n\n7. See Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 125.\n\n8. James, _Pages from Hopi History_ , 123\u201324.\n\n9. Natalie Curtis, \"An American Indian Composer,\" 627\u201328.\n\n10. Natalie Curtis, \"Indian Song on a Desert Path,\" 344.\n\n11. Natalie Curtis, _The Indians' Book_ , 474\u201378.\n\n###  Chapter 3\n\nElizabeth Shepley Sergeant, _Willa Cather: A Memoir_ , 71.\n\n1. Bridgham Curtis, \"Memorial.\"\n\n2. Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 5\u20136.\n\n3. Natalie Curtis, \"Mr. Roosevelt and Indian Music,\" 399\u2013400.\n\n4. Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 5\u20136.\n\n5. Natalie Curtis, \"Mr. Roosevelt and Indian Music,\" 399\u2013400.\n\n6. Theodore Roosevelt to Indian Agents, July 22, 1903, quoted in Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 149.\n\n7. Theodore Roosevelt to Ethan Allen Hitchcock, July 22, 1903, quoted in Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 150.\n\n8. See Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 149\u201350.\n\n9. Natalie Curtis to Theodore Roosevelt, July 29, 1903, NCBA.\n\n10. Natalie Curtis, \"The Winning of an Indian Reservation,\" 327.\n\n11. Natalie Curtis to Charles Lummis, June 5, 1903, LC\/BRL. Mead and Lummis did connect, and Mead would found the East Coast chapter of the Sequoyah League. But even the defiantly unconventional Lummis considered Mead unusual and questioned his \"mental ballast.\" When Roosevelt asked Lummis about Mead, Lummis told him that Mead was \"a good deal saner than he looks, and I am inclined to trust him; but still would hardly _vouch_ for his balance.\" Lummis quoted in Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 151.\n\n12. Natalie Curtis, \"The Plight of the Mojave-Apache Indians,\" 30.\n\n13. Natalie Curtis to Theodore Roosevelt from Wave Crest, August 31, 1903, LOC.\n\n14. Natalie Curtis, \"The Winning of an Indian Reservation,\" 329\u201330.\n\n15. Natalie Curtis, \"Mr. Roosevelt and Indian Music,\" 399.\n\n16. George DeClyver Curtis, \"Biography of Natalie Curtis Burlin.\"\n\n17. Natalie Curtis, _The Indians' Book_ , 475.\n\n18. Ibid., 477\u201378. Natalie's mood reflected the prevalent attitude among Anglo America that the language, traditions, village life, identity, and sovereignty of Native America would not endure another half century. Natalie's recording and collecting of Indian music, chants, and stories was considered \"salvage\" ethnology. Although she vigorously defended the innate value and intrinsic importance of the art and culture of native people, in the twenty years that Natalie would work and live among the Indians of North America, she did not personally or professionally challenge the popular contemporary notion among whites that Native America was in its twilight years.\n\n19. Ibid., 474. A former Mennonite minister and one of the first missionaries to work among the Hopis, Rev. H. R. Voth translated Lololomai's song and explained its content and nuances for Natalie. Natalie called upon Voth, then at the Field Columbian Museum, Chicago, for assistance with all her Hopi recordings. See ibid., 478.\n\n20. Bridgham Curtis, \"Memorial.\"\n\n21. Natalie Curtis, _The Indians' Book_ , 490.\n\n22. Ibid., 347.\n\n23. Natalie Curtis, \"Mr. Roosevelt and Indian Music,\" 400.\n\n24. Ibid.\n\n###  Chapter 4\n\nAnsel Adams and Mary Austin, _Taos Pueblo_.\n\n1. George DeClyver Curtis, diary from St. Louis World's Fair, 1904.\n\n2. Natalie Curtis, _The Indians' Book_ , 323\u201324.\n\n3. Natalie Curtis, \"The Winning of an Indian Reservation,\" 330.\n\n4. Natalie Curtis to Charles Lummis, April 4, 1905, LC\/BRL.\n\n5. Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 123, 159.\n\n6. Charles Lummis to Natalie Curtis, December 14, 1905, LC\/BRL.\n\n7. Theodore Roosevelt to Natalie Curtis, May 7, 1906, _The Indians' Book_ , xx.\n\n8. Angel De Cora was born in a wigwam on the Winnebago Reservation in Dakota (Thurston) County, Nebraska, on May 3, 1871. At the age of twelve, she went to study at the Hampton Institute. She began her career as an artist when she entered the art department of Smith College, from which she graduated in 1896. After studying at the Drexel Institute in Philadelphia, De Cora became a professional illustrator.\n\n9. Natalie Curtis, _The Indians' Book_ , xii.\n\n10. George DeClyver Curtis, diary, January 6, 1908.\n\n11. The Sherman Institute, opened in 1902 as a grammar and junior high school, boarded Indian students from all over the West. The school was modeled on Commissioner Francis Leupp's Indian policy that concentrated on preparing Indian youth for farm and ranch work. At Sherman, Native Americans from a dozen tribes learned English, math, American history; attended classes on Christianity; and received instruction in agricultural and industrial arts. By 1908, 550 students were enrolled.\n\n12. See James, _Pages From Hopi History_ , 132\u201339.\n\n13. Theodore Roosevelt to Leonard Wood, June 16, 1903, quoted in Hagan, _Theodore Roosevelt & Six Friends of the Indian_, 159. Natalie was acquainted with Leupp, who was the former chief of the _New York Evening Post_ 's Washington bureau. Although she did not leave a written record of her reaction to the 1906 events at Oraibi, it is not difficult to imagine her disappointment with Leupp's decision to exile Tawakwaptiwa and his people.\n\n14. Natalie Curtis, _The Indians' Book_ , 480.\n\n15. Talayesva, _Sun Chief_ , 134.\n\n###  Chapter 5\n\nGregory Marwood (George DeClyver Curtis), _The Wooing of a Recluse_ , 110.\n\n1. Heyneman, _Arthur Putnam Sculptor_ , 11i.\n\n2. Although he would always struggle financially, Putnam did become a celebrated sculptor in his lifetime. Rodin, upon seeing Putnam's bronze sculptures of wild cats on exhibit in Paris in 1907, remarked, \"This is the work of a master.\" See Carol M. Osborne, \"Arthur Putnam, Animal Sculptor,\" _American Art Review_ 3 (September\u2013October 1976).\n\n3. Laura wed a bridge engineer named Julius Wangenheim, who joined the family business and was soon chairman of the board of the renamed Klauber-Wangenheim business. By 1900 Wangenheim was among San Diego's most influential visionaries and community builders, and the already prestigious Klauber family now had members and in-laws at the forefront of virtually every important civic undertaking. Wangenheim established banks and title companies, helped found the Scripps Institution of Oceanography, built the San Diego Library, and would serve on numerous boards including the first board of trustees of Scripps College.\n\n4. See Scharff, \"Introduction: Women Envision the West.\"\n\n5. Details of Alice's trip from Diary \"My Vacation,\" SDMA; and \"Italian Trip, Chase Notes,\" SDHC.\n\n6. Robinson, _Georgia O'Keeffe_ , 57, 59.\n\n7. Klauber, \"My Vacation,\" June 9, 1907.\n\n8. Klauber, Diary, July 25, 1907.\n\n9. Gertrude Stein in Perlman, _Robert Henri_ , 3.\n\n10. Walter Pach to Alice Klauber, December 24, 1924, SDMA.\n\n11. Alice Klauber, Diary, September 1, 1907.\n\n12. Ibid., October 26, 1907.\n\n13. Walter Pach to Alice Klauber, November 16, 1907, SDMA.\n\n14. Ibid., August 19, 1908. See Walter Pach letters to Alice Klauber, November 1907\u2013November 1909, SDMA.\n\n15. Walter Pach to Alice Klauber, November 2, 1908, SDMA.\n\n16. Bridgham Curtis, \"Memorial.\"\n\n17. Natalie Curtis to Ale\u0161 Hrdli\u010dka, March 30, 1916, in Patterson, _Natalie Curtis Burlin_ , 193.\n\n18. The Chicago Kindergarten Institute, considered a pioneer in social settlement education, partnered with local churches and universities, including the University of Chicago and Northwestern University. See Johnston, \"The Chicago Kindergarten Institute,\" 573\u201374. For Mary Walton Free Kindergarten for Colored Children, see Patterson, _Natalie Curtis Burlin_ , 218.\n\n19. Frank and Walter Damrosch were the German-born sons of Leopold Damrosch, the former general manager and chief conductor of the Metropolitan Opera. Walter Damrosch, the \"youthful Adonis\" conductor over whom Natalie swooned as a teenager, was in 1911 director of both the Oratorio Society and the New York Symphony Society (which merged with the New York Philharmonic in 1928). During a trans-Atlantic voyage in 1877, Walter began a conversation with a young newlywed named Andrew Carnegie. By the end of the crossing, Walter had inspired Carnegie to build New York's Carnegie Hall. His older brother, Frank, founded the New York Institute of Musical Art in 1905, which merged with the Juilliard Graduate School to become the Juilliard School. Walter still holds the record for Carnegie Hall appearances. From opening night in 1891, when he shared conducting duties with Tchaikovsky, until his final performance conducting the New York Philharmonic on March 27, 1942, the maestro played a stamina-busting 850 performances at Carnegie Hall.\n\n20. The story goes that Douglas's recovery made a dramatic turn when a fellow priest, Father Irving P. Johnson, came to the hospital to give him his last rites. Douglas managed to a make a feeble joke about his own demise that caused Pastor Johnson and a very weak Douglas such hysterics that the deepest abscess in Douglas's lung punctured, thus saving his life. See Ellinwood and Douglas, _To Praise God_ , 14.\n\n21. Gertrude House, a settlement house named for Pestalozzi's ideal woman, opened its doors in 1904. It was the headquarters for the Chicago Kindergarten Institute, and by 1910 it was also a boarding house for over sixty women training for social settlement work. Plainsong refers to a style of liturgical vocal music developed in Europe early in the first millennium.\n\n22. Ellinwood and Douglas, _To Praise God_ , 20.\n\n23. Bridgham Curtis, \"Memorial,\" 9.\n\n24. Marwood, _The Wooing of a Recluse_ , 110.\n\n25. Natalie Curtis, \"A New Type of Architecture in the Southwest,\" 334.\n\n26. Guest book of Wheeler Bailey, April 28, 1913, collection of David N. Reynolds.\n\n27. The Palomar incorporated the exotic North African architectural elements Mead loved\u2014arches and balconies, inlaid tile, and plants and fountains in an oasis-like atrium enclosed and private from the street. Future residents of the Palomar Apartments included Wallis Warfield Simpson and Charles Lindbergh.\n\n28. George DeClyver Curtis, Diary, July 18, 1913.\n\n29. Ibid., various entries, July 1913. Natalie Curtis to Charles Lummis, December 1913, LC\/BRL.\n\n###  Chapter 6\n\nVirginia Scharff, \"Introduction: Women Envision the West,\" 5.\n\n1. In 1870 the Nahant town officers\u2014the town clerk, selectmen, assessors, and the town treasurer\u2014were _all_ Johnsons, and twenty-five of eighty-seven households on the island were headed by someone named Johnson. See _Essex-Country History_.\n\n2. In author interviews and correspondence with Caroline Bishop Stanley's living relatives and with individuals who knew her, there was no evidence that she ever mentioned to anyone that she was born a twin.\n\n3. Programme, Graduating Exercises, Town of Nahant School Report, 1893\u201394, 16\u201317, NHS.\n\n4. Offord, \"The Foremost American School of Music.\"\n\n5. New England Conservatory of Music's 1905 yearbook; author correspondence with Vicky R. Harrison, archivist, Alpha Chi Omega.\n\n6. \"Italian Count Elopes with Miss M. F. Johnson,\" _Lynn (MA) Daily Item_ , January 7, 1907.\n\n7. \"Cardinal in Frederick,\" _Baltimore Sun_ , November 18, 1906.\n\n8. \"Hannah More Academy Opens,\" _Baltimore Sun_ , October 6, 1907.\n\n9. _Lynn (MA) Daily Evening Item_ , June 9, 1913.\n\n10. See Natalie Curtis, \"Value of Music School Settlements in Cities.\" Details of Carol Stanley in Boston ca. 1913 from interview with Harriet Cabot Ropes, MCW\/MW.\n\n###  Chapter 7\n\nWilla Cather, _The Selected Letters_ , 262.\n\n1. See Heller and Rudnick, _1915_ , 169.\n\n2. Hutchins Hapgood quoted in Heller and Rudnick, _1915_ , 69.\n\n3. Rudnick, _Mabel Dodge Luhan_ , 170.\n\n4. Martin Green, \"The New Art,\" in Heller and Rudnick, _1915_ , 159. The name \"Ashcan\" referred to the \"mildly squalid scenes\" their art portrayed. See Robinson, _Georgia O'Keeffe_ , 96.\n\n5. The Eight were Henri, William Glackens, John Sloan, George Luks, and Everett Shinn, who had all been pupils of Henri's, plus Arthur B. Davies, Ernest Lawson, and Maurice Prendergast.\n\n6. See Notebooks of Alice Klauber, \"Bits of the Summer 1912,\" and \"Robert Henri, Class in Spain.\"\n\n7. Klauber, \"Bits of the Summer 1912,\" June 22\u201326.\n\n8. Henri, _The Art Spirit_ , 15.\n\n9. Klauber, \"Robert Henri lectures,\" 22.\n\n10. Henri, _The Art Spirit_ , 16, 111.\n\n11. Margery Ryerson's book written with Henri, the classic _The Art Spirit_ , appeared in 1923 while Henri was still alive. Alice's transcriptions of Henri's lectures and notes, known among art history scholars as the Klauber Manuscript, were published in their entirety in Bennard B. Perlman's _Robert Henri: His Life and Art_.\n\n12. See Fowler, _A Laboratory for Anthropology_ , 272; and Chapman and Barrie, _Kenneth Milton Chapman_ , 87. \"San Diego Exposition News,\" October 1912, quoted in Chauvenet, _Hewett and Friends_ , 102.\n\n13. Fowler, _A Laboratory for Anthropology_ , 274.\n\n14. Natalie Curtis to Alice Klauber, March 11, 1915, SDMA.\n\n15. \"New Mexico's Exhibit at San Diego in 1915,\" _El Palacio_ , November 1913, 5.\n\n16. Robert Henri to Alice Klauber, March 11, 1914, SDMA.\n\n17. In the collection of SDMA.\n\n18. Robert Henri to Edgar Hewett, October 25, 1914, SDMA.\n\n19. Petersen, _San Diego's First Lady of the Arts_ , 21, 32. Henri's California visit has been the subject of several articles. See Henri, \"My People,\" _Craftsmen_ 27 (February 1915), 459\u201369; Martin Petersen, \"Henri's California Visit,\" _FASM Newsletter_ I (February 1971); Petersen, \"Modern Art Goes to California,\" _Southwest Art_ 2 (September 1972); Jean Stern, \"Robert Henri and the 1915 San Diego Exposition,\" _American Art Review_ 2 (September\u2013October 1975).\n\n20. Trenton, _Independent Spirits_ , 75.\n\n###  Chapter 8\n\nJohn Wetherill, quoted in Gillmor and Wetherill, _Traders to the Navajos_ , 256.\n\n1. Charles Winfred Douglas to Kurt Schindler, March 8, 1913. Details of the Douglas-Schindler relationship from correspondence, Charles Winfred Douglas to Kurt Schindler, 1913, KS\/NYPL.\n\n2. Diary of George DeClyver Curtis, September 21, 1913.\n\n3. Charles Winfred Douglas to Kurt Schindler, March 8, July 22, and August 2, 1913, KS\/NYPL.\n\n4. Natalie Curtis Burlin, \"Theodore Roosevelt in Hopi-land,\" 87. Natalie and Roosevelt must have corresponded about their Arizona itineraries because their individual accounts make reference to their rendezvous at Walpi as an anticipated and orchestrated event, including rooms reserved for their meeting at the government agency below Third Mesa.\n\n5. The assassin's bullet was thwarted when it became lodged in the steel spectacle case Roosevelt carried in his right breast pocket. Following his near miss with death, Roosevelt roared the famous line \"It takes more than that to kill a Bull Moose!\"\n\n6. Morris, _Colonel Roosevelt_ , 267. In April 1913 Roosevelt's daughter Ethel was married at Christ Church in Oyster Bay. The ceremony was attended by Nahant resident Henry Cabot Lodge and by Roosevelt's old friend, and Carol Stanley's grandmother's cousin, the former ambassador to Great Britain Joseph Choate.\n\n7. Mellow, _Charmed Circle_ , 171; Morris, _Colonel Roosevelt_ , 268.\n\n8. Theodore Roosevelt, \"Across the Navajo Desert,\" 309.\n\n9. Charles Winfred Douglas to Kurt Schindler, July 22, and August 2, 1913, KS\/NYPL.\n\n10. Diary of Alice Klauber, August 10, 1913, SDMA.\n\n11. Diary of Alice Klauber, August 13, 1913, SDMA.\n\n12. Natalie Curtis Burlin, \"Theodore Roosevelt in Hopi-land,\" 87.\n\n13. Diary of Alice Klauber, August 12, 1913, SDMA.\n\n14. See Froeschauer-Nelson, _Cultural Landscape Report_.\n\n15. Ellinwood and Douglas, _To Praise God_ , 20.\n\n16. Diary of Alice Klauber, August 13 and August 18, 1913, SDMA.\n\n17. Kirk owned two stores near the canyon. He would be bought out by George Kennedy in 1916. Kennedy's business would be bought in 1919 by \"Cozy\" McSparron and partners and become the trading post and dude ranch known as the Thunderbird Lodge.\n\n18. Diary of Alice Klauber, August 14, 1913, SDMA.\n\n19. Mummy Cave, named for the dozens of mummified skeletons found here by archaeologists, was the site of a massacre of the Navajos by the Spanish in 1804.\n\n20. Natalie Curtis Burlin, \"Theodore Roosevelt in Hopi-land,\" 87.\n\n###  Chapter 9\n\nZane Grey, foreword, _The Rainbow Trail_.\n\n1. James, _Pages from Hopi History_ , 165, 169.\n\n2. Diary of Alice Klauber, \"Arizona Trip Summer 1913,\" August 18\u201319, 1913.\n\n3. This quotation and the following quotations in the next several paragraphs are from Natalie Curtis Burlin, \"Theodore Roosevelt in Hopi-land,\" 87\u201388.\n\n4. Morris, _Colonel Roosevelt_ , 199.\n\n5. N. C. Burlin, \"Theodore Roosevelt in Hopi-land,\" 88.\n\n6. Theodore Roosevelt, \"Across the Navajo Desert,\" 316.\n\n7. N. C. Burlin, \"Theodore Roosevelt in Hopi-land,\" 87\u201388.\n\n8. Nicholas Roosevelt Diary, August 22, 1913, AC 194-p, Fray Ang\u00e9lico Ch\u00e1vez History Library, New Mexico History Museum, Santa Fe, New Mexico.\n\n9. N. C. Burlin, \"Theodore Roosevelt in Hopi-land,\" 88.\n\n10. T. Roosevelt, \"The Hopi Snake Dance,\" 366.\n\n11. Ibid., 367. Douglas was referring to his extended time among the Hopi in the first years of the century when he had spent six months in the Southwest outback.\n\n12. All excerpts from Diary of Nicholas Roosevelt, August 22, 1913.\n\n13. T. Roosevelt, \"The Hopi Snake Dance,\" 368\u201369.\n\n14. Diary of Nicholas Roosevelt, August 22, 1913.\n\n15. N. C. Burlin, \"Theodore Roosevelt in Hopi-land,\" 88.\n\n16. Ibid., 88\u201389. Roosevelt wrote three articles for _Outlook_ about his six weeks in Arizona. \"Across the Navajo Desert\" was the second of the series.\n\n17. Diary of Nicholas Roosevelt, August 22, 1913.\n\n18. Petersen, _San Diego's First Lady of the Arts_ , 13.\n\n19. Diary of Nicholas Roosevelt, August 22, 1913.\n\n20. N. C. Burlin, \"Theodore Roosevelt in Hopi-land,\" 89.\n\n21. Diary of Nicholas Roosevelt, August 22, 1913.\n\n22. Natalie wrote that the brief introduction to Hopi culture helped prepare the crowd for the astonishing ceremony they witnessed that day. Nicholas Roosevelt, however, was appalled by her suggestion that Roosevelt be involved and blamed what he saw as Natalie's poor judgment on Douglas's influence. See Natalie's \"Theodore Roosevelt in Hopi-land,\" and Diary of Nicholas Roosevelt, August 22, 1913.\n\n23. The Snake Dance is the grand finale of Hopi ceremonies that pray for rain. The Hopis believe their ancestors originated in an underworld, and that their gods and the spirits of their ancestors live there. Snakes are their brothers, and the Hopis believe that snakes will carry their prayers for rain to the Rainmakers beneath the earth. Thus the dancers carry snakes in their mouths to impart prayers to them.\n\n24. Petersen, _San Diego's First Lady of the Arts_ , 13.\n\n25. Alice did not record her Hopi name or its translation. Natalie referenced the name and addressed Alice's Christmas cards in 1913 and 1914 to \"My dear friend Koiyahoinim\" or Koyahoynim. Alice Klauber archive, SDMA.\n\n26. Alice Klauber, \"Arizona Trip Summer 1913,\" August 26, 1913; Ellinwood and Douglas, _To Praise God_ , 21.\n\n27. Douglas's obituary in _The Explorers Journal_ , Spring 1945, stated, \"He located the ancient trail leading up the Enchanted Mesa near the pueblo of Acoma and he was the first to describe the flute altar at Shimgopavi.\"\n\n28. Alice Klauber, \"Arizona Trip Summer 1913,\" September 4, 1913.\n\n29. Diary of George DeClyver Curtis, September 16 and 23, 1913.\n\n30. T. Roosevelt, \"The Hopi Snake Dance,\" 365, 367\u201368.\n\n###  Chapter 10\n\nWilla Cather, _The Song of the Lark_ , 278.\n\n1. Milne, _George William Curtis & the Genteel Tradition_, 188.\n\n2. See Pack, _We Called It Ghost Ranch_ , 23; Poling-Kempes, _Ghost Ranch_ , 19.\n\n3. Bernheimer, _Rainbow Bridge_ , 44; T. Roosevelt, \"Across the Navajo Desert,\" 314.\n\n4. T. Roosevelt, \"Across the Navajo Desert,\" 316.\n\n5. Bernheimer, _Rainbow Bridge_ , 29.\n\n6. Hegemann, _Navaho Trading Days_ , 228, 239.\n\n7. Roosevelt, \"Across the Navajo Desert,\" 311.\n\n8. Hegemann, _Navaho Trading Days_ , 224.\n\n9. Gillmor and Wetherill, _Traders to the Navajos_ , 34. Details about Kidder's early career are from Chapman and Barrie, _Kenneth Milton Chapman_ , 62\u201364.\n\n10. Gillmor and Wetherill, _Traders to the Navajos_ , 219.\n\n11. Diary of Louisa Wetherill, June 26\u2013July 8, 1915.\n\n12. Several years after his acquittal and return to his father's tribe, young Tse-ne-gat died of tuberculosis. See Frank Pyle, \"The Uprising of Polk and Posey in 1915,\" in Berkholz, _Old Trading Posts of the Four Corners_ , 21\u201322; and Gillmor and Wetherill, _Traders to the Navajos_ , 211\u201316.\n\n###  Chapter 11\n\nAlice Corbin, \"Pedro Montoya of Arroyo Hondo,\" _Red Earth_ , 43.\n\n1. _Santa Fe New Mexican_ , January 17, 1916, in La Farge, _Santa Fe_ , 222.\n\n2. _Santa Fe New Mexican_ , July 8, 1916.\n\n3. Ibid., April 11, 1916.\n\n4. _Chicago Daily Tribune_ , October 31 and November 1, 1915.\n\n5. Bridgham Curtis, \"Memorial,\" 9.\n\n6. Patterson, _Natalie Curtis Burlin_ , 269.\n\n7. Natalie Curtis, \"Busoni's Indian Fantasy,\" 538. See also Bredenberg, \"Natalie Curtis Burlin,\" 7.\n\n8. Weigle and Fiore, _Santa Fe and Taos_ , 5\u201310.\n\n9. Tobias and Woodhouse, _Santa Fe_ , 53.\n\n10. Weigle and Fiore, _Sante Fe and Taos_ , 10.\n\n11. Henderson, \"Early Impressions of Santa Fe, 1916\u20131934,\" ACH\/HRC.\n\n12. Alice Corbin quoted in Hildegard Hawthorne, \"The Literary Digest International Book Review, 1916,\" ACH\/HRC.\n\n13. E. Lewis, _Willa Cather Living_ , 95\u201396.\n\n14. Ina Sizer Cassidy quoted by Robertson and Nestor, _Artists of the Canyons and Caminos_ , 31.\n\n15. Ibid., 48.\n\n16. Rothman, _On Rims and Ridges_ , 138.\n\n17. Richard Leroy Pf \u00e4ffle was called by various names in his lifetime, including Richard, Dick, and Roy. Except where the name is quoted otherwise, he is Roy throughout this text.\n\n18. Jack Lambert to Peggy Pond Church, January 3, 1980, LAHM\/PPC.\n\n19. Diary of George DeClyver Curtis, April 19, 1916.\n\n20. Ibid., May 20\u201323, 1916.\n\n21. Ibid., May 27, 1916.\n\n22. Ibid., May 29, May 30, 1916.\n\n23. Natalie Curtis to Roy D. Chapin, July 14, 1916, LAHM\/PPC.\n\n24. Long, _Roy D. Chapin_ , xix; Roy D. Chapin to Natalie Curtis, July 7, 1916, LAHM\/PPC.\n\n25. Natalie Curtis to Roy Chapin, July 5, 1916, LAHM\/PPC.\n\n26. Carol Stanley to Roy D. Chapin, August 7, 1916, LAHM\/PPC.\n\n27. Natalie Curtis to Roy D. Chapin, September 1916; Roy D. Chapin to Ashley Pond, August 23, 1916, LAHM\/PPC.\n\n28. Natalie Curtis to Roy Chapin, July 14, 1916; Roy D. Chapin to Natalie Curtis, July 20, 1916, LAHM\/PPC.\n\n29. Roy D. Chapin to partners, August 22, 1916, LAHM\/PPC.\n\n30. Ashley Pond's daughter, Peggy Pond Church, became a highly respected poet and writer who researched the relationship between her father and the Detroit partners. See Peggy Pond Church to Francesca, May 7, 1979, LAHM\/PPC. With the exception of Roy Chapin, who visited the Ramon Vigil Ranch and later sent his son, Roy Jr., to Ashley Pond's Los Alamos Ranch School in 1929, the Detroit partners were woefully ignorant about their ranch's natural resources, climate, cultural context, and remote location, and thus about how these factors were as responsible for the failure of the Pajarito Club as Pond's business ineptitude. Peggy Pond Church believed the \"scheme of those two women [Carol Stanley and Natalie Curtis] to form a cultural Indian art center was even more hair-brained\" than any of her father's ideas (ibid.). During World War 11 the Pajarito Club was seized by the U.S. government for the Manhattan Project. The headquarters area became a fenced and guarded \"radioactivity site\" and later a facility for remotely controlled critical assemblies. The main lodge was torn down, but the log house built by Pond in 1915 was still standing in 1979. See Peggy Pond Church to Francesca, May 7 and 21, 1979, LAHM\/PPC. For the complete story of Peggy Pond Church and her family at Los Alamos, see Sharon Snyder, _At Home on the Slopes of Mountains_.\n\n###  Chapter 12\n\nMarjorie Lambert quoted in Chapman and Barrie, _Kenneth Milton Chapman_ , 69.\n\n1. See Traugott, _The Art of New Mexico_.\n\n2. Alice visited Santa Fe in March 1915, but her diary did not record the details of this trip. Diary of Alice Klauber, August 12, 1916, SDMA.\n\n3. Carol Stanley to Roy Chapin, August 7, 1916, LAHM\/PPC.\n\n4. McCarthy, _Walter Pach (1883\u20131958)_ , 80, 81.\n\n5. Diary of Alice Klauber, August 14, 1916, SDMA.\n\n6. Crescensio and his brother Julian Martinez would exhibit their paintings in New York and, with artists Fred Kabotie, Awa Tsireh, and others, set style precedents that would dominate Indian painting for fifty years. See Chapman and Barrie, _Kenneth Milton Chapman_ , 177.\n\n7. Diary of Alice Klauber, August 16, 1916, SDMA.\n\n8. Marjorie Lambert quoted in Chapman and Barrie, _Kenneth Milton Chapman_ , 69.\n\n9. Ibid.\n\n10. Diary of Alice Klauber, August 23\u201326, 1916, SDMA.\n\n11. Rainbow Bridge Registry, 1916, WACC\/NPS.\n\n12. _Santa Fe New Mexican_ , October 5, 1916; Journals of C. W. Douglas, July, 1921, 10, CWD\/DMA.\n\n13. Carol Stanley to Roy D. Chapin, August 22, 1916, LAHM\/PPC.\n\n14. Peterson, _Alice Klauber_ , 27.\n\n15. Udall, _Modernist Painting in New Mexico_ , 20.\n\n16. Diary of Alice Klauber, September 4, 1916, SDMA.\n\n17. Natalie Curtis to Roy Chapin, September 19, 1916, LAHM\/PPC; _Santa Fe New Mexican_ , \"Personals,\" Tuesday, September 26, 1916.\n\n18. Natalie Curtis to Roy Chapin, September 19, 1916, LAHM\/PPC.\n\n19. _Santa Fe New Mexican_ , \"Personals,\" September 13, 1916.\n\n20. _Santa Fe New Mexican_ , October 23, 1916.\n\n21. Details of the Pf\u00e4ffles' trip to Kayenta and Rainbow Bridge in ibid.\n\n###  Chapter 13\n\nAlice Corbin, \"A Litany in the Desert,\" in Powell, _The Spirit of Democracy_ , 27.\n\n1. Annual Report of the School of American Research, 1918, quoted in Lewis and Hagan, _A Peculiar Alchemy_ , 24.\n\n2. Udall, _Modernist Painters in New Mexico_ , 28.\n\n3. _Santa Fe New Mexican_ , January 13, 1917.\n\n4. _Avoca News_ , September 29, 1907.\n\n5. Udall, _Modernist Painters in New Mexico_ , 26; Patterson, _Natalie Curtis Burlin_ , 282.\n\n6. Augusta Curtis to George DeClyver Curtis, August 18, 1917, NCBA.\n\n7. Diary of George DeClyver Curtis, March 2, 1917, UCLA.\n\n8. Natalie Curtis to Aunt Natalie, May 27, 1917, NCBA; \"after first youth\" is from Sergeant, _Willa Cather_ , 150.\n\n9. Natalie Curtis to Aunt Natalie, May 27, 1917, NCBA.\n\n10. Robert Henri to Alice Klauber, June 18, 1917, SDMA.\n\n11. Ibid., August 2, 1917, SDMA.\n\n12. Ibid., June 18, 1917, SDMA.\n\n13. Ibid., August 2, 1917, SDMA.\n\n14. Simmons, _Charles. F. Lummis, Author and Adventurer_ , 34; Charles Lummis to Natalie Curtis Burlin, August 2, 1917, LC\/BRL.\n\n15. Augusta Curtis to George DeClyver Curtis, August 18, 1917, NCBA.\n\n16. Faunce, _Desert Wife_ , 242, 278.\n\n17. Carol Pf\u00e4ffle to Roy Chapin, December 13, 1917, LAHM\/PPC.\n\n18. Carol Pf\u00e4ffle to Roy Chapin, November 30, 1917, LAHM\/PPC.\n\n19. _Santa Fe New Mexican_ , November 26, 1917.\n\n20. Robertson and Nestor, _Artists of the Canyons and Caminos_ , 55.\n\n21. Chauvenet, _Hewett and Friends_ , 128.\n\n22. Diary of George DeClyver Curtis, September 8, 1917, UCLA.\n\n23. Robert Henri to Alice Klauber, September 22, 1917, and October 9, 1917, SDMA.\n\n24. Robert Henri quoted in Robertson and Nestor, _Artists of the Canyons and Caminos_ , 53.\n\n25. _Santa Fe New Mexican_ , November 26, 1917.\n\n26. LC\/BRL.\n\n###  Chapter 14\n\nMabel Dodge Luhan, _Edge of Taos Desert_ , 17.\n\n1. Luhan, _Edge of Taos Desert_ , 19\u201320.\n\n2. Ibid., 22.\n\n3. Ibid., 22\u201323.\n\n4. Ibid., 3.\n\n5. Udall, _Modernist Painters in New Mexico_ , 34, 41.\n\n6. Hartley, \"Aesthetic Sincerity,\" 333.\n\n7. Weigle and Fiore, _Santa Fe and Taos_ , 13. Mary Austin's Santa Fe Little Theater, founded in 1922 (today the Santa Fe Playhouse), incorporated the same open-door policy for amateurs and professional adopted by the Museum of Art a few blocks away on the plaza.\n\n8. Cronyn, _The Path on the Rainbow_ , xiii, xxii.\n\n9. Peggy Pond Church, \"Early Days,\" 2, LAHM\/PPC.\n\n10. Carol Pf\u00e4ffle to Roy Chapin, March 6, 1918, LAHM\/PPC; Diary of Margaret Pf\u00e4ffle, December 15, 16, 17, 1917.\n\n11. Diary of Margaret Pf \u00e4ffle, January 24\u201331, February 12, 22, March 28, April 23, May 4, 1918. Frederic \"Eric\" H. Douglas (born October 29, 1897) received a BA degree from the University of Colorado in 1921 and then did five years of graduate work in fine arts at the University of Michigan and the Pennsylvania Academy of Fine Arts. In 1929 he joined the Denver Art Museum as curator of Indian art and was director of the museum from 1940 to 1942. In 1947 Douglas became curator of the Department of Native Art, a position he held until his death in 1956. Douglas brought a new approach to the art of Native American peoples and was credited with single-handedly creating a new benchmark for Indian art in the contemporary art world. See \"Frederic Douglas, His Story,\" Douglas Society newsletter, Denver Art Museum, October 2006.\n\n12. Carol Pf\u00e4ffle to Roy Chapin, February 26, 1918, LAHM\/PPC.\n\n13. See Jenkins and Thorpe, _From Retreat to Resort_.\n\n14. Carol Pfaffle to Roy Chapin, February 26 and March 6, 1918, LAHM\/PPC.\n\n15. Jack Lambert biography from Peggy Pond Church Collection, undated notes, \"Lambert & Pf\u00e4ffle,\" LAHM\/PPC; Love, \"A Cowboy Recollects,\" _Santa Fean_ magazine, December 1981; King, \"Jack Lambert\"; author interviews with Henry McKinley and Peggy Pack McKinley.\n\n16. Diary of Margaret Pf \u00e4ffle, July 26, October 2 and 3, December 16, 1918. Roy Chapin would serve as Hoover's secretary of commerce in 1932\u201333. In 1933 Chapin helped draft Franklin D. Roosevelt's automobile industry code for the National Industrial Recovery Act. Chapin's son Roy Jr., born in 1915, was the first of Chapin's six children and grew up to become the chairman and CEO of American Motors. See Long, _Roy D. Chapin_.\n\n17. _Santa Fe New Mexican_ , November 11, 1918; Diary of Margaret Pf \u00e4ffle, November 11, 1918.\n\n18. Sze, \"The Witter Bynner House,\" 8.\n\n19. Natalie Curtis Burlin to George Curtis, March 9, 1921, NCBA.\n\n20. Robertson and Nestor, _Artists of the Canyons and Caminos_ , 74.\n\n21. Pach, \"John Sloan,\" 3.\n\n22. Robertson and Nestor, _Artists of the Canyons and Caminos_ , 76.\n\n23. Ibid., 78. The Davey's home on Upper Canyon Road became the Randall Davey Audubon Center in 1983.\n\n24. Patterson, _Natalie Curtis Burlin_ , 307.\n\n25. Lois Pf\u00e4ffle to Margaret Pf\u00e4ffle, September 23, 1919.\n\n26. Dairy of Lois Pf\u00e4ffle, January 3, 20, 21, February 8, 20, March 5, 1920; Dorothy Kent to Rockwell Kent, January 13, 1919, RK\/AAA.\n\n27. Details of Pf\u00e4ffle purchase from Pf\u00e4ffle papers, Francis O. Wilson files, SRA. At least one of the ranch underwriters, Florence Dibell Bartlett of Chicago, began construction on her own summer house in 1922, on property leased from the Pf\u00e4ffles. Pf\u00e4ffle papers, Wilson files, SRA.\n\n###  Chapter 15\n\nAlice Corbin, \"Sunlight,\" _Red Earth_ , 40.\n\n1. Natalie Curtis to Augusta Curtis, June 18, 20, 1921, NCBA.\n\n2. Natalie Curtis Burlin to George Peabody, July 1, 1921, NCBA.\n\n3. Natalie Curtis Burlin to friends, July 27, 1921, Peabody Papers, LOC.\n\n4. Natalie Curtis Burlin to Augusta Curtis, July 4, 1921, NCBA.\n\n5. Natalie Curtis Burlin, \"A Futurist Experiment.\"\n\n6. Natalie Curtis Burlin to friends, July 13, 1921, Peabody papers, LOC.\n\n7. Leo Stein quoted in Rudnick, _Mabel Dodge Luhan_ , 168.\n\n8. Stein told Hemingway, \"You are all a ' _g\u00e9n\u00e9ration perdue._ '. . . That is what you are. That's what you all are . . . all of you young people who served in the war. You are a lost generation.\" See Hemingway, _A Moveable Feast_ , 61.\n\n9. Diary of Alice Klauber, September 11, 1921; Walter Pach to Alice Klauber, December 24, 1924, May 6, 1925, February 15, 1944, SDMA.\n\n10. Diary of Alice Klauber, October 11, 1921, SDMA.\n\n11. Natalie Curtis Burlin to friends, October 16, 1921, NCBA.\n\n12. Tappert, _Out of the Background_.\n\n13. Natalie Curtis Burlin to Peabody and friends, October 16, 1921, NCBA.\n\n14. Ibid.\n\n15. Diary of Alice Klauber, October 3, 1921, SDMA.\n\n16. Hemingway, _A Moveable Feast_ , 15.\n\n17. Diary of Alice Klauber, October 24, 1921, SDMA.\n\n18. Diary of George DeClyver Curtis, October 26, 1921.\n\n19. Diary of Alice Klauber, October 24, 1921, SDMA.\n\n20. Alice Klauber to Augusta Curtis, October 24, 1921, NCBA.\n\n21. Details of the memorial service and procession from Diary of Alice Klauber, October 29, 31, 1921, SDMA; Paul Burlin to Augusta Curtis, November 2, 1921, NCBA.\n\n22. George DeClyver Curtis to Bridgham Curtis, October 29, 30, 1921, NCBA; Diary of George DeClyver Curtis, October 30, 1921.\n\n23. Augusta Curtis to George Curtis, October 27, 1921, NCBA.\n\n24. Paul Burlin to Augusta Curtis, November 2, 1921, NCBA.\n\n25. Diary of Alice Klauber, November 17, 1921, SDMA.\n\n26. George DeClyver Curtis, \"Natalie Curtis Burlin.\"\n\n27. Natalie Curtis Burlin, \"Theodore Roosevelt in Hopi-land,\" 87.\n\n###  Chapter 16\n\nMabel Dodge Luhan, \"Taos and Individualism.\"\n\n1. Douglas, Journals, July\u2013August 1921, 33, 36. Richard Leroy Pf \u00e4ffle reached Rainbow Bridge at least four times between 1916 and 1923, including twice in 1918 (May and August). Because the registry at the bridge was lost or stolen several times after it was started in 1909, and many years are missing, it is impossible to say how many trips Roy, or Carol, made to the bridge. On July 10, 1921, John Wetherill wrote in the registry, \"Every trip I see new beauties. It improves on acquaintance.\" Roy, Winfred and Eric Douglas, and Albert Smith's signatures follow Wetherill's on July 29, 1921. It was Smith's second trek to the bridge in two weeks. Rainbow Bridge Registry transcribed by Harvey Leake.\n\n2. Carol Pf\u00e4ffle to Elbridge Adams, December 16, 1921, NCBA.\n\n3. Paul Burlin to Augusta Curtis, November 2, 1921, NCBA.\n\n4. Bynner, \"Alice and I,\" 36; Rudnick, _Mabel Dodge Luhan_ , 243; Bynner, \"A City of Change,\" 45\u201346.\n\n5. Witter Bynner to Alice Long, in Sze, \"The Witter Bynner House,\" 8, 14.\n\n6. Dietrich, _New Mexico Recollections_. Dietrich's properties in Santa Fe included several historic homes on Canyon Road including El Zaguan (the Johnson house), the Borrego House, and the Juan Jose Prada House, where Dietrich made her home.\n\n7. Witter Bynner quoted in Museum of New Mexico, _Window on the West_ , 191.\n\n8. Sze, \"The Witter Bynner House,\" 4\u20135.\n\n9. D. H. Lawrence, _New Mexico_ , quoted in Hillerman, _The Spell of New Mexico_ , 30\u201331.\n\n10. _Santa Fe New Mexican_ , May 3, 1922.\n\n11. Klauber, \"In Hospital,\" _Poems_.\n\n12. Inside back cover, Diary of Alice Klauber, 1923, SDHC.\n\n13. Alice Klauber, untitled, _Poems_.\n\n14. Carol Pf\u00e4ffle to Elbridge Adams, December 16, 1921; copied by Augusta Curtis for George D. Curtis, December 13, 1922, NCBA.\n\n15. Parsons, _Social Organization of the Tewa_ , 32.\n\n16. Pf\u00e4ffle lease agreements, April and September 1926, Francis Wilson papers, SRA. The address is given as both 537 #2 and 547 Canyon Road.\n\n17. Robertson and Nestor, _Artists of the Canyons and Caminos_ , vii, 106; Ina Sizer Cassidy, _Santa Fe New Mexican_ , October 29, 1921. In 1924 Will Shuster created the giant puppet effigy Zozobra, Old Man Gloom, burned each year at the Santa Fe Fiesta.\n\n18. See Mullin, _Culture in the Marketplace_ , 61.\n\n19. Sergeant, _Shadow-Shapes_ , 205.\n\n20. Sergeant, \"Journal of a Mud House, IV,\" 61.\n\n21. Sergeant, \"Journal of a Mud House, 111,\" 778.\n\n22. In 1922 Harry P. Mera's simple Zia sun symbol, based on the historic design motif found on local ceramics, won the competition for the New Mexico state flag.\n\n23. Chapman and Barrie, _Kenneth Milton Chapman_ , 333.\n\n###  Chapter 17\n\nElsie Clews Parsons, _Social Rule_ , 55.\n\n1. Maud Cabot Morgan, July 22, 1985, tape transcription, 1, MCW\/ WM. Mary's mother, Sarah Perkins Cabot Wheelwright, was related to Thomas Handyside Perkins, the prominent Boston businessman who built the first resort hotel on Nahant island in 1823.\n\n2. Ibid.; Wheelwright, \"Journey Towards Understanding,\" 2.\n\n3. Wheelwright, \"Journey Towards Understanding,\" 8.\n\n4. Wheelwright, tape transcription, July 2003, reel 1, 12, 13, MCW\/ WM.\n\n5. Ibid., 13.\n\n6. Newcomb, _Hosteen Klah_ , 157\u201358.\n\n7. Wheelwright, \"Journey Towards Understanding,\" 7.\n\n8. Wheelwright, tape transcription, July 2003, reel 1, 7\u20138, MCW\/ WM.\n\n9. Wheelwright, \"Journey Towards Understanding,\" 17; Wheelwright, tape transcription, July 2003, reel 1, 8, MCW\/ WM. Franc Newcomb recounts Mary Wheelwright's first visit to Nava differently and wrote that the San Gabriel group, guided by Cox, arrived on horseback. Cox never went on horse expeditions, and only went into the outback as an automobile driver and guide. Wheelwright's version was given in a taped interview and repeated in her unpublished memoir, \"Journey Towards Understanding.\" See Newcomb, _Hosteen Klah_ , 159\u201360.\n\n10. Wheelwright, \"Journey Towards Understanding,\" 18, 19. Family historians claim that when Mary was a very young child her father had tied her to the mast of his boat to force her to overcome her fear of water. She did and she became a confident sailor and competent navigator\u2014unusual for a woman of her time\u2014and was known to urge captains into rough water. See interview with Mrs. W. Rodman \"Gertrude\" Fay (daughter of the founder of the Schirmer Music Co.), Northeast Harbor, Maine, August 1, 1985, MCW\/ WM.\n\n11. Newcomb, \"Symbols in Sand,\" 2.\n\n12. Klah's apprentice, his nephew Beyal (or Beaal) Begay, died in 1931. Klah knew that he did not have enough time and energy to begin the training with another apprentice. See McGreevy, \"Why Navajo Studies?\"\n\n13. In her memoir, Mary dismissed references to Klah being a transvestite. See \"Journey Towards Understanding,\" 23; Newcomb, _Hosteen Klah_ , 97.\n\n14. Wheelwright, \"Journey Towards Understanding,\" 31, 53, 68.\n\n15. Ibid., 14. Matthews's notebooks of medicinal plants, diaries of ceremonies attended, and files were moved from the University of California, Berkeley, to the Wheelwright Museum in 1951.\n\n16. Wheelwright, \"Journey Towards Understanding,\" 9.\n\n17. King, \"Jack Lambert.\"\n\n18. Wheelwright, \"Journey Towards Understanding,\" 42.\n\n19. Grey, _Tales of Lonely Trails_ , 16\u201317.\n\n20. Rainbow Bridge Registry, May 8, 1923, NPS.\n\n21. Wheelwright, \"Journey Towards Understanding,\" 5.\n\n22. Phillip Blood, interview with author, April 2012; Church, \"notes,\" LAHM\/PPC. Several years between 1909 and 1925 are missing from the Rainbow Bridge Registry held in the National Park Service collection. Not everyone who visited the bridge signed in, and John Wetherill often later recorded names based on his recollections. Harvey Leake to author.\n\n23. Boas, \"Elsie Clews Parsons,\" 480; Lavender, _Scientists and Storytellers_ , 51.\n\n24. Parsons, _The Old-Fashioned Woman_ , 50.\n\n25. Parsons, _Tewa Tales_ , ix.\n\n26. Lavender, _Scientists and Storytellers_ , 55. Parsons donated thousands of dollars to the society and aided the work of Ruth Benedict, Franz Boas, Ruth Bunzel, Esther Goldfrank, Morris Opler, Gladys Reichard, Ruth Underhill, and many others.\n\n27. Robert Herrick wrote two novels, _Chimes_ (1926) and _The End of Desire_ (1932), based on his relationship with Elsie Clews Parsons. See Deacon, _Elsie Clews Parsons_ , 279.\n\n28. Parsons, _Social Organization of the Tewa_ , 7.\n\n29. Elsie Clews Parsons to Herbert Parsons, November 17, December 4, 1923, quoted in Deacon, _Elsie Clews Parsons_ , 286.\n\n30. Elsie Clews Parsons, quoted by Gladys Reichard, in Lavender, _Scientists and Storytellers_ , 68.\n\n31. Elsie (Lissa) Parsons Kennedy, in Lavender, _Scientists and Storytellers_ , 67.\n\n32. Elsie Clews Parsons to Lissa, November 25, 1925, in Deacon, _Elsie Clews Parsons_ , 300.\n\n33. Diary of Margaret Pf\u00e4ffle, December 7, 1925\u2013January 10, 1926.\n\n###  Chapter 18\n\nVirginia Scharff, \"Introduction: Women Envision the West,\" I.\n\n1. _Grand Canyon MacLeans_ , 8.\n\n2. Willa Cather to Elizabeth Shepley Sergeant, June 23, 1925, and Willa Cather to Harriet Fox Whicher, October 16, 1925, in Cather, _Selected Letters_ , 370\u201371, 374.\n\n3. John D. Rockefeller 111 and his family rented Canjilon Camp for an entire month in the summer of 1930. See _Grand Canyon MacLeans_ , 10.\n\n4. Diary of Margaret Pf\u00e4ffle, July 19, 1926.\n\n5. Pen LaFarge, in Stark and Rayne, _El Delirio_ , 106.\n\n6. _Santa Fe New Mexican_ , July 6, 1927.\n\n7. Francis Wilson papers.\n\n8. Love, \"A Cowboy Recollects.\"\n\n9. Richard Leroy Pf\u00e4ffle, December 13, 1927, Lambert papers, Fray Angelico Chavez History Library.\n\n10. King, \"Jack Lambert.\"\n\n11. Lois Lambert returned to New Mexico by 1930 or 1931 and managed the dining room run by the Fred Harvey Company at the Puye Cliff Dwellings at Santa Clara Pueblo. While working at Puye, Lois met a forest ranger named Tom Buchanan, whom she married and lived with near Santa Fe the rest of her life.\n\n12. _Grand Canyon MacLeans_ , 8.\n\n13. The American Southwest, especially the Spanish-speaking region of New Mexico, was still an untamed frontier to the folks around Nahant. In the minds of the attorneys in Lynn, Mass., who were assembling the legal documents for sale of the Johnson property on Nahant, they were managing an international real estate deal because Carol Bishop Stanley had moved to a foreign country. In a letter to Francis Wilson, Esq., Joseph D. A. Healey, Esq., suggested that Carol Pf\u00e4ffle needed to travel to the nearest American consul to sign papers before a notary public. Wilson responded to Healey via Healey's real estate partner Fred A. Wilson (no relation) that Mrs. Pf\u00e4ffle wanted it known that the nearest American consul was in Juarez, Old Mexico, a distance greater than from Lynn to New York City. Wilson went on to explain to Mr. Healey that New Mexico had been admitted to the Union in 1912, and that Santa Fe had been settled by Europeans \"about twelve years before the Pilgrims set foot on Plymouth Rock.\" Before closing, Wilson mentioned that he himself had been born and raised in Massachusetts, and had \"spent four years in the shade of the old elms in the Harvard yard.\" Francis C. Wilson to Fred A. Wilson, December 23, 1919, SRA.\n\n14. Francis Wilson papers, Pf\u00e4ffle files, various correspondence from 1929 to 1930.\n\n15. Francis Wilson papers, Mary Cabot Wheelwright files, Pf\u00e4ffle files, SRA; Dorthy Fredericks to author.\n\n16. Mary Cabot Wheelwright to Francis Wilson, June 19, 1931, MCW\/ WM.\n\n17. Willa Cather to Elizabeth Shepley Sergeant, August 14, 1912, _Selected Letters_ , 166.\n\n###  Chapter 19\n\nWilla Cather, _Selected Letters_ , 209.\n\n1. Harriet Ropes Cabot interview, MCW\/WM.\n\n2. The Night Riders terrorized Rio Arriba County in the 1920s, stealing cattle, burning barns, and robbing private homes, including Mary Wheelwright's. One of gang's leaders, Big Bill McKinley, was the older brother of the three McKinleys who worked for Carol at San Gabriel and Ghost Ranch. Big Bill, convicted as the leader of the notorious gang, was pursued and finally apprehended by a private detective, Bill Martin, after the Night Riders' sensational burglary of Wheelwright's valuable Navajo rugs collection kept at Los Luceros. See Poling-Kempes, _Ghost Ranch_ , 45.\n\n3. Carol Stanley Accounts Ledger, 1928\u20131935, Peabody family collection.\n\n4. Bennett, \"Reminiscences from the Thirties\"; Joe Fitzgibbon and Dorthy Fredericks to author.\n\n5. Pack, _We Called it Ghost Ranch_ , 21.\n\n6. Ibid., 23.\n\n7. Pack was a childhood friend of John D. Rockefeller, and took John D.'s college-aged son, Laurence, on at least one Canadian expedition. Pack may have introduced the Rockefellers to the Pf\u00e4ffles and their Canjilon Camp, where the Rockefellers spent a month in 1930.\n\n8. The John Bartol family included Charlotte Cabot Bartol, Lucy Cabot, and Harriet Ropes Cabot. Harriet Ropes Cabot interview, MCW\/ WM. On Fred Harvey family at Ghost Ranch, Ed Groesbeck to author.\n\n9. Edward H. Bennett Jr. and Dorthy Fredericks to author.\n\n10. O'Keeffe, quoted in Tryk, \"O'Keeffe,\" 20; Francis C. Wilson correspondence to Mrs. Garland, March 4, 1931, regarding land transactions 1927\u201328, Francis C. Wilson papers.\n\n11. Georgia O'Keeffe to Rebecca Strand [James], April 26, 1934, in Cowart, Hamilton, and Greenough, _Georgia O'Keeffe_ , 221.\n\n12. Georgia O'Keeffe to Alfred Stieglitz, September 20, 1937, in O'Keeffe, _Catalogue of the 14th Annual Exhibition_ , 10.\n\n13. Georgia O'Keeffe quoted in Tryk, \"O'Keeffe,\" 25.\n\n14. Georgia O'Keeffe quoted in Wallach, \"Georgia O'Keeffe.\"\n\n15. Georgia O'Keeffe quoted in Tryk, \"O'Keeffe,\" 20. In 1940 O'Keeffe purchased Rancho de los Burros from Arthur Pack.\n\n16. Francis I. Proctor quoted in Stark and Rayne, _El Delirio_ , 99.\n\n17. Alice Corbin to Amelia White, November 1930, quoted in ibid., 99.\n\n18. AP story, March 3, 1937, quoted in Newcomb, _Hosteen Klah_ , 211, 214.\n\n19. Bennett, \"Reminiscences from the Thirties.\"\n\n20. Dorthy Fredericks to author; Carol Stanley Account Book, Peabody family collection.\n\n21. Harriet Ropes Cabot interview, MCW\/WM.\n\n22. Wetherill Guest Registry, October 14 and 19, 1935, Harvey Leake collection. The Coltons founded the Museum of Northern Arizona, and were in Kayenta with the museum's curator, Katharine Bartlett.\n\n23. Dick (Roy) Pf\u00e4ffle to Lois and Tom Buchanan, November 7, 1955, Peg Terrell and Patricia Hibbard collection.\n\n###  Afterword\n\nAlice Corbin, \"Untitled,\" _Red Earth_ , 3.\n\n1. From _The Sun Dial_ , quoted in Petersen, _San Diego's First Lady of the Arts_ , 44.\n\n2. Among Alice's donations to the Museum of Art were the cowboy sketches and drawings given to her long ago by a young, handsome, and unknown artist named Arthur Putnam. See Petersen, _San Diego's First Lady of the Arts_ , 23. Surgery to remove a brain tumor in 1911 had left Putnam partially paralyzed and unable to sculpt. He died in 1930 at the age of forty-seven, but he enjoyed success as an animal sculptor in the first decade of the twentieth century. The face of one of Putnam's sculptures, _The Mermaid_ , bore an uncanny resemblance to Alice, or so his wife, Grace, told Alice in a letter. In 1903 a commission of five large bronze sculptures for the San Diego estate of E. W. Scripts introduced Putnam to the writer Jack London, who became a close friend. Among Putnam's best-known pieces is the Jack London Writing Tablet carved from California Redwood. When London died in 1916, he owned the tablet, which features Old Buck from _Call of the Wild_. The tablet was lost from 1917 until 2003 and is today considered a California treasure. See tablet owner Christian Chaffee's website, jacklondontablet .com, and _National Geographic_ 's _America's Lost Treasures_ series website.\n\n3. Petersen, _San Diego's First Lady of the Arts_ , 25.\n\n4. Klauber, \"Untitled,\" _Poems_.\n\n5. Marjorie Lambert interview, March 3, 1985, MCW\/WM.\n\n6. Aimee and Rosamond Lamb interview, July 21, 1985, MCW\/ WM.\n\n7. Comments from Memorial Service, MCW\/WM. \n\n# SELECTED BIBLIOGRAPHY\n\nAdams, Ansel, and Mary Austin. _Taos Pueblo_. Grabhorn Press, 1930.\n\nBabcock, Barbara A., and Nancy J. Parezo. _Daughters of the Desert: Women Anthropologists and the Native American Southwest, 1880\u20131980_. 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ACH\/HRC.\n\nCurtis, George DeClyver [Gregory Marwood, pseud.]. Diaries of George DeClyver Curtis. UCLA.\n\n\u2014\u2014\u2014. \"In Memory of Natalie.\" n.d. NCBA.\n\n\u2014\u2014\u2014. Letter to Bridgham Curtis, October 29, 30, 1921. NCBA.\n\n\u2014\u2014\u2014. \"Biography of Natalie Curtis Burlin.\" 1957. NCBA.\n\nCurtis, Natalie [see also Natalie Curtis Burlin]. \"An American Indian Composer.\" _Harper's Magazine_ 107, September 1903.\n\n\u2014\u2014\u2014. \"Busoni's Indian Fantasy.\" _Southern Workman_ 44, 1915.\n\n\u2014\u2014\u2014. _The Indians' Book_. Dover Publications, 1968.\n\n\u2014\u2014\u2014. \"Indian Song on a Desert Path.\" _Southern Workman_ 33, June 1904.\n\n\u2014\u2014\u2014. \"Mr. Roosevelt and Indian Music: A Personal Reminiscence.\" _Outlook_ 121, March 5, 1919.\n\n\u2014\u2014\u2014. \"A New Type of Architecture in the Southwest.\" _Craftsman_ 25, January 1914.\n\n\u2014\u2014\u2014. \"The Plight of the Mojave-Apache Indians.\" _Outlook_ 129, September 7, 1921.\n\n\u2014\u2014\u2014. \"Value of Music School Settlements in Cities.\" _Craftsman_ 21, December 1911.\n\n\u2014\u2014\u2014. \"A Western Reverie.\" _The Nation_ , November 24, 1920.\n\n\u2014\u2014\u2014. \"The Winning of an Indian Reservation.\" _Outlook_ 122, June 25, 1919.\n\n\u2014\u2014\u2014. \"The Words of Hiparopai: A Leaf from a Traveler's Diary.\" _Craftsman_ 13, December 1907.\n\nDeacon, Deasley. _Elsie Clews Parsons: Inventing Modern Life_. University of Chicago Press, 1999.\n\nDietrich, Margretta Stewart. _New Mexico Recollections_. Vergara Print Company, 1961.\n\nDinesen, Isak. _Seven Gothic Tales_. Vintage Books, 1972.\n\nDouglas, Charles Winfred. Journals of C. W. Douglas. CWD\/DMA.\n\nEllinwood, Leonard, and Anne Woodward Douglas. _To Praise God: The Life and Work of Charles Winfred Douglas_. Hymn Society of America, 1958.\n\n_Essex-Country History_. Boston: C. A. & J. F. Wood, 1870.\n\nFaunce, Hilda. _Desert Wife_. Bison Books, 1981.\n\nFoote, Lorien. _Seeking the One Great Remedy: Francis George Shaw and Nineteenth-Century Reform_. Ohio University Press, 2003.\n\nFowler, Don D. _A Laboratory for Anthropology: Science and Romanticism in the American Southwest, 1846\u20131930_. University of New Mexico Press, 2000.\n\nFroeschauer-Nelson, Peggy. _Cultural Landscape Report: Hubbell Trading Post National Historic Site, Ganado, Arizona_. U.S. Department of the Interior, 1998.\n\nGillmor, Frances, and Louisa Wade Wetherill. _Traders to the Navajos: The Wetherills of Kayenta_. University of New Mexico Press, 1979.\n\n_Grand Canyon MacLeans_. No. 3, July 1996. Leonard and Millan, 1996.\n\nGrey, Zane. _The Rainbow Trail_. Harper & Brothers, 1915.\n\n\u2014\u2014\u2014. _Tales of Lonely Trails_. Wildside Press, 2004\n\nHagan, William T. _Theodore Roosevelt & Six Friends of the Indian_. University of Oklahoma Press, 2002.\n\nHartley, Marsden. \"Aesthetic Sincerity.\" _El Palacio_ , December 2, 1918.\n\n\"Hawthorne, Hildegard. \"The Literary Digest International Book Review, 1916.\" ACH\/HRC.\n\nHegemann, Elizabeth Compton. _Navaho Trading Days_. University of New Mexico Press, 1987.\n\nHeilbrun, Carolyn G. _Writing a Woman's Life_. W. W. Norton, 1988.\n\nHeller, Adele, and Lois Rudnick, eds. _1915: The Cultural Moment, The New Politics, the New Woman, the New Psychology, the New Art, and the New Theater in America_. Rutgers University Press, 1991.\n\nHemingway, Ernest. _A Moveable Feast: The Restored Edition_. Scribner, 2010.\n\nHenri, Robert. _The Art Spirit_. Compiled by Margery Ryerson. Westview Press, 1984.\n\nHeyneman, Julie Helen. _Arthur Putnam, Sculptor_. Johnck and Seeger, 1932.\n\nHillerman, Tony. _The Spell of New Mexico_. University of New Mexico Press, 1984.\n\nHoard, Dorothy. _Historic Roads of Los Alamos_. Los Alamos Historical Society, 2009.\n\nJames, Harry C. _Pages from Hopi History_. University of Arizona Press, 1974.\n\nJenkins, Myra Ellen, and James R. Thorpe. _From Retreat to Resort: A History of Bishop's Lodge_. Bishop's Lodge Ranch and Resort, 2008.\n\nJohnston, Bertha. \"The Chicago Kindergarten Institute.\" _Kindergarten Magazine_ 12, September 1899\u2013June 1900.\n\nKelley, Klara Bonsack, and Harris Francis. _Navajo Sacred Places_. Indiana University Press. 1994.\n\nKent, Dorothy. Letter to Rockwell Kent, January 13, 1919. RK\/AAA.\n\nKing, Mary Scott. \"Jack Lambert: The Last of His Kind.\" _Santa Fe New Mexican_. February 17, 1974.\n\nKlauber, Alice. \"Arizona Trip Summer 1913.\" SDMA.\n\n\u2014\u2014\u2014. \"Bits of the Summer 1912.\" SDMA.\n\n\u2014\u2014\u2014. Diaries, 1907\u20131924. SDMA.\n\n\u2014\u2014\u2014. _Poems_. Denrich Press, 1928.\n\n\u2014\u2014\u2014. \"Robert Henri, Class in Spain.\" June 25, 1912. SDMA.\n\n\u2014\u2014\u2014. \"Robert Henri lectures.\" SDMA.\n\nLa Farge, Oliver. _Santa Fe: The Autobiography of a Southwestern Town_. University of Oklahoma Press, 1959.\n\nLaird, Carobeth. _Encounter with an Angry God: Recollections of My Life with John Peabody Harrington_. University of New Mexico Press, 1975.\n\nLavender, Catherine J. _Scientists and Storytellers: Feminist Anthropologists and the Construction of the American Southwest_. University of New Mexico Press, 2006.\n\nLewis, Edith. _Willa Cather Living: A Personal Record_. University of Nebraska Press, 2000.\n\nLewis, Nancy Owen, and Kay Leigh Hagan. _A Peculiar Alchemy: A Centennial History of SAR 1907\u20132007_. SAR Press, 2007.\n\nLong, J. C., and Charles K. Hyde. _Roy D. Chapin: The Man Behind the Hudson Motor Company_. Wayne State University Press, 2004.\n\nLove, Marian F. \"A Cowboy Recollects.\" _Santa Fean_ , December 1981.\n\nLuhan, Mabel Dodge. _Edge of Taos Desert: An Escape to Reality_. University of New Mexico Press, 1993.\n\n\u2014\u2014\u2014. \"Taos and Individualism, a Brief R\u00e9sum\u00e9 of an Environment.\" _New Mexico Quarterly_ 21 (Summer 1951): 135\u201380.\n\nLummis, Charles F. _The Land of Poco Tiempo_. University of New Mexico Press, 1973.\n\nMarwood, Gregory [George DeClyver Curtis]. _The Wooing of a Recluse_. Devin-Adair, 1914.\n\nMcCarthy, Laurette E. _Walter Pach (1883\u20131958): The Armory Show and the Untold Story of Modern Art in America_. Penn State University Press, 2012.\n\nMcGreevy, Susan. \"Why Navajo Studies? The Example of Hastiin Klah and Mary Cabot Wheelwright.\" MCW\/ WM.\n\nMcPherson, Bruce. _Measure by Measure: A History of the New England Conservatory of Music since 1867_. The Trustees of the New England Conservatory of Music, 1995.\n\nMellow, James R. _Charmed Circle: Gertrude Stein & Company_. Praeger, 1974.\n\nMilne, Gordon. _George William Curtis & the Genteel Tradition_. Indiana University Press, 1956.\n\nMorris, Edmund. _Colonel Roosevelt_. Random House, 2010.\n\nMullin, Molly H. _Culture in the Marketplace: Gender, Art, and Value in the American Southwest_. Duke University Press, 2001.\n\nMuseum of New Mexico. _Window on the West: The Collector's_ El Palacio. Museum of New Mexico Foundation, 1989.\n\nNewcomb, Franc Johnson. _Hosteen Klah, Navajo Medicine Man and Sand Painter_. University of Oklahoma Press, 2012.\n\n\u2014\u2014\u2014. \"Symbols in Sand.\" New Mexico Association on Indian Affairs, 1936.\n\nNorwood, Vera, and Janice Monk, eds. _The Desert Is No Lady: Southwestern Landscapes in Women's Writing and Art_. Yale University Press, 1987.\n\nOfford, John A. \"The Foremost American School of Music\u2014the New England Conservatory.\" _New York Observer_ , August 2, 1906.\n\nO'Keeffe, Georgia. _Catalogue of the 14th Annual Exhibition of Paintings, with some Recent O'Keeffe Letters: December 27\u2013February 11, 1938._ An American Place, 1937.\n\nPach, Walter. Letters to Alice Klauber, December 24, 1924, May 6, 1925, February 15, 1944. Alice Klauber Collection, SDMA.\n\n\u2014\u2014\u2014. \"John Sloan.\" _New Mexico Artists_. New Mexico Artists Series, no. 3. University of New Mexico Press, 1952.\n\nPack, Arthur Newton. _We Called It Ghost Ranch_. Ghost Ranch Conference Center, 1979.\n\nParsons, Elsie Worthington Clews. _The Old-Fashioned Woman: Primitive Fancies about the Sex_. G. P. Putnam's Sons, 1913.\n\n\u2014\u2014\u2014. _Social Organization of the Tewa_. American Anthropological Association, 1929.\n\n\u2014\u2014\u2014. _Social Rule: A Study of the Will to Power_. G. P. Putnam's Sons, 1916.\n\nPatterson, Michelle Wick. _Natalie Curtis Burlin: A Life in Native and African American Music_. University of Nebraska Press, 2010.\n\nPerlman, Bennard B. _Robert Henri: His Life and Art_. Dover Publications, 1991.\n\nPetersen, Martin E. _San Diego's First Lady of the Arts: Alice Ellen Klauber & Friends_. SDMA.\n\nPf\u00e4ffle, Carol Stanley. Correspondence with Augusta Curtis; Elbridge Adams. NCBA.\n\n\u2014\u2014\u2014. Correspondence with Roy Chapin. LAHM\/PCC.\n\nPf\u00e4ffle, Lois. Diaries and letters. Collection of Patricia Hibbard.\n\nPf\u00e4ffle, Margaret. Diaries. Collection of Patricia Hibbard.\n\nPf\u00e4ffle, Richard Leroy. Correspondence with Lois and Tom Buchanan, November 7, 1955. Collection of Patricia Hibbard.\n\nPoling-Kempes, Lesley. _Ghost Ranch_. University of Arizona Press, 2005.\n\nPowell, Gertrude Wilson. _The Spirit of Democracy_. Rand, McNally, 1918.\n\nRainbow Bridge Registry. Various years 1909\u20131926. National Park Service, and Collection of Harvey Leake\/Wetherill family.\n\nRobertson, Edna, and Sarah Nestor. _Artists of the Canyons and Caminos: Santa Fe, the Early Years_. Ancient City Press, 1996.\n\nRobinson, Roxana. _Georgia O'Keeffe: A Life_. Harper Collins, 1989.\n\nRoosevelt, Nicholas. Diary, July\u2013August 1913. Nicholas Roosevelt Collection, Fray Ang\u00e9lico Ch\u00e1vez Historical Library, New Mexico History Museum, Santa Fe, New Mexico.\n\nRoosevelt, Theodore. \"Across the Navajo Desert: Prayer to the Dawn.\" _Outlook_ 105, October 11, 1913.\n\n\u2014\u2014\u2014. \"A Cougar Hunt on the Rim of the Grand Canyon.\" _Outlook_ 105, October 4, 1913.\n\n\u2014\u2014\u2014. \"The Hopi Snake Dance.\" _Outlook_ 105, October 18, 1913.\n\nRothman, Hal K. _On Rims and Ridges: The Los Alamos Area since 1880_. University of Nebraska Press, 1997.\n\nRudnick, Lois Palken. _Mabel Dodge Luhan: New Woman, New Worlds_. University of New Mexico Press, 1987.\n\nScharff, Virginia. \"Introduction: Women Envision the West: 1890\u20131945.\" In _Independent Spirits: Women Painters of the American West, 1890\u20131945_ , edited by Patricia Trenton, I\u20137. University of California Press, 1995.\n\n\u2014\u2014\u2014. _Twenty Thousand Roads: Women, Movement, and the West_. University of California Press, 2002.\n\nSchindler, Kurt. Correspondence. KS\/NYPL.\n\nSergeant, Elizabeth Shepley. \"Earth Horizon.\" _The Nation_ , June 29, 1927.\n\n\u2014\u2014\u2014. \"The Journal of a Mud House.\" _Harper's Magazine_ , March, April, May, June 1922.\n\n\u2014\u2014\u2014. _Shadow-Shapes: The Journal of a Wounded Woman, October 1918\u2013May 1919_. Houghton Mifflin, 1920.\n\n\u2014\u2014\u2014. _Willa Cather: A Memoir_. Ohio University Press, 1992.\n\nSimmons, Marc. _Charles. F. Lummis, Author and Adventurer_. Sunstone Press, 2008.\n\nSnyder, Sharon. _At Home on the Slopes of Mountains: The Story of Peggy Pond Church_. Los Alamos Historical Society Publications, 2011.\n\nStark, Gregor, and E. Catherine Rayne. _El Delirio: The Santa Fe World of Elizabeth White_. SAR Press, 1998.\n\nStarr, Kevin. _Inventing the Dream: California through the Progressive Era_. Oxford University Press, 1986.\n\nSze, Corinne P. \"The Witter Bynner House.\" _Bulletin of the Historic Santa Fe Foundation_ 20, no. 2 (September 1992).\n\nTalayesva, Don C. _Sun Chief: The Autobiography of a Hopi Indian_. Edited by Leo W. Simmons. Yale University Press, 1963.\n\nTappert, Tara Leigh. _Out of the Background: Cecilia Beaux and the Art of Portraiture_. Traditional Fine Arts Organization, Inc., 2009.\n\nTobias, Henry J., and Charles E. Woodhouse. _Santa Fe: A Modern History, 1880\u20131990_. University of New Mexico Press, 2001.\n\nTraugott, Joseph. _The Art of New Mexico: How the West Was One_. University of New Mexico Press, 2001.\n\nTryk, Sheila. \"O'Keeffe.\" _New Mexico Magazine_ , January\u2013February 1973.\n\nUdall, Sharyn Rohlfsen. _Modernist Painting in New Mexico, 1913\u20131935_. University of New Mexico Press, 1984.\n\nWallach, Amei. \"Georgia O'Keeffe.\" _Newsday_ , October 30, 1977.\n\nWeigle, Marta, and Kyle Fiore. _Santa Fe and Taos: The Writer's Era, 1916\u20131941_. Ancient City Press, 1982.\n\nWetherill, Louisa Wade. Diary of Louisa Wetherill. Collection of Harvey Leake.\n\nWetherill Guest Registry. Collection of Harvey Leake.\n\nWheelwright, Mary Cabot. \"Journey Towards Understanding.\" Unpublished manuscript, Wheelwright Museum of the American Indian.\n\nWilson, Francis O., Esq. Papers and correspondence. SRA.\n\nWineapple, Brenda. _Sister Brother: Gertrude and Leo Stein_. Putnam Adult, 1996. \n\n#  ILLUSTRATION CREDITS\n\nFigure 1. Photograph by Charles C. Pierce. California Historical Society Collection at the University of Southern California. Title Insurance and Trust\/C.C. Pierce Photography Collection, CHS-1426.\n\nFigure 2. Courtesy Natalie Curtis Burlin Archive.\n\nFigure 3. Photograph by John K. Hillers. Collection of the Smithsonian Institution, NARA record: 3028457.\n\nFigure 4. Courtesy Natalie Curtis Burlin Archive.\n\nFigure 5. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 6. Archives of American Art, #4903.\n\nFigure 7. Gertrude Stein and Alice B. Toklas Papers, Yale Collection of American Literature. Beinecke Rare Book and Manuscript Library, Yale University.\n\nFigure 8. Rotograph Co.\n\nFigure 9. Nahant Historical Society.\n\nFigure 10. _The Bay State Monthly_ , vol. 2, December 1884.\n\nFigure 11. _The Neume Yearbook_ , 1905, New England Conservatory of Music.\n\nFigure 12. Carol Stanley Collection.\n\nFigure 13. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 14. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 15. Carleton Monroe Winslow, _The Architecture and Gardens of the San Diego Exposition_ (San Francisco: Paul Elder and Company, 1916).\n\nFigure 16. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 17. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 18. Photograph by Ben Wittick. NPS photo, HUTR Neg. RP-312.\n\nFigure 19. Photograph by Timothy H. O'Sullivan. Library of Congress digital ID ppmsca.10057.\n\nFigure 20. Photograph by John K. Hillers, National Archives and Records Administration, NARA record: 3028457.\n\nFigure 21. Courtesy John and Louisa Wetherill Collection.\n\nFigure 22. Theodore Roosevelt Collection, #560.8, Houghton Library, Harvard University.\n\nFigure 23. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 24. Photograph by Emery Clifford Kolb. Northern Arizona University, Cline Library Emery Kolb Collection, NAU 98183 Kolb.\n\nFigure 25. Still from the motion picture \"Hopi Indians dance for TR at [Walpi, Ariz.] 1913.\" Library of Congress.\n\nFigure 26. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 27. Photograph by George Wharton James. California Historical Society Collection, CHS-4624.\n\nFigure 28. Courtesy John and Louisa Wetherill Collection.\n\nFigure 29. Arizona State Museum, University of Arizona, Wetherill Collection.\n\nFigure 30. Courtesy John and Louisa Wetherill Collection.\n\nFigure 31. Carol Stanley Collection.\n\nFigure 32. Photograph by T. Harmon Parkhurst. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #10786.\n\nFigure 33. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #059757.\n\nFigure 34. Jesse Nusbaum Collection, Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #046554.\n\nFigure 35. San Diego Museum of Art Archives, Alice Klauber Collection.\n\nFigure 36. Courtesy Natalie Curtis Burlin Archive.\n\nFigure 37. Carol Stanley Collection.\n\nFigure 38. Carol Stanley Collection.\n\nFigure 39. Carol Stanley Collection.\n\nFigure 40. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #006713.\n\nFigure 41. Courtesy Natalie Curtis Burlin Archive.\n\nFigure 42. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #016755.\n\nFigure 43. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #014232.\n\nFigure 44. Carol Stanley Collection.\n\nFigure 45. Carol Stanley Collection.\n\nFigure 46. Carol Stanley Collection.\n\nFigure 47. Project Canterbury.\n\nFigure 48. Carol Stanley Collection.\n\nFigure 49. Courtesy Natalie Curtis Burlin Archive.\n\nFigure 50. Carol Stanley Collection.\n\nFigure 51. Collection of Patricia Hibbard.\n\nFigure 52. Collection of the Wheelwright Museum of the American Indian.\n\nFigure 53. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #132146.\n\nFigure 54. Collection of the Wheelwright Museum of the American Indian.\n\nFigure 55. Photograph by Edward Kemp. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #151372.\n\nFigure 56. Collection of the American Philosophical Society.\n\nFigure 57. Collection of Patricia Hibbard.\n\nFigure 58. Collection of Patricia Hibbard.\n\nFigure 59. Betty Bartlett Seals Collection, Ghost Ranch Archive.\n\nFigure 60. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #089626.\n\nFigure 61. Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #89665.\n\nFigure 62. Betty Bartlett Seals Collection, Ghost Ranch Archive.\n\nFigure 63. John Candelario Collection, Courtesy Palace of the Governors Photo Archives (NMHM\/DCA), #165666. \n\n#  INDEX\n\nPage numbers in _italics_ represent illustrations.\n\nAbiqui\u00fa (NM), , , ,\n\nAcoma Pueblo (NM), , 102\u20133, 144\u201345, , 275\u201376, 334n27\n\nAdams, Elbridge, , , , ,\n\nAlbuquerque (NM), , ,\n\nAlcalde (NM), , , , , , , , , , , 276\u201377, 282\u201383, , , , , 303\u20134, ,\n\nAntelope Dance (Hopi), 136\u201337\n\nApplegate, Frank,\n\nArboles (CO), 311\u201312\n\nArchaeological Institute of America (AIA), 52\u201354, , , , , ,\n\nArchuleta brothers, ,\n\nArmory Show, , 92\u201393, , 104\u20135, 110\u201311, , 227\u201328, ,\n\nArt Students League (San Francisco Art Association), 61\u201362\n\nAshcan painters, 94\u201395, , , , 331n4\n\nAssociation of American Painters and Sculptors,\n\nAustin, Mary Hunter, , , , , 258\u201359,\n\nBakos, Jozef,\n\nBailey, Wheeler J., , , 75\u201376, ,\n\nBalboa Park, , , 98\u2013106, __ , __ , , , , , 171\u201372, 178\u201379, , , , , ,\n\nBandelier, Adolph F., , , ,\n\nBandelier National Monument, , __\n\nBarney, Esther Stevens, ,\n\nBartlett, Florence Dibell, 288\u201389, , 340n27\n\nBaumann, Gustave,\n\nBayreuth (Germany), , ,\n\nBeacon Hill, 81\u201382, ,\n\nBeauregard, Donald, , ,\n\nBeaux, Cecilia, , , ,\n\nBegay, Beyal (Beaal), , , 343n12\n\nBellows, George, , , , ,\n\nBennett, Edward H., 301\u20132\n\nBennett, Edward H. (Ted), Jr., 301\u20132,\n\nBishop's Lodge, , ,\n\nBishop's Ranch, 222\u201325, , 231\u201332, , , ,\n\nBluff (UT), , 155\u201356, 158\u201360,\n\nBlumenschein, Ernest L., ,\n\nBoas, Franz, , , 343n26\n\nBond, Frank, ,\n\nBoston (MA), 11\u201312, , , , , , 81\u201386, __ , 88\u201389, , , 148\u201350, , 163\u201364, , 194\u201395, 200\u2013201, , 219\u201320, , , , 260\u201361, 268\u201369, , , , 316\u201317\n\nBoston Brahmins, , , , 85\u201386, , , , ,\n\nBrancusi, Constantin, ,\n\nBrook, H. H., ,\n\nBryn Mawr, , 256\u201358,\n\nBuckman (NM), , , , ,\n\nBuckman Bridge, , ,\n\nBuffalo Bill's Wild West Show,\n\nBureau of Indian Affairs (BIA), , , , , , 42\u201343, , , , , , , , 326n4\n\nBurlin, Natalie Curtis. _See_ Curtis, Natalie\n\nBurlin, Paul, , , 191\u201394, __ , , 202\u20133, 205\u20136, , , 213\u201314, 216\u201317, 219\u201320, 226\u201328, 334\u201336, , 242\u201345, 248\u201349\n\nBurnham, Dave, , ,\n\nBurroughs, John,\n\nBursum Bill, ,\n\nBurton, Charles E., 27\u201329, 33\u201334, ,\n\nBusoni, Ferruccio, ,\n\nBynner, Witter, 248\u201349, ,\n\nCabot, Harriet Ropes, , 345n8\n\nCamino del Monte Sol, , ,\n\nCamp Neosho, , ,\n\nCanjilon Camp, , __ , , 287\u201388, , , 344n3, 345n7\n\nCa\u00f1on del Muerto,\n\nCanyon de Chelly, , 120\u201322, __ , , , , , , __ , 265\u201366, , ,\n\nCarlisle Indian School (PA), ,\n\nCassidy, Gerald, , , ,\n\nCather, Willa, , , , 256\u201358, 282\u201383,\n\nC\u00e9zanne, Paul, , ,\n\nChaco Canyon, , , , , ,\n\nChaffee, Alice E., ,\n\nChama River, , , , , , 293\u201394, ,\n\nChanler, Robert,\n\nChapin, Roy D., , , 179\u201383, , , 193\u201395, , , 222\u201323, , 336n30, 339n16\n\nChapman, Kenneth, , , , , , , ,\n\nChase, William Merritt, , 64\u201365, 67\u201368, 94\u201395\n\nChase School of Art. _See_ New York School of Art\n\nChicago (IL), , , , , , , 157\u201358, , , , , , , , 228\u201329, , , , , , , 327n19, 340n27\n\nChicago Kindergarten Institute, , , , 329n18, 330n21\n\nChief Polk (Billy Hatch), 156\u201357, 160\u201361\n\nChile Line, , ,\n\nChiparopai, 22\u201324\n\nChiricahua Apaches, ,\n\nChoate, Ebenizer,\n\nChoate, Joseph Hodges, , , 333n6\n\nChurch, Peggy Pond, 336n30\n\nCivil War, , 10\u201311,\n\nCochan. _See_ Yumas\n\nCollier, Charles,\n\nCollier, John, ,\n\nColorado Plateau, , , ,\n\nColorado River, 6\u20137, , , , , 118\u201319, ,\n\nCommunity of St. Mary (St. Mary's), , , 107\u20138, , , , 232\u201333,\n\nCottonwood Gulch, , ,\n\nCottonwood Ranch,\n\nCorbin, Alice, 4\u20135, 168\u201370, 198\u201399, 213\u201314, , , 247\u201348, , 256\u201359, , , , 306\u20138\n\nCox, Orville, , 264\u201365, , , , 297\u201398\n\nCrane, Leo, ,\n\nCummings, Byron,\n\nCurtis, Augusta (Mimsey), , 9\u201310, 12\u201313, , , , 35\u201336, , , 74\u201376, , , , 202\u20133, 206\u20137\n\nCurtis, Bridgham, 7\u20138, , , , , , , 244\u201345,\n\nCurtis, Constance (Connie), , 244\u201345\n\nCurtis, Edward (Bogey), 7\u201311, , , , 35\u201336, , , ,\n\nCurtis, George DeClyver, 6\u20137, , 14\u201317, 19\u201320, 23\u201330, __ , , , , , , 49\u201351, , , , 73\u201376, , , , , , , , , 177\u201380, 202\u20133, 207\u201310, , , , , 244\u201345,\n\nCurtis, George William, 10\u201312, 16\u201317, , 36\u201337, , , ,\n\nCurtis, Julia,\n\nCurtis, Marian, ,\n\nCurtis, Natalie, ; and AIA, 52\u201353; and Alice Klauber, , 75\u201376, 237\u201342; and Balboa Park, 101\u20132; and Bogey's illness, , 73\u201374; collapse, 14\u201316; death of, 242\u201348, 250\u201353, ; _The Deer Dance_ , ; education, 6\u201310, 13\u201314; and federal assimilationist policies, ; first trip to Southwest, 16\u201321; home in Santa Fe, , 226\u201328, 248\u201349, ; Hopi name Tawimana, ; and Indian rights, 35\u201338; _The Indians' Book_ , 54\u201356, __ ; International Congress of Art History, Paris, 234\u201342, __ ; in La Jolla, 75\u201377; and Mabel Dodge Luhan, 213\u201317; marriage to Paul Burlin, 202\u20137; and New Women, , 256\u201357, 277\u201378, 316\u201318; _The Path on the Rainbow_ , ; and Paul Burlin, 191\u201393, ; and Ramon Vigil Ranch, 172\u201383, 193\u201394, ; records Indian music, 22\u201334, __ , 42\u201346, 48\u201351; and Santa Fe Museum of Art, , 208\u201312, __ ; and settlement house movement, 70\u201371; _Songs and Tales from the Dark Continent_ , ; and Theodore Roosevelt, 35\u201343, 46\u201347, , 128\u201332, 134\u201343, __ , __ , __ , 146\u201347\n\nDavies, Arthur B., , , , 331n5\n\nDamrosch, Frank, , 329n19\n\nDamrosch, Walter, , , 71\u201372, 329n19\n\nDasburg, Andrew,\n\nDavey, Randall, , __ , 227\u201328, , , 339n23\n\n_Death Comes for the Archbishop_ ,\n\nDe Cora, Angel, , , 328n8\n\nDenver (CO), , , , ,\n\nDenver and Rio Grande Railroad (D & RG), ,\n\nDietrich, Margretta Stewart, , 248\u201349, , , , 341n6\n\nDin\u00e9. _See_ Navajo\n\nDodge, Mabel. _See_ Luhan, Mabel Dodge\n\nDorothy Cove (MA), , , , , , , ,\n\nDouglas, Charles Winfred, 71\u201373, 86\u201387, __ , , , 107\u20139, 112\u201322, __ , , , , , 133\u201334, 136\u201339, 142\u201345, __ , 149\u201351, 154\u201356, , , , 164\u201366, , , , , , , 189\u201391, 193\u201394, , , , 229\u201330, __ , 232\u201333, , , , , , , , 330n20, 334n11, 334n22, 334n27, 341n1\n\nDouglas, Frederic (Eric) H., , , 112\u201316, __ , , , , , , 144\u201345, , __ , 339n11, 341n1\n\nDouglas, Mary Josepha Williams (Dr. Jo), 72\u201373, , , , , ,\n\nDozier, Pete, ,\n\nDuchamp, Marcel, ,\n\nDurango (CO), , , , ,\n\nDust Bowl,\n\nDvo\u0159\u00e1k, Anton\u00edn,\n\nEl Alisal, 16\u201319, __\n\nEl Delirio, 283\u201384,\n\nEllis, Fremont, 255 Enchanted Mesa (Mesa Encantada), , __ , 334n27\n\nEpiscopal Church, ,\n\nEspa\u00f1ola (NM), , , , , ,\n\nExplorers Club, , 334n27\n\nFewkes, J. Walter,\n\nFirst Mesa (AZ), , , , , 126\u201327, __ , , , 135\u201338, , __ , , ,\n\nFletcher, Alice C., , ,\n\nFlorence (Italy), , 66\u201367, , , ,\n\nFlute Ceremony (Hopi), , 143\u201344\n\nFort McDowell, ,\n\nFour Corners, 151\u201352, 156\u201358, , ,\n\nFred Harvey Company, , , , , 344n11\n\nFriedham, Arthur,\n\nFriere-Marreco, Barbara,\n\nFrijoles Canyon (NM), , , 176\u201377, , , , ,\n\nFrissell, Hollis, , ,\n\nGallegos, Juan de Dios,\n\nGallup (NM), , , , , 194\u201395, , , , , ,\n\nGanado (AZ), 114\u201319, , , , , ,\n\nGarland, Marie Tudor, 273\u201374, 278\u201379, , 303\u20134\n\nGates, Gertrude L., 28\u201329\n\nGehring, Meta, ,\n\n_gen\u00edzaros_ ,\n\nGeronimo, ,\n\nGhost Dance,\n\nGhost Ranch, 3\u20134, 288\u201391, 293\u2013304, __ , __ , __ , __ , 309\u201312, , 345n2\n\nGill, Irving J., , , ,\n\nGleizes, Albert, , 236\u201337, , , 242\u201344\n\nGleizes, Juliette, , , 243\u201344\n\nGrainger, Percy,\n\nGrand Canyon, , , , , , 193\u201394, , , ,\n\nGray, David, , , ,\n\nGray, Paul, , , ,\n\nGreat Depression,\n\nGreenwich Village, , 6\u20137, 10\u201312, , 91\u201392, , , 274\u201375\n\nGrey, Zane, , , , ,\n\nGuernsey, Samuel James, ,\n\nHampton Institute, , , , , , 328n8\n\nH & M Ranch, ,\n\nHannah More Girls' Academy, , 86\u201388\n\nHano, , ,\n\n_Harper's Weekly_ , 11\u201312, 36\u201337, ,\n\nHarrington, John P.,\n\nHartley, Marsden, , , 216\u201318, __\n\nHarvard, 6\u20137, 10\u201311, , , , , , , , , ,\n\nHemingway, Ernest, , , , 340n8\n\nHenderson, Alice Corbin. _See_\n\nCorbin, Alice\n\nHenderson, William Penhallow, , __ , , , , , , , , , , , , __ , , , , 307\u20138\n\nHenri, Marjorie Organ (Margie \"O\"), , , ,\n\nHenri, Robert, , , 94\u201399, __ , __ , 104\u20135, , , , 184\u201387, , , , 204\u20136, , 210\u201311, , 227\u201328, , ,\n\nHewett, Donizetta, ,\n\nHewett, Edgar Lee, , 99\u2013105, , 166\u201367, 170\u201378, , 184\u201389, , , 208\u20139, , 227\u201328, , , 256\u201358, , , ,\n\nHiamovi,\n\nHilero, , 75\u201376, ,\n\nHill, Edward Burlingame, , 239\u201342\n\nHiparopai. _See_ Chiparopai\n\nHitchcock, Ethan Allen, , 38\u201339\n\nHodge, Frederick Webb, , , ,\n\nHolbrook (AZ), 112\u201315, 117\u201318,\n\nHopi, 26\u201334, , 43\u201346, 56\u201359, , 100\u2013101, 109\u201311, 119\u2013147, , , , , , 207\u20138, , 246\u201347, , , , , , , 327n19\n\nHotel De Vargas (Santa Fe), , __ , , 171\u201372, , , 185\u201386,\n\nHotevilla (AZ), ,\n\nHouck (AZ), 24\u201326\n\nHouse of Navajo Religion,\n\nHrdli\u010dka, Ale\u0161, , 100\u2013101\n\nHubbell, John (Don Lorenzo), 117\u201319, __ , , , , , ,\n\nHubbell, Lorenzo, Jr.,\n\nHubbell's trading post, , 116\u201319, __\n\nHunt, George W. P., , , __\n\nIndian Arts Fund, , 283\u201384,\n\n_The Indians' Book_ , 54\u201356, , , , , , , 165\u201366, , ,\n\nIndian Wells (AZ), ,\n\nInternational Congress of Art\n\nHistory, ,\n\nInternational Exhibition of Modern Art. _See_ Armory Show\n\nIsleta Pueblo,\n\nJames, Henry, ,\n\nJarmon, Alfred \"Slim,\" , ,\n\nJemez Mountains, , , , , , , ,\n\nJohnson, Almira Choate Estes, 78\u201379, , 88\u201390,\n\nJohnson, Fannie (Mrs. Edwin Stanley), , ,\n\nJohnson, J. Bishop, 78\u201379, , ,\n\nJohnson, Mary Frothingham,\n\nJohnson, Otis, ,\n\nJohnson, Robert Wood, 301\u20132\n\nJohnson, Seward,\n\nJohnson, Willard \"Spud,\" ,\n\nJones, Paul,\n\nJoy, Henry B., , , ,\n\nKayenta (AZ), , , , , __ , , 151\u201359, __ , __ , , 164\u201365, , 196\u201397, , , , , , , , , , 346n22\n\nKeams, Thomas,\n\nKeam's Canyon (AZ), 27\u201328, , , 124\u201325\n\nKent, Dorothy, , 186\u201387, , 193\u201395, 229\u201330, , , , 282\u201383, ,\n\nKent, Rockwell, , , ,\n\nKidder, Alfred V. (Ted), , , , , , 335n9\n\nKlah, Hostiin, 264\u201369, __ , 286\u201387, 305\u20138, , 343n12, 343n13\n\nKlauber, Abraham,\n\nKlauber, Alice Ellen, , __ ; and Arthur Putman, 62\u201363; and Balboa Park, 99\u2013106, __ ; civic activities, 62\u201363; death of, 314\u201316; and death of Natalie Curtis, 242\u201345, 250\u201351; desert expedition, 107\u20139, 112\u201328, 132\u201333, __ , 137\u201339, 142\u201346, __ ; education and childhood, 60\u201361; friendship with Natalie Curtis, , 75\u201376; and Gertrude, Leo, and Michael Stein, 65\u201369, ; Hopi name Koiyahoinim, , 334n25; and International Congress of Art History, 237\u201340, 242\u201345; in New Mexico, , 185\u201393, __ , 191\u201393, 208\u201310, 276\u201377, ; and Robert Henri, 91\u201399, __ , 104\u20135, 185\u201387; as San Diego's first lady of the arts, 314\u201316; and Walter Pach, 65\u201369; and William Merritt Chase, 64\u201369\n\nKlauber, Ella,\n\nKlauber, Laura, , 329n3\n\nKlauber, Leda,\n\nKlauber, Stella,\n\nKlauber, Theresa,\n\nKolp, Belle Axtell, 28\u201329\n\nKuhn, Walt, ,\n\nLaboratory of Anthropology, 305\u20138\n\nLa Farge, Oliver,\n\nLaguna Pueblo (NM), , 144\u201345, 275\u201376\n\nLa Jolla (CA), , 75\u201376, 97\u201398, , , , ,\n\nLamb, Aimee, 316\u201317\n\nLamb, Rosamond, 316\u201317\n\nLambert, Everett Vey (Jack), __ , 223\u201324, __ , , 229\u201333, __ , 270\u201371, __ , 281\u201382, 285\u201386,\n\nLambert, Lois. _See_ Pf\u00e4ffle, Lois\n\nLambert, Louise, , , __ ,\n\nLambert, Margery Ferguson,\n\nLamy (NM), , , , , , ,\n\nLamy, Archbishop Jean-Baptiste, ,\n\nLawrence, D. H., 249\u201350, ,\n\nLawrence, Frieda,\n\nLeupp, Francis, , , 328n11\n\nLewis, Edith, , 282\u201383,\n\nLodge, Henry Cabot, , , 333n6\n\nLololomai, 32\u201334, , , , , , , 327n19\n\nLongfellow, Henry Wadsworth, , ,\n\nLos Alamos Ranch School, , __ , 336n30\n\nLos Angeles (CA), , , , , , , 105\u20136, , , ,\n\nLos Cinco Pintores (The Five Painters),\n\nLos Gallos,\n\nLos Luceros, 272\u201373, __ , , , , , , , 345n2\n\nLouisiana Purchase Exposition,\n\nLuhan, Mabel Dodge, , 91\u201392, 213\u201317, __ , , , 274\u201375, 278\u201379, , , , ,\n\nLuhan, Tony, , , ,\n\nLuks, George, , 331n5\n\nLummis, Charles Fletcher, 16\u201320, , 27\u201328, , , 52\u201353, , , , , , , , 187\u201388, , , , , 224\u201325\n\nMacDowell Chorus, ,\n\nMacLean, Sandy, , 281\u201382,\n\nMancos (CO), , , , ,\n\nManhattan Project, 336n30\n\nMarin, John, ,\n\nMartinez, Crescensio, , 337n6\n\nMartinez, Julian, , , __ , 188\u201389, __\n\nMartinez, Maria, , , , 188\u201389, __\n\nMarword, Gregory. _See_ Curtis, George DeClyver\n\nMason, Charlotte Osgood, 40\u201343, 50\u201352, , , ,\n\nMatisse, Henri, 66\u201369, , , , ,\n\nMatthews, Washington, , ,\n\nMcAlpin, David,\n\nMcElmo Canyon, , , 160\u201361\n\nMcKinley, Jack, , ,\n\nMcKinley, Lester, ,\n\nMcKinley, Marvin,\n\nMcKinley, William \"Big Bill,\" 345n2\n\nMead, Frank, 40\u201342, 50\u201351, , 75\u201376, , , , 146\u201347, 179\u201382, 327n11\n\nMeem, John Gaw, 305\u20136\n\nMera, Frank E.,\n\nMera, Harry P., Jr., , , , 342n22\n\nMesa Verde, , , , , , , , ,\n\nMetropolitan Opera House, , , 329n19\n\nMiller, Lloyd, , __ , 310\u201311,\n\nMonument Valley, , , , , , , ,\n\nMoqui. _See_ Hopi\n\nMorley, Sylvanus, , ,\n\nMount Holyoke College,\n\nMruk, Wladyslaw (Walter),\n\n\"Mugwumps,\"\n\nMuir, John,\n\nMuseum of Art (Santa Fe), , , 208\u201312, __ , , 338n7\n\nMuseum of Navajo Ceremonial Art, 307\u20138,\n\nMuseum of New Mexico, , , ,\n\nMusic School Settlement for Colored People,\n\nNahant (MA), , 78\u201383, __ , 86\u201390, 148\u201350, 161\u201362, , , , , , , , 311\u201312, 333n6, 344n13\n\nNampayo, , 126\u201327, , ,\n\nNash, Willard, ,\n\nNavajo Reservation, , , , , , ,\n\nNavajos, , , , , , , , 119\u201320, , , 151\u201352, , , , 266\u201369, , , 333n19\n\nNewcomb, Arthur, 265\u201366,\n\nNewcomb, Franc, 265\u201366, , , 342n9\n\nNewcomb trading post,\n\nNew England Conservatory of Music, 82\u201385, __ , __\n\nNew Mexico Building (Balboa Park), , __ , , ,\n\nNew Women, , , , ,\n\nNew York (NY), 7\u20138, 51\u201352, , , 70\u201374, 92\u201393, , , , , , , 202\u20133, , , , , 256\u201357, , , ,\n\nNew York School of Art, 64\u201365, , ,\n\nNight Riders, , , 345n2\n\nNonnezoshe. _See_ Rainbow Bridge\n\nNordfeldt, J. O.,\n\nNusbaum, Jesse, , , , , , , , , , ,\n\nOakland (CA), ,\n\nO'Keeffe, Georgia, , 64\u201365, 302\u20134, __ , 309\u201310,\n\nOld Oraibi. _See_ Oraibi\n\nO\u00f1ate Hotel, , 199\u2013200, ,\n\nOraibi (AZ), 26\u201335, __ , , , , 56\u201359, , , , 142\u201343, 328n13\n\nOraibi Split, , 56\u201358,\n\n_Outlook_ , , , 110\u201311, , , 136\u201337, , , 334n16\n\n_Out West_ , 18\u201319, 28\u201329,\n\nPach, Walter, 65\u201369, __ , , 95\u201397, , , , , , , , ,\n\nPack, Arthur Newton, 3\u20134, 299\u2013302, 309\u201310\n\nPack, Charles Lathrop,\n\nPack, Eleanor (Brownie), 300\u2013301\n\nPainted Desert, , , 100\u2013101, , , , ,\n\nPaiutes, , 156\u201357,\n\nPajarito (Little Bird) Plateau (NM), , , , , , , , , , , , 207\u20138, 219\u201320, , , 230\u201332, , , ,\n\nPajarito Club, 172\u201373, 176\u201377, , 181\u201382, , , , 207\u20138, , 336n30\n\nPalace of the Governors, , , 171\u201372, , 184\u201385, , , , , 198\u201399, , , __ ,\n\nPanama-California Exposition. _See_ Balboa Park\n\nParis (France), , , , 66\u201369, 91\u201394, 96\u201397, , , 235\u201346, , , , , 329n2\n\nParsons, Elsie Clews, , 274\u201380, , , 343nn26\u201327\n\nParsons, Sheldon, , , ,\n\nPasadena (CA), 16\u201317, , , 28\u201329,\n\nPeabody, George Foster, 42\u201343, , 70\u201371, , , , , , ,\n\nPeabody, Ted, , , 299\u2013300\n\nPedernal Mountain (Cerro Pedernal), , , , __ , 299\u2013300, 303\u20134\n\nPelia, 40\u201342,\n\nPenitentes,\n\nP\u00e8re Lachaise Cemetery,\n\nPerkins, Colonel Thomas H., ,\n\nPf\u00e4ffle, Carol. _See_ Stanley, Carol (Caroline) Bishop\n\nPf\u00e4ffle, Lois, , , , 225\u201326, 229\u2013230, __ , , , , , __ , , , , , 344n11\n\nPf\u00e4ffle, Margaret (Gaga), , 201\u20132, , , 222\u201323, 225\u201326, , , , 279\u201380, __ , 285\u201386, ,\n\nPf\u00e4ffle, Marie, , , ,\n\nPf\u00e4ffle, Richard LeRoy (Roy), , 189\u201390, 193\u201395, 199\u2013202, 205\u20138, 219\u201320, 222\u201326, __ , 229\u201332, __ , , , 251\u201353, , , , 279\u201380, 282\u201392, , , , 310\u201311, , , 336n17, 341n1\n\nPicasso, Pablo, , , ,\n\nPiedra Lumbre, 287\u201390, , , 299\u2013301, ,\n\nplainsong (plain-chant), , , , 330n21\n\nPolacca, , ,\n\nPond, Ashley, , , 180\u201381, , 336n30\n\nPond, Peggy. _See_ Church, Peggy Pond\n\nPonianomsi, ,\n\nPrendergast, Maurice, , 331n5\n\nPring, Alice, , , , ,\n\nPueblo Pottery Fund,\n\nPutnam, Arthur, 62\u201363, 329n2, 346n2\n\nQuechan Indians, , ,\n\nRainbow Bridge, , , , , , , , __ , __ , , , 271\u201372, , 341n1, 343n22\n\nRainbow Trail, , , , , , 270\u201372, ,\n\nRamona (CA), ,\n\nRamon Vigil Ranch, , 172\u201373, 176\u201377, , 181\u201382, , , 336n30\n\nRancho de los Brujos (Ranch of the Witches). _See_ Ghost Ranch\n\nRancho Ramon Vigil, , __ , 222\u201323, , ,\n\nRapp, Isaac H., 102\u20133,\n\nRapp, William M., 102\u20133,\n\nRenehan, A. B., 289\u201390\n\nRio Arriba County (NM), , 287\u201388, , 345n2\n\nRio Grande, , , , , , , , , , , , , , , 233\u201334, 248\u201349, , , , , 272\u201374, 277\u201378, , 285\u201386, , , , ,\n\nRito de Los Frijoles (NM), , , 172\u201373, __ ,\n\nRito del Yeso, 287\u201389\n\nRockefeller, John D., , , , 344n3, 345n7\n\nRocky Mountain Camp Company, , , , 194\u201396,\n\nRodakiewicz, Henwar, ,\n\nRoosevelt, Archie, , 129\u201330, , __ , __\n\nRoosevelt, Nicholas (Nick), , , 138\u201339, 281\u201382, 334n22\n\nRoosevelt, Quentin, , , , __ , __\n\nRoosevelt, Theodore, , , 27\u201329, 36\u201339, 41\u201343, 46\u201347, 49\u201350, 52\u201354, , , , , 110\u201312, , , 127\u201342, __ , __ , __ , __ , 146\u201347, 151\u201352, 154\u201355, 158\u201359, , , , , 245\u201346, , , 327n11, 332n4, 332n5, 333n6, 334n22\n\nRuggles, C. B., ,\n\nSagamore Hill, 36\u201338, , , , ,\n\nSalazar, Alfredo,\n\nSan Diego (CA), 60\u201364, 68\u201370, 75\u201376, , 97\u201399, 102\u20137, , , , , 177\u201378, , , 202\u20133, , 208\u201311, , 314\u201315, , 329n3\n\nSan Diego Museum of Art, 314\u201315\n\nSan Francisco (CA), , ,\n\nSan Gabriel Ranch, 232\u201333, __ , , 251\u201355, __ , , 263\u201364, 266\u201367, , 271\u201372, 274\u201379, __ , __ , 281\u201383, 285\u201393, , , , , 311\u201312, 342n9, 345n2\n\nSangre de Cristo Mountains, 167\u201368, , , , , , ,\n\nSan Ildefonso Pueblo, , , , 187\u201389, __ , , , , ,\n\nSan Juan Pueblo, , , , , , , ,\n\nSanta Clara Pueblo, , , , 344n11\n\nSanta Fe (NM), , , , , 101\u20133, 112\u201313, , 163\u201372, __ , 174\u201380, , 184\u201387, 189\u201391, 193\u201395, , 201\u201319, __ , __ , 222\u201323, 226\u201328, 230\u201332, 234\u201335, 238\u201339, , 248\u201350, 253\u201359, , 277\u201379, 282\u201385, 289\u201391, , 297\u201399, 301\u201313, 317\u201318\n\nSanta Fe Railway, 15\u201316, , , , , , , , 166\u201367, , , ,\n\nSchindler, Kurt, , 71\u201373, 107\u20139, 112\u201316, 119\u201322, , , , 135\u201339, 142\u201344, __ , 229\u201330,\n\nSchirmer Music Co. (G. Schirmer), , , , , 343n10\n\nSchola Cantorum, , , , ,\n\nSchool of American Archaeology, , , 166\u201367, ,\n\nSchool of American Research (SAR), ,\n\nSeaman's Bethel (Boston), ,\n\nSecond Mesa (AZ), , , , ,\n\nSeidl, Anton, , 13\u201314\n\nSequoyah League, , , , 327n11\n\nSergeant, Elizabeth Shepley (Elsie), , 248\u201349, 256\u201359, , , , ,\n\nSharp, J. H.,\n\nShaw, Robert Gould, 11\u201312, 325n9\n\nSherman Institute (Sherman Indian School), , 58\u201359, 328n11\n\nShongopovi, ,\n\nShore, Henrietta, 105\u20136\n\nShuster, William Howard (Will), , , 342n17\n\nSioux, ,\n\nSloan, John, , __ , 227\u201328, , , , , 331n5\n\nSnake Dance (Hopi), , 110\u201311, , , , , , 138\u201341, __ , __ , , , 334n23\n\nSmith, Albert, , 341n1\n\nSorbonne, 237\u201342, ,\n\nSouth End Music School (Boston), , , ,\n\nSpringer, Frank,\n\nStanley, Carol (Caroline) Bishop, 3\u20134; and Alice Klauber, 185\u201387, ; and Arthur Newton Pack, 299\u2013301; and Bishop's Ranch, 222\u201326; in Boston, 88\u201389, 148\u201350; childhood on Nahant island, 78\u201382, __ ; death of, , 331n2, 331n10, 333n6, 336n30; and death of Natalie Curtis, , 251\u201353; and Elsie Clews Parsons, 277\u201378; first journey to the Southwest, 150\u201351, __ ; and Georgia O'Keeffe, ; and Ghost Ranch (Rancho de los Brujos), 287\u201391, 293\u2013304, __ , __ , ; and Indian Arts Fund, ; and Louisa Wade Wetherill, 151\u201362, __ , __ , __ ; marriage to Lloyd Miller, 310\u201311; marriage to Roy Pf\u00e4ffle, 194\u201395; and Mary Cabot Wheelwright, , , 262\u201367, 272\u201374; music teacher, 84\u201388; and Natalie Curtis, 162\u201383; at New England Conservatory of Music, 82\u201384; in New Mexico, 164\u201367, 170\u201377, , 189\u201394, 199\u2013208; and O\u00f1ate Hotel, 200\u2013202, 206\u20137; and Rainbow Bridge, 196\u201397, __ , __ , ; and Ramon Vigil Ranch, 177\u201383, ; and Rancho Ramon Vigil, 219\u2013223, __ ; and San Gabriel Ranch, 231\u201333, __ , 253\u201354, __ , 272\u201386, 309\u201311; Tewa name T'apokwj,\n\nStanley, Edward, ,\n\nStanley, Edward C., 85\u201386,\n\nStanley, Edwin,\n\nStein, Gertrude, 65\u201369, __ , 92\u201393, 95\u201397, , 238\u201339,\n\nStein, Leo, 65\u201369, __ , 216\u201317, 238\u201339,\n\nStein, Michael, , __ , 238\u201339, ,\n\nStein, Sally, , 238\u201339, ,\n\nStern, Nathan B. (Nate), 207\u20138,\n\nSterne, Mabel Dodge. _See_ Luhan, Mabel Dodge\n\nSterne, Maurice, 213\u201315, ,\n\nStevenson, Matilda Coxe, ,\n\nStieglitz, Alfred, ,\n\nStokowski, Leopold,\n\nSt. Vincent's Sanatorium, , , ,\n\nSunmount, 168\u201370, , , ,\n\nSutton Island (ME), 316\u201317\n\nTaos (NM), , , , , , 214\u201317, , , , , , 276\u201377, , , , ,\n\nTaos Pueblo, ,\n\nTawakwaptiwa, 56\u201359, __ , , , __ , 328n13\n\nTesuque (NM), , , 222\u201323, , 257\u201358, , ,\n\nThird Mesa (AZ), , 28\u201330, 33\u201335, 43\u201344, , , , , , 132\u201334, , , 332n4\n\nThorpe, James R., 222\u201323,\n\nToklas, Alice B., 95\u201396, 238\u201339\n\nTse-ne-gat (Everett Hatch), 156\u201358, 160\u201362, 335n12\n\nTusayan (Hopiland), 26\u201327, , ,\n\nTwitchell, Ralph Emerson, 102\u20133\n\nUtes, , 151\u201352, 156\u201358, __ , 160\u201362, , ,\n\nVierra, Carlos, 101\u20132, , , , ,\n\nVilla, Pancho,\n\nVoth, H. R., 327n19\n\nWagner, Cosima,\n\nWalden Pond, ,\n\nWalpi, , 109\u201311, 125\u201326, __ , 129\u201330, __ , 134\u201335, 137\u201339, __ , __ , , , , , 332n4\n\nWard, E. J., ,\n\nWashington, DC, , , , , , , 52\u201353, , , , , , , , , , , , ,\n\nWashington Place, 7\u20138, , , 15\u201316, , , , , , , ,\n\nWashington Square, , , , , 91\u201392,\n\nWave Crest, , 39\u201340\n\nWetherill, Ben,\n\nWetherill, John, , , , , 137\u201339, , 150\u201356, 159\u201362, , , , , , , 341n1, 343n22\n\nWetherill, Louisa Wade, , , 151\u201362, __ , __ , , , , , 310\u201311\n\nWetherill, Richard,\n\nWetherill Trading Post, __ , , , , , , , , , ,\n\nWheelwright, Mary Cabot, ; childhood, ; death of, , 316\u201318, 342n9, 342n10, 345n2; and Ghost Ranch, ; and Hostiin Klah, 264\u201369, 306\u20138; and Jack Lambert, 270\u201371, __ ; and Laboratory of Anthropology, 305\u20137; and Los Luceros, 272\u201373, __ ; and Museum of Navajo Ceremonial Art (Wheelwright Museum of the American Indian), 307\u20139, ; in New Mexico, 259\u201360; and Rainbow Trail, 270\u201371; at Rancho Ramon Vigil, ; at San Gabriel Ranch, 262\u201363, __ , , , 282\u201383, , 288\u201389,\n\nWhitaker, Ted, , , 114\u201317, , , 136\u201337, __ , 144\u201345\n\nWhite, Amelia Elizabeth, , 258\u201359, , 283\u201386, 305\u20137\n\nWhite, Martha Root, , 258\u201359, , 283\u201386,\n\nWhite House, , , , , ,\n\nWhite Swan Ranch,\n\nWhitman, Walt,\n\nWilliam Cody's Wild West Show. _See_ Buffalo Bill's Wild West Show\n\nWilson, Francis, , 289\u201390, 344n13\n\nWinnebago Reservation,\n\nWorld War I, ,\n\nWoy, Maude,\n\nYavapai, 40\u201342,\n\nYeso Canyon (NM), , 287\u201388,\n\nYoukeoma, 56\u201357\n\nYuma (AZ), , , 22\u201323,\n\nYumas (Yuma Indians), ,\n\nZuni Pueblo (NM), , , , , , , 275\u201376\n\n#  ABOUT THE AUTHOR\n\n**Lesley Poling-Kempes** is the award-winning author of six books about the American Southwest. Her novel _Bone Horses_ won both the Tony Hillerman Award for Best Fiction and the WILLA Award for Contemporary Fiction. Her books of nonfiction include _The Harvey Girls: Women Who Opened the West_ (Zia Award for Excellence), and _Ghost Ranch_ (Southwest Books of the Year \"Top Choice\" award), and she co-authored _Georgia O'Keeffe & New Mexico: A Sense of Place_. Poling-Kempes's first novel, _Canyon of Remembering_ , was a Western Writers of America Spur Award finalist for Best First Novel. _Valley of Shining Stone_ is the history of her adopted home, Abiqui\u00fa and the Chama River Valley in northern New Mexico, where she has lived with her husband and family for several decades.\n\n# Table of Contents\n\n  1. Cover Page\n  2. Title Page\n  3. Copyright Page\n  4. Contents\n  5. Prologue: Ladies of The Canyons\n  6. Chapter 1. Imperfectly Independent\n  7. Chapter 2. Land of Ineffable Light and Sudden Shadow\n  8. Chapter 3. Capturing The Wind\n  9. Chapter 4. The Indians' Book\n  10. Chapter 5. \"Time and The World\"\n  11. Chapter 6. Secrets and Dualities\n  12. Chapter 7. Creating a Brave New World\n  13. Chapter 8. Desert Euphoria\n  14. Chapter 9. Walpi Rendezvous\n  15. Chapter 10. \"A Fine Woman To Be Out With\"\n  16. Chapter 11. The City Different\n  17. Chapter 12. Remarkable Time, Remarkable Place\n  18. Chapter 13. A Rare Thing: Becoming Oneself\n  19. Chapter 14. \"A Country of Things In Light\"\n  20. Chapter 15. Death In Paris\n  21. Chapter 16. City of Ladies\n  22. Chapter 17. New Women of The Rio Grande \n  23. Chapter 18. \"Roughneck and Low-brow\"\n  24. Chapter 19. Everything and Nothing\n  25. Afterword: Something of The Spirit\n  26. Acknowledgments\n  27. Abbreviations\n  28. Notes\n  29. Selected Bibliography\n  30. Illustration Credits\n  31. Index\n\n## List of Pages\n\n  1. i\n  2. iii\n  3. iv\n  4. v\n  5. vi\n  6. vii\n  7. viii\n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337. \n  338. \n  339. \n  340. \n  341. \n  342. \n  343. \n  344. \n  345. \n  346. \n  347. \n  348. \n  349. \n  350. \n  351. \n  352. \n  353. \n  354. \n  355. \n  356. \n  357. \n  358. \n  359. \n  360. \n  361. \n  362. \n  363. \n  364. \n  365. \n  366. \n  367. \n  368. \n  369. \n  370. \n  371. \n  372. \n  373. \n  374. \n  375. \n  376.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nCopyright(c) 2012 Jo Franks\n\nTable of Contents\n\n## deg deg deg deg deg\n\n### Table of Contents\n\n### Table of Contents .................................................................................................................... iii\n\n### Preface ............................................................................................................................................v\n\nChapter 1\n\n### A-M ................................................................................................................................................. 1\n\n4 Cheese Stuffed Shells.................................................................................. 1\n\nAntipasto With Red Pepper Tapenade............................................................. 2\n\nAsiago Bagels ................................................................................................ 3\n\nAsiago Dip with Crostini ................................................................................ 5\n\nAsiago Soft Grits............................................................................................ 5\n\nBaked Tomatoes with Basil and Parmesan ...................................................... 6\n\nBalsamic Pasta Salad ...................................................................................... 7\n\nBlack Angus Garlic-Cheese Bread .................................................................. 7\n\nBreakfast Popovers With Parmesan................................................................. 8\n\nBroccoli With Garlic And Asiago ..................................................................... 9\n\nCanederli With Speck ..................................................................................... 9\n\nCavatappi With Spinach, Beans, & Asiago Cheese ......................................... 10\n\nCheese Puffs ............................................................................................... 11\n\nCheese Special............................................................................................. 12\n\nCheesy Sloppy Joe Pizza............................................................................... 13\n\nCheesy Turkey Casserole ............................................................................. 13\n\nChicken Enchiladas With Tomatillo Sauce ..................................................... 14\n\nCreamy Herb Pasta Sauce............................................................................. 16\n\nDandelion Green Fettuccini .......................................................................... 17\n\nFava Beans, Shrimp & Savory W\/ Orecchiette Pasta ....................................... 18\n\nFettucine With Lemon (Paradiso Perduto) ..................................................... 19\n\nFrico Tacos With Mache ............................................................................... 21\n\nGreen Gnocchi With Caduta Di Formaggio.................................................... 22\n\nGrilled 4 Cheese Sandwiches........................................................................ 23\n\nGrilled Stuffed Pasilla Chile With Avocado .................................................... 24\n\nHearts Of Romaine, Palm, And Artichoke ..................................................... 25\n\nHerb Cheesecake ......................................................................................... 26\n\niii\n\nItalian Mini Meat Loaves With Mac And Three Cheeses ................................. 27\n\nKids' Fast, Fake-Baked Ziti ........................................................................... 28\n\nLinguine Fine With French Lentils, Escarole And Asiago ................................ 30\n\nChapter 2\n\n### N-Z ................................................................................................................................................31\n\nOrecchiette Pasta W\/Red Wine Braised Chicken And Aromatic Vegetables .... 31\n\nOysters Randall............................................................................................ 32\n\nParmesan Straws.......................................................................................... 33\n\nPasta Fontina ............................................................................................... 35\n\nPrawns Provencal ......................................................................................... 35\n\nQuail Spiedini With Sage Polenta And Asiago................................................ 36\n\nQuick Creamy Tomato Soup......................................................................... 37\n\nRabbit With Asiago Grits .............................................................................. 38\n\nRicotta-Stuffed Zucchini Flowers.................................................................. 39\n\nRigatoni With Vodka Tomato Sauce .............................................................. 40\n\nRoast Garlic Mashed Potatoes ...................................................................... 41\n\nRoasted Garlic Linguini A La Cafe Tosca....................................................... 42\n\nRoasted Potato And Leek Frittata ................................................................. 43\n\nRomaine And Fennel Salad ........................................................................... 44\n\nSauted Apples In Amarone With Asiago ........................................................ 45\n\nSmoky Mushroom Tart................................................................................. 46\n\nSpinach-Ricotta Gnocchi .............................................................................. 47\n\nStovetop Spinach And Artichoke Dip ............................................................ 49\n\nStuffed Cherry Peppers ................................................................................ 50\n\nTagliatelle With Prosciutto And Radicchio..................................................... 51\n\nTagliatelle With Sausage And Radicchio ....................................................... 52\n\nThree-Herb Fettuccine With Vegetables and Shaved Asiago ......................... 53\n\nTuscany Beef & Spinach ............................................................................... 54\n\n### Index .............................................................................................................................................56\n\niv  \nPreface\n\n## deg deg deg deg deg\n\n### Preface\n\nNotice of Rights\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.\n\nNotice of Liability\n\nThe information in this book is distributed on an \"As Is\" basis without warranty. While every precaution has been taken in the preparation of the book, neither the author nor the publisher shall have any liability to any person or entity with respect to any loss or damage caused or alleged to be caused directly or indirectly by the instructions contained in this book or by the products described in it.\n\nTrademarks\n\nMany of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations appear as requested by the owner of the trademark. All other product names and services identified throughout this book are used in editorial fashion only and for the benefit of such companies with no intention of infringement of the trademark. No such use, or the use of any trade name, is intended to convey endorsement or other affiliation with this book.\n\nJo Franks\n\nv  \nAsiago Greats\n\n## Chapter 1\n\n### A-M\n\n## 4 CHEESE STUFFED SHELLS\n\n4 servings\n\n### Source: Asiago Greats\n\n### Salt, as needed\n\n### 8 pieces jumbo pasta shells\n\n### 1 1\/2 lbs. ricotta cheese\n\n### = (or part skim ricotta cheese)\n\n### 1 lb. mozzarella, diced\n\n### 1\/2 cup grated Parmigiano-Reggiano\n\n### 1 cup shredded Asiago\n\n### 1\/4 cup chopped flat-leaf parsley\n\n### 2 tablespoons extra-virgin olive oil\n\n### 3 cloves garlic, chopped\n\n### 1 small onion, finely chopped\n\n### 1 can crushed tomatoes - (28 oz)\n\n### Salt, to taste\n\n### Freshly-ground black pepper, to taste\n\n### 6 fresh basil leaves - (to 7), torn or shredded\n\nPreheat oven or broiler to 450 degrees.  \nBring a large pot of water to a boil. Salt water and add pasta. Cook shells 12 to 15 minutes, they should be softened but still undercooked at the center. Drain pasta and cool.  \nCombine ricotta, 1\/2 of the diced mozzarella, a couple of handfuls of Parmigiano and 1\/2 of the Asiago. Add parsley to the cheeses and stir to combine.  \n1  \nChapter 1: A-M  \nTo a small saucepot over moderate heat add oil, garlic and onions. Saute onions and garlic 5 minutes. Add tomatoes and season sauce with salt and pepper. Simmer sauce 5 minutes and stir in basil leaves  \nPour a little sauce into the bottom of a shallow medium-sized casserole dish. Fill shells with rounded spoonfuls of cheese mixture and arrange them seam-side down in casserole dish. Top shells with remaining sauce and remaining mozzarella and Asiago cheeses. Place shells in very hot oven or 8 inches from hot broiler and cook 6 to 8 minutes to melt cheeses and bubble sauce.  \nThis recipe yields 4 servings.\n\n## ANTIPASTO WITH RED PEPPER TAPENADE\n\n4 servings\n\n### Source: Asiago Greats\n\n### == RED PEPPER TAPENADE & CHARRED BREAD ==\n\n### 1\/2 loaf day-old crusty bread, sliced\n\n### 2 large garlic cloves, cracked\n\n### Extra-virgin olive oil, for drizzling\n\n### 1 jar roasted red peppers - (16 to 18), drained\n\n### 1\/4 cup parsley leaves\n\n### 1\/4 cup black oil-cured olives - (20 olives)\n\n### 2 tablespoons capers, drained\n\n### === ANTIPASTO === Italian tuna in oil, 4 ounces\n\n### 1 jar marinated artichokes - (6 oz)\n\n### Hot peppers, cherry peppers, banana\n\n### peppers or pepperoncini\n\n### 1\/2 lb. sliced Italian deli meats\n\n### = (such as hot or sweet sopressata, Genoa\n\n### salami)\n\n### 1\/2 lb. assorted Italian cheeses, sliced or cubed\n\n### = (such as asiago, provolone, pepato\n\n### black pepper cheese, smoked mozzarella,\n\n2  \nAsiago Greats\n\n### red pepper flake sheep's milk cheese)\n\nIf your slices are wider than a few inches, cut each slice in 1\/2. Char pieces of bread under hot broiler to toast and crisp. Rub with cracked garlic and drizzle bread with extra-virgin olive oil.  \nPlace roasted red peppers in a food processor with parsley. Place a few olives at a time under the flat of a knife and whack with the heal of your hand. Discard olive pits. Add olives to the food processor. Drain a few spoonfuls of capers and add to processor. Pulse the processor and grind into a paste. Transfer to a small dish.  \nArrange tuna, artichokes, hot peppers, meats and cheeses on a platter and serve with charred bread and red pepper spread for all-evening snacking.  \nThis recipe yields 4 servings.\n\n## ASIAGO BAGELS\n\n1 servings\n\n### Source: Asiago Greats\n\n### 2 packages active dry yeast\n\n### 3 tablespoons sugar\n\n### 5 1\/2 teaspoons salt\n\n### 6 cups all-purpose flour\n\n### 1\/4 lb. asiago or parmesan cheese\n\n### 1 large egg yolk\n\n1. Combine yeast and 2 cups warm (110deg) water in a large bowl; let stand 5 minutes. Stir in sugar, salt, and 51\/2 cups flour until moistened.  \n2. Spread 1\/2 cup flour on a board and scrape dough onto it. Knead dough until very smooth and elastic, 10 to 15 minutes; add more  \n3  \nChapter 1: A-M  \nflour as required to prevent sticking.  \n3. Place dough in a lightly oiled bowl, cover with plastic wrap, and let rise in a warm place until almost doubled, about 40 minutes.  \n4. When dough is almost risen, bring 3 quarts water to boiling in a covered 4-to 5-quart pan over high heat.  \n5. Meanwhile, knead dough on a lightly floured board to expel air, then divide into 12 equal pieces. To shape bagels, knead each piece into a smooth ball. Holding the ball with both hands, poke your thumbs through the center to form a hole. With 1 thumb in the hole, work around the bagel to make it smooth and evenly thick.  \n6. Set bagels on lightly floured 12-by 15-inch baking sheets; let stand, covered with plastic wrap, just until they are slightly puffy, about 10 minutes.  \n7. With your hand, gently slip bagels into boiling water, adding 4 at a time to pan. Adjust heat to maintain a gentle boil and turn bagels often until they feel firm but are still slightly spongy, 3 to 5 minutes. Chill remaining bagels until you are ready to boil them.  \n8. With a slotted spoon, transfer bagels to a towel; drain briefly, then place bagels slightly apart on 2 lightly oiled 12-by 15-inch baking sheets.  \n9. In a food processor or with a knife, coarsely chop cheese.  \n10. Beat egg yolk to blend with 1 tablespoon water and brush mixture over bagels. Then pat cheese onto bagels, pressing gently to make  \nthe cheese stick.  \n11. Bake bagels in a 375deg oven until browned, 30 to 35 minutes. Cool on racks. Serve warm or cool.  \nPer bagel: 289 cal., 12% (34 cal.) from fat; 9.8 g protein; 3.8 g fat (2.3  \n4  \nAsiago Greats  \ng sat.); 51 g carbo (2 g fiber); 385 mg sodium; 24 mg chol. Makes 12.\n\n## ASIAGO DIP WITH CROSTINI\n\n32 servings\n\n### Source: Asiago Greats\n\n### 1 cup light mayonnaise\n\n### 1\/2 cup green onions, thinly sliced\n\n### 1\/3 cup grated Asiago, (about 1\n\n### 1\/2 ounces) OR\n\n### 1\/3 cup grated Parmesan cheese, (about 1 1\/2 ounces)\n\n### 1\/4 cup sliced mushrooms\n\n### 1\/4 cup sun-dried tomato sprinkles\n\n### 1 (8-ounce) ca low-fat sour cream\n\n### 1 tablespoon grated Asiago, (1\/4 ounce)\n\n### OR\n\n### 1 tablespoon grated Parmesan cheese, (1\/4 ounce)\n\n### 32 slices French bread baguette, (about 2 baguettes),\n\nPreheat oven to 350degF.  \nCombine first 6 ingredients in a bowl; spoon into a 1-quart casserole. Sprinkle with 1 tablespoon cheese. Bake at 350[?] for 30 minutes or until bubbly. Serve with toasted bread.  \n(Serving size: 1 tablespoon dip and 1 crostino).\n\n## ASIAGO SOFT GRITS\n\n12 servings\n\n### Source: Asiago Greats\n\n### 3 3\/4 cups water 2 1\/4 cups low-salt chicken broth\n\n5  \nChapter 1: A-M\n\n### 1\/4 teaspoon salt\n\n### 1\/8 teaspoon pepper\n\n### 3 cloves garlic, chopped\n\n### 1 cup uncooked regular grits\n\n### 1\/2 cup 2% low-fat milk\n\n### 6 ounces grated Asiago cheese, (1-1\/2 cups)\n\nBring first 5 ingredients to a boil in a large saucepan; gradually stir in grits. Cover, reduce heat, and simmer 15 minutes or until thickened, stirring occasionally. Remove from heat; stir in milk and cheese. Yield:  \n6 cups (serving size: 1\/2 cup).\n\n## BAKED TOMATOES WITH BASIL AND PARMESAN\n\n6 servings\n\n### Source: Asiago Greats\n\n### 3 lg vine-ripened tomatoes\n\n### (about 1 1\/2 pounds)\n\n### 1\/4 cup minced fresh herbs\n\n### (basil, parsley, marjor\n\n### 1\/2 cup dry grated bread crumbs\n\n### 1\/2 cup shredded Parmesan or\n\n### Asiago cheese\n\n### 2 garlic cloves, finely minced\n\n### 1 Pinch salt\n\n### 1 Pinch freshly ground black pepper\n\n### 3 tablespoons extra-virgin olive oil\n\nPreheat the oven to 350degF. Put the tomatoes in a nonstick baking dish, cut sides up. Combine the herbs, bread crumbs, cheese, garlic, salt, pepper, and oil in a small bowl.  \nSprinkle each tomato with an equal portion of the mixture.  \nBake for 30 minutes or until crusty. Tomatoes will be soft yet hold their shape.\n\n6  \nAsiago Greats\n\n## BALSAMIC PASTA SALAD\n\n5 servings\n\n### Source: Asiago Greats\n\n### 1\/4 cup balsamic vinegar\n\n### 3 tablespoons water\n\n### 1 1\/2 teaspoons olive oil\n\n### 1\/4 teaspoon salt\n\n### 1\/4 teaspoon pepper\n\n### 1 clove garlic, minced\n\n### 1\/2 cup small broccoli flowerets\n\n### 1\/2 cup small cauliflower flowerets\n\n### 1\/2 cup carrot, julienne\n\n### 1\/2 cup red bell pepper, julienne\n\n### 4 cups farfalle*, cooked cooked without salt\n\n### 2 tablespoons fresh basil, thinly sliced\n\n### 1\/4 cup asiago cheese, grated\n\n* bow tie pasta  \nCombine vinegar and next 5 ingredients in a jar; cover tightly, and shake vigorously. Set aside.  \nDrop broccoli, cauliflower, and carrot into a large saucepan of boiling water; cook 30 seconds. Drain. Pour cold water over vegetables; drain.  \nCombine drained vegetables, bell pepper, and next 3 ingredients in a large bowl.  \nAdd vinegar mixture, and toss gently. Cover and chill.\n\n## BLACK ANGUS GARLIC-CHEESE BREAD\n\n10 servings\n\n### Source: Asiago Greats\n\n7  \nChapter 1: A-M\n\n### 1 loaf French bread\n\n### 1\/2 cup butter, softened\n\n### 1 cup shredded Asiago cheese\n\n### 1 cup shredded Jack cheese\n\n### 1 cup mayonnaise\n\n### 1 bunch green onions, chopped\n\n### 2 cloves garlic, pureed\n\nSplit French bread loaf into halves horizontally. In bowl mix butter, cheeses, mayonnaise, green onions and garlic. Blend well.  \nSpread cut side of bread halves with garlic cheese spread.  \nBake at 350degF 7 minutes, then place under broiler about 3 minutes longer. Cut into slices and serve.\n\n## BREAKFAST POPOVERS WITH PARMESAN\n\n8 servings\n\n### Source: Asiago Greats\n\n### 1\/2 cup Liquid Egg Substitute\n\n### 1 1\/4 cups whole grain flour\n\n### 1 cup Fat-Free Milk\n\n### 2 tablespoons Fat-Free Milk\n\n### 1 tablespoon Trans-Free\n\n### Margarine [melted]\n\n### 3 tablespoons Parmesan or\n\n### Asiago Cheese [grated]\n\nPreheat the oven to 375degF. Coat 8 custard cups or popover-pan cups with cooking spray.  \nWhisk the egg substitute in a medium bowl. Add the flour, milk, and margarine, and whisk until the ingredients are combined. Stir in the cheese.  \nEvenly divide the batter among the prepared cups. Place the cups on a large baking sheet.  \n8  \nAsiago Greats  \nBake for 30 minutes, or until the popovers are puffed and golden. Remove the popovers from the cups and serve hot.\n\n## BROCCOLI WITH GARLIC AND ASIAGO\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1 large head broccoli, cut long thin spears\n\n### 2 tablespoons extra-virgin olive oil\n\n### 4 cloves garlic, finely chopped\n\n### 1\/4 lb. Asiago cheese, shaved\n\nSimmer broccoli in 1\/2-inch boiling, salted water in a shallow covered skillet for 6 or 7 minutes, until tender. Drain broccoli and remove  \nfrom pan.  \nReturn pan to stove top over medium heat. Add oil and garlic. Saute garlic in oil 3 minutes. Add broccoli spears to the skillet and coat them with garlic oil. Transfer broccoli to a serving dish and garnish with curls of Asiago cheese.  \nThis recipe yields 4 servings.\n\n## CANEDERLI WITH SPECK\n\n1 servings\n\n### Source: Asiago Greats\n\n### 3\/4 lbs. day-old bread, cut 1\/2\" cubes\n\n### 1 1\/4 cups milk\n\n### 4 ounces pancetta, finely chopped\n\n### 2 ounces liver salami, cut small cubes\n\n### 1\/2 cup flour\n\n### 1\/2 teaspoon nutmeg\n\n### 1\/2 teaspoon freshly-ground black pepper\n\n9  \nChapter 1: A-M\n\n### 1\/2 teaspoon salt\n\n### 2 eggs\n\n### 4 ounces butter\n\n### 2 ounces speck, cut 1\/8\" julienne\n\n### 1\/2 cup grated asiago cheese\n\nPlace bread in mixing bowl and douse with milk, mixing well, and let stand 5 minutes.  \nMeanwhile, place pancetta into a 12-inch to 14-inch saut? pan and place over medium heat. Cook until fat is melting, about 5 minutes, and add bread and milk mixture, liver salami, flour, nutmeg, pepper and salt and stir to mix well, about 1 minute. Remove and allow to cool, about 5 minutes. Add eggs and mix well.  \nForm into 15 to 20 quarter-sized balls. Bring 6 quarts water to boil and add 2 tablespoons salt. Place balls in water and cook 6 to 7 minutes, until tender.  \nMeanwhile, place butter and speck in a 12-inch to 14-inch saut? pan and melt over medium heat. Draw canederli well and toss into pan. Add grated cheese and toss to coat. Serve immediately.\n\n## CAVATAPPI WITH SPINACH, BEANS, & ASIAGO CHEESE\n\n4 servings\n\n### Source: Asiago Greats\n\n### 8 cups spinach leaves, coarsely chopped\n\n### 6 ounces spiral shaped pasta, uncooked\n\n### 2 tablespoons olive oil\n\n### 1\/4 teaspoon salt\n\n### 1\/4 teaspoon pepper\n\n### 19 ounces canned cannellini beans, drained\n\n### or\n\n### other white beans\n\n### 2 cloves garlic, crushed\n\n### 1\/2 cup Asiago cheese, shredded\n\n### Fresh ground black pepper, optional\n\n10  \nAsiago Greats  \nCombine all ingredients in a large bowl, and toss well.\n\n## CHEESE PUFFS\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1 tablespoon Butter\n\n### 1 tablespoon Olive oil\n\n### 1\/2 teaspoon Salt\n\n### 1\/2 teaspoon Cayenne pepper\n\n### 1 cup Flour\n\n### 2 lg Eggs\n\n### 4 lg Egg whites\n\n### 3\/4 cups Plus\n\n### 2 tablespoons Freshly grated\n\n### Asiago\n\n### or Parmesan\n\nLightly coat 2 baking sheets with nonstick cooking spray or line with parchment paper. In a medium-sized saucepan, combine 1 cup water, butter, oil, salt and cayenne; bring to a boil over medium heat.  \nRemove from heat and add flour all at once. Stir with a wooden spoon until it forms a smooth paste. Return the mixture to low heat and cook, stirring, for about 3 minutes, to dry the paste slightly.  \nRemove from heat and cool for 2 minutes. Lightly whisk together eggs and 3 of the whites. With a wooden spoon, beat 1\/4 of the egg mixture into the flour paste until absorbed. Repeat with 3 more additions of the egg mixture, until the paste is smooth and glossy.  \nStir in 3\/4 c of the cheese. Using a pastry bag or spoon, drop 1\" mounds 1 1\/2\" apart on the prepared baking sheets. Lightly beat the remaining egg white and brush it over each puff. Sprinkle with the remaining 2 T cheese.  \n11  \nChapter 1: A-M  \nBake, one sheet at a time, for 20 to 25 minutes, or until the puffs are firm and well browned. Serve warm. (The puffs be frozen, for up to 2 months and reheated for 10 minutes at 350 degrees).  \nPer puff: 24 calories, 1 g protein, 1 g fat, 2 g carbohydrate, 59 mg sodium, 10 mg cholesterol.\n\n## CHEESE SPECIAL\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1 cup onions\n\n### 2 tablespoons olive oil\n\n### 1 lb. ground round\n\n### 1 teaspoon seasoned salt\n\n### 1 cup chopped spinach, squeezed dry\n\n### 6 whole eggs\n\n### 1\/4 cup grated asiago cheese,\n\n### * see note\n\n### 1\/4 cup grated parmesan cheese\n\nSaute the onions in olive oil over medium high heat. Add ground round and season with salt.  \nCook until medium well. Break up with fork.  \nAdd spinach and whole eggs and stir and cook until done, but not dry.  \nServe on individual plates. Sprinkle with combined cheeses.  \n* You can substitute Romano cheese if you desire.\n\n12  \nAsiago Greats\n\n## CHEESY SLOPPY JOE PIZZA\n\n4 servings\n\n### Source: Asiago Greats\n\n### 3\/4 lbs. lean ground beef\n\n### 3\/4 cups spicy spaghetti sauce\n\n### 1 10 inch pizza crust, pre-baked, round\n\n### 1\/2 cup green onions, chopped\n\n### 1 cup Monterey Jack, shredded\n\nIn large skillet, heat 1 tsp vegetable oil over medium heat; add beef and cook, breaking up meat with back of spoon, for 4-5 minutes or until meat is no longer pink. Drain off fat. Stir in spaghetti sauce, 1\/2 tsp salt and 1\/4 tsp pepper.  \nPlace pizza crust on baking sheet; spread sauce over crust. Sprinkle with green onions; scatter shredded cheese over top. Bake in 400degF  \n200C oven for 20-25 minutes or until cheese is bubbly and golden.  \nVegetables may be added to the sauce. Corn is colorful. Asiago cheese may be used in lieu of Monterey Jack.  \nServe with a crisp, fresh lettuce and crunchy cucumber salad tossed with Ranch dressing.\n\n## CHEESY TURKEY CASSEROLE\n\n4 servings\n\n### Source: Asiago Greats\n\n### 3\/4 cups dry bread crumbs\n\n### 2 tablespoons parmesan, freshly grated\n\n### 1\/4 teaspoon salt\n\n### 1\/4 teaspoon pepper\n\n### 1\/4 teaspoon dried basil\n\n### 2 egg whites\n\n### 1 lb. turkey scaloppine\n\n### 2 teaspoons olive oil\n\n### 2 teaspoons butter\n\n### 1\/2 cup asiago cheese, shredded\n\n### -eggplant tomato sauce---\n\n13  \nChapter 1: A-M\n\n### 1 eggplant\n\n### 1 tablespoon olive oil\n\n### 1 onion, chopped\n\n### 2 garlic cloves, minced\n\n### 28 ounces canned stewed tomatoes\n\n### 1 teaspoon dried basil\n\n### 3\/4 teaspoons granulated sugar\n\n### 1\/2 teaspoon dried oregano\n\n### 1\/4 teaspoon pepper\n\nEggplant Tomato Sauce: Peel and chop eggplant. In heavy saucepan, heat oil over medium heat; cook eggplant, onion and garlic, stirring, for about 5 minutes or until softened. Add tomatoes, mashing with fork, basil, sugar, oregano and pepper; bring to boil. Reduce heat; simmer, stirring occasionally, for about 15 minutes or until thickened.  \nSauce can be made ahead, covered and refrigerated for up to 3 days. In bowl, combine bread crumbs, Parmesan, salt, pepper and basil. In separate bowl, whisk egg whites. cut turkey into serving-size portions; dip into egg whites, then into crumb mixture to coat on both sides. In large nonstick skillet, heat 1 ts each of the oil and butter over medium-high heat; cook turkey in batches, adding remaining butter and oil as necessary, for 4 minutes or until browned on both sides.  \nPour sauce into shallow 10-cup casserole dish; arrange turkey in single layer over sauce. Sprinkle with cheese. Cover with foil. Bake, covered, in 350degF 180degC oven for about 25 minutes or until turkey is tender and sauce is bubbly. Remove foil; broil for 2 minutes or until cheese is bubbly.\n\n## CHICKEN ENCHILADAS WITH TOMATILLO SAUCE\n\n4 servings\n\n### Source: Asiago Greats\n\n### \\--sauce---\n\n### 1 1\/4 lbs. fresh tomatillos\n\n### 1 to...\n\n### 2 jalapeno peppers\n\n14  \nAsiago Greats\n\n### 1 small onion, peeled and finely\n\n### 1 medium garlic clove, peeled and minced\n\n### 1 tablespoon vegetable oil\n\n### 2 cups low-sodium chicken broth\n\n### 1\/4 teaspoon salt\n\n### \\--enchiladas---\n\n### 2 whole chicken breasts, boneless and skinless\n\n### 2 tablespoons minced onion\n\n### 1\/3 cup sour cream\n\n### 1\/4 teaspoon salt\n\n### 1\/3 cup vegetable oil\n\n### 8 corn tortillas\n\n### 1 cup sharp cheddar cheese, coarsely grated\n\n### 1\/2 cup crumbled asiago cheese, or more jack cheese\n\nTO MAKE THE SAUCE. Husk and wash the tomatillos. Bring a pan of water to the boil, add the tomatillos and jalapeno peppers and time for 10 minutes. Drain and remove the stem ends of the peppers. Put the tomatillos, peppers, onion and garlic in a food processor and process to a coarse puree.  \nIn a large pan heat the vegetable oil over medium heat. Add the vegetable puree and simmer 2 minutes. Stir in the broth and salt; simmer 15 minutes, stirring occasionally. Set aside.  \nTO MAKE THE ENCHILADAS. Place the chicken in a large saucepan and cover with cold water. Bring just to a boil, reduce the heat, cover and simmer until cooked through. Remove the chicken from the water and cool slightly. Shred the chicken and set aside.  \nCombine the cooled, shredded chicken with the minced onion, sour cream, and salt. Stir in 1\/4 cup of the tomatillo sauce. Set aside.  \nIn a 9-to 10-inch frying pan heat the oil over medium-high heat. Put  \n1 tortilla at a time in the hot oil and fry about 30 seconds on each side. Drain on paper towels. Cool slightly.  \nSpread a little of the sauce in a 9-by 13-inch baking dish. Spoon  \n15  \nChapter 1: A-M  \nsome of the filling down the center of each tortilla and roll. Place in the baking dish; it will be a tight fit. Spoon the sauce over the enchiladas and sprinkle with the cheddar and asiago cheeses.  \nBake in a preheated 350degF oven 25 minutes. Cool 5 minutes before serving.\n\n## CREAMY HERB PASTA SAUCE\n\n4 servings\n\n### Source: Asiago Greats\n\n### 2 cups half-and-half\n\n### 1\/2 cup butter or margarine, cut into 8 chunks\n\n### 2\/3 cups parmesan, asiago or romano cheese, freshly grated\n\n### 3\/4 cups fresh herbs, chopped salt and freshly ground pepper, to taste\n\nNote: Fresh herbs can be any combination of parsely, chives, basil, mint and oregano.  \nCombine half-and-half and butter in a saucepan. Boil gently about 20 minutes, until mixture has reduced to 1-1\/2 to 1-2\/3 cups.  \nRemove pan from heat; whisk in cheese and herbs. Use immediately or cool to room temperature and refrigerate or freeze (see Timesaver Tip).  \nTo serve: Cook 1-1\/4 to 1-1\/3 pounds fresh or fresh-frozen fettucine, or 12 ounces dried, following package directions. Drain. Toss with sauce.  \nMakes 2 cups (4 main dish or 6 appetizer servings).  \n16  \nAsiago Greats  \n* Timesaver Tip: Sauce can be made up to 6 hours ahead, covered and refrigerated, or it can be frozen. To freeze, spoon into a freezer container, cover, label, and freeze at 0F up to 2 months. To serve, defrost sauce 15 to 20 minutes in microwave oven on Defrost setting, stirring several times, or thaw in refrigerator 8 to 24 hours. Gently warm sauce in a saucepan or microwave oven on 50% power until warmed through, about 3 minutes, whisking frequently and  \nvigorously to bring it to a creamy consistency.\n\n## DANDELION GREEN FETTUCCINI\n\n1 servings\n\n### Source: Asiago Greats\n\n### 2 cups dandelion greens\n\n### 2 eggs\n\n### 1 1\/2 cups flour\n\n### 1\/2 teaspoon salt\n\nSome people find dandelion greens too strong simply added to a salad, but here is a basic pasta recipe that uses them to the best effect. It also works for other greens; beet greens, arugula, orach, or chicory. It is so tasty that it needs only butter and grated Asiago cheese. When making the pasta, adjust the amount of flour to the moisture of the greens.  \nIn a blender put dandelion greens and eggs, blend until smooth. Transfer to a bowl, add salt and start adding flour while beating with  \na spoon. Keep adding until dough is stiff.  \nTurn out onto floured surface and knead until smooth (approximately  \n5 minutes).  \nRoll out with rolling pin to 1\/8\"-1\/4\" thickness or thinner.  \n17  \nChapter 1: A-M  \nAllow to stand and dry 1 hour, then cut into strips. Drop into boiling water and cook 1-2 minutes.\n\n## FAVA BEANS, SHRIMP & SAVORY W\/ ORECCHIETTE PASTA\n\n4 servings\n\n### Source: Asiago Greats\n\n### 2 1\/2 lbs. fresh fava beans, in the pod\n\n### 3 tablespoons olive oil\n\n### 2 teaspoons slivered garlic\n\n### 2 ounces pancetta cut into tiny dice, about 1\/3 cup\n\n### 3\/4 lbs. medium size shelled shrimp\n\n### 1 big pinch red chile flakes\n\n### 1 tablespoon chopped fresh savory OR\n\n### 2 tablespoons chopped fresh basil\n\n### 1\/2 cup ripe tomato, seeded, diced\n\n### 2 teaspoons grated lemon zest\n\n### 8 ounces dry orecchiette pasta, cooked al dente\n\n### rich chicken, shellfish or vegetable stock, if desired\n\n### salt and freshly ground pepper\n\n### \\--garnish---\n\n### shaved parmesan or asiago cheese\n\n### savory or basil sprigs\n\nRemove beans from the pod and drop into boiling salted water for  \n1-2 minutes. Strain and immediately drop into ice water to stop the cooking. When cool, drain and remove the tough bitter outer skin. Set beans aside.  \nHeat olive oil in a skillet and Saute garlic over moderate heat until soft but not brown, add pancetta and Saute 1 minute longer until translucent. Add shrimp, fava beans, chile flakes and savory and toss  \n18  \nAsiago Greats  \nuntil shrimp just turn pink, 1-2 minutes.  \nAdd tomato, lemon zest and pasta and toss to warm through. If mixture seems dry, add a little stock. Season with salt and pepper and garnish with shavings of cheese and savory or basil sprigs.\n\n## FETTUCINE WITH LEMON (PARADISO PERDUTO)\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1\/2 recipe fresh fettucine, (below)\n\n### 1\/2 recipe fresh green fettucine, (below)\n\n### 4 tablespoons extra-virgin olive oil\n\n### 4 shallots, finely chopped\n\n### 1 treviso radicchio head, chopped 1\" pieces\n\n### 1 1\/2 cups cream\n\n### 4 lemons, plus , zest of\n\n### 1 lemon, juice of\n\n### 1\/2 cup grated asiago\n\n### === fresh fettucine ===\n\n### 4 eggs, extra-large\n\n### 3 1\/2 cups unbleached all-purpose flour,\n\n### plus\n\n### 1\/2 cup unbleached all-purpose flour,\n\n### for dusting\n\n### 1\/2 teaspoon olive oil\n\n### === fresh green fettucine ===\n\n### 4 eggs, extra-large\n\n### 6 ounces frozen chopped spinach,\n\n### defrosted, and squeezed very dry\n\n### 3 1\/2 cups unbleached all-purpose flour,\n\n### plus\n\n### 1\/2 cup unbleached all-purpose flour,\n\n### for dusting\n\n### 1\/2 teaspoon olive oil\n\nTo Prepare Fresh Fettucine: Make a mound of the flour in the center of a large wooden cutting board. Make a well in the middle of the  \n19  \nChapter 1: A-M  \nflour and add the eggs and oil. Using a fork, beat together the eggs and oil and begin to incorporate the flour starting with the inner rim of the well. As you expand the well, keep pushing the flour up to  \nretain the well shape. Do not worry that this initial phase looks messy. The dough will come together when half of the flour is incorporated.  \nStart kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up any leftover crusty bits. Discard these bits. Lightly flour the board and continue kneading for 3 more minutes. The dough should be elastic and a little sticky. Continue to knead for another 3 minutes, remembering to dust your board when necessary. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature. Do not skip the kneading or resting portion of this recipe. They are essential for a light pasta.  \nRoll out pasta to thinnest setting on pasta machine. Cut pasta into  \n1\/4-inch thick noodles by hand or with machine and set aside under a moist towel.  \nTo Prepare Green Fettucine: Follow the regular fettucine instructions above.  \nTo Prepare Dish: Bring 6 quarts water to boil and add 2 tablespoons salt.  \nIn a 12-to 14-inch saute pan, heat oil over the heat. Add shallots and saute until light golden-brown, about 5 to 6 minutes. Add radicchio and cook 30 seconds. Add cream, reduce by half and remove from heat. Drop pasta into boiling water and cook until tender yet al dente. Drain pasta well and toss into pan with cream mixture. Return to heat, add lemon zest and juice and toss to coat. Add half the cheese and toss again. Serve immediately in warm bowl, making extra cheese available on the side.\n\n20  \nAsiago Greats\n\n## FRICO TACOS WITH MACHE\n\n24 servings\n\n### Source: Asiago Greats\n\n### 4 teaspoons fresh lemon juice\n\n### 1 small shallot, minced\n\n### 1 teaspoon Dijon mustard\n\n### 1\/4 cup extra-virgin olive oil\n\n### Kosher salt, to taste\n\n### Freshly-ground black pepper, to taste\n\n### 2 cups mache leaves\n\n### === ASIAGO FRICO ===\n\n### 9 ounces Asiago cheese\n\n### 1 tablespoon all-purpose flour\n\nFor the asiago frico: Grate the cheese on the fine holes of a box grater to yield 3 1\/2 cups. In a medium bowl, toss cheese and flour  \ntogether. Heat a medium nonstick skillet over medium-low heat. Sprinkle 1 1\/2 tablespoons of the cheese mixture into the pan to form a circle about 4 inches in diameter. Cook until the cheese is  \nsomewhat melted but not firm, about 2 minutes. Using a spatula, turn the cheese over, and cook until it is visibly firm and just taking on a little color, 30 seconds to 1 minute.  \nTransfer the frico from the pan to a paper towel, and hold it in the palm of your hand. Press an offset spatula down the middle to form a taco shape. Transfer to paper towels to cool. Repeat with remaining cheese. Store at room temperature in an airtight container for 2 to 3 days. (Makes 2 dozen 4-inch fricos) NOTE: Can also be made with cheddar, Gruyere or Parmesan cheese.  \nFor the tacos: In a small bowl, whisk together lemon juice, shallot, and mustard. Whisk in olive oil. Add salt and pepper to taste. Gently toss mache with the vinaigrette. Carefully place mache into the frico tacos and serve. Assemble as close to serving time as possible.  \nThis recipe yields 2 dozen.  \nComments: Frico means \"little trifle\" in Italian. Fricos are delicate, lacy melted-cheese wafers that originated in Friuli, a small Italian region  \n21  \nChapter 1: A-M  \nthat borders Austria and Slovenia. Traditionally made with Montasio cheese, these crisps can be molded into a wide range of shapes, then filled with greens such as mache or arugula. The wafers are molded while still warm.  \nTo create a taco-shell shape, the wafers can be draped over rolling pins or small bowls, or turned upward in the palm of your hand. Shaped this way, they make unusual and attractive hors d'oeuvres. Parmesan, Gruyere, cheddar, and Asiago are acceptable substitutes for Montasio cheese, but keep in mind that the level of heat and cooking time will vary, depending on the type you use.  \nGrate the cheese at the last minute, as grating it too soon will result in a loss of moisture; dry cheese won't melt or spread properly and will make the molding process difficult. Kept in an airtight container, these delicate cheese hors d'oeuvres will keep for two to three days.\n\n## GREEN GNOCCHI WITH CADUTA DI FORMAGGIO\n\n4 servings\n\n### Source: Asiago Greats\n\n### 4 ounces gorgonzola\n\n### 2 ounces butter\n\n### 1 ounce grappa\n\n### 1 recipe green gnocchi,\n\n### 1\/2 cup grated asiago\n\n### 1\/4 cup chopped chives\n\nBring 6 quarts water to boil and add 2 tablespoons salt.  \nIn a 12-to 14-inch saute pan, mash together Gorgonzola and butter until smooth. Add grappa and simmer 4 minutes. Place Green Gnocchi in water and cook at low boil until they float.  \nDrain well and toss into pan. Toss over medium heat, add Asiago and chives, pour into warmed bowl and serve.\n\n22  \nAsiago Greats\n\n## GRILLED 4 CHEESE SANDWICHES\n\n4 servings\n\n### Source: Asiago Greats\n\n### 2 tablespoons extra-virgin olive oil\n\n### 3 tablespoons butter\n\n### 1 clove garlic, cracked away from\n\n### skin\n\n### 8 slices crusty Italian semolina bread\n\n### 1 cup shredded provolone\n\n### 1 cup shredded mozzarella\n\n### 1\/2 cup grated Parmesan or\n\n### Romano\n\n### 1 cup shredded Asiago\n\nIn a small skillet over medium-low heat, combine oil, butter and garlic. Cook garlic in butter and oil gently for 2 or 3 minutes. Remove garlic butter mixture from heat.  \nPlace a large nonstick skillet on the stove over medium-high heat. Using a pastry brush, brush 1 side of 4 slices of bread with garlic oil and place buttered-side down in skillet. Top each slice with equal amounts of the 4 cheeses, distributing them equally over the 4 slices. Top each sandwich with another slice of bread brushed with garlic butter, buttered-side up.  \nFlip the grill cheese sandwiches a few times until cheeses are melted and gooey and bread is toasty and golden. Cut grilled 4 cheese sandwiches from corner to corner and serve.  \nThis recipe yields 4 servings.  \nComments: All of these cheeses are available in specialty cheese case and dairy aisle already shredded\n\n23  \nChapter 1: A-M\n\n## GRILLED STUFFED PASILLA CHILE WITH AVOCADO\n\n6 servings\n\n### Source: Asiago Greats\n\n### 6 fresh pasilla, (poblano), chiles, Anaheim's or\n\n### 1\/2 cup grated fontina cheese\n\n### 1\/2 cup grated Monterey jack cheese\n\n### 1\/2 cup grated jarisberg cheese\n\n### 1 cup grated white cheddar cheese\n\n### 1\/2 cup grated asiago or parmesan,\n\n### cheese\n\n### 12 cilantro sprigs\n\n### ***AVOCADO SALSA***\n\n### 2 ripe avocados\n\n### 1\/2 jalapeno chile, roasted, seeded and, up to 1\n\n### 1\/2 small red onion, minced\n\n### 3 scallions, minced (white and 2 inches of green)\n\n### 3 tablespoons roughly chopped cilantro\n\n### 2 tablespoons rice vinegar\n\n### 6 tablespoons olive oil kosher or sea salt freshly ground pepper\n\nPrepare the chiles by cutting all the way around the stem so that you can remove the stem and seed base in 1 piece. Keeping the stem and top intact, trim off any seeds and membranes, and if any get left behind in the pepper, remove them, too. Blanch the chiles for 2 minutes in lightly salted boiling water. Remove the chiles, refresh in ice water, and drain.  \nCombine the grated cheeses in a bowl and mix well. Stuff each chile with about 1\/2 cup of the cheese mixture. Do not overstuff or pack the cheese in too densely. Replace the stem and top of the chile.  \nCook on a grill over a medium fire, turning frequently until the cheeses are melted and the chiles are hot all the way through.  \nAvocado Salsa: Cut each avocado in half. Remove the seed, make dice-sized crisscrosses in the avocado flesh with a knife, and scoop out the dice with a spoon. Place in a mixing bowl and add the chile,  \n24  \nAsiago Greats  \nonion, scallions and chopped cilantro. Whisk together the vinegar and oil in a separate bowl, add salt and pepper to taste, and pour over the avocado mixture. Mix gently, being careful not to mash the avocado  \nor the salsa will look like guacamole and taste \"muddy.\" Place on a bed of Avocado Salsa and garnish with the cilantro sprigs.\n\n## HEARTS OF ROMAINE, PALM, AND ARTICHOKE\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1 romaine lettuce heart, shredded\n\n### 1 cup picked flat-leaf parsley leaves\n\n### = (pick the greens of half a bundle from\n\n### produce department)\n\n### 1 can hearts of palm - (14 oz), drained\n\n### 1\/4 lb. prosciutto\n\n### 1 can quartered artichoke hearts in water, drained\n\n### 1\/4 lb. wedge pecorino Romano or Asiago\n\n### Balsamic vinegar, for drizzling\n\n### Extra-virgin olive oil, for drizzling\n\n### Coarse salt, to taste\n\n### Freshly-ground black pepper, to taste\n\nPlace romaine on a platter and toss with parsley tops.  \nWrap hearts of palm in prosciutto and cut into bite-size pieces on an angle. Arrange prosciutto wrapped palm sections and quartered artichoke hearts over romaine greens.  \nShred cheese with a vegetable peeler into short ribbons, working over the salad plate. Drizzle the salad with balsamic vinegar and oil;\n\nseason with salt and pepper. This recipe yields 4 servings.  \n25  \nChapter 1: A-M\n\n## HERB CHEESECAKE\n\n1 servings\n\n### Source: Asiago Greats\n\n### 3 packages cream cheese, softened (8 ounces each)\n\n### 2 cups sour cream, divided\n\n### 1 can cream of celery\n\n### or cream, (10-3\/4 ounces) of asparagus soup\n\n### 3 eggs\n\n### 1\/2 cup grated Romano or asiago\n\n### cheese\n\n### 2 cloves garlic, minced\n\n### 1 tablespoon cornstarch\n\n### 2 tablespoons finely chopped fresh\n\n### basil leaves or 2, crushed teaspoons dried basil leaves\n\n### 1 tablespoon finely chopped thyme\n\n### leaves or 1 teaspoon, crushed dried thyme leaves\n\n### 1 teaspoon finely chopped fresh\n\n### tarragon leaves or, crushed\n\n### 1\/4 teaspoon dried tarragon leaves\n\n### 1\/2 teaspoon cracked pepper sweet red pepper strips, lemon peel twists and assorted fresh herbs for garnish\n\n### crackers, melba toast or fresh cut-up vegetables\n\nPreheat oven to 350degF. Grease side and bottom of 9-inch Springform pan.  \nIn covered food processor * or large mixer bowl, combine cream cheese, 1 cup of the sour cream and the soup. Blend in food processor or beat with mixer at medium speed until smooth. Add eggs, Romano cheese, garlic, cornstarch, basil, thyme, tarragon and pepper. Blend or beat until smooth. Turn into prepared Springform pan; place on jelly-roll pan.  \nBake 1 hour or until light brown (top may crack). Turn off oven; let stand in oven 30 minutes more. Cool in pan on wire rack. Cover; refrigerate until serving time, at least 4 hours or overnight.  \n26  \nAsiago Greats  \nCarefully remove from pan. Spread remaining sour cream over cheesecake. Garnish with red pepper strips, lemon peel twists and fresh herbs, if desired. Serve with crackers.  \n*Quantity of mixture requires an 8-cup capacity food processor.\n\n## ITALIAN MINI MEAT LOAVES WITH MAC AND THREE CHEESES\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1 1\/3 lbs. ground sirloin\n\n### 1 large egg, beaten\n\n### 2\/3 cup Italian bread crumbs\n\n### 1\/4 cup grated Parmigiano\n\n### 1\/2 small green bell pepper, chopped\n\n### 1 small yellow onion, finely chopped\n\n### 4 cloves garlic, chopped\n\n### 3 tablespoons tomato paste\n\n### Salt, to taste\n\n### Freshly-ground black pepper, to taste\n\n### 2 tablespoons extra-virgin olive oil\n\n### 1 lb. corkscrew shaped pasta with lines, (cavatappi)\n\n### = (or small shell pasta)\n\n### === THREE CHEESE SAUCE\n\n### ===\n\n### 2 tablespoons butter\n\n### 2 tablespoon all-purpose flour\n\n### 2 cups milk\n\n### 2 cups shredded sharp Cheddar\n\n### 1\/2 cup shredded asiago cheese\n\n### 1\/2 cup grated Parmigiano\n\n### 1 teaspoon coarsely-ground black pepper\n\n### 1\/2 cup Italian bread crumbs\n\n### === ASSEMBLY ===\n\n### 2 tablespoons extra-virgin olive oil\n\n### 2 tablespoons chopped fresh thyme leaves\n\n### 2 tablespoons chopped rosemary leaves - (to 3)\n\n### 1\/4 cup chopped flat-leaf parsley\n\n27  \nChapter 1: A-M  \nPut a large pot of water on to boil for pasta. Preheat oven to 425 degrees.  \nPlace meat in a bowl. Make a well in meat. Fill well with egg, bread crumbs, 1\/4 cup cheese, peppers, onions, garlic, tomato paste, salt and pepper. Mix meat and breading and form 4 individual oval meat loaves, 1-inch thick. Coat loaves with extra-virgin olive oil and arrange on a baking sheet. Roast 18 to 20 minutes.  \nWhen the pasta water boils, salt water and add pasta. Cook until slightly undercooked -- pasta will continue to cook when combined with cheese sauce.  \nMelt butter in a saucepan and add flour. Whisk in milk. Bring to a bubble. Thicken milk, 2 to 3 minutes. Add cheeses: Cheddar, asiago,  \n1\/2 cup Parmesan and black pepper. Stir to melt cheeses.  \nDrain pasta and combine with melted cheese sauce and transfer to a baking dish or casserole. Place bread crumbs in a bowl and add  \nextra-virgin olive oil and herbs to bread crumbs. Top pasta with bread crumbs. Place under a broiler and brown bread crumbs. Serve cheese and pasta alongside meat loaf.  \nThis recipe yields 4 servings.\n\n## KIDS' FAST, FAKE-BAKED ZITI\n\n6 servings\n\n### Source: Asiago Greats\n\n### 3 tablespoons extra-virgin olive oil\n\n### 3 cloves garlic, finely chopped\n\n### 1 can whole peeled tomatoes - (28 oz)\n\n### = (San Marzano recommended)\n\n### 1 can crushed tomatoes - (14 oz)\n\n### = (San Marzano recommended, if San\n\n### Marzanos are not available, add a\n\n28  \nAsiago Greats\n\n### teaspoon sugar to your sauce, optional)\n\n### Coarse salt, to taste\n\n### A handful fresh basil leaves, torn\n\n### 1 lb. ziti rigate with ridges\n\n### 2 tablespoons butter\n\n### 2 tablespoons all-purpose flour\n\n### Freshly-ground black pepper, to taste\n\n### A generous grating nutmeg\n\n### 2 cups whole milk\n\n### 1\/2 cup shredded asiago cheese\n\n### 1\/2 cup shredded Parmigiano-Reggiano\n\n### 1 cup sliced fresh mozzarella\n\n### Crusty bread, to pass at table\n\nPut pasta water on to boil.  \nIn a medium sauce pot over medium heat, saute garlic in extra-virgin olive oil. Chop whole tomatoes and add them to the pan. Add crushed tomatoes and salt and simmer 10 minutes, add basil and simmer over low heat 10 minutes more.  \nAdd salt and pasta to boiling water and cook 6 minutes, leaving pasta a little chewy. While pasta cooks, melt butter in a small pot over medium heat. Whisk in flour, then cook 1 minute, adding salt and pepper and nutmeg to flour. Add milk and bring sauce to a bubble. Reduce 5 minutes.  \nDrain pasta and transfer to a large casserole dish. Pour the tomato and basil sauce over the pasta and turn to coat the pasta. Pour the bechamel over the already coated pasta. Cover the top of the pasta with Asiago, Parmesan and mozzarella. Place the casserole under hot broiler and melt the 3 cheeses until brown and bubbly, 3 to 5 minutes. Serve immediately with Spinach And Artichoke Salad and bread.  \nThis recipe yields 6 servings.\n\n29  \nChapter 1: A-M\n\n## LINGUINE FINE WITH FRENCH LENTILS, ESCAROLE AND ASIAGO\n\n4 servings\n\n### Source: Asiago Greats\n\n### 4 tablespoons extra-virgin olive oil\n\n### 1 large red onion, cut 1\/8\" julienne\n\n### 1 medium escarole head\n\n### -, (abt 3\/4 lb ), washed, spun dry\n\n### chopped 1\/2\" pieces to yield 3 cups\n\n### 1 cup cooked tiny green French lentils\n\n### salt, to taste\n\n### freshly-ground black pepper, to taste\n\n### 1 lb. linguine fine pasta\n\n### 1 cup freshly-grated asiago cheese\n\nBring 6 quarts Water to boil and add 2 tablespoons salt.  \nIn a l2-to 14-inch saute pan, saute onion in olive oil until translucent (about 7 to 8 minutes). Toss in escarole and cook over medium heat until wilted yet still bright green. Add lentils and toss to heat through. Season with salt and pepper and remove from heat.  \nDrop linguine into water and boil according to package instructions until al dente. Drain in colander over sink and pour hot pasta into pan with escarole. Toss, stirring well to mix, over medium-high heat  \nabout 1 minute. Add grated asiago and toss to mix. Pour into heated  \nserving dish and serve immediately.\n\n30  \nAsiago Greats\n\n## Chapter 2\n\n### N-Z\n\n## ORECCHIETTE PASTA W\/RED WINE BRAISED CHICKEN AND AROMATIC VEGETABLES\n\n6 servings\n\n### Source: Asiago Greats\n\n### 6 large chicken thighs, boned and skinned\n\n### salt and freshly ground black pepper, to taste\n\n### 2 tablespoons olive or canola oil\n\n### 1\/2 lb. fresh shiitake mushrooms,\n\n### stem'd and thick slice\n\n### 1 cup yellow onion, sliced\n\n### 3 tablespoons slivered garlic\n\n### 1\/2 cup diced carrots\n\n### 1\/2 cup thinly sliced celery\n\n### 2 cups hearty red wine\n\n### 2 cups ripe tomatoes, diced, seeded\n\n### 1 teaspoon chopped fresh thyme\n\n### 1 teaspoon chopped fresh sage\n\n### 4 cups chicken stock balsamic vinegar, (optional)\n\n### 1\/3 cup finely chopped parsley\n\n### 1\/2 lb. orecchiette pasta, uncooked\n\n### 1\/4 cup chopped fresh basil\n\n### 1\/4 cup drained sun dried tomatoes\n\n### in oil, finely -garnish--- fresh basil sprigs\n\n### freshly shaved asiago or parmesan cheese\n\nSeason chicken liberally with salt and pepper. In a large, heavy-bottomed saucepan, heat 2 tablespoons oil and quickly brown  \n31  \nChapter 2: N-Z  \nchicken over high heat.  \nRemove and set aside. Add mushrooms, onion, garlic, carrots, and celery and saute until very lightly browned. Return chicken to pan and add wine, tomatoes, thyme, sage, and the stock and bring to a simmer. Cover and simmer until chicken is tender, 25 to 30 minutes.  \nRemove chicken and cut meat into bite-sized slices and set aside. Strain braising liquid reserving vegetables and return liquid to pot.  \nBring to a boil and cook over high heat for 8 to 10 minutes to reduce  \nand thicken lightly. This also concentrates the flavors.  \nTaste and adjust seasoning with salt, pepper and drops of balsamic vinegar if using. Add reserved meat and vegetables and heat through.  \nStir in parsley and keep warm. Meanwhile, bring a large pot of water to a boil, add salt to taste and pasta. Cook pasta until just tender, drain and toss with chopped basil and sun dried tomatoes.  \nPlace pasta in center of warm, large flat pasta bowls and ladle braised chicken sauce around. Garnish with basil springs and shaved cheese and serve immediately. Yield: 6 servings\n\n## OYSTERS RANDALL\n\n12 servings\n\n### Source: Asiago Greats\n\n### 10 ounces spinach, frozen, chopped\n\n### 1\/2 cup white wine\n\n### 2 ounces bacon, chopped\n\n### 1\/4 cup onion, chopped fine\n\n### 1 garlic clove, minced\n\n### 12 medium oysters, *see note\n\n### 1\/2 cup heavy cream\n\n### 1\/2 teaspoon thyme\n\n### 1 ounce hazelnut liqueur\n\n### 4 ounces Swiss cheese, grated\n\n### 2 ounces asiago cheese, grated\n\n32  \nAsiago Greats\n\n### white pepper, to taste\n\n### 1 egg, beaten slightly\n\n### 1 tablespoon water\n\n### 1\/2 package puff pastry\n\nIn a medium saucepan, add frozen spinach and wine. Cook covered over low heat until done. Cool and drain by squeezing spinach with hands or cheesecloth, reserving liquid.  \nIn a medium saute pan, cook bacon until almost done, then add onions and garlic, continue cooking until onions are golden. Remove bacon mixture from pan and add cooking liquid from spinach. Open oysters (be sure to keep the 1\/2 shell you will not be using near to  \n1\/2 shell containing oyster) draining liquid into pan also.  \nAdd thyme and boil over medium heat until reduced to about 1\/4 cup. Add cream and hazelnut liqueur; continue cooking until mixture is thickened and reduced a bit. Season to taste with white pepper. In a medium mixing bowl, combine spinach with bacon mixture and cream sauce. Stir until thoroughly combined. Top each oyster with 1-2 tablespoons of spinach mixture.  \nSprinkle with Swiss cheese and half as much asiago cheese. Roll puff pastry accordingly. Using the 1\/2 shell as a template, set on top of pastry and cut out, then lift the piece of pastry and put on top of corresponding oyster. Continue doing this until all oysters are topped with pastry, discard unneeded 1\/2 shells.  \nBeat egg with 1 tablespoon of water and with a pastry brush, brush tops of each oyster. Place oysters on a baking sheet and bake in a  \n400degF oven for 30-40 minutes or until pastry is a dark golden brown. Serve immediately.\n\n## PARMESAN STRAWS\n\n5 servings\n\n### Source: Asiago Greats\n\n33  \nChapter 2: N-Z\n\n### 1 lg Egg white\n\n### 2 tablespoons Olive oil\n\n### 1\/4 teaspoon Salt\n\n### 1\/8 teaspoon Cayenne (opt)\n\n### 6 Sheets Phyllo dough, 14 x\n\n### 18\"\n\n### 3\/4 cups Freshly grated\n\n### Parmesan\n\n### or Asiago cheese\n\nPreheat oven to 400degF. Lightly coat 2 baking sheets with nonstick cooking spray or line with parchment paper. In a small bowl, whisk together egg white, oil, salt and cayenne. Lay a sheet of phyllo on a work surface with a short side toward you.  \nKeep remaining phyllo sheets covered with plastic wrap or wax paper. With a pastry brush, lightly coat the lower half of the sheet with the egg-white mixture and sprinkle with about 2 t cheese. Fold the upper half over to cover the lower half. Brush the right half of the folded sheet with egg-white mixture, sprinkle with a heaping teaspoon of cheese and fold the left half over the cheese.  \nBrush the bottom half of the folded sheet with the egg-white mixture, sprinkle with 1 t cheese and fold the upper half over the lower half. Finally, brush the top with the egg-white mixture and sprinkle with 1  \nt cheese. Cut into ten 1\/2\" strips using a knife or serrated pastry cutter. With a wide spatula, transfer the strips to the prepared baking sheet, placing them about 1\/2\" apart. Repeat the procedure with the remaining 5 sheets of phyllo, egg-white mixture and cheese.  \nBake the straws for 8 to 10 minutes, or until golden and crisp. Transfer to a rack to cool. (The straws be stored in an airtight container at room temp for 1 week.)  \nPer straw: 19 calories, 1 g protein, 1 g fat, g carbohydrate, 38 mg sodium, 1 mg cholesterol.\n\n34  \nAsiago Greats\n\n## PASTA FONTINA\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1\/4 lb. asiago cheese, grated\n\n### 1 cup unsalted butter\n\n### 1\/4 cup minced fresh parsley\n\n### 1\/2 lb. fontina cheese, grated\n\n### 1 lb. fresh linguine\n\nWhile you are cooking the pasta, melt butter in a large skillet and cook, stirring, until it just turns brown. Meanwhile, toss the cheeses together (if you can't find Asiago, use Romano or Parmesan, not quite the same but still good).  \nDrain pasta, place in a large warmed pasta bowl. Sprinkle with cheeses, drizzle with browned butter and toss well. Serve at once with parsley sprinkled over the top.\n\n## PRAWNS PROVENCAL\n\n6 servings\n\n### Source: Asiago Greats\n\n### 2 lbs. Jumbo Prawns\n\n### 2 cups Croutons, coarsely ground\n\n### 6 Lemons\n\n### 1 cup Asiago Cheese, grated\n\n### 1\/2 cup Parsley, chopped\n\n### 2 teaspoons Garlic, finely chopped\n\n### 1 ds Salt\n\n### 1 ds Pepper\n\n### 6 ounces Olive Oil, enough to make a paste\n\nPreheat the oven to 375degF. Remove the zest from the lemons and chop, you should have about 1\/4 cup chopped zest. Reserve the lemons. Peel, clean and butterfly the prawns. Mix the croutons, zest, Asiago, parsley, garlic, sat and pepper together, then drizzle enough of the oz of olive oil to make a paste.  \n35  \nChapter 2: N-Z  \nSmear the butterflied opening of the prawns with the mixture. Bake for 8-10 minutes or until the tails stand tall. When they come out of the oven squeeze with the zested lemons. Serve hot or at room temperature.  \nPer Serving: 359 calories, 32 g protein, 11 g carbohydrate, 21 g fat, g saturated fat, 204 mg cholesterol, 572 mg sodium, no fiber.\n\n## QUAIL SPIEDINI WITH SAGE POLENTA AND ASIAGO\n\n1 servings\n\n### Source: Asiago Greats\n\n### 4 skewers, soaked in water 8 hours\n\n### 8 quail, boneless\n\n### 4 pieces pancetta, into 1\" cubes\n\n### 2 ounces balsamic vinegar\n\n### 2 ounces virgin olive oil\n\n### 2 tablespoons honey\n\n### 1 tablespoon black pepper\n\n### 1 medium red onion, in 1\/4\" dice\n\n### 4 cups water\n\n### 10 sage leaves, chopped\n\n### 1 cup polenta\n\n### 1\/2 cup asiago, freshly grated\n\nCheck quail for bones or feathers and place in mixing bowl. Add pancetta, vinegar, olive oil, honey and black pepper and toss to coat. Set aside and preheat grill.  \nIn a 3-quart saucepan, place onion, water and sage and bring to a boil. Thread one quail, followed by one piece of pancetta, followed by one quail on each of four skewers and place on hottest part of grill. Cook 4 to 5 minutes on each side, until just pink at the leg bones.  \nMeanwhile, pour polenta slowly in a thin stream into boiling water, until all is incorporated and polenta thickens, about 1 to 2 minutes. Switch to a wooden spoon, add Asiago and cook another minute, until  \n36  \nAsiago Greats  \nas thick as paste. Remove from heat and pour on to a service-ready cutting board. Pile skewers on top of polenta and serve.\n\n## QUICK CREAMY TOMATO SOUP\n\n4 servings\n\n### Source: Asiago Greats\n\n### 2 containers chicken or vegetable stock\/brot, (15 oz ea)\n\n### = (preferred brand: Kitchen\n\n### Basics)\n\n### 1 can concentrated crushed tomatoes - (28 oz)\n\n### 1 cup heavy cream\n\n### Coarse salt, to taste\n\n### Freshly-ground black pepper, to taste\n\n### 20 fresh basil leaves, cut into chiffonade\n\n### === SOUP TOPPERS ===\n\n### 4 slices crusty bread, 4\" across by 1\/4\" thick\n\n### = (artisan breads at counter in market)\n\n### 2 fresh garlic cloves, cracked optional\n\n### Extra-virgin olive oil\n\n### Freshly-ground black pepper, to taste\n\n### 1\/2 lb. shredded cheese, to compliment soup\n\n### = (provolone, Asiago, cheddar, Gruyere,\n\n### Swiss, etc.)\n\nCombine broth and tomatoes in a medium saucepan over moderate heat. When soup bubbles, stir in heavy cream and reduce heat to low. Season with a little salt and pepper and simmer gently 15 minutes, stirring occasionally. With an immersion blender, puree soup. Serve bowls of soup with basil chiffonade and floating soup toppers.  \nSoup Toppers: Toast bread slices in your toaster until golden. Rub the toasted bread rounds with crushed garlic, if using. Arrange bread on cookie or baking sheet. Drizzle each slice with extra-virgin oil and  \ntop with a grind of black pepper and a layer of shredded cheese. Place  \n37  \nChapter 2: N-Z  \nunder hot broiler to melt cheese until toppers are golden brown in color.\n\n## RABBIT WITH ASIAGO GRITS\n\n12 servings\n\n### Source: Asiago Greats\n\n### 1\/2 lb. fresh portabella mushrooms, (2 large)\n\n### 1\/4 lb. fresh shiitake mushrooms\n\n### 1\/4 lb. fresh morel or chanterelle mushrooms\n\n### 7 teaspoons olive oil, divided\n\n### 3 young rabbits, (2-1\/\n\n### 4-pound)\n\n### (portioned into loins and legs by your butcher)\n\n### 6 medium carrots each cut into 4 pieces\n\n### 4 small onions, quartered\n\n### 4 stalks celery\n\n### each cut into 4 pieces\n\n### 2 medium leeks\n\n### each trimmed and quartered\n\n### 4 bay leaves\n\n### 1 bunch fresh thyme, (1\/2 ounce)\n\n### 4 cups low-salt chicken broth\n\n### 1\/3 cup chopped shallot\n\n### 2 teaspoons chopped fresh thyme\n\n### Asiago Soft Grits Parsley leaves Fresh chives\n\nUsing a sharp knife, remove brown gills from the undersides of portabella mushrooms; discard gills. Remove and reserve stems from portabella, shiitake, and more mushrooms. Slice mushroom caps; set aside.  \nHeat 1 teaspoon olive oil in a large nonstick skillet over medium heat. Add half of rabbit legs; saute 8 minutes or until browned. Place in a large 8-quart Dutch oven or stockpot; set aside. Repeat procedure  \n38  \nAsiago Greats  \nwith 2 teaspoons olive oil and remaining rabbit legs.  \nHeat 1 teaspoon olive oil in skillet over medium-high heat. Add carrot pieces and next 5 ingredients; saute 12 minutes or until vegetables are lightly browned. Add carrot mixture to Dutch oven. Stir in  \nreserved mushroom stems and chicken broth, and bring to a boil. Cover, reduce heat, and simmer 1 hour. Remove rabbit legs; set aside, and keep warm. Strain stock through a colander into a large bowl, and set stock aside. Discard solids.  \nHeat 1 teaspoon olive oil in skillet over medium heat. Add rabbit loins, and cook 2-1\/2 minutes on each side or until browned. Cover and cook for 10 minutes or until done, turning loins after 5 minutes. Remove from skillet, and cut loins in half crosswise. Set aside, and keep warm.  \nHeat remaining 2 teaspoons olive oil in skillet over medium heat. Add shallot and 2 teaspoons thyme; saute 1 minute. Add sliced mushroom caps; saute 5 minutes. Add rabbit stock; bring to a boil, and cook, uncovered, 7 minutes or until sauce is reduced to 6 cups.  \nSpoon 1\/2 cup Asiago Soft Grits into each of 12 large shallow bowls. Divide meat among bowls, and top each with 1\/2 cup sauce. Yield: 12 servings (serving size: about 4 ounces rabbit, 1\/2 cup grits, and 1\/2 cup sauce).\n\n## RICOTTA-STUFFED ZUCCHINI FLOWERS\n\n10 servings\n\n### Source: Asiago Greats\n\n### 1 lb. Ricotta cheese\n\n### 1 Onion, minced\n\n### 1\/2 cup Toasted almonds or pine nuts (finely chopped)\n\n### 1\/2 cup Grated Italian Asiago cheese OR-Parmesan cheese\n\n### 1\/2 teaspoon Ground pepper\n\n### 1 teaspoon Seasoning salt\n\n39  \nChapter 2: N-Z\n\n### 2 tablespoons Minced fresh basil, -=OR=-\n\n### 1 teaspoon -Dried basil\n\n### 2 tablespoons Minced parsley\n\n### 1 teaspoon Butter, melted\n\n### 20 md Zucchini flowers, or more OR-any squash\n\n### flowers, Freshly picked and rinsed in cold water\n\n### Nasturtiums for garnish\n\nMix together all ingredients except butter and flowers. With filling at room temperature, use a pastry tube to carefully stuff flowers; do not overfill.  \nDrizzle melted butter over flowers and cook in microwave on medium power for 3 minutes, or at 350degF in regular oven for about 15\n\nminutes. Be careful not to let filling ooze out of flowers. Garnish with nasturtiums stuffed with extra filling.\n\n## RIGATONI WITH VODKA TOMATO SAUCE\n\n2 servings\n\n### Source: Asiago Greats\n\n### 2 tablespoons olive oil\n\n### 1 cup finely chopped shallots\n\n### 1\/4 teaspoon dried crushed red pepper\n\n### 1\/2 cup vodka\n\n### 3\/4 cups whipping cream\n\n### 3\/4 cups tomato sauce\n\n### 8 ounces rigatoni pasta\n\n### 4 ounces thinly sliced prosciutto,\n\n### chopped\n\n### 1\/3 cup grated asiago cheese, (about, 2 oz)\n\n### 2 tablespoons chopped fresh parsley\n\n### 2 tablespoons chopped fresh basil or\n\n### 2 t, dried\n\n40  \nAsiago Greats  \nHeat oil in heavy large skillet over medium heat. Add shallots and crushed red pepper; saute until shallots are translucent, about 5 minutes. Add vodka and ignite with long match. Simmer until flames subside, shaking pan occasionally, about 2 minutes. Increase heat to high, add cream until mixture thickens, about 3 minutes.  \nAdd tomato sauce; boil until sauce thickens and coats back of spoon, about 2 minutes. (Can be made 1 day ahead; chill.) Cook pasta in pot of boiling salted water until tender but still firm to bite. Drain; reserve  \n1\/4 C cooking liquid. Bring sauce to simmer. Add pasta, prosciutto,  \n1\/3 C cheese, parsley and basil to skillet and toss to coat. Add reserved pasta cooking liquid if mixture is too dry. Season to taste with pepper. Transfer to large bowl. Sprinkle with remaining 1\/3 C cheese and serve.\n\n## ROAST GARLIC MASHED POTATOES\n\n14 servings\n\n### Source: Asiago Greats\n\n### 8 to 10 cloves garlic, peeled\n\n### 1 cup olive oil\n\n### 4 russet potatoes\n\n### 2 tablespoons butter\n\n### 1\/3 cup heavy cream\n\n### 1\/4 cup Asiago cheese, grated 2 tbsp.\n\n### Parmigiano-Reggiano cheese, grated\n\n### Salt and pepper, to taste\n\nPut the garlic and olive oil in a heavy saucepan over lowest possible heat and simmer until soft; 30 to 40 minutes.  \nDrain off oil (reserve for marinades or vinaigrettes). Puree garlic; set aside.  \nMeanwhile, prick potatoes with a fork and bake in a 400degF oven for 1  \nhour, or until soft. While still hot, peel and mash, or pass through a  \n41  \nChapter 2: N-Z  \npotato ricer.  \nMelt butter in heavy cream; whisk in pureed garlic. Stir into potatoes.  \nStir in cheeses and season with salt and pepper. Spoon into a gratin dish.  \nPlace in a 400degF oven for 12 to 15 minutes or until browned and bubbling.\n\n## ROASTED GARLIC LINGUINI A LA CAFE TOSCA\n\n2 servings\n\n### Source: Asiago Greats\n\n### 10 ounces linguini, a quality one\n\n### oil\n\n### 20 whole garlic cloves, peeled and trimmed\n\n### 1\/2 cup chopped parsley\n\n### 2 pn kosher salt\n\n### 2 pn fresh ground black pepper\n\n### 2 ounces white wine\n\n### 1 tablespoon fresh lemon juice\n\n### 2 ounces sweet butter asiago or any other good dry cheese, (s)\n\nCook, drain, rinse and chill pasta. Oil when cool. Refrigerate and  \ncover what you don't use. You can bag and use for several more days.  \nOil garlic. Cover and oven roast at 425 until tender, about 30 minutes. Try 10 minutes in the microwave. Cool and reserve.  \nHeat medium fry pan. Brown garlic. Add parsley, salt and pepper.  \n42  \nAsiago Greats  \nSaute 30 seconds. Add wine and lemon juice. Bring back to heat.  \nDip al dente pasta in hot water for 1 minute. Drain and put in pasta bowl.  \nAdd butter to garlic pan. Add a touch more wine if it has cooked  \naway. Swirl butter et al around quickly. Add to cooked pasta. Toss and top with cheese(s). Best served with toasted Italian bread.\n\n## ROASTED POTATO AND LEEK FRITTATA\n\n4 servings\n\n### Source: Asiago Greats\n\n### 5 eggs\n\n### 1\/4 teaspoon salt\n\n### Freshly-ground black pepper, to taste\n\n### 1 tablespoon olive oil\n\n### 1 leek, white part and 1\" green, washed well,\n\n### and coarsely chopped - (abt 1 cup)\n\n### 2 ounces prosciutto, 2 thin slices\n\n### 3\/4 lbs. boiling potatoes - (to 1 lb), roasted\n\n### 1 teaspoon minced fresh rosemary\n\n### 1\/3 cup grated Asiago cheese\n\n### === GARNISH ===\n\n### 1 small tomato, (optional), diced\n\n### Chopped Italian parsley, (optional)\n\nLightly beat eggs with salt and pepper to taste. Slice prosciutto crosswise into 1\/4-inch strips. Cut potatoes into 1\/4-inch dice to measure 1 1\/2 cups.  \nHeat oil in 10-inch nonstick, ovenproof skillet over medium heat. Add leek and saute until just starting to soften, about 2 minutes. Add prosciutto and saute until leek is soft, 1 to 2 minutes. Remove half of leek mixture and set aside.  \n43  \nChapter 2: N-Z  \nAdd potatoes and rosemary to skillet and toss to heat potatoes, 1 minute. Reduce heat to medium-low and add eggs. Do not stir. Cook just until bottom of eggs is set, 3 to 4 minutes. Top should still be wet.  \nSprinkle with cheese and reserved leek mixture. Broil 4 inches from heat source until frittata is golden and sizzling, about 2 minutes. Invert frittata onto cutting board, sprinkle with tomato and parsley, and cut into wedges.  \nThis recipe yields 4 servings.  \nEach serving: 265 calories; 586 mg sodium; 279 mg cholesterol; 13 grams fat; 6 grams carbohydrates; 15 grams protein; 0.62 gram fiber.  \nComments: New potatoes work best for this recipe. If your nonstick skillet is not ovenproof, wrap the handle with several layers of foil before placing it in the oven.\n\n## ROMAINE AND FENNEL SALAD\n\n4 servings\n\n### Source: Asiago Greats\n\n### 2 romaine hearts, torn or rough cut\n\n### 1 large fennel bulb, trimmed of fronds and tops\n\n### 1\/4 red onion, peeled, and thinly sliced\n\n### Several sprigs flat-leaf parsley\n\n### Balsamic vinegar, as needed\n\n### Extra-virgin olive oil, as needed\n\n### Coarse salt, to taste\n\n### Freshly-ground black pepper, to taste\n\n### Asiago or Parmigiano\n\n### Reggiano, shaved for garnish\n\n44  \nAsiago Greats  \nPlace chopped or torn romaine in a large salad bowl.  \nCut fennel bulb down the center, then again, quartering it. Remove the core with angled cuts on each quarter bulb. Thinly slice quarter bulbs across and add to salad bowl. Combine romaine and fennel with parsley sprigs and red onion.  \nPlace one fingertip over the top of balsamic vinegar to control the pour and splash a few tablespoons into the bowl. Drizzle the salad generously with extra-virgin olive oil and toss lightly to coat salad evenly with vinegar and oil.  \nSeason with salt and pepper, to taste, and top salads with curls of\n\nAsiago or Parmigiano, created by shaving cheese with your knife. This recipe yields 4 servings.\n\n## SAUTED APPLES IN AMARONE WITH ASIAGO\n\n4 servings\n\n### Source: Asiago Greats\n\n### 4 tablespoons cold unsalted butter, divided\n\n### 4 medium apples, (granny smith, or macintosh), peeled, cored,\n\n### and quartered\n\n### 4 tablespoons sugar\n\n### 1 pn freshly-grated nutmeg\n\n### 6 ounces amarone wine or other\n\n### full bodied wine from veneto\n\n### 1\/2 lb. fresh asiago cheese, the youngest possible\n\nHeat a large 12-to 14-inch saute pan over medium-high heat and add 2 tablespoons of the butter. The butter should sputter and begin to brown immediately. When the butter stops sputtering, but before it turns very dark brown, add the apples. Continue to cook 3 to 4  \n45  \nChapter 2: N-Z  \nminutes until softened yet still retaining their shape. Add sugar, nutmeg and wine and cook 1 minute. Add 2 tablespoons cold butter and shake the pan to completely blend the butter into the wine mixture, forming a creamy emulsion.  \nSpoon the apples into a shallow serving dish. Shave the Asiago cheese into paper thin shards over the apple mixture using a peeler or a shovel nose cheese knife. Serve warm.  \nThis recipe yields 4 servings as dessert or cheese course.\n\n## SMOKY MUSHROOM TART\n\n8 servings\n\n### Source: Asiago Greats\n\n### 1\/3 butter pastry dough\n\n### 1 egg white, lightly beaten\n\n### 2 tablespoons butter\n\n### 10 ounces fresh mushrooms, (1 package), sliced\n\n### 7 ounces shitake mushrooms, (1, package), stems disc\n\n### and mushrooms sliced\n\n### 1 tablespoon minced fresh garlic\n\n### 2 teaspoons dried oregano, crushed\n\n### 1\/8 teaspoon ground black pepper\n\n### 1\/2 lb. smoked mozzarella cheese,\n\n### thinly sliced\n\n### 2 tablespoons grates asiago or parmesan,\n\n### cheese\n\n### 1\/3 cup walnut pieces\n\n### 1 tablespoon chopped flat-leaf, (Italian), parsley\n\nPreheat oven to 400degF. On lightly floured surface roll dough to a  \n14-inch round.  \nTransfer to an 11-inch tart pan with removable bottom. Trim edges;  \n46  \nAsiago Greats  \nprick bottom with tines of fork. Line pastry shell with foil and and pastry weights, dried beans, or raw rice. Bake 15 minutes. Remove foil and weights.  \nBake 5 to 6 minutes longer or just until pastry starts to turn golden. Brush with egg white; bake 1 minute longer. Cool completely on wire rack. In large skillet, melt butter over medium-low heat.  \nAdd mushrooms, garlic, oregano, and pepper. Saute until mushrooms are golden and liquid has evaporated, about 8 minutes; cool to room temperature. Cover bottom of tart shell with mozzarella, cutting slices to fill in spaces.  \nTop with mushroom mixture then sprinkle with asiago and walnuts. Bake 20 minutes. Cool 5 minutes on wire rack before removing outer ring.  \nServe warm. Makes 8 servings.\n\n## SPINACH-RICOTTA GNOCCHI\n\n6 servings\n\n### Source: Asiago Greats\n\n### 2 large baking potatoes\n\n### 6 ounces fresh spinach\n\n### 1 cup all-purpose flour, divided\n\n### 1\/4 cup part-skim ricotta cheese\n\n### 1\/2 teaspoon salt\n\n### 1\/8 teaspoon pepper\n\n### 1\/8 teaspoon ground nutmeg\n\n### 1 egg\n\n### Vegetable cooking spray\n\n### 2 tablespoons butter\n\n### 1 cup seeded and diced peeled red tomato\n\n### 1 cup seeded and diced peeled yellow tomato\n\n### 1 clove garlic, minced\n\n### 1 tablespoon red wine vinegar\n\n47  \nChapter 2: N-Z\n\n### 3\/4 cup grated Asiago cheese, (3 ounces)\n\n### 1 tablespoon chopped fresh chives\n\nWrap potatoes in foil; bake at 400 degrees for 1 hour. Let cool; peel and mash.  \nRemove stems from spinach, and wash leaves. Place a large Dutch oven over medium heat, and add spinach; cover and cook 2 minutes  \nor until spinach wilts (do not add water). Press spinach between paper towels; chop.  \nCombine mashed potato, chopped spinach, 1\/2 cup flour, and next 5 ingredients; stir to form a soft dough. Turn dough out onto a well-floured surface; knead in remaining flour. Divide dough into 4 portions, shaping each into a 12-inch rope. Cut each rope into 12  \n(1-inch) pieces; roll each piece into a ball. Drag tines of a fork through half of each ball, forming a concave gnocchi shape. Place on a baking sheet coated with cooking spray; set aside.  \nBring 14 cups of water to a boil in a Dutch oven. Add half of the gnocchi, and cook 1-1\/2 minutes. Remove gnocchi with a slotted spoon; place in a colander. Repeat with remaining gnocchi.  \nMelt butter in a skillet over medium heat, and cook until nutty-brown color. Add tomatoes and garlic; saute 2 minutes. Remove from heat; stir in vinegar.  \nPlace gnocchi in a 2-quart baking dish coated with cooking spray. Spoon tomato mixture over gnocchi; sprinkle with cheese. Bake at  \n350 degrees for 30 minutes. Sprinkle with chives. Yield: 6 servings  \n(serving size: 8 gnocchi).\n\n48  \nAsiago Greats\n\n## STOVETOP SPINACH AND ARTICHOKE DIP\n\n8 servings\n\n### Source: Asiago Greats\n\n### 2 tablespoons extra virgin olive oil\n\n### 1 tablespoon butter\n\n### 3 cloves garlic, chopped\n\n### 1\/2 onion, chopped\n\n### 2 tablespoons chopped fresh thyme leaves\n\n### = (or 2 tspns dried thyme)\n\n### 1\/2 small red bell pepper, seeded, chopped\n\n### 2 tablespoons all-purpose flour\n\n### 1\/2 cup dry white wine\n\n### 1 cup chicken or vegetable stock\n\n### 1\/2 cup half-and-half or heavy cream\n\n### 1 can artichoke hearts in water\n\n### \\- (15 oz), drained, and coarsely chopped\n\n### 2 boxes chopped spinach - (10 oz ea), defrosted, and\n\n### squeezed dry in kitchen towel\n\n### 1 1\/2 cups shredded Italian 4 cheese blend\n\n### = (provolone, Parmesan, mozzarella and\n\n### Asiago)\n\n### Salt, to taste\n\n### Freshly-ground black pepper, to taste\n\n### 1 round loaf crusty bread, top removed and\n\n### cubed, bottom hollowed out to use as\n\n### serving bowl\n\n### 1 loaf multigrain or whole wheat baguette, sliced at\n\n### bakery\/bread counter in market\n\nTo a medium saucepot preheated over moderate heat, add about 2 tablespoons extra-virgin olive oil, 2 turns of the pan in a slow stream. Add butter to oil. When butter melts, add garlic and onions to the pot. Sprinkle in thyme leaves. Saute 2 minutes, add chopped red pepper. Saute mixture a minute more.  \nSprinkle in flour, stir to coat vegetables. Cook flour 1 minute. Whisk in wine and reduce by half. Whisk in stock and thicken sauce 1 minute. Stir in half-and-half or cream. When sauce returns to a bubble, add artichokes, spinach and cheeses. Keep stirring until  \n49  \nChapter 2: N-Z  \ncheeses melt and sauce is well combined. Add salt and pepper and adjust seasonings, to your taste.  \nPour dip into hollow bread bowl on serving platter and surround with sliced multi or whole grain bread for dipping. Reserve extra dip to warm and refill as necessary. Cut up bread bowl when baguette slices are gone -- the bowl is the best part as it absorbs juices from the dip.  \nThis recipe yields 8 servings.\n\n## STUFFED CHERRY PEPPERS\n\n4 servings\n\n### Source: Asiago Greats\n\n### === HOT STUFFED CHERRY PEPPERS ===\n\n### 8 cubes pepperoncino - (1\/2\"), see * Note\n\n### 4 slices cappicola (hot Italian ham), cut in half\n\n### 8 hot cherry peppers, tops cut off,\n\n### and seeds removed\n\n### === SWEET STUFFED CHERRY PEPPERS ===\n\n### 2 slices Prosciutto di Parma\n\n### 3 ounces herb and garlic spreadable cheese\n\n### = (or goat cheese)\n\n### 1 celery rib, cut 1\/2\" pieces\n\n### 8 sweet peppers, tops removed,\n\n### and seeded\n\n* Note: Crushed red pepper Italian table cheese, available in specialty cheese case. Asiago or pepato may be substituted.  \nFor hot peppers: Wrap cheese cubes in half of one slice of hot ham. Stuff ham wrapped cheese cubes inside the cherry peppers and reset caps with a toothpick.  \nFor sweet cherry peppers: Spread each slice of prosciutto with a thin  \n50  \nAsiago Greats  \nlayer of herb and garlic cheese. Cut each slice of ham in four pieces, then wrap a piece of celery inside each strip of ham. Stuff the meat and cheese wrapped celery into each cherry pepper.  \nThis recipe yields 16 stuffed peppers, 4 servings.\n\n## TAGLIATELLE WITH PROSCIUTTO AND RADICCHIO\n\n6 servings\n\n### Source: Asiago Greats\n\n### 1 lb. fettucine, or tagliatelle\n\n### 6 1\/2 ounces prosciutto, or turkey bacon\n\n### 1 tablespoon canola oil, butter flavored\n\n### 1 tablespoon virgin olive oil\n\n### 6 tablespoons basil leaves, shredded\n\n### 1 large radicchio, head, shredded\n\n### salt and pepper, to taste\n\n### 3 tablespoons asiago cheese, shavings\n\nPASTA POT - cook pasta until al dente. Drain well. Toss with butter flavored canola. LARGE SKILLET or WOK - Fry prosciutto or bacon until crisp; add a little olive oil, as needed, to prevent sticking.  \nOption #1: Combine the herb(s), radicchio, cheese, pepper and meat in a serving bowl and toss. Add hot \"buttered\" pasta and toss. Distribute servings to heated bowls. Top with cheese.  \nOption #2: Push meat aside in the wok or skillet and stir fry radicchio and herbs for about 3 minutes, until just wilted and the \"cranberry\" color darkens (\"maroon\"). Add the pasta to the wok. Toss. Salt and pepper to taste. Transfer to a heated serving bowl. Top with cheese: some shavings, some grated.\n\n51  \nChapter 2: N-Z\n\n## TAGLIATELLE WITH SAUSAGE AND RADICCHIO\n\n4 servings\n\n### Source: Asiago Greats\n\n### 1 recipe basic fresh egg pasta dough\n\n### 2 tablespoons salt\n\n### 1 lb. sweet fennel sausage, removed from casing\n\n### and broken into pea-size pieces\n\n### 1\/2 medium red onion, thinly sliced\n\n### 1 cup dry white wine\n\n### 4 ounces sweet butter\n\n### 2 radicchio\n\n### heads, core removed, cut into 1\/2\" ribbons\n\n### 4 ounces asiago cheese, grated\n\n### (very young cheese if possible)\n\n### 8 drops balsamic vinegar\n\nBring 6 quarts water to boil and add 2 tablespoons salt.  \nIn a 12-to 14-inch skillet heat the crumbled sausage over medium heat to render the fat. Cook the meat 8 to 10 minutes. Drain fat from pan and return to heat. Add onion and saute with sausage for 1 minute. Add the white wine and butter. Bring to boil. Remove from heat and let stand.  \nRoll out the pasta dough, cut into ribbons on largest setting (about  \n3\/8-inch). Drop pasta into boiling water and cook 1 to 2 minutes until al dente. Drain in colander over sink.  \nPour hot pasta into pan with sausage mixture. Add radicchio and cook over medium heat 1 minute, just to wilt the radicchio. Do not  \novercook the radicchio or it will darken in color. Add grated cheese, stir to mix and divide evenly onto 4 warm dinner plates. Put 2 drops  \nof balsamic vinegar on each serving. Serve immediately.\n\n52  \nAsiago Greats\n\n## THREE-HERB FETTUCCINE WITH VEGETABLES AND SHAVED ASIAGO\n\n6 servings\n\n### Source: Asiago Greats\n\n### 2 cups sugar snap peas\n\n### 1 tablespoon olive oil\n\n### 4 cloves garlic, minced\n\n### 2 1\/2 cups chopped spinach, divided\n\n### 1\/2 cup sliced green onions, (1\/\n\n### 2-inch)\n\n### 2 medium zucchini\n\n### halved lengthwise and thinly sliced, (3 cups)\n\n### 1 cup canned vegetable broth\n\n### 2 medium green tomatoes peeled, seeded, and cut into\n\n### thin wedges\n\n### 1\/2 cup thinly sliced fresh basil\n\n### 2 tablespoons chopped fresh chives\n\n### 2 tablespoons chopped fresh oregano\n\n### 1\/4 teaspoon salt\n\n### 1\/4 teaspoon freshly ground pepper\n\n### 6 cups hot cooked fresh fettuccine\n\n### (1 pound uncooked pasta)\n\n### 3 ounces shaved Asiago cheese\n\n### Fresh herbs, (optional)\n\nTrim ends from peas, and remove strings.  \nDrop peas into a Dutch oven of boiling water; cook 30 seconds. Drain and rinse under cold water.  \nHeat oil in pan over medium heat. Add garlic; saute 1 minute. Add peas, 1-1\/4 cups spinach, green onions, and zucchini; saute 2 minutes. Add broth and tomatoes; cook 2 minutes, stirring occasionally.  \nAdd remaining spinach, basil, chives, oregano, salt, and pepper; cook  \n1 minute, stirring occasionally. Yield: 6 servings (serving size: 1 cup pasta, 1 cup vegetable mixture, and 1\/2 ounce cheese).\n\n53  \nChapter 2: N-Z\n\n  Serving Ideas : Garnish with fresh herbs, if desired.NOTES : From Chef Charles Palmer of Aureole. Over the last six years, Aureole has become a New York institution with a worldwide reputation for its progressive American cuisine. Before opening Aureole, Charles Palmer was executive chef at the River Cafe in Brooklyn, where he earned a reputation as one of America's most creative culinary talents. Set in a townhouse overlooking a garden, Aureole was rated number one for top American cuisine by the 1994 Zagat Survey. Chef's note: \"I really like the edge that the sharp Asiago adds when shaved over the pasta. However, you can use whatever grating cheese you like, or use none at all! P.S. If you use dried pasta, remember to adjust the cooking time.\" To serve, spoon vegetable mixture over pasta, and top with cheese.\n\n## TUSCANY BEEF & SPINACH\n\n4 servings\n\n### Source: Asiago Greats\n\n### 4 boneless beef chuck top blade, or chuck eye steaks, cut 3\/4 inch thick (about\n\n### 1-1\/2 pounds)\n\n### 2 cloves garlic, minced\n\n### Salt and pepper\n\n### 2 tablespoons olive oil\n\n### 1 package (6 ounces) fresh baby spinach (6 to 7 cups)\n\n### 1\/2 teaspoon dried rosemary leaves, crushed\n\n### 1 cup shredded Asiago cheese, divided\n\n1. Combine garlic, salt and pepper; press evenly onto beef steaks. Heat 1 tablespoon oil in large nonstick skillet over medium heat until hot. Place steaks in skillet; cook top blade steaks 10 to 12 minutes (chuck eye steaks 9 to 11 minutes) for medium rare to medium doneness, turning once. Remove; keep warm.  \n2. Heat remaining 1 tablespoon oil in same skillet over medium heat  \n54  \nAsiago Greats  \nuntil hot. Add spinach, rosemary, salt and pepper. Cook and stir 1 to  \n2 minutes or until spinach is just wilted. Remove from heat. Add 3\/4 cup of the cheese to spinach; toss.  \n3. Serve steaks on spinach. Sprinkle with remaining 1\/4 cup cheese.\n\n* Start to finish: 30 Minutes\n\n55  \nIndex\n\n## deg deg deg deg deg\n\n### Index\n\n, 18\n\n(1 pound uncooked pasta), 53\n\n(6 ounces) fresh baby spinach (6 to 7 cups), 54\n\n(8-ounce) ca low-fat sour cream, 5 (about, 40\n\n(below), 19, 19\n\n(portioned into loins and legs, 38 (very young cheese if, 52\n\n* see note, 12\n\n***avocado salsa***, 24\n\n-, 30\n\n\\--enchiladas---, 15\n\n\\--garnish---, 18\n\n\\--sauce---, 14\n\n-dried basil, 40\n\n10 inch pizza crust, 13\n\n2% low-fat milk, 6\n\n= (artisan breads at counter in market), 37\n\n= (or 2 tspns dried thyme), 49\n\n= (or goat cheese), 50\n\n= (or part skim ricotta cheese), 1\n\n= (or small shell pasta), 27\n\n= (pick the greens of half a bundle from, 25\n\n= (preferred brand: kitchen basics), 37\n\n= (provolone, 37, 49\n\n= (san marzano recommended, 28\n\n= (san marzano recommended), 28\n\n= (such as asiago, 2\n\n= (such as hot or sweet sopressata, 2\n\n== red pepper tapenade & charred bread ==, 2\n\n=== antipasto ===, 2\n\n=== asiago frico ===, 21\n\n=== assembly ===, 27\n\n=== fresh fettucine ===, 19\n\n=== fresh green fettucine ===, 19\n\n=== garnish ===, 43\n\n=== hot stuffed cherry peppers ===,\n\n50\n\n=== soup toppers ===, 37\n\n=== sweet stuffed cherry peppers\n\n===, 50\n\n=== three cheese sauce ===, 27 a generous grating nutmeg, 29\n\na handful fresh basil leaves, 29\n\nall-purpose flour, 3, 21, 27, 29, 47, 49 amarone wine or other, 45\n\nand 2 inches of green), 24 and broken into pea-size, 52\n\nand coarsely chopped - (abt 1 cup), 43 and mushrooms sliced, 46\n\nand quartered, 45 and seeded, 50\n\nand seeds removed, 50\n\nartichoke hearts in water - (15 oz), 49 asiago, 36\n\nasiago cheese, 7, 9, 10, 13, 21, 32,\n\n35, 35, 41, 51, 52\n\nasiago or any other good, 42 asiago or parmesan cheese, 3 asiago or parmigiano reggiano, 44 asiago soft grits, 38\n\n56  \nAsiago Greats\n\nasiago), 49\n\nassorted fresh herbs for, 26 assorted italian cheeses, 2 bacon, 32\n\nbalsamic vinegar, 7, 25, 31, 36, 44 basil leaves, 51\n\nbasil leaves or 2, 26 bay leaves, 38\n\nbig pinch red chile flakes, 18\n\nblack oil-cured olives - (20 olives), 2 black pepper, 31, 36\n\nblack pepper cheese, 2\n\nboiling potatoes - (to 1 lb), 43 boned and skinned, 31 boneless and skinless, 15\n\nboneless beef chuck top blade, 54 boxes chopped spinach - (10 oz ea), 49\n\nbunch fresh thyme, 38 bunch green onions, 8\n\nbutter, 8, 10, 11, 13, 22, 23, 27, 29,\n\n40, 41, 46, 47, 49\n\nbutter or margarine,, 16 butter pastry dough, 46 by your butcher), 38\n\ncan cream of celery, 26 canned cannellini beans, 10 canned stewed tomatoes, 14 canned vegetable broth, 53 canola oil, 51\n\ncapers, 2 carrot, 7\n\ncayenne (opt), 34 cayenne pepper, 11 celery rib, 50\n\ncheese, 18, 24, 26, 41, 46 cheese or-parmesan cheese, 39 chicken or vegetable stock, 49 chicken stock, 31\n\nchopped, 40\n\nchopped 1\" pieces, 19 chopped 1\/2\" pieces to, 30 chopped chives, 22\n\nchopped flat-leaf, 46\n\nchopped flat-leaf parsley, 1, 27 chopped fresh basil, 18, 31 chopped fresh basil or, 40 chopped fresh chives, 48, 53 chopped fresh oregano, 53 chopped fresh parsley, 40 chopped fresh sage, 31 chopped fresh savory or, 18 chopped fresh thyme, 31, 38\n\nchopped fresh thyme leaves, 27, 49 chopped italian parsley, 43\n\nchopped parsley, 42\n\nchopped rosemary leaves - (to 3), 27 chopped shallot, 38\n\nchopped spinach, 12, 53 cilantro sprigs, 24\n\nclove garlic, 7 cloves garlic, 26\n\ncoarse salt, 25, 29, 37, 44 coarsely chopped, 49 coarsely grated, 15\n\ncoarsely-ground black pepper, 27 cold unsalted butter, 45\n\nconcentrated crushed tomatoes - (28 oz), 37\n\ncontainers chicken or vegetable stock\/\n\nbrot, 37\n\ncooked tiny green french, 30 cooked without salt, 7\n\ncorkscrew shaped pasta with lines, 27 corn tortillas, 15\n\ncornstarch, 26 counter in market, 49 cracked pepper, 26 crackers, 26\n\ncream, 19 croutons, 35\n\ncrumbled asiago cheese,, 15 crushed dried thyme leaves, 26 crushed tomatoes - (14 oz), 28 crushed tomatoes - (28 oz), 1 crusty bread, 29\n\n57  \nIndex\n\ncubed, 49\n\ncubes pepperoncino - (1\/2\"), 50 cut into 1\/2\" ribbons, 52\n\ncut into 8 chunks, 16 dandelion greens, 17 day-old bread, 9 defrosted, 19\n\ndice, 18\n\ndiced carrots, 31 dijon mustard, 21\n\ndrained sun dried tomatoes, 31 dried, 40\n\ndried basil, 13, 14\n\ndried crushed red pepper, 40 dried oregano, 14, 46\n\ndried rosemary leaves, 54 drops balsamic vinegar, 52 dry bread crumbs, 13\n\ndry cheese, 42\n\ndry grated bread crumbs, 6 dry orecchiette pasta,, 18 dry white wine, 49, 52\n\nds pepper, 35 ds salt, 35\n\neach cut into 4 pieces, 38, 38 each trimmed and quartered, 38 egg, 33, 47\n\negg white, 46 egg whites, 13 eggplant, 14\n\neggs, 10, 17, 19, 19, 26, 43 extra virgin olive oil, 49\n\nextra-virgin olive oil, 1, 2, 6, 9, 19,\n\n21, 23, 25, 27, 27, 28, 30, 37, 44 farfalle*, 7\n\nfat-free milk, 8, 8 fettucine, 51\n\nfinely chopped fresh, 26, 26 finely chopped parsley, 31 finely chopped shallots, 40 finely chopped thyme, 26 flour, 9, 11, 17\n\nflowers, 40\n\nfontina cheese, 35 for dusting, 19, 19 french bread baguette, 5 fresh asiago cheese,, 45 fresh basil, 7\n\nfresh basil leaves, 37\n\nfresh basil leaves - (to 7), 1 fresh basil sprigs, 31\n\nfresh chives, 38 fresh fava beans, 18 fresh garlic cloves, 37\n\nfresh ground black pepper, 10 fresh herbs, 16, 53\n\nfresh lemon juice, 21, 42 fresh linguine, 35\n\nfresh morel or chanterelle mushrooms,\n\n38\n\nfresh mushrooms, 46 fresh pasilla, 24\n\nfresh portabella mushrooms, 38 fresh shiitake mushrooms, 38 fresh shiitake mushrooms,, 31 fresh spinach, 47\n\nfresh tomatillos, 14 freshly grated asiago, 11 freshly grated parmesan, 34 freshly ground pepper, 24, 53 freshly shaved asiago, 31\n\nfreshly-grated asiago cheese, 30 freshly-ground black pepper, 1, 9, 21, 25, 27, 29, 37, 37, 43, 44, 49 freshly-ground black pepper,, 30 from veneto, 45\n\nfronds and tops, 44\n\nfrozen chopped spinach,, 19 full bodied wine, 45\n\ngarlic, 1, 6, 8, 9, 10, 23, 27, 28, 35,\n\n47, 49, 53, 54 garlic clove, 32 garlic cloves, 6, 14 garnish, 26 gorgonzola, 22\n\ngranulated sugar, 14\n\n58  \nAsiago Greats\n\ngrappa, 22\n\ngrated asiago, 5, 5, 19, 22\n\ngrated asiago cheese, 6, 10, 43, 48 grated asiago cheese,, 12, 40 grated asiago or parmesan,, 24 grated fontina cheese, 24\n\ngrated italian asiago, 39 grated jarisberg cheese, 24 grated lemon zest, 18\n\ngrated monterey jack cheese, 24 grated parmesan cheese, 5, 5, 12 grated parmesan or romano, 23 grated parmigiano, 27, 27\n\ngrated parmigiano-reggiano, 1 grated romano or asiago, 26 grated white cheddar cheese, 24 grates asiago or parmesan,, 46 green onions, 5, 13\n\nground black pepper, 46 ground nutmeg, 47 ground pepper, 39 ground round, 12\n\nground sirloin, 27 half-and-half, 16\n\nhalf-and-half or heavy cream, 49 halved lengthwise and thinly sliced, 53 hazelnut liqueur, 32\n\nheads, 52\n\nhearts of palm - (14 oz), 25 hearty red wine, 31\n\nheavy cream, 32, 37, 41\n\nherb and garlic spreadable cheese, 50 honey, 36\n\nhot cherry peppers, 50\n\nhot cooked fresh fettuccine, 53 hot peppers, 2\n\nin oil, 31\n\nitalian bread crumbs, 27, 27 italian tuna in oil, 2\n\njalapeno chile, 24 jalapeno peppers, 14 jumbo prawns, 35 kosher or sea salt, 24\n\nkosher salt, 21\n\nlarge baking potatoes, 47 large chicken thighs,, 31 large egg, 27\n\nlarge egg yolk, 3 large fennel bulb, 44 large garlic cloves, 2 large head broccoli, 9 large radicchio, 51 large red onion, 30 lean ground beef, 13\n\nleaves or 1 teaspoon,, 26\n\nleek, white part and 1\" green, 43 lemon, 19\n\nlemon peel twists and, 26 lemons, 19, 35\n\nlentils, 30\n\nlg egg white, 34 lg egg whites, 11 lg eggs, 11\n\nlg vine-ripened tomatoes (about 1 1\/2 pounds), 6\n\nlight mayonnaise, 5 linguine fine pasta, 30 linguini, 42\n\nliquid egg substitute, 8 liver salami, 9\n\nloaf day-old crusty bread, 2 loaf french bread, 8\n\nloaf multigrain or whole wheat baguette, 49\n\nlow-salt chicken broth, 5, 38 low-sodium chicken broth, 15 mache leaves, 21\n\nmarinated artichokes - (6 oz), 2 marzanos are not available, 28 mayonnaise, 8\n\nmd zucchini flowers,, 40 medium apples, 45 medium carrots, 38 medium escarole head, 30 medium garlic clove,, 15\n\nmedium green tomatoes, 53\n\n59  \nIndex\n\nmedium leeks, 38 medium oysters, 32 medium red onion, 36, 52\n\nmedium size shelled shrimp, 18 medium zucchini, 53\n\nmilk, 9, 27\n\nminced fresh basil, 40 minced fresh garlic, 46 minced fresh herbs (basil, 6 minced fresh parsley, 35 minced fresh rosemary, 43 minced onion, 15\n\nminced parsley, 40 monterey jack, 13 mozzarella, 1\n\nnasturtiums for garnish, 40 nutmeg, 9\n\nnuts (finely chopped), 39 oil, 42\n\nolive oil, 7, 10, 11, 12, 13, 14, 18, 19,\n\n19, 24, 34, 35, 38, 40, 41, 43, 53, 54 olive or canola oil, 31\n\nonion, 14, 32, 39, 49 onions, 12\n\nor, 5, 5, 10\n\nor asiago cheese, 34 or cream, 26\n\nor more jack cheese, 15\n\nor more or-any squash, 40 or parmesan, 11\n\nor parmesan cheese, 31 orecchiette pasta, 31 other white beans, 10 package puff pastry, 33\n\npackages active dry yeast, 3 packages cream cheese,, 26 pancetta, 9\n\npancetta cut into tiny, 18 parmesan, 13, 16\n\nparmesan or asiago cheese [grated], 8 parmigiano-reggiano, 41\n\nparsley, 35\n\nparsley leaves, 2, 38\n\npart-skim ricotta cheese, 47 pasta dough, 52\n\npeeled, 53\n\npeeled and minced, 15 peeled and trimmed, 42\n\npepper, 6, 7, 10, 13, 14, 16, 18, 47 peppers or pepperoncini, 2\n\npicked flat-leaf parsley leaves, 25 pieces, 52\n\npieces jumbo pasta shells, 1 pieces pancetta, 36\n\npinch freshly ground black pepper, 6 pinch salt, 6\n\nplus, 11, 19, 19\n\npn fresh ground black pepper, 42 pn freshly-grated nutmeg, 45\n\npn kosher salt, 42 polenta, 36 possible), 52\n\nproduce department), 25 prosciutto, 25, 43, 51 quail, 36\n\nquartered artichoke hearts in water, 25 radicchio, 52\n\nrecipe basic fresh egg, 52 recipe fresh fettucine,, 19\n\nrecipe fresh green fettucine,, 19 recipe green gnocchi,, 22\n\nred bell pepper, 7 red onion, 44\n\nred pepper flake sheep's milk cheese),\n\n3\n\nred wine vinegar, 47 removed from casing, 52 rice vinegar, 24\n\nrich chicken, 18 ricotta cheese, 1, 39 rigatoni pasta, 40 ripe avocados, 24 ripe tomato, 18\n\nripe tomatoes, 31\n\nroasted red peppers - (16 to 18), 2 romaine hearts, 44\n\n60  \nAsiago Greats\n\nromaine lettuce heart, 25 roughly chopped cilantro, 24 round loaf crusty bread, 49 russet potatoes, 41\n\nsage leaves, 36 salami), 2\n\nsalt, 1, 1, 3, 6, 7, 10, 10, 11, 13, 15,\n\n15, 17, 27, 30, 34, 43, 47, 49, 52, 53 salt and freshly ground, 16, 18, 31 salt and pepper, 41, 51, 54\n\nsavory or basil sprigs, 18 scallions, 24\n\nseasoned salt, 12 seasoning salt, 39\n\nseeded and diced peeled red tomato,\n\n47\n\nseeded and diced peeled yellow tomato, 47\n\nserving bowl, 49\n\nseveral sprigs flat-leaf parsley, 44 shallots, 19\n\nsharp cheddar cheese,, 15 shaved asiago cheese, 53 shaved parmesan or asiago, 18 sheets phyllo dough, 34\n\nshitake mushrooms, 46 shredded asiago, 1, 23\n\nshredded asiago cheese, 8, 27, 29, 54 shredded cheese, 37\n\nshredded italian 4 cheese blend, 49 shredded jack cheese, 8\n\nshredded mozzarella, 23\n\nshredded parmesan or asiago cheese,\n\n6\n\nshredded parmigiano-reggiano, 29 shredded provolone, 23\n\nshredded sharp cheddar, 27 skewers, 36\n\nskin, 23\n\nsliced fresh mozzarella, 29 sliced green onions, 53 sliced italian deli meats, 2 sliced mushrooms, 5\n\nslices cappicola (hot italian ham), 50 slices crusty bread, 37\n\nslices crusty italian semolina bread, 23 slices prosciutto di parma, 50\n\nslivered garlic, 18, 31 small broccoli flowerets, 7 small cauliflower flowerets, 7 small green bell pepper, 27 small onion, 1, 15\n\nsmall onions, 38\n\nsmall red bell pepper, 49 small red onion, 24\n\nsmall shallot, 21 small tomato, 43\n\nsmall yellow onion, 27\n\nsmoked mozzarella cheese,, 46 softened (8 ounces each), 26 sour cream, 15, 26\n\nspeck, 10\n\nspicy spaghetti sauce, 13 spinach, 32\n\nspinach leaves, 10 spiral shaped pasta, 10\n\nsqueezed dry in kitchen towel, 49 squeezed very dry, 19\n\nstalks celery, 38\n\nstem'd and thick slice, 31 sugar, 3, 45\n\nsugar snap peas, 53\n\nsun-dried tomato sprinkles, 5 sweet butter, 42, 52\n\nsweet fennel sausage,, 52 sweet peppers, 50\n\nsweet red pepper strips,, 26 swiss, 37\n\nswiss cheese, 32 tarragon leaves or, 26\n\nteaspoon sugar to your sauce, 29 the youngest possible, 45\n\nthinly sliced, 44, 46 thinly sliced celery, 31 thinly sliced fresh basil, 53 thinly sliced prosciutto,, 40\n\n61  \nIndex\n\nthyme, 32\n\nto 10 cloves garlic, 41 to taste, 30\n\nto..., 14\n\ntoasted almonds or pine, 39 tomato paste, 27\n\ntomato sauce, 40\n\ntrans-free margarine [melted], 8 treviso radicchio head,, 19 turkey scaloppine, 13\n\nunbleached all-purpose flour,, 19, 19,\n\n19, 19\n\nuncooked regular grits, 6 unsalted butter, 35 vegetable cooking spray, 47 vegetable oil, 15, 15\n\nvirgin olive oil, 36, 51\n\nvodka, 40\n\nwalnut pieces, 46 water, 5, 7, 33, 36\n\nwedge pecorino romano or asiago, 25 whipping cream, 40\n\nwhite pepper, 33 white wine, 32, 42\n\nwhole chicken breasts,, 15 whole eggs, 12\n\nwhole garlic cloves,, 42 whole grain flour, 8 whole milk, 29\n\nwhole peeled tomatoes - (28 oz), 28 yellow onion, 31\n\nyield 3 cups, 30 young rabbits, 38\n\nziti rigate with ridges, 29\n\n62\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nMore Food of My Friends\n\nTheir Favorite Recipes\n\nJudith Shepard\n\nNew York\nThanks\n\nTo all cooks everywhere and especially to those who contributed so enthusiastically to this book.\n\nTo Karen Sukoneck, who again designed such a nice cover.\n\nTo Mary Cummings, who again edited for me.\n\nTo my husband, Marty, who somehow gets it all together and who loves to test the recipes.\nContents\n\nFOREWORD\n\nINTRODUCTION\n\nNOTES AND GUIDELINES\n\n1. Shopping\n\n2. Meat\n\n3. Additives\n\n4. Butter, Oils and Margarines\n\n5. Sweeteners\n\n6. Flour\n\n8. Grains, Beans, and So Forth\n\n9. Milk, Cream, Yogurt, and Cheese\n\n10. Breakfast Cereals\n\n11. Salt\n\n12. Health Food Suppliers\n\n13. Cooking Methods\n\n14. Quantity\n\n15. Time and Procedures\n\nHINTS\n\nAPPETIZERS AND SIDE DISHES\n\nSOUPS\n\nSALADS AND DRESSINGS\n\nRICE, POTATOES, NOODLES, BARLEY, AND CORNMEAL\n\nPASTA\n\nVEGETABLES\n\nMEAT\n\nPOULTRY\n\nFISH\n\nDESSERTS\n\nBREADS\n\nSANDWICH LIST\n\nGood and Useful Cookbooks\n\nINDEX\nForeword\n\nIn 1981, I wrote a cookbook called _Food Of My Friends-The Best Meals in Town._ It came about because of a remark my husband Marty made. He said that he was spoiled. He didn't like to eat in restaurants anymore because \"our home and the homes of our friends served the best meals in town.\" My book was, as the title implies, a collection of recipes from my friends as well as some of my own.\n\nWhen I decided to write a second cookbook, I thought it would be a little more difficult and would take much longer to gather a sufficient number of dishes. I felt I might have exhausted my own supply and I would have to find more friends, relatives and even strangers who would be willing to share their favorite recipes. I was wrong. People were so willing to make contributions and were so helpful and supportive that this became the easiest part of the book.\n\nI had a grand time. It was quite exciting to be given a recipe and then test it out on my family, definitely another bonus for writing a book like this. It made mealtime a bit of an adventure.\n\nAnd the recipes came from all over. Some of my friends invented new ones especially for this book and I was inspired to create some myself\u2014I who usually consider myself a recipe follower. Even my husband joined in by creating his own pasta sauce.\n\nOther people dug up old family recipes. I found some of mine that were written on paper so old and yellow that it practically disintegrated before my eyes.\n\nMy mother gave me one that had been my grandmother's. It became a favorite of mine. I passed it on to my daughter, Liza, and now there will have been three generations of us preparing \"Quilting Bees.\"\n\nEveryone seemed to be on the lookout for me. Gay Tarlo came back from a party in New York City and called to tell me about a chicken salad she had enjoyed so much that she jotted down the ingredients so I could include it here. An acquaintance who lives on a commune sent me a whole sheaf of recipes telling me to use whatever I wanted. Franklin Heller wrote a P.S. on the bottom of a business letter asking if I would like an old family recipe for Liptaur Cheese which I eagerly accepted.\n\nI think I am the most partial to family recipes, the ones that have been handed down. It may be only an illusion, of course, but it speaks of a kind of continuity that is very appealing to me. And if they weren't family recipes, I liked the idea that people were so eager to share. After all, it doesn't cost anything and it contributes to a whole culinary network\u2014a cultural menu of our times, if you will.\n\nSo, I consider this book first of all a collaborative effort and an indication of the essential willingness of people to share something that gives us all pleasure.\n\nJudith Shepard\n\nSag Harbor, N.Y.\nIntroduction\n\nAbout 19 years ago, I picked up a book by Gaylord Hauser which changed my ideas about food and nutrition and set me upon a path that I am still trying to follow. Mr. Hauser talked about being attentive to what you eat and how you prepare it, and about the benefits this can bring for your health and general well being. It made a lot of sense to me then and does even more so now that my knowledge and experience in this subject has increased.\n\nAt the time, my ex-husband and I were traveling across country in a play called \"Who's Afraid of Virginia Wolfe?\" The tour was ten months long; we had our three children, Aaron, Liza, and Caleb with us, all of whom were under five, and anything that I could do to make this venture easier and less fatiguing was very welcome. After reading Mr. Hauser's book, I became excited by the idea that I might have some measure of control over my family's health and, of course, my own. I began changing some of my cooking concepts and trying Mr. Hauser's recipes. When I began to notice a definite change in our physical and mental state, I became a true convert.\n\nAt that time, people were not nearly as health and diet-conscious as they are now. In fact, certain ideas that are quite germaine to sound nutrition now might have been denounced as fads then. I find the whole emphasis and investigation into the importance of eating wisely, by both laymen and the medical community, a welcome sign of progress.\n\nI also find that I continue to change and refine my thinking on this subject as I become older. I have watched my friends change as they modify their diets. It's not unusual to be invited to a dinner party and have the hostess announce that she has been cooking without much salt and that there is a salt shaker on the table for those who might want to add more. More and more people either don't allow smoking in their homes or set aside a separate room for smokers. And, while this is not an exercise book, I would point out that, for many of us, diet has gone hand-in-hand with exercise. Just as people have become more aware of good nutrition, they have also realized the importance of keeping their bodies in good shape.\n\nI continue to be appalled at the way food is manipulated for a longer shelf life and more profit. The preservatives, additives, bleaches, emulsifiers, mono and diglycerides, other chemicals, and vast amounts of sugar and salt are not for me. I feel it is up to us, the consumers, to show that we will not be force fed these inferior foods and we should refuse to buy such products. If enough people boycott these items, maybe those who are responsible for producing them will be re-educated. It is our health that counts, not their profits.\n\nI do have a fair number of recipes in my book calling for meat. I also have some rich and luscious desserts and recipes that do not adhere to strict dietary rules. Since many of these contributions are very tasty, I have included them. What I do strive for, however, is some kind of balance. I don't make them too often. I try to stick to main courses of vegetables, poultry or fish, salads, fruits, and simple desserts. However, I do eat meat and rich dishes and desserts occasionally. As I said in first cookbook, \"Everything in moderation, including moderation.\"\n\nOne of the newer things I am working on these days is improving the appearance of the food when it reaches the table. One only has to think about Japanese dishes or look through some of the cookbooks that have expensive, glossy pictures to see what I mean. They practically make your mouth water. I am convinced that to see a beautifully laid table with balanced colors and textures must do something for one's psyche. Since I like the ritual of dining with all that it implies\u2014the sharing of ideas, the coming together at a certain time of day or evening\u2014setting an attractive table seems like a necessary component.\n\nI am also trying to strike a balance between rich and lean, piquant and bland, solid and liquid when preparing menus. Of course, weather plays a part too. A chilled fruit soup is superb during a hot summer evening, but is not so desirable in December. Hearty stews and black bread are great during blizzards, but not during a muggy July evening. So it's worthwhile and rather intriguing to let the weather help you plan your menu and to see where your creativity might lead you.\n\nOne last thing concerns the people who have contributed these recipes. Half the fun of writing my first book was that I could write a little bit about my friends. Since many of the same people have contributed recipes for this book too, I hope they will understand that I didn't want to repeat myself. My new contributors I have described briefly. I guess it's a way of sharing and spreading our community to other places and people. I hope that any of the recipes that especially please you will be passed along too.\nNotes and Guidelines\n\nThis may be somewhat repetitious for those of you who have read my first book, but I feel that there are some basic rules that one should be aware of in planning meals. Most of the information I'm passing on has come from various books and sources that I have read over the years, starting from the first one by Gaylord Hauser, on to Adele Davis, and more recently the New York Times which cited a report by an expert panel of the National Academy of Sciences in which the kind of dietary recommendations that so-called food faddists have been advocating for years are confirmed. Many well informed people, very concerned with the relationship between diet and health, have been treated with a certain derision by the scientific and medical community. Now it is beginning to look as though that community is catching up with them. And I hope new doctors and nurses in training are beginning to take diet and food intake very seriously, so we all can begin to practice preventive health. Preventive health is a silly way of putting it since, after all, what we want is to prevent sickness before it happens. However, it is the term that both doctors and dentists seem to be using. There are many fine source books that you can read if you would like more comprehensive information than what I am giving you in this book.\n\nMy own notes and guidelines change as new information becomes available. A few years ago, I tried cutting down on my salt intake; now I cut down even more. My menu planning becomes more selective and I see the same thing occurring with my friends. We are all watching our weight more carefully. Fewer and fewer of us are serving lots of cheeses with cocktails. Rich pastries are being replaced with light desserts.\n\nI have mentioned that I am not a purist, so I try to be somewhat flexible while not indulging myself too much. Although a Pavlova is an exquisite dessert, I serve it only on special occasions. I dearly love all kinds of pasta, but I am not going to make all those dishes that contain huge amounts of butter and cream too often.\n\nI hope that you will follow some of these guidelines while realizing that knowledge about food, its preparation and its effects on health is always evolving. People talk about adventures in good eating; they could also talk about adventures in good health.\n\n1. Shopping\n\nMy first rule of thumb is to shop for quality. I would rather have less of a good thing than more of a bad thing. I would rather have a smaller cut of prime meat and have it less often. I would rather buy good, fresh fish a couple of times a month than any of the frozen fish found in supermarkets. I try to buy vegetables and fruits in season and when that's not possible, I still look for the best quality.\n\nI believe that it is important to find a good butcher, fishmonger and green grocer. They can be your best friends. It is true that it is very convenient to buy all your foods at one market. Unfortunately, it seems that the quality of certain foods just won't be as good. To travel to those shopkeepers who offer the highest quality takes a certain degree of organization and a desire to buy only the best. I try to stay away from canned and frozen foods as much as possible. Fresh fruits and vegetables simply taste better and they're better for you.\n\nI do not buy convenience foods. As far as I'm concerned, they're only convenient for the manufacturer; they usually taste awful and cost much more than they're worth.\n\nSo, shop in season, shop for quality, and find good sources for your food.\n\n2. Meat\n\nThe panel of the National Academy of Sciences has advised Americans to eat less fat and very little salt-cured, pickled and smoked foods. Try to cut down your meat intake. We have meat a couple of times a month and then I try to find the leanest pieces I can. Try not to buy meat that contains nitrates and nitrites. Don't buy smoked meat or fish. They are all said to contain cancer causing substances. We never eat ham, but once in a while I indulge and buy bacon. If you look hard, you can find bacon and hotdogs that do not contain nitrates and nitrites. Try to wean yourself to fish and poultry. It's not as hard as it may seem.\n\n3. Additives\n\nI don't buy anything that contains additives, preservatives, artificial colors and dyes, calcium proprionate, dough conditioners, (which may contain sodium stearoyl-2 lactylate, mono and diglycerides, ethoxylated mono and diglycerides, and succinylated mono glycerides). I don't buy breads that contain yeast conditioners. It may sound innocuous, but yeast conditioners are ammonium chloride and potassium bromate and, who needs that in his bread? If a bread says, \"no preservatives,\" look very carefully because chances are it has something in it that shouldn't be there. All this means that you must read your labels each time. The manufacturers do try to fool you.\n\nIt may also mean that you might want to start baking your own bread because good healthy breads are hard to find. Recently, I heard a doctor who had been given a great deal of recognition for his study on cancer viruses. He said that if there were one thing that he would like to tell the American public about their health it was this: Eat more bread, but good healthy bread, bread made with unbleached, stoneground flours, flours with bran and wheat germ\u2014the kind of bread your grandmother probably made.\n\n4. Butter, Oils, and Margarines\n\nI am aware of the dilemma over choosing between butter and margarine. Butter is saturated fat and the less we have of it, the better. Margarines, on the other hand, usually are unsaturated fats, but they contain additives and food dyes. In the past I have opted for sweet butter. The salt in salted butters can mask rancidity and I don't want that extra salt anyway. However, now I have come upon a new variation\u2014a combination of butter and safflower oil that you blend yourself. It has the advantage of having less saturated fat and spreading like margarine without the harmful additives. I am very pleased to have this substitute, my family likes it, and it is easy to prepare. I call it butterblend for want of a proper name, and I have included the recipe (page 29).\n\nI try to substitute vegetable oils for butter whenever I can. However, hydrogenated oils contain preservatives to increase their shelf life. Some vegetable oils contain cottonseed oil. Cotton is not considered an edible crop and is heavily sprayed with pesticides which I'm sure make their way into the oil so, here again, you must read your labels carefully.\n\nMy favorite oil is a cold-pressed poly-unsaturated safflower oil. It's light and can be used for salads, sauteing and dishes cooked in a wok. I occasionally use olive, sesame and walnut oils.\n\nIn cutting down one's use of oil, I find if you use heavy pans and let them get quite hot, you can reduce the amount of oil. Use a rubber spatula to see that the whole surface is lubricated.\n\n5. Sweeteners\n\nI don't use white sugar. It has been sneaked in so many foods that probably I'm getting a lot of it anyway. It causes obesity, tooth decay, and depletes B vitamins in the body. Raw sugar is not too much better, but I use it when I need to because at least it hasn't gone through the refining process, and it may contain some of the original nutrients. Chemicals have not been added to make it anti-caking and it hasn't been exposed to the whitening process.\n\nThe best sugar substitutes are honey, molasses and maple syrup. Maple syrup should be pure and unrefined. The commercial brands contain artificial flavor, sodium benzoate and sorbic acid.\n\n6. Flour\n\nIf I have to use white flour, I make sure it's unbleached. I do try to use stoneground, unrefined whole wheat flour as much as possible or at least a combination of whole wheat and white flours. Whole wheat flour can certainly be used for thickening sauces and gravies. It makes delicious breads and muffins and is one of the best ways to include more of the nutrients we need for good health. Desserts made with whole wheat flour can be a little heavy so I often try to use a combination of unbleached white and whole wheat flours. Stay away from flours that have been bromated or phosphated.\n\n7. Rice\n\nI like both brown and white rice, but brown rice is better for you. It contains all the natural ingredients many of which white rice loses in the processing. When I do use white rice, I choose the converted brands because they may contain some of the original nutrients. I never buy quick-cooking or seasoned rice. Quick-cooking rice contains chemicals which have been added to reduce to cooking time; seasoned rice is replete with additives.\n\n8. Grains, Beans, and So Forth\n\nI am trying to become more knowledgeable about the various grains available to us. Rice is only one of them. Others include cracked wheat, kasha, barley, millet, corn meal, lentils, wild rice, rye and hominy. There seems to be a tremendous variety of recipes using these ingredients and I'm sure eating more of them could do much to improve our diets. In the back I've listed some of my favorite cookbooks. Two of them, \"Whole Grain Cookery\" by Stella Standard and \"Future Foods\" by Colin Tudge should give all of us enough recipes to experiment with for two lifetimes. There are too many nutritious foodstuffs to restrict ourselves to just rice, noodles and potatoes.\n\n9. Milk, Cream, Yogurt, and Cheese\n\nHigh fat dairy products are simply not that good for you. It is very hard for me to cut down on cheese and I love cream. One of the ways I try to compensate is by using dry, skim milk. Whenever a recipe calls for milk, you can simply mix dry milk with the prescribed amount of water and use that as a substitute. You can make skim milk and mix it half and half with regular milk for drinking and you can make yogurt with any kind of substitution. You can even make it with all skim milk. In this way, you not only are saving money, but you are cutting down on dairy fats.\n\nIn buying yogurt, beware of those that contain fruit. They usually contain incredible amounts of sugar. It's better to buy the plain kind and mix it with fresh fruit.\n\nWhen buying cheeses, check for additives. Because all ingredients are not labeled, I often buy imported kinds. Since some countries are very proud of their cheeses, they often do not add preservatives, not wanting to tamper with a good thing.\n\n10. Breakfast Cereals\n\nThis year I have been trying to use a variety of both hot and cold breakfast cereals. I seldom buy cold cereals in supermarkets because they are usually quite dreadful and loaded with sugar. I either make my own or buy them at my local health food store. When buying cereals, you must read all the labels very carefully.\n\nSome of the hot cereals you might try besides oatmeal and cream of wheat are kasha, cream of rice, wheatena, cornmeal, cracked wheat, cream of rye, and whole wheat cereal.\n\nDo not buy quick cooking cereals because they have been over processed.\n\nI also try to add untoasted wheatgerm to both hot and cold cereals because it is so high in nutrients. Keep it in the refrigerator so that it does not spoil and become rancid.\n\n11. Salt\n\nI think everyone knows these days that we consume far too much salt and that it is not good for you in such quantities. It is a natural ingredient in all foods, but has been added by the food industry in overwhelming amounts. I have listed the standard \"salt and pepper\" in many of my recipes but I have not given the amounts, preferring to leave that to individual tastes. However, more and more, I find myself eliminating the salt entirely. For those of us who use tamari (soy sauce) as a substitute, I must add that it is also loaded with salt. When I do use salt, I buy vegetable or sea salt. Too many of the salts sold in supermarkets have many additives, including aluminum compounds and agents to make the salt pour easily.\n\nOne final word\u2014I think we have become addicted to salt and also sugar. As with all addictions, it is hard to cut down or eliminate them. But I do believe if you stick with it, you become more appreciative of the taste of the food itself and that idea may help you when you automatically reach for the salt shaker or find yourself starting to add an extra teaspoon of sugar.\n\n12. Health Food Supplies\n\nMany supermarkets now carry health foods and have special sections allotted to those products. I think they are worth looking into and one should also find a convenient health food store. As people have become more and more conscious of the quality of the food they buy, health food sections have expanded and today you can find many different kinds of flour, breakfast cereals, unsulphured dried fruits, good selections of tamari, nut butters, bouillon cubes, crackers, juices without added sugar, and even hot dogs and bacon preserved without nitrates and nitrites. Some are exorbitantly priced, so you must be knowledgeable and shop wisely. Be aware that these items do not have preservatives and many must be refrigerated. Since some of these products cost more than the usual supermarket variety, you may have to be a more skillful juggler with your food budget.\n\n13. Cooking Methods\n\nI try to steam vegetables or at least to use as little water as possible. I like to prepare dishes in a wok. Baking is preferable to frying and, when I do fry or saute, I cheat a little and use less oil than is called for. I seldom deep fry, although once in a while, as with tempura, there is no way to get around it.\n\nA good set of pots and pans is a wise investment. I do not use teflon or aluminum pans as they are made with soft materials that can dissolve and be ingested. Stainless steel, heavy cast iron or clay pots seem to be the best.\n\n14. Quantity\n\nMost of the recipes in this book serve four to six people.\n\n15. Time and Procedures\n\nGood healthful cooking takes more time; there's no doubt about it. It's a choice you have to make. Sometimes, I find that I have overdone it. I am so busy preparing meals that I don't have time to do anything else. I am solving this dilemma by more careful planning. If I am having a dinner that takes quite a lot of time, lunch must be simple. If I make enchiladas for dinner, the accompaniments must take only moments to prepare. If I decide to bake bread, maybe I can get my husband to cook something for dinner. For if I'm not careful, I am like the child whose eyes are too big for her stomach, only I am the cook who has such wonderful fantasies of all the delectable things I'm going to make, that I am literally a slave to my kitchen.\n\nIn order to make planning easier, I have starred every recipe. One star represents a dish that is easy or quick to prepare, a recipe with two stars takes more time, and three stars tells you that this is something that requires longer preparation.\n\nIt is very useful to read the recipe all the way through before you begin. This way, you won't be caught without one of the ingredients or find at the end that the dish is supposed to marinate overnight.\n\nA lot of the best cooks I know all have the same habit. They taste as they go along, so it seems to me that it would be well to adopt this practice.\n\nLast but not least, cooking is a creative act. It helps to think of it that way, especially since it's something many of us must do every day. If it is done with a sense of adventure, if you delight in tempting the senses, and if you think of yourself as an artist, you will be well rewarded by friends and family, and what is a chore for many, will be a pleasure for you.\n\nHints\n\nThese are a few hints that you might want to glance through. I have jotted them down as I have written this book as experience has taught me that they are easy and very useful.\n\n1. Fresh herbs and spices are always preferable to dried ones, when available.\n\n2. My Italian food store sells me chunks of Romano cheese instead of Parmesan and the owner, Savario, says it is fine in Italian dishes and much less expensive.\n\n3. I prefer to cook my pasta _al dente_ which means that it is a tiny bit firm. Otherwise, by the time you get it to the table, the pasta is too mushy.\n\n4. I see no reason to add salt to all boiling water, such as that for vegetables and pasta. People can always add more at the table.\n\n5. Mushrooms absorb water easily, so they should be cleaned with a damp cloth.\n\n6. Many dishes that have tomatoes will taste better if the sauce is well cooked down, especially pasta sauces. If you wish, you can drain the tomatoes before adding them. The point is to have a rich, thick sauce, not a watery one. Save the liquid for a soup. I might add that when I make a tomato sauce for pasta dishes, it it's summer and fresh tomatoes are plentiful and inexpensive, I use them skinned and chopped as a substitute for canned tomatoes. Because tomatoes _are_ so watery, I like to toss a salad first and add them at the last moment.\n\n7. When you are cooking a delicate fish, be sure you have plenty of room in the pan so you are able to turn the fillets easily. Otherwise, they will flake apart.\n\n8. All roasts should stand 10 to 15 minutes before carving for easier slicing and juicier meat.\n\n9. I have a standard rule for tossing salads (when you are not using a mixed dressing). Always toss the leaves with the oil first. This way the leaves are well coated before the vinegar or lemon juice is added.\n\n10. It is very difficult to find a good cream that does not contain stabilizers and emulsifiers. If you find one, make a note of it and don't settle for anything else.\n\n11. Mix dry milk with the prescribed amount of water to substitute for regular milk in all baking.\n\n12. Eggplant soaks up enormous amounts of oil. When a recipe calls for sauteed or fried eggplant, try putting the slices in a well oiled baking dish to brown. They should occasionally be turned.\n\n13. When a recipe calls for stir frying, you don't have to go through all your Chinese cookbooks or take a cooking class. It simply means to cook food that has been cut up in a large wok or frying pan in a small amount of oil over high heat. As it cooks, keep flipping the pieces over and over using two utensils until they are cooked or coated as the recipe requires.\n\n14. Bouillon cubes should definitely be purchased in health food stores. Otherwise, you are getting pure salt and chemicals.\n\n15. Prepare fish the same day it is caught or bought. It should not have any kind of a fishy smell.\n\n16. Store brown sugar in a jar with a lid and include a small piece of apple to prevent the sugar from hardening.\n\n17. Generally speaking, use a third or a half as much of dried herbs as you would if they were fresh and buy both kinds in small quantities as they lose their delicate flavor quickly.\n\n18. Store soups that have been made with meat or poultry in the refrigerator so that you are able to skim off the grease easily before heating and serving.\n\n19. It is nice to be able to store pie crusts in your freezer in case you want to make a quick quiche or fruit pie. If you don't want to make your own there are some frozen brands available with pure ingredients, so look for those.\nAppetizers and Side Dishes\n\nMy good friend, Bill Tarlo, is a fine creative cook, blending and tasting as he goes along. He is especially creative at artful combinations, as you can see by this invention. This is easy and quick to prepare. You can substitute yogurt for sour cream, if you're counting calories.\n\nBill Tarlo's Cabbage & Beet Appetizer *\n\n1 16 oz. jar beets, drained\n\n1 16 oz. jar red cabbage, drained\n\n\u00be cup sour cream\n\n3 tbsp. orange juice\n\n1. Mix all ingredients together and serve as a side dish or even as a salad.\n\nMy husband said I should put this in my cookbook. I had never thought of it; it seemed so simple. This is what most of the Greek women who lived in my neighborhood in Silver Spring, Md. did with olives. I know you can buy them at delicatessens, but there's something nice about making your own. The longer they marinate, the better they'll be. I happen to like the olives with pits, but you can use pitted ones, if you wish.\n\nGreek Olives *\n\n3 cups black or green olives\n\n1 cup olive oil\n\n\u00bd cup wine vinegar\n\n2 cloves garlic\n\n1 bay leaf\n\n1 tsp. oregano\n\n1. Make a small cut in each of the olives.\n\n2. Put olives in a jar with a top and cover with all the rest of the ingredients.\n\n3. Store in refrigerator.\n\nFranklin Heller says that this is an old Viennese recipe passed down to them and that the trick is the crushed caraway seeds. However, that's very hard to do. I believe they can be purchased in some specialty shops. I used regular seeds and found it perfectly acceptable. I also think it's absolutely delicious.\n\nFranklin and Banne Heller's Liptauer Cheese Spread*\n\n2 8 oz. packages cream cheese\n\n1 4 oz. stick butter\n\n1 tbsp. caraway seeds, crushed if possible\n\n1 tbsp. paprika\n\n1 tbsp. prepared mustard\n\n\u00bd tsp. white pepper\n\n\u00bc tsp. salt\n\n2 anchovies minced (optional)\n\n2 tbsp. capers, chopped\n\n3 tbsp. fresh chives, chopped\n\n1 tsp. fresh dill, chopped\n\n2 tbsp. gin or brandy\n\n1. Allow cream cheese and butter to soften and then mix together.\n\n2. Add all the other ingredients and blend thoroughly.\n\n3. Put in a container filled to the top and cover tightly.\n\n4. Refrigerate for three days before serving.\n\n5. When serving, use at room temperature with crackers or sliced pumpernickel.\n\nThis was a dip served to us at Cynthia Epstein's house. She is my husband's cousin, a marvelous cook in her own right. However, this was made by one of her house guests who gives cooking classes in New York City. I like it because it gives you another way to use tahini (sesame butter).\n\nEggplant Tahini Dip *\n\n1 eggplant, steamed\n\n3 tsp. tahini\n\n2 tsp. olive oil\n\n\u00bc cup onion, minced\n\n2 garlic cloves, pressed\n\nsalt & pepper\n\njuice from 1 lemon\n\n2 tbsp. parsley, minced\n\n1. Peel eggplant, chop it very well and let it drain in a colander. Remove.\n\n2. Add tahini and olive oil and mix well.\n\n3. Add onion, garlic, salt and pepper and stir.\n\n4. Add the lemon juice, stir again and sprinkle parsley on top.\n\n5. Serve with crackers.\n\nMy friend Ginger showed me how to make a nice simple chutney. It's less expensive than buying it and goes well with not only curries, but all kinds of roasts and vegetarian meals.\n\nGinger's Orange Chutney *\n\n1 18-oz. jar orange marmalade\n\n\u00bc cup green pepper, seeded and chopped in small pieces\n\n\u00bc cup onions, chopped fine\n\n1 large apple, peeled, seeded and chopped\n\n\u00bc cup raisins\n\n1 tbsp. cider vinegar\n\ndash of cayenne pepper\n\n1. Mix all ingredients together and store in refrigerator until ready to serve.\n\nBernice Hunt's Caviar Pie **\n\nThis is an elegant, very easy, very impressive appetizer.\n\n8 hardboiled eggs\n\n1 small onion, chopped fine salt and pepper mayonnaise\n\n3\u20134 tbsp. sour cream\n\n8 ounces cream cheese, room temperature\n\n3\u20134 ounces black caviar\n\n1. Make a hardboiled egg salad by mashing the eggs with finely chopped onion, salt and pepper to taste, and just enough mayonnaise to hold the mixture together. (You can make it quickly in a food processor.)\n\n2. Spread the egg salad evenly over the bottom of an 8- or 9-inch spring form. Blend enough sour cream into the cream cheese to make it easily spreadable (use the processor again) and distribute it evenly over the egg mixture. Cover with plastic wrap and refrigerate until just before serving. (You can do this much the day before.)\n\n3. Run a knife around the rim of the springform and remove it. Leave the pie on the base of the pan but set it on a large plate. Cover the top of the pie with a layer of caviar and place a wreath of watercress or parsley around the edge of the plate.\n\n4. Serve with melba toast or small slices of French bread. Enough for at least 12 people.\n\nRuth and Arthur Steel are childhood sweethearts who have been happily enjoying each other for over 40 years. They live on the ocean in Bridgehampton and seem to be ready to experience just about anything from dancing at the local discotheque to embarking on a camel ride through Turkey. Attractive, well informed, and energetic, they entertain graciously and easily. I buy an inexpensive caviar from the local supermarket for this spread of Ruth's.\n\nRuth Steel's Caviar Spread *\n\n1 cup whipped cream cheese\n\n2 tbsp. minced onion\n\n3 tbsp. lemon juice\n\n8 tsp. black caviar\n\n1. Mix the first three ingredients.\n\n2. Fold the caviar in gently and serve immediately with melba toast or crackers.\n\nPickled Carrot Sticks *\n\n4 good sized carrots\n\n1 large garlic clove, sliced\n\n\u00bd tsp. red pepper flakes\n\n3 peppercorns\n\n\u00bd tsp. dry mustard\n\n2 slices of fresh ginger about the size of a dime.\n\n\u00bc cup apple cider vinegar\n\n1. Slice carrots in julienne strips.\n\n2. Place in bowl with all the rest of the ingredients and marinate overnight.\n\nKathy Fisher is a potter and makes beautiful tiles. One night she brought this spread to one of our communal meals and it was gone in an instant. If you are on a diet, you better pass it up.\n\nKathy's Herbal Butter Spread *\n\n1 cup softened butter\n\n\u2153 cup parsley, minced\n\n2 large basil leaves, minced\n\n1 large fat clove of garlic, pressed\n\n1. Mix all ingredients well and serve with an especially good bread.\n\nThe lecithin included in this recipe may be found in health food stores. This butter\/margarine substitute is healthier than either butter or margarine and always spreads easily, but it must be kept in the refrigerator when not in use. I do not include salt, but you may want to add it. It is also very easy to add herbs, garlic and spices if you want to make a special spread.\n\nButter Blend *\n\n2 sticks sweet butter, softened\n\n1 cup safflower oil\n\n2 tbsp. dried milk\n\n\u00bc tsp. lecithin\n\n1. Place all ingredients in blender and whip.\n\n2. Place in a covered container and keep in the refrigerator.\n\nI like this hot dip because it's unusual and easy to make.\n\nHot Artichoke Dip *\n\n1 cup artichoke hearts chopped into small pieces\n\n1 cup parmesan cheese\n\n\u00bd cup scallions, chopped\n\n\u00be cup mayonnaise\n\n\u00bd tsp. garlic salt\n\n\u00bd tsp. parsley, chopped\n\n\u00bd tsp. pepper paprika\n\n1. Mix all ingredients except paprika.\n\n2. Spread in a small baking dish.\n\n3. Sprinkle with paprika.\n\n4. Bake at 350 degrees for 20 to 25 minutes.\n\n5. Serve with raw vegetables, crackers, or chips.\n\nBlack Bean Dip **\n\n2 cups onions, chopped\n\n\u2153 cup peanut oil\n\n2 cups black beans, cooked\n\n1 large clove garlic, pressed with a garlic press\n\n2 tsp. cumin\n\n1 tsp. tabasco sauce\n\n3 tsp. wine vinegar\n\n1\u00bd tsp. salt\n\n\u00bd tsp. pepper\n\nCooking water from the beans\n\n1. Saute onions in oil until soft.\n\n2. Place onions in bowl and stir in beans.\n\n3. Add garlic puree, cumin, tabasco, vinegar, salt and pepper and stir.\n\n4. Place in blender and add enough of the cooking water, so that you are able to blend the mixture to a good consistency for dipping.\n\n5. Blend well and serve with corn chips.\n\n_Note_ : Beans should be soaked overnight in a large pot of water and cooked until tender the next day.\n\nSally Maruscak's Clam Dip *\n\n1 cup minced clams\n\n1\u00bd cups cottage cheese\n\n\u00bd cup sour cream or yogurt\n\n3 tbsp. minced onions\n\n2 tsp. Worcestershire Sauce juice of 1 lemon\n\n\u00bc tsp. salt\n\n\u00bd tsp. pepper\n\n2 tbsp. chives or scallions, minced\n\n1. Mix all ingredients except chives or scallions.\n\n2. Sprinkle chives or scallions on top and serve at room temperature.\n\nWho has ever heard of warm, pickled onions? I never had until they were served to me down in Memphis, Tenn. as an accompaniment to fried catfish.\n\nPickled Onions *\n\n4 cups onions, thinly sliced\n\n\u00be cups water\n\n\u00bc cup white vinegar\n\n1. Simmer the onions in water and vinegar for 5 minutes.\n\n2. Serve as a side dish, preferably with fried fish.\n\nTo make this dip, you need to save some of the water the beans were cooked in or the liquid from the can, if you used canned beans.\n\nJala-Pinto Dip **\n\n2 tbsp. oil\n\n1 medium onion, diced\n\n1\u00bd tsp. garlic powder\n\n2\u00bd tsp. chili powder\n\n\u00be tsp. cumin\n\n\u00bc tsp. oregano\n\n\u00bd cup tomato sauce\n\n2 cups cooked jala-pinto beans\n\n\u00bd cup bean juice\n\n2 tsp. jalapeno peppers, minced\n\n\u00be tsp. salt\n\n3 tsp. raw onion, minced\n\n1. Heat oil in a skillet and saute onion with garlic powder, chili powder, cumin and oregano for 4 minutes, stirring constantly.\n\n2. Add tomato sauce, beans, bean juice, peppers and salt and simmer over low heat for 15 minutes, stirring occasionally.\n\n3. Place dip in bowl, sprinkle raw onion on top and serve with corn chips, crackers, or raw vegetables.\nSoups\n\nI finally realized, after all these years, why the soups I used to make were never as good, or as hearty and tasty as I would have liked. It was because I had always believed in that old saw about good soups being made out of a few bones and whatever leftovers were around\u2014a few potatoes, a carrot etc. One night I tasted a delicious soup made by Armand and thought, \"Why can't I make something like that?\" When he donated the recipe to me, I realized that I had been scrimping too much. It may have been economical, but it didn't taste special. So here is Armand's soup, the soup that changed my style.\n\nArmand La Macchia's Chicken Soup **\n\n1 chicken, either whole or cut in pieces\n\n1 tsp. salt\n\n\u00bd tsp. pepper\n\n\u00bd tsp. poultry seasoning\n\n1 medium sized onion, chopped\n\n3 carrots, sliced\n\n\u00bd stalk celery, chopped\n\n\u00bd lb. fresh spinach\n\n1 egg. beaten\n\n4 bouillon cubes\n\n1 lb. macaroni such as shells, corkscrews etc.\n\n\u00bd lb. sweet sausage meat, removed from casings and chopped into small pieces\n\ndash of tabasco sauce\n\n1 cup Romano cheese as a side dish\n\n1. Stew the chicken in a large pot with salt, pepper, and poultry seasoning, and enough water to cover the whole chicken for about 45 minutes or until chicken is done.\n\n2. Remove any foam that has accumulated and set chicken aside.\n\n3. Add all vegetables to broth and cook for 15 minutes.\n\n4. In the meantime, take all the chicken meat from the bones and cut into small pieces. Do not use the skin.\n\n5. Return chicken pieces to pot along with the beaten egg, bouillon cubes, macaroni, sausage and tabasco sauce.\n\n6. Bring to a boil and lower heat; cook for 15 minutes.\n\n7. Check for seasonings.\n\n8. Serve in large, individual soup bowls with cheese on the side.\n\nThis soup is easy to make and a good way to use up leftover corn on the cob. Dorothy says she cuts the corn off the cob in the summer when the corn is fresh and the price is low. Then she freezes the kernels in freezer bags to be used throughout the rest of the year.\n\nDorothy's Corn Chowder *\n\n1 small onion, minced\n\n1 tbsp. butter\n\n4 cups rich chicken broth\n\n1\u00bc cups corn\n\n2 tbsp. dried skim milk\n\n\u00be tbsp. curry powder\n\nsalt and fresh ground black pepper\n\n1. Saute onion in butter until soft.\n\n2. Put onions, 1 cup chicken broth, 1 cup of the corn, skim milk, and spices in blender and blend at low speed until smooth.\n\n3. Place in pan with the rest of the broth and heat thoroughly.\n\n4. When ready to serve, garnish with the remaining corn.\n\nI have done a lot of experimenting with this particular recipe because it happens to be one of my favorite soups and I always order it when I eat at a Thai restaurant. It is not terribly difficult, but there are three crucial ingredients which must be purchased in a specialty store. I can't find any American substitutions, so if you can't buy citrus leaves, lemon grass, and galangal roots, you probably should pass this one up. I buy the coconut in a health food store so I can get a pure product. Save the pulp for making the coconut cake in the dessert section, or any other recipe.\n\nThailand Coconut Soup **\n\n2 cups coconut\n\n2 cups milk\n\n4 cups rich chicken broth\n\n2 dried citrus leaves (by maglood)\n\n2 heaping tsp. dried lemon grass (da kry)\n\n1 piece of dried galangal root(ka)\u2014about the size of a quarter\n\n2 tbsp. fish sauce (found in many Chinese sections)\n\n1 cup cooked chicken, minced\n\n\u00be cup mushrooms, sliced in small pieces\n\njuice of one nice sized lime\n\n1\u00bd tbsp. chili powder\n\n\u00bd cups milk\n\n\u00bd cup cream\n\n1 tbsp. fresh coriander, chopped, if available. Otherwise use flat Italian parsley.\n\n1. Bring the coconut and milk to a boil.\n\n2. Let sit for 20 minutes and then press through strainer. You should have \u00bc to \u00bd cup of coconut milk. Reserve.\n\n3. Place chicken broth, citrus leaves, lemon grass, galangal root and fish sauce in a soup pot.\n\n4. Bring to a boil, cover, lower heat and simmer 10 minutes.\n\n5. Add chicken and mushrooms and simmer 4 or 5 minutes.\n\n6. Add lime juice, chili powder, the coconut milk, and cream; bring to a boil, and remove from heat.\n\n7. Sprinkle with coriander or parsley and serve.\n\nAn unusual soup and especially tasty in hot summer weather, this comes from Mrs. Acheson, the mother of my good friend, Chris Italiani.\n\nMrs. Acheson's Cucumber and Buttermilk Soup *\n\n2 tbsp. oil\n\n2 lbs. cucumber, peeled and cubed\n\n1 cup onion, diced\n\n1 large potato, cooked, peeled and cubed\n\n2 cups chicken broth\n\n2 cups buttermilk\n\nsalt and pepper\n\nchives and radish slices for garnish.\n\n1. Saute cucumbers, onions, and potatoes in oil for 5 minutes.\n\n2. Add broth, bring to a boil, cover, lower heat and simmer for 20 minutes.\n\n3. Cool and then blend in blender or food processor.\n\n4. Add buttermilk, salt and pepper and cool for at least two hours in refrigerator.\n\n5. Garnish with chopped chives and radish slices.\n\nTomato-Clam Soup Oriental Style **\n\n1\u00bd dozen little neck clams\n\n4 cups rich chicken broth\n\n2 tbsp. oil\n\n3 tomatoes\n\n3 tbsp. Chinese sesame seed oil\n\nsalt and pepper\n\n1\u00bd tbsp. corn starch\n\n3 tbsp. water\n\n\u00bd cup slivered scallion tops\n\n1. Steam clams in a small amount of boiling water until the clams open.\n\n2. Remove clams from shells and set aside.\n\n3. Skin tomatoes by placing in boiling water for a minute or two and then peeling with a sharp knife.\n\n4. Chop tomatoes and saute in oil for five minutes over a low flame.\n\n5. Transfer tomatoes to a soup pot and add clams, chicken broth, sesame seed oil, salt and pepper.\n\n6. Blend corn starch with 3 tbsp. water and add to soup.\n\n7. Simmer gently for 20 minutes.\n\n8. Place in soup tureen or individual soup bowls, garnish with the scallion tops and serve.\n\nMy mother-in-law, Marcia, makes a wonderful borscht. She, in turn, says that she got the recipe from her mother, Chaya, and that \"they don't make borscht like that anymore.\" This recipe calls for dried lima beans and if you prefer to use frozen ones, omit the directions for dry lima beans and add the frozen ones at the end with the potatoes. Marcia says that the secret of this borscht is to get the right blend of lemon and sugar. Borscht is a hearty soup that is delicious on a cold night with black bread.\n\nChaya's Borscht **\n\n\u00bd cup dried lima beans\n\n2\u00bd qts. water\n\n2 or 3 small veal or beef soup bones\n\n1 small piece of flanken, chuck steak, or beef short ribs\u2014about 2 lbs.\n\n1 large onion, diced\n\n1 8-oz, can tomato sauce\n\n1 lb. head of cabbage, shredded and grated\n\n1 bunch of beets, peeled and grated juice from 1 lemon, more if necessary salt and pepper\n\n\u00bc cup sugar\n\n1 large potato, peeled and cut in quarters\n\n2 cloves garlic, peeled\n\n1. Soak the lima beans overnight in a large pot of water.\n\n2. Drain beans and place in a soup pot with 2\u00bd quarts of water, the soup bones and flanken or other cut of meat.\n\n3. Bring water to a boil, skim off all foam; add remaining ingredients except for potatoes and garlic.\n\n4. Cook over low heat for 2 to 3 hours, or until lima beans are done.\n\n5. Add the potatoes and garlic and cook 20 minutes more.\n\n6. Check to see if you might need more lemon and\/or sugar and serve piping hot.\n\nMy husband Marty and I were in Zurich, Switzerland for a few days and having a frustrating time finding a nice place to eat. We made a long trip to one restaurant specializing in fondu and found that they were \"all out,\" if you can imagine that in Switzerland! We were staying in a charming place on a hill run by the Ladies Temperance Society, so no wine, of course, in _their_ dining room. They had posted a very emphatic announcement that no liquor would be served \"even if prescribed by a doctor.\" Finally someone suggested a place right around the corner and, surprisingly enough, right under our noses, there was an excellent restaurant. Marty had a lovely soup and when I returned home, I tried to duplicate it. It must have a very rich chicken broth as a base.\n\nZurich Avocado Soup **\n\n1 medium onion, minced\n\n1 medium green pepper, seeded and minced\n\n1 medium red pepper, seeded and minced\n\n1\u00bd cup mushrooms, sliced\n\n3 tbsp. butter\n\n6 cups rich chicken broth\n\n1 large avacado, ripe but firm, peeled and cut in small chunks\n\nkernels from two large ears of corn, or \u00bd to 1 cup frozen corn\n\n\u00bd tsp. salt\n\n\u00bc tsp. black pepper\n\n\u00bc tsp. cayenne pepper\n\n2 fresh basil leaves, minced\n\n2 tbsp. flour\n\n4 tbsp. water\n\njuice from 1 lemon\n\n\u00bd cup heavy cream\n\n1. Saute onions, peppers and mushrooms in butter until soft.\n\n2. Transfer onions, peppers, and mushrooms to a large soup pot and add chicken stock, avocado, corn and all the spices; cook over low heat for \u00bd hour.\n\n3. Mix flour and water together, add to soup and cook 10 minutes.\n\n4. Stir in lemon juice and heavy cream and cook for 10 more minutes.\n\n5. Serve very hot with crusty French bread.\nSalads and Dressings\n\nAn excellent and healthy recipe for tofu, this comes from a lady named Rhoda whom I have never met. She did some pasteup work for my first cookbook and sent me this to include in my second one. For those people who love egg salad, but don't want to include too many eggs in their diet, tofu salad is an awfully good substitute. It can also be used as a sandwich spread on a good bread.\n\nRhoda's Tofu Salad *\n\n12 oz. tofu\n\n1 stalk celery, chopped\n\n\u00bc cup minced onion\n\n1 tbsp. minced parsley\n\n\u00bc tsp turmeric\n\n\u00bd tsp. dill leaves\n\n\u00bd tsp. salt\n\n\u00bc tsp. dry mustard\n\n\u00bc tsp. celery salt\n\n\u00bc tsp. pepper\n\nsalad greens\n\n1\u00bd cups alfalfa sprouts\n\n1\u00bd tbsp. mayonnaise\n\n1. Combine all ingredients in blender except sprouts and greens and mix well.\n\n2. Serve on greens and sprinkle sprouts on top.\n\nAt one of our communal meals, a young man spoke with great enthusiasm about how he loved to create dishes and asked if he could send me some of his creations for my cookbook. It was with a certain surprise that I discovered that the dish that intrigued me the most at this dinner was his carrot salad.\n\nJohn's Carrot Creation **\n\n3 cups raw carrots, shredded\n\n\u00bd cup feta cheese, broken up in small bits\n\n\u00bd cup olives without the pimentoes\n\n2 scallions, cut in 1-inch slivers\n\n\u2153 cup pignoli nuts\n\n\u00bc cup olive oil\n\njuice of one large lemon\n\n\u00bc tsp. garlic salt\n\n\u00bc tsp. white pepper\n\n1. Mix carrots, cheese, olives, scallions, and nuts with the olive oil until well coated.\n\n2. Add lemon juice, garlic salt and pepper; stir.\n\nJudy Wolfe, or just plain Wolfe as she likes to be called, has also collected her friends' recipes and lent me her collection when I was doing this one. These are two nice side dishes that she got from her friend Carlos Lindgrin, so you can see how recipes get passed around out here. I don't know Carlos, but Judy is a very special friend. Not only is she invaluable in helping my husband and me in our publishing business, but she is great fun to have around. She has a unique way of expressing herself and always seems to say the unexpected. She has often done the unexpected too. When she was younger, she decided that she would have a child at 20 and then one at 40. After raising son Daniel, she gave birth to Conor 21 years later, only one year off schedule.\n\nCarlos' Spicy Chinese Cukes *\n\n2 cucumbers, peeled and sliced very thin\n\n4 tbsp. light sesame oil\n\n4 tbsp. tamari\n\n2 tbsp. sugar\n\n\u00bd tbsp. salt\n\n\u00bd tbsp. tabasco\n\n2 tbsp. rice vinegar\n\n1. Mix cucumbers with the sesame oil until well coated.\n\n2. Mix the tamari, sugar, salt and tabasco in a small bowl and add to the cucumbers.\n\n3. Last, add the vinegar, mix, and chill before serving.\n\nCarlos' Bean Sprout Salad *\n\n3 cups fresh bean sprouts\n\n\u00bc cup scallions, chopped\n\n\u00bc cup pimento, cut in thin strips\n\n1 garlic clove, minced\n\n\u00bc cup peanut oil\n\n\u00bd tbsp. salt\n\n1 tbsp. pepper\n\n2 tbsp. tamari\n\n2 tbsp. rice vinegar\n\n2 tbsp. ground, toasted sesame seeds\n\n1. Mix sprouts, scallions, pimento, and garlic.\n\n2. Add oil, salt and pepper and mix well.\n\n3. Add tamari and vinegar and mix again.\n\n4. Sprinkle ground sesame seeds on top and chill for 20 minutes before serving.\n\nThis is best prepared one day in advance. Yogurt can be substituted for the sour cream.\n\nMrs. Acheson's Beet and Apple Salad *\n\n\u00be tsp. dry mustard\n\n1 tbsp. sugar\n\n\u00bd tsp. Worcestershire Sauce\n\n2 tbsp. wine vinegar\n\n\u00bc tsp. salt\n\n\u00bc tsp. pepper\n\n\u00be cup sour cream\n\n1 16 oz. jar of pickled beets, drained\n\n2 medium apples, peeled and cubed\n\n1 medium cucumber, peeled and cubed\n\n1. Mix mustard, sugar, Worcestershire sauce, vinegar, salt, pepper and sour cream.\n\n2. In another bowl, mix the drained beets with apples and cucumber.\n\n3. Add dressing to the beet mixture and stir well.\n\n4. Chill overnight before serving.\n\nDuring a visit to Baltimore we were given an unexpected dinner invitation. The fine meal was cooked by a psychiatrist named Rita Lautz. She graciously donated this salad recipe. It should be served as soon as you add the hot dressing.\n\nRita Lautz's Hot Salad **\n\n8 slices lean bacon\n\n1 small onion, minced\n\n2 eggs beaten\n\n2 tbsp. cornstarch\n\n\u00bd cup cider vinegar\n\n\u00bd cup sugar\n\n1 tsp. salt\n\n1 tsp. pepper\n\n1 head of lettuce or other salad greens, washed, dried, and torn into pieces\n\n1. Fry bacon and onions over low heat until bacon is done.\n\n2. Drain bacon and onions, cool and cut bacon into small pieces. Reserve.\n\n3. Add cornstarch to beaten eggs and stir well.\n\n4. Put into small saucepan and add vinegar, sugar, salt and pepper.\n\n5. Stir over medium heat until thickened.\n\n6. Put greens into a serving bowl and toss with dressing.\n\n7. Add bacon and onion and toss lightly.\n\n8. Serve immediately.\n\nWe have a friend named Michael Barrett who doesn't cook at all. He bought a very good restaurant in the city so he could eat nicely. One day, out at his country house, he said, \"I'm going to have lunch at my friend Alan's house. Why don't you all come with me?\" He then called Alan and said, \"I'm bringing some friends for lunch. Put more apples in the chicken salad.\" Alan is a most gracious host. Michael brought at least eight of us. We spent all day in his beautiful pool and ate a lovely chicken salad with apples cooked by a young man who was just learning to be a chef. You can leave out the vodka as I usually do. It's just as good.\n\nA Special Chicken Salad ***\n\n2 chicken breasts, skinned, boned, and cut into bite sized pieces\n\n1 tbsp. vodka\n\n5 tbsp. orange juice\n\n2 tbsp. olive oil\n\n2 tbsp. dry vermouth\n\n\u00bd cup diced celery\n\n\u00bc cup diced onions\n\n\u00bd cup diced cucumber\n\n2 apples, skinned and diced\n\n\u00bd cup mayonnaise\n\n\u00bc cup yogurt\n\n3 tbsp. Dijon mustard\n\nsalt and pepper\n\n\u00bd cup parsley, minced\n\n1. Marinate chicken pieces in vodka and orange juice overnight, or at least 4 hours.\n\n2. Drain chicken and saute in olive oil over medium heat until done (when chicken has turned white and lost its pink color).\n\n3. Put chicken in a bowl with vermouth, celery, onion, cucumber, and apples.\n\n4. Make a sauce of mayonnaise, yogurt, mustard, salt and pepper.\n\n5. Mix with the chicken and garnish with parsley.\n\nA couple of years ago, I was in New York City studying acting with Uta Hagen, a fine actress, an excellent teacher and a wonderful cook, who, incidentally, has written a very fine cookbook of her own. In each of her classes two people would donate food for all to share during the break. I thought that this donation was a nice variation of a common dish.\n\nCurried Tuna Salad *\n\n2 6\u00bd cans tunafish, drained\n\n1 small onion, minced\n\n\u00be cup mayonnaise\n\n1 tbsp. curry powder\n\n1 cup cooked peas, either canned or frozen, but well drained.\n\nsalad greens\n\n1. Chop tunafish well and add minced onions.\n\n2. Mix mayonnaise with the curry powder and add to tunafish. Mix well.\n\n3. Gently add peas to mixture and serve on bed of salad greens.\n\nChris Italiani is a good friend who teaches languages at the high school out here. She is warm, independent and has a good sense of humor. Widowed when her son was only two years old, she has had to learn to pinch pennies at various times in her life and this recipe is a good example of how she did it. I don't like any of the potato salads made in delicatessens, but I love this one that Chris makes.\n\nChris' Potato Salad **\n\n6 cups peeled, cubed potatoes\n\n\u00bd cup finely diced carrots\n\n\u00bd cup finely diced celery\n\n1 cup minced onion\n\n\u00bd cup sliced olives\n\n\u00bc cup safflower oil\n\n2 tsp. dry mustard\n\n3 tbsp. dried dill (or 1 cup fresh dill)\n\n\u2153 cup minced parsley\n\nsalt and pepper\n\n5 tbsp. vinegar\n\n1. Put potatoes, carrots, celery, onions, and olives in a large salad bowl.\n\n2. Make a dressing of the oil, mustard, dill, parsley, salt, and pepper; gently toss the potato salad to coat well.\n\n3. Add the vinegar and toss briefly again.\n\nMy husband was determined to grow his own potatoes out here on Long Island. What we harvested were lots of tiny and odd shaped potatoes, too small or strange to peel. I remembered having some kind of hot potato salad with bacon out in Indiana many years ago. Hence this dish. I used unskinned new potatoes, but most people like them skinned.\n\nHot Potato Salad **\n\n2 lbs. potatoes\n\n5 slices of bacon\n\n2 tbsp. bacon drippings\n\n1\u00bd tbsp. raw sugar\n\n1\u00bd tsp. flour\n\nsalt and pepper\n\n\u2153 cup cider vinegar\n\n\u2154 cup water\n\n1 large onion, chopped\n\n2 tbsp. chopped dill\n\n2 tbsp. chopped parsley\n\n1. Boil potatoes until done; drain and slice.\n\n2. Fry bacon until crisp, drain well and crumble.\n\n3. Remove all but about 2 tbsp. of bacon drippings from pan.\n\n4. Add sugar, flour, salt, pepper, vine gar, and water and cook slightly over low heat stirring to make a smooth sauce.\n\n5. Add onion and cook 3 minutes.\n\n6. Add potatoes and bacon and cook to heat through.\n\n7. Place in serving bowl and sprinkle with dill and parsley.\n\nI sometimes see pictures of dishes that look so pretty that I cut them out to try and duplicate them at a later time. This was such a salad and all I have is the picture. The different ingredients are not mixed together, but put on the plate separately and arranged in a pretty pattern. Then I pour the dressing over it. I like a simple dressing best, made with lemon juice and olive oil.\n\nSuper Salad *\n\n1 head Boston lettuce, separated into leaves\n\n2 Belgian endives, separated into leaves\n\n1 cup radishes, sliced\n\n1 large avocado, skinned and cubed\n\n2 cups orange sections _Dressing_\n\n4 tbsp. olive oil\n\njuice from 1 to 2 lemons\n\nsalt and pepper\n\n1. Arrange all ingredients on a platter.\n\n2. Pour dressing over it and serve immediately.\n\n1. Put in small jar and shake well before pouring over salad.\n\nToo often, the zucchini in my refrigerator ends up being thrown away because it has started to spoil. One of the reasons I have it in the first place is because it seems to grow abundantly in so many of our friend's gardens and they don't know what to do with it either. The following recipe is my own invention and came out of my determination not to let another zucchini get relagated to the garbage can.\n\nMarinated Zucchini Salad **\n\n3 tbsp. safflower oil\n\n3 zucchini, sliced\n\n2 stalks celery, sliced\n\n3 tbsp. water\n\n2 cloves garlic, minced\n\n1 medium onion, diced\n\n2 tomatoes, diced\n\n1\u00bd cups any kind of cooked beans (green beans, kidney beans, fava beans or chickpeas). Freshly cooked beans are best. Frozen or canned will do.\n\n3 tbsp. fresh parsley\n\n\u00bd tsp. basil\n\n\u00bd tsp. chervil\n\n6 tbsp. olive oil\n\n6 tbsp. wine vinegar\n\nsalt and pepper\n\n1. Saute zucchini and celery in safflower oil for a few moments.\n\n2. Add 3 tbsp. water, cover, turn heat low and cook until the zucchini is done. Do not overcook; it should be crunchy.\n\n3. Drain zucchini and celery and place in a salad bowl with garlic, on ions, and tomatoes.\n\n4. Drain beans well and add to bowl along with parsley, basil and chervil.\n\n5. Add olive oil and toss salad thoroughly.\n\n6. Add vinegar, salt and pepper and mix well.\n\n7. Refrigerate for at least 3 hours before serving.\n\nAlong with the soup that we had in Zurich, Switzerland, which is included in the last chapter, we also had a lovely salad. This can be used as a main dish for a summer meal or as a very fancy accompaniment to dinner.\n\nZurich Shrimp and Avocado Salad ***\n\ngreens for 4 people\n\n1 cup alfalfa sprouts\n\n1 large avocado, sliced\n\n2 tomatoes, sliced\n\n1 large tart apple, peeled and sliced\n\n\u00bd lb. cooked shrimp, cleaned\n\n1 cup mayonnaise\n\n\u00bc cup catsup\n\n\u00bc cup milk\n\n3 tbsp. horseradish\n\nsalt and pepper\n\n2 scallions, chopped fine\n\n1. Arrange greens and sprouts in individual salad bowls.\n\n2. Divide avocado, tomatoes, apple and shrimp and put some in each bowl.\n\n3. When ready to serve, mix mayonnaise, catsup, milk, horseradish, salt and pepper and put some on top of each salad.\n\n4. Garnish with scallions.\n\nChicken Salad a la Balducci **\n\n4 chicken breasts\n\n1 small onion, sliced\n\n1 stalk celery, sliced\n\n1 cup cooked artichoke hearts, sliced\n\n6 cherry tomatoes\n\n\u00be cup sour cream\n\n\u00bd cup mayonnaise\n\n\u00bd cup yogurt\n\n4 tbsp. tarragon\n\nsalt and pepper\n\n1. Simmer chicken breasts, onion, car rots and celery in 2 quarts water until chicken is done.\n\n2. Reserve broth for soup and prepare chicken by de-boning and removing skin.\n\n3. Cut chicken into bite-sized chunks.\n\n4. Mix chicken, artichoke hearts, and tomatoes.\n\n5. Mix sour cream, mayonnaise, yogurt, tarragon, salt and pepper in a separate bowl.\n\n6. Place chicken in a serving bowl and pour the dressing over it, stirring well. 6. Chill before serving.\n\nGreek Marinated Vegetables **\n\n1 small head of cauliflower, trimmed and broken into flowerets\n\n\u00bd pound mushrooms, wiped off and sliced\n\n1 8\u00bd oz. can artichokes, drained and cut in half\n\n\u2153 cup olive oil\n\n1 large clove garlic, minced\n\n\u00bd tsp. oregano\n\n\u00bd tsp. basil\n\nsalt and pepper\n\n\u2153 cup wine vinegar\n\n1. Steam cauliflower until done. It should be crunchy, not mushy. Drain.\n\n2. Place in salad bowl with mushrooms and artichokes.\n\n3. Add olive oil, garlic, herbs, salt and pepper; toss well.\n\n4. Add vinegar; toss again and serve chilled.\n\nI usually like some kind of salad with dinner, but an ordinary salad doesn't seem to fit in if the meal is Chinese. I had this one time at someone's house and rushed home to add it to my collection.\n\nChinese Cabbage Salad *\n\n1 small cabbage, shredded\n\n1 onion, sliced thin\n\n4 tbsp. sesame oil\n\n3 tbsp. rice wine vinegar\n\nsalt\n\n1. Parboil the cabbage and onions in boiling water until they are softened but still crunchy. Drain.\n\n2. Mix the cabbage and onions with the sesame oil to coat well.\n\n3. Add the vinegar and salt and let sit a couple of hours before serving.\n\nThis is another simple salad to accompany Chinese dishes which I concocted because I like something cold and crisp with Chinese food. You could substitute almost anything as long as you use rice wine vinegar and sesame oil.\n\nChinese Salad *\n\n1 cucumber, peeled and sliced very thin\n\n1 tomato, sliced thin\n\n1 green pepper, seeded and cut into thin strips\n\n1 onion, sliced and separated into rings\n\n\u00bc tsp. celery salt\n\nsalt and pepper\n\n3 tbsp. dark sesame oil\n\n3 tbsp. rice wine vinegar\n\n4 tbsp. sesame seeds\n\n1. Mix all vegetables in a salad bowl.\n\n2. Add celery salt, salt, pepper and oil; toss well.\n\n3. Add vinegar and mix.\n\n4. Sprinkle with sesame seeds.\n\nA pretty and vivacious woman named Belle contributed a rather difficult but very good oriental noodle dish elsewhere in this book. However, she also gave me this simple salad dressing to balance it out. I make enough to have on hand for a couple of days.\n\nBelle's Mustard Dressing *\n\n\u00be cup olive oil\n\n1\u00bc tsp. dry mustard\n\n1 tsp. sugar\n\n\u00bd tsp. salt\n\n\u00bd tsp. pepper\n\n1 tsp. Worcestershire sauce\n\n\u00bc cup good wine vinegar\n\n1. Place all ingredients in a covered jar and shake well before using it on salad.\n\nSour Cream\/Roquefort Dressing *\n\n1 \u00bc oz. Roquefort cheese\n\n1 tbsp. oil\n\n\u00bd cup sour cream\n\n2 tbsp. lemon juice\n\nsalt and pepper\n\n1. Crumble cheese and mix well with oil.\n\n2. Add the rest of the ingredients and stir.\n\nSour Cream Dressing *\n\n\u00bd cup sour cream or yogurt\n\n3 tbsp. wine vinegar\n\n\u00bd tsp. sugar\n\n\u00bc tsp. salt\n\n\u00bc tsp. paprika\n\n\u00bd tsp. dry mustard\n\n1\u00bd tsp. tomato paste or puree freshly ground black pepper\n\n1. Mix all ingredients and pour on salad; then add freshly ground black pepper.\n\nTomato Dressing *\n\n\u2154 cup safflower oil\n\n\u2153 cup red wine vinegar\n\n\u00bc cup tomato juice\n\n1 tsp. grated onion\n\n1 tsp. grated green pepper\n\n\u00bc tsp. dry mustard\n\n\u00bd tsp. Worcestershire sauce\n\ndash of tabasco sauce\n\n1 clove garlic\n\nsalt and pepper\n\n1. Mix all ingredients in a jar and store in refrigerator.\n\n2. Shake well before using on salads.\nRice, Potatoes, Noodles, Barley, Cornmeal, and So Forth\n\nJeanne Humphries had a couple of nice recipes in my last cookbook and willingly donated another to this one. The pilaf can be the basis for a whole meal and it's particularly good with a roast pork loin, I found.\n\nJeanne's Pilaf *\n\n5 tbsp. butter\n\n1 large clove garlic, minced\n\n2 medium onions, chopped\n\n1\u00bd stalks celery, sliced diagonally\n\n1 small green pepper, seeded and chopped\n\n1 cup brown rice\n\n2\u2154 cup boiling water\n\n2 bouillon cubes\n\n\u00bd tsp. salt\n\n\u00bc tsp. black pepper\n\ndash of cayenne pepper\n\n1 generous cup raisins\n\n1. Saute garlic, onions, celery and green pepper in butter until onions are transparent.\n\n2. Add rice and saute 5 minutes more over low heat, stirring frequently.\n\n3. Add boiling water, bouillon cubes, salt, black pepper, cayenne pepper and raisins.\n\n4. Stir over high heat until the mixture comes to a boil, then reduce heat, cover, and simmer until rice is done (50 to 60 minutes).\n\nPriscilla Bollard was my speech teacher many years ago when I attended The Neighborhood Playhouse School of the Theatre in New York. I have always remained friends with her because she is a very special person, warm and caring. When her husband died tragically, she raised her three young children and went back to college receiving a doctorate in speech pathology. She served me this rice stuffing years ago and it's a nice change from the usual bread and giblets. It should be enough to stuff a 4 to 6 lb. chicken.\n\nPriscilla's Rice and Apricot Stuffing *\n\n2 cups cold rice\n\n3 tbsp. melted butter\n\n3 tbsp. chopped parsley\n\n3 tbsp. chives\n\n\u00bd cup finely chopped celery tops\n\n\u215b tsp. thyme\n\n\u215b tsp. cloves\n\n\u215b tsp. mace\n\nsalt and pepper\n\n1 cup finely chopped dried apricots\n\n1. Mix all ingredients together.\n\n2. Chill in refrigerator for one hour before stuffing.\n\nMy friend, Don Mathews, from Argentina had a recipe for a delicious stew in my first cookbook. However, he said that what I listed as Argentine Potato Pie in that book was really a \"tortilla\" and he would send me the recipe for the authentic Argentine Potato Pie. He also remarked that it was a good dish to make ahead of time and it could be stored nicely in the freezer. He wished us all \"Buen provecho.\"\n\nDon Mathew's Argentine Potato Pie **\n\n1 tbsp. oil\n\n1 clove garlic, chopped\n\n1 medium onion, chopped\n\n2 lbs. ground lean beef\n\nsalt and pepper to taste\n\n\u215b tsp. red pepper\n\n1 bay leaf\n\n12 green olives, chopped\n\n1 tbsp. seedless raisins\n\n2 hardboiled eggs, sliced\n\n2 lbs. mashed potatoes seasoned with\n\nsalt, pepper, and a little nutmeg\n\n1 egg yolk, beaten\n\n1. Heat oil in skillet and saute garlic and onion until transparent.\n\n2. Add ground meat, salt, pepper, red pepper, and bay leaf and stir fry until meat has lost its pink color.\n\n3. Remove from heat and add olives and raisins.\n\n4. Pour into a large deep casserole and arrange the sliced eggs over the meat.\n\n5. Cover with mashed potatoes.\n\n6. Smooth the surface and then scratch with a fork to decorate.\n\n7. Baste with the egg yolk and place in a 350 degree oven until warm\u2014about 20 minutes. If you have frozen it, of course, it will take longer to heat.\n\nI always have leftover rice somehow, so was happy to be given this dish by my friend, Ginger. It can be made ahead of time and used for any meal, including breakfast for a change.\n\nRice Pancakes *\n\n1 cup cooked rice, white or brown\n\n1 tbsp. melted butter\n\n1 cup skim milk\n\n1 tsp. salt\n\n1 tsp. sugar\n\n1 egg, beaten\n\n1 cup flour, sifted\n\n2 tsp. baking powder, sifted\n\n4 tbsp. safflower oil\n\n1. Mix all ingredients together.\n\n2. Heat oil in heavy frying pan.\n\n3. When oil sizzles, drop by large spoonfuls in pan and fry until nicely browned.\n\n4. Flip onto other side and brown.\n\n5. Serve plain with salt and pepper, or with sour cream or yogurt.\n\nWhen my friend, Craig Braun came out to visit last summer he brought the delightful Belle (whose mustard dressing appeared earlier) with him. She proceeded to make us a delicious and exotic dish. Later on, she sent me the recipe; meticulously written on three pages of paper with all kinds of advice and hints. Her dish was very, very hot and I have modified it somewhat. You may even want to make it milder. There are many things you can add, says Belle, but to simplify it, I am just giving the standard version. It is not as difficult as it may seem, especially if you assemble everything first. If it becomes a little too dry, add more sesame oil.\n\nBelle's Chinese Noodles ***\n\n\u00bc cup hoisin sauce (found in the Chinese section of most supermarkets)\n\n1 tbsp. sesame oil\n\n1 tsp. soy sauce\n\n1 tsp. Chinese chili sauce with garlic\n\n\u00bc cup raw sugar\n\n\u00be tsp. cayenne pepper\n\n1 chicken breast, cooked and cut in small pieces\n\n\u00bd lb. bean curd, cut in small pieces\n\n10 oz. thin noodles, preferably whole wheat\n\n\u215b cup sesame oil\n\n2 eggs, beaten\n\n1 bunch of scallions, chopped in 2 in. pieces.\n\n1 cup peanuts, ground\n\n\u00bc cup sesame seeds\n\n1. Mix hoisin sauce, sesame oil, soy sauce and Chinese chili sauce together.\n\n2. Mix sugar with the cayenne pepper.\n\n3. Cook and chop chicken into small pieces; cut bean curd into small pieces.\n\n4. Heat a large pot of water to boiling and drop noodles into water.\n\n5. Cook until _al dente_ and drain well.\n\n6. Heat the \u215b cup sesame oil in wok and then add noodles.\n\n7. Stir fry for a moment just to coat them.\n\n8. Make a hole in the center of noodles, pour in eggs and scramble briefly.\n\n9. Add hoisin sauce mixture and stir well.\n\n10. Add raw sugar and cayenne and stir again.\n\n11. Add chicken, bean curd, scallions, and peanuts and stir fry until everything is well mixed.\n\n12. Sprinkle with sesame seeds and serve hot or cold.\n\nSesame Noodles is an unusual and interesting dish given to me by Judy Wolfe. It has taken me a number of times to get the measurements correct since Judy just lists ingredients. You may want to doctor them a little to suit your taste. When I finally made it successfully for one of our music sessions, four of my friends asked for the recipe. I am not sure of the geographic origins of this dish, but it is unique, can be served hot or cold and is a little easier to prepare than Belle's Chinese Noodles. You can add diced, cooked white chicken if you wish.\n\nSesame Noodles **\n\n1 lb. thin linguine or whole wheat noodles\n\n2 tbsp. dark sesame seed oil with chilis\n\n\u00bd cup sesame seed paste (tahini)\n\n1 cup chunky peanut butter\n\n2 tbsp. rice wine vinegar\n\n2 tbsp. raw sugar\n\n2 tbsp. soy sauce (tamari) water\n\n2 tbsp. dark sesame seed oil\n\n2 tbsp. peanut oil\n\n\u2153 cup minced scallions\n\n1. Cook noodles in large pot of boiling water.\n\n2. While noodles are cooking, make a smooth sauce of dark sesame seed oil with chilis, sesame seed paste, peanut butter, vinegar, sugar, soy sauce and enough water to give it a smooth consistency.\n\n3. Drain noodles and coat well with the remaining dark sesame seed oil and the peanut oil.\n\n4. Toss the noodles with the sauce. You may have to add a little more water in order to toss it well.\n\n5. Sprinkle with scallions.\n\nIf you want to make your own granola and avoid all the sugar in the supermarket brands, here's one way to do it.\n\nHomemade Granola **\n\n4 cups oats\n\n1 cup sesame seeds\n\n1 cup cornmeal\n\n1\u00bd cups whole wheat flour\n\n1 cup wheat germ\n\n\u00bd to 1 cup chopped, unsalted nuts\n\n\u00bd cup honey\n\n\u00bd cup safflower oil\n\n1 cup cold water\n\n1. Mix all dry ingredients.\n\n2. Mix honey, oil and water and stir thoroughly into dry ingredients.\n\n3. Spread on bottom of roasting pan or cookie sheet and place in a 250 degree oven.\n\n4. Heat for 1\u00bd hours and stir every 15 minutes or so in order to prevent burning.\n\n5. Place in a covered jar to store.\n\nThis is Marty's version of nachos. We had it in a splendid Mexican restaurant up near Lake Placid, N.Y.\n\nMarty's Cheese Nachos **\n\n10 corn tortillas (usually available in supermarkets)\n\ncorn oil\n\n1\u00bd Montery Jack cheese, grated\n\n\u2154 to 1 cup hot jalapeno peppers, sliced. (The amount depends on how hot you like your food.)\n\n1. Stack tortillas on top of each other and cut in sixths.\n\n2. Add enough corn oil to cover the bottom of a large frying pan. You might have to cook these in shifts; you should have enough oil so that it bubbles around the tortillas.\n\n3. Fry over medium heat until the tortillas get a little stiff, turning once. This should take only a few minutes.\n\n4. Drain on paper towels and then spread in a large, flat baking pan. The whole bottom of the pan should be covered, but the tortillas should not overlap, so you may have to use 2 pans.\n\n5. Preheat oven to 375 degrees.\n\n6. Sprinkle cheese over tortillas and then add the peppers. You should try to see that each tortilla has at least one slice of pepper.\n\n7. Cook in oven for about 10 minutes or until cheese is melted and bubbly.\n\n8. Serve immediately. It's very nice with ice cold beer in frosted glasses with a slice of lemon or lime.\n\nPolenta with Gorgonzola **\n\n\u00bc tsp. salt\n\n2 cups cornmeal, yellow or white\n\n3 tbsp. butter\n\n\u00bc lb. gorgonzola cheese\n\n\u00bc cup parmesan cheese\n\n1. Bring 2 quarts water to a boil and add salt.\n\n2. Gradually stir in cornmeal with a wire whisk.\n\n3. Lower heat and cook for 40 minutes, stirring often with a wooden spoon. It _will_ stick to the sides and bottom of the pan.\n\n4. Very lightly grease a square Pyrex dish and pour the cornmeal into it.\n\n5. Cool for a couple of hours or overnight.\n\n6. When ready to cook, melt butter in a saucepan.\n\n7. Crumble gorgonzola cheese over top of cornmeal.\n\n8. Then drizzle butter and parmesan cheese over it and bake in a 350 degree oven for about 20 minutes or until heated thoroughly.\n\n9. Cool slightly, cut into squares and serve.\n\nThis is another way to use up cooked rice which my family likes. I use hot chilis. If you don't like spicy foods, use green chilis.\n\nHot Chili Rice *\n\n2 large hot chilis or 1 4-oz. can green chilis\n\n1\u00bd cups yogurt\n\n1\u00bd cups sour cream\n\n3 cups cooked rice\n\n\u00be pound Montery Jack cheese.\n\n1. Chop chilis into small pieces.\n\n2. Mix chilis with yogurt and sour cream.\n\n3. Place half of the rice in a buttered casserole and pour half of the sauce over it.\n\n4. Cut half of the cheese into strips and place over casserole mixture.\n\n5. Add the rest of the rice, the rest of the sauce and cut the remaining cheese into strips and place on top.\n\n6. Bake at 350 degrees, covered for 30 minutes.\n\n7. Remove cover and bake 10 minutes more.\n\nI love french fries, but I don't want to eat them; there's too much fat from deep frying and too much salt. Here's a terrific substitute.\n\nBaked Potato Crisps *\n\n6 nice sized potatoes\n\ngarlic salt\n\npaprika\n\npepper\n\n1. Peel potatoes and cut into thin round slices.\n\n2. Sprinkle with garlic salt, paprika and pepper.\n\n3. Place slices in a large, oiled, flat baking dish, taking care that they don't overlap.\n\n4. Bake in a 400 degree oven for 30 to 40 minutes until nicely browned.\n\nMy grandmother used to make this hearty dish and only recently I found that its proper name is \"Bubble and Squeak.\" It comes from England and is properly made with bacon drippings, but I prefer to use safflower oil.\n\nMy Grandmother's Bubble & Squeak **\n\n\u2153 cup safflower oil\n\n4 cups cabbage, shredded and cooked\n\n2 cups potatoes, cooked, skinned, but not mashed\n\nsalt and pepper\n\n1. Mix cabbage well with the cooked potato. The potatoes should be slightly mushy to mix well with the cabbage.\n\n2. Add salt and plenty of pepper.\n\n3. Heat oil in a large, heavy frying pan and then add the mixture.\n\n5. Press it down to make a flat cake and let the bottom get nicely browned. Do not turn.\n\n6. Serve it piping hot and cut in wedges.\n\nHow's this for an unusual way to use oatmeal?\n\nOatcakes *\n\n1 onion, minced\n\n1 clove garlic, minced\n\n1 small green pepper, minced\n\n1 green chili, chopped fine\n\n1\u00bd cups oatmeal\n\n1\u2153 cups yogurt\n\n\u00bc tsp. salt\n\n\u00bc cup safflower oil (more if needed)\n\n1. Mix all ingredients except oil.\n\n2. Heat the oil in a large frying pan and drop the oatcake mixture onto pan as you would pancakes.\n\n3. Fry until brown; then brown on the other side.\n\nI haven't cooked many dishes made with barley. A friend of mine named Rose knew that my husband loves it, so she donated this to my cookbook and whenever I make it, Marty never fails to remark on how good it is.\n\nMarty's Favorite Barley Dish **\n\n3\u00bd cups chicken broth\n\n3 cups beef broth\n\n1 cup barley\n\n\u00bd cup onion, chopped\n\n2 stalks celery, chopped\n\n3 tbsp. safflower oil\n\n\u00bd cup white wine\n\n\u00bd cup parmesan cheese\n\n\u00bd cup mushrooms, sliced salt and pepper\n\n1. Bring 3 cups of the chicken broth and all of the beef broth to a boil and add barley.\n\n2. Turn heat down to low; cover and simmer 1 hour.\n\n3. Saute onions and celery in oil for a few moments.\n\n4. Add wine and continue cooking until the wine is reduced to about half.\n\n5. Add onion mixture to barley along with the extra half cup of chicken broth and simmer 10 minutes.\n\n6. Stir in parmesan cheese, mushrooms, salt and lots of freshly ground black pepper.\n\n7. Cook about 4 minutes more.\nPasta\n\nBernice Hunt's Pasta Alla Carbonara **\n\n1 pound spaghetti or fettuchine\n\n4 slices bacon\n\n2 eggs\n\n1 cup heavy cream (more, if needed; see note)\n\n\u00bd cup freshly grated parmesan cheese\n\n\u00bd stick butter\n\nsalt and pepper\n\n1. Boil the pasta in salted water according to package directions; do not overcook.\n\n2. Fry the bacon until crisp; drain on paper towels and cut or crumble into small pieces.\n\n3. Beat the eggs in a bowl until frothy; then beat in the cream and cheese.\n\n4. Melt the butter in a very large skillet and add the egg mixture. Cook, stirring, until eggs are barely beginning to set.\n\n5. Add the hot, drained pasta at once and stir in the bacon.\n\n6. Season to taste and serve with additional cheese at the table.\n\n_Note_ : If pasta seems too dry you can add a little more cream.\n\nCarol Lindemann called one late afternoon and said, \"Let's make a dinner together. I have a friend who can make pasta and I'll bring artichokes.\" Carol is a psychologist who specializes in phobias. She lives in New York, but comes out to her country place on weekends. She's nice to spend time with, so I said sure. Over she came with bags of groceries and a young man who proceeded to make a lovely pasta in a matter of minutes. This is it.\n\nA Young Man's Pasta *\n\n1 lb. spaghetti, cooked\n\n5 cloves garlic, minced\n\n5 tbsp. olive oil\n\n1 2-lb. can Italian plum tomatoes\n\n\u00bd cup pitted black olives, sliced\n\nsalt and pepper to taste\n\n1 cup grated parmesan cheese\n\n1. Cook pasta and drain\n\n2. Saute the garlic in olive oil until golden, not brown.\n\n3. Drain the tomatoes well and add them to the pan.\n\n4. Mash and stir the tomatoes with a wooden spoon for a few minutes.\n\n5. Add the black olives, salt, and pepper and cook gently for 20 minutes.\n\n6. Add the parmesan cheese to sauce; mix well and serve over pasta.\n\nMarty experimented with this pasta after being smitten with something like it prepared by my sister-in-law, Linda. The cream and butter make it delicious, but rich, so one shouldn't have it once a week. Save it for special occasions.\n\nMarty and Linda's Pasta **\n\n\u00be lb. good sweet sausage\n\n1 pint of cream\n\n\u00bd cup grated parmesan cheese\n\n\u00bd stick butter, melted\n\n1 tbsp. cracked pepper\n\n\u00be cup minced parsley\n\n1 lb. spaghetti\n\n1 egg, beaten\n\n1. Take the casings off the sausage and brown in a heavy skillet, using a spatula to crumble and making sure the sausage is completely done. It should probably take about 15 minutes.\n\n2. Pour off any excess grease and add the cream, parmesan cheese, butter, cracked pepper and parsley. Cook over low heat for 10 minutes.\n\n3. In the meantime, cook the pasta in a large pot of boiling water.\n\n4. When pasta is _al dente,_ drain well and pour into a large serving bowl; immediately add the beaten egg and mix well.\n\n5. Pour the sauce over the pasta and mix again.\n\n6. Serve right away with a bowl of extra cheese on the side.\n\nChris Thompson is a longtime friend of ours who is many things. She's pretty, bright, and a talented artist as well. She's married to Richard Pannebianco, one of the most handsome and nicest doctors I know. To top it off, they are the proud parents of extraordinary six-year-old twins, Jace and Nova. Richard loves to go scalloping when the season opens and Chris cooked this pasta dish for us one night. It too is extraordinary.\n\nChris's Pasta with Scallops **\n\n2 tbsp. chopped garlic\n\n3 tbsp. minced parsley\n\n\u00bd cup olive oil\n\n\u00bd lb. scallops, rinsed and patted dry\n\n\u00bd tsp. red pepper flakes salt and pepper\n\n\u00bc tsp. basil\n\n1 lb. pasta\n\n\u00bd cup Italian breadcrumbs, toasted\n\n\u00bd cup grated parmesan cheese\n\n1. Saute garlic and parsley in olive oil until garlic is golden.\n\n2. Add scallops, red pepper, salt, pepper, and basil and cook over low heat until scallops turn white.\n\n3. Meanwhile, cook pasta, drain and put in large serving bowl.\n\n4. Add scallop mixture and toss well.\n\n5. Add breadcrumbs and parmesan cheese; mix again.\n\n6. Serve immediately with more parmesan cheese on the side.\n\nAn absolutely succulent seafood pasta, this is easy to make and not too expensive, considering the shrimp and clams in it.\n\nSeafood Pasta **\n\n1 dozen littleneck clams (they seem to be the most tender)\n\n\u00bd cup butter\n\n\u00bd cup olive oil\n\n6 large garlic cloves, minced\n\n1 tsp. oregano\n\n1 tsp. rosemary\n\n1 cup dry white wine\n\n\u00bd lb. shrimp, peeled, de-veined and cut in pieces\n\n1 cup tomatoes, peeled and chopped\n\n\u00bd cup minced flat parsley salt and pepper\n\n1 cup grated parmesan cheese\n\n1 lb. thin spaghetti\n\n1. Cook clams in a covered pot with a small amount of boiling water until the shells have opened. Reserve clams and discard shells.\n\n2. Melt butter and olive oil in a frying pan over medium heat until bubbly.\n\n3. Add garlic and cook until golden.\n\n4. Add oregano, rosemary and wine and cook until wine has cooked down.\n\n5. Reduce heat and add clams, shrimp, tomatoes, parsley, salt and pepper and cook over low heat for 25 minutes or until the tomatoes have cooked down and the sauce is thick, not watery.\n\n6. In the meantime, cook pasta until done and drain.\n\n7. Mix pasta, sauce and parmesan cheese together and serve.\n\nThis is a delicious and unusual dish that I had at Bernice Hunt's house in Bedford, N.Y. When I came back to the Hamptons, I tried to figure out how she had made it. Since she called it Greek Pasta, I decided it must have a little oregano and lemon juice. You want to make sure that your sauce is ample, so it might be necessary to add a little tomato puree, juice, or paste if it has cooked down too much.\n\nGreek Pasta **\n\n3 cloves garlic, minced\n\n3 tbsp. olive oil\n\n6 nice ripe tomatoes, skinned and chopped (or 1 16-oz. can of tomatoes, drained).\n\n\u00bd tsp. sugar\n\n1 tbsp. lemon juice\n\n1 tsp. oregano salt and pepper\n\n\u00be to 1 lb. shrimp, peeled and de-veined (either whole or cut in pieces).\n\n1 lb. rigatoni\n\n\u00be lb. feta cheese\n\n1. Saute garlic in olive oil until lightly browned.\n\n2. Add tomatoes, sugar, lemon juice, oregano, salt and pepper and simmer for 45 minutes.\n\n3. Add shrimp and cook 10 minutes more or until it has lost its pink color.\n\n4. In the meantime, cook the rigatoni in a large pot of boiling water and drain.\n\n5. Place rigatoni in a large flat casserole and mix it with the shrimp sauce.\n\n6. Cut up the feta cheese and crumble it over the pasta.\n\n7. Place under broiler until cheese is melted and slightly browned.\n\nI often look at the menu offerings discussed in restaurant reviews. Many of them look delicious but, frankly, we would probably not get to sample too many of them. Either the restaurant is inaccessible or else, just too expensive. I do the next best thing; I try to figure out the recipe myself. Here is a recent and very good experiment if you're a cheese aficionado.\n\nThree Cheese Pasta **\n\n2 tbsp. butter\n\n1 cup heavy cream\n\n\u00bc lb. gorgonzola cheese, crumbled\n\n\u00bc lb. bel paesa cheese, shredded\n\n\u00bd lb. fontina cheese, shredded\n\n1 lb. corkscrew pasta salt\n\nfreshly ground black pepper\n\n\u00be cup grated parmesan cheese\n\n1. In a large heavy pot, melt butter and put in \u2154 cup of cream.\n\n2. When butter has melted in the cream, add the gorgonzola and stir over heat until mixed well.\n\n3. Add the rest of the shredded cheeses and stir slowly until you have a nice creamy sauce.\n\n4. Meanwhile, cook the pasta in a large pot of boiling, salted water and drain.\n\n5. Add the pasta to the pot and mix thoroughly with the cheese sauce.\n\n6. Add the rest of the cream, the salt and the pepper and mix again.\n\n7. Serve with a bowl of parmesan on the side for those who want it.\n\nAnother recipe inspired by a menu, this one with salmon and dill is unusual.\n\nItalian Salmon Pasta **\n\n3 tbsp. butter\n\n6 tbsp. olive oil\n\n2 cloves garlic, minced\n\n1 8\u00bd oz. can salmon, drained and well chopped\n\n1\u00bd to 2 cups light cream\n\n2 tbsp. dill, chopped salt and pepper\n\n1 lb. linguine\n\ngrated parmesan cheese to serve on the side\n\n1. Melt butter with 3 tbsp. of the olive oil in a large frying pan over low heat.\n\n2. Add garlic and saute until golden.\n\n3. Add salmon and cook for a few minutes, making sure the salmon is well chopped.\n\n4. Slowly add cream until the sauce is well heated and of a nice consistency.\n\n5. Stir in dill, salt and pepper.\n\n6. In the meantime, cook linguine in a large pot of boiling water and drain.\n\n7. Add the other 3 tbsp. olive oil to linguine and mix until linguine is well coated.\n\n8. Pour sauce over linguine; mix again and serve with parmesan cheese on the side.\n\nOur local Italian pork store is run by a man named Savario and it is one of my favorite stores because everything in it is so good. Savario grinds his own sausage, makes bracciole, various antipastos and even some of the cheeses. Best of all, he is always ready to tell you how to improve on your own Italian dishes, what to add, what to leave out and even how to experiment. Once he mentioned a particular salami that made a fine addition to pasta. So I went home and concocted this.\n\nPasta with Salami **\n\n1 stick butter\n\n1 lb. mushrooms, sliced\n\n1\u00bd cups mild salami, cut into small pieces\n\n\u00bd cup minced scallions\n\nsalt and pepper\n\n3 egg yolks\n\n1 cup heavy cream\n\n\u00bd cup grated parmesan cheese\n\n1 lb. spaghetti\n\n1. Melt butter in a heavy skillet and cook mushrooms over high heat until liquid has evaporated.\n\n2. Lower heat, add salami, scallions, salt and pepper and cook for about 4 to 5 minutes.\n\n3. In a separate bowl, beat eggs, cream, and cheese.\n\n4. Meanwhile, cook pasta and drain.\n\n5. Put pasta in serving bowl; add cream sauce and mix well.\n\n6. Add salami sauce and mix well again.\n\n7. Serve immediately with extra cheese on the side.\n\nI love this pasta! I think that it looks pretty, it's easy to make and was served to me in a small unpretenious cafe in Italy, a pleasant surprise.\n\nPasta with Green Beans **\n\n1 lb. string beans, cut in two inch pieces\n\n1 cup olive oil\n\n1 cup bacon, minced\n\n4 cloves garlic, minced\n\nsalt and pepper\n\n\u00bd cup olive oil\n\n1 lb. fettuccine, green\n\n3 tbsp. minced flat Italian parsley\n\n1 cup grated parmesan cheese\n\n1. Steam the green beans until tender.\n\n2. Cook the bacon slowly until crisp, drain on paper towels and then crumble.\n\n3. Saute beans, garlic, salt and pepper in olive oil for 3 minutes.\n\n4. Add to cooked fettuccine along with bacon.\n\n5. Sprinkle with parsley and parmesan and serve.\n\nThis is a lovely pasta, one of my favorites and so easy to make.\n\nPasta with Sausage *\n\n1 lb. mild, sweet sausage\n\n\u00bd cup olive oil\n\n\u2154 cups scallion tops, minced\n\n2 tbsp. raisins\n\nsalt and pepper\n\n1 lb. taglionini (or other pasta such as corkscrews, ziti etc.)\n\n1. With scissors, cut the casings away from the sausage and saute in the olive oil.\n\n2. Chop the sausage into fine pieces as it cooks.\n\n3. When meat is brown and well chopped, add the scallions, raisins, salt, and pepper and cook over low heat until scallions are tender and meat is completely cooked.\n\n4. In the meantime, cook the taglionini and drain.\n\n5. Add the parsley to the sausage and stir.\n\n6. Pour sauce over the hot pasta, mix well, and serve.\n\nMy husband and I visited Italy a few years ago and one of our most cherished memories is visiting the exquisite medieval town of San Gimignamo. It's hard to explain the feeling we had as we sat having a perfect lunch in a restaurant high on a mountain top overlooking lush vineyards. Food cooking in an open hearth, the snugness of being inside a small, walled village, a mist rising from the valley\u2014all these things contributed to a lovely sense of continuity and pleasure. The food was superb and I have tried to approximate both our meals, not knowing now, in retrospect, how close I might have come, but the memories never change. Incidentally, the secret to this dish, I'm sure, is just coordination. Steps 7\u201310 should take only a minute.\n\nSan Gimignamo Pasta ***\n\n4 tbsp. butter\n\n2 tbsp. minced onions\n\n\u00bd lb. mushrooms, cleaned and cut in small pieces\n\n1\u00bd tsp. salt\n\n\u00be tsp. coarse, freshly ground black pepper\n\n\u00be cup minced, unsmoked ham\n\n\u00be cup peas, preferably small ones\n\n1\u00bd cups cream\n\n1 lb. fettuccine\n\n\u00be cup grated parmesan cheese\n\n1 egg, beaten\n\n1. In a medium pan, melt 3 tbsp. of the butter and add onions, mushrooms, \u00be tsp. of the salt, pepper, and saute gently.\n\n2. When you begin to see lots of liquid from the mushrooms, turn heat high and cook for 3 minutes.\n\n3. Lower heat, add ham and peas and cook 1 minute.\n\n4. Add \u00be cup of the cream and cook a couple of minutes more.\n\n5. Put remaining butter and cream in a large serving dish and place in a 275 degree oven along with the pan containing the mushroom sauce.\n\n6. Meanwhile, cook pasta according to directions and drain.\n\n7. Take butter and cream sauce out of the oven and put on a very low flame on top of the stove.\n\n8. Add pasta to butter and cream sauce and mix well.\n\n9. Add cheese and beaten egg and mix well again.\n\n10. Remove mushroom sauce from oven, add and give pasta a final mix.\n\n11. Check for seasonings and serve immediately.\nVegetables\n\nMarcia, my mother-in-law, has a wonderful way with cabbage and every time she visits us, she says, \"If you have a little piece of cabbage I'll make something for you.\" This dish I loved and seemed to take no time at all. She said that she always cuts her cabbage with a knife and doesn't use a shredder. She advises tasting a lot and checking to see whether you want to add more sweet (sugar) or sour (lemon juice).\n\nSweet and Sour Cabbage *\n\n1 large onion, diced\n\n1 medium cabbage, cut in chunks\n\n1\u00bd cups tomato puree\n\n1\u00bd cups water\n\n2 tbsp. sugar\n\n1 clove garlic, minced\n\njuice from 1 lemon\n\n1 tart, green apple, peeled and sliced\n\nsalt and pepper\n\n1. Put all ingredients in a large frying pan or heavy pot and cook until cabbage is tender.\n\nAnother cabbage recipe, this one was donated by my friend, Chris Italiani who came for dinner one night when I was testing a very unsuccessful cabbage dish. She said, \"I'll have my Mother send you her recipe. It's much better.\"\n\nMrs. Acheson's Red Cabbage *\n\n2 tbsp. safflower oil\n\n2 medium onions, chopped\n\n\u00bd head red cabbage, finely sliced\n\n2 apples, peeled, cored and sliced\n\n1 to 2 bay leaves\n\n2 tbsp. vinegar\n\n2 tbsp. brown sugar\n\n\u00bd cup water\n\nsalt and pepper\n\n1. Saute onions in oil until transparent.\n\n2. Add remaining ingredients and cook over low flame until cabbage is tender, approximately \u00bd hour.\n\nOne day I was looking for a new way to make eggplant. I happened to remember what Gene Friedman had said about his wife Dorothy's cooking. I had asked him why he thought she was such a good cook and one of the things he replied was \"substitution.\" In other words, if she saw some nice turnips she might say, \"Since I make potatoes au gratin, why not try turnips au gratin?\" So, I decided, \"Since I make a meat lasagna, why not make an eggplant lasagna?\".\n\nEggplant Lasagna **\n\n1 eggplant cut in very small chunks\n\n3 cloves garlic, sliced\n\n1 medium onion, diced\n\n6 tbsp. peanut oil\n\n6 large tomatoes, skinned and chopped\n\n1 green pepper, seeded and cut in strips\n\n1\u00bd cups tomato puree\n\n1 basil leaf, minced\n\n2 tbsp. fresh parsley, minced\n\nsalt and pepper\n\n\u00bd package lasagna noodles (about 14)\n\n1 16-oz. package part skim mozzarella cheese, sliced\n\n\u00be cup grated parmesan cheese\n\n1. Saute eggplant, garlic and onions in oil until eggplant is soft.\n\n2. Add tomatoes, green pepper, puree, herbs and spices and cook over low heat for 40 minutes.\n\n3. Meanwhile, cook lasagna noodles in large pot of boiling water and drain.\n\n4. Take a large flat casserole and place half of the needles in bottom of dish.\n\n5. Pour half of the eggplant tomato sauce over them and spread with half of the mozzarella cheese.\n\n6. Layer with the rest of the noodles.\n\n7. Pour the remaining sauce over them and then the rest of the mozzarella cheese.\n\n8. Sprinkle parmesan over the casserole; cover with tinfoil and bake for 40 minutes in a 350 degree oven.\n\n9. Take foil off and continue baking for 10 to 15 minutes more.\n\nStuffed Eggplant ***\n\n3 small eggplants\n\n2 small red peppers\n\n6 tbsp. olive oil\n\n1 large clove garlic, minced salt and pepper\n\n2 small onions, chopped\n\n3 tomatoes, chopped\n\n4 tbsp. minced fresh parsley\n\n1 tsp. chopped fresh basil or \u00bd tsp. dried basil\n\n\u00bd cup wheat germ\n\n\u00be cup grated parmesan cheese\n\n\u00bc cup chopped walnuts\n\n2 tbsp. butter\n\n\u00bc cup milk\n\n1. Slice eggplants lengthwise and scoop out pulp.\n\n2. Dice pulp.\n\n3. Seed and dice red peppers.\n\n4. Heat 3 tbsp. of the oil and saute eggplant and peppers and garlic until soft.\n\n5. Add salt and pepper and divide mixture among eggplant shells, pressing it down to make room for the next ingredients.\n\n6. Heat the rest of the oil and saute the onions, tomatoes, parsley, and basil until the onions are soft.\n\n7. Spread this mixture on top of the eggplants.\n\n8. Mix wheat germ, parmesan cheese and walnuts.\n\n9. Melt butter over low heat in the milk.\n\n10. Stir milk into wheat germ mixture to make a paste.\n\n11. Spread on top of shells.\n\n12. Place in oiled baking dish and cook in 350\u00b0 oven for 45 minutes.\n\nEsalen is a humanistic center for learning situated out in the Big Sur, a truly magnificent part of our country. The people at Esalen grow and cook all their own food high above the Pacific. The gardens are beautiful and I must say that the four days my husband and I spent there last year were four of the best eating experiences of our lives. Right after we came home, I tried to copy all the dishes we had had there.\n\nI happen to like eggplant very much and make eggplant parmesan quite often, but that takes a lot of time.\n\nI was pleased to find this recipe at Esalen. It takes much less time and is delicious. You may need to use more oil than I quoted. Eggplant seems to soak up so much of it. Also, you can try browning the eggplant by placing slices in an oiled baking pan and putting the pan in a hot oven.\n\nEsalen Eggplant **\n\n1 medium eggplant, sliced\n\n2 eggs, beaten\n\n1\u00bd cups toasted breadcrumbs\n\n1 cup peanut oil\n\nsalt and pepper\n\n\u00bc tsp. garlic salt\n\n3 nice, good sized tomatoes, sliced\n\n\u00be cup grated parmesan cheese\n\n1. Dip eggplant slices in egg and then in breadcrumbs.\n\n2. Saute the slices in hot oil until brown and then turn over to brown on the other side.\n\n3. Drain on paper towels and place in large flat baking dish.\n\n4. Sprinkle with salt, pepper, and garlic salt.\n\n5. Cover with tomato slices.\n\n6. Sprinkle with parmesan cheese; cover with tinfoil and bake at 350 degrees for 40 minutes.\n\nMy Mother sent this to me and since I seem to be making lots of souffles these days, this seemed like a good one to include. Also, I had never heard of a souffle made with eggplant and this is very tasty.\n\nMother's Eggplant Souffle **\n\n1 large eggplant or two smaller ones, peeled and cubed\n\n1 onion, thickly sliced\n\n4 tbsp. butter\n\n\u00bc cup bread crumbs\n\n\u00bd cup milk\n\n3 egg yolks\n\n\u00bc tsp. nutmeg\n\nsalt and pepper\n\n3 egg whites\n\n2 tbsp. breadcrumbs\n\n2 tbsp. slivered almonds\n\n1. Cook eggplant and onion in 2 cups of boiling water until tender. Discard onion.\n\n2. Drain eggplant well and mash.\n\n3. Add 2 tbsp. butter, breadcrumbs, milk, egg yolks, nutmeg, salt and pepper to eggplant and mix well.\n\n4. Pour into a lightly buttered souffle dish and let cool.\n\n5. When ready to bake, beat egg-whites until stiff and fold into eggplant mixture.\n\n6. Saute breadcrumbs and almonds in the remaining 2 tbsp. butter for a few moments.\n\n7. Sprinkle over souffle and bake in a 375 degree oven for 30 minutes.\n\nSue Mermelstein is a reporter for the East Hampton Star. A petite woman, bright and a terrific volleyball player, she took pity on us one night because we had been without a kitchen for two weeks and invited us for dinner. If you like eggplant and spiciness, this can't be beat.\n\nSue's Spicy Eggplant **\n\n2 tsp. Chinese chili paste with garlic\n\n5 tsp. thin soy sauce\n\n2 tsp. sugar\n\n\u00bd cup water\n\n1 medium eggplant, cubed\n\n4 tbsp. peanut oil\n\n2 cloves garlic, sliced\n\n1 large onion, cubed\n\n4 cups rice, cooked\n\n1. Mix chili paste, soy sauce, sugar, and water.\n\n2. Heat oil in wok or large frying pan and stir fry eggplant until soft.\n\n3. Remove eggplant, add garlic and stir fry for 1 minute.\n\n4. Add onion and stir fry over lowered heat until soft.\n\n5. Return eggplant to wok along with sauce and heat thoroughly before servserving over warm rice.\n\nI saw eggplants something like these in that marvelous Italian case at Balducci's in New York City and came home to try and duplicate them. The inside of the eggplant should be saved, of course, and can be used in a variety of recipes.\n\nRicotta Eggplant **\n\n4 baby eggplants\n\n2 tbsp. olive oil\n\n1 clove garlic, minced\n\n1 small onion, diced\n\n2 tbsp. chopped fresh parsley\n\n1 basil leaf\n\n\u00bd tsp. oregano\n\nsalt and pepper\n\n1 8-oz. can tomato puree\n\n1 16-oz. carton of ricotta cheese\n\n4 tbsp. grated parmesan cheese\n\n1. Wash the eggplants; place on a cookie sheet and bake in 350 degree oven until a fork pierces them easily.\n\n2. Remove from oven; slice lengthwise and cool.\n\n3. Heat oil in a large frying pan and saute garlic and onion until golden.\n\n4. Add the rest of the ingredients except the parmesan and ricotta cheese and simmer over a low heat for 20 minutes.\n\n5. Scrape out eggplant pulp and save for another dish.\n\n6. Place eggplant shells in a baking dish and divide the ricotta cheese among the eggplant shells.\n\n7. Spoon tomato sauce over the shells.\n\n8. Sprinkle with parmesan cheese and bake for 20 minutes in a 350 degree oven.\n\nSpinach Provencale *\n\n1 large onion, chopped\n\n1 large clove garlic, minced\n\n2 tbsp. olive oil\n\n3 lbs. fresh washed spinach\n\n2 eggs, beaten\n\n\u00be cups grated parmesan cheese\n\nsalt and pepper\n\n1. Place onion and garlic in a large frying pan and saute in olive oil until soft.\n\n2. Add spinach, cover and cook 4 minutes, or until spinach wilts.\n\n3. Place spinach mixture on cutting board and chop.\n\n4. When spinach has cooled, place in a buttered baking dish and stir in eggs, \u00bd cup of the parmesan cheese, salt and pepper.\n\n5. Sprinkle with the remaining \u00bc cup of cheese and bake at 375 degrees for 10 to 15 minutes.\n\nI think this is probably my favorite squash dish and you can use butternut, or hubbard squash instead of acorn if you wish. You can also substitute sour cream for yogurt, but the yogurt's healthier, of course.\n\nBest ever Squash *\n\n2 nice sized acorn squashes\n\n2 scallions, chopped\n\n1 tbsp. butter\n\n5 tbsp. yogurt\n\nsalt and pepper\n\n1. Bake whole squashes until tender in a 350 degree oven (approximately 45 minutes to 1 hour).\n\n2. When cooked, cut the squashes in half; carefully scrape out the pulp from the shells and reserve both squash and shells.\n\n3. Saute scallions for a few minutes in butter.\n\n4. Mash squash in bowl.\n\n5. Add scallions, yogurt, salt and pepper and mix well.\n\n6. Fill shells with the mixture and return to the oven to heat before serving.\n\nHere's a nice pleasant complement to pasta or fish. Someone suggested that it is also served cold with the condiments when you are serving a curry dish.\n\nString Beans with Almonds *\n\n1 pound string beans\n\n1 small package of slivered almonds\n\n2 scallion tops, minced\n\n3 tbsp. olive oil\n\nsalt and pepper\n\n1 tbsp. vinegar\n\n1. Steam whole string beans until tender, but not soft; drain.\n\n2. Place in bowl with the almonds, scallions, oil, salt and pepper and mix well.\n\n3. Add the vinegar and mix again.\n\nWhat can one say about Minsky and still confine it to a few sentences? Richard Minsky is a master bookbinder, extraordinaire. The last book he bound was one that he wrote himself. It's called _Minsky in London_ and sells for $1800. One is in the Victoria and Albert Museum, one in the rare books room of the New York Public Library, so you can see what kind of books they are. Richard himself is unique. He plays the violin and seems to have an informed opinion about everything topped with a good sense of humor. But as he says, \"I'm just a regular guy.\" The following is a dish that he makes just about every night with variations. The variations depend on what's fresh at the market. I will give a standard recipe and you can substitute whatever is fresh, just as Minsky does.\n\nMinsky's Vegetables **\n\n2 large cloves garlic, minced\n\n1 tsp. minced fresh ginger\n\n3 tbsp. peanut oil\n\n1 eggplant, cut in small cubes\n\n1 large onion, sliced\n\n1 cup celery, sliced\n\n2 zucchini, sliced\n\n2 carrots, shredded\n\n1 tsp. cornstarch\n\n2 tsp. water\n\n2 tbsp. tamari\n\n2 scallion tops cut in two inch pieces\n\n4 cups cooked rice, brown or white\n\n1. In a wok or large frying pan saute garlic and ginger in oil for a few minutes.\n\n2. Add all the vegetables and stir fry until tender, but not mushy. (You will probably have to add more oil if you are using eggplant since it absorbs so much oil)\n\n3. Dissolve cornstarch in water and add tamari.\n\n4. Cook over low heat in a small sauce pan until thickened.\n\n5. Add to vegetables, cover, and heat through for a few minutes.\n\n6. Serve over the rice with scallions sprinkled on top.\n\nGreek Zucchini *\n\n2 lbs. zucchini, sliced\n\n2 large tomatoes, cut in chunks\n\n1 medium onion, chopped\n\n2 cloves garlic, minced 1 tsp. oregano\n\nsalt and pepper\n\n3 tbsp. olive oil\n\n1. Place all ingredients in a saucepan with a cover.\n\n2. Cook over low heat until zucchini is tender.\n\nArmand LaMacchia brought this to one of our communal meals. I thought it was a nice way to use parsnips as I don't have a great variety of recipes for them. I like the combination of these two vegetables.\n\nArmand's Broccoli and Parsnip Casserole **\n\n1 head of broccoli, trimmed and cut into stalks.\n\n6 parsnips, peeled and sliced\n\n2 tbsp. butter\n\n2 tbsp. flour\n\n\u00bc cup water\n\n\u00be cup milk\n\nsalt\n\n\u00be pound of sharp cheddar cheese, shredded.\n\n1. Steam broccoli until crisp-tender.\n\n2. Cook parsnips in a small amount of water.\n\n3. Drain both vegetables.\n\n4. Melt butter over low heat in a saucepan.\n\n5. Make a flour and water paste and add to butter, stirring constantly.\n\n6. Add milk gradually and stir until sauce is thickened.\n\n7. Add salt and shredded cheese and keep stirring to make a nice cheese sauce.\n\n8. Place vegetables in a casserole and cover with cheese sauce.\n\n9. Bake covered for 30 minutes in a 350 degree oven.\n\nThe usual ways to prepare cauliflower are to steam it or, for the more ambitious, perhaps to bake it in a cheese sauce. This unusual dish can be served hot or cold and was given to me by a friend of a friend at one of our communal meals.\n\nSpicy Cauliflower **\n\n1 cauliflower, separated into flowerets\n\n1 clove garlic, minced\n\n1 tsp. peppercorns\n\n3 tbsp. pine nuts\n\n\u00bd tsp. red pepper flakes\n\n\u00bd tsp. salt\n\n\u00bc tsp. sugar\n\n\u00bd cup olive oil\n\n\u00bc cup red wine vinegar\n\n1. Steam cauliflower until crisp-tender.\n\n2. Grind garlic, peppercorns, and pine nuts together; I use a mortar and pestle.\n\n3. Place in a small bowl and add the red pepper flakes and sugar.\n\n4. Add oil and mix thoroughly.\n\n5. Pour sauce over cauliflower and stir well to coat.\n\n6. Add vinegar last and mix all ingredients.\n\nAt our marvelous Italian pork store in East Hampton, which I have mentioned before, everything is made on the premises by the owners, Savario and his wife, Carmella. My husband got this recipe from him and as it came to him causally over the counter, the measurements are a little difficult and often depend on the size of the eggplant. Experiment!\n\nSavario's Eggplant ***\n\n1 long, narrow eggplant\n\n3 eggs, beaten\n\n1 cup flour\n\n\u00bd cup breadcrumbs\n\n1 cup olive oil 3 cloves garlic\n\n2 cups canned tomatoes, drained and chopped\n\n\u00bc cup fresh basil or 1 tsp. dry basil\n\nsalt and pepper\n\n8 oz. package mozzarella cheese\n\n1 8 oz. carton ricotta cheese\n\n1. Skin eggplant and slice lengthwise into slices about 1 inch thick.\n\n2. Dip eggplant into eggs.\n\n3. Mix flour and breadcrumbs and coat each slice with the mixture.\n\n4. Pour \u2154 cup of olive oil in a large flat casserole. You may have to use another dish because the eggplant shouldn't overlap.\n\n5. Place eggplant in dish and cook in a 400 degree oven until done, turning once. This should take about half an hour. Drain on paper towels.\n\n6. Put the remaining oil in a frying pan and saute garlic until golden.\n\n7. Add tomatoes, basil, salt and pepper and cook over low heat for 20 minutes.\n\n8. Slice mozzarella and mix with ricotta and spinach.\n\n9. Place about 2 tbsp. of cheese mixture on end of each slice of eggplant and roll up so that it looks like a hot dog roll or manicotti.\n\n10. Arrange in a smaller baking dish and cover with the tomato sauce.\n\n11. Bake at 350 degrees for \u00bd hour.\n\nMy husband doesn't really like tofu, but I keep buying it because I think it's good for you and it's not expensive. I finally invented one dish that he liked and he said, \"Quick, write it down for your next cookbook before you forget it.\"\n\nTofu with Apples *\n\n1 tsp. fresh ginger, minced\n\n2 large cloves garlic, minced\n\n1 tbsp. safflower oil\n\n1 large piece of tofu (around 8 oz.).\n\n1 large apple, peeled and sliced\n\n3 tbsp. tamari\n\n2 scallions, slivered in two inch pieces\n\n1. Saute ginger and garlic in oil for a few minutes, but do not let garlic brown.\n\n2. Place tofu and apples in a small baking dish.\n\n3. Add garlic, ginger and tamari.\n\n4. Sprinkle with scallions, cover and bake at 350 degrees for \u00bd hour.\n\nVegetable Curry **\n\n2 tbsp. salad oil\n\n2 tsp. ground coriander\n\n2 tsp. cumin\n\n2 tsp. tumeric\n\n\u00bc tsp. ground cloves\n\n\u00bc tsp. cinnamon\n\n3 garlic cloves, minced\n\n\u00bc to \u00bd tsp. red pepper flakes\n\n4 larges potatoes, peeled and cubed\n\n2 cups chick peas\n\n3 tomatoes, cut in chunks\n\n1 larges green pepper, seeded and cut in strips\n\n1\u00bd cup peas\n\n1\u00bd tsp. salt\n\n2 cups water\n\n1\u00bd cups yogurt\n\n\u00bd cup minced onion\n\n1. Heat oil until hot and add garlic and spices.\n\n2. Cook 2 minutes, stirring so it does not stick.\n\n3. Add potatoes, chick peas, 2 of the tomatoes, green pepper and peas and mix well with the spices.\n\n4. Add water and salt and heat until boiling. Reduce heat, cover and cook until potatoes are done.\n\n5. Garnish with the remaining tomato and serve with yogurt mixed with onion.\n\nI have wanted to include a recipe for soybeans because they are so healthy and they are inexpensive. I also think you have to cultivate a taste for them. I hope this helps. Remember to soak beans overnight first.\n\nBaked Soybean Casserole **\n\n1 cup soybeans\n\n3 tbsp. butter\n\n1 small onion, chopped\n\n1 small green pepper, sliced in thin strips\n\n1 small red pepper, cut in strips\n\n2 cloves garlic, sliced\n\n\u00bd tsp. thyme\n\n2 tbsp. parsley\n\n\u00bd tsp. dill week\n\n1 tbsp. safflower oil\n\n3 tomatoes sliced\n\n2 medium squashes, either yellow crookneck or zucchini, sliced in rounds.\n\n\u00bd cup grated parmesan cheese salt and pepper to taste\n\n1. Place beans in a large pot with 6 cups water and bring to a boil.\n\n2. Cook 2 minutes, turn off heat, cover, and soak overnight.\n\n3. In the morning, cook beans over low heat, uncovered, for about 2 to 3 hours or until done. Soybeans are usually slightly crunchy.\n\n4. When done, drain, but reserve 1 cup of the cooking water.\n\n5. Melt butter in a large skillet and saute onion, pepper, and garlic for a couple of minutes.\n\n6. Add thyme, parsley, dill, soybeans and cooking water and cook over low heat for 15 minutes.\n\n7. Grease a casserole with the safflower oil and add the soybeans along with tomatoes, squash, parmesan cheese, salt and pepper.\n\n8. Mix well, cover, and bake at 300 degrees for 2 hours.\n\nThis sounds strange, but it's surprisingly good.\n\nChristina's Geetam Tofu Steaks *\n\n2 tbsp. tahini\n\n2 tbsp. honey\n\n6 tofu slabs, \u00bc inch deep and cut into squares, 2 inches by 3 inches.\n\n\u00bc cup tamari\n\n3 tbsp. oil\n\n1. Mix tahini, honey and tamari in a large flat pan.\n\n2. Add tofu slabs and marinate for two hours.\n\n3. Heat oil in large frying pan.\n\n4. Fry tofu for a few minutes on each side.\n\nGreen Chilis with Eggs and Cheese *\n\n3 cups grated Montery Jack or Swiss cheese\n\n2 cans green chilis, drained and chopped\n\n4 eggs, beaten\n\n1. Pre-heat oven to 300 degrees.\n\n2. Grease a square baking pan.\n\n3. Add cheese to pan.\n\n4. Sprinkle chopped chilis evenly over cheese.\n\n5. Pour eggs into pan.\n\n6. Bake 1 hour.\n\n7. Cool and then cut into squares.\n\nA quiche that is lighter than the usual kind, yogurt cheese pie is an interesting variation, has fewer calories and can be used with just about any kind of vegetables.\n\nYogurt Cheese Pie **\n\n\u00bd cup grated Montery Jack cheese\n\n\u00bd cup grated cheddar cheese\n\n1 unbaked pie shell\n\n\u00bc cup onion, chopped\n\n1 cup chopped vegetables, such as broccoli, cauliflower, zucchini, mushrooms, spinach etc.\n\n4 eggs, beaten\n\n\u00bd cup cottage cheese\n\n2 cups yogurt\n\n\u00bd cup milk\n\n\u00bd tsp. salt\n\n\u00bd tsp. white pepper\n\n1 tsp. minced parsley dash of cayenne\n\n1. Mix the first two cheeses and place in the unbaked pie shell.\n\n2. Add onions and vegetables to shell.\n\n3. Mix remaining ingredients in a bowl.\n\n4. Pour yogurt mixture into pie shell.\n\n5. Bake at 350 degrees for 1 hour.\n\nSpaghetti squash is a special favorite of a lot of my friends out here in the Hamptons. The only problem is that it is made with a squash called, appropriately enough, spaghetti squash and I don't know how readily available it is in other parts of the country. It is a large, pale yellow squash. When it is baked and cut open, the strands are scraped out and look very much like spaghetti strands. I suppose that if you can't find this type of squash, you can use Dorothy Friedman's maxim and substitute another type of squash and use the sauce over it.\n\nSpaghetti Squash with Tomato Sauce **\n\n1 good sized spaghetti squash\n\n2 cloves garlic, minced\n\n1 small green pepper, seeded and minced\n\n1 celery stalk, cut in small pieces\n\n2 tbsp. olive oil\n\n6 large tomatoes, skinned and cut in pieces\n\n2 tbsp. tomato paste\n\n\u2153 cup water\n\n1 tsp. minced fresh basil salt and pepper\n\n\u00bc tsp. red pepper flakes\n\n1 cup grated parmesan cheese\n\n1. Place the squash in a pan with a tiny bit of water in the bottom and bake at 350 degrees until done. This may take an hour to an hour and a half depending on the size of the squash. A fork should puncture it easily.\n\n2. While the squash is cooking, saute the garlic, green pepper, and celery in the olive oil.\n\n3. When garlic is lightly colored, add remaining ingredients except for cheese and simmer for about 1 hour.\n\n4. When squash is done, cut it open, take out seeds and gently scoop out squash strands with a fork.\n\n5. Pour the tomato sauce over the squash and serve with the parmesan cheese on the side.\n\nStuffed Tomatoes **\n\n4 good sized, ripe tomatoes salt and pepper\n\n1 clove garlic, minced\n\n3 tbsp. minced scallions\n\n3 tbsp. minced parsley\n\n1 tsp. fresh basil or \u00bc tsp. dried basil\n\n\u2153 tsp. thyme\n\n\u00bd cup fresh bread crumbs\n\n3 tbsp. olive oil\n\n2 tbsp. butter\n\n1. Cut the tomatoes in half horizontally.\n\n2. Scoop out the seeds and pulp and save for soups or sauces.\n\n3. Sprinkle the tomato shells with salt and pepper and turn over to drain.\n\n4. Combine the remaining ingredients, except butter.\n\n5. When you are ready to cook the tomatoes, pre-heat the oven to 400 degrees.\n\n6. Fill the shells with the breadcrumb mixture, dot with butter and bake in a lightly greased flat baking pan for 15 minutes.\n\nBroccoli Frittata **\n\n6 large cloves garlic, sliced\n\n3 tbsp. olive oil\n\n1 medium head broccoli, trimmed and chopped in small pieces.\n\n6 eggs, well beaten\n\n\u00bd cup parmesan cheese salt and pepper, to taste\n\n3 tbsp. butter\n\n1. Saute garlic in olive oil until almost golden.\n\n2. Add chopped broccoli and stir fry a few minutes until broccoli is coated with oil.\n\n3. Add \u00bc cup of water, cover, lower heat and simmer until broccoli is tender. Drain all liquid off.\n\n4. In a bowl, mix the broccoli with eggs, parmesan cheese, salt and pepper.\n\n5. Melt butter in a large frying pan.\n\n6. Pour egg\/broccoli mixture in frying pan and cover.\n\n7. Cook until eggs are set.\n\n8. Place pan under broiler and cook until the frittata is nicely browned on top.\n\n9. Cut in pie shaped wedges and serve hot or cold.\n\nTurnip Casserole *\n\n6 cups turnips or rutabaga, peeled and cubed\n\n\u00bd tsp. salt\n\n\u00bc tsp. caraway seeds\n\n2 tsp. fresh basil or \u00bd tsp. dried basil\n\n\u00bc cup lemon juice\n\n1 cup yogurt freshly ground black pepper\n\n2 tbsp. minced parsley\n\n1. Steam turnips until tender and put into a casserole.\n\n2. Mix salt, caraway seeds, basil, lemon juice and yogurt.\n\n3. Pour over turnips and sprinkle with black pepper and parsley.\n\n4. Bake, covered, in a 375 degree oven for 20 minutes.\n\nAnna Marie Frisbee is a friend who lives out here in the Hamptons. She has a lovely face, a good sense of humor and is struggling to care for her very active young son, Matthew, who I'm sure doesn't give her much time to cook. However, she is very conscious of preparing healthy foods, as the following recipe for a delicious quiche will attest.\n\nAnna Marie's Spinach Quiche ***\n\n4 tbsp. butter\n\n3 tbsp. wholewheat flour\n\n1\u00bc cups milk\n\n2 tbsp. soy sauce\n\n\u00bc cup white wine\n\n3 tbsp. chopped parsley\n\n4 eggs (at room temperature)\n\n1 cup well packed grated cheddar cheese\n\n1 medium onion, chopped\n\n\u00be cup sliced mushrooms\n\n3 tbsp. oil\n\n3 cups chopped spinach\n\n2 8-inch pie shells a few grindings of fresh nutmeg\n\n1. Melt butter over low heat and add flour.\n\n2. Slowly add the milk, a little at a time and cook for 5 minutes, stirring constantly so the sauce doesn't get lumpy.\n\n3. Cool slightly and then add soy sauce, wine, and parsley.\n\n4. When it is cool, add one egg at a time, stirring after each addition.\n\n5. Add cheese, stir and set aside.\n\n6. In a wok or large frying pan, saute onions and mushrooms for 4 or 5 minutes in the oil.\n\n7. Add spinach, tossing just until it starts to wilt.\n\n8. Combine spinach with cheese sauce.\n\n9. Prick the bottoms of the pie shells and place in a pre-heated 400 degree oven for five minutes.\n\n10. Turn heat down to 350 degrees.\n\n11. Divide spinach-cheese mixture between the two shells.\n\n12. Grate a little nutmeg over the tops.\n\n13. Bake the quiches for 35 to 45 minutes or until a knife inserted into the center comes out clean.\n\nWhen we were at Esalen and this dish was posted on the bulletin board as part of the evening menu, everyone groaned. Later they all went back for seconds, so don't write it off. Actually, you could probably use other types of greens, such as spinach, collard greens etc. successfully.\n\nSwiss Chard Cutlets **\n\n3 cups coarsely chopped swiss chard\n\n\u00bc cup chopped onion\n\n\u00bc cup breadcrumbs\n\n\u00bd cup grated parmesan cheese\n\n2 eggs\n\n2 tbsp. lemon juice salt and pepper\n\n2 tbsp. oil\n\n1. Steam Swiss chard until tender.\n\n2. Place all ingredients in blender and blend slightly.\n\n3. Heat oil in a large frying pan and when it is hot, pour the mixture on the hot griddle as if you were making pancakes. Make sure they are nicely browned before you flip them over.\n\nMy sister-in-law Linda made this for us and I adopted it. A nice easy quiche to make, I like to prepare some extra ones to have on hand in the freezer. You can use a prepared pie shell to save time.\n\nLinda's easy Quiche **\n\n1\u2153 cups flour\n\n1 stick butter, softened\n\n1 cup heavy cream\n\n1\u00bd tsp. sugar\n\n2 eggs\n\n\u2153 tsp. cayenne pepper\n\n8 oz. Swiss cheese, shredded\n\n1 cup thinly sliced onion\n\n1. Mix flour and butter and press into a pie pan.\n\n2. Mix cream, sugar, eggs, and pepper in the blender for only 30 seconds.\n\n3. Put cheese and onions in the unbaked pie shell.\n\n4. Pour the cream mixture over the cheese and onions and bake for 20 minutes in a 425 degree oven.\n\n5. Lower heat to 350 degrees and bake for another 30 minutes.\n\nWe have a lot of beautiful white and purple cauliflower piled up in great mounds along with the pumpkins at roadside stands in the fall out here in the Hamptons. When a friend remarked that she never quite knew what to do with cauliflower except cover it with a cheese sauce, I tried to come up with a new dish for her.\n\nItalian Cauliflower *\n\n1 medium head of cauliflower, trimmed and broken into flowerets.\n\n1 pound of pasta (rigatoni, shells or bows)\n\n\u00bd cup olive oil\n\n1 tsp. minced garlic\n\n3 tbsp. chopped parsley\n\nsalt and pepper\n\n1 cup grated cheese\n\nsprinkling of red pepper flakes\n\n1. Steam cauliflower until tender and drain.\n\n2. While cauliflower is steaming, cook pasta in boiling water until _al dente;_ drain.\n\n3. Saute garlic and parsley in olive oil until garlic turns slightly golden.\n\n4. Place cauliflower and pasta in serving bowl and pour the olive oil with garlic and parsley over it.\n\n5. Add salt, pepper, grated cheese and mix well.\n\n6. Sprinkle with a small amount of red pepper flakes and serve.\n\nIt probably goes without saying that I only put dishes in this book that I like. In compiling the recipes, some stand out in my mind as being extra special. This stuffed zucchini comes from Judy Wolfe's friend Carlos, a person I don't even know. Delicious!\n\nCarlos' Stuffed Zucchini ***\n\n4 medium zucchini\n\n\u00be cup chopped onion\n\n1 clove garlic, minced\n\n\u00be cup skinned and chopped tomatoes\n\n\u00bd cup chopped mushrooms\n\n\u00bd cup breadcrumbs\n\n2 tbsp. chopped capers\n\n3 tbsp. minced parsley\n\n2 leaves fresh basil, minced\n\nsalt and pepper\n\n\u00bd cup olive oil\n\n1. Steam zucchini over boiling water for 4 minutes.\n\n2. Slice zucchini in half lengthwise and scoop out pulp.\n\n3. Chop zucchini pulp into small pieces and place in bowl.\n\n4. Add all the remaining ingredients with the exception of the olive oil. Add only \u00bc cup of the oil and mix well.\n\n5. Place zucchini shells in a large slightly oiled baking dish and stuff with the filling.\n\n6. Drizzle the remaining olive oil over the zucchini shells and bake in a 350 degree oven for \u00bd hour.\n\nChristina's Cauliflower and Tomato Special *\n\n1 small onion, chopped\n\n1 small cauliflower, trimmed and broken into small flowerets\n\n4 tbsp. tomato puree\n\n3 medium sized tomatoes, chopped\n\n1\u00bd tsp. rosemary\n\n1 tsp. oregano\n\n\u00bd tsp. garlic salt\n\nsalt and freshly ground black pepper\n\n2 tbsp. tamari\n\n1. Lightly saute onion in oil.\n\n2. Add cauliflower and saute a few minutes more.\n\n3. Add all remaining ingredients and mix thoroughly.\n\n4. Cover and and cook over low heat, stirring occasionally until cauliflower is done, but not mushy.\n\nThis is another one from my mother-in-law, Marcia, volunteered enthusiastically. It is a dish that can be made with either squash or corn.\n\nMarcia's Squash or Corn Souffle *\n\n2 cups yellow crookneck squash, cut in small cubes. (See note.)\n\n2 tbsp. melted butter\n\n2 tbsp. sugar\n\n\u00bc tsp. salt\n\n2 eggs, beaten\n\n\u00bd cup flour\n\n1 cup milk\n\n1. Cook squash in a small amount of boiling water until tender, but not mushy; drain.\n\n2. Melt butter in an ovenproof casserole.\n\n3. Add sugar, salt and eggs to dish.\n\n4. Stir in flour and milk.\n\n5. Add squash; stir and bake 1 hour in a 350 degree oven.\n\n_Note_ : If you wish to make a corn souffle, substitute 1 cup of creamed corn for squash and use only 1 tbsp. sugar.\n\nThis dish was given to me by Christina, an Australian lady we met at one of our communal meals. I have a dessert recipe of hers in the dessert section that is probably one of my all-time favorites. But this is healthier.\n\nChristina's Rice and Spinach Casserole **\n\n\u00bd cup sunflower seeds (sesame seeds can be substituted)\n\n2 tbsp. butter\n\n2 cups tomato sauce or juice\n\n1 pound fresh spinach, cooked and well chopped\n\n6 cups cooked brown rice\n\n\u00bd tsp. oregano\n\n\u00bd tsp. thyme\n\n4 oz. hard cheddar cheese, grated\n\n\u00be tsp. paprika\n\n1. Saute sunflower seeds or sesame seeds lightly in butter for a few minutes.\n\n2. Mix tomato sauce with spinach, rice, oregano, and thyme and simmer until most of the liquid is absorbed.\n\n3. Add the seeds and mix well.\n\n4. Place in a baking dish and sprinkle with the cheese and paprika.\n\n5. Bake for 30 minutes at 375 degrees.\n\nChristina is a disciple of the Indian guru, Rajneesh and is now living on one of his ashrams in Vermont. One day I received in the mail a package of recipes from her to test for my book. Mushroom Special is a very nice and different way to use mushrooms. Her recipe called for sour cream. I used yogurt. Both work nicely.\n\nMushroom Special **\n\n3 tbsp. sesame seeds\n\n3 medium onions, sliced and separated into rings\n\n2 tbsp. butter\n\n\u2154 cup sour cream or yogurt\n\n2 tbsp. tahini\n\n1 tbsp. tamari\n\n\u00bd tsp. celery salt\n\n\u00bd tsp. black pepper\n\n\u00bd lb. mushrooms, sliced\n\n3 sticks celery, sliced in small pieces\n\n\u00bd cup raisins\n\n1. Toast sesame seeds lightly in a frying pan and set aside.\n\n2. In a good sized frying pan, lightly fry the onions in butter and remove to a plate.\n\n3. Mix sour cream or yogurt, tahini, tamari, celery salt, and pepper.\n\n4. Add remaining ingredients to the frying pan and stir in the yogurt sauce.\n\n5. Simmer over low heat, stirring occasionally.\n\n6. Add onions and heat a few minutes more.\n\n7. Put in serving dish and sprinkle sesame seeds on top.\n\nMarty's cousin Cynthia whipped something like this up so quickly one night that I felt I certainly should include it, especially because I think that chickpeas are a good staple.\n\nCynthia Epstein's easy Chickpea Stew *\n\n2 tbsp. safflower oil\n\n1 large onion, sliced\n\n1 green pepper, chopped\n\n1 clove garlic, minced\n\n1 lb. 12 oz. can tomatoes\n\n1 20 oz. can chickpeas, drained\n\n1 tbsp. curry powder\n\n\u00bd tsp. chili powder\n\n1 tsp. oregano salt and pepper\n\n1. In the oil, saute onions, green pepper, and garlic until onions are golden.\n\n2. Add tomatoes, chickpeas and spices; cover and cook over low heat for 20 minutes.\n\nSesame Asparagus *\n\n2 lbs. asparagus, cut and trimmed into\n\n2-inch pieces\n\n2 tbsp. safflower oil\n\n1 tbsp. dark sesame oil\n\n1 tbsp. lemon juice\n\n1 tbsp. sesame seeds\n\n1. Heat safflower oil in wok or heavy frying pan.\n\n2. Add asparagus and stir fry over high heat for a couple of minutes, so that asparagus is well coated with oil\n\n3. Lower heat; add 2 tbsp. water; cover and cook until asparagus is done. It should be crunchy. Do not overcook.\n\n4. Transfer to serving bowl and toss with sesame oil.\n\n5. Add lemon juice; stir and sprinkle with sesame seeds.\nMeat\n\nShort Rib Stew ***\n\n4 lbs. lean short ribs, cut in 3-inch pieces\n\n2 medium onions, chopped\n\n\u00bd cup chopped carrots\n\n3 cups rich beef broth\n\n\u00bd tsp. marjoram\n\n1 tsp. caraway seeds\n\n1 bay leaf\n\n\u00bd tsp. savory\n\n1\u00bd tsp. salt\n\n1 tsp. pepper\n\ngrated rind of one lemon\n\n\u00bc cup wine vinegar\n\n3 tbsp. whole wheat flour\n\n\u00bc cup water\n\n\u00be lb. mushrooms, sliced\n\n1 small green pepper cut in thin strips\n\n2 tbsp. oil\n\n1. Heat a shallow roasting pan in a 500 degree oven for 5 minutes.\n\n2. Put ribs in pan and brown for about 15 minutes, turning in order to brown all sides.\n\n3. Transfer ribs to a large heavy pot; lower heat to 350 degrees and cook for 30 minutes.\n\n4. Pour off fat; place pot on top of stove; add onions and carrots and cook until onions are transparent.\n\n5. Add beef broth, seasoning, grated lemon, and vinegar.\n\n6. Cover, put back in oven and cook for 1\u00bd hours.\n\n7. Make a paste of the flour and water and cook on top of the stove until thick.\n\n8. Add to stew in oven.\n\n9. Saute mushrooms and peppers in oil until tender.\n\n10. Add to stew and cook 15 minutes more.\n\nFor anyone who loves sauerkraut as I do, this goulash is a must. I happen to use a pork tenderloin, but a less expensive cut of meat such as pork shoulder will do. It takes about 2\u00bd to 3 hours, so give yourself enough time.\n\nCompany Goulash ***\n\n3 slices of bacon\n\n2 cloves garlic, minced\n\n2 large onions, sliced\n\n2 cans (1 lb., 15 oz.) sauerkraut, rinsed and drained\n\n2 tbsp. caraway seeds\n\n14 peppercorns\n\n6 juniper berries, bruised\n\n1 cup white wine\n\n2 cups chicken broth\n\n2 tbsp. oil\n\n3 lbs. pork tenderloin cut in cubes\n\n2 tbsp. paprika\n\n1 1 lb., 12-oz. can tomatoes\n\nsalt and pepper, to taste\n\n1 cup sour cream\n\n1 tbsp. grated lemon rind\n\n1. Slowly cook bacon until almost done.\n\n2. Pour off all but 3 tbsp. grease; add garlic and onions and cook until soft.\n\n3. Add sauerkraut, stir and cook for a few minutes.\n\n4. Pour into large casserole and add caraway seeds, peppercorns, juniper berries, wine and chicken broth.\n\n5. Cover and cook in a 325 degree oven for 1 hour.\n\n6. Sprinkle pork with paprika and brown in a frying pan in the 2 tbsp. oil.\n\n7. Add pork, canned tomatoes, salt and pepper to sauerkraut; stir well, cover, and return to oven for another hour and a half.\n\n8. Stir in sour cream, sprinkle with lemon peel and serve.\n\nI happen to like dishes from Thailand, Vietnam, and Indonesia very much. We have a favorite Thai restaurant in New York and have often tried to inquire about the recipes for certain dishes. The owner is charming, but doesn't speak English well, so we're not very successful. This particular dish was given to me by a friend who knew that I liked this type of food. It's quite unusual, I think.\n\nMary's Pork Sate ***\n\n4 teaspoons small walnut pieces\n\n1 medium onion, minced\n\n1 tbsp. grated ginger\n\n2 cloves of garlic, minced juice from 1 lemon or lime\n\nsalt and pepper\n\n1\u00bd lb. lean, boneless pork, cut in cubes\n\n1. Blend all ingredients except pork.\n\n2. Mix pork cubes with this marinade and marinate for at least 4 hours.\n\n3. Broil pork on a charcoal grill or under the broiler until done.\n\n4. Serve with dipping sauce. (I think this sauce could be used with all kinds of meat.)\n\n_Dipping Sauce_\n\n\u00be cup roasted peanuts\n\n2 tbsp. peanut oil\n\n2 scallions, slivered\n\n\u00bd tsp. dried chilis\n\n1 small onion, minced\n\n\u00bd tsp. salt\n\nboiling water\n\n1\u00bd tbsp. tamari\n\njuice of 1 lime or lemon\n\n1. Grind the peanuts in a blender.\n\n2. In the oil, saute the slivered scallions for 2 minutes.\n\n3. Add chili and onion and fry 2 minutes more.\n\n4. Place mixture in a bowl and add peanuts and salt.\n\n5. Add enough boiling water to make a smooth paste.\n\n6. Stir in the tamari and lemon or lime juice and serve in a dipping bowl.\n\npork chops with raisin stuffing **\n\n6 nice sized pork chops\n\n2 tbsp. oil\n\n\u00bd tsp. pepper\n\n3 cups toasted bread crumbs\n\n1\u00bd cups chopped peeled apples\n\n\u00bd cup raisins\n\n\u00bd cup celery\n\n\u00bd cup onions\n\n\u00bd tsp. poultry seasoning\n\n\u00bc tsp. pepper\n\n1 cup beef or vegetable broth\n\n3 tart red apples, cored and halved\n\n1. Brown chops slowly on both sides in oil.\n\n2. Place in a heavy baking dish and sprinkle with \u00bc tsp. pepper.\n\n3. Combine bread crumbs, chopped apples, raisins, celery, onions, poultry seasoning and the other \u00bc tsp. pepper.\n\n4. Add broth and mix well.\n\n5. Spread stuffing evenly over pork chops.\n\n6. Lay 1 apple half on top of each pork chop.\n\n7. Cover with foil or a lid and bake in a 350 degree hour for 1 hour.\n\nMy husband and I stopped in Portofino, Italy and were very fortunate to find not only a lovely pension to stay in, but one that was presided over by a superb and joyous cook. His son waited on us for dinner. We were the only guests as it was off-season, so his father went off happily to prepare something for us. When we praised the meal to his son, our waiter, the young son proudly replied, \"If you like this, you should taste my Father's cooking when he really cooks.\" I have tried to approximate our main dish. I have experimented with it a number of times and this is as close as I can get.\n\nPortofino Pork With Sauce ***\n\n4 to 5 lb. pork loin\n\nfresh black pepper\n\n1 tsp. oregano\n\n1 tbsp. minced parsley\n\n2 cloves garlic, crushed\n\n1 cup white wine\n\n2 cups water\n\n1. Preheat oven to 400 degrees.\n\n2. Put pork in roasting pan and sprinkle with pepper, oregano, and parsley.\n\n3. Roast in oven for \u00bd hour.\n\n4. Add garlic, wine and water, cover and reduce heat to 350 degrees; cook for 1\u00bd hours or until pork is done, basting occasionally.\n\n5. Slice meat and spoon sauce over each serving.\n\n_Sauce_\n\n2 tbsp. olive oil\n\n2 tbsp. onion, finely chopped\n\n\u00bd cup carrots, cut into very fine julienne strips, about 2 inches long\n\n\u00bd cup celery, cut into very fine julienne strips, about 2 inches long\n\n\u2153 tsp. thyme\n\nsalt and pepper\n\n\u00bc cup white wine\n\n\u00bc cup water\n\na few tbsp. sauce from roasting meat near the end of it's roasting time (to add flavor)\n\n1. In the oil, saute onions, carrots, and celery for a few minutes.\n\n2. Add spices, wine and water.\n\n3. Cover, turn heat low and cook until done. The carrots and celery should be very soft.\n\n4. Toward the end of the cooking time, add two to three tbsp. of broth from the roasting pan to add flavor.\n\nAthos Zacharias is a fine painter who lives in New York and the Hamptons. For years now, he has been working on a house which he built himself and it is becoming as much a work of art as his paintings. Recently, we went to his home for dinner and to consider which of five paintings we were going to purchase. Our mutual friend, Richard Minsky had also been invited. The evening was prepared with great care\u2014the dinner, the viewing of the paintings, and the four of us playing a little music on assorted instruments afterwards. One of the highlights, however, was Zach's shy announcement that he had created a new meatloaf for my second cookbook. Zach has come newly to cooking, so I was a little apprehensive. It's hard to turn a friend's dish down. To my delight and surprise, it was wonderful, pretty to look at, delicious to eat, unlike the usual meatloaves, and a tribute to Zach's everpresent creativity.\n\nZach's Mint Meatloaf **\n\n3 tbsp. butter\n\n2 large white onions, chopped fine\n\n2 cloves garlic, minced\n\n2 lbs. ground chuck\n\n2 eggs, beaten\n\n\u00bc cup water\n\n\u00be cups Italian bread crumbs\n\n2 sweet red peppers, chopped fine\n\n\u00bd tsp. tarragon\n\n\u00bd generous cup of minced fresh mint or 3 level tsp. dried mint (if fresh is unavailable)\n\nsalt and pepper\n\n\u2153 cup wheat germ\n\n4 tsp. honey\n\n1. In the butter, saute onions and garlic until soft.\n\n2. Place ground chuck in a mixing bowl and add the onions and garlic.\n\n3. Add eggs, water, bread crumbs, sweet peppers, tarragon, mint, salt, and pepper; mix well with your hands.\n\n4. Place ingredients in a baking dish and shape into a nice loaf.\n\n5. Sprinkle wheat germ on top and drizzle honey over the wheat germ. (Zach wrote that one should drizzle the honey in an interesting pattern. Why not?)\n\n5. Bake in a 350 degree oven for about 1 hour.\n\nGinger beef stew is one of the those recipes that I have found along the way and I have not the slightest idea where it came from. My husband suggested a variation as he likes vegetarian meals.\n\nGinger Beef Stew **\n\n4 medium onions, chopped fine\n\n4 cloves garlic, minced\n\n1 tsp. salt\n\n1 tsp. tumeric\n\n\u00bd tsp. crushed chili pepper\n\n2 tsp. grated fresh ginger\n\n1 flank steak or an equally good cut of beef\n\n\u00bd cup sesame oil\n\n2 cups chopped tomatoes\n\n2 cups beef broth\n\n1. Mix the first six ingredients.\n\n2. If you are using flank steak, cut it in 4-inch strips. If you are using another kind of beef, cut it in small cubes.\n\n3. Marinate the meat in the spices for 3 hours.\n\n4. Heat sesame oil in large frying pan and brown meat quickly.\n\n5. Add tomatoes and beef broth.\n\n6. Lower heat, cover and stew until meat is tender.\n\n_Marty's Variation **_\n\n1. Instead of using flank steak, peel and cube a large eggplant.\n\n2. Coat the eggplant with the first six ingredients. You do not need to marinate the eggplant for any amount of time.\n\n3. Heat the oil and brown the eggplant.\n\n4. Add only 1 cup of beef broth with the tomatoes, turn heat down, cover and simmer until tender.\n\nA marvelous dish from Judy Wolfe's friend Carlos again. Every recipe of his has been especially good.\n\nCarlos' Liver and Noodle Casserole **\n\n\u00bd lb. chicken livers\n\n\u00bd lb. mushrooms, sliced\n\n3 tbsp. butter\n\n\u00bd pint sour cream\n\n\u00bc cup dry white wine\n\n1 lb. noodles (any size or kind)\n\n\u00bd cup minced parsley\n\n3 tbsp. chives\n\n\u2153 cup grated parmesan cheese\n\n1. Saute chicken livers and mush rooms in butter until livers are brown on the outside.\n\n2. Stir in sour cream, wine, salt and pepper and mix well.\n\n3. Meanwhile, cook noodles in boiling water until _al dente_ ; drain.\n\n4. Place noodles and liver\/mushroom mixture in a casserole and mix well.\n\n5. Sprinkle parsley, chives, and parmesan cheese on top.\n\n6. Cover with lid or foil and bake in a 350 degree oven for 20 to 25 minutes.\n\nMy mother must have given me this recipe at least 25 years ago and after I typed this, the original paper simply fell apart. Since veal is so terribly expensive, my husband suggested chicken cutlets, a perfectly acceptable substitute. If you use chicken, of course, you must cook it a little longer than veal.\n\nscallopini lucia **\n\n6 tbsp. butter\n\n\u00bd lb. mushrooms, thinly sliced\n\n\u00bd cup boiling water\n\n\u00bd cup white wine\n\n\u215b tsp. freshly ground nutmeg\n\n\u00be cup flour\n\nsalt and pepper\n\n2 tbsp. tomato paste\n\n\u00bc cup sour cream\n\n1. Melt 3 tbsp. butter and saute mush rooms for 5 minutes.\n\n2. Add the boiling water, wine and nutmeg and let simmer 20 minutes.\n\n3. Lightly coat the veal with flour, salt and pepper and saute in the other 3 tbsp. butter until light brown. This takes only a few minutes or the veal will turn tough.\n\n4. Add the juices from the veal pan to the mushroom sauce along with the tomato paste.\n\n5. Simmer 5 minutes longer.\n\n6. Add the veal slices and the sour cream and simmer just a few seconds until hot.\n\n7. Serve immediately.\n\nJudy Wolfe said to me wonderingly, \"What, you've never heard of picadillo? Come over to my house next week and I'll make it for you.\" We did indeed go over and eat picadillo. I liked it and thought that perhaps if I hadn't heard of it, some other enthusiastic cooks might not know about it either.\n\nJudy Wolfe's Picadillo **\n\n\u00bd cup raisins or currents\n\n2 tbsp. peanut oil\n\n1 large onion, finely chopped\n\n2 cloves garlic, minced\n\n1 large green pepper, seeded and chopped finely\n\n1 cup stuffed green olives\n\n1 3 oz. jar capers\n\n2 tbsp. vinegar\n\nsalt and pepper\n\n\u00bd tsp. cinnamon\n\n\u00bd tsp. allspice\n\n1 bay leaf\n\na dash of tabasco sauce\n\n2 lbs. ground meat\n\n1 16-oz. can tomatoes, partially drained.\n\n\u00bd cup chili sauce\n\n1. Place raisins or currants in a bowl of warm water so that they will \"plump up.\"\n\n2. Heat 1 tbsp. of the oil in a frying pan, add onions, garlic, and green pep per and cook until soft.\n\n3. Add olives, capers, vinegar, salt, pep per, spices, bay leaf, and tabasco sauce and cook 5 minutes.\n\n4. Cook ground meat separately in a skillet until it just loses its pink color.\n\n5. Pour off any grease and add the olive and spice mixture to the ground meat, stirring to blend.\n\n6. Drain raisins and add.\n\n7. Add tomatoes and chili sauce and cook over low heat for about 45 minutes.\n\n8. Serve with saffron rice, sour cream and chopped hard boiled egg as Judy does, if you wish.\n\nA perfect dish to make in chilly weather, choucroute garnie is a rather well known dish and everyone must have a favorite recipe. It is simply various cuts of pork and sausages baked with sauerkraut and either juniper berries or gin to give it a special flavor. You can substitute other kinds of pork from what I have used in this dish if you wish.\n\nChoucroute Garnie **\n\n2 large onions, chopped\n\n1 large carrot, chopped\n\n3 tbsp. peanut oil\n\n1 large clove garlic, minced\n\n2 tart apples, peeled, cored, and chopped\n\n2 1 lb., 11-oz. cans sauerkraut\n\n2\u00bd cups chicken broth\n\n1 cup dry white wine\n\n1 bay leaf\n\n8 whole peppercorns\n\n8 juniper berries or 2 tbsp. gin\n\nsalt and pepper\n\n1\u00bd lb. kielbasa sausage, cut in three inch pieces\n\n\u00bd lb. baked ham, sliced\n\n1\u00bd lbs. boneless smoked pork tender loin, sliced\n\n1. In a large frying pan, saute onion and carrots until soft.\n\n2. Add garlic and apple and saute a few minutes more.\n\n3. In the meantime, drain sauerkraut and rinse with cold water.\n\n4. Add to pan along with chicken broth and wine.\n\n5. When the stew begins to simmer, add the bay leaf, peppercorns, juniper berries or gin, salt and pepper and stir well.\n\n6. Transfer to a large deep baking dish, cover, and bake at 325 degrees for 2 hours.\n\n7. Add kielbasa and cook 20 minutes more. If broth has evaporated, add more broth or wine.\n\n8. Add the ham and pork, cover and cook 20 minutes more.\n\n9. Serve with hot mustard.\n\nWhen I was a newlywed, I complained that although chicken livers were cheap and fit my budget, I didn't like them. A neighbor from upstairs said, \"Nonsense, you just don't know how to prepare them and I will show you how.\" He did\u2014so here they are. I really did like them, much to my surprise.\n\nAl Maline's Chicken Livers *\n\n3 tbsp. olive oil\n\n2 cloves garlic, minced\n\n1 cup sliced onions\n\n1 cup green pepper, sliced\n\n1 pound chicken livers, trimmed and cut in pieces\n\n\u00bd tsp. rosemary\n\n\u00bd cup white wine\n\nsalt and lots of pepper\n\n1. In the oil, saute garlic for a few minutes.\n\n2. Add onions and peppers and saute a few minutes more.\n\n3. In the meantime, pat the chicken livers dry and add to the frying pan. Cook until almost done.\n\n4. Turn heat low and add the rosemary, wine, salt and pepper.\n\n5. Cover and cook for five minutes.\n\nMy husband says this is his favorite way to prepare liver. It couldn't be simpler.\n\nCalves Liver with Lemon *\n\n1 lb. calves liver\n\n\u2153 cup flour\n\n2 tbsp. butter\n\n2 tbsp. oil\n\njuice from 2 lemons\n\n2 tbsp. finely chopped fresh parsley\n\n1. Trim liver well and cut in thin strips.\n\n2. Dredge liver on both sides lightly in the flour.\n\n3. Heat butter and oil in a skillet and, when hot, saute liver until done, turning once. It should be brown on the out side, but still pink inside.\n\n4. Remove to serving platter.\n\n5. Add lemon juice and parsley to pan and for 1 minute scrape the sides and bottom of the pan well with a wooden spoon.\n\n6. Pour sauce over liver and serve immediately.\nPoultry\n\nAn easy baked chicken recipe given to me by Mrs. Feiman when I was first married, this seems to please children everywhere, and adults have no complaints either. You can purchase cornflake crumbs, but I prefer to buy my own cornflakes in a health food store and grind them in my blender.\n\nMrs. Feiman's Baked Chicken *\n\n\u00bc cup safflower oil\n\n\u00bc tsp. salt\n\n\u00bc tsp. garlic salt\n\n\u00bd tsp. pepper\n\n1 frying chicken, cut in pieces\n\n2 cups cornflakes crumbs\n\n1. Mix oil, salt, garlic salt, and pepper in a small dish.\n\n2. With a pastry brush, coat each piece of chicken with seasoned oil.\n\n3. Dip chicken pieces in cornflake crumbs and place in a lightly greased flat baking dish, making sure that pieces don't overlap.\n\n4. Cover with foil and bake for 45 minutes in a 350 degree oven.\n\n5. Remove foil and bake 15 minutes more until chicken is brown and crisp.\n\nThe only dish I know from Pakistan, this spicy chicken has a unique flavor.\n\nPakistani Chicken *\n\n4 chicken breasts\n\n6 cloves garlic, minced\n\n2 onions, chopped fine\n\n\u00bd cup ground, blanched almonds\n\n1 tsp. cardamon\n\n\u00bd tsp. red pepper flakes\n\n\u00bd tsp. cloves\n\n1\u00bd tsp. salt\n\n1 tsp. pepper\n\n1 piece of fresh ginger the size of a quarter, minced\n\n4 tbsp. melted butter\n\n2 tbsp. water\n\n1. Take the skin off the breasts and prick the chicken all over with a fork so that it will absorb the spices.\n\n2. Mix remaining ingredients to make a smooth paste.\n\n3. Spread paste all over the chicken.\n\n4. Place in a buttered baking dish making sure that the pieces don't overlap.\n\n4. Roast in a 350 degree oven, covered, for 1 hour.\n\n5. Remove cover and roast 20 minutes more.\n\nWhen we drove through the Swiss Alps, the first region we entered in Italy was the Piedmont region and this was our first meal there or, at least, this is how I tried to copy it. It's especially nice for dinner parties because it can be prepared in advance and just stuck under the broiler at the last moment.\n\nPiedmont Chicken *\n\n4 boneless chicken breasts\n\n\u00be cup flour\n\nsalt and pepper\n\n4 tbsp. butter\n\n1 bouillon cube\n\n\u00be cup hot water\n\n\u00bc cup white wine\n\n1 cup grated parmesan cheese\n\n3 tbsp. fresh parsley, chopped\n\n1. Pound chicken breasts until quite flat and cut in half.\n\n2. Mix flour, salt and pepper and coat chicken lightly.\n\n3. Heat butter in frying pan and brown chicken pieces.\n\n4. Turn heat to medium and cook chicken until done, turning every so often.\n\n5. Set chicken aside and keep warm.\n\n5. Dissolve bouillon in hot water.\n\n6. Place bouillon and wine in the same skillet and cook until liquid is reduced by half.\n\n7. Place chicken in a roasting dish, pour liquid over it and sprinkle with parmesan cheese.\n\n8. Place under broiler and broil until cheese is melted.\n\nAn exotic dish, this one is a little too rich for my blood, so I try not to overdo it. It's best made with cream, but you can always try half-and-half and serve it a little more often. It comes from Thailand and is quite delicious. Save the coconut for the coconut cake in my dessert section.\n\nCoconut Chicken *\n\n2 cups heavy cream\n\n1\u00bd cups dried, grated coconut\n\n6 chicken breasts\n\n4 cloves garlic, minced\n\n\u00bd tsp. dried red pepper flakes\n\n1 small onion, minced\n\n1 tbsp. grated lemon rind\n\n1 tsp. coriander\n\n1 tbsp. soy sauce\n\n3 tbsp. ground peanuts\n\n1. Put cream and coconut in a saucepan; place over low heat and bring to a boil, then remove from fire and let sit for 30 minutes.\n\n2. Drain and save coconut for another use.\n\n3. Place chicken and coconut milk in a heavy saucepan; cover and cook for 40 minutes over low heat, turning chicken occasionally.\n\n4. Add remaining ingredients and cook for another 15 minutes, again turning chicken to coat with sauce.\n\n5. Serve with rice.\n\nMost people I know hardly ever use a popular Greek pasta called orzo. Fortunately, we knew a fine Greek cook and friend named Bessie Ahouris from Silver Spring, Md., who fed us this. Look for orzo in the pasta section. It actually looks a bit like rice.\n\nBessie's Chicken and Orzo **\n\n2 small chickens, cut in small pieces salt and pepper\n\n2 tbsp. butter\n\n1 16-oz. can tomatoes, drained\n\n\u00bd tsp. garlic powder\n\n2 cups orzo\n\n\u00bd to 1 cup water\n\n1. Place chickens in a roasting pan, sprinkle with salt and pepper and dot with butter.\n\n2. Bake in a 350 degree oven until nicely browned (about 1 to 1\u00bd hours).\n\n3. Remove chicken and set aside.\n\n4. Place tomatoes sprinkled with garlic powder in the roasting pan and brown in oven. As tomatoes start to brown on the edges, keep scraping the sides of the pan and cook for around 20 minutes.\n\n5. In the meantime, cook orzo according to directions on box and drain.\n\n6. When tomatoes are brown, return chicken to pan; add orzo and enough water to make a good consistency.\n\n7. Place pan in oven and cook an additional 10 minutes to blend all flavors.\n\n8. Check for seasonings and serve.\n\nAn experiment\u2014slightly different with a nice spiciness!\n\nChicken Piquant *\n\n1 oz. dried Italian mushrooms\n\n2 cups canned tomatoes\n\n3 tbsp. oil\n\n2 tbsp. paprika\n\n1 chicken, cut into frying pieces\n\n1 cup hard salami, cut in thin 3 inch strips\n\n2 cloves garlic, chopped\n\n\u00bd cup dry white wine\n\n1 cup sliced fresh mushrooms\n\n1 tsp. red pepper flakes\n\nsalt and pepper\n\n1. Put dry mushrooms in a bowl; cover with warm water and let sit one hour.\n\n2. Put tomatoes in a saucepan and simmer until they are reduced by about half.\n\n3. Heat oil in heavy skillet, sprinkle chicken with paprika, and brown well on both sides.\n\n4. Add salami and garlic and cook for 3 minutes.\n\n5. Pour off fat, add wine and reduced tomato sauce to skillet.\n\n6. Drain and squeeze moisture out of soaked Italian mushrooms.\n\n7. Add the Italian mushrooms, fresh mushrooms, red pepper flakes, salt, and pepper and cover.\n\n8. Cook over low heat for about one hour until chicken is done.\n\nWhen I was in college, I had a Greek boyfriend who used to take me home for Sunday dinners. I was fascinated with his Greek family, their friends, their rituals, and especially their cooking. At my own home, Sunday chicken dinner usually meant a roast chicken with mashed potatoes. At my girlfriend's home, stewed chicken with boiled potatoes was the order of the day. This Greek lady did it differently and I've tried to approximate it here. I believe she cooked everything in the pan from the beginning, but I find it easier to parboil the vegetables first, so that everything gets done on time. If you find there is too little juice to baste with, add a little more wine.\n\nSunday Greek Dinner **\n\n1 frying chicken, washed, patted dry, and cut in pieces\n\n2 tbsp. oil\n\njuice from 1 large lemon\n\n1 large clove garlic, minced\n\n\u2153 cup oregano\n\nsalt and pepper\n\n\u00be to 1 cup white wine\n\n4 potatoes, peeled and halved\n\n4 large carrots, cut in strips\n\n8 small onions, peeled\n\n1. Lay the chicken in a large flat baking pan.\n\n2. With a pastry brush, dab the oil on the chicken pieces to coat them.\n\n3. Pour the lemon juice over the chicken.\n\n4. Place the garlic on top of the chicken pieces and then sprinkle generously with the oregano, saving some of it for the vegetables.\n\n5. Add salt and pepper and pour the wine into the bottom of the pan.\n\n6. Cover with foil and place in a 350 degree oven.\n\n7. Cook the chicken for 45 minutes.\n\n8. Parboil the potatoes and carrots in boiling water for 10 minutes and then add the onions for 10 minutes more. Drain. (Save cooking water for soup.)\n\n9. When the chicken has cooked for 45 minutes, place the parboiled vegetables in the pan and sprinkle with the rest of the oregano, salt and pepper. Cover and bake 15 minutes more.\n\n10. Remove foil and bake for the last 15 minutes, basting with the pan juices.\n\nValdi Sherwood is a very attractive woman who possesses boundless optimism and deals with assorted children, animals, jobs and traveling with great cheerfulness and equanimity. She and her mate, Cliff Irving, are friends whose company we have enjoyed for a number of years. Valdi laughs easily and I am always impressed with her resourcefulness and zest for life. When you prepare her dish, make sure you choose a rice pilaf that has pure ingredients.\n\nvaldi's chicken pilaf **\n\n1 9 oz. package of rice pilaf\n\n\u00bd cup raisins.\n\n\u00bd cup slivered almonds\n\n1 8\u00bd oz. can artichoke hearts, drained\n\n2 tbsp. safflower oil\n\n1 medium onion, chopped\n\n2 cloves garlic, sliced\n\n2 chicken breasts, skinned and cut into thin strips\n\n1 cup white wine\n\n1 tsp. dry mustard\n\nsalt and pepper\n\n\u00bd cup sour cream\n\n1. Prepare a package of rice pilaf according to directions.\n\n2. When done, add \u00bd cup raisins, \u00bd cup slivered almonds, the artichoke hearts, and stir well.\n\n3. Heat oil in a large frying pan, and saute onions and garlic until golden.\n\n4. Add chicken and stir fry over high heat until chicken has lost its pink color.\n\n5. Add white wine, and stir for a few minutes in order to deglaze pan, scraping often.\n\n6. Add mustard, salt, pepper, and sour cream and heat through.\n\n7. Reheat pilaf.\n\n8. Pour chicken over pilaf and serve immediately.\n\nOne of the easiest of chicken curry recipes, this doesn't take days to prepare. You can serve it with rice and all the standard condiments, like coconut, chutney, chopped hard-boiled eggwhites, mashed hard-boiled egg yolks, and peanuts.\n\nEasy Chicken Curry **\n\n1 whole chicken (4 to 5 pounds)\n\n2 quarts water\n\n3 tbsp. peanut oil\n\n3 tbsp. curry powder\n\n1 clove garlic, minced\n\n1 cup onion, diced\n\n2 apples, peeled and diced\n\n1 tomato, chopped\n\n1 banana, sliced\n\ndash of cayenne pepper\n\n1 bay leaf\n\nsalt and pepper\n\n2 cups chicken broth\n\n1. Put chicken into 2 quarts water and simmer until done. Save broth.\n\n2. Skin and bone chicken and shred chicken finely.\n\n3. Saute curry powder in oil for a few minutes.\n\n4. Add garlic and onions and cook a few minutes more.\n\n5. Add the shredded chicken and stir to coat well.\n\n6. Add apples, tomatoes, bananas, cayenne pepper, bay leaf, salt and pepper.\n\n7. Cook 1 minute.\n\n8. Add 2 cups of the chicken broth, cover and simmer 20 minutes.\n\n9. Remove cover and cook 15 minutes more.\n\nBaked chicken salad is a marvelous party recipe from Dorothy and Wilson Stone, our neighbors in East Hampton. I always enjoy their Sunday lunches held in their sun-filled dining room\u2014flowers on the tables, music softly playing in the background, interesting guests, many laughs. They entertain with great style and are warm, gracious hosts. Dorothy prepared this for us on one such Sunday, but said that it was actually given to her by Wilson's Mother. It's such a convenient dish because it can be made way ahead and just popped into the oven when guests arrive.\n\nBaked Chicken Salad ***\n\n4 cups cooked, salted chicken, cubed\n\n2 cups celery, diced\n\n\u00be cup slivered almonds\n\n1 8-oz. can water chestnuts, sliced\n\n\u00be cup cheddar cheese, grated\n\n1\u00bd cups mayonnaise, preferably homemade\n\n\u00bd can cream of mushroom soup\n\n3 tbsp. lemon juice\n\n6 slices thin white bread\n\n\u00bd stick butter\n\n1. Mix all ingredients except bread and butter and place in a flat baking dish.\n\n2. Trim crusts from bread and cut in cubes.\n\n3. Melt butter in skillet and saute bread cubes until brown and crunchy.\n\n4. Sprinkle cubes over chicken and bake in a 350 degree oven for 45 minutes.\nFish\n\nA recipe from Cambodia, this is called michha trong kroeung. It is quite easy to prepare and although I like a rather delicate white fleshed fish, I imagine you could use a heavier kind like codfish or monkfish for both economy or variety, making sure to cook the fish longer before adding the sauce.\n\nCambodian Fish with Scallion Sauce **\n\n5 tbsp. peanut oil\n\n1\u00bd lbs. fish fillets\n\n4 cloves garlic, sliced\n\n12 scallions, halved and sliced into 3 inch pieces\n\n1 tbsp. paprika\n\n2 tsp. fennel or anise seeds\n\n1 tbsp. lemon juice\n\ndash of salt and pepper\n\n\u00bd cup water\n\n\u00bd cup chopped parsley\n\n1. Heat oil in large skillet and add fish.\n\n2. Cook just a few moments until fish is almost done, turning once.\n\n3. Remove fish to baking dish and set aside.\n\n4. Add remaining ingredients to skillet, except parsley, and simmer over low flame for 15 minutes, adding a little more water if necessary.\n\n5. Place fish in baking dish, spoon sauce over it and bake in 350 degree oven for 15 minutes.\n\n6. Sprinkle with fresh parsley and serve.\n\nThis particular recipe coming from my step-mother is unusual. She used sour cream, but I use half yogurt and half sour cream. You can use all yogurt or all sour cream.\n\nFern's Chinese Tuna *\n\n2 tbsp. peanut oil\n\n\u00bd cup onion, minced\n\n\u00bd cup green pepper, minced\n\n1 10 oz. can tuna, well drained\n\n\u00bd can tomato paste\n\n2 cups water\n\n\u00bc tsp. black pepper\n\n\u00bc tsp. red pepper flakes\n\n\u00bd pint yogurt\n\n\u00bd pint sour cream\n\n4 cups Chinese noodles\n\n1. Heat oil in frying pan and add onions and green peppers.\n\n2. Cook until onions become transparent.\n\n3. Add tuna and mix well with a fork.\n\n4. Stir tomato paste in water and mix well.\n\n5. Add tomato paste mixture to tuna along with both black and red pepper and stir.\n\n6. Lower heat and simmer for 15 minutes.\n\n7. Just before serving, stir in the yogurt and sour cream.\n\n8. Pour over warmed Chinese noodles.\n\nA lovely and different way to prepare shrimp, this is a good party dinner if you're not having too many people since shrimp is rather expensive. It can be made early in the day and reheated before dinner. It's a very old recipe of mine and comes from Ceylon.\n\nCeylon Shrimp Curry **\n\n1 lb. shrimp, shelled and deveined\n\n1 medium onion, sliced in rings\n\n1 tsp. red pepper\n\n\u215b tsp. tumeric\n\n2 cloves garlic, sliced\n\n2 slices fresh ginger or \u00bd tsp. dry ginger\n\n1 tsp. salt\n\n2 inch piece cinnamon stick\n\n\u00be cup milk\n\n1 tbsp. oil\n\njuice of \u00bd lime\n\n1 tsp. cornstarch\n\n\u00bc cup water\n\n1. Place shrimp, \u00bd of the onion, red pepper, tumeric, garlic, ginger, salt, cinnamon stick and milk in a saucepan and boil gently for 10 minutes, stirring occasionally.\n\n2. Brown the rest of the onions in the oil.\n\n3. Add onions and lime juice to shrimp and simmer 10 more minutes.\n\n4. Mix cornstarch with water, add to shrimp and cook until slightly thickened.\n\n5. Serve with boiled rice.\n\nJudy Wolfe came over one morning and said, \"I had the best fish dish last night. You'll have to try it.\" So, the next night I tried it out on my husband and mother-in-law. They both felt it was an interesting and unusual dish, so I'm passing it along like a chain letter.\n\nJudy's Fish with Mushrooms *\n\n4 fish fillets (bluefish, flounder, fluke etc.)\n\n\u00bc cup mayonnaise\n\n4 tbsp. yogurt\n\n2 tbsp. Dijon mustard\n\n2 tbsp. lemon juice\n\nsalt and pepper\n\n1 large onion, sliced\n\n\u00bd lb. mushrooms, sliced\n\n2 tbsp. butter\n\n1. Place fish fillets in a flat baking dish.\n\n2. Mix mayonnaise, yogurt, mustard, lemon juice, salt, and pepper.\n\n3. Spread sauce over fish and bake in a 350 degree oven until fish flakes easily.\n\n4. In the meantime, saute onions and mushrooms in butter until soft.\n\n5. When fish is done, pour the mush room and onion sauce over the fish; place the fish under the broiler for a few minutes and serve.\n\nMy good friend Florence Grey invented this dish, invited us to dinner to sample it, and then donated it to my cookbook. Now, that's a true friend. I found that the directions on preparing couscous differ on each brand, so you need to follow the directions on your particular box. Florence uses \"Harissa,\" a hot sauce found in gourmet shops, but you can use another brand or simply cayenne or red pepper flakes.\n\nFlorence Grey's Scallops with Couscous **\n\n1 1-lb. box couscous\n\n\u00bd tbsp. \"Harissa\" or comparable hot sauce\n\n3 tbsp. butter\n\n1 clove garlic crushed\n\n1 lb. scallops\n\n2 tbsp. parsley, minced\n\n1. Cook couscous according to directions.\n\n2. Mix it well with the hot sauce. You may want to add more or less sauce depending on how hot you like your food.\n\n3. Keep the couscous warm while you cook the scallops.\n\n4. Heat butter with the garlic and saute scallops very quickly for just a few minutes until they turn from pink to white. Do not overcook.\n\n5. Pour scallops into couscous, scraping the pan and mix well.\n\n6. Garnish with parsley and serve immediately.\n\nI have never really been able to make sole amandine that pleases me. One evening our good friends Bernice and Morton Hunt treated us to dinner at their favorite restaurant in the Hamptons. Everyone ordered sole amandine and it was truly superb. \"Bernice,\" I said, \"what do you think is the secret of this dish\"? Bernice is such a superior cook that I thought if anyone knew, she would. \"I think they dip it lightly in flour and then cook it quickly over a very high heat in lots of butter,\" she replied. The next day, I went and bought sole fillets and proceeded to make sole amandine for dinner. Bernice was right. Incidentally, you can use any thin white fillet of fish; flounder is a good choice but it must be very fresh.\n\nSole Amandine *\n\n4 fillets of sole\n\n\u00bd cup flour\n\n\u2153 cup butter\n\n\u2153 cup slivered almonds\n\nlemon wedges\n\nparsley\n\n1. Lightly flour the fish fillets on both sides.\n\n2. Heat butter over high flame in a heavy frying pan and saute fillets in butter until done. This should take only a few moments and the fish should be turned once.\n\n3. Put fish on a platter, lower heat, quickly saute the almonds lightly and pour almonds and butter over the fish.\n\n4. Serve with lemon wedges and parsley.\n\nA nice way to prepare a cheaper kind of fish, this is a simple and tasty fish stew. You can use a thick filleted fish like anglerfish, monkfish, or I suppose cod. Serve with crusty french bread to dip into the sauce.\n\nGood Cheap Fish *\n\n3 tbsp. olive oil\n\n4 fish fillets\n\n2 tbsp. safflower oil\n\n1 large onion, minced\n\n1 scallion, cut in pieces\n\n3 tomatoes, skinned and quartered\n\n\u00be cup white wine\n\nsalt and pepper\n\n1 tsp. basil\n\na sprinkling of red pepper flakes\n\n2 tbsp. minced parsley\n\n1. Heat olive oil in frying pan, add fish and fry for 5 minutes, turning once. Set aside.\n\n2. In a large pan, saute onions and scallions in safflower oil for a few minutes.\n\n3. Add remaining ingredients and bring to a boil.\n\n4. Turn heat low and simmer for about 20 minutes, so that the sauce cooks down and is not watery.\n\n5. Add the fish to the sauce and cook over low heat until fish is done and it flakes easily.\n\nI had never used a Chinese spice called star anise, so when I found an old recipe in my files calling for it, I decided to try it. It's a very pretty spice and indeed looks like a star. It imparts a very delicate flavor to this dish.\n\nChinese Flounder **\n\n\u00bc cup safflower oil\n\n1 lb. flounder fillets or any kind of delicate white fish.\n\n1 clove garlic, minced\n\n1 tsp. ginger, minced\n\n2 scallions, cut in 1-inch lengths\n\n\u00bd cup sliced mushrooms\n\n1 tsp. salt\n\n4 tsp. tamari\n\n2 tsp. dry sherry\n\n1 cup chicken stock\n\n1 clove star anise\n\n1. Heat oil in a large pan and saute the fillets for \u00bd minute on each side.\n\n2. Add garlic, ginger, scallions, mush rooms, salt, tamari, and sherry and cook 1 minute.\n\n3. Slowly add the chicken stock.\n\n4. Turn heat on high and add the star anise.\n\n5. When the stock begins to boil, turn the heat to low, cover and cook 5 to 10 minutes or until the fish is done.\n\nA concoction that came about when I wanted to use some leftovers and I didn't have enough for a meal. Kasha Imperial was created by adding sausage and a few other ingredients. It's reminiscent of a paella and of course you can substitute different kinds of fish, including mussels, lobster etc.\n\nKasha Imperial **\n\n3 cloves garlic, mashed\n\n2 onions, sliced in rings\n\n1 green pepper, seeded and cut in thin strips\n\n\u00bc cup olive oil\n\n2 tbsp. curry powder\n\n3 lbs. fish, such as weakfish, blackfish etc.\n\n2 lbs. Italian hot sausage\n\n1 cup dry white wine\n\n2 bay leaves\n\n1 2-lb. can Italian plum tomatoes\n\n4 cups cooked kasha\n\n1 dozen small clams\n\n\u00bd lb. scallops\n\n1 fresh red tomato, cut in cubes\n\n1. Saute garlic, onions, and green pepper in olive oil until the onion is golden.\n\n2. Add curry powder and cook 2 minutes.\n\n3. Push vegetables aside, add fish and brown lightly.\n\n4. Remove fish from pan and set aside.\n\n5. Add sausages and brown well on all sides, pricking to release any grease.\n\n6. Remove grease from pan and add wine, bay leaves, canned tomatoes and simmer gently until slightly reduced. To this point, everything can be made well in advance.\n\n7. Return fish to pan along with the cooked kasha and simmer until both fish and sausage is done.\n\n8. Add clams and scallops; cover and cook until clams have opened.\n\n9. Place in large serving bowl and garnish with freshly chopped tomatoes.\n\nFlorence Grey made this tangy fish dish for us one evening with the help of her husband Jerry, who sliced all the scallions beautifully like a fine oriental chef. He made such delicate slivers of scallions that the dish was very pretty to look at as well as tasting very, very good. While she broiled bluefish fillets, you can substitute other kinds of fish and it can be steamed instead of baked. The secret is the sauce. Make sure most of the liquid boils away, so it isn't runny when you pile it on the fish.\n\nFlorence's Tangy Fish *\n\n2 tbsp. butter\n\n2 bunches of scallion tops, slivered very thin and cut in 3 inch pieces\n\n1 piece of fresh ginger, about the size of a quarter and sliced in small thin pieces\n\n\u00bd tsp. Chinese hot chili paste (found in oriental stores)\n\n\u2153 cup white wine\n\n4 bluefish fillets\n\n2 tbsp. olive oil\n\n1. Melt butter in skillet, add scallion tops and ginger and saute for a few minutes.\n\n2. Add the Chinese chili and wine; stir and simmer over low heat for about 10 to 15 minutes until the sauce has cooked down.\n\n3. In the meantime, brush the fillets with olive oil and broil until done.\n\n4. Spoon some of the hot sauce on top of each fillet and serve immediately.\nDesserts\n\nThis is a delicious dessert that came from my former mother-in-law. It can be served plain or with fresh fruit.\n\nRussian Cream *\n\n1 cup cream\n\n\u00be cup sugar\n\n2 tsp. gelatin\n\n1 cup cold water\n\n1 cup sour cream\n\n\u00bc tsp. salt\n\n\u00bd tsp. vanilla\n\n1. Heat cream and sugar in top of double boiler until luke warm.\n\n2. Soak gelatin in 1 cup cold water for 5 minutes and then add to cream.\n\n3. Stir very well until dissolved and then place in refrigerator to chill.\n\n4. When mixture begins to thicken, add sour cream, salt and vanilla.\n\n5. Stir well and put into mold or individual dishes to set.\n\nA very simple dessert, easy to prepare and not very fattening, this comes from Pam Topham.\n\nApplesauce Fluff *\n\n5 eggwhites, whipped into peaks\n\n1 quart applesauce\n\nsprinkling of fresh nutmeg\n\n1. Fold eggwhites gently into applesauce.\n\n2. Sprinkle with nutmeg and serve.\n\nWhen Marty was interviewing people for a biography of the psychiatrist, Fritz Perls, one of Dr. Perl's relatives served us this rich and wonderful dessert during a coffee break.\n\nBavarian Apple Slice ***\n\n1\u00bd cups flour\n\n\u00bd tsp. salt\n\n3 tbsp. sugar\n\n\u00bd cup butter\n\n1 egg yolk\n\n5 tbsp. milk\n\n1 to 1\u00bd lbs. tart apples, peeled and sliced\n\n2 tbsp. seedless raisins\n\n2 tbsp. golden raisins\n\n3 tbsp. fresh bread crumbs\n\n1 tbsp. brown sugar\n\n1 tsp. cinnamon\n\nwhipped cream or sour cream as a side dish if desired.\n\n1. Sift flour, salt and sugar into a mixing bowl.\n\n2. Cut butter into mixture with a pastry fork or knives.\n\n3. Mix egg yolk with milk and stir into pastry to bind together. (You may have to add more milk.)\n\n4. Chill 30 minutes.\n\n5. Mix apples, both kinds of raisins, bread crumbs, brown sugar and cinnamon.\n\n6. Knead pastry lightly on a floured board.\n\n7. Divide into two parts\u2014a \u2154 piece, and a \u2153 piece.\n\n8. Roll out the larger piece so that it will fit an 8-inch pie tin.\n\n9. Place it gently in the tin and fill with the fruit.\n\n10. Roll out the smaller piece and cut into \u00bc-inch strips.\n\n11. Cover the pie with the strips to make a lattice work, using a little milk or water to seal the edges.\n\n12. Bake in a 375 degree over for about 35 minutes.\n\n13. Serve with whipped or sour cream.\n\nA wonderful dessert from Indiana again, pecan surprise squares are easy to prepare and take very little time.\n\nPecan Surprise Squares *\n\n1 egg, beaten\n\n1 tsp. vanilla\n\n1 cup brown sugar, firmly packed\n\n\u00bd cup flour\n\n\u00bc tsp. salt\n\n\u00bc tsp. baking soda\n\n1 cup pecans, coarsely chopped\n\n1. Stir egg, vanilla, and brown sugar together.\n\n2. Sift flour, salt and baking soda.\n\n3. Add to brown sugar and stir well.\n\n4. Stir in pecans.\n\n5. Spread in a buttered square dish. Use your fingers to press mixture down, but do not worry if it doesn't completely cover, as it will spread in the baking.\n\n6. Bake at 350 degrees for 18 to 20 minutes. Mixture will be soft when you take it from the oven.\n\n7. Let it cool slightly and then cut in squares.\n\nA dish I concocted, pears with ginger can be served with sour cream instead of yogurt\u2014not as healthy and slimming, but very nice.\n\nPears with Ginger *\n\n4 fresh pears, peeled and cut in quarters\n\n1 cup water\n\n\u00bc cup raw sugar\n\n1 cup yogurt\n\n\u00bc cup honey\n\n2 tsp. crystallized ginger, cut in small pieces\n\n1. Stir pears in water and sugar over low heat until done.\n\n2. Combine yogurt and honey and mix well.\n\n3. Divide pears and a little bit of the syrup into four dessert dishes.\n\n4. Spoon yogurt mixture on top and sprinkle each with the ginger.\n\nWhat an elegant dessert! We were having friends over for a communal meal and evening of music when Christina showed up with a dish that had everyone ooing and aahing. She said that no barbecue was complete in her native Australia without a \"Pavlova.\" She graciously gave me the recipe and I tried it out the next night for my husband and two friends. It was consumed in a moment with great relish and gusto. You can substitute any kinds of fruit.\n\nPavlova **\n\n4 egg whites\n\npinch of salt\n\npinch of cream of tartar\n\n\u00bd cup sugar\n\n\u00bd tsp. vanilla\n\n\u00bd cup slivered almonds or walnut bits.\n\n\u00bd pint whipping cream\n\n2 tbsp. brown sugar\n\n1 large pear, peeled and sliced\n\n1 banana, sliced\n\n1\u00bd cups strawberries, sliced\n\n1. Place eggwhites, salt, and cream of tartar in a bowl and beat until eggwhites are stiff.\n\n2. Slowly beat in sugar.\n\n3. Add vanilla and fold in nuts.\n\n4. Place meringue on a large piece of buttered foil and fold up edges of foil slightly to contain meringue.\n\n5. Bake in a 200 degree oven for one hour.\n\n6. Allow it to cool in the oven, so the meringue won't crack.\n\n7. Whip cream into peaks.\n\n8. Fold in sugar and fruit.\n\n9. Spread whipped cream and fruit on meringue and cut in pie-shaped wedges.\n\nOrange Delight *\n\n4 oranges, skinned and cut into bite sized pieces\n\n\u00bd cup triple sec liquor\n\n\u00bd cup marmalade\n\n\u00bd cup almonds, blanched and slivered\n\n1. Marinate orange pieces in triple sec for 3 hours.\n\n2. Pour off liquid and combine it with marmalade.\n\n3. Mix the marmalade and triple sec in a blender for a few minutes.\n\n4. In 4 individual dessert bowls, layer the orange pieces with the almonds.\n\n5. Pour the sauce over each dessert and chill before serving.\n\nEsalen grows many of its own fruits and berries and this dessert was a fine way to end one of their marvelous meals. I am going to list the fruits that they used, but you can adapt it to whatever you see at your market that looks fresh and delectable. I like to serve the dressing on the side because the fruit looks so pretty, but of course you can mix it beforehand.\n\nEsalen Fruit Ambrosia **\n\n1 big chunk of watermelon, seeded and cut in bite-sized chunks\n\n1 bunch of grapes, seeded and separated\n\n2 apples, skinned and cut in slices\n\n4 nectarines or peaches, sliced\n\n3 bananas, peeled and sliced\n\na handful of either sunflower seeds or slivered almonds\n\n1. Place all the fruit in a bowl and serve with the topping on the side.\n\n_Ambrosia Topping_\n\n\u00be lb. of whipped cream cheese at room temperature\n\n\u00bd cup yogurt\n\n\u2153 cup honey\n\n\u00bc tsp. cinnamon\n\ndash of powdered cloves\n\n1. Mix cream cheese and yogurt well.\n\n2. Add honey, cinnamon and cloves and blend into cream cheese mixture.\n\nThis is a traditional recipe out in Indiana and made superbly by Grandma Reiber. No Christmas was complete without it. I thought that I had never eaten suet before and it was strange to think of putting it into a dessert until I remembered how mince pie was made.\n\nGrandma Reiber's Suet Pudding **\n\n\u00bd cup suet, ground\n\n\u00bd cup molasses\n\n\u00bd cup milk\n\n1\u00bd cups flour\n\n\u00bd tsp. baking powder\n\n\u00bd tsp. nutmeg\n\n\u00bd tsp. cinnamon\n\n\u00bd tsp. baking soda\n\n1 cup raisins\n\n1. Mix suet, molasses and milk.\n\n2. Sift all dry ingredients and add to suet mixture.\n\n3. Fold in raisins.\n\n4. Place in a pudding mold with a cover and boil in rapidly boiling water until done\u20141\u00bd to 2 hours. If necessary, add more boiling water.\n\n_Note_ : If you don't have a pudding mold, I have heard that you can put the dough in a wet towel, tie it securely and then boil as usual in a pan of water.\n\nPam Topham's Wonderful Chestnut Mousse **\n\n7 egg whites\n\npinch of cream of tartar\n\n\u00bd cup sugar\n\n1 cup chestnut (marron) puree\n\n3 tbsp. cream\n\n1 tbsp. vanilla\n\n6 egg yolks\n\n1 heavy cream\n\nnutmeg\n\n1. Whip egg whites, cream of tartar, and \u00bc cup of the sugar to form soft peaks.\n\n2. In a double boiler, stir the other \u00bc cup of the sugar, chestnut puree, the 3 tbsp. cream and vanilla until smooth.\n\n3. Whip the pint of cream into peaks.\n\n4. Mix the whipped egg whites, the chestnut puree sauce and the whipped cream gently together and sprinkle with nutmeg.\n\n5. Chill 8 hours before serving.\n\nMy mother-in-law Marcia always brings goodies when she comes to visit. This is one of them and I think that it's one of the best recipes for carrot cake that I have tasted.\n\nMarcia's Carrot Cake **\n\n1 cup flour\n\n\u00be cup sugar\n\n1 tsp. salt\n\n1 tsp. baking soda\n\n1 tsp. cinnamon\n\n2 eggs, beaten\n\n\u00bd cup corn oil\n\n1 tbsp. lemon juice\n\n1\u00bd cups carrots, grated\n\n\u00bd cup walnuts, chopped\n\n\u00bd cup raisins\n\n1 tsp. grated lemon peel\n\n1. Sift dry ingredients into a large bowl.\n\n2. Add beaten eggs and oil.\n\n3. Stir in lemon juice.\n\n4. Add carrots, nuts, raisins, grated lemon peel, and mix thoroughly.\n\n5. Pour into a greased loaf pan and bake at 350 degrees about 50 to 60 minutes until done.\n\nFrom a neighbor Shirley to Aunt Florrie to my mother-in-law, that's how this noodle pudding came to me. I try to use cornflakes from the health food store and low fat cream cheese.\n\nNoodle Pudding **\n\n6 eggs\n\n1 pt. sour cream\n\n1 cup sugar (reserve 2 tbsp. for topping)\n\n\u00bd lb. melted butter (reserve 2 tbsp. for topping)\n\n12 oz. package of cream cheese, softened\n\n1 lb. flat noodles, cooked\n\n2 cups cornflakes, crushed\n\n1. Beat eggs with sour cream.\n\n2. Add butter and sugar.\n\n3. Add cream cheese and noodles and mix well.\n\n4. Place in a 9 x 13 greased baking dish.\n\n5. Mix the cornflakes with the re served melted butter and sugar.\n\n6. Sprinkle on top and put in refrigerator overnight.\n\n7. Bake at 325 degrees for 1 hour.\n\nA dessert not to have too often for obvious reasons, I like this cake of my mother-in-law's because though it's not sickeningly sweet, it certainly satisfies all sweet tooths. She started making this in the 1930's, incidentally, so it's a 50 year old recipe.\n\nMarcia's Chocolate Creole Cake ***\n\n\u00bd cup shortening\n\n1\u00bd cups brown sugar, well packed\n\n3 eggs, separated\n\n1 tsp. baking powder, sifted\n\n2 cups flour, sifted\n\n1 tsp. baking soda dissolved in 1 tsp. boiling water\n\n1 cup warm milk\n\n2 squares unsweetened Baker's chocolate, shaved.\n\n1. Cream shortening and sugar.\n\n2. Add egg yolks and stir.\n\n3. Add the baking powder and flour alternately with milk.\n\n4. Add dissolved baking soda.\n\n5. Add chocolate and mix well.\n\n6. Beat egg whites until stiff and fold in cake mixture.\n\n7. Bake in a greased tube pan at 350 degrees for 40 to 45 minutes.\n\n_Mocha Creamy Frosting_\n\n1 3-oz. package cream cheese\n\n\u00be cup confectionary sugar, sifted\n\n1 tsp. vanilla\n\n1 square melted chocolate\n\n2 tbsp. strong coffee\n\n1. Blend cream cheese with sugar.\n\n2. Add vanilla.\n\n3. Stir in melted chocolate and mix well.\n\n4. Add the coffee and stir, using more coffee if you need a better spreading consistency.\n\n5. Spread over the tops and sides of the cooled cake.\n\nThe most beautiful island in the Carribean, my husband and I agree, is Dominica, not to be confused with the Dominican Republican. It's a shangri la, showering intermittently throughout the day with profusions of rainbows, beaches of black sand, mountain peaks ringed with clouds and very few tourists. This is a dessert from Dominica, rich, but special.\n\nDominican Custard Pudding *\n\n3\u00bd cups milk\n\n6 tbsp. butter\n\n1\u00bd cups brown sugar\n\n4 tbsp. cornstarch\n\n2 eggs\n\n\u00bd tsp. vanilla\n\n1. Scald milk in a pan with the butter.\n\n2. Mix with remaining ingredients; stir well and pour into a buttered baking dish.\n\n3. Place dish in a pan of water and bake at 325 degrees for about an hour or until an inserted knife comes out clean.\n\n4. Cool and serve.\n\nThis recipe was given to me about 25 years ago out in Clinton, Indiana, by a friend named Mary Alice Osmon. It was interesting because it used mayonnaise. Whoever heard of that! Surprisingly, my husband, who doesn't like chocolate cake, loved this because he felt that it wasn't \"dry\" like other cakes. Here it is.\n\nMary Alice Osmon's Chocolate Mayonnaise Cake **\n\n1 cup flour\n\n\u00be cup sugar\n\n\u00bc cup cocoa\n\n1 tsp. baking soda\n\n\u00bd cup warm water\n\n\u00bd cup mayonnaise\n\n1. Mix all the dry ingredients.\n\n2. Add warm water and mix.\n\n3. Add mayonnaise and mix well.\n\n4. Pour into a lightly oiled flat baking pan about 9'' by 13'' in size and bake in a 350 degree oven about 30 to 35 minutes until done.\n\n_Frosting for Mayonnaise Cake_\n\n\u00bd box sifted powdered sugar\n\n\u00bc cup cocoa\n\n\u00bd stick butter, softened\n\n\u00bc cup karo syrup\n\n1 tbsp. milk or cream\n\n1 tsp. vanilla\n\n1. Mix powdered sugar and cocoa making sure that all lumps are dissolved.\n\n2. Add butter and cream well.\n\n3. Add syrup, cream or milk and vanilla and mix until smooth.\n\n4. Spread on cooled cake.\n\nMy grandmother made only three different kinds of cookies\u2014peanut butter, chocolate chip and oatmeal\u2014but she never failed to make one large batch of them every Saturday morning. This is her recipe for peanut butter cookies.\n\nOld Fashioned Peanut Butter Cookies **\n\n2\u00bd cups sifted all purpose flour\n\n1 tsp. baking powder\n\n1\u00bd tsp. baking soda\n\n\u00bd tsp. salt\n\n\u00bd cup butter (1 stick)\n\n\u00bd cup Crisco\n\n1 cup sugar\n\n1 cup brown sugar, lightly packed\n\n2 eggs, beaten\n\n1 tsp. vanilla\n\n1\u00bd cups peanut butter\n\n1. Sift flour, baking powder, soda and salt.\n\n2. In another bowl, cream butter, Crisco and sugars until smooth.\n\n3. Add eggs and vanilla.\n\n4. Add peanut butter and stir until it is well mixed.\n\n5. Gradually add dry ingredients.\n\n6. Divide dough in two balls, wrap in wax paper and chill for several hours or overnight.\n\n7. When ready to bake, roll small amounts of dough between your palms to make little balls.\n\n8. Place balls on cookie sheet and flat ten with a fork to make ridges.\n\n9. Bake at 350 degrees for 8 to 10 minutes.\n\nThis is a delicious coconut cake, very different from your usual kind. Since I like to make coconut milk for some Thai and Cambodian dishes, this gives me a use for the leftover coconut.\n\nCoconut Cake **\n\n1 stick butter\n\n1 cup sugar\n\n4 egg yolks\n\n\u00be cup farina, sifted\n\n\u00bd cup flour, sifted\n\n1 tsp. baking powder, sifted\n\n1\u00bd cups grated coconut\n\n1 tsp. vanilla\n\n4 egg whites, stiffly beaten\n\n1. Cream butter and sugar.\n\n2. Stir in egg yolks.\n\n3. Add dry ingredients and mix well.\n\n4. Add grated coconut and vanilla.\n\n5. Fold in egg whites.\n\n6. Pour into lightly buttered or greased square pan and bake for 30 to 40 minutes or until a toothpick comes out clean.\n\nIt seems that whenever I want strawberry shortcake, I can only find those tasteless, preservative laden cakes displayed so prominently in the stores during strawberry season. The only alternative is to use Bisquick and make your own. The problem is that Bisquick has preservatives also. My Mother solved this dilemma for me by sending me a recipe from her childhood. This is what shortcake should taste like. Of course, you can smother it with any fruit in season before topping it with heavy cream or whipped cream.\n\nYou can also simplify this by just pressing dough into a flat baking dish and cutting it in wedges.\n\nMrs. Biffar's Shortcake **\n\n2 cups flour\n\n4 tsp. baking powder\n\n\u00bd tsp. salt\n\n1 tbsp. sugar\n\n\u00bc cup shortening\n\n1 egg, well beaten\n\n\u2154 cup milk\n\n1. Sift dry ingredients.\n\n2. Cut in shortening until mixture has the consistency of coarse crumbs.\n\n3. Add egg and milk and stir just to mix.\n\n4. Put dough on a floured board and pat it down so that it's about \u2153 inch thick.\n\n5. With a round cookie cutter, cut out rounds for the shortcake.\n\n6. Place rounds on cookie sheet and bake in a 425 degree oven for 20 minutes.\n\nWhenever Ron Millican comes to dinner or we dine at his house, we have one of a variety of lovely fruit, pies made by him. Ron is a nice looking, articulate man who recently married a vivacious young actress named Laurie Heineman. They are both fun to spend time with. The other night Ron mentioned how easy this dessert was to prepare and I thought it would be a good addition to this book. It can be served with yogurt, whipped cream or ice cream, if you wish.\n\nRon's Fruit Pie *\n\n1 unbaked pieshell\n\n2 cups fresh fruit in season (like apples, peaches, pears or berries) or a combination of fruits, peeled and sliced\n\n\u00bc to \u00bd cup pure maple syrup\n\n1. Bake pie shell in a 400\u00b0 oven until slightly browned.\n\n2. Arrange fruit slices in a nice pattern in the pie shell.\n\n3. Drizzle maple syrup over fruit.\n\n4. Bake in a 350 degree oven for 15 minutes or until fruit is done.\n\nA holiday topping from Jane Wilson's Party Box, it's easy to see why this is one of New York City's best catering services. Serve this over vanilla ice cream.\n\nJane's Cranberry Jubilee *\n\n2 cups raw cranberries\n\n\u00be cup water\n\n12 oz. red currant jelly\n\n\u00bc cup raw sugar\n\n\u00bd cup rum\n\n\u00bc cup brandy\n\ngrated rind of 1 orange\n\n1. Cook cranberries in water until they pop.\n\n2. Add the rest of the ingredients and simmer for 10 minutes.\n\n3. Cool and store in refrigerator until ready to use.\n\nJudy Wolfe exclaimed one day, \"If you're writing a cookbook, you must include the mousse that my friend Judy Long makes. It's so delicious and doesn't take any time at all. You just put everything in a blender.\" However, I must caution you to use only the regular chocolate chips. The first time I tried it, my husband picked up milk chocolate chips and the mousse wouldn't jell.\n\nJudy Long's Chocolate Mousse *\n\n1 6-oz. package of semi-sweet chocolate chips\n\n2 eggs\n\n2 tbsp. triple sec or rum\n\n3 tbsp. hot strong coffee\n\n\u00be cup milk, scalded not boiled\n\n1. Put all these ingredients in a blender and whip at high speed for two minutes.\n\n2. Pour into individual dishes and chill until set. This takes a couple of hours. Easy!\n\nMy Mother gave this recipe of my Grandmother Honey's to me when I was starting to write this book. When I tried it out, I felt a very warm and satisfying feeling to know that I was cooking one of my grandmother's recipes. I liked the continuity and a few days later passed the recipe along to my daughter, Liza. Now, there is a third generation preparing it. Incidentally, this turned out to be my favorite cupcake recipe, much better than your ordinary yellow cupcakes with chocolate or vanilla frosting.\n\nHoney's \"Quilting Bees\" *\n\n2 cups raisins\n\n2 cups water\n\n1\u00bd cups sugar\n\n\u00bd cup shortening\n\n2 eggs\n\n2\u00bd cups flour\n\n2 tsp. cinnamon\n\n\u00bd tsp. cloves\n\n1 tsp. nutmeg\n\n\u00bc tsp. salt\n\n2 tsp. baking powder\n\n1 cup walnut pieces\n\n1. Boil raisins in water for 10 minutes and drain.\n\n2. Cream sugar and shortening.\n\n3. Add eggs and stir.\n\n4. Add all dry ingredients and mix well.\n\n5. Stir in nuts and raisins.\n\n6. Put in a lightly greased muffin tin and bake in a 350 degree oven until done (30 to 40 minutes, depending on the size muffin tins you are using).\nBreads\n\nMrs. Corner's Lemon-Sesame Bread *\n\n\u00bc cup oil\n\n\u00bd cup honey\n\n2 eggs\n\njuice of one lemon\n\ngrated rind of two lemons\n\n1 cup whole wheat flour\n\n\u00bc cup soy flour (you can substitute white flour if you don't have any soy)\n\n2 tbsp. dried milk\n\n2 tbsp. baking powder\n\n\u00bd cup sesame seeds\n\njuice of one lemon (for topping)\n\n3 tbsp. honey\n\n1. Beat oil and honey until blended.\n\n2. Stir in eggs, lemon juice and grated rind.\n\n3. Mix all dry ingredients and sesame seeds and then stir them into the honey mixture.\n\n4. Bake in a small oiled loaf pan (preferably smaller than 5x9) at 350 degrees for 45 minutes.\n\n5. Combine lemon juice and honey.\n\n6. Poke holes in cake and pour topping over it while it's still warm.\n\n7. Cool for 10 minutes and then remove from pan.\n\nI used to use Bisquick quite a lot for making biscuits, pancakes, waffles, and shortbread. When I finally read the label one day, I found that it was preserved with BHA and BHT and used hydrogenated oils beside. So I now make my own and store it in a jar and use it as I would the commerical brand.\n\nA Better Biscuit Mix *\n\n3 cups flour\n\n1 tsp. salt\n\n\u00bc tsp. cream of tartar\n\n1\u00bd tbsp. baking powder\n\n\u2154 cups shortening\n\n1. Mix all dry ingredients.\n\n2. Cut in shortening until mix is crumbly.\n\n3. Store in refrigerator and use as a base for biscuits, waffles, pancakes, shortbread, and coffeecake.\n\nWalnut bread is an easy recipe and good to have for breakfast in the morning. If you have already made your biscuit mix and have it on hand, it takes about 2 minutes to prepare.\n\nWalnut Bread *\n\n1 egg, beaten\n\n\u00be cup milk\n\n\u00bd cup sugar\n\n3 cups biscuit mix\n\n1 tbsp. grated orange peel\n\n1 cup chopped walnuts\n\n1. Stir egg, milk, sugar and biscuit mix.\n\n2. Add grated peel and walnuts.\n\n3. Pour into a greased loaf pan and bake at 350 degrees for about 1 hour.\n\nI get discouraged trying to find breakfast coffee cakes and muffins in the supermarket that are not made with preservatives, artificial coloring and artificial flavors and I also get tired of the usual toast each morning. So, I was happy to be given a nice, easy recipe for pumpkin muffins by Judy Wolfe. I usually make a batch and freeze some of them to be used later.\n\nJudy's Pumpkin Muffins *\n\n1\u00bd cups white flour, sifted\n\n2 tsp. baking powder, sifted\n\n\u00bd cup brown sugar\n\n\u00bd tsp. salt\n\n\u00bd tsp. cinnamon\n\n\u00bd tsp. nutmeg\n\n\u00bd cup unsweetened pumpkin\n\n1 egg\n\n\u00bc cup melted butter\n\n\u00bd cup raisins\n\n\u00bc cup sugar\n\n1. Mix all ingredients. Batter may be a little lumpy.\n\n2. Pour batter into 12 greased medium sized muffin cups.\n\n3. Sprinkle a little sugar over the top of each muffin.\n\n4. Bake in a pre-heated oven of 400 degrees for 18 to 20 minutes.\n\nSouthern Chili Cornbread *\n\nbacon drippings or butter\n\n2 cups yellow cornmeal\n\n1\u00bd cups flour\n\n1 tsp. sugar\n\n1 tsp. salt\n\n2\u00bd tsp. baking powder\n\n1 egg, beaten\n\n4 tbsp. melted butter\n\n1 tbsp. safflower oil\n\n2 cups milk (jpreferably skim milk)\n\n\u2153 cup green chilis, cut in small pieces\n\n1. Place bacon drippings or butter in large, heavy skillet and put in 400 degree oven to melt.\n\n2. Sift all dry ingredients in bowl.\n\n3. Add beaten egg, melted butter, safflower oil and mix.\n\n4. Stir in milk and chili pieces and mix well.\n\n5. Place in hot skillet in oven and cook for 45 minutes or until the top is nicely browned.\n\nMother's Zucchini Bread *\n\n2 cups zucchini, grated\n\n3 eggs, beaten\n\n1 cup safflower oil\n\n2 cups sugar\n\n3 tsp. vanilla 3 cups flour\n\n1 tsp. baking soda\n\n\u00bc tsp. baking powder\n\n1 tsp. salt\n\n3 tsp. cinnamon\n\n\u00bd cup chopped walnuts\n\n1. In a bowl, lightly mix zucchini, eggs, oil and sugar.\n\n2. In another bowl, mix all the rest of the ingredients.\n\n3. Combine all ingredients and pour into two lightly greased loaf pans.\n\n4. Bake at 350 degree for approximately 1 hour and 15 minutes or until a toothpick comes out clean.\n\nI can't tell you how many times people have said to me that Rosalie Gwathmey is the best cook in the Hamptons. The wife of a painter, the mother of an architect, Rosalie is an artist in her own right and she is also an artist in the kitchen. When I asked her if she would like to give me a recipe, she said she didn't know why people didn't make batterbread any more. Now we will. There is another thing about Rosalie, she has a beautiful smile, her whole face lights up.\n\nRosalie Gwathmey's Batterbread *\n\n\u00bd cup cornmeal\n\n1\u00bd cups milk\n\n4 tbsp. butter\n\n4 eggs at room temperature\n\n1 tsp. salt\n\n1. Cook cornmeal and milk in a sauce pan stirring constantly until thick. Set aside.\n\n2. Heat oven to 375 degrees. Put butter in a 2-qt. casserole and melt in oven.\n\n3. Separate eggs and beat the yolks into the cornmeal mixture.\n\n4. Pour into the casserole and mix with the butter.\n\n5. Sprinkle salt over the egg whites and beat into soft peaks.\n\n6. Add a little of the egg whites to the batter and stir.\n\n7. Fold the rest of the whites into the batter.\n\n8. Bake about 40 minutes or until golden.\nSandwiches\n\nThis is just a list, not recipes, but if you're like me and never know what to make for lunch, it might come in handy. I actually like leftovers for lunch, but they're not always available. Then, I'm in a quandry. I don't care for coldcuts and I get very tired of the standard tuna and egg salad. I decided to make a list of sandwiches, so I wouldn't waste my time trying to always figure out what to have for lunch. I hope by listing the ingredients, they will be easy to follow. Needless to say, good bread is a must.\n\n1. Pocket veggies\u2014pita bread with raw zucchini slices, alfalfa sprouts, shreds of purple cabbage, carrots, celery, and Swiss cheese put in oven wrapped in foil until cheese melts.\n\n2. Pita bread filled with tahini, tomatoes, onions, sprouts and avocado.\n\n3. A breakfast pita sandwich\u2014pita filled with apple slices and tahini.\n\n4. Pita bread filled with cottage cheese, scallions, radishes, cucumber, celery and sprinkled with celery salt.\n\n5. Cream cheese mixed with chopped scallions, sprouts, watercress and sprinkled with thyme.\n\n6. Cream cheese, sliced olives and alfalfa sprouts on whole wheat.\n\n7. Cream cheese, chopped olives, scallions and pimento on black bread.\n\n8. Bagels with cream cheese, lox, tomatoes and onions.\n\n9. My mother-in-law's invention\u2014bagels, cream cheese, tomatoes, onions and sliced, black olives.\n\n10. Bacon, cream cheese and red onion.\n\n11. Open faced sandwich\u2014onion, tomato, Montery jack and green chilis, broiled.\n\n12. Grilled cheese\u2014always with pickles on the side.\n\n13. Grilled cheese with bacon.\n\n14. Open faced grilled cheese with tomato slices and onion.\n\n15. Ham and Swiss cheese with mayonnaise and butter, grilled like a regular grilled cheese sandwich.\n\n16. Broiled open faced sandwich with Swiss cheese and apple on rye.\n\n17. Open faced sandwich with Camenbert cheese, sliced ham and tomato on whole wheat.\n\n18. Mozzarella cheese and anchovy pieces on Italian bread, placed under broiler until cheese melts.\n\n19. Mortadella, provolone and marinated red peppers on Italian bread.\n\n20. Cottage cheese, tomato, cucumber and onion on rye.\n\n21. Cheese, green chilis and scallions on whole wheat bread.\n\n22. Grilled tomato and onion slices with crumbled tofu.\n\n23. Avocado, tomato, chopped scallions and Russian dressing on whole wheat.\n\n24. Peanut butter and sliced bananas on toasted whole wheat.\n\n25. Egg salad with onions, celery, and minced carrots.\n\n26. Rhoda's tofu salad (see chapter on salads).\n\n27. Bacon and egg.\n\n28. Bacon, lettuce and tomato.\n\n29. My Father's favorite\u2014bacon and avocado.\n\n30. Club sandwiches, bacon, lettuce, tomato, and sliced chicken with mayonnaise.\n\n31. Bacon and chicken salad.\n\n32. Roast beef on rye with sliced, cooked beets and horseradish sauce.\n\n33. Roast beef with red onion, lettuce and mayonnaise mixed with mustard and horseradish sauce on pumpernickel.\n\n34. Cornbeef on rye with Swiss cheese, sauerkraut, Dijon mustard and Russian dressing.\n\n35. Tuna, diced cooked potatoes, chickpeas and onions mixed with mayonnaise.\n\n36. Tuna, tiny diced carrots, onions, and olives with mayonnaise.\n\n37. Canned salmon mixed with mayonnaise and asparagus, tomato and cucumber on rye or cornbread.\n\n38. Canned salmon with mayonnaise and dill, lettuce and cucumber on toast.\nGood and Useful Cookbooks\n\n_The Natural Foods Cookbook_ by Beatrice Trum Hunter\n\n_The Supermarket Cookbook_ by Nikki and David Goldbeck\n\n_Yoga Natural Foods Cookbook_ by Richard Hittleman\n\n_Whole Grain Cookery_ by Stella Standard\n\n_Soups For All Seasons_ by Stella Standard\n\n_Future Food_ by Colin Tudge\n\n_Great Bread_ by Bernice Hunt\n\n_Vegetarian Epicure_ by Anna Thomas\n\n_World of the East-Vegetarian Cooking_ by Madhur Jaffrey\n\n_Moosewood Cookbook_ Mollie Katsen\n\n_Diet For A Small Planet_ by Frances Moore Lappe\n\n_Recipes For A Small Planet_ by Ellen Buchman Ewald\nIndex\n\nAPPLES\n\nBavarian apple slice\n\nMrs. Acheson's beet and apple salad\n\nTofu with apples\n\nApplesauce fluff\n\nAPRICOTS\n\nPriscilla's rice and apricot stuffing\n\nASPARAGUS\n\nSesame asparagus\n\nAVOCADOS\n\nZurich avocado soup\n\nZurich shrimp and avocado salad\n\nARTICHOKES\n\nHot artichoke dip\n\nBARLEY\n\nMarty's favorite barley dish\n\nBEANS\n\nBlack bean dip\n\nJala-pinto dip\n\nBEAN SPROUTS\n\nCarlos' bean sprout salad\n\nBEEF\n\nGinger beef stew\n\nJudy Wolfe's picadillo\n\nBEETS\n\nBill Tarlo's cabbage and beet appetizer\n\nMrs. Acheson's beet and apple salad\n\nBREADS\n\nA better biscuit mix\n\nMother's zucchini bread\n\nMrs. Corner's lemon-sesame bread\n\nRosalie Gwathmey's batterbread\n\nSouthern chili cornbread\n\nWalnut bread\n\nBROCCOLI\n\nArmand's broccoli and parsnip casserole\n\nBroccoli fitttata\n\nBUBBLE AND SQUEAK\n\nBUTTER\n\nButter blend\n\nCathy's herbal butter spread\n\nCABBAGE\n\nBill Tarlo's cabbage and beet appetizer\n\nBubble and squeak\n\nChinese cabbage salad\n\nMrs. Acheson's red cabbage\n\nSweet and sour cabbage\n\nCAKE\n\nCoconut cake\n\nMarcia's carrot cake\n\nMarcia's chocolate Creole cake\n\nMary Alice Osmon's chocolate mayonnaise cake\n\nMrs. Biffar's Shortcake\n\nCARROTS\n\nJohn's carrot creation\n\nPickled carrot sticks\n\nCAULIFLOWER\n\nChristina's cauliflower and tomato special\n\nItalian cauliflower\n\nSpicy cauliflower\n\nCAVIAR\n\nBernice Hunt's caviar pie\n\nRuth Steel's caviar spread\n\nCHEESE\n\nGreen chilis with\n\neggs and cheese\n\nMarty's cheese nachos\n\nFranklin and Banne Heller's\n\nLiptauer cheese spread\n\nYogurt cheese pie\n\nCHICKEN\n\nArmand LaMacchia's chicken soup\n\nA special chicken salad\n\nBaked chicken salad\n\nBessie's chicken and orzo\n\nChicken piquant\n\nChicken salad a la Balducci\n\nCoconut chicken\n\nEasy chicken curry\n\nMrs. Feiman's baked chicken\n\nPakistani chicken\n\nPiedmont chicken\n\nSunday Green dinner\n\nValdi's chicken pilaf\n\nCHICKEN LIVERS\n\nAl Maline's chicken livers\n\nCarlos' chicken liver and noodle casserole\n\nCHICKPEAS\n\nCynthia Epstein's easy chickpea stew\n\nCHOUCROUTE GARNIE\n\nCHUTNEY\n\nGinger's orange chutney\n\nCLAMS\n\nSally Maruscak's clam dip\n\nTomato-clam soup oriental style\n\nCOCONUT\n\nCoconut cake\n\nThailand coconut soup\n\nCOOKIES\n\nOld fashioned peanut butter cookies\n\nCORN\n\nDorothy's corn chowder\n\nMarcia's squash or corn souffle\n\nCOUSCOUS\n\nFlorence Grey's scallops with couscous\n\nCRANBERRIES\n\nJane's cranberry jubilee\n\nCUCUMBERS\n\nCarlos' spicy Chinese cukes\n\nCUPCAKES\n\nHoney's \"Quilting Bees\"\n\nDRESSINGS\n\nBelle's mustard dressing\n\nSour cream dressing\n\nSour cream\/roquefort dressing\n\nTomato dressing\n\nEGGS\n\nGreen chilis with eggs and cheese\n\nEGGPLANT\n\nEggplant lasagna\n\nEggplant tahini dip\n\nEsalen eggplant\n\nMother's eggplant souffle\n\nRicotta eggplant\n\nSavario's eggplant\n\nStuffed eggplant\n\nSue's spicy eggplant\n\nESALEN FRUIT AMBROSIA\n\nFISH\n\nCambodian fish with scallion sauce\n\nChinese flounder\n\nFlorence's tangy fish\n\nGood, cheap fish\n\nJudy's fish with mushrooms\n\nKasha imperial\n\nSole amandine\n\nGOULASH\n\nCompany goulash\n\nGRANOLA\n\nHomemade granola\n\nKasha imperial\n\nLIVER\n\nAl Maline's chicken livers\n\nCalves liver with lemon\n\nCarlos' liver and\n\nnoodle casserole\n\nMEATLOAF\n\nZach's mint meatloaf\n\nMOUSSE\n\nJudy Long's chocolate mousse\n\nPam Topham's wonderful chestnut mousse\n\nMUFFINS\n\nJudy's pumpkin muffins\n\nMUSHROOMS\n\nMushroom special\n\nNACHOS\n\nMarty's cheese nachos\n\nNOODLES\n\nBelle's Chinese noodles\n\nCarlos' chicken liver and noodle casserole\n\nSesame noodles\n\nOATCAKES\n\nOLIVES\n\nGreek olives\n\nONIONS\n\nPickled onions\n\nORANGE DELIGHT\n\nPARSNIPS\n\nArmand's broccoli and parsnip casserole\n\nPASTA\n\nA young man's pasta\n\nBernice Hunt's pasta alla carbonara\n\nChris' pasta with scallops\n\nGreek pasta\n\nItalian salmon pasta\n\nMarty and Linda's pasta\n\nPasta with green beans\n\nPasta with salami\n\nPasta with sausage\n\nSeafood pasta\n\nSan Gimignamo pasta\n\nThree-cheese pasta\n\nPAVLOVA\n\nPEARS\n\nPears with ginger\n\nPECAN SURPRISE SQUARES\n\nPIE\n\nRon's fruit pie\n\nPOLENTA\n\nPolenta with gorgonzola\n\nPORK\n\nMary's pork sate\n\nPork chops with raisin stuffing\n\nPortofino pork with sauce\n\nPOTATOES\n\nBaked potato crisps\n\nBubble and squeak\n\nChris' potato salad\n\nHot potato salad\n\nDon Mathew's Argentine potato pie\n\nPUDDING\n\nDominican custard pudding\n\nGrandma Reiber's\n\nsuet pudding\n\nNoodle pudding\n\nQUICHE\n\nAnna Marie's spinach quiche\n\nLinda's easy quiche\n\nRICE\n\nChristina's rice and\n\nspinach casserole\n\nHot chili rice\n\nJeanne's pilaf\n\nPriscilla's rice and apricot stuffing\n\nRice pancakes\n\nValdi's chicken pilaf\n\nRUSSIAN CREAM\n\nSALADS\n\nA special chicken salad\n\nBaked chicken salad\n\nCarlos' bean sprout salad\n\nCarlos' spicy Chinese cukes\n\nChicken salad a la Balducci\n\nChinese cabbage salad\n\nChinese salad\n\nChris' potato salad\n\nCurried tuna salad\n\nGreek marinated vegetables\n\nHot potato salad\n\nJohn's carrot creation\n\nMarinated zucchini salad\n\nMrs. Acheson's beet and apple salad\n\nRhoda's tofu salad\n\nRita Lautz's hot salad\n\nSuper salad\n\nZurich shrimp and avocado salad\n\nSALAMI\n\nPasta with salami\n\nSALMON\n\nItalian salmon pasta\n\nSANDWICHES ,\n\nSAUSAGE\n\nPasta with sausage\n\nSCALLOPS\n\nChris' pasta with scallops\n\nFlorence Grey's scallops with couscous\n\nSHORT RIB STEW\n\nSHRIMP\n\nCeylon shrimp curry\n\nZurich shrimp and avocado salad\n\nSOUPS\n\nArmand LaMacchia's\n\nchicken soup\n\nChaya's borscht\n\nDorothy's corn chowder\n\nMrs. Acheson's cucumber-buttermilk soup\n\nThailand coconut soup\n\nTomato-clam soup oriental style\n\nZurich Avocado Soup\n\nSOYBEANS\n\nBaked soybean casserole\n\nSPINACH\n\nAnna Marie's spinach quiche\n\nChristina's rice and spinach casserole\n\nSpinach provencale\n\nSQUASH\n\nBest ever squash\n\nMarcia's squash and corn souffle\n\nSpaghetti squash with tomato sauce\n\nSTUFFING\n\nPriscilla's rice and apricot stuffing\n\nSTRING BEANS\n\nPasta with green beans\n\nString beans with almonds\n\nSWISS CHARD CUTLETS\n\nTOFU\n\nChristina's geetam tofu steaks\n\nRhoda's tofu salad\n\nTofu with apples\n\nTOMATOES\n\nChristina's cauliflower and tomato special\n\nStuffed tomatoes\n\nTomato-clam soup oriental style\n\nTUNA\n\nCurried tuna salad\n\nFern's Chinese tuna\n\nTURNIPS\n\nTurnip casserole\n\nVEAL\n\nScallopini Lucia\n\nVEGETABLES\n\nMinsky's vegetables\n\nVegetable curry\n\nYOGURT CHEESE PIE\n\nZUCCHINI\n\nCarlos' stuffed zucchini\n\nGreek zucchini\n\nMarinated zucchini salad\n\nMother's zucchini bread \nAll rights reserved, including without limitation the right to reproduce this ebook or any portion thereof in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nCopyright \u00a9 1983 by Judith Shepard\n\nISBN: 978-1-5040-2860-8\n\nThe Permanent Press\n\n4170 Noyac Road\n\nSag Harbor, NY 11963\n\nwww.thepermanentpress.com\n\nDistributed by Open Road Distribution\n\n180 Maiden Lane\n\nNew York, NY 10038\n\nwww.openroadmedia.com\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n## Contents\n\n  1. Prologue\n  2. Chapter 1\n  3. Chapter 2\n  4. Chapter 3\n  5. Chapter 4\n  6. Chapter 5\n  7. Chapter 6\n  8. Chapter 7\n  9. Chapter 8\n  10. Chapter 9\n  11. Chapter 10\n  12. Chapter 11\n  13. Chapter 12\n  14. Chapter 13\n  15. Chapter 14\n  16. Chapter 15\n  17. Chapter 16\n  18. Chapter 17\n  19. Chapter 18\n  20. Chapter 19\n  21. Chapter 20\n  22. Chapter 21\n  23. Chapter 22\n  24. Chapter 23\n  25. Chapter 24\n  26. Chapter 25\n  27. Chapter 26\n  28. Chapter 27\n  29. Chapter 28\n  30. Chapter 29\n  31. Chapter 30\n  32. Chapter 31\n  33. Chapter 32\n  34. Chapter 33\n  35. Chapter 34\n  36. Chapter 35\n  37. Chapter 36\n  38. Chapter 37\n  39. Chapter 38\n  40. Chapter 39\n  41. Chapter 40\n  42. Chapter 41\n  43. Chapter 42\n  44. Chapter 43\n  45. Chapter 44\n  46. Chapter 45\n  47. Chapter 46\n  48. Chapter 47\n  49. Chapter 48\n  50. Chapter 49\n  51. Chapter 50\n  52. Acknowledgments\n  53. About the Author\n  54. Credits\n  55. Copyright\n  56. About the Publisher\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Chapter 1\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. \n  317. \n  318. \n  319. \n  320. \n  321. \n  322. \n  323. \n  324. \n  325. \n  326. \n  327. \n  328. \n  329. \n  330. \n  331. \n  332. \n  333. \n  334. \n  335. \n  336. \n  337.\n\n## Prologue\n\nMY NAME IS CHESS, and I was born inside a cage.\n\nImagine a wooden platform jutting from a mountain cliff. Now picture a chain falling from that platform and vanishing into the Fog, a deadly white mist that covers the entire Earth.\n\nThat's where I was born: locked in a cage, at the end of a chain, inside the Fog.\n\nAnd I would've died there, too, if Mrs. E hadn't saved me.\n\nWhen she saw my face for the first time, wisps of Fog swirled inside my right eye, shimmering white shapes that marked me as a freak. That's why I've spent thirteen years keeping my head down, staying quiet and afraid\u2014but now Mrs. E needs help, now _she_ needs saving.\n\nIt's time to stop hiding. Everything is going to change.\n\n##\n\nAFTER A LONG MORNING searching the woods, I spotted a school bus through the Fog. The broken windows looked like rotten teeth as I edged closer, hoping to salvage hubcaps or engine parts.\n\nThen a growl rumbled through the swirling mist. A low, warning sound, maybe a mountain lion or a jaguar. Probably just telling me to stay away.\n\nI wasn't about to argue.\n\nMy heart clenched and I reached for the hand brake on the harness buckled around my chest. A long cord\u2014my \"tether\"\u2014rose three hundred yards upward from the harness. If I squeezed the brake, my crew would reel in the tether, heaving me to the safety of our raft, which floated in the clear blue sky high above the Fog.\n\nBut when I touched the hand brake, the growling stopped.\n\nHm. I didn't know if that was a good sign or a bad one. I peered toward the rusty school bus, but even I couldn't see ten feet through the dense foggy whiteness. And the Fog muffled sounds, so for all I knew the big cat was padding closer, paws crunching through the leaves.\n\nPeople died in the swirling mist, but animals thrived. They ran wild in the forests and the rubble. Packs of ferocious boars and troops of rowdy monkeys didn't even notice the whiteness. Only _humans_ were blind and deaf in the Fog, stumbling around like accidents waiting to happen.\n\nOr tasty treats.\n\nSweat trickled down my forehead and pooled on my goggles. I wanted to squeeze the hand brake and flee, but I needed to stay in the Fog. I needed to stay strong and brave. My crew was counting on me. So I took one slow step backward, then another and another. Two minutes later, I slumped in relief. I really needed to find a better place to search for valuables than a school bus where a jaguar made her den.\n\nMy crew and I lived in the slums of a mountain-peak empire called the Rooftop, one of the few places not covered by Fog. We flew our rickety air-raft over the endless white vapor every day. As the \"tetherboy,\" I dove into the Fog and searched the ancient wreckage for stuff we could sell back in the slum for food and clothes and rent.\n\nBut these days, we needed more. These days, we flew deep into uncharted Fog, hoping to find something big\u2014something _huge_ \u2014to save Mrs. E's life. We were running out of time.\n\nI spent the rest of the afternoon prowling through Fog-covered hills, keeping my tether free of tree branches, rummaging in heaps of concrete and searching the husks of pod-cars. I wasted an hour digging through rotting planks, hoping to find pipes or plastic, but all I unearthed was thousands of beetles. Then the hand brake on my tether jerked: three fast yanks.\n\nIt was a message from Hazel, the captain of our ramshackle raft, signaling me from above the Fog. _Come back,_ she was saying. It was getting late, and nobody survived a night in the Fog\u2014even I was afraid to stay after sunset.\n\nI started to sign back _okay,_ then stopped when I heard something in the distance, a muffled _eee-huuurk_.\n\nI smiled at the sound and squeezed the hand brake. _Not yet_.\n\n_Come back_.\n\n_Not yet,_ I told Hazel again.\n\nNot the most fascinating conversation, but cut me a little slack. We couldn't say much with a hand-brake cable connected to a bell on the raft deck.\n\nAfter a minute, she signaled, _Okay_.\n\n\"Cool.\" I peered at the sky. \"Dinner's on me.\"\n\nI didn't expect to find salvage this late in the day, but I hoped to find food. Meals were scarce in the slum, and that _eee-huuurk_ had sounded like a goose. Like delicious roast goose for dinner.\n\nAdjusting my goggles, I headed downhill through the underbrush. Wisps of whiteness surrounded me. The Fog felt like cool breath against my skin, with the faint pressure of air before a big storm. Leaves crunched under my boots, and my tether unspooled with a _whirr-click-whirrr_.\n\nEyes wide and ears pricked, I stalked through the Fog. I crossed a meadow full of dandelions and smelled water. I listened for the burbling of a stream as I edged past some brambles . . . and an animal lunged at me through the mist.\n\nMy pulse rocketed. I yelped and leaped straight upward\u2014eight feet into the air\u2014and spun like a bat chasing a moth, feeling the world slow down around me.\n\nOn the raft or in the slum, I moved like an ordinary kid, but inside the Fog, I was _fast_. Rattlesnake fast. I jumped like a kangaroo, tumbled like a monkey, and climbed like a squirrel. That was me, a rattlaroo squirbat.\n\nMy body felt weightless as I flung myself through the mist, tracking the dark shape of my attacker with my gaze. Then I fell to the ground in a crouch and saw it clearly. It wasn't a mountain lion or bear or baboon\u2014I'd been assaulted by an angry goose.\n\n\"Okay, feather-face,\" I said. \"Come and get me.\"\n\nThe goose glared with beady eyes\u2014it didn't even notice the billowing Fog\u2014and made a hissing sound like a broken valve: _hhhhhhh, hhhhhhh_. Uncoiling its neck, it beat the air and snapped at my face.\n\nBut this time, I was ready, and I trapped its long feathery neck under my arm.\n\nThe goose struggled and thumped my chest with its wings. _Mee-hurrrrk-ee!_\n\n\"Ha!\" I clamped its wings tight. \"Gotcha.\"\n\nIt hissed and wriggled, and its webbed feet pedaled in the air.\n\n\"Sorry,\" I said as I started to wring its neck. \"But we're hungry, and you're dinner.\"\n\nThen I heard a faint _eep_.\n\n_Eep eep eep eep!_\n\nI looked down and saw four fuzzy little goslings waddling toward me through the haze. The one in front tilted its downy yellow head upward and stared at me with big eyes, like it was begging for mercy.\n\n\"You're out of luck,\" I told him, gripping the mother goose's neck harder. \"We've got to eat.\"\n\n_Eeep,_ he informed me.\n\n\"Easy for you to say,\" I muttered. \"Shoo!\"\n\n_Eeep,_ he repeated.\n\n\"Go away!\" I stomped, trying to scare the goslings off. \"I can't do this with you watching!\"\n\n_Honk,_ the mother goose cried.\n\n_Eeep eep eep!_ the little ones said, bumbling closer like puffballs on webbed feet.\n\nThe crew needed food\u2014we _always_ needed food\u2014and we never ate anything as tasty as roast goose. But something about four defenseless babies who needed a mother stopped me cold.\n\n\"Fine!\" I sighed, loosening my grip. \"But if we starve, it's your fault.\"\n\nI set the mother goose down near her babies, and she said _hooooork!_ and whacked me so hard with one of her wings that I fell on my butt. Then she led her goslings away, honking and hissing.\n\n\"You're welcome!\" I called after her.\n\nI sat there feeling like an idiot. We were hungry all the time, and I'd let a perfectly tasty goose get away. I didn't even want to _think_ about what Swedish\u2014our raft pilot\u2014would say. And I couldn't stand the thought of watching Bea\u2014our mechanic, and the youngest member of the crew\u2014go to sleep hungry again.\n\nAfter a while, I pushed to my feet and started plucking dandelion greens from the meadow. They were bitter, but they'd fill our stomachs. I was shoving one last handful into a sack when a breeze blew a perfumed scent toward me. _Flowers? Maybe roses_.\n\nA grin tugged at my lips. I'd learned that roses meant fancy gardens and houses, which were good places to scavenge.\n\nI followed the scent uphill, and a shape loomed through the Fog: dark bars in the whiteness. I edged closer and saw an iron gate, a row of black posts with sharp points. Good, thick, valuable iron, only slightly rusty.\n\nI smiled. \"Now we're talking.\"\n\nI reached for the hacksaw in my leg-sheath, and the hand brake on my tether jerked three times: Hazel was saying _Come back_.\n\n_Not yet_.\n\n_Come back,_ she signaled. _Come back, COME BACK!_\n\nI frowned. That was pretty bossy, even for Hazel.\n\nThen I noticed the Fog darkening around me. I'd lost track of time. Dusk was falling and long shadows were creeping across the field.\n\n\"Yikes,\" I muttered. Hazel was right, of course. Sometimes she was more \"boss\" than \"bossy.\"\n\nI signaled back: _Ready_.\n\nA moment later, my tether straightened in the air above me. With a tug at my harness, it lifted me off my feet and reeled me upward.\n\nI rose into the air as white clouds billowed around me. Higher and higher until finally, in an instant, the Fog fell away and my full weight returned. The harness dug into my chest, my arms and legs turned to lead, and even my boots felt heavy, like they were suddenly filled with mud.\n\nThe endless Fog spread below me, touched by the rays of the setting sun. It looked more like a cool mist than the plague that had almost destroyed humanity. And that still hid the treasures we needed to survive.\n\n##\n\nOUR SALVAGE RAFT WAS a hodgepodge of mismatched parts and tattered patches. It floated above the Fog, dangling from three balloons lashed together with fraying ropes. A wicker basket swayed in the tangled rigging beneath the balloons, the \"crow's nest\" where Hazel usually stood.\n\nBelow that, the raft itself was a weather-beaten deck of canvas, wooden floorboards, and copper pipes. The winch for my tether rattled beside my diving plank and, farther back, Swedish spun the ship's wheel and clattered at the steam organ that controlled the rudders and propellers.\n\nUnder the deck, Bea tinkered with the clockwork engine that powered the fans and pistons and vents. And hanging below the whole thing, the empty cargo net swayed in the stiff breeze. It wouldn't be empty for long. That iron gate would cover food for a week or two, even after the bosses took most of the money.\n\nBut we still needed a much bigger score to help Mrs. E.\n\nThe tether winched me toward the underside of the raft, bringing me close enough to grab the boarding ladder.\n\n\"Chess!\" Bea called to me, with a smile on her freckled face. \"You're okay!\"\n\n\"Of course!\" I struck a pose on the ladder. \"For I am Freakula, Lord of the Fog.\"\n\nShe giggled. \"You're a chucklebutt! How come you stayed down so long?\"\n\n\"I found something,\" I told her. \"Float a buoy!\"\n\nShe flashed a salute and disappeared into the gearwork. It was too late to grab that iron gate today, so she'd mark the site with a buoy and we'd return tomorrow, after drifting all night. We couldn't hover in one place overnight, not without a pilot at the wheel, and even Swedish needed to sleep.\n\nI climbed on deck, shoved my goggles to the top of my head, and started unbuckling my harness.\n\n\"Next time come up when I tell you to!\" Hazel called from the crow's nest. \"Look at the sun.\"\n\nFrom above, the Fog usually looked like an endless ocean with motionless white waves. But now, the orange light of sunset brushed the high crests.\n\n\"Sorry,\" I said. \"I got busy.\"\n\n\"Doing what?\" she asked. \"You look messed up.\"\n\n\"Chess always looks messed up,\" Swedish said from the wheel.\n\n\"I got into a fight,\" I admitted.\n\nHazel frowned. \"Are you okay?\"\n\n\"Was it wolves?\" Swedish asked. \"Baboons?\"\n\n\"Worse,\" I told him.\n\n\"Not driftsharks,\" Hazel said, giving me a worried look.\n\n\"Of course not,\" I said. \"You don't _fight_ driftsharks, you just . . . die.\"\n\n\"Hyenas?\" Swedish guessed.\n\n\"Um . . .\" I didn't want to tell them I'd lost a perfectly tasty goose. \"I found an iron gate that's in good shape. And I'm pretty sure I smelled roses nearby.\"\n\n_\"Finally!\"_ Hazel brightened, forgetting that I looked a mess. \"I knew that flying this far would pay off. We'll grab everything in sight, and make enough to help Mrs. E!\"\n\n\"If the troopers don't arrest us first,\" Swedish grumbled.\n\n\"Nobody's going to arrest us,\" Hazel told him.\n\n\"They will if Lord Kodoc hears about Chess.\"\n\nI swallowed. Lord Kodoc was more than the tyrant who ruled the Rooftop and commanded the roof-troopers\u2014he was the reason we lived small and quiet in a remote corner of the slum. _Everyone_ with half a brain was scared of Lord Kodoc, but it was different for us. He was the bogeyman Mrs. E had scared us with since we were little. The monster in all our nightmares. She said that if Kodoc found us, he'd tear us apart.\n\n\"Yeah,\" Hazel said, \"but he won't. He doesn't even know Chess exists.\"\n\n\"Not yet,\" Swedish muttered darkly. \"You've heard the rumors.\"\n\nI ducked my head. Swedish was right. We needed to escape the slum to find a cure for Mrs. E's fogsickness\u2014but also to get far away from Lord Kodoc. He thought I'd died thirteen years ago, after he'd lowered my mother into the Fog. Recently, though, we'd heard terrifying rumors about a kid with a Fog-eye. If Kodoc found out I'd survived, he'd hunt me down. He'd lock me to a tether and dangle me in the white until the Fog killed me.\n\nHazel shot Swedish a dirty look, then turned back to me. \"So what'd you get in a fight with?\"\n\nApparently she _hadn't_ forgotten about me looking messed up.\n\n\"Here.\" I tossed my sack to Swedish. \"Dinner.\"\n\nHe peered inside. \"You got in a fight with dandelion greens?\"\n\nI sighed. \"Fine! It was a goose. I got in a fight with a goose.\"\n\nFrom behind me I heard Bea's familiar giggle, and she teased me while Swedish started dinner. He soaked the greens in rainwater, tossed the last of our seagull jerky into the broth, and simmered the whole thing over an exhaust vent.\n\nI plopped down under the balloons while the soup cooked, feeling drowsy and content. I'd follow the scent of roses tomorrow, and with any luck I'd find a drawer of silverware or even\u2014in my wildest dreams\u2014a cabinet full of unbroken wineglasses. Rich people on the upper slopes of the Rooftop paid huge for stuff like that.\n\nHazel sat beside me, gazing at the sunset, her braids falling around her shoulders, as Bea fiddled with a handful of wires. She made figurines out of cables and wire, what she called \"twistys,\" miniature people, airships, and animals.\n\nShe handed me a twisty of a cute little bird. \"Here!\"\n\n\"What's this?\" I asked, even though I already knew the answer. \"A chickadee?\"\n\n\"A silly goose,\" she said. \"Like you!\"\n\nSwedish poured the soup into bowls and handed them around. \"A _rabid_ goose. Nothing else could beat Chess.\"\n\n\"It probably had fangs,\" Hazel said. \"And glowing red eyes.\"\n\nBea wiped her mouth with her sleeve and asked, \"What really happened?\"\n\n\"She had baby geese with her,\" I admitted. \"They kept looking at me with their big eyes. I just . . . couldn't.\"\n\n\"You fog-face,\" Swedish grumbled. \"We could be eating roast goose right now.\"\n\n\"Of course Chess couldn't kill her!\" Bea told him. \"She's a mother! She had babies.\"\n\nSwede grunted. \"They're probably all adopted.\"\n\n\" _We're_ all adopted!\" Hazel reminded him.\n\n\"Sure,\" he said. \"But we're not delicious.\"\n\n##\n\nNIGHT FELL, AND a million stars freckled the dark sky. We climbed into our hammocks to sleep as the engine ticked and the rigging fluttered. When a cool breeze rose, my hammock began swaying.\n\n\"In the old days,\" I said, pulling my blanket to my chin, \"before the Fog came, people used to see shapes in the stars.\"\n\n\"You already told us that,\" Swedish said. \"They called them constipations.\"\n\n\"Constellations,\" Hazel said with a soft laugh.\n\n\"That's what I said!\"\n\nGazing at the stars, I almost told them the old tale of \"Skywalker Trek,\" about a space war between the Klingons and the Jedi, set in a future when people lived on distant planets, and fought Tribbles, Ewoks, and Borgs. I decided to stick with constellations, though, because sometimes my stories got a little garbled.\n\n\"They saw archers and bulls and foxes,\" I said. \"They gave the stars names, like 'Elvis Parsley' and 'Greta Garbo' and 'Michael Jackson.'\"\n\n\"There's no way anyone was ever named 'Garbo!'\" Bea said in the darkness. \"That's too silly, even for the old days.\"\n\n\"That's what it says in my dad's scrapbook.\" When my father died, he'd left me a notebook filled with historical facts he'd pieced together. \"Greta Garbo.\"\n\n\"'Greta' is nice,\" Hazel said after a minute.\n\n\"'Garbo' sounds like the noise you said bullfrogs make,\" Swedish told me. \" _Gar_ bo, _gar_ bo.\"\n\n\"It sounds bossy,\" I said. \"Like a command. Swab the deck! Garbo the sails!\"\n\nI turned toward Hazel's hammock. \"That could be your name. Hazel Garbo.\"\n\nBea giggled. \"I'm Bea Parsley!\"\n\n\"Swedish Jackson,\" Swede said. \"I kind of like that.\"\n\nThe raft rocked in the breeze, and we fell silent. Snug in our hammocks, safe and together, far from our troubles. The rich green scent of trees and meadows rose through the Fog, so much sweeter and cleaner than the stink of the slum where we lived.\n\nMy eyes closed and my mind began to drift\u2014\n\n\"I can't sleep!\" Bea called out. \"Tell me the story again.\"\n\n\"No!\" Hazel and Swedish said at the same time.\n\n\"Pretty please?\" Bea pleaded. \"With pigeon on top?\"\n\n\"Be quiet,\" Swedish grumbled.\n\n\"Count the stars, sweetie,\" Hazel told her. \"Until you fall asleep.\"\n\n\"Pretty please, with churro on top?\" Bea asked. \"Pretty please with _cucumber_?\"\n\n\"Would you tell her already, Chess?\" Swedish smacked the bootball he used as a pillow. \"I'm getting hungry just listening to her beg.\"\n\nI yawned. \"You've heard it a hundred times.\"\n\n\"I don't care,\" Bea said. \"It's _our_ story. Nobody else even knows it, because Mrs. E only told _us_.\"\n\n\"Some people on the Rooftop know,\" Hazel said. \"A few of them, at least.\"\n\n\"You know what I mean! None of the other slumkids. Pretty please, with frog legs on top? Pretty please with\u2014\"\n\n\"Just tell her,\" Swedish grumbled.\n\n\"Sure.\" I rolled over in my hammock, gathering my thoughts. \"Um . . .\"\n\n\"'Before the Fog rose . . . ,'\" Bea prompted, in a singsong voice.\n\n\"Before the Fog rose,\" I said, \"there was something called the Smog. The Smog covered the whole Earth, like the Fog does now, except it made _everything_ sick. Not just people\u2014but also grass and trees, and every animal in the sky and sea and land. The Smog choked the entire Earth, slowly killing every single living thing.\"\n\n\"So the gearslingers . . . ,\" Bea prompted again.\n\n\"They called them engineers back then,\" Hazel told her for the hundredth time. \"Nanotech engineers. They made tiny machines to clean the Smog. So tiny that a million of them could fit into your smallest freckle. But\u2014\"\n\n\"It worked!\" Bea interrupted. \"You can always count on gearslingers.\"\n\n\"Engineers,\" I said. \"And yeah, it worked. Sort of. At first. The tiny machines\u2014\"\n\n\"Nanites,\" Swedish grumbled. \"She knows they're called nanites\u2014she's heard the story a thousand times.\"\n\n\"Do you want to tell it?\" I asked.\n\n\"I want to sleep. Hurry up and finish.\"\n\n\"Fine,\" I said, trying not to smile. Swedish loved to complain, but I could tell that he was listening to the story, too. \"The nanites fixed the Smog and healed the Earth. They cleaned the water and the air, they scrubbed the poison and pollution from every crack and crevice. Only there was one problem. . . .\"\n\n\"They didn't stop cleaning after they finished,\" Hazel said, taking over the story. \"They were designed to attack any new sources of pollution, and they calculated that because we humans _made_ the Smog, they needed to stop us. The nanites turned themselves into the Fog.\"\n\n\"You mean they _created_ the Fog,\" Bea said like she always did.\n\n\"No, they _became_ the Fog. Some of the engineers had been afraid the nanites might go haywire, so they built a 'command password,' a secret code inside each nanite that automatically killed it after three months.\"\n\n\"That way,\" I said, \"if the nanites went loco, the engineers could just shut down the nano factories, and boom. A few months later, no more nanites.\"\n\n\"What they didn't expect . . . ,\" Bea prompted.\n\n\"They didn't expect the nanites to build their _own_ factories. Nanites started making more nanites. More than humans ever built, zillions of machines so small that they floated like droplets of fog.\"\n\n\"And we had no way to stop them,\" Bea breathed.\n\n\"The nanites programmed themselves not to bother animals,\" I said after a moment's silence. \"They don't block the sunshine\u2014plants grow like crazy in the whiteness\u2014but they keep _us_ out. They only target _human_ brains. If we enter the Fog, we become blind and deaf. And if we stay too long, we die of fogsickness.\"\n\n\"But not _you,_ \" Bea said.\n\n\"Not yet,\" I said.\n\nBea shook her head. \"You're different.\"\n\n\"I'm a freak.\"\n\n\"Yeah,\" Swedish said. \"You get beat up by birds.\"\n\nI went on, ignoring him. \"Nobody knows exactly when the Fog started rising. A hundred years ago? Two hundred? We only know that the nanites rose slowly, inch by inch, foot by foot, until they covered the entire Earth in a thick white mist, and nothing remained but a few scattered mountaintops. The Rooftop and Port Oro.\"\n\nI fell silent. We lived on the Rooftop\u2014except slumkids weren't allowed on the mountain itself\u2014but we'd never even seen Port Oro. The Rooftop troopers didn't let anyone get that far from home.\n\nWe'd heard stories, though, and we knew that on Port Oro we'd find a cure for Mrs. E . . . and would finally be safe from Lord Kodoc. Ever since the mutineers of Port Oro broke away from the Rooftop years ago, airship skirmishes between the two settlements were common, so the Port was the only place where Kodoc couldn't follow us.\n\n\"Do you think there are people on other mountaintops?\" Bea asked.\n\n\"I _know_ there are,\" Hazel said from her hammock. \"Somewhere across the Fog, if you fly for months or years. There must be.\"\n\n\"If they're too far to reach, it doesn't matter,\" Swedish said. \"It's like they don't even exist.\"\n\n\"I wonder,\" Hazel said with a wistful tone in her voice, \"if they also think they're the last people on Earth.\"\n\nI watched the stars through the rigging and imagined distant mountaintops across the Fog. Foreign places where everything was different, where nobody feared Lord Kodoc.\n\nAlthough he'd been born into a minor branch of the Five Families, Kodoc became a captain of the roof-troopers at a young age. In those days, the scientists known as the \"Subassembly\" worked for the Five Families, but Kodoc suspected that they were hoarding information. So he attacked them. He seized their labs and research, he took hostages and lives. And as the tattered remains of the Subassembly fled to Port Oro, Captain Kodoc used his newfound knowledge to become _Lord_ Kodoc, head of the Five Families and ruler of the Rooftop.\n\n\"Tell the rest of the story, Chess!\" Bea demanded. \"The part where there's gear inside the Fog that controls the nanites.\"\n\n\"You just told it,\" I said. \"The myths talk about ancient machines that can lower the Fog.\"\n\n\"Do you honestly think they had these machines and didn't use them?\" Swedish asked. \"They just sat around and watched the Fog rise? That's loco. That's beyond loco.\"\n\n\"Not if they didn't know about them at first.\" I swayed in my hammock. \"What if the nanites built them? Either way, Mrs. E says Lord Kodoc is obsessed with finding them. He thinks if he controls the Fog, he controls the world.\"\n\n\"He's right about that,\" Hazel said. \"If anyone messed with him, he'd bury them in Fog. Entire families, whole neighborhoods. All of Port Oro.\"\n\n\"That's why he'll kidnap Chess if he finds him.\" Bea rolled toward me in her hammock. \"And force you to search for the machines.\"\n\nI swallowed. \"Well, yeah. . . .\"\n\n\"Because nobody dives as well as you.\"\n\n\"I'm the only freak around.\"\n\n\"You're not a freak,\" Hazel told me. \"You're a garbo.\"\n\n\"He's a total garbo,\" Swedish said. \"Now go to sleep, Bea, before I toss you overboard.\"\n\n\"Tell me the story of Robbing Hood first!\" she insisted.\n\nShe loved the story of the hoodie-wearing thief who lived in Sherlock Forest and stole from the rich. She loved the Hood and his crew: a robot called Made Marian and a monkey named Fryer Tuck. We used to play Robbing Hood when we were little, with Hazel as the Hood, Bea as Made Marian, and me as Fryer Tuck. We'd make Swedish be the evil Sheriff of Nodding Ham, of course.\n\n\"Bea, let Chess sleep,\" Hazel told her firmly. \"He needs to stay sharp for tomorrow. This is our big break.\"\n\n\"Oh, okay,\" Bea said in a small voice. \"Sorry.\"\n\n\"Night, everyone,\" I said.\n\n\"Night, Chess,\" Bea said. \"Night, Swede. G'night, Hazy.\"\n\n\"Good night, honeybee,\" Hazel said, a smile in her voice.\n\nI smiled, too, then snuggled under my blanket as the scent of pine trees washed over the raft. The gyroscope spun, the gears clicked, and I swayed in my hammock and gazed at the sky. So many constellations: archer, crab, Oprah. Was there a constellation for us? For salvage crews and slumkids?\n\nWas there a star for _me_? A tetherboy and a Fog-eyed freak?\n\n##\n\nTHE NEXT MORNING, I walked the plank. The cool air gave me goose bumps, and my boots tingled from the ticking of the engine under the deck. My heart beat even faster than usual before a dive. This time I was searching uncharted Fog, following the scent of roses.\n\n\"Settle down, Chess!\" Hazel called from the crow's nest. \"We're almost there!\"\n\n\"I am settled,\" I yelled back.\n\n\"You are not!\" she laughed. \"Go check your tether.\"\n\nI mock saluted her. \"Yes, _sir_!\"\n\n\"That's me!\" Hazel gestured with her battered spyglass as her skirt whipped in the wind. \"Admiral of the fleet!\"\n\nSome fleet. Our rigging looked like tangled ropes, and goofy-looking whales were painted on the bulging balloons. Still, things worked better when Hazel was in charge, so I stepped off the plank and checked my tether while she scanned for the buoy Bea had floated the night before.\n\n\"I don't see it,\" Hazel muttered a minute later. \"It should be right here.\"\n\nAt the back of the raft, Swedish spun the ship's wheel a few inches. \"I have a bad feeling about this.\"\n\n\"You have a bad feeling about _everything,_ \" I told him. \"If you ever had a good feeling, you'd get a bad feeling about it.\"\n\n\"Keep laughing,\" he grumbled. \"That's what _they_ want.\"\n\n\"They who?\"\n\n_\"Them,\"_ Swedish said ominously.\n\nSwedish was convinced that _they_ were watching us\u2014not Kodoc, some other _they_. He'd been paranoid about it for years, though he'd never managed to explain who _they_ were or why _they_ were so fascinated by a bunch of slumkids.\n\n\"You always say _they_ 're watching,\" I told him. \"But you never say who _they_ are.\"\n\nHis eyes narrowed cunningly. \"If I knew who they were, they wouldn't be _them_.\"\n\nArguing with Swedish always made me dizzy. \"So\u2014because you know absolutely nothing about them, you're absolutely sure they exist.\"\n\nSwedish nodded. \"Now you're getting it.\"\n\n\"That doesn't make any sense!\"\n\n\"That's what they _want_ you to think,\" Swedish said.\n\n\"A few wisps to the left, Swede!\" Hazel called.\n\n\"Can you see the buoy?\" I asked her.\n\n\"Not yet,\" she said. \"But it's close.\"\n\nI didn't ask how she knew. Hazel charted the Fog better than anyone. Just like Swedish was the best pilot, and Bea kept our raft in the air better than anyone could.\n\n\"Close is good,\" I said.\n\nI tugged my goggles over my eyes. Bubbles of excitement rose in my chest and I started rolling my shoulders.\n\n\"Take a hard left, Swedish,\" Hazel said. \"And stop fidgeting, Chess!\"\n\n\"I'm not fidgeting,\" I informed her. \"I'm limbering.\"\n\n\"Then stop limbering! You're making me nervous.\"\n\n\"Why?\" I asked innocently. \"Because if there isn't great salvage even this far from home, we'll never escape the slum?\"\n\n\"And you'll never afford those pink boots you want,\" Swedish called to Hazel, tapping on the steam organ.\n\nShe glared at him. \"I don't want pink boots.\"\n\n\"Yeah, Swede,\" I said, crossing to the plank, \"she wants _yellow_ boots and pink _ribbons_.\"\n\nWe teased Hazel about ribbons and dresses because she was such a weird combination of \"girly\" and \"commanding.\" She wore long, flowing skirts, dreamed of fancy dances, loved pretty sunsets . . . and could bark out orders faster than the toughest junkyard boss. She was about fifteen, a few years older than me and a dozen times smarter. And pretty, with light brown eyes, dark brown skin, and dozens of silky braids.\n\nI looked like every other tetherkid who ran with a salvage crew. I was compact, wiry, and undersized. My boots were stained and my goggles were scraped, and the leather bracers I wore on my wrists to catch the tether were scarred. The only difference was that I always kept my head down and my hair over my freak-eye.\n\nSwedish looked more like a thug or a bootball player than a raft pilot. He was so burly, shaggy, and bearish that he barely fit in the thoppers\u2014sleek, narrow airships\u2014that he flew in drag races to earn extra money. And Bea was our kid sister, with short red hair, big green eyes, and smears of grease on her face. She didn't dream of roast meat and conspiracies like Swedish, but of gears and pistons and building crazy new thoppers that looked like demented dragonflies and flew like hunting eagles.\n\n\"On Port Oro,\" Swedish said, trying to mimic Hazel's voice, \"everyone wears yellow boots.\"\n\n\"And there are no junkyard bosses,\" I added.\n\n\"Pigeons lay scrambled eggs,\" Swedish said.\n\n\"It rains soup,\" I added, \"and snows rice!\"\n\n\"And apples grow on trees!\"\n\nI gave him a look. \"Um, actually, they kind of do. . . .\"\n\n\"Oh,\" Swedish muttered. \"Right.\"\n\n\"If you two are done,\" Hazel said, eyeing us, \"can we get back to looking for the buoy?\"\n\nI laughed and bounced slightly on the plank. Maybe today we'd finally catch a break. Those roses were a great sign. I started scanning the Fog for the buoy again, and\u2014\n\nA deafening _POP!_ shattered the calm morning. It came from beneath the deck, from the engine where Bea was working, adjusting a faulty propeller.\n\nMy blood froze. \"Bea?\" I called. \"Are you okay?\"\n\nNo answer.\n\n\"Chess, _go_!\" Hazel shouted. \"Swedish, cut the engines!\"\n\nI dove from the plank, caught a cable with one hand and swung under the deck as a bubble of fear expanded in my chest. What if something had happened to Bea?\n\n##\n\nI SCRAMBLED PAST THE side rudder and the vents, and spotted Bea beside an exhaust pipe.\n\n\"Bea!\" I said, slumping in relief. \"What happened?\"\n\nShe didn't answer, her usually pale face so dark with soot that she looked like she was wearing a mask. She tapped a bolt with her wrench and told the engine, \"Not funny.\"\n\nShe talked to the machinery, the rotors, cables, and gears. That wasn't so weird, except she was sure that they talked _back_. Of course I couldn't argue with the results. No other gearslinger could've kept this scruffy raft in the air.\n\nWhen I tapped her shoulder, she jerked in surprise.\n\n\"Chess!\" she shouted, even though I was only a foot away. \"What are you doing down here?\"\n\n\"We heard a huge pop and\u2014\"\n\n\"What?\" she shouted. \"I can't hear you! There was this huge pop!\"\n\nI eyed her. \"Are you okay?!\"\n\nShe eyed me back. \"Are you okay?\"\n\n\"Hoo boy,\" I muttered. \"I'm fine!\" I shouted. \"Are. You. Okay?\"\n\nShe gave me a thumbs-up. \"No problem! The hydraulic valve's just mad because I didn't adjust him yesterday!\"\n\n\"Right,\" I said.\n\n\"Valves are moody,\" she explained.\n\nI shot her a dubious look, and she grinned back. She knew we all thought she was whackadoo, but she didn't care.\n\nI tapped her leather cap twice, saying good-bye, then climbed back onto the deck.\n\n\"She's fine,\" I told Hazel. \"Just bickering with the spark plugs.\"\n\nHazel rubbed her face. \"Do _other_ captains have these problems?\"\n\n\"Other captains have airships,\" Swedish told her. \"You have a floating rattrap.\"\n\n\"That's what I have _for now,_ \" she said.\n\nSwedish and I shared a bemused look at Hazel and her big dreams.\n\n\"But I can't find the buoy,\" she continued. \"Chess, help me look.\"\n\nI shoved my goggles to the top of my head, started to brush my hair away from my freak-eye, then hesitated. Like I was afraid that someone might burst out of the clouds and spot the white wisps drifting across my right eye.\n\nThis was the fear that never left me. The clouds of nanites in my eye helped me see farther, hear more, and move faster in the Fog than anyone else, but they also marked me as a freak. As Kodoc's freak. He wasn't just my enemy, he was also my _creator_. Millions of tiny machines swarmed through my brain because of him. Cobblers made shoes and weavers made cloth and Kodoc made me. Like I was nothing more than a tool he'd crafted to help him find those ancient fog-machines\u2014so he could kill his enemies in the silent rise of white.\n\nI'd felt his power every day of my life, before I'd even heard his name. Not just because of the big things, like not having a mother. Kodoc was also the reason I'd worn an eye patch as a little kid. My dad was the one who'd given it to me. He knew I had to hide my eye, but he'd hated how ashamed I felt. So after he died, I vowed that I'd never wear a patch again. And I hadn't: I'd just kept my hair long, my head down, and my mouth shut.\n\nOf course the crew didn't care about my freak-eye, because they were my family. The only family I had left. So I brushed my hair away and scanned the sky again for the buoy.\n\n\"No luck,\" I called. \"I don't see it anywhere.\"\n\nHazel swung down from the crow's nest. \"It's gone.\"\n\nI stared at her in disbelief. Buoys didn't just disappear. \"What do you mean, it's gone?\"\n\n\"It should be right there,\" she said, pointing to an empty stretch of Fog. \"But it's not. Full stop, Swedish.\"\n\nSwedish clattered on the organ keyboard, and the raft jerked to a standstill.\n\nI shaded my eyes. \"How can it be gone?\"\n\n\"Only two ways,\" Swedish muttered.\n\nEither the buoy malfunctioned or someone messed with it. And if someone messed with it, they might start messing with _us_. A worried silence fell . . . then a hatch slammed open and Bea popped through, her leather cap askew.\n\n\"Look at that!\" she cried happily. \"She still stops on a thumbnail! This raft is purple as a real airship.\" For some reason, Bea considered \"purple\" the highest praise.\n\nHazel quirked a grin. \"I don't know what we'd do without you, Bea.\"\n\n\"We'd sink,\" Swedish muttered. \"We probably still will.\"\n\n\"She _likes_ stopping,\" Bea said, her soot-smudged face flushing as she looked past us. \"Um, guys, where's the buoy?\"\n\n\"Gone,\" Swedish told her.\n\n\"Oh.\" Bea fiddled with her tool belt. \"Where'd he go?\"\n\n\"Now that,\" Hazel told her, \"is a good question.\"\n\n\"I've got a _feeling_ about this,\" Swedish declared.\n\n\"Yeah?\" I said. \"What _kind_ of feeling? Surely not a paranoid, doomed sort of feeling?\"\n\n\"No,\" he said. \"A realistic one. Someone stole our buoy.\"\n\n\"Who'd steal a buoy?\"\n\n\"Who knows?\" Swedish said as he nervously squeezed the bootball he kept beside the wheel. \"We're too far from home this time. Probably in mutineer airspace.\"\n\nMy stomach clenched. Everyone knew that the mutineers sometimes shot trespassers on sight. \"Don't even say that!\"\n\nHazel tucked a braid behind her ear. \"We're not in mutineer territory. We're in no-man's-land.\"\n\n\"Oh, that's much better,\" Swedish muttered.\n\n\"Maybe the buoy just deflated?\" I asked.\n\n\"It's possible,\" Hazel said. \"Bea, check it out.\"\n\n\"Sure thing, Cap'n!\" Bea knelt at the bin where we kept the buoys. \"Remember that buoy who went out yesterday?\" she asked the spare buoys. She \"listened\" for a moment, then nodded. \"That's right, Bumbleboy.\"\n\nHazel and I exchanged a glance. Now Bea was naming them?\n\n\"How was he feeling? Hmm? Purple as the day he was stitched.\" Bea frowned, then raised her head to Hazel. \"The buoy's fine. I don't know where he is now, but he definitely didn't deflate.\"\n\n\"So somebody took it,\" Swedish said. \"We're not alone out here.\"\n\n##\n\nTHE DAY SUDDENLY FELT cooler. The raft swayed in the breeze, the rigging creaked under the balloons, and foggium whooshed through the copper pipes.\n\n\"You mean mutineers?\" Bea finally asked.\n\n\"They're not so bad,\" Hazel said.\n\nSwedish snorted. \"Yeah, they just shoot down any airship that gets too close to Port Oro.\"\n\n\"They're defending themselves,\" Hazel said. \"You know why they're called 'mutineers'?\"\n\n\"Because they rose up against the Rooftop?\" Bea asked. \"A long time ago?\"\n\nHazel nodded. \"They were ordinary people, like us. Not nobles or merchants. They got fed up with paying the Five Families for the chance to breathe clean air. They gathered on Port Oro, fought off the Rooftop, and started ruling themselves.\"\n\n\"Roof-troopers.\" Swedish snorted. \"They're the worst.\"\n\n\"They're not as bad as Lord Kodoc,\" Bea said, fiddling with her cap.\n\n\"Kodoc's not worse than the roof-troopers,\" Hazel told her as a breeze rose around us. \"He _is_ the roof-troopers. They're his private army. Nothing's worse than him.\"\n\n\"Except driftsharks,\" I said. \"But they stay in the Fog.\"\n\nSwedish tapped the steam organ keyboard to keep the raft in place, but nobody spoke. They knew how much driftsharks scared me, swimming through the air with lashing tails and gaping jaws.\n\nDriftsharks were about the size and shape of real sharks I'd seen in pictures, except they didn't have solid bodies. They were dense swirls in the Fog, with bulky heads, misshapen fins, and wispy tails. They were made of billions of tiny nanites, but they seemed like a single animal, driven to destroy any human in the Fog.\n\nA year earlier, I'd seen a driftshark kill another tetherboy right in front of me. I still heard his screams in my nightmares.\n\n\"Um,\" Bea said after a second, \"now _I'm_ getting an itchy feeling.\"\n\n\"Let's find a different site,\" Swedish said. \"Something closer to the junkyard.\"\n\nThe \"junkyard\" was what everyone called the slum where we lived, because it was a sprawl of floating platforms jam-packed with shacks and trash, held aloft by rusty fans and creaky balloons. It was shaped like a huge, uneven ring and encircled the base of the craggy Rooftop mountains that towered above the Fog.\n\nThe Five Families lived high on the rich green peaks of the Rooftop, descendants of the people who'd grabbed the best land when the Fog started rising. The farther down the mountain you lived, the less powerful you were, until you came to the junkyard, where the air stank of garbage. Rats swarmed the streets in the night, at least until someone trapped them for dinner. It was dangerous and dirty, and the only home we'd ever known.\n\nWe lived in a run-down shack crammed in among thousands of other run-down shacks. And we didn't even _own_ our shack, or the raft. We weren't allowed to. Instead, we rented them from the junkyard bosses, who called us \"bottom-feeders\" and took all of our best salvage. They answered to Lord Kodoc, and we were lucky if they left us enough salvage to buy a sack of rice every week. Some neighborhoods weren't so bad\u2014like the ones where servants lived, crossing the bridges every day to work on the Rooftop\u2014but most were shaky, violent, and mean.\n\nSo, after swallowing my nervousness over the missing buoy, I pleaded my case. \"We _need_ this. There's an iron gate waiting, and probably more. We didn't come this far to turn back now.\"\n\n\"Forget it,\" Swedish said. \"There's enough salvage near home to buy food, if we look hard.\"\n\n\"We need more than food,\" I said.\n\n\"Tell that to my stomach.\"\n\n\"If we stick close to home, we'll never raise the money to reach Port Oro.\"\n\nSwedish snorted. \"You really think they can heal Mrs. E?\"\n\n\"They can!\" I said. \"I know they can.\"\n\n\"And there's food in Port Oro, Swedish,\" Hazel said. \"Enough food that you won't be hungry.\"\n\n\"We do okay in the slum,\" he said stubbornly.\n\n\"Enough food,\" she said, \"that _Bea_ won't go hungry.\"\n\nSwedish looked at Bea and his expression softened. Ever since she'd joined the family, we'd tried to protect her from the worst of the slum. Maybe we were used to it, but she deserved better.\n\n\"There are real houses, instead of living in shacks,\" Hazel continued more gently. \"Real jobs, instead of dropping Chess into the Fog. And no Lord Kodoc. Port Oro is\u2014\"\n\n\"It's _paradise,_ \" Swedish scoffed, tossing his bootball from hand to hand. \"They'll heal Mrs. E and we'll live on a cloud. You keep saying that, but you don't know what it's like there. Nobody does. All we know is that the mutineers will blow us from the sky if we get too close.\"\n\nI started to answer, then hesitated. He was right. Port Oro reminded me of a magical island in one of Hazel's storybooks, a golden land of fruit trees and freshwater springs . . . guarded by a ferocious sea serpent. Maybe the island was great, but the serpent would eat you before you even saw the beach. And it didn't help that the roof-troopers arrested anyone who tried to move to the Port. They didn't care if you lived or died, but they'd be damned if they'd let you join the enemy.\n\nStill, we needed to try. For Bea. For Mrs. E. For all of us.\n\n\"I'm diving.\" I grabbed my harness. \"What's the worst that could happen?\"\n\n\"You could slice yourself on broken glass and bleed gallons,\" Swedish said. \"Again.\"\n\n\"That was a scratch!\"\n\n\"Hazel gave you twenty stitches,\" Bea reminded me. \"And I heard alligators are coming back.\"\n\n\"You don't even know what alligators are,\" I said.\n\n\"Like snapping turtles,\" she said. \"Only bigger.\"\n\n\"And what are snapping turtles?\"\n\n\"Turtles,\" she said. \"That snap.\"\n\n\"It's not animals I'm worried about,\" Swedish said. \"Someone's out here. They're probably watching us right now.\"\n\nI didn't bother answering. I just turned to Hazel and waited.\n\nShe tapped her spyglass against her leg, her forehead furrowing. I knew she was thinking that we needed a big find, to save Mrs. E\u2014and ourselves. But she also worried that we couldn't afford to take stupid chances, and that every day I spent in the Fog, I came closer to losing my strength and my mind.\n\n\"That missing buoy means trouble.\" Hazel rubbed her nose. \"Are you sure this is a good site?\"\n\n\"Well, nobody's scavenged here in forever,\" I told her.\n\n\"Nobody else is dumb enough to dive in the middle of nowhere,\" Swedish grumbled.\n\n\"Plus, I smelled roses,\" I said.\n\n\"What does that mean?\" Bea asked.\n\n\"They used to plant flowers around fancy houses, so it means this is a good site.\" I looked to Hazel. \"There are roses on the Rooftop, right? On the upper slopes, near the fancy houses?\"\n\n\"Red and white and pink,\" she said with a wistful smile.\n\n\"How would you know?\" Swedish growled at her. \"You've never been to the Rooftop. Born and raised in the junkyard. If it weren't for Mrs. E, we'd all be working in a refinery for a bowl of gruel and a beating.\"\n\n\"I know,\" Hazel told him, \"because I _read_.\"\n\nSwedish cracked his knuckles. \"You can't believe those stories. _They_ control the books.\"\n\n\" _They_ don't control my dad's scrapbook,\" I told him.\n\n\"That's just odds and ends from before the Fog.\"\n\n\"It's history,\" I said a little defensively, because the scrapbook was my only connection to my father. \"How else would we know about the weird animals that used to exist? Spelling bees? Hello Kitties?\"\n\n\"How _did_ kitties say hello?\" Bea asked.\n\n\"The same way bees spelled, I guess,\" I said.\n\n\"Someone took the buoy,\" Swedish said. \"And even if the roses\u2014\"\n\n\"I'm diving,\" I interrupted. \"This site is special.\"\n\nHazel took a breath. \"Promise you'll run like a cockroach at the first sign of trouble?\"\n\n\"With all six legs,\" I told her.\n\n\"And hope you don't get squashed,\" Swedish muttered.\n\n\"You know me,\" I told him, crossing toward the diving plank. \"I'm unsquashable.\"\n\nSwedish jostled me as he headed for the wheel. \"Just watch out for geese, okay?\"\n\nI strapped my harness around my chest and attached my tether to the winch on the deck. Bea checked my buckles; then I stepped onto the diving plank, my heart beating fast and hard.\n\n\"Goggles down,\" Hazel called from the winch.\n\nI lowered my goggles. \"Goggles down!\"\n\n\"Tether free?\"\n\nI checked my tether. \"Tether free!\"\n\n\"Dive at will, Chess, and\u2014\" Hazel paused. \"Come back safe.\"\n\n\"Every time,\" I told her.\n\nBehind me, Bea asked, \"What do roses smell like?\"\n\n\"Like a full belly and a warm bed,\" I said, and dove into the Fog below.\n\n##\n\nMY TETHER HISSED THROUGH the air, uncoiling from the raft.\n\nThe winch squeaked as I fell\u2014then I hit the Fog, and the world disappeared into a silent blur of white. Wind whipped my face and ruffled my pants, and I whooped in excitement, forgetting my worries as I somersaulted through the mist.\n\nI loved diving. Sure, the Fog scared me half to death\u2014but it was also the only place I felt fully alive.\n\nI'd been a salvage diver for three years, and most divers didn't even survive one. There were too many dangers in the Fog: wild dogs, driftsharks, and\u2014worst of all\u2014the jagged branches of trees and the spiky edges of crumbling buildings. I knew Hazel had nightmares about hoisting my tether above the Fog and finding an empty harness.\n\nThe Fog was treacherous and brutal, but it was beautiful, too. Shimmering shapes tumbled in the mist, and my body felt different. Buoyant, weightless. I jumped impossibly high, I twirled and flipped and sprang.\n\nI never talked about how much I loved diving. I knew the Fog was terrible, I knew that millions\u2014billions\u2014of people had died when it rose. I hated the Fog for that, and I hated that the Fog was inside me. Most of all, I despised Kodoc for making me like this: afraid of strangers, afraid of exposure, afraid of myself. Scarred for life by a freak-eye that affected me\u2014or _infected_ me\u2014in ways I didn't understand.\n\nBut I loved how the Fog carried me like a leaf in the wind, how it freed me. And I loved that diving would help me care for Mrs. E.\n\nThe junkyard bosses had taken my father's shack after he died, so I'd been sleeping under a pile of molding roof shingles when a gray-haired lady with sharp eyes and a hawk nose started bringing me food. I'd spent my days stealing and my nights trying to spear rats with a stick. I'd almost turned into a rat myself, filthy and skinny, and driven by exactly two things: fear and hunger.\n\nSometimes I'd hang around marketplaces, keeping my right eye closed and whining for food, but the beggar gangs usually chased me off. One day, though, I'd found a bag of potato peels. I'd scampered into a gloomy alley, and shoved handfuls into my mouth.\n\n\"I've been looking for you!\" a woman's voice rang out.\n\nShe stepped toward me from the end of the alley, and I stopped chewing, my gaze darting everywhere, looking for escape.\n\n\"Don't be frightened,\" she said soothingly. \"I knew your father.\"\n\nI wasn't loco enough to believe that. I figured I'd stolen from her, and she wanted to get close enough to kick me.\n\n\"I have food.\" She reached into her bag. \"Grilled pigeon feet.\"\n\nEven from ten feet away, the meat smelled delicious. My mouth watered and my stomach gurgled, but I looked at her kindly face and followed my instincts: I ran.\n\nTwo days later, she found me again. She put a pickled egg on the ground and walked away. The day after that, she left me a bunch of toasted grasshoppers, and a pair of shoes.\n\nOver the next few weeks, she kept feeding me . . . and I kept mistrusting her. What could she possibly want? What horrible thing was she planning for me?\n\nShe treated me like a skittish wild animal, slowly guiding me across the slum toward her shack. She'd already adopted Swedish and Hazel, which reassured me. She hadn't cooked _them_ in a pot and chewed on their bones. I figured that was a good sign.\n\nFinally one day, after giving me a bowl of corn mush, she told me, \"When you're finished, bring the bowl inside, please.\"\n\nAgainst my better judgment, I went inside and put the bowl on the cabinet. And from behind me, she said, \"Chess?\"\n\nI spun toward her, tensed to flee.\n\n\"Welcome home.\"\n\nHer name was Ekaterina, but we called her Mrs. E, and she raised us like some combination of mother, teacher, and boss. She made me practice my tetherskills until my fingers bled. She made Swedish fly thoppers in under-slum drag races. When Bea came, she made her fix engines in the dark, until she knew every valve with her eyes closed. She was the hardest on Hazel, though; she made Hazel make the rest of us do our jobs. And at the end of the long, exhausting days, she taught us to read and told us stories of the time before the Fog.\n\nShe was the kindest thing in our lives. I once woke up in the middle of the night and caught her sewing scraps of cloth by candlelight, making a poufy dress for Hazel's doll. She did the same years later for Bea\u2014except instead of a dress, she made overalls and a tiny wrench. And when Mrs. E teased him, Swedish used to laugh, this big booming laugh.\n\nThen the fogsickness began to sap her strength. And nobody could cure the sickness except the Subassembly. They were our only hope. We just had to find them.\n\nOnly a few \"fogheads\"\u2014Subassembly members\u2014had stayed on the Rooftop after Kodoc's attack. They lived in hiding, though, and studied the Fog in secret. If they were discovered, they were sentenced to hard labor in the stinking refineries that produced foggium.\n\nSomehow, Hazel had tracked down one of their hideouts in the junkyard. And after Mrs. E started getting feeble and confused, she'd dragged me to a Subassembly meeting in a dank chamber that dangled under the floating slum platforms.\n\nWe'd covered ourselves in cloaks, but the fogheads still stared at me like they knew all my secrets.\n\n\"We can't cure your friend,\" a hard-faced woman told Hazel. \"We don't have the right gear.\"\n\n\"Who does?\" Hazel had asked.\n\n\"Our brothers and sisters on Port Oro,\" a younger foghead had told her, eying my cloaked face. \"They're your only chance.\"\n\n\"But you'll never reach Port Oro by yourselves,\" the woman said. \"Now, tell us how you found us. We never meet in the same place twice.\"\n\n\"You always meet in whichever hideout is closest to the Fog,\" Hazel told her. \"I looked for the highest peaks of Fog under the slum.\"\n\nThe woman inhaled sharply. \"You read the Fog as well as one of us! Who are you?\"\n\nShe reached toward me and brushed my hood back a few inches before I knocked her hand away. The younger foghead gasped at my roughness, but I didn't care. At least he hadn't seen my eye . . . or so I'd thought.\n\n\"We're nobody,\" Hazel had said, and led me away.\n\nAs I swooped through the Fog, I somersaulted at the end of my tether until I felt the ground nearing. I slowed suddenly and dangled in midair as a line of metal spears came into focus five feet below. Any one of them would've skewered me like a kabob.\n\n\"Ha!\" I said, though the thick mist swallowed my voice. \"Direct hit, Hazel.\"\n\nShe'd dropped me right over the iron gate I'd spotted the previous evening, even _without_ a buoy.\n\nExcept . . . where was the buoy? If Bea wasn't wrong\u2014and she never was, about mechanical things\u2014then some airship had stolen it. Hopefully not the roof-troopers, who'd sink us on suspicion of trying to escape the Rooftop. Of course, the muties were just as deadly, even if all they cared about was protecting Port Oro.\n\nI unspooled my tether until I landed on the ground, then wiped my goggles and got to work. First I cleared the shrubs twining through the rusting gateposts, enjoying the rich scent of greenery, so unlike the stink of the slum. Then I pulled my hacksaw and started cutting the gate from the earth. Still spooked by the missing buoy, I stopped to peer into the Fog every few minutes, but after an hour the gate finally toppled.\n\nI attached a cargo tether, then squeezed the hand brake on my harness, telling the raft to raise the cargo.\n\nNothing happened for a few seconds; then I felt four jerks on the hand brake. It was Hazel, asking me to confirm. When I repeated the message, the gate lifted into the air and winched toward the raft, tilting and swaying.\n\nIn a minute, it was ten feet above me, out of sight.\n\nI squeezed the hand brake, telling them I was safe and wanted to scout for more salvage. The iron would pay for food, but we needed a big score. So big that even after the bosses took most of the money, we'd still have enough to hire \"coyotes\"\u2014smugglers who secretly transported people from the Rooftop to Port Oro. That was our plan. Dive like maniacs in uncharted Fog until we raised enough money to pay for coyotes, then dodge roof-trooper warships and angry mutineers to sneak onto Port Oro and cure Mrs. E.\n\n\"Easy as falling off a pie,\" as they used to say. Which didn't make any sense to me, but they also used to say \"May the horse be with you,\" so as far as I could tell, they talked gibberish pretty much all the time.\n\nWhen Hazel signaled that it was okay, I followed the crumbling asphalt uphill, and a sudden humming sounded through the muffled quiet. A flowering bush loomed into view, covered in bees. Even insects thrived in the Fog\u2014it was only the humans who died.\n\nSometimes fogsickness struck a few days after contact with the Fog, or sometimes it took a few weeks or months. Sometimes nothing happened for over a decade. Like with Mrs. E: she was dying now because she'd braved the Fog to save my life, thirteen years ago.\n\nMost people got fogsick if they spent even a few hours in the white. Only tetherkids built up resistance as little kids, so we could dive all day and stay healthy for years\u2014though we didn't usually live that long. Of course I'd been diving over three years and still hadn't gotten sick, but I was born in the Fog itself, with nanites in my brain.\n\nWhen I stepped closer to the bush, dozens of peach-colored roses came into view. I stood there for a moment, clearing my mind. I loved diving, and I loved _this_ : exploring the Fog-hidden Earth. Nobody in the slum ever saw roses or deer tracks or moss-covered rocks in burbling streams. Heck, none of them even saw _trees_.\n\nHumming to myself, I snipped a big blossom for Bea.\n\nI tucked the flower in my pocket, sat down on a pile of stones, and ate a snack of dried locusts. I didn't like eating bugs, but a swarm had flown over the slum a few weeks back, and food was scarce. One thing every slumkid knew was hunger, the kind of hunger that made you dizzy and desperate.\n\nWhen I started off again, a slithery motion caught my eye. I lifted my head and saw a swirl of Fog thickening in the air as the nanites formed dense white cords above me. My breath caught and my mind blanked with terror: a driftshark.\n\n##\n\nI STARED INTO THE whiteness, eyes wide and heart pounding. Horror tightened my throat and terrifying images rose in my mind\u2014memories of a tetherboy, limp in his harness.\n\nLast year, we'd teamed with another crew to salvage a load of broken glass I'd found in a half-buried truck. I worked alongside a tetherkid named Bilal, a scrawny boy with a goofy smile. We didn't say much at first, because silence felt natural in the Fog. At the end of the second day, though, he grabbed my arm.\n\n\"What's wrong with you?\" he demanded.\n\nI gulped, worried that he'd seen my eye. \"You mean other than my personality?\"\n\n\"No, I mean . . . you're not afraid.\"\n\n\"Are you kidding? I'm scared of everything. I'm even scared of _butterflies_.\" I shuddered. \"With their creepy wings.\"\n\nBilal smiled but somehow still looked serious. \"You're not scared in the Fog, Chess. I know you're not.\"\n\n\"Yeah,\" I muttered, ducking my head. \"I don't know.\"\n\n\"And you're . . .\" He eyed me suspiciously. \"You're too good at this. You _like_ it.\"\n\n\"Right,\" I scoffed. \"I'm a huge fan of broken glass.\"\n\nHe didn't mention it again, and by the end of the week we'd become friends. Spending days in the Fog forged a special bond. Nobody else knew the muffled quiet of that whiteblind world.\n\nThen early one day, after we'd loaded the cargo crate, Bilal stretched his arms overhead. \"Almost done.\"\n\n\"Two more loads,\" I said.\n\n\"Maybe we can keep diving together,\" he said almost shyly. \"I mean, when we're done with this. If you want.\"\n\n\"That sounds cool,\" I said. \"If our crews are okay with it\u2014\"\n\nA driftshark lashed forward from the mist, thick cords of nanites forming a sleek fishlike body with wispy fins and foggy jaws that opened way too wide. Trillions of tiny mindless machines acting in unison\u2014but it looked like a fraying, hazy shark diving through the Fog.\n\nAs I shouted a warning, the knot of whiteness closed around Bilal's head, the nanites invading his brain. There wasn't a crunch of bones or splatter of blood. Just one thin scream, and Bilal sagged in his tether. By the time his crew winched him back to his raft, he was dead without a mark on him, except for his flat, white pupils.\n\nI'd watched helplessly from only a few feet away as the driftshark had twirled into the distance, leaving me limp with grief and guilt. The same Fog that had killed my friend swirled inside my own eye.\n\nThe memory faded as the haze of nanites above me dwindled into nothing. Just another puff of Fog. Not a shark at all. I exhaled in relief. I should've known better. Driftsharks usually appeared only in ruined cities and towns, and I was in the middle of nowhere.\n\nAfter catching my breath, I headed uphill . . . and almost fell in a ditch.\n\nInstead, I sprang upward, spinning in midair. I landed in a crouch on the other side of a wide rectangular hole that looked like what Mrs. E called a \"swimming pool.\" She said that people used to swim for fun, but c'mon: even in the old days nobody would waste good drinking water like that. A huge puddle in the yard, just to splash around in? The whole idea was ridiculous.\n\nWhen I searched the hole, I found pillbugs and black ants and beetle shells. And then, finally, a scattering of broken blue-and-white tiles.\n\n\"Nice,\" I said.\n\nI stuffed the tiles into my bag, jumped from the pool, and caught sight of a dark doorway half covered with thick green ivy.\n\nI frowned. I hated untethering in the Fog, because if things went wrong I'd never get back to the raft. Not unless I learned to fly. But searching inside a building was tough with a tether, and this site looked worth the risk.\n\nSo I squeezed my hand brake, telling Hazel I was untethering and going inside the house.\n\nThe hand brake jerked twice, giving me the answer: _No_.\n\n\"Thinks she's an admiral,\" I muttered, and sent the same message again. _I'm going inside_.\n\nNothing happened for a while, and I pictured them as they argued on the raft above. Then, finally: _Okay_.\n\n\"Ha,\" I said, and sent another message: _I'm detaching my tether_.\n\nThe reply came immediately. _No_.\n\nI grumbled and sent the message again.\n\n_No. No. No. NO!_\n\n_Okay,_ I told her, and muttered aloud, \"Hazel Garbo.\"\n\nI pushed aside the curtain of ivy. My tether scraped against the doorway for a second; then the Fog muffled the sound as I stepped inside.\n\nThe darkness smelled of rust and mud and hungry wolves. I mean, I wasn't sure what hungry wolves smelled like, but the shadowy gloom looked like the perfect den for _something_ with fangs and claws. I stood very still, trying not to look tasty, until I was able to make out a small room, with rotting leaves carpeting the floor.\n\nCompletely empty. No fangs or claws. Just how I liked my shadowy dens.\n\nAnother misty doorframe led me into a room with the remains of an altar, what Mrs. E called a \"teevee,\" a screen that told everyone what to wear and eat, and showed them pictures of royalty like the Burger King and Dairy Queen. The teevee itself lay on the floor, a shattered shell of plastic, with wires and circuit boards inside. Not worth much, except as building material.\n\nDown a hallway, the Fog glowed brighter, like the house was missing a wall or part of the roof, letting sunlight in. I clambered over the rubble toward a staircase that rose into the white\u2014and something fast and gray blurred through the mist.\n\nI jumped backward, my heart pounding. I didn't carry a weapon, because if you fought anything in the Fog, you lost. Instead, you ran: you tugged on your tether and rose into the sky. Unless of course you were trapped inside a building with a hungry wolf pack.\n\nHolding my breath, I scanned the misty whiteness. A scrabbling sounded behind me and I sprang upward until I was jerked back down\u2014my tether snagged in the other room.\n\nThe tether jerked tight, but I kept moving, like a ball at the end of a chain, and smashed straight into a crumbling brick chimney.\n\nPain burst in my side, and the bricks collapsed around me. I moaned, half covered in rubble. I should've untethered, no matter what Hazel told me. Now I was trapped, dazed and defenseless, while the wolves prowled closer\u2014\n\nA gray squirrel dashed across the stairs.\n\nOh. Not wolves. I'd almost killed myself running from a _squirrel_. Maybe I wouldn't mention that to the others, especially after yesterday's epic goose failure.\n\nWith a groan, I began pushing the bricks aside. Then a glint of metal caught my eye, a dented box that must've been hidden in the fireplace. Still half covered in rubble, I grabbed the box and opened the lid.\n\n\"Nice!\" I said when I saw a stack of what Mrs. E called \"cash,\" faded green rectangles with weird faces printed on them.\n\nAnd numbers, too. Most of _this_ cash said \"100,\" though some said \"1000.\" But that didn't really matter. What mattered was that cash made excellent toilet paper\u2014the best in the Rooftop. The 1000s weren't any softer than the 100s, though.\n\nI shoved the \"cash\" into my pocket, and saw the second box. Smaller, and almost cube shaped, with a tattered felt cover. Inside, I found a ring nestled on a puff of discolored satin.\n\nA gold ring with a sparkly stone. A diamond.\n\nMy breath caught in my throat. \"No way.\"\n\nImpossible. We'd spent weeks diving for a jackpot in uncharted Fog, and it was a _squirrel_ that finally led me to the biggest haul ever? Completely impossible. I mean, I'd never seen a diamond, but this looked exactly like one of the pictures in Hazel's books.\n\nAnd the Rooftop paid huge for diamonds. Heck, only members of the Five Families were even allowed to own them. They were illegal for slum-dwellers and lower slope crafters and even for the merchants on the middle slopes. In fact, I'd heard that if you got caught keeping a diamond for yourself, Lord Kodoc would personally cut off your hand.\n\nBut that didn't mean you couldn't sell them.\n\nI closed my eyes for a second, then peered again at the ring. Still there.\n\n\"You're not dreaming,\" I told myself. \"You're not fogsick. That is one big honking diamond.\"\n\nI was grinning like a garbo as I tucked the ring into my belt pouch. I wanted to shout and sing, but I dug myself out of the bricks instead. I almost scrambled outside to return to the raft, because we didn't need any more salvage, not with a diamond like this in my pouch. But I'd been a tetherboy for too long, and I couldn't leave good salvage behind. So I filled the bag with bricks from the chimney, smiling the whole time.\n\nI couldn't wait to see everyone's faces when I showed them the diamond. Bea would jump up and down, Swedish would spin his bootball on his finger, and Hazel would make a triumphant speech. And to think we'd been _worried_ about today's dive! No driftsharks, no accidents\u2014no problems at all.\n\nAll our work, rewarded. All our troubles, solved.\n\nPort Oro, here we come.\n\n##\n\nAS THE CARGO TETHER rose, I gazed into the Fog, daydreaming about life on Port Oro. Imagining a place without hunger, without Kodoc, without junkyard bosses or roving gangs. Imagining a cure for fogsickness.\n\nChewing my lip in excitement, I squeezed the hand brake. A moment later, my harness yanked me upward. My boots left the ground and the whiteness blurred around my face as I rose toward the clear world above.\n\nI laughed as I passed through the highest fringes of the Fog, eager to show everyone what I'd found. When the blue sky burst into sight and my full weight returned, it felt like I was carrying a fifty-pound backpack, but for once I didn't mind.\n\nTwirling toward the underside of the raft, I saw pistons chugging and vapor hissing from valves. Before I reached the boarding ladder, Bea's face popped over the side.\n\nHer leather cap was askew and her green eyes were wide. She gestured wildly toward a peak of Fog.\n\n\"What?\" I called.\n\nShe raised a finger to her lips, shushing me, then disappeared from view.\n\nI almost yelled, _Don't shush me\u2014I found a diamond!_\n\nThe winch stopped suddenly, with me still dangling twenty feet below the deck. The propellers whirled and the raft jerked forward.\n\nI yelped as I trailed behind the speeding raft, dragged along by my tether.\n\nWhat was going on? Were we running? From what?\n\nI climbed my tether, hand over hand, swinging sideways as the raft turned in crazy angles. I reached the deck just in time to catch a glimpse of Bea vanishing into a hatch. At the wheel, Swedish handled the lumbering three-ballooned raft like a racing thopper, playing hide-and-seek behind white waves of Fog.\n\nI climbed toward the crow's nest. \"What's going\u2014\"\n\n\"Mutineers,\" Hazel said without lowering her spyglass.\n\nAbove us, the middle balloon creaked in protest as Swedish veered suddenly to the left.\n\n_\"What?\"_ I shivered in the sunlight. Not now, not with a diamond in my belt pouch. \"Where?\"\n\nShe nodded at the Fog. \"On the far side of that peak.\"\n\n\"Did they see us?\"\n\n\"I don't know. They came into sight a mile away and turned toward us. Maybe a coincidence, but . . .\"\n\n\"They'll leave us alone, right?\" I patted my pouch nervously. \"I mean, mutineers hardly ever bother salvage rafts.\"\n\n\" _Hardly_ ever,\" Hazel said.\n\nWhen the mutineers revolted, they'd taken over the most distant mountaintop that belonged to the Rooftop. Port Oro was less than a quarter of the size of the Rooftop, but they'd beaten back every attack for decades. That was the good news. The bad news was that being outnumbered made the muties defensive and quick-tempered, so sometimes they shot first and asked questions later.\n\nAnd by \"sometimes,\" I meant \"usually.\"\n\n\"Bring her lower,\" Hazel called to Swedish. \"Along that ravine.\"\n\nSwedish scowled at the wide, wispy crack in the Fog. \"That's a tight fit.\"\n\nI swallowed nervously. The raft was fueled by refined Fog, called \"foggium,\" which didn't work inside the Fog itself. That's why machinery didn't run in the white. If the engine dipped below the surface, we'd lose power immediately. The foggium would stop heating the air inside the balloons, and we'd tumble hundreds of feet and crash.\n\n\"You've got a foot on either side,\" Hazel told him. \"You can do this.\"\n\n\"I can kiss my butt good-bye,\" Swedish muttered, plunging the raft into the ravine.\n\nThe deck jerked as the middle balloon groaned, then hissed like an angry cat: the sound of a balloon about to pop.\n\nI froze, straining to hear. Wind whistled through the rigging, and a valve released pressure with a _click-shhhht-click_. Nothing more from the middle balloon.\n\nAfter a few seconds, I unclenched my hands from the rail. \"The cargo netting's inside the Fog, Hazel. If it snags anything, we're dead.\"\n\n\"I _know_ that,\" she snapped.\n\n\"Then why are we\u2014\"\n\n\"Mutineers never come this close to the Rooftop, Chess. Something's not right. I don't know why they're here, but if they don't see us, they won't sink us, so\u2014\"\n\nA shadow fell over us from behind.\n\nI turned and saw an airship with a zeppelin\u2014a long cigar-shaped balloon\u2014that dwarfed our three patched balloons completely. Whirling fans droned on cranes that sprouted from the airship like a water bug's legs, and runoff pipes spat clouds of exhaust.\n\nIt was a mutineer ship, with dull brass and predatory fins, her name gleaming on the prow: _Night Tide_.\n\nA _war_ ship. We were so dead.\n\n##\n\nA VOICE BOOMED FROM the mutineer ship. \"Stop your props or we'll blow you from the sky!\"\n\nHazel swung down from the rigging. \"Do as he says, Swedish.\"\n\n_\"Muties,\"_ Swedish mumbled as he slowed the raft to a halt. \"Now they're after salvage rafts?\"\n\n\"I've heard of the _Night Tide,_ \" Bea said in a quavering voice. \"I heard she opened fire on a trading post.\"\n\n\"That's a lie,\" I said, though I wasn't so sure. \"Just the roof-troopers trying to make them sound bad.\"\n\nSwedish snorted. \"Did the roof-troopers make them threaten to blow us from the sky?\"\n\n\"They're not driftsharks.\" Hazel took a shaky breath. \"They're just people.\"\n\n\"People in a _warship,_ \" Swedish said.\n\n\"They're from Port Oro.\"\n\n\"Which has been at war with the Rooftop for years,\" I reminded her.\n\n\"We'll get through this,\" she told me.\n\n\"Or we'll spiral down and die in the white,\" Swedish said with a shudder. \"Our bodies eaten by dogs and\u2014\"\n\nBea squeaked in alarm, and Hazel said, \"Swede!\"\n\n\"Sorry,\" Swedish mumbled. \"I'm just saying.\"\n\n\"Well, _stop,_ \" Hazel said, inspecting the warship with her spyglass.\n\nThe voice boomed again from the mutineer ship. \"Higher still, my poppets\u2014up where we can see you!\"\n\nAnother mutie shouted from the _Night Tide_ : \"Where we can shoot you!\"\n\n\"Board you!\" a different mutie called.\n\n\"Sink you!\"\n\n\"Well, you were right, Hazel,\" Swedish muttered. \"They sound like sweethearts.\"\n\n\"Take us up,\" Hazel told him.\n\n\"They'll kill us!\" Bea cried.\n\n\"We'll be okay, honeybee,\" Hazel reassured her. \"I have an idea.\"\n\nI propped my goggles on my forehead with shaky hands and tugged my hair lower. After a few days on the raft, I almost forgot that I needed to hide my freak-eye. But the moment strangers approached, a hot flush of shame reminded me.\n\nAs I ducked my head, Swedish hunched at the wheel, Bea whispered to the foggium array, and Hazel stood at the prow, chin high and gaze steady.\n\nOn the upper deck of the mutineer airship, an almost elegant stretch of narrow railings was spiked with cannons and harpoons and lined with a dozen airsailors with sharp eyes and curved blades. A tall man with a scarred face lifted his hand, and silence fell. He strode to the quarterdeck, his cloak billowing behind him, revealing the cutlass and dagger at his belt. Ruby and emerald rings flashed on his fingers.\n\nLooked like mutineering paid pretty well. Still, I didn't see any diamonds, so maybe bottom-feeding paid even better.\n\nI patted my belt pouch to reassure myself that the diamond was still there, then quickly changed the gesture into a scratch. I didn't want to call any attention to myself. Not only because I hated being seen, but because I couldn't let them find the ring.\n\nOn the other hand . . . what if I just _gave_ them the diamond? Would they take us to Port Oro? Maybe, but what about Mrs. E? We couldn't exactly ask them to pick her up in the slum. Plus, they were mutineers\u2014outlaws and pirates. They'd probably just _take_ the diamond.\n\n\"Is that an airship,\" the mutineer captain called, \"or has a chunk of the junkyard floated away?\"\n\nA few mutineers jeered and one of them swiveled a harpoon toward me. My heart clenched, but a second later the mutie moved on, targeting the props instead.\n\n\"And what,\" the captain continued, frowning at the raft's balloons, \"are _those_?\"\n\n\"Our balloons, sir,\" Hazel called back.\n\n\"I mean the designs. Those paintings.\"\n\nHazel's eyes narrowed. \"They're whales!\"\n\n\"With fangs?\"\n\n\"Those aren't fangs! They're smiling.\"\n\n\"They're enough to give _me_ nightmares,\" the mutineer captain said. \"And that's not easy. Who are you?\"\n\n\"Just a salvage crew, sir, with some scrap metal and\u2014\" She cocked an eyebrow at me. \"What else?\"\n\nI lowered my head. \"Bricks and cash.\"\n\n\"Rusted iron, broken bricks, and toilet paper.\" Then, a little sarcastically, she continued: \"A poor haul for such a grand ship as yours.\"\n\nThe mutineer captain frowned at her tone. \"You're renting that trash heap from the junkyard bosses and you're mocking _my_ ship?\"\n\n\"Your aft propellers are off-kilter,\" Hazel told him with the barest tremor in her voice. \"The _Night Tide_ is waddling like a toddler in a diaper.\"\n\nAn angry murmur sounded from the warship, and I held my breath and clutched my belt pouch. You didn't mess with mutineers\u2014you fled or you groveled. Was Hazel _trying_ to make them angry?\n\nThe scar-faced mutineer raised his telescope and swept our raft scornfully. \"And _you_ are the captain?\" he asked Hazel.\n\n\"I am,\" she said.\n\n\"You look more like a dancing girl,\" he said. \"Permission to board?\"\n\n\"Permission denied,\" Hazel answered.\n\n\"Prepare to fire,\" he told his crew, and the clatter of harpoons filled the air, accompanied by the whistle of grappling hooks.\n\n\"On second thought,\" Hazel said, \"we'd be honored if you'd pay us a visit.\"\n\n\"I thought you'd see it my way,\" he said.\n\nThe mutineer captain muttered a command, and a plank slid from the _Night Tide_ and slammed onto our deck. The raft swayed when he crossed from his warship, and for a moment, I thought I heard that angry-cat hissing again from the middle balloon.\n\nThree burly mutineers followed the mutineer captain across the plank, and the raft suddenly seemed even smaller and junkier than before.\n\nHazel curtsied. \"Welcome aboard, Vidious.\"\n\nI exhaled shakily. Figured that she'd know the mutineer's name.\n\n\"That's _Captain_ Vidious,\" a red-haired mutie grunted.\n\nVidious just laughed. \"You have the advantage of me, girl. What's your name?\"\n\n\"Hazel.\"\n\n\" _Captain_ Hazel,\" Swedish muttered.\n\nThe red-haired mutie shoved him. \"Shut your yap.\"\n\nSwedish raised a hand to shove back, and Hazel said, \"No, Swedish. They're our guests.\"\n\n\"Plus,\" Bea added with a bright nervousness, \"there's a hundred of them and only four of us, and our foggium array isn't feeling well.\"\n\nI didn't say anything and tried to look small and boring. I had the crew's future in my belt pouch and couldn't give the mutineers any excuse to search me. This guy Vidious would definitely steal the diamond and leave us behind in the slum.\n\n\"Enough,\" Vidious told his airsailors. \"If they touch you, teach them a lesson. Otherwise, don't bother. Now, then\u2014\" He eyed Hazel. \"You're right that I don't care about salvage. I'm looking for reprisal against the Rooftop.\"\n\nHazel frowned. \"Oh.\"\n\n\"What does that mean?\" Bea asked in a small voice. \"'Reprisal'?\"\n\n\"Like revenge.\" Hazel eyed the mutineer captain. \"Is that why you shot down our buoy?\"\n\nBea gasped. \"Poor Bumbleboy. . . .\"\n\n\"We won't tolerate _any_ Rooftop vessel at the moment.\" The quiet anger in Vidious's voice made me shiver. \"Not even a buoy.\"\n\n\"Sorry!\" Hazel said brightly. \"Our mistake! We'll just be going now!\"\n\n\"You're not going anywhere yet. You look like roof-trooper spies to me.\"\n\n\"We do?\" she asked, her brow furrowing. \"Don't we kind of look more like . . . slumkids?\"\n\n\"Maybe you're both.\" He inspected us coldly. \"And I'm going to make the roof-troopers pay for ambushing my sister.\"\n\n\"Well, what did you expect?\" Hazel asked. \"You're way too close to the Rooftop.\"\n\nVidious ran his hand along a frayed rope. \"That's no excuse.\"\n\n\"But you're days from Port Oro. What're you doing this far from home?\"\n\nVidious quirked an eyebrow. \"Minding our own business, until a roof-trooper airship attacked us. And you're part of the Rooftop, aren't you?\"\n\n\"Not really. We're from the slum.\"\n\n\"The slum of the Rooftop,\" he said, his voice sharp with accusation.\n\n\"Well, yes, but\u2014\"\n\n\"Cut their cargo tether!\" Vidious called.\n\n##\n\nA HARPOON FLASHED FROM the _Night Tide_ and sliced through the ropes holding the cargo netting. The bag of bricks and rusted iron gate tumbled away into the Fog, and the raft rocked violently. For a third time, I thought I heard a faint hiss from the middle balloon.\n\n\"We should take their geargirl, too,\" Vidious said as an afterthought. \"If she can keep this junkpile in the air, she's some kind of genius.\"\n\n\"Over my dead body,\" Hazel said, and swung her fist at Vidious.\n\nSwedish kneed one of the airsailors and shoved another before the red-haired mutineer smashed him in the head with the flat of his sword. As Swedish crumpled to the deck, Vidious sidestepped Hazel's punch and twirled her. Almost like a dance, until he clamped her to his chest, his sword pricking her neck next to the seashell she wore on a leather cord.\n\nMy heart shrank three sizes. I froze, my hand on my belt pouch.\n\n\"I usually sink trespassers,\" Vidious purred into Hazel's ear, \"even if my sister sulks for a week.\"\n\n\"We're not\u2014\" Hazel gulped. \"We're not trespassers! We're in no-man's-land!\"\n\n\"You're probably working for the roof-troopers. Still, I'll take the geargirl and leave the rest\u2014\"\n\n\"If you take Bea,\" Hazel said, her voice soft but strong, \"I swear by the Fog that I'll find you. I'll claw my way back from the grave and haunt your dreams. I swear by the silence and the white that I'll ruin you and everyone you love. I swear by all the high places that\u2014\"\n\n\"Stop teasing her, Vid,\" a woman called from the _Night Tide_. \"Let them go.\"\n\nJittery with nerves, I peeked at the warship and saw a woman on the quarterdeck, dressed in black with a pair of daggers at her hips. Her blond hair flowed down her back, and strings of beads looped around her neck.\n\nCaptain Vidious cocked his head. \"Why should I do that, Nisha?\"\n\n\"The girl is playing to _me,_ \" the woman said. \"She must've seen me with her spyglass. I gather she knows my reputation.\"\n\nMy fingers tightened on the belt pouch. Had Hazel seen the woman\u2014Nisha\u2014on deck? Is that why she'd tried to anger the captain?\n\n\"Besides,\" the woman continued, \"none of my sailors were hurt, and the _Anvil Rose_ is almost fixed. Plus, we _do_ have more important tasks than reprisal.\"\n\n\"True enough,\" Vidious said, shoving Hazel away. \"Though I'd still like to get my hands on Kodoc.\"\n\nHazel stumbled a few steps. \"L-Lord Kodoc?\"\n\n\"You've heard of him?\" Vidious asked.\n\n\"Everyone in the slum knows his name,\" she said a little too quickly. \"He's the leader of the roof-troopers. Is he . . . out here?\"\n\nVidious gave a careless shrug. \"Maybe.\"\n\nI peered uneasily across the Fog. A few puffy clouds floated overhead, and I suddenly worried that Kodoc's airship was hiding inside them. What if I actually met Kodoc, face-to-face? Would he recognize me as his creation? Would he control me somehow, like a falconer controls his bird? Could I resist?\n\n\"Why do you care?\" Vidious asked.\n\n\"They say he\u2014\" Hazel swallowed. \"He works tetherkids to death. They say he experiments on them. They say he feeds kids to driftsharks to see what happens.\"\n\n\"He's a dangerous man.\"\n\nHazel tugged on a braid. \"Um, there's one thing I don't get. . . .\"\n\n\"Spit it out, poppet.\"\n\n\"Well, you just happened to be patrolling this close to the Rooftop?\" Hazel asked too dubiously. \"Minding your own business?\"\n\n\"Exactly,\" Nisha said from the railing of the _Night Tide_.\n\nHazel nodded toward the warship's hull. \"Then what are you doing with _that_?\"\n\nI followed her gaze and didn't see anything except a few exhaust vents and a row of lifeboats.\n\n\"Those are lifeboats,\" Nisha told her.\n\n\"I beg your pardon,\" Hazel said politely. \"For a moment, I thought one of them was a cargo raft disguised as a lifeboat. My mistake.\"\n\nAn ache throbbed in my stomach when I looked closer. Hazel was right. One of the boats attached to the hull _was_ a cargo raft: bigger, sturdier, and more maneuverable than a lifeboat.\n\nVidious scowled, and I silently begged Hazel to change the subject before he lost his temper completely\u2014but Bea widened her eyes at the cargo raft.\n\n\"Ooh, he's got lovely turnbuckles!\" she chirped. \"Swede could fly him upside down through a gnat's eye!\"\n\n\"Er,\" Hazel said quickly. \"She means if it _wasn't_ a lifeboat.\"\n\nVidious and Nisha exchanged a glance that seemed to chill the air, and I patted my belt pouch for reassurance. I definitely didn't want these two finding the diamond.\n\n\"But obviously it _is_ a lifeboat.\" Hazel gave Bea a quick look. \"Isn't that right, Bea?\"\n\n\"Yes!\" Bea squeaked, her cheeks flushing red. \"Lifeboat!\"\n\nVidious touched the hilt of his dagger. \"We should sink them,\" he told his sister.\n\n\"You're probably right,\" she agreed. \"But we're not going to.\"\n\nI wasn't exactly sure what was going on, but Hazel sounded frightened as she said, \"Kodoc is no friend of ours. He's no friend of anyone in the slum, except the bosses.\"\n\nNisha leaned on the railing of the _Night Tide,_ then vaulted overboard, her long hair streaming behind her. I gaped in shock, but she landed on the boarding plank that stretched between the warship and the raft.\n\n\"You remind me of myself at your age,\" she told Hazel, strolling closer.\n\n\"When you were her age,\" Vidious told his sister, \"you couldn't have captained an empty boot, much less a salvage raft.\"\n\nNisha's mouth quirked. \"Well, at her age _someone_ got tangled in the rigging, and hung upside down for an hour.\"\n\n\"I was sick!\" Vidious said.\n\n\"Of course you were,\" Nisha said mock soothingly. \"Why do you think Hazel mentioned the . . . odd-looking lifeboat to us?\"\n\n\"Because she's a dumb kid.\"\n\n\"Look closer,\" Nisha told him. \"This is a tight-run ship, even if it's patched together from junk. She's telling us she knows what we're after. She's offering to give us information.\"\n\nVidious narrowed his eyes at Hazel. \"What are we after?\"\n\n\"Um, you're smuggling something into the Rooftop?\" Hazel gulped. \"That's why you've got that cargo raft.\"\n\n\"You're half right.\" Vidious nodded to his sister. \"Fine. Talk to her.\"\n\n\"Tell me about the roof-trooper patrols,\" Nisha told Hazel. \"Do they search all rafts approaching the Rooftop?\"\n\nHazel nodded. \"But anyone can land on the _slum_. Well, if you find a dock or slipway.\"\n\n\"So getting into the slum is easy.\" Nisha absently rattled her beaded necklace. \"How about going from the slum to the Rooftop?\"\n\n\"Not easy at all,\" Hazel said. \"You need a pass to cross the bridges, and there are armed guards watching everyone.\"\n\nNisha fired off a dozen questions about roof-trooper patrols, surface-to-air defenses, and pursuit strategies, and Hazel answered them all.\n\nI gaped at her. How did she know all that?\n\nFinally Nisha clapped Hazel on the shoulder in thanks and sauntered back across the boarding plank. \"C'mon, Vid,\" she called over her shoulder. \"We've learned as much as we're going to.\"\n\n\"Maybe you have,\" Vidious said. \"But I still don't know why that tetherboy keeps patting his belt pouch.\"\n\n##\n\nA KNOT TIGHTENED IN my chest. I hadn't even _touched_ the pouch! Well, maybe once or twice. But how had he noticed?\n\nVidious stalked toward me. \"Hand it over.\"\n\n\"It's just\u2014\" I swallowed. \"It's nothing.\"\n\n\"Don't test my patience,\" Vidious said.\n\nMy mouth went dry and I nervously shifted my weight. After a second, I grubbed around in my pants pocket for the wire goose \"twisty\" that Bea had made.\n\n\"This is, um, all I have,\" I said, my voice shaky.\n\nVidious narrowed his eyes. \"Your belt pouch.\"\n\n\"I don't have anything!\" I blurted. \"Just a . . . a rose.\"\n\n\"Stop scaring the tetherboy,\" Nisha called from the boarding plank.\n\n\"Then you won't mind sharing,\" Vidious told me, ignoring his sister.\n\n\"Give it to him, Chess,\" Hazel said. \"We're all friends here.\"\n\n\"No!\" Whatever happened, I couldn't lose that diamond. \"I dove for it, and I'm not going to\u2014\"\n\n\"Boy!\" Vidious barked at me. \"Give me the pouch.\"\n\nWhen I shook my head, Vidious stepped toward me\u2014and a sudden spurt of fear gave me strength.\n\nI flicked the twisty goose into the air, and when Vidious followed it with his eyes, I vaulted off the deck. I dropped five feet, grabbed a rung of the boarding ladder, and swung under the raft. My heart pounding, I hauled myself past gears and rudders, then stopped below a hatch that opened to the deck and fiddled desperately with my pouches.\n\nTen seconds later, the hatch slammed open above me, and a rough hand yanked me onto the deck.\n\nThe red-haired airsailor tossed me across the raft. I landed hard on my elbow, and my goggles flew from my head. I moaned, lying in a heap at Vidious's feet.\n\n\"Let's see what you're hiding,\" Vidious said, slashing at my belt with his cutlass.\n\nThe tip of his blade flicked my pouch into the air. He caught it and pulled out a rose. That big peach blossom from the bush in the Fog.\n\n\"That's mine!\" I whined, trying to hide my relief that he hadn't found the diamond I'd just tucked into my boot pocket. \"It's for . . . for Valentine's Day!\"\n\n\"For _what_?\"\n\n\"An old holiday,\" I said, trying to distract him with facts from my father's scrapbook. \"When they used to wear green and say 'be mine' and kiss under a shamrock.\"\n\nVidious frowned. \"Why?\"\n\n\"They gave flowers to their sweethearts,\" I added, trying to look embarrassed.\n\nNisha's laughter sounded from the _Night Tide_. \"The tetherboy is a romantic,\" she said. \"I think I may swoon.\"\n\n\"Oh-ho, poppet!\" Vidious said with a mocking smile. \"Is this for your girlfriend?\"\n\nI glanced at Bea. \"She's not my girlfriend.\"\n\n\"You brought me a rose!\" Bea exclaimed, her eyes suddenly bright.\n\nI wasn't sure if she was playing along or just excited to see the flower. No telling with Bea.\n\nThe mutineer captain looked from scrawny Bea to beautiful Hazel and his lip quirked. \"You're giving the wrong girl flowers, tetherboy.\"\n\n\"Go fog your head,\" I muttered. \"You garbo.\"\n\nVidious tapped me with the toe of his boot. \"Mind your tongue.\"\n\n\"Don't tease the boy, Vid,\" Nisha said. \"Young love is hard enough.\"\n\nMy head throbbed, my elbow burned, and they thought I was in love with Bea\u2014but I didn't care. I still had the diamond.\n\n\"You're lucky I'm in a good mood,\" Vidious said, and crushed the rose in his fist.\n\n\"If this is a good mood,\" I muttered, \"I'd hate to see what you look like cranky.\"\n\nHe sprinkled the rose petals onto the deck, then strolled across the boarding plank. His three airsailors followed, and a moment later the plank vanished into a slot in the _Night Tide_.\n\n\"What do you think you're doing?\" Hazel asked me, too softly for the muties to hear.\n\n\"Getting my butt kicked,\" I muttered, still splayed on the deck.\n\n\"In that case, you did a pretty good job.\"\n\n\"It's my special gift,\" I told her.\n\n\"Owww,\" Swedish groaned, waking up and rubbing his head. \"Why aren't we dead yet?\"\n\n\"Scurry back to the junkyard,\" Vidious shouted from the deck of the _Night Tide_. \"And if I see you again, I'll drop you into the Fog.\"\n\nThe warship veered away, fins slicing the air. The back draft of its fans shook our balloons, and the deck jerked and swayed. I exhaled in relief but stayed still until the _Night Tide_ disappeared behind a towering spray of Fog. Then a smile crept across my face\u2014I still had the diamond!\n\n\"Got 'em!\" Bea announced before I could say anything.\n\n\"Got what?\" Swedish asked.\n\nShe showed him the crushed rose petals she'd collected from the deck. \"These! They're so purple!\"\n\n\"Looks more like pink to me.\"\n\n\"Not that kind of purple, Swede! And they're softer than silk.\"\n\n\"Chess!\" Hazel said. \"Would you tell me what\u2014\"\n\nA slithery hissing sounded above us, louder than before, and the terrifying _put-put_ of air escaping from a hole.\n\nWe all froze, staring up past the rigging. For an instant, I refused to believe my ears. Then the raft lurched, the deck slanted, and the middle balloon popped with a terrifying _bang_.\n\n##\n\n\"SWEDISH, TAKE THE WHEEL!\" Hazel shouted. \"Chess, the balloons! Bea\u2014we need altitude, _now_!\"\n\nBea flung herself at the ballast tank as shreds of fabric cascaded around us. I raced into the rigging and wove my tether through the main lines: if I didn't hook the two remaining balloons together, they'd slip from the rigging and we'd fall from the sky.\n\nWhile I tugged the lines closer, Bea crooned to the overtaxed engine.\n\nSwedish muttered under his breath for three long minutes, then said, \"Would someone tell me what happened?\"\n\n\"Our balloon popped,\" I told him, \"because you fly this thing like it's a racing thopper.\"\n\nWhich wasn't fair of me, but what can I say? Getting tossed around by mutineers made me grumpy.\n\n\"I mean, how come we're still alive?\" Swedish asked with a scowl in my direction. \"What happened after I got knocked out?\"\n\n\"Hazel saved us!\" Bea called to him from the gyroscope. \"She saw a lady on the mutie ship who she knew was nice, only they dumped our cargo and wanted to take me prisoner\u2014I mean, they said I'm a good geargirl, and then Hazel saw a cargo raft and realized they're trying to smuggle something onto the Rooftop and\u2014\"\n\n\"The woman is Captain Nisha,\" Hazel called from the crow's nest. \"She's Vidious's sister. They say she's three kinds of awesome, so I knew she wouldn't actually hurt us. Watch that Fog!\"\n\nThe raft jerked sideways, avoiding a crest of Fog, and for the next few frenzied minutes, I tugged and jerked, forcing the lines closer, trying to secure the balloons.\n\n\"C'mon,\" I whispered to myself. \"We're too rich to die.\"\n\n\"Get her five feet higher!\" Hazel shouted beneath me. \" _Now,_ Bea!\"\n\n\"She can't,\" Bea wailed. \"She can't!\"\n\n\"We're touching Fog,\" Hazel snarled. \"You promised me five more feet.\"\n\n\"That was ten feet ago!\"\n\n\"Give me five feet or we're dead.\"\n\n\"Fine!\" Bea pulled a cable cutter from her belt and started apologizing to the raft.\n\nHazel gritted her teeth. \"One notch to the left, Swedish.\"\n\nSwedish frowned, his knuckles white on the wheel. \"Poor Mrs. E. She's going to starve without us.\"\n\nThe raft swayed and bobbed, missing a ridge of Fog by inches. My arms ached from the effort of squeezing the lines tighter, and the bruise on my elbow throbbed. I gave one final heave, straining to pull the ropes together . . . and couldn't.\n\nHazel bellowed, \"The aft balloon's slipping, Chess! Steady them _now_!\"\n\n\"What do you think I'm doing?\" I shouted, and threw my arms around the lines.\n\nLinking the leather bracers on my wrists, I trapped the lines inside a desperate hug. The ropes chafed my arms, and the pain brought tears to my eyes, but the rigging started to steady. I wanted to call for help but closed my eyes instead. If Swedish moved the wheel an inch too far, the raft would overturn, if Bea didn't keep the fans spinning, we'd sink, and if Hazel didn't plot the shortest course, we'd shake apart in the air.\n\nSo I breathed slowly, trying to think of other things. Like switching the ring with the rose to trick the mutineers. Like selling the diamond to the bosses, then hiring coyotes to smuggle us into Port Oro. Finally we'd have more than a few scraps of food and a rickety shack. And we'd find a cure for Mrs. E.\n\nFor twelve years after saving me from the Fog, Mrs. E hadn't felt a twinge of fogsickness. Then last year, she'd started getting weaker. Only for a few hours, at first, but soon the sickness struck deeper. About six months ago, she'd spent two weeks straight sick in bed. Unable to stand, unable to feed herself. We'd known then that if we didn't find a cure, we'd lose her forever. That's when Hazel took me to meet the fogheads in the chamber under the platforms. And when they told us they couldn't help, we decided to smuggle Mrs. E\u2014and all of us\u2014into Port Oro.\n\nWe gathered in the kitchen and told her our plan. \"We'll start scavenging farther from home,\" Hazel explained. \"Until we raise enough money. There are places nobody's searched in a hundred years.\"\n\nMrs. E pursed her lips. \"And there are dangers nobody's seen in that long, too. This is a foolish idea.\"\n\n\"There's no other way to get you to Port Oro,\" Hazel told her. \"We need money. Lots of money.\"\n\n\"I forbid it!\" Mrs. E said, her voice stronger than usual.\n\n\"Too bad.\" Swedish crossed his arms. \"I'm not going to sit around and watch you die.\"\n\n\"Me neither,\" Bea said.\n\nI wiped my face with my palm. \"None of us are.\"\n\nHazel took Mrs. E's hand. \"You fed us when we were hungry, you loved us when we were alone.\"\n\n\"And now it's _our_ turn to take care of _you,_ \" I said.\n\nMrs. E's eyes welled with tears. \"The four of you . . . you're all I have. Every time you step onto that raft, I know I might never see you again. That's worse than fogsickness.\"\n\n\"We'll always come back to you,\" Bea promised.\n\n\"And if Chess finds something big, we can hire smugglers no problem,\" Hazel said with an eager smile. \"Heck, if he finds something _really_ big, I'll haggle with the rooftoppers directly. Even Lord Kodoc himself might let us leave if we\u2014\"\n\nMrs. E jerked like she'd been bitten. \"No! Not Kodoc. You stay away from Kodoc.\"\n\n\"I was just saying!\" Hazel protested.\n\n\"Promise you'll stay away from him,\" Mrs. E demanded. \"Promise me.\"\n\n\"Fine. I promise.\"\n\n\"I thought I could protect you, but now . . . I'm not so sure,\" Mrs. E said, her voice wavering but intense. \"There's something I haven't told you.\"\n\n\"Something about Kodoc?\" Hazel asked.\n\n\"Yes,\" Mrs. E said, her lips thinning. \"If he ever finds us, he'll kidnap Chess and work him to death.\"\n\nThe walls tilted around me and I stared at Mrs. E, not quite believing my ears. That had been the first time I heard about the link between me and Kodoc.\n\n\"M-me?\" I spluttered.\n\n\"He'll chain you to a tether, Chess, and he'll drag you through the Fog, scraping you against rocks and ruins in his search for the ancient machines\u2014\"\n\n\"The machines that lower the Fog?\" Bea interrupted.\n\n\"If they exist.\" Mrs. E took a trembling breath. \"If Kodoc catches Chess, he'll kill him. He's killed dozens of tetherkids already, experimenting on them, keeping them in the Fog for days on end\u2014\"\n\nHazel frowned. \"Does he know about Chess's eye?\"\n\n\"He doesn't even know Chess is alive. But Chess wasn't the first child born with Fog in his eye. There are old stories of another child with the same white spots\u2014\"\n\n\"What stories?\" I said. \"I never heard any.\"\n\n\"Secret stories, Chess.\"\n\n\"You could've told me!\"\n\nMrs. E raised a trembling hand. \"It was safer this way.\"\n\nAnger and confusion swirled in my head. I knew Mrs. E kept secrets, but not about _me_. Not about the one thing that shaped my whole life.\n\nHazel touched my arm. \"Let her talk.\"\n\nWhen I nodded, Mrs. E continued: \"The stories say that a hundred years ago a child was born inside the Fog with white in her eye. A shifting, shimmering cloud of nanites. Like a pearly mist.\"\n\n\"But Chess wasn't born in the Fog,\" Bea said.\n\n\"Maybe he was,\" Hazel said.\n\n\"Lord Kodoc spent decades trying to create a baby like the girl in the story,\" Mrs. E said. \"He thinks that any child born with nanites will have a special affinity for the Fog.\"\n\n\"What does that mean?\" Bea asked. \"Affinity?\"\n\n\"A special connection to the Fog,\" Mrs. E explained. \"Special abilities.\"\n\n\"Like how Chess doesn't get fogsick even though he dives way too much?\"\n\nMrs. E nodded grimly. \"Kodoc wanted to harness that power. So he locked pregnant women inside cages and lowered them into the Fog to give birth, hoping one child would be born with eyes of Fog.\" Grief deepened the furrows on her face. \"But they were born with fogsickness instead. Every one of them died, until . . .\"\n\n\"Until me.\" I touched the hair over my eye. \"So _Kodoc_ is the reason I'm like this?\"\n\nShe nodded again. \"When I was young, I worked with him. This is\u2014\"\n\n\"You _what_?\"\n\n\"W-with Kodoc?\" Bea stammered.\n\n\"No way,\" Swedish said. \"How could you?\"\n\nA wistful expression flickered on Mrs. E's face. \"We had big plans, big dreams. We were going to change the world, build shining cities on the mountaintops. Then I realized that his true goal was to turn the Fog into a weapon. I saw how cruel he was, how obsessed. He'd do anything to find the ancient machines.\" Mrs. E closed her eyes briefly. \"Including experimenting on innocent people. I started to spy on him, looking for a way to stop him. But he found out. . . .\"\n\nShe fell silent, and my anger and confusion gave way to curiosity. And respect. Mrs. E had been fighting this battle before I was born; she'd done things I couldn't even imagine.\n\n\"I had a friend named Tiara,\" she continued. \"She was younger than me, and I used to dress her in silly costumes and braid her hair. We were like sisters.\" For a moment, Mrs. E seemed lost in her memories. \"She met a man from the lower slopes. A poor man. A good man, Chess. They got married, and soon she was pregnant with you.\" Her dull eyes shone with unshed tears. \"When Kodoc realized I'd been spying on him, he went after your mother. He forced her into the Fog with a dozen other women. And I lost my head. . . .\"\n\nHer voice faded, and she swayed. In a flash, Swedish was standing behind her, his hands on her shoulders to steady her.\n\n\"My mother,\" I said, blinking back tears of my own. \"You went into the Fog after her.\"\n\n\"I waited until the middle of the night,\" Mrs. E replied in barely more than a whisper. \"I snuck past Kodoc's guards. And I climbed down the chains leading to the cages.\"\n\nMy throat tightened as I thought about my mother, trapped like an animal in a cage in the darkening Fog. She must've been so scared and alone.\n\n\"Most of the women were beyond help, but finally, _finally_ I heard a noise. I must've only been a foot away\u2014\" She smiled sadly. \"Or maybe you had really strong lungs. I heard you crying. You were so little that I managed to pull you between the bars. Your mother, though . . .\"\n\nShe shook her head, and then I knew. My mother had died giving birth to me in the Fog.\n\n\"When Kodoc raised the cages the next day,\" she continued, \"he must have thought you'd squirmed away.\"\n\n\"So you grabbed him,\" Bea said, biting her lower lip. \"And brought him to his dad.\"\n\nMrs. E smiled softly at me. \"You should've seen your father's face, Chess. He thought he'd lost both of you. He mourned your mother, but the way he took you in his arms . . . I knew from the first moment that he loved you.\"\n\nI wanted to say something, but my throat felt tight and thick. That's why Mrs. E was sick: because she'd tried to save my mother and me. That's why I was a freak: because Kodoc had lowered my mother into the white. And that's why I loved the Fog, why I moved faster and saw farther than anyone: because I'd been born there.\n\n\"So that's when you got sick,\" Swedish said.\n\nMrs. E exhaled slowly. \"By the time I carried Chess from the Fog, it was almost dawn. I'd spent too many hours in the white.\"\n\nFor a moment, nobody spoke. Then Swedish said, \"I don't get it. Why hasn't Kodoc been searching the slums for a kid with a Fog-eye?\"\n\n\"He thought all the test subjects died,\" Mrs. E told him. \"He thought his experiment failed. I burned his records, burned his entire estate to the ground. Then Chess's father and I disappeared into the junkyard. We kept our distance from each other, in case Kodoc caught one of us.\" She looked to me. \"I didn't even know that he'd died until months afterward.\"\n\nI nodded. I didn't blame her for that. I didn't blame her for anything, really. But my mind was still whirling. I hadn't known the truth about myself until that day. I didn't only _look_ like a freak, I _was_ one. The leader of the Rooftop had killed my mother and given me this eye so I'd lead him to these ancient machines? It was terrifying. But I also felt a wash of relief, like a darkness in my mind had brightened. Because I finally _knew_ why I was like this. I finally knew that my connection to the Fog was real, not a crazy delusion.\n\n\"And Kodoc never heard about my eye?\" I asked after a long moment.\n\n\"You know what the junkyard's like,\" Mrs. E said. \"There are a thousand new rumors every day. A cloudy eye would've sounded like any other disfigurement.\"\n\n\"If he'd heard about you,\" Hazel said, \"the bosses would've lined up every kid in the junkyard and checked our eyes.\"\n\n\"Indeed.\" Mrs. E started trembling. \"And now I think you're right. You must go to Port Oro.\"\n\n\"Yes,\" Hazel said. \"We'll find someone to cure you and\u2014\"\n\n\"For Chess!\" Mrs. E said, her voice suddenly high and frightened. \"Before Kodoc finds him!\"\n\n\"Kodoc won't find him,\" Hazel assured her. \"We'll dive for salvage and hire coyotes\u2014\"\n\nMrs. E slid sideways, almost toppling from the chair. Swedish steadied her, but she'd already slipped back into the sickness. She kept repeating \"before Kodoc finds him\" over and over as Swedish carried her to bed.\n\nAt first, I dismissed her fears about Kodoc finding me. After all, he hadn't found me in thirteen years. But a few weeks later, we caught wind of a rumor about a kid with a Fog-eye living on the Rooftop. We'd never heard anything like that before. Then a new rumor claimed that Lord Kodoc himself was searching for the Fog-eyed kid\u2014which terrified us.\n\nHow had the rumors started? Who was spreading them? We didn't know. We just spent more time than ever on the raft, searching the Fog for our ticket to Port Oro.\n\nMeanwhile, Mrs. E spent most days asleep. She'd wake for a few hours and talk loco, her clouded eyes wild. I hated that; the real Mrs. E was strong and fierce, not pathetic and weepy. But one time, she woke and was herself again, a skinny woman whose beak-like nose and sharp alertness reminded me of a hawk.\n\nShe eyed me. \"How long have you been sitting there?\"\n\n\"Not long,\" I said.\n\n\"Liar,\" she whispered with a smile.\n\n\"Well, not _that_ long.\"\n\nI would've sat there forever if I'd known she'd sound like this when she woke, instead of babbling nonsense like usual. The thought of her going mad was too much\u2014I couldn't handle Mrs. E acting like a stranger.\n\nShe must've seen something in my face, because she suddenly took my hand. \"It's not your fault, Chess.\"\n\nI shrugged like I didn't know what she was talking about. \"That you sleep so much?\"\n\n\"That I'm fogsick.\"\n\nI'd never told her that I blamed myself for her sickness, but somehow she knew. Still, I wasn't sure what to say. So I just sat there as the sound of a bootball game drifted through the wall, and the distant _thrip_ of a rivet gun echoed from Bea's workshop.\n\nAfter a while, I said, \"I barely remember my dad. I guess I was too little when he died. But I remember the way he sang me to sleep, and how he read from his scrapbook. And one other thing.\"\n\n\"What's that?\"\n\n\"A story he told me about a lady who risked her life trying to save me when I was a baby. Me and my mom.\"\n\nA faraway smile rose on her face. \"Oh, Chess, I still don't know how I found you in all that Fog.\"\n\nI took a deep breath. \"But you\u2014you're dying, Mrs. E. There's no way to help, at least not on the Rooftop.\"\n\n\"Shhh,\" she said, her eyes closing. \"Hush, Chess. . . .\"\n\n\"We're not even sure _Port Oro_ can heal you. The fogheads said so, but who knows if they're lying?\"\n\n\"On the Port,\" she said, her voice faint, \"many things are possible.\"\n\nShe fell asleep before saying more, and I watched her slow, steady breathing. Usually I hated when she slept, because it felt like she was already leaving us, already gone. But this time was different. This time I had hope.\n\n##\n\nTIME LIMPED PAST AS I clung to the rigging beneath the leaking balloon. I drifted in a painful haze until Hazel swung down beside me, tied the main lines together with leather straps, and said, \"You can let go now, Chess.\"\n\n\"Do I have to?\" I gasped as I unhooked my bracers and sagged into the rigging. \"I was just starting to enjoy myself.\"\n\nShe flashed an uneasy smile, then scanned the Fog with her spyglass.\n\nI rubbed my aching arms and followed her gaze. Between Bea's frantic efforts with the engine and my painful grappling with the rigging, the raft could now fly with only two balloons. But not for long\u2014we were still too low, and any strong gust of wind might shove us down into the white.\n\nFor hours, I waited for that deadly gust, yet our luck held. I wanted to tell them about the diamond, but Bea was busy with the engine, Swedish with the wheel, and Hazel scanning the Fog, plotting our route to the inch.\n\nI decided to wait until things calmed down a little. And finally, squinting toward the setting sun, I caught a glimpse of the Rooftop, a jagged mountain range with foothill \"islands\" breaking through the Fog all around. Green fields seemed to glow in the high meadows, and distant waterfalls caught the fading sunlight. Fancy mansions dotted the high peaks, where Mrs. E had lived as a little girl, among the parks and estates of the upper slopes, with lakes full of catfish and forests full of deer. She said that a busy city rose on the third-highest peak, packed with shops selling fresh-baked naan, smoked sausages, and unrotten fruit.\n\nWe didn't completely believe her stories, but Hazel dreamed of flying over the Rooftop for a closer look. She ached to see the elegant clothes, to hear the fancy music, and maybe even to feel solid ground under her feet for the first time in her life. But the roof-troopers didn't let slum-dwellers onto\u2014or even over\u2014the mountain. Patrolling airships would've shot us from the sky.\n\nBelow the highest peaks, ramshackle buildings of salvaged alumina and concrete crowded the lower slopes, along with smoldering smokestacks and ratty windmill-blimps. Even lower, the junkyard surrounded the mountain, floating fifty feet above the Fog on dilapidated platforms. Trash and shacks covered every foot, except for a few sparse patches where the junkyard bosses had \"ditched\" a neighborhood by overturning a platform and dumping everything into the deadly Fog below. That was the punishment for not paying rent.\n\nStill, as the raft spluttered closer to the slum, I felt myself smile. The junkyard was horrible, but it was still home.\n\n\"Sight for sore eyes?\" Hazel asked.\n\n\"Sight for sore _everything,_ \" I told her, rubbing my bruises.\n\nShe chewed her lower lip. \"Do you think this whole thing is a mistake? I mean, trying to sneak onto Port Oro? Nobody ever makes it that far.\"\n\n\"Nah,\" I said. \"Port Oro's going to be great. I hear apples grow on trees.\"\n\n\"And it rains _water_!\" she said.\n\nI laughed. \"I'm just glad we made it back.\"\n\n\"We haven't yet,\" Swedish grumbled from the wheel. \"We're going to crash and burn.\"\n\n\"Swede's right,\" Bea called from beside the condenser, a smudge of grease across her nose. \"We're not going to reach our dock, not even close.\"\n\n\"How about the nearest slipway?\" Hazel asked. \"Can we reach that?\"\n\nSlipways were makeshift ramps with mooring masts and space for undercarriage engines. Not as safe as our dock, and farther from home, but better than dying.\n\n\"She might hold together that long.\" Bea nibbled her lower lip. \"Barely.\"\n\n\"How barely?\" Hazel asked.\n\n\" _Barely_ barely,\" Bea said. \"With only two balloons, she's flying on dreams and dandelions.\"\n\nHazel set a course, and Swedish threaded the wallowing raft through high crests of Fog, toward the outer edge of the junkyard, a fringe of welded scaffolding, rusty chains, and plastic bags woven into sheets.\n\n\"Speaking of dreams,\" I said, patting my boot pocket.\n\nHazel glanced at me, a curious glint in her eyes. \"Yeah, what were you doing picking a fight over a flower?\"\n\n\"Not that it wasn't a very _nice_ flower,\" Bea called from under the raft.\n\n\"Probably poisonous,\" Swedish muttered, angling the raft toward the slipway.\n\nA clank sounded from below. \"It was not!\"\n\n\"Would you two hush?\" Hazel said. \"Chess is trying to tell us something.\"\n\n\"Probably that it's dangerous to talk to muties about smuggling stuff onto the Rooftop,\" Swedish grumbled, adjusting the rudder to catch a breeze. \"If the troopers capture them\u2014\"\n\n\"I found a diamond,\" I broke in.\n\nSilence fell, and I laughed at the hope and disbelief flickering on their faces.\n\nBea popped from the hatch and everyone spoke at once:\n\n\"A _what_?\" Hazel asked with a shocked laugh. \"Are you sure?\"\n\n\"A diamond?\" Swedish said. \"Probably fake. Probably _cursed_.\"\n\n\"Oooh,\" Bea said. \"Can I see? Is it pretty?\"\n\n\"Chess, if you found a real diamond\u2014\"\n\n\"Mrs. E says they sparkle like the stars.\"\n\n\"You know what _they_ do with diamonds?\" Swedish demanded. \"Why nobody's allowed to own one?\"\n\n\"What do they do?\" I asked, widening my eyes in fake fascination.\n\n\" _They_ are collecting diamonds to build a bomb\u2014a mountain buster big enough to blow the whole Rooftop into gravel.\"\n\n\"Right,\" I scoffed. \"Because that's what the roof-troopers want more than anything. Gravel.\"\n\n\"A diamond!\" Hazel crowed. \"You know what this means? We're halfway to Port Oro already.\"\n\n\"Let's see!\" Bea said. \"I want to see it! A real diam\u2014\"\n\nThe raft gave a violent shudder, and the engine coughed and spat.\n\n\"We're not home yet!\" Hazel yelled. \"Bea, keep her in the air. Chess, take the crow's nest.\"\n\nAs I climbed the rigging, she shouted, \"And don't drop that rock!\"\n\nHazel leaned over the prow of the raft like a figurehead, ready to call a warning if an outcropping of Fog threatened the engine. As we flew closer, the stink of the junkyard rose in the air and mixed with the sharp tang of the overheated engine.\n\nA crowd watched us wheeze toward the slipway's mooring mast, a tottering tower of rusty automobile engines and bathroom fixtures. A stringy-haired vendor shouted her prices for rat kabobs, as a frenzy of trading started around her. Almost nobody used coins in the slums. Real money was tucked away for special occasions. Instead, we bartered lengths of wire for plasteel buttons, a bag of potting soil for a bone needle, a skewer of crab apples for some grilled squirrel.\n\nAt first, I didn't understand why they were trading. Then I almost laughed. \"They're _gambling,_ \" I called to the others. \"They're betting about us reaching the slipway.\"\n\n\"Wish _I_ could throw down a little something,\" Swedish said.\n\n\"Me too!\" Bea chirped. \"We're going to get there for sure!\"\n\nHazel grunted. \"That's not how Swedish would bet, Bea.\"\n\n\"He'd bet _against_ us?\" Bea's voice sharpened. \"Swedish, you're horrible!\"\n\n\"Listen to the engine,\" he said, his fingers dancing over the steam organ keyboard. \"The gears are snapping. There's no way we're staying in the air.\"\n\nA series of loud pops shook the raft. Gray smoke puffed from an intake valve.\n\n\"We _will_ reach that slipway,\" Bea said, her green eyes narrowing as she spliced cables together.\n\nA hundred feet from the slipway, sparks started shooting from the propeller. Hazel's grip tightened on the prow and she leaned forward. \"We can't sink now,\" she said. \"Think of Port Oro!\"\n\nSeventy feet away, the forward balloon began flapping and hissing. \"C'mon, c'mon,\" Hazel muttered. \"Stay in the air.\"\n\nAt fifty feet, the deck was shaking like a rattlesnake's tail, and Hazel yelled, \"Almost there, Bea! Almost there!\"\n\nForty feet, and I grabbed a mooring strap and raced for the prow. I needed to hitch us to the mooring mast before the engine died.\n\nHazel shouted commands and Swedish muttered dire predictions. Bea clamped a valve shut with her foot and crooned to the engine, which answered with a screech. Catcalls sounded from the crowd, the odds of our survival falling fast.\n\nOnly I remained silent, twirling the mooring strap overhead. Not close enough. Not yet. . . .\n\n\"Bea!\" Hazel yelled, running toward her. \"Abandon ship, abandon ship!\"\n\n\"No! She needs me!\"\n\nHazel grabbed Bea's wrist and dragged her forward. \"We have to jump! Swedish, _run_! Throw the strap, Chess.\"\n\n\"Not yet,\" I said.\n\nThe rear balloon exploded into scraps, and the deck tilted crazily. \"Now!\" Hazel shouted. \"Now, now!\"\n\n\"Not yet\u2014\"\n\n_\"Throw it now!\"_ she screamed, and I did.\n\n##\n\nTHE STRAP SPUN IN the air, a loop of nylon and wire.\n\nTime slowed, freezing the faces of the shouting crowd, freezing the snarl on Swedish's lips and Bea's horrified expression as Hazel shoved her toward the prow.\n\nThe mooring mast\u2014the target of my throw\u2014glowed in the light of the setting sun. The loop twirled closer and closer . . . and fell three feet short.\n\nBut one thing a tetherboy knew was _tethers,_ and a strap wasn't all that different. Three feet short? No way.\n\nI whipcracked the strap, and the loop flicked higher. It still missed the mooring hook, but it snagged a fender that jutted just below it. The fender was crimped and rusty, definitely not strong enough to keep the raft in the air. Still, it might hold the weight of the _crew,_ so I screamed\u2014\"Grab the strap! Grab the strap!\"\u2014as I pulled the hacksaw from my leg sheath.\n\nTen feet away from landing safely on the slipway, the raft engine died.\n\nThe world moved in slow motion. The raft tumbled downward. The rigging snapped, the pistons screamed. The deck angled crazily as I furiously slashed and hacked at the strap.\n\nThe raft fell five feet. My breath caught and my boots scrambled for a foothold.\n\nThe raft fell ten feet, and fear built in my lungs. In seconds, the weight of the raft would tear that rusty fender from the mooring mast and we'd all die. Swedish, Hazel, Bea\u2014\n\n_No_. With one final, terrified slash, I sliced the strap from the raft's deck.\n\nThrough fear-widened eyes, I caught a glimpse of Hazel and the others above me clinging to the mooring line\u2014now attached only to the fender, not the raft. The loop tightened around the fender, and they slammed into the mast, bruised and battered but dangling safely below the slipway.\n\nFrantic and trembling, I hitched the line to my tether, and an instant later it jerked at my harness, stopping my fall. I swung like a pendulum and slammed into the slipway.\n\nOuch.\n\nBelow me, the raft dropped into the white and disappeared. No crash sounded, no explosion flashed. The Fog simply swallowed the raft whole.\n\nI dangled there, limp and aching and defeated. We'd come so far. We'd dived in uncharted territory and found a diamond, we'd survived mutineers and reached the junkyard\u2014but we'd lost our raft to the Fog, and that ruined everything.\n\nAfter a minute, I felt myself being dragged higher, until Swedish pulled me onto the slipway and gave me a fierce back-pounding hug, ignoring the groans of the people who'd wagered against us.\n\n\"Too small,\" Swedish said gruffly. \"I should throw you back.\"\n\nI brushed my hair over my freak-eye. \"You sound like that mutineer captain.\"\n\n\"My poor raft,\" Bea sniffled.\n\n\"We got lucky, Bea!\" Swedish said. \"We crashed and didn't die. If Chess hadn't tossed that line, we'd be lumps of meat right now.\"\n\n\" _We_ didn't crash,\" Bea said, removing her cap respectfully. \" _She_ crashed. The raft gave her life for ours.\"\n\n\"And I lost my lucky bootball,\" Swedish grumbled.\n\nHe shouldered through the crowd, leading us toward the slipway owner, a skinny man eating his dinner in a darkened doorway.\n\n\"Who's going to pay for the damage to my slipway?\" the slipway owner barked at Swedish, digging in his bowl with chopsticks. \"You bottom-feeders dented my mast\u2014and you owe me docking fees. Either you pay, or I take the pretty one as a chambermaid.\"\n\nSwedish cracked his knuckles. \"Just _try_.\"\n\nI took a breath and stepped beside him. I didn't like fighting, but I'd lived my whole life in the junkyard\u2014I'd fight if I had to. Then a rough-looking woman in a stained sari appeared behind the owner, cradling a makeshift crossbow. We couldn't fight _that_.\n\n\"We'll pay,\" Hazel announced, stepping forward. \"For the damage, the fees, _and_ for salvage rights to our raft.\"\n\nThe slipway owner pointed his chopsticks at her. \"Your raft is smashed into a thousand pieces.\"\n\n\"That's not your problem,\" she told him. \"We'll dive from your slipway, salvage the wreckage, and rebuild.\"\n\nBea pulled me closer. \"I can't rebuild the raft, not after _that_ crash.\"\n\nI shushed her. I didn't know what Hazel was doing, but I knew she was doing something. She was _always_ doing something.\n\n\"You can't give this guy money,\" Swedish murmured. \"We barely have enough for food.\"\n\n\"We don't have a choice,\" Hazel muttered back. \"We crashed the raft, and we don't _own_ the raft.\"\n\nMy stomach soured. That was the problem. We'd lost the junkyard bosses' raft, and they'd make us pay for a new one\u2014with the diamond. Our only way of getting to Port Oro.\n\n\"There won't be much worth salvaging,\" Bea said in a small voice. \"Not enough to rebuild\u2014\"\n\n\"Shut your mouth, Bea!\" Hazel snarled between clenched teeth. \"Do as you're told.\"\n\nBea's lips trembled. She pulled a wire from her pocket and started making a twisty\u2014to keep herself from crying.\n\nHazel and the slipway owner bickered for a while; then she turned to me. \"Give him the toilet paper.\"\n\nI took the folded stack of \"cash\" from my pocket. \"The thousands are softer than the hundreds,\" I told him.\n\nHe didn't seem to care. Maybe he knew it wasn't true. He kept telling Hazel that he needed more coin, and they haggled and argued until we finally set off for home along a trash-littered road skirting an open sewer.\n\nSwedish scowled at Hazel. \"You made Bea cry.\"\n\n\"I'll explain when we get home,\" she told him. \"I wouldn't talk like that without a good reason.\"\n\nBea peered at her. \"You\u2014you didn't mean it?\"\n\n\"You know better than that,\" Hazel said. I could tell her feelings were a little hurt that they'd thought she'd really lost her temper. \"The both of you.\"\n\nSwedish shot Hazel an apologetic look, then kicked at an imaginary bootball.\n\nBea said, \"Sorry, Cap'n.\"\n\n\"Have some faith,\" I said, tapping her leather cap.\n\nShe wrinkled her nose at me and slipped the twisty she'd just made into her tool belt.\n\n\"What'd you make?\" I asked.\n\n\"Nothing,\" she said.\n\n\"C'mon!\" I said. \"Let's see.\"\n\nShe scuffed her feet. \"It's nothing.\"\n\n\"Hey, I told you about the goose.\"\n\nShe glared but handed it over. I angled the twisty to catch the moonlight and saw that she'd wrapped the wire into the shape of a sloppy face, with a big, open mouth and snakes instead of hair.\n\n\"What is it?\" I asked.\n\nSwedish peered closer. \"A startled porcupine.\"\n\n\"Or a really angry mop,\" I said.\n\n\"It's Hazel,\" Bea said. \"Yelling at me.\"\n\nHazel glanced at the twisty. \"My hair doesn't do that.\"\n\n\"I can't believe we lost the raft!\" Bea blurted. \"Well, at least we still have the\u2014\"\n\n_\"Crew,\"_ Hazel interrupted, before Bea could say _diamond_.\n\n\"What?\" Bea asked.\n\n\"The crew,\" Hazel said, nodding toward the hostile faces watching us from shadowy doorways. \"At least we have each other.\"\n\n\"Oh!\" Bea flushed. \"Right.\"\n\nAfter that, we kept quiet as we headed through darkening passages and garbage-filled clearings. We didn't normally dock in the slipways, so none of us knew this neighborhood. Strange noises rose in the gloom\u2014the grinding of unseen clockwork, the sickly wheeze of the fans keeping the slum aloft. Rats and roaches scuttled in the corners, and a baby cried out from a shadowy archway.\n\nThen a hoarse voice shouted, \"The world didn't end with a bang! It began with a slow rise of mist! The Fog is the healing breath of the Earth!\"\n\n\"Fogheads,\" Swedish muttered. \"They're all whackadoo.\"\n\n\"Let's go around him,\" Hazel said.\n\nI pointed toward a detour. \"That way.\"\n\nSwedish knew that Mrs. E called fogheads the \"Subassembly,\" and that they'd been scientists working for the Five Families before Kodoc betrayed them and stole their research. But he insisted that they were complete lug nuts, always skulking around and muttering about the Fog.\n\nBefore we turned away, the foghead stepped from the shadows, a small man wearing a gray robe. \"A thousand ticktocks will rise from the lowest Fog,\" he told us, \"to cleanse and burn and purify!\"\n\n\"Let's go,\" Hazel said, and headed off.\n\n\"They come not to bury us, but to warn us!\" the man yelled, edging closer.\n\nBea and Swedish followed Hazel, but when I started after them, the foghead grabbed my wrist. My pulse rocketed and I raised my boot to stomp on his foot as he searched my face.\n\n\"Wait!\" he blurted, releasing me. \"Are you\u2014are you him?\"\n\nHunching my shoulders, I started to follow the others.\n\n\"The boy with Fog in his eye?\"\n\nMy stomach dropped, and the night suddenly felt cold and unfriendly.\n\n\"I\u2014I don't know what you're talking about,\" I said, ducking my head.\n\nHis gaze softened a little with recognition. \"Your name is Chess, isn't it?\"\n\n\"What do you want?\"\n\n\"You're more important than you know,\" he said, sidling closer. \"Your eye isn't a sign of shame, but hope.\"\n\nSharp needles of fear pricked my skin, but I didn't move. A stranger knew my secret. How did he know? It was impossible. And what did he mean, a sign of hope?\n\nI swallowed. \"Wh-what are you talking about?\"\n\n\"You're marked by the Fog, Chess. You belong to the Fog, and maybe the Fog belongs to you. But you're in danger. Grave danger. And we are to blame.\"\n\n\"You are? Fogheads?\"\n\nHe pursed his lips. \"The Subassembly. You came to us for help, months ago\u2014you and the girl.\"\n\nMy breath caught in my throat. \"I\u2014what if I did?\"\n\n\"One of us saw your eye.\" The man bowed his head. \"And he mentioned it\u2014in public. The rumors are our fault. . . .\"\n\n\"Hey!\" Swedish yelled, stepping from the alley. \"Get away from him!\"\n\n\"Kodoc knows,\" the foghead whispered to me. \"Kodoc knows you're alive.\"\n\n##\n\nTHE DARKNESS GATHERED AROUND me, and nausea rose in my stomach. _Kodoc_. The name sounded like a death sentence. I steadied myself on a dingy wall, dizzy with fear. One of the fogheads saw my eye? What else did they know about me? What other secrets had they spread\u2014\n\n\"You!\" Swedish roared, stepping closer. \"Foghead! Back off!\"\n\nFor a moment, the man stood there staring at me. Then Hazel joined Swedish, and he slunk away into the shadows.\n\n\"Are you okay?\" Hazel asked me.\n\nI swallowed. \"Y-yeah.\"\n\n\"What was that about?\"\n\n\"I don't know. That guy, he . . .\" I shrugged, still jittery and afraid. \"I don't know.\"\n\n\"What did he say?\"\n\n\"Uh, he told me . . .\" The truth caught in my throat. \"He mostly rambled.\"\n\nI ducked my head and slouched toward the alley. Why had I lied to them? If Kodoc really knew I'd survived the cage, I needed to tell the crew. But something strangled the words before I could speak them. Fear, I guess. Like if I said Kodoc's name out loud, I'd summon him. Like if he knew I was alive, he'd come and claim me.\n\nHis creation. His creature. His freak.\n\n\"Chess!\" Hazel touched my arm. \"What's wrong?\"\n\n\"Nothing. He just . . . I guess he saw my freak-eye.\"\n\nWhen I was younger, if anyone spotted my eye, I'd claim that the white shimmers were floating scars. Sometimes they'd still give me a thrashing, just for being ugly. It stopped happening when I'd gotten older and more careful, but the memories still hurt.\n\n\"Oh,\" Hazel said.\n\n\"I like your eye,\" Bea said loyally. \"It's cool, like clouds reflected in a puddle.\"\n\n\"So now you're saying I'm a muddy puddle,\" I told her, trying to sound normal as we headed away.\n\n\"A _cool_ muddy puddle,\" she insisted.\n\n\"Yeah,\" I muttered. \"That's like an _awesome_ toenail clipping.\"\n\n\"Fogheads.\" Swedish spat. \"They give me the creeps. If they like the Fog so much, they ought to live there.\"\n\n\"They're not so bad,\" Hazel said. \"You're just mad we lost the raft.\"\n\n\"There's no way we're getting to Port Oro now,\" he grumbled. \" _They_ saw to that. We have to use the . . . rock to repay the bosses for the raft.\"\n\nHazel shushed him. \"Let's talk about that when we get home.\"\n\nThe slum shifted underfoot as a breeze swirled through the Fog, and the miles-wide patchwork of floats and balloons and hoverfans heaved and creaked and popped.\n\nI didn't mention the foghead again, even though I knew I had to tell them what he'd said. _Kodoc knows you're alive_. Those weren't just rumors we'd been hearing, they were rumors about _me_. I kept trying to say the words aloud, but I couldn't. I was too scared. Sometimes I thought that when Kodoc had made me a freak, he'd also made me a coward.\n\nSo I kept my head down, feeling ashamed as we picked our way through a thicket of chains tethering balloons to the slum platforms. They clattered so loudly that we had no warning. We had just turned the corner when we found ourselves facing down a street gang playing bootball in the dusky light.\n\nA boy missing his front teeth grabbed a sharp-edged brick and snarled, \"What're you doing here? This is _our_ patch.\"\n\n\"Just passing through,\" Hazel said, her voice almost steady.\n\nA scar-faced girl touched the pipe strapped to her leg. Gangs like this battled over food and turf and money. They fought over insults and respect, and sometimes just for fun. That was life in the slum\u2014life and death.\n\nI felt the others tense behind me. We were salvage rats, not street fighters, but I didn't see any way out of this without bloodshed. For once it didn't scare me\u2014 _anything_ was better than worrying about Kodoc.\n\nEven this.\n\n##\n\nSWEDISH STEPPED IN FRONT of me and glared at the toothless boy. The gang tensed and reached for their weapons. When Swedish scowled, he looked pretty mean, and he was bigger than the biggest of the gang kids.\n\nThen Swedish toed a bootball from the ground into the air, catching it on his foot.\n\nHe bobbled the ball from his foot to his knee, caught it with his ankle, then kicked it above his head. He hunched his shoulders, and the bootball landed on the back of his neck, balanced there for a second, then plopped to the ground.\n\nThe gang kids laughed, and relief bubbled through my chest.\n\nSwedish popped the ball straight upward, then smacked it with his forehead toward the goal.\n\nThe goalie swatted it away, and his team cheered.\n\n\"What do you play?\" the scar-faced girl asked.\n\n\"Goalie,\" Swedish said. \"Or fixo.\"\n\n\"You're kind of big for it.\"\n\n\"He used to be smaller,\" I told her.\n\nHazel gave a quick smile and nudged Swedish, \"Well, we'll get out of your way,\" she told the gang kids, and started across the clearing.\n\n\"If you're ever looking for a game,\" the first guy called after Swedish, \"you know where to find us.\"\n\nHalfway down the next alley, Swedish started grinning like a goon. \"Did you see that?\" he said. \"If everyone played bootball, there'd be no problems. No hunger, no fighting. Bootball\u2014I'm telling you\u2014it's the solution.\"\n\n\"To what?\" Bea asked.\n\n\"To everything!\"\n\n\"In the old days,\" I said, trying to take my mind off the foghead, \"they played a game called 'golf.' You knocked a ball the size of an egg into a little hole in the ground, using a club.\"\n\nSwedish cocked his head. \"You couldn't use your feet?\"\n\n\"No, just the club.\"\n\n\"What'd they call it?\" Bea asked.\n\n\"Golf,\" I repeated.\n\n\"Not that,\" she said. \"I mean, what'd they call the club?\"\n\n\"Oh. Just a club, I think.\"\n\nShe giggled. \"They did _not_ call it a club! Might as well call it a cudgel or a beating stick.\"\n\n\"Well, that's what it says in the scrapbook.\"\n\n\"Huh,\" Swedish said, scratching his cheek. \"So each team had a hole? It sounds too easy to guard. You just put your foot over the hole.\"\n\n\"Until the other team starts beating you with their clubs,\" Hazel said.\n\n\"Oh!\" Swedish nodded, satisfied. \"Yeah, that makes sense.\"\n\nBea made a face. \"Golfball sounds pretty violent.\"\n\n\"And it's not just the other team you had to watch out for,\" I told her. \"The field also had sand traps and hazards and windmills.\"\n\n\"Windmills?\" Hazel asked, quirking an eyebrow.\n\n\"I'm not sure where those came in,\" I admitted.\n\n\"Maybe they milled grain while they played,\" she suggested. \"Makes more sense than whacking an egg with a cudgel.\"\n\nWe started walking faster as we neared our neighborhood, following a trash-ridden trail around the Spew\u2014the river of sludge that seeped from mounds of garbage, winding its way around the slum before cascading into the Fog.\n\nFinally Bea sang out, \"We're home!\"\n\nShe disabled the booby trap we always set when we left Mrs. E alone, then flung open the narrow door. The entryway was cluttered with splintery crate shelves and baskets of plastic bags from before the Fog. Whenever we weren't on salvage runs, we wove the bags into sheets and sold them for handfuls of rice flour or lambs' feet.\n\n\"I'll check if Mrs. E's awake,\" Bea said.\n\nSwedish grunted. \"Not likely.\"\n\n\"She was fine when we left!\"\n\n\"That was yesterday,\" Swedish said. \"And she wasn't _fine,_ she was _awake_. There's a difference.\"\n\nBea glared at him. \"Well, I'll check if she's\u2014\"\n\n\"Hold on a second,\" Hazel told her.\n\n\"What?\"\n\n\"First we need to talk.\" Hazel rubbed her face. \"We lost the raft. That's seriously bad news. We can't\u2014\"\n\nMrs. E's voice floated from the main room. \"Children? Is that you?\"\n\nBea squealed and shot into the shack, with Swedish close on her heels. Hazel and I exchanged a hopeful look. Mrs. E was awake and didn't even sound confused!\n\nThe main room was slightly higher than the cramped entryway, with a rickety table, a woodstove, and a catch barrel for rain. Chopsticks and plastic bowls cluttered the cabinet, along with what Mrs. E called a \"milk jug.\" She insisted that people used to drink _cow's_ milk, which always made Bea giggle.\n\nEveryone knew that milk came from sheep and goats and camels.\n\nOnce I saw Mrs. E sitting at the table, I forgot all about the foghead. She almost never left her bedroom these days\u2014she almost never left her _bed_ \u2014so I felt myself smile at the sight of her. The scent of the four slices of honey bread on the table didn't hurt, either.\n\n\"Mrs. E!\" Bea hugged her fiercely. \"We've got so much to tell you! The muties boarded us and they shot the cargo tether and\u2014wait! First Chess went diving, and you'll never guess what he found!\"\n\n\"Mutineers?\" Mrs. E's clouded eyes narrowed. \"They boarded the raft?\"\n\n\"They wanted to kidnap me,\" Bea said, \"but Hazel stopped them, and then a balloon popped, but we kept the raft afloat until\u2014\"\n\n\"Slow down, sweetie.\" Mrs. E smiled, her voice gentle. \"Have a bite of honey bread. Swedish, where are you? I can't see you.\"\n\n\"I'm here.\" He took one of her frail hands. \"How did you afford the bread? You know _they_ 're watching us\u2014\"\n\n\"Stop fretting,\" she told him. \"A friend brought it.\"\n\n\"Who? You don't have friends. You're afraid friends might . . .\" Swedish trailed off, glancing at me. \"Y'know.\"\n\n\"Expose our secrets,\" Hazel finished.\n\n\"He's a very old friend,\" Mrs. E said with a strange note in her voice. \"He just found me again, after many years.\"\n\nHazel twisted a braid around her finger. \"Is something wrong? What is it?\"\n\n\"Stop _thinking_ so much,\" Mrs. E scolded Hazel. \"I'll tell you everything after we eat. Now where's Chess? Lurking in the background?\"\n\n\"I don't lurk,\" I told her, stepping forward. \"I skulk.\"\n\n\"Then we're all here,\" she said. \"Now sit down and eat.\"\n\nSwedish grabbed two slices. He gave one to Bea as I started wolfing down a third. The sweetness melted on my tongue and warmth spread in my stomach, but Hazel just toyed with her slice, eyeing Mrs. E.\n\n\"Did your friend tell you something?\" she asked, taking a bite. \"What happened?\"\n\n\"He brought news.\"\n\n\"What kind of news?\"\n\nMrs. E took a shaky breath. \"Terrible news.\"\n\nThe sweet bread in my mouth turned to mud. \"Is it . . . Kodoc?\"\n\nMrs. E nodded. \"Kodoc heard that one of his 'test subjects' survived the cages as an infant\u2014he's on the hunt for Chess.\"\n\n\"How?\" Swedish demanded while Hazel gasped, \"No!\"\n\n\"He's suspected for weeks,\" Mrs. E said, swaying in her seat. \"Because of the rumors. He's been scouring the mountain, checking every child between the ages of ten and fifteen. And once he's done there . . .\"\n\n\"After he finishes the Rooftop,\" Hazel said. \"He'll start searching in the slum.\"\n\n\"He'll find Chess,\" Mrs. E whispered, her face paling. \"You need to hide, you need to run. . . .\"\n\nHer voice faded, her head dipped, and she slumped sideways, falling toward the floor.\n\n##\n\nSWEDISH CAUGHT MRS. E as she fainted, and the rest of us jumped from our chairs.\n\n\"C'mon,\" Hazel told Swedish. \"Bring her to bed.\"\n\nShe followed him through the traffic-sign door while I picked at a crack on the tabletop. I thought about Mrs. E climbing into the Fog to save me, about my mother dying, my father being handed a baby with one freakish eye. And about being Kodoc's creation: not just _born_ a freak, but _made_ a freak.\n\nNow he was coming for me.\n\n\"At least we're already planning to leave,\" Bea said. \"I mean, Kodoc knows about you, but that doesn't change anything.\"\n\n\"Except that there are a thousand roof-troopers hunting for us now.\"\n\n\"Other than that,\" she agreed. \"But we were trying to get to Port Oro anyway.\"\n\n\"Which would be easy if we could sell the ring. But we owe the bosses for the raft, and they won't let us leave without paying. Our only way to pay is\u2014\"\n\n\"The diamond!\" she blurted, brightening. \"Show me, show me!\"\n\nI almost smiled at her enthusiasm. But when I reached to unsnap my boot pocket, Swedish and Hazel returned to the main room, looking grim.\n\n\"Is she okay?\" Bea asked them, her brightness fading.\n\nHazel fiddled with her braid. \"She's not great.\"\n\n\"Not _great_?\" Swedish rubbed his face. \"More like 'terrible.'\"\n\n\"Yeah, well . . . at least she's sleeping comfortably.\"\n\n\"For now,\" Swedish said.\n\nHazel nodded. \"Let's talk in the workshop, so we don't bother her.\"\n\nWe went into the front room, where Swedish opened a hatch in the floor. Inside, a rope ladder dangled down a shaft that led to the shadowy underside of the slum, beneath the platforms where fans roared and sludge dripped, fifty feet above the Fog.\n\nThe workshop was bolted under the platform, a square room with telefoam poles for a frame, tightly woven cables for walls, and scraps of alumina drywall for a floor. It was where Bea tinkered, designing and building the thoppers that Swedish flew in drag races.\n\nWhen she fiddled with a knob, a soft glow shone over dinged tools, scratched dials, and a battered foggium compressor. A workbench sat against one wall, with tidy racks of bolts, wire, and old lenses, and a large rectangular hole opened in the middle of the floor. The thopper\u2014about the size of an overgrown camel, with three wings on each side and a wide fantail\u2014dangled below, hanging from sturdy chains. There was nothing between the thopper and the Fog but empty air.\n\n\"We have lots to talk about\u2014\" Hazel started.\n\n\"Wait,\" I interrupted. \"I need to say something.\"\n\n\"Stop talking and show us the diamond!\" Bea said. \"Let's see that purple rock!\"\n\n\"Yeah,\" Swedish said with a tired smile. \"We've been waiting for hours.\"\n\nSo I leaned against the workbench and pulled the box from my boot pocket. I waited a moment\u2014enjoying their excitement\u2014then lifted the lid. The diamond caught the dim light in the workshop and cast rainbow sparkles onto the wall.\n\nBea gasped, her big green eyes shining. \"Is it really real?\"\n\n\"Definitely fake,\" Swedish said in a soft, awed voice.\n\nHazel carefully lifted the ring from the box. Silence fell. That diamond was worth more than everything we owned, more than everything we'd _ever_ owned.\n\n\"Give me your hand,\" she told Bea.\n\n\"Sure,\" Bea said. \"Why?\"\n\nHazel raised the ring toward her finger. \"I want to see how it looks.\"\n\n\"Oh, no!\" Bea squeaked, and stuck her hand behind her back. \" _You_ should wear it, not me!\"\n\n\"Go on, Bea,\" I said.\n\nI knew that for Hazel, the diamond was like a window into the world of the upper slopes, with silk dresses and rolling parks. And she didn't just dream of that life for herself, she dreamed of it for all of us\u2014especially Bea.\n\nAfter Hazel slipped the ring on her finger, Bea wrinkled her nose. \"It looks dumb on my dirty hand.\"\n\n\"Nah,\" Swedish said. \"Nothing goes better with a diamond than an honest smear of grease. Well, except roast goose. _Everything_ goes better with roast goose.\"\n\nI threw a rivet at him, then Bea climbed onto the workbench and locked the ring in a hidden compartment in the ceiling. \"Why'd you promise that slipway guy money?\" she asked Hazel, hopping down. \"And why'd you tell me to shut up?\"\n\n\"Hold on.\" I looked at my scuffed boots. \"Let me go first.\"\n\n\"You're the Prince of Diamonds, Chess,\" Hazel said. \"You can say anything.\"\n\n\"Remember that foghead who grabbed me?\"\n\n\"What did he say?\" Bea asked.\n\nI swallowed the lump in my throat. \"Uh, he kind of told me why Kodoc knows I'm alive.\" I explained about a young foghead blurting the truth in public. \"And that's how the rumors started.\"\n\n\"But you didn't say anything,\" Bea said, sounding confused. \"You didn't tell us.\"\n\n\"Chess!\" Hazel snapped. \"He _recognized_ you? What if he'd told the roof-troopers?\"\n\n\"I\u2014I didn't think\u2014\"\n\n\"What if they followed us home to Mrs. E? I know how you feel about your eye, but\u2014\"\n\n\"You don't know!\" I snapped back, my face flaming. \"You don't.\"\n\nHer jaw clenched. \"Having Fog in your eye doesn't mean you can lie to us.\"\n\n\"It means I spend every minute lying. That's all I ever do\u2014I pretend I'm normal. My whole life is one big lie.\"\n\n\"Not to _us_!\"\n\nI glared at her. \"I've got a billion machines in my brain, and . . . and Kodoc _customized_ me, like Bea fixing an engine. No, like Bea building a weapon. You don't know what it's like, Hazel. He made me. He created me.\"\n\n\"So did your mom, and your dad, and Mrs. E.\" Hazel returned my glare for a second, then sort of smiled. \"And all of us.\"\n\nI ducked my head. \"I guess.\"\n\n\"We're crew,\" Hazel said. \"You don't need to keep secrets from us.\"\n\nMy anger faded. \"Sorry. I was . . . scared.\"\n\n\"Course you were,\" Swedish grunted. \"I would've done the same.\"\n\nI blinked at him. \"You would?\"\n\n\"Me too,\" Bea said. \" _Everyone_ would be scared of that.\" She peered at Hazel. \"Well, almost everyone.\"\n\nIn a girly voice, Swedish said, \"My name is Hazel, and I'm _soooo_ brave.\"\n\n\"I eat roof-troopers for breakfast!\" Bea chimed in as she did a gawky version of one of Hazel's ballerina spins. \"And I dance with mutineers!\"\n\n\"Oh, shut your valves,\" Hazel grumbled. \" _Anyway_. I told you to keep quiet at the slipway, Bea, because you started saying that we can't salvage the raft.\"\n\n\"We can't! My poor raft is in a thousand pieces!\"\n\n\"And if the bosses know that, they'll make us pay,\" Hazel said. \"Or they'll ditch us into the Fog as an example.\"\n\nBea's forehead wrinkled. \"So we have to give them the diamond?\"\n\n\"But they _don't_ know we can't fix the raft,\" I told Bea as I realized. \"That's what Hazel was doing\u2014buying us time.\"\n\n\"Time for what?\" Swedish asked.\n\n\"For something sneaky,\" I said.\n\nHazel tucked a few braids behind her ear. \"Forget the raft and forget the bosses. We're giving the diamond straight to a coyote, as payment for smuggling us to the Port.\"\n\nMy heart skipped a beat. \"And we're not paying the bosses for crashing their raft?\"\n\n\"That's right,\" she said.\n\nSwedish whistled in awe. \"You want to cheat the bosses?\"\n\n\"Whoa,\" I said a little shakily. Nobody messed with the bosses\u2014not if they wanted to live.\n\n\"It's the only way.\" Hazel exhaled. \"Kodoc knows about Chess. We're running out of time.\"\n\n\"Are we . . .\" Bea bit her lower lip. \"Are we ready?\"\n\n\"No,\" Hazel told her. \"But this is it, honeybee. Either we cheat the bosses or we're finished.\"\n\nThe clockwork engines in the workshop ticked and tocked until I broke the silence, \"Well, I'm not standing around picking my nose until Mrs. E dies. Let's do it.\"\n\n\"I'm in,\" Bea said, her eyes shining. \"Port Oro, here we come!\"\n\n\"We're going to regret this,\" Swedish grumbled. \"When the bosses find out, they'll ditch us into the Fog. This is a stupid move. A huge mistake.\"\n\nNobody said anything.\n\n\"But,\" he continued, \"you're my crew. Whatever huge mistakes we make, we make together.\"\n\n\"Always,\" Hazel said, her eyes clearer than any diamond. \"Okay, first we'll talk to some coyotes I know, then we'll\u2014\"\n\nA bell chimed faintly from above.\n\n\"That's Mrs. E!\" Bea squealed, and scrambled up the rope ladder.\n\nHazel glanced at the ceiling, smiling. \"I guess what we'll do,\" she said, \"is plan our escape later.\"\n\n##\n\nI WAS THE LIGHTEST sleeper, so I bedded down near the front door, ready to raise the alarm if anything woke me. Swedish slept in the main room\u2014the only place he could stretch in the morning without knocking holes in the ceiling. Hazel and Bea shared a lumpy acrylicloth mattress in a nook behind a beaded curtain. And Mrs. E had the only real room, a cramped space behind a door made of rusted traffic signs, the floor heaped with mildewed notebooks and smudged charts and my father's scrapbook on a special shelf against the wall.\n\nI followed Bea into the room and found Mrs. E curled in bed, toying with the bell she'd rung.\n\nWhen she'd first found me in the alleys, she'd seemed like a force of nature, striding along with her chin high and her spine straight. But now she was bent and wizened, with clouded eyes and clawed fingers. Still, she looked better than usual. At least she was awake.\n\n\"Hey, Mrs. E!\" Bea chirped. \"Do you need something?\"\n\n\"How _very_ kind of you to ask,\" Mrs. E told Bea in an almost girlish voice. \"I want ices. Raspberry ices.\"\n\n\"Ices?\" Bea repeated, the happiness draining from her face. \"You mean, um\u2014\"\n\nMrs. E gasped. \"You're not my governess!\"\n\nI wanted to cry at the confusion on Mrs. E's face. She'd recognized us twenty minutes earlier, but now she thought she was a child again, living in the upper slopes, surrounded by servants and sweets.\n\nBea hung her head. \"I'll get her, ma'am.\"\n\n\"Why did she call me _ma'am_?\" Mrs. E asked me, her fear fading into a pout. \"You're the gardener's boy, aren't you? Would you like a glass of lemonade?\"\n\nI backed toward the door. I didn't know what to say, and I couldn't stand seeing her like this, so feeble and confused.\n\n\"My grandpapa says he once drank _real_ lemonade as a child. He says _his_ grandfather was alive when the Fog started rising. Just a fine dew on the ground, in the lowest places. Then it started rising an inch every week, then a foot. . . .\" She widened her cloudy eyes at Hazel when she entered. \"Where _is_ that coachman? You promised me a ride in the camel cart.\"\n\n\"Not in this snow,\" Hazel said, her voice calm, even though I knew how much she hated playing along with the sickness. \"I'll read you a story, instead.\"\n\n\"Read me one about the Fog!\" Mrs. E grabbed a weathered notebook from the floor. \"My favorite! _Gasiform Nanites and White Blood Cells: An Analogy_.\"\n\nHazel grimaced. She read better than Swedish or me\u2014Bea was still learning\u2014but Mrs. E's technical notebooks were beyond her. So instead of reading, she sat on the bed and spoke in low, soothing tones.\n\nI squared my shoulders and started collecting the dirty laundry while Swedish stood in the doorway, blinking away tears.\n\nHours later, I snuggled into my threadbare blanket on the floor, thinking about the dive, the diamond, and Hazel's desperate plan. We'd have to sell the ring to the coyotes without the bosses catching us, leave the Rooftop without Kodoc catching us, and then sneak into Port Oro without the mutineers catching us.\n\nWe sure were counting on a whole lot of people not catching us.\n\nGetting there sounded impossible, but I still daydreamed about our new lives on Port Oro. Maybe we'd work as a shuttle crew. Maybe we'd buy a fishing airship. Or maybe, knowing Hazel, we'd join the mutineers ourselves. And of course we'd find a cure for Mrs. E.\n\nBare feet padded closer, then Hazel plopped down beside me.\n\n\"Can't sleep?\" I asked.\n\n\"No,\" she said. \"You?\"\n\nI shook my head. \"My mind's racing like a thopper.\"\n\n\"Worried about selling the diamond?\"\n\n\"That and, y'know, saving Mrs E.\"\n\n\"And avoiding Kodoc,\" she said. \"But no pressure.\"\n\nI smiled, then asked the question I'd been wondering about. \"The bosses hear everything that happens in the slum. How can we hire a coyote without them finding out?\"\n\n\"Carefully,\" she said.\n\nI shot her a look. \"Oh, good, thanks. Now I feel much better.\"\n\n\"I've been talking to coyotes for months, Chess. As long as nothing cockeyed happens, we're in good shape.\"\n\n\"You're telling _me_ about cockeyed?\"\n\nShe smiled at first; then two little lines appeared between her eyebrows. \"Actually, I'm worried about everything.\" She looked toward where Swedish and Bea were sleeping. \"What if something happens to one of them? To one of _you_.\"\n\nI nudged her with my elbow. Sometimes I felt sorry for Hazel, like we all depended on her too much. She worried more about us than herself.\n\n\"To one of _us,_ \" I corrected her.\n\n\"I know,\" she said, swallowing. \"But I worry most about losing Bea.\"\n\n\"It'd be like losing our hearts.\" I said.\n\n\"Yeah.\"\n\n\"If we lost you,\" I told her, \"it'd be like losing our dreams. And if we lost Swedish, it'd be like losing our\u2014\"\n\n\"Goalie,\" she said, and we laughed softly. \"No,\" she said. \"He's our anchor. He keeps us grounded.\"\n\n\"Yeah.\"\n\nHazel rested her head on my shoulder. \"Our heart, our dreams, our anchor. . . . Do you know what _you_ are?\"\n\n\"Our killer instinct?\" I asked. \"Our deadly warrior?\"\n\nShe raised her eyebrow at me and thumped my arm. \"Our hope,\" she said.\n\nWe sat in silence, listening to the distant fans whirr. After a while, Hazel said, \"On the raft today, at the slipway?\"\n\n\"Mm?\"\n\n\"You threw that mooring strap, cut the line, and ran up the deck in two seconds.\"\n\n\"Well,\" I explained, \"I was in a rush.\"\n\nShe smiled. \"I've never seen you move that fast.\"\n\n\"I move a lot faster than that in the Fog.\"\n\nHazel stopped smiling. \"Nobody else does.\"\n\n\"Yeah.\" I shrugged. \"Well, I'm a frea\u2014\"\n\n_\"Different,\"_ she interrupted. \"And you weren't in the Fog when you cut the mooring line, Chess. What if the nanites are changing you? What if you're getting sick?\"\n\n\"Fogsickness doesn't make you faster.\"\n\n\"You're not like everyone else.\"\n\n\"Yeah, I'm not just a freak, I'm an experiment.\"\n\n\"You're mostly a garbo.\"\n\n\"At least I don't look like a startled porcupine,\" I said, remembering Bea's twisty of an angry Hazel.\n\nShe gave me a shove. \"Let me see your eye.\"\n\nFor most of my life, accidentally revealing my eye meant a beating, so the idea of showing it on purpose made me a little sick. But this was Hazel; she checked my eye after every long dive, to make sure it wasn't getting worse.\n\nI wiped my hair off my forehead and she examined me. I watched her for signs of disgust but didn't see any. I never had, not from her.\n\n\"Bea's right,\" she finally said.\n\n\"It looks like a muddy puddle?\"\n\n\"Like white clouds in a dark sky. It's kind of beautiful.\"\n\n\"Yeah, I'm gorgeous.\"\n\n\"You're scruffy as an old boot. But your eye is cool. Almost hypnotic.\"\n\n\"You're getting sleepy . . . ,\" I said in a dopey voice.\n\nShe laughed\u2014a little. \"At least it looks the same as always. I'm probably worrying for no reason.\"\n\n\"You always worry that I'm getting sick,\" I told her. \"Now you're worrying that I'm not?\"\n\nShe chewed on her lower lip. \"Nobody spends that much time in the Fog and stays healthy, Chess.\"\n\n\"Don't be so impatient,\" I said, pretending to scold her. \"Give me a few years.\"\n\n\"It's not funny. I have nightmares about you getting fogsick, but it's almost as scary that you haven't.\"\n\nI tugged my hair over my eye. \"I guess I must have an affinity.\"\n\n\"Mrs. E says the nanites won't protect you forever.\"\n\n\"It doesn't matter. Once you talk to a coyote, we're out of here. I won't need to dive after we get to the Port . . . as long as nothing cockeyed happens.\"\n\n\"Yeah,\" she said, but she still looked worried.\n\n##\n\nI WOKE TO THE scuffle of footsteps outside the shack.\n\nThey didn't sound like the sewer worker down the alley or the family with the rust stall. No, they sounded quick, purposeful, and dangerous. Like a gang closing in.\n\nMy pulse spiked, and I rolled to the side and threw a bootball at Swedish, who was snoring in the main room. He snorted awake. Without a word, I crawled behind the crate shelves, grabbed the lever that operated the trapdoor, and waited.\n\nWhispers sounded from the main room, and the beaded curtain rattled. I stayed hidden, my back to the wall, as the footsteps outside scuffed closer.\n\nHow many of them were there? Five? Ten? Did they know about my eye? Had Kodoc sent them to grab me? Was this the end?\n\nI couldn't do anything except take a shaky breath and try not to cry. So I did both as I watched Swedish grab a shovel and Hazel reach for Mrs. E's rapier. I squeezed the lever tight. The gang would burst inside, then stop at the sight of Swedish. If Hazel couldn't convince them to leave, I'd open the trapdoor beneath them.\n\nAt least, that was the plan.\n\nInstead, I felt the prick of a knife in my back. Someone had cut a hole in the chickenwire-and-plastic wall and crept inside to wait for me.\n\n\"Hands off the lever,\" a girl whispered in my ear. \"Don't yell! You'll scare Perry and he'll start breaking things.\"\n\n\"Loretta?\" I gasped, recognizing the voice.\n\nLoretta was a gang girl Swedish had a crush on. She was short and stocky, with spiky black hair and a tattooed face. Swede liked her gap-toothed smile, narrow eyes, and fearlessness. Hazel claimed she liked him back, but I'd never seen any sign of it. And her knife in my back wasn't exactly winning me over.\n\n\"Bosses' orders,\" she murmured, almost apologetic. \"We ain't killing you, though. Well, as long as you don't open that trap.\"\n\nMy hand dropped from the lever. \"What do they want? They sent Perry? He's the worst kneebreaker they've got\u2014\"\n\n\"Wake-wakey, bottom-feeders,\" a shrill voice called from outside. \"And open this door!\"\n\nHazel lowered her rapier. \"Perry? Is that you?\" She sounded scared, like she thought they'd come for me. \"What do you want?\"\n\nPerry didn't answer. Instead, two thuds sounded. The door shuddered, then snapped from the hinges and crashed to the floor.\n\nWhen I shifted closer to the lever, Loretta tapped me harder with her knife. I ignored her. I wasn't the bravest guy in the world, but you didn't need courage to protect your family. I'd open that trapdoor in a second if the thugs attacked, despite the fear bubbling up in my stomach.\n\nPerry stomped inside, a pale creep with a lot of yellow hair. He worked for his uncle, one of the bosses who ran the junkyard for Kodoc, as an enforcer. Rumor was he'd ditched his own parents into the Fog, that he showed no mercy and felt no pain. His voice sounded like a five-year-old's, but he still scared the goggles off me.\n\n\"When I knock, you _run_ to open your door,\" he squeaked at Hazel. \"My uncle owns everything from here to the Spew, including your nasty little shack.\"\n\nShe narrowed her eyes. \"That's why we pay him rent.\"\n\n\"Shut it, Hazel, or I'll break more than your door.\"\n\nHe glared as if daring her to speak, but she just lifted her chin and remained silent. I knew how scared she was, but she managed to look completely unafraid.\n\n\"I've got a message from the bosses,\" he said as two thugs stepped beside him.\n\nMy breath caught in my throat. A message? So they weren't here for me? Did that mean the bosses knew about the diamond?\n\n\"Pay up,\" Perry continued.\n\nThe shack tilted around me. They _did_ know about the diamond!\n\n\"Give me your raft rent,\" Perry sneered. \"Right now.\"\n\nThe shack stopped tilting. Wait a second. They only wanted raft rent? So they _didn't_ know about the diamond! And they didn't know about me. I would've slumped in relief if I hadn't had a knife in my ribs.\n\n\"Rent's not due for five days, Perry,\" Hazel said. \"And, um, you probably haven't heard. Our raft crashed.\"\n\n\"It's not your raft,\" he squeaked. \"It's _our_ raft. And of course I heard, that's why we're collecting early, as penalty for losing the raft.\"\n\n\"It's not lost. We're salvaging it. We'll be back on the Fog in no time.\"\n\n\"You ain't fixing that raft. Not even _your_ geargirl can raise the dead.\"\n\n\"We'll see.\" Hazel smoothed her kimono. \"In five days, we'll\u2014\"\n\n\"You don't get it.\" Perry kicked the basket of plastic bags. \"You think you're so clever, but you don't know nothing. We've had orders from Lord Kodoc himself.\"\n\nHazel frowned. \"From . . . from Kodoc?\"\n\n\"That's right, you _should_ look scared,\" Perry told her. \"His lordship's hunting for some deformed kid in the junkyard.\"\n\n\"A what? Why? What kind of deformed kid?\"\n\nPerry shrugged. \"They ain't told us yet. All you need to know is that my uncle's raising money for the search. That's why you're handing over your raft rent right now.\"\n\n\"We don't\u2014\"\n\n\"Hey!\" Perry interrupted, his eyes narrowing. \"You've got a deformed kid in your crew.\"\n\nMy heart stopped.\n\n\"I\u2014what?\" Hazel said.\n\n\"Yeah, your geargirl.\" Perry laughed scornfully. \"All those freckles. Now get me that money, or we'll tear down the walls and stomp the roof flat as a cockroach.\"\n\nSwedish grunted. \"Cockroaches aren't flat.\"\n\n\"They are after you stomp 'em,\" Perry said in his high-pitched voice. His thugs laughed cruelly. \"Draw your swords, boys, maybe we'll stick a few holes in the bootball player first.\"\n\nLoretta stiffened behind me, but her knife stayed steady.\n\n\"Wait!\" Hazel said. \"Look around. We don't have anything worth taking.\"\n\n\"How about in that workshop?\" Perry stomped a few times. \"Under the floor? I'm betting you've got a hidey-hole down there with some valuables.\"\n\n\"We're not hiding anything\u2014\" Swedish started.\n\nQuick as a snake, Perry punched him in the stomach. \"Don't mess with me, bottom-feeder.\"\n\n\"Chess,\" Swedish wheezed. \"Show 'em.\"\n\nThat was my cue. I was supposed to threaten Perry with the trapdoor, but instead Loretta pushed me forward, showing everyone the knife at my back. I saw a tightening in Hazel's eyes.\n\n\"Loretta?\" Swedish stared up at her in disbelief. \"What're you doing?\"\n\n\"Orders is orders,\" she said.\n\n\"You work for _Perry_ now?\"\n\n\"I work for whoever feeds me, don't I?\"\n\nHe shot her a look of disgust. \"I thought you were better than that.\"\n\n\"I guess I'm not,\" she said, and I heard a hitch in her voice.\n\nSwedish curled his lip. \"Don't need to _guess_.\"\n\n\"Shut up,\" Perry shouted at Swedish, \"unless you want to tell me about your stash. Every crew has one. Maybe I'll ask that geargirl of yours.\"\n\nA new voice cut through the shack. \"You'll do no such thing!\"\n\nMrs. E stepped into the main room, with one frail arm around Bea's shoulders for support. She looked wan and sickly, but her dull white eyes sparked with anger.\n\n\"I know your uncle, Periwinkle,\" she snapped at Perry. \"I've seen him stare into the Fog and cry like a baby. Look around you, boy. Hazel runs the best salvage crew in the slum, worth a hundred times more than any raft. So why don't the bosses keep them under lock and key, make them work for free?\"\n\n\"Because they're too ugly,\" Perry squeaked.\n\n\"Because the bosses need to keep _me_ happy,\" Mrs. E said. \"I've forgotten more about the upper slopes than they ever knew. Who do you think tells them how to deal with the Rooftop? How to hold a fork and ask a favor?\" Her lips thinned. \"Tell your bosses that Ekaterina says we have five days to pay our rent. Until then, you _will_ leave us in peace.\"\n\nThe spark in Mrs. E's face seemed to kindle a blaze in Hazel. She suddenly looked a foot taller than Perry.\n\n\"Thank you, ma'am,\" she said to Mrs. E. Then she nodded to Bea, who helped Mrs. E back into the bedroom, and looked at me. \"Chess, come here.\"\n\nWith a knife in my ribs, I was almost too scared to move, but Hazel's confidence gave me courage. I stepped forward on wobbly legs. A moment later, I was beside Swedish, my pulse pounding in my ears.\n\n\"We'll see you in five days,\" Hazel told Perry.\n\nPerry sneered. \"I ain't scared of a sick old lady.\"\n\n\"Of course not,\" she said soothingly. \"Still, you should probably talk to your uncle.\"\n\n\"Maybe I will.\" He kicked the crates. \"But there's one thing I've got to tell you first.\"\n\n\"What's that?\"\n\n\"Don't even _think_ about running. We start searching for this deformed kid tomorrow, but that doesn't mean you're off the hook. My uncle already spread the word, and there ain't a coyote in the junkyard who'll even talk to you, not anymore.\"\n\nI almost whimpered, but Hazel just said, \"Why would we want to talk to a coyote?\"\n\nPerry glared at her for a few seconds and then stomped away. His thugs swaggered after him, as Loretta faded into the shadows and a sharp-edged silence fell in the shack. Then Swedish exhaled and I leaned against the table to support my wobbly knees.\n\n\"Are you okay?\" Hazel asked me.\n\n\"Not really. We can't hire a coyote now\u2014we're dead.\"\n\n\"I was talking about the knife in your back.\"\n\n\"Oh. Yeah.\" I squinched up my face. \"Sorry about that. Loretta's a sneaky one.\"\n\n\"She's a gutter rat,\" Swedish said, wiping his face with his hand. \"I should've known. Can't trust a gang girl.\"\n\nHazel shot him a sympathetic look. \"Can you believe Mrs. E, though? She was wonderful. She\u2014\"\n\n\"Come quick, something's wrong!\" Bea cried, from inside the bedroom. \"She's hot and shivery and can't hardly breathe.\"\n\n##\n\nI RUSHED INTO THE bedroom behind Hazel and saw Mrs. E lying on her back, her face flushed and her eyes blank. She raised her trembling hands like she was reaching for the ceiling, and her entire body started shivering.\n\n\"She's feverish,\" Hazel said, feeling her forehead. \"Chess, soak a pair of socks in water.\"\n\nI grabbed a pair of limp gray socks, ran into the main room, and shoved them into a bowl of water from the rain barrel. Then I rushed back into Mrs. E's room and stood there with the wet socks in my hands.\n\n\"Put them on her feet!\" Hazel snapped, looking up from washing Mrs. E's face.\n\n\"How come I always get her feet?\" I grumbled.\n\nI wrestled the cold, wet socks onto Mrs. E's knobby feet despite her pitiful whines. We bathed her hands with moist rags, massaged her scalp, and finally, by late morning, the fever subsided.\n\nWith the crisis over, we gathered, exhausted, in the main room. Suddenly famished, I emptied the last bag of dried locusts onto the table. Hazel sighed and Bea wrinkled her nose. Swedish offered to soak them for a while, then mash them into paste, but that only sounded worse.\n\n\"In the old days,\" I said, trying to distract them from the locusts, \"people ate gross stuff, too. Like mud and string and sticks.\"\n\nSwedish squinted at me. \"They did not.\"\n\n\"They did! They just called it 'mud pie' and 'string cheese' and 'fish sticks.' Trying to make it sound better.\" I looked at the locust in my hand. \"Maybe we should call these 'bug nuggets.'\"\n\n\"Or sky candy,\" Swedish said.\n\n\"Candy doesn't have hairy legs,\" Bea grumbled, rubbing her eyes.\n\nWe crunched in silence until Hazel said, \"She's dying. She can't survive another fever.\"\n\n\"How long does she have?\" Swedish asked.\n\n\"Maybe a week,\" Hazel said. \"And now I can't even talk to the coyotes.\"\n\nBea started crying. \"So what're we going to do?\"\n\nSwedish wrapped his arm around her narrow shoulders, and I asked, \"How did the bosses know we were going to run?\"\n\n\"They're mean,\" Hazel told me, \"but they're clever. And tomorrow they'll get a description of the kid they're looking for. Even Perry might realize he's never seen Chess's eye.\"\n\n\"There are plenty of one-eyed kids,\" Bea said, wiping tears from her face.\n\n\"And they'll check them all,\" Hazel said. \"We have to leave today.\"\n\n\"What?\" Bea asked, sniffling. \"Leave for where?\"\n\n\"The Rooftop.\"\n\nI choked on a locust leg. \"Today?\"\n\n\"This morning. We need to sneak onto the lower slopes, find a jeweler, and sell the diamond. The bosses scared off all the _junkyard_ coyotes, so we need a _Rooftop_ coyote to smuggle us to the Port.\"\n\n\"Why sell the diamond?\" I asked. \"Why not just give it to the coyote?\"\n\n\"We'll need money when we reach the Port.\" Hazel rubbed her eyes. \"The trouble is, I don't know any smugglers on the Rooftop.\"\n\n\"What about Mrs. E?\" Bea asked.\n\n\"She'll sleep for days,\" Hazel said. \"We'll come back for her once we hire a coyote.\"\n\n\"And, uh, how do we even get to the Rooftop without passes?\"\n\n\"We bribe the guards,\" Hazel said with a sudden glint in her eyes.\n\nHazel loaded the leather cord of her necklace with dyed feathers, fused-glass gems, and colorful nanowire twistys. She tucked the diamond ring in the middle, hidden amid all the brightness and sparkle, and hung the whole thing around Bea's neck.\n\n\" _I_ don't want it!\" Bea said, her green eyes wide with alarm.\n\n\"It's safest there,\" Hazel told her. \"You won't call any attention to yourself.\"\n\nPlus it made sense to keep the two most precious things together: Bea and the diamond. Of course, nobody would notice Bea, not with Hazel wearing a shimmery tank top, flowing skirt, and gauzy half veil. And Swedish looked dangerous with his hair in a thuggish topknot.\n\nI wore what I always wore: scuffed boots, patched cargo pants, and a worn jacket, but Hazel made Bea change into a patchwork dress and striped stockings.\n\n\"Dresses are stupid,\" Bea said, wrinkling her nose.\n\n\"You look more like a maid now,\" Hazel explained.\n\n\"How about me?\" I asked. \"Do I look okay?\"\n\n\"Perfect. You're a natural.\"\n\n\"I am?\"\n\n\"Yeah, you _always_ look like a downtrodden servant.\" She laughed as she dodged the plastic carton I threw at her. \"Bea, cut a hole in the wall so we can slip out without Perry seeing.\"\n\nWhile Bea snipped the chicken wire, Hazel wrote a letter on a sheet of fancy paper she'd been saving for a special occasion. We checked on Mrs. E one last time, then squeezed through the hole into the garbage-strewn alley.\n\nBea glanced at the letter Hazel was tucking into her sleeve. \"Is this going to work?\"\n\n\"It'll get us killed,\" Swedish groused. \"Kodoc's searching for Chess, and we're running _toward_ him? I'm starting to think this whole thing's a\u2014\"\n\n\"Conspiracy?\" I asked, slipping into the maze of alleys.\n\nSwedish glared at me. \"A bad idea.\"\n\n\"Of course it'll work!\" Hazel told Bea. \"We don't need the guards to believe what the letter says. We only need them to believe that their bosses will believe them.\"\n\n\"Oh.\" Bea wrinkled her nose. \"What?\"\n\nHazel shot me a sidelong look, amused by Bea's confusion. \"The letter gives them an excuse to take the bribe, that's all.\"\n\nWe followed the alleys for a while, then skirted a hill of packed garbage, ignoring the squawks of seagulls and the mutters of the trash pickers.\n\nAs we passed a gloomy alley, a memory sparked in my mind. \"Hey,\" I said, stopping. \"That's where Mrs. E found me. Down there.\"\n\nBea peered into the darkness. \"Yuck.\"\n\n\"She found me on the other side of the Spew,\" Hazel said, chewing on a braid.\n\nMrs. E had plucked Hazel from a crew of child soldiers, she'd snatched Swedish from a work gang, and she'd bought Bea from a factory. We'd all lost families before she adopted us. Swede's mom had gotten a job on the Rooftop and never come back for him. Bea's parents had had too many kids to feed and sold her to the factory. Hazel's dads had died in a refinery, after they couldn't pay rent. Mrs. E had done more than save our lives when she'd taken us in. She'd given us a reason to live\u2014a family.\n\nTwenty minutes later, we'd almost reached the Rooftop . . . but we couldn't just stroll from the junkyard to the mountains. A wide gap separated the floating slum platforms from solid ground, and roof-trooper checkpoints stood sentry at the bridges that spanned the distance.\n\nNone of us had ever set foot on the mountain, but between slum-dwellers who worked on the lower slopes and rich toughs who slummed in the junkyard, traffic flowed across the bridges pretty steadily. A marketplace had sprung up around the checkpoint, and when we got closer, I watched the crowd shuffle past bored-looking roof-troopers.\n\nBored looking, but heavily armed. I suppressed a shiver of worry. What if they were checking kids' eyes? Would a letter and bribe really work?\n\nI watched for a few seconds, and didn't see any of the troopers stopping kids. Maybe the search hadn't reached the slum yet\u2014or maybe they didn't think I was stupid enough to head straight for a checkpoint.\n\nWell, I'd show them. I was _plenty_ stupid enough. \"Chucklebutt\" was my middle name.\n\n\"Something's wrong,\" Bea said suddenly.\n\n\"What?\" I asked, scanning the street. \"Where? Who?\"\n\n\"It's the ground.\" Bea stomped on the floor. \"It feels solid. It's all . . . weird.\"\n\n\" _You're_ weird,\" I said, exhaling in relief.\n\nHazel laughed softly. \"The slum platforms don't move much this close to the mountain, honeybee. That's what you're feeling, the lack of motion.\"\n\n\"Yeah!\" Bea said, wrinkling her nose. \"The ground's all stiff and funny.\"\n\n\"Because it's supported by stilts and columns,\" Hazel told her. \"They run through the Fog into the ground.\"\n\n\"But there are balloons, too, right?\" Bea looked worried until she spotted a blimp not too far away, attached to the slum with massive ropes. \"Oh, there's one! I don't how anyone can sleep at night without balloons nearby.\"\n\n\"Soon,\" Hazel said, her brown eyes bright above the veil, \"we'll sleep every night on solid ground\u2014on the Port.\"\n\n\"I don't know,\" Bea said dubiously. \"Solid ground is so . . . groundy.\"\n\nHazel took her arm and started toward the bridge. \"Once we get through the marketplace and past the checkpoints, we'll be fine.\"\n\n\"As long as nothing cockeyed happens,\" I muttered.\n\nShe shot me a look. \"Everyone ready?\"\n\n\"Ready to die,\" Swedish said.\n\n\"This is going to be so _purple_!\" Bea squeaked a little nervously.\n\n\"Let's go,\" I said, and started across the road toward the marketplace.\n\n\"There they are!\" a voice called out. \"Get them!\"\n\nFive gang kids jogged toward us, with Loretta in front, leading the pack.\n\n##\n\n\"GO!\" SWEDISH BARKED AT us. \"Get out of here!\"\n\nHazel and I exchanged a glance. \"We're not leaving you,\" she said.\n\n\"There's five of them and three of us.\" I squared my shoulders. \"We'll steam their butts.\"\n\n\"No, there's _four_ of us!\" Bea said, like she was going to start brawling.\n\n\"Do as I say,\" Swedish snarled, his eyes fierce. \" _Now!_ Get Bea out of here.\"\n\nWithout another word, Swedish stalked toward Loretta's gang. He looked so angry that pedestrians quickly stepped aside. I'm not much of a fighter, but I started to follow him, because win or lose, I had his back.\n\nThen I felt Hazel's hand on my elbow. \"He's right,\" she said. \"We've got to go.\"\n\nI tore my arm from her grip. \"But he's _crew_.\"\n\n\"Yeah,\" she said, her eyes angry, \"so why don't you try trusting him?\"\n\nI glared at her but followed as she dragged Bea through the marketplace toward the bridge. Behind us, someone yelled: _\"Hey! What're you kids doing?\"_\n\nA sick feeling coiled in my stomach. \"We can't just leave him there. What if he needs us?\"\n\n\"Now you know how we feel,\" Hazel told me, \"every time you dive.\"\n\nI stopped short. \"I never thought of that.\"\n\n\"Because you're a dunderbunny,\" Bea explained.\n\nShe was kind of right. Whenever I dove, I left them on the raft to worry about me. \"Okay,\" I said. \"Well, let me take a peek, at least?\"\n\n\"Quickly,\" Hazel said.\n\nI clambered over a quilter's table in the marketplace, then pulled myself onto a rickety roof. I shaded my eyes, scanned the snarls of foot traffic, and\u2014there!\n\nSwedish was circling a gang kid in the middle of a crowd near a gambling tent. The mah-jongg players from inside had gathered to watch the fight. And gamble on it, of course. I couldn't see clearly, but the thug he was facing looked short and spiky-haired. Like Loretta.\n\n\"Is he okay?\" Bea asked after I hopped down.\n\n\"I don't know, it looks like he and Loretta are _talking_.\"\n\n\"That doesn't sound right.\"\n\n\"Keep moving,\" Hazel said with a weak smile. \"If Swede can't cross, we'll meet him at home later.\"\n\nWe trotted through the marketplace, and ten yards from the roof-trooper checkpoint, Hazel bowed her head. When she straightened, she looked different, with a soft smile and dull eyes. She strolled toward the checkpoint, while Bea and I trailed behind like drudges. I kept my head down and my hair over my freak-eye, suddenly thankful for all the years of practice.\n\nOne of the roof-trooper guards eyed Hazel. \"Papers?\"\n\n\"My pleasure, sir.\" Hazel handed him the letter. \"We have jobs lined up.\"\n\nThe guard scanned the letter. \"Says that this fellow needs a scullery maid, a shoeshine, a doorman, and an entertainer.\"\n\n\"That's right, sir,\" Hazel said in a breathy voice. \"This is the maid and the shoeshine boy. And I'm\u2014\"\n\n\"The doorman?\"\n\n\"Oh, no, sir!\" Hazel fluttered her eyelashes, which almost made me laugh. \"I'm the dancer!\" She spun, and her skirt flared. \"I dance.\"\n\n\"I mean, where's the doorman?\"\n\n\"Oh! He's behind us. A tall boy with a topknot. Impossible to miss.\"\n\nThe guard scratched his cheek. \"Is this all you have for me?\"\n\n\"No, sir! I also have this.\"\n\nCoins clinked as she handed the guard a pouch containing all the money we'd saved in the past three years. \"Looks fine,\" he said, weighing it in his hand. \"Go on, then.\"\n\n\"Thanks so much!\" Hazel gushed. \"And the doorman? You won't forget him?\"\n\n\"We'll see,\" the guard said.\n\nHazel's jaw clenched, like she was going to argue, so I cleared my throat, still looking at my boots. After a tense moment, she exhaled. \"Well, we can't keep our new boss waiting.\"\n\nHazel lowered her eyes demurely as she stepped onto the bridge. Bea bustled along, looking young and overwhelmed.\n\nWe paused in the middle of the bridge and glanced backward. Still no sign of Swedish. After a minute, Hazel shook her head.\n\nWe continued across the bridge, twenty or thirty feet above the terraced fields on the mountainside just over the fogline. The freshly plowed earth smelled so rich that it made my stomach rumble.\n\n\"Look at that,\" Bea gasped.\n\nAt first I thought she was staring at the stone building on the other end of the bridge, but then I realized she was looking at the _trees_. She dashed over to one that was growing alongside the bridge and strummed her fingers across the bark.\n\n\"It feels like a cat's tongue,\" she said, \"but a thousand times rougher!\" Her eyes shone. \"Look at the leaves! There are so many of them, and they're all juicy and green and _purple_!\"\n\nShe'd never seen a tree this close before. Sometimes I forgot that the others had never walked on the solid earth, never splashed in a stream or sat in the grass. So after we crossed the bridge, I took charge, leading Hazel and Bea past the stone building\u2014another roof-trooper checkpoint\u2014toward a neighborhood of cramped houses with plastic-thatched roofs. Chickens clucked, goats bleated, and pushcarts rattled across cobblestones.\n\nA flush of triumph warmed me. We still needed Swedish, still needed to sell the diamond and hire a coyote, but we'd gotten this far. We'd made it to the Rooftop.\n\nOne step closer to Port Oro. Also one step closer to Lord Kodoc, and surrounded by thousands of roof-troopers. But we were on our way to finding a cure for Mrs. E. On our way to a new life and a new world, free from the shadow of fear.\n\n##\n\nTHE HOUSES OF THE lower slope were bigger and sturdier than junkyard shacks. Some were built with timber and tin, some had clay-and-laptop walls\u2014a few of them even had two stories. Camel carts heaved past them, laden with carrots and radishes, and a grimy boy walked along, eating a shiny apple.\n\nMy mouth watered. In the slum, a bigger kid would've snatched that apple in a second\u2014and this was still the _poor_ section of the Rooftop.\n\nWe wandered down the street, gaping at the sights, until Hazel stopped so suddenly that a pinch-faced woman almost crashed into her.\n\n\"Oh!\" Hazel said. \"Sorry.\"\n\nThe woman glared. \"Watch where you're walking, miss fancy-piece\u2014and that goes for your urchins, too.\"\n\nShe bustled past, and Hazel nudged me. \"See? I told you, people are the same everywhere.\"\n\n\"Yeah,\" I said. \"Grumpy.\"\n\n\"We're _urchins_!\" Bea told me proudly.\n\nI couldn't tell if she knew what \"urchins\" meant, and I almost laughed. \"We're really moving up in the world.\"\n\n\"Why'd you stop?\" Bea asked Hazel.\n\n\"We need to stay at the bridge,\" she said. \"So we'll see Swedish when he crosses.\"\n\n\"Oh, right.\" Keeping my head lowered, I scanned the area. \"We can wait outside that glassblowing shop.\"\n\nBea brightened. \"Glassblowing! Do you think they'll show me how?\"\n\n\"No,\" I said. \"And you can't ask.\"\n\n\"Why not?\" she demanded. \"I'll show them how to make twistys.\"\n\n\"They'd probably like it better if you showed them how to fix a foggium engine with a rusty bottle cap and a little chitchat.\"\n\nShe stuck her tongue out at me. \"Why can't I at least talk to them?\"\n\n\"Because we don't want to attract any attention while we wait.\"\n\n\"You're not waiting,\" Hazel told me.\n\n\"I'm not? What am I doing?\"\n\n\"Finding us a jeweler.\"\n\nI felt a tingle of excitement with an edge of fear. \"Right now?\"\n\nShe nodded. \"I need to stay close in case Swedish has trouble crossing. And look at that. . . .\"\n\nDown the street, a patrol of roof-troopers marched through the crowd. They wore armored uniforms and carried swords and steam-bows. One plump woman didn't see them coming and was knocked to the ground. When the patrol disappeared into the guard station, the entire world breathed a sigh of relief.\n\n\"You can't stand around next to the guard stations,\" Hazel continued.\n\nI took a deep breath. \"Yeah, okay. So where should I look for a jeweler?\"\n\n\"I don't know.\" Hazel unclasped the veil from her face. \"Not this close to the slum. Look higher on the mountain, and meet us back at the bridge.\"\n\n\"Gotcha.\"\n\n\"Then we'll sell the diamond together,\" she told me. \"That's not the kind of thing you want to do alone.\"\n\n\"Okay.\"\n\n\"And don't go chasing after coyotes. We'll find them once we get the money.\"\n\n\"All right.\"\n\n\"And Chess? If you get into trouble, just bow and grovel and run away.\"\n\n\"Sure.\"\n\nShe smoothed my hair farther over my eye. \"One more thing.\"\n\n\"Yes, Admiral Bossy-pants?\"\n\nShe poked me in the chest. \"Don't steal any apples.\"\n\nBea giggled, and I took a ragged breath and left them. I slouched along the street, past chugging generators shooting sparks and street vendors bellowing prices. I couldn't believe how rich everyone was. Nobody fought over the pencil stub on the ground or the pile of soggy acrylicloth scraps\u2014nobody even noticed them.\n\nSwallowing my amazement, I followed a switchback trail uphill. I scanned the crowd for someone to ask about a jeweler, and a shadow fell across the street. When I looked upward, I saw a patrol airship swooping directly toward me.\n\n\"No,\" I gasped.\n\nI froze, rooted to the spot as the ship angled closer . . . then flew past.\n\nThe troopers hadn't seen me, they were just patrolling. Dizzy with relief, I watched the guardship roar over the smokestacks and chimneys. And _whoa,_ it was one sweet ship. I mean, Rooftop patrols flew over the slum sometimes, but never that low and slow, showing off every detail.\n\nI hoped Bea had a good view.\n\nWhen I looked back to the street, I spotted a skinny man with a friendly-looking face. \"Uh, excuse me, sir?\" I said, falling into step with him. \"I'm looking for a jewelry store?\"\n\nTo my surprise, he smiled at me. \"Sure. Take a left at the mill, head up-mountain for a few minutes, and it's in the square past the hydrostatic station.\"\n\nFive minutes later, I was lost in a junction of walkways and stairs near the mill. Five minutes after that, I was lost at a stone quarry. But five minutes after _that_ I crossed a square and saw the sign: JEWELRY BOO-TEAK.\n\nStrange name, but the shop itself looked perfect, with flowers blooming beside the door and thick iron bars on the windows. I turned in a circle to get my bearings, then headed down-mountain, back to the bridge.\n\nHalfway there, I paused on a stairway beside a hillside airfield dotted with gearslinger workshops, oversized thoppers, and half-built frames. Looked like gearhead heaven. Bea would go peanuts over all the top-notch gear\u2014and so would Swedish, if he got across.\n\nThe thought worried me. Had he made it? I rushed down the flight of wooden steps, then trotted toward the bridge in search of the crew. No sign of them. Not outside the glass shop, not under the white-barked trees.\n\nWhere _were_ they? Had something happened?\n\nI drifted toward a row of empty pushcarts, feeling an itch of fear. Then a hand clamped my mouth from behind, and a knife jabbed my ribs.\n\n##\n\n\"SHUT YOUR FACE!\" a voice breathed as the hand dragged me into an alley.\n\nThe attacker had struck without warning, like a coil of Fog thickening suddenly into a driftshark. At least this was just a person. Half blind with fear, I stomped with my heel, aiming for the knife wielder's foot.\n\nI hit the cobblestones instead, and pain burst through my leg. So I lunged backward, but the attacker sidestepped and my head smacked the wall. I slumped sideways, seeing double. Two walls beside me, two skies above me, two Lorettas in front of me.\n\n\"L-Loretta?\" I stammered, rubbing my aching head. \"What are you doing here? Where's Perry? Where's Swedish?\"\n\n\"What do you think I'm doing here?\" She grinned crookedly. \"I came to buy some pretty hijabs, like Hazel.\"\n\nMy fear edged into anger. I wasn't good at fighting, but I wasn't afraid of it, either. Especially after she put her knife away.\n\n\"Where _is_ Hazel?\" I demanded, stepping closer.\n\n\"Whoa, there.\" Loretta raised her hands in surrender, though her fingerless gloves only made her look even more thuggish. \"Me and Swede came over together.\"\n\n\"Together?\"\n\n\"Uh-huh.\"\n\nI glared. \"You and Swedish?\"\n\n\"Yeah. And Perry's nowhere. Well, he's s _ome_ where, but not here. He doesn't even know we're here, not yet.\"\n\n\"You came with Swedish? Why?\"\n\n\"Well, I kind of, um . . .\" Loretta toed the ground. \"I got to thinking about what he said? About me being worse than he thought? And, um\u2014yeah.\"\n\n\"Yeah, what? Last I saw, your thugs were trying to stomp him.\"\n\nHer broad face broke into a smile. \"You should've seen him, Chess! For a sweet guy, he really knows how to handle himself. He mowed down a couple of Perry's best hitters before you could say 'cockroach relish.'\"\n\nI rubbed my head again. \"And then what?\"\n\n\"Then I fought Swede down the block, and maybe I could've stabbed him in the knee, but I didn't.\" She considered. \"Of course, he could've broke my arm and _he_ didn't, either. So I thought, you know, he still likes me.\"\n\nI blinked at her. \"Because he didn't break your arm?\"\n\n\"Ain't that sweet?\" she said, sighing.\n\n\"It's adorable,\" I said.\n\n\"And anyway, here we are.\"\n\n\"No,\" I said. \"Here _you_ are. Where's he?\"\n\n\"With Hazel and Bea. They're hiding.\"\n\n\"What? Why?\"\n\n\"You can't drop Hazel in a crowded street and think nobody's going to notice! Dressed like _that_? The boys started swarming like flies on a skinned rat.\"\n\n\"Oh. So they sent you to look for me?\"\n\n\"That's right,\" she said with her gap-toothed smile. \"Nobody notices me. I ain't pretty like Hazel, or big like Swede. Me and you, Chess, you know what stands out about _us_?\"\n\n\"What?\"\n\n\"Nothing.\"\n\n\"Yeah,\" I said.\n\n\"Except . . .\" She peered at the hair falling over my face. \"I bet your eye's messed up. I've never seen it.\"\n\n\"Good.\"\n\n\"What is it, an empty socket?\"\n\n\"Maybe.\"\n\nShe made a disgusted face. \"I bet it's gross.\"\n\n\"It's hypnotic,\" I told her. \"Are you going to take me to Hazel?\"\n\n\"Of course I am! What do you think? C'mon, stop wasting time.\"\n\nShe led me past a few round buildings with kangaroo-hide walls, and I felt a slow simmer of suspicion. I didn't trust Loretta. I hadn't trusted her last week or last year, and I sure didn't trust her after last night.\n\n\"How did you get past the guards?\" I asked.\n\n\"I borrowed a pass,\" she said.\n\n\"Someone lent you a pass?\"\n\nShe shrugged. \"Well, me and my knife borrowed it.\"\n\n\"Oh.\"\n\n\"And I still had to knock the guy out.\" She shook her head sadly and stepped into a farm shack. \"Some people, huh?\"\n\n\"Right,\" I said, my suspicion rising as I followed her inside.\n\n\"Chess!\" Bea bounded from the shadows. \"You're okay!\"\n\nI yelped in surprise. \"I'm fine! But what's Loretta doing here?\"\n\n\"She's with us now,\" Hazel said. \"Tell me you found a jeweler.\"\n\nI frowned. What did that mean, _she's with us now_? What about my freak-eye? If Loretta joined the crew, I'd need to start hiding my eye in private, too. And this was my family. You don't run around inviting new people into your family, not without asking everyone first.\n\n\"What are you talking about?\" I said, my voice rising. \"Loretta's here\u2014that doesn't mean she's _with_ us.\"\n\n\"She is,\" Hazel told me. \"She's crew.\"\n\n\"No,\" I said, clamping down on my anger. \"We're crew. The four of us.\"\n\nHazel rubbed her eyes, then asked me very quietly, \"Do you trust me?\"\n\n\"Do you trust Loretta?\" I shot back. \"Do any of us trust her?\"\n\n\"Far as I can throw her,\" Swedish said.\n\nBea cocked her head. \"How far is that?\"\n\n\"I don't know,\" Swedish said, glancing at Loretta. \"I haven't tried yet.\"\n\nLoretta toed the dirt again and for some reason looked almost pleased.\n\n\"She helped us, Chess,\" Hazel told me. \"She got Swedish across the bridge, and she\u2014\"\n\n\"She mugged some poor slob for\u2014\"\n\n_\"And,\"_ Hazel interrupted, \"she told the other thugs not to report us to the bosses.\"\n\n\"She did?\"\n\n\"Uh-huh.\" Hazel nodded. \"Loretta's friends won't even tell Perry we're on the Rooftop.\"\n\n\"Oh, they'll tell him,\" Loretta said. \"They'll take their lazy time is all.\"\n\n\"How much time?\" I asked.\n\n\"They'll probably wait till nightfall. The gangs are going to be busy searching for this kid Kodoc wants\u2014\"\n\n\"Yeah,\" I cut in. \"Whoever _that_ is.\"\n\n\"But once the bosses hear you ran,\" Loretta continued, \"they'll make an example of you. They'll ditch your whole block into the Fog.\"\n\n\"They'll ditch your block, too,\" I said.\n\n\"I don't have a block, I don't have anything that matters except . . .\" She glanced at Swedish. \"Anyway, if Hazel's making a break for the Rooftop, she's got a plan. And you know what?\"\n\n\"What?\" Swedish asked.\n\n\"I don't want to die for the bosses, just another worthless slumgirl. I don't want to die fighting over a pair of rat-skin gloves. You four are different. You've got that gleam in your eyes\u2014you always have. So whatever scam you're running, I want in. Plus you need a fighter.\"\n\n\"We never needed one before,\" I told her.\n\n\"After the bosses ditch your shack, they'll keep hunting you. They won't ever stop. They answer to Lord Kodoc himself. You know anything about him?\"\n\nI glanced at Hazel. \"We've heard the name.\"\n\n\"Well, he's not a fellow who lets things slide. They'll keep coming until they toss you in the Fog.\"\n\n\"If they can find us.\"\n\n\"He's a Rooftop lord,\" Loretta said, \"and we're slum trash. My money's on him.\"\n\n\"You don't have any money,\" Swedish said.\n\nLoretta's bold gaze flicked to Hazel. \"You sent Chess for a jeweler?\"\n\n\"Yeah,\" Hazel said.\n\n\"Why? You found some gold or something?\"\n\nInstead of answering, Hazel looked at me, like she was asking my permission to tell Loretta about the diamond. Because once we did that, she was part of the crew, like it or not. We'd have to tell her about everything else. Including my eye.\n\n\"If you're smart,\" I told Loretta, \"you'll head back to the slum and forget all about this.\"\n\nShe scratched the burn scar on her arm. \"Well, nobody ever accused me of being a deep thinker.\"\n\nI frowned at the floor. Our lives were changing fast\u2014too fast. Maybe that was scaring me more than telling Loretta our secrets. But some things never changed: of course I trusted Hazel. I trusted her more than I trusted _myself_.\n\nSo after a second, I nodded.\n\nHazel flicked a smile at me, then said, \"We found a diamond.\"\n\nLoretta gaped at her. \"A di\u2014a diam\u2014no _way_.\"\n\n\"The purplest diamond you ever saw,\" Bea said.\n\n\"Like a . . . a _diamond_ diamond?\"\n\n\"Almost exactly,\" Swedish told her, his eyes crinkling.\n\n\"Now tell me you found a jeweler,\" Hazel said to me.\n\n\"Yep,\" I told her. \"Fifteen minutes up the mountain.\"\n\n\"So we have until tonight to sell the diamond, hire a Rooftop coyote, and pick up Mrs. E.\"\n\n\"Coyote?\" Loretta said. \"What're you smuggling?\"\n\n\"Us. Into Port Oro. After we sell the diamond.\"\n\n\"Port Oro,\" she breathed.\n\n\" _If_ we sell the diamond,\" Swedish muttered.\n\nLoretta scratched her nose. \"Um, can I see it?\"\n\n\"Maybe later,\" I said quickly. \"It's in my boot pocket.\"\n\nLoretta didn't need to know that it was actually on Bea's necklace. Maybe I'd learn to trust her eventually, but there was no reason to rush into things.\n\nSwedish shot me a look. \"Chess, you don't have to\u2014\"\n\n\"Time to go,\" Hazel interrupted.\n\nHa. Looked like Hazel wasn't so sure about Loretta either.\n\n\"Keep your eyes down and your hands to yourselves,\" Hazel continued. \"Stay together. We'll meet here if we're separated. No fighting, no stealing, no shoving. No shouting, no swearing, no\u2014\"\n\n\"Fun,\" Loretta said.\n\n##\n\nI LED THEM UP the winding streets of the mountain toward the jeweler.\n\nWhen we reached the stairway beside the airfield, Bea stared at the thoppers and babbled about pressure valves and rotation speeds. Then she stopped short, halfway up the hill. \"That's a twelve-spark repeating tes-array!\"\n\n\"Bea!\" Hazel tugged her arm. \"C'mon.\"\n\n\"But Cap'n, _look_ at her.\" She pointed to a twin-hulled thopper with a gleaming engine and exposed gearworks. \"She's purple as a mayfly!\"\n\n\"As a _what_?\" I said.\n\n\"She looks like a mayfly.\"\n\nActually, she looked like a thopper, but I knew better than to argue. Once Bea gave an airship a nickname, she refused to budge.\n\nSwedish nodded. \"She's a beaut.\"\n\n\"Except for that condenser,\" Bea said.\n\nLoretta said, \"Huh?\"\n\n\"The condenser,\" Bea explained. \"He's grumpy as an upside-down sundial.\"\n\n\"You mean the pilot?\"\n\n\"She means the condenser,\" Swedish told her.\n\n\"It's _grumpy_?\"\n\nI hunched my shoulders, disgusted at Loretta's confusion. How could she be a crew member if she didn't even understand Bea? \"We're not far now,\" I said. \"Let's go.\"\n\n\"One day,\" Bea announced, still gazing at the twin-hulled craft, \"I'm going to build a thopper as pretty as that mayfly.\"\n\n\"Bea, come on! Does Swedish have to carry you?!\" Hazel asked her.\n\n\"I'm coming, I'm coming!\"\n\nWhen we reached the square, we clustered under the awning of a general store and eyed the Jewelry Boo-Teak\u2014and the bakery and blacksmith and a handful of vendors with pushcarts.\n\n\"Ooh,\" Bea said. \"Look at all the shops!\"\n\n\"Now you sound like Hazel,\" Swedish told her. \"If you see any pretty skirts\u2014\"\n\n\"Shht!\" Hazel hissed, staring across the square. \"Look at that.\"\n\nI followed her gaze toward a small crowd standing around a water pump. At first, I didn't see the problem. Then I realized that one of them was a bull-necked guy with armored plates on his chest and forearms, and a wooden cudgel at his hip.\n\nA roof-trooper. He was only filling a jug of water from the pump, but what if he could tell we were slumkids? What if he checked our eyes?\n\n\"What do we do?\" Bea whispered.\n\n\"Look natural,\" Hazel said. \"He'll leave in a minute.\"\n\nBea started whistling, I ducked my head, Loretta cleaned her fingernails with her knife, and Swedish pretended he was kicking an invisible bootball.\n\n\"Stop it!\" Hazel hissed. \"Not like _that_! Just stay still!\"\n\nWe all stopped exactly where we were.\n\n\"Don't _freeze_! Would you\u2014\" She gritted her teeth in frustration. \"Loretta and Swede, pretend you're . . . into each other. Bea, show Chess one of your twistys.\"\n\nLoretta grabbed one of Swedish's hands and beamed up at him. \"Start being into me, big guy. I'm all ears.\"\n\nHe flushed. \"Well, that's good, because your ears are as cute as berry pie.\"\n\nI pretended to barf as Bea tugged my sleeve. She showed me a twisty in her palm, a jagged shape that looked like a rat crossed with a stepladder.\n\n\"What is it?\" I asked, glancing at the bull-necked roof-trooper.\n\nHe was scanning the square as he chatted with a woman in overalls, so I quickly turned my attention to the twisty.\n\n\"A snapping turtle,\" Bea said. \"I even made hinges on the shell.\"\n\nI looked closer. \"Um, I'm not sure the shell actually snaps.\"\n\n\"Of course it does! A snapping turtle is a turtle that snaps, like a bobcat is a cat that bobs. It says so in the name.\"\n\n\"Sure,\" I said. \"And grizzly bears loooove to grizz.\"\n\n\"I bet they do,\" she told me. \"Everyone loves to grizz.\"\n\n\"He's leaving!\" Hazel whispered as she turned from the general store window. \"Okay, let's sell this diamond. How should we\u2014 Swede?!\"\n\nSwedish was still chatting with Loretta, a few steps away. \"Hm?\" he asked.\n\n\"You can stop now.\"\n\n\"Oh.\" He dropped Loretta's hand and flushed again. \"Um.\"\n\n\"He's a real sweet-talker,\" Loretta said with a wide grin.\n\n\"And so _cute_!\" I said in a fake, gushy voice\u2014then dodged Swede's fist.\n\nHazel took a shaky breath. \"How should we do this? We've never sold anything on the mountain before. Anyone have any ideas?\"\n\nI shrugged. \"Not me.\"\n\n\"Just march in there,\" Loretta said. \"Easy as pigeon potpie.\"\n\n\"I hope so.\" Hazel thought for a second. \"Chess, come with me. Everyone else stay here. We don't want to spook the jeweler.\"\n\n\"Take Bea if you don't want to spook them,\" I said.\n\n\"Just keep your head down.\"\n\nI nodded, figuring she didn't want to take Swedish because he was so big and didn't want to take Bea because it might get dangerous.\n\n\"Okay, Bea,\" she continued, \"give me the ring.\"\n\nBea pulled the necklace off and fiddled with the trinkets, carefully removing the diamond from the strap.\n\nLoretta glared at me. \"You said it was in your boot pocket.\"\n\n\"I lied.\"\n\n\"I'm on your side now!\"\n\n\"Someone once told me that you work for whoever feeds you. Hmm, who was that?\" I paused for dramatic effect. \"Oh, that's right, it was _you_.\"\n\n\"Jerk,\" she muttered.\n\n\"Maybe after you go a whole day without pulling your knife on me, I'll trust you.\"\n\n\"Ought to stick my knife _in_ you.\"\n\n\"Not helping, 'Retta,\" Swedish told her.\n\n\"I don't like it here.\" She eyed the shoppers in the square. \"It's too pretty.\"\n\n\" _They're_ trying to lull us,\" Swedish said. \"That's how _they_ work.\"\n\n\"What does that even mean?\" I asked.\n\nHazel slipped the ring into the little box. \"You ready?\" she asked me.\n\n\"Not even close.\"\n\n\"Then let's go,\" she said, and we started across the square.\n\n##\n\nA BELL JINGLED AS Hazel opened the jewelry shop door. I followed her inside, then stopped short: cabinets built from hubcaps and iSlates lined the wall, displaying rings, necklaces, and bracelets. I'd never seen so much wealth in one place before.\n\nThe jeweler flashed a fake smile at us. \"Welcome, welcome! Looking to buy a trinket?\"\n\nI lowered my head and stayed behind Hazel, desperately hoping this would work. If the sale went smoothly, we'd have enough money to find a coyote and escape the Rooftop.\n\n\"Actually, sir,\" Hazel said, \"we're looking to _sell_ something.\"\n\n\"Is that so?\" The smile vanished. \"Wire jewelry you made? Bead earrings?\"\n\nThe bell over the door jingled again before Hazel could answer, and when I turned to look, I almost whimpered. So much for this sale going smoothly.\n\nThe roof-trooper from the square stepped inside, his water jug in one hand and his cudgel in the other. \"What are you kids doing here?\" he barked.\n\n\"Oh! I beg your pardon, sir,\" Hazel said with a nervous little curtsy. \"We'll, uh, we'll come back when you're done.\"\n\n\"I asked you a question!\" the trooper snarled. \"What're you doing here?\"\n\n\"We\u2014we came to sell some jewelry, sir.\"\n\n\"What're you doing on the _mountain_?\" He set his clay jug down on the counter with a clunk. \"Couple of slumkids sneaking into a jewelry store? That's not right.\"\n\n\"They're from the junkyard?\" The jeweler sniffed. \"Well! I don't deal with filth like that.\"\n\n\"We have day passes, sir,\" Hazel told the trooper tightly. \"For a job.\"\n\nThe trooper stuck his hand out, and Hazel gave him the letter she'd written.\n\nHe frowned at the letter and I studied the floor, trying to make myself invisible. _Let us go. Just read the letter and let us go_. . . .\n\n\"Huh,\" he finally said. \"This looks okay.\"\n\n\"Yes, sir,\" Hazel said.\n\n\"Now get out of here,\" he said. \"Back to work.\"\n\n\"Yes, sir,\" she said again, and stepped past him toward the door.\n\nI followed\u2014until the trooper blocked me with one thick arm.\n\n\"But first,\" he said, \"let me have a look at you.\"\n\n\"At me?\" I whispered, hanging my head even lower than normal.\n\n\"Um, is\u2014is something wrong?\" Hazel asked.\n\n\"He keeps his eye hidden,\" the trooper told her. \"That's not allowed on the mountain. Not for a kid his age. Raise your head, boy.\"\n\n\"Okay,\" I said, but I didn't move.\n\nI _couldn't_. The idea of exposing my freak-eye had always terrified me, and I always worried that the fear of calling attention to myself made me a coward. But this time, the threat was real.\n\n\"Please, sir,\" Hazel pleaded. \"He's scarred and he's shy. Don't make him\u2014\"\n\n\"I ain't asking again!\" The trooper grabbed my shoulders in a tight grip. \"Show me your face or I'll\u2014\"\n\n_Crash!_\n\nWater splashed my cheeks, and the trooper howled. He shoved me away and spun toward Hazel, who was holding the shattered water jug in her hands.\n\nShe'd broken it over his head. Wow.\n\nHe lunged at Hazel and she dodged, her eyes wild and her braids flying everywhere. In her shimmery tank top and flowing skirt, she looked totally defenseless as the trooper swung his cudgel at her head.\n\nA flush of rage rose in my chest. I wasn't going to let this guy hurt Hazel. I wasn't going to let him touch her.\n\nI dove at his knees and slammed him hard, my shoulder jolting. He grunted and stumbled, his cudgel flying from his grip and clattering across the floor.\n\n\"Guards!\" the jeweler screeched. \"Help!\"\n\n\"Run!\" I yelled, wrapping my arms around the trooper's knees. \"Hazel, _go_!\"\n\nThe trooper jerked his legs, but I only clung tighter. The bell over the door jingled, and I knew Hazel was safe. My sense of relief was cut short when the trooper started punching one of my legs so hard that I almost screamed.\n\nInstead, I sank my teeth into his thigh. What can I say? I fought like a slumkid.\n\n\"You little roach,\" he snarled, and boxed my ears.\n\nMy head spun, and he grabbed my hair and twisted. Pain shot through my scalp and tears sprang to my eyes\u2014then the bell jingled again, and a loud crash came from across the room.\n\nA second later, a meaty smack sounded, and the trooper collapsed beside me.\n\nLoretta stood over him, his cudgel in her hand. She started to hit him again, but Hazel said, \"Enough,\" and caught her arm midswing.\n\nWith my head still throbbing, I pushed to my feet and saw Swedish standing over the groaning jeweler.\n\n\"Look at me!\" Loretta cried, grabbing some jewelry from the floor. She held a bright blue earring to her nose. \"I'm a princess!\"\n\n\"People are coming,\" Bea squeaked from the doorway. \"They heard the noise.\"\n\n\"Are you okay?\" Hazel asked me. \"Can you walk?\"\n\n\"We can run.\" Swedish slung an arm around me. \"C'mon, 'Retta! Leave the rest.\"\n\n\"Sure,\" she said, snatching another handful of pearls.\n\nIn the square, shoppers gawked as we dashed out of the jewelry shop. A deliverywoman shouted, \"Stop them!\" as a bald man yelled, \"Guards!\"\n\nSwedish helped me as we raced along a row of rusting vans converted into homes, then skidded through a garage door into a factory. A deafening clatter filled the room, and my head pounded as we wove past tables where little kids were assembling flywheels. Then we burst from the other side of the factory into an open yard\u2014and stopped short.\n\nThe throb of an approaching engine sounded above the clamor.\n\n\"What's that?\" I asked.\n\n\"A roof-trooper guardship,\" Bea said, pointing overhead. \"They've spotted us!\"\n\n##\n\n\"OVER THERE!\" I POINTED across the yard. \"Hide in that alley!\"\n\nWe dashed into a narrow passageway as the growl of the airship engine became a roar. Clouds of dust billowed around us. My eyes watered, and I could barely see Hazel throwing herself at a flimsy door.\n\n\"Swede!\" Hazel yelled. \"Kick this down.\"\n\nSwedish took two steps and slammed his foot beside the latch.\n\nThe door flung open and we tumbled into a yard where wet clothes dangled on the line. From there, we dashed into a neighborhood of cramped houses, slipping under crooked stairways and past smokestacks. We raced blindly until we turned a corner and almost crashed into a mariachi band playing to a packed crowd. We were in a clearing above a sheer thirty-foot drop. This whole neighborhood was built on a rise, and we'd reached the end. The _dead_ end\u2014a granite cliff side.\n\nA flash of motion blurred above the band as a wide bronze hull sped through the air.\n\n\"The guardship!\" I yelled, shading my eyes as the wind ripped through my hair.\n\nHazel tugged Bea into the alley. \"Get back, before they see us!\"\n\nToo late. The ship swooped closer, and the troopers hopped to the ground, pulling swords and steam-bows as they chased us.\n\nWe dashed past a trash fire, then veered down a half-hidden stairway into a courtyard where old women sat on planks, coiling copper wire around spindles. Another dead end, with four high wooden walls and heavily shuttered windows.\n\n\"Oh, no!\" Bea cried.\n\n\"Don't panic,\" I told her, trying to sound calm.\n\n\"Why not?\" Loretta asked.\n\n\"Because if she panics,\" I said, \"I'll freak out.\"\n\nThat got a trembling smile from Bea . . . but it didn't last long, as the shouts of the troopers grew closer.\n\n\"Let me think,\" Hazel muttered, tugging on a braid. \"Let me think, let me think. . . .\"\n\n\"Looks like you're in a spot of trouble,\" said one of the old women, taking a corncob pipe from between her lips.\n\nHazel spun toward her. \"Please, ma'am\u2014please help us.\"\n\n\"You're polite for a slumgirl,\" the woman said.\n\n\"Listen, you old bat,\" Loretta snarled. \"She asked you to help, not\u2014\"\n\nA trooper shouted into the courtyard from the top of the stairs: \"Halt! Hands on your heads or we'll fire!\"\n\n\"That's it,\" Swedish muttered. \"We're dead. _They_ always win in the end.\"\n\nThe old woman pointed her pipe at a curtain on the courtyard wall. \"Through there.\"\n\nIn a flash, Hazel grabbed Bea's hand and dragged her toward the curtain.\n\nBoots sounded on the stairway, and the roar of the guardship engine grew louder. Someone whimpered\u2014maybe me\u2014as we raced toward the wall. I was two steps behind Hazel when she pulled back the curtain . . . and I suddenly wanted to cry.\n\nThere was nothing behind it but another plank-shuttered window.\n\nClosed tight. No escape.\n\n\"On your knees!\" a trooper shouted from the top of the stairway. \"Down, now!\"\n\n##\n\nAS THE TROOPERS THUNDERED down the stairway toward us, the shutters behind the curtain flung open, slamming against the wall with a solid _thunk_.\n\n\"Come!\" a voice whispered from inside the dark window. \"Quickly now!\"\n\n\"Bea, go!\" Hazel boosted her through the opening. \"Now you, Loretta! Inside! Chess, move!\"\n\nThe minute Loretta disappeared, I dove headfirst through the window and landed with her elbow in my gut. She squirmed away, and Swedish fell on me like a bear.\n\nI yelped and the roof-troopers sounded loud and close in the courtyard.\n\n\"There!\" one yelled. \"She's climbing in that window.\"\n\n\"Get her!\" another yelled. \"Grab her!\"\n\nThe shutters slammed closed. They must've been three inches thick, and the room turned black as iron bolts slid shut, locking the soldiers out in the courtyard along with\u2014\n\n\"No!\" I gasped. \"Hazel's still out there! Open it up!\"\n\n\"I'm here,\" her voice said in the darkness.\n\nI almost sobbed in relief. \"Oh, thank the peaks.\"\n\n\"Save your thanks,\" said a rough whisper in the darkness. \"And follow me.\"\n\nAs soldiers hammered on the other side of the shutters, a hooded lantern sprang to life in the gloom. A dingy light hovered in the air, and I caught a glimpse of a cloaked man.\n\n\"This whole thing stinks like a trap,\" Swedish muttered.\n\n\"What stinks,\" I said, shoving him off me, \"is your armpit in my face.\"\n\n\"At least we're not dead,\" Hazel said. \"Follow the light. Stay together.\"\n\nThe glow drifted from the room, and we trailed after the cloaked man along a mazelike hallway. When we heard the soldiers smash through the shutters behind us, Bea squeaked in the darkness and Loretta swore.\n\nThe man didn't react. He just led us through a sliding panel, then whispered, \"Watch your step.\"\n\nThe sound of the soldiers faded as we followed a stairway downward. A moist draft washed over us, and the lantern painted the rough-hewn stairs with a sickly light. We headed into a dirt-walled corridor, and the sound of dripping water echoed all around.\n\n\"Where are we?\" Hazel asked, sounding as lost as I felt.\n\n\"In the mines,\" the voice whispered. \"They won't find us here.\"\n\n\"How do you know?\"\n\n\"This section was condemned as unsafe years ago.\"\n\n\"Oh, good,\" I muttered, glancing at the ceiling. \"I feel much better now.\"\n\nSwedish snorted. \"Yeah, I always wanted to die in a cave-in.\"\n\n\"Holy goalie!\" Bea blurted. \"Are we _underground_?\"\n\n\"Yes,\" the cloaked man whispered.\n\nWith a whoop, Bea started jumping up and down, stomping on the floor. \"It's dirt! I never thought there was this much dirt in the whole world! Look at all this\u2014\"\n\nI grabbed her arm. \"Would you _stop_ that?\"\n\n\"What?\"\n\n\"It's been condemned as unsafe!\"\n\nShe peered down the dark hallway. \"What does 'condemned' mean?\"\n\n\"That it'll fall on our heads if you keep hopping around.\"\n\n\"Oops.\"\n\n\"It is pretty cool, though,\" I admitted. \"I've never been underground before.\"\n\n\"Underground smells like camel butt,\" Loretta muttered.\n\n\"Shhh,\" the cloaked man said.\n\nWe climbed the uneven stone stairs in the gloom. Silence fell around us, except for the scuffing of our footsteps and the drips of water echoing through the stairwell.\n\nFinally Hazel said, \"Who are you?\"\n\nThe cloaked figure stopped on a landing. \"My name is Turning.\"\n\n\"Why did you help us?\"\n\n\"Cover your eyes.\" He fiddled with a rusted gearwork mechanism on the wall. \"We're stepping into the light.\"\n\nA door opened and sunlight poured into the dark stairway. I squinted as I followed Loretta into a cluttered room. Not an ordinary room: more like a laboratory.\n\nCabinets lined the walls, and a workbench divided the room, littered with half-built instruments and strange devices. Rows of tools were sorted in racks, and a clockwork machine as big as Swedish hunched in one corner, half covered with a cloth. Charts filled a bookcase, and round windows, shining with bright sunlight, overlooked the lower slope and the slum.\n\nBea gasped, gazing at a huge scope tilted toward one window. \"Look at that spyglass!\"\n\n\"It's a telescope,\" Turning told her in his chilly whisper.\n\nHe lowered his hood, and I saw his face: tired eyes, a crooked nose, and a braided beard. Cloudy-white gems glinted in his earrings, the color of Fog.\n\n\"I never thought I'd see you so close,\" he said, surveying us. Then he frowned at Loretta. \"And _you're_ a surprise.\"\n\n\"We weren't exactly expecting _you,_ either,\" she said.\n\nTurning's laugh sounded like his whisper, scratchy and thin. \"I suppose not, but\u2014\"\n\n\"You never thought you'd see us this _close_?\" Hazel asked, frowning. \"So you've seen us from afar?\"\n\n\"Indeed I have,\" he said. \"Through the telescope.\"\n\n\"He's been watching us.\" Swedish turned to me. \"I _told_ you someone was watching us.\"\n\n\"Yeah, well\u2014\" I couldn't think of a snappy reply. \"You were right.\"\n\n\"Ha!\" he crowed. \"I knew it!\"\n\n\"Oh, Mr. Turning, can I look in the telescope?\" Bea pleaded. \"Please? Pretty please?\"\n\n\"Of course,\" he said.\n\nShe scampered across the room, greeted the telescope politely, then peeked through. \"Ooh, everything looks so close! There's a waterwheel. Oh, there's the airfield with that mayfly! That is _such_ a purple thopper!\"\n\n\"Why have you been watching us?\" Hazel asked Turning, her voice sharp. \"How did you find us? Was that old lady in the courtyard expecting us?\"\n\n\"We monitor the roof-troopers,\" Turning said. \"We saw them chasing you.\"\n\nI felt a chill of worry as I scanned the room. Who _was_ this guy, with his secret passages and hidden laboratory?\n\n\"You monitor them?\" Hazel asked. \"Why?\"\n\n\"For our own safety.\"\n\nHazel's eyes narrowed. \"Who _are_ you?\"\n\n\"Look!\" Bea squealed. \"The slum's so close I could touch it.\"\n\n\"He's a foghead.\" I pointed to a charcoal sketch on an easel. \"That's too freaky for anyone else.\"\n\n\"What\u2014\" Loretta made a face. \"What is it?\"\n\nThe drawing looked like an evil thopper with legs instead of wings, and a spiny shell of shattered bricks. I said, \"A ticktock, I bet.\"\n\n\"Only in my imagination,\" Turning said with a slight smile. \"I've never seen one. Nobody has. Not in a hundred years.\"\n\n\"So it's true?\" Hazel asked Turning. \"You're a foghead?\"\n\n\"I suppose I am,\" Turning said, his scratchy voice calm. \"Though that's not what we call ourselves. We are the Subassembly, and I'm a leader\u2014a 'cog.' My full name is Cog Turning.\"\n\nSo he wasn't just a foghead, he was a foghead _boss_? And he'd brought us to his hidden lair, with illegal books and creepy drawings? We needed a coyote to take us to Port Oro, and instead we got a mist-sniffing fog-freak. A sick sense of dread uncoiled inside me like a driftshark: we needed to get out of there. We needed to sell the diamond and get away from the Rooftop before everything fell apart.\n\n##\n\nLORETTA'S HAND DRIFTED TOWARD her knife, and Swedish stepped closer to Hazel. The room thickened with suspicion, everyone clenched and edgy. Well, everyone except Bea, who kept looking into the telescope, completely oblivious.\n\n\"Don't be afraid,\" Turning said, his voice calm. \"I'm a friend.\"\n\nSwedish scowled. \"That's what _they_ always say. You fogheads are the ones who started spreading rumors about Chess.\"\n\n\"I'm sorry about that,\" Turning said. \"However, if the Assemblers you met in the junkyard hadn't foolishly spread those rumors, I never would've found Ekaterina. And I couldn't have saved you just now.\"\n\n\"Why _did_ you save us?\" Hazel asked. \"How do you know Mrs. E? What do you want from us?\"\n\n\"There's the market!\" Bea called from the window. \"Oooh! That's the watering hole. _Our_ watering hole.\"\n\n\"I'll answer all your questions,\" Turning said, stroking his braided beard. \"But would you like to eat first? I have cucumbers and chorizo stew.\"\n\nDespite my fear, my mouth watered. I'd never even seen a whole cucumber, only the peels.\n\n\"You have meat?\" Swedish asked.\n\n\"We're not hungry,\" Hazel lied. \"Answer our questions.\"\n\n\"I'll do more than that,\" Turning said. \"I'll help you reach Port Oro.\"\n\nA chill silence fell. How did he know we were trying to reach Port Oro?\n\n\"Port Oro?\" Hazel's eyes narrowed. \"What makes you think we're trying to get to Port Oro? I don't know what game you're playing, but\u2014\"\n\nHe lifted his gnarled hand. \"I've known Ekaterina for a very long time.\"\n\n\"You're her friend!\" Hazel shook her head in surprise. \"Her mysterious friend.\"\n\n\"No way,\" Swedish muttered. \"This foghead?\"\n\nTurning arched a bushy eyebrow. \"I'd offer you honey bread, but I'm fresh out.\"\n\n\"It _is_ you!\" I said. Nobody else would know about the honey bread.\n\n\"Indeed it is, Chess.\" He looked at me intently. \"And I'd very much like to see your eye.\"\n\nMy stomach dropped. Of course he knew about my eye\u2014it seemed like the entire Subassembly knew\u2014but I still felt sick and scared, like I was naked in the middle of a crowd.\n\n\"What do you want to see his eye for?\" Loretta asked. \"I heard it's gross.\"\n\nTurning smiled faintly but didn't answer her question. \"I'd been looking for Ekaterina for a long time. She changed her name thirteen years ago\u2014she wasn't born 'Ekaterina,' you know\u2014and she vanished off the face of the Rooftop.\"\n\n\"This is so purple!\" Bea announced from the telescope, totally unaware of the conversation. \"I can see everything. Where's our house?\"\n\n\"I thought she'd fled into the lower slopes,\" Turning continued, \"or even Port Oro. Not the junkyard.\"\n\n\"There's no better hiding place on the Rooftop,\" Hazel said.\n\nHe nodded again. \"When I tracked the rumors down, I finally realized where to look. But she was so sick. I\u2014I hadn't expected that. She didn't even recognize me at first.\"\n\n\"She told us you're old friends,\" Hazel said.\n\nA spark glinted in Turning's tired eyes. \"The last time I saw her, years ago, she was standing at the edge of a firestorm in the middle of the night.\"\n\n\"When she burned Kodoc's house down?\" I asked.\n\n\"She's a remarkable woman.\" He steepled his fingers. \"However, since Kodoc learned about the unexpected results of his long-ago experiment\u2014\"\n\n\"What experiment?\" Loretta blurted. \"What're you talking about?\"\n\n\"A science experiment,\" I told her.\n\n\"What I'm trying to say,\" Turning continued, \"is that while Kodoc's been searching for you, I've been arranging passage for you, to Port Oro.\"\n\n\"No!\" Hazel said, her breath catching. \"Truly?\"\n\n\"Oh, look!\" Bea suddenly called, from the telescope. \"There's our alley! I bet I can find our shack.\"\n\n\"Truly,\" Turning told Hazel. \"I sent a carrier pigeon last week and asked the Port to come pick up a package. The mutineers aren't overly fond of the Subassembly, but we still work together. They know that we understand the Fog better than anyone alive. And we find them quite useful.\"\n\n\"A package?\" Swedish asked. \"What kind of package?\"\n\n\"The five of you.\" Turning eyed Loretta. \"Six, now.\"\n\n\"Whoa,\" Hazel said. \"A ship from the Port? For us?\"\n\nTurning nodded. \"I think my message was lost, but fortunately I found a coyote willing to take you.\"\n\n\"You didn't!\" Hazel said, her eyes dancing. \"Get _out_! Really?\"\n\n\"This foghead's too nice,\" Loretta muttered, scratching the tattoo on her cheek. \"Nobody's this nice without wanting something.\"\n\n\"If he can heal Mrs. E's fogsickness,\" I told her, \"I don't care about anything else.\"\n\n\"Hey, that's Perry!\" Bea fiddled with the telescope's focus. \"He's tiny, like an ant. I could crush him!\"\n\n\"Can you really cure Mrs. E?\" Hazel asked Turning.\n\n\" _I_ can't, no.\" He smiled softly. \"But for my friends on Port Oro, many things are possible.\"\n\n\"That's what she always says.\"\n\n\"She's right. You'll get to the Port, you'll cure Ekaterina, and you'll escape Kodoc.\"\n\n\"But why?\" Hazel demanded. \"Loretta's right. You must want something.\"\n\n\"What are they doing?\" Bea asked the telescope. \"That's weird. . . .\"\n\n\"I want to help my old friend.\" Turning tugged at his earring. \"And I will do anything to keep Kodoc from getting his hands on Chess.\"\n\nHazel's lips narrowed. \"Why is that?\"\n\n\"Because Chess is the only fog diver who can find the Compass.\"\n\n\"You want a compass?\" Loretta asked. \" _I_ can find you a compass.\"\n\n\"Not a compass,\" Turning said. \" _The_ Compass. An ancient machine, buried in the Fog, that controls the nanites.\"\n\n\"Control the what knights?\" Loretta had never heard Mrs. E's stories. \"The naan knights? You fogheads are even dippier than I\u2014\"\n\n\"It doesn't matter now,\" Turning interrupted. \"What matters is that I'll send word to the coyote and you'll leave for Port Oro tonight.\"\n\n\"Tonight?\" Hazel's sudden smile brightened the room. \"That soon?\"\n\n\"We're really going!\" I said, feeling a tingle of excitement. \"After all this time\u2014\"\n\n\"Hazel!\" Bea cried at the telescope. \"They're ditching our block! They're ditching our shack, and Mrs. E is still inside!\"\n\n##\n\n\"THEY'RE DOING WHAT?\" TURNING asked, sounding uncertain. \"Ditching?\"\n\nI barely heard him over the blood thundering in my ears. If the bosses ditched our block, Mrs. E would fall into the Fog. This couldn't be happening, not now, not when we were so close to getting to Port Oro. No, no\u2014\n\n\"No,\" Swedish gasped.\n\nHazel bent over the telescope, and I found myself standing beside her, watching her face. Hazel would say Bea was wrong, and we'd tease her for scaring us. Because she _was_ wrong. She had to be.\n\nBut just then Hazel made a horrified sound. \"They're prepping the platform to ditch. They're halfway done already,\" she said.\n\n\"How\u2014how long do we have?\" Bea asked.\n\n\"I don't know,\" Hazel said. \"Thirty minutes?\"\n\n\"We're too far away,\" Bea wailed. \"What do we do? How do we stop them?\"\n\nBefore I knew what I was doing, I shoved Loretta.\n\nShe stumbled backward. \"Hey!\"\n\n\"Your friends did this!\" I snarled at her. I'd told them we couldn't trust her. I'd told them, and now look what'd happened.\n\n\"I'm sorry,\" she said, her face falling. \"I thought they'd wait till nightfall.\"\n\nI shoved her again, dizzy with fury and fear. \"I knew you were bad news. I knew it. If they ditch her, I swear by the Fog that I'll\u2014\"\n\nA hand clamped my shoulder. \"Please, Chess,\" Turning said. \"Make no vows in anger.\"\n\nI jerked away. \"If anything happens to Mrs. E\u2014\"\n\n\"Quiet!\" Hazel snapped. \"Let me think.\"\n\nI closed my mouth so fast that my teeth clicked. Anger drifted through me in a red haze, but I kept still. Hazel was our only chance.\n\nTurning frowned. \"Would someone please tell me what 'ditching'\u2014\"\n\n\"She said _quiet,_ \" Swedish barked, slamming his palm on the workbench.\n\nHazel paced, chewing on a knuckle. I watched her. Swedish watched her. We all watched her.\n\nShe crossed the floor three times, then stopped and looked at Bea. \"The mayfly!\"\n\n\"The what?\" Bea said, her eyes glossy with tears.\n\n\"That thopper you called a 'mayfly.' On the hillside airfield. You said you saw it through the telescope. Find it again. Hurry!\"\n\nBea bent over the eyepiece, softly begging the scope for help.\n\n\"Ditching,\" Hazel told Turning, \"is when the junkyard bosses flip a section of the slum, dropping everything into the Fog.\"\n\nHe frowned. \"That sounds dangerous.\"\n\n\"That's the point,\" Swedish said.\n\n\"They do that on purpose?\" Turning asked, a flash of horror in his eyes. \"I've seen it, but I thought it was an accident. That's terrible\u2014\"\n\n\"We need your help,\" Hazel interrupted, and I'd never been happier to hear the bossy tone in her voice. \"Bea's looking for an airfield we saw. Once she finds it, we need you to take us there\u2014fast.\"\n\n\"An airfield? What are you going to do?\"\n\n\"The same as always,\" Hazel told him. \"Whatever we can.\"\n\n\"Found it!\" Bea called.\n\nTurning peered through the eyepieceand muttered fretfully. \"Yes,\" he finally said. \"I can get you there.\"\n\n\"But you _must_ return.\" Cog Turning's voice echoed in the gloomy mine tunnel. \"After you save Ekaterina, you must return to the Rooftop and find me.\"\n\n\"We will,\" Hazel told him.\n\n\"There's no other way to cure her, there's no other way to save yourselves. There's no other way to get Chess to Port Oro. Promise me you'll return.\" Turning hesitated at a dark junction. \"I should bring you straight to the coyote and send you off.\"\n\nMy breath caught. If he let the bosses ditch Mrs. E, forget about Port Oro\u2014I'd ditch _him_.\n\n\"But I can't do that to Ekaterina.\" Turning looked at each of us in turn. \"Promise me you'll come back.\"\n\n\"We promise,\" Bea said.\n\nTurning nodded, then led us up a rickety ladder into what he called a root cellar. At the far end, he opened a trapdoor in the planked wooden ceiling.\n\n\"Turn right on the street,\" he told us. \"The thopper field is down the block.\"\n\n\"Thank you,\" Hazel told him. \"We'll come back with Mrs. E.\"\n\nI nodded uneasily. We'd snuck into the Rooftop once that day\u2014I wasn't sure we'd get lucky twice.\n\n\"I'll speak to the coyote.\" Turning receded into the darkness. \"Take care. You are more important than you know.\"\n\nWe climbed up through the trapdoor into an empty hut that must've belonged to a farmer or laborer. As we clustered around the door to the street, Bea said, \"What did _that_ mean? More important than we know?\"\n\n\"Because of Chess,\" Hazel said. \"And this Compass machine.\"\n\n\"Because he's a fog-sniffing lug nut,\" Swedish said.\n\n\"Yeah,\" Loretta agreed. \"He's as cracked as a glass trampoline.\"\n\n\"What's a trampoline?\" Bea asked.\n\nLoretta shrugged. \"I don't know, but that guy is a cracked one.\"\n\n\"I like him.\" I told them over my shoulder. \"Sure, that stuff about me is loco, but he said they'll cure Mrs. E.\"\n\n\"Yeah,\" Swedish said. \"And he hired us a coyote.\"\n\n\"Let's go,\" Hazel said, grabbing Bea's hand.\n\nShe headed onto the bustling street, and Swedish and Loretta followed a second later. I patted my hair down, took a deep breath, then stepped into the sunlight.\n\n\"Is everyone clear on the plan?\" Hazel asked.\n\nBea nodded. \"I jump-start the mayfly.\"\n\n\"I bust heads,\" Loretta said.\n\n\"And we all go down in a blaze of glory,\" Swedish said as we jogged past a cobbler's shed.\n\nI surprised myself with a laugh. \"Our favorite kind of blaze.\"\n\nHazel glanced at me. I guess I'd surprised her, too. Usually I stayed quiet in public, afraid that someone might notice my freak-eye. But even though I was scared for Mrs. E, I was also excited: we finally had a real chance to get to Port Oro.\n\nWe brushed past workers hauling crates and clambered over a fence onto the hillside airfield. A few of the gearslingers fiddling with engines and steam organs glanced at us, but nobody raised an alarm. We looked harmless, I guess, just a bunch of kids playing around.\n\nBut we weren't playing.. We were fighting for Mrs. E's life.\n\nBea and Hazel jogged toward the mayfly\u2014that sleek thopper\u2014while the rest of us ambled along behind. In the tool-cluttered outdoor workshop, two gearheads tinkered with the mayfly engine, and a woman in a fancy coat stood nearby, watching with a sour expression.\n\n\"She's lovely,\" Hazel said when she got close enough. \"Is she fit to fly?\"\n\n\"We're fine-tuning the piston array,\" one of the gearheads told her.\n\nHazel cocked her head. \"Does she _fly,_ though?\"\n\n\"Oh, yes,\" the woman in the fancy coat said. \"She's quite\u2014\"\n\n\"She flies!\" Hazel called, and everyone sprang into action.\n\nLoretta touched her knife to the woman's throat, whispering threats, and a thump sounded from the other side of the thopper as Swedish clobbered one of the gearheads. I put my arm around the other one and said, \"My friends have nasty tempers. Let's keep this quiet.\"\n\n\"You're s-stealing the thopper?\" the gearhead stammered. \"That's crazy! The guardships will shoot you down. There's nowhere to go.\"\n\n\"In that case,\" I told him, \"there's no reason to make a fuss.\"\n\nBea monkeyed with the pistons while Swedish ran his fingers over the controls\u2014then the engine roared to life. Hazel swung into position, Loretta grabbed for a boarding strap, and I scrambled on deck.\n\n\"Help!\" the woman in the fancy coat screamed. \"Somebody, help!\"\n\nWith a blast of air, the thopper lifted off\u2014and Loretta shouted, \"Wait, wait!\"\n\nSure, she could slink through shadows like a hunting cat, but she didn't know anything about boarding airships. She was still dangling halfway off the deck, her boots scrabbling against the hull, trying to get a toehold.\n\nThe thopper rose, and I looked down at Loretta and froze. We needed to save Mrs. E, not Loretta. We couldn't even trust her not to stab us.\n\n\"Chess!\" Hazel snapped. \"Get her. Now.\"\n\nSo I hooked my boots into the railing and snagged Loretta as she started to slip off the hull. I didn't _really_ want her to break any bones.\n\n\"Climb me like a ladder,\" I told her.\n\nShe grabbed a handful of my jacket, then my pants, pulling herself upward until Hazel heaved her on board. I lay there for a moment, draped over the edge, watching the hillside blur beneath us, praying that Mrs. E was still okay.\n\nAfter I grabbed the railing and hopped on deck, I found Loretta sitting with her arms around her knees. \"Th-thanks,\" she told me.\n\n\"It's nothing,\" I said, trying to ignore Hazel's dirty look.\n\n\"No, I mean it.\" Loretta swallowed. \"I don't like heights. And I know you don't trust me, so\u2014thanks.\"\n\n\"Sure,\" I said, then turned away to stare anxiously toward the junkyard.\n\nI couldn't spot our neighborhood in the endless sprawl, and my jaw clenched as I pictured Mrs. E drowsing in her bed before the walls started collapsing around her. _We're coming, Mrs. E. Stay safe for three more minutes_.\n\n\"Hang on,\" Hazel called, grabbing a strap behind Swedish. \"We've got company.\"\n\n\"C-company?\" Loretta gulped.\n\n\"A guardship.\" Hazel grabbed Swedish's shoulder. \"Swede, they're behind that rise.\"\n\nI scanned the sky and saw a guardship veering toward us, a sleek craft with jointed fins, coming fast. There was nowhere to hide.\n\n\"We can't outrun _that,_ \" I said.\n\n##\n\n\"HE'S FAST, BUT HE can't turn,\" Swedish told me, yanking a lever. \"No rooftop puddle-jumper is keeping me away from Mrs. E.\"\n\nHe swung the mayfly into the air, rising toward the clouds. As we climbed higher, the guardship drew closer. Loretta moaned while Bea danced around the engine, adjusting gears. Swedish hammered a soundless tune on the steam organ keyboard, his fingers a blur, and Hazel stood behind him, her braids flying.\n\n\"The shack's three wisps to your left, Swede,\" she shouted. \"The guardship's five hundred yards and closing.\"\n\n\"That's what he thinks,\" Swedish said, aiming the thopper higher and higher.\n\nFlying slower and slower as we rose.\n\n\"Four hundred yards,\" Hazel said, and shot me a look.\n\nSometimes with Hazel there was no need for words. \"Bea!\" I yelled. \"Fasten down, this is going to get bumpy!\"\n\nLoretta looked at me with frightened eyes, so I unspooled a jack line and handed it to her, saying \"Wrap this around yourself.\"\n\n\"Three hundred fifty yards!\" Hazel shouted.\n\n\"Wh-why?\" Loretta asked me.\n\n\"Because any second now,\" I said, \"Swedish is going to start _flying_ this thing.\"\n\n\"What's he doing now?\" she moaned, and started winding the rope around her arm.\n\n\"Three hundred!\"\n\nI yanked the rope from Loretta's hands and strapped her down properly. The wind whipped my hair from my face, and she gasped. \"Your _eye_! It's full of Fog\u2014and it's _moving_. That's freaky!\"\n\n\"Yeah,\" I said.\n\n\" _Totally_ freaky!\"\n\n\"Yup,\" I said.\n\n\"Dude,\" she said. \"You're a Fog-eye.\"\n\n\"I know!\"\n\n\"Wait, _you're_ the kid Kodoc is looking for! You're the most wanted person in the entire Rooft\u2014\" she shrieked as the mayfly jerked.\n\n\"Two hundred yards!\" Hazel shouted. \"Swede, they've got harpoons cocked.\"\n\nThe deck was almost vertical. I tugged my hair into place, wedged myself against a boiler vent, and watched the guardship climbing after us, my heart thundering.\n\n\"One fifty!\" Hazel shouted. \"Swede, stop showing off!\"\n\nSwedish laughed and flicked a valve open.\n\nThe engine sputtered, and stalled. Then stopped.\n\nA weightless silence surrounded us. No pistons pumped, no foggium flowed. We hung motionless in the air, aimed almost directly upward.\n\nThen the pneumatic _pop-pop_ of harpoons sounded behind us. Airtroopers shouted, and the mayfly dropped like a stone. The harpoons missed us by five feet. Wind lashed the hulls, and through my fear I felt a tug of excitement: it reminded me of diving into the Fog.\n\nWhen the guardship roared past, our little mayfly rocked in its wake. In five seconds, the airship was a hundred yards above us, and heading away, while we were still plummeting toward the lower slope.\n\nSwedish wrestled the wheel, and Bea stepped on a hose to adjust the flow of foggium. Loretta screamed, Hazel whooped, and the engine roared to life. The rudders caught the wind and we swooped toward the slum, ten feet over the rooftops. The bridge whirred past, then shacks and alleys blurred beneath us.\n\n\"Swedish did that on _purpose,_ \" Loretta said, through chattering teeth. \"If we live, I'm going to kill him.\"\n\nThen a terrible wrenching noise overtook us: the sound of a junkyard neighborhood being ditched. Cracks appeared along the edges of a platform and all the shanties and trash\u2014and Mrs. E, still in her bedroom\u2014started sliding downhill. I leaned forward, urging the mayfly on.\n\nI scanned the slum, my heart clenched in my chest. A hill of garbage sped past beneath us. A flock of startled seagulls screamed around us. I spotted the clearing where we'd stumbled onto gang kids playing bootball\u2014but the cluster of balloons was gone. The bosses weren't trying to keep this part of the slum aloft, not anymore.\n\nMy throat dried and sweat stung my eyes. I saw the water hole, the alley stalls . . . then our half-collapsed block came into view, and the shack was still standing! I exhaled in relief\u2014and saw a shadow flickering on the roofs beneath us.\n\nI glanced over my shoulder and gasped.\n\nThe guardship was a hundred yards behind us, closing fast. Two cannon ports slid open along the prow.\n\n\"Cannons behind us!\" I screamed. \"The guardship, they're firing _now_!\"\n\nSwedish hammered the keyboard and the thopper rolled to the left so hard that a rudder snapped. A second later, the cannons roared like a thunderclap. The barrage slammed into the slum, blasting tents into scraps . . . and one cannonball clipped the mayfly.\n\nBehind me, Bea gave an agonized shout that turned my blood to ice.\n\nDespite the wild wobbling of the deck, I spun and staggered toward her. But she didn't look hurt. As I got closer, I saw that she hadn't yelled from pain. She'd yelled because the engine was on fire. Orange flames spewed from a broken hose, and smoke poured behind the mayfly like a black river.\n\n\"How bad is it?\" I yelled.\n\n\"She's dead!\" Bea called back. \"We're going down!\"\n\n##\n\n\"SWEDE,\" I SHOUTED, TURNING toward the cockpit. \"The engine's dead!\"\n\nTrash tumbled along the alleys below us as the slum platform tilted. The slimy river of the Spew changed course, sweeping away a cluster of stalls.\n\n\"Crash-land at home,\" Hazel bellowed to Swedish.\n\n\"Are you peanuts?\" Loretta asked. \"That whole block's going to ditch!\"\n\n\"Mrs. E's in there,\" I told her.\n\n\"Everyone up front!\" Hazel yelled, above the sound of the shrieking engine. \"Bea, quick!\"\n\nWith the deck shuddering beneath me, I dragged Loretta to the prow. When Bea arrived a second later, I swung her between me and Hazel, so we'd cushion her when we crashed.\n\nPlumes of dust rose from collapsing shanties, twenty feet below the thopper.\n\nTen feet.\n\nFive feet.\n\nWhen the rooftops were two feet below us, Swedish veered into an alley like it was a landing strip. Trash flew everywhere, shacks started falling like dominos, and we hit the platform so hard that my knees buckled. We slid fifty feet, spun twice, and stopped.\n\nFor a long moment, nobody spoke, as the neighborhood shook apart around us.\n\n\"That wasn't so bad,\" Hazel said.\n\nBea giggled a little hysterically.\n\n\"You all,\" Loretta moaned, \"are insane.\"\n\n\"At least the roof-troopers are writing us off as dead,\" I said, my voice oddly steady as I watched the guardship returning to the Rooftop.\n\n\"Look at that,\" Swedish said, swinging to the ground. \"Twenty feet from the shack, and she landed sweet as berry pie.\"\n\n\"Let's go!\" Hazel called. \"We've got to get to Mrs. E.\"\n\nShe and Bea hopped down from the thopper, stumbling on the tilting platform.\n\n\"C'mon,\" I said, giving Loretta my hand.\n\nShe swung to the quaking ground, then looked up at my face. \"Your eye's not _that_ freaky,\" she said apologetically.\n\n\"Sure,\" I said. \"Everyone's got one.\"\n\n\"Those white swirls are kind of neat,\" she said as we started trotting toward the shack. \"Like, um, snowballs.\"\n\n\"Snowballs?\"\n\nShe flashed a gap-toothed smile. \"Except totally freaky.\"\n\nI gave a huff of laughter; then a wall buckled nearby and we jumped away. Debris tumbled along the tilting platform as we scrambled after Hazel and the others. An avalanche of noise surrounded us\u2014buildings collapsing, metal screaming\u2014and trash battered us until we shoved into the shack.\n\nSwedish staggered from the bedroom with a limp Mrs. E in his arms.\n\n\"How is she?\" I yelled.\n\n\"Sleeping!\" he grunted.\n\nShe was only half asleep, actually. She peered around the crumbling room and mumbled, \"Look at this mess\u2014no dessert for you!\" then closed her eyes.\n\n\"Well, we're here,\" Loretta said as she wiped blood from her split lip. \"Now what?\"\n\nSwedish plowed toward the entryway. \"C'mon!\"\n\n\"Hurry!\" Hazel followed Swedish, struggling uphill as the shack floor tilted. \"C'mon!\"\n\n\"What are we doing?\" Loretta asked, grabbing the table for support.\n\n\"Going to the thopper,\" I said.\n\n\"The thopper crashed.\"\n\nI pushed her after Hazel. \"Not _that_ thopper.\"\n\nLoretta took off, but I hesitated. I wanted to run into Mrs. E's room and grab my dad's scrapbook from the shelf. I couldn't leave it behind. But the platform jerked so hard that the rain barrel jounced across the room, and I knew I was out of luck\u2014the entire shack was about to collapse. I scrambled after Loretta into the entryway. Plastic jugs and broken crates tumbled everywhere as we helped Swedish lower Mrs. E through the hatch into the workshop under the shaking platform.\n\nThe slum gave another sharp shudder and I slammed against the wall, then slid to the ground in a painful daze. Groaning, I crawled back to the hatch and wriggled through, into Bea's workshop. The floor was at a steep diagonal. Tools cascaded off the workbench. The foggium compressor rolled into the hole where the thopper dangled by heavy chains, then disappeared into the Fog below.\n\nA racing thopper didn't have room for passengers, so Swedish was hunched in the cockpit with Bea on his lap and Mrs. E squeezed beside them. Hazel straddled the forward hull, holding the butterfly valve. On the far side of the thopper, Loretta desperately wound an inner tube around her wrist, strapping herself to a pump vent.\n\nThe ticking and whirring of the thopper engine grew louder. The wings spread to catch the wind, and foggium flowed through ignition chambers.\n\n\"Chess!\" Hazel shouted. \"Jump!\"\n\nI scrambled toward the hole in the floor\u2014or what _used_ to be the floor. I balanced on the edge, aimed for the fattest part of the thopper\u2014\n\nAnd Hazel screamed, \" _Wait!_ Stop!\"\n\n##\n\nMY ARMS WINDMILLED, AND I barely caught myself. A shower of tiny springs ricocheted past my head and tumbled through the hole into the Fog.\n\n\"The chain!\" Hazel pointed to the rear of the thopper. \"The chain's stuck!\"\n\nUsually Swedish released the support chains from inside the cockpit, but with the platform slanting, the rear chain had snagged in an ugly snarl of metal. And if the thopper was stuck, we'd be ditched into the Fog in just a few seconds.\n\nI blocked out the roar of the demolition and jumped. I caught the chain, slid down, and stomped at the snarl. It unraveled immediately, and I felt a flash of satisfaction. Ha! Can't beat a tetherboy!\n\nThen it hit me: I'd just freed the chain from the thopper, but I was still _hanging from_ the chain.\n\nHuh. Maybe I should've thought that through.\n\nBefore the chain swung me too far away, I dropped. I hit the thopper near the fantail and couldn't get a grip. Rivets fell from the workshop and jabbed my face as I slid backward, digging my fingernails into a seam in the metal. Just when I started to fall, Swedish rolled the thopper sideways, which lifted me upward until the hull was beneath me.\n\nI grabbed a hitch and tried not to faint as a downpour of trash roared into the Fog a few feet behind me. Bea had designed the thopper for speed and handling, but it was a one-person craft. Weighed down by six of us, it wallowed beneath the wildly tilting platform.\n\nWith a grind of protest, the thopper started lumbering forward, and soon we were flying just under the slum, in the fifty-foot gap above the white froth of Fog. Slime dripped from the platforms overhead, and huge fans whirred as I straddled the rear section of the hull. My head throbbed, and my shoulders and hands burned.\n\nA prop crew stopped working as we flew past, and shouted jeers at us. One girl started climbing a ladder\u2014probably to tell the bosses she'd seen us\u2014and fear squeezed my heart like a fist. If we didn't move fast, the bosses would toss nets over the edges of the platforms to trap us. That way they could ditch us personally.\n\nDid the roof-troopers know that the kids from the jewelry store were the same ones who'd swooped over the junkyard? Had Kodoc found out my name yet? Did he know my face and my crew? I could barely breathe until we emerged from the shadow of the junkyard.\n\nThen I slumped in relief\u2014we'd made it! Except for the fact that we were stuck on the Fog, in an overloaded thopper, without anywhere to land.\n\nI looked over my shoulder and watched the slum recede into the distance. The only home I'd ever known had been tossed like garbage into the Fog. My neighborhood was gone forever\u2014the clearing where I'd played bootball, the corner where Mrs. E once kissed my scraped elbow. My bracers were gone, my harness and my bedroll. So was the table where we ate, where we'd laughed and talked and argued. And my father's scrapbook.\n\nEverything was gone. I didn't have a home anymore. None of us did.\n\n\"When I was a girl,\" Mrs. E said suddenly, \"everyone called me Kat. Short for Katherine.\"\n\nHazel leaned toward her. \"Yes, Mrs. E?\"\n\n\"Don't interrupt! Everyone called me Kat, but I also _had_ a cat. I called my cat Me.\"\n\nI glanced at Bea. We hadn't heard that one before.\n\n\"So we were Kat and Me,\" Mrs. E said. \"Except I was Kat and the cat was Me. Her real name was Meow.\"\n\n\"That's sweet,\" Hazel said. \"Now stay still and\u2014\"\n\n\"Back then,\" Mrs. E said, \"the Subassembly and the roof-troopers worked together. Not like today.\"\n\n\"The fogheads worked with the army?\" Loretta asked.\n\n\"The roof-troopers are more than just an army,\" Hazel told her. \"They do everything for the Five Families. They're in charge of schools, banks\u2014\"\n\n\"Some are scientists,\" Mrs. E interrupted in a lilting tone that sounded nothing like her normal voice. \"Trying to find a way to control the Fog. In the old days, they wanted to harness the power of the Fog into engines and machines.\"\n\n\"You mean foggium?\" Loretta asked.\n\nMrs. E giggled, which made my stomach sink. I hated when she started acting like a little girl. It felt like the fogsickness was mocking her.\n\n\"The Subassembly discovered foggium,\" she said, \"but the roof-troopers built the refineries. Did I ever tell you about my cat?\"\n\n\"Her name was Me,\" Hazel said.\n\n\"Don't be silly!\" Mrs. E snapped. \"Her name was Petunia. Now stop your chattering. I'm sleepy.\"\n\nAn uncomfortable silence fell over us. None of us wanted to face the fact that Mrs. E was slipping away. Swedish wiped his eyes, Hazel gazed into the distance, and I clung to the thopper, trying to keep my mind blank as we soared over the Fog, the valves clicking and the wings heaving us through the air.\n\nHeading where? Nowhere.\n\nThe thought chilled me. \"How much fuel do we have?\" I asked.\n\n\"An hour's worth,\" Hazel told me. \"Maybe two, with a tailwind.\"\n\n\"Shouldn't we head back?\" Loretta asked, her face buried in her arms.\n\n\"Back to what?\"\n\n\"The slum?\"\n\n\"The bosses would kill us.\"\n\nBea bit her lip. \"How about the Rooftop? We can find Mr. Turning again.\"\n\n\"The guardships would kill us.\" Hazel rested her palm on Mrs. E's forehead, checking for a fever. \"And then Kodoc would kill us all over again.\"\n\n\"Yeah,\" I said. \"But if we stay out here, the crash will kill us first.\"\n\n##\n\nNOBODY SAID ANYTHING FOR a long time. The quiet felt ominous.\n\n\"Okay,\" Hazel finally said. \"I've got a plan.\"\n\n\"What kind of plan?\" I asked.\n\n\"The regular kind.\"\n\n\"You mean desperate and loco?\"\n\n\"Not necessarily,\" Swedish told me. \"She could mean risky and doomed.\"\n\n\"You're both right!\" Hazel said, brushing braids from her face. \"This one is desperate _and_ risky.\"\n\n\"In that case,\" I said, \"I'm in.\"\n\nSwedish snorted. \"Yeah. What's the worst that could happen?\"\n\n\" _This_ is the worst that could happen.\" Loretta raised her head and looked faintly green behind her tattoos. \"I'd sell my nose for a block of solid slum.\"\n\nFor some reason, her misery cheered me. At least I wasn't turning green. \"Maybe not a _whole_ block, Loretta\u2014your nose is pretty small.\"\n\nShe glared at me, then dropped her head again, wrapped in her private despair.\n\nI scooted forward. \"What's the plan, Hazel? We all grow wings?\"\n\n\"We'll be angels soon enough,\" Swedish muttered, \"once we crash.\"\n\n\"We're not crashing,\" Hazel said. \"We're heading for the shipping lanes.\"\n\nI steadied myself. \"You mean the lanes that Mrs. E always told us to stay away from, because the merchants might report us to Lord Kodoc?\"\n\nShe nodded. \"Exactly.\"\n\n\"That whole 'let's grow wings' thing is sounding better and better,\" Swedish said.\n\n\"Don't worry,\" Hazel said. \"We'll keep out of sight until we can intercept a merchant ship.\"\n\nI gaped at her. \"Until we can _what_?\"\n\n\"Intercept a merchant ship,\" she repeated. \"We'll offer them the diamond in exchange for passage back to the Rooftop.\"\n\n\"You want to swoop down on a merchant convoy? And before they open fire, you'll start _haggling_?\"\n\n\"That's right.\"\n\n\"Then they'll bring us back to the Rooftop? To find Cog Turning?\"\n\n\"Yup. And he'll bring us to his coyote friend.\"\n\n\"Wow,\" I said. \"There's a word for that.\"\n\n\"Deranged?\" Loretta asked. \"Suicidal?\"\n\n\"Brilliant,\" I said, and Hazel's laugh rang out across the Fog.\n\n\"I thought you'd like it,\" she said.\n\nA gust of wind rocked the thopper, and the gyroscopes whirred. Bea worried that the clockwork sounded draggy, so I held her legs as she dangled overboard, fiddling with the torsion cables and grousing about the dress Hazel had made her wear. The breeze calmed as Swedish angled the thopper around a crooked pillar of Fog and we drifted over a misty white ravine.\n\n\"Slow her down, Swede,\" Hazel finally said. \"This is the shipping lane.\"\n\n\"How does she know?\" Loretta asked me. \"Fog all looks the same to me.\"\n\n\"She knows Fog like you know fighting,\" I told her. \"If Hazel says we're here, we're here.\"\n\nLoretta scratched the scar on her arm. \"When do we run out of fuel?\"\n\nI shrugged. \"Forty minutes? Maybe an hour.\"\n\n\"That's all we've got?\"\n\n\"No,\" I said. \"We've also got Hazel.\"\n\n\"That's why she's the captain,\" Bea told Loretta, clasping my arm and pulling herself upright. \"She'll get us out of this.\"\n\nTwenty minutes later, though, even Bea looked nervous, scanning a thick cloud bank\u2014of real clouds, not Fog\u2014hoping for a glimpse of a merchant convoy. We drifted in aimless circles, but there were no airships anywhere.\n\n\"I wonder what a cucumber tastes like,\" I said to break the silence.\n\n\"Watermelon rind,\" Swedish immediately answered. \"Except sour.\"\n\n\"Nah,\" Bea said. \"I bet it's more like pigeon.\"\n\n\"Pigeon?\" Loretta asked.\n\n\"Sure. Everything tastes like pigeon.\"\n\n\"But pigeon's a bird, and cucumber's a fruit.\"\n\n\"Cucumber's not a fruit,\" Swedish said. \"Now _berry_ is a fruit.\"\n\n\"I'm pretty sure cucumber's a fruit,\" Bea said.\n\n\"Hey, Swede,\" Loretta said. \"Have you ever tasted a berry pie?\"\n\n\"I saw one once,\" he told her, \"when I was a kid. Some rooftoppers came around with free food. Mostly slop, but right there in the middle . . . a berry pie.\"\n\n\"You get a taste?\"\n\n\"Well, the neighborhood boss had sent me begging, y'know? So I grabbed the pie and ran it back to him.\"\n\n\"Did he give you a bite?\"\n\n\"Gave me a smack,\" Swedish said. \"But I still remember how it smelled.\"\n\n\"There!\" Hazel pointed into the thick mist. \"A merchant ship!\"\n\nA dark speck grew in the cloud bank, then turned into a sweeping airship with a massive cylindrical balloon. Cannons and portholes dotted the hull, nanofiber propellers shone in the sunlight, and two smaller airships prowled alongside\u2014armored gunships with chain guns and flamethrowers.\n\n\"That's no merchant ship,\" Swedish said. \"Check out the diving platform.\"\n\nI shaded my eyes and looked closer. Underneath the ship, a dozen tethers and winches were spaced across a massive scaffolding, like hundreds of ladders lashed together.\n\n\"That's the ultimate salvage raft,\" I said.\n\n\"What?\" Loretta asked. \"I can't see anything.\"\n\n\"They've got ten or fifteen tetherkids diving at once,\" I told her. \"Look at all those winches.\"\n\n\"Diving for salvage?\" she asked.\n\n\"I guess,\" I said. \"They must be looking for something _big_.\"\n\n\"She looks like a Five Family ship,\" Hazel said, her voice tight.\n\n\"Who cares?\" Loretta said. \"Either they help us or we crash.\"\n\n\"At least she's not a warship.\" Hazel twisted one of her braids. \"Approach them slowly, Swede. Everyone look desperate and harmless.\"\n\n\"We _are_ desperate and harmless,\" I reminded her.\n\n\"Then it'll be easy.\"\n\nSwedish swung the thopper toward the convoy, and I read the name on the big ship's hull: _Teardrop_.\n\n\"Weird name,\" I said.\n\n\"Maybe they're merchants and that's their motto.\" Loretta deepened her voice. \"'On the _Teardrop,_ we're crying for a sale.'\"\n\n\"'Prices so low,'\" I intoned, \"'you'll weep with joy.'\"\n\n\"Would you two shut up?\" Hazel said.\n\nOne of the armored gunships swooped forward, and Hazel waved her veil like a white flag and shouted, \"We're a salvage crew!\"\n\nThe gunship heaved to alongside us. Sheets of riveted metal blurred closer, and fumes seeped from brass nozzles. The wind from the gunship's gearwork fans rocked the thopper, and armored panels slid open to reveal airtroopers pointing weapons at us.\n\n\"We're a salvage crew,\" Hazel repeated. \"Don't shoot!\"\n\n\"That's no salvage raft,\" said a grizzled woman in an officer's jacket. She peered closer. \"And what're you doing with that old lady?\"\n\n\"Um, we lost our raft, ma'am, but\u2014\"\n\n\"On second thought, I don't care about your sad story,\" the woman said. \"All I care about is that you're interrupting Lord Kodoc's search.\"\n\nMy breath caught and my heart clamped tight. _Kodoc_. This was Kodoc's ship.\n\nDarkness crept in at the edges of my vision, and the day turned murky. I felt dizzy and doomed, like in nightmares when I couldn't escape from a monster. Bea whimpered and Hazel went speechless for a long horrifying moment. Even Loretta fell silent.\n\nThe grizzled woman didn't seem to notice. \"I'll give you thirty seconds to start running. Then we'll open fire.\"\n\n##\n\n\"WE'RE ALMOST OUT OF fuel!\" Loretta cried. \"We can't reach the slum!\"\n\n\"Too bad,\" the woman said. \"His lordship's time is more important than your lives.\"\n\n\"Wait, ma'am.\" Hazel straightened up, trying to look official while perched on the wobbly thopper. \"We, um, have something to sell.\"\n\n\"Hazel, no!\" Bea whispered. \"Not to _Kodoc_.\"\n\n\"We don't have a choice,\" I croaked.\n\nOne of the airtroopers laughed at Hazel. \"What are slumkids selling? A handful of rust?\"\n\n\"A ring,\" Hazel said. \"A diamond.\"\n\nThe woman paused. \"Show me,\" she said. \"Come closer.\"\n\nHazel glanced at me, biting her lower lip. If Kodoc and his troops didn't buy the diamond, we'd crash. But what if Kodoc saw us? He might recognize Mrs. E\u2014though she was so frail that she hardly looked like herself. Or worse\u2014he might check my eye.\n\nMrs. E's words of warning echoed in my mind: _he'll kidnap Chess and work him to death_. He'd drop me in the white, in an endless search for some ancient machine that might not exist, and I'd never see the light of day again. Even if I succeeded, Kodoc would toss me overboard when he was through with me. But we had to take the chance. I'd rather become one of Kodoc's lab rats than lose my whole family to the Fog.\n\n\"Go on, Swede.\" I exhaled shakily. \"She can't see the ring, not this far away.\"\n\nSwedish edged the thopper nearer to the gunship, and when Hazel opened the lid of the box, the diamond glinted in the sun.\n\nThe officer muttered to her soldiers. \"Keep still,\" she told Hazel. \"We'll tow you to his lordship.\"\n\n\"Let's do this here,\" Hazel said tightly. \"I'll give you the diamond, you bring us to the Rooftop. No need to bother his lordship.\"\n\n\"You'd trust me with your diamond?\"\n\nHazel bowed her head. \"Only because I don't have a choice.\"\n\n\"Slumgirl's got spirit,\" an airtrooper laughed.\n\n\"I don't have a choice, either,\" the officer told Hazel before turning to a soldier. \"Heliograph his lordship and inform him of the situation.\"\n\nA moment later, a polished copper plate\u2014the heliograph\u2014flashed a pattern of reflected sunlight at the bigger ship, passing the message along.\n\nThe thopper trembled beneath me, and I trembled right back. We were trapped on the Fog, with Lord Kodoc only a hundred feet away. The engine coughed, and I wondered how long we had before we ran out of fuel. Ten minutes? Twenty?\n\nThen a light flashed from the _Teardrop,_ and the officer said, \"This is your lucky day. Lord Kodoc will grant you an audience.\"\n\n\"Great,\" Hazel said.\n\n\"Watch your tongue, though. You don't want to make him angry.\"\n\nI cringed at the thought. A perfectly happy Kodoc scared me to death\u2014I didn't even want to _think_ about an angry one.\n\nSwedish glanced at Hazel, as if he expected her to tell him to fly away. But she only nodded, so he angled the thopper toward the _Teardrop_. The big airship was already rising below us, like one of Hazel's drawings of a whale breaching the surface of the ocean.\n\nFirst the massive steel-ribbed balloon loomed into sight, then came the mainsails, and after that an endless sweep of rigging. A few seconds later, the quarterdeck drew level with us, and I gasped at its grandeur. Polished floorboards gleamed in the sun, tidy sailors adjusted shining valves, and tables laden with charts and food scattered the deck.\n\nA man with a long face stepped to the railing. His fancy green jacket shone in the sunlight. His glossy hair sparkled, and even his skin shone. He was the cleanest thing I'd ever seen, and when he inspected us through a monocle, I felt like a dirty ant on the wrong side of a magnifying glass.\n\nHazel gave a weird seated curtsy and whispered, \"Bow! Everyone, heads down!\"\n\nI bowed, happy for an excuse to hide my face. Of course, there was no way that Kodoc would recognize me. He'd never seen me, not even as a baby. And I knew he wouldn't _personally_ check the eyes of every kid he met\u2014he'd leave that to his troopers. Definitely. No question about it. I wasn't scared of being discovered, my skin wasn't clammy, and my mind wasn't blanking. No, I was trembling for completely different reasons.\n\nSure I was.\n\nThen he spoke. \"Am I to understand that you\u2014of all people\u2014have in your possession a diamond?\"\n\n\"Yes, Lord Kodoc,\" Hazel said, straightening from her curtsy.\n\n\"And you expect me to purchase it?\"\n\n\"We hope you will, your lordship.\"\n\n\"You're hardly in a position to bargain, girl.\" His cold gaze flicked toward Mrs. E's slumped form, and he cocked an eyebrow. \"If you keep the diamond, you'll be committing a crime punishable by death.\"\n\n\"That's why we flew into the Fog, your lordship. With, um, our grandmother. To sell the diamond to a person of good standing, a nobleman who'd treat us fairly, instead of a lower-slope\u2014\"\n\n\"Yes, yes, I take your point,\" Lord Kodoc said, a hint of cold amusement in his voice. \"So you wish to sell a diamond. In exchange for what consideration?\"\n\nI didn't know what \"consideration\" meant, but Hazel said, \"We beg safe passage to the Rooftop, your lordship, and permission to remain on the lower slopes.\"\n\nLord Kodoc laughed, a slithery sound. \" _That_ is your dream? To live on the lower slopes, packed together like animals?\"\n\n\"Yes, your lordship.\"\n\nLord Kodoc swept us with his monocle. \"Very well. Consider the diamond sold. I will drop you in the lower slopes.\"\n\n\"Thank you!\" Hazel gushed. \"Thank you, your lord\u2014\"\n\n\"Now bring that pathetic ship to the landing dock,\" Kodoc interrupted, turning away to grab a wineglass from a tray.\n\n\"See, Loretta?\" Bea said. \"I told you Hazel would save us.\"\n\nSwedish swiveled the thopper toward the landing dock, and I clenched the hull so hard that my hands ached. \"Save us\"? Docking on Kodoc's ship was like a mouse hiding inside a cat's mouth.\n\n\"Wait!\" Lord Kodoc said as he turned to inspect us. \"You're a salvage crew? Which one's the tetherkid?\"\n\nAfter a terrified second, I raised my hand.\n\n\"I lost a diver yesterday.\" He swirled the wine in his glass. \"Fifth one this month. You'll take her place.\"\n\n\"Yes, sir,\" I said in a tiny voice.\n\n\"Junkyard tetherkids are worthless until my people train them,\" he observed, \"but an empty harness is even worse than an incompetent diver\u2014\"\n\nHe stopped suddenly, staring at me. Like he saw past my hair, straight to my eye. No, like he saw past my _face,_ straight into my _thoughts_.\n\nMy throat clenched and my skin felt too small for my body. Seconds ticked past. I hunched my shoulders, listening to the engines, feeling the chill wind of the propellers.\n\nThen Kodoc called, \"Helmsman! Bring me closer!\"\n\nThe big airship swept toward the thopper, like a massive hand swatting at a fly. The hull stopped ten feet away, and Lord Kodoc's slithery voice said, \"Look at me when I talk to you, boy.\"\n\nMy stomach dropped, but I lifted my head a bit. \"Sorry.\"\n\n\"Higher,\" Kodoc told me.\n\nI moved my chin an inch.\n\n\"I know you,\" he said, almost too softly for me to hear.\n\n\"I\u2014I don't think so, my lord.\"\n\n\"You're him.\" His gaze sharpened. \"You're mine. You _survived_.\"\n\nThe world narrowed into a tunnel between Kodoc and me. Nothing else existed. My mind screamed, _You don't know me, you don't know anything about me!_ but my mouth refused to form words, and I just sat there, silent and frozen.\n\n\"Show me your right eye,\" Kodoc said.\n\n\"No,\" I breathed.\n\nLord Kodoc smashed his wineglass against the deck. \"Show me the eye that I gave you!\"\n\nBea made a frightened noise that I barely heard over the rush of blood in my ears. I couldn't help myself: with a shaking hand, I brushed the hair from my freak-eye.\n\nFor a terrible second, Kodoc stared at me. His face burned with a feverish hunger as he whispered, \"You. You're finally mine.\"\n\nMy arms started trembling, and I swallowed hard, too scared to speak.\n\n\"I'd hoped to intercept some mutineers who'd been pestering me,\" he said offhandedly, though his glittering eyes never left my face. \"I never dreamed you'd fly out here to meet me. What a cooperative boy. What is your name?\"\n\nI swallowed again. \"Ch-Chess.\"\n\n\"No.\" His smile made me shiver. \"Your name is whatever I call you.\"\n\nI trembled harder, so scared that all I could say was \"Yes, s-sir.\"\n\n\"Bring that thopper to the landing dock,\" Kodoc ordered Swedish. _\"Now!\"_\n\n##\n\nSWEDISH GLANCED AT HAZEL for instructions. She bit her lip, and I saw in her face that she'd run out of ideas, which scared me as much as Kodoc . . . almost.\n\n\"Do what he says,\" she told him.\n\n\"No way,\" Swedish snarled. \"I'd rather crash.\"\n\n\"Me, too\u2014but we're not going to lose Bea.\"\n\n\"Don't blame me!\" Bea squeaked. \" _I'd_ rather crash.\"\n\n\"I wouldn't,\" Loretta muttered.\n\n\"You don't get a vote,\" I told Bea shakily. \"Head for the landing dock, Swedish.\"\n\nHe started to answer, and a blast of cannon fire roared so loud that my ears rang. Wood splintered, metal shrieked. Airtroopers screamed in pain and fear.\n\n\"We're under attack!\" a trooper shouted. \" _Mutineers!_ Return fire, return fire!\"\n\n\"Grab those slumkids!\" Lord Kodoc snarled. \"Get me that thopper!\"\n\n\"I can't bring her around, m'lord,\" the helmsman answered. \"Not with mutineers attacking.\"\n\n\"No excuses!\" Lord Kodoc ordered. \"I want that boy!\"\n\n\"Dive!\" Hazel screamed at Swedish. \"Dive!\"\n\nThat was the best idea I'd heard all year. \"Now!\" I yelled. \"Go-go- _go_!\"\n\nBea hugged Mrs. E to keep her steady, and the thopper plunged downward. A ripple of flame shot through the air where we'd been a moment before. Chains whirled and harpoons hissed, and Lord Kodoc shouted commands as the _Teardrop_ swung closer, almost batting the thopper from the sky.\n\nTwo airtroopers on a lower deck hurled grappling hooks at us, but they missed when Swedish corkscrewed into a cloud of cannon smoke. Voices cried, hoses hissed, and splinters rained around us. I hugged the hull as we swooped lower. Hazel screamed at Swedish, and Bea screamed at the engine, and Loretta just screamed.\n\nThe thopper shuddered, then leveled out. With the noise behind us, I lifted my head and saw that we were skimming along the jagged surface of the Fog. The battle still raged, but all I heard were Kodoc's words: _you're finally mine_.\n\nHe scared me more than Perry, more than the junkyard bosses. Maybe even more than driftsharks. He'd looked at me like I was an object, a _thing_. Something in his eyes made me shrivel, made my bones turn to twigs. Something in his voice when he'd shouted, _I want that boy!_\n\nMy mind stuttered over my fears until a _boom_ made me look upward.\n\nI scanned the smoke and flame, then spotted a mutineer warship that must've slunk into attack range while the roof-trooper ships were surrounding us. Looking closer, I recognized the cigar-shaped zeppelin of the mutineer airship, and the armored bronze bands.\n\n\"It's Vidious!\" I said, not quite believing my eyes. \"It's the _Night Tide_!\"\n\nSwedish glanced higher. \"What's he doing?\"\n\n\"Probably calling Kodoc a 'poppet.'\"\n\n\"He's not alone.\" Hazel pointed to the sun shining above the battle. \"There's Captain Nisha.\"\n\nI squinted into the light but didn't see anything. Then a shadow detached from the sun\u2014and a second mutineer warship dropped from the sky to blast the _Teardrop_ from above.\n\n\"Never thought I'd be glad to see mutineer warships,\" I said.\n\nKodoc's ship was quick for her size, but not quick enough. Cannon fire pocked her quarterdeck and harpoons stabbed the steel-ribbed balloon that kept her aloft. The armored gunships returned fire, but a few mutie airsailors still slid down the harpoon lines from Nisha's warship, almost like tetherkids, and landed on the _Teardrop_ 's balloon.\n\nHazel shaded her eyes and looked up in awe. \"Nisha's flying the _Anvil Rose_. Now _that's_ a beautiful ship\u2014\"\n\n\"I hate to interrupt,\" Swedish said, his voice tight, \"but we're into the dregs of the foggium.\"\n\nLoretta gasped, and I said, \"How long've we got?\"\n\n\"Five minutes.\"\n\nHazel closed her eyes. Three seconds passed. Then ten. Then she opened her eyes and said, \"Turn around and head back to the battle.\"\n\n_Back to the battle?_ I gulped. That meant back to Kodoc.\n\nLoretta groaned and Bea's eyes widened, but Swedish just tilted the thopper into a curve and started climbing toward the noise and smoke and danger.\n\n\"His fancy lordship wants Chess,\" Loretta told Hazel. \"What are you going to do, hand him over?\"\n\n\"You _can't_!\" Bea said.\n\n\"It's okay,\" I said, despite the lump in my throat. \"We don't have a choice. We're running out of fuel.\"\n\n\"We're not flying to Kodoc.\" Hazel nudged Swedish. \"Head higher, toward the _Anvil Rose_.\"\n\nFor a moment, nobody spoke. Then Loretta asked, her voice faint, \"You want to start chatting with mutineers in the middle of a sky battle?\"\n\n\"No,\" I said, suddenly awed. \"She wants to sell them a diamond.\"\n\n\"It's worth a try,\" Hazel said. \"We're not letting that creep get his hands on Chess.\"\n\nI looked away from the fiery combat to stare at Hazel, because she was even more impressive than a flying death match.\n\nShe never panicked. When _I_ got scared, my mind blanked and I froze. Not Hazel. Instead, she accepted every new problem and immediately calculated the best response. Maybe we'd only have a 50 percent chance of success, maybe only 10. Heck, if we only had a _1_ percent chance of survival, she'd just grit her teeth and grab at that tiny sliver of hope.\n\nI'd dived into uncharted Fog for three years and seen things nobody else had even imagined, but I'd never seen anything like Hazel.\n\n##\n\nAN UGLY RIPPING SOUND yanked my attention back to the battle. Mutineers and roof-troopers fought hand to hand on the _Teardrop_ 's ribbed balloon, and through the smoke I glimpsed figures tossing buckets of sand on the flames. As Vidious blasted the big warship, Nisha poured fire at the escorting gunships. War chains whirled, harpoons spiderwebbed the sky . . . and the _Teardrop_ spun and retreated.\n\nA cheer rang out from the mutineer ships.\n\n\"Kodoc's running!\" Loretta said. \"The muties must've hit something important.\"\n\n\"But they didn't,\" Bea said, a worried note in her voice.\n\n\"Then why's he taking off?\"\n\n\"It doesn't make sense,\" Bea told Loretta. \"What's he doing, Cap'n?\"\n\n\"I don't know.\" Hazel wiped her braids away from her face. \"But there's no way he's giving up. You heard how much he wants Chess.\"\n\n\"Yeah, I've got a few questions about that,\" Loretta said.\n\n\"I'll tell you all about it,\" Swedish said, \"if we're still alive in five minutes.\"\n\nThe thopper engine sputtered as we sped higher, watching the Rooftop ships retreat toward the cloud bank. Repair crews swarmed the mutineer warships, and everything ran so smoothly that I couldn't help feeling a spark of admiration. But mostly I was afraid that our five minutes of fuel were already down to a few seconds.\n\nAs we approached the _Anvil Rose,_ a few mutineers glanced at the thopper, but nobody said anything until Vidious noticed us.\n\n\"Oy, Nisha!\" he called to his sister's airship. \"Look what washed up on our shore!\"\n\n\"Who are they?\" she yelled back.\n\n\"That salvage crew from yesterday. They're flying a thopper now, clinging to the hull like fleas on a dog.\"\n\n\"We're in our thopper,\" Hazel told Vidious as we hovered beside his airship, \"because we had a spot of trouble with one of our balloons.\"\n\n\"Did you hear, Nisha? 'A spot of trouble' with that junky raft.\" Vidious laughed. \"Now, my poppets, you find me on the horns of a dilemma. On the one hand\u2014\" He stopped abruptly, and told his sister, \"The little one is staring at my forehead, looking for horns.\"\n\n\"Sorry,\" Bea mumbled.\n\n\"The 'horns of a dilemma' means having two bad choices, poppet. On the one hand\u2014\"\n\nThe thopper jerked as a piston misfired, and Swedish muttered, \"We're running out of time. . . .\"\n\n\"On the one hand,\" Vidious repeated, \"I'm in a generous mood, as we just repaid Kodoc for attacking my sister. And we needed to chase him off, in any case. To keep him from getting in the way while we take care of our . . . real task.\"\n\nNisha swung onto the railing of the _Anvil Rose,_ her blond hair wild in the wind. \"Those slumkids helped us win, Vid,\" she called. \"With their well-timed distraction.\"\n\n\"On the other hand,\" Vidious said, ignoring his sister, \"I recall saying that if I saw you again, I'd send you crashing into the Fog.\"\n\n\"That was before the whole 'well-timed distraction' thing!\" I reminded him.\n\n\"True,\" he said. \"But I hate to go back on my word.\"\n\n\"If you drop us,\" Hazel told him, lifting her left hand, \"you drop _this,_ too.\"\n\nShe twirled her wrist, and the ring glinted on her finger.\n\nI held my breath as the captains exchanged a glance, and felt a glimmer of hope. Maybe we would actually pull this off. Then the thopper engine gasped, and my hope was replaced by anxiety. _Hurry up!_\n\n\"So _that's_ what the tetherboy was hiding yesterday,\" Nisha said, raking me with her gaze.\n\n\"Give me the diamond,\" Vidious told Hazel, \"and I'll refrain from sinking you.\"\n\n\"The diamond is yours,\" Hazel said, \"if you'll take us to Port Oro.\"\n\nNisha laughed, a silvery sound. \"You want to switch sides? Join the mutineers?\"\n\n\"Yes, ma'am,\" Hazel told her.\n\nThe thopper trembled again, and I saw Swedish wince. We were running on fumes.\n\n\"I'll bring you to the Port, poppet, when we're done here,\" Vidious called to Hazel, rubbing the scar on his cheek. \"But if the diamond's fake, I'll toss you overboard. Is that a deal?\"\n\n\"No,\" Hazel told him. \"If the diamond's fake, Captain _Nisha_ will throw us overboard. I'm not bringing my crew onto the _Night Tide_.\"\n\n\"Your engine is gasping,\" Vidious said. \"You're not in a position to\u2014\"\n\n\"Permission to come aboard?\" Hazel asked Nisha.\n\n\"You seem to have a talent for infuriating my brother,\" Nisha told her. \"I like that in a girl. Of course you're welcome aboard.\"\n\nHazel immediately said, \"Go, Swede, _now_!\"\n\nWhen Swedish tapped the keyboard, the thopper sputtered forward.\n\nThen, just twenty feet from the _Anvil Rose,_ the wings twitched.\n\nVapor spat from the vents.\n\nThe clockwork engine squealed.\n\nOur fuel ran out, and we fell from the sky.\n\n##\n\nTHE THOPPER PLUMMETED. I gasped, and Bea grabbed my hand. I wanted to tell everyone that I was sorry. Sorry for finding the diamond too late, sorry for not curing Mrs. E. Sorry for the Fog in my eye. I wanted to tell them that they weren't just my crew, they were my family.\n\nAnd . . . _WHAM!_\n\nA silver bolt slammed into the hull.\n\nThe thopper bucked, and I squeezed Bea's hand. Loretta screamed and flew overboard\u2014and kept screaming as she dangled from her inner-tube-knotted wrist.\n\nSwedish yelled, \"'Retta!\" and we stopped falling. The silver bolt was a harpoon. The _Anvil Rose_ had speared us from above.\n\nAfter the mutineers heaved the thopper onto the warship's deck, Hazel hopped down from her perch as I untied Loretta with trembling fingers. Swedish checked Mrs. E\u2014still somehow asleep in the cockpit\u2014and Bea stared around the _Anvil Rose_ in awe.\n\nI guess I did the same, once my heart stopped pounding. I'd never been on a warship before. Everything slotted smoothly together, the gyroscopes and gunwales, the sky sails and the airsailors swarming the ship.\n\n\"This,\" Bea whispered, \"is the _purplest_.\"\n\n\"Doesn't get any purpler,\" I agreed.\n\nShe giggled. \"That's not a word!\"\n\n\"Oh, but everyone says 'purplest.'\"\n\nShe stuck her tongue out at me. \"Listen to that engine, Chess! She's ticking as strong as a metal monkey.\"\n\nWe were so busy gaping at the rigging and wheelhouse that I almost missed Hazel curtsying to Captain Nisha, then thanking her for such a well-timed\u2014and accurate\u2014harpoon shot.\n\nHazel looked overwhelmed, so I lowered my head and crossed the deck to stand beside her. She tugged the ring from her finger and eyed the diamond for a minute, like she didn't want to say good-bye. Then she took a breath and handed it to the captain.\n\n\"I've never seen one so big,\" Nisha purred, holding the diamond to the light. \"But I'll give you a choice.\"\n\n\"What's that, ma'am?\" Hazel asked.\n\n\"You can pay for the trip with the ring, or you can work for your passage and keep it. A diamond will come in handy when we reach the Port.\"\n\n\"We can keep it?\" Hazel's smile looked like sunshine. \"Wow. Yes, please. Yes. Thanks!\"\n\n\"I warn you, I'm a demanding boss,\" the captain said, returning the diamond.\n\n\"We're a working crew, ma'am. In the slum, if you don't work, you don't eat. Oh, except for Mrs. E. She's fogsick, so one of us needs to stay with her.\"\n\n\"She slept through the air battle?\" Nisha toyed with her string of beads. \"That's not a good sign.\"\n\nI felt a flash of worry, but Hazel said more mildly than ever, \"Yes, Captain.\"\n\nNisha cocked an eyebrow at her. \"I'm starting to suspect that there's more to you than a bunch of junkyard bottom-feeders.\"\n\n\"Of course, ma'am,\" Hazel said. \"We're _mutineer_ bottom-feeders now.\"\n\nThe captain laughed softly. \"In that case, the old lady can have a cot in the surgery. You must bring her to the healers on the Port as soon as we land.\"\n\n\"Whoa!\" Loretta called, stepping toward a harpoon. \"Check it out, Swedish! This is one serious pig-sticker.\"\n\nSwedish looked up from Mrs. E. \"That'll come in handy . . . when pigs fly.\"\n\n\"What are you talking about?\" Loretta asked, patting the gleaming brass.\n\n\"It's a saying from Chess's old scrapbook,\" Swedish told her. \"It means something impossible, like 'a bunch of slumkids will escape from Lord Kodoc when pigs fly.'\"\n\n\"But pigs _don't_ fly.\"\n\n\"That's the point!\"\n\n\"Your point is that pigs don't fly?\" Loretta shot him a look. \"How about gerbils? Do they dance?\"\n\n\"Gerbils will dance when pigs fly.\"\n\n\"Now you're just messing with me.\"\n\n\"Um, Captain?\" I said as they happily bickered.\n\nNisha looked at me with her too-blue eyes. \"Yes?\"\n\n\"Uh, well . . .\" I wanted to ask if Kodoc was really beaten. I wanted to ask if she knew he was hunting me. But all I said was \"Um, about that diving platform on the bottom of Kodoc's airship . . .\"\n\nHazel came to my rescue. \"We're wondering what Lord Kodoc is looking for with all those tetherkids.\"\n\nCaptain Nisha's lips thinned. \"A fairy tale.\"\n\n\"You mean like 'Hairy Otter'?\" Loretta asked, looking up from the harpoon. \"Or 'Little Red Gliding Hood'?\"\n\n\"I mean a story about an impossible machine, hidden in the Fog, that Kodoc calls the Compass.\"\n\nI ducked my head. Did _everyone_ know about that? Well, of course the mutineers knew\u2014the fogheads must've told them. But what else did they know?\n\n\"Kodoc is obsessed with finding it,\" Nisha continued. \"And terrified that the Subassembly will get there first. Whoever finds the Compass controls the Fog . . . and whoever controls the Fog rules the world. Kodoc thinks he's close. He just needs divers.\"\n\nI didn't want Nisha thinking about Fog divers, so I jumped in, \"He sounds as cracked as a trampoline.\"\n\n\"Huh?\" she asked.\n\n\"Wait, that's not right. Um . . . glass! Cracked as a _glass_ trampoline.\"\n\n\"What's a trampoline?\" Captain Nisha asked.\n\n\"I'm not sure,\" I said. \"But Kodoc is a cracked one.\"\n\nShe looked at me oddly, then turned back to Hazel. \"I'll have one of the sailors show you around.\"\n\n\"Sheesh,\" I muttered. \"It was funny when Loretta said it.\"\n\n\"Cap'n?\" Bea called from the railing, her voice squeaking.\n\n\"Yes?\" Hazel and Nisha said at the same time.\n\n\"Over there.\" Bea pointed over the Fog. \"Look at Kodoc's airship.\"\n\nI followed them to the railing and squinted toward the _Teardrop,_ which was hovering just inside the cloud bank. She didn't seem any different. Airtroopers bustled around repairing damage from the fight, the propellers twirled, and the sails billowed.\n\n\"Looks like they're securing the deck.\" Hazel frowned. \"Bolting everything down.\"\n\n\"That's not all they're doing,\" Bea said.\n\nCaptain Nisha peered at Kodoc's ship through her spyglass. \"Hm. They're engaging some kind of mechanism. . . . What _is_ that? He's diverting the power of the engine into . . .\" She gasped. \"The\u2014the hull is cracking in half.\"\n\nHazel shaded her eyes. \"The propellers are separating.\"\n\n\"Vidious!\" Nisha shouted, still watching through her glass. \"Look to the _Teardrop_!\"\n\n\"Maybe he's sinking,\" I said hopefully. \"Maybe you damaged him worse than you thought.\"\n\n\"The ship's not sinking,\" Bea told me with a hint of awe in her voice. \"She's _changing_.\"\n\n\"What are you talking about?\"\n\n\"The props are unfolding, the hull's getting sleeker and longer. The balloon's the same, but the rigging is moving. Kodoc's bringing armor to bear and the guns\u2014are those guns? I can't see.\"\n\n\"Those are guns,\" Captain Nisha said tightly. \"Big guns. The geargirl's right, he's changing that ship into a . . .\"\n\n\"A warship,\" Bea said.\n\n\"I've never seen one so big,\" Hazel breathed.\n\n\"Even the name is changing,\" Nisha said. \"The letters are shifting around.\"\n\n\"The letters of _Teardrop_?\" I asked.\n\n\"Yes,\" Nisha said. \"Now they spell _Predator_.\"\n\n##\n\nHAZEL TURNED TO BEA. \"How long before the ship is done changing?\"\n\n\"Five or six hours,\" Bea said.\n\nNisha cocked an eyebrow. \"How would you know?\"\n\n\"I\u2014um.\" Bea blushed a bright pink. \"I can sort of tell.\"\n\n\"How fast is she?\" Hazel asked. \"Can Kodoc catch us?\"\n\nBea wrinkled her nose. \"These mutie ships\u2014\" She flushed again, glancing to the captain. \"I mean, these _mutineer_ ships are pretty fast, but with those props and that hull? The _Predator_ is faster.\"\n\n\"We'll see about that.\" Nisha turned to her brother and called, \"Looks like we're racing that warship back to the Port!\"\n\n\"Not until we get the package!\" Vidious replied from the deck of his ship.\n\n\"We can't beat _that_.\" She gestured at the _Predator_ with her spyglass. \"Check out those guns!\"\n\nVidious shook his head. \"We need the package.\"\n\n\"The package?\" Hazel asked, her brow furrowing. \"You're not smuggling anything _to_ the Rooftop, you're smuggling a package _from_ the Rooftop?\"\n\n\"We're trying to,\" Nisha said. \"Why?\"\n\nA crazy thought occurred to me. \"No way!\" I blurted, turning to Hazel. \"It can't be.\"\n\n\"Ten strips of kangaroo jerky says I'm right.\" Hazel looked at the captain. \"Um, does the name Cog Turning mean anything to you?\"\n\nNisha narrowed her eyes. \"You know Turning? If you know Turning . . .\" She fell silent for a moment, then slowly turned to me, a smile curling at her mouth. \"Wait. You always keep your eye hidden. . . .\"\n\nI ducked my head.\n\n\"Are you the kid?\" she asked, a hint of wonder in her voice. \"You are! You're the package!\"\n\n\"Um,\" I said, caught between my fear of admitting anything and my relief that the mutineers actually wanted to help us.\n\n\"But you're a _boy,_ \" she said, sounding disappointed.\n\n\"Ye-es,\" I said. \"Um. Sorry?\"\n\nShe reached into her vest and tossed a pouch to Hazel. \"A message came, ripped into shreds, mostly unreadable. Something about a girl with a . . . special eye. On the highest slopes. Cog Turning asked for a ship to bring her to the Port.\"\n\n\"You're the ship!\" Bea said, clapping her hands. \"You're the ship coming to smuggle us!\"\n\nHazel nodded. \"That's why they've been hanging around. I don't understand about this 'girl,' though.\"\n\n\"Cog Turning must've meant Mrs. E.\" I turned to the captain. \"He helped us because he wants to cure her.\"\n\n\"She might be one reason,\" Nisha said. \"But you're the other. We must keep you out of Kodoc's hands . . . if you truly are the child.\"\n\nI hunched a shoulder. \"I guess.\"\n\n\"May I see it?\" she asked. \"The mark?\"\n\nA flush of humiliation rose in my chest and I looked at my scuffed boots. I hated showing strangers my freak-eye.\n\n\"Chess.\" Hazel put a hand on my arm. \"Just this once.\"\n\nI kept my head down. My stomach soured at the thought of letting Nisha see my darkest secret, my deepest shame.\n\n\"Pretend she's me,\" Hazel said softly.\n\nNeither of them spoke for a few seconds, and I took a deep breath. Just this once. No big deal. Maybe I was a coward, but I could do this.\n\nI took a breath, then raised my head and brushed my hair from my face.\n\nWhen Captain Nisha saw the mark, surprise flashed on her face, mixed with disgust and maybe fascination. The Fog that had killed billions, the Fog that blanketed the earth, the Fog that swirled in an ocean of whiteness also glimmered inside my eye.\n\n\"You're _him,_ \" she said, a strange hitch in her voice. \"Born in the Fog.\"\n\nThe shame thickened in my throat as I tugged my hair back into place.\n\n\"Yes, ma'am.\" Hazel stepped closer to me. \"And we appreciate everything you're doing but, um, could you ask that sailor to show us around now? And could you not tell anyone about Chess?\"\n\n\"Only my brother,\" Nisha promised. \"He needs to know that we got the package.\"\n\n\"But Captain Nisha?\" Bea said, slipping beside Hazel. \"What're we going to do? We can't beat the _Predator_ to Port Oro.\"\n\n\"Of course we can,\" Nisha said.\n\nBea blinked her big green eyes at Nisha.\n\n\"Okay, the _Predator_ 's faster,\" Nisha admitted. \"But if we get close enough to the Port, other mutineers will help us fight Kodoc.\"\n\n\"Can we get close enough?\" Hazel asked.\n\n\"If I push the _Rose_ faster than she's ever flown?\" Nisha said. \"Maybe.\"\n\n##\n\nTHE MUTINEER WARSHIP WAS divided into four levels, with engine rooms, cargo holds, an armory, a surgery, and workshops for the mechanics and carpenters. Narrow barracks lined both sides of the hull, and a steamy kitchen huddled in the stern, where massive engines rattled the pots and pans.\n\nA grizzled sailor brought us to a cramped hallway in the bowels of the ship. The _whir-tick_ of the props sounded through the floor as he opened a wooden chest and rasped, \"After nightfall, you can string these hammocks across the corridor for beds.\"\n\n\"Are there any regular bunks?\" Loretta asked, biting her lower lip. \"I mean, that don't swing around?\"\n\n\"You're on an airship now, girl,\" he cackled.\n\n\"Don't worry,\" I told her. \"You'll get the _hang_ of it pretty soon.\"\n\nBea giggled and told Loretta, \"It's kind of fun, actually.\"\n\n\"Come along,\" the sailor said. \"The quartermaster will assign you jobs.\"\n\n\"How come we've got to work?\" Swedish grumbled as we followed the sailor across the ship.\n\n\"They let us keep the rock,\" Hazel told him.\n\n\"But they were looking for us anyway. We saved them a trip to the Rooftop.\"\n\nLoretta scratched her cheek. \"I guess that Cog Turning guy really called in some favors.\"\n\n\"Yeah,\" I said. \"I guess.\"\n\n\"Except . . . how much of this is about you?\"\n\nI hunched a shoulder. That part made me nervous. \"Not sure.\"\n\n\"Can I see your eye again?\"\n\n\"No.\"\n\n\"I'll show you the scar on my butt.\"\n\n\"Ew,\" I said.\n\n\"They should at least let me fly this heap,\" Swedish muttered.\n\n\"Just do your job,\" Hazel told him, and headed into the quartermaster's office.\n\nInside, a bald guy with brown teeth sat behind a cluttered desk, reading a ledger. He flipped a page. Then another. Finally, he looked up and inspected us. He didn't seem impressed.\n\n\"You. Big fella.\" He glowered at Swedish. \"What's your name?\"\n\n\"Swedish.\"\n\n\"Are you as strong as you look? Find the master carpenter. They're fixing some damage\u2014you'll fetch and carry.\" He eyed Loretta. \"Are you good for anything?\"\n\n\"I'm awesome at harpooning,\" she told him.\n\nHe quirked an eyebrow. \"Have you ever touched one?\"\n\n\"Sure.\"\n\n\"Before today?\"\n\n\"Well, no.\" She sighed. \"I'm good with a knife, though.\"\n\n\"Fine. Then you're on kitchen duty.\"\n\nLoretta grumbled.\n\n\"How about you, little one?\" he asked Bea. \"I hear you're quite the geargirl.\"\n\n\"She's staying with Mrs. E,\" Hazel said brusquely. Then she softened her tone. \"If you please, sir.\"\n\nThe quartermaster frowned at Bea. \"We could use you in the engine shaft\u2014some of the valves are hard to reach for anyone bigger'n a chipmunk. And you've never seen a sweeter clockwork than the one driving the _Anvil Rose_.\"\n\nBea shot a pleading look at Hazel. \"Can I, please? Pretty please?\"\n\n\"Mrs. E needs you,\" Hazel told her.\n\nBea sighed and said, \"No, thank you, sir.\"\n\nThe quartermaster scowled at Hazel. \"Fancy yourself in charge, do you? You like giving orders? I bet you're expecting I'll put you to work as a first lieutenant.\"\n\n\"I'm fairly good with a needle,\" Hazel told him. \"And a glue gun.\"\n\n\"Oh, you're a clever girl,\" he grumbled. \"Not getting above yourself. Fine, to the sailmakers with you.\" He turned to me. \"You're a tetherboy?\"\n\n\"Yup,\" I said. \"I mean, yes, sir.\"\n\n\"A little old for it, ain't you?\"\n\n\"Only because I haven't died yet.\"\n\nI hadn't meant it as a joke, but the quartermaster laughed. \"You'll work for the riggers, then.\"\n\nSo Swedish wandered off to haul planks while Hazel disappeared into the sailmakers' workshop. A toothless sailor helped Bea bring Mrs. E to the surgeon, while Loretta headed for the kitchen and I returned to the deck.\n\nShading my eyes against the sudden brightness, I scanned the distance behind us. A few clouds smudged the blue horizon . . . and a black dot hung in the air like a splinter in the sky. The _Predator_. She was so far back, I couldn't see her propellers or guns. But she was still there, still changing. Getting ready for the chase.\n\nWith a shaky breath, I turned away. I found the rigging crew easily enough, because they all looked like me, compact and wiry. The rigging master was a swarthy woman of about my height, even though she must've been twenty or something.\n\n\"You're Chess?\" she asked, eying me dubiously. \"What're you, a bottom-feeder?\"\n\n\"I prefer 'salvage engineer.'\"\n\nShe didn't smile. \"You scavenge in the Fog?\"\n\n\"Yes, ma'am,\" I said.\n\n\"Well, if you find yourself in the Fog in this job, it means you slipped off your rig and you ain't _never_ coming back.\"\n\n\"I'll try not to\u2014\"\n\n\"It would serve you right,\" she said. \"It's your fault we've got Kodoc after us, you and that crew of yours.\"\n\nI wanted to stick up for my crew, but I just ducked my head. Afraid to act, afraid to call attention to myself. Worrying that maybe Kodoc hadn't just given me nanites, he'd also given me cowardice.\n\nThe rigging master glared toward the distant speck. \"I know riggings, boy, and that ship's rigged to catch us. Just a matter of time.\"\n\nThen she sent me overboard in a harness to clean exhaust vents, and I spent the rest of the day scrubbing.\n\nAfter a dinner of tortilla and kimchi with the rigging crew, I checked the sky. The little black dot was twice as big, and as I watched, the setting sun glinted off the _Predator_ 's armor plating.\n\nComing closer.\n\nWhen I returned to the hallway, I found Hazel mending clothes, and everyone else gently swinging in their hammocks. Bea was fiddling with wire, making a twisty, while Loretta and Swedish held hands as they rocked. Part of me wanted to talk about Kodoc and the _Predator,_ but more of me wanted a break, wanted to pretend that my life wasn't a total mess.\n\nSo I flopped down beside Hazel and loosened my bootstraps. \"You want a hand? You know I sew better than you.\"\n\n\"You're good with a glue gun, too,\" she said.\n\nSwedish snorted a laugh.\n\n\"What?\" Loretta asked.\n\n\"A couple years ago,\" Swedish told her, \"Chess glued his thumb to his face.\"\n\n\"I was a little kid! I fell asleep holding a glue gun.\" I threw my boot at Swedish, then asked Bea, \"How's Mrs. E?\"\n\nShe wrinkled her nose. \"The same.\"\n\n\"Was she rambling again?\" Hazel asked.\n\n\"No,\" Bea said. \"She slept all afternoon.\"\n\nI rubbed my face. \"Can we see her?\"\n\n\"The doctor says she needs peace and quiet. No visitors.\"\n\n\"Maybe tomorrow,\" Hazel told me.\n\n\"There's no reason for me to sit there!\" Bea blurted. \"She's always sleeping, and I want to see the engine. I want to _do_ something.\"\n\n\"I'm sorry,\" Hazel said. \"But no.\"\n\n\"It's not fair! Just because I'm the littlest, you don't let me do anything! I won't hurt myself, I promise.\"\n\n\"I'm not afraid you'll hurt yourself,\" Hazel said, looking up from her sewing. \"I'm afraid they'll see how good you are and never let you leave.\"\n\n\"Captain Nisha would never do that.\"\n\nShe was probably right, but I knew the real reason Hazel wouldn't let her work on the engine. We'd almost died 287 times in the past few days, and Kodoc was still behind us, madder than ever. The last thing we needed was to put Bea in any more danger.\n\n\"And anyway,\" Bea continued, \"I'm not _that_ good.\"\n\n\"You're worth more than the diamond,\" Hazel told her.\n\n\"Worth almost as much as berry pie,\" Swedish said. \"With berries on top.\"\n\n\"Plus,\" I added, \"you're completely whackadoo. Talking to engines all the time. They'd probably lock you up.\"\n\n\"Fine!\" Bea scowled. \"I'll stay at the surgery.\"\n\nAfter a short pause, Hazel said, \"How about you, Loretta? How was the kitchen?\"\n\nLoretta sighed. \"I spent all afternoon grinding fish bones into powder, then mixing the powder into paste.\" She scratched her scarred arm morosely. \"Do you know what they've got in the kitchen?\"\n\n\"Fish paste?\" Swedish guessed.\n\n_\"Food.\"_ Loretta flashed her gap-toothed smile. \"Oh, it's like a beautiful dream, Swede! You've never seen so much food.\"\n\nHazel and I climbed into our hammocks as Loretta described mounds of pickled eel, stacks of roti, and simmering pots of Chungking pigeon. When she finished, we listened to the distant shanties of the airsailors celebrating their victory over Kodoc.\n\nThen, after a lull, a voice carried to us: \". . . until that _Predator_ swoops down and kills every last sailor on board. Don't kid yourself, she'll catch us easy, long before we reach the Port. . . .\"\n\nThe voice faded, but the air in the hallway suddenly felt thick and itchy. The sailor was right: the mutie warships were fast, and fighting for every bit of speed, but the _Predator_ looked even faster.\n\n\"Tell us a story, Chess,\" Bea said in a small voice. \"From the scrapbook.\"\n\n\"What's a scrapbook?\" Loretta asked.\n\n\"My dad collected notes from the time before the Fog,\" I told her. \"Any information he could find. I guess . . . I guess it's gone now.\"\n\nFor a second, I wanted to cry. Not only was Kodoc closing in, but the scrapbook\u2014my only real link to my father\u2014had been ditched along with the shack. It was in the Fog now, lost forever.\n\n\"You memorized the whole thing anyway,\" Swedish said.\n\nI took a breath. \"Mostly.\"\n\n\"Tell us the one about the sailor,\" Hazel said in the darkness.\n\nI swayed in my hammock as I tried to remember. It was better than thinking about Kodoc. \"Okay. This is from a diary.\"\n\n\"A what?\" Loretta asked.\n\n\"Like a logbook. The guy who wrote it said, 'In the town where I was born, there lived a man who sailed the seas. He lived beneath the waves in a yellow submarine.' Then there's a bit about tangerine trees and marmalade skies and a girl with colliding scope eyes.\"\n\n\"What's marmalade?\" Loretta asked.\n\n\"Some kind of poison?\" I guessed. \"Like the Fog? Maybe that's why they lived beneath the sea.\"\n\nShe eyed me in disgust. \"You don't believe that, do you?\"\n\n\"Well, it's possible.\"\n\n\"There's no such thing as a 'sea.' There's never been that much water.\"\n\n\"Yeah,\" Bea agreed. \"That part's silly.\"\n\n\"There _were_ seas,\" Hazel told them. \"Full of whales and squids and squarepants.\"\n\nWe'd had this argument a hundred times, so I changed the subject. I told them how in the old days, people searched \"the googol\" for answers to their questions.\n\n\"So the googol was a fortune-teller?\" Loretta asked. \"Like palm reading?\"\n\n\"The googol is a number,\" I told her. \"A one with a hundred zeros after it.\"\n\n\"One zero zero zero zero . . . a hundred times?\"\n\nI nodded. \"They wrote down the googol and stared at it until they found the answer they needed.\"\n\n\"Didn't work for you,\" Swedish said.\n\n\"You tried it?\" Loretta asked me.\n\n\"Once,\" I said. \"But the only thing I found was a headache.\"\n\nHazel turned down the lantern a few minutes later, and I closed my eyes and listened to the unfamiliar whir of the mutie propellers. Every time I started to drift off, I imagined the _Predator_ sweeping through the night like a driftshark through Fog.\n\nFinally Bea said, \"I can't sleep. Tell me the story.\"\n\n\"Not again!\" Swedish grumbled from the darkness.\n\n\"Please?\" Bea pleaded. \"Pretty please with pickled eels?\"\n\n\"What story?\" Loretta asked.\n\n\"The secret history of the world,\" Bea told her. \"Mrs. E told us.\"\n\n\"Before the Fog rose,\" I started, \"there was something called the Smog, which covered the whole earth and made _everything_ sick. Not just people. . . .\"\n\nWhen I finished, Bea made Hazel tell the story of the red-caped hero named Superbowl, who leaped mountains in a single bound and threw pigskins.\n\nThen a new mutie song started above us, softer and slower than the others, and we listened in silence. We were in the depths of a warship, heading to an unknown city, fleeing from Kodoc and trying to save Mrs. E. We didn't have the raft or the shack\u2014or even the slum. But we still had each other. And, as we swayed in the dark hallway, that thought lulled me to sleep.\n\n##\n\nTHE NEXT MORNING I woke before dawn. I swayed in my hammock and heard a faint metallic whine. The gears were straining for speed, the pistons firing hot and fast. Captain Nisha was pushing the _Anvil Rose_ hard, trying to stay ahead of Kodoc.\n\nMaybe too hard. If the engine failed, we were dead in the air.\n\nI rolled from my hammock and crept away barefoot so I wouldn't wake the others. Around the corner, I tugged my boots onto my feet, then crossed the ship and slipped into the surgery.\n\nThe doctor snored in her hammock, bottles of medicine rattled in racks, and Mrs. E slept in a cot with high railings to keep her from falling out. Her skin was an unhealthy white, with red smudges on her cheeks. I stood there for a long time, watching her sleep. She'd given me everything I'd ever loved, everything good in my life since my dad died, and I still needed her. We all needed her.\n\nWhen I returned to the hammocks, Hazel and Bea were lying on their stomachs in the middle of the hallway. Heads together, chatting softly, with dozens of scraps of paper scattered on the floor between them.\n\n\"What're those?\" I asked.\n\n\"They're bits of the message that Turning sent to the Port,\" Hazel told me.\n\n\"Ripped into a hundred pieces,\" Loretta said from her hammock. \"Looks like a hawk caught the carrier pigeon.\"\n\n\"Sounds tasty.\" Swedish yawned. \"Lucky hawk.\"\n\n\"Look at this, Chess.\" Hazel pointed to an uneven rectangle of paper scraps. \"This is the part they already deciphered.\"\n\nThe paper was crumpled and dirty, with a few splotches of what must've been pigeon blood. Still, some of the words were clear:\n\n_. . . found the child with . . . eye. Kodoc in pursuit . . . girl, on the highest slopes. . . . Send a ship to bring . . . to the Port . . . igh alert_.\n\n\"Turning said I was a girl?\" I rubbed the back of my neck. \"I thought he knew all about me.\"\n\n\"Yeah,\" Hazel said. \"I don't know how he got that wrong.\"\n\nSwedish squatted beside Bea. \"Maybe he didn't.\"\n\n\"You mean Chess really is a girl?\" Loretta asked.\n\nI ignored her as a spark of hope ignited in my chest. \"He means there might be _two_ of us! _Another_ kid with a freak-eye!\"\n\nThat'd be awesome. If some upper slopes girl had a Fog-eye, I wouldn't be such a freak.\n\n\"We'll find out tonight,\" Hazel said, \"when we put the rest of the pieces together.\"\n\n\"Wait a second.\" Swedish messed with the bits of paper. \"I'm good at puzzles.\"\n\n\"Bea and I have been at this since we woke up, Swede,\" Hazel said. \"You're not going to figure it out before breakfast.\"\n\n\"So, uh, you all know how to read, huh?\" Loretta scratched the tattoo on her cheek. \"Like, whole words and everything?\"\n\n\"They do,\" Bea said. \"I'm still learning.\"\n\nLoretta glowered. \"Reading is stupid.\"\n\n\"You can learn with me, if you want,\" Bea told her. \"Swedish will teach us together.\"\n\nLoretta squinted at Swedish, who was shuffling scraps of paper on the floor. \"He will?\"\n\n\"Course he will,\" Bea said.\n\n\"That way you can read his love notes,\" I teased.\n\nLoretta scowled, but I could tell she liked the idea. \"You really think I can learn?\"\n\n\"Sure you can,\" I said.\n\n\"It's easier than piecing together a hundred scraps of paper,\" Hazel told her. \"C'mon, Swede, let's go.\"\n\n\"I'm almost done,\" he told her, moving one last piece of paper into place. \"There!\"\n\nHe looked at the message on the floor and read aloud:\n\n_. . . found the child with the eye. Kodoc in pursuit . . . he doesn't even know if the child is a boy or girl, on the highest slopes or . . . lowest. Send a ship to bring . . . to the Port. Find me. I'll lead you . . . nd take care. Kodoc on high alert. Noti . . ._\n\n\"Oh,\" Bea said. \"He was saying that _Kodoc_ didn't know if Chess was a girl.\"\n\nHazel gaped at Swedish. \"How did you do that?\"\n\n\"Everything fits together,\" Swedish loftily informed her, \"if you know how to look.\"\n\n\"So I guess I'm the only freak,\" I muttered unhappily.\n\n\"You're not the only freak.\" Loretta elbowed me. \"Did you see Swedish solve that puzzle? Now _that_ was freaky.\"\n\nAs I unstrung my hammock, I thought about the message. Bits were still missing, but it looked like Cog Turning hadn't even mentioned Mrs. E. He'd only told them about me. Why did the Port care so much? Did they just want to keep me out of Kodoc's hands, so he couldn't find the Compass? Or did they want the Compass for themselves?\n\n\"C'mon!\" Bea said. \"We'll miss breakfast.\"\n\nWe headed to the sweep deck, where the rising sun brushed the Fog with yellow fire and the breeze blew fresh and cold. I shivered, then opened my jacket so I could wrap Bea in half of it. We tore into the bread that Loretta had swiped the previous day and eyed the black smear hanging in the air. The _Predator_ looked bigger than a thumbnail this morning. More like a fist.\n\n\"Kodoc's getting closer,\" I said.\n\n\"Only a little,\" Loretta said. \"Maybe he's not as fast as you think.\"\n\n\"He's even faster-er now,\" Bea said.\n\nSwedish eyed her. \"What do you mean, 'now'?\"\n\n\"Listen to the engine, Swede.\" Bea's lower lip trembled. \"She's all trembly and overheated. She can't keep up this speed.\"\n\nHazel put a hand on Bea's shoulder. \"She'll be okay, Bea. The captain knows what she's doing.\"\n\n\"She's trying to get away from Kodoc.\" Bea's voice softened into a whisper. \"But even pushing the _Rose_ this hard, she can't.\"\n\n##\n\nI SPENT THE REST of the morning dangling over the edge of the warship, scrubbing vents. Waves of vaporous Fog swirled and rippled beneath me, and part of me wanted to dive\u2014but the other part enjoyed the break. Scrubbing at splattered insects wasn't exactly _fun,_ but at least it was easy.\n\nAt lunchtime, I unhitched my harness and headed for the sailmakers to find Hazel.\n\nHalfway there, I heard a shout from the other side of the ship: Loretta's shout. _Uh-oh_. I trotted across the side deck, raced around a corner . . . and skidded to a halt.\n\nLoretta was standing at the harpoon with a huge gap-toothed smile on her face.\n\n\"That's enough for today.\" Captain Nisha clapped her on the back. \"Off to the kitchen with you.\"\n\nLoretta saluted a few times, then jogged over to me. \"The captain's showing me how to fire the harpoon!\"\n\n_\"Why?\"_ I asked in horror. \"You're bad enough with a knife.\"\n\n\"Hazel asked her to!\" Loretta beamed. \"She's the best.\"\n\n\"Sometimes I wish she'd mind her own business.\"\n\n\"You should see the captain shoot!\" Loretta said, scratching her spiky hair. \"She can put a harpoon through a goose's eye at a hundred yards.\"\n\n\"Don't say that in front of Chess,\" Swedish told her, falling into step with us. \"He loves geese.\"\n\n\"Swede!\" Loretta grabbed his hand, her face shining with excitement. \"Captain says I'm really good for a beginner. She's says I've got the eye for it.\"\n\n\"Oh, great,\" I muttered. \"Now _everyone_ wants the eye.\"\n\nSwedish snorted a laugh, and we headed belowdecks as Loretta babbled about her harpoon lessons. We met Hazel and Bea at the foggium tanks and waited in line behind a couple of welders for bowls of oxtail stew. We spent five minutes stuffing our faces, then another five chatting about the name. Why call it \"oxtail stew\"? Everyone knew it was made from camel tails.\n\nAfter we licked our bowls clean, we headed outside and scanned the sky. An uneasy silence fell, but nobody spotted the _Predator_.\n\n\"Where is he?\" Bea asked, frowning into the distance.\n\nFor some reason, not seeing Kodoc at all was even scarier than watching him get closer.\n\n\"He's doing something.\" Swedish scowled. \"He's _planning_ something.\"\n\n\"Maybe we lost him,\" Loretta said.\n\n\"Sure, I'll believe that when gerbils dance.\"\n\nHazel nodded toward the crow's nest at the top of the rigging. \"The captain's got two lookouts on duty. They'll spot Kodoc when he gets close.\"\n\n\"When he gets _too_ close,\" Bea said, wrinkling her nose. \"And . . . and I can't stand it!\"\n\n\"We're all scared, honeybee.\" Hazel gave Bea a squeeze. \"Knowing he's coming, even though we can't see him\u2014\"\n\n\"Not that! I mean the engine. Listen to the _Rose_ \u2014she's crying. She's too hot. She's sweating and crying and she needs help.\"\n\n\"Captain Nisha knows what she's doing.\"\n\nBea clamped her jaw. \"I know what _I'm_ doing, too.\"\n\n\"You're looking after Mrs. E.\"\n\n\"Would you tell her?\" Bea demanded, turning to me. \"Tell Hazel I should help with the engine.\"\n\nBut I barely heard her. I just kept looking into the distance, where I'd last seen Kodoc's ship. Thinking of his face when he'd said, _Your name is whatever I call you_.\n\nAnd the tremor in my voice when I'd replied, _Yes, sir_.\n\nAt sunset, I met the crew in the surgery and we watched Mrs. E sleep. She looked so weak that a lump rose in my throat. The surgeon treated Mrs. E kindly and promised she'd keep her alive until we reached Port Oro, but she didn't _know_ Mrs. E. All she saw was a tiny, helpless old lady. She didn't know that Mrs. E was a giant. She didn't know Mrs. E was the biggest thing in our lives.\n\nAn hour later, we swayed in our hammocks in the dark hallway, but I couldn't sleep. My mind whirled with worries and questions.\n\nFinally I spoke. \"How did Kodoc recognize me?\"\n\nNobody answered.\n\nThen Bea's soft voice said, \"He knew your mom and dad, I guess.\"\n\nI pulled my blanket tighter. \"He said he knew me.\"\n\n\"Well, you have to remember,\" Swedish said, \"he's whackadoo.\"\n\n\"He probably noticed you hiding your face,\" Bea said.\n\n\"What's the story with your eye?\" Loretta asked me. \"I mean, Swedish told me about Mrs. E saving you, but . . . can you feel the white gunk in there?\"\n\n\"'Retta,\" Swedish said warningly. \"He doesn't like talking about it.\"\n\n\"No, that's okay.\" I turned toward the dark shape of Loretta's hammock. \"It feels just like my normal eye.\"\n\n\"Can you see out of it?\"\n\n\"Yeah, I don't even notice the white gunk.\"\n\n\"The gunk isn't so bad, but the way it moves around like it's alive . . .\" Loretta's voice trailed off, and I could almost hear her shudder. _\"Gross.\"_\n\n\"Yeah,\" I said.\n\n\"It works, though,\" she said. \"That's all that matters.\"\n\nLike that settled it. Like my freak-eye wasn't worth talking about. Sure, I had clouds of Fog drifting around my pupil, but so what? I found myself smiling in the darkness. Only Loretta could call you gross and make you feel better at the same time.\n\n\"At least I don't have a scar on my butt,\" I said.\n\n\"That's the most important lesson I ever learned,\" she said. _\"Never sit on a barbed-wire fence.\"_ A wind buffeted the airship, and she groaned. \"Well, that and _stay away from hammocks_.\"\n\nSwedish said, \"I still don't understand why Kodoc wants Chess in the first place.\"\n\n\"For his search,\" Loretta said. \"For this fog-machine.\"\n\n\"But he's already got tetherkids,\" Bea said. \"Chess isn't _that_ much better.\"\n\n\"Yeah, he is,\" Hazel said, her voice rough in the darkness. \"When Kodoc forced his mom to give birth in the Fog, he gave Chess powers, he made Chess into\u2014\"\n\n\"His freak.\" I frowned in the darkness. \"I'm, like, his creation.\"\n\n\"Well,\" Swedish said, \"Kodoc can scream all he wants that you're his\u2014\"\n\n\"But you're not,\" Hazel finished. \"You're _ours_.\"\n\n##\n\nOUR THIRD DAY ON the _Anvil Rose_ dawned gray and misty. Life on the warship was starting to feel almost normal, and I contentedly cleaned vents for hours, until the rigging master told me to scrub the lifeboats lashed to the hull.\n\nThree of them were fastened to each side of the airship, stripped-down craft attached to foggium tanks for emergency inflation. Except one was different. One was a disguised cargo raft, just like on the _Night Tide_.\n\nI swung over for a closer look. Then I whistled. It was totally sweet, like a one-balloon version of our salvage raft, but built for speed and stealth\u2014for smuggling.\n\nFor smuggling _me_. For snatching me from the Rooftop to the safety of Port Oro.\n\n\"Which is weird,\" I muttered.\n\nI'd always known I was a freak, but I'd never suspected I was an _important_ freak. Still, at least life on the warship was pretty good. I scrubbed, Hazel sewed, Swedish hauled, Loretta washed pots and fired harpoons. And Bea sat with Mrs. E and made dozens of twistys.\n\nBefore lunch that day, the rest of us joined Bea in the cramped surgery and gathered around Mrs. E.\n\n\"How's she doing?\" Hazel asked the surgeon.\n\n\"She's getting worse. She needs the Subassembly.\"\n\n\"They can really help her?\" Swedish asked.\n\n\"You heard Cog Turning,\" Bea told him.\n\n\"Yeah, but he's a foghead. They _always_ talk loco.\"\n\n\"They know the sickness better than anyone,\" the surgeon said. \"You need to bring her straight to them when we reach the Port.\"\n\n\"We will,\" I promised.\n\n\" _If_ we reach the Port,\" the surgeon muttered under her breath.\n\nMrs. E tossed fitfully and made a whimpering sound. \"Chess . . . ?\"\n\nLeaning closer, I took her cool hand. \"I'm right here.\"\n\n\"Under my . . . pillow,\" she whispered, then drifted back to sleep.\n\n\"What'd she say?\" Bea asked.\n\nI eased my hand under the pillow and felt a hard edge. \"There's something here. . . .\"\n\nWhen I pulled out my scrapbook, my breath caught. I opened the first page and saw my mother's name, and my father's, and mine. Written in my dad's handwriting. I blinked a few times, fighting back tears.\n\n\"She must've hidden it in her coat,\" Bea said, her voice soft.\n\nI cleared my throat. \"We've got to get her to Port Oro.\"\n\n\"We will,\" Hazel promised me.\n\n\"Unless Kodoc stops us,\" Swedish said.\n\n\"We'll get there,\" Hazel said, glaring at him, \"But for now, we just keep our heads down and do our jobs.\"\n\nKeeping my head down sounded smart. Not only because I _always_ kept my head down, but also because the mutineers were scary. Even the friendly ones didn't enjoy being hunted by Lord Kodoc, and some of the less friendly ones blamed us outright.\n\nAfter we grabbed bowls of stockfish and rice for lunch, a big sailor shouldered Swedish\u2014hard. Swedish's face flushed, but he kept walking.\n\nThe big sailor shoved him again. \"Watch where you're going, roof rat.\"\n\n\"I don't want any trouble,\" Swedish mumbled.\n\n\"Neither did we, but thanks to you we've got a deathship chasing us across the sky.\"\n\n\"We'll reach Port Oro in time\u2014\"\n\n\"We _won't_!\" the sailor snarled. \"Listen to the _Anvil Rose_ \u2014she's breaking apart. The captain's melting the engine into slag to save your sorry hides.\"\n\n\"Then talk to her,\" Swedish said, turning away. \"Not me.\"\n\nThe sailor shoved Swedish from behind, sending him sprawling to the ground. \"I'll melt _you_ into slag.\"\n\nThe other mutineers jeered, eager for a fight, and I tugged my hair lower. _Okay, then_. I wasn't much of a brawler, but I'd rather get stomped than sit there and watch some guy beat up Swedish.\n\n\"Chess,\" Hazel hissed. \"Stay put. He doesn't need your help.\"\n\nI frowned in surprise. Usually she'd throw herself into a fight if anyone messed with one of us. But I stayed put.\n\nThe big sailor loomed over him. \"Get up, bottom-feeder.\"\n\n\"The reason he doesn't want to fight,\" Loretta called, pushing to her feet, \"is because we're here as guests.\"\n\n\"Mind your own business, girl,\" the big sailor growled.\n\n\"See, we don't want to offend the captain.\" Loretta strolled toward him, though the top of her head didn't even reach his chin. \"Which might happen if we beat one of her dogs.\"\n\nThe big sailor glared. \"You calling me a dog?\"\n\n\"Nah,\" she told him. \"Up close you look more like a cockroach.\"\n\n\"Are you blind? I'm twice as big as you.\"\n\nLoretta put her hands in her pockets and shrugged. \"You're bigger, I'll give you that.\"\n\n\"I'm twice as strong.\"\n\n\"Lots stronger,\" she agreed.\n\n\"And twice as mean,\" he said.\n\nLoretta flashed her gap-tooth smile, pulled her hands from her pockets and threw a fistful of crushed fishbones into the sailor's eyes.\n\nWhen he grabbed his face, she slipped to the side and kicked him in the knee with her metal-toed boot. The sailor grunted and collapsed to the deck\u2014and Loretta slammed his ear with her forehead, then punched him in the neck.\n\nAs she lunged forward again, Swedish grabbed her arm to stop her.\n\nAn uneasy silence fell on the deck, broken only by the gasping of the injured sailor. Then the other mutineers muttered angrily, and I felt a flutter of fear.\n\n\"You think that's bad?\" Hazel asked, her voice cutting through the silence. \"You should see her _cook_!\"\n\nWhen one of the soldiers laughed, the tension faded. Two other airsailors helped the big guy limp to the surgery, and everyone else returned to their meals.\n\nI was still hunched nervously over my bowl five minutes later when a metallic shriek sounded. Then voices shouted, boots tromped against the deck, and a cloud of stinking exhaust surrounded us.\n\nHazel touched my arm and glanced toward Bea, who was gripping her bowl so tightly that her fingers were white. That shriek must've sounded like a cry of agony to her.\n\nI knew what Hazel wanted. \"Come on, Bea,\" I said, pushing to my feet. \"Let's find the captain.\"\n\nBea's wide green gaze cut to Hazel. \"Really?\"\n\nHazel nodded. \"The _Rose_ needs you.\"\n\nThe three of us headed below, past a dozen frantic gearslingers, and found Nisha in a smoky corridor outside the engine room. \"Captain,\" Hazel said, \"I think you might\u2014\"\n\n\"Not now!\" Nisha snapped, her face sweaty and her blond hair limp. \"I've got flywheels tearing my ship apart.\"\n\n\"Bea can fix this,\" Hazel said.\n\n\"She's a child. Get out of here before I\u2014\"\n\n\"She knows engines the way Chess knows Fog,\" Hazel told Nisha. \"I swear to you, by the wind and the water, that if Bea cannot fix this, it cannot be fixed.\"\n\nNisha eyed Hazel for two seconds, then looked at Bea. \"Get in there and prove your captain right.\"\n\nBea scampered into the engine room, Nisha followed, and I stared at Hazel.\n\n\"'By the wind and the water?'\" I asked, disbelieving. \"How do you come up with these things?\"\n\n\"I don't know,\" she said, flashing a grin. \"It just came out.\"\n\nWhen we returned to the deck, the _Night Tide_ was hovering alongside the _Anvil Rose_. \"You can't evacuate onto my ship,\" Vidious was yelling to one of Nisha's officers. \"My clockwork's ready to blow, too. I can't keep up this speed much longer.\"\n\nThe clatter of gears from the engine room drowned out the officer's reply.\n\n\"Either we slow down and let Kodoc catch us,\" Vidious answered, \"or we think of a way to cool the engine that I've never seen before!\"\n\nAn explosion shook the air.\n\nA hole ripped through the side of the _Anvil Rose,_ splintering the wood and belching gray smoke. I rushed to the railing through a crowd of shocked airsailors, and a little voice came from the cloud of exhaust.\n\n\"Sorry!\" Bea called. \"I had to move things around a little!\"\n\nTwo seconds later, the grind of the engine softened. The clockwork ticked smoothly, and the unhealthy whine of overheated machinery quieted. The _Anvil Rose_ was running cool again.\n\n\"Well, _that's_ something I've never seen before,\" Vidious said, a lopsided grin on his scarred face. \"Send that geargirl to blow holes in my ship when she's done over there.\"\n\nBea fixed the _Night Tide_ after she finished with the _Rose_ \u2014then the airsailors on both ships cheered her, and she flushed as red as a beet.\n\nWe told the story of her gearslinging triumph to a sleeping Mrs. E, then retold it twice after dinner, embarrassing Bea a little more each time. So we reenacted the whole thing, with me playing Bea and Swedish and Loretta playing the airships.\n\nFinally, we stopped teasing her and settled into our hammocks. Bea swayed contentedly, and I eyed Hazel.\n\n\"What?\" Hazel asked.\n\n\"At lunchtime? You _wanted_ Loretta to fight that big mutineer.\"\n\nHer bright brown gaze flicked toward me. \"Why did I want that?\"\n\n\"Because they think Loretta's just a pint-sized kid. So losing to her is worse than losing to Swede. Now they won't bother us again.\"\n\nHazel tied a ribbon around one of her braids. \"She's like Bea. Easy to underestimate.\"\n\n\"Ain't that the truth!\" Loretta crowed.\n\n\"But you cheated!\" Bea told her. \"Throwing that powder in his face.\"\n\n\"Look who's talking. You blew a hole in the ship.\"\n\n\"I had to! You cheated on purpose.\"\n\n\"Of course I did,\" Loretta said scornfully. \"I never could've beat him in a _fair_ fight.\"\n\n##\n\nON THE MORNING OF the fourth day, excitement buzzed through the ship. Snatches of cheerful shanties sounded from passing airsailors, and the cook heaped extra eel in our bowls. Nisha and Vidious started flying side by side, with the _Night Tide_ almost close enough to touch. The crews shouted to each other, friendly jeers and rude taunts\u2014and more laughter than I'd ever imagined on a mutineer warship.\n\nNot only were the ships flying fast and strong, thanks to Bea's fixes, but today we'd reach Port Oro. And the _Predator_ still hadn't reappeared.\n\nDangling in my harness beside the hull, I found myself smiling. A cure for Mrs. E, freedom from Kodoc, and a big honking diamond: life didn't get any better than that.\n\nThe wind fluttered my hair, and the Fog pooled and rippled fifty yards below me. After four days in a warship, I missed diving. It was stupid and dangerous but I longed for the freedom, the speed, and the thrill.\n\nI pulled the scrapbook from inside my jacket and ran my hand over the cover, thinking about my father. Then my fingers snagged on a curl of plastic sticking between two pages, like a bookmark.\n\nHuh. Had Mrs. E left that for me?\n\nI opened to the marked page, and found the scraps about \"Skywalker Trek.\" I knew the whole thing was just a story . . . but sometimes, late at night, I still prayed that a fleet of spaceships would rescue us. They'd send the _X-Wing Enterprise_ to gather everyone who'd survived the Fog and bring us to a lush, green world where clear land stretched to the horizon.\n\nI'd told Mrs. E about that once, as we'd sat outside the shack. We'd watched the starry sky for a long time, and then she'd said, \"What if the _X-Wing Enterprise_ already landed? What if the rescue party is already here?\"\n\n\"Swedish is totally a Klingon,\" I said, \"and nobody's more Princess Solo than Hazel.\"\n\nShe'd ruffled my hair. \"And you're my young Fogwalker.\"\n\nSmiling to myself, I tucked the scrapbook away. I missed Mrs. E. And now that we were just a few hours away from Port Oro, the fogheads would heal her, and we'd be a family again.\n\nA sharp cry cut into my reverie: \"Ship on the horizon! The _Predator_ 's closing fast, Captain!\"\n\nMy smile died. Not _now_. Not when we were so close. My skin prickled and I felt Kodoc across the miles. His hot, hungry breath touched my neck like a jaguar about to sink its teeth into a jackrabbit.\n\nI spun in my harness, my breath ragged. Without a spyglass, I couldn't see anything except Fog and storm clouds, so I just slumped there, limp and shivering.\n\n\"Tetherboy?\" the rigging master called down. \"I'm bringing you topside!\"\n\nMy mouth was too dry to answer.\n\n\"Tetherboy?\" she shouted again.\n\nI licked my lips. \"R-ready!\"\n\n\"Lose the harness,\" she said after winching me up. \"Captain Nisha's looking for you.\"\n\n\"What does she want?\"\n\n\"Whatever she wants, you say 'yes, ma'am.'\"\n\n\"Yes, ma'am,\" I said, trying to cover my fear with humor.\n\n\"You're not bad, for a bottom-feeder. Now get moving.\"\n\nNumb and frightened, I trotted along the gangway. I stumbled down the steps from the spar deck and found the crew clustered around the captain.\n\nI slunk into place beside Bea, my shoulders hunched.\n\n\"Have a look,\" Captain Nisha was saying, handing her spyglass to Hazel. \"Do you see him?\"\n\nHazel scanned the distance. \"No.\"\n\n\"Neither did I. Look higher.\"\n\nHazel tilted the spyglass upward. \"Where?\"\n\n\"Higher.\"\n\nHazel gasped. \"Oh, _no_.\"\n\n\"That's why we haven't spotted him,\" Nisha told us. \"Kodoc camouflaged the underside of his ship, then spent the last few days climbing. Higher and higher, into the air that's almost too thin to breathe. Then all through last night . . .\"\n\n\"He dove like a bird of prey.\" The spyglass trembled in Hazel's hand. \"Like a peregrine falcon.\"\n\nMy breath caught. That was a fact from the scrapbook. A peregrine falcon normally flies at fifty miles per hour, but when it dives for the kill? It reaches two _hundred_ miles per hour.\n\nHazel didn't say anything for a few terrible seconds. \"He'll hit us in an hour. Two hours if you overcrank the props. But your engine's still weak. . . .\"\n\nNisha tapped the hilt of her dagger. \"You're a born airsailor, Hazel. I'd offer to make you a lieutenant when we reach the Port\u2014\"\n\n\" _If_ we reach it,\" Swedish muttered.\n\n\"\u2014but you'd say no, wouldn't you?\" Nisha finished.\n\nHazel nodded. \"Yes, ma'am.\"\n\n\"Because you already have a crew.\" Nisha glanced at the rest of us.\n\n\"Born to bottom-feed,\" Loretta said.\n\nSwedish nudged her. \"Doomed to dive.\"\n\nThe captain ignored them. \"I _am_ overcranking the props, Hazel. I'm pouring everything into speed and hoping we don't melt down. He'll still hit us in two hours.\"\n\n\"How far are we from the Port?\"\n\n\"Three hours.\" Nisha jerked her thumb toward the opposite railing. \"Maybe a little less. You see that smoke on the horizon? That's the Port.\"\n\nMy heart started racing. Three hours from Port Oro! We'd come so _close_. . . .\n\n\"Um,\" Bea said in a little voice. \"So we're three hours from the Port, and Kodoc is only two hours from us?\"\n\nCaptain Nisha nodded. \"He'll catch us an hour from home.\"\n\n\"Will your friends help?\" Bea asked.\n\n\"She means the mutineer navy,\" Hazel said. \"Can't they fly out to stop Kodoc?\"\n\n\"They're too far away,\" Captain Nisha said, her blond hair half covering her face. \"By the time they realize what's happening, Kodoc will be on us.\"\n\n\"And they're too timid, poppet,\" Vidious called to Hazel. \"You might as well ask a turtle to leave his shell.\"\n\n\"They'll try to help,\" Nisha insisted. \"I'll signal them. But there's no way they'll reach us in time.\"\n\n\"So what are we gonna do?\" Loretta asked.\n\n\"There's only one thing _to_ do,\" Vidious told his sister.\n\nNisha went completely motionless. Her stillness and silence made me nervous. What was Vidious talking about? Fighting? Surrendering? Handing me over?\n\n\"We could split up,\" she said.\n\n\"He's fast enough to catch us both,\" Vidious said. \"And you've got the package.\"\n\n\"Don't do anything stupid, Vid.\"\n\n\"Stupid is our only choice.\" Vidious raised a hand to his pilot, and his airship swerved away from the _Anvil Rose_. \"Take care, little sister!\"\n\n\"We're twins, you idiot!\" Nisha shouted after Vidious. \"You're older by ten minutes!\"\n\nVidious's laugh reached us faintly as he stalked to the prow of his ship, his cloak billowing behind him.\n\n\"What are they talking about?\" Swedish asked Hazel. \"What's going on?\"\n\n\"He's going to attack the _Predator,_ \" she said.\n\n\"The _Night Tide_ can't beat Kodoc,\" Bea said, her green eyes wide. \"Not even close.\"\n\n\"He's not trying to beat him,\" Hazel explained. \"He's trying to buy us time.\"\n\n##\n\n\"MY BROTHER'S TOUGH,\" CAPTAIN Nisha said, her worried gaze on the _Night Tide_. \"He'll come through.\"\n\nHazel rubbed her face with her hand. \"But Kodoc won't waste his time on the _Night Tide_. He'll stop her, then come for us.\"\n\n\"Well, _that's_ comforting,\" Loretta muttered.\n\nCaptain Nisha eyed Hazel. \"I hope you're worth all this trouble.\" She glanced at me. \"I hope _he_ is.\"\n\n\"I\u2014we'll try to be.\" Hazel offered the spyglass back. \"Thank you. For everything.\"\n\n\"Keep the spyglass,\" Nisha said, crossing toward the prow. \"And keep your crew out of my way . . . _Captain_ Hazel.\"\n\nDespite the danger, a spark of pleasure glowed in Hazel's eyes. She seemed to stand straighter as the wind tugged at her braids. She tightened her lips, trying not to smile, but I knew her too well: being called \"Captain\" by Nisha made her want to spin and shout.\n\n\"C'mon,\" Hazel said, heading toward the stern.\n\n\"Captain Nisha's awesome,\" Bea announced as she followed.\n\n\"Yeah,\" Swedish said. \"But scary.\"\n\n\"Like a snake,\" Loretta agreed. \"With claws.\"\n\n\"Snakes don't have claws,\" Bea told Loretta, starting down a ladder.\n\n\"Some do,\" Loretta said. \"The kind _they_ don't want you to know about.\"\n\nSwedish gave her a look like he wanted to kiss her\u2014then the deck started trembling as the airship accelerated.\n\nWhen we reached the sweep deck, Bea frowned. \"We're on the wrong side of the ship\u2014we can't see the _Predator_ from here.\"\n\n\"Doesn't matter.\" Hazel lifted her new spyglass to her eye. \"We can't do anything about Kodoc anyway.\"\n\nWhile she scanned the Fog, Bea and Swedish and I looked at one another. What did that mean? _Hazel_ could always do something.\n\n\"Shouldn't we, y'know . . .\" I gestured feebly. \"Spring into action?\"\n\n\"What action?\" Hazel said. \"Vidious is already trying to stop Kodoc, and Nisha is already flying at top speed.\"\n\n\"But Kodoc's ship is faster.\" Bea cocked her head, listening to the engine. \"He'll win.\"\n\nHazel nodded curtly. \"I know.\"\n\nBea's lower lip started trembling, so I grabbed her hand. Loretta sidled beside Swede and wrapped his arm around her shoulders. When Bea started shivering, I looked down at her sweet face, usually so bright and hopeful, and saw nothing but dismay.\n\nI glanced at Hazel over Bea's tousled head, and she met my gaze. I saw that she felt just as dismayed as Bea, just as hopeless. She also felt responsible, like she'd failed us. I looked back at her and tried to tell her with my eyes that whatever happened next, I was proud of her. We were all proud of her.\n\nShe bowed her head and scanned the distance with her spyglass. Nobody spoke for a long time. Twenty minutes, thirty, forty. . . . The _Anvil Rose_ trembled beneath us. Foggium hissed through hoses, and the props strained for speed.\n\nFinally Hazel said, \"There she is. Port Oro.\"\n\nI trotted to the railing and shaded my eyes. A million peaks of whiteness caught the light of the sun and the wide blue sky. In the distance, a tumble of puffy clouds mixed with the mist of the Fog . . . and I spotted a dot on the horizon, a jagged green-and-gold crown rising through the vapor.\n\n\"We did it!\" I said, feeling a dumb grin spread across my face. \"We actually did it.\"\n\n\"Except for the whole Kodoc-blowing-us-from-the-sky thing,\" Swedish grumbled.\n\nBea fidgeted beside me. \"I'm scared.\"\n\nAfter a moment, Hazel said, \"We're all scared, but . . . do you know what they're calling us, back in the slum?\"\n\n\"Dead meat?\" Swedish asked.\n\n_\"Legends,\"_ Hazel told him with a sudden, wolfish smile. \"We snatched a diamond and got away. Who does that? Nobody. Who beats the bosses? Nobody. Who outruns the troopers?\"\n\n\"Nobody?\" Bea said.\n\n\"Nobody,\" Hazel agreed.\n\nHearing the strength in Hazel's voice, I felt something expanding in my chest, something fierce and happy. Something like hope.\n\n\"Who snuck onto the Rooftop?\" Hazel asked.\n\n\"We did,\" Bea said.\n\n\"Who escaped Kodoc? Who haggled with mutineers?\"\n\n\"We did! We did!\"\n\n\"Yeah,\" Hazel said. \"And next, we're going to cure Mrs. E.\"\n\n\"How?\" Bea chewed her lower lip. \"There's no way we'll reach the Port now.\"\n\nHazel turned to look at the growing speck of Port Oro. \"I don't know how.\"\n\n\"Neither do I,\" Swedish said. \"But only an idiot would bet against _legends_.\"\n\nIn the silence that followed, I gaped at him, shocked by this sudden burst of optimism.\n\n\"Are there fishing ships?\" Bea asked, a minute later. \"Can you see them?\"\n\n\"The lake's on the other side of the mountain,\" Hazel told her.\n\nWe'd heard that a fleet of airships lowered nets into a massive Fog-shrouded lake every day, trawling for fish hundreds of yards below. We'd heard that farms and mines dotted outlying mountaintop \"islands,\" and even that a few bits of Port Oro were built on the tops of ancient skyscrapers that rose from the white. Which was completely loco. How could super-tall buildings still rise above the white?\n\n\"What about skyscratchers?\" Loretta asked.\n\n\"Scrapers,\" Hazel corrected. \"I don't see them. Not yet.\"\n\n\"You don't actually believe in those,\" Swedish said. \"Do you?\"\n\n\"In the old days,\" Hazel told him, \"there were hundreds of them.\"\n\n\"There were big bens, too,\" I said. At least, according to my dad's scrapbook. \"And wall streets.\"\n\nLoretta squinted at me. \"What's a big ben?\"\n\n\"A huge clock,\" I said. \"Named after Big Ben Franklin.\"\n\n\"The guy whose face is on fancy toilet paper?\" she asked.\n\n\"That's the one.\"\n\n\"There!\" Hazel pointed. \"Warships. Nine mutie warships, coming toward us.\"\n\n\"How far away are they?\" I asked, my grip tightening on the rail.\n\n\"Too far,\" she said. \"Unless Vidious slows Kodoc down.\"\n\nAs if in answer, a sudden _BOOM_ echoed across the sky.\n\nAirsailors shouted, and I spun toward the noise. The gun deck blocked my view, so I climbed the mizzen rigging until I spotted the _Night Tide_.\n\nHigh above and far behind us, Vidious's airship seemed tiny in the shadow of the _Predator_. Kodoc's steel-ribbed balloon cut through the air as his warship swooped in for the attack, her big guns thundering again.\n\nThe crackle of Vidious's cannons answered\u2014too softly, like a cricket's chirp after a lion's roar.\n\nHazel climbed beside me, and we watched the one-sided air battle. Three times in ten minutes the _Predator_ blasted the _Night Tide_ with a terrible volley, then swiveled to chase the _Anvil Rose_. To chase _me_. But every time, Vidious managed to fire one more harpoon or war chain at Kodoc's ship. Just enough to make him turn back again.\n\nJust enough to slow him down.\n\nIn the end, Vidious didn't have a chance. The _Predator_ was too fast, too armored, too powerful. Flames spat from Kodoc's guns, and cannonballs tore into the _Night Tide_. Even from miles away, I almost _felt_ the damage, almost heard the hiss of foggium and smelled the bitter stench of smoke.\n\nVidious's cigar-shaped zeppelin buckled, then sagged in the middle. The _Night Tide_ tilted, and for one heart-clenching moment, I thought the entire airship was going to flip over, hurling everyone into the Fog. Instead, the hull cracked. The elegant stretch of railings splintered and the rigging twisted into a gnarled knot. Broken rudders and charred fans tumbled into the white.\n\nI held my breath, waiting for the ruined ship to plummet. It slumped under the limp zeppelin, drifted helplessly in the air . . . but stayed aloft. Hanging broken in the sky, under a cloud of smoke.\n\nThe _Predator_ pivoted toward us. She hovered for two heartbeats, then rocketed forward, propellers blurring.\n\n\"How long before Kodoc reaches us?\" I asked Hazel.\n\n\"Thirty minutes,\" she said. \"Maybe forty.\"\n\n\"And the mutie ships from the Port?\"\n\nShe looked through her spyglass. \"An hour. At least.\"\n\n\"So he'll catch us.\"\n\n\"Yeah,\" she said. \"He'll catch us.\"\n\n##\n\nI FLOPPED ONTO THE sweep deck and stared toward Port Oro. So close\u2014yet we'd never reach it. The nine mutineer warships still looked like toys, while the _Predator_ loomed ever larger.\n\nMinutes slipped past, way too fast. I wasn't ready to lose, not yet. I wasn't ready to say good-bye to my dreams, I wasn't ready to say good-bye to my crew.\n\nIf only I'd done things differently. If only I'd been born normal. If only the _X-Wing Enterprise_ would fall from the stars to save us all. I touched the scrapbook in my jacket and remembered Mrs. E's words. _What if the_ X-Wing Enterprise _already landed? What if the rescue party is already here?_\n\n\"Oh,\" I said under my breath.\n\nMaybe Mrs. E meant that nobody would save us . . . so we needed to save ourselves. But how? I didn't know. It wasn't possible. I closed my eyes and lay back in defeat. The deck shuddered beneath me. The shouts of the airsailors sounded far away. The glow of sunlight on my eyelids reminded me of the Fog, and I daydreamed about diving\u2014about the freedom and the speed.\n\nThen a terrifying thought rose in my mind, like a jaguar's growl rumbling through the Fog. _No_. No, it was a dumb idea. A deadly idea.\n\nI swallowed a few times. \"Uh, Hazel? We can't outrun Kodoc, right?\"\n\n\"Right,\" she said.\n\n\"So either he pounds Nisha's ship and _then_ grabs me . . . or he just grabs me. We have to sneak off the ship and hand me over.\"\n\nHer eyes narrowed. \"Give you to Kodoc?\"\n\n\"Yeah. To keep the _Rose_ out of it.\"\n\n\"No.\"\n\n\"Do you have a better idea?\"\n\nShe didn't answer.\n\n\"It's our only choice,\" I told her. \"Either I surrender, or I dive without a tether.\"\n\nHazel cocked her head and gazed past me, like she was reading our future in the fields of Fog.\n\n\"I'm no expert,\" Loretta said, \"but isn't 'diving without a tether' the same as 'jumping overboard'?\"\n\n\"That's what Chess is saying.\" Swedish glared at me. \"Even though he knows I'll stuff him in a chest before I let that happen.\"\n\n\"Wait,\" Hazel murmured, her eyes dark with thought. \"Maybe that's it.\"\n\n\"Maybe what's it?\" Loretta asked.\n\n\"Diving without a tether.\"\n\nSwedish squinted at Hazel. \"What're you talking about?\"\n\n\"I think,\" she said slowly, \"I have a plan.\"\n\n\"Is it better than Chess jumping overboard?\"\n\n\"No,\" Hazel said. \"It's not.\"\n\n\"Captain Nisha is going to kill us,\" Swedish grunted twenty minutes later.\n\n\"She'll thank us,\" Loretta said. \"Are you done with those bolts yet?\"\n\nSwedish raised his head from the harpoon. \"She's going to thank us for stealing her lifeboat?\"\n\n\"You're not a pokey little _lifeboat,_ are you?\" Bea crooned, fiddling with the cargo raft's engine. \"No, you're my smuggle-buggy!\"\n\nI was too scared to smile at Bea, too numb and nervous. My mind kept snagging on Hazel's plan, especially the part where I surrendered to Kodoc. It had seemed like a better idea before we'd started trying to make it happen.\n\nI spun on my tether to look at Hazel, but she didn't notice. She was perched in the access hatch that opened beside the lifeboats, her eyes half closed. Plotting and planning as Swedish bolted a stolen harpoon\u2014which I'd lowered alongside the _Anvil Rose_ \u2014onto the cargo raft deck.\n\n\"That's right,\" Bea murmured to the pistons. \"You're fast as a greasy cheetah.\"\n\nSwinging beside Hazel, I unlatched from the tether and closed my jacket around the harness we'd lifted, along with a pair of goggles. My hands shook too much to work the buttons\u2014because pretty soon I'd be alone on the _Predator,_ face-to-face with Kodoc.\n\n\"You can do this, Chess,\" Hazel said, fastening my buttons. \"Swedish, are you done?\"\n\nHe patted the harpoon. \"The pig-sticker is in place.\"\n\n\"Bea?\" Hazel hopped from the warship onto the still-deflated cargo raft. \"How's the engine look?\"\n\n\"One minute, Cap'n! The spark plugs are a weensy bit scared.\"\n\n\"Well, sweet-talk them\u2014quickly.\"\n\nI crouched at the base of the harpoon and pretended to check the long rope coiled there. Trying to look busy instead of petrified.\n\n\"How about you, Loretta?\" Hazel asked.\n\n\"I am awesome.\" Loretta flashed a gap-toothed smile. \"I get to shoot a harpoon directly up Kodoc's nose.\"\n\n\"Loretta!\" Hazel snapped, grabbing a strap near the mast. \"Do not aim at Kodoc. You only get one shot.\"\n\n\"I know, I know,\" she grumbled. \"I'm just kidding.\"\n\n\"The plugs are happy!\" Bea chimed, lifting her head from the hatch.\n\n\"Swedish?\" Hazel called.\n\nSwede ran his fingers over the steam organ keyboard. \"She's just like the salvage raft,\" he said with a satisfied grunt. \"But slicker than a snail's sneeze.\"\n\n\"Then everyone fasten down\u2014we're going for a ride!\" Hazel reached for the emergency inflation handle . . . then paused. \"Um, Chess? Would you please keep Loretta from plunging to her death?\"\n\n\"I'm fastened!\" Loretta insisted, gripping the railing tighter. \"Stop picking on me!\"\n\n\"Well, just in case,\" I said.\n\n\"Thanks,\" she muttered. \"How come all this airship stuff always happens in the _air_?\"\n\n\"In three,\" Hazel said. \"Two, one . . .\"\n\nShe turned the handle, and the tank of foggium coughed to life. The hose stiffened for a few seconds, then the cargo raft woke. The mast unfolded on pneumatic hinges, pulling the rigging taut. The propellers straightened, the deck snapped straight, and _whoosh,_ the raft's balloon inflated like a frog's throat.\n\nLoretta stumbled, and I steadied her as we drifted free from the _Anvil Rose_. A shiver of excitement ran across my skin, along with an edge of terror. I was on a raft again, flying over the Fog with my crew . . . and heading for Kodoc's ship.\n\nI lowered Bea through a hatch, where she started scolding and tinkering as the cargo raft dropped through the air. Swedish was taking us beneath the _Anvil Rose,_ because we needed to get to the other side, closer to the _Predator_.\n\n\"Okay,\" Hazel said, chewing her lip nervously. \"So much for the easy part.\"\n\n\"Look on the bright side,\" I told her. \"Now we're stealing from muties. We've come a long way.\"\n\nShe raised her spyglass to look behind us. \"What's next, mugging the Five Families?\"\n\n\"Either that,\" I said, \"or winning a fight with a goose.\"\n\nI followed her gaze and caught a glimpse of the nine warships approaching from the Port, closer than ever. \"How long before they get here?\"\n\n\"Fifteen minutes,\" she said.\n\n\"That's great! We can wait for them to catch up instead of\u2014\"\n\n\"No,\" she interrupted, pointing her spyglass in the other direction. \"Look.\"\n\nAs we flew out from beneath the _Anvil Rose,_ the _Predator_ swooped into firing range.\n\n##\n\nTHE PREDATOR'S BIG GUNS swiveled toward Nisha's airship, and a roar slammed across the sky. A volley of cannonballs barely missed the _Anvil Rose'_ s zeppelin\u2014a warning shot.\n\n\"Give me the boy!\" Kodoc shouted into the sudden silence. \"Or the next blast guts your ship!\"\n\n\"Come get him, you slime-sniffing roof rat!\" Nisha shouted back. \"If you can.\"\n\n\"Sweet,\" Loretta murmured beside her harpoon. \"The captain knows how to sling an insult.\"\n\nThe _Predator_ fired again. The sound of the blast almost split my head\u2014and the impact almost split the _Anvil Rose_ 's propellers. Smoke billowed and bits of wood and copper rained around us.\n\n\"Watch out!\" I shouted to Swedish. \"It's going to squash us!\"\n\n\"Not today,\" he said between gritted teeth. \"That's what _they_ want.\"\n\nHis fingers danced across the keyboard organ, and we zoomed from the shadow of the _Anvil Rose_ as flaming debris plunged around us.\n\n\"Whoa,\" Swedish said. \"This thing really moves.\"\n\nHe spun the wheel and clattered on the organ. With a hiss of foggium, the propellers sliced through the air as explosions and screams sounded high above. A quick glance at the _Anvil Rose_ made me wince: jagged holes pocked the hull, smoke poured from two gun emplacements, and the rigging crew frantically patched the balloon.\n\n\"Lord Kodoc!\" a roof-trooper cried from the _Predator_. \"Down there! There are three boys in that raft! Is that him?\"\n\n\"Heh-heh,\" Loretta snickered to Hazel. \"They think you're a boy.\"\n\nIn middle of the fear and the danger, Hazel and I looked at each other and almost smiled. Did Loretta really think that the trooper had mistaken _Hazel_ for the third boy?\n\nThen the cannons roared again. \"Catch him!\" Kodoc roared. \"I need him alive!\"\n\nSwedish swooped low, speeding away from the Port, away from the _Rose_ \u2014away from the safety of the nine incoming mutineer warships\u2014and dove under the _Predator_. The cargo raft zigzagged, swerving in sharp angles to avoid harpoons and nets, and I closed my mind against the racket, trying to tame my fear.\n\nThen I heard Kodoc's voice directly above me, cutting through the air. \"You will never escape me, Chess. Not ever.\"\n\nMy heart beat so hard in my chest that my ribs ached. I raised my head and saw Kodoc standing on the bridge of the _Predator_ forty feet away, pinning me with his stare. I couldn't do this. Hazel's plan was loco. How could I walk onto Kodoc's ship . . . and then defy him?\n\nWhat if I couldn't stand up to Kodoc? He'd created me. He'd transformed me from a normal baby into a cowardly freak, just like he'd transformed the _Teardrop_ into the _Predator,_ and I couldn't do this.\n\nThe wind of his propellers blew the hair from my face, but I was too scared to even raise my hand to cover my eye. I wanted to scream at Hazel\u2014 _Run! Hide! Get me away from him!_ \u2014but fear paralyzed me.\n\nSwedish slammed the organ and we veered forward, past the _Predator,_ heading away from the _Anvil Rose_. We needed to draw Kodoc as far as possible from Nisha. The cargo raft spun and swooped, swerving so hard that the rudders squealed, staying fifty yards over the highest crests of Fog. Loretta gripped the harpoon, Bea peeked from a hatch, and Hazel scanned the sky, her jaw clenched.\n\n\"It's time,\" she said.\n\nI swung from the rigging to the deck as the _Predator_ dove from the sky in front of us. Swedish spun the wheel and hammered the organ, and we slewed to a halt.\n\nWe were ten feet from Kodoc, with nowhere to run.\n\n##\n\nA HEAVY BOARDING PLANK shot from the _Predator_ and slammed onto the deck of the cargo raft. The _crack_ sounded like a bone snapping.\n\nKodoc stood at the far end of the plank, watching me with hungry eyes. \"Get over here, boy. You've caused enough trouble.\"\n\n\"Y-yes, sir,\" I said.\n\nMy stomach ached as I edged onto the plank with quivering legs. A sour taste rose in my throat, and I forced myself to step forward.\n\nFrom the crow's nest, Hazel said, too softly for Kodoc to hear, \"Goggles down, Chess. Tether free.\"\n\nThe wind stilled. I paused, wanting to look at her but afraid I'd start crying.\n\n\"Dive at will,\" she whispered. \"And come back safe.\"\n\nI ducked my head and crossed the rest of the boarding plank.\n\nThe instant I stepped onto the _Predator's_ deck, the plank retracted with an ominous scrape. I almost whimpered: utterly alone, cut off from my crew, stranded on a warship with Lord Kodoc, as his troopers hunched over their wicked-looking harpoons and cannons.\n\nThe big warship wheeled in the air, turning away from the crew, away from the Port . . . away from the nine mutineer ships that slipped into view behind the _Anvil Rose_.\n\n\"The Port Oro navy is ten minutes from firing range, m'lord,\" an airtrooper said, watching the warships through a spyglass.\n\n\"They're too slow,\" Kodoc said. \"They can't catch us now.\"\n\nMy teeth chattered from fear, and darkness dimmed my vision. What if I fainted? If I fainted, I was finished. So I took slow, shaky breaths as the _Predator_ picked up speed, starting the long journey back to the Rooftop.\n\nKodoc looked down at me. \"You will find the Compass. My other divers failed, but not you, you were born for this. Now show me your eye.\"\n\n\"Y-y-yes, s-sir\u2014\"\n\n\"Stop stammering and show me!\"\n\nMy mind writhed with terror. Kodoc was the monster who'd chased me every night while I slept, the beast who'd killed me a thousand times. Now he was standing two feet away. He was _real_. I saw the faint lines on his forehead and the cruelty in his eyes. I smelled the sickly sweetness of his breath.\n\n\"Now!\" he screamed.\n\nI flinched, then pushed my hair aside.\n\nHe grabbed my head and stared at my eye. \"So you lived in the junkyard?\"\n\n\"Y-yes.\"\n\n\"Hidden by that coward Katherine,\" he sneered. \"What does she call herself now? I'll find her next.\"\n\nDespite my fear, a spark caught fire in my heart. \"N-no. You won't.\"\n\n\"Never say no to me!\" Kodoc cuffed me, and my goggles clattered across the deck. \"You're not a tetherboy anymore, you're not a slumkid or a bottom-feeder. You're not even human, not anymore. You're just a tool. And I'm going to beat you into shape.\"\n\nAt first, I almost wet myself with fear. But as he kept ranting, a strange thing happened: my terror started to fade.\n\nSeeing him this close, I realized that Kodoc _wasn't_ a monster, he wasn't a nightmare beast who'd created me by snapping his fingers. Spittle flew from his mouth when he shouted, and one nostril hair dangled from his nose like a curly wire. He wasn't a monster\u2014he was just a cruel Rooftop lord with too much power.\n\nFor my whole life, I'd been afraid to call attention to myself. My whole life I'd worried that being a freak also made me a coward. But Kodoc was just an evil man, not my creator. Which meant I was not his creature. Just a kid with a freak-eye.\n\nAnd I was no coward.\n\n\"You will obey me,\" he snarled. \"You _will_ dive. You _will_ find the Compass. And once the Fog is mine\u2014\"\n\nA shrill whistle interrupted him, a sharp note from across the Fog.\n\nKodoc turned at the sound, and I peered past him. The cargo raft hovered a hundred yards away. Hazel watched from the crow's nest, while Swedish stood at the wheel, and Loretta slouched beside the harpoon, two fingers in her mouth.\n\nShe stopped whistling, and Hazel gestured with her spyglass, like she was showing us the entire sky. High above her, the smoldering _Night Tide_ limped toward the Port. And far behind her, the nine mutineer warships guarded the _Anvil Rose_.\n\nWhatever happened now, Mrs. E was safe. _Suck pigeon eggs, Kodoc_.\n\nKodoc narrowed his eyes. \"Nine warships could beat me, boy\u2014but they can't catch me.\"\n\n\"They don't have to,\" I said, and kicked him in the shin.\n\nHe gave a shout of pain, then lunged at me, but I ducked and raced across the deck.\n\n\"Grab him!\" he bellowed.\n\nI dodged an airsailor and scrambled onto the railing, balancing at the edge of the airship as Hazel shouted, _\"NOW!\"_ and Loretta fired her harpoon, attached to a long, coiled rope\u2014a tether.\n\nThe world moved in slow motion. My heart thudded to a stop and my breath faded to nothing as a silver spear arced through the air toward me, trailed by the tether. A good shot, straight and true.\n\nI tracked the motion with my gaze, my knees bent, my focus absolute. The harpoon flashed under the _Predator,_ the rope streaming behind . . . and I jumped.\n\n\"Turn the ship!\" Kodoc screamed behind me. \"Battle stations! Catch him! He's _mine,_ he\u2014\"\n\nThe rush of air in my ears silenced him. The wind whipped my face and ruffled my jacket, and my heart pounded with sudden, thrilling hope. I'd done it. I'd walked onto Kodoc's airship of my own free will, I'd kicked him in the shin, and I'd walked back off again.\n\nWell, I'd _jumped_ off. Without a tether. Maybe it wasn't time to celebrate yet.\n\nBut everything was going according to Hazel's plan . . . so far. \"You're going to dive without a tether,\" she'd told me. \"You'll jump from an airship a half mile above the ground and catch a rope in midair.\"\n\n\"What\u2014what airship?\" I'd stammered.\n\n\"The _Predator,_ \" she'd explained. \"We'll fly you to Kodoc and hand you over. That'll buy time for the mutie warships to reach us. Then you'll jump overboard and catch the rope.\"\n\n\"What rope?\" Bea asked.\n\n\"The one Loretta's going to shoot, attached to a harpoon. Like a tether.\"\n\n\"No way.\" Bea chewed her lower lip. \"Shoot from where?\"\n\n\"From the cargo raft we're about to steal.\"\n\nI'd gaped at Hazel. \"Are you trying to kill me? You want me to jump from a ship and catch a rope in midair? There's no way I can do that.\"\n\n\"You listen to me, Chess,\" Hazel had said. \"I know _exactly_ what you can do.\"\n\nAnd I'd seen the truth in her face: she trusted me the way I trusted her, she saw me the way I saw her, as someone special, someone extraordinary. She believed in me, which made me believe in myself.\n\nExcept now, falling through the air, I couldn't see the rope. Without my goggles, the wind brought tears to my eyes, and the endless white crests of Fog, five hundred feet beneath me, reflected the sunlight too brightly.\n\nWhere was that tether? C'mon, c'mon. . . .\n\n\"Yes!\" I breathed, spotting the rope below me.\n\nHa! Can't stop the tetherboy. I spread my arms to catch the tether, and a blow slammed into my shoulder from behind, spinning me off course.\n\nThen a grappling hook sheared past.\n\nKodoc was trying to fish me from the air.\n\nI lashed out with my left leg and barely caught the tether on the tip of my boot. But when I curled in midair to bring it to my hands, a harpoon shot from the sky and grazed my leg.\n\nI flinched at the pain, the tether slipped from my boot\u2014and another harpoon flashed past. Kodoc was trying to _spear_ fish me from the air.\n\nWhich was definitely _not_ part of the plan.\n\nTwisting wildly, free-falling hundreds of feet above the Fog, I lunged for the tether. A dozen more spears fired at me from the _Predator_ as Kodoc attempted to spike me and drag me back on deck. The harpoons burned through the sky. Too many too fast. Too _accurate_. One drilled directly toward my stomach. With a panicked yelp, I arched in the air, and the razor-edged blade only sliced my hip.\n\nI'd lost the cargo raft tether completely, and the next three harpoons were dead on target: two shooting at my chest and one about to sink into my leg. Moving too fast to miss. I couldn't dodge them.\n\nThey were three seconds from skewering me. Two seconds, one second\u2014\n\n##\n\nI HIT THE FOG, and the world disappeared into a silent blur of mist. Time stopped. Coolness touched my face and quicksilver filled my veins.\n\nThe closest harpoon sliced toward my chest\u2014and I snatched it from the air.\n\nWhirling as fast as a hummingbird, I swung the harpoon like a sword and deflected the second two. The clash of metal sounded soft inside the Fog, and the harpoons tumbled away into the thick mist . . . but more scythed at me from above.\n\nI dodged and twirled, yanking hard on the harpoon in my hands.\n\nThe line immediately tightened. The harpooner must've thought she'd caught me, so she stopped the rope from unspooling from the _Predator_. It hung firmly in the air. Which meant that the roof-troopers weren't just shooting harpoons at me: they were shooting _tethers_.\n\nDespite everything, I laughed. Maybe I wasn't good at fighting or fixing or flying\u2014but nobody beat me in the Fog.\n\n_Nobody_.\n\nWithin seconds, dozens of harpoon ropes\u2014all fired from the _Predator_ \u2014dangled around me, like a forest of tethers in the mist. A Fog diver's dream. Grinning wildly, I yanked on a nearby rope, then another. Anchored on Kodoc's warship, I swung between them like a monkey on a vine, cartwheeling to dodge the harpoons still slicing downward as I scanned the billows of Fog.\n\n_There!_ The tether from Loretta's harpoon was a sideways slash across all the vertical lines, falling slowly through the white vapor below me.\n\nI swung hard, released my grip on a harpoon line, and somersaulted into the blankness. Fog surrounded me. White on white. Free-falling. No sounds except my ragged breathing.\n\nSo peaceful that I almost wanted to stay forever.\n\nThen the cargo raft tether rose through the whiteness and curled gently into my fist.\n\n##\n\nWITH MY HEART PUMPING and my fingers clumsy, I hitched the tether to the harness under my jacket. Safe. Connected. I slumped in relief, inhaled the cool whiteness . . . and was jerked sideways through the Fog.\n\n\"Ow!\" I shouted toward the sky. \"Swedish!\"\n\nThe tether jolted me again\u2014to the left, then the right\u2014then it whipped me in a wide circle. Which meant that high above, Swedish was speeding away from the _Predator,_ racing toward the mutineer ships.\n\nTowing me to safety.\n\nPain flared in my slashed leg as the tether whiplashed, and I slumped in my harness, dazed and drained and exhilarated. The ripples and billows of Fog swam in my vision. My breath sounded loud, the world faded into a blur. . . .\n\nWhen the tether started winching me upward, my head jerked and my pulse pounded. I must've fainted for a minute or two. Or five. I shook my head as I rose higher and higher, until in an instant, the Fog fell away.\n\nMy body felt leaden and my leg throbbed with a steady ache, but my vision suddenly extended for miles. I swiveled in my harness and saw the mutineer warships standing guard nearby. Dozens of big guns were aimed at the _Predator,_ which hovered in the distance\u2014too far to pose a threat.\n\nMrs. E was safe, and Kodoc would never catch me now. He didn't control me anymore\u2014maybe he never had. Because Hazel and Swedish were right. I didn't belong to Kodoc, not even close.\n\nThe sun warmed my face as I spun to look at the cargo raft flying above me. Bea beamed from the engine, waving her cap wildly and hopping around like a flea. I lifted a hand to her as the tether dragged me upward. A minute later, Swedish dragged me on deck and into a back-thumping bear hug.\n\nYeah, I knew _exactly_ where I belonged.\n\n\"For a second there,\" he muttered in a choked voice, \"I thought _they_ got you.\"\n\n\" _They_ wouldn't dare,\" Loretta said. \"You've got a fighter in the crew now.\"\n\n\"Ow!\" I said as Swedish thumped me again. \"Ouch!\"\n\n\"He's bleeding, Swede!\" Hazel snapped. \"Let him go.\"\n\n\"Oops!\" he said, releasing me. \"Sorry.\"\n\nWhen I swayed, Hazel grabbed me\u2014and hugged me even tighter.\n\nShe didn't say anything. Neither did I. We just held each other as the deck shifted beneath us and the propellers spun. Maybe she cried a little\u2014maybe we both did\u2014but for once I didn't care if anyone was watching. For once, I wasn't afraid of being seen.\n\n\"You did it,\" she finally said.\n\n\" _We_ did it,\" I told her. \"How long before we reach the Port?\"\n\nHer smile was the brightest thing in the world. \"Look behind you.\"\n\nI turned and there it was: the jagged green gleaming peaks of Port Oro, framed by the sun. We'd fought the bosses and the troopers and Lord Kodoc himself, and we'd won.\n\n\"We made it,\" Hazel said. \"We're home.\"\n\n## Acknowledgments\n\nMany thanks to Caitlin Blasdell, Alyson Day, Ren\u00e9e Cafiero, Toni Markiet, Joel Tippie, Jenna Lisanti, Gina Rizzo, and Tina Wexler.\n\n## About the Author\n\n**JOEL ROSS** is the author of two World War II thrillers for adults ( _Double Cross Blind_ and _White Flag Down_ ). This is his tween debut. He lives in Santa Barbara, California, with his wife, Lee Naftali, who is also a full-time writer, and his son, Ben, who is a full-time kid.\n\nDiscover great authors, exclusive offers, and more at hc.com.\n\n## Credits\n\nCover art \u00a9 2015 by Matt Rockefeller\n\nCover design by Joel Tippie\n\n## Copyright\n\nTHE FOG DIVER. Copyright \u00a9 2015 by Joel Ross. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.\n\nwww.harpercollinschildrens.com\n\n* * *\n\nLibrary of Congress Cataloging-in-Publication Data\n\nRoss, Joel N., date\n\nThe Fog diver \/ Joel Ross.\n\npages cm\n\nSummary: \"In this futuristic high-stakes adventure, humanity clings to cities on the highest mountain peaks above the deadly Fog, and airships transport the pirates of the skies. Daring 13-year-old tetherboy Chess and his salvage crew must face the dark plans of Lord Kodoc and work to save their beloved Mrs. E\"\u2014Provided by publisher.\n\nISBN 978-0-06-235293-4 (hardback)\n\nEPub Edition \u00a9 April 2015 ISBN 9780062352965  \nVersion 08152017\n\n[1. Adventure and adventurers\u2014Fiction. 2. Survival\u2014Fiction. 3. Recycling (Waste, etc.)\u2014Fiction. 4. Environmental degradation\u2014Fiction. 5. Orphans\u2014Fiction. 6. Science fiction.] I. Title.\n\nPZ7.1.R677Fog 2015 |  2014034154  \n---|---  \n[Fic]\u2014dc23 |  CIP  \n|  AC\n\n* * *\n\n15 16 17 18 19 CG\/RRDH 10 9 8 7 6 5 4 3 2 1\n\nFIRST EDITION\n\n## About the Publisher\n\n**Australia**\n\nHarperCollins Publishers Australia Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\nwww.harpercollins.com.au\n\n**Canada**\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON M4W 1A8, Canada\n\nwww.harpercollins.ca\n\n**New Zealand**\n\nHarperCollins Publishers New Zealand\n\nUnit D1, 63 Apollo Drive\n\nRosedale 0632\n\nAuckland, New Zealand\n\nwww.harpercollins.co.nz\n\n**United Kingdom**\n\nHarperCollins Publishers Ltd.\n\n1 London Bridge Street\n\nLondon SE1 9GF, UK\n\nwww.harpercollins.co.uk\n\n**United States**\n\nHarperCollins Publishers Inc.\n\n195 Broadway\n\nNew York, NY 10007\n\nwww.harpercollins.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nWHY  \nSIZE  \nMATTERS\n\n# WHY  \nSIZE  \nMATTERS\n\nFROM BACTERIA TO BLUE WHALES\n\n_John Tyler Bonner_\n\nCopyright \u00a9 2006 by Princeton University Press\n\nPublished by Princeton University Press, 41 William Street,  \nPrinceton, New Jersey 08540  \nIn the United Kingdom: Princeton University Press,  \n6 Oxford Street, Woodstock, Oxfordshire OX20 1TW  \npress.princeton.edu  \nAll Rights Reserved\n\nSixth printing, and first paperback printing, 2012  \nPaperback ISBN: 978-0-691-15233-2\n\n_The Library of Congress has cataloged the cloth edition of this book as follows_  \nBonner, John Tyler  \nWhy size matters : from bacteria to blue whales \/ John Tyler Bonner.  \np. cm.  \nIncludes bibliographical references and index.  \nISBN-13: 978-0-691-12850-4 (hardcover : alk. paper)  \nISBN-10: 0-691-12850-2 (hardcover : alk. paper)  \n1. Body size. I. Title.  \nQL799.B66 2006  \n578.4'1\u2014dc22 2006004945\n\nBritish Library Cataloging-in-Publication Data is available  \nThis book has been composed in Bembo and Helvetica Neue  \nPrinted on acid-free paper. \u221e  \nPrinted in the United States of America  \n7 9 10 8 6\n_for Slawa_\n\n## CONTENTS\n\n_Preface_\n\n1 Introduction\n\n2 The Human View of Size\n\n3 The Physics of Size\n\n4 The Evolution of Size\n\n5 Size and the Division of Labor\n\n6 Size and Time\n\n7 Envoi\n\n_Notes_\n\n_Index_\nOne can live in the shadow of an idea  \nwithout grasping it.  \n\u2014 _Elizabeth Bowen_\n\n## PREFACE\n\nOur interest in the size of things is entrenched in the human psyche. It reveals itself in literature from _Gulliver's Travels_ , to the Grimms' fairy tales, to _Alice in Wonderland_. We see it in our daily thoughts of our growing children, of the people who are around us, of our pets, of the fish we catch, of the portions of the food we are served, of the clothes we buy\u2014are you small, medium, or large?\u2014and one could go on and on. There is hardly anything we observe in daily life that we, either consciously or unconsciously, do not take measure of its size. We love to measure everything with rulers and scales and clocks.\n\nI began to think of the matter of biological size years ago when I first read that glorious chapter in D'Arcy Wentworth Thompson's _On Growth and Form_ called \"On Magnitude.\" It is a model of insight, erudition, and beautiful prose. He showed me that size and shape are indeed interrelated and that the reason that this is so is a matter of physics that underlies the biology. From this initial inspiration there slowly grew inside me the feeling that there was a hidden other dimension of the subject that was eluding me. That inner feeling persisted for many years, and slowly something began to take shape. I am finally putting it all together in this book\u2014I feel as though those shadowy thoughts have erupted through the surface.\n\nThis book is a summary of those thoughts. It is an enormous subject that I try to bring down to reasonable dimensions so that I can include it all. As will be clear, I am interested in painting the big picture on a small canvas.\n\nIf we are a bacterium, or an elephant, or a human being, we have our own size worlds, and for each of us there are things smaller and larger than ourselves. But no one can escape the universal rules imposed by size.\n\nIn looking at the subject of biological size in its entirety, from large to small, from plant to animal to microbe, it will be evident that everything is interconnected. An examination of the effects of size is a way of bringing all life together.\n\nJust as the content of this book has almost taken a lifetime to mature, the actual writing has been an equally painful and slow process with innumerable adjustments and corrections in my course as I proceeded. These were greatly helped by the kindness and wisdom of numerous individuals to whom I am deeply indebted. Before I even began the book, my colleague Henry Horn was enormously helpful (as he has always been over the years) in purifying my thoughts about size. At a very early stage of writing I had the help of my friend Jonathan Weiner, who urged on me the need for a sense of direction. The first complete draft of this book (still in its underwear!) was prematurely sent to two anonymous readers, and while the comments of one of them were highly critical, they gave me the needed jolt at just the right moment. Later drafts were greatly improved due to the comments of Sam Elworthy, Brian Hall, Slawa Lamont, Mary Jane West-Eberhard, and another anonymous reviewer. Also I want to thank David Kirk for his help with the section on _Volvox_ and its relatives, and my colleague Ted Cox on some matters of physics. I almost feel as though they all ought to be listed as co-authors. My special thanks to Alice Calaprice and Deborah Tegarden for their skill and great help in seeing the book through its final stages of preparation. Finally I would like to thank Hannah Bonner for applying her superb illustrator skills to produce seven of the original drawings.\n\nMARGAREE HARBOUR  \n _Cape Breton, Nova Scotia_\n\n## WHY  \nSIZE  \nMATTERS\n\n##   CHAPTER 1\n\n## INTRODUCTION\n\nIn the seventeenth century it was held by some that inside a human sperm there was a minute human being\u2014a homunculus\u2014that was planted inside the womb. Development consisted of the miniature homunculus enlarging and passing through birth and on to maturity\u2014just like inflating a balloon. There were others, going back to the early ideas of Aristotle and the many who followed him, who took the view that vast changes in shape occurred between egg and adult, for it could be plainly seen that the early stages of development of any animal bore no resemblance to what came later. These two views frame the point I want to make in this book. In the case of the homunculus, shape is totally unconnected to size; as size increases shape remains unaltered. In the other case\u2014now totally accepted\u2014as size increases from egg to adult, the shape must change; there is no alternative.\n\nLet me put the matter in another way. If an engineer is commissioned to build two bridges, one across the Hudson River and the other across a brook no more than 30 feet wide, it is quite obvious that the two bridges will be very different in their appearance. Even more importantly, they will differ in their construction and materials. These differences will have nothing to do with the artistic whims of the engineer, at least for the larger bridge: they are absolute requirements. Any attempt to build the Hudson River bridge with wooden planks would collapse into the water long before it was finished. The elaborate steel trusses and the carefully designed architecture of the huge bridge are demanded by the width of the Hudson\u2014it is dictated by its large size. As we shall see, this perfectly mirrors what happens in living organisms; they too cannot escape the conditions set by size; they have no choice.\n\nWith these thoughts in mind, let me state the main argument of this book. Changes in size are not a consequence of changes in shape, but the reverse: changes in size often require changes in shape. To put it another way, size is a supreme regulator of all matters biological. No living entity can evolve or develop without taking size into consideration. Much more than that, size is a prime mover in evolution. There is abundant evidence for the natural selection of size, for both increases and decreases. Those size changes have the remarkable effect that they guide and encourage novelties in the structure of all organisms. Size is not just a by-product of evolution, but a major player. Size increase requires changes in structure, in function, and, as we will see, in other familiar evolutionary innovations. It requires them because they are needed for the individual to exist. Life would be impossible without the appropriate size-related modifications.\n\nThe subject of size has not been ignored in the past. Quite to the contrary, and as will be clear in the pages to come, there is a great literature on matters of size, beginning with the Greeks and bursting into flower with Galileo. This is true for the West, and no doubt there are similar traditions in other cultures.\n\nHowever, the subject is always to some degree fragmented because it is generally introduced as an adjunct to some other biological phenomenon or property. For instance, the topic might be running speed, or rate of metabolism, or one of many other possibilities, and in the discussion of each of these phenomena the crucial role of size would be included. Many of the themes treated in this book can be found elsewhere. Here I wish to look at them from a different point of view\u2014from the other end of the telescope\u2014and show that the biological world revolves around size.\n\nThe mindset that size is not a central issue is quite understandable. To say an elephant is big says nothing about all the things that make an elephant: its anatomy, its physiology, and even its behavior. These are the aspects that draw our attention and the matters we want to study. Yet size is an overarching issue. Its effect is something that no organism, from the smallest bacteria to the largest whale, can escape. It governs their shape and all their activities in a way that is of fundamental significance. Size dictates the characteristics of all living forms. It is the supreme and universal determinant of what any organism can be and can do. Therefore, why is it a subject that always resides in the wings rather than center stage?\n\nThe main reason is that organisms are material objects while size is a bloodless geometric construct. Any object, whether animate or inanimate, will have a size. Airplanes, boats, or musical string instruments vary in size just like animals and plants, and in all cases their size and their material construction are totally different matters even though they affect one another.\n\nThat the role of size has been to some degree neglected in biology may lie in its simplicity. Size may be a property that affects all of life, but it seems pallid compared to the matter which makes up life. Yet size is an aspect of the living that plays a remarkable, overreaching role that affects life's matter in all its aspects. It is a universal frame from which nothing escapes.\n\nThere are many things one wants to know about size, in particular those that concern its evolution. For instance, what is the evidence for my contention that size differences are a prime object of natural selection and are followed by changes in construction? What is the relation between size and internal complexity\u2014that is, the division of labor\u2014and, again, what is the evidence for which came first? What is the relation between size and the timing of all living activities such as the speed of movement of animals, or life span; and does size impose the timing, or the reverse? As we shall see, it is generally true that size is the prime mover: if size changes occur through the agency of natural selection, all those other matters must follow.\n\n### SIZE RULES\n\nIn the pages to come we will see many examples of where size rules life. They are supported by correlations in which various properties of organisms vary with size. It is these correlations that provide the foundation, the underpinning, for my contention that size rules life. The correlations can be stated in the form of five rules that will be briefly mentioned here and expanded and explained later. The rules are as follows:\n\n**RULE 1** Strength varies with size.\n\n**RULE 2** Surfaces that permit diffusion of oxygen, of food, and of heat in and out of the body, vary with size.\n\n**RULE 3** The division of labor (complexity) varies with size.\n\n**RULE 4** The rate of various living processes varies with size, such as metabolism, generation time, longevity, and the speed of locomotion.\n\n**RULE 5** The abundance of organisms in nature varies with their size.\n\nEach of these rules will be put in its proper context. Some are physical or engineering principles; rules that apply for size differences occur in the inanimate as well as the animate. A central issue is the role of size in evolution, and this can be seen in numerous manifestations. Size also affects in many fundamental ways the physiology of animals, plants, and other organisms; in fact, this is true for all aspects of living things that involve time or rates of activity. And the human interest in the matter of size (including my own) will not be neglected.\n\n##   CHAPTER 2\n\n## THE HUMAN VIEW OF SIZE\n\nIt is only natural that we should measure everything in the world around us in terms of our own size. An elephant is bigger than we are, and a mouse is smaller. Some years ago, TimeLife was publishing a series of illustrated books on various subjects, and they called me up to ask for advice on a book they were doing on the general subject of growth\u2014would I please come in to New York to discuss the matter with their editors, for they had numerous questions they wanted to ask an interested biologist. I no longer remember what those questions were, but at the end of our conference they said they were having difficulty thinking of a suitable photograph for the cover of the book. I thought about it as they were talking, and suggested they should have the large open hand of a man and in his palm have the hand of an infant. They did not seem very enthusiastic about the idea, but when they sent me the finished book, that was exactly what they had on the cover. We see and are conscious of the size of everything that surrounds us, whether it is smaller or larger, and nothing makes the point more clearly than the growth of children. Who has not remarked upon seeing\u2014after an absence\u2014the child of a friend or family, \"My, you are so much taller than when I saw you last.\" Once I visited Louis Pasteur's house outside of Paris, and one of the doorways still had pencil marks recording the annual growth of his children, something that will sound familiar to everyone. Either consciously or, because it is so much part of our natures, unconsciously, we are forever taking note of the size of things and gauging any increase or decrease.\n\nOur world is the world we see with our naked eyes, and that is what we use for our everyday measuring stick. We are also aware that there are worlds that are larger and smaller than we can see in our normal existence\u2014in fact, how to see the things above and below our vision was among the great discoveries in our history. The telescope was one of the profound technical advances in our civilization. The first to see huge, distant bodies was Galileo, who in the seventeenth century devised an improved way to put lenses together to greatly magnify the heavenly planets and stars (fig. 1). Following this discovery of great importance and consequence, there has been a continuous improvement in telescopes to explore the sky. Today we have the Hubble (and similar) telescopes carried by a satellite orbiting the earth which not only has an enormously powerful telescope, but it can operate free of the optical disturbances created by the earth's atmosphere. The universe is unimaginably large, yet with this tool we are learning things about it that are totally beyond the reach of unaided human eyes.\n\nFigure 1. Galileo's telescope. (Drawing by Hannah Bonner)\n\nThe microworld is too small for us to see without help. That help first came in the seventeenth century when Anton van Leeuwenhoek ground very small lenses to greatly magnify objects (fig. 2). He not only invented the microscope, but he went on to illuminate a whole world that had never before been known to exist. He described for the first time all sorts of microscopic animals and plants that live in ponds and rain barrels: he even, for the first time, described bacteria, which are exceedingly small. He was the discoverer of spermatozoa in human semen, although it was not until much later that their nature was correctly understood. The technical advances in microscopes over the years has been no less remarkable than those of telescopes; their complexity and their power bears little resemblance to the tiny lenses ground out by van Leeuwenhoek many years ago.\n\nFigure 2. Van Leeuwenhoek's microscope. (Drawing by Hannah Bonner)\n\nIt has often struck me that, conscious as we are of our own size and of those around us, we do not ordinarily think that once we were microscopic. It is possible to believe that at one time we were infants, and there are all those old family photographs to prove it. It is even possible to imagine that earlier we were a foetus, but the idea that we were once a microscopic, single cell\u2014a fertilized egg\u2014is not something that ordinarily crosses our mind. In our own growth we go from the world of van Leeuwenhoek to the world we see around us every day. Perhaps if we were to have eyes and senses and a memory at that single cell stage of our life\u2014an absurd possibility\u2014we would look at the world differently, for our sense of size does not come to us until we are a small child. (In any event, an egg does not have much to look at in the darkness of a Fallopian tube.) One of my early recollections is being so small that I could not reach a light switch on the wall and had to get help from some giant adult. I had a psychologist friend who wanted to apply for a grant to build a huge room with gigantic furniture exactly as a small child would see it, and study the effect it might have on an adult who reenters his childhood world. The university committee that had to pass on research done on human subjects turned the request down: the effect might be too dangerous for the human psyche!\n\nClearly children are especially conscious of their size as they are surrounded by huge adults. This is reflected in fairy stories and children's tales. A good example is Lewis Carroll's _Alice in Wonderland_ , where Alice keeps changing her size to go down rabbit burrows and back to her normal large self. She begins with the bottle which says \"Drink Me\" and shrivels to a size that allows her to join the underground world. Later, the hookah-smoking caterpillar puts her on to a magic mushroom: a bite from one side makes her larger, and a bite from the other smaller. What a perfect child's dream to go back and forth from one size world to another. At one point she worries that she will shrink to nothing, which has special resonance for me. When I was a very small boy I had a terrifying recurrent nightmare that I would enter a room with adults and I would instantly shrink to something minute, like an untied balloon letting out its air. I never knew my final size\u2014I always woke up shrieking. Perhaps that was the first spark of an early interest in the role of size in biology, a subject that has pursued me these many years.\n\nAnother way in which we show our fascination with size is our sustained interest in human dwarfs and giants. In the nineteenth century, Barnum had great success in his circus exhibiting the midget Tom Thumb along with the giant Angus MacAskill, who were so extraordinarily small and large compared to everyday human beings. MacAskill came from Cape Breton in Nova Scotia, now an hour's drive from where I write these words. He was unusual in that he was not only huge\u2014he was 7 feet and 9 inches tall\u2014but he was phenomenally strong. His palms were almost eight inches across and he could hold Tom Thumb, who was a mere 3 feet tall, in his hand.\n\nBoth Tom Thumb and Angus MacAskill suffered from a malfunctioning pituitary gland: in Tom Thumb's case there was a deficiency in the secretion of growth hormone, and in MacAskill's case an oversecretion. It is a powerful hormone that controls our size during our growing years, and if it is not tuned just right, as it normally is, a dwarf or a giant will be the result. Today if the pituitary malfunctions, abnormal size can be prevented by controlling the amount of growth hormone. In the case of pituitary giants it is rarely the case that the individual will have the strength of MacAskill, and in general sufferers of the disease do not live to an old age. (MacAskill died at the age of 38 of \"brain fever,\" presumably meningitis.) The excess of growth occurs mostly by stimulating the cartilage at the ends of the long bones, so it is possible to produce gigantism only in a growing youth. If too much hormone is produced after the long bones become sealed off and their layer of cartilage is gone, which usually occurs around the age of 20, only those parts of the human anatomy that still have cartilage will have a growth spurt. This will happen mainly in the hand and the face\u2014a disease called acromegaly\u2014and those unfortunates stricken with it will have enlarged jaws, broadened noses, and distorted finger joints. The cause is usually a tumor on the pituitary gland.\n\nThe fascination with giants and dwarfs goes way back in literature. Children's stories are replete with them, from Jack the Giant Killer to the Seven Dwarfs and innumerable others. The wonderful Icelandic sagas have diminutive, evil trolls as well as heroes of huge proportions who pitilessly slice off parts of the enemy with their gigantic broad swords.\n\nPerhaps one of the best known and most enduring tales is _Gulliver's Travels_ by Jonathan Swift. Lemuel Gulliver was about twelve times taller than the Lilliputians, and the Brobdingnagians were that amount again taller than he was. Swift sensed some of the problems, and in order to feed Gulliver, the Lilliputians cubed their difference in height and for their half pint of wine they gave him 1,728 (123) half pints. We know today that this calculation does not take into account the relative differences in metabolic rate, and it would have made Gulliver roaring drunk, but Swift was taking a step in the right direction. In other things he made pure fictions, but interesting ones. According to Swift, being smaller meant being able to see smaller things; everything in the natural world of Lilliput was to scale\u2014not just the people. The horses, the chickens, the sheep, the trees, the corn in the fields were all miniature so that their world appeared to them as ours does to us. From this perspective, the Lilliputians' ability to see small things was far greater than ours. \"I have been much pleased with observing a Cook pulling a Lark, which is not so large as a common Fly, and a young Girl threading an invisible Needle with invisible Silk.\" A Lilliputian microbiologist would not need much of a microscope to help her see microbes. The corresponding difference is seen between Gulliver and the Brobdingnagians; he viewed their huge bodies as though through a magnifying lens. He watched the gigantic queen's maids of honor undress, \"before their naked Bodies; which, I am sure, to me was far from being a tempting Sight, or from giving me any other Motions than those of Horror and Disgust. Their skins appeared so coarse and uneven, so variously coloured when I saw them near, with a Mole here and there as broad as a Trencher, and Hairs hanging from it thicker than Pack-threads, to say nothing further concerning the rest of their persons.\"\n\nAs we shall see, none of this is physically possible; no matter how big or how small a living object, the same physical laws will govern it, and those laws will make for differences in shape and form with differences in size. So the three size levels in Gulliver's world, with man and all the beasts being of similar proportions no matter their size, is impossible; it is the world of fantasy. Yet Gulliver is correct when he says, \"Undoubtedly Philosophers are in the Right when they tell us, nothing is great or little except by Comparison.\"\n\n### THE SIZES OF LIVING THINGS\n\nMy plan is first to give the reader some feeling for the enormous variety of animals and plants and all the lower forms. Then in later chapters I will go on to examine what these size differences mean for all aspects of their existence: for their shape, for their complexity and their division of labor, for their evolution, for their abundance in nature, and for all those time-related activities such as metabolic rates, generation times, speed of movement, and even the pitch of their voices and songs.\n\nBig animals and plants have a special fascination, beginning for us all in our childhood. My first biology book was a huge volume by H. G. Wells, Julian Huxley, and G. P. Wells called _The Science of Life_ (1931). It was exciting reading for a small boy, and in fact it trapped me into becoming a biologist. In particular I was fascinated by a section on the size of organisms, which no doubt psychologically imprinted me so that after all these years I am writing this book! I still have my battered copy, and to this day I find the figures on the extreme size of a variety of plants and animals engrossing. They have a rather antique appearance, but they send a clear message.\n\nConsider their figure of the largest forms (fig. 3). To make it easy to understand relative sizes, a man is included alongside the largest of all animals, a blue whale (1). As Galileo understood, it is to be expected that the champion would be an aquatic beast because the whale's specific gravity is close to that of water, and therefore it is essentially weightless and need not have supporting limbs.\n\nWell-known dinosaurs, such as _Tyrannosaurus_ (2) and _Diplodocus_ (3), were among the biggest terrestrial animals, but there were larger ones. In fact, it was known since the early part of the twentieth century\u2014well before the book was published in 1931\u2014that the biggest of them is not among the popular ones shown in the figure. The record giant is _Brachiosaurus_ , first discovered in Colorado in 1900, but a complete skeleton was not unearthed until a dozen or so years later in what was then German East Africa, now Rwanda and Burundi. The reconstructed skeleton can be found in the Humboldt museum in Berlin\u2014standing in imperial elegance in a huge hall built especially to house it. It stands 40 feet tall, and its weight has been estimated to be about 80 tons; this compares to approximately 30 tons for the _Diplodocus_ we see in the figure. The tallest living mammal is the giraffe.\n\nFigure 3. The largest animals and plants, from H. G. Wells, J. S. Huxley, and G. P. Wells, _The Science of Life_ (1931). The names of the organisms are given in the text.\n\nThe largest living animals that can fly are condors, and albatrosses (5) have a record wingspan of 11 feet, but they seem very small compared to the fossil pterosaurs (4) (see also fig. 4). Pterosaur remains were discovered in the late eighteenth century, and the celebrated French anatomist Georges Cuvier showed that they were reptiles and gave them their name. They appeared long before birds and bats and existed for about 215 million years, then became extinct at roughly the same time as the dinosaurs at the end of the Cretaceous era. Even though there were some small ones of birdlike proportions, the larger ones are truly huge (fig. 4). The record holder was discovered in Texas a few decades ago: it was estimated to have a wingspan of at least 36 feet, far bigger than a small airplane. The bones of these huge fliers were hollow and very light; their musculature must have been formidable. These record holders were not known to exist in 1931 when _The Science of Life_ was published; if we were to add these giants they would have a much larger wingspan than the pterosaur shown in figure 3. It is tempting to imagine that the mythical, immense flying bird, the roc of the _Arabian Nights_ , was a pterosaur that lifted Sinbad to safety. However, our credibility is stretched beyond the limit when Sinbad says the roc could lift an elephant in its talons.\n\nFigure 4 (facing page). Soarers. (a) The magnificent frigate bird has a wingspan of more than 6.5 feet; (b) the wandering albatross can have a wingspan of as much as 11 feet, the greatest of any living bird; (c) the largest flying animal of any age was the pterosaur, with a wingspan estimated at 36 to 39 feet. (From T. A. McMahon and J. T. Bonner, _On Size and Life_ , 1983)\n\nWe are all familiar with the fact that flightless birds have become much larger then their flying relatives. There are many examples: emus, rheas, cassowaries, and the champion of them all, the ostrich (7), which seems especially large if we compare it in our minds to a chicken (8). But all the real record breakers are now extinct: the moas and the huge _Aepyornis_ (6). Some recent evidence reinforces the view that their extinction was caused by the advent of early colonization by human beings of the Pacific islands where they lived. Large, flightless birds are not only easily captured by wily hunters, but they are big enough to provide a substantial meal. There is the story of Charles Darwin, who, on his famous voyage on the HMS _Beagle_ , shot a rhea on the Pampas and only after he and his compatriots had half eaten it did he realize that it was a new species and had to carefully resurrect what was left of the uneaten parts.\n\nThe largest snake is a fossil form (9) that exceeded 50 feet in length. I am particularly fond of the drawing that appears to be no more than a straight line (10), showing no anatomical detail, for it would be too slim to depict on this scale. It is a record tapeworm found in a human being, and the fact it is so long means that our intestine is also long and highly convoluted.\n\nLizards can be very big, the largest being an extinct, fossil species (14), which is considerably bigger than a record crocodile from West Africa (12). The huge and ferocious Komodo dragon (13) is modest by comparison, but seen alongside a man or a sheep (11) it appears fairly large. Those who have seen nature films of these inhabitants of Komodo Island near Bali, and watched them tear a carcass apart in a viscous frenzy, do not think of them as being small; they are indeed dragons.\n\nAmong invertebrates the largest jellyfish is _Cyanea_ (16), which was made famous in one of the Sherlock Holmes stories, _The Lion's Mane_ , where it turned out to be the murderer. The biggest specimen known was found in the North Sea: its tentacles spread about 8 feet below the huge bell of jelly. No doubt it is a threat with all its innumerable tiny, poisonous nettle-like stinging cells; but so far as I know, Conan Doyle's fictional account is the only one in which it has actually killed a man. The largest related polyp is of comparable size (15). The giant clam (18) of the southern Pacific Ocean is very big compared to the ones we normally eat\u2014all the invertebrates depicted here are in the same size bracket as a horse (17). Clams feed by bringing a current of water in between the valves and consuming the microscopic plankton they draw in; the valves are held together with a strong muscle that shuts them when the beast is disturbed. It has often been claimed that they can be accidental murderers if a diver inadvertently happens to get a foot inside the closing valves of a giant clam, but apparently it is no more than a tall tale, for no such lethal accident has ever been recorded.\n\nTo continue with the invertebrates, crustaceans can become quite large, such as the Japanese spider crab (20), which towers over a large Atlantic lobster (21). Even bigger is an extinct sea scorpion, or _Eurypterid_ (22), a common fossil form. The biggest invertebrate of them all is a mollusk, the giant squid (25) that lives deep in the ocean and is a prey of the sperm whale.\n\nAmong fishes, a large tarpon (19) seems small alongside the huge whale shark (23), yet it in turn seems modest in size compared to a blue whale (1), which, of course, is a mammal.\n\nThe plant kingdom is less well represented in this figure, although this kingdom produces the biggest organisms of them all. A huge sequoia (26) is so big that only less than a third of it will fit on the page, and for comparison a 100-foot larch tree is superimposed. However, the size of trees is special in that a large part of their bulk consists of dead wood and the living tissue is a relatively thin layer under the bark, besides of course the leaves.\n\nThe largest flower is _Rafflesia_ (24), which is found in Malaysia and Indonesia (fig. 5). Sir Thomas Raffles, the extraordinary British explorer and statesman who founded Singapore, discovered it in 1818. He encountered the flower in Sumatra and wrote in a letter,\n\nFigure 5. The largest flower, _Ralesia_. (Drawing by Hannah Bonner)\n\nThe most important discovery . . . was a gigantic flower, of which I can hardly attempt to give anything like a just description. It is perhaps the largest and most magnificent flower in the world . . . its dimensions will astonish you\u2014it measured across from the extremities of the petals rather more than a yard . . . and the weight of the whole flower fifteen pounds.\n\nIt is a parasitic plant that feeds off the roots of vines\u2014it has no leaves. Not surprisingly, considering its weight, the flower lies on the ground, and it blooms for a brief but glorious four days. It has huge red petals that look like seat cushions, and produces a wonderfully offensive odor of putrefying flesh\u2014irresistible to flies that pollinate the flowers.\n\nAlso not shown in the figure are some very large brown algae. These are the very big marine forms known as kelp. They have a holdfast that anchors them to the bottom of the ocean, and a long stem leads to the blades near the surface where they can bask in the sun for photosynthesis. There is an old account from one of the early sea voyages that a species of kelp ( _Macrocystis_ ) can be over a 1000 feet long, and here lies an interesting tale: in the early part of the twentieth century, the longest measured was 138 feet, which is not inconsiderable.\n\nThe older account of vastly exaggerated length died hard after the modern measurements were made in 1914. When I was a student, textbooks of elementary biology and botany still carried the tall tale even though quite a few years earlier it had been shown to be untrue. It was not until midcentury that the fiction began to fade, proving that we like large organisms, and if nature does not provide ones large enough we manufacture improvements.\n\nA few years ago there was a big splash in the newspapers announcing that the biggest organism ever had been discovered. It was a fungus that parasitized the roots of evergreen trees and spread huge distances in the forest, attacking and killing the trees as it advanced. It traveled over thousands of acres about 3 feet underground, and all the tips of the spreading fan of fungal filaments were shown to be part of the same individual; all were genetically identical. This is, however, a spurious individual organism; in no way does it qualify as a record giant organism. Soil fungi start from a germinated spore at one point and spread outward. It is commonly observed that they give rise to \"fairy rings,\" a circle of mushrooms that form along the edge of the spreading filaments. On aerial photos of the Salisbury Plain in England they can be seen as discolored circles, and knowing how far they grow in a year one can estimate that some are 500 years old. The parasitic fungus mentioned above spread much farther and is estimated to be well over a 1000 years old. The difficulty is that the peripheral filaments become separated from their common stem\u2014they are clones, not one giant beast. They are like groves of aspen (and some other plants) that spread by runners, but we do not consider the entire grove one individual.\n\nWells, Huxley, and Wells also examined the other end of the scale and show a rather old-fashioned drawing of a variety of small organisms that surrounded us: the microscopic world that van Leeuwenhoek opened up with his microscope (fig. 6). In a pond or in a swamp we are likely to find a number of protozoa that are single-cell animals. Mostly they are free swimming, although sometimes they are attached to the bottom, as in _Vorticella_ (8). The one we remember from our elementary biology course is _Paramecium_ (9), and the largest ciliate protozoan is _Bursaria_ (1); note its huge nucleus that looks like a long piece of rope. Number (11) is a human liver cell, and (10) an amoeba that causes dysentery. A very small wasp is seen in (4), a species of rotifer, which is a very small multicellular animal, in (5), and a cheese mite in (6). A human sperm (7) seems small alongside a human egg (6). Also shown is the foreleg of a flea (2). If this figure were done today it would include the recently discovered smallest vertebrate, a fish in Australia that is about 7 millimeters long and lives for only two weeks.\n\nFigure 6. Some smaller organisms, from H. G. Wells, J. S. Huxley, and G. P. Wells, _The Science of Life_ (1931). The names of the organisms are given in the text.\n\nWhile all of these organisms are very small, they are by no means the smallest. The most abundant of all the microorganisms are bacteria, and they would be about the size of the stipple dots in the figure. There are even smaller organisms, the so-called mycoplasmas. Unlike bacteria, they lack cell walls and are a fraction of the size of bacteria. At the other extreme is a very large and curious bacterium that is big enough to be seen with the naked eye and lives in the intestine of some fish.\n\nIt is interesting to note that at one point in the early history of our Earth bacteria were the only organisms, and they have not only failed to become extinct, but are found everywhere today in incredibly vast numbers\u2014a genuine case of multizillions. They have even been found in the deepest trenches in the ocean floor, and most remarkably in the crevices of rocks in the deepest mines. They also exist in great abundance in the intestines of animals. It is estimated that we harbor in our own gut 1013 bacteria (ten trillion), which is considerably more than the number of cells in our bodies. It is mostly the big beasts, such as the dinosaurs, that die out; the smallest ones started off as a success and have remained so ever since.\n\n##   CHAPTER 3\n\n## THE PHYSICS OF SIZE\n\n### WEIGHT-STRENGTH\n\nSize and shape are inextricably connected. One of the most elementary ways we can show that size is the supreme arbiter and has a profound effect on shape is to consider the relation between weight and strength. Galileo first demonstrated the principle elegantly in his book _Dialogues concerning Two New Sciences_ , which he wrote after he was tried and condemned by the court of the Inquisition in Rome for having committed heresy in supporting Copernicus's heliocentric view of the heavens\u2014that our world was not the center of everything. He wrote the book while in house arrest in the palace of the archbishop of Sienna, but he was forbidden to publish anything in Italy. An opportunity arose when the Dutch publisher Louis Elzevir came to Italy in 1636 and managed to take with him, on his return home, the then completed part of the manuscript. The subjects, mechanics and motions, were ones that Galileo had been working and lecturing on for many years, and now it was brought together in what he (and others) considered his most important work.\n\nThe book was written in Italian, and there is an excellent translation in English. The science is presented in the form of a dialogue between three men: Salvati, Sagredo, and Simplicio. Today it is so unlike the way we present science that it seems rather contrived, but the main points are made with the utmost clarity, often illustrated by diagrams and drawings. There are four books, or Days, and in Day Two Galileo expounds the basic relation between size and shape. His main point is that the weight of any body is a cubic function, that is, the cube of its linear dimension, be it height or width. Strength, on the other hand, is a function of the cross-section area, that is, the square of the linear dimensions. Let me give you the consequences of this difference in his own words, which he relays through Salvati:\n\nFrom what has been already demonstrated, you can plainly see the impossibility of increasing the size of structures to vast dimensions either in art or in nature; likewise the impossibility of building ships, palaces, or temples of enormous size in such a way that their oars, yards, beams, iron-bolts, and, in short, all their other parts will hold together; nor can nature produce trees of extraordinary size because the branches would break under their own weight; so also it would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only be employing a material which is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animal suggests a monstrosity. (Fig. 7.)\n\nSalvati then goes on to say that because of this weight- strength relationship, a smaller animal will have a greater relative strength, which he illustrates by pointing out that \"a small dog could probably carry on his back two or three dogs of his own size, but I believe that a horse could not carry even one of his own size.\"\n\nFigure 7. A drawing From Galileo's _Dialogues concerning Two New Sciences_ , showing how a normal bone (top) would need to be increased in thickness (bottom) if the weight of the animal were much greater.\n\nSimplicio then says he doubts this statement is true because of \"the enormous size of certain fish, such as the whale.\" He was a bit off calling a whale a fish, but Salvati has a ready answer to this welcome question: by being submerged in the water, a whale is \"making their bones and muscles not merely light but entirely devoid of weight.\"\n\nThe consequences of the weight-strength ratio being influenced by size are large, and here I will stress the living world rather than that of the engineer, although both are equally affected. Beginning with the animal world, the fact that a small gazelle has slender, graceful legs, and an elephant is endowed with great stumps is precisely what Galileo would have expected. If there were elephants on the moon, which is smaller than the earth and therefore the gravity is less, one could imagine that they would have evolved into gigantic gazelle-like beasts. Yet another way to look at the same point may be found in that splendid chapter in D'Arcy Thompson's book _On Growth and Form_. He points out that the size of a terrestrial vertebrate is reflected in the relative weight of its bones, and consequently, \"bones make up some 8 per cent of the body of a mouse or wren, 13 or 14 per cent of goose or dog, and 17 or 18 per cent of the body of a man.\"\n\nAs Galileo makes clear, the shape of trees is equally affected by size, and we expect a giant sequoia to have, like the elephant's foot, a huge wide and solid base compared to a smaller tree. The difference can even be seen in the growth of a single tree: a maple starts off as a slender, willowy sapling that, with increasing size, becomes disproportionately thick in its trunk and limbs. The same principle can be seen in fruit of different sizes. The strength of the stem of each fruit will depend on its cross-section area, so the stem of an apple will be disproportionately thick compared to that of a cherry. The stems of coconuts are very thick and sturdy, while watermelons have solved the problem by lying on the ground; their solution is somewhat analogous to that of whales, for in different ways both have largely circumvented the pull of gravity. The relationship between strength and weight can be expressed in a simple equation (to which we will return):\n\n(1) Strength \u221d Weight2\/3  \n(\u221d= varies as).\n\n### BACK TO GULLIVER\n\nTo return to _Gulliver's Travels_ , let us reexamine Jonathan Swift's work in the light of our discussion; it is a good way of illustrating the principles laid out here.\n\nLet us assume that Lemuel Gulliver was 5 feet and 8 inches tall. The Lilliputians were one-twelfth his height, which would make them just shy of 6 inches tall\u2014roughly the size of a mouse. This would mean that their legs and their bones could be relatively slender compared to normal human beings; they could have stick legs like a sandpiper that runs along the edge of the water on the beach. After all, if Gulliver weighed about 150 pounds, then the similarly constructed Lilliputian would weigh less than a pound. The result would be that he would not be similarly proportioned to Gulliver, for legs like match sticks would be quite sufficient to support him.\n\nThe speed of running of Lilliputians, while no doubt slower than that of Gulliver, would not be that far behind because their legs will move much faster (like the sandpiper), and for one of Gulliver's normal steps they would have taken many. All the movements of the smaller homunculus would be more rapid than our movements, just as the wing beat of a small hummingbird is far quicker than that of a swan.\n\nWe know that rats and shore birds have lungs, a heart, gut, and circulatory system as we do. The only difference is that the proportions of these parts of their anatomy will differ from ours, for they have relatively less work to do because of their smaller size: the heart pumps the blood into a smaller set of blood vessels and therefore can be smaller. Because the Lilliputians have smaller vocal chords their voices will be high and squeaky, and they would talk with great rapidity, so much so that it might be difficult for Gulliver to hear and understand them.\n\nThe reverse of these arguments would apply to the huge Brobdingnagians, who, to begin with, would have extremely deep and slow voices. Swift describes the first of the giants he encountered this way:\n\nHe appeared as Tall as an ordinary Spire-steeple; and took about ten Yards in each Stride, as near as I could guess. I was struck with the utmost Fear and Astonishment, and ran to hide my self in the Corn, from whence I saw him at the Top of the stile, looking back to the next Field on the right Hand; and heard him in a Voice many Degrees louder than a speaking Trumpet; but the Noise was so High in the Air, that at first I certainly thought it was Thunder.\n\nIf their proportions are those of a normal human being and they are twelve times taller than we are, which would make them 68 feet tall, they would weigh between 12 and 13 tons. This makes them lighter than the larger dinosaurs, but about ten times heavier than a large elephant. Like these large beasts, the Brobdingnagians would require greatly thickened legs and leg bones, so much so that they would no longer appear like normal human beings, but more like victims of advanced elephantiasis of the legs.\n\n### SIZE AND DIFFUSION\n\nStrength is not the only feature that affects the shape of an organism as its size changes. Another most important one is diffusion, which, as we shall see, has a powerful effect on shape. Diffusion is always through surfaces, and since the surface is a square of the linear dimensions and the weight is the cube, we can again write a simple equation to express the relation of surface to size:\n\n(2) Surface \u221d Weight2\/3.\n\nMost organisms, including ourselves, depend on oxygen for their existence; every cell of our body needs to have oxygen diffuse into it in order for it to function and stay alive. Diffusion is the random movement of molecules from regions of high concentration to low, so that ultimately, at equilibrium, they are evenly spread out. Molecules zip around at great speed when they collide with one another. In regions where they are numerous, they collide often and have more luck spreading when there are fewer ones to bump into. A way to look at this might be to imagine a tea bag placed in a glass of cold water and all currents, convection and otherwise, are eliminated. The brown color released from the tea bag is first concentrated in and around the bag itself, but eventually the brown color will be evenly distributed; it has diffused from the bag.\n\nIf oxygen has to get into an animal or to its cells it must diffuse through a surface that, like strength, is a function of the square of the linear dimensions. This can be illustrated in a simple fashion. Consider two hypothetical spherical organisms, one the size of a sand grain a millimeter in diameter, and the other the size of a golf ball. Oxygen is a small molecule that diffuses rapidly, and in a matter of seconds it can penetrate a millimeter into a living body. The middle of the tiny sphere is close enough to the surface so that as soon as it uses up its oxygen it will be replenished by a steady diffusion of more from the surface. But in the golf-ball beast it would take an hour or so to reach the interior, and we all know that would not do\u2014if we are deprived of oxygen we die. In fact, we are so tuned in to this problem we cannot even hold our breath for more than a few minutes; we have a built-in mechanism that forces us to start breathing again.\n\nThe only way an organism larger than a millimeter can survive is to bring its interior near the surface everywhere so that no part is farther away than one millimeter. This can be managed by radically increasing the amount of surface, making many convolutions and folds so that every point in the interior is close to the surface. Instead of a perfect, smooth sphere, we now have a golf ball with an incredibly wrinkled surface (fig. 8). Or to put it another way: because the enlarged spherical organism cannot exist in it original form, it must grow more skin, and the result will be a radical change in shape.\n\nShape changes caused by diffusion problems can be far more revolutionary than this transformation. Much of the shape of human beings is molded by the diffusion of oxygen and the problem of getting it to our inner cells. To achieve this end, we have a marvelously complex set of devices. We have lungs that bring the oxygen close to our blood vessels where the hemoglobin can pick it up; we have a circulatory system with a heart, a most ingenious and durable pump that pushes the oxygen-containing blood into the minute vessels, or capillaries. They permeate all parts of our bodies and lie close\u2014well within the one millimeter limit\u2014to each of our deep-seated cells. Note that the lungs and the blood vessels have vast surfaces; like our wrinkled golf ball, they keep the surface-volume ratio constant despite our large size. More than that, they have set up a system of motion so that the lungs move the air in and out, and the beating heart keeps the blood circulating. The movements supplement diffusion so that fresh oxygen can be rapidly picked up in the lungs and delivered to the cells.\n\nFigure 8. The shape of a hypothetical golf-ball-size organism that requires every internal cell to be near the surface. (Drawing by Hannah Bonner)\n\nOxygen is not the only bit of fuel needed to keep a large animal going. Food to burn for energy is needed as well, and it also must be taken in through a surface. Our food canal, from our mouth to our anus, is a tube, a surface through which the broken-down, dissolved food has to pass. The greater the surface of our gut, the greater our ability to absorb food. Because surface increases as the square, the larger the size of an animal (which increases as the cube), the greater must be the surface of the gut to keep pace in order to maintain a constant surface-volume ratio. This change can be seen clearly if we compare ourselves to minute nematode worms that are little bigger than an eyelash. Their gut is a straight tube, but ours is a tortuous, highly convoluted tube many times our length. Furthermore, our small intestine is not a simple tube, but has a forest of tiny papillae that stick up into its inside, each one greatly increasing the gut surface to absorb more food. We have also developed glands: the salivary glands, the pancreas, and the liver that produce batteries of enzymes to break down the food into small molecules such as simple sugars and amino acids, so that they can diffuse through the gut wall more easily.\n\nAll these devices found in a large animal are present to get energy from food by combusting it with oxygen in all the cells of the body. This job calls for lungs, a circulatory system, and a highly elaborate gut and its associated glands. From this necessity it is clear that size increase requires quite extraordinary changes in shape. In no way do we resemble a smoothly spherical organism, nor even a wrinkled golf ball; we have evolved into a far more complex shape largely because of the increase in our size.\n\nViewed through the eyes of Lemuel Gulliver, the Lilliputians need only a comparatively small number of blood vessels to reach all the tissues of their bodies. The Brobdingnagians, on the other hand, need a vast number, and their guts will have to be disproportionately extended with an immense increase in their surface area to absorb sufficient food to accommodate their giant size.\n\n### SMALL WORLDS\n\nIt is possible to put large and minute organisms in quite separate baskets; there is a fundamental rift between the living worlds of the large and very small, even though it is obviously a continuum if one goes from large animals and plants to small ones; organisms come in all sizes in between. The reason for arguing for such an artificial separation of two size worlds has to do with the forces affecting the totally different size levels. The effect of gravity becomes progressively insignificant the smaller an organism, while molecular forces come to the fore and play an increasingly important role. Even though these forces decline and rise smoothly as the size changes, one can artificially designate a critical point where one jumps from one world into the other. Presently I will give an example, but first let us look at a simple graph (which I have taken from a splendid paper by F. W. Went) in which the force of gravity and the force of molecular cohesion are plotted on a size scale (fig. 9). In terrestrial organisms, gravity varies with the weight, and therefore with the cube of the linear dimensions, while the force of attraction between surface molecules varies roughly as the linear dimension. This approximation takes into account the forces of attraction between molecules and the distance between the molecules on the two surfaces. The important point is that the cohesion curve is less steep than the curve for gravity. The figure is drawn in such a way that the lines for gravity and cohesion arbitrarily intersect for an organism roughly a millimeter in length; above that point, the force of gravity increases more rapidly than that of cohesion, while below that point smaller organisms see a declining effect of gravity compared to that of cohesion. In other words, minute beasts hardly feel the effects of gravity at all; rather, the forces of cohesion affect every aspect of their existence.\n\nFigure 9 (facing page). A synthetic graph comparing how different size terrestrial organisms are affected by gravity and cohesive forces. Forces due to weight vary as the length cubed, while the forces due to molecular cohesion average out very roughly to be proportional to the length. The lines are arbitrarily placed so that they cross for organisms 1 mm in length, and it can be seen that in larger organisms the forces of gravity (which vary with the mass) play a far more significant role than the forces of cohesion, while for smaller organisms cohesion forces dominate. (This figure is based on one in F. W. Went, _American Scientist_ 56 [1968]: 400\u2013413)\n\nA fly or a gecko can easily walk up a wall, something we humans cannot even begin to manage. The reason is that these creatures' weight is negligible and the bottoms of their feet are specially constructed with many hairs so that they can have very intimate contact with all the minute nooks and crannies of the rough surface of the wall. This feature can be seen especially clearly in the foot of a small gecko (fig. 10), indicating that the molecules in the feet and those in the wall become very close to one another and the force of that molecular attraction increases the closer the molecules of the two surfaces are. The smaller the beast, the greater the effect of cohesion compared to that of gravity.\n\nFrom this we see that small organisms live in a very different world than large ones; it is physically different. Besides the increase in the importance of molecular forces, it is also the world of low Reynolds numbers, a world that does not exist for larger forms.\n\nThe Reynolds number is a convenient way of describing the immediate environment of a moving body. It is a way of showing that a swimming bacterium is in a very different environment than a swimming whale even though both are swimming through water. The vast difference in their size, for purely physical reasons, affects their locomotion. A whale has a far greater inertial force, and if it suddenly stops swimming it will glide on for some time. Because of its great mass it is endowed with a large amount of momentum. On the other hand, when a minute bacterium stops swimming, its mass is so small that it stops in a split second.\n\nFigure 10. The foot of a gecko and a high-power microscopic view showing the mass of very fine hairs that can make intimate contact with a solid surface. (Drawing by Hannah Bonner)\n\nIn the latter part of the nineteenth century, Osborne Reynolds, a distinguished engineer, came to understand this problem in a deep way. But here I will describe his sophisticated principle in a very informal way that will no doubt make a physicist cringe. Nevertheless, its essence can be expressed as follows:\n\nBoth the whale and the bacterium swim in water so the denominator in the equation above will be the same for both. However, the inertial forces for the whale will be huge compared to that of the bacterium, with the result that the whale will have a correspondingly large Reynolds number while the bacterium's will be small. Microscopic organisms live in a world of low Reynolds numbers.\n\nTo give one a feel of what this means for minuscule organisms, the noted physicist Edward Purcell painted the following picture. Imagine a man swimming in a liquid that produces the same low Reynolds number as would exist for his own sperm. It would be like trying to swim in thick molasses in which one was not allowed to move one's arms or legs faster than the hands of a clock. Under these rules, if one moved a couple of meters in a couple of weeks, one would qualify as a low Reynolds number swimmer. This vivid picture gives one an idea of how the changes in the inertial forces affect a bacterium; it is indeed a different world. Purcell says, \"Motion at low Reynolds numbers is very slow, majestic and regular.\"\n\nSwimming microbes have whiplike structures called cilia or flagella that propel the organisms through the water. In single-cell protozoa, cilia move the organism by behaving like sophisticated oars; they are flexible and bend on the back stroke, but are stiff when pushing forward and provide the cell with a powerful thrust (fig. 11a).\n\nLet me interject an aside here. When I was a student and first saw, through the microscope, these little whiplike structures move one-cell organisms across the field of vision, I marveled at how fast they traveled. Later I realized that they were in fact moving very slowly and the magnification of the microscope had fooled me. After all, speed is distance divided by time (as in miles per hour), and the high power of the microscope made the minute distance appear very large. For me it was part of growing up: I had not realized that what I was seeing\u2014those protozoa zipping around at an incredible rate\u2014is an optical illusion created by the magnification of the microscope. It was rather disappointing to learn that the microworld was not dazzlingly fast but rather sedate.\n\nBacteria are very much smaller and for that reason have a different set of problems. Not only are they in an environment of even lower Reynolds numbers, but also they are so small they cannot construct the relatively huge and complex oarlike cilia or flagella of larger microbes. They are in the nanoworld, having very thin flagella of a simple construction that produces locomotion in quite a different way. Each flagellum is attached to a rotor that goes around and around, causing the flagella themselves to rotate like a ship's propeller, and this action thrusts the bacterium forward (fig. 11b). There are many remarkable aspects of this ingenious method of dealing with the difficulty of moving a minute body in conditions of very low Reynolds number.\n\nFigure 11. Microbe propulsion. (a) The flexible oar of ciliate. The solid curves show successive positions of the cilium beating downwards for forward movement; the broken curves show the cilium on the reverse stroke (from T. A. McMahon and J. T. Bonner, _On Size and Life_ , 1983). (b) A bacterium ( _E. coli_ ) swimming forward (top); and how a sudden reversal of rotation causes the flagella to splay out and the cell to tumble (bottom). (After J. Adler and H. C. Berg; from Bonner, _The Evolution of Culture in Animals_ , 1980)\n\nThe discovery that bacteria move this way was made not that many years ago. It was first predicted on theoretical arguments based exactly on what we have just seen: the special problems connected with low Reynolds numbers. This prediction was followed by a very clever experiment in which the flagella were trapped on to a specially prepared sticky microscope slide with the amazing result that the bodies of the bacteria could be seen, through the microscope, to rotate continuously. They possessed a true rotor, something not previously known to exist in the living world.\n\nWe often congratulate ourselves on our inventiveness that places us above all other animals. We not only discovered how to control fire and make use of it, but we also invented the wheel, that clever device that allows us to move heavy loads. It is usually claimed that no other animal can manage this feat, but as we now realize, it was in fact devised billions of years before _Homo sapiens_ appeared on the globe. As we have just seen, bacteria invented the wheel, and exploited theirs in a most sophisticated manner.\n\nTo illustrate this sophistication, let me briefly explain how bacteria can orient themselves in their aqueous environment. It is well known that swimming bacteria will go towards food and away from noxious substances; now we know how they do it. They have molecules on their surface that can adhere to specific external chemicals, and if they happen to be swimming in the direction towards food, then there is a molecular signaling mechanism that tells the rotor to keep going. If, on the other hand, the food molecules become increasingly less abundant because they are swimming away from the food, then the rotor gets a new message and abruptly reverses its direction. This action causes the whole cell to stop, to tumble, and when it starts up again there is a good chance it will be pointed in a new direction, which, with luck, might be the direction of the energy-rich food. So bacteria not only can move, but they are blessed with the ability to move in a direction that is to their advantage. And they have managed all this without a brain!\n\nA low Reynolds number environment also influences very small flying insects. Their force of inertia is insignificant, with the result that air is also like molasses. As a consequence in more than one group of insects, their wings have become transformed into what look like minute feather dusters (fig. 12). These so-called fairy flies (which are in fact minute parasitic wasps) use their wings the way ciliate protozoa use their cilia; they swim through the air.\n\nFigure 12. Among the smallest insects are fairy flies, which are actually tiny wasps. Their body lengths can be less than half a millimeter. The wings are slender stalks, whose margins are fringed with hairs. (From T. A. McMahon and J. T. Bonner, _On Size and Life_ , 1983)\n\nIt is clear that size, from the pure point of view of physics, has an enormous effect on living organisms. It constrains, or even controls, the shape of an organism, including its internal structure. It determines what is possible and what is impossible. And as we have also seen, it can affect the behavior of animals as well as their locomotion. In the next chapter we will see that size is also directly related to the degree of division of labor.\n\n### MEASURING PROPORTIONS: ALLOMETRY\n\nNote that if one compares small and large animals or plants, one aspect of their relative shapes is a change in proportions. This is not surprising because, as we have seen, in order to exist those properties that are related to the square of the linear dimensions, such as strength and the diffusion processes related to energy utilization, must, with increased size, rise more rapidly to keep pace with an increase in volume or mass. It shows itself by a disproportionate increase in the thickness of limbs and a more convoluted gut in larger animals.\n\nThere is a simple and elegant way to describe the changes in proportion to size, known as allometry. Consider our earlier example of the bones of small gazelles and large elephants. If they were similar in all respects, then an elephant would have legs that looked like a photographic enlargement of a gazelle. If this were so and we plotted the height of animals of different sizes against the diameter of their bones, we would get a straight line with a slope of 1 (fig. 13). That is, the diameter varies as (\u221d) the height:\n\nDiameter \u221d Height.\n\nAs we learned from Galileo, they are not similar because the overall size (weight) of the animal rises as the cube of the linear dimensions, while the strength of the bones rises as the square of the linear dimensions. That is,\n\n(2) Strength \u221d Weight2\/3.\n\nThis equation neatly expresses how the strength-weight affects the proportions. It is convenient to plot this relationship on the graph as logarithms, for the exponent of 2\/3 will produce a straight line whose slope is 2\/3 (fig. 13).\n\nFigure 13. A log-log graph showing allometric relations. (Line a) If diameter is plotted against height in geometrically similar organisms of different sizes, then the slope of the line equals 1. (Line b) If strength (which is diameter squared) is plotted against weight, which is the cube of the linear dimensions, the slope of the line equals 2\/3.\n\nThere are many ways to illustrate the usefulness of allometry, and I shall pick two: size changes during human growth, and relative brain sizes.\n\nProportions change as animals and plants grow. For instance, a human baby is quite different in its shape from an adult\u2014there is a continual change in proportions as we grow. The best way to show this is to draw the various stages of human growth adjusted to the same height (fig. 14). If one looks at a small baby its head seems enormous compared to that of a grown-up. In fact, a baby's head accounts for 34 percent of its height, but for a mature person the head is only 13 percent of its height. The body grows in length about twice as fast as the head in the growth of a human being.\n\nBrain size is a subject especially dear to us human beings because we are so pleased and self-satisfied with our very large brains. If one plots the weight of the brain against the body weight of various vertebrates, one again finds a 2\/3 relationship (fig. 15). The bigger the animal, the bigger the brain, although it is not directly proportional, for as the body size increases the brain size increases less (roughly 2\/3 of the increase in body size). It is not clear why the brain of large animals should be relatively smaller than those of little animals.\n\nBigger brains generated an interesting idea by a distinguished German evolutionary biologist named Bernard Rensch some years ago. Rensch argued that being bigger meant having a larger brain, which meant that larger animals must be smarter. Increased size automatically conferred such an advantage, and this was one of the reasons there are often evolutionary trends towards larger size. Rensch used the ability to learn, and retention of what is learned in the form of memory, as his measure of intelligence. To examine his idea he compared similar birds of different sizes\u2014bantam chickens, normal-sized ones, and giant breeds. He gave them a simple learning procedure followed by periodic testing to see how long they remembered what they had learned. He found that the larger birds could learn more and retained the newly acquired information for a longer period of time. Having a big brain did seem to provide an advantage. He realized that all of this fits in with the old adage that an elephant never forgets, and proceeded to make some interesting observations on circus elephants, and indeed they had a remarkably long memory for some card puzzles he taught them. I am reminded of a story my father used to tell us children about the circus elephant that recognized his old trainer in the audience and lifted him up with his trunk and moved him from the fifty-cent seats to the dollar seats.\n\nFigure 14. Changes in body proportions in a human being with increasing age. (After Wells, Huxley, and Wells, 1931; from T. A. McMahon and J. T. Bonner, _On Size and Life_ , 1983)\n\nFigure 15. Brain size of 200 species of vertebrates plotted against body size on a log-log graph. Primates are open squares; other mammals are solid dots; birds are solid triangles; bony fishes are open circles; and reptiles are open triangles. (After H. J. Jerison, _The Evolution of the Brain and Intelligence_ , 1973)\n\nRensch once came to our university to lecture on all this, and in the discussion period I asked him if perhaps it was not intelligence, but because the animal's metabolism is slower the larger the animal, the elephant was simply slow to forget. Professor Rensch got quite irritated with this fresh question from a young man, but to this day I wonder if I did not have a point. Later I plan to talk about size and metabolic rate in some detail. I cannot believe that Rensch's intriguing idea has been an important factor in evolution. There are too many small animals of remarkable intelligence, such as parrots and crows among numerous others. One cannot imagine that the size of elephants increased because of a selection advantage of being better at learning and remembering things.\n\nIf we turn to primates, we see a similar relationship: the larger the monkey or ape, the larger the brain. However, if we look to the evolution of human beings, from _Australopithicus_ to _Homo_ , then we see an amazingly steep rise of brain size relative to body size. Instead of the brain enlarging at a slower rate (2\/3 or less) than the body size, it rises at a much greater rate\u2014almost two times that of the increase in the body size (fig. 16). Note that if we compare _Australopithicus_ species of different sizes, the allometric relation of their brain size increase is modest, mirroring that of apes. In other words, in our evolution from a common ancestor with apes, one great difference from them is our remarkable increase in brain size. We did indeed become the brainy animal. The only other animals that had a similar increase in relative brain size are porpoises and whales; we are only beginning to understand something of their complex behavior.\n\nFigure 16. The endocranial volume plotted against the body weight of great apes, Australopithecines, and the _Homo_ line on a logarithmic scale. For the fossil forms the weight is, of course, only an estimate. (Redrawn from D. Pilbeam and S. J. Gould, _Science_ 186 [1974]: 892\u2013901)\n\nAs one might expect, there has been a considerable amount of speculation as to why brain size might have made these sudden spurts. One of the earlier hypotheses centered around the idea that it had something to do with needed extra brain power to make and use tools. For various reasons this notion has fallen in disfavor, largely because tool making has come to light not only in great apes but in many animals. For instance, Caledonian crows will not only use a piece of wire to extract food from a long, narrow glass vessel, but after a few unsuccessful tries they will carefully bend the tip of the wire to hook the food at the bottom. It seems unlikely that devising and using tools would drive brain evolution to any great degree; those Caledonian crows have the same size brain as other crows. Another idea is that the brain increase is associated with the rise of human language, which implies that in terms of Darwinian natural selection, being able to speak more effectively conferred a selective advantage, and the better communicators had more offspring to whom they passed on those skills. The difficulty with this suggestion is that an increase in brain size along with communication skills is merely a correlation; it is hard to see what comes first\u2014what caused what.\n\nA related and more general argument comes from the modern study of primate social groups, from lemurs to chimpanzees. Clearly, reproductive success in those groups does not just depend on brute strength, but on careful manipulation of one's position within the group. This in return requires managing alliances with others in the group to strengthen one's social position; it is what Frans de Waal has called, based on his fascinating studies, \"chimpanzee politics.\" This whole idea has now taken hold and has even been given a name: \"Machiavellian intelligence.\" According to this hypothesis, it is the social interactions that are key, and the individuals who can influence others to help them in their power struggle are the ones blessed with reproductive success. It is of special interest that recent work on the large-brain dolphins shows some strikingly elaborate social behavior; maybe they too have become big brained due to Machiavellian intelligence.\n\nBecause a large brain is such a striking character of _Homo sapiens_ , great efforts have been put into attempts to tell us something about differences among humans. Using head or cranium size as a measure of brain size, there is great variation in our species. Those who have some sort of dubious agenda have especially loved this: those who scrounge for evidence to prove that a race with bigger heads is superior in intelligence to another with smaller ones. It has even been used to argue that men are superior to women because they have, on average, larger bodies and therefore larger heads!\n\nOver and above the fact that this approach is distasteful, for many reasons it has proved to be a fruitless and muddled enterprise. To begin with, variation in brain size, as measured by volume within the cranium, very likely is too rough an approximation of the number of neurons in the brain and their pattern of interconnections. Variation in how closely packed the nerve cells are in different parts of the brain, and the degree of folding of the brain surface, will not be reflected in head size. In a splendid play about the famous mathematician Alan Turing, who did so much to launch computers and artificial intelligence, there is a scene where he goes back to his old school to give a lecture about the brain, which he describes to the boys as looking like a lump of porridge. In this chaos we know little about how the millions of neurons, and even more importantly how their interconnections, lead to intelligence. To make the point in a forceful way: thanks to the pioneering work of Karl von Frisch, we know that honeybees have an extraordinary ability to memorize information and even communicate that information to their fellow workers, yet the brain of a bee is minute with a correspondingly minute fraction of the neurons found in any mammal, let alone human beings. This is a fascinating area of inquiry, and despite recent considerable advances, we have yet to scratch the surface.\n\nFurthermore, our crude methods of measuring human intelligence are woefully inadequate. IQ tests tell us something about thinking, but one never quite knows what. There is a strong suspicion among many, including myself, that they do not say anything about what we would normally recognize as intelligence, but merely tell us how good we are at taking that kind of a test. I have already pointed out that brain size varies with body size and therefore it is to be expected that women, who are on the average smaller than men, should have smaller brains, but there is no evidence for any significant difference in their mental abilities, even their ability in taking IQ tests.\n\nAs one might imagine, there is a great literature on this subject, often polemical. To my mind it can be brushed aside in a simple way. There are many well-known savants who had small skulls containing great minds. An example of such a small-headed person is Anatole France, the French author who is famous for his novels and for having exposed the anti-Semitic injustice of putting Dreyfus in the penal colony on Devil's Island at the turn of the twentieth century. In looking this matter up, I was delighted to discover that Jonathan Swift of _Gulliver's Travels_ had an exceptionally large head\u2014twice the size of Anatole France's!\n\n### FIVE RULES\n\nLet us pause for a moment to comment on the two rules that have just been discussed. Both strength and surfaces vary with the weight in the same way:\n\n(1) Strength \u221d Weight2\/3\n\n(2) Surface \u221d Weight2\/3.\n\nLater we will see that other important properties of living organisms vary in a similar way, although the exponents, and therefore the slopes of the lines when plotted logarithmically, vary. The three examples to come are\n\n(3) Complexity \u221d Weighta\n\n(4) Abundance \u221d Weight\u2212b\n\n(5) Metabolic rate \u221d Weightc.\n\nAll five reflect the fact that size differences, as indicated by their weight, affect different properties: for a given weight, an appropriate strength is required for support and movement; an appropriate surface, so that an adequate diffusion of oxygen and food substances can reach the inner tissues; an appropriate division of labor, so that the body of the animal or plant can function; and the appropriate rate of metabolism for that functioning. Finally, size affects the abundance of animal and plants in nature: the bigger the organism, the more space it needs. These five rules are central to the argument of this book; they are the pillars that underpin the effects of size on living organisms.\n\n##   CHAPTER 4\n\n## THE EVOLUTION OF SIZE\n\n### SIZE INCREASE OVER GEOLOGICAL TIME\n\nIn looking at the history of the earth, there is general agreement that at some early era all organisms were unicellular; first bacterial (prokaryotic) cells, and later cells with a nucleus (eukaryotic). This was followed by the advent of multicellularity, and ultimately those multicellular animals and plants became the huge organisms I have just described. Over the years, from Aristotle on, some form of this trajectory was interpreted in terms of progress. Aristotle called it the scale of nature, which went from vegetables and worms to human beings. Because we were the pinnacle of this progress, it was not a perfect progression in size increase but rather one of inching towards the complexities and the great virtues of our own species.\n\nLargely as the result of many discoveries of fossils, a similar idea sprang up in the nineteenth century. Fossils of some animals were well preserved, and it could clearly be shown that as geological time advanced the descendants of those animals became larger. One of the best known examples was the evolution of horses, where the dawn horse is the size of a fox, and with time the fossil remains showed increasingly large animals until we reach the large modern horse. There are similar fossil series for numerous other animals, such as the camel.\n\nBy noting these size-increase series over time, it was common, at least at the end of the nineteenth century, for people to consider that there was a mystical evolutionary force that governed these trends. The process was even given the fancy name of \"orthogenesis\"\u2014it was an innate tendency in evolution for progress. The remarkable thing is that this continued many years after Charles Darwin's publication of his theory of natural selection that was quite incompatible with the idea of orthogenesis. Darwin pointed out that all species showed variation\u2014inherited variation\u2014and that the variants that were best adapted to a particular environment were the most successful ones in producing offspring. This was not a goaldirected process; there is no built-in mechanism that drives evolution in one direction; there is no \"progress.\" The fact is that there were very few biologists until the 1930s who believed that Darwin's natural selection was the whole story; there had to be more, and for this they assumed some innate notion of progress, apparently undismayed by the fact that this was no explanation, but just a name for an unknown, assumed mechanism. Furthermore, it was not until much later into the twentieth century that animal series were discovered that decreased in size over geological time: size trends could go up or down.\n\nThat natural selection is all that is needed to explain the evolution of size increase from single-celled bacteria to blue whales and giant sequoias is taken for granted today. There is no longer a psychological need to assume unknown forces; we are comfortable with the simple and totally undirected mechanistic explanation. There has been a selection for larger organisms; under some circumstances size increase can confer advantages and promote reproductive success. This argument applies at the smallest level from unicellular to multicellular organisms. If there is an advantage to increased size, one of the simplest ways to achieve it is to become larger by becoming multicellular.\n\nIt is important at this point to understand the nature of natural selection for size; there is an ecological principle. If one looks at any environment, from fields to forests, from ponds to oceans, there is always an array of organisms of different sizes, from the smallest bacteria to the largest trees or vertebrates and all the middle-sized forms in between. The organisms at each size level are competing with one another for food, either by being more successful at catching or gobbling up the smaller prey if they are animals, or by being more successful at catching the sun's rays if they are photosynthetic plants. In a balanced environment, each size level will have certain advantages; in a sense they have their own house, their own right-sized niche that they can inhabit. If such a niche is vacated by an extinction, then it will soon be filled, for natural selection will favor the small individuals of a large species to become smaller, or larger individuals of a small species to become larger (or, of course another species of the same size). So each size level in the ecological environment will fill up if vacated. The process that achieves this is Darwinian natural selection.\n\nIf put in this way it is immediately evident that the size level that is always open is the largest\u2014there is always room at the top. So if one follows the course of evolution over time, one would expect that the number of the size levels would steadily expand since the one at the top is always open and available to be filled. This simple fact explains why the world has evolved from one in which there was nothing larger than bacteria to today's, in which we have blue whales and giant sequoias. Let us examine this trajectory more carefully by looking at the fossil record to see if it conforms to the principle that there will be an expansion of the upper size limit throughout geological time. One can plot the largest organism known at any one time in earth history, and it is clear that this upper limit steadily increases from the beginning of life through today (fig. 17).\n\nFigure 17. A log-log graph showing a rough estimate of the maximum sizes of organisms at different periods of life on Earth. (Data for the larger forms comes from Bonner, _Size and Cycle_ , 1965; for data on the five smaller forms I am indebted to Shuhai Xiao)\n\n### WHAT CHANGES CAME FIRST: SIZE  \nOR CONSTRUCTION NOVELTIES?\n\nThis brings us to the central argument of this book. If we look at the rising curve in figure 17 we see two things happening: there is an increase in maximum size over time, and there are great changes in the construction of the organisms involved. This observation gives us an opportunity to ask which is the prime mover. What we find is that the universal selection for size is ever present, but in the case of size increase it is always limited by the internal construction of the organism. One could not build an organism even the size of a small mouse with a pile of amoebae. On the other hand\u2014and we are thinking in terms of geological time involving great numbers of generations\u2014there will be mutations that affect the internal structure. Those changes might be of such a nature that a larger size can be achieved and remain structurally sound. If that mass of amoebae trying to be a mouse developed some bone and muscle, it could not only exist but would have the basic building materials needed for the construction of an elephant. In other words, an increase in size requires adequate building materials, and until those are invented there is a strict limit to how big an organism can become through natural selection. This restriction means that size increase encourages and exploits innovations in structure, and those changes in structure do not cause but permit a greater increase in size.\n\nTo give another example of such innovations, the beginnings of vascular tissue in plants presumably served the function of supporting small stalks for more effective dispersal of spores or seeds and for that reason was selectively advantageous. Once a vascular system was established with a sound genetic basis, it could, through progressive modifications, ultimately produce giant trees. Therefore, it is not strictly true that size always precedes structure; clearly, the reverse is also true, and during the course of evolution there must have been a seesawing back and forth. Structure keeps pace with size, and size takes advantage and exploits every change in structure. When size changes lead the way, the structural changes may be relatively minor, often involving mere changes in proportions. However, when structural changes come first, they are often major changes that permit large increases in size, as can be seen when we follow the changes in maximum size in different major groups of animals and plants through the ages.\n\nOne thing is obvious in looking at the figure: the upward ride is in reality rather bumpy and not a smooth curve. This is so because the world does not consist of one species of different sizes, but many species with different body plans. Each of the separate groups of organisms will always find room at the top and therefore show a trend towards a greater maximum size, but each will be limited by its ancestral body plan. Think of the construction differences between trees and mammals or, on a less obvious level, the differences between birds and mammals. Selection for size will operate in each group, and for each efficiency will be maintained by natural selection. Note that the curve in figure 17 levels off as one approaches the world of today. Does this mean that we have reached some sort of upper size limit and that new or drastically modified body plans will be needed to evolve even larger animals and plants? (Before jumping to such a conclusion one must take into account that the graph is on a logarithmic scale.)\n\nSize is a key object of selection, requiring changes in shape that will vary for different body plans. If there is a selection for a change in shape alone, then a change in size does not necessarily follow. For instance, an evolutionary increase in the thickness of the legs of a mammal would not require a larger body, while a larger body absolutely requires thick legs.\n\n### CELL SIZE\n\nDespite the extremes one sees in the cell size of various small, unicellular organisms in figure 6, there is a remarkable constancy in the size of cells for most organisms. The biggest exception is bacteria; they indeed are much smaller than nucleated cells (eukaryotic cells), but then cyanobacteria (what used to be called blue-green algae) have cells comparable in size to eukaryotic ones. An exception for the largest cell in figure 6 is _Bursaria_ , the huge ciliate. However, ciliates are special and in some sense can be considered equivalent to multicellular organisms. One can see the enormous \"C\" shaped nucleus in the figure, and this \"macronucleus\" has the genetic material duplicated many times. So it is functionally multinucleate, which has made it possible for ciliate protozoa to become large cells in this unique fashion.\n\nSometimes cells appear larger for other reasons. For instance, before development, bird eggs are enormous, but like most eggs they are loaded with yolk to feed the emerging embryo. Many plant cells seem exceptionally big because they have a huge vacuole containing water. In both cases the amount of cytoplasm is appropriately small for the size of the nucleus.\n\nThis brings us to an interesting point. The size of the nucleus usually reflects the size of a cell; there is a tendency to keep the nucleo-cytoplasmic ratio constant. For many years I had a gifted colleague, Gerhard Fankhauser, who did some fascinating work on newts. By treating their eggs with temperature shocks at the beginning of their development, he was able to produce newts that had a different number of sets of chromosomes. Normally, most animals, including ourselves, have two sets, one from our mother and one from our father. We are diploids (2n), but Fankhauser could produce newts of different ploidies, from haploid (n) up to pentaploids (5n). For each there was a corresponding cell size (keeping the nucleo-cytoplasmic ratio constant), but the surprising thing was that the bodies of the grown-up larvae were all the same size, regardless of their cell size. This prompted him to look at the cells of these different individuals, and he discovered the remarkable fact that since the organs were all the same size, too, they were built with many small cells in diploids, compared to pentaploids where few, large cells were involved (fig. 18).\n\nOne day I suggested to him that it would be interesting to test newts of different ploidies in a learning maze since they would have the same size brains and therefore a different number of brain cells (neurons). With the help of a clever undergraduate they tested diploid and triploid newts in a \"Y\" maze; if they took the wrong turn they received a mild shock. And indeed, they found that the diploids, which had relatively more (and smaller) brain cells, were better at learning this simple test than the triploids with fewer, larger cells. We really do need those neurons!\n\nFigure 18. Two larvae of the salamander _Triturus viridescens_. The pentaploid larva with five sets of chromosomes in each cell (left) is the same size as the diploid larva with two sets (right). As can be seen from the circular cross section of a kidney tubule (middle) and a covering layer of cells over the lens (bottom), the tissues are also the same size despite the striking differences in cell size. (After G. Fankhauser; from Bonner, _The Evolution of Culture in Animals_ , 1980; drawing by Margaret LaFarge)\n\n### SIZE INCREASE BY BECOMING MULTICELLULAR\n\nAll larger organisms, which evolved originally from unicellular ancestors, are multicellular. There are a number of reasons for this. For starters, the cell is a functional metabolic unit; it has all the necessary features to stay alive. By being small it can keep a manageable surface-volume ratio to permit diffusion. It also is an appropriate size not only for carrying the genes on the chromosomes, but for those genes to produce proteins in the cytoplasm. Cells can be thought of as building blocks, like a house made of bricks where each brick is alive, and together they make it so the individual can digest and process energy, move, orient itself in the environment, and in the case of animals, even behave and think.\n\nI argued that there has been, during the course of evolution, a selection for larger and larger organisms because there is always room for bigger ones to escape the competition with the smaller ones. This is the case both within a species and among species. (In animals, size differences can also be driven by sexual selection and produce considerable size differences between the sexes, as is found, for instance, among some species of seals. To give an extreme example, elephant seal males are ten times the weight of females.)\n\nThe top of the scale is always an open ecological niche. This is the case right from the very beginning of multicellularity in early earth history: the easiest way to become bigger has always been to add more cells, and the size of those cells has always remained roughly the same from minute algae and invertebrates to giant animals and plants. The former may have as few as a dozen or a few hundred cells; there are considerably more than a trillion cells in our bodies, and since the number of cells is determined by the volume of an animal, imagine how many cells there must be in _Brachiosaurus_ or in a blue whale.\n\nIt has been of great interest to me for some time that the invention of multicellularity did not just occur once but many times in the history of our earth. I was raised on the idea that it occurred once, or at the most twice, for the origin of animals and the origin of plants. My early student career in a laboratory was concerned exclusively with lower organisms, which forced me to realize the obvious: fungi, algae, slime molds, slime bacteria (myxobacteria), cyanobacteria, and a number of others could not possibly have come from one ancestor but from separate ones.\n\nFurthermore, there are two different ways of becoming multicellular: the most common way would be for cells to divide and the daughter cells to stick together. Such a beginning would have been the norm in aquatic organisms and found in most of the groups mentioned above. However, there are a number of organisms that have another way. For them, feeding occurs as separate cells that then come together to form a multicellular organism. This is true for cellular slime molds, where the separate amoebae feed by engulfing bacteria; after they have cleaned an area, they aggregate to form a multicellular, unified, and coordinated individual. The equivalent method is found in other organisms: the myxobacteria, most fungi, and even in a unique multicellular ciliate. The interesting thing is that all these cases of multicellularity by aggregation are terrestrial, so there is a fundamental difference between them and the way aquatic forms became multicellular. This must have been the first conquest of land, long before it occurred in vascular plants, insects, and amphibians.\n\nEach example has its own particular details of its construction. For terrestrial forms the cells differ; from rod-shaped bacteria, to amoebae, to fungal filaments, to ciliate protozoan cells. (In the case of fungi and true slime molds\u2014myxomycetes\u2014there are no cell membranes around each nucleus, as is the case in most other organisms. They are multinucleate, but as the great nineteenth-century botanist Julius Sachs pointed out, each nucleus, and its surrounding cytoplasm, was a functional cell, an \"energid\" as he called it.) In aquatic forms, the cells may be bacterial, such as in the cyanobacteria, similar to the vast multitude of algae that also have stiff cell walls and are photosynthetic, or amoeboid cells with thin cell membranes, as in sponges and all the more complex animals. Then there are differences in how the cells stick together: in some cases their entire surface is sticky; in others there are specific sticky spots that affect the arrangement of the cells and therefore the architecture of the organism. One could go on with the details, but the message is that these differences are the basis of the great variety of living designs of multicellular organisms\u2014large and small\u2014that exist today.\n\nIt is important to ask what might be the advantage for these unicellular organisms to become multicellular, especially considering how successful they have been as single cells. It is true that it is an easy way to get larger, but what might be the immediate advantages for them? This is not quite like asking what might be the benefit to an elephant or a large tree; the reasons for the first steps towards multicellularity must be quite different. In a few cases we have some clues as to what some of them might be. Let me give three examples.\n\nFirst, there are the myxobacteria, which are multicellular; what possible advantage might they have over their unicellular relatives? In some species they stick up into the air, making it possible to spread their cysts, which would be an advantage in dispersal. The more obvious advantage comes during their feeding stage. The individual feeding cells are present in multicellular swarms, and each bacterial rod secretes extra cellular digestive enzymes that break down complex food substances such as starches and proteins into their component sugars and amino acids in the same way our gut enzymes process the food we eat. They then can directly absorb those smaller energy-giving molecules. By having many such cells in a swarm, they can collectively produce enough of the enzymes to digest the particles around them; their large number means they can overpower potentially indigestible sources of food. This process has been called the wolf pack principle whereby many can capture the prey while lone cells will fail.\n\nA second example among bacteria are the species that form clumps. These species live in places where there is no oxygen; oxygen will kill them. It is conjectured that if some oxygen should appear around a clump it will not get to the cells in the center, which will be protected and be able to survive.\n\nA third example is illustrated by an interesting experiment on a unicellular alga that was grown in culture in the laboratory. If one adds another species, a predator of the alga in the culture, then all the small cells will be eaten. However, occasionally a mutation occurs in the vulnerable alga that produces a sticky substance on its surface, with the result that the cells form clumps that are too big to be engulfed by the predator, allowing them to survive.\n\nThere are many other ways in which togetherness produces obvious advantages: the important point is that reasons for becoming multicellular are easy to conceive, and all depend on the advantage of a collective behavior\u2014a size increase.\n\n### BACK TO LEMUEL GULLIVER\n\nBecause of our modern understanding of the microscopic, cellular nature of animals and plants, we see that there are some biological problems with Jonathan Swift's delightful size worlds. The cells of the Lilliputians, of Gulliver, and of the Brobdingnagians would all have to be the same size; therefore, as one went up the size scale, the characters would have been made up of more and more cells. This would mean, among other things, that the Brobdingnagians would be brainier than the lesser mortals\u2014following the arguments of Bernard Rensch discussed earlier\u2014but such a progression of intelligence is not clear from the book. The eyes of the Lilliputians would have the same size cells as their larger brethren, and therefore their eyes would be made up of fewer cells. Unfortunately this would not mean that they could see smaller things, as though their vision was through a microscope: they would see small objects with no more acuity than Gulliver or the Brobdingnagians. If anything, they would have a smaller field of vision that would put them at a relative disadvantage. Their plight is clearly the reverse of Gerhard Fankhauser's newts of different chromosome numbers: in Jonathan Swift's world the cells stay the same size but the bodies vary from huge to small, and in the newts body size is constant while the cell size varies.\n\nAs we have seen, there are many important consequences to being small, and the cells found in all organisms fall in that lower size realm. However, by sticking together they have been able to produce large multicellular animals and plants, and in doing so they have crossed from one size world to another.\n\n##   CHAPTER 5\n\n## SIZE AND THE DIVISION OF LABOR\n\nFor many years I have been struggling to understand the relation between size and the division of labor in multicellular organisms. One wants to know why it is generally true that the bigger the organism, the greater the division of labor. It is a principle that to some degree we all accept as given; let us examine it closely.\n\nThe division of labor, which is a reflection of biological complexity, is a venerable subject. The history of the idea has a few years ago been admirably discussed by Camille Limoges. He points out that the pioneer for considering the matter within organisms goes back to Henri Milne-Edwards, who as early as 1827 said that animals had a \"division of physiological labor.\" Limoges goes on to say that Darwin was favorably impressed by the writings of Milne-Edwards, who in turn influenced the pioneer in sociology, Emile Durkheim. The conclusion of Limoges is that there has always been a close association between the concept of division of labor as it applies to individual organisms and as it applies to societies (a matter to which I will return). However, none of these earlier authors were particularly interested in size as a significant correlate. This tradition continues to this day, and the matter of size is not central to our current thinking. What I wish to show here is that significant increases or decreases in size during the course of evolution are accompanied by increases or decreases in the division of labor.\n\nWe now come to another of our basic size rules. The rules discussed so far are the relation between (1) size and strength, along with (2) size and surface areas. There is a similar relation between size and complexity. Like the others, it also can be expressed as an allometric relation:\n\n(3) Complexity \u221d Weighta.\n\nAs we proceed from the strength-size rule, and particularly to the surface-size rule, there is a direct trend towards the complexity-size rule. As I pointed out before, in a larger aerobic animal it is imperative to get oxygen to the internal tissues. To do this it is necessary to produce a circulatory system with a pumping heart, innumerable capillaries, and lungs or gills with huge surfaces to trap the oxygen that is then carried by the red blood cells to the tissues. Plainly, as size increases the demands of the surface-size rule has dictated an enormous increase in complexity. The three rules are indeed connected to one another and they are all directly related to changes in size.\n\nA convenient way to measure biological complexity is to think of it in terms of the number of cell types. A mammal will have a large number of distinct cell types, such as muscle cells, liver cells, brain cells, blood cells, and so forth. A parallel kind of list can be made for all other organisms, including plants. The number of cell types can be considered a rough reflection of the division of labor, or complexity; it is a useful indicator.\n\nFor each rule, an increase in size requires an increase, be it in strength, or in surface, or in the division of labor. On the other hand, a decrease in size generally does not necessarily require a decrease in these three properties; rather, they are simply permitted, for the smaller organism is likely to be able to manage even though it might have a surfeit of strength, or surface, or division of labor.\n\nEstimates of the number of cell types for different size organisms have been gathered from the literature by Graham Bell and Arne Mooers. Their results are an excellent way of illustrating the size-complexity rule as can be seen when they are brought together in a graph (fig. 19). The larger the organism, the greater the number of cell types.\n\nNote that despite the clear trend there is a wide scatter of the points in the figure. The variation might have a number of causes. It is very difficult to make accurate measurements of the number of cell types; all these estimates are rough. The data come from different authors who undoubtedly have differences of opinion as to what constitutes a discrete cell type.\n\nFigure 19. A log-log graph of the number of cell types plotted against the total number of cells for a vast array of organisms from very small colonial forms to the largest animals and plants. (Redrawn from G. Bell and A. Mooers, _Biol. J. Linn. Soc_. 60 [1997]: 345\u2013363)\n\nIt should be noted here that the size-complexity rule also applies during development. Both plants and animals normally begin at a single-cell stage and become increasingly multicellular with time. During this process there is a proliferation of cell types, which ultimately become adults that reach the number shown on the curve in figure 19. Since during development the organism becomes progressively larger, it would be very interesting to know how many cell types there are at each developmental size. Unfortunately, I have been unable to find any such measurements in the literature, but it is obvious that the size-complexity relation must hold. However, there is one point to remember: many of the cell types will not be functional when they first appear. For instance, all the cell types associated with digestion in the gut in an animal are passive until the first meal. Before that they are merely preparing for the passage of food. So the size-complexity rule is somewhat different during the course of development, yet the rule holds.\n\n### THE ORIGIN OF DIVISION OF LABOR\n\nMy plan here is to examine the size-complexity rule across a huge spectrum of sizes, from small and lowly primitive organisms to huge societies. The former are of particular interest because they give some insight into how the division of labor might have arisen in early evolution. There is no better place to examine the matter than at the origin of differentiation, where one finds closely related species with either one cell type or two. An example is found among the green alga _Volvox_ and its relatives.\n\n#### _Volvocine Algae_\n\nSpecies of _Volvox_ are common in freshwater ponds, first described by van Leeuwenhoek in the seventeenth century using his simple microscope. It consists of a sphere of a single layer of some thousands of cells at the surface of a sphere (rather like a minute tennis ball with its hollow center) about the size of a small pinhead (fig. 20). Most of the cells have two flagella that stick outward and beat in coordinated waves so that the whole ball rotates like a globe that can spin around its axis. What concerns us here is that in _Volvox_ there is a division of labor: most of the cells are the ones just mentioned that move the colony and are capable of photosynthesis, but there are a few cells that are reproductive and have no flagella\u2014they develop into daughter colonies.\n\nFigure 20. The ranges of sizes in the volvocine algae. From left to right, the ancestral single cell _Chlamydomonas_ , the 16-cell _Gonium, Eudorina_ , and _Volvox_. (Selected from W. H. Brown, _The Plant Kingdom_ , 1935)\n\nA number of other species of volvocine algae are smaller and are made up of fewer cells (fig. 20). The whole group has been studied in exemplary detail by David Kirk, and among many of the things he and his co-workers discovered using DNA analysis, the multicellular forms arose from a single-cell ancestor known as _Chlamydomonas_ , as had been suspected before based on morphological grounds. Another interesting bit of information has come out of Kirk's DNA analysis of the ancestral tree of the volvocine algae. The old, conventional idea that they evolved by a simple, linear progression from small to large has been shown to be false. There is indeed an obvious span of sizes from the unicellular _Chlamydomonas_ to the largest species of _Volvox_ , which may consist of 50,000 or more cells (fig. 20). However, the history of the group is more complicated than was previously believed. _Volvox_ itself has arisen independently more than once during the course of evolution. Even more relevant to this discussion, it appears that certain species of _Volvox_ are ancestral to smaller forms, such as _Pleodorina_ , as well as the reverse. So there apparently have been evolutionary decreases as well as increases in size during volvocine evolution.\n\nHere I want to focus on the two cell types that are present in the asexual life cycle of _Volvox_. The biflagellate, nonreproductive (somatic) cells are responsible for the movement of the colony, and the reproductive cells (gonidia) produce daughter colonies. The former are terminally differentiated and are incapable of dividing, while the latter become cell-division specialists and are incapable of locomotion.\n\nThere is another important bit of information concerning this dichotomy. We know that very few genes are involved in this transition from one to two cell types. The initial discovery was the mutation of one gene that would transform a normal _Volvox_ so that all the cells that would be somatic became reproductive and divided very rapidly, producing thousands of minicolonies in one large individual, similar to what occurs in small volvocine species. Subsequently, it was discovered that a few other genes were involved, but clearly the invention of this division of labor appears to involve only a small number of genetic steps.\n\nThis two-cell-type state is correlated with size. At one end of the size range, _Volvox_ , which contains thousands of cells, always has both cell types. At the other end of the spectrum, _Gonium_ , which never has more than 16 cells, has no division of labor: all _Gonium_ cells are initially biflagellate and involved in locomotion, but later all of them shed their flagella and undergo reproduction by successive divisions to produce daughter colonies.\n\nThere is a particularly interesting feature concerning the invention of the division of labor among the volvocines. Some species can exist in different sizes, and only the larger forms will have more of the two cell types. They apparently can asses their size: the larger they are, the greater the division of labor. This ability is known as _quorum sensing_.\n\n_Eudorina_ can have either 16 or 32 cells, depending on environmental conditions. In the 16-cell colonies, like _Gonium_ , all of the cells participate first in motility and then subsequently in reproduction. However, in 32-cell colonies, the four cells at the anterior end of the colony often remain terminally differentiated nonreproductive cells that continue beating their flagella, while the other 28 cells reproduce. In another genus, _Pleodorina_ , which always has a division of labor, the ratio of nonreproductive cells to reproductive cells increases as the total cell number is increased. In colonies with 32 cells, 25 percent of the cells are sterile nonreproductive cells, while 50 percent of the cells are so in colonies of 128 cells. Clearly, within this critical intermediate size range the proportions of the two cell types varies depending on the size of the colony.\n\nA central question is what selective advantage might volvocine algae have gained by producing terminally differentiated somatic cells that are incapable of reproduction. It might be simply that this dichotomy enables them to reproduce while simultaneously swimming and gaining energy through photosynthesis. If it is a small species that has only one cell type, in its nonmotile reproductive phase it will sink to the dark bottom of the pond where photosynthesis would be impossible. In _Eudorina_ and _Pleodorina_ we see an intermediate situation: in the larger forms there is an increase in ability to carry on photosynthesis while reproducing. This is the basis for suggesting a quorum-sensing mechanism where a group of cells take into account their number so that they behave differently when they are large as compared to small. It means that in these instances there must be a genetic mechanism for setting apart the two cell lineages, a system that is not rigid but plastic and capable of responding to size. It is puzzling to imagine in this case what the selective advantage of quorum sensing might be; perhaps there is none and it is simply a byproduct of the genetic activity.\n\n#### _Slime Molds_\n\nThe cellular slime molds provide another interesting example of the borderline between one cell type and two. They have an unconventional life cycle in that the separate amoebae eat bacteria in the soil, and upon starvation they aggregate to form a multicellular individual that ultimately produces a small fruiting body: a mass of encapsulated spores held up by a delicate stalk. Unlike most other organisms, they feed first and then become multicellular. Most species of slime molds have two cell types: spores and stalk cells. The exceptions are the species of _Acytostelium_ that make only spores; their stalk consists of a delicate strand of cellulose extruded by all the cells before they become encapsulated into spores (fig. 21). From some work directed by S. Baldauf and P. Schaap using DNA methods to build a slime mold ancestral tree, we now know that _Dictyostelium discoideum_ (including some of the other common _Dictyostelium_ species) and _Acytostelium_ are far removed from one another and only distantly related. So, unlike what we saw in the volvocine algae, the two genera of slime molds did not go back and forth in size during their evolution.\n\nNevertheless, the different species of _Dictyostelium_ that have two cell types are in general larger than _Acytostelium_ that has no stalk cells and therefore only one cell type. There is a separation between the smaller species with one cell type and the larger ones with two. In these organisms, size is not determined by multicellular growth, as it is in most organisms including _Volvox_ , but by the number of amoebae that enter an aggregate. Therefore, this size-complexity relation holds regardless of how size increase occurs. Despite this difference, both the volvocine algae and the cellular slime molds are similar in having a size threshold for the number of cell types.\n\nQuorum sensing is also found in one species of cellular slime mold; the case of _Dictyostelium lacteum_ is particularly interesting. When very small fruiting bodies arise, a partially acellular stalk is produced; it is as though there is a threshold concentration of a key substance or substances that stimulate a cellular stalk, but if there are very few cells, the concentration is insufficient to be wholly effective in producing a stalk with cells (fig. 22).\n\nFigure 21. A diagrammatic comparison of a rising fruiting body of _Acytostelium_ with only one cell type (left) and _Dictyostelium_ with two\u2014the dark spores and the stalk and prestalk cells (right). A particularly small _Dictyostelium_ is drawn so that the cell structure of the two can be compared. (From Bonner, _The Evolution of Complexity_ , 1988)\n\nFigure 22. A very small fruiting body of _Dictyostelium lacteum_ showing an acellular tip. (From Bonner and Dodd, _Biol_. _Bull_. 122 1[1962]: 13\u201324)\n\nWhy is there quorum sensing involving the division of labor? It might be adaptive, for it reduces the cost of construction in those forms that are of intermediate size. In the case of _D. lacteum_ , for instance, if a cellular stalk is not needed for support, then, by making part of the stalk acellular, a few extra spores are produced. In larger species, very small fruiting bodies can be produced in which the stalk is cellular; the smallest one known is a fruiting body of one species ( _P. pallidum_ ), made up of three stalk cells and four spores. In other words, quorum sensing is not universal and not found in larger species but only in one species near the borderline between large and small.\n\n#### _Cyanobacteria_\n\nA final example of going from one to two cell types may be seen in the filamentous cyanobacteria, known to be very ancient organisms. In many species, besides spores they have two vegetative cell types: the photosynthetic cells and the heterocysts that fix nitrogen (fig. 23). The two activities cannot be carried on in the same cell because the enzymes that are responsible for the incorporation of nitrogen cannot function in the presence of oxygen, which is produced by photosynthesis. Nitrogen is essential for all living cells for many key substances that contain nitrogen, such as proteins and nucleic acids and many others. With heterocysts, both of the essential biochemical activities can occur at the same time.\n\nNot all species have heterocysts: some have a temporal separation of the two activities, a temporal division of labor. All the cells photosynthesize during the day and fix nitrogen at night.\n\nIn cyanobacteria, the ratio of vegetative cells to heterocysts remains constant regardless of the length of the filaments. The important size constant is the physiologically functional units; they are modules. This is a form of quorum sensing, for a heterocyst senses the number of neighboring vegetative cells, and a proportion is retained in each module. In this case, the selective advantage is obvious: to produce a balance between photosynthesis and nitrogen fixation.\n\nFigure 23. A cyanobacterium filament showing evenly spaced heterocysts. (From G. M. Smith, _The Freshwater Algae of the United_ States, 1933)\n\n#### _Quorum Sensing_\n\nQuorum sensing is a newly recognized phenomenon that has become an exciting field of research in microbiology. The idea is that when cells come together, a critical number is required to be able to perform activities that would not be possible if there were fewer cells. I have already described a somewhat similar phenomenon in the wolf-pack feeding of myxobacteria, where feeding can be effectively achieved only if large swarms of cells exist; only then can they produce enough extracellular enzyme to effectively digest external food. Bonnie Bassler made the key discovery that luminescent bacteria are only capable of giving off light if they are present in a critical mass\u2014when that number is attained, the light goes on. This has turned out to be of great importance in medicine because the same situation is found among pathogenic bacteria that can only produce their undesirable effects when they are present in sufficient numbers: a quorum is needed before they can do their damage.\n\nBoth the volvocine algae and the cellular slime molds provide good examples of size quorum sensing. In each case they occur in species that are intermediate in the size range, and the individuals of those species can vary in size. If the individuals are large they will have two cell types, if small only one. They can sense their size and provide and respond accordingly when they are above or below a certain threshold.\n\nIn all the cases we have discussed with one or two cell types, including those cases of quorum sensing, it is evident that size is tightly correlated with\u2014and very likely the direct cause of\u2014the number of cell types, and not the reverse. Changing the number of cell types does not dictate the size of the colony or group\u2014it is the size that dictates the number of cell types.\n\n### SIZE DECREASE IN LARGER ORGANISMS\n\nThe lesson to be gained from this kind of quorum sensing is that there can be a tight control between size and the number of cell types. If we now turn to larger organisms with many cell types, we find a somewhat more general correlation, but nevertheless the size-complexity rule holds. We have already seen that when there has been a size increase during the course of evolution, it has often been accompanied by an increase in the number of cell types, but even more revealing in providing evidence that a change in size can result in a change in the division of labor are those cases where there has been a decrease in size and a corresponding decrease in the number of cell types. Here an example will be given for animals and another for plants.\n\n#### _Rotifers_\n\nRotifers are very small animals that live in ponds and lakes and other wet places (see the rotifer in fig. 6). They feed on small unicellular forms, especially bacteria. The majority of rotifers have a conventional gut tube, while others have a central part of their body into which food vacuoles are formed at one end and excreted at the other, very similar to the digestive system found in ciliate protozoa (fig. 24). The no-lumen rotifers lack a cellular stomach and on this basis are considered to have fewer cell types.\n\nFigure 24. A diagrammatic view of the gut of a species of rotifer, one with a lumen (left), and one without a lumen (right). (From T. A. McMahon and J. T. Bonner, _On Size and Life_ , 1983)\n\nIn the past it has been my assumption that their ciliate-like digestive system is related to small size. This was just asserted; is it possible to find any evidence for the assertion? Fortunately, Josef Donner in his exhaustive monograph of the Bdelloid rotifers gives the length of large numbers of species, and I have extracted this information for forty-eight species without lumen and seventy-four with. It is clear that the mean lengths of the former are significantly smaller than the latter, and the assertion is justified.\n\nHowever, the overlap is considerable: some species without a lumen are larger than the smaller ones that have one. This raises the question of which anatomy came first during the course of evolution. At the moment there is no molecular ancestral tree evidence, but it would be reasonable to assume that the gut with a lumen is ancestral and that there has been a subsequent size reduction in some species. If this were so, it would be another example of size affecting the number of cell types. The fact that there is not a perfect correlation between size and the degree of cell differentiation is another indication that the rule is approximate.\n\n#### _Duckweeds_\n\nA parallel example can be given for plants. Ordinary angiosperms are estimated to have up to forty or more cell types while the two small duckweeds have far fewer: _Lemna_ has eighteen and the minute _Wolfia_ about eight (fig. 25). There is little doubt that these smaller forms are descendants of larger ancestors and that there has been a reduction in the number of cell types accompanying the size decrease. The key question is, why are there fewer cell types? The most obvious explanation is that there has been a selection for size decrease, and that mechanically all the ancestral cell types are neither needed, nor is there room for them in the small body. Again, the size and division-of-labor rule is followed among higher plants, albeit again in an approximate way.\n\nFigure 25. Two species of duckweed that are the smallest flowering plants. Floating on the surface of water, the minute _Wolffia_ (about 1 mm long) surrounds the larger _Lemna_. (Drawing by Hannah Bonner)\n\n### MORE ON THE DIVISION OF LABOR AND SIZE\n\nLet us turn now to the matter of the effect of size increase on complexity. As I pointed out earlier, from the evolutionary point of view we know that the maximum size of both animals and plants has increased progressively over geological time (fig. 17) and we can now say that this also must be true for the number of cell types. This is because, as we have just seen, with size increase there is a corresponding increase in the division of labor. In fact, James Valentine and his colleagues have plotted the maximum number of cell types on an evolutionary timescale and show that they rise with time in the same manner as size (Fig. 26).\n\nThe relation between size and the number of cell types being approximate, one may ask what sets the upper and lower limits of the size range for organisms with a particular number of cell types. If, in the selection for size changes, an animal or a plant becomes too big or too small to be physiologically efficient and be able to compete effectively, it will become extinct. The range of size among mammals is extensive\u2014from mice and shrews to elephants and whales\u2014yet each in its own environment can successfully exist, and they do not differ to any great extent in their number of cell types. This great span of sizes of similarly built and physiologically efficient animals has taken millions of years to achieve\u2014each step requires adjustments for stability.\n\nFigure 26. Estimated numbers of cell types of early members of various animal groups. Only members of the groups that are believed to have been rather near the upper limit of cell type numbers when they originated are included. (From Valentine, et al., _Paleobiology_ 20 [1994]: 131\u2013142)\n\nA large size span can, however, be achieved very quickly through artificial selection. Consider dogs, from the miniature Chihuahua to a huge Great Dane. Here it is unlikely that there is any difference in the number of their cell types, and the change in size has taken mere thousands of years, beginning with the domestication of the wolf (fig. 27). However, this is not natural selection but artificial selection and breeds with less successful physiological constitutions like the larger breeds of dogs have a significantly shorter life span than smaller ones.\n\nFigure 27. Showing the span of the sizes of dogs, the result of artificial selection for size from the ancestral wolf. A large Great Dane alongside a small Chihuahua. (Drawing by Hannah Bonner)\n\n#### _Energy and Size_\n\nThis leads to an important point that has always puzzled me. Why do mammals with the same number of cell types have a huge size range? The answer comes from an important insight of Megan McCarthy and Brian Enquist, who show that it has to do with their levels of metabolism. Size is not the only factor to influence the number of cell types, but metabolic energy is an additional key factor that correlates with cell type numbers. The metabolic rate of a mouse is much greater than that of an elephant, and for this reason it also supports a large number of cell types. A related question is, why do animals have more cell types for a given size than plants? The reason is the same: plants have a much lower metabolic rate than animals. In the next chapter I will return to this relation of metabolism to size.\n\n#### _Societies_\n\nThe whole idea that there is a relation between size and the division of labor was first realized in human societies. It is an ancient principle that we associate with the eighteenthcentury economist Adam Smith. The larger a nation or a community, the greater the division of labor, often expressed in terms of an increase in specialized trades or occupations. In a small village, individuals have to be jacks-of-all-trades, while in larger communities there is a cobbler, a butcher, a smith, a tailor, and so forth. In even larger cities or towns, there is a further division of labor where we find lawyers, bankers, policemen, doctors, and many other specialized occupations. Even institutions as they increase in size will divide the labor: large businesses will have a whole hierarchy of managers; large universities will have layers of administrators and teaching faculty; an army has a multitude of officers and enlisted men of different ranks, and so forth. The rule clearly holds for human societies, and there are a few examples where measurements have been made (fig. 28a, b).\n\nThe rule also applies to insect societies, which are quite remarkable for their size and diversity. Some ant colonies will have one queen with millions of workers attending to all the activities and functions of the nest, such as feeding and caring for the young, foraging for food, and protecting the colony. At the other end of the spectrum there are certain primitive wasps that have very small colonies, consisting sometimes of less than a dozen individuals.\n\nFigure 28 (facing page). Log-log graphs showing the relation of size to the division of labor for societies. (a) The number of occupations found in each of the states in India (graph of N. V. Joshi in Bonner, _Current Science_ 64 [1993]: 459\u2013466). (b) The number of organizational traits (specific crafts and occupations) plotted against the size of single-community societies. (Five of the latter are labeled.) (From Carniero, _Southwest Journal of Anthropology_ 23 [1967]: 234\u2013243)\n\nIn these simple social wasps there are no morphological differences among the females that come together to form a small colony. There is, however, a dominance hierarchy that can be seen in behavioral differences. One female will remain queen by lording it over the others, and she alone will lay eggs for the next generation: she will forcibly prevent other females from doing so. The other females do not all do the same thing: some concentrate on food gathering, others on protection of the nest, and some just loaf. In other words, these primitive social wasps do divide the labor, but only by using subtle, behavioral means of doing so.\n\nBy contrast, in large colonies of ants and termites there are morphological differences between the workers. They may not only differ in size, but in shape as well, and those shapes are associated with specific tasks within the colony. The smallest workers are primarily occupied with the care of the young, the middle-sized workers are busy foraging for food, while the largest workers, the soldiers, will guard and protect the nest. To see if one could measure the extent of this morphological division of labor in relation to the size of the colony, I took a table from B. H\u00f6ldobler and E. O. Wilson's great tome on ants, which gave the colony size for a large number of ant species. I broke this up into groups based on colony size, and Wilson was kind enough to tell me how many morphological castes there were for each of these species. He added that for some species this task was easy, for the castes are discrete in their shape; but for others there is a continuum, and therefore assigning a number for the different castes is difficult and subjective. I was fortunate to be able to rely on his great experience in observing ants and was able to plot colony size versus caste number to show that, as expected, there is a clear and significant increase in the number of castes with an increase in the size of the colony, despite a considerable scatter of the points (fig. 29).\n\nFigure 29. The number of ant castes plotted against the size of the colony. The line is a regression showing a significant trend (r2 = .25). (From Bonner, _Current Science_ 64 [1993]: 459-466)\n\nThe results for societies can be compared with similar plots for multicellular organisms (see fig. 19). In all cases the trend is highly significant. Note that societies also show a considerable scatter of points; here again the rule is a statistical correlation and not an exact one. If a village grows, there is no sharp, critical point that dictates an increase in the division of labor.\n\nWhat multicellular organisms and societies have in common is the need to increase their efficiency. In organisms this is done by genetic variations subjected to natural selection; in human societies the efficiency is measured in economic success. Both involve competition, but differ in the nature of their variation. In one, the variation is inherited and culled by conventional Darwinian principles; in the other, the variation is not genetic but cultural, and it is culled by economic rather than biological efficiency.\n\nThus far we have examined three size rules: strength and diffusion keep pace with size increase, as does the division of labor. The latter is not only true for the number of cell types within an organism, but for the number of specialized activities in societies. The three rules are closely interrelated and show the ubiquitous importance of the role of size.\n\nIn looking at the relation between size and the division of labor, the important question is the effect they have upon each other. Natural selection will act on both size changes and the internal physiological efficiency. If there is selection for a significant increase in size, selection will in turn change the internal structure to maintain physiological efficiency, and this effect may involve an increase in the division of labor. If there is a selection for a size decrease, again the efficiency must be maintained. Usually this process does not require a decrease in the division of labor, although, as has been shown, it can occur. Going in either direction there are presumably thresholds beyond which efficiency (involving the appropriate division of labor) must change to keep pace with size.\n\nThis means that within these limits there can be changes in size without changes in complexity, and conversely changes in complexity without changes in size. Because of the latitude of change within the limits for both size and complexity, one can expect\u2014and finds, as we have seen\u2014large variation in the size-complexity rule.\n\nIf we now ask why human societies seem to follow the same size-division of labor principle as cells, the answer is somewhat different from that of cells. There are none of the obvious physical constraints we saw for cells; but even more important, there is a totally new element, namely behavior. Individual organisms devoid of a brain can only pass information through their genes, but in the evolution of animals there arose an entirely new method of passing information, and that is by behavioral signals between one individual and another. These signals have been called \"memes\" by Richard Dawkins. Genes can be passed on only through egg and sperm (or asexual bodies such as spores) from one generation to the next, and therefore many generations are required over a long period of time for a particular gene to spread through a population. Memes, on the other hand, can pass from one individual to another in an instant of time, and as a result can spread very rapidly through a population.\n\nThere is, of course, much more to behavior than the passing of information between individuals; for instance, we can solve problems. Animals can also solve problems, but of all animals we are undoubtedly the most skilled in doing so. It is this problem-solving ability that is the key to how the labor is divided in societies.\n\nA good example may be found in ant societies. Their division of labor is ultimately genetically based\u2014including their ability to respond to nutritional and environmental conditions\u2014and therefore subject to natural selection. This is certainly true, but insects have some flexible behavior as well, and perhaps even a modest ability to solve problems. Those who work with social insects can give many examples; here is one that makes my point. If, in an ant colony where different castes perform different labors, one removes one of the castes, then some of the workers from other castes will take over the tasks that they never performed previously. In other words, they show a behavioral flexibility; in a way, this could be considered an example of minor problem solving. By being flexible they see to it that all functions, all the labors, required for the welfare of the whole colony are carried out, despite the loss of one of the castes. From this one might conclude that in social insects there can be a mixture of behavior in a background of a strong genetic basis for the division of labor.\n\nFor human beings, the genetic component takes a back seat. No doubt the structure and the abilities of our brains have a genetic basis, but what we do with our brains in terms of thoughts, including problem solving, is way beyond our genes. Our skills may have a genetic basis, but what we do with those skills opens up a new world.\n\nToday the whole subject of division of labor has taken on a new cast for human societies. With the tremendous increase in the size of our population, standard ways of increasing that labor, such as we have been discussing, have become increasingly complicated. This is true in all ways: in commerce and in economics in general, and in government. We are in an era of world trade, of globalization, of amalgamation of businesses into mega corporations; size alone has created vast complexity, and new ways of remaining efficient are being created. This has been helped, even fueled, by the extraordinary increase in the efficiency and rapidity of our methods of communication, beginning with the telephone, the radio, television, fax, and then the extraordinary burst created by computers with all the lightning powers of e-mail and the internet. World population and these technological advances have brought us beyond the straightforward size\/division-of-labor relation I have been describing. Now we are in the next phase of unimaginable complexity. It is as though we have gone from the brain of an insect to that of a human, where not only has there been an enormous increase in the number of cell types, but the sheer number of neurons and their interconnections have risen astronomically. Just as our brains are capable of doing things that an insect cannot do, and does everything in a different way, so our modern society deals with all matters that preserve efficiency in a way that is different from what we did before.\n\nFrom the many examples of different magnitudes that we have examined\u2014from two cell types to huge societies\u2014we see that the change in size comes first, followed by the degree of the division of labor. This process is especially clear-cut when there is a size increase that directly leads to a greater division of labor. But we also saw that a decrease in size can be followed by a corresponding reduction in the number of cell types, as in rotifers and duckweed. The most compelling evidence of size being the arbiter of complexity is from the quorum sensing in volvocine algae and in cellular slime molds, where only two cell types are involved. There is either one cell type or two types, depending on a small shift in size.\n\n### THE SIZE-ABUNDANCE RULE\n\nWe now go to a big canvas: to huge ecological communities. There we have organisms of all sizes living together and interacting with one another either in competition or in cooperation. In many ways it is similar to a community of cells that make up a multicellular organism: the component parts in each case are continually communicating and interacting with one another so that in both cases there is a collective wholeness. In one case it is within an individual multicellular organism, and in the other it is between the individual organisms in a patch of nature. One major difference is of size: individual organisms get no bigger than whales and giant sequoias, while ecological communities can spread over huge geographic areas.\n\nIt is obvious that small organisms will be more abundant than large ones\u2014we are surrounded by evidence of this fact. The number of bacteria per square mile in the African veldt is unimaginally larger than the number of elephants.\n\nBeginning with plants, it is well known that trees in a growing forest first are a dense thicket of saplings, but as they grow they do not all survive; there is a \"die-off\" or \"self-thinning\" of the less fortunate individuals, so that a forest or patch begins with many small young trees, and ends up with a few large ones. The same is true of crop plants if sown too close together. Clearly, as the plants grow they need greater resources, such as leaf space to catch the sun for photosynthesis and root space to gather water and minerals in the soil; they compete with one another, and as a result there are winners and losers.\n\nThe same rule applies to animals; the main difference is that animals are mobile. The spacing of animals is also governed by energy considerations and the availability of resources in the form of food. Since animals do not use the sun's energy for nutrition but eat vegetable matter or meat, in general the consumer is larger than the consumed. Many of the larger animals, such as giraffes or moose, are plant eaters, and they certainly are larger than the salads they consume. This is true at the other end of the size scale, too: minute rotifers eat even smaller bacteria and unicellular forms. There are exceptions to this very general rule. For instance, large plants can also be consumed by small insects. And parasites have fiendish ways of attacking hosts many times their size, either by devouring them from within, such as parasitic wasps infesting huge caterpillars, or living within a large animal as a nonlethal partner, as for example our intestinal bacteria that help to process our food.\n\nEarlier I laid down five size-related rules, and the relation between the abundance of organisms and their size was one of them. The rule can be expressed as\n\n(4) abundance \u221d Weight-b,\n\nwhich simply says the greater the size of an organism, the fewer individuals there are in a geographic area. (Note that contrary to the other rules, the slope of the line goes in the opposite direction.) This is an old subject that has been measured and treated in detail. Following the equation above, one can plot the logarithm of the density against the logarithm of the size of the organism. R. H. Peters has collected a large amount of information on animals from the literature and put it in one graph (fig. 30). A straight line can be drawn through all the points, although as Peters discusses there is a considerable scatter of the points, making it difficult to be certain of any general mathematical principle. The important message for us is that the size-abundance rule holds for a tremendously wide range of organisms.\n\nFigure 30. A log-log plot of the size of animals and their abundance in nature. Solid dots are mammals; squares are birds; hollow circles are invertebrates; x's are reptiles and amphibians. (From R. H. Peters, _The Ecological Implications of Size_ , 1983)\n\nThe rule can be illustrated in another way. If one divides the size of an organism (expressed in linear dimensions, such as its approximate diameter), which we will call D, by the average distance between individuals of its own kind, or B, the values for this B\/D ratio will be in the same order of magnitude from bacteria to elephants.\n\nA glance at the table shows that terrestrial organisms, over a huge size range from bacteria to large mammals\u2014seven orders of magnitude\u2014all have B\/D values that are quite close to one another, a range from roughly 7 to 70. This means that if these organisms were evenly spaced\u2014which they are not\u2014then the distance between them would be more or less the same number of body diameters apart, no matter their size.\n\nThe same is undoubtedly true for plants, as one would expect from their \"die-off.\" To give an example, my colleague Henry Horn has measured the trunk diameters and the distance between trees in five different woodland areas, and the B\/D values are all within the same range as those of the animals.\n\nMeasuring the ratio of the distance between organisms and their linear size and finding that it is roughly constant over a great range of sizes is no more than another way of illustrating the size-abundance rule. Note that since both measurements are distance, that is, linear, the B\/D ratio is a dimensionless number.\n\n#### ANOTHER WAY OF VISUALIZING ANIMAL ABUNDANCE\n\n**By dividing the distance between organisms (B) and their diameter (D) it can be seen that over a size span of 7 orders of magnitude the ratio is moderately constant.**\n\n##   CHAPTER 6\n\n## SIZE AND TIME\n\nAt the time the new millennium was about to come upon us, I was asked by the Buddhist magazine _Tricycle_ to write a few paragraphs on what the year 2000 might mean to my slime molds. This seemed to me like a fun idea and this is what I wrote:\n\nTIME FROM THE POINT OF VIEW  \nOF A SLIME MOLD\n\nTime and life are intertwined in so many different ways, something all biologists are acutely aware of. Consider a few extremes: a single cell bacterium may have its entire life cycle in half an hour, but a generation for an elephant takes 12 years and a giant sequoia takes 60 years. One reason I work with slime molds, which are soil amoebae that start off as single cells, and then come together to form a multicellular organism, is that their generations are short, so that if I start an experiment on Monday, I will know the result by Wednesday or Thursday. This kind of biological time\u2014life cycle time\u2014is at the middle of the time scale of living phenomena.\n\nAt the faster end of that scale is physiological time: how many beats does a heart have in a minute, or how long does it take to swerve the car in order to avoid a squirrel on the road. As with life cycles, these rapid living activities are greatly affected by size, so a huge elephant will have about 25 heartbeats per minute, while a tiny shrew's heart goes at the amazing rate of over 600 beats every minute. The elephant will step to one side with slow deliberation compared with a small sparrow on the window ledge with its lightning movements. We can combine the concept of the time required for a life cycle and the time required for rapid physiological processes in an interesting way. A shrew will live only a year or two, but an elephant will average 40 to 50 years; yet they have one thing in common: the total number of heartbeats they have in their whole lifetime will be approximately the same. So life for the small beast goes faster because its engine is racing along compared to the larger beast, and the total budgets for their actions are the same.\n\nEvolutionary time is another time scale in the realm of life. Now we are no longer dealing with one generation, but with a great series of generations going way back in time to the beginning of life on earth. We are no longer dealing with minutes or a few scores of years, but with millions, and even billions of years. We find in the fossil record an era when all life consisted of single cells, which later were followed by simple multicellular organisms, leading ultimately to all the great variety of animals and plants that we see on the earth today. This has been an exceedingly slow and an exceedingly grand evolution that has taken up a vast quantity of time\u2014so much so that it is difficult to comprehend the magnitude of geological time. And we know that even these time spans are modest compared with those of the astronomer, who thinks in terms of light-years.\n\nSo what does a biologist think of this second millennium? It is too short a time for major changes in evolution, but time enough for many generations. Every 1000 years will allow some 50 human generations, but the shrew will have a new generation each year, which means a 1000 each millennium. So for slime molds and shrews the second millennium has meant waiting impatiently for a huge number of generations, while for elephants and ourselves\u2014the wait barely tries our patience.\n\nThe important thing to note here is that size differences affect not only physiological processes, such as the speed of contraction of muscles, but these very effects are another facet of longevity, that is, the average time an animal lives. This is shown by the rather startling statement that the total number of heartbeats in the life of a hummingbird will be very roughly the same as that of an elephant or a whale. Increase in size means all life processes go more slowly and therefore take longer to happen. And this is all the result of living things clinging to efficiency rather than clinging to preserve the volume\/surface ratio no matter what the size. Efficiency is the prime result of natural selection and by slowing down the metabolism with increase size, every aspect of the organism's existence will be affected by time, from the speed its motor is running to how long it will run. Time and size are inseparable for the great diversity of the animals and plants on this earth.\n\nNote that all the processes we are considering here involve time. So if one divides the life span by the number of heart beats of an animal, it is time over time: another dimensionless number. And like the B\/D ratio it remains roughly constant regardless the size of the animal. This will also be true of metabolic rate.\n\n### SIZE AND METABOLISM\n\nOne of the most discussed of all aspects of the size of organisms\u2014and here we mean primarily animals\u2014is the way size affects the rate at which the internal motor runs, that is, the metabolism. Large animals obviously need more total energy simply because they are bigger; after all, a huge truck uses more gasoline than a minicar driving over a given distance. While this is undeniably the case, it is also true that there is an important difference between large and small animals. The rate at which they burn their fuel\u2014be it sugar or some other food\u2014in an ounce of flesh differs greatly: an elephant consumes much less energy-providing food in that ounce than does a mouse.\n\nWhy should that be so? The simple answer is that the components needed to keep the motor running have to enter by diffusion: the sugars, the amino acids, and all the other elements of energy-providing food must come in through the surface of the gut. And oxygen is needed to combust that food in the cells of the body and turn it into useful energy such as muscle contraction; it also must come in through a surface, that of a lung, or in the case of fish, a gill. So the running of the living motor is constrained by the extent of the organism's surfaces, by the square of the linear dimensions. On the other hand, the controlled combustion that takes place within the cells is limited by the number of cells making up an animal, that is, its total weight, which varies as the cube of the linear dimensions.\n\nAs we have seen from the earlier discussion of the effect of size on the shape of organisms, size is expressed as weight or volume, and when it increases in comparing two animals of different sizes, then surface structures increase at an even greater rate to keep up. However, there is a paradox: if the surface increase did keep pace exactly with volume, so that the ratio of surface to volume remained constant, then one would expect the metabolism for each cell to remain constant, regardless of the size of the animal it inhabits. Clearly this is not the case: each cell of an elephant combusts its fuel, its food, at a slower rate than that of a mouse.\n\nConventionally the relation between metabolism and size is expressed allometrically. That is,\n\nAs before, this relation can be expressed logarithmically:\n\nIf this information is now plotted on a graph, it will give a straight line whose slope will equal _a_ (fig. 31). This curve, the so-called mouse-to-elephant curve, was the brainchild of Max Kleiber in the 1930s, and he calculated that the slope was about 3\/4 or 0.75. This tells us that in larger animals the extent of the increase of the surface does not keep pace with the increase in volume.\n\nFollowing Kleiber there have been many others who have confirmed this mouse-to-elephant curve and extended the observations to include cold-blooded vertebrates, invertebrates, and even down to small bacteria. There is a big literature on the significance of the allometric exponent of 0.75 and the variations of this slope and the theory that must lie behind it. The arguments have been ingenious, but it is a topic that is beside the main point I want to make here. I am looking simply for the explanation of why the slope does not equal 1. Why and how does an animal manage to keeps its motor going if its surfaces that absorb oxygen and food lag with size increase?\n\nFigure 31. The \"mouse-to-elephant curve,\" showing the relation of metabolic rate to body weight. (From Schmidt-Nielsen, _Scaling: Why Is Animal Size So Important?_ , 1984, after Benedict, 1938)\n\nThe why, like so many things in biology, can be answered in terms of Darwinian natural selection. We have gone over some of the arguments why there has been a selection for size increase, but there is also a constant selection, no matter what the size, for greater efficiency. Any animal must function in a way so that it is successful in producing offspring, and that may not just involve its size, but all aspects of it functioning: its physiological efficiency.\n\nThis brings us to the \"how.\" To begin with, it is reasonable to assume that if a way can be found to use less fuel, less food and oxygen, and still function effectively, this will be an advantage. If an elephant's cells were burning its food with the same intensity as those of a mouse, it would require vastly more food than an elephant consumes. Furthermore, combusting food generates heat, and in order to keep its temperature at a reasonable level it would need a far greater surface for dissipating the heat, for cooling. Of course, if the surface\/volume ratio is kept at 1, it would indeed have a greater surface, but it would no longer look like an elephant but have great convolutions of its skin and internal organs, rather like some sort of monstrous walnut. The solution is to slow down its metabolism just enough so that it can exist and look like an elephant. This is the main way an animal can and does manage size increase and remain efficient. That is perhaps not a sufficiently strong way of putting it: a larger animal could not even exist unless its cells had a reduced rate of metabolism. It would either starve or burst into flames, or both.\n\nOne consequence of the differences of metabolism for different size organisms is that small warm-blooded animals must eat constantly to keep their intense internal fires burning. In the case of a shrew, if it cannot find food there will be irreversible internal damage after a few hours and it will die of starvation, while we humans can go on a hunger strike and survive many days without perishing. Presumably elephants can go for even longer, although the information on how long is a bit less reliable because they are wise enough not to go on voluntary hunger strikes. There are other metabolism-related consequences, such as longevity, that we will examine shortly.\n\nLet us return for a moment to _Gulliver's Travels_ and consider where the various-size characters might fit on the mouse-to-elephant curve (fig. 31). Lilliputians are roughly the size of a mouse, Gulliver is already on the curve, and the Brobdingnagians are roughly the size of an elephant. This means that, like a mouse or shrew, the Lilliputians' heart is racing, while by comparison the heart of a Brobdingnagian pumps at an exceedingly slow rate. Furthermore, the movements of a Lilliputian will be as rapid as a blink, while those of a Brobdingnagian will be deliberate and slow. The differences in their sizes means big differences in all their activities.\n\n### SIZE AND SONG\n\nOne of the most curious effects of size is on the noises animals make, from speech to songs, from alarm cries to love calls; every form of animal communication. Sound does, after all, involve time for it depends on frequency. The larger the animal, the lower the frequency, the lower the sound. In the case of human beings, large individuals tend to have deeper voices; we expect a basso profundo to be a bigger man than a tenor, although this often turns out not to be true. It has to do with the length of the vocal chords: the longer they are, the deeper the voice. Think of the range of sounds of a violin compared with that of a huge bass viol whose longer strings can make much lower, very profundo sounds.\n\nThe point is well illustrated in frogs, where the larger males have a deeper croak that is particularly alluring to the females. There was a flurry of excitement at one time because it was suggested that if a small male sat in very cold water it would make his voice deeper for it would affect the muscles in his sound box: the undersized were freezing to death, all for love. Whether or not this romantic idea is true is uncertain, but the thought is rather nice.\n\nThe sound range of different animals has only been appreciated in the last sixty years. We are intimately aware of what human beings can hear, all the way from the piccolo to the boom of a kettle drum in a symphony orchestra, and for centuries we were totally unaware that we could not hear the sounds of some animals because they were too high or low pitched for our human ears. The first surprise came when it was found that bats made a whole spectrum of sounds above the upper limits of what we could hear. Not only that, but they used those sounds to \"see\" their environment: they sent out the sound signals and then they caught them as they bounced back to their ears from surfaces or objects. This was called _echolocation_ , and with this amazingly sensitive system bats could catch tiny insects in the dead of night.\n\nThere has been an interesting recent report on the differences between large and small forms within one species of bat, the long-eared horseshoe bat of Southeast Asia and Australia. As one would expect, the larger ones produce lower frequency sounds than the smaller ones with the result that they eat different size foods: the smaller bats can capture the smaller insects that the larger ones cannot find with their echolocation system. Because of these differences, the size forms may have become, or are becoming, separate species entirely because of their size difference and its effect on their vocal cords and the pitch of their calls.\n\nThis phenomenon was first revealed using instruments that could hear and record the high sounds to which we are deaf. This is another example where, like the microscope and the telescope, we can extend our faculties to not only see beyond our visual capacities, but our hearing capacities as well. Those sound recording devices also helped us at the other end of the scale to hear sounds at a very low register that we did not know existed. It turned out that whales made sounds of very low frequency, and they, like birds, used those sounds for courtship, for signaling between parents and young, and for disclosing their location. These sounds are in our hearing range, and some of the whale songs are quite haunting; at one time, whale music records were very popular. One trick I found very interesting was to play the whale song recordings much faster than normal; doing so made the low-frequency sounds higher. Now they no longer sounded like whales, but like the song of small, chirpy birds, making the point that size is indeed correlated with the frequency of sound waves.\n\nA parallel story is now known for elephants. In the 1970s the zoologist Katy Payne was gazing at some elephants in a zoo and noticed that periodically the folds around their mouth would move as though they were exhaling, and she wondered if it was a sound she could not hear. With the help of a microphone and a recording instrument she found that indeed they were making very low-frequency sounds inaudible to us. She then transferred her studies to Africa and eavesdropped on quite remarkable conversations of elephants in the wild. One of the things she noted, and indeed this is equally true for whale calls, is that low-frequency sounds, unlike high-frequency ones, will travel great distances. So elephants\u2014or whales\u2014that are miles apart can keep in touch.\n\nThe lesson from all this is that each of us hears in a frequency range corresponding to our size: bats are small and elephants and whales huge and we are somewhere in between. The same must be true for Gulliver: the Lilliputians must have squeaked like a mouse and there is a good chance that neither they nor Gulliver himself could even hear the speech of the Brobdingnagians.\n\n### SIZE AND GENERATION TIME\n\nOne of the consequences of an increase in size is that it takes longer to build a big animal or plant than it does a smaller one. Let us examine this fairly obvious point in more detail. For living things, the building process means growth, so the period of size increase is the period of development from the single-cell stage to that of maturity when it in turn can start a new generation. This period is the part of the organism's life cycle called the _generation time_. In human beings the generation time is about twenty years so that in a century one would expect about five generations for _Homo sapiens_. On the average it takes us that much time to grow up from conception to adults that produce children. We can reproduce at an earlier age, as all those teen-age pregnancies attest, nevertheless twenty years is closer to the norm. The timing of maturity in human beings is not a perfect match with the ability to produce egg and sperm. It is true our growth stops when we are about twenty and the growth zones in our long bones congeal, so maturity occurs at roughly the same time we begin, on the average, to have children.\n\nThere is a great difference between our generation time and that of smaller and larger organisms. At one extreme, bacteria\u2014which are single cells\u2014may divide every half hour. Compared to us, the number of their generations that can be packed into one of ours is positively enormous. On the other hand, a giant sequoia needs sixty years for a generation\u2014it takes that long for it to produce cones with seeds. We can produce three generations during their one. That there is a clear relationship between the size of an organism and its generation time can be shown on a graph that plots the length of a plant or animal against its generation time (fig. 32). Admittedly there is considerable variation between species, but in a general sort of way, if one doubles the length of an organism, the time it takes to produce a generation doubles as well. No doubt there are many reasons why all species do not fall exactly on the line. It might have to do with the particular ecological conditions to which they are adapted; it may have to do with the way they are constructed; or, in the case of higher animals, it may even have to do with differences in behavior.\n\nThere is nothing that is not plain in this relationship. The bigger the plant or animal, the more time is needed to build it; there is no way of getting around this very obvious fact.\n\nFigure 32. A log-log graph showing the length of an organisms at the time of reproduction plotted against its generation time. (From Bonner, _Size and Cycle_ , 1965)\n\nIt applies to the nonliving world as well. A rowboat might be built in a day, but building an aircraft carrier will be a long-term enterprise. One need only think of the immense number of parts that need to be put into place in a carrier compared to the few screws required to construct a rowboat; no wonder there is a big difference in their construction time. Similarly, if one compares the time it takes for a minute alga made up of a few hundred cells in all to grow to its adult size, with that of a comparatively huge human being that has more than a trillion cells, it is not surprising that the alga can come to reproductive maturity in a matter of days, while we take a number of years before we can manufacture egg and sperm. Size increase or growth in living organisms consists largely of cell divisions, and although most cells in multicellular organisms do not divide as rapidly as bacteria, they may take hours, or sometimes days, between one division and the next. Since we both start as single cells it obviously takes a shorter time for the hundreds of cell divisions of the alga to mature compared to our trillions; no wonder we take longer to grow up.\n\n### LONGEVITY AND SIZE\n\nThere is another important aspect of life cycles that correlates with size: larger animals and plants live longer. This means that over and above the period of growth, the organism may live for a long time after its first reproduction\u2014after the end of its generation time. Some organisms, such as ourselves, provide good examples. After the cessation of growth at age 20 we might continue to live many more years, perhaps three- or fourscore years beyond the first.\n\nMany organisms do not stop growing after they start reproducing. Trees, for instance, keep growing for many years after the first time they bear fruit. Fish and many reptiles do the same. Being a fisherman, I cherish the continued growth of fish; I might someday still catch the ancient giant I've dreamed about. One summer some friends who rented a boat to sail among the Hebrides in Scotland came to a small fishing harbor. Looking over the boats was a statue of Jesus with his arms spread out, blessing the fleet. A fisherman saw them looking at it and said, \"We call it, 'The one that got away.' \" Even some mammals, such as the elephant, continue to grow for many years after they are capable of reproduction. In all such cases of continued growth, however, the rate of the growth slowly declines with age. Whether there is growth during maturity or not, the life span correlates well with size: bigger organisms live longer.\n\nLife span is controlled in two basic ways. One, which is found in animals and in many other organisms including unicellular ones, is genetic: there are genes that control the life span. This is an intrinsic control mechanism; the other is extrinsic and is particularly relevant to the main message of this book. It is what controls the size of trees\u2014it is what makes an old tree come to the end of its growing. There is good evidence that a tree starts declining in its growth when it becomes so tall that the water and nutrients can no longer effectively reach the growing tips of the outer stems. This has been shown in a recent study in which four species of trees were measured. For each species, the growth of the terminal stems of large trees in the field were compared with growth in cuttings from those same trees, and only the outer stems on the large trees showed a decline. If young shoots are grafted into the crown of an old tree, they will decline in their growth rate in their new environment comparable to that of their neighboring stem tips. Parenthetically, this probably explains why Bonsai trees, which are continually trimmed to keep them small, are known to live for exceedingly long times\u2014one tree is reputed to be over seven hundred years old. Tree growth is a perfect example of how size can exert supreme control over life.\n\nIf one plots the size of an organism, in this case mammals, against their life span (in the sense of the maximum length of life known for individuals of a species) one will have a curve that will reflect what I have just described, that larger organisms live longer (fig. 33a). Note that the scatter of points is considerable, much as they were with generation time and size in figure 32, although in both cases the trend is clear. As before, there are no doubt many reasons for the scatter, but none of them overshadow the obvious trend. The puzzling fact is that for the same mammals shown in the figure, if instead of body size the brain size is plotted against life span, the scatter is greatly reduced (fig. 33b). Why can this possibly be? Why is brain size a more reliable indicator of life span than body size? Perhaps it is because body size can be greatly affected by environmental conditions, such as the food supply, while the brain is permanently encased in a bone prison.\n\nFigure 33. (a, above) A log-log graph showing the relation of life span to body weight for mammals. Open circles are primates and lemurs; solid circles are rodents and insectivores; crosses are carnivores; solid triangles are ungulates and elephants; the open circle at the top is a human being. (b, facing page) A graph for the same animals, in which life span is plotted against brain weight. (From G.A. Sacher, _CIBA Foundation Colloquia on Aging_ 5 [1959]: 115\u2013141)\n\nOne point is above the curve (the top circles in figs. 33, a and b) and is far above the average. This is the point for human beings\u2014we have an unusually long life span for our size. It has been suggested that the reason for this is that our comparatively large brain means that our elders may help the general welfare of the group through their storage of wisdom, which has resulted in selection for a long postreproductive span of years.\n\nThere are reasons to believe that life span is related directly to metabolism. Larger animals live longer and their internal fires burn at a slower, less frenetic rate. As we saw, each cell of an elephant has a much lower rate of combustion than that of a mouse, and the animal the cell is part of lives longer. Some experimental evidence supports this idea that longevity is affected by metabolism. It is well established that keeping an animal on a minimal diet, thereby artificially reducing its metabolic activity, will result in longer life. There is current interest in the genetics of aging, and mutations have been found in yeast, nematode worms, and fruit flies that cause the life span to be either longer or shorter than normal. There is a mutant in a nematode that lives twice as long as a normal worm, and it has been shown in this particular case that its metabolic rate is half that of normal; this appears to be the reason it lives longer. Needless to say, the great interest in longevity comes from our permanent fascination with the idea of extending human life and our endless quest for the fountain of youth. It is doubtful that one could extend human life by altering our genes, and even if one could it might be that we would end up with people who were extremely slow and lethargic, which sounds like too high a price to pay for becoming a Methuselah.\n\nIt is also instructive to think of the mouse-size Lilliputians. They would live only a year or two compared to Gulliver's sixty to a hundred. If we compare the huge Brobdingnagians with Gulliver, we would find that these giants\u2014larger than elephants\u2014might live about twice as long as we do.\n\n### SIZE AND SPEED\n\nAs we have seen, this reduction of metabolism with size increase is intimately tied in with the fact that a larger animal does everything at a slow pace. A hummingbird's wings move so fast that one cannot see them, while a great blue heron gracefully waves its wings in what, by comparison, appears to be slow motion. We can observe this relation between size and movement in our own bodies. The very small muscles in our eyelids that are responsible for blinking can contract in a split second, while by comparison we move our legs slowly. In this case the difference is the direct result of the size of the muscle.\n\nIn 1950 an important paper was published by the noted English physiologist A.V. Hill in which he examined this very point. He concluded that animals of exactly the same proportions should run at the same speed: for example, a mouse and some similarly constructed large mammal ten times taller would tie in a race. The mouse's legs would make a step that is one-tenth of the length of a stride of the larger animal, but it would move ten times faster, so they should run at the same speed. As examples he compared a whippet, a greyhound, and a horse whose maximum speeds were 34, 38, and 42 mph, respectively.\n\nHill's general principle is particularly apt for Jonathan Swift's people of three different sizes. Each is ten times taller than the other: there are three orders of magnitude from Lilliputians to Brobdingnagians. Moreover, they are similarly constructed; all are shaped and proportioned like human beings, however improbable that might be, as we have seen. This means they perfectly fulfill Hill's conditions and therefore they should be able to run at exactly the same speed. One step of a Brobdingnagian would take the same amount of time as ten rapid steps of Gulliver, and one hundred lightning steps for a Lilliputian. The latter's small muscles would vibrate like the wings of a hummingbird, while each step of the giant would be stately and slow, but they would all reach the finish line together. The principle is undoubtedly sound, but clearly it is not the whole story; larger animals do run faster, as one can even see from his example of racing speeds in dogs and horses.\n\nSome years ago in looking for the possible advantages an animal might have by being larger I searched the literature for information on the speed of various animals, not only those that ran, but also those that swam or flew. It turned out to be an interesting exercise in which the conclusion is that, in general, bigger meant faster regardless of whether the animals were similarly constructed or not (fig. 34).\n\nThe smallest runners I could think of were mites, but I could not find anything on their running speed. I had met Paul Winston at the University of Colorado who works with mites, so I wrote him to ask if he had any information. He said he did not, but that he and his colleagues would run some mite races of different-size mites, and the results are shown on the figure. The next bigger runners were ants measured by the distinguished astronomer Harlow Shapley on Mount Palomar. He was interested in the speed of two species as they crossed a path on the mountainside and how that speed varied with temperature as the day progressed. He found that he could determine the temperature accurately by their speed. Speed increases with temperature. I have always thought this was an inspired daytime activity for a nocturnal astronomer. After this, there is a big jump on the graph, to mice, lizards, other small mammals, a fox, and the fastest of them all, the cheetah. Note that even a bipedal ostrich does remarkably well. Since the figure is plotted logarithmically and the slope is approximately 1.0, this means that if one doubles the length of an animal it runs roughly twice as fast. This is the case for animals of vastly different construction, from eight-legged mites to two-legged ostriches.\n\nFigure 34. The speed of various animals\u2014swimmer, runners, and flyers\u2014plotted against their length on a log-log graph. (From Bonner, _Size and Cycle_ , 1965)\n\nSwimmers follow the same trajectory as runners, as can be seen in the figure. The big difference is that the smallest swimmers, namely bacteria, are much smaller than mites. Protozoa come next, and then a water mite that swims about the same speed as a running mite of comparable size. Fish come in all sizes from larvae up to mature tuna and the related wahoo. All seem pretty close to the line, with the possible exceptions of the penguin and dolphin; the only animal that clearly does not fit is the blue whale; the reason for this divergence is poorly understood.\n\nThe smallest flyer about which I could find information on speed was the fruit fly ( _Drosophila_ ), so important as a laboratory animal. There are much smaller flying insects, such as fairy flies (which are parasitic wasps; see figs. 6 and ), but no one seems to have clocked their speed. Going up the curve in the figure, we see that horse flies are faster than fruit flies. Next we find hummingbirds and dragonflies, a bat, a warbler. A swift matches its name, and it is fast for its size, while a flying fish, a curlew (whimbrel), and a duck fall on the line. The fastest flyers are large: the swan and the pelican.\n\nIf we stand back and look at some of the implications of this graph, for starters, it is obvious that, like running speed, swimming speed is proportional to length; if the length is doubled, so is the speed. The advantage of increased size for a flying animal is less dramatic, although it is clearly there. The problem encountered in flying has to do with the extra horsepower needed for flight compared to swimmers and runners.\n\nAnother particularly interesting fact that is revealed by the figure is that the three lines join in roughly the same area. This means that the fastest animals are in the general neighborhood of one meter in length, and this is true for swimmers, runners, and flyers. A great excess in length, such as found in the blue whale, does not produce any further advantage in speed.\n\n### SIZE AND SPEED FROM THE COSMOS  \nTO ELEMENTARY PARTICLES\n\nMotion is everywhere and not just in living organisms. Planets and stars race about at great speeds, as do atoms and elementary particles within those atoms. Everything in the universe moves. If their size and their speeds are compared to those of living organisms, the latter fall into an intermediate range between the largest and smallest physical entities (fig. 35). Why electrons and other subatomic particles rush around at such incredible speeds, as indeed do celestial bodies, is a great mystery to me. I am neither a physicist nor an astronomer and therefore it is fortunate that the matter is beyond the province of this book. However, let us examine the left-hand part of the figure more closely: the world of molecules and living organisms.\n\nFigure 35. A diagrammatic graph on a logarithmic scale showing the relation of speed of movement to size, from elementary particles to celestial bodies. (From Bonner, _The Evolution of Culture in Animals_ , 1980)\n\nWe know from those early science classes at school that molecules move and that their motion is affected by temperature. If they are in a solid state, as are water molecules in ice, they are closely packed and their motion is greatly restricted. At warmer temperatures, when the ice turns into liquid, the molecules are released from prison and now move around with agility; at still higher temperatures they turn into water vapor, a gas, and now those molecules zip around at a furious pace with great distances between them. In fact, the three states\u2014solid, liquid, and gas\u2014are defined by the movement of molecules, and those states are determined by the temperature. It is this same molecular motion that is responsible for diffusion, as we discussed earlier: diffusion of oxygen through the lung surface or the diffusion of amino acids or sugars through the gut surface of animals. Those processes are totally dependent upon molecular motion.\n\nOne feature of this motion is that the smaller the molecule, the faster it moves, as is clearly shown in figure 35. Note on the graph that the largest molecules and the smallest bacteria meet: the point of slowest motion of all of life and of all nonliving physical bodies.\n\nIn this region of minimal speed we are in the realm of Brownian motion. In 1827 Robert Brown, an English botanist, was looking inside some plant pollen and saw small bodies bouncing about. He correctly pointed out that the movement could not be a manifestation of life because the pollen was at least one hundred years old and therefore totally dead. With the rise of physical chemistry that followed as the century progressed, it became evident that molecules in the solution were bombarding the small visible particles in the liquid. Those molecules were too small to be seen in a light microscope, but their effect could be observed in the movement of larger visible bodies that were being rammed into, giving them a drunken, zigzag motion. The subject culminated in a paper by Albert Einstein published in 1905 in which he established the theoretical basis of the phenomenon.\n\nIn terms of size and speed, large molecules are more or less in the same league with living bacteria, but from an evolutionary point of view the gap is enormous. The long, slow origin of life must have spanned many millions of years. However, here we are only concerned with speed and size, and large molecules and small bacteria move at roughly the same pace, a pace that is the slowest of everything that moves in the universe. Bacteria move, as was discussed earlier, by an elaborate and clever rotary propelling of flagella; molecules, on the other hand, do not have a motor but are propelled by inner energies that are part of their fundamental makeup. The larger the molecule becomes, the slower it moves. Motile organisms are just the opposite, for as we have seen, the bigger they are, the faster they travel. (The living organism balloon drawn on the figure encompasses all of the previous figure [fig. 34], the one that shows the speeds of different-size animals.)\n\nWhat came first: did the invention of a means of propelling the organism come first and cause the evolution of size increase, or the reverse? Selection for size changes had to come first and become the prime mover, for if the selection for movement followed size increase, then one would expect different mechanical ways of dealing with a particular size, and that is exactly what has happened. The smallest motile beasts, namely bacteria, have the extraordinary mini-rotary motor that can effectively move in the world of extremely low Reynolds numbers; in larger single-cell organisms, flagella do not rotate but contain contractile proteins that allow them to become active whips; in larger unicells, cilia beat like oars to move the cell (fig. 11); this is followed by the invention of muscle and bones to deal with running, swimming, and flying in larger animals. The progression of size during the course of evolution leads to a series of different inventions for locomotion, each appropriate for the dimensions of the organism.\n\nObviously, this effect of size only applies to organisms that move, and there are many that do not: many species of bacteria, algae, fungi, many invertebrates such as sponges and corals, and all vascular plants. Only some kinds of organisms need to move. In some of the larger immobile forms, the gametes may be motile, single-flagellated cells. Even the largest animals, elephants and whales, have motile sperm, but note that the sperm's mechanical means of locomotion is appropriate for their small size: the single-cell sperm of all larger animals do not move using muscle and bones, but they have a flagellum. Here, too, size dictates the locomotion machinery, and the reason that this is so is purely physical; a whale could not manage to move at all if its bulk had to rely on a furry coat of beating cilia. Again we see evidence of the overreaching role of size in all of life.\n\n##   CHAPTER 7\n\n## ENVOI\n\nSize and the actual organisms that are measured are rather like shadow and substance. The organisms are material objects. In correlation with their size, they come in many shapes and have different physiologies, degrees of internal complexity, generation times, life spans, speeds, and frequencies to their songs. A beast's size is like a shadow: it has no substance. It is simply a statement of how much matter makes up a particular living entity.\n\nWhat is remarkable is that although size is no more than a shadow, no more than a description, a property, it nevertheless exerts enormous power over the living world; it drives the form and function of everything that lives. How can this be so?\n\nThe root reason is natural selection. The size of an organism is under constant selective surveillance. Nature is made up of a vast network of size niches, and all living forms are always faced with the possibility that it may be advantageous to have their descendants become larger or smaller. If they are animals, size may be important for avoiding predators, or catching prey; for plants it may be important for success in the competition to catch the rays of the sun for photosynthesis. In microorganisms the most obvious advantages in becoming larger may also involve greater speed to pursue prey or escape predators. Becoming larger by multicellularity may mean more effective dispersal of spores, or more effective digesting of food, where the presence of more digestive enzymes are insured by increasing numbers of cells. The list could go on, as I have argued in much of this book. The key point is that size changes\u2014increases or decreases\u2014will have advantages or disadvantages and will be encouraged or discouraged by natural selection. Selection is the motive force for size change.\n\nAnd if the size is changed there will be a myriad of consequences, especially if it is increased. It will mean various constraints bear down: size is volume, yet life's activities require the appropriate surface to go with the volume and the result will be different shapes for different sizes. This will have many ramifications affecting properties such as metabolism and locomotion. It is particularly interesting that as the volume increases, there is an increase in the division of labor in the form of the number of cell and tissue types. Unless these and many of the associated changes occurred, larger organisms simply could not manage. They would not exist. Size increase has forced all these changes. The shadow is not really just a shadow: it is much more. Even though it is not substance, it is a dictator that holds complete sway over what on organism will look like and how it will function. Size rules life.\n\nIn the cases of size decreases, size plays a somewhat different role\u2014it is no longer the absolute dictator. This is because having less bulk may permit a decrease in the structures that are involved in matters of strength or surfaces for diffusion, but they do not necessarily require it. It could be that with continued size decrease, they may not have room for all the structures their larger ancestor possessed and may ultimately have to shed some of them; clearly size decrease is far less demanding than size increase.\n\nIf we stand back even farther, we see a principle here that goes beyond life. In the inanimate physical world, size has the same effect as it does on living organisms: Galileo made this clear right from the start. A small pendulum swings faster than a large one; a large boat such as an eight-oared shell will move faster than a one-man single. I pointed out earlier that it takes longer to build an aircraft carrier than a rowboat, but also that the larger ship has a vastly larger division of labor in its parts. The proportions of bridges or of buildings must differ with size increase because of the weight-strength relationship. The engineer who builds airplanes, engines, or bridges must make size one of his prime concerns; if it is neglected, all his endeavors will fail.\n\nOne could say that what makes life unique is natural selection that culls so that only the efficient constructions survive, while the equivalent for human constructions is whether or not they work. Mechanical or physical efficiency in engineering structures simply has a different method of selection, just as the selection forces of economic efficiency governed the division of labor in human societies. Size is ubiquitous and casts its shadow on all material things, both the living and the nonliving.\n\nPutting all these matters concerning size together leads to another insight. The size rules that have been laid out show that there is a connection between strength, surface activities, division of labor, and all activities that involve rates of processes such as metabolism, generation times, longevity, speed of locomotion, and even the abundance of organisms in nature. What connects them all is size. A change in weight will either require, or be correlated with, a change in strength, in surfaces, in the division of labor, in all the time-related rate processes, and even in the density of the distribution of organisms in nature. Each of the five size-related categories is interdependent with all the others: one changes and they all change. It is much the same thing as one finds in the gas law where changes in temperature will change the pressure or the volume of a gas: a similar kind of connectivity is found in the size rules.\n\nAll the thoughts in this book reflect the human preoccupation with size. This is so for Galileo and all the scientists who have followed him up to this very minute as I write these words. It is equally so in literature from Sinbad the Sailor through the delights of _Gulliver's Travels_ , right up to many artists and writers today. And one can add to them a multitude of aspects of human existence: architecture, engineering, business, transportation, and innumerable others. There is no corner of human endeavor and human thought that escapes the tentacles of size. Couple size with the evolution of living organisms\u2014and this becomes the book I have written.\n\n## NOTES\n\n. Frye, T. C., G. B. Rigg, and W. C. Randall. The size of kelps on the Pacific coast of North America. _Botanical Gazette_ 60: 473\u2013482 (1915).\n\n. Ferguson, B. A., T. A. Dreisbach, C. G. Parks, G. M. Filip, and C. L. Schmitt. Coarse-scale population structure of pathogenic _Armillaria_ species in a mixed-conifer forest in the Blue Mountains of northeast Oregon. _Canadian Journal of Forest Research_ 33: 612\u2013623 (2003).\n\n. See Random Samples. _Science_ 305: 472 (2004).\n\n. Galileo, Galilei. _Dialogues concerning Two New Sciences_. New York: Dover Publications, 1914.\n\n. Went, F. W. The size of man. _American Scientist_ 56: 400\u2013418. (1968).\n\n. For a more rigorous discussion of Reynolds number, see McMahon, T. A. and J. T. Bonner. _On Size and Life_. New York: Scientific American Library, 1983.\n\n. Purcell, E. M. Life at low Reynolds number. _American Journal of Physics_ 45: 3\u201311. (1977).\n\n. The pioneer of allometry was Julian Huxley with his path- breaking book, _Problems in Relative Growth_. London: Methuen,1932.\n\n. Rensch, B. Increase in learning with increase in brain size. _American Naturalist_ 90: 81\u201395 (1956).\n\n. Weir, A. A., J. Chappell, and A. Kacelnik. Shaping of hooks in New Caledonian crows. _Science_ 297: 981 (2002).\n\n. de Waal, F. _Chimpanzee Politics_. New York: Harper and Row, 1982.\n\n. Byrne, R. W., and A. Whiten, eds. _Machiavellian Intelligence_. New York: Oxford University Press, 1988; II, New York: Cambridge University Press, 1997.\n\n. Fankhauser, G. The effect of changes in chromosome numbers on amphibian development. _Quarterly Review of Biology_ 20: 20\u201378 (1945).\n\n. Fankhauser, G., J. A. Vernon, W. H. Frank, and W. V. Slack. Effect of size and the number of brain cells on learning of larvae of the salamander _Triturus viridescens. Science_ 122: 692\u2013693 (1955).\n\n. Borass, M. E., D. B. Seale, and J. E. Boxhorn. Phagotrophy by a flagellate selects for colony prey: a possible origin of multicellularity. _Evolutionary Ecology_ 12: 153\u2013164 (1998).\n\n. Limoges, C. Milne-Edwards, Darwin, Durkheim, and the division of labor: a case study in the reciprocal conceptual exchanges between the social and the natural sciences. In _The Natural and the Social Sciences_ , ed. I. B. Cohen. Cambridge, Mass.: MIT Press, 1940.\n\n. Bell, G., and A. O. Mooers. Size and complexity among multicellular organisms. _Biological Journal of the Linnaean Society_ 60: 345\u2013363 (1997).\n\n. Kirk, D. _Volvox_. Cambridge, U. K.: Cambridge University Press, 1998.\n\n. Schaap, P., T. Winckler, M. Nelson, E. Alvarez-Curto, B. Elgie, H. Hagiwara, J. Cavender, A. Milano-Curto, D. E. Rozen, T. Dingermann, R. Mutzel, and S. L. Baldauf. Molecular phylogeny and evolution of morphology in the social amoebas (submitted).\n\n. Bonner, J. T., and M. R. Dodd. Aggregation territories in the cellular slime molds. _Biological Bulletin_ 122: 13\u201324 (1962).\n\n. Waters, C. M., and B. Bassler. Quorum sensing: cell-to-cell communication in bacteria. _Annual Review of Cell and Developmental Biology_ 21: 319\u2013346 (2005).\n\n. Donner, J. _Ordnung Bdelloidea_. Berlin: Akademie-Verlag, 1965.\n\n. Valentine, J. W., A. G. Collins, and C. P. Meyer. Morphological complexity increase in metazoans. _Paleobiology_ 20: 131\u2013142 (1994).\n\n. McCarthy, M. C., and B. J. Enquist. Organismal size, metabolism and the evolution of complexity in metazoans. _Evolutionary Ecology Research_ 7: 681\u2013696 (2005).\n\n. H\u00f6lldobler, B., and E. O. Wilson. _The Ants_. Cambridge, Mass.: Harvard University Press, 1990.\n\n. Dawkins, R. _The Selfish Gene_. New York: Oxford University Press, 1976.\n\n. Peters, R. H. _The Ecological Implications of Body Size_. Cambridge, U. K.: Cambridge University Press, 1983.\n\n. _Tricycle_ 34 (Winter): 64\u201365 (1999).\n\n. For an excellent treatment of metabolism and body size, see K. Schmidt-Nielson, _Scaling: Why Is Animal Size So Important?_ Cambridge, U. K.: Cambridge University Press, 1984.\n\n. Kingston, T., and S. J. Rossiter. Harmonic-hopping in Wallacea's bats. _Nature_ 429: 654\u2013657 (2004).\n\n. Payne, K. B., M. Thompson, and L. Kramer. Elephant calling patterns as indicators of group size and composition: the basis for an acoustic monitoring system. _African Journal of Ecology_ 41: 99\u2013107 (2003).\n\n. Pe\u00f1uelas, J. A big issue for trees. _Nature:_ 437: 965\u2013966 (2005); Mencuccini, M., J. Martinez-Vilalta, D. Vanderklein, H. A. Hamid, S. Lee, and B. Michiels. Size-mediated ageing reduces vigour in trees. _Ecology Letters_ 8: 1183\u20131190 (2005).\n\n. Sacher, G. A. The relation of lifespan to brain weight and body weight in mammals. _CIBA Foundation Colloquia on Aging_ 5: 115\u2013141 (1959).\n\n. Van Voorhies, W. A., and S. Ward. Genetic and environmental conditions that increase longevity in Caenorhabditis elegans decrease metabolic rate. _Proceedings of the National Academy of Sciences_ 96: 11399\u201311403 (1999).\n\n. Hill, A. V. The dimensions of mammals and their muscular dynamics. _Science Progress_ 38: 209\u2013230 (1950).\n\n. Shapley, H. Thermokinetics of _Liometupum apiculatum_ Mayr. _Proceedings of the National Academy of Sciences_ 6: 204\u2013211 (1920); Note on the thermokinetics of Dolichoderine ants. _PNAS_ 10: 436\u2013439 (1924).\n\n## INDEX\n\nacromegaly,\n\n_Acytostelium_ , \u201390\n\n_Aepyornis_ ,\n\nalbatross,\n\nAlice in Wonderland,\n\nallometry, \u201356\n\namoeba,\n\n_Arabian Nights_ ,\n\nAristotle, ,\n\nartificial selection, \u2013100\n\n_Australopithicus_ , brain size,\n\nB\/D ratio, \u2013115\n\nbacteria, discovery of, ;\n\nswimming, ,\n\ngeneration time, ,\n\nspeed of,\n\nBaldauf, S.,\n\nBarnum, Phineas T.,\n\nBassler, Bonnie,\n\nbat, long-eared horseshoe,\n\nBonsai trees,\n\n_Brachiosaurus_ , ,\n\nbrain size, \u201360\n\nbridges, construction of, ,\n\nBrobdingnagians, \u201315, \u201334, , , , ,\n\nBrownian motion,\n\n_Bursaria_ , ,\n\nCaledonian crows,\n\nCarroll, Lewis,\n\ncell size, \u201372\n\ncell type number, as measure of complexity, \u201383\n\nchimpanzee politics,\n\n_Chlamydomonas_ , \u201385\n\ncilia,\n\nclam, giant,\n\ncondor,\n\nCopernicus,\n\ncrocodile,\n\nCuvier, Georges,\n\n_Cyanea_ ,\n\ncyanobacteria, , ,\n\nDarwin, Charles, , ,\n\nDawkins, Richard,\n\nde Waal, Franz,\n\n_Dialogues concerning Two New Sciences_ ,\n\n_Dictyostelium_ , \u201391;\n\n_D. lacteum_ ,\n\ndie-off,\n\ndiffusion, and size, \u201339\n\n_Diplodocus_ ,\n\ndivision of labor, origin of, \u201393\n\nand size, \u2013100\n\nand energy,\n\nin societies, \u2013110\n\nDonner, Josef,\n\nduckweeds, \u201398\n\nDurkheim, \u00c9mile,\n\ndwarfs, human,\n\necholocation,\n\nEinstein, Albert,\n\nelephant seal, males larger than females,\n\nelephant, size of,\n\non moon,\n\ngeneration time of,\n\nmetabolism of,\n\nsounds of,\n\nElzevir, Louis,\n\nenergid,\n\nEnquist, Brian,\n\n_Eudorina_ , ,\n\n_Eurypterid_ ,\n\nfairy fly,\n\nfairy rings,\n\nFallopian tube,\n\nFankhauser, Gerhard,\n\nflagella,\n\nflea,\n\nfrigate bird,\n\nfrog, calls of,\n\nfungus, giant,\n\nGalileo, , , \u201332, ,\n\ngecko, feet of, \u201343\n\ngeneration time, and size, \u2013131\n\ngiants, human,\n\n_Gonium_ , ,\n\nGrimms' fairy tales, ix\n\ngrowth hormone,\n\n_Gulliver's Travels_ (Swift), ix, \u201315, \u201334, , , , , ,\n\nheart beats, and size, ,\n\nheterocysts,\n\nHill, A. V., \u2013138\n\nHMS _Beagle_ ,\n\nH\u00f6ldobler, B.,\n\nHomo, brain size,\n\nhomunculus,\n\nHubble telescope,\n\nHuxley, Julian,\n\ninertia, \u201344\n\ninsect societies, division of labor in, \u2013105\n\nIQ tests,\n\nJack the Giant Killer,\n\nkelp, length of,\n\nKirk, David,\n\nKleiber, Max,\n\nKomodo dragon,\n\nlarch,\n\n_Lemna_ , \u201397\n\nlemurs,\n\nlife span, genetic and physiological control of, \u2013133\n\nand metabolism,\n\nLilliputians, \u201315, \u201333, , , , ,\n\nLimoges, Camille,\n\nlizard, largest,\n\nlobster,\n\nlongevity and size, \u2013137\n\nMacAskill, Angus,\n\nMachiavellian intelligence,\n\n_Macrocystis_ , length of,\n\nmacronucleus,\n\nMcCarthy, Megan,\n\nmemes,\n\nmicroscope, first,\n\nMilne-Edwards, Henri,\n\nmites, , \u2013139\n\nmouse-to-elephant curve,\n\nmulticellularity, \u201377;\n\nmore than one origin of,\n\nmycoplasma,\n\nmyxobacteria,\n\nand \"wolf pack\" method of feeding,\n\nnatural selection, for size, , , \u201365,\n\nfor efficiency, , , ,\n\nnematode,\n\nnewts,\n\nnucleo-cytoplasmic ratio,\n\n_On Growth and Form_ (Thompson), ix,\n\northogenesis,\n\nostrich,\n\n_Paramecium_ ,\n\nPasteur, Louis,\n\nPayne, Katy,\n\nPeters, R. H.,\n\npituitary gland,\n\nplants, evolution of vascular tissue in,\n\n_Pleodorina_ , \u201388\n\n_Polysphondylium pallidum_ ,\n\nproportions, change in, \u201356\n\nPurcell, Edward,\n\nquorum sensing, , \u201391, \u201394,\n\nRaffles, Sir Thomas,\n\n_Rafflesia_ , \u201324\n\nRensch, Bernard, \u201355\n\nReynolds number, \u201349,\n\nrhea,\n\nroc,\n\nrotifers, \u201396,\n\nSacher, G. A.,\n\nSachs, Julius,\n\nscale of nature,\n\nSchaap, P.,\n\nself-thinning,\n\nsequoia, , ,\n\ngeneration time of, ,\n\nSeven Dwarfs,\n\nShapely, Harlow,\n\nSherlock Holmes,\n\nSinbad the Sailor,\n\nsize decrease, \u201398\n\nsize rules, ,\n\nsize, and metabolism, \u2013124\n\nsize-abundance rule, \u2013115\n\nsize-complexity rule, \u2013110\n\nduring development, \u201383\n\nslime molds, , \u201391\n\nSmith, Adam,\n\nsnake, largest,\n\nspeed and size, \u2013146\n\nspermatozoa, discovery of,\n\nsquid, giant,\n\nSwift, Jonathan, \u201315, \u201334. _See also_ Gulliver's Travels (Swift)\n\ntarpon,\n\ntelescope, invention of,\n\nGalileo's,\n\n_The Science of Life_ (Well, Huxley, Wells), \u201327\n\nThompson, D'Arcy Wentworth, ix;\n\nTom Thumb,\n\nTuring, Alan,\n\n_Tyrannosaurus_ ,\n\nValentine, James,\n\nvan Leeuwenhoek, Anton, \u201310\n\nvolvocine algae, \u201388\n\n_Volvox_ , \u201386,\n\nVon Frisch, Karl,\n\n_Vorticella_ ,\n\nWells, G. P.,\n\nWells, H. G.,\n\nWent, F. W.,\n\nwhale shark,\n\nwhale, blue, , ,\n\nsongs of,\n\nWilson, E. O.,\n\n_Wolfia_ , \u201397\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nDedication\n\nTo my writing friends, who dragged me out from under my rock and made me put words on the screen. You guys are the best. To Bethany, and all at Samhain for taking a chance on a weird post-apocalyptic reaper thing.\nChapter One\n\n\"They say she roams the old roads, looking for her victims. Human, paranormal...she don't care. She'll kill anything.\"\n\nThe raspy voice echoed around the half-empty bar. Like a chill wind it reached into the corners of the candle-lit room, found all those who were listening and pulled a shiver from the very depths of their souls. Forget the bogeyman, since the war people had learnt there were far more dangerous things than the pretend monster under the bed.\n\nThose in the room listened anyway. Chair legs scuffed on the worn wooden floor, disturbing the sawdust scattered over its battered surface. Shadows crowded against the steel-grilled windows that showed evidence of the bars use as a last line of defense.\n\nAt the back of the bar, Mason snorted into his whiskey. Good old Fred. He sure did like his ghost stories, even if they did scare the crap out of the customers. Not that they had many. Normally the place was filled with locals. It had been that way since the world went to shit in a storm of fire and brimstone.\n\nMason rolled a mouthful of the amber fluid around his mouth, before letting it burn its way down to his stomach. Stuff would rot your gut, but hell did it have a kick, exactly what a man needed at times. He leaned back against the wall and swirled the rich colored liquid around in his glass as he watched the bar as a whole. After wandering in here six years ago, badly busted up from an altercation with a couple of Lycans, he'd gone from recovering patient to self-appointed town protector.\n\nRight now his attention was on a group of youngsters at the other end of the bar who'd waltzed into town earlier. After scoring refills on their water bottles and a pack-load of supplies, they were living it up in the bar, full of bravado, and crazy-ass stories of escaping a nest of Vamps.\n\nMason had been up close and personal with a couple of Vamps, and they were tough bastards. It'd be hard, not impossible to escape a whole nest of the bloodsuckers, but you'd have to be a combination of Bruce Lee, Rambo and the terminator.\n\nTwo girls, three boys. Way too young to have survived on the roads. He pursed his lips, feeling the scar at the corner pull slightly. He remembered his days as a wanderer. It was a tough life. Practically everything out there wanted to screw you over, and eat you. Or screw you, and eat you. Or screw you whilst eating you...there were some kinky-ass critters out there.\n\nHe lifted his glass to his lips again. He never sat at one of the middle tables, preferring to keep his back to the wall with his gun free and easy by his side. Even in the supposed safety of the town Mason carried, locked and loaded. The light from the candle on his table caught the rim of his glass, twinkling in the corner of his eye.\n\nHe looked at the group again and caught his breath. Between the light spots from the candle and the flickering shadows cast by the other candles in the room, the group of visitors looked different. Changed.\n\nIt was a spilt-second, as though their masks had slipped a little and allowed him a glimpse at the creatures beneath. The one Mason had tagged as the leader turned around, looking around the bar with an assessing eye. Too assessing, and way too hungry.\n\nMason took another drink as though nothing had changed. But this time he didn't savor the drink as it went down. Before the war Mason had been a soldier. A damn good one. In that life he'd seen lots of corpses, but he'd never seen one walk and talk until ten years ago. Like the one scanning the bar as if it was an all-you-can-eat buffet.\n\nGhouls. How the fuck had they gotten past the defenses? They had everything from warding sigils carved into the plaster of the walls under the hangings to a demon-trap painted on the underside of the floorboards by the door. The first drink of water visitors to the town were given contained a dash of holy water, and the cutlery they got was silver-plated. With all their precautions, the town was loaded for everything but bear.\n\nHis gaze flicked around the room. Just his luck, most of the locals were in tonight. A heavy sigh escaped his chest as he drained his glass. He fucking hated Ghouls.\n\nGetting rid of them was always messy.\n\nHis lips compressed as he placed his glass on the table in front of him. Seeing an empty, Valerie, the bar-owner, headed over. Her hips swayed as she walked, and there was a smile of invitation on her lips as she approached.\n\n\"All done there, sugar-bun? Want me to get you another?\"\n\n\"No, thank you, I'm good. Listen, V, I need you to start moving people out.\" His voice was low and firm--completely at odds with the brilliant smile he flashed her. Anyone looking their way would assume the two were flirting, which was exactly what he wanted.\n\nValerie's face dropped. \"Aww, no... Those kids? You gotta be kidding me, Mason. They're barely old enough to be out on their own.\"\n\n\"Exactly. They aren't old enough, or anywhere near tough enough. They're not human, V. Don't be fooled by the cute mask they're wearing. Otherwise next week it might be your Suzie's.\"\n\nValerie's face went stony. The only thing that made her madder than a bunch of wet hens was a threat to her daughter. Mason had no idea what had happened to the kid's father. He'd never asked. Out here, no one did. Everyone had a past they would rather forget, things they'd had to do to survive. Most of the time those things didn't make for a good night's sleep.\n\n\"Give me five minutes, I'll get it cleared.\" She started to turn, and then paused to look back at him. \"Make it quick and clean, would ya? The girls look scared, like they don't want to be here. Plus, it took me weeks to get the bloodstains out of the floor last time, and I'm all out of sawdust.\"\n\nMason nodded, pitching his voice a little louder. \"Sure thing, sweets. Leave it to me.\"\n\nHis tone was friendly and flirtatious. For good measure he leaned over and swatted Val on the ass as she passed him. She squealed in delight, wagging her finger at him as she headed back to the bar.\n\nThroughout the room the locals carried on drinking, or talking, but he knew they'd caught the signal. The last time he'd flirted with a woman and meant it, mankind had a future, not this squalid mess they were trying to survive in.\n\nThe evacuation didn't take long. As soon as Valerie had dropped another glass off at his table with a wink, there was a slow, but determined exodus. The non-combatants left as the rest found reasons to go talk to friends on the edge of the room.\n\nExcept old Fred, who sat on the table next to the Ghouls. It was his favorite spot. The table never moved. It couldn't, it was riveted to the floor. Even now Mason knew that Fred's finger was on the trigger of the shotgun rigged underneath it.\n\n\"Lady of Death? Ohhhh purlease, old man, that's just a kid's tale. Like the bogeyman,\" one of the Ghouls scoffed. Mason had already tagged him as the leader. Loud and obnoxious he was the sort Mason would want to waste even if he was human.\n\n\"You wanna be careful, sonny,\" Fred warned. He was an odd choice for a front man, but he was one of Mason's best. A crack-shot with that gun, he looked like everyone's favorite grandfather. No one suspected Fred, even if his life depended on it.\n\n\"Oh yeah, Pops. Why's that?\"\n\nThe Ghoul spun his chair around to face Fred and straddled it. The skin between Mason's spine itched. He'd seen what Ghouls could do. They could rip a man apart with their bare hands without breaking a sweat. The instant the kid looked like he was even thinking of making a move towards Fred, Mason would put a bullet between his eyes.\n\n\"They say the lady sees all, and knows all. There's no way to escape her. Don't matter if you don't believe in her, as long as she believes in _you._ \"\n\nMason watched Valerie busying herself at the bar out the corner of his eye. She was wiping glasses, her attention seeming to be half on the conversation going on and half on her work. He knew better. By her left knee, next to the defunct cash register there was a fully loaded rifle. When the shit hit the fan, she would have it in her hands and firing within a heartbeat.\n\nHe almost felt sorry for the Ghouls. Almost.\n\nHe looked back. The mask had slipped completely now. Confident that the dumb humans they'd found themselves in the middle of couldn't tell what they were, they'd let their cover slip.\n\nMason _could_ see through their human disguises, and right down to the foulness beneath. They had the same matted hair and rotten flesh of every other corpse he'd seen. Added to that were blackened, claw-like fingers, and the guy on the left had a bad drool problem. Looked like he'd gotten a busted jaw before he'd become what he was now.\n\n\"Don't be fucking stupid, old man.\" The Ghoul started to stand. Fred didn't move an inch, still smiling even though Mason knew he had to have the trigger down to first pressure. Interesting that, when they had their balls to the wall, humanity could rock it against creatures of darkness in the bloody violence stakes.\n\n\"No, Johnny, don't. They've been nice to us.\" The girl at his side caught his arm, stopping Johnny's headlong rush into certain suicide. _Do us all a favor, lass, and let Fred blow his head off. Might be something human in you left to save._\n\nThe kid turned on her like a Rottweiler, a low snarl in his throat. \"Shut it, bitch. You're here to look pretty, not question me.\"\n\nHe had had enough. Even though she was a Ghoul, if the kid hit her, Mason was taking his hand off at the wrist. \"What do you believe in, Johnny?\"\n\n\"What's it got to do with you?\" The Ghoul's head whipped around, isolating the new voice in the conversation with the instinct of a predator.\n\nMason leaned forwards. \"If you don't believe in the Lady of Death, what do you believe in? Vamps? Lycans? The Abominable Snowman? Santa Claus? Ghouls? How about Ghouls?\"\n\nJohnny's face went blank, his eyes wary and flat. Mason levered himself from the bench in one lithe movement, his rifle held loosely in his hands. Even though he was armed, in this day and age that wasn't a threat. Everyone carried, if you didn't, you were dead.\n\n\"You believe in Ghouls, Johnny boy?\"\n\nJohnny shrugged. \"Lots of weird shit out there. Could be Ghouls too.\"\n\nMason made it to the end of the bar. If Johnny had been anything close to clever, he'd have realized by now all the locals were quiet. More to the point, he'd have twigged that all the friendly locals of earlier weren't looking half so friendly now.\n\n\"Oh, I know there are Ghouls. Now, because I'm a fair man I'm going to give you and your dream team one chance. Get up, get your stuff and walk out of my town.\"\n\nJohnny looked around the room. Mason watched as the penny dropped _._\n\n\"Come on, man,\" he wheedled, injecting a pathetic note into his voice. That was what disgusted Mason about these creatures. Ghouls were tricksters, playing up the sympathy vote to get into a town, and get a free meal. \"It's after nightfall. You can't turn us out onto the road at night. God knows what's out there.\"\n\nMason shrugged. \"We got a simple rule in this town, kid. You ain't human, you ain't staying. Now walk, before we perform a little...vermin control.\"\n\n\"Johnny, they're onto us. We should go...\"\n\n\"No.\" Johnny shook her arm off with a snarl as he glared at Mason. \"I'm not being told what to do by a fucking human. We're the top of the food chain man, and no one can stop us. We're gonna eat your heart and laugh while we chow down.\"\n\nMason didn't bother with any more dialogue. His dad had always said, when you face down a man with a gun you'd better hope that man is a bad man. A bad man will keep you talking before he shoots you so, if you're smart, you can find a way to escape. A good man though...he won't bother chatting you up, he'll just shoot you without a word. He wasn't sure his dad had gotten those definitions the right way around, because he sure as hell wasn't a good man.\n\nHis rifle was on his shoulder, and the Ghoul in his sights within a heartbeat. The next a blackened rose spread over the kid's forehead as Mason's double tap blew out what was left of his rotten brain.\n\nThe bar erupted into gunfire. Bullets and shotgun slugs tore into the small group of Ghouls without mercy, making them dance like marionettes. It wasn't a shooting. It was an execution.\n\n\"Cease fire,\" Mason yelled over the noise. The Ghouls were done for. The guys had all received double taps to the head courtesy of Mason, and Johnny had pretty much gotten shredded.\n\nHis footsteps were light as he approached, to check if the Ghouls were dead. Of course, with their kind, dead was a relative term. He needed to make sure they weren't the snacking-on-your-guts kind of dead.\n\n\"Fucking hell, just the stink'd put you off.\"\n\nFred was right by Mason's side as he reached what remained of the lead Ghoul, Johnny. Just as he suspected, the thing was old and rotten to the core. Rifle trained on the mass of torn flesh and rags, he kicked the leg nearest to him.\n\n\"That's Ghouls for you. Corpses too dumb to die.\"\n\nThere was no resistance. If the thing was playing possum it was good at it. He took one last look at the sightless eyes and moved his attention. The other two males were just as bad. From the sheer amount of damage, Mason figured they'd copped at least half the payload from the townsfolk. The girls, by comparison, looked almost untouched.\n\nHe scooted around one side of the bodies on the floor as Fred went the other way. They both saw movement at the same time. An arm twitched and then one of the girls, the one who had spoken earlier, groaned.\n\n\"We got a live one.\" Mason barely heard the end of his own sentence over the sound of rifles and pistols being cocked.\n\nHis world shrank to what he could see through his sights. Gaze firmly fixed on the girl's face, he watched as her eyes fluttered open and she slowly turned her head to look at him. Black blood covered one side of her face, and he didn't want to think about the thicker fluid oozing out from behind her ear.\n\n\"Please...\" she begged, her pale eyes fixed on him. \"I didn't want this. Make it quick...please.\"\n\nShe held his gaze, the moment stretching out. In that instant she ceased to be a monster, and became just a young girl wanting the nightmare to go away. Throat thick, he didn't trust himself to speak so he just nodded.\n\n\"Thank you,\" she whispered. He pulled the trigger and sent her into the afterlife.\nChapter Two\n\nTen years, three months and four days. That was how long it had been since the worst day of Andy's life. Of course, since that date coincided with the Apocalypse--Doomsday, Armageddon or whatever you wanted to call it--it had been a pretty shitty day the world over.\n\nThings hadn't got much better. She settled her backpack more securely on her shoulders and studied the road ahead. She'd walked these roads since that day, always on the move, never stopping for more than a night or two. She'd tried to in the beginning, but she was just too different to hide amongst humanity for long.\n\nShe trudged along the road, the tightening in her calf muscles telling her she was heading up an incline. She wouldn't have known otherwise, after a while everything looked the same. Dust and fuck-all else leading into foothills and mountains in the distance. Apart from yesterday...yesterday she'd passed a tree. It had provided hours of entertainment.\n\nReaching the top of the incline something new caught her eye. Pausing to rest her booted foot on the bumper of an abandoned car, she shielded her eyes and squinted. Despite her dark glasses the bright sun foiled her vision, making the dark smudge on the horizon dance and waver.\n\nShe growled under her breath. Why the hell couldn't she have gotten useful abilities like some other paranormals? The ability to change form and run like a Lycan, or the night vision of a Vampire...either would have been useful. At least, far more useful than what she did have, dangled on the end of a chain at the disposal of fate, chasing silver threads only she could see. It sucked, big time.\n\nOf course, most people would tell her to look on the upside--she couldn't die. Would've helped if she'd known that _before_ she'd tried to commit suicide. Three times. That had been the year after the war. She'd been way unstable back then. Mind you, when you were forced to kill your family, friends--hell, everyone you knew, then it was bound to knock a few cogs loose upstairs. Since then she'd come to terms with what she was, somewhat, and just did her job.\n\nThe smudge on the horizon resolved itself into a plume of smoke. Five silver lines, the sort only she could see, flickered and lit up in the corner of her vision. They headed off straight towards the smoke.\n\nShe sighed. _Another job. No rest for the wicked_.\n\nThe small black mark on the horizon grew larger and larger as she walked. Eventually it became a small town. Andy studied it as she trudged closer. Most humans lived in places like these. Towns fortified against any sort of attack--be that attacks from other humans looking for supplies, or attacks from any of the paranormal types.\n\nThis one had particularly good defenses. The person who'd put them together had really known what they were doing. She passed an outer redoubt of steel and iron barricades, nodding at the stony-faced guard stationed at the lookout post.\n\nThe silver lines she was following didn't lead into the town. Instead they branched off to the right. Like a good little puppy she followed them. The skin between her shoulder blades itched as she walked. Within seconds more armed figures appeared on the main wall, silent and watching. She was impressed. These people were on the ball.\n\nTurning the corner she found what she was looking for. A funeral pyre smoldered away, billowing black smoke high into the air. The wind changed direction for a second. Wrinkling her nose she tried to breathe through her mouth. Humans smelt bad enough when cremated, but Ghouls were even worse.\n\nShe didn't need to count the bodies on the pyre. Five silver lines fed straight into what she was looking for. Five souls, the ones belonging to the remains on the pyre, stood waiting for her. Standing in a nice little line, ready and waiting for her to reap them.\n\nUsed to the drill Andy took a deep breath, and let her spirit slip into the Shade. The layer between life and the afterlife, it was where the souls waited for a Reaper to come along and send them into the afterlife.\n\nThe world changed hue, painted in shades of black and grey. There was no color here, no life to speak of, and the truly alive couldn't enter this place. She looked over her shoulder at the figures on the wall watching her. To them she would appear to be looking at the pyre. She could step bodily into the Shade if she had to, but figured that would freak them out too much if she just disappeared.\n\nAs it was, they wouldn't see her reach around and under her pack, drawing the twin sickles sheathed there with practiced movements. A good thing, because she didn't fancy being hit with enough lead to drop a rhino. She'd only had this jacket a couple of weeks, and the last thing it needed was ventilation.\n\nThe weapons filled her hands, their well-used handles as smooth as silk under her callused palms as she walked towards all that remained of the Ghouls. Twirling the twin blades around, she sliced her way through the souls with ease.\n\nWith the souls already separated from their bodies, it was easy. They only put up a fight when she had to both kill them and reap their souls at the same time. She could understand that, most people were kind of attached to breathing. Sometimes she caught a break--death by old age or something like that--but most went down fighting, all the way.\n\nAs soon as her blades touched them the souls shattered, disintegrating like smoke blown away by a stiff breeze. As the last soul broke up, a tendril separated itself and wrapped around Andy's wrist for a moment. A surge of relief and gratitude filled her, welling in her chest and bringing dampness to her eyes. She smiled, knowing it was the soul's way of saying thank you.\n\n\"You're welcome.\" She flicked her blades back along her forearms and sheathed them with a small _snick_ under the pack again. \"I hope you find the peace you're looking for.\"\n\nJob done, Andy returned to the spot she'd entered the Shade. It was easy to spot. Behind her, stretching back into the world of the living was her own lifeline. It wasn't the silver of the Ghouls, or the gold of humanity, or any other color that marked a creature that was truly alive...instead her lifeline was black as pitch.\n\nShe snorted in amusement as she stepped back between the worlds. Twin sickles, which when you looked at them were just very small scythes, black on black clothing and a black lifeline... All she needed was to become anorexic, wear a black cloak and she could really rock the Grim Reaper image.\n\nShe ignored the watchers on top of the walls as she headed back towards the main gates. Like every other town she'd visited there was another guard on the gate.\n\n\"Morning, ma'am, what's your business in Sanctuary?\"\n\nThe greeting from the guard was polite enough--even if the smile didn't extend to his eyes and there was absolutely no way he could hide the threat in the casually held shotgun. To his credit he didn't even try.\n\n\"Just a quick stop off to trade for supplies and water. Perhaps a night's sleep without having to keep one eye open.\"\n\nThe guard nodded. It was a familiar story, probably one he'd heard countless times. Still his eyes swept her in a quick but thorough assessment. Without asking, she knew he'd be able to describe her from the weaponry she was carrying but wouldn't be able to remember her hair color.\n\nBlack. Fitting for a Reaper.\n\n\"Okay, step under the arch please, ma'am, just there on the red cross. Thanks.\"\n\nWith a frown she did as she was told, standing right in the middle of the crude cross sprayed in red on the rough concrete. A shiver ran down her spine as magic surrounded her. The guard's gaze flicked up, so Andy followed suit. There was nothing on the underside of the arched gateway. Frowning, she flicked her vision back into the Shade, and was almost blinded by the devil's trap painted up there.\n\nClever. Someone here knew how to mix holy water with PVA glue to make paint invisible to the living eye but capable of use in a spell. She kept her expression level and slightly puzzled, as though she wasn't sure what the kid was looking at. No need to alert anyone to the fact she wasn't _homo sapiens_.\n\n\"That it?\" She injected a bored note into her voice. \"Or would you like me to dance a jig as well?\"\n\nHe chuckled, and a rueful smile spread over his face. Andy grinned back. Under the wariness and the worry on his face, the kid was rather good looking.\n\n\"No, you're okay, ma'am. You're good to go.\" He nodded through the gates into the town. \"Best place to go is Val's. Just head up the road and it's on your left, can't miss it.\"\n\n\"Thank you.\" She inclined her head, shrugged her pack to a more comfortable position and walked into Sanctuary.\n\nInside the walls with their impressive defenses it was pretty much the same as every other town she'd visited. Dilapidated buildings lined the streets, and as she walked further she passed inhabited houses like small fortresses.\n\nIt didn't take her long to reach Val's bar. Like the rest of the town it was a little worse for wear. Right at the moment though, it looked like heaven. She couldn't remember the last time she'd stopped. Sitting in an actual chair rather than on the ground would be a novelty, and if they had running water and a bath, she might just think she'd died and gone to heaven.\n\nThe hackles on the back of her neck rose a second before the door in front of her burst open. A young lad stumbled out, glaring over his shoulder to snarl. \"You'll regret saying no. Jed doesn't like people who don't play ball.\"\n\nAndy stopped. The kid righted himself, his yellowish gaze sweeping over her, and all her instincts went into high alert. He wasn't human, not with eyes like that. Her palms itched, but she fought the compulsion to draw her sickles.\n\n\"What you lookin' at?\" he snarled, recoiling a little from her presence. Without realizing what he was doing he kept her in front of him as he skirted around her. He couldn't know what she was, but most paranormal creatures recognized the ultimate predator when they saw one.\n\nShrugging, she carried on into the bar. Perhaps they had coffee. Her brain all but shut down at the thought. She could smell it already, taste the stuff on her tongue. It was her only vice, if you discounted bloody murder and reaping souls. She'd run out weeks ago and cold turkey was no fun. It made her cranky.\n\nShe caught the swinging door on the way out, halting it with a flat hand on the wood and pushing back inwards. Her fingertips came away tingling. A devil's trap at the gate, and active ash-wood inlaid in the door? This place got any more interesting, and she'd have to raid an old cinema for popcorn.\n\nAfter the bright sunlight outside, entering the room was like being plunged into cool water. Andy stood in the doorway and let her eyesight adjust. Her sense of smell kicked in first. The stench of blood under pine floor cleaner was unmistakable--as was the fact someone had tried to scrub more than one kind of blood off the wooden floor. Under the Ghoul she could make out Vampire, Were and...Brownie? Holy hell, if they'd managed to see off a Brownie infestation then these people were serious players.\n\nThe room resolved itself into a large, almost cavernous space with a long bar at one end. At one time four pool tables had stood near the door, rings on the floor marking their positions like invisible sentinels. Sauntering down the center of the room, she headed towards the bar. Every set of eyes in the room was on her, which didn't surprise her after the Werewolf.\n\nShe reached the bar. Elbows on the wooden surface, she planted a booted foot on the rail and smiled at the woman wiping glasses behind the bar. Andy knew the smile didn't reach her eyes, but it didn't matter. The dark glasses did more than shield her eyes from the sun.\n\n\"Howdy. What'll it be?\"\n\nThe barmaid put a glass away next to an army of its cohorts lining the shelves. Andy wasn't fooled by the easy manner. If the woman didn't have at least three weapons within easy reach she was a monkey's uncle. Or aunt. Whatever.\n\n\"A refill of water, and if you've got some coffee I'll love you forever.\"\n\nAndy pitched her voice to polite and friendly as she put her water bottle on the bar. Invulnerable she might be, but being shot hurt. Since she had no active job in this town, all she wanted was to resupply, and perhaps get a good night's sleep.\n\n\"Room for a night, if there's one available. If not, I'll kip down in here...with your permission, of course.\"\n\nThe woman, Val presumably, inclined her head. \"One water and a meal on the house, coffee you gotta pay for.\"\n\n\"You trade?\" It was what Andy had expected. Her hands were already in her pockets as she withdrew a few trinkets she'd collected on her travels. Lip-salve, a box of old plasters with smurfs on them and a couple of disposable lighters. All suitable payment for the supplies she needed.\n\nVal's keen eyes assessed the offerings, and she nodded towards a table nearby. \"Can do, have a seat and I'll bring them over. No rooms, you sleep in here. There's someone on the bar all night.\"\n\nAndy's lips quirked as she turned and headed towards the table indicated. In other words, there will be someone with a gun on you all night, so don't try anything funny. Good policy.\n\nReaching the table, she slid her pack off her shoulders and shoved it under the table, out of the way. The intake of breath behind her warned Andy that her sickles had been spotted. She ignored it and sat. Since the whole thing was spelled to be inconspicuous, then that meant there were spells here that countered illusion. She really _was_ going to have to pick up some popcorn.\n\nShe lounged back in the chair and took her time looking around the room. She didn't bother with the people, instead she scanned the walls and the ceiling. She couldn't see them, but she knew the enchantments were there.\n\nHer water and food arrived. Andy gave up her search. With tricks like holy water and PVA glue to draw devil traps she was sure they'd gotten inventive here too. If she wasn't very much mistaken then her cutlery was silver plate, and she'd bet her bottom dollar that the water had a drop of the holy stuff in it as well. _Boiling Vamps from the inside was a new approach. Nice._\n\nShe studied the food in front of her for a while. One of the figures the other side of the room peeled himself from the wall and headed her way. Andy watched him from the corner of her eye. He didn't walk, he stalked...a predator like her. The rifle in his hand seemed an extension of his being, like her sickles were, and a pistol played peek-a-boo from his shoulder holster.\n\nMost men didn't bother with holsters, just shoved their pistols into the waistband of their pants. Andy had always wondered how many had done the gene pool a favor and castrated themselves. Stupidity like that didn't deserve to breed.\n\n\"Good work with the Ghouls. I'm impressed,\" she commented as he reached the table and spun a chair around to straddle it. He looked back, his blue-grey eyes as blank as his expression.\n\nCute and hot. Very hot.\n\nShe might not be human, but she _was_ female. Everything about this guy--from the blond velvet-like stubble on his scalp to the solid, ripped body the tight T-shirt hinted at--called to every feminine instinct she had. Worse, in the middle of a cruelly handsome face sat a perfectly straight nose and a sinful pair of lips that made even a reaper think wanton thoughts.\n\n\"The Brownies impressed me more though.\" Andy leaned back in her chair and studied him more closely. To his credit he didn't flinch. \"How'd you manage to waste the little freaks? They're worse than a bad dose of the clap.\"\n\nMr. Tall, blond, and less-than-chatty shrugged but didn't answer her question. Instead he nodded at the plate in front of her. \"Not hungry?\"\n\nThe voice didn't match the rest of the package. Andy had traveled the length and breadth of the land. She'd tracked and reaped virtually every creature within living legend and a fair few that weren't. The guy in front of her was human, but the voice. _Ohmygod,_ the voice was something else.\n\nIf Andy didn't know better, couldn't _see_ better, she'd swear he was a Vamp, or even a fae...some being with the ability to hypnotize with sound alone. Smooth as silk, it went down like a good whiskey, making her think of languorous nights in front of a roaring fire. Then the bite kicked in, like the burn of a good shot as it slid down her throat. Satin over a core of pure steel. Of all the creatures Andy had come across on the roads, her instincts warned her that this one, this mere human, was the most dangerous.\n\n\"Not particularly.\"\n\nShe met him look for look. Foolish perhaps but she found she rather enjoyed baiting him. She knew that, even at this moment, he was making the decision on her. One signal, and she'd be treated to the same fate as the Ghoul's whose blood had decorated the floor beneath her feet.\n\n\"Neat.\" She nodded to the items on the table. \"Silver for Weres. Splash of holy water in the glass by any chance?\"\n\nHe started, and Andy's lips quirked again. She'd surprised him. Lifting the glass she took a long swallow then put it back on the table pointedly.\n\n\"Just proves you ain't a Vamp. Take off the glasses.\"\n\nHer eyebrow winged up. There was no way to take that as anything other than an order. As a rule she didn't take well to orders...\n\n\"Tell me your name.\" She hooked a finger around the arm of her glasses and slid them down her nose to look at him over the top. Dark eyes met light, and her breath caught for a moment. Andy kicked herself. _Oh for heaven 's sake, get over it girl. What do you think this is...some kind of great romance novel?_\n\n\"Mason.\"\n\nOh my, the sparkling conversation was going to be the death of her. She just hoped his talents lay in...other directions. Her mind hit the gutter level as she wondered what all those tight muscles under his T-shirt felt like. She sighed, and tried to get her raging libido under control.\n\n\"Pleased to meet you, Mason. I'm Andy. You're not part-siren by any chance, are you?\"\n\nHe recoiled, disgust written over his features. \"I'm not part anything. I'm human through and through.\"\n\nAndy kept her skepticism to herself. There was something about him--she just couldn't put her finger on it.\n\n\"So...Andy. How about you? You going to drop the charade, or pick up the silver so we can see what you really are? I warn you though, Old Fred behind you is a crack shot with that sawn-off.\"\n\nAmusement rolled through her as she took her glasses off, and placed them precisely on the table in front of her. If Fred was going to shoot her in the back, then she'd rather not be wearing them. Good sunglasses were damn hard to find these days.\n\n\"A little unsporting, wouldn't you say? What happens if I pick the silver up, and nothing happens? Do I pass all your tests?\"\n\nHis gaze hardened. \"What makes you think you passed them all?\"\n\nA chuckle of amusement escaped her lips. She hadn't had this much fun in years. \"Let me see... Ash-wood in the door to bar witches...wouldn't stop a warlock or a sorcerer though. The door-step has an iron strip, and you'd be hard pressed to find a real living horse this far out so those...\" she nodded to the horseshoes behind the bar. \"...are definitely imports.\n\n\"I'm guessing you have another devil's trap by the door, but I can't see it.\" She scanned about the room, picking up each layer of protection she could see. \"Wall hangings to cover warding sigils and the ones you have carved in the windowsills are some serious mojo. I'd recommend adding a couple to stop Banshee's though...passed a band of them a couple of days ago heading this way.\"\n\nPicking up the silver fork she speared a lump of beef from the stew and popped it into her mouth, busting the theory she was a Were or a Ghoul. Silver burned the first, and the other couldn't stomach cooked meat. She looked at him in expectation for a second and grinned at his shell-shocked expression. \"Did I miss anything?\"\n\nHis surprise disappeared under a glower. \"What the fuck are you?\"\n\n\"Let's just say I'm not a threat to you and your town. In fact, I'm intrigued. Where did you learn all this stuff?\" She waved her hand, indicating the silver, the wards and the other protections in the place. \"I've never come across a place so well prepared.\"\n\n\"Playin' dangerous games, sweetheart,\" Mason bit out, holding his hand up in warning as someone in the shadows behind her cocked a weapon. \"People around here get jumpy when it comes to non-humans. I'll give you the same warning I did those Ghouls on the barbeque outside. Get your kit, and walk. You ain't human, you ain't stayin'.\"\n\nAndy rose to her feet. Below average in height, she'd long ago given up trying to attain the grace her mother had possessed. Finding out at the end of the world she was not only adopted but a different species altogether sure explained a lot.\n\n\"Can't say fairer than that I guess.\"\n\nSwiping a razor-sharp nail over the pad of a finger Andy watched blood well up from the small cut. Leaning forwards she drew a symbol onto the table in blood.\n\n\"What the fuck are you doing?\" Mason demanded, his gaze sharp as it flicked from her to the symbol and back again.\n\nAndy shrugged her pack on and picked up her glasses. \"Copy the ward, wash the table and do the same with the water as you did with the symbol on the gate. It'll keep those banshees out.\" With that, she slid her glasses on and walked out the door.\nChapter Three\n\nThe day had been a long one. Mason walked into his room, and shut the door behind him with a quiet click. Still fully clothed he rolled onto the bed, and closed his eyes with a grateful sigh.\n\nHe was exhausted. As normal, he'd been up before sunrise to join the work-parties that kept the town running. He should have been exhausted, but sleep was elusive. Ordinarily it was an insistent bed partner, often claiming him before he managed to shrug out of his clothes. Tonight it danced just out of reach. Every time he closed his eyes he saw Andy, the woman from earlier.\n\nHe ground his teeth, and squeezed his eyes together until fireworks lit up the back of his eyelids. Correction, she wasn't a woman. She was a paranormal. Not human, not a woman, no one he should be bothered about. A thing he shouldn't worry himself about. With those wicked-looking blades strapped crosswise on her back then no doubt she could take care of herself. For someone to carry blades instead of a gun was a statement all in itself.\n\nIt said that the carrier was either a) stupid and about to buy the farm or b) they were good enough to get those blades into anything they came up against. To do that they'd have to be faster than a Werewolf, able to hear even a Vampire creeping up, and be able to outwit a shade. She'd looked too human, and way too cute, to be capable of half that.\n\nMason groaned as images of her lying in the darkness, her skin pale as she bled out, filled his head. His male protective instincts, rusty from inactivity, rushed to the fore. He shouldn't have let her leave. Unlike other non-humans, she hadn't offered them any harm. She'd even drawn them a ward...and as soon as he'd looked at it, he'd known she was telling the truth. The thing had hummed with power.\n\nHow he knew stuff like that, he had no idea. He just did. Since the war he'd been able to spot all manner of paranormals, and magic users, under their disguises. Not just non-humans either. Souls black with sin...that was the saying. He'd always expected a soul ready for hell to be as black as pitch, rotten through and through. They weren't though. If they were human, they were bright silver. Paranormals, as always, were different. Their souls were a whole rainbow of different colors. Amber for wolves, red for Vampires, black for the undead.\n\nHis eyes snapped open as he ran through his memories again. He'd done the usual cursory check to see if he could detect anything hidden behind the pretty face and had come up with nothing. Looked human on the outside, no monster lurking inside to leap out like a freaky jack-in-the-box. Still, Mason's instincts had warned him there was something not right about her, and now it hit him.\n\nHer soul had been black.\n\nAt three forty-five he gave up on sleep, and rolled out of bed, sat on the edge of the narrow cot and rubbed his eyes. They were like piss-holes in the snow--gritty and hot. He cast a baleful glance back at the lumpy mattress. He needed a bottle of whiskey, fourteen hours sleep and a new mattress. Preferably in that order.\n\nThe stink of sweat assaulted his nostrils. Mason grimaced and risked a quick sniff at his armpits. Shower time for sure. He walked out of his clothes on his way to the bathroom. Even here, in the safety of his own room, his rifle was the last thing to leave his hand. Propping it up against the wall beside the door Mason switched the water on.\n\n\"Jesus _fucking_ Christ...that's cold.\"\n\nTeeth chattering, he forced himself under the spray of frigid water. This time of morning, he had no chance of it warming up. The town attracted a lot of waifs and strays so they had most occupations, from a former hairdresser to a Hollywood gynecologist, but could they find a damn plumber? Until they did, the ailing boiler was only run for short periods, and carefully baby-sat in case it decided to go critical.\n\nPity it couldn't do that with Jed and his pathetic pack of dogs in the room. He shook his head as he scrubbed down quickly with the hard soap produced in town. He still could not believe that animal had had the audacity to send one of his mutts into town, and demand a tithe.\n\n_\" Five women, old enough to fuck and not too old to have kids. So we don't want any dried up old-grandma's. If the tithe is suitable, your town will be spared our wrath.\"_\n\n\"Spared our wrath.\" Mason snorted, washing his sack and crack with quick economical movements. Who did the guy think he was? He didn't give a flying fuck about Jed's wrath. He tried anything in Mason's town, and he'd end up like every other para that tried it...with Val scraping his brains off the bar's floor.\n\nFinally he stepped out of the shower and grabbed the thin towel draped over the rail. Like everything else here it showed signs of hard use. He dried off quickly and dropped the towel and his dirty clothes in the washing basket.\n\nThe clean ones he put on were nearly identical. He pulled an old T-shirt over his head, covering the multitude of scars that marred his torso, and grabbed his combat pants. Once they'd been black but now they were a faded grey. Clean socks went on next, after he gave his feet a quick check.\n\nThey said an army marched on its stomach, which was true. What was also true was that a soldier never neglected his feet or his boots. He laced up quickly and tucked the ends back into the top of the boots. Pulling his pant leg up, he strapped his knife to his calf. A legacy from his army days, it was Mason's last line of defense. If he ever had to draw it, then the shit had hit the fan in a big way. Shrugging his shoulder holster on, he grabbed his rifle and headed for the door. If he hurried, he might be able to catch the morning hunting party.\n\nAn hour later, Mason was walking point. The small group of hunters fanned out behind him. Their faces were grim and professional. A sense of pride filled him. In this, at least, his former life had been of some help. Whenever they left the town to hunt, they took their lives into their hands. Out here, things fell into one of two categories--things that ran away from them, and things to run away _from._\n\nHe pulled his rifle tighter into his shoulder, steps soundless as the group walked. Concentration wrapped around them so tightly it was like a cloak. Every gaze was alert to the smallest movement. The only major cover for half mile or so, this was where they were likely to find either game or become it.\n\nTension wound through his frame as he paused, raising his arm with a clenched fist. At the signal the group stopped, and assumed kneeling positions facing outwards from the group. Mason crouched to study the tree line.\n\nAll was quiet. There weren't many birds around these days, anything bigger than a sparrow was an immediate candidate for the stew pot. But this was _too_ quiet, as though nature herself was holding her breath, and watching the scene around the copse unfold.\n\nWay too quiet.\n\nDecision made, he stood and signaled to the group to skirt around. The likelihood of bagging some sort of game within the trees was good, but the risk was too high. They had a few hours left to find something elsewhere. He wouldn't risk good hunters.\n\nHe sighed softly as they moved out. When the scientists had unleashed hell on earth, and changed the fabric of reality, why couldn't they have created cows the size of trucks? He'd happily trade in Ghouls and wolves for a guaranteed good meal. And steak...he had re-occurring fantasies about a good steak.\n\n\"Keep it tight.\"\n\nHis order was low as James, one of the newest hunters, edged into his line of sight. Not a place he wanted to be when Mason was on patrol. Anything that moved was getting shot on the basis that it was either food, or it was hostile.\n\nThe patrol continued in silence as they walked their pre-arranged route. Frustration began to mount in his chest as they walked further and there was still no sign of prey. Pickings were getting increasingly slim. Each hunting team came back in with smaller and smaller catches until recently, when most times they came back with nothing at all.\n\nIf they didn't bring something back in today Val was going to have to dip into the canned goods secured in bar's kitchen. That was something he tried to avoid. Canned goods would last for years. They were his fail-safe in case the whole situation went tits-up and they had to move out of town. On the road, they would be too busy defending themselves to risk hunting for food.\n\nUnease itched in the space between his shoulder blades. It started off a small itch, then grew and grew until his shoulders were tight with tension. Someone...no, some _thing,_ was watching them. Mason'd bet his last bullet there wasn't just one of them either.\n\nBefore he could alert the team, they were attacked. Hard and fast, everything happened at once. \"Contact. Full right, six o'clock,\" he yelled, rifle on his shoulder and already squeezing rounds off as the first Werewolf burst out of cover to his left.\n\n\"Weres!\"\n\n\"No shit, Sherlock,\" he muttered as his double tap took the first wolf between the eyes and splattered its brains over the one right behind it. The second jumped over the carcass, red eyes fixed on Mason as it snarled. Saliva dripped from its yellowed teeth as it crouched low, stalking him.\n\nHe aimed and fired. His bullet ripped through its right front leg, shattering the joint. Squealing in pain it crashed to the ground, and tried to shift to heal the massive damage to its leg.\n\nMason had seen Weres change before, and it turned his stomach. All that bone snapping and changing, not to mention skin that melted and reformed to the new shape. Bile rose in his throat as he approached the creature where it writhed on the ground.\n\n\"Down boy.\" He lifted the rifle and put a bullet through its reforming brain.\n\n\"Keep in. Controlled bursts, conserve your ammo,\" he yelled over the sound of firing as the small group of humans clustered back to back for protection. The wolves circled them, flitting in and out of cover.\n\nThey'd already lost Julian. The wolves had gotten to him within seconds of the initial attack. He snorted. The stupid cunt had tried to play the hero, ignoring all orders in favor of doing his own thing like some kind of post-apocalyptic Rambo. These days you didn't want to be the hero. Playing the hero just got you dead. Fast.\n\nThe kid had deserved to suffer for the stunt he'd pulled, but Mason had done the decent thing and put a bullet between his eyes when the wolves had ripped into his stomach. No one deserved to be breathing through something like that.\n\nJulian's body, or what was left of it, lay off to the left. His abdomen was torn open, his intestines strewn around him like the stuffing out of a battered teddy bear. Steam rose off the slimy red tubes in the cold morning air.\n\nSausages, Mason thought absently as he rattled off a couple more rounds at a Werewolf that dared to poke its head over the top of an old concrete pipe. He hadn't had sausages in years. He could still remember how they tasted. Little bites of pure, fatty pleasure that burst on the tongue.\n\nNext to him, Julia kept a sharp eye out as he reloaded. The drill was smoothly executed, and instinctive. It was his last magazine. He didn't need to ask to know they were all running low on ammunition.\n\nWhat the fuck where they going to do when the bullets ran out?\n\n\"Single shot,\" he ordered. \"Make them count. If you can't get a head shot, blow a leg out. I don't care how, but I want these fuckers on the ground. If we're going down, we're taking them with us.\"\n\nThe bushes around them rustled as the wolves closed in. Mason knew they were closing the net. Within minutes, they'd launch their final attack and the humans were screwed six ways to Sunday.\n\nAhead of him, an ear poked up over one of the concrete tubes the bastards were hiding behind. Mason grinned. It was less an ear, and more a furry suggestion of an ear. Disregarding his own order Mason aimed and squeezed the trigger, and grinned at the squeal of pain and fury that emanated from behind the pipe.\n\n\"What was that?\" He cupped his ear as though listening hard for something. \"Sorry you flea-bitten mutt, you'll have to come a little closer. I can't _hear_ you!\"\n\nThe group behind him chuckled softly. Within seconds, a low rumble overwhelmed the sounds of mirth, a rumble that coalesced into vicious snarling. He centered himself. This was it. They were about to die. He knew that, the men and women with him knew that. Miracles didn't happen. Not anymore, not for anyone. Mason rolled his shoulders, checked his safety catch was off and waited for furry vengeance.\n\nToday was a good day to die.\n\n\"Incoming,\" he yelled as the wolves swarmed out of cover. Gunshots sounded around him as battle was joined. He emptied the last of his rounds into the face of a wolf stalking towards him and dropped into a crouch to yank the big blade from the sheath on his calf.\n\n\"Come on, you bastard. Come get me.\" Mason's voice was thick with fury as he faced off against a lean, pissed-off-looking wolf with a tattered ear and murder in his amber eyes. Mason grinned, showing all his teeth. \"Let's see if I can't make that other ear match...\"\n\nThe wolf curled its lip back, and snarled a low warning of pain and terror to come. As it bunched powerful legs underneath its body, Mason prepared for his last battle. Adrenalin sang in his veins. There was nothing like the imminent threat of death to make a man feel alive. He felt no fear. In fact, he didn't feel anything at all. Except a small measure of regret about kicking Andy out of town last night without even trying for so much as a kiss.\n\nThe wolf lunged at him. Mason was quicker. He sidestepped as the creature rushed him, letting its momentum and weight carry it past him. As it did, he stepped back into its side, easily avoiding the slash of vicious teeth and landed a solid back-fist on the side of its skull. The wolf howled as Mason grimaced in pain. It was like punching bloody granite.\n\nThe creature turned, and Mason knew he was done for. The group was scattered, wolves closing in on each of them. Julie, his fire team partner screamed as a wolf tumbled her to the ground, standing over her and slowly licking her face.\n\n\"Come on then. Come and get me.\" Mason's lip curled back as he snarled. \"I hope I give you the shits.\"\n\nThe wolf grinned, eyeing him up as though deciding which tasty portion of Mason's anatomy he was snacking on first. Mason's grip tightened on the blade in his hand. No matter where the creature struck, his knife was going through its heart.\n\n\"Want a hand there, handsome?\"\n\nThe seductive voice took him by surprise. Andy. He slid a glance sideways, trying to look at her and keep an eye on the wolf too.\n\n\"What the hell are you doing here, woman?\"\n\nShe was already injured. Blood coated one of her arms, soaking through the light material of her T-shirt. The bullet hole at the shoulder and the tattered skin beneath told him the blood was hers. \"You'll get hurt.\"\n\nMason wasn't quite sure how the events of the next thirty seconds unfolded. As the snarling Werewolf launched itself towards them he tried to shove her out of the way, and protect her with his own body. Only she wasn't there anymore.\nChapter Four\n\nGrim determination flowed through Andy's veins as she stepped towards the Were. Her non-human instincts were in full force. Its lifeline was bright red, ready to be reaped. Trouble was, the soul was still firmly embedded in the body, and she knew from the look in its eye that it wasn't giving it up without a fight.\n\nShe grinned. Just the way she liked them.\n\nTime slowed to a crawl as the creature barreled towards her. She dropped to one knee, the other leg stretched out for balance, as it sailed over her. Spinning the sickle in her right hand she sliced across its belly. The contents of its abdomen evacuated the premises in a torrent of blood and heavier things.\n\nAndy ignored it as it thudded to the ground to twitch its last. The soul and the body were separate, and that was all she cared about. Most corpses tended to twitch a little before they realized they were dead.\n\nShe turned her attention on the other wolves. Some of the humans were already dead--there was nothing she could do about that. All she could do was protect those that still lived. Muscles filled with the power of her calling, Andy swept through them like a hurricane.\n\nHer blades sliced and diced, ripped and danced, as she caught fur and bone alike, slicing through the body to catch the soul inside.\n\nWithin seconds she stood in a pile of lupine bodies. Her eyes were flat and unemotional as she watched the remaining wolves beat a hasty retreat. She didn't blame them. Vamps and Weres--both thought they were the top of the food chain. Did them good to come across something more powerful...something they couldn't beat, couldn't kill.\n\nSlowly she became aware of her ragged breathing, and the pain streaking like wildfire down her arm. She wrinkled her nose as she looked at her shoulder. \"Dammit...\"\n\nThe bullet had ripped through the fabric, and from the feel of it, was still lodged in her flesh. Great, surgery before lunch. She sighed. She needed a vacation. Somewhere hot. She'd heard Hawaii was nice this time of year.\n\n\"Here, you're bleeding. Let me.\"\n\nA voice at her side made Andy jump a little. She turned and looked up into warm eyes. A smear of blood marked his cheek, but Andy was caught instead by the small laughter lines at the corners of his eyes. He must have smiled a lot at some point. The desire to get him to smile at her was nearly overwhelming.\n\n\"This? It's just a scratch.\"\n\nShe flinched as he slapped a field dressing over her arm and applied pressure. To her, it was little more than a hole in her skin. She wouldn't call it a wound. Such a name gave it an importance that was unwarranted. Whatever she did--dressed it or allowed it to bleed--nothing untoward would happen. No matter how much blood she lost, she wouldn't bleed out.\n\n\"Just a scratch, huh?\"\n\nHis lips quirked into a half smile. The expression transformed his harshly handsome face into something that took Andy's breath away. _Oh, for God 's sake girl, get a grip. He's a good-looking guy and you're an intelligent woman. No need to start thinking with your damn ovaries._\n\n\"I suppose if you lost your leg, you'd claim it was a mere flesh wound, eh?\"\n\n\"Something like that.\"\n\nHer smile turned into a hiss of pain as he pressed hard onto her arm. It hurt like a bitch, but her reaper physiology only needed a little help. Already the bleeding had stopped, and she could feel the skin starting to close. She could feel something in there, which meant she was going to have to open it up later to get the damn thing out. Trust her to get hit by the only shotgun in the group, just her luck.\n\n\"How's it feel?\" His pale eyes studied her with a perception Andy found disturbing and thrilling at the same time. It was like he could see right through to her soul. If she even had one...the jury was still out on that. Still she had a feeling that, of all the people she'd come across in the last ten years, Mason was the one who would see past her human disguise to the real woman beneath.\n\nShe wiped her blades off on the nearest furry carcass with swift swipes and sheathed them with efficient movements. They slid back into place with a satisfying click. Waving his hand away, she gingerly peeled back the field dressing to look. As she'd suspected the ragged tear in her skin was closed and fresh, pink skin had taken its place. She flashed him a smile as she dropped the dressing back into place.\n\n\"Good as new, thanks.\"\n\nMason arched an eyebrow. \"What, no pain? Feeling dizzy, anything like that?\"\n\nShe gave him a long look. \"Not human, remember?\"\n\n\"Hmm...you still bleed like the rest of us.\"\n\nHis voice was low as he looked around the small group. They were scattered around the scene of their showdown with the wolves. A couple just lay on their backs, staring up at the sky as they dragged harsh breaths of air into their lungs.\n\n\"Yeah. So do wolves.\"\n\nMason frowned, looking at the small pile of lupine bodies in front of them. \"That's odd. They normally turn human again when you kill them.\"\n\nSurprise filled Andy, but she hid it as she looked sideways at him. She'd under-estimated him. Again. After seeing the way he had the town set up, she should know better. \"They do. Well spotted. I take it you've killed a few in your time?\"\n\nHe nodded. \"Yeah. Spent a couple of years on the road before settling down. Traveled with a witch for six months...seen just about everything that's out there.\"\n\nA witch. It made sense now. The defenses at the town weren't random--they were planned with experience of protecting such a location.\n\n\"I saw. Nice touch with the devil's trap on the main gate. Almost didn't spot that. Good setup, I'm guessing you're ex-military?\"\n\nSurprise filled his eyes. \"You saw that? It's invisible.\"\n\n\"To a human maybe. As we've already established, I'm not human.\"\n\n\"Good point. What are you?\"\n\nBlunt and to the point. She liked that in a guy. The smile that had been trying to break free for the last couple of minutes spread across her lips. She tapped the side of her nose. \"That's for me to know, and you to find out.\"\n\nMason looked at her for a long moment, the corners of his lips twitching suspiciously. Then, as though he couldn't hold it in any longer, he burst into laughter. \"Oh, I intend to, sweetheart, I intend to.\"\n\nAndy's heart, an organ she'd long thought dead, skipped at the warm smile and flirtatious look on his face. Oh hell, this man could charm the birds from the trees if he wanted. A second later though, it was gone as he flicked a glance down at her arm. His jaw tightened, and she could sense the inner battle raging from the tenseness of his body.\n\n\"You should come back with us. You know...in case that gets infected.\"\n\n\"Breaking your own rules, Mason? _Tsk_ , _tsk_. Shame on you.\" She softened her refusal with a smile. \"I'll be okay. You get your guys back to safety before the rest of this lot come looking for their friends.\"\n\n\"Of all the stubborn, mule-headed, bloody stupid...\"\n\n_Bloody women. Can 't live with them, can't live without them and most of the time they drove normal, sensibly minded guys completely insane._\n\n\"You aright, hun?\" Cleaning tables on the other side of the room, Valerie paused and looked up at his outburst.\n\nHe grimaced as he leaned back in his chair. Maps of the local area spread over the table in front of him, highlighted by a shaft of sunlight from a nearby window. He watched, his thoughts in turmoil, as dust-motes danced in the sun.\n\nHe couldn't concentrate. All he could see was the blood running down Andy's arm. She was paranormal. What type Mason didn't know--his usual acuity in spotting what a person really was seemed absent in this particular case. Whatever she was, so far she hadn't harmed any of them.\n\nIn fact, she not only had _not_ harmed anyone in town but she'd come riding to the rescue when they were in trouble like some kind of white knight. Mason sighed and ran his hands across the short stubble on his scalp. The way she moved... Pure lethality and grace in motion. Like a combination of a ninja and the hottest super model he'd ever seen all rolled into one uber-sexy package. His perfect woman...and he'd let her go injured into the wilds alone. What kind of fucking idiot was he?\n\n\"Nothing. Just a little tired, that's all.\"\n\nValerie looked less than convinced. Giving the table in front of her a last swipe with her cloth she picked up the bottle of cleaner and headed towards him.\n\n\"Mister-I-don't-need-sleep-invincible-Mason-Callahan is tired?\" She plonked the bottle and cloth down on the table and flopped onto the seat in front of him. \"Yeah, right. Try something believable...like...you moonlight as Santa Claus. _That_ I might believe.\"\n\nHe gave her a blank look, but the expression on her face said she wasn't having any of it. He swore. The vicious curse did nothing but elicit laughter from the blond bar-keeper.\n\n\"It's that girl, isn't it? The one that was in here last night. What was her name...Andrea or something?\"\n\n\"Andy,\" he replied begrudgingly. No point in trying to hide anything. If V was confronting him about it, then she already had the answers. Just like a woman to work that way, and V was as manipulative as they came.\n\nHe sighed and rearranged his maps for the tenth time since opening them up. He was trying to plot a route to the nearest town. Something that wouldn't leave them in the open too long, was on the least blocked highways and avoided both the local Werewolf packs and Vampire nests. That just left the other stuff to worry about--wandering Were-packs, nomadic creatures like banshees and plain weird shit he didn't have a name for.\n\n\"Andy. I knew it was A something. Julie says she was out there today.\" V ran her finger along a deep grove in the tabletop. \"Said you lot would have bought the farm if she hadn't turned up.\"\n\nMason just nodded.\n\n\"Had us surrounded. Must have been stalking us for a while. I just didn't see it. Julian broke formation, so they got to him first. They were about to take us all out when she showed up. Took on three herself, like there was no stopping her.\"\n\nHe shook his head, still amazed at what he'd seen. \"The rest ran. Like they were scared shitless of her. Don't blame them, I would've been too.\"\n\nValerie looked up, her expression serious. \"So, what do we think? Vampire?\"\n\nHe shook his head. \"Not the right vibe for a Vamp.\"\n\n\"Something worse?\"\n\n\"Worse than a Vamp?\"\n\nMason chuckled, but Valerie's words struck a chord deep within him. There _was_ something dangerous about Andy. Something Werewolves ran from, something he was sure would scare the un-dead crap out of Vampires too. It was the same thing that called out like a siren to him.\n\nHe stood in a lithe movement and gathered his maps.\n\n\"She helped us, and she got hurt. Wouldn't come into town because of our rules. That doesn't sit right with me. Human or not, the last thing anyone needs out there is to be bleeding from a fresh wound. I'm gonna go look for her...she's on foot. Can't have gotten far.\"\n\n\"Fucking... _hell! \"_\n\nThe sound of Andy's curses reverberated back at her, echoed nicely by the concrete of the old bridge she'd made camp under. There was a small tent in the backpack next to her sleeping bag but she rarely bothered to unpack it unless the weather was crappy.\n\nMore curses spilled from her lips as she glared at the bloody mess that was her shoulder. Small knife in her hand, not one of her sickles, she squinted and tried to spot the pellet she knew was in there.\n\n\"Stupid cow. You should have dug it out there and then, not let it heal over.\"\n\nGritting her teeth, she probed with the knife again. Just because she couldn't die didn't mean she couldn't feel pain. Right now, she was feeling shed-loads of it. Her blade scraped against bone, sending razor sharp needles of pain through her, and twisted her gut into a cat's cradle of bile and nausea.\n\n_Not throwing up, not throwing up, s_ he chanted the mantra in her head as she tried to force the feeling back down. She hated being sick. Hated it with a passion. Always had, and always would. Leaning back against the rough concrete wall, she tried to use the cold surface to leech some of the heat from her body as she waited for the feeling to subside. By slow degrees it did, until she could think straight again. Grimly she gathered the courage to try again. Finally she lifted her head to study the edges of the wound again.\n\n\"What the _hell_ do you think you're doing?\"\n\nAndy jumped, and lost her grip on the knife.\n\n\"Oh bollocks! Look at what you made me do,\" she snapped, searching in the folds of her sleeping bag for the bloodied knife. It came away from the fabric with a wet smear that made her wince. \"Great, that's gonna need sponging off or the flies will have a field day. What the hell are you doing out here anyway? Thought you'd be tucked up all nice and safe in your little town?\"\n\nMason stalked into the circle of light cast by the small fire. \"Looking for you, that's what. Wanted to make sure you weren't doing anything stupid...like trying to carve up your arm with a god-damned butcher's knife!\"\n\nShe frowned at the blade in her hand. Sure, it was a little on the long side but it was the only straight blade she had. Her sickles were sharper, but knowing her luck, she'd probably amputate her own arm.\n\n\"Huh? This thing? It's nowhere near heavy enough for a butcher's knife.\"\n\nIgnoring him as he squatted in front of her, she dug the knife into her arm again. The pellet was still in there...she could feel it.\n\nAfter her second bout of swearing Mason reached out and plucked the knife from her hand. \"Here. Let me. I'll make less of a mess than you are. You trying to end up with a scar?\"\n\nAndy shrugged. \"Wouldn't make a difference. I don't scar.\"\n\n\"Yeah, right. Everyone scars...only things that don't are Vamps. And you aren't a Vamp. I might be human, but I'm not stupid.\"\n\nHis voice was amused as he studied her arm. Light from the fire behind him caught the tips of his cropped hair, casting the hard plains of his face into shadow and giving him a fiery-red halo. She leaned her head back and closed her eyes. Felt nice to let someone look after her for a change.\n\n\"I could be a day-walker Vampire.\"\n\nHe chuckled. \"Sorry sweetheart. This is reality, not a Hollywood blockbuster. Only Vamps we got are the ones who don't tan well.\"\n\nFingers moving gently, he probed the ragged edges of the wound in her arm. Then he paused. \"This looks different. What have you been doing to it?\"\n\n\"Doing to it? You mean, other than digging around in it with a sharp implement?\"\n\nIt was sarcastic and she knew it. But Andy couldn't help it. She'd always had a sharp tongue and spending years traveling with only herself for company...that was bound to warp even the healthiest of minds. She didn't want to think too closely on her mental state. If a psychiatrist assessed her, she was sure the words homicidal and possibly fruit-loop would feature heavily in their report.\n\nHe shot her a look.\n\nAndy grinned, unrepentant. \"What? Ask a stupid question, get a stupid answer. Owww! Lay off with the squeezing, would you? That bloody hurts.\"\n\n\"Cry-baby. It's just a scratch. Remember?\"\n\nHe ignored her glare with an innocent expression and carried on manipulating around the wound. Blood oozed thickly down her arm.\n\n\"It _was_ just a scratch. Earlier. That was before I opened it up to dig a hole the size of freaking China to find the pellet one of your friends shot me with.\"\n\n\"It's the wrong shape.\"\n\n\"What do you mean it's the wrong shape?\"\n\nAndy winced and tried to wriggle away as he put more pressure on her upper arm and dug in with the knife. Bloody hell, with friends like this, who needed enemies? Pain lanced down her arm as he went after the pellet. Fire and ice shot through her body and brought a slick of sweat to her skin. She clamped her jaw shut to keep from crying out until she was sure the pressure was going to crack her teeth.\n\n\"Ahh, there it is.\" Mason sat back on his heels, the small pellet between his bloodied thumb and forefinger. With satisfaction, he threw it into the fire.\n\n\"It's the wrong shape for China. So, what do you mean by you opened it up? Had it already scabbed over?\"\n\nShe recognized a leading question when she heard one. Humans healed slowly but still, most paranormal races would show some sign of such a recent injury--a healing wound, or a scar--something.\n\n\"Not scabbed over, no.\" She paused. Was she ready to tell him what she was? He already knew she wasn't a Vampire so she couldn't give that excuse. Was being the physical embodiment of death worse than being a blood-sucking fiend from beyond the grave? What the hell...he already knew she wasn't human. \"It had healed over.\"\n\nThe silence between them stretched out. His gaze locked with hers, and she couldn't look away. Even if the bombs had fallen all over again, she wouldn't have been able to move a muscle.\n\nHis expression was level and emotionless, apart from his eyes. His eyes blazed with suspicion and a deep, dark something Andy wasn't sure she was reading right. If she didn't know better, she'd say it was...interest?\n\nThat couldn't be right, even when she'd been human men had never found her interesting, much less attractive. Too short, too skinny, pale skin, dark hair. An all-round plain Jane.\n\nGreat. The first guy she was interested in and who just _might_ be interested in her, and she was about to tell him she was the female version of the Grim Reaper.\n\nThe battle continued to wage in his eyes, the knuckles of the hand that held the knife white with pressure. His voice was tight and contained.\n\n\"I'll ask again, for the last time. What are you?\"\n\nMason's heart did a tango in his chest as he waited for her answer. He'd come out to look for her because he'd been worried. Concerned about her safety out here all alone and injured. Concerned about a woman who'd taken down three fully shifted Lycan's without so much as breaking a sweat.\n\n_Well done, Mason. That 's what thinking with your prick does for you._\n\nHer dark eyes flicked to the blade in his hand. One by one he forced his fingers to relax. She might be about to tear his face from his skull but he felt a strange need not to scare her. She was small, female...and that delicate curl in the curve of her neck was driving him nuts. Of all the women he'd met--why did the only one that sparked an interest in him have to be a paranormal?\n\n\"You sure you want to know?\"\n\nHer arched eyebrow was a challenge he couldn't ignore. Throwing caution to the wind he flicked his wrist and half threw-half thrust the knife into the dirt by his knee. With the speed she'd displayed earlier, no knife was going to help if she wanted to hurt him.\n\n\"The truth,\" he said firmly. He'd had enough of being given the run around by this one. However pretty she was, with her masses of dark hair and soft lips that made a man think wicked things, he had a limited supply of patience.\n\nAgain that maddening half smile flirted with the corner of her lips. \"The truth? You can't handle the truth.\"\n\nAmusement rolled through him at her quick-fire reply. For a moment all of it fell away--the war, the hard fight for survival since, the fact the town was running out of food. For a few blessed seconds he could forget and just enjoy being in her company.\n\n\"You ever going to give me a straight answer, or just more movie quotes?\"\n\n\"Depends.\" She grinned, a quick flash of teeth against her pale skin.\n\nHe knew she wasn't a Vampire, but still Mason did a quick dental check. No elongated canines or anything that could be remotely described as a fang.\n\n\"Depends on what?\"\n\nAt his words the atmosphere between them changed. Her eyes darkened and he was stuck, caught in a web of fascination. Heat and smoke coiled in puppy-dog brown, changing her eyes from pretty to breathtaking. Need and desire hit him low down in the gut, sending shivers along every inch of his skin.\n\n\"On whether you kiss me or not.\"\nChapter Five\n\nThe bold statement threw a shudder along Mason's spine. His gaze dropped to her lips. Plump and perfect with a slight sheen across the lower one. A cupid's bow of temptation. He shouldn't, he knew that...there were a thousand reasons why he shouldn't.\n\n\"Is that so?\"\n\nHe leaned forwards by slow degrees, teasing them both with anticipation. Her breath caught as she went still, her gaze locked on his mouth. His lips brushed hers. An introductory touch that whispered over the surface of her skin.\n\nShe moaned. A soft sound in the back of her throat, but it fuelled his male pride. With a groan he claimed her mouth fully. Crushing her lips under his, he dominated the kiss, demanding a response from her.\n\nHis hand delved into the thick hair at the nape of her neck and held her head still as he explored every inch of her lips. _This is heaven_. Her lips were soft and seductive, clinging to his as she wrapped her arms around his neck. He'd heard flowery descriptions from old books about this moment...about women tasting like champagne and strawberries and other such nonsense.\n\nAndy didn't taste like that. What she did taste like was indescribable. Like a taste that extended through his whole body and affected each part of him from the top of his head right down to his toes, and certain portions of his anatomy in between. Bollocks to champagne and strawberries--she tasted of heaven and coming home.\n\nA taste he wanted more of.\n\nWrapping his arms more securely around her he pulled her forwards onto his lap. She whimpered again, that little half-moan in the back of her throat that called out to everything male within him. Sweeping his tongue over her lower lip he demanded entry. Like a flower opening to the sun she let him in, and then it was Mason's turn to bite back a moan.\n\nHe slid his tongue against hers, teasing and tempting, and then, when she responded to him, he took control again. His free hand roamed down her back, and the part of his brain that was functioning noted how well she fit against him. Her slender curves were a perfect match for the harder plains of his body.\n\nWithin seconds the kiss wasn't enough, was never going to be enough. The need to roll her under him and slide deep into her welcoming softness almost overwhelmed him, and banished all other thoughts. The crotch of his jeans tented, the savage ache there reminding him just how long it had been since he'd been with a woman. How long it had been since he'd even _wanted_ to be.\n\nGentling his movements with an iron control he didn't know he had, Mason lowered her to her back and stretched over her. One large hand looped about her wrists and he pulled them slowly over her head, alert for any sign of discomfort.\n\nHer breathing caught, a delightful little hitch as her eyes darkened another notch. Her back arched as he held her hands captive. Her breasts brushed against his chest, and caused a cascade of fire to roll down his spine.\n\nShe was smaller than he'd thought. Now, with that sassy tongue quiet and her eyes watching him with an unreadable expression, he realized just how delicate she really was.\n\nA warm breeze gusted over them, catching a stray curl and blowing it over her face. Slowly, gently, he reached out and brushed it back. The look of wary trust in her eyes nearly unmanned him.\n\n\"If you don't want this, say something now.\" His demand was raspy and rough-edged, but he didn't care.\n\nSwooping in, he claimed her lips again, letting go of her hands. He didn't know, didn't want to think, what he was going to do if she said no. Okay, so that was a lie. Somehow he would find the strength to release her from his arms. He'd never forced a woman before, and there was no way he intended to start.\n\nHer lips pursed in disappointment when he pulled away to look at her.\n\n\"As I recall I asked you to kiss me.\"\n\nShe stroked a soft hand up his arm, finding the edge of the worn T-shirt and sliding under it. He sucked in a ragged breath. He was harder than he could ever recall. What was it about her that drew such a reaction from him?\n\n\"Yeah, but a kiss isn't the same as...um...\"\n\nJust like that Mason found himself speechless. What the hell did he say next? Having sex...fucking...doing the dirty? Making love? Startled, he damped down on that last thought, not sure where it had come from. He'd known her all of forty-eight hours. Love didn't come into it in any way, shape, or form.\n\n\"Mason?\"\n\n\"Yeah?\"\n\n\"Shut up, and get your kit off.\"\n\n\"Yes, ma'am.\"\n\nThe speed Mason scrambled to obey her softly spoken order massaged Andy's feminine ego. Within a heartbeat he hauled his shirt over his head to reveal a muscled and scarred torso that made her mouth water. His T-shirt had barely hit the ground before he started on his belt buckle.\n\nShe chuckled softly. \"Slow down, tiger. We have all night.\"\n\nHis second boot thudded into the dirt next to the first. Andy paid it no mind as Mason crawled towards her. The inferno in her veins threatened to engulf her, burning everything else away apart from the arousal that clenched her stomach and made her ache.\n\nHer skin was too hot and tight. Pulling at her top she fought down the urge to tear it away from her. She needed to feel the cool night air against her skin, needed to cool the fires that raged through her at the look in his eyes.\n\n\"Oh, I know. I'm counting on it.\"\n\nHis voice was a silken whisper as he reached her. Then he was in her space, something she guarded jealously, not allowing anyone within it. He didn't seem to care, crowding her with a dominance, which took her breath away. \"I'm planning on using that time to explore you. Taste you...and make you scream my name.\"\n\nA soft moan of need and arousal escaped her before she could stop it.\n\nHis hands bracketed the sides of her head, capturing her between the hard ground under her and his hard male body above. His nostrils flared as a tiny muscle in his jaw pulsed tightly. He wanted her but still he held back. The air around them swirled with sexual tension, yet he was taking things slow rather than falling on her like an animal.\n\n\"If you want me to stop, say something,\" he begged hoarsely. \"Because if you don't say something in the next two seconds, then it's gonna be too late.\"\n\nShe smiled--the mysterious little smile of a woman who knows she has her man. Arching her back she let her breasts brush against the hardness of his chest. He was lean and solid, with the kind of muscle that said he didn't spend his days sitting comfortable at a guard post. No, he had muscles that could only have been built by hard work and hunting.\n\nThe scent of warm, clean man filled her senses. Andy lifted her hand and trailed her fingers down his chest and then beyond to explore the ridge at the front of his jeans. He shuddered and closed his eyes.\n\nShe smiled, enjoying her sensual hold over this powerful man. Again that disturbing feeling, that he was more than human, came back to haunt her. She squashed it in favor of dipping the tips of her fingers under his waistband.\n\n\"How about you quit talking and make good on your promises?\"\n\nHis lips crashed down on hers. Without thinking she opened up, allowing him access. His tongue thrust past her lips, and he kissed her like there was no tomorrow.\n\nTheir hands were everywhere, stroking and caressing, touching and teasing. Clothing fell away as though it were no more substantial than mist. His skin slid against hers and she was lost. They moved in concert, part of a dance as old as time yet made new all over again because this was _their_ dance. Heat and need welled up from her core as his hair-roughened knee slid between her thighs. They parted automatically, his weight cradled by her hips.\n\nHer teeth worried at her lower lip as he moved. The broad head of his arousal brushed against the entrance to her body, slick and ready for him. With a moan that was echoed a second later by hers, he sank into her welcoming warmth.\n\nSleep receded in comfortable waves and, for the first time in years, Andy woke feeling refreshed. Grunting in complaint as the chill dampness of the morning air whispered across her exposed cheeks, she snuggled back against the warm, male chest behind her. Waking up next to someone was a novel feeling, something new, but she liked it.\n\nShe hadn't woken up next to anyone for a long time. Sure, she'd had lovers...she wasn't an innocent, no matter how people decided to interpret her looks... There was no way she'd have trusted any of them enough to sleep next to them though. She knew better than to put herself in such a vulnerable position.\n\nSome people said having sex was the most vulnerable position a woman could be in. That allowing such intimate access to her body left her open to all kinds of emotional and physical violence. Andy disagreed. If someone was going to hurt you then they were going to do it anyway. Or at the least, try.\n\nAwake, you could fight back. You could see the situation developing and manage it to the best of your ability. Anyone that couldn't see a violent situation brewing these days quite honestly didn't deserve to live. It was survival of the fittest. Asleep a person, male or female, was truly defenseless.\n\nA solitary bird chirped the dawn chorus with entirely too much enthusiasm. Andy yawned and opened her eyes. The campsite was a mess. Clothes and boots lay in a tangled mess, thrown willy-nilly in their passion last night. The only things laid out neatly within easy reach of the sleeping bag were their weapons.\n\nShe wriggled again, trying to get more comfortable. It was a roomy single but with Masons large frame crammed into it as well, it was a little snug. The smell of warm bodies, sleep and sex wafted up as she wriggled.\n\nShe felt deliciously sore in places she had begun to think didn't exist anymore. Hell, a nun would have seen more action than she had the last couple of years. He'd woken her a couple of times in the night and their coupling had gone from tender and touching to fast and frenzied, as though they couldn't get enough of each other.\n\nOrdinarily that sort of link would have scared the crap out of her, but with Mason it felt right. Amused, she snorted a little. Who would have thought it...the big, bad Reaper feeling safe and protected in the arms of a human?\n\nBehind her Mason stirred, and his strong arm looped crosswise over her body to pull her closer into him. He sighed and buried his face against the nape of her neck, as if she were a human-sized teddy bear.\n\nShe nuzzled her cheek against his hand on her shoulder. What was it about him? As soon as she'd seen him in the bar the other day something had pricked her interest. Both her feminine _and_ her non-human instincts had stood up and taken notice. At first she'd thought the latter was because he was due to make a decision that might bring him under her blades but no, a look at his lifeline had revealed a dull, flat line.\n\nShe turned her head to study him. He was older than she was. Physically at least. Inside Andy felt ancient. If she didn't know better, she'd believe the claim her grandmother had always made.\n\n_\" You're an old soul, Andrea, my love. You've been here before.\"_\n\nThe first part might have been true but not the latter. She didn't have a soul, ancient or otherwise, just a dead lifeline. She snorted in wry amusement. Dead lifeline for a Reaper. Fitting.\n\nMid-thirties, she decided, still studying Mason. His muscled frame bore no puppy fat and the lines on his face were telling. Even before the war he'd had a hard life. His skin bore the scars of violence...both new and old...one gunshot wound looked like it had been treated surgically, which put it pre-war.\n\nYes, his lifeline _was_ different. Now she had the chance and the leisure to study it properly she realized it was the grey of humanity but flatter than normal. Most lifelines pulsated with life, the potential of a thousand decisions taking them closer or further away from the touch of the Reaper. His didn't move, no pulse of life even though her other senses, and her experiences last night, told her he was very much alive.\n\n_How odd_ , she thought to herself. _Just what are you, Mason?_\n\nMason woke as a cold draft chased down his spine. Pulling the sleeping bag tighter around his bare torso he tried to recapture sleep. Then it occurred to him that he didn't sleep naked and didn't own a sleeping bag.\n\nMovement nearby had his eyes snapping open. For a second he was treated to an extremely nice view of Andy's backside before she pulled her jeans up over her hips. Everything slammed back in a second...going after her, her wounded arm, and everything else. No wonder he was tired, the woman was insatiable.\n\n\"Nice view to wake to.\" His voice was rough and raspy with sleep. Propping himself on one elbow Mason cleared his throat and waited for her answer.\n\nShe was gorgeous, all slender curves and satin skin. His cock stirred as he thought of what they'd done. By rights he should be exhausted, but he wanted her again already.\n\nA frown settled onto his brow at her quick movements. This was no leisurely dressing, she was moving with economy and purpose, as though she needed to be somewhere quick. That or she wanted to be away from somewhere quickly. Or someone...\n\n\"Going someplace?\"\n\nHis voice was light-hearted, but concealed the doubt that seethed in his gut. She regretted last night...that was it, and she was trying to get away from him as fast as she could.\n\n\" _Your soul is dark Mason, black and steeped in blood. I can 't be around you. Not anymore._ \" __ Julietta's words, never far away, escaped from the locked box he kept them in and swam up to haunt him. Once the first part of the memory was free, the rest wriggled loose and filled his mind.\n\n\" _Babe, you know I 've only been protecting us. That girl wasn't human...she was going to kill us,_\" _he argued, shock filled his chest. He needed Julie. She was the only thing that made sense in a world gone mad._\n\n_She shook her head, her palm soft against his cheek as she looked into his eyes. Looked down into his very soul._ \" _It 's not that Mason. It's not even recent. Your soul...it's dark, black. I've never seen anything like it. You have blood and lives on your hands. I'm sorry, I can't be around someone like that. It's not good for my karma._ \"\n\nMason's heart stilled as lead lined his gut. Perhaps his soul was too corrupt for any woman to want him for more than a night and a quick fuck. His heart ached. He didn't want that, didn't want to be a throwaway lover. He knew he wasn't a good prospect for any woman, but for Andy, he wanted to be.\n\n\"Huh? What?\"\n\nThe exasperated look she shot him as she shrugged on her jacket and reached for her curved blades reassured him. It wasn't the hunted look of a woman trying to get away but the look of someone with a task on their mind.\n\n\"Yeah, time to get up. Early bird catches the worm and all that.\"\n\nShe slammed home the buckles of the sheath and tested how easily the wicked-looking blades pulled free. Nodding in satisfaction she looked at him and smiled. His breath caught in his throat. The expression transformed her from pretty to breathtaking.\n\nTheir gazes locked. She knew what he was going to ask, and he knew she knew it. Before he could open his mouth though, she spoke.\n\n\"I...help people. It's part of what I am,\" she explained, as though the words were dragged from her. \"Like I helped you yesterday.\"\n\n\"You save people from Werewolves?\"\n\nHer dark hair danced on her shoulders as she shook her head. \"Not just wolves. Any predator, paranormal or human. Far too many of them out there.\"\n\n\"How can you tell someone needs help?\"\n\nMason was relentless in his questioning but he had to know. They were his people in the town and damned if he was going to put them...or her in danger.\n\n\"I just can.\"\n\nShe leaned forwards and brushed a lingering kiss against his lips. Carnal thoughts ambushed Mason as the urge to drag her back down onto the sleeping bag almost over-whelmed him. Possessiveness filled him. Somehow, in the last two days, she'd wormed her way past his guard and under his skin. This was _his_ woman, damn it, and no one was stealing her away, not as long as he had breath in his body.\n\nIt took them less than five minutes to pack the campsite up. Whilst she packed the sleeping back, Mason dealt with everything else. A few minutes later he did the top of the pack and looked around. All that remained of their cozy little camp was the remains of the fire. Slinging the backpack over one shoulder he kicked the ashes over to make sure it was out.\n\nHe stamped again, trying to shake the ash, which clung in a black tidemark on the battered leather. There was no sense in making tracking them any easier. Leaning down he grabbed his rifle, and smiled at Andy.\n\n\"Ready?\"\n\n\"Yeah. Let's go.\"\nChapter Six\n\nThoughts swirled around his head as they walked in silence. Whilst they were alone, them being together seemed right, normal. Now, faced with their return to the town, he could reflect on the strangeness of it all and on the strange puzzle she presented. Trouble was, even after the long walk that got him back in sight of the town walls, he was no closer to solving it than he had been when they started.\n\n\"Wait. There's something wrong.\"\n\nInstantly Mason slowed his pace and slid her a sideways glance. Everything looked fine. The walls were quiet. Automatically he glanced at the town walls as he approached, looking for damage or smoke that would indicate the town had been over-run whilst he was gone. Nothing was out of place.\n\n\"You sure?\" Everything looked fine but, after yesterday, he wasn't going to argue. She had a serious set of non-human skills on her, and he'd be nuts to ignore that.\n\nShe stood motionless, her eyes not on the walls ahead, but out of focus, as though she could see something he couldn't. \"Something bad has happened. I'm going to scout around. Be careful.\"\n\nBefore he could say another word she stepped backwards, and pulled a Houdini on him. Mason looked at the space she'd been just a moment before for a long moment.\n\n\"Great. Chicks that disappear. What next?\"\n\nWith nothing better to do and not wanting to stand around like a lemon with a thumb up his ass, Mason turned towards the town and resettled Andy's backpack over his shoulder. Disappearing act or not, he knew she was coming back. If only to collect her belongings.\n\n\"Morning, Joe,\" he nodded to the guard on the outer redoubt as he passed. A man of few words, Joe nodded back and let him pass. Mason didn't think anything of the long, sweeping glance as he walked by. All his people knew that anyone coming in was suspect. For all they knew he could have run into a Vampire or worse whilst out there and been changed into something else.\n\nThere wasn't a guard on the main gate. Damn it! How many times had he told them to get cover when they had to take a leak? Irritation flashed white hot in Mason's chest as he stormed down the main street. All he wanted was to make sure they were safe, and they disregarded the most basic safety principles. It was enough to make a guy lose it...\n\nHis footsteps slowed as he clocked the crowd outside Val's bar. His gaze moved from one grim face to the next, noting reddened eyes and tear tracks down dust-covered cheeks. Then he realized the children were gone.\n\n\"What happened?\"\n\nStony silence met his question. Dread curled its icy fingers around Mason's gut as he looked from one stony face to the other. Instead of the warmth he was used to, their eyes were hard and unforgiving. He'd seen that look before. Right before they drew down on something non-human.\n\n\"Well come on. I'm no mindreader. Someone tell me what bloody well happened!\"\n\nWariness thrilled through his veins like a fine concerto played by an orchestra. They were armed. Thanks to him they were always armed but this was different. The subtle change in their body language said he needed to be careful, very careful.\n\n\"Where've you been, Mason?\"\n\nHe half turned at the voice behind him to find Joe, his rifle held loosely. Surprise joined the wariness and worry fighting for dominance in his veins. He needed Andy here, something was very wrong.\n\n\"What are you playing at? Please don't tell me the gate's unguarded...\"\n\nJoe shrugged. \"Seems to me we got more worries in here than we have from out there. Where were you last night?\"\n\n\"I went after Andy, the girl who saved our asses against the wolves yesterday, remember? She was wounded...\" He trailed off as the silent tension in the group mounted.\n\n\"Andy...the _paranormal_ chick we threw out a couple of days ago?\"\n\nMason's brow furrowed at the hostility in Joe's voice. He turned a little more, trying to keep an eye on the weapons in the group. His instincts screamed at him to un-sling his rifle and take cover, but he stayed where he was. That the crowd had itchy trigger fingers was easy enough to see. Just one move would be enough to set them off.\n\n\"Well, it wasn't really a case of letting her anything. If she hadn't shown up then we'd all be dead, you know that Joe.\" He looked around, searching the group for another face. \"So do you Julia.\"\n\nJulia's expression wasn't as understanding as he'd expected. Instead fear and hatred filled her eyes. \"Aye, but better us dead than those creatures take our kids.\"\n\n\"Aye, she's right.\"\n\n\"Should have let be, like Fred...God rest his soul...said.\"\n\n\"His fault... We should never have listened to him.\"\n\nThe chorus of agreement from the parents in the group staggered Mason. They saw him as the enemy? Nausea rolled in his gut and clawed its way up his throat. In his travels he'd seen what happened when a community turned on one of its own, for whatever reason. The result was never pretty.\n\n\"What...the _wolves_ took the kids?\" he demanded, desperate to get to the bottom of this. Instinctively he looked for Valerie. _Please don 't let them have taken Suzie. _His heart dropped when he located her at the back. Her reddened eyes and tear-marked cheeks were answer enough.\n\n\"Aye. They took the kids. Stole in here in the middle of the night whilst we were sleeping and took them.\"\n\n\"What about the guards?\"\n\n\"Took out Katie on the outer wall with a crossbow. We don't know what happened to Victor on the gate, no sign of him. It was only old Fred, God rest his soul, that tried to stop them.\"\n\nMason was stunned. They'd protected the town for years against creatures, how could their methods fail so dramatically in one night? He'd only left to go after Andy because he was sure that the town was adequately defended.\n\n\"This is _your_ fault.\"\n\nThe shrill accusation came from behind him. Mason turned again, feeling like he was watching a tennis match.\n\n\"How?\"\n\n\"If you hadn't told that wolf no then they wouldn't have taken our kids. God knows what they're doing to them. Jamie's only three--\" Tears streamed down Julia's face as her voice cracked.\n\nGuilt hit him like a truck at Mach-one. This was his fault. If he hadn't gone, then he'd have been here to stop them. He opened his mouth to argue. What would they have preferred? To lose five women, including some of the mothers of the children who'd been taken...\n\nHe shut his jaw with a click. He knew the answer. Of course they would have. Any parent would lay their life down if it meant his or her child lived. That was what being a parent was about.\n\n\"I'm sorry. I didn't know...\" He trailed off into silence, not sure what else to say. What else _could_ he say? Then something prickled at the back of his mind.\n\n_God rest his soul._ That wasn't a phrase people used for someone who was alive. It was used for someone who was six foot under or shortly to head that way. Anguish clawed at his soul, both for the loss of a friend and the communities' loss.\n\n\"Where's Fred?\"\n\n\"You'd better come through,\" Valerie said quietly from the back of the group. Her voice was hoarse from crying. \"He doesn't have long left, so we've made him comfortable. It's all we can do...\"\n\nThe crowd parted, the hostile stares and the fingers curled around triggers making Mason jumpy as he walked through them. They hadn't tried to relieve him of his weaponry yet, but he knew it was coming. Unless they planned on just turning him out of town, then someone was going to make a play for his weapons pretty soon. With the way things were going, that wasn't going to turn out pretty.\n\nHe didn't think they were going to let him walk. Tension was running too high, and mob mentality had already set in. With their children stolen right from under their noses, they needed someone to blame. Mason was no idiot. He was going to be the scapegoat. It was already cut and dried.\n\nThe trouble was, he couldn't blame them. He was in charge of making sure the town was safe. He'd told them they could trust him, rely on him. This was his fault. He shouldn't have left them alone last night.\n\n\"V,\" he started as he drew level, and tried to catch her eye. Of all the people in town, he was closest to Valerie. Something deep inside needed her to understand how sorry he was for letting them down. How bad he felt that their kids, that Suzie, had been taken.\n\n\"We'll get them back. _I 'll_ get them back... I'll get Suzie back. I promise.\"\n\n\"He's in there. You might want to hurry. State he's in, he won't last long.\"\n\nShe gave the door a sharp nod, and turned to walk away quickly. Mason watched her for a long moment. Pain lanced through him. Valerie didn't cry in front of people. Not ever. Blanking his expression, Mason took a deep breath, steeled himself, and walked into the bar.\n\nDarkness enveloped him in its welcoming embrace. Mason paused to let his vision adjust, and regretted it. Out of the dark the smell of death hit him like a sucker punch to the gut. Bile rose and hammered on the back of his throat. He swallowed it down, and tried to breathe through his mouth, his ears...tried to breathe through anything if it meant he didn't have to suffer through the smell that was packed into the dark room like a crowd hundreds deep.\n\nHe'd smelt death before, so the coppery tang of blood mixed with a fouler stench was nothing new. The smell of a gut wound was unmistakable. He'd know it anywhere. The contents of his stomach rose sharply again, bile burning up his throat. He'd smelt it before, but it was a hundred times worse with the realization that the person whose guts were leaking out over the floor was a friend.\n\nHis footsteps rang out against the floorboards in measured treads as he walked towards the bar. His eyes adjusted, and the room was transformed from blackness to shades of gray. Within seconds, Mason wished it hadn't.\n\nFred sat slumped in front of the bar, head dropped forwards onto his chest, with his back to the wooden surface. Sat was a relative term. Most of Fred sat in front of the bar, in more or less a humanoid shape, but there were other bits Mason didn't need, or want, to identify scattered around the room.\n\nHis sight had adjusted completely to the dim light in the room as he rounded the last table--the one riveted to the floor--and looked down at the slumped figure. Blood decorated the bar, the floor, and other surfaces around him in thick, black arcs that spoke of violence and pain. Mason couldn't help the wince as he looked at what remained of his friend.\n\nHe was in a worse condition than Mason had thought. A red towel covered his midsection, from chest to mid-thigh, covering the worst of the damage. It was sticky with blood and other fluids, but that wasn't the worst of it. Fred had always been a well-built man, retaining an impressive beer gut despite the recent heavy rationing. However, now Fred's abdomen looked as lean as his own. Mason gritted his teeth. He would not be sick. He wouldn't dishonour the older man in that way. How Fred was still alive was beyond him, _if_ he was still alive.\n\n\"Wondered how long it would take you.\"\n\nThe voice was little more than a wheeze, but Mason caught it. He dropped to his knees next to Fred, not caring that his jeans were instantly soaked, and grabbed for the hand the older man lifted.\n\n\"Sorry, mate, traffic's a bitch.\"\n\nFred's hand was cold, the skin icy to the touch. Mason didn't need to press on the nail beds to know Fred's circulation was severely compromised. The wounded man chuckled. The sound degenerated into a cough, which was more a death rattle. For long moments he struggled to breathe.\n\nA tiny muscle in the corner of Mason's jaw jumped. His throat was thick with emotion as he fought back the hot prickles at the backs of his eyes. In all his time on the road, all the times he'd nearly died, and all the times he'd watched others die, he'd never felt as helpless as he did now.\n\n\"What happened, Fred?\"\n\nFred rolled his head back to lean it against the bar. Mason winced. The skin was shredded to the bone. A gaping hole in one cheek revealed the whiteness of teeth, and the eye socket furthest from Mason was empty.\n\n\"Jesus...\"\n\n\"Yeah. I'm not gonna win any beauty contests, am I?\"\n\nThe self-deprecating comment startled a laugh out of Mason. \"No, now that you mention it, I doubt it's on the cards. What happened, man?\"\n\nFred started to speak but another coughing fit hit him. Mason looked away as blood and spittle flew. Good thing he seemed to be immune to the Lycan infection, or Fred's tainted blood would have done him for sure.\n\n\"Sorry 'bout that. Anyways...Jed happened. Or them bastard wolves of his did. They come out of nowhere, Mason. Not like they normally do...this was sneaky. They were in human form, got the kids real quiet like. One of them had on Victor's coat...you know, that long jacket he always wore.\"\n\nMason bowed his head, shaking with anger.\n\n\"Go on.\"\n\nHis voice was tight and controlled. The sheer force of feeling surging through him was like a tidal wave. One tiny slip, one little crack in the iron wall of his control, and it would spill over. Then whoever was in the way would be in for a world of hurt.\n\n\"I was in here, heard something out the back.\" Fred shrugged. \"Went to investigate, didn't see the one sneaking up on me and, well, the rest you can see... Bastards thought they'd done me for, and didn't bother to finish me off. I need to warn you--\"\n\nMason shook his head. That didn't make sense, he already knew about the attack. \"Warn me about what? We already know the kids are gone.\"\n\n\"No, not that.\" Fred flicked a glance to the door at the front of the bar. \"You gotta get out of here, Mason. The wolves left a message. Jed'll give the kids back--\"\n\n\"Good. He'd better if he knows what's good for him. I'm gonna take great pleasure in ripping his guts out,\" Mason growled, his anger almost overwhelming him. That alpha was a dead wolf. Even if they did give the kids back, which he highly doubted, there was no way he was leaving Jed alive. Not after this.\n\n\"No, you don't get it. Jed'll give the kids back...if your head is on a stake by sunrise.\"\n\nMason didn't get time to flee. He didn't even get time to process Fred's words before the door to the bar burst open, and the townsfolk spilled in. Hard faces, and harder eyes watched him down the barrels of several rifles, and he knew. They were here for him. They weren't waiting for sundown, never mind sunrise.\n\nInstead of going for his own rifle he remained where he was, looking down at the friend who'd tried to warn him. The look in Fred's remaining blue eye was compassionate. \"Don't blame them. They're only protecting their kids.\"\n\nHe nodded and shook Fred's hand. \"Thank you, my friend. Go easy.\"\n\nFred smiled and clasped his other hand over Mason's in a final farewell. \"Was just waiting for you. Sorry it was too late.\"\n\n\"No worries. Catch you on the flipside.\"\n\nHe watched Fred's hand slide from his. After a second he reached out and closed the single, sightless eye with gentle fingers then he stood to face his execution party.\n\n\"This is it then?\"\n\nMason couldn't believe his voice was so level, or calm. These people were going to kill him, and with so many rifles trained on him, there was nothing he could do about it.\n\n\"So you're just going to kill me in cold blood?\"\n\n\"Here, this way. Let's get him to the walls.\"\n\nMason groaned. His right side was on fire, blood soaking through what remained of his T-shirt, the rags of the fabric a match for the torn skin underneath. They'd shot him, several times. So much for not shooting an unarmed man.\n\nNone of his arguments had saved him. As soon as the first trigger had been pulled, more had followed. He had no idea how much lead he'd taken but it had to be enough to drop a damn rhino. It fucking hurt.\n\nClamping his eyes shut, he tried to breathe through the pain as they dragged him through the streets. He knew what fate awaited him on the walls of the town. Locking his legs and planting his feet, he pit his weight against his captors. His heavy boots dug into the dirt as he put up a fight. He was bigger built than either of the men trying to drag him, something they seemed to have overlooked. There was no way he was going anywhere _near_ that wall. Not if he could help it. They'd have to drag him kicking and screaming, and he'd make them pay for every step.\n\n\"Christ, he's like a bull...some help here.\"\n\nHe slammed a hard elbow into the side of the guy on his right, freeing that hand. Fist clenched, he twisted and drove it upwards into the jaw of the guy on his left. The vicious uppercut dropped the smaller man without a word.\n\nMason moved back, his guard snapping up. It should have been a light dance on his feet, as he'd been trained in the ring, except that his side felt as if it had been caved in by a sledgehammer. He started to turn to face his next opponent when there was the distinctive click of a hammer being drawn back, and something hard was pressed against the back of his head. He froze.\n\n\"Now I've seen just about everything out there on the road.\" Valerie's voice was cool and calm, with an edge of suppressed hysteria no man in his right mind would mess with. \"I've seen men get up after being filled with lead like you. But I ain't ever seen any of them get up after a couple of rounds to the brain. What's the betting you don't either?\"\n\nCrap. Mason held his hands up in the universal gesture of surrender.\n\n\"Only one problem with that darlin'. Jed wants my head on a stake. You put a bullet through my brain with that, and there ain't gonna be much of my head left. What do you think your chances of getting the kids back are then?\"\n\nMason counted silently as he waited for Valerie to make a decision. Finally the pressure let up, and he heard the hammer click back into place. Relief and cold sweat slithered down his spine as he released a breath he didn't realize he was holding.\n\n\"He has a point. Get him up on that wall. And make sure you have a good hold on him this time, for pity's sake.\"\n\nHe held in his curses as he was grabbed again, and Donny, the victim of the uppercut, took the opportunity to dig an elbow into the wounds that peppered his side.\n\n_Bastard, you 'll pay for that._ Mason glared at his tormentor as they shoved him roughly forwards.\n\n\"I suggest you let him go. Now.\"\n\nMason's head snapped up as the townsfolk went for their guns. There was Andy, as large as life, standing in the middle of the town gates. In their hurry to lure him in like a lamb to the slaughter the residents of Sanctuary had forgotten Town Defense 101. _Shut the fucking gate._\n\nIf he weren't so messed up, he'd kiss each and every one of them.\n\n\"I suggest you take your paranormal ass, and hightail it out of here, missy. Unless you want your head to join his on a stake,\" Donny snapped, high on a cocktail of anger, pain, and adrenalin.\n\n\"Is that so?\"\n\nA chill wind blew through the town, whipping the dust of the road and her long, black coat up into a frenzy around her. Andy slid the dark glasses down her nose to look at Donny, and then at the rest of the townsfolk, her gaze as hard and unblinking as a rattlesnake. That was when Mason realized what was different about her.\n\nWhen she'd arrived in Sanctuary she'd walked and talked human. Even with his talents he'd had to see some serious evidence she wasn't human before he believed her claim to be something paranormal. The speed and sheer lethality she'd displayed with her blades during the rescue yesterday had nearly done it, but Mason himself was proof that a human could outwit and outfight the majority of paranormals.\n\nIt wasn't until he'd dug a bullet out of her arm, and seen for himself how quickly she healed that he'd truly believed her claims. And even now he wasn't sure what she was, just that there was no way she could be plain old _homo sapiens_.\n\nNow though, with the sky churning and angry behind her where it had been calm a moment ago, and the wind driving up a storm, there was no mistaking the fact she was something on the darker side of paranormal. Lightening cracked, and thunder rolled, leaving her highlighted in silhouette. The crowd around him drew a collective breath as just for a second she was cast into shadow, her silhouette stretching into a familiar cloaked and hooded figure carrying a scythe.\n\n\"The Lady of Death.\"\n\nThe whisper ran through the crowd, from one pair of lips to another. She couldn't be...could she? The Lady wasn't a paranormal. They said she was Death itself made flesh and set upon the Earth to hunt down the unworthy. Which, when you thought about it, amounted to ninety percent of the current population.\n\n\"Don't be ridiculous. That's just a story, made up to scare the kids.\" Donny had his rifle in his shoulder, but if he could aim and fire accurately Mason was a monkey's uncle.\n\n\"You sure about that, Donald? Just because you don't believe in the bogeyman, doesn't mean the bogeyman doesn't believe in you.\" Andy took her sunglasses off and slid them into her inside pocket.\n\n\"H-how did you know my name?\"\n\nShe shrugged, her voice calm and pleasant when she spoke. \"Same way I know Valerie's, and John's, and Eric's. Believe me, I'm not like anything you've ever met before. And you might as well put that away...\" She motioned to Donny's rifle. \"Shooting me'll only piss me off.\"\n\n\"Para's have threatened to kill us before. It's all been hot air and illusion. None have managed it,\" Valerie broke in, clapping Donny on the shoulder as she walked forwards to face Andy. Mason had to admit, she had guts.\n\nAndy smiled. It wasn't a nice expression. The wind rose again, more ferocious this time, moaning as it whirled through the gate and around them. A chill wind, which whispered bad things directly into the souls of all who heard it.\n\n\"You know what other names I know? How about Katie, Victor and Frederick?\"\n\nLightening spilt the sky behind her. Mason shook his head and blinked as three figures materialized around her. Something was wrong with his eyes. He could swear the three dead townspeople stood there, as pale and opaque as ghosts. Hell...they were dead. If this wasn't a trick, some illusion, and Andy really was who they said she was, then they could be ghosts.\n\nAs his brain tried to find some logical and scientific rationale for the appearance of the three figures, he knew. He'd felt the tug inside as she'd raised her hands and knew that she'd called the souls of the dead to her.\n\nJaws dropped, and rifles clattered to the dirt. Her voice, cold and terrible, cut through the mournful moan of the wind.\n\n\"You called me, and you named me. I am Death, and that man--\" she lifted her hand to point at him, \"--is the only thing standing between you and me. Kill him and you won't have to worry about the Werewolves. I'll drag you to hell myself.\"\nChapter Seven\n\n\"They're freaking out six ways to Sunday.\"\n\nStill unable to believe what had happened and his luck, Mason leaned back against the cool stone wall of the town, and watched as the townsfolk went about their business. Mostly they were keeping their distance from Mason, and the object of their fear, Andy, who knelt in front of him.\n\nShe snorted as she lifted his shirt to assess his wounds. \"So they should be. I wasn't kidding when I said I'd drag them to hell.\"\n\nHe watched her. She was beautiful. Scary, but beautiful.\n\nShe looked up, a frown on her face. \"Okay, handsome. Want to tell me why you're not dead yet? You should be.\"\n\n\"Sorry, sweetheart. Not a clue.\" Mason shrugged as she dropped the shirt back into place. Soaked in blood, it clung wetly to his torso.\n\nShe sighed. \"You're not dead, Mason. I don't know how or why. You look human. You walk, talk and act human. But you're not. You can't be.\"\n\n\"Don't talk rubbish,\" he scoffed. \"Of course I'm human. What else would I be?\"\n\n\"That's what worries me.\" Andy stood and held her hand out to help him up. \"I don't feel the call to reap your soul, and your body is already healing the damage. You're not human.\"\n\nMason grabbed her wrist and hauled himself to his feet in one lithe movement. An hour ago he'd felt like he was at death's door but now he felt energized, and raring to go. Hand still wrapped around Andy's he looked down into her eyes.\n\n\"I'm as human as they come,\" he told her firmly. \"Nasty, devious...yes, but plain old human. And those wolves are about to find out why we were the top of the food chain for thousands of years.\"\n\n\"Listen up. Are we putting together a rescue plan for the kids, or am I doing this on my own?\"\n\nAndy watched as Mason walked away from her, his shoulders squared as he challenged the very people who, less than an hour ago, had been ready to kill him. Admiration filled her. That had to hurt. She knew how much the town, and the safety of the people within it, meant to him. To insist on helping them after what they'd done to him? It said good things about his soul.\n\nSilence met his challenge as people looked away. Andy didn't blame them. She'd forced them to actually look at what they were doing. Forced them to admit to themselves they were about to kill one of their own.\n\nThey called the other races monsters. But those monsters, herself included, had no choice in what they were, how they acted. Vampires craved blood to the exclusion of all else, Lycans were born for the hunt and the kill. They couldn't help their instincts. Humans though, they were different. They could choose to kill, or not. Most of the time, when pushed, they chose to kill rather than walk away. In her book, that made them worse than any of the monsters out there.\n\nUnable to bear the silence she sighed, brushed the dirt off her denim-covered jeans and stood.\n\n\"I'm in,\" she said simply, moving to stand next to Mason.\n\nHe smiled quickly, shooting her a sideways glance filled with thanks. She didn't need the thanks. Whilst she understood the needs and instincts, which drove them, kidnapping and extortion were not natural Lycan drives. She and that pack Alpha would be having words, and they weren't going to be pleasant ones.\n\nShe looked around the assembled crowd, most of whom looked embarrassed.\n\n\"Now I'm sure you're not going to let us go in alone to rescue _your_ children, are you?\" she prompted, and took half a step into the Shade.\n\nThe world lost its color. She reached out and called the lifelines of every living creature within a hundred meters to her. The sparkling silver ropes leapt into her palms like puppies eager to do her bidding. Only two didn't. Her own, which stretched out flat and black behind her, and Mason's.\n\nShe paused to look at it. The grey of humanity had started to fade. Already it was a lot darker than the silver strands she held in her hands. Puzzlement filled her, as a frown creased her brow. Just what the hell was he?\n\nNo way was he human, but he wasn't a Reaper either, despite the darkness of his lifeline. Chiding herself for getting distracted, she pulled on the lifelines, and stepped back through into the mortal world. As expected she had the attention of everyone in the crowd. Including Mason. She wasn't surprised. She'd just reminded them she held their lives in her hands. Sure, she couldn't actually _kill_ them, or reap their souls until they were ready to go, but they didn't need to know that.\n\n\"Just as I thought. Let's plan this thing then, shall we?\"\n\n\"Ugh. That was like pulling teeth,\" Andy announced as, three hours later, the pair made it back to Mason's room. \"Are they always so obstructive?\"\n\nHe closed the door behind them, and, for good measure, threw the bolt. It didn't take a genius to work out that he didn't trust the townsfolk any more than she did. She didn't blame him. Without their stabilizing influence, or her there to scare the crap out of them, there was no saying that they wouldn't revert to plan A, and try to stick Mason's head on a stake again.\n\nThat happened and she wasn't bothering with the softly-softly approach anymore. She might not be able to drag their souls to hell until they were dead, but all it took was one little decision and their lifelines were ready to reap. One little decision and those threads would light up like a Fourth of July display. One little decision, like deciding to kill Mason, and she'd go to work with her sickles and carve a bloody swathe through the town.\n\nShe rolled her shoulders as she wandered around the small room. Small was the only word to describe it, and bare. A small window opposite the door allowed bright sunlight to stream in and provided a stage for the ever-present dust motes to dance in.\n\nA small bed took up one wall, a battered chest of drawers next to it. A clock sat on it. Even without the _tick-tock_ in the room she knew it was broken. Batteries were a precious commodity these days, so valuable no one would waste one on a clock.\n\nShe turned to find Mason with his back to the door, watching her. His eyes were intense, and his expression patient. There was something different about him, but she couldn't put her finger on it. The hungry look on his face sent a shiver down her spine. He looked like a starving man faced with a sumptuous banquet, and she had a feeling she knew what was on the menu.\n\n\"Come over here.\" His voice was low, with a rough, husky timbre to it that sent a shiver down her spine. Hands clasped behind her back she looked at him, and raised an eyebrow.\n\n\"Why?\"\n\nHis eyes darkened dangerously, the heat of hell itself contained there. It was an inferno that licked at Andy's skin, heated the blood in her veins and made her panties damp.\n\n\"Because I said so. Don't make me come over there.\"\n\nShe chuckled, enjoying the light-hearted banter. They had an hour or so before sundown and the planned start of the mission. There was time. More than time for what he obviously had in mind.\n\n\"Not dressed like that you won't, Mister.\" She motioned towards his torn and bloodied shirt. \"You need a shower and a change, then we'll think about it. Let me look at how those wounds are doing.\"\n\nHe dropped his chin and looked up at her through his eyelashes. It should have been a feminine gesture but on the very masculine Mason it became something totally different. She sucked a shaky breath in, feeling like she'd been punched in the gut. Only no punch she'd ever taken before had left her with a tingly feeling through her entire body and an ache that made her want to clench her thighs together.\n\n\"All healed. See? How, I have no fucking clue. But they are.\"\n\nHe looped a hand under his ragged T-shirt and smoothed it upwards over his toned and muscled torso. Despite a very feminine interest in those muscles, Andy still had enough about her to check for damage. Where there had been wounds, bloody tears in his skin where bullets had torn through it on their way into his body, there was nothing. No marks, no healing scars, no old scars. Nothing.\n\nJust like her.\n\nA rush of feeling hit her. After all these years believing herself alone, she wasn't. Mason should be dead. He had been dead, he had to have been, no one could survive being shot that many times and not pass over. Except a Reaper.\n\nPerhaps that was it. Perhaps because he _had_ died...maybe that was the trigger? The bombs of the war had done their work and twisted humanity's DNA a decade ago. Some had changed immediately and become whatever their twisted DNA dictated. For the first time in history Vampires, Werewolves and their brethren, the creatures of legend and myth, had walked the earth. And the world had gone to hell in a hand-basket since.\n\nOthers didn't change. Somehow they carried the potential for that change deep within them. Waiting for something like a bite, or death itself, to trigger what was locked inside. Her world tilted on its axis as she crossed the small room.\n\nHer hand smoothed over the undamaged skin, but her gaze was locked with his. From the moment she'd seen him in the bar, she'd known something was different about him. Now she knew what. Like called to like.\n\n\"Reckon we'll both fit in that shower,\" she whispered against his lips, and then claimed them with her own.\n\nDarkness blanketed the world, the moon playing peek-a-boo through the thick clouds above. Andy knelt in the shadows next to Mason and watched as he organized his team with silent gestures. That he'd been military was obvious, as was the fact that he'd trained these people to work together. A fan of the march-in-and-kick-ass school of thought, she tried to stay still and not jitter impatiently.\n\nThen it was game on. The attack started, not with a bang, but a whisper as the human forces approached the Lycan camp on silent feet. Mason had made them all shower to try and erase the smell of human sweat. Like soldiers from before the war, their faces were smeared with dirt, and they carried their rifles as if they meant business.\n\nThis was Mason's gig, and she was happy to let him lead. She was used to working alone and had no idea how to get a group of people working together without half scaring them out of their minds. He didn't need to terrify them to get them to work well. Even though they'd tried to kill him earlier they were following his orders now.\n\nWraiths in the night, they flitted from cover to cover, moving towards their objective. She thought back to the briefing. Over a scale model of the terrain made from dirt and anything lying to hand Mason had given each team directions and orders about how the attack was to go down. As plans went, it was a good one. She just knew from experience no plan survived first contact with the enemy. Double that when you were dealing with paranormals.\n\nThe camp appeared before them like an island rising from a foggy sea. The light from the campfire cast sky above and around in a warm orange glow. It should have been a welcoming scene; the makeshift camp with sleeping bags lay haphazardly around the fire warding off the chill of the night. It wasn't too much of a stretch to imagine marshmallows being toasted or steaks sizzling on a grill plate...only this was ten years after the end of the world. Marshmallows had gone the way of the dodo, and Lycans preferred their meat uncooked.\n\nThe wolves lounged around the fire in human form. She'd noticed that about them, unless they needed to hunt or fight Lycans preferred their two-legged forms. She'd never understood that. If she had the freedom to shift forms and run with the wind like they could, she'd never switch back to being human.\n\nHer breath plumed on the night air as she dropped into cover behind a group of old barrels next to Mason. He flicked her a glance and smiled, a quick flash of white teeth in the darkness, before he reached out to squeeze her arm quickly. A brief touch but one that spoke volumes as his fingers lingered a second on her skin.\n\nShe could hear his breathing and her own heart as they waited for the signal from the other teams. Every second stretched out to infinity as she studied the layout of the camp. To the side of the fire was the makeshift cage they had the kids in. Sneaky bastards. They'd put it between the fire and an outcrop of rock, which meant the human teams were going to have to go through the Lycan pack.\n\nNext to her, Mason muttered under his breath as he did a headcount. Andy didn't bother. Her reaper senses told her that all the kids were alive. Whether they made it through the night that way though, that one was anyone's guess. Life or death hung in such a precarious balance that perhaps only a Reaper could appreciate it. Every day a multitude of decisions took each soul nearer or further away from death. Right now, the decisions the humans made, and those of the Lycans, would affect whether those kids lived or died tonight.\n\nThe moon flitted in and out of view, allowing her a brief glimpse of the terrain on the other side of the camp. Mason's teams were good. She could see neither hide nor hair of them with her normal vision. Feeling Mason's attention on her, she switched her vision to look into the Shade. The mingled gray and silver lifelines decorated the landscape like tinsel on a Christmas tree.\n\nHer eyes narrowed. The first two teams were in position, but the third had yet to reach its mark. She waited until they jockeyed into position behind the tree stump Mason had indicated earlier, and nodded. Cooking with gas now.\n\nShe watched as Mason checked his weaponry, every movement calm as he watched the camp below for movement. Nothing happened. Most of the Lycans appeared to be asleep, their lifelines sparkling red to Andy. Her palms itched to draw the blades sheathed across her back, but she held off. The breeze picked up, bringing the scents of the campsite to them. She wrinkled her nose at the smell of stale beer and unwashed bodies.\n\nMason leaned down and selected one of the pebbles by his knee. A thoughtful look on his face, he hefted it in his hand a couple of times, as though considering its potential. Reeling his arm back he launched it into the middle of the camp. She caught her breath, wondering what the hell he was playing at as it landed in the center of a group of empty beer cans.\n\nWith a loud clatter the projectile scattered the cans. The noise woke the snoozing Weres, bringing them to their feet with shouts of alarm. Andy glared at Mason. The idiot had just given away the element of surprise, which was pretty much all they'd had going for them.\n\nHe winked at her as he pulled something from the side of his vest. A grenade. Pulling the pin with a vicious twist of his wrist he launched it into the middle of the Lycan camp.\n\n\"Fire in the hole.\"\nChapter Eight\n\nThe pair twisted away as the small canister sailed through the air and landed with a hop, skip and a jump by the campfire. Andy clapped her hands over her ears as it went off with a balance-destroying bang, and lit the night sky up like a sunny day.\n\nThere was no rest for the wicked though. Immediately she and Mason were on their feet and moving. Adrenalin surged through her body as they covered the distance between their hiding place and the camp.\n\nThe wolves caught in the grenade's radius staggered around as though drunk, blinking as they tried to regain their vision. In wolf form Lycans could see a mouse taking a shit at six hundred meters, but shine a big enough torch in anything's face and it was blind as a bat for a couple of seconds.\n\nThey were still as dangerous as hell, probably more so when blinded and panicking. Leaving the worst affected to the humans, she went for the Lycans who'd been on the edge of the blast radius. Stun grenades were non-lethal; a percussive explosion and a bright light enough to render most incapable long enough for the attacking force to control whatever was going on. For humans it was sufficient but Lycans recovered fast.\n\nA feral howl of fury erupted from the center of the camp, the sound peppered with the sound of gunshots as Mason's teams picked their targets. The human forms of the Lycans were swallowed up as their wolf forms exploded from them faster than the eye could see. Instead of a group of teenagers out on a camping trip, they were faced with a pack of snarling and angry predators.\n\nShe grinned to herself, an expression without either humor or mercy as her blades cleared leather and lined up her first target, a scraggy-looking wolf already turning towards her. Its lips curled into a snarl as it gathered itself to leap.\n\n\"Com'on then, fugly,\" Andy taunted, waggling her blades in warning. She fought with them in opposing grips, one blade up and one with the blade curved along her forearm. Matched with a healthy dose of martial arts training, it was a lethal combination. \"Let me give you a good belly scratch.\"\n\nMassive paws skittered in the dust as it launched at her. The thrill of the fight singing through her, Andy waited until the creature was almost upon her then leapt into action. Time slowed to a crawl as a hop put her in the air, and her back foot lashed out. Her steel-toe cap caught the wolf under the chin, and slammed its mouth shut over the vicious canines within. Satisfaction surged through her as she felt bone crunch. Even if she had busted the creature's jaw, it wouldn't make much difference. Already its freaky physiology would be repairing the damage.\n\n\"Awww, does that hurt? Lemme give you something for that.\"\n\nIt staggered away, trying to put some distance between them so it could heal the damage. She didn't let it and followed. Her blade flashed in the light of the campfire as she slashed it through the air towards the furry throat. The razor-sharp edge cut through fur, skin and muscle like a hot knife through butter. Gritting her teeth, she rammed the blade deeper until it scraped against bone. Triumph surged as she felt her blade hook the edge of the soul. With a vicious yank she pulled it back, and severed the soul from the body. Blood spurted as the beast shuddered, and slumped dead at her feet.\n\nOther howls filled the air as the human teams picked their targets, and overwhelmed the Lycan pack. Mixed amongst the furry death-calls were human screams as the creatures defended themselves. So many lifelines lit up that Andy was almost blinded. Knowing she was going to have her work cut out for her later, she flicked her vision firmly to human and looked about for her next victim.\n\nTo her left a group was baiting a big wolf. It was bleeding from numerous bullet wounds, but was still on its feet and lashing out with razor-sharp teeth. A body lay under the massive paws, throat a bloody mess and sightless eyes staring up at the night sky. Donny. Try as she might, Andy couldn't bring herself to feel sorrow that the mouthy human had met his end in such a violent way, just annoyance that his lifeline had been cut short unnecessarily.\n\n\"Oscar...left knee. Now!\" Valerie yelled the order as she lifted her rifle. In concert, shots rang out. The wolf ate dirt as its front knees disintegrated in a shower of blood and bone. Without a word Andy slid between the humans, and slashed her blades across its neck to free the soul.\n\n\"Nice move.\"\n\nShe nodded at the other woman as she danced back, impressed by the teamwork. Pity this lot had tried to kill Mason, she might have actually grown to like them at some point.\n\nValerie just smiled in reply, a small tight nod as she looked around at the cage the children were being kept in. Three wolves surrounded it, guarding their captives. The team who was supposed to be freeing the children was dead, scattered in pieces around the wolve's paws.\n\nThe two women locked eyes as understanding flowed between them. The children were paramount. Without a word, both women, human and reaper, turned on a dime and stalked towards the cage and its furry guards.\n\nAndy was impressed all over again. Unlike her, Valerie was just plain old human. Easily caught, and just as easy to kill. Despite that, she had a look of grim determination and hard vengeance on her face. Left to herself, Andy had no doubt that if anything happened to her kid, the woman would spend the rest of her life tracking those responsible and taking them apart piece by piece. Slowly.\n\nShe could admire vengeance like that.\n\n\"Yo, heads up.\"\n\nAndy grinned as she approached the cage, calling out to attract the wolves' attention to herself. However you cut the cards, she was the one who could hold out longest against the creatures, and when it came to it, survive a wolf attack. She might wish like hell they'd kill her, but at some point the pain would end and she'd heal. Then hunt herself some wolves and make a nice winter coat from their hides.\n\nThe three wolves turned their attention on her, eyes picking up the light from the fire as she walked past it and shining brightly. Lips curled back from their teeth as they snarled, drool dripping on the dirt beneath them. Unlike the kid in town the other day, these were fully shifted Lycans which meant any human survival instincts they had were buried under layers and layers of predatory desires. Desires and needs that marked the two-legged, soft-skinned creature walking towards them firmly as prey.\n\n\"How many wolves does it take to change a light bulb?\"\n\nFrom the corner of her eye she saw Valerie step back into the darkness and work her way towards the cage. Good girl, she'd gotten the message. The three wolves looked at each other and back at her. Puzzlement crossed their furry faces as they tried to figure out why she wasn't scared or running away.\n\nShe deliberately didn't look behind them to where Valerie was freeing the kids. The cage door was secured with a length of rope, the knots too tight for young fingers but no match for the knife Valerie carried.\n\n\"Com'on...it's a killer.\"\n\nShe hid her grin and walked closer, waggling her blades to make them catch what little light there was. Reading them was like reading a book. They weren't used to someone who looked human, and more importantly _smelt_ human, not running the other way in terror.\n\nThe trouble was, at some point in the next minute, that puzzlement was going to turn into irritation, which was then going to turn into violence. By the time that happened she needed to be within range. Calmness filled her, radiating out from her core to fill the rest of her being. Input from all of her senses was heightened, and sharper.\n\nShe could hear the rasp of breathing as the massive chests rose and fell, the stink of their fetid breath as they bared their teeth and the smell of unkempt fur, like they hadn't groomed properly for weeks. She'd seen Lycan packs before and this wasn't a pack. This was a bunch of filthy scavengers who were a disgrace to their species.\n\nWithout warning she slipped into the Shade fully, leaving them staring at empty space.\n\n\"Actually, who gives a shit?\"\n\nHer voice was conversational as she re-appeared less than a heartbeat later between the largest of the wolves. Both turned and lunged at her in the same instant, their teeth slicing the air where she had been. Instead of the soft target they had been expecting though, she was gone back into the Shade, and two sets of teeth encountered only the furred hides of their pack mates.\n\nIgnoring the howls of pain and fury, Andy stepped from the Shade again by the third Lycan. Slightly smaller than the other two, she'd guess it was female, but that was only a guess. She'd never been sufficiently interested in the different forms Lycans took to learn to identify their genders. As long as they bled, that was all she cared about.\n\nIt sensed her by its shoulder at the same time she raised the wickedly curved sickle, and froze. The eye nearest to her rolled back in the socket to try and get a bead on her, but it was too late. A soft whimper escaped the creature's lips as Andy slammed the blade down hard. The point parted the fur and popped through the skin with a small _snick_. Using her weight and momentum the sickle powered through the Lycan's throat, and exited the other side in less time than it took to think about it.\n\nBlood pulsed from the ruin of its throat to splatter the dry dirt below. With something almost akin to grace, the creature's knees folded and it collapsed to the ground. Deftly Andy reversed the blade as she dropped into the Shade and dispatched the soul on the way back in.\n\nShe turned her attention to the other two wolves, nipping in and out of the Shade to harass and harry them. They whirled and danced, trying to catch her as she appeared and disappeared. Their extra abilities were no use against a creature who could simply chose not to be in their plane of existence. She kept an eye on how Valerie was doing rescuing the children, and as the seconds ticked by, the gray lifelines active to her reaper senses started to wink out one by one. A signal that some of those kids would survive.\n\nFor tonight at least.\n\nThe shit had well and truly hit the fan. As Mason pounded across hard dirt, his teams unleashed seven levels of hell on the Lycans. They didn't have long until the creatures recovered from the flash-bang, so he'd told them to make every second count. Advice he was taking to heart himself.\n\nThey hit the outer perimeter of the camp, and he paused for a second as Andy peeled away. Pulling his rifle tight into his shoulder, he selected his targets rapidly and fired in short, economical bursts. Each time his muzzle spat bullets he hit his target. A knee here, through a throat there, even taking the teeth right out of the mouth of one wolf as it lunged at one of the humans. Suppressive fire, designed to put as many wolves down as possible so the others could get close.\n\nHis aim was lethal, punching holes through furry hides and making the turned Lycans bellow with pain. Without silver shot though, any damage he inflicted would be temporary. Where was a fifty cal when you needed one? He could really level the playing field with something like that.\n\nHis rifle clicked, the bolt holding open in a warning that his magazine was empty. No time to reload. Casting it aside, Mason pulled the Glock from his shoulder holster and moved further into the camp. He skirted between the groups already fighting, his eyes peeled for one figure. Screams from his left distracted him for a moment, as the group attacking the cage went down. Grimly he ignored them. They'd all known the risks before they'd come on the mission. He'd made sure of it. And he was damned if, after trying to kill him, he was going out of his way to cover their asses anymore. Nope, he was back into mission mode. He'd complete the objective, and then he was out of here.\n\n\"Jed. Get your furry ass out here and face me like a fucking man, rather than a whipped puppy,\" he yelled, trying to taunt the alpha into the open. Truth be told, he had no fucking clue what the guy looked like, as a man or a wolf. None of them did, a fact Mason hadn't been too happy about. Going into a hot situation without intel was usually a good way to get people killed. However, so was taking on a pack of Lycans on their own turf. But then, Mason had never claimed to be sane, Andy wasn't even mortal and the people of Sanctuary were just plain desperate.\n\nHe didn't have to wait long. A rumbling snarl sounded behind him, the sound full of menace. Of course, any Lycan snarl was full of menace but this was a very personalized sort of malevolence. Mason whipped around, the Glock already leveled and aiming right between the eyes of the large wolf behind him.\n\n\"Jed, I presume?\" he asked, as though this was a society dinner, and they'd just been introduced. Regardless of his nonchalant attitude, the muzzle of the gun didn't waver. If Jed looked like he was going to leap, Mason would double-tap him right between the eyes before taking his next breath. Unfortunately, now he'd actually seen Jed, Mason didn't think that would put the bastard down for long.\n\nHe was huge. Possibly the biggest wolf-type Lycan Mason had ever seen, and he'd hunted more than enough of the creatures. Gray and black fur covered a frame straight out of a pre-war horror film, and his teeth wouldn't look out of place on a sabertooth. Amber and black eyes fixed on Mason, hatred and anger burning in their depths. A silver bar cut through the creature's eyebrow.\n\nMason's brow winged up in surprise. One of the only things the old films had gotten right about Werewolves was silver. It burned them, ate at their skin like acid. Most avoided it like plague, or like the average Vampire did sunlight. It took a twisted SOB-Were to actually pierce himself with the stuff. Jed dropped his head lower to the ground and curled his lips from his teeth.\n\n\"I'll take that as a yes then. Christ, you are one _ugly_ fucker, aren't you?\"\n\nThe lips curled back further, the massive paws shifting in the dust. It was all the warning Mason got before the massive Werewolf leapt. He pulled the trigger before his eyes and brain got their action together, his instincts kicking in. The Glock spat fire, but Jed was faster. He twisted to the side as the 9mm rounds tore through his shoulder instead of going through his brain.\n\nMason's heart pounded, driving adrenalin around his veins in a survival-driven chaotic race. It galvanized every cell in his body as the age-old fight or flight instinct took over. Time slowed to a crawl as the Lycan launched towards him. Its jaws opened wide, giving him a good view of toothy death as it came for him.\n\nHe bellowed an incoherent war cry, falling backwards and still firing at the oncoming behemoth. His shoulder hit the dirt, barrel tracking the creature's head as it sailed over him. His finger carried on, pulling the trigger independent of thought until the magazine was empty. The bullets slammed into Jed's body, punching holes through his skin on the way. He snarled in fury and pain, landing on paws bigger than dinner plates and stumbled, face planted in the dirt by the still blazing campfire.\n\nMason was on his feet in the blink of an eye. He tossed the useless Glock aside. He wouldn't get a chance to reload it anyway and went for the fighting knife on his leg. Most people, if they knew Mason had been a soldier at all, assumed he was American. An infiltration expert, he only had an accent when he wanted, and he could choose which accent that was.\n\nOnly those who had seen the winged dagger tattoo on one arse-cheek, done by his squad-mates whilst he was insensible, and clocked the Fairbairn-Sykes knife he carried, realized he wasn't only not American, but also something a cut above normal for the British army.\n\nMan and the beast who'd once been a man eyed each other up. Mason had always wondered, would he have become something like this? If his DNA had been slightly different? It was a thought that had plagued him for years. At first, in the first bloody years after the war, when those who had turned preyed on those who hadn't, he'd gone to sleep each night expecting it to be his last. Expecting to wake the next morning as something else, something not human.\n\nIt had never happened. Whatever evils he'd done in his former life, they hadn't been visited on him in this version of hell on earth. Something Mason wasn't sure whether he was grateful for or pissed off about. Was it reward or punishment to live in a world gone mad?\n\n\"Come on then, you fucker. Let's be having you,\" he roared, his native accent out in full force.\n\nThe hilt felt good in his hand as he faced down Jed. The familiar grip fit his palm like a glove, the weapon an extension of his being as he circled Jed. Everything else...the sounds of the other groups fighting, the pitiful whimpering behind him as a team finished off one of the Lycans...all fell away as he concentrated on one thing, and one thing alone. Killing something considered unkillable.\n\nThere was nothing Mason liked more than a challenge. He grinned slowly, the expression making Jed blink and falter a little in surprise.\n\nJed feinted to the left, paws kicking up dust, and then to the right. Mason kept to his low crouch, knife held along his forearm and glinting dully in the light cast by the fire. He was only going to get one shot at this, so he had to make sure it was a good one. His gaze focused in on a small spot on Jed's broad, lupine chest. Any second now, confident that Mason didn't have another gun on him, the wolf was going to go for the kill, then Mason had him.\n\nHis FS was dipped in silver nitrate, something guaranteed to give any Were a bad day for all of ten seconds. Unfortunately for said Were it would be the last ten seconds of its life. Unfortunately for Mason, getting close enough was also close enough to get his throat torn out. Something he hadn't planned on doing this morning but, hey, shit happened.\n\nHis gaze still locked with Jed's, he saw the moment the Lycan made the decision. Then the air was full of pouncing wolf, fur and sharp teeth. He didn't even try to avoid the creature as it bowled him over. His breath was knocked from his body in a savage whoosh as he hit the deck hard. A grin of fury and triumph crossed his face as his head slammed against a large stone near the campfire and stars filled his vision.\n\nTeeth clamped around his throat, the sharp points popping through the skin like fingernails though a balloon. Warm blood streamed as they drove in, but Mason was already there. Even as he felt his jugular and more tear, he rammed the knife between two ribs and right into Jed's heart.\nChapter Nine\n\nLycan blood coated her blades and ran down her arms, but Andy didn't bother to wipe it away. She'd only get bloody again. Using both hands she dispatched the wolf in front of her with a vicious, cross-handed swipe and watched without emotion as it dropped lifeless to the ground. Blood surged from the ruin of its throat onto the thirsty dirt below. Idly she wondered what would grow there, with the soil being watered with wolf's blood. _Plants with a tendency to bite and snarl at other plants?_\n\nThen it happened. She looked up at the howl behind her, and started to turn. They'd heard plenty of howls during the attack, but this was different. The fury and defiance, the general bad-assed, I'm-a-wolf tone was gone, replaced by a sound of unimaginable torment so complete it made even Andy shudder.\n\nShe whipped around. A huge wolf was on its side, writhing in pain. Its shape-shifting ability seemed to be completely out of control. Its body was in a continuous state of flux, each part shifting from human to wolf independently of any other. Bones popped and cracked, breaking only to reform themselves, then start the cycle all over again. She winced as its ribcage inflated to the size of a barrel but its shoulders turned human. Even from here she could hear muscles tearing and bones creaking as they tried to settle into a form even the current, twisted version of nature had never intended.\n\nThe skin was worse. Sickeningly worse. It bubbled and slid, flowing over the monstrous forms beneath, but like everything else, it couldn't seem to make up its mind what to be. One second it was hide and fur, and the next it was soft human skin regardless of what lay underneath. A massive paw formed from the end of a human arm, then covered itself in skin before the bones within, too large for the casing, ripped through it.\n\nShe turned her head away for a second, covering her mouth as bile rose, and fought the urge to lose her lunch. She hated to be sick but just looking at that was enough to make her forget the habit of a lifetime.\n\nThe sound of torment rose into a plaintive plea for mercy. A death howl which called to every non-human instinct she had. All emotion leeching from her, Andy stepped forwards and into the Shade, her blades already raised for the killing blow.\n\nWarmth and color disappeared as she left the land of the living behind. Here things were calmer, so much simpler. Spectral souls, human and Lycan, waited by their bodies, nice and orderly, for her to go along and reap them. She ignored them. They were dispatched. First she needed to deal with the tortured wolf.\n\nWithout asking, she knew it was the alpha, Jed. The power rolling off him was unmistakable. Just as she knew what had killed him. The silver-treated knife lodged into his heart glowed white hot in her enhanced vision. Even if she'd been of a mind to kill him, he was done for. After the blade had entered his heart to deliver its fatal payload, the continuous form changes had locked it within his flesh, putting protective layer after layer over it. The only way they were getting it out would be with a gifted surgeon, or a chainsaw.\n\nShe sure hoped that wasn't Mason's favorite knife.\n\nJed's soul stood to one side, watching his body contort with an expression of fascinated horror on his face. Registering movement he looked up and at her. His face blanched, which considering where they were was an achievement.\n\n\"No! Not me, not yet. I'm not done,\" he protested, and tried to run. Andy didn't hurry her measured pace, nor crack a smile. She'd seen this before. It was rare, admitted, but sometimes when a person was caught between life and death, they could see her in the Shade.\n\nShe watched as Jed ran two steps then fell, yanked back to his body by the unbreakable thread of his own lifeline. There was only one creature on the planet that could cut a lifeline. Her. He looked up over his shoulder, whimpering as he tried to scramble away.\n\nHer lips compressed into a thin straight line as she looked deep into his heart. Normally she didn't bother. Whether they went up or down was no business of hers. This time though, she was interested. If Jed wasn't headed down, so help her God, she was going to drag him to the fiery pit herself.\n\nShe needn't have worried. His heart was black, rotten to the core with evil deeds and disgusting impulses. She blinked as every act of violence he'd committed, every rape, every murder, every evil deed right back to his childhood flashed before her eyes. Here, in her world, she didn't feel sick. She just felt angry.\n\nShe stalked him, murder in her eyes. Apart from the fact he was already dead... Kind of hard to murder someone who was already dead. He leapt over his own body, trying to escape again. She sighed and changed direction. If there had been railings nearby, she'd have clicked her blades along them as she approached, just to scare him more.\n\n\"I can do this all day, you know? Now stand still like a good little doggie.\"\n\n\"Fuck you, bitch,\" he sneered.\n\n\"Not without a week long bath, a change of species, and a personality transplant,\" she snarled back as he flipped to his back and bared his teeth at her. \"Might as well put them away, sunshine. This is my __ world and here we play by _my_ rules.\"\n\n\"Bitch. I'm gonna rip your fucking heart out.\"\n\nSliding one of her sickles home into its sheath, she reached down to the lifeline from the shadowy mound that represented his body in the mortal world. Like everything else, here it was indistinct and lacking in features, but she knew what it was. The pulsing scarlet thread attached to it was a dead giveaway.\n\n\"Oh, you can try.\"\n\nShe started to wrap the thread around her fist, pulling him closer by slow increments. He struggled and fought, to no avail. She dragged him closer and closer until he was within range.\n\nFinally close enough, he took a swipe at her. Vicious claws erupted from the ends of his fingers as they flashed towards Andy's face in an unstoppable arc. She stood and waited, an impassive look on her face, as they passed right through her and exited the other side, turning the enraged soul into something akin to a spinning top.\n\nAs soon as he turned she slammed her booted foot into the small of his back at the same time as she looped his own lifeline around his throat, then pulled him hard against her.\n\n\"How's it feel to be helpless, Jedediah?\" she whispered into his ear, picking his hated full name from the memories she'd gleaned. \"To be able to do nothing whilst it all goes to shit around you? Impotent, useless...at the whim of someone else. Terrified, like so many of the people you killed.\"\n\nHer breath whispered over his neck as she tightened the red cord around his throat. Of course, his soul couldn't be killed by strangulation, but Jed didn't know that and he reacted as though Andy was throttling the very life out of him. His fingers scrabbled at her hands, as he kicked and bucked in her hold, but Andy was unshakeable. Her lips twisted as she pulled the line tighter.\n\n\"Rot in hell, Jed,\" she whispered, lover-like, before she snapped the cord.\n\nShe stepped out of the Shade to a much quieter scene than she'd left. The fire popped and crackled merrily to itself. Soft sobbing and the odd whimper broke the silence periodically. Half the attacking humans were dead. She nodded slightly to herself, the numbers were better than she'd expected. If she was honest, she hadn't expected any of them to survive. Period. There was one way to describe humans attacking a Lycan camp.\n\nSuicidal.\n\nShe stood for a moment in the center of the camp, just by the hideously twisted corpse of the former alpha and scanned the survivors. They sat in shell-shocked silence, most of them covered in blood. She couldn't see Mason but she wasn't overly worried. Something told her that, if anyone could survive attacking a Lycan camp, the hard-nosed, ex-soldier turned town-guardian could.\n\nShe turned and looked behind her. The cage the children had been kept in was empty. Valerie had done her job well. Andy flicked her vision to the Shade for a moment, noting that all the lifelines that led away from the cage into the darkness were flat and healthy, including Valerie's. Good. Some of the parents hadn't survived the attack, and those kids would need someone strong to look after them.\n\nScanning around her in the Shade she looked for Mason's peculiar lifeline. She'd never seen another one like it, so it would be hard to miss. However hard she looked though, she couldn't spot it. Which was impossible. Wherever a person went, their line followed behind them, like a scent trail. It was what Andy followed when she tracked a soul ready to be reaped. Unlike scent though, it was something that was impossible to mask, and Andy was no bloodhound to be put off the scent easily.\n\nWhere the hell was he? A familiar sense of irritation rising, Andy stomped around the fire to the nearest survivor, one of the young men from the town. He sat in a trance next to another of the furry corpses, his face splattered with blood.\n\n\"Have you seen Mason?\" she demanded, waving her hand in front of his face. \"Hey! Wakey-wakey!\"\n\nAt the sound, the guy snapped out of it somewhat. Still wearing a stunned expression he transferred his attention to her face. \"Huh?\"\n\n\"Mason? Tall, shaved head. Kick-ass attitude,\" she said shortly. \"Seen him?\"\n\nHe lifted his hand slowly to point behind her. A sickening feeling wormed its way into Andy's heart as she turned to look at what he was pointing at. There, hidden behind the bulk of Jed's twisted body, caught somewhere between human and Lycan, was another, smaller one. Andy's heart shuddered to a halt as ice filled her veins.\n\n\"No...it can't be.\" Her voice was a low whisper, a plea to any God who might be listening. Unable to tear her eyes away from the fallen figure, she took a couple of steps forwards and stopped, her feet refusing to carry her like they were stuck in quicksand. It couldn't be Mason, it just couldn't.\n\nThe broad shoulders were too familiar though, as was the tattered T-shirt stretched over them. He lay on his side, unmoving. Too still. Her breath caught in her throat as she raced around the fire and slid to her knees in the dirt next to him.\n\n\"Oh God, no. Please. No,\" she moaned as she tugged on a shoulder to roll him onto his back. There was no resistance. Mason flopped onto his back, his eyes wide and unfocused as they stared sightlessly up at the night sky. His throat was a bloody mass of mangled flesh, the bones of his spine glinting white in the gaping wound.\n\nAndy screamed, a mingled sound of agony and fury as her heart broke in her chest with a sharp crack. She fussed, gathering him to her, not caring that his blood soaked through her top. Tears streamed down her cheeks as she cradled him. She'd only just found him, the other half of her soul, and now he was gone. Snatched from her by a cruel twist of fate and the snapping of a Lycan's jaws.\n\nDead.\n\nHer head lifted slowly.\n\nDead. Mason was dead.\n\n\"No.\"\n\nIt was a simple word but one spoken with utter conviction. She looked at the man in her arms. He couldn't be dead because she wouldn't allow it. Death did not scare her, nor did it hold any sway over her. She was a Reaper...perhaps the _only_ Reaper. The Grim Reaper herself.\n\nDeath would bow to her, be bound by her rules and her will.\n\nFighting back panic she reached deep inside herself. Deep within, in a place she avoided thinking about or calling on too much, was a dark place devoid of anything. The center of her reaper powers, the core whatever that made her what she was. For years she'd avoided it, tried to forget it was there, as she carried on pretending to be human.\n\nBut she wasn't human, and she never would be.\n\nThis time she didn't just brush the void, or embrace it, she did a run up and went for a full-on base jump right into it. The void enveloped her, consumed her. She threw her head back and screamed again, as the memories of a billion or more deaths slammed into her. Time lost meaning as she tumbled, bounced from death to death throughout the ages. Her voice expanded, power added to the feminine scream as she accepted her destiny and took on the full power of a Grim Reaper.\n\nHer mouth snapped shut--the sound cut off as she lowered her chin and opened her eyes to look at the world around her anew. Jed still lay in front of her, his body locked in everlasting torment, the fire still blazed merrily, and the man she loved still lay dead in her arms.\n\nOnly, like her, the man she loved wasn't human, and never had been.\n\n\"Can't be having that, now can we?\" she said to no one in particular, and leaned down to kiss him. At the first touch of her lips, power flowed between them. A spark of life that she gently blew against his lips. Nothing happened for a long moment, then he coughed.\n\nShe lifted her head. As she watched, the hideous mess that was his throat started to heal. Flesh filled out as structures started to form. Veins slithered across the open space and started to pulse with blood as skin crept over the wound to cover it. Within a minute it was gone, his skin as unmarked as it was this morning.\n\nHe opened his eyes to look up at her. She smiled as a pitch-black lifeline flared into life in the corner of her vision.\n\n\"Welcome back, my love.\"\n\nHe'd been dead six months, and it had been the best six months of his life. Sighing with contentment Mason leaned back in his chair and took a sip from his drink as his gaze followed the small woman behind the bar.\n\nSix months pregnant, Andy was starting to waddle but she still insisted on pulling her weight. Like pulling the late shift in the bar whilst Valerie was settling the kids. With so many orphaned the night of the Lycan attack, looking after them all was a full-time job.\n\nShe wiped glasses with efficient and swift movements, placing them neatly on the shelves behind the bar. Every so often she would cut a glance his way, heat in her dark eyes which set an answering heat to smoldering within him.\n\n_\" Just you wait until later, woman,\" _he promised silently, knowing she could hear him. Quite how the link between them worked, or exactly what he was now Mason hadn't figured out. All he knew was that he had been dead, and now he wasn't. And that now he could see things...lines he couldn't see before. Like the three amber lines which blinked into existence in the corner of his vision. Andy calmly wiped the last glass, and placed it on the shelf.\n\n\"Incoming,\" she announced as the door opened and three Vampires swaggered in.\n\nMason smiled and caressed the trigger of the shotgun rigged under his table. Vampires before nine in the evening? Looked like it was going to be a busy night.\nAbout the Author\n\nMina Carter was born and raised in Middle Earth (otherwise known as the Midlands, England). After a slew of careers ranging from logistics to land-surveying she can now be found in the wilds of Leicestershire with her husband and young daughter...the true boss of the family.\n\nSuffering the curse of eternal curiosity Mina never tires of learning new skills which has led to Aromatherapy, Corsetry, Chain-maille making, Welding, Canoeing, Shooting, and pole-dancing to name but a few.\n\nShe juggles being a mum, working full time and writing, tossing another ball in the air with her cover artwork. For Mina, writing time is the wee hours of the morning before anyone wakes up and starts making demands, or any spare minute that can be begged, bought or conned.\n\nHer first stories were penned at age 11, when she used a stationery set meant for Christmas thank you letters to write stories instead. More recently, she wrote for her own amusement and to save on outrageous monthly book bills. Now she's totally addicted and needs her daily writing fix or heads roll!\n\nYou can find Mina at:\n\nWebsite: www.mina-carter.com\n\nTwitter: @minacarter\n\nFacebook: www.facebook.com\/mina.carter\n_A one night stand turns into a partnership for survival..._\n\nNeon Chaos\n\n_(C) 2011 Karen Erickson_\n\nParty girl Samantha Sanders is celebrating her thirtieth birthday in style--a road trip to Las Vegas for a long weekend with her girlfriends. Staying in the swankiest hotel on the strip, she's naked and in bed with the sexiest guy she's ever laid eyes on when the clock strikes twelve. Who knew the world would really end December 21, 2012? Those Mayans, they weren't lyin'...\n\nRuss Weaver is above all else a soldier. Looking for a little fun before spending Christmas with his crazy family, a weekend on leave in Vegas unexpectedly turns into his newest mission----protecting the sweet, scared woman he happens to be in bed with when the world as they know it is over.\n\nWandering throughout the virtually abandoned streets of Vegas, they fine city has turned into a war zone. Their only hope to get out alive is to stick together--and fight those who wish to cease their endeavors. Permanently.\n\n_Enjoy the following excerpt for_ Neon Chaos:\n\n\"Where the hell could they be?\" he roared, throwing the flashlight with all his might. It crashed against the wall before it dropped to the floor, denting the drywall, and she watched in horror as her personal hero crumpled to the ground and beat his fist upon it.\n\nSamantha ran to him, kneeling by his side and slipping her arm around his trembling shoulders. He was upset, he had every right to be and it scared the ever-living crap out of her.\n\nHe'd been the strong one from the beginning. If he lost it now how would they ever make it? On her skills and gumption alone?\n\n_Yeah, right._\n\n\"They must be dead.\" His voice was shredded, and he sniffed. \"My parents must be dead. I don't know where else they could be.\"\n\nShe wrapped her arms around him and brought him to her, his head resting against her chest. Smoothing a hand over his thick, dark hair, she offered soothing words, her eyes closing when she felt him shudder.\n\nIt broke her heart to see him like this. His arms banded around her waist, and he held her tight, his face buried against her neck. She swore she felt the hot dampness of tears, but she would never ask him. Had a feeling he was a bit of a macho man who would never admit to such a thing.\n\n\"What should we do, Russ?\" She hated asking, but they needed some sort of plan.\n\nHe lifted his head, his gaze meeting hers. The discarded flashlight was still on, throwing a shaft of light across his face, and she saw all the despair there, all the weariness.\n\n\"I don't know,\" he croaked, sounding good and truly lost. \"What do you think?\"\n\nShe pushed his hair back with her fingers. \"You're tired. Maybe we should try and rest for a bit.\"\n\n\"They'll find us.\"\n\n\"We'll stay only for a little while. You need to sleep.\"\n\nHis eyes slid closed for the briefest moment, and he grabbed hold of her hand, bringing it to his mouth so he could press a gentle kiss to the back of it. A full body shiver moved through her at the contact and when his eyes opened, she leaned in, kissing him soundly on the lips.\n\nHe looked like he really needed it.\n\n\"I fucking hate this,\" he whispered. \"I don't know what to do.\"\n\n\"I don't know either, but we can't go on like this. We're both running ragged. That's why we should rest. Even for just an hour.\"\n\nHe sighed. \"You're right. Let's stock up and hide in the detached garage in the back. Sleep for a while and then get the hell out of here.\"\n\n\"Stock up? I have food.\" She shook her purse, which she'd brought in with her.\n\n\"I'm not just talking food, sweetheart, though we should grab more, good idea. I'm talking about _useful_ stuff.\" He kissed her again, as if he needed the bit of physical contact to keep going. \"I'm talking weapons. My father has a stockpile of them.\"\n\nThe sadness had slowly left him, replaced by a simmering, potent rage that bubbled just beneath the surface. His parents were fucking gone--hell the entire family had probably disappeared, and he could do nothing about it.\n\nNothing.\n\nHe'd cried, not like a blubbering baby but tears had slipped out and he'd struggled not to ball. Samantha had offered comfort, holding him close, not saying a bunch of stupid words to try and make him feel better because nothing would've made him feel better at that particular moment. Not even now.\n\nNope, she hadn't said anything at all, just offered him a shoulder to sniffle on. He appreciated her more than she could imagine.\n\nAfter gaining control over his emotions he forced himself to get over the sadness. He was pissed. He wanted to kick some ass.\n\nHe wanted revenge.\n\nFirst though, he needed to remain calm and rational and gather supplies.\n\nSamantha filled a few grocery bags with food from the pantry while he went to his father's den. The gun case was locked. Samantha had the flashlight so he kicked his booted foot through the glass door, smashing it to bits. The broken glass fell to the floor in a tinkling pile.\n\nHis mother would've had a fit if she could see him.\n\nReaching through the broken glass pane, he unlocked the door and it swung open. He grabbed two shotguns and two handguns, a set each for himself and for Samantha. They needed to be prepared for whatever might come at them. He had a sneaking suspicion it would be pretty damn bad.\n\nRuss hoped like hell she wasn't scared of guns.\n\nReaper\n\n__\n\n__\n\n__\n\n_Mina Carter_\n\n__\n\n__\n\n__\n\n_The World after the End of the World will never be the same again..._\n\nSanctuary. Cliched name, but the sentiment is still the same. Ten years after the end of the world, ex-soldier Mason and a small group of humans defend their fortified town against creatures of myth and legend made real. But with dwindling game to hunt and a lycan pack in the area looking for an easy meal, just surviving is getting harder every day.\n\nAndy has a few screws loose, and she knows it. She's been on the road since the bombs fell and changed humanity forever. Driven by inhuman instincts, she tracks the newly and soon-to-be dead and dispatches their souls to the afterlife. Sometimes they go quietly, most put up a fight. She doesn't care either way. Her ambition in life is to find her next hit of coffee and one day, maybe, sleep in a real bed again.\n\nThen Andy's instincts bring her to Sanctuary and its enigmatic leader, Mason, and even the world after the end of the world will never be the same again...\n\nWarning: Contains a snarky female Reaper with a hair-trigger temper and a caffeine addiction, a hot ex-commando with an attitude and a twisted sense of humour and a happily ever after that defies death itself.\n\n**eBooks are _not_ transferable.**\n\n**They cannot be sold, shared or given away as it is an infringement on the copyright of this work.**\n\nThis book is a work of fiction. The names, characters, places, and incidents are products of the writer's imagination or have been used fictitiously and are not to be construed as real. Any resemblance to persons, living or dead, actual events, locale or organizations is entirely coincidental.\n\nSamhain Publishing, Ltd.\n\n11821 Mason Montgomery Road Suite 4B\n\nCincinnati OH 45249\n\nReaper\n\nCopyright (C) 2011 by Mina Carter\n\nISBN: 978-1-60928-630-9\n\nEdited by Bethany Morgan\n\nCover by Kanaxa\n\nAll Rights Are Reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations embodied in critical articles and reviews.\n\nFirst Samhain Publishing, Ltd. electronic publication: November 2011\n\nwww.samhainpublishing.com\n\n# Table of Contents\n\nDedication\n\nChapter One\n\nChapter Two\n\nChapter Three\n\nChapter Four\n\nChapter Five\n\nChapter Six\n\nChapter Seven\n\nChapter Eight\n\nChapter Nine\n\nAbout the Author\n\nAlso Available from Samhain Publishing, Ltd.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n_The Litter of the Law_ is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, or locales is entirely coincidental.\n\nCopyright \u00a9 2013 by American Artists, Inc.\n\nExcerpt from _Nine Lives to Die_ copyright \u00a9 2014 by American Artists, Inc.\n\nAll rights reserved.\n\nPublished in the United States by Bantam Books, an imprint of The Random House Publishing Group, a division of Random House LLC, a Penguin Random House Company, New York.\n\nBANTAM BOOKS and the HOUSE colophon are registered trademarks of Random House LLC.\n\nIllustrations copyright \u00a9 2013 by Michael Gellatly\n\nLIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA  \nBrown, Rita Mae.  \nThe litter of the law : a Mrs. Murphy mystery \/ Rita Mae Brown & Sneaky Pie Brown; Illustrated by Michael Gellatly.  \npages cm  \nISBN 978-0-345-53048-6  \neBook ISBN: 978-0-345-53857-4  \n1. Haristeen, Harry (Fictitious character)\u2014Fiction.  \n2. Murphy, Mrs. (Fictitious character) \u2014Fiction.  \n3. Women detectives\u2014Virginia\u2014Fiction. I. Title.  \nPS3552.R698L58 2013  \n813'.54\u2014dc23  \n2013007940\n\nwww.bantamdell.com\n\nCover design: Beverly Leung  \nCover illustrations: \u00a9 Daniel Pelvin (cat),  \n\u00a9 Shutterstock\/MisterElements (yarn)\n\nv3.1_r2\n\n# Contents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\nCast of Characters\n\nThe Really Important Characters\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\n_Dedication_\n\n_Other Books by these Authors_\n\n_About the Authors_\n\n_Excerpt from_ Nine Lives to Die\n\n# Cast of Characters\n\n**Mary Minor Haristeen** \u2014\"Harry,\" just forty-one, a Smith graduate who wound up being Crozet, Virginia's, postmistress for sixteen years, is now trying to make some money by farming. She survived breast cancer and prefers not to think about it. She more or less lives on the surface of life until her curiosity pulls her deeper.\n\n**Phararond Haristeen, D.V.M**.\u2014\"Fair\" specializes in equine reproduction. After graduating from Auburn he married his childhood sweetheart, Harry. He reads people's emotions much better than his wife does. He is a year older than Harry.\n\n**Susan Tucker** \u2014Outgoing, adept at any and all social exchange, she's Harry's best friend since cradle days. She loves Harry but worries about how Harry just blunders into things.\n\n**The Very Reverend Herbert Jones** \u2014A Vietnam veteran, Army, he is pastor at St. Luke's Lutheran Church, which is well over two hundred years old. He is a man of deep conviction and feeling. He's known Harry since her childhood.\n\n**Deputy Cynthia Cooper** \u2014Tall, lean, and Harry's next-door neighbor as she rents the adjoining farm, she loves law enforcement. Harry meddles in Cooper's business from time to time but the Smith graduate has an uncanny knack of finding important information.\n\n**Tazio Chappars** \u2014She's an architect in her early thirties often assisted by her yellow Lab rescue, Brinkley. Italian African heritage gives her dazzling good looks. She hails from St. Louis, Missouri.\n\n**Buddy Janss\u2014** A huge fellow and a successful farmer; he can be counted on to pitch in for fund-raisers. He and Harry like to discuss crops, weather, and new equipment.\n\n**Hester Martin** \u2014A middle-aged graduate of Mary Baldwin, she runs a roadside produce stand. Odd, opinionated, but blessed with a good heart, she drags people into her projects.\n\n**Aunt Tally Urquhart\u2014** This 101-year-old aunt of Marilyn Sanburne, Sr., does what she wants when she wants. She's not in too much evidence in this volume, which gives everyone a rest.\n\n**Marilyn Sanburne, Sr.\u2014** \"Big Mim,\" known as The Queen of Crozet. She runs everything and everyone except her aunt. Big Mim is a political animal.\n\n**Marilyn Sanburne, Jr**.\u2014\"Little Mim\" has just had a baby, Roland. Her mother doesn't like the name. Often Little Mim doesn't like her mother.\n\n**Miranda Hogendobber** \u2014A second mother to Harry, a devout member of the evangelical Church of the Holy Light, she, too, isn't much in evidence in this volume. Like Big Mim, she's in her seventies and has no idea how she got there so fast.\n\n**Sheriff Rick Shaw** \u2014The sheriff of Albemarle County, he is overburdened, underfunded, and overworked. Despite that, he likes law enforcement and has learned to trust Cooper. Originally, he wasn't thrilled having a woman in the department.\n\n**Neil Jordan** \u2014The treasurer for St. Luke's, Neil can be picky, picky, picky. He drives Harry crazy and vice versa but they have to work together as both are on the vestry board. He owns a fertilizer business that is lucrative.\n\n**Wesley Speer** \u2014He's also on the vestry board and brings a business perspective. He works sometimes with Neil since Wesley owns an upscale realty firm. He'll often refer clients to Neil. Wesley, like any high-end Realtor, is slick.\n\n**BoomBoom Craycroft** \u2014Another childhood friend of Harry's, she had an affair with Harry's husband years back. It was a mess, of course. Everyone has recovered and in many ways is the better for it. BoomBoom runs her late husband's concrete business. She is conventionally beautiful.\n\n**Alicia Palmer** \u2014Now here's a showstopper. Alicia was a movie star in the fifties, whipped through a few husbands, affairs, etc., made pots of money, inherited more from an old flame. She returned to Crozet, fell in love with BoomBoom, and is blissfully happy.\n\n**Sarah Price** \u2014Hester Martin's niece comes up to Crozet from Houston.\n\n**Paul Diaz** \u2014Tazio's boyfriend, who trains Big Mim's horses.\n\n# The Really Important Characters\n\n**Mrs. Murphy** \u2014She's a tiger cat who is usually cool, calm, and collected. She loves her humans, Tucker the dog, and even Pewter, the other cat, who can be a pill.\n\n**Pewter** \u2014She's self-centered, rotund, intelligent when she wants to be. Selfish as this cat is, she often comes through at the last minute to help and then wants all the credit.\n\n**Tee Tucker** \u2014This corgi could take your college boards. She is devoted to Harry, Fair, and Mrs. Murphy. She is less devoted to Pewter.\n\n**Simon** \u2014He's an opossum who lives in the hayloft of the Haristeens' barn.\n\n**Matilda** \u2014She's a large blacksnake with a large sense of humor. She also lives in the hayloft.\n\n**Flatface** \u2014This great horned owl lives in the barn cupola. She irritates Pewter, but the cat realizes the bird could easily pick her up and carry her off.\n\n**Shortro** \u2014A young Saddlebred in Harry's barn who is being trained as a foxhunter. He's very smart, young, good-natured.\n\n**Tomahawk** \u2014Harry's older Thoroughbred. They've been friends a long time.\n\nThe Lutheran Cats\n\n**Elocution** \u2014She's the oldest of the St. Luke's cats and cares a lot about the \"Rev,\" as his friends sometimes call the Very Reverend Herbert Jones.\n\n**Cazenovia** \u2014This cat watches everybody and everything.\n\n**Lucy Fur** \u2014She's the youngest of the kitties. While ever playful, she obeys her elders.\n\n#\n\nFair Haristeen, doctor of veterinary medicine, and his wife, Mary Minor \"Harry\" Haristeen, loved to steal a Saturday and cruise the back roads of central Virginia. It reminded them of their courting days, back in high school, when Fair, bruised from Friday night's football game, would pick up Harry, dirty from the stable, and they'd drive around in his 1958 Chevy pickup. Now, over two decades later, Fair was at the wheel of their station wagon, Harry beside him, the pets in the back seat, as they rode through the countryside.\n\nMrs. Murphy, the tiger cat, Pewter, her gray, overweight sidekick, and Tucker, the corgi, usually accompanied their people everywhere except in high heat. On a mild day like today, windows down a crack, the three could sleep or chat while the humans talked.\n\n\"Perfect weather,\" Fair declared.\n\nOctober 12 was indeed a ravishing fall day\u2014early fall, for the summer warmth lingered late this year. Forests looked spray-painted with yellow, orange, flaming red, deep red, old gold.\n\n\"Hey, Miranda got the respiratory flu.\" Harry mentioned a former co-worker and dear friend. \"She's swearing that drinking electrolytes will cure her. She saw it on TV.\"\n\nFair shook his head. \"Electrolytes will help, but our beloved Miranda seems susceptible to quacks.\"\n\nWatching the passing scenery, Pewter noticed a lovely yellow clapboard farmhouse. _\"Quack\u2014duck. Why call a crook a quack?\"_\n\n_\"I don't know,\"_ Tucker replied. The corgi was well used to Pewter's inquiring mind. _\"They also use the term 'snake oil.' A quack sells snake oil. It's confusing.\"_\n\n_\"Ha!\"_ Pewter whooped. _\"If they'll buy snake oil, maybe we can get them hooked on catnip.\"_\n\n_\"Humans don't sniff catnip,\"_ Tucker replied with dignity.\n\n_\"They can learn.\"_ The gray cat spoke with conviction.\n\n_\"Pewter, sometimes I think you're cracked as well as fat,\"_ the dog unwisely said.\n\n_\"Fat!\"_ Pewter raged.\n\n_\"You need a seat all your own. Every time we take a turn, the flab on your belly sways,\"_ Tucker teased.\n\nPewter lashed out, a quick right to the shoulder.\n\nTucker growled, showing her fangs.\n\n\"That is enough!\" Harry turned around.\n\n_\"I haven't done a thing.\"_ Mrs. Murphy distanced herself from the combatants, who now rounded on her.\n\n_\"Brown-noser!\"_ Pewter whacked the tiger cat, who gave as good as she got.\n\nThe hissing and barking irritated Fair to the point where he pulled over to the side of the road, near where Hester Martin's vegetable and fruit stand was located.\n\nHarry got out of the car, opened the back door. \"I am going to give you such a smack.\"\n\nAll three animals jumped to the far back of the Volvo station wagon. Harry walked around to the rear of the car and opened the hatch door; the animals jumped back into their original seats.\n\nSlamming both doors shut, Harry cursed as Fair couldn't help but laugh. She walked over to the driver's side; he had the window down.\n\n\"They know how to pluck your last nerve,\" said Fair, laughing.\n\n\"Yours, too. I'm not the one who pulled the car over.\" Harry looked down the road at the produce stand, a small white clapboard building with a large overhang, goods displayed in orderly, colorful rows. \"Hey, let's get some pattypan squash. Bet Hester still has some.\" She walked around the car, getting in the passenger side before turning to face her animal tormentors. \"If I hear one peep, one sniff, one hiss while I am shopping, no food tonight. Got it?\"\n\n_\"Hateful.\"_ Pewter turned her back on Harry.\n\nAs Tucker hung her head, Mrs. Murphy, the tiger cat, loudly defended herself. _\"I didn't do one thing.\"_\n\n_\"Of course not, the perfect puss.\"_ Pewter curled her upper lip.\n\nFair coasted to the stand, where Hester\u2014orange apron, black jeans, and an orange shirt\u2014was talking to customers, most of whom lived in Crozet or nearby.\n\n\"I'll stay here.\" Fair knew how Hester could go on, plus Buddy Janss was there, all three hundred pounds of him, and he could outtalk Hester.\n\nOrange and black bunting festooned the roof overhang. Scarecrows flanked the outdoor wooden cartons overflowing with squashes, pumpkins, every kind of apple imaginable. Inside, one could buy a good sandwich. Little ghosts floated from the rafters; big green eyes glowed in the room's upper corners. Brilliantly gold late corn and huge mums and zinnias added to the color.\n\nAlmost as big as Buddy, a sign sat catty-cornered to the entrance, announcing the community Halloween Hayride to raise money for the Crozet Library. No doubt Tazio Chappars, an architect, had designed the impressive sign. She worked hard for the library and the sign really grabbed you: From a large drawn skeleton, one bony arm actually reached out to get your attention.\n\nHester looked up. \"Harry Haristeen, I haven't seen you in weeks.\"\n\nBuddy turned. \"How'd you do with your sunflowers?\"\n\nBuddy, a farmer who rented thousands of acres along with cultivating his own holdings, enjoyed Harry's foray into niche farming. Who knew better than Buddy the cost of equipment and implements for wheat, corn, soybeans? Harry had made a wise choice in focusing on her field of sunflowers, her quarter acre of Petit Manseng grapes, and the ginseng she grew down by the strong deep creek that divided her property from the old Jones farm.\n\n\"Pretty good,\" she said, not wanting to brag that this year's field of sunflowers was her biggest yet. \"How's your year so far?\"\n\nHe hooked his thumbs in his overalls. \"Tell you what, girl, that mini-drought thinned out my corn crop. I did better than most because my lower acres received enough rain. Others didn't. Never saw anything like it. On one side of the road the corn would be twisted right up, and on the other just as plump as you'd please. The corn behind the old schoolhouses looks poorly.\"\n\nHester jumped in. \"Government's fault. All that stuff they have circling around up there in space. Gotta affect us.\"\n\nBoth Harry and Buddy nodded politely, for Hester was a little in space herself. Sometimes a lot out there. Middle-aged, good-looking, with glossy light brown hair hanging to her shoulders, she applied just enough makeup to draw attention to her symmetry and health. Every small town as well as big city has its Hesters, it's just they can't hide in the small towns. Good-looking people, often bright, but they don't quite fit in and often they never marry. Hester had gone to Mary Baldwin, excelled in her studies, but came back over the Blue Ridge Mountains to run this roadside stand. Her brother, more ambitious, moved to Houston right out of the College of William and Mary. He had perfect timing, hitting Texas on the cusp of a building boom and making the most of it. Her parents had built the stand more as a hobby than a business, but it flourished. Her father had been a banker; her mother had run the stand. These days Hester seemed happy enough, engaged with a steady stream of regulars and tourists.\n\nBuddy kindly semi-agreed. \"What scares me is what we don't know. I mean, just in general, look at this drought and, hey, we came out a lot better off than they did in the Midwest, where everything burned up. Right now our water table is good. I planted more Silver Queen corn because I think the weather will stay warm longer. I'll get it harvested and if not, I'll make a lot of critters happy.\" He let out a booming laugh.\n\nHester asked, \"You've got crop coverage, Buddy? After the drought of 1988, surely you started paying for an insurance policy, revenue protection.\"\n\n\"I do. I elected an eighty percent revenue protection policy. Yes, I did learn from 1988 but, girl, every time I turn around I'm writing another check and I see my return diminish. Farming gets harder and harder,\" said the well-organized man, a true steward of the land. \"Just to keep up, I have to plant more acreage. Plant an early crop, then come back and throw soybeans down. I feel like I'm running to stay in place.\"\n\n\"Think we all do,\" Hester agreed.\n\n\"Only way I can buy or rent\u2014and renting makes sense in the short term\u2014is to sell some of my land closer in to Crozet or Charlottesville.\"\n\nHester's shoulders snapped back. \"Don't do that, Buddy. Don't ever do that.\"\n\n\"Before I forget, Hester, do you have any pattypan squash?\" Harry didn't want to keep Fair or the arguing animals in limbo.\n\n\"I do. Wait until you see it.\" Hester nodded to Buddy, who winked at Harry.\n\nThe two women walked inside, where there was crooknecked squash, acorn squash, and Harry's favorite, cream-white pattypan squash that looked like scalloped discuses.\n\n\"Beautiful! And the right size.\"\n\n\"Right about now the pattypan is usually over, but this year with the long, long summer, I'm still getting some,\" said Hester. \"The melons are over, though. I do so love melons. Before I forget, now, you and Fair are buying tickets for the hayride. You must. The library is built but there's a lot to be done. We need $59,696 just for adult computers and, oh my, the adult area needs tables and we need furniture for a meditative reading room. The list is endless.\"\n\n\"Of course we'll buy tickets. I'll even buy tickets for Mrs. Murphy, Pewter, and Tucker.\"\n\n\"If that gray cat of yours gets any fatter, I'll have to find a special wagon and pony just for her.\" Hester laughed.\n\n\"You're looking pretty Halloweeny yourself, all orange and black.\"\n\n\"Oh, this is just my warm-up. Next week I'll be out here in my witch's costume.\"\n\n\"So long as you don't scare customers away.\"\n\n\"I could be a Halloween fairy except I've never seen a Halloween fairy.\"\n\nThey kept chatting as Harry picked out two succulent squashes, then paid at the cash register run by Lolly Currie, a young woman looking for a better job but making ends meet at Hester's stand until then.\n\nBack on the road, Fair grinned. \"That is the shortest time you have ever spent at Martin's Stand.\"\n\n\"Buddy Janss helped me out, because as soon as I paid for my squash, he came back to chat up Hester, about late produce deliveries. I swear, Buddy has put on more weight. His chins now have chins.\"\n\n\"Buddy may be fat but he's light on his feet. He was a hell of a football player in high school and college. It's a pity that retired linemen run to fat so often.\"\n\n\"Boxers, too.\" She watched rolling hills pass by.\n\n_\"Maybe you should go live with Buddy. The two of you could be Team Tubby.\"_ Tucker knew this would start a fight.\n\n_\"Don't,\"_ Mrs. Murphy counseled in vain.\n\n_\"Bubble Butt. Poop Breath!\"_ Pewter hissed loudly.\n\nHarry twisted around in the front seat just in time to see Pewter hook the dog's shoulder with one claw.\n\n_\"Ouch,\"_ Tucker yelped.\n\n_\"Next, your eyes.\"_\n\n\"Pull over, honey. There will be fur all over the car if I don't stop this right now.\"\n\nHe pulled over on the side of the road. The field on the north side of the two-lane road was jammed with corn. Morrowdale Farm usually put these fields in good hay, but this year row after row of healthy corn filled them. They had somehow escaped the small drought.\n\nOpening the door to again castigate the backseat passengers, Harry remarked, \"This has to be one of the best-run and prettiest farms in Albemarle County.\"\n\n\"Sure is.\"\n\nThey looked out to the scarecrow in the middle of the field, currently being mobbed by crows.\n\n\"I thought scarecrows were supposed to frighten crows,\" Fair said.\n\n\"Those crows are having a party. Look at that. Pulling on the wig under the hat.\" Harry laughed. \"What are all those birds doing?\"\n\nFair stepped out of the car to stare intently as a crow plucked out an eyeball.\n\n\"Honey, that's not a scarecrow.\"\n\n#\n\n\"Tucker, come back!\" Harry called to the corgi as the dog raced across the cornfield.\n\nFair, in his shock, hadn't closed the station wagon door, so all three animals had rushed out after deciding to see what was going on.\n\nThe corn rustled as the strong little dog bounded through.\n\nThe two cats also sped down a row, curiosity raging.\n\n\"Selective hearing.\" Harry shook her head as she followed, starting into a corn row.\n\n\"Honey, they'll be back. You should stay where you are, otherwise you might destroy footprints or some other kind of evidence.\"\n\nShe stopped, turned to face her husband. \"You're right.\"\n\n\"I'm not sure you want to see the corpse anyway.\"\n\nHarry leaned up against the Volvo. \"Death really is ugly and this one is probably especially so. But, Fair, why truss someone up like a scarecrow?\"\n\nHe folded his arms across his chest. \"Clever, really. How many people passed by this field on Garth Road? Plenty, I bet, and still no one stopped or called the sheriff's department. The only reason we did was because of the ruckus raised by our passengers, and then the crows caught our eye, and... well.\"\n\nAs the married couple waited for the sheriff's department to arrive, the three investigating animals reached the base of the scarecrow.\n\nA blue-black crow perched on the straw hat looked down. _\"Beat it!\"_ he squawked.\n\nMrs. Murphy knew she could climb the dead man's leg if need be, so she stood on her hind legs reaching far up, feeling the cold flesh under the faux scarecrow's pants. _\"I can climb up and shoo all of you away,\"_ she threatened the birds.\n\nA second crow in this mob, on an outstretched arm, gibed, _\"Go ahead. We'll fly away, circle, and come right on back.\"_\n\nThe first crow opened his wings to their full span, the light picking up the blue highlights. _\"What do you want with this feast? Cats don't eat carrion.\"_\n\nPewter ignored the question and asked one of her own: _\"Did you see the scarecrow being set up?\"_\n\nThe second crow spoke. _\"No, but he hasn't been here long. We caught a whiff as we flew over this cornfield on our way to Shelford Farm. When we tear off a juicy piece of meat, some blood still drips.\"_\n\nFew scarecrows are well dressed. Neither was this one. It wore a drab, wrinkled shirt over a red undershirt. Worn, old pants, rope for a belt, took care of his bottom half. Old work boots, the sole separated from the left one, covered his feet. The straw hat, edges frayed, hatband missing, gave the fellow the final country touch.\n\nAs blood pools in the extremities, the crows provided valuable information. The scarecrow wouldn't show the signs of rigor mortis because the body was tied, arms outstretched, legs tied down, too. No blood was moving, the body temperature had cooled down, but this was a fresh kill, relatively.\n\nMrs. Murphy noticed that the eyes had been plucked out and a lot of flesh had already been eaten off his face and hands. Eventually, the crows would have torn through the clothing.\n\n_\"Did you smell another human?\"_ the tiger cat asked.\n\n_\"No. The sun had been up about an hour. What we smelled was him,\"_ the first crow reported, his olfactory powers acute, especially for blood and meat.\n\n_\"Without his eyes, I can't tell if he was strangled,\"_ Pewter matter-of-factly announced. _\"They'd be bulging and bloodshot.\"_\n\n_\"Eyes are so tasty.\"_ A smaller crow opened his beak wide. _\"A real delicacy.\"_\n\n_\"Any idea how he was killed?\"_ asked Mrs. Murphy.\n\n_\"You didn't hear him scream, did you?\"_ Pewter, normally not interested in much besides her own meals, was oddly thrilled at having discovered such an unusual murder.\n\n_\"How could we have heard him scream?\"_ a young crow replied. _\"He was dead and gone by the time we found him.\"_\n\n_\"Eat what you can, because the sheriff is on his way. He'll cut him down,\"_ Mrs. Murphy advised.\n\nTucker sniffed the bottom of the stake, sniffed the corpse's shoes, then picked up the diminishing odor of a set of rubber boots. Raising her nose, she sensed the smell moving away from the body, then, nose to ground, she began to track, the cats in her wake. As the three friends stuck to their trail of the pair of rubber boots, presumably those of the person who had carried the body, the crows burst out singing a song whose refrain was _\"Oh, those beautiful eyes, those great big beautiful eyes.\"_ Then they burst into raucous laughter.\n\n_\"Gross,\"_ Tucker said.\n\n_\"Yeah.\"_ Pewter looked back. _\"Twisted. They're really twisted.\"_\n\n_\"It's the killer who's twisted,\"_ Mrs. Murphy sensibly replied as she, too, kept her nose down.\n\nThe three followed the line until it came out to the side of the road, where there was a small stain that smelled like motor oil.\n\n_\"Every third person wears rubber boots around here when it's wet.\"_ Tucker sat down. _\"But I think this is the spot where the scarcrow's companion parked, then carried out his body from here.\"_\n\n_\"A strong person. They don't call it dead weight for nothing,\"_ Mrs. Murphy noted.\n\n_\"She's red in the face,\"_ Pewter said, referring to Harry, calling their names in the distance with increasing frustration. _\"We'd better go back to the wagon.\"_\n\nWhen they got back to the Volvo, Harry scooped them up, put them in the back, and closed the door. \"Curiosity killed the cat,\" she huffed, unaware of the irony of Harry Haristeen making such a statement.\n\n_\"Yeah, yeah.\"_ Pewter put her paws on the window just to make a smear.\n\n_\"She's upset.\"_ Tucker put her head on her paws.\n\n_\"Pop is, too. Humans can't face death.\"_ Pewter was right about that.\n\n_\"This is murder. Worse.\"_ Mrs. Murphy heard cars coming closer.\n\n_\"I found a head in a pumpkin, remember?\"_ Pewter reminisced.\n\n_\"We've heard that story a hundred times,\"_ Tucker grumbled, heading her off. _\"This is just as weird. And we were first on the scene. I mean, after the crows and the killer.\"_\n\nThe sheriff's car rolled up. Sheriff Rick Shaw stepped out from the driver's side and Deputy Cynthia Cooper emerged from the other. Cooper\u2014Harry never called her Cynthia\u2014rented the farm next to Harry's farm, the old Jones homeplace. The two women had become friends.\n\nThe two law enforcement officers carefully pushed through the late-maturing corn, the leaves rattling, ears full on the stalks. They looked downward as they walked but were rows away from the footprints that Tucker had found.\n\nHarry and Fair stayed with their station wagon as instructed. They could see how carefully Rick and Cooper looked about, conferred, looked down. Then the two circled the scarecrow. The crows flew in loops around them.\n\nOne crow dive-bombed. _\"Leave us alone!\"_\n\nCooper ducked, then waved her hands at the noisy birds. \"Damn.\"\n\nRick, tempted to take out his sidearm and fire, did not. No need to alert the residents of Morrowdale or anyone else at this moment.\n\nAfter twenty minutes, they returned.\n\n\"Do you know who it is?\" Harry asked.\n\nCooper shook her head. \"The face is pretty well gone. But he's youngish, and had been in fairly good shape. Look, why don't you two go on home? I'll get a statement from you later. If there's anything of immediate importance, tell me now. Otherwise, you'll get caught up in the removal team, the forensic team, and, of course, the news team, as they know where we are every minute thanks to being able to listen in to all our calls.\"\n\nTucker barked from the car. _\"There's a drip of oil just up the road. And footprints in a corn row.\"_\n\n_\"Save your breath,\"_ Pewter, paws on the windowsill, counseled.\n\n_\"They'll find the footprints,\"_ Mrs. Murphy said. _\"The humans will crawl over that cornfield and the two of them will be down at Morrowdale questioning everyone and going through the barns and sheds.\"_\n\nHarry and Fair drove west down Garth Road, then turned toward Crozet, heading south. The Blue Ridge Mountains were now on their right. They passed a large cattle farm, Dunrovin, with Herefords in the pastures; they passed by rolling acres of grapes, the land dotted here and there with old farmhouses and the occasional new structure, always sited for the view.\n\n\"You okay?\" Fair asked.\n\n\"Yeah. You?\"\n\n\"Yeah.\" They passed the apple shed now housing Chuck Pinell's leather shop. \"Yeah, but...\" His voice trailed off.\n\n\"Creepy.\" Harry shivered.\n\n\"People kill for lust, for love, in a fit of anger, or for money, and some because they are plain nuts,\" Fair said.\n\n\"You'd need to be pretty demented to take someone you've just killed and tie them up as a scarecrow, especially with Halloween just around the corner,\" Harry said. \"Or it could be a side show designed to cover up another crime. Think about it.\"\n\nFair couldn't take his eyes off the road, because it was two-lane and treacherous. \"I'd rather not.\" As he continued, his voice was firm, for his wife was more curious than the cats. \"You don't need to think that much about it either. It was a shock. An unfortunate discovery. We can say a prayer for the victim and then go about our business.\"\n\n\"Prayers are wonderful. So are results. Who speaks for an innocent victim? Until I know more, I'm assuming he's innocent.\"\n\nKnowing he was losing the battle against his wife's curiosity, he calmly replied, \"Just leave this to Rick and Cooper.\"\n\n\"Of course.\"\n\n_\"Boy, was that a fib.\"_ Pewter giggled.\n\nThe others laughed with her.\n\nHarry then said, \"Whoever did it has quite the imagination.\"\n\n\"The last thing this county or state needs is an imaginative killer,\" said Fair, \"especially if you're one of the victims.\"\n\n\"Fair, think about this: Don't most murderers try to dispose of their victim's bodies so no one finds them? Or if it's a crime of anger or passion, they run away and leave it, but they don't turn the corpse into a scarecrow or a public display. Whoever did this had time to plan it out.\"\n\n\"I guess.\"\n\n\"So I don't think it's a crime of passion.\"\n\n\"Unless the killer meant to make a mockery of the corpse.\" Fair braked at the stop sign at the Amoco station in Crozet. \"Dammit. Now you've got me thinking about it. Let's just let it all go.\"\n\n\"Mmm.\" Harry was already off and running.\n\n#\n\nOn the kitchen table, Pewter flopped on her side, her tail gently swaying. She thought this her best angle. Mrs. Murphy and Tucker thought otherwise.\n\nAs Harry opened the oven door to pop in a casserole, Pewter lifted her head.\n\n_\"I know you're making that for me.\"_ Her voice hit the dulcet-tone register.\n\nMrs. Murphy and Tucker, each curled up in their faux-sheepskin-lined animal beds by the door to the back porch, observed with amusement.\n\nMrs. Murphy imitated the gray cat's voice: _\"I am the most loving kitty in the world.\"_\n\nPointedly ignoring this, Pewter again sweetly meowed. _\"I could use a little tuna until the casserole is ready.\"_\n\nHarry closed the door, set the timer, then turned to behold the cat, whose head was now raised, tail moving a bit faster. \"Does smell good, doesn't it, Pewts?\" Harry said. She caressed the cat's silken fur.\n\n_\"I have suffered a terrible shock,\"_ Pewter panted, pushing her head into Harry's hand. _\"The sight of a shredded face. Crows devouring human fleshbefore being impertinent to me. If one of those vile birds had dropped even two feet, I could have leapt up and torn it to bits.\"_\n\n_\"You're laying it on a little thick.\"_ The dog raised her head.\n\n_\"Shut up, Bubble Butt. If she breaks out the cookies, you owe me big-time.\"_ Pewter rolled onto her back, cocking her head to one side.\n\n\"All right.\" Harry opened the treat cabinet, counted out two greenies, and gave them to Tucker. Next she opened a bag of cat treats in the shape of fishes. She gave half of these to Pewter, then walked over and gave the rest to Mrs. Murphy.\n\n_\"You owe me!\"_ Pewter cried in triumph as she gulped her tiny yellow fish.\n\nHarry\u2014unaware of the exchange, it sounded like meows and catcalling to her\u2014walked back to her husband's small office in the old farmhouse.\n\n\"Forty-five minutes,\" she told him.\n\n\"Huh.\" He looked up from the screen. \"Okay.\"\n\n\"Work?\"\n\nFair was the best equine veterinarian in central Virginia.\n\nHe smiled sheepishly. \"No. That's the trouble with the Internet. Easy to get sidetracked.\"\n\n\"And?\" She came up behind him, placing her hands on his broad shoulders.\n\nNot an inch of fat on the man.\n\n\"Uh, well, I've been kind of reading about bizarre murders. This website has examples going back to the eighteenth century. Really weird things, like duels fought in costumes or heads put on London Bridge with fake crowns. I guess that's political. But here's one from Wisconsin in the 1850s that caught my eye: A guy would kill men for no particular reason, or at least one no one could find, and he'd put them in a boat, push it out onto Lake Michigan, and set it afire. A Viking funeral. His victims were all men he had admired.\"\n\n\"Sometimes I wonder when I hear or read these things whether anyone is normal.\"\n\nFair leaned back in his chair. \"I guess that's debatable.\" He rolled his chair around to face her, the rollers clicking on the hardwood floor. \"I guess I can't fuss at you. Sometimes I'm a little too curious myself.\"\n\nShe kissed his cheek. \"Makes me feel better,\" she said, then headed to the kitchen.\n\nHe followed the wonderful aroma of her chicken casserole, her mother's recipe.\n\n\"That scent brings back so many memories,\" Harry said. \"And, hey, Halloween is what, two and a half weeks away? More memories.\"\n\n_\"Heads in pumpkins,\"_ Pewter blathered.\n\nTucker listened, then put her head back on her paws. _\"I thought they were about to discuss food. They'd be much better off focusing on things that matter rather than random corpses.\"_\n\nThe tiger cat silently agreed as she left her own bed to curl up with the corgi.\n\nBoth animals felt the chill of premonition.\n\n#\n\nThe day after the grisly discovery, the temperature dropped twenty degrees and rains came. Like all farmers, Harry had a rain plan. There were the chores that one did no matter the weather, and then there were those set aside for downpours.\n\nThe tack room in the old barn doubled as her office. If she had fixed up an office in the house, she knew she'd bother Fair or vice versa. The tack room made sense plus she could smell the leather, the horses, and their sweet feed. She liked sitting in the old knotty-pine room, the size of two good stalls, twenty by twenty-four feet. One wall held saddle racks and bridle holders. Under those items rested her personal tack trunk, as well as her husband's. Each horse stall also had a tack trunk in front of it, carrying items Harry felt should be separated from the main tack room. And each tack trunk hid treats: dried apples, special horse cookies. When a lid was lifted, the nickering started.\n\nAt fifty-two degrees Fahrenheit, the temperature hit the perfect mark for the horses. Most of them were turned out in the rain, which was now steady but soft. Once on the other side of the equinox, Harry switched her schedule, bringing the horses in at night and turning them out during the day. Horses needed to move about.\n\nPewter, splayed out on Harry's tack trunk, which was covered with a lush saddle pad, had no such inclination. Tiny snores emitted from her body. Mrs. Murphy, wide awake, sat on the desk surface just inches from her human, who was trolling the Internet and considering seed purchases for the spring. If Harry ordered now, she would benefit with a ten percent discount from Southern States, the large supplier. She would always double-check with Augusta Co-op to see if those prices were better.\n\nBrow furrowed, chin resting on her hand, Harry scrolled through various seed types as the tiger cat peered at the screen, too.\n\nTucker, knocked out in her plaid bed under the desk, was as oblivious as Pewter.\n\nAn old massive teacher's desk, painted hunter green, a tall wooden file cabinet, and two director's chairs as well as the wooden teacher's chair in front of the desk took up the space opposite the saddle wall.\n\n\"I can't believe this,\" Harry said to Mrs. Murphy. \"They say they've developed a winter-resistant Bermuda grass and it's only $123.50 per hundredweight. First, I don't believe it. Second, that is an outrageous price. Bermuda grass isn't as good as alfalfa or orchard grass.\"\n\n_\"Then why use it?\"_ The cat had a practical turn of mind.\n\nHearing the clear meow, Harry looked into the bright green eyes. \"I love you, pussycat.\"\n\n_\"I love you, too,\"_ the cat replied as the attractive forty-one-year-old woman returned to her task.\n\n\"It's the terrible summers we're having, Murphy. That's what makes Bermuda grass useful. We now need some kind of forage that can withstand the heat and drought conditions. Unfortunately, it dies in the winter. It looks as though fescue, orchard grass, and timothy die in summer's searing heat, but they do not. They burn off, or wilt. The pastures are brown, but with a bit of moisture or a snowy winter, those grasses pop back up. Of course, clover really holds water in nodules.\" She nattered on, captivated with grass crops, as she had been since she was a tiny girl following her father around the farm.\n\nWhile not enraptured by grasses, legumes, or corn, the tiger cat proved a good listener. Corn appealed to her because it brought in mice, foxes, and other animals seeking the high calories. Then she remembered the scarecrow and the crows.\n\nBoth Harry and Mrs. Murphy looked up when they heard a motor, then a door slam. Harry hurried outside to catch whomever it was before they ran through the rain to the back door.\n\n\"Coop, I'm in the barn,\" Harry hollered.\n\nThe tall blonde deputy smiled and hurried into the barn. \"Can you believe how much cooler it is all of a sudden?\"\n\nAs she walked down the center aisle to the tack room, Harry replied, \"October.\"\n\nOnce inside, Cooper sank into a director's chair. She leaned over to peer under the desk.\n\n\"So much for Tucker being a guard dog.\"\n\nHarry laughed. \"She really is dead to the world, isn't she?\" Then she indicated the fat gray cat on the tack trunk. \"Another one.\"\n\n\"You need to tie a roller skate under Pewter's stomach.\"\n\n\"Coop, that's a great idea.\"\n\nMrs. Murphy giggled.\n\n\"How was church this morning?\" Cooper inquired. \"I overslept.\"\n\n\"Herb gave a really good sermon, as always. He talked about harvesttime and read some passages from the New Testament about gathering. He always holds my interest.\"\n\n\"He makes it real. Not a bunch of rules.\" Cooper rented the Reverend Herb Jones's homeplace, as the pastor of St. Luke's Lutheran Church had moved in to the beautiful vicarage on the church grounds.\n\n\"Can you imagine building St. Luke's? This used to be the Wild West. The Monacans\"\u2014Harry mentioned an Indian tribe\u2014\"weren't happy to see us.\"\n\n\"Still aren't, I bet,\" Coop said.\n\n\"Small wonder.\" Harry inhaled. \"Anyway, Albemarle County didn't really start rolling until after the Revolutionary War. That's when the first stone was laid for St. Luke's. Don't you love the church building that evolved?\"\n\n\"I do. I love people that evolve, too.\" Coop sighed.\n\n\"Okay, what's on your mind?\" Harry knew her friend and neighbor well enough to tell from the tilt of the conversation that Coop was turning something over in her mind.\n\n\"University of Virginia football, for one. Every time there's a home game, it's one scrape after another, plus we have to really keep our eyes out for the kids who are flat-out loaded. Hey, I was in college once, too. I don't mind if you get drunk. Everyone has to learn that one, how to handle the bottle, but I don't want them behind the wheel of a car.\"\n\n\"That's not going to change. Do you know who the scarecrow is yet?\"\n\n\"No. We found a class ring.\" Cooper leaned closer. \"The crows had eaten the flesh from his fingers and hands. It slipped off. Actually, crows like shiny things. If we hadn't gotten there when we did, that Virginia Tech ring would be in a nest somewhere.\"\n\n\"Did it have initials and a year inscribed?\"\n\n\"J.H., 1998.\"\n\n\"Did you find anything else?\"\n\n\"Nothing. Pockets empty. But we'll get an ID soon enough. Well, I speak too soon. But the faster you have an ID, the easier some links of inquiry are. For all we know, the killer is in Paraguay by now.\"\n\n\"I don't think so.\" Harry leaned forward.\n\n\"Actually, Harry, I don't either.\"\n\n\"I'll see that scarecrow forever. The sight itself was unpleasant enough, but the whole idea of it is really disturbing, you know?\"\n\n\"I do.\" Cooper sat quietly for a moment. \"Pewter snores.\"\n\n\"Yes, she does.\" Harry laughed.\n\n_\"No one wants to sleep next to her,\"_ Mrs. Murphy informed them to no avail.\n\n\"Where's Fair?\"\n\n\"He got an emergency call. Sometimes that man works around the clock. It's a good thing he loves what he does.\"\n\n\"Me, too,\" Cooper chimed in.\n\n\"The horrible part of police work, like finding a corpse scarecrow, doesn't get to you?\" Harry wondered.\n\n\"I can't say that finding murder victims thrills me, but finding their killer does.\"\n\n\"You know I read the paper, magazines. There are articles claiming that there are identifiable traits in children who grow up to become violent. Some writers even suggest putting them away before a crime has been committed.\"\n\n\"Even if we could violate individual rights that way, there would still be murders,\" Cooper stated.\n\n\"The human condition?\"\n\n\"Unfortunately, yes.\"\n\n#\n\nPulling his veterinary truck up to the house that Sunday evening, Fair opened the truck's door, then got out and wearily leaned against it.\n\nTucker, hearing the motor, dashed out the house's animal door to greet him. _\"Hi, Pop. I missed you. I'm glad you're home.\"_\n\nThe tall man bent down to pet the dog. \"Hey, buddy.\"\n\n_\"You're covered in blood and you're sad and tired. Can I help?\"_ Tucker implored him with her soft brown eyes.\n\nFair stroked the smooth head once more before standing. Taking a deep breath, he walked to the house.\n\nIn the kitchen, Harry heard his footfall but didn't look up as she stuffed a Cornish hen. \"First frost tonight, I think.\"\n\n\"Feels like it.\"\n\nShe turned and took in his bloodied, bedraggled appearance. \"Oh, no! Honey, is the horse all right?\"\n\nHe sank into a kitchen chair. \"Couldn't save her. She'd nicked her aorta. By the time Paul Diaz found her out in the pasture, she'd already lost so much blood. What a beautiful filly.\" He rested his head in his hands. \"The Medaglia d'Oro filly.\"\n\n\"Oh, no.\" Harry washed her hands. \"Big Mim had such hopes for her.\"\n\nMedaglia d'Oro was a Thoroughbred stallion with a big career. Even in these hard economic times, his stud fees had been creeping up, and Big Mim had selected a mare to breed to him. He'd been siring winners on the track. The Queen of Crozet, as she was called behind her back and even to her face, had a knack for breeding, whether for steeplechasing or flat racing. It ran in her family. Her mother had it, too, and Big Mim passed it on to her daughter, Little Mim, who had recently given birth to a boy. Perhaps the magic would pass to him.\n\n\"That filly was one of the most correct horses I've ever seen. We all thought she was bound for greatness.\"\n\nComing from Fair, that meant something.\n\n\"Is Big Mim okay?\"\n\nHe thought a moment. \"She's a horsewoman. She accepts fate. But she's upset. Seeing any animal you love die...\" He shrugged.\n\nHarry put her arms around him. \"I know you did your best. I'm so sorry, honey.\"\n\n\"The poor girl was down in the pasture. She'd lost so much blood, she couldn't stand up, so I ran out, cleaned the wound, and she died while I was stitching her up. If she'd lived, I think we could have rolled her onto a canvas and dragged her into the barn, gotten her in a stall. I was prepared to give her massive transfusions and drip antibiotics into her. Whatever it took.\"\n\n\"Big Mim would have sat up with you.\" Harry warmed at the thought of the svelte septuagenarian sitting in the aisle, wrapped in a blanket.\n\n\"She would; Paul would, too. I think even Jim\"\u2014he named Big Mim's husband, who was not a horse person\u2014\"would have taken a turn.\"\n\n\"Me, too.\" She kissed him. \"You are such a good veterinarian. Such a good man. I love that you care.\"\n\nHe kissed her hand. \"Most of us do. A person should only go into medicine, veterinary or human, if they really care.\"\n\n\"Well, that's a subject for a long discussion, and my money is on the vets.\" She kissed his cheek again.\n\n\"Let me get out of these clothes, shower. I'll throw them in the washer.\"\n\n\"Fair, how did it happen?\"\n\n\"No idea. Honestly, honey, if I knew how half my patients did the stuff they did to themselves, I would be a genius. Horses are pretty careful animals but they can do the dumbest things sometimes, and she was young.\" He smiled. \"That doesn't help.\"\n\n\"Doesn't for us either.\" Harry stopped. \"Except now that I am officially middle-aged, I pray the young will be a little wild, take some crazy chances, think the unthinkable.\"\n\nHe stood up. \"You still do. Every now and then, I really have no idea what's going on upstairs.\" He tapped his head with his forefinger.\n\n_\"He's right,\"_ Pewter, in her kitchen bed, remarked to Mrs. Murphy, who sprawled in Tucker's bed as the dog followed Fair out of the room.\n\n_\"Poor Fair.\"_ Mrs. Murphy ignored Pewt's comment about Harry\u2014not because it would start an argument, but because she knew it was true.\n\n_\"Tucker will cheer him up,\"_ said Pewter.\n\n_\"We could, too,\"_ said the tiger cat. _\"We could take our catnip mouse in the bedroom and throw it around. Fair always laughs when we do that.\"_\n\nPewter was firm. _\"I'm not getting out of this bed unless food is involved.\"_\n\n_\"Right.\"_ Mrs. Murphy smiled at her friend.\n\nHarry slid the two small Cornish hens into the oven. She'd made a salad earlier. Neither she nor Fair ate heavy rich foods and this would be a good supper for them. Also, Harry lacked the time to prepare complicated meals.\n\nThe wall phone rang.\n\nShe wiped her hands on a dish towel and picked it up. \"Haristeen.\"\n\n\"Cooper.\"\n\n\"Hey, if you haven't eaten supper, come on back. I'll have plenty.\"\n\n\"Date tonight.\"\n\n\"You stopped by here and you didn't tell me that? I am wounded, deeply wounded,\" Harry teased.\n\n\"Forgot. It's a first date. We'll see. I'm just glad I have a night off. I've worked the last three weekends.\"\n\n\"The county really needs to hire more people, don't they?\"\n\n\"No money. I called to tell you, since you and Fair found him: We have an ID on the scarecrow.\"\n\n\"That was fast.\"\n\n\"The ring really helped, and we have super people sitting behind those computers and making calls. I don't think people in the county have any idea how good their sheriff's department really is.\"\n\n\"It's kind of like making a will. No one thinks about it until they have to, I guess. So?\"\n\n\"Joshua Hill, graduated from Tech in 1998. Accounting major. Worked for a large firm in Richmond for four years, then hung out his own shingle in Farmville, where he quickly built up a large clientele. Unmarried. Hobbies: fly-fishing, country music concerts.\"\n\n\"How did you get so much information so quickly?\"\n\n\"Caitlin did,\" said Coop, referring to one of the criminologists on the staff, a fantastic researcher. \"She went online, got the 1998 yearbook, and started looking. Even though our victim was torn up, we had a decent description of height, weight, approximate age, and hair color, and she narrowed it down to a few possibilities. Then she started calling places where the potential matchups worked. Josh didn't come into his office on Friday, nor did he call, which his assistant found odd but she wasn't overly concerned. I'm going down there Tuesday. Haven't seen Farmville in a long time, and I hear Longwood University has grown. It's a pretty school.\"\n\n\"Yes, it is.\" Harry paused. \"Accountants don't get themselves murdered too often, do they?\"\n\n\"No. This is a curious case.\"\n\n\"Who's the date?\" Harry just had to know.\n\n\"Barry Betz, new batting coach for UVA. First year here. This guy has the sweetest smile. He lights up a room.\"\n\n\"Hope it's fun. I'd go out with him just because of his name.\" Harry smiled. \"Thanks for calling me.\"\n\nFair walked in, clean, wearing a T-shirt and sweatpants. When he sat down, she told him about Josh, the dead accountant.\n\n\"Maybe he was cooking the books,\" said Fair, after devouring half of the Cornish hen Harry had cooked for dinner.\n\n\"If that's a motive, wouldn't there be so many more dead people in America, especially in certain professions and industries?\" she remarked, gazing at him across the table.\n\n\"You've got a point there.\" Fair was feeling better and so was Tucker, wedged between his slippered feet.\n\n\"Sometimes I think about why people commit crimes, not the impulsive ones but the premeditated kind,\" Harry said. \"I bet once you're free from society's rules or an ideology, anything is possible. The world is your oyster.\"\n\n\"Never thought of it that way.\" He stopped for a second. \"This hen is wonderful.\"\n\n\"Oh, thanks. Miranda's recipe.\" Harry knew any recipe from Miranda would be delicious. \"It's kind of like offense and defense. The criminal is the offense, so that split-second advantage is his. He knows what he will do. The law has to react.\"\n\nShe neglected to add that the law could only react if they knew what was going on.\n\n#\n\n\"I ought to arrest you, throw the book at you!\" Cooper shouted at Harry two days later, on the street outside Joshua Hill's office in Farmville.\n\n\"For what?\"\n\n\"Stalking?\"\n\n\"I came to shop. You have no grounds for suspicion.\"\n\nThe attractive police officer shook her head. \"Harry, how can you look in the mirror after a lie like that?\"\n\n\"Farmville is famous for its furniture warehouses. I especially like Number 9, so named since all the warehouses had numbers on the outside, easy to see. And come on, Coop, you know I've been wanting to get down here for months. It's been one thing after another.\"\n\nCalmed down a bit, Cooper replied, \"You didn't have to come today. You want to know what I found out at Hill's office, and it was a big zero.\"\n\n\"His assistant wouldn't talk?\"\n\n\"No.\" The lean woman put her elbows on the hood of her unmarked car, called a slicktop. \"She kind of just answers the phone. She has a million pictures of her grandchildren on her desk. It's fair to say she isn't overly involved in her work.\"\n\n\"But she did note that her boss hadn't called in?\"\n\n\"Yes, but she also said a lot of times he worked at home, or he called on clients at their offices.\"\n\n\"Did she give you a client list?\"\n\n\"Harry!\"\n\n\"Hey, you wouldn't be on this case if I hadn't found the body.\" Though she knew this wasn't one hundred percent true, Harry still pressed her point. \"And murder is a lot more exciting than picking up drunk frat boys who then puke all over the back of your squad car.\"\n\n\"No drunk has ever puked in my car.\"\n\n\"Now who's the liar?\"\n\nCooper took out her service revolver. \"I put this to their temple, and tell them if they throw up in my car, I will blow their brains out.\"\n\n\"I can see how that might work.\"\n\n\"One time,\" the cop mused, \"I had to pull over 'cause a guy pretended he was going to be sick, and then he ran into the woods.\"\n\n\"What did you do?\"\n\n\"Followed until he tripped and fell. It was pitch-black. Then he threw up. Drunks are truly disgusting.\"\n\n\"Mmm. Anything of interest in Hill's office?\"\n\n\"You enter into a small waiting room. There are a few nature prints, both there and in his office. His desk didn't have a single paper on it. I'll get the Prince Edward County Sheriff's Office to go over it all.\" Cooper gave out just enough information to tease Harry.\n\n\"You'd think an accountant would have piles of papers on his desk,\" said Harry.\n\n\"Or some, anyway,\" Coop agreed. \"Though clients send so much stuff through email.\"\n\n\"Was his computer in the office?\"\n\n\"Yes. Obviously, I can't take it without jumping through all the proper legal hoops, but I've already set that in motion. A forensic accountant could find out if Hill was doing anything suspicious. Anything you put into a computer can be dug out. Best to not put it there in the first place. Of course, I don't know that this murder has anything to do with numbers. Truth is, I don't know what this murder is about at all. Usually, I get a hunch.\"\n\n\"Don't you want to know what I found in Number 9 warehouse?\"\n\nCooper gave her a sharp look. \"If you did find something, you'll complain about the price.\"\n\n\"Follow me.\"\n\nHarry climbed back in her Volvo station wagon. She'd parked outside Hill's office once she'd cruised through her favorite warehouse so her excuse for being in Farmville wouldn't be a total fib.\n\n\"Why should I follow you to see furniture?\"\n\n\"Cooper, please follow me.\"\n\nSomething in Harry's voice made Cooper close the door to the slicktop, turn the key, and tag behind her neighbor.\n\nOnce at the warehouse, Harry opened the showroom door for Cooper. \"Ready?\"\n\nCooper stepped inside and stood a moment. \"This place is huge.\"\n\n\"Four big floors, I think. But what I want you to see is right here on the first floor, the flashy showroom floor.\"\n\nBriskly walking, Harry reached the middle of the cavernous building. Pumpkins, mums, and Halloween witches overhead all drew attention to a stunning country kitchen with a solid oak table, easily seating twelve.\n\n\"Look.\"\n\nCooper followed Harry's forefinger to a figure in the corner of the display room, among ghosts, more pumpkins, and little goblins popping out of the pumpkins.\n\n\"Jesus.\" The officer whistled.\n\nIn the middle of this lively display was a scarecrow with a drab undershirt, old pants with a rope belt, worn-out work boots, the sole separating from the left one. A straw hat topped it off. An exact replica of Joshua Hill.\n\n#\n\nSquares of fabric and seed catalogues littered Cooper's desk at police headquarters. Peering at her computer screen, she scanned a list of clothing manufacturers.\n\nEyes watering, she clicked off her computer, got up, stretched, and went outside. She tapped out a long coffin nail, lit it, watched the sky shift colors.\n\nSometimes taking a smoke break cleared her mind. Along with an increased chance of lung cancer, she'd get better ideas. After a few thoughtful minutes, she ground out the stub, walked back in, tidied up her desk, and left for the day. She'd worked overtime this Tuesday but didn't mind.\n\nDriving home, she noticed the last of the day birds returning to their nests. A few bats were already out and about. She liked bats; they kept the bug population down.\n\nThinking about bugs and pests snapped Coop back to this strange scarecrow case. After Harry had pointed out the unsettling look-alike dummy, Coop had sent her on her way, then had gone to the store's office and asked to see their display person. In Coop's mother's day, that individual would have been called a window dresser regardless of what part of the store they decorated.\n\nThe display person was named Melinda and the young lady worked in the store full-time. Well groomed, well spoken, she was well dressed, good-looking.\n\nCooper was good at relaxing people. She sat the light-blonde-haired woman down in the middle of one of her own displays. After a bit of chitchat, she asked Melinda about the scarecrow.\n\n\"Don't you love the boots?\" she gushed. \"They aren't as easy to find as you might think and...\" She trailed off. \"Sorry. I get enthusiastic about these things.\"\n\n\"No, no, that's fine. But did you work with anyone else on that particular display?\"\n\n\"No. I do all the work here in Number 9. The guys help me move the furniture but that's it. Actually, I like that I can let my imagination go.\"\n\n\"Would you have any way of knowing if someone asked specifically about that scarecrow?\"\n\n\"No. No one asked me, but, Officer Cooper, there's a great volume of foot traffic through these warehouses, especially on the weekends. People come from all over Virginia, even from out of state. Lots of volume.\"\n\n\"Did you have any particular inspiration for the clothing?\"\n\nThe young lady thought for a moment. \"No. I mean, just the scarecrows I saw as a child.\" She grinned. \"Can't be a scarecrow without a big straw hat.\"\n\n\"Would you mind giving me a list of where you bought the pants, the shirt, the boots, and the hat?\"\n\n\"Not at all. If you wait a minute, I can print it out for you.\" Melinda hurried to her office, returning to Cooper in ten minutes, just enough time for the deputy to see an end table she wanted.\n\n\"Thank you,\" Cooper said as she quickly perused the list Melinda had given her. \"I guess anyone could buy this clothing.\"\n\n\"That's the idea.\" Melinda smiled.\n\nCooper smiled back. \"I won't take up much more of your time. Did you know Joshua Hill?\"\n\n\"No.\"\n\n\"Do you live in Farmville?\"\n\n\"No. I live on the eastern edge of Buckingham County and drive in. I'd like to move down here when I can afford it. May I ask you a question? What's this about?\"\n\n\"Friday, October 11, Josh Hill, who had an accounting business near here, was murdered in Albemarle County. His dead body was found on Saturday. It was a big football weekend. Didn't make the front page of the papers until Monday.\"\n\n\"That's horrible.\"\n\n\"He was dressed just like your scarecrow here.\"\n\nMelinda's eyes popped. \"How awful.\"\n\n\"Well, yes, but it must have been a lot of work to dress him up, I would think.\"\n\nMelinda paused. \"The straw man took me about an hour to get all the clothes on, then make a head, paint on features. He's not exactly scary or gruesome. I mean, a scarecrow isn't, but I think a human one would be awful. I wish I could help you, but anyone could have come in and seen my scarecrow, and the clothing isn't unique or anything.\"\n\n\"Well, I've taken up enough of your time.\" Cooper handed her her card. \"If you should think of something or if someone as to come in here and seem really focused on your display, let me know.\"\n\n\"I will.\"\n\nLater, Cooper replayed that conversation. She felt the young woman was telling the truth. It had been a shot in the dark, but that's what she did. Lots of shots in the dark, lots of scraps of paper, old receipts, chewing gum wrappers, computer records, if she could get them. The endless gathering of data, most of it useless. But it only takes one perfect clue to point you in the right direction.\n\nAfter being sent home, Harry drove leisurely through Dillwyn, then west through Scottsville on Route 6. At Route 151 she turned right, right again on Route 250 to head home to Crozet. The late afternoon sun's magical light enhanced every field and stone wall, even those raggedy few gas stations on the way.\n\nTuesday, October 15, was just a gorgeous day, the kind that makes one forget the suffocating heat of summer or the soon-to-come frigid winter days.\n\nHester Martin's stand stood up ahead, decorated even more lavishly than usual. Two farm trucks were unloading produce and there stood Hester, in witch's costume, directing the men. Next to Hester a well-groomed black mini schnauzer kept an eye on the men.\n\nLaughing, Harry parked in the lot.\n\n\"Woo woo,\" Harry called out as she got out of the car.\n\nHester turned and picked up her broom, to shake at her.\n\nThe deliveryman closest to Hester ducked.\n\n\"Olin, don't you dare drop one apple,\" she admonished the man.\n\nJake, in the bed of his truck, leaned over with another wooden crate of plump green apples. \"You fly with that broom, Hester?\"\n\n\"I've got my pilot's license,\" she sassed. Then, to Harry, she said, \"Girl, you just go on and pick out whatever you want. I can't leave these boys for one minute. They need constant supervision. Heidi is helping.\" Hester indicated the schnauzer. \"She belongs to my old friend Cindy Walters, who's around here somewhere.\"\n\nJake rolled his eyes, saying nothing.\n\nThe other fellow on the ground, Greg Perez, carefully carried pumpkins to a pile he was building. No point in having Hester cuss him out, plus she might give him a big tip.\n\nInside the other truck, his partner, Stafford Schikel, groaned as he lifted another major pumpkin. \"Hester, these are the best pumpkins you've ever seen.\"\n\n\"Big,\" she replied simply. \"Did you cheat and put grow dust on them?\"\n\n\"No. You know we do everything organic. We lose a lot to worms, birds, and rats because of it and you.\" He grinned. \"You only buy organic.\"\n\n\"You charge me enough.\" She hadn't put her broom back.\n\n\"You want organic pumpkins, you pay.\"\n\n\"Yeah. Yeah.\" She pushed her pointed hat a bit to the side. \"You know I go by your fields, I get out and smell everything. I run my fingers over the skins.\"\n\n\"We know.\"\n\nShe'd told them this ad infinitum.\n\nOlin picked up another apple crate and asked, \"Ever find anything?\"\n\n\"Not at either of your farms, but let us never forget the owner of the organic store in Charlottesville who got busted for lying. His stuff was no different than Food Lion's.\"\n\nGreg couldn't help but tweak her. \"Food Lion is a good company, and, Hester, not everybody can afford organic produce. It is more expensive.\"\n\n\"You are what you eat,\" she forcefully replied. \"If you care about your body, you eat right. No processed foods. No foods that have suffered chemical sprays. That's that.\"\n\nHarry joined the conversation. \"Well, I am eating this fresh lettuce and I will buy one of your super pumpkins. Think I'll make a big jack-o'-lantern.\" She smiled.\n\n\"Fellas, excuse me.\" Hester walked back inside to the cash register.\n\nLolly Currie quickly put down her e-book reader.\n\n\"Now, girl,\" said Hester to Harry, \"it's time to buy those hayride tickets.\" Hester reached next to the old cash register as Lolly slipped out two large glossy tickets decorated with an illustration of goblins riding on a hay wagon drawn by spectral horses. \"Two?\" Hester asked.\n\nLolly piped up: \"We have lots.\"\n\n\"I bought my tickets,\" Harry reminded Hester, \"but I'll see if I can sell some.\"\n\n\"Hmm. So you did.\"\n\n\"Shameless.\" Cindy Walters laughed as she pointed her forefinger at Hester. \"You'll chain people to the pumpkin stand before you'll let them go without a ticket.\"\n\n\"Well.\" Hester blushed, then introduced Cindy. \"Harry, this is Cindy Walters from Florida and this little tyke is Heidi.\"\n\n_\"Hello.\"_ The schnauzer barked.\n\n\"We met at an environmental conference years ago. Been friends ever since.\" Cindy checked the clock on the wall. \"I'd better push off.\"\n\n\"Oh, stay the night,\" Hester offered.\n\n\"If I get into North Carolina I can make Florida the next day.\"\n\n\"Wait! You can't go without buying a Halloween Hayride ticket.\"\n\n\"And I want to buy your best pumpkin,\" Harry quickly interjected.\n\n\"I'll have Greg pick out the biggest and put it in your station wagon. How's the mileage on that?\"\n\n\"Good,\" said Harry. \"Of course, newer cars get even better.\"\n\n\"Save where you can so you can help the library.\" She grinned. \"Our two-year book fund is $175,000.\"\n\nHarry laughed. \"Hester, you're relentless.\"\n\nShe nodded. \"I know, but this is critical to not just Crozet but the western part of the county. For years Crozet was always the weak sister, but we're coming up. This library means a lot. Mike Marshall, the Crozet reporter, is coming on the hayride. You know he'll write about Mrs. Murphy, Pewter, and Tucker. Cindy, don't you dare walk out of here without a ticket.\"\n\nMike Marshall was the editor and publisher of the _Crozet Gazette_ , so if he covered it, it was news.\n\n\"Hester, I'm touched that you remember my cats' and dog's names.\"\n\n\"Family.\" She exited with Harry. \"Greg, pick the biggest and prettiest pumpkin for Harry.\"\n\n\"You bet.\"\n\nCindy walked with them, ticket in her hand.\n\nJust then Buddy Janss pulled up front, crates of late sweet corn in the back of his well-used pickup. A minute later, he walked toward them.\n\nHester turned to Buddy and fired away: \"How many acres of late-maturing sweet corn did you plant?\"\n\n\"Just like I told you last week and the week before, I planted two hundred acres. And those cool September nights just make the late sweet corn taste like candy. I harvest it twenty days after the first silks appear, I put it in huge tanks of ice water, and I bring it to you.\"\n\n\"Mmm?\" She raised her carefully plucked eyebrows. \"Long, long summer. You were smart to plant so late.\"\n\n\"Well, I planted corn every two weeks throughout the summer, but I waited extra long for the Silver Queen. Read my _Farmers' Almanac_. Better than the National Weather Service.\" He grinned, revealing a slight gap between his front teeth, as he winked at Harry.\n\n\"Odd, isn't it?\" Harry agreed. \"I find the same thing and I read it every morning.\"\n\nA few cars rolled by. One turned in, lured by the display.\n\n\"A new customer.\" Hester beamed as she walked over to welcome the young man.\n\nBuddy shook his head, smiling. \"She asks me the same thing again and again.\"\n\n\"Yep.\" Harry crossed her arms over her chest. \"If there's one tiny deviation in your story or one of mine from one week to the other, she's like a chicken after a grub.\"\n\n\"Just her way.\" He shrugged his massive shoulders.\n\n\"Some people can't fully trust,\" said Harry. \"They can like you but they can't accept what you tell them. They have to see it for themselves or check and double-check, just like Hester. Buddy, can you imagine how exhausting that is?\"\n\n\"Never thought about it.\"\n\n\"Means you can't learn from other people, your world becomes very narrow. I guess I thought about it as a kid because I had a great-uncle like that. After all these years, I now believe that trust is the bedrock of a community and it's the only way we can progress. Each of us doesn't have to invent the wheel.\"\n\n\"Harry, how do you think of this stuff?\" He took off his ball cap, revealing tightly curled jet-black hair.\n\n\"On my tractor. Bouncing along jogs my brain.\"\n\nHe chuckled. \"I'm on my tractor more than you, and I think about how dry the soil is, what is the soil temperature, should I check it, and what's the chance of rain.\"\n\n\"Yeah, but your tractor is like a Rolls-Royce. Doesn't bounce.\" She grabbed his hand and squeezed it while they both had a laugh.\n\nHester returned just as Tazio Chappars drove onto the crushed-stone parking lot. Brinkley, her yellow Lab, sat in the car, with the windows rolled down.\n\nTazio, now in her early thirties, became more lovely with each year. Harry liked her very much but had to wonder what her secret was: great bone structure or an unerring fashion sense? Whatever it was, Harry felt she didn't have it, but she muddled along and in extremis would smack a full coat of makeup on her face. Fortunately, she, too, had good bone structure.\n\n\"Taz!\" Harry waved as the gorgeous architect walked in. \"It is true! Hester attracts the best people.\"\n\nUnder his breath, Buddy muttered, \"And the most beautiful.\"\n\n\"Why, thank you, Buddy.\" Harry punched him lightly in the stomach.\n\n\"Violence! No violence at my stand.\" Hester joined in the fun.\n\n\"You look happy,\" Harry remarked to Taz.\n\n\"I just got the green light to redesign the Western Albemarle High School library.\" As she looked Hester's way, Tazio's gorgeous features displayed the beauty of her African Italian heritage. \"And I thank you for that,\" she said.\n\nThe older lady smiled. \"I didn't do a thing.\"\n\n\"Yes, you did. You fought for me to work on the new Crozet Library, and what a difference that has made in my career. I thought I'd be knocking out development houses forever and then I prayed I would be once the crash came. I can never repay you, Hester.\"\n\n\"Sure you can. Buy a pumpkin.\"\n\nThe neighbors and friends laughed, and Brinkley barked from the car, which set off Tucker in the Volvo.\n\nMore cars pulled in.\n\n\"Wow. Big day,\" Harry noted.\n\n\"I'd better go inside and help out Lolly at the cash register,\" Hester said. \"She's easily overwhelmed.\"\n\nWith a pumpkin in the back of the Volvo, Harry smiled all the way home. She glanced in her rearview mirror to see Hester still talking to Cindy and Heidi. Obviously, the good woman hadn't made it back into her store nor could Cindy get in her car.\n\nLater that evening, Cooper turned in to the driveway to Harry's farm. The lights were still on. She stopped her SUV at the barn just as Tucker rushed out to greet her. Both Harry's station wagon and Fair's truck were parked by the barn.\n\nAs Tucker accompanied her, she opened the screened porch door, then knocked on the kitchen door.\n\nWithin a minute, Fair, smiling, opened the door. \"Hello, neighbor.\"\n\n\"Hi, Fair. Forgive me for not calling first.\"\n\n\"Are you just coming home from work?\"\n\n\"Long day, but you know all about long days.\"\n\nHe nodded. \"Madam is in the living room and I'm on my way to the barn, if you need me.\"\n\nCooper entered the simple well-proportioned room with high ceilings. The fireplace gave off an inviting scent: burning pear-wood. The cats lounged on the back of the sofa.\n\nHarry, board on her lap, was drawing.\n\nSetting that aside on the coffee table, she asked, \"Are you still mad at me?\"\n\n\"No. I got over being mad at you when you showed me the scarecrow at Number 9.\"\n\n\"Any luck?\"\n\nThe taller woman shrugged. \"No, but I didn't expect the decorator to come at me with a meat-ax. She was quite nice, actually.\"\n\n\"Isn't that weird? I mean, the exact outfit.\"\n\n\"Yes, it is.\"\n\n\"So what's next?\"\n\n\"Tracking down clients. Rick questioned everyone at Morrowdale the day you and Fair found the body on their property. They were horrified, but no one had driven out that way, so they hadn't seen it. Everyone on the farm was questioned.\" She sat down across from her friend.\n\n\"You have a lot of patience,\" Harry complimented her. \"You're dogged and determined.\"\n\n\"I have to be. I live next to you.\"\n\nHarry laughed. \"Hey, look at my little garden drawing. Maybe I'll get it right this time.\"\n\n\"You taught me to plan my garden in the fall or winter, and so I have. Along with trying to focus on this case, I'm going through the possibilities.\"\n\n\"Is there anything I can do to help?\"\n\n\"With my garden, I can always use your help. With the case, I live in fear.\"\n\n\"I'm not that bad.\"\n\n_\"Yes, you are,\"_ Mrs. Murphy said.\n\n\"Let's say you have a way of stumbling onto things like the Number 9 scarecrow today,\" said Coop. \"What if the killer had been the decorator? Or someone in there observed you studying the scarecrow? Someone involved in this murder. You can't take chances like that.\"\n\nHarry didn't reply.\n\n\"Harry?\"\n\n\"I know you're right.\"\n\n_\"Knowing Cooper is right doesn't mean she'll stop,\"_ said Pewter.\n\n\"We know so little,\" said Coop. \"This could be a revenge killing. If you dig enough into people's lives, you eventually find someone who can't stand them or someone who is unbalanced.\"\n\n\"Kind of scary,\" Harry mumbled.\n\n_\"Well, it's almost Halloween.\"_ Pewter giggled.\n\n\"I dug down a little deeper.\" Cooper crossed one leg over the other, pulled at her anklet to stretch her leg. \"Hill loved fly-fishing. Had a two-thousand-dollar fly rod. How can a fishing rod cost that much?\"\n\n\"Beats me.\" Harry held up her hands.\n\n_\"Me, too. I don't need one,\"_ Pewter bragged, lifting her head off her arm.\n\nTucker was astonished at such a bold-faced lie. _\"What? You can't fish.\"_\n\n_\"I didn't say I fished. I wait for Mom to open a can of tuna. Now, that's real fishing.\"_\n\nMrs. Murphy laughed at her gray friend.\n\n\"Two thousand dollars.\" Cooper dropped her crossed leg. \"Boxes of flies. He was very organized and had quite a few books on fishing, according to the team that searched his house today. Fishing was his passion in life, so it seems. On his computer we found what you would expect\u2014a list of clients, a list of other accounting firms and legal firms as well as IRS agents for his area. Pretty cut and dried. Oh, one strange thing: Hester Martin's name. No tagline, nothing, just 'Hester Martin' and the farm stand address, phone number, and her email.\"\n\n\"Did you talk to Hester about him?\"\n\n\"Drove over earlier today. She said she knew him some. He was a member of the Upper Mattaponi tribe. She seemed a little bit resentful at being questioned. I don't know. Maybe she was in a bad mood. That was the extent of it.\"\n\n\"Hester does attend those annual powwows. Every year she says she's going. Never says why or what happens.\" Harry thought a moment. \"She's not a tribe member, but she's so interested in proper use of the land, I think that's the real draw.\"\n\nCooper furrowed her brow. \"She told me she's always had an interest in the Virginia tribes. Her mother was a Sessoms, which is a Cherokee name.\"\n\nHarry drew a long breath. \"That's right. I think my mother once mentioned it back when I was in fifth grade.\"\n\n\"How can you remember fifth grade?\"\n\n\"Because that's when we learn about the peoples who were here before we came. I loved it.\"\n\n\"Anyway, that was that. Hester was shocked that such a nice fellow, as she put it, was killed. As far as she knew, he wasn't a crook. Said Josh Hill always treated her kindly.\"\n\n\"Coop, did you know there are eleven Virginia tribes?\"\n\n\"I do now. I started looking this stuff up to see if there might be any connection at all to Hill's bizarre death. It's terrible.\"\n\n\"His murder, sure,\" Harry responded.\n\n\"No. The way the Virginia tribes are pushed around. The Commonwealth only recognizes eight tribes and the federal government doesn't recognize Virginia tribes at all. It really stinks.\"\n\n\"That and much else.\" Harry sighed. \"I guess the feds didn't take responsibility until after the 1870s. After all the Indian Wars, they had to do something. And Virginians with Native American blood were denied official status by the federal government. Since the seventeenth century, many Virginia Indians had intermarried with European descendants. Every Sessoms I know has blue eyes, bright blue eyes like Hester. Anyway, this gets the feds out of any form of repayment or protections as near as I can tell.\"\n\n\"That doesn't let the Commonwealth off the hook,\" Cooper shrewdly said.\n\n\"Doesn't.\" Harry returned to the murder. \"Well, you know a bit more than yesterday.\"\n\n\"Just enough to make this more confusing.\" Cooper suddenly smiled. \"But you know, sooner or later, a picture emerges. You get a feeling. For all the legwork in the world, for all the computer checks and cross-checks, I still rely on that hunch. It will come.\"\n\n\"Maybe it has something to do with fish. A two-thousand-dollar fly rod.\"\n\n\"Harry, you can be really awful.\"\n\n\"I know.\"\n\n_\"So true!\"_ Pewter sat up to give her remark emphasis.\n\n#\n\nThat same evening, glittering stars pierced the night. Looking out the tall, high windows of the old schoolhouse, Hester Martin could vaguely make out the obelisk in the cemetery a mile down the tertiary state road. Occasionally a truck would pass. She thought she heard a coyote.\n\nSitting down at one of the small school desks, she took out her Moleskine notebook, flipped it open, pulled out a pen. Before she could write down her thoughts, a car pulled up outside, its headlights illuminating the windows, then both the motor and the lights turned off.\n\nWithin seconds, Tazio entered the lovely room accompanied by her yellow Labrador retriever, the popular Brinkley.\n\n\"Good to see you,\" said Hester. \"I know you're busy, what with your job and various committees.\"\n\n\"Hester, I always have time for you and it's important we run through the Halloween Hayride.\" Tazio sat in the desk across the aisle from Hester's.\n\nThe desks remained in rows just as they were in 1965, when the school was abandoned.\n\n\"Gets dark so early now,\" Hester remarked. \"Somehow it always affects me. Makes me sleepy.\" She laughed at herself.\n\n\"Makes me fat.\" Tazio ruefully smiled. \"I always put on weight in the winter. This year I am determined not to do it.\"\n\n\"Natural. It's a natural cycle.\"\n\n\"You never gain weight. Neither does Harry,\" Tazio said.\n\n\"With me it's high metabolism. That or worry. As to Harry, both her mother and father stayed slim. They worked hard, those people. So does Harry and so do you.\"\n\n\"Hard enough, but most of the time I'm sitting on my butt. If I go to a building site or walk through construction, that's about it. I need to join a gym.\"\n\n\"You look just fine. I wanted you to see these buildings from the inside. This one, the middle one, was literally the middle school. Has some lab equipment, not much. Everything these students got was already used, passed down. The books especially were worn.\" She thought a moment. \"Your outstanding work for the library is almost done. We still have to raise money but your architectural work is complete and so practical. That's why I wanted you to see this.\"\n\n\"Funny, I've driven by these schoolhouses from time to time but never stopped. I always wanted to.\"\n\n\"They're built to last.\" She pointed to the windows. \"So much natural light saved lighting money. When my mother was small, each of these buildings had a wood-burning stove smack in the middle. You can't see the hole for the flue, as when the stoves were removed the workmen patched the ceiling. Put in oil-fired heat. And since the county still pays the electric bill and fills up those old tanks, there is low heat here throughout the cold weather. The pipes don't freeze.\"\n\nTazio rose, walked to the back of the room, with Brinkley following her, and opened a door. \"Well, they put a bathroom in, too.\"\n\n\"Right around the time of World War One. That's what my mother said.\" Hester smiled, pulling an old-fashioned long key from her coat pocket, a grosgrain ribbon attached. \"Come here.\"\n\nTazio and Brinkley walked back. \"Wow, that's really old.\"\n\n\"Hold out your hand,\" Hester ordered, dropping the key into the young woman's outstretched palm. \"This key opens all the doors. It's the master key. I've had it for years.\" She held up her hand. \"Long story made short: I took it back in the eighties. Didn't trust the county commissioners or anyone else, really. Tazio, consider bringing these buildings back to life. Oh, it will take time, money, and lots of political organizing, but you of all people can imagine the possibilities if we saved the buildings' best features. No lowered ceilings.\"\n\nTazio looked around. \"For what purpose? It won't be used as a school again.\"\n\n\"I don't know about that. It's possible it could be the basis for a small private school or a museum. You'll have to fight for it.\"\n\n\"And you are assigning me this task?\" Tazio asked, eyebrows raised. \"Aren't you going to help?\"\n\n\"Yes, but\"\u2014she smiled weakly\u2014\"my brother died six years ago. Sometimes I think I'm not long for this world.\"\n\n\"Hester, I hope not.\" Tazio's voice registered concern.\n\nHester waved her hand. \"No one knows, do they? I could live to one hundred or be gone tomorrow. Now come along with me. Bring the doggy.\"\n\n_\"Thank you,\"_ Brinkley replied.\n\nThe three of them piled into Hester's SUV. \"Let me just review with you, briefly, the Halloween Hayride. I'll be in wagon one. You're the ringmaster. You've got to make sure our actors are in costume, go to their proper places. If anything is amiss, you fix it. I'm going to sit in hay and enjoy the show.\" She grinned.\n\nDriving slowly, Hester headed north on the winding road, Buddy's cornfields on her left behind and around the schoolhouses.\n\nShe turned to Tazio in the passenger seat and said, \"Okay, Frankenstein and Dr. Frankenstein will be in the schoolhouse. Goblins and ghosts that fly around will be in the dried-out cornfields. That ought to be scary, hearing the rustle.\"\n\n\"And it will be dark, too,\" said Tazio. \"I checked my calendar. It's a couple of days after the new moon.\"\n\n\"As you know, the ghosts and goblins will be lit from within. Oh, this ought to scare the devil out of people.\" Hester stopped between two huge trees on each side of the road. \"Jeepers Creepers will fly between the trees.\"\n\n\"Right.\" Tazio knew the order of events, but riding with Hester through the outdoor fright stations amplified how dramatic this year's hayride would be.\n\n\"We've got a cable to run between the trees. For the Headless Horseman\u2014but you know him. Your boyfriend.\"\n\n\"Well, yes,\" Tazio laughed.\n\n\"Now, here's a good one.\" The middle-aged lady stopped on the road, the stone retaining wall of the graveyard standing out against the night sky. \"Jason with his chain saw battles Count Dracula. Don't forget to make a convincing arm to come out of a grave.\"\n\n\"Already have it. We aren't using a real grave. I imagine the family would be upset.\"\n\n\"Mmm.\" Hester cast her eyes toward the obelisk and a few other tall statues. \"The big monuments\u2014families had big money back then, and you know, every one of them came to wrack and ruin. The names still fill the county voting registers but not much else. Ever notice how sometimes money makes people stupid?\"\n\nTazio laughed again. \"Among other things. But if things are too easy, I guess people lose their ambition.\"\n\n\"Oh, the Villions, the Huntleys, the Yosts, they either gambled it away, drank it away, or made really bad business decisions.\"\n\n\"Wine, women, and song?\" Tazio raised one eyebrow.\n\n\"I will give those families some credit. The women were beautiful, all married well. It was the men who went to hell in a handbasket. Well, anyway, on the other side of the graveyard you repel Dracula with a cross, and then just past you, Reverend Jones will be a monk with an electric torch guiding people to Mount Carmel Church. You can just see the little spire. Mount Carmel only has but so much money.\" She stopped. \"We are paying them well for the use of the rec hall. Always a good idea to keep on the sunny side of any church.\" She stopped talking, turned the truck down the farm road on the north side of the graveyard, backed out to return to the schoolhouses.\n\n\"Aren't there something like twenty-two thousand Christian sects, each with different ideas?\" Tazio asked.\n\n\"I don't know. I'm Catholic myself. Sometimes I believe it. Sometimes I don't. Mostly I love mass. But out here looking at the stars, that's my true church,\" Hester replied with great feeling.\n\n_\"It's mine,\"_ Brinkley piped up.\n\n\"I know what you mean,\" Tazio quietly agreed.\n\n\"Where'd you put the key?\"\n\n\"In my pants pocket. It's deeper than my coat's.\"\n\n\"Don't you lose that key. That's the master.\"\n\n\"No, ma'am, I won't.\"\n\n\"Good girl to call me ma'am. You always call a lady older than yourself ma'am.\"\n\nTazio laughed again. \"Hester, we do that in St. Louis, too.\"\n\n\"Miss it?\"\n\n\"Not really. I go home to see Mom and Dad but I don't think I could live in a big city anymore.\"\n\n\"Millions upon millions do.\" She pulled in to the parking lot, a flat dirt place, next to Tazio's car. \"That's another thing that scares me: How can the newcomers appreciate these buildings? They move down here to escape the city, but they bring their ways and they want efficiency, services, bottom-line kind of thinking. Spending money on these historic buildings would seem stupid to them.\"\n\n\"Maybe not,\" said Tazio, taking the bait. \"It's part of our history. No matter where you come from.\"\n\n\"Well.\" Hester slowed her speech. \"I hope you're right. My true dream is that eventually we can buy these from the county. Ha!\" She clapped her hands. \"Won't that be a fight! Will it help that you're mixed race? Yes. Politically it will help. You have such wonderful gifts, gifts few people possess. You could bring these buildings back to life. Wouldn't it be glorious to hear laughter inside them?\"\n\n\"Yes, it would,\" Tazio agreed.\n\nHester cut the motor, turned to face her. \"I know people think I'm weird.\"\n\nTazio didn't quite know what to say. \"You're different from most Virginia ladies.\"\n\n\"I speak my mind. I don't have the time for the minuet of politeness. Bores me.\"\n\n\"I certainly understand that.\" Tazio smiled, remembering what a jolt it was to move from Missouri to Virginia.\n\n\"Maybe I am weird. I get worked up about things, history, getting books into childrens' hands, bringing buildings back to life, righting old wrongs.\" She inhaled deeply. \"I get ideas like everyone worries about carbon emissions. What about the billions of people breathing out CO2? That has to damage the environment. I blurt out this stuff and then people think I'm weird. They don't want to think. That's the problem.\"\n\n\"It's painful to think, Hester.\"\n\nHester stared at her. \"But you do.\"\n\n\"Only after I've exhausted every other alternative.\"\n\nThis made Hester giggle. \"Sometimes I do that, too. Well, girl, I railroaded you into designing parts of the Crozet Library and now I'm railroading you again.\"\n\n\"You are,\" Tazio said honestly.\n\n\"Will you take this on?\"\n\n\"You know I will. But you have to work with me.\"\n\n\"I will, Taz, but things can happen. If something happens to me, you carry the ball, hear?\"\n\n\"Don't say that, Hester.\" She breathed deeply. \"But if anything happens to you, I will carry the ball and vice versa.\"\n\n\"Deal,\" Hester quickly answered.\n\nWhen Tazio and Brinkley drove away, Hester returned to the middle schoolhouse. Inside, she sat down, took out her notebook, and started to write, then paused. She walked up to the teacher's desk in the front of the room, pulled open a drawer, took out a yellowed square of paper, and wrote a name on it with her fountain pen. She'd just dropped a lot on Tazio. She knew that when the younger woman read this, her natural curiosity would do the rest. She returned it to the middle drawer, turned out the lights, and shut the door and locked it, for she had a backup key to the outside door. Hearing the satisfying click, she walked to her truck, then stopped a moment to study the stars. They'd been up there long before she was born and they'd be there long after she was gone. She found that comforting.\n\nTazio, driving home with Brinkley next to her, wrestled with emotion. Hester had touched her. She had a strong feeling that Hester must have a premonition of her death, and that the kind-hearted eccentric was passing the torch on to her.\n\n#\n\nTucker, left behind, mournfully watched as Susan Tucker, Harry's best friend, picked her up at six in the evening on Wednesday, October 16. Rushing from the house, Harry jumped into the Audi station wagon's passenger seat. Joined at the hip, friends since cradle days, these two discussed everything and everybody with each other daily. Of course, Harry had already told Susan about the scarecrow down at Farmville.\n\nAnd Susan, naturally, had commanded her friend to stay out of it.\n\n\"Thanks for picking me up,\" said Harry, closing the door to the station wagon. They almost always rode together to the St. Luke's vestry meeting.\n\n\"Gives us more time. Anyway, I'm not sure I trust you by yourself.\" Susan smiled.\n\n\"You never make a mistake.\"\n\n\"Finally you've realized that.\" Susan reached the state road at the end of the long gravel drive, looking both ways. \"Uh-oh, here comes Aunt Tally. Let's give her a wide berth.\"\n\nThe almost-101-year-old indomitable woman behind the wheel of her old Bronco beeped and waved, swerving slightly to the right but correcting herself, much to the gratitude, no doubt, of her passenger, her great-niece, Little Mim.\n\n\"Wonder where those two are going,\" said Harry. \"This has got to be the first time Little Mim has left the baby.\"\n\n\"She's a new mother. He's only three months old. It's good she's left him with Blair.\"\n\n\"He's a good father. Bet Big Mim is there.\"\n\n\"Harry, you got that right. The new grandmother\u2014wait, the _only_ grandmother in the world\u2014doesn't believe men can take care of babies. Well, in her defense, she said her husband never changed a diaper.\"\n\n\"They didn't back then.\" Harry knew that prior generations led more gender-defined lives.\n\n\"Plenty don't now, but in the main I think young men want to be involved. I remember my father at the end crying that he barely knew his children until he retired. Poor Dad. He did what men did. He worked his ass off and came home after we were asleep.\"\n\n\"Your father did work hard. He ran that lumberyard, and when you own the business, it owns you.\"\n\n\"You were lucky that you could farm with your father,\" Susan said without envy.\n\n\"I was. Mom was the librarian, so she would come home a little after I got home from school. We didn't have much money but we had a lot of love. My parents taught me a lot.\" She looked at Mount Tabor Presbyterian Church as they passed. \"They taught me not to be a Presbyterian.\"\n\n\"Harry!\"\n\nHarry laughed. \"Mom and Dad were generally open-minded people, but they did have their share of religious prejudice. If there had been a Church of England here, that's where they would have worshipped.\"\n\n\"Reminds me. You have your building-and-grounds notes for the vestry meeting?\"\n\n\"Yeah. Why?\"\n\n\"Because if there's ever an outlay of expenses, that's where it is,\" Susan said as she, too, eyed the lovely white Mount Tabor Church.\n\nThere was a bit of traffic on the narrow two-lane highway. People were driving home from work. In the morning and the evening, going could be slow.\n\n\"I never noticed that,\" Susan said, her voice rising.\n\n\"What? Mount Tabor is Mount Tabor. Really pretty.\"\n\n\"No. There's a Halloween scene on the grounds nearest the road.\"\n\nTwisting to look back, sure enough, Harry saw pumpkins, tied-up cornstalks, baskets of harvest, and a jolly-looking witch on a broomstick over a sickle moon. Actually, the broomstick used the midpoint of the moon for stability.\n\n\"So?\" Harry shrugged.\n\n\"Harry, pagan. Halloween is a pagan festival.\"\n\n\"It might have started that way,\" Harry said, \"but then, Christmas is a co-opted pagan festival. So this became All Hallows' Eve and the early church fathers could sleep soundly at night.\"\n\nSusan, who knew her history, maintained, \"Pagan. When have you ever seen a church with a Halloween display?\"\n\nHarry shrugged. \"I give Mount Tabor credit. It's fun.\"\n\n\"Yeah, I guess it is.\"\n\n\"Here we are,\" said Harry as Susan pulled the Audi into the Lutheran church's parking lot. \"No Halloween display at St. Luke's. Leaves are raked. Grounds are sleek. Am I doing a good job or what?\"\n\n\"Divine, darling.\"\n\n\"Well, it is our church.\"\n\nThe meeting took place in a twenty-by-thirty-foot room with a high ceiling. Built of native stone in the eighteenth century, St. Luke's emitted a feeling of peace, of thoughtfulness. And being Lutheran, it excelled in good works.\n\nThe Very Reverend Herbert C. Jones, a Vietnam veteran, deeply believed anyone could talk about Christ. One had to comfort one's fellow man. And he did, personally. His congregants did, too. St. Luke's provided tutoring for children in need. Also, their small soup kitchen had recently grown with the depression.\n\nThe Reverend Jones used innovative techniques to draw in his flock. Every October 4th, on St. Francis's Day, people brought their animals to the church for a blessing. This year's service had proved especially unusual in that a young woman brought a jar of worms. She had started a worm farm and wanted the reverend's blessing. The clergyman dutifully picked up the jar and blessed those worms as though they had been devoted dogs. We are all God's creatures.\n\nWhen reprimanded by a parishioner for keeping a saint's day\u2014\"leave that to the Catholics\"\u2014Herb took no offense. He merely replied that the saints were as good a model for Lutherans as they were for Catholics. And who could be a better example for all than St. Francis of Assisi?\n\nThe St. Luke's vestry board, six people, usually met the second Wednesday of each month.\n\nElocution, Cazenovia, and Lucy Fur, the Lutheran cats, invariably joined the proceedings.\n\nThese meetings generally ran smoothly until the group had to consider expenditures. Last year, the old church truck breathed its last. The unplanned expense of a new four-wheel-drive truck had sent the church treasurer, Neil Jordan, into a tailspin. Now in his second year on the board, Neil wanted to impress his fellow parishioners as a fiscal conservative. He was always seeking new ways to save money. But also, he was beginning to discover if it wasn't one thing, it was another, and the bills piled up.\n\nNeil read the treasurer's report. He looked straight at Harry over his tortoiseshell glasses, expensive ones from Ben Silver in Charleston. \"You're the one I worry about.\"\n\nBoomBoom Craycroft, another childhood friend of Harry's as well as Susan's, laughed a little, as did the reverend.\n\n\"Neil, all the church's equipment is in good order,\" Harry assured him. \"We can get one more year out of the leaf blower, and I fixed the zero-turn mower this summer. We're in good shape except for one small necessity.\"\n\nNeil froze. \"Yes?\"\n\n\"There's a small portion of the slate roof right on that northwest corner which gets hit the hardest,\" said Harry. \"Two of the shingles have dislodged, and with one more hard blow, I'm afraid we will lose a larger section than need be. I climbed up there.\"\n\n\"Harry, I wish you wouldn't do that,\" Reverend Jones admonished her.\n\n\"Rev, I'm in charge of buildings and grounds.\"\n\nThe good pastor said, \"Surely there's a man who can do some of this.\"\n\n\"Are we going to have the Sexism 101 talk?\" BoomBoom laughed. \"We are supposed to be long past that caveman talk, Rev.\"\n\n\"No, but...\" The Reverend Jones sighed heavily.\n\nLooking at the beloved clergyman, Harry compromised. \"How about next time I go up on the roof, I take someone with me, possibly someone male who is actually a roofer? I can do it but we need a professional up there.\"\n\nThis truly alarmed Neil. \"Roofing costs an arm and a leg!\"\n\nKeeping her tone level, Harry replied, \"It does. However, if we don't attend to this right now before the weather turns\u2014and you know how bad the winds get in winter straight through March\u2014we will most certainly lose more slate shingles. The water will run into the beams and travel from there. I doubt it will come through the ceiling, though, unless a huge hole is blown into the roof. We'd need a big tree limb to accomplish that\u2014I think. But that sort of silent water damage will come back to haunt us, years from now when the building's wood beams rot out. Pay now or pay later, big-time.\"\n\nThe members of the vestry board were persuaded, as was the reverend.\n\nNeil, defeated, asked, \"Do you have a ballpark figure?\"\n\n\"That's why I would like to get a professional roofer up there. My personal guess is if we do this now, we can do it for about thirty-five hundred dollars.\"\n\n\"Thirty-five hundred! For two shingles?\" Neil's voice cracked.\n\n\"No. I estimate we will need to replace about a two-foot square. True slate shingles are expensive, plus we have to try and match shingles that are over two hundred and forty years old. The labor is expensive, too, but once we have the materials, it's not an all-day job. Oh, and I expect there will be a gas surcharge if the gas prices go up again.\"\n\n\"That's the truth. Everyone's doing it.\" BoomBoom, her beautiful mane of blonde hair catching the light, nodded. \"There's no way business, small business, can absorb those prices and still make a profit.\"\n\n\"How about the post office?\" asked Harry, the former postmistress. \"I can't even imagine what a one-cent increase in gasoline does to their budget. Delivery trucks, cars in every state in this nation\u2014it has to be mind-boggling.\"\n\nWesley Speer shook his head. Also new to the board, he owned a high-end realty firm. They'd felt the downturn but not nearly as badly as Realtors in Las Vegas or other large cities. Wesley believed once the foreclosure mess cleared up, maybe in another two years, the economy might pick up some momentum. He knew he'd never again see the craziness, the flipping of houses and farms, that he saw in, say, 2007, but he was confident sales would rebound. If people can get work, they want to own a home. He'd built a life's career on that.\n\nSusan joined the conversation. \"Did you read in the paper where our area, Richmond in particular, has the worst postal service in the country?\"\n\n\"Your husband's in the House of Delegates,\" Neil teased her. \"Make him fix it.\"\n\n\"Virginia's state government can't fix the federal postal system, but you know Ned would if he could. I swear that man is a glutton for punishment. He actually loves being in the House of Delegates.\" Susan threw her hands up in the air.\n\n\"He must be a glutton for punishment, he married you.\" Harry smiled so sweetly.\n\n\"Well, I think this meeting is over, unless there's more discussion concerning the roof,\" the Reverend Jones said, enjoying the banter between the two old friends.\n\nThe committee members zoomed toward the kitchen, some heading straight for the bar. The vestry board enjoyed getting together after the meeting, catching up on one another's news and listening to Reverend Jones, who had an appropriate Bible quote at the ready for just about any topic of conversation.\n\nLucy Fur wove between people's legs. _\"Do drop something, will you?\"_ the cat urged the thoughtless humans munching away on ham biscuits.\n\nElocution, the senior cat, curled up in Reverend Jones's lap.\n\nCazenovia picked out Susan for her victim. Susan couldn't take a step without Cazenovia sticking right next to her, meowing.\n\n\"Suffer the little kitties to come unto you,\" the reverend joked.\n\nLibations in hand, all sat in chairs around the coffee table, books pushed to the middle.\n\n\"Harry,\" Wesley said, \"you've known Buddy Janss longer than I have. Why won't he sell the one hundred acres behind the old abandoned schoolhouses? Be very profitable for him.\"\n\n\"More to the point,\" BoomBoom interjected, \"why doesn't the county finally do something about those schoolhouses? Old well-built buildings can still be useful. Instead, the county blows millions on new construction. Everything has to be new.\"\n\n\"Plans for repairs tend to languish in the county budget.\" Reverend Jones's deep voice rumbled. \"But it's much as you say, BoomBoom, everything has to be new and, to my way of thinking, antiseptic. Those three wooden buildings, with their big tall windows, beckon one to learn.\" He smiled. \"Can't you imagine sitting at one of the old desks with the flip-top lid we all used, staring out the windows on an early spring day? If nothing else, might make you want to learn about the environment.\"\n\nThe group laughed.\n\nHarry answered Wesley's question as best she could: \"Buddy fears development, and for good reason. The school buildings and those acres are in a prime spot.\"\n\nWesley tried not to sound too judgmental, even though he was. \"He would make so much money. At least three point two million. These are hard times. That profit would allow him to buy or rent much more land farther west in the county or he could invest it in bonds or something.\"\n\n\"Buddy isn't averse to profit, but like I said, he fears development,\" said Harry. \"He thinks good soil, good farmland, should stay farmland. Wesley, he isn't going to sell.\"\n\n\"Mmm.\" Wesley heard Harry's words but he still hoped he could find a way to pry those one hundred acres from Buddy.\n\n\"Don't forget, Wesley, the schoolhouses could create a problem for any development.\" Susan Tucker kicked off her shoes. \"Sorry, my feet hurt.\"\n\n\"Yeah, you guys should be forced to wear heels just once in your life,\" Harry said. \"Torture,\" she declared, grinning.\n\n\"Well, these are low heels but I've had enough.\" Susan rubbed her right foot. \"Okay, the problem: County land remains county land. And those school buildings might be considered historically significant. Now, the county can elect to sell its land. There must be a public hearing advertised in the newspaper each week for a month. These days the announcement has to go on the county website, too.\"\n\nNeil spoke up. \"You think the fear is that if the one hundred acres go, whoever purchases same will have no use for the schoolhouses and demolish them. That's jumping the gun, I'd say.\"\n\nHarry's mouth fell open a little, then she said, \"I hadn't thought that far ahead, but I guess it could be a real concern.\"\n\n\"Those schoolhouses would make welcome living quarters for the aged,\" said BoomBoom, thinking out loud. \"Nice setting, pretty grounds, easy access from the state road and not really far from Route 250.\"\n\n\"Or condominiums, which would bring the county more revenue than what we used to call the poorhouse.\" Reverend Jones drained his glass.\n\n\"What do you mean?\" Harry, one of the younger people in the room, asked him.\n\nThe reverend clarified, \"Honey, back when the earth was cooling, every county in this state had its own poorhouse, and it was usually a farm. Those down on their luck worked the farm. We don't have that anymore, but a place for the aged is somewhat like that in that anyone dependent on government-subsidized living is usually poor. 'Course, at Random Row, they wouldn't have to do farm work.\"\n\n\"Random Row?\" BoomBoom repeated. \"I remember Mother saying that once or twice, but I figured she was just forgetting the actual name of the place.\"\n\n\"It's a great name,\" Neil said, nodding to Reverend Jones. \"Be a great name for condominiums.\"\n\n\" 'Tis, but I doubt they would be called that,\" Reverend Jones quickly replied.\n\n\"All right, Rev, what's the story?\" Harry plied him.\n\n\"Well,\" he said, \"those schoolhouses were built for the African American children. When I was little, you didn't use terms like 'African American.' The polite word was 'colored'\u2014polite among white folks, anyway. Really, back then no one said things like 'Italian American' either. Well, I'm getting off the track, but I do think about these things sometimes. Anyway, so many of the children at the schoolhouses had white fathers. Rarely did the men visit the school, because often they were married to white women, but many of those men did support their children, the random children, economically.\"\n\nA silence followed this, then Susan said, \"Nothing is simple, is it?\"\n\n\"Not when it comes to human beings.\" Reverend Jones smiled. \"What always strikes me is how most of us try to normalize an abnormal situation. I guess I learned this in Vietnam. We just struggled to keep things tied down. I mean, all I could think about apart from staying alive was, how were the Baltimore Orioles doing? We'd all try to get baseball scores and then football scores. If my team lost and one of my buddies gloated, fistfight. Here we were in a war in a different world, and we'd fight over a football score. But it felt safe in a way. Random Row was kind of the same thing, people normalizing a difficult situation.\"\n\n\"The desegregation act was enforced in 1965,\" Susan informed them. \"That resolved it.\"\n\n\"You weren't born yet.\" Reverend Jones smiled at her. \"It resolved political issues; it did not nor could not resolve personal issues. If your father is white and doesn't claim you, you may not be thinking about desegregation the same way.\"\n\nNeil looked at the reverend. \"What's the old saying, 'The personal is political'?\"\n\n\"Is and isn't.\" BoomBoom was firm about this. \"But in 1965, what happened to these so-called random kids' schools?\"\n\n\"Abandoned,\" said Reverend Jones. \"It was a political victory but it came at an unintended price. At least, I think it did. Basically, the children from Random Row were crammed into the white schools with no support system. The assumption was and still is that white ways are better. I don't exactly see this as black and white, I see it more in class terms, but the reality of the children at Random Row was most of them were African American or mixed race, and poor. They were thrown into schools with children from a higher socioeconomic group and with vastly different needs. Not a good thing, in my mind.\"\n\n\"Well, it's a done deal,\" Wesley replied with no emotion.\n\n\"It is.\" The reverend nodded. \"But that doesn't mean we can't learn from it, and not repeat our mistakes. We have to think of people's emotions.\"\n\n\"What happened to the teachers at Random Row?\" asked BoomBoom; at forty-one, she was the same age as Harry and Susan.\n\n\"I suspect they were bought off. You know, early retirement or something like that. I guess what really gets my blood up is the assumption those teachers weren't as well educated. Howard? Grambling? The list of excellent black colleges can go on. The teachers may have gone to segregated colleges, but tell you what, I never met any graduate of those colleges who wasn't well educated.\"\n\n\"Racism is subtle and not so subtle.\" Susan grabbed a cookie from the plate that Reverend Jones had placed on the table.\n\n\"So is sexism,\" said BoomBoom.\n\n\"Yes, but you ladies have so many ways to even the score,\" Neil teased her.\n\n\"Don't forget that, Neil,\" BoomBoom teased him right back.\n\n\"Not to ignore this fascinating history,\" interjected Wesley, \"but what I deduce from this is that the county, which could realize high profits on those buildings, won't for political reasons?\"\n\n\"Don't you think it depends on the budget?\" said Harry. \"We're in hard times. If the Board of County Commissioners wants to let those buildings and the land go, this would be the time.\"\n\n\"If they're willing to put up with the protest that our history is being demolished,\" said Neil, in between bites of a chocolate chip cookie. \"Here's what I think. Use them or sell them. To keep the land idle, just sitting there, is stupid.\"\n\n\"Do you think Buddy would sell his hundred acres to the county?\" Wesley asked.\n\n\"They wouldn't need it,\" Harry replied.\n\n\"Probably not,\" said Wesley. \"I mean, if a housing development was part of the plan, yes. Otherwise, no. It does seem wasteful, though. The schoolhouses just abandoned and going to ruin.\"\n\nSusan simply said, \"I say restore the buildings as a museum. It could be a good lesson for all and those rooms have cozy, lovely proportions. I'm like Reverend Jones, everything today is too antiseptic and big. I'm really tired of big.\"\n\nAfter an hour of lively talk, the group began to trickle out. Harry, Susan, and BoomBoom stayed longest, helping Herb clean up, washing dishes and glasses.\n\nDrying her hands, BoomBoom remarked to Harry, \"I was so sorry to hear that Big Mim lost the Medaglia d'Oro filly. Stunning, that filly.\"\n\n\"Fair was devastated. He thought she was one of the most perfectly formed horses he had ever seen. Big Mim was, well, in tears according to Fair, but still took it like a trooper. Being a grandmother was a comfort, I think.\"\n\n\"Heard she doesn't like the boy's name, Roland.\"\n\n\"Little Mim and Blair love the ancient history of Roland at the Roncevaux Pass. 'Course, Big Mim just ignored it and calls him Roy.\"\n\nBoomBoom laughed. \"Those two never have gotten along. Still, they do love each other.\"\n\n\"Would you want to be Big Mim's daughter?\" asked Harry.\n\n\"No,\" the blonde replied. \"I love being her friend, though.\"\n\n\"Me, too. She's one hell of a horsewoman.\"\n\nThey nattered on, Susan chiming in as she wiped down the coffee table, Herb as he put away glasses. Each lady kissed Herb's cheek, then each other as they left St. Luke's.\n\nOn the drive home, the stars glittered in pale silver light. Friday would be a full moon.\n\n\"Good meeting.\" Susan hit her brights. \"At least Neil isn't talking about algorithms anymore. He gave a straightforward treasurer's report.\"\n\n\"He's figuring out that simple is better,\" Harry said. \"Always is, too, no matter what the subject.\"\n\n\"Ned says what drives him crazy about his fellow politicians in Richmond is how they complicate things to make themselves look smarter. He also said the level of discourse is so low a man of average intelligence has to stoop to match it.\" She smiled. \"But there are some good people there in both chambers. He likes working with David Toscano and he likes working with some of the people from the farming counties. He's not too thrilled with the ones from northern Virginia.\"\n\n\"Like I said before, he's a glutton for punishment.\"\n\nThey rounded the curve, Mount Tabor up ahead.\n\n\"My tire light went on. Reach into the glove compartment and get out my pressure check, will you?\" Susan turned in to Mount Tabor's parking lot. \"Oh, dear, Witchy Woo has fallen over. Come on, let's set her up. They put a lot of work into this. Then I'll check my tire.\"\n\nThe two kept the brights on in the direction of the display and walked over.\n\nThey reached the corn bundles, all the arranged pumpkins, some cut as jack-o'-lanterns, the large baskets overflowing with gourds and squashes. Witchy Woo had fallen uphill onto the uneven ground.\n\nHarry sniffed. \"Something's gone off.\"\n\nSusan inhaled. \"Just. Probably a dead gopher somewhere. He'll be gone tomorrow.\"\n\nHarry bent over to pick up the tumbled-over figure, noticing that its rubber mask was complete with a long, warty witch's nose. She stood up again. \"It's not a gopher.\"\n\nBoth women stared at the black-clad figure.\n\n\"Good God.\" Susan put her hand to her face.\n\nHarry reached down to pull off the mask.\n\nSusan shouted, grabbing her arm, \"Don't!\"\n\n#\n\n\"Aren't you off work tonight?\" Harry, still upset, asked as Cooper crossed the parking lot in her civilian clothing.\n\n\"I am, but Dabny called me from headquarters and said that you, once again, have found a strange corpse. So here I am. Anyway, I'd like to see this before the body goes to the state medical examiner. You two stay here. I _mean_ it.\"\n\n\"All right,\" Susan firmly agreed. \"I'll take charge of Harry.\"\n\nCooper turned her back to walk away, then faced them again. \"Are you two doing okay?\"\n\nHarry shrugged, and fibbed a tad. \"Yeah. It's gross but...\" She shrugged again as Susan nodded in assent.\n\nThe police investigative team circled the display. The photographer snapped, stepping out of the way of the officers.\n\nDorothy Maddox, chief of forensics, had only been with the team a year. She was kneeling down, surgical gloves on, carefully touching the corpse's arm. In the temporary lights now shining on the scene, she studied a swollen hand, and a forearm with purple splotches.\n\nCooper stood behind Dorothy. \"Thirty-six hours at most, my guess.\"\n\n\"Your guess isn't far wrong. The nights have been cold, the days in the seventies. She's on the other side of maximum rigor mortis, obviously, but intact, and that's a huge help.\" Dorothy stood up. \"Is Rick on the way?\"\n\nNo sooner was his name spoken than the sheriff pulled off the road and into the church driveway. He, too, was in his civilian clothes and driving his personal vehicle. He glanced toward Harry, then walked over to the scene.\n\n\"We'll need to dust everything. The pumpkins, the baskets, every single thing.\" He took a deep breath, then coughed slightly. \"I hated to leave my ball game, but this is, well, original.\" He paused. \"There's nothing like the odor of death, is there? It isn't even that bad yet. I'm surprised the body wasn't damaged.\"\n\n\"Boss, can I remove the victim's mask?\" Dorothy asked. \"I waited for you.\"\n\n\"Yes, of course.\" Rick motioned for all the lights to shine on the witch's face.\n\nCarefully, Dorothy removed the rubber mask with the hooked nose.\n\n\"My God,\" Susan exclaimed, as she could see the face with the flashlight focused on it. \"It's Hester Martin!\"\n\nHarry recognized her all the way from where she stood in the parking lot. She covered her mouth with her hand, then let it fall. \"Hester Martin. She never did a thing to anybody.\"\n\nTears filled Susan's eyes for the middle-aged lady. Her mind flashed to Hester proudly showing off produce, filling her specially decorated wooden wheelbarrows, some worn and painted green, some barn red, some faded marine blue. Large wheels were yellow with a pinstripe matching the color of the painted display cart. Hester had a good eye for proportion and color. The produce gleamed, as she had misted it, too. Susan's tears rolled faster now. She met Harry's eyes. \"Remember when Hester declared that black gum trees were conspiring against humans? Well, everyone gets a free pass for a few crackbrained ideas. Hester's seemed more imaginative than most.\"\n\n\"I can't believe this!\" exclaimed Harry. She, too, cried a bit.\n\nRick was as surprised as they at the victim's identity. \"Dorothy, get the body out of here as soon as you can. We're lucky it's night.\"\n\n\"Sure. I've done what I can do without disturbing the rest of this Halloween scene.\"\n\n\"Here's Ted. Excuse me.\"\n\nRev. Ted Foster had driven over as soon as the sheriff's office called him. He lived about twenty minutes up Route 810. Along the way, he'd had the presence of mind to pick up Bunky Fouche, the church groundskeeper.\n\nSeeing Hester laid out in the witchy garb, Bunky had to be steadied.\n\nRick escorted both men directly to the corpse.\n\nBunky shook uncontrollably. \"Oh, Sheriff, I can't look at dead bodies.\"\n\n\"Bunky, tell me who this is.\"\n\n\"It's Hester Martin, God rest her sweet soul. She was good to me.\"\n\n\"Reverend Foster.\" Rick turned to the minister, who also appeared shaken by the grisly sight and rank odor. \"When did you put up this Halloween cr\u00e8che, for lack of a better word?\"\n\n\"Three days ago,\" answered Reverend Foster, his voice low. \"The witch was a manikin and she had straw hair.\"\n\n\"And did you look at the display each day?\"\n\n\"No, sir, I didn't. From a distance, it all looked fine to me.\"\n\n\"It was fine.\" Bunky's voice quavered.\n\n\"So, neither of you has any idea when Hester Martin's body was placed here?\" Frustration edged into the sheriff's tone.\n\n\"No,\" both answered.\n\nRick put his hand under Bunky's elbow to steady him, walking him away from the eerie but all-too-real vision.\n\nCooper watched the men's departure, then said to Dorothy, \"It's a lot of work to carry a body, dress it up, place it on the broom.\"\n\nHarry had inched closer from the parking lot, and piped up. \"Maybe Hester's body was already dressed up when it was brought here. The killer first observed the manikin's witch outfit and dressed her just the same. I mean, it makes sense the killer would make it easy on himself.\"\n\nCooper stared at Harry, thought a moment, then replied, \"A possibility.\"\n\nSusan said, \"If Hester's body had been here for any real length of time, dogs would have already gotten at it, crows, flies. We're all country people. We know the stages of death.\"\n\nThe three women stood silent.\n\n\"We've got a real sicko,\" Cooper replied simply, saying what they were all thinking: The killer had kept Hester's body somewhere else and placed it here once rigor mortis decreased and the muscles relaxed.\n\n\"Coop, let me get poor Hester out of here.\" Dorothy motioned for the stretcher and the body bag. \"I need to get her in the cooler at the morgue before there's more damage. And maybe when I get the costume off, I'll know how she was murdered.\"\n\n\"If the modus operandi is the same as our scarecrow accountant, we've got a major problem,\" said Coop.\n\nDorothy whispered, \"No matter what, we've got a major problem.\"\n\n#\n\nFair ran out to the drive at 11:10 P.M., when his wife finally drove up with Susan.\n\n\"Why didn't you let me come to you?\" Harry's husband asked, his voice betraying his concern.\n\nHarry opened the door to get out as Susan rolled down the window. The dog and the two cats also ran out of the house at Harry's arrival.\n\n\"Honey, there was enough confusion,\" she said. \"Susan and I were together. We're all right.\"\n\n\"We are,\" Susan reassured Fair. \"I mean, as all right as you can be after finding something like that.\" She took a deep breath. \"Let me get on home. My dog needs to go to the bathroom.\"\n\n\"Where's Ned?\" Fair asked.\n\n\"In Richmond.\"\n\n\"Why don't you stay here?\" he invited her.\n\n\"I appreciate that but Owen needs to go out and I'll feel better with him, at home, with a hot bath and bed.\"\n\n\"Sure?\" he asked, his eyebrows raised.\n\n\"Sure. And when I get there, if I can't do as well as I think I can, I'll bring Owen back with me and we'll bunk up with you all.\"\n\n\"Okay,\" he agreed.\n\nHarry leaned in the window, gave Susan a kiss. \"I'll call you tomorrow.\"\n\n\"Better.\" Susan rolled up the window and turned the car around. Her two friends and their animal companions watched her motor down the long drive.\n\nSlipping his arm around her waist, Fair walked Harry back into the house.\n\n_\"Must be good,\"_ Pewter said, meaning another big mess Harry had stumbled into.\n\n_\"If only we could have heard the phone call,\"_ Mrs. Murphy said.\n\n_\"Yeah. I hate not knowing.\"_ Tucker followed right on Harry's heels, resisting the urge to slightly bite them.\n\nOnce inside, closing the kitchen door against the chill, Harry surprised Fair as she sat at the table. \"I'd like a drink. What should I have?\"\n\n\"Oh, how about if I make you a simple scotch and soda? Not too strong.\"\n\n\"You know I can't drink. But I need something,\" she said, stating the obvious.\n\n\"This will relax you and help you sleep. No nightmares.\" He opened the pie safe, where the liquor bottles were lined up like orderly soldiers.\n\nNeither husband nor wife was much of a drinker, but there were spirits on hand for guests. Fair occasionally liked a cold beer at night in the summer and a scotch in the winter, but he could go days without a drink.\n\nTucker lifted her head. _\"Scotch has an interesting smell. Not bad.\"_\n\n_\"Tuna is better,\"_ Pewter remarked, patrolling the kitchen counter.\n\n_\"Can't drink tuna.\"_ Mrs. Murphy jumped up on the corner.\n\nPushed by Fair, Harry took a sip of the scotch, then recounted everything. When she finished her tale, she sighed, then said, \"Tell me about your day. I don't want to think about this anymore.\"\n\n\"I'm sorry, honey. I'm just worried and I want you to carry the old .38. Your father's Smith and Wesson is as good as the day it was made.\"\n\n\"Why? I'm in no danger.\"\n\n\"Probably not, but you found two bodies and it seems likely both were done in by the same killer or killers. You will be in the paper and on TV. A bad guy might wonder why Harry Haristeen has a nose for murder.\"\n\n\" 'Nose' is the right word.\" Harry wrinkled hers.\n\n_\"Humans can't smell worth squat.\"_ Pewter leapt onto the counter with Mrs. Murphy.\n\n_\"Even a human can smell a corpse that's been dead for a bit, the days being warm. It's not the full-blown effect but they can tell.\"_ With her phenomenal powers of smell, Tucker knew of what she spoke.\n\n_\"Sometimes I wonder how they survive.\"_ Pewter looked out the window over the sink, where the last moth of the year fluttered.\n\nUnaware of the animals' condescending observations, Fair leaned back in his chair. \"Pretty easy day today, so I actually got a little research done. Stem cell stuff.\"\n\n\"Haven't you used stem cell therapies?\"\n\n\"Not much, honey, though I'd like to. It's complicated but those treatments really work for horses' musculoskeletal injuries. Vets have been using stem cell transplants since 2005. The problem is that there are bogus firms on the Internet that claim stem cell therapies can treat laminitis and neurological conditions, and that's not true.\"\n\nHarry knew that \"laminitis\" meant an inflammation of the sensitive tissue of a horse's hoof.\n\n\"I guess there are scam artists in every profession,\" she lamented.\n\n\"There are, but when they cause suffering to living creatures, my blood boils. Someone without veterinary knowledge or degree, but with every good intention in the world to help their horse, gets on the Internet, finds a bogus product, buys it, and their horse continues to suffer.\"\n\n\"Do you think the stem cell transplants used for horses will work for people?\"\n\nHe folded his hands together. \"Yes. But there again, as the science progresses, you will have doctors making wild claims or dishonest companies doing so.\"\n\nFair was such a fount of knowledge that sometimes Harry fired question after question at him. \"You think in time the obstacles for using human stem cells will be removed?\"\n\n\"In some cases they already have, and while I believe in relieving suffering, I have to think about this one. It truly is a complex moral issue.\"\n\n\"Maybe everything is, honey,\" Harry said.\n\n\"Yes. Take murder. It seems cut and dried, doesn't it? And yet surely there are times when murder is justified. A wife defends herself and her children against a rampaging husband. I couldn't find it in my heart to condemn such a woman. Throughout all of nature, mothers kill or die defending their young.\"\n\n\"Nothing is really that clear cut, is it?\" Harry agreed.\n\n\"Well, the Ten Commandments make it seem so, but I guess there are exceptions to every rule, even those. Maybe that means I'm going to hell.\" He half-smiled.\n\n\"You're the best man I know,\" she said, smiling his way. \"And there have to be millions of people who ask questions, who wonder. What helps me is talking to Reverend Jones. I don't know what I would do without him.\"\n\n\"I should talk to him about this.\" Fair unfolded his hands. \"That and genetic engineering. We may be on the cusp of creating a super horse, and if we do that, people aren't far behind.\"\n\n\"That's a terrifying thought.\"\n\n\"It will start out safely enough. A tag on a gene sequence will be discovered to cause some kind of cancer. Doctors will get in there and manipulate the sequence. It sounds far-fetched but it isn't. Just look at the genetic manipulations you've seen in crops.\"\n\nShe took a long sip of scotch. \"You know, when I was a kid, Dad and I would sit down with the seed catalogues. We'd try and figure out which corn could survive a drought, too much rain, which one had the sweetest taste. You had so many choices and now, well, you really don't. I look in my catalogues and there's just one page for corn, and every offering has disease resistance, a list of qualifications. Me, I just want Silver Queen,\" she said, citing an especially delicious corn usually available at central Virginia vegetable stands in August. Her eyes misted. \"Hester sold the best Silver Queen.\"\n\n\"Brave new world.\" He smiled at her. \"I am so sorry you found Hester. So sorry.\"\n\n\"If it's a brave new world, that means we have to be brave to face it. But I remember the Law of Unintended Consequences. You never know what you're stirring up.\"\n\n\"By God, that's the truth,\" he said, slipping his arm around her again.\n\n\"But these murders have very intended consequences. They were carefully planned and enacted, and inflicted on the rest of us. That means there's a message. You don't do something as elaborate as this unless it's an attempt to make a statement of some sort.\"\n\n\"And that's why you'd better carry your father's snubnose .38.\" Fair's voice was firm.\n\n#\n\nFriday the air sparkled, the leaves exploded with color, and the temperature hung at about forty-seven degrees. Fall ushered in many changes. Fur-covered animals now had their thick undercoat, the outer coat shining luxuriously. With the waning of daylight, chickens laid less eggs. All those creatures slowed down by summer's heat now surged with energy. Robins and some ducks and geese had already departed on their southern journey, as had the monarch butterflies. Everyone else busied themselves with nest repair. Turtles readied for sleep, along with other amphibians and reptiles. Toads lined their shallow nests with straw, hay, anything that could insulate, as did mice, who could unravel a sweater quickly.\n\nHarry had once left a thick wool sweater in the tack room only to come back the next morning to find mice had chewed big holes in it, all of that fine wool now lining their nests behind the wall. Those mice lived good.\n\nThis perfect October morning, having just finished the barn chores, Harry tossed up some jelly beans for the possum in the hayloft, then shook the hay bits out of her hair.\n\n_\"Don't shake that on me,\"_ complained a perfectly groomed Pewter, languishing below.\n\n\"All right, who's ready to go?\" asked Harry.\n\n_\"Me!\"_ Tucker ran in from the back of the barn.\n\n_\"Me, too,\"_ Mrs. Murphy echoed her friend.\n\nBoth eagerly sat in the center aisle as Harry hung up her big wide sweeper. \"That's two. Let me check on Pewts.\"\n\n_\"Leave her here,\"_ Tucker advised. _\"She's such a priss and a pain.\"_\n\nPewter lifted her head from her paws. _\"I heard that.\"_\n\nShe'd been sleeping in the tack room, disturbed only when Harry had shaken out the hay while looking in the big mirror.\n\n\"Come on,\" Harry urged her.\n\n_\"I have nothing to wear,\"_ Pewter replied facetiously.\n\n_\"Just leave her,\"_ Tucker practically begged.\n\nMuch as Pewter wished to languish in the tack room, the prospect of irritating the corgi held greater allure. She rose, stretching fore and aft, then daintily leapt to the floor and sauntered out the tack room door.\n\n_\"Peon,\"_ the gray cat remarked to the sitting dog as she passed.\n\n_\"Pissant,\"_ Tucker fired back.\n\nTucker flattened her ears and readied herself to lunge after the large cat, but Mrs. Murphy whispered, _\"Cool it.\"_\n\n_\"How can I let her get away with that?\"_ asked Tucker.\n\n_\"If you growl or chase after her, Mom will leave you here. She hates fights in the truck or car, you know that.\"_\n\nTucker's ears drooped, her expression saddened. _\"That cat gets away with murder.\"_\n\nPewter, full sashay working\u2014a swing to the right, a swing to the left\u2014called over her shoulder, _\"I am fascinating. Harry never likes to go anywhere without me. You, on the other hand, are a mere drooling dog. So eager to please. Peon. You really are a peon.\"_\n\nMrs. Murphy walked tightly next to her canine friend. _\"Ignore her.\"_\n\nHarry opened the door to the 1978 Ford F-150, a half-ton pickup truck you couldn't kill on Judgment Day. The two cats jumped onto the floorboard while the human bent over to pick up the solid dog. \"Onf.\"\n\n_\"Don't call_ me _fat.\"_ Pewter grinned as the dog was placed on the seat.\n\n_\"I'm a lot bigger than you,\"_ Tucker said, defending her weight.\n\n_\"Oh, la,\"_ the cat sang out, then crawled onto Harry's lap once she was behind the wheel.\n\nHarry could easily drive with a cat in her lap, so she fired up the old engine, listened to the melodic deep rumble, then pulled around and down the long drive.\n\n_\"Where are we going?\"_ Pewter asked.\n\n\"Okay, you all, we're going to Cr\u00e8me de la Cr\u00e8me. I am finally going to break down and buy two of those heavy Italian mugs.\"\n\n_\"She understood?\"_ Tucker's ears shot up.\n\n_\"Of course she didn't,\"_ Pewter laughed. _\"She just likes to hear herself yak.\"_\n\n_\"Look who's talking,\"_ Tucker said in a low voice to Mrs. Murphy.\n\n_\"I have good ears, you know,\"_ Pewter said.\n\n_\"You do,\"_ Mrs. Murphy agreed.\n\n_\"It's your attitude that's not so good.\"_ Tucker had to say it.\n\nThe gray cat turned her back on the other two on the bench seat, rested her chin on Harry's left forearm, and watched the passing scenery out the side window.\n\nSlowing for a turn, Harry could see the houses on the ridge at the Old Trail development. Below, she spotted Buddy Janss on his huge tractor, harvesting his soybeans. The other side of the road was filled with corn: some rows cut, others left standing to dry. On the road, yellow metal signs about the size of the old Burma Shave signs marked the rows. Each sign showed a golden ear of corn with two green leaves folded back. Below that, the seed company Demeter was identified in red letters, and under that in black Arabic numerals were the seed ID numbers. Buddy Janss had worked with Demeter for years and this was his test field. His other acres were dedicated to revenue crops.\n\nHarry pulled off the road as Buddy cut the motor of his tractor. He climbed down to check something on the attachment. Satisfied, he turned to climb back up.\n\n\"Buddy!\" Harry called out as she walked toward him, with three four-footed friends in tow.\n\n\"Harry.\" He smacked his baseball hat on his leg. \"After hearing the news, I was going to call you, girl, but I figured you'd had enough.\"\n\nHe wrapped his massive arms around her, giving her a hug as she kissed his cheek.\n\n\"I appreciate that. It was crazy.\"\n\nHe put his hand on her shoulder. \"I can't believe it, just can't believe it.\"\n\n\"Me neither.\"\n\n\"Now, Hester was peculiar, no doubt, but she was a good girl.\" He wiped a tear from his eye with his handkerchief.\n\n\"I've been racking my brain to think who could do such a thing.\"\n\n\"Can't think of a soul, can you? Who would want to hurt Hester?\" He looked into her eyes. \"She could get in your face about things, stuff she really cared about, like ethanol, but you don't kill someone over ethanol. And there were certain people she just wouldn't do business with, but how much money would a farmer lose by not having his produce sold at Hester's? I'm like you, racking my brain.\"\n\n\"I've been thinking over her many pet projects, pet peeves,\" said Harry. \"We all know she loved the Crozet Library. She loved history and wanted to preserve as much of our history as she could. She also cared about farming practices.\" Harry laughed as the tears rolled down her face. \"I don't think Hester ever saw an abandoned building she didn't want to save. Like some people save animals, she tried to save existing buildings or raise funds for a building the community needs. Remember when she was afraid the old Coca-Cola building would be torn down? And I think she checked into the three abandoned school buildings there by your one hundred acres. Oh, also she wanted to have designated as a historical spot the house where Georgia O'Keeffe lived ever so briefly. You don't murder someone over any of that.\" Harry cried more, which made Buddy join in.\n\nBuddy, like most powerful men in this part of the world, readily showed emotion. Bighearted to a fault, they wanted to hold babies, pet your dogs, take your arm as though you needed guidance, and to help any lady, old, young, pretty or not.\n\nHarry wiped her eyes, then reached up and wiped Buddy's. \"I hate that someone made a mockery of her in her death,\" she said.\n\n\"You know, girl, sooner or later that S.O.B. will make a mistake, and I want to be there.\"\n\nLittle did Buddy or Harry know, he would get his wish.\n\n#\n\nThat afternoon, Harry peered down at Mrs. Murphy, Pewter, and Tucker, all sitting at her feet in the tack room. \"Remind me never to buy an Italian desk lamp again.\"\n\nAfter two years, the high-intensity bulb had burned out. The lamp boasted appealingly sleek design, but getting its bulb out was proving infuriating. She had to flip back the head of the angular lamp, figure out how to remove the frosted-glass square, then dislodge the small cylindrical bulb.\n\nA rustle of mice behind the tack box irritated Mrs. Murphy. She left her human and jumped on top of the box, squinting behind it. _\"Stay put,\"_ she warned.\n\nMartha, the savvy mother of many, giggled. _\"We've never heard so much cussing in our lives.\"_\n\nThe tiger cat smiled. _\"You know our bargain.\"_\n\n_\"I do, but think of my children. Such language.\"_ Martha looked up into the predator's green eyes.\n\nYears ago the two cats had made a deal with the mice who lived behind the baseboard in the tack room. The insulated tack room walls provided toasty winter lodging, along with the yarn, hay bits, and old rag pieces the mice brought in to further line their nests. So long as the mice didn't show themselves or chew tack or saddle pads, they could eat whatever fell in the horses' stalls. This way the cats didn't look as though they'd fallen down on the job and the mice could tidy up the horses' mess. Also, mice heard things domestic animals did not. They occasionally provided useful information.\n\nAbout once a month, Mrs. Murphy or Pewter would dispatch a field mouse or mole and dutifully drop it at Harry's feet, after which the attractive woman praised them lavishly. She never knew the difference, bragging to her friends about how she never saw a mouse in her house or barn. Well, she never did.\n\nHaving finally pried out the oddly shaped light bulb, Harry turned it around in her hand. \"How am I supposed to find something like this?\"\n\n_\"Go to Eck,\"_ Tucker said, sensibly suggesting an electrical supply firm because Harry would never find such a replacement bulb at Wal-Mart.\n\nHarry glanced down as the dog offered unintelligible advice, then looked up again.\n\n_\"Car!\"_ Tucker immediately charged out the tack room's animal door, then charged back in. _\"Coop!\"_\n\n_\"I could have told you that,\"_ Pewter said, sprawled on the desk behind the offending lamp.\n\n_\"It's my job to announce any intruder or visitor,\"_ Tucker said. _\"I am good at my work.\"_ The corgi pouted for a moment.\n\n_\"You are,\"_ Mrs. Murphy complimented the dog, then looked behind the tack trunk and addressed Martha the mouse. _\"I can't control what she says. Cover your children's ears.\"_\n\nCooper entered the tack room and took a look around. \"Is this another Haristeen project?\"\n\nHarry motioned for her to sit in one of the director's chairs. \"You can call it that. I will never buy anything based on design again.\"\n\nCooper, studying the lamp on its side, said, \"Pain in the ass. All this fabulous-looking stuff. Like Gucci high heels that torture the feet. Just a royal pain. I'm ready to break out the oil lamps.\"\n\n\"I have them for emergencies. The smell isn't all that bad but the little plume of smoke will have you scrubbing ceilings and walls.\"\n\n\"If it's dark, you won't see it,\" Cooper laughed. \"Less light now anyway. Every day gets shorter until December twenty-first, the solstice.\"\n\nHarry dropped her hands into her lamp. \"Always gets me a little.\" Then she handed the light bulb to her neighbor. \"Look at this.\"\n\nBringing the tiny bulb close to her eyes, Cooper said, \"To get another one of these, you'll have to go into town, spend time and burn gas. Oil lamps, I'm telling you, and think what we'd save on electricity.\"\n\n\"My darned electric bill for the house, the barn, and the big shed ran me over five hundred dollars last month, and you know that figure will go up with the darkness. Electric bills never get cheaper.\"\n\n\"Nope,\" Cooper said. \"Of course, our entire society is dependent on it, and I'm as dependent as the next guy. Sometimes I wonder what kind of corner we have painted ourselves into.\"\n\n\"Me, too.\" Harry took the bulb back, placing it in the long desk drawer.\n\nPewter would knock it on the floor if Harry didn't hide it. She had to remember where she put it, though. Sometimes when a lot was happening all at once, Harry would forget the little things.\n\n\"I brought my seed book, thinking I'd swing by on my way home,\" said Coop.\n\n\"Where's the book?\"\n\n\"Out in the car,\" Cooper said, standing. \"I figured I'd ask first if you had the time.\"\n\n\"You don't have to ask. Go get it.\"\n\nWithin a few minutes the lanky police officer returned with a large, fat seed catalogue.\n\n_\"If they go through that whole thing, we will never get supper,\"_ said Pewter, mildly alarmed.\n\n_\"Do you some good.\"_ Tucker mischievously grinned.\n\nPewter sat up. _\"I'm laying for you, Bubble Butt.\"_\n\nThe dog ignored her as Mrs. Murphy left the tack trunk to sit next to Tucker, just in case.\n\nHarry flipped through the pages, the glossy photos tempting her to think she, too, could grow such specimens. \"So, what are you looking for, flowers or vegetables?\"\n\n\"Buddy Janss promised me some corn seed and Miranda is giving me rose cuttings. And she said I could dig up that one Italian lilac bush she has.\"\n\n\"If it's Italian, don't do it,\" Harry laughed. \"This damned lamp is Italian.\"\n\nCooper laughed with her. \"I'll bear that in mind. You know the best varieties of okra, lettuce, all kinds of tomatoes. I don't know too much.\"\n\nHarry read copy. \"Okay. My advice\"\u2014she picked up a pencil and began circling vegetables\u2014\"is to go with the hardiest. Also, the old varieties often taste better but they're harder to grow sometimes. So, here.\" Harry pointed to a green pea. \"A little water, a little sun. Tough. And so is this squash.\"\n\nAs Harry flipped through pages, circling types of vegetables, Cooper talked. \"Both of our recent murder victims' bodies are with the medical examiner, but we already know how they were killed: bullet through the heart. No struggle, and the killer faced Josh and Hester. So it's likely the killer seemed unthreatening, or they knew who it was and didn't fear him\u2014or her.\"\n\n\"Face-to-face. Damn, that's cold-blooded.\"\n\n\"It is. I'm telling you because you found them. The paper will report the gunshot wounds, but we're holding back details, like face-to-face. No true bruises on the forearms, no teeth knocked out. You'd be surprised how many people fight for their lives, but neither Josh nor Hester fought, so I hope it was quick. Finding those victims always affects me. I wonder, were they afraid or was the adrenaline too high for them to act defensively? I guess it varies from person to person. One confusing moment of recognition when the gun was pulled, then _bam_.\"\n\nHarry shivered. \"It's still an awful thought.\"\n\n\"Well, this is more awful: If Hester knew her murderer, so do we. I think of that a lot, how many killers do I pass each day and don't know it?\"\n\nHarry rested her chin in her hand. \"Never thought about that, but then, I'm not a deputy.\"\n\nCooper sighed. \"Well, didn't mean to sound so negative. Back to the seeds.\"\n\n\"I marked everything you need, because I know your soils,\" said Harry. \"You and I are both right up on the base of the mountains. We have our own little weather system.\"\n\nCooper took the catalogue back. \"Thanks.\"\n\n\"This isn't negative exactly, but I can't erase the sight of both those people's grisly ends, and I didn't know how much I liked Hester until, well, until she wasn't with us,\" admitted Harry. \"What could she have ever done to provoke being killed? I didn't know that accountant, but Hester wouldn't hurt anyone. Oh, she might make you check your watch, but she was okay.\"\n\n\"Best roadside stand in the county, and she'd always try to give me something for free,\" said Coop. \"I'd tell her I can't take anything when I'm on duty. I mean, really, she wasn't bribing me, but the rules are rigid and people these days are so quick to find fault. If anyone had seen me take a cantaloupe and not pay, I bet you Rick or the newspaper would have heard about it.\"\n\n\"I can see the headline now: 'Scandalous Melon Payoff.' \" Harry laughed.\n\n\"I'm starting to wonder if these aren't some sort of thrill killings. Usually sex is involved in those cases, or some sort of dominance or power play. This doesn't exactly fit the pattern, but then again, maybe we, the public, are supposed to feel a thrill, a ripple of fear.\"\n\nHarry thought a long time. \"So the killer is really warped or really smart, or both.\"\n\n#\n\n\"P _oppy, she's not listening to you,\"_ Elocution warned as she sat in the chancery window the next morning, observing Harry outside.\n\nCazenovia and Lucy Fur meowed in unison.\n\nRev. Herb Jones, wedged behind his desk, glanced up, opening his mouth to quiet the kitties, then he heard a metallic _clink_. Pushing the chair away from his desk, he rose, hurrying to the window where Elocution fussed.\n\n_\"See! See!\"_ the cat spoke louder.\n\n\"I will bless her.\" Reverend Jones hustled out of his beautiful office, grabbing his coat and dashing out the back door. \"Harry, what are you doing?\" he said, finding her next to a ladder propped against the building's wall.\n\n\"Waiting for the roofer?\" she half-fibbed.\n\n\"You were going to climb up there, I know it.\" Reverend Jones's face reddened.\n\n\"Well, eventually.\" She flashed her brightest smile.\n\nInside, Lucy Fur turned to Elocution. _\"He can never resist her smile.\"_\n\nCazenovia agreed. _\"It's amazing.\"_\n\nOutside, Harry, hands in pockets for the air had chilled, headed off a lecture. \"Seth Isman will be here in a minute. I just know bad weather is around the corner, so I figured we'd better hop on this.\"\n\n\"Uh-huh.\" The reverend crossed his arms over his chest. \"Hop on, hop up.\"\n\n\"Don't worry about me. I've got two feet on the ground and am looking at the most wonderful Lutheran minister in Virginia.\"\n\nHe burst out laughing. \"You stinker.\" Then he put his arm around her. \"Sweetie, I do thank you for taking on building and grounds and for doing this so soon after finding Hester's body. We would all have understood if you'd waited.\" He took a deep breath. \"God rest her soul.\"\n\n\"I'd already made the appointment for today, Reverend, and truthfully, I feel better if I'm busy.\"\n\n\"You know, Hester's service still isn't organized.\" The pastor shook his head. \"Hester's brother died years ago and her niece lives in Houston. I called over to St. Francis in Staunton, where Hester worshipped, but so far, no plans. We are all distressed. If I knew her niece I'd offer help, but Hester's priest told me the young lady\u2014her name is Sarah Price\u2014is doing all she can and he feels things will be properly done. She'll get here from Texas next week. Terrible. Such a terrible thing.\" He turned as the roofing van drove up and parked in the rear of the church lot. \"Now, see here, Harry, don't you get up on that roof. Look me in the face and promise.\"\n\nTaking a deep breath, she promised, \"I won't.\"\n\n\"You can't dissemble to my face.\" He laughed. \"Well, you can't really lie anyway. Never could, but that doesn't prevent you from withholding information or wiggling just a little.\"\n\n\"You've known me too long.\"\n\n\"Remember that. I've got my eye on you.\"\n\nTrue to her word, Harry remained earthbound while wiry Seth scrambled around up on the roof. After a few minutes, he backed down the ladder.\n\n\"Is it worse than I thought?\" asked Harry of the short young man.\n\n\"No,\" he replied. \"A two-foot-square area, just like you thought, should fix the problem and prevent more. The workmanship on that old roof is something, just something.\"\n\n\"Our ancestors knew what they were doing and they weren't deluded by technology. It still takes good materials and a good man who knows how to use them.\"\n\nSeth smiled, which enlivened his strong face. \"Yeah. We're losing it, though. Losing hand skills.\"\n\n\"You haven't lost yours,\" Harry complimented him.\n\n\"Thanks, Harry. Once I decided to concentrate on older structures, things just fell into place. I don't work with cheap materials. After I've repaired a roof, I don't get calls back about leaking. I understand that most folks don't know about construction. And they only have so much money, so they buy footage and flash instead of maybe something smaller that is well built. Being able to get up on this roof, seeing how those shingles were laid... I don't know. Kinda gives me chills. Like I'm part of something that goes way back.\"\n\n\"I know what you mean. Well, you know I have to report to the board, so as soon as you can write up an estimate, I will deliver it. They already agreed for the work to be done, but if I can present an estimate, that makes everyone feel better.\"\n\n\"You'll save money because I can do this with just one other man. We can work pretty fast together. The really good news is I have a source of slate shingles that should closely match yours. A huge old house was dismantled in Cumberland County. The heirs just let it go. Built in 1719.\" He paused. \"A little bit of history slips away but it takes money to restore and keep those old places going. I understand, but if you can't do it, sell it to someone who can. Don't wait until it falls apart.\"\n\n\"Good advice, but when there's more than one heir, things tend to get dragged out.\"\n\n\"Boy, that's the truth. Anyway, I can get on this next Monday. Figure a full day just in case. If all goes well, half a day. I don't think the bill will go over four thousand dollars, and I will try to do it for less. Preacher's price.\" He smiled broadly.\n\n\"Seth, you're a good egg.\"\n\n\"You don't know me,\" he devilishly replied. \"Want me to put the ladder back?\"\n\n\"Sure. Thank you. I'll walk with you to the shed.\"\n\nAs Harry and Seth strolled away, chatting about SEC football, she noticed Neil Jordan drive up, followed soon after by Wesley Speer.\n\nAfter Seth drove away, Harry returned to the chancery.\n\n_\"There's a lift to her step,\"_ Lucy Fur noted. _\"Must be good news.\"_\n\n_\"We'll see.\"_ Cazenovia hopped off the windowsill to hurry out of the room and down the hall to greet Harry, whom she very much liked.\n\nPushing open the back door, Harry beheld the beautiful longhaired calico cat already on her hind legs.\n\n\"Caz.\" Harry knelt down and scooped her up. \"Such a religious kitty. And such a good concierge.\"\n\n_\"I am.\"_\n\nCarrying the contented cat, Harry peered into the large office. She didn't want to disturb the reverend, as he seemed to be in the middle of a meeting. Neil and Wesley sat in the chairs around the coffee table. The reverend was standing at his desk, papers in hand. He looked up at her.\n\n\"I can come back,\" said Harry.\n\n\"No, come on in.\"\n\nPutting Cazenovia down, Harry pulled off her work gloves. \"Hi.\"\n\nNeil and Wesley stood up to greet her.\n\n\"Should I brace for the worst?\" Neil joked.\n\n\"No. Good news. Seth can start on the roof next Monday, should finish the same day, and\u2014here's the good part\u2014he's got a source of old slate shingles and he feels sure he can keep the bill under four thousand. What luck.\"\n\n\"I suppose if a bill can be said to be good, that is,\" Neil solemnly replied.\n\n\"Neil, slate costs an arm and a leg,\" said Wesley. \"Old slate, especially. We'll make up the shortfall if we have one, but I bet we don't.\" He beamed.\n\n\"Harry, sit down. Let's all have a hot cup of cocoa or whatever. Betty!\" Reverend Jones called.\n\nA middle-aged woman stuck her head in the room. \"Yes? Oh, hi, everyone.\"\n\n\"How about cocoa?\" The reverend looked at his small gathering.\n\n\"Cocoa sounds perfect.\" Harry smiled.\n\n\"I'll go with that,\" Wesley agreed, as did Neil.\n\nThe young, pretty secretary usually outside the reverend's office was now in her last months of pregnancy and on leave. Filling in was Betty Maddox, cousin to Dorothy, the sheriff's department's chief of forensics. You had to be careful what you said about people in Crozet, as most folks were related.\n\n\"While we wait, it's good you're here, Harry,\" said Neil. \"I'd like to give the church lawns a good dressing of fertilizer and put it down before mid-November. Give it plenty of time to get into the ground. Checked the soil. Good pH, selenium. Potassium okay. Needs a little magnesium.\"\n\n\"Neil, that's wonderful.\" Harry smiled. \"The little mini-drought that we had didn't affect our lawns too much, but fertilizer always helps, and I'll come on back in springtime and drill in some wonderful lush grass seed. I always throw some rye in, too. Give the clover and bluegrass early cover.\"\n\nThanks to his real estate company, Wesley kept up with farming news. His largest sales were big estates and he had to know something about soil conditions and crop yields if selling to a true farmer, or even a new person who would lease out the land. Most new people wanted to live on a grand estate but didn't want to actually farm, which was wise since they weren't raised to it.\n\n\"Harry, did you see where the USDA\"\u2014he used the initials for the United States Department of Agriculture\u2014\"predicts the drought reduced our economic growth by almost half a percentage point? That's extremely serious.\"\n\n\"Sure is,\" she agreed. \"But I was talking to Buddy Janss and he said what was so bizarre was that sometimes fields on one side of a road twisted up while on the other side the crops were healthy. What crazy weather. Buddy has suffered some losses, though.\"\n\nNeil didn't much like Buddy, in part because the large fellow didn't buy his fertilizer. Buddy was so smart he'd worked out a deal years ago with horse owners to remove their manure and straw for free. This he put in piles, let it cook, then the next year used it himself, selling the extra for fertilizer. The horse owners, most of them owning but a few horses, gladly paid him to haul off the muck. Buddy used commercial fertilizers if a field needed extra potassium or another nutrient. That drove Neil crazy, but then, the two possessed such differing personalities they would have struggled to like each other no matter what. Neil was detail oriented and picky, whereas Buddy was expansive, and did his best but didn't fret.\n\n\"Wasn't Buddy friends with Hester?\" Neil asked, as he had only lived in the area a few years.\n\n\"For years and years.\" Harry smiled. \"You know Hester wouldn't sell anything that was sprayed or if the seeds had been genetically modified.\"\n\n\"She was a crank,\" Neil said. \"Not that I wished her dead, but really.\"\n\n\"Hester was an eccentric,\" the reverend said in his most diplomatic tone, \"but she worked hard for causes she believed in, she mentored younger people like Tazio, and I expect any of us could be considered a crank at one time or another.\"\n\n\"Not you.\" Harry grinned and the men laughed.\n\n_\"You should live with him,\"_ Elocution called out from her fuzzy den on the floor.\n\n_\"He feeds us Fancy Feast and he even tried to see if we'd chew on greenies,\"_ Cazenovia chided her from the windowsill. _\"He's the best.\"_\n\n_\"Yeah, Elo,\"_ Lucy Fur chimed in. _\"Button your lip.\"_\n\n_\"All right, all right,\"_ the Lutheran cat said, giving in.\n\nBetty arrived with a tray of hot cocoa and sugar cookies. The Reverend Jones jumped up to carry it and place it on the table.\n\nWesley returned to the subject of Hester. \"Horrible. Harry, you have endured two shocks. Finding that young man and then Hester.\"\n\n\"Did,\" she agreed. \"As I didn't know the fellow who was killed, it was a shock and that was all, but Hester, that hurt. Yes, she had her ways, but she was a good soul and really pretty smart. I mean a lot smart, actually.\"\n\n\"That she was,\" Wesley agreed. \"The last time I stopped by the stand, we got on the subject of crop irrigation. I don't remember how we did get on it\u2014you know with Hester, one thing didn't lead to another, it jumped to another. But anyway, she was telling me that farmers have been pulling water out of the Ogallala Aquifer since the early 1950s and some of those irrigation booms are a half mile long. A half mile!\"\n\n\"Great day,\" Reverend Jones exclaimed.\n\n\"It's hard to imagine, isn't it?\" said Wesley. \"A half-mile boom spinning around a fixed water pipe? But there's a lot of talk, consideration in a lot of the affected states, about cutting back on irrigation because the droughts are dropping water levels, as is all the population growth.\"\n\n\"Where are water levels dropping? Which states?\" Neil asked.\n\nFortified by the cocoa, Wesley leapt in. \"Neil, it's eastern Wyoming, about all of Nebraska, southern South Dakota, eastern Colorado and New Mexico, a huge swath of Kansas, Oklahoma's Panhandle, and a chunk of Texas. The water shortage is huge.\"\n\n\"The breadbasket,\" Harry thought out loud.\n\n\"For us. For the world, too, really,\" Wesley said. \"Hester had been reading up on it, just like she was always reading about chemicals, her history interests, that sort of thing. I was so impressed at the facts she had at her fingertips. She felt if farmers didn't cut back, wells would run dry and that would become a disaster, a true disaster. Irrigation accounts for one-third of our nation's annual water demand. I told her that genetic engineering could create more drought-tolerant corn, soybeans, etc. We could reduce our irrigation, but she didn't want to hear that.\"\n\nThey laughed.\n\nHarry stood up. \"It was good to see you all and I'm glad I have what I think is good news about the roof. Tell you what we could do for Hester: Let's sell all those tickets for the Halloween Hay-ride. The funds go to the library, and we know how much Hester loved the Crozet Library. Will you all help me?\"\n\n\"It would be an honor,\" Wesley immediately replied.\n\n\"Of course,\" Neil agreed.\n\nIn his gravelly voice, the Reverend Jones said, \"I can preach a good sermon on this. We'll sell those tickets. We'll sell out! Thou hast put gladness in my heart.\" He smiled. \"Psalm 4:7. If we sell those tickets, it will put gladness in all our hearts.\"\n\n#\n\n\"How many miles have you racked up on this car?\" Harry asked later that day. The two friends were headed back to that shopping mecca, Warehouse Number 9.\n\nIn the driver's seat, Susan glanced down at her Audi's odometer. \"Let's see... 131,839. I'm averaging about 40,000 miles per year. Engines are so well made these days they aren't even broken in until 100,000 miles.\"\n\nSwaying slightly as they turned right at the stoplight in Dillwyn on Route 20, Harry said, \"True. The advances in engine longevity are pretty fabulous. 'Course, my old '78 rolls along, but I baby that truck, as you know. Actually, Susan, I really like driving without computer chips.\"\n\n\"That's you. I don't care.\" Susan smiled. \"Is everyone asleep back there? It's so quiet without the cats.\"\n\nHarry twisted to look. \"Owen is curled up with his sister Tucker. That was one of the best litters you ever bred.\"\n\n\"It was.\" Susan nodded. \"I loved breeding corgis, but it was so much work, and part of that work was making sure the puppies found the right homes. I love all dogs but I especially love corgis.\"\n\n\"I love Tucker. Sometimes I think about the German shepherd Mom and Dad had when I was a kid. That was a great dog. Funny how you can measure your life by animal lives.\"\n\n\"Wonder if that scarecrow fellow had any pets.\"\n\n\"No. Coop told me he lived an unencumbered life.\"\n\n\"Sad,\" Susan replied simply.\n\n\"I think so, too. What's the purpose of being alive if you don't have husbands, friends, cats, dogs, horses, birds, possums, more friends, and friends' children? It just goes on and on. Mother used to say that if everyone in Virginia studied their bloodlines, we'd find out we are related. No one is all black, no one is all white. We're all part of one another and that includes the Indians.\"\n\n\"Can't say 'Indian' anymore.\"\n\n\"Sure about that? These labels we give ourselves are always changing.\"\n\n\"Now, Harry. You look just like your father when you get muley.\"\n\n\"Do I?\"\n\n\"Exactly.\"\n\n\"Oh, dear.\" Harry slumped in her seat slightly. \"Daddy could be...\"\n\n\"Yes, he could.\"\n\nThey both laughed, remembering Harry's much-loved and very original father.\n\n\"Today is going to be a mob scene at the store. People are feeling that change in the seasons, the holidays looming. Those credit cards start burning in one's pocket.\"\n\n\"Not mine,\" Harry staunchly declared.\n\n\"It is possible to be too tight. I mean, Harry, you don't have voice messaging on your phone because it costs an extra three dollars a month. That's silly. Three dollars!\"\n\n\"A penny saved is a penny earned,\" Harry countered.\n\n\" 'Tis, but you can carry it too far. Hey, where do I turn?\"\n\n\"Left up ahead.\"\n\nSusan turned onto the Farmville main drag, then turned left again at Harry's direction, and shortly the Audi station wagon was parked in the lot closest to Number 9 warehouse. As predicted, the place was packed.\n\nSusan cracked the windows for the two dogs, although the day was brisk. \"You all go back to sleep. We won't be long.\" She hoped this wasn't a fib. She loved looking at furniture, fabrics, even lampshades.\n\nThe two women walked into Number 9, and Harry immediately pulled Susan to the Halloween display.\n\n\"Exactly the same,\" Harry declared.\n\n\"As I didn't see the corpse in the cornfield, I can only imagine what a human looked like as opposed to this.\"\n\n\"But that's just it. From a distance, they look exactly the same.\"\n\nSusan stood next to the ghosts. \"No witches.\"\n\n\"Not in this display. It's ghosts, little goblins, pumpkins, and the scarecrow.\" Harry sat down in a kitchen chair for a moment. \"You know, I think of Hester hanging all that black and orange bunting at the stand, then last week trussing herself up in a witch costume while unloading produce. She did have such a funny sense of things, and you had to laugh. Why would anyone kill her? I've thought of everything, including her being a Russian spy. I mean wild stuff. Nah.\" Harry shook her head.\n\n\"Me, too.\" Susan looked around. \"I'm going to walk through the floor. Won't be long.\"\n\n\"I'll tag along.\" Harry did.\n\n\"Look at this.\" Susan pointed out a distressed bureau painted a sky blue. \"That would look good in my workroom.\"\n\n\"What do you need with a bureau in your workroom?\"\n\n\"Store papers in it. Better looking than a file cabinet.\"\n\n\"Yeah, I guess. Here.\" Harry handed her a notebook and pencil from her coat pocket. \"Write down the particulars. If you decide you want it, you can call. They deliver.\"\n\n\"I forgot about that.\" Susan scribbled down the item number and manufacturer.\n\nHarry again tacked over to the Halloween display. \"Whoever killed Josh Hill had to have seen this scarecrow. It looks exactly the same. So the killer is someone who comes through here regularly.\"\n\n\"Maybe. It could also be someone with a good memory or someone who took a picture on their phone. That could be just about anyone.\"\n\n\"You're right. I'm jumping to conclusions.\"\n\n\"They should make that an Olympic sport.\" Susan put her hand under Harry's elbow to steer her out of the store.\n\nThey walked over to Number 8, which had a courtyard featuring large outdoor sculptures for sale.\n\n\"I like the large horse.\" Harry stood next to an almost life-sized horse resembling the horses of Piazza San Marco. \"Can you imagine what would happen if I put one by the barn?\"\n\nBoth women laughed. They knew the statue would spook the real horses, although eventually they would adjust.\n\nSusan flipped the price tag over. \"You'll need smelling salts.\"\n\nHarry bent over to peer at the tag. \"Nine thousand dollars!\"\n\n\"You pay for your pleasures.\" Susan checked her watch. \"Speaking of which, if I stay here, I am going to spend money, and I don't have it right now. Ned isn't making as much at the law firm. He's in session and there goes the income. He thought he could swing it, but there's so much to do down in Richmond, so many meetings and so much material to master, plus he had to rent an apartment. It's overwhelming. Yet he loves being in our House of Delegates. Anyway, I'm thinking of finding a job.\"\n\nOnce they were back in the car and heading home, Harry said, \"Your kids are out of the house. No reason you can't work full-time.\"\n\n\"When we graduated from college and I got my first job as a legal assistant, I remember shopping in the supermarket, seeing the women at the checkout counters and wondering what went wrong. You know what I mean? How did they wind up in that job?\"\n\n\"I never thought about it. You were, _are_ , better about that stuff than I am.\" Harry put on her sunglasses.\n\n\"Well, I thought maybe those cashiers had picked the wrong man. He'd left them high and dry and with children. Or they were people who didn't plan ahead and one day woke up at forty. As years rolled along, I realized that sometimes bad luck rolls over someone like a tide. I felt less superior after that. Now I look at those women and think it could be me, you know?\"\n\nHarry thought for a long time. \"I don't. Susan, I always knew I would farm.\"\n\n\"But what if the crop failed year after year? What if you became injured?\"\n\n\"I have had crops fail and I survived, on not much. I figure whatever happens, I can deal with it. 'Course, it's easier now with Fair. My hardest times were without him.\"\n\n\"It preys on my mind, finding Hester like that,\" said Susan. \"She never expressed fears. But I think maybe her ideas\u2014like aliens being responsible for crop circles, stuff like that\u2014maybe that was how she expressed fear.\"\n\n\"Susan, you might be right. I don't know. I don't look into other people like you do or like Fair does; I kind of take everyone at face value.\"\n\n_\"What if they're hiding behind a mask?\"_ Tucker wondered.\n\n#\n\nAlthough two hundred and twenty years old, the organ at St. Luke's sounded as good as the day it was installed, twenty years after the church's cornerstone was laid, and possibly even better, for time had enriched the sounds. The early small congregations had worked tirelessly to afford such a wonderful organ. Subsequent generations of worshippers continued to give thanks.\n\nThis Sunday morning, even Cazenovia, Lucy Fur, and Elocution luxuriated in the deep reverberations of the low notes, the sparkle of the high. While Reverend Jones delivered his sermon, the three cats sat in the balcony along with the organist and the robed choir. Occasionally the choir would sing down below, but somehow their voices always sounded better from the balcony. As the cats often attended services, none of the choir members paid them any mind.\n\nLucy Fur listened as her human's deep voice filled the church. _\"Poppy worked so hard on this sermon.\"_\n\n_\"He likes any story about people helping people,\"_ said Elocution. _\"I don't think there are Bible stories about people helping animals.\"_\n\nCazenovia added her two cents: _\"There's lots of stories about us helping them. All the animals in the stable, and how about the donkey that carried Jesus on his last journey?\"_\n\n_\"We helped build Christianity,\"_ said Lucy Fur. _\"I mean, how about all the disciples? They had to travel. Donkeys and mules carried them or carted them, too. Dogs protected them and cats kept the grain supply free from pests. All the saints would have died young if it weren't for us.\"_\n\n_\"That's too long ago and far away,\"_ said Elocution. _\"How about all we did to create the United States? We saved the colonists time and again, and then when we went to war against Great Britain, animals fought and suffered, too. Cats are perfect spies. The problem was the humans didn't listen. We could have shortened the war.\"_\n\nCazenovia smiled. _\"Poppy reads aloud, and remember when he read about those battles lost in South Carolina? But it all turned out all right. We're here.\"_\n\nAs the Reverend Jones preached about humans seeing beyond one another's superficial differences and helping others, the cats convinced themselves of their own superiority.\n\nOnce the service ended, Reverend Jones walked down the center aisle to the door at the back of the church. As the congregants exited, he shook the hand of each one, chatting with them a few moments.\n\nHarry had always liked this part of the service.\n\nAs this was the eleven o'clock service, the crowd of worshippers poured out onto the grounds a bit after twelve noon. The temperature had risen to sixty-two degrees; it was a gorgeous October day, the leaves in flaming color.\n\nBoomBoom and her partner, Alicia, chatted with Susan and Ned. Groups formed and re-formed as different folks caught up with one another.\n\nNeil Jordan moved from group to group, selling tickets to the Halloween Hayride. Whether they intended to go or not, everybody bought one.\n\n\"Harry, Fair,\" he greeted them. \"Tickets?\"\n\n\"We bought five,\" said Harry. \"I bought them from Hester.\" She thought about that for a moment, then recovered. \"But tell you what. If you give me a handful, I'll sell them this week.\"\n\n\"Harry, I only have twenty left.\" He beamed.\n\n\"Ah. I'll get more from the library, then. That's wonderful, Neil.\"\n\n\"Eighty tickets.\" He couldn't resist telling the number he'd sold. \"I'll have these twenty sold in no time.\"\n\n\"You're a big success.\" Fair slapped him on the back.\n\nBefore he left them, Neil said, \"Harry, I want to bring you materials on some different kinds of fertilizer I have. If you fertilize now in the fall, it's perfect. And if you've planted winter wheat or cold-resistant rye, you will be amazed at the yield. I know you're busy, but I'm really high on these new types of fertilizer applications. The normal corn yield in a good year is about 207 bushels per acre without irrigation. My fields yielded 250 and my irrigated fields averaged 320 per acre. And this wasn't a particularly good year. If you like my products, we can work out a payment plan.\"\n\n\"Sure. I'll call you.\"\n\nHe persisted. \"I rented two thousand acres in Nelson County to show what these fertilizers can do. I numbered the strips just like you do with corn varieties. You just wait. Next year's numbers will soar.\" With that and a big smile, Neil left.\n\nWalking with her husband toward BoomBoom and Alicia, Harry remarked, \"He'll soon have as many acres under cultivation at Buddy Janss.\"\n\nFair shrugged. \"Neil seems to thrive on competition, on a task, I guess.\"\n\n\"Hey, wasn't that a great sermon?\" Alicia, a former movie star now in her fifties, hugged Fair.\n\nThe old friends all started talking at once as soon as Susan and Ned joined them.\n\n\"My wife told me you'll start on the roof tomorrow,\" Ned said to Harry.\n\n\"Be done before nightfall.\"\n\n\"Susan said that Seth has old slate shingles,\" said the Richmond politician, with respect.\n\n\"Ned, I think he has everything. He doesn't even bother with the salvage yards. He has lines to those small companies dismantling old buildings or rebuilding historic ones that can be saved.\"\n\n\"Smart. I think there's a real niche for that kind of business. It's not all big companies. I keep trying to push in the House for the small businessman, the artisans, and little by little some of my colleagues are getting it.\"\n\nFair smiled. \"You can't always shoot the stag, but you can still eat if you bring home a lot of rabbits.\"\n\nThe group smiled and nodded.\n\nBoomBoom then said, \"Sometimes I think small is better. I go to the bank now, the same bank I have used for twenty-five years, it's been bought up and amalgamated so many times that even though the tellers all know me, I have to go through hoops! I can't even transfer money from my personal account to my business account without paperwork. My money!\"\n\nAlicia put her hand on BoomBoom's forearm, since she knew a tirade was dangerously close. \"If it's too big to fail, it's too big to exist,\" she said.\n\nThat got them all going.\n\nOn the way home in Fair's vet truck, Harry fluffed her plaid wrap skirt. \"Don't you love our friends? We can talk about anything and I always learn something. And I love that we can agree to disagree.\"\n\n\"Me, too.\"\n\n\"Yesterday at a meeting in the chancery, Neil described Hester as a wacko. Well, maybe he didn't put it that strongly, but we let him know in the nicest way that, yes, she was a little strange, but she was part of Crozet and she did much good. Now he's hawking those tickets. Maybe this is atonement.\"\n\n\"Well, that would be nice,\" said Fair. \"My experience of Neil is he assumes we're all dumb rednecks.\"\n\n\"If he wants his fertilizer business to thrive, he'd better get used to it. And you don't go into fertilizer if you want to discuss Raphael,\" said Harry, an art history major at Smith.\n\nFair laughed. \"You have a lot to answer for.\"\n\n\"Oh,\" she mused, \"Mother thought whatever I did was fine, but Dad sure was surprised. He'd say, 'How can you make a dime being an art history major?' And I'd say, 'Dad, this is the only time in my life when I can study, when I don't have to make money. I'll come back to the farm.' \" She paused. \"Little did I know they'd both be gone by the beginning of my junior year.\"\n\n\"You never know. I loved your parents. We all did.\"\n\nThey drove along in silence, then Harry thought out loud, \"Do you think anyone ever loved Hester like that?\"\n\nA long silence followed. \"No,\" he answered at last. \"But she was part of all of us, she was valued. That counts for something.\"\n\nTurning down the long gravel driveway, Harry added, \"Alicia was telling me to read a book about the environment. And then she told me to pick up one that's a few years old, _The Great Warming.\"_\n\n\"She's always been a big reader,\" said Fair.\n\n\"She said that back in her acting days, there was so much downtime on the sets that she made up for not going to college with one book after another.\" Harry saw a redheaded woodpecker dart along the fence line. \"I hope our fences don't have bugs.\"\n\n\"I doubt it. That pesky fellow is heading for the next tree. You know, sometimes I look at Alicia and I think what a terrific vet she would have made, or a professor. She does read all the time and she wants to learn. A real passion for knowledge.\"\n\n\"She always told me she hated Hollywood. She felt like a piece of meat. The money was great, but through her, I learned about the sorrows of great beauty.\"\n\nHe turned to look at her. \"You don't have any?\"\n\nAt this she let out a war whoop.\n\nSunday afternoon, Harry and Fair, accompanied by Mrs. Murphy, Pewter, and Tucker, tackled laying out all the heavy-duty extension cords, hooking them up at various places, and running them out to large water troughs in the smaller fenced pastures. The water troughs had heaters in them, and though conditions were mild, it was best to do this before heavy frosts arrived.\n\nThe largest pasture, seventy good acres, lay along the creek between Harry's farm and Coop's, so it didn't need a trough.\n\nThe horses happily drank from the clear flowing mountain waters, for Harry had extended some fencing onto Coop's side so they could easily wade around, which they enjoyed. Coop thought that was just fine. Most country people worked at accommodation. The trouble began when an outsider bought an old farm and for whatever reason felt no need to share. They seemed to think that people wanted to take advantage of them and that boundary lines were sacrosanct. Naturally, this created problems and often the newcomers found they had few friends except for other newcomers. Then something awful would happen and their neighbors would show up to help out. It usually changed their attitude. They figured out why their neighbors, whom they had usually disturbed or offended in some fashion, showed up to help. It was the country way. Most learned to be a little country themselves. A few did not and returned to where they had originated or moved on, looking for the next wondrous place. Perhaps this happened all over the country, but it happened in Virginia a lot, probably due to the state's great beauty. People wanted to live there.\n\nHarry thought all this as she checked her lines. If Coop couldn't eventually buy the old Jones place and if Reverend Jones one day had to sell, she could be facing this problem.\n\nThere were problems enough for now.\n\nShe and Fair finished the day's work, ate supper, then took a sunset walk all the way back to the vast walnut groves Susan owned on this side of the Blue Ridge Mountains.\n\nA barred owl flew from one tree to another.\n\n\"So silent,\" Harry noted.\n\n\"And such an efficient hunter,\" Fair added.\n\n_\"I'm just as good,\"_ Pewter bragged from below, never one to tolerate compliments of others.\n\nFrom a perch high above, the owl looked down at this groundling. _\"Dream on.\"_\n\n_\"I'll show you,\"_ Pewter sassed, _\"and furthermore, you don't come around the barn because Flatface is bigger than you.\"_\n\nFlatface, the great horned owl, lived in the cupola. She was two and a half feet high. The barred owl was about a foot and a half, with a wingspan just under four feet, impressive enough on his own terms.\n\n_\"Pewter, I wouldn't start a fight with an owl, even a screech owl,\"_ Tucker wisely admonished.\n\n_\"Hoo!\"_ the barred owl replied.\n\nAnd down below, Harry, once again thinking of who could have killed Hester, thought, \"Who, indeed?\"\n\n#\n\n\"W _hy are we going this way?\"_ Pewter asked as she stared out the station wagon window Monday evening. _\"I'm ready to go home.\"_\n\n_\"We've only been on the road for fifteen minutes,\"_ Mrs. Murphy replied. _\"Plus you just ate fresh tuna. Don't be crabby.\"_\n\n_\"I'm not crabby!\"_ Pewter snapped. _\"I just want to know what she's doing, that's all. She checked the roof work at St. Luke's and now she's heading west\u2014the wrong direction. You know how she can get if she sees a friend or passes one on the road.\"_ The gray cat referred to Harry's conviviality; her human was always stopping to chew the fat with another local.\n\nAlso staring out the window, Tucker said, _\"She had to get to St. Luke's before sunset. She wanted to recheck the roof work.\"_\n\n_\"The roof work is fine,\"_ Pewter spoke louder.\n\nThe three watched as Harry slowed, then turned in to the old gravel driveway to the three abandoned school buildings.\n\n_\"Hey, there's Brinkley.\"_ Tucker stood on her hind legs as she saw her yellow Lab friend sitting in front of the faded clapboard building with paint peeling.\n\nAfter parking, Harry stepped out, then opened the door for the animals, all of whom rushed to the big sweet dog.\n\n\"Hey,\" Tazio Chappars called out as Harry stepped through the schoolhouse door, which creaked.\n\nHarry looked around. \"I've never been in here.\"\n\n\"Few people have after 1965, I guess.\" Tazio dropped her hand to pet Brinkley's head. \"What do you think?\"\n\n\"Has character. Public buildings don't anymore. Plus they look so cheap. Ugly boxes.\"\n\n\"You're talking to an architect.\" Tazio laughed. \" 'Ugly' is too kind a word. And these three distinguished buildings were built for the underclass, for lack of a better word. We have beautiful examples throughout the state of what was built for the middle classes and the rich. Maybe builders had a better feel back then for space, light, warm materials. I don't find reinforced concrete warm.\" She smiled. \"Hester railroaded me. Now I'm going to railroad you, girl.\"\n\n\"Let me sit down.\" Harry sat at one of the old-fashioned desks and took a deep breath. \"I'm ready. Have at me.\"\n\nTazio sat at the desk across from Harry, as she once had done with Hester. \"You know so many people. Your people have been here since the Revolutionary War. They've worshipped at St. Luke's since that time.\"\n\nHarry crossed her arms over her chest. \"With a lead-in like that, this is going to be a biggie. I know it.\"\n\n\"Uh, yes.\" Tazio leaned toward Harry a bit. \"I believe Hester knew she was going to die.\" Tazio held up her hand, sensing that Harry was about to interrupt her. \"She knew she was in danger. When she asked me to take on the fight\u2014the project of bringing these buildings back to life\u2014I said I would only do it if she led the charge. She agreed but then she made me promise before we parted that if something happened to her, I would carry on.\"\n\n\"Dear God.\" Harry's hand flew to her face.\n\n\"It's a promise I must fulfill. I, well, I just must.\"\n\n\"Of course, Tazio. It's a debt of honor, and think of how much she trusted you.\"\n\n\"I do.\"\n\n\"You told this to Cooper?\"\n\n\"I did. Gives her not one more solid fact, but she did say it's possible Hester knew more than she was telling. We'll never know, but what I want to know is, will you work with me, Harry, use your contacts to help save the schoolhouses?\"\n\nHarry thought a bit, then replied, \"I will, but you and I have to be clear about the future use of the buildings. That means involving other people, asking their opinions, and, well, I don't want to put the cart before the horse. Let's do the Halloween Hayride first. I see you've started on Frankenstein's table.\"\n\nHarry looked at the red lights that accentuated the fake pools of plastic cut-out blood on the table and floor. Strangely cut lampshades cast ominous shadows with low light.\n\nA flat table, straps across it, stood in the middle of the classroom. Tazio had moved some of the desks aside to make room for it. \"This is the mad doctor's operating room,\" she stated with faux solemnity.\n\n\"Sure looks convincing,\" said Harry.\n\n\"Good. I want to make this year special,\" said Tazio. \"This has to be the best Halloween Hayride ever. Raise tons of money for the library.\"\n\n\"Who is going to be Frankenstein, or will he be a cutout figure?\" Harry inquired.\n\n\"Buddy Janss volunteered to be the monster. Wesley Speer said he'd be the doctor. Has a lab coat, sort of, and clothes they wore back in Mary Shelley's time.\"\n\n\"Wesley Speer. Good for him.\" Harry smiled at the thought of her fellow vestry board member being Dr. Frankenstein.\n\n\"I heard that Neil Jordan has sold one hundred hayride tickets in just a few days,\" said Tazio. \"That's something. He must be twisting every arm he knows.\"\n\n\"He can be persuasive, and it is a tradition. Also, in a sick way, the scarecrow and the witch deaths have kind of promoted the horror aspect, driving up sales.\" Harry looked around. \"Built solid, this schoolhouse.\"\n\n\"All three of them have stood the test of time. One for the little children, then the middle school, and the last building was for the big kids. I went through drawers and found old test tubes and stuff. I'm going to set it all up, see if I can't get some things smoking and bubbling and then backlight it.\"\n\n\"Creepy and perfect. However, don't let Brinkley in. That tail could be lethal.\"\n\n\"I'll put everything over his head. I learned the hard way, he can clean off a coffee table. He'd make a real mess in here.\"\n\nAs though on cue, Brinkley pushed open the front door, letting in a rush of cold air. _\"I'm here. I'm watching everything.\"_\n\n_\"What can you see?\"_ Pewter said, marching in behind. _\"My eyes are a lot better than yours.\"_\n\nBrinkley, a natural diplomat, replied, _\"They are. I wish I could see as good in the dark as you and Mrs. Murphy do.\"_\n\nHarry walked to the door to close it. \"Boy, that temperature drops with the sun.\"\n\n\"That's another thing,\" said Tazio. \"This old heating system works. I checked it out, cast iron. The boiler is enormous but solid cast iron. The boiler room was installed right about the time of World War One.\"\n\nHarry wondered, \"Who would know how to repair the boiler?\"\n\n\"Same company's been servicing it since installing it in 1915. Couldn't stand it\u2014I hopped on my computer, and sure enough, the information is online.\"\n\n\"That's a piece of luck.\" Harry smiled.\n\nTazio agreed. \"It is. Harry, thank you for signing on. You and I will make a great team. I hope Hester's looking down on us and giving a cheer.\"\n\n\"Me, too, but she might be saying, 'Not Harry!' \"\n\nTazio smiled. \"Not a chance. Well, I think we've got the ride in good order.\"\n\n\"This ride scares me before I even get on the hay wagon.\" Harry's eyes widened. \"It's going to be spectacular.\"\n\n\"Let me show you the little bathroom.\" Tazio stood up.\n\nThe two walked to the door at the back of the large room. Tazio opened it.\n\n\"Water still runs.\" Harry turned the faucet on and off. \"The old towel dispenser still works, too.\" She gave the white towel a tug and more came down as the used portion fed up into the metal dispensing box. \"Gets me excited. The quality of the workmanship, the layout.\"\n\nThey closed the door and walked to the front of the room. Harry, always curious, sat behind the large teacher's desk, which was set on a dais so the teacher could view the entire classroom.\n\n\"You will now recite your ABC's,\" Harry ordered.\n\nTazio, before her, ran through them quickly, then shoved Harry from the seat.\n\n\"Harry Haristeen, what is twelve times twelve?\"\n\n\"One hundred and forty-four,\" Harry victoriously answered.\n\n\"I gave you an easy one,\" Tazio teased as she pulled out the middle desk drawer. \"Hey, look.\"\n\nHarry stepped back up on the dais. \"Pencils, a hand sharpener, a wooden ruler.\"\n\n\"Grandpa's Tar Soap,\" Tazio said, reading the advertising printed on the ruler. \"And here's an old piece of paper.\"\n\nHarry read out the name printed on the paper: \"Walter Ashby Plecker.\"\n\n\"If Walter's name was in the teacher's drawer, he must have been a bad boy,\" said Tazio.\n\n#\n\nThursday, October 24, the service for Hester Martin was finally held at St. Francis Catholic Church in Staunton. Harry quietly sat in the pew, next to Susan, BoomBoom, Alicia, Big Mim, and Miranda Hogendobber. Fair, up in Leesburg at a veterinary conference, couldn't attend, but most everyone else who knew Hester was there. Wearing a suit, Buddy Janss made people look twice, since the portly farmer was nearly always seen wearing overalls.\n\nHarry appreciated the dignity of the Catholic service. She thought that being a Lutheran, as she was, was sort of like being a Catholic but without the incense. In her mind, people divided up into high church and low church. She admired Miranda, staunch member of the Church of the Holy Light, a charismatic church, for her strong feeling of a personal relationship with God. But Harry needed the liturgy, the ritual. Obviously, Hester had needed it, too.\n\nFortunately, Hester's niece, Sarah Price, raised Catholic, had made sure the ceremony was done just right. She had spoken at length with the priest and had picked out appropriate hymns. A woman in her mid-thirties, Sarah quite resembled her eccentric aunt.\n\nHester's niece had relied on Susan Tucker to help her with the other necessary arrangements after finding her name in Hester's address book. She'd placed a gold star next to Susan's name. Hester used different colored stars and gold meant the best.\n\nSarah also had the presence of mind to give the address book to the sheriff.\n\nAs the mourners filed out after the service, they walked down steep steps to the parking lot below. Wesley Speer and Buddy assisted the elderly down the hazardous steps, the older folks grasping the railing for all they were worth.\n\nSlowly descending next to Harry, Big Mim said, \"Staunton is a town of hills. One can find a wonderful view for a reasonable price.\"\n\n\"True,\" Harry replied.\n\n\"Mary Baldwin has the best spot in town,\" remarked BoomBoom, just behind them.\n\nMary Baldwin College did indeed have a wonderful setting. The prestigious school had been continually graduating women since 1842, and most of those alumnae had flourished, often bucking the odds against women.\n\nWoodrow Wilson's house rested not far from the college, and Harry wondered whether as a boy he had watched the girls walk by. It was hard to imagine the former president as a man being dazzled by women. In photographs, he appeared rather cold.\n\n\"Well, on to the cemetery. It's really beautiful,\" Alicia noted. On the west side of town, the graveyard was a refuge for the living to think and reflect, and a fitting place for the departed.\n\nThe graveyard was glowing with October sunlight when Hester's Crozet friends reached it. Again, the burial service for the dead was dignified and brief.\n\nThe reception that followed was held in Hester's home and started at four. It took most of the crowd about forty-five minutes to drive to the simple brick two-story house, a graceful structure that had belonged to Hester's grandparents. The paint on the brick, a creamy yellow, had flaked in spots, and the soft paprika of old brick shone through. The old place felt warm and lived in.\n\nHaving never been inside Hester's house, Harry was curious to see it, and paused in the entryway.\n\nCooper, right next to her, also paused a moment. \"Some of this furniture has to go back to the Revolution.\"\n\n\"Heppelwhite,\" Big Mim, close by, crisply filled her in. \"And the silver is Georgian, but not just any George. George II.\"\n\n\"I had no idea,\" Harry exclaimed.\n\n\"That was her way.\" Big Mim removed her hat. \"Hester lived simply. She wanted it that way.\"\n\nAlways proper, Big Mim wore a hat in church, as did most of the older women. Harry and Susan also wore hats, mostly because their mothers had long ago drummed it into them. Neither woman much liked hats.\n\nBig Mim knew Hester better than the others. \"She inherited most of what one needs in life. Not an ounce of the snob in her; she would never have called attention to the quality of the furnishings, the fabrics, and, of course, the elegant silver. I will miss her.\" The older woman smiled sorrowfully, then began moving about, a pure political animal regardless of circumstance. Big Mim was of that generation that worked through men. Her husband, Jim, was mayor of Crozet.\n\n\"Well, old girl, ready for the shake and howdy?\" Harry teased Cooper, who had not been born and bred in the region.\n\n\"I'm getting ready.\" Cooper followed Harry.\n\nSusan stood next to Sarah, introducing her to the guests.\n\n\"Sarah, please meet my best and oldest friend in the world, Harry Haristeen, and with her, one of our sheriff's department deputies, Cynthia Cooper.\"\n\nSarah shook their hands. \"Thank you so much for helping to celebrate my aunt's life.\"\n\n\"The service became her: simple and elegant,\" Harry complimented her.\n\nCooper stepped up to the plate. \"And the gravesite is so beautiful.\"\n\n\"Thank you. Please have some refreshments,\" said Sarah. \"Buddy Janss made the punch. He said it was my aunt's favorite.\"\n\nThe two moved on, glancing at each other with raised eyebrows.\n\nHarry pushed Cooper through the crowd. \"You first.\"\n\n\"I am not drinking that stuff.\"\n\n\"A sip. Come on, girl. You can do it.\"\n\nThey arrived at an enormous silver scalloped punch bowl; the family initials in elegant script were intertwined on its front.\n\nBetween laughter and tears, Buddy ladled out a full silver cup.\n\n\"Buddy,\" warned Coop, not yet committed to this alcoholic endeavor.\n\n\"Come on, Coop. You're not on duty.\"\n\n\"If I drink this, I will pass out,\" Coop protested.\n\n\"First your legs will lock up. But I'll carry you home,\" he promised.\n\nHe was irresistible, so Coop took a too-big swig. Harry wisely sipped hers.\n\nCoop gasped. \"My throat is on fire.\"\n\nBuddy laughed. \"Well, go on and talk to people. That will cool you down. Neil, come on, your turn.\"\n\nNeil Jordan accepted a silver cup, drank a bit. His eyes watered. Reverend Jones squeezed in next to him and laughed.\n\n\"Did Hester really drink this stuff?\" Neil sputtered. \"My God, what a tough broad.\"\n\n\"You're just now figuring that out?\" Reverend Jones slapped him on the back.\n\nNeil didn't spill a drop. He reached into his pockets, pulled out tickets to the Halloween Hayride, and began moving through the crowd\u2014with difficulty, but he was selling those tickets.\n\n\"Reverend, did you have a clue that Hester had such impeccable taste in home furnishings?\" Harry asked as he was now pushed next to her.\n\n\"Well, I'd been here once or twice. Knew her people, of course, as did you. All of them quiet living. Well, you knew her mother and father and her older brother.\"\n\n\"I was pretty little and they seemed so old. I don't remember her brother except that he was tall,\" Harry responded.\n\nThe party grew louder as the punch took effect. Faces red, people in the crowd talked over one another as they each recalled their favorite Hester stories. Some burst into tears, but that's the way of a Virginia celebration. Emotions rise right up to the surface.\n\n\"She was not lonely,\" said the reverend. \"People thought she was, because in this part of the world you march in twos. Crozet is a Noah's ark.\" The preacher took a sip, peered over the silver rim. \"And, Coop, you'll be walking side by side with someone before you know it.\"\n\n\"Oh, I don't know.\"\n\n\"A good-looking woman like you? Just you wait.\" He beamed at her, then returned to the subject of Hester. \"She threw herself into good works. I believe she was a fulfilled person, a good person. Granted, sometimes the notions about black gum trees or the fact that food modification would make us all idiots took me aback, but we all have our pet peeves.\"\n\nAs the conversation continued, Wesley Speer moved toward Buddy. A funeral gathering is as good a place as any to patch up hard feelings.\n\nBuddy held out a full cup. \"Wesley.\"\n\n\"Buddy, I've been pushing you a little hard about those one hundred acres. I'm overanxious.\"\n\nBuddy took a deep breath. \"Wesley, in these times I think we are all overanxious. Let's just set it aside for now and we can talk maybe after Thanksgiving. I can't sell rich soil without replacing it, you see?\"\n\n\"I do, Buddy, I really do.\"\n\nThe two men clinked cups and Buddy then nodded to the next person pressing at the punch bowl.\n\n\"I can hardly breathe,\" Cooper whispered.\n\n\"What?\" Harry inclined her ear toward her.\n\nA bit louder, Coop repeated herself.\n\n\"It's the punch,\" said Harry. \"It'll stay with you for a while. Don't drink any more,\" she advised.\n\n\"I'm sorry I drank what I did. This can't be legal.\" Cooper ruefully smiled.\n\n\"Well, dear Deputy, if you run a roadside stand and you've lived here all your life and your people have lived here since way back, your friends know where to find the best country waters to see you off with.\"\n\nCooper laughed as she saw her boss, Rick Shaw, the sheriff, come into the room. \"He knows about the hooch, of course.\"\n\n\"Always did.\" Harry laughed. \"It's a wise law enforcement officer who knows when to turn a blind eye.\"\n\n\"Ain't that the truth? Let's see if we can work our way over to the library. Doesn't look like so many people there.\"\n\nThe two edged their way through the crowd toward the mahogany-shelved library, chatting as they did so, which meant the short walk into the next room took a half hour.\n\nJust before reaching the library, Harry bumped into Cindy Walters, whom she introduced to Cooper.\n\n\"This is so terrible.\" Cindy spoke above the crowd noise. \"I no sooner reached home than I turned around to come back. She would have done the same for me.\"\n\n\"Yes,\" Harry simply agreed.\n\nCindy looked at Cooper. \"I don't know if this will help you but Hester told me she was stepping on toes. Her refusal to sell sprayed crops, her opposition to development. She mentioned this in passing.\"\n\n\"Any names?\" Cooper was accustomed to people providing information.\n\n\"No.\"\n\n\"Did she seemed frightened?\"\n\n\"Officer, I don't really know. Hester hid a lot.\"\n\n\"Thank you, Miss Walters.\"\n\n\"Where's Heidi?\" Harry asked.\n\n\"Upstairs. Couldn't live without her.\" The short, trim lady smiled.\n\nOnce inside the library, they looked at the books, many old, bound in Moroccan leather of deep colors.\n\nHarry found a shelf dealing with agriculture. \"She's got books dating back to World War One; she's got books released year by year from the U.S. Department of Agriculture.\"\n\nCoop bent down to read spines on the lower shelves. \"Some of these are in French. Hey, here's one on Percherons and it looks very, very old.\"\n\nHarry knelt down. \"Percherons are French draft horses. I had no idea. I mean, I knew that Hester had a college degree, but look at all these books.\"\n\n\"Here're two rows on Indian affairs.\" Cooper squinted to see better.\n\nHarry joined her and looked closely at the titles. \"She must have everything Virginia and the U.S. government ever released on the subject.\"\n\n\"She's got newer stuff, too,\" said Coop. \"Custer and Little Big Horn. _Bury My Heart at Wounded Knee_.\" The law officer read a few more titles out loud, then moved to the next shelf at her eye level. She stopped cold. \"Harry.\"\n\n\"Yeah?\" Harry was utterly absorbed in her examination of the old volumes.\n\n\"Come here.\"\n\nHarry did as she was asked and beheld a photograph of Hester in an old silver frame. She had a fly rod in her hand, and her surroundings looked to be Bath or Highland County, Virginia counties adjacent to the state of West Virginia. Standing next to her in a stream was a man considerably younger, fishing rod also in hand. He held up a lovely trout.\n\nHarry leaned closer.\n\n\"It's Josh Hill,\" Cooper said, her voice low.\n\nHarry swallowed. \"I never saw his face, I mean intact.\"\n\nCooper had seen plenty of photographs of the accountant during her ongoing research. \"It's him all right, which makes me wonder: Were these two fishing for trouble together?\"\n\n#\n\nSunlight flooded Hester's kitchen, which faced east. The morning after the reception, Cooper and Rick Shaw sat at the kitchen's small square wooden table with Sarah Price. After apologizing for troubling her at such a time, the two law enforcement officers began their questioning of Hester Martin's niece.\n\n\"Did you ever meet the young man in the photograph?\" Rick Shaw asked Sarah. The silver-framed photo of Hester and Josh Hill sat on the table in front of them.\n\n\"Not that I remember,\" the pleasant woman replied. \"I don't know who he is.\"\n\n\"She never mentioned Josh Hill?\" asked Rick.\n\nSarah looked again at the photo. \"No.\"\n\n\"Did she talk about fishing?\" Cooper asked.\n\n\"Some. Aunt Hester and I would speak over the phone about once a week. She wouldn't text me or email. She said she wanted to hear my voice, then she'd know if I was okay.\"\n\n\"Did she talk about her other interests?\" Rick folded his hands then unfolded them.\n\n\"Aunt Hester loved to lecture! That is, once she had inquired about my health, boyfriend status\u2014I'm divorced\u2014and my career advancement or lack thereof.\" Sarah smiled. \"After all that, I would be treated to discussions about the global food crisis, why agribusiness couldn't meet the demand, and why she refused to sell foods treated with pesticides. She admitted organic farming was less efficient. A lot of goods are lost to bugs and stuff. You didn't so much talk with my late aunt as you listened.\"\n\nBoth Rick and Cooper smiled before Cooper spoke up. \"Did your aunt ever talk about what she was reading? That's a gorgeous library. All those books from the nineteenth century and the early twentieth. She must have loved reading or at least collecting.\"\n\n\"Much of that library she inherited, but she was an avid reader. Often she read in French, especially plays and novels. We would laugh about something she quoted from Moli\u00e8re. But mostly, with me, anyway, she would talk about something she'd read in English about farming or about human impact on wildlife.\"\n\n\"It's funny, Miss Price,\" said Cooper. \"I have stopped at your aunt's roadside stand for years and I never knew she could read in French, never knew she owned such beautiful things.\" She looked around the kitchen, her eyes resting on the old wooden cupboards.\n\n\"That doesn't surprise me. I don't know as I would classify Aunt Hester as secretive so much as, uh, compartmentalized.\" She leaned to her left, toward Rick. \"Her friends and interests fell into categories, which didn't overlap.\"\n\n\"Did she talk about them with you?\" Rick inquired.\n\n\"Not much. Most of what I knew came from my dad, who died about six years ago from lung cancer. He was older than Hester by two years. They got along but weren't close. Too different.\"\n\n\"How?\" Cooper often found that an offhand comment, a recollection, pointed in the right direction.\n\n\"Oh, Dad was sophisticated, driven. And social. Houston is a great city in which to be social. He married very well. Both my parents loved fine things, evening-gown parties. You know the type. Aunt Hester thought he was superficial.\"\n\n\"Was he?\" Rick's eyebrows lifted.\n\nA silence followed this. \"By Virginia standards, he was. He talked about money too openly. His suits were too flashy and he wore a big gold Rolex, which Aunt Hester called a Texas timepiece. But Daddy had a heart of gold, so if he wanted to wear a little gold, okay. He made sure I got the best education possible. He went to Houston to make money and he did. Sure, he indulged Mother and he indulged me, but he also made sure I knew right from wrong, and he could be tough. Can you tell? I loved my dad.\"\n\n\"He sounds like a good fellow.\" Rick nodded. \"And Virginians can be snobs. Aunt Hester might have filled those shoes.\"\n\n\"Oh, she didn't mean anything by it. Dad took it with a grain of salt. He called her the Old Maid and declared if she'd find a good man she'd be much less judgmental. I remember Aunt H, as I would call her, used to say to my mother, 'He's my brother, I love him, but how can you live with him?' Mom would laugh.\"\n\n\"What did you think?\" Cooper shrewdly asked.\n\n\"I guess in some ways I agreed with Dad, but you never got the full picture with Aunt H. Her interests were passionate but compartmentalized, as I said. Like the fishing, for instance. She rarely talked to me about it but she would talk about it for hours to Mom, who liked to fish, too. Once they went together to the Snake River in Wyoming. Dad paid for everything. Aunt Hester was appalled that Mother put on makeup to fish.\" Sarah laughed, a tinkling, engaging laugh.\n\n\"Maybe the fish liked it.\" Rick laughed with her.\n\n\"They must have, because Mom caught more than Aunt H, and that didn't sit well.\"\n\n\"Do you know if she traveled to other places?\" Cooper kept on.\n\n\"I do know, again through Mom, that Aunt Hester usually fished in Bath or Highland County in Virginia. They both swore it was the best fishing on the East Coast.\"\n\n\"I've heard that,\" Cooper said. \"Do you fish?\"\n\n\"No. I'm a golfer. Love being out there surrounded by such green vistas, sometimes all by myself. Other times in a foursome. Houston has some wonderful courses. Of course, Charlottesville, for such a small place, does, too.\"\n\n\"Farmington?\" Rick raised his voice as a question.\n\n\"Those long fairways. Keswick Club is a challenge. Glenmore. A short drive to the Country Club of Virginia. And in four hours I can drive down to Pinehurst, North Carolina, to one of the most fabled golf courses in the country.\"\n\n\"Did Hester golf?\" Cooper pressed on.\n\n\"No. Her interests were, as you know, varied: fishing, farming, the library, old buildings. She loved the Library of Virginia in Richmond. Loved Monument Avenue. She had a quiet, longstanding interest in the Virginia tribes.\"\n\nCooper sat up straighter. \"Why do you think she was interested in Virginia Indians?\"\n\n\"Our maternal line is Sessoms, a Cherokee name,\" Sarah explained. \"But they adapted so well to the early colonists\u2014I mean early as in eighteenth century\u2014that the Sessoms farmed, wore English clothing, spoke English, and intermarried with Europeans. Over time they became so much like the English that they didn't have the trouble the other tribes did, including the Cherokees in the more southern states. Those people went through hell. Sessoms is a common last name in the tribe, just like Adams is a common last name for the Upper Mattaponi.\"\n\n\"Did you know that Josh Hill was an Upper Mattaponi?\" Cooper felt that little buzz when she knew she was finding her way on a case.\n\nWhere it was going, she didn't know.\n\nSarah shook her head. \"I knew nothing about this fellow, but he appears to have been a fishing buddy, and if he was a member of a Virginia tribe, Aunt H would have been fascinated.\"\n\n\"She never spoke to you about this? About the Cherokee connection?\"\n\n\"No. It was Dad who told me about that part of our ancestry.\" She thought a moment. \"Once I mentioned something about a bracelet I saw that had been made by the Pueblos. Aunt H said southwestern Indians, Texas Indians, were much different from the East Coast tribes but all were fascinating to study.\"\n\n\"Anything else?\" Cooper persisted.\n\n\"She said\u2014and this I do remember, because I heard it at odd times, not a lot but enough to remember, and I heard it repeated by my dad, too\u2014the Indians never raped the land.\"\n\nBoth sheriff and deputy sat quietly for a moment, then Rick asked, \"Were either of your parents ever involved in environmental causes or perhaps trying to return tribal lands to their original owners?\"\n\n\"They supported the Nature Conservancy. They made vacations to go on field trips, wonderful places like southern Chile, Moosehead Lake in Maine. They really pitched in with the Nature Conservancy and Ducks Unlimited. We had to live in the now. Mom and Dad wholeheartedly believed that. Dad said that he thought in some ways Aunt H lived too much in the past.\"\n\n\"And Aunt Hester's attitude?\" Cooper asked.\n\n\"That we needed to make amends. We needed to preserve the past and also make amends to other peoples, to wildlife. As you see, she preserved the past in this house. Aunt H was consistent and she really did care about providing good food at her stand, about taking care of the land. Maybe that's why she liked fishing. She could get away but still be part of nature. I suspect she threw back whatever she caught. Mom always did.\"\n\nBoth interrogators smiled.\n\nCooper then said, \"We combed her roadside stand. Lolly\u2014you met Lolly?\" When Sarah nodded yes, Cooper continued. \"Showed us the back rooms, everything. She opened the cash register, lifted the tray where Hester kept notes, odd returned checks, stuff like that.\"\n\nSarah quietly interrupted, \"And you found that my aunt Hester was organized about everything but her financial records. I've found old bank statements in kitchen drawers, in the visor of her truck.\"\n\nCooper smiled. \"She appears to have used a system unique to herself. What interested us was a check for a thousand dollars written to Joshua Hill. Written on the bottom memo line was 'Research.' It had been cashed.\"\n\nSarah registered surprise. \"She wouldn't write a check that size on a whim.\"\n\n\"We don't know what the research was, and his records are sparse,\" Rick said.\n\n\"So there is a connection between the murders.\" Sarah spoke lowly.\n\n\"It certainly seems possible,\" Cooper answered.\n\n\"May I ask you a question?\" said Sarah.\n\n\"Anything.\" Rick liked her, obviously.\n\n\"Do you have any idea who would kill my aunt, or why?\"\n\n\"I won't b.s. you,\" he answered. \"We don't but I promise you, Sarah, we will find out and we will bring them to justice. Your aunt was a good woman.\"\n\nCooper looked Sarah in the eye. \"I apologize for pushing you with questions, but you knew Hester as well as anyone, perhaps better. The rest of us took her at face value, and even those who had been in her home, like Mim Sanburne, only knew a fraction of who and what she was.\"\n\n\"You don't think she was involved in anything illegal, do you?\" asked Sarah, distressed. \"I mean, I can't imagine her doing something illegal. Aunt H was a straight arrow.\"\n\n\"No. But my hunch, and it is just a hunch, is she may have stumbled onto something someone else was doing that was illegal.\"\n\nSarah's hand covered her heart for a moment.\n\n\"I wonder if she knew she was in danger. She would have kept it to herself. She was so independent, had lived her whole life alone\u2014if she did think she was in danger, she would have thought she could handle it.\" Sarah swallowed. \"I make her sound unrealistic but who could have foreseen something like this? Aunt H never did understand evil.\"\n\n#\n\nThe wind rustled through the dried cornstalks in Buddy Janss's one hundred acres behind the three abandoned schoolhouses. Buddy walked through the fields with an insurance agent. The crop insurer, on overload thanks to the drought, worked every day but Sunday. This Friday, October 25, he had already visited three farms before noon.\n\nThe U.S. government underpinned about sixty percent of insurance premiums. Until now, the premiums farmers paid for various farming insurance had more than covered the payouts, but this year it looked as though the government payouts would be greater than the intake.\n\nLooking down at his clipboard, Drake Stoneman, thirty-two, traced acreage numbers with his index finger. \"Why do you insure some acres and not others for the same crop?\"\n\nBuddy, fifty-two, didn't much like the tone of this younger man nor the assumption, so prevalent these days, that one must explain one's self exhaustively. \"Because for years I carted some vegetables and corn to a roadside food stand where the owner was fierce about organic foods.\"\n\n\"Given the losses to fungus, birds, and insects, you must have gotten top dollar.\" Stoneman looked up from his figures into Buddy's dark brown eyes.\n\n\"I did not,\" the large man replied firmly. \"I did business with the lady who owned the stand and my father did business with her people. It's just something I did.\"\n\n\"Hard to farm if you don't put profit first,\" the college-educated fellow smugly said.\n\n\"You don't have much of a life if profit is all you care about.\" Buddy, feeling his anger rise, then followed with, \"I make enough money elsewhere.\"\n\nStoneman nodded at the equipment parked nearby. \"This your boom sprayer?\"\n\n\"Is. It's calibrated for each nozzle to spray 1\/128 of an acre. Been doing this all my life. Divide the tank capacity by the gallons per acre pesticide or liquid fertilizer application rate. Do the math. Set your nozzles correctly given the crop, figure your normal spraying speed, and record the travel time in seconds. It's the only way to get the application correct. Otherwise, I waste my money.\"\n\nDrake looked at Buddy, slowly realizing it might be prudent not to lecture him on broadcast applications. \"I see.\"\n\n\"You asked about my organic corn, which is how I think of it,\" said Buddy. \"Silver Queen is in this field. As you are a bright fellow from Virginia Tech...?\"\n\nQuickly Drake replied, \"North Carolina State.\"\n\n\"Uh-huh. You know Silver Queen is hard to grow.\"\n\n\"That I do.\"\n\n\"My untreated acres did better than this one hundred. Look. Lost everything and I sprayed this patch every two days. In this climate, an insect like the corn borer can span three generations in one crop cycle.\" He yanked an ear off the stalk. \"No corn borers.\"\n\nJust to prove he was on the ball, Drake twisted off an ear himself, separate from the one in Buddy's hand, and inspected it. \"I see.\"\n\n\"This entire crop is worthless. Couldn't even use it for forage.\"\n\nDrake couldn't miss the nasty signs of smut on the now dried-up ears. \"It is possible to be fooled. This stuff sometimes just seems to appear overnight.\"\n\n\"I kept after these acres. I shouldn't have any disease in here, including the smut. I sprayed for that, too. I mean, I did everything. Sure, I want my crop insurance. That's why I pay the premiums, but I want to know how this happened.\"\n\n\"I would, too.\" Drake was becoming a little more sympathetic. \"Corn means big bucks. Our country's exports for fresh corn alone, not counting frozen or canned corn, should bring in about $47 million for farmers, and this year it's nowhere close.\"\n\n\"Got to cut growing forty percent of the crop for ethanol when people go hungry,\" Buddy firmly stated.\n\n\"That's a hot potato, Mr. Janss.\" He paused. \"Do you mind if I look at your acres, the ones you used for the roadside stand?\"\n\n\"Not at all. Everything's off of them.\"\n\n\"Plow under the stalks?\"\n\n\"No, I leave stuff up until the first hint of spring. Gleanings for birds, bunnies, foxes. Brings 'em right in.\"\n\n\"Why don't you plow this under now so it can start breaking down?\"\n\nBuddy wondered what this kid had learned in college.\n\nHe calmly explained, \"Smut spores can live in the soil, for one. But we do share the earth and the old corn helps wildlife. If I plow it under, the soil will compact by spring and I'll have to break it up again\u2014duplication of effort, hours, and cost, plus I just may have made it easy for the smut to regenerate.\"\n\n\"Uh, yes.\"\n\n\"A ripper and shredder, John Deere 2720, can just tear this up, save time, do a terrific job. But a seventeen-foot with forty disc blades costs almost fifty thousand dollars. The twenty-seven-foot with sixty-six disc blades starts at eighty-nine thousand dollars. Can't begin to afford that. I use my old disc\u2014well, it's ancient, really, and sometimes I even pull my old York rake when the stubble's down. I can't just figure crop profit. I have to factor in equipment costs, hours, wear and tear, and then there's wear and tear on me, too.\"\n\nDrake knelt down, pulled out a little brown paper bag and a small sharp trowel from his pocket. He dug out a sample of soil. He then walked twenty-five yards, took another one.\n\nBuddy watched. He took his own soil samples each fall and then again in the spring. In this part of the world, a farmer needed to keep checking for magnesium deficiency. Out west the lack was often selenium, not a problem in central Virginia.\n\nDrake came back, Buddy hopped in his new Ford while Drake followed in a serviceable old Ram.\n\nThey reached the harvested fields, where the younger man again took soil samples.\n\nBuddy watched, his anger somewhat subsiding. \"I can send you my soil tests for all my acres. I'll shoot it to you from my computer.\"\n\n\"That would be helpful. You'll receive your check. Like you said, Mr. Janss, that's why you pay the premiums, and your taxes, too. We need the government for a lot of this. You can imagine the stress on farmers in the Midwest and the Southwest.\"\n\n\"I pray for those people.\" Buddy meant it, too.\n\nDrake shook his head. \"Don't see how people can argue with climate change.\"\n\n\"Actually, Mr. Stoneman, I do see how they can and I have some real questions about our ability to predict and plan how to handle Mother Earth, but I know something's changing. I just want to know more and I'd sure like the politics to be taken out of it.\"\n\nFor a flash Drake dropped his professional mask. \"Politics is in everything.\"\n\nBuddy looked right into his eyes. \"You got that right, brother.\"\n\nAs Drake drove away, Buddy folded his arms across his chest, looking over some of his well-tended acreage. It was odd that the untreated Silver Queen corn had done better this year than the treated. He never planted more than two hundred acres with this type of corn, mostly because it took so much management. It was susceptible to about every pest and problem known to corn and needed steady water, too. But, oh, how sweet that Silver Queen was and how people looked forward to it come August, September, and this year, even early October.\n\nEvery now and then Buddy would think he'd just plant every field with orchard grass and the hell with it. But he couldn't. He loved his corn crops, loved the squash, and _really_ loved the small orchard he tended of the old Alberta pears. People would come from all around to get those pears, not much seen anymore.\n\nSighing, he stepped on the running board and got into the Ford's cab. He drove back to the disease-ridden crop, parked at the three schoolhouses, and looked for a moment, then drove into the middle of the field.\n\nHarry, pets along with her, happened to be cruising by at that moment to double-check the Halloween Hayride route that Tazio had given her.\n\nShe saw Buddy. So she drove out to the field, her old 1978 Ford churning through the hardened, bumpy earth with ease.\n\nAs she opened the door, the dog and two cats shot out.\n\n\"Buddy. Hey. Saw you here and wanted to thank you again for serving at Hester's house and, well, for being a pallbearer.\"\n\n\"It was an honor to be a pallbearer. Something hits you when you hoist that coffin on your shoulders along with five other men. Can't explain it.\"\n\n\"Well, thank you for honoring a special person.\"\n\nHe leaned against his truck, having stepped out since she left her truck. \"Here's something funny goin' on. Wish Hester were with us, she'd crow with delight. These acres, sprayed every two days. My Silver Queen. Well, smut ruined everything. The untreated Silver Queen acres\u2014healthy. Oh, sure, a worm here and there, and always corn spiders, but they do no harm. The birds got some of the corn, but mostly my untreated acres are really healthy, even with our brief drought. I can't figure it out.\"\n\n\"What about your irrigated acres?\"\n\n\"Same story. I irrigated half my plots. The irrigated rows did a little better but not as much as one would think, but, you know\"\u2014he swept his hand westward\u2014\"so many creeks, and I think some of that moisture in the air from them helped. But like I said to you before, my crop yield varied from one side of the road to the other, but this, this is smut and I don't know what's going on. I read all the time and I haven't read anywhere where smut has become resistant to treatment. Now, we know some _insects_ are becoming resistant. I'll get my insurance check but I want to figure this out. I'm here to grow good crops, not to collect money for failure.\"\n\n\"I hear you.\" Harry studied the midsized John Deere tractor parked fifty yards off, boom on the back. \"Mind if I look? I'd love to get a boom and I'd sure love to get a drill seeder, too.\"\n\n\"I'll tell you if I see anything used that's in good shape. Stuff new costs as much as a car or a house these days.\"\n\n\"Yeah, I know.\"\n\nThey reached the green and yellow tractor. It had a tank affixed to the back, with a long boom off that. \"I used to rely on a gravity feed, but now I've got a small pump to suck out any stuff on the bottom,\" Buddy told her.\n\n\"You thought of everything.\" Harry admired practical solutions.\n\nHe smiled. \"Try. I need to flush this baby out and take her back to the shed for the winter. You know, Harry, I kind of lost heart. Since I pulled down that first husk and saw the damage, I haven't done much in here.\"\n\nTucker, right under the machine's tank, called out, _\"No chemicals. Can't smell a one. What's he spraying with?\"_\n\nThe two cats joined her.\n\nPewter sniffed. _\"Smells, though.\"_\n\n_\"Does,\"_ Mrs. Murphy agreed, then jumped on the back of the boom hitch and up onto the tank, where she balanced herself and tapped at the screw-on cap.\n\nHarry laughed. \"Looks like we've got a Future Farmer of America here.\"\n\n_\"Open it up. Come on. There's no chemicals in here. We'd know,\"_ the cat pleaded.\n\nSmiling, Harry did untwist the cap as Mrs. Murphy jumped off. Harry peered into the tank. \"Buddy,\" she said, sniffing, \"Buddy, look at this.\"\n\nHe dutifully did and immediately became enraged at the sight. \"Goddammit to hell!\" Then he apologized. \"Sorry.\"\n\n\"Buddy, I'd have said worse.\"\n\nThe tank had smut in the bottom. With his own system, Buddy had infected his crop.\n\n\"Harry, I calibrated the gallons per acre. I flushed the system clean, I checked every screw, nozzle, everything. And I refilled this tank each morning.\"\n\n\"Well, Buddy, someone drained your tank halfway, put a smut slurry in, then refilled it. How would you know? And I bet they cleaned it after you left the day's work. Someone who knows farming also knows your schedule. And smut spores are easy to grow. You can do it in your kitchen.\"\n\nHis face blanched, then turned scarlet. \"Why? Who would do such a thing?\" He paused, color deepening. \"I'll kill the S.O.B.\"\n\nHarry said nothing. Any talk of killing right now gave her a chill.\n\n#\n\nAdjustable wrench in hand, Fair frowned as he worked in the big red shed near the barn. \"That's strange,\" he said, squinting at the dismantled ATV in front of him.\n\n\"Honey, there's so much weird stuff going on around here, this is just one more thing, but to deliberately put smut in a spray tank...\" Harry shook her head. \"Why?\"\n\n\"You don't think it could have occurred naturally?\"\n\n\"No.\"\n\n\"Well, it will all come out in the wash.\" He checked the sun, now low in the sky above the fields. \"I'd better start putting this back together.\"\n\n\"Did you find the problem?\"\n\n\"The first problem was the fuel line clogged. The second problem\u2014I'm not sure but this generator isn't going off.\"\n\nHarry peered down into the ATV's engine. \"It's a bitty thing.\"\n\n\"Anything compared to the engine in your '78 Ford is a bitty thing. Well, let me put this back together. We need it.\"\n\n\"If we can't get it back working, I'll call Wayne's Cycle.\" She mentioned the place where they had bought the ATV years back in Waynesboro, then realized her husband didn't want to hear that.\n\n\"It will run,\" he loudly announced.\n\nAs she walked back to the house, Tucker beside her, she looked over her fields, the sunflowers all harvested. \"Time to plow stuff under,\" she said to the corgi.\n\n_\"If you leave it alone, rabbits will come in,\"_ Tucker said. She liked to chase rabbits.\n\nPushing open the screen door, Harry heard a frantic scramble on the kitchen countertops.\n\n_\"You forgot to completely close the toaster oven,\"_ said the corgi. _\"I smell the corn bread.\"_\n\nStepping into the kitchen, no cats in sight, Harry noticed corn bread crumbs strewn across the counter in front of the toaster oven.\n\n\"Those boogers!\"\n\nThe cats had hooked the corn bread inside the toaster oven, tearing pieces off, pulling them out of the oven and onto the counter, where they ate them. However, they had been interrupted in their thievery, so crumbles\u2014golden evidence\u2014lay scattered on the counter and on the floor.\n\nSince some was on the floor, Tucker ate it. No point in letting food go to waste.\n\nBefore Harry could cuss, the phone rang.\n\n\"Susan,\" Harry greeted her.\n\n\"I got a job,\" came her enthusiastic voice.\n\n\"Where?\"\n\n\"At Ivy Nurseries. I'll be making arrangements and stuff like that.\"\n\n\"Wonderful.\"\n\n\"Well, I learned a lot from you.\"\n\n\"You learned more from Miranda.\"\n\nMiranda Hogendobber, a passionate gardener and former co-worker at the post office, possessed a gift for arranging height, color, breadth. If it involved a flower, Miranda could grow it, then display it.\n\nSusan replied warmly, \"How about I give you both credit? I need to do more than I've been doing.\"\n\nHarry then told her about the corn smut and Buddy. \"Never saw him so mad.\"\n\n\"Remind me, what's corn smut?\"\n\n\"It's a fungus. It can survive during the winter if it finds the right place to hide. It can survive in old cornstalks, but usually the wind has blown spores all over the place after the swollen infected kernels explode. Not a lot left in the stalks. You and I could grow smut ourselves in corn. The later-maturing corn varieties are more susceptible to it. Has a lot to do with the change in nighttime temperature from midsummer. And when kernels explode, you can see the stuff. It's actually not that hard to control if you spray before you get it. Once you get it, though, you might as well forget it, and sweet corn is pretty vulnerable.\"\n\n\"Doesn't make sense.\"\n\n\"No, it doesn't. Before I forget, when do you start your job?\"\n\n\"Monday.\"\n\n\"I'll drop by the nursery around quitting time.\"\n\n\"Great.\"\n\nThey hung up. Harry looked out the window over the sink. She could tell from her husband's walk that he hadn't fixed the ATV. She wouldn't bring it up but she would make sure the magnetic card for Wayne's Cycle was moved to the front of the refrigerator.\n\nGiven the scowl on his face, she thought she'd better distract him. She disappeared into the small workroom and turned on the computer.\n\nMrs. Murphy and Pewter smelled the computer. Humans couldn't detect the smell computers gave off when they were working, but for the cats the odor was coppery, distinctive.\n\n_\"She gets wrapped up with that nonsense,\"_ Pewter gloated. _\"She'll forget what we did.\"_\n\n_\"She won't forget but she will be occupied. That corn bread, oh, full of butter.\"_ Mrs. Murphy smiled.\n\nThe two cats wiggled out from under the bed where they'd been hiding and silently made their way into Fair's small office, where Harry peered at the screen.\n\nFair, calmer now, stuck his head in. \"What's cooking?\"\n\n\"Lasagna,\" answered Harry.\n\n\"No, I mean, what's cooking here?\" He pointed to the computer.\n\n_\"Lasagna,\"_ Pewter said, sounding crushed. _\"Not my favorite but it's okay.\"_\n\n_\"You'll eat. You'll eat anything,\"_ said Mrs. Murphy.\n\nThe culprits tiptoed to one side of the desk, sitting to listen.\n\n\"I'm looking up corn stuff,\" Harry explained to her husband. \"Like, did you know that people living in what is now Mexico domesticated corn fourteen thousand years ago?\"\n\n\"Isn't corn basically a grass crop?\"\n\n\"Yeah, but says here that the original plant didn't look anything like modern corn. They called it _teosinte_.\"\n\nHe stood next to her now. \"Fourteen thousand years ago. Imagine if you got a toothache back then. Ouch.\"\n\n\"Hurts enough now.\" She looked up at him, then back at the screen.\n\n\"Says here that what we call sweet corn was first grown in Pennsylvania in the mid-1700s. The first commercial variety was introduced in 1779.\" She scrolled up more stuff. \"Hey, hey, honey, how about this?\"\n\nHe leaned down and read along with her. \"Corn invaded by corn smut is considered a delicacy in Mexico. Infested corn was cooked even before Columbus.\"\n\n\"Guess every culture enjoys its delicacies.\" She touched his hand. \"But maybe Buddy can make a little money. I'm going to call him.\"\n\n\"Okay. I'll shower.\"\n\n\"Thirty minutes to supper at the most.\"\n\nHe kissed her on the cheek.\n\nShe dialed Buddy Janss, launched right in with her discovery.\n\n\"They eat that stuff?\" replied an incredulous Buddy.\n\n\"Buddy, if you go to your computer, Metapathogen.com has a little section on corn smut, under its Latin name, Ustilago maydis. Yeah, Mexican restaurants think it's terrific.\"\n\n\"Well, I already walked the insurance agent through.\"\n\n\"You did, but if you call around to some really fancy Mexican restaurants, maybe you can figure prices. Obviously, if they'll pay more than the crop insurance, that's an easy decision.\"\n\n\"You bet.\" His voice picked up energy.\n\n\"Before it slips my mind, when Cooper and I walked through Hester's library looking at her beautiful books, we found some on fishing, and a picture of her with a friend fishing. She ever talk about this with you?\" Harry pointedly did not mention the friend was scarecrow Josh Hill.\n\n\"Oh, well now, over the years maybe once or twice. Hester and I mostly stuck to business.\" He chuckled. \"Her version of business.\"\n\n\"Had you ever been in her house before the reception?\"\n\n\"No. What about you?\"\n\n\"Me neither. I was surprised at how lovely it was. And the expensive things she owned.\"\n\n\"Life is full of surprises.\"\n\n#\n\nSaturdays Harry and Fair liked to join their friends for foxhunting. As the fox was chased, not killed, they especially enjoyed riding behind hounds, land rolling before them like green waves, Blue Ridge Mountains behind, a splendid theatrical backdrop.\n\nToday's hunt lasted three hours. Once back at their trailers, people wiped down horses and threw sweat sheets over them, since it was warm, in the mid-fifties. After putting out buckets of water, they hurried to join everyone else at the tailgate. Literally it was a tailgate: The tailgates on trucks were dropped, a few card tables were put out and little oil tablecloths were tossed over them.\n\nThe talk always began with the day's sport before rapidly moving to other subjects. Many of today's hunters had also attended Hester's service.\n\nBig Mim, hot coffee in hand, mentioned, \"I believe Sarah Price will take over Hester's house.\"\n\n\"Wonderful,\" Wesley said, nodding.\n\n\"I'd think you'd feel otherwise,\" said Neil with a hint of sarcasm. He was a non-rider who'd come to join the group, as did others, food and drink being a reliable magnet.\n\n\"Why? It's a piece of old Virginia, and better that such places stay in the family.\"\n\n\"Ah.\" Neil swilled his scotch. \"You're right. I was thinking of the commission on a sale. Would sell for a lot, that place.\"\n\nHarry said, \"I couldn't possibly afford my farm today. It's kind of crazy.\"\n\n\"Prices go up and down,\" said Wesley, \"but when it comes to beautiful farms in Virginia, they have held steady despite all. Now, I'm not saying I've sold a lot lately, mind you, but we are in a better position than most of the country.\"\n\n\"Not the boom towns,\" Neil pressed.\n\n\"Like Oklahoma City?\" Fair asked. \"You know, it's exciting when something hits like the boom in the Dakotas and Oklahoma. Hope, energy, jobs, but you wonder how it will all turn out down the road.\"\n\n\"Honey, that's true for everything.\" Harry smiled, then focused on Neil. \"How about fertilizer samples? Just enough to, say, put on three small patches, four feet by four feet. I'll make little squares back behind the sunflowers.\"\n\n\"Be happy to. I know if you have a good experience and endorse my products, others will follow.\" Neil was right about that. \"Have you thought about what you would be growing?\"\n\n\"Have.\"\n\nTazio and her boyfriend, Paul Diaz, joined them. As Paul rode and trained Big Mim's horses, Tazio had realized she'd better learn to ride.\n\nTo Paul's credit, he was studying architecture, and the two, on his weekday off, would drive to Richmond, Washington, and other places to look at buildings constructed at different periods in our history. He found he liked it, just as Tazio found she liked riding.\n\n\"She's going to move up to Second Flight,\" Paul bragged of Tazio, referring to the foxhunting group closer to the action.\n\nTazio rode in the back on an adorable babysitter of a horse, but as she gained skill and confidence, she would move up a notch.\n\n\"Never doubted that for a minute,\" Fair told her.\n\n\"How's it coming for the Halloween Hayride?\" Neil asked.\n\n\"Frankenstein will be ready,\" said Tazio. \"He'll snap the restraining belts, climb off the table, attack the good doctor\"\u2014she nodded at Wesley\u2014\"then run out the door.\"\n\n\"I'm scared already,\" Harry said.\n\nNeil laughed. \"It's going to be the scariest hayride ever, and we will raise a bundle. I'm committed to that and others are, too.\"\n\n\"I think a room in the library should be named for Hester,\" Harry thought out loud.\n\n\"You're right, honey,\" Fair agreed.\n\n\"After the hayride, we can bring it before the library board. I'm getting excited about this.\" Neil smiled.\n\n\"You get excited about anything that makes money,\" Wesley teased him.\n\n\"Profit motive. Built this country,\" Neil fired back.\n\nBig Mim, who had left the group, sailed back into their conversation, changing the subject. \"Given the dryness, not a bad hunt. We do need rain, though. Desperately.\"\n\n\"That we do,\" Fair said. \"The ground is so hard it's like running on brick.\"\n\n\"Tazio,\" Big Mim addressed the architect, who looked stunning in hunt kit, \"you've been over there at the school buildings. Are they salvageable?\"\n\nWith a big grin, Tazio replied, \"They are in great shape. The real expense in fixing them up would be plumbing, heating, air-conditioning. But those buildings were solidly built, well sited, and there's not even a leak in those roofs. You could actually still use the huge cast-iron furnaces.\"\n\n\"Good,\" Big Mim said. \"Lot of history there.\"\n\n\"I wish older people would write down what they lived through\u2014the good, the bad, and the ugly,\" Harry said with some emotion. \"History books can be dry or filled with speculation about this world force and that armament technology. I want to hear what people who lived through it all thought and felt.\"\n\n\"Good point.\" Tazio rested her hand on Harry's shoulder for a moment.\n\n\"Speaking of knowing, the TV reporters and the newspaper say that Hester was shot,\" said Neil. \"And so was that fellow you found in the Morrowdale field. But how and where were they shot, exactly?\" he asked, not realizing that Harry might not wish to recall any of this.\n\n\"I don't know,\" she replied.\n\nFair stepped in. \"When we found the scarecrow, he was fully dressed. Hester was, too. No wounds were evident.\"\n\n\"Neil, I don't really _want_ to know,\" Harry lied. Cooper had told her they were shot through the heart. Cooper had also told her the sheriff's department was withholding the exact M.O. \"They're both gone, a young man and a neighbor. That's enough.\"\n\nNeil shrugged. \"I guess I get too curious. Too many crime shows on television.\"\n\n\"It's always so antiseptic, those shows. No faces frozen in horror.\" Tazio reached for Paul's hand. \"What I want to know is why our society is so enthralled by crime and violence. Why can't we be enthralled by beauty, harmony, or perfect proportion?\"\n\n\"Because they demand sensitivity.\" Fair surprised them by coming right out with this. \"Anyone can see a beautiful sunrise or hear great music, but not everyone can _feel_ it. Yet everyone can feel violence.\"\n\n\"I never thought of that,\" Wesley remarked.\n\n\"And I suppose everyone can kill,\" Tazio said, \"but how many people can compose a symphony?\"\n\n\"I'm not sure everyone can kill,\" Neil replied. \"Then again, I don't want to find out.\"\n\nTo change the subject, Harry asked Tazio, \"That old slip of paper you found\u2014did you by any chance check to see if it was a student? I mean, I wonder if they have the old rolls.\"\n\n\"I didn't find out yet.\"\n\n\"What was the name?\" Wesley was nosy.\n\n\"Walter Ashby Plecker,\" Tazio answered.\n\n#\n\nLater that afternoon, after Harry finished her barn chores, she set up shop at the computer in the tack room. Outside, the sun was already setting as Simon, the possum, peeped over the hayloft.\n\nPatrolling the barn's center aisle while the horses munched away, Mrs. Murphy heard the possum's squeak.\n\n_\"Murphy?\"_\n\n_\"What, Simon?\"_\n\n_\"What does she do in there? I see that bluish light. She sits there for hours! It's unnatural for people to sit still that long.\"_\n\n_\"Ha.\"_ Pewter, faking her patrol, stopped to look up. _\"Millions of people sit on their butts for weeks and years. After a while, part of them is in the next zip code.\"_\n\n_\"Look who's talking,\"_ sassed Tucker, plonked down on an aisle tack trunk.\n\nFor a fat girl, Pewter could move. She flew down the aisle, jumped onto the tack trunk, batted the corgi with an extended claw, then leapt off in an attempt to flee the barn, Tucker in pursuit.\n\n_\"I loathe violence.\"_ Simon closed his eyes.\n\n_\"Mmm,\"_ was the tiger cat's reply, since she often considered batting Pewter, as well as Tucker. Well, more Pewter than Tucker\u2014she could reason with Tucker.\n\nHeavyset though she was, Pewter easily flummoxed the dog. She could zig and zag so quickly that Tucker would skid out trying to catch her. Then Pewter would run straightaway, Tucker would make up lost ground, and once again the cat would turn. She even stopped dead in her tracks, faced the onrushing dog, then soared right over Tucker, who by now was barking nonstop.\n\n_\"I hate you!\"_ barked the corgi. _\"I really, really hate you.\"_\n\n_\"Peon!\"_ Pewter gleefully tormented the dog.\n\n\"What now?\" Hearing the clamor, Harry pushed away from the computer and walked outside. \"All right, you two. Calm down.\"\n\n_\"Kill. I want to kill!\"_ Tucker practically foamed at the mouth.\n\n_\"Bubble Butt, Tailless Wonder!\"_ Pewter was merciless as she climbed a gum tree, then spread out on a lower branch like a courtesan, tail swaying to and fro. _\"You'll never catch me,\"_ she taunted.\n\n_\"You have to sleep sometime.\"_ Tucker stood on her hind legs, reaching as high as she could with her front paws on the thick ridged bark.\n\n_\"I sleep with one eye open,\"_ Pewter called down in a singsong voice.\n\n_\"What a liar she is,\"_ laughed Mrs. Murphy, now with the human.\n\nGrabbing Tucker by her rolled leather collar, Harry pulled the enraged dog away from the tree. Pewter watched from above, enjoying the spectacle.\n\n\"Tucker, leave it,\" Harry ordered.\n\n_\"Really, Tucker,\"_ Mrs. Murphy counseled. _\"She's not worth this much emotion.\"_\n\nTucker stared imploringly at Harry. _\"You don't know how awful she is. You don't know how I suffer.\"_ She thought a moment, searching for further damning ideas. _\"I think she's a member of a Confederate underground. She's gray, you know. She wants to restore the old ways. She's really, really awful.\"_\n\nMrs. Murphy laughed, while poor Simon, who had run to view this chase from the opened upper hayloft door, wrung his front paws. _\"Tucker, she would be the same no matter if it was the old days or these days,\"_ he said.\n\n_\"She'd be worse. I know it.\"_ Tucker still stared at Harry, who reached down to pat her silky head.\n\nAs though singing an aria, Pewter meowed, _\"She can dish it out but she can't take it.\"_\n\n\"Pewter, that's the worst screeching ever,\" Harry insulted the cat. \"Now, here's the deal. If you don't behave, it's lockdown. Separate rooms. Closed doors. No treats. Hear that? No treats.\"\n\nTucker growled low. _\"I'd starve to get even.\"_\n\n_\"I wouldn't.\"_ Pewter hastily backed down the tree, circled Tucker so she would be behind Harry, then rubbed the human's legs while purring mightily.\n\n_\"How can she fall for this?\"_ Crestfallen, Tucker lowered her head.\n\n_\"Because she likes me better.\"_ Pewter kept rubbing.\n\n\"I can't concentrate when you all carry on like this,\" Harry complained. \"Too much noise. If we were in the house, God only knows what would have been smashed to bits. Now come on. Settle down.\" She turned to go back to the barn.\n\nDutifully, Tucker stuck by the human's heels while Pewter, in a flash of glory, or so she thought, raced ahead, tail straight up. She paused for a moment, then Mrs. Murphy zoomed up next to her and the two cats chased each other, in good fun, to the barn.\n\nHarry loved watching animals play. \"Tucker, cats are, well, cats. They'll chase each other, play-hiss, howl\u2014it's just dumb stuff. You, being a sober and responsible dog, are above it.\"\n\nTucker considered this and thought for a fraction of a moment that maybe Harry did understand. To some extent, she did. Anyone who lives with cats figures out soon enough they will do what they want.\n\nBack in her tack room chair, Harry wiggled to get comfortable. The lamp she was using until she could buy the Italian light bulb\u2014which is how she thought of it\u2014couldn't shine its light as precisely as the designer one, but it was okay.\n\nTucker flopped at her feet. This made Harry happy because she always enjoyed reporting her progress to the dog, who invariably perked her ears at Harry's voice.\n\n\"Tucker, I have gotten into the county records for students, but the records for Random Row are spotty at best. I'm trying to find a student's name that was on a piece of paper in the teacher's desk.\" She scrolled through the years. \"The years before 1918 aren't even entered. They microfilmed the written records back in the 1960s. Maybe the handwritten records are in a forgotten vault somewhere in the county building.\" She kept clicking the mouse. \"Oh, hey, they actually scanned them in. The handwriting is beautiful. I can't make some of this out, but there does not appear to be a student named Walter Ashby Plecker.\"\n\nMissing his wife, Fair walked into the barn, looked up, and saw Simon. \"Hey, fella.\"\n\n_\"Hey,\"_ said Simon, then scuttled away.\n\nFair entered the tack room. \"Simon is such a scaredy-cat. 'Course, most possums are.\"\n\n\"It helps if you feed him.\" Harry leaned her chin on her hand. \"Molasses on bread or molasses in the snow.\"\n\n\"I know, but when am I going to have time to feed a possum? When do you have time?\"\n\nShe smiled up at him. \"I do it every day.\"\n\n_\"Feeds us, too,\"_ Shortro, the athletic Saddlebred horse in the stall next to the tack room, called out.\n\nTomahawk, Harry's beloved Thoroughbred, also nickered. _\"We love Harry,\"_ he declared.\n\nAll the horses agreed, and up in his nest even Simon squeaked, _\"Me, too.\"_\n\nThe two cats entered the tack room just as Harry finished telling Fair about her failure to find any information on the mysterious name.\n\n\"Here.\" He leaned over, typed a bit, then stood back. \"You did the logical thing. You assumed Walter was a student's name because the paper was found in the teacher's desk. I just punched in his name to see what would show up. There you go.\" Fair started to read over her shoulder. \"Hmm, not so good,\" he said.\n\n\"Why didn't I think of that?\" Harry, delighted that her husband was smart, was equally put out by her own slowness on this subject.\n\nShe read along with him. \" ' _Paper genocide_ is often the term used to describe the actions of Walter Ashby Plecker, the government employee who was head of Vital Statistics in Virginia from 1912 to 1946.' \"\n\nFair continued. \" 'Plecker replaced the term _Indian_ with the term _colored_ on all official documents, birth and death certificates, marriage licenses, and voter registration forms.' \" He stepped back a moment. \"Paper genocide.\"\n\nShe looked up at her husband. \"Fair, what does it mean exactly?\"\n\n\"I'm not sure but I think it means that if everyone is jammed under one umbrella, you can treat or mistreat everyone the same. This is bizarre.\" He read more as she scrolled down the text. \" 'Members of Virginia Indian tribes are severely handicapped in proving they are indeed Indians according to federal standards. They can't apply for scholarships or receive federal funds for housing, health care, or economic development.' \" He stopped looking at the screen, looked at his wife. \"And, of course, they can't open casinos, which brings in big bucks. Wait a minute, here. Says the Virginia tribes do not want to open casinos and have signed away those rights.\" He'd returned his gaze to the computer screen.\n\n\"Fair, this is a terrible thing.\" She read more on the subject. \" 'Seven Virginia governors, irrespective of party affiliation, have supported federal recognition of Virginia's Indians.' But, in so many words, they've been told to sit on a tack.\"\n\n\"Hey, look at this. No Virginia tribe member can return their ancestors' bones to a rightful and respectful burial. Harry, this is outrageous. I mean, I had no idea. This is one of the most disgusting things I have ever read.\"\n\n\"Well,\" Harry said, \"a bill, H.R. 783, the Thomasina E. Jordan Indian Tribes of Virginia Federal Recognition Act of 2011, is presently working its way through Congress. Yeah, right. And how many bills prior to this have died in one subcommittee or another?\"\n\nFair rarely swore but he let it fly. \"Bastards.\"\n\n\"Walter Ashby Plecker appears to have been the biggest bastard of the bunch, and it's just rolled on from there.\" Stunned and deeply disturbed, Harry clicked off her computer. \"Let's call Coop,\" she said, standing.\n\n\"Why?\"\n\n\"Come on. Let's go inside and use the house line. Anything ever spoken on a cellphone is out there somewhere.\"\n\nIntrigued, he followed his wife into the kitchen, as did the two cats and the dog.\n\nHarry rang Cooper up on the kitchen wall phone and explained what she and Fair had discovered about Walter Ashby Plecker. \"It's only a scrap of paper but maybe you should ask Sarah Price if you can go through Hester's desk to see if there's a link. After all, Hester had Cherokee blood on her mother's side, and Josh was a member of the Upper Mattaponi tribe.\"\n\n\"I've asked Sarah to go through Hester's things on Monday, and I'll be there with her.\"\n\nUrgency in her voice, Harry prodded. \"Move it up. Go tomorrow, Sunday, if she'll do it.\"\n\n\"Harry, this isn't much to go on.\"\n\n\"It's a long shot, a really long shot, Cooper, but right now it's one of the few links between the two murders, other than both corpses were dressed for Halloween, cleverly disguised.\"\n\nA long, long pause followed this. Over the phone line, Harry could almost hear Cooper's mind whirring. \"All right.\"\n\nHusband and wife remained silent after Harry hung up.\n\nFinally, Fair said, \"While you were on your computer in the barn, I was on mine. Come on. I have something to show you.\"\n\nOnce inside his small, tidy office, he showed her an article reporting that jobs exposing women to plastics and man-made chemicals greatly elevate a woman's chance of developing breast cancer.\n\nHe pointed to the screen. \"A long-term study of more than two thousand women in Ontario found those who worked for at least ten years in food canning and automotive plastics developed cancer at a rate five times higher than women in other jobs. Chemicals such as BPA\u2014bisphenol A\u2014are to blame.\"\n\nShe replied, \"I'm a farmer. Well, I worked in the post office after college. I've checked out fine at every exam for the last two years, and after my next checkup, I won't have to go back for six months. I'm okay.\"\n\nFair, who had felt shocked, frightened, and useless when his wife was diagnosed with stage I breast cancer two years ago, still carried with him the fear that it might return. Because of this, he was vigilant concerning cancer research and treatments. \"Read on.\"\n\n\"Pesticide exposure is also elevated for female farm workers.\" She quieted for a moment. \"I rarely use much of anything.\"\n\n\"What about your father?\"\n\nShe nodded. \"He used more. Cut back later on, but he said that in the beginning they thought those things were a godsend.\"\n\n\"Our air, food, and water are loaded with chemicals unheard of even fifty years ago. BPA and phthalates, to name a few, are known hormone disrupters. I see so much more cancer in horses than I did when I started practicing as a vet.\" Harry's husband looked stricken.\n\n\"Honey, my cancer is not coming back,\" she assured him.\n\n\"I know.\" He kissed her cheek. \"But now that you've been through it, I want to keep abreast of recent research, forgive the pun.\"\n\nThey both laughed and she hugged him when he stood up. \"You're stuck with me.\"\n\n\"Better be, which reminds me: Carry your father's old snubnosed .38.\"\n\n\"I can't shoot cancer cells.\"\n\n\"No, but you can't keep your nose out of those two murders. I know you. Just carry the gun.\"\n\n#\n\n\"A Montblanc Diplomat,\" Cooper said the next afternoon, holding up the fancy pen, which she had fetched from the drawer of Hester's desk.\n\n\"She didn't have much or spend much, but what she had was the very best.\" Sarah smiled, remembering her aunt's lectures on prudent expenditures. \"If she was going to fork out cash, it had to be for something that would last.\"\n\n\"This certainly will.\" Cooper studied the gold point. \"Medium.\"\n\n\"How do you know so much about pens?\"\n\nCooper laughed. \"How do you?\"\n\n\"Drilled into my head: Write in your own hand on good paper. Always write a thank-you and a condolence, and, of course, the condolence should never be on pastel paper. The rules, but I'm glad I know them. Of course, who in my generation practices such etiquette?\"\n\n\"We're close in age. I kind of think these things are coming back. I mean good stationery, fountain pens, elegant clothing, and hats for men, too. Cycles. Then again, how many butt cracks can you observe before you decide that maybe low-rise trousers are not the way to go?\"\n\nSarah, emitting peals of laughter, snatched a heavy wooden ruler from the desk. \"This would solve a lot of problems in that regard.\"\n\nCooper laughed, too, as they both kept rooting through the long drawer of the antique desk.\n\nSarah pulled out a Smythson leather-bound day calendar. \"Probably you should keep this, and take a lot of time with it.\"\n\n\"Right.\" Cooper ran her forefinger over the textured leather. \"I don't see a computer anywhere in here.\"\n\n\"Aunt H didn't have one at home, just one at the stand, which you know about. I'd bug her about it but she said when she came home she didn't want to think about work.\"\n\n\"We've already got someone working on the store computer with Lolly. Lolly still needs it to transact business. Working in the evening, our computer whiz found sales, purchases, and a soil map for Albemarle County with the farms she did business with clearly marked. Wherever she bought anything from anyone, she checked their soils. She really was a very thorough person.\"\n\n\"About most everything. Dad was that way, too. 'If you're going to do it, do it right.' Must have heard that a thousand times.\" Sarah sighed and smiled. \"I miss them. I will miss their generation and my grandparents' generation. I never really thought about it much before.\" She blinked. \"Sorry. You're here to go through Aunt H's effects and I'm babbling on.\"\n\n\"Not babble. It does kind of dawn on you that nobody is here forever. Then you have to realize you won't be here forever either. I see enough death in my line of work to give me great respect for life.\"\n\nSarah tilted her head. \"Officer, that's a wonderful thought.\"\n\n\"Will you call me Cooper? I'm not even in uniform. Thank God. I mean, have you ever seen a law enforcement uniform that looked good on a woman?\"\n\n\"Now that you mention it, there's never quite enough room for...\" Sarah made a rolling motion over her breasts.\n\nAgain they both laughed.\n\n\"Nothing here,\" Cooper said, glancing at an empty drawer. \"There's still this big one on the bottom, the double drawer. Let me get on my knees and hand you the files.\" She did just that.\n\nSarah arranged the files in neat piles. \"Hmm,\" she said, reading the tabs. \"A lot on fertilizer, pesticides, wildlife studies. County soil and water maps.\"\n\n\"Probably hard copies of the maps that are on the work computer.\" Cooper pulled out the map file, flipped it open, and unfolded a large county map.\n\nSarah reached into the open folder. \"Here's the info key.\"\n\nCooper studied the numbers and the outlines, all in different colors. \"Number one is her own holdings. Pretty good soil. Number two is eastern Albemarle. Hmm, not as much produce over there. She has most of it marked as hay.\" Cooper ran her finger back to western Albemarle. \"Morrowdale, that's a beautiful farm out on Garth Road, where we found the first body. According to Hester, good soil, but she has the pesticide sign near some of the acres. I guess that means they were sprayed.\"\n\n\"Look at all three yellow outlines. Someone owns a lot of property,\" Sarah remarked.\n\nCooper checked the key again. \"That's Buddy Janss, the big man who served punch at the reception. He rents most of it.\"\n\n\"He must be rich.\"\n\n\"Land poor might be another way to put it.\" Cooper smiled. \"He owns maybe eight hundred acres outright. Rents the rest, so he has a lot of money sunk into soil improvement. That's the conundrum of renting land. You usually need to improve it. Most landowners, especially suburban or city people who buy in the country, don't want to spend money on their fields and don't realize how important it is. Buddy dutifully fertilizes, weeds, tests soil. He doesn't want to sell his acres if he can help it. And he'd buy what he rents if he could.\"\n\n\"Who is in chartreuse?\"\n\nCooper ran her finger down the list. \"Neil Jordan. Hester marked what's owned, what's rented, and what's for sale. She must have updated this weekly.\"\n\nCooper folded the map and placed it back in the manila folder, then pulled out the file on fertilizers. \"This is full of equations,\" she exclaimed.\n\nSarah studied the figures. \"Aunt H took high school chemistry. Maybe she took more when she was at Mary Baldwin.\"\n\n\"Bet she did. This stuff is complicated.\" Cooper shook her head.\n\n\"The brochures aren't.\" Sarah handed her a pile of glossy brochures for fertilizer products, featuring photos of lush fields of corn, wheat, soybeans, even orchard rows.\n\nCooper examined a pamphlet. \"Here's one that Neil Jordan wrote on the back of, telling your aunt to call him if she had questions. It's for a new fertilizer.\"\n\n\"If it was a natural product, she probably did call him, but she was wary, as you know, and she was really opposed to anything petroleum based,\" said Sarah. \"She would say something about it to me every now and then, but not too much since I don't understand agriculture.\"\n\n\"Did she ever ask you about your work?\"\n\n\"A little. Enough to tell me it's boring, which it is.\" Sarah looked directly at Cooper. \"Insurance is a good thing to have but so much of it is oversold on fears. I've done well, but I, well...\" She shrugged. \"I think I can walk away from it now, thanks to Aunt H.\"\n\n\"She must have loved you very much.\"\n\nSarah's eyes teared up, then she laughed her tinkling laugh. \"I was her only heir. I suppose she had to love me.\"\n\n\"Big Mim put the word out that you're going to move here. Live in Hester's house. Keep the family place alive. Oh, Big Mim runs Crozet, which I should have told you or someone should have told you before the reception.\"\n\n\"Susan Tucker filled me in on all the locals. I learned who, what, when, and where, and sometimes why. Aunt H, when I'd come on visits, didn't much talk about other people.\"\n\nChin on her hand now, Cooper pulled out another folder, flipping it open.\n\nShe sat up straighter. \"Here are the procedures for officially establishing an American Indian group as an Indian tribe.\" She read a footnote. \"The seven criteria are presented here in abbreviated form. 'For the complete federal text, refer to 25 CFR Part 83.' Huh?\"\n\nSarah rummaged through the files. \"She's got some really old stuff here. Stuff before computers took over. It looks to be clearly presented. It's all about the Virginia Indian tribes. What about the criteria you have, is it more recent stuff?\"\n\n\"Number one is, 'The petitioner has been identified as an American Indian entity on a substantially continuous basis since 1900.' \"\n\n\"My list says, 'Be identified from historic times until the present on a substantially continuous basis as _American Indian or aboriginal_.' \"\n\nEach reviewed the other's list and the language.\n\n\"Your list is clearer,\" Sarah said.\n\n\"It is. Number two: 'Prove that a substantial portion of members lives in a specific area or lives as a community viewed as American Indian and distinct from other populations of the area; and prove that members of this community are descendants of an Indian tribe.' \"\n\n\"That's easy to do in the western states.\" Sarah read on for Cooper. \"Number three: 'Prove that it has maintained tribal political influence or other authority over its members as an autonomous entity throughout history until the present.' \"\n\nCooper read on, then threw up her hands. \"All this crap about documentation. If you've been moved around or removed, how can you provide documentation to the very same government that's screwing you?\" She stopped herself. \"Sorry.\"\n\n\"No, no. I understand, but this is even worse.\" While flipping through the papers, Sarah had plucked out the criteria needed for an individual to prove he was Indian, as described by the Bureau of Indian Affairs. \"This is real betrayal. Worse than 'screwing you.' Look at this.\"\n\nCooper read the passage, then said, \"This is flat-out impossible. You have to give the maiden names of all women listed on the request for the Certificate of Degree of Indian Blood. So you need, at the least, your mother and grandmother's maiden names. You can abbreviate that to CDIB. How helpful.\"\n\nSarah continued in Cooper's line. \"Birth certificates? Okay, that's reasonable. 'Delayed birth certificates,' what in the devil does that mean? Death certificates... and 'the Indian tribe must have a duly adopted tribal ordinance concerning the issuance of such documents.' So you can't be born or die according to your customs, you have to do it the U.S. government way and prove it? And your degree of Indian blood can only be computed if there are records of ancestors of Indian blood who were listed on an official roll.\" Sarah caught her breath. \"This is a bureaucratic paper nightmare.\"\n\n\"Paper genocide,\" Cooper whispered.\n\n\"What?\"\n\nCooper explained what Harry had told her about Walter Ashby Plecker and the peculiar phrase linked to him.\n\nSarah was silent for a long time, then said, \"It is genocide. Dear God, it is. Why did Aunt H have all this stuff?\"\n\n\"Do you think she might have wanted to prove her own Cherokee blood?\"\n\n\"No. There would be no way to prove it and by now it's so intermingled with, for lack of a better description, British Isles blood.\" Sarah shook her head.\n\n\"Yeah.\" Cooper stared at the folder, then turned more pages. \"Obviously, Hester was fascinated with this. And you know about the Virginia tribes not being recognized by the federal government?\"\n\nAs Sarah did not, Cooper explained. Sarah exploded, \"They're totally ripped off. It isn't even the loans and all that stuff. It's needing to be recognized for surviving at all, for keeping their cultural integrity intact. This may sound odd but one of the main reasons I go to Mass is for the liturgy, for the tradition. It's my culture. It has been intact for two thousand years. I am saying and praying what people before me prayed throughout the centuries. I need that. Doesn't it make sense that those of Indian blood need their customs, spiritual solace?\"\n\nCooper returned to the file. \"It does make sense. It's more important than the money, but I'm willing to bet the reason this is all so complicated and difficult is the government doesn't want thousands of people applying for scholarships, loans, you name it.\"\n\n\"Everything comes down to money. Like divorce. A relationship devolves into fighting over money and who gets the couch. Pretty much the whole thing repulses me.\"\n\n\"Divorce?\" Cooper half-smiled at her.\n\n\"Yeah, but what I'm really talking about is bean counting. That's what I do at the insurance company: I count beans.\"\n\nCooper took in what the young woman said, then returned her gaze to the file in front of her. \"Says here that tribes may purchase or reacquire traditional lands and have the property placed in trust status, exempt from state or county rules, including, in some cases, zoning restrictions.\" She read more. \"This states the possible land base for Virginia Indians is too small here.\"\n\n\"I don't believe that,\" Sarah said. \"It's just that the land has been in other hands for centuries. White folks got here in 1607. Well, earlier if you count the city of St. Augustine down in Florida.\"\n\n\"Here's something else,\" Cooper said. \"If a close affiliation with a property can be shown, the tribe might reclaim the lands or demand damages for its 'illegal' usage\u2014'illegal' is in quotes\u2014many years ago. Means church and school lands used by the Catholic Church, Quakers, etc., could possibly be reclaimed, bought back, and some sort of reparation deal structured.\"\n\n\"What a mess, except it isn't because it's hidden,\" Sarah said. \"Squelched. People nowadays are too busy downloading the latest film to even think about something like this.\"\n\nCooper smiled again at Hester's niece. \"I don't think being self-centered is unique to our time. We simply have more ways to pursue it.\" She then looked down before exclaiming, \"Big bucks!\"\n\nSarah took the paper from Cooper's hand and read, \" 'The Mashantucket Pequots of Connecticut run the biggest moneymaker in the western hemisphere with their casino.' All the other tribal-owned casinos are listed here with profits. And there's a note at the bottom.\"\n\nCooper read out loud: \" 'Dear Hester, Per your request. It's fascinating. Think of the fly rods one could buy with even a sliver of those profits. Ha. Your buddy, Josh.' \"\n\nCooper's face turned pale. \"He was onto something. He had to be. He couldn't have been killed in such a bizarre way for finding an accounting error in the books of the local convenience store.\"\n\n\"Maybe he wanted to push for a casino.\"\n\n\"But the Virginia Indians signed away that possibility.\"\n\n\"Laws can be overturned. It must be something like that, Cooper, because my aunt would never, ever be involved in something underhanded. She wanted just enough to live decently and wished the same for others.\"\n\n\"Sarah, I know that's the truth, but your aunt Hester discovered something dangerous, and I'll bet she shared it with Josh.\"\n\n#\n\nBrilliant sunshine flooded Harry's fields and pastures Monday afternoon, turning golden the sunflower stubble and the plowed fields.\n\nSoil map in hand, Neil Jordan by her side, Harry stepped over a gray fox den nestled between a deep fold in her back quarter acre where she'd planted Petit Manseng grapes. Foxes love grapes, or any sweet things.\n\nTucker, his snout smushed down in the opening, announced, _\"Vixen! I can smell old chicken bones.\"_\n\nA fox's voice from within called, _\"I killed that chicken fair and square, Bubble Butt.\"_\n\nPewter, daintily following the dog and Mrs. Murphy, paw midair, shrieked with delight at the female fox's declaration. _\"Bubble Butt! She called you Bubble Butt. See, I'm not the only one that recognizes your more ridiculous qualities.\"_\n\n_\"Don't worry, she heard you,\"_ Tucker snapped, a little embarrassed.\n\n_\"I did,\"_ said the fox. _\"You can hear Fatty Screwloose for acres when she runs her big flannel mouth.\"_\n\nAppalled at the insult, Pewter flashed to the den's opening, pushing aside the dog. _\"How dare you? Come out of there and I'll pull every whisker from your pointy face.\"_\n\n_\"Come in here and say that to my pointy face,\"_ the gray fox challenged.\n\nEnraged, Pewter stuck her head farther into the den.\n\nSensible as usual, Mrs. Murphy warned, _\"Don't go in there. She has every advantage.\"_\n\n_\"For one thing, I'm not fat,\"_ the fox sang out, enjoying herself.\n\nPewter moved in a bit farther, to stare into two golden eyes. She hit reverse in a hurry, waddling back out comically.\n\nTucker held her tongue.\n\nPewter sat for a moment, licking her paw.\n\n_\"Come on, Pewts,\"_ Mrs. Murphy told her. _\"Time to keep up.\"_ Neither she nor Tucker was fooled by the cat's studied nonchalance.\n\nThe three animals scampered up to be with Harry and Neil.\n\nThe gray fox emerged from her den to watch. Ever curious about humans and domesticated animals, she wondered how on earth they could get along, but they seemed to do just fine. She liked Harry's grapes, so she somewhat liked Harry. Sometimes seeds dropped; some corn kernels from a little patch were tasty. This was a perfect location for a den. Occasionally, Tucker would drop and forget a bone nearby. Those were treasures. But still, spending your life following an animal lurching around on two legs? Seemed odd, and perhaps undignified. Humans were so slow, but the two cats and the dog willingly poked along.\n\n\"Right here I have a pH of seven.\" Harry turned over some dirt with her boot tip. \"I'd like to get it to six point five.\"\n\n\"Let me give you Centerpoint,\" said Neil. \"It's one of my best products. Given that you want to experiment with a small portion\u2014what, four by four feet?\"\n\n\"Right,\" Harry answered.\n\n\"Spread it by hand, or if you have one of those walking spreaders, the type people use for small lawn areas, that will do it. Mark your corners and I promise you next harvest you'll see a difference.\"\n\n\"All right.\"\n\n\"You're fortunate to have good soil. Well, good soil for Virginia. Davis loam. Some alluvial deposits.\"\n\n\"Most of the lower fields are like that, but Dad really kept at them and when I was little we could use muck from the Bay.\" She meant the Chesapeake Bay. \"Can't do that now, but that really helped here. We could also use crushed oyster shells.\"\n\n\"Calcium,\" Neil said, nodding. \"Well, that's the best, but since those things are off-limits now, these commercial applications do provide the same things: calcium, selenium, potassium, magnesium, and on and on the list goes. Soil tests are so accurate today they can pinpoint the exact application you need for your specific crop. Much more cost-effective.\"\n\n\"Until you hit red clay.\" She scuffed some dirt with the toe of her boot.\n\n\"Harry, we can even enrich that these days. Clay has important uses. The reason so many early Virginian homes are brick is thanks to that red clay. It's the devil to dig up. But I mean it, fertilizer today helps even that.\"\n\nThey turned to walk back to Harry's house, about a half mile away, glowing in gorgeous afternoon light.\n\n\"Neil, how'd you wind up selling fertilizer and other ag products? You didn't go to ag school.\"\n\n\"My college major was business and I liked it okay, but an old girlfriend, premed, goaded me into organic chemistry. She said it was the washout course for premed and that included vet premed, too. First, I had to take regular chemistry. Liked it. Then I took organic and found I loved it. But what could I do with it? I didn't want to be a medical student.\"\n\n\"I think you are the only person I have ever heard say that they loved organic chemistry.\" She smiled.\n\n\"Actually, a lot of people do, but you have to have a feeling for it, because it's not always logical like, say, mathematics is. Magic happens in those equations.\" He grinned. \"Anyway, I graduated from Amherst with a business degree and starting working at a Monsanto satellite company outside of Minneapolis\u2014great city, by the way. That's when I realized that, much as I did like the business end of the company, I truly liked the hands-on, using the products. Monsanto gets attacked all the time for their genetic engineering of seeds, etc., but I learned a lot, and, Harry, how are we going to feed billions? Twenty-five million babies are born annually in India and seventeen million are born each year in China. They want Western foods, technology, all our goodies.\"\n\n\"What's the number of births annually in the U.S., if you know it?\"\n\n\"Four point three million.\"\n\n\"Ah.\" The implications were becoming dreadfully clear. \"All those people in developing countries... if they can't get enough to eat, seems to me there will be tremendous instability.\"\n\n\"There already is,\" he said with conviction. \"Anyway, I started thinking about what to do to make money based on what I was learning, and I decided to go out on my own selling fertilizer. There are good companies that I can call on, get the best products, and, of course, buy in bulk from Mosaic, PotashCorp. I work with the best. Even companies like Bayer, which made their fortunes in other areas in the late nineteenth century, the early twentieth, now have ag arms. Food is the future, not doctors.\"\n\n\"I believe that.\"\n\n\"Harry, in the next forty years, farmers will need to produce more than they have in the past ten thousand, and the biggest grain producer in the world is still the United States and it will always be. Huge corn producer, too.\"\n\n\"That I know. No other nation in the world has the soils, the variety of climate, that we do. But that huge midwestern belt is gold, pure gold. And we are bound by two huge oceans and have the Gulf right in the middle of the country. Fishing alone is worth billions.\"\n\n\"Like everything else, it's politicized. Do I think we need to carefully manage our resources? I do. It's a tightrope walk, so either some of the environmentalists give a little ground or ultimately millions will starve to death and then a billion or so. There will be nine billion people by 2050 and I think you and I might still be alive then.\"\n\n\"Who knows?\" Harry shrugged. \"But I can see you do your homework.\"\n\n\"I wouldn't be in business if I didn't.\"\n\n_\"He kind of smells like fertilizer, don't you think?\"_ Tucker said.\n\n_\"It's a medley of smells.\"_ Mrs. Murphy flared her nostrils.\n\nThey reached the back porch.\n\n\"Come on in for a Co'Cola or maybe something stiffer,\" offered Harry.\n\n\"Too early for that.\" He smiled. \"But I could use something cold.\"\n\n\"I have some iced tea. Always have it, even in winter.\"\n\n\"Sounds good.\"\n\nThe animals walked into the kitchen, where Pewter took up her post next to her bowl, just in case.\n\nAs Neil sipped his beverage, Harry unfolded the soil map with the plan of her farm.\n\nNeil studied it again. \"You're very fortunate.\"\n\n\"I am. Given your business and your outlook for the future, I mean that population growth, how do you feel about housing development?\"\n\n\"In the next ten years Richmond and its surrounding area is supposed to grow by four hundred thousand people. That means good farmland goes under.\"\n\n\"Does. Well, it's happening here.\"\n\n\"Yes, it is, but if developers will plan entire communities with common gardens, wild spaces, stuff like that, or even build more row houses that share an exterior wall, maybe we can limit the damage. People have to live somewhere and not everyone is meant for a high-rise, especially in Virginia. People don't move to Albemarle County to live in a skyscraper. It's much easier to make money as a developer than by farming. We need to find that balance, but profit is what drives all business. There's going to be more development everywhere.\"\n\n\"Wouldn't it make sense for, say, Wesley to buy and build on those acres where the soil is poor?\"\n\nNeil leaned back. \"Sure. But he also has to consider access to I-64, Route 29, good roads in general. Water. Can wells be dug in places where there is no city water? The whole permitting process is complicated, and like every other county in America, there are people on the board of commissioners dead set against any form of growth or development.\"\n\n\"Right,\" said Harry. \"So why do you and Wesley want Buddy's hundred acres so much?\"\n\n\"Harry, look where it is! Close to Crozet, a hop to Route 250, maybe a twenty-minute commute to Charlottesville or thirty to Waynesboro, forty minutes to Staunton. Perfect location and the soils seem to be decent, which I don't like to see built upon, but Buddy's got corn smut. Those spores have to be in the soil.\"\n\n\"If he sprays for corn borer, it will somewhat cut down on the smut, not a lot.\"\n\n\"There is no seed treatment for corn smut,\" Neil said, warming to the subject. \"You can remove the galls in a home garden, but not for one hundred acres. So he has to burn or plow under the diseased stalks. Burn, then plow, that's the surest way.\"\n\n\"He can plant a more resistant variety.\"\n\n\"Not for Silver Queen.\"\n\n\"True, but Mexican restaurants like cooking with those galled ears, think it's a delicacy. And they'll pay good money for it.\"\n\n\"Really?\" His eyebrows shot up.\n\n\"I told Buddy and he called around. Just talked to him this morning, actually. He says he can make more selling to them than he can just selling straight old Silver Queen. It's too late for this crop\u2014it's been on the stalk too long\u2014but he's kind of excited.\"\n\n\"Harry, you never cease to surprise me. Maybe you saved those one hundred acres from development.\"\n\n\"Hey, if I can help out a friend, I will. I don't know if this will help Buddy and his hundred acres, but Tazio and I are going to work to save Random Row. Before Hester was killed, she asked Tazio to head that project. It's the last thing we can do for Hester, and it's the best thing we can do for our special history.\" She glanced at the calendar on the wall. \"Three days until Halloween. Any tickets left?\"\n\n\"Sold every one.\" Finishing his drink, he thanked her, then said, \"Well, I've got to push off. If you come on out with me, I'll give you the Centerpoint sample now. Have it in the truck. And I'll drop some more by later.\"\n\n\"Great.\"\n\nHarry happily followed Neil, Tucker in her wake. The dog didn't much like the odor of fertilizer, and said so to the cats.\n\n_\"Powdery stuff,\"_ Mrs. Murphy chimed in. _\"Goes right up your nose.\"_\n\n_\"Always smells a little like dead stuff, which I usually like,\"_ said the dog, _\"but you can detect other things, man-made odors.\"_\n\n_\"Like car exhaust?\"_ Pewter wrinkled her nose and the others laughed.\n\nIvy Nursery, just west of Charlottesville and the Boar's Head complex, contained long greenhouses as well as trees and other plants in rows outside the buildings.\n\nHarry pulled into the almost full parking lot right before 6 P.M., quitting time for Susan. She walked inside the main building and there to the side was Susan, creating a wonderful boxwood topiary.\n\n\"Harry!\"\n\n\"Said I'd see you on your first day of work. That looks interesting.\"\n\n\"My inspiration is the gardens of Harvey Ladew in Harford County, Maryland. So I'm making this little fox.\" She put down her shears, pulled off her protective but flexible gloves.\n\n\"Let me be your first customer. I'll buy your fox.\"\n\n\"Harry, you don't have to do that.\"\n\n\"I'd love to. He's cute as a button.\" She touched his boxwood nose.\n\n\"My boss saw right away that if I could make foxes and hounds, we'd do a big business. It's taking me some time.\"\n\n\"What do you do, outline it first?\"\n\n\"That's the thing, you can't really make a good three-dimensional outline. I have all these photographs.\" She swept her hand in front of six fox photographs leaning against the back of the long table at which she worked. A deep metal sink stood in the corner with glazed pots, terra-cotta pots, and square redwood containers on lined shelves. Ribbons\u2014every color imaginable, including the fashionable gauze ones\u2014lined another shelf, the spools affixed to the edge, a bit of ribbon hanging down from each. The trimming and cutting tools hung above the worktable on a magnetic strip. Filling the workroom were the fragrances of potted plants, small trees, and cut flowers.\n\n\"You just eyeballed it?\" Harry was incredulous.\n\n\"Did.\"\n\n\"You always got A's in art class. Not me.\"\n\nSusan wiped her hands with a small terry-cloth towel. \"You got them in physics.\"\n\n\"I like that he's running,\" said Harry of the topiary fox.\n\n\"I thought it would be easier to do than a sitting fox. But I'll get the hang of it. Hounds, too.\"\n\n\"Susan, you could do dachshunds, Labs, cats. People could give you pictures of their pets.\"\n\n\"Great idea. I'll see if my boss will go for it.\"\n\n\"Who is the boss?\"\n\n\"Karen Corriss, you remember her? She was three grades ahead of us.\"\n\n\"You mean Karen Dillard?\"\n\n\"Yeah. She married Rudy Corriss.\" She lowered her voice. \"Apparently he's not doing too well. Real estate.\"\n\n\"I can believe that, but, well\"\u2014Harry shrugged\u2014\"Ivy Nursery has to be an interesting place to work.\"\n\n\"For one day, it is.\" Susan laughed. \"Okay, let me take pictures of this on my cellphone so I can show Karen my work and my first sale. You're a peach, you know that?\"\n\nAs Susan took pictures from every angle on her cellphone, Harry beamed. \"You never called me a peach before.\"\n\n\"Oh, come on. I have so.\"\n\n\"Sometimes you've called me a good egg. I like peach better.\"\n\nThe two old friends laughed. Susan wrapped a beautiful gauze bow around the fox's neck. \"Charlie in gold,\" she said, calling the fox by his English name. The French use \"Reynard.\"\n\nHarry held up the creation. \"Let me get this through the cash register and I'll meet you outside.\"\n\nThey met by Harry's truck, animals in it.\n\nOpening the door, Harry placed the topiary fox on the passenger-seat floor. \"Touch that and I pull your whiskers out.\"\n\n_\"Why would I touch it? It's not food.\"_ Pewter tossed her head.\n\n\"Tucker?\" Harry stared right into her wonderful corgi's brown eyes.\n\n_\"Not me,\"_ the intrepid dog replied.\n\n_\"Don't even ask, I don't chew greenery,\"_ Mrs. Murphy said.\n\n_\"No, you just wrecked last year's Christmas tree,\"_ Pewter reminded her.\n\n_\"I had help.\"_ The tiger cat's pupils enlarged as she growled at Pewter.\n\n\"Enough.\" Harry shut the door, the window open a crack. \"These last two months all those cats have done is fight. And Pewter chases Tucker, too.\"\n\n\"Not for long, I assume,\" Susan laughed.\n\n\"She does need Weight Watchers, doesn't she?\"\n\nThey stood there in the faltering light, coolness coming on. Shoppers left the nursery, purchases in hand.\n\nWesley Speer emerged, two huge amaryllises in his arms. Harry, on seeing him, ran up. \"Let me help. These are beautiful.\"\n\n\"Thanks, Harry. Hey, Susan. We could have a vestry board meeting right here.\"\n\n\"No quorum,\" Susan said as she walked with them to Wesley's Lexus SUV.\n\nHe opened the back.\n\n\"You bought my chest of drawers!\" Susan exclaimed.\n\n\"Huh?\" He stepped back from the inside of the Lexus, a beautiful vehicle and an expensive one.\n\n\"My chest of drawers!\" repeated Susan. \"Baby blue. You went down to Farmville, Number 9, didn't you?\"\n\n\"I wanted to get something for Rebecca,\" Wesley said, mentioning his wife. \"We often go down there.\"\n\n\"Well, I wanted that one.\"\n\n\"Susan, I bet if you call down there, they'll have another one or can get it for you. This just shows what good taste we both have. Rebecca dragged me down there, oh, weeks ago and she fell in love with this. I sneaked back today and bought it.\"\n\n\"She'll love you for this.\" Harry smiled.\n\nHe smiled back. \"Every now and then it does a husband good to surprise his wife. I thought I'd carry it up to her dressing room, put it by the window, and place the two amaryllises on it.\"\n\n\"Lovely.\" Susan nodded. \"Well, home to Owen. He needs to go outside. I'll see you...?\"\n\n\"Halloween Hayride, if not before,\" Wesley answered.\n\nThe two women friends returned to Susan's station wagon, parked not far from the old Ford.\n\nSusan kissed Harry on the cheek. \"Thank you for making my day.\"\n\n\"You made mine. I have a fox.\"\n\nSusan put her hand on the door handle. \"I can't believe he bought my bureau.\"\n\n\"Honey, Wesley's right. Call them up. It's kind of funny\u2014I'm telling you to spend money and I just spent a little. I didn't even make a fuss.\"\n\n\"Maybe we're exchanging personalities.\"\n\n\"Nah.\" Harry shook her head. \"You can smooth over ruffled feathers, make people feel good. I say what I mean too often and suffer the consequences.\"\n\nSusan hugged Harry, whom she truly loved. \"Harry, I know we are both getting older, because nine times out of ten you now hold your tongue.\"\n\n\"Mmm, five out of ten.\"\n\n\"Six.\"\n\n\"Go on.\"\n\n\"I am,\" Susan said, slipping behind the wheel.\n\n#\n\nThe hum of an overhead fluorescent light distracted Cooper. Outside the police station's windows, the sun was setting. Opening her metal briefcase, she laid out the papers from Hester's folder onto the long table in the conference room.\n\n\"Precise,\" observed Rick, the sheriff, rubbing his chin as he noted a large map with parcels of land and soil types outlined in different colors.\n\n\"Sarah was helpful. We went through Hester's papers a few times, and she reiterated that her aunt didn't speak of these much, except to say she was extra vigilant about where and from whom she bought her produce.\"\n\nHe read the numbered key, checked the land to which it referred. \"She had properties' sales reports, too, with the price and when land changed hands, which wasn't often.\"\n\n\"From everything we know of Hester, that makes sense. She liked to do business with people she'd done business with for years, preferably whose parents did business with her parents. If a family farm was sold, she wanted to know the new owner's intentions.\"\n\n\"That's one thing about this part of the world. There are times when you can see five generations at a crack.\"\n\n\"Makes you see quite clearly where bad blood galloped from generation to generation.\" Cooper laughed.\n\nHe laughed, too. \"Does, but other things, little things\u2014like have you ever noticed that Tally Urquhart, Big Mim, Little Mim, all favor bright blue? Blue signs, blue dresses, blue cars.\"\n\n\"Now that you mention it, they do. Looks like Aunt Tally is going to bust a hundred and one this winter.\" She paused at the thought of the wise old woman. \"I think I'll call on Aunt Tally. She knows these farms, knew them before most of the roads were paved. Maybe there's something we're missing.\"\n\nThe sheriff pushed his chair back from the table. \"Coop, any time spent with the old dragon is time well spent, but I don't see the connection between the farms on this map and Hester's death.\"\n\nShe sat staring at the paper, thin where it had been repeatedly folded and unfolded over the years. Pointing to Buddy Janss's holding, she said, \"Hester dealt with Buddy for decades. Land value and use are changing. Buddy is certainly aware of it. For one thing, the sale of even one of his parcels would be worth seven figures.\"\n\n\"To Buddy, not to Hester.\"\n\n\"She was opposed to development,\" said Cooper.\n\n\"So is Buddy, up to a point.\" Tired, Rick's voice betrayed it.\n\n\"Chief, I told you about the tie between Josh Hill and Hester. The Virginia tribes connection.\"\n\n\"So they went fishing together,\" said Rick, his voice now skeptical.\n\n\"Here's Morrowdale, where Josh was found,\" said Cooper, now on a roll. \"Hester has a soil outline for it. Also on this map, Hester has all of Neil Jordan's purchases outlined and soil-typed. I keep coming back to this. It has to do with soil, crops, and organic farming, but as far as I know, no one has ever been killed for applying fertilizer.\"\n\nHe smiled. \"Not yet anyway.\"\n\n\"But here's where I'm heading. Some of these farms had to have been formerly owned by the Monacans.\"\n\nHe shifted in his seat. \"Wouldn't Hester have marked that as well?\"\n\n\"I don't know. All the stuff Sarah and I found in a folder on land issues, Indian scholarship funds, housing funds, all hinged on proving one's Indian blood to the federal government. I think she might have paid Josh Hill to research who was a Monacan tribe member. We just haven't found his research.\"\n\n\"That's a big jump,\" he said, holding up his hand. \"There is no solid connection between the murders themselves that we know of other than the costumes\u2014no connection to farming practices and no connection to who owned what when.\"\n\n\"Here's where I'm going. The Virginia tribes are not recognized by the federal government, but that doesn't mean they couldn't work to repurchase such lands, nor does it mean they can't continue their pressure on the government as well as the state. The state can't deny them as easily as the feds, but the state would have a difficult time assisting in any purchases.\"\n\nHe threw up his hands. \"So what?\"\n\n\"These lands are worth millions, but they sure wouldn't be if returned or repurchased by a tribe, because they would never come on the market again.\"\n\n\"But if the land was sold now, wouldn't the tribe have to pay fair market value?\"\n\nCooper leaned toward him. \"Maybe. Maybe not. There are so many possibilities. Let's say a landowner dies without any heirs. If the state takes over and the tribes create an outcry, they could possibly win the right to reclaim the land or maybe the right to purchase it at a reduced rate.\"\n\nThe lawman remained unconvinced. \"But first the tribe has to come up with the money.\"\n\n\"They can open it for fishing and hunting, find new ways to create revenue based on their history and culture, and lastly, even though the Virginia tribes signed away the right to create casinos, they could possibly fight to reclaim those rights. As they are not now recognized by the federal government, think of the fight\u2014a long fight, national attention, with billions at stake! The federal government would look really bad, and what if the timing was just right, say, during a presidential election? A great case could be made that the Monacans' rights were signed away and stolen under duress. This could be huge.\" She paused. \"In Alexandria, preservationists want to save Carver School\u2014built in 1944 for African American children, which means tribal children, too. The purchase price is $675,000. For Alexandria, right outside of D.C., that is not bad. If they succeed, it will energize other preservationists. Obviously, the focus is African American. They may not know about Plecker.\"\n\nHe drummed his fingers on the table. \"Billions of dollars, that seems to me to be more motivation for murder than protecting an ancestral school or home.\"\n\n\"To us, but we're white.\"\n\nThis struck the sheriff. \"Coop, you're grasping at straws, but one of those straws might be the right one. I'll call the chief of the Upper Mattaponi. I don't fully understand the hopes of the various Virginia tribes.\"\n\nCooper refolded the map. \"Had another thought. Halloween is Thursday. We might want to assign extra people to the hayride.\"\n\n\"We're stretched thin as it is,\" he said. \"I've assigned extra officers to the downtown mall. It's bound to get rowdy. If a private function gets out of control, we'll be called, but the mall is public. I can't really spare anyone.\"\n\n\"If our killer is as arrogant as I think he is, the hayride could be a perfect venue for more of his showmanship.\"\n\n\"Arrogant, yes, but we still haven't got a compelling motive for these murders, and I don't see how the Halloween Hayride, a fund-raiser for the library, ties to the two killings, other than people will be in scary costumes.\"\n\n#\n\nThe fact that October 29 was forever marked as Black Friday hovered in Harry's consciousness. Not much of a history buff, she was reminded by the media, as were most people, of the dark day when Wall Street went to hell in a handbasket. Of course, panics and depressions had occurred before but, thanks to radio and newsreels, 1929 slid the whole country into debt, disillusionment, and death in front of the entire world. The news media fed the panic.\n\nJust lately, a second depression, called the Great Recession, was broadcast live by cable TV, radio, the Internet, and every possible form of instant communication. Same old, same old, no matter how you sliced and diced it. That was Harry's logic as she pondered life from her tractor seat. She plowed under her harvested barley field. She allowed her sunflowers and the small patch of cornstalks to stand as she thought of how history is a shadow dogging your every step.\n\nFinishing up her work in bright sunshine, the temperature a remarkable sixty-four degrees, she bumped along in the old reliable machine, kicking up dust, driving back to the big shed. Next to her, Tucker had a little seat with a seatbelt. At every lurch, her corgi ears would jangle, then straighten out. Made Harry laugh. Sometimes her animals or the wild animals she watched proved so comical, tears would roll down her cheeks.\n\n\"Got to check the vital signs,\" she said, turning off the engine. After unhooking Tucker's seatbelt, she held the dog in one arm, not easy, and climbed down backward off the vehicle.\n\n_\"You are very particular,\"_ the dog accurately noted.\n\nHarry checked the tractor's oil, the fuel line, and the hydraulic line. The only thing she didn't check was tire pressure. She rarely worried about that, although if a big temperature shift occurred, she'd do it.\n\n\"Tucker, waste not, want not,\" said Harry. \"If you keep things tip-top, saves money and saves worry, too.\"\n\n_\"Right, Mom,\"_ answered the corgi. _\"I don't waste bones or greenies.\"_\n\nA flutter overhead made both domesticated creatures look up. A brilliantly colored male goldfinch perched on a rafter, showing off his plumage. _\"Whatcha doing?\"_ the little fellow asked.\n\n_\"Mom's going over her tractor,\"_ said Tucker. _\"She does this every time she finishes a job. Careful.\"_\n\n_\"Tell her to put out more seed, will you? The bird feeder's getting low.\"_ Thinking he saw something to eat, he pecked at the rafter. Nope. He spat it out.\n\n_\"I'll try, believe me,\"_ Tucker promised, _\"but she doesn't understand. Hey, how come you're on your own? You goldfinches are so social.\"_\n\n_\"Too much chatter in the flock. I needed a break. Who needs to know the sordid details of everyone's nest? I like being in the tractor shed. Nice and quiet. No one else is here, although in the summers the swallowtails build nests. Pushy, those birds.\"_\n\n_\"Don't the cats harass you?\"_ the dog wondered.\n\n_\"The gray cat is too fat to harass anybody,\"_ he chirped with pleasure. _\"Now, the tiger, got to keep my eye out for her. Fortunately, she's usually occupied by something else.\"_\n\n_\"Do you talk regularly to any other birds? Birds that fly around farther away than you do?\"_\n\nAs Harry wiped off her greasy hands on an old red cloth, the happy little fellow hopped down on the tractor seat and looked down at Tucker. _\"Sure,\"_ answered the bird. _\"Birds do love to gossip.\"_\n\n_\"Ever run into those crows who ate the scarecrow?\"_ asked Tucker.\n\nThe goldfinch hopped down to a back fender to get closer. _\"I heard you were there with the cats. The crows complained you all spoiled a great find, but they didn't know who killed the human, if that's what you want to know. They said he was fresh meat. So he was killed close by, maybe even killed in a car or something.\"_\n\n_\"We kind of stumbled upon it. Did you talk to anyone who knew about the murdered witch at the church?\"_\n\n_\"No, but I heard about it. When you live in a flock, news travels fast. And I also socialize with birds other than goldfinches,\"_ he said proudly, bragging at how cosmopolitan he was. _\"Anyway, I don't know anything about that witch except people seem to like killing one another, so what's it to you?\"_\n\n_\"My human is curious, plus she liked the witch lady. She wants to catch whoever did it.\"_\n\nThe bird cocked a snapping black eye at the corgi. _\"You need better control over her. She's wasting her time. Now, spending more time on her fields, that will yield something of value.\"_\n\nTucker's attention was diverted by a vehicle coming down the driveway. The sound sent her flying out of the tractor shed to bark warnings.\n\n_\"Dogs are idiots.\"_ The goldfinch laughed as he hopped to the other rear fender, getting closer to Harry.\n\nThe slender woman turned around, hands cleaner, beheld the small brilliantly colored bird. \"Hello,\" she said.\n\n_\"Hello back at you,\"_ he chirped, which sounded to her ears like \"potato chips.\"\n\nHearing the familiar goldfinch call, she smiled. \"You're a bold fellow.\"\n\nShe hung up the red rags, walked out of the shed, glanced over her shoulder. The goldfinch chirped, then flew up to the rafter.\n\n\"Hey,\" Cooper called out, Tucker at her heels.\n\n\"Is this a courtesy call or business? I never know when you're in uniform,\" Harry said.\n\n\"I was down at Rose Hill. Thought I'd stop by on my way back to town.\"\n\nThe two fell in side by side as they made their way to the house.\n\n\"How's Aunt Tally?\" Harry asked. Rose Hill was the old lady's estate.\n\n\"Never changes. She's besotted with the baby.\"\n\nThe two walked into the sunny kitchen.\n\n\"Is she dispensing advice?\" Harry asked, raising an eyebrow.\n\n\"No, she's leaving that up to Big Mim, who, by all accounts, is a treasure trove of childrearing wisdom.\"\n\nThey laughed.\n\nBefore they sat, Harry asked, \"Coffee, tea, Co'Cola, hot chocolate?\"\n\n\"Nothing, thanks.\" Cooper took a seat at the table. \"Harry, I asked Aunt Tally a lot of questions about Hester, about ownership of old farms, stuff like that. Someone should record what she remembers before it goes with her.\" Cooper felt that every old person was a library.\n\n\"Good idea,\" said Harry, sitting across from her. \"Learn anything new?\"\n\nThe tall deputy nodded. \"More than I imagined. Who married whom. How many children, legitimate and non. Who hated whom and who was smart and not. She said such things run in families. I showed her a county map with farms outlined. She knew the history of every single one, and her memory started with the end of World War One. Got really clear in the twenties, but she said with her parents, her friends' parents, and their grandparents\u2014well, the living memories recounted to her when she was young, all put together, reached back to the 1830s. She could just rattle stuff off.\n\n\"She pointed out which farms had been well managed, even through the Depression, World War Two, and up to today. She knew where some of the old slave graveyards were, and even where there are Indian mounds, which may or may not be graveyards. Most of that history has been lost. She said if you have an ancestor in a graveyard, you have a legal right to tend to that graveyard once a year. She noted farms that had endured ups and downs, and much of that seemed to tie in to drinking. Then there were those who lost everything.\" Cooper sighed. \"I hit a dead end. Learned a lot, but...\"\n\n\"Any disputes?\" asked Harry. Aunt Tally could always tell a good story.\n\n\"Not as many as you would think. A lot of squabbling among heirs in certain families. Any dispute she recounted seemed to involve wine, women, moonshine.\" Cooper laughed.\n\n\"Never a bad place to start.\"\n\nCooper, hat off, ran her fingers through her ash-blonde hair. \"I'm pretty frustrated.\"\n\n\"I can understand that,\" said Harry. \"Look, I know I get in the way, but let me make a suggestion. Hester, though well educated, was country. I'm a Smith graduate but country. I can help.\"\n\nCooper shook her head at her neighbor and friend. \"You? Harry, one of these days you'll either fall into a well or get yourself killed.\"\n\n\"Now, just a minute here. Hear me out. Let's go to Hester's house. Let me go through her truck, the outbuildings where she kept equipment. If we don't find anything of interest, then let's go to her roadside stand.\"\n\n\"You don't want to go into the house?\" Cooper's curiosity rose. Despite herself, she wanted to understand Harry's logic.\n\n\"No. Like I said, I'm country, Hester was country. I can look at equipment, tools, trucks, tractors, and see things you don't.\"\n\n\"Like what?\"\n\n\"Let's find out,\" Harry egged her on.\n\nChallenged face-to-face, Cooper had to say yes.\n\nCooper called ahead and asked permission from Sarah to comb the outside grounds and equipment. Sarah readily agreed.\n\nIn the barn, pulling open every drawer of Hester's freestanding toolbox, Harry found old tools but nothing that proved helpful. She ran her hands along the inside walls of the shed\u2014no false boards or hiding places. She opened the small box behind the tractor seat, finding the manual. She lifted up the seat. Carefully searching each outbuilding, examining each piece of equipment, she even looked into the bins holding birdseed, sticking a broom handle down into them and twirling it around.\n\nCooper admired Harry's thoroughness. \"Hiding something in a seed bin\u2014now, that's clever, though there's nothing here.\"\n\n\"Moonshine is often hidden that way, just like when it's trucked, it's generally hidden in the middle of another shipment, like furniture boxes. Or before flyovers with infrared cameras, when the boys would grow marijuana, it would usually be in the middle of a corn crop. That doesn't work anymore.\" Harry looked around the shed. \"Let me check her truck. She spent a lot of time in it; she told me it was on its third set of tires.\"\n\nBefore going into the cab, Harry pulled off the hubcaps. \"Cooper, pull the ones on the other side. Hubcaps can be good storage if you're careful.\"\n\nUsing her penknife, Cooper popped off the hubcaps, took a look, then replaced them. Next she opened the truck's passengerside door as Harry opened the one on the driver's side.\n\n\"Our team went over the truck,\" said Cooper.\n\n\"I know,\" said Harry, though she continued searching, undeterred.\n\nIn the glove box, they found the usual: manual, registration, insurance information, old pens, a box of Altoids. From the side pocket, Harry fished out one earring, a notebook, which she leafed through, a powerful LED flashlight, and bits of leaf, dirt. Leaning over the steering wheel, she ran her palm over the dash.\n\n\"Damn,\" Harry cursed low.\n\n\"I told you we went over it.\"\n\n\"Pull up the floor mat.\" Harry did it on her side of the truck and tossed the rubber mat outside. \"There's the covering that came with the vehicle underneath. Take your penknife and slowly work around that to see if anything lifts up.\" As Harry did this on the driver's side, excitement crept into her voice. \"I think I've got something.\"\n\nShe raised the loose edge of the original mat and carefully slid her hand underneath it. \"Aha\u2014here's something...,\" she said, and pulled out a flat manila envelope with a clasp. \"Let's see what we've got.\" She hurried to the back of the truck and dropped the tailgate.\n\nNow next to her, Cooper watched as Harry drew papers and a map from the envelope and gently spread the map on the truck bed.\n\n\"Same as the other map...\" Cooper's voice trailed off.\n\nHarry squinted. \"Some properties have parcels that are outlined in purple.\"\n\nCooper grabbed some papers. \"Here's the key. The purple signifies ancestral land.\" She pointed to a spot on a nearby farm. \"Here's a Quaker school near Midway Farm. That school's gone now. Boy, is this thorough!\"\n\nCooper flipped through the other pages. \"This is Josh Hill's research,\" she said, pointing to his name on a document. \"Look here. Says the Virginia tribes cared greatly about education.\"\n\nHarry exclaimed, \"Some of this goes back to right after the Revolutionary War. He has a note here, 'Many of us took Quaker names. The Quakers have been consistently helpful identifying and helping the Monacans and other tribes to reclaim our lands.' \"\n\nShe picked up an old newspaper clipping. \"Here's an article about how the Upper Mattaponi purchased land and restored Sharon Indian School, which is the only Indian school on the Virginia Landmarks Register and the National Register.\" Harry checked the map again. \"Hill drew double lines around old church schools, Random Row, and also where old churches used to stand. And hey, about forty acres of Buddy's one hundred acres are marked off in purple.\"\n\nCooper leaned over and studied this after setting down a paper she'd just been examining. \"Hill wrote out a plan for raising money to purchase Random Row and part of Buddy's land, saying it would be easier to prove tribal usage there, even with Walter Ashby Plecker's paper genocide.\"\n\nHarry straightened up at the ramifications of Hill's report.\n\nCooper continued. \"He wrote that while church lands were carefully recorded, the buildings had usually disappeared after a century and more, but, and he underlined 'but' in red, with the School Desegregation Act of 1965, the records for schools are much more recent.\" She put her forefinger to her lips. \"The problem goes back to how to prove you're a Virginia Indian.\"\n\n\"If someone like Josh or Hester could arouse interest among the African American community to jointly preserve history\u2014say, at Random Row\u2014working together would render that less important,\" said Harry. \"Here's a list of local people Hill thought might help their efforts.\"\n\n\"Your name is on there,\" Cooper remarked.\n\nHarry stared at her name, along with the names of professors, businesspeople, and community leaders, white, black, and tribal. \"Tazio's name is on here, too. Hill did his homework.\"\n\n\"And he paid for it,\" said Cooper.\n\n\"Hester did, too. She must've hid these documents because she was afraid. She didn't want this information to fall into the wrong hands and all of this research to be destroyed.\"\n\nCooper whistled. \"Those that would gain by this being destroyed are Buddy, Wesley, and maybe Neil, as they often work together on land purchases.\"\n\n\"They don't seem like murderers,\" said Harry, mulling it over. \"Wesley and Neil are on the vestry board. Buddy is the sweetest man ever.\"\n\n\"People can fool you.\" Cooper thought a long time. \"I can't arrest three men on suspicion of murder with only a map and these papers to go on. And there is the good possibility they have nothing to do with it. But I'm nervous about what might happen at the hayride, and it's only two days away.\"\n\n\"Can you assign extra security to it?\"\n\n\"I can try again.\" Cooper called her boss and pleaded her case once more.\n\nFinally Rick relented and said he'd assign Dabny to work with her.\n\n\"Thanks, boss.\" Cooper clicked off her cellphone. \"Dabny.\"\n\n\"One more is better than none.\" Harry folded her arms across her chest. \"Thursday night is going to be interesting.\"\n\n\"I hope not,\" Cooper said, though she feared the worst. Halloween had never seemed so frightening.\n\n#\n\nHalloween colors, orange and black, gave way to shimmering slate on Thursday night as twilight fell over the rolling Virginia countryside. Those trees without leaves appeared outlined in charcoal, and the conifers swayed blue and silver. The pin oaks, dried leaves still attached, rustled in the light breeze. When the wind lifted their leaves upward, the pale underside contrasted with the tree's dark bark. Then as the wind died down, they turned right side up.\n\nAs the sun set, the actors for the Halloween Hayride met at Random Row's middle schoolhouse, along with the starter, Lolly Currie; Neil, who would keep reports on traffic midway through the hayride; and two boys charged with keeping the goblins lit. The number of people totaled thirteen, the ideal number for a Halloween drama. While they reviewed their parts, a molten sky faded to blue velvet.\n\n\"Darkening of the moon,\" said Wesley. \"What luck. We can scare the pants off everyone.\" He laughed as he glanced out the long windows.\n\n\"You do that anyway, Wes,\" Neil remarked.\n\nTazio focused on the task at hand, once more reviewing the night's plans. At her side was Lolly, with a duplicate schedule on her clipboard. \"Okay,\" said Tazio. \"First hay wagon leaves from the barn at seven P.M. After that, the wagons leave at ten-minute intervals and we have...\"\n\n\"Ten wagons plus foot followers,\" Paul reminded her, and Tazio was grateful for the strong man's presence at her side.\n\n\"Why are people going on foot?\" Wesley wondered.\n\n\"Some people like to walk and some groups won't all fit in the wagons. We'll put the children in first and have the adults walk alongside the wagons,\" Tazio replied.\n\n\"How many people can we expect tonight?\" Cooper had just finished putting on her Jeepers Creepers costume.\n\n\"Three hundred and twenty-one,\" Tazio said. \"That's how many bought tickets. That doesn't mean all will come out. The library also received contributions from over one hundred people who won't be here. After expenses, we net over twenty thousand dollars. Pretty good. Paul, we can't thank you enough for getting the horse-drawn wagons and drivers to contribute their services, and Reverend Jones, you organized the truck-drawn wagons. We have four of those. They'll be good backups if a problem arises with the horses.\"\n\n\"I trust the horses more than engines,\" Reverend Jones replied.\n\n\"Okay. First scare, after Lolly gives the initial go-ahead. The wagon rolls by the schoolhouse, Dr. Frankenstein has the monster on the table.\" She glanced over at Buddy Janss, a credible if rotund Frankenstein's Monster. \"Buddy breaks the bands, rises up, chokes Dr. F after a suitable struggle, then flees the building, running into the cornfield, threatening folks on the wagon. Then he runs back into the corn and sneaks into the schoolhouse, gets on the table, and does it all over again.\"\n\nLolly read from her schedule, \"After that, 'glowing goblins and ghosts flutter through the cornfield as the wagon progresses.' How you guys made those things work, I'll never know.\"\n\n\"Pretty much the same way you make a jack-o'-lantern.\" Paul smiled at the two high school boys who had created the goblins and ghosts. \"They're lit by LEDs, and they go up and down, back and forth on wires, using the tiny battery packs in their backs.\"\n\nBrows wrinkling for a moment, Tazio asked Neil, \"No one using candles tonight, or torches?\"\n\n\"No, too dangerous,\" said Neil. \"We have a fake torch at the end, when the monk\"\u2014he nodded to Reverend Jones\u2014\"calls the spirits to order and points the wagons toward the old Mount Carmel Church, where it all ends festively.\"\n\n\"And safely,\" the reverend reminded them.\n\n\"All right,\" said Tazio. \"After the goblins and ghosts, Frankenstein's Monster should be back in the schoolhouse, ready for the next wagon. Okay. Now, the first wagon goes between the two big trees on either side of the dirt lane.\"\n\n\"That's my cue.\" Cooper smiled in her Jeeper Creepers costume. \"And given that I need to get hooked up to a darned cable, I'm leaving now.\" What she didn't say was that Dabny, who would fasten her to the guy wire between the trees, would be observing from an old farmside road along the edge of the cornfield, ready in case trouble occurred.\n\nFrom her perch in either of the trees she would swing between, Cooper would have as good a view as possible, given it was five days after the dark of the moon. Both on-duty officers would have cellphones, but should a cellphone not properly work, Cooper and Dabny also carried a piercing whistle.\n\n\"Once people recover from the Jeepers Creepers scare,\" said Tazio, \"they round the bend and the Headless Horseman gallops toward them before going into the open-air shed, disappearing from the hay wagon's sight. Is there any way to throw a pumpkin for a head from there?\" Tazio asked her boyfriend.\n\n\"It would scare the driving horses,\" said Paul. Despite his assigned role, his head was always screwed on straight. \"Basically, I'll be running toward them, then veering off. We did set out a jack-o'-lantern on a fence post about a quarter of a mile down the road from me and Dinny.\" He named his horse, a lovely old hunting fellow who had done it all, seen it all.\n\n\"Neil, you're all in black near the shed,\" said Tazio. \"If the wagons need to move along, you call Lolly. Hopefully, no one will see you as our secret traffic manager.\"\n\n\"Right.\" Neil nodded to Tazio.\n\n\"The wagons pass the hayfield, large rolls stacked together. There will be a green-eyed goblin on top\u2014again courtesy of our high-tech guys. Then, the evil Jason and his chainsaw, meaning you, Blair\"\u2014she tipped her paper toward Little Mim's husband\u2014\"will jump out from behind the tall obelisk in the graveyard.\"\n\n\"And I'll turn on the ghost noises and wails from the graveyard before I jump out,\" Blair added with enthusiasm.\n\n\"We've got lights on one bony arm coming up from a fake grave\u2014we didn't want to desecrate a real one,\" said Tazio. \"We were going to have a witch in the graveyard, but after Hester's death that seemed insensitive.\"\n\n\"Good thinking,\" Neil complimented her. \"Jason beheads the vampire, Count Dracula, who is the undead, so of course he just puts his head back on.\"\n\n\"You've always had a good head on your shoulders.\" Reverend Jones laughed at Barry Betz, the batting coach whom Cooper had started dating. Barry would portray Dracula this evening.\n\nAs Cooper had talked him into it, everyone assumed the relationship between them was heating up. They were right.\n\n\"Any questions?\" asked Tazio.\n\n\"We aren't allowed to drink inside the meeting room at Mount Carmel, right?\" Wesley asked.\n\n\"Now, Wesley, when has anyone been allowed to drink inside a church building?\" Reverend Jones winked at him.\n\n\"Right.\" Wesley smiled.\n\n\"The second hay wagon, the one pulled by the team of Belgians, has a full bar under the driver's seat, since we knew how parched some of you can get from your labors.\" Tazio smiled broadly.\n\n\"Belgians...?\" Neil looked for direction.\n\n\"The cream-colored draft horses,\" Paul informed him.\n\n\"Ah.\" Neil nodded. \"I'm glad to know the Belgian is a horse of a different color.\"\n\n\"We've got this covered,\" said Tazio. \"It's going to be the most exciting Halloween Hayride this county has ever seen.\" She laughed. \"And I'm the Fallen Angel who appears as the hay wagon reaches Mount Carmel, holding up a crucifix to our vampire, head back on, who shrinks and screams in horror, vanquished at last. Hey, we've thought of everything.\"\n\n\"What about the kids who wet their pants?\" Wesley joked.\n\n\"That's up to Mom and Dad. By now they should know to bring Handi Wipes and towels.\"\n\n#\n\nWaiting back in the last wagon, which was being pulled by the Haristeens' truck, Pewter complained as she shifted to find a comfortable spot on the hay bale. _\"By the time we get to the church, there won't be any food left. We're the caboose.\"_\n\n_\"Sure there will be,\"_ said Tucker, an expert on dropped treats. She confidently predicted, _\"Stuff falls on the floor.\"_\n\n_\"Tucker, here's a frightening Halloween idea: I'll jump on the table and make everyone scream, 'Get the cat off the table!' Now, that's really scary.\"_\n\nMrs. Murphy laughed. _\"Pewter, you jump on that table and Mom will smack you. Then you'll be the one screaming and that will scare the children.\"_\n\n_\"I'm going inside the truck,\"_ said Pewter, who leapt from the wagon's hay bale to the truck bed, also full of hay. She started smacking the sliding window. _\"Release me from these trolls!\"_\n\nHarry moved forward in the fluffed hay to also knock on the window. In the driver's seat, Fair turned around and slid it open.\n\n\"Honey, Pewter's being a pill. Will you take her with you?\" Then she said to Susan's husband, Ned, sitting in the passenger seat, \"Ned, she'll be on your lap.\"\n\n\"Fine with me as long as she doesn't drive.\"\n\nHarry grunted as she picked up the fat cat, passing her through the open back window.\n\nFair asked, \"Do you have the .38?\"\n\n\"I do.\"\n\nNed looked at Fair with alarm. \"What's going on?\"\n\nHarry nonchalantly replied, \"Better to have it and not need it than need it and not have it.\"\n\nSusan, sitting next to Miss Mona in the wagon bed, patted the old lady's hand. Miss Mona's walker was strapped to the back tailgate. A stroke had affected her mobility, which sometimes embarrassed her, but the group of old friends pretended she was the Miss Mona they knew in childhood. She was, except that she couldn't get around like she used to.\n\nBoomBoom tended to Colonel Friend, also very old. Colonel Friend was a bemedaled World War II vet. Bunny Biedecke, another World War II vet, leaned against a small hay bale as he sat in the sweet-smelling fluffed hay. Dear old Bunny was already asleep. BoomBoom's partner, Alicia, sat next to him. Bunny had to have been tired, because he was the kind of fellow who would normally revel in the experience of sitting next to a beautiful woman. Come to think of it, most of the men in Crozet fit into that category.\n\nTazio had interspersed the trucks with the horse-drawn wagons. If a horse threw a shoe, the people could get back to a truck wagon; these held fewer people but there was also room in the truck's bed. Most people wanted to ride in the horse-drawn wagons, but some enjoyed the truck-drawn ride, mostly because all the kids were clustered together, noisy with excitement, in the horse-drawn wagons.\n\nAunt Tally hotly refused getting stuck with \"old people,\" as she called them. She sat in the first horse-drawn wagon up with the driver, regaling him, so she thought, with stories of her youth and her own \"excellent\" driving abilities. Big Mim grimly sat behind, fearing Aunt Tally had hidden a small flask in her heavy cardigan sweater.\n\nAs the mercury dropped, the screams rose up. Waiting to move forward, Harry and the passengers in the last wagon could hear them as they pierced the night in sequence. The temperature fell into the mid-forties. The beautiful stillness of this velvet black Halloween meant they could even hear the bellows of fright from the graveyard, a good mile and a half distant.\n\nChecking his texts, Ned noted when the seventh wagon had been challenged by the Headless Horseman. \"Okay, Fair, roll,\" he ordered. \"Lolly says we're up. Neil said wagon number seven just passed him.\"\n\nWatch in hand, Lolly was standing just outside the first schoolhouse. She had wanted to do something to honor her boss's memory, so she had volunteered for the job of starter in Hester's pet project. Lolly was good with details. Dressed in a skeleton costume, the young woman checked and rechecked her watch and various texts. Naturally, some wagons clattered along more slowly than others, but in the main, the evening's planned event was running quite smoothly. All seriousness, Lolly would call out each passing wagon's number and say, \"Move out.\"\n\nIn the last wagon, driving at fifteen miles an hour, the cats, dogs, and humans passed the middle schoolhouse. Eerie lights showed a large beaker bubbling froth in green light. Dr. Frankenstein's Monster lifted his bulky head. The doctor poised over him, gigantic hypodermic needle in hand. _Snap_ , the bonds broke, falling away as the monster reached up with his right hand, grabbing Dr. Frankenstein by the throat. The furious struggle was enhanced by the green light. The contents of the giant needle shot upward in the air as the doctor helplessly sank to his knees. The monster threw up his hands in triumph, not unlike a football player in the end zone. He whipped his head around as best he could, despite the spike in his neck, then crashed out the side door, roaring as he did, rushing at the wagon. Hearing the riders' screams, he then turned to disappear into the cornfield. As the wagon moved forward, one could hear the cornstalks bending and rustling.\n\n\"Good scream,\" Susan complimented Harry.\n\nPewter, on Fair's lap, pupils wide, meowed, _\"I don't like the monster's face.\"_\n\nThe back window, left open so Harry and Susan could holler at their husbands, allowed Mrs. Murphy and Tucker to hear Pewter. It sounded like a tiny meow to the two men in the cab.\n\n_\"Scaredy-cat,\"_ the two animals teased.\n\n_\"Piffle,\"_ the gray cat replied. _\"I just don't like Frankenstein's face. I'm not scared.\"_\n\nThe truck crept forward and for a moment they heard the far-off clip-clop of the draft horses pulling the cart in front of them, so still was the night.\n\n\"Don't you love the sound of hoofbeats?\" Shawl wrapped around her shoulders, Miss Mona smiled.\n\n\"I do.\" Harry held one hand while Susan held the other. \"Miss Mona, your hand is cold. Let's put on your gloves.\"\n\nAs they did that, two glowing goblins and two ghosts fluttered above the cornfield, moving from side to side, then up, only to sink back down.\n\nAhead, they heard an explosive scream of fright.\n\n\"Must be really scary,\" Miss Mona said to Colonel Friend.\n\n\"We'll see.\" His voice quavered, but as he'd fought his way through Europe, it's doubtful too much could rattle the colonel.\n\nAs they were poised between the two trees, branches twisting into the night, out flew Jeepers Creepers. So sudden and silent was this attacking nasty bird\/human, wings outstretched, strange face looking down with a snarl, that even Harry drew deeper into the hay.\n\n_\"Kill! That bird wants to kill us!\"_ Pewter screamed.\n\nEven Tucker barked in surprise, then breathed out in relief. _\"It's Coop!\"_ The corgi had recognized Cooper's scent.\n\nMrs. Murphy inhaled the crisp air. _\"So it is. She scared me.\"_\n\nHigh in the tree, Cooper folded her wings and gazed over the scene. She could see a little bit around the curve, back to the schoolhouses, which stood like clapboard rectangles in the darkness. Buddy was running through the cornfield, charged by Count Dracula. This seemed to be an impromptu scare as the last wagon rolled by. Now that Harry, Fair, and the others had passed, Cooper prepared to push off hard on the zipline and swoop in the opposite direction to get back into the tree and climb down.\n\nAs Fair slowly took his passengers around the big curve, out charged the Headless Horseman with a menacing howl, cape flying behind him, hoofbeats clattering.\n\nThrough the fake neck, Paul could see pretty well. His horse, Dinny, wondered why they just kept going into the shed again and again. His job was to chase hounds who were chasing foxes. This back-and-forth stuff was boring, but being a good soul, he did as he was asked, ears twitching as people screamed. What a racket!\n\nNext to Fair in the cab, Ned remarked, \"That horse could be in a movie.\"\n\nFair, Dinny's physician, chuckled. \"Dinny is dipped in gold.\"\n\nSighing, Ned absentmindedly stroked Pewter's head as she chose to grace his lap with her large presence. \"I've been a horse husband for twenty-three years,\" said Ned. \"I'll bet I've spent more money on horses, tack, and membership fees to hunt clubs than I did on my children's college educations.\"\n\n\"No doubt, but you have a happy wife,\" said Fair. \"Think of the men who don't.\"\n\nNed laughed. \"Point well taken.\"\n\nMiss Mona, ears still keen, said to Susan, \"I hear screams behind us.\"\n\nSusan nodded. \"Over in the hayfield.\"\n\nThe jack-o'-lantern flickered ominously, on the fence post at the end of the hayfield.\n\nBuddy Janss, still dressed as Frankenstein's Monster, clambered up one of the hay rolls, kicking the man dressed as Dracula as he tried to follow. Finally, the faux vampire grasped his ankle, pulling down the huge fellow. From a distance, Dracula appeared to bite the monster in the neck as Frankenstein bellowed, then fell still. Dracula opened his cape, slipping his knife into his belt. He also carried a small pistol but didn't use it on Frankenstein. The noise would have proven too distracting. As it was, those viewing the drama thought Frankenstein had been bitten, which he was. He was stabbed, too.\n\nBloodcurdling screams filled the air, the perfect cover for real mayhem. This Halloween Hayride was topping all prior ones for thrills.\n\nThe big square churchyard, hand-laid stone fence surrounding it, hove into view. The obelisk shone silver. As they approached, Harry could read the name on the monument: VILLION.\n\nFrom behind the obelisk, movie villain Jason appeared, chain saw in hand, white mask in place. Swinging the saw around like a hammer throw, he advanced toward them. The chain saw was not turned on, but that didn't lessen the startling effect. As he rushed them, wailing ghost noises from the graveyard added to the drama.\n\nFrom the hayfield, Count Dracula ran hard, jumping the low stone wall at the other end of the graveyard, fangs dripping blood as he headed toward Jason.\n\n\"Oh, look at the bony arm reaching out from the grave.\" Miss Mona shivered a moment.\n\nThe colonel nodded, for he had seen this sort of thing in real life. He'd seen much, maybe too much.\n\nDespite the screams, Bunny Biedecke remained asleep.\n\nJason turned to meet his attacker and swept the chain saw at Dracula, whose head tumbled off backward. But being undead, the Count picked it up, put it right back on. The two creatures struggled; the Count grappled with Jason, biting him in the neck. Jason fell to the ground, grabbing his neck, fake blood shooting through his fingers.\n\nUsing one hand, Dracula vaulted over the graveyard stone wall to disappear. As the wagon slowly passed, Jason rose up, returning to the graveyard.\n\nHearing a strangled cry, Little Mim's husband, Blair, pulled off his Jason mask and walked to peer over the other side of the graveyard wall. On his side, trying to clear his throat and his head, was Barry Betz, the original Dracula.\n\n\"Barry! Barry!\" Blair said, hopping over.\n\nThe young man couldn't yet speak. Blair looked over the field and beheld another person dressed as Count Dracula running toward Tazio, as had been planned.\n\n\"Watch out!\" Blair shouted to Tazio.\n\nShe turned, holding up her cross as the Fallen Angel, but this final time Dracula did not shrink back as scripted. Instead, he struck her hard enough to knock her sideways. Grabbing her, he pulled her up; she struggled to escape until he put a gun to her temple. He dragged her to a dip in the land where he had hidden a dirt bike.\n\n\"Get on the bike,\" he ordered.\n\nTazio did as she was told. He sat behind her, gun still to her temple. He started the dirt bike. Given that the gas was on the right handlebar, he had to slip the gun into his belt alongside the knife. As the bike picked up speed, Tazio sat still.\n\nFrom her perch, Cooper could only dimly see the unplanned drama. She didn't have time to punch in numbers on her cellphone, and pulled out the whistle instead.\n\nThe piercing note carried across the fields, over the assembled wagons. Dabny fired up the truck, roaring out of the side farm road. He screeched to a halt beneath a tree.\n\n\"Get me outta here!\" said Cooper. \"Something's wrong.\"\n\nDabny backed the truck under the tree and stepped up into the bed. He lifted himself from the bed onto the cab and reached a limb. Swinging himself up, he climbed toward Cooper, who was on her way down.\n\n\"It's these damned wings.\"\n\nDabny unfastened her wings as she tore off the mask. Then they both backed down the tree and got into the truck.\n\n\"Graveyard,\" was all she said.\n\nWithin minutes they reached the graveyard, where Blair was helping out an injured Barry.\n\nFrom the Haristeens' truck bed, Harry shouted to Cooper, \"Some maniac's got Taz!\" Harry hopped off the truck and started running after the dirt bike, now churning away from the graveyard. Mrs. Murphy and Tucker leapt off to follow. In the cab, Pewter thought they were crazy. She stayed put, hoping Ned wouldn't join in the ridiculous tumult.\n\nDabny turned the truck around and headed in the direction in which they last saw the fleeing vampire, with Cooper sitting next to him, straining to see anything.\n\nRiding Dinny back to the schoolhouse, Paul caught a glimpse of Dracula on his bike, carrying the Fallen Angel. The abductor dipped down the other side of a swale in the hayfield.\n\nLeaning forward on the solid horse, Paul galloped toward the spot. When he reached the cusp of the swale, he saw below him what he assumed to be a crazed idiot stop to position Tazio so he could hold her tighter. Without a second of hesitation Paul charged down the low rise and came alongside the dirt bike, which hadn't picked up speed with its cargo. Leaning over, he tried to climb on, and grabbed Dracula's shoulder; Dracula reached for his gun. Paul slid off Dinny like a calf roper and the horse stopped cold.\n\nPaul wrestled the fake vampire to the ground, the bike's wheels spinning as they went over. The young man shoved Dracula away, then grabbed Tazio. Dracula retrieved his gun and remounted his bike. He buzzed off.\n\nTazio's eyes fluttered as Paul lifted her into his arms.\n\n\"It's all right, honey. You're safe.\" Paul hoisted her onto Dinny's saddle and Tazio slumped forward on the animal's neck. Holding the reins, Paul walked them up the rise, across the northern end of the hayfield. Ahead, he saw Buddy Janss as Frankenstein's Monster, sprawled on a hay bale.\n\nFlagging down Cooper and Dabny, Paul asked for help for Buddy.\n\n\"I'll call an ambulance,\" said Dabny. \"Which way did Dracula go?\" he asked Paul, who pointed west.\n\nDabny drove alongside the field in the direction Paul pointed.\n\nNow in the hayfield, Harry ran toward the fiend dressed as Dracula, who pushed the dirt bike for more speed. Evidently, he was still a bit wobbly from Paul's blow.\n\nAfter loading Barry into the truck, Fair ran, keeping pace about two hundred yards behind Harry, but she was lighter and faster.\n\nAfter looking over his shoulder to see Harry in pursuit, the attacker circled in the hayfield. He steered his bike behind the hay bales, cutting his engine.\n\nHarry flew through the hay stubble faster than she'd ever run when she was on her college track team.\n\nThe dirt-bike Dracula saw that Harry was high up and behind him, but he couldn't see Fair even farther behind. Slowly, he pushed his bike around the hay bales, darkness shielding him.\n\nThe darkness also shielded Taz on Dinny's back, but as the attacker pushed his bike he now saw them.\n\nThe vampire pulled out a gun and leveled it.\n\nHarry boomed out, \"Paul, move!\"\n\nTazio's abductor stopped for a moment as Harry barely touched the earth, heading straight at him, Tucker and Mrs. Murphy in front of her.\n\nFiring, he missed Tazio, then turned and restarted the bike. Switching his gun to his left hand, he revved the engine. He roared straight for Harry, cape flying, mask in place, fangs showing.\n\nAs the bike hurtled at her, he fired his revolver with his left hand, wide of the mark. Harry hit the ground and rolled. Cunning, she pulled out the .38. She figured he would be ineffective firing the gun with his left hand.\n\nThen Dracula spotted Fair running at him, but he slowed and made a one-eighty, again barreling straight for Harry, now back on her feet.\n\nShe kept her eyes on his right hand. He had to take his hand off the throttle and switch the gun to his right hand to accurately fire the weapon. That gave her a ray of hope. Bending down, she picked up a handful of dirt and hay stubble with her left hand.\n\nNearly upon her, he slowed and pulled out his gun.\n\nHarry fired one shot from the .38 at his torso and sent another bullet for the front wheel, too. The tire blew; the bike lurched, then toppled over.\n\nAs the rider screamed, Mrs. Murphy leapt onto his arm. Harry threw dirt in the eyeholes of his mask. With her front claws, the cat ripped the mask. He coughed, trying to hold on to the gun. With everything she had, Harry threw herself onto him, smashing the butt of the .38 into his masked head.\n\nIn her jaws, Tucker grabbed the man's neck. Blood was running inside his mask where the animals had torn through. Bright crimson blood now poured from the deep fang marks the corgi made, and more blood oozed from Harry's bullet, which had hit a lung. The dog continued biting as Mrs. Murphy nipped and nipped, her sharp claws lethal on small prey and painful on large.\n\nFair caught up with them. Hauling the man to his feet as he started to come to, Fair hit him so hard Harry heard the man's jaw break.\n\nHaving followed the bike's high revs, Cooper and Dabny now arrived, driving across the hayfield, dirt churning behind the truck.\n\nCooper pulled out her service revolver as Dabny took the snubnose .38 from Harry.\n\n\"Tucker, Mrs. Murphy, enough!\" Harry commanded.\n\nThe cat sank her claws into the mask and pulled at it before it snapped back.\n\nCooper leaned down and yanked it off.\n\nHarry couldn't believe it. \"Neil Jordan!\"\n\nHis jaw hung, the break obvious, teeth missing.\n\n\"You could have been killed!\" Fair exclaimed to his wife. \"Why would you do something like that?\"\n\n\"He was going to kill Taz,\" she said to her husband, Cooper, and Dabny. \"And I had Dad's .38. You told me to use it if necessary. I always do what you tell me.\"\n\n#\n\n\"Four people taken away to the hospital,\" said Harry. \"What a Halloween.\" She, Fair, and Cooper were sitting at the kitchen table.\n\n_\"Buddy, Taz, Barry, and Neil\u2014and Neil under guard,\"_ growled Tucker, sitting on the floor nearby. _\"If I could have had just a little more time, I could have killed him,\"_ bragged the corgi. Perhaps it was true.\n\nMrs. Murphy sat on Harry's lap. _\"He's blind in one eye.\"_\n\n_\"If I'd been with you, I'd have gotten the other one,\"_ said Pewter, in Fair's lap.\n\nTucker laughed at her. _\"Chicken, you stayed in the truck.\"_\n\n_\"I jumped in the bed because Miss Mona had a spell while tending to Barry,\"_ Pewter explained, or tried to. _\"She used to be a nurse, you know, back in the 1950s, but it was too much for her.\"_ The gray paused. _\"I had to help.\"_\n\nWhile this was wishful thinking, neither Mrs. Murphy nor Tucker teased her, because Cooper was now telling the assembled humans just what had happened after the ambulances left.\n\n\"We walked into Mount Carmel, told everyone gathered there in the church to stay still, and we counted heads. Wesley Speer was missing. So we drove to the schoolhouse. No Wesley. Put out an alert and picked him up the next morning, boarding a 6:10 A.M. flight to Phoenix out of Dulles. Wesley admits to being an accessory for the first two killings. Said he opposed killing Tazio and Harry. When he heard the shots, he knew Neil was out there. He got scared and ran. He swears he tried to talk Neil out of it on their cellphones. Swears Neil threatened him. Said he turned into a maniac after killing Josh and Hester. Mind you, I take this with a grain of salt. He wants to save his own skin.\" Cooper paused. \"I told Rick we should give Dinny, Mrs. Murphy, and Tucker an award for service to the public.\"\n\n_\"Hey, hey, I helped Miss Mona.\"_ Pewter put her paws on the table, which Fair removed, settling her back in his lap. _\"I did my part,\"_ she insisted, vainly and in vain.\n\nIt was Monday morning, and Cooper had stopped by the farm before going to work.\n\n\"How's Buddy?\" asked Harry. \"I don't want to call Georgia. She's got to be on overload.\" Georgia was Buddy's wife.\n\n\"He'll be okay. Neil stabbed him in the back and must've thought he'd killed him, but Buddy is big, has a lot of muscle as well as fat, and the knife didn't puncture anything vital. That deep wound will take time to heal, but he'll be fine. He goes home today. He had no idea what any of this was about.\"\n\n\"Did Neil confess?\" Fair asked.\n\n\"Not so far, but his jaw is wired shut. He's blind in one eye. One lung has collapsed. He'll live. Can't talk. We'll get him to write answers to our questions once he's not so drugged up. If he cooperates, that is. I expect he'll lie like a rug.\" Cooper smiled. \"Mrs. Murphy and Tucker did as much damage as you did, Harry. I have never seen anyone run as fast as you once I caught sight of you in the hayfield. Go, girl!\"\n\n\"Luckily I had my .38. It's amazing what you can do when you have to.\"\n\n\"Brave. And foolish. He had a gun, too.\" Cooper admired Harry, even though her neighbor had once again taken a wild risk.\n\nHarry shrugged. \"So did Wesley say anything else?\"\n\n\"He did,\" said Cooper. \"Wesley said they stood to realize between twenty to twenty-four million dollars if they could have developed that property behind Random Row. They thought they could get the schoolhouses and develop them, too.\"\n\n\"But Buddy hadn't agreed to sell,\" Fair said. \"Or had he?\"\n\n\"He was weighing the options,\" Cooper said. \"They kept throwing money at Buddy and figured he'd cave at two and a half million. They'd pay the taxes. That's a big chunk of change.\"\n\n\"Sure is.\" Harry's eyes widened. \"But why kill Hester and Josh? They hadn't yet raised money or gathered people who might raise opposition.\" Harry knew she would always miss Hester.\n\n\"One, they certainly would have done so. With Tazio on board, they would have been a formidable team. Hard to deny the worth of their cause. Each could make a strong argument for return of the schoolhouses and a portion of the lands. There goes the profit from a potential development. That didn't sit well with Wesley and Neil, who were figuring they could snap those schoolhouses up, especially as the county tax base continues to shrink. Wesley said when we've reached the point where some factions start talking about selling off Yellowstone Park, anything is up for grabs. Neil and Wesley thought they could get the whole package by June of next year. They already had the money, and they removed their two greatest obstacles.\"\n\n_\"What matters to humans? Is it always money?\"_ Mrs. Murphy wondered.\n\n_\"Money certainly drove Wesley and Neil,\"_ Tucker answered.\n\nCooper continued, \"Wesley is already turning against Neil, obviously. He says Buddy was out there in the field because he was going to walk to Mount Carmel in costume to scare more people after the last wagon took off. I don't believe it for a minute. Buddy says Wesley offered him another big sum for his land, Buddy refused, and Wesley threatened him. Wesley pulled a gun, Buddy deflected it, then ran out of the schoolhouse. Wesley, shrewdly, called Neil to intercept Buddy. At least that's what I believe. If they killed him, the men figured they could break down Buddy's wife by throwing money at her. After all, being a widow would probably have reduced her income. Wesley swears that when he heard the shots, he fled. He knew Neil had lost it. Once Neil can talk, we'll have his side of the story.\"\n\n\"Idolatrous capital,\" Harry said, then noticed their blank stares. \"Mother used to say that. People worship capital, money. Things are more important than people.\"\n\n\"That's nothing new,\" Fair quietly said.\n\n\"Tazio and Harry really were the last impediments. Tazio might not have been, except they killed Hester. Tazio picked up the torch,\" Cooper said. \"As for Harry, Wesley knew she was on the trail and getting way too close.\"\n\nHarry said, \"Taz's okay, by the way. A knot on the head and the doctors think she might have had a slight concussion, so she has to be a little careful.\"\n\n\"Barry will be fine, too,\" Coop reported. \"He said he heard the dirt bike, heard it stop somewhere behind him, but not close to him. He didn't hear the footsteps, no surprise given all the screams. You know what I think about? They might have gotten away with it. They killed and dressed both Josh and Hester. Wesley says they did it to create confusion, make us think the killer was mad, nuts. But you know, I think they both got caught up in it, the theater of it. It became a high for them. Neil, getting arrogant, figured he could pull this off without too much trouble. Wesley says he tried to talk him out of it, but I doubt he tried too hard.\"\n\nCooper exhaled. \"Anyway, I expect they've brought about what they most feared. Once all this is out in the open, the public will clamor for Random Row to become a museum. The lynchpin will be Buddy. Will he give up the forty acres that used to belong to the Monacans?\"\n\n\"Buddy usually does the right thing.\" Harry stroked Mrs. Murphy's soft fur. \"I do think the animals should get something.\"\n\n\"I'll work on it.\" Cooper smiled.\n\n\"You know what surprised me the most?\" Harry suddenly said. \"Neil and Wesley are Lutherans. Really.\"\n\nFair burst out laughing. \"Honey, I'm certain Lutherans have committed heinous crimes in the past.\"\n\nThe humans laughed, then Pewter piped up, _\"Murphy, see if you can get them to understand you don't want a medal. Tuna! We want tuna and catnip.\"_\n\n_\"Or bones,\"_ Tucker remarked.\n\n_\"Tuna is better. You can eat tuna. Bones you just chew.\"_\n\n_\"Pewter, who says I'm going to share?\"_ Mrs. Murphy replied.\n\n_\"Of course you will, because I am your very best friend.\"_ Pewter purred this with uncommon sweetness.\n\nThe tiger cat and corgi just looked at each other.\n\nFinally, Mrs. Murphy said, _\"If you promise to help Mom and Tazio fulfill Hester's dream, save the schoolhouses and the land, I will share lots.\"_\n\nWithout hesitation, Pewter passionately agreed. _\"Of course I will. You know very well Mom can't do anything without me.\"_\n\n#\n\nDear Reader,\n\nThe Monacan Indian Nation celebrates an annual powwow usually during high spring. Kathleen King, a friend made way-back-when through basset hunting, often attends. The dancing, food, storytelling, paintings, carvings, baskets, and more underscore this old culture that has survived despite all.\n\nThanks to Kathleen and her research, I became interested. The federal government's refusal to acknowledge the Virginia tribes took me by surprise. Then I got mad. Perhaps you will, too.\n\nMany nations on earth have subjugated others at one time in their history and often with astonishing brutality. Each government, each nation, has to study its errors and learn from it all, and redress the sorrows where possible.\n\nAs Americans we are in a better position than most to do this. We live and work in the light of the Declaration of Independence, the Constitution, and the Bill of Rights.\n\nLet's live up to them.\n\nAlways and Ever,\n\n#\n\nDear Reader,\n\nWhile it is true that Kathleen King provided much information, I really did all the work.\n\nYours,\n\n#\n\nDear Reader,\n\nLiar!\n\nThe true artist,\n\n#\n\nTo Whom It May Concern,\n\nCats are crazy.\n\nYours,\n\nFor  \nKathleen King  \nWho knows the past is always with us.\n_Books by Rita Mae Brown & Sneaky Pie Brown_\n\nWISH YOU WERE HERE \u2022 REST IN PIECES \u2022 MURDER AT MONTICELLO  \nPAY DIRT \u2022 MURDER, SHE MEOWED \u2022 MURDER ON THE PROWL  \nCAT ON THE SCENT  \nSNEAKY PIE'S COOKBOOK FOR MYSTERY LOVERS  \nPAWING THROUGH THE PAST \u2022 CLAWS AND EFFECT  \nCATCH AS CAT CAN \u2022 THE TAIL OF THE TIP-OFF  \nWHISKER OF EVIL \u2022 CAT'S EYEWITNESS \u2022 SOUR PUSS  \nPUSS 'N CAHOOTS \u2022 THE PURRFECT MURDER \u2022 SANTA CLAWED  \nCAT OF THE CENTURY \u2022 HISS OF DEATH \u2022 THE BIG CAT NAP  \nSNEAKY PIE FOR PRESIDENT \u2022 THE LITTER OF THE LAW\n\n_Books by Rita Mae Brown featuring \"Sister\" Jane Arnold_\n\nOUTFOXED \u2022 HOTSPUR \u2022 FULL CRY \u2022 THE HUNT BALL  \nTHE HOUNDS AND THE FURY \u2022 THE TELL-TALE HORSE  \nHOUNDED TO DEATH \u2022 FOX TRACKS\n\n_The Mags Rogers Books_\n\nMURDER UNLEASHED \u2022 A NOSE FOR JUSTICE\n\n_Books by Rita Mae Brown_\n\nANIMAL MAGNETISM: MY LIFE WITH CREATURES GREAT AND SMALL  \nTHE HAND THAT CRADLES THE ROCK  \nSONGS TO A HANDSOME WOMAN \u2022 THE PLAIN BROWN RAPPER  \nRUBYFRUIT JUNGLE \u2022 IN HER DAY \u2022 SIX OF ONE  \nSOUTHERN DISCOMFORT \u2022 SUDDEN DEATH \u2022 HIGH HEARTS  \nSTARTED FROM SCRATCH: A DIFFERENT KIND OF WRITER'S MANUAL  \nBINGO \u2022 VENUS ENVY  \nDOLLEY: A NOVEL OF DOLLEY MADISON IN LOVE AND WAR  \nRIDING SHOTGUN \u2022 RITA WILL: MEMOIR OF A LITERARY RABBLE-ROUSER  \nLOOSE LIPS \u2022 ALMA MATER \u2022 SAND CASTLE\n\n# About the Authors\n\nRITA MAE BROWN has written many bestsellers and received two Emmy nominations. In addition to the Mrs. Murphy series, she has authored _A Nose for Justice_ and _Murder Unleashed_ , the first two mysteries in a new dog series, and the Sister Jane foxhunting series, as well as many other acclaimed books. She and Sneaky Pie live with several other rescued animals.\n\nSNEAKY PIE BROWN, a tiger cat rescue, has written many mysteries\u2014witness the list at the front of this novel. Having to share credit with the above-named human is a small irritant, but she manages it. Anything is better than typing, which is what \"Big Brown\" does for the series. Sneaky calls her human that name behind her back, after the wonderful Thoroughbred racehorse. As her human is rather small, it brings giggles among the other animals. Sneaky's main character\u2014Mrs. Murphy, a tiger cat\u2014is a bit sweeter than Miss Pie, who can be caustic.\n\nwww.ritamaebrown.com\n\n#\n\nTurn the page for an exclusive sneak peek at Rita Mae Brown and Sneaky Pie Brown's next Mrs. Murphy Mystery\n\n_**Nine Lives to Die**_\n\nAvailable in hardcover and eBook from Bantam\n\n#\n\n_\"G_ in!\"\n\n\"I don't believe it.\" Susan Tucker stared at the cards that her childhood friend, Harriett Haristeen, \"Harry,\" had smacked down.\n\nThe six other women in the room, all slack-jawed, came over to view the winning card.\n\n\"Well, Susan, she did,\" BoomBoom Craycroft, another childhood friend, said and smiled.\n\n\"Harry can't play cards worth squat,\" Susan complained.\n\n\"Well, I did tonight.\" Harry beamed. \"Susan, mark your calendar, Tuesday, December third, my best friend Harry knocked the stuffing out of me at gin.\"\n\nJessica Hexham was petite and middle-aged, well dressed even though the evening was relaxed. She murmured, \"Maybe something less exuberant for the calendar\u2014just a red-letter day?\"\n\n\"Do you remember when Miss Donleavey lectured us about red-letter days on the ancient Roman calendar?\" Susan rolled her eyes.\n\nBoomBoom, Susan, and Harry had been in the same class at old Crozet High School. While the buildings still stood, students now attended Western Albemarle High School, a large school consolidating former small community schools. Jessica Hexham, Alicia Palmer, Charlene Vavilov, Arden Higham had not. Jessica had attended Miss Porter's; Alicia, Orange High School; Arden, Buckingham High; and Charlene, older than the others, attended St. Catherine's in Richmond.\n\nWith the exception of Jessica, all were central Virginia natives. Jessica, born and raised in Concord, Massachusetts, often found them amusing while contradictory at times, and they were reliably solid friends.\n\n_\"Alea jacta est,\"_ Susan pronounced with emphasis.\n\nHarry translated. \"The die is cast. Said when Julius Caesar crossed the Rubicon in 49 B.C. at the head of the Thirteenth Legion. He knew civil war would follow.\"\n\n\"Talk about a red-letter day,\" said Boom Boom.\n\n\"Isn't it something, though, how a device thousands of years old still works, I mean, a red-letter day? God bless Miss Donleavey. She taught us well.\"\n\nJessica also recalled her Latin teacher at the expensive private school, perhaps less fondly. \"I would never bless Miss Greely.\"\n\nThe others laughed.\n\n\"Charlene, bet you took Latin at St. Catherine's,\" Alicia wondered.\n\n\"You couldn't go to college without two years of it,\" said Charlene. \"I took four. It's helped me more than I could know when I hated memorizing those conjugations.\" She laughed.\n\n\"Funny, isn't it?\" the uncommonly beautiful Alicia said. \"What we use? What we remember?\"\n\n\"What I remember, apart from _amo, amas, amat_ , was Miss Donleavey's mysterious disappearance. Never found her.\" Harry picked up the cards to shuffle.\n\nSusan reached across the card table, placing her hand on Harry's forearm. \"Don't you dare.\"\n\n\"Huh?\" Harry blinked.\n\n\"I'll shuffle.\"\n\n\"Are you calling me a cheat?\" Harry's voice rose.\n\n\"No, but you won the last hand, so it's my turn to shuffle. Plus, what if you have a hot hand?\" Susan used the gambling term.\n\n\"I'd better tell that to my husband.\"\n\nThis evoked more laughter.\n\nThe lights flickered, once, twice, then no light.\n\n\"Dammit,\" Susan cursed the dark. \"Stay put, ladies. I'll get the candles.\"\n\n\"You need my little flashlight.\" Harry reached into her pocket, pulling out a two-and-a-half-inch LED flashlight made in China.\n\nSusan pressed the button. \"Wow.\"\n\n\"What else do you have in your pocket?\" Jessica asked.\n\n\"One pocketknife,\" BoomBoom answered for Harry. \"She always has a pocketknife and a little money.\"\n\n\"The emphasis is on _little_ ,\" said Harry, emptying her pockets onto the card table as Susan returned with candles.\n\n\"Let me help you.\" Thanks to the tiny LED flashlight, Alicia could see. She reached for some candles.\n\n\"There's a hurricane glass lamp. Well, here, let's do it together. Girls, we'll be right back.\"\n\nTrue to her word, Susan and Alicia returned to the living room with small brass candleholders, which they placed about and lit. The large hurricane candleholder glowed on the card table. All held six- to eight-inch candles.\n\nSusan noticed the small pile of debris.\n\n\"Harry, what's your stuff doing on the card table?\"\n\n\"Jessica wanted to know what was in my pocket.\"\n\n\"In the dark?\" Susan questioned.\n\n\"We knew there'd be light,\" Harry shot back.\n\nJessica dutifully investigated the contents: one Case pocketknife, a folded cotton handkerchief, twenty-two dollars in small bills, one dog cookie.\n\nHarry pointed out the cookie. \"Never know when I might get hungry.\"\n\nThe ladies laughed again as Alicia walked to the large triple-sash windows. \"Girls, we're in for it.\"\n\n\"No kidding?\" Harry hurried over, as did the others.\n\n\"The storm's early.\" BoomBoom, like all country people, paid intense attention to the weather.\n\n\"We have a little time before we need to worry about the roads,\" Harry confidently predicted. \"Everyone has four-wheel drive, right?\"\n\n\"If not, I'm happy to sell you one.\" Charlene smiled. She and her husband, Pete, owned the Ford dealership.\n\n\"We're good,\" the others replied.\n\n\"Well, let's not play cards by candlelight. Ladies, I whipped up vegetable hors d'oeuvres, and they're really tasty, if I do say so myself. I can't eat them all. You have to help me. Harry, use your flashlight again and let's bring the food out from the kitchen. BoomBoom, you know where the bar is. Give the girls what they want.\"\n\nBoomBoom picked up a candle as she glided to the well-stocked bar. Susan's husband, Ned, was a delegate to the state legislature in Richmond, and the couple entertained frequently. In this part of the world, good liquor was considered an essential by any host and hostess. Southerners did drink wine, but many still preferred a high-octane bourbon or scotch, and then there were the legions of vodka drinkers who believed it didn't linger on their breath.\n\nOnce settled in the living room, comfortable in decidedly not-modern d\u00e9cor, Jessica, curious, asked, \"So what did happen to your Latin teacher?\"\n\n\"Nobody knows.\" BoomBoom shrugged. \"She disappeared after a Friday-night football game. Her car was in the parking lot. Monday, she didn't come to school.\"\n\n\"We played the Louisa Dragons that night,\" Harry recalled. \"Good game. Miss Donleavey never missed a football game.\"\n\n\"She dated the coach, Mr. Toth,\" Susan filled in. \"Handsome, handsome, handsome.\"\n\n\"Coach Toth? That Toth?\" Jessica asked. \"Silver Linings?\" She mentioned a youth organization the coach supported, as did all the husbands of the women in the room. Apart from helping young men, business leaders and former athletes ran Silver Linings. To belong was beneficial to one's career.\n\n\"Jessica, this must be irritating, being in the middle of a bunch of old friends.\" Harry handed her a napkin.\n\n\"No, it's fascinating. A vanished Latin teacher.\"\n\n\"You know the stereotype of the old-maid Latin teacher? Well, not Miss Donleavey. She was voluptuous, raven-haired, so pretty,\" BoomBoom noted, herself voluptuous.\n\n\"Suspects?\" Jessica's eyebrows raised.\n\nMiranda answered. \"At first, people thought it might have been a rival of the coach's. Men were crazy for her.\"\n\nSusan added, \"Lots of men were questioned. Everyone had an alibi.\"\n\n\"Anyone else?\" Jessica persisted.\n\n\"Esther Mercier. Hated Miss Donleavey, just hated her.\" Harry bit into a carrot incised with a tiny trench filled with rich cream cheese.\n\n\"In love with Coach Toth.\" BoomBoom filled in facts. \"An attractive enough woman, but not in Miss Donleavey's league.\"\n\n\"What was her first name?\" Jessica asked. \"Miss Donleavey?\"\n\n\"Uh, Margaret. It's funny, but I still have a hard time calling my teachers by their first names. I mean, Coach Toth is always Coach Toth.\" Susan smiled. \"And eventually he did marry Miss Mercier, one of the math teachers.\"\n\n\"You'd think someone would have known something. Crozet is still a small place,\" Charlene wondered.\n\n\"If they did, no one noticed. Crozet, like any place anywhere in the world, is full of secrets which people take to their graves,\" Harry remarked. \"Miss Donleavey's kin, all older, are gone. It's one of those persistent small-town mysteries.\"\n\n\"Well, people don't just disappear off the face of the earth.\" Alicia twirled a fresh bit of broccoli.\n\n\"The Black Dahlia,\" BoomBoom countered.\n\n\"You're right, to a degree,\" said Alicia. \"'Course, I wasn't in Hollywood then. And she didn't disappear, Sweetie. They never found the killer.\"\n\n\"You're right.\" BoomBoom got up and walked over to the window, nose almost on the windowpane. \"It's really coming down now. We'd all better head home.\"\n\n\"Let me help you clean up,\" Harry offered.\n\n\"A tray of vegetables and a couple of glasses? Anyway, no power, no water. Go on. If your cellphones don't work you can still text if you have a Droid.\"\n\nArden said, \"I hope the Silver Linings fund-raiser isn't canceled.\"\n\n\"We'll cross our fingers.\" Charlene crossed hers.\n\n\u2022 \u2022 \u2022\n\nAfter a long, careful drive, Harry slowly finally drove down her long farm driveway, windshield wipers flipping as fast as they could. She pulled in front of the old white frame farmhouse, cut the motor, the lights with it.\n\nGolden candlelight cascaded over the snow. The frosted windows glowed pale gold, the wavy imperfections of the hand-blown glass all the more obvious with the candles behind her.\n\n_\"Mom's home.\"_ Inside the house, Tucker the corgi barked joyfully.\n\nPewter flopped on the kitchen table, lifted her head. _\"About time.\"_\n\nMrs. Murphy, the tiger cat, walked alongside Fair, Harry's husband, as he opened the kitchen door to the porch. He carried a huge flashlight, which he focused on the path to the back porch, screened-in in summer, glassed-in in winter.\n\n\"Honey, I'm glad you're home.\" He stepped into the snow.\n\n\"Fair, get back inside. I can see.\"\n\nHe didn't, of course, kissing her as she hurried onto the porch, Tucker and Mrs. Murphy at her feet.\n\nPewter considered a welcoming meow when Harry walked into the kitchen, then thought better of it. It's never wise to indulge humans.\n\nHarry stamped her feet again. \"Boy, it's really snowing.\"\n\n\"I'll get the generator going. Just got home myself about ten minutes ago. Buried in paperwork today.\"\n\nHanging up her coat on one of the Shaker pegs inside by the kitchen door, she shook her head free of snow. \"Honey, do you have your Droid?\"\n\n\"Yeah, sure.\"\n\n\"May I borrow it?\"\n\nFair retrieved the device, which he'd placed on a kitchen counter when he walked into the kitchen, handing it to his wife.\n\nHarry texted Susan, \"I won.\"\n\nShe then recounted her small triumph with her husband, who celebrated with her.\n\nTucker also laughed, for she knew how frequently Harry lost at cards.\n\n_\"I shredded a pack of cards once,\"_ Pewter crowed. _\"Good cards, they had Susan's initials on them.\"_\n\n_\"We know,\"_ both Mrs. Murphy and Tucker replied. _\"Thanks to you, the girls won't play cards here,\"_ Tucker added.\n\n_\"Who cares?\"_ Pewter saucily called down from the table.\n\n_\"I do,\"_ the intelligent corgi said. _\"The girls always drop food.\"_\n\n_\"Well\u2014\"_ Pewter had no comeback for that.\n\nFire blazing, Harry and Fair cuddled on the sofa.\n\n\"First big snow of the season. Even though it creates all manner of problems, I do love it.\" Fair smiled.\n\n\"Any horses at the clinic?\" Harry asked. Fair was an equine vet.\n\n\"No, which is why I can enjoy the snow. I don't have to drive back there until the roads are plowed.\" He glanced out the window. \"They'll have their work cut out for them.\"\n\n\"At least the refrigerator is running with the generator, and the stove is on propane gas.\"\n\nFair pulled Harry closer. \"I like the candlelight.\"\n\n\"Me, too, and I like the silence, especially when the fridge cuts off. Say, we got to talking about Miss Donleavey.\"\n\n\"Haven't thought of her in years. That was a good game the night she disappeared. We creamed Louisa. And someone got away with murder.\"\n\n\"Maybe.\" Harry's voice, light, lifted a bit higher.\n\n\"Oh, honey, she's gone forever.\"\n\nBut she wasn't.\n\n# _What's next on  \nyour reading list?_\n\n[Discover your next  \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9780345538574\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nGILLES DELEUZE\n\nSPINOZA\n\nET LE PROBL\u00c8ME\n\nDE L'EXPRESSION\n\n_ARGUMENTS_\n\nLES \u00c9DITIONS DE MINUIT\n\u00a9 1968 by LES \u00c9DITIONS DE MINUIT pour l'\u00e9dition papier\n\n\u00a9 2013 by LES \u00c9DITIONS DE MINUIT pour la pr\u00e9sente \u00e9dition \u00e9lectronique\n\nwww.leseditionsdeminuit.fr\n\nISBN 9782707330222\n\n# Sommaire\n\nAVANT-PROPOS\n\nINTRODUCTION - R\u00d4LE ET IMPORTANCE DE L'EXPRESSION\n\nPREMI\u00c8RE PARTIE - LES TRIADES DE LA SUBSTANCE\n\nCHAPITRE PREMIER - DISTINCTION NUM\u00c9RIQUE ET DISTINCTION R\u00c9ELLE\n\nCHAPITRE II - L'ATTRIBUT COMME EXPRESSION\n\nCHAPITRE III - ATTRIBUTS ET NOMS DIVINS\n\nCHAPITRE IV - L'ABSOLU\n\nCHAPITRE V - LA PUISSANCE\n\nDEUXI\u00c8ME PARTIE - LE PARALL\u00c9LISME ET L'IMMANENCE\n\nCHAPITRE VI - L'EXPRESSION DANS LE PARALL\u00c9LISME\n\nCHAPITRE VII - LES DEUX PUISSANCES ET L'ID\u00c9E DE DIEU\n\nCHAPITRE VIII - EXPRESSION ET ID\u00c9E\n\nCHAPITRE IX - L'INAD\u00c9QUAT\n\nCHAPITRE X - SPINOZA CONTRE DESCARTES\n\nCHAPITRE XI - L'IMMANENCE ET LES \u00c9L\u00c9MENTS HISTORIQUES DE L'EXPRESSION\n\nTROISI\u00c8ME PARTIE - TH\u00c9ORIE DU MODE FINI\n\nCHAPITRE XII - L'ESSENCE DE MODE : PASSAGE DE L'INFINI AU FINI\n\nCHAPITRE XIII - L'EXISTENCE DU MODE\n\nCHAPITRE XIV - QU'EST-CE QUE PEUT UN CORPS ?\n\nCHAPITRE XV - LES TROIS ORDRES ET LE PROBL\u00c8ME DU MAL\n\nCHAPITRE XVI - VISION \u00c9THIQUE DU MONDE\n\nCHAPITRE XVII - LES NOTIONS COMMUNES\n\nCHAPITRE XVIII - VERS LE TROISI\u00c8ME GENRE\n\nCHAPITRE XIX - B\u00c9ATITUDE\n\nCONCLUSION - TH\u00c9ORIE DE L'EXPRESSION CHEZ LEIBNIZ ET SPINOZA (L'expressionnisme en philosophie)\n\nAPPENDICE - \u00c9TUDE FORMELLE DU PLAN DE L'\u00c9THIQUE ET DU R\u00d4LE DES SCOLIES DANS LA R\u00c9ALISATION DE CE PLAN : LES DEUX \u00c9THIQUES\n\nINDEX DES NOMS PROPRES\n\nTABLE DES MATI\u00c8RES\n\n# _AVANT-PROPOS_\n\n _Nous d\u00e9signons les \u0153uvres de Spinoza par des abr\u00e9viations_ : CT ( _pour_ Court Trait\u00e9), TRE ( _pour le_ Trait\u00e9 de la r\u00e9forme), PPD ( _pour les_ Principes de la philosophie de Descartes), PM ( _pour les_ Pens\u00e9es m\u00e9taphysiques), TTP ( _pour le_ Trait\u00e9 th\u00e9ologico-politique), \u00c9 ( _pour_ \u00c9thique), TP ( _pour le_ Trait\u00e9 politique).\n\n _Quant aux textes que nous citons : chaque fois que les chiffres sont assez d\u00e9taill\u00e9s et permettent de retrouver facilement le passage dans les \u00e9ditions courantes, nous ne donnons pas d'autres indications. Mais pour les lettres et pour le_ Trait\u00e9 th\u00e9ologico-politique _, nous indiquons la r\u00e9f\u00e9rence \u00e0 l'\u00e9dition Van Vloten et Land, en quatre tomes r\u00e9unis en deux volumes. C'est nous qui soulignons certains passages ou certains mots dans les citations._\n\n _Sauf exceptions, les traductions sont emprunt\u00e9es \u00e0 A. Gu\u00e9rinot pour l'_ \u00c9thique _(\u00e9d. Pelletan), \u00e0 A. Koyr\u00e9 pour le_ Trait\u00e9 de la r\u00e9forme _(\u00e9d. Vrin), \u00e0 C. Appuhn (\u00e9d. Garnier) pour les autres \u0153uvres._\n\n _Ce livre a \u00e9t\u00e9 pr\u00e9sent\u00e9 comme th\u00e8se compl\u00e9mentaire sous le titre \u00ab L'id\u00e9e d'expression dans la philosophie de Spinoza_ \u00bb.\n\n# INTRODUCTION\n\n# R\u00d4LE ET IMPORTANCE DE L'EXPRESSION\n\nDans le premier livre de l' _\u00c9thique_ , l'id\u00e9e d'expression appara\u00eet d\u00e8s la d\u00e9finition 6 : \u00ab Par Dieu j'entends un \u00eatre absolument infini, c'est-\u00e0-dire une substance consistant en une infinit\u00e9 d'attributs, dont chacun _exprime_ une essence \u00e9ternelle et infinie. \u00bb Cette id\u00e9e prend par la suite une importance de plus en plus grande. Elle est reprise dans des contextes vari\u00e9s. Tant\u00f4t Spinoza dit : chaque attribut exprime _une certaine essence_ \u00e9ternelle et infinie, une essence correspondant au genre de l'attribut. Tant\u00f4t : chaque attribut exprime _l'essence_ de la substance, son \u00eatre ou sa r\u00e9alit\u00e9. Tant\u00f4t enfin : chaque attribut exprime l'infinit\u00e9 et la n\u00e9cessit\u00e9 de _l'existence_ substantielle, c'est-\u00e0-dire l'\u00e9ternit\u00e91. Et, sans doute, Spinoza montre bien comment l'on passe d'une formule \u00e0 l'autre. Chaque attribut exprime une essence, mais en tant qu'il exprime en son genre l'essence de la substance ; et l'essence de la substance enveloppant n\u00e9cessairement l'existence, il appartient \u00e0 chaque attribut d'exprimer, avec l'essence de Dieu, son existence \u00e9ternelle2. Il n'en reste pas moins que l'id\u00e9e d'expression r\u00e9sume toutes les difficult\u00e9s concernant l'unit\u00e9 de la substance et la diversit\u00e9 des attributs. La nature expressive des attributs appara\u00eet alors comme un th\u00e8me fondamental dans le premier livre de l' _\u00c9thique_.\n\nLe mode, \u00e0 son tour, est expressif : \u00ab Tout ce qui existe exprime la nature de Dieu, autrement dit son essence, d'une fa\u00e7on certaine et d\u00e9termin\u00e9e \u00bb (c'est-\u00e0-dire sous un mode d\u00e9fini)3. Nous devons donc distinguer un second niveau de l'expression, une sorte d'expression de l'expression. En premier lieu, la substance s'exprime dans ses attributs, et chaque attribut exprime une essence. Mais, en second lieu, les attributs s'expriment \u00e0 leur tour : ils s'expriment dans les modes qui en d\u00e9pendent, et chaque mode exprime une modification. Nous verrons que le premier niveau doit \u00eatre compris comme une v\u00e9ritable constitution, presque une g\u00e9n\u00e9alogie de l'essence de la substance. Le second doit \u00eatre compris comme une v\u00e9ritable production des choses. En effet, Dieu produit une infinit\u00e9 de choses parce que son essence est infinie ; mais parce qu'il a une infinit\u00e9 d'attributs, il produit n\u00e9cessairement ces choses en une infinit\u00e9 de modes dont chacun renvoie \u00e0 l'attribut dans lequel il est contenu4. L'expression n'est pas en elle-m\u00eame une production, mais le devient \u00e0 son second niveau, quand c'est l'attribut qui s'exprime \u00e0 son tour. Inversement, l'expression-production trouve son fondement dans une expression premi\u00e8re. Dieu s'exprime par soi-m\u00eame \u00ab avant \u00bb de s'exprimer dans ses effets ; Dieu s'exprime en constituant par soi la nature naturante, avant de s'exprimer en produisant en soi la nature natur\u00e9e.\n\nLa notion d'expression n'a pas seulement une port\u00e9e ontologique, mais aussi gnos\u00e9ologique. On ne s'en \u00e9tonnera pas, puisque l'id\u00e9e est un mode de la pens\u00e9e : \u00ab Les pens\u00e9es singuli\u00e8res, autrement dit cette pens\u00e9e-ci, ou celle-l\u00e0, sont des modes qui expriment la nature de Dieu d'une fa\u00e7on certaine et d\u00e9termin\u00e9e5. \u00bb Mais ainsi la connaissance devient une esp\u00e8ce de l'expression. La connaissance des choses a le m\u00eame rapport avec la connaissance de Dieu que les choses en elles-m\u00eames, avec Dieu : \u00ab Puisque rien ne peut \u00eatre ni \u00eatre con\u00e7u sans Dieu, il est certain que tous les \u00eatres de la nature _enveloppent et expriment_ le concept de Dieu, en proportion de leur essence et de leur perfection ; il est donc certain que, plus nous connaissons de choses dans la nature, plus grande et plus parfaite est la connaissance de Dieu que nous acqu\u00e9rons6. \u00bb L'id\u00e9e de Dieu s'exprime dans toutes nos id\u00e9es, comme leur source et leur cause, si bien que l'ensemble des id\u00e9es reproduit exactement l'ordre de la nature enti\u00e8re. Et l'id\u00e9e, \u00e0 son tour, exprime l'essence, la nature ou perfection de son objet : la d\u00e9finition ou l'id\u00e9e sont dites exprimer la nature de la chose telle qu'elle est en elle-m\u00eame. Les id\u00e9es sont d'autant plus parfaites qu'elles expriment d'un objet plus de r\u00e9alit\u00e9 ou de perfection ; les id\u00e9es que l'esprit forme \u00ab absolument \u00bb expriment donc l'infinit\u00e97. L'esprit con\u00e7oit les choses sous l'esp\u00e8ce de l'\u00e9ternit\u00e9, mais parce qu'il poss\u00e8de une id\u00e9e qui, sous cette esp\u00e8ce, exprime l'essence du corps8. Il semble que la conception de l'ad\u00e9quat chez Spinoza ne se s\u00e9pare pas d'une telle nature expressive de l'id\u00e9e. D\u00e9j\u00e0 le _Court Trait\u00e9_ \u00e9tait \u00e0 la recherche d'un concept capable de rendre compte de la connaissance, non pas comme d'une op\u00e9ration qui resterait ext\u00e9rieure \u00e0 la chose, mais comme d'une r\u00e9flexion, d'une expression de la chose dans l'esprit. L' _\u00c9thique_ t\u00e9moigne toujours de cette exigence, bien qu'elle l'interpr\u00e8te d'une nouvelle mani\u00e8re. De toutes fa\u00e7ons, il ne suffit pas de dire que le vrai est pr\u00e9sent dans l'id\u00e9e. Nous devons demander encore : qu'est-ce qui est pr\u00e9sent dans l'id\u00e9e vraie ? Qu'est-ce qui s'exprime dans une id\u00e9e vraie, qu'est-ce qu'elle exprime ? Si Spinoza d\u00e9passe la conception cart\u00e9sienne du clair et du distinct, s'il forme sa th\u00e9orie de l'ad\u00e9quat, c'est toujours en fonction de ce probl\u00e8me de l'expression.\n\nLe mot \u00ab exprimer \u00bb a des synonymes. Les textes hollandais du _Court Trait\u00e9_ emploient _uytdrukken-uytbeelden_ (exprimer), mais pr\u00e9f\u00e8rent _vertoonen_ (\u00e0 la fois manifester et d\u00e9montrer) : la chose pensante _s'exprime_ en une infinit\u00e9 d'id\u00e9es correspondant \u00e0 une infinit\u00e9 d'objets ; mais aussi bien l'id\u00e9e d'un corps _manifeste_ Dieu imm\u00e9diatement ; et les attributs _se manifestent_ eux-m\u00eames par eux-m\u00eames9. Dans le _Trait\u00e9 de la r\u00e9forme_ , les attributs manifestent l'essence de Dieu : _ostendere_10. Mais les synonymes ne sont pas le plus important. Plus importants sont les corr\u00e9latifs, qui pr\u00e9cisent, accompagnent l'id\u00e9e d'expression. Ces corr\u00e9latifs sont _explicare_ et _involvere._ Ainsi, la d\u00e9finition n'est pas seulement dite exprimer la nature de la chose d\u00e9finie, mais _l'envelopper_ et _l'expliquer_11. Les attributs n'expriment pas seulement l'essence de la substance, tant\u00f4t ils l'expliquent, tant\u00f4t ils l'enveloppent12. Les modes enveloppent le concept de Dieu en m\u00eame temps qu'ils l'expriment, si bien que les id\u00e9es correspondantes enveloppent elles-m\u00eames l'essence \u00e9ternelle de Dieu13.\n\nExpliquer, c'est d\u00e9velopper. Envelopper, c'est impliquer. Les deux termes pourtant ne sont pas contraires : ils indiquent seulement deux aspects de l'expression. D'une part, l'expression est une explication : d\u00e9veloppement de ce qui s'exprime, manifestation de l'Un dans le multiple (manifestation de la substance dans ses attributs, puis, des attributs dans leurs modes). Mais d'autre part, l'expression multiple enveloppe l'Un. L'Un reste envelopp\u00e9 dans ce qui l'exprime, imprim\u00e9 dans ce qui le d\u00e9veloppe, immanent \u00e0 tout ce qui le manifeste : en ce sens, l'expression est un enveloppement. Entre les deux termes il n'y a pas d'opposition, sauf dans un cas pr\u00e9cis que nous analyserons plus tard, au niveau du mode fini et de ses passions14. Mais, en r\u00e8gle g\u00e9n\u00e9rale, l'expression enveloppe, implique ce qu'elle exprime, en m\u00eame temps qu'elle l'explique et le d\u00e9veloppe.\n\nImplication et explication, enveloppement et d\u00e9veloppement, sont des termes h\u00e9rit\u00e9s d'une longue tradition philosophique, toujours accus\u00e9e de panth\u00e9isme. Pr\u00e9cis\u00e9ment parce que ces concepts ne s'opposent pas, ils renvoient eux-m\u00eames \u00e0 un principe synth\u00e9tique : la _complicatio._ Dans le n\u00e9o-platonisme, il arrive souvent que la _complication_ d\u00e9signe \u00e0 la fois la pr\u00e9sence du multiple dans l'Un et de l'Un dans le multiple. Dieu, c'est la Nature \u00ab complicative \u00bb ; et cette nature explique et implique Dieu, enveloppe et d\u00e9veloppe Dieu. Dieu \u00ab complique \u00bb toute chose, mais toute chose l'explique et l'enveloppe. Cet embo\u00eetement de notions constitue l'expression ; en ce sens il caract\u00e9rise une des formes essentielles du n\u00e9o-platonisme chr\u00e9tien et juif, tel qu'il \u00e9volue durant le Moyen Age et la Renaissance. On a pu dire, de ce point de vue, que l'expression \u00e9tait une cat\u00e9gorie fondamentale de la pens\u00e9e de la Renaissance15. Or, chez Spinoza la Nature comprend tout, contient tout, en m\u00eame temps qu'elle est expliqu\u00e9e et impliqu\u00e9e par chaque chose. Les attributs enveloppent et expliquent la substance, mais celle-ci comprend tous les attributs. Les modes enveloppent et expliquent l'attribut dont ils d\u00e9pendent, mais l'attribut contient toutes les essences de modes correspondantes. Nous devons demander comment Spinoza s'ins\u00e8re dans la tradition expressionniste, dans quelle mesure il en est tributaire et comment il la renouvelle.\n\nCette question est d'autant plus importante que Leibniz lui-m\u00eame fait de l'expression un de ses concepts fondamentaux. Chez Leibniz comme chez Spinoza, l'expression a une port\u00e9e \u00e0 la fois th\u00e9ologique, ontologique et gnos\u00e9ologique. Elle anime la th\u00e9orie de Dieu, des cr\u00e9atures et de la connaissance. Ind\u00e9pendamment l'un de l'autre, les deux philosophes semblent compter sur l'id\u00e9e d'expression pour d\u00e9passer les difficult\u00e9s du cart\u00e9sianisme, pour restaurer une philosophie de la Nature, et m\u00eame pour int\u00e9grer les acquis de Descartes dans des syst\u00e8mes profond\u00e9ment hostiles \u00e0 la vision cart\u00e9sienne du monde. Dans la mesure o\u00f9 l'on peut parler d'un anticart\u00e9sianisme de Leibniz et de Spinoza, cet anticart\u00e9sianisme se fonde sur l'id\u00e9e d'expression.\n\nNous supposons que l'id\u00e9e d'expression est importante, \u00e0 la fois pour la compr\u00e9hension du syst\u00e8me de Spinoza, pour la d\u00e9termination de son rapport avec le syst\u00e8me de Leibniz, pour les origines et la formation des deux syst\u00e8mes. D\u00e8s lors, pourquoi les meilleurs commentateurs n'ont-ils pas tenu compte (ou gu\u00e8re) d'une telle notion dans la philosophie de Spinoza ? Les uns n'en parlent pas du tout. D'autres y attachent une certaine importance, mais indirecte ; ils y voient le synonyme d'un terme plus profond. Expression ne serait qu'une fa\u00e7on de dire \u00ab \u00e9manation \u00bb. Leibniz le sugg\u00e9rait d\u00e9j\u00e0, reprochant \u00e0 Spinoza d'avoir interpr\u00e9t\u00e9 l'expression dans un sens conforme \u00e0 la Kabbale et de l'avoir r\u00e9duite \u00e0 une sorte d'\u00e9manation16. Ou bien exprimer serait un synonyme d' _expliquer._ Les postkantiens semblaient les mieux plac\u00e9s pour reconna\u00eetre dans le spinozisme la pr\u00e9sence d'un mouvement de gen\u00e8se et d'autod\u00e9veloppement dont ils cherchaient partout le signe pr\u00e9curseur. Mais le terme \u00ab expliquer \u00bb les confirme dans l'id\u00e9e que Spinoza n'a pas su concevoir un v\u00e9ritable d\u00e9veloppement de la substance, pas plus qu'il n'a su penser le passage de l'infini dans le fini. La substance spinoziste leur para\u00eet morte : l'expression spinoziste leur parait intellectuelle et abstraite ; les attributs leur paraissent \u00ab attribu\u00e9s \u00bb \u00e0 la substance par un entendement lui-m\u00eame explicatif17. M\u00eame Schelling, quand il \u00e9labore sa philosophie de la manifestation _(Offenbarung)_ , ne se r\u00e9clame pas de Spinoza, mais de Boehme : c'est de Boehme, et non de Spinoza ni m\u00eame de Leibniz, que lui vient l'id\u00e9e d'expression _(Ausdruck)._\n\nOn ne r\u00e9duit pas l'expression \u00e0 une simple explication de l'entendement sans tomber dans un contresens historique. Car expliquer, loin de d\u00e9signer l'op\u00e9ration d'un entendement qui reste ext\u00e9rieur \u00e0 la chose, d\u00e9signe d'abord le d\u00e9veloppement de la chose en elle-m\u00eame et dans la vie. Le couple traditionnel _explicatio-complicatio_ t\u00e9moigne historiquement d'un vitalisme toujours proche du panth\u00e9isme. Loin qu'on puisse comprendre l'expression \u00e0 partir de l'explication, il nous semble au contraire que l'explication, chez Spinoza comme chez ses devanciers, suppose une certaine id\u00e9e de l'expression. Si les attributs renvoient essentiellement \u00e0 un entendement qui les per\u00e7oit ou les comprend, c'est d'abord parce qu'ils expriment l'essence de la substance, et que l'essence infinie n'est pas exprim\u00e9e sans se manifester \u00ab objectivement \u00bb dans l'entendement divin. C'est l'expression qui fonde le rapport \u00e0 l'entendement, non pas l'inverse. Quant \u00e0 l'\u00e9manation, il est certain qu'on en trouvera des traces, non moins que de la participation, chez Spinoza. Pr\u00e9cis\u00e9ment la th\u00e9orie de l'expression et de l'explication, \u00e0 la Renaissance comme au Moyen \u00c2ge, s'est form\u00e9e chez des auteurs fort inspir\u00e9s de n\u00e9o-platonisme. Reste qu'elle avait pour but et pour effet de transformer profond\u00e9ment ce n\u00e9o-platonisme, de lui ouvrir des voies toutes nouvelles, \u00e9loign\u00e9es de celles de l'\u00e9manation, m\u00eame quand les deux th\u00e8mes coexistaient. De l'\u00e9manation aussi, nous dirions donc qu'elle n'est pas apte \u00e0 nous faire comprendre l'id\u00e9e d'expression. Au contraire, c'est l'id\u00e9e d'expression qui peut montrer comment le n\u00e9o-platonisme \u00e9volua jusqu'\u00e0 changer de nature, en particulier comment la cause \u00e9manative tendit de plus en plus \u00e0 devenir une cause immanente.\n\nCertains commentateurs modernes consid\u00e8rent directement l'id\u00e9e d'expression chez Spinoza. Kaufmann y voit un fil pour le \u00ab labyrinthe spinoziste \u00bb, mais insiste sur l'aspect mystique et esth\u00e9tique de la notion prise en g\u00e9n\u00e9ral, ind\u00e9pendamment de l'usage qu'en fait Spinoza18. D'une autre mani\u00e8re, Darbon consacre \u00e0 l'expression une page tr\u00e8s belle, mais d\u00e9clare finalement qu'elle reste inintelligible. \u00ab Pour expliquer l'unit\u00e9 de la substance, Spinoza nous dit seulement que chacun des attributs exprime son essence. Loin de nous \u00e9clairer, l'explication soul\u00e8ve un monde de difficult\u00e9s. D'abord _ce qui est exprim\u00e9_ devrait \u00eatre distinct de _ce qui s'exprime... \u00bb_ , et Darbon conclut : \u00ab Les attributs expriment tous l'essence infinie et \u00e9ternelle de Dieu ; encore ne pouvons-nous distinguer entre _ce qui est exprim\u00e9_ et _ce qui l'exprime._ On comprend que la t\u00e2che du commentateur soit difficile, et que la question des rapports de la substance et des attributs dans le spinozisme ait donn\u00e9 lieu \u00e0 beaucoup d'interpr\u00e9tations diff\u00e9rentes19. \u00bb\n\nSans doute y a-t-il une raison \u00e0 cette situation du commentaire. C'est que l'id\u00e9e d'expression chez Spinoza n'est objet ni de d\u00e9finition ni de d\u00e9monstration, et ne peut pas l'\u00eatre. Elle appara\u00eet dans la d\u00e9finition 6 ; mais elle n'est pas plus d\u00e9finie qu'elle ne sert \u00e0 d\u00e9finir. Elle ne d\u00e9finit ni la substance ni l'attribut, puisque ceux-ci le sont d\u00e9j\u00e0 (3 et 4). Pas davantage Dieu, dont la d\u00e9finition peut se passer de toute r\u00e9f\u00e9rence \u00e0 l'expression. Dans le _Court Trait\u00e9_ comme dans les lettres, Spinoza dit souvent que Dieu est une substance consistant en une infinit\u00e9 d'attributs dont chacun est infini20. Il semble donc que l'id\u00e9e d'expression surgisse seulement comme la d\u00e9termination du rapport dans lequel entrent l'attribut, la substance et l'essence, quand Dieu pour son compte est d\u00e9fini comme une substance consistant en une infinit\u00e9 d'attributs eux-m\u00eames infinis. L'expression ne concerne pas la substance ou l'attribut en g\u00e9n\u00e9ral, dans des conditions ind\u00e9termin\u00e9es. Quand la substance est absolument infinie, quand elle poss\u00e8de une infinit\u00e9 d'attributs, alors, et alors seulement, les attributs sont dits exprimer l'essence, parce que la substance aussi bien s'exprime dans les attributs. Il serait inexact d'invoquer les d\u00e9finitions 3 et 4 pour en d\u00e9duire aussit\u00f4t la nature du rapport entre la substance et l'attribut tel qu'il doit \u00eatre en Dieu, puisque Dieu suffit \u00e0 \u00ab transformer \u00bb ce rapport, l'\u00e9levant \u00e0 l'absolu. _Les d\u00e9finitions 3 et 4 sont seulement nominales ; seule la d\u00e9finition 6 est r\u00e9elle_ et nous dit ce qui s'ensuit pour la substance, l'attribut et l'essence. Mais que signifie \u00ab transformer le rapport \u00bb ? On le comprendra mieux si l'on demande pourquoi l'expression n'est pas davantage objet de d\u00e9monstration.\n\n\u00c0 Tschirnhaus qui s'inqui\u00e8te de la c\u00e9l\u00e8bre proposition 16 (livre I de l' _\u00c9thique_ ), Spinoza fait une importante concession : il y a une diff\u00e9rence certaine entre le d\u00e9veloppement philosophique et la d\u00e9monstration math\u00e9matique21. \u00c0 partir d'une d\u00e9finition, le math\u00e9maticien ne peut conclure ordinairement qu'une seule propri\u00e9t\u00e9 ; pour en conna\u00eetre plusieurs, il doit multiplier les points de vue et rapprocher \u00ab la chose d\u00e9finie d'autres objets \u00bb. La m\u00e9thode g\u00e9om\u00e9trique est donc soumise \u00e0 deux limitations : l'ext\u00e9riorit\u00e9 des points de vue, le caract\u00e8re distributif des propri\u00e9t\u00e9s. Hegel ne disait pas autre chose lorsqu'il soutenait, pensant \u00e0 Spinoza, que la m\u00e9thode g\u00e9om\u00e9trique \u00e9tait inapte \u00e0 comprendre le mouvement organique ou l'autod\u00e9veloppement qui convient seul avec l'absolu. Soit la d\u00e9monstration des trois angles = deux droits, o\u00f9 l'on commence par prolonger la base du triangle. Il est clair que cette base n'est pas comme une plante qui pousserait toute seule : il faut le g\u00e9om\u00e8tre pour la prolonger, le g\u00e9om\u00e8tre encore doit consid\u00e9rer d'un nouveau point de vue le c\u00f4t\u00e9 du triangle auquel il m\u00e8ne une parall\u00e8le, etc. On ne peut pas penser que Spinoza lui-m\u00eame ait ignor\u00e9 ces objections ; ce sont celles de Tschirnhaus.\n\nLa r\u00e9ponse de Spinoza risque de d\u00e9cevoir : quand la m\u00e9thode g\u00e9om\u00e9trique s'applique \u00e0 des \u00eatres r\u00e9els et, \u00e0 plus forte raison, \u00e0 l'\u00eatre absolu, nous avons le moyen de d\u00e9duire \u00e0 la fois plusieurs propri\u00e9t\u00e9s. Sans doute avons-nous l'impression que Spinoza s'accorde ce qui est en question. Mais nous ne sommes d\u00e9\u00e7us que parce que nous confondons des probl\u00e8mes tr\u00e8s divers soulev\u00e9s par la m\u00e9thode. Spinoza demande : Y a-t-il un moyen par lequel des propri\u00e9t\u00e9s, conclues une par une, peuvent \u00eatre consid\u00e9r\u00e9es collectivement, et par lequel des points de vue, pris \u00e0 l'ext\u00e9rieur d'une d\u00e9finition, peuvent \u00eatre situ\u00e9s \u00e0 l'int\u00e9rieur de la chose d\u00e9finie ? Or, dans la _R\u00e9forme de l'entendement_ , Spinoza a montr\u00e9 que les figures en g\u00e9om\u00e9trie pouvaient \u00eatre d\u00e9finies par une cause prochaine ou faire l'objet de d\u00e9finitions g\u00e9n\u00e9tiques22. Le cercle n'est pas seulement le lieu des points situ\u00e9s \u00e0 \u00e9gale distance d'un m\u00eame point appel\u00e9 centre, mais une figure d\u00e9crite par toute ligne dont une extr\u00e9mit\u00e9 est fixe, et l'autre, mobile. De m\u00eame la sph\u00e8re est une figure d\u00e9crite par tout demi-cercle qui tourne autour de son axe. Il est vrai qu'en g\u00e9om\u00e9trie ces causes sont fictives : _fingo ad libitum._ Comme dirait Hegel, mais comme Spinoza le dit aussi, le demi-cercle ne tourne pas tout seul. Mais si ces causes sont fictives ou imagin\u00e9es, c'est dans la mesure o\u00f9 elles n'ont de v\u00e9rit\u00e9 que parce qu'elles sont inf\u00e9r\u00e9es \u00e0 partir de leurs effets. Elles se pr\u00e9sentent comme des moyens, des artifices, des fictions, parce que les figures sont ici des \u00eatres de raison. Il n'en est pas moins vrai que les propri\u00e9t\u00e9s, qui sont r\u00e9ellement conclues une \u00e0 une par le g\u00e9om\u00e8tre, acqui\u00e8rent un \u00eatre collectif par rapport \u00e0 ces causes et au moyen de ces fictions23. Or, dans le cas de l'absolu, il n'y a plus rien de fictif : la cause n'est plus inf\u00e9r\u00e9e de son effet. En affirmant que l'Absolument infini est cause, nous n'affirmons pas, comme pour la rotation du demi-cercle, quelque chose qui ne serait pas contenu dans son concept. Il n'est donc pas besoin de fiction pour que les modes en leur infinit\u00e9 soient assimil\u00e9s \u00e0 des propri\u00e9t\u00e9s collectivement conclues de la d\u00e9finition de la substance, et les attributs, \u00e0 des points de vue int\u00e9rieurs \u00e0 cette substance sur laquelle ils ont prise. D\u00e8s lors, si la philosophie est justiciable des math\u00e9matiques, c'est parce que les math\u00e9matiques trouvent dans la philosophie la suppression de leurs limites ordinaires. La m\u00e9thode g\u00e9om\u00e9trique ne rencontre pas de difficult\u00e9 quand elle s'applique \u00e0 l'absolu ; au contraire, elle trouve le moyen naturel de surmonter les difficult\u00e9s qui grevaient son exercice, tant qu'elle s'appliquait \u00e0 des \u00eatres de raison.\n\nLes attributs sont comme des points de vue sur la substance ; mais, dans l'absolu, les points de vue cessent d'\u00eatre ext\u00e9rieurs, la substance comprend en soi l'infinit\u00e9 de ses propres points de vue. Les modes se d\u00e9duisent de la substance, comme les propri\u00e9t\u00e9s se d\u00e9duisent d'une chose d\u00e9finie ; mais, dans l'absolu, les propri\u00e9t\u00e9s acqui\u00e8rent un \u00eatre collectif infini. Ce n'est plus l'entendement fini qui conclut des propri\u00e9t\u00e9s une par une, qui r\u00e9fl\u00e9chit sur la chose et l'explique en la rapportant \u00e0 d'autres objets. C'est la chose qui s'exprime, c'est elle qui s'explique. Alors les propri\u00e9t\u00e9s toutes ensemble \u00ab tombent sous un entendement infini \u00bb. L'expression n'a donc pas \u00e0 \u00eatre objet de d\u00e9monstration ; c'est elle qui met la d\u00e9monstration dans l'absolu, qui fait de la d\u00e9monstration la _manifestation imm\u00e9diate_ de la substance absolument infinie. Il est impossible de comprendre les attributs sans d\u00e9monstration ; celle-ci est la manifestation de ce qui n'est pas visible, et aussi le regard sous lequel tombe ce qui se manifeste. C'est en ce sens que les d\u00e9monstrations, dit Spinoza, sont des yeux de l'esprit par lesquels nous percevons24.\n\n* * *\n\n1.  Les formules correspondantes sont, dans l' _\u00c9thique_ : 1o) _aeternam et infinitam certam essentiam exprimit_ (I, 10, sc.). 2o) _divinae substantiae essentiam exprimit_ (I, 19 dem.) ; _realitatem sive esse substantiae exprimit_ (I, 10, sc.). 3o) _existentiam exprimunt_ (I, 10, c.). Les trois types de formules se trouvent r\u00e9unis en I, 10, sc. Ce texte comporte \u00e0 cet \u00e9gard des nuances et des glissements extr\u00eamement subtils.\n\n2.  _\u00c9_ , I, 19 et 20, dem.\n\n3.  _\u00c9_ , I, 36, dem. (et 25, cor. : _Modi quibus Dei attributa certo et determinato modo exprimuntur._ )\n\n4.  _\u00c9_ , I, 16, dem.\n\n5.  _\u00c9_ , II, 1, dem.\n\n6.  _TTP_ , ch. 4 (II, p. _136)._\n\n7.  _TRE_ , 108 ( _infinitatem exprimunt_ ).\n\n8.  _\u00c9_ , V, 29, prop. et dem.\n\n9.  Cf. _CT_ , II, ch. 20, 4 ( _uytgedrukt_ ) ; I, second dialogue, 12 ( _vertoonen_ ) ; I, ch. 7, 10 ( _vertoond_ ).\n\n10.  _TRE_ , 76.\n\n11.  _\u00c9_ , I, 8, sc. 2 : ... _Veram uniuscujusque rei definitionem nihil involvere neque exprimere praeter definitae naturam. TRE_ , 95, _Definitio, ut dicatur perfecta, debebit intimam essentiam rei explicare._\n\n12.  _\u00c9_ , I, 19, dem. ; 20, dem.\n\n13.  _\u00c9_ , II, 45 et 46, dem.\n\n14.  Cf. chapitre IX.\n\n15.  Cf. A. KOYR\u00c9, _La Philosophie de Jacob Boehme_ (Vrin, 1929), et surtout _Mystiques, spirituels, alchimistes du XVIe si\u00e8cle allemand_ (Armand Colin, 1947).\n\n16.  Cf. Foucher de CAREIL, _Leibniz, Descartes et Spinoza_ (1862). Parmi les interpr\u00e8tes r\u00e9cents, \u00c9. LASBAX est un de ceux qui poussent le plus loin l'identification de l'expression spinoziste avec une \u00e9manation n\u00e9o-platonicienne : _La Hi\u00e9rarchie dans l'Univers chez Spinoza_ (Vrin, 1919.)\n\n17.  C'est sous l'influence de Hegel que E. Erdmann interpr\u00e8te les attributs spinozistes tant\u00f4t comme des formes de l'entendement, tant\u00f4t comme des formes de la sensibilit\u00e9 ( _Versuch einer wissenschaftlichen Darstellung der neueren Philosophie_ , 1836 ; _Grundriss der Geschichte der Philosophie_ , 1866).\n\n18.  Fritz KAUFMANN, _Spinoza's system as theory of expression_ (Philosophy and phenomenological research, Universit\u00e9 de Buffalo, sept. 1940).\n\n19.  Andr\u00e9 DARBON, _\u00c9tudes spinozistes_ (P.U.F., 1946, pp. 117-118).\n\n20.  _Lettres 2 et 4, \u00e0 Oldenburg_ (III, p. 5 et p. 11). Et _CT_ , I, ch. 2, 1.\n\n21.  _Lettres 82, de Tschirnhaus, et 83, \u00e0 Tschirnhaus._\n\n22.  _TRE_ , 72 et 95.\n\n23.  _TRE_ , 72 : \u00ab Pour former le concept de la sph\u00e8re, je forme arbitrairement la fiction d'une cause, \u00e0 savoir qu'un demi-cercle tourne autour de son centre et que la sph\u00e8re est comme engendr\u00e9e par sa rotation. Cette id\u00e9e est certainement vraie et, bien que nous sachions qu'aucune sph\u00e8re ne fut jamais engendr\u00e9e ainsi dans la nature, c'est l\u00e0 n\u00e9anmoins une perception vraie, et la mani\u00e8re la plus facile de former le concept de la sph\u00e8re. Il faut noter en outre que cette perception affirme que le demi-cercle tourne, affirmation qui serait fausse si elle n'\u00e9tait jointe au concept de la sph\u00e8re... \u00bb\n\n24.  _\u00c9_ , V, 23, sc. _TTP_ , ch. 13 (II, p. 240) : \u00ab Dira-t-on qu'il n'est pas besoin de conna\u00eetre les attributs de Dieu, mais uniquement de croire, simplement et sans d\u00e9monstration ? C'est pure frivolit\u00e9. Car les choses invisibles, et qui sont objets de la seule pens\u00e9e ne peuvent \u00eatre vues par d'autres yeux que les d\u00e9monstrations. Qui donc n'a pas de d\u00e9monstrations ne voit absolument rien de ces choses. \u00bb\n\n# PREMI\u00c8RE PARTIE\n\n# LES TRIADES DE LA SUBSTANCE\n\n## CHAPITRE PREMIER\n\n## DISTINCTION NUM\u00c9RIQUE\n\n## ET DISTINCTION R\u00c9ELLE\n\nL'expression se pr\u00e9sente comme une triade. Nous devons distinguer la substance, les attributs, l'essence. La substance s'exprime, les attributs sont des expressions, l'essence est exprim\u00e9e. L'id\u00e9e d'expression reste inintelligible tant qu'on voit seulement deux termes dans le rapport qu'elle pr\u00e9sente. Nous confondons substance et attribut, attribut et essence, essence et substance, tant que nous ne tenons pas compte de la pr\u00e9sence et de l'interm\u00e9diaire du troisi\u00e8me. La substance et les attributs se distinguent, mais en tant que chaque attribut exprime une certaine essence. L'attribut et l'essence se distinguent, mais en tant que chaque essence est exprim\u00e9e comme essence de la substance et non de l'attribut. L'originalit\u00e9 du concept d'expression se manifeste ici : l'essence, en tant qu'elle existe, n'existe pas hors de l'attribut qui l'exprime ; mais, en tant qu'elle est essence, elle ne se rapporte qu'\u00e0 la substance. Une essence est exprim\u00e9e par chaque attribut mais comme essence de la substance elle-m\u00eame. Les essences infinies se distinguent dans les attributs o\u00f9 elles existent, mais s'identifient dans la substance \u00e0 laquelle elles se rapportent. Nous retrouverons toujours la n\u00e9cessit\u00e9 de distinguer trois termes : la substance qui s'exprime, l'attribut qui l'exprime, l'essence qui est exprim\u00e9e. C'est par les attributs que l'essence est distingu\u00e9e de la substance, mais c'est par l'essence que la substance elle-m\u00eame est distingu\u00e9e des attributs. La triade est telle que chacun de ses termes, en trois syllogismes, est apte \u00e0 servir de moyen par rapport aux deux autres.\n\nL'expression convient avec la substance, en tant que la substance est absolument infinie ; elle convient avec les attributs, en tant qu'ils sont une infinit\u00e9 ; elle convient avec l'essence, en tant que chaque essence est infinie dans un attribut. Il y a donc une nature de l'infini. Merleau-Ponty a bien marqu\u00e9 ce qui nous para\u00eet aujourd'hui le plus difficile \u00e0 comprendre dans les philosophies du XVIIe si\u00e8cle : l'id\u00e9e de l'infini positif comme \u00ab secret du grand rationalisme \u00bb, \u00ab une mani\u00e8re innocente de penser \u00e0 partir de l'infini \u00bb, qui trouve sa perfection dans le spinozisme1. L'innocence, il est vrai, n'exclut pas le travail du concept. Il fallait \u00e0 Spinoza toutes les ressources d'un \u00e9l\u00e9ment conceptuel original pour exposer la puissance et l'actualit\u00e9 de l'infini positif. Si l'id\u00e9e d'expression remplit ce r\u00f4le, c'est dans la mesure o\u00f9 elle porte dans l'infini certaines distinctions qui correspondent \u00e0 ces trois termes : substance, attributs, essence. Quel est le type de distinction dans l'infini ? Quel type de distinction peut-on porter dans l'absolu, dans la nature de Dieu ? Tel est le premier probl\u00e8me pos\u00e9 par l'id\u00e9e d'expression ; il domine le premier livre de l' _\u00c9thique._\n\nD\u00e8s le d\u00e9but de l' _\u00c9thique_ Spinoza demande comment deux choses, au sens le plus g\u00e9n\u00e9ral du mot, peuvent se distinguer, puis comment deux substances, au sens pr\u00e9cis du mot, doivent se distinguer. La premi\u00e8re question pr\u00e9pare la seconde. La r\u00e9ponse \u00e0 cette seconde question semble sans \u00e9quivoque : s'il est vrai que deux choses en g\u00e9n\u00e9ral diff\u00e8rent par les attributs des substances ou bien par les modes, deux substances \u00e0 leur tour ne peuvent pas se distinguer par le mode, mais seulement par l'attribut. Donc il est impossible qu'il y ait deux ou plusieurs substances de m\u00eame attribut2. Que Spinoza prenne ici son point de d\u00e9part dans un domaine cart\u00e9sien n'est pas douteux. Mais ce qu'il accepte de Descartes, ce qu'il en refuse, et surtout ce qu'il accepte pour le retourner contre Descartes, tout cela doit \u00eatre \u00e9valu\u00e9 soigneusement.\n\nQu'il n'existe que des substances et des modes, le mode \u00e9tant en autre chose et la substance en soi, on en trouve le principe explicite chez Descartes3. Et si les modes supposent toujours une substance qu'ils suffisent \u00e0 nous faire conna\u00eetre, c'est par l'interm\u00e9diaire d'un attribut principal qu'ils impliquent et qui constitue l'essence de la substance elle-m\u00eame : ainsi deux ou plusieurs substances se distinguent et sont connues distinctement par leurs attributs principaux4. Descartes en conclut que nous concevons une distinction r\u00e9elle entre deux substances, une distinction modale entre la substance et le mode qui la suppose sans r\u00e9ciprocit\u00e9, une distinction de raison entre la substance et l'attribut sans lequel nous ne pourrions en avoir une connaissance distincte5. L'exclusion, l'implication unilat\u00e9rale et l'abstraction sont les crit\u00e8res correspondants dans l'id\u00e9e, ou plut\u00f4t les donn\u00e9es \u00e9l\u00e9mentaires de la repr\u00e9sentation qui permettent de d\u00e9finir et de reconna\u00eetre ces types de distinction. La d\u00e9termination et l'application de ces types jouent un r\u00f4le essentiel dans le cart\u00e9sianisme. Et sans doute Descartes profitait de l'effort pr\u00e9c\u00e9dent de Suarez pour mettre de l'ordre dans un probl\u00e8me si compliqu\u00e96. Mais l'usage qu'il fait lui-m\u00eame des trois distinctions semble, par sa richesse, comporter encore de nombreuses \u00e9quivoques.\n\nUne premi\u00e8re ambigu\u00eft\u00e9, de l'aveu de Descartes, concerne la distinction de raison, la distinction modale et leur rapport. Elle appara\u00eet d\u00e9j\u00e0 dans l'emploi des mots \u00ab mode \u00bb, \u00ab attribut \u00bb, \u00ab qualit\u00e9 \u00bb. Un attribut quelconque \u00e9tant donn\u00e9, il est qualit\u00e9 parce qu'il qualifie la substance comme telle ou telle, mais il est aussi bien mode en tant qu'il la diversifie7. Quelle est de ce point de vue la situation de l'attribut principal ? Je ne peux s\u00e9parer la substance de cet attribut que par abstraction. Mais aussi je peux distinguer cet attribut de la substance, \u00e0 condition de ne pas en faire quelque chose de subsistant par soi, \u00e0 condition d'en faire seulement la propri\u00e9t\u00e9 que la substance a de changer (c'est-\u00e0-dire d'avoir des figures variables ou des pens\u00e9es diverses). C'est pourquoi Descartes dit que l'extension et la pens\u00e9e peuvent \u00eatre con\u00e7ues distinctement de deux fa\u00e7ons : \u00ab en tant que l'une constitue la nature du corps, et l'autre celle de l'\u00e2me \u00bb ; mais aussi en les distinguant de leurs substances, en les prenant simplement pour des \u00ab modes \u00bb ou des \u00ab d\u00e9pendances \u00bb8. Or, si dans le premier cas les attributs distinguent des substances qu'ils qualifient, il semble bien que, dans le second cas, les modes distinguent des substances de m\u00eame attribut. Ainsi des figures variables renvoient \u00e0 tel ou tel corps r\u00e9ellement distinct des autres ; et les pens\u00e9es diverses, \u00e0 une \u00e2me r\u00e9ellement distincte. L'attribut constitue l'essence de la substance qu'il qualifie, mais n'en constitue pas moins aussi l'essence des modes qu'il rapporte aux substances de m\u00eame attribut. Ce double aspect soul\u00e8ve de grandes difficult\u00e9s dans le cart\u00e9sianisme9. Nous pouvons seulement en retenir la cons\u00e9quence : _qu'il y a des substances de m\u00eame attribut. En d'autres termes, il y a des distinctions num\u00e9riques qui sont en m\u00eame temps r\u00e9elles ou substantielles_.\n\nLa seconde difficult\u00e9 concerne la distinction r\u00e9elle en elle-m\u00eame. Celle-ci, non moins que les autres, est une donn\u00e9e de la repr\u00e9sentation. Deux choses sont r\u00e9ellement distinctes quand on peut concevoir clairement et distinctement l'une en excluant tout ce qui appartient au concept de l'autre. C'est en ce sens que Descartes explique \u00e0 Arnauld que le crit\u00e8re de la distinction r\u00e9elle est seulement l' _id\u00e9e_ comme compl\u00e8te. Il rappelle \u00e0 bon droit qu'il n'a jamais confondu les choses con\u00e7ues comme r\u00e9ellement distinctes avec les choses r\u00e9ellement distingu\u00e9es. Pourtant, le passage des unes aux autres lui para\u00eet n\u00e9cessairement l\u00e9gitime ; ce n'est qu'une question de moment. Il suffit, dans l'ordre des M\u00e9ditations, d'en arriver au Dieu cr\u00e9ateur, pour conclure qu'il manquerait singuli\u00e8rement de v\u00e9racit\u00e9 s'il cr\u00e9ait les choses autrement qu'il ne nous en donne l'id\u00e9e claire et distincte. La distinction r\u00e9elle ne poss\u00e8de pas en soi la raison du distingu\u00e9 ; mais cette raison se trouve fournie par la causalit\u00e9 divine, ext\u00e9rieure et transcendante, qui cr\u00e9e les substances conform\u00e9ment \u00e0 la mani\u00e8re dont nous les concevons comme possibles. L\u00e0 encore, toutes sortes de difficult\u00e9s naissent en rapport avec l'id\u00e9e de cr\u00e9ation. L'ambigu\u00eft\u00e9 principale est dans la d\u00e9finition de la substance : \u00ab une chose qui peut exister par soi-m\u00eame \u00bb10. N'y a-t-il pas contradiction \u00e0 poser l'existence par soi comme n'\u00e9tant en soi qu'une simple possibilit\u00e9 ? Nous pouvons ici retenir une seconde cons\u00e9quence : le Dieu cr\u00e9ateur nous fait passer des substances con\u00e7ues comme r\u00e9ellement distinctes aux substances r\u00e9ellement distingu\u00e9es. _La distinction r\u00e9elle_ , soit entre substances d'attributs diff\u00e9rents, soit entre substances de m\u00eame attribut, _s'accompagne d'une division des choses, c'est-\u00e0-dire d'une distinction num\u00e9rique qui lui correspond._\n\nC'est en fonction de ces deux points que s'organise le d\u00e9but de l' _\u00c9thique._ Spinoza demande : En quoi consiste l'erreur, quand nous posons plusieurs substances de m\u00eame attribut ? Cette erreur, Spinoza la d\u00e9nonce de deux fa\u00e7ons, suivant un proc\u00e9d\u00e9 qui lui est cher. D'abord dans une d\u00e9monstration par l'absurde, puis dans une d\u00e9monstration plus complexe. S'il y avait plusieurs substances de m\u00eame attribut, elles devraient se distinguer par les modes, ce qui est absurde, puisque la substance par nature est ant\u00e9rieure \u00e0 ses modes et ne les implique pas : telle est la voie br\u00e8ve, en I, 5. Mais la d\u00e9monstration positive appara\u00eet plus loin, dans un scolie de 8 : deux substances de m\u00eame attribut seraient seulement distinctes _in numero_ ; or les caract\u00e8res de la distinction num\u00e9rique excluent la possibilit\u00e9 d'en faire une distinction r\u00e9elle ou substantielle.\n\nD'apr\u00e8s ce scolie, une distinction ne serait pas num\u00e9rique si les choses n'avaient pas le m\u00eame concept ou la m\u00eame d\u00e9finition ; mais ces choses ne seraient pas distinctes s'il n'y avait hors de la d\u00e9finition une cause ext\u00e9rieure par laquelle elles existent en tel nombre. Deux ou plusieurs choses num\u00e9riquement distinctes supposent donc autre chose que leur concept. C'est pourquoi des substances ne pourraient \u00eatre num\u00e9riquement distinctes qu'en renvoyant \u00e0 une causalit\u00e9 externe capable de les produire. Or, quand nous affirmons que des substances sont produites, nous avons beaucoup d'id\u00e9es confuses \u00e0 la fois. Nous disons qu'elles ont une cause, mais que nous ne savons pas comment cette cause proc\u00e8de ; nous pr\u00e9tendons avoir de ces substances une id\u00e9e vraie, puisqu'elles sont con\u00e7ues par elles-m\u00eames, mais nous doutons que cette id\u00e9e soit vraie, puisque nous ne savons pas par elles-m\u00eames si elles existent. On retrouve ici la critique de l'\u00e9trange formule cart\u00e9sienne : ce qui _peut_ exister par soi. La causalit\u00e9 externe a un sens, mais seulement \u00e0 l'\u00e9gard des modes existants finis : chaque mode existant renvoie \u00e0 un autre mode, pr\u00e9cis\u00e9ment parce qu'il _ne peut pas_ exister par soi. Quand nous appliquons cette causalit\u00e9 aux substances, nous la faisons jouer hors des conditions qui la l\u00e9gitiment et la d\u00e9terminent. Nous l'affirmons, mais dans le vide, en lui retirant toute d\u00e9termination. Bref, la causalit\u00e9 externe et la distinction num\u00e9rique ont un sort commun : elles s'appliquent aux modes et seulement aux modes.\n\nL'argument du scolie 8 se pr\u00e9sente donc sous la forme suivante : 1o) la distinction num\u00e9rique exige une cause ext\u00e9rieure \u00e0 laquelle elle renvoie ; 2o) or il est impossible d'appliquer une cause ext\u00e9rieure \u00e0 une substance, en raison de la contradiction contenue dans un tel usage du principe de causalit\u00e9 ; 3o) deux ou plusieurs substances ne peuvent donc pas se distinguer _in numero_ , il n'y a pas deux substances de m\u00eame attribut. L'argument des huit premi\u00e8res d\u00e9monstrations n'a pas la m\u00eame structure : 1o) deux ou plusieurs substances ne peuvent pas avoir le m\u00eame attribut, parce qu'elles devraient se distinguer par les modes, ce qui est absurde ; 2o) une substance ne peut donc pas avoir une cause externe, elle ne peut pas \u00eatre produite ou limit\u00e9e par une autre substance, car toutes deux devraient avoir la m\u00eame nature ou le m\u00eame attribut ; 3o) il n'y a donc pas de distinction num\u00e9rique dans une substance de quelque attribut, \u00ab toute substance est n\u00e9cessairement infinie \u00bb11.\n\nTout \u00e0 l'heure, de la nature de la distinction num\u00e9rique, on concluait son impuissance \u00e0 s'appliquer \u00e0 la substance. Maintenant, de la nature de la substance, nous concluons son infinit\u00e9, donc l'impossibilit\u00e9 de lui appliquer des distinctions num\u00e9riques. De toutes fa\u00e7ons, la distinction num\u00e9rique ne distingue jamais des substances, mais seulement des modes enveloppant le m\u00eame attribut. Car le nombre exprime \u00e0 sa fa\u00e7on les caract\u00e8res du mode existant : la composition des parties, la limitation par autre chose de m\u00eame nature, la d\u00e9termination externe. En ce sens il peut aller \u00e0 l'infini. Mais la question est : peut-il \u00eatre port\u00e9 dans l'infini lui-m\u00eame ? Ou, comme dit Spinoza : m\u00eame dans le cas des modes, est-ce de la multitude des parties que nous concluons qu'elles sont une infinit\u00e912 ? Quand nous faisons de la distinction num\u00e9rique une distinction r\u00e9elle ou substantielle, nous la portons dans l'infini, ne serait-ce que pour assurer la conversion devenue n\u00e9cessaire entre l'attribut comme tel et l'infinit\u00e9 de parties finies que nous y distinguons. En sortent de grandes absurdit\u00e9s : \u00ab Si une quantit\u00e9 infinie est mesur\u00e9e en parties \u00e9gales \u00e0 un pied, elle devra consister en une infinit\u00e9 de telles parties ; et de m\u00eame si elle est mesur\u00e9e en parties \u00e9gales \u00e0 un doigt ; et par suite un nombre infini sera douze fois plus grand qu'un autre nombre infini13. \u00bb L'absurdit\u00e9 ne consiste pas, ainsi que le croyait Descartes, \u00e0 hypostasier l'\u00e9tendue comme attribut, mais au contraire \u00e0 la concevoir comme mesurable et compos\u00e9e de parties finies avec lesquelles on pr\u00e9tend la convertir. La physique, ici, vient confirmer les droits de la logique : qu'il n'y ait pas de vide dans la nature signifie seulement que la division des parties n'est pas une distinction r\u00e9elle. La distinction num\u00e9rique est une division, mais la division n'a lieu que dans le mode, seul le mode est divis\u00e914.\n\n_Il n'y a pas plusieurs substances de m\u00eame attribut._ D'o\u00f9 l'on conclut, du point de vue de la relation, qu'une substance n'est pas produite par une autre ; du point de vue de la modalit\u00e9, qu'il appartient \u00e0 la nature de la substance d'exister ; du point de vue de la qualit\u00e9, que toute substance est n\u00e9cessairement infinie15. Mais ces r\u00e9sultats sont comme envelopp\u00e9s dans l'argument de la distinction num\u00e9rique. C'est lui qui nous ram\u00e8ne au point de d\u00e9part : \u00ab Il n'existe qu'une seule substance de m\u00eame attribut16. \u00bb Or, \u00e0 partir de la proposition 9, il semble que Spinoza change d'objet. Il s'agit de d\u00e9montrer, non plus qu'il y a seulement une substance par attribut, mais qu'il y a seulement une substance pour tous les attributs. L'encha\u00eenement des deux th\u00e8mes semble difficile \u00e0 saisir. Car, dans cette nouvelle perspective, quelle port\u00e9e faut-il accorder aux huit premi\u00e8res propositions ? Le probl\u00e8me gagne en clart\u00e9 si nous consid\u00e9rons que, pour passer d'un th\u00e8me \u00e0 l'autre, il suffit d'op\u00e9rer ce qu'on appelle en logique la conversion d'une universelle n\u00e9gative. La distinction num\u00e9rique n'est jamais r\u00e9elle ; r\u00e9ciproquement, la distinction r\u00e9elle n'est jamais num\u00e9rique. L'argument de Spinoza devient le suivant : les attributs sont r\u00e9ellement distincts ; or la distinction r\u00e9elle n'est pas num\u00e9rique ; donc il n'y a qu'une substance pour tous les attributs.\n\nSpinoza dit que les attributs sont \u00ab con\u00e7us comme r\u00e9ellement distincts \u00bb17. Dans cette formule on ne verra pas un usage affaibli de la distinction r\u00e9elle. Spinoza ne sugg\u00e8re pas que les attributs sont autres qu'on ne les con\u00e7oit, ni qu'ils soient de simples conceptions qu'on se fait de la substance. Pas davantage on ne croira que Spinoza fasse de la distinction r\u00e9elle un usage seulement hypoth\u00e9tique ou pol\u00e9mique18. La distinction r\u00e9elle, au sens le plus strict, est toujours une donn\u00e9e de la repr\u00e9sentation : deux choses sont r\u00e9ellement distinctes lorsqu'elles sont _con\u00e7ues_ comme telles, c'est-\u00e0-dire \u00ab l'une sans le secours de l'autre \u00bb, de telle mani\u00e8re que l'on _con\u00e7oive_ l'une en niant tout ce qui appartient au _concept_ de l'autre. \u00c0 cet \u00e9gard Spinoza ne diff\u00e8re nullement de Descartes : il en accepte le crit\u00e8re et la d\u00e9finition. Le seul probl\u00e8me est de savoir si la distinction r\u00e9elle ainsi comprise s'accompagne ou non d'une division dans les choses. Chez Descartes, seule l'hypoth\u00e8se d'un Dieu cr\u00e9ateur fondait cette concomitance. Suivant Spinoza, on ne fera correspondre une division \u00e0 la distinction r\u00e9elle qu'en faisant de celle-ci une distinction num\u00e9rique au moins possible, donc en la confondant d\u00e9j\u00e0 avec la distinction modale. Or il est impossible que la distinction r\u00e9elle soit num\u00e9rique ou modale.\n\nQuand on demande \u00e0 Spinoza comment il arrive \u00e0 l'id\u00e9e d'une seule substance pour tous les attributs, il rappelle qu'il a propos\u00e9 deux arguments : plus un \u00eatre a de r\u00e9alit\u00e9, plus il faut lui reconna\u00eetre d'attributs ; plus on reconna\u00eet d'attributs \u00e0 un \u00eatre, plus il faut lui accorder l'existence19. Or, aucun de ces arguments ne serait suffisant s'il n'\u00e9tait garanti par l'analyse de la distinction r\u00e9elle. Seule cette analyse, en effet, montre qu'il est _possible_ d'accorder tous les attributs \u00e0 un \u00eatre, donc de passer de l'infinit\u00e9 de chaque attribut \u00e0 l'absoluit\u00e9 d'un \u00eatre qui les poss\u00e8de tous. Et ce passage, \u00e9tant possible ou n'impliquant pas contradiction, se r\u00e9v\u00e8le n\u00e9cessaire, selon la preuve de l'existence de Dieu. Bien plus, c'est encore l'argument de la distinction r\u00e9elle qui montre que _tous_ les attributs sont une infinit\u00e9. Car nous ne pourrions pas passer par l'interm\u00e9diaire de trois ou quatre attributs sans r\u00e9introduire dans l'absolu cette m\u00eame distinction num\u00e9rique que nous venons d'exclure de l'infini20.\n\nSi l'on divisait la substance conform\u00e9ment aux attributs, il faudrait la traiter comme un genre, et les attributs comme des diff\u00e9rences sp\u00e9cifiques. La substance serait pos\u00e9e comme un genre qui ne nous ferait rien conna\u00eetre en particulier ; alors elle serait distincte des attributs, comme le genre de ses diff\u00e9rences, et les attributs seraient distincts des substances correspondantes, comme les diff\u00e9rences sp\u00e9cifiques et les esp\u00e8ces elles-m\u00eames. C'est ainsi qu'en faisant de la distinction r\u00e9elle entre attributs une distinction num\u00e9rique entre substances, on porte de simples _distinctions de raison_ dans la r\u00e9alit\u00e9 substantielle. Il ne peut y avoir de n\u00e9cessit\u00e9 d'exister pour une substance de m\u00eame \u00ab esp\u00e8ce \u00bb que l'attribut ; une diff\u00e9rence sp\u00e9cifique ne d\u00e9termine que l'existence possible d'objets qui lui correspondent dans le genre. Voil\u00e0 toujours la substance r\u00e9duite \u00e0 une simple possibilit\u00e9 d'exister, l'attribut n'\u00e9tant que _l'indication, le signe_ d'une telle existence possible. La premi\u00e8re critique \u00e0 laquelle Spinoza soumet la notion de signe dans l' _\u00c9thique_ appara\u00eet pr\u00e9cis\u00e9ment \u00e0 propos de la distinction r\u00e9elle21. La distinction r\u00e9elle entre attributs n'est pas plus \u00ab signe \u00bb d'une diversit\u00e9 de substances que chaque attribut n'est le caract\u00e8re sp\u00e9cifique d'une substance qui lui correspondrait ou pourrait lui correspondre. Ni la substance n'est genre, ni les attributs ne sont des diff\u00e9rences, ni les substances qualifi\u00e9es ne sont des esp\u00e8ces22. Sont \u00e9galement condamn\u00e9es chez Spinoza la pens\u00e9e qui proc\u00e8de par genre et diff\u00e9rence, la pens\u00e9e qui proc\u00e8de par signes.\n\nDans un livre o\u00f9 il d\u00e9fend Descartes contre Spinoza, R\u00e9gis invoque l'existence de deux sortes d'attributs, les uns \u00ab sp\u00e9cifiques \u00bb, qui distinguent les substances d'esp\u00e8ce diff\u00e9rente, les autres \u00ab num\u00e9riques \u00bb, qui distinguent des substances de m\u00eame esp\u00e8ce23. Mais c'est pr\u00e9cis\u00e9ment ce que Spinoza reproche au cart\u00e9sianisme. Selon Spinoza, l'attribut n'est jamais sp\u00e9cifique ni num\u00e9rique. Il semble que nous puissions ainsi r\u00e9sumer la th\u00e8se de Spinoza : 1o) quand nous posons plusieurs substances de m\u00eame attribut, nous faisons de la distinction num\u00e9rique une distinction r\u00e9elle, mais alors nous confondons la distinction r\u00e9elle et la distinction modale, nous traitons les modes comme des substances ; 2o) et quand nous posons autant de substances qu'il y a d'attributs diff\u00e9rents, nous faisons de la distinction r\u00e9elle une distinction num\u00e9rique, nous confondons la distinction r\u00e9elle non seulement avec une distinction modale mais encore avec des distinctions de raison.\n\nDans ce contexte, _il semble difficile de consid\u00e9rer que les huit premi\u00e8res propositions n'aient qu'un sens hypoth\u00e9tique._ On fait parfois comme si Spinoza commen\u00e7ait par raisonner dans une hypoth\u00e8se qui n'\u00e9tait pas la sienne, comme s'il partait d'une hypoth\u00e8se qu'il avait l'intention de r\u00e9futer. On laisse ainsi \u00e9chapper le sens cat\u00e9gorique des huit premi\u00e8res propositions. Il n'y a pas plusieurs substances de m\u00eame attribut, la distinction num\u00e9rique n'est pas r\u00e9elle : nous ne nous trouvons pas devant une hypoth\u00e8se provisoire, valable tant qu'on n'a pas encore d\u00e9couvert la substance absolument infinie ; au contraire nous sommes en pr\u00e9sence d'une gen\u00e8se qui nous conduit n\u00e9cessairement \u00e0 la position d'une telle substance. Et le sens cat\u00e9gorique des premi\u00e8res propositions n'est pas seulement n\u00e9gatif. Comme dit Spinoza, \u00ab il n'existe qu'une substance de m\u00eame nature \u00bb. L'identification de l'attribut \u00e0 une substance infiniment parfaite, aussi bien dans l' _\u00c9thique_ que dans le _Court Trait\u00e9_ , n'est pas elle-m\u00eame une hypoth\u00e8se provisoire. Elle doit s'interpr\u00e9ter positivement du point de vue de la _qualit\u00e9._ Il y a une substance par attribut du point de vue de la qualit\u00e9, mais une seule substance pour tous les attributs du point de vue de la _quantit\u00e9._ Que signifie cette multiplicit\u00e9 purement qualitative ? Cette formule obscure marque les difficult\u00e9s de l'entendement fini s'\u00e9levant \u00e0 la compr\u00e9hension de la substance absolument infinie. Elle est justifi\u00e9e par le nouveau statut de la distinction r\u00e9elle. Elle veut dire : les substances qualifi\u00e9es se distinguent qualitativement, non pas quantitativement. Ou, mieux encore, elles se distinguent \u00ab formellement \u00bb, \u00ab quidditativement \u00bb, non pas \u00ab ontologiquement \u00bb.\n\nL'anti-cart\u00e9sianisme de Spinoza trouve une de ses sources dans la th\u00e9orie des distinctions. Dans les _Pens\u00e9es m\u00e9taphysiques_ , Spinoza exposait la conception cart\u00e9sienne : \u00ab Il y a trois sortes de distinctions entre les choses, r\u00e9elle, modale et de raison \u00bb. Et il semblait l'approuver : \u00ab Nous n'avons cure d'ailleurs du fatras des distinctions des P\u00e9ripat\u00e9ticiens \u00bb24. Mais ce qui compte, c'est moins la liste des distinctions reconnues que leur sens et leur distribution d\u00e9termin\u00e9e. \u00c0 cet \u00e9gard, il n'y a plus rien de cart\u00e9sien chez Spinoza. Le nouveau statut de la distinction r\u00e9elle est essentiel : purement qualitative, quidditative ou formelle, la distinction r\u00e9elle exclut toute division. N'est-ce pas, sous un nom cart\u00e9sien, le retour d'une de ces distinctions p\u00e9ripat\u00e9ticiennes en apparence m\u00e9pris\u00e9es ? Que la distinction r\u00e9elle n'est pas num\u00e9rique et ne peut pas l'\u00eatre, nous semble un des motifs principaux de l' _\u00c9thique_. En suit un bouleversement profond des autres distinctions. Non seulement la distinction r\u00e9elle ne renvoie plus \u00e0 des substances _possibles_ distingu\u00e9es _in numero_ , mais la distinction modale, \u00e0 son tour, ne renvoie plus \u00e0 des accidents comme \u00e0 des d\u00e9terminations _contingentes._ Chez Descartes une certaine contingence des modes fait \u00e9cho \u00e0 la simple possibilit\u00e9 des substances. Descartes a beau rappeler que les accidents ne sont pas r\u00e9els, la r\u00e9alit\u00e9 substantielle n'en a pas moins des accidents. Les modes, pour \u00eatre produits, ont besoin d'autre chose que de la substance \u00e0 laquelle ils se rapportent, soit d'une autre substance qui les mette dans la premi\u00e8re, soit de Dieu qui cr\u00e9e la premi\u00e8re avec ses d\u00e9pendances. Tout autre est la vision spinoziste : il n'y a pas plus de contingence du mode par rapport \u00e0 la substance, que de possibilit\u00e9 de la substance par rapport \u00e0 l'attribut. Tout est n\u00e9cessaire, ou bien par son essence ou bien par sa cause : la N\u00e9cessit\u00e9 est la seule affection de l'\u00catre, la seule modalit\u00e9. La distinction de raison, \u00e0 son tour, en est elle-m\u00eame transform\u00e9e. Nous verrons qu'il n'est pas un axiome cart\u00e9sien (le n\u00e9ant n'a pas de propri\u00e9t\u00e9s, etc.) qui ne prenne un nouveau sens, hostile au cart\u00e9sianisme, \u00e0 partir de la nouvelle th\u00e9orie des distinctions. Cette th\u00e9orie trouve son principe dans le statut qualitatif de la distinction r\u00e9elle. Dissoci\u00e9e de toute distinction num\u00e9rique, la distinction r\u00e9elle est port\u00e9e dans l'absolu. Elle devient capable d'exprimer la diff\u00e9rence dans l'\u00eatre, elle entra\u00eene en cons\u00e9quence le remaniement des autres distinctions.\n\n* * *\n\n1.  Cf. M. MERLEAU-PONTY, in _Les Philosophes c\u00e9l\u00e8bres_ (Mazenod \u00e9d., p. 136).\n\n2.  _\u00c9_ , I, 5, prop et dem.\n\n3.  Spinoza expose ainsi la th\u00e8se cart\u00e9sienne, _PM_ , II, 5 : \u00ab ... Il faut se rappeler ce que Descartes a indiqu\u00e9 dans les _Principes de philosophie_ (partie I, articles 48 et 49), \u00e0 savoir qu'il n'y a rien dans la nature en dehors des substances et de leurs modes ; d'o\u00f9 est d\u00e9duite une triple distinction (articles 60, 61 et 62), c'est-\u00e0-dire la r\u00e9elle, la modale et la distinction de raison. \u00bb\n\n4.  DESCARTES, _Principes_ , I, 53.\n\n5.  DESCARTES, _Principes_ , I, 60, 61 et 62.\n\n6.  Cf. SUAREZ, _Metaphysicarum disputationum_ , D VII. Suarez ne reconna\u00eet que les distinctions r\u00e9elle, modale et de raison, et critique la distinction formelle de Duns Scot dans des termes tr\u00e8s voisins de ceux que Descartes utilisera.\n\n7.  DESCARTES, _Principes_ , I, 56.\n\n8.  DESCARTES, _Principes_ , I, 63 et 64.\n\n9.  Sur ces paragraphes 63 et 64, cf. la discussion entre F. ALQUI\u00c9 et M. GU\u00c9ROULT, _Descartes, Cahiers de Royaumont_ (\u00e9d. de Minuit, 1967), pp. 32-56.\n\n10.  DESCARTES, _R\u00e9ponses aux quatri\u00e8mes objections_ (AT, IX, p. 175).\n\n11.  Cette division tripartite est expos\u00e9e dans la _Lettre 2, \u00e0 Oldenburg_ (III, p. 5).\n\n12.  _Lettre 81, \u00e0 Tschirnhaus_ (III, p. 241). Cf. aussi _Lettre 12, \u00e0 Meyer_ (III, p. 41) : le nombre n'exprime pas ad\u00e9quatement la nature des modes en tant qu'ils sont une infinit\u00e9, c'est-\u00e0-dire en tant qu'ils d\u00e9coulent de la substance.\n\n13.  _\u00c9_ , I, 15, sc.\n\n14.  _CT_ , I, ch. 2, 19-22.\n\n15.  _\u00c9_ , I, 5, 6, 7 et 8, prop.\n\n16.  _\u00c9_ , I, 8, sc. 2.\n\n17.  _\u00c9_ , I, 10, sc.\n\n18.  Cf. l'interpr\u00e9tation de P. LACHI\u00c8ZE-REY, _Les Origines cart\u00e9siennes du Dieu de Spinoza_ (Vrin, 2e \u00e9d., p. 151) : \u00ab L'usage fait ainsi de cette distinction n'implique d'ailleurs nullement son admission de la part de Spinoza ; elle reste uniquement un moyen de d\u00e9monstration utilis\u00e9 en partant de l'hypoth\u00e8se d'une pluralit\u00e9 de substances et destin\u00e9 \u00e0 annuler les effets possibles de cette hypoth\u00e9tique pluralit\u00e9. \u00bb\n\n19.  _Lettre 9, \u00e0 De Vries_ (III, p. 32). Dans l' _\u00c9thique_ , le premier argument se retrouve presque litt\u00e9ralement en I, 9 ; le second, moins nettement, en I, 11, sc.\n\n20.  Cf. _Lettre 64, \u00e0 Schuller_ (III, p. 206).\n\n21.  _\u00c9_ , I, 10, sc. \u00ab Si quelqu'un demande maintenant \u00e0 quel signe nous pourrons reconna\u00eetre la diversit\u00e9 des substances, qu'il lise les propositions suivantes, qui montrent qu'il n'existe dans la Nature qu'une substance unique et qu'elle est absolument infinie, ce pourquoi on chercherait en vain le signe en question. \u00bb\n\n22.  _CT_ , I, ch. 7, 9-10.\n\n23.  Cf. R\u00c9GIS, _R\u00e9futation de l'opinion de Spinoza touchant l'existence et la nature de Dieu_ , 1704.\n\n24.  _PM_ , II, 5.\n\n## CHAPITRE II\n\n## L'ATTRIBUT COMME EXPRESSION\n\nSpinoza ne dit pas que les attributs existent par soi, ni qu'ils soient con\u00e7us de telle fa\u00e7on que l'existence suive ou d\u00e9coule de leur essence. Il ne dit pas non plus que l'attribut est en soi et est con\u00e7u par soi, comme la substance. Il dit seulement que l'attribut est con\u00e7u par soi et en soi1. Le statut de l'attribut s'\u00e9bauche \u00e0 travers les formules tr\u00e8s complexes du _Court Trait\u00e9._ Si complexes, il est vrai, que le lecteur a le choix entre plusieurs hypoth\u00e8ses : pr\u00e9sumer des dates diverses de leur r\u00e9daction ; rappeler de toutes fa\u00e7ons l'imperfection des manuscrits ; ou m\u00eame invoquer l'\u00e9tat encore h\u00e9sitant de la pens\u00e9e de Spinoza. Toutefois, ces arguments ne peuvent intervenir que s'il est av\u00e9r\u00e9 que les formules du _Court Trait\u00e9_ ne s'accordent pas entre elles, et ne s'accordent pas davantage avec les donn\u00e9es ult\u00e9rieures de l' _\u00c9thique_. Or il ne semble pas en \u00eatre ainsi. Les textes du _Court Trait\u00e9_ ne seront pas d\u00e9pass\u00e9s par l' _\u00c9thique_ , mais plut\u00f4t transform\u00e9s. Et cela, gr\u00e2ce \u00e0 une utilisation plus syst\u00e9matique de l'id\u00e9e d'expression. Inversement donc, ils peuvent nous renseigner sur le contenu conceptuel inform\u00e9 par cette id\u00e9e d'expression chez Spinoza.\n\nCes textes disent tour \u00e0 tour : 1o) \u00ab \u00e0 l'essence des attributs appartient l'existence, en sorte qu'en dehors d'eux n'existe aucune essence ou aucun \u00eatre \u00bb ; 2o) \u00ab nous les concevons seulement dans leur essence et non dans leur existence, nous ne les concevons pas de telle sorte que l'existence d\u00e9coule de leur essence \u00bb ; \u00ab tu ne les con\u00e7ois pas comme subsistant par eux-m\u00eames \u00bb ; 3o) ils existent \u00ab formellement \u00bb et \u00ab en acte \u00bb ; \u00ab nous d\u00e9montrons a priori qu'ils existent \u00bb2.\n\nD'apr\u00e8s la premi\u00e8re formule, l'essence en tant qu'essence n'existe pas hors des attributs qui la constituent. L'essence se distingue donc _dans_ les attributs o\u00f9 elle existe. Elle existe toujours en un genre, en autant de genres qu'il y a d'attributs. Chaque attribut d\u00e8s lors est l'existence d'une essence \u00e9ternelle et infinie, d'une \u00ab essence particuli\u00e8re \u00bb3. C'est en ce sens que Spinoza peut dire : \u00e0 l'essence des attributs il appartient d'exister, mais pr\u00e9cis\u00e9ment d'exister dans les attributs. Ou m\u00eame : \u00ab L'existence des attributs ne diff\u00e8re pas de leur essence4. \u00bb L'id\u00e9e d'expression, dans l' _\u00c9thique_ , recueillera ce premier moment : l'essence de la substance n'existe pas hors des attributs qui l'expriment, si bien que chaque attribut exprime une certaine essence \u00e9ternelle et infinie. L'exprim\u00e9 n'existe pas hors de ses expressions, chaque expression est comme l'existence de l'exprim\u00e9. (C'est ce m\u00eame principe qu'on retrouve chez Leibniz, si diff\u00e9rent que soit le contexte : chaque monade est l'expression du monde, mais le monde exprim\u00e9 n'existe pas hors des monades qui l'expriment.)\n\nComment peut-on dire que les attributs expriment, non seulement une certaine essence, mais l'essence de la substance ? L'essence est exprim\u00e9e comme essence de la substance, et non de l'attribut. Les essences sont donc distinctes dans les attributs o\u00f9 elles existent, mais ne font qu'un dans la substance dont elles sont l'essence. La r\u00e8gle de convertibilit\u00e9 affirme : toute essence est essence de quelque chose. Les essences sont r\u00e9ellement distinctes du point de vue des attributs, mais l'essence est une du point de vue de l'objet avec lequel elle se r\u00e9ciproque. Les attributs ne sont pas attribu\u00e9s \u00e0 des substances correspondantes, de m\u00eame genre ou de m\u00eame esp\u00e8ce qu'eux-m\u00eames. Au contraire, ils attribuent leur essence \u00e0 _autre chose_ , qui reste donc la m\u00eame pour tous les attributs. C'est pourquoi Spinoza va jusqu'\u00e0 dire : \u00ab Aussi longtemps qu'une substance est con\u00e7ue \u00e0 part, il s'ensuit qu'elle ne peut \u00eatre une chose qui existe \u00e0 part, mais doit \u00eatre une chose telle qu'un attribut d'une _autre_ , qui est l'\u00eatre unique ou le tout... Aucune substance existant en acte ne peut \u00eatre con\u00e7ue comme existant en elle-m\u00eame, mais elle doit appartenir \u00e0 quelque _autre chose_5. \u00bb Toutes les essences existantes sont donc exprim\u00e9es par les attributs dans lesquels elles existent, mais comme l'essence d'autre chose, c'est-\u00e0-dire d'une seule et m\u00eame chose pour tous les attributs. Nous demandons alors : Qu'est-ce qui existe par soi, de telle fa\u00e7on que l'existence d\u00e9coule de son essence ? Il est clair que c'est la substance, le corr\u00e9lat de l'essence, et non l'attribut dans lequel l'essence existait seulement comme essence. On ne confondra pas l'existence de l'essence avec l'existence de son corr\u00e9lat. Toutes les essences existantes sont rapport\u00e9es ou attribu\u00e9es \u00e0 la substance, mais comme au seul \u00eatre dont l'existence d\u00e9coule n\u00e9cessairement de l'essence. La substance a le privil\u00e8ge d'exister par soi : existe par soi, _non pas l'attribut_ , mais ce \u00e0 quoi chaque attribut rapporte son essence, de telle mani\u00e8re que l'existence d\u00e9coule n\u00e9cessairement de l'essence ainsi constitu\u00e9e. Des attributs consid\u00e9r\u00e9s en eux-m\u00eames, Spinoza dira donc de mani\u00e8re parfaitement coh\u00e9rente : \u00ab Nous les concevons seulement dans leur essence et non dans leur existence, nous ne les concevons pas de telle sorte que l'existence d\u00e9coule de leur essence. \u00bb Ce second type de formule ne contredit pas le pr\u00e9c\u00e9dent, mais mesure l'approfondissement d'un probl\u00e8me ou son changement de perspective.\n\nL'exprim\u00e9 n'existe pas hors de son expression, mais il est exprim\u00e9 comme l'essence de ce qui s'exprime. Nous retrouvons toujours la n\u00e9cessit\u00e9 de distinguer ces trois termes : la substance qui s'exprime, les attributs qui sont des expressions, l'essence exprim\u00e9e. Mais enfin, s'il est vrai que les attributs expriment l'essence de la substance, comment n'exprimeraient-ils pas aussi l'existence qui en d\u00e9coule n\u00e9cessairement ? Ces m\u00eames attributs auxquels on refuse l'existence par soi n'en ont pas moins, en tant qu'attributs, une existence actuelle et n\u00e9cessaire. Bien plus, en d\u00e9montrant que quelque chose est attribut, nous d\u00e9montrons a priori qu'il existe. La diversit\u00e9 des formules du _Court Trait\u00e9_ doit donc s'interpr\u00e9ter ainsi : elles concernent tour \u00e0 tour _l'existence de l'essence, l'existence de la substance, l'existence de l'attribut lui-m\u00eame._ Et dans l' _\u00c9thique_ , c'est l'id\u00e9e d'expression qui recueille ces trois temps, leur donnant une forme syst\u00e9matique.\n\nLe probl\u00e8me des attributs de Dieu fut toujours en \u00e9troit rapport avec celui des noms divins. Comment pourrions-nous nommer Dieu si nous n'en avions pas une connaissance quelconque ? Mais comment le conna\u00eetrions-nous s'il ne se faisait lui-m\u00eame conna\u00eetre en quelque fa\u00e7on, se r\u00e9v\u00e9lant et s'exprimant ? La Parole divine, le Verbe divin, scelle l'alliance des attributs et des noms. Les noms sont des attributs, pour autant que les attributs sont des expressions. Il est vrai que toute la question est de savoir ce qu'ils expriment : la nature m\u00eame de Dieu telle qu'elle est en soi, ou seulement des actions de Dieu comme cr\u00e9ateur, ou m\u00eame de simples qualit\u00e9s divines extrins\u00e8ques, relatives aux cr\u00e9atures ? Spinoza ne manque pas de recueillir ce probl\u00e8me traditionnel. Trop habile grammairien pour n\u00e9gliger la parent\u00e9 des noms et des attributs. Le _Trait\u00e9 th\u00e9ologico-politique_ demande sous quels noms ou par quels attributs Dieu \u00ab se r\u00e9v\u00e8le \u00bb dans l'\u00c9criture ; il demande ce qu'est la parole de Dieu, quelle valeur expressive il faut reconna\u00eetre \u00e0 la voix de Dieu. Et quand Spinoza veut illustrer ce qu'il entend personnellement par attribut, lui vient \u00e0 l'esprit l'exemple des noms propres : \u00ab J'entends par Isra\u00ebl le troisi\u00e8me patriarche, et par Jacob le m\u00eame personnage auquel ce nom a \u00e9t\u00e9 donn\u00e9 parce qu'il a saisi le talon de son fr\u00e8re6. \u00bb Le rapport du spinozisme avec la th\u00e9orie des noms doit \u00eatre \u00e9valu\u00e9 de deux fa\u00e7ons. Comment Spinoza s'ins\u00e8re-t-il dans la tradition ? Mais, surtout, comment la renouvelle-t-il ? On peut d\u00e9j\u00e0 pr\u00e9voir qu'il la renouvelle doublement : parce qu'il con\u00e7oit autrement ce qu'est le nom ou l'attribut, parce qu'il d\u00e9termine autrement ce qui est attribut.\n\nLes attributs chez Spinoza sont des formes dynamiques et actives. Et voil\u00e0 bien ce qui parait essentiel : l'attribut n'est plus attribu\u00e9, il est en quelque sorte \u00ab attributeur \u00bb. Chaque attribut exprime une essence, et l'attribue \u00e0 la substance. Toutes les essences attribu\u00e9es se confondent dans la substance dont elles sont l'essence. Tant que nous concevons l'attribut comme quelque chose d'attribu\u00e9, nous concevons par-l\u00e0 m\u00eame une substance qui serait de m\u00eame esp\u00e8ce ou de m\u00eame genre que lui ; cette substance, alors, n'a par soi qu'une existence possible, puisqu'il d\u00e9pend de la bonne volont\u00e9 d'un Dieu transcendant de la faire exister conform\u00e9ment \u00e0 l'attribut qui nous la fait conna\u00eetre. Au contraire, d\u00e8s que nous posons l'attribut comme \u00ab attributeur \u00bb, nous le concevons en m\u00eame temps comme attribuant son essence \u00e0 quelque chose qui reste identique pour tous les attributs, c'est-\u00e0-dire \u00e0 une substance qui existe n\u00e9cessairement. L'attribut rapporte son essence \u00e0 un Dieu immanent, \u00e0 la fois principe et r\u00e9sultat d'une n\u00e9cessit\u00e9 m\u00e9taphysique. En ce sens les attributs chez Spinoza sont de v\u00e9ritables _verbes_ , ayant une valeur expressive : dynamiques, ils ne sont plus attribu\u00e9s \u00e0 des substances variables, ils attribuent quelque chose \u00e0 une substance unique.\n\nMais qu'est-ce qu'ils attribuent, qu'est-ce qu'ils expriment ? Chaque attribut exprime une essence infinie, c'est-\u00e0-dire une qualit\u00e9 illimit\u00e9e. Ces qualit\u00e9s sont substantielles, parce qu'elles qualifient toutes une m\u00eame substance ayant tous les attributs. Aussi bien y a-t-il deux mani\u00e8res de reconna\u00eetre ce qui est attribut : ou bien l'on cherche a priori quelles sont les qualit\u00e9s que l'on con\u00e7oit comme illimit\u00e9es. Ou bien, partant de ce qui est limit\u00e9, nous cherchons a posteriori quelles qualit\u00e9s sont susceptibles d'\u00eatres port\u00e9es \u00e0 l'infini, qui sont comme \u00ab envelopp\u00e9es \u00bb dans les limites du fini : \u00e0 partir de cette pens\u00e9e-ci ou de cette pens\u00e9e-l\u00e0, nous concluons \u00e0 la pens\u00e9e comme attribut infini de Dieu ; \u00e0 partir de tel ou tel corps, \u00e0 l'\u00e9tendue comme attribut infini7.\n\nCette derni\u00e8re m\u00e9thode, a posteriori, doit \u00eatre \u00e9tudi\u00e9e de pr\u00e8s : elle pose tout le probl\u00e8me d'un enveloppement de l'infini. Elle consiste \u00e0 nous faire conna\u00eetre les attributs de Dieu \u00e0 partir des \u00ab cr\u00e9atures \u00bb. Mais, dans cette voie, elle ne proc\u00e8de ni par abstraction, ni par analogie. Les attributs ne sont pas abstraits des choses particuli\u00e8res, encore moins transf\u00e9r\u00e9s \u00e0 Dieu de mani\u00e8re analogique. _Les attributs sont directement atteints comme des formes d'\u00eatre communes aux cr\u00e9atures et \u00e0 Dieu, communes aux modes et \u00e0 la substance._ On voit bien le pr\u00e9tendu danger d'un tel proc\u00e9d\u00e9 : l'anthropomorphisme, et plus g\u00e9n\u00e9ralement la confusion du fini et de l'infini. Dans une m\u00e9thode d'analogie, on se propose explicitement d'\u00e9viter l'anthropomorphisme : suivant saint Thomas, les qualit\u00e9s qu'on attribue \u00e0 Dieu n'impliquent pas une communaut\u00e9 de forme entre la substance divine et les cr\u00e9atures, mais seulement une analogie, une \u00ab convenance \u00bb de proportion ou de proportionnalit\u00e9. Tant\u00f4t Dieu poss\u00e8de formellement une perfection qui reste extrins\u00e8que dans les cr\u00e9atures, tant\u00f4t il poss\u00e8de \u00e9minemment une perfection qui convient formellement aux cr\u00e9atures. Or l'importance du spinozisme, ici, doit \u00eatre jug\u00e9e \u00e0 la mani\u00e8re dont il renverse le probl\u00e8me. Chaque fois que nous proc\u00e9dons par analogie, nous empruntons certains caract\u00e8res aux cr\u00e9atures, pour les attribuer \u00e0 Dieu soit de mani\u00e8re \u00e9quivoque, soit de mani\u00e8re \u00e9minente. Dieu aurait Vouloir et Entendement, Bont\u00e9 et Sagesse, etc., mais \u00e9quivoquement ou \u00e9minemment8. L'analogie ne peut se passer ni de l'\u00e9quivocit\u00e9 ni de l'\u00e9minence et, par l\u00e0, contient un anthropomorphisme subtil, aussi dangereux que l'anthropomorphisme na\u00eff. Il va de soi qu'un triangle, s'il pouvait parler, dirait que Dieu est \u00e9minemment triangulaire. La m\u00e9thode d'analogie nie qu'il y ait des formes communes \u00e0 Dieu et aux cr\u00e9atures ; mais, loin d'\u00e9chapper au danger qu'elle d\u00e9nonce, elle confond constamment les essences de cr\u00e9atures et l'essence de Dieu. Tant\u00f4t elle supprime l'essence des choses, r\u00e9duisant leurs qualit\u00e9s \u00e0 des d\u00e9terminations qui ne conviennent intrins\u00e8quement qu'\u00e0 Dieu. Tant\u00f4t elle supprime l'essence de Dieu, lui pr\u00eatant \u00e9minemment ce que les cr\u00e9atures poss\u00e8dent formellement. Au contraire, Spinoza affirme l'identit\u00e9 de forme entre les cr\u00e9atures et Dieu, mais s'interdit toute confusion d'essence.\n\nLes attributs constituent l'essence de la substance, mais ils ne constituent nullement l'essence des modes ou des cr\u00e9atures. _Ce sont pourtant des formes communes_ , parce que les cr\u00e9atures les impliquent dans leur propre essence comme dans leur existence. D'o\u00f9 l'importance de la r\u00e8gle de convertibilit\u00e9 : l'essence n'est pas seulement ce sans quoi la chose ne peut ni \u00eatre ni \u00eatre con\u00e7ue, mais r\u00e9ciproquement ce qui ne peut, sans la chose, ni \u00eatre ni \u00eatre con\u00e7u. C'est d'apr\u00e8s cette r\u00e8gle que les attributs sont bien l'essence de la substance, mais ne sont nullement l'essence des modes, par exemple de l'homme : ils peuvent fort bien \u00eatre con\u00e7us sans les modes9. Reste que les modes les enveloppent ou les impliquent, et _les impliquent pr\u00e9cis\u00e9ment sous cette forme qui leur est propre en tant qu'ils constituent l'essence de Dieu._ Ce qui revient \u00e0 dire que les attributs \u00e0 leur tour contiennent ou comprennent les essences de mode, et les comprennent formellement, non pas \u00e9minemment. _Les attributs sont donc des formes communes \u00e0 Dieu dont ils constituent l'essence, et aux modes ou cr\u00e9atures qui les impliquent essentiellement._ Les m\u00eames formes s'affirment de Dieu et des cr\u00e9atures, bien que les cr\u00e9atures et Dieu diff\u00e9rent en essence autant qu'en existence. Pr\u00e9cis\u00e9ment, la diff\u00e9rence consiste en ceci : les modes sont seulement compris sous ces formes, qui se r\u00e9ciproquent au contraire avec Dieu. Cette diff\u00e9rence n'affecte pas la raison formelle de l'attribut prise en tant que telle.\n\nSpinoza, sur ce point, est fort conscient de son originalit\u00e9. Sous pr\u00e9texte que les cr\u00e9atures diff\u00e8rent de Dieu tant par l'essence que par l'existence, on veut que Dieu n'ait rien de commun formellement avec les cr\u00e9atures. En v\u00e9rit\u00e9, c'est tout le contraire : les m\u00eames attributs se disent de Dieu qui s'explique en eux, et des modes qui les impliquent \u2013 qui les impliquent sous la m\u00eame forme que celle qui convient \u00e0 Dieu. Bien plus : tant qu'on refuse la communaut\u00e9 formelle, on se condamne \u00e0 confondre les essences ; on les confond par analogie. Mais d\u00e8s qu'on pose la communaut\u00e9 formelle, on se donne le moyen de les distinguer. C'est pourquoi Spinoza ne se vante pas seulement d'avoir r\u00e9duit \u00e0 l'\u00e9tat de cr\u00e9atures des choses que l'on consid\u00e9rait jusqu'\u00e0 lui comme des attributs de Dieu, mais en m\u00eame temps d'avoir \u00e9lev\u00e9 \u00e0 l'\u00e9tat d'attributs de Dieu des choses que l'on consid\u00e9rait comme des cr\u00e9atures10. En r\u00e8gle g\u00e9n\u00e9rale, Spinoza ne voit aucune contradiction entre l'affirmation d'une communaut\u00e9 de forme et la position d'une distinction d'essences. Il dira dans des textes voisins : 1o) si des choses n'ont rien de commun entre elles, l'une ne peut \u00eatre la cause de l'autre ; 2o) si une chose est cause de l'essence et de l'existence d'une autre, elle doit en diff\u00e9rer tant en raison de l'essence qu'en raison de l'existence11. La conciliation de ces textes ne nous para\u00eet soulever aucun probl\u00e8me particulier dans le spinozisme. Quand les correspondants de Spinoza s'\u00e9tonnent, Spinoza s'\u00e9tonne \u00e0 son tour : il rappelle qu'il a toutes raisons pour dire \u00e0 la fois que les cr\u00e9atures diff\u00e8rent de Dieu en essence et en existence, _et_ que Dieu a quelque chose de commun formellement avec les cr\u00e9atures12.\n\nLa m\u00e9thode de Spinoza n'est ni abstraite ni analogique. C'est une m\u00e9thode formelle et de communaut\u00e9. Elle op\u00e8re par notions communes ; or toute la th\u00e9orie spinoziste des notions communes trouve pr\u00e9cis\u00e9ment son principe dans ce statut de l'attribut. S'il faut enfin donner un nom \u00e0 cette m\u00e9thode, comme \u00e0 la th\u00e9orie sous-jacente, on y reconna\u00eetra facilement la grande tradition de l'univocit\u00e9. Nous croyons que _la philosophie de Spinoza reste en partie inintelligible, si l'on n'y voit pas une lutte constante contre les trois notions d'\u00e9quivocit\u00e9, d'\u00e9minence et d'analogie._ Les attributs, selon Spinoza, sont des formes d'\u00eatre univoques, qui ne changent pas de nature en changeant de \u00ab sujet \u00bb, c'est-\u00e0-dire quand on les pr\u00e9dique de l'\u00eatre infini et des \u00eatres finis, de la substance et des modes, de Dieu et des cr\u00e9atures. Nous croyons n'\u00f4ter rien \u00e0 l'originalit\u00e9 de Spinoza en le repla\u00e7ant dans une perspective qui \u00e9tait d\u00e9j\u00e0 celle de Duns Scot. Comment Spinoza pour son compte interpr\u00e8te la notion d'univocit\u00e9, comment il la comprend d'une tout autre fa\u00e7on que Duns Scot, nous devons remettre \u00e0 plus tard cette analyse. Il nous suffit pour le moment de r\u00e9unir les premi\u00e8res d\u00e9terminations de l'attribut. Les attributs sont des formes d'\u00eatre infinies, des raisons formelles illimit\u00e9es, ultimes, irr\u00e9ductibles ; ces formes sont communes \u00e0 Dieu dont elles constituent l'essence et aux modes qui les impliquent dans leur propre essence. Les attributs sont des verbes exprimant des qualit\u00e9s illimit\u00e9es ; ces qualit\u00e9s sont comme envelopp\u00e9es dans les limites du fini. Les attributs sont des expressions de Dieu ; ces expressions de Dieu sont univoques, elles constituent la nature m\u00eame de Dieu comme Nature naturante, elles sont envelopp\u00e9es dans la nature des choses ou Nature natur\u00e9e qui, d'une certaine fa\u00e7on, les r\u00e9-exprime \u00e0 son tour.\n\nSpinoza, d\u00e8s lors, est en mesure de distinguer les attributs et les propres. Le point de d\u00e9part est aristot\u00e9licien : le propre est ce qui appartient \u00e0 une chose, mais n'explique jamais ce qu'elle est. Les propres de Dieu sont donc seulement des \u00ab adjectifs \u00bb qui ne nous font rien conna\u00eetre substantiellement ; Dieu ne serait pas Dieu sans eux, mais n'est pas Dieu par eux13. Spinoza peut, conform\u00e9ment \u00e0 une longue tradition, donner aux propres le nom d'attributs ; il n'y en aura pas moins, selon lui, diff\u00e9rence de nature entre deux sortes d'attributs. Mais que veut dire Spinoza, quand il ajoute que les propres de Dieu ne sont que \u00ab des modes qui peuvent lui \u00eatre imput\u00e9s14 \u00bb ? Mode, ici, ne doit pas \u00eatre pris au sens particulier que Spinoza lui donne souvent, mais en un sens plus g\u00e9n\u00e9ral, au sens scolastique de \u00ab modalit\u00e9 de l'essence \u00bb. Infini, parfait, immuable, \u00e9ternel sont des propres qui se disent de tous les attributs. Omniscient, omnipr\u00e9sent, des propres qui se disent d'un attribut d\u00e9termin\u00e9 (la pens\u00e9e, l'\u00e9tendue). En effet, tous les attributs expriment l'essence de la substance, chaque attribut exprime une essence de substance. Mais les propres n'expriment rien : \u00ab Nous ne pouvons pas savoir par ces propres quelle est l'essence et quels sont les attributs de l'\u00eatre auxquels appartiennent ces propres15. \u00bb Ils ne constituent pas la nature de la substance, mais se disent de ce qui constitue cette nature. Ils ne forment donc pas l'essence d'un \u00catre, mais seulement la modalit\u00e9 de cette essence telle qu'elle est form\u00e9e. Infini est le propre de la substance, c'est-\u00e0-dire la modalit\u00e9 de chaque attribut qui en constitue l'essence. Omniscient est le propre de la substance pensante, c'est-\u00e0-dire la modalit\u00e9 infinie de cet attribut pens\u00e9e qui exprime une essence de substance. Les propres ne sont pas des attributs, \u00e0 proprement parler, pr\u00e9cis\u00e9ment parce qu'ils ne sont pas _expressifs._ Ils seraient plut\u00f4t comme des \u00ab notions impresses \u00bb, comme des caract\u00e8res imprim\u00e9s, soit dans tous les attributs, soit dans tel ou tel d'entre eux. L'opposition des attributs et des propres porte donc sur deux points. Les attributs sont des verbes exprimant des essences ou des qualit\u00e9s substantielles ; mais les propres sont seulement des adjectifs indiquant la modalit\u00e9 de ces essences ou de ces qualit\u00e9s. Les attributs de Dieu sont des formes communes, communes \u00e0 la substance qui se r\u00e9ciproque avec elles, et aux modes qui les impliquent sans r\u00e9ciprocit\u00e9 ; mais les propres de Dieu sont vraiment propres \u00e0 Dieu, ils ne se disent pas des modes mais seulement des attributs.\n\nUne seconde cat\u00e9gorie de propres concerne Dieu comme cause, en tant qu'il agit ou produit : non plus infini, parfait, \u00e9ternel, immuable, mais cause de toutes choses, pr\u00e9destination, providence16. Or, puisque Dieu produit dans ses attributs, ces propres sont soumis au m\u00eame principe que les pr\u00e9c\u00e9dents. Certains se disent de tous les attributs ; d'autres, de tel ou tel. Ces seconds propres sont encore des adjectifs ; mais au lieu d'indiquer des modalit\u00e9s, ils indiquent des relations, relations de Dieu \u00e0 ses cr\u00e9atures ou \u00e0 ses produits. Enfin, une troisi\u00e8me cat\u00e9gorie d\u00e9signe des propres qui n'appartiennent m\u00eame pas \u00e0 Dieu : Dieu comme souverain bien, comme mis\u00e9ricordieux, comme juste et charitable17. \u00c0 cet \u00e9gard, c'est surtout le _Trait\u00e9 th\u00e9ologico-politique_ qui peut nous \u00e9clairer. Ce Trait\u00e9 parle de la justice et de la charit\u00e9 divines comme d'\u00ab attributs qui peuvent servir de mod\u00e8le \u00e0 une certaine mani\u00e8re de vivre18 \u00bb. Ces propres n'appartiennent pas \u00e0 Dieu comme cause ; il ne s'agit plus d'un rapport de Dieu avec ses cr\u00e9atures, mais de d\u00e9terminations extrins\u00e8ques qui indiquent seulement la fa\u00e7on dont les cr\u00e9atures imaginent Dieu. Il est vrai que ces d\u00e9nominations ont des sens et des valeurs extr\u00eamement variables : on va jusqu'\u00e0 pr\u00eater \u00e0 Dieu des \u00e9minences en tous genres, une bouche et des yeux divins, des qualit\u00e9s morales et des passions sublimes, des montagnes et des cieux. Mais, m\u00eame \u00e0 s'en tenir \u00e0 la justice et \u00e0 la charit\u00e9, on n'atteint rien de la nature de Dieu, ni de ses op\u00e9rations comme Cause. Adam, Abraham, Mo\u00efse ignorent non seulement les vrais attributs divins, mais aussi la plupart des propres de la premi\u00e8re et de la seconde esp\u00e8ce19. Dieu se r\u00e9v\u00e8le \u00e0 eux sous des d\u00e9nominations extrins\u00e8ques qui leur servent d'avertissements, de commandements, de r\u00e8gles ou de mod\u00e8le de vie. Plus que jamais, il faut dire que ces troisi\u00e8mes propres n'ont rien d'expressif. Ce ne sont pas des expressions divines, mais des notions imprim\u00e9es dans l'imagination pour nous faire ob\u00e9ir, nous faire servir un Dieu dont nous ignorons la nature.\n\n* * *\n\n1.  _Lettre 2, \u00e0 Oldenburg_ (III, p. 5) : _quod concipitur per se et in se._ Il ne semble donc pas que Delbos soit fond\u00e9 \u00e0 dire que, dans cette lettre 2, l'attribut se d\u00e9finit comme la substance (cf. \u00ab La Doctrine spinoziste des attributs de Dieu \u00bb, _Ann\u00e9e philosophique_ , 1912).\n\n2.  Cf. 1o) _CT_ , Appendice I, 4, cor. 2o) _CT_ , I, ch. 2, 17 et note 5 ; et premier dialogue, 9. 3o) _CT_ , I, ch. 2, passim et 17 (note 5).\n\n3.  _CT_ , I, ch. 2, 17.\n\n4.  _Lettre 10, \u00e0 De Vries_ (III, p. 34).\n\n5.  _CT_ , I, ch. 2, 17, note 5.\n\n6.  _Lettre 9, \u00e0 De Vries_ (III, p. 33).\n\n7.  _\u00c9_ , II, 1 et 2 : Spinoza d\u00e9montre que la pens\u00e9e et l'\u00e9tendue sont des attributs. Le proc\u00e9d\u00e9 a posteriori appara\u00eet dans la d\u00e9monstration m\u00eame, le proc\u00e9d\u00e9 a priori, dans le scolie.\n\n8.  Sur la critique de l'\u00e9quivocit\u00e9, cf. _\u00c9_ , I, 17, cor. 2. (Si la volont\u00e9 et l'entendement s'attribuaient essentiellement \u00e0 Dieu, ce serait de mani\u00e8re \u00e9quivoque, donc toute verbale, \u00e0 peu pr\u00e8s comme le mot \u00ab chien \u00bb d\u00e9signe une constellation c\u00e9leste.) Sur la critique de l'\u00e9minence, cf. _Lettre 56, \u00e0 Boxel_ , III, p. 190. (Si le triangle pouvait parler, il dirait que Dieu est triangulaire \u00e9minemment... Spinoza r\u00e9pond ici \u00e0 Boxel qui pensait que l'\u00e9minence et l'analogie \u00e9taient seules capables de nous sauver de l'anthropomorphisme.)\n\n9.  _\u00c9_ , II, 10, scolie du corollaire. La d\u00e9finition insuffisante de l'essence (ce sans quoi la chose ne peut ni \u00eatre ni \u00eatre con\u00e7ue) se trouve dans Suarez : cf. \u00c9. GILSON, _Index scolastico-cart\u00e9sien_ , pp. 105-106.\n\n10.  _Lettre 6, \u00e0 Oldenburg_ (III, p. 25).\n\n11.  Cf. 1o) _\u00c9_ , I, 3, prop. ; 2o) _\u00c9_ , I, 17, sc. [Pour concilier ces textes, on a parfois cherch\u00e9 des diff\u00e9rences de points de vue (causalit\u00e9 immanente et causalit\u00e9 transitive, etc.) : cf. LACHI\u00c8ZE-REY, _op. cit._ , pp. 156-159, note.]\n\n12.  _Lettre 4, \u00e0 Oldenburg_ (III, p. 11) : \u00ab Quant \u00e0 ce que vous dites, que Dieu n'a rien de commun formellement avec les choses cr\u00e9\u00e9es, j'ai pos\u00e9 le contraire dans ma d\u00e9finition \u00bb (il s'agit de la d\u00e9finition de Dieu comme substance consistant en une infinit\u00e9 d' _attributs_ ). _Lettre 64, \u00e0 Schuller_ (III, p. 206) : \u00ab Est-il possible qu'une chose soit produite par une autre dont elle diff\u00e8re tant par l'essence que par l'existence ? Et, en effet, des choses diff\u00e9rant ainsi l'une de l'autre _semblent_ n'avoir rien de commun. _Mais_ comme toutes les choses singuli\u00e8res, \u00e0 part celles qui sont produites par leurs semblables, diff\u00e9rent de leurs causes tant par l'essence que par l'existence, je ne vois ici rien de douteux \u00bb. (Spinoza renvoie alors \u00e0 la d\u00e9finition du _mode_ , _\u00c9_ , I, 25, cor.)\n\n13.  _CT_ , I, ch. 7, 6 (cf. aussi I, ch. 1, 9, note 4 ; ch. 3, 1, note 1).\n\n14.  _CT_ , I, ch. 7, 1, note 1.\n\n15.  _CT_ , I, ch. 7, 6.\n\n16.  Cf. _CT_ , I, chapitres 3, 4, 5, et 6.\n\n17.  _CT_ , I, ch. 7.\n\n18.  _TTP_ , ch. 13 (III, p. 241).\n\n19.  _TTP_ , ch. 2 (II, p. 115) : Adam par exemple sait que Dieu est cause de toutes choses, mais il ne sait pas que Dieu est omniscient et omnipr\u00e9sent.\n\n## CHAPITRE III\n\n## ATTRIBUTS ET NOMS DIVINS\n\nD'apr\u00e8s une longue tradition, les noms divins se rapportent \u00e0 des manifestations de Dieu. Inversement, les manifestations divines sont des paroles par lesquelles Dieu se fait conna\u00eetre sous tel ou tel nom. Il revient donc au m\u00eame de demander si les noms qui d\u00e9signent Dieu sont des affirmations ou des n\u00e9gations, si les qualit\u00e9s qui le manifestent et les attributs qui lui conviennent sont positifs ou n\u00e9gatifs. Le concept d'expression, \u00e0 la fois parole et manifestation, lumi\u00e8re et son, semble avoir une logique propre qui favorise les deux hypoth\u00e8ses. On insistera tant\u00f4t sur la positivit\u00e9, c'est-\u00e0-dire sur l'immanence de l'exprim\u00e9 dans l'expression, tant\u00f4t sur la \u00ab n\u00e9gativit\u00e9 \u00bb, c'est-\u00e0-dire sur la transcendance de ce qui s'exprime par rapport \u00e0 toutes les expressions. Ce qui cache exprime aussi, mais ce qui exprime cache encore. C'est pourquoi, dans le probl\u00e8me des noms divins ou des attributs de Dieu, tout est question de nuance. La th\u00e9ologie dite n\u00e9gative admet que des affirmations sont capables de d\u00e9signer Dieu comme cause, sous des r\u00e8gles d'immanence qui vont du plus proche au plus lointain. Mais Dieu comme substance ou essence ne peut \u00eatre d\u00e9fini que n\u00e9gativement, suivant des r\u00e8gles de transcendance o\u00f9 l'on nie tour \u00e0 tour les noms les plus lointains, puis les plus proches. Et enfin, la d\u00e9it\u00e9 suprasubstantielle ou suressentielle se tient splendide, aussi loin des n\u00e9gations que des affirmations. La th\u00e9ologie n\u00e9gative combine donc la m\u00e9thode n\u00e9gative avec la m\u00e9thode affirmative, et pr\u00e9tend les d\u00e9passer toutes deux. Comment saurait-on ce qu'il faut nier de Dieu comme essence, si l'on ne savait d'abord ce qu'on doit en affirmer comme cause ? On ne peut donc d\u00e9finir la th\u00e9ologie n\u00e9gative que par son dynamisme : les affirmations se d\u00e9passent dans des n\u00e9gations, les affirmations et les n\u00e9gations se d\u00e9passent dans une \u00e9minence t\u00e9n\u00e9breuse.\n\nUne th\u00e9ologie d'ambition plus positive, comme celle de saint Thomas, compte sur l'analogie pour fonder de nouvelles r\u00e8gles affirmatives. Les qualit\u00e9s positives ne d\u00e9signent pas seulement Dieu comme cause, mais lui conviennent substantiellement, \u00e0 condition de subir un traitement analogique. Dieu est bon ne signifie pas que Dieu est non mauvais ; ni qu'il est cause de bont\u00e9. Mais en v\u00e9rit\u00e9 : ce que nous appelons bont\u00e9 dans les cr\u00e9atures \u00ab pr\u00e9existe \u00bb en Dieu, suivant une modalit\u00e9 plus haute qui convient avec la substance divine. L\u00e0 encore, c'est un dynamisme qui d\u00e9finit la nouvelle m\u00e9thode. Ce dynamisme, \u00e0 son tour, maintient les droits du n\u00e9gatif et de l'\u00e9minent, mais les comprend dans l'analogie : on remonte d'une n\u00e9gation pr\u00e9alable \u00e0 un attribut positif, cet attribut s'appliquant \u00e0 Dieu _formaliter eminenter_1.\n\nLa philosophie arabe, la philosophie juive se heurtaient au m\u00eame probl\u00e8me. Comment des noms s'appliqueraient-ils, non seulement \u00e0 Dieu comme cause, mais \u00e0 l'essence de Dieu ? Faut-il les prendre n\u00e9gativement, les nier d'apr\u00e8s certaines r\u00e8gles ? Faut-il les affirmer, d'apr\u00e8s d'autres r\u00e8gles ? Or, si nous nous pla\u00e7ons du point de vue du spinozisme, les deux tendances paraissent \u00e9galement fausses, parce que le probl\u00e8me auquel elles se rapportent est lui-m\u00eame enti\u00e8rement faux.\n\nIl est \u00e9vident que la division tripartite des propres chez Spinoza reproduit une classification traditionnelle des attributs de Dieu : 1o) d\u00e9nominations symboliques, formes et figures, signes et rites, m\u00e9tonymies du sensible au divin ; 2o) attributs d'action ; 3o) attributs d'essence. Soit une liste ordinaire d'attributs divins : bont\u00e9, essence, raison, vie, intelligence, sagesse, vertu, b\u00e9atitude, v\u00e9rit\u00e9, \u00e9ternit\u00e9 ; ou bien grandeur, amour, paix, unit\u00e9, perfection. On demande si ces attributs conviennent avec l'essence de Dieu ; s'il faut les comprendre comme des affirmations conditionnelles, ou comme des n\u00e9gations qui marqueraient seulement l'ablation d'un privatif. Mais selon Spinoza, ces questions ne se posent pas, parce que la plupart de ces attributs sont seulement des propres. Et ceux qui ne le sont pas sont des \u00eatres de raison. Ils n'expriment rien de la nature de Dieu, ni n\u00e9gativement ni positivement. _Dieu n'est pas plus cach\u00e9 en eux qu'exprim\u00e9 par eux_. Les propres ne sont ni n\u00e9gatifs ni affirmatifs ; en style kantien, on dirait qu'ils sont ind\u00e9finis. Quand on confond la nature divine avec les propres, il est in\u00e9vitable qu'on ait de Dieu une id\u00e9e elle-m\u00eame ind\u00e9finie. On oscille alors entre une conception \u00e9minente de la n\u00e9gation et une conception analogique de l'affirmation. Chacune, dans son dynamisme, implique un peu de l'autre. On se fait une fausse conception de la n\u00e9gation parce qu'on introduit l'analogie dans l'affirm\u00e9. Mais l'affirmation n'en est plus une quand elle cesse d'\u00eatre univoque, ou de s'affirmer formellement de ses objets.\n\nQue la nature de Dieu n'a jamais \u00e9t\u00e9 d\u00e9finie, parce qu'elle fut toujours confondue avec les \u00ab propres \u00bb, c'est une des th\u00e8ses principales de Spinoza. Elle explique son attitude \u00e0 l'\u00e9gard des _th\u00e9ologiens_. Mais les philosophes ont suivi la th\u00e9ologie : Descartes lui-m\u00eame croit que la nature de Dieu consiste dans l'infiniment parfait. L'infiniment parfait, pourtant, n'est qu'une modalit\u00e9 de ce qui constitue la nature divine. Seuls les attributs au vrai sens du mot, la pens\u00e9e, l'\u00e9tendue, sont les \u00e9l\u00e9ments constitutifs de Dieu, ses expressions constituantes, ses affirmations, ses raisons positives et formelles, en un mot sa nature. Mais pr\u00e9cis\u00e9ment, n'\u00e9tant pas cach\u00e9s par vocation, on se demandera pourquoi ces attributs furent ignor\u00e9s, pourquoi Dieu fut d\u00e9natur\u00e9, confondu avec ses propres qui en donnaient une image ind\u00e9finie. Il faut trouver une raison capable d'expliquer pourquoi, malgr\u00e9 tout leur g\u00e9nie, les pr\u00e9d\u00e9cesseurs de Spinoza s'en tinrent aux propri\u00e9t\u00e9s et ne surent pas d\u00e9couvrir la nature de Dieu.\n\nLa r\u00e9ponse de Spinoza est simple : on manquait d'une m\u00e9thode historique, critique et interne, capable d'interpr\u00e9ter l'\u00c9criture2. On ne se demandait pas quel \u00e9tait le projet des textes sacr\u00e9s. On les consid\u00e9rait comme la Parole de Dieu, la mani\u00e8re dont Dieu s'exprimait. Ce qu'ils disaient de Dieu nous en paraissait tout l'\u00ab exprim\u00e9 \u00bb, ce qu'ils ne disaient pas semblait inexprimable3. \u00c0 aucun moment nous ne demandions : la r\u00e9v\u00e9lation religieuse porte-t-elle sur la nature de Dieu ? A-t-elle pour but de nous faire conna\u00eetre cette nature ? Est-elle justiciable des traitements, positif ou n\u00e9gatif, qu'on pr\u00e9tend lui appliquer pour achever la d\u00e9termination de cette nature ? En v\u00e9rit\u00e9, la r\u00e9v\u00e9lation concerne seulement certains propres. Elle ne se propose nullement de nous faire conna\u00eetre la nature divine et ses attributs. Sans doute les donn\u00e9es de l'\u00c9criture sont-elles h\u00e9t\u00e9rog\u00e8nes : nous nous trouvons tant\u00f4t devant des enseignements rituels particuliers, tant\u00f4t devant des enseignements moraux universels, tant\u00f4t m\u00eame devant un enseignement sp\u00e9culatif, le minimum de sp\u00e9culation n\u00e9cessaire \u00e0 l'enseignement moral. Mais nul attribut de Dieu n'est jamais r\u00e9v\u00e9l\u00e9. Rien que des \u00ab signes \u00bb variables, d\u00e9nominations extrins\u00e8ques qui garantissent un commandement divin. Au mieux, des \u00ab propres \u00bb, comme l'existence divine, l'unit\u00e9, l'omniscience et l'omnipr\u00e9sence, qui garantissent un enseignement moral4. Car le but de l'\u00c9criture est de nous soumettre \u00e0 des mod\u00e8les de vie, de nous faire ob\u00e9ir et de fonder l'ob\u00e9issance. Il serait absurde, alors, de croire que la connaissance puisse se substituer \u00e0 la r\u00e9v\u00e9lation : comment la nature divine suppos\u00e9e connue pourrait-elle servir de r\u00e8gle pratique dans la vie quotidienne ? Mais il est plus absurde encore de croire que la r\u00e9v\u00e9lation nous fasse conna\u00eetre quelque chose de la nature ou de l'essence de Dieu. Cette absurdit\u00e9 pourtant traverse toute la th\u00e9ologie. _Et, de l\u00e0, elle compromet la philosophie tout enti\u00e8re_. Tant\u00f4t l'on fait subir aux propres de la r\u00e9v\u00e9lation un traitement sp\u00e9cial qui les r\u00e9concilie avec la raison ; tant\u00f4t m\u00eame on d\u00e9couvre des propres de la raison, distincts de ceux de la r\u00e9v\u00e9lation. Mais on ne sort pas ainsi de la th\u00e9ologie ; toujours on compte sur des propri\u00e9t\u00e9s pour exprimer la nature de Dieu. On m\u00e9conna\u00eet leur diff\u00e9rence de nature avec les v\u00e9ritables attributs. Or il est in\u00e9vitable que Dieu soit toujours \u00e9minent par rapport \u00e0 ses propres. D\u00e8s qu'on leur pr\u00eate une valeur expressive qu'ils n'ont pas, on pr\u00eate \u00e0 la substance divine une nature inexprimable qu'elle n'a pas davantage.\n\nJamais ne fut pouss\u00e9 plus loin l'effort pour distinguer deux domaines : la r\u00e9v\u00e9lation et l'expression. Ou deux relations h\u00e9t\u00e9rog\u00e8nes : celle du signe et du signifi\u00e9, celle de l'expression et de l'exprim\u00e9. _Le signe se rattache toujours \u00e0 un propre_ ; il signifie toujours un commandement ; et il fonde notre ob\u00e9issance. _L'expression concerne toujours un attribut_ ; elle exprime une essence, c'est-\u00e0-dire une nature \u00e0 l'infinitif ; elle nous la fait conna\u00eetre. Si bien que la \u00ab Parole de Dieu \u00bb a deux sens tr\u00e8s divers : une Parole expressive, qui n'a pas besoin de mots ni de signes, mais seulement de l'essence de Dieu et de l'entendement de l'homme. Une Parole impresse, imp\u00e9rative, op\u00e9rant par signe et commandement : elle n'est pas expressive, mais frappe notre imagination et nous inspire la soumission n\u00e9cessaire5. Dira-t-on au moins que les commandements \u00ab expriment \u00bb les volont\u00e9s de Dieu ? Ce serait encore pr\u00e9juger de la volont\u00e9 comme appartenant \u00e0 la nature de Dieu, prendre un \u00eatre de raison, une d\u00e9termination extrins\u00e8que, pour un attribut divin. Tout m\u00e9lange des deux domaines est ruineux. Chaque fois qu'on fait d'un signe une expression, on voit des myst\u00e8res partout, d'abord et y compris dans l'\u00c9criture elle-m\u00eame. Tels les Juifs qui pensent que tout exprime Dieu, sans condition6. On se fait alors une conception mystique de l'expression : celle-ci ne nous para\u00eet pas moins cacher que r\u00e9v\u00e9ler ce qu'elle exprime. Les \u00e9nigmes, les paraboles, les symboles, les analogies, les m\u00e9tonymies viennent ainsi troubler l'ordre rationnel et positif de l'expression pure. En v\u00e9rit\u00e9 l'\u00c9criture est bien Parole de Dieu, mais parole de commandement : imp\u00e9rative, elle n'exprime rien, parce qu'elle ne fait conna\u00eetre aucun attribut divin.\n\nL'analyse de Spinoza ne se contente pas de marquer l'irr\u00e9ductibilit\u00e9 des domaines. Elle propose une explication des signes, qui est comme la gen\u00e8se d'une illusion. Il n'est pas faux de dire, en effet, que chaque chose exprime Dieu. L'ordre de la nature enti\u00e8re est expressif. Mais il suffit de mal comprendre une loi naturelle pour la saisir comme un imp\u00e9ratif ou un commandement. Quand Spinoza illustrera les diff\u00e9rents genres de connaissance par l'exemple fameux des nombres proportionnels, il montrera que, au plus bas degr\u00e9, nous ne comprenons pas la r\u00e8gle de proportionnalit\u00e9 : alors nous en retenons un signe, qui nous dit quelle op\u00e9ration nous _devons_ faire sur ces nombres. M\u00eame les r\u00e8gles techniques prennent un aspect moral quand nous ignorons leur sens et n'en retenons qu'un signe. \u00c0 plus forte raison les lois de nature. Dieu r\u00e9v\u00e8le \u00e0 Adam que l'ingestion de la pomme aurait pour lui des cons\u00e9quences funestes ; mais Adam, impuissant \u00e0 saisir les rapports constitutifs des choses, imagine cette loi de nature comme une loi morale qui lui d\u00e9fend de manger du fruit, et Dieu lui-m\u00eame comme un souverain qui le sanctionne parce qu'il en a mang\u00e97. Le signe est la chose des proph\u00e8tes ; mais pr\u00e9cis\u00e9ment les proph\u00e8tes ont l'imagination forte et l'entendement faible8. Les expressions de Dieu ne tombent jamais dans l'imagination ; celle-ci saisit tout sous l'aspect du signe et du commandement.\n\nDieu ne s'exprime ni par des signes, ni dans des propres. Quand nous lisons dans l' _Exode_ que Dieu s'est r\u00e9v\u00e9l\u00e9 \u00e0 Abraham, \u00e0 Isaac et \u00e0 Jacob, mais comme _Dieu Sada\u00ef_ (suffisant aux besoins de chacun) et non comme J\u00e9hovah, nous ne devons pas conclure au myst\u00e8re du t\u00e9tragramme, ni \u00e0 la sur\u00e9minence de Dieu pris dans sa nature absolue. Nous devons conclure plut\u00f4t que la r\u00e9v\u00e9lation n'a pas pour objet d'exprimer cette nature ou essence9. En revanche, la connaissance naturelle implique l'essence de Dieu ; et elle l'implique, parce qu'elle est connaissance des attributs qui expriment effectivement cette essence. Dieu s'exprime dans ses attributs, les attributs s'expriment dans les modes qui en d\u00e9pendent : c'est par l\u00e0 que l'ordre de la nature manifeste Dieu. Les seuls noms expressifs de Dieu, les seules expressions divines sont donc les attributs : formes communes qui se disent de la substance et des modes. Si nous n'en connaissons que deux, c'est pr\u00e9cis\u00e9ment parce que nous sommes constitu\u00e9s par un mode de l'\u00e9tendue et un mode de la pens\u00e9e. Du moins ces attributs ne supposent-ils aucune r\u00e9v\u00e9lation ; ils renvoient \u00e0 la lumi\u00e8re naturelle. Nous les connaissons tels qu'ils sont en Dieu, dans leur \u00eatre commun \u00e0 la substance et aux modes. Spinoza insiste sur ce point, en citant un texte de saint Paul dont il fait presque un manifeste de l'univocit\u00e9 : \u00ab Les choses divines cach\u00e9es depuis les fondements du monde sont aper\u00e7ues par l'entendement dans les cr\u00e9atures de Dieu10... \u00bb Il semble que l'univocit\u00e9 des attributs se confonde avec leur expressivit\u00e9 : de mani\u00e8re indissoluble, les attributs sont expressifs et univoques.\n\nLes attributs ne servent pas \u00e0 nier, pas plus qu'on ne les nie de l'essence. Pas davantage on ne les affirme de Dieu par analogie. Une affirmation par analogie ne vaut pas mieux qu'une n\u00e9gation par \u00e9minence (il y a encore de l'\u00e9minence dans le premier cas, d\u00e9j\u00e0 de l'analogie dans le second). Il est vrai, dit Spinoza, qu'un attribut est _ni\u00e9_ d'un autre11. Mais en quel sens ? \u00ab Si l'on dit que l'\u00e9tendue n'est pas limit\u00e9e par l'\u00e9tendue, mais par la pens\u00e9e, cela ne revient-il pas \u00e0 dire que l'\u00e9tendue n'est pas infinie absolument, mais seulement en tant qu'\u00e9tendue12 ? \u00bb La n\u00e9gation, ici, n'implique donc aucune opposition ni privation. L'\u00e9tendue comme telle ne souffre d'aucune imperfection ou limitation qui d\u00e9pendrait de sa nature ; aussi bien est-il vain d'imaginer un Dieu qui poss\u00e9derait \u00ab \u00e9minemment \u00bb l'\u00e9tendue13. Inversement, en quel sens l'attribut est-il affirm\u00e9 de la substance ? Spinoza insiste souvent sur ce point : les substances ou les attributs existent _formellement_ dans la Nature. Or, parmi les nombreux sens du mot \u00ab formel \u00bb, nous devons tenir compte de celui par lequel il s'oppose \u00e0 \u00ab \u00e9minent \u00bb ou \u00e0 \u00ab analogue \u00bb. Jamais la substance ne doit \u00eatre pens\u00e9e comme comprenant \u00e9minemment ses attributs ; \u00e0 leur tour, les attributs ne doivent pas \u00eatre pens\u00e9s comme contenant \u00e9minemment les essences de mode. Les attributs s'affirment formellement de la substance. Les attributs se disent formellement de la substance dont ils constituent l'essence, et des modes dont ils contiennent les essences. Spinoza ne cesse de rappeler le caract\u00e8re affirmatif des attributs qui d\u00e9finissent la substance, comme la n\u00e9cessit\u00e9 pour toute bonne d\u00e9finition d'\u00eatre elle-m\u00eame affirmative14. Les attributs sont des affirmations. Mais l'affirmation, dans son essence, est toujours formelle, actuelle, univoque : c'est en ce sens qu'elle est expressive.\n\nLa philosophie de Spinoza est une philosophie de l'affirmation pure. L'affirmation est le principe sp\u00e9culatif dont toute l' _\u00c9thique_ d\u00e9pend. \u00c0 ce point, nous pouvons chercher comment Spinoza rencontre, pour s'en servir, une id\u00e9e cart\u00e9sienne. Car la distinction r\u00e9elle tendait \u00e0 donner au concept d'affirmation une v\u00e9ritable logique. En effet, la distinction r\u00e9elle telle que Descartes l'utilisait nous mettait sur la voie d'une d\u00e9couverte profonde : les termes distingu\u00e9s conservaient toute leur positivit\u00e9 respective, au lieu de se d\u00e9finir par opposition l'un avec l'autre. _Non opposita sed diversa_ , telle \u00e9tait la formule de la nouvelle logique15. La distinction r\u00e9elle semblait annoncer une nouvelle conception du n\u00e9gatif, sans opposition ni privation, mais aussi une nouvelle conception de l'affirmation, sans \u00e9minence et sans analogie. Or si cette voie n'aboutit pas dans le cart\u00e9sianisme, c'est pour une raison que nous avons vue pr\u00e9c\u00e9demment : Descartes donne encore \u00e0 la distinction r\u00e9elle une valeur num\u00e9rique, une fonction de division substantielle dans la nature et dans les choses. Il con\u00e7oit toute qualit\u00e9 comme positive, toute r\u00e9alit\u00e9 comme perfection ; mais tout n'est pas r\u00e9alit\u00e9 dans une substance qualifi\u00e9e et distingu\u00e9e, tout n'est pas perfection dans la nature d'une chose. C'est \u00e0 Descartes, entre autres, que Spinoza pense quand il \u00e9crit : \u00ab Dire que la nature de la chose exigeait la limitation et par suite ne pouvait \u00eatre autrement, c'est ne rien dire, car la nature d'une chose ne peut rien exiger tant qu'elle n'est pas16. \u00bb Chez Descartes, il y a des limitations que la chose \u00ab exige \u00bb en vertu de sa nature, des id\u00e9es qui ont si peu de r\u00e9alit\u00e9 qu'on pourrait presque dire qu'elles proc\u00e8dent du n\u00e9ant, des natures auxquelles manque quelque chose. Par l\u00e0 se r\u00e9introduit tout ce que la logique de la distinction r\u00e9elle \u00e9tait cens\u00e9e chasser : la privation, l'\u00e9minence. Nous verrons que _l'\u00e9minence, l'analogie, m\u00eame une certaine \u00e9quivocit\u00e9 restent des cat\u00e9gories presque spontan\u00e9es de la pens\u00e9e cart\u00e9sienne_. Au contraire, pour d\u00e9gager les cons\u00e9quences extr\u00eames de la distinction r\u00e9elle con\u00e7ue comme logique de l'affirmation, il fallait s'\u00e9lever jusqu'\u00e0 l'id\u00e9e d'une seule substance ayant tous les attributs r\u00e9ellement distincts. Il fallait d'abord \u00e9viter toute confusion, non seulement des attributs et des modes, mais des attributs et des propres.\n\nLes attributs sont les affirmations de Dieu, les _logoi_ ou les vrais noms divins. Revenons au texte o\u00f9 Spinoza invoque l'exemple d'Isra\u00ebl, ainsi nomm\u00e9 comme patriarche, mais appel\u00e9 Jacob par rapport \u00e0 son fr\u00e8re17. D'apr\u00e8s le contexte, il s'agit d'illustrer la distinction de raison telle qu'elle est entre la substance et l'attribut : Isra\u00ebl est dit Jacob ( _Supplantator_ ) par rapport \u00e0 son fr\u00e8re, comme \u00ab plan \u00bb est dit \u00ab blanc \u00bb par rapport \u00e0 un homme qui le regarde, comme la substance est dite telle ou telle par rapport \u00e0 l'entendement qui lui \u00ab attribue \u00bb telle ou telle essence. Il est certain que ce passage favorise une interpr\u00e9tation intellectualiste ou m\u00eame id\u00e9aliste des attributs. Mais un philosophe est toujours amen\u00e9 \u00e0 simplifier sa pens\u00e9e dans certaines occasions, ou \u00e0 la formuler partiellement. Spinoza ne manque pas de souligner l'ambigu\u00eft\u00e9 des exemples qu'il cite. En v\u00e9rit\u00e9 l'attribut n'est pas une simple fa\u00e7on de voir ou de concevoir ; sa relation avec l'entendement est bien fondamentale, mais s'interpr\u00e8te autrement. C'est parce que les attributs sont eux-m\u00eames des _expressions_ qu'ils renvoient n\u00e9cessairement \u00e0 l'entendement comme \u00e0 la seule instance qui per\u00e7oit _l'exprim\u00e9_. C'est parce que les attributs expliquent la substance, qu'ils sont, par l\u00e0-m\u00eame, relatifs \u00e0 un entendement dans lequel toutes les explications se reproduisent, ou \u00ab s'expliquent \u00bb elles-m\u00eames objectivement. Alors le probl\u00e8me tend \u00e0 se pr\u00e9ciser : les attributs sont des expressions, mais comment des expressions diff\u00e9rentes peuvent-elles d\u00e9signer une seule et m\u00eame chose ? Comment des noms diff\u00e9rents peuvent-ils avoir un m\u00eame d\u00e9sign\u00e9 ? \u00ab Vous d\u00e9sirez que je montre par un exemple comment une seule et m\u00eame chose peut \u00eatre d\u00e9sign\u00e9e ( _insigniri_ ) par deux noms. \u00bb\n\nLe r\u00f4le de l'entendement est celui qui lui revient dans une logique de l'expression. Cette logique est le r\u00e9sultat d'une longue tradition, sto\u00efcienne et m\u00e9di\u00e9vale. On distingue dans une expression (par exemple dans une proposition) ce qu'elle exprime et ce qu'elle d\u00e9signe18. L'exprim\u00e9 est comme le sens qui n'existe pas hors de l'expression ; il renvoie donc \u00e0 un entendement qui le saisit objectivement, c'est-\u00e0-dire id\u00e9alement. Mais il se dit de la chose, et non de l'expression elle-m\u00eame ; l'entendement le rapporte \u00e0 l'objet d\u00e9sign\u00e9, comme l'essence de cet objet. On con\u00e7oit d\u00e8s lors que des noms puissent se distinguer par leur sens, mais que ces sens diff\u00e9rents soient rapport\u00e9s au m\u00eame objet d\u00e9sign\u00e9 dont ils constituent l'essence. Il y a dans la conception spinoziste des attributs une sorte de transposition de cette th\u00e9orie du sens. Chaque attribut est un nom ou une expression distincte ; ce qu'il exprime est comme son sens ; mais s'il est vrai que l'exprim\u00e9 n'existe pas hors de l'attribut, il n'en est pas moins rapport\u00e9 \u00e0 la substance comme \u00e0 l'objet d\u00e9sign\u00e9 par tous les attributs ; ainsi tous les sens exprim\u00e9s forment l'\u00ab exprimable \u00bb ou l'essence de la substance. Celle-ci sera dite \u00e0 son tour s'exprimer dans les attributs.\n\nIl est vrai qu'en assimilant la substance \u00e0 l'objet d\u00e9sign\u00e9 par diff\u00e9rents noms, nous ne r\u00e9solvons pas le probl\u00e8me essentiel, celui de la diff\u00e9rence entre ces noms. Bien plus, la difficult\u00e9 augmente dans la mesure o\u00f9 ces noms sont univoques et positifs, donc s'appliquent formellement \u00e0 ce qu'ils d\u00e9signent : leur sens respectif semble introduire dans l'unit\u00e9 du d\u00e9sign\u00e9 une multiplicit\u00e9 n\u00e9cessairement actuelle. Il n'en est pas ainsi dans une vision analogique : les noms s'appliquent \u00e0 Dieu par analogie, leur sens \u00ab pr\u00e9existe \u00bb en lui sur un mode \u00e9minent qui en assure l'inconcevable unit\u00e9, l'inexprimable unit\u00e9. Mais que faire si les noms divins ont le m\u00eame sens, tels qu'ils sont appliqu\u00e9s \u00e0 Dieu et tels qu'ils sont impliqu\u00e9s dans les cr\u00e9atures, c'est-\u00e0-dire dans tous les emplois qu'on en fait, si bien que leur distinction ne peut plus se fonder sur les choses cr\u00e9\u00e9es, mais doit \u00eatre fond\u00e9e dans ce Dieu qu'ils d\u00e9signent ? On sait que Duns Scot, au Moyen \u00c2ge, avait pos\u00e9 ce probl\u00e8me et lui avait donn\u00e9 une solution profonde. Duns Scot sans doute est celui qui mena le plus loin l'entreprise d'une th\u00e9ologie positive. Il d\u00e9nonce \u00e0 la fois l'\u00e9minence n\u00e9gative des n\u00e9o-platoniciens, la pseudo-affirmation des thomistes. Il leur oppose l'univocit\u00e9 de l'\u00catre : _l'\u00eatre se dit au m\u00eame sens_ de tout ce qui est, infini ou fini, bien que ce ne soit pas sous la m\u00eame \u00ab modalit\u00e9 \u00bb. Mais pr\u00e9cis\u00e9ment, l'\u00eatre ne change pas de nature en changeant de modalit\u00e9, c'est-\u00e0-dire quand son concept est pr\u00e9diqu\u00e9 de l'\u00eatre infini et des \u00eatres finis (d\u00e9j\u00e0 chez Scot, l'univocit\u00e9 n'entra\u00eene donc aucune confusion d'essences)19. Et l'univocit\u00e9 de l'\u00eatre entra\u00eene elle-m\u00eame l'univocit\u00e9 des attributs divins : le concept d'un attribut qui peut \u00eatre \u00e9lev\u00e9 \u00e0 l'infini est commun lui-m\u00eame \u00e0 Dieu et aux cr\u00e9atures, \u00e0 condition d'\u00eatre pris dans sa raison formelle ou dans sa quiddit\u00e9, car \u00ab l'infinit\u00e9 ne supprime nullement la raison formelle de ce \u00e0 quoi on l'ajoute20 \u00bb. Mais, se disant formellement et positivement de Dieu, comment les attributs infinis ou les noms divins n'introduiraient-ils pas en Dieu une pluralit\u00e9 correspondante \u00e0 leurs raisons formelles, \u00e0 leurs quiddit\u00e9s distinctes ?\n\nC'est \u00e0 ce probl\u00e8me que Scot applique un de ses concepts les plus originaux, qui vient compl\u00e9ter celui de l'univocit\u00e9 : l'id\u00e9e de la distinction formelle21. Celle-ci concerne l'appr\u00e9hension de quiddit\u00e9s distinctes qui n'en appartiennent pas moins \u00e0 un m\u00eame sujet. Elle renvoie \u00e9videmment \u00e0 un acte de l'entendement. Mais l'entendement ne se contente pas ici d'exprimer une m\u00eame r\u00e9alit\u00e9 sous deux aspects qui pourraient exister \u00e0 part dans d'autres sujets, ni d'exprimer une m\u00eame chose \u00e0 divers degr\u00e9s d'abstraction, ni d'exprimer quelque chose analogiquement par rapport \u00e0 d'autres r\u00e9alit\u00e9s. Il appr\u00e9hende objectivement des formes actuellement distinctes, mais qui, comme telles, composent un seul et m\u00eame sujet. Entre animal et raisonnable, il n'y a pas seulement une distinction de raison comme entre _homo-humanitas_ ; il faut que la chose elle-m\u00eame soit d\u00e9j\u00e0 \u00ab structur\u00e9e selon la diversit\u00e9 pensable du genre et de l'esp\u00e8ce22 \u00bb. La distinction formelle est bien une distinction r\u00e9elle, parce qu'elle exprime les diff\u00e9rentes couches de r\u00e9alit\u00e9s qui forment ou constituent un \u00eatre. En ce sens elle est dite _formalis a parte rei_ ou _actualis ex natura rei_. Mais elle est un minimum de distinction r\u00e9elle, parce que les deux quiddit\u00e9s r\u00e9ellement distinctes se coordonnent et composent un \u00eatre unique23. _R\u00e9elle et pourtant non num\u00e9rique_ , tel est le statut de la distinction formelle24. Encore doit-on reconna\u00eetre que, dans le fini, deux quiddit\u00e9s comme animal et raisonnable ne communiquent que par le troisi\u00e8me terme auquel elles sont identiques. Mais il n'en est pas de m\u00eame dans l'infini. Deux attributs port\u00e9s \u00e0 l'infini seront encore formellement distincts, tout en \u00e9tant ontologiquement identiques. Comme dit \u00c9. Gilson, \u00ab parce qu'elle est une modalit\u00e9 de l'\u00eatre (et non un attribut), l'infinit\u00e9 peut \u00eatre commune \u00e0 des raisons formelles quidditativement irr\u00e9ductibles, et leur conf\u00e9rer l'identit\u00e9 dans l'\u00eatre sans supprimer leur distinction dans la formalit\u00e925. \u00bb Deux attributs de Dieu, par exemple Justice et Bont\u00e9, sont donc des noms divins qui d\u00e9signent un Dieu absolument un, tout en signifiant des quiddit\u00e9s distinctes. Il y a l\u00e0 comme deux ordres, l'ordre de la raison formelle et l'ordre de l'\u00eatre, la pluralit\u00e9 de l'un se conciliant parfaitement avec la simplicit\u00e9 de l'autre.\n\nC'est ce statut qui trouve en Suarez un adversaire d\u00e9clar\u00e9. Celui-ci ne voit pas comment la distinction formelle ne se r\u00e9duirait pas soit \u00e0 une distinction de raison, soit \u00e0 une distinction modale26. Elle en dit trop ou pas assez : trop pour une distinction de raison, mais pas assez pour une distinction r\u00e9elle. Descartes, \u00e0 l'occasion, a la m\u00eame attitude27. Nous retrouvons toujours chez Descartes la m\u00eame r\u00e9pugnance \u00e0 concevoir une distinction r\u00e9elle entre choses qui ne seraient pas dans des sujets diff\u00e9rents, c'est-\u00e0-dire qui ne s'accompagneraient pas d'une division dans l'\u00eatre ou d'une distinction num\u00e9rique. Or, il n'en est pas de m\u00eame chez Spinoza : dans sa conception d'une distinction r\u00e9elle non num\u00e9rique, on n'aura pas de peine \u00e0 retrouver la distinction formelle de Scot. Bien plus, la distinction formelle cesse avec Spinoza d'\u00eatre un minimum de distinction r\u00e9elle, elle devient toute la distinction r\u00e9elle, donnant \u00e0 celle-ci un statut exclusif.\n\n1o) Les attributs chez Spinoza sont r\u00e9ellement distincts, ou con\u00e7us comme r\u00e9ellement distincts. En effet, ils ont des raisons formelles irr\u00e9ductibles ; chaque attribut exprime une essence infinie comme sa raison formelle ou sa quiddit\u00e9. Les attributs se distinguent donc \u00ab quidditativement \u00bb, formellement : ce sont bien des substances, en un sens purement qualitatif ; 2o) Chacun attribue son essence \u00e0 la substance comme \u00e0 _autre chose_. Fa\u00e7on de dire que, \u00e0 la distinction formelle entre attributs, ne correspond aucune division dans l'\u00eatre. La substance n'est pas un genre, les attributs ne sont pas des diff\u00e9rences sp\u00e9cifiques : il n'y a donc pas de substances de _m\u00eame_ esp\u00e8ce que les attributs, il n'y a pas de substance qui serait _la m\u00eame chose_ ( _res_ ) que chaque attribut ( _formalitas_ ) ; 3o) Cette \u00ab autre chose \u00bb est donc _la m\u00eame pour_ tous les attributs. Bien plus : elle est _la m\u00eame que_ tous les attributs. Cette derni\u00e8re d\u00e9termination ne contredit nullement la pr\u00e9c\u00e9dente. Tous les attributs formellement distincts sont rapport\u00e9s par l'entendement \u00e0 une substance ontologiquement une. Mais l'entendement ne fait que reproduire objectivement la nature des formes qu'il appr\u00e9hende. Toutes les essences formelles forment l'essence d'une substance absolument une. Toutes les substances qualifi\u00e9es forment une seule substance du point de vue de la quantit\u00e9. Si bien que les attributs eux-m\u00eames ont \u00e0 la fois l'identit\u00e9 dans l'\u00eatre, la distinction dans la formalit\u00e9 ; ontologiquement un, formellement divers, tel est le statut des attributs.\n\nMalgr\u00e9 son allusion au \u00ab fatras des distinctions p\u00e9ripat\u00e9ticiennes \u00bb, Spinoza restaure la distinction formelle, lui assurant m\u00eame une port\u00e9e qu'elle n'avait pas chez Scot. _C'est la distinction formelle qui donne un concept absolument coh\u00e9rent de l'unit\u00e9 de la substance et de la pluralit\u00e9 des attributs, c'est elle qui donne \u00e0 la distinction r\u00e9elle une nouvelle logique_. On demandera alors pourquoi Spinoza n'emploie jamais ce terme, mais parle seulement de distinction r\u00e9elle. C'est que la distinction formelle est bien une distinction r\u00e9elle. Ensuite, Spinoza avait tout avantage \u00e0 utiliser un terme que Descartes, par l'emploi qu'il en avait fait, avait en quelque sorte neutralis\u00e9 th\u00e9ologiquement ; le terme \u00ab distinction r\u00e9elle \u00bb permettait alors les plus grandes audaces, sans ressusciter d'anciennes pol\u00e9miques que Spinoza jugeait sans doute inutiles et m\u00eame nuisibles. Nous ne croyons pas que le pr\u00e9tendu cart\u00e9sianisme de Spinoza aille plus loin : toute sa th\u00e9orie des distinctions est profond\u00e9ment anticart\u00e9sienne.\n\n\u00c0 proposer l'image d'un Spinoza scotiste et non cart\u00e9sien, nous risquons de tomber dans certaines exag\u00e9rations. En fait, nous voulons dire que les th\u00e9ories scotistes furent certainement connues de Spinoza, et qu'elles particip\u00e8rent, avec d'autres th\u00e8mes, \u00e0 la formation de son panth\u00e9isme28. Le plus int\u00e9ressant d\u00e8s lors est la mani\u00e8re dont Spinoza utilise et renouvelle les notions de distinction formelle et d'univocit\u00e9. Qu'est-ce que Duns Scot, en effet, appelait \u00ab attribut \u00bb ? Justice, bont\u00e9, sagesse, etc., bref des propres. Sans doute reconnaissait-il que l'essence divine peut \u00eatre con\u00e7ue sans ces attributs ; mais il d\u00e9finissait l'essence de Dieu par des perfections intrins\u00e8ques, entendement et volont\u00e9. Scot \u00e9tait \u00ab th\u00e9ologien \u00bb et, \u00e0 ce titre, restait aux prises avec des propres et des \u00eatres de raison. C'est pourquoi, chez lui, la distinction formelle n'avait pas toute sa port\u00e9e, s'exer\u00e7ant toujours sur des \u00eatres de raison, comme les genres et les esp\u00e8ces, comme les facult\u00e9s de l'\u00e2me, ou bien sur des propres, comme ces pr\u00e9tendus attributs de Dieu. Plus encore, l'univocit\u00e9 chez Scot semblait compromise par le souci d'\u00e9viter le panth\u00e9isme. Car la perspective th\u00e9ologique, c'est-\u00e0-dire \u00ab cr\u00e9ationniste \u00bb, le for\u00e7ait \u00e0 concevoir l'\u00catre univoque comme un concept _neutralis\u00e9, indiff\u00e9rent_. Indiff\u00e9rent au fini et \u00e0 l'infini, au singulier et \u00e0 l'universel, au parfait et \u00e0 l'imparfait, au cr\u00e9\u00e9 et \u00e0 l'incr\u00e9\u00e929. Chez Spinoza au contraire, l'\u00catre univoque est parfaitement d\u00e9termin\u00e9 dans son concept comme ce qui se dit en un seul et m\u00eame sens de la substance qui est en soi, et des modes qui sont en autre chose. Avec Spinoza, l'univocit\u00e9 devient l'objet d'affirmation pure. La m\u00eame chose, _formaliter_ , constitue l'essence de la substance et contient les essences de mode. C'est donc l'id\u00e9e de cause immanente qui, chez Spinoza, prend le relais de l'univocit\u00e9, lib\u00e9rant celle-ci de l'indiff\u00e9rence et de la neutralit\u00e9 o\u00f9 la maintenait la th\u00e9orie d'une cr\u00e9ation divine. Et c'est dans l'immanence que l'univocit\u00e9 trouvera sa formule proprement spinoziste : Dieu est dit cause de toutes choses _au sens m\u00eame_ ( _eo sensu_ ) o\u00f9 il est dit cause de soi.\n\n* * *\n\n1.  Sur tous ces points, cf. M. de GANDILLAC, _Introduction aux \u0153uvres compl\u00e8tes du Pseudo-Denys_ (Aubier, 1941) ; et _La Philosophie de Nicolas de Cues_ (Aubier, 1943). Dans ce dernier ouvrage, M. de Gandillac montre bien comment la th\u00e9ologie n\u00e9gative d'une part, l'analogie d'autre part combinent toutes deux les affirmations et les n\u00e9gations, mais dans un rapport inverse : \u00ab \u00c0 l'inverse donc de Denys, qui r\u00e9duisait les affirmations elles-m\u00eames \u00e0 des n\u00e9gations d\u00e9guis\u00e9es, saint Thomas... usera surtout de l'apophase pour remonter de telle ou telle n\u00e9gation pr\u00e9alable \u00e0 quelque attribut positif. De l'impossibilit\u00e9 du mouvement divin, il tirera par exemple une preuve de l'\u00c9ternit\u00e9 divine ; de l'exclusion de la mati\u00e8re, il fera un argument d\u00e9cisif en faveur de la co\u00efncidence en Dieu de l'essence et de l'existence \u00bb (p. 272.)\n\n2.  _TTP_ , ch. 7 (II ; p. 185) \u00ab ... La voie que (cette m\u00e9thode) enseigne, qui est la droite et la vraie, n'a jamais \u00e9t\u00e9 suivie ni fray\u00e9e par les hommes, de sorte qu'\u00e0 la longue elle est devenue tr\u00e8s ardue et presque impraticable. \u00bb Et ch. 8 (II, p. 191) : \u00ab Je crains toutefois que ma tentative ne survienne trop tard... \u00bb\n\n3.  _TTP_ , ch. 2 (II, p. 113) : \u00ab Avec une surprenante pr\u00e9cipitation, tout le monde s'est persuad\u00e9 que les proph\u00e8tes ont eu la science de tout ce que l'entendement humain peut saisir. Et, bien que certains passages de l'\u00c9criture nous disent le plus clairement que les proph\u00e8tes ont ignor\u00e9 certaines choses, on aime mieux d\u00e9clarer qu'on n'entend pas ces passages que d'accorder que les proph\u00e8tes aient ignor\u00e9 quelque chose, ou bien l'on s'efforce de torturer les textes de l'\u00c9criture pour lui faire dire ce que manifestement elle ne veut pas dire. \u00bb\n\n4.  Cf. _TTP_ , ch. 14 : la liste des \u00ab dogmes de la foi \u00bb. On remarquera que, m\u00eame du point de vue des \u00ab propres \u00bb, la r\u00e9v\u00e9lation reste limit\u00e9e. Tout est centr\u00e9 sur justice et charit\u00e9. L'infinit\u00e9, notamment, ne semble pas r\u00e9v\u00e9l\u00e9e dans l'\u00c9criture ; cf. ch. 2, o\u00f9 Spinoza expose les ignorances d'Adam, d'Abraham et de Mo\u00efse.\n\n5.  Sur les deux sens de la \u00ab Parole de Dieu \u00bb, cf. _TTP_ , ch. 12. D\u00e9j\u00e0 le _Court Trait\u00e9_ opposait la communication imm\u00e9diate \u00e0 la r\u00e9v\u00e9lation par signes : II, ch. 24, 9-11.\n\n6.  _TTP_ , ch. 1 (II, p. 95).\n\n7.  _TTP_ , ch. 4 (II, p. 139). _Lettre 19, \u00e0 Blyenbergh_ (III, p. 65).\n\n8.  Cf. _TTP_ , chapitres 2 et 3.\n\n9.  _TTP_ , ch. 13 (II, pp. 239-240).\n\n10.  _TTP_ , ch. 4 (II, p. 144).\n\n11.  _\u00c9_ , I, def. 6, expl. : \u00ab De ce qui est infini seulement en son genre, nous pouvons nier une infinit\u00e9 d'attributs. \u00bb\n\n12.  _Lettre 4, \u00e0 Oldenburg_ (III, p. 10).\n\n13.  _CT_ , II, ch. 19, 5.\n\n14.  Cf. les formules constantes du _Court Trait\u00e9_ (surtout I, ch. 2) d'apr\u00e8s lesquelles les attributs s'affirment, et s'affirment d'une Nature elle-m\u00eame positive. Et _TRE_ 96 : \u00ab Toute d\u00e9finition doit \u00eatre affirmative. \u00bb\n\n15.  Cf. les remarques de Lewis Robinson \u00e0 cet \u00e9gard, et les textes des Cart\u00e9siens qu'il cite : _Kommentar zu Spinozas Ethik_ , Leipzig, 1928.\n\n16.  _CT_ , I, ch. 2, 5, note. Sur l'imperfection de l'\u00e9tendue selon Descartes, cf. par exemple _Principes_ , I, 23.\n\n17.  _Lettre 9, \u00e0 De Vries_ (III, p. 33).\n\n18.  La distinction de \u00ab l'exprim\u00e9 \u00bb (sens) et du \u00ab d\u00e9sign\u00e9 \u00bb ( _designatum, denominatum_ ) n'est pas r\u00e9cente dans une logique des propositions, bien qu'elle r\u00e9apparaisse chez beaucoup de philosophes modernes. L'origine en est dans la logique sto\u00efcienne, qui distingue l' _exprimable_ et l'objet. Ockham \u00e0 son tour distingue la chose en tant que telle ( _extra animam_ ) et la chose comme _exprim\u00e9e_ dans la proposition ( _declaratio, explicatio, significatio_ sont des synonymes d' _expressio_ ). Certains disciples d'Ockham poussent encore plus loin la distinction, et rejoignent les paradoxes sto\u00efciens, faisant de \u00ab l'exprim\u00e9 \u00bb une entit\u00e9 non existante, irr\u00e9ductible \u00e0 la chose et \u00e0 la proposition : cf. H. \u00c9LIE, _Le Complexe significabile_ (Vrin, 1936). Ces paradoxes de l'expression jouent un grand r\u00f4le dans la logique moderne (Meinong, Frege, Husserl), mais leur source est ancienne.\n\n19.  Duns SCOT, _Opus oxoniense_ (\u00e9d. Viv\u00e8s) : sur la critique de l'\u00e9minence et de l'analogie, I. D3, q. 1, 2 et 3 ; sur l'univocit\u00e9 de l'\u00eatre, I, D8, q. 3. On a souvent remarqu\u00e9 que l'\u00catre univoque laisse subsister la distinction de ses \u00ab modes \u00bb : quand on le consid\u00e8re, non plus dans sa nature en tant qu'\u00catre, mais dans ses modalit\u00e9s individuantes (infini, fini), il cesse d'\u00eatre univoque. Cf. \u00c9. GILSON, _Jean Duns Scot_ , Vrin, 1952, pp. 89, 629.\n\n20.  _Op. ox_ , I, D8, q. 4 (a. 2, n. 13).\n\n21.  _Op. ox_ , I, D2, q. 4 ; D8, q. 4 (cf. \u00c9. GILSON, ch. 3).\n\n22.  M. de GANDILLAC, \u00ab Duns Scot et la Via antiqua \u00bb, in _Le Mouvement doctrinal du IXe au XIVe si\u00e8cle_ (Bloud et Gay, 1951), p. 339.\n\n23.  _Op. ox._ , I, D2, q. 4 (a. 5, n. 43) : La distinction formelle est _minima in suo ordine, id est inter omnes quae praecedunt intellectionem._\n\n24.  _Op. ox._ , II, D3, q. 1 : La forme distincte a une entit\u00e9 r\u00e9elle, _ista unitas est realis, non autem singularis nel numeralis._\n\n25.  \u00c9. Gilson, p. 251.\n\n26.  SUAREZ, _Metaphysicarum Disputationum_ , D7.\n\n27.  CATERUS, dans les _Premi\u00e8res objections_ , avait invoqu\u00e9 la distinction formelle \u00e0 propos de l'\u00e2me et du corps. Descartes r\u00e9pond : \u00ab Pour ce qui regarde la distinction formelle que ce tr\u00e8s docte th\u00e9ologien dit avoir prise de Scot, je r\u00e9ponds bri\u00e8vement qu'elle ne diff\u00e8re point de la modale, et qu'elle ne s'\u00e9tend que sur les \u00eatres incomplets... \u00bb (AT, IX, pp. 94-95).\n\n28.  Il n'y a vraiment pas lieu de se demander si Spinoza a lu Duns Scot. Il est peu vraisemblable qu'il l'ait lu. Mais nous savons, ne serait-ce que par l'inventaire de ce qui restait de sa biblioth\u00e8que, le go\u00fbt de Spinoza pour les trait\u00e9s de m\u00e9taphysique et de logique, du type _quaestiones disputatae_ ; or, ces trait\u00e9s comportent toujours des expos\u00e9s de l'univocit\u00e9 et de la distinction formelle scotistes. De tels expos\u00e9s font partie des lieux communs de la logique et de l'ontologie des XVIe et XVIIe si\u00e8cles (cf. par exemple Heereboord dans son _Collegium logicum_ ). Nous savons aussi, gr\u00e2ce aux travaux de Gebhardt et de R\u00e9vah, l'influence probable de Juan de Prado sur Spinoza ; or Juan de Prado avait une connaissance certaine de Duns Scot (cf. I.S. R\u00c9VAH, _Spinoza et Juan de Prado_ , \u00e9d. Mouton, 1959, p. 45).\n\nOn ajoutera que les probl\u00e8mes d'une th\u00e9ologie n\u00e9gative ou positive, d'une analogie ou d'une univocit\u00e9 de l'\u00eatre, et d'un statut correspondant des distinctions, ne sont nullement propres \u00e0 la pens\u00e9e chr\u00e9tienne. On les trouve, aussi vivaces, dans la pens\u00e9e juive du Moyen \u00c2ge. Certains commentateurs ont soulign\u00e9 l'influence de Hasda\u00ef Crescas sur Spinoza, en ce qui concerne la th\u00e9orie de l'\u00e9tendue. Mais plus g\u00e9n\u00e9ralement, Crescas semble avoir \u00e9labor\u00e9 une th\u00e9ologie positive, comportant l'\u00e9quivalent d'une distinction formelle entre attributs de Dieu (cf. G. VADJA, _Introduction \u00e0 la pens\u00e9e juive du Moyen \u00c2ge_ , Vrin, 1947, p. 174).\n\n29.  _Op. ox._ , I, D3, q. 2 (a. 4, n. 6) : _Et ita neuter ex se, sed in utroque illorum includitur ; ergo univocus._\n\n## CHAPITRE IV\n\n## L'ABSOLU\n\nSpinoza d\u00e9montre avec soin que toute substance (qualifi\u00e9e) doit \u00eatre illimit\u00e9e. L'ensemble des arguments du _Court Trait\u00e9_ et de l' _\u00c9thique_ se pr\u00e9sente ainsi : si une substance \u00e9tait limit\u00e9e, elle devrait l'\u00eatre ou bien par elle-m\u00eame, ou bien par une substance de m\u00eame nature, ou bien par Dieu qui lui aurait donn\u00e9 une nature imparfaite1. Or elle ne peut \u00eatre limit\u00e9e par elle-m\u00eame, car \u00ab elle aurait d\u00fb changer toute sa nature \u00bb. Ni par une autre, car il y aurait deux substances de m\u00eame attribut. Ni par Dieu, parce que Dieu n'a rien d'imparfait ni de limit\u00e9, et \u00e0 plus forte raison ne se trouve pas devant des choses qui \u00ab exigeraient \u00bb ou impliqueraient une limitation quelconque avant d'\u00eatre cr\u00e9\u00e9es. L'importance de ces th\u00e8mes est indiqu\u00e9e par Spinoza, mais de mani\u00e8re elliptique : \u00ab Si nous pouvons d\u00e9montrer qu'il ne peut y avoir aucune substance limit\u00e9e, toute substance doit alors appartenir sans limitation \u00e0 l'\u00eatre divin. \u00bb La transition semble \u00eatre celle-ci : si toute substance est illimit\u00e9e, nous devons reconna\u00eetre que chacune en son genre ou dans sa forme est infiniment parfaite ; il y a donc _\u00e9galit\u00e9_ entre toutes les formes ou tous les genres d'\u00eatre ; aucune forme d'\u00eatre n'est inf\u00e9rieure \u00e0 une autre, aucune n'est sup\u00e9rieure. C'est cette transition que Spinoza formule explicitement dans un autre texte : \u00ab Il n'y a entre les attributs aucune sorte d'in\u00e9galit\u00e9. \u00bb2\n\nD\u00e8s lors on ne pourra pas penser que Dieu contienne la r\u00e9alit\u00e9 ou perfection d'un effet sous une forme meilleure que celle dont d\u00e9pend l'effet ; car il n'y a pas de forme qui soit meilleure qu'une autre. On en conclut : toutes les formes \u00e9tant \u00e9gales (attributs), Dieu ne peut pas poss\u00e9der l'une sans poss\u00e9der les autres ; il ne peut pas en poss\u00e9der une qui vaudrait pour une autre \u00e9minemment. Toutes les formes d'\u00eatre \u00e9tant _infiniment parfaites_ , elles doivent sans limitation appartenir \u00e0 Dieu comme \u00e0 un \u00catre _absolument infini_.\n\nCe principe d'une \u00e9galit\u00e9 des formes ou des attributs n'est qu'un autre aspect du principe d'univocit\u00e9, et du principe de distinction formelle. Il n'en a pas moins une application particuli\u00e8re : il nous force \u00e0 passer de l'Infini \u00e0 l'Absolu, de l'infiniment parfait \u00e0 l'absolument infini. Les formes d'\u00eatre \u00e9tant toutes parfaites et illimit\u00e9es, donc infiniment parfaites, elles ne peuvent constituer des substances in\u00e9gales qui renverraient \u00e0 l'infiniment parfait comme \u00e0 un \u00eatre distinct jouant le r\u00f4le d'une cause \u00e9minente et efficiente. Pas davantage elles ne peuvent former des substances elles-m\u00eames \u00e9gales ; car des substances \u00e9gales ne pourraient l'\u00eatre que num\u00e9riquement, elles devraient avoir la m\u00eame forme, \u00ab l'une devrait n\u00e9cessairement limiter l'autre et ne pourrait par suite \u00eatre infinie3. \u00bb Les formes \u00e9galement illimit\u00e9es sont donc les attributs d'une seule substance qui les poss\u00e8de toutes, et les poss\u00e8de actuellement. Mais alors, la plus grande erreur serait de croire que l'infiniment parfait suffise \u00e0 d\u00e9finir la \u00ab nature \u00bb de Dieu. L'infiniment parfait est la modalit\u00e9 de chaque attribut, c'est-\u00e0-dire le \u00ab propre \u00bb de Dieu. Mais la nature de Dieu consiste en une infinit\u00e9 d'attributs, c'est-\u00e0-dire dans l'absolument infini.\n\nOn peut pr\u00e9voir d\u00e9j\u00e0 la transformation que Spinoza, contre Descartes, va faire subir aux preuves de l'existence de Dieu. Car toutes les preuves cart\u00e9siennes proc\u00e8dent par l'infiniment parfait. Et non seulement elles proc\u00e8dent ainsi, mais elles se meuvent dans l'infiniment parfait, l'identifiant \u00e0 la nature de Dieu. La preuve a posteriori, dans sa premi\u00e8re formulation, dit : \u00ab L'id\u00e9e que j'ai d'un \u00eatre plus parfait que le mien doit n\u00e9cessairement avoir \u00e9t\u00e9 mise en moi par un \u00eatre qui soit en effet plus parfait. \u00bb La seconde formulation est : \u00ab De cela seul que j'existe, et que l'id\u00e9e d'un \u00eatre souverainement parfait (c'est-\u00e0-dire de Dieu) est en moi, l'existence de Dieu est tr\u00e8s \u00e9videmment d\u00e9montr\u00e9e4. \u00bb La preuve ontologique ou _a priori_ s'\u00e9nonce enfin : \u00ab Ce que nous concevons clairement et distinctement appartenir \u00e0 la nature ou \u00e0 l'essence, ou \u00e0 la forme immuable et vraie de quelque chose, cela peut \u00eatre dit ou affirm\u00e9 avec v\u00e9rit\u00e9 de cette chose ; mais apr\u00e8s que nous avons assez soigneusement recherch\u00e9 ce que c'est que Dieu, nous concevons clairement et distinctement qu'il appartient \u00e0 sa vraie et immuable nature qu'il existe ; donc alors nous pouvons affirmer avec v\u00e9rit\u00e9 qu'il existe5. \u00bb Or, dans la mineure, la recherche \u00e0 laquelle Descartes fait allusion consiste pr\u00e9cis\u00e9ment \u00e0 d\u00e9terminer le \u00ab souverainement parfait \u00bb comme la forme, l'essence ou la nature de Dieu. L'existence, \u00e9tant une perfection, appartient \u00e0 cette nature. Gr\u00e2ce \u00e0 la majeure, on en conclut que Dieu existe effectivement.\n\nLa preuve ontologique elle-m\u00eame implique donc l'identification de l'infiniment parfait avec la nature de Dieu. En effet, consid\u00e9rons les secondes objections faites \u00e0 Descartes. On lui reproche de ne pas avoir d\u00e9montr\u00e9, dans la mineure, que la nature de Dieu \u00e9tait possible ou n'impliquait pas contradiction. On lui oppose : Dieu existe, _s'il est possible_. Leibniz reprendra l'objection dans des textes c\u00e9l\u00e8bres6. Descartes r\u00e9pond : La difficult\u00e9 qu'on pr\u00e9tend d\u00e9noncer dans la mineure est d\u00e9j\u00e0 r\u00e9solue dans la majeure. Car la majeure ne signifie pas : Ce que nous concevons clairement et distinctement appartenir \u00e0 la nature d'une chose peut \u00eatre dit avec v\u00e9rit\u00e9 appartenir \u00e0 la nature de cette chose. Ce serait une simple tautologie. La majeure signifie : \u00ab Ce que clairement et distinctement nous concevons appartenir \u00e0 la nature de quelque chose, cela peut \u00eatre dit ou affirm\u00e9 avec v\u00e9rit\u00e9 _de cette chose_. \u00bb Or cette proposition garantit la possibilit\u00e9 de tout ce que nous concevons clairement et distinctement. Si l'on r\u00e9clame un autre crit\u00e8re de possibilit\u00e9, qui serait comme une raison suffisante du c\u00f4t\u00e9 de l'objet, nous avouons notre ignorance, comme l'impuissance de l'entendement \u00e0 atteindre cette raison7.\n\nIl semble que Descartes pressente le sens de l'objection, et pourtant ne le comprenne pas ou ne veuille pas le comprendre. On lui reproche de ne pas avoir d\u00e9montr\u00e9 la possibilit\u00e9 de la nature d'un \u00eatre _dont l'_ \u00ab _infiniment parfait_ \u00bb _ne peut \u00eatre que le propre._ Une telle d\u00e9monstration, peut-\u00eatre, n'est pas possible elle-m\u00eame : mais en ce cas, l'argument ontologique n'est pas concluant8. De toute mani\u00e8re, l'infiniment parfait ne nous fait rien conna\u00eetre de la nature de l'\u00eatre auquel il appartient. Si Descartes pense avoir r\u00e9solu toutes les difficult\u00e9s dans la majeure, c'est d'abord parce qu'il confond la nature de Dieu avec un propre : il pense alors que la conception claire et distincte du propre suffit \u00e0 garantir la possibilit\u00e9 de la _nature_ correspondante. Sans doute arrive-t-il \u00e0 Descartes d'opposer l'aspect sous lequel Dieu est pr\u00e9sent\u00e9 dans l'\u00c9criture (\u00ab fa\u00e7ons de parler... qui contiennent bien quelque v\u00e9rit\u00e9, mais seulement en tant qu'elle est rapport\u00e9e aux hommes \u00bb) et l'aspect sous lequel Dieu appara\u00eet lui-m\u00eame \u00e0 la lumi\u00e8re naturelle9. Mais ainsi, il oppose seulement des propres d'une esp\u00e8ce \u00e0 ceux d'une autre. Concernant un \u00eatre qui a pour propri\u00e9t\u00e9 rationnelle d'\u00eatre infiniment parfait, la question subsiste enti\u00e8rement : un tel \u00eatre est-il possible ? Si l'on demande enfin pourquoi Descartes, de son point de vue, peut identifier le propre avec la nature de Dieu, nous croyons que, l\u00e0 encore, la raison en est dans sa mani\u00e8re d'invoquer l'\u00e9minence et l'analogie. Descartes rappelle que, \u00ab des choses que nous concevons \u00eatre en Dieu et en nous \u00bb, aucune n'est _univoque_10. Or, pr\u00e9cis\u00e9ment dans la mesure o\u00f9 l'on admet une in\u00e9galit\u00e9 fonci\u00e8re entre les formes d'\u00eatre, l'infiniment parfait peut d\u00e9signer une forme sup\u00e9rieure qui se confond avec la nature de Dieu. D\u00e9finissant Dieu, Descartes donne une liste de propri\u00e9t\u00e9s : \u00ab Par le nom de Dieu, j'entends une substance infinie, \u00e9ternelle, immuable, ind\u00e9pendante, toute-connaissante, toute-puissante11... \u00bb C'est dans la brume de leur \u00e9minence que ces propri\u00e9t\u00e9s, consid\u00e9r\u00e9es toutes ensemble, peuvent para\u00eetre assimilables \u00e0 une nature simple.\n\nChez Leibniz, deux th\u00e8mes sont profond\u00e9ment li\u00e9s : l'infiniment parfait ne suffit pas \u00e0 constituer la nature de Dieu ; l'id\u00e9e claire et distincte ne suffit pas \u00e0 garantir sa propre r\u00e9alit\u00e9, c'est-\u00e0-dire la possibilit\u00e9 de son objet. Les deux th\u00e8mes se rejoignent dans l'exigence d'une raison suffisante ou d'une d\u00e9finition r\u00e9elle. Infini et parfait sont seulement des marques distinctives ; la connaissance claire et distincte que nous en avons ne nous apprend nullement si ces caract\u00e8res sont compatibles ; peut-\u00eatre y a-t-il contradiction dans l' _ens perfectissimum_ comme dans \u00ab le plus grand nombre \u00bb ou \u00ab la plus grande vitesse \u00bb. L'essence d'un tel \u00eatre est seulement conjectur\u00e9e ; toute d\u00e9finition de Dieu par la simple perfection reste donc une d\u00e9finition nominale. D'o\u00f9 la critique extr\u00eame de Leibniz : Descartes en g\u00e9n\u00e9ral ne d\u00e9passe pas Hobbes, il n'y a pas de raisons de se fier \u00e0 des crit\u00e8res de la conscience psychologique (le clair et le distinct), plut\u00f4t qu'\u00e0 de simples combinaisons de mots12. Il semble, que dans un tout autre contexte, ces th\u00e8mes sont aussi ceux de Spinoza. On ne s'\u00e9tonnera pas qu'il y ait des points communs fondamentaux dans la r\u00e9action anticart\u00e9sienne de la fin du XVIIe si\u00e8cle. Selon Spinoza, l'infiniment parfait n'est qu'un propre. Cette propri\u00e9t\u00e9 ne nous apprend rien sur la nature de l'\u00eatre auquel elle appartient ; elle ne suffit pas \u00e0 d\u00e9montrer que cet \u00eatre n'enveloppe pas contradiction. Tant qu'une id\u00e9e claire et distincte n'est pas saisie comme \u00ab ad\u00e9quate \u00bb, on pourra douter de sa r\u00e9alit\u00e9 comme de la possibilit\u00e9 de son objet. Tant qu'on ne donne pas une d\u00e9finition r\u00e9elle, portant sur l'essence d'une chose et non sur des _propria_ , on reste dans l'arbitraire de ce qui est simplement con\u00e7u, sans rapport avec la r\u00e9alit\u00e9 de la chose telle qu'elle est hors de l'entendement13. Chez Spinoza non moins que chez Leibniz, il semble donc que la raison suffisante fasse valoir ses exigences. Spinoza posera l'ad\u00e9quation comme raison suffisante de l'id\u00e9e claire et distincte, l'absolument infini comme raison suffisante de l'infiniment parfait. La preuve ontologique chez Spinoza ne portera plus sur un \u00eatre ind\u00e9termin\u00e9 qui serait infiniment parfait, mais sur l'absolument infini, d\u00e9termin\u00e9 comme ce qui consiste en une infinit\u00e9 d'attributs. (L'infiniment parfait sera seulement le mode de chacun de ces attributs, la modalit\u00e9 de l'essence exprim\u00e9e par chaque attribut.)\n\nToutefois, si notre hypoth\u00e8se est juste, on est en droit de s'\u00e9tonner de la mani\u00e8re dont Spinoza d\u00e9montre _a priori_ que l'absolument infini, c'est-\u00e0-dire une substance consistant en une infinit\u00e9 d'attributs, existe n\u00e9cessairement14. Une premi\u00e8re d\u00e9monstration dit : S'il n'existait pas, ce ne serait pas une substance, puisque toute substance existe n\u00e9cessairement. Seconde d\u00e9monstration : Si l'\u00eatre absolument infini n'existait pas, il devrait y avoir une raison de cette non-existence ; cette raison devrait \u00eatre interne, l'absolument infini devrait donc impliquer contradiction ; \u00ab or il est absurde d'affirmer cela de l'\u00eatre absolument infini et souverainement parfait \u00bb. Il est clair que ces raisonnements proc\u00e8dent encore par l'infiniment parfait. L'absolument infini (substance consistant en une infinit\u00e9 d'attributs) existe n\u00e9cessairement, sinon ce ne serait pas une substance ; sinon il ne serait pas infiniment parfait. Mais le lecteur est en droit de r\u00e9clamer une d\u00e9monstration plus profonde, et pr\u00e9alable. Il faut d\u00e9montrer qu'une substance, qui existe n\u00e9cessairement, a pour nature de consister en une infinit\u00e9 d'attributs ou, ce qui revient au m\u00eame, que l'infiniment parfait a pour raison l'absolument infini.\n\nMais ce que le lecteur est en droit de demander, Spinoza l'a pr\u00e9cis\u00e9ment fait. L'id\u00e9e selon laquelle, dans l' _\u00c9thique_ , Spinoza \u00ab s'installe \u00bb en Dieu et \u00ab commence \u00bb par Dieu, n'est qu'une id\u00e9e approximative, litt\u00e9ralement inexacte. Nous verrons d'ailleurs que, selon Spinoza, il est tout \u00e0 fait impossible de _partir_ de l'id\u00e9e de Dieu. La d\u00e9monstration de l'existence de Dieu appara\u00eet dans la 11e proposition. Or les dix premi\u00e8res ont montr\u00e9 ceci : _la distinction num\u00e9rique n'\u00e9tant pas r\u00e9elle, toute substance r\u00e9ellement distincte est illimit\u00e9e et infiniment parfaite ; inversement, la distinction r\u00e9elle n'\u00e9tant pas num\u00e9rique, toutes les substances infiniment parfaites composent une substance absolument infinie dont elles sont les attributs ; l'infiniment parfait est donc le propre de l'absolument infini, et l'absolument infini, la nature ou raison de l'infiniment parfait._ D'o\u00f9 l'importance de ces premi\u00e8res d\u00e9monstrations, qui n'ont rien d'hypoth\u00e9tique. D'o\u00f9 l'importance des consid\u00e9rations sur la distinction num\u00e9rique et la distinction r\u00e9elle. C'est seulement \u00e0 ces conditions que la proposition 11 est en droit de conclure : la substance absolument infinie, n'impliquant pas contradiction, existe n\u00e9cessairement ; si elle n'existait pas, elle n'aurait pas pour propri\u00e9t\u00e9 l'infiniment parfait, pas plus qu'elle ne serait une substance.\n\nLe plan du d\u00e9but de l' _\u00c9thique_ est donc le suivant : 1o) _D\u00e9finitions 1-5_ : ce sont de simples d\u00e9finitions nominales, n\u00e9cessaires au m\u00e9canisme des d\u00e9monstrations futures ; 2o) _D\u00e9finition 6_ : c'est la d\u00e9finition r\u00e9elle de Dieu, comme \u00catre absolument infini, c'est-\u00e0-dire \u00ab substance consistant en une infinit\u00e9 d'attributs dont chacun exprime une essence \u00e9ternelle et infinie \u00bb. Cette d\u00e9finition reprend les termes substance et attribut, pour leur donner un statut r\u00e9el. Mais que cette d\u00e9finition soit elle-m\u00eame r\u00e9elle ne signifie pas qu'elle montre imm\u00e9diatement la possibilit\u00e9 de son objet. Pour qu'une d\u00e9finition soit r\u00e9elle, il suffit qu'on puisse d\u00e9montrer la possibilit\u00e9 de l'objet tel qu'il est d\u00e9fini. On prouve en m\u00eame temps la r\u00e9alit\u00e9 ou v\u00e9rit\u00e9 de la d\u00e9finition ; 3o) _Propositions 1-8_ , premi\u00e8re \u00e9tape de la d\u00e9monstration de la r\u00e9alit\u00e9 de la d\u00e9finition : la distinction num\u00e9rique n'\u00e9tant pas r\u00e9elle, chaque attribut r\u00e9ellement distinct est infiniment parfait, chaque substance qualifi\u00e9e est unique, n\u00e9cessaire et infinie. Cette s\u00e9rie, \u00e9videmment, doit s'appuyer seulement sur les cinq premi\u00e8res d\u00e9finitions ; 4o) _Propositions 9 et 10_ , seconde \u00e9tape : la distinction r\u00e9elle n'\u00e9tant pas num\u00e9rique, les attributs distincts ou substances qualifi\u00e9es forment une seule et m\u00eame substance ayant toutes les qualifications, c'est-\u00e0-dire tous les attributs. Cette seconde s\u00e9rie s'ach\u00e8ve dans le scolie de 10 ; celui-ci constate qu'une substance absolument infinie n'implique pas contradiction. La d\u00e9finition 6 est donc une d\u00e9finition r\u00e9elle15 ; 5o) _Proposition 11_ : l'absolument infini existe n\u00e9cessairement ; sinon il ne pourrait pas \u00eatre une substance, il ne pourrait pas avoir comme propri\u00e9t\u00e9 l'infiniment parfait.\n\nUne contre-\u00e9preuve serait donn\u00e9e par l'examen du _Court Trait\u00e9_. Car ce qu'on dit \u00e0 tort de l' _\u00c9thique_ s'applique bien au _Court Trait\u00e9_ : celui-ci commence par Dieu, s'installe dans l'existence de Dieu. Spinoza, \u00e0 ce moment, croyait encore qu'il \u00e9tait possible de partir d'une id\u00e9e de Dieu. L'argument a priori re\u00e7oit donc une premi\u00e8re formulation enti\u00e8rement conforme \u00e0 l'\u00e9nonc\u00e9 de Descartes16. Mais, ainsi, l'argument se meut tout entier dans l'infiniment parfait, et ne nous donne aucun moyen de conna\u00eetre la nature de l'\u00eatre correspondant. Telle qu'elle est plac\u00e9e en t\u00eate du _Court Trait\u00e9_ , la preuve ontologique ne sert strictement \u00e0 rien. Aussi voit-on Spinoza y joindre un second \u00e9nonc\u00e9 fort obscur (\u00ab L'existence de Dieu est essence \u00bb)17. Nous croyons que cette formule, prise litt\u00e9ralement, ne peut s'interpr\u00e9ter que du point de vue de l'absolument infini, non plus de l'infiniment parfait. En effet, pour que l'existence de Dieu soit essence, il faut que les m\u00eames \u00ab attributs \u00bb qui constituent son essence constituent en m\u00eame temps son existence. C'est pourquoi Spinoza joint une note explicative, anticipant sur la suite du _Court Trait\u00e9_ , et invoquant d\u00e9j\u00e0 les attributs d'une substance absolument infinie : \u00ab \u00c0 la nature d'un \u00eatre qui a des attributs infinis appartient un attribut qui est \u00catre18. \u00bb Les diff\u00e9rences du _Court Trait\u00e9_ avec l' _\u00c9thique_ nous semblent les suivantes : 1o Le _Court Trait\u00e9_ commence par \u00ab Que Dieu est \u00bb, avant toute d\u00e9finition r\u00e9elle de Dieu. Il ne dispose donc en droit que de la preuve cart\u00e9sienne. Alors, il est forc\u00e9 de juxtaposer \u00e0 l'\u00e9nonc\u00e9 orthodoxe de cette preuve un tout autre \u00e9nonc\u00e9 qui anticipe sur le second chapitre (\u00ab Ce que Dieu est \u00bb) ; 2o Au lieu de juxtaposer deux \u00e9nonc\u00e9s, l'un proc\u00e9dant par l'infiniment parfait, l'autre par l'absolument infini, l' _\u00c9thique_ propose une preuve qui proc\u00e8de encore par l'infiniment parfait, mais qui est tout enti\u00e8re subordonn\u00e9e \u00e0 la position pr\u00e9alable et bien fond\u00e9e de l'absolument infini. Alors le deuxi\u00e8me \u00e9nonc\u00e9 du _Court Trait\u00e9_ perd sa n\u00e9cessit\u00e9, en m\u00eame temps que son caract\u00e8re obscur et d\u00e9sordonn\u00e9. Il aura son \u00e9quivalent dans l' _\u00c9thique_ : mais non plus comme preuve de l'existence, simplement comme preuve de l'immutabilit\u00e9 de Dieu19.\n\n\u00c0 ce point, nous ne pouvons faire aucune diff\u00e9rence entre les exigences de Leibniz et celles de Spinoza : m\u00eame r\u00e9clamation d'une d\u00e9finition r\u00e9elle pour Dieu, d'une nature ou raison pour l'infiniment parfait. M\u00eame subordination de la preuve ontologique \u00e0 une d\u00e9finition r\u00e9elle de Dieu, et \u00e0 la d\u00e9monstration que cette d\u00e9finition est bien r\u00e9elle. On s'\u00e9tonnera d'autant plus de la mani\u00e8re dont Leibniz raconte l'histoire. Nous disposons de deux textes \u00e0 cet \u00e9gard. D'abord une note jointe au manuscrit _Quod Ens perfectissimum existit_ , o\u00f9 Leibniz parle de ses entrevues avec Spinoza de 1676 : \u00ab J'ai montr\u00e9 \u00e0 Spinoza, lorsque j'\u00e9tais \u00e0 La Haye, cette argumentation qu'il jugea solide. Comme il l'avait d'abord contredite, j'\u00e9crivis et lui lus ce feuillet20. \u00bb D'autre part les annotations de Leibniz concernant l' _\u00c9thique_ : il reproche \u00e0 la d\u00e9finition 6 de _ne pas_ \u00eatre une d\u00e9finition r\u00e9elle. Elle ne montre pas l'\u00e9quivalence des termes \u00ab absolument infini \u00bb et \u00ab consistant en une infinit\u00e9 d'attributs \u00bb ; elle ne montre pas la compatibilit\u00e9 des attributs entre eux ; elle ne montre pas la possibilit\u00e9 de l'objet d\u00e9fini21. Ou bien Leibniz veut dire que la d\u00e9finition 6 ne montre pas imm\u00e9diatement la possibilit\u00e9 du d\u00e9fini ; mais Leibniz lui-m\u00eame ne croit pas plus que Spinoza \u00e0 l'existence d'une telle intuition de Dieu. Ou bien il veut dire que Spinoza ne s'est pas aper\u00e7u qu'il fallait d\u00e9montrer la r\u00e9alit\u00e9 de la d\u00e9finition ; alors cette critique m\u00e9conna\u00eet enti\u00e8rement le projet g\u00e9n\u00e9ral de l' _\u00c9thique_ et le sens des dix premi\u00e8res propositions. En fait, si l'on consid\u00e8re les formules par lesquelles Leibniz d\u00e9montre lui-m\u00eame la possibilit\u00e9 de Dieu, on n'y verra au premier abord aucune diff\u00e9rence avec celles de Spinoza.\n\nSuivant Leibniz, Dieu est possible, parce que l'infiniment parfait est le propre d'un \u00ab \u00catre absolu \u00bb, qui renferme en soi tous les \u00ab attributs \u00bb, \u00ab toutes les formes simples absolument prises \u00bb, toutes les \u00ab natures qui sont susceptibles du dernier degr\u00e9 \u00bb, \u00ab toutes les qualit\u00e9s positives exprimant une chose sans aucune limite22 \u00bb. Comment ces formes suffisent-elles \u00e0 d\u00e9montrer la possibilit\u00e9 de Dieu ? Chacune est simple et irr\u00e9ductible, con\u00e7ue par soi, _index sui_. Leibniz dit : c'est leur disparit\u00e9 m\u00eame qui assure leur compatibilit\u00e9 (l'impossibilit\u00e9 de leur contradiction) ; c'est leur compatibilit\u00e9 qui assure la possibilit\u00e9 de l'\u00catre auquel elles appartiennent. Rien n'oppose ici Leibniz et Spinoza. Tout leur est commun litt\u00e9ralement, y compris l'utilisation de l'id\u00e9e d'expression, y compris la th\u00e8se selon laquelle les formes expressives sont \u00ab la source des choses \u00bb. \u00c0 cet \u00e9gard au moins, Leibniz ne pouvait rien apprendre \u00e0 Spinoza. Nous devons penser que Leibniz ne rapporte pas exactement la conversation de La Haye. Ou bien Spinoza \u00e9couta, et parla peu, constatant \u00e0 part soi la co\u00efncidence des id\u00e9es de Leibniz avec les siennes. Ou bien le d\u00e9saccord se d\u00e9clara, mais portant sur la mani\u00e8re respective dont chacun interpr\u00e9tait les formes ou qualit\u00e9s positives infinies. Car Leibniz con\u00e7oit celles-ci comme de premiers possibles dans l'entendement de Dieu. D'autre part ces premiers possibles, \u00ab notions absolument simples \u00bb, \u00e9chappent \u00e0 notre connaissance : nous savons qu'ils sont n\u00e9cessairement compatibles, sans savoir ce qu'ils sont. Ils semblent ant\u00e9rieurs et sup\u00e9rieurs \u00e0 toute relation logique : la connaissance atteint seulement des \u00ab notions relativement simples \u00bb qui servent de termes \u00e0 notre pens\u00e9e, et dont, peut-\u00eatre, on dirait au mieux qu'elles symbolisent avec les premiers simples23. Par l\u00e0 Leibniz \u00e9chappe \u00e0 la n\u00e9cessit\u00e9 absolue qu'il d\u00e9nonce comme le danger du spinozisme : il emp\u00eache la n\u00e9cessit\u00e9 \u00ab m\u00e9taphysique \u00bb de sortir de Dieu et de se communiquer aux cr\u00e9atures. Il introduit une esp\u00e8ce de finalit\u00e9, un principe du maximum, dans la preuve ontologique elle-m\u00eame. D\u00e8s ses rencontres avec Spinoza, Leibniz pense que la n\u00e9cessit\u00e9 absolue est l'ennemi. Mais, inversement, Spinoza ne pouvait-il penser que, pour sauver les cr\u00e9atures et la cr\u00e9ation, Leibniz conservait toutes les perspectives de l'\u00e9minence, de l'analogie, du symbolisme en g\u00e9n\u00e9ral ? Peut-\u00eatre Leibniz ne d\u00e9passe-t-il qu'en apparence l'infiniment parfait, peut-\u00eatre n'atteint-il qu'en apparence une nature ou raison.\n\nSpinoza pense que la d\u00e9finition de Dieu telle qu'il la donne est une d\u00e9finition r\u00e9elle. Par d\u00e9monstration de la r\u00e9alit\u00e9 de la d\u00e9finition, il faut entendre une v\u00e9ritable gen\u00e8se de l'objet d\u00e9fini. Tel est le sens des premi\u00e8res propositions de l' _\u00c9thique : non pas hypoth\u00e9tique, mais g\u00e9n\u00e9tique._ Parce que les attributs sont r\u00e9ellement distincts, irr\u00e9ductibles les uns aux autres, ultimes dans leur forme respective ou dans leur genre, parce que chacun est con\u00e7u par soi, ils ne peuvent pas se contredire. Ils sont n\u00e9cessairement compatibles, et la substance qu'ils forment est possible. \u00ab Il est de la nature de la substance que chacun de ses attributs soit con\u00e7u par soi, puisque tous les attributs qu'elle poss\u00e8de ont toujours \u00e9t\u00e9 en m\u00eame temps en elle, et que l'un n'a pu \u00eatre produit par l'autre, mais que chacun exprime la r\u00e9alit\u00e9 ou l'\u00eatre de la substance. Il s'en faut donc de beaucoup qu'il soit absurde d'attribuer plusieurs attributs \u00e0 une seule et m\u00eame substance24. \u00bb Avec les attributs nous atteignons les \u00e9l\u00e9ments premiers et substantiels, notions irr\u00e9ductibles de la substance unique. Appara\u00eet ici l'id\u00e9e d'une constitution logique de la substance, \u00ab composition \u00bb qui n'a rien de physique. L'irr\u00e9ductibilit\u00e9 des attributs non seulement prouve mais constitue la non-impossibilit\u00e9 de Dieu comme substance unique ayant tous les attributs. Il ne peut y avoir contradiction qu'entre termes dont l'un, au moins, n'est pas con\u00e7u par soi. Et la compatibilit\u00e9 des attributs ne se fonde pas, chez Spinoza, dans une r\u00e9gion de l'entendement divin sup\u00e9rieure aux relations logiques elles-m\u00eames, mais _dans une logique propre \u00e0 la distinction r\u00e9elle_. C'est la nature de la distinction r\u00e9elle entre attributs qui exclut toute division de substances ; c'est cette nature de la distinction r\u00e9elle qui conserve aux termes distincts toute leur positivit\u00e9 respective, interdisant de les d\u00e9finir en opposition l'un avec l'autre, et les rapportant tous \u00e0 une m\u00eame substance indivisible. Spinoza semble \u00eatre celui qui va le plus loin dans la voie de cette nouvelle logique : logique de l'affirmation pure, de la qualit\u00e9 illimit\u00e9e, et, par l\u00e0, de la totalit\u00e9 inconditionn\u00e9e qui poss\u00e8de toutes les qualit\u00e9s, c'est-\u00e0-dire logique de l'absolu. Les attributs doivent \u00eatre compris comme les \u00e9l\u00e9ments de cette composition dans l'absolu.\n\nLes attributs comme expressions ne sont pas seulement des \u00ab miroirs \u00bb. La philosophie expressionniste nous rapporte deux m\u00e9taphores traditionnelles : celle du miroir qui refl\u00e8te ou r\u00e9fl\u00e9chit une image, mais aussi celle du germe qui \u00ab exprime \u00bb l'arbre tout entier. Les attributs sont l'un et l'autre, suivant le point de vue o\u00f9 l'on se place. D'une part, l'essence se r\u00e9fl\u00e9chit et se multiplie dans les attributs, les attributs sont des miroirs dont chacun exprime en son genre l'essence de la substance : ils renvoient n\u00e9cessairement \u00e0 l'entendement, comme les miroirs \u00e0 l'\u0153il qui voit l'image. Mais, aussi bien, l'exprim\u00e9 est envelopp\u00e9 dans l'expression, comme l'arbre dans le germe : l'essence de la substance est moins r\u00e9fl\u00e9chie dans les attributs que constitu\u00e9e par les attributs qui l'expriment ; les attributs sont moins des miroirs que des \u00e9l\u00e9ments dynamiques ou g\u00e9n\u00e9tiques.\n\nLa nature de Dieu ( _nature naturante_ ) est expressive. Dieu s'exprime dans les fondements du monde, qui forment son essence, avant de s'exprimer dans le monde. Et l'expression n'est pas manifestation sans \u00eatre aussi constitution de Dieu lui-m\u00eame. La Vie, c'est-\u00e0-dire l'expressivit\u00e9, est port\u00e9e dans l'absolu. Il y a dans la substance une unit\u00e9 du divers, dans les attributs une diversit\u00e9 actuelle de l'Un : la distinction r\u00e9elle s'applique \u00e0 l'absolu, parce qu'elle r\u00e9unit ces deux moments et les rapporte l'un \u00e0 l'autre. Aussi ne suffit-il pas de dire que Spinoza privil\u00e9gie l' _Ens necessarium_ sur l' _Ens perfectissimum._ En v\u00e9rit\u00e9, l'essentiel, c'est l' _Ens absolutum. Perfectissimum_ est seulement un propre, propre dont on part comme de la modalit\u00e9 de chaque attribut. _Necessarium_ est encore un propre, propre auquel on arrive comme \u00e0 la modalit\u00e9 de la substance ayant tous les attributs. Mais entre les deux, se fait la d\u00e9couverte de la nature ou de l'absolu : substance \u00e0 laquelle on rapporte la pens\u00e9e, l'\u00e9tendue, etc., toutes les formes d'\u00eatre univoques. C'est pourquoi, dans ses lettres, Spinoza insiste sur la n\u00e9cessit\u00e9 de ne pas perdre de vue la d\u00e9finition 6, d'y revenir constamment25. Cette d\u00e9finition est la seule qui nous livre une nature, cette nature est la nature expressive de l'absolu. Revenir \u00e0 cette d\u00e9finition n'est pas seulement la garder dans sa m\u00e9moire, mais y revenir comme \u00e0 une d\u00e9finition dont, entre-temps, on a d\u00e9montr\u00e9 qu'elle \u00e9tait r\u00e9elle. Et cette d\u00e9monstration n'est pas comme une op\u00e9ration de l'entendement qui resterait ext\u00e9rieure \u00e0 la substance ; elle se confond avec la vie de la substance elle-m\u00eame, avec la n\u00e9cessit\u00e9 de sa constitution a priori.\n\n\u00ab Quand je d\u00e9finis Dieu : l'\u00catre souverainement parfait, comme cette d\u00e9finition n'exprime pas une cause efficiente (j'entends une cause efficiente tant interne qu'externe), je ne pourrai en d\u00e9duire toutes les propri\u00e9t\u00e9s de Dieu. Au contraire, quand je d\u00e9finis Dieu : un \u00catre, etc. (voir _\u00c9thique_ , partie I, d\u00e9finition 6)26. \u00bb Telle est la transformation de la preuve a priori : Spinoza d\u00e9passe l'infiniment parfait vers l'absolument infini, dans lequel il d\u00e9couvre la Nature ou Raison suffisante. _Cette d\u00e9marche aboutit \u00e0 une seconde triade de la substance_. 1o) toutes les formes d'\u00eatre sont \u00e9gales et \u00e9galement parfaites, il n'y a pas d'in\u00e9galit\u00e9 de perfection entre les attributs ; 2o) chaque forme est donc illimit\u00e9e, chaque attribut exprime une essence infinie ; 3o) toutes les formes appartiennent donc \u00e0 une seule et m\u00eame substance, tous les attributs s'affirment \u00e9galement sans limitation d'une substance absolument infinie. La premi\u00e8re triade \u00e9tait : attribut-essence-substance. La seconde : parfait-infini-absolu. La premi\u00e8re se fondait sur un argument pol\u00e9mique : la distinction r\u00e9elle ne peut pas \u00eatre num\u00e9rique. Et sur un argument positif : la distinction r\u00e9elle est une distinction formelle entre attributs qui s'affirment d'une seule et m\u00eame substance. La seconde triade a pour argument pol\u00e9mique : les propres ne constituent pas une nature. Et pour argument positif : tout est perfection dans la nature. Il n'y a pas de \u00ab nature \u00bb \u00e0 laquelle manque quelque chose ; toutes les formes d'\u00eatre s'affirment sans limitation, donc s'attribuent \u00e0 quelque chose d'absolu, l'absolu dans sa nature \u00e9tant infini sous toutes les formes. La triade de l'absolu compl\u00e8te ainsi celle de la substance : elle la relaie, nous menant \u00e0 la d\u00e9couverte d'une troisi\u00e8me et derni\u00e8re d\u00e9termination de Dieu.\n\n* * *\n\n1.  _CT_ , I, ch. 2, 2-5 (et notes 2 et 3). _\u00c9_ , I, 8, dem.\n\n2.  _CT_ , Appendice II, 11.\n\n3.  _CT_ , I, ch. 2, 6. Qu'il n'y ait pas \u00ab deux substances \u00e9gales \u00bb ne contredit pas l'\u00e9galit\u00e9 des attributs : les deux th\u00e8mes s'impliquent.\n\n4.  DESCARTES, _M\u00e9ditation III_ , AT IX, p. 38, p. 40.\n\n5.  _R\u00e9ponses aux premi\u00e8res objections_ , AT IX, p. 91.\n\n6.  Les premiers textes de Leibniz, \u00e0 cet \u00e9gard, sont de 1672 ( _Leibnitiana_ , \u00e9d. Jagodinsky, p. 112). Cf. aussi la note de 1676, _Quod ens perfectissimum existit_ (Gerhardt VII, p. 261).\n\n7.  _R\u00e9ponses aux secondes objections_ , AT, IX, p. 118 : \u00ab ... Ou bien vous feignez quelque autre possibilit\u00e9, de la part de l'objet m\u00eame, laquelle, si elle ne convient pas avec la pr\u00e9c\u00e9dente, ne peut jamais \u00eatre connue par l'entendement humain... \u00bb\n\n8.  Telle semble \u00eatre la position des auteurs des secondes objections (cf. AT, IX, p. 101).\n\n9.  _R\u00e9ponses aux secondes objections_ , AT, IX, p. 112.\n\n10.  _R\u00e9ponses aux secondes objections_ , AT, IX, p. 108. C'est un des principes fondamentaux du thomisme : _De Deo et creaturis nil univoce praedicatur._\n\n11.  _M\u00e9ditation_ III, AT, IX, p. 36.\n\n12.  Cf. LEIBNIZ, _Lettre \u00e0 la princesse \u00c9lisabeth_ , 1678, et _M\u00e9ditations sur la connaissance, la v\u00e9rit\u00e9 et les id\u00e9es_ , 1684.\n\n13.  Sur le caract\u00e8re nominal d'une d\u00e9finition de Dieu par l'infiniment parfait, cf. _Lettre 60, \u00e0 Tschirnhaus_ (III, p. 200).\n\n14.  _\u00c9_ , I, 11, les deux premi\u00e8res d\u00e9monstrations.\n\n15.  _\u00c9_ , I, 10, sc. : \u00ab Il s'en faut de beaucoup qu'il soit absurde d'attribuer plusieurs attributs \u00e0 une seule substance... \u00bb\n\n16.  _CT_ , I, ch. 1, 1.\n\n17.  _CT_ , I, ch. 1, 2 (Sur l'ambigu\u00eft\u00e9 de la formule et sa traduction, cf. la note d'Appuhn, \u00e9d. Garnier, p. 506).\n\n18.  _CT_ , I, ch. 1, 2, note 2.\n\n19.  _\u00c9_ , I, 20, dem. et cor.\n\n20.  Cf. G. FRIEDMANN, _Leibniz et Spinoza_ (NRF, 1946), pp. 66-70.\n\n21.  LEIBNIZ, _Ad \u00c9thicam..._ (Gerhardt, I), pp. 139-152.\n\n22.  Cf. _Quod ens..., Lettre \u00e0 la princesse \u00c9lisabeth, M\u00e9ditations sur la connaissance..._\n\n23.  Cf. LEIBNIZ, _Elementa calculi, Plan de la science g\u00e9n\u00e9rale, Introductio ad Encyclopaediam Arcanam_ (\u00e9d. Conturat). Sur les absolument simples, qui sont de purs \u00ab disparates \u00bb, ant\u00e9rieurs aux relations logiques, cf. M. GU\u00c9ROULT, \u00ab La Constitution de la substance chez Leibniz \u00bb ( _Revue de m\u00e9taphysique et de morale_ , 1947).\n\n24.  _\u00c9_ , I, 10, sc.\n\n25.  _Lettre 2, \u00e0 Oldenburg_ (III, p. 5) ; _Lettre 4, \u00e0 Oldenburg_ (III, pp. 10-11) ; _Lettres 35 et 36, \u00e0 Hudde_ (III, pp. 129-132).\n\n26.  _Lettre 60, \u00e0 Tschirnhaus_ (III, p. 200).\n\n## CHAPITRE V\n\n## LA PUISSANCE\n\nDans toutes les critiques de Leibniz contre Descartes, un th\u00e8me revient constamment : Descartes va \u00ab trop vite \u00bb. Descartes a cru que la consid\u00e9ration de l'infiniment parfait suffisait dans l'ordre de l'\u00eatre, que la possession d'une id\u00e9e claire et distincte suffisait dans l'ordre de la connaissance, que l'examen des quantit\u00e9s de r\u00e9alit\u00e9 ou de perfection suffisait \u00e0 nous faire passer de la connaissance \u00e0 l'\u00eatre. Leibniz aime \u00e0 retourner contre Descartes l'accusation de pr\u00e9cipitation. Toujours Descartes est entra\u00een\u00e9, par sa rapidit\u00e9, \u00e0 confondre le relatif avec l'absolu1. Si nous cherchons encore ce qu'il y a de commun dans la r\u00e9action anticart\u00e9sienne, nous voyons que Spinoza, de son c\u00f4t\u00e9, s'en prend \u00e0 la _facilit\u00e9_ chez Descartes. La complaisance de Descartes \u00e0 faire un usage philosophique des notions de \u00ab facile \u00bb et de \u00ab difficile \u00bb avait inqui\u00e9t\u00e9 d\u00e9j\u00e0 beaucoup de ses contemporains. Quand Spinoza se heurte \u00e0 l'emploi cart\u00e9sien du mot facile, il perd sa s\u00e9r\u00e9nit\u00e9 de professeur qui s'\u00e9tait promis d'exposer les _Principes_ sans rien dire qui en differ\u00e2t de \u00ab la largeur d'un ongle \u00bb ; il manifeste m\u00eame une sorte d'indignation2. Sans doute n'est-il pas le premier \u00e0 d\u00e9noncer cette facilit\u00e9, pas plus que Leibniz n'est le premier \u00e0 d\u00e9noncer cette rapidit\u00e9. Mais c'est avec Leibniz et avec Spinoza que la critique prend son aspect le plus complet, le plus riche et le plus efficace.\n\nDescartes donne deux \u00e9nonc\u00e9s de la preuve a posteriori de l'existence de Dieu : Dieu existe, parce que son id\u00e9e est en nous ; et aussi parce que nous-m\u00eames, qui avons son id\u00e9e, existons. La premi\u00e8re d\u00e9monstration se fonde imm\u00e9diatement sur la consid\u00e9ration des quantit\u00e9s de perfection ou de r\u00e9alit\u00e9. Une cause doit avoir au moins autant de r\u00e9alit\u00e9 que son effet ; la cause d'une id\u00e9e doit avoir au moins autant de r\u00e9alit\u00e9 formelle que cette id\u00e9e contient de r\u00e9alit\u00e9 objective. Or j'ai l'id\u00e9e d'un \u00eatre infiniment parfait (c'est-\u00e0-dire une id\u00e9e qui contient \u00ab plus de r\u00e9alit\u00e9 objective qu'aucune autre \u00bb)3. La seconde d\u00e9monstration est plus complexe, parce qu'elle proc\u00e8de d'une hypoth\u00e8se absurde : Si j'avais le pouvoir de me produire, il me serait encore plus facile de me donner les propri\u00e9t\u00e9s dont j'ai l'id\u00e9e ; et il ne me serait pas plus difficile de me conserver que de me produire ou de me cr\u00e9er4. Cette fois, le principe est : Qui peut le plus peut le moins. \u00ab Ce qui peut faire le plus ou le plus difficile peut aussi faire le moins5. \u00bb Or s'il est plus difficile de cr\u00e9er ou de conserver une substance que de cr\u00e9er ou de conserver des propri\u00e9t\u00e9s, c'est parce que la substance a plus de r\u00e9alit\u00e9 que les propri\u00e9t\u00e9s elles-m\u00eames. On objectera que la substance se confond avec ses propri\u00e9t\u00e9s consid\u00e9r\u00e9es collectivement. Mais \u00ab distributivement \u00bb, les attributs sont comme les parties d'un tout ; c'est en ce sens qu'ils sont plus faciles \u00e0 produire. On objecte en second lieu que la comparaison ne peut pas se faire entre une substance (par exemple finie) et les attributs d'une autre substance (par exemple infinie). Mais pr\u00e9cis\u00e9ment, si j'avais le pouvoir de me produire comme substance, les perfections dont j'ai l'id\u00e9e feraient partie de moi-m\u00eame, il me serait donc plus facile de me les donner que de me produire ou de me conserver tout entier. On objecte enfin qu'une cause d\u00e9termin\u00e9e, destin\u00e9e par nature \u00e0 produire tel effet, ne peut pas produire \u00ab plus facilement \u00bb un autre effet, m\u00eame de quantit\u00e9 moindre. Mais, du point de vue d'une cause premi\u00e8re, les quantit\u00e9s de r\u00e9alit\u00e9 qui correspondent aux attributs et aux modes entrent dans des rapports de tout \u00e0 partie qui permettent de d\u00e9terminer le plus et le moins, le plus difficile et le plus facile6.\n\nIl est clair qu'un m\u00eame argument anime les deux d\u00e9monstrations. Ou bien Descartes rapporte des quantit\u00e9s de r\u00e9alit\u00e9 objective \u00e0 des quantit\u00e9s de r\u00e9alit\u00e9 formelle, ou bien il fait entrer les quantit\u00e9s de r\u00e9alit\u00e9 dans des rapports de tout \u00e0 partie. L'ensemble de la preuve a posteriori proc\u00e8de en tout cas par examen des quantit\u00e9s de r\u00e9alit\u00e9 ou de perfection prises en tant que telles. Quand Spinoza expose Descartes, il ne se retient pas d'attaquer la seconde d\u00e9monstration ; il retrouve ou reprend les objections contre la notion de \u00ab facile \u00bb. Mais la mani\u00e8re dont il le fait laisse penser que, lorsqu'il parle pour son compte, il n'a pas plus d'indulgence pour la premi\u00e8re d\u00e9monstration. En effet, l'on trouve dans l'\u0153uvre de Spinoza beaucoup de versions d'une preuve a posteriori de l'existence de Dieu. Nous croyons qu'elles ont toutes quelque chose de commun, les unes enveloppant une critique de la premi\u00e8re d\u00e9monstration cart\u00e9sienne, les autres enveloppant une critique de la seconde, _mais toutes ayant pour but de substituer un argument des puissances \u00e0 l'argument des quantit\u00e9s de r\u00e9alit\u00e9_. Tout se passe comme si, de fa\u00e7ons tr\u00e8s diverses, Spinoza sugg\u00e9rait toujours une m\u00eame critique : Descartes a pris le relatif pour l'absolu. Dans la preuve a priori, Descartes a confondu l'absolu avec l'infiniment parfait ; mais l'infiniment parfait n'est qu'un relatif. Dans la preuve a posteriori, Descartes prend la quantit\u00e9 de r\u00e9alit\u00e9 ou de perfection pour un absolu ; mais celle-ci encore n'est qu'un relatif. L'absolument infini comme raison suffisante et nature de l'infiniment parfait ; la puissance comme raison suffisante de la quantit\u00e9 de r\u00e9alit\u00e9 : telles sont les transformations corr\u00e9latives que Spinoza fait subir aux preuves cart\u00e9siennes.\n\nLe _Court Trait\u00e9_ ne contient aucune trace du deuxi\u00e8me \u00e9nonc\u00e9 cart\u00e9sien ; mais il conserve le premier, dans des termes semblables \u00e0 ceux de Descartes : \u00ab Si une id\u00e9e de Dieu est donn\u00e9e, la cause de cette id\u00e9e doit exister formellement et contenir en elle tout ce que l'id\u00e9e contient objectivement ; or une id\u00e9e de Dieu est donn\u00e9e7. \u00bb Toutefois, c'est la d\u00e9monstration de ce premier \u00e9nonc\u00e9 qui se trouve profond\u00e9ment modifi\u00e9e. On assiste \u00e0 une multiplication de syllogismes, t\u00e9moignant d'un \u00e9tat de la pens\u00e9e de Spinoza, si obscur soit-il, o\u00f9 celui-ci tente d\u00e9j\u00e0 de d\u00e9passer l'argument de la quantit\u00e9 de r\u00e9alit\u00e9 pour y substituer un argument fond\u00e9 sur la puissance. Le raisonnement est le suivant : Un entendement fini n'a pas par lui-m\u00eame \u00ab le pouvoir \u00bb de conna\u00eetre l'infini, ni de conna\u00eetre ceci plut\u00f4t que cela ; or il \u00ab peut \u00bb conna\u00eetre quelque chose ; il faut donc qu'un objet existe formellement, qui le d\u00e9termine \u00e0 conna\u00eetre ceci plut\u00f4t que cela ; et il \u00ab peut \u00bb concevoir l'infini ; il faut donc que Dieu lui-m\u00eame existe formellement. En d'autres termes, Spinoza demande : pourquoi la cause de l'id\u00e9e de Dieu doit-elle contenir formellement tout ce que cette id\u00e9e contient objectivement ? C'est dire que l'axiome de Descartes ne le satisfait pas. L'axiome cart\u00e9sien \u00e9tait : il doit y avoir \u00ab au moins autant \u00bb de r\u00e9alit\u00e9 formelle dans la cause d'une id\u00e9e que de r\u00e9alit\u00e9 objective dans l'id\u00e9e elle-m\u00eame. (Ce qui suffisait \u00e0 garantir qu'il n'y en avait pas \u00ab plus \u00bb, dans le cas d'une quantit\u00e9 de r\u00e9alit\u00e9 objective infinie). Or nous pouvons pressentir que Spinoza cherche une raison plus profonde. Le texte du _Court Trait\u00e9_ pr\u00e9pare d\u00e9j\u00e0 certains \u00e9l\u00e9ments qui feront partie d'un _axiome des puissances_ : l'entendement n'a pas plus de puissance pour conna\u00eetre que ses objets pour exister et agir ; la puissance de penser et de conna\u00eetre ne peut pas \u00eatre plus grande qu'une puissance d'exister n\u00e9cessairement corr\u00e9lative.\n\nS'agira-t-il, \u00e0 proprement parler, d'un _axiome_ ? Un autre texte du _Court Trait\u00e9_ , certainement plus tardif, \u00e9nonce : \u00ab Il n'y a aucune chose dont l'id\u00e9e ne soit dans la chose pensante, et aucune id\u00e9e ne peut \u00eatre sans que la chose ne soit aussi8. \u00bb Cette formule sera fondamentale dans tout le spinozisme. Pour autant qu'elle peut \u00eatre d\u00e9montr\u00e9e, elle m\u00e8ne \u00e0 l'\u00e9galit\u00e9 de deux puissances. Il est vrai que la premi\u00e8re partie de la formule est difficilement d\u00e9montrable, si l'on ne se donne pas d\u00e9j\u00e0 l'existence de Dieu. Mais la seconde partie se laisse ais\u00e9ment d\u00e9montrer. Une id\u00e9e qui ne serait pas l'id\u00e9e de quelque chose d'existant n'aurait aucune distinction, ne serait pas l'id\u00e9e de ceci ou de cela. Ou bien, meilleure d\u00e9monstration \u00e0 laquelle Spinoza arrivera : Conna\u00eetre, c'est conna\u00eetre par la cause, donc aucune chose ne peut \u00eatre connue sans une cause qui la fait \u00eatre, en existence ou en essence. De cet argument, on peut d\u00e9j\u00e0 conclure que la puissance de penser, dont participent toutes les id\u00e9es, n'est pas sup\u00e9rieure \u00e0 une puissance d'exister et d'agir dont toutes les choses participent. Et c'est l'essentiel, du point de vue d'une preuve a posteriori.\n\nNous avons une id\u00e9e de Dieu ; nous devons donc affirmer une puissance infinie de penser comme correspondant \u00e0 cette id\u00e9e ; or la puissance de penser n'est pas plus grande que la puissance d'exister et d'agir ; nous devons donc affirmer une puissance infinie d'exister comme correspondant \u00e0 la nature de Dieu. De l'id\u00e9e de Dieu, on n'inf\u00e8re plus imm\u00e9diatement l'existence de Dieu ; _on passe par le d\u00e9tour des puissances_ pour trouver, dans la puissance de penser, la raison de la r\u00e9alit\u00e9 objective contenue dans l'id\u00e9e de Dieu, et dans la puissance d'exister, la raison de la r\u00e9alit\u00e9 formelle en Dieu lui-m\u00eame. Le _Court Trait\u00e9_ nous semble d\u00e9j\u00e0 pr\u00e9parer les \u00e9l\u00e9ments d'une preuve de ce genre. Puis le _Trait\u00e9 de la r\u00e9forme_ en donne une formule explicite9. Mais c'est dans une lettre que Spinoza r\u00e9v\u00e8le le plus nettement ce qu'il cherchait depuis le _Court Trait\u00e9_ : substituer un axiome des puissances \u00e0 l'axiome cart\u00e9sien des quantit\u00e9s de r\u00e9alit\u00e9, jug\u00e9 obscur. \u00ab La puissance de penser n'est pas plus grande pour penser que la puissance de la nature, pour exister et agir. C'est l\u00e0 un axiome clair et vrai, \u00e0 partir duquel l'existence de Dieu suit de son id\u00e9e, de mani\u00e8re tr\u00e8s claire et efficace10. \u00bb\n\nNous devons remarquer toutefois que Spinoza parvient tardivement \u00e0 la possession de son \u00ab axiome \u00bb. Bien plus, il ne lui donne pas son plein \u00e9nonc\u00e9, lequel impliquerait une stricte \u00e9galit\u00e9 des deux puissances. Bien plus, il pr\u00e9sente comme axiome une proposition qu'il sait d\u00e9montrable en partie. \u00c0 toutes ces ambigu\u00eft\u00e9s, il y a une raison. L'\u00e9galit\u00e9 des puissances est d'autant mieux d\u00e9montr\u00e9e qu'on part d'un Dieu d\u00e9j\u00e0 existant. Donc, \u00e0 mesure que Spinoza parvient \u00e0 une possession plus parfaite de cette formule d'\u00e9galit\u00e9, il cesse de s'en servir pour \u00e9tablir a posteriori l'existence de Dieu ; il lui r\u00e9serve un autre usage, un autre domaine. En effet, l'\u00e9galit\u00e9 des puissances jouera un r\u00f4le fondamental dans le livre II de l' _\u00c9thique_ ; mais ce r\u00f4le, elle l'aura comme facteur d\u00e9cisif dans la d\u00e9monstration du parall\u00e9lisme, une fois prouv\u00e9e l'existence de Dieu.\n\nOn ne doit donc pas s'\u00e9tonner que la preuve a posteriori de l' _\u00c9thique_ soit d'une autre esp\u00e8ce que celle du _Court Trait\u00e9_ et du _Trait\u00e9 de la r\u00e9forme_. Elle n'en proc\u00e8de pas moins par la puissance. Mais elle ne passe plus par l'id\u00e9e de Dieu, ni par une puissance de penser correspondante, pour conclure \u00e0 une puissance infinie d'exister. Elle op\u00e8re imm\u00e9diatement dans l'existence, par la puissance d'exister. En ce sens l' _\u00c9thique_ se sert des indications que Spinoza donnait d\u00e9j\u00e0 dans sa version remani\u00e9e des _Principes_. Dans les _Principes_ , Spinoza exposait la premi\u00e8re d\u00e9monstration cart\u00e9sienne, sans commentaires ni corrections ; mais c'\u00e9tait la seconde d\u00e9monstration qui se trouvait profond\u00e9ment remani\u00e9e. Spinoza s'en prenait violemment \u00e0 l'emploi du mot \u00ab facile \u00bb chez Descartes. Il proposait un tout autre raisonnement : 1o) Plus une chose a de r\u00e9alit\u00e9 ou de perfection, plus grande est l'existence qu'elle enveloppe (existence possible correspondant aux degr\u00e9s finis de perfection, existence n\u00e9cessaire correspondant \u00e0 l'infiniment parfait). 2o) Qui a la _puissance_ ( _potentiam_ ou _vim_ ) de se conserver n'a besoin d'aucune cause pour exister, non seulement pour exister \u00ab d'existence possible \u00bb, mais \u00ab d'existence n\u00e9cessaire \u00bb. Qui a la puissance de se conserver existe donc n\u00e9cessairement. 3o) Je suis imparfait, donc n'ai pas l'existence n\u00e9cessaire, donc n'ai pas la puissance de me conserver ; je suis conserv\u00e9 par un autre, mais par un autre qui a n\u00e9cessairement le pouvoir de se conserver lui-m\u00eame, donc qui existe n\u00e9cessairement11.\n\nDans le _Court Trait\u00e9_ , il n'y a pas trace du deuxi\u00e8me \u00e9nonc\u00e9 de Descartes ; le premier est conserv\u00e9, mais d\u00e9montr\u00e9 d'une tout autre fa\u00e7on. Dans l' _\u00c9thique_ au contraire, il n'y a plus trace du premier (pr\u00e9cis\u00e9ment parce que l'argument des puissances est r\u00e9serv\u00e9 maintenant pour un meilleur usage). Mais on trouve, dans l' _\u00c9thique_ , une version de la preuve a posteriori qui renvoie au second \u00e9nonc\u00e9 de Descartes, ne serait-ce que par les critiques implicites et les remaniements qu'elle propose. Spinoza d\u00e9nonce ceux qui pensent que, plus il appartient \u00e0 une chose, plus elle est difficile \u00e0 produire12. Mais il va plus loin que dans les _Principes_. L'expos\u00e9 des _Principes_ ne disait pas le plus important : l'existence, possible ou n\u00e9cessaire, est elle-m\u00eame puissance ; _la puissance est identique \u00e0 l'essence elle-m\u00eame_. C'est pr\u00e9cis\u00e9ment parce que l'essence est puissance que l'existence possible (dans l'essence) est autre chose qu'une \u00ab possibilit\u00e9 \u00bb. L' _\u00c9thique_ pr\u00e9sentera donc l'argument suivant : 1o) Pouvoir exister, c'est puissance (il s'agit de l'existence possible envelopp\u00e9e dans l'essence d'une chose finie). 2o) Or un \u00eatre fini existe d\u00e9j\u00e0 n\u00e9cessairement (c'est-\u00e0-dire en vertu d'une cause ext\u00e9rieure qui le d\u00e9termine \u00e0 exister). 3o) Si l'\u00catre absolument infini n'existait pas n\u00e9cessairement lui aussi, il serait moins puissant que les \u00eatres finis : ce qui est absurde. 4o) Mais l'existence n\u00e9cessaire de l'absolument infini ne peut pas \u00eatre en vertu d'une cause ext\u00e9rieure ; c'est donc par lui-m\u00eame que l'\u00eatre absolument infini existe n\u00e9cessairement13. Ainsi fond\u00e9e sur la puissance d'exister, la preuve a posteriori donne lieu \u00e0 une nouvelle preuve a priori : plus il appartient de r\u00e9alit\u00e9 ou de perfection \u00e0 la nature d'une chose, plus elle a de puissance, c'est-\u00e0-dire de forces pour exister ( _virium... ut existat_ ) ; \u00ab Dieu a donc par soi-m\u00eame une puissance absolument infinie d'exister, et par suite il existe absolument14. \u00bb\n\nL'argument de la puissance a donc deux aspects chez Spinoza, l'un renvoyant \u00e0 la critique du premier \u00e9nonc\u00e9 de Descartes, l'autre \u00e0 la critique du second. Mais dans les deux cas, et surtout dans le deuxi\u00e8me qui repr\u00e9sente l'\u00e9tat d\u00e9finitif de la pens\u00e9e de Spinoza, nous devons chercher la signification de cet argument. On attribue \u00e0 un \u00eatre fini une puissance d'exister comme identique \u00e0 son essence. Sans doute un \u00eatre fini n'existe-t-il pas par sa propre essence ou puissance, mais en vertu d'une cause externe. Il n'en a pas moins une puissance qui lui est propre, bien que cette puissance soit n\u00e9cessairement effectu\u00e9e sous l'action de choses ext\u00e9rieures. Raison de plus pour demander : \u00e0 quelle condition attribuons-nous \u00e0 un \u00eatre fini, qui n'existe pas par soi, une _puissance d'exister et d'agir_ identique \u00e0 son essence15 ? La r\u00e9ponse de Spinoza semble \u00eatre celle-ci : Nous affirmons cette puissance d'un \u00eatre fini dans la mesure o\u00f9 nous consid\u00e9rons cet \u00eatre comme la partie d'un tout, comme le mode d'un attribut, la modification d'une substance. Donc cette substance a pour son compte une puissance infinie d'exister, d'autant plus de puissance qu'elle a plus d'attributs. Le m\u00eame raisonnement vaut pour la puissance de penser : nous attribuons \u00e0 une id\u00e9e distincte une puissance de conna\u00eetre, mais dans la mesure o\u00f9 nous consid\u00e9rons cette id\u00e9e comme la partie d'un tout, le mode d'un attribut pens\u00e9e, la modification d'une substance pensante qui, pour son compte, poss\u00e8de une puissance infinie de penser16.\n\nAppara\u00eet plus clairement comment la preuve a posteriori de l' _\u00c9thique_ donne lieu \u00e0 une preuve a priori. Il suffit de constater que Dieu, ayant tous les attributs, poss\u00e8de a priori _toutes les conditions_ sous lesquelles on affirme de quelque chose une puissance : il a donc une puissance \u00ab absolument infinie d'exister, il _existe_ \u00ab absolument \u00bb et par soi. Bien plus, nous verrons que Dieu, ayant un attribut qui est la pens\u00e9e, poss\u00e8de \u00e9galement une puissance absolue infinie de penser17. En tout ceci les attributs semblent avoir un r\u00f4le essentiellement dynamique. Non pas qu'ils soient eux-m\u00eames puissances. Mais, pris collectivement, ce sont les conditions sous lesquelles on attribue \u00e0 la substance absolue une puissance absolument infinie d'exister et d'agir, identique \u00e0 son essence formelle. Pris distributivement, ce sont les conditions sous lesquelles on attribue \u00e0 des \u00eatres finis une puissance identique \u00e0 leur essence formelle, en tant que cette essence est contenue dans tel ou tel attribut. D'autre part, l'attribut pens\u00e9e pris en lui-m\u00eame est la condition sous laquelle on rapporte \u00e0 la substance absolue une puissance absolument infinie de penser identique \u00e0 son essence objective ; sous laquelle, aussi, on attribue aux id\u00e9es une puissance de conna\u00eetre identique \u00e0 l'essence objective qui les d\u00e9finit respectivement. C'est en ce sens que les \u00eatres finis sont conditionn\u00e9s, \u00e9tant n\u00e9cessairement modifications de la substance ou modes d'un attribut ; la substance est comme la totalit\u00e9 inconditionn\u00e9e, parce qu'elle poss\u00e8de ou remplit a priori l'infinit\u00e9 des conditions ; les attributs sont des conditions communes, communes \u00e0 la substance qui les poss\u00e8de collectivement et aux modes qui les impliquent distributivement. Comme dit Spinoza, ce n'est pas par des attributs humains (bont\u00e9, justice, charit\u00e9...) que Dieu \u00ab communique \u00bb aux cr\u00e9atures humaines les perfections qu'elles poss\u00e8dent18. Au contraire, c'est par ses propres attributs que Dieu communique \u00e0 toutes les cr\u00e9atures la puissance qui leur est propre.\n\nLe _Trait\u00e9 politique_ expose une preuve a posteriori de la m\u00eame famille que celle des _Principes_ et de l' _\u00c9thique_ ; les \u00eatres finis n'existent et ne se conservent pas par leur propre puissance ; pour exister et se conserver, ils ont besoin de la puissance d'un \u00eatre capable de se conserver soi-m\u00eame et d'exister par soi ; la puissance par laquelle un \u00eatre fini existe, se conserve et agit est donc la puissance de Dieu lui-m\u00eame19. \u00c0 certains \u00e9gards, on pourrait croire qu'un tel texte tend \u00e0 supprimer toute puissance propre aux cr\u00e9atures. Il n'en est rien. Tout le spinozisme s'accorde pour reconna\u00eetre aux \u00eatres finis une puissance d'exister, d'agir et de pers\u00e9v\u00e9rer ; et le contexte m\u00eame du _Trait\u00e9 politique_ souligne que les choses ont une puissance propre, identique \u00e0 leur essence et constitutive de leur \u00ab droit \u00bb. Spinoza ne veut pas dire qu'un \u00eatre qui n'existe pas par soi n'a pas de puissance ; il veut dire qu'il n'a de puissance propre qu'en tant qu'il est la partie d'un tout, c'est-\u00e0-dire une partie de la puissance d'un \u00eatre qui, lui, existe par soi. (Toute la preuve a posteriori repose sur ce raisonnement, qui va du conditionn\u00e9 \u00e0 l'inconditionn\u00e9.) Spinoza dit, dans l' _\u00c9thique_ : la puissance de l'homme est \u00ab une partie de la puissance infinie de Dieu20 \u00bb. Mais la partie se r\u00e9v\u00e8le irr\u00e9ductible, degr\u00e9 de puissance original et distinct de tous les autres. Nous sommes une partie de la puissance de Dieu, mais pr\u00e9cis\u00e9ment dans la mesure o\u00f9 cette puissance est \u00ab expliqu\u00e9e \u00bb par notre essence elle-m\u00eame21. La participation chez Spinoza sera toujours pens\u00e9e comme une participation des puissances. Mais jamais la participation des puissances ne supprime la distinction des essences. Jamais Spinoza ne confond une essence de mode et une essence de substance : ma puissance reste ma propre essence, la puissance de Dieu reste sa propre essence, au moment m\u00eame o\u00f9 ma puissance est une partie de la puissance de Dieu22.\n\nComment est-ce possible ? Comment concilier la distinction des essences et la participation des puissances ? Si la puissance ou l'essence de Dieu peut \u00eatre \u00ab expliqu\u00e9e \u00bb par une essence finie, c'est parce que les attributs sont des formes communes \u00e0 Dieu dont ils constituent l'essence, et aux choses finies dont ils contiennent les essences. La puissance de Dieu se divise ou s'explique dans chaque attribut suivant les essences comprises dans cet attribut. _C'est en ce sens que le rapport tout-partie tend \u00e0 se confondre avec le rapport attribut-mode, substance-modification._ Les choses finies sont des parties de la puissance divine parce qu'elles sont les modes des attributs de Dieu. Mais la r\u00e9duction des \u00ab cr\u00e9atures \u00bb \u00e0 l'\u00e9tat de modes, loin de leur retirer toute puissance propre, montre au contraire comment une part de puissance leur revient en propre, conform\u00e9ment \u00e0 leur essence. L'identit\u00e9 de la puissance et de l'essence s'affirme \u00e9galement (sous les m\u00eames conditions) des modes et de la substance. Ces conditions sont les attributs, par lesquels la substance poss\u00e8de une toute-puissance identique \u00e0 son essence, sous lequel les modes poss\u00e8dent une part de cette puissance, identique \u00e0 leur essence. C'est pourquoi les modes, impliquant ces m\u00eames attributs qui constituent l'essence de Dieu, sont dits \u00ab expliquer \u00bb ou \u00ab exprimer \u00bb la puissance divine23. R\u00e9duire les choses \u00e0 des modes d'une substance unique n'est pas un moyen d'en faire des apparences, des fant\u00f4mes, comme Leibniz le croyait ou feignait de le croire, mais au contraire le seul moyen, selon Spinoza, d'en faire des \u00eatres \u00ab naturels \u00bb, dou\u00e9s de force ou de puissance.\n\nL'identit\u00e9 de la puissance et de l'essence signifie ceci : la puissance est toujours acte ou, du moins, en acte. Une longue tradition th\u00e9ologique affirmait d\u00e9j\u00e0 l'identit\u00e9 de la puissance et de l'acte, non seulement en Dieu, mais dans la nature24. D'autre part, une longue tradition physique et mat\u00e9rialiste affirmait, dans les choses cr\u00e9\u00e9es elles-m\u00eames, le caract\u00e8re actuel de toute puissance : \u00e0 la distinction de la puissance et de l'acte, on substituait la corr\u00e9lation d'une puissance d'agir et d'une puissance de p\u00e2tir, toutes deux actuelles25. Chez Spinoza les deux courants se rejoignent, l'un renvoyant \u00e0 l'essence de la substance, l'autre \u00e0 l'essence du mode. C'est que, dans le spinozisme, toute puissance entra\u00eene un pouvoir d'\u00eatre affect\u00e9 qui lui correspond et en est ins\u00e9parable. Or ce pouvoir d'\u00eatre affect\u00e9 est toujours et n\u00e9cessairement rempli. \u00c0 la _potentia_ correspond une _aptitudo_ ou _potestas_ ; mais il n'y a pas d'aptitude ou de pouvoir qui ne soit effectu\u00e9s, donc pas de puissance qui ne soit actuelle26.\n\nUne essence de mode est puissance ; lui correspond dans le mode un certain pouvoir d'\u00eatre affect\u00e9. Mais parce que le mode est une partie de la nature, son pouvoir est toujours rempli, soit par des affections produites par les choses ext\u00e9rieures (affections dites passives), soit par des affections qui s'expliquent par sa propre essence (affections dites actives). Ainsi la distinction de la puissance et de l'acte, au niveau du mode, dispara\u00eet au profit d'une corr\u00e9lation entre deux puissances \u00e9galement actuelles, puissance d'agir et puissance de p\u00e2tir, qui varient en raison inverse, mais dont la somme est constante et constamment effectu\u00e9e. C'est pourquoi Spinoza peut pr\u00e9senter la puissance du mode, tant\u00f4t comme un invariant identique \u00e0 l'essence, puisque le pouvoir d'\u00eatre affect\u00e9 reste constant, tant\u00f4t comme sujette \u00e0 des variations, puisque la puissance d'agir (ou force d'exister) \u00ab augmente \u00bb ou \u00ab diminue \u00bb suivant la proportion des affections actives qui contribuent \u00e0 remplir ce pouvoir \u00e0 chaque instant27. Reste que le mode, de toute mani\u00e8re, n'a d'autre puissance qu'actuelle : \u00e0 chaque instant il est tout ce qu'il peut \u00eatre, sa puissance est son essence.\n\n\u00c0 l'autre p\u00f4le, l'essence de la substance est puissance. Cette puissance absolument infinie d'exister entra\u00eene un pouvoir d'\u00eatre affect\u00e9 d'une infinit\u00e9 de fa\u00e7ons. Mais cette fois, le pouvoir d'\u00eatre affect\u00e9 ne peut \u00eatre rempli que par des affections actives. Comment la substance absolument infinie aurait-elle une puissance de p\u00e2tir, celle-ci supposant \u00e9videmment une limitation de la puissance d'agir ? \u00c9tant toute-puissante en elle-m\u00eame et par elle-m\u00eame, la substance est n\u00e9cessairement capable d'une infinit\u00e9 d'affections, et cause active de toutes les affections dont elle est capable. Dire que l'essence de Dieu est puissance, c'est dire que Dieu produit une infinit\u00e9 de choses, en vertu de cette m\u00eame puissance par laquelle il existe. _Il les produit donc comme il existe._ Cause de toutes choses \u00ab au m\u00eame sens \u00bb que cause de soi, il produit toutes choses dans ses attributs, puisque ses attributs constituent \u00e0 la fois son essence et son existence. Il ne suffira donc pas de dire que la puissance de Dieu est actuelle : elle est n\u00e9cessairement active, elle est acte. L'essence de Dieu n'est pas puissance sans qu'une infinit\u00e9 de choses n'en d\u00e9coulent, et n'en d\u00e9coulent pr\u00e9cis\u00e9ment dans ces attributs qui la constituent. Aussi bien les modes sont-ils les affections de Dieu ; mais jamais Dieu ne p\u00e2tit de ses modes, il n'a d'affections qu'actives28.\n\nToute essence est essence de quelque chose. On distinguera donc : l'essence comme puissance ; ce dont elle est l'essence ; le pouvoir d'\u00eatre affect\u00e9 qui lui correspond. Ce dont l'essence est l'essence, c'est toujours une quantit\u00e9 de r\u00e9alit\u00e9 ou de perfection. Mais une chose a d'autant plus de r\u00e9alit\u00e9 ou de perfection qu'elle peut \u00eatre affect\u00e9e d'un plus grand nombre de fa\u00e7ons : _la quantit\u00e9 de r\u00e9alit\u00e9 trouve toujours sa raison dans une puissance identique \u00e0 l'essence._ La preuve a posteriori part de la puissance propre aux \u00eatre finis : on cherche la condition sous laquelle un \u00eatre fini a une puissance, on s'\u00e9l\u00e8ve jusqu'\u00e0 la puissance inconditionn\u00e9e d'une substance absolument infinie. En effet, une essence d'\u00eatre fini n'est puissance que par rapport \u00e0 une substance dont cet \u00eatre est le mode. Mais cette d\u00e9marche a posteriori n'est qu'une fa\u00e7on, pour nous, d'arriver \u00e0 une d\u00e9marche a priori plus profonde. L'essence de la substance absolument infinie est toute-puissance, parce que la substance poss\u00e8de a priori toutes les conditions sous lesquelles on attribue la puissance \u00e0 quelque chose. Or, s'il est vrai que les modes, en vertu de leur puissance, se disent par rapport \u00e0 la substance, la substance, en vertu de la sienne, se dit en rapport avec les modes : elle n'a pas une puissance absolument infinie d'exister sans remplir, par une infinit\u00e9 de choses en une infinit\u00e9 de modes, le pouvoir d'\u00eatre affect\u00e9 qui correspond \u00e0 cette puissance.\n\nC'est en ce sens que Spinoza nous conduit \u00e0 une derni\u00e8re triade de la substance. Partant des preuves de la puissance, la d\u00e9couverte de cette triade occupe toute la fin du premier livre de l' _\u00c9thique_. Elle se pr\u00e9sente ainsi : l'essence de la substance comme puissance absolument infinie d'exister ; la substance comme _ens realissimum_ existant par soi ; un pouvoir d'\u00eatre affect\u00e9 d'une _infinit\u00e9 de fa\u00e7ons_ , correspondant \u00e0 cette puissance, n\u00e9cessairement rempli par des affections dont la substance m\u00eame est la cause active. Cette troisi\u00e8me triade vient prendre place \u00e0 c\u00f4t\u00e9 des deux pr\u00e9c\u00e9dentes. Elle ne signifie pas, comme la premi\u00e8re, la n\u00e9cessit\u00e9 d'une substance ayant tous les attributs ; ni, comme la seconde, la n\u00e9cessit\u00e9 pour cette substance d'exister absolument. Elle signifie la n\u00e9cessit\u00e9, pour cette substance existante, de produire une infinit\u00e9 de choses. Et elle ne se contente pas de nous faire passer aux modes, elle s'applique ou se communique \u00e0 eux. Si bien que le mode lui-m\u00eame pr\u00e9sentera la triade suivante : essence de mode comme puissance ; mode existant d\u00e9fini par sa quantit\u00e9 de r\u00e9alit\u00e9 ou de perfection ; pouvoir d'\u00eatre affect\u00e9 d'un _grand nombre_ de fa\u00e7ons. Ainsi le premier livre de l' _\u00c9thique_ est comme le d\u00e9veloppement de trois triades, qui trouvent leur principe dans l'expression : la substance, l'absolu, la puissance.\n\n* * *\n\n1.  LEIBNIZ, _Lettre \u00e0 la princesse \u00c9lisabeth_ , 1678 : \u00ab II faut avouer que ces raisonnements [les preuves cart\u00e9siennes de l'existence de Dieu] sont un peu suspects parce qu'ils vont trop vite et nous font violence sans nous \u00e9clairer. \u00bb Le th\u00e8me \u00ab trop vite \u00bb revient constamment : contre Descartes, Leibniz invoque son propre go\u00fbt pour l'esprit lent et pesant, son go\u00fbt du continu qui interdit les \u00ab bonds \u00bb, son go\u00fbt des d\u00e9finitions r\u00e9elles et des polysyllogismes, son go\u00fbt pour un art d'inventer qui prend du temps. Quand Leibniz reproche \u00e0 Descartes d'avoir cru que la quantit\u00e9 de mouvement se conservait, il faut voir dans cette critique un cas particulier (sans doute particuli\u00e8rement important) d'une objection tr\u00e8s g\u00e9n\u00e9rale : Descartes, dans tous les domaines, prend le relatif pour l'absolu, \u00e0 force d'aller trop vite.\n\n2.  _PPD_ , I, 7, sc. : \u00ab Je ne sais pas ce qu'il veut dire par l\u00e0. Qu'appelle-t-il facile et difficile en effet ?... L'araign\u00e9e tisse facilement une toile que les hommes ne pourraient tisser sans de tr\u00e8s grandes difficult\u00e9s... \u00bb\n\n3.  DESCARTES, _M\u00e9ditation III_ et _Principes_ I, 17-18.\n\n4.  _M\u00e9ditation III_ et _Principes_ , I, 20-21 (le texte des _Principes_ , toutefois, \u00e9vite oute r\u00e9f\u00e9rence explicite aux notions de facile et de difficile).\n\n5.  _Abr\u00e9g\u00e9 g\u00e9om\u00e9trique des secondes r\u00e9ponses_ , axiome 8, AT IX, p. 128.\n\n6.  Sur toutes ces objections faites \u00e0 Descartes par certains de ses correspondants, et sur les r\u00e9ponses de Descartes, cf. l' _Entretien avec Burman_ , et aussi la _Lettre 347, \u00e0 Mesland_ , AT, IV, p. 111.\n\n7.  _CT_ , I, ch. 1, 3-9.\n\n8.  _CT_ , II, ch. 20, 4, note 3.\n\n9.  _TRE_ , 76 et note 2 : \u00ab Comme la source de la Nature... ne peut s'\u00e9tendre dans l'entendement plus loin que dans la r\u00e9alit\u00e9..., on ne doit craindre aucune confusion au sujet de son id\u00e9e... \u00bb ; \u00ab Si un tel \u00eatre n'existait pas, il ne pourrait jamais \u00eatre produit, et l'esprit pourrait comprendre plus que la Nature ne pourrait pr\u00e9senter. \u00bb\n\n10.  _Lettre 40, \u00e0 Jelles_ , mars 1667 (III, p. 142).\n\n11.  _PPD_ , I, 7, lemmes 1 et 2, et d\u00e9monstration de 7.\n\n12.  _\u00c9_ , I, 11, sc.\n\n13.  _\u00c9_ , 1, 11 3e dem.\n\n14.  _\u00c9_ , I, 11, sc.\n\n15.  Sans doute Spinoza parle-t-il plus souvent d'un effort de pers\u00e9v\u00e9rer dans l'\u00eatre. Mais ce _conatus_ est lui-m\u00eame _potentia agendi._ Cf. _\u00c9_ , III, 57, dem. : _potentia seu conatus. \u00c9_ , III, d\u00e9finition g\u00e9n\u00e9rale des affects : _agendi potentia sive existendi vis. \u00c9_ , IV, 29, dem. : _hominis potentia qua existet et operatur._\n\n16.  _CT_ , II, ch. 20, 4, note 3 : \u00ab Cette id\u00e9e, isol\u00e9e, consid\u00e9r\u00e9e en dehors des autres id\u00e9es, ne peut rien \u00eatre de plus qu'une id\u00e9e d'une certaine chose, et elle ne peut avoir une id\u00e9e de cette chose ; attendu qu'une id\u00e9e ainsi consid\u00e9r\u00e9e, _n'\u00e9tant qu'une partie_ , ne peut avoir d'elle-m\u00eame et de son objet aucune connaissance claire et distincte ; cela n'est possible qu'\u00e0 la chose pensante qui, seule, est la Nature enti\u00e8re, car un fragment consid\u00e9r\u00e9 en dehors du tout auquel il appartient ne peut, etc. \u00bb.\n\n17.  _\u00c9_ , II, _5_ , dem.\n\n18.  _Lettre 21_ , _\u00e0 Blyenbergh_ (III, p. 86).\n\n19.  _TP_ , ch. 2, 2-3.\n\n20.  _\u00c9_ , IV, 4, dem.\n\n21.  _\u00c9_ , IV, 4, dem.\n\n22.  _\u00c9_ , IV, 4, dem. : \u00ab La puissance de l'homme, en tant qu'elle s'explique par _son_ essence actuelle, est une partie de la puissance infinie de Dieu ou de la Nature, c'est-\u00e0-dire de _son_ essence. \u00bb\n\n23.  _\u00c9_ , I, 36, dem.\n\n24.  L'identit\u00e9 de la puissance et de l'acte, au moins dans le No\u00fbs, est un th\u00e8me fr\u00e9quent du n\u00e9o-platonisme. On le retrouve dans la pens\u00e9e chr\u00e9tienne aussi bien que dans la pens\u00e9e juive. Nicolas de Cues en tire la notion de _Possest_ , qu'il applique \u00e0 Dieu ( _\u0152uvres choisies_ , Aubier \u00e9d., pp. 543-546 ; et M. de Gandillac, _La Philosophie de Nicolas de Cues_ , pp. 298-306). Cette identit\u00e9 de l'acte et de la puissance en Dieu, Bruno l'\u00e9tend au \u00ab simulacre \u00bb, c'est-\u00e0-dire \u00e0 l'univers ou \u00e0 la Nature ( _Cause, Principe, Unit\u00e9_ , Alcan \u00e9d., 3e dialogue).\n\n25.  Cette tradition trouve d\u00e9j\u00e0 un aboutissement chez Hobbes (cf. _De Corpore_ , ch. X).\n\n26.  Spinoza parle souvent d'une _aptitude_ du corps, qui correspond \u00e0 sa puissance : le corps est apte ( _aptus_ ) \u00e0 agir et \u00e0 p\u00e2tir ( _\u00c9_ , II, 13, sc.) ; il _peut_ \u00eatre affect\u00e9 d'un grand nombre de fa\u00e7ons ( _\u00c9_ , III, postulat 1), l'excellence de l'homme vient de ce que son corps est \u00ab apte au plus grand nombre de choses \u00bb ( _\u00c9_ , V, 39). D'autre part, une _potestas_ correspond \u00e0 la puissance de Dieu ( _potentia_ ) ; Dieu peut \u00eatre affect\u00e9 d'une infinit\u00e9 de fa\u00e7ons, et produit n\u00e9cessairement toutes les affections dont il a le pouvoir ( _\u00c9_ , I, 35).\n\n27.  Sur les variations de la _vis existendi_ , cf. _\u00c9_ , III, d\u00e9f. g\u00e9n\u00e9rale des affects.\n\n28.  _CT_ , I, ch. 2, 22-25. _\u00c9_ , I, 15, sc.\n\n# DEUXI\u00c8ME PARTIE\n\n# LE PARALL\u00c9LISME\n\n# ET L'IMMANENCE\n\n## CHAPITRE VI\n\n## L'EXPRESSION DANS LE PARALL\u00c9LISME\n\nPourquoi Dieu produit-il ? Le probl\u00e8me d'une raison suffisante de la production ne dispara\u00eet pas dans le spinozisme ; au contraire il gagne en urgence. Car la nature de Dieu est expressive en elle-m\u00eame, comme nature naturante. Cette expression est tellement naturelle ou essentielle \u00e0 Dieu qu'elle ne se contente pas de refl\u00e9ter un Dieu tout fait, mais forme une esp\u00e8ce de d\u00e9veloppement du divin, une constitution logique et g\u00e9n\u00e9tique de la substance divine. Chaque attribut exprime une essence formelle ; toutes les essences formelles sont exprim\u00e9es comme l'essence absolue d'une seule et m\u00eame substance dont l'existence d\u00e9coule n\u00e9cessairement ; cette existence elle-m\u00eame est donc exprim\u00e9e par les attributs. Ces moments sont les v\u00e9ritables moments de la substance ; l'expression est, en Dieu, la vie m\u00eame de Dieu. D\u00e8s lors on ne pourra pas dire que Dieu produise le monde, l'univers ou la nature natur\u00e9e, _pour_ s'exprimer. Non seulement la raison suffisante doit \u00eatre n\u00e9cessaire, excluant tout argument de finalit\u00e9, mais Dieu s'exprime en lui-m\u00eame, dans sa propre nature, dans les attributs qui le constituent. Il n'a nul \u00ab besoin \u00bb de produire, ne manquant de rien. Il faut prendre \u00e0 la lettre une m\u00e9taphore de Spinoza qui montre que le monde produit n'ajoute rien \u00e0 l'essence de Dieu : quand un artisan sculpte des t\u00eates et des poitrines, puis joint une poitrine \u00e0 une t\u00eate, cette addition n'ajoute rien \u00e0 l'essence de la t\u00eate1. Celle-ci garde la m\u00eame essence, _la m\u00eame expression._ Si Dieu s'exprime en lui-m\u00eame, l'univers ne peut \u00eatre qu'une expression au second degr\u00e9. La substance s'exprime d\u00e9j\u00e0 dans les attributs qui constituent la nature naturante, mais les attributs s'expriment \u00e0 leur tour dans les modes, qui constituent la nature natur\u00e9e. Raison de plus pour demander : pourquoi ce second niveau ? Pourquoi Dieu produit-il un univers modal ?\n\nPour rendre compte a priori de la production, Spinoza invoque un premier argument. Dieu agit, ou produit, comme il se comprend ( _seipsum intelligit_ ) : se comprenant n\u00e9cessairement, il agit n\u00e9cessairement2. Un deuxi\u00e8me argument appara\u00eet tant\u00f4t comme d\u00e9pendant du premier, tant\u00f4t comme distinct et conjoint. Dieu produit comme il existe ; existant n\u00e9cessairement, il produit n\u00e9cessairement3.\n\nQuel est le sens du premier argument ? Que signifie \u00ab se comprendre \u00bb ? Dieu ne con\u00e7oit pas des _possibilit\u00e9s_ dans son entendement, mais comprend la n\u00e9cessit\u00e9 de sa propre nature. L'entendement infini n'est pas le lieu des possibles, mais la forme de l'id\u00e9e que Dieu a n\u00e9cessairement de lui-m\u00eame ou de sa propre essence. La science de Dieu n'est pas une science des possibles, mais la science que Dieu a de soi-m\u00eame et de sa propre nature. Comprendre s'oppose donc \u00e0 concevoir quelque chose comme possible. Mais comprendre, en ce sens, c'est d\u00e9duire des propri\u00e9t\u00e9s \u00e0 partir de ce qu'on appr\u00e9hende comme n\u00e9cessaire. Ainsi, \u00e0 partir de la d\u00e9finition du cercle, nous d\u00e9duisons plusieurs propri\u00e9t\u00e9s qui suivent r\u00e9ellement de cette d\u00e9finition. Dieu se comprend lui-m\u00eame ; une infinit\u00e9 de propri\u00e9t\u00e9s s'ensuivent, qui tombent n\u00e9cessairement sous l'entendement de Dieu. Dieu ne peut pas comprendre sa propre essence sans produire une infinit\u00e9 de choses, qui d\u00e9coulent de celle-ci _comme les propri\u00e9t\u00e9s d\u00e9coulent d'une d\u00e9finition._ On voit que, dans cet argument, les modes sont assimil\u00e9s \u00e0 des propri\u00e9t\u00e9s logiquement n\u00e9cessaires qui suivent de l'essence de Dieu telle qu'elle est comprise. Quand Spinoza f\u00e9licite certains H\u00e9breux d'avoir entrevu que Dieu, l'entendement de Dieu et les choses comprises par lui \u00e9taient une seule et m\u00eame chose, il veut dire \u00e0 la fois que l'entendement de Dieu est la science que Dieu a de sa propre nature, et que cette science comprend une infinit\u00e9 de choses qui en d\u00e9coulent n\u00e9cessairement4.\n\nMais pourquoi Dieu se comprend-il ? Il arrive \u00e0 Spinoza de pr\u00e9senter cette proposition comme une sorte d'axiomes5. Cet axiome renvoie \u00e0 des conceptions aristot\u00e9liciennes : Dieu se pense lui-m\u00eame, il est lui-m\u00eame objet de sa pens\u00e9e, sa science n'a d'autre objet que lui-m\u00eame. Tel est le principe qu'on oppose \u00e0 l'id\u00e9e d'un entendement divin qui penserait des \u00ab possibles \u00bb. Et beaucoup de commentateurs pouvaient r\u00e9unir des arguments convaincants pour montrer que le Dieu d'Aristote, se pensant lui-m\u00eame, pense aussi toutes les autres choses qui en d\u00e9coulent n\u00e9cessairement : la tradition aristot\u00e9licienne tendait ainsi vers un th\u00e9isme, parfois m\u00eame vers un panth\u00e9isme, identifiant le connaissant, la connaissance et le connu (les H\u00e9breux invoqu\u00e9s par Spinoza sont des philosophes juifs aristot\u00e9liciens).\n\nToutefois la th\u00e9orie spinoziste de l'id\u00e9e de Dieu est trop originale pour se fonder sur un simple axiome ou se r\u00e9clamer d'une tradition. Que Dieu se comprenne lui-m\u00eame, cela doit suivre de la n\u00e9cessit\u00e9 de la nature divine6. Or, de ce point de vue, la notion d'expression joue un r\u00f4le d\u00e9terminant. Dieu ne s'exprime pas sans se comprendre en tant qu'il s'exprime. Dieu ne s'exprime pas formellement dans ses attributs sans se comprendre objectivement dans une id\u00e9e. L'essence de Dieu n'est pas exprim\u00e9e dans les attributs comme essence formelle, sans \u00eatre exprim\u00e9e dans une id\u00e9e comme essence objective. C'est pourquoi, d\u00e8s la d\u00e9finition de l'attribut, Spinoza se r\u00e9f\u00e9rait \u00e0 un entendement capable de percevoir. Non pas que l'attribut soit \u00ab attribu\u00e9 \u00bb par l'entendement : le mot \u00ab percevoir \u00bb indique suffisamment que l'entendement ne saisit rien qui ne soit dans la nature. Mais les attributs n'expriment pas l'essence de la substance sans se rapporter n\u00e9cessairement \u00e0 un entendement qui les comprend objectivement, c'est-\u00e0-dire qui per\u00e7oit ce qu'ils expriment. Ainsi l'id\u00e9e de Dieu se trouve fond\u00e9e dans la nature divine elle-m\u00eame : parce que Dieu a pour nature une infinit\u00e9 d'attributs dont chacun \u00ab exprime \u00bb une essence infinie, il suit de cette nature expressive que Dieu se comprenne et, se comprenant, produise toutes les choses qui \u00ab tombent \u00bb sous un entendement infini7. Les expressions sont toujours des explications. Mais les explications faites par l'entendement sont seulement des perceptions. Ce n'est pas l'entendement qui explique la substance, mais les explications de la substance renvoient n\u00e9cessairement \u00e0 un entendement qui les comprend. Dieu se comprend n\u00e9cessairement, comme il s'explique ou s'exprime.\n\nConsid\u00e9rons le deuxi\u00e8me argument : Dieu produit comme il existe. Les modes, ici, ne sont plus assimil\u00e9s \u00e0 des propri\u00e9t\u00e9s logiques, mais plut\u00f4t \u00e0 des affections physiques. Le d\u00e9veloppement autonome de cet argument est donc fond\u00e9 sur la puissance : plus une chose a de puissance, plus elle peut \u00eatre affect\u00e9e d'un grand nombre de fa\u00e7ons ; or nous avons d\u00e9montr\u00e9, soit a posteriori, soit a priori, que Dieu avait une puissance absolument infinie d'exister. Dieu a donc un pouvoir d'\u00eatre affect\u00e9 d'une infinit\u00e9 de fa\u00e7ons, _potestas_ qui correspond \u00e0 sa puissance ou _potentia._ Ce pouvoir est n\u00e9cessairement rempli, mais ne peut pas l'\u00eatre par des affections qui viendraient d'autre chose que de Dieu ; donc Dieu produit n\u00e9cessairement et activement une infinit\u00e9 de choses qui l'affectent d'une infinit\u00e9 de fa\u00e7ons.\n\nQue Dieu produise n\u00e9cessairement nous dit en m\u00eame temps comment il produit. Se comprenant comme substance compos\u00e9e d'une infinit\u00e9 d'attributs, existant comme substance compos\u00e9e d'une infinit\u00e9 d'attributs, Dieu agit comme il se comprend et comme il existe, donc _dans_ ces attributs qui expriment \u00e0 la fois son essence et son existence. Il produit une infinit\u00e9 de choses, mais \u00ab en une infinit\u00e9 de modes \u00bb. C'est-\u00e0-dire : les choses produites n'existent pas hors des attributs qui les contiennent. Les attributs sont les conditions univoques sous lesquelles Dieu existe, mais aussi sous lesquelles il agit. Les attributs sont des formes univoques et communes : ils se disent, sous la m\u00eame forme, des cr\u00e9atures et du cr\u00e9ateur, des produits et du producteur, constituant formellement l'essence de l'un, contenant formellement l'essence des autres. Le principe de la n\u00e9cessit\u00e9 de la production renvoie donc \u00e0 une double univocit\u00e9. Univocit\u00e9 de la cause : Dieu est cause de toutes choses _au m\u00eame sens_ que cause de soi. Univocit\u00e9 des attributs : Dieu produit par et dans ces m\u00eames attributs qui constituent son essence. C'est pourquoi Spinoza poursuit une constante pol\u00e9mique : il ne cesse de montrer l'absurdit\u00e9 d'un Dieu qui produirait par des attributs moraux, comme bont\u00e9, justice ou charit\u00e9, ou m\u00eame par des attributs humains, comme entendement et volont\u00e9.\n\nSupposons, par _analogie_ avec l'homme, que l'entendement et la volont\u00e9 soient des attributs de Dieu lui-m\u00eame8. Nous aurons beau faire, nous n'attribuons \u00e0 Dieu volont\u00e9 et entendement que de mani\u00e8re \u00e9quivoque : en vertu de la distinction d'essence entre l'homme et Dieu, la volont\u00e9 et l'entendement divins n'auront avec l'humain qu'une \u00ab communaut\u00e9 de nom \u00bb, comme le Chien-constellation avec le chien-animal aboyant. En sortent de nombreuses absurdit\u00e9s, d'apr\u00e8s lesquelles Dieu devra contenir \u00e9minemment les perfections sous lesquelles il produit les cr\u00e9atures. 1o) Du point de vue de l'entendement : on dira que Dieu est \u00ab tout-puissant \u00bb pr\u00e9cis\u00e9ment parce qu'il \u00ab ne peut pas \u00bb cr\u00e9er les choses avec les m\u00eames perfections qu'il les entend, c'est-\u00e0-dire sous les m\u00eames formes que celles qui lui appartiennent. Ainsi l'on pr\u00e9tend prouver la toute-puissance de Dieu par une impuissance9. 2o) Du point de vue de la volont\u00e9 : on dira que Dieu aurait pu vouloir autre chose, ou que les choses auraient pu \u00eatre d'une autre nature si Dieu l'avait voulu. On attribue \u00e0 Dieu la volont\u00e9, on en fait donc l'essence de Dieu ; mais on suppose en m\u00eame temps que Dieu aurait pu avoir une autre volont\u00e9, donc une autre essence (\u00e0 moins de faire de la volont\u00e9 divine un pur \u00eatre de raison, auquel cas les contradictions redoublent) ; d\u00e8s lors on suppose que deux ou plusieurs dieux pourraient \u00eatre donn\u00e9s. Cette fois, on met en Dieu variabilit\u00e9 et pluralit\u00e9 pour prouver son \u00e9minence10.\n\nNous simplifions les critiques de Spinoza. Mais chaque fois qu'il s'attaque \u00e0 l'image d'un Dieu qui serait essentiellement dou\u00e9 d'entendement et de volont\u00e9, nous croyons qu'il d\u00e9veloppe les implications critiques de sa th\u00e9orie de l'univocit\u00e9. Il veut montrer que l'entendement et la volont\u00e9 ne pourraient \u00eatre consid\u00e9r\u00e9s comme attributs de Dieu que par analogie. Mais l'analogie ne r\u00e9ussit pas \u00e0 cacher l'\u00e9quivocit\u00e9 dont elle part, l'\u00e9minence \u00e0 laquelle elle arrive. Or des perfections \u00e9minentes en Dieu, comme des attributs \u00e9quivoques, entra\u00eenent toutes sortes de contradictions. Seules sont attribu\u00e9es \u00e0 Dieu ces formes qui sont aussi parfaites dans les cr\u00e9atures qui les impliquent et en Dieu qui les comprend. Dieu ne produit pas parce qu'il veut, mais parce qu'il est. Il ne produit pas parce qu'il con\u00e7oit, c'est-\u00e0-dire parce qu'il con\u00e7oit des choses comme possibles, mais parce qu'il se comprend lui-m\u00eame, parce qu'il comprend n\u00e9cessairement sa propre nature. En un mot, Dieu agit \u00ab d'apr\u00e8s les seules lois de sa nature \u00bb : il n'aurait pu produire autre chose, ni produire les choses dans un autre ordre, sans avoir une autre nature11. On remarquera que Spinoza, en g\u00e9n\u00e9ral, n'a gu\u00e8re besoin de d\u00e9noncer directement les incoh\u00e9rences de l'id\u00e9e de cr\u00e9ation. Il lui suffit de demander : Comment Dieu produit-il, dans quelles conditions ? Les conditions m\u00eames de la production font de celle-ci autre chose qu'une cr\u00e9ation, et des \u00ab cr\u00e9atures \u00bb autre chose que des cr\u00e9atures. Dieu produisant n\u00e9cessairement, et dans ses propres attributs, les produits sont n\u00e9cessairement des modes de ces attributs qui constituent la nature de Dieu.\n\nTout se passe comme si l'expression avait une logique qui l'amenait \u00e0 se redoubler. Spinoza est trop soucieux de grammaire pour qu'on puisse n\u00e9gliger les origines linguistiques de l'\u00ab expression \u00bb. Nous avons vu que les attributs \u00e9taient des noms : des verbes plut\u00f4t que des adjectifs. Chaque attribut est un verbe, une proposition premi\u00e8re infinitive, une expression dou\u00e9e d'un sens distinct ; mais tous les attributs d\u00e9signent la substance comme une seule et m\u00eame chose. La distinction traditionnelle entre le sens exprim\u00e9 et l'objet d\u00e9sign\u00e9 (qui s'exprime) trouve donc dans le spinozisme un champ d'application imm\u00e9diate. Mais cette distinction fonde n\u00e9cessairement un certain mouvement de l'expression. Car il faut que le sens d'une proposition primaire devienne \u00e0 son tour le d\u00e9sign\u00e9 d'une proposition secondaire, ayant elle-m\u00eame un nouveau sens, etc. Ainsi la substance d\u00e9sign\u00e9e s'exprimait dans les attributs, les attributs exprimaient une essence. Maintenant les attributs s'expriment \u00e0 leur tour : ils s'expriment dans les modes qui les d\u00e9signent, ces modes expriment une modification. Les modes sont de v\u00e9ritables propositions \u00ab participiales \u00bb, qui d\u00e9rivent des propositions infinitives principales. C'est en ce sens que l'expression, par son propre mouvement, engendre une expression du second degr\u00e9. L'expression poss\u00e8de en soi la raison suffisante d'une re-expression. Ce second degr\u00e9 d\u00e9finit la production elle-m\u00eame : Dieu est dit produire, en m\u00eame temps que ses attributs s'expriment. Si bien qu'en derni\u00e8re instance c'est toujours Dieu qui se trouve d\u00e9sign\u00e9 par toutes choses, \u00e0 la diff\u00e9rence de niveau pr\u00e8s. Les attributs d\u00e9signent Dieu, mais les modes le d\u00e9signent encore sous l'attribut dont ils d\u00e9pendent. \u00ab Ce que certains H\u00e9breux semblent avoir vu comme \u00e0 travers un nuage, puisqu'ils admettent que Dieu, l'entendement de Dieu et les choses comprises par lui sont une seule et m\u00eame chose12. \u00bb\n\nIl y a un _ordre_ dans lequel Dieu produit n\u00e9cessairement. Cet ordre est celui de l'expression des attributs. D'abord chaque attribut s'exprime dans sa nature absolue : un mode infini imm\u00e9diat est donc la premi\u00e8re expression de l'attribut. Puis l'attribut modifi\u00e9 s'exprime, dans un mode infini m\u00e9diat. Enfin l'attribut s'exprime \u00ab d'une fa\u00e7on certaine et d\u00e9termin\u00e9e \u00bb, ou plut\u00f4t d'une infinit\u00e9 de fa\u00e7ons qui constituent les modes existants finis13. Ce dernier niveau resterait inexplicable si les modes infinis, dans le genre de chaque attribut, ne contenaient des lois ou des principes de lois d'apr\u00e8s lesquels les modes finis correspondants sont eux-m\u00eames d\u00e9termin\u00e9s et ordonn\u00e9s.\n\nS'il existe un ordre de production, cet ordre est le m\u00eame pour tous les attributs. En effet, Dieu produit \u00e0 la fois dans tous les attributs qui constituent sa nature. Les attributs s'expriment donc dans un seul et m\u00eame ordre : jusqu'aux modes finis, qui doivent avoir le m\u00eame ordre dans les attributs divers. Cette identit\u00e9 d'ordre d\u00e9finit une correspondance des modes : \u00e0 tout mode d'un attribut correspond n\u00e9cessairement un mode de chacun des autres attributs. Cette identit\u00e9 d'ordre exclut tout rapport de causalit\u00e9 r\u00e9elle. Les attributs sont irr\u00e9ductibles et r\u00e9ellement distincts ; aucun n'est cause de l'autre, ni d'une chose quelconque dans l'autre. Les modes enveloppent donc exclusivement le concept de leur attribut, non pas celui d'un autre14. L'identit\u00e9 d'ordre, la correspondance entre modes d'attributs diff\u00e9rents, exclut donc tout rapport de causalit\u00e9 r\u00e9elle active entre ces modes, comme entre ces attributs. Et \u00e0 cet \u00e9gard, nous n'avons aucune raison s\u00e9rieuse de croire \u00e0 un changement dans la pens\u00e9e de Spinoza : les textes c\u00e9l\u00e8bres du _Court Trait\u00e9_ o\u00f9 Spinoza parle d'une action d'un attribut sur un autre, d'un effet d'un attribut dans un autre, d'une interaction de modes d'attributs diff\u00e9rents, ne semblent pas devoir s'interpr\u00e9ter en termes de causalit\u00e9 r\u00e9elle15. Le contexte pr\u00e9cise que deux attributs (la pens\u00e9e et l'\u00e9tendue) agissent l'un sur l'autre quand ils sont \u00ab pris tous deux ensemble \u00bb, ou que deux modes d'attributs diff\u00e9rents (l'\u00e2me et le corps) agissent l'un sur l'autre dans la mesure o\u00f9 ils forment \u00ab les parties d'un tout \u00bb. Rien, ici, ne d\u00e9passe r\u00e9ellement l'affirmation d'une correspondance : deux choses \u00e9tant les parties d'un tout, rien ne peut changer dans l'une qui n'ait son correspondant dans l'autre, et aucune ne peut changer sans que le tout lui-m\u00eame ne change16. Tout au plus verra-t-on dans ces textes la marque d'un moment o\u00f9 Spinoza n'exprime pas encore ad\u00e9quatement la diff\u00e9rence de sa propre doctrine avec des doctrines en apparence voisines (causalit\u00e9 occasionnelle, causalit\u00e9 id\u00e9ale). Il ne semble pas que Spinoza ait jamais admis une causalit\u00e9 r\u00e9elle active pour rendre compte du rapport entre modes d'attributs diff\u00e9rents.\n\nLes principes pr\u00e9c\u00e9dents m\u00e8nent \u00e0 un r\u00e9sultat dans lequel on reconna\u00eetra la premi\u00e8re formule du parall\u00e9lisme de Spinoza : il y a une _identit\u00e9 d'ordre_ ou _correspondance_ entre modes d'attributs diff\u00e9rents. On peut appeler \u00ab parall\u00e8les \u00bb, en effet, deux choses ou deux s\u00e9ries de choses qui sont dans un rapport constant, tel qu'il n'y ait rien dans l'une qui n'ait dans l'autre un correspondant, toute causalit\u00e9 r\u00e9elle entre les deux se trouvant exclue. Mais on se m\u00e9fiera du mot \u00ab parall\u00e9lisme \u00bb, qui n'est pas de Spinoza. C'est, Leibniz qui le cr\u00e9e, semble-t-il, et qui l'emploie pour son propre compte, afin de d\u00e9signer cette correspondance entre s\u00e9ries autonomes ou ind\u00e9pendantes17. Nous devons donc penser que l'identit\u00e9 d'ordre ne suffit pas \u00e0 distinguer le syst\u00e8me spinoziste ; en un sens, elle se retrouve plus ou moins dans toutes les doctrines qui refusent d'interpr\u00e9ter les correspondances en termes de causalit\u00e9 r\u00e9elle. Si le mot parall\u00e9lisme d\u00e9signe ad\u00e9quatement la philosophie de Spinoza, c'est parce qu'il implique lui-m\u00eame autre chose qu'une simple identit\u00e9 d'ordre, autre chose qu'une correspondance. Et, en m\u00eame temps, parce que Spinoza ne se contente pas de cette correspondance ou de cette identit\u00e9 pour d\u00e9finir le lien qui unit les modes d'attributs diff\u00e9rents.\n\nPr\u00e9cis\u00e9ment Spinoza donne deux autres formules qui prolongent la premi\u00e8re : _identit\u00e9 de connexion ou \u00e9galit\u00e9 de principe, identit\u00e9 d'\u00eatre ou unit\u00e9 ontologique._ La th\u00e9orie proprement spinoziste s'\u00e9nonce donc ainsi : \u00ab Un seul et m\u00eame ordre, c'est-\u00e0-dire une seule et m\u00eame connexion de causes, autrement dit les m\u00eames choses se suivant les unes les autres18. \u00bb Surtout l'on ne doit pas se h\u00e2ter de consid\u00e9rer l'ordre et la connexion ( _connexio_ ou _concatenatio_ ) comme strictement synonymes. Il est certain que, dans le texte que nous venons de citer, l'affirmation d'une identit\u00e9 d'\u00eatre dit quelque chose de plus que la simple identit\u00e9 de connexion ; il est donc vraisemblable que la connexion, d\u00e9j\u00e0, implique quelque chose de plus que l'ordre. En effet, l'identit\u00e9 de connexion ne signifie pas seulement une autonomie des s\u00e9ries correspondantes, mais une isonomie, c'est-\u00e0-dire une \u00e9galit\u00e9 de principe entre s\u00e9ries autonomes ou ind\u00e9pendantes. Supposons deux s\u00e9ries correspondantes, mais dont les principes soient in\u00e9gaux, le principe de l'une \u00e9tant en quelque mani\u00e8re \u00e9minent par rapport \u00e0 celui de l'autre : entre un solide et sa projection, entre une ligne et l'asymptote, il y a bien identit\u00e9 d'ordre ou correspondance, il n'y a pas \u00e0 proprement parler \u00ab identit\u00e9 de connexion \u00bb. Les points d'une courbe ne s'encha\u00eenent pas ( _concatenantur_ ) comme ceux d'une droite. Dans de tels cas, on ne pourra parler de parall\u00e9lisme qu'en un sens tr\u00e8s vague. Les \u00ab parall\u00e8les \u00bb, au sens pr\u00e9cis, exigent une \u00e9galit\u00e9 de principe entre les deux s\u00e9ries de points correspondants. Lorsque Spinoza affirme que les modes d'attributs diff\u00e9rents n'ont pas seulement le m\u00eame ordre, mais aussi la m\u00eame connexion ou concat\u00e9nation, il veut dire que les principes dont ils d\u00e9pendent sont eux-m\u00eames \u00e9gaux. D\u00e9j\u00e0, dans les textes du _Court Trait\u00e9_ , si deux attributs ou deux modes d'attributs diff\u00e9rents sont \u00ab pris ensemble \u00bb c'est parce qu'ils forment les parties \u00e9gales ou les moiti\u00e9s d'un tout. C'est l'\u00e9galit\u00e9 des attributs qui donne au parall\u00e9lisme son sens strict, garantissant que la connexion est la m\u00eame, entre choses dont l'ordre est le m\u00eame.\n\nAinsi Leibniz cr\u00e9e le mot \u00ab parall\u00e9lisme \u00bb, mais, pour son compte, il l'invoque de mani\u00e8re tr\u00e8s g\u00e9n\u00e9rale et peu ad\u00e9quate : le syst\u00e8me de Leibniz implique bien une correspondance entre s\u00e9ries autonomes, substances et ph\u00e9nom\u00e8nes, solides et projections, mais les principes de ces s\u00e9ries sont singuli\u00e8rement in\u00e9gaux. (Aussi bien Leibniz, quand il parle plus pr\u00e9cis\u00e9ment, invoque-t-il l'image de la projection plut\u00f4t que celle des parall\u00e8les.) Inversement, Spinoza n'emploie pas le mot \u00ab parall\u00e9lisme \u00bb ; mais ce mot convient \u00e0 son syst\u00e8me, parce qu'il pose l'\u00e9galit\u00e9 des principes dont d\u00e9coulent les s\u00e9ries ind\u00e9pendantes et correspondantes. On voit bien, l\u00e0 encore, quelles sont les intentions pol\u00e9miques de Spinoza. Par son strict parall\u00e9lisme, Spinoza refuse toute analogie, toute \u00e9minence, toute forme de sup\u00e9riorit\u00e9 d'une s\u00e9rie sur l'autre, toute action id\u00e9ale qui supposerait une pr\u00e9\u00e9minence : il n'y a pas plus de sup\u00e9riorit\u00e9 de l'\u00e2me sur le corps que de l'attribut pens\u00e9e sur l'attribut \u00e9tendue. Et la troisi\u00e8me formule du parall\u00e9lisme, celle qui affirme l'identit\u00e9 d'\u00eatre, ira encore plus loin dans le m\u00eame sens : les modes d'attributs diff\u00e9rents n'ont pas seulement le m\u00eame ordre et la m\u00eame connexion, mais le m\u00eame \u00eatre ; ce sont les _m\u00eames choses_ qui se distinguent seulement par l'attribut dont elles enveloppent le concept. Les modes d'attributs diff\u00e9rents sont une seule et m\u00eame modification qui diff\u00e8re seulement par l'attribut. Par cette identit\u00e9 d'\u00eatre ou unit\u00e9 ontologique, Spinoza refuse l'intervention d'un Dieu transcendant qui mettrait en accord chaque terme d'une s\u00e9rie avec un terme de l'autre ou, m\u00eame, qui accorderait les s\u00e9ries l'une sur l'autre en fonction de leurs principes in\u00e9gaux. La doctrine de Spinoza est bien nomm\u00e9e \u00ab parall\u00e9lisme \u00bb, mais parce qu'elle exclut toute analogie, toute \u00e9minence, toute transcendance. Le parall\u00e9lisme, \u00e0 strictement parler, ne se comprend ni du point de vue d'une cause occasionnelle, ni du point de vue d'une causalit\u00e9 id\u00e9ale, mais seulement du point de vue d'un Dieu immanent et d'une causalit\u00e9 immanente.\n\nL'essence de l'expression se trouve en jeu dans tout ceci. Car le rapport d'expression d\u00e9borde le rapport de causalit\u00e9 : il vaut pour des choses ind\u00e9pendantes ou des s\u00e9ries autonomes, qui n'en ont pas moins l'une avec l'autre une correspondance d\u00e9termin\u00e9e, constante et r\u00e9gl\u00e9e. Si la philosophie de Spinoza et celle de Leibniz trouvent un terrain d'affrontement naturel, c'est dans l'id\u00e9e d'expression, dans l'usage qu'ils font respectivement de cette id\u00e9e. Or nous verrons que le mod\u00e8le \u00ab expressif \u00bb de Leibniz est toujours celui de l'asymptote ou de la projection. Tout autre est le mod\u00e8le expressif qui se d\u00e9gage de la th\u00e9orie de Spinoza : mod\u00e8le \u00ab parall\u00e9liste \u00bb, il implique l'\u00e9galit\u00e9 de deux choses qui en expriment une m\u00eame troisi\u00e8me, et l'identit\u00e9 de cette troisi\u00e8me telle qu'elle est exprim\u00e9e dans les deux autres. L'id\u00e9e d'expression chez Spinoza recueille et fonde \u00e0 la fois les trois aspects du parall\u00e9lisme.\n\nLe parall\u00e9lisme doit se dire des modes, et seulement des modes. Mais il se fonde sur la substance et les attributs de la substance. Dieu produit en m\u00eame temps dans tous les attributs : il produit dans le m\u00eame ordre, il y a donc correspondance entre modes d'attributs diff\u00e9rents. Mais, parce que ces attributs sont r\u00e9ellement distincts, cette correspondance ou identit\u00e9 d'ordre exclut toute action causale des uns sur les autres. Parce que ces attributs sont tous \u00e9gaux, il y a identit\u00e9 de connexion entre ces modes qui diff\u00e9rent par l'attribut. Parce que ces attributs constituent une seule et m\u00eame substance, ces modes qui diff\u00e8rent par l'attribut forment une seule et m\u00eame modification. On voit en quelque sorte la triade de la substance \u00ab descendre \u00bb dans les attributs et se communiquer aux modes. La substance s'exprimait dans les attributs, chaque attribut \u00e9tait une expression, l'essence de la substance \u00e9tait exprim\u00e9e. Maintenant, chaque attribut s'exprime, les modes qui en d\u00e9pendent sont des expressions, une modification est exprim\u00e9e. On se rappelle que l'essence exprim\u00e9e n'existait pas hors des attributs, mais \u00e9tait exprim\u00e9e comme l'essence absolue de la substance, la m\u00eame pour tous les attributs. Il en est de m\u00eame ici : la modification n'existe pas hors du mode qui l'exprime dans chaque attribut, mais elle est exprim\u00e9e comme modification de la substance, la m\u00eame pour tous les modes qui diff\u00e9rent par l'attribut. Une seule et m\u00eame modification se trouve donc exprim\u00e9e dans l'infinit\u00e9 des attributs sous \u00ab une infinit\u00e9 de modes \u00bb, qui ne diff\u00e8rent que par l'attribut. C'est pourquoi nous devons attacher de l'importance aux termes \u00ab mode \u00bb et \u00ab modification \u00bb. En principe, le mode est une affection d'un attribut, la modification une affection de la substance. L'un se comprend formellement, l'autre ontologiquement. Tout mode est la forme d'une modification dans un attribut, toute modification est l'\u00eatre en soi des modes qui diff\u00e8rent par l'attribut (l'\u00eatre en soi ne s'oppose pas ici \u00e0 un \u00eatre pour nous, mais \u00e0 un \u00eatre formel). Leur corr\u00e9lation s'\u00e9nonce ainsi : les modes qui diff\u00e9rent par l'attribut expriment une seule et m\u00eame modification, mais cette modification n'existe pas hors des modes qui s'expriment dans les attributs divers. D'o\u00f9 une formule que Spinoza lui-m\u00eame pr\u00e9sente comme obscure : \u00ab Dieu est r\u00e9ellement cause des choses comme elles sont en soi ( _ut in se sunt_ ), en tant qu'il consiste en une infinit\u00e9 d'attributs ; et je ne puis pour le moment expliquer cela plus clairement \u00bb19. \u00ab En soi \u00bb ne signifie \u00e9videmment pas que les choses produites par Dieu sont des substances. La _res in se_ est la modification substantielle ; or Dieu ne produit pas cette modification hors des modes qui l'expriment \u00e0 la fois dans tous les attributs. Nous voyons donc la triade de la substance se prolonger dans une triade du mode (attribut-mode-modification). Et c'est bien ainsi que, dans la scolie de II 7, Spinoza d\u00e9montre le parall\u00e9lisme : De m\u00eame qu'une seule et m\u00eame substance est \u00ab comprise \u00bb sous les divers attributs, une seule et m\u00eame chose (modification) est \u00ab exprim\u00e9e \u00bb dans tous les attributs ; comme cette chose n'existe pas hors du mode qui l'exprime dans chaque attribut, les modes qui diff\u00e8rent par l'attribut ont un m\u00eame ordre, une m\u00eame connexion, un m\u00eame \u00eatre en soi.\n\n* * *\n\n1.  _CT_ , I, second dialogue, 5.\n\n2.  _\u00c9_ , II, 3, sc.\n\n3.  _\u00c9_ , I, 25, sc. : \u00ab Au sens o\u00f9 Dieu est dit cause de soi, il doit \u00eatre dit aussi cause de toutes choses. \u00bb II, 3, sc. : \u00ab Il nous est aussi impossible de concevoir Dieu n'agissant pas que Dieu n'existant pas. \u00bb IV, pr\u00e9face : \u00ab Dieu ou la nature agit avec la m\u00eame n\u00e9cessit\u00e9 qu'il existe \u00bb.\n\n4.  _\u00c9_ , II, 7, sc.\n\n5.  _\u00c9_ , II, 3, sc. : \u00ab Ainsi que tous l'admettent d'une voix unanime... \u00bb (De m\u00eame _Lettre 75, \u00e0 Oldenburg, III_ , p. 228).\n\n6.  C'est d\u00e9j\u00e0 ce qui appara\u00eet dans la _d\u00e9monstration_ de II, 3, qui se r\u00e9clame de I, 16. Et la scolie elle-m\u00eame souligne cette r\u00e9f\u00e9rence (\u00ab _Il suit de la n\u00e9cessit\u00e9 de la nature_ divine... que Dieu se comprenne lui-m\u00eame. \u00bb).\n\n7.  _\u00c9_ , I, 16, prop. et dem.\n\n8.  _\u00c9_ , I, 17, sc.\n\n9.  _\u00c9_ , I, 17, sc., et I, 33, sc. 2. _CT_ , I, ch. 4, 1-5.\n\n10.  _\u00c9_ , I, 33, dem. et sc. 2. _CT_ , I, ch. 4, 7-9.\n\n11.  _\u00c9_ , I, 17 et 33, prop. et dem.\n\n12.  _\u00c9_ , II, 7, sc. Nous avons vu pr\u00e9c\u00e9demment (ch. III) comment Spinoza, dans sa th\u00e9orie de l'expression, retrouvait certains th\u00e8mes d'une logique des propositions, d'origine sto\u00efcienne et renouvel\u00e9e par l'\u00e9cole d'Ockham. Mais il faudrait tenir compte d'autres facteurs ; et notamment de la langue h\u00e9bra\u00efque. Dans son _Compendium grammatices linguae hebrae_ , Spinoza d\u00e9gage certains caract\u00e8res qui forment une v\u00e9ritable logique de l'expression d'apr\u00e8s les structures grammaticales de l'h\u00e9breu, et qui fondent une th\u00e9orie des propositions. Faute d'une \u00e9dition comment\u00e9e, ce livre est peu compr\u00e9hensible pour le lecteur qui ne conna\u00eet pas la langue. Nous ne pouvons donc en saisir que certaines donn\u00e9es simples : 1o) le caract\u00e8re intemporel de l'infinitif (ch. 5, ch. 13) ; 2o) le caract\u00e8re participial des modes (ch. 5, ch. 33) ; 3o) la d\u00e9termination de diverses esp\u00e8ces d'infinitifs, dont l'une exprime l'action rapport\u00e9e \u00e0 une cause principale (l'\u00e9quivalent de _constituere aliquem regnantem_ ou _constitui ut regnaret_ , cf. ch. 12).\n\n13.  _\u00c9._ , I, 21-23. prop. et dem.\n\n14.  _\u00c9_ , II, 6, dem.\n\n15.  _CT_ , II, ch. 19, 7 sq., ch. 20, 4-5. (D\u00e9j\u00e0 Albert L\u00e9on montrait que les textes du _Court Trait\u00e9_ n'impliquaient pas n\u00e9cessairement l'hypoth\u00e8se d'une causalit\u00e9 r\u00e9elle entre attributs, ou entre l'\u00e2me et le corps : cf. _Les \u00e9l\u00e9ments cart\u00e9siens de la doctrine spinoziste sur les rapports de la pens\u00e9e et de son objet_ , Alcan, 1907, p. 200.)\n\n16.  _CT_ , II, ch. 20, 4, note 3 : \u00ab L'objet ne peut \u00e9prouver de changement que l'id\u00e9e n'en \u00e9prouve un, et vice versa... \u00bb.\n\n17.  Par \u00ab parall\u00e9lisme \u00bb, Leibniz entend une conception de l'\u00e2me et du corps, qui les rend ins\u00e9parables d'une certaine mani\u00e8re, tout en excluant un rapport de causalit\u00e9 r\u00e9elle entre les deux. Mais c'est _sa propre_ conception qu'il d\u00e9signe ainsi. Cf. _Consid\u00e9rations sur la doctrine d'un Esprit universel_ , 1702, \u00a7 12.\n\n18.  _\u00c9_ , II, 7, sc.\n\n19.  _\u00c9_ , II, 7, sc.\n\n## CHAPITRE VII\n\n## LES DEUX PUISSANCES ET L'ID\u00c9E DE DIEU\n\nIl semblerait donc que le parall\u00e9lisme soit facile \u00e0 d\u00e9montrer. Il suffirait de transf\u00e9rer l'unit\u00e9 de la substance \u00e0 la modification, et le caract\u00e8re expressif des attributs aux modes. Ce transfert se fonderait sur la n\u00e9cessit\u00e9 de la production (second niveau d'expression). Mais, consid\u00e9rant l'ensemble de la proposition 7 du livre II, nous sommes d\u00e9concert\u00e9s parce que nous nous trouvons devant une op\u00e9ration beaucoup plus complexe. 1o) Le texte de la proposition, la d\u00e9monstration et le corollaire affirment bien une identit\u00e9 d'ordre, de connexion et m\u00eame d'\u00eatre ; mais non pas entre des modes qui exprimeraient la m\u00eame modification dans chaque attribut. La triple identit\u00e9 s'affirme seulement de l'id\u00e9e, qui est un mode de la pens\u00e9e, et de la chose repr\u00e9sent\u00e9e, qui est un mode d'un certain attribut. Ce parall\u00e9lisme est donc _\u00e9pist\u00e9mologique_ : il s'\u00e9tablit entre l'id\u00e9e et son \u00ab objet \u00bb ( _res ideata, objectum ideae_ ). 2o) En revanche, le scolie suit la d\u00e9marche indiqu\u00e9e pr\u00e9c\u00e9demment : elle conclut \u00e0 un parall\u00e9lisme _ontologique_ entre tous les modes qui diff\u00e8rent par l'attribut. Mais il n'arrive lui-m\u00eame \u00e0 cette conclusion que par la voie de la d\u00e9monstration et du corollaire : il g\u00e9n\u00e9ralise le cas de l'id\u00e9e et de son objet, il l'\u00e9tend \u00e0 _tous_ les modes qui diff\u00e8rent par l'attribut1.\n\nPlusieurs questions se posent. D'une part, \u00e0 supposer que les deux parall\u00e9lismes s'accordent, pourquoi faut-il passer d'abord par le d\u00e9tour \u00ab \u00e9pist\u00e9mologique \u00bb ? Est-ce seulement un d\u00e9tour ? Quel en est le sens et l'importance dans l'ensemble de l' _\u00c9thique_ ? Mais surtout, les deux parall\u00e9lismes sont-ils conciliables ? Le point de vue \u00e9pist\u00e9mologique signifie : un mode \u00e9tant donn\u00e9 dans un attribut, une id\u00e9e lui correspond dans l'attribut pens\u00e9e, qui repr\u00e9sente ce mode et ne repr\u00e9sente que lui2. Loin de nous mener \u00e0 l'unit\u00e9 d'une \u00ab modification \u00bb exprim\u00e9e par tous les modes d'attributs diff\u00e9rents, le parall\u00e9lisme \u00e9pist\u00e9mologique nous conduit \u00e0 la simple unit\u00e9 d'un \u00ab individu \u00bb form\u00e9 par le mode d'un certain attribut et l'id\u00e9e qui repr\u00e9sente exclusivement ce mode3. Loin de nous mener \u00e0 l'unit\u00e9 de tous les modes qui diff\u00e8rent par leur attribut, il nous conduit \u00e0 la multiplicit\u00e9 des id\u00e9es qui correspondent aux modes d'attributs diff\u00e9rents. C'est en ce sens que le parall\u00e9lisme \u00ab psycho-physique \u00bb est un cas particulier du parall\u00e9lisme \u00e9pist\u00e9mologique : l'\u00e2me est l'id\u00e9e du corps, c'est-\u00e0-dire l'id\u00e9e d'un certain mode de l'\u00e9tendue, et seulement de ce mode. Le point de vue \u00e9pist\u00e9mologique se pr\u00e9sente donc ainsi : un seul et m\u00eame individu est exprim\u00e9 par un certain mode et par l'id\u00e9e qui lui correspond. Mais le point de vue ontologique : une seule et m\u00eame modification est exprim\u00e9e par tous les modes correspondants qui diff\u00e8rent par l'attribut. De tous les \u00e9l\u00e8ves et amis de Spinoza, Tschirnhaus est celui qui souligne le mieux la difficult\u00e9, s'apercevant qu'elle est au c\u0153ur du syst\u00e8me de l' _expression_4. Comment concilier les deux points de vue ? D'autant plus que l'\u00e9pist\u00e9mologie nous force \u00e0 conf\u00e9rer \u00e0 l'attribut pens\u00e9e un singulier privil\u00e8ge : cet attribut doit contenir autant d'id\u00e9es irr\u00e9ductibles qu'il y a de modes d'attributs diff\u00e9rents, bien plus, autant d'id\u00e9es qu'il y a d'attributs. Ce privil\u00e8ge appara\u00eet en contradiction flagrante avec toutes les exigences du parall\u00e9lisme ontologique.\n\nIl est donc n\u00e9cessaire d'examiner en d\u00e9tail la d\u00e9monstration et le corollaire de la proposition 7 : \u00ab L'ordre et la connexion des id\u00e9es sont les m\u00eames que l'ordre et la connexion des choses. \u00bb La d\u00e9monstration est simple ; elle se contente d'invoquer un axiome, \u00ab la connaissance de l'effet d\u00e9pend de la connaissance de la cause et l'enveloppe \u00bb. Ce qui nous renvoie encore \u00e0 un principe aristot\u00e9licien : conna\u00eetre, c'est conna\u00eetre par la cause. Dans une perspective spinoziste, on conclut : 1o) \u00e0 toute id\u00e9e correspond quelque chose (en effet, aucune chose ne peut \u00eatre connue sans une cause qui la fait \u00eatre, en essence ou en existence) ; 2o) l'ordre des id\u00e9es est le m\u00eame que l'ordre des choses (une chose n'est connue que par la connaissance de sa cause).\n\nToutefois, cette perspective proprement spinoziste n'implique pas seulement l'axiome d'Aristote. On ne comprendrait pas pourquoi Aristote lui-m\u00eame et beaucoup d'autres n'arriv\u00e8rent pas \u00e0 la th\u00e9orie du parall\u00e9lisme. Spinoza le reconna\u00eet volontiers : \u00ab Nous avons montr\u00e9 que l'id\u00e9e vraie... manifeste comment et pourquoi quelque chose est ou a \u00e9t\u00e9 fait, et que ses effets objectifs dans l'\u00e2me proc\u00e9dent en conformit\u00e9 avec l'essence formelle de l'objet. Ce qui est la m\u00eame chose que ce qu'ont dit les Anciens, \u00e0 savoir que la science vraie proc\u00e8de de la cause aux effets. \u00c0 cela pr\u00e8s que jamais, autant que je sache, ils n'ont con\u00e7u, comme nous l'avons fait ici, l'\u00e2me agissant selon des lois d\u00e9termin\u00e9es et comme un automate spirituel5. \u00bb \u00ab Automate spirituel \u00bb signifie d'abord qu'une id\u00e9e, \u00e9tant un mode de la pens\u00e9e, ne trouve pas sa cause (efficiente et formelle) ailleurs que dans l'attribut pens\u00e9e. De m\u00eame, un objet quel qu'il soit ne trouve sa cause efficiente et formelle que dans l'attribut dont il est le mode et dont il enveloppe le concept. Voil\u00e0 donc ce qui s\u00e9pare Spinoza de la tradition antique : toute causalit\u00e9 efficiente ou formelle (\u00e0 plus forte raison mat\u00e9rielle et finale) est exclue entre les id\u00e9es et les choses, les choses et les id\u00e9es. Cette double exclusion ne renvoie pas \u00e0 un axiome, mais est objet de d\u00e9monstrations qui occupent le d\u00e9but du livre II de l' _\u00c9thique_6. Spinoza peut donc affirmer l'ind\u00e9pendance des deux s\u00e9ries, s\u00e9rie des choses et s\u00e9rie des id\u00e9es. Que, \u00e0 toute id\u00e9e, corresponde quelque chose est dans ces conditions un premier \u00e9l\u00e9ment du parall\u00e9lisme.\n\nMais seulement un premier \u00e9l\u00e9ment. Pour que les id\u00e9es aient la m\u00eame connexion que les choses, il faut encore qu'\u00e0 toute chose corresponde une id\u00e9e. Nous retrouvons les deux formules du _Court Trait\u00e9_ : \u00ab Aucune id\u00e9e ne peut \u00eatre sans que la chose ne soit \u00bb, mais aussi \u00ab il n'y a aucune chose dont l'id\u00e9e ne soit dans la chose pensante7 \u00bb. Or, pour d\u00e9montrer que toute chose est l'objet d'une id\u00e9e, nous ne nous heurtons plus aux difficult\u00e9s qui nous avaient arr\u00eat\u00e9s dans la preuve a posteriori. Car nous partons maintenant d'un Dieu existant. Nous savons que ce Dieu se comprend lui-m\u00eame : il forme une id\u00e9e de soi-m\u00eame, il poss\u00e8de un entendement infini. Mais il suffit que ce Dieu se comprenne pour qu'il produise et, produisant, comprenne tout ce qu'il produit.\n\nDans la mesure o\u00f9 Dieu produit comme il se comprend, tout ce qu'il produit \u00ab tombe \u00bb n\u00e9cessairement sous son entendement infini. Dieu ne se comprend pas, lui-m\u00eame et sa propre essence, sans comprendre aussi tout ce qui d\u00e9coule de son essence. C'est pourquoi l'entendement infini comprend tous les attributs de Dieu, mais aussi toutes les affections8. L'id\u00e9e que Dieu forme est l'id\u00e9e de sa propre essence ; mais aussi l'id\u00e9e de tout ce que Dieu produit formellement dans ses attributs. Il y a donc autant d'id\u00e9es que de choses, toute chose est l'objet d'une id\u00e9e. On appelle \u00ab chose \u00bb, en effet, tout ce qui suit formellement de la substance divine ; la chose s'explique par tel attribut dont elle est le mode. Mais parce que Dieu comprend tout ce qu'il produit, une id\u00e9e dans l'entendement de Dieu correspond \u00e0 chaque mode qui suit d'un attribut. C'est en ce sens que les id\u00e9es elles-m\u00eames d\u00e9coulent de l'id\u00e9e de Dieu, _comme_ les modes suivent ou d\u00e9coulent de leur attribut respectif ; l'id\u00e9e de Dieu sera donc cause de toutes les id\u00e9es, comme Dieu lui-m\u00eame est cause de toutes choses.\n\n\u00c0 toute id\u00e9e correspond quelque chose, et \u00e0 toute chose une id\u00e9e. C'est bien ce th\u00e8me qui permet \u00e0 Spinoza d'affirmer une _\u00e9galit\u00e9 de principe_ : il y a en Dieu deux puissances \u00e9gales. Dans la proposition 7, le corollaire s'encha\u00eene avec la d\u00e9monstration, en reconnaissant pr\u00e9cis\u00e9ment cette \u00e9galit\u00e9 de puissances : \u00ab Il suit de l\u00e0 que la puissance de penser de Dieu est \u00e9gale \u00e0 sa puissance actuelle d'agir. \u00bb L'argument des puissances ne sert donc plus \u00e0 prouver a posteriori l'existence de Dieu, mais joue un r\u00f4le d\u00e9cisif dans la d\u00e9termination du parall\u00e9lisme \u00e9pist\u00e9mologique. Il nous permet d'aller encore plus loin, d'affirmer enfin une _identit\u00e9 d'\u00eatre_ entre les objets et les id\u00e9es. Telle est la fin du corollaire : la m\u00eame chose suit formellement (c'est-\u00e0-dire dans tel ou tel attribut) de la nature infinie de Dieu, et suit objectivement de l'id\u00e9e de Dieu. Un seul et m\u00eame \u00eatre est formel dans l'attribut dont il d\u00e9pend sous la puissance d'exister et d'agir, objectif dans l'id\u00e9e de Dieu dont il d\u00e9pend sous la puissance de penser. Un mode d'un attribut et l'id\u00e9e de ce mode sont une seule et m\u00eame chose exprim\u00e9e de deux fa\u00e7ons, sous deux puissances. Dans l'ensemble de la d\u00e9monstration et du corollaire, nous retrouvons donc les trois temps du parall\u00e9lisme : identit\u00e9 d'ordre, identit\u00e9 de connexion ou \u00e9galit\u00e9 de principe, identit\u00e9 d'\u00eatre, mais qui ne s'appliquent ici qu'aux rapports de l'id\u00e9e et de son objet.\n\nLe Dieu de Spinoza est un Dieu qui est et qui produit tout, comme l'Un-Tout des Platoniciens ; mais aussi un Dieu qui se pense et qui pense tout, comme le Premier moteur d'Aristote. D'une part nous devons attribuer \u00e0 Dieu une puissance d'exister et d'agir identique \u00e0 son essence formelle ou correspondant \u00e0 sa nature. Mais d'autre part nous devons \u00e9galement lui attribuer une puissance de penser, identique \u00e0 son essence objective ou correspondant \u00e0 son id\u00e9e. Or ce principe d'\u00e9galit\u00e9 des puissances m\u00e9rite un examen minutieux, parce que nous risquons de la confondre avec un autre principe d'\u00e9galit\u00e9, qui concerne seulement les attributs. Pourtant, _la distinction des puissances et des attributs a une importance essentielle dans le spinozisme._ Dieu, c'est-\u00e0-dire l'absolument infini, poss\u00e8de deux puissances \u00e9gales : puissance d'exister et d'agir, puissance de penser et de conna\u00eetre. Si l'on peut se servir d'une formule bergsonienne, l'absolu a deux \u00ab c\u00f4t\u00e9s \u00bb, deux moiti\u00e9s. Si l'absolu poss\u00e8de ainsi deux puissances, c'est en soi et par soi, les enveloppant dans son unit\u00e9 radicale. Il n'en est pas de m\u00eame des attributs : l'absolu poss\u00e8de une infinit\u00e9 d'attributs. Nous n'en connaissons que deux, l'\u00e9tendue et la pens\u00e9e, mais parce que notre connaissance est limit\u00e9e, parce que nous sommes constitu\u00e9s par un mode de l'\u00e9tendue et un mode de la pens\u00e9e. La d\u00e9termination des deux puissances, au contraire, n'est nullement relative aux limites de notre connaissance, pas plus qu'elle ne d\u00e9pend de l'\u00e9tat de notre constitution. La puissance d'exister que nous affirmons de Dieu est une puissance absolument infinie : Dieu existe \u00ab absolument \u00bb, et produit une infinit\u00e9 de choses dans \u00ab l'infinit\u00e9 absolue \u00bb de ses attributs (donc en une infinit\u00e9 de modes)9. De m\u00eame la puissance de penser est absolument infinie. Spinoza ne se contente pas de dire qu'elle est infiniment parfaite ; Dieu se pense absolument, et pense une infinit\u00e9 de choses en une infinit\u00e9 de modes10. D'o\u00f9 l'expression _absoluta cogitatio_ pour d\u00e9signer la puissance de penser ; _intellectus absolute infinitus_ , pour d\u00e9signer l'entendement infini ; et la th\u00e8se selon laquelle, de l'id\u00e9e de Dieu, suivent (objectivement) une infinit\u00e9 de choses en une infinit\u00e9 de modes11. Les deux puissances n'ont donc rien de relatif : ce sont les moiti\u00e9s de l'absolu, les dimensions de l'absolu, les puissances de l'absolu. Schelling est spinoziste quand il d\u00e9veloppe une th\u00e9orie de l'absolu, repr\u00e9sentant Dieu par le symbole A3 qui comprend le r\u00e9el et l'id\u00e9al comme ses puissances12.\n\nOn demandera : _\u00c0 quelles conditions_ affirme-t-on de Dieu une puissance absolument infinie d'exister et d'agir qui correspond \u00e0 sa nature ? \u00c0 condition qu'il ait une infinit\u00e9 d'attributs formellement distincts qui, tous ensemble, constituent cette nature elle-m\u00eame. Il est vrai que nous ne connaissons que deux attributs. Mais nous savons que la puissance d'exister ne se confond pas avec l'attribut \u00e9tendue : une id\u00e9e n'existe pas moins qu'un corps, la pens\u00e9e n'est pas moins que l'\u00e9tendue forme d'existence ou \u00ab genre \u00bb. Et la pens\u00e9e et l'\u00e9tendue prises ensemble ne suffisent pas davantage \u00e0 \u00e9puiser ni \u00e0 remplir une puissance absolue d'exister. Nous atteignons ici la raison positive pour laquelle Dieu a une infinit\u00e9 d'attributs. Dans un texte important du _Court Trait\u00e9_ , Spinoza affirme que \u00ab nous trouvons en nous quelque chose qui nous r\u00e9v\u00e8le clairement l'existence non seulement d'un plus grand nombre, mais encore d'une infinit\u00e9 d'attributs parfaits \u00bb ; les attributs inconnus \u00ab nous disent qu'ils sont sans nous dire ce qu'ils sont13 \u00bb. En d'autres termes : le fait m\u00eame de notre existence nous r\u00e9v\u00e8le que l'existence ne se laisse pas \u00e9puiser par les attributs que nous connaissons. L'infiniment parfait n'ayant pas sa raison en lui-m\u00eame, Dieu doit avoir une infinit\u00e9 d'attributs infiniment parfaits, tous \u00e9gaux entre eux, chacun constituant une forme d'existence ultime ou irr\u00e9ductible. Nous savons qu'aucun n'\u00e9puise cette puissance absolue d'exister qui revient \u00e0 Dieu comme raison suffisante.\n\nL'absolument infini consiste d'abord en une infinit\u00e9 d'attributs formellement ou r\u00e9ellement distincts. Tous les attributs sont \u00e9gaux, aucun n'est sup\u00e9rieur ou inf\u00e9rieur \u00e0 l'autre, chacun exprime une essence infiniment parfaite. Toutes ces essences formelles sont exprim\u00e9es par les attributs comme l'essence absolue de la substance, c'est-\u00e0-dire s'identifient dans la substance ontologiquement une. L'essence formelle est l'essence de Dieu telle qu'elle existe dans chaque attribut. L'essence absolue est la m\u00eame essence, telle qu'elle se rapporte \u00e0 une substance dont l'existence d\u00e9coule n\u00e9cessairement, substance qui poss\u00e8de donc tous les attributs. L'expression se pr\u00e9sente ici comme le rapport de la forme et de l'absolu : chaque forme exprime, explique ou d\u00e9veloppe l'absolu, mais l'absolu contient ou \u00ab complique \u00bb une infinit\u00e9 de formes. L'essence absolue de Dieu est puissance absolument infinie d'exister et d'agir ; mais, pr\u00e9cis\u00e9ment, si nous affirmons cette premi\u00e8re puissance comme identique \u00e0 l'essence de Dieu, c'est _sous la condition_ d'une infinit\u00e9 d'attributs formellement ou r\u00e9ellement distincts. La puissance d'exister et d'agir est donc l'essence formelle-absolue. Et c'est ainsi qu'il faut comprendre le principe d'\u00e9galit\u00e9 des attributs : tous les attributs sont \u00e9gaux par rapport \u00e0 cette puissance d'exister et d'agir qu'ils conditionnent.\n\nMais l'absolu a une seconde puissance, comme une seconde formule ou \u00ab p\u00e9riode \u00bb de l'expression : Dieu se comprend ou s'exprime objectivement. L'essence absolue de Dieu est formelle dans les attributs qui constituent sa nature, objective dans l'id\u00e9e qui repr\u00e9sente n\u00e9cessairement cette nature. C'est pourquoi l'id\u00e9e de Dieu repr\u00e9sente tous les attributs formellement ou r\u00e9ellement distincts, au point qu'une \u00e2me ou une id\u00e9e distincte correspond \u00e0 chacun14. Les m\u00eames attributs qui se distinguent formellement en Dieu se distinguent objectivement dans l'id\u00e9e de Dieu. Mais cette id\u00e9e n'en est pas moins absolument une, comme la substance constitu\u00e9e par tous les attributs15. L'essence objective-absolue est donc la seconde puissance de l'absolu lui-m\u00eame : nous ne posons pas un \u00eatre comme la cause de toutes choses sans que son essence objective ne soit aussi la cause de toutes les id\u00e9es16. L'essence absolue de Dieu est objectivement puissance de penser et de conna\u00eetre, comme elle est formellement puissance d'exister et d'agir. Raison de plus pour demander, dans ce nouveau cas : \u00e0 quelles conditions attribuons-nous \u00e0 Dieu cette puissance absolument infinie de penser comme identique \u00e0 l'essence objective ?\n\n _Pas plus que l'attribut \u00e9tendue ne se confond avec la puissance d'exister, l'attribut pens\u00e9e ne se confond en droit avec la puissance de penser._ Pourtant, un texte de Spinoza semble dire express\u00e9ment le contraire, identifiant l'attribut pens\u00e9e avec _l'absoluta cogitatio_17. Mais Spinoza pr\u00e9cisera en quel sens cette identification doit \u00eatre interpr\u00e9t\u00e9e : c'est seulement parce que la puissance de penser n'a pas d'autre condition que l'attribut pens\u00e9e. En effet, il arrive \u00e0 Spinoza de s'interroger sur la condition de la puissance de penser ou, ce qui revient au m\u00eame, sur la _possibilit\u00e9_ de l'id\u00e9e de Dieu : pour que Dieu puisse penser une infinit\u00e9 de choses en une infinit\u00e9 de modes, pour qu'il ait la possibilit\u00e9 de former une id\u00e9e de son essence et de tout ce qui s'ensuit, il faut et il suffit qu'il ait un attribut qui est la pens\u00e9e18. Ainsi, l'attribut pens\u00e9e suffit \u00e0 conditionner une puissance de penser \u00e9gale \u00e0 la puissance d'exister, laquelle est pourtant conditionn\u00e9e par tous les attributs (y compris la pens\u00e9e). On ne se h\u00e2tera pas de d\u00e9noncer les incoh\u00e9rences du spinozisme. Car on ne trouve d'incoh\u00e9rence qu'\u00e0 force de confondre, chez Spinoza, deux principes d'\u00e9galit\u00e9 tr\u00e8s diff\u00e9rents. D'une part tous les attributs sont \u00e9gaux ; mais cela doit se comprendre par rapport \u00e0 la puissance d'exister et d'agir. D'autre part, cette puissance d'exister n'est qu'une moiti\u00e9 de l'absolu, l'autre moiti\u00e9 est une puissance de penser qui lui est \u00e9gale : c'est par rapport \u00e0 cette seconde puissance que l'attribut pens\u00e9e jouit de privil\u00e8ges. \u00c0 lui seul il conditionne une puissance \u00e9gale \u00e0 celle que tous les attributs conditionnent. Il n'y a l\u00e0, semble-t-il, aucune contradiction, mais plut\u00f4t un _fait ultime_. Ce fait ne concerne nullement notre constitution, ni la limitation de notre connaissance. Ce fait serait plut\u00f4t celui de la constitution divine ou du d\u00e9veloppement de l'absolu. \u00ab Le fait est \u00bb qu'aucun attribut ne suffit \u00e0 remplir la puissance d'exister : quelque chose peut exister et agir, sans \u00eatre \u00e9tendu ni pensant. Au contraire, rien ne peut \u00eatre connu sauf par la pens\u00e9e ; la puissance de penser et de conna\u00eetre est effectivement remplie par l'attribut pens\u00e9e. Il y aurait contradiction si Spinoza posait d'abord l'\u00e9galit\u00e9 de tous les attributs, puis, d'un m\u00eame point de vue, donnait \u00e0 l'attribut pens\u00e9e des pouvoirs et des fonctions contraires \u00e0 cette \u00e9galit\u00e9. Mais Spinoza ne proc\u00e8de pas ainsi : c'est l'\u00e9galit\u00e9 des puissances qui conf\u00e8re \u00e0 l'attribut pens\u00e9e des pouvoirs particuliers, dans un domaine qui n'est plus celui de l'\u00e9galit\u00e9 des attributs. _L'attribut pens\u00e9e est \u00e0 la puissance de penser ce que tous les attributs (y compris la pens\u00e9e) sont \u00e0 la puissance d'exister et d'agir._\n\nDu rapport (donc aussi de la diff\u00e9rence) entre la puissance de penser et l'attribut pens\u00e9e, trois cons\u00e9quences d\u00e9coulent. D'abord la puissance de penser s'affirme, par nature ou participation, de tout ce qui est \u00ab objectif \u00bb. L'essence objective de Dieu est puissance absolument infinie de penser ; et tout ce qui d\u00e9coule de cette essence participe \u00e0 cette puissance. _Mais l'\u00eatre objectif ne serait rien s'il n'avait lui-m\u00eame un \u00eatre formel dans l'attribut pens\u00e9e._ Non seulement l'essence objective de ce qui est produit par Dieu, mais aussi les essences objectives d'attributs, l'essence objective de Dieu lui-m\u00eame, sont soumises \u00e0 la condition d'\u00eatre \u00ab form\u00e9es \u00bb dans l'attribut pens\u00e9e19. _C'est en ce sens que l'id\u00e9e de Dieu n'est qu'un mode de la pens\u00e9e_ et fait partie de la nature natur\u00e9e. Ce qui est mode de l'attribut pens\u00e9e, ce n'est pas, \u00e0 proprement parler, l'essence objective ou l'\u00eatre objectif de l'id\u00e9e comme tel. Ce qui est mode ou produit, c'est toujours l'id\u00e9e prise dans son \u00eatre formel. C'est pourquoi Spinoza prend grand soin de donner au premier mode de la pens\u00e9e le nom d'entendement infini : car l'entendement infini, ce n'est pas l'id\u00e9e de Dieu sous n'importe quel point de vue, c'est pr\u00e9cis\u00e9ment l'\u00eatre formel de l'id\u00e9e de Dieu20. Il est vrai, et nous devons insister sur ce point, que l'\u00eatre objectif ne serait rien s'il n'avait cet \u00eatre formel par lequel il est un mode de l'attribut pens\u00e9e. Ou, si l'on pr\u00e9f\u00e8re, il serait seulement en puissance sans que cette puissance soit effectu\u00e9e.\n\nReste que nous devons distinguer deux points de vue : _d'apr\u00e8s sa n\u00e9cessit\u00e9_ , l'id\u00e9e de Dieu se trouve fond\u00e9e dans la nature naturante. Car il appartient \u00e0 Dieu, pris dans sa nature absolue, de se comprendre n\u00e9cessairement. Lui revient une puissance absolue de penser identique \u00e0 son essence objective ou correspondant \u00e0 son id\u00e9e. L'id\u00e9e de Dieu est donc principe objectif, principe absolu de tout ce qui suit objectivement en Dieu. Mais _d'apr\u00e8s sa possibilit\u00e9_ , l'id\u00e9e de Dieu n'est fond\u00e9e que dans la nature natur\u00e9e \u00e0 laquelle elle appartient. Elle ne peut \u00eatre \u00ab form\u00e9e \u00bb que dans l'attribut pens\u00e9e, elle trouve dans l'attribut pens\u00e9e le principe formel dont elle d\u00e9pend, pr\u00e9cis\u00e9ment parce que cet attribut est la condition sous laquelle on affirme de Dieu la puissance absolument infinie de penser. La distinction des deux points de vue, n\u00e9cessit\u00e9 et possibilit\u00e9, nous para\u00eet importante dans la th\u00e9orie de l'id\u00e9e de Dieu21. La nature de Dieu, \u00e0 laquelle correspond la puissance d'exister et d'agir, est fond\u00e9e _\u00e0 la fois_ en n\u00e9cessit\u00e9 et en possibilit\u00e9 : sa possibilit\u00e9 se trouve \u00e9tablie par les attributs formellement distincts, et sa n\u00e9cessit\u00e9 par ces m\u00eames attributs pris ensemble, ontologiquement \u00ab un \u00bb. Il n'en est pas de m\u00eame de l'id\u00e9e de Dieu : sa n\u00e9cessit\u00e9 objective est \u00e9tablie dans la nature de Dieu, mais sa possibilit\u00e9 formelle dans le seul attribut pens\u00e9e, auquel, d\u00e8s lors, elle appartient comme un mode. On se souvient que la puissance divine est toujours acte ; mais justement, la puissance de penser qui correspond \u00e0 l'id\u00e9e de Dieu ne serait pas actuelle si Dieu ne produisait l'entendement infini comme l'\u00eatre formel de cette id\u00e9e. Aussi bien l'entendement infini est-il appel\u00e9 le fils de Dieu, le Christ22. Or, dans l'image fort peu chr\u00e9tienne que Spinoza propose du Christ, comme Sagesse, Parole ou Voix de Dieu, on distingue un aspect par lequel il s'accorde objectivement avec la nature absolue de Dieu, un aspect par lequel il d\u00e9coule formellement de la nature divine envisag\u00e9e sous le seul attribut pens\u00e9e23. C'est pourquoi la question de savoir si le Dieu spinoziste se pense lui-m\u00eame en lui-m\u00eame est une question d\u00e9licate, qui n'est pas r\u00e9solue pour autant qu'on rappelle que l'entendement infini n'est qu'un mode24. Car, si Dieu a une sagesse ou une science, c'est une science de soi-m\u00eame et de sa propre nature ; s'il se comprend n\u00e9cessairement, c'est en vertu de sa propre nature : la puissance de penser, et de se penser, lui appartient donc en propre absolument. Mais cette puissance resterait en puissance si Dieu ne cr\u00e9ait pas dans l'attribut pens\u00e9e l'\u00eatre formel de l'id\u00e9e dans laquelle il se pense. C'est pourquoi l'entendement de Dieu n'appartient pas \u00e0 sa nature, alors que la puissance de penser appartient \u00e0 cette nature. Dieu produit comme il se comprend objectivement ; mais se comprendre a n\u00e9cessairement une forme qui, elle, est un produit25.\n\nTel est le premier privil\u00e8ge de l'attribut pens\u00e9e : il contient formellement des modes qui, pris objectivement, repr\u00e9sentent les attributs eux-m\u00eames. On ne confondra pas ce premier privil\u00e8ge avec un autre, qui en d\u00e9coule. Un mode qui d\u00e9pend d'un attribut d\u00e9termin\u00e9 est repr\u00e9sent\u00e9 par une id\u00e9e dans l'attribut pens\u00e9e ; mais un mode qui diff\u00e8re du pr\u00e9c\u00e9dent par l'attribut doit \u00eatre repr\u00e9sent\u00e9 par _une autre id\u00e9e._ En effet, tout ce qui participe \u00e0 la puissance d'exister et d'agir, sous tel ou tel attribut, participe aussi \u00e0 la puissance de penser, mais dans le m\u00eame attribut pens\u00e9e. Comme dit Schuller, \u00ab l'attribut de la pens\u00e9e a une extension bien plus grande que les autres attributs26 \u00bb. Si l'on suppose une modification substantielle, elle sera exprim\u00e9e une seule fois dans chacun des autres attributs, mais une infinit\u00e9 de fois dans l'entendement infini, donc dans l'attribut pens\u00e9e27. Et chaque id\u00e9e qui l'exprimera dans la pens\u00e9e repr\u00e9sentera le mode de tel attribut, non d'un autre. Si bien qu'entre ces id\u00e9es il y aura autant de distinction qu'entre les attributs eux-m\u00eames ou les modes d'attributs diff\u00e9rents : elles n'auront \u00ab aucune connexion28 \u00bb. Il y aura donc une distinction objective entre id\u00e9es, \u00e9quivalente \u00e0 la distinction r\u00e9elle-formelle entre attributs ou modes d'attributs diff\u00e9rents. Bien plus, cette distinction entre id\u00e9es sera elle-m\u00eame objective-formelle, pour autant qu'on la rapportera \u00e0 l'\u00eatre formel des id\u00e9es elles-m\u00eames. Il y aura donc dans la pens\u00e9e des modes qui, appartenant \u00e0 un m\u00eame attribut, ne se distingueront pourtant pas modalement, mais formellement ou r\u00e9ellement. L\u00e0 encore, ce privil\u00e8ge resterait inintelligible si l'on ne faisait intervenir le rapport particulier de l'attribut pens\u00e9e avec la puissance de penser. _La distinction objective-formelle est dans l'id\u00e9e de Dieu le corr\u00e9lat n\u00e9cessaire de la distinction r\u00e9elle-formelle_ , telle qu'elle est dans la nature de Dieu ; elle d\u00e9signe l'acte de l'entendement infini quand il saisit des attributs divers ou des modes correspondants d'attributs divers.\n\nEn troisi\u00e8me lieu, tout ce qui existe formellement a une id\u00e9e qui lui correspond objectivement. Mais l'attribut pens\u00e9e est lui-m\u00eame une forme d'existence, et toute id\u00e9e a un \u00eatre formel dans cet attribut. C'est pourquoi toute id\u00e9e, \u00e0 son tour, est l'objet d'une id\u00e9e qui la repr\u00e9sente ; cette autre id\u00e9e, l'objet d'une troisi\u00e8me, \u00e0 l'infini. En d'autres termes : s'il est vrai que toute id\u00e9e qui participe \u00e0 la puissance de penser appartient formellement \u00e0 l'attribut pens\u00e9e, inversement toute id\u00e9e qui appartient \u00e0 l'attribut pens\u00e9e est l'objet d'une id\u00e9e qui participe \u00e0 la puissance de penser. D'o\u00f9 ce dernier privil\u00e8ge apparent de l'attribut pens\u00e9e, qui fonde une capacit\u00e9 de l'id\u00e9e de se r\u00e9fl\u00e9chir \u00e0 l'infini. Il arrive \u00e0 Spinoza de dire que l'id\u00e9e de l'id\u00e9e a, avec l'id\u00e9e, le m\u00eame rapport que l'id\u00e9e avec son objet. On s'en \u00e9tonne, dans la mesure o\u00f9 l'id\u00e9e et son objet sont une m\u00eame chose con\u00e7ue sous deux attributs, tandis que l'id\u00e9e de l'id\u00e9e et l'id\u00e9e sont une m\u00eame chose sous un seul attribut29. Mais l'objet et l'id\u00e9e ne renvoient pas seulement \u00e0 deux attributs, ils renvoient aussi \u00e0 deux puissances, puissance d'exister et d'agir, puissance de penser et de conna\u00eetre. De m\u00eame l'id\u00e9e et l'id\u00e9e de l'id\u00e9e : sans doute renvoient-elles \u00e0 un seul attribut, mais aussi \u00e0 deux puissances, puisque l'attribut pens\u00e9e est d'une part une forme d'existence, d'autre part la condition de la puissance de penser.\n\nOn comprend, d\u00e8s lors, que la th\u00e9orie de l'id\u00e9e de l'id\u00e9e se d\u00e9veloppe dans deux directions diff\u00e9rentes. Car l'id\u00e9e et l'id\u00e9e de l'id\u00e9e se distinguent pour autant que nous consid\u00e9rons l'une dans son \u00eatre formel, par rapport \u00e0 la puissance d'exister, et l'autre dans son \u00eatre objectif, par rapport \u00e0 la puissance de penser : le _Trait\u00e9 de la r\u00e9forme_ pr\u00e9sentera l'id\u00e9e de l'id\u00e9e comme une autre id\u00e9e, distincte de la premi\u00e8re30. Mais d'autre part, toute id\u00e9e se rapporte \u00e0 la puissance de penser : m\u00eame son \u00eatre formel n'est que la condition sous laquelle elle participe \u00e0 cette puissance. Appara\u00eet de ce point de vue l'unit\u00e9 de l'id\u00e9e et de l'id\u00e9e de l'id\u00e9e, en tant qu'elles sont donn\u00e9es en Dieu avec la m\u00eame n\u00e9cessit\u00e9, _de la m\u00eame puissance de penser_31. D\u00e8s lors, il n'y a plus qu'une distinction de raison entre les deux id\u00e9es : l'id\u00e9e de l'id\u00e9e, c'est la forme de l'id\u00e9e, rapport\u00e9e comme telle \u00e0 la puissance de penser.\n\nLes pseudo-contradictions du parall\u00e9lisme s'\u00e9vanouissent si l'on distingue deux arguments tr\u00e8s diff\u00e9rents : celui des puissances et de leur \u00e9galit\u00e9, celui des attributs et de leur \u00e9galit\u00e9. Le parall\u00e9lisme \u00e9pist\u00e9mologique d\u00e9coule de l'\u00e9galit\u00e9 des puissances. Le parall\u00e9lisme ontologique d\u00e9coule de l'\u00e9galit\u00e9 des attributs (par rapport \u00e0 la puissance d'exister). Pourtant, une difficult\u00e9 subsiste encore. La scolie de II 7 passe du parall\u00e9lisme \u00e9pist\u00e9mologique au parall\u00e9lisme ontologique. Dans ce passage, il proc\u00e8de par simple g\u00e9n\u00e9ralisation : \u00ab Et je l'entends de m\u00eame pour les autres attributs. \u00bb Mais comment rendre compte de ce passage ? De ce qu'un objet (dans un attribut quelconque) et une id\u00e9e (dans l'attribut pens\u00e9e) sont une seule et m\u00eame chose (individu), Spinoza conclut que des objets dans tous les attributs sont une seule et m\u00eame chose (modification). Or il semblerait que l'argumentation d\u00fbt nous conduire, non pas \u00e0 l'unit\u00e9 d'une modification, mais au contraire \u00e0 une pluralit\u00e9 irr\u00e9ductible et infinie de couples \u00ab id\u00e9e-objet \u00bb.\n\nLa difficult\u00e9 ne se r\u00e9sout que si l'on consid\u00e8re le statut complexe de l'id\u00e9e de Dieu. Du point de vue de sa n\u00e9cessit\u00e9 objective, l'id\u00e9e de Dieu est principe absolu, et n'a pas moins d'unit\u00e9 que la substance absolument infinie. Du point de vue de sa possibilit\u00e9 formelle, elle est seulement un mode qui trouve son principe dans l'attribut pens\u00e9e. Voil\u00e0 donc que l'id\u00e9e de Dieu est apte \u00e0 communiquer aux modes quelque chose de l'unit\u00e9 substantielle. En effet il y aura une unit\u00e9 proprement modale dans les id\u00e9es qui d\u00e9coulent de l'id\u00e9e de Dieu elle-m\u00eame, c'est-\u00e0-dire dans les modes de penser qui font partie de l'entendement infini. C'est donc une m\u00eame modification qui s'exprimera en une infinit\u00e9 de mani\u00e8res dans l'entendement infini de Dieu. _D\u00e8s lors_ , les objets que ces id\u00e9es repr\u00e9sentent seront des objets ne diff\u00e9rant que par l'attribut : de m\u00eame que leurs id\u00e9es, ils exprimeront une seule et m\u00eame modification. Un mode dans tel attribut forme avec l'id\u00e9e que le repr\u00e9sente un \u00ab individu \u00bb irr\u00e9ductible ; et aussi une id\u00e9e, dans l'attribut pens\u00e9e, avec l'objet qu'elle repr\u00e9sente. Mais cette infinit\u00e9 d'individus correspondent, en ce qu'ils expriment une seule modification. Ainsi la m\u00eame modification n'existe pas seulement en une infinit\u00e9 de modes, mais en une infinit\u00e9 d'individus, dont chacun est constitu\u00e9 par un mode et par l'id\u00e9e de ce mode.\n\nMais pourquoi fallait-il passer par le parall\u00e9lisme \u00e9pist\u00e9mologique ? Pourquoi ne pas conclure directement de l'unit\u00e9 de la substance \u00e0 l'unit\u00e9 d'une modification substantielle ? C'est que Dieu produit dans les attributs formellement ou r\u00e9ellement distincts ; il est certain que les attributs s'expriment, mais chacun s'exprime pour son compte, en tant que forme ultime et irr\u00e9ductible. Sans doute, tout nous fait penser que la production b\u00e9n\u00e9ficiera d'une unit\u00e9 d\u00e9rivant de la substance elle-m\u00eame. Car, si chaque attribut s'exprime pour son compte, Dieu n'en produit pas moins dans tous les attributs \u00e0 la fois. Tout laisse donc pr\u00e9voir qu'il y aura dans les diff\u00e9rents attributs des modes exprimant la m\u00eame modification. Pourtant, nous n'en avons pas une certitude absolue. \u00c0 la limite, on pourrait concevoir autant de mondes qu'il y a d'attributs. La Nature serait une dans sa substance, mais multiple dans ses modifications, ce qui est produit dans un attribut restant absolument diff\u00e9rent de ce qui est produit dans un autre. C'est parce que les modes ont une consistance propre, une sp\u00e9cificit\u00e9, que nous sommes forc\u00e9s de chercher une raison particuli\u00e8re de l'unit\u00e9 dont ils sont capables. Kant reprochait au spinozisme de ne pas avoir cherch\u00e9 un principe sp\u00e9cifique pour l'unit\u00e9 du divers dans le mode32. (Il pensait \u00e0 l'unit\u00e9 des modes dans un m\u00eame attribut, mais le m\u00eame probl\u00e8me se pose pour l'unit\u00e9 d'une modification par rapport aux modes d'attributs diff\u00e9rents.) Or l'objection ne semble pas l\u00e9gitime. Spinoza fut parfaitement conscient d'un probl\u00e8me particulier de l'unit\u00e9 des modes, et de la n\u00e9cessit\u00e9 de faire appel \u00e0 des principes originaux pour rendre compte du passage de l'unit\u00e9 substantielle \u00e0 l'unit\u00e9 modale.\n\nC'est bien l'id\u00e9e de Dieu qui nous donne un tel principe, en vertu de son double aspect. On passe de l'unit\u00e9 de la substance, constitu\u00e9e par tous les attributs qui en expriment l'essence, \u00e0 l'unit\u00e9 d'une modification comprise dans l'entendement infini, mais constitu\u00e9e par des modes qui l'expriment dans chaque attribut. \u00c0 la question : pourquoi n'y a-t-il pas autant de mondes que d'attributs de Dieu ?, Spinoza r\u00e9pond seulement en renvoyant le lecteur au scolie de II, 733. Or, pr\u00e9cis\u00e9ment, ce texte implique un argument qui proc\u00e8de par l'entendement infini (d'o\u00f9 l'importance de l'allusion \u00e0 \u00ab certains H\u00e9breux \u00bb) : l'entendement de Dieu n'a pas moins d'unit\u00e9 que la substance divine, d\u00e8s lors les choses comprises par lui n'ont pas moins d'unit\u00e9 que lui-m\u00eame.\n\n* * *\n\n1.  _\u00c9_ , II, 7, sc. : \u00ab Et je l'entends de m\u00eame pour les autres attributs... \u00bb.\n\n2.  Ainsi l'\u00e2me est une id\u00e9e qui repr\u00e9sente exclusivement un certain mode de l'\u00e9tendue : cf. _\u00c9_ , II, 13, prop.\n\n3.  Sur cet emploi du mot \u00ab individu \u00bb signifiant l'unit\u00e9 d'une id\u00e9e et de son objet, cf. _\u00c9_ , II, 21, sc.\n\n4.  _Lettre 65, de Tschirnhaus_ (III, p. 207).\n\n5.  _TRE_ , 85.\n\n6.  _\u00c9_ , II, 5 et 6.\n\n7.  _CT_ , II, ch. 20, 4, note 3.\n\n8.  _\u00c9_ , I, 30, prop.\n\n9.  Cf. _\u00c9_ , I, 16, dem. : _infinita absolute attributa._\n\n10.  _\u00c9_ , II, 3, prop. et dem\n\n11.  Cf. _\u00c9_ , I, 31, dem. : _absoluta cogitatio. Lettre 64, \u00e0 Schuller_ (III, p. 206) : _intellectus absolute infinitus._\n\n12.  SCHELLING, \u00ab Conf\u00e9rences de Stuttgart \u00bb, 1810 (tr. fr. in _Essais_ , Aubier \u00e9d., pp. 309-310) : \u00ab Les deux unit\u00e9s ou puissances se trouvent \u00e0 nouveau unies dans l'Unit\u00e9 absolue, la position commune de la premi\u00e8re et de la deuxi\u00e8me puissances sera donc A3... Les puissances sont d\u00e9sormais pos\u00e9es \u00e9galement comme des p\u00e9riodes de la _r\u00e9v\u00e9lation_ de Dieu. \u00bb\n\n13.  _CT_ , I, ch. 1, 7, note 3.\n\n14.  _CT_ , Appendice II, 9 : \u00ab Tous les attributs infinis qui ont une \u00e2me aussi bien que l'\u00e9tendue... \u00bb\n\n15.  _\u00c9_ , II, 4, prop. et dem.\n\n16.  _TRE_ , 99 : Il faut que \u00ab nous recherchions s'il y a un \u00catre, et aussi quel il est, qui soit la cause de toutes choses, de telle sorte que son essence objective soit aussi la cause de toutes nos id\u00e9es \u00bb.\n\n17.  _\u00c9_ , I, 31, d\u00e9m. : L'entendement, \u00e9tant un mode de penser, \u00ab doit \u00eatre con\u00e7u par la _pens\u00e9e absolue_ , autrement dit il doit \u00eatre con\u00e7u par quelque _attribut_ de Dieu qui exprime l'essence \u00e9ternelle et infinie de Dieu, de telle sorte que sans cet attribut il ne puisse ni \u00eatre ni \u00eatre con\u00e7u \u00bb.\n\n18.  _\u00c9_ , II, 1, sc. \u00ab Un \u00eatre qui peut penser une infinit\u00e9 de choses en une infinit\u00e9 de modes est n\u00e9cessairement infini par la vertu de penser. \u00bb (C'est-\u00e0-dire : un \u00eatre qui a une puissance _absolue_ de penser a n\u00e9cessairement un attribut _infini_ qui est la pens\u00e9e.) _\u00c9_ , II, 5, dem. : \u00ab Nous concluions que Dieu peut former l'id\u00e9e de son essence et de tout ce qui en suit n\u00e9cessairement, et nous le concluions de cela seul que Dieu est chose pensante. \u00bb\n\n19.  Cf. _\u00c9_ , II, 5, dem. : _Deum ideam suae essentiae... formare posse._\n\n20.  C'est l'entendement infini, non pas l'id\u00e9e de Dieu, qui est dit un mode : _\u00c9_ , I, 31 prop. et dem. ; _CT_ , I, ch. 9, 3.\n\n21.  Les commentateurs ont souvent distingu\u00e9 plusieurs aspects de l'id\u00e9e de Dieu _ou_ de l'entendement infini. Georg Busolt a \u00e9t\u00e9 le plus loin, posant que l'entendement infini appartient \u00e0 la nature natur\u00e9e comme principe des modes intellectuels finis, mais \u00e0 la nature naturante en tant qu'on le consid\u00e8re en lui-m\u00eame ( _Die Grundz\u00fcge der Erkenntnisstheorie und Metaphysik Spinoza's_ , Berlin, 1895, II, pp. 127 sq.). Cette distinction toutefois nous para\u00eet mal fond\u00e9e, car, en tant que principe de ce qui suit objectivement en Dieu, l'id\u00e9e de Dieu devrait au contraire appartenir \u00e0 la nature naturante. C'est pourquoi nous croyons plus l\u00e9gitime une distinction entre l'id\u00e9e de Dieu, prise objectivement, _et_ l'entendement infini, pris formellement.\n\n22.  Cf. _CT_ , I, ch. 9, 3. _Lettre 73, \u00e0 Oldenburg_ (III, p. 226).\n\n23.  Cf. _CT_ , II, ch. 22, 4, note 1 : \u00ab L'entendement infini, que nous appelions le fils de Dieu, doit \u00eatre de toute \u00e9ternit\u00e9 dans la nature, car, puisque Dieu a \u00e9t\u00e9 de toute \u00e9ternit\u00e9, son id\u00e9e aussi doit \u00eatre aussi dans la chose pensante ou en lui-m\u00eame \u00e9ternellement, _laquelle id\u00e9e s'accorde_ objectivement avec lui. \u00bb\n\n24.  Victor Brochard exprimait d\u00e9j\u00e0 des doutes \u00e0 cet \u00e9gard : cf. _Le Dieu de Spinoza_ (\u00c9tudes de philosophie ancienne et de philosophie moderne, Vrin), pp. 332-370.\n\n25.  Aux deux th\u00e8ses pr\u00e9c\u00e9demment expos\u00e9es \u2013 Dieu produit comme il se comprend, Dieu comprend tout ce qu'il produit \u2013, il faut donc ajouter cette troisi\u00e8me : Dieu produit la forme sous laquelle il se comprend et comprend tout. Les trois s'accordent sur un point fondamental : l'entendement infini n'est pas un lieu qui contiendrait des possibles.\n\n26.  _Lettre 70, de Schuller_ (III, p. 221).\n\n27.  _Lettre 66, \u00e0 Tschirnhaus_ (III, p. 207).\n\n28.  _Lettre 66, \u00e0 Tschirnhaus_ (III, p. 208).\n\n29.  Cf. _\u00c9_ , II, 21, sc. Albert L\u00e9on r\u00e9sume la difficult\u00e9 : \u00ab Comment sortir de ce dilemme ? Ou bien l'id\u00e9e et l'id\u00e9e de l'id\u00e9e sont dans le m\u00eame rapport qu'un objet \u00e9tranger \u00e0 la pens\u00e9e et l'id\u00e9e qui le repr\u00e9sente, et elles sont alors deux expressions d'un m\u00eame contenu sous des attributs diff\u00e9rents ; ou leur contenu commun est exprim\u00e9 sous un seul et m\u00eame attribut, et alors l'id\u00e9e de l'id\u00e9e est absolument identique \u00e0 l'id\u00e9e consid\u00e9r\u00e9e, la conscience absolument identique \u00e0 la pens\u00e9e, et celle-ci ne saurait se d\u00e9finir en dehors de celle-l\u00e0. \u00bb ( _Les \u00c9l\u00e9ments cart\u00e9siens de la doctrine spinoziste sur les rapports de la pens\u00e9e et de son objet_ , p. 154.)\n\n30.  _TRE_ , 34-35 : _altera idea_ ou _altera essentia objectiva_ sont dits trois fois. La distinction de l'id\u00e9e et de l'id\u00e9e de l'id\u00e9e est m\u00eame assimil\u00e9e \u00e0 celle de l'id\u00e9e de triangle et de l'id\u00e9e de cercle.\n\n31.  _\u00c9_ , II, 21, sc. (sur l'existence d'une simple distinction de raison entre l'id\u00e9e de l'id\u00e9e et l'id\u00e9e. Cf. _\u00c9_ , IV, _8_ , dem. et V. 3, dem.).\n\n32.  Kant, _Critique du Jugement_ , \u00a7 73.\n\n33.  C'est Schuller qui posait la question, _Lettre 63_ (III, p. 203).\n\n## CHAPITRE VIII\n\n## EXPRESSION ET ID\u00c9E\n\nLa philosophie de Spinoza est une \u00ab logique \u00bb. La nature et les r\u00e8gles de cette logique font l'objet de la m\u00e9thode. La question de savoir si la m\u00e9thode et la logique du _Trait\u00e9 de la r\u00e9forme_ sont tout enti\u00e8res conserv\u00e9es dans l' _\u00c9thique_ est une question importante ; mais elle ne peut \u00eatre r\u00e9solue que par l'examen du _Trait\u00e9_ lui-m\u00eame. Or celui-ci nous pr\u00e9sente deux parties distinctes. La premi\u00e8re partie concerne le but de la m\u00e9thode ou de la philosophie, le but final de la pens\u00e9e : elle porte en principe sur la forme de l'id\u00e9e vraie1. La seconde partie concerne surtout les moyens d'atteindre ce but ; elle porte sur le contenu de l'id\u00e9e vraie2. La premi\u00e8re partie anticipe n\u00e9cessairement sur la seconde, comme le but pr\u00e9d\u00e9termine les moyens gr\u00e2ce auxquels on l'atteint. C'est chacun de ces points qu'il faut analyser.\n\nLe but de la philosophie ou la premi\u00e8re partie de la m\u00e9thode ne consistent pas \u00e0 nous faire conna\u00eetre quelque chose ; mais \u00e0 nous faire conna\u00eetre notre puissance de comprendre. Non pas nous faire conna\u00eetre la Nature, mais nous faire concevoir et acqu\u00e9rir une nature humaine sup\u00e9rieure3. C'est dire que la m\u00e9thode, sous son premier aspect, est essentiellement r\u00e9flexive : elle consiste dans la seule connaissance de l'entendement pur, de sa nature, de ses lois et de ses forces4. \u00ab La m\u00e9thode n'est rien d'autre que la connaissance r\u00e9flexive ou l'id\u00e9e de l'id\u00e9e \u00bb5. \u00c0 cet \u00e9gard, on ne verra nulle diff\u00e9rence entre l' _\u00c9thique_ et le _Trait\u00e9 de la r\u00e9forme_. L'objet de la m\u00e9thode est aussi bien le but final de la philosophie. Le livre V de l' _\u00c9thique_ d\u00e9crit ce but, non pas comme la connaissance de quelque chose, mais comme la connaissance de notre puissance de comprendre ou de notre entendement ; on en d\u00e9duit les conditions de la b\u00e9atitude, comme pleine effectuation de cette puissance. D'o\u00f9 le titre du livre V : _De potentia intellectus seu de libertate humana._\n\n\u00ab Puisque la m\u00e9thode est la connaissance r\u00e9flexive elle-m\u00eame, ce principe qui doit diriger nos pens\u00e9es ne peut \u00eatre rien d'autre que la connaissance de ce qui constitue la _forme_ de la v\u00e9rit\u00e96. \u00bb En quoi consiste ce rapport de la forme et de la r\u00e9flexion ? La connaissance r\u00e9flexive est l'id\u00e9e de l'id\u00e9e. Nous avons vu que l'id\u00e9e de l'id\u00e9e se distinguait de l'id\u00e9e, pour autant que nous rapportions celle-ci dans son \u00eatre formel \u00e0 la puissance d'exister, celle-l\u00e0 dans son \u00eatre objectif \u00e0 la puissance de penser. Mais, d'un autre point de vue, l'id\u00e9e prise dans son \u00eatre formel se rapporte d\u00e9j\u00e0 \u00e0 la puissance de penser. En effet, l'\u00eatre formel de l'id\u00e9e, c'est son existence dans l'attribut pens\u00e9e. Or cet attribut n'est pas seulement un genre d'existence, c'est aussi la condition sous laquelle on rapporte \u00e0 quelque chose une puissance de penser, de comprendre ou de conna\u00eetre. Dieu sous l'attribut pens\u00e9e a une puissance absolument infinie de penser. Une id\u00e9e dans l'attribut pens\u00e9e a une puissance d\u00e9termin\u00e9e de conna\u00eetre ou de comprendre. La puissance de comprendre qui appartient \u00e0 une id\u00e9e, c'est la puissance de penser de Dieu lui-m\u00eame, en tant qu'elle \u00ab s'explique \u00bb par cette id\u00e9e. On voit donc que l'id\u00e9e de l'id\u00e9e, c'est l'id\u00e9e consid\u00e9r\u00e9e dans sa forme, en tant qu'elle poss\u00e8de une puissance de comprendre ou de conna\u00eetre (comme partie de la puissance absolue de penser). En ce sens, forme et r\u00e9flexion s'impliquent.\n\nLa forme est donc toujours forme d'une id\u00e9e que nous _avons_. Encore faut-il pr\u00e9ciser : il n'y a de forme que de la v\u00e9rit\u00e9. Si la fausset\u00e9 avait une forme, il nous serait impossible de prendre le faux pour le vrai, donc de nous tromper7. La forme est donc toujours forme d'une id\u00e9e vraie que nous avons. Il suffit d'avoir une id\u00e9e vraie pour qu'elle se r\u00e9fl\u00e9chisse, et r\u00e9fl\u00e9chisse sa puissance de conna\u00eetre ; il suffit de savoir pour savoir qu'on sait8. C'est pourquoi la m\u00e9thode suppose qu'on ait une id\u00e9e vraie quelconque. Elle suppose une \u00ab force inn\u00e9e \u00bb de l'entendement qui ne peut pas manquer, parmi toutes ses id\u00e9es, _d'en avoir une au moins qui soit vraie_9. La m\u00e9thode n'a nullement pour but de nous faire acqu\u00e9rir une telle id\u00e9e, mais de nous faire \u00ab r\u00e9fl\u00e9chir \u00bb celle que nous avons, ou de nous faire comprendre notre puissance de conna\u00eetre.\n\nMais en quoi consiste cette r\u00e9flexion ? La forme ne s'oppose pas au contenu en g\u00e9n\u00e9ral. L'\u00eatre formel s'oppose \u00e0 l'\u00eatre objectif ou repr\u00e9sentatif : l'id\u00e9e de l'id\u00e9e, c'est l'id\u00e9e dans sa forme, ind\u00e9pendamment de l'objet qu'elle repr\u00e9sente. En effet, comme tous les attributs, la pens\u00e9e est autonome ; les modes de la pens\u00e9e, les id\u00e9es, sont donc automates. C'est dire qu'ils d\u00e9pendent uniquement de l'attribut pens\u00e9e dans leur \u00eatre formel : ils sont consid\u00e9r\u00e9s \u00ab sans relation \u00e0 un objet10 \u00bb. La forme de l'id\u00e9e s'oppose donc \u00e0 son contenu objectif ou repr\u00e9sentatif. Mais elle ne s'oppose nullement \u00e0 un autre contenu que l'id\u00e9e poss\u00e9derait elle-m\u00eame ind\u00e9pendamment de l'objet qu'elle repr\u00e9sente. En fait, nous devons nous garder d'une double erreur concernant le contenu, mais aussi la forme de l'id\u00e9e. Soit la d\u00e9finition de la v\u00e9rit\u00e9 comme correspondance d'une id\u00e9e et de son objet. Il est certain qu'elle ne nous apprend rien sur la forme de l'id\u00e9e vraie : d'o\u00f9 pourrait-on savoir qu'une id\u00e9e convient avec l'objet ? Mais elle ne nous apprend rien, non plus, sur le contenu de l'id\u00e9e vraie ; car une id\u00e9e vraie, d'apr\u00e8s cette d\u00e9finition, n'aurait pas plus de r\u00e9alit\u00e9 ou de perfection interne qu'une fausse11. La conception de la v\u00e9rit\u00e9 comme correspondance ne nous donne aucune d\u00e9finition du vrai, ni mat\u00e9rielle ni formelle ; elle nous propose seulement une d\u00e9finition nominale, une d\u00e9nomination extrins\u00e8que. Or on pensera peut-\u00eatre que \u00ab le clair et le distinct \u00bb nous donnent une meilleure d\u00e9termination, c'est-\u00e0-dire une caract\u00e9ristique interne du vrai tel qu'il est _dans_ l'id\u00e9e. En fait, il n'en est rien. Pris en eux-m\u00eames, le clair et le distinct portent bien sur le contenu de l'id\u00e9e, mais seulement sur son contenu \u00ab objectif \u00bb ou \u00ab repr\u00e9sentatif \u00bb. Ils portent \u00e9galement sur la forme, mais seulement sur la forme d'une \u00ab conscience psychologique \u00bb de l'id\u00e9e. _Ainsi ils nous permettent de reconna\u00eetre une id\u00e9e vraie, pr\u00e9cis\u00e9ment celle que la m\u00e9thode suppose_ , mais ne nous donne aucune connaissance du contenu mat\u00e9riel de cette id\u00e9e ni de sa forme logique. Bien plus, le clair et le distinct sont incapables de d\u00e9passer la dualit\u00e9 de la forme et du contenu. La clart\u00e9 cart\u00e9sienne n'est pas une, mais double ; Descartes nous invite lui-m\u00eame \u00e0 distinguer une \u00e9vidence mat\u00e9rielle, qui serait comme la clart\u00e9 et la distinction du contenu objectif de l'id\u00e9e, et une \u00e9vidence formelle, clart\u00e9 qui porte sur la \u00ab raison \u00bb de notre croyance \u00e0 l'id\u00e9e12. C'est ce dualisme qui se prolongera dans la division cart\u00e9sienne de l'entendement et de la volont\u00e9. Bref, il manque au cart\u00e9sianisme, non seulement de concevoir le vrai contenu, comme contenu mat\u00e9riel, et la vraie forme, comme forme logique de l'id\u00e9e, mais de s'\u00e9lever jusqu'\u00e0 la position de \u00ab l'automate spirituel \u00bb impliquant l'identit\u00e9 des deux.\n\nIl y a un formalisme logique, qui ne se confond pas avec la forme de la conscience psychologique. Il y a un contenu mat\u00e9riel de l'id\u00e9e, qui ne se confond pas avec un contenu repr\u00e9sentatif. Il suffit d'acc\u00e9der \u00e0 cette vraie forme et \u00e0 ce vrai contenu pour concevoir en m\u00eame temps l'unit\u00e9 des deux : l'\u00e2me ou l'entendement comme \u00ab automate spirituel \u00bb. _La forme, en tant que forme de v\u00e9rit\u00e9, ne fait qu'un avec le contenu de l'id\u00e9e vraie quelconque_ : c'est en pensant le contenu d'une id\u00e9e vraie que nous avons que nous r\u00e9fl\u00e9chissons l'id\u00e9e dans sa forme, et que nous comprenons notre puissance de conna\u00eetre. D\u00e8s lors, on voit pourquoi la m\u00e9thode comporte une seconde partie, et pourquoi la premi\u00e8re anticipe n\u00e9cessairement sur cette seconde. La premi\u00e8re partie de la m\u00e9thode ou le but final concerne la forme de l'id\u00e9e vraie, l'id\u00e9e de l'id\u00e9e ou l'id\u00e9e r\u00e9flexive. La seconde partie concerne le contenu de l'id\u00e9e vraie, c'est-\u00e0-dire l'id\u00e9e ad\u00e9quate. Cette seconde partie est comme le moyen subordonn\u00e9 au but, mais aussi comme le moyen dont d\u00e9pend la r\u00e9alisation du but. Elle demande : En quoi consiste le contenu de l'id\u00e9e, c'est-\u00e0-dire l'id\u00e9e comme ad\u00e9quate ?\n\nL'id\u00e9e vraie est, du point de vue de la forme, l'id\u00e9e de l'id\u00e9e ; et du point de vue de la mati\u00e8re, l'id\u00e9e ad\u00e9quate. De m\u00eame que l'id\u00e9e de l'id\u00e9e se d\u00e9finit comme _id\u00e9e r\u00e9flexive_ , l'id\u00e9e ad\u00e9quate se d\u00e9finit comme _id\u00e9e expressive_. Le terme \u00ab ad\u00e9quat \u00bb, chez Spinoza, ne signifie jamais la correspondance de l'id\u00e9e avec l'objet qu'elle repr\u00e9sente ou d\u00e9signe, mais la convenance interne de l'id\u00e9e avec quelque chose qu'elle exprime. Qu'est-ce qu'elle exprime ? Consid\u00e9rons d'abord l'id\u00e9e comme la connaissance de quelque chose. Elle n'est une vraie connaissance que dans la mesure o\u00f9 elle porte sur l'essence de la chose : elle doit \u00ab expliquer \u00bb cette essence. Mais elle n'explique l'essence que dans la mesure o\u00f9 elle comprend la chose par sa cause prochaine : elle doit \u00ab exprimer \u00bb cette cause m\u00eame, c'est-\u00e0-dire \u00ab envelopper \u00bb la connaissance de la cause13. Tout est aristot\u00e9licien dans cette conception de la connaissance. Spinoza ne veut pas dire simplement que les effets connus d\u00e9pendent des causes. Il veut dire \u00e0 la mani\u00e8re d'Aristote que la connaissance d'un effet d\u00e9pend elle-m\u00eame de la connaissance de la cause. Mais ce principe aristot\u00e9licien se trouve renouvel\u00e9 par l'inspiration parall\u00e9liste : que la connaissance aille ainsi de la cause \u00e0 l'effet doit se comprendre comme la loi d'une pens\u00e9e autonome, l'expression d'une puissance absolue dont toutes les id\u00e9es d\u00e9pendent. Il revient donc au m\u00eame de dire que la connaissance de l'effet, prise objectivement, \u00ab enveloppe \u00bb la connaissance de la cause, ou que l'id\u00e9e, prise formellement, \u00ab exprime \u00bb sa propre cause14. _L'id\u00e9e ad\u00e9quate, c'est pr\u00e9cis\u00e9ment l'id\u00e9e comme exprimant sa cause_. C'est en ce sens que Spinoza rappelle que sa m\u00e9thode se fonde sur la possibilit\u00e9 d'encha\u00eener les id\u00e9es les unes aux autres, l'une \u00e9tant \u00ab cause compl\u00e8te \u00bb d'une autre15. Tant que nous en restons \u00e0 une id\u00e9e claire et distincte, nous n'avons que la connaissance d'un effet ; ou si l'on pr\u00e9f\u00e8re, nous ne connaissons qu'une propri\u00e9t\u00e9 de la chose16. Seule l'id\u00e9e ad\u00e9quate, en tant qu'expressive, nous fait conna\u00eetre par la cause, ou nous fait conna\u00eetre l'essence de la chose.\n\nOn voit d\u00e8s lors en quoi consiste la seconde partie de la m\u00e9thode. Nous sommes toujours suppos\u00e9s avoir une id\u00e9e vraie, nous la reconnaissons \u00e0 sa clart\u00e9. Mais, m\u00eame si la \u00ab force inn\u00e9e \u00bb de l'entendement nous assure \u00e0 la fois cette reconnaissance et cette possession, nous restons encore dans le simple \u00e9l\u00e9ment du hasard ( _fortuna_ ). Nous n'avons pas encore une id\u00e9e ad\u00e9quate. Tout le probl\u00e8me de la m\u00e9thode devient celui-ci : Comment arracher au hasard nos pens\u00e9es vraies ? C'est-\u00e0-dire : comment faire d'une pens\u00e9e vraie une id\u00e9e ad\u00e9quate, qui s'encha\u00eene avec d'autres id\u00e9es ad\u00e9quates ? Nous partons donc d'une id\u00e9e vraie. _Nous avons m\u00eame avantage, en fonction de notre dessein, \u00e0 choisir une id\u00e9e vraie, claire et distincte, qui d\u00e9pende en toute \u00e9vidence de notre puissance de penser, n'ayant aucun objet dans la nature, par exemple l'id\u00e9e de sph\u00e8re_ ( _ou de cercle_ )17. Cette id\u00e9e, nous devons la rendre ad\u00e9quate, donc la rattacher \u00e0 sa propre cause. Il ne s'agit pas, comme dans la m\u00e9thode cart\u00e9sienne, de conna\u00eetre la cause \u00e0 partir de l'effet ; une telle d\u00e9marche ne nous ferait rien conna\u00eetre de la cause, sauf pr\u00e9cis\u00e9ment ce que nous consid\u00e9rons dans l'effet. Il s'agit, au contraire, de comprendre la connaissance que nous avons de l'effet par une connaissance elle-m\u00eame plus parfaite de la cause.\n\nOn objectera que, de toutes mani\u00e8res, nous partons d'un effet connu, c'est-\u00e0-dire d'une id\u00e9e suppos\u00e9e donn\u00e9e18. Mais nous n'allons pas des propri\u00e9t\u00e9s de l'effet \u00e0 certaines propri\u00e9t\u00e9s de la cause, lesquelles seraient seulement comme des conditions n\u00e9cessaires en fonction de cet effet. Partant de l'effet, nous d\u00e9terminons la cause, _serait-ce par_ \u00ab _fiction_ \u00bb, comme la _raison suffisante_ de toutes les propri\u00e9t\u00e9s que nous concevons qu'il poss\u00e8de19. C'est en ce sens que nous connaissons _par_ la cause, ou que la cause est mieux connue que l'effet. La m\u00e9thode cart\u00e9sienne est une m\u00e9thode r\u00e9gressive et analytique. La m\u00e9thode spinoziste est une m\u00e9thode r\u00e9flexive et synth\u00e9tique : r\u00e9flexive parce qu'elle comprend la connaissance de l'effet par la connaissance de la cause ; synth\u00e9tique parce qu'elle engendre toutes les propri\u00e9t\u00e9s de l'effet \u00e0 partir de la cause connue comme raison suffisante. Nous avons une id\u00e9e ad\u00e9quate dans la mesure o\u00f9, de la chose dont nous concevons clairement certaines propri\u00e9t\u00e9s, nous donnons une d\u00e9finition _g\u00e9n\u00e9tique_ , d'o\u00f9 d\u00e9coulent au moins toutes les propri\u00e9t\u00e9s connues (et m\u00eame d'autres que nous ne connaissions pas). On a souvent remarqu\u00e9 que les math\u00e9matiques chez Spinoza avaient exclusivement le r\u00f4le d'un tel processus g\u00e9n\u00e9tique20. La cause comme raison suffisante est ce qui, \u00e9tant donn\u00e9, fait que toutes les propri\u00e9t\u00e9s de la chose le sont aussi, et, \u00e9tant supprim\u00e9, fait que les propri\u00e9t\u00e9s le sont toutes \u00e9galement21. Nous d\u00e9finissons le plan par le mouvement de la ligne, le cercle par le mouvement d'une ligne dont une extr\u00e9mit\u00e9 est fixe, la sph\u00e8re par le mouvement d'un demi-cercle. Dans la mesure o\u00f9 la d\u00e9finition de la chose exprime la cause efficiente ou la gen\u00e8se du d\u00e9fini, l'id\u00e9e m\u00eame de la chose exprime sa propre cause : nous avons fait de l'id\u00e9e quelque chose d'ad\u00e9quat. C'est en ce sens que Spinoza dit que la seconde partie de la m\u00e9thode est d'abord une th\u00e9orie de la d\u00e9finition : \u00ab Le point principal de toute cette seconde partie de la m\u00e9thode se rapporte exclusivement \u00e0 la connaissance des conditions d'une bonne d\u00e9finition22... \u00bb\n\nD'apr\u00e8s ce qui pr\u00e9c\u00e8de, la m\u00e9thode spinoziste se distingue d\u00e9j\u00e0 de toute d\u00e9marche analytique ; pourtant elle n'est pas sans une apparence r\u00e9gressive. La r\u00e9flexion emprunte la m\u00eame apparence que l'analyse, puisque nous \u00ab supposons \u00bb d'abord une id\u00e9e, puisque nous partons de la connaissance suppos\u00e9e d'un effet. Nous supposons connues clairement certaines propri\u00e9t\u00e9s du cercle ; nous nous \u00e9levons jusqu'\u00e0 la raison suffisante d'o\u00f9 toutes les propri\u00e9t\u00e9s d\u00e9coulent. Mais en d\u00e9terminant la raison du cercle comme le mouvement d'une ligne autour d'une de ses extr\u00e9mit\u00e9s nous n'atteignons pas encore une pens\u00e9e qui serait form\u00e9e par elle-m\u00eame ou \u00ab absolument \u00bb. En effet, un tel mouvement n'est pas contenu dans le concept de ligne, il est lui-m\u00eame _fictif_ et r\u00e9clame une cause qui le d\u00e9termine. C'est pourquoi, si la seconde partie de la m\u00e9thode consiste d'abord dans la th\u00e9orie de la d\u00e9finition, elle ne se r\u00e9duit pas \u00e0 cette th\u00e9orie. Un dernier probl\u00e8me se pr\u00e9sente : _Comment conjurer la supposition dont on est parti ?_ Par-l\u00e0 m\u00eame, comment sortir d'un encha\u00eenement fictif ? Comment construire le r\u00e9el lui-m\u00eame, au lieu d'en rester au niveau des choses math\u00e9matiques ou des \u00eatres de raison ? Nous arrivons \u00e0 la position d'un principe \u00e0 partir d'une hypoth\u00e8se ; _mais il faut que le principe soit d'une nature telle qu'il s'affranchisse enti\u00e8rement de l'hypoth\u00e8se_ , qu'il se fonde lui-m\u00eame et fonde le mouvement par lequel nous y arrivons ; il faut qu'il rende caduc, _aussit\u00f4t que possible_ , le pr\u00e9suppos\u00e9 dont nous sommes partis pour le d\u00e9couvrir. La m\u00e9thode spinoziste, dans son opposition \u00e0 Descartes, pose un probl\u00e8me tr\u00e8s analogue \u00e0 celui de Fichte dans sa r\u00e9action contre Kant23.\n\nSpinoza reconna\u00eet qu'il ne peut pas imm\u00e9diatement exposer \u00ab les v\u00e9rit\u00e9s de la nature \u00bb dans l'ordre d\u00fb24. C'est-\u00e0-dire : il ne peut pas imm\u00e9diatement encha\u00eener les id\u00e9es comme elles doivent l'\u00eatre pour que le R\u00e9el soit reproduit par la seule puissance de la pens\u00e9e. On n'y verra pas une insuffisance de la m\u00e9thode, mais une exigence de la m\u00e9thode spinoziste, sa mani\u00e8re \u00e0 elle de prendre du temps. Car, en revanche, Spinoza reconna\u00eet aussi qu'il peut _tr\u00e8s vite_ arriver au principe absolu dont toutes les id\u00e9es d\u00e9coulent dans l'ordre d\u00fb : la m\u00e9thode ne sera parfaite que lorsque nous poss\u00e9derons l'id\u00e9e de l'\u00catre parfait ; \u00ab d\u00e8s le d\u00e9but donc il nous faudra veiller \u00e0 ce que nous arrivions _le plus rapidement possible_ \u00e0 la connaissance d'un tel \u00eatre \u00bb. Il faut que \u00ab nous commencions, _aussit\u00f4t que faire se peut_ , par les premiers \u00e9l\u00e9ments, c'est-\u00e0-dire par la source et l'origine de la Nature \u00bb ; \u00ab conform\u00e9ment \u00e0 l'ordre, et pour que toutes nos perceptions soient ordonn\u00e9es et unifi\u00e9es, il faut que, _aussi rapidement que faire se peut et que la raison l'exige_ , nous recherchions s'il y a un \u00catre, et aussi quel il est, qui soit la cause de toutes choses, afin que son essence objective soit aussi la cause de toutes nos id\u00e9es25 \u00bb. Il arrive que les interpr\u00e8tes d\u00e9forment ces textes. Il arrive aussi qu'on les explique comme s'ils se rapportaient \u00e0 un moment imparfait dans la pens\u00e9e de Spinoza. Il n'en est pas ainsi : qu'on ne puisse pas partir de l'id\u00e9e de Dieu, qu'on ne puisse pas d\u00e8s le d\u00e9but s'installer en Dieu, est une constante du spinozisme. Les diff\u00e9rences de l' _\u00c9thique_ avec le _Trait\u00e9 de la r\u00e9forme_ sont r\u00e9elles, mais elles ne portent pas sur ce point (elles porteront seulement sur les moyens utilis\u00e9s pour arriver le plus vite possible \u00e0 l'id\u00e9e de Dieu).\n\nQuelle est la th\u00e9orie du _Trait\u00e9 de la r\u00e9forme ?_ Si nous consid\u00e9rons une r\u00e9gression \u00e0 l'infini, c'est-\u00e0-dire un encha\u00eenement infini de choses qui n'existent pas par leur propre nature ou dont les id\u00e9es ne sont pas form\u00e9es par elles-m\u00eames, nous reconnaissons que le concept de cette r\u00e9gression n'a rien d'absurde. Mais en m\u00eame temps, et c'est l\u00e0 le vrai sens de la preuve a posteriori classique, il serait absurde de ne pas reconna\u00eetre ceci : que les choses qui n'existent pas par leur nature sont d\u00e9termin\u00e9es \u00e0 exister (et \u00e0 produire leur effet) par une chose qui, elle, existe n\u00e9cessairement et produit ses effets par soi. C'est toujours Dieu qui d\u00e9termine une cause quelconque \u00e0 produire son effet ; aussi bien Dieu n'est-il jamais, \u00e0 proprement parler, cause \u00ab lointaine \u00bb ou \u00ab \u00e9loign\u00e9e \u00bb26. Nous ne partons donc pas de l'id\u00e9e de Dieu, mais nous y arrivons tr\u00e8s vite, d\u00e8s le d\u00e9but de la r\u00e9gression ; sinon, nous ne pourrions m\u00eame pas comprendre la possibilit\u00e9 d'une s\u00e9rie, son efficience et son actualit\u00e9. _Peu importe, d\u00e8s lors, que nous passions par une fiction_. Et m\u00eame, il peut \u00eatre avantageux d'invoquer une fiction pour arriver le plus vite possible \u00e0 l'id\u00e9e de Dieu sans tomber dans les pi\u00e8ges d'une r\u00e9gression infinie. Par exemple, nous concevons la sph\u00e8re, nous formons une id\u00e9e \u00e0 laquelle aucun objet ne correspond dans la nature. Nous l'expliquons par le mouvement du demi-cercle : cette cause est bien fictive, puisqu'il n'y a rien dans la nature qui soit produit de cette fa\u00e7on ; elle n'en est pas moins une \u00ab perception vraie \u00bb, mais dans la mesure o\u00f9 elle est jointe \u00e0 l'id\u00e9e de Dieu comme au principe qui d\u00e9termine id\u00e9alement le demi-cercle \u00e0 se mouvoir, c'est-\u00e0-dire qui d\u00e9termine cette cause \u00e0 produire l'id\u00e9e de sph\u00e8re.\n\nOr tout change d\u00e8s que nous arrivons ainsi \u00e0 l'id\u00e9e de Dieu. Car cette id\u00e9e, nous la formons par elle-m\u00eame et absolument. \u00ab S'il y a un Dieu ou quelque \u00eatre omniscient, il ne peut former absolument aucune fiction27. \u00bb \u00c0 partir de l'id\u00e9e de Dieu, nous d\u00e9duisons toutes les id\u00e9es les unes des autres dans \u00ab l'ordre d\u00fb \u00bb. Non seulement l'ordre est maintenant celui d'une synth\u00e8se progressive ; mais, prises dans cet ordre, les id\u00e9es ne peuvent plus consister en \u00eatres de raison, et excluent toute fiction. Ce sont n\u00e9cessairement des id\u00e9es de \u00ab choses r\u00e9elles ou vraies \u00bb, des id\u00e9es auxquelles correspond quelque chose dans la nature28. \u00c0 partir de l'id\u00e9e de Dieu, la production des id\u00e9es est en elle-m\u00eame une reproduction des choses de la nature ; l'encha\u00eenement des id\u00e9es n'a pas \u00e0 copier l'encha\u00eenement des choses, il reproduit automatiquement cet encha\u00eenement, dans la mesure o\u00f9 les id\u00e9es sont produites, elles-m\u00eames et pour leur compte, \u00e0 partir de l'id\u00e9e de Dieu29.\n\nIl est certain que les id\u00e9es \u00ab repr\u00e9sentent \u00bb quelque chose, mais pr\u00e9cis\u00e9ment, elles ne repr\u00e9sentent quelque chose que parce qu'elles \u00ab expriment \u00bb leur propre cause, et expriment l'essence de Dieu qui d\u00e9termine cette cause. Toute les id\u00e9es, dit Spinoza, expriment ou enveloppent l'essence de Dieu, et, en tant que telles, sont des id\u00e9es de choses r\u00e9elles ou vraies30. Nous ne sommes plus dans le processus r\u00e9gressif qui rattache une id\u00e9e vraie \u00e0 sa cause, m\u00eame par fiction, pour s'\u00e9lever aussi vite que possible \u00e0 l'id\u00e9e de Dieu : ce processus d\u00e9terminait seulement en droit le contenu de l'id\u00e9e vraie. Nous suivons maintenant une d\u00e9marche progressive, excluant toute fiction, allant d'un \u00eatre r\u00e9el \u00e0 un autre, d\u00e9duisant les id\u00e9es les unes des autres \u00e0 partir de l'id\u00e9e de Dieu : alors, les id\u00e9es s'encha\u00eenent conform\u00e9ment \u00e0 leur contenu propre ; mais aussi leur contenu se trouve d\u00e9termin\u00e9 par cet encha\u00eenement ; _nous saisissons l'identit\u00e9 de la forme et du contenu_ , nous sommes certains que l'encha\u00eenement des id\u00e9es reproduit la r\u00e9alit\u00e9 comme telle. Comment cette d\u00e9duction se fait en d\u00e9tail, nous le verrons plus tard. Il nous suffit pour le moment de consid\u00e9rer que l'id\u00e9e de Dieu, comme principe absolu, s'affranchit de l'hypoth\u00e8se dont nous \u00e9tions partis pour nous \u00e9lever jusqu'\u00e0 elle, et fonde un encha\u00eenement des id\u00e9es ad\u00e9quates identique \u00e0 la construction du r\u00e9el. Donc la seconde partie de la m\u00e9thode ne se contente pas d'une th\u00e9orie de la d\u00e9finition g\u00e9n\u00e9tique, mais doit s'achever dans la th\u00e9orie d'une d\u00e9duction productive.\n\nLa m\u00e9thode de Spinoza comporte donc trois grands chapitres, chacun \u00e9troitement impliqu\u00e9 par les autres. La premi\u00e8re partie de la m\u00e9thode concerne la fin de la pens\u00e9e : celle-ci consiste moins \u00e0 conna\u00eetre quelque chose qu'\u00e0 conna\u00eetre notre puissance de conna\u00eetre. De ce point de vue, la pens\u00e9e est consid\u00e9r\u00e9e dans sa forme : la forme de l'id\u00e9e vraie, c'est l'id\u00e9e de l'id\u00e9e ou l'id\u00e9e r\u00e9flexive. La d\u00e9finition formelle de la v\u00e9rit\u00e9 est la suivante : l'id\u00e9e vraie, c'est l'id\u00e9e _en tant qu'elle s'explique par notre puissance de conna\u00eetre_. La m\u00e9thode, sous ce premier aspect, est elle-m\u00eame r\u00e9flexive.\n\nLa deuxi\u00e8me partie de la m\u00e9thode concerne le moyen de r\u00e9aliser cette fin : une id\u00e9e vraie quelconque est suppos\u00e9e donn\u00e9e, mais nous devons en faire une id\u00e9e ad\u00e9quate. L'ad\u00e9quation constitue la mati\u00e8re du vrai. La d\u00e9finition de l'id\u00e9e ad\u00e9quate (d\u00e9finition mat\u00e9rielle de la v\u00e9rit\u00e9) se pr\u00e9sente ainsi : l'id\u00e9e _en tant qu'elle exprime sa propre cause, et en tant qu'elle exprime l'essence de Dieu comme d\u00e9terminant cette cause_. L'id\u00e9e ad\u00e9quate, c'est donc l'id\u00e9e expressive. Sous ce second aspect, la m\u00e9thode est g\u00e9n\u00e9tique : on d\u00e9termine la cause de l'id\u00e9e comme la raison suffisante de toutes les propri\u00e9t\u00e9s de la chose. C'est cette partie de la m\u00e9thode qui nous m\u00e8ne \u00e0 la plus haute pens\u00e9e, c'est-\u00e0-dire qui nous conduit le plus vite possible \u00e0 l'id\u00e9e de Dieu.\n\nLa seconde partie s'ach\u00e8ve dans un troisi\u00e8me et dernier chapitre, qui concerne l'unit\u00e9 de la forme et du contenu, du but et du moyen. Il en est chez Spinoza comme chez Aristote, o\u00f9 la d\u00e9finition formelle et la d\u00e9finition mat\u00e9rielle en g\u00e9n\u00e9ral morcellent l'unit\u00e9 r\u00e9elle d'une d\u00e9finition compl\u00e8te. Entre l'id\u00e9e et l'id\u00e9e de l'id\u00e9e, il n'y a qu'une distinction de raison : l'id\u00e9e r\u00e9flexive et l'id\u00e9e expressive sont une seule et m\u00eame chose en r\u00e9alit\u00e9.\n\nComment comprendre cette unit\u00e9 derni\u00e8re ? Jamais une id\u00e9e n'a pour cause l'objet qu'elle repr\u00e9sente ; au contraire, elle repr\u00e9sente un objet parce qu'elle exprime _sa propre_ cause. Il y a donc un contenu de l'id\u00e9e, contenu expressif et non repr\u00e9sentatif, qui renvoie seulement \u00e0 la puissance de penser. Mais la puissance de penser est ce qui constitue la forme de l'id\u00e9e comme telle. L'unit\u00e9 concr\u00e8te des deux se manifeste quand toutes les id\u00e9es se d\u00e9duisent les unes des autres, mat\u00e9riellement \u00e0 partir de l'id\u00e9e de Dieu, formellement sous la seule puissance de penser. De ce point de vue, la m\u00e9thode est d\u00e9ductive : la forme, comme forme logique, et le contenu, comme contenu expressif, se r\u00e9unissent dans l'encha\u00eenement des id\u00e9es. On remarquera combien Spinoza insiste sur cette unit\u00e9 dans l'encha\u00eenement. Au moment m\u00eame o\u00f9 il dit que la m\u00e9thode ne se propose pas de nous faire conna\u00eetre quelque chose mais de nous faire conna\u00eetre notre puissance de comprendre, il ajoute que nous ne connaissons celle-ci que dans la mesure o\u00f9 nous connaissons le plus de choses possible li\u00e9es les unes aux autres31. Inversement, quand il montre que nos id\u00e9es sont causes les unes des autres, il en conclut que toutes ont pour cause notre puissance de comprendre ou de penser32. Mais surtout, le terme \u00ab automate spirituel \u00bb t\u00e9moigne de l'unit\u00e9. L'\u00e2me est une esp\u00e8ce d'automate spirituel, c'est-\u00e0-dire : En pensant nous n'ob\u00e9issons qu'aux lois de la pens\u00e9e, lois qui d\u00e9terminent \u00e0 la fois la forme et le contenu de l'id\u00e9e vraie, qui nous font encha\u00eener les id\u00e9es d'apr\u00e8s leurs propres causes et suivant notre propre puissance, si bien que nous ne connaissons pas notre puissance de comprendre sans conna\u00eetre par les causes toutes les choses qui tombent sous cette puissance33.\n\nEn quel sens l'id\u00e9e de Dieu est-elle \u00ab vraie \u00bb ? On ne dira pas d'elle qu'elle exprime sa propre cause : form\u00e9e absolument, c'est-\u00e0-dire sans l'aide d'autres id\u00e9es, elle _exprime l'infini_. C'est donc \u00e0 propos de l'id\u00e9e de Dieu que Spinoza d\u00e9clare : \u00ab La forme de la pens\u00e9e vraie doit r\u00e9sider dans cette pens\u00e9e m\u00eame, sans aucun rapport avec d'autres pens\u00e9es34. \u00bb Il peut para\u00eetre \u00e9trange, cependant, que Spinoza ne r\u00e9serve pas l'application d'un tel principe \u00e0 l'id\u00e9e de Dieu, mais l'\u00e9tende \u00e0 toutes les pens\u00e9es. Au point qu'il ajoute : \u00ab Il ne faut pas dire que la diff\u00e9rence (du vrai et du faux) provient de ce que la pens\u00e9e vraie consiste \u00e0 conna\u00eetre les choses par leurs causes premi\u00e8res, en quoi certes elle diff\u00e9rerait d\u00e9j\u00e0 beaucoup de la fausse. \u00bb Nous croyons que ce texte obscur doit s'interpr\u00e9ter ainsi : Spinoza reconna\u00eet que la connaissance vraie se fait par la cause, mais il estime qu'il n'y a l\u00e0 encore qu'une d\u00e9finition mat\u00e9rielle du vrai. L'id\u00e9e ad\u00e9quate, c'est l'id\u00e9e comme exprimant la cause ; mais nous ne savons pas encore ce qui constitue la forme du vrai, ce qui donne de la v\u00e9rit\u00e9 m\u00eame une d\u00e9finition formelle. L\u00e0 comme ailleurs, nous ne devons pas confondre absolument _ce qui s'exprime_ et _l'exprim\u00e9_ : l'exprim\u00e9, c'est la cause, mais ce qui s'exprime, c'est toujours notre puissance de conna\u00eetre ou de comprendre, la puissance de notre entendement. C'est pourquoi Spinoza dit : \u00ab Ce qui constitue la forme de la pens\u00e9e vraie doit \u00eatre cherch\u00e9 dans cette pens\u00e9e m\u00eame et \u00eatre d\u00e9duit de la nature de l'entendement35. \u00bb C'est pourquoi aussi il dira que le troisi\u00e8me genre de connaissance n'a pas d'autre cause formelle que l'\u00e2me ou l'entendement lui-m\u00eame36. Il en est de m\u00eame de l'id\u00e9e de Dieu : l'exprim\u00e9, c'est l'infini, mais ce qui s'exprime est la puissance absolue de penser. Il fallait donc joindre le point de vue de la forme \u00e0 celui de la mati\u00e8re, pour concevoir finalement l'unit\u00e9 concr\u00e8te des deux telle que la manifeste l'encha\u00eenement des id\u00e9es. C'est seulement de cette fa\u00e7on que nous parvenons \u00e0 la d\u00e9finition compl\u00e8te du vrai, et que nous comprenons le ph\u00e9nom\u00e8ne total de l'expression dans l'id\u00e9e. Non seulement l'id\u00e9e de Dieu mais toutes les id\u00e9es s'expliquent formellement par la puissance de penser. Le contenu de l'id\u00e9e se r\u00e9fl\u00e9chit dans la forme, exactement comme l'exprim\u00e9 se rapporte ou s'attribue \u00e0 ce qui s'exprime. C'est en m\u00eame temps que toutes les id\u00e9es d\u00e9coulent mat\u00e9riellement de l'id\u00e9e de Dieu, et formellement de la puissance de penser : leur encha\u00eenement traduit l'unit\u00e9 des deux d\u00e9rivations.\n\nNous n'avons une puissance de conna\u00eetre, de comprendre ou de penser que dans la mesure o\u00f9 nous participons \u00e0 la puissance absolue de penser. Ce qui implique \u00e0 la fois que notre \u00e2me est un mode de l'attribut pens\u00e9e, et une partie de l'entendement infini. Ces deux points concernent et renouvellent un probl\u00e8me classique : Quelle est la nature de notre id\u00e9e de Dieu ? Selon Descartes, par exemple, nous ne \u00ab comprenons \u00bb pas Dieu, mais nous n'en avons pas moins une id\u00e9e claire et distincte ; car nous \u00ab entendons \u00bb l'infinit\u00e9, ne serait-ce que n\u00e9gativement, et nous \u00ab concevons \u00bb la chose infinie de fa\u00e7on positive, ne serait-ce que partiellement. Notre connaissance de Dieu n'est donc limit\u00e9e que de deux fa\u00e7ons : parce que nous ne connaissons pas Dieu tout entier, parce que nous ne savons pas comment ce que nous connaissons de lui se trouve compris dans son \u00e9minente unit\u00e937. Il n'est certes pas question de dire que Spinoza supprime toute limitation. Mais, bien qu'il s'exprime parfois d'une mani\u00e8re proche de celle de Descartes, il interpr\u00e8te les limites de notre connaissance dans un contexte enti\u00e8rement nouveau.\n\nD'une part, la conception cart\u00e9sienne pr\u00e9sente ce m\u00e9lange de n\u00e9gation et d'affirmation qu'on retrouve toujours dans les m\u00e9thodes d'analogie (on se rappelle les d\u00e9clarations explicites de Descartes contre l'univocit\u00e9). Chez Spinoza, au contraire, la critique radicale de l'\u00e9minence, la position de l'univocit\u00e9 des attributs, ont une cons\u00e9quence imm\u00e9diate : notre id\u00e9e de Dieu n'est pas seulement claire et distincte, mais ad\u00e9quate. En effet, les choses que nous connaissons de Dieu appartiennent \u00e0 Dieu sous cette m\u00eame forme o\u00f9 nous les connaissons, c'est-\u00e0-dire sous une forme commune \u00e0 Dieu qui les poss\u00e8de et aux cr\u00e9atures qui les impliquent et les connaissent. Il n'en reste pas moins que, chez Spinoza comme chez Descartes, nous ne connaissons qu'une partie de Dieu : nous ne connaissons que deux de ces formes, deux attributs seulement, puisque notre corps n'implique rien d'autre que l'attribut \u00e9tendue, notre id\u00e9e, rien d'autre que l'attribut pens\u00e9e. \u00ab Et par suite l'id\u00e9e du corps enveloppe la connaissance de Dieu en tant seulement qu'il est consid\u00e9r\u00e9 sous l'attribut de l'\u00e9tendue... et par suite l'id\u00e9e de cette id\u00e9e enveloppe la connaissance de Dieu en tant qu'il est consid\u00e9r\u00e9 sous l'attribut de la pens\u00e9e et non sous un autre38. \u00bb Bien plus, chez Spinoza, l'id\u00e9e m\u00eame de parties de Dieu est mieux fond\u00e9e que chez Descartes, l'unit\u00e9 divine \u00e9tant parfaitement concili\u00e9e avec une distinction r\u00e9elle entre attributs.\n\nPourtant, m\u00eame sur ce second point, la diff\u00e9rence entre Descartes et Spinoza reste fondamentale. Car, avant de conna\u00eetre une partie de Dieu, notre \u00e2me est elle-m\u00eame \u00ab une partie de l'entendement infini de Dieu \u00bb : nous n'avons en effet de puissance de comprendre ou de conna\u00eetre que dans la mesure o\u00f9 nous participons \u00e0 la puissance absolue de penser qui correspond \u00e0 l'id\u00e9e de Dieu. D\u00e8s lors, _il suffit qu'il y ait quelque chose de commun au tout et \u00e0 la partie pour que ce quelque chose nous donne de Dieu une id\u00e9e, non seulement claire et distincte, mais ad\u00e9quate_39. Cette id\u00e9e qui nous est donn\u00e9e n'est pas l'id\u00e9e de Dieu tout enti\u00e8re. Elle est pourtant ad\u00e9quate, parce qu'elle est dans la partie comme dans le tout. On ne s'\u00e9tonnera donc pas qu'il arrive \u00e0 Spinoza de dire que l'existence de Dieu ne nous est pas connue par elle-m\u00eame : il veut dire que cette connaissance nous est n\u00e9cessairement donn\u00e9e par des \u00ab notions communes \u00bb, sans lesquelles elle ne serait m\u00eame pas claire et distincte, mais gr\u00e2ce auxquelles elle est ad\u00e9quate40. Quand Spinoza rappelle au contraire que Dieu se fait conna\u00eetre imm\u00e9diatement, qu'il est connu par lui-m\u00eame et non par autre chose, il veut dire que la connaissance de Dieu n'a besoin ni de signes, ni de proc\u00e9d\u00e9s analogiques : cette connaissance est ad\u00e9quate parce que Dieu poss\u00e8de toutes les choses que nous connaissons lui appartenir, et les poss\u00e8de sous la forme m\u00eame o\u00f9 nous les connaissons41. Quel rapport y a-t-il entre ces notions communes qui nous donnent la connaissance de Dieu et ces formes elles-m\u00eames communes ou univoques sous lesquelles nous connaissons Dieu ? Nous devons encore remettre \u00e0 plus tard cette derni\u00e8re analyse : elle d\u00e9borde les limites du probl\u00e8me de l'ad\u00e9quation.\n\n* * *\n\n1.  Cf. _TRE_ , 39 : _Una methodi pars_ ; 106 : _Praecipua nostrae methodi pars._ Suivant les d\u00e9clarations de Spinoza, l'expos\u00e9 de cette premi\u00e8re partie se termine en 91-94.\n\n2.  _TRE_ , 91 : _Secundam partem._ Et 94.\n\n3.  _TRE_ , 37 (et 13 : _Naturam aliquam humanam sua multo firmiorem_ ).\n\n4.  _TRE_ , 106 : _Vires et potentiam intellectus. Lettre 37, \u00e0 Bouwmeester_ (III, p. 135) : \u00ab On voit clairement quelle doit \u00eatre la vraie m\u00e9thode, et en quoi elle consiste essentiellement, \u00e0 savoir dans la seule connaissance de l'entendement pur, de sa nature et de ses lois. \u00bb\n\n5.  _TRE_ , 38.\n\n6.  _TRE_ , 105.\n\n7.  Cf. _\u00c9_ , II, 33, dem.\n\n8.  _\u00c9_ , II, 43, prop. (Ce texte se concilie parfaitement avec celui du _Trait\u00e9 de la R\u00e9forme_ , 34-35, selon lequel, inversement, il n'est pas besoin de savoir qu'on sait pour savoir.)\n\n9.  _TRE_ , 33 : \u00ab L'id\u00e9e vraie, car nous avons une id\u00e9e vraie... \u00bb ; 39 : \u00ab Avant toute chose doit exister en nous une id\u00e9e vraie, comme _instrument inn\u00e9_... \u00bb Cette id\u00e9e vraie suppos\u00e9e par la m\u00e9thode ne pose pas de probl\u00e8me particulier : nous l'avons et la reconnaissons par la \u00ab force inn\u00e9e de l'entendement \u00bb ( _TRE_ , 31) c'est pourquoi Spinoza peut dire que la m\u00e9thode n'exige rien d'autre qu'une \u00ab toute petite connaissance de l'esprit \u00bb ( _mentis historialam_ ), du genre de celle qu'enseigne Bacon : cf. _Lettre 37, \u00e0 Bouwmeester_ (III, p. 135).\n\n10.  _\u00c9_ , II, _21_ sc.\n\n11.  Cf. _CT_ , II, ch. 15, 2.\n\n12.  Dans ses _R\u00e9ponses aux secondes objections_ , Descartes pr\u00e9sente un principe g\u00e9n\u00e9ral : \u00ab Il faut distinguer entre la _mati\u00e8re_ ou la chose \u00e0 laquelle nous donnons notre cr\u00e9ance, et la _raison formelle_ qui meut notre volont\u00e9 \u00e0 la donner \u00bb (AT, IX, p. 115). Ce principe explique selon Descartes que la mati\u00e8re \u00e9tant obscure (mati\u00e8re de religion), nous n'en ayons pas moins une raison claire de donner notre adh\u00e9sion (lumi\u00e8re de la gr\u00e2ce). Mais il s'applique aussi dans le cas de la connaissance naturelle : la mati\u00e8re claire et distincte ne se confond pas avec la raison formelle, elle-m\u00eame claire et distincte, de notre croyance (lumi\u00e8re naturelle).\n\n13.  La d\u00e9finition (ou le concept) _explique_ l'essence et _comprend_ la cause prochaine : _TRE_ , 95-96. Elle _exprime_ la cause efficiente : _Lettre 60, \u00e0 Tschirnhaus_ (III, p. 200). La connaissance de l'effet (id\u00e9e) _enveloppe_ la connaissance de la cause : _\u00c9_ , I, axiome 4, et II, 7, dem.\n\n14.  _TRE_ , 92 : \u00ab La connaissance de l'effet consiste exclusivement \u00e0 acqu\u00e9rir une connaissance plus parfaite de la cause. \u00bb\n\n15.  _Lettre 37, \u00e0 Bouwmeester_ (III, p. 135). Telle est la _concatenatio intellectus_ ( _TRE_ , 95).\n\n16.  _TRE_ , 19 et 21 (sur cette insuffisance de l'id\u00e9e claire et distincte, cf. chapitre suivant).\n\n17.  _TRE_ , 72.\n\n18.  Par exemple nous avons l'id\u00e9e du cercle comme d'une figure dont tous les rayons sont \u00e9gaux : ce n'est que l'id\u00e9e claire d'une \u00ab propri\u00e9t\u00e9 \u00bb du cercle ( _TRE_ , 95). De m\u00eame dans la recherche finale d'une d\u00e9finition de l'entendement, nous devons partir des _propri\u00e9t\u00e9s_ de l'entendement _clairement_ connues : _TRE_ , 106-110. Tel est, nous l'avons vu, le _requisit_ de la m\u00e9thode.\n\n19.  Ainsi, _\u00e0 partir du_ cercle comme figure aux rayons \u00e9gaux, nous formons la _fiction_ d'une cause, \u00e0 savoir qu'une ligne droite se meut autour d'une de ses extr\u00e9mit\u00e9s : _fingo ad libitum_ ( _TRE_ , 72).\n\n20.  Ce qui int\u00e9resse Spinoza dans les math\u00e9matiques n'est nullement la g\u00e9om\u00e9trie analytique de Descartes mais la m\u00e9thode synth\u00e9tique d'Euclide et les conceptions g\u00e9n\u00e9tiques de Hobbes : Cf. LEWIS ROBINSON, _Kommentar zu Spinozas Ethik_ , Leipzig, 1928, pp. 270-273.\n\n21.  _TRE_ , 110.\n\n22.  _TRE_ , 94.\n\n23.  Fichte, non moins que Kant, part d'une \u00ab hypoth\u00e8se \u00bb. Mais, contrairement \u00e0 Kant, il pr\u00e9tend arriver \u00e0 un principe absolu qui fait dispara\u00eetre l'hypoth\u00e8se de d\u00e9part : ainsi, d\u00e8s que le principe est d\u00e9couvert, le donn\u00e9 doit faire place \u00e0 une construction du donn\u00e9, le \u00ab jugement hypoth\u00e9tique \u00bb \u00e0 un \u00ab jugement th\u00e9tique \u00bb, l'analyse \u00e0 une gen\u00e8se. M. Gu\u00e9roult dit fort bien : \u00ab \u00c0 quelque moment que ce soit, [ _La Doctrine de la science_ ] affirme toujours que, le principe devant valoir \u00e0 lui seul, la m\u00e9thode analytique ne doit pas poursuivre d'autre fin que sa propre suppression ; elle entend donc bien que toute l'efficacit\u00e9 doit rester \u00e0 la seule m\u00e9thode constructive. \u00bb ( _L'\u00c9volution et la structure de la Doctrine de la science chez Fichte_ , Les Belles-Lettres, 1930, t. I, p. 174.)\n\n24.  Spinoza a invoqu\u00e9 \u00ab l'ordre d\u00fb \u00bb ( _debito ordine_ ) _TRE_ , 44. En 46, il ajoute : \u00ab Si par hasard quelqu'un demande pourquoi moi-m\u00eame n'ai pas tout d'abord et avant tout expos\u00e9 dans cet ordre les v\u00e9rit\u00e9s de la Nature, une fois dit que la v\u00e9rit\u00e9 se manifeste elle-m\u00eame, je lui r\u00e9ponds, et en m\u00eame temps lui conseille... de bien vouloir d'abord consid\u00e9rer l'ordre de notre d\u00e9monstration. \u00bb [La plupart des traducteurs supposent qu'il y a une lacune dans ce dernier texte. Et ils consid\u00e8rent que Spinoza se fait \u00e0 lui-m\u00eame une \u00ab objection pertinente \u00bb. Ils consid\u00e8rent que, plus tard, dans l' _\u00c9thique_ , Spinoza aurait trouv\u00e9 le moyen d'exposer les v\u00e9rit\u00e9s \u00ab dans l'ordre d\u00fb \u00bb (Cf. KOYR\u00c9, trad. du _TRE_ , Vrin, p. 105). Il ne nous semble pas qu'il y ait la moindre lacune : Spinoza dit qu'il ne peut pas _d\u00e8s le d\u00e9but_ suivre l'ordre d\u00fb, parce que cet ordre ne peut \u00eatre atteint qu' _\u00e0 un certain moment_ dans l'ordre des d\u00e9monstrations. Et loin que l' _\u00c9thique_ corrige ce point, elle le maintient rigoureusement, comme nous le verrons au chapitre XVIII.]\n\n25.  _TRE_ , 49, 75, 99. [Dans ce dernier texte aussi, beaucoup de traducteurs d\u00e9placent _et ratio postulat_ pour le faire porter sur l'ensemble de la phrase.]\n\n26.  _\u00c9_ , I, _26_ , prop.\n\n27.  _TRE_ , 54.\n\n28.  Cf. _\u00c9_ , V, 30, dem. : \u00ab ... Concevoir les choses en tant qu'elles se con\u00e7oivent par l'essence de Dieu comme des \u00eatres r\u00e9els. \u00bb\n\n29.  _TRE_ , 42.\n\n30.  _\u00c9_ , II, 45, prop. : \u00ab Toute id\u00e9e de quelque corps ou de chose singuli\u00e8re existant en acte enveloppe n\u00e9cessairement l'essence \u00e9ternelle et infinie de Dieu. \u00bb (Dans le scolie, et aussi dans le scolie de V, 29, Spinoza pr\u00e9cise que les choses existant en acte d\u00e9signent ici les choses comme \u00ab vraies ou r\u00e9elles \u00bb telles qu'elles d\u00e9coulent de la nature divine, leurs id\u00e9es sont donc les id\u00e9es ad\u00e9quates.)\n\n31.  _TRE_ , 40-41.\n\n32.  _Lettre 37 \u00e0 Bouwmeester_ (III, p. 135).\n\n33.  L'\u00ab automate spirituel \u00bb appara\u00eet en _TRE_ , 85. Leibniz pour son compte, qui n'emploie pas l'expression avant le _Syst\u00e8me nouveau_ de 1695, semble bien l'emprunter \u00e0 Spinoza. Et malgr\u00e9 la diff\u00e9rence des deux interpr\u00e9tations, l'automate spirituel a un aspect commun chez Leibniz et chez Spinoza : il d\u00e9signe la nouvelle forme logique de l'id\u00e9e, le nouveau contenu expressif de l'id\u00e9e, et l'unit\u00e9 de cette forme et de ce contenu.\n\n34.  Cf. _TRE_ , 70-71.\n\n35.  _TRE_ , 71.\n\n36.  _\u00c9_ , V, 31, prop.\n\n37.  Sur la distinction entre l'infinit\u00e9 (entendue n\u00e9gativement) et la chose infinie (con\u00e7ue positivement, mais non pas enti\u00e8rement), cf. DESCARTES, _R\u00e9ponses aux premi\u00e8res objections_ , AT, IX, p. 90. La distinction cart\u00e9sienne des _quatri\u00e8mes r\u00e9ponses_ , entre conception compl\u00e8te et conception enti\u00e8re s'applique aussi dans une certaine mesure au probl\u00e8me de la connaissance de Dieu : la _M\u00e9ditation IV_ parlait de l'id\u00e9e de Dieu comme celle d'un \u00ab \u00eatre complet \u00bb (AT, IX, p. 42), bien que nous n'en ayons pas une connaissance enti\u00e8re.\n\n38.  _Lettre 64, \u00e0 Schuller_ (III, p. 205).\n\n39.  _\u00c9_ , II, 46, dem. : \u00ab Ce qui donne la connaissance de l'essence \u00e9ternelle et infinie de Dieu est commun \u00e0 toutes choses et est \u00e9galement dans la partie et dans le tout ; par cons\u00e9quent cette connaissance sera ad\u00e9quate. \u00bb\n\n40.  _TTP_ , ch. 6 (II, p. 159) : \u00ab L'existence de Dieu, n'\u00e9tant pas connue par elle-m\u00eame, doit n\u00e9cessairement se conclure de notions dont la v\u00e9rit\u00e9 soit si ferme et in\u00e9branlable... \u00bb et la note 6 du _TTP_ (II, p. 315) rappelle que ces notions sont les notions communes.\n\n41.  Cf. _CT_ , II, ch. 24, 9-13.\n\n## CHAPITRE IX\n\n## L'INAD\u00c9QUAT\n\nQuelles sont les cons\u00e9quences de cette th\u00e9orie spinoziste de la v\u00e9rit\u00e9 ? Nous devons d'abord en chercher la contre-\u00e9preuve dans la conception de l'id\u00e9e inad\u00e9quate. _L'id\u00e9e inad\u00e9quate, c'est l'id\u00e9e inexpressive_. Mais comment est-il possible que nous ayons des id\u00e9es inad\u00e9quates ? Cette possibilit\u00e9 n'appara\u00eet que si l'on d\u00e9termine les conditions sous lesquelles nous avons des id\u00e9es en g\u00e9n\u00e9ral.\n\nNotre \u00e2me elle-m\u00eame est une id\u00e9e. En ce sens, notre \u00e2me est une affection ou modification de Dieu sous l'attribut pens\u00e9e, comme notre corps, une affection ou modification de Dieu sous l'attribut \u00e9tendue. Cette id\u00e9e qui constitue notre \u00e2me ou notre esprit est donn\u00e9e en Dieu. Il la poss\u00e8de, mais il la poss\u00e8de pr\u00e9cis\u00e9ment en tant qu'il est affect\u00e9 d'une autre id\u00e9e, cause de celle-ci. Il l'a, mais en tant qu'il a \u00ab conjointement \u00bb une autre id\u00e9e, c'est-\u00e0-dire l'id\u00e9e d'autre chose. \u00ab La cause de l'id\u00e9e d'une chose particuli\u00e8re, c'est une autre id\u00e9e, autrement dit Dieu en tant qu'il est consid\u00e9r\u00e9 comme affect\u00e9 d'une autre id\u00e9e, et de celle-ci encore il est cause en tant qu'il est affect\u00e9 d'une autre, et ainsi \u00e0 l'infini1. \u00bb Non seulement Dieu poss\u00e8de toutes les id\u00e9es, autant qu'il y a de choses ; mais toutes les id\u00e9es, telles qu'elles sont en Dieu, expriment leur propre cause et l'essence de Dieu qui d\u00e9termine cette cause. \u00ab Toutes les id\u00e9es sont en Dieu, et en tant qu'elles sont rapport\u00e9es \u00e0 Dieu, sont vraies et ad\u00e9quates2. \u00bb En revanche, nous pouvons d\u00e9j\u00e0 pressentir que cette id\u00e9e qui constitue notre \u00e2me, _nous nel'avons pas_. Du moins nous ne l'avons pas imm\u00e9diatement ; car elle est en Dieu, mais seulement en tant qu'il poss\u00e8de aussi l'id\u00e9e d'autre chose.\n\nCe qui est mode participe \u00e0 la puissance de Dieu : de m\u00eame que notre corps participe \u00e0 la puissance d'exister, notre \u00e2me participe \u00e0 la puissance de penser. Ce qui est mode est en m\u00eame temps partie, partie de la puissance de Dieu, partie de la Nature. Il subit donc n\u00e9cessairement l'influence des autres parties. N\u00e9cessairement les autres id\u00e9es agissent sur notre \u00e2me, comme les autres corps sur notre corps. Apparaissent ici des \u00ab affections \u00bb d'une seconde esp\u00e8ce : il ne s'agit plus du corps lui-m\u00eame, mais de ce qui arrive dans le corps ; il ne s'agit plus de l'\u00e2me (id\u00e9e du corps), mais de ce qui arrive dans l'\u00e2me (id\u00e9e de ce qui arrive dans le corps)3. Or, c'est en ce sens que _nous avons_ des id\u00e9es ; car les id\u00e9es de ces affections sont en Dieu, mais en tant qu'il s'explique par notre \u00e2me seule, ind\u00e9pendamment des autres id\u00e9es qu'il a ; elles sont donc en nous4. Si nous avons une connaissance des corps ext\u00e9rieurs, de notre propre corps, de notre \u00e2me elle-m\u00eame, c'est uniquement par ces id\u00e9es d'affections. Elles seules nous sont donn\u00e9es : nous ne percevons les corps ext\u00e9rieurs qu'en tant qu'ils nous affectent, nous ne percevons notre corps qu'en tant qu'il est affect\u00e9, nous percevons notre \u00e2me par l'id\u00e9e de l'id\u00e9e de l'affection5. Ce que nous appelons \u00ab objet \u00bb, c'est seulement l'effet qu'un objet a sur notre corps ; ce que nous appelons \u00ab moi \u00bb, c'est seulement l'id\u00e9e que nous avons de notre corps et de notre \u00e2me en tant qu'ils subissent un effet. Le donn\u00e9 se pr\u00e9sente ici comme la relation la plus intime et la plus v\u00e9cue, la plus confuse aussi, entre la connaissance des corps, la connaissance du corps et la connaissance de soi.\n\nConsid\u00e9rons ces id\u00e9es que nous avons, et qui correspondent \u00e0 l'effet d'un objet sur notre corps. D'une part, elles d\u00e9pendent de notre puissance de conna\u00eetre, c'est-\u00e0-dire de notre \u00e2me ou de notre esprit, comme de leur cause formelle. Mais nous n'avons pas l'id\u00e9e de notre corps, ni de notre \u00e2me, ind\u00e9pendamment de l'effet subi. Nous ne sommes donc pas en \u00e9tat de nous comprendre comme la cause formelle des id\u00e9es que nous avons ; elles apparaissent au plus haut comme le fruit du hasard6. D'autre part, elles ont pour causes mat\u00e9rielles des id\u00e9es de choses ext\u00e9rieures. Mais ces id\u00e9es de choses ext\u00e9rieures, nous ne les avons pas davantage ; elles sont en Dieu, mais non pas en tant qu'il constitue notre \u00e2me ou notre esprit. Donc, nous ne poss\u00e9dons pas nos id\u00e9es dans des conditions telles qu'elles puissent exprimer leur propre cause (mat\u00e9rielle). Sans doute nos id\u00e9es d'affections \u00ab enveloppent-elles \u00bb leur propre cause, c'est-\u00e0-dire l'essence objective du corps ext\u00e9rieur ; mais elles ne l'\u00ab expriment \u00bb pas, ne \u00ab l'expliquent \u00bb pas. De m\u00eame, elles enveloppent notre puissance de conna\u00eetre, mais elles ne s'expliquent pas par elle, et renvoient au hasard. _Voil\u00e0 que, dans ce cas, le mot_ \u00ab _envelopper_ \u00bb _n'est plus un corr\u00e9latif de_ \u00ab _expliquer_ \u00bb _ou_ \u00ab _exprimer_ \u00bb, mais s'oppose \u00e0 eux, d\u00e9signant le m\u00e9lange du corps ext\u00e9rieur et de notre corps dans l'affection dont nous avons l'id\u00e9e. La formule employ\u00e9e le plus souvent par Spinoza est la suivante : nos id\u00e9es d'affections _indiquent_ un \u00e9tat de notre corps, mais _n'expliquent_ pas la nature ou l'essence du corps ext\u00e9rieur7. C'est dire que les id\u00e9es que nous avons sont des signes, des images indicatives imprim\u00e9es en nous, non pas des id\u00e9es expressives et form\u00e9es par nous ; des perceptions ou des imaginations, non pas des compr\u00e9hensions.\n\nAu sens le plus pr\u00e9cis, l'image est l'empreinte, la trace ou l'impression physique, l'affection du corps elle-m\u00eame, l'effet d'un corps sur les parties fluides et molles du n\u00f4tre ; au sens figur\u00e9, l'image est l'id\u00e9e de l'affection, qui nous fait seulement conna\u00eetre l'objet par son effet. Mais une telle connaissance n'en est pas une, tout au plus une r\u00e9cognition. De l\u00e0 d\u00e9coulent les caract\u00e8res de l'indication en g\u00e9n\u00e9ral : le premier \u00ab indiqu\u00e9 \u00bb n'est jamais notre essence, mais un \u00e9tat momentan\u00e9 de notre constitution variable ; l'indiqu\u00e9 secondaire (indirect) n'est jamais l'essence ou la nature d'une chose ext\u00e9rieure, mais l'apparence qui nous permet seulement de reconna\u00eetre la chose \u00e0 partir de son effet, donc d'en affirmer la simple pr\u00e9sence, \u00e0 tort ou \u00e0 raison8. Fruits du hasard et des rencontres, servant \u00e0 la r\u00e9cognition, purement indicatives, les id\u00e9es que nous avons sont inexpressives, c'est-\u00e0-dire inad\u00e9quates. L'id\u00e9e inad\u00e9quate n'est ni privation absolue ni ignorance absolue : elle enveloppe une privation de connaissance9.\n\nCette connaissance dont nous sommes priv\u00e9s est double : connaissance de nous-m\u00eames, et de l'objet qui produit en nous l'affection dont nous avons l'id\u00e9e. L'id\u00e9e inad\u00e9quate est donc une id\u00e9e qui enveloppe la privation de la connaissance de sa propre cause, tant formellement que mat\u00e9riellement. C'est en ce sens qu'elle reste inexpressive : \u00ab tronqu\u00e9e \u00bb, comme une cons\u00e9quence sans ses pr\u00e9misses10. Or l'essentiel est que Spinoza ait montr\u00e9 _comment_ une cons\u00e9quence pouvait, ainsi, \u00eatre s\u00e9par\u00e9e de ses deux pr\u00e9misses. Nous sommes naturellement dans une situation telle que les id\u00e9es qui nous sont donn\u00e9es sont n\u00e9cessairement inad\u00e9quates, parce qu'elles ne peuvent pas exprimer leur cause ni s'expliquer par notre puissance de conna\u00eetre. Sur tous les points, connaissance des parties de notre corps et de notre corps lui-m\u00eame, connaissance des corps ext\u00e9rieurs, connaissance de notre \u00e2me ou de notre esprit, connaissance de notre dur\u00e9e et de celle des choses, nous n'avons que des id\u00e9es inad\u00e9quates11. \u00ab Lorsque nous regardons le soleil, nous imaginons qu'il est \u00e9loign\u00e9 de nous de deux cents pieds environ ; et cette erreur ne consiste pas dans cette seule imagination, mais en ce que, tandis que nous imaginons ainsi le soleil, nous ignorons sa vraie distance, ainsi que _la cause de cette imagination_12. \u00bb L'image, en ce sens, est une id\u00e9e qui ne peut pas exprimer sa propre cause, c'est-\u00e0-dire l'id\u00e9e dont elle d\u00e9rive en nous, qui ne nous est pas donn\u00e9e : il s'agit de la cause mat\u00e9rielle. Mais l'image n'exprime pas davantage sa cause formelle, et ne peut pas s'expliquer par notre puissance de conna\u00eetre. C'est pourquoi Spinoza dit que l'image, ou l'id\u00e9e d'affection, est comme une cons\u00e9quence sans ses pr\u00e9misses : il y a bien deux pr\u00e9misses, mat\u00e9rielle et formelle, dont l'image enveloppe la privation de connaissance.\n\nNotre probl\u00e8me se transforme donc. La question n'est plus : Pourquoi avons-nous des id\u00e9es inad\u00e9quates ? Mais au contraire : Comment arriverons-nous \u00e0 former des id\u00e9es ad\u00e9quates ? Chez Spinoza, il en est de la v\u00e9rit\u00e9 comme de la libert\u00e9 : elles ne sont pas donn\u00e9es en principe, mais apparaissent comme le r\u00e9sultat d'une longue activit\u00e9 par laquelle nous produisons des id\u00e9es ad\u00e9quates, \u00e9chappant \u00e0 l'encha\u00eenement d'une n\u00e9cessit\u00e9 externe13. Par l\u00e0, l'inspiration spinoziste est profond\u00e9ment empiriste. Il est toujours frappant de constater la diff\u00e9rence d'inspiration entre les empiristes et les rationalites. Les uns s'\u00e9tonnent de ce qui n'\u00e9tonne pas les autres. \u00c0 entendre les rationalistes, la v\u00e9rit\u00e9 et la libert\u00e9 sont avant tout des droits ; ils se demandent comment nous pouvons d\u00e9choir de ces droits, tomber dans l'erreur ou perdre la libert\u00e9. C'est pourquoi le rationalisme a trouv\u00e9 dans la tradition adamique, posant en principe l'image d'un Adam libre et raisonnable, un th\u00e8me qui convenait particuli\u00e8rement avec ses pr\u00e9occupations. Dans une perspective empiriste, tout est renvers\u00e9 : l'\u00e9tonnant, c'est que les hommes arrivent parfois \u00e0 comprendre le vrai, parfois \u00e0 se comprendre entre eux, parfois \u00e0 se lib\u00e9rer de ce qui les encha\u00eene. \u00c0 la vigueur avec laquelle Spinoza s'oppose constamment \u00e0 la tradition adamique, on reconna\u00eet d\u00e9j\u00e0 l'inspiration empiriste, qui con\u00e7oit la libert\u00e9 et la v\u00e9rit\u00e9 comme des produits ultimes surgissant \u00e0 la fin. Un des paradoxes de Spinoza, et ce n'est pas le seul cas o\u00f9 nous le verrons s'exercer, est d'avoir retrouv\u00e9 les forces concr\u00e8tes de l'empirisme pour les mettre au service d'un nouveau rationalisme, un des plus rigoureux qu'on ait jamais con\u00e7us. Spinoza demande : Comment arriverons-nous \u00e0 former et produire des id\u00e9es ad\u00e9quates, alors que tant d'id\u00e9es inad\u00e9quates nous sont n\u00e9cessairement donn\u00e9es, qui distraient notre puissance et nous s\u00e9parent de ce que nous pouvons ?\n\nNous devons distinguer deux aspects dans l'id\u00e9e inad\u00e9quate : elle \u00ab enveloppe la privation \u00bb de la connaissance de sa cause, mais aussi elle est un effet qui \u00ab enveloppe \u00bb cette cause en quelque mani\u00e8re. Sous son premier aspect, l'id\u00e9e inad\u00e9quate est fausse ; mais, sous le deuxi\u00e8me, elle contient _quelque chose de positif_ , donc quelque chose de vrai14. Par exemple nous imaginons que le soleil est distant de deux cents pieds. Cette id\u00e9e d'affection n'est pas en \u00e9tat d'exprimer sa propre cause : elle n'explique pas la nature ou l'essence du soleil. Reste qu'elle enveloppe cette essence \u00ab en tant que le corps en est affect\u00e9 \u00bb. Nous aurons beau savoir la vraie distance du soleil, il continuera \u00e0 nous affecter dans de telles conditions que nous le verrons toujours \u00e0 deux cents pieds : comme dit Spinoza, l'erreur sera supprim\u00e9e, mais non l'imagination. Il y a donc quelque chose de positif dans l'id\u00e9e inad\u00e9quate, une sorte d'indication qu'on peut saisir clairement. C'est m\u00eame ainsi que nous pouvons avoir quelque id\u00e9e de la cause : apr\u00e8s avoir saisi clairement les conditions sous lesquelles nous voyons le soleil, nous en inf\u00e9rons clairement qu'il est un objet suffisamment \u00e9loign\u00e9 pour para\u00eetre petit, non pas un objet petit qui serait vu de pr\u00e8s15. Si l'on ne tient pas compte de cette positivit\u00e9, plusieurs th\u00e8ses de Spinoza deviennent inintelligibles : d'abord, que l'on puisse avoir naturellement une id\u00e9e vraie, conform\u00e9ment \u00e0 ce que la m\u00e9thode exige avant son exercice. Mais, surtout, le faux n'ayant pas de forme, on ne comprendrait pas que l'id\u00e9e inad\u00e9quate donne lieu elle-m\u00eame \u00e0 une id\u00e9e de l'id\u00e9e, c'est-\u00e0-dire ait une forme qui renvoie \u00e0 notre puissance de penser16. La facult\u00e9 d'imaginer se d\u00e9finit par les conditions sous lesquelles nous avons naturellement des id\u00e9es, donc des id\u00e9es inad\u00e9quates ; elle n'en est pas moins une _vertu_ par un de ses aspects ; elle enveloppe notre puissance de penser, bien qu'elle ne s'explique pas par elle ; l'image enveloppe sa propre cause, bien qu'elle ne l'exprime pas17.\n\nCertes, pour avoir une id\u00e9e ad\u00e9quate, il ne suffit pas de saisir ce qu'il y a de positif dans une id\u00e9e d'affection. C'est pourtant le premier pas. Car, \u00e0 partir de cette positivit\u00e9, nous pourrons former l'id\u00e9e de _ce qui est commun_ au corps affectant et au corps affect\u00e9, au corps ext\u00e9rieur et au n\u00f4tre. Or nous verrons que cette \u00ab notion commune \u00bb, elle, est n\u00e9cessairement ad\u00e9quate : elle est dans l'id\u00e9e de notre corps comme elle est dans l'id\u00e9e du corps ext\u00e9rieur ; elle est donc en nous comme elle est en Dieu ; elle exprime Dieu et s'explique par notre puissance de penser. Mais de cette notion commune d\u00e9coule \u00e0 son tour une id\u00e9e d'affection, elle-m\u00eame ad\u00e9quate : la notion commune est n\u00e9cessairement cause d'une id\u00e9e ad\u00e9quate d'affection, qui ne se distingue que par une \u00ab raison \u00bb de l'id\u00e9e d'affection dont nous \u00e9tions partis. Ce m\u00e9canisme complexe ne consistera donc pas \u00e0 supprimer l'id\u00e9e inad\u00e9quate que nous avons, mais \u00e0 utiliser ce qu'il y a de positif en elle pour former le plus grand nombre possible d'id\u00e9es ad\u00e9quates, et faire que les id\u00e9es inad\u00e9quates subsistantes n'occupent enfin que la plus petite partie de nous-m\u00eames. Bref, nous devons nous-m\u00eames acc\u00e9der \u00e0 des conditions telles que nous puissions produire des id\u00e9es ad\u00e9quates.\n\nNotre but n'est pas encore d'analyser ce m\u00e9canisme par lequel nous arrivons aux id\u00e9es ad\u00e9quates. Notre probl\u00e8me \u00e9tait seulement : Qu'est-ce que l'id\u00e9e ad\u00e9quate ? Et, par contre-\u00e9preuve : Qu'est-ce que l'id\u00e9e inad\u00e9quate ? L'id\u00e9e ad\u00e9quate, c'est l'id\u00e9e qui exprime sa propre cause et qui s'explique par notre propre puissance. L'id\u00e9e inad\u00e9quate, c'est l'id\u00e9e inexpressive et non expliqu\u00e9e : l'impression qui n'est pas encore expression, l'indication qui n'est pas encore explication. Se d\u00e9gage ainsi l'intention qui pr\u00e9side \u00e0 toute la doctrine spinoziste de la v\u00e9rit\u00e9 : _il s'agit de substituer la conception de l'ad\u00e9quat \u00e0 la conception cart\u00e9sienne du clair et du distinct._ Et sans doute, \u00e0 cet \u00e9gard, Spinoza utilise une terminologie variable : tant\u00f4t il emploie le mot \u00ab ad\u00e9quat \u00bb pour marquer l'insuffisance du clair et du distinct, soulignant ainsi la n\u00e9cessit\u00e9 de d\u00e9passer les crit\u00e8res cart\u00e9siens ; tant\u00f4t il se sert pour son compte des mots \u00ab clair et distinct \u00bb, mais les applique seulement \u00e0 des id\u00e9es qui d\u00e9coulent d'une id\u00e9e elle-m\u00eame ad\u00e9quate ; tant\u00f4t enfin il s'en sert pour d\u00e9signer cette id\u00e9e ad\u00e9quate, mais, \u00e0 plus forte raison, leur donne alors une signification implicite tout \u00e0 fait diff\u00e9rente de celle de Descartes18.\n\nDe toute mani\u00e8re, la doctrine de la v\u00e9rit\u00e9 chez Spinoza ne se s\u00e9pare pas d'une pol\u00e9mique, directe ou indirecte, dirig\u00e9e contre la th\u00e9orie cart\u00e9sienne. Pris en eux-m\u00eames, le clair et le distinct nous permettent tout au plus de reconna\u00eetre une id\u00e9e vraie que nous avons, c'est-\u00e0-dire ce qu'il y a de positif dans une id\u00e9e encore inad\u00e9quate. Mais former une id\u00e9e ad\u00e9quate nous entra\u00eene au-del\u00e0 du clair et du distinct. L'id\u00e9e claire et distincte par elle-m\u00eame ne constitue pas une v\u00e9ritable connaissance, pas plus qu'elle ne contient en elle-m\u00eame sa propre raison : le clair et le distinct ne trouvent leur raison suffisante que dans l'ad\u00e9quat, l'id\u00e9e claire et distincte ne forme une v\u00e9ritable connaissance que dans la mesure o\u00f9 elle d\u00e9coule d'une id\u00e9e elle-m\u00eame ad\u00e9quate.\n\nNous trouvons \u00e0 nouveau un point commun entre Spinoza et Leibniz, qui contribue \u00e0 d\u00e9finir la r\u00e9action anticart\u00e9sienne. Le mot de Leibniz : _la connaissance est une esp\u00e8ce de l'expression_ , pourrait \u00eatre sign\u00e9 de Spinoza19. Sans doute ne con\u00e7oivent-ils pas de la m\u00eame mani\u00e8re la nature de l'ad\u00e9quat, parce qu'ils ne comprennent ni n'utilisent de la m\u00eame mani\u00e8re le concept d'expression. Mais, sur trois chapitres essentiels, leur accord involontaire est r\u00e9el. D'une part, Descartes, dans sa conception du clair et du distinct, s'en est tenu au contenu repr\u00e9sentatif de l'id\u00e9e ; il ne s'est pas \u00e9lev\u00e9 jusqu'\u00e0 un contenu expressif infiniment plus profond. Il n'a pas con\u00e7u d'ad\u00e9quat comme raison n\u00e9cessaire et suffisante du clair et du distinct : c'est-\u00e0-dire l'expression comme fondement de la repr\u00e9sentation. D'autre part, Descartes n'a pas d\u00e9pass\u00e9 la forme d'une conscience psychologique de l'id\u00e9e ; il n'a pas atteint la forme logique par laquelle l'id\u00e9e s'explique, d'apr\u00e8s laquelle aussi les id\u00e9es s'encha\u00eeneront les unes les autres. Enfin, Descartes n'a pas con\u00e7u l'unit\u00e9 de la forme et du contenu, c'est-\u00e0-dire l'\u00ab automate spirituel \u00bb qui reproduit le r\u00e9el en produisant ses id\u00e9es dans l'ordre d\u00fb. Descartes nous a appris que le vrai \u00e9tait pr\u00e9sent dans l'id\u00e9e. Mais de quoi nous sert ce savoir tant que nous ne savons pas ce qui est pr\u00e9sent dans l'id\u00e9e vraie ? L'id\u00e9e claire et distincte est encore inexpressive, et reste inexpliqu\u00e9e. Bonne pour la r\u00e9cognition, mais incapable de fournir un v\u00e9ritable principe de connaissance.\n\nNous avons vu les trois acquis principaux de la th\u00e9orie de l'id\u00e9e chez Spinoza : le contenu repr\u00e9sentatif n'est qu'une apparence, en fonction d'un contenu expressif plus profond ; la forme d'une conscience psychologique est superficielle, par rapport \u00e0 la vraie forme logique ; l'automate spirituel tel qu'il se manifeste dans l'encha\u00eenement des id\u00e9es est l'unit\u00e9 de la forme logique et du contenu expressif. Or ces trois points forment aussi les grandes th\u00e8ses de Leibniz. C'est pourquoi Leibniz aime le mot de Spinoza, \u00ab automate spirituel \u00bb. Pour son compte, il l'interpr\u00e8te au sens d'une autonomie des substances pensantes individuelles. Mais, m\u00eame pour Spinoza, l'automatisme d'un mode de la pens\u00e9e n'exclut pas une sorte d'autonomie de sa puissance de comprendre (en effet, la puissance de comprendre est une partie de la puissance absolue de penser, mais en tant que celle-ci s'explique par celle-l\u00e0). Toutes les diff\u00e9rences entre Leibniz et Spinoza n'\u00f4tent rien \u00e0 leur accord sur ces th\u00e8ses fondamentales, qui constituent la r\u00e9volution anticart\u00e9sienne par excellence.\n\nLa critique de Descartes par Leibniz est c\u00e9l\u00e8bre : le clair et le distinct en eux-m\u00eames nous permettent seulement de _reconna\u00eetre_ un objet, mais ne nous donnent pas de cet objet une connaissance v\u00e9ritable ; ils n'atteignent pas l'essence, mais portent seulement sur des apparences externes ou sur des caract\u00e8res extrins\u00e8ques \u00e0 travers lesquels nous ne pouvons que \u00ab conjecturer \u00bb l'essence ; ils n'atteignent pas la cause qui nous montre pourquoi la chose est n\u00e9cessairement ce qu'elle est20. Pour \u00eatre moins remarqu\u00e9e, la critique spinoziste n'en proc\u00e8de pas moins de la m\u00eame fa\u00e7on, d\u00e9non\u00e7ant avant tout l'insuffisance de l'id\u00e9e cart\u00e9sienne : pris en eux-m\u00eames, le clair et le distinct ne nous donnent qu'une connaissance ind\u00e9termin\u00e9e ; ils n'atteignent pas l'essence de la chose, mais portent seulement sur des _propria_ ; ils n'atteignent pas une cause dont toutes les propri\u00e9t\u00e9s de la chose d\u00e9couleraient \u00e0 la fois, mais nous font seulement reconna\u00eetre un objet, la pr\u00e9sence d'un objet, d'apr\u00e8s l'effet qu'il a sur nous ; l'id\u00e9e claire et distincte n'exprime pas sa propre cause, elle ne nous fait rien comprendre de la cause \u00ab en dehors de ce que nous consid\u00e9rons dans l'effet21. \u00bb En tout ceci, Spinoza et Leibniz m\u00e8nent un combat commun, la continuation de celui qui les opposait d\u00e9j\u00e0 \u00e0 la preuve ontologique cart\u00e9sienne, la recherche d'une raison suffisante qui manque singuli\u00e8rement dans tout le cart\u00e9sianisme. L'un et l'autre, par des processus diff\u00e9rents, d\u00e9couvrent _le contenu expressif de l'id\u00e9e, la forme explicative de l'id\u00e9e._\n\n* * *\n\n1.  _\u00c9_ , II, 9, dem. Et II, 11, cor. : Dieu \u00ab en tant qu'il poss\u00e8de _conjointement_ avec l'esprit humain l'id\u00e9e d'une autre chose... \u00bb ; III, 1, dem. : Dieu \u00ab en tant qu'il contient en m\u00eame temps les esprits des autres choses \u00bb.\n\n2.  _\u00c9_ , II, 36, dem.\n\n3.  _\u00c9_ , II, 9, cor. : \u00ab ce qui arrive ( _contingit_ ) dans l'objet singulier d'une id\u00e9e quelconque... \u00bb.\n\n4.  _\u00c9_ , II, 12, dem. : \u00ab Tout ce qui arrive dans l'objet de l'id\u00e9e constituant l'esprit humain, la connaissance en est n\u00e9cessairement donn\u00e9e en Dieu en tant qu'il constitue la nature de l'esprit humain ; c'est-\u00e0-dire la connaissance de cette chose sera n\u00e9cessairement dans l'esprit, autrement dit l'esprit la per\u00e7oit. \u00bb\n\n5.  _\u00c9_ , II, 19, 23 et 26.\n\n6.  Sur le r\u00f4le du hasard ( _fortuna_ ) dans les perceptions qui ne sont pas ad\u00e9quates, cf. _Lettre 37, \u00e0 Bouwmeester_ (III, p. 135).\n\n7.  _Indicare : \u00c9_ , II, 16, cor. 2 ; IV, 1 sc. _Indiquer_ ou _envelopper_ s'opposent alors \u00e0 _expliquer_. Ainsi l'id\u00e9e de Pierre telle qu'elle est en Paul \u00ab indique l'\u00e9tat du corps de Paul \u00bb, tandis que l'id\u00e9e de Pierre en elle-m\u00eame \u00ab explique directement l'essence du corps de Pierre \u00bb (II, 17, sc.). De m\u00eame les id\u00e9es \u00ab qui ne font qu'envelopper la nature de choses ext\u00e9rieures au corps humain \u00bb s'opposent aux id\u00e9es \u00ab qui expliquent la nature de ces m\u00eames choses \u00bb (II, 18, sc.).\n\n8.  Sur l'indiqu\u00e9 principal : nos id\u00e9es d'affections indiquent en premier lieu la constitution de notre corps, constitution _pr\u00e9sente_ et variable _(\u00c9_ , II, 16, cor. 2 ; III, d\u00e9f. g\u00e9n\u00e9rale des affects ; IV, 1, sc.). Sur l'indiqu\u00e9 secondaire ou indirect : nos id\u00e9es d'affections enveloppent la nature d'un corps ext\u00e9rieur, mais indirectement, de telle fa\u00e7on que nous ne faisons que croire \u00e0 la _pr\u00e9sence_ de ce corps tant que notre affection dure ( _\u00c9_ , II, 16, dem. ; II, 17, prop., dem. et cor.).\n\n9.  _\u00c9_ , II, 35, prop. et dem.\n\n10.  _\u00c9_ , II, 28, dem.\n\n11.  _\u00c9_ , II, 24, 25, 27, 28, 29, 30, 31.\n\n12.  _\u00c9_ , II, 35, sc.\n\n13.  Il y a en effet un encha\u00eenement ( _ordo_ et _concatenatio_ ) des id\u00e9es inad\u00e9quates entre elles, qui s'oppose \u00e0 l'ordre et \u00e0 l'encha\u00eenement de l'entendement. Les id\u00e9es inad\u00e9quates, s'encha\u00eenent dans l'ordre o\u00f9 elles s'impriment en nous. C'est l'ordre de la M\u00e9moire. Cf. _\u00c9_ , II, 18, sc.\n\n14.  _\u00c9_ , II, 33, prop. et dem. ; II, 35, sc. ; IV, 1, prop., dem. et sc.\n\n15.  Exemple analogue, _TRE_ , 21.\n\n16.  Cf. _\u00c9_ , II, 22 et 23.\n\n17.  _\u00c9_ , II, 17, sc. : \u00ab Car si l'esprit, pendant qu'il imagine comme lui \u00e9tant pr\u00e9sentes des choses qui n'existent pas, savait en m\u00eame temps que ces choses n'existent pas en r\u00e9alit\u00e9, il regarderait certes cette puissance d'imaginer comme une vertu de sa nature et non comme un vice, surtout si cette facult\u00e9 d'imaginer d\u00e9pendait de sa nature seule. \u00bb (C'est-\u00e0-dire : _si cette facult\u00e9 ne se contentait pas d'envelopper notre puissance de penser, mais s'expliquait par elle._ )\n\n18.  Cf. la _Lettre 37, \u00e0 Bouwmeester_ , o\u00f9 Spinoza se sert des mots \u00ab clair et distinct \u00bb pour d\u00e9signer l'ad\u00e9quat lui-m\u00eame. En un sens plus pr\u00e9cis, Spinoza entend par \u00ab clair et distinct \u00bb ce qui suit de l'ad\u00e9quat, donc ce qui doit trouver sa raison dans l'ad\u00e9quat : \u00ab Tout ce qui suit d'une id\u00e9e qui est ad\u00e9quate en nous, nous le comprenons clairement et distinctement \u00bb ( _\u00c9_ , V, 4, sc.). Mais ce texte se r\u00e9clame de II, 40, qui disait que tout ce qui suit d'une id\u00e9e ad\u00e9quate est ad\u00e9quat aussi.\n\n19.  LEIBNIZ, _Lettre \u00e0 Arnauld_ (Janet, t. I, p. 593) : \u00ab L'expression est commune \u00e0 toutes les formes, et c'est un genre dont la perception naturelle, le sentiment animal et la connaissance intellectuelle sont des esp\u00e8ces. \u00bb\n\n20.  Cf. LEIBNIZ, _M\u00e9ditations sur la connaissance..._ ; Discours de m\u00e9taphysique, \u00a7 24.\n\n21.  La critique de l'id\u00e9e claire est men\u00e9e par Spinoza de mani\u00e8re explicite en _TRE_ , 19 et note, 21 et note. Il est vrai que Spinoza ne dit pas \u00ab clair et distinct \u00bb. Mais c'est qu'il se r\u00e9serve d'employer ces mots pour son compte en un tout autre sens que Descartes. Nous verrons dans le chapitre suivant comment la critique spinoziste porte en fait sur l'ensemble de la conception cart\u00e9sienne.\n\n## CHAPITRE X\n\n## SPINOZA CONTRE DESCARTES\n\nLe cart\u00e9sianisme repose sur une certaine suffisance de l'id\u00e9e claire et distincte. Cette suffisance fonde la m\u00e9thode de Descartes, mais d'autre part est prouv\u00e9e par l'exercice de cette m\u00e9thode elle-m\u00eame. Descartes affirme sa pr\u00e9f\u00e9rence pour l' _analyse_. Dans un texte important, il dit que la m\u00e9thode analytique a le m\u00e9rite de nous faire voir \u00ab comment les effets d\u00e9pendent des causes1 \u00bb. Cette d\u00e9claration pourrait sembler paradoxale, pr\u00eatant \u00e0 l'analyse ce qui revient \u00e0 la synth\u00e8se, si l'on n'en mesurait pas l'exacte port\u00e9e. Suivant Descartes, nous avons une connaissance claire et distincte d'un effet _avant_ d'avoir une connaissance claire et distincte de la cause. Par exemple, je sais que j'existe comme \u00eatre pensant avant de conna\u00eetre la cause par laquelle j'existe. Sans doute, la connaissance claire et distincte de l'effet suppose une certaine connaissance de la cause, mais seulement une connaissance confuse. \u00ab Si je dis 4 + 3 = 7, cette conception est n\u00e9cessaire, parce que nous ne concevons pas distinctement le nombre 7 sans y inclure 3 et 4 _confusa quadam ratione_2. \u00bb La connaissance claire et distincte de l'effet suppose donc une connaissance confuse de la cause, mais en aucun cas ne d\u00e9pend d'une connaissance plus parfaite de la cause. Au contraire, c'est la connaissance claire et distincte de la cause qui d\u00e9pend de la connaissance claire et distincte de l'effet. Telle est la base des _M\u00e9ditations_ , de leur ordre en particulier et de la m\u00e9thode analytique en g\u00e9n\u00e9ral : m\u00e9thode d'inf\u00e9rence ou d'implication.\n\nD\u00e8s lors, si cette m\u00e9thode nous fait voir comment les effets d\u00e9pendent des causes, c'est de la mani\u00e8re suivante : \u00e0 partir d'une connaissance claire de l'effet, nous tirons au clair la connaissance de la cause qu'elle impliquait confus\u00e9ment, et par l\u00e0, nous montrons que l'effet ne serait pas ce que nous le connaissons \u00eatre s'il n'avait telle cause dont il d\u00e9pend n\u00e9cessairement3. Chez Descartes, donc, deux th\u00e8mes sont fondamentalement li\u00e9s : la suffisance th\u00e9orique de l'id\u00e9e claire et distincte, la possibilit\u00e9 pratique d'aller d'une connaissance claire et distincte de l'effet \u00e0 une connaissance claire et distincte de la cause.\n\nQue l'effet d\u00e9pende de la cause n'est pas en question. La question porte sur la meilleure mani\u00e8re de le montrer. Spinoza dit : Il nous est possible de partir d'une connaissance claire d'un effet ; mais ainsi nous ne parviendrons qu'\u00e0 une connaissance claire de la cause, nous ne conna\u00eetrons rien de la cause en dehors de ce que nous consid\u00e9rons dans l'effet, jamais nous n'obtiendrons une connaissance ad\u00e9quate. Le _Trait\u00e9 de la r\u00e9forme_ contient une critique fondamentale de la m\u00e9thode cart\u00e9sienne, du proc\u00e9d\u00e9 d'inf\u00e9rence ou d'implication dont elle se sert, de la pr\u00e9tendue suffisance du clair et du distinct dont elle se r\u00e9clame. L'id\u00e9e claire ne nous donne rien, sinon une certaine connaissance des propri\u00e9t\u00e9s de la chose, et ne nous m\u00e8ne \u00e0 rien, sinon \u00e0 une connaissance n\u00e9gative de la cause. \u00ab Il y a une perception o\u00f9 l'essence d'une chose est inf\u00e9r\u00e9e d'une autre chose, mais non pas de mani\u00e8re ad\u00e9quate \u00bb ; \u00ab Nous ne comprenons rien de la cause en dehors de ce que nous consid\u00e9rons dans l'effet : ce qui se voit suffisamment du fait que la cause, alors, n'est d\u00e9sign\u00e9e que par les termes les plus g\u00e9n\u00e9raux, comme _il y a donc quelque chose_ , _il y a donc quelque puissance_ , etc. Ou aussi du fait qu'on la d\u00e9signe d'une fa\u00e7on n\u00e9gative, _par cons\u00e9quent ce n'est pas ceci ou cela, etc._ \u00bb ; \u00ab Nous inf\u00e9rons une chose d'une autre de la mani\u00e8re suivante : apr\u00e8s avoir clairement per\u00e7u que nous sentons tel corps et nul autre, nous en inf\u00e9rons clairement, dis-je, que l'\u00e2me est unie au corps, et que cette union est la cause d'une telle sensation. Mais quelle est cette sensation et cette union, nous ne pouvons pas le comprendre par l\u00e0 d'une fa\u00e7on absolue \u00bb ; \u00ab Une telle conclusion, bien que certaine, n'est pas assez s\u00fbre4. \u00bb Dans ces citations, il n'y a pas une ligne qui ne soit dirig\u00e9e contre Descartes et sa m\u00e9thode. Spinoza ne croit pas \u00e0 la suffisance du clair et du distinct, parce qu'il ne croit pas qu'on puisse de mani\u00e8re satisfaisante aller d'une connaissance de l'effet \u00e0 une connaissance de la cause.\n\nIl ne suffit pas d'une id\u00e9e claire et distincte, il faut aller jusqu'\u00e0 l'id\u00e9e ad\u00e9quate. C'est-\u00e0-dire : il ne suffit pas de montrer comment les effets d\u00e9pendent des causes, il faut montrer comment la connaissance vraie de l'effet d\u00e9pend elle-m\u00eame de la connaissance de la cause. Telle est la d\u00e9finition de la m\u00e9thode _synth\u00e9tique_. Sur tous ces points, Spinoza se retrouve aristot\u00e9licien, contre Descartes : \u00ab Ce qui est la m\u00eame chose que ce que les Anciens ont dit, \u00e0 savoir que la science vraie proc\u00e8de de la cause aux effets5. \u00bb Aristote montrait que la connaissance scientifique est par la cause. Il ne disait pas seulement que la connaissance doit d\u00e9couvrir la cause, s'\u00e9lever jusqu'\u00e0 la cause dont un effet connu d\u00e9pend ; il disait que l'effet n'est connu que dans la mesure o\u00f9 la cause est elle-m\u00eame et d'abord mieux connue. La cause n'est pas seulement ant\u00e9rieure \u00e0 l'effet parce qu'elle en est la cause, elle est ant\u00e9rieure aussi du point de vue de la connaissance, devant \u00eatre plus connue que l'effet6. Spinoza reprend cette th\u00e8se : \u00ab La connaissance de l'effet n'est en r\u00e9alit\u00e9 rien d'autre que l'acquisition d'une connaissance plus parfaite de la cause7. \u00bb Entendons : non pas \u00ab plus parfaite \u00bb que celle que nous avions d'abord, mais plus parfaite que celle que nous avons de l'effet lui-m\u00eame, ant\u00e9rieure \u00e0 celle que nous avons de l'effet. La connaissance de l'effet peut \u00eatre dite claire et distincte, mais la connaissance de la cause est plus parfaite, c'est-\u00e0-dire ad\u00e9quate ; et le clair et le distinct ne sont fond\u00e9s que pour autant qu'ils d\u00e9coulent de l'ad\u00e9quat en tant que tel.\n\nConna\u00eetre par la cause est le seul moyen de conna\u00eetre l'essence. La cause est comme le moyen terme qui fonde la connexion de l'attribut avec le sujet, le principe ou la raison dont d\u00e9coulent toutes les propri\u00e9t\u00e9s qui reviennent \u00e0 la chose. C'est pourquoi, suivant Aristote, se confondent la recherche de la cause et la recherche de la d\u00e9finition. D'o\u00f9 l'importance du syllogisme scientifique dont les pr\u00e9misses nous donnent la cause ou la d\u00e9finition formelles d'un ph\u00e9nom\u00e8ne, et la conclusion, la cause ou la d\u00e9finition mat\u00e9rielles. La d\u00e9finition totale est celle qui r\u00e9unit la forme et la mati\u00e8re dans une \u00e9nonciation continue, de telle fa\u00e7on que l'unit\u00e9 de l'objet ne soit plus fragment\u00e9e, mais au contraire affirm\u00e9e dans un concept intuitif. Sur tous ces points, Spinoza reste en apparence aristot\u00e9licien : il souligne l'importance de la th\u00e9orie de la d\u00e9finition, il pose l'identit\u00e9 de la recherche de la d\u00e9finition et de la recherche des causes, il affirme l'unit\u00e9 concr\u00e8te d'une d\u00e9finition totale englobant la cause formelle et la cause mat\u00e9rielle de l'id\u00e9e vraie.\n\nDescartes n'ignore pas les pr\u00e9tentions d'une m\u00e9thode synth\u00e9tique de type aristot\u00e9licien : la preuve qu'elle contient, dit-il, est souvent \u00ab des effets par les causes8 \u00bb. Descartes veut dire : la m\u00e9thode synth\u00e9tique pr\u00e9tend toujours conna\u00eetre par la cause, mais elle n'y r\u00e9ussit pas toujours. L'objection fondamentale est la suivante : Comment la cause elle-m\u00eame serait-elle connue ? En g\u00e9om\u00e9trie, nous pouvons conna\u00eetre _par_ la cause, mais parce que la mati\u00e8re est claire et convient avec les sens. Descartes l'admet (d'o\u00f9 son emploi du mot \u00ab souvent \u00bb)9. De m\u00eame Aristote : le point, la ligne, m\u00eame l'unit\u00e9, sont des principes ou des \u00ab genres-sujets \u00bb, des indivisibles atteints par l'intuition ; leur existence est connue, en m\u00eame temps que leur signification, comprise10. Mais que se passe-t-il dans les autres cas, par exemple en m\u00e9taphysique, quand il s'agit d'\u00eatres r\u00e9els ? Comment la cause, le principe ou le moyen-terme sont-ils trouv\u00e9s ? Il semble bien qu'Aristote nous renvoie lui-m\u00eame \u00e0 un processus inductif, qui ne se distingue gu\u00e8re d'une abstraction et qui trouve son point de d\u00e9part dans une perception confuse de l'effet. En ce sens, c'est l'effet qui est le plus connu, le plus connu pour nous, par opposition au \u00ab plus connu absolument \u00bb. Quand Aristote d\u00e9taille les moyens de parvenir au moyen-terme ou \u00e0 la d\u00e9finition causale, il part d'un ensemble confus pour en abstraire un universel \u00ab proportionn\u00e9 \u00bb. C'est pourquoi la cause formelle est toujours un caract\u00e8re sp\u00e9cifique abstrait, qui trouve son origine dans une mati\u00e8re sensible et confuse. De ce point de vue, l'unit\u00e9 de la cause formelle et de la cause mat\u00e9rielle reste un pur id\u00e9al chez Aristote, au m\u00eame titre que l'unit\u00e9 du concept intuitif.\n\nLa th\u00e8se de Descartes se pr\u00e9sente donc ainsi : la m\u00e9thode synth\u00e9tique a une ambition d\u00e9mesur\u00e9e ; mais elle ne nous donne aucun moyen de conna\u00eetre les causes r\u00e9elles. En fait, elle part d'une connaissance confuse de l'effet, et s'\u00e9l\u00e8ve \u00e0 des abstraits qu'elle nous pr\u00e9sente \u00e0 tort comme des causes ; c'est pourquoi, malgr\u00e9 ses pr\u00e9tentions, elle se contente d'examiner les causes par les effets11. La m\u00e9thode analytique, au contraire, est d'intention plus modeste. Mais, parce qu'elle d\u00e9gage d'abord une perception claire et distincte de l'effet, elle nous donne le moyen d'inf\u00e9rer de cette perception une connaissance v\u00e9ritable de la cause ; c'est pourquoi elle est apte \u00e0 montrer comment les effets eux-m\u00eames d\u00e9pendent des causes. La m\u00e9thode synth\u00e9tique n'est donc l\u00e9gitime qu'\u00e0 une condition : quand elle n'est pas livr\u00e9e \u00e0 elle-m\u00eame, quand elle vient apr\u00e8s la m\u00e9thode analytique, quand elle s'appuie sur une connaissance pr\u00e9alable des causes r\u00e9elles. La m\u00e9thode synth\u00e9tique ne nous fait rien conna\u00eetre par elle-m\u00eame, elle n'est pas une m\u00e9thode d'invention ; elle trouve son utilit\u00e9 dans l'exposition de la connaissance, dans l'exposition de ce qui est d\u00e9j\u00e0 \u00ab invent\u00e9 \u00bb.\n\nOn remarquera que jamais Descartes ne songe \u00e0 d\u00e9partager les deux m\u00e9thodes en rapportant la synth\u00e8se \u00e0 l'ordre de l'\u00eatre, l'analyse, \u00e0 l'ordre de la connaissance. Pas davantage Spinoza. Il serait donc insuffisant, et inexact, d'opposer Descartes \u00e0 Spinoza en disant que le premier suit l'ordre de la connaissance, le second l'ordre de l'\u00eatre. Sans doute d\u00e9coule-t-il de la d\u00e9finition de la m\u00e9thode synth\u00e9tique qu'elle co\u00efncide avec l'\u00eatre. Mais cette cons\u00e9quence a peu d'importance. Le seul probl\u00e8me est de savoir si la m\u00e9thode synth\u00e9tique est capable, d'abord et par elle-m\u00eame, de nous faire conna\u00eetre les principes qu'elle suppose. Peut-elle, en v\u00e9rit\u00e9, nous _faire conna\u00eetre_ ce qui est ? Le seul probl\u00e8me est donc : Quelle est la vraie m\u00e9thode du point de vue de la connaissance12 ? Alors l'anti-cart\u00e9sianisme de Spinoza se manifeste pleinement : selon Spinoza, la m\u00e9thode synth\u00e9tique est la seule m\u00e9thode d'invention v\u00e9ritable, la seule m\u00e9thode qui vaille dans l'ordre de la connaissance13. Or, une telle position n'est tenable que si Spinoza estime avoir les moyens, non seulement de retourner les objections de Descartes, mais aussi de surmonter les difficult\u00e9s de l'aristot\u00e9lisme. Pr\u00e9cis\u00e9ment, quand il pr\u00e9sente ce qu'il appelle le troisi\u00e8me \u00ab mode de perception \u00bb dans le _Trait\u00e9 de la r\u00e9forme_ , il groupe sous ce mode ou dans ce genre _imparfait_ deux proc\u00e9d\u00e9s tr\u00e8s divers, dont il d\u00e9nonce \u00e9galement l'insuffisance14. Le premier consiste \u00e0 inf\u00e9rer une cause \u00e0 partir d'un effet clairement per\u00e7u : on reconna\u00eet ici la m\u00e9thode analytique de Descartes et son processus d'implication. Mais le second consiste \u00e0 \u00ab tirer une conclusion d'un universel qui est toujours accompagn\u00e9 d'une certaine propri\u00e9t\u00e9 \u00bb : on reconna\u00eet la m\u00e9thode synth\u00e9tique d'Aristote, son processus d\u00e9ductif \u00e0 partir du moyen-terme con\u00e7u comme caract\u00e8re sp\u00e9cifique. Si Spinoza, non sans ironie, peut ainsi r\u00e9unir Descartes et Aristote, _c'est parce qu'il revient au m\u00eame, \u00e0 peu pr\u00e8s, d'abstraire un universel \u00e0 partir d'une connaissance confuse de l'effet, ou d'inf\u00e9rer une cause \u00e0 partir d'une connaissance claire de l'effet_. Aucun de ces proc\u00e9d\u00e9s ne m\u00e8ne \u00e0 l'ad\u00e9quat. La m\u00e9thode analytique de Descartes est insuffisante, mais Aristote n'a pas su davantage concevoir la suffisance de la m\u00e9thode synth\u00e9tique.\n\nCe qui manque aux Anciens, dit Spinoza, c'est de concevoir l'\u00e2me comme une esp\u00e8ce d'automate spirituel, c'est-\u00e0-dire la pens\u00e9e comme d\u00e9termin\u00e9e par ses propres lois15. C'est donc le parall\u00e9lisme qui donne \u00e0 Spinoza le moyen de d\u00e9passer les difficult\u00e9s de l'aristot\u00e9lisme. La cause formelle d'une id\u00e9e n'est jamais un universel abstrait. Les universaux, genres ou esp\u00e8ces, renvoient bien \u00e0 une puissance d'imaginer, mais cette puissance diminue au fur et \u00e0 mesure que nous comprenons plus de choses. La cause formelle de l'id\u00e9e vraie, c'est notre puissance de comprendre ; et plus nous comprenons de choses, moins nous formons ces fictions de genres et d'esp\u00e8ces16. Si Aristote identifie la cause formelle avec l'universel sp\u00e9cifique, c'est parce qu'il en reste _au plus bas degr\u00e9_ de la puissance de penser, sans d\u00e9couvrir les lois qui permettent \u00e0 celle-ci d'aller d'un \u00eatre r\u00e9el \u00e0 un autre r\u00e9el \u00ab sans passer par les choses abstraites \u00bb. D'autre part, la cause mat\u00e9rielle d'une id\u00e9e n'est pas une perception sensible confuse : une id\u00e9e de chose particuli\u00e8re trouve toujours sa cause dans une autre id\u00e9e de chose particuli\u00e8re d\u00e9termin\u00e9e \u00e0 la produire.\n\nFace au mod\u00e8le aristot\u00e9licien, Descartes ne pouvait saisir les possibilit\u00e9s de la m\u00e9thode synth\u00e9tique. Il est vrai que celle-ci, sous un de ses aspects, ne nous fait pas conna\u00eetre quelque chose ; mais on aurait tort d'en conclure qu'elle a seulement un r\u00f4le d'exposition. Sous son premier aspect, la m\u00e9thode synth\u00e9tique est r\u00e9flexive, c'est-\u00e0-dire nous fait conna\u00eetre notre puissance de comprendre. Il est vrai aussi que la m\u00e9thode synth\u00e9tique _forge ou feint_ une cause en fonction d'un effet ; mais loin d'y voir une contradiction, nous devons reconna\u00eetre ici le minimum de r\u00e9gression qui nous permet, _le plus vite possible_ , d'atteindre \u00e0 l'id\u00e9e de Dieu comme \u00e0 la source de toutes les autres id\u00e9es. Sous ce second aspect, la m\u00e9thode est constructive ou g\u00e9n\u00e9tique. Enfin, les id\u00e9es qui d\u00e9coulent de l'id\u00e9e de Dieu sont des id\u00e9es d'\u00eatres r\u00e9els : leur production est en m\u00eame temps la d\u00e9duction du r\u00e9el, la forme et la mati\u00e8re du vrai s'identifient dans l'encha\u00eenement des id\u00e9es. La m\u00e9thode, sous ce troisi\u00e8me aspect, est d\u00e9ductive. R\u00e9flexion, gen\u00e8se et d\u00e9duction, ces trois moments constituent tous ensemble la m\u00e9thode synth\u00e9tique. C'est sur eux que Spinoza compte, \u00e0 la fois pour d\u00e9passer le cart\u00e9sianisme et pour pallier les insuffisances de l'aristot\u00e9lisme.\n\nConsid\u00e9rons maintenant la th\u00e9orie de l'\u00eatre : nous voyons que l'opposition de Spinoza \u00e0 Descartes se d\u00e9place, mais ne cesse pas d'\u00eatre radicale. Aussi bien serait-il \u00e9tonnant que la m\u00e9thode analytique et la m\u00e9thode synth\u00e9tique impliquent une m\u00eame conception de l'\u00eatre. L'ontologie de Spinoza est domin\u00e9e par les notions de _cause de soi, en soi_ et _par soi_. Ces termes \u00e9taient pr\u00e9sents chez Descartes lui-m\u00eame ; mais les difficult\u00e9s qu'il rencontrait dans leur emploi doivent nous renseigner sur les incompatibilit\u00e9s du cart\u00e9sianisme et du spinozisme.\n\nContre Descartes, Caterus et Arnauld objectaient d\u00e9j\u00e0 : \u00ab par soi \u00bb se dit n\u00e9gativement et ne signifie que l'absence de cause17. M\u00eame en admettant avec Arnauld que, si Dieu n'a pas de cause, c'est en raison de la pleine positivit\u00e9 de son essence et non pas en fonction de l'imperfection de notre entendement, on n'en conclura pas qu'il est par soi \u00ab positivement comme par une cause \u00bb, c'est-\u00e0-dire, qu'il est cause de soi. Descartes, il est vrai, estime que cette pol\u00e9mique est surtout verbale. Il demande seulement qu'on lui accorde la pleine positivit\u00e9 de l'essence de Dieu : d\u00e8s lors, on reconna\u00eetra que cette essence joue un r\u00f4le _analogue_ \u00e0 celui d'une cause. Il y a une raison positive pour laquelle Dieu n'a pas de cause, donc une cause formelle par laquelle il n'a pas de cause efficiente. Descartes pr\u00e9cise sa th\u00e8se dans les termes suivants : Dieu est cause de soi, mais _en un autre sens_ qu'une cause efficiente est cause de son effet ; il est cause de soi au sens o\u00f9 son essence est cause formelle ; et son essence est dite cause formelle, non pas directement, mais par analogie, dans la mesure o\u00f9 elle joue par rapport \u00e0 l'existence un r\u00f4le analogue \u00e0 celui d'une cause efficiente par rapport \u00e0 son effet18.\n\nCette th\u00e9orie repose sur trois notions intimement li\u00e9es : _l'\u00e9quivocit\u00e9_ (Dieu est cause de soi, mais en un autre sens qu'il n'est cause efficiente des choses qu'il cr\u00e9e ; d\u00e8s lors, l'\u00eatre ne se dit pas au m\u00eame sens de tout ce qui est, substance divine et substances cr\u00e9\u00e9es, substances et modes, etc.) ; _l'\u00e9minence_ (Dieu contient donc toute la r\u00e9alit\u00e9, mais \u00e9minemment, sous une autre forme que celle des choses qu'il cr\u00e9e) ; _l'analogie_ (Dieu comme cause de soi n'est donc pas atteint en lui-m\u00eame, mais par analogie : c'est par analogie avec la cause efficiente que Dieu peut \u00eatre dit cause de soi, ou par soi \u00ab comme \u00bb par une cause). Ces th\u00e8ses sont moins formul\u00e9es explicitement par Descartes qu'elles ne sont re\u00e7ues et accept\u00e9es comme un h\u00e9ritage scolastique et thomiste. Mais, pour n'\u00eatre jamais discut\u00e9es, elles n'en ont pas moins une importance essentielle, partout pr\u00e9sentes chez Descartes, indispensables \u00e0 sa th\u00e9orie de l'\u00eatre, de Dieu et des cr\u00e9atures. Sa m\u00e9taphysique ne trouve pas son sens en elles ; mais, sans elles elle perdrait beaucoup de son sens. C'est pourquoi les Cart\u00e9siens pr\u00e9sentent si volontiers une th\u00e9orie de l'analogie : plus qu'ils ne tentent de r\u00e9concilier l'\u0153uvre du ma\u00eetre et le thomisme, ils d\u00e9veloppent alors une pi\u00e8ce essentielle du cart\u00e9sianisme qui restait implicite chez Descartes lui-m\u00eame.\n\nIl est toujours possible d'imaginer, entre Descartes et Spinoza, des filiations fantaisistes. Par exemple, dans une d\u00e9finition cart\u00e9sienne de la substance (\u00ab ce qui n'a besoin que de soi-m\u00eame pour exister \u00bb), on pr\u00e9tend d\u00e9couvrir une tentation moniste et m\u00eame panth\u00e9iste. C'est n\u00e9gliger le r\u00f4le implicite de l'analogie dans la philosophie de Descartes, qui suffit \u00e0 la pr\u00e9munir contre toute tentation de ce genre : comme chez saint Thomas, l'acte d'exister sera par rapport aux substances cr\u00e9\u00e9es quelque chose d'analogue \u00e0 ce qu'il est par rapport \u00e0 la substance divine19. Et il semble bien que la m\u00e9thode analytique d\u00e9bouche naturellement dans une conception analogique de l'\u00eatre ; son proc\u00e9d\u00e9 lui-m\u00eame conduit spontan\u00e9ment \u00e0 la position d'un \u00eatre analogue. On ne s'\u00e9tonnera donc pas que le cart\u00e9sianisme retrouve \u00e0 sa mani\u00e8re une difficult\u00e9 d\u00e9j\u00e0 pr\u00e9sente dans le thomisme le plus orthodoxe : malgr\u00e9 ses ambitions, l'analogie n'arrive pas \u00e0 se d\u00e9gager de l'\u00e9quivocit\u00e9 dont elle part, de l'\u00e9minence \u00e0 laquelle elle arrive.\n\nSelon Spinoza, Dieu n'est pas cause de soi en un autre sens que cause de toutes choses. _Au contraire, il est cause de toutes choses au m\u00eame sens que cause de soi_20. Descartes en dit trop ou n'en dit pas assez : trop pour Arnauld, mais pas assez pour Spinoza. Car il n'est pas possible d'employer \u00ab par soi \u00bb positivement, tout en usant de \u00ab cause de soi \u00bb par simple analogie. Descartes reconna\u00eet que, si l'essence de Dieu est cause de son existence, c'est au sens de cause formelle, non pas de cause efficiente. La cause formelle, pr\u00e9cis\u00e9ment, est l'essence immanente, coexistante \u00e0 son effet, ins\u00e9parable de son effet. Encore faut-il une raison positive pour laquelle l'existence de Dieu n'a pas de cause efficiente et ne fait qu'un avec l'essence. Or, cette raison, Descartes la trouve dans une simple propri\u00e9t\u00e9 : l'immensit\u00e9 de Dieu, sa surabondance ou son infinit\u00e9. Mais une telle propri\u00e9t\u00e9 ne peut jouer que le r\u00f4le d'une r\u00e8gle de proportionnalit\u00e9 dans un jugement analogique. Parce que cette propri\u00e9t\u00e9 ne d\u00e9signe rien de la nature de Dieu, Descartes en reste \u00e0 une d\u00e9termination indirecte de la cause de soi : celle-ci se dit en un autre sens que la cause efficiente, mais aussi se dit par analogie avec elle. Ce qui manque chez Descartes, c'est donc une raison sous laquelle la cause de soi puisse \u00eatre atteinte en elle-m\u00eame et directement fond\u00e9e dans le concept ou dans la nature de Dieu. C'est cette raison que Spinoza d\u00e9couvre quand il distingue la nature divine et les propres, l'absolu et l'infini. Les attributs sont les \u00e9l\u00e9ments formels immanents qui constituent la nature absolue de Dieu. Ces attributs ne constituent pas l'essence de Dieu sans en constituer l'existence ; ils n'expriment pas l'essence sans exprimer l'existence qui en d\u00e9coule n\u00e9cessairement ; c'est pourquoi l'existence ne fait qu'un avec l'essence21. Ainsi les attributs constituent la raison formelle qui fait de la substance en elle-m\u00eame une cause de soi, directement, non plus par analogie.\n\nLa cause de soi est d'abord atteinte en elle-m\u00eame ; c'est \u00e0 cette condition que \u00ab en soi \u00bb et \u00ab par soi \u00bb prennent une signification parfaitement positive. En d\u00e9coule la cons\u00e9quence suivante : La cause de soi ne se dit plus _en un autre sens_ que la cause efficiente, c'est la cause efficiente au contraire qui se dit au m\u00eame sens que la cause de soi. Dieu produit donc comme il existe : d'une part il produit n\u00e9cessairement, d'autre part il produit n\u00e9cessairement dans ces m\u00eames attributs qui constituent son essence. On retrouve ici les deux aspects de l'univocit\u00e9 spinoziste, l'univocit\u00e9 de la cause et l'univocit\u00e9 des attributs. Depuis le commencement de nos analyses, il nous a sembl\u00e9 que le spinozisme n'\u00e9tait pas s\u00e9parable du combat qu'il menait contre la th\u00e9ologie n\u00e9gative, mais aussi contre toute m\u00e9thode proc\u00e9dant par \u00e9quivocit\u00e9, \u00e9minence et analogie. Non seulement Spinoza d\u00e9nonce l'introduction du n\u00e9gatif dans l'\u00eatre, mais toutes les fausses conceptions de l'affirmation dans lesquelles le n\u00e9gatif survit. Ce sont ces survivances que Spinoza retrouve et combat, chez Descartes et chez les Cart\u00e9siens. Le concept spinoziste d'immanence n'a pas d'autre sens : il exprime la double univocit\u00e9 de la cause et des attributs, c'est-\u00e0-dire l'unit\u00e9 de la cause efficiente avec la cause formelle, l'identit\u00e9 de l'attribut tel qu'il constitue l'essence de la substance et tel qu'il est impliqu\u00e9 par les essences de cr\u00e9atures.\n\nOn ne croira pas qu'en r\u00e9duisant ainsi les cr\u00e9atures \u00e0 des modifications ou \u00e0 des modes Spinoza leur retire toute essence propre ou toute puissance. L'univocit\u00e9 de la cause ne signifie pas _que la cause de soi et la cause efficiente aient un seul et m\u00eame sens, mais que toutes deux se disent au m\u00eame sens de ce qui est cause_. L'univocit\u00e9 des attributs ne signifie pas que la substance et les modes aient le m\u00eame \u00eatre ou la m\u00eame perfection : la substance est en soi, les modifications sont dans la substance comme dans autre chose. Ce qui est en autre chose et ce qui est en soi ne se disent pas au m\u00eame sens, mais l'\u00eatre se dit formellement au m\u00eame sens de ce qui est en soi et de ce qui est en autre chose : les m\u00eames attributs, pris au m\u00eame sens, constituent l'essence de l'un et sont impliqu\u00e9s par l'essence de l'autre. Bien plus, cet \u00eatre commun n'est pas chez Spinoza, comme chez Duns Scot, un \u00catre neutralis\u00e9, indiff\u00e9rent au fini et \u00e0 l'infini, \u00e0 l' _in-se_ et \u00e0 l' _in-alio_. Au contraire, c'est l'\u00catre qualifi\u00e9 de la substance, dans lequel la substance reste en soi, mais aussi dans lequel les modes restent comme dans autre chose. L'immanence est donc la nouvelle figure que prend la th\u00e9orie de l'univocit\u00e9 chez Spinoza. La m\u00e9thode synth\u00e9tique conduit naturellement \u00e0 la position de cet \u00eatre commun ou de cette cause immanente.\n\nDans la philosophie de Descartes, certains axiomes reviennent constamment. Le principal est que le n\u00e9ant n'a pas de propri\u00e9t\u00e9s. Il en d\u00e9coule, du point de vue de la quantit\u00e9, que toute propri\u00e9t\u00e9 est propri\u00e9t\u00e9 d'un \u00eatre : donc, tout est \u00eatre ou propri\u00e9t\u00e9, substance ou mode. Et aussi, du point de vue de la qualit\u00e9, toute r\u00e9alit\u00e9 est perfection. Du point de vue de la causalit\u00e9, il doit y avoir au moins autant de r\u00e9alit\u00e9 dans la cause que dans l'effet ; sinon quelque chose serait produit par le n\u00e9ant. Enfin, du point de vue de la modalit\u00e9, il ne peut y avoir d'accident \u00e0 proprement parler, l'accident \u00e9tant une propri\u00e9t\u00e9 qui n'impliquerait pas n\u00e9cessairement l'\u00eatre auquel on le rapporte. De tous ces axiomes, il appartient \u00e0 Spinoza de donner une interpr\u00e9tation nouvelle, conforme \u00e0 la th\u00e9orie de l'immanence et aux exigences de la m\u00e9thode synth\u00e9tique. Et il semble bien \u00e0 Spinoza que Descartes n'a pas saisi le sens et les cons\u00e9quences de la proposition : Le n\u00e9ant n'a pas de propri\u00e9t\u00e9s. D'une part, toute pluralit\u00e9 de substances devient impossible : il n'y a ni substances in\u00e9gales et limit\u00e9es, ni substances illimit\u00e9es \u00e9gales, car \u00ab elles devraient tirer quelque chose du n\u00e9ant22 \u00bb. D'autre part, on ne se contentera pas de dire que toute r\u00e9alit\u00e9 est perfection. On doit reconna\u00eetre aussi que tout dans la nature d'une chose est r\u00e9alit\u00e9, c'est-\u00e0-dire perfection ; \u00ab dire \u00e0 ce sujet que la nature d'une chose exigeait (la limitation) et par suite ne pouvait \u00eatre autrement, c'est ne rien dire, car la nature d'une chose ne peut rien exiger tant qu'elle n'est pas23. \u00bb On \u00e9vitera donc de croire qu'une substance subisse une limitation de nature en vertu de sa propre possibilit\u00e9.\n\nPas plus qu'il n'y a de possibilit\u00e9 d'une substance en fonction de son attribut, il n'y a de contingence des modes par rapport \u00e0 la substance. Il ne suffit pas de montrer, avec Descartes, que les accidents ne sont pas r\u00e9els. Les modes d'une substance restent accidentels chez Descartes, parce qu'ils ont besoin d'une causalit\u00e9 externe qui, en quelque mani\u00e8re, les \u00ab mette \u00bb dans cette substance elle-m\u00eame. Mais, en v\u00e9rit\u00e9, l'opposition du mode et de l'accident montre d\u00e9j\u00e0 que la n\u00e9cessit\u00e9 est la seule affection de l'\u00eatre, sa seule modalit\u00e9 : Dieu est cause de toutes choses au m\u00eame sens que cause de soi ; donc tout est n\u00e9cessaire, par son essence ou par sa cause. Enfin, il est vrai que la cause est plus parfaite que l'effet, la substance plus parfaite que les modes ; mais, bien qu'elle ait plus de r\u00e9alit\u00e9, jamais la cause ne contient la r\u00e9alit\u00e9 de son effet sous une autre forme ni d'une autre mani\u00e8re que celle dont d\u00e9pend l'effet lui-m\u00eame. Avec Descartes, on passe de la sup\u00e9riorit\u00e9 de la cause \u00e0 la sup\u00e9riorit\u00e9 de certaines formes d'\u00eatre sur d'autres, donc \u00e0 l'\u00e9quivocit\u00e9 ou \u00e0 l'analogie du r\u00e9el (puisque Dieu contient la r\u00e9alit\u00e9 sous une forme sup\u00e9rieure \u00e0 celle qui se trouve impliqu\u00e9e dans les cr\u00e9atures). C'est ce passage qui fonde le concept d'\u00e9minence ; mais ce passage est radicalement ill\u00e9gitime. Contre Descartes, Spinoza pose l'\u00e9galit\u00e9 de toutes les formes d'\u00eatre, et l'univocit\u00e9 du r\u00e9el qui d\u00e9coule de cette \u00e9galit\u00e9. De tous les points de vue, la philosophie de l'immanence appara\u00eet comme la th\u00e9orie de l'\u00catre-un, de l'\u00catre-\u00e9gal, de l'\u00catre univoque et commun. Elle cherche les conditions d'une affirmation v\u00e9ritable, d\u00e9non\u00e7ant tous les traitements qui retirent \u00e0 l'\u00eatre sa pleine positivit\u00e9, c'est-\u00e0-dire sa communaut\u00e9 formelle.\n\n* * *\n\n1.  DESCARTES, _R\u00e9ponses aux secondes objections_ , AT, IX, p. 121. Ce texte, qui n'existe que dans la traduction fran\u00e7aise de Clerselier, suscite de grandes difficult\u00e9s : F. Alqui\u00e9 les souligne dans son \u00e9dition de Descartes (Garnier, t. II, p. 582). Nous demandons toutefois, dans les pages suivantes, si le texte ne peut pas \u00eatre interpr\u00e9t\u00e9 \u00e0 la lettre.\n\n2.  DESCARTES, _Regulae_ , R\u00e8gle 12 (AT, X, p. 421). Constamment chez Descartes, une connaissance claire et distincte implique, en tant que telle, une perception confuse de la cause ou du principe. Laporte donne toutes sortes d'exemples, dans _Le Rationalisme de Descartes_ (P.U.F. 1945, pp. 98-99). Quand Descartes dit : \u00ab J'ai _en quelque fa\u00e7on_ premi\u00e8rement en moi la notion de l'infini que du fini \u00bb ( _M\u00e9d._ III), il faut entendre que l'id\u00e9e de Dieu est impliqu\u00e9e par celle du moi, mais confus\u00e9ment ou implicitement ; un peu comme 4 et 3 sont impliqu\u00e9s dans 7.\n\n3.  Par exemple, _M\u00e9ditation_ III, AT, IX, p. 41 : \u00ab ... je reconnais qu'il ne serait pas possible que ma nature f\u00fbt telle qu'elle est, c'est-\u00e0-dire que j'eusse en moi l'id\u00e9e d'un Dieu, si Dieu n'existait pas v\u00e9ritablement. \u00bb\n\n4.  _TRE_ , 19 (\u00a7 III) et 21 (et notes correspondantes). Tous ces textes d\u00e9crivent une partie de ce que Spinoza appelle le troisi\u00e8me \u00ab mode de perception \u00bb. _Il ne s'agit pas d'un proc\u00e9d\u00e9 d'induction_ : l'induction appartient au deuxi\u00e8me mode, et se trouve d\u00e9crite et critiqu\u00e9e en _TRE_ , 20. Ici au contraire, il s'agit d'un proc\u00e9d\u00e9 d'inf\u00e9rence ou d'implication de type cart\u00e9sien.\n\n5.  _TRE_ , 85.\n\n6.  Cf. ARISTOTE, _Seconds Analytiques_ , I, 2, 71 b, 30.\n\n7.  _TRE_ , 92.\n\n8.  DESCARTES, _R\u00e9ponses aux secondes objections_ , AT, IV, p. 122 (l\u00e0 encore, ce texte est de la traduction Clerselier).\n\n9.  _Ibid._\n\n10.  Cf. ARISTOTE, _Seconds Analytiques_ , I, 32, 88 b, 25-30.\n\n11.  DESCARTES, _R\u00e9ponses aux secondes objections_ , AT, IX, p. 122 : \u00ab La synth\u00e8se au contraire, par une voie tout autre, et comme en examinant les causes par leurs effets (bien que la preuve qu'elle contient soit souvent aussi des effets par les causes)... \u00bb\n\n12.  F. Alqui\u00e9, dans une intervention orale au sujet de Descartes, met bien ce point en lumi\u00e8re : \u00ab Je ne vois nulle part que l'ordre synth\u00e9tique soit l'ordre de la chose... La chose est vraiment l'unit\u00e9 ; c'est l'\u00eatre, c'est l'unit\u00e9 confuse ; c'est moi qui mets un ordre lorsque je connais. Et ce qu'il faut \u00e9tablir, c'est que l'ordre de ma connaissance, et qui est toujours un ordre de connaissance, _qu'il soit synth\u00e9tique ou analytique_ , est vrai \u00bb. ( _Descartes, Cahiers de Royaumont_ , \u00e9d. de Minuit, 1957, p. 125.)\n\n13.  _TRE_ , 94 : \u00ab La voie correcte de l' _invention_ consiste \u00e0 former les pens\u00e9es en partant de quelque d\u00e9finition donn\u00e9e. \u00bb\n\n14.  _TRE_ , 19, \u00a7 III.\n\n15.  _TRE, 85._\n\n16.  _TRE_ , 58 : \u00ab L'esprit poss\u00e8de une puissance d'autant plus grande de former des fictions qu'il comprend moins..., et plus il comprend, plus cette puissance diminue. \u00bb En effet, plus l'esprit imagine, plus sa puissance de comprendre reste _envelopp\u00e9e_ , donc moins il comprend effectivement.\n\n17.  Cf. _Premi\u00e8res objections_ , AT, IX, p. 76 ; _Quatri\u00e8mes objections_ , AT, IX, pp. 162-166.\n\n18.  DESCARTES, _R\u00e9ponses aux premi\u00e8res objections_ , AT, IX, pp. 87-88 : Ceux qui ne s'attachent \u00ab qu'\u00e0 la propre et \u00e9troite signification d'efficient \u00bb, \u00ab ne remarquent ici aucun autre genre de cause qui ait _rapport et analogie_ avec la cause efficiente \u00bb. Ils ne remarquent pas que \u00ab il nous est tout-\u00e0-fait loisible de penser que (Dieu) fait _en quelque fa\u00e7on_ la m\u00eame chose \u00e0 l'\u00e9gard de soi-m\u00eame que la cause efficiente \u00e0 l'\u00e9gard de son effet. \u00bb _R\u00e9ponses aux quatri\u00e8mes objections_ , AT, IX, pp. 182-188 (\u00ab toutes ces mani\u00e8res de parler qui ont rapport et analogie avec la cause efficiente... \u00bb)\n\n19.  DESCARTES, _Principes_ , I, 51 (\u00ab Ce que c'est que la substance ; et que c'est un nom qu'on ne peut attribuer \u00e0 Dieu et aux cr\u00e9atures en m\u00eame sens \u00bb).\n\n20.  _\u00c9_ , I, 25, sc. Il est curieux que P. Lachi\u00e8ze-Rey, citant ce texte de Spinoza, en inverse l'ordre. Il fait comme si Spinoza avait dit que Dieu \u00e9tait cause de soi au sens o\u00f9 il \u00e9tait cause des choses. Dans la citation ainsi d\u00e9form\u00e9e, il n'y a pas un simple lapsus, mais la survivance d'une perspective \u00ab analogique \u00bb, invoquant _d'abord_ la causabilit\u00e9 efficiente. (Cf. _Les Origines cart\u00e9siennes du Dieu de Spinoza_ , pp. 33-34).\n\n21.  _\u00c9_ , I, 20, d\u00e9m.\n\n22.  _CT_ , I, ch. 2, 2, note 2.\n\n23.  _CT_ , I. ch. 2, 5, note 3.\n\n## CHAPITRE XI\n\n## L'IMMANENCE ET LES \u00c9L\u00c9MENTS HISTORIQUES\n\n## DE L'EXPRESSION\n\nDeux probl\u00e8mes se posent. Quels sont les liens logiques de l'immanence et de l'expression ? Et : Comment l'id\u00e9e d'une immanence expressive s'est-elle form\u00e9e historiquement dans certaines traditions philosophiques ? Il n'est pas exclu que ces traditions ne soient complexes et ne r\u00e9unissent elles-m\u00eames des inspirations tr\u00e8s diverses.\n\nTout commence, semble-t-il, avec le probl\u00e8me platonicien de la participation. Platon pr\u00e9sentait, \u00e0 titre d'hypoth\u00e8ses, plusieurs sch\u00e9mas de participation : participer, c'est prendre part ; mais aussi, c'est imiter ; et encore, c'est recevoir d'un d\u00e9mon... Suivant ces sch\u00e9mas, la participation se trouve interpr\u00e9t\u00e9e tant\u00f4t de fa\u00e7on mat\u00e9rielle, tant\u00f4t de mani\u00e8re imitative, tant\u00f4t de mani\u00e8re \u00ab d\u00e9monique \u00bb. Mais, dans tous les cas, les difficult\u00e9s semblent avoir une m\u00eame raison : chez Platon, le principe de participation est avant tout cherch\u00e9 du c\u00f4t\u00e9 du participant. La participation appara\u00eet le plus souvent comme une aventure qui survient du dehors au particip\u00e9, comme une violence subie par le particip\u00e9. Si la participation consiste \u00e0 prendre part, on voit mal comment le particip\u00e9 ne souffrirait pas d'une division ou d'une s\u00e9paration. Si participer, c'est imiter, il faut un artiste ext\u00e9rieur qui prend l'Id\u00e9e pour mod\u00e8le. Et l'on voit mal enfin quel est le r\u00f4le d'un interm\u00e9diaire en g\u00e9n\u00e9ral, artiste ou d\u00e9mon, sinon forcer le sensible \u00e0 reproduire l'intelligible, mais aussi forcer l'Id\u00e9e \u00e0 se laisser participer par quelque chose qui r\u00e9pugne \u00e0 sa nature. M\u00eame lorsque Platon traite de la participation des Id\u00e9es entre elles, la puissance correspondante est saisie comme puissance de participer, plut\u00f4t que d'\u00eatre particip\u00e9e.\n\nLa t\u00e2che post-platonicienne par excellence exige un renversement du probl\u00e8me. On cherche un principe qui rende possible la participation, mais qui la rende possible du point de vue du particip\u00e9 lui-m\u00eame. Les N\u00e9o-platoniciens ne partent plus des caract\u00e8res du participant (multiple, sensible, etc.) pour se demander sous quelle violence la participation devient possible. Ils tentent de d\u00e9couvrir, au contraire, le principe et le mouvement interne qui fondent la participation dans le particip\u00e9 comme tel, du c\u00f4t\u00e9 du particip\u00e9 comme tel. Plotin reproche \u00e0 Platon d'avoir vu la participation du petit c\u00f4t\u00e91. En v\u00e9rit\u00e9, ce n'est pas le particip\u00e9 qui passe dans le participant. Le particip\u00e9 reste en soi ; il est particip\u00e9 pour autant qu'il produit, il produit pour autant qu'il donne. Mais il n'a pas \u00e0 sortir de soi pour donner ni produire. Tel est le programme formul\u00e9 par Plotin : partir du plus haut, subordonner l'imitation \u00e0 une gen\u00e8se ou production, substituer l'id\u00e9e d'un _don_ \u00e0 celle d'une violence. Le particip\u00e9 ne se divise pas, n'est pas imit\u00e9 du dehors, ni contraint par des interm\u00e9diaires qui feraient violence \u00e0 sa nature. La participation n'est ni mat\u00e9rielle, ni imitative, ni d\u00e9monique : elle est \u00e9manative. \u00c9manation signifie \u00e0 la fois cause et don : causalit\u00e9 par donation, mais aussi donation productrice. La v\u00e9ritable activit\u00e9 est celle du particip\u00e9 ; le participant n'est qu'un effet, et re\u00e7oit ce que la cause lui donne. La cause \u00e9manative est la Cause qui donne, le Bien qui donne, la Vertu qui donne.\n\nQuand nous cherchons le principe interne de participation _du c\u00f4t\u00e9_ du particip\u00e9, nous devons n\u00e9cessairement le trouver \u00ab au-del\u00e0 \u00bb ou \u00ab au-dessus \u00bb. Il n'est pas question que le principe qui rend la participation possible soit lui-m\u00eame particip\u00e9 ou participable. De ce principe, tout \u00e9mane ; il donne tout. Mais il n'est pas lui-m\u00eame particip\u00e9, car la participation se fait seulement suivant ce qu'il donne, et \u00e0 ce qu'il donne. C'est en ce sens que Proclus \u00e9laborait sa profonde th\u00e9orie de l'Imparticipable ; il n'y a de participation que par un principe lui-m\u00eame imparticipable, mais qui donne \u00e0 participer. Et d\u00e9j\u00e0 Plotin montrait que l'Un est n\u00e9cessairement sup\u00e9rieur \u00e0 ses dons, qu'il donne ce qu'il n'a pas, ou qu'il n'est pas ce qu'il donne2. L'\u00e9manation en g\u00e9n\u00e9ral se pr\u00e9sentera sous forme d'une triade : le donateur, ce qui est donn\u00e9, ce qui re\u00e7oit. Participer, c'est toujours participer suivant ce qui est donn\u00e9. Donc nous ne devons pas seulement parler d'une gen\u00e8se du participant, mais d'une gen\u00e8se du particip\u00e9 lui-m\u00eame, qui rend compte de ce fait qu'il est particip\u00e9. Double gen\u00e8se, du donn\u00e9 et de ce qui re\u00e7oit : l'effet qui re\u00e7oit d\u00e9termine son existence quand il poss\u00e8de pleinement ce qui lui est donn\u00e9 ; mais il ne le poss\u00e8de pleinement qu'en se retournant vers le donateur. Le donateur est sup\u00e9rieur \u00e0 ses dons comme \u00e0 ses produits, participable d'apr\u00e8s ce qu'il donne, imparticipable en lui-m\u00eame ou selon lui-m\u00eame ; et, par l\u00e0, fondant la participation.\n\nNous pouvons d\u00e9j\u00e0 d\u00e9terminer des caract\u00e8res d'apr\u00e8s lesquels la cause \u00e9manative et la cause immanente ont logiquement quelque chose de commun, mais aussi des diff\u00e9rences profondes. Leur caract\u00e8re commun, c'est qu'elles ne sortent pas de soi : _elles restent en soi_ pour produire3. Quand Spinoza d\u00e9finit la cause immanente, il insiste sur cette d\u00e9finition qui fonde une certaine assimilation de l'immanence et de l'\u00e9manation4. Mais la diff\u00e9rence porte sur la mani\u00e8re dont les deux causes produisent. _Si la cause \u00e9manative reste en soi, l'effet produit n'est pas en elle et ne reste pas en elle._ De l'Un comme premier principe ou comme cause des causes, Plotin dit : \u00ab C'est parce que _rien n'est en lui_ que tout vient de lui5. \u00bb Quand il rappelle que l'effet n'est jamais s\u00e9par\u00e9 de la cause, il pense \u00e0 la continuit\u00e9 d'un flux et d'un rayonnement, non pas \u00e0 l'inh\u00e9rence actuelle d'un contenu. La cause \u00e9manative produit d'apr\u00e8s ce qu'elle donne, mais elle est au-del\u00e0 de ce qu'elle donne : si bien que l'effet sort de la cause, n'existe qu'en sortant de la cause, et ne d\u00e9termine son existence qu'en se retournant vers la cause dont il est sorti. C'est pourquoi la d\u00e9termination de l'existence de l'effet n'est pas s\u00e9parable d'une conversion, o\u00f9 la cause appara\u00eet comme Bien dans une perspective de finalit\u00e9 transcendante. Une cause est immanente, au contraire, quand l'effet lui-m\u00eame est \u00ab imman\u00e9 \u00bb dans la cause au lieu d'en \u00e9maner. Ce qui d\u00e9finit la cause immanente, c'est que l'effet est en elle, sans doute comme dans autre chose, mais est et reste en elle. L'effet ne reste pas moins dans la cause que la cause ne reste en elle-m\u00eame. De ce point de vue, jamais la distinction d'essence entre la cause et l'effet ne sera interpr\u00e9t\u00e9e comme une d\u00e9gradation. Du point de vue de l'immanence, la distinction d'essence n'exclut pas, mais implique une \u00e9galit\u00e9 d'\u00eatre : c'est le m\u00eame \u00eatre qui reste en soi dans la cause, mais aussi dans lequel l'effet reste comme dans autre chose.\n\nPlotin dit encore : l'Un n'a \u00ab rien de commun \u00bb avec les choses qui viennent apr\u00e8s lui6. Car la cause \u00e9manative n'est pas seulement sup\u00e9rieure \u00e0 l'effet, mais \u00e0 ce qu'elle donne \u00e0 l'effet. Mais pourquoi la cause premi\u00e8re est-elle pr\u00e9cis\u00e9ment l'Un ? Donnant l'\u00eatre \u00e0 tout ce qui est, elle est n\u00e9cessairement au-del\u00e0 de l'\u00eatre ou de la substance. Aussi l'\u00e9manation, dans son \u00e9tat pur, n'est-elle pas s\u00e9parable d'un syst\u00e8me de l'Unsup\u00e9rieur \u00e0 l'\u00eatre ; la premi\u00e8re hypoth\u00e8se du _Parm\u00e9nide_ domine tout le n\u00e9o-platonisme7. Et l'\u00e9manation n'est pas davantage s\u00e9parable d'une th\u00e9ologie n\u00e9gative, ou d'une m\u00e9thode d'analogie qui respecte l'\u00e9minence du principe ou de la cause. Proclus montre que, dans le cas de l'Un lui-m\u00eame, la n\u00e9gation est g\u00e9n\u00e9ratrice des affirmations qui s'appliquent \u00e0 ce que l'un donne et \u00e0 ce qui proc\u00e8de de l'Un. Bien plus, \u00e0 chaque stade de l'\u00e9manation, on doit reconna\u00eetre la pr\u00e9sence d'un imparticipable dont les choses proc\u00e8dent et auquel elles se convertissent. L'\u00e9manation sert donc de principe \u00e0 un univers hi\u00e9rarchis\u00e9 ; la diff\u00e9rence des \u00eatres en g\u00e9n\u00e9ral y est con\u00e7ue comme diff\u00e9rence hi\u00e9rarchique ; chaque terme est comme l'image du terme sup\u00e9rieur qui le pr\u00e9c\u00e8de, et se d\u00e9finit par le degr\u00e9 d'\u00e9loignement qui le s\u00e9pare de la cause premi\u00e8re ou du premier principe.\n\nEntre la cause \u00e9manative et la cause immanente, une seconde diff\u00e9rence appara\u00eet donc. L'immanence implique pour son compte une pure ontologie, une th\u00e9orie de l'\u00catre o\u00f9 l'Un n'est que la propri\u00e9t\u00e9 de la substance et de ce qui est. Et encore, l'immanence \u00e0 l'\u00e9tat pur exige le principe d'une \u00e9galit\u00e9 de l'\u00eatre ou la position d'un \u00catre-\u00e9gal : non seulement l'\u00eatre est \u00e9gal en soi, mais l'\u00eatre appara\u00eet \u00e9galement pr\u00e9sent dans tous les \u00eatres. Et la Cause, \u00e9galement proche partout : il n'y a pas de cause \u00e9loign\u00e9e. Les \u00eatres ne sont pas d\u00e9finis par leur rang dans une hi\u00e9rarchie, ne sont pas plus ou moins \u00e9loign\u00e9s de l'Un, mais chacun d\u00e9pend directement de Dieu, participant \u00e0 l'\u00e9galit\u00e9 de l'\u00eatre, recevant imm\u00e9diatement tout ce qu'il peut en recevoir d'apr\u00e8s l'aptitude de son essence, ind\u00e9pendamment de toute proximit\u00e9 et de tout \u00e9loignement. Bien plus, l'immanence \u00e0 l'\u00e9tat pur exige un \u00catre univoque qui forme une Nature, et qui consiste en formes positives, communes au producteur et au produit, \u00e0 la cause et \u00e0 l'effet. Nous savons que l'immanence ne supprime pas la distinction des essences ; mais il faut des formes communes qui constituent l'essence de la substance comme cause, alors qu'elles contiennent les essences de modes en tant qu'effets. C'est pourquoi la sup\u00e9riorit\u00e9 de la cause subsiste du point de vue de l'immanence, mais n'entra\u00eene aucune \u00e9minence, c'est-\u00e0-dire aucune position d'un principe au-del\u00e0 des formes qui sont elles-m\u00eames pr\u00e9sentes dans l'effet. L'immanence s'oppose \u00e0 toute \u00e9minence de la cause, \u00e0 toute th\u00e9ologie n\u00e9gative, \u00e0 toute m\u00e9thode d'analogie, \u00e0 toute conception hi\u00e9rarchique du monde. Tout est affirmation dans l'immanence. La Cause est sup\u00e9rieure \u00e0 l'effet, mais non pas sup\u00e9rieure \u00e0 ce qu'elle donne \u00e0 l'effet. Ou plut\u00f4t, elle ne \u00ab donne \u00bb rien \u00e0 l'effet. La participation doit \u00eatre pens\u00e9e de mani\u00e8re enti\u00e8rement positive, non pas \u00e0 partir d'un don \u00e9minent, mais \u00e0 partir d'une communaut\u00e9 formelle qui laisse subsister la distinction des essences.\n\nS'il y a tant de diff\u00e9rence entre l'\u00e9manation et l'immanence, comment peut-on les assimiler historiquement, ne serait-ce que de fa\u00e7on partielle ? C'est que, dans le n\u00e9o-platonisme lui-m\u00eame, et sous des influences sto\u00efciennes, une cause v\u00e9ritablement immanente se joint en fait \u00e0 la cause \u00e9manative8. D\u00e9j\u00e0 au niveau de l'Un, la m\u00e9taphore de la sph\u00e8re et du rayonnement corrigent singuli\u00e8rement la stricte th\u00e9orie de la hi\u00e9rarchie. Mais surtout, la premi\u00e8re \u00e9manation nous donne l'id\u00e9e d'une cause immanente. De l'Un, \u00e9mane l'Intelligence ou l'\u00catre ; or il n'y a pas seulement immanence mutuelle de l'\u00eatre et de l'intelligence, mais l'intelligence contient toutes les intelligences et tous les intelligibles, comme l'\u00eatre contient tous les \u00eatres et tous les genres d'\u00eatre. \u00ab Pleine des \u00eatres qu'elle a engendr\u00e9s, l'intelligence les engloutit en quelque sorte en les retenant en elle-m\u00eame9. \u00bb Sans doute, de l'intelligence \u00e0 son tour, \u00e9mane une nouvelle hypostase. Mais l'intelligence n'agit ainsi comme cause \u00e9manative que dans la mesure o\u00f9 elle atteint son point de perfection ; et elle l'atteint seulement comme cause immanente. L'\u00eatre et l'intelligence sont encore l'Un, mais l'Un qui est et qui conna\u00eet, l'Un de la seconde hypoth\u00e8se du _Parm\u00e9nide_ , Un dans lequel le multiple est pr\u00e9sent et qui, lui-m\u00eame, est pr\u00e9sent dans le multiple. Plotin montre que l'\u00eatre est identique au nombre \u00e0 l'\u00e9tat d'union, les \u00eatres, identiques au nombre \u00e0 l'\u00e9tat de d\u00e9veloppement (c'est-\u00e0-dire au nombre \u00ab expliqu\u00e9 \u00bb)10. D\u00e9j\u00e0 chez Plotin, il y a une \u00e9galit\u00e9 de l'\u00catre qui se conjugue avec la sur\u00e9minence de l'Un11. Damascius pousse tr\u00e8s loin la description de cet \u00e9tat de l'\u00catre, o\u00f9 le multiple est ramass\u00e9, concentr\u00e9, _compris_ dans l'Un, mais aussi o\u00f9 l'Un _s'explique_ dans les plusieurs.\n\nTelle est l'origine d'un couple de notions qui prendra une importance de plus en plus grande \u00e0 travers les philosophies du Moyen \u00c2ge et de la Renaissance : _complicare-explicare_12. Toutes choses sont pr\u00e9sentes \u00e0 Dieu qui les complique, Dieu est pr\u00e9sent \u00e0 toutes choses qui l'expliquent et l'impliquent. \u00c0 la s\u00e9rie des \u00e9manations successives et subordonn\u00e9es se substitue la co-pr\u00e9sence de deux mouvements corr\u00e9latifs. Car les choses ne restent pas moins en Dieu, en tant qu'elles l'expliquent ou l'impliquent, que Dieu ne reste en soi pour compliquer les choses. La pr\u00e9sence des choses \u00e0 Dieu constitue l'inh\u00e9rence, comme la pr\u00e9sence de Dieu aux choses constitue l'implication. \u00c0 la hi\u00e9rarchie des hypostases se substitue l'\u00e9galit\u00e9 de l'\u00eatre ; car c'est le m\u00eame \u00eatre auquel les choses sont pr\u00e9sentes et qui, lui-m\u00eame, est pr\u00e9sent dans les choses. L'immanence se d\u00e9finit par l'ensemble de la complication et de l'explication, de l'inh\u00e9rence et de l'implication. Les choses restent inh\u00e9rentes au Dieu qui les complique, comme Dieu reste impliqu\u00e9 par les choses qui l'expliquent. C'est Dieu compliquant qui s'explique \u00e0 travers toutes choses : \u00ab Dieu est la complication universelle, en ce sens que tout est en lui ; et l'universelle explication, en ce sens qu'il est en tout13. \u00bb La participation trouve son principe, non plus dans une \u00e9manation dont l'Un serait la source plus ou moins proche, mais dans l'expression imm\u00e9diate et ad\u00e9quate d'un \u00catre absolu qui comprend tous les \u00eatres et s'explique par l'essence de chacun. L'expression comprend tous ces aspects : complication, explication, inh\u00e9rence, implication. Ces aspects de l'expression sont aussi les cat\u00e9gories de l'immanence ; l'immanence se r\u00e9v\u00e8le expressive, l'expression immanente, dans un syst\u00e8me de relations logiques o\u00f9 les deux notions sont corr\u00e9latives.\n\nDe ce point de vue, l'id\u00e9e d'expression rend compte de la v\u00e9ritable activit\u00e9 du particip\u00e9, et de la possibilit\u00e9 de la participation. C'est dans l'id\u00e9e d'expression que le nouveau principe d'immanence s'affirme. L'expression appara\u00eet comme l'unit\u00e9 du multiple, comme la complication du multiple et l'explication de l'Un. Dieu s'exprime lui-m\u00eame dans le monde ; le monde est l'expression, l'explication d'un Dieu-\u00eatre ou de l'Un qui est. Le monde est promu en Dieu, de telle mani\u00e8re qu'il perd ses limites ou sa finitude, et participe imm\u00e9diatement de l'infinit\u00e9 divine. La m\u00e9taphore du cercle dont le centre est partout et la circonf\u00e9rence nulle part convient au monde lui-m\u00eame. Entre Dieu et le monde le rapport d'expression fonde, non pas une identit\u00e9 d'essence, mais une \u00e9galit\u00e9 d'\u00eatre. Car le m\u00eame \u00eatre est pr\u00e9sent en Dieu qui complique toutes choses suivant sa propre essence, et dans les choses qui l'expliquent suivant leur propre essence ou leur mode. Si bien que Dieu doit \u00eatre d\u00e9fini comme identique \u00e0 la Nature _complicative_ , et la Nature comme identique \u00e0 Dieu _explicative_. Mais cette \u00e9galit\u00e9, ou identit\u00e9 dans la distinction, constitue deux moments pour l'ensemble de l'expression : Dieu s'exprime dans son Verbe, son Verbe exprime l'essence divine ; mais le Verbe s'exprime \u00e0 son tour dans l'univers, l'univers exprimant toutes choses suivant le mode qui revient \u00e0 chacune essentiellement. Le Verbe est l'expression de Dieu, expression-langage ; l'Univers est l'expression de cette expression, expression-figure ou physionomie. (Ce th\u00e8me classique d'une double expression se retrouve chez Eckhart : Dieu s'exprime dans le Verbe, qui est parole int\u00e9rieure et silencieuse ; le Verbe s'exprime dans le monde, qui est figure ou parole ext\u00e9rioris\u00e9e14.)\n\nNous avons essay\u00e9 de montrer comment une immanence expressive de l'\u00catre se greffait sur la transcendance \u00e9manative de l'Un. Toutefois, chez Plotin et ses successeurs, cette cause immanente reste subordonn\u00e9e \u00e0 la cause \u00e9manative. Il est vrai que l'\u00eatre ou l'intelligence \u00ab s'expliquent \u00bb ; mais seul s'explique ce qui est d\u00e9j\u00e0 multiple, et qui n'est pas premier principe. \u00ab L'intelligence s'explique. C'est qu'elle veut poss\u00e9der tous les \u00eatres ; mais il e\u00fbt \u00e9t\u00e9 meilleur pour elle de ne pas le vouloir, car ainsi elle devient second principe15. \u00bb L'\u00eatre immanent, la pens\u00e9e immanente ne peuvent former un absolu mais supposent un premier principe, cause \u00e9manative et fin transcendante dont tout d\u00e9coule et \u00e0 quoi tout se convertit. Sans doute ce premier principe, l'Un sup\u00e9rieur \u00e0 l'\u00eatre, contient-il virtuellement toutes choses : _il est expliqu\u00e9_ , mais il _ne s'explique pas lui-m\u00eame_ , contrairement \u00e0 l'intelligence, contrairement \u00e0 l'\u00eatre16. Il n'est pas affect\u00e9 par ce qui l'exprime. Aussi faut-il attendre l'extr\u00eame \u00e9volution du n\u00e9o-platonisme pendant le Moyen \u00c2ge, la Renaissance et la R\u00e9forme, pour voir la cause immanente prendre une importance de plus en plus grande, l'\u00catre rivaliser avec l'un, l'expression rivaliser avec l'\u00e9manation et parfois tendre \u00e0 la supplanter. On a souvent cherch\u00e9 ce qui faisait de la philosophie de la Renaissance une philosophie moderne ; nous suivons pleinement la th\u00e8se d'Alexandre Koyr\u00e9, pour qui la cat\u00e9gorie sp\u00e9cifique de l'expression caract\u00e9rise le mode de penser de cette philosophie.\n\nEt pourtant il est certain que cette tendance expressionniste n'aboutit pas pleinement. C'est le christianisme qui la favorise, par sa th\u00e9orie du Verbe, et surtout par ses exigences ontologiques qui font du premier principe un \u00catre. Mais c'est lui qui la refoule, par l'exigence encore plus puissante de maintenir la transcendance de l'\u00eatre divin. Aussi voit-on toujours l'accusation d'immanence et de panth\u00e9isme menacer les philosophes, et les philosophes se soucier avant tout d'\u00e9chapper \u00e0 cette accusation. D\u00e9j\u00e0 dans Scot \u00c9rig\u00e8ne il faut admirer les combinaisons philosophiquement subtiles o\u00f9 se trouvent concili\u00e9s les droits d'une immanence expressive, d'une transcendance \u00e9manative et d'une cr\u00e9ation exemplaire _ex nihilo_. En fait, la transcendance d'un Dieu cr\u00e9ateur est sauv\u00e9e gr\u00e2ce \u00e0 une conception analogique de l'\u00catre, ou du moins gr\u00e2ce \u00e0 une conception \u00e9minente de Dieu qui limite la port\u00e9e de l'\u00catre-\u00e9gal. Le principe d'\u00e9galit\u00e9 de l'\u00catre est lui-m\u00eame interpr\u00e9t\u00e9 de mani\u00e8re analogique ; toutes les ressources du symbolisme pr\u00e9servent la transcendance. L'inexprimable est donc maintenu au sein de l'expression elle-m\u00eame. Non pas qu'on revienne \u00e0 Plotin ; non pas qu'on revienne \u00e0 la position de l'Un ineffable et sup\u00e9rieur \u00e0 l'\u00catre. Car c'est le m\u00eame Dieu, le m\u00eame \u00eatre infini, qui s'affirme et s'exprime dans le monde comme cause immanente, et qui reste inexprimable et transcendant comme objet d'une th\u00e9ologie n\u00e9gative qui nie de lui tout ce qu'on affirmait de son immanence. Or, m\u00eame dans ces conditions, l'immanence appara\u00eet comme une th\u00e9orie-limite, corrig\u00e9e par les perspectives de l'\u00e9manation et de la cr\u00e9ation. La raison en est simple : l'immanence expressive ne peut se suffire \u00e0 elle-m\u00eame tant qu'elle ne s'accompagne pas d'une pleine conception de l'univocit\u00e9, d'une pleine affirmation de l'\u00catre univoque.\n\nL'immanence expressive vient se greffer sur le th\u00e8me de l'\u00e9manation, qui la favorise en partie, qui la refoule en partie. Mais elle n'interf\u00e8re pas moins, dans des conditions analogues, avec le th\u00e8me de la cr\u00e9ation. La Cr\u00e9ation, sous un de ses aspects, semble r\u00e9pondre au m\u00eame souci que l'\u00c9manation ; il s'agit toujours de trouver un principe de participation du c\u00f4t\u00e9 du particip\u00e9 lui-m\u00eame. On met les Id\u00e9es en Dieu : au lieu de renvoyer \u00e0 une instance inf\u00e9rieure qui les prendrait pour mod\u00e8les ou les forcerait \u00e0 descendre dans le sensible, elles ont elles-m\u00eames une valeur exemplaire. Repr\u00e9sentant l'\u00eatre infini de Dieu, elles repr\u00e9sentent aussi tout ce que Dieu veut faire et peut faire. Les Id\u00e9es en Dieu _sont des similitudes exemplaires_ ; les choses cr\u00e9\u00e9es _ex nihilo_ sont des _similitudes imitatives_. La participation est une imitation, mais le principe de l'imitation se trouve du c\u00f4t\u00e9 du mod\u00e8le ou de l'imit\u00e9 : les Id\u00e9es ne se distinguent pas par rapport \u00e0 Dieu, mais se distinguent par rapport aux choses dont elles fondent la participation possible \u00e0 Dieu lui-m\u00eame. (Malebranche d\u00e9finira les Id\u00e9es en Dieu comme principes d'expression, repr\u00e9sentant Dieu participable ou imitable.)\n\nCette voie fut trac\u00e9e par saint Augustin. Or, l\u00e0 encore, le concept d'expression surgit pour d\u00e9terminer \u00e0 la fois le statut de la similitude exemplaire et de la similitude imitative. Saint Bonaventure, \u00e0 la suite de saint Augustin, est celui qui donne le plus d'importance \u00e0 cette double d\u00e9termination : les deux similitudes forment l'ensemble concret de la similitude \u00ab expressive \u00bb. Dieu s'exprime dans son Verbe ou dans l'Id\u00e9e exemplaire ; mais l'Id\u00e9e exemplaire exprime la multiplicit\u00e9 des choses cr\u00e9ables et cr\u00e9\u00e9es. Tel est le paradoxe de l'expression comme telle : intrins\u00e8que et \u00e9ternelle, elle est une par rapport \u00e0 ce qui s'exprime, multiple par rapport \u00e0 l'exprim\u00e917. L'expression est comme une radiation qui nous conduit de Dieu, qui s'exprime, aux choses exprim\u00e9es. \u00c9tant elle-m\u00eame exprimante (et non exprim\u00e9e), elle s'\u00e9tend \u00e9galement \u00e0 tout, sans limitation, comme l'essence divine elle-m\u00eame. Nous retrouvons un principe d'\u00e9galit\u00e9 d'apr\u00e8s lequel saint Bonaventure nie toute hi\u00e9rarchie entre les Id\u00e9es telles qu'elles sont en Dieu. En effet, la th\u00e9orie d'une similitude expressive implique une certaine immanence. Les id\u00e9es sont en Dieu ; donc les choses sont en Dieu, d'apr\u00e8s leurs similitudes exemplaires. Mais ne faut-il pas encore que les choses elles-m\u00eames soient en Dieu, comme imitations ? N'y a-t-il pas une certaine inh\u00e9rence de la copie au mod\u00e8le18 ? On ne peut \u00e9chapper \u00e0 cette cons\u00e9quence qu'en maintenant une conception strictement analogique de l'\u00eatre. (Saint Bonaventure lui-m\u00eame oppose constamment la similitude expressive et la similitude univoque ou d'univocation.)\n\nLa plupart des auteurs invoqu\u00e9s pr\u00e9c\u00e9demment se rattachent aux deux traditions \u00e0 la fois : \u00e9manation et imitation, cause \u00e9manative et cause exemplaire, Pseudo-Denys et saint Augustin. Mais l'important, c'est que ces deux voies se rejoignent dans le concept d'expression. On le voit d\u00e9j\u00e0 chez Scot \u00c9rig\u00e8ne, qui forge une philosophie de l'expression tant\u00f4t \u00ab similitudinaire \u00bb, tant\u00f4t \u00ab \u00e9manative \u00bb. _L'\u00e9manation nous m\u00e8ne \u00e0 une expression-explication. La cr\u00e9ation nous m\u00e8ne \u00e0 une expression-similitude._ Et l'expression, en effet, a ce double aspect : d'une part elle est miroir, mod\u00e8le et ressemblance ; d'autre part germe, arbre et rameau. Mais jamais ces m\u00e9taphores n'aboutissent. L'id\u00e9e d'expression se trouve refoul\u00e9e d\u00e8s qu'elle est suscit\u00e9e. C'est que les th\u00e8mes de la cr\u00e9ation ou de l'\u00e9manation ne peuvent pas se passer d'un minimum de transcendance, qui emp\u00eache l'\u00ab expressionnisme \u00bb d'aller jusqu'au bout de l'immanence qu'il implique. L'immanence est pr\u00e9cis\u00e9ment le vertige philosophique, ins\u00e9parable du concept d'expression (double immanence de l'expression dans ce qui s'exprime, et de l'exprim\u00e9 dans l'expression).\n\nLa signification du spinozisme nous semble la suivante : affirmer l'immanence comme principe ; d\u00e9gager l'expression de toute subordination \u00e0 l'\u00e9gard d'une cause \u00e9manative ou exemplaire. _L'expression elle-m\u00eame cesse d'\u00e9maner, comme de ressembler_. Or, un tel r\u00e9sultat ne peut \u00eatre obtenu que dans une perspective d'univocit\u00e9. Dieu est cause de toutes choses au m\u00eame sens que cause de soi ; il produit comme il existe formellement, ou comme il se comprend objectivement. Il produit donc les choses dans les formes m\u00eames qui constituent sa propre essence, et les id\u00e9es dans l'id\u00e9e de sa propre essence. Mais les m\u00eames attributs qui constituent formellement l'essence de Dieu contiennent toutes les essences formelles de modes, l'id\u00e9e de l'essence de Dieu comprend toutes les essences objectives ou toutes les id\u00e9es. Les choses en g\u00e9n\u00e9ral sont des modes de l'\u00eatre divin, c'est-\u00e0-dire impliquent les m\u00eames attributs que ceux qui constituent la nature de cet \u00eatre. En ce sens, toute similitude est d'univocation, se d\u00e9finissant par la pr\u00e9sence d'une qualit\u00e9 commune \u00e0 la cause et \u00e0 l'effet. Les choses produites ne sont pas des imitations, pas plus que les id\u00e9es ne sont des mod\u00e8les. M\u00eame l'id\u00e9e de Dieu n'a rien d'exemplaire, \u00e9tant elle-m\u00eame produite dans son \u00eatre formel. Inversement, les id\u00e9es n'imitent pas les choses. Dans leur \u00eatre formel elles suivent de l'attribut pens\u00e9e ; et si elles sont repr\u00e9sentatives, c'est seulement dans la mesure o\u00f9 elles participent \u00e0 une puissance absolue de penser qui, par elle-m\u00eame est \u00e9gale \u00e0 la puissance absolue de produire ou d'agir. Ainsi toute similitude imitative ou exemplaire est exclue du rapport expressif. Dieu s'exprime dans les formes qui constituent son essence, comme dans l'Id\u00e9e qui r\u00e9fl\u00e9chit cette essence. L'expression se dit \u00e0 la fois de l'\u00eatre et du conna\u00eetre. Mais seul l'\u00eatre univoque, seule la connaissance univoque est expressive. La substance et les modes, la cause et les effets ne sont et ne sont connus que par les formes communes qui constituent actuellement l'essence de l'une, et qui contiennent actuellement l'essence des autres.\n\nC'est pourquoi Spinoza oppose deux domaines, toujours confondus dans les traditions pr\u00e9c\u00e9dentes : celui de l'expression, et de la connaissance expressive, seule ad\u00e9quate ; celui des signes, et de la connaissance par signes, par apophase ou par analogie. Spinoza distingue diff\u00e9rentes sortes de signes : des signes indicatifs, qui nous font conclure \u00e0 quelque chose d'apr\u00e8s l'\u00e9tat de notre corps ; des signes imp\u00e9ratifs, qui nous font saisir des lois comme lois morales ; des signes de r\u00e9v\u00e9lation, qui nous font eux-m\u00eames ob\u00e9ir et qui, tout au plus, nous d\u00e9couvrent certains \u00ab propres \u00bb de Dieu. Mais, de toutes fa\u00e7ons, la connaissance par signes n'est jamais expressive, et reste du premier genre. L'indication n'est pas une expression, mais un \u00e9tat confus d'enveloppement dans lequel l'id\u00e9e reste impuissante \u00e0 s'expliquer ou \u00e0 exprimer sa propre cause. L'imp\u00e9ratif n'est pas une expression, mais une impression confuse qui nous fait croire que les vraies expressions de Dieu, les lois de la nature, sont autant de commandements. La r\u00e9v\u00e9lation n'est pas une expression, mais une culture de l'inexprimable, une connaissance confuse et relative par laquelle nous pr\u00eatons \u00e0 Dieu des d\u00e9terminations analogues aux n\u00f4tres (Entendement, Volont\u00e9), quitte \u00e0 sauver la sup\u00e9riorit\u00e9 de Dieu dans une \u00e9minence en tous genres (l'Un sur\u00e9minent, etc.). Gr\u00e2ce \u00e0 l'univocit\u00e9, Spinoza donne un contenu positif \u00e0 l'id\u00e9e d'expression, l'opposant aux trois sortes de signes. L'opposition des expressions et des signes est une des th\u00e8ses fondamentales du spinozisme.\n\nEncore fallait-il lib\u00e9rer l'expression de toute trace d'\u00e9manation. Le n\u00e9o-platonisme tirait une partie de sa force de la th\u00e8se suivante : que la production ne se fait pas par composition (addition d'esp\u00e8ce au genre, r\u00e9ception d'une forme dans une mati\u00e8re), mais par distinction et diff\u00e9renciation. Mais le n\u00e9o-platonisme se trouvait pris dans des exigences diverses : il fallait que la distinction f\u00fbt produite \u00e0 partir de l'Indistinct ou de l'absolument Un, et pourtant qu'elle f\u00fbt actuelle ; il fallait qu'elle f\u00fbt actuelle, et pourtant non num\u00e9rique. Ces exigences expliquent les efforts du n\u00e9o-platonisme pour d\u00e9finir l'\u00e9tat de distinctions indistinctes, de divisions indivises, de pluralit\u00e9s implurifiables. Spinoza, au contraire, trouve une autre voie dans sa th\u00e9orie des distinctions. En rapport avec l'univocit\u00e9, l'id\u00e9e d'une distinction formelle, c'est-\u00e0-dire d'une distinction r\u00e9elle qui n'est pas et ne peut pas \u00eatre num\u00e9rique, lui permet de concilier imm\u00e9diatement l'unit\u00e9 ontologique de la substance avec la pluralit\u00e9 qualitative des attributs. Loin d'\u00e9maner d'une Unit\u00e9 \u00e9minente, les attributs r\u00e9ellement distincts constituent l'essence de la substance absolument une. La substance n'est pas comme l'Un dont proc\u00e9derait une distinction paradoxale ; les attributs ne sont pas des \u00e9manations. L'unit\u00e9 de la substance et la distinction des attributs sont des corr\u00e9latifs qui constituent l'expression dans son ensemble. La distinction des attributs ne fait qu'un avec la composition qualitative d'une substance ontologiquement une ; la substance se distingue en une infinit\u00e9 d'attributs, qui sont comme ses formes actuelles ou ses qualit\u00e9s composantes. Avant toute production, il y a donc une distinction, mais cette distinction est aussi bien composition de la substance elle-m\u00eame.\n\nIl est vrai que la production des modes se fait par diff\u00e9renciation. Mais il s'agit alors d'une diff\u00e9renciation purement quantitative. Si la distinction r\u00e9elle n'est jamais num\u00e9rique, inversement la distinction num\u00e9rique est essentiellement modale. Sans doute le nombre convient-il mieux aux \u00eatres de raison qu'aux modes eux-m\u00eames. Reste que la distinction modale est quantitative, m\u00eame si le nombre exprime mal la nature de cette quantit\u00e9. On le voit bien dans la conception spinoziste de la participation19. Les th\u00e9ories de l'\u00e9manation et de la cr\u00e9ation s'accordaient pour refuser \u00e0 la participation tout sens mat\u00e9riel. Au contraire, chez Spinoza, c'est le principe m\u00eame de la participation qui nous oblige \u00e0 interpr\u00e9ter celle-ci comme une participation mat\u00e9rielle et quantitative. Participer, c'est avoir part, \u00eatre une part. Les attributs sont comme des qualit\u00e9s dynamiques auxquelles correspond la puissance absolue de Dieu. Un mode dans son essence est toujours un certain degr\u00e9, une certaine quantit\u00e9 d'une qualit\u00e9. Par-l\u00e0 m\u00eame il est, dans l'attribut qui le contient, comme une partie de la puissance de Dieu. \u00c9tant des formes communes, les attributs sont les conditions sous lesquelles la substance poss\u00e8de une toute-puissance identique \u00e0 son essence, sous lesquelles aussi les modes poss\u00e8dent une partie de cette puissance identique \u00e0 leur essence. La puissance de Dieu s'exprime ou s'explique modalement, mais seulement par et dans cette diff\u00e9renciation quantitative. C'est pourquoi dans le spinozisme l'homme perd tous les privil\u00e8ges qu'il devait \u00e0 une pr\u00e9tendue qualit\u00e9 propre, et qui lui appartenaient seulement du point de vue d'une participation imitative. Les modes se distinguent quantitativement : chaque mode exprime ou explique l'essence de Dieu, pour autant que cette essence s'explique elle-m\u00eame par l'essence du mode, c'est-\u00e0-dire se divise suivant la quantit\u00e9 correspondant \u00e0 ce mode20.\n\nLes modes d'un m\u00eame attribut ne se distinguent pas par leur rang, par leur proximit\u00e9 ou leur \u00e9loignement de Dieu. Ils se distinguent quantitativement, par la quantit\u00e9 ou capacit\u00e9 de leur essence respective qui participe toujours directement \u00e0 la substance divine. Sans doute une hi\u00e9rarchie semble subsister chez Spinoza entre le mode infini imm\u00e9diat, le mode fini m\u00e9diat et les modes finis. Mais Spinoza rappelle constamment que _Dieu n'est jamais, \u00e0 proprement parler, cause \u00e9loign\u00e9e_21. Dieu, consid\u00e9r\u00e9 sous tel attribut, est cause prochaine du mode infini imm\u00e9diat correspondant. Quant au mode infini que Spinoza appelle m\u00e9diat, il d\u00e9coule de l'attribut d\u00e9j\u00e0 modifi\u00e9 ; _mais la premi\u00e8re modification n'intervient pas comme une cause interm\u00e9diaire dans un syst\u00e8me d'\u00e9manations_ , elle se pr\u00e9sente comme la modalit\u00e9 sous laquelle Dieu lui-m\u00eame produit en lui-m\u00eame la seconde modification. Si nous consid\u00e9rons les essences de modes finis, nous voyons qu'elles ne forment pas un syst\u00e8me hi\u00e9rarchique o\u00f9 les moins puissantes d\u00e9pendraient des plus puissantes, mais une collection actuellement infinie, un syst\u00e8me d'implications mutuelles o\u00f9 chaque essence convient avec toutes les autres, et o\u00f9 toutes les essences sont comprises dans la production de chacune. Ainsi Dieu produit directement chaque essence avec toutes les autres. Enfin, les modes existants eux-m\u00eames ont Dieu pour cause directe. Sans doute un mode existant fini renvoie-t-il \u00e0 autre chose que l'attribut ; il trouve une cause dans un autre mode existant ; celui-ci \u00e0 son tour dans un autre, \u00e0 l'infini. Mais, pour chaque mode, Dieu est la puissance qui d\u00e9termine la cause \u00e0 avoir tel effet. Jamais nous n'entrons dans une r\u00e9gression \u00e0 l'infini ; il suffit de consid\u00e9rer un mode avec sa cause pour arriver directement \u00e0 Dieu comme au principe qui d\u00e9termine cette cause \u00e0 avoir tel effet. C'est en ce sens que Dieu n'est jamais cause \u00e9loign\u00e9e, m\u00eame des modes existants. D'o\u00f9 la c\u00e9l\u00e8bre formule spinoziste \u00ab en tant que... \u00bb. C'est toujours Dieu qui produit directement, mais sous des modalit\u00e9s diverses : en tant qu'il est infini, en tant qu'il est modifi\u00e9 d'une modification elle-m\u00eame infinie ; en tant qu'il est affect\u00e9 d'une modification particuli\u00e8re. \u00c0 la hi\u00e9rarchie des \u00e9manations se substitue une hi\u00e9rarchie des modalit\u00e9s en Dieu lui-m\u00eame ; mais, sous chaque modalit\u00e9, Dieu s'exprime imm\u00e9diatement, ou produit directement ses effets. C'est pourquoi tout effet est en Dieu et reste en Dieu, c'est pourquoi Dieu lui-m\u00eame est pr\u00e9sent dans chacun de ses effets.\n\n _La substance s'exprime d'abord en soi-m\u00eame_. Cette premi\u00e8re expression est formelle ou qualitative. La substance s'exprime dans les attributs formellement distincts, qualitativement distincts, r\u00e9ellement distincts ; chaque attribut exprime l'essence de la substance. On retrouve ici le double mouvement de la complication et de l'explication : la substance \u00ab complique \u00bb les attributs, chaque attribut explique l'essence de la substance, la substance s'explique par tous les attributs. Cette premi\u00e8re expression, avant toute production, est comme la constitution de la substance elle-m\u00eame. Appara\u00eet ici la premi\u00e8re application d'un principe d'\u00e9galit\u00e9 : non seulement la substance est \u00e9gale \u00e0 tous les attributs, mais tout attribut est \u00e9gal aux autres, aucun n'est sup\u00e9rieur ou inf\u00e9rieur. _La substance s'exprime pour soi-m\u00eame_. Elle s'exprime dans l'id\u00e9e de Dieu, qui comprend tous les attributs. Dieu ne s'exprime pas, ne s'explique pas sans se comprendre. Cette seconde expression est objective. Elle implique un nouvel usage du principe d'\u00e9galit\u00e9 : la puissance de penser, qui correspond \u00e0 l'id\u00e9e de Dieu, est \u00e9gale \u00e0 la puissance d'exister, qui correspond aux attributs. L'id\u00e9e de Dieu (le Fils ou le Verbe) a un statut complexe : objectivement \u00e9gale \u00e0 la substance, elle n'est qu'un produit dans son \u00eatre formel. Aussi nous conduit-elle \u00e0 une troisi\u00e8me expression : _La substance se r\u00e9-exprime, les attributs s'expriment \u00e0 leur tour dans les modes_. Cette expression est la production des modes eux-m\u00eames : Dieu produit comme il se comprend ; il ne se comprend pas sans produire une infinit\u00e9 de choses, sans comprendre aussi tout ce qu'il produit. Dieu produit dans ces m\u00eames attributs qui constituent son essence, il pense tout ce qu'il produit dans cette m\u00eame id\u00e9e qui comprend son essence. Ainsi tous les modes sont expressifs, et les id\u00e9es qui correspondent \u00e0 ces modes. Les attributs \u00ab compliquent \u00bb les essences de modes, et s'expliquent par elles, comme l'Id\u00e9e de Dieu comprend toutes les id\u00e9es et s'explique par elles. Cette troisi\u00e8me expression est quantitative. Aussi aura-t-elle deux formes, comme la quantit\u00e9 elle-m\u00eame : intensive dans les essences de modes, extensive quand les modes passent \u00e0 l'existence. Le principe d'\u00e9galit\u00e9 trouve ici sa derni\u00e8re application : non pas que les modes soient \u00e9gaux \u00e0 la substance elle-m\u00eame, mais la sup\u00e9riorit\u00e9 de la substance n'entra\u00eene aucune \u00e9minence. Les modes sont pr\u00e9cis\u00e9ment expressifs en tant qu'ils impliquent les m\u00eames formes qualitatives que celles qui constituent l'essence de la substance.\n\n* * *\n\n1.  PLOTIN, VI, 6, IV, \u00a7 2, 27-32 : \u00ab Nous [c'est-\u00e0-dire les Platoniciens], nous posons l'\u00eatre dans le sensible, puis nous mettons _l\u00e0-bas_ ce qui devrait \u00eatre partout ; alors, imaginant le sensible comme quelque chose de grand, nous nous demandons comment cette nature qui est l\u00e0-bas peut venir s'\u00e9tendre dans une chose aussi grande. Mais en fait, ce que l'on nomme grand est petit ; et ce que l'on croit petit est grand, puisque tout entier il arrive le premier pr\u00e8s de chaque partie du sensible... \u00bb Plotin souligne ici la n\u00e9cessit\u00e9 de renverser le probl\u00e8me platonicien, et de partir du particip\u00e9, et m\u00eame de ce qui fonde la participation dans le particip\u00e9.\n\n2.  Cf. PLOTIN, VI, 7, \u00a7 17, 3-6. La th\u00e9orie de l'Imparticipable, du donateur et du don, est constamment d\u00e9velopp\u00e9e et approfondie par Proclus et par Damascius, dans leurs commentaires du Parm\u00e9nide.\n\n3.  Sur la Cause ou Raison qui \u00ab reste en soi \u00bb pour produire, et sur l'importance de ce th\u00e8me chez Plotin, cf. R. ARNOU, _Praxis et Theoria_ , Alcan, 1921, pp. 8-12.\n\n4.  Le _Court Trait\u00e9_ d\u00e9finit la cause immanente comme agissant _en elle-m\u00eame_ (I, ch. 2, 24). Par l\u00e0 elle est semblable \u00e0 une cause \u00e9manative, et Spinoza rapproche les deux dans son \u00e9tude des cat\u00e9gories de la cause ( _CT_ , I, ch. 3, 2). M\u00eame dans l' _\u00c9thique_ , il emploiera _effluere_ pour indiquer la mani\u00e8re dont les modes suivent de la substance (I, 17, sc.) ; et dans la lettre 43, \u00e0 Osten (III, p. 161) _omnia necessario a Dei natura emanare._ Spinoza semble en retrait sur une distinction traditionnelle qu'il conna\u00eet bien : on dit que la cause immanente a une causalit\u00e9 qui se distingue de son existence, tandis que la causalit\u00e9 \u00e9manative ne se distingue pas de l'existence de la cause (cf. HEEREBOORD, _Meletemata philosophica_ , t. II, p. 229). Mais, pr\u00e9cis\u00e9ment, Spinoza ne peut pas accepter cette diff\u00e9rence-l\u00e0.\n\n5.  PLOTIN, V, II, \u00a7 1, 5.\n\n6.  PLOTIN, V, 5, \u00a7 4. Sans doute y a-t-il, selon Plotin, une forme commune \u00e0 toutes choses ; mais c'est une forme de finalit\u00e9, la forme du Bien, qui doit s'interpr\u00e9ter en un sens analogique.\n\n7.  Cf. \u00c9. GILSON, _L'\u00catre et l'essence_ , Vrin, 1948, p. 42 : \u00ab Dans une doctrine de l'\u00catre, l'inf\u00e9rieur n'est qu'en vertu de l'\u00eatre du sup\u00e9rieur. Dans une doctrine de l'Un, c'est au contraire un principe g\u00e9n\u00e9ral que l'inf\u00e9rieur n'est qu'en vertu de ce que le sup\u00e9rieur n'est pas ; en effet le sup\u00e9rieur ne donne jamais que ce qu'il n'a pas, puisque, pour pouvoir donner cette chose, il faut qu'il soit au-dessus d'elle. \u00bb\n\n8.  M. de GANDILLAC a analys\u00e9 ce th\u00e8me dans _La Philosophie de Nicolas de Cues_ , Aubier, 1942.\n\n9.  PLOTIN, V, 1, \u00a7 7, 30.\n\n10.  PLOTIN, VI, 6, \u00a7 9. Le terme _exelittein_ (expliquer, d\u00e9velopper) a une grande importance chez Plotin et ses successeurs, au niveau d'une th\u00e9orie de l'\u00catre et de l'Intelligence.\n\n11.  Cf. PLOTIN, VI, 2, \u00a7 11, 15 : \u00ab Une chose peut n'avoir pas moins d'\u00eatre qu'une autre, tout en ayant moins d'unit\u00e9. \u00bb\n\n12.  Bo\u00e8ce applique \u00e0 l'\u00catre \u00e9ternel les termes _comprehendere, complectiri_ (cf. _Consolation de la philosophie_ , prose 6). Le couple de substantifs _complicatio \u2013 explicatio_ , ou d'adverbes _complicative \u2013 explicative_ , prend une grande importance chez les commentateurs de Bo\u00e8ce, et notamment dans l'\u00e9cole de Chartres au XIIe si\u00e8cle. Mais c'est surtout avec Nicolas de Cues et avec Bruno que ces notions acqui\u00e8rent un statut philosophique rigoureux : cf. M. de GANDILLAC, _op. cit._\n\n13.  Nicolas de CUES, _Docte Ignorance_ , II, ch. 3.\n\n14.  Sur la cat\u00e9gorie de l'expression chez Eckhart, cf. LOSSKY, _Th\u00e9ologie n\u00e9gative et connaissance de Dieu chez ma\u00eetre Eckhart_ (Vrin, 1960).\n\n15.  PLOTIN, III, 8, \u00a7 8. Et V, 3 \u00a7 10 : \u00ab Ce qui s'explique est multiple \u00bb.\n\n16.  PLOTIN, VI, 8, \u00a7 18, 18 : \u00ab Le centre se manifeste \u00e0 travers les rayons, tel qu'il est, tel qu'il est expliqu\u00e9, mais sans s'expliquer soi-m\u00eame. \u00bb\n\n17.  Saint Bonaventure d\u00e9veloppe une triade de l'expression, qui comprend la V\u00e9rit\u00e9 qui s'exprime, la chose exprim\u00e9e, l'expression m\u00eame : _In hac autem expressione est tria intelligere, scilicet ipsam veritatem, ipsam expressionem et ipsam rem. Veritas exprimens una sola est et re et ratione ; ipsae autem res quae exprimuntur habent multiformitatem vel actualem vel possibilem ; expressio vero, secundum id quod est, nihil aliud est quam ipsa veritas ; sed secundum id ad quod est, tenet se ex parte rerum quae exprimuntur_ ( _De Scienta Christi_ , Opera omnia, V, 14 a). Sur les mots \u00ab exprimer \u00bb, \u00ab expression \u00bb chez saint Augustin et saint Bonaventure, cf. \u00c9. GILSON, _La Philosophie de saint Bonaventure_ (Vrin, 3e \u00e9d.), pp. 124-125.\n\n18.  C'est en ce sens que Nicolas de CUES remarque : \u00ab Il faut bien que l'image soit contenue dans son mod\u00e8le, sans quoi elle ne serait pas v\u00e9ritablement image... Le mod\u00e8le par cons\u00e9quent est dans toutes les images, et en lui sont toutes les images. Ainsi aucune image n'est ni plus ni moins que le mod\u00e8le. Et c'est pourquoi toutes les images sont images d'un unique mod\u00e8le \u00bb (\u00ab Le Jeu de la boule \u00bb, _\u0152uvres choisies_ , Aubier, p. 530).\n\n19.  Le mot et la notion de Participation (participation \u00e0 la nature de Dieu, \u00e0 la puissance de Dieu) forment un th\u00e8me constant de l' _\u00c9thique_ et des _Lettres._\n\n20.  Cf. _\u00c9_ , IV, 4, dem.\n\n21.  Chaque fois que Spinoza parle de \u00ab cause derni\u00e8re ou \u00e9loign\u00e9e \u00bb, il pr\u00e9cise que la formule ne doit pas \u00eatre prise \u00e0 la lettre : cf. _CT_ , I, ch. 3, 2 ; _\u00c9_ , I, 28, sc.\n\n# TROISI\u00c8ME PARTIE\n\n# TH\u00c9ORIE DU MODE FINI\n\n## CHAPITRE XII\n\n## L'ESSENCE DE MODE :\n\n## PASSAGE DE L'INFINI AU FINI\n\nOn retrouve chez Spinoza l'identit\u00e9 classique de l'attribut et de la qualit\u00e9. Les attributs sont des qualit\u00e9s \u00e9ternelles et infinies : c'est en ce sens qu'ils sont indivisibles. L'\u00e9tendue est indivisible, en tant que qualit\u00e9 substantielle ou attribut. Chaque attribut est indivisible en tant que qualit\u00e9. Mais aussi chaque attribut-qualit\u00e9 a une quantit\u00e9 infinie, qui, elle, est divisible sous certaines conditions. Cette quantit\u00e9 infinie d'un attribut forme une mati\u00e8re, mais une mati\u00e8re seulement modale. Un attribut se divise donc modalement, non pas r\u00e9ellement. Il a des parties qui se distinguent modalement : des parties modales, non pas r\u00e9elles ou substantielles. Ceci est valable pour l'\u00e9tendue comme pour les autres attributs : \u00ab N'y a-t-il pas des parties dans l'\u00e9tendue avant qu'il y ait des modes ? En aucune fa\u00e7on, dis-je1. \u00bb\n\nMais il appara\u00eet dans l' _\u00c9thique_ que le mot \u00ab partie \u00bb doit \u00eatre compris de deux fa\u00e7ons. Tant\u00f4t il s'agit de parties de puissance, c'est-\u00e0-dire de parties intrins\u00e8ques ou intensives, v\u00e9ritables degr\u00e9s, degr\u00e9s de puissance ou d'intensit\u00e9. Ainsi les essences de modes se d\u00e9finissent comme des degr\u00e9s de puissance (Spinoza retrouve une longue tradition scolastique, selon laquelle _modus intrinsecus = gradus = intensio_ )2. Mais, tant\u00f4t aussi, il s'agit de parties extrins\u00e8ques ou extensives, ext\u00e9rieures les unes aux autres, agissant du dehors les unes sur les autres. C'est ainsi que les corps les plus simples sont les ultimes divisions modales extensives de l'\u00e9tendue. (On \u00e9vitera de croire que l'extension soit un privil\u00e8ge de l'\u00e9tendue : les modes de l'\u00e9tendue se d\u00e9finissent essentiellement par des degr\u00e9s de puissance, et inversement un attribut comme la pens\u00e9e a lui-m\u00eame des parties modales extensives, des id\u00e9es qui correspondent aux corps les plus simples3.)\n\nTout se passe donc comme si chaque attribut \u00e9tait affect\u00e9 de deux quantit\u00e9s elles-m\u00eames infinies, mais divisibles sous certaines conditions, chacune \u00e0 sa mani\u00e8re : une quantit\u00e9 intensive, qui se divise en parties intensives ou en degr\u00e9s ; une quantit\u00e9 extensive, qui se divise en parties extensives. On ne s'\u00e9tonnera donc pas que, outre l'infini qualitatif des attributs qui se rapportent \u00e0 la substance, Spinoza fasse allusion \u00e0 deux infinis quantitatifs proprement modaux. Il \u00e9crit, dans la lettre \u00e0 Meyer : \u00ab Certaines choses sont (infinies) par la vertu de la cause dont elles d\u00e9pendent, et toutefois, quand on les con\u00e7oit abstraitement, elles peuvent \u00eatre divis\u00e9es en parties et \u00eatre consid\u00e9r\u00e9es comme finies ; _certaines autres_ enfin peuvent \u00eatre dites infinies ou, si vous pr\u00e9f\u00e9rez, ind\u00e9finies, parce qu'elles ne peuvent \u00eatre \u00e9gal\u00e9es par aucun nombre, bien qu'on puisse les concevoir comme plus grandes ou plus petites4. \u00bb Mais alors beaucoup de probl\u00e8mes se posent : En quoi consistent ces deux infinis ? Comment et sous quelles conditions se laissent-ils diviser en parties ? Quels sont leurs rapports, et quels sont les rapports de leurs parties respectives ?\n\nQu'est-ce que Spinoza appelle une essence de mode, essence particuli\u00e8re ou singuli\u00e8re ? Sa th\u00e8se se r\u00e9sume ainsi : les essences de modes ne sont ni des possibilit\u00e9s logiques, ni des structures math\u00e9matiques, ni des entit\u00e9s m\u00e9taphysiques, mais des r\u00e9alit\u00e9s physiques, des _res physicae_. Spinoza veut dire que l'essence, en tant qu'essence, a une existence. _Une essence de mode a une existence qui ne se confond pas avec l'existence du mode correspondant._ Une essence de mode existe, elle est r\u00e9elle et actuelle, m\u00eame si n'existe pas actuellement le mode dont elle est l'essence. D'o\u00f9 la conception que Spinoza se fait du mode non-existant : celui-ci n'est jamais quelque chose de possible, mais un objet dont l'id\u00e9e est n\u00e9cessairement comprise dans l'id\u00e9e de Dieu, comme son essence est n\u00e9cessairement contenue dans un attribut5. L'id\u00e9e d'un mode inexistant est donc le corr\u00e9lat objectif n\u00e9cessaire d'une essence de mode. Toute essence est essence de quelque chose ; une essence de mode est l'essence de quelque chose qui doit \u00eatre con\u00e7u dans l'entendement infini. De l'essence elle-m\u00eame, on ne dira pas qu'elle est un possible ; on ne dira pas davantage que le mode non-existant tende, en vertu de son essence, \u00e0 passer \u00e0 l'existence. Sur ces deux points, l'opposition est radicale entre Spinoza et Leibniz : chez Leibniz, l'essence ou la notion individuelle est une possibilit\u00e9 logique, et ne se s\u00e9pare pas d'une certaine r\u00e9alit\u00e9 m\u00e9taphysique, c'est-\u00e0-dire d'une \u00ab exigence d'existence \u00bb, d'une tendance \u00e0 l'existence6. Il n'en est pas ainsi chez Spinoza : l'essence n'est pas une possibilit\u00e9, mais poss\u00e8de une existence r\u00e9elle qui lui revient en propre ; le mode non-existant ne manque de rien et n'exige rien, mais est con\u00e7u dans l'entendement de Dieu comme le corr\u00e9lat de l'essence r\u00e9elle. Ni r\u00e9alit\u00e9 m\u00e9taphysique ni possibilit\u00e9 logique, _l'essence de mode est pure r\u00e9alit\u00e9 physique._\n\nC'est pourquoi les essences de modes n'ont pas moins une cause efficiente que les modes existants. \u00ab Dieu n'est pas seulement cause efficiente de l'existence des choses, mais encore de leur essence7. \u00bb Lorsque Spinoza montre que l'essence d'un mode n'enveloppe pas l'existence, certes il veut dire d'abord que l'essence n'est pas cause de l'existence du mode. Mais il veut dire aussi que l'essence n'est pas cause de sa propre existence8.\n\nNon pas qu'il y ait une distinction r\u00e9elle entre l'essence et _sa propre_ existence ; la distinction de l'essence et de l'existence est suffisamment fond\u00e9e d\u00e8s qu'on accorde que l'essence a une cause elle-m\u00eame distincte. Alors, en effet, l'essence existe n\u00e9cessairement, mais elle existe en vertu de sa cause (et non par soi). On reconna\u00eet ici le principe d'une th\u00e8se c\u00e9l\u00e8bre de Duns Scot et, plus lointainement, d'Avicenne : l'existence accompagne n\u00e9cessairement l'essence, mais en vertu de la cause de celle-ci ; elle n'est donc pas incluse ou envelopp\u00e9e dans l'essence ; elle s'y ajoute. Elle ne s'y ajoute pas comme un acte r\u00e9ellement distinct, mais seulement comme une sorte de d\u00e9termination ultime qui r\u00e9sulte de la cause de l'essence9. En un mot, l'essence a toujours l'existence qu'elle m\u00e9rite en vertu de sa cause. C'est pourquoi, chez Spinoza, s'unissent les deux propositions suivantes : _Les essences ont une existence ou r\u00e9alit\u00e9 physique ; Dieu est cause efficiente des essences._ L'existence de l'essence ne fait qu'un avec l'\u00eatre-caus\u00e9 de l'essence. Donc on ne confondra pas la th\u00e9orie spinoziste avec une th\u00e9orie cart\u00e9sienne en apparence analogue : lorsque Descartes dit que Dieu produit m\u00eame les essences, il veut dire que Dieu n'est assujetti \u00e0 aucune loi, qu'il cr\u00e9e tout, m\u00eame le possible. Spinoza veut dire au contraire que les essences ne sont pas des possibles, mais qu'elles ont une existence pleinement actuelle qui leur revient en vertu de leur cause. Les essences de modes ne peuvent \u00eatre assimil\u00e9es \u00e0 des possibles que dans la mesure o\u00f9 nous les consid\u00e9rons abstraitement, c'est-\u00e0-dire o\u00f9 nous les s\u00e9parons de la cause qui les pose comme des choses r\u00e9elles ou existantes.\n\nSi toutes les essences _conviennent_ , c'est pr\u00e9cis\u00e9ment parce qu'elles ne sont pas causes les unes des autres, mais, toutes ont Dieu pour cause. Quand nous les consid\u00e9rons concr\u00e8tement, les rapportant \u00e0 la cause dont elles d\u00e9pendent, nous les posons toutes ensemble, coexistantes et convenantes10. Toutes les essences conviennent par l'existence ou r\u00e9alit\u00e9 qui r\u00e9sulte de leur cause. Une essence ne peut \u00eatre s\u00e9par\u00e9e des autres qu'abstraitement, quand on la consid\u00e8re ind\u00e9pendamment du principe de production qui les comprend toutes. C'est pourquoi les essences forment un syst\u00e8me total, un ensemble actuellement infini. De cet ensemble on dira, comme dans la _Lettre \u00e0 Meyer_ , qu'il est infini par sa cause. Nous devons donc demander : Comment les essences de modes se distinguent-elles, elles qui sont ins\u00e9parables les unes des autres ? Comment sont-elles singuli\u00e8res, alors qu'elles forment un ensemble infini ? Ce qui revient \u00e0 demander : En quoi consiste la r\u00e9alit\u00e9 physique des essences en tant que telles ? On sait que ce probl\u00e8me, \u00e0 la fois de l'individualit\u00e9 et de la r\u00e9alit\u00e9, soul\u00e8ve beaucoup de difficult\u00e9s dans le spinozisme.\n\nIl ne semble pas que Spinoza ait eu d\u00e8s le d\u00e9but une solution claire, ni m\u00eame une claire position du probl\u00e8me. Deux textes c\u00e9l\u00e8bres du _Court Trait\u00e9_ soutiennent que, tant que les modes eux-m\u00eames n'existent pas, leurs essences ne se distinguent pas de l'attribut qui les contient, pas davantage ne se distinguent les unes des autres ; qu'elles n'ont donc en elles-m\u00eames aucun principe d'individualit\u00e911. L'individuation se ferait seulement par l'existence du mode, non par son essence. (Pourtant le _Court Trait\u00e9_ a d\u00e9j\u00e0 besoin de l'hypoth\u00e8se d'essences de modes singuli\u00e8res en elles-m\u00eames, et utilise pleinement cette hypoth\u00e8se.)\n\nMais peut-\u00eatre les deux textes du _Court Trait\u00e9_ sont-ils ambigus, plut\u00f4t qu'ils n'excluent radicalement toute singularit\u00e9 et toute distinction des essences en tant que telles. Car le premier texte semble dire ceci : Tant qu'un mode n'existe pas, son essence existe seulement comme contenue _dans_ l'attribut ; or l'id\u00e9e de l'essence ne peut pas avoir elle-m\u00eame une distinction qui ne serait pas dans la nature ; elle ne peut donc pas repr\u00e9senter le mode non-existant comme s'il se distinguait _de_ l'attribut et des autres modes. De m\u00eame le second texte : Tant qu'un mode n'existe pas, l'id\u00e9e de son essence ne peut pas envelopper une existence distincte ; tant que la muraille est toute blanche, on ne peut rien appr\u00e9hender qui se distingue d'elle ou qui se distingue en elle. (M\u00eame dans l' _\u00c9thique_ , cette th\u00e8se n'est pas abandonn\u00e9e : tant qu'un mode n'existe pas, son essence est contenue dans l'attribut, son id\u00e9e est comprise dans l'id\u00e9e de Dieu ; cette id\u00e9e ne peut donc pas envelopper une existence distincte, ni se distinguer des autres id\u00e9es)12.\n\nEn tout ceci, \u00ab se distinguer \u00bb s'oppose brutalement \u00e0 \u00ab \u00eatre contenu \u00bb. \u00c9tant seulement contenues dans l'attribut, les essences de modes ne s'en distinguent pas. _La distinction est donc prise au sens de distinction extrins\u00e8que._ L'argumentation est la suivante. Les essences de modes sont contenues dans l'attribut ; tant qu'un mode n'existe pas, aucune distinction extrins\u00e8que n'est possible entre son essence et l'attribut, ni entre son essence et les autres essences ; donc aucune id\u00e9e ne peut repr\u00e9senter ou appr\u00e9hender les essences de modes comme des parties extrins\u00e8ques de l'attribut, ni comme des parties ext\u00e9rieures les unes aux autres. Cette th\u00e8se peut para\u00eetre \u00e9trange, puisqu'elle suppose inversement que la distinction extrins\u00e8que ne r\u00e9pugne pas aux modes existants, et m\u00eame est exig\u00e9e par eux. Nous remettons \u00e0 plus tard l'analyse de ce point. Remarquons seulement que le mode existant a une dur\u00e9e ; et pour autant qu'il dure, il cesse d'\u00eatre simplement contenu dans l'attribut, comme son id\u00e9e cesse d'\u00eatre simplement comprise dans l'id\u00e9e de Dieu13. C'est par la dur\u00e9e (et aussi, dans le cas des modes de l'\u00e9tendue, par la figure et par le lieu) que les modes existants ont une individuation proprement extrins\u00e8que.\n\nTant que la muraille est blanche, aucune figure ne se distingue d'elle ni en elle. C'est-\u00e0-dire : dans cet \u00e9tat, la qualit\u00e9 n'est pas affect\u00e9e par quelque chose qui se distinguerait d'elle extrins\u00e8quement. Mais la question subsiste de savoir s'il n'y a pas un autre type de distinction modale, comme un principe intrins\u00e8que d'individuation. Bien plus, tout laisse penser qu'une individuation par l'existence du mode est insuffisante. Nous ne pouvons distinguer les choses existantes que dans la mesure o\u00f9 leurs essences sont suppos\u00e9es distinctes ; de m\u00eame, toute distinction extrins\u00e8que semble bien supposer une distinction intrins\u00e8que pr\u00e9alable. Il est donc probable qu'une essence de mode est singuli\u00e8re en elle-m\u00eame, m\u00eame quand le mode correspondant n'existe pas. Mais comment ? Revenons \u00e0 Duns Scot : la blancheur, dit-il, a des intensit\u00e9s variables ; celles-ci ne s'ajoutent pas \u00e0 la blancheur comme une chose \u00e0 une autre chose, comme une figure s'ajoute \u00e0 la muraille sur laquelle on la trace ; les degr\u00e9s d'intensit\u00e9 sont des d\u00e9terminations intrins\u00e8ques, des modes intrins\u00e8ques de la blancheur, qui reste univoquement la m\u00eame sous quelque modalit\u00e9 qu'on la consid\u00e8re14.\n\nIl semble bien qu'il en soit ainsi chez Spinoza : les essences de modes sont des modes intrins\u00e8ques ou des quantit\u00e9s intensives. L'attribut-qualit\u00e9 reste univoquement ce qu'il est, contenant tous les degr\u00e9s qui l'affectent sans en modifier la raison formelle ; les essences de modes se distinguent donc de l'attribut comme l'intensit\u00e9 de la qualit\u00e9, et se distinguent entre elles comme les divers degr\u00e9s d'intensit\u00e9. Nous pouvons penser que, sans d\u00e9velopper explicitement cette th\u00e9orie, Spinoza s'oriente vers l'id\u00e9e d'une distinction ou d'une singularit\u00e9 propre aux essences de modes en tant que telles. La diff\u00e9rence des \u00eatres (essences de modes) est \u00e0 la fois intrins\u00e8que et purement quantitative ; car la quantit\u00e9 dont il s'agit ici, c'est la quantit\u00e9 intensive. Seule une distinction quantitative des \u00eatres se concilie avec l'identit\u00e9 qualitative de l'absolu. Mais cette distinction quantitative n'est pas une apparence, c'est une diff\u00e9rence interne, une diff\u00e9rence d'intensit\u00e9. Si bien que chaque \u00eatre fini doit \u00eatre dit _exprimer l'absolu_ , suivant la quantit\u00e9 intensive qui en constitue l'essence, c'est-\u00e0-dire suivant son degr\u00e9 de puissance15. L'individuation chez Spinoza n'est ni qualitative ni extrins\u00e8que, elle est quantitative-intrins\u00e8que, intensive. En ce sens, il y a bien une distinction des essences de modes, \u00e0 la fois par rapport aux attributs qui les contiennent et les unes par rapport aux autres. Les essences de modes ne se distinguent pas de mani\u00e8re extrins\u00e8que, \u00e9tant contenues dans l'attribut ; elles n'en ont pas moins un type de distinction ou de singularit\u00e9 qui leur est propre, dans l'attribut qui les contient.\n\nLa quantit\u00e9 intensive est une quantit\u00e9 infinie, le syst\u00e8me des essences est une s\u00e9rie actuellement infinie. Il s'agit d'un infini \u00ab par la cause \u00bb. C'est en ce sens que l'attribut contient, c'est-\u00e0-dire complique toutes les essences de modes ; il les contient comme la s\u00e9rie infinie des degr\u00e9s qui correspondent \u00e0 sa quantit\u00e9 intensive. Or on voit bien que cet infini, en un sens, n'est pas divisible : on ne peut pas le diviser en parties extensives ou extrins\u00e8ques, sauf par abstraction. (Mais par abstraction, nous s\u00e9parons les essences de leur cause et de l'attribut qui les contient, nous les consid\u00e9rons comme de simples possibilit\u00e9s logiques, nous leur retirons toute r\u00e9alit\u00e9 physique). En v\u00e9rit\u00e9, les essences de modes sont donc ins\u00e9parables, elles se d\u00e9finissent par leur convenance totale. Mais elles n'en sont pas moins singuli\u00e8res ou particuli\u00e8res, et distinctes les unes des autres par une distinction intrins\u00e8que. Dans leur syst\u00e8me concret, toutes les essences sont comprises dans la production de chacune : non seulement les essences de degr\u00e9 inf\u00e9rieur, mais aussi de degr\u00e9 sup\u00e9rieur, puisque la s\u00e9rie est actuellement infinie. Pourtant, dans ce syst\u00e8me concret, chaque essence est produite comme un degr\u00e9 irr\u00e9ductible, n\u00e9cessairement appr\u00e9hend\u00e9 comme unit\u00e9 singuli\u00e8re. Tel est le syst\u00e8me de la \u00ab complication \u00bb des essences.\n\nLes essences de modes sont bien les parties d'une s\u00e9rie infinie. Mais en un sens tr\u00e8s sp\u00e9cial : parties intensives ou intrins\u00e8ques. On \u00e9vitera de donner des essences particuli\u00e8res spinozistes une interpr\u00e9tation leibnizienne. Les essences particuli\u00e8res ne sont pas des microcosmes. Elles ne sont pas toutes contenues dans chacune, mais toutes sont contenues dans la production de chacune. Une essence de mode est une _pars intensiva_ , non pas une _pars totalis_16. Comme telle, elle a un pouvoir expressif, mais ce pouvoir expressif doit \u00eatre compris d'une mani\u00e8re tr\u00e8s diff\u00e9rente de celle de Leibniz. Car le statut des essences de modes renvoie \u00e0 un probl\u00e8me proprement spinoziste, dans la perspective d'une substance absolument infinie. Ce probl\u00e8me est celui du passage de l'infini au fini. La substance est comme l'identit\u00e9 ontologique absolue de toutes les qualit\u00e9s, la puissance absolument infinie, puissance d'exister sous toutes les formes et de penser toutes les formes ; les attributs sont les formes ou qualit\u00e9s infinies, comme telles indivisibles. Le fini n'est donc ni substantiel ni qualitatif. Mais il n'est pas davantage apparence : il est modal, c'est-\u00e0-dire quantitatif. Chaque qualit\u00e9 substantielle a une quantit\u00e9 modale-intensive, elle-m\u00eame infinie, qui se divise actuellement en une infinit\u00e9 de modes intrins\u00e8ques. Ces modes intrins\u00e8ques, contenus tous ensemble dans l'attribut, sont les parties intensives de l'attribut lui-m\u00eame. Par-l\u00e0 m\u00eame, ils sont les parties de la puissance de Dieu, sous l'attribut qui les contient. C'est en ce sens, d\u00e9j\u00e0, que nous avions vu que les modes d'un attribut divin participaient n\u00e9cessairement \u00e0 la puissance de Dieu : leur essence m\u00eame est une partie de la puissance de Dieu, c'est-\u00e0-dire un degr\u00e9 de puissance ou partie intensive. L\u00e0 encore, la r\u00e9duction des cr\u00e9atures \u00e0 l'\u00e9tat de modes appara\u00eet comme la condition sous laquelle leur essence est puissance, c'est-\u00e0-dire partie irr\u00e9ductible de la puissance de Dieu. Ainsi les modes dans leur essence sont expressifs : ils expriment l'essence de Dieu, chacun selon le degr\u00e9 de puissance qui constitue son essence. L'individuation du fini chez Spinoza ne va pas du genre ou de l'esp\u00e8ce \u00e0 l'individu, du g\u00e9n\u00e9ral au particulier ; elle va de la qualit\u00e9 infinie \u00e0 la quantit\u00e9 correspondante, qui se divise en parties irr\u00e9ductibles, intrins\u00e8ques ou intensives.\n\n* * *\n\n1.  _CT_ , I, ch. 2, 19, note 6.\n\n2.  Le probl\u00e8me de l'intensit\u00e9 ou du degr\u00e9 joue un r\u00f4le important, notamment aux XIIIe et XIVe si\u00e8cles : une qualit\u00e9 peut-elle, sans changer de raison formelle ou d'essence, \u00eatre affect\u00e9e par des degr\u00e9s divers ? Et ces affections appartiennent-elles \u00e0 l'essence elle-m\u00eame, ou seulement \u00e0 l'existence ? La th\u00e9orie du mode intrins\u00e8que ou du degr\u00e9 est particuli\u00e8rement d\u00e9velopp\u00e9e dans le scotisme.\n\n3.  Cf. _\u00c9_ , II, 15, prop. et dem.\n\n4.  _Lettre 12, \u00e0 Meyer_ (III, p. 42).\n\n5.  _\u00c9_ , II, 8, prop. et cor. (Et I, 8, sc. 2 : nous avons des id\u00e9es vraies des modifications non-existantes, parce que \u00ab leur essence est comprise en une autre chose, de telle sorte qu'elles peuvent \u00eatre con\u00e7ues par cette chose \u00bb.)\n\n6.  LEIBNIZ, _De l'origine radicale des choses_ : \u00ab Il y a dans les choses possibles, c'est-\u00e0-dire dans la possibilit\u00e9 m\u00eame ou l'essence, quelque exigence d'existence ou, pour ainsi dire, quelque pr\u00e9tention \u00e0 l'existence, et pour le renfermer en un mot, l'essence tend d'elle-m\u00eame \u00e0 l'existence. \u00bb\n\n7.  _\u00c9_ , I, 26, prop.\n\n8.  En _\u00c9_ , I, 24, prop. et dem., Spinoza dit que \u00ab l'essence des choses produites par Dieu n'enveloppe pas l'existence \u00bb. C'est-\u00e0-dire : l'essence d'une chose n'enveloppe pas l'existence de cette chose. Mais, dans le corollaire de I, 24, il ajoute : \u00ab Soit que les choses existent, soit qu'elles n'existent pas, toutes les fois que nous portons notre attention sur leur essence, nous trouvons qu'elle n'enveloppe ni l'existence ni la dur\u00e9e ; par cons\u00e9quent _leur essence ne peut \u00eatre cause ni de sa propre existence ni de sa propre dur\u00e9e (neque suae existentiae neque suae durationis)_ \u00bb. Les traducteurs, semble-t-il, font un contresens \u00e9tonnant lorsqu'ils font dire \u00e0 Spinoza : \u00ab Par cons\u00e9quent leur essence (l'essence des choses) ne peut \u00eatre cause ni de _leur_ existence ni de _leur_ dur\u00e9e. \u00bb M\u00eame si cette version \u00e9tait possible, ce qu'elle n'est absolument pas, on ne comprendrait plus ce que le corollaire apporte de nouveau, par rapport \u00e0 la d\u00e9monstration. Sans doute ce contresens est-il provoqu\u00e9 par l'allusion \u00e0 la dur\u00e9e. Comment Spinoza pourrait-il parler de la \u00ab dur\u00e9e \u00bb de l'essence, puisque l'essence ne dure pas ? Mais, que l'essence ne dure pas, on ne le sait pas encore en I, 24. Et m\u00eame quand Spinoza l'aura dit, il lui arrivera encore d'employer ce mot dur\u00e9e d'une mani\u00e8re tr\u00e8s g\u00e9n\u00e9rale, en un sens litt\u00e9ralement inexact : cf. V, 20 sc. L'ensemble de I, 24, nous para\u00eet donc s'organiser ainsi : 1o) l'essence d'une chose produite n'est pas cause de l'existence de la chose (d\u00e9monstration) ; 2o) mais elle n'est pas davantage cause de sa propre existence en tant qu'essence (corollaire) ; 3o) _d'o\u00f9_ I, 25, Dieu est cause, m\u00eame de l'essence des choses.\n\n9.  Dans des pages d\u00e9finitives, \u00e0 propos d'Avicenne et de Duns Scot, \u00c9. Gilson a montr\u00e9 comment la distinction de l'essence et de l'existence n'\u00e9tait pas n\u00e9cessairement une distinction r\u00e9elle (cf. _L'\u00catre et l'essence_ , Vrin, 1948, p. 134, p. 159).\n\n10.  Sur la convenance des essences, cf. _\u00c9_ , I, 17, sc.\n\n11.  a) _CT_ , App. II, 1 : \u00ab Ces modes, en tant qu'ils n'existent pas r\u00e9ellement, sont n\u00e9anmoins tous compris dans leurs attributs ; et comme il n'y a entre les attributs aucune sorte d'in\u00e9galit\u00e9, non plus qu'entre les essences des modes, il ne peut y avoir dans l'Id\u00e9e aucune distinction, puisqu'elle ne serait pas dans la nature. Mais si quelques-uns de ces modes rev\u00eatent leur existence particuli\u00e8re et se distinguent ainsi en quelque mani\u00e8re de leurs attributs (parce que l'existence particuli\u00e8re qu'ils ont dans l'attribut est alors sujet de leur essence), alors une distinction se produit entre les essences des modes, et par suite aussi entre leurs essences objectives qui sont n\u00e9cessairement contenues dans l'Id\u00e9e. \u00bb\n\nb) _CT_ , II, ch. 20, 4, note 3 : \u00ab En tant que, en d\u00e9signant une chose, on con\u00e7oit l'essence sans l'existence, l'id\u00e9e de l'essence ne peut \u00eatre consid\u00e9r\u00e9e comme quelque chose de particulier ; cela est possible seulement quand l'existence est donn\u00e9e avec l'essence, et cela parce que, alors, existe un objet qui auparavant n'existait pas. Si par exemple la muraille est toute blanche, on ne distingue en elle ni ceci ni cela. \u00bb\n\n12.  _\u00c9_ , II, 8, prop. et sc.\n\n13.  _\u00c9_ , II, 8, cor. : \u00ab Lorsqu'on dit que des choses singuli\u00e8res existent, non seulement en tant qu'elles sont comprises dans les attributs de Dieu, mais encore en tant qu'elles sont dites durer, leurs id\u00e9es aussi enveloppent l'existence par laquelle on dit qu'elles durent. \u00bb (Et II, 8, sc. : Lorsqu'on trace effectivement certains des c\u00f4t\u00e9s d'angles droits compris dans le cercle, \u00ab alors leurs id\u00e9es existent aussi, non seulement en tant qu'elles sont comprises dans l'id\u00e9e du cercle, mais encore en tant qu'elles enveloppent l'existence de ces c\u00f4t\u00e9s d'angles droits ; ce qui fait qu'elles se distinguent des autres id\u00e9es des autres c\u00f4t\u00e9s d'angles droits. \u00bb\n\n14.  Cf. Duns SCOT, _Opus exoniense_ , I, D3, q. 1 et 2, a. 4, n. 17. Le rapprochement de Spinoza avec Duns Scot ne porte ici que sur le th\u00e8me des quantit\u00e9s intensives ou des degr\u00e9s. La th\u00e9orie de l'individuation, que nous exposons dans le paragraphe suivant comme celle de Spinoza, est tout \u00e0 fait diff\u00e9rente de celle de Duns Scot.\n\n15.  On trouverait chez Fichte et chez Schelling un probl\u00e8me analogue, de la _diff\u00e9rence quantitative et de la forme de quantitabilit\u00e9_ dans leurs rapports avec la _manifestation_ de l'absolu (cf. Lettre de Fichte \u00e0 Schelling, octobre 1801, _Fichte's Leben_ II, Zweite Abth. IV, 28, p. 357).\n\n16.  On a parfois donn\u00e9, des essences selon Spinoza, une interpr\u00e9tation exag\u00e9r\u00e9ment leibnizienne. Ainsi HUAN, _Le Dieu de Spinoza_ , 1914, p. 277 : les essences \u00ab embrassent chacune d'un point de vue particulier l'infinit\u00e9 du r\u00e9el et pr\u00e9sentent dans leur nature intime une image microscopique de l'Univers entier. \u00bb\n\n## CHAPITRE XIII\n\n## L'EXISTENCE DU MODE\n\nNous savons que l'existence d'une essence de mode n'est pas l'existence du mode correspondant. Une essence de mode existe, sans que le mode lui-m\u00eame existe : l'essence n'est pas cause de l'existence du mode. L'existence du mode a donc pour cause un autre mode, lui-m\u00eame existant1. Mais cette r\u00e9gression \u00e0 l'infini ne nous dit nullement en quoi consiste l'existence. Toutefois, s'il est vrai qu'un mode existant \u00ab a besoin \u00bb d'un grand nombre d'autres modes existants, nous pouvons d\u00e9j\u00e0 pressentir qu'il est lui-m\u00eame compos\u00e9 d'un grand nombre de parties, parties qui lui viennent d'ailleurs, qui commencent \u00e0 lui appartenir d\u00e8s qu'il existe en vertu d'une cause ext\u00e9rieure, qui se renouvellent sous le jeu des causes, tant qu'il existe, et qui cessent de lui appartenir d\u00e8s qu'il meurt2. Alors, nous pouvons dire en quoi consiste l'existence du mode : _exister, c'est avoir actuellement un tr\u00e8s grand nombre de parties (plurimae)._ Ces parties composantes sont ext\u00e9rieures \u00e0 l'essence du mode, ext\u00e9rieures les unes aux autres : ce sont des parties extensives.\n\nNous croyons que, chez Spinoza, il n'y a pas de mode existant qui ne soit actuellement compos\u00e9 d'un tr\u00e8s grand nombre de parties extensives. Il n'y a pas de corps existant, dans l'\u00e9tendue, qui ne soit compos\u00e9 d'un tr\u00e8s grand nombre de corps simples. Et l'\u00e2me, en tant qu'elle est l'id\u00e9e d'un corps existant, est elle-m\u00eame compos\u00e9e d'un grand nombre d'id\u00e9es qui correspondent aux parties composantes du corps, et qui se distinguent extrins\u00e8quement3. Bien plus, les facult\u00e9s que l'\u00e2me poss\u00e8de en tant qu'elle est l'id\u00e9e d'un corps existant sont de v\u00e9ritables parties extensives, qui cessent d'appartenir \u00e0 l'\u00e2me d\u00e8s que le corps cesse lui-m\u00eame d'exister4. Voici donc, semble-t-il, les premiers \u00e9l\u00e9ments du sch\u00e9ma spinoziste : une essence de mode est un degr\u00e9 d\u00e9termin\u00e9 d'intensit\u00e9, un degr\u00e9 de puissance irr\u00e9ductible ; le mode existe, lorsqu'il poss\u00e8de actuellement un tr\u00e8s grand nombre de parties extensives qui correspondent \u00e0 son essence ou degr\u00e9 de puissance.\n\nQue signifie \u00ab un tr\u00e8s grand nombre \u00bb ? La _Lettre \u00e0 Meyer_ donne une indication pr\u00e9cieuse : il y a des grandeurs qu'on appelle infinies ou, mieux, ind\u00e9finies, parce qu'\u00ab on ne peut en d\u00e9terminer ou repr\u00e9senter les parties par aucun nombre \u00bb ; \u00ab elles ne peuvent \u00eatre \u00e9gal\u00e9es \u00e0 aucun nombre, mais d\u00e9passent tout nombre assignable5 \u00bb. Nous reconnaissons ici le second infini modal-quantitatif dont parle la lettre \u00e0 Meyer : il s'agit d'un infini proprement extensif. Spinoza donne un exemple g\u00e9om\u00e9trique : la somme des in\u00e9galit\u00e9s de distance comprises entre deux cercles non concentriques d\u00e9passe tout nombre assignable. Cette quantit\u00e9 infinie a trois caract\u00e8res originaux, n\u00e9gatifs, il est vrai, plut\u00f4t que positifs. En premier lieu, elle n'est pas constante ni \u00e9gale \u00e0 elle-m\u00eame : on peut la concevoir comme plus grande ou plus petite (Spinoza pr\u00e9cisera dans un autre texte : \u00ab Dans l'espace total compris entre deux cercles ayant des centres diff\u00e9rents, nous concevons une multitude de parties deux fois plus grande que dans la moiti\u00e9 de cet espace, et cependant le nombre des parties, aussi bien de la moiti\u00e9 de l'espace total, est plus grand que tout nombre assignable \u00bb)6. L'infini extensif est donc un infini n\u00e9cessairement con\u00e7u comme plus ou moins grand. Mais en second lieu, il n'est pas \u00e0 proprement parler \u00ab illimit\u00e9 \u00bb : il se rapporte en effet \u00e0 quelque chose de limit\u00e9 ; il y a un maximum et un minimum des distances comprises entre les deux cercles non concentriques, ces distances se rapportent \u00e0 un espace parfaitement limit\u00e9 et d\u00e9termin\u00e9. Enfin, en troisi\u00e8me lieu, cette quantit\u00e9 n'est pas infinie par la multitude de ses parties ; car, \u00ab si l'infinit\u00e9 se concluait de la multitude des parties, nous ne pourrions en concevoir une multitude plus grande, leur multitude devant \u00eatre plus grande que toute multitude donn\u00e9e \u00bb. Ce n'est pas par le nombre de ses parties que cette quantit\u00e9 est infinie ; au contraire, c'est parce qu'elle est toujours infinie qu'elle se divise en une multitude de parties qui d\u00e9passent tout nombre.\n\nOn remarquera que le nombre n'exprime jamais ad\u00e9quatement la nature des modes. Il peut \u00eatre utile d'identifier la quantit\u00e9 modale et le nombre ; c'est m\u00eame n\u00e9cessaire, par opposition \u00e0 la substance et aux qualit\u00e9s substantielles. Nous l'avons fait chaque fois que nous avons pr\u00e9sent\u00e9 la distinction modale comme une distinction num\u00e9rique. Mais en v\u00e9rit\u00e9 le nombre n'est qu'une mani\u00e8re d'imaginer la quantit\u00e9, ou une fa\u00e7on de penser abstraitement les modes. Les modes, en tant qu'ils d\u00e9coulent de la substance et des attributs, sont autre chose que des fant\u00f4mes de l'imagination, autre chose aussi que des \u00eatres de raison. Leur \u00eatre est quantitatif, non pas num\u00e9rique \u00e0 proprement parler. Si l'on consid\u00e8re le premier infini modal, l'infini intensif, il n'est pas divisible en parties extrins\u00e8ques. Les parties intensives qu'il comporte intrins\u00e8quement, les essences de modes, ne sont pas s\u00e9parables les unes des autres ; le nombre les s\u00e9pare les unes des autres, et du principe de leur production, donc les saisit abstraitement. Si l'on consid\u00e8re le second infini, l'infini extensif, sans doute est-il divisible en parties extrins\u00e8ques qui composent les existences. Mais ces parties extrins\u00e8ques vont toujours par ensembles infinis ; leur somme d\u00e9passe toujours tout nombre assignable. Quand nous les expliquons par le nombre, nous laissons \u00e9chapper l'\u00eatre r\u00e9el des modes existants, nous ne saisissons que des fictions7.\n\nLa _Lettre \u00e0 Meyer_ , entre autres choses, expose donc le cas sp\u00e9cial d'un infini modal extensif, variable et divisible. Cette exposition est importante en elle-m\u00eame ; Leibniz, \u00e0 ce propos, f\u00e9licitait Spinoza d'avoir \u00e9t\u00e9 plus loin que beaucoup de math\u00e9maticiens8. Mais, du point de vue du spinozisme lui-m\u00eame, la question est : \u00e0 quoi se rapporte cette th\u00e9orie du deuxi\u00e8me infini modal, dans l'ensemble du syst\u00e8me ? La r\u00e9ponse nous semble \u00eatre : l'infini extensif concerne l'existence des modes. En effet, lorsque Spinoza affirme, dans l' _\u00c9thique_ , que le mode compos\u00e9 a un _tr\u00e8s grand nombre_ de parties, il entend bien par \u00ab tr\u00e8s grand nombre \u00bb un nombre inassignable, c'est-\u00e0-dire une multitude qui d\u00e9passe tout nombre. L'essence d'un tel mode est elle-m\u00eame un degr\u00e9 de puissance ; mais quel que soit le degr\u00e9 de puissance qui constitue son essence, le mode n'existe pas sans avoir actuellement une infinit\u00e9 de parties. Si l'on consid\u00e8re un mode dont le degr\u00e9 de puissance est double du pr\u00e9c\u00e9dent, son existence est compos\u00e9e d'une infinit\u00e9 de parties, elle-m\u00eame double de la pr\u00e9c\u00e9dente. \u00c0 la limite, il y a une infinit\u00e9 d'ensembles infinis, un ensemble de tous les ensembles, qui est comme l'ensemble de toutes les choses existantes, simultan\u00e9es et successives. Bref, les caract\u00e8res que Spinoza, dans la _Lettre \u00e0 Meyer_ , assigne au deuxi\u00e8me infini modal, ne trouvent leur application que dans la th\u00e9orie du mode existant telle qu'elle appara\u00eet dans l' _\u00c9thique_ ; et l\u00e0, ils trouvent leur pleine application. C'est le mode existant qui a une infinit\u00e9 de parties (un tr\u00e8s grand nombre) ; c'est son essence ou degr\u00e9 de puissance qui forme toujours une limite (un maximum et un minimum) ; c'est l'ensemble des modes existants, non seulement simultan\u00e9s mais successifs, qui constitue le plus grand infini, lui-m\u00eame divisible en infinis plus ou moins grands9.\n\nEncore faudrait-il savoir d'o\u00f9 viennent ces parties extensives, et en quoi elles consistent. Ce ne sont pas des atomes : non seulement les atomes impliquent le vide, mais une infinit\u00e9 d'atomes ne pourrait pas se rapporter \u00e0 quelque chose de limit\u00e9. Ce ne sont pas non plus les termes virtuels d'une divisibilit\u00e9 \u00e0 l'infini : ceux-ci ne pourraient pas former des infinis plus ou moins grands. De l'hypoth\u00e8se de l'infinie divisibilit\u00e9 \u00e0 celle des atomes, on tombe \u00ab de Charybde en Scylla10 \u00bb. En v\u00e9rit\u00e9, les ultimes parties extensives sont les parties infiniment petites actuelles d'un infini lui-m\u00eame actuel. La position d'un infini actuel dans la Nature n'a pas moins d'importance chez Spinoza que chez Leibniz : il n'y a aucune contradiction entre l'id\u00e9e de parties ultimes absolument simples et le principe d'une division infinie, pour peu que cette division soit _actuellement infinie_11. Nous devons penser qu'un attribut n'a pas seulement une quantit\u00e9 intensive, mais une quantit\u00e9 extensive infinie. C'est cette quantit\u00e9 extensive qui est actuellement divis\u00e9e en une infinit\u00e9 de parties extensives. Ces parties sont des parties extrins\u00e8ques, agissant du dehors les unes sur les autres et se distinguant du dehors. _Toutes ensemble et sous tous leurs rapports, elles forment un univers infiniment changeant, correspondant \u00e0 la toute-puissance de Dieu. Mais sous tel ou tel rapport d\u00e9termin\u00e9, elles forment des ensembles infinis plus ou moins grands, qui correspondent \u00e0 tel ou tel degr\u00e9 de puissance, c'est-\u00e0-dire \u00e0 telle ou telle essence de mode._ Elles vont toujours par infinit\u00e9s : une infinit\u00e9 de parties correspondent toujours \u00e0 un degr\u00e9 de puissance, si petit soit-il ; l'ensemble de l'univers correspond \u00e0 la Puissance qui comprend tous les degr\u00e9s.\n\nC'est en ce sens que nous devons comprendre l'analyse des modes de l'\u00e9tendue. L'attribut \u00e9tendue a une quantit\u00e9 extensive modale qui se divise actuellement en une infinit\u00e9 de corps simples. Ces corps simples sont des parties extrins\u00e8ques qui ne se distinguent les unes des autres, et ne se rapportent les unes aux autres, que par le mouvement et le repos. Mouvement et repos sont pr\u00e9cis\u00e9ment la forme de la distinction extrins\u00e8que et des rapports ext\u00e9rieurs entre les corps simples. Les corps simples sont d\u00e9termin\u00e9s du dehors au mouvement ou au repos, \u00e0 l'infini, et se distinguent par le mouvement ou le repos auxquels ils sont d\u00e9termin\u00e9s. Ils se groupent toujours par ensembles infinis, _chaque ensemble \u00e9tant d\u00e9fini par un certain rapport de mouvement et de repos._ C'est sous ce rapport qu'un ensemble infini correspond \u00e0 telle essence de mode (c'est-\u00e0-dire \u00e0 tel degr\u00e9 de puissance), donc constitue dans l'\u00e9tendue l'existence du mode lui-m\u00eame. Si l'on consid\u00e8re l'ensemble de tous les ensembles infinis sous tous les rapports, on a \u00ab la somme de toutes les variations de la mati\u00e8re en mouvement \u00bb, ou \u00ab la figure de l'univers entier \u00bb sous l'attribut \u00e9tendue. Cette figure ou cette somme correspondent \u00e0 la toute-puissance de Dieu en tant que celle-ci comprend tous les degr\u00e9s de puissance ou toutes les essences de modes dans ce m\u00eame attribut de l'\u00e9tendue12.\n\nCe sch\u00e9ma para\u00eet en mesure de dissiper certaines contradictions qu'on a cru voir dans la physique de Spinoza, ou plut\u00f4t qu'on a cru voir dans l' _\u00c9thique_ , entre la physique des corps et la th\u00e9orie des essences. Ainsi Rivaud remarquait qu'un corps simple est d\u00e9termin\u00e9 au mouvement et au repos, mais toujours et seulement d\u00e9termin\u00e9 du dehors : ce corps renvoie \u00e0 un ensemble infini de corps simples. Mais alors, comment concilier cet \u00e9tat des corps simples avec le statut des essences ? \u00ab Un corps particulier, au moins un corps simple n'a donc pas d'essence \u00e9ternelle. Sa r\u00e9alit\u00e9 para\u00eet se r\u00e9sorber dans celle du syst\u00e8me infini des causes \u00bb ; \u00ab Nous cherchions une essence particuli\u00e8re, nous ne trouvons qu'une cha\u00eene infinie de causes dont nul terme ne para\u00eet avoir de r\u00e9alit\u00e9 essentielle propre \u00bb ; \u00ab Cette solution impos\u00e9e, semble-t-il, par les textes que l'on vient de citer, para\u00eet en contradiction avec les principes les plus certains du syst\u00e8me de Spinoza. Que devient alors l'\u00e9ternit\u00e9 des essences, affirm\u00e9e sans restriction \u00e0 tant de reprises ? Comment un corps, si petit soit-il, si fugitif que soit son \u00eatre, peut-il exister sans une nature propre, faute de laquelle il ne pourrait ni arr\u00eater ni transmettre le mouvement qu'il re\u00e7oit ? Ce qui n'a pas d'essence du tout ne peut exister et toute essence est, par d\u00e9finition, immuable. La bulle de savon, qui existe \u00e0 un moment donn\u00e9, a n\u00e9cessairement une essence \u00e9ternelle, sans quoi elle ne serait pas13. \u00bb\n\nIl nous semble au contraire qu'il n'y a pas lieu de chercher une essence pour chaque partie extensive. Une essence est un degr\u00e9 d'intensit\u00e9. Or les parties extensives et les degr\u00e9s d'intensit\u00e9 (parties intensives) ne se correspondent nullement terme \u00e0 terme. \u00c0 tout degr\u00e9 d'intensit\u00e9, si petit soit-il, correspondent une infinit\u00e9 de parties extensives, ayant entre elles et devant avoir des rapports uniquement extrins\u00e8ques. Les parties extensives vont par infinit\u00e9s plus ou moins grandes, mais toujours par infinit\u00e9 ; il n'est pas question que chacune ait une essence, puisqu'une infinit\u00e9 de parties correspondent \u00e0 la plus petite essence. La bulle de savon a bien une essence, mais non pas chaque partie de l'ensemble infini qui la compose sous un certain rapport. En d'autres termes, chez Spinoza, _il n'y a pas de mode existant qui ne soit actuellement compos\u00e9 \u00e0 l'infini_ , quelle que soit son essence ou son degr\u00e9 de puissance. Spinoza dit que les modes compos\u00e9s ont un \u00ab tr\u00e8s grand nombre \u00bb de parties ; mais ce qu'il dit du mode compos\u00e9, il faut l'entendre de tout mode existant, car il n'y a de mode existant que compos\u00e9, toute existence est par d\u00e9finition compos\u00e9e. D\u00e8s lors, dira-t-on que les parties simples extensives existent ? Dira-t-on que les corps simples existent dans l'\u00e9tendue ? Si l'on veut dire qu'ils existent un par un, ou par un nombre, l'absurdit\u00e9 est \u00e9vidente. \u00c0 strictement parler, les parties simples n'ont ni essence ni existence qui leur soient propres. Elles n'ont pas d'essence ou de nature interne ; elles se distinguent extrins\u00e8quement les unes des autres, se rapportent extrins\u00e8quement les unes aux autres. Elles n'ont pas d'existence propre, mais composent l'existence : exister, c'est avoir actuellement une infinit\u00e9 de parties extensives. Par infinit\u00e9s plus ou moins grandes, elles composent, sous des rapports divers, l'existence de modes dont l'essence est d'un degr\u00e9 plus ou moins grand. Non seulement la physique de Spinoza, mais le spinozisme tout entier deviennent inintelligibles si l'on ne distingue pas ce qui revient aux essences, ce qui revient aux existences, et le type de leur correspondance qui n'est nullement terme \u00e0 terme.\n\nNous avons des \u00e9l\u00e9ments pour r\u00e9pondre \u00e0 la question : comment une infinit\u00e9 de parties extensives composent-elles l'existence d'un mode ? Par exemple, un mode existe dans l'\u00e9tendue lorsqu'une infinit\u00e9 de corps simples lui appartiennent actuellement, qui correspondent \u00e0 son essence. Mais comment peuvent-elles correspondre \u00e0 son essence, ou lui appartenir ? Depuis le _Court Trait\u00e9_ , la r\u00e9ponse de Spinoza est constante : _sous un certain rapport de mouvement et de repos_. Tel mode \u00ab vient \u00e0 exister \u00bb, il passe \u00e0 l'existence, quand une infinit\u00e9 de parties extensives entrent sous _tel_ rapport ; il continue d'exister tant que ce rapport est effectu\u00e9. C'est donc sous des rapports gradu\u00e9s que les parties extensives se groupent en ensembles vari\u00e9s, correspondant \u00e0 diff\u00e9rents degr\u00e9s de puissance. Des parties extensives forment un ensemble infini plus ou moins grand, en tant qu'elles entrent dans tel ou tel rapport ; sous ce rapport, elles correspondent \u00e0 telle essence de mode et composent l'existence de ce mode lui-m\u00eame ; prises sous un autre rapport, elles font partie d'un autre ensemble, correspondent \u00e0 une autre essence de mode, composent l'existence d'un autre mode. Telle est d\u00e9j\u00e0 la doctrine du _Court Trait\u00e9_ , concernant le passage des modes \u00e0 l'existence14. L' _\u00c9thique_ dit encore plus clairement : peu importe que les parties composantes d'un mode existant se renouvellent \u00e0 chaque instant ; l'ensemble demeure le m\u00eame, tant qu'il est d\u00e9fini par un rapport sous lequel ses parties quelconques appartiennent \u00e0 telle essence de mode. Le mode existant est donc sujet \u00e0 des variations consid\u00e9rables et continuelles : peu importe aussi que la r\u00e9partition du mouvement et du repos, de la vitesse et de la lenteur, change entre les parties. Tel mode continue d'exister tant que le m\u00eame rapport subsiste dans l'ensemble infini de ses parties15.\n\nIl faut donc reconna\u00eetre qu'une essence de mode (degr\u00e9 de puissance) s'exprime \u00e9ternellement dans un certain rapport gradu\u00e9. Mais le mode ne passe pas \u00e0 l'existence avant qu'une infinit\u00e9 de parties extensives ne soient actuellement d\u00e9termin\u00e9es \u00e0 entrer sous ce m\u00eame rapport. Ces parties peuvent \u00eatre d\u00e9termin\u00e9es \u00e0 entrer sous un autre rapport ; elles s'int\u00e8grent alors \u00e0 un ensemble infini, plus grand ou plus petit, qui correspond \u00e0 une autre essence de mode et compose l'existence d'un autre mode. La th\u00e9orie de l'existence chez Spinoza comporte donc trois \u00e9l\u00e9ments : _l'essence singuli\u00e8re_ , qui est un degr\u00e9 de puissance ou d'intensit\u00e9 ; _l'existence particuli\u00e8re_ , toujours compos\u00e9e d'une infinit\u00e9 de parties extensives ; _la forme individuelle,_ c'est-\u00e0-dire le rapport caract\u00e9ristique ou expressif, qui correspond \u00e9ternellement \u00e0 l'essence du mode, mais aussi sous lequel une infinit\u00e9 de parties se rapportent temporairement \u00e0 cette essence. Dans un mode existant, l'essence est un degr\u00e9 de puissance ; ce degr\u00e9 s'exprime dans un rapport ; ce rapport subsume une infinit\u00e9 de parties. D'o\u00f9 la formule de Spinoza : les parties, comme \u00e9tant \u00ab sous la domination d'une seule et m\u00eame nature \u00bb, oblig\u00e9es de s'ajuster les unes aux autres suivant que l'exige cette nature16 \u00bb.\n\nUne essence de mode s'exprime \u00e9ternellement dans un rapport, mais nous ne devons pas confondre l'essence et le rapport dans lequel elle s'exprime. Une essence de mode n'est pas cause de l'existence du mode lui-m\u00eame : cette proposition reprend, en termes spinozistes, un vieux principe d'apr\u00e8s lequel l'existence d'un \u00eatre fini ne d\u00e9coule pas de son essence. Mais quel est le nouveau sens de ce principe dans les perspectives de Spinoza ? Il signifie ceci : une essence de mode a beau s'exprimer dans un rapport caract\u00e9ristique, ce n'est pas elle qui d\u00e9termine une infinit\u00e9 de parties extensives \u00e0 entrer sous ce rapport. (Ce n'est pas la nature simple qui \u00e9tablit par elle-m\u00eame sa domination, ni qui oblige elle-m\u00eame des parties \u00e0 s'ajuster conform\u00e9ment au rapport dans lequel elle s'exprime.) Car les parties extensives se d\u00e9terminent les unes les autres, du dehors et \u00e0 l'infini ; elles n'ont pas d'autre d\u00e9termination qu'extrins\u00e8que. Un mode passe \u00e0 l'existence, non pas en vertu de son essence, mais en vertu de lois purement m\u00e9caniques qui d\u00e9terminent une infinit\u00e9 de parties extensives quelconques \u00e0 entrer sous tel rapport pr\u00e9cis, dans lequel son essence s'exprime. Un mode cesse d'exister d\u00e8s que ses parties sont d\u00e9termin\u00e9es \u00e0 entrer sous un autre rapport, correspondant \u00e0 une autre essence. Les modes passent \u00e0 l'existence, et cessent d'exister, en vertu de lois ext\u00e9rieures \u00e0 leurs essences.\n\nQuelles sont ces lois m\u00e9caniques ? Dans le cas de l'\u00e9tendue, il s'agit en derni\u00e8re instance des lois de la communication du mouvement. Si nous consid\u00e9rons l'infinit\u00e9 des corps simples, nous voyons qu'ils se groupent en ensembles infinis toujours variables. Mais l'ensemble de tous ces ensembles reste constant, cette constance \u00e9tant d\u00e9finie par la quantit\u00e9 de mouvement, c'est-\u00e0-dire par la proportion totale qui contient une infinit\u00e9 de rapports particuliers, rapports de mouvement et de repos. Les corps simples ne sont jamais s\u00e9parables d'un de ces rapports quelconques, sous lequel ils appartiennent \u00e0 un ensemble. Or, la proportion totale restant toujours constante, ces rapports se font et se d\u00e9font, suivant des lois de composition et de d\u00e9composition.\n\nSupposons deux corps compos\u00e9s ; chacun poss\u00e8de, sous un certain rapport, une infinit\u00e9 de corps simples ou de parties. Quand ils se rencontrent, il peut arriver que les deux rapports soient directement composables. Alors les parties de l'un s'ajustent aux parties de l'autre, sous un troisi\u00e8me rapport compos\u00e9 des deux pr\u00e9c\u00e9dents. Il y a ici formation d'un corps encore plus compos\u00e9 que ceux dont nous \u00e9tions partis. Dans un texte c\u00e9l\u00e8bre, Spinoza montre comment le chyle et la lymphe composent leur rapport respectif pour former, sous un troisi\u00e8me rapport, le sang17. Mais, dans des conditions plus ou moins complexes, ce processus est celui de toute naissance ou de toute formation, c'est-\u00e0-dire de tout passage \u00e0 l'existence : des parties se rencontrent sous deux rapports diff\u00e9rents ; chacun de ces rapports correspond d\u00e9j\u00e0 \u00e0 une essence de mode ; les deux rapports se composent de telle mani\u00e8re que les parties qui se rencontrent entrent sous un troisi\u00e8me rapport, qui correspond \u00e0 une autre essence de mode ; le mode correspondant passe alors \u00e0 l'existence. Mais il peut arriver que les deux rapports ne soient pas directement composables. Les corps qui se rencontrent, ou bien sont indiff\u00e9rents l'un \u00e0 l'autre ; ou bien l'un, sous son rapport, d\u00e9compose le rapport de l'autre, donc d\u00e9truit l'autre corps. Il en est ainsi d'un toxique ou d'un poison, qui d\u00e9truit l'homme en d\u00e9composant le sang. Il en est ainsi de l'alimentation, mais en sens inverse : l'homme force les parties du corps dont il se nourrit \u00e0 entrer sous un nouveau rapport qui convient avec le sien, mais qui suppose la destruction du rapport sous lequel ce corps existait pr\u00e9c\u00e9demment.\n\nIl y a donc des lois de composition et de d\u00e9composition de rapports qui d\u00e9terminent le passage \u00e0 l'existence des modes et aussi la fin de leur existence. Ces lois \u00e9ternelles n'affectent nullement la v\u00e9rit\u00e9 \u00e9ternelle de chaque rapport : chaque rapport a une v\u00e9rit\u00e9 \u00e9ternelle, en tant qu'une essence s'exprime en lui. Mais les lois de composition et de d\u00e9composition d\u00e9terminent les conditions sous lesquelles un rapport est effectu\u00e9, c'est-\u00e0-dire subsume actuellement des parties extensives, ou au contraire cesse d'\u00eatre effectu\u00e9. C'est pourquoi nous ne devons surtout pas confondre les essences et les rapports, ni la loi de production des essences et la loi de composition des rapports. Ce n'est pas l'essence qui d\u00e9termine l'effectuation du rapport dans lequel elle s'exprime. Les rapports se composent et se d\u00e9composent suivant des lois qui sont les leurs. L'ordre des essences se d\u00e9finit par une convenance totale. Il n'en est pas de m\u00eame de l'ordre des rapports : sans doute tous les rapports se combinent-ils \u00e0 l'infini, mais non pas n'importe comment. N'importe quel rapport ne se compose pas avec n'importe quel autre. Ces lois de composition qui sont propres aux rapports caract\u00e9ristiques, et qui r\u00e8glent le passage des modes \u00e0 l'existence, posent de multiples probl\u00e8mes. Ces lois ne sont pas contenues dans les essences elles-m\u00eames. Spinoza pensait-il \u00e0 elles quand il parlait d\u00e9j\u00e0, dans le _Trait\u00e9 de la R\u00e9forme_ , de lois inscrites dans les attributs et dans les modes infinis \u00ab comme dans leurs vrais codes18 \u00bb ? La complexit\u00e9 de ce texte nous interdit de l'utiliser pour le moment. D'autre part, connaissons-nous ces lois, et comment ? Spinoza semble bien admettre que nous devons passer par une \u00e9tude empirique des corps pour savoir quels sont leurs rapports et comment ils se composent19. Quoi qu'il en soit, il nous suffit provisoirement de marquer l'irr\u00e9ductibilit\u00e9 d'un ordre des rapports \u00e0 l'ordre des essences elles-m\u00eames.\n\nL'existence d'un mode ne d\u00e9coule donc pas de son essence. Lorsqu'un mode passe \u00e0 l'existence, il est d\u00e9termin\u00e9 \u00e0 le faire par une loi m\u00e9canique qui compose le rapport dans lequel il s'exprime, c'est-\u00e0-dire qui contraint une infinit\u00e9 de parties extensives \u00e0 entrer sous ce rapport. Chez Spinoza, le passage \u00e0 l'existence ne doit jamais \u00eatre compris comme un passage du possible au r\u00e9el : pas plus qu'une essence de mode n'est un \u00ab possible \u00bb, un mode existant n'est la r\u00e9alisation d'un possible. Les essences existent n\u00e9cessairement, en vertu de leur cause ; les modes dont elles sont les essences passent n\u00e9cessairement \u00e0 l'existence en vertu de causes qui d\u00e9terminent des parties \u00e0 entrer sous les rapports qui correspondent \u00e0 ces essences. Partout la n\u00e9cessit\u00e9 comme seule modalit\u00e9 de l'\u00eatre, mais cette n\u00e9cessit\u00e9 a deux \u00e9tages. Nous avons vu que la distinction d'une essence et de sa propre existence ne devait pas s'interpr\u00e9ter comme une distinction r\u00e9elle ; pas davantage, la distinction d'une essence et de l'existence du mode lui-m\u00eame. Le mode existant, c'est l'essence m\u00eame en tant qu'elle poss\u00e8de actuellement une infinit\u00e9 de parties extensives. De m\u00eame que l'essence existe en vertu de sa cause, le mode lui-m\u00eame existe en vertu de la cause qui d\u00e9termine des parties \u00e0 lui appartenir. Mais les deux formes de causalit\u00e9 que nous sommes ainsi amen\u00e9s \u00e0 consid\u00e9rer nous forcent \u00e0 d\u00e9finir deux types de position modale, deux types de distinction modale.\n\nTant que nous consid\u00e9rions les essences de modes, nous les d\u00e9finissions comme des r\u00e9alit\u00e9s intensives. Elles ne se distinguaient pas de l'attribut, elles ne se distinguaient pas les unes des autres, sauf sous un type de distinction tr\u00e8s sp\u00e9ciale (distinction intrins\u00e8que). Elles n'existaient que comme contenues dans l'attribut, leurs id\u00e9es n'existaient que comme comprises dans l'id\u00e9e de Dieu. Toutes les essences \u00e9taient \u00ab compliqu\u00e9es \u00bb dans l'attribut ; c'est sous cette forme qu'elles existaient, et qu'elles exprimaient l'essence de Dieu, chacune suivant son degr\u00e9 de puissance. Mais quand les modes passent \u00e0 l'existence, ils acqui\u00e8rent des parties extensives. Ils acqui\u00e8rent une grandeur et une dur\u00e9e : chaque mode dure tant que les parties restent sous le rapport qui le caract\u00e9rise. Par l\u00e0, il faut bien reconna\u00eetre que les modes existants se distinguent _extrins\u00e8quement_ de l'attribut, et se distinguent _extrins\u00e8quement_ les uns des autres. Les _Pens\u00e9es m\u00e9taphysiques_ d\u00e9finissaient \u00ab l'\u00eatre de l'existence \u00bb comme \u00ab l'essence m\u00eame des choses en dehors de Dieu \u00bb, par opposition \u00e0 \u00ab l'\u00eatre de l'essence \u00bb qui d\u00e9signait les choses telles qu'elles sont \u00ab comprises dans les attributs de Dieu20 \u00bb. Peut-\u00eatre cette d\u00e9finition correspond-elle plus qu'on ne croit \u00e0 la pens\u00e9e de Spinoza lui-m\u00eame. Elle pr\u00e9sente \u00e0 cet \u00e9gard plusieurs caract\u00e8res importants.\n\nElle nous rappelle d'abord que la distinction de l'essence et de l'existence n'est jamais une distinction r\u00e9elle. L'\u00eatre de l'essence (existence de l'essence), c'est la position de l'essence dans un attribut de Dieu. L'\u00eatre de l'existence (existence de la chose elle-m\u00eame), c'est encore une position de l'essence, mais une position extrins\u00e8que, _hors de_ l'attribut. Or, nous ne voyons pas que l' _\u00c9thique_ renonce \u00e0 cette th\u00e8se. L'existence d'une chose particuli\u00e8re, c'est la chose m\u00eame, non plus seulement en tant qu'elle est contenue dans l'attribut, non plus seulement en tant qu'elle est comprise en Dieu, mais en tant qu'elle dure, en tant qu'elle est en relation avec un certain temps et un certain lieu distincts extrins\u00e8quement21. On objectera qu'une telle conception s'oppose radicalement \u00e0 l'immanence. Car, du point de vue de l'immanence, les modes en passant \u00e0 l'existence ne cessent pas d'appartenir \u00e0 la substance et d'\u00eatre contenus en elle. Ce point est tellement \u00e9vident qu'il faut chercher plus loin. Spinoza ne dit pas que les modes existants cessent d'\u00eatre contenus dans la substance, mais qu'ils \u00ab ne sont plus seulement \u00bb contenus dans la substance ou dans l'attribut22. La difficult\u00e9 se r\u00e9sout ais\u00e9ment _si nous consid\u00e9rons que la distinction extrins\u00e8que est encore et toujours une distinction modale._ Les modes ne cessent pas d'\u00eatre des modes quand ils sont pos\u00e9s hors de l'attribut, car cette position extrins\u00e8que est purement modale et non substantielle. Si l'on nous permet un rapprochement accidentel avec Kant, on se souviendra que Kant explique que l'espace est la forme de l'ext\u00e9riorit\u00e9, mais que cette forme d'ext\u00e9riorit\u00e9 n'est pas moins int\u00e9rieure au moi que la forme d'int\u00e9riorit\u00e9 : elle pr\u00e9sente des objets comme ext\u00e9rieurs \u00e0 nous-m\u00eames et comme ext\u00e9rieurs les uns aux autres, sans aucune illusion, mais elle-m\u00eame nous est int\u00e9rieure et nous reste int\u00e9rieure23. De m\u00eame chez Spinoza, dans un tout autre contexte, \u00e0 un tout autre sujet : la quantit\u00e9 extensive n'appartient pas moins \u00e0 l'attribut que la quantit\u00e9 intensive ; mais elle est comme une forme d'ext\u00e9riorit\u00e9 proprement modale. Elle pr\u00e9sente les modes existants comme ext\u00e9rieurs \u00e0 l'attribut, comme ext\u00e9rieurs les uns aux autres. Elle n'en est pas moins contenue, ainsi que tous les modes existants, dans l'attribut qu'elle modifie. L'id\u00e9e d'une distinction modale-extrins\u00e8que ne contredit nullement le principe de l'immanence.\n\nD\u00e8s lors, que signifie cette distinction modale extrins\u00e8que ? Quand les modes font l'objet d'une position extrins\u00e8que, ils cessent d'exister sous la forme _compliqu\u00e9e_ qui est la leur tant que leurs essences sont seulement contenues dans l'attribut. Leur nouvelle existence est une _explication_ : ils expliquent l'attribut, chacun l'explique \u00ab d'une mani\u00e8re certaine et d\u00e9termin\u00e9e \u00bb. C'est-\u00e0-dire : chaque mode existant explique l'attribut sous le rapport qui le caract\u00e9rise, d'une mani\u00e8re qui se distingue extrins\u00e8quement des autres mani\u00e8res sous d'autres rapports. C'est en ce sens que le mode existant n'est pas moins expressif que l'essence de mode, mais l'est d'une autre fa\u00e7on. L'attribut ne s'exprime plus dans les essences de mode qu'il complique ou contient, conform\u00e9ment \u00e0 leurs degr\u00e9s de puissance ; il s'exprime en outre dans des modes existants, qui l'expliquent d'une mani\u00e8re certaine et d\u00e9termin\u00e9e, c'est-\u00e0-dire conform\u00e9ment aux rapports qui correspondent \u00e0 leurs essences. L'expression modale enti\u00e8re est constitu\u00e9e par ce double mouvement de la complication et de l'explication24.\n\n* * *\n\n1.  _\u00c9_ , I, 28, prop. et dem.\n\n2.  L'id\u00e9e d'un grand nombre de causes ext\u00e9rieures et celle d'un grand nombre de parties composantes forment deux th\u00e8mes qui s'encha\u00eenent : Cf. _\u00c9_ , II, 19, dem.\n\n3.  _\u00c9_ , II, 15, prop. et dem. C'est ce point, entre autres, qui g\u00eane Blyenbergh ( _Lettre 24, de Blyenbergh_ , III, p. 107) : l'\u00e2me, \u00e9tant un compos\u00e9, ne se dissoudrait pas moins que le corps apr\u00e8s la mort. C'est oublier que l'\u00e2me, et aussi le corps, ont une essence intensive d'une tout autre nature que leurs parties extensives.\n\n4.  Ainsi l'imagination, la m\u00e9moire, la passion : cf. _\u00c9_ , V, 21 et 34 ; et V, 40, cor. \u00ab Quant \u00e0 cette _partie_ que nous avons montr\u00e9 qui p\u00e9rit... \u00bb)\n\n5.  _Lettre 12, \u00e0 Meyer_ (III, pp. 41-42).\n\n6.  _Lettre 81, \u00e0 Tschirnhaus_ (III, p. 241). Sur cet exemple des cercles non concentriques, et la somme des \u00ab in\u00e9galit\u00e9s de distance \u00bb, cf. M. GU\u00c9ROULT, \u00ab La Lettre de Spinoza sur l'infini \u00bb, _Revue de m\u00e9taphysique et de morale_ , octobre 1966, no 4.\n\n7.  Cf. _Lettre 12, \u00e0 Meyer_ (III, pp. 40-41).\n\n8.  Leibniz avait eu connaissance de la plus grande partie de la _Lettre \u00e0 Meyer_. Il fait des critiques de d\u00e9tail ; mais, \u00e0 propos de l'infini plus ou moins grand, il commente : \u00ab Ceci, que la plupart des math\u00e9maticiens ignor\u00e8rent, et particuli\u00e8rement Cardan, est remarquablement observ\u00e9 et tr\u00e8s soigneusement inculqu\u00e9 par notre auteur. \u00bb (Cf. Gerhardt, I, p. 137, n. 21.)\n\n9.  L'exemple g\u00e9om\u00e9trique de la _Lettre \u00e0 Meyer_ (somme des in\u00e9galit\u00e9s de distance comprises entre deux cercles) n'est pas de m\u00eame nature que celui de l' _\u00c9thique_ II, 8, sc. (ensemble des c\u00f4t\u00e9s d'angles droits compris dans un cercle). Dans le premier cas, il s'agit d'illustrer _l'\u00e9tat des modes existants_ , dont les parties forment des infinis plus ou moins grands, l'ensemble de tous ces infinis correspondant \u00e0 la Figure de l'Univers. C'est pourquoi la _Lettre \u00e0 Meyer_ assimile la somme des in\u00e9galit\u00e9s de distance \u00e0 la somme des variations de la mati\u00e8re (III, p. 42). Mais dans le second cas, celui de l' _\u00c9thique_ , il s'agit d'illustrer _l'\u00e9tat des essences_ de modes telles qu'elles sont contenues dans l'attribut.\n\n10.  _Lettre 12, \u00e0 Meyer_ (III, p. 41). De m\u00eame, la _Lettre 6, \u00e0 Oldenburg_ , refuse \u00e0 la fois le progr\u00e8s \u00e0 l'infini et l'existence du vide (\u00ab De la fluidit\u00e9 \u00bb, III, p. 22.)\n\n11.  Nous ne comprenons pas pourquoi A. Rivaud, dans son \u00e9tude sur la physique de Spinoza, voyait ici une contradiction : \u00ab Comment, dans une \u00e9tendue o\u00f9 la division actuelle est infinie, parler de corps tr\u00e8s simples ! De tels corps ne peuvent \u00eatre r\u00e9els qu'au regard de notre perception. \u00bb (\u00ab La physique de Spinoza \u00bb, _Chronicon Spinozanum_ , IV, p. 32). 1o) Il n'y aurait contradiction qu'entre l'id\u00e9e de corps simples et le principe d'une divisibilit\u00e9 \u00e0 l'infini. 2o) Les corps simples ne sont r\u00e9els qu'en de\u00e7\u00e0 de toute perception possible. Car la perception n'appartient qu'\u00e0 des modes compos\u00e9s d'une infinit\u00e9 de parties, et ne saisit elle-m\u00eame que de tels compos\u00e9s. Les parties simples ne sont pas per\u00e7ues, mais appr\u00e9hend\u00e9es par le raisonnement : cf. _Lettre 6, \u00e0 Oldenburg_ (III, p. 21).\n\n12.  L'expos\u00e9 de la physique appara\u00eet en _\u00c9_ , II, apr\u00e8s la proposition 13. (Pour \u00e9viter toute confusion, nos r\u00e9f\u00e9rences \u00e0 cet expos\u00e9 sont pr\u00e9c\u00e9d\u00e9es d'un ast\u00e9risque.) La th\u00e9orie des corps simples occupe : * axiomes 1 et 2, lemmes 1, 2 et 3, axiomes 1 et 2. Spinoza y insiste sur la d\u00e9termination purement extrins\u00e8que ; il est vrai qu'il parle de la \u00ab nature \u00bb du corps, au niveau des corps simples, mais la \u00ab nature \u00bb signifie seulement ici l'\u00e9tat pr\u00e9c\u00e9dent.\n\n13.  A. RIVAUD, _op. cit._ , pp. 32-34.\n\n14.  _CT_ , II, pr\u00e9face, note 1, \u00a7\u00a7 VII-XIV.\n\n15.  _\u00c9_ , II, * lemmes 4, 6 et 7.\n\n16.  _Lettre 32, \u00e0 Oldenburg_ (III, pp. 120-121).\n\n17.  _Lettre 32, \u00e0 Oldenburg._\n\n18.  _TRE_ , 101.\n\n19.  _Lettre 30, \u00e0 Oldenburg : \u00ab .._. J'ignore comment chacune de ces parties s'accorde avec le tout, comment elle se rattache aux autres \u00bb (III, p. 119.)\n\n20.  _PM_ , I, ch. 2.\n\n21.  _\u00c9_ , II, 8, cor. : distinction entre \u00ab exister en durant \u00bb et \u00ab exister en \u00e9tant _seulement_ contenu dans l'attribut. \u00bb _\u00c9_ , V, 29, cor. : distinction entre \u00ab exister en relation avec un certain temps et un certain lieu \u00bb et \u00ab exister comme contenu en Dieu et d\u00e9coulant de la n\u00e9cessit\u00e9 de sa nature. \u00bb\n\n22.  Cf. _\u00c9_ , II, 8, cor. et sc. : _non tantum... sed etiam..._\n\n23.  Cf. KANT, _Critique de la raison pure_ , 1re \u00e9dition (\u00ab critique du quatri\u00e8me paralogisme... \u00bb) : La mati\u00e8re \u00ab est une esp\u00e8ce de repr\u00e9sentations (intuitions) qu'on appelle _ext\u00e9rieures_ , non parce qu'elles se rapportent \u00e0 des objets ext\u00e9rieurs en soi, mais parce qu'elles rapportent les perceptions \u00e0 l'espace, o\u00f9 toutes choses existent les unes en dehors des autres, _tandis que l'espace lui-m\u00eame est en nous_... L'espace lui-m\u00eame, avec tous ses ph\u00e9nom\u00e8nes comme repr\u00e9sentations, n'existe qu'en moi ; dans cet espace pourtant, le r\u00e9el, ou la mati\u00e8re de tous les objets de l'intuition ext\u00e9rieure m'est donn\u00e9e v\u00e9ritablement et ind\u00e9pendamment de toute fiction. \u00bb\n\n24.  Les essences de modes, en tant qu'elles sont comprises dans l'attribut, sont d\u00e9j\u00e0 des \u00ab explications \u00bb. Ainsi Spinoza parle de l'essence de Dieu en tant qu'elle _s'explique_ par l'essence de tel ou tel mode : _\u00c9_ , IV, 4, dem. Mais il y a deux r\u00e9gimes d'explication, et le mot _expliquer_ convient particuli\u00e8rement au second.\n\n## CHAPITRE XIV\n\n## QU'EST-CE QUE PEUT UN CORPS ?\n\nLa triade expressive du mode fini se pr\u00e9sente ainsi : l'essence comme degr\u00e9 de puissance ; le rapport caract\u00e9ristique dans lequel elle s'exprime ; les parties extensives subsum\u00e9es sous ce rapport, et qui composent l'existence du mode. Mais nous voyons que, dans l' _\u00c9thique_ , un strict syst\u00e8me d'\u00e9quivalences nous conduit \u00e0 une seconde triade du mode fini : l'essence comme degr\u00e9 de puissance ; un certain pouvoir d'\u00eatre affect\u00e9 dans lequel elle s'exprime ; des affections qui remplissent \u00e0 chaque instant ce pouvoir.\n\nQuelles sont ces \u00e9quivalences ? Un mode existant poss\u00e8de actuellement un tr\u00e8s grand nombre de parties. Or, la nature des parties extensives est telle qu'elles \u00ab s'affectent \u00bb les unes les autres \u00e0 l'infini. On en conclut que le mode existant est affect\u00e9 d'un tr\u00e8s grand nombre de fa\u00e7ons. Spinoza va des parties \u00e0 leurs affections, de leurs affections aux affections du mode existant tout entier1. Les parties extensives n'appartiennent \u00e0 tel mode que sous un certain rapport. De m\u00eame, les affections d'un mode se disent en fonction d'un certain pouvoir d'\u00eatre affect\u00e9. Un cheval, un poisson, un homme, ou m\u00eame deux hommes compar\u00e9s l'un avec l'autre, n'ont pas le m\u00eame pouvoir d'\u00eatre affect\u00e9 : ils ne sont pas affect\u00e9s par les m\u00eames choses, ou ne sont pas affect\u00e9s par la m\u00eame chose de la m\u00eame fa\u00e7on2. Un mode cesse d'exister quand il ne peut plus maintenir entre ses parties le rapport qui le caract\u00e9rise ; de m\u00eame, il cesse d'exister quand \u00ab il n'est plus apte \u00e0 pouvoir \u00eatre affect\u00e9 d'un grand nombre de fa\u00e7ons3 \u00bb. Bref, un rapport n'est pas s\u00e9parable d'un pouvoir d'\u00eatre affect\u00e9. Si bien que Spinoza peut consid\u00e9rer comme \u00e9quivalentes deux questions fondamentales : _Quelle est la structure_ (fabrica) _d'un corps ? Qu'est-ce que peut un corps ?_ La structure d'un corps, c'est la composition de son rapport. Ce que peut un corps, c'est la nature et les limites de son pouvoir d'\u00eatre affect\u00e94.\n\nCette seconde triade du mode fini montre bien comment le mode exprime la substance, participe de la substance et, m\u00eame, la reproduit \u00e0 sa mani\u00e8re. Dieu se d\u00e9finissait par l'identit\u00e9 de son essence et d'une puissance absolument infinie _(potentia) ;_ comme tel, il avait une _potestas_ , c'est-\u00e0-dire un pouvoir d'\u00eatre affect\u00e9 d'une infinit\u00e9 de fa\u00e7ons ; ce pouvoir \u00e9tait \u00e9ternellement et n\u00e9cessairement rempli, Dieu \u00e9tant cause de toutes choses au m\u00eame sens que cause de soi. De son c\u00f4t\u00e9, le mode existant a une essence identique \u00e0 un degr\u00e9 de puissance ; comme tel, il a une aptitude \u00e0 \u00eatre affect\u00e9, un pouvoir d'\u00eatre affect\u00e9 d'un tr\u00e8s grand nombre de fa\u00e7ons ; tant qu'il existe, ce pouvoir est rempli de mani\u00e8re variable, mais toujours et n\u00e9cessairement rempli sous l'action des modes ext\u00e9rieurs.\n\nQuelle est, de tous ces points de vue, la diff\u00e9rence entre le mode existant et la substance divine ? En premier lieu, on ne confondra pas \u00ab infinit\u00e9 de fa\u00e7ons \u00bb et \u00ab tr\u00e8s grand nombre de fa\u00e7ons \u00bb. Un tr\u00e8s grand nombre, c'est une infinit\u00e9, mais d'un type sp\u00e9cial : infini plus ou moins grand, qui se rapporte \u00e0 quelque chose de limit\u00e9. Dieu, au contraire, est affect\u00e9 d'une infinit\u00e9 de fa\u00e7ons : c'est un infini par la cause, puisque Dieu est cause de toutes ses affections ; c'est un infini proprement illimit\u00e9, qui comprend toutes les essences de modes et tous les modes existants.\n\nSeconde diff\u00e9rence : Dieu est cause de toutes ses affections, donc ne peut pas en p\u00e2tir. On aurait tort, en effet, de confondre affection et passion. Une affection n'est une passion que quand elle ne s'explique pas par la nature du corps affect\u00e9 : sans doute l'enveloppe-t-elle, mais elle s'explique par l'influence d'autres corps. Si nous supposons des affections qui s'expliquent enti\u00e8rement par la nature du corps affect\u00e9, ces affections sont actives, sont elles-m\u00eames des actions5. Appliquons \u00e0 Dieu le principe de cette distinction : il n'y a pas de causes ext\u00e9rieures \u00e0 Dieu ; Dieu est n\u00e9cessairement cause de toutes ses affections, toutes ses affections s'expliquent par sa nature, donc sont des actions6. Il n'en est pas de m\u00eame des modes existants. Ceux-ci n'existent pas en vertu de leur propre nature ; leur existence est compos\u00e9e de parties extensives qui sont d\u00e9termin\u00e9es et affect\u00e9es du dehors, \u00e0 l'infini. Il est donc forc\u00e9 que chaque mode existant soit affect\u00e9 par des modes ext\u00e9rieurs, qu'il subisse des changements qui ne s'expliquent pas par sa seule nature. Ses affections sont d'abord et avant tout des passions7. Spinoza remarque que l'enfance est un \u00e9tat mis\u00e9rable, mais un \u00e9tat commun o\u00f9 nous d\u00e9pendons \u00ab au plus haut degr\u00e9 des causes ext\u00e9rieures8 \u00bb. La grande question qui se pose \u00e0 propos du mode existant fini est donc : Arrivera-t-il \u00e0 des affections actives, et comment ? Cette question est \u00e0 proprement parler la question \u00ab \u00e9thique \u00bb. Mais, m\u00eame \u00e0 supposer que le mode arrive \u00e0 produire des affections actives, tant qu'il existe il ne supprimera pas en lui toute passion, mais fera seulement que ses passions n'occupent plus qu'une petite partie de lui-m\u00eame9.\n\nUne derni\u00e8re diff\u00e9rence concerne le contenu m\u00eame du mot \u00ab affection \u00bb, suivant qu'on le rapporte \u00e0 Dieu ou aux modes. Car les affections de Dieu sont les modes eux-m\u00eames, essences de modes et modes existants. Leurs id\u00e9es expriment l'essence de Dieu comme cause. Mais les affections des modes sont comme des affections au second degr\u00e9, des affections d'affections : par exemple, une affection passive que nous \u00e9prouvons n'est que l'effet d'un corps sur le n\u00f4tre. L'id\u00e9e de cette affection n'exprime pas la cause, c'est-\u00e0-dire la nature ou l'essence du corps ext\u00e9rieur : elle indique plut\u00f4t la constitution pr\u00e9sente de notre corps, donc la mani\u00e8re dont notre pouvoir d'\u00eatre affect\u00e9 se trouve rempli \u00e0 tel moment. L'affection de notre corps est seulement une image corporelle, et l'id\u00e9e d'affection telle qu'elle est dans notre esprit, une id\u00e9e inad\u00e9quate ou une imagination. Nous avons encore une autre esp\u00e8ce d'affections. D'une id\u00e9e d'affection qui nous est donn\u00e9e d\u00e9coulent n\u00e9cessairement des \u00ab affects \u00bb ou sentiments _(affectus)_10. Ces sentiments sont eux-m\u00eames des affections, ou plut\u00f4t des id\u00e9es d'affections de nature originale. On \u00e9vitera de pr\u00eater \u00e0 Spinoza des th\u00e8ses intellectualistes qui ne furent jamais les siennes. En fait, une id\u00e9e que nous avons indique l'\u00e9tat actuel de la constitution de notre corps ; tant que notre corps existe, il dure et se d\u00e9finit par la dur\u00e9e ; son \u00e9tat actuel n'est donc pas s\u00e9parable d'un \u00e9tat pr\u00e9c\u00e9dent avec lequel il s'encha\u00eene dans une dur\u00e9e continue. C'est pourquoi, _\u00e0 toute id\u00e9e qui indique un \u00e9tat de notre corps est n\u00e9cessairement li\u00e9e une autre esp\u00e8ce d'id\u00e9e qui enveloppe le rapport de cet \u00e9tat avec l'\u00e9tat pass\u00e9_. Spinoza pr\u00e9cise : on ne croira pas qu'il s'agit d'une op\u00e9ration intellectuelle abstraite, par laquelle l'esprit comparerait deux \u00e9tats11. Nos sentiments, par eux-m\u00eames, sont des id\u00e9es qui enveloppent le rapport concret du pr\u00e9sent avec le pass\u00e9 dans une dur\u00e9e continue : ils enveloppent les variations d'un mode existant qui dure.\n\nLes affections donn\u00e9es d'un mode sont donc de deux sortes : \u00e9tats du corps ou id\u00e9es qui indiquent ces \u00e9tats, variations du corps ou id\u00e9es qui enveloppent ces variations. Les secondes s'encha\u00eenent avec les premi\u00e8res, varient en m\u00eame temps qu'elles : on devine comment, \u00e0 partir d'une premi\u00e8re affection, nos sentiments s'encha\u00eenent avec nos id\u00e9es, de mani\u00e8re \u00e0 remplir \u00e0 chaque instant tout notre pouvoir d'\u00eatre affect\u00e9. Mais, surtout, nous sommes toujours ramen\u00e9s \u00e0 une certaine condition du mode, qui est celle de l'homme en particulier : les id\u00e9es qui lui sont donn\u00e9es d'abord sont des affections passives, des id\u00e9es inad\u00e9quates ou des imaginations ; les affects ou sentiments qui en d\u00e9coulent sont donc des passions, des sentiments eux-m\u00eames passifs. On ne voit pas comment un mode fini, surtout au d\u00e9but de son existence, pourrait avoir autre chose que des id\u00e9es inad\u00e9quates ; on ne voit pas d\u00e8s lors comment il pourrait \u00e9prouver autre chose que des sentiments passifs. Le lien des deux est bien marqu\u00e9 par Spinoza : l'id\u00e9e inad\u00e9quate est une id\u00e9e dont nous ne sommes pas cause (elle ne s'explique pas formellement par notre puissance de comprendre) ; cette id\u00e9e inad\u00e9quate est elle-m\u00eame cause (mat\u00e9rielle et efficiente) d'un sentiment ; nous ne pouvons donc pas \u00eatre cause ad\u00e9quate de ce sentiment ; or un sentiment dont nous ne sommes pas cause ad\u00e9quate est n\u00e9cessairement une passion12. Notre pouvoir d'\u00eatre affect\u00e9 se trouve donc rempli, d\u00e8s le d\u00e9but de notre existence, par des id\u00e9es inad\u00e9quates et des sentiments passifs.\n\nUn lien aussi profond se v\u00e9rifierait entre des id\u00e9es suppos\u00e9es ad\u00e9quates et des sentiments actifs. Une id\u00e9e ad\u00e9quate en nous se d\u00e9finirait formellement comme une id\u00e9e dont nous serions cause ; elle serait cause mat\u00e9rielle et efficiente d'un sentiment ; nous serions cause ad\u00e9quate de ce sentiment lui-m\u00eame ; or un sentiment dont nous sommes cause ad\u00e9quate est une action. C'est en ce sens que Spinoza peut dire : \u00ab Dans la mesure o\u00f9 notre esprit a des id\u00e9es ad\u00e9quates, il est n\u00e9cessairement actif en certaines choses, et dans la mesure o\u00f9 il a des id\u00e9es inad\u00e9quates, il est n\u00e9cessairement passif en certaines choses \u00bb ; \u00ab Les actions de l'esprit naissent des seules id\u00e9es ad\u00e9quates ; et les passions d\u00e9pendent des seules id\u00e9es inad\u00e9quates13. \u00bb D\u00e8s lors, la question proprement \u00e9thique se trouve li\u00e9e \u00e0 la question m\u00e9thodologique : Comment arriverons-nous \u00e0 \u00eatre actifs ? Comment arriverons-nous \u00e0 produire des id\u00e9es ad\u00e9quates ?\n\nOn peut d\u00e9j\u00e0 pressentir l'importance extr\u00eame d'un domaine de l' _\u00c9thique_ , celui des variations existentielles du mode fini, variations expressives. Ces variations sont de plusieurs sortes, et doivent s'interpr\u00e9ter \u00e0 plusieurs niveaux. Soit un mode ayant telle essence et tel pouvoir d'\u00eatre affect\u00e9. Ses affections passives (id\u00e9es inad\u00e9quates et sentiments-passions) changent constamment. Toutefois, tant que son pouvoir d'\u00eatre affect\u00e9 se trouve rempli par des affections passives, ce pouvoir lui-m\u00eame se pr\u00e9sente comme une _force ou puissance de p\u00e2tir._ On appelle puissance de p\u00e2tir le pouvoir d'\u00eatre affect\u00e9, en tant qu'il se trouve actuellement rempli par des affections passives. La puissance de p\u00e2tir du corps a pour \u00e9quivalent dans l'\u00e2me la puissance d'imaginer et d'\u00e9prouver des sentiments passifs.\n\nSupposons maintenant que le mode, \u00e0 mesure qu'il dure, arrive \u00e0 remplir (au moins partiellement) son pouvoir d'\u00eatre affect\u00e9 par des affections actives. Sous cet aspect, ce pouvoir se pr\u00e9sente comme _force ou puissance d'agir._ La puissance de comprendre ou de conna\u00eetre est la puissance d'agir propre \u00e0 l'\u00e2me. Mais, pr\u00e9cis\u00e9ment, _le pouvoir d'\u00eatre affect\u00e9 reste constant, quelle que soit la proportion des affections passives et des affections actives._ Nous arrivons donc \u00e0 l'hypoth\u00e8se suivante : La proportion des affections passives et actives serait susceptible de varier, pour un m\u00eame pouvoir d'\u00eatre affect\u00e9. Si nous arrivons \u00e0 produire des affections actives, nos affections passives diminuent d'autant. Tant que nous restons dans des affections passives, notre puissance d'agir est \u00ab emp\u00each\u00e9e \u00bb d'autant. Bref, pour une m\u00eame essence, pour un m\u00eame pouvoir d'\u00eatre affect\u00e9, la puissance de p\u00e2tir et la puissance d'agir seraient susceptibles de varier en raison inversement proportionnelle. Toutes deux constituent le pouvoir d'\u00eatre affect\u00e9, dans des proportions variables14.\n\nEn second lieu, il faut faire intervenir un autre niveau de variations possibles. Car le pouvoir d'\u00eatre affect\u00e9 ne reste pas constant toujours, ni sous tous les points de vue. En effet, Spinoza sugg\u00e8re que le rapport qui caract\u00e9rise un mode existant dans son ensemble est dou\u00e9 d'une sorte d'\u00e9lasticit\u00e9. Bien plus, sa composition passe par tant de moments, et aussi sa d\u00e9composition, qu'on peut presque dire qu'un mode change de corps ou de rapport en sortant de l'enfance, ou en entrant dans la vieillesse. Croissance, vieillissement, maladie : nous avons peine \u00e0 reconna\u00eetre un m\u00eame individu. Et encore, est-ce bien ce m\u00eame individu ? Ces changements, insensibles ou brusques, dans le rapport qui caract\u00e9rise un corps, nous les constatons aussi dans son pouvoir d'\u00eatre affect\u00e9, comme si pouvoir et rapport jouissaient d'une marge, d'une limite dans laquelle ils se forment et se d\u00e9forment15. Certains passages de la _Lettre \u00e0 Meyer_ prennent ici tous leur sens, qui font allusion \u00e0 l'existence d'un maximum et d'un minimum.\n\nPr\u00e9c\u00e9demment, nous avons fait comme si la puissance de p\u00e2tir et la puissance d'agir formaient deux principes distincts, dont l'exercice \u00e9tait inversement proportionnel pour un m\u00eame pouvoir d'\u00eatre affect\u00e9. Et c'est vrai, mais seulement dans le cadre des limites extr\u00eames de ce pouvoir. C'est vrai, tant que nous consid\u00e9rons les affections abstraitement, sans consid\u00e9rer concr\u00e8tement l'essence du mode affect\u00e9. Pourquoi ? Nous sommes au seuil d'un probl\u00e8me qui trouve son d\u00e9veloppement chez Leibniz aussi bien que chez Spinoza. Ce n'est pas par hasard que Leibniz, \u00e0 sa premi\u00e8re lecture de l' _\u00c9thique_ , d\u00e9clare admirer la th\u00e9orie des affections chez Spinoza, la conception spinoziste de l'action et de la passion. Plus qu'\u00e0 une influence de Spinoza sur Leibniz, il faut penser \u00e0 une co\u00efncidence dans le d\u00e9veloppement de leurs philosophies respectives16. Cette co\u00efncidence est d'autant plus remarquable. \u00c0 un certain niveau, Leibniz pr\u00e9sente la th\u00e8se suivante : la force d'un corps, dite \u00ab force d\u00e9rivative \u00bb, est double ; elle est force d'agir et force de p\u00e2tir, force active et force passive ; la force active reste \u00ab morte \u00bb ou devient \u00ab vive \u00bb, suivant les obstacles ou les sollicitations qu'elle rencontre, enregistr\u00e9s par la force passive. Mais, \u00e0 un niveau plus profond, Leibniz demande : la force passive doit-elle \u00eatre con\u00e7ue comme distincte de la force active ? Est-elle autonome dans son principe, a-t-elle une positivit\u00e9 quelconque, affirme-t-elle quelque chose ? La r\u00e9ponse est : seule la force active est r\u00e9elle en droit, positive et affirmative. La force passive n'affirme rien, n'exprime rien sauf l'imperfection du fini. Tout se passe comme si la force active avait h\u00e9rit\u00e9 de tout ce qui est r\u00e9el, positif ou parfait dans le fini lui-m\u00eame. La force passive n'est pas une force autonome, mais la simple limitation de la force active. Elle ne serait pas une force sans la force active qu'elle limite. Elle signifie la limitation inh\u00e9rente \u00e0 la force active ; et finalement la limitation d'une force encore plus profonde, c'est-\u00e0-dire d'une _essence_ qui s'affirme et s'exprime uniquement dans la force active en tant que telle17.\n\nSpinoza aussi pr\u00e9sente une premi\u00e8re th\u00e8se : la puissance de p\u00e2tir et la puissance d'agir sont deux puissances qui varient corr\u00e9lativement, le pouvoir d'\u00eatre affect\u00e9 restant le m\u00eame ; la puissance d'agir est morte ou vive (Spinoza dit : emp\u00each\u00e9e ou aid\u00e9e) suivant les obstacles ou les occasions qu'elle trouve du c\u00f4t\u00e9 des affections passives. Mais cette th\u00e8se, vraie physiquement, n'est pas vraie m\u00e9taphysiquement. Chez Spinoza d\u00e9j\u00e0, \u00e0 un niveau plus profond, la puissance de p\u00e2tir n'exprime rien de positif. Dans toute affection passive, il y a quelque chose d'imaginaire qui l'emp\u00eache d'\u00eatre r\u00e9elle. Nous ne sommes passifs et passionn\u00e9s qu'en raison de notre imperfection, par notre imperfection m\u00eame. \u00ab Car il est certain que l'agent agit par ce qu'il a, et que le patient p\u00e2tit par ce qu'il n'a pas \u00bb ; \u00ab Le p\u00e2tir, dans lequel l'agent et le patient sont distincts, est une imperfection palpable18. \u00bb Nous p\u00e2tissons d'une chose ext\u00e9rieure, distincte de nous-m\u00eames ; nous avons donc nous-m\u00eames une force de p\u00e2tir et une force d'agir distinctes. Mais notre force de p\u00e2tir est seulement l'imperfection, la finitude ou la limitation de notre force d'agir en elle-m\u00eame. Notre force de p\u00e2tir n' _affirme_ rien, parce qu'elle n' _exprime_ rien du tout : elle \u00ab enveloppe \u00bb seulement notre impuissance, c'est-\u00e0-dire la limitation de notre puissance d'agir. En v\u00e9rit\u00e9, notre puissance de p\u00e2tir est notre impuissance, notre servitude, c'est-\u00e0-dire _le plus bas degr\u00e9 de notre puissance d'agir_ : d'o\u00f9, le titre du livre IV de l' _\u00c9thique_ , \u00ab De la servitude de l'homme \u00bb. La puissance d'imaginer est bien une puissance ou une vertu, dit Spinoza, mais le serait encore plus si elle d\u00e9pendait de notre nature, c'est-\u00e0-dire si elle \u00e9tait active, au lieu de signifier seulement la finitude ou l'imperfection de notre puissance d'agir, bref, notre impuissance19.\n\nNous ne savons pas encore comment nous arriverons \u00e0 produire des affections actives ; nous ne savons donc pas quelle est notre puissance d'agir. Et pourtant, nous pouvons dire ceci : la puissance d'agir est la seule forme r\u00e9elle, positive et affirmative d'un pouvoir d'\u00eatre affect\u00e9. Tant que notre pouvoir d'\u00eatre affect\u00e9 se trouve rempli par des affections passives, il est r\u00e9duit \u00e0 son minimum, et manifeste seulement notre finitude ou notre limitation. Tout se passe comme si, dans l'existence du mode fini, se produisait une disjonction : le n\u00e9gatif tombe du c\u00f4t\u00e9 des affections passives, les affections actives exprimant tout le positif du mode fini. En effet, les affections actives sont les seules \u00e0 remplir r\u00e9ellement et positivement le pouvoir d'\u00eatre affect\u00e9. La puissance d'agir, \u00e0 elle seule, est identique au pouvoir d'\u00eatre affect\u00e9 tout entier ; la puissance d'agir, \u00e0 elle seule, exprime l'essence, et les affections actives, elles seules, affirment l'essence. Dans le mode existant, l'essence ne fait qu'un avec la puissance d'agir, la puissance d'agir ne fait qu'un avec le pouvoir d'\u00eatre affect\u00e9.\n\nOn voit se concilier chez Spinoza deux inspirations fondamentales. D'apr\u00e8s l'inspiration physique : un pouvoir d'\u00eatre affect\u00e9 reste constant pour une m\u00eame essence, qu'il soit rempli par des affections actives ou des affections passives ; le mode est donc toujours aussi parfait qu'il peut l'\u00eatre. Mais d'apr\u00e8s l'inspiration \u00e9thique, le pouvoir d'\u00eatre affect\u00e9 n'est constant que dans des limites extr\u00eames. Tant qu'il est rempli par des affections passives, il est r\u00e9duit \u00e0 son minimum ; nous restons alors imparfaits et impuissants, nous sommes en quelque sorte s\u00e9par\u00e9s de notre essence ou de notre degr\u00e9 de puissance, s\u00e9par\u00e9s de ce que nous pouvons. Il est bien vrai que le mode existant est toujours aussi parfait qu'il peut l'\u00eatre : mais seulement en fonction des affections qui appartiennent actuellement \u00e0 son essence. Il est bien vrai que les affections passives que nous \u00e9prouvons remplissent notre pouvoir d'\u00eatre affect\u00e9 ; mais elles l'ont d'abord r\u00e9duit \u00e0 son minimum, elles nous ont d'abord s\u00e9par\u00e9s de ce que nous pouvions (puissance d'agir). Les variations expressives du mode fini ne consistent donc pas seulement en variations m\u00e9caniques des affections \u00e9prouv\u00e9es, elles consistent encore en variations dynamiques du pouvoir d'\u00eatre affect\u00e9, et en variations \u00ab m\u00e9taphysiques \u00bb de l'essence elle-m\u00eame : tant que le mode existe, son essence m\u00eame est susceptible de varier suivant les affections qui lui appartiennent \u00e0 tel moment20.\n\nD'o\u00f9 l'importance de la question \u00e9thique. _Nous ne savons m\u00eame pas ce que peut un corps_ , dit Spinoza21. C'est-\u00e0-dire : _Nous ne savons m\u00eame pas de quelles affections nous sommes capables, ni jusqu'o\u00f9 va notre puissance_. Comment pourrions-nous le savoir \u00e0 l'avance ? D\u00e8s le d\u00e9but de notre existence, nous sommes n\u00e9cessairement remplis d'affections passives. Le mode fini na\u00eet dans des conditions telles que, \u00e0 l'avance, il est s\u00e9par\u00e9 de son essence ou de son degr\u00e9 de puissance, s\u00e9par\u00e9 de ce qu'il peut, de sa puissance d'agir. La puissance d'agir, nous pouvons savoir par raisonnement qu'elle est la seule expression de notre essence, la seule affirmation de notre pouvoir d'\u00eatre affect\u00e9. Mais ce savoir reste abstrait. Nous ne savons pas quelle est cette puissance d'agir, ni comment l'acqu\u00e9rir ou la retrouver. Et sans doute ne le saurons-nous jamais, si nous ne tentons pas concr\u00e8tement de devenir actifs. L' _\u00c9thique_ se termine sur le rappel suivant : la plupart des hommes ne se sentent exister que quand ils p\u00e2tissent. Ils ne supportent l'existence qu'en p\u00e2tissant ; \u00ab sit\u00f4t qu'il cesse de p\u00e2tir, (l'ignorant) cesse en m\u00eame temps d'\u00eatre \u00bb22.\n\nLeibniz prit l'habitude de caract\u00e9riser le syst\u00e8me de Spinoza par l'impuissance o\u00f9 les cr\u00e9atures se trouveraient r\u00e9duites : la th\u00e9orie des modes ne serait qu'un moyen de retirer aux cr\u00e9atures toute activit\u00e9, tout dynamisme, toute individualit\u00e9, toute r\u00e9alit\u00e9 authentique. Les modes seraient seulement des fantasmes, des fant\u00f4mes, des projections fantastiques d'une Substance unique. Et, de ce caract\u00e8re pr\u00e9sent\u00e9 comme crit\u00e8re, Leibniz se sert pour interpr\u00e9ter d'autres philosophies, pour y d\u00e9noncer soit les appr\u00eats d'un spinozisme \u00e9bauch\u00e9, soit les s\u00e9quelles d'un spinozisme cach\u00e9 : ainsi Descartes est le p\u00e8re du spinozisme parce qu'il croit \u00e0 l'existence d'une \u00e9tendue inerte et passive ; les occasionalistes sont spinozistes sans le vouloir dans la mesure o\u00f9 ils retirent aux choses toute action et tout principe d'agir. Cette critique d'un spinozisme g\u00e9n\u00e9ralis\u00e9e est habile ; mais il n'est pas s\u00fbr que Leibniz lui-m\u00eame y ait cru. (Sinon, pourquoi aurait-il tant admir\u00e9 la th\u00e9orie spinoziste de l'action et de la passion dans le mode ?)\n\nIl est clair, en tout cas, que tout d\u00e9ment cette interpr\u00e9tation dans l'\u0153uvre de Spinoza. Spinoza rappelle constamment qu'on ne peut pas, sans les d\u00e9naturer, confondre les modes avec des \u00eatres de raison ou avec des \u00ab auxiliaires de l'imagination \u00bb. Quand il parle des modifications, il cherche des principes sp\u00e9cifiquement modaux, soit pour conclure de l'unit\u00e9 de la substance \u00e0 l'unit\u00e9 ontologique des modes qui diff\u00e8rent par l'attribut, soit pour conclure de l'unit\u00e9 de la substance \u00e0 l'unit\u00e9 syst\u00e9matique des modes contenus dans un seul et m\u00eame attribut. Et surtout l'id\u00e9e m\u00eame de mode n'est jamais un moyen de retirer toute puissance propre \u00e0 la cr\u00e9ature : au contraire, selon Spinoza, c'est le seul moyen de montrer comment les choses \u00ab participent \u00bb \u00e0 la puissance de Dieu, c'est-\u00e0-dire sont des parties de la puissance divine, mais des parties singuli\u00e8res, des quantit\u00e9s intensives ou des degr\u00e9s irr\u00e9ductibles. Comme dit Spinoza, la puissance de l'homme est une \u00ab partie \u00bb de la puissance ou de l'essence de Dieu, mais seulement en tant que l'essence de Dieu _s'explique_ elle-m\u00eame par l'essence de l'homme23.\n\nEn fait Leibniz et Spinoza ont un projet commun. Leurs philosophies constituent les deux aspects d'un nouveau \u00ab naturalisme \u00bb. Ce naturalisme est le vrai sens de la r\u00e9action anticart\u00e9sienne. Dans des pages tr\u00e8s belles, Ferdinand Alqui\u00e9 a montr\u00e9 comment Descartes avait domin\u00e9 la premi\u00e8re moiti\u00e9 du XVIIe si\u00e8cle en poussant jusqu'au bout l'entreprise d'une science math\u00e9matique et m\u00e9canicienne ; le premier effet de celle-ci \u00e9tait de d\u00e9valoriser la Nature, en lui retirant toute virtualit\u00e9 ou potentialit\u00e9, tout pouvoir immanent, tout \u00eatre inh\u00e9rent. La m\u00e9taphysique cart\u00e9sienne compl\u00e8te la m\u00eame entreprise, parce qu'elle cherche l'\u00eatre hors de la nature, dans un sujet qui la pense et dans un Dieu qui la cr\u00e9e24. Dans la r\u00e9action anticart\u00e9sienne, au contraire, il s'agit de restaurer les droits d'une Nature dou\u00e9e de forces ou de puissance. Mais, aussi, il s'agit de conserver l'acquis du m\u00e9canisme cart\u00e9sien : toute puissance est actuelle et en acte ; les puissances de la nature ne sont plus des virtualit\u00e9s faisant appel \u00e0 des entit\u00e9s occultes, \u00e0 des \u00e2mes ou des esprits qui les r\u00e9alisent. Leibniz formule parfaitement ce programme : contre Descartes, redonner \u00e0 la Nature sa force d'agir et de p\u00e2tir, mais sans retomber dans une vision pa\u00efenne du monde, dans une idol\u00e2trie de la Nature25. Le programme de Spinoza est tout \u00e0 fait semblable (\u00e0 ceci pr\u00e8s qu'il ne compte pas sur le christianisme pour nous sauver de l'idol\u00e2trie). Spinoza et Leibniz s'en prennent \u00e0 Boyle comme au repr\u00e9sentant d'un m\u00e9canisme content de soi. Si Boyle voulait seulement nous apprendre que tout, dans les corps, se fait par figure et mouvement, la le\u00e7on serait mince, \u00e9tant bien connue depuis Descartes26. Mais, pour tel corps, quelles figures et quels mouvements ? Pourquoi _telle_ figure, _tel_ mouvement ? Alors on verra que le m\u00e9canisme n'exclut pas l'id\u00e9e d'une nature ou essence de chaque corps, mais l'exige au contraire, comme la raison suffisante de telle figure, de tel mouvement, de telle proportion de mouvement et de repos. La r\u00e9action anti-cart\u00e9sienne est partout recherche d'une raison suffisante : raison suffisante pour l'infiniment parfait, raison suffisante pour le clair et le distinct, enfin raison suffisante pour le m\u00e9canisme lui-m\u00eame.\n\nLe nouveau programme se r\u00e9alise, chez Leibniz, \u00e0 travers trois niveaux distincts. D'abord tout se passe dans les corps m\u00e9caniquement, par figure et mouvement. Mais les corps sont des \u00ab agr\u00e9gats \u00bb, actuellement et infiniment compos\u00e9s, r\u00e9gis par des lois. Or le mouvement ne contient aucune marque distinctive d'un corps \u00e0 un moment donn\u00e9 ; les figures qu'il constitue ne sont donc pas davantage discernables en divers moments. Ce sont les mouvements eux-m\u00eames qui supposent des forces, de p\u00e2tir et d'agir, sans lesquelles les corps ne se distingueraient pas plus que leurs figures. Ou, si l'on pr\u00e9f\u00e8re, ce sont les lois m\u00e9caniques elles-m\u00eames qui supposent une nature intime des corps qu'elles r\u00e9gissent. Car ces lois ne seraient pas \u00ab ex\u00e9cutables \u00bb si elles conf\u00e9raient aux corps une simple d\u00e9termination extrins\u00e8que et s'imposaient \u00e0 eux ind\u00e9pendamment de ce qu'ils sont : c'est en ce sens que l'effet d'une loi ne peut pas seulement \u00eatre compris dans la volont\u00e9 de Dieu, comme le croient les Occasionnalistes, mais doit l'\u00eatre aussi _dans le corps lui-m\u00eame._ D\u00e8s lors, aux agr\u00e9gats comme tels, il faut attribuer des forces d\u00e9rivatives : \u00ab la nature inh\u00e9rente aux choses ne se distingue pas de la force d'agir et de p\u00e2tir27. \u00bb Mais la force d\u00e9rivative \u00e0 son tour ne contient pas sa propre raison : elle est purement instantan\u00e9e, bien qu'elle relie l'instant aux pr\u00e9c\u00e9dents et aux futurs. Elle renvoie donc \u00e0 une loi de s\u00e9rie des instants, qui est comme une force primitive ou _essence_ individuelle. Simples et actives, ces essences sont la source des forces d\u00e9rivatives qui s'attribuent aux corps. Elles forment enfin une v\u00e9ritable m\u00e9taphysique de la nature, qui n'intervient pas dans la physique, mais qui correspond \u00e0 cette physique elle-m\u00eame.\n\nOr, chez Spinoza, la r\u00e9alisation du programme naturaliste est fort analogue. Le m\u00e9canisme r\u00e9git des corps existants infiniment compos\u00e9s. Mais ce m\u00e9canisme renvoie d'abord \u00e0 une th\u00e9orie dynamique du pouvoir d'\u00eatre affect\u00e9 (puissance d'agir et de p\u00e2tir) ; et, en derni\u00e8re instance, \u00e0 une th\u00e9orie de l'essence particuli\u00e8re, qui s'exprime dans les variations de cette puissance d'agir et de p\u00e2tir. Chez Spinoza comme chez Leibniz, trois niveaux se distinguent : m\u00e9canisme, force, essence. C'est pourquoi la v\u00e9ritable opposition des deux philosophes ne doit pas \u00eatre cherch\u00e9e dans la critique tr\u00e8s g\u00e9n\u00e9rale de Leibniz, quand il soutient que le Spinozisme prive les cr\u00e9atures de tout pouvoir et de toute activit\u00e9. Leibniz lui-m\u00eame r\u00e9v\u00e8le les vraies raisons de son opposition, bien qu'il les relie \u00e0 ce pr\u00e9texte. En v\u00e9rit\u00e9, il s'agit de raisons pratiques, portant sur le probl\u00e8me du mal, de la providence et de la religion, portant sur la conception pratique du r\u00f4le de la philosophie dans son ensemble.\n\nToutefois, il est certain que ces divergences ont aussi une forme sp\u00e9culative. Nous croyons que l'essentiel, \u00e0 cet \u00e9gard, concerne la notion de _conatus_ chez Spinoza et chez Leibniz. Suivant Leibniz, _conatus_ a deux sens : physiquement, il d\u00e9signe la tendance d'un corps au mouvement ; m\u00e9taphysiquement, la tendance d'une essence \u00e0 l'existence. Il ne peut en \u00eatre ainsi chez Spinoza. Les essences de modes ne sont pas des \u00ab possibles \u00bb ; elles ne manquent de rien, elles sont tout ce qu'elles sont, m\u00eame quand les modes correspondants n'existent pas. Elles n'enveloppent donc aucune tendance \u00e0 passer \u00e0 l'existence. Le _conatus_ est bien l'essence du mode (ou degr\u00e9 de puissance), _mais une fois que le mode a commenc\u00e9 d'exister._ Un mode vient \u00e0 exister quand des parties extensives sont d\u00e9termin\u00e9es du dehors \u00e0 entrer sous le rapport qui le caract\u00e9rise : alors et alors seulement, son essence elle-m\u00eame est d\u00e9termin\u00e9e comme _conatus_. Le _conatus_ chez Spinoza n'est donc que l'effort de pers\u00e9v\u00e9rer dans l'existence, une fois celle-ci donn\u00e9e. Il d\u00e9signe la fonction existentielle de l'essence, c'est-\u00e0-dire l'affirmation de l'essence dans l'existence du mode. C'est pourquoi, si nous consid\u00e9rons un corps existant, le _conatus_ ne peut pas davantage \u00eatre une tendance au mouvement. Les corps simples sont d\u00e9termin\u00e9s du dehors au mouvement ; ils ne pourraient pas l'\u00eatre s'ils n'\u00e9taient \u00e9galement d\u00e9terminables au repos. Chez Spinoza se retrouve constamment la th\u00e8se antique selon laquelle le mouvement ne serait rien si le repos n'\u00e9tait pas quelque chose aussi28. Le _conatus_ d'un corps simple ne peut \u00eatre qu'un effort de conserver l'\u00e9tat auquel il a \u00e9t\u00e9 d\u00e9termin\u00e9 ; le _conatus_ d'un corps compos\u00e9, un effort pour conserver le rapport de mouvement et de repos qui le d\u00e9finit, c'est-\u00e0-dire pour maintenir des parties toujours nouvelles sous ce rapport qui d\u00e9finit son existence.\n\nLes caract\u00e8res dynamiques du _conatus_ s'encha\u00eenent avec les caract\u00e8res m\u00e9caniques. Le _conatus_ d'un corps compos\u00e9 est aussi bien l'effort de maintenir ce corps apte \u00e0 \u00eatre affect\u00e9 d'un grand nombre de fa\u00e7ons29. Or, comme les affections passives remplissent \u00e0 leur mani\u00e8re notre pouvoir d'\u00eatre affect\u00e9, nous nous effor\u00e7ons de pers\u00e9v\u00e9rer dans l'existence, non seulement en tant que nous sommes suppos\u00e9s avoir des id\u00e9es ad\u00e9quates et des sentiments actifs, mais en tant que nous avons des id\u00e9es inad\u00e9quates et que nous \u00e9prouvons des passions30. Le _conatus_ d'un mode existant n'est donc pas s\u00e9parable des affections que le mode \u00e9prouve \u00e0 chaque instant. Deux cons\u00e9quences s'ensuivent.\n\nUne affection, quelle qu'elle soit, est dite d\u00e9terminer le _conatus_ ou l'essence. Le _conatus_ , en tant que d\u00e9termin\u00e9 par une affection ou un sentiment qui nous est actuellement donn\u00e9, s'appelle \u00ab d\u00e9sir \u00bb ; comme tel, il s'accompagne n\u00e9cessairement de conscience31. \u00c0 la liaison des sentiments avec les id\u00e9es, nous devons joindre une nouvelle liaison, des d\u00e9sirs avec les sentiments. Tant que notre pouvoir d'\u00eatre affect\u00e9 reste rempli par des affections passives, le _conatus_ est d\u00e9termin\u00e9 par des passions ou, comme dit Spinoza, nos d\u00e9sirs eux-m\u00eames \u00ab naissent \u00bb de passions. Mais, m\u00eame dans ce cas, notre puissance d'agir est mise en jeu. En effet, l'on doit distinguer ce qui nous d\u00e9termine et ce \u00e0 quoi nous sommes d\u00e9termin\u00e9s. Une affection passive \u00e9tant donn\u00e9e, elle nous d\u00e9termine \u00e0 faire ceci ou cela, \u00e0 penser \u00e0 ceci ou \u00e0 cela, par quoi nous nous effor\u00e7ons de conserver notre rapport ou de maintenir notre pouvoir. Tant\u00f4t nous nous effor\u00e7ons d'\u00e9carter une affection qui ne nous convient pas, tant\u00f4t de retenir une affection qui nous convient, et toujours avec un d\u00e9sir d'autant plus grand que l'affection m\u00eame est grande32. Mais \u00ab ce \u00e0 quoi \u00bb nous sommes ainsi d\u00e9termin\u00e9s s'explique par notre nature ou notre essence, et renvoie \u00e0 notre puissance d'agir33. Il est vrai que l'affection passive t\u00e9moigne de notre impuissance et nous s\u00e9pare de ce que nous pouvons ; mais il est vrai aussi qu'elle _enveloppe_ un degr\u00e9, si bas soit-il, de notre puissance d'agir. Si nous sommes en quelque sorte s\u00e9par\u00e9s de ce que nous pouvons, c'est parce que notre puissance d'agir est immobilis\u00e9e, fix\u00e9e, d\u00e9termin\u00e9e \u00e0 investir l'affection passive. Mais, en ce sens, le _conatus_ est toujours identique \u00e0 la puissance d'agir elle-m\u00eame. Les variations du _conatus_ en tant qu'il est d\u00e9termin\u00e9 par telle ou telle affection sont les variations dynamiques de notre puissance d'agir34.\n\nQuelle est la vraie diff\u00e9rence entre Leibniz et Spinoza, d'o\u00f9 d\u00e9coulent aussi toutes les oppositions pratiques ? Chez Spinoza non moins que chez Leibniz l'id\u00e9e d'une Nature expressive est \u00e0 la base du nouveau naturalisme. Chez Spinoza non moins que chez Leibniz l'expression dans la Nature signifie que le m\u00e9canisme est d\u00e9pass\u00e9 de deux fa\u00e7ons. Le m\u00e9canisme renvoie, d'une part, \u00e0 un dynamisme du pouvoir d'\u00eatre affect\u00e9, d\u00e9fini par les variations d'une puissance d'agir et de p\u00e2tir ; d'autre part, \u00e0 la position d'essences singuli\u00e8res d\u00e9finies comme des degr\u00e9s de puissance. Mais les deux philosophes ne proc\u00e8dent pas du tout de la m\u00eame mani\u00e8re. Si Leibniz reconna\u00eet aux choses une force inh\u00e9rente et propre, c'est en faisant des essences individuelles autant de substances. Au contraire, chez Spinoza, c'est en d\u00e9finissant les essences particuli\u00e8res comme des essences de modes, et plus g\u00e9n\u00e9ralement, en faisant des choses m\u00eames les modes d'une substance unique. Mais cette diff\u00e9rence reste impr\u00e9cise. En v\u00e9rit\u00e9, chez Leibniz, le m\u00e9canisme renvoie \u00e0 ce qui le d\u00e9passe comme aux exigences d'une finalit\u00e9 qui reste en partie transcendante. Si les essences sont d\u00e9termin\u00e9es comme substances, si elles ne sont pas s\u00e9parables d'une tendance \u00e0 passer \u00e0 l'existence, c'est parce qu'elles sont prises dans un ordre de finalit\u00e9 en fonction duquel elles sont choisies par Dieu, ou m\u00eame sont simplement soumises \u00e0 ce choix. Et la finalit\u00e9 qui pr\u00e9side ainsi \u00e0 la constitution du monde se retrouve dans le d\u00e9tail de ce monde : les forces d\u00e9rivatives t\u00e9moignent d'une harmonie analogue, en vertu de laquelle le monde est le meilleur jusque dans ses parties elles-m\u00eames. Et non seulement il y a des principes de finalit\u00e9 qui r\u00e9gissent les substances et les forces d\u00e9rivatives, mais il y a un accord final entre le m\u00e9canisme lui-m\u00eame et la finalit\u00e9. D\u00e8s lors, la Nature expressive chez Leibniz est une nature dont les diff\u00e9rents niveaux se hi\u00e9rarchisent, s'harmonisent et, surtout, \u00ab symbolisent entre eux \u00bb. L'expression suivant Leibniz ne sera jamais s\u00e9par\u00e9e d'une symbolisation dont le principe est toujours la finalit\u00e9 ou l'accord final.\n\nChez Spinoza, le m\u00e9canisme renvoie \u00e0 ce qui le d\u00e9passe, mais comme aux exigences d'une causalit\u00e9 pure absolument immanente. Seule la causalit\u00e9 nous fait penser l'existence ; elle suffit \u00e0 nous la faire penser. Du point de vue de la causalit\u00e9 immanente, les modes ne sont pas des apparences d\u00e9nu\u00e9es de force et d'essence. Spinoza compte sur cette causalit\u00e9 bien comprise pour doter les choses d'une force ou puissance propre, qui leur revient pr\u00e9cis\u00e9ment en tant qu'elles sont des modes. \u00c0 la diff\u00e9rence de Leibniz, le dynamisme et l'\u00ab essentialisme \u00bb de Spinoza excluent d\u00e9lib\u00e9r\u00e9ment toute finalit\u00e9. La th\u00e9orie spinoziste du _conatus_ n'a pas d'autre fonction : montrer ce qu'est le dynamisme, en lui retirant toute signification finaliste. Si la Nature est expressive, ce n'est pas au sens o\u00f9 ses diff\u00e9rents niveaux symboliseraient les uns avec les autres ; signe, symbole, harmonie sont exclus des vraies puissances de la Nature. _La triade compl\u00e8te du mode se pr\u00e9sente ainsi_ : une essence de mode s'exprime dans un rapport caract\u00e9ristique ; ce rapport exprime un pouvoir d'\u00eatre affect\u00e9 ; ce pouvoir est rempli par des affections variables, comme ce rapport, effectu\u00e9 par des parties qui se renouvellent. Entre ces diff\u00e9rents niveaux de l'expression, on ne trouvera nulle correspondance finale, nulle harmonie morale. On ne trouvera que l'encha\u00eenement n\u00e9cessaire des diff\u00e9rents effets d'une cause immanente. Aussi bien chez Spinoza n'y a-t-il pas une m\u00e9taphysique des essences, une dynamique des forces, une m\u00e9canique des ph\u00e9nom\u00e8nes. Tout est \u00ab physique \u00bb dans la Nature : physique de la quantit\u00e9 intensive qui correspond aux essences de modes ; physique de la quantit\u00e9 extensive, c'est-\u00e0-dire m\u00e9canisme par lequel les modes eux-m\u00eames passent \u00e0 l'existence ; physique de la force, c'est-\u00e0-dire dynamisme d'apr\u00e8s lequel l'essence s'affirme dans l'existence, \u00e9pousant les variations de la puissance d'agir. Les attributs s'expliquent dans les modes existants ; les essences de modes, elles-m\u00eames contenues dans les attributs, s'expliquent dans des rapports ou des pouvoirs ; ces rapports sont effectu\u00e9s par des parties, ces pouvoirs par des affections qui les expliquent \u00e0 leur tour. L'expression dans la Nature n'est jamais une symbolisation finale, mais toujours et partout une _explication_ causale.\n\n* * *\n\n1.  Cf. _\u00c9_ , II, 28, dem. : \u00ab Les affections sont les fa\u00e7ons dont les parties du corps humain, et cons\u00e9quemment le corps tout entier est affect\u00e9. \u00bb De m\u00eame II, * postulat 3.\n\n2.  _\u00c9_ , III, 51, prop. et dem., et 57 sc.\n\n3.  _\u00c9_ , IV, 39, dem.\n\n4.  _\u00c9_ , III, 2, sc. : \u00ab Personne en effet n'a jusqu'\u00e0 pr\u00e9sent d\u00e9termin\u00e9 _ce que peut_ le corps... Car personne, jusqu'\u00e0 pr\u00e9sent, n'a connu la _structure_ du corps. \u00bb\n\n5.  _\u00c9_ , III, d\u00e9f. 1-3.\n\n6.  _CT_ , II, ch. 26, 7-8.\n\n7.  _\u00c9_ , IV, 4, prop., dem. et cor.\n\n8.  _\u00c9_ , V, 6, sc. ; et 39, sc.\n\n9.  Cf. _\u00c9_ , V, 20, sc.\n\n10.  L'affect, le sentiment, suppose une id\u00e9e et en d\u00e9coule : _CT_ , Appendice II, 7 ; _\u00c9_ , II, axiome 3.\n\n11.  _\u00c9_ , III, d\u00e9finition g\u00e9n\u00e9rale des affects : \u00ab Je n'entends pas que l'esprit compare la pr\u00e9sente constitution du corps avec une pass\u00e9e, mais que l'id\u00e9e qui constitue la forme de l'affect affirme du corps quelque chose qui enveloppe \u00e0 la v\u00e9rit\u00e9 plus ou moins de r\u00e9alit\u00e9 qu'auparavant. \u00bb\n\n12.  _Ad\u00e9quat_ et _inad\u00e9quat_ qualifient d'abord des id\u00e9es. Mais, en second lieu, ce sont les qualifications d'une cause : nous sommes \u00ab cause ad\u00e9quate \u00bb d'un sentiment qui suit d'une id\u00e9e ad\u00e9quate que nous avons.\n\n13.  _\u00c9_ , III, 1 et 3.\n\n14.  Le pouvoir d'\u00eatre affect\u00e9 se d\u00e9finit comme l'aptitude d'un corps, aussi bien \u00e0 p\u00e2tir qu'\u00e0 agir : cf. _\u00c9_ , II, 13, sc. (\u00ab plus un corps est apte, par rapport aux autres, \u00e0 agir et \u00e0 p\u00e2tir de plus de fa\u00e7ons \u00e0 la fois... \u00bb) ; IV, 38, prop. (\u00ab plus le corps est rendu apte \u00e0 \u00eatre affect\u00e9 et \u00e0 affecter d'autres corps de plusieurs fa\u00e7ons... \u00bb)\n\n15.  _\u00c9_ , IV, 39, sc. : \u00ab Il arrive parfois qu'un homme subit de tels changements que je ne dirais pas ais\u00e9ment qu'il est le m\u00eame. C'est ce que j'ai entendu raconter de certain po\u00e8te espagnol... Et si cela semble incroyable, que dirons-nous des enfants ? Un homme d'\u00e2ge avanc\u00e9 croit leur nature si diff\u00e9rente de la sienne qu'il ne pourrait se persuader qu'il a jamais \u00e9t\u00e9 enfant s'il ne conjecturait de lui-m\u00eame d'apr\u00e8s les autres. \u00bb\n\n16.  Les notes de Leibniz t\u00e9moignent d'un int\u00e9r\u00eat constant pour la th\u00e9orie de l'action et de la passion chez Spinoza : cf. par exemple un texte post\u00e9rieur \u00e0 1704, \u00e9d. Grua, t. II, pp. 667 sq. Leibniz s'exprimera souvent dans des termes analogues \u00e0 ceux de Spinoza : G. Friedmann l'a bien montr\u00e9, _Leibniz et Spinoza_ , N.R.F., 1946, p. 201.\n\n17.  Cf. LEIBNIZ, _De la nature en elle-m\u00eame..._ (1698), \u00a7 11. Ce rapport de la force passive et de la force active est analys\u00e9 par M. GU\u00c9ROULT, _Dynamique et m\u00e9taphysique leibniziennes_ , Les Belles Lettres 1934, pp. 166-169.\n\n18.  _CT_ , II, ch. 26, 7 ; et I, ch. 2, 23. Cf. _\u00c9_ , III, 3, sc. : \u00ab Les passions ne se rapportent \u00e0 l'esprit qu'en tant qu'il a quelque chose qui enveloppe une n\u00e9gation. \u00bb\n\n19.  _\u00c9_ , II, 17, sc.\n\n20.  C'est pourquoi Spinoza, en _\u00c9_ , III (d\u00e9finition du d\u00e9sir), emploie les mots : \u00ab affection _de_ l'essence \u00bb, _affectionem humanae essentiae._\n\n21.  _\u00c9_ , III, 2, sc. : \u00ab On ne sait pas ce que peut le corps ou ce que l'on peut d\u00e9duire de la seule consid\u00e9ration de sa nature. \u00bb\n\n22.  _\u00c9_ , V, 42, sc.\n\n23.  _\u00c9_ , IV, 4, dem.\n\n24.  Cf. F. ALQUI\u00c9, _Descartes, l'homme et l'\u0153uvre_ , Hatier-Boivin, 1956, pp. 54-55. Il est vrai que Descartes, dans ses derni\u00e8res \u0153uvres, revient \u00e0 des consid\u00e9rations naturalistes, mais n\u00e9gatives plut\u00f4t que positives. (F. ALQUI\u00c9, _La D\u00e9couverte m\u00e9taphysique de l'homme chez Descartes_ , P.U.F., 1950, pp. 271-272).\n\n25.  LEIBNIZ, _De la Nature en elle-m\u00eame..._ , \u00a7 2. Et \u00a7 16 : faire une philosophie \u00ab \u00e9galement distante du formalisme et du mat\u00e9rialisme. \u00bb\n\n26.  Cf. la critique de Boyle par Leibniz : _De la nature en elle-m\u00eame..._ , \u00a7 3. Par Spinoza : _Lettres 6 et 13, \u00e0 Oldenburg_ (\u00ab Pour moi je n'ai jamais pens\u00e9, et en v\u00e9rit\u00e9 il me serait impossible de croire que ce savant homme n'ait eu d'autre dessein, dans son _Trait\u00e9 du Nitre_ , que de montrer la fragilit\u00e9 de cette doctrine enfantine et ridicule des formes substantielles... \u00bb cf. _Lettre_ 13, III, p. 45).\n\n27.  LEIBNIZ, _De la nature en elle-m\u00eame..._ , \u00a7 9.\n\n28.  _CT_ , II, ch. 19, 8, note 3 : \u00ab ... deux modes parce que le repos n'est pas un pur n\u00e9ant. \u00bb Si l'on peut parler d'une \u00ab tendance \u00bb au mouvement, selon Spinoza, c'est seulement au cas o\u00f9 un corps est emp\u00each\u00e9 de suivre le mouvement auquel il est d\u00e9termin\u00e9 du dehors, par d'autres corps non moins ext\u00e9rieurs qui contrarient cette d\u00e9termination. C'est en ce sens d\u00e9j\u00e0 que Descartes parlait d'un _conatus_ : cf. _Principes_ , III, 56 et 57.\n\n29.  _\u00c9_ , IV, 38 et 39 (les deux expressions : \u00ab ce qui dispose le corps humain \u00e0 ce qu'il puisse \u00eatre affect\u00e9 de plus de fa\u00e7ons \u00bb, et \u00ab ce qui fait que le rapport de mouvement et de repos qu'ont entre elles les parties du corps humain soit conserv\u00e9. \u00bb)\n\n30.  _\u00c9_ , III, 9, prop. et dem.\n\n31.  Sur cette d\u00e9termination de l'essence et du _conatus_ par une affection _quelconque_ , cf. _\u00c9_ , III, 56, fin de la dem. ; et III, d\u00e9finition du d\u00e9sir. En III, 9, sc., Spinoza avait simplement d\u00e9fini le d\u00e9sir comme le _conatus_ ou l'app\u00e9tit \u00ab avec conscience de soi \u00bb. C'\u00e9tait une _d\u00e9finition nominale_. Au contraire, lorsqu'il montre que le _conatus_ est n\u00e9cessairement d\u00e9termin\u00e9 par une affection _dont nous avons l'id\u00e9e_ (m\u00eame inad\u00e9quate), il donne une d\u00e9finition r\u00e9elle, impliquant \u00ab la cause de la conscience \u00bb.\n\n32.  _\u00c9_ , III, 37, dem.\n\n33.  _\u00c9_ , III, 54, prop.\n\n34.  _\u00c9_ , III, 57, dem. : _potentia seu conatus_ ; III, d\u00e9f. g\u00e9n\u00e9rale des affects, explication : _Agendi potentia sive existendi vis_ ; IV, 24, prop. : _Agere, vivere, suum esse conservare, haec tria idem significant._\n\n## CHAPITRE XV\n\n## LES TROIS ORDRES ET LE PROBL\u00c8ME DU MAL\n\nUn attribut s'exprime de trois fa\u00e7ons : il s'exprime dans sa nature absolue (mode infini imm\u00e9diat), il s'exprime en tant que modifi\u00e9 (mode infini m\u00e9diat), il s'exprime d'une mani\u00e8re certaine et d\u00e9termin\u00e9e (mode infini existant)1. Spinoza cite lui-m\u00eame les deux modes infinis de l'\u00e9tendue : le mouvement et le repos, la figure de l'univers entier2. Quelle en est la signification ?\n\nNous savons que les rapports de mouvement et de repos doivent eux-m\u00eames \u00eatre consid\u00e9r\u00e9s de deux fa\u00e7ons : en tant qu'ils expriment \u00e9ternellement des essences de modes ; en tant qu'ils subsument temporairement des parties extensives. Du premier point de vue, le mouvement et le repos ne comprennent pas tous les rapports sans contenir aussi toutes les essences telles qu'elles sont dans l'attribut. C'est pourquoi Spinoza, dans le _Court Trait\u00e9_ , affirme que le mouvement et le repos comprennent les essences m\u00eames de choses qui n'existent pas3. Plus nettement encore, il soutient que le mouvement affecte l'\u00e9tendue avant qu'elle n'ait des parties modales extrins\u00e8ques. Pour admettre que le mouvement est bien dans le \u00ab tout infini \u00bb, il suffit de se rappeler qu'il n'y a pas de mouvement seul, mais \u00e0 la fois du mouvement et du repos4. Ce rappel est platonicien : les N\u00e9o-platoniciens insistaient souvent sur l'immanence simultan\u00e9e du mouvement et du repos, sans laquelle le mouvement lui-m\u00eame serait impensable dans le tout.\n\nDu second point de vue, les divers rapports groupent des ensembles infinis variables de parties extensives. Ils d\u00e9terminent alors les conditions sous lesquelles les modes passent \u00e0 l'existence. Chaque rapport effectu\u00e9 constitue la forme d'un individu existant. Or il n'y a pas de rapport qui ne se compose dans un autre pour former, sous un troisi\u00e8me rapport, un individu de degr\u00e9 sup\u00e9rieur. \u00c0 l'infini : si bien que l'univers tout entier est un seul individu existant, d\u00e9fini par la proportion totale du mouvement et du repos, comprenant tous les rapports qui se composent \u00e0 l'infini, subsumant l'ensemble de tous les ensembles sous tous les rapports. Cet individu, d'apr\u00e8s sa forme, est le _\u00ab facies totius universi_ , qui demeure toujours le m\u00eame bien qu'il change en une infinit\u00e9 de mani\u00e8res5 \u00bb.\n\nTous les rapports se composent \u00e0 l'infini pour former ce _facies._ Mais ils se composent suivant des lois qui leur sont propres, lois contenues dans le mode infini m\u00e9diat. C'est dire que les rapports ne se composent pas n'importe comment, n'importe quel rapport ne se compose pas avec n'importe quel autre. En ce sens les lois de composition nous ont paru pr\u00e9c\u00e9demment \u00eatre aussi des lois de d\u00e9composition ; et quand Spinoza dit que le _facies_ demeure le m\u00eame en changeant d'une infinit\u00e9 de mani\u00e8res, il ne fait pas seulement allusion aux compositions de rapports, mais \u00e0 leurs destructions ou d\u00e9compositions. Toutefois ces d\u00e9compositions (pas plus que les compositions) n'affectent la v\u00e9rit\u00e9 \u00e9ternelle des rapports ; un rapport est compos\u00e9, quand il commence \u00e0 subsumer des parties ; il se d\u00e9compose quand il cesse d'\u00eatre ainsi effectu\u00e96. D\u00e9composer, d\u00e9truire signifient donc seulement : deux rapports ne se composant pas directement, les parties subsum\u00e9es par l'un d\u00e9terminent (conform\u00e9ment \u00e0 une loi) les parties de l'autre \u00e0 entrer sous un nouveau rapport qui, lui, se compose avec le premier.\n\nNous voyons que, d'une certaine mani\u00e8re, tout est composition dans l'ordre des rapports. Tout est composition dans la Nature. Quand le poison d\u00e9compose le sang, c'est seulement d'apr\u00e8s la loi qui d\u00e9termine les parties du sang \u00e0 entrer sous un nouveau rapport qui se compose avec celui du poison. La d\u00e9composition n'est que l'envers d'une composition. Mais la question se pose toujours : Pourquoi cet envers ? Pourquoi les lois de composition s'exercent-elles aussi comme des lois de destruction ? La r\u00e9ponse doit \u00eatre : c'est parce que les corps existants ne se rencontrent pas _dans l'ordre_ o\u00f9 leurs rapports se composent. Dans toute rencontre, il y a composition de rapports, mais les rapports qui se composent ne sont pas n\u00e9cessairement ceux des corps qui se rencontrent. Les rapports se composent _suivant des lois_ ; mais les corps existants, \u00e9tant eux-m\u00eames compos\u00e9s de parties extensives, se rencontrent _de proche en proche._ Les parties d'un des corps peuvent donc \u00eatre d\u00e9termin\u00e9es \u00e0 prendre un nouveau rapport exig\u00e9 par la loi en perdant celui sous lequel elles appartenaient \u00e0 ce corps.\n\nSi nous consid\u00e9rons l'ordre des rapports en lui-m\u00eame, nous voyons que c'est un pur ordre de composition. S'il d\u00e9termine aussi des destructions, c'est parce que les corps se rencontrent suivant un ordre qui n'est pas celui des rapports. D'o\u00f9 la complexit\u00e9 de la notion spinoziste \u00ab ordre de la Nature \u00bb. Dans un mode existant, nous devions distinguer trois choses : l'essence comme degr\u00e9 de puissance ; les rapports dans lequel elle s'exprime ; les parties extensives subsum\u00e9es sous ce rapport. \u00c0 chacun de ces niveaux correspond un ordre de la Nature.\n\nEn premier lieu, il y a un ordre des essences, d\u00e9termin\u00e9 par les degr\u00e9s de puissance. Cet ordre est un ordre de convenance totale : chaque essence convient avec toutes les autres, toutes les essences \u00e9tant comprises dans la production de chacune. Elles sont \u00e9ternelles, et l'une ne pourrait pas p\u00e9rir sans que les autres ne p\u00e9rissent aussi. L'ordre des rapports est fort diff\u00e9rent : c'est un ordre de composition suivant des lois. Il d\u00e9termine les conditions \u00e9ternelles sous lesquelles les modes passent \u00e0 l'existence, et continuent d'exister tant qu'ils conservent la composition de leur rapport. Tous les rapports se composent \u00e0 l'infini, mais non pas tout rapport avec tout autre. Nous devons consid\u00e9rer, en troisi\u00e8me lieu, un ordre des rencontres. C'est un ordre de convenances et de disconvenances partielles, locales et temporaires. Les corps existants se rencontrent par leurs parties extensives, de proche en proche. Il se peut que les corps qui se rencontrent aient pr\u00e9cis\u00e9ment des rapports qui se composent d'apr\u00e8s la loi (convenance) ; mais il se peut que, les deux rapports ne se composant pas, l'un des deux corps soit d\u00e9termin\u00e9 \u00e0 d\u00e9truire le rapport de l'autre (disconvenance). Cet ordre des rencontres d\u00e9termine donc effectivement le moment o\u00f9 un mode passe \u00e0 l'existence (quand les conditions fix\u00e9es par la loi sont remplies), la dur\u00e9e pendant laquelle il existe, le moment o\u00f9 il meurt ou est d\u00e9truit. Spinoza le d\u00e9finit \u00e0 la fois comme \u00ab l'ordre commun de la Nature \u00bb, comme l'ordre des \u00ab d\u00e9terminations extrins\u00e8ques \u00bb et des \u00ab rencontres fortuites \u00bb, comme l'ordre des passions7.\n\nEn effet, c'est l'ordre commun, puisque tous les modes existants y sont soumis. C'est l'ordre des passions et des d\u00e9terminations extrins\u00e8ques, puisqu'il d\u00e9termine \u00e0 chaque instant les affections que nous \u00e9prouvons, produites par les corps ext\u00e9rieurs que nous rencontrons. Enfin, il est dit \u00ab fortuit \u00bb _(fortuitus occursus)_ sans que Spinoza r\u00e9introduise ici la moindre contingence. L'ordre des rencontres, pour son compte, est parfaitement d\u00e9termin\u00e9 : sa n\u00e9cessit\u00e9 est celle des parties extensives et de leur d\u00e9termination externe \u00e0 l'infini. Mais il est fortuit par rapport \u00e0 l'ordre des rapports ; pas plus que les essences ne d\u00e9terminent les lois d'apr\u00e8s lesquelles leurs rapports se composent, les lois de composition ne d\u00e9terminent elles-m\u00eames les corps qui se rencontrent et la mani\u00e8re dont ils se rencontrent. L'existence de ce troisi\u00e8me ordre pose toutes sortes de probl\u00e8mes chez Spinoza. Car, pris dans son ensemble, il co\u00efncide avec l'ordre des rapports. Si l'on consid\u00e8re l'ensemble infini des rencontres dans la dur\u00e9e infinie de l'univers, chaque rencontre entra\u00eene une composition de rapports, et tous les rapports se composent avec toutes les rencontres. Mais les deux ordres ne co\u00efncident nullement dans le d\u00e9tail : si nous consid\u00e9rons un corps ayant tel rapport pr\u00e9cis, il rencontre n\u00e9cessairement des corps dont le rapport ne se compose pas avec le sien, et finira toujours par en rencontrer un dont le rapport d\u00e9truira le sien. Ainsi il n'y a pas de mort qui ne soit _brutale, violente et fortuite_ ; mais pr\u00e9cis\u00e9ment parce qu'elle est enti\u00e8rement _n\u00e9cessaire_ dans cet ordre des rencontres.\n\nDeux cas de \u00ab rencontres \u00bb doivent \u00eatre distingu\u00e9s. D'apr\u00e8s le premier cas, je rencontre un corps dont le rapport se compose avec le mien. (Cela m\u00eame peut s'entendre de plusieurs fa\u00e7ons : tant\u00f4t le corps rencontr\u00e9 a un rapport qui se compose naturellement avec un de mes rapports composants, et par l\u00e0 m\u00eame contribue \u00e0 maintenir mon rapport global ; tant\u00f4t les rapports de deux corps conviennent si bien dans leur ensemble qu'ils forment un troisi\u00e8me rapport sous lequel les deux corps se conservent et prosp\u00e8rent.) Quoi qu'il en soit, le corps dont le rapport se conserve avec le mien est dit \u00ab convenir avec ma nature \u00bb : il m'est \u00ab bon \u00bb, c'est-\u00e0-dire \u00ab utile8 \u00bb. Il produit en moi une affection qui, elle-m\u00eame, est bonne ou convient avec ma nature. Cette affection est passive parce qu'elle s'explique par le corps ext\u00e9rieur ; l'id\u00e9e de cette affection est une passion, un sentiment passif. Mais c'est un sentiment de joie, puisqu'il est produit par l'id\u00e9e d'un objet qui nous est bon ou qui convient avec notre nature9. Or, quand Spinoza se propose de d\u00e9finir \u00ab formellement \u00bb cette joie-passion, il dit : elle augmente ou aide notre puissance d'agir, elle est notre puissance d'agir elle-m\u00eame en tant qu'augment\u00e9e ou aid\u00e9e par une cause ext\u00e9rieure10. (Et nous ne connaissons le bon qu'en tant que nous percevons qu'une chose nous affecte de joie11.)\n\nQue veut dire Spinoza ? Il n'oublie certes pas que nos passions, quelles qu'elles soient, sont toujours la marque de notre impuissance : elles ne s'expliquent pas par notre essence ou puissance, mais par la puissance d'une chose ext\u00e9rieure ; par l\u00e0 elles enveloppent notre impuissance12. Toute passion nous s\u00e9pare de notre puissance d'agir ; tant que notre pouvoir d'\u00eatre affect\u00e9 est rempli par des passions, nous sommes s\u00e9par\u00e9s de ce que nous pouvons. C'est pourquoi Spinoza dit : la joie-passion n'est une passion qu'en tant que \u00ab la puissance d'agir de l'homme n'est pas augment\u00e9e jusqu'au point qu'il se con\u00e7oive lui-m\u00eame et ses actions de fa\u00e7on ad\u00e9quate13. C'est-\u00e0-dire : notre puissance d'agir n'est pas encore augment\u00e9e \u00e0 un point tel que nous soyons actifs. Nous sommes encore impuissants, encore s\u00e9par\u00e9s de notre puissance d'agir.\n\nMais notre impuissance est seulement la limitation de notre essence et de notre puissance d'agir elle-m\u00eame. Enveloppant notre impuissance, nos sentiments passifs enveloppent un degr\u00e9, si bas soit-il, de notre puissance d'agir. En effet un sentiment, quel qu'il soit, d\u00e9termine notre essence ou _conatus._ Il nous d\u00e9termine donc \u00e0 d\u00e9sirer, _c'est-\u00e0-dire \u00e0 imaginer et \u00e0 faire quelque chose qui d\u00e9coule de notre nature_. Quand le sentiment qui nous affecte convient lui-m\u00eame avec notre nature, notre puissance d'agir se trouve donc n\u00e9cessairement augment\u00e9e ou aid\u00e9e. Car cette joie _s'ajoute_ elle-m\u00eame au d\u00e9sir qui s'ensuit, si bien que la puissance de la chose ext\u00e9rieure favorise et augmente notre propre puissance14. Le _conatus_ , \u00e9tant notre effort de pers\u00e9v\u00e9rer dans l'existence, est toujours recherche de ce qui nous est utile ou bon ; il comprend toujours un degr\u00e9 de notre puissance d'agir auquel il s'identifie : cette puissance augmente donc, quand le _conatus_ est d\u00e9termin\u00e9 par une affection qui nous est utile ou bonne. Nous ne cessons pas d'\u00eatre passifs, nous ne cessons pas d'\u00eatre s\u00e9par\u00e9s de notre puissance d'agir, mais nous tendons \u00e0 en \u00eatre moins s\u00e9par\u00e9s, nous nous rapprochons de cette puissance. Notre joie passive est et reste une passion : elle ne \u00ab s'explique \u00bb pas par notre puissance d'agir, mais elle \u00ab enveloppe \u00bb un plus haut degr\u00e9 de cette puissance.\n\nEn tant que le sentiment de joie augmente la puissance d'agir, il nous d\u00e9termine \u00e0 d\u00e9sirer, \u00e0 imaginer, \u00e0 faire tout ce qui est notre pouvoir pour conserver cette joie m\u00eame et l'objet qui nous la procure15. C'est en ce sens que l'amour s'encha\u00eene avec la joie, d'autres passions avec l'amour, si bien que notre pouvoir d'\u00eatre affect\u00e9 est enti\u00e8rement rempli. Si nous supposons ainsi une ligne d'affections joyeuses, d\u00e9coulant les unes des autres \u00e0 partir d'un premier sentiment de joie, nous voyons que notre pouvoir d'\u00eatre affect\u00e9 se trouve rempli de telle mani\u00e8re que notre puissance d'agir augmente toujours16. Mais jamais elle n'augmente suffisamment pour que nous la poss\u00e9dions r\u00e9ellement, pour que nous soyons actifs, c'est-\u00e0-dire cause ad\u00e9quate des affections qui remplissent notre pouvoir d'\u00eatre affect\u00e9.\n\nPassons maintenant au second cas de rencontre. Je rencontre un corps dont le rapport ne se compose pas avec le mien. Ce corps ne convient pas avec ma nature, il est contraire \u00e0 ma nature, mauvais ou nuisible. Il produit en moi une affection passive qui, elle-m\u00eame, est mauvaise ou contraire \u00e0 ma nature17. L'id\u00e9e de cette affection est un sentiment de tristesse, cette tristesse-passion se d\u00e9finit par la diminution de ma puissance d'agir. Et nous ne connaissons le mauvais qu'en tant que nous percevons une chose qui nous affecte de tristesse. Toutefois, on objectera qu'il est n\u00e9cessaire de distinguer plusieurs cas. Il semble que, dans une telle rencontre, tout d\u00e9pende de l'essence ou de la puissance respective des corps qui se rencontrent. Si mon corps poss\u00e8de essentiellement un plus grand degr\u00e9 de puissance, c'est lui qui d\u00e9truira l'autre ou d\u00e9composera le rapport de l'autre. Au contraire, s'il poss\u00e8de un plus petit degr\u00e9. Les deux cas ne semblent pas pouvoir se ramener au m\u00eame sch\u00e9ma.\n\nEn v\u00e9rit\u00e9, l'objection reste abstraite. Car, dans l'existence, nous ne pouvons pas tenir compte des degr\u00e9s de puissance absolument consid\u00e9r\u00e9s. Quand nous consid\u00e9rons les essences ou les degr\u00e9s de puissance en eux-m\u00eames, nous savons qu'aucun ne peut en d\u00e9truire un autre, et que tous conviennent. Au contraire, quand nous consid\u00e9rons les luttes et les incompatibilit\u00e9s entre modes existants, nous devons faire intervenir toutes sortes de facteurs concrets, qui nous emp\u00eachent de dire que le mode dont l'essence ou le degr\u00e9 de puissance est le plus fort gagnera certainement. En effet, les corps existants qui se rencontrent ne sont pas seulement d\u00e9finis par le rapport global qui leur est propre : se rencontrant parties par parties, de proche en proche, ils se rencontrent n\u00e9cessairement sous certains de leurs rapports partiels ou composants. Il se peut qu'un corps moins fort que le mien soit plus fort qu'un de mes composants : il suffira \u00e0 me d\u00e9truire, pour peu que ce composant me soit vital.\n\nC'est en ce sens que Spinoza rappelle que la lutte des modes, _d'apr\u00e8s_ leur degr\u00e9 de puissance, ne doit pas s'entendre de ces degr\u00e9s pris en eux-m\u00eames : il n'y a pas de lutte entre essences comme telles18. Mais inversement, quand Spinoza montre qu'il y a toujours dans l'existence des corps plus puissants que le mien qui peuvent me d\u00e9truire, il ne faut pas croire n\u00e9cessairement que ces corps aient une essence dont le degr\u00e9 de puissance est plus grand, ou qu'ils aient une plus grande perfection. Un corps peut \u00eatre d\u00e9truit par un corps d'essence moins parfaite si les conditions de la rencontre (c'est-\u00e0-dire le rapport partiel sous lequel elle se fait) sont favorables \u00e0 cette destruction. Pour savoir \u00e0 l'avance l'issue d'une lutte, il faudrait savoir exactement sous quel rapport les deux corps se rencontrent, sous quel rapport s'affrontent les rapports incomposables. Il faudrait un savoir infini de la Nature, que nous n'avons pas. En tout cas, dans toute rencontre avec un corps qui ne convient pas avec ma nature intervient toujours un sentiment de tristesse, au moins partiel, qui vient de ce que le corps me l\u00e8se toujours dans un de mes rapports partiels. Bien plus, ce sentiment de tristesse est la seule mani\u00e8re dont nous savons que l'autre corps ne convient pas avec notre nature19. Que nous devions triompher ou non, cela ne change rien : nous ne le savons pas \u00e0 l'avance. Nous triomphons, si nous arrivons \u00e0 \u00e9carter ce sentiment de tristesse, donc \u00e0 d\u00e9truire le corps qui nous affecte. Nous sommes vaincus si la tristesse nous gagne de plus en plus, sous tous nos rapports composants, marquant alors la destruction de notre rapport global.\n\nOr, comment, \u00e0 partir d'un premier sentiment de tristesse, notre pouvoir d'\u00eatre affect\u00e9 se trouve-t-il rempli ? Non moins que la joie, la tristesse d\u00e9termine le _conatus_ ou l'essence. C'est-\u00e0-dire : de la tristesse na\u00eet un d\u00e9sir, qui est la haine. Ce d\u00e9sir s'encha\u00eene avec d'autres d\u00e9sirs, d'autres passions : antipathie, d\u00e9rision, m\u00e9sestime, envie, col\u00e8re, etc. Mais l\u00e0 encore, en tant qu'elle d\u00e9termine notre essence ou _conatus_ , la tristesse enveloppe quelque chose de notre puissance d'agir. D\u00e9termin\u00e9 par la tristesse, le _conatus_ ne cesse pas d'\u00eatre recherche de ce qui nous est utile ou bon : nous nous effor\u00e7ons de triompher, c'est-\u00e0-dire de faire en sorte que les parties du corps qui nous affecte de tristesse prennent un nouveau rapport qui se concilie avec le n\u00f4tre. Nous sommes donc d\u00e9termin\u00e9s \u00e0 tout faire pour \u00e9carter la tristesse et d\u00e9truire l'objet qui en est cause20. Et pourtant, dans ce cas, notre puissance d'agir est dite \u00ab diminuer \u00bb. C'est que le sentiment de tristesse ne s'ajoute pas au d\u00e9sir qui s'ensuit : ce d\u00e9sir est au contraire emp\u00each\u00e9 par ce sentiment, si bien que la puissance de la chose ext\u00e9rieure _se soustrait_ de la n\u00f4tre21. Les affections \u00e0 base de tristesse s'encha\u00eenent donc les unes les autres et remplissent notre pouvoir d'\u00eatre affect\u00e9. Mais elles le font de telle mani\u00e8re que notre puissance d'agir diminue de plus en plus et tend vers son plus bas degr\u00e9.\n\nNous avons fait jusqu'\u00e0 maintenant comme si deux lignes d'affections, joyeuses et tristes, correspondaient aux deux cas de rencontres, bonnes rencontres et mauvaises rencontres. Mais cette vue reste encore abstraite. Si l'on tient compte des facteurs concrets de l'existence, on voit bien que les deux lignes interf\u00e8rent constamment : les relations extrins\u00e8ques sont telles qu'un objet peut toujours \u00eatre cause de tristesse ou de joie par accident22. Non seulement en vertu de ces relations, mais en vertu de la complexit\u00e9 des rapports qui nous composent intrins\u00e8quement, nous pouvons \u00e0 la fois aimer et ha\u00efr un m\u00eame objet23. Bien plus : une ligne de joie peut toujours \u00eatre interrompue par la destruction, ou m\u00eame par la simple tristesse de l'objet aim\u00e9. Inversement la ligne de tristesse sera interrompue par la tristesse ou la destruction de la chose ha\u00efe : \u00ab Celui qui imagine que ce qu'il a en haine est d\u00e9truit se r\u00e9jouira \u00bb, \u00ab Celui qui imagine ce qu'il a en haine comme affect\u00e9 de tristesse se r\u00e9jouira24. \u00bb Nous sommes toujours d\u00e9termin\u00e9s \u00e0 chercher la destruction de l'objet qui nous rend tristes ; mais d\u00e9truire, c'est donner aux parties de l'objet un nouveau rapport qui convient avec le n\u00f4tre ; nous \u00e9prouvons donc une joie qui augmente notre puissance d'agir. C'est ainsi que, les deux lignes interf\u00e9rant constamment, notre puissance d'agir ne cesse pas de varier.\n\nNous devons encore tenir compte d'autres facteurs concrets. Car le premier cas de rencontres, les bonnes rencontres avec des corps dont le rapport se compose directement avec le n\u00f4tre, reste tout \u00e0 fait hypoth\u00e9tique. La question est la suivante : Une fois que nous existons, _avons-nous des chances de faire naturellement de bonnes rencontres et d'\u00e9prouver les affections joyeuses qui s'ensuivent ?_ En v\u00e9rit\u00e9, ces chances sont peu nombreuses. Quand nous parlons de l'existence, nous ne devons pas consid\u00e9rer absolument les essences ou degr\u00e9s de puissance ; nous ne devons pas davantage consid\u00e9rer abstraitement les rapports dans lesquels elles s'expriment. Car un mode existant existe toujours comme d\u00e9j\u00e0 affect\u00e9 par des objets, sous des rapports partiels et particuliers ; il existe comme d\u00e9j\u00e0 d\u00e9termin\u00e9 \u00e0 ceci ou \u00e0 cela. S'est fait un am\u00e9nagement des rapports partiels, entre les choses ext\u00e9rieures et lui-m\u00eame, tel que son rapport caract\u00e9ristique est \u00e0 peine saisissable ou singuli\u00e8rement d\u00e9form\u00e9. C'est ainsi que, en principe, l'homme devrait parfaitement convenir avec l'homme. Mais, en r\u00e9alit\u00e9, les hommes conviennent fort peu en nature les uns avec les autres ; et cela parce qu'ils sont tellement d\u00e9termin\u00e9s par leurs passions, par des objets qui les affectent de diverses fa\u00e7ons, qu'ils ne se rencontrent pas naturellement sous les rapports qui se composent en droit25. \u00ab Comme ils sont soumis \u00e0 des sentiments qui surpassent de beaucoup la puissance ou la vertu humaine, ils sont donc diversement entra\u00een\u00e9s et sont contraires les uns aux autres26. \u00bb Ces entra\u00eenements vont si loin qu'un homme peut, en quelque sorte, \u00eatre contraire \u00e0 lui-m\u00eame : ses rapports partiels peuvent faire l'objet de tels am\u00e9nagements, se transformer si bien sous l'action de causes ext\u00e9rieures insensibles qu'il \u00ab rev\u00eat lui-m\u00eame une autre nature contraire \u00e0 la premi\u00e8re \u00bb, autre nature qui le d\u00e9termine \u00e0 supprimer la premi\u00e8re27.\n\nNous avons donc fort peu de chances de faire naturellement de bonnes rencontres. Il semble que nous soyons d\u00e9termin\u00e9s \u00e0 beaucoup lutter, beaucoup ha\u00efr, et \u00e0 n'\u00e9prouver que des joies partielles ou indirectes qui ne rompent pas suffisamment l'encha\u00eenement de nos tristesses et de nos haines. Les joies partielles sont des \u00ab chatouillements \u00bb, qui n'augmentent en un point notre puissance d'agir qu'en la diminuant partout ailleurs28. Les joies indirectes sont celles que nous \u00e9prouvons \u00e0 voir l'objet ha\u00ef triste ou d\u00e9truit ; mais ces joies restent empoisonn\u00e9es par la tristesse. La haine en effet est une tristesse, elle enveloppe elle-m\u00eame la tristesse dont elle proc\u00e8de ; les joies de la haine recouvrent cette tristesse, elles l'emp\u00eachent, mais jamais ne la suppriment29. Voil\u00e0 donc que nous semblons plus \u00e9loign\u00e9s que jamais d'acqu\u00e9rir la possession de notre puissance d'agir : notre pouvoir d'\u00eatre affect\u00e9 est rempli non seulement par des affections passives, mais surtout par les passions tristes, qui enveloppent de la puissance d'agir un degr\u00e9 de plus en plus bas. On ne s'en \u00e9tonnera pas, la Nature n'\u00e9tant pas faite pour notre utilit\u00e9, mais en fonction d'un \u00ab ordre commun \u00bb auquel l'homme est soumis comme partie de la Nature.\n\nEt pourtant nous avons fait un progr\u00e8s, m\u00eame si ce progr\u00e8s reste abstrait. Nous \u00e9tions partis d'un premier principe du Spinozisme : l'opposition des passions et des actions, des affections passives et des affections actives. Ce principe se pr\u00e9sentait lui-m\u00eame sous deux aspects. Sous un premier aspect, il s'agissait presque d'une opposition r\u00e9elle : affections passives et affections actives, donc puissance de p\u00e2tir et puissance d'agir, variaient en raison inverse pour un m\u00eame pouvoir d'\u00eatre affect\u00e9. Mais, plus profond\u00e9ment, l'opposition r\u00e9elle \u00e9tait une simple n\u00e9gation : les affections passives t\u00e9moignaient seulement de la limitation de notre essence, elles enveloppaient notre impuissance, elles ne se rapportaient \u00e0 l'esprit qu'en tant qu'il enveloppait lui-m\u00eame une n\u00e9gation. Sous cet aspect, les affections actives \u00e9taient les seules capables de remplir effectivement ou positivement notre pouvoir d'\u00eatre affect\u00e9 ; la puissance d'agir \u00e9tait donc identique \u00e0 ce pouvoir lui-m\u00eame : quant aux affections passives, elles nous s\u00e9paraient de ce que nous pouvions.\n\nLes affections passives s'opposent aux affections actives parce qu'elles ne s'expliquent pas par notre puissance d'agir. Mais, enveloppant la limitation de notre essence, elles enveloppent en quelque sorte les plus bas degr\u00e9s de cette puissance. \u00c0 leur mani\u00e8re, elles sont notre puissance d'agir, mais \u00e0 l'\u00e9tat envelopp\u00e9, non exprim\u00e9, non expliqu\u00e9. \u00c0 leur mani\u00e8re, elles remplissent notre pouvoir d'\u00eatre affect\u00e9, mais en le r\u00e9duisant au minimum : plus nous sommes passifs, moins nous sommes aptes \u00e0 \u00eatre affect\u00e9s d'un grand nombre de fa\u00e7ons. Si les affections passives nous s\u00e9parent de ce que nous pouvons, _c'est parce que notre puissance d'agir se trouve r\u00e9duite \u00e0 en investir la trace_ , soit pour les conserver quand elles sont joyeuses, soit pour les \u00e9carter quand elles sont tristes. Or, en tant qu'elles enveloppent une puissance d'agir r\u00e9duite, tant\u00f4t elles augmentent cette puissance, tant\u00f4t elles la diminuent. L'augmentation peut se poursuivre ind\u00e9finiment, nous ne serons jamais en possession formelle de notre puissance d'agir tant que nous n'aurons pas des affections actives. Mais l'opposition des actions et des passions ne doit pas nous cacher cette autre opposition qui constitue le second principe du Spinozisme : celle des affections passives joyeuses, et des affections passives tristes, les unes augmentent notre puissance, les autres la diminuent. Nous nous rapprochons de notre puissance d'agir pour autant que nous sommes affect\u00e9s de joie. La question \u00e9thique, chez Spinoza, se trouve donc d\u00e9doubl\u00e9e : _Comment arriverons-nous \u00e0 produire des affections actives ?_ Mais d'abord : _Comment arriverons-nous \u00e0 \u00e9prouver un maximum de passions joyeuses ?_\n\nQu'est-ce que le mal ? Il n'y a pas d'autres maux que la diminution de notre puissance d'agir et la d\u00e9composition d'un rapport. Encore la diminution de notre puissance d'agir n'est-elle un mal que parce qu'elle menace et r\u00e9duit le rapport qui nous compose. On retiendra donc du mal la d\u00e9finition suivante : c'est la destruction, la d\u00e9composition du rapport qui caract\u00e9rise un mode. D\u00e8s lors, le mal ne peut se dire que du point de vue particulier d'un mode existant : il n'y a pas de Bien ni de Mal dans la Nature en g\u00e9n\u00e9ral, mais _il y a du bon et du mauvais_ , de l'utile et du nuisible pour chaque mode existant. Le mal est le mauvais du point de vue de tel ou tel mode. \u00c9tant nous-m\u00eames hommes, nous jugeons le mal de notre point de vue ; et Spinoza rappelle souvent qu'il parle du bon et du mauvais en consid\u00e9rant la seule utilit\u00e9 de l'homme. Par exemple, nous ne songeons gu\u00e8re \u00e0 parler d'un mal lorsque, pour nous nourrir, nous d\u00e9truisons le rapport sous lequel existe un animal. Mais en deux cas, nous parlons de \u00ab mal \u00bb : lorsque notre corps est d\u00e9truit, notre rapport d\u00e9compos\u00e9, sous l'action d'autre chose ; ou bien lorsque nous-m\u00eames d\u00e9truisons un \u00eatre semblable \u00e0 nous, c'est-\u00e0-dire un \u00eatre dont la ressemblance suffit \u00e0 nous faire penser qu'il convenait en principe avec nous, et que son rapport en principe \u00e9tait composable avec le n\u00f4tre30.\n\nLe mal \u00e9tant ainsi d\u00e9fini de notre point de vue, nous voyons qu'il en est de m\u00eame de tous les autres points de vue : _le mal est toujours une mauvaise rencontre_ , le mal est toujours une d\u00e9composition de rapport. Le type de ces d\u00e9compositions, c'est l'action d'un poison sur notre corps. Selon Spinoza, le mal subi par un homme est toujours du _type indigestion, intoxication, empoisonnement._ Et le mal fait par quelque chose sur l'homme, ou par un homme sur un autre homme, agit toujours comme un poison, comme un \u00e9l\u00e9ment toxique ou indigeste. Spinoza insiste sur ce point, quand il interpr\u00e8te un exemple c\u00e9l\u00e8bre : Adam a mang\u00e9 le fruit d\u00e9fendu. Il ne faut pas croire, dit Spinoza, que Dieu ait d\u00e9fendu quelque chose \u00e0 Adam. Simplement, il lui a r\u00e9v\u00e9l\u00e9 que ce fruit \u00e9tait capable de d\u00e9truire son corps et d'en d\u00e9composer le rapport : \u00ab C'est ainsi que nous savons par lumi\u00e8re naturelle qu'un poison donne la mort31. \u00bb La th\u00e9orie du mal chez Spinoza resterait obscure si les questions d'un de ses correspondants, Blyenbergh, ne l'avait amen\u00e9 \u00e0 pr\u00e9ciser ses th\u00e8mes. Non pas que Blyenbergh ne fasse des contresens ; et ces contresens impatientent Spinoza qui renonce \u00e0 les dissiper. Mais, sur un point essentiel, Blyenbergh comprend fort bien la pens\u00e9e de Spinoza : \u00ab Vous vous abstenez de ce que j'appelle les vices... comme on laisse de c\u00f4t\u00e9 un aliment dont notre nature a horreur32. \u00bb Le _mal-mauvaise rencontre_ , le _mal-empoisonnement_ , constituent le fond de la th\u00e9orie spinoziste.\n\nAlors, si l'on demande : En quoi le mal consiste-t-il dans l'ordre des rapports ?, on doit r\u00e9pondre que le mal n'est rien. Car, dans l'ordre des rapports, il n'y a rien d'autre que des compositions. On ne dira pas qu'une composition de rapports quelconques soit un mal : toute composition de rapports est bonne, du point de vue des rapports qui se composent, c'est-\u00e0-dire du seul point de vue positif. Quand un poison d\u00e9compose mon corps, c'est parce qu'une loi naturelle d\u00e9termine les parties de mon corps, au contact du poison, \u00e0 prendre un nouveau rapport qui se compose avec celui du corps toxique. Rien n'est un mal, ici, du point de vue de la Nature. Dans la mesure o\u00f9 le poison est d\u00e9termin\u00e9 par une loi \u00e0 avoir un effet, cet effet n'est pas un mal, puisqu'il consiste en un rapport qui se compose lui-m\u00eame avec le rapport du poison. De m\u00eame quand je d\u00e9truis un corps, f\u00fbt-il semblable au mien, c'est parce que, _sous le rapport et dans les circonstances o\u00f9 je le rencontre_ , il ne convient pas avec ma nature : je suis donc d\u00e9termin\u00e9 \u00e0 faire tout ce qui est en mon pouvoir pour imposer aux parties de ce corps un nouveau rapport sous lequel elles me conviendront. Le m\u00e9chant, comme le vertueux, cherche donc ce qui lui est utile ou bon (s'il y a une diff\u00e9rence entre les deux, la diff\u00e9rence n'est pas l\u00e0). D'o\u00f9 un premier contresens de Blyenbergh qui consiste \u00e0 croire que, d'apr\u00e8s Spinoza, le m\u00e9chant est d\u00e9termin\u00e9 au mal. Il est vrai que nous sommes toujours d\u00e9termin\u00e9s ; notre _conatus_ lui-m\u00eame est d\u00e9termin\u00e9 par les affections que nous \u00e9prouvons. Mais nous ne sommes jamais d\u00e9termin\u00e9s au mal ; nous sommes d\u00e9termin\u00e9s \u00e0 chercher ce qui nous est bon d'apr\u00e8s les rencontres que nous faisons, et d'apr\u00e8s les circonstances de ces rencontres. Dans la mesure o\u00f9 nous sommes d\u00e9termin\u00e9s \u00e0 avoir un effet, cet effet se compose n\u00e9cessairement avec sa cause, et ne contient rien qu'on puisse appeler \u00ab mal \u00bb33. Bref, le mal n'est rien parce qu'il n'exprime aucune composition de rapports, aucune loi de composition. Dans toute rencontre, que je d\u00e9truise ou que je sois d\u00e9truit, se fait une composition de rapports qui, en tant que telle, est bonne. Donc, si l'on consid\u00e8re l'ordre total des rencontres, on dira qu'il co\u00efncide avec l'ordre total des rapports. Et l'on dira que le mal n'est rien dans l'ordre des rapports eux-m\u00eames.\n\nNous demandons en second lieu : En quoi le mal consiste-t-il dans l'ordre des essences ? L\u00e0 encore, il n'est rien. Soit notre mort ou destruction : notre rapport est d\u00e9compos\u00e9, c'est-\u00e0-dire cesse de subsumer des parties extensives. Mais les parties extensives ne constituent rien de notre essence ; notre essence m\u00eame, ayant en elle-m\u00eame sa pleine r\u00e9alit\u00e9, n'a jamais pr\u00e9sent\u00e9 la moindre tendance \u00e0 passer \u00e0 l'existence. Sans doute, une fois que nous existons, notre essence est-elle un _conatus_ , un effort de pers\u00e9v\u00e9rer dans l'existence. Mais le _conatus_ est seulement l'\u00e9tat que l'essence _est d\u00e9termin\u00e9e_ \u00e0 prendre dans l'existence, en tant que cette essence ne d\u00e9termine pas l'existence elle-m\u00eame ni la dur\u00e9e de l'existence. Donc, \u00e9tant effort de pers\u00e9v\u00e9rer dans l'existence ind\u00e9finiment, le _conatus_ n'enveloppe aucun temps d\u00e9fini : l'essence ne sera ni plus ni moins parfaite suivant que le mode aura r\u00e9ussi \u00e0 pers\u00e9v\u00e9rer plus ou moins de temps dans l'existence34. Ne manquant de rien quand le mode n'existe pas encore, l'essence n'est priv\u00e9e de rien quand il cesse d'exister.\n\nSoit au contraire le mal que nous faisons, quand nous d\u00e9truisons un corps semblable au n\u00f4tre. Si l'on consid\u00e8re l'action de frapper (par exemple lever le bras, serrer le poing, mouvoir le bras de haut en bas), on voit qu'elle exprime quelque chose d'une essence pour autant que le corps humain peut la faire en conservant le rapport qui le caract\u00e9rise. Mais en ce sens, cette action \u00ab est une vertu qui se con\u00e7oit par la structure du corps humain35. \u00bb Maintenant, que cette action soit agressive, qu'elle menace ou d\u00e9truise le rapport qui d\u00e9finit un autre corps, cela manifeste bien une rencontre entre deux corps dont les rapports sont incompatibles sous cet aspect, mais n'exprime rien d'une essence. On dira que l'intention elle-m\u00eame \u00e9tait m\u00e9chante. Mais _la m\u00e9chancet\u00e9 de l'intention consiste en ceci seulement que j'ai joint l'image de cette action \u00e0 l'image d'un corps dont le rapport est d\u00e9truit par cette action_36. Il n'y a \u00ab mal \u00bb que dans la mesure o\u00f9 cette action prend pour objet quelque chose ou quelqu'un dont le rapport ne se combine pas avec celui dont elle d\u00e9pend. Il s'agit toujours d'un cas analogue \u00e0 celui d'un poison.\n\nLa diff\u00e9rence entre deux matricides c\u00e9l\u00e8bres, N\u00e9ron tuant Agrippine et Oreste tuant Clytemnestre, est susceptible de nous \u00e9clairer. On estime qu'Oreste n'est pas coupable parce que Clytemnestre, ayant commenc\u00e9 par tuer Agamemnon, s'est mise elle-m\u00eame dans un rapport qui ne pouvait plus se composer avec celui d'Oreste. On estime N\u00e9ron coupable parce qu'il lui fallut de la m\u00e9chancet\u00e9 pour appr\u00e9hender Agrippine sous un rapport absolument incomposable avec le sien, et pour lier l'image d'Agrippine \u00e0 l'image d'une action qui la d\u00e9truirait. Mais en tout cela, rien n'exprime une essence37. Seule appara\u00eet la rencontre de deux corps sous des rapports incomposables, seule appara\u00eet la liaison de l'image d'un acte avec l'image d'un corps dont le rapport ne se compose pas avec celui de l'acte. Le m\u00eame geste est une vertu s'il prend pour objet quelque chose dont le rapport se compose avec le sien (c'est ainsi qu'il y a des b\u00e9n\u00e9dictions qui ont l'air de frapper). D'o\u00f9 le second contre sens de Blyenbergh. Celui-ci croit que, selon Spinoza, le mal devient un bien, le crime une vertu, pour autant qu'il exprime une essence, serait-ce celle de N\u00e9ron. Et Spinoza ne le d\u00e9trompe qu'\u00e0 moiti\u00e9. Non seulement parce que Spinoza s'impatiente des exigences maladroites ou m\u00eame insolentes de Blyenbergh, mais surtout parce qu'une th\u00e8se \u00ab amoraliste \u00bb comme celle de Spinoza ne peut se faire comprendre qu'\u00e0 l'aide d'un certain nombre de provocations38. En v\u00e9rit\u00e9, le crime n'exprime rien d'une essence, il n'exprime aucune essence, pas m\u00eame celle de N\u00e9ron.\n\nLe Mal appara\u00eet donc seulement dans le troisi\u00e8me ordre, l'ordre des rencontres. Il signifie seulement que les rapports qui se composent ne sont pas toujours ceux des corps qui se rencontrent. Et encore, nous avons vu que le mal n'\u00e9tait rien dans l'ordre total des rencontres. De m\u00eame, il n'est rien dans le cas extr\u00eame o\u00f9 le rapport est d\u00e9compos\u00e9, puisque cette destruction n'affecte ni la r\u00e9alit\u00e9 de l'essence en elle-m\u00eame, ni la v\u00e9rit\u00e9 \u00e9ternelle du rapport. Reste donc un seul cas o\u00f9 le mal semble \u00eatre quelque chose. Durant son existence, d'apr\u00e8s les rencontres qu'il fait, tel mode existant passe par des variations qui sont celles de sa puissance d'agir ; or, quand la puissance d'agir diminue, le mode existant _passe \u00e0 une moindre perfection_39. Le mal n'est-il pas dans cet \u00ab acte de passer \u00e0 une perfection moindre \u00bb ? Comme dit Blyenbergh, il faut bien que le mal existe, quand on est priv\u00e9 d'une condition meilleure40. La r\u00e9ponse c\u00e9l\u00e8bre de Spinoza est : Il n'y a aucune privation dans le passage \u00e0 une moindre perfection, la privation est une simple n\u00e9gation. Le mal n'est rien m\u00eame dans ce dernier ordre. Un homme devient aveugle ; un homme, tout \u00e0 l'heure anim\u00e9 par le d\u00e9sir du bien, est saisi par un app\u00e9tit sensuel. Nous n'avons aucune raison de dire qu'il soit priv\u00e9 d'un \u00e9tat meilleur, puisque cet \u00e9tat n'appartient pas plus \u00e0 sa nature, \u00e0 l'instant consid\u00e9r\u00e9, qu'\u00e0 celle de la pierre ou du diable41.\n\nOn pressent les difficult\u00e9s de cette r\u00e9ponse. Blyenbergh reproche vivement \u00e0 Spinoza d'avoir confondu deux types de comparaison tr\u00e8s distincts : la comparaison entre choses qui n'ont pas la m\u00eame nature et la comparaison entre diff\u00e9rents \u00e9tats d'une seule et m\u00eame chose. Il est vrai qu'il n'appartient pas \u00e0 la nature de la pierre de voir ; mais la vision appartenait \u00e0 la nature de l'homme. D'o\u00f9, l'objection principale est celle-ci : Spinoza pr\u00eate \u00e0 l'essence d'un \u00eatre une instantan\u00e9it\u00e9 qu'elle ne saurait avoir ; \u00ab suivant votre opinion n'appartient \u00e0 l'essence d'une chose que ce que, _au moment consid\u00e9r\u00e9_ , on per\u00e7oit qui est en elle42. \u00bb D\u00e8s lors toute progression, toute r\u00e9gression dans le temps deviennent inintelligibles.\n\nBlyenbergh fait comme si Spinoza disait qu'un \u00eatre est toujours aussi parfait qu'il peut l'\u00eatre, _en fonction de l'essence qu'il poss\u00e8de \u00e0 tel moment._ Mais pr\u00e9cis\u00e9ment, c'est l\u00e0 son troisi\u00e8me contresens. Spinoza dit tout autre chose : Un \u00eatre est toujours aussi parfait qu'il peut l'\u00eatre, _en fonction des affections qui, \u00e0 tel moment, appartiennent \u00e0 son essence_. Il est clair que Blyenbergh confond \u00ab appartenir \u00e0 l'essence \u00bb et \u00ab constituer l'essence \u00bb. \u00c0 chaque moment les affections que j'\u00e9prouve appartiennent \u00e0 mon essence, en tant qu'elles remplissent mon pouvoir d'\u00eatre affect\u00e9. Tant qu'un mode existe, son essence elle-m\u00eame est aussi parfaite qu'elle peut l'\u00eatre en fonction des affections qui remplissent \u00e0 tel moment le pouvoir d'\u00eatre affect\u00e9. Si telles affections remplissent \u00e0 tel moment mon pouvoir, celui-ci ne peut pas au m\u00eame moment se trouver rempli par d'autres affections : il y a incompatibilit\u00e9, exclusion, n\u00e9gation, non pas privation. Reprenons l'exemple de l'aveugle. Ou bien l'on imagine un aveugle qui aurait encore des sensations lumineuses ; mais il est aveugle en ceci qu'il ne peut plus agir d'apr\u00e8s ces sensations ; ses affections lumineuses subsistantes sont enti\u00e8rement passives. Dans ce cas, seule aura vari\u00e9 la proportion des affections actives et des affections passives pour un m\u00eame pouvoir d'\u00eatre affect\u00e9. Ou bien l'on imagine un aveugle ayant perdu toute affection lumineuse ; dans ce cas, son pouvoir d'\u00eatre affect\u00e9 se trouve effectivement r\u00e9duit. Mais la conclusion est la m\u00eame : un mode existant est aussi parfait qu'il peut l'\u00eatre en fonction des affections qui remplissent son pouvoir d'\u00eatre affect\u00e9 et qui le font varier dans des limites compatibles avec l'existence. Bref, chez Spinoza, nulle contradiction n'appara\u00eet entre l'inspiration \u00ab n\u00e9cessitariste \u00bb selon laquelle le pouvoir d'\u00eatre affect\u00e9 se trouve \u00e0 chaque instant n\u00e9cessairement rempli ; et l'inspiration \u00ab \u00e9thique \u00bb selon laquelle \u00e0 chaque instant il est rempli de telle mani\u00e8re que la puissance d'agir augmente ou diminue, et qu'il varie lui-m\u00eame avec ces variations. Comme dit Spinoza, il n'y a aucune privation, mais il n'y en a pas moins des passages \u00e0 des perfections plus ou moins grandes43.\n\nEn aucun sens le Mal n'est quelque chose. \u00catre, c'est _s'exprimer, ou exprimer, ou \u00eatre exprim\u00e9._ Le mal n'est rien, n'\u00e9tant expressif en rien. Et surtout il n'exprime rien. Il n'exprime aucune loi de composition, aucune composition de rapports ; il n'exprime aucune essence ; il n'exprime aucune privation d'un \u00e9tat meilleur dans l'existence. Pour \u00e9valuer l'originalit\u00e9 de cette th\u00e8se, il faut l'opposer \u00e0 d'autres mani\u00e8res de _nier_ le mal. On peut appeler \u00ab moralisme rationaliste \u00bb (optimisme) une tradition qui trouve ses sources chez Platon, et son plein d\u00e9veloppement dans la philosophie de Leibniz ; le Mal n'est rien, parce que seul le Bien est, ou, mieux encore, parce que le Bien, sup\u00e9rieur \u00e0 l'existence, d\u00e9termine tout ce qui est. Le Bien, ou le Meilleur _font \u00eatre._ La th\u00e8se spinoziste n'a rien \u00e0 voir avec cette tradition : elle forme un \u00ab amoralisme \u00bb rationaliste. Car, selon Spinoza, le Bien n'a pas plus de sens que le Mal : il n'y a ni Bien ni Mal dans la Nature. Spinoza le rappelle constamment ; \u00ab si les hommes naissaient libres, ils ne formeraient aucun concept du bien et du mal aussi longtemps qu'ils seraient libres \u00bb44. La question de l'ath\u00e9isme de Spinoza manque singuli\u00e8rement d'int\u00e9r\u00eat tant qu'elle d\u00e9pend de l'arbitraire des d\u00e9finitions th\u00e9isme-ath\u00e9isme. Aussi cette question ne peut-elle \u00eatre pos\u00e9e qu'en fonction de ce que la plupart des gens appellent Dieu du point de vue de la religion : c'est-\u00e0-dire un Dieu ins\u00e9parable d'une _ratio boni_ , proc\u00e9dant par la loi morale, agissant comme un juge45. En ce sens, Spinoza, de toute \u00e9vidence, est ath\u00e9e : la pseudo-loi morale mesure seulement nos contresens sur les lois de la nature ; l'id\u00e9e des r\u00e9compenses et des ch\u00e2timents t\u00e9moigne seulement de notre ignorance du vrai rapport entre un acte et ses cons\u00e9quences ; le Bien et le Mal sont des id\u00e9es inad\u00e9quates, et ne sont con\u00e7us par nous que dans la mesure o\u00f9 nous avons des id\u00e9es inad\u00e9quates46.\n\nMais qu'il n'y ait ni Bien ni Mal ne signifie pas que toute diff\u00e9rence disparaisse. Il n'y a pas de Bien ni de Mal dans la Nature, mais il y a du bon et du mauvais pour chaque mode existant. L'opposition morale du Bien et du Mal dispara\u00eet, mais cette disparition ne rend pas toutes les choses \u00e9gales, ni tous les \u00eatres. Comme Nietzsche le dira, \u00ab _Par-del\u00e0 le Bien et le Mal_ , cela du moins ne veut pas dire _par-del\u00e0 le bon et le mauvais_47 \u00bb. Il y a des augmentations de la puissance d'agir, des diminutions de la puissance d'agir. La distinction du bon et du mauvais servira de principe pour une v\u00e9ritable diff\u00e9rence \u00e9thique, qui doit se substituer \u00e0 la fausse opposition morale.\n\n* * *\n\n1.  _\u00c9_ , I, 21-25.\n\n2.  _Lettre 64_ , _\u00e0 Schuller_ (III, p. 206).\n\n3.  _CT_ , Appendice I, 4, dem. : \u00ab ... Toutes les essences de choses que nous voyons qui, auparavant, quand elles n'existaient pas, \u00e9taient comprises dans l'\u00e9tendue, le mouvement et le repos... \u00bb\n\n4.  _CT_ , I, ch. 2, note 6 : \u00ab Mais, objectez-vous, s'il y a du mouvement dans la mati\u00e8re, ce mouvement doit \u00eatre dans une partie de la mati\u00e8re, non dans le tout, puisque le tout est infini ; dans quelle direction en effet pourrait-il se mouvoir, puisque rien n'existe en dehors de lui ? C'est donc dans une partie. R\u00e9pondons : il n'y a pas de mouvement seul, mais \u00e0 la fois du mouvement et du repos, et ce mouvement est dans le tout... \u00bb\n\n5.  _Lettre 64, \u00e0 Schuller_ , (III, p. 206).\n\n6.  En effet, les parties qui entrent sous un rapport existaient auparavant sous d'autres rapports. Il a fallu que ces rapports se composent pour que les parties qu'ils subsumaient eux-m\u00eames soient soumises au nouveau rapport. Celui-ci, en ce sens, est donc _compos\u00e9._ Inversement, il se _d\u00e9compose_ quand il perd ses parties, qui entrent n\u00e9cessairement dans d'autres rapports.\n\n7.  _\u00c9_ , II, 29, cor. _ex communi Naturae ordine._ II, 29, sc. : _Quoties (mens) ex communi Naturae ordine res percipit, hoc est quoties externe, ex rerum nempe fortuito occursu, determinatur..._ F. Alqui\u00e9 a soulign\u00e9 l'importance de ce th\u00e8me de la rencontre ( _occursus_ ) dans la th\u00e9orie spinoziste des affections : cf. _Servitude et libert\u00e9 chez Spinoza_ , cours publi\u00e9, C.D.U., p. 42.\n\n8.  _\u00c9_ , IV, def. 1 ; IV, 31, prop. ; et surtout IV, 38 et 39, prop.\n\n9.  _\u00c9_ , IV, 8.\n\n10.  Cf. _\u00c9_ , III, 57, dem.\n\n11.  _\u00c9_ , IV, 8, prop. : \u00ab La connaissance du bon et du mauvais n'est rien d'autre qu'un sentiment de joie ou de tristesse, en tant que nous en sommes conscients. \u00bb\n\n12.  _\u00c9_ , IV, _5_ , prop. : \u00ab La force et l'accroissement d'une _passion quelconque_ , et sa pers\u00e9v\u00e9rance \u00e0 exister, ne sont pas d\u00e9finis par la puissance par laquelle nous nous effor\u00e7ons de pers\u00e9v\u00e9rer dans l'existence, mais par la puissance d'une cause ext\u00e9rieure compar\u00e9e avec la n\u00f4tre. \u00bb\n\n13.  _\u00c9_ , IV, 59, dem.\n\n14.  _\u00c9_ , IV, 18, dem. : \u00ab Le d\u00e9sir qui na\u00eet de la joie est aid\u00e9 ou augment\u00e9 par ce sentiment m\u00eame de joie... Et, par cons\u00e9quent, la force du d\u00e9sir qui na\u00eet de la joie doit \u00eatre d\u00e9finie \u00e0 la fois par la puissance humaine et par la puissance d'une cause ext\u00e9rieure. \u00bb\n\n15.  _\u00c9_ , III, 37, dem.\n\n16.  En effet, l'amour est lui-m\u00eame une joie, qui s'ajoute \u00e0 la joie dont il proc\u00e8de... (cf. _\u00c9_ , III, 37, dem.).\n\n17.  Cf. _\u00c9_ , V, 10, prop. et dem. : \u00ab affects contraires \u00e0 notre nature. \u00bb\n\n18.  _\u00c9_ , V, 37, sc.\n\n19.  _\u00c9_ , IV, 8, prop. et dem.\n\n20.  _\u00c9_ , III, 13, prop. ; III, 28, prop. Et III, 37, dem. \u00ab La puissance d'agir par laquelle l'homme, en retour, s'efforcera d'\u00e9carter la tristesse... \u00bb\n\n21.  _\u00c9_ , IV, 18, dem. : \u00ab Le d\u00e9sir qui na\u00eet de la tristesse est diminu\u00e9 ou emp\u00each\u00e9 par ce sentiment m\u00eame de tristesse. \u00bb\n\n22.  _\u00c9_ , III, 15 et 16. Pas plus que \u00ab fortuit \u00bb, \u00ab accidentel \u00bb ne s'oppose ici \u00e0 \u00ab n\u00e9cessaire \u00bb.\n\n23.  Cf. le \u00ab flottement de l'\u00e2me \u00bb, _\u00c9_ , III, 17, prop. et sc. (Il y a deux cas de _flottement_ : l'un, d\u00e9fini dans la d\u00e9monstration de cette proposition 17, s'explique par les relations extrins\u00e8ques et accidentelles entre objets ; l'autre, d\u00e9fini dans le scolie, s'explique par la diff\u00e9rence des rapports qui nous composent intrins\u00e8quement.\n\n24.  _\u00c9_ , III, 20 et 23, prop.\n\n25.  _\u00c9_ , IV, 32, 33 et 34.\n\n26.  _\u00c9_ , IV, 37, sc. 2.\n\n27.  Cf. _\u00c9_ , IV, 20, sc., l'interpr\u00e9tation spinoziste du suicide : \u00ab ... ou bien enfin, c'est parce que des causes ext\u00e9rieures cach\u00e9es disposent l'imagination et affectent le corps, de sorte qu'il rev\u00eat une autre nature contraire \u00e0 la premi\u00e8re, et dont l'id\u00e9e ne peut \u00eatre donn\u00e9e dans l'esprit. \u00bb\n\n28.  _\u00c9_ , IV, 43, prop. et dem.\n\n29.  _\u00c9_ , III, 45, dem. : \u00ab La tristesse qu'enveloppe la haine. \u00bb III, 47, prop. : \u00ab La joie qui na\u00eet de ce que nous imaginons qu'une chose est d\u00e9truite, ou affect\u00e9 d'un autre mal, ne na\u00eet pas sans quelque tristesse de l'\u00e2me. \u00bb\n\n30.  Cf. _\u00c9_ , III, 47, dem.\n\n31.  _Lettre 19, \u00e0 Blyenbergh_ (III, p. 65). M\u00eame argument en _TTP_ , ch. 4 (II, p. 139). La seule diff\u00e9rence entre cette r\u00e9v\u00e9lation divine et la lumi\u00e8re naturelle est que Dieu a r\u00e9v\u00e9l\u00e9 \u00e0 Adam la _cons\u00e9quence_ , c'est-\u00e0-dire l'empoisonnement qui r\u00e9sulterait de l'ingestion du fruit, mais ne lui a pas r\u00e9v\u00e9l\u00e9 la _n\u00e9cessit\u00e9_ de cette cons\u00e9quence ; ou du moins Adam n'avait pas l'entendement assez fort pour comprendre cette n\u00e9cessit\u00e9.\n\n32.  _Lettre 22, de Blyenbergh_ (III, p. 96).\n\n33.  Ce que Spinoza appelle les \u00ab \u0153uvres \u00bb, dans la correspondance avec Blyenbergh, ce sont pr\u00e9cis\u00e9ment les effets auxquels nous sommes d\u00e9termin\u00e9s.\n\n34.  _\u00c9_ , III, 8, prop. : \u00ab L'effort par lequel chaque chose s'efforce de pers\u00e9v\u00e9rer dans son \u00eatre n'enveloppe aucun temps fini, mais un temps ind\u00e9fini. \u00bb _\u00c9_ , IV, pr\u00e9face : \u00ab _Aucune chose singuli\u00e8re ne peut \u00eatre dite plus parfaite parce qu'elle a pers\u00e9v\u00e9r\u00e9 plus de temps dans l'existence._ \u00bb\n\n35.  _\u00c9_ , IV, 59, sc.\n\n36.  _\u00c9_ , IV, 59, sc. : \u00ab Si un homme, pouss\u00e9 par la col\u00e8re ou par la haine, est d\u00e9termin\u00e9 \u00e0 serrer le poing ou \u00e0 mouvoir le bras, _cela provient de ce qu'une seule et m\u00eame action peut \u00eatrejointe \u00e0 n'importe quelles images dechoses._ \u00bb\n\n37.  _Lettre 23, \u00e0 Blyenbergh_ (III, p. 99) : _nihil horum aliquid essentiae exprimere._ C'est l\u00e0 que Spinoza commente le cas d'Oreste et celui de N\u00e9ron.\n\n38.  Cf. objection de Blyenbergh, _Lettre 22_ (III, p. 96) : \u00ab Se pose donc la question de savoir si, au cas o\u00f9 il y aurait une \u00e2me \u00e0 la nature singuli\u00e8re de laquelle conviendrait la qu\u00eate des plaisirs et des crimes au lieu de lui r\u00e9pugner, si en pareil cas, dis-je, il existerait un argument de vertu, qui p\u00fbt d\u00e9terminer un pareil \u00eatre \u00e0 agir vertueusement et \u00e0 s'abstenir du mal ? \u00bb R\u00e9ponse de Spinoza, _Lettre 23_ (III, p. 101) : \u00ab C'est, \u00e0 mes yeux, comme si l'on demandait : Peut-il mieux convenir \u00e0 la nature de quelqu'un de se pendre, ou bien peut-on donner des raisons pour qu'il ne se pende pas ? Supposons cependant qu'une telle nature puisse exister..., je l'affirme alors, si quelqu'un voit qu'il peut vivre plus commod\u00e9ment suspendu au gibet qu'assis \u00e0 sa table, il agirait en insens\u00e9 en ne se pendant pas ; et de m\u00eame, qui verrait clairement qu'il peut jouir d'une vie ou d'une essence meilleure en commettant des crimes plut\u00f4t qu'en s'attachant \u00e0 la vertu, il serait insens\u00e9 lui aussi s'il ne le faisait pas. Car les crimes, au regard d'une nature humaine aussi perverse, seraient vertu. \u00bb\n\n39.  Cf. _\u00c9_ , III, d\u00e9finition de la tristesse.\n\n40.  _Lettre 20, de Blyenbergh_ (III, p. 72).\n\n41.  _Lettre 21, \u00e0 Blyenbergh_ (III, pp. 87-88).\n\n42.  _Lettre 22, de Blyenbergh_ (III, p. 94).\n\n43.  _\u00c9_ , III, d\u00e9finition de la tristesse, explic. : \u00ab Et nous ne pouvons pas dire que la tristesse consiste dans la privation d'une plus grande perfection, car une privation n'est rien, tandis que le sentiment de tristesse est un acte, qui pour cette raison ne peut \u00eatre autre que l'acte de passer \u00e0 une perfection moindre. \u00bb\n\n44.  _\u00c9_ , IV, 68, prop.\n\n45.  C'\u00e9taient les crit\u00e8res de Leibniz, et de tous ceux qui reproch\u00e8rent \u00e0 Spinoza son ath\u00e9isme.\n\n46.  _\u00c9_ , IV, 68, dem.\n\n47.  NIETZSCHE, _G\u00e9n\u00e9alogie de la morale_ , I, 17.\n\n## CHAPITRE XVI\n\n## VISION \u00c9THIQUE DU MONDE\n\nLorsque Spinoza dit : Nous ne savons m\u00eame pas ce que peut un corps, cette formule est presque un cri de guerre. Il ajoute : Nous parlons de la conscience, de l'esprit, de l'\u00e2me, du pouvoir de l'\u00e2me sur le corps. Nous bavardons ainsi, mais nous ne savons m\u00eame pas ce que peut un corps1. Le bavardage moral remplace la vraie philosophie.\n\nCette d\u00e9claration est importante \u00e0 plusieurs \u00e9gards. Tant que nous parlons d'un pouvoir de l'\u00e2me sur le corps, nous ne pensons pas vraiment en termes de pouvoir ou de puissance. Nous voulons dire en effet que l'\u00e2me, en fonction de sa nature \u00e9minente et de sa finalit\u00e9 particuli\u00e8re, a des \u00ab devoirs \u00bb sup\u00e9rieurs : elle doit faire ob\u00e9ir le corps, conform\u00e9ment \u00e0 des lois auxquelles elle est elle-m\u00eame soumise. Quant au pouvoir du corps, ou bien c'est un pouvoir d'ex\u00e9cution, ou bien c'est un pouvoir de distraire l'\u00e2me et de la d\u00e9tourner de ses devoirs. En tout ceci nous pensons moralement. La vision morale du monde appara\u00eet dans un principe qui domine la plupart des th\u00e9ories de l'union de l'\u00e2me et du corps : l'un des deux n'agirait pas sans que l'autre ne p\u00e2tisse. Tel est notamment le principe de l'action r\u00e9elle chez Descartes : le corps p\u00e2tit quand l'\u00e2me agit, le corps n'agit pas sans que l'\u00e2me ne p\u00e2tisse \u00e0 son tour2. Or, bien qu'ils nient l'action r\u00e9elle, les successeurs de Descartes ne renoncent pas \u00e0 l'id\u00e9e de ce principe : l'harmonie pr\u00e9\u00e9tablie, par exemple, maintient entre l'\u00e2me et le corps une \u00ab action id\u00e9ale \u00bb d'apr\u00e8s laquelle, toujours, l'un p\u00e2tit quand l'autre agit3. Dans de telles perspectives, nous n'avons aucun moyen de _comparer_ la puissance du corps et la puissance de l'\u00e2me ; n'ayant pas le moyen de les comparer, nous n'avons aucune possibilit\u00e9 de les _\u00e9valuer_ respectivement4.\n\nSi le parall\u00e9lisme est une doctrine originale, ce n'est pas parce qu'il nie l'action r\u00e9elle de l'\u00e2me et du corps. C'est parce qu'il renverse le principe moral d'apr\u00e8s lequel les actions de l'un sont les passions de l'autre. \u00ab L'ordre des actions et des passions de notre corps va, par nature, de pair avec l'ordre des actions et des passions de l'esprit5. \u00bb Ce qui est passion dans l'\u00e2me est _aussi_ passion dans le corps, ce qui est action dans l'\u00e2me est _aussi_ action dans le corps. C'est en ce sens que le parall\u00e9lisme exclut toute \u00e9minence de l'\u00e2me, toute finalit\u00e9 spirituelle et morale, toute transcendance d'un Dieu qui r\u00e9glerait une s\u00e9rie sur l'autre. C'est en ce sens que le parall\u00e9lisme s'oppose pratiquement, non seulement \u00e0 la doctrine de l'action r\u00e9elle, mais aux th\u00e9ories de l'harmonie pr\u00e9\u00e9tablie et de l'occasionnalisme. Nous demandons : Qu'est-ce que peut un corps ? De quelles affections est-il capable, passives aussi bien qu'actives ? Jusqu'o\u00f9 va sa puissance ? Alors, et alors seulement, nous pourrons savoir ce que peut une \u00e2me en elle-m\u00eame et quelle est sa puissance. Nous aurons les moyens de \u00ab comparer \u00bb la puissance de l'\u00e2me avec la puissance du corps ; d\u00e8s lors, nous aurons les moyens d'\u00e9valuer la puissance de l'\u00e2me consid\u00e9r\u00e9e en elle-m\u00eame.\n\nPour arriver \u00e0 l' _\u00e9valuation_ de la puissance de l'\u00e2me en elle-m\u00eame, il fallait passer par la _comparaison_ des puissances : \u00ab Pour d\u00e9terminer en quoi l'esprit humain diff\u00e8re des autres et en quoi il l'emporte sur les autres, il nous est n\u00e9cessaire de conna\u00eetre la nature de son objet, c'est-\u00e0-dire du corps humain... Je dis en g\u00e9n\u00e9ral que plus un corps est apte, par rapport aux autres, \u00e0 agir et \u00e0 p\u00e2tir de plus de fa\u00e7ons \u00e0 la fois, plus son esprit est apte \u00e0 percevoir plus de choses \u00e0 la fois ; et plus les actions d'un corps d\u00e9pendent de lui seul, et moins d'autres corps concourent avec lui \u00e0 une action, plus son esprit est apte \u00e0 comprendre distinctement6. \u00bb Pour penser vraiment en termes de puissance, il fallait d'abord poser la question \u00e0 propos du corps, il fallait en premier lieu lib\u00e9rer le corps du rapport inversement proportionnel qui rend impossible toute comparaison de puissances, donc aussi qui rend impossible toute \u00e9valuation de la puissance de l'\u00e2me prise en elle-m\u00eame. Il fallait prendre pour mod\u00e8le la question : Qu'est-ce que peut un corps ? _Ce mod\u00e8le n'implique aucune d\u00e9valorisation de la pens\u00e9e par rapport \u00e0 l'\u00e9tendue, mais seulement une d\u00e9valorisation de la conscience par rapport \u00e0 la pens\u00e9e._ On se souvient que Platon disait que les mat\u00e9rialistes, s'ils \u00e9taient intelligents, parleraient de puissance au lieu de parler du corps. Mais il est vrai inversement que les dynamistes, quand ils sont intelligents, parlent d'abord du corps pour \u00ab penser \u00bb la puissance. La th\u00e9orie de la puissance, d'apr\u00e8s laquelle les actions et passions du corps vont de pair avec les actions et passions de l'\u00e2me, forme une vision \u00e9thique du monde. La substitution de l'\u00e9thique \u00e0 la morale est la cons\u00e9quence du parall\u00e9lisme, et en manifeste la v\u00e9ritable signification.\n\nLa question \u00ab Qu'est-ce que peut un corps ? \u00bb a un sens en elle-m\u00eame, parce qu'elle implique une nouvelle conception de l'individu corporel, de l'esp\u00e8ce et du genre. Sa signification biologique, nous le verrons, ne doit pas \u00eatre n\u00e9glig\u00e9e. Mais, _prise comme mod\u00e8le_ , elle a d'abord une signification juridique et \u00e9thique. Tout ce que peut un corps (sa puissance), est aussi bien son \u00ab droit naturel \u00bb. Si nous arrivons \u00e0 poser le probl\u00e8me du droit au niveau des corps, nous transformons toute la philosophie du droit, par rapport aux \u00e2mes elles-m\u00eames. Corps et \u00e2me, chacun cherche ce qui lui est utile ou bon. S'il arrive \u00e0 quelqu'un de rencontrer un corps qui se compose avec le sien sous un rapport favorable, il cherche \u00e0 s'unir avec lui. Quand quelqu'un rencontre un corps dont le rapport ne se compose pas avec le sien, un corps qui l'affecte de tristesse, il fait tout ce qui est en son pouvoir pour \u00e9carter la tristesse ou d\u00e9truire ce corps, c'est-\u00e0-dire pour imposer aux parties de ce corps un nouveau rapport qui convient avec sa propre nature. \u00c0 chaque instant donc, les affections d\u00e9terminent le _conatus_ ; mais \u00e0 chaque instant le _conatus_ est recherche de ce qui est utile en fonction des affections qui le d\u00e9terminent. _C'est pourquoi un corps va toujours aussi loin qu'il le peut, en passion comme en action_ ; et ce qu'il peut est son droit. La th\u00e9orie du droit naturel implique la double identit\u00e9 du pouvoir et de son exercice, de cet exercice et du droit. \u00ab Le droit de chacun s'\u00e9tend jusqu'aux bornes de la puissance limit\u00e9e dont il dispose7. \u00bb Le mot _loi_ n'a pas d'autre sens : la loi de nature n'est jamais une r\u00e8gle de devoirs, mais la norme d'un pouvoir, l'unit\u00e9 du droit, du pouvoir et de son effectuation8. \u00c0 cet \u00e9gard, on ne fera nulle diff\u00e9rence entre le sage et l'insens\u00e9, le raisonnable et le d\u00e9ment, le fort et le faible. Sans doute diff\u00e8rent-ils par le genre d'affections qui d\u00e9terminent leur effort \u00e0 pers\u00e9v\u00e9rer dans l'existence. Mais l'un et l'autre s'efforcent \u00e9galement de se conserver, ont autant de droit que de puissance, en fonction des affections qui remplissent actuellement leur pouvoir d'\u00eatre affect\u00e9. L'insens\u00e9 lui-m\u00eame est de la nature, et n'en trouble aucunement l'ordre9.\n\nCette conception du droit naturel est directement h\u00e9rit\u00e9e de Hobbes. (La question des diff\u00e9rences essentielles de Spinoza avec Hobbes se pose \u00e0 un autre niveau.) Ce que Spinoza doit \u00e0 Hobbes, c'est une conception du droit de nature qui s'oppose profond\u00e9ment \u00e0 la th\u00e9orie classique de la loi naturelle. Si nous suivons Cic\u00e9ron, qui recueille \u00e0 la fois des traditions platoniciennes, aristot\u00e9liciennes et sto\u00efciennes, nous voyons que la th\u00e9orie antique de la loi naturelle pr\u00e9sente plusieurs caract\u00e8res : 1o) Elle d\u00e9finit la nature d'un \u00eatre par sa perfection, conform\u00e9ment \u00e0 un ordre des fins (ainsi l'homme est \u00ab naturellement \u00bb raisonnable et sociable). 2o) Il s'ensuit que l'\u00e9tat de nature pour l'homme n'est pas un \u00e9tat qui pr\u00e9c\u00e9derait la soci\u00e9t\u00e9, ne serait-ce qu'en droit, mais au contraire une vie conforme \u00e0 la nature dans une \u00ab bonne \u00bb soci\u00e9t\u00e9 civile. 3o) Donc ce qui est premier et inconditionnel dans cet \u00e9tat, ce sont les \u00ab devoirs \u00bb ; car les pouvoirs naturels sont seulement en puissance, et ne sont pas s\u00e9parables d'un acte de la raison qui les d\u00e9termine et les r\u00e9alise en fonction de fins auxquelles ils doivent servir. 4o) La comp\u00e9tence du sage, par l\u00e0 m\u00eame, est fond\u00e9e ; car le sage est le meilleur juge de l'ordre des fins, des devoirs qui en d\u00e9coulent, des offices et des actions qu'il revient \u00e0 chacun de faire et de remplir. On devine quel parti le christianisme devait tirer de cette conception de la loi de nature. Avec lui, cette loi devenait ins\u00e9parable de la th\u00e9ologie naturelle et m\u00eame de la R\u00e9v\u00e9lation10.\n\nIl revient \u00e0 Hobbes d'avoir d\u00e9gag\u00e9 quatre th\u00e8ses fondamentales, qui s'opposent \u00e0 celles-l\u00e0. Ces th\u00e8ses originales transforment le probl\u00e8me philosophique du droit, mais pr\u00e9cis\u00e9ment parce qu'elles prennent le corps pour mod\u00e8le m\u00e9canique et dynamique. Spinoza adopte ces th\u00e8ses, les int\u00e9grant dans son propre syst\u00e8me o\u00f9 elles trouvent des perspectives nouvelles. 1o) La loi de nature n'est plus rapport\u00e9e \u00e0 une perfection finale, mais au premier d\u00e9sir, \u00e0 \u00ab l'app\u00e9tit \u00bb le plus fort ; dissoci\u00e9e d'un ordre des fins, elle se d\u00e9duit de l'app\u00e9tit comme de sa cause efficiente. 2o) De ce point de vue la raison ne jouit d'aucun privil\u00e8ge : l'insens\u00e9 ne s'efforce pas moins que l'\u00eatre raisonnable de pers\u00e9v\u00e9rer dans son \u00eatre, et les d\u00e9sirs ou les actions qui naissent de la raison ne manifestent pas plus cet effort que les d\u00e9sirs ou les passions de l'insens\u00e9 lui-m\u00eame. Bien plus, _personne ne na\u00eet raisonnable._ Il se peut que la raison utilise et conserve la loi de nature, en aucun cas elle n'en est le principe ou le mobile. De m\u00eame _personne ne na\u00eet citoyen_11. Il se peut que l'\u00e9tat civil conserve le droit de nature, mais l'\u00e9tat de nature en lui-m\u00eame est pr\u00e9-social, pr\u00e9-civil. Et encore, _personne ne na\u00eet religieux_ : \u00ab L'\u00e9tat de nature, par nature et dans le temps, est ant\u00e9rieur \u00e0 la religion ; la nature n'a jamais enseign\u00e9 \u00e0 personne qu'il est oblig\u00e9 d'ob\u00e9ir \u00e0 Dieu...12 \u00bb 3o) Ce qui est premier et inconditionn\u00e9, c'est donc le pouvoir ou le droit. Les \u00ab devoirs \u00bb, quels qu'ils soient, sont toujours secondaires, relatifs \u00e0 l'affirmation de notre puissance, \u00e0 l'exercice de notre pouvoir, \u00e0 la conservation de notre droit. Et la puissance ne renvoie plus \u00e0 un acte qui la d\u00e9termine et la r\u00e9alise en fonction d'un ordre des fins. Ma puissance elle-m\u00eame est en acte, parce que les affections que j'\u00e9prouve \u00e0 chaque instant la d\u00e9terminent et la remplissent de plein droit, quelles que soient ces affections. 4o) Il s'ensuit que nul n'a comp\u00e9tence pour d\u00e9cider de mon droit. Chacun dans l'\u00e9tat de nature, qu'il soit sage ou insens\u00e9, est juge de ce qui est bon et mauvais, de ce qui est n\u00e9cessaire \u00e0 sa conservation. D'o\u00f9, le droit de nature n'est contraire \u00ab ni aux luttes, ni aux haines, ni \u00e0 la col\u00e8re, ni \u00e0 la tromperie, ni \u00e0 rien absolument de ce que l'app\u00e9tit conseille13. \u00bb Et s'il arrive que nous soyons amen\u00e9s \u00e0 renoncer \u00e0 notre droit naturel, ce ne sera pas en reconnaissant la comp\u00e9tence du sage, mais en consentant par nous-m\u00eames \u00e0 ce renoncement, par crainte d'un plus grand mal ou par espoir d'un plus grand bien. Le principe du consentement (pacte ou contrat) devient principe de la philosophie politique et se substitue \u00e0 la r\u00e8gle de comp\u00e9tence.\n\nAinsi d\u00e9fini, l'\u00e9tat de nature manifeste en soi ce qui le rend invivable. L'\u00e9tat de nature n'est pas viable, tant que le droit naturel qui lui correspond reste th\u00e9orique et abstrait14. Or, dans l'\u00e9tat de nature, je vis au hasard des rencontres. Il est bien vrai que ma puissance est d\u00e9termin\u00e9e par les affections qui remplissent \u00e0 chaque instant mon pouvoir d'\u00eatre affect\u00e9 ; il est bien vrai que j'ai toujours toute la perfection dont je suis capable en fonction de ces affections. Mais pr\u00e9cis\u00e9ment, dans l'\u00e9tat de nature, mon pouvoir d'\u00eatre affect\u00e9 se trouve rempli dans de telles conditions que j'\u00e9prouve non seulement des affections passives qui me s\u00e9parent de ma puissance d'agir, mais encore des affections passives \u00e0 pr\u00e9dominance de tristesse qui ne cessent de diminuer cette puissance elle-m\u00eame. Je n'ai aucune chance de rencontrer des corps qui se composent directement avec le mien. J'aurais beau l'emporter dans plusieurs rencontres avec des corps qui me sont contraires ; ces triomphes ou ces joies de la haine ne supprimeront pas la tristesse que la haine enveloppe ; et surtout, je ne serai jamais s\u00fbr d'\u00eatre encore vainqueur \u00e0 la prochaine rencontre, je serai donc affect\u00e9 d'une crainte perp\u00e9tuelle.\n\nIl n'y aurait qu'un moyen de rendre viable l'\u00e9tat de nature : en s'effor\u00e7ant _d'organiser les rencontres._ Quel que soit le corps rencontr\u00e9, je cherche l'utile. Mais il y a une grande diff\u00e9rence entre chercher l'utile au hasard (c'est-\u00e0-dire s'efforcer de d\u00e9truire les corps qui ne conviennent pas avec le n\u00f4tre) et chercher une organisation de l'utile (s'efforcer de rencontrer les corps qui conviennent en nature avec nous, sous les rapports o\u00f9 ils conviennent). Seul ce deuxi\u00e8me effort d\u00e9finit _l'utile propre ou v\u00e9ritable_15. Sans doute cet effort a-t-il des limites : nous serons toujours d\u00e9termin\u00e9s \u00e0 d\u00e9truire certains corps, ne serait-ce que pour subsister ; nous n'\u00e9viterons pas toute mauvaise rencontre, nous n'\u00e9viterons pas la mort. Mais nous nous effor\u00e7ons de nous unir \u00e0 ce qui convient avec notre nature, de composer notre rapport avec des rapports qui se combinent avec le n\u00f4tre, de joindre nos gestes et nos pens\u00e9es \u00e0 l'image de choses qui s'accordent avec nous. D'un tel effort nous sommes en droit d'attendre, par d\u00e9finition, un maximum d'affections joyeuses. Notre pouvoir d'\u00eatre affect\u00e9 sera rempli dans de telles conditions que notre puissance d'agir augmentera. Et si l'on demande en quoi consiste ce qui nous est le plus utile, on voit bien que c'est l'homme. Car l'homme, en principe, convient en nature avec l'homme ; il compose son rapport avec le sien ; l'homme est utile \u00e0 l'homme absolument ou v\u00e9ritablement. Chacun, cherchant ce qui lui est v\u00e9ritablement utile, cherche donc aussi ce qui est utile \u00e0 l'homme. Ainsi l'effort d'organiser les rencontres est d'abord l'effort de former l'association des hommes sous des rapports qui se composent16.\n\nIl n'y a pas de Bien ni de Mal dans la Nature, il n'y a pas d'opposition morale, mais il y a une diff\u00e9rence \u00e9thique. Cette diff\u00e9rence \u00e9thique se pr\u00e9sente sous plusieurs formes \u00e9quivalentes : entre le raisonnable et l'insens\u00e9, entre le sage et l'ignorant, entre l'homme libre et l'esclave, entre le fort et le faible17. Et en v\u00e9rit\u00e9, la sagesse ou la raison n'ont pas d'autre contenu que la force, la libert\u00e9. Cette diff\u00e9rence \u00e9thique ne porte pas sur le _conatus_ , puisque l'insens\u00e9 non moins que le raisonnable, le faible non moins que le fort s'efforce de pers\u00e9v\u00e9rer dans son \u00eatre. _Elle porte sur le genre d'affections qui d\u00e9terminent le conatus_. \u00c0 la limite, l'homme libre, fort et raisonnable se d\u00e9finira pleinement par la possession de sa puissance d'agir, par la pr\u00e9sence en lui d'id\u00e9es ad\u00e9quates et d'affections actives ; au contraire, l'esclave, le faible n'ont que des passions qui d\u00e9rivent de leurs id\u00e9es inad\u00e9quates, et qui les s\u00e9parent de leur puissance d'agir.\n\nMais la diff\u00e9rence \u00e9thique s'exprime d'abord \u00e0 ce niveau plus simple, pr\u00e9paratoire ou pr\u00e9liminaire. Avant d'en arriver \u00e0 la possession formelle de sa puissance, _l'homme libre et fort se_ reconna\u00eet \u00e0 ses passions joyeuses, \u00e0 ses affections qui augmentent cette puissance d'agir ; _l'esclave ou le faible_ se reconnaissent \u00e0 leurs passions tristes, aux affections \u00e0 base de tristesse qui diminuent leur puissance d'agir. Tout se passe donc comme si l'on devait distinguer deux moments de la raison ou de la libert\u00e9 : augmenter la puissance d'agir en s'effor\u00e7ant d'\u00e9prouver le maximum d'affections passives joyeuses ; et ainsi, passer au stade final o\u00f9 la puissance d'agir a si bien augment\u00e9 qu'elle devient capable de produire des affections elles-m\u00eames actives. Il est vrai que l'encha\u00eenement des deux temps nous reste encore myst\u00e9rieux. Du moins la pr\u00e9sence du premier temps n'est-elle pas douteuse. L'homme qui devient raisonnable, fort et libre, commence par faire tout ce qui est en son pouvoir pour \u00e9prouver des passions joyeuses. C'est donc lui qui s'efforce de s'arracher au hasard des rencontres et \u00e0 l'encha\u00eenement des passions tristes, d'organiser les bonnes rencontres, de composer son rapport avec les rapports qui se combinent directement avec le sien, de s'unir avec ce qui convient en nature avec lui, de former l'association raisonnable entre les hommes ; tout cela, de mani\u00e8re \u00e0 \u00eatre affect\u00e9 de joie. Dans l' _\u00c9thique_ la description du livre IV, concernant l'homme libre et raisonnable, identifie l'effort de la raison avec cet art d'organiser les rencontres, ou de former une _totalit\u00e9_ sous des rapports qui se composent18.\n\nChez Spinoza, la raison, la force ou la libert\u00e9 ne sont pas s\u00e9parables d'un devenir, d'une formation, d'une culture. Personne ne na\u00eet libre, personne ne na\u00eet raisonnable19. Et personne ne peut faire pour nous la lente exp\u00e9rience de ce qui convient avec notre nature, l'effort lent pour d\u00e9couvrir nos joies. L'enfance, dit souvent Spinoza, est un \u00e9tat d'impuissance et d'esclavage, un \u00e9tat insens\u00e9 o\u00f9 nous d\u00e9pendons au plus haut point des causes ext\u00e9rieures, et o\u00f9 nous avons n\u00e9cessairement plus de tristesses que de joies ; jamais nous ne serons autant s\u00e9par\u00e9s de notre puissance d'agir. Le premier homme, Adam, est l'enfance de l'humanit\u00e9. C'est pourquoi Spinoza s'oppose avec tant de force \u00e0 la tradition chr\u00e9tienne, puis rationaliste, qui nous pr\u00e9sente avant la faute un Adam raisonnable, libre et parfait. Au contraire, il faut imaginer Adam comme un enfant : triste, faible, esclave, ignorant, livr\u00e9 au hasard des rencontres. \u00ab Il faut reconna\u00eetre qu'il n'\u00e9tait pas au pouvoir du premier homme d'user droitement de la raison, mais qu'il a \u00e9t\u00e9, comme nous le sommes, soumis aux passions20. \u00bb C'est-\u00e0-dire : _Ce n'est pas la faute qui explique la faiblesse, c'est notre faiblesse premi\u00e8re qui explique le mythe de la faute._ Spinoza pr\u00e9sente trois th\u00e8ses concernant Adam, qui forment un ensemble syst\u00e9matique : 1o) Dieu n'a rien d\u00e9fendu \u00e0 Adam, mais lui a simplement r\u00e9v\u00e9l\u00e9 que le fruit \u00e9tait un poison qui d\u00e9truirait son corps s'il entrait en contact avec lui. 2o) Son entendement \u00e9tant faible comme celui d'un enfant, Adam a per\u00e7u cette r\u00e9v\u00e9lation comme une d\u00e9fense ; il d\u00e9sob\u00e9it comme un enfant, ne comprenant pas la n\u00e9cessit\u00e9 naturelle du rapport action-cons\u00e9quence, croyant que les lois de la nature sont des lois morales qu'il est possible de violer. 3o) Comment imaginer un Adam libre et raisonnable, alors que le premier homme est n\u00e9cessairement affect\u00e9 de sentiments passifs, et n'a pas eu le temps d'entreprendre cette longue formation que la raison suppose non moins que la libert\u00e9 ?21.\n\nL'\u00e9tat de raison, tel qu'il appara\u00eet d\u00e9j\u00e0 sous son premier aspect, a un rapport complexe avec l'\u00e9tat de nature. D'une part, l'\u00e9tat de nature n'est pas soumis aux lois de la raison : la raison concerne l'utilit\u00e9 propre et v\u00e9ritable de l'homme, et tend uniquement \u00e0 sa conservation ; la nature au contraire n'a nul \u00e9gard \u00e0 la conservation de l'homme et comprend une infinit\u00e9 d'autres lois qui concernent l'univers entier, dont l'homme n'est qu'une petite partie. Mais d'autre part, l'\u00e9tat de raison n'est pas d'un autre ordre que l'\u00e9tat de nature lui-m\u00eame. M\u00eame dans ses \u00ab commandements \u00bb, la raison ne demande rien qui soit contraire \u00e0 la nature : elle demande seulement que chacun s'aime soi-m\u00eame, cherche ce qui lui est utile en propre et s'efforce de conserver son \u00eatre en augmentant sa puissance d'agir22. Aussi n'y a-t-il pas d'artifice ou de convention dans l'effort de la raison. La raison ne proc\u00e8de pas par artifice, mais par composition naturelle de rapports ; elle ne proc\u00e8de pas \u00e0 des calculs, mais \u00e0 une esp\u00e8ce de reconnaissance directe de l'homme par l'homme23. La question de savoir si des \u00eatres suppos\u00e9s raisonnables, ou en train de le devenir, ont besoin de s'engager mutuellement par une sorte de contrat, est tr\u00e8s complexe ; mais m\u00eame s'il y a contrat \u00e0 ce niveau, ce contrat n'implique aucun renoncement conventionnel au droit naturel, aucune limitation artificielle. L'\u00e9tat de raison ne fait qu'un avec la formation d'un corps sup\u00e9rieur et d'une \u00e2me sup\u00e9rieure, jouissant du droit naturel correspondant \u00e0 leur puissance : en effet, si deux individus composent enti\u00e8rement leurs rapports, ils forment naturellement un individu deux fois plus grand, ayant lui-m\u00eame un droit de nature deux fois plus grand24. L'\u00e9tat de raison ne supprime ni ne limite en rien le droit naturel, il l'\u00e9l\u00e8ve \u00e0 une puissance sans laquelle ce droit resterait irr\u00e9el et abstrait.\n\n\u00c0 quoi donc se r\u00e9duit la diff\u00e9rence entre l'\u00e9tat de raison et l'\u00e9tat de nature ? Dans l'ordre de la nature, chaque corps en rencontre d'autres, mais son rapport ne se compose pas n\u00e9cessairement avec ceux des corps qu'il rencontre. La co\u00efncidence des rencontres et des rapports se fait seulement au niveau de la nature enti\u00e8re ; elle a lieu d'ensemble \u00e0 ensemble dans le mode infini m\u00e9diat. Toutefois, quand nous nous \u00e9levons dans la s\u00e9rie des essences, nous assistons \u00e0 un effort qui pr\u00e9figure celui de la nature enti\u00e8re. Les plus hautes essences, dans l'existence, s'efforcent d\u00e9j\u00e0 de faire co\u00efncider _leurs propres_ rencontres avec des rapports qui se composent avec _le leur._ Cet effort, qui ne peut aboutir compl\u00e8tement, constitue l'effort de la raison. C'est en ce sens que l'\u00eatre raisonnable, \u00e0 sa fa\u00e7on, peut \u00eatre dit reproduire et exprimer l'effort de la nature enti\u00e8re.\n\nMais comment les hommes arriveront-ils \u00e0 se rencontrer sous des rapports qui se composent et \u00e0 former ainsi une association raisonnable ? Si l'homme convient avec l'homme, c'est seulement en tant qu'on le suppose _d\u00e9j\u00e0_ raisonnable25. Tant qu'ils vivent au hasard des rencontres, tant qu'ils sont affect\u00e9s de passions fortuites, les hommes sont diversement entra\u00een\u00e9s, et pr\u00e9cis\u00e9ment n'ont aucune chance de se rencontrer sous des rapports qui conviennent : ils sont contraires les uns aux autres26. Il est vrai que nous \u00e9chappons \u00e0 la contradiction dans la mesure o\u00f9 nous invoquons une tr\u00e8s lente exp\u00e9rience, une tr\u00e8s lente formation empirique. Mais nous tombons aussit\u00f4t dans une autre difficult\u00e9. D'une part, le poids des rencontres pr\u00e9sentes est toujours l\u00e0, qui risque d'annihiler l'effort de la raison. Et surtout cet effort, au mieux, n'aboutira qu'\u00e0 la fin de la vie ; \u00ab or il faut bien vivre jusque-l\u00e027. \u00bb Voil\u00e0 pourquoi la raison ne serait rien et ne conquerrait jamais sa propre puissance si elle ne trouvait une aide dans une puissance d'un autre genre, qui se joint \u00e0 elle, qui en pr\u00e9pare et en accompagne la formation. Cette puissance d'un autre genre est celle de l'\u00c9tat ou de la cit\u00e9.\n\nEn effet, la cit\u00e9 n'est nullement une association raisonnable. Elle s'en distingue de trois fa\u00e7ons. 1o) Le mobile de sa formation n'est pas une affection de la raison, c'est-\u00e0-dire une affection produite en nous par un autre homme sous un rapport qui se compose parfaitement avec le n\u00f4tre. Le mobile en est la crainte, ou l'angoisse de l'\u00e9tat de nature, l'espoir d'un plus grand bien28. 2o) Le tout comme id\u00e9al de la raison est constitu\u00e9 par des rapports qui se composent directement et naturellement, par des puissances ou des droits qui s'additionnent naturellement. Il n'en est pas ainsi de la cit\u00e9 : les hommes n'\u00e9tant pas raisonnables, il faut que chacun \u00ab renonce \u00bb \u00e0 son droit naturel. Seule cette ali\u00e9nation rend possible la formation d'un tout qui b\u00e9n\u00e9ficie lui-m\u00eame de la somme de ces droits. Tel est le \u00ab pacte \u00bb ou le \u00ab contrat \u00bb civil29. Alors la cit\u00e9 souveraine a assez de puissance pour instituer des rapports indirects et conventionnels sous lesquels les citoyens _sont forc\u00e9s_ de convenir et de s'accorder. 3o) La raison est au principe d'une distinction \u00e9thique, entre \u00ab ceux qui vivent sous sa conduite \u00bb et ceux qui restent sous la conduite du sentiment, ceux qui se lib\u00e8rent et ceux qui restent esclaves. Mais l'\u00e9tat civil distingue seulement les justes et les injustes, suivant l'ob\u00e9issance \u00e0 ses lois. Ayant renonc\u00e9 \u00e0 leur droit de juger ce qui est bon et mauvais, les citoyens s'en remettent \u00e0 l'\u00c9tat, qui r\u00e9compense et qui ch\u00e2tie. P\u00e9ch\u00e9-ob\u00e9issance, justice-injustice sont des cat\u00e9gories proprement sociales ; l'opposition morale elle-m\u00eame a pour principe et pour milieu la soci\u00e9t\u00e930.\n\nPourtant, entre la cit\u00e9 et l'id\u00e9al de la raison, une grande ressemblance existe. Chez Spinoza comme chez Hobbes, le souverain se d\u00e9finit par son droit naturel, \u00e9gal \u00e0 sa puissance, c'est-\u00e0-dire \u00e9gal \u00e0 tous les droits dont les contractants se sont dessaisis. Mais ce souverain n'est pas, comme chez Hobbes, un tiers au b\u00e9n\u00e9fice duquel se ferait le contrat des particuliers. Le souverain est le tout ; le contrat se fait entre individus, mais qui transf\u00e8rent leurs droits au tout qu'ils forment en contractant. C'est pourquoi la cit\u00e9 est d\u00e9crite par Spinoza comme une personne collective, corps commun et \u00e2me commune, \u00ab masse conduite en quelque sorte par une m\u00eame pens\u00e9e31 \u00bb. Que son proc\u00e9d\u00e9 de formation soit tr\u00e8s diff\u00e9rent de celui de la raison, qu'il soit pr\u00e9-rationnel, n'emp\u00eache pas que la cit\u00e9 n'imite et ne pr\u00e9pare la raison. En effet, il n'y a pas et il ne peut pas y avoir de totalit\u00e9 irrationnelle, contraire \u00e0 la raison. Sans doute le souverain est-il en droit de commander tout ce qu'il veut, autant qu'il a de puissance ; il est seul juge des lois qu'il institue et ne peut ni p\u00e9cher ni d\u00e9sob\u00e9ir. Mais, pr\u00e9cis\u00e9ment parce qu'il est un tout, il ne peut se conserver comme tel que dans la mesure o\u00f9 \u00ab il tend au but que la saine raison enseigne \u00e0 tous les hommes d'atteindre \u00bb : le tout ne peut se conserver que s'il tend \u00e0 quelque chose ayant au moins l'apparence de la raison32. Le contrat par lequel les individus ali\u00e8nent leur droit n'a pas d'autre ressort que l'int\u00e9r\u00eat (la crainte d'un plus grand mal, l'espoir d'un plus grand bien) ; si les citoyens se mettent \u00e0 craindre la cit\u00e9 plus que tout, ils se retrouvent \u00e0 l'\u00e9tat de nature, en m\u00eame temps que la cit\u00e9 perd de sa puissance, en butte aux factions qu'elle a suscit\u00e9es. C'est donc sa propre nature qui d\u00e9termine la cit\u00e9 \u00e0 viser autant que possible l'id\u00e9al de la raison, \u00e0 s'efforcer de conformer \u00e0 la raison l'ensemble de ses lois. Et la cit\u00e9 conviendra d'autant plus avec la raison qu'elle produira moins de passions tristes chez les citoyens (crainte ou m\u00eame espoir), s'appuyant plut\u00f4t sur des affections joyeuses33.\n\nTout cela doit se comprendre de la \u00ab bonne \u00bb cit\u00e9. Car il en est de la cit\u00e9 comme de l'individu : beaucoup de causes interviennent, parfois insensibles, qui en pervertissent la nature et en provoquent la ruine. Mais, du point de vue de la bonne cit\u00e9, deux autres arguments s'ajoutent aux pr\u00e9c\u00e9dents. D'abord, que signifie pour le citoyen \u00ab renoncer \u00e0 son droit naturel \u00bb ? Non pas \u00e9videmment renoncer \u00e0 pers\u00e9v\u00e9rer dans l'\u00eatre. Mais : renoncer \u00e0 se d\u00e9terminer d'apr\u00e8s des _affections personnelles quelconques._ Abandonnant son droit de juger personnellement ce qui est bon et mauvais, le citoyen s'engage donc \u00e0 recevoir des _affections communes et collectives._ Mais, en fonction de ces affections, il continue personnellement \u00e0 pers\u00e9v\u00e9rer dans son \u00eatre, \u00e0 faire tout ce qui est en son pouvoir pour conserver son existence et veiller \u00e0 ses int\u00e9r\u00eats34. C'est en ce sens que Spinoza peut dire que chacun renonce \u00e0 son droit naturel suivant la r\u00e8gle de la cit\u00e9, et pourtant conserve enti\u00e8rement ce droit naturel dans l'\u00e9tat civil35. D'autre part, les _affections de la raison_ ne rel\u00e8vent pas de la cit\u00e9 : la puissance de conna\u00eetre, de penser et d'exprimer sa pens\u00e9e reste un droit naturel inali\u00e9nable, que la cit\u00e9 ne peut pas compromettre sans r\u00e9tablir entre elle et ses sujets des rapports de simple violence36.\n\nLa \u00ab bonne \u00bb cit\u00e9 tant\u00f4t tient lieu de raison \u00e0 ceux qui n'en ont pas ; tant\u00f4t pr\u00e9pare, pr\u00e9figure et imite \u00e0 sa mani\u00e8re l'\u0153uvre de la raison. C'est elle qui rend possible la formation de la raison m\u00eame. On ne consid\u00e9rera pas comme preuves d'un optimisme exag\u00e9r\u00e9 les deux propositions de Spinoza : finalement et malgr\u00e9 tout, la cit\u00e9 est le meilleur milieu o\u00f9 l'homme puisse devenir raisonnable ; et aussi le meilleur milieu o\u00f9 l'homme raisonnable puisse vivre37.\n\nDans une vision \u00e9thique du monde, il est toujours question de pouvoir et de puissance, et il n'est pas question d'autre chose. La loi est identique au droit. Les vraies lois naturelles sont les normes du pouvoir, non pas des r\u00e8gles de devoir. C'est pourquoi la loi morale, qui pr\u00e9tend interdire et commander, implique une sorte de mystification : moins nous comprenons les lois de la nature, c'est-\u00e0-dire les normes de vie, plus nous les interpr\u00e9tons comme des ordres et des d\u00e9fenses. Au point que le philosophe doit h\u00e9siter \u00e0 se servir du mot loi, tant ce mot garde un arri\u00e8re-go\u00fbt moral : il vaut mieux parler de \u00ab v\u00e9rit\u00e9s \u00e9ternelles \u00bb. En v\u00e9rit\u00e9, les lois morales ou les devoirs sont purement civils, sociaux : seule la soci\u00e9t\u00e9 ordonne et d\u00e9fend, menace et fait esp\u00e9rer, r\u00e9compense et ch\u00e2tie. Sans doute la raison comprend-elle pour son compte une _pietas_ et une _religio_ ; sans doute y a-t-il des pr\u00e9ceptes, des r\u00e8gles ou des \u00ab commandements \u00bb de la raison. Mais la liste de ces commandements suffit \u00e0 montrer qu'il ne s'agit pas de devoirs, mais de normes de vie, concernant la \u00ab force \u00bb de l'\u00e2me et sa puissance d'agir38. Sans doute aussi peut-il arriver que ces normes co\u00efncident avec des lois de la morale ordinaire ; mais d'une part ces co\u00efncidences ne sont pas nombreuses ; d'autre part, quand la raison recommande ou d\u00e9nonce quelque chose d'analogue \u00e0 ce que la morale commande ou interdit, c'est toujours pour des raisons tr\u00e8s diff\u00e9rentes de celles de la morale39. L' _\u00c9thique_ juge des sentiments, des conduites et des intentions en les rapportant, non pas \u00e0 des valeurs transcendantes mais \u00e0 des modes d'existence qu'ils supposent ou impliquent : il y a des choses qu'on ne peut faire ou m\u00eame dire, croire, \u00e9prouver, penser, qu'\u00e0 condition d'\u00eatre faible, esclave, impuissant ; d'autres choses qu'on ne peut faire, \u00e9prouver, etc., qu'\u00e0 condition d'\u00eatre libre ou fort. _Une m\u00e9thode d'explication des modes d'existences immanents_ remplace ainsi le recours aux valeurs transcendantes. De toutes mani\u00e8res, la question est : Tel sentiment, par exemple, augmente-t-il ou non notre puissance d'agir ? Nous aide-t-il \u00e0 acqu\u00e9rir la possession formelle de cette puissance ?\n\nAller jusqu'au bout de ce qu'on peut, est la t\u00e2che proprement \u00e9thique. C'est par l\u00e0 que l' _\u00c9thique_ prend mod\u00e8le sur le corps ; car tout corps \u00e9tend sa puissance aussi loin qu'il le peut. En un sens, tout \u00eatre, \u00e0 chaque moment, va jusqu'au bout de ce qu'il peut. \u00ab Ce qu'il peut \u00bb, c'est son pouvoir d'\u00eatre affect\u00e9, qui se trouve n\u00e9cessairement et constamment rempli par la relation de cet \u00eatre avec les autres. Mais, en un autre sens, notre pouvoir d'\u00eatre affect\u00e9 peut \u00eatre rempli de telle fa\u00e7on que nous soyons s\u00e9par\u00e9s de notre puissance d'agir, et que celle-ci ne cesse de diminuer. En ce second sens, il arrive que nous vivions s\u00e9par\u00e9s de \u00ab ce que nous pouvons \u00bb. C'est m\u00eame le sort de la plupart des hommes, la plupart du temps. _Le faible, l'esclave, n'est pas quelqu'un dont la force est moindre, prise absolument._ Le faible est celui qui, quelle que soit sa force, reste s\u00e9par\u00e9 de sa puissance d'agir, maintenu dans l'esclavage ou l'impuissance. Aller jusqu'au bout de son pouvoir signifie deux choses : Comment remplir notre pouvoir d'\u00eatre affect\u00e9 de telle fa\u00e7on que notre puissance d'agir augmente ? Et comment augmenter cette puissance au point que, enfin, nous produisions des affections actives ? Il y a donc des faibles et des forts, des esclaves et des hommes libres. Il n'y a pas de Bien ni de Mal dans la Nature, il n'y a pas d'opposition morale, mais il y a une diff\u00e9rence \u00e9thique. Cette diff\u00e9rence est celle des modes d'existence immanents, envelopp\u00e9s dans ce que nous \u00e9prouvons, faisons, pensons.\n\nCette conception \u00e9thique a un aspect critique fondamental. Spinoza s'inscrit dans une tradition c\u00e9l\u00e8bre : la t\u00e2che pratique du philosophe consiste \u00e0 d\u00e9noncer tous les mythes, toutes les mystifications, toutes les \u00ab superstitions \u00bb, quelle qu'en soit l'origine. Cette tradition, croyons-nous, ne se s\u00e9pare pas du naturalisme comme philosophie. La _superstition_ , c'est tout ce qui nous maintient s\u00e9par\u00e9 de notre puissance d'agir et ne cesse de diminuer celle-ci. Aussi la source de la superstition est-elle l'encha\u00eenement des passions tristes, la crainte, l'espoir qui s'encha\u00eene \u00e0 la crainte, l'angoisse qui nous livre aux fant\u00f4mes40. Comme Lucr\u00e8ce, Spinoza sait qu'il n'y a pas de mythe ou de superstition joyeuse. Comme Lucr\u00e8ce, il dresse l'image d'une Nature positive contre l'incertitude des dieux : _ce qui s'oppose \u00e0 la Nature n'est pas la culture, ni l'\u00e9tat de raison, ni m\u00eame l'\u00e9tat civil, mais seulement la superstition qui menace toutes les entreprises de l'homme._ Comme Lucr\u00e8ce encore, Spinoza assigne au philosophe la t\u00e2che de d\u00e9noncer tout ce qui est tristesse, tout ce qui vit de la tristesse, tous ceux qui ont besoin de la tristesse pour asseoir leur pouvoir. \u00ab Le grand secret du r\u00e9gime monarchique et son int\u00e9r\u00eat vital consistent \u00e0 tromper les hommes en travestissant du nom de religion la crainte, dont on veut les tenir en bride ; de sorte qu'ils combattent pour leur servitude comme s'il s'agissait de leur salut... \u00bb41. La d\u00e9valorisation des passions tristes, la d\u00e9nonciation de ceux qui les cultivent et qui s'en servent, forment l'objet pratique de la philosophie. Peu de th\u00e8mes dans l' _\u00c9thique_ apparaissent aussi constamment que celui-ci : tout ce qui est triste est mauvais, et nous rend esclave ; tout ce qui enveloppe la tristesse exprime un tyran.\n\n\u00ab Aucune puissance divine, nul autre qu'un envieux ne prend plaisir \u00e0 mon impuissance et \u00e0 mon d\u00e9sagr\u00e9ment, et ne nous tient pour vertu les larmes, les sanglots, la crainte et autres manifestations de ce genre, qui sont des signes d'une \u00e2me impuissante. Mais au contraire, d'autant nous sommes affect\u00e9s d'une plus grande joie, d'autant nous passons \u00e0 une perfection plus grande, c'est-\u00e0-dire d'autant plus il est n\u00e9cessaire que nous participions de la nature divine. \u00bb \u00ab Celui qui sait bien que toutes choses suivent de la n\u00e9cessit\u00e9 de la nature divine et se font selon les lois et les r\u00e8gles \u00e9ternelles de la Nature, celui-l\u00e0 certes ne trouvera rien qui soit digne de haine, de ris\u00e9e ou de m\u00e9pris, et il n'aura piti\u00e9 de qui que ce soit ; mais autant que le permet la vertu humaine, il s'efforcera d'agir bien, comme on dit, et de se r\u00e9jouir. \u00bb \u00ab Les superstitieux, qui savent reprocher les vices plut\u00f4t qu'enseigner les vertus, et qui s'appliquent non \u00e0 conduire les hommes par la raison, mais \u00e0 les contenir par la crainte de fa\u00e7on qu'ils fuient le mal plut\u00f4t qu'ils n'aiment les vertus, ne tendent \u00e0 rien d'autre qu'\u00e0 rendre les hommes aussi malheureux qu'eux-m\u00eames ; aussi n'est-il pas \u00e9tonnant que la plupart du temps ils soient importuns et odieux aux hommes. \u00bb \u00ab Ceux qui sont mal re\u00e7us de leur ma\u00eetresse ne pensent \u00e0 rien qu'\u00e0 l'inconstance des femmes, \u00e0 leur esprit de tromperie et \u00e0 leurs autres vices dont on parle tant, toutes choses qu'ils oublient aussit\u00f4t d\u00e8s qu'ils sont de nouveau accueillis par leur ma\u00eetresse. C'est pourquoi celui qui s'applique \u00e0 r\u00e9gler ses sentiments et ses app\u00e9tits d'apr\u00e8s le seul amour de la libert\u00e9 s'efforcera, autant qu'il peut, de conna\u00eetre les vertus et leurs causes, et de remplir son \u00e2me du contentement qui na\u00eet de leur connaissance vraie ; mais pas du tout de consid\u00e9rer les vices des hommes, de ravaler les hommes et de se r\u00e9jouir d'une fausse apparence de libert\u00e9. \u00bb \u00ab L'homme libre ne pense \u00e0 aucune chose moins qu'\u00e0 la mort, et sa sagesse est une m\u00e9ditation non de la mort, mais de la vie42. \u00bb\n\n\u00c0 travers les scolies du livre IV, on voit Spinoza former une conception proprement \u00e9thique de l'homme, fond\u00e9e sur la joie et les passions joyeuses. Il l'oppose \u00e0 une conception superstitieuse ou satirique, qui se fonde seulement sur des passions tristes : \u00ab la plupart, au lieu d'une _\u00c9thique_ , ont \u00e9crit une satire43 \u00bb. Plus profond\u00e9ment Spinoza d\u00e9nonce les puissances oppressives qui ne peuvent r\u00e9gner qu'en inspirant \u00e0 l'homme des passions tristes dont elles profitent (\u00ab ceux qui ne savent que briser les \u00e2mes des hommes... \u00bb)44. Sans doute certaines passions tristes ont-elles une utilit\u00e9 sociale : ainsi la crainte, l'espoir, l'humilit\u00e9, m\u00eame le repentir. Mais c'est dans la mesure o\u00f9 nous ne vivons pas sous la conduite de la raison45. Reste que toute passion est mauvaise par elle-m\u00eame, en tant qu'elle enveloppe de la tristesse : m\u00eame _l'espoir_ m\u00eame _la s\u00e9curit\u00e9_46. Une cit\u00e9 est d'autant meilleure qu'elle s'appuie davantage sur des affections joyeuses ; l'amour de la libert\u00e9 doit l'emporter sur l'espoir, la crainte et la s\u00e9curit\u00e947. Le seul commandement de la raison, la seule exigence de la _pietas_ et de la _religio_ , c'est d'encha\u00eener un maximum de joies passives avec un maximum de joies actives. Car seule la joie est une affection passive qui augmente notre puissance d'agir ; et seule la joie peut \u00eatre une affection active. L'esclave se reconna\u00eet \u00e0 ses passions tristes, et l'homme libre \u00e0 ses joies, passives et actives. Le sens de la joie appara\u00eet comme le sens proprement \u00e9thique ; il est \u00e0 la pratique ce que l'affirmation elle-m\u00eame est \u00e0 la sp\u00e9culation. Le naturalisme de Spinoza se d\u00e9finit par l'affirmation sp\u00e9culative dans la th\u00e9orie de la substance, par la joie pratique dans la conception des modes. Philosophie de l'affirmation pure, l' _\u00c9thique_ est aussi philosophie de la joie qui correspond \u00e0 cette affirmation.\n\n* * *\n\n1.  _\u00c9_ , III, 2, sc. Ce texte fondamental ne doit pas \u00eatre s\u00e9par\u00e9 de II, 13, sc., qui le pr\u00e9pare, et de V, pr\u00e9face, qui en d\u00e9veloppe les cons\u00e9quences.\n\n2.  DESCARTES, _Trait\u00e9 des passions_ , I, 1 et 2.\n\n3.  Leibniz explique souvent que sa th\u00e9orie de l' _action id\u00e9ale_ respecte \u00ab les sentiments \u00e9tablis \u00bb, et laisse enti\u00e8rement subsister la r\u00e9partition de l'action ou de la passion dans l'\u00e2me et dans le corps suivant la r\u00e8gle du rapport inverse. Car, de deux substances qui \u00ab symbolisent \u00bb, telles que l'\u00e2me et le corps, on doit attribuer l'action \u00e0 celle dont _l'expression_ est plus distincte, la passion \u00e0 l'autre. C'est un th\u00e8me constant des _Lettres \u00e0 Arnauld._\n\n4.  _\u00c9_ , II, 13 sc.\n\n5.  _\u00c9_ , III, 2, sc.\n\n6.  _\u00c9_ , II, 13, sc.\n\n7.  _TTP_ , ch. 16 (II, p. 258).\n\n8.  Sur l'identit\u00e9 de la \u00ab loi d'institution naturelle \u00bb avec le droit de nature, cf. _TTP_ , ch. 16, et _TP_ , ch. 2, 4.\n\n9.  _TTP_ , ch. 16 (II, pp. 258-259) ; _TP_ , ch. 2, 5.\n\n10.  Ces quatre th\u00e8ses, ainsi que les quatre th\u00e8ses contraires que nous indiquons dans le paragraphe suivant, sont bien marqu\u00e9es par L\u00e9o Strauss, dans son livre _Droit naturel et Histoire_ (tr. fr., Plon, 1953). Strauss confronte la th\u00e9orie de Hobbes, dont il souligne la nouveaut\u00e9, aux conceptions de l'antiquit\u00e9.\n\n11.  _TP_ , ch. 5, 2 : \u00ab Les hommes ne naissent pas citoyens, mais le deviennent. \u00bb\n\n12.  _TTP_ , ch. 16 (II, p. 266).\n\n13.  _TP_ , ch. 2, 8 ( _\u00c9_ , IV, 37, sc. 2 : \u00ab Par le droit supr\u00eame de la nature, chacun juge de ce qui est bon, de ce qui est mauvais... \u00bb.)\n\n14.  _TP_ , ch. 2, 15 : \u00ab Aussi longtemps que le droit naturel humain est d\u00e9termin\u00e9 par la puissance de chacun, et est la chose de chacun, ce droit est en r\u00e9alit\u00e9 inexistant, plus th\u00e9orique que r\u00e9el, puisqu'on n'a nulle assurance d'en profiter. \u00bb\n\n15.  Cf. _\u00c9_ , IV, 24, prop. : _proprium utile._\n\n16.  Cf. _\u00c9_ , IV, 35, prop., dem., cor. 1 et 2, sc.\n\n17.  _\u00c9_ , IV, 66, sc. (l'homme libre et l'esclave) ; IV, 73, sc. (l'homme fort) ; V, 42, sc. (le sage et l'ignorant).\n\n18.  Cf. _\u00c9_ , IV, 67-73.\n\n19.  _\u00c9_ , IV, 68.\n\n20.  _TP_ , ch. 2, 6.\n\n21.  En _\u00c9_ , IV, 68, sc., Spinoza fait remonter \u00e0 Mo\u00efse la tradition adamique : le mythe d'un Adam raisonnable et libre s'explique dans la perspective d'une \u00ab hypoth\u00e8se \u00bb abstraite, o\u00f9 l'on consid\u00e8re Dieu \u00ab non pas en tant qu'il est infini, mais en tant seulement qu'il est la cause pour laquelle l'homme existe. \u00bb\n\n22.  _\u00c9_ , IV, 18, sc.\n\n23.  L'id\u00e9e d'un devenir ou d'une formation de la raison avait d\u00e9j\u00e0 \u00e9t\u00e9 d\u00e9velopp\u00e9e par Hobbes (Cf. les commentaire de R. POLIN, _Politique et philosophie chez Thomas Hobbes_ , P.U.F., 1953, pp. 26-40). Hobbes et Spinoza con\u00e7oivent tous deux l'acte de la raison comme une sorte d'addition, comme la formation d'un tout. Mais chez Hobbes, il s'agit d'un calcul ; chez Spinoza, d'une composition de rapports qui, au moins en droit, est objet d'intuition.\n\n24.  _\u00c9_ , IV, 18, sc.\n\n25.  _\u00c9_ , IV, 35.\n\n26.  _\u00c9_ , IV, 32-34.\n\n27.  _TTP_ , ch. 16 (II, p. 259).\n\n28.  _TP_ , ch. 6, 1.\n\n29.  Cf. _TTP_ , ch. 16 (Et _\u00c9_ , IV, 37, sc. 2). Quel que soit le r\u00e9gime d'une soci\u00e9t\u00e9, la d\u00e9l\u00e9gation contractuelle suivant Spinoza se fait toujours, non pas au b\u00e9n\u00e9fice d'un tiers (comme chez Hobbes), mais au b\u00e9n\u00e9fice du Tout, c'est-\u00e0-dire de la totalit\u00e9 des contractants. Mme Franc\u00e8s a raison de dire en ce sens que Spinoza annonce Rousseau (bien qu'elle minimise l'originalit\u00e9 de Rousseau dans la mani\u00e8re de concevoir la formation de ce tout) : cf. \u00ab Les R\u00e9miniscences spinozistes dans le Contrat social de Rousseau \u00bb, _Revue philosophique_ , janvier 1951, pp. 66-67. Mais s'il est vrai que le contrat transf\u00e8re la puissance \u00e0 l'ensemble de la cit\u00e9, les conditions de cette op\u00e9ration, sa diff\u00e9rence avec une op\u00e9ration de la raison pure, exigent la pr\u00e9sence d'un second moment par lequel l'ensemble de la cit\u00e9, \u00e0 son tour, transf\u00e8re sa puissance \u00e0 un roi, \u00e0 une assembl\u00e9e aristocratique ou d\u00e9mocratique. Est-ce un _second_ contrat, r\u00e9ellement distinct du premier, comme le sugg\u00e8re _TTP_ , ch. 17 ? (Spinoza en effet dit que les H\u00e9breux form\u00e8rent un tout politique en transf\u00e9rant leur puissance \u00e0 Dieu, _puis_ transf\u00e9r\u00e8rent la puissance du tout \u00e0 Mo\u00efse, pris comme interpr\u00e8te de Dieu, cf. II, p. 274). Ou bien le premier contrat n'existe-t-il qu'abstraitement, comme fondement du second ? (Dans le _Trait\u00e9 politique_ , l'\u00c9tat semble bien ne pas exister sous sa forme _absolue, absolutum imperium_ , mais \u00eatre toujours repr\u00e9sent\u00e9 dans une forme monarchique, aristocratique ou d\u00e9mocratique, la d\u00e9mocratie \u00e9tant le r\u00e9gime qui se rapproche le plus de l'\u00c9tat absolu.)\n\n30.  _\u00c9_ , IV, 37, sc. 2 ; _TP_ , ch. 2, 18, 19 et 23.\n\n31.  _TP_ , ch. 3, 2.\n\n32.  _TTP_ , ch. 16 (II, pp. 262-263). Et _TP_ , ch. 2, 21 ; ch. 3, 8 ; ch. 4, 4 ; ch. 5, 1.\n\n33.  Le mobile de formation de la cit\u00e9 est toujours la crainte et l'espoir, crainte d'un plus grand mal, espoir d'un plus grand bien. Mais ce sont des passions essentiellement tristes (cf. _\u00c9_ , IV, 47, dem.). La cit\u00e9, une fois \u00e9tablie, doit susciter _l'amour de la libert\u00e9_ plut\u00f4t que la crainte des ch\u00e2timents ou m\u00eame l'espoir des r\u00e9compenses. \u00ab C'est aux esclaves, non pas aux hommes libres, qu'on donne des prix de vertu \u00bb ( _TP_ , ch. 10, 8).\n\n34.  _TP_ , ch. 3, 3 et 8.\n\n35.  En deux textes importants ( _Lettre 50, \u00e0 Jelles_ , III, p. 172, et _TP_ , ch. 3, 3), Spinoza dit que sa th\u00e9orie politique a pour caract\u00e9ristique de maintenir le droit naturel dans l'\u00e9tat civil lui-m\u00eame. Cette d\u00e9claration s'interpr\u00e8te diff\u00e9remment dans les deux cas : tant\u00f4t c'est le souverain qui se d\u00e9finit par son droit naturel, ce droit \u00e9tant \u00e9gal \u00e0 la somme des droits auxquels les sujets renoncent ; tant\u00f4t ce sont les sujets qui conservent leur droit naturel \u00e0 pers\u00e9v\u00e9rer dans l'\u00eatre, bien que ce droit soit maintenant d\u00e9termin\u00e9 par des affections communes.\n\n36.  _TTP_ , ch. 20 (III, pp. 306-307). Et _TP_ , ch. 3, 10 : \u00ab L'\u00e2me, dans la mesure o\u00f9 elle use de la raison, ne rel\u00e8ve pas du souverain, mais d'elle-m\u00eame. \u00bb\n\n37.  _\u00c9_ , IV, 35, sc. ; IV, 73, prop. et dem.\n\n38.  Sur la _pietas_ et la _religio_ , toujours relatives \u00e0 notre puissance d'agir, cf. _\u00c9_ , IV, 37, sc. 1, et V, 41. Sur les \u00ab commandements \u00bb de la raison ( _dictamina_ ), cf. _\u00c9_ , IV, 18, sc.\n\n39.  Par exemple, la raison d\u00e9nonce la haine et tout ce qui s'y rapporte : _\u00c9_ , IV, 45 et 46. Mais c'est uniquement parce que la haine ne se s\u00e9pare pas de la _tristesse_ qu'elle enveloppe. L'espoir, la piti\u00e9, l'humilit\u00e9, le repentir ne seront pas moins d\u00e9nonc\u00e9s, puisqu'ils enveloppent aussi de la tristesse : _\u00c9_ , IV, 47, 50, 53, 54.\n\n40.  L'analyse que Spinoza fait de la superstition, dans la pr\u00e9face du _TTP_ , est tr\u00e8s proche de celle de Lucr\u00e8ce : la superstition se d\u00e9finit essentiellement par un m\u00e9lange d'avidit\u00e9 et d'angoisse. Et la cause de la superstition n'est pas une id\u00e9e de Dieu confuse, mais la crainte, les passions tristes et leur encha\u00eenement ( _TTP_ , pr\u00e9face, II, p. 85).\n\n41.  _TTP_ , pr\u00e9face (II, p. 87).\n\n42.  Cf. _\u00c9_ , IV, 45, sc. 2 ; IV, 50, sc. ; IV, 63, sc. ; V, 10, sc. ; IV, 67.\n\n43.  _TP_ , ch. 1, 1.\n\n44.  _\u00c9_ , IV, appendice, 13.\n\n45.  _\u00c9_ , IV, 54, sc.\n\n46.  _\u00c9_ , IV, 47, sc.\n\n47.  _TP_ , ch. 10, 8.\n\n## CHAPITRE XVII\n\n## LES NOTIONS COMMUNES\n\nLe spinozisme n'est nullement une philosophie qui s'installe en Dieu, ni qui trouve dans l'id\u00e9e de Dieu son point de d\u00e9part naturel. Au contraire : les conditions sous lesquelles nous avons des id\u00e9es semblent nous condamner \u00e0 n'avoir que des id\u00e9es inad\u00e9quates ; les conditions sous lesquelles nous sommes affect\u00e9s semblent nous condamner \u00e0 n'\u00e9prouver que des affections passives. Les affections qui remplissent naturellement notre pouvoir d'\u00eatre affect\u00e9 sont des passions qui le r\u00e9duisent au minimum, qui nous s\u00e9parent de notre essence ou de notre puissance d'agir.\n\nDans cette \u00e9valuation pessimiste de l'existence, un premier espoir appara\u00eet pourtant : la distinction radicale de l'action et de la passion ne doit pas nous faire n\u00e9gliger une distinction pr\u00e9alable entre deux sortes de passions. Sans doute toute passion nous maintient-elle s\u00e9par\u00e9s de notre puissance d'agir ; mais plus ou moins. Tant que nous sommes affect\u00e9s de passions, nous n'avons pas la possession formelle de notre puissance d'agir. Mais les passions joyeuses nous rapprochent de cette puissance, c'est-\u00e0-dire l'augmentent ou l'aident ; les passions tristes nous en \u00e9loignent, c'est-\u00e0-dire la diminuent ou l'emp\u00eachent. La premi\u00e8re question de l' _\u00c9thique_ est donc : Que faire pour \u00eatre affect\u00e9 d'un maximum de passions joyeuses ? La Nature ne nous favorise pas \u00e0 cet \u00e9gard. Mais nous devons compter sur l'effort de la raison, effort empirique et tr\u00e8s lent qui trouve dans la cit\u00e9 les conditions qui le rendent possible : la raison, dans le principe de sa gen\u00e8se ou sous son premier aspect, est l'effort d'organiser les rencontres de telle mani\u00e8re que nous soyons affect\u00e9s d'un maximum de passions joyeuses. En effet, les passions joyeuses augmentent notre puissance d'agir ; la raison est puissance de comprendre, puissance d'agir propre \u00e0 l'\u00e2me ; les passions joyeuses conviennent donc avec la raison, nous conduisent \u00e0 comprendre ou nous d\u00e9terminent \u00e0 devenir raisonnables1.\n\nMais il ne suffit pas que notre puissance d'agir augmente. Elle pourrait augmenter ind\u00e9finiment, les passions joyeuses pourraient s'encha\u00eener avec les passions joyeuses ind\u00e9finiment, nous n'aurions pas encore la possession formelle de notre puissance d'agir. Une sommation de passions ne fait pas une action. Il ne suffit donc pas que les passions joyeuses s'accumulent ; il faut que, \u00e0 la faveur de cette accumulation, nous trouvions le moyen de conqu\u00e9rir notre puissance d'agir pour \u00e9prouver enfin des affections actives dont nous serons la cause. La deuxi\u00e8me question de l' _\u00c9thique_ sera donc : Que faire pour produire en soi des affections actives ?\n\n1o) Des affections actives, si elles existent, sont n\u00e9cessairement des affections de joie : il n'y a pas de tristesse active, puisque toute tristesse est diminution de notre puissance d'agir ; seule la joie peut \u00eatre active2. En effet, si notre puissance d'agir augmente au point que nous en ayons la possession formelle, des affections s'ensuivent, qui sont n\u00e9cessairement des joies actives3. 2o) La joie active est \u00ab un autre \u00bb sentiment que la joie passive4. Et pourtant Spinoza sugg\u00e8re que, entre les deux, la distinction n'est que de raison5. C'est que les deux sentiments se distinguent seulement par la cause ; la joie passive est produite par un objet qui convient avec nous, dont la puissance augmente notre puissance d'agir, mais dont nous n'avons pas encore une id\u00e9e ad\u00e9quate. La joie active est produite par nous-m\u00eames, elle d\u00e9coule de notre puissance d'agir elle-m\u00eame, elle suit d'une id\u00e9e ad\u00e9quate en nous. 3o) Dans la mesure o\u00f9 les joies passives augmentent notre puissance d'agir, elles _conviennent avec_ la raison. Mais, la raison \u00e9tant la puissance d'agir de l'\u00e2me, les joies suppos\u00e9es actives _naissent_ de la raison. Quand Spinoza sugg\u00e8re que ce qui convient avec la raison peut aussi en na\u00eetre, il veut dire que toute joie passive peut donner lieu \u00e0 une joie active qui s'en distingue seulement par la cause6.\n\nSupposons deux corps qui conviennent enti\u00e8rement, c'est-\u00e0-dire qui composent tous leurs rapports : ils sont comme les parties d'un tout, le tout exerce une _fonction g\u00e9n\u00e9rale_ par rapport \u00e0 ces parties, ces parties ont une _propri\u00e9t\u00e9 commune_ par rapport au tout. Deux corps qui conviennent enti\u00e8rement ont donc une identit\u00e9 de structure. Parce qu'ils composent tous leurs rapports, ils ont une analogie, similitude ou communaut\u00e9 de composition. Supposons maintenant des corps qui conviennent de moins en moins, ou qui sont contraires : leurs rapports constitutifs ne se composent plus directement, mais pr\u00e9sentent de telles diff\u00e9rences que toute ressemblance entre ces corps para\u00eet exclue. Pourtant, il y a encore similitude ou communaut\u00e9 de composition, mais _d'un point de vue de plus en plus g\u00e9n\u00e9ral_ qui, \u00e0 la limite, met en jeu la Nature enti\u00e8re. Il faut tenir compte en effet du \u00ab tout \u00bb que ces deux corps forment, non pas directement l'un avec l'autre, mais avec tous les interm\u00e9diaires qui nous permettent de passer de l'un \u00e0 l'autre. Comme tous les rapports se composent dans la Nature enti\u00e8re, la Nature pr\u00e9sente du point de vue le plus g\u00e9n\u00e9ral une similitude de composition valable pour tous les corps. On passera d'un corps \u00e0 un autre, si diff\u00e9rent soit-il, par simple variation du rapport entre les parties ultimes de l'un. Car seuls varient les rapports, dans l'ensemble de l'univers o\u00f9 les parties restent identiques.\n\nVoil\u00e0 ce que Spinoza appelle une \u00ab notion commune \u00bb. La notion commune est toujours l'id\u00e9e d'une similitude de composition dans les modes existants. Mais en ce sens, il y a diff\u00e9rents types de notions. Spinoza dit que les notions communes sont plus ou moins utiles, plus ou moins faciles \u00e0 former ; et aussi plus ou moins universelles, c'est-\u00e0-dire qu'elles s'organisent suivant des points de vue plus ou moins g\u00e9n\u00e9raux7. En fait, on distinguera deux grandes esp\u00e8ces de notions communes. Les moins universelles (mais aussi les plus utiles) sont celles qui repr\u00e9sentent une similitude de composition entre corps qui conviennent directement et de leur propre point de vue. Par exemple, une notion commune repr\u00e9sente \u00ab ce qui est commun \u00e0 un corps humain et \u00e0 _certains_ corps ext\u00e9rieurs8 \u00bb. Ces notions nous font donc comprendre les convenances entre modes : elles n'en restent pas \u00e0 une perception externe des convenances observ\u00e9es fortuitement, mais trouvent dans la similitude de la composition une raison interne et n\u00e9cessaire de la convenance des corps.\n\n\u00c0 l'autre p\u00f4le, les notions communes les plus universelles repr\u00e9sentent une similitude ou communaut\u00e9 de composition, mais entre corps qui conviennent d'un point de vue tr\u00e8s g\u00e9n\u00e9ral et non pas de _leur propre_ point de vue. Elles repr\u00e9sentent donc \u00ab ce qui est commun \u00e0 toutes choses \u00bb, par exemple l'\u00e9tendue, le mouvement et le repos, c'est-\u00e0-dire l'universelle similitude dans les rapports qui se composent \u00e0 l'infini du point de vue de la nature enti\u00e8re9. Ces notions ont encore leur utilit\u00e9 ; car elles nous font comprendre les disconvenances elles-m\u00eames, et nous en donnent une raison interne et n\u00e9cessaire. Elles nous permettent en effet de d\u00e9terminer le point de vue \u00e0 partir duquel cesse la convenance la plus g\u00e9n\u00e9rale entre deux corps ; elles montrent comment et pourquoi la contrari\u00e9t\u00e9 appara\u00eet quand nous nous pla\u00e7ons du point de vue \u00ab moins universel \u00bb de ces deux corps eux-m\u00eames. Nous pouvons, par une exp\u00e9rience de pens\u00e9e, faire varier un rapport jusqu'au point o\u00f9 le corps correspondant rev\u00eat en quelque sorte une nature \u00ab contraire \u00bb \u00e0 la sienne ; nous pouvons par l\u00e0 comprendre la nature des disconvenances entre corps dont les rapports sont tels ou tels. C'est pourquoi, quand il assigne le r\u00f4le de toutes les notions communes prises ensemble, Spinoza dit que l'esprit est d\u00e9termin\u00e9 de l'int\u00e9rieur \u00e0 comprendre les convenances entre les choses, et aussi les diff\u00e9rences et les oppositions10.\n\nSpinoza distingue avec soin les Notions communes d'une part, d'autre part les Termes transcendantaux (\u00catre, chose, quelque chose) ou les Notions universelles (genres et esp\u00e8ces, Homme, Cheval, Chien)11. Pourtant les notions communes elles-m\u00eames sont universelles, \u00ab plus ou moins \u00bb universelles suivant leur degr\u00e9 de g\u00e9n\u00e9ralit\u00e9 ; il faut donc penser que Spinoza n'attaque pas l'universel, mais seulement une certaine conception de l'universel abstrait. De m\u00eame, Spinoza ne critique pas les notions de genre et d'esp\u00e8ce en g\u00e9n\u00e9ral ; pour son compte, il parle du Cheval ou du Chien comme de types naturels, de l'Homme lui-m\u00eame comme d'un type ou d'un mod\u00e8le normatif12. L\u00e0 encore, il faut penser que Spinoza s'attaque seulement \u00e0 une certaine d\u00e9termination abstraite des genres et des esp\u00e8ces. En effet, une id\u00e9e abstraite a deux aspects qui t\u00e9moignent de son insuffisance. D'abord, elle ne retient entre les choses que des diff\u00e9rences sensibles et grossi\u00e8res : nous choisissons un caract\u00e8re sensible, facile \u00e0 imaginer ; nous distinguons les objets qui le poss\u00e8dent et ceux qui ne le poss\u00e8dent pas ; nous identifions tous ceux qui le poss\u00e8dent ; quant aux petites diff\u00e9rences, nous les n\u00e9gligeons, pr\u00e9cis\u00e9ment parce que les objets se confondent d\u00e8s que leur nombre d\u00e9passe la capacit\u00e9 de notre imagination. D'autre part, le caract\u00e8re diff\u00e9rentiel sensible est par nature extr\u00eamement variable : il est fortuit, d\u00e9pendant de la mani\u00e8re dont les objets affectent chacun de nous au hasard des rencontres. \u00ab Ceux qui ont plus souvent consid\u00e9r\u00e9 avec admiration la stature des hommes entendront sous le nom d'homme un animal de stature droite, tandis que ceux qui ont accoutum\u00e9 de consid\u00e9rer autre chose se formeront des hommes une autre image commune : par exemple l'homme est un animal capable de rire, un animal \u00e0 deux pieds sans plumes, un animal raisonnable13. \u00bb Et le caract\u00e8re retenu ne varie pas seulement avec chaque individu, mais aussi suivant les objets qui affectent un m\u00eame individu : certains objets seront d\u00e9finis par leur forme sensible, d'autres par leur usage ou leur fonction suppos\u00e9e, par leur mani\u00e8re d'\u00eatre, etc. De toutes fa\u00e7ons, l'id\u00e9e abstraite est profond\u00e9ment inad\u00e9quate : c'est une image qui ne s'explique pas par notre puissance de penser, mais au contraire enveloppe notre impuissance ; et qui n'exprime pas la nature des choses, mais indique plut\u00f4t l'\u00e9tat variable de notre constitution.\n\nIl est clair en tout ceci que Spinoza n'attaque pas seulement les proc\u00e9d\u00e9s du sens commun, mais aussi la tradition aristot\u00e9licienne. C'est dans la biologie aristot\u00e9licienne qu'appara\u00eet l'effort pour d\u00e9finir les genres et les esp\u00e8ces par des diff\u00e9rences ; et encore ces diff\u00e9rences sensibles sont de nature tr\u00e8s variable suivant les animaux consid\u00e9r\u00e9s. Contre cette tradition, Spinoza sugg\u00e8re un grand principe : consid\u00e9rer les structures, et non plus les formes sensibles ou les fonctions14. Mais que signifie \u00ab structure \u00bb ? C'est un syst\u00e8me de rapports entre les parties d'un corps (ces parties n'\u00e9tant pas des organes, mais les \u00e9l\u00e9ments anatomiques de ces organes). On cherchera comment les rapports varient dans tel ou tel autre corps ; on aura le moyen de d\u00e9terminer directement les ressemblances entre deux corps, si \u00e9loign\u00e9s soient-ils. La forme et la fonction d'un organe, dans un animal d\u00e9termin\u00e9, d\u00e9pendent uniquement des rapports entre parties organiques, c'est-\u00e0-dire entre \u00e9l\u00e9ments anatomiques constants. \u00c0 la limite, la Nature tout enti\u00e8re est un m\u00eame Animal o\u00f9, seuls, varient les rapports entre les parties. \u00c0 l'examen des diff\u00e9rences sensibles, s'est substitu\u00e9 un examen des similitudes intelligibles, apte \u00e0 nous faire comprendre \u00ab de l'int\u00e9rieur \u00bb les ressemblances, et aussi les diff\u00e9rences entre les corps. Les notions communes chez Spinoza sont des id\u00e9es biologiques, plus encore que des id\u00e9es physiques ou math\u00e9matiques. Elles jouent v\u00e9ritablement le r\u00f4le d'Id\u00e9es dans une philosophie de la Nature dont toute finalit\u00e9 se trouve exclue. (Sans doute les indications de Spinoza sont-elles rares sur cet aspect des notions communes. Mais en v\u00e9rit\u00e9, elles sont rares sur tous les aspects des notions communes ; nous verrons pourquoi. Les indications de Spinoza suffisent toutefois \u00e0 faire de lui un pr\u00e9curseur de Geoffroy Saint-Hilaire, sur la voie du grand principe d'unit\u00e9 de composition15.)\n\nLes notions communes sont des id\u00e9es _g\u00e9n\u00e9rales_ , non pas des id\u00e9es _abstraites._ Or, en tant que telles, elles sont n\u00e9cessairement \u00ab ad\u00e9quates \u00bb. Soit le cas des notions les moins universelles : ce qui est commun \u00e0 mon corps et \u00e0 certains corps ext\u00e9rieurs est \u00ab \u00e9galement \u00bb dans chacun de ces corps ; l'id\u00e9e en est donc donn\u00e9e en Dieu, non seulement en tant qu'il a l'id\u00e9e des corps ext\u00e9rieurs, mais aussi en tant qu'il a simplement l'id\u00e9e de mon corps ; j'ai donc moi-m\u00eame l'id\u00e9e de ce quelque chose de commun, et je l'ai telle qu'elle est en Dieu16. Quant aux notions les plus universelles : ce qui est commun \u00e0 toutes choses est \u00ab \u00e9galement \u00bb dans la partie et dans le tout, l'id\u00e9e en est donc donn\u00e9e en Dieu, etc17. Ces d\u00e9monstrations fondent les deux aspects sous lesquels les notions communes en g\u00e9n\u00e9ral sont n\u00e9cessairement ad\u00e9quates ; en d'autres termes, _les notions communes sont des id\u00e9es qui s'expliquent formellement par notre puissance de penser et qui, mat\u00e9riellement, expriment l'id\u00e9e de Dieu comme leur cause efficiente._ Elles s'expliquent par notre puissance de penser parce que, \u00e9tant en nous comme elles sont en Dieu, elles tombent sous notre propre puissance comme elles tombent sous la puissance absolue de Dieu. Elles expriment l'id\u00e9e de Dieu comme cause parce que, Dieu les poss\u00e9dant comme nous les poss\u00e9dons, elles \u00ab enveloppent \u00bb n\u00e9cessairement l'essence de Dieu. En effet, lorsque Spinoza dit que toute id\u00e9e de chose particuli\u00e8re enveloppe n\u00e9cessairement l'essence \u00e9ternelle et infinie de Dieu, il s'agit des choses particuli\u00e8res telles qu'elles sont en Dieu, donc des id\u00e9es de choses telles que Dieu les poss\u00e8de18. Parmi les id\u00e9es que nous _avons_ , les seules qui puissent exprimer l'essence de Dieu, ou envelopper la connaissance de cette essence, sont donc des id\u00e9es qui sont en nous comme elles sont en Dieu : bref, les notions communes19.\n\nD'o\u00f9 plusieurs cons\u00e9quences importantes : 1o) Nous demandions comment nous pourrions arriver \u00e0 des id\u00e9es ad\u00e9quates. Tout dans l'existence nous condamnait \u00e0 n'avoir que des id\u00e9es inad\u00e9quates : nous n'avions ni l'id\u00e9e de nous-m\u00eames, ni l'id\u00e9e des corps ext\u00e9rieurs, mais seulement des id\u00e9es d'affections, indiquant l'effet d'un corps ext\u00e9rieur sur nous. Mais pr\u00e9cis\u00e9ment, \u00e0 partir de cet effet, nous pouvons former l'id\u00e9e de ce qui est commun \u00e0 un corps ext\u00e9rieur et au n\u00f4tre. Compte tenu des conditions de notre existence c'est pour nous la seule voie capable de nous mener \u00e0 une id\u00e9e ad\u00e9quate. _La premi\u00e8re id\u00e9e ad\u00e9quate que nous ayons_ , c'est la notion commune, l'id\u00e9e de ce \u00ab quelque chose de commun \u00bb. 2o) Cette id\u00e9e s'explique par notre puissance de comprendre ou de penser. Or la puissance de comprendre, c'est la puissance d'agir de l'\u00e2me. Nous sommes donc actifs en tant que nous formons des notions communes. La formation de la notion commune marque le moment o\u00f9 nous entrons en _possession formelle_ de notre puissance d'agir. Par l\u00e0 m\u00eame, elle constitue le second moment de la raison. La raison, dans sa gen\u00e8se, est l'effort d'organiser les rencontres en fonction des convenances et des disconvenances per\u00e7ues. La raison dans son activit\u00e9 m\u00eame est l'effort de concevoir les notions communes, donc de comprendre intellectuellement les convenances et les disconvenances elles-m\u00eames. Quand nous formons une notion commune, notre \u00e2me est dite \u00ab se servir de la raison \u00bb : nous parvenons \u00e0 la possession de notre puissance d'agir ou de comprendre, nous sommes devenus des \u00eatres raisonnables. 3o) Une notion commune est notre premi\u00e8re id\u00e9e ad\u00e9quate. Mais quelle qu'elle soit, elle nous m\u00e8ne imm\u00e9diatement \u00e0 une autre id\u00e9e ad\u00e9quate. L'id\u00e9e ad\u00e9quate est expressive et, ce qu'elle exprime, c'est l'essence de Dieu. Une notion commune quelconque nous donne imm\u00e9diatement la connaissance de l'essence \u00e9ternelle et infinie de Dieu. Nous n'avons pas une id\u00e9e ad\u00e9quate, c'est-\u00e0-dire expressive, sans que cette id\u00e9e ne nous donne la connaissance de ce qu'elle exprime, donc la connaissance ad\u00e9quate de l'essence m\u00eame de Dieu.\n\nToutefois, la notion commune risque d'intervenir comme un miracle tant que nous n'expliquons pas comment nous arrivons \u00e0 la former. _Comment vient-elle rompre l'encha\u00eenement des id\u00e9es inad\u00e9quates auxquelles nous semblions condamn\u00e9s ? \u00ab_ Commun \u00bb, sans doute, ne signifie pas seulement quelque chose de commun \u00e0 deux ou plusieurs corps, mais aussi commun aux esprits capables d'en former l'id\u00e9e. Mais, d'abord, Spinoza rappelle que les notions communes sont plus ou moins communes \u00e0 tous les esprits20. Et, m\u00eame si on les assimile \u00e0 des id\u00e9es inn\u00e9es, l'inn\u00e9it\u00e9 n'a jamais dispens\u00e9 d'un effort de formation, d'une _causa fiendi_ n\u00e9cessaire pour nous faire retrouver ce qui n'est donn\u00e9 qu'en droit. Que les notions communes soient en nous comme elles sont en Dieu signifie seulement que, si nous les formons, nous les _avons_ comme Dieu les _a_. Mais pr\u00e9cis\u00e9ment, comment les formons-nous, dans quelles circonstances favorables ? Comment arrivons-nous \u00e0 notre puissance d'agir ?\n\nTant que nous en restons \u00e0 un point de vue sp\u00e9culatif, ce probl\u00e8me reste insoluble. Deux erreurs d'interpr\u00e9tation nous semblent dangereuses dans la th\u00e9orie des notions communes : n\u00e9gliger leur sens biologique au profit de leur sens math\u00e9matique ; mais surtout n\u00e9gliger leur fonction pratique au profit de leur contenu sp\u00e9culatif. Or cette derni\u00e8re erreur trouve peut-\u00eatre son occasion dans la mani\u00e8re dont Spinoza lui-m\u00eame introduit le syst\u00e8me des notions communes. Le livre II de l' _\u00c9thique_ consid\u00e8re en effet ces notions du point de vue de la pure sp\u00e9culation ; il les expose donc dans un ordre logique, qui va des plus universelles aux moins universelles21. Mais alors, Spinoza montre seulement que, _si_ nous formons des notions communes, celles-ci sont n\u00e9cessairement des id\u00e9es ad\u00e9quates. La cause et l'ordre de leur formation nous \u00e9chappent encore ; de m\u00eame, la nature de leur fonction pratique, qui se trouve seulement sugg\u00e9r\u00e9e dans le livre II22.\n\nIl est vrai que tous les corps ont quelque chose de commun, ne serait-ce que l'\u00e9tendue, le mouvement et le repos. Les corps qui ne conviennent pas et qui sont contraires n'en ont pas moins quelque chose de commun, c'est-\u00e0-dire une similitude de composition tr\u00e8s g\u00e9n\u00e9rale qui met en jeu la Nature enti\u00e8re sous l'attribut de l'\u00e9tendue23. C'est m\u00eame pourquoi l'exposition des notions communes, dans l'ordre logique, se fait \u00e0 partir des plus universelles : donc, \u00e0 partir des notions qui s'appliquent \u00e0 des corps tr\u00e8s \u00e9loign\u00e9s les uns des autres et contraires les uns aux autres. Mais, s'il est vrai que deux corps contraires ont quelque chose de commun, jamais en revanche un corps ne peut \u00eatre contraire \u00e0 l'autre, mauvais pour l'autre, par ce qu'il a de commun avec lui : \u00ab Nulle chose ne peut \u00eatre mauvaise par ce qu'elle a de commun avec notre nature, mais dans la mesure o\u00f9 elle est mauvaise pour nous, elle nous est contraire24. \u00bb Quand nous \u00e9prouvons une affection mauvaise, une affection passive triste produite en nous par un corps qui ne nous convient pas, _rien ne nous induit \u00e0 former l'id\u00e9e_ de ce qui est commun \u00e0 ce corps et au n\u00f4tre. Au contraire, quand nous \u00e9prouvons une affection joyeuse : une chose nous \u00e9tant bonne dans la mesure o\u00f9 elle convient avec notre nature, l'affection joyeuse elle-m\u00eame nous induit \u00e0 former la notion commune correspondante. Les premi\u00e8res notions communes que nous formons sont donc les moins universelles, c'est-\u00e0-dire celles qui s'appliquent \u00e0 notre corps et \u00e0 un autre corps, qui convient directement avec le n\u00f4tre et qui l'affecte de joie. Si nous consid\u00e9rons l'ordre de formation des notions communes, nous devons partir des notions les moins universelles ; car les plus universelles, s'appliquant \u00e0 des corps qui sont contraires au n\u00f4tre, ne trouvent aucun _principe inducteur_ dans les affections que nous \u00e9prouvons.\n\nEn quel sens prenons-nous \u00ab induire \u00bb ? Il s'agit d'une sorte de _cause occasionnelle._ L'id\u00e9e ad\u00e9quate s'explique formellement par notre puissance de comprendre ou d'agir. Or tout ce qui s'explique par notre puissance d'agir d\u00e9pend de notre seule essence, donc est \u00ab inn\u00e9 \u00bb. Mais d\u00e9j\u00e0 chez Descartes l'inn\u00e9 renvoyait \u00e0 une esp\u00e8ce d'occasionalisme. L'inn\u00e9 est actif ; mais pr\u00e9cis\u00e9ment, il ne peut devenir actuel que sil trouve une occasion favorable dans les affections qui viennent du dehors, affections passives. Il semble donc que le sch\u00e9ma de Spinoza soit le suivant :\n\nQuand nous rencontrons un corps qui convient avec le n\u00f4tre, quand nous \u00e9prouvons une affection passive joyeuse, nous sommes induits \u00e0 former l'id\u00e9e de ce qui est commun \u00e0 ce corps et au n\u00f4tre. Voil\u00e0 pourquoi, dans le livre V de l' _\u00c9thique_ , Spinoza est amen\u00e9 \u00e0 reconna\u00eetre le privil\u00e8ge des passions joyeuses dans la formation des notions communes : \u00ab _Aussi longtemps que nous ne sommes pas tourment\u00e9s par des sentiments qui sont contraires \u00e0 notre nature_ [sentiments de tristesse, provoqu\u00e9s par des objets contraires qui ne nous conviennent pas], aussi longtemps la puissance de l'esprit, par laquelle il s'efforce de comprendre les choses, n'est pas emp\u00each\u00e9e, et par cons\u00e9quent aussi longtemps il a le pouvoir de former des id\u00e9es claires et distinctes25. \u00bb En effet, il suffit que l'emp\u00eachement soit \u00f4t\u00e9 pour que la puissance d'agir passe \u00e0 l'acte, et que nous entrions en possession de ce qui nous est inn\u00e9. On voit pourquoi il ne suffisait pas d'accumuler les passions joyeuses pour devenir actif. L'amour-passion s'encha\u00eene \u00e0 la joie-passion, d'autres sentiments et d\u00e9sirs s'encha\u00eenent \u00e0 l'amour. Tous augmentent notre puissance d'agir ; mais jamais jusqu'au point o\u00f9 nous devenions actifs. Il fallait d'abord que ces sentiments fussent \u00ab assur\u00e9s \u00bb ; il nous fallait d'abord \u00e9viter les passions tristes qui diminuaient notre puissance d'agir ; tel \u00e9tait le premier effort de la raison. Mais ensuite, il fallait sortir d'un simple encha\u00eenement des passions, m\u00eame joyeuses. Car celles-ci ne nous donnent pas encore la possession de notre puissance d'agir ; nous n'avons pas l'id\u00e9e ad\u00e9quate de l'objet qui convient en nature avec nous ; les passions joyeuses elles-m\u00eames naissent d'id\u00e9es inad\u00e9quates, qui indiquent seulement l'effet d'un objet sur nous. Il faut donc que, _\u00e0 la faveur des passions joyeuses_ , nous formions l'id\u00e9e de ce qui est commun entre le corps ext\u00e9rieur et le n\u00f4tre. Car cette id\u00e9e seule, cette notion commune, est ad\u00e9quate. Tel est le deuxi\u00e8me moment de la raison ; alors, et alors seulement, nous comprenons et agissons, nous sommes raisonnables : non pas par l'accumulation des passions joyeuses en tant que passions, mais par un v\u00e9ritable \u00ab saut \u00bb, qui nous met en possession d'une id\u00e9e ad\u00e9quate, \u00e0 la faveur de cette accumulation.\n\nPourquoi devenons-nous actifs dans la mesure o\u00f9 nous formons une notion commune ou avons une id\u00e9e ad\u00e9quate ? L'id\u00e9e ad\u00e9quate s'explique par notre puissance de comprendre, donc par notre puissance d'agir. Elle nous met en possession de cette puissance, mais de quelle mani\u00e8re ? Il faut se rappeler qu'une id\u00e9e ad\u00e9quate, \u00e0 son tour, n'est pas s\u00e9parable d'un encha\u00eenement d'id\u00e9es qui en d\u00e9coulent. L'esprit qui forme une _id\u00e9e_ ad\u00e9quate est _cause_ ad\u00e9quate des id\u00e9es qui d\u00e9coulent de celle-ci : c'est en ce sens qu'il est actif26. Quelles sont donc ces id\u00e9es, qui suivent de la notion commune que nous formons \u00e0 la faveur des passions joyeuses ? Les passions joyeuses sont les id\u00e9es des affections produites par un corps qui convient avec le n\u00f4tre ; notre esprit lui seul forme l'id\u00e9e de ce qui est commun \u00e0 ce corps et au n\u00f4tre ; _en d\u00e9coule une id\u00e9e d'affection, un sentiment qui n'est plus passif, mais actif._ Ce sentiment n'est plus une passion, parce qu'il suit d'une id\u00e9e ad\u00e9quate en nous ; il est lui-m\u00eame id\u00e9e ad\u00e9quate. Il se distingue du sentiment passif dont nous \u00e9tions partis, mais s'en distingue seulement par la cause : il a pour cause, non plus l'id\u00e9e inad\u00e9quate d'un objet qui convient avec nous, mais l'id\u00e9e n\u00e9cessairement ad\u00e9quate de ce qui est commun \u00e0 cet objet et \u00e0 nous-m\u00eames. C'est pourquoi Spinoza peut dire : \u00ab Un sentiment qui est une passion cesse d'\u00eatre une passion sit\u00f4t que nous en formons une id\u00e9e claire et distincte (ad\u00e9quate)27. \u00bb Car nous en formons une id\u00e9e claire et distincte dans la mesure o\u00f9 nous le rattachons \u00e0 la notion commune comme \u00e0 sa cause ; alors, il est actif et d\u00e9pend de notre puissance d'agir. Spinoza ne veut pas dire que toute passion disparaisse : ce qui dispara\u00eet n'est pas la joie passive elle-m\u00eame, mais toutes les passions, tous les d\u00e9sirs qui s'encha\u00eenent avec elle, li\u00e9s \u00e0 l'id\u00e9e de la chose ext\u00e9rieure (amour-passion, etc.)28.\n\nUn sentiment quelconque d\u00e9termine le _conatus_ \u00e0 faire quelque chose en fonction d'une id\u00e9e d'objet ; le _conatus_ , ainsi d\u00e9termin\u00e9, s'appelle un d\u00e9sir. Mais tant que nous sommes d\u00e9termin\u00e9s par un sentiment de joie passive, nos d\u00e9sirs sont encore irrationnels, puisqu'ils naissent d'une id\u00e9e inad\u00e9quate. Or, maintenant, \u00e0 la joie passive s'ajoute une joie active qui s'en distingue seulement par la cause ; de cette joie active naissent des d\u00e9sirs qui appartiennent \u00e0 la raison, parce qu'ils proc\u00e8dent d'une id\u00e9e ad\u00e9quate29. \u00ab Tous les app\u00e9tits ou d\u00e9sirs sont des passions dans la mesure o\u00f9 ils naissent d'id\u00e9es inad\u00e9quates, et ils s'adjoignent \u00e0 la vertu quand ils sont provoqu\u00e9s ou engendr\u00e9s par des id\u00e9es ad\u00e9quates ; car tous les d\u00e9sirs par lesquels nous sommes d\u00e9termin\u00e9s \u00e0 faire quelque chose peuvent na\u00eetre aussi bien d'id\u00e9es ad\u00e9quates que d'id\u00e9es inad\u00e9quates30. \u00bb Donc, des d\u00e9sirs de la raison remplacent les d\u00e9sirs irrationnels, ou plut\u00f4t un encha\u00eenement rationnel se substitue \u00e0 l'encha\u00eenement irrationnel des d\u00e9sirs : \u00ab Nous avons le pouvoir d'ordonner et d'encha\u00eener les affections du corps suivant un ordre conforme \u00e0 l'entendement31. \u00bb\n\nL'ensemble de l'op\u00e9ration d\u00e9crite par Spinoza pr\u00e9sente quatre moments : 1o) Joie passive qui augmente notre puissance d'agir, d'o\u00f9 d\u00e9coulent des d\u00e9sirs ou des passions, en fonction d'une id\u00e9e encore inad\u00e9quate ; 2o) \u00c0 la faveur de ces passions joyeuses, formation d'une notion commune (id\u00e9e ad\u00e9quate) ; 3o) Joie active, qui suit de cette notion commune et qui s'explique par notre puissance d'agir ; 4o) Cette joie active s'ajoute \u00e0 la joie passive, mais _remplace_ les d\u00e9sirs-passions qui naissaient de celle-ci par des d\u00e9sirs qui appartiennent \u00e0 la raison, et qui sont de v\u00e9ritables actions. Ainsi se r\u00e9alise le programme de Spinoza : non pas supprimer toute passion, mais \u00e0 la faveur de la passion joyeuse, faire que les passions n'occupent plus que la plus petite partie de nous-m\u00eames et que notre pouvoir d'\u00eatre affect\u00e9 soit rempli par un maximum d'affections actives32.\n\nAu d\u00e9but du Livre V de l' _\u00c9thique_ , Spinoza montre qu'un sentiment cesse d'\u00eatre une passion sit\u00f4t que nous en formons une id\u00e9e claire et distincte (ad\u00e9quate) ; et que nous en formons une id\u00e9e claire et distincte sit\u00f4t que nous le rattachons \u00e0 une notion commune comme \u00e0 sa cause. Toutefois, Spinoza ne r\u00e9serve pas cette th\u00e8se au sentiment de joie, il l'affirme valable pour _tout_ sentiment : \u00ab Il n'est aucune affection du corps dont nous ne puissions former quelque concept clair et distinct33. \u00bb La d\u00e9monstration de cette proposition est fort concise : \u00ab Les choses qui sont communes \u00e0 toutes ne peuvent \u00eatre con\u00e7ues, sinon de fa\u00e7on ad\u00e9quate ; et par cons\u00e9quent... \u00bb Prenons donc le cas de la tristesse. \u00c9videmment, Spinoza ne veut pas dire que la tristesse, \u00e9tant une passion in\u00e9vitable, est elle-m\u00eame commune \u00e0 tous les hommes ou \u00e0 tous les \u00eatres. Spinoza n'oublie pas que la notion commune est toujours l'id\u00e9e de quelque chose de positif : rien n'est commun par simple impuissance ou par imperfection34. Spinoza veut dire que, m\u00eame dans le cas d'un corps qui ne convient pas avec le n\u00f4tre et nous affecte de tristesse, nous pouvons former l'id\u00e9e de ce qui est commun \u00e0 ce corps et au n\u00f4tre ; simplement, cette notion commune sera tr\u00e8s universelle, impliquant un point de vue beaucoup plus g\u00e9n\u00e9ral que celui des deux corps en pr\u00e9sence. Elle n'en a pas moins une fonction pratique : elle nous fait comprendre pourquoi les deux corps pr\u00e9cis\u00e9ment, ne conviennent pas de _leur_ propre point de vue. \u00ab Nous voyons que la tristesse provenant de la perte de quelque bien s'adoucit, sit\u00f4t que l'homme qui a perdu ce bien consid\u00e8re qu'il n'aurait pu \u00eatre conserv\u00e9 d'aucune mani\u00e8re35. \u00bb (L'homme, en effet, comprend que son propre corps et le corps ext\u00e9rieur n'auraient pu composer leurs rapports de mani\u00e8re durable que dans d'autres circonstances : si les interm\u00e9diaires avaient \u00e9t\u00e9 donn\u00e9s, mettant en jeu la Nature enti\u00e8re, du point de vue de laquelle une telle composition devenait possible.) Mais quand une notion commune tr\u00e8s universelle nous fait comprendre une disconvenance, l\u00e0 encore un sentiment de joie active en d\u00e9coule : _toujours une joie active suit de ce que nous comprenons_. \u00ab Dans la mesure o\u00f9 nous comprenons les causes de la tristesse, elle cesse d'\u00eatre une passion, c'est-\u00e0-dire qu'elle cesse d'\u00eatre tristesse36. \u00bb Il semble donc que, m\u00eame si nous partons d'une passion triste, l'essentiel du sch\u00e9ma pr\u00e9c\u00e9dent se retrouve : tristesse ; formation d'une notion commune ; joie active qui en d\u00e9coule.\n\nDans le livre II de l' _\u00c9thique_ , Spinoza consid\u00e8re les notions communes dans leur contenu sp\u00e9culatif ; il les suppose donn\u00e9es, ou donnables ; il est donc normal qu'il aille des plus universelles aux moins universelles, suivant un ordre logique. Au d\u00e9but du livre V de l' _\u00c9thique_ , Spinoza analyse la fonction pratique des notions communes suppos\u00e9es donn\u00e9es : cette fonction consiste en ceci que la notion est cause d'une id\u00e9e ad\u00e9quate d'affection, c'est-\u00e0-dire d'une joie active. Cette th\u00e8se est valable pour les notions communes les plus universelles comme pour les moins universelles : on peut donc consid\u00e9rer toutes les notions communes prises ensemble, dans l'unit\u00e9 de leur fonction pratique.\n\nMais tout change quand Spinoza demande au cours du livre V : Comment arrivons-nous \u00e0 former une notion commune, nous qui semblons condamn\u00e9s aux id\u00e9es inad\u00e9quates et aux passions ? Alors, nous voyons que les premi\u00e8res notions sont n\u00e9cessairement les moins universelles. Les moins universelles, en effet, sont celles qui s'appliquent \u00e0 mon corps et \u00e0 un autre corps qui convient avec lui (ou \u00e0 quelques autres corps) ; _elles seules trouvent dans les joies passives que j'\u00e9prouve l'occasion d'\u00eatre form\u00e9es._ Les plus universelles, au contraire, s'appliquent \u00e0 tous les corps ; elles s'appliquent donc \u00e0 des corps tr\u00e8s diff\u00e9rents, contraires les uns aux autres. Mais la tristesse, ou la contrari\u00e9t\u00e9, produite en nous par un corps qui ne convient pas avec le n\u00f4tre, n'est jamais _l'occasion de former_ une notion commune. Si bien que le processus de formation des notions se pr\u00e9sente ainsi : Nous cherchons d'abord \u00e0 \u00e9prouver un maximum de passions joyeuses (premier effort de la raison). Nous cherchons donc \u00e0 \u00e9viter les passions tristes, \u00e0 \u00e9chapper \u00e0 leur encha\u00eenement, \u00e0 conjurer les mauvaises rencontres. Puis, en second lieu, nous nous servons des passions joyeuses pour former la notion commune correspondante, d'o\u00f9 d\u00e9coulent des joies actives (deuxi\u00e8me effort de la raison). Une telle notion commune est parmi les moins universelles, puisqu'elle s'applique seulement \u00e0 mon corps et \u00e0 des corps qui conviennent avec lui. Mais elle nous rend encore plus forts pour \u00e9viter les mauvaises rencontres ; et surtout elle nous met en possession de notre puissance d'agir et de comprendre. _Alors, en troisi\u00e8me lieu_ , nous sommes devenus capables de former des notions communes plus universelles, qui s'appliquent \u00e0 tous les cas, m\u00eame aux corps qui nous sont contraires ; nous sommes devenus capables de comprendre m\u00eame nos tristesses et de tirer de cette compr\u00e9hension une joie active. Nous sommes capables de faire face aux mauvaises rencontres que nous ne pouvons pas \u00e9viter, de r\u00e9duire les tristesses qui subsistent n\u00e9cessairement en nous. Mais on n'oubliera pas que, malgr\u00e9 l'identit\u00e9 g\u00e9n\u00e9rale de leur fonction pratique (produire des joies actives), les notions communes sont d'autant plus utiles, d'autant plus efficaces qu'elles proc\u00e8dent de passions joyeuses et sont moins universelles37.\n\nToutes les notions communes ont un m\u00eame contenu sp\u00e9culatif : elles impliquent une certaine g\u00e9n\u00e9ralit\u00e9 sans abstraction. Elles ont une m\u00eame fonction pratique : id\u00e9es n\u00e9cessairement ad\u00e9quates, elles sont telles qu'une joie active en d\u00e9coule. Mais leur r\u00f4le, sp\u00e9culatif et pratique, n'est pas du tout le m\u00eame si nous consid\u00e9rons les conditions de leur formation. Les premi\u00e8res notions communes que nous formons, ce sont les moins universelles, parce qu'elles trouvent dans nos passions joyeuses un principe inducteur efficace. C'est au niveau du \u00ab moins universel \u00bb que nous conqu\u00e9rons notre puissance d'agir : nous accumulons les joies passives, nous y trouvons l'occasion de former des notions communes, d'o\u00f9 d\u00e9coulent des joies actives. En ce sens, l'augmentation de notre puissance d'agir nous donne l'occasion de conqu\u00e9rir cette puissance, ou de devenir actif effectivement. Ayant conquis notre activit\u00e9 sur certains points, nous devenons capables de former des notions communes, m\u00eame dans les cas moins favorables. Il y a tout un apprentissage des notions communes, ou du _devenir-actif_ : on ne doit pas n\u00e9gliger dans le Spinozisme l'importance du probl\u00e8me d'un processus de formation ; il faut partir des notions communes les moins universelles, les premi\u00e8res que nous ayons l'occasion de former.\n\n* * *\n\n1.  _\u00c9_ , IV, 59, dem. : \u00ab Dans la mesure o\u00f9 la joie est bonne, elle convient avec la raison, car elle consiste en ce que la puissance d'agir de l'homme est augment\u00e9e ou aid\u00e9e. \u00bb\n\n2.  _\u00c9_ , III, 59, prop. et dem.\n\n3.  _\u00c9_ , III, 58, prop. et dem. ; IV, 59, dem.\n\n4.  _\u00c9_ , III, 58, prop.\n\n5.  Le sentiment actif et le sentiment passif se distinguent comme l'id\u00e9e ad\u00e9quate et l'id\u00e9e inad\u00e9quate. Mais entre une id\u00e9e inad\u00e9quate et une id\u00e9e ad\u00e9quate d'affection, la distinction est seulement de raison : _\u00c9_ , V, 3, dem.\n\n6.  Cf. _\u00c9_ , IV, 51, dem.\n\n7.  Plus ou moins utiles, plus ou moins faciles \u00e0 d\u00e9couvrir ou \u00e0 former : _\u00c9_ , II, 40, sc. 1. Plus ou moins universelles ( _maxime universales, minime universalia_ ) : _TTP_ , ch. 7, II, p. 176.\n\n8.  Cas des notions communes les moins universelles : _\u00c9_ , II, 39, prop.\n\n9.  Cas des notions communes les plus universelles : _\u00c9_ , II, 37 et 38, prop.\n\n10.  _\u00c9_ , II, 29, sc. : \u00ab Toutes les fois que l'esprit est d\u00e9termin\u00e9 de l'int\u00e9rieur, c'est-\u00e0-dire parce qu'il consid\u00e8re plusieurs choses en m\u00eame temps, \u00e0 comprendre leurs convenances, leurs diff\u00e9rences et leurs oppositions, toutes les fois en effet qu'il est dispos\u00e9 de l'int\u00e9rieur de telle ou telle mani\u00e8re, alors il consid\u00e8re les choses clairement et distinctement, comme je le montrerai plus bas. \u00bb\n\n11.  _\u00c9_ , II, 40, sc. 1.\n\n12.  Cf. _\u00c9_ , IV, pr\u00e9face.\n\n13.  _\u00c9_ , II, 40, sc. 1.\n\n14.  _\u00c9_ , 2, sc. : \u00ab Car personne jusqu'ici n'a connu la structure ( _fabrica_ ) du corps si exactement qu'il ait pu en expliquer toutes les fonctions. \u00bb\n\n15.  _\u00c9tienne Geoffroy Saint-Hilaire_ d\u00e9finit sa \u00ab philosophie de la Nature \u00bb par le principe d'unit\u00e9 de composition. Il oppose sa m\u00e9tbhode \u00e0 la m\u00e9thode classique issue d'Aristote, qui consid\u00e8re les formes et les fonctions. Au-del\u00e0 de celles-ci, il se propose de d\u00e9terminer les rapports variables entre \u00e9l\u00e9ments anatomiques constants : les animaux diff\u00e9rents correspondent aux variations de rapport, de situation respective et de d\u00e9pendance de ces \u00e9l\u00e9ments, si bien qu'ils se r\u00e9duisent tous aux modifications d'un seul et m\u00eame Animal en soi. Aux ressemblances de formes et aux analogies de fonctions, qui restent toujours ext\u00e9rieures, Geoffroy substitue ainsi le point de vue intrins\u00e8que d'une unit\u00e9 de composition ou d'une similitude de rapports. Il aime \u00e0 se r\u00e9clamer de Leibniz, et d'un principe d'unit\u00e9 dans le divers. Il nous semble pourtant encore plus spinoziste ; car sa philosophie de la Nature est un monisme, et elle exclut radicalement tout principe de finalit\u00e9, externe ou interne. Cf. _Principes de philosophie zoologique_ , 1830, et _\u00c9tudes progressives d'un naturaliste_ , 1835.\n\n16.  _\u00c9_ , II, 39, prop. et dem.\n\n17.  _\u00c9_ , II, 38, prop. et dem.\n\n18.  _\u00c9_ , II, 45, prop. et sc.\n\n19.  _\u00c9_ , II, 46, dem. : \u00ab Donc, ce qui donne la connaissance de l'essence \u00e9ternelle et infinie de Dieu est commun \u00e0 toutes choses, et est \u00e9galement dans une partie comme dans le tout. \u00bb\n\n20.  _\u00c9_ , II, 40, sc. 1 : Par notre m\u00e9thode, on \u00e9tablirait \u00ab quelles notions sont communes, et quelles sont claires et distinctes pour ceux-l\u00e0 seulement qui ne s'embarrassent pas de pr\u00e9jug\u00e9s... \u00bb.\n\n21.  Cf. _\u00c9_ , II, 38 et 39. De m\u00eame _TTP_ , ch. 7, o\u00f9 l'on part des notions les plus universelles (II, pp. 176-177).\n\n22.  Cf. _\u00c9_ , II, 39, dem. : De la notion commune d\u00e9coule une id\u00e9e d'affection (telle est sa fonction pratique).\n\n23.  _\u00c9_ , IV, 29, prop. : \u00ab Et, d'une fa\u00e7on absolue, aucune chose ne peut \u00eatre bonne _ou mauvaise_ pour nous, \u00e0 moins qu'elle n'ait quelque chose de commun avec nous. \u00bb\n\n24.  _\u00c9_ , IV, 30, prop.\n\n25.  _\u00c9_ , V, 10, dem.\n\n26.  _\u00c9_ , III, 1, dem.\n\n27.  _\u00c9_ , V, 3, prop. Et la proposition suivante pr\u00e9cise le moyen de former cette id\u00e9e claire et distincte : rattacher le sentiment \u00e0 une notion commune comme \u00e0 sa cause.\n\n28.  Cf. _\u00c9_ , V, 2, prop. et dem. Et V, 4, sc. : Ce qui est d\u00e9truit n'est pas la joie passive elle-m\u00eame, mais les amours qui en proc\u00e8dent.\n\n29.  _\u00c9_ , IV, 63, dem. du cor. : \u00ab Le d\u00e9sir qui na\u00eet de la raison peut seulement na\u00eetre d'un sentiment de joie qui n'est pas une passion. \u00bb\n\n30.  _\u00c9_ , V, 4, sc.\n\n31.  _\u00c9_ , V, 10, prop. et dem.\n\n32.  Cf. _\u00c9_ , V, 20 sc.\n\n33.  _\u00c9_ , V, prop. et cor.\n\n34.  _\u00c9_ , IV, 32, prop. : \u00ab Dans la mesure o\u00f9 les hommes sont soumis aux passions, on ne peut dire qu'ils conviennent par nature. \u00bb Et le scolie pr\u00e9cise : \u00ab Les choses qui conviennent dans la seule n\u00e9gation, autrement dit en ce qu'elles n'ont pas, ne conviennent en r\u00e9alit\u00e9 en aucune chose. \u00bb\n\n35.  _\u00c9_ , V, 6, sc.\n\n36.  _\u00c9_ , V, 18, sc.\n\n37.  C'est l'ordre que pr\u00e9sente _\u00c9_ , V, 10. 1o) Dans la mesure o\u00f9 \u00ab nous ne sommes pas tourment\u00e9s par des sentiments contraires \u00e0 notre nature \u00bb, nous avons le pouvoir de former des id\u00e9es claires et distinctes (notions communes), et d'en d\u00e9duire les affections qui s'encha\u00eenent les unes aux autres conform\u00e9ment \u00e0 la raison. Ce sont donc les passions joyeuses (sentiments convenant avec notre nature) qui servent d'occasion premi\u00e8re \u00e0 la formation des notions communes. Nous devons _s\u00e9lectionner_ nos passions, et m\u00eame quand nous rencontrons quelque chose qui ne convient pas avec nous, nous devons nous efforcer de r\u00e9duire la tristesse au minimum (cf. scolie). 2o) Quand nous avons form\u00e9 les premi\u00e8res notions communes, nous sommes d'autant plus forts pour \u00e9viter les mauvaises rencontres et les sentiments qui nous sont contraires. Et dans la mesure o\u00f9 nous \u00e9prouvons n\u00e9cessairement encore de tels sentiments, nous sommes capables de former de nouvelles notions communes, qui nous font comprendre ces disconvenances et ces contrari\u00e9t\u00e9s elles-m\u00eames (cf. scolie).\n\n## CHAPITRE XVIII\n\n## VERS LE TROISI\u00c8ME GENRE\n\nLes genres de connaissance sont aussi des mani\u00e8res de vivre, des modes d'existence. Le premier genre (imagination) est constitu\u00e9 par toutes les id\u00e9es inad\u00e9quates, par les affections passives et leur encha\u00eenement1. Ce premier genre correspond d'abord \u00e0 _l'\u00e9tat de nature_ : je per\u00e7ois les objets au hasard des rencontres, d'apr\u00e8s l'effet qu'ils ont sur moi. Cet effet n'est qu'un \u00ab signe \u00bb, une \u00ab indication \u00bb variable. Cette connaissance est par _exp\u00e9rience vague_ ; et vague, selon l'\u00e9tymologie, renvoie au caract\u00e8re hasardeux des rencontres2. Ici, nous ne connaissons de la Nature que son \u00ab ordre commun \u00bb, c'est-\u00e0-dire l'effet des rencontres entre parties suivant des d\u00e9terminations purement extrins\u00e8ques.\n\nMais l' _\u00e9tat civil_ , lui aussi, appartient au premier genre de connaissance. D\u00e8s l'\u00e9tat de nature, l'imagination forme des id\u00e9es universelles abstraites, qui retiennent de l'objet tel ou tel caract\u00e8re sensible. Ce caract\u00e8re sera d\u00e9sign\u00e9 par un nom, qui servira de signe soit par rapport \u00e0 des objets qui ressemblent au premier, soit par rapport \u00e0 des objets qui sont li\u00e9s d'habitude avec le premier3. Mais avec le langage et l'\u00e9tat civil se d\u00e9veloppe une seconde sorte de signes : non plus indicatifs, mais imp\u00e9ratifs. Des signes nous paraissent dire ce qu'il _faut_ faire pour obtenir tel r\u00e9sultat, pour r\u00e9aliser telle fin : cette connaissance est par _ou\u00ef-dire._ Ainsi, dans l'exemple fameux de Spinoza, un signe repr\u00e9sente l'op\u00e9ration que nous \u00ab devons \u00bb faire sur trois nombres pour trouver le quatri\u00e8me. Lois de la nature ou r\u00e8gles techniques, il est in\u00e9vitable que toute loi nous apparaisse sous une forme morale, pr\u00e9cis\u00e9ment dans la mesure o\u00f9 nous n'en avons pas une connaissance ad\u00e9quate ; une loi nous para\u00eet morale, ou de type moral, chaque fois que nous en faisons d\u00e9pendre l'effet d'un signe imp\u00e9ratif (et non des rapports constitutifs des choses).\n\nCe qui forme l'unit\u00e9 du premier genre de connaissance, ce sont les signes. Ils d\u00e9finissent l'\u00e9tat d'une pens\u00e9e qui reste inad\u00e9quate, envelopp\u00e9e, non expliqu\u00e9e. \u00c0 ce premier genre, il faudrait m\u00eame joindre l' _\u00e9tat de religion_ , c'est-\u00e0-dire l'\u00e9tat de l'homme par rapport \u00e0 un Dieu qui lui donne une r\u00e9v\u00e9lation. Cet \u00e9tat ne diff\u00e8re pas moins de l'\u00e9tat de nature que l'\u00e9tat civil lui-m\u00eame : \u00ab La nature n'a jamais enseign\u00e9 \u00e0 personne que l'homme est oblig\u00e9 d'ob\u00e9ir \u00e0 Dieu ; aucun raisonnement m\u00eame ne saurait le lui apprendre. Seule la r\u00e9v\u00e9lation, confirm\u00e9e par des signes, le fait conna\u00eetre \u00e0 chacun4 \u00bb. Cet \u00e9tat de religion n'en est pas moins dans le premier genre : pr\u00e9cis\u00e9ment parce qu'il fait partie de la connaissance inad\u00e9quate, parce qu'il est fond\u00e9 sur des signes et se manifeste sous forme de lois qui commandent et ordonnent. La R\u00e9v\u00e9lation s'explique elle-m\u00eame par le caract\u00e8re inad\u00e9quat de notre connaissance, et porte uniquement sur certains propres de Dieu. Les signes de la r\u00e9v\u00e9lation constituent une troisi\u00e8me sorte de signes et d\u00e9finissent la religion des proph\u00e8tes, religion du premier genre ou de l'imagination.\n\nLe second genre de connaissance dans l' _\u00c9thique_ correspond \u00e0 l'\u00e9tat de raison : c'est une connaissance des notions communes, et par notions communes. C'est l\u00e0, dans l' _\u00c9thique_ , qu'appara\u00eet la v\u00e9ritable rupture entre les genres de connaissance : \u00ab La connaissance du second et du troisi\u00e8me genre, et non celle du premier, nous enseigne \u00e0 distinguer le vrai du faux5. \u00bb Avec les notions communes, nous entrons dans le domaine de _l'expression_ : ces notions sont nos premi\u00e8res id\u00e9es ad\u00e9quates, elles nous arrachent au monde des signes inad\u00e9quats. Et parce que toute notion commune nous conduit \u00e0 l'id\u00e9e de Dieu dont elle exprime l'essence, le second genre de connaissance implique lui aussi une religion. Cette religion n'est plus de l'imagination, mais de l'entendement ; l'expression de la Nature remplace les signes, l'amour remplace l'ob\u00e9issance ; ce n'est plus la religion des proph\u00e8tes, mais, \u00e0 des degr\u00e9s divers, la religion de Salomon, la religion des Ap\u00f4tres, la v\u00e9ritable religion du Christ, fond\u00e9e sur les notions communes6.\n\nMais pr\u00e9cis\u00e9ment, qu'est-ce que nous connaissons par ces notions ? Il est entendu que les notions communes ne constituent l'essence particuli\u00e8re d'aucune chose. Pourtant, il ne suffit pas de les d\u00e9finir par leur g\u00e9n\u00e9ralit\u00e9. _Les notions s'appliquent aux modes existants particuliers, et n'ont pas de sens ind\u00e9pendamment de cette application._ Repr\u00e9sentant (de points de vue plus ou moins g\u00e9n\u00e9raux) la similitude de composition des modes existants, elles sont pour nous le seul moyen d'arriver \u00e0 la connaissance ad\u00e9quate des rapports caract\u00e9ristiques des corps, de la composition de ces rapports et de leurs lois de composition. L\u00e0 encore, on le voit bien dans l'exemple des nombres : dans le deuxi\u00e8me genre de connaissance, nous n'appliquons plus une r\u00e8gle connue par ou\u00ef-dire, comme on ob\u00e9it \u00e0 une loi morale ; comprenant la r\u00e8gle de proportionnalit\u00e9 dans une notion commune, nous saisissons la mani\u00e8re dont se composent les rapports constitutifs des trois nombres donn\u00e9s. C'est pourquoi les notions communes nous font conna\u00eetre l'ordre positif de la Nature au sens de : ordre des rapports constitutifs ou caract\u00e9ristiques sous lesquels les corps conviennent et s'opposent. Les lois de la Nature n'apparaissent plus comme des commandements et des d\u00e9fenses, mais pour ce qu'elles sont, v\u00e9rit\u00e9s \u00e9ternelles, normes de composition, r\u00e8gles d'effectuation des pouvoirs. C'est cet ordre de la Nature qui exprime Dieu comme source ; et plus nous connaissons les choses suivant cet ordre, plus nos id\u00e9es elles-m\u00eames expriment l'essence de Dieu. Toute notre connaissance exprime Dieu, quand elle est dirig\u00e9e par les notions communes.\n\nLes notions communes sont une des d\u00e9couvertes fondamentales de l' _\u00c9thique_. Nous devons, \u00e0 cet \u00e9gard, attacher la plus grande importance \u00e0 la chronologie. Ferdinand Alqui\u00e9 insista r\u00e9cemment sur ce point : l'introduction des notions communes dans l' _\u00c9thique_ marque un moment d\u00e9cisif du spinozisme7. En effet, ni le _Court Trait\u00e9_ ni le _Trait\u00e9 de la r\u00e9forme_ n'en font \u00e9tat. Le _Court Trait\u00e9_ sait d\u00e9j\u00e0 que les choses ont des rapports caract\u00e9ristiques, mais se fie seulement au \u00ab raisonnement \u00bb pour les d\u00e9couvrir ; nulle mention des notions communes8. Aussi le correspondant du deuxi\u00e8me genre de connaissance dans le _Court Trait\u00e9_ (deuxi\u00e8me \u00ab mode de conscience \u00bb) ne constitue-t-il pas une connaissance ad\u00e9quate, mais une simple croyance droite. Dans le _Trait\u00e9 de la r\u00e9forme_ , le correspondant du deuxi\u00e8me genre (troisi\u00e8me \u00ab mode de perception \u00bb) ne constitue encore qu'une connaissance claire, non pas une connaissance ad\u00e9quate : il ne se d\u00e9finit nullement par les notions communes, mais par des inf\u00e9rences de type cart\u00e9sien et des d\u00e9ductions de type aristot\u00e9licien9.\n\nPourtant, _dans un tout autre contexte_ , on trouve dans le _Trait\u00e9 de la r\u00e9forme_ un pressentiment et une approximation de ce que seront les notions communes. Un passage c\u00e9l\u00e8bre parle, en effet, des \u00ab choses fixes et \u00e9ternelles \u00bb qui, en raison de leur omnipr\u00e9sence, sont \u00e0 notre \u00e9gard \u00ab comme des universaux ou des genres pour la d\u00e9finition des choses singuli\u00e8res changeantes \u00bb : on reconna\u00eet ici les notions les plus universelles, \u00e9tendue, mouvement, repos, qui sont communes \u00e0 toutes choses. Et la suite du texte r\u00e9clame encore d'autres \u00ab adjuvants \u00bb n\u00e9cessaires pour comprendre les choses singuli\u00e8res changeantes : on pressent alors le r\u00f4le des notions communes moins universelles10. Mais si ce texte soul\u00e8ve beaucoup de difficult\u00e9s, c'est parce qu'il est \u00e9crit du point de vue du mode de perception ou du genre de connaissance supr\u00eame, portant sur les essences elles-m\u00eames : dans les choses fixes et \u00e9ternelles, dit Spinoza, des lois sont inscrites comme dans leurs vrais codes ; or ces lois semblent aussi bien des lois de production des essences que des lois de composition des rapports11.\n\nComment expliquer que Spinoza assimile ici des sortes de lois si diff\u00e9rentes ? Nous supposons qu'il n'eut le pressentiment des notions communes qu'en avan\u00e7ant dans la r\u00e9daction du _Trait\u00e9 de la r\u00e9forme._ Or, \u00e0 ce moment, il avait d\u00e9j\u00e0 d\u00e9fini autrement le troisi\u00e8me mode de perception (correspondant au second genre de connaissance). D'o\u00f9, les choses fixes et \u00e9ternelles faisant fonction d'universaux ne trouvaient de place qu'au niveau du genre ou du mode supr\u00eame : elles \u00e9taient confondues d\u00e8s lors avec le principe de la connaissance des essences. Une autre place \u00e9tait possible ; mais il aurait fallu que Spinoza rev\u00eent en arri\u00e8re, et repr\u00eet la description des modes de perception en fonction de sa nouvelle id\u00e9e. Cette hypoth\u00e8se explique en partie pourquoi Spinoza renonce \u00e0 terminer le _Trait\u00e9 de la r\u00e9forme_ , pr\u00e9cis\u00e9ment quand il arrive \u00e0 l'expos\u00e9 de ce qu'il appelle lui-m\u00eame une propri\u00e9t\u00e9 commune. Cette hypoth\u00e8se permettrait aussi de dater la pleine formation de la th\u00e9orie des notions communes par Spinoza entre l'abandon du _Trait\u00e9 de la r\u00e9forme_ et la r\u00e9daction de l' _\u00c9thique._ Or cette pleine possession devait lui inspirer le d\u00e9sir de modifier le _Trait\u00e9_ , de refaire la th\u00e9orie du second genre ou troisi\u00e8me mode de perception, en donnant aux notions communes leur d\u00e9veloppement autonome et distinct ; c'est pourquoi Spinoza, dans l' _\u00c9thique_ , parle d'un Trait\u00e9 o\u00f9 il se propose de d\u00e9velopper ces points12.\n\nQuand Spinoza d\u00e9couvre que les notions communes sont nos premi\u00e8res id\u00e9es ad\u00e9quates, un hiatus s'\u00e9tablit donc entre le premier et le second genre de connaissance. L'existence de ce hiatus ne doit pas nous faire oublier pourtant tout un syst\u00e8me de correspondances entre ces deux genres, sans lesquelles la formation d'une id\u00e9e ad\u00e9quate ou d'une notion commune resterait incompr\u00e9hensible. D'abord nous avons vu que l'\u00e9tat civil tenait lieu de raison, pr\u00e9parait la raison et l'imitait. Cela serait impossible si les lois morales et les signes imp\u00e9ratifs, malgr\u00e9 le contresens qu'ils impliquent, ne co\u00efncidaient d'une certaine mani\u00e8re avec l'ordre v\u00e9ritable et positif de la Nature. Ainsi ce sont bien les lois de la Nature que les proph\u00e8tes saisissent et transmettent, bien qu'ils les comprennent inad\u00e9quatement. De m\u00eame, le plus grand effort de la soci\u00e9t\u00e9 est de choisir des signes et d'instituer des lois dont l'ensemble co\u00efncide au maximum avec l'ordre de la nature et, surtout, avec la subsistance de l'homme dans cet ordre. \u00c0 cet \u00e9gard, la variabilit\u00e9 des signes devient un avantage, et nous ouvre des possibilit\u00e9s que l'entendement n'a pas par lui-m\u00eame, possibilit\u00e9s propres \u00e0 l'imagination13. Mais, de plus, la raison n'arriverait pas \u00e0 former des notions communes, c'est-\u00e0-dire \u00e0 entrer en possession de sa puissance d'agir, si elle ne se cherchait elle-m\u00eame au cours de ce premier effort qui consiste \u00e0 s\u00e9lectionner les passions joyeuses. Avant de devenir actifs, il faut s\u00e9lectionner et encha\u00eener les passions qui augmentent notre puissance d'agir. Or ces passions se rapportent \u00e0 l'image d'objets qui conviennent en nature avec nous ; ces images elles-m\u00eames sont encore des id\u00e9es inad\u00e9quates, simples indications qui ne nous font conna\u00eetre les objets que par l'effet qu'ils ont sur nous. La raison ne se \u00ab trouverait \u00bb donc pas si son premier effort ne se dessinait dans le cadre du premier genre, utilisant toutes les ressources de l'imagination.\n\nEnvisag\u00e9es dans leur origine, les notions communes trouvent dans l'imagination les conditions m\u00eames de leur formation. Mais bien plus : envisag\u00e9es dans leur fonction pratique, elles ne s'appliquent qu'\u00e0 des choses qui peuvent \u00eatre imagin\u00e9es. C'est pourquoi elles sont elles-m\u00eames, \u00e0 certains \u00e9gards, assimilables \u00e0 des images14. _L'application des notions communes en g\u00e9n\u00e9ral implique une curieuse harmonie entre la raison et l'imagination,entre les lois de la raison et les lois de l'imagination_. Spinoza analyse diff\u00e9rents cas. Les livres III et IV de l' _\u00c9thique_ avaient montr\u00e9 sous quelles lois sp\u00e9cifiques de l'imagination une passion devient plus o\u00f9 moins intense, plus ou moins forte. Ainsi, le sentiment envers une chose que nous imaginons en elle-m\u00eame est plus fort que le sentiment que nous \u00e9prouvons, quand nous croyons qu'elle est n\u00e9cessaire ou n\u00e9cessit\u00e9e15. Or la loi sp\u00e9cifique de la raison consiste pr\u00e9cis\u00e9ment \u00e0 consid\u00e9rer les choses comme n\u00e9cessaires : les notions communes nous font comprendre la n\u00e9cessit\u00e9 des convenances et des disconvenances entre corps. La raison profite ici d'une disposition de l'imagination : plus nous comprenons les choses comme n\u00e9cessaires, moins les passions fond\u00e9es sur l'imagination ont de force ou d'intensit\u00e916. L'imagination, suivant sa propre loi, commence toujours par affirmer la pr\u00e9sence de son objet ; ensuite, elle est affect\u00e9e par des causes qui excluent cette pr\u00e9sence ; elle entre dans une sorte de \u00ab flottement \u00bb, et ne croit plus \u00e0 son objet que comme possible ou m\u00eame contingent. L'imagination d'un objet contient donc, avec le temps, le principe de son affaiblissement. Mais la raison, d'apr\u00e8s sa propre loi, forme des notions communes, c'est-\u00e0-dire l'id\u00e9e de propri\u00e9t\u00e9s \u00ab que nous consid\u00e9rons toujours comme pr\u00e9sentes17. \u00bb Ici la raison satisfait \u00e0 l'exigence de l'imagination mieux que celle-ci ne peut le faire elle-m\u00eame. L'imagination, entra\u00een\u00e9e par son propre destin qui l'affecte de causes diverses, n'arrive pas \u00e0 maintenir la pr\u00e9sence de son objet. Seule la raison ne se contente pas de diminuer relativement la force des passions : \u00ab eu \u00e9gard au temps \u00bb, les sentiments actifs qui naissent de la raison ou de la notion commune sont plus forts en eux-m\u00eames que tous les sentiments passifs qui naissent de l'imagination18. D'apr\u00e8s la loi de l'imagination, un sentiment est d'autant plus fort qu'il est provoqu\u00e9 par plus de causes agissant ensemble19. Mais, d'apr\u00e8s sa propre loi, la notion commune s'applique ou se rapporte \u00e0 plusieurs choses ou images de choses qui se joignent facilement \u00e0 elles : elle est donc fr\u00e9quente et vivace20. En ce sens, elle diminue l'intensit\u00e9 du sentiment de l'imagination, parce qu'elle d\u00e9termine l'esprit \u00e0 consid\u00e9rer plusieurs objets. Mais aussi, ces objets qui se joignent \u00e0 la notion sont comme des causes favorisant le sentiment de la raison qui d\u00e9coule de celle-ci21.\n\nN\u00e9cessit\u00e9, pr\u00e9sence et fr\u00e9quence sont les trois caract\u00e8res des notions communes. Or ces caract\u00e8res font qu'elles s'imposent en quelque sorte \u00e0 l'imagination, soit pour diminuer l'intensit\u00e9 des sentiments passifs, soit pour assurer la vivacit\u00e9 des sentiments actifs. Les notions communes se servent des lois de l'imagination pour nous lib\u00e9rer de l'imagination m\u00eame. Leur n\u00e9cessit\u00e9, leur pr\u00e9sence, leur fr\u00e9quence leur permettent de s'ins\u00e9rer dans le mouvement de l'imagination, et d'en d\u00e9tourner le cours \u00e0 leur profit. Il n'est pas exag\u00e9r\u00e9 de parler ici d'une _libre harmonie_ de l'imagination avec la raison.\n\n_La majeure partie de l'_ \u00c9thique _, exactement jusqu'en V 21, est \u00e9crite dans la perspective du second genre de connaissance._ Car c'est seulement par les notions communes que nous arrivons \u00e0 avoir des id\u00e9es ad\u00e9quates, et une connaissance ad\u00e9quate de Dieu lui-m\u00eame. Il n'y a pas l\u00e0 une condition de toute connaissance, mais une condition de notre connaissance, en tant que nous sommes des modes existants finis compos\u00e9s d'une \u00e2me et d'un corps. Nous qui n'avons d'abord que des id\u00e9es inad\u00e9quates et des affections passives, nous ne pouvons conqu\u00e9rir notre puissance de comprendre et d'agir qu'en formant des notions communes. Toute notre connaissance passe par ces notions. C'est pourquoi Spinoza peut dire que l'existence m\u00eame de Dieu n'est pas connue par elle-m\u00eame, mais \u00ab doit se conclure de notions dont la v\u00e9rit\u00e9 soit si ferme et si in\u00e9branlable qu'il ne puisse y avoir ni \u00eatre con\u00e7u de puissance capable de les changer22 \u00bb. M\u00eame aveu dans l' _\u00c9thique_ : le premier livre nous fait conna\u00eetre Dieu, et toutes choses comme d\u00e9pendant de Dieu ; or cette connaissance est elle-m\u00eame du second genre23.\n\nTous les corps conviennent en certaines choses, \u00e9tendue, mouvement, repos. Les id\u00e9es d'\u00e9tendue, de mouvement, de repos sont pour nous des notions communes tr\u00e8s universelles, puisqu'elles s'appliquent \u00e0 tous les corps existants. Nous demandons : faut-il consid\u00e9rer l'id\u00e9e de Dieu elle-m\u00eame comme une notion commune, la plus universelle de toutes ? Beaucoup de textes semblent le sugg\u00e9rer24. Pourtant il n'en est pas ainsi : notre id\u00e9e de Dieu est en relation \u00e9troite avec les notions communes, mais n'est pas une de ces notions. En un sens, l'id\u00e9e de Dieu s'oppose aux notions communes, parce que celles-ci s'appliquent toujours \u00e0 des choses qui peuvent \u00eatre imagin\u00e9es, tandis que Dieu ne peut pas l'\u00eatre25. Spinoza dit seulement que les notions communes nous m\u00e8nent \u00e0 l'id\u00e9e de Dieu, qu'elles nous _\u00ab donnent \u00bb_ n\u00e9cessairement la connaissance de Dieu, et que, sans elles, nous n'aurions pas cette connaissance26. En effet, une notion commune est une id\u00e9e ad\u00e9quate ; l'id\u00e9e ad\u00e9quate, c'est l'id\u00e9e comme expressive ; et ce qu'elle exprime, c'est l'essence m\u00eame de Dieu. L'id\u00e9e de Dieu est donc en relation d'expression avec les notions communes. Les notions communes expriment Dieu comme la source de tous les rapports constitutifs des choses. En tant que rapport\u00e9e \u00e0 ces notions qui l'expriment, l'id\u00e9e de Dieu fonde la religion du second genre. Car des sentiments actifs, des joies actives d\u00e9coulent des notions communes ; pr\u00e9cis\u00e9ment, elles en d\u00e9coulent \u00ab avec accompagnement de l'id\u00e9e de Dieu \u00bb. L'amour de Dieu n'est que cette joie avec cet accompagnement27. Le plus haut effort de la raison, en tant qu'elle con\u00e7oit des notions communes, est donc de conna\u00eetre Dieu et de l'aimer28. (Mais ce Dieu rapport\u00e9 aux notions communes n'a pas \u00e0 r\u00e9pondre \u00e0 notre amour : Dieu impassible, qui ne nous paie pas de retour. Car, si actives qu'elles soient, les joies d\u00e9coulant des notions ne sont pas s\u00e9parables de joies passives ou de donn\u00e9es de l'imagination qui, d'abord, ont augment\u00e9 notre puissance d'agir et nous ont servi de causes occasionnelles. Or Dieu lui-m\u00eame est exempt de passions : il n'\u00e9prouve aucune joie passive, ni m\u00eame aucune joie active du genre de celles qui supposent une joie passive29.)\n\nOn se souvient des exigences m\u00e9thodologiques du _Trait\u00e9 de la r\u00e9forme_ : nous ne pouvons pas partir de l'id\u00e9e de Dieu, mais nous devons y arriver aussi vite que possible. Or \u00ab le plus vite possible \u00bb, dans le _Trait\u00e9_ , se pr\u00e9sentait ainsi : nous devions partir de ce qui \u00e9tait positif dans une id\u00e9e que nous avions ; nous nous efforcions de rendre cette id\u00e9e ad\u00e9quate ; elle \u00e9tait ad\u00e9quate quand elle \u00e9tait rattach\u00e9e \u00e0 sa cause, quand elle exprimait sa cause ; mais elle n'exprimait pas sa cause sans exprimer aussi l'id\u00e9e de Dieu qui d\u00e9terminait cette cause \u00e0 produire un tel effet. Ainsi, nous ne risquions pas d'entrer dans une r\u00e9gression infinie de cause en cause : c'est \u00e0 chaque niveau que Dieu \u00e9tait exprim\u00e9 comme ce qui d\u00e9terminait la cause.\n\nIl nous semble inexact d'opposer sur ce point l' _\u00c9thique_ au _Trait\u00e9 de la r\u00e9forme. Pas plus que le_ Trait\u00e9, _l'_ \u00c9thique _ne commence par Dieu comme substance absolument infinie._ L' _\u00c9thique_ ne part nullement de l'id\u00e9e de Dieu comme d'un inconditionn\u00e9 ; nous avons vu \u00e0 cet \u00e9gard le r\u00f4le des premi\u00e8res propositions. L' _\u00c9thique_ a le m\u00eame projet que le _Trait\u00e9 de la r\u00e9forme_ : s'\u00e9lever aussi vite que possible \u00e0 l'id\u00e9e de Dieu sans tomber dans une r\u00e9gression infinie, sans faire de Dieu lui-m\u00eame une cause \u00e9loign\u00e9e. Si l' _\u00c9thique_ se distingue du _Trait\u00e9 de la r\u00e9forme_ , ce n'est donc pas par un changement de m\u00e9thode, encore moins par un changement de principe, mais seulement parce que l' _\u00c9thique_ a trouv\u00e9 des moyens moins artificiels et plus concrets. Ces moyens, ce sont les notions communes (jusqu'en V, 21). Nous ne partons plus de ce qui est positif dans une id\u00e9e quelconque pour essayer de former une id\u00e9e ad\u00e9quate : un tel proc\u00e9d\u00e9 est peu s\u00fbr et reste ind\u00e9termin\u00e9. Nous partons de ce qu'il y a de positif dans une passion joyeuse ; nous sommes alors d\u00e9termin\u00e9s \u00e0 former une notion commune, notre premi\u00e8re id\u00e9e ad\u00e9quate. Puis nous formons des notions communes de plus en plus g\u00e9n\u00e9rales, qui constituent le syst\u00e8me de la raison ; mais chaque notion commune, \u00e0 son propre niveau, exprime Dieu et nous conduit \u00e0 la connaissance de Dieu. Chaque notion commune exprime Dieu comme la source des rapports qui se composent dans les corps auxquels la notion s'applique. On ne dira donc pas que les notions plus universelles expriment Dieu mieux que les moins universelles. On ne dira surtout pas que l'id\u00e9e de Dieu soit elle-m\u00eame une notion commune, la plus universelle de toutes : en v\u00e9rit\u00e9, chaque notion nous y conduit, chaque notion l'exprime, les moins universelles comme les plus universelles. Dans le syst\u00e8me de l'expression, jamais Dieu n'est une cause \u00e9loign\u00e9e.\n\nC'est pourquoi l'id\u00e9e de Dieu, dans l' _\u00c9thique_ , va jouer le r\u00f4le d'un pivot. Tout tourne autour d'elle, tout change avec elle. Spinoza annonce que, \u00ab outre \u00bb le second genre de connaissance, un troisi\u00e8me est donn\u00e930. Bien plus, il pr\u00e9sente le second genre comme \u00e9tant la cause motrice du troisi\u00e8me : c'est le second qui nous d\u00e9termine \u00e0 entrer dans le troisi\u00e8me, \u00e0 \u00ab former \u00bb le troisi\u00e8me31. La question est : Comment le second genre nous d\u00e9termine-t-il ainsi ? _Seule l'id\u00e9e de Dieu peut expliquer ce passage_ , qui appara\u00eet dans l' _\u00c9thique_ en V 20-21. 1o) Chaque notion commune nous conduit \u00e0 l'id\u00e9e de Dieu. Rapport\u00e9e aux notions communes qui l'expriment, l'id\u00e9e de Dieu fait elle-m\u00eame partie du second genre de connaissance. Dans cette mesure, elle repr\u00e9sente un Dieu impassible, mais cette id\u00e9e accompagne toutes les joies qui d\u00e9coulent de notre puissance de comprendre (en tant que cette puissance proc\u00e8de par notions communes). L'id\u00e9e de Dieu, en ce sens, est la pointe extr\u00eame du second genre. 2o) Mais, bien qu'elle se rapporte n\u00e9cessairement aux notions communes, l'id\u00e9e de Dieu n'est pas elle-m\u00eame une notion commune. C'est pourquoi elle nous pr\u00e9cipite dans un nouvel \u00e9l\u00e9ment. Nous ne pouvons atteindre \u00e0 l'id\u00e9e de Dieu que par le second genre ; mais nous ne pouvons pas y atteindre sans \u00eatre d\u00e9termin\u00e9s \u00e0 sortir de ce second genre pour entrer dans un nouvel \u00e9tat. Dans le second genre, c'est l'id\u00e9e de Dieu qui sert de fondement au troisi\u00e8me ; par \u00ab fondement \u00bb, il faut entendre la vraie cause motrice, la _causa fiendi_32. Cette id\u00e9e de Dieu elle-m\u00eame changera donc de contenu, prendra un autre contenu, dans le troisi\u00e8me genre auquel elle nous d\u00e9termine.\n\nUne notion commune a deux caract\u00e8res : elle s'applique \u00e0 plusieurs modes existants ; elle nous fait conna\u00eetre les rapports sous lesquels les modes existants conviennent ou s'opposent. \u00c0 la limite, on comprend qu'une id\u00e9e d'attribut nous apparaisse d'abord comme une notion commune : l'id\u00e9e d'\u00e9tendue est une notion tr\u00e8s universelle en tant qu'elle s'applique \u00e0 tous les corps qui existent ; et l'id\u00e9e des modes infinis de l'\u00e9tendue nous fait conna\u00eetre la convenance de tous les corps du point de vue de la Nature enti\u00e8re. Mais l'id\u00e9e de Dieu, qui s'ajoute \u00e0 toutes les notions communes ou les \u00ab accompagne \u00bb, nous inspire une nouvelle appr\u00e9ciation des attributs et des modes. L\u00e0 encore, il en est dans l' _\u00c9thique_ comme dans le _Trait\u00e9 de la r\u00e9forme_ : l'id\u00e9e de Dieu nous introduit dans le domaine des \u00ab \u00eatres r\u00e9els \u00bb et de leur encha\u00eenement. L'attribut ne sera plus seulement compris comme une _propri\u00e9t\u00e9 commune_ \u00e0 tous les _modes existants_ qui lui correspondent, mais comme ce qui constitue _l'essence singuli\u00e8re_ de la substance divine et comme ce qui contient toutes les _essences particuli\u00e8res_ de modes. Le troisi\u00e8me genre de connaissance est ainsi d\u00e9fini : il s'\u00e9tend \u00ab de l'id\u00e9e ad\u00e9quate de l'essence formelle de certains attributs de Dieu \u00e0 la connaissance ad\u00e9quate de l'essence des choses \u00bb33. L'attribut est encore une forme commune mais, ce qui a chang\u00e9, c'est le sens du mot \u00ab commun \u00bb. Commun ne signifie plus g\u00e9n\u00e9ral, c'est-\u00e0-dire applicable \u00e0 plusieurs modes existants ou \u00e0 tous les modes existants d'un certain genre. Commun signifie univoque : l'attribut est univoque, ou commun \u00e0 Dieu dont il constitue l'essence singuli\u00e8re et aux modes dont il contient les essences particuli\u00e8res. Bref, une diff\u00e9rence fondamentale appara\u00eet entre le deuxi\u00e8me et le troisi\u00e8me genre : les id\u00e9es du second genre se d\u00e9finissent par leur fonction g\u00e9n\u00e9rale, elles s'appliquent aux modes existants, elles nous font conna\u00eetre la composition des rapports qui caract\u00e9risent ces modes existants. Les id\u00e9es du troisi\u00e8me genre se d\u00e9finissent par leur nature singuli\u00e8re, elles repr\u00e9sentent l'essence de Dieu, elles nous font conna\u00eetre les essences particuli\u00e8res telles qu'elles sont contenues en Dieu lui-m\u00eame34.\n\nNous sommes nous-m\u00eames des modes existants. Notre connaissance est soumise \u00e0 la condition suivante : nous devons passer par les notions communes pour atteindre aux id\u00e9es du troisi\u00e8me genre. Loin de pouvoir d\u00e9duire le rapport qui caract\u00e9rise un mode \u00e0 partir de son essence, nous devons d'abord conna\u00eetre le rapport pour arriver \u00e0 conna\u00eetre l'essence. De m\u00eame nous devons concevoir l'\u00e9tendue comme une notion commune avant de la comprendre comme ce qui constitue l'essence de Dieu. Le second genre est pour nous cause efficiente du troisi\u00e8me ; et dans le second genre, c'est l'id\u00e9e de Dieu qui nous fait passer du second au troisi\u00e8me. Nous commen\u00e7ons par former des notions communes qui expriment l'essence de Dieu ; alors seulement nous pouvons comprendre Dieu comme s'exprimant lui-m\u00eame dans les essences. Cette condition de notre connaissance n'est pas une condition pour toute connaissance : le vrai Christ ne passe pas par les notions communes. Il adapte, il conforme aux notions communes l'enseignement qu'il nous donne ; mais sa propre connaissance est imm\u00e9diatement du troisi\u00e8me genre ; l'existence de Dieu lui est donc connue par elle-m\u00eame, ainsi que toutes les essences, et l'ordre des essences35. C'est pourquoi Spinoza dit : contrairement au Christ, nous ne connaissons pas l'existence de Dieu par elle-m\u00eame36. Dans la situation naturelle de notre existence, nous sommes remplis d'id\u00e9es inad\u00e9quates et d'affections passives ; nous n'arriverons jamais \u00e0 une id\u00e9e ad\u00e9quate quelconque ni \u00e0 une joie active, si nous ne formons d'abord des notions communes. Pourtant, on n'en conclura pas que Dieu ne nous soit connu qu' _indirectement._ Les notions communes n'ont rien \u00e0 voir avec des signes ; elles constituent seulement les conditions sous lesquelles nous atteignons nous-m\u00eames au troisi\u00e8me genre de connaissance. Ainsi les preuves de l'existence de Dieu ne sont pas des preuves indirectes : l'id\u00e9e de Dieu y est encore saisie dans son rapport avec les notions communes, mais elle nous d\u00e9termine pr\u00e9cis\u00e9ment \u00e0 \u00ab former \u00bb le troisi\u00e8me genre, ou \u00e0 conqu\u00e9rir une vision directe.\n\n* * *\n\n1.  _\u00c9_ , II, 41, dem.\n\n2.  _TRE_ , 19.\n\n3.  Sur la liaison par m\u00e9moire ou habitude : _\u00c9_ , II, 18, sc. Sur la liaison par ressemblance, qui d\u00e9finit une connaissance par signes : _\u00c9_ , II, 40, sc. 1 et sc. 2.\n\n4.  _TTP_ , ch. 16 (II, p. 266).\n\n5.  _\u00c9_ , II, 42, prop. Et V, 28, prop.\n\n6.  Cette religion du second genre ne se confond pas avec ce que Spinoza, dans le _Trait\u00e9 th\u00e9ologique-politique_ , appelle la \u00ab foi universelle \u00bb, \u00ab commune \u00e0 tous les hommes \u00bb. Telle qu'elle est d\u00e9crite ch. 14 (II, pp. 247-248) la foi universelle concerne encore l'ob\u00e9issance, et utilise abondamment les concepts moraux de faute, repentir et pardon : en fait elle m\u00e9lange des id\u00e9es du premier genre et des notions du second genre. La vraie religion du second genre, uniquement fond\u00e9e sur les notions communes, n'a d'expos\u00e9 syst\u00e9matique qu'en _\u00c9_ , V, 14-20. Mais le _TTP_ , donne des indications pr\u00e9cieuses : c'est d'abord la religion de Salomon, qui sut se guider sur la lumi\u00e8re naturelle (ch. 4, II, pp. 142-144). En un autre sens, c'est la religion du Christ : non pas que le Christ ait besoin de notions communes pour conna\u00eetre Dieu, mais il conforme son enseignement aux notions communes, au lieu de le r\u00e9gler sur des signes (il va de soi que la Passion et la R\u00e9surrection font partie du premier genre, cf. ch. 4, II, pp. 140-141, p. 144). Enfin c'est la religion des Ap\u00f4tres, mais seulement dans une part de leur enseignement et de leur activit\u00e9 (ch. 11, _passim_ ).\n\n7.  Cf. F. ALQUI\u00c9, _Nature et V\u00e9rit\u00e9 dans la philosophie de Spinoza_ , cours publi\u00e9, C.D.U. pp. 30 sq.\n\n8.  _CT_ , II, ch. 1, 2-3.\n\n9.  _TRE_ , 19-21 (cf. notre chapitre X).\n\n10.  _TRE_ , 101-102. Et le _Trait\u00e9 de la r\u00e9forme_ se termine au moment o\u00f9 Spinoza cherche une propri\u00e9t\u00e9 commune ( _aliquid commune_ ) dont d\u00e9pendraient tous les caract\u00e8res positifs de l'entendement : 110.\n\n11.  Spinoza en effet dit que \u00ab les choses fixes et \u00e9ternelles \u00bb doivent nous donner la connaissance de \u00ab l'essence intime \u00bb des choses ; nous sommes ici dans le dernier genre de connaissance. Mais d'autre part, les choses fixes doivent aussi servir d'\u00ab universaux \u00bb par rapport aux modes existants variables : nous sommes alors dans le second genre, et dans le domaine de la composition des rapports, non plus de la production des essences. Les deux ordres sont donc m\u00e9lang\u00e9s. Cf. _TRE_ , 101.\n\n12.  _\u00c9_ , II, 40, sc. 1 : \u00e0 propos du probl\u00e8me des notions, et des diff\u00e9rentes esp\u00e8ces de notions, Spinoza dit qu'il a \u00ab autrefois m\u00e9dit\u00e9 sur ces choses \u00bb. Il s'agit \u00e9videmment du _Trait\u00e9 de la r\u00e9forme_. Mais il _ajoute_ qu'il a \u00ab r\u00e9serv\u00e9 ces sujets pour un autre trait\u00e9 \u00bb : nous supposons qu'il s'agit alors d'un remaniement du _Trait\u00e9 de la r\u00e9forme_ , en fonction de la fin qui obligeait Spinoza \u00e0 tout reprendre.\n\n13.  _TTP_ , ch. 1 (II, p. 106) : \u00ab \u00c0 partir de paroles et d'images on peut combiner beaucoup plus d'id\u00e9es qu'\u00e0 partir des seuls principes et notions sur lesquels toute notre connaissance naturelle est construite. \u00bb\n\n14.  En _\u00c9_ , II, 47, sc., Spinoza signale express\u00e9ment l'affinit\u00e9 des notions communes avec les choses qui peuvent \u00eatre imagin\u00e9es, c'est-\u00e0-dire les corps. C'est m\u00eame pourquoi l'id\u00e9e de Dieu est ici distingu\u00e9e des notions communes. Spinoza parlera des propri\u00e9t\u00e9s communes que nous \u00ab imaginons \u00bb toujours de la m\u00eame fa\u00e7on ( _\u00c9_ , V, 7, dem.), ou bien \u00ab des images qui se rapportent aux choses comprises clairement et distinctement \u00bb ( _\u00c9_ , V, 12, prop.).\n\n15.  _\u00c9_ , IV, 49 ; V, 5.\n\n16.  _\u00c9_ , V, 6, prop. et dem.\n\n17.  _\u00c9_ , V, 7, dem. : \u00ab Un sentiment de la raison se rapporte n\u00e9cessairement aux propri\u00e9t\u00e9s communes des choses, que nous consid\u00e9rons toujours comme pr\u00e9sentes (car rien ne peut \u00eatre donn\u00e9 qui en exclue l'existence pr\u00e9sente) et que nous imaginons toujours de la m\u00eame fa\u00e7on. \u00bb\n\n18.  _\u00c9_ , V, 7, prop. (Ce texte se r\u00e9f\u00e8re seulement aux sentiments de l'imagination qui concernent des choses \u00ab consid\u00e9r\u00e9es comme absentes \u00bb. Mais, compte tenu du temps, il arrive _toujours_ \u00e0 l'imagination d'\u00eatre d\u00e9termin\u00e9e \u00e0 consid\u00e9rer son objet comme absent).\n\n19.  _\u00c9_ , V, 8, prop. et dem.\n\n20.  _\u00c9_ , V, 11, 12 et 13.\n\n21.  Cf. _\u00c9_ , V, 9 et 11.\n\n22.  _TTP_ , ch. 6 (II, p. 159). Cf. aussi la note jointe \u00e0 ce texte (II, p. 315).\n\n23.  _\u00c9_ , V, 36, sc.\n\n24.  En _\u00c9_ , II, 45-47. Spinoza passe des notions communes \u00e0 l'id\u00e9e de Dieu (cf. surtout 46, dem.). En V, 14-15, passage analogue : ayant montr\u00e9 qu'un grand nombre d'images se joignaient facilement \u00e0 la notion commune, Spinoza conclut que nous pouvons joindre et rapporter tourtes les images \u00e0 l'id\u00e9e de Dieu.\n\n25.  _\u00c9_ , II, 47, sc. : \u00ab Que les hommes n'aient pas une connaissance \u00e9galement claire de Dieu et des notions communes, cela provient de ce qu'ils ne peuvent imaginer Dieu comme ils imaginent les corps. \u00bb\n\n26.  _\u00c9_ , II, 46, dem. ( _id quod dat_ ).\n\n27.  _\u00c9_ , V, 15, dem.\n\n28.  _\u00c9_ , IV, 28, dem.\n\n29.  Cf. _\u00c9_ , V, 17 et 19. Spinoza rappelle explicitement que Dieu ne peut \u00e9prouver aucune augmentation de sa puissance d'agir, donc aucune joie passive. Mais il trouve ici l'occasion de nier que Dieu puisse \u00e9prouver une joie quelconque en g\u00e9n\u00e9ral : en effet, les seules joies actives qui sont connues _\u00e0 ce moment_ de l' _\u00c9thique_ sont celles du second genre. Or ces joies supposent des passions, et sont exclues de Dieu au m\u00eame titre que les passions.\n\n30.  _\u00c9_ , II, 40, sc. 2.\n\n31.  _\u00c9_ , V, 28, prop. : \u00ab L'effort ou le d\u00e9sir de conna\u00eetre les choses par le troisi\u00e8me genre de connaissance ne peut na\u00eetre du premier genre, mais bien du second genre de connaissance. \u00bb\n\n32.  En _\u00c9_ , V, 20 sc., Spinoza parle du \u00ab fondement \u00bb du troisi\u00e8me genre. Ce fondement est \u00ab la connaissance de Dieu \u00bb. Il ne s'agit pas, \u00e9videmment, de la connaissance de Dieu telle que le troisi\u00e8me genre nous la livrera. Comme le contexte le prouve (V, 15 et 16), il s'agit d'une connaissance de Dieu donn\u00e9e par les notions communes. De m\u00eame, en II, 47, sc., Spinoza dit que nous \u00ab formons \u00bb le troisi\u00e8me genre de connaissance \u00e0 partir d'une connaissance de Dieu. L\u00e0 encore, le contexte (II, 46, dem.) montre qu'il s'agit de la connaissance de Dieu telle qu'elle fait partie du second genre.\n\n33.  _\u00c9_ , II, 40, sc. 2 (cf. aussi V, 25, dem.).\n\n34.  Dans quelle mesure les id\u00e9es du deuxi\u00e8me et du troisi\u00e8me genre sont-elles les m\u00eames ? Se distinguent-elles seulement par leur fonction ou leur usage ? Le probl\u00e8me est complexe. Il est certain que les notions communes les plus universelles co\u00efncident avec les id\u00e9es des attributs. Comme notions communes, elles sont saisies dans la fonction g\u00e9n\u00e9rale qu'elles exercent par rapport aux _modes existants._ Comme id\u00e9es du troisi\u00e8me genre, elles sont pens\u00e9es dans leur essence objective, et en tant qu'elles contiennent objectivement les _essences de modes._ Toutefois, les notions communes moins universelles ne co\u00efncident pas pour leur compte avec les id\u00e9es des essences particuli\u00e8res (les rapports ne se confondent pas avec les essences, bien que les essences s'expriment dans des rapports).\n\n35.  _TTP_ , ch. 4 (II, pp. 140-141).\n\n36.  _TTP_ , ch. 1 (II, pp. 98-99).\n\n## CHAPITRE XIX\n\n## B\u00c9ATITUDE\n\nLe premier genre de connaissance a seulement pour objet les rencontres entre parties, d'apr\u00e8s leurs d\u00e9terminations extrins\u00e8ques. Le second genre s'\u00e9l\u00e8ve jusqu'\u00e0 la composition des rapports caract\u00e9ristiques. Mais seul le troisi\u00e8me genre concerne les essences \u00e9ternelles : connaissance de l'essence de Dieu, et des essences particuli\u00e8res telles qu'elles sont en Dieu et sont con\u00e7ues par Dieu. (Ainsi, dans les trois genres de connaissance, nous retrouvons les trois aspects de l'ordre de la Nature : ordre des passions, ordre de composition des rapports, ordre des essences elles-m\u00eames.) Or les essences ont plusieurs caract\u00e8res. D'abord, elles sont particuli\u00e8res, donc irr\u00e9ductibles les unes aux autres : chacune est un \u00eatre r\u00e9el, une _res physica_ , un degr\u00e9 de puissance ou d'intensit\u00e9. C'est pourquoi Spinoza peut opposer le troisi\u00e8me genre au second, en disant que le second genre nous montre en g\u00e9n\u00e9ral que toutes les choses qui existent d\u00e9pendent de Dieu, mais que seul le troisi\u00e8me genre nous fait comprendre la d\u00e9pendance de telle essence en particulier1. Pourtant, d'autre part, chaque essence convient avec toutes les autres. C'est que toutes les essences sont comprises dans la production de chacune. Il ne s'agit plus de convenances relatives, plus ou moins g\u00e9n\u00e9rales, entre modes existants, mais d'une convenance \u00e0 la fois singuli\u00e8re et absolue de chaque essence avec toutes les autres2. D\u00e8s lors, l'esprit ne conna\u00eet pas une essence, c'est-\u00e0-dire une chose sous l'esp\u00e8ce de l'\u00e9ternit\u00e9, sans \u00eatre d\u00e9termin\u00e9 \u00e0 conna\u00eetre encore plus de choses et \u00e0 d\u00e9sirer en conna\u00eetre de plus en plus3. Enfin, les essences sont expressives : non seulement chaque essence exprime toutes les autres dans le principe de sa production, mais elle exprime Dieu comme ce principe lui-m\u00eame qui contient toutes les essences et dont chacune d\u00e9pend en particulier. Chaque essence est une partie de la puissance de Dieu, donc est con\u00e7ue par l'essence m\u00eame de Dieu, mais en tant que l'essence de Dieu s'explique par _cette_ essence4.\n\nLa connaissance supr\u00eame comprend donc trois donn\u00e9es. Une id\u00e9e ad\u00e9quate de nous-m\u00eames ou de notre propre essence (id\u00e9e qui exprime l'essence de notre corps sous l'esp\u00e8ce de l'\u00e9ternit\u00e9) : chacun forme l'id\u00e9e de sa propre essence, et c'est \u00e0 cette id\u00e9e que Spinoza pense lorsqu'il dit que le troisi\u00e8me genre montre comment une essence _en particulier_ d\u00e9pend de Dieu5. Une id\u00e9e ad\u00e9quate du plus grand nombre de choses possibles, toujours dans leur essence ou sous l'esp\u00e8ce de l'\u00e9ternit\u00e9. Une id\u00e9e ad\u00e9quate de Dieu, en tant que Dieu contient toutes les essences, et les comprend toutes dans la production de chacune (donc dans la production de la n\u00f4tre en particulier).\n\nLe moi, les choses et Dieu sont les trois id\u00e9es du troisi\u00e8me genre. En d\u00e9coulent des joies, un d\u00e9sir et un amour. Les joies du troisi\u00e8me genre sont des joies actives : en effet, elles s'expliquent par notre propre essence et \u00ab s'accompagnent \u00bb toujours de l'id\u00e9e ad\u00e9quate de cette essence. Tout ce que nous comprenons sous le troisi\u00e8me genre, y compris l'essence des autres choses et celle de Dieu, nous le comprenons du fait que nous concevons notre essence (l'essence de notre corps) sous l'esp\u00e8ce de l'\u00e9ternit\u00e96. C'est en ce sens que le troisi\u00e8me genre n'a pas d'autre _cause formelle_ que notre puissance d'agir ou de comprendre, c'est-\u00e0-dire la puissance de penser de Dieu lui-m\u00eame en tant qu'elle s'explique par notre propre essence7. Dans le troisi\u00e8me genre, toutes les id\u00e9es ont pour cause formelle notre puissance de comprendre. Toutes les affections qui suivent de ces id\u00e9es sont donc par nature des affections actives, des joies actives8. Il faut concevoir que l'essence de Dieu affecte la mienne, et que les essences s'affectent les unes les autres ; mais il n'y a pas d'affections d'une essence qui ne s'expliquent formellement par cette essence elle-m\u00eame, donc qui ne s'accompagnent de l'id\u00e9e de soi comme cause formelle ou de la consid\u00e9ration de la puissance d'agir.\n\nDe cette joie d\u00e9coulant de l'id\u00e9e ad\u00e9quate de nous-m\u00eames na\u00eet un _d\u00e9sir_ , d\u00e9sir de conna\u00eetre toujours plus de choses dans leur essence ou sous l'esp\u00e8ce de l'\u00e9ternit\u00e9. Mais, surtout, na\u00eet un _amour._ Car, dans le troisi\u00e8me genre, l'id\u00e9e de Dieu \u00e0 son tour est comme la cause mat\u00e9rielle de toutes les id\u00e9es. Toutes les essences expriment Dieu comme ce par quoi elles sont con\u00e7ues : l'id\u00e9e de ma propre essence repr\u00e9sente ma puissance d'agir, mais ma puissance d'agir n'est que la puissance de Dieu lui-m\u00eame en tant qu'elle s'explique par mon essence. Il n'y a donc pas de joie du troisi\u00e8me genre qui ne s'accompagne de l'id\u00e9e de Dieu comme cause mat\u00e9rielle : \u00ab Du troisi\u00e8me genre de connaissance na\u00eet n\u00e9cessairement l'Amour intellectuel de Dieu ; car de ce genre de connaissance na\u00eet la joie qu'accompagne l'id\u00e9e de Dieu en tant que cause9 _. \u00bb_\n\nOr, comment les joies actives du troisi\u00e8me genre se distinguent-elles de celles du second ? Les joies du second genre sont d\u00e9j\u00e0 actives, parce qu'elles s'expliquent par une id\u00e9e ad\u00e9quate que nous avons. Elles s'expliquent donc par notre puissance de comprendre ou d'agir. Elles impliquent que nous ayons la possession formelle de cette puissance. Mais bien que celle-ci ne paraisse plus susceptible d'augmentation, lui manque encore une certaine qualit\u00e9, nuance qualitative individuelle qui correspond au degr\u00e9 de puissance ou d'intensit\u00e9 de notre essence propre. En effet, tant que nous en restons au second genre de connaissance, l'id\u00e9e ad\u00e9quate que nous avons n'est pas encore une id\u00e9e de nous-m\u00eames, de notre essence, de l'essence de notre corps. Cette restriction para\u00eetra importante si l'on se rappelle quel est le point de d\u00e9part du probl\u00e8me de la connaissance : nous n'avons pas imm\u00e9diatement l'id\u00e9e ad\u00e9quate de nous-m\u00eames ou de notre corps parce que celle-ci n'est en Dieu qu'en tant qu'il est affect\u00e9 par des id\u00e9es d'autres corps ; nous ne connaissons donc notre corps que par des id\u00e9es d'affections, n\u00e9cessairement inad\u00e9quates, et nous ne nous connaissons nous-m\u00eames que par les id\u00e9es de ces id\u00e9es ; quant aux id\u00e9es de corps ext\u00e9rieurs, quant \u00e0 l'id\u00e9e de notre propre corps ou de notre propre esprit, nous ne les _avons_ pas, dans les conditions imm\u00e9diates de notre existence. Or le second genre de connaissance nous donne bien des id\u00e9es ad\u00e9quates ; mais ces id\u00e9es sont seulement celles de propri\u00e9t\u00e9s communes \u00e0 notre corps et \u00e0 des corps ext\u00e9rieurs. Elles sont ad\u00e9quates parce qu'elles sont dans la partie comme dans le tout et parce qu'elles sont en nous, dans notre esprit, comme elles sont dans les id\u00e9es des autres choses. Mais elles ne constituent nullement une id\u00e9e ad\u00e9quate de _nous-m\u00eames_ , ni une id\u00e9e ad\u00e9quate d'une _autre chose_10. Elles s'expliquent par notre essence, mais ne constituent pas elles-m\u00eames une id\u00e9e de cette essence. Au contraire, avec le troisi\u00e8me genre de connaissance, nous formons des id\u00e9es ad\u00e9quates de nous-m\u00eames et des autres choses telles qu'elles sont en Dieu et sont con\u00e7ues par Dieu. Les joies actives qui d\u00e9coulent des id\u00e9es du troisi\u00e8me genre sont donc d'une autre nature que celles qui d\u00e9coulent des id\u00e9es du deuxi\u00e8me. Et, plus g\u00e9n\u00e9ralement, Spinoza est en droit de distinguer deux formes d'activit\u00e9 de l'esprit, deux modes sous lesquels nous sommes actifs et nous nous sentons actifs, deux expressions de notre puissance de comprendre : \u00ab ... Il est de la nature de la raison de concevoir les choses sous une esp\u00e8ce d'\u00e9ternit\u00e9 [second genre], et il appartient aussi \u00e0 la nature de l'esprit de concevoir l'essence du corps sous une esp\u00e8ce d'\u00e9ternit\u00e9 [troisi\u00e8me genre] ; et \u00e0 part ces _deux choses_ , rien d'autre n'appartient \u00e0 l'essence de l'esprit11. \u00bb\n\nToutes les affections, passives ou actives, sont des affections de l'essence dans la mesure o\u00f9 elles remplissent le pouvoir d'\u00eatre affect\u00e9 dans lequel l'essence s'exprime. Mais les affections passives, tristesses ou joies, sont _adventices_ puisqu'elles sont produites du dehors ; les affections actives, les joies actives sont _inn\u00e9es_ parce qu'elles s'expliquent par notre essence ou notre puissance de comprendre12. Pourtant tout se passe comme si l'inn\u00e9 avait deux dimensions diff\u00e9rentes, qui rendent compte des difficult\u00e9s que nous \u00e9prouvons \u00e0 le rejoindre ou \u00e0 le retrouver. En premier lieu, les notions communes sont elles-m\u00eames inn\u00e9es, comme les joies actives qui en d\u00e9coulent. Ce qui ne les emp\u00eache pas de devoir \u00eatre form\u00e9es, et d'\u00eatre form\u00e9es plus ou moins facilement, donc d'\u00eatre plus ou moins communes aux esprits. L'apparente contradiction dispara\u00eet, si l'on consid\u00e8re que nous naissons s\u00e9par\u00e9s de notre puissance d'agir ou de comprendre : nous devons, dans l'existence, conqu\u00e9rir ce qui appartient \u00e0 notre essence. Pr\u00e9cis\u00e9ment, nous ne pouvons former des notions communes, m\u00eame les plus g\u00e9n\u00e9rales, que si nous trouvons un point de d\u00e9part dans des passions joyeuses qui augmentent d'abord notre puissance d'agir. C'est en ce sens que les joies actives qui d\u00e9coulent des notions communes trouvent en quelque sorte leurs causes occasionnelles dans des affections passives de joie : inn\u00e9es en droit, elles n'en d\u00e9pendent pas moins d'affections adventices comme de causes occasionnelles. Mais Dieu lui-m\u00eame dispose imm\u00e9diatement d'une puissance d'agir infinie qui n'est susceptible d'aucune augmentation. Dieu n'\u00e9prouve donc aucune passion, m\u00eame joyeuse, pas plus qu'il n'a d'id\u00e9es inad\u00e9quates. Mais se pose aussi la question de savoir si les notions communes, et les joies actives qui en d\u00e9coulent, sont en Dieu. \u00c9tant des id\u00e9es ad\u00e9quates, les notions communes sont bien en Dieu, mais seulement en tant qu'il a d'abord d'autres id\u00e9es qui les comprennent n\u00e9cessairement (ces autres id\u00e9es seront pour nous celles du troisi\u00e8me genre)13. Si bien que jamais Dieu, ni le Christ qui est l'expression de sa pens\u00e9e, ne pensent _par_ notions communes. Les notions communes ne peuvent donc pas, en Dieu, servir de principes \u00e0 des joies correspondantes \u00e0 celles que nous \u00e9prouvons dans le second genre : Dieu est exempt de joies passives, mais il n'\u00e9prouve m\u00eame pas les joies actives du second genre qui supposent une augmentation de la puissance d'agir comme cause occasionnelle. C'est pourquoi, _d'apr\u00e8s l'id\u00e9e du second genre_ , Dieu n'\u00e9prouve aucun sentiment de joie14.\n\nLes id\u00e9es du troisi\u00e8me genre ne s'expliquent pas seulement par notre essence, elles consistent dans l'id\u00e9e de cette essence elle-m\u00eame et de ses relations (relation avec l'id\u00e9e de Dieu, relations avec les id\u00e9es des autres choses, sous l'esp\u00e8ce de l'\u00e9ternit\u00e9). \u00c0 partir de l'id\u00e9e de notre essence comme cause formelle, \u00e0 partir de l'id\u00e9e de Dieu comme cause mat\u00e9rielle, nous concevons toutes les id\u00e9es telles qu'elles sont en Dieu. Sous le troisi\u00e8me genre de connaissance, nous formons des id\u00e9es et des sentiments actifs qui sont en nous comme ils sont imm\u00e9diatement et \u00e9ternellement en Dieu. Nous pensons comme Dieu pense, nous \u00e9prouvons les sentiments m\u00eames de Dieu. Nous formons l'id\u00e9e de nous-m\u00eames telle qu'elle est en Dieu, et au moins en partie nous formons l'id\u00e9e de Dieu telle qu'elle est en Dieu lui-m\u00eame : les id\u00e9es du troisi\u00e8me genre constituent donc une dimension plus profonde de l'inn\u00e9, et les joies du troisi\u00e8me genre sont les seules vraies affections de l'essence en elle-m\u00eame. Sans doute avons-nous l'air d' _arriver_ au troisi\u00e8me genre de connaissance15. Mais, ici, ce qui nous sert de cause occasionnelle, ce sont les notions communes elles-m\u00eames, donc quelque chose d'ad\u00e9quat et d'actif. Le \u00ab passage \u00bb n'est plus qu'une apparence ; en v\u00e9rit\u00e9, nous nous retrouvons tels que nous sommes imm\u00e9diatement et \u00e9ternellement en Dieu. \u00ab L'esprit poss\u00e8de \u00e9ternellement ces m\u00eames perfections que nous nous sommes figur\u00e9 lui arriver16. \u00bb C'est pourquoi les joies qui suivent des id\u00e9es du troisi\u00e8me genre sont les seules \u00e0 m\u00e9riter le nom de _b\u00e9atitude_ : ce ne sont plus des joies qui augmentent notre puissance d'agir, ni m\u00eame des joies qui supposent encore une telle augmentation, ce sont des joies qui d\u00e9rivent absolument de notre essence, telle qu'elle est en Dieu et est con\u00e7ue par Dieu17.\n\nNous devons encore demander : Quelle est la diff\u00e9rence entre l'id\u00e9e de Dieu du second genre et celle du troisi\u00e8me ? L'id\u00e9e de Dieu n'appartient au second genre que dans la mesure o\u00f9 elle est rapport\u00e9e aux notions communes qui l'expriment. Et les conditions de notre connaissance sont telles que nous \u00ab arrivons \u00bb \u00e0 l'id\u00e9e de Dieu par les notions communes. Mais l'id\u00e9e de Dieu n'est pas en elle-m\u00eame une de ces notions. C'est elle, donc, qui nous fait sortir du second genre de connaissance et nous r\u00e9v\u00e8le un contenu ind\u00e9pendant : non plus des propri\u00e9t\u00e9s communes, mais l'essence de Dieu, mon essence et toutes les autres qui d\u00e9pendent de Dieu. Or, tant que l'id\u00e9e de Dieu se rapporte aux notions communes, elle repr\u00e9sente un \u00eatre souverain qui n'\u00e9prouve aucun amour, aucune joie. Mais, nous d\u00e9terminant au troisi\u00e8me genre, elle re\u00e7oit elle-m\u00eame de nouvelles qualifications qui correspondent \u00e0 ce genre. Les joies actives que nous \u00e9prouvons dans le troisi\u00e8me genre de connaissance sont des joies que Dieu lui-m\u00eame \u00e9prouve, parce que les id\u00e9es dont elles suivent sont en nous comme elles sont \u00e9ternellement et imm\u00e9diatement en Dieu. On ne verra donc nulle contradiction entre les deux amours successivement d\u00e9crites dans le livre V de l' _\u00c9thique_ : amour pour un Dieu qui ne peut pas nous aimer, puisqu'il n'\u00e9prouve aucune joie ; amour pour un Dieu lui-m\u00eame joyeux, qui s'aime et nous aime du m\u00eame amour que celui dont nous l'aimons. Il suffit, comme le contexte l'indique, de rapporter les premiers textes au second genre de connaissance, les autres au troisi\u00e8me genre18.\n\nProc\u00e9dant de l'id\u00e9e de nous-m\u00eames telle qu'elle est en Dieu, nos joies actives sont une partie des joies de Dieu. Notre joie est la joie de Dieu lui-m\u00eame en tant qu'il s'explique par notre essence. Et l'amour du troisi\u00e8me genre que nous \u00e9prouvons pour Dieu est \u00ab une partie de l'amour infini dont Dieu s'aime lui-m\u00eame \u00bb. L'amour que nous \u00e9prouvons pour Dieu, c'est l'amour que Dieu \u00e9prouve pour soi en tant qu'il s'explique par notre propre essence, donc l'amour qu'il \u00e9prouve pour notre essence elle-m\u00eame19. La _b\u00e9atitude_ ne d\u00e9signe pas seulement la possession d'une joie active telle qu'elle est en Dieu, mais celle d'un amour actif tel qu'il est en Dieu20. En tout ceci, le mot partie doit toujours s'interpr\u00e9ter de mani\u00e8re explicative ou expressive : n'est pas une partie ce qui compose, mais ce qui exprime et explique. Notre essence est une partie de Dieu, l'id\u00e9e de notre essence est une partie de l'id\u00e9e de Dieu, mais pour autant que l'essence de Dieu s'explique par la n\u00f4tre. Et c'est dans le troisi\u00e8me genre que le syst\u00e8me de l'expression trouve sa forme finale. La forme finale de l'expression, c'est l'identit\u00e9 de l'affirmation sp\u00e9culative et de l'affirmation pratique, l'identit\u00e9 de l'\u00catre et de la Joie, de la Substance et de la Joie, de Dieu et de la Joie. La joie manifeste le d\u00e9veloppement de la substance elle-m\u00eame, son explication dans les modes et la conscience de cette explication. L'id\u00e9e de Dieu n'est plus simplement exprim\u00e9e par les notions communes en g\u00e9n\u00e9ral, c'est elle qui s'exprime et s'explique dans toutes les essences suivant la loi de production qui leur est propre. Elle s'exprime dans chaque essence en particulier, mais chaque essence comprend toutes les autres essences dans sa loi de production. La joie que nous \u00e9prouvons est la joie que Dieu lui-m\u00eame \u00e9prouve en tant qu'il a l'id\u00e9e de notre essence ; la joie que Dieu \u00e9prouve est celle que nous \u00e9prouvons nous-m\u00eames en tant que nous avons des id\u00e9es telles qu'elles sont en Dieu.\n\nD\u00e8s notre existence dans la dur\u00e9e, donc \u00ab durant \u00bb notre existence elle-m\u00eame, nous pouvons acc\u00e9der au troisi\u00e8me genre de connaissance. Mais nous n'y r\u00e9ussissons que dans un ordre strict, qui repr\u00e9sente la meilleure mani\u00e8re dont notre pouvoir d'\u00eatre affect\u00e9 puisse \u00eatre rempli : 1o) Id\u00e9es inad\u00e9quates qui nous sont donn\u00e9es, et affections passives qui s'ensuivent, les unes augmentant notre puissance d'agir, les autres la diminuant. 2o) Formation des notions communes \u00e0 l'issue d'un effort de s\u00e9lection portant sur les affections passives elles-m\u00eames ; les joies actives du second genre suivent des notions communes, un amour actif suit de l'id\u00e9e de Dieu telle qu'elle se rapporte aux notions communes. 3o) Formation des id\u00e9es ad\u00e9quates du troisi\u00e8me genre, joies actives et amour actif qui suivent de ces id\u00e9es (b\u00e9atitude). Mais, tant que nous existons dans la dur\u00e9e, il est vain d'esp\u00e9rer n'avoir que des joies actives du troisi\u00e8me genre ou, seulement, des affections actives en g\u00e9n\u00e9ral. Nous aurons toujours des passions, et des tristesses avec nos joies passives. Notre connaissance passera toujours par les notions communes. Tout ce \u00e0 quoi nous pouvons nous efforcer, c'est \u00e0 avoir proportionnellement plus de passions joyeuses que de tristesses, plus de joies actives du second genre que de passions, et le plus grand nombre possible de joies du troisi\u00e8me genre. Tout est question de proportion dans les sentiments qui remplissent notre pouvoir d'\u00eatre affect\u00e9 : il s'agit de faire en sorte que les id\u00e9es inad\u00e9quates et les passions n'occupent que _la plus petite partie de nous-m\u00eames_21.\n\nLa dur\u00e9e se rapporte \u00e0 l'existence des modes. On se rappelle que l'existence d'un mode est constitu\u00e9e par des parties extensives qui, sous un certain rapport, sont d\u00e9termin\u00e9es \u00e0 appartenir \u00e0 l'essence de ce mode. C'est pourquoi la dur\u00e9e se mesure par le temps : un corps existe aussi longtemps qu'il poss\u00e8de des parties extensives sous le rapport qui le caract\u00e9rise. D\u00e8s que les rencontres en disposent autrement, le corps lui-m\u00eame cesse d'exister, ses parties formant d'autres corps sous de nouveaux rapports. Il est donc \u00e9vident que nous ne pouvons pas supprimer toute passion durant notre existence : les parties extensives, en effet, sont d\u00e9termin\u00e9es et affect\u00e9es du dehors \u00e0 l'infini. Aux parties du corps, correspondent des _facult\u00e9s_ de l'\u00e2me, facult\u00e9s d'\u00e9prouver des affections passives. Aussi l'imagination correspond-elle \u00e0 l'empreinte actuelle d'un corps sur le n\u00f4tre, la m\u00e9moire \u00e0 la succession des empreintes dans le temps. M\u00e9moire et imagination sont de v\u00e9ritables parties de l'\u00e2me. L'\u00e2me a des parties extensives, qui ne lui appartiennent que dans la mesure o\u00f9 elle est l'id\u00e9e d'un corps, lui-m\u00eame compos\u00e9 de parties extensives22. L'\u00e2me \u00ab dure \u00bb, dans la mesure o\u00f9 elle exprime l'existence actuelle d'un corps qui dure. Et les facult\u00e9s de l'\u00e2me renvoient elles-m\u00eames \u00e0 une puissance, puissance de p\u00e2tir, puissance d'imaginer les choses d'apr\u00e8s les affections qu'elles produisent dans notre corps, donc puissance de concevoir les choses dans la dur\u00e9e et en relation avec le temps23.\n\nLes parties extensives appartiennent \u00e0 l'essence sous un certain rapport et pendant un certain temps ; mais elles ne constituent pas cette essence. L'essence elle-m\u00eame a une tout autre nature. L'essence en elle-m\u00eame est un degr\u00e9 de puissance ou d'intensit\u00e9, une partie intensive. Rien ne nous para\u00eet plus inexact qu'une interpr\u00e9tation math\u00e9matique des essences particuli\u00e8res chez Spinoza. Il est vrai qu'une essence s'exprime dans un rapport, mais elle ne se confond pas avec ce rapport. Une essence particuli\u00e8re est une r\u00e9alit\u00e9 physique ; c'est pourquoi les affections sont des affections de l'essence, et l'essence elle-m\u00eame une essence de corps. Cette r\u00e9alit\u00e9 physique est une r\u00e9alit\u00e9 intensive, une existence intensive. On con\u00e7oit, d\u00e8s lors, que l'essence ne dure pas. La dur\u00e9e se dit en fonction des parties extensives et se mesure au temps pendant lequel ces parties appartiennent \u00e0 l'essence. Mais l'essence en elle-m\u00eame a une r\u00e9alit\u00e9 ou une existence \u00e9ternelle ; elle n'a pas de dur\u00e9e, ni de temps qui marque l'ach\u00e8vement de cette dur\u00e9e (aucune essence ne peut en d\u00e9truire une autre). Spinoza dit exactement que l'essence est _con\u00e7ue_ \u00ab avec une certaine n\u00e9cessit\u00e9 \u00e9ternelle24 \u00bb. Mais cette formule, \u00e0 son tour, n'autorise aucune interpr\u00e9tation intellectualiste ou id\u00e9aliste. Spinoza veut dire seulement qu'une essence particuli\u00e8re n'est pas \u00e9ternelle par elle-m\u00eame. Seule la substance divine est \u00e9ternelle en vertu de soi-m\u00eame ; mais une essence n'est \u00e9ternelle qu'en vertu d'une cause (Dieu), d'o\u00f9 d\u00e9rive son existence ou sa r\u00e9alit\u00e9 d'essence. Elle est donc n\u00e9cessairement con\u00e7ue par cette cause ; elle est donc con\u00e7ue avec la n\u00e9cessit\u00e9 \u00e9ternelle qui d\u00e9rive de cette cause. On ne s'\u00e9tonnera pas que Spinoza, d\u00e8s lors, parle de \u00ab l'id\u00e9e qui exprime l'essence de tel ou tel corps humain sous l'esp\u00e8ce de l'\u00e9ternit\u00e9 \u00bb. Il ne veut pas dire que l'essence du corps n'existe qu'en id\u00e9e. Le tort de l'interpr\u00e9tation id\u00e9aliste est de retourner contre le parall\u00e9lisme un argument qui en fait partie int\u00e9grante, ou de comprendre comme une preuve de l'id\u00e9al un argument de la pure causalit\u00e9. Si une id\u00e9e en Dieu exprime l'essence de tel ou tel corps, c'est parce que Dieu est cause des essences ; il s'ensuit que l'essence est n\u00e9cessairement con\u00e7ue par cette cause25.\n\nLe corps existe et dure pour autant qu'il poss\u00e8de actuellement des parties extensives. Mais il a une essence, qui est comme une partie intensive \u00e9ternelle (degr\u00e9 de puissance). L'\u00e2me elle-m\u00eame a des parties extensives, en tant qu'elle exprime l'existence du corps dans la dur\u00e9e. Mais elle a aussi une partie intensive \u00e9ternelle, qui est comme l'id\u00e9e de l'essence du corps. L'id\u00e9e qui exprime l'essence du corps constitue la partie intensive ou l'essence de l'\u00e2me, n\u00e9cessairement \u00e9ternelle. Sous cet aspect, l'\u00e2me poss\u00e8de une facult\u00e9, c'est-\u00e0-dire une puissance qui s'explique par sa propre essence : puissance active de comprendre, et de comprendre les choses par le troisi\u00e8me genre sous l'esp\u00e8ce de l'\u00e9ternit\u00e9. En tant qu'elle exprime l'existence actuelle du corps dans la dur\u00e9e, l'\u00e2me a la puissance de concevoir les autres corps dans la dur\u00e9e ; en tant qu'elle exprime l'essence du corps, l'\u00e2me a la puissance de concevoir les autres corps sous l'esp\u00e8ce de l'\u00e9ternit\u00e926.\n\nLe Spinozisme affirme donc une distinction de nature entre la dur\u00e9e et l'\u00e9ternit\u00e9. Si Spinoza, dans l' _\u00c9thique_ , \u00e9vite d'employer le concept d' _immortalit\u00e9_ , c'est que celui-ci lui para\u00eet impliquer les plus f\u00e2cheuses confusions. Trois arguments se retrouvent, \u00e0 des titres divers, dans une tradition de l'immortalit\u00e9 qui va de Platon \u00e0 Descartes. En premier lieu, la th\u00e9orie de l'immortalit\u00e9 repose sur un certain postulat de la simplicit\u00e9 de l'\u00e2me : seul le corps est con\u00e7u comme divisible ; l'\u00e2me est immortelle parce qu'indivisible, ses facult\u00e9s n'\u00e9tant pas des parties. En second lieu, l'immortalit\u00e9 de cette \u00e2me absolument simple est con\u00e7ue dans la dur\u00e9e : l'\u00e2me existait d\u00e9j\u00e0 quand le corps n'avait pas commenc\u00e9 d'exister, elle dure lorsque le corps a cess\u00e9 de durer. C'est pourquoi la th\u00e9orie de l'immortalit\u00e9 entra\u00eene souvent l'hypoth\u00e8se d'une m\u00e9moire purement intellectuelle, par laquelle l'\u00e2me s\u00e9par\u00e9e du corps peut \u00eatre consciente de sa propre dur\u00e9e. Enfin, l'immortalit\u00e9 ainsi d\u00e9finie ne peut \u00eatre l'objet d'une exp\u00e9rience directe tant que dure le corps. Sous quelle forme survit-elle au corps, quelles sont les modalit\u00e9s de la survie, quelles sont les facult\u00e9s de l'\u00e2me une fois d\u00e9sincarn\u00e9e ? Seule une _r\u00e9v\u00e9lation_ pourrait nous le dire maintenant.\n\nCes trois th\u00e8ses rencontrent chez Spinoza un adversaire d\u00e9clar\u00e9. La th\u00e9orie de l'immortalit\u00e9 ne se s\u00e9pare pas d'une confusion de la dur\u00e9e et de l'\u00e9ternit\u00e9. D'abord, le postulat d'une simplicit\u00e9 absolue de l'\u00e2me ne se s\u00e9pare pas lui-m\u00eame de l'id\u00e9e confuse d'une union de l'\u00e2me et du corps. Rapportant l'\u00e2me au corps, on oppose la simplicit\u00e9 de l'\u00e2me prise en un tout et la divisibilit\u00e9 du corps pris lui-m\u00eame en un tout. On comprend que le corps a des parties extensives en tant qu'il existe, mais on ne comprend pas que l'\u00e2me poss\u00e8de aussi de telles parties pour autant qu'elle est l'id\u00e9e du corps existant. On comprend (plus ou moins bien) que l'\u00e2me a une partie intensive absolument simple et \u00e9ternelle qui constitue son essence, mais on ne comprend pas qu'elle exprime ainsi l'essence du corps, non moins simple et \u00e9ternelle. En second lieu, l'hypoth\u00e8se de l'immortalit\u00e9 nous invite \u00e0 penser en termes de succession, et nous rend incapables de concevoir l'\u00e2me comme un compos\u00e9 de coexistences. Nous ne comprenons pas que, tant que le corps existe, la dur\u00e9e et l'\u00e9ternit\u00e9 \u00ab coexistent \u00bb elles-m\u00eames dans l'\u00e2me comme deux \u00e9l\u00e9ments qui diff\u00e9rent en nature. L'\u00e2me dure en tant que lui appartiennent des parties extensives qui ne constituent pas son essence. L'\u00e2me est \u00e9ternelle en tant que lui appartient une partie intensive qui d\u00e9finit son essence. Nous ne devons pas penser que l'\u00e2me dure au-del\u00e0 du corps : elle dure autant que le corps dure lui-m\u00eame, elle est \u00e9ternelle en tant qu'elle exprime l'essence du corps. Tant que l'\u00e2me est l'id\u00e9e du corps existant, coexistent en elle des parties extensives qui lui appartiennent dans la dur\u00e9e, et une partie intensive qui la constitue dans l'\u00e9ternit\u00e9. Enfin, nous n'avons besoin de nulle r\u00e9v\u00e9lation pour savoir sous quels modes et comment l'\u00e2me survit. L'\u00e2me reste \u00e9ternellement ce qu'elle est d\u00e9j\u00e0 dans son essence, durant l'existence du corps : partie intensive, degr\u00e9 de puissance ou puissance de comprendre, id\u00e9e qui exprime l'essence du corps sous l'esp\u00e8ce de l'\u00e9ternit\u00e9. Aussi l'\u00e9ternit\u00e9 de l'\u00e2me est-elle objet d'une exp\u00e9rience directe. Pour sentir et exp\u00e9rimenter que nous sommes \u00e9ternels, il suffit d'entrer dans le troisi\u00e8me genre de connaissance, c'est-\u00e0-dire de former l'id\u00e9e de nous-m\u00eames telle qu'elle est en Dieu. Cette id\u00e9e est pr\u00e9cis\u00e9ment celle qui exprime l'essence du corps ; dans la mesure o\u00f9 nous la formons, dans la mesure o\u00f9 nous l'avons, nous _exp\u00e9rimentons_ que nous sommes \u00e9ternels27.\n\nQue se passe-t-il quand nous mourons ? La mort est une soustraction, un retranchement. Nous perdons toutes les parties extensives qui nous appartenaient sous un certain rapport ; notre \u00e2me perd toutes les facult\u00e9s qu'elle ne poss\u00e9dait qu'en tant qu'elle exprimait l'existence d'un corps lui-m\u00eame dou\u00e9 de parties extensives28. Mais ces parties et ces facult\u00e9s avaient beau appartenir \u00e0 notre essence, elles ne _constituaient_ rien de cette essence : notre essence en tant que telle ne perd rien en perfection quand nous perdons en extension les parties qui composaient notre existence. De toutes mani\u00e8res, la partie de nous-m\u00eames qui demeure, quelle qu'en soit la grandeur (c'est-\u00e0-dire le degr\u00e9 de puissance ou la quantit\u00e9 intensive), est plus parfaite que toutes les parties extensives qui p\u00e9rissent, et conserve toute sa perfection quand disparaissent ces parties extensives29. Bien plus, quand notre corps a cess\u00e9 d'exister, quand l'\u00e2me a perdu toutes ses parties qui se rapportent \u00e0 l'existence du corps, nous ne sommes plus en \u00e9tat d'\u00e9prouver des affections passives30. Notre essence cesse d'\u00eatre maintenue dans un \u00e9tat d'enveloppement, nous ne pouvons plus \u00eatre s\u00e9par\u00e9s de notre puissance : seule demeure en effet notre puissance de comprendre ou d'agir31. Les id\u00e9es que nous avons sont n\u00e9cessairement des id\u00e9es ad\u00e9quates du troisi\u00e8me genre, telles qu'elles sont en Dieu. Notre essence exprime ad\u00e9quatement l'essence de Dieu, les affections de notre essence expriment ad\u00e9quatement cette essence. _Nous sommes devenus totalement expressifs_ , plus rien ne subsiste en nous qui soit \u00ab envelopp\u00e9 \u00bb ou simplement \u00ab indiqu\u00e9 \u00bb. Tant que nous existions, nous ne pouvions avoir qu'un certain nombre d'affections actives du troisi\u00e8me genre, elles-m\u00eames en relation avec des affections actives du second genre, elles-m\u00eames en relation avec des affections passives. Nous ne pouvions esp\u00e9rer qu'une b\u00e9atitude partielle. Mais tout se passe comme si la mort nous mettait dans une situation telle que nous ne pouvions plus \u00eatre affect\u00e9s que par des affections du troisi\u00e8me genre, qui s'expliquent elles-m\u00eames par notre essence.\n\nIl est vrai que ce point soul\u00e8ve encore beaucoup de probl\u00e8mes. 1o) En quel sens, apr\u00e8s la mort, sommes-nous encore affect\u00e9s ? Notre \u00e2me a perdu tout ce qui lui appartient en tant qu'elle est l'id\u00e9e d'un corps existant. Mais demeure l'id\u00e9e de l'essence de notre corps existant. Mais demeure l'id\u00e9e de l'essence de notre corps telle qu'elle est en Dieu. Nous avons nous-m\u00eames l'id\u00e9e de cette id\u00e9e telle qu'elle est en Dieu. Notre \u00e2me est donc affect\u00e9e par l'id\u00e9e de soi, par l'id\u00e9e de Dieu, par les id\u00e9es des autres choses sous l'esp\u00e8ce de l'\u00e9ternit\u00e9. Comme toutes les essences conviennent avec chacune, comme elles ont pour cause Dieu qui les comprend toutes dans la production de chacune, les affections qui d\u00e9coulent des id\u00e9es du troisi\u00e8me genre sont n\u00e9cessairement des affections actives et intenses, qui s'expliquent par l'essence de celui qui les \u00e9prouve, en m\u00eame temps qu'elles expriment l'essence de Dieu. 2o) Mais si, apr\u00e8s la mort, nous sommes encore affect\u00e9s, n'est-ce pas que notre pouvoir d'\u00eatre affect\u00e9, notre rapport caract\u00e9ristique, subsistent eux-m\u00eames avec notre essence ? En effet, notre rapport peut \u00eatre dit d\u00e9truit ou d\u00e9compos\u00e9, mais seulement en ce sens qu'il ne subsume plus de parties extensives. Les parties extensives qui nous appartenaient sont maintenant d\u00e9termin\u00e9es \u00e0 entrer sous d'autres rapports incomposables avec le n\u00f4tre. Mais le rapport qui nous caract\u00e9rise n'en a pas moins une v\u00e9rit\u00e9 \u00e9ternelle en tant que notre essence s'exprime en lui. C'est le rapport dans sa v\u00e9rit\u00e9 \u00e9ternelle qui demeure avec l'essence. (C'est pourquoi les notions communes restent comprises dans les id\u00e9es des essences.) De m\u00eame, notre pouvoir d'\u00eatre affect\u00e9 peut \u00eatre dit d\u00e9truit, mais dans la mesure o\u00f9 il ne peut plus \u00eatre effectu\u00e9 par des affections passives32. Il n'en a pas moins une puissance \u00e9ternelle, qui est identique \u00e0 notre puissance d'agir ou de comprendre. C'est le pouvoir d'\u00eatre affect\u00e9, dans sa puissance \u00e9ternelle, qui demeure avec l'essence.\n\nMais comment concevoir que, de toutes fa\u00e7ons, nous jouissions apr\u00e8s la mort d'affections actives du troisi\u00e8me genre, comme si nous retrouvions n\u00e9cessairement ce qui nous est \u00e9ternellement inn\u00e9 ? Leibniz adresse plusieurs critiques \u00e0 la conception spinoziste de l'\u00e9ternit\u00e9 : il lui reproche son g\u00e9om\u00e9trisme, les id\u00e9es d'essences \u00e9tant analogues \u00e0 des formes ou figures math\u00e9matiques ; il lui reproche d'avoir con\u00e7u une \u00e9ternit\u00e9 sans m\u00e9moire et sans imagination, tout au plus l'\u00e9ternit\u00e9 d'un cercle ou d'un triangle. Mais une troisi\u00e8me critique de Leibniz nous semble plus importante, parce qu'elle pose le vrai probl\u00e8me final du Spinozisme : si Spinoza avait raison, il n'y aurait pas lieu de se perfectionner pour laisser apr\u00e8s soi une essence \u00e9ternelle d'autant plus parfaite (comme si cette essence ou id\u00e9e platonique \u00ab n'\u00e9tait d\u00e9j\u00e0 dans la nature, soit que je t\u00e2che de lui ressembler ou non, et comme s'il me servirait apr\u00e8s ma mort, si je ne suis plus rien, d'avoir ressembl\u00e9 \u00e0 une telle id\u00e9e \u00bb)33. En effet, la question est : de quoi nous sert l'existence si, de toutes mani\u00e8res, nous rejoignons notre essence apr\u00e8s la mort, dans de telles conditions que nous \u00e9prouvons intens\u00e9ment toutes les affections actives qui lui correspondent ? Nous ne perdons rien en perdant l'existence : nous ne perdons que des parties extensives. Mais de quoi sert notre effort durant l'existence si notre essence est de toutes fa\u00e7ons ce qu'elle est, degr\u00e9 de puissance indiff\u00e9rent aux parties extensives qui ne lui furent rapport\u00e9es que du dehors et temporairement ?\n\nEn fait, selon Spinoza, notre pouvoir d'\u00eatre affect\u00e9 ne sera pas rempli (apr\u00e8s la mort) par des affections actives du troisi\u00e8me genre si nous n'avons pas r\u00e9ussi durant l'existence elle-m\u00eame \u00e0 \u00e9prouver proportionnellement un maximum d'affections actives du second genre et, d\u00e9j\u00e0, du troisi\u00e8me. C'est en ce sens que Spinoza peut estimer qu'il conserve enti\u00e8rement le contenu positif de la notion de salut. L'existence m\u00eame est encore con\u00e7ue comme une sorte d'\u00e9preuve. Non pas une \u00e9preuve morale, il est vrai, mais une \u00e9preuve physique ou chimique, comme celle des artisans qui v\u00e9rifient la qualit\u00e9 d'une mati\u00e8re, d'un m\u00e9tal ou d'un vase.\n\nDans l'existence, nous sommes compos\u00e9s d'une partie intensive \u00e9ternelle, qui constitue notre essence, et de parties extensives qui nous appartiennent dans le temps sous un certain rapport. Ce qui compte est l'importance respective de ces deux sortes d'\u00e9l\u00e9ments. Supposons que nous r\u00e9ussissions, d\u00e8s notre existence, \u00e0 \u00e9prouver des affections actives : nos parties extensives elles-m\u00eames sont affect\u00e9es par des affections qui s'expliquent par notre seule essence ; les passions subsistantes sont proportionnellement moindres que les affections actives. C'est-\u00e0-dire : notre pouvoir d'\u00eatre affect\u00e9 se trouve proportionnellement rempli par un plus grand nombre d'affections actives que d'affections passives. Or les affections actives s'expliquent par notre essence ; les affections passives s'expliquent par le jeu infini de d\u00e9terminations extrins\u00e8ques des parties extensives. On en conclut que, des deux \u00e9l\u00e9ments qui nous composent, la partie intensive de nous-m\u00eames a pris relativement beaucoup plus d'importance que les parties extensives. \u00c0 la limite, quand nous mourons, ce qui p\u00e9rit n'est \u00ab d'aucune importance eu \u00e9gard \u00e0 ce qui persiste34 \u00bb. D'autant plus nous connaissons de choses par le deuxi\u00e8me et troisi\u00e8me genres, d'autant plus grande est relativement la partie de nous-m\u00eames \u00e9ternelle35. Il va de soi que cette partie \u00e9ternelle, prise en elle-m\u00eame ind\u00e9pendamment des parties extensives qui s'y ajoutent pour composer notre existence, est un absolu. Mais supposons que, durant notre existence, nous restions remplis et d\u00e9termin\u00e9s par des affections passives. Des deux \u00e9l\u00e9ments qui nous composent, les parties extensives auront relativement plus d'importance que la partie intensive \u00e9ternelle. Nous perdons d'autant plus en mourant ; c'est pourquoi seul craint la mort celui qui a quelque chose \u00e0 en craindre, celui qui perd relativement davantage en mourant36. Notre essence n'en reste pas moins l'absolu qu'elle est en elle-m\u00eame ; l'id\u00e9e de notre essence n'en reste pas moins ce qu'elle est absolument en Dieu. Mais le pouvoir d'\u00eatre affect\u00e9 qui lui correspond \u00e9ternellement reste vide : ayant perdu nos parties extensives, nous avons perdu toutes nos affections qui s'expliquaient par elles. Or nous n'avons pas d'autres affections. Quand nous mourons, notre essence demeure, mais comme un abstrait ; notre essence reste inaffect\u00e9e.\n\nAu contraire, quand nous avons su faire de la partie intensive l'\u00e9l\u00e9ment le plus important de nous-m\u00eames. En mourant, nous perdons peu de choses : nous perdons les passions qui subsistaient en nous, puisque celles-ci s'expliquaient par les parties extensives ; dans une certaine mesure, nous perdons aussi les notions communes et les affections actives du second genre, qui n'ont de valeur autonome, en effet, qu'en tant qu'elles s'appliquent \u00e0 l'existence ; enfin, les affections actives du troisi\u00e8me genre ne peuvent plus s'imposer aux parties extensives puisque celles-ci ne nous appartiennent plus. Mais notre pouvoir d'\u00eatre affect\u00e9 subsiste \u00e9ternellement, accompagnant notre essence et l'id\u00e9e de notre essence ; or ce pouvoir est n\u00e9cessairement et absolument rempli par les affections du troisi\u00e8me genre. Durant notre existence, nous avons fait de notre partie intensive la part relativement la plus importante de nous-m\u00eames ; apr\u00e8s notre mort, les affections actives qui s'expliquent par cette partie remplissent absolument notre pouvoir d'\u00eatre affect\u00e9 ; ce qui reste de nous-m\u00eames est absolument effectu\u00e9. Notre essence telle qu'elle est en Dieu, et l'id\u00e9e de notre essence telle qu'elle est con\u00e7ue par Dieu, se trouvent enti\u00e8rement affect\u00e9es.\n\nIl n'y a jamais de sanctions morales d'un Dieu justicier, ni ch\u00e2timents ni r\u00e9compenses, mais des cons\u00e9quences naturelles de notre existence. Il est vrai que, durant notre existence, notre pouvoir d'\u00eatre affect\u00e9 se trouve toujours et n\u00e9cessairement rempli : mais soit par des affections passives, soit par des affections actives. Or si notre pouvoir, tant que nous existons, est enti\u00e8rement rempli par des affections passives, il restera vide, et notre essence abstraite, une fois que nous aurons cess\u00e9 d'exister. Il sera absolument effectu\u00e9, par des affections du troisi\u00e8me genre, si nous l'avons proportionnellement rempli par un maximum d'affections actives. D'o\u00f9 l'importance de cette \u00ab \u00e9preuve \u00bb de l'existence : existant, nous devons s\u00e9lectionner les passions joyeuses, car seules elles nous introduisent aux notions communes et aux joies actives qui en d\u00e9coulent ; et nous devons nous servir des notions communes comme d'un principe qui nous introduit d\u00e9j\u00e0 aux id\u00e9es et aux joies du troisi\u00e8me genre. Alors, apr\u00e8s la mort, notre essence aura toutes les affections dont elle est capable ; et toutes ces affections seront du troisi\u00e8me genre. Telle est la voie difficile du salut. La plupart des hommes, la plupart du temps, restent fix\u00e9s aux passions tristes, qui les s\u00e9parent de leur essence et la r\u00e9duisent \u00e0 l'\u00e9tat d'abstraction. La voie du salut est la voie m\u00eame de l'expression : devenir expressif, c'est-\u00e0-dire devenir actif \u2013 exprimer l'essence de Dieu, \u00eatre soi-m\u00eame une id\u00e9e par laquelle l'essence de Dieu s'explique, avoir des affections qui s'expliquent par notre propre essence et qui expriment l'essence de Dieu.\n\n* * *\n\n1.  En _\u00c9_ , V, 36, sc., Spinoza oppose la d\u00e9monstration _g\u00e9n\u00e9rale_ du second genre \u00e0 la conclusion _singuli\u00e8re_ du troisi\u00e8me genre.\n\n2.  _\u00c9_ , V, 37, sc. Seuls des modes existants peuvent se d\u00e9truire, aucune essence ne peut en d\u00e9truire une autre.\n\n3.  Cf. _\u00c9_ , V, 25-27.\n\n4.  _\u00c9_ , V, 22, dem., et 36, prop.\n\n5.  Cf. _\u00c9_ , V, 36, sc. (tout le contexte prouve qu'il s'agit pour chacun de sa propre essence, de l'essence de son propre corps : cf. V, 30, prop. et dem.).\n\n6.  _\u00c9_ , V, 29, prop.\n\n7.  _\u00c9_ , V, 31, prop : \u00ab Le troisi\u00e8me genre de connaissance d\u00e9pend de l'esprit comme de sa cause formelle, en tant que l'esprit lui-m\u00eame est \u00e9ternel. \u00bb\n\n8.  _\u00c9_ , V, 27, dem. Celui qui conna\u00eet par le troisi\u00e8me genre \u00ab est affect\u00e9 de la plus grande joie ( _summa laetitia_ ) \u00bb.\n\n9.  _\u00c9_ , V, 32, cor.\n\n10.  C'est pourquoi les notions communes en tant que telles ne constituent l'essence d'aucune chose singuli\u00e8re : cf. _\u00c9_ , II, 37, prop. Et en V, 41, dem., Spinoza rappelle que le second genre ne nous donne aucune id\u00e9e de l'essence \u00e9ternelle de l'esprit.\n\n11.  _\u00c9_ , V, 29, dem. Il y a donc ici deux esp\u00e8ces d'\u00e9ternit\u00e9, l'une d\u00e9finie par la _pr\u00e9sence_ de la notion commune, l'autre par l' _existence_ de l'essence singuli\u00e8re.\n\n12.  Sur les affections de l'essence en g\u00e9n\u00e9ral, et sur l'adventice et l'inn\u00e9, cf. _\u00c9_ , III, explication de la d\u00e9finition du d\u00e9sir.\n\n13.  D'apr\u00e8s _\u00c9_ , II, 38 et 39, dem., les notions communes sont bien en Dieu. Mais seulement en tant qu'elles sont comprises dans les id\u00e9es de choses singuli\u00e8res (id\u00e9es de nous-m\u00eames et des autres choses) qui sont elles-m\u00eames en Dieu. Il n'en est pas ainsi pour nous : les notions communes sont premi\u00e8res dans l'ordre de notre connaissance. C'est pourquoi elles sont en nous source d'affections sp\u00e9ciales (les joies du second genre). Dieu au contraire n'\u00e9prouve d'affections que du troisi\u00e8me genre.\n\n14.  Cf. _\u00c9_ , V, 14-20.\n\n15.  _\u00c9_ , V, 31, sc. : \u00ab Bien que nous soyons maintenant certains que l'esprit est \u00e9ternel en tant qu'il con\u00e7oit les choses sous l'esp\u00e8ce de l'\u00e9ternit\u00e9, cependant, pour expliquer plus facilement et faire mieux comprendre ce que nous voulons montrer, nous le consid\u00e9rons comme s'il commen\u00e7ait maintenant d'\u00eatre, et de comprendre les choses sous l'esp\u00e8ce de l'\u00e9ternit\u00e9... \u00bb\n\n16.  _\u00c9_ , V, 33, sc.\n\n17.  _\u00c9_ , V, 33, sc.\n\n18.  Amour envers Dieu, du second genre : _\u00c9_ , V, 14-20. Amour de Dieu, du troisi\u00e8me genre : _\u00c9_ , V, 32-37.\n\n19.  _\u00c9_ , V, 36, prop. et cor.\n\n20.  _\u00c9_ , V, 36, sc.\n\n21.  Cf. _\u00c9_ , V, 20, sc. ; 38, dem.\n\n22.  Sur les parties de l'\u00e2me, cf. _\u00c9_ , II, 15. Sur l'assimilation des facult\u00e9s \u00e0 des parties, cf. _\u00c9_ , V, 40, cor.\n\n23.  _\u00c9_ , V, 23, sc., et 29, dem. (cette facult\u00e9 de p\u00e2tir, d'imaginer ou de concevoir dans la dur\u00e9e est bien une _puissance_ , parce qu'elle \u00ab enveloppe \u00bb l'essence ou la puissance d'agir).\n\n24.  _\u00c9_ , V, 22, dem.\n\n25.  _\u00c9_ , V, 22, dem. Cette d\u00e9monstration se r\u00e9clame pr\u00e9cis\u00e9ment de l'axiome du parall\u00e9lisme suivant lequel la connaissance de l'effet d\u00e9pend de la connaissance de la cause et l'enveloppe. La formule de Spinoza _species aeternitatis_ d\u00e9signe \u00e0 la fois l' _esp\u00e8ce_ d'\u00e9ternit\u00e9 qui d\u00e9coule d'une cause, et la _conception_ intellectuelle qui en est ins\u00e9parable.\n\n26.  _\u00c9_ , V, 29, prop. et dem.\n\n27.  _\u00c9_ , V, 23, sc. Cette exp\u00e9rience appartient n\u00e9cessairement au troisi\u00e8me genre ; car le second genre ne poss\u00e8de pas l'id\u00e9e ad\u00e9quate de l'essence de notre corps, et ne nous fait pas encore savoir que notre esprit est \u00e9ternel (cf. V, 41, dem.).\n\n28.  _\u00c9_ , V, 21, prop. : \u00ab L'esprit ne peut rien imaginer et ne peut se souvenir des choses pass\u00e9es si ce n'est pendant la dur\u00e9e du corps. \u00bb\n\n29.  _\u00c9_ , V, 40, cor. : \u00ab La partie de l'esprit qui persiste, de quelque grandeur qu'elle soit, est plus parfaite que l'autre. \u00bb\n\n30.  _\u00c9_ , V, 34, prop. : \u00ab L'esprit n'est soumis que pendant la dur\u00e9e du corps aux sentiments qui se rapportent \u00e0 des passions. \u00bb\n\n31.  _\u00c9_ , V, 40, cor. : \u00ab La partie \u00e9ternelle de l'esprit est l'entendement, par lequel seul nous sommes dits agir. Quant \u00e0 cette partie que nous avons montr\u00e9 qui p\u00e9rit, c'est l'imagination elle-m\u00eame, par laquelle seule nous sommes dits p\u00e2tir. \u00bb\n\n32.  En _\u00c9_ , IV, 39, dem. et sc., Spinoza dit que la mort d\u00e9truit le corps, donc \u00ab nous rend tout \u00e0 fait inaptes \u00e0 pouvoir \u00eatre affect\u00e9 \u00bb. Mais, comme le contexte l'indique, il s'agit des affections passives produites par d'autres corps existants.\n\n33.  LEIBNIZ, _Lettre au Landgrave_ , 14 ao\u00fbt 1683. Cf. Foucher de Careil, _R\u00e9futation in\u00e9dite de Spinoza par Leibniz_ (Paris 1854). En faisant comme si l'\u00e9ternit\u00e9 de l'\u00e2me spinoziste \u00e9tait semblable \u00e0 celle d'une v\u00e9rit\u00e9 math\u00e9matique, Leibniz n\u00e9glige toutes les diff\u00e9rences entre le troisi\u00e8me genre et le second.\n\n34.  _\u00c9_ , V, 38 sc. Notre effort durant l'existence est ainsi d\u00e9fini, _\u00c9_ , V, 39, sc. : former notre corps de telle mani\u00e8re qu'il se rapporte \u00e0 un esprit au plus haut point conscient de soi-m\u00eame, de Dieu et des choses. Alors, ce qui concerne la m\u00e9moire et l'imagination sera \u00ab \u00e0 peine de quelque importance eu \u00e9gard \u00e0 l'entendement \u00bb.\n\n35.  _\u00c9_ , V, 38, dem. : \u00ab D'autant plus de choses comprend l'esprit par le deuxi\u00e8me et le troisi\u00e8me genres de connaissance, d'autant plus grande en est la partie qui reste indemne. \u00bb\n\n36.  _\u00c9_ , V, 38, prop. et sc.\n\n# CONCLUSION\n\n# TH\u00c9ORIE DE L'EXPRESSION\n\n# CHEZ LEIBNIZ ET SPINOZA\n\n# _L'expressionnisme en philosophie_\n\nLa force d'une philosophie se mesure aux concepts qu'elle cr\u00e9e, ou dont elle renouvelle le sens, et qui imposent un nouveau d\u00e9coupage aux choses et aux actions. Il arrive que ces concepts soient appel\u00e9s par le temps, charg\u00e9s d'un sens collectif conforme aux exigences d'une \u00e9poque, et soient d\u00e9couverts, cr\u00e9\u00e9s ou recr\u00e9\u00e9s par plusieurs auteurs \u00e0 la fois. Il en est ainsi pour Spinoza et Leibniz, et le concept d'expression. Ce concept prend sur soi la force d'une r\u00e9action anticart\u00e9sienne men\u00e9e par ces deux auteurs, de deux points de vue tr\u00e8s diff\u00e9rents. Il implique une red\u00e9couverte de la Nature et de sa puissance, une re-cr\u00e9ation de la logique et de l'ontologie : un nouveau \u00ab mat\u00e9rialisme \u00bb et un nouveau \u00ab formalisme \u00bb. Le concept d'expression s'applique \u00e0 l'\u00catre d\u00e9termin\u00e9 comme Dieu, pour autant que Dieu s'exprime dans le monde. Il s'applique aux id\u00e9es d\u00e9termin\u00e9es comme vraies, pour autant que les id\u00e9es vraies expriment Dieu et le monde. Il s'applique enfin aux individus d\u00e9termin\u00e9s comme essences singuli\u00e8res, pour autant que les essences singuli\u00e8res s'expriment dans les id\u00e9es. Si bien que les trois d\u00e9terminations fondamentales : _\u00eatre, conna\u00eetre, agir ou produire_ , sont mesur\u00e9es et syst\u00e9matis\u00e9es sous ce concept. \u00catre, conna\u00eetre, agir sont les esp\u00e8ces de l'expression. C'est l'\u00e2ge de la \u00ab raison suffisante \u00bb : les trois branches de la raison suffisante, _ratio essendi, ratio cognoscendi, ratio fiendi ou agendi_ , trouvent dans l'expression leur racine commune.\n\nLe concept d'expression, pourtant, tel qu'il est red\u00e9couvert par Spinoza et par Leibniz, n'est pas nouveau : il a d\u00e9j\u00e0 une longue histoire philosophique. Mais une histoire un peu cach\u00e9e, un peu maudite. Nous avons essay\u00e9 de montrer en effet comment le th\u00e8me de l'expression se glissait dans les deux grandes traditions th\u00e9ologiques de l'\u00e9manation et de la cr\u00e9ation. Il n'intervient pas comme un troisi\u00e8me concept rivalisant du dehors avec ceux-l\u00e0. Il intervient plut\u00f4t \u00e0 un moment de leur d\u00e9veloppement, et risque toujours de les d\u00e9tourner, de les confisquer \u00e0 son profit. Bref c'est un concept proprement philosophique, au contenu immanent, qui s'immisce dans les concepts transcendants d'une th\u00e9ologie \u00e9manative ou cr\u00e9ationniste. Il apporte avec lui le \u00ab danger \u00bb proprement philosophique : le panth\u00e9isme ou l'immanence \u2013 immanence de l'expression dans ce qui s'exprime, et de l'exprim\u00e9 dans l'expression. Il pr\u00e9tend p\u00e9n\u00e9trer dans le plus profond, dans les \u00ab arcanes \u00bb, suivant un mot que Leibniz aimait. Il redonne \u00e0 la nature une \u00e9paisseur qui lui est propre, et en m\u00eame temps rend l'homme capable de p\u00e9n\u00e9trer dans cette \u00e9paisseur. Il rend l'homme ad\u00e9quat \u00e0 Dieu, et d\u00e9tenteur d'une nouvelle logique : automate spirituel, \u00e9gal \u00e0 la combinatoire du monde. N\u00e9 dans les traditions de l'\u00e9manation et de la cr\u00e9ation, il s'en fait deux ennemies, parce qu'il conteste aussi bien la transcendance d'un Un sup\u00e9rieur \u00e0 l'\u00eatre que la transcendance d'un \u00catre sup\u00e9rieur \u00e0 la cr\u00e9ation. Tout concept poss\u00e8de en soi virtuellement un appareil m\u00e9taphorique. L'appareil m\u00e9taphorique de l'expression, c'est le miroir et le germe1. L'expression comme _ratio essendi_ , se r\u00e9fl\u00e9chit dans le miroir comme _ratio cognoscendi_ , et se reproduit dans le germe comme _ratio fiendi._ Mais voil\u00e0 que le miroir semble absorber, et l'\u00eatre qui se r\u00e9fl\u00e9chit en lui, et l'\u00eatre qui regarde l'image. Le germe, ou le rameau, semble absorber, et l'arbre dont il provient, et l'arbre qui en provient. Et quelle est cette \u00e9trange existence, telle qu'elle est \u00ab prise \u00bb dans le miroir, telle qu'elle est impliqu\u00e9e, envelopp\u00e9e dans le germe \u2013 bref _l'exprim\u00e9_ , entit\u00e9 dont on peut dire \u00e0 peine qu'elle existe ? Nous avons vu que le concept d'expression avait comme deux sources : l'une ontologique, qui concerne _l'expression de Dieu_ , qui na\u00eet \u00e0 l'abri des traditions de l'\u00e9manation et de la cr\u00e9ation, mais qui les conteste profond\u00e9ment ; l'autre, logique, qui concerne _l'exprim\u00e9 des propositions_ , qui na\u00eet \u00e0 l'abri de la logique aristot\u00e9licienne, mais la conteste et la bouleverse. Toutes deux se rejoignent dans le probl\u00e8me des Noms divins, du Logos ou du Verbe.\n\nSi Leibniz et Spinoza au XVIIe si\u00e8cle, l'un \u00e0 partir d'une tradition chr\u00e9tienne, l'autre \u00e0 partir d'une tradition juive, retrouvent le concept d'expression et lui donnent une nouvelle lumi\u00e8re, c'est \u00e9videmment dans un contexte qui est celui de leur temps, en fonction de probl\u00e8mes qui sont ceux de leur syst\u00e8me respectif. Essayons d'abord de d\u00e9gager ce qu'il y a de commun dans les deux syst\u00e8mes, et pour quelles raisons ils r\u00e9inventent le concept d'expression.\n\nCe qu'ils reprochent \u00e0 Descartes, concr\u00e8tement, c'est d'avoir fait une philosophie trop \u00ab rapide \u00bb ou trop \u00ab facile \u00bb. Dans tous les domaines, Descartes va si vite qu'il laisse \u00e9chapper la raison suffisante, l'essence ou vraie nature : partout, il en reste au relatif. _D'abord \u00e0 propos de Dieu_ : la preuve ontologique de Descartes repose sur l'infiniment parfait, et se h\u00e2te de conclure ; mais l'infiniment parfait est un \u00ab propre \u00bb, tout \u00e0 fait insuffisant pour montrer quelle est la \u00ab nature \u00bb de Dieu et comment cette nature est possible. De m\u00eame les preuves a posteriori de Descartes reposent sur la consid\u00e9ration des quantit\u00e9s de r\u00e9alit\u00e9 donn\u00e9es, et ne s'\u00e9l\u00e8vent pas jusqu'\u00e0 un principe dynamique dont celles-ci d\u00e9pendent. _Ensuite \u00e0 propos des id\u00e9es_ : Descartes d\u00e9couvre les crit\u00e8res du clair et du distinct ; mais le \u00ab clair-et-distinct \u00bb est encore un propre, une d\u00e9termination extrins\u00e8que de l'id\u00e9e qui ne nous renseigne pas sur la nature et la possibilit\u00e9 de la chose en id\u00e9e, ni de la pens\u00e9e comme telle. Descartes en reste au contenu repr\u00e9sentatif de l'id\u00e9e et \u00e0 la forme de la conscience psychologique qui la pense : il rate ainsi le v\u00e9ritable contenu immanent de l'id\u00e9e, comme la vraie forme logique, et l'unit\u00e9 des deux (l'automate spirituel). Il nous dit que le vrai est pr\u00e9sent dans l'id\u00e9e claire et distincte, mais qu'est-ce qui est pr\u00e9sent dans l'id\u00e9e vraie ? \u00c0 quel point d\u00e9j\u00e0 ce second courant critique rejoint le premier, on le voit ais\u00e9ment : car, si l'on en reste au clair-et-distinct, on ne peut mesurer les id\u00e9es entre elles et les comparer aux choses que par la consid\u00e9ration des quantit\u00e9s de r\u00e9alit\u00e9. Ne disposant que d'un caract\u00e8re extrins\u00e8que de l'id\u00e9e, on n'atteint dans l'\u00catre que des caract\u00e8res eux-m\u00eames extrins\u00e8ques. Bien plus, la distinction comme norme de l'id\u00e9e pr\u00e9juge de l'\u00e9tat des distinctions entre choses repr\u00e9sent\u00e9es dans l'id\u00e9e : c'est en rapport avec le crit\u00e8re du clair et du distinct que Descartes, dans tout le tr\u00e9sor des distinctions scolastiques, ne retient que la distinction r\u00e9elle, selon lui n\u00e9cessairement num\u00e9rique, la distinction de raison, selon lui n\u00e9cessairement abstraite, la distinction modale, selon lui n\u00e9cessairement accidentelle. _Enfin \u00e0 propos des individus et de leurs actions_ : Descartes interpr\u00e8te l'individu humain comme le compos\u00e9 r\u00e9el d'une \u00e2me et d'un corps, c'est-\u00e0-dire de deux termes h\u00e9t\u00e9rog\u00e8nes cens\u00e9s agir r\u00e9ellement l'un sur l'autre. N'est-il pas in\u00e9vitable, alors, que tant de choses soient \u00ab incompr\u00e9hensibles \u00bb selon Descartes ? Non seulement ce compos\u00e9 lui-m\u00eame, mais le processus de sa causalit\u00e9, et aussi l'infini, et aussi la libert\u00e9 ? C'est dans un seul et m\u00eame mouvement qu'on r\u00e9duit l'\u00eatre \u00e0 la platitude de l'infiniment parfait, les choses \u00e0 la platitude des quantit\u00e9s de r\u00e9alit\u00e9, les id\u00e9es \u00e0 la platitude de la causalit\u00e9 r\u00e9elle \u2013 et qu'on red\u00e9couvre toute l'\u00e9paisseur du monde, mais, d\u00e8s lors, sous une forme incompr\u00e9hensible.\n\nOr quelles que soient les diff\u00e9rences entre Leibniz et Spinoza, et notamment leurs diff\u00e9rences dans l'interpr\u00e9tation de l'expression, c'est un fait qu'ils se servent tous deux de ce concept pour d\u00e9passer, \u00e0 tous les niveaux pr\u00e9c\u00e9dents, ce qu'ils estiment \u00eatre l'insuffisance ou la facilit\u00e9 du cart\u00e9sianisme, pour restaurer l'exigence d'une raison suffisante op\u00e9rant dans l'absolu. Non pas qu'ils reviennent en de\u00e7\u00e0 de Descartes. Il y a pour eux des acquis du cart\u00e9sianisme qui ne peuvent \u00eatre remis en question : ne serait-ce, pr\u00e9cis\u00e9ment, que les propri\u00e9t\u00e9s de l'infiniment parfait, de la quantit\u00e9 de r\u00e9alit\u00e9, du clair et du distinct, du m\u00e9canisme, etc. Spinoza et Leibniz sont des post-cart\u00e9siens, au sens o\u00f9 Fichte, Schelling, Hegel sont des post-kantiens. Il s'agit pour eux d'atteindre au fondement dont d\u00e9coulent toutes ces propri\u00e9t\u00e9s pr\u00e9c\u00e9demment \u00e9num\u00e9r\u00e9es, de red\u00e9couvrir un absolu qui soit \u00e0 la mesure du \u00ab relativisme \u00bb cart\u00e9sien. Comment proc\u00e8dent-ils, et pourquoi le concept d'expression est-il le meilleur pour cette t\u00e2che ?\n\nL'infiniment parfait comme propre doit \u00eatre d\u00e9pass\u00e9 vers l'absolument infini comme nature. Et les dix premi\u00e8res propositions de l' _\u00c9thique_ montrent que Dieu existe n\u00e9cessairement, mais parce que l'absolument infini est possible ou non contradictoire : telle est la d\u00e9marche spinoziste o\u00f9, parmi toutes les d\u00e9finitions du d\u00e9but de l' _\u00c9thique_ , qui sont nominales, il est d\u00e9montr\u00e9 que la d\u00e9finition 6 est r\u00e9elle. Or cette r\u00e9alit\u00e9 m\u00eame est constitu\u00e9e par la coexistence de toutes les formes infinies, qui introduisent leur distinction dans l'absolu sans y introduire le nombre. Ces formes constitutives de la nature de Dieu, et qui n'ont l'infiniment parfait que comme propri\u00e9t\u00e9, sont l'expression de l'absolu. Dieu est repr\u00e9sent\u00e9 comme infiniment parfait, mais il est constitu\u00e9 par ces formes plus profondes, il _s'exprime_ dans ces formes, dans ces attributs. La d\u00e9marche de Leibniz est formellement semblable : m\u00eame d\u00e9passement de l'infini vers l'absolu. Non pas certes que l'\u00catre absolu de Leibniz soit le m\u00eame que celui de Spinoza. Mais l\u00e0 encore, il s'agit de d\u00e9montrer la r\u00e9alit\u00e9 d'une d\u00e9finition, et d'atteindre \u00e0 une nature de Dieu par-del\u00e0 la propri\u00e9t\u00e9. L\u00e0 encore cette nature est constitu\u00e9e par des formes simples et distinctes, dans lesquelles Dieu s'exprime et qui expriment elles-m\u00eames des qualit\u00e9s positives infinies2. De m\u00eame, chez Spinoza comme chez Leibniz, nous l'avons vu, c'est la d\u00e9couverte de quantit\u00e9s intensives ou de quantit\u00e9s de puissance, comme plus profondes que les quantit\u00e9s de r\u00e9alit\u00e9, qui transforment les proc\u00e9d\u00e9s a posteriori, en y introduisant l'expressivit\u00e9.\n\nPassons au second point, qui concerne la connaissance et l'id\u00e9e. Ce qui est commun \u00e0 Leibniz et \u00e0 Spinoza, c'est la critique du clair-et-distinct cart\u00e9sien, comme convenant \u00e0 la r\u00e9cognition et aux d\u00e9finitions nominales plut\u00f4t qu'\u00e0 la vraie connaissance par d\u00e9finitions r\u00e9elles. Or la vraie connaissance est d\u00e9couverte comme une _esp\u00e8ce_ de l'expression : c'est dire \u00e0 la fois que le contenu repr\u00e9sentatif de l'id\u00e9e est d\u00e9pass\u00e9 vers un contenu immanent, proprement expressif, et que la forme de la conscience psychologique est d\u00e9pass\u00e9e vers un formalisme logique, \u00ab explicatif \u00bb. Et l'automate spirituel pr\u00e9sente l'identit\u00e9 de cette nouvelle forme et de ce nouveau contenu. Nous sommes nous-m\u00eames des id\u00e9es, en vertu de notre pouvoir expressif ; \u00ab et on pourrait appeler notre essence ou id\u00e9e, ce qui comprend tout ce que nous exprimons, et comme elle exprime notre union avec Dieu m\u00eame, elle n'a point de limites et rien ne la passe. \u00bb3\n\nQuant au troisi\u00e8me point, nous devons repenser l'individu d\u00e9fini comme le compos\u00e9 d'une \u00e2me et d'un corps. C'est que l'hypoth\u00e8se d'une causalit\u00e9 r\u00e9elle est peut-\u00eatre le moyen le plus simple d'interpr\u00e9ter les ph\u00e9nom\u00e8nes de ce compos\u00e9, les actions et les passions, mais n'est pas pour cela le moyen le plus convaincant ni le plus intelligible. On n\u00e9glige en effet un monde riche et profond : celui des _correspondances non causales._ Bien plus, il se peut que la causalit\u00e9 r\u00e9elle s'\u00e9tablisse et veille seulement dans certaines r\u00e9gions de ce monde des correspondances non causales, et le suppose en v\u00e9rit\u00e9. La causalit\u00e9 r\u00e9elle serait seulement un cas particulier d'un principe plus g\u00e9n\u00e9ral. On a \u00e0 la fois l'impression que l'\u00e2me et le corps ont une quasi-identit\u00e9 qui rend la causalit\u00e9 r\u00e9elle inutile entre eux, et une h\u00e9t\u00e9rog\u00e9n\u00e9it\u00e9, une h\u00e9t\u00e9ronomie qui la rend impossible. L'identit\u00e9 ou quasi-identit\u00e9 est celle d'un \u00ab invariant \u00bb ; l'h\u00e9t\u00e9ronomie est celle de deux s\u00e9ries variables, l'une corporelle, l'autre spirituelle. Or la causalit\u00e9 r\u00e9elle intervient bien dans chacune des s\u00e9ries pour son compte ; mais le rapport des deux s\u00e9ries, et leur rapport \u00e0 l'invariant, d\u00e9pend d'une correspondance non causale. Si nous demandons maintenant quel est le concept capable de rendre compte d'une telle correspondance, il appara\u00eet que c'est celui d'expression. Car s'il est vrai que le concept d'expression s'applique ad\u00e9quatement \u00e0 la causalit\u00e9 r\u00e9elle, au sens o\u00f9 l'effet exprime la cause, et o\u00f9 la connaissance de l'effet exprime une connaissance de la cause, ce concept n'en d\u00e9borde pas moins la causalit\u00e9 puisqu'il fait correspondre et r\u00e9sonner des s\u00e9ries tout \u00e0 fait \u00e9trang\u00e8res l'une \u00e0 l'autre. Si bien que la causalit\u00e9 r\u00e9elle est une esp\u00e8ce de l'expression, mais seulement une esp\u00e8ce subsum\u00e9e sous un genre plus profond. Ce genre traduit imm\u00e9diatement la possibilit\u00e9 pour des s\u00e9ries distinctes h\u00e9t\u00e9rog\u00e8nes (les expressions) d'exprimer un m\u00eame invariant (l'exprim\u00e9), en \u00e9tablissant dans chaque s\u00e9rie variable un m\u00eame encha\u00eenement de causes et d'effets. L'expression s'installe au c\u0153ur de l'individu, dans son corps et dans son \u00e2me, dans ses passions et ses actions, dans ses causes et ses effets. Et par _monade_ Leibniz, mais aussi par _mode_ Spinoza n'entendent rien d'autre que l'individu comme centre expressif.\n\nSi le concept d'expression a bien cette triple importance, du point de vue de l'\u00eatre universel, du conna\u00eetre sp\u00e9cifique, de l'agir individuel, on ne peut \u00e0 cet \u00e9gard exag\u00e9rer l'importance de la communaut\u00e9 Spinoza-Leibniz. M\u00eame s'ils divergent en chaque point, dans leur usage et leur interpr\u00e9tation du concept. Et d\u00e9j\u00e0 les diff\u00e9rences formelles, les diff\u00e9rences de ton pr\u00e9figurent les diff\u00e9rences de contenu. Nous disions qu'on ne trouve pas chez Spinoza une d\u00e9finition ni une d\u00e9monstration explicites de l'expression (bien que cette d\u00e9finition, cette d\u00e9monstration soient constamment impliqu\u00e9es dans l'\u0153uvre). Chez Leibniz, au contraire, on trouve des textes qui traitent explicitement de la compr\u00e9hension et de l'extension de la cat\u00e9gorie d'expression. Mais, bizarrement, c'est Leibniz qui donne \u00e0 cette cat\u00e9gorie une extension telle qu'elle finit par tout recouvrir, y compris le monde des signes, des similitudes, des symboles et des harmonies4 \u2013 tandis que Spinoza m\u00e8ne la plus s\u00e9v\u00e8re \u00e9puration, et oppose strictement les expressions aux signes ou aux analogies.\n\nUn des textes les plus nets de Leibniz est _Quid est idea_5. Apr\u00e8s avoir d\u00e9fini l'expression par une correspondance d' _habitus_ entre deux choses, Leibniz distingue deux grands types d'expressions naturelles : celles qui impliquent une certaine similitude (un dessin par exemple), celles qui enveloppent une certaine loi ou causalit\u00e9 (une projection). Mais il appara\u00eet que, de toutes mani\u00e8res, un des termes du rapport d'expression est toujours sup\u00e9rieur \u00e0 l'autre : soit qu'il jouisse de l'identit\u00e9 reproduite par le second, soit qu'il enveloppe la loi que l'autre d\u00e9veloppe. Et dans tous les cas il \u00ab concentre \u00bb dans son unit\u00e9 ce que l'autre \u00ab disperse dans la multitude \u00bb. L'expression selon Leibniz fonde dans tous les domaines un tel rapport de l'Un et du Multiple : ce qui s'exprime est \u00ab dou\u00e9 d'une v\u00e9ritable unit\u00e9 \u00bb par rapport \u00e0 ses expressions ; ou ce qui revient au m\u00eame, l'expression est une, par rapport \u00e0 l'exprim\u00e9 multiple et divisible6. Mais ainsi une certaine zone obscure ou confuse est toujours introduite dans l'expression : le terme sup\u00e9rieur, en raison de son unit\u00e9, exprime _plus distinctement_ ce que l'autre exprime _moins distinctement_ dans sa multitude. C'est m\u00eame en ce sens qu'on r\u00e9partit les causes et les effets, les actions et les passions : lorsqu'on dit qu'un corps nageant est cause d'une \u00ab infinit\u00e9 de mouvements de parties de l'eau \u00bb, et non l'inverse, c'est parce que le corps a une unit\u00e9 qui permet d'expliquer plus distinctement ce qui arrive7. Bien plus, comme le second terme est exprim\u00e9 dans le premier, celui-ci taille en quelque sorte son expression distincte dans une r\u00e9gion obscure qui l'entoure de toutes parts et dans laquelle il plonge : ainsi chaque monade trace son expression partielle distincte sur fond d'une expression totale confuse ; elle exprime confus\u00e9ment la totalit\u00e9 du monde, mais n'en exprime clairement qu'une partie, pr\u00e9lev\u00e9e ou d\u00e9termin\u00e9e par le rapport lui-m\u00eame expressif qu'elle entretient avec son corps. Le monde exprim\u00e9 par chaque monade est un continuum pourvu de singularit\u00e9s, et c'est autour de ces singularit\u00e9s que les monades se forment elles-m\u00eames en tant que centres expressifs. Il en est de m\u00eame pour les id\u00e9es : \u00ab Notre \u00e2me ne fait r\u00e9flexion que sur les ph\u00e9nom\u00e8nes plus singuliers qui se distinguent des autres, ne pensant distinctement \u00e0 aucun lorsqu'elle pense \u00e9galement \u00e0 tous8. \u00bb C'est pourquoi notre pens\u00e9e n'atteint pas \u00e0 l'absolument ad\u00e9quat, aux formes absolument simples qui sont en Dieu, mais en restent \u00e0 des formes et \u00e0 des termes relativement simples (c'est-\u00e0-dire simples relativement \u00e0 la multitude qu'ils enveloppent). Et cela est encore vrai de Dieu, \u00ab des diff\u00e9rentes vues de Dieu \u00bb, dans les r\u00e9gions de son entendement qui concernent la cr\u00e9ation possible : les diff\u00e9rents mondes cr\u00e9ables forment ce fond obscur \u00e0 partir duquel Dieu cr\u00e9e le meilleur, en cr\u00e9ant les monades ou expressions qui l'expriment pour le mieux. M\u00eame en Dieu, ou du moins dans certaines r\u00e9gions de son entendement, l'Un se combine avec un \u00ab z\u00e9ro \u00bb qui rend la cr\u00e9ation possible. Nous devons donc tenir compte de deux facteurs fondamentaux dans la conception leibnizienne de l'expression : _l'Analogie_ , qui exprime surtout les diff\u00e9rents types d'unit\u00e9, en rapport avec les multiplicit\u00e9s qu'ils enveloppent ; _l'Harmonie_ , qui exprime surtout la mani\u00e8re dont une multiplicit\u00e9 correspond dans chaque cas \u00e0 son unit\u00e9 de r\u00e9f\u00e9rence9.\n\nTout ceci forme une philosophie \u00ab symbolique \u00bb de l'expression, o\u00f9 l'expression n'est jamais s\u00e9par\u00e9e des signes de ses variations, pas plus que des zones obscures o\u00f9 elle plonge. Le distinct et le confus varient dans chaque expression (l'entre-expression signifie notamment que, ce qu'une monade exprime confus\u00e9ment, une autre l'exprime distinctement). _Une telle philosophie symbolique est n\u00e9cessairement une philosophie des expressions \u00e9quivoques._ Et, plut\u00f4t que d'opposer Leibniz \u00e0 Spinoza en rappelant l'importance des th\u00e8mes leibniziens du possible et de la finalit\u00e9, il nous semble n\u00e9cessaire de d\u00e9gager ce point concret qui concerne la mani\u00e8re dont Leibniz interpr\u00e8te et vit le ph\u00e9nom\u00e8ne de l'expression, parce que tous les autres th\u00e8mes et les autres concepts en d\u00e9coulent. Tout se passe comme si Leibniz, \u00e0 la fois pour sauver la richesse du concept d'expression et pour conjurer le \u00ab danger \u00bb panth\u00e9iste attenant, trouvait une nouvelle formule d'apr\u00e8s laquelle la cr\u00e9ation et l'\u00e9manation \u00e9taient les deux esp\u00e8ces r\u00e9elles de l'expression, ou correspondaient \u00e0 deux dimensions de l'expression : la _cr\u00e9ation_ , dans la constitution originaire des unit\u00e9s expressives analogues (\u00ab combinaisons de l'unit\u00e9 avec le z\u00e9ro \u00bb) ; l _'\u00e9manation_ , dans la s\u00e9rie d\u00e9riv\u00e9e qui d\u00e9veloppe les multiplicit\u00e9s exprim\u00e9es dans chaque type d'unit\u00e9 (les enveloppements et d\u00e9veloppements, les \u00ab transproductions \u00bb, les \u00ab m\u00e9tasch\u00e9matismes \u00bb)10.\n\nOr Spinoza donne de l'expression une interpr\u00e9tation vivante tout \u00e0 fait diff\u00e9rente. Car l'essentiel, pour Spinoza, c'est de s\u00e9parer le domaine des signes, toujours \u00e9quivoques, et celui des expressions dont la r\u00e8gle absolue doit \u00eatre l'univocit\u00e9. Nous avons vu en ce sens comment les trois types de signes (signes indicatifs de la perception naturelle, signes imp\u00e9ratifs de la loi morale et de la r\u00e9v\u00e9lation religieuse) \u00e9taient radicalement rejet\u00e9s dans l'inad\u00e9quat ; et ce qui tombe avec eux, c'est tout le langage de l'analogie aussi bien celui qui pr\u00eate \u00e0 Dieu un entendement et une volont\u00e9 que celui qui pr\u00eate aux choses une fin. Et du m\u00eame coup l'id\u00e9e absolument ad\u00e9quate peut \u00eatre atteinte et form\u00e9e par nous, dans la mesure o\u00f9 elle re\u00e7oit ses conditions du strict r\u00e9gime de l'univocit\u00e9 : l'id\u00e9e ad\u00e9quate, c'est l'id\u00e9e expressive, c'est-\u00e0-dire l'id\u00e9e distincte en tant qu'elle a conjur\u00e9 ce fond obscur et confus dont elle ne se s\u00e9parait pas chez Leibniz. (Nous avons essay\u00e9 de montrer comment Spinoza op\u00e9rait concr\u00e8tement cette s\u00e9lection, dans le proc\u00e9d\u00e9 de formation des notions communes o\u00f9 l'id\u00e9e cesse d'\u00eatre un signe pour devenir une expression univoque.) Quels que soient les termes mis en jeu dans le rapport d'expression, on ne dira pas que l'un exprime distinctement ce que l'autre exprime confus\u00e9ment. Surtout ce n'est pas ainsi qu'on r\u00e9partira l'actif et le passif, l'action et la passion, la cause et l'effet ; car, contrairement au principe traditionnel, les actions vont de pair avec les actions, les passions avec les passions. Si l'harmonie pr\u00e9\u00e9tablie de Leibniz et le parall\u00e9lisme de Spinoza ont en commun de rompre avec l'hypoth\u00e8se d'une causalit\u00e9 r\u00e9elle entre l'\u00e2me et le corps, leur diff\u00e9rence fondamentale n'en est pas moins celle-ci : la r\u00e9partition des actions et des passions reste chez Leibniz ce qu'elle \u00e9tait dans l'hypoth\u00e8se traditionnelle (le corps p\u00e2tissant quand l'\u00e2me agit, et inversement) \u2013 tandis que Spinoza bouleverse toute la r\u00e9partition pratique, en affirmant la parit\u00e9 des passions de l'\u00e2me avec celles du corps, des actions du corps avec celles de l'\u00e2me. C'est que, chez Spinoza, le rapport d'expression ne s'\u00e9tablit qu'entre \u00e9gaux. C'est l\u00e0 le vrai sens du parall\u00e9lisme : il n'y a jamais \u00e9minence d'une s\u00e9rie. Certes la cause dans sa s\u00e9rie reste plus parfaite que l'effet, la connaissance de la cause dans sa s\u00e9rie reste plus parfaite que celle de l'effet ; mais, loin que la perfection implique une \u00ab analogie \u00bb, une \u00ab symbolisation \u00bb, d'apr\u00e8s laquelle le plus parfait existerait _sur_ un autre mode qualitatif que le moins parfait, elle implique seulement un processus quantitatif immanent d'apr\u00e8s lequel le moins parfait existe _dans_ le plus parfait, c'est-\u00e0-dire _dans_ cette forme et _sous_ cette m\u00eame forme univoque qui constitue l'essence du plus parfait. (C'est en ce sens aussi, nous l'avons vu, qu'on doit opposer la th\u00e9orie de l'individuation qualitative chez Leibniz et la th\u00e9orie de l'individuation quantitative chez Spinoza, sans pouvoir en conclure certes que le mode ait moins d'autonomie que la monade.)\n\nChez Spinoza comme chez Leibniz, le rapport d'expression concerne essentiellement l'Un et le Multiple. Mais dans l' _\u00c9thique_ , on chercherait en vain un signe par lequel le Multiple, en tant qu'imparfait, implique une certaine confusion par rapport \u00e0 la distinction de l'Un qui s'exprime en lui. Le plus ou moins de perfection, selon Spinoza, n'implique jamais un changement de forme. Ainsi la multiplicit\u00e9 des attributs est strictement \u00e9gale \u00e0 l'unit\u00e9 de la substance : par cette stricte \u00e9galit\u00e9, nous devons entendre que les attributs sont _formellement_ ce que la substance est _ontologiquement._ Au nom de cette \u00e9galit\u00e9, les formes d'attributs n'introduisent aucune distinction num\u00e9rique entre substances ; au contraire, leur propre distinction formelle est \u00e9gale \u00e0 toute la diff\u00e9rence ontologique de la substance unique. Et si nous consid\u00e9rons la multitude des modes dans chaque attribut, ces modes enveloppent l'attribut, mais sans que cet enveloppement signifie que l'attribut prenne une autre forme que celle sous laquelle il constitue l'essence de la substance : les modes enveloppent et expriment l'attribut _sous cette forme m\u00eame_ o\u00f9 il enveloppe et exprime l'essence divine. C'est pourquoi le spinozisme s'accompagne d'une extraordinaire th\u00e9orie des distinctions qui, m\u00eame lorsqu'elle emprunte la terminologie cart\u00e9sienne, parle un tout autre langage : ainsi la distinction r\u00e9elle y est en v\u00e9rit\u00e9 une distinction formelle non num\u00e9rique (cf. les attributs) ; la distinction modale y est une distinction num\u00e9rique intensive ou extensive (cf. les modes) ; la distinction de raison y est une distinction formelle-objective (cf. les id\u00e9es). Dans sa propre th\u00e9orie, Leibniz multiplie les types de distinction, mais pour assurer toutes les ressources de la symbolisation, de l'harmonie et de l'analogie. Chez Spinoza, au contraire, le seul langage est celui de l'univocit\u00e9 : d'abord _univocit\u00e9 des attributs_ (en tant que les attributs, sous la m\u00eame forme, sont ce qui constitue l'essence de la substance et ce qui contient les modes et leurs essences) ; ensuite _univocit\u00e9 de la cause_ (en tant que Dieu est cause de toutes choses au m\u00eame sens que cause de soi) ; ensuite _univocit\u00e9 de l'id\u00e9e_ (en tant que la notion commune est la m\u00eame dans la partie et dans le tout). Univocit\u00e9 de l'\u00eatre, univocit\u00e9 du produire, univocit\u00e9 du conna\u00eetre ; forme commune, cause commune, notion commune \u2013 telles sont les trois figures de l'Univoque qui se r\u00e9unissent absolument dans l'id\u00e9e du troisi\u00e8me genre. Loin que l'expression chez Spinoza se r\u00e9concilie avec la cr\u00e9ation et l'\u00e9manation, elle les expulse au contraire, elle les rejette du c\u00f4t\u00e9 des signes inad\u00e9quats ou du langage \u00e9quivoque. Spinoza accepte le \u00ab danger \u00bb proprement philosophique impliqu\u00e9 dans la notion d'expression : l'immanence, le panth\u00e9isme. Bien plus, il parie pour ce danger-l\u00e0. Chez Spinoza _toute la th\u00e9orie de l'expression est au service de l'univocit\u00e9_ ; et tout son sens est d'arracher l'\u00catre univoque \u00e0 son \u00e9tat d'indiff\u00e9rence ou de neutralit\u00e9, pour en faire l'objet d'une affirmation pure, effectivement r\u00e9alis\u00e9e dans le panth\u00e9isme ou l'immanence expressive. Voil\u00e0, nous semble-t-il, la v\u00e9ritable opposition de Spinoza et de Leibniz : _la th\u00e9orie des expressions univoques de l'un s'oppose \u00e0 la th\u00e9orie des expressions \u00e9quivoques de l'autre._ Toutes les autres oppositions (la n\u00e9cessit\u00e9 et la finalit\u00e9, le n\u00e9cessaire et le possible) en d\u00e9coulent, et sont abstraites par rapport \u00e0 celle-ci. Car il y a bien une origine concr\u00e8te des diff\u00e9rences philosophiques, une certaine mani\u00e8re d' _\u00e9valuer_ un ph\u00e9nom\u00e8ne : ici celui de l'expression.\n\nMais quelle que soit l'importance de l'opposition, nous devons revenir \u00e0 ce qu'il y a de commun entre Leibniz et Spinoza, dans cet usage de la notion d'expression qui manifeste toute la force de leur r\u00e9action anticart\u00e9sienne. Cette notion d'expression est essentiellement triadique : on doit distinguer ce qui s'exprime, l'expression m\u00eame et l'exprim\u00e9. Or le paradoxe est que, \u00e0 la fois, l'\u00ab exprim\u00e9 \u00bb _n'existe pas_ hors de l'expression, et pourtant ne lui ressemble pas, mais est _essentiellement_ rapport\u00e9 \u00e0 ce qui s'exprime, comme distinct de l'expression m\u00eame. Si bien que l'expression est le support d'un double mouvement : ou bien on enveloppe, on implique, on enroule l'exprim\u00e9 dans l'expression, pour ne retenir que le couple \u00ab exprimant-expression \u00bb ; ou bien on d\u00e9veloppe, on explique, on d\u00e9roule l'expression de fa\u00e7on \u00e0 restituer l'exprim\u00e9 (\u00ab exprimant-exprim\u00e9 \u00bb). Ainsi chez Leibniz, il y a bien d'abord une expression divine : Dieu s'exprime dans des formes absolues ou des notions absolument simples, comme dans un divin Alphabet ; ces formes expriment des qualit\u00e9s illimit\u00e9es qui se rapportent \u00e0 Dieu comme son essence. Puis Dieu se r\u00e9-exprime, au niveau de la cr\u00e9ation possible : il s'exprime alors dans des notions individuelles ou relativement simples, monades, qui correspondent \u00e0 chacune des \u00ab vues \u00bb de Dieu ; et ces expressions \u00e0 leur tour expriment le monde entier, c'est-\u00e0-dire la totalit\u00e9 du monde choisi, qui se rapporte \u00e0 Dieu comme la manifestation de sa \u00ab gloire \u00bb et de sa volont\u00e9. On voit bien chez Leibniz que le monde n'existe pas hors des monades qui l'expriment, et que pourtant Dieu fait exister le monde plut\u00f4t que les monades11. Ces deux propositions ne sont nullement contradictoires, mais t\u00e9moignent du double mouvement par lequel le monde exprim\u00e9 s'enveloppe dans les monades qui l'expriment, et par lequel inversement les monades se d\u00e9veloppent et restituent cette continuit\u00e9 d'un fond pourvu de singularit\u00e9s autour desquelles elles se sont constitu\u00e9es. Sous toutes les r\u00e9serves pr\u00e9c\u00e9dentes, on en dirait autant de Spinoza. Dans la triade de la substance, Dieu s'exprime dans les attributs, les attributs expriment des qualit\u00e9s illimit\u00e9es qui constituent son essence. Dans la triade du mode, Dieu se r\u00e9-exprime, ou les attributs s'expriment \u00e0 leur tour : ils s'expriment dans des modes, les modes expriment des modifications comme modifications de la substance, constitutives d'un m\u00eame monde \u00e0 travers tous les attributs. C'est en fonction de ce caract\u00e8re toujours triadique que le concept d'expression ne se laisse rapporter ni \u00e0 la causalit\u00e9 dans l'\u00eatre, ni \u00e0 la repr\u00e9sentation dans l'id\u00e9e, mais les d\u00e9borde toutes deux, et en fait deux de ses cas particuliers. Car, \u00e0 la dyade de la cause et de l'effet, ou \u00e0 celle de l'id\u00e9e et de son objet, se joint toujours un troisi\u00e8me terme qui les transforme. Il est certain que l'effet exprime sa cause ; mais plus profond\u00e9ment la cause et l'effet forment une s\u00e9rie qui doit exprimer quelque chose, et quelque chose d'identique (ou de semblable) \u00e0 ce qu'exprime une autre s\u00e9rie. Ainsi la causalit\u00e9 r\u00e9elle se trouve localis\u00e9e dans des s\u00e9ries expressives qui b\u00e9n\u00e9ficient entre elles de correspondances non causales. De m\u00eame l'id\u00e9e repr\u00e9sente un objet, et d'une certaine mani\u00e8re l'exprime ; mais plus profond\u00e9ment l'id\u00e9e et son objet expriment quelque chose qui leur est commun, et pourtant propre \u00e0 chacun : la puissance, ou l'absolu sous deux puissances, qui sont celles de penser ou de conna\u00eetre, d'\u00eatre ou d'agir. Ainsi la repr\u00e9sentation se trouve localis\u00e9e dans un certain rapport extrins\u00e8que de l'id\u00e9e et de l'objet, chacun b\u00e9n\u00e9ficiant pour son compte d'une expressivit\u00e9 par-del\u00e0 la repr\u00e9sentation. Bref, partout l'exprim\u00e9 intervient comme un tiers qui transforme les dualismes. Par-del\u00e0 la causalit\u00e9 r\u00e9elle, par-del\u00e0 la repr\u00e9sentation id\u00e9elle, on d\u00e9couvre l'exprim\u00e9 comme le tiers qui rend les distinctions infiniment plus r\u00e9elles, l'identit\u00e9 infiniment mieux pens\u00e9e. L'exprim\u00e9, c'est le sens : plus profond que le rapport de causalit\u00e9, plus profond que le rapport de repr\u00e9sentation. Il y a un m\u00e9canisme des corps suivant la r\u00e9alit\u00e9, il y a un automatisme des pens\u00e9es suivant l'id\u00e9alit\u00e9 ; mais nous apprenons que la m\u00e9canique corporelle et l'automate spirituel sont le plus _expressifs_ lorsqu'ils re\u00e7oivent leur \u00ab sens \u00bb et leur \u00ab correspondance \u00bb, comme cette raison n\u00e9cessaire qui manquait partout dans le cart\u00e9sianisme.\n\nNous ne pouvons dire ce qui est le plus important : les diff\u00e9rences de Leibniz et de Spinoza dans leur \u00e9valuation de l'expression ; ou leur commun appel \u00e0 ce concept pour fonder une philosophie post-cart\u00e9sienne.\n\n* * *\n\n1.  Sur ces deux th\u00e8mes, du miroir et du germe (ou du rameau), en rapport essentiel avec la notion d'expression, cf. par exemple le proc\u00e8s d'Eckhart. En effet, ces th\u00e8mes font partie des principaux chefs d'accusation : cf. _\u00c9dition critique des pi\u00e8ces relatives au proc\u00e8s d'Eckhart_ , par G. TH\u00c9RY, Archives d'histoire doctrinale et litt\u00e9raire du Moyen \u00c2ge (Vrin \u00e9d., 1926-1927).\n\n2.  Sur \u00ab les formes simples absolument prises \u00bb, \u00ab attributs m\u00eames de Dieu \u00bb, \u00ab causes premi\u00e8res et raison derni\u00e8re des choses \u00bb, cf. _Lettre \u00e0 \u00c9lisabeth_ , 1678, et _M\u00e9ditations sur la connaissance_ , 1684. Dans la note de 1676, _Quod ens perfectissimum existit_ , la perfection se d\u00e9finit par une qualit\u00e9 positive absolue _seu quae quicquid exprimit, sine ullis limitibus exprimit_ (Gerhardt VII, pp. 261-262). Leibniz fait allusion, dans les _Nouveaux Essais_ , aux \u00ab qualit\u00e9s originales ou connaissables distinctement \u00bb qui peuvent \u00eatre port\u00e9es \u00e0 l'infini.\n\n3.  LEIBNIZ, _Discours de m\u00e9taphysique_ , \u00a7 16.\n\n4.  Cf. _Lettre de Leibniz \u00e0 Arnauld_ (Janet I, p. 594) : \u00ab L'expression est commune \u00e0 toutes les formes, et c'est un genre dont la perception naturelle, le sentiment animal et la connaissance intellectuelle sont des esp\u00e8ces. \u00bb\n\n5.  \u00c9d. Gerhardt, VII, pp. 263-264.\n\n6.  LEIBNIZ, _Lettre \u00e0 Arnauld_ (Janet I, p. 594) : \u00ab Il suffit que ce qui est divisible et mat\u00e9riel, et se trouve divis\u00e9 en plusieurs \u00eatres, soit exprim\u00e9 ou repr\u00e9sent\u00e9 dans un seul \u00eatre indivisible, ou dans la substance qui est dou\u00e9e d'une v\u00e9ritable unit\u00e9. \u00bb Et _Nouveaux Essais_ III, 6, \u00a7 24 : L'\u00e2me et la machine \u00ab s'accordent parfaitement, et, quoiqu'elles n'aient point d'influence imm\u00e9diate l'une sur l'autre, elles s'expriment mutuellement, l'une ayant concentr\u00e9 dans une parfaite unit\u00e9 tout ce que l'autre a dispers\u00e9 dans la multitude \u00bb.\n\n7.  _Projet d'une lettre \u00e0 Arnauld_ (Janet I, pp. 552-553).\n\n8.  _Lettre \u00e0 Arnauld_ (Janet I, p. 596).\n\n9.  Cf. Grua \u00e9d., p. 126 : \u00ab Comme tous les esprits sont des unit\u00e9s, on peut dire que Dieu est l'unit\u00e9 primitive, exprim\u00e9e par toutes les autres suivant leur port\u00e9e... C'est de quoi r\u00e9sulte l'op\u00e9ration en la cr\u00e9ature, laquelle est vari\u00e9e selon les diff\u00e9rentes combinaisons de l'unit\u00e9 avec le z\u00e9ro, ou bien du positif avec le privatif. \u00bb Ce sont ces diff\u00e9rents types d'unit\u00e9 qui _symbolisent_ les uns avec les autres : par exemple les notions relativement simples de notre entendement, avec les absolument simples de l'entendement divin (cf. \u00e9d. Couturat, _Elementa Calculi_ , et _Introductio ad Encyclopaediam Arcanam_ ). Un type d'unit\u00e9 est toujours cause finale par rapport \u00e0 la multiplicit\u00e9 qu'il subsume. Et Leibniz emploie particuli\u00e8rement le mot \u00ab harmonie \u00bb pour d\u00e9signer cette r\u00e9f\u00e9rence du multiple \u00e0 l'un ( _Elementa verae pietatis_ , Grua, p. 7).\n\n10.  Il arrive que Leibniz emploie le mot \u00ab \u00e9manation \u00bb pour d\u00e9signer la cr\u00e9ation des unit\u00e9s et leurs combinaisons : cf. par exemple _Discours de m\u00e9taphysique_ , \u00a7 14.\n\n11.  Th\u00e8me constant dans les _Lettres \u00e0 Arnauld_ : Dieu n'a pas cr\u00e9\u00e9 Adam p\u00e9cheur, mais le monde o\u00f9 Adam a p\u00e9ch\u00e9.\n\n# APPENDICE\n\n# \u00c9TUDE FORMELLE DU PLAN DE L' _\u00c9THIQUE_\n\n# ET DU R\u00d4LE DES SCOLIES DANS LA R\u00c9ALISATION\n\n# DE CE PLAN :\n\n# LES DEUX _\u00c9THIQUES_ | TH\u00c8ME | CONS\u00c9QUENCE | CONCEPT EXPRESSIF CORRESPONDANT\n\n---|---|---|---\n\nLIVRE I\n\n|  |  |\n\nL'Affirmation sp\u00e9culative.\n\n1-8 | Il n'y a pas plusieurs substances de m\u00eame attribut, la distinction num\u00e9rique n'est pas r\u00e9elle. | Ces 8 propositions ne sont pas hypoth\u00e9tiques, mais cat\u00e9goriques ; il est donc faux que l' _\u00c9thique_ \u00ab commence \u00bb par l'id\u00e9e de Dieu. | Premi\u00e8re triade de la substance : attribut, essence, substance.\n\n9-14 | La distinction r\u00e9elle n'est pas num\u00e9rique, il n'y a qu'une substance pour tous les attributs. | L\u00e0 seulement est atteinte l'id\u00e9e de Dieu comme celle d'une substance absolument infinie ; et il est d\u00e9montr\u00e9 que la d\u00e9finition 6 est r\u00e9elle. | Seconde triade de la substance : parfait, infini, absolu.\n\n15-36 | La puissance ou la production : les proc\u00e9d\u00e9s de la production et la nature des produits (modes). | L'immanence signifie \u00e0 la fois l'univocit\u00e9 des attributs et l'univocit\u00e9 de la cause (Dieu est cause de toutes choses au m\u00eame sens que cause de soi). | Troisi\u00e8me triade de la substance : l'essence comme puissance, ce dont elle est l'essence, le pouvoir d'\u00eatre affect\u00e9 (par des modes).\n\nLIVRE II\n\n|  |  |\n\nL'Id\u00e9e expressive.\n\n1-7 | Parall\u00e9lisme \u00e9pist\u00e9mologique de l'id\u00e9e et de son objet, parall\u00e9lisme ontologique de l'\u00e2me et du corps. | De la substance aux modes, transfert de l'expressivit\u00e9 : r\u00f4le de l'id\u00e9e de Dieu dans ce transfert. | Triade modale : attribut, mode, modification.\n\n8-13 | Les conditions des id\u00e9es : les id\u00e9es que Dieu a en fonction de sa nature, les id\u00e9es que nous avons en fonction de notre nature et de notre corps. | Les aspects de Dieu par rapport aux id\u00e9es : Dieu en tant qu'infini, en tant qu'affect\u00e9 de beaucoup d'id\u00e9es, en tant qu'il a seulement telle id\u00e9e. | L'ad\u00e9quat et l'inad\u00e9quat.\n\nExpos\u00e9 de la physique | Le mod\u00e8le du corps. | Les parties extensives, les rapports de mouvement et de repos, la composition et la d\u00e9composition de ces rapports. | Premi\u00e8re triade individuelle du mode : l'essence, le rapport caract\u00e9ristique les parties extensives.\n\n14-36 | Les conditions dans lesquelles nous avons des id\u00e9es font que celles-ci sont n\u00e9cessairement inad\u00e9quates : id\u00e9e de soi-m\u00eame, id\u00e9e de son corps, id\u00e9e des autres corps. | L'id\u00e9e inad\u00e9quate est \u00ab indicative \u00bb, \u00ab enveloppante \u00bb, par opposition \u00e0 l'id\u00e9e ad\u00e9quate, qui est expressive et explicative : le hasard, les rencontres et le premier genre de connaissance. | Caract\u00e8re inexpressif de l'id\u00e9e inad\u00e9quate.\n\n37-49 | Comment les id\u00e9es ad\u00e9quates sont-elles possibles ? Ce qui est commun \u00e0 tous les corps, ou \u00e0 plusieurs corps. | Les notions communes, par opposition aux id\u00e9es abstraites. Comment les notions communes m\u00e8nent \u00e0 l'id\u00e9e de Dieu : le second genre de connaissance et la raison. | Caract\u00e8re expressif de l'id\u00e9e ad\u00e9quate, du point de vue de sa forme et de sa mati\u00e8re.\n\nLIVRE III\n\n|  |  |\n\nLa joie pratique.\n\n1-10 | Ce qui suit des id\u00e9es : les affections ou sentiments. Le \u00ab conatus \u00bb, en tant que d\u00e9termin\u00e9 par ces affections. | Distinction de deux sortes d'affections : les actives et les passives ; les actions qui suivent des id\u00e9es ad\u00e9quates et les passions qui suivent des id\u00e9es inad\u00e9quates. | Seconde triade individuelle du mode : l'essence, le pouvoir d'\u00eatre affect\u00e9, les affections qui remplissent ce pouvoir.\n\n11-57 | La distinction de deux sortes d'affections, actives et passives, ne doit pas faire n\u00e9gliger la distinction de deux sortes d'affections passives, les unes joyeuses et les autres tristes. | Les deux lignes, de joie et de tristesse : leurs d\u00e9veloppements, leurs variations et leurs recoupements. | Augmenter et diminuer la puissance d'agir.\n\n58-59 | Possibilit\u00e9 d'une joie active, distincte de la joie passive : poss\u00e9der la puissance d'agir. | Critique de la tristesse. | Le concept complet de joie.\n\nLIVRE IV\n\n|  |  |\n\nLe bon et le mauvais.\n\n1-18 | Les rapports de force entre affections : les facteurs de leurs puissances respectives. | Le bon et le mauvais, par opposition au Bien et au Mal. | Les d\u00e9terminations du \u00ab conatus \u00bb.\n\n19-45 | Premier aspect de la raison : s\u00e9lectionner les affections passives, \u00e9liminer les tristesses, organiser les rencontres, composer les rapports, augmenter la puissance d'agir, \u00e9prouver le maximum de joies. | Utilit\u00e9 et n\u00e9cessit\u00e9 relatives de la soci\u00e9t\u00e9, comme rendant possible, pr\u00e9parant et accompagnant ce premier effort de la raison. | Critique d\u00e9velopp\u00e9e de la tristesse.\n\n46-73 | Le bon et le mauvais d'apr\u00e8s ce crit\u00e8re de la raison. | Suite de la critique de la tristesse. | L'homme libre et l'esclave, le fort et le faible, le raisonnable et l'insens\u00e9.\n\nLIVRE V\n\n|  |  |\n\nJoie pratique et affirmation sp\u00e9culative.\n\n1-13 | Comment nous arrivons en fait \u00e0 former des id\u00e9es ad\u00e9quates (notions communes). Comment les affections passives joyeuses nous y am\u00e8nent. Et comment, par l\u00e0, nous diminuons les tristesses, et formons une id\u00e9e ad\u00e9quate de toutes les affections passives. | Nous arrivons donc au second genre de connaissance, \u00e0 la faveur de certaines occasions fournies par le premier genre. | Second aspect de la raison : former les notions communes, et les affections actives de joie qui s'ensuivent. Devenir actif.\n\n14-20 | L'id\u00e9e de Dieu, \u00e0 l'extr\u00e9mit\u00e9 du second genre de connaissance. | Des notions communes \u00e0 l'id\u00e9e de Dieu. | Le Dieu impassible tel qu'il est compris dans le second genre.\n\n21-42 | Cette id\u00e9e de Dieu, \u00e0 son tour, nous fait sortir du second genre, et acc\u00e9der \u00e0 un troisi\u00e8me genre de connaissance : le Dieu r\u00e9ciproque du troisi\u00e8me genre, l'id\u00e9e ad\u00e9quate de soi-m\u00eame, du corps et des autres corps. | Il y a autant de parties de l'\u00e2me que de types d'affections. Non seulement des affections passives de tristesse et de joie, mais aussi des affections actives de joie du second genre ; et encore des affections actives de joie du troisi\u00e8me genre. D'o\u00f9 l'on conclut \u00e0 ce qui est mortel et \u00e0 ce qui est \u00e9ternel dans l'\u00e2me : le c\u00f4t\u00e9 qui meurt et le c\u00f4t\u00e9 qui subsiste, les parties extensives et l'essence intensive. | L' _\u00c9thique_ proc\u00e9dait jusqu'ici par notions communes ; uniquement par notions communes. Mais elle change, et parle maintenant au nom du troisi\u00e8me genre. Unit\u00e9, dans ce troisi\u00e8me genre, de la joie pratique et de l'affirmation sp\u00e9culative : devenir expressif, la b\u00e9atitude, la r\u00e9ciprocit\u00e9, l'univocit\u00e9.\n\nIl faudrait une longue \u00e9tude des proc\u00e9d\u00e9s formels de l' _\u00c9thique_ et du r\u00f4le de chaque \u00e9l\u00e9ment (d\u00e9finitions, axiomes, postulats, etc.). Nous voudrions seulement consid\u00e9rer la fonction particuli\u00e8re et complexe des scolies.\n\nLe premier grand scolie de l' _\u00c9thique_ est celui de I, 8 (scolie 2). Il se propose de donner une _autre_ d\u00e9monstration de la proposition 5, d'apr\u00e8s laquelle il ne peut pas y avoir plusieurs substances de m\u00eame attribut. Comme nous l'avons vu dans notre premier chapitre, il proc\u00e8de ainsi : 1o) la distinction num\u00e9rique implique une causalit\u00e9 externe ; 2o) or il est impossible d'appliquer une cause ext\u00e9rieure \u00e0 une substance, parce que toute substance est en soi et est con\u00e7ue par soi ; 3o) deux ou plusieurs substances ne peuvent donc pas se distinguer num\u00e9riquement, sous un m\u00eame attribut.\n\nLa proposition 5 proc\u00e9dait autrement, et plus bri\u00e8vement : deux substances de m\u00eame attribut devraient se distinguer par les modes, ce qui est absurde. Mais apr\u00e8s 5, la proposition 6 d\u00e9montrait que la causalit\u00e9 externe ne peut _donc_ pas convenir \u00e0 la substance. Et 7, qu'une substance est _donc_ cause de soi. Et 8 concluait qu'une substance est _donc_ n\u00e9cessairement infinie.\n\nLe groupe des propositions 5-8, et le scolie 8, proc\u00e8dent \u00e0 l'inverse l'un de l'autre. Les propositions partent de la nature de la substance, et concluent \u00e0 son infinit\u00e9, c'est-\u00e0-dire \u00e0 l'impossibilit\u00e9 de lui appliquer des distinctions num\u00e9riques. Le scolie part de la nature de la distinction num\u00e9rique, et conclut \u00e0 l'impossibilit\u00e9 de l'appliquer \u00e0 la substance.\n\nOr on peut croire que le scolie, pour prouver que la substance r\u00e9pugne \u00e0 la causalit\u00e9 externe, aurait avantage \u00e0 invoquer les propositions 6 et 7. En fait c'est impossible. Car 6 et 7 supposent 5 ; le scolie ne serait donc pas une autre d\u00e9monstration. Il invoque pourtant, et longuement, la proposition 7. Mais en un sens tout nouveau : il en retient un contenu purement axiomatique, et la d\u00e9tache enti\u00e8rement de son contexte d\u00e9monstratif. \u00ab Si les hommes portaient leur attention sur la substance, ils ne douteraient nullement de la v\u00e9rit\u00e9 de la proposition 7, bien plus cette proposition serait pour tous un axiome et serait compt\u00e9e parmi les notions communes... \u00bb Alors le scolie peut op\u00e9rer lui-m\u00eame une d\u00e9monstration tout \u00e0 fait ind\u00e9pendante du groupe d\u00e9monstratif 5-8.\n\nNous pouvons d\u00e9gager trois caract\u00e8res d'un tel scolie : 1o) Il propose une seconde d\u00e9monstration, et cette d\u00e9monstration est _positive et intrins\u00e8que,_ par rapport \u00e0 la premi\u00e8re qui op\u00e9rait n\u00e9gativement, extrins\u00e8quement. (En effet, la proposition 5 se contentait d'invoquer l'ant\u00e9riorit\u00e9 de la substance pour conclure \u00e0 l'impossibilit\u00e9 d'assimiler la distinction modale \u00e0 une distinction substantielle. Le scolie de 8 conclut bien \u00e0 l'impossibilit\u00e9 d'assimiler la distinction num\u00e9rique \u00e0 la distinction substantielle, mais \u00e0 partir des caract\u00e8res intrins\u00e8ques et positifs du nombre et de la substance). 2o) Le scolie est _ostensif_ puisque, ind\u00e9pendant des d\u00e9monstrations pr\u00e9c\u00e9dentes, il doit se substituer \u00e0 elles, et retient seulement certaines propositions de mani\u00e8re axiomatique, en les d\u00e9tachant de leur encha\u00eenement d\u00e9monstratif. (Bien s\u00fbr, il arrive qu'un scolie invoque des d\u00e9monstrations, mais non pas celles du groupe qu'il est charg\u00e9 de \u00ab doubler \u00bb.) 3o) D'o\u00f9 vient alors l'\u00e9vidence qui permet de traiter les propositions reprises comme des axiomes, ind\u00e9pendamment de leur premier contexte et de leur d\u00e9monstration ? Cette \u00e9vidence nouvelle leur vient d'arguments _pol\u00e9miques_ , o\u00f9 Spinoza attaque, souvent avec violence, ceux qui ont l'esprit trop confus pour comprendre, ou m\u00eame qui ont int\u00e9r\u00eat \u00e0 entretenir la confusion. (D\u00e8s le scolie de 8 sont vivement d\u00e9nonc\u00e9s ceux qui ne comprennent pas la proposition 7 en elle-m\u00eame, et qui sont pr\u00eats \u00e0 croire aussi bien que les arbres parlent ou que les hommes naissent des pierres).\n\nBref, les scolies sont en g\u00e9n\u00e9ral positifs, ostensifs et agressifs. En vertu de leur ind\u00e9pendance \u00e0 l'\u00e9gard des propositions qu'ils doublent, on dirait que l' _\u00c9thique_ a \u00e9t\u00e9 simultan\u00e9ment \u00e9crite deux fois, sur deux tons, sur un double registre. En effet, il y a une mani\u00e8re, discontinue, dont les scolies sautent des uns aux autres, se font \u00e9cho, se retrouvent dans la pr\u00e9face de tel livre de l' _\u00c9thique_ ou dans la conclusion de tel autre, formant une ligne bris\u00e9e qui traverse toute l'\u0153uvre en profondeur, mais qui n'affleure qu'en tel ou tel point (les points de brisure). Par exemple, le scolie de I, 8 constitue une telle ligne avec celui de I, 15, puis celui de I, 17, puis celui de I, 33, puis celui de II, 3, enfin celui de II, 10 : il s'agit des diff\u00e9rents modes de d\u00e9figuration que l'homme fait subir \u00e0 Dieu. De m\u00eame le scolie de II, 13, qui \u00e9rige le mod\u00e8le du corps, saute dans le scolie de III, 2, pour aboutir dans la pr\u00e9face du livre V. De m\u00eame une ligne bris\u00e9e de scolies forme une sorte d'hymne \u00e0 la joie, toujours interrompu, et o\u00f9 se trouvent violemment d\u00e9nonc\u00e9s ceux qui vivent de tristesse, ceux qui ont int\u00e9r\u00eat \u00e0 nos tristesses, ceux qui ont besoin de la tristesse humaine pour assurer leur pouvoir : IV, 45, sc. 2 ; IV, 50, sc. ; IV, 63 sc. ; V, 10, sc. De m\u00eame encore, le couple homme libre-esclave de IV, 66, sc., se retrouve dans le couple fort-faible de IV, 73, sc., puis sage-ignorant de V, 42, sc. sur lequel se termine l' _\u00c9thique_. Ou enfin V, 4, sc. ; V, 20 sc., qui forment la cha\u00eene royale nous conduisant au troisi\u00e8me genre.\n\nD\u00e8s lors les grands \u00ab tournants \u00bb de l' _\u00c9thique_ sont forc\u00e9ment pr\u00e9sent\u00e9s dans les scolies. Car la continuit\u00e9 des propositions et d\u00e9monstrations ne peut recevoir des points remarquables, des impulsions diverses, des changements de directions, que par l'\u00e9mergence de quelque chose qui s'exprime dans les scolies, pierre-scolie, remous-scolie, provoquant cette brisure en \u00e9mergeant. Exemples de tels tournants : II, 13, sc. (l'appel au mod\u00e8le du corps) ; III, 57, sc. (l'appel au mod\u00e8le des joies actives) ; IV, 18, sc. (l'appel au mod\u00e8le de la raison) ; V, 20, sc. et 36, sc. (l'appel au troisi\u00e8me genre).\n\nIl y a donc comme deux _\u00c9thiques_ coexistantes, l'une constitu\u00e9e par la ligne ou le flot continus des propositions, d\u00e9monstrations et corollaires, l'autre, discontinue, constitu\u00e9e par la ligne bris\u00e9e ou la cha\u00eene volcanique des scolies. L'une, avec une rigueur implacable, repr\u00e9sente une sorte de terrorisme de la t\u00eate, et progresse d'une proposition \u00e0 l'autre sans se soucier des cons\u00e9quences pratiques, \u00e9labore ses _r\u00e8gles_ sans se soucier d'identifier les _cas_. L'autre recueille les indignations et les joies du c\u0153ur, manifeste la joie pratique et la lutte pratique contre la tristesse, et s'exprime en disant \u00ab c'est le cas \u00bb. En ce sens l' _\u00c9thique_ est un livre double. Il peut \u00eatre int\u00e9ressant de lire la seconde _\u00c9thique_ sous la premi\u00e8re, en sautant d'un scolie \u00e0 l'autre.\n\nRevenons aux trois caract\u00e8res du scolie : positif, ostensif, agressif. Il est \u00e9vident que ces caract\u00e8res empi\u00e8tent les uns sur les autres, \u00e0 l'int\u00e9rieur d'un m\u00eame scolie. Nous pouvons toutefois les consid\u00e9rer s\u00e9par\u00e9ment.\n\nQue le scolie proc\u00e8de positivement, cela peut vouloir dire, nous l'avons vu, qu'il s'appuie sur des caract\u00e8res intrins\u00e8ques, alors que la d\u00e9monstration correspondante reposait seulement sur des propri\u00e9t\u00e9s extrins\u00e8ques. Un exemple particuli\u00e8rement net est donn\u00e9 en III, 7, \u00e0 propos du \u00ab flottement de l'\u00e2me \u00bb : celui-ci, dans la d\u00e9monstration de la proposition, est d\u00e9fini par le jeu des causes ext\u00e9rieures qui le provoquent, mais dans le scolie, par la diversit\u00e9 des rapports internes qui nous composent. Cela peut vouloir dire aussi que le scolie proc\u00e8de a priori, tandis que la d\u00e9monstration est a posteriori : ainsi en II, 1, o\u00f9 la d\u00e9monstration passe par les modes, mais le scolie repose sur la possibilit\u00e9 de penser directement une qualit\u00e9 comme infinie. De m\u00eame en I, 11, le scolie propose une d\u00e9monstration a priori fond\u00e9e \u00ab sur le m\u00eame principe \u00bb que le proc\u00e9d\u00e9 a posteriori de la d\u00e9monstration. Ou bien encore le scolie si important du parall\u00e9lisme, en II, 7 : alors que la d\u00e9monstration va de l'effet \u00e0 la cause pour conclure que l'ordre de la connaissance est le m\u00eame que celui des choses, alors que l'ensemble de la d\u00e9monstration et du corollaire s'\u00e9l\u00e8ve de cette identit\u00e9 d'ordre dans les modes \u00e0 une \u00e9galit\u00e9 de puissances en Dieu, le scolie au contraire part de l'unit\u00e9 ontologique de la substance pour conclure \u00e0 l'\u00e9galit\u00e9 des puissances et \u00e0 l'identit\u00e9 d'ordre. (Entre les deux proc\u00e9d\u00e9s, nous l'avons vu, il y a un d\u00e9calage, qui ne peut \u00eatre combl\u00e9 que dans la mesure o\u00f9 Spinoza, dans le scolie lui-m\u00eame, invoque l'id\u00e9e de Dieu d'une mani\u00e8re ostensive : ce qui nous renvoie d\u00e9j\u00e0 au second caract\u00e8re des scolies.)\n\nMais, pour en finir avec le premier caract\u00e8re, nous devons dire que la positivit\u00e9 des scolies se manifeste encore d'une autre fa\u00e7on particuli\u00e8rement complexe : il se peut que le scolie op\u00e8re dans l'\u00e9l\u00e9ment d'une d\u00e9finition r\u00e9elle, tandis que la proposition et la d\u00e9monstration tiraient les cons\u00e9quences de d\u00e9finitions nominales : c'est ainsi que, dans le livre I, les propositions 9 et 10 \u00e9tablissent la possibilit\u00e9 simplement logique d'un m\u00eame \u00eatre ayant une infinit\u00e9 d'attributs dont chacun est con\u00e7u par soi, mais se contentent d'invoquer les d\u00e9finitions 3 et 4, qui sont les d\u00e9finitions nominales de la substance et de l'attribut. Le scolie, au contraire, invoque la d\u00e9finition 6, dont nous avons vu qu'elle \u00e9tait la seule r\u00e9elle de toutes celles qui ouvrent le livre I. Bien plus, comme une d\u00e9finition r\u00e9elle est une d\u00e9finition dont on doit pouvoir _d\u00e9montrer_ qu'elle est r\u00e9elle, c'est-\u00e0-dire qu'elle fonde la possibilit\u00e9 \u00ab r\u00e9elle \u00bb de son objet (possibilit\u00e9 transcendantale par opposition \u00e0 la possibilit\u00e9 seulement logique), le scolie de 10 se charge effectivement de cette t\u00e2che, et d\u00e9montre que la d\u00e9finition 6 est bien r\u00e9elle : en effet la distinction des attributs, en vertu de ses caract\u00e8res positifs, ne peut pas \u00eatre num\u00e9rique. L\u00e0 encore il faut un usage ostensif de la proposition 9, s\u00e9par\u00e9e de son contexte.\n\nLe caract\u00e8re positif des scolies a donc trois aspects : intrins\u00e8que, a priori ou r\u00e9el. Consid\u00e9rons le second caract\u00e8re, ostensif. Lui aussi a plusieurs aspects, dont nous avons vu le principal. Cet aspect principal est axiomatique : il consiste, pour le scolie, \u00e0 invoquer le th\u00e8me d'une proposition pr\u00e9c\u00e9dente en l'extrayant de la cha\u00eene continue des propositions et d\u00e9monstrations, en lui donnant une force nouvelle directement pol\u00e9mique : ainsi dans les scolies de I, 8 (usage de la proposition 7) ; de I, 10 (usage de la proposition 9) ; de II, 3 (invocation de l'id\u00e9e de Dieu) ; de II, 7 (invocation des H\u00e9breux)... Le deuxi\u00e8me aspect, il est vrai, semble en retrait sur celui-ci ; car il arrive que les scolies se contentent de pr\u00e9senter un simple exemple de la proposition correspondante : ainsi en II, 8 (l'exemple des lignes dans le cercle) ; en IV, 40 (l'exemple si curieux de l'action de frapper) ; en IV, 63 (l'exemple du valide et du malade)... Mais il semble que la plupart des exemples chez Spinoza se d\u00e9passent dans deux directions, vers deux fonctions plus hautes et essentielles : l'une paradigmatique, l'autre casuistique. Ainsi en II, 13, sc., puis en III, 2, sc., se trouve dress\u00e9 le _mod\u00e8le_ du corps : non pas que le corps serve de mod\u00e8le \u00e0 la pens\u00e9e, et rompe le parall\u00e9lisme ou l'autonomie respective de la pens\u00e9e et de l'\u00e9tendue, mais il intervient comme un exemple d\u00e9veloppant une fonction paradigmatique, pour montrer \u00ab parall\u00e8lement \u00bb combien il y a de choses dans la pens\u00e9e m\u00eame qui d\u00e9passe la conscience. De m\u00eame le mod\u00e8le de la nature humaine, annonc\u00e9 en IV, 18, sc., d\u00e9velopp\u00e9 en V, 10, sc., et 20, sc. Enfin le mod\u00e8le du troisi\u00e8me genre, annonc\u00e9 en II, 40, sc., puis dans les derni\u00e8res lignes de V, 20 sc., et formul\u00e9 en V, 36, sc.\n\nD'autre part la fonction casuistique du pseudo-exemple appara\u00eet dans tous les scolies qui s'expriment, par rapport \u00e0 la d\u00e9monstration pr\u00e9c\u00e9dente, sous la forme d'un \u00ab c'est pr\u00e9cis\u00e9ment le cas \u00bb. L\u00e0 encore il ne s'agit pas d'un simple exemple, mais d'une stricte assignation des conditions sous lesquelles l'objet de la d\u00e9monstration correspondante se trouve effectivement r\u00e9alis\u00e9 : le scolie d\u00e9termine le cas subsum\u00e9 par la r\u00e8gle contenue dans la d\u00e9monstration correspondante, non pas comme un cas parmi d'autres, mais comme le cas qui remplit cette r\u00e8gle et satisfait \u00e0 toutes les conditions. Il arrive que les conditions soient restrictives, et qu'un scolie, parfois tr\u00e8s loin de la proposition correspondante, rappelle que cette proposition et la d\u00e9monstration devaient s'entendre en un sens restreint : II, 45, sc. ; IV, 33, sc. ; etc. Mais plus profond\u00e9ment il y a dans cet aspect des scolies quelque chose qui vient recouper le proc\u00e9d\u00e9 positif, puisque, au moins pour les erreurs et les passions, il est impossible d'obtenir une d\u00e9finition r\u00e9elle ind\u00e9pendamment des conditions qui effectuent l'objet pr\u00e9alablement indiqu\u00e9 dans la proposition et la d\u00e9monstration, impossible aussi de d\u00e9gager ce qu'il y a de positif dans l'erreur ou la passion si ces conditions ne sont pas d\u00e9termin\u00e9es dans le scolie. C'est pourquoi les scolies de ce type proc\u00e8dent sous la forme d'un \u00ab fiat \u00bb : voil\u00e0 comment la chose se produit... Ainsi le scolie de II, 35 explique comment l'erreur, d\u00e9finie comme une privation dans la proposition, se produit effectivement, et n'en a pas moins d\u00e9j\u00e0 une certaine positivit\u00e9 dans ces conditions o\u00f9 elle se produit. Ainsi encore, II, 44, ayant \u00e9nonc\u00e9 et d\u00e9montr\u00e9 que seule l'imagination consid\u00e8re les choses comme contingentes, le scolie se propose \u00e0 son tour de d\u00e9montrer \u00ab dans quelle condition cela a lieu \u00bb ( _qua ratione fiat_ ). Le livre III g\u00e9n\u00e9ralise ce proc\u00e9d\u00e9 : quand les propositions et les d\u00e9monstrations tracent dans leur progression continue le mouvement par lequel les affections s'encha\u00eenent et d\u00e9rivent les unes des autres, les scolies introduisent un arr\u00eat, comme une photo tout d'un coup prise, un figeage, une immobilit\u00e9 provisoire, un instantan\u00e9, qui montre que telle affection ou telle facult\u00e9 bien connues r\u00e9pondent effectivement, et dans telles conditions, \u00e0 ce dont parlait la proposition. Il en \u00e9tait d\u00e9j\u00e0 ainsi dans le livre II, avec la m\u00e9moire (II, 18, sc.), avec les notions communes (II, 40, sc. 1). Mais dans le livre III, se multiplient les formules des scolies du type : \u00ab Nous connaissons par l\u00e0 comment il peut arriver... \u00bb, \u00ab Nous voyons qu'il peut arriver \u00bb, \u00ab Cela a lieu parce que... \u00bb Et en m\u00eame temps les affections ou facult\u00e9s trouvent leur nom : non seulement M\u00e9moire, Notions communes dans le livre II, mais dans le livre III tous les noms d'affections, qui seront recueillies dans les d\u00e9finitions finales, comme dans un \u00e9cho de tous les scolies, Joie, Tristesse, Amour, Haine, etc. Comme si le mouvement des propositions, d\u00e9monstrations et corollaires poussait contin\u00fbment le flot des affections, mais que celui-ci ne formait ses vagues et ses cr\u00eates que dans les scolies. Comme si les propositions, d\u00e9monstrations et corollaires parlaient le plus haut langage, impersonnel et peu soucieux d'identifier ce dont il parle, puisque ce qu'il dit est de toutes fa\u00e7ons fond\u00e9 dans une v\u00e9rit\u00e9 sup\u00e9rieure \u2013 tandis que les scolies baptisent, donnent un nom, identifient, d\u00e9signent et d\u00e9noncent, sondant en profondeur ce que l'\u00ab autre \u00bb langage \u00e9talait et faisait avancer.\n\nLe second caract\u00e8re du scolie, ostensif, a donc \u00e0 son tour trois aspects principaux : axiomatique, paradigmatique et casuistique. Or ils mettent d\u00e9j\u00e0 constamment en jeu le dernier caract\u00e8re des scolies, pol\u00e9mique ou agressif. Ce caract\u00e8re ultime a lui aussi des aspects divers : tant\u00f4t il s'agit d'analyser la _confusion sp\u00e9culative_ ou la stupidit\u00e9 intellectuelle de ceux qui d\u00e9figurent Dieu, qui le traitent comme un \u00ab roi \u00bb, qui lui pr\u00eatent entendement et volont\u00e9, finalit\u00e9 et projet, figure et fonction, etc. (surtout les scolies du livre I). Tant\u00f4t il s'agit de d\u00e9terminer les conditions dans lesquelles se produisent l' _erreur sensible_ et les passions qui en d\u00e9coulent (surtout les scolies des livres II et III). Tant\u00f4t il s'agit de d\u00e9noncer le _mal pratique,_ c'est-\u00e0-dire les passions tristes, la contagion de ces passions, l'int\u00e9r\u00eat de ceux qui en profitent \u2013 cette d\u00e9nonciation se faisant surtout dans le livre IV, mais en rapport avec le projet le plus g\u00e9n\u00e9ral de l' _\u00c9thique_ tel qu'il est rappel\u00e9 dans les pr\u00e9faces ou conclusions de certaines parties. La pol\u00e9mique a donc pour son compte trois aspects, sp\u00e9culatif, sensible et pratique. Comment s'\u00e9tonner que tous ces aspects, et tous les caract\u00e8res dont ils d\u00e9pendent, se confirment et empi\u00e8tent les uns sur les autres ? Les grands scolies les r\u00e9unissent tous. Le scolie a toujours une intention positive ; mais il ne peut la remplir qu'\u00e0 l'aide d'un proc\u00e9d\u00e9 ostensif ; et il ne peut fonder celui-ci qu'en impliquant une pol\u00e9mique. Le proc\u00e9d\u00e9 ostensif, \u00e0 son tour, se trouve partag\u00e9 entre l'argumentation pol\u00e9mique qui lui donne sa pleine valeur, et le principe positif qu'il sert. On se demandera comment concilier la d\u00e9marche positive du scolie avec son argument pol\u00e9mique, critique et n\u00e9gateur. C'est que, inversement, la puissance pol\u00e9mique si vive de Spinoza se d\u00e9veloppe en silence, loin des discussions, au service d'une affirmation sup\u00e9rieure et d'une \u00ab ostensivit\u00e9 \u00bb sup\u00e9rieure. Selon Spinoza, la n\u00e9gation ne sert que pour nier le n\u00e9gatif, pour nier ce qui nie et ce qui obscurcit. La pol\u00e9mique, la n\u00e9gation, la d\u00e9nonciation ne sont l\u00e0 que pour nier ce qui nie, ce qui trompe et ce qui cache : ce qui profite de l'erreur, ce qui vit de la tristesse, ce qui pense dans le n\u00e9gatif. C'est pourquoi les scolies les plus pol\u00e9miques r\u00e9unissent, dans un style et sur un ton particuliers, les deux go\u00fbts supr\u00eames de l'affirmation sp\u00e9culative (celle de la substance) et de la joie pratique (celle des modes) : le double langage, pour une double lecture de l' _\u00c9thique_. \u00c0 la fois la pol\u00e9mique est le plus important dans les plus grands scolies, mais sa puissance se d\u00e9veloppe d'autant plus qu'elle est au service de l'affirmation sp\u00e9culative et de la joie pratique, et les fait se rejoindre dans l'\u00e9l\u00e9ment de l'univocit\u00e9.\n\n# INDEX\n\n# DES NOMS PROPRES\n\nAlqui\u00e9, , , , , , .\n\nAppuhn, .\n\nAristote, , , -143, .\n\nArnou, .\n\nBo\u00e8ce, .\n\nBonaventure, .\n\nBrochard, .\n\nBruno, , .\n\nBusolt, .\n\nCrescas, .\n\nDarbon, .\n\nDelbos, .\n\nDescartes, -24, , , -62, -75, , , -152, , , -302.\n\nDuns Scot, -58, , .\n\nEckhart, , .\n\n\u00c9lie, .\n\nErdmann, .\n\nFichte, , .\n\nFoucher de Careil, , .\n\nFranc\u00e8s, .\n\nFriedmann, , .\n\nGandillac, , , , , .\n\nGeoffroy-Saint-Hilaire, .\n\nGilson, , , , , , .\n\nGu\u00e9roult, , , , , .\n\nHeereboord, , .\n\nHobbes, , , .\n\nHuan, .\n\nKant, , .\n\nKaufmann, .\n\nKoyr\u00e9, , , .\n\nLachi\u00e8ze-Rey, , , .\n\nLaporte, .\n\nLasbax, .\n\nLeibniz, , -63, -68, , -97, , -139, , , , -209, -212, , , -311.\n\nL\u00e9on, , .\n\nLossky, .\n\nLucr\u00e8ce, .\n\nMerleau-Ponty, .\n\nNicolas de Cues, , , .\n\nNietzsche, .\n\nPlaton, -154.\n\nPlotin, -158, -161.\n\nPolin, .\n\nR\u00e9gis, .\n\nR\u00e9vah, .\n\nRivaud, , .\n\nRobinson, , .\n\nRousseau, .\n\nSchelling, , , .\n\nStrauss, .\n\nSuarez, , , .\n\nVajda, .\n\n# TABLE DES MATI\u00c8RES\n\nINTRODUCTION : R\u00d4LE ET IMPORTANCE DE L'EXPRESSION... -18\n\n> Importance du mot \u00ab exprimer \u00bb chez Spinoza. Son triple emploi : exprimer une essence, exprimer l'essence, exprimer l'existence. \u2013 Caract\u00e8re expressif de l'attribut, du mode et de l'id\u00e9e. \u2013 Exprimer : expliquer ou d\u00e9velopper ; impliquer ou envelopper ; compliquer, contenir ou comprendre. \u2013 Leibniz et Spinoza comptent sur l'id\u00e9e d'expression pour d\u00e9passer les difficult\u00e9s du cart\u00e9sianisme. \u2013 Pourquoi les commentateurs n'ont gu\u00e8re retenu l'id\u00e9e d'expression chez Spinoza. \u2013 Pourquoi l'id\u00e9e d'expression chez Spinoza n'est ni objet de d\u00e9finition, ni objet de d\u00e9monstration. Expression et d\u00e9monstration.\n\nPREMI\u00c8RE PARTIE : LES TRIADES DE LA SUBSTANCE\n\n_CHAPITRE I : Distinction num\u00e9rique et distinction r\u00e9elle_... -32\n\n> L'expression comme triade. Premi\u00e8re triade de l'expression : substance, attribut, essence.\n> \n> Le probl\u00e8me des distinctions chez Descartes. \u2013 Selon Descartes, il y a des substances de m\u00eame attribut : distinctions num\u00e9riques qui sont r\u00e9elles. \u2013 Et il y a des substances d'attribut diff\u00e9rent : distinctions r\u00e9elles qui sont num\u00e9riques. \u2013 Th\u00e9orie de Spinoza : il n'y a pas plusieurs substances de m\u00eame attribut, la distinction num\u00e9rique n'est jamais r\u00e9elle. \u2013 Cons\u00e9quence : la distinction r\u00e9elle n'est jamais num\u00e9rique, il n'y a pas plusieurs substances correspondant aux attributs diff\u00e9rents. \u2013 Les huit premi\u00e8res propositions de l' _\u00c9thique_ n'ont pas un sens simplement hypoth\u00e9tique. Gen\u00e8se ou constitution de la substance.\n> \n> Opposition de Spinoza avec Descartes, du point de vue de la th\u00e9orie des distinctions. Signification de la distinction r\u00e9elle chez Spinoza.\n\n _CHAPITRE II : L'attribut comme expression_... -43\n\n> Le statut de l'attribut et son caract\u00e8re expressif. Les textes du _Court Trait\u00e9._\n> \n> Probl\u00e8me des noms divins. \u2013 Attribut, attribution et qualit\u00e9. \u2013 Les attributs sont des formes communes \u00e0 Dieu et aux \u00ab cr\u00e9atures \u00bb. \u2013 Comment cette th\u00e8se ne supprime nullement la distinction d'essence entre Dieu et les choses. \u2013 Spinoza, partisan de l'univocit\u00e9 : contre l'\u00e9quivocit\u00e9, contre l'\u00e9minence, contre l'analogie. \u2013 Univocit\u00e9 des attributs et noms divins.\n> \n> Opposition des attributs et des propres. \u2013 Les trois esp\u00e8ces de propres. \u2013 Les propres ne sont pas expressifs.\n\n _CHAPITRE III : Attributs et noms divins_... -58\n\n> Th\u00e9ologie n\u00e9gative et m\u00e9thode d'analogie. \u2013 L'une et l'autre impliquent une confusion des attributs avec les propres. Confusion de la nature de Dieu avec de simples propri\u00e9t\u00e9s, confusion de l'expression avec la \u00ab r\u00e9v\u00e9lation \u00bb. \u2013 Pourquoi ces confusions sont constantes dans la th\u00e9ologie. \u2013 Oppositions du signe et de l'expression. \u2013 Noms expressifs et mots imp\u00e9ratifs. \u2013 Les attributs comme affirmations pures. \u2013 Distinction r\u00e9elle et affirmation.\n> \n> Comment des \u00ab expressions \u00bb diverses d\u00e9signent une seule et m\u00eame chose. La logique du sens. \u2013 Th\u00e9ologie positive et univocit\u00e9. \u2013 Distinction formelle selon Duns Scot et distinction r\u00e9elle selon Spinoza. \u2013 De l'univocit\u00e9 \u00e0 l'immanence.\n\n _CHAPITRE IV : L'absolu_... -71\n\n> L'\u00e9galit\u00e9 des attributs. \u2013 L'infiniment parfait et l'absolument infini.\n> \n> L'infiniment parfait comme \u00ab nerf \u00bb des preuves cart\u00e9siennes de l'existence de Dieu. \u2013 Sens des objections dirig\u00e9es contre la preuve ontologique de Descartes. \u2013 Leibniz et Spinoza : insuffisance de l'infiniment parfait. \u2013 Spinoza : l'absolument infini comme raison de l'infiniment parfait. \u2013 La preuve ontologique chez Spinoza ; plan du d\u00e9but de l' _\u00c9thique_. \u2013 Diff\u00e9rences du _Court Trait\u00e9_ et de l' _\u00c9thique. \u2013_ Leibniz et Spinoza du point de vue de la preuve ontologique. \u2013 La d\u00e9finition 6 est une d\u00e9finition r\u00e9elle.\n> \n> Seconde triade de l'expression : le parfait, l'infini, l'absolu.\n\n _CHAPTRE V : La puissance_... -84\n\n> Descartes, accus\u00e9 de rapidit\u00e9 ou de facilit\u00e9. \u2013 Les formulations de la preuve a posteriori chez Descartes : la notion de \u00ab facile \u00bb. \u2013 La quantit\u00e9 de r\u00e9alit\u00e9 ou de perfection, comme nerf de la preuve a posteriori de Descartes. \u2013 Insuffisance de la quantit\u00e9 de r\u00e9alit\u00e9 : la puissance comme raison.\n> \n> La preuve a posteriori dans le _Court Trait\u00e9. \u2013_ Formation d'un argument des puissances. \u2013 Les deux puissances : de penser et de conna\u00eetre, d'exister et d'agir. \u2013 La preuve a posteriori dans l' _\u00c9thique_ : la puissance d'exister, consid\u00e9r\u00e9e directement.\n> \n> Les attributs : conditions sous lesquelles on attribue \u00e0 quelque chose une puissance. \u2013 Cas de la substance absolument infinie, cas des \u00eatres finis. \u2013 Puissance et essence. \u2013 Les choses sont des modes, c'est-\u00e0-dire ont une puissance.\n> \n> Puissance et pouvoir d'\u00eatre affect\u00e9. \u2013 Troisi\u00e8me triade de l'expression : l'essence comme puissance, ce dont elle est l'essence, le pouvoir d'\u00eatre affect\u00e9.\n\nDEUXI\u00c8ME PARTIE : LE PARALL\u00c9LISME ET L'IMMANENCE\n\n_CHAPITRE VI : L'expression dans le parall\u00e9lisme_... -98\n\n> La production comme re-expression. \u2013 Dieu produit comme il se comprend, Dieu produit comme il existe. \u2013 Univocit\u00e9 de la cause : Dieu, cause de toutes choses au m\u00eame sens que cause de soi. \u2013 Contre l'analogie. \u2013 Logique du sens et re-expression.\n> \n> Ordre de production. \u2013 Exclusion d'une causalit\u00e9 r\u00e9elle entre modes d'attribut diff\u00e9rent. \u2013 Le parall\u00e9lisme : identit\u00e9 d'ordre, identit\u00e9 de connexion, identit\u00e9 d'\u00eatre. \u2013 L'identit\u00e9 de connexion et le principe d'\u00e9galit\u00e9. \u2013 L'identit\u00e9 d'\u00eatre : mode et modification. \u2013 Nouvelle triade de l'expression. : attribut, mode et modification.\n\n _CHAPITRE VII : Les deux puissances et l'id\u00e9e de Dieu_... -113\n\n> Complexit\u00e9 de la d\u00e9monstration du parall\u00e9lisme : l'id\u00e9e et son objet. \u2013 Parall\u00e9lisme \u00e9pist\u00e9mologique et parall\u00e9lisme ontologique.\n> \n> \u00c0 toute id\u00e9e correspond quelque chose : influence d'Aristote. \u2013 \u00c0 toute chose correspond une id\u00e9e. \u2013 Pourquoi Dieu se comprend n\u00e9cessairement. \u2013 \u00ab N\u00e9cessit\u00e9 \u00bb de l'id\u00e9e de Dieu. \u2013 La puissance de penser est n\u00e9cessairement \u00e9gale \u00e0 la puissance d'exister et d'agir.\n> \n> Les deux puissances et leur \u00e9galit\u00e9. \u2013 Distinction de la puissance et de l'attribut. \u2013 Les attributs et la puissance d'exister. \u2013 L'attribut pens\u00e9e et la puissance de penser. \u2013 Source des \u00ab privil\u00e8ges \u00bb de l'attribut pens\u00e9e.\n> \n> \u00ab Possibilit\u00e9 \u00bb de l'id\u00e9e de Dieu. \u2013 Pourquoi l'entendement infini est un produit. \u2013 Les trois privil\u00e8ges de l'attribut pens\u00e9e.\n> \n> Pourquoi il \u00e9tait n\u00e9cessaire de passer par le parall\u00e9lisme \u00e9pist\u00e9mologique. \u2013 Seule l'id\u00e9e de Dieu permet de conclure de l'unit\u00e9 de la substance \u00e0 l'unit\u00e9 d'une modification. Transfert de l'expression.\n\n _CHAPITRE VIII : Expression et id\u00e9e_... -129\n\n> Premier aspect de la m\u00e9thode, formel ou reflexif : l'id\u00e9e de l'id\u00e9e, l'id\u00e9e qui s'explique par notre puissance de comprendre. \u2013 Forme et r\u00e9flexion.\n> \n> Passage au deuxi\u00e8me aspect. \u2013 Deuxi\u00e8me aspect de la m\u00e9thode, mat\u00e9riel ou g\u00e9n\u00e9tique : le contenu de l'id\u00e9e vraie, l'id\u00e9e ad\u00e9quate, l'id\u00e9e qui exprime sa propre cause. \u2013 Id\u00e9e ad\u00e9quate et d\u00e9finition g\u00e9n\u00e9tique. \u2013 R\u00f4le de la fiction. \u2013 Comment la gen\u00e8se nous conduit \u00e0 l'id\u00e9e de Dieu. \u2013 Passage au troisi\u00e8me aspect : arriver le plus vite possible \u00e0 l'id\u00e9e de Dieu. \u2013 Troisi\u00e8me aspect de la m\u00e9thode : unit\u00e9 de la forme et du contenu, l'automate spirituel, la concat\u00e9nation. \u2013 Expression et repr\u00e9sentation.\n> \n> D\u00e9finition mat\u00e9rielle et d\u00e9finition formelle de la v\u00e9rit\u00e9. \u2013 L'expression, l'id\u00e9e ad\u00e9quate et l'id\u00e9e r\u00e9flexive. \u2013 Caract\u00e8re ad\u00e9quat de l'id\u00e9e de Dieu.\n\n _CHAPITRE IX : L'inad\u00e9quat_... -139\n\n> Comment nous \u00ab avons \u00bb des id\u00e9es. \u2013 Les conditions sous lesquelles nous avons des id\u00e9es ne semblent pas permettre que ces id\u00e9es soient ad\u00e9quates. \u2013 En quel sens \u00ab envelopper \u00bb s'oppose \u00e0 \u00ab exprimer \u00bb. \u2013 L'id\u00e9e inad\u00e9quate est inexpressive. \u2013 Probl\u00e8me de Spinoza : comment arriverons-nous \u00e0 avoir des id\u00e9es ad\u00e9quates ? \u2013 Quelque chose de positif dans l'id\u00e9e inad\u00e9quate.\n> \n> L'insuffisance du clair et du distinct. \u2013 Le clair et le distinct servent seulement \u00e0 la recognition. \u2013 Ils manquent d'une raison suffisante. \u2013 Descartes en reste au contenu repr\u00e9sentatif, il n'atteint pas le contenu expressif de l'id\u00e9e. Il en reste \u00e0 la forme de la conscience psychologique, il n'atteint pas la forme logique. \u2013 Le clair et le distinct laissent \u00e9chapper l'essence et la cause. \u2013 Leibniz et Spinoza, du point de vue de la critique de l'id\u00e9e claire et distincte.\n\n _CHAPITRE X : Spinoza contre Descartes_... -152\n\n> En quel sens la m\u00e9thode de Descartes est analytique. \u2013 Insuffisance de cette m\u00e9thode, selon Spinoza. \u2013 M\u00e9thode synth\u00e9tique. \u2013 Aristote et Spinoza : conna\u00eetre par la cause. \u2013 Comment la cause elle-m\u00eame est connue.\n> \n> Dieu comme cause de soi, selon Descartes : \u00e9quivocit\u00e9, \u00e9minence, analogie. \u2013 Dieu comme cause de soi, selon Spinoza : univocit\u00e9. \u2013 Univocit\u00e9 et immanence. \u2013 Les axiomes cart\u00e9siens et leur transformation chez Spinoza.\n\n _CHAPITRE XI : L'immanence et les \u00e9l\u00e9ments historiques de l'expression_... -169\n\n> Probl\u00e8me de la participation dans le n\u00e9o-platonisme. \u2013 Don et \u00e9manation. \u2013 Double diff\u00e9rence entre la cause \u00e9manative et la cause immanente, Comment, dans le n\u00e9o-platonisme, une cause immanente se joint \u00e0 la cause \u00e9manative : l'\u00eatre ou l'intelligence. \u2013 _Complicare-explicare. \u2013_ Immanence et principe d'\u00e9galit\u00e9. \u2013 L'id\u00e9e d'expression dans l'\u00e9manation. \u2013 L'id\u00e9e d'expression dans la cr\u00e9ation : expression et similitude. \u2013 Comment, dans la th\u00e9orie de la cr\u00e9ation, une cause immanente se joint \u00e0 la cause exemplaire.\n> \n> L'expression, selon Spinoza, cesse d'\u00eatre subordonn\u00e9e aux hypoth\u00e8ses de la cr\u00e9ation et de l'\u00e9manation. \u2013 Opposition de l'expression et du signe. \u2013 Immanence : distinction et univocit\u00e9 des attributs. \u2013 Th\u00e9orie spinoziste de la hi\u00e9rarchie. \u2013 L'expression et les diff\u00e9rents sens du principe d'\u00e9galit\u00e9.\n\nTROISI\u00c8ME PARTIE : TH\u00c9ORIE DU MODE FINI\n\n_CHAPITRE XII : L'essence de mode : passage de l'infini au fini_... -182\n\n> Sens du mot \u00ab partie \u00bb. \u2013 Qualit\u00e9, quantit\u00e9, intensive, quantit\u00e9 extensive. \u2013 Les deux infinis modaux, dans la _Lettre \u00e0 Meyer._\n> \n> L'essence de mode comme r\u00e9alit\u00e9 physique : degr\u00e9 de puissance ou quantit\u00e9 intensive. \u2013 Statut du mode non-existant. \u2013 Essence et existence. \u2013 Essence et existence de l'essence. \u2013 Probl\u00e8me de la distinction des essences de modes. \u2013 Th\u00e9orie de la distinction ou de la diff\u00e9renciation quantitative. \u2013 La production des essences : essence de mode et complication.\n> \n> L'expression quantitative.\n\n _CHAPITRE XIII : L'existence du mode_... -196\n\n> En quoi consiste l'existence du mode : existence et parties extensives. \u2013 La quantit\u00e9 extensive, seconde forme de la quantit\u00e9. \u2013 Difference de la quantit\u00e9 et du nombre. \u2013 Les corps simples. \u2013 Il n'y a pas lieu de chercher des essences correspondant aux corps les plus simples.\n> \n> Premi\u00e8re triade de l'expression dans le mode fini : essence, rapport caract\u00e9ristique, parties extensives. \u2013 Lois de composition et de d\u00e9composition des rapports.\n> \n> Sens de la distinction de l'essence et de l'existence du mode. \u2013 Probl\u00e8me de la distinction des modes existants. \u2013 Comment le mode existant se distingue de l'attribut de mani\u00e8re extrins\u00e8que. \u2013 Mode existant et explication.\n\n _CHAPITRE XIV : Qu'est-ce que peut un corps ?_... -213\n\n> Seconde triade de l'expression dans le mode fini : essence, pouvoir d'\u00eatre affect\u00e9, affections qui remplissent ce pouvoir. \u2013 Affections de la substance et affections du mode. \u2013 Affections actives et affections passives. \u2013 Les affects ou sentiments. \u2013 Nous semblons condamn\u00e9s aux id\u00e9es inad\u00e9quates et aux sentiments passifs. \u2013 Les variations existentielles du mode fini. \u2013 Force active et force passive chez Leibniz, puissance d'agir et puissance de p\u00e2tir chez Spinoza. \u2013 En quoi la puissance d'agir est seule positive et r\u00e9elle. \u2013 Inspiration physique : notre pouvoir d'\u00eatre affect\u00e9 est toujours rempli. \u2013 Inspiration \u00e9thique : nous sommes s\u00e9par\u00e9s de ce que nous pouvons.\n> \n> Critique du spinozisme par Leibniz, caract\u00e8re ambigu de cette critique. \u2013 Ce qui est commun \u00e0 Leibniz et \u00e0 Spinoza : le projet d'un nouveau naturalisme, contre Descartes. \u2013 Les trois niveaux chez Leibniz et chez Spinoza. \u2013 La v\u00e9ritable opposition de Leibniz et de Spinoza : le _conatus. \u2013_ L'affection comme d\u00e9termination du _conatus_. \u2013 En quel sens la passion nous s\u00e9pare de ce que nous pouvons. \u2013 La nature expressive : naturalisme finalis\u00e9, ou naturalisme sans finalit\u00e9 ?\n\n _CHAPITRE XV : Les trois ordres et le probl\u00e8me du mal_... -233\n\n>  _Facies totius universi_. \u2013 En quel sens deux rapports peuvent ne pas se composer. \u2013 Les trois ordres, correspondant \u00e0 la triade du mode : l'ordre des essences, l'ordre des rapports, l'ordre des rencontres. \u2013 Importance du th\u00e8me de la rencontre fortuite chez Spinoza.\n> \n> Rencontre entre corps dont les rapports se composent. \u2013 Augmenter ou aider la puissance d'agir. \u2013 Comment la distinction des passions joyeuses et des passions tristes vient se joindre \u00e0 celle des affections actives et des affections passives. \u2013 Rencontre entre corps dont les rapports ne se composent pas. \u2013 Passion triste et \u00e9tat de nature. \u2013 Comment arriverons-nous \u00e0 \u00e9prouver des passions joyeuses ?\n> \n> Pas de bien ni de mal, mais du bon et du mauvais. Le mal comme mauvaise rencontre ou d\u00e9composition d'un rapport. \u2013 M\u00e9taphore de l'empoisonnement. \u2013 Le mal n'est rien dans l'ordre des rapports ; le premier contresens de Blyenbergh. \u2013 Le mal n'est rien dans l'ordre des essences : deuxi\u00e8me contresens de Blyenbergh. \u2013 Le mal et l'ordre des rencontres ; l'exemple de l'aveugle et le troisi\u00e8me contresens de Blyenbergh.\n> \n> Sens de la th\u00e8se : le mal n'est rien. \u2013 Substitution de la diff\u00e9rence \u00e9thique \u00e0 l'opposition morale.\n\n _CHAPITRE XVI : Vision \u00e9thique du monde_... -251\n\n> Principe du rapport inverse de l'action et de la passion dans l'\u00e2me et dans le corps. \u2013 Opposition de Spinoza \u00e0 ce principe : la signification pratique du parall\u00e9lisme.\n> \n> Le droit naturel : pouvoir et droit. \u2013 Les quatre oppositions du droit naturel avec la loi naturelle antique. \u2013 \u00c9tat de nature et hasard des rencontres. \u2013 La raison sous son premier aspect : effort pour organiser les rencontres. \u2013 La diff\u00e9rence \u00e9thique : l'homme raisonnable, libre ou fort. \u2013 Adam. \u2013 \u00c9tat de nature et raison. \u2013 N\u00e9cessit\u00e9 d'une instance qui favorise l'effort de la raison. \u2013 La cit\u00e9 : diff\u00e9rences et ressemblances entre l'\u00e9tat civil et l'\u00e9tat de raison.\n> \n> L'\u00e9thique pose les probl\u00e8mes en termes de pouvoir et de puissance. \u2013 Opposition de l'\u00e9thique et de la morale. \u2013 Aller jusqu'au bout de ce qu'on peut. \u2013 Signification pratique de la philosophie. \u2013 D\u00e9noncer la tristesse et ses causes. \u2013 Affirmation et joie.\n\n _CHAPITRE XVII : Les Notions communes_... -267\n\n> Premi\u00e8re question : Comment arriverons-nous \u00e0 \u00e9prouver un maximum de passions joyeuses ? \u2013 Deuxi\u00e8me question : Comment arriverons-nous \u00e0 \u00e9prouver des affections actives ? \u2013 Joie passive et joie active.\n> \n> Convenance des corps, composition des rapports et communaut\u00e9 de composition. \u2013 Points de vue plus ou moins g\u00e9n\u00e9raux. \u2013 Les notions communes : leurs vari\u00e9t\u00e9s, suivant leur g\u00e9n\u00e9ralit\u00e9. \u2013 Les notions communes sont des id\u00e9es g\u00e9n\u00e9rales, mais non des id\u00e9es abstraites. \u2013 Critique de l'id\u00e9e abstraite. \u2013 De Spinoza \u00e0 Geoffroy St Hilaire. \u2013 Les notions communes sont n\u00e9cessairement ad\u00e9quates. \u2013 R\u00e9ponse \u00e0 la question : Comment arriverons-nous \u00e0 former des id\u00e9es ad\u00e9quates ? \u2013 Notion commune et expression.\n> \n> L'ordre de formation des notions communes va des moins g\u00e9n\u00e9rales aux plus g\u00e9n\u00e9rales. \u2013 La joie passive nous induit \u00e0 former une notion commune. \u2013 La raison sous son deuxi\u00e8me aspect : formation des notions communes. \u2013 Sens pratique de la notion commune : nous donner des joies actives. \u2013 Comment, \u00e0 partir des notions communes les moins g\u00e9n\u00e9rales, nous formons les plus g\u00e9n\u00e9rales. \u2013 Comprendre les tristesses in\u00e9vitables.\n\n _CHAPITRE XVIII : Vers le troisi\u00e8me genre_... -281\n\n> Complexit\u00e9 du premier genre de connaissance : \u00e9tat de nature, \u00e9tat civil, \u00e9tat de religion. \u2013 Les signes et le premier genre.\n> \n> Le second genre et l'\u00e9tat de raison. \u2013 Application des notions communes aux modes existants. \u2013 Les notions communes comme d\u00e9couverte de l' _\u00c9thique_. Pressentiments dans le _Trait\u00e9 de la r\u00e9forme. \u2013_ Harmonies entre le premier genre de connaissance et le second. \u2013 Harmonies de la raison et de l'imagination.\n> \n> Les notions communes comme conditions de notre _connaissance. \u2013_ Des notions communes \u00e0 l'id\u00e9e de Dieu : en quel sens elle appartient au second genre, en quel sens elle nous fait passer au troisi\u00e8me. \u2013 Notions communes et formes communes. \u2013 Le troisi\u00e8me genre et l'ordre des essences.\n\n _CHAPITRE XIX : B\u00e9atitude_... -298\n\n> Les trois d\u00e9terminations du troisi\u00e8me genre. \u2013 Joies actives du troisi\u00e8me genre.\n> \n> Diff\u00e9rence entre la joie active du troisi\u00e8me genre et celle du second. \u2013 L'id\u00e9e de nous-m\u00eames. \u2013 Affections adventices et affection inn\u00e9es. \u2013 L'inn\u00e9 du second genre et l'inn\u00e9 du troisi\u00e8me. \u2013 Le Dieu du second genre et le Dieu du troisi\u00e8me. \u2013 Le troisi\u00e8me genre et l'expression.\n> \n> Comment nous acc\u00e9dons au troisi\u00e8me genre, durant notre existence. \u2013 Limites de cet acc\u00e8s. \u2013 Contre l'interpr\u00e9tation math\u00e9matique et id\u00e9aliste des essences. \u2013 Diff\u00e9rence de nature entre la dur\u00e9e et l'\u00e9ternit\u00e9 : critique du concept d'immortalit\u00e9. \u2013 La mort. \u2013 Les affections du troisi\u00e8me genre ne remplissent enti\u00e8rement notre pouvoir d'\u00eatre affect\u00e9 qu'apr\u00e8s la mort. \u2013 En quel sens l'existence est une \u00e9preuve : l'id\u00e9e de salut chez Spinoza. \u2013 Partie intensive et parties extensives : leur importance respective du point de vue de l'expression. Devenir expressif.\n\nCONCLUSION : TH\u00c9ORIE DE L'EXPRESSION CHEZ LEIBNIZ ET CHEZ SPINOZA _(l'expressionnisme en philosophie)_... -311\n\n> Exprimer : \u00eatre, conna\u00eetre, agir ou produire. \u2013 Sens historique de ce concept. \u2013 Sens que lui donnent Leibniz et Spinoza : le triple aspect de la r\u00e9action contre Descartes. \u2013 La diff\u00e9rence Leibniz-Spinoza : les expressions \u00e9quivoques et l'analogie, les expressions univoques et l'univocit\u00e9. \u2013 Les trois figures de l'Univoque selon Spinoza. \u2013 Le paradoxe de l'expression : l'exprim\u00e9.\n\nAPPENDICE : \u00c9tude formelle du plan de l' _\u00c9thique_ et du r\u00f4le des scolies dans la r\u00e9alisation de ce plan... -322\nDU M\u00caME AUTEUR\n\nPR\u00c9SENTATION DE SACHER-MASOCH, _1967_ (\u00ab Reprise \u00bb, no 15)\n\nSPINOZA ET LE PROBL\u00c8ME DE L'EXPRESSION, _1968_\n\nLOGIQUE DU SENS, _1969_\n\nL'ANTI-\u0152DIPE (avec F\u00e9lix Guattari), _1972_\n\nKAFKA \\- Pour une litt\u00e9rature mineure (avec F\u00e9lix Guattari), _1975_\n\nRHIZOME (avec F\u00e9lix Guattari), _1976_ (repris dans _Mille plateaux_ )\n\nSUPERPOSITIONS (avec Carmelo Bene), _1979_\n\nMILLE PLATEAUX (avec F\u00e9lix Guattari), _1980_\n\nSPINOZA \\- PHILOSOPHIE PRATIQUE, _1981_ (\u00ab Reprise \u00bb, no 4)\n\nCIN\u00c9MA 1 - L'IMAGE-MOUVEMENT, _1983_\n\nCIN\u00c9MA 2 - L'IMAGE-TEMPS, _1985_\n\nFOUCAULT, _1986_ (\u00ab Reprise \u00bb, no 7)\n\nP\u00c9RICL\u00c8S ET VERDI. La philosophie de Fran\u00e7ois Ch\u00e2telet, _1988_\n\nLE PLI. Leibniz et le baroque, _1988_\n\nPOURPARLERS, _1990_ (\u00ab Reprise \u00bb, no 6)\n\nQU'EST-CE QUE LA PHILOSOPHIE ? (avec F\u00e9lix Guattari), _1991_ (\u00ab Reprise \u00bb, no 13)\n\nL'\u00c9PUIS\u00c9 ( _in_ Samuel Beckett, _Quad_ ), _1992_\n\nCRITIQUE ET CLINIQUE, _1993_\n\nL'\u00ceLE D\u00c9SERTE. Textes et entretiens 1953-1974 (\u00e9dition pr\u00e9par\u00e9e par David Lapoujade), _2002_\n\nDEUX R\u00c9GIMES DE FOUS. Textes et entretiens 1975-1995 (\u00e9dition pr\u00e9par\u00e9e par David Lapoujade), _2003_\n\n_Aux P.U.F._\n\nEMPIRISME ET SUBJECTIVIT\u00c9, _1953_\n\nNIETZSCHE ET LA PHILOSOPHIE, _1962_\n\nLA PHILOSOPHIE DE KANT, _1963_\n\nPROUST ET LES SIGNES, _1964_ \\- \u00e9d. augment\u00e9e, _1970_\n\nNIETZSCHE, _1965_\n\nLE BERGSONISME, _1966_\n\nDIFF\u00c9RENCE ET R\u00c9P\u00c9TITION, _1969_\n\n_Aux \u00c9ditions Flammarion_\n\nDIALOGUES (en collaboration avec Claire Parnet), _1977_\n\n_Aux \u00c9ditions du Seuil_\n\nFRANCIS BACON : LOGIQUE DE LA SENSATION, _(1981), 2002_\nCette \u00e9dition \u00e9lectronique du livre _Spinoza et le probl\u00e8me de l'expression_ de Gilles Deleuze a \u00e9t\u00e9 r\u00e9alis\u00e9e le 17 d\u00e9cembre 2013 par les \u00c9ditions de Minuit \u00e0 partir de l'\u00e9dition papier du m\u00eame ouvrage dans la collection \u00ab Arguments \u00bb\n\n(ISBN 9782707300072, n\u00b0 d'\u00e9dition 4856, n\u00b0 d'imprimeur 100095, d\u00e9p\u00f4t l\u00e9gal mars 2010).\n\nLe format ePub a \u00e9t\u00e9 pr\u00e9par\u00e9 par ePagine.  \nwww.epagine.fr\n\nISBN 9782707330222\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nA woman is forced to question her own identity in this riveting and emotionally charged thriller by the blockbuster bestselling author of The Good Girl, Mary Kubica\n\nJessie Sloane is on the path to rebuilding her life after years of caring for her ailing mother. She rents a new apartment and applies for college. But when the college informs her that her social security number has raised a red flag, Jessie discovers a shocking detail that causes her to doubt everything she's ever known.\n\nFinding herself suddenly at the center of a bizarre mystery, Jessie tumbles down a rabbit hole, which is only exacerbated by grief and a relentless lack of sleep. As days pass and the insomnia worsens, it plays with Jessie's mind. Her judgment is blurred, her thoughts are hampered by fatigue. Jessie begins to see things until she can no longer tell the difference between what's real and what she's only imagined.\n\nMeanwhile, twenty years earlier and two hundred and fifty miles away, another woman's split-second decision may hold the key to Jessie's secret past. Has Jessie's whole life been a lie or have her delusions gotten the best of her?\nPraise for Mary Kubica\n\n\" _When the Lights Go Out_ will keep you questioning everything\u2014and everyone\u2014until the riveting conclusion. A twisty, captivating, edge-of-your-seat read.\" \u2014MEGAN MIRANDA\n\n\"A page-turning whodunit and a moving account of grief.\" \u2014RUTH WARE\n\n\"Mary Kubica is a must-read.\" \u2014LISA SCOTTOLINE\n\n\"Mary Kubica has a knack for crafting engrossing psychological thrillers.\" \u2014 _I_ N _STYLE_\n\n\"Kubica [is] a writer of vise-like control.\" \u2014 _CHICAGO TRIBUNE_\n\n\"Kubica leads the pack when it comes to her genre.\" \u2014 _KIRKUS REVIEWS_\n\n\"[ _Pretty Baby_ ] raises the ante on the genre and announces the welcome second coming of a talent well worth watching.\" \u2014 _LA TIMES_\n\n\"Hypnotic.\" \u2014 _PEOPLE_\n\n\"The twists you expect aren't the ones that arrive.\" \u2014NPR\n\n\"Suspense done well.\" \u2014 _NEW YORK MAGAZINE_\n\n\"Intricately wrought suspense.\" \u2014 _VULTURE_\n\n\"[Kubica] practically has you holding your breath for all 300 pages.\" \u2014 _BUSTLE.COM_\n\n\"Will encourage comparisons to _Gone Girl_.\" \u2014 _PUBLISHERS WEEKLY, STARRED REVIEW_\n\n\"A twisty roller coaster ride.\" \u2014LISA GARDNER\n\n\"Lots of twists and turns.... Comparisons to _Gone Girl_ and _The Silent Wife_ are deserved.\" \u2014 _HUFFINGTON POST_\n\n\"An utterly mesmerizing tale.... Haunting, psychologically deft and full of hairpin turns.\" \u2014MEGAN ABBOTT\n\n\"Brilliant, intense, and utterly addictive. Be prepared to run a gauntlet of emotions!\" \u2014B. A. PARIS\nALSO BY MARY KUBICA\n\nThe Good Girl\n\nPretty Baby\n\nDon't You Cry\n\nEvery Last Lie\nWHEN THE LIGHTS GO OUT\n\nMary Kubica\n\nAbout the Author\n\n**Mary Kubica** is the _New York Times_ bestselling author of several novels, including the blockbuster _The Good Girl_ , which has sold over a million copies. She holds a bachelor of arts degree in history and American literature from Miami University in Oxford, Ohio. She lives outside of Chicago with her husband and two children. Follow Mary on Twitter, @marykubica.\n\nwww.MaryKubica.com\nFor\n\nDick & Eloise\n\nRudy & Myrtle\n\"The mind is its own place,\n\nand in itself can make\n\na heaven of hell,\n\na hell of heaven.\"\n\n\u2014John Milton, _Paradise Lost_\nContents\n\nPrologue\n\nChapter 1: Jessie\n\nChapter 2: Eden\n\nChapter 3: Jessie\n\nChapter 4: Eden\n\nChapter 5: Jessie\n\nChapter 6: Eden\n\nChapter 7: Jessie\n\nChapter 8: Eden\n\nChapter 9: Jessie\n\nChapter 10: Eden\n\nChapter 11: Jessie\n\nChapter 12: Eden\n\nChapter 13: Jessie\n\nChapter 14: Eden\n\nChapter 15: Jessie\n\nChapter 16: Eden\n\nChapter 17: Jessie\n\nChapter 18: Eden\n\nChapter 19: Jessie\n\nChapter 20: Eden\n\nChapter 21: Jessie\n\nChapter 22: Eden\n\nChapter 23: Jessie\n\nChapter 24: Eden\n\nChapter 25: Jessie\n\nChapter 26: Eden\n\nChapter 27: Jessie\n\nChapter 28: Eden\n\nChapter 29: Jessie\n\nChapter 30: Eden\n\nChapter 31: Jessie\n\nChapter 32: Eden\n\nChapter 33: Jessie\n\nChapter 34: Eden\n\nChapter 35: Eden\n\nChapter 36: Jessie\n\nChapter 37: Eden\n\nChapter 38: Jessie\n\nChapter 39: Eden\n\nAcknowledgments\nprologue\n\nThe city surrounds me. A panorama. With arms outstretched, I can't help but spin, taking it all in. Enjoying the view, knowing fully well this may be the last thing my eyes ever see.\n\nI stare at the four metal steps before me, aware of how frail and broken-down they look. They're orange with rust, paint flaking, some of the slats loose so that when I press my foot to the first step, it buckles beneath me and I fall.\n\nStill, I have no choice but to climb.\n\nI pull myself back up, set my hands on the rails and scale the steps. The sweat bleeds from my palms so that the metal beneath them is slippery, slick. I can't hold tight. I slip from the second step, try again. I call out, voice cracking, a voice that doesn't sound like mine.\n\nAs I reach the roof's ledge, my knees give. It takes everything I have not to topple over the edge of the building and onto the street below. Seventeen floors.\n\nI'm so high I could touch the clouds, I think. The sense of vertigo is overpowering. The ground whooshes up and at me, the skyscrapers, the trees starting to sway until I no longer know what's moving: them or me. Little yellow matchbooks soar up and down the city streets. Cabs.\n\nIf I was standing at street level, the ledge would feel plenty wide. But up here it's not. Up here it's a thread and on it, I'm trying to balance my two wobbly feet.\n\nI'm scared. But I've come this far. I can't go back.\n\nThere's a moment of calm that comes and goes so quickly I almost don't notice it. For one split second the world is still. I'm at peace. The sun moves higher and higher into the sky, yellow-orange glaring at me through the buildings, making me peaceful and warm. My hands rise beside me as a bird goes soaring by. As if my hands are wings, I think in that moment what it would be like to fly.\n\nAnd then it comes rushing back to me.\n\nI'm hopelessly alone. Everything hurts. I can no longer think straight; I can no longer see straight; I can no longer speak. I don't know who I am anymore. If I am anyone.\n\nAnd I know in that moment for certain: I am no one.\n\nI think what it would feel like to fall. The weightlessness of the plunge, of gravity taking over, of relinquishing control. Giving up, surrendering to the universe.\n\nThere's a flicker of movement beneath me. A flash of brown, and I know that if I wait any longer, it will be too late. The decision will no longer be mine. I cry out one more time.\n\nAnd then I go.\njessie\n\nI don't have to see myself to know what I look like.\n\nMy eyes are fat and bloated, so bloodshot the sclera is bereft of white. The skin around them is red and raw from rubbing. They've been like this for days. Ever since Mom's body began shutting down, her hands and feet cold, blood no longer circulating there. Since she began to drift in and out of consciousness, refusing to eat. Since she became delirious, speaking of things that aren't real.\n\nOver the last few days, her breathing has changed too, becoming noisier and unstable, developing what the doctor called Cheyne-Stokes respiration where, for many seconds at a time, she didn't breathe. Short, shallow breaths followed by no breaths at all. When she didn't breathe, I didn't breathe. Her nails are blue now, the skin of her arms and legs blotchy and gray. \"It's a sign of imminent death _,_ \" __ the doctor said only yesterday as he set a firm hand on my shoulder and asked if there was someone they could call, someone who could come sit with me until she passed.\n\n\"It won't be long now,\" he'd said.\n\nI had shaken my head, refusing to cry. It wasn't like me to cry. I've sat in the same armchair for nearly a week now, in the same rumpled clothes, leaving only to collect coffee from the hospital cafeteria. \"There's no one,\" I said to the doctor. \"It's only Mom and me.\"\n\nOnly Mom and me as it's always been. If I have a father somewhere out there in the world, I don't know a thing about him. Mom didn't want me to know anything about him.\n\nAnd now this evening, Mom's doctor stands before me again, taking in my bloated eyes, staring at me in concern. This time offering up a pill. He tells me to take it, to go lie down in the empty bed beside Mom's and sleep.\n\n\"When's the last time you've slept, Jessie?\" he asks, standing there in his starch white smock, tacking on, \"I mean, really _slept_ ,\" before I can lie. Before I can claim that I slept last night. Because I did, for a whole thirty minutes, at best.\n\nHe tells me the longest anyone has gone without sleep. He tells me that people can die without sleep. He says to me, \"Sleep deprivation is a serious matter. You need to sleep,\" though he's not my doctor but Mom's. I don't know why he cares.\n\nBut for whatever reason, he goes on to list for me the consequences of not sleeping. Emotional instability. Crying and laughing for no sound reason at all. Behaving erratically. Losing concept of time. Seeing things. Hallucinating. Losing the ability to speak.\n\nAnd then there are the physical effects of insomnia: heart attack, hypothermia, stroke.\n\n\"Sleeping pills don't work for me,\" I tell him, but he shakes his head, tells me that it's not a sleeping pill. Rather a tranquilizer of some sort, used for anxiety and seizures. \"It has a sedative effect,\" he says. \"Calming. It will help you sleep without all the ugly side effects of a sleeping pill.\"\n\nBut I don't need to sleep. What I need instead is to stay awake, to be with Mom until she makes the decision to leave.\n\nI push myself from my chair, strut past the doctor standing in the doorway. \"Jessie,\" he says, a hand falling gently to my arm to try and stop me before I can go. His smile is fake.\n\n\"I don't need a pill,\" I tell him briskly, plucking my arm away. My eyes catch sight of the nurse standing in the hallway beside the nurses' station, her eyes conveying only one thing: pity. \"What I need is coffee,\" I say, not meeting her eye as I slog down the hallway, feet heavy with fatigue.\n\n* * *\n\nThere's a guy I see in the cafeteria every now and then, a little bit like me. A weak frame lost inside crumpled-up clothes; tired, red eyes but doped up on caffeine. Like me, he's twitchy. On edge. He has a square face; dark, shaggy hair; and thick eyebrows that are sometimes hidden behind a pair of sunglasses so that the rest of us can't see he's been crying. He sits in the cafeteria with his feet perched on a plastic chair, a red sweatshirt hood pulled over his head, sipping his coffee.\n\nI've never talked to him before. I'm not the kind of girl that cute guys talk to.\n\nBut tonight, for whatever reason, after I get my cup of coffee, I drop down into the chair beside him, knowing that under any other circumstance, I wouldn't have the nerve to do it. To talk to him. But tonight I do, mostly, I think, to delay going back to Mom's room, to give the doctor his chance to examine her and leave.\n\n\"Want to talk about it?\" I ask, and at first his look is surprised. Incredulous, even. His gaze rises up from his own coffee cup and he stares at me, his eyes as blue as a blue morpho butterfly's wings.\n\n\"The coffee,\" he says after some time, pushing his cup away. \"It tastes like shit,\" he tells me, as though that's the thing that's bothering him. The only thing. Though I see well enough inside the cup to know that he drank it down to the dregs, so it couldn't have been that bad.\n\n\"What's wrong with it?\" I ask, sipping from my cup. It's hot and so I peel back the plastic lid and blow on it. Steam rises to greet me as I try again and take another sip. This time, I don't burn my mouth.\n\nThere's nothing wrong with the hospital's coffee. It's just the way I like it. Nothing fancy. Just plain old coffee. But still, I dump four packets of Equal in and swirl it around because I don't have a stir stick or spoon.\n\n\"It's weak and there are grounds in it,\" he tells me, giving his abandoned cup the stink eye. \"I don't know,\" he says, shrugging. \"Guess I just like my coffee stronger than this.\"\n\nAnd yet, he reaches again for the cup before remembering there's nothing left in it.\n\nThere's an anger in his demeanor. A sadness. It doesn't have anything to do with the coffee. He just needs something to take his anger out on. I see it in his blue eyes, how he wishes he was somewhere else, anywhere else but here.\n\nI too want to be anywhere else but here.\n\n\"My mother's dying,\" I tell him, looking away because I can't stand to stare into his eyes when I say the words aloud. Instead I gaze toward a window where outside the world has gone black. \"She's going to die.\"\n\nSilence follows. Not an awkward silence, but just silence. He doesn't say he's sorry because he knows, like me, that sorry doesn't mean a thing. Instead, after a minute or two, he says that his brother's been in a motorcycle accident. That a car cut him off and he went flying off the bike, headfirst, into a utility pole.\n\n\"There's no saying if he'll make it,\" he says, talking in euphemisms because it's easier that way than just saying there's a chance he'll die. Kick the bucket. Croak. \"Odds are good we'll have to pull the plug sometime soon. The brain damage.\" He shakes his head, picks at the skin around his fingernails. \"It's not looking good,\" he tells me, and I say, \"That sucks,\" because it does.\n\nI rub at my eyes and he changes topics. \"You look tired,\" he tells me, and I admit that I can't sleep. That I haven't been sleeping. Not for more than thirty minutes at a time, and even that's being generous. \"But it's fine,\" I say, because my lack of sleep is the least of my concerns.\n\nHe knows what I'm thinking.\n\n\"There's nothing more you can do for your mom,\" he says. \"Now you've got to take care of you. You've got to be ready for what comes next. You ever try melatonin?\" he asks, but I shake my head and tell him the same thing I told Mom's doctor.\n\n\"Sleeping pills don't work for me.\"\n\n\"It's not a sleeping pill,\" he says as he reaches into his jeans pocket and pulls out a handful of pills. He slips two tablets into the palm of my hand. \"It'll help,\" he says to me, but any idiot can see that his own eyes are bloodshot and tired. It's obvious this melatonin didn't help him worth shit. But I don't want to be rude. I slip the tablets into the pocket of my own jeans and say thanks.\n\nHe stands from the table, chair skidding out from beneath him, and says he'll be right back. I think that it's an excuse and that he's going to take the opportunity to split. \"Sure thing,\" I say, looking the other way as he leaves. Trying not to feel sorry for myself as I'm hit with that sudden sense of being alone. Trying not to think about my future, knowing that when Mom finally dies, I'll be alone forever.\n\nHe's gone now and I watch other people in the cafeteria. New grandparents. A group of people sitting at a round table, laughing. Talking about old times, sharing memories. Some sort of hospital technician in blue scrubs eating alone. I reach for my now-empty cup of coffee, thinking that I too should split. Knowing that the doctor is no doubt done with Mom by now, and so I should get back to her.\n\nBut then the guy comes back. In his hands are two fresh cups of coffee. He returns to his chair and states the obvious. \"Caffeine is the last thing either of us needs,\" he tells me, saying that it's decaf, and it occurs to me then that this has nothing to do with the coffee, but rather the company.\n\nHe digs into his pocket and pulls out four rumpled packets of Equal, dropping them to the table beside my cup. I manage a thanks, flat and mumbled to hide my surprise. He was watching me. He was paying attention. No one ever pays attention to me, aside from Mom.\n\nBeside me he hoists his feet back onto the empty seat across from him, crosses them at the ankles. Drapes the red hood over his head.\n\nI wonder what he'd be doing right now if he wasn't here. If his brother hadn't been in that motorcycle accident. If he wasn't close to dying.\n\nI think that if he had a girlfriend, she'd be here, holding his hand, keeping him company. Wouldn't she?\n\nI tell him things. Things I've never told anyone else. I don't know why. Things about Mom. He doesn't look at me as I talk, but at some imaginary spot on the wall. But I know he's listening.\n\nHe tells me things too, about his brother, and for the first time in a while, I think how nice it is to have someone to talk to, or to just share a table with as the conversation in time drifts to quiet and we sit together, drinking our coffees in silence.\n\n* * *\n\nLater, after I return to Mom's room, I think about him. The guy from the cafeteria. After the hospital's hallway lights are dimmed and all is quiet\u2014well, mostly quiet save for the ping of the EKG in Mom's room and the rattle of saliva in the back of her throat since she can no longer swallow\u2014I think about him sitting beside his dying brother, also unable to sleep.\n\nIn the hospital, Mom sleeps beside me in a drug-induced daze, thanks to the steady drip, drip, drip of lorazepam and morphine into her veins, a solution that keeps her both pain-free and fast asleep at the same time.\n\nSometime after nine o'clock, the nurse stops by to turn Mom one last time before signing off for the night. She checks her skin for bedsores, running a hand up and down Mom's legs. I've got the TV in the room turned on, anything to drown out that mechanical, metallic sound of Mom's EKG, one that will haunt me for the rest of my life. It's one of those newsmagazine shows\u2014 _Dateline_ , _60 Minutes_ , I don't know which\u2014the one thing that was on when I flipped on the TV. I didn't bother channel surfing; I don't care what I watch. It could be home shopping or cartoons, for all I care. It's just the noise I need to help me forget that Mom is dying. Though, of course, it isn't as easy as that. There isn't a thing in the world that can make me forget. But for a few minutes at least, the news anchors make me feel less alone.\n\n\"What are you watching?\" the nurse asks, examining Mom's skin, and I say, \"I don't even know.\"\n\nBut then we both listen together as the anchors tell the story of some guy who'd assumed the identity of a dead man. He lived for years posing as him, until he got caught.\n\nLeave it to me to watch a show about dead people as a means of forgetting that Mom is dying.\n\nMy eyes veer away from the TV and to Mom. I mute the show. Maybe the repetitive ping of the EKG isn't so bad after all. What it says to me is that Mom is still alive. For now.\n\nUlcers have already formed on her heels and so she lies with feet floating on air, a pillow beneath her calves so they can't touch the bed. \"Feeling tired?\" the nurse asks, standing in the space between Mom and me. I am, of course, feeling tired. My head hurts, one of those dull headaches that creeps up the nape of the neck. There's a stinging pain behind my eyes too, the kind that makes everything blur. I dig my palms into my sockets to make it go away, but it doesn't quit. My muscles ache, my legs restless. There's the constant urge to move them, to not sit still. It gnaws at me until it's all I can think about: moving my legs. I uncross them, stretch them out before me, recross my legs. For a whole thirty seconds it works. The restlessness stops.\n\nAnd then it begins again. That prickly urge to move my legs.\n\nIf I let it, it'll go on all night until, like last night, when I finally stood and paced the room. All night long. Because it was easier than sitting still.\n\nI think then about what the guy in the cafeteria said. About taking care of myself, about getting ready for what comes next. I think about what comes next, about Mom's and my house, vacant but for me. I wonder if I'll ever sleep again.\n\n\"Doc left some clonazepam for you,\" the nurse says now, as if she knows what I'm thinking. \"In case you changed your mind.\" She says that it could be our little secret, hers and mine. She tells me Mom is in good hands. That I need to take care of myself now, again just like the guy in the cafeteria said.\n\nI relent. If only to make my legs relax. She steps from the room to retrieve the pills. When she returns, I climb onto the empty bed beside Mom and swallow a single clonazepam with a glass of water and sink beneath the covers of the hospital bed. The nurse stays in the room, watching me. She doesn't leave.\n\n\"I'm sure you have better things to do than keep me company,\" I tell her, but she says she doesn't.\n\n\"I lost my daughter a long time ago,\" she says, \"and my husband's gone. There's no one at home waiting for me. None other than the cat. If it's all right with you, I'd rather just stay. We can keep each other company, if you don't mind,\" she says, and I tell her I don't mind.\n\nThere's an unearthly quality to her, ghostlike, as if maybe she's one of Mom's friends from her dying delusions, come to visit me. Mom had begun to talk to them the last time she was awake, people in the room who weren't in the room, but who were already dead. It was as if Mom's mind had already crossed over to the other side.\n\nThe nurse's smile is kind. Not a pity smile, but authentic. \"The waiting is the hardest part,\" she tells me, and I don't know what she means by it\u2014waiting for the pill to kick in or waiting for Mom to die.\n\nI read something once about something called terminal lucidity. I didn't know if it's real or not, a fact\u2014scientifically proven\u2014or just some superstition a quack thought up. But I'm hoping it's real. Terminal lucidity: a final moment of lucidity before a person dies. A final surge of brainpower and awareness. Where they stir from a coma and speak one last time. Or when an Alzheimer's patient who's so far gone he doesn't know his own wife anymore wakes up suddenly and remembers. People who have been catatonic for decades get up and for a few moments, they're normal. All is good.\n\nExcept that it's not.\n\nIt doesn't last long, that period of lucidity. Five minutes, maybe more, maybe less. No one knows for sure. It doesn't happen for everyone.\n\nBut deep inside I'm hoping for five more lucid moments with Mom.\n\nFor her to sit up, for her to speak.\n\n\"I'm not tired yet,\" __ I confess to the nurse after a few minutes, sure this is a waste of time. I can't sleep. I won't sleep. The restlessness of my legs is persistent, until I have no choice but to dig the melatonin out of my pocket when the nurse turns her back and swallow those too.\n\nThe hospital bed is pitted, the blankets abrasive. I'm cold. Beside me, Mom's breathing is dry and uneven, her mouth gaping open like a robin hatchling. Scabs have formed around her lips. She jerks and twitches in her sleep. \"What's happening?\" I ask the nurse, and she tells me Mom is dreaming.\n\n\"Bad dreams?\" I ask, worried that nightmares might torment her sleep.\n\n\"I can't say for sure,\" the nurse says. She repositions Mom on her right side, tucking a rolled-up blanket beneath her hip, checking the color of her hands and feet. \"No one even knows for sure why we dream,\" the nurse tells me, adding an extra blanket to my bed in case I catch a draft in my sleep. \"Did you know that?\" she asks, but I shake my head and tell her no. \"Some people think that dreams serve no purpose,\" she adds, winking. \"But I think they do. They're the mind's way of coping, of thinking through a problem. Things we saw, felt, heard. What we're worried about. What we want to achieve. You want to know what I think?\" she asks, and without waiting for me to answer, she says, \"I think your mom is getting ready to go in that dream of hers. Packing her bags and saying goodbye. Finding her purse and her keys.\"\n\nI can't remember the last time I'd dreamed.\n\n\"It can take up to an hour to kick in _,_ \" __ the nurse says, and this time I know she means the medicine.\n\nThe nurse catches me staring at Mom. \"You can talk to her, you know?\" she asks. \"She can hear you,\" she says, but it's awkward then. Talking to Mom while the nurse is in the room. And anyway, I'm not convinced that Mom can really hear me, so I say to the nurse, \"I know,\" but to Mom, I say nothing. I'll say all the things I need to say if we're ever alone. The nurses play Mom's records some of the time because, as they've told me, hearing is the last thing to go. The last of the senses to leave. And because they think it might put her at ease, as if the soulful voice of Gladys Knight & the Pips can penetrate the state of unconsciousness where she's at, and become part of her dreams. The familiar sound of her music, those records I used to hate when I was a kid but now know I'll spend the rest of my life listening to on repeat.\n\n\"This must be hard on you,\" the nurse says, watching me as I stare mournfully at Mom, taking in the shape of her face, her eyes, for what might be the last time. Then she confesses, \"I know what it's like to lose someone you love.\" I don't ask the nurse who, but she tells me anyway, admitting to the little girl she lost nearly two decades ago. Her daughter, only three years old when she died. \"We were on vacation,\" she says. \"My husband and me with our little girl.\" He's her ex-husband now because, as she tells me, their marriage died that day too, same day as their little girl. She tells me how there was nothing Madison loved more than playing in the sand, searching for seashells along the seashore. They'd taken her to the beach that summer. \"My last good memories are of the three of us at the beach. I still see her sometimes when I close my eyes. Even after all these years. Bent at the waist in her purple swimsuit, digging fat fingers into the sand for seashells. Funny thing is that I have a hard time remembering her face, but clear as day I see the ruffles of that purple tulle skirt moving in the air.\"\n\nI don't know what to say. I know I should say something, something empathetic. I should commiserate. But instead I ask, \"How did she die?\" because I can't help myself. I want to know, and there's a part of me convinced she wants me to ask.\n\n\"A hit-and-run,\" she admits while dropping into an empty armchair in the corner of the room. Same one that I've spent the last few days in. She tells me how the girl wandered into the street when she and her husband weren't paying attention. It was a four-lane road with a speed limit of just twenty-five as it twisted through the small seaside town. The driver rounded a bend at nearly twice that speed, not seeing the little girl before he hit her, before he fled.\n\n\"He,\" she says then. \"He.\" And this time, she laughs, a jaded laugh. \"I'll never know one way or the other if the driver was male or female, but to me it's always been _he_ because for the life of me I can't see a woman running her car into a child and then fleeing. It goes against our every instinct, to nurture, to protect,\" she says.\n\n\"It's so easy to blame someone else. My husband, the driver of the car. Even Madison herself. But the truth is that it was my fault. I was the one not paying attention. I was the one who let my little girl waddle off into the middle of the street.\"\n\nAnd then she shakes her head with the weariness of someone who's replayed the same scene in her life for many years, trying to pinpoint the moment when it all went wrong. When Madison's hand slipped from hers, when she fell from view.\n\nI don't mean for them to, but still, my eyes fill with tears as I picture her little girl in her purple swimsuit, lying in the middle of the road. One minute gathering seashells in the palm of a hand, and the next minute dead. It seems so tragic, so catastrophic, that my own tragedy somehow pales in comparison to hers. Suddenly cancer doesn't seem so bad.\n\n\"I'm sorry,\" I say. \"I'm so sorry,\" but she shoos me off and says no, that she's the one who should be sorry. \"I didn't mean to make you sad,\" she says, seeing my watery eyes. \"Just wanted you to know that I can empathize. That I can relate. It's never easy losing someone you love,\" she says again, and then stands quickly from the armchair, gets back to tending to Mom. She tries to change the subject. \"Feeling tired yet?\" she asks again, and this time I tell her I don't know. My body feels heavy. That's as much as I knew. But heavy and tired are two different things.\n\nShe suggests then, \"Why don't I tell you a story while we wait? I tell stories to all my patients to help them sleep.\"\n\nMom used to tell me stories. We'd lie together under the covers of my twin-size bed and she'd tell me about her childhood. Her upbringing. Her own mom and dad. But she told it like a fairy tale, like a _once upon a time_ kind of story, and it wasn't Mom's story at all, but rather the story of a girl who grew up to marry a prince and become queen.\n\nBut then the prince left her. Except she always left that part out. I never knew if he did or if he didn't, or if he was never there to begin with.\n\n\"I'm not your patient,\" I remind the nurse but she says, \"Close enough,\" while dimming the overhead lights so that I can sleep. She sits down on the edge of my bed, pulling the blanket clear up to my neck with warm, competent hands so that for one second I envy Mom her care.\n\nThe nurse's voice is low, her tone flat so she doesn't wake Mom from her deathbed. Her story begins somewhere just outside of Moab, though it doesn't go far.\n\nAlmost at once, my eyelids grow heavy; my body becomes numb. My mind fills with fog. I become weightless, sinking into the pitted hospital bed so that I become one with it, the bed and me. The nurse's voice floats away, her words themselves defying gravity and levitating in the air, out of reach but somehow still there, filling my unconscious mind. I close my eyes.\n\nIt's there, under the heavy weight of two thermal blankets and at the sound of the woman's hypnotic voice, that I fall asleep. The last thing I remember is hearing about the snarling paths and the sandstone walls of someplace known as the Great Wall.\n\nWhen I wake up in the morning, Mom is dead.\n\nI slept right through it.\neden\n\nMay 16, _1996  \n_ Egg Harbor\n\nAaron showed me the house today. I'm in love with it already\u2014a cornflower blue cottage perched on a forty-five-foot cliff that overlooks the bay. Pine floors and whitewashed walls. A screened-in porch. A long wooden staircase that leads down to the dock at the water's edge where the Realtor promised majestic sunsets and fleets of sailboats floating by. _Quaint_ , _charming_ and _serene_. Those are the words the Realtor used. Aaron, as always, didn't say much of anything, just stood on the balding lawn with his hands in the pockets of his jeans, staring out at the bay, thinking. He's recently taken a job as a line cook at one of the restaurants in town, a chophouse in Ephraim. The cottage will more than cut his commute time in half. It's also a steal compared to our current mortgage, and set on two acres of waterfront land that spans the heavily wooded backcountry to the rocky shores of Green Bay.\n\nAnd there's a garden. A ten-by-twenty-or thirty-foot space overrun with brambles and weeds. It's in need of work, but already Aaron has promised raised beds. There is a greenhouse, a sorry sight if I've ever seen one, set in a sunnier patch of the yard where the grass still grows. Small, shedlike, with aged glass windows and some sort of clear, corrugated roof meant to attract the sun. The door hangs cockeyed, one of its hinges broken. Aaron took a look and said that he can fix it, which comes as no surprise to me. There isn't a thing in this world that Aaron can't fix. Cobwebs cling to the corners of the room like lace. Already I'm imagining rows and rows of peat pots of soil and seed soaking up the sun, waiting to be transported into the garden.\n\nNearby, a swing hangs from the mighty branch of a burr oak tree. It was the tree that cinched it for me. Or maybe not the tree itself, but the promise of the tree, the notion of children one day causing ruckus and mayhem on the tree's swing, three feet of lumber fastened to the branch with a sturdy rope. I envision them climbing deep into the divots of the tree's trunk and laughing. I can hear them already, Aaron's and my unborn children. Laughing and screaming in delight.\n\nAaron asked if I loved it as much as him, and I didn't know if he meant whether I loved the cottage as much as I love _him_ , or if I loved the cottage as much as he loves the cottage, but either way I told him I did.\n\nAaron left the Realtor with our bid. It's a buyer's market, he said, trying to finagle the asking price down a good 10 percent. Me, I would have paid asking price, too afraid to lose the cottage otherwise. Tomorrow we'll know if it's ours.\n\nTonight I won't sleep. How is it possible to love something so much, to want something so badly, when only hours ago I didn't know it existed?\n\nJuly 1, 1996 _  \n_Egg Harbor\n\nThe boxes are plentiful. There is no end to the number of cardboard boxes the movers carry through the front door, delivering them to their marked rooms\u2014living room, bedroom, master bath\u2014stomping across our home in dusty work boots. Sixteen hundred square feet of space needing to be filled as Aaron and I divvied up our gender-appropriate tasks, he directing the movers with couches and beds while I unpacked and washed the dishes by hand and placed them in the cabinets. I watched the many laps they took, each man's head beginning to glimmer with sweat. Aaron's too, though he hardly carried a thing, and yet the authority in his voice, the obvious clout as grown men trailed him through our home, heeding his every word, was enough to catch my eye. I watched him round the home time and again, wondering how I was so lucky to have him all to my own.\n\nIt wasn't like me to be lucky in love. Not until I met Aaron. The men who came before him were deadbeats and drifters, bottom-feeders. But not Aaron. We dated for a year before he proposed. Tomorrow we celebrate two years. Soon there will be kids, a whole gaggle of little ones spinning circles at our feet. As soon as we're settled, Aaron always said, and now, as my eyes assess the new home, the sprawling landscape, the sixteen hundred square feet of space, three bedrooms\u2014two vacant and left to fill\u2014I realize the time has come and like clockwork, something inside me starts to tick.\n\nWhen the movers' backs were turned, Aaron kissed me in the kitchen, pinning me against the cabinets, hands gripping my hips. It was unasked for and yet very much wanted as he kissed with his eyes closed, whispering that all of our dreams were finally coming true. Aaron isn't one to be sentimental or romantic, and yet it was true: the cottage, his job, leaving the city. We'd both wanted to get away from Green Bay since the day we were married, his hometown and my hometown, so that two sets of parents couldn't show up at our door on any given day, unsolicited, waging a secret battle as to which in-law could occupy the most of our time. We hadn't gone far, sixty-seven miles to be precise, but enough that visits would be preempted with a simple phone call.\n\nTonight we made love on the living room floor to the glow of candlelight. The electricity had yet to be turned on and so, other than the dance of candlelight on the whitewashed walls, the house was dark.\n\nAaron was the first to suggest it, discontinuing my birth control pill, as if he knew what I was thinking, as if he could read my mind. It was as we lay together on the wide wooden floorboards staring out the open windows at the stars, Aaron's prowling hand moving across my thigh, contemplating a second go. That's when he said it. I told him _yes!_ that I am ready for a family. That _we_ are ready. Aaron is twenty-nine. I am twenty-eight. His paycheck isn't extravagant, and yet it's enough. We aren't spendthrifts; we've been saving for years.\n\nAnd even though I knew it wasn't possible yet, the pill in my system nipped any possibility of pregnancy in the bud, I still imagined a creature no bigger than a speck starting to take form as Aaron again let himself inside me.\n\nJuly 9, 1996 _  \n_Egg Harbor\n\nOur days begin with coffee on the dock, bare feet dangling over the edge, downward toward the bay. The water is cold, and our feet don't reach anyway. But as promised, there are sailboats. Aaron and I spend hours watching them pass by, as well as sandpipers and other shorebirds that come to call, their long legs wading through the shallow water for a meal. We stare at the birds and the sailboats, watching the sun rise higher into the sky, warming our skin, burning off the early-morning fog. Heaven on earth, Aaron says.\n\nAs we sit on the dock, Aaron tells me about his nights at the chophouse that steals him from me for ten hours at a time. About the heat of the kitchen, and the persistent noise. The rumble of voices calling out orders in sync. The sputter of boneless rib eye on the grill, the dicing and hashing of vegetables.\n\nHis voice is placid. He doesn't complain because Aaron, ever easygoing Aaron, isn't one to complain. Rather he tells me about it, describing it for me so that I can see in my mind's eye what he's doing when he's away from me for half the day. He wears a white chef jacket and black chef pants and a cap, something along the lines of a beanie that is also white. Aaron's been assigned the role of _saucier_ , or sauce chef, one that's new to him, but no doubt comes with ease. Because this is the way it is with Aaron. No matter what he tries his hand at, things always come with ease.\n\nOur property is fringed by trees so that as we sit on the deck's edge, Aaron and me, it feels as if we're all alone, partitioned from society by the lake and the trees. If we have neighbors, we've never seen them. Never laid eyes on them. Never spied another home through the canopy of trees. Never are we disturbed by the sound of voices, but only the colloquy of birds as they perch in the trees and yammer back and forth about whatever it is birds talk about. On occasion the helmsmen will wave a hearty hello from behind the steering wheel of their sailboats, but more often than not they're too far away to see Aaron and me at the dock's edge, feet dangling southward, holding hands, sitting in silence, listening to the breeze through the trees.\n\nWe're marooned on an island, stranded and shipwrecked, but we don't mind. It's just the way it should be.\n\nAaron's work shift begins at two in the afternoon and ends when the last customer leaves and the kitchen is clean, most nights stumbling into bed around midnight or after, smelling of sweat and grease.\n\nBut the days are ours to do with as we please.\n\nLast week, Aaron repaired the greenhouse door and we stripped it of cobwebs and bugs. We spent days cultivating the garden and Aaron made good on his promise of raised beds, three feet by five feet by ten inches deep, made of white cedar that will one day house cucumbers and zucchini. But not this year. It's far too late in the season to grow produce this year and so for now, we buy it from any number of tatty roadside farm stands. We live two miles from town and even though the population around here expands sevenfold in the summer months thanks to a healthy tourist population, outside town it's still mainly rural, long stretches of open country roads that intersect with nothing but sky.\n\nInstead of planting produce this year, Aaron and I sowed perennial seeds to enjoy next year: baby's breath and lavender and hollyhocks because all the fences and cottages around here, it seems, are flanked with hollyhocks. We placed them in peat pots of gardening soil in the greenhouse and set them in the sunniest spot we could find. In a month or so, we'll transplant them to the garden. They won't bloom for some time, not until next spring. But still, I stand hopeful in the greenhouse, staring at the peat pots, imagining what might be happening beneath the soil's surface, whether the seeds' roots are taking hold, pushing down into the soil to anchor the seedling to this world, or if the seed has merely shriveled up and died in there, a dead embryo in its mother's womb.\n\nAs I clear out the last of my birth control pills and run a hand across what I imagine to be my uterus, I wonder what is happening inside there too.\njessie\n\nI had Mom cremated at her request. I carry her around now in a rhubarb-glazed clay urn with a cork in the top, one she bought for herself when the cancer spread. It's cylindrical and inconspicuous, the cork stuck on with an ample amount of Gorilla Glue so I don't lose Mom by chance.\n\nMom had two wishes when she died, ones she let slip in the last brief moments of consciousness before she drifted off to sleep, a sleep from which she would never wake up. One, that she be cremated and lobbed from the back end of the Washington Island Ferry and into Death's Door. And two, that I find myself and figure out who I am. The second hinged on the esoteric and didn't make obvious sense. I blamed the drugs for it, that and the imminence of death.\n\nI'm nowhere near accomplishing either, though I filled out a college application online. But I have no plans of parting with Mom's remains anytime soon. She's the only thing of value I have left.\n\nI haven't slept in four days, not since some doctor took pity on me and offered me a pill. Three if you count the one where I nearly nodded off at the laundromat waiting for clothes to dry, anesthetized by the sound of sweaters tumbling around a dryer. The effects are obnoxious. I'm tired. I'm grumpy. I can focus on nothing and my reaction time is slow. I've lost the ability to think.\n\nYesterday, a package arrived from UPS and the driver asked me to sign for it. He stood before me, shoving a pen and a slip of paper up under my nose and I could only stare, unable to put two and two together. He said it again. _Can you sign for it?_ He forced the pen into my hand. He pointed at the signature line. For a third time, he asked me to sign.\n\nAnd even then I scribbled with the cap still on the pen. The man had to snatch it from my hand and uncap it.\n\nI'm pretty sure I've begun to see things too. Things that might not be real, that might not be there. A millipede dashing across the tabletop, an ant on the kitchen floor. Sudden movements, immediate and quick, but the minute I turn, they're gone.\n\nI keep track of the sleepless nights in the notched lines beneath my eyes, like the annual rings of a tree. One wrinkle for each night that I don't sleep. I stare at myself in the mirror each day, counting them all. This morning there were four. The surface effects of insomnia are even worse than what's going on on the inside. My eyes are red and swollen. My eyelids droop. Overnight, wrinkles appear by the masses, while I lie in bed counting sheep. I could go to the clinic and request something else to help me sleep. Some more of the clonazepam. But with the pills in my system, I slept right on through Mom's death. I don't want to think about what else I'd miss.\n\nAt McDonald's, I'm asked if I want ketchup with my fries, but I can only stare at the worker dumbly because what I heard was _It's messed up when boats capsize_ , and I nod lamely because it is disastrous and sad, and yet so out of left field I can't respond with words.\n\nIt's only when he drops a stack of ketchup packets on my tray that my brain makes the translation, too late it seems because I hate ketchup. I dump them on the table when I go, the mother lode for someone who likes it. On the way out the door I trip, because coordination is also affected by a lack of sleep.\n\nTwo hours ago I dragged my heavy body from bed after another sleepless night, and now I stand in the center of Mom's and my house, deciding which of our belongings to take and which to leave. I can't stand to stay here much longer, a decision I've come to quickly over the last four days. I've spoken to a Realtor already, figured out next steps. First I'm to pack up what I want to keep, and then everything else will be sold in an estate sale before some junk removal service tosses the rest of our stuff in the trash.\n\nThen some other family will move in to the only home I've ever known.\n\nI'm eyeing the sofa, wondering if I should take it or leave it, __ when the phone rings. \"Hello?\" I ask.\n\nA voice on the other end informs me that she's calling from the financial aid office at the college. \"There's a problem with your application,\" she says to me.\n\n\"What problem?\" I ask the woman on the phone, afraid I'm about to be cited for tax evasion. It's a likely possibility; I'd left blank every question on the FAFSA form that asked about adjusted gross income and tax returns. I might have lied on the application too. There was a question that asked if both of my parents were deceased. I said yes to that, though I don't know if it's true.\n\nIs my father dead?\n\nOn the other end of the line, the woman asks me to verify my social security number for her and I do. \"That's what I have,\" she says, and I ask, \"Then what's the problem? Has my application been denied?\" My heart sinks. How can that be? It's only a community college. It's not like I registered for Yale or Harvard.\n\n\"I'm sure it's just a weird mix-up with vital statistics,\" she says.\n\n\"What mix-up?\" I ask, feeling relieved for a mix-up as opposed to a denied application. A mix-up can be fixed.\n\n\"It's the strangest thing,\" she says. \"There was a death certificate on file for a Jessica Sloane, from seventeen years ago. With your birth date and your social security number. By the looks of this, Ms. Sloane,\" she says, and I amend _Jessie_ , because Ms. Sloane is Mom. \"By the looks of this, Jessie,\" she says, and the words that follow punch me so hard in the gut they make it almost impossible to breathe. \"By the looks of this, you're already dead.\"\n\nAnd then she laughs as if somehow or other this is funny.\n\n* * *\n\nToday I'm looking for a new place to live. Staying in our old home is no longer a viable option because of the residual ghosts of Mom that remain in every corner of the home. The smell of her Crabtree & Evelyn hand cream that fills the bathroom. The feel of the velvet-lined compartments in the mahogany dresser. The chemo caps. The cartons of Ensure on the refrigerator shelf.\n\nI perch in the back seat of a Kia Soul, trying hard not to think too much about the call from the financial aid office. This is easier said than done. Just thinking about it makes my stomach hurt. A mix-up, the woman claimed, but still, it's hard to grapple with the words _you_ and _dead_ in the same sentence. Though I try to, I can't push them from my mind. The way she and I left things, I'm to provide a copy of my social security card to the college before they'll take another look at my application for a loan, which is a problem because I don't have the first clue where the card is. But it's more than that too. Because the woman also told me about some death index my name was found on. A _death index_. My name on a database maintained by the Social Security Administration of millions of people who have died, nullifying their social security numbers so that no one else can use them, so that I can't use my own social security number. Because, according to the Social Security Administration, I'm dead.\n\n_You might want to look into that_ , __ she'd suggested before ending our call, and I couldn't help but feel shaken up by it even now, hours later. My name on a death database. Though it's a mistake, of course.\n\nBut still I pray this isn't some sort of foresight. A prophecy of what's to come.\n\nI gaze out the window as some woman sits behind the wheel of the Kia, steering us through the streets of Chicago. Her name is Lily and she calls herself an _apartment finder_. The first I'd heard of Lily was days ago, when I'd come home from a cleaning job\u2014hating the feeling of coming home to Mom's and my empty house alone, wishing she was there but knowing she would never be again, making a flip decision to sell the home and leave. I came home, leaving my bike on the sidewalk, and there, hanging on the handle of our front door, was an ad for Lily's efficient and cost-free services. An apartment finder. I'd never heard of such a thing, and yet she was just the thing I needed. The door hanger was in-your-face marketing, the kind I couldn't recycle with the rest of the junk mail. And so I called Lily and we made an appointment to meet.\n\nLily's parallel parking skills are second to none, though it seems easy enough for someone like me who's never driven a car before. Growing up in an old brick bungalow in Albany Park, there was never a need to drive a car. We didn't have one. The Brown Line or the bus took us everywhere we needed to go. Either that or our own two feet. I also have my Schwinn, Old Faithful, which is surprisingly resilient in even the worst weather, except for, of course, three feet of snow.\n\nI was fifteen when Mom was diagnosed with cancer, which meant that for the time being, my life was on hold, anything that wasn't essential set aside. I went to school. I worked. I helped with the mortgage and saved as much as I could. And I held Mom's hair for her when she puked.\n\nShe found the lump herself, slim fingers palpating her own breast because she knew sooner or later this would happen. She didn't tell me about the lump until after she'd been diagnosed with cancer, one mammogram and a biopsy later. She didn't want to worry me. They removed the breast first, followed by months of chemotherapy. But it wasn't long before the cancer returned, in the chest and in the bones this time. The lungs. Back for vengeance.\n\n_Jessie, I'm dying. I'm going to die_ , she had said to me then. We were sitting on the front porch, hand in hand, the day she learned the cancer was back. At that point, her five-year survival rate took a nosedive. She only lived for two more, and none of them great.\n\nThe cancer, it's hereditary. Some aberrant gene that runs through our family line, red pegs lined up in my battleship already. Like Mom and her mom before her, it's only a matter of time before I too will sink.\n\nI claimed the back seat of the Kia after Lily dropped her purse into the passenger's chair. She drives with one hand on the horn at all times, so she can scare pedestrians out of the way, those she hollers at from behind safety glass to _shake a leg_ and _scoot your boot_. I have no credit history and no bank account, which I've confessed to Lily, and instead carry a pocketful of cash. Her eyes grew wide when I showed her my money, thirty hundred-dollar bills folded in half and stuck inside a wristlet.\n\n\"This might be a problem,\" Lily said, shrugging her shoulders not at the cash but rather the shortage of credit, the absence of a bank account, \"but we'll see.\"\n\nShe suggested I offer a landlord more up front to offset the fact that I'm one of those people who keeps all my money in a fireproof safe box beneath my bed. The checks I earn cleaning houses get cashed at Walmart for a three-dollar service fee, and then deposited into my trusty box. I considered signing on with a temp agency once, but thought better of it. There are perks to my job I won't find anywhere else. Because I'm cleaning houses, I don't have to pay taxes to Uncle Sam. I'm an independent contractor. At least that's the way I've always rationalized it in my head, though, for all I know, IRS agents are hot on my heels, planning to nab me for tax evasion.\n\nAnd still, I load my cleaning supplies into a basket on the back end of Old Faithful __ each day and pedal off to work, earning as much as two hundred dollars some days by cleaning someone else's home. I do it in peace with my headphones on. I don't have to make small talk. No one supervises me. It's the best job in the world.\n\n\"Either that,\" said Lily as she easily navigated the streets of Chicago, pulling in to an alley behind a high-rise on Sheridan and putting the car in Park, \"or you'll need to find someone to cosign on the loan,\" which isn't an option for me. I have no one to cosign on the loan.\n\nThe apartment search is nearly an abject failure.\n\nLily shows me apartment after apartment. A third-floor unit in a high-rise in Edgewater. A mid-rise on Ashland, newly rehabbed, in my price range though at the high end of it. Unit after unit of boxlike rooms enclosed by four thin gypsum walls, foggy windows that inhibit the light from coming in. The window screens are torn, one stuffed full with an air-conditioning unit, which is supposed to make me happy because, as Lily points out, renters usually have to buy them themselves, those repulsive window units that bar any natural light from entering the room.\n\nThe kitchens are tight. The stoves are old and electric. Freckles of mold grow in the showers' grout. The closets smell like urine. Lightbulbs have burned out.\n\nBut it isn't the mold or the windows that bother me. It's the noise and the neighbors\u2014strange people just on the other side of drywall, their domestic life partitioned from mine by a paltry combination of plaster and paper. The sense of claustrophobia that settles under my skin as I pretend to listen to Lily as she goes on and on about the two hundred and eighty square feet in the unit. The laundry facilities. The high-speed internet. But all I hear is the noise of someone's hair dryer. Women laughing. Men upstairs screaming at a ball game on TV. A phone conversation streaming through the walls. The ding of a microwave, the smell of someone's lunch.\n\nFour days without sleep. My body is tired, my mind like soup. I lean against the wall, feeling the force of gravity as it threatens to tug my heavy body to the ground.\n\n\"What do you think?\" Lily asks over the noise of the hair dryer, and I can't help myself.\n\n\"I hate it,\" I say, for the eighth or ninth time in a row, one for as many apartments as we've seen. Insomnia does that too. It keeps us honest because we don't have the energy to manufacture a lie.\n\n\"How come?\" she asks, and I tell her about the hair dryer next door. How it's loud.\n\nLily keeps composed, though inside her patience with me must be wearing thin. \"Then we keep looking,\" she says as I follow her out the door. I'd love to believe that she wants me to be happy, that she wants me to find the perfect place to live. But ultimately it comes down to one thing: my signature on a dotted line. What a lease agreement means for Lily is that an afternoon with me isn't a complete waste of time.\n\n\"I have one more to show you,\" she says, promising something different from the last umpteen apartments we've seen. We return to the Kia and I buckle up in the back seat, behind the purse that's already riding shotgun. We drive. Minutes later the car pulls to a sluggish stop before a greystone on Cornelia, gliding easily into a parking spot. The street is residential, lacking completely in communal living structures. No apartments. No condominiums. No high-rises with elevators that overlook crappy convenient marts. No strangers milling around on street corners.\n\nThe house is easily a hundred years old, beautiful and yet overwhelming for its grandeur. It's three stories tall and steep, with wide steps that lead to a front porch. A bank of windows lines each floor. There's a flat-as-a-pancake roof. Beneath the first floor there's a garden apartment, peeking up from beneath concrete.\n\n\"This is a three flat?\" I ask as we step from the car, envisioning stacks of independent units filling the home, all united by a common front door. I expect Lily to say yes.\n\nBut instead she laughs at me, saying, \"No, this is a private residential home. It's not for sale, not that you could afford it if it was. Easily a million and a half,\" she says. \"Dollars, that is,\" and I pause beneath a tree to ask what we're doing here. The day is warm, one of those September days that holds autumn at bay. What we want is to climb into sweaters and jeans, sip cocoa, wrap ourselves in blankets and watch the falling leaves. But instead we drip with sweat. The nights grow cold, but the days are hot, thirty-degree variants from morning to night. It won't last long. According to the weatherman, a change is coming, and it's coming soon. But for now, I stand in shorts and a T-shirt, a sweatshirt wrapped around my waist. When the sun goes down, the temperature will too.\n\n\"This way,\" Lily says with a slight nod of the head. I hurry along after her, but before we round the side of the greystone, something catches my eye. A woman walking down the sidewalk in our direction. She's a good thirty feet away, but moving closer to us. I don't see her face at first because of the force of the wind pushing her dark hair forward and into her eyes. But it doesn't matter. It's the posture that does it for me. That and the tiny feet as they shuffle along. It's the unassuming way she holds herself upright, curved at the shoulders just so. It's her shape, the height and width of it. The shade and texture of a periwinkle coat, a parka, midthigh length with a drawstring waist and a hood, though it's much too warm for a coat with a hood.\n\nThe coat is the same one as Mom had.\n\nI feel my heart start to beat. My mouth opens and a single word forms there on my lips. _Mom_. Because that's exactly who it is. It's her; it's Mom. She's here, alive, in the flesh, coming to see me. My arm lifts involuntarily and I start to wave, but with the hair in her eyes, she can't see me standing there on the sidewalk six feet away, waving.\n\nMom doesn't look at me as she passes by. She doesn't see me. She thinks I'm someone else. I call to her, my voice catching as the word comes out, so that it doesn't come out. Instead it gets trapped somewhere in my throat. Tears pool in my eyes and I think that I'm going to lose her, that she's going to keep walking by. And so my hand reaches out and latches on to her arm. A knee-jerk reaction. To stop her from walking past. To prevent her from leaving.\n\nMy hand grabs a hold of her forearm, clamping down. But just as it does, the woman frees her face of the hair and casts a glance at me. And I see then what I failed to see before, that this woman is barely thirty years old, much too young to be my mother. And that her face is covered in an enormity of makeup, unlike Mom, who wore her face bare.\n\nHer coat is not periwinkle at all but darker, more like eggplant or wine. And it has no hood. As she nears, I see more clearly. It isn't a coat after all, but a dress.\n\nShe looks nothing like Mom.\n\nFor a second I feel like I can't breathe, the wind knocked out of me. The woman tugs her arm free. She gives me a dirty look, scooting past me as I slip from the sidewalk, my feet falling on grass.\n\n\"I'm sorry,\" I whisper as she skirts eye contact, avoids my stare. She moves to the far edge of the sidewalk where she'll be two feet away, where I can't reach her. \"I thought you were someone else,\" I breathe as my eyes turn to find Lily with her arms folded, trying to pretend that this didn't just happen.\n\n_Of course it's not Mom_ , I tell myself as I watch the woman in the eggplant dress move on\u2014faster now, no longer shuffling along but now walking at a clipped pace to get away from me.\n\nOf course it's not Mom, because Mom is dead.\n\n\"You coming?\" Lily asks, and I say yes.\n\nI follow Lily as we sneak along a brick paver patio and into the backyard. My heart still beats hard. My nerves are rattled. The backyard opens up to reveal a patio and a yard, and behind that, a red brick garage with a jade green door. \"This is why we're here,\" says Lily, gesturing to the garage, and I stop where I am and ask, \"You want me to live in a garage?\"\n\n\"It's a carriage house,\" she says, explaining how there's living space up above, as is apparently evidenced by a window or two on the second and third floors. \"These are quite the find. Some people love them. The minute they come on the market, they're usually gone. This listing just came in this morning,\" she says, telling me how carriage houses used to be just that in the olden days, a place to park a horse and buggy and for the carriage driver to live. Servants' quarters. They're tucked away on an alley, camouflaged behind a far less humble house, living in the shadows of something bigger and better than them.\n\nWhich seems to me to be just the thing I need. To be camouflaged, to live hermit-like in seclusion, in the shadows of something grand.\n\n\"Can we see?\" I ask, meaning the inside, and Lily lets us in through a tall, tapered front door and immediately up a flight of rickety stairs.\n\nIt's larger than anything we've yet seen, nearly five hundred square feet of living space that is dilapidated and old, everything painted a hideous brown. The wooden floors have taken a beating. The boards are squeaky and uneven, with square-cut nails that lift right up out of the floorboards to a toe-stubbing height. The kitchen lines a living room wall, if it can even be called a kitchen. An old stove, an old refrigerator and a small bank of cabinets lined in a row beside where a TV should go. The lighting fixtures are archaic, giving off a scant amount of light. The place is minimally furnished; just a couple pieces of furniture that look to be about as decrepit as the home.\n\nThe bathroom appears to have had minor renovations. The fixtures, the paint are new, but the floor tile looks to be older than me. \"You won't hear a neighbor's hair dryer from here,\" Lily says. The so-called bedroom is up a second flight of precarious stairs, a loftlike space with an arched ceiling that follows the low roofline.\n\nOn the top floor I can't stand upright. I have to hunch.\n\n\"This is hardly suitable living space,\" says Lily, bent at the neck so she doesn't hit her head. Her wedge sandals struggle down the wooden steps, her hand clinging to the banister lest she fall. She doesn't think I will like it, but I do.\n\nCarriage homes like these, Lily says, don't follow the same rules as prescribed in the city's landlord-tenant ordinance. I wouldn't be protected in the same way. They're overlooked when it comes to regular safety inspections. There's only one door, which generally goes against fire codes that require two. Because garbage bins are relegated to the alley that abuts this home, it can be loud. The smell, especially in the summer months, can be sickening, she says.\n\n\"Rats are bent on eating from garbage bins, which means...\" she begins, but I hold up a hand and stop her there. She doesn't need to tell me. I know exactly what she means.\n\n\"What do you think?\" Lily asks.\n\nI listen for the sound of women's laughter. For rowdy men screaming at a TV. There are none.\n\n\"How do I apply?\" I ask.\n\nLily takes care of the paperwork. The landlord is a woman by the name of Ms. Geissler, a widow who lives alone in the greystone. We never meet, though Lily provides her with my completed application, a list of references\u2014ladies whose homes I clean\u2014and a letter of recommendation from a former high school guidance counselor. I kiss three grand goodbye, enough to cover first and last months' rent, plus two more for good measure. As they say, money speaks.\n\nAt Lily's suggestion, I wait in the car while she goes inside to meet with the landlord. I hold my breath, knowing it's liable the landlord will soon discover the same slipup as the college's financial aid office. That my social security number belongs to a dead girl. And she'll deny my application.\n\nBut, to my great relief, she doesn't. It takes less than fifteen minutes for Lily to emerge through the front door of the greystone, a key ring in hand. The keys to the carriage home. I breathe a sigh of relief. As it turns out, Lily let on about my mom and for that reason, Ms. Geissler approved the application without vetting me first. Out of sympathy and pity. Because she felt sorry for me, which is fine by me, so long as I have a place to live. A place that doesn't remind me of Mom.\n\nAs we pull away, I stare out the window and toward the imposing home. It's masked in shadows now, the sun slipping down on the opposite side of the street, burying the greystone in shade. The house is dignified but solemn. Sad. The house itself is sad.\n\nFrom the third story, I watch the window shade slowly peel back, though what's on the other side I can't see because it's shadowy and dim. But I imagine a woman, a widow, standing on the other side, watching until our car disappears from view.\neden\n\nJuly 26, _1996  \n_ Egg Harbor\n\nIt just so happens that we do have neighbors.\n\nThey came this afternoon after Aaron had gone off to work, a pregnant Miranda and her two boys, five-year-old Jack and two-year-old Paul. They came trudging down our gravel drive, Miranda pulling both boys in a red Radio Flyer wagon so that by the time they arrived she was sweaty and spent. She'd come to deliver a welcoming gift.\n\nIt was the sound of wheels on gravel that caught my attention as I stood on a ladder, painting the living room walls a pale gray, the windows and doors open to expel chemical scents from the air. This is how I now spend my days when Aaron is away. Unpacking boxes of belongings. Cleaning the insides of closets and cabinets. Painting the home.\n\nI saw them through the window first, heard the tired woman growl at the boys to _stop crying_ and to _behave_ , her cheeks flushed red from the heat and the pregnancy and, I guessed, the desire to impress. Her blond hair blew around her face and into her eyes as she walked. Her body was cemented with a short maternity dress, fastened to her with sweat. On her feet were Birkenstocks. In her eyes, exhaustion and discontent. From the moment I first spied her out the open window I knew one thing: motherhood did not suit her well.\n\nI set down my painting supplies and met them on the porch. Dropping the wagon's handle, Miranda introduced herself first and then the kids, neither of whom said hello, for they were far too busy clawing their way out of the wagon, elbowing one another for room on the porch step. I didn't mind. They had blond hair like their mother, and if it weren't for the apparent age difference could have easily been twins. They fought one another, vying for the right to their mother's free hand. The bigger of the two won out in the end and as he slipped his hand inside Miranda's, the little guy fell to the ground in a puddle of tears. \"Get up,\" __ Miranda commanded, her sharp voice jabbing through the placid air, apologizing to me for their manners as she tried hard to raise Paul from the ground. But Paul was a deadweight and wouldn't stand, and as she tugged on his underarms he cried out in pain that she'd hurt him. Tears came pouring from his eyes _._\n\n\"Damn it, Paul,\" she said, pulling again roughly on those underarms. \"Get up.\"\n\nWhat she saw were naughty children making a fuss, embarrassing her, making her feel humiliated and ashamed. But not me. I saw something else entirely. I dropped down beside little Paul and held out a hand to him. \"There's a tree swing in the backyard. Let's go have a ride on it, and let Mommy rest awhile?\" __ I said. His pale green eyes rose to mine, snot gathering along his nostrils, running downward toward his lips. He wiped at his nose with the back of a dirty hand and nodded his sweet little head.\n\nMiranda had walked far to bring us a blueberry loaf, more than a block in the heat. The pits of her dress were damp with sweat, the cotton pulled taut across the baby bump. When she spoke, her voice was breathless, exhausted, burned-out from the energy it took to raise two boys on her own, and she confessed to me that this time\u2014while running a hand over that baby\u2014she was hoping for a girl.\n\nShe sat on a patio chair, kicking off her Birkenstocks and resting her swollen ankles on another seat as I poured us each a glass of lemonade, conscious of the dried paint on the backs of my hands.\n\nMiranda's husband, she told me, is employed by the Department of Public Works. She stays at home with Jack and Paul, though what she always wanted to be\u2014what she used to be in her life before kids\u2014was a medical malpractice attorney. She asked how long Aaron and I have been married and when I told her, her eyebrows rose up in curiosity and she asked about kids.\n\nDo we have them?\n\nDo we plan to have them?\n\nIt seemed an intimate conversation to have with someone I hardly knew, and yet there was a great thrill at saying the words aloud, as if cementing them to reality. I felt my cheeks redden as I thought of that morning before dawn when Aaron rose, dreamlike, above me, lifting my nightgown up over my head. Outside it was dark, just after four o'clock in the morning, and our eyes were still drowsy, heavy with sleep, our minds not yet preoccupied by the thoughts that arrive with daylight. We moved together there on the bed, sinking into the aging mattress. And then later, while grinning at each other over mugs of coffee on the dock, watching as the fleets of sailboats went floating by on the bay, I had to wonder if it happened at all, or if it was only a dream.\n\nWhen Miranda asked, I told her that we're trying. Trying to have a child, trying to start a family. An odd choice of words for creating a baby, if you ask me. _Trying_ is how one learns to ride a bike. To knit, to sew. To write poetry.\n\nAnd yet it was exactly what we were doing as Aaron and I made love with reckless abandon, and then followed it up a week or two later with a home pregnancy test. The tests were all negative thus far, that lone pink line on the display screen notifying me again and again that I wasn't yet pregnant. I tried not to let it get the best of me, and yet it was hard to do. It wasn't as though Aaron and I minded the time spent _trying_ ; in fact, we enjoyed it quite a bit, but with every passing month I yearned exponentially more for a baby. For a baby to have, a baby to hold.\n\nI never mentioned to Aaron that I was taking the pregnancy tests.\n\nI took them while he was at work, watching out the cottage window as his car slipped from view and then, when he was out of sight, rushing to the bathroom, where I closed and locked the door in case he mistakenly left something behind and had to return for it.\n\nAnd then, when the single pink line appeared on the display screen as it always did, I wrapped the negative pregnancy test sticks up in tissue and discarded them discreetly in the garbage bins.\n\nMiranda beamed when I told her that we're trying. \"How exciting!\" __ she told me, her smile mirroring the one on my own face.\n\nAnd then, helping herself to a slice of her own blueberry loaf and running a hand over her bump for a second time, she said that her baby and my baby could one day go to school together.\n\nThat they could one day be friends.\n\nAnd it was a thought that filled me with consummate joy. I grinned.\n\nI'd been a lone wolf for much of my life. An introvert. The kind of woman who never felt comfortable in her own skin. Aaron changed that for me.\n\nThe idea thrilled me to bits and, in turn, I instinctively stroked my own empty womb and thought how much I wanted my baby to have a friend.\njessie\n\nTonight makes five days since I've been asleep. It's my first night in my new place. I spend it not sleeping, but rather imagining myself dead. I think of what it must be like for Mom, being dead. Is there blackness all around her, a pit of nothingness, the blackest of the black holes? Or has time simply stopped for her, and there's no such thing anymore as the living and the dead? Sometimes I wonder if she's not dead at all but rather alive in the clay urn of hers, screaming to get out. I wonder if there's enough oxygen in the urn. Can Mom breathe? But then I remember it doesn't matter anyway.\n\nMom is dead.\n\nI wonder if it hurts when you die. If it hurt when Mom died. And I think, in frightening detail, what it feels like when you can't breathe. I find myself holding my breath until my lungs begin to hurt, to burn. It's a prickling pain that stretches from my throat to my torso. It's reflexive, automatic when my mouth gapes open, and I suck in all the oxygen I can to soothe the burn.\n\nIt hurts, I decide. It hurts to die.\n\nThere's a clock on the wall, one that came with the house. _Tick, tock, tick, tock,_ it goes all night long, keeping track of the minutes I don't sleep. Keeping count for me. It's loud, a conga drum pounding in my ear, and though I try and remove the batteries, the _tick, tock_ doesn't go away. It stays.\n\nI feel out of place in this strange place. The house smells different than what I'm used to, an earthy smell like pine. It's older than Mom's and my old home, where I lived my entire life. One of the windows doesn't close tight so that when the wind whips its way around the house as it does tonight, air sneaks in. I can't feel it but I hear it, the hiss of the wind forcing its way in through a gap.\n\nI lie there in bed, trying hard to catch my breath, to not think about dying, to will myself to do the impossible and sleep. Beside me, on the floor, are four boxes, the only ones I brought from the old home. Some clothes, a few picture frames, and a box of random paperwork Mom kept, just an old white bankers box, kept closed with a string and button. It seemed important enough for Mom to keep, and so I kept it. A thought comes to me now: _Could my social security card be in that box, tucked away with Mom's financial paperwork?_\n\nI climb out of bed and turn on a light, dropping to the floor beside the box. I loosen the string and lift the lid, meeting reams of paper head-on. If there's any sort of method to the madness, I don't see it.\n\nI search through the paperwork for my social security card, to be sure the numbers I dashed off on the FAFSA form weren't incorrect. That I didn't write the wrong ones down by mistake. Because never in my life have I been asked to give my social security number, and so it's conceivable, I think, that I have the numbers mixed-up. I look for the card itself, grabbing stacks of paper by the handful and flipping through them one sheet at a time, hoping the card falls out. But instead I find the deed to our home, an old checkbook ledger. Gas and electric bills. Years' worth of tax returns that gives me pause, because if I know one thing, it's that Uncle Sam isn't about to pay out tax refunds without a social security number.\n\nI set everything else aside except for the tax returns. My eyes go straight to the exemptions, the spot where someone would list their dependents and their dependents' social security numbers, meaning me and my social security number. Except that when I come to it, I find the line blank. Mom didn't list me as a dependent and, though I double-check the year of the form to be sure I was alive at the time, I see that I was. That I was eleven years old at the time the form was completed.\n\nAnd though I don't know much about income taxes, I do know it would have saved Mom a buck or two if she had thought to use me as a tax deduction. A baby gift from Uncle Sam.\n\nI wonder why Mom, who was frugal to a fault, didn't claim me as a dependent that year.\n\nIt was a mistake, I think. An oversight only. I dig through to find another 1040 in the tower of paperwork\u2014this one older, when I was four years old\u2014and search there for my name and social security number, finding it nowhere. Another year that Mom didn't claim me.\n\nI sift through them all, six tax return forms that I can find\u2014my movements becoming faster, more frantic as I dig\u2014and discover that never once did Mom claim me as a dependent. Not one single time.\n\nI turn off the light and get back into bed. I lie there, wondering why Mom didn't claim me as a dependent. What did she know about the IRS that I don't know? Probably a lot, I reason. I don't pay taxes. I've never once been sent a check from them. My only knowledge comes from hearsay, from eavesdropping on clients like Mr. and Mrs. Ricci, discussing whether they could claim Mrs. Ricci's shopping binges as exemptions, all those fancy clothes she toted home in the trunks of cabs.\n\nMom must've had a good reason for what she did.\n\nI listen to the clock, _tick, tock_. I don't bother closing my eyes except to blink, because I know that I won't sleep. I pull the blanket up clear to my neck because it's cold in the room. Though the thermostat downstairs is set to sixty-eight degrees, I have yet to hear the heat kick on.\n\nFall is here and winter is coming soon.\n\nI'm rubbing my hands together for friction, to try and create heat. To make myself warm. I rub them together and then press them to my cheeks. Rub and then press, rub and then press. And that's when I hear a noise.\n\nIt's sudden, the kind of noise that makes me sit up straighter in bed, that makes me hold my breath to listen.\n\nThe only way to describe it is a ping. A ping, and then nothing. Ping, and then nothing. It's a piercing noise when it comes, like some sort of mechanical bleep or chime, the second or two between each ping a welcome reprieve. I rub at my ears, certain at first that the noise originates there, in my own eardrums. That it's merely tinnitus, a ringing in the ears, something only I can hear.\n\nBut then I realize it's not coming from my ears.\n\nIt's coming from somewhere on the other side of the room.\n\nI stare though the blackness but see nothing. It's too dark to see much of anything, aside from my own hand when it's pressed all the way up to my face. And so I push the blanket from me and rise, following the noise. I move blindly, feet guiding me, my steps small because I don't know what's in front of me. Where the bedroom ends and the stairs begin. I have to be careful so that I don't fall.\n\nI skirt around the edge of the bed, where I find myself on the other side of the room, hunched at the shoulders because the squat ceiling doesn't allow me to stand upright. From there, the noise rises up from the floor to greet me.\n\nI drop to my knees, running my hands over a metal grate by accident. There I discover a floor register, one of those metal contraptions that attaches to the end of an air duct and leads somewhere under the floor, to some other room in the home. That's where the ping is coming from, from some other room in the home. In my imagination, I see a mallet being tapped against the slats of another register in another room, because that's what it sounds like to me. Like metal on metal, rhythmic and fixed.\n\nI lie on the floor, pressing an ear to the grate so I can hear it more clearly. The ping. Which makes me think only of sonar emitting pulses underwater and then waiting for them to return, to see if there's anything out there, anything like whales or submarines. Except the only thing here is me.\n\nI'm overcome with the strangest thought then. An irrational thought but one that somehow makes sense.\n\nSomeone is trying to speak to me. To communicate with me.\n\nI press my lips again to the cold metal grate and call out, \"Hello?\"\n\nAt first there's no reply. The ping disappears, and as I sit there, waiting foolishly for someone to respond to me through the floor register, I realize this is ridiculous. Of course there's no one at the other end of the floor register speaking to me.\n\nBecause if there was, that would mean they're in the carriage home with me.\n\nA chill rises up my spine, one vertebra at a time.\n\n_Is there someone in the carriage home with me?_\n\nI rise to my feet and scurry across the room\u2014quicker this time, forgetting altogether about falling down stairs. I reach out to flip on the bedroom light. A yellow glare spreads over the room, obliterating the darkness. I stand at the top of the steps, staring down over the rest of the carriage home, listening for sounds, watching for movement. But there are none.\n\n\"Is anyone there?\" I call over the stairwell, my voice timid and afraid. My heart beats hard; my hands begin to sweat. For three or four minutes, no one appears and in time, logic begins to watch over me. I shake my head, feeling stupid.\n\nOf course no one is here.\n\nIt's the newness of the home that's to blame. That's what has me on edge. Because for the first time in my entire life, I'm alone and somewhere new. I feel lost without Mom, not knowing who I am or where I belong. If I belong anywhere.\n\nI turn off the bedroom light, and the room is once again plunged into darkness. It's darker now than it was before because my eyes have adjusted to the light. I creep across the room and back toward the bed, reminding myself that this house is old. Old homes come with all sorts of strange but innocuous noises. Rats living in the insulation, the settling of the home, water moving through the pipes. That's all that it is.\n\nAs I reach for the bed, I almost have myself convinced.\n\nUntil seconds later when the voices come. Female voices by the pitch of it, higher than that of a man. I suck in a gulp of air and hold it in, not believing my own ears.\n\nSomeone is there.\n\nThe voices are hard to hear, as if they're a million miles away, the sound dampened by distance and the network of aluminum tubes that make up the ductwork. At first it's only sounds, the cadence of women speaking, but no words that I can make out.\n\nUntil I do.\n\n\"It won't be long now,\" I hear, and at first I'm scared. My knees buckle. My throat constricts. My hands go to my throat without meaning to, pressing hard against my vocal cords. My tongue turns to sandpaper and though I'm cold, sweat breeds on my skin.\n\nI see women in some sort of insulated room, by the sound of it. Patients in a psych ward, the walls covered with plastic and foam; a door, padded on the inside, but reinforced with steel. No knob on the door. No way to leave. That's where I imagine the women are.\n\nI stagger back to the floor register, setting myself down over it. I press my ear to the grate, willing the voices to return again, but at the same time hoping they won't. Because I pray that no one is here.\n\nI call into the floor register, my voice mousy at first, scared, \"What? What won't be long now?\" Though my words are a whisper only, and if they were standing in the very same room as me, two feet away, they wouldn't hear.\n\nI cup my hands around my lips, pressing them flush to the floor register this time, so close I taste the bitter metal in my mouth. I call out, voice louder and more emphatic than it was before, \"Can you hear me? Is anyone there?\"\n\nThe only words I hear are low and plaintive. \"She's dead to the world.\" But to my question there is no reply. Whoever is there can't hear me.\n\nThe voices are hollow at first before they go silent. They disappear completely as I sit there, pressing my ear to the floor register in vain. But the only sound that I hear now is the _tick, tock_ of the wall clock.\n\nMy pulse is going at a breakneck speed. It pounds hard through my temple, my wrist. Wind rattles the carriage home, hissing its way in through the window's gap.\n\nA noise returns from the floor just then and I think that they are back. The women, the voices. The ping. I press my ear to the metal grate and listen.\n\nBut this time the only thing that comes is a rush of lukewarm air blasting into me.\n\nThe heat. The heat has finally kicked on.\n\nI think of the maze of tubes that work their way through the home and into this room from the furnace. The pipes and fittings and ducts. The ductwork, which, for a home this old, whimpers at every bend like the high pitch of female voices speaking, a whimper that my tired mind only doctored into words. There were never any women there.\n\nIt was the furnace's burners igniting, starting to produce heat. The furnace spurting air into the home. It comes out with a whine this time, and I press my hands to the grate to thaw them out.\n\nI'm aware suddenly of just how much my entire body aches.\n\nThe insomnia has taken my sleep from me, and now it's taking my mind. Turning the gray matter to sludge. How long can I go on, I wonder, without sleep?\n\nI return to bed and lie down on the mattress, staring out the open window at the sky. It's turned black now, though before I can sleep, dawn will be here. Not in the blink of an eye because that's not the way it is with insomnia.\n\nTime is as slow as the three-toed sloth when you can't sleep.\neden\n\nAugust 2, _1996  \n_ Egg Harbor\n\nThe days have grown longer now that the task of getting settled into the cottage is through. The walls are painted; the unpacking is done. The garden has become a waiting game, staring at the soil, waiting for something to appear. Always waiting.\n\nEvery day, once Aaron has gone off to work, the next ten hours last a lifetime to me. Ten hours with nothing to do but wait until Aaron comes home to keep me company. Afternoons alone are lonely; dinners alone are lonely. I can't fall asleep until Aaron, completely tuckered out from another work shift, drops into bed beside me, nor can I bring myself to admit to him that I am lonely and bored.\n\nBefore leaving Green Bay I worked reception for a local pediatrician. It wasn't anything glamorous or ambitious, answering phones, greeting customers, coding medical records, tallying up bills, but it was _something_. But now Aaron has suggested that I not work, that I stay home, that soon enough we'll have a baby to raise and then I'll have something to do.\n\nOn occasion Miranda and her boys stop by for a visit, their afternoons long and lonely as well. We sit in the backyard, watching Jack and Paul wreak havoc on the tree swing, and as we do, I listen to Miranda depreciate parenthood, complain about her husband and her kids, knock the tedium of her everyday routine: the frozen waffles, the syrup in the hair, the messy bath times and all the books that she and her children are meant to read but never do because it's far too easy to just let them watch TV. Her husband\u2014Joe\u2014wants her to limit TV time to an hour a day, and Miranda laughed at this, saying Joe didn't have the first clue what it was like to be pregnant, what it was like to raise rough-and-tumble boys like Jack and Paul. She'd take any quiet time she could get, even if it meant they sat perched in front of the TV for five hours at a time, so close they were liable to go deaf and blind. She didn't care. Anything so long as they were quiet.\n\nThese were the words Miranda used and I stared at her openmouthed; I could hardly believe my ears. I agreed with Joe, treading carefully, delicately, saying how I'd read that too much TV can lead to obesity in children, to aggressiveness among other things, and she made light of this, saying I didn't know the first thing about being a mother.\n\n\"Just wait until you're a mother,\" she said. \"Then you'll see,\" __ while hoisting her bare feet onto my patio chair and drinking her lemonade.\n\nAnd then when adorable little Paul ran over and made every attempt to scamper onto her lap, hot and sweating, Miranda shirked away, saying, \"Come on, buddy, it's too hot for laps today,\" while pushing him off as if he was some sort of bug who'd landed on her legs.\n\nIn that moment, what I wanted to do\u2014what I ought to have done\u2014was pull him up onto my own lap. Let him rest his tiny head on my shoulder for a while. He was tired too, in addition to hot and sweating, his eyes begging for a cool bath and his afternoon nap, though Miranda was too busy whining about the drudgery of motherhood and wasn't yet ready to leave.\n\nSuddenly I wanted to feel the weight of him on my own two legs; I craved the heat of his skin on mine. I wanted to press those blond curls away from his eyes.\n\nI've started noticing kids with more frequency lately. Little kids, big kids. Babies. Kids at the park. Kids at the market. Kids walking down the streets of town, holding hands with their fathers and mothers. It seems everyone in the world suddenly had kids, everyone but Aaron and me.\n\nHad they been there all along and I failed to notice?\n\nOr did they arrive just then and there the moment Aaron and I decided to conceive?\n\nI didn't welcome Paul onto my lap as I wanted to do, but watched instead as he pouted and walked away, forced off Miranda's lap with her own two hands. His eyes were downcast, his bottom lip thrust out. He cried, not big crocodile tears but rather quiet and ashamed tears, the tears of someone who'd been told one too many times not to cry.\n\nAnd, as he disappeared to a corner of the yard to be sad, Miranda released a massive sigh of relief, grateful Paul was gone and she could once again breathe.\n\nAugust 14, 1996 _  \n_Egg Harbor\n\nIt's starting to become apparent that sex alone doesn't lead to a baby.\n\nWhen I woke this morning with blood dotting the inside of my underwear, my belly seized by a cramp, I knew another month had come and gone without a child. After the second month of trying, that blood in my underwear came as a startling blow, and there in the bathroom, hunched over the toilet seat, staring at the candy-apple red flecks on the lining of my favorite lace underwear, I began to sob. I kept it quiet and stifled, so that Aaron, in the kitchen brewing our morning coffee, couldn't hear. I didn't want him to know that I was upset. For whatever reason, I'd convinced myself over the last few days that every single twitch and prick I felt were the earliest signs of pregnancy. The tenderness in my chest, the desire to lay waste to almost everything I could find in the pantry, especially that which was high calorie, high fat.\n\nThese weren't signs of pregnancy after all, but rather signs of my period. The same ones I'd felt every single month for the last fifteen years of my life, since the middle of seventh grade when I started my period in science class, red blood seeping through a pair of white jean shorts. And now my biological clock had only convinced me I was pregnant. I'd been sure I was nauseated, morning sickness _already_ , when what it was was a change in hormones, my uterus clearing the decks, paving the way, getting ready to welcome a life that wouldn't be.\n\nI feel empty now, robbed of something that was mine, but why?\n\nHow can I grieve for something I never had?\n\nAfter Aaron left for work, I scrubbed my underwear clean with detergent and bleach and headed into town. I couldn't bring myself to tell him about the blood. Of course we don't much talk about babies and pregnancy or use words like _ovulation_ or _conceive_. Ostensibly, we just have sex, though inwardly what I'm thinking about, what we're _both_ thinking about, as we lie together in the moments afterward, my head draped across his chest, his warm hands massaging me, moving yo-yo-like up and down my back, is the end product, our handiwork, our creation, Aaron and me coming together, the best of him and the best of me fusing to create a baby.\n\nI know he wants this as much as I do.\n\nOnly one time did Aaron whisper to me as we lay there in the darkness of the bedroom, still trying hard to catch our breaths after we were through, that he wondered what she would look like and when I asked, \"Who?\" he said, \"Our baby girl. Our baby girl.\" I beamed there from ear to ear and when I told him I didn't know, he said, \"I bet she'll look like you.\"\n\nAnd then he kissed me slowly and deeply, the kind of kiss I felt all the way to my every extremity, and though he didn't say it, I knew that in Aaron's eyes if our baby girl looked like me, that she'd be the most beautiful girl in the world.\n\nIn all my life no one has ever made me feel as special as Aaron makes me feel.\n\nI've watched him garden, watched the way he carefully carries the peat pots from greenhouse to garden, his every move screaming of paternal instinct; the way he digs the perfect holes, assessing their dimensions twice for accuracy; the way he lays the tiny biodegradable contraptions inside as if setting an infant in a crib, scattering soil over the top as gently as drawing a blanket to a sleeping child's chin. He waters and watches and waits, and as he does, I watch him, this solid figure who, by his stature alone should be anything but gentle and soft-spoken, and yet he is. He wears his chestnut hair short these days, easier to hide behind the chef's cap so there can be no false claims of hair in food, at least not from him, his hands and forearms marked with a selection of scratches and burns. For as long as I remember, he's had them, those scratches and burns: badges of honor, war wounds dating back to his culinary school years.\n\nThere are times I find that I can't take my eyes off those scars.\n\nEach time he steps carefully through the garden, tending the seeds, careful not to step on our seedlings, it strikes me what a good father Aaron will one day be, so patient, so protective, so loving, the way he is with me.\n\nAnd so, to say the words aloud now, to tell him I've started my period, would be to confess to Aaron that though we _tried_ again this month, _tried_ to conceive a baby, we failed.\n\nAfter I wiped my eyes, I joined him on the dock for coffee and together we watched the boats pass by and shortly before two o'clock, as always, he left for work and again I was alone.\njessie\n\nEverything changes with the break of day.\n\nAs the sun rises, gliding over the horizon, the world turns bright. The oppressive burden of night disappears. For the first time in eight long hours, I can breathe.\n\nIn daylight, I find myself standing above the floor register on the bedroom floor, feet straddling it. I stare down at the black rectangle between my legs. There's nothing ominous about it; it's just an ordinary metal grate, cold now, the furnace no longer producing heat. I rub at my arms in an effort to warm them up.\n\nI shower and dress and head out into the day. Outside it's a cold start, no more than forty degrees that will rise up to sixty-five by midday. The sky is blue for now, though there's rain in the forecast. The grass is wet with dew. My fingers are cold as I lock the door.\n\nFrom where I stand, I catch a glimpse of my landlord through the window of her own kitchen. It's the back of her, just a pouf of hair and the ribs of a blue sweater before they meet with the wooden slats of a chair. It is a distorted image at best, muddled by the reflection of the outside world on glass. She doesn't see me.\n\nI could knock on the door, make an introduction, but that really isn't my thing.\n\nI round the side of the carriage home, gathering Old Faithful from the alleyway where I left her, leaned up against the side of the home. Ivy grows up the brick of the garage, the leaves starting to turn red. The alley is abandoned. There is nothing more than garage doors and Dumpsters here. City of Chicago garbage bins. No people. No rats. No feral cats. No signs of life anywhere. I settle Mom and her urn into the basket on back, nothing more than a metal milk crate that I keep secure with bungee cords. We set off down the street.\n\nIt's no secret that Chicago is the alley capital of the country, with over a thousand miles of shadowy backstreets. The kind of darkened corridors where people like to hide their trash and vermin, and nobodies like me.\n\nMorning traffic, as always, is a mess. Millions of people move this way and that like cattle in a cattle drive. My first stop is the same as always: coffee. I take it to go with a sugar twist from the bakery, where the donuts are fresh and the coffee is hot and cheap. I don't have six bucks a day to spend on coffee, and the owner knows me, sort of. She always says hello and calls me Jenny, and I don't have the heart to tell her that, after all these years, she's got it wrong. I set my coffee in the cup holder, pedaling away, making my way toward the Loop. I take my time, moving in wide circles around cars and trucks illegally parked in the bike lines, careful to avoid the city's sewage grates. I stay away from potholes.\n\nHaving no luck finding my social security card in the box of Mom's paperwork, I started the day with an idea in mind: getting a new one. That and figuring out how to get my name removed from this inauspicious death index it's on. I head toward the Social Security Office and there, wait in line for a mind-numbing hour, only to learn that in order to get a new social security card, I need to prove who I am. Something more legitimate than just my word. I need to provide some sort of official identifying documentation like a driver's license or a birth certificate that says I'm Jessica Sloane, neither of which I have.\n\nOn the advice of an employee at the Social Security Office, I head next to the Cook County Clerk's Office in the Richard J. Daley Center\u2014the Bureau of Vital Records\u2014in the hopes of tracking my birth certificate down.\n\nWhen I arrive at the Daley Center, the plaza is teeming with people. I tie Old Faithful up to the bike rack outside, watching as men and women in business suits take wide strides across the plaza. I rush past the Picasso and into the imposing lobby, where I wait in line to pass through security, looking on as others empty their pockets with the speed of a snail. I make it through the X-ray machine and the contents of my bag are searched. When I'm deemed harmless, the guard sends me on my way to the clerk's office, which is in the lower level of the building.\n\nA surge of people wait before the elevator doors and so I take the stairs alone, heading down where I take my place in a long line, sighing in solidarity with those who also wait, avoiding eye contact, losing patience.\n\nWhen it's my turn, an employee beckons, \"Next,\" with a hand held up in the air so that I see her there, hunched over a computer screen, shoulders sagging. I go to her, telling her what I need.\n\nSuddenly it dawns on me all the information I'm liable to find when the woman locates my birth certificate. Not only the documentation I need to prove I'm Jessica Sloane, but the place where I was born. The exact time I slipped from Mom's womb. The name of the obstetrician who stood below, waiting to catch me as I fell.\n\nMy father's name.\n\nIn just a few short minutes, I'll know once and for all who he is. Not only will I have proof of my own identity, but of my father's as well.\n\nI would never have done something as flagrant as seek out my birth certificate from vital records if Mom were still alive. That would have broken her heart, my having access to all these things she never wanted me to have. Searching our home seemed innocent enough, but tracking down my birth certificate feels like a really egregious act were she still here.\n\nBut Mom told me to _find myself_ , and that's what I'm trying to do. To get into college, to make something of myself. To do something that would make Mom proud, all of which I can't do without a social security card.\n\n\"I need to get a copy of my birth certificate,\" I say to the employee. My heart quickens as she slides a request form across the counter. She tells me to fill it out. I reach for a pen, completing as much of the form as I can. It isn't much. I can't answer the question that pertains to place of birth or anything having to do with my dad\u2014what his name is, where he was born.\n\nIt's only as I pause in my writing that the worker takes pity on me. Her eyes soften ever so slightly and she says, \"You don't have to fill it all in,\" while staring uncomfortably at the urn in the crook of my arm, seeing the way the pen in my hand hovers above the words _father's name_. \"Just as much as you know,\" she adds, telling me she can try and look it up with what little I know. I slide the form back to her, half-complete, and she says she'll just need the payment and to see a photo ID.\n\n_A photo ID._\n\nIt's easy to explain why I don't have a photo ID. Because by this point in most people's lives, they have a driver's license, which is something I also don't have. Because the cancer came the year I turned fifteen, the year I was meant to enroll in my high school's after-school drivers' education program. Because after we learned that Mom had an invasive tumor in her left breast, knowing how to drive a car\u2014in a city where we didn't need or own a car\u2014didn't take top priority. Because my afternoons were tied up with Mom from then on, riding the bus with her to bajillions of doctor appointments or working to help pay for our home and her care. Because once I knew there was a good chance Mom would die, I wanted to spend every minute I could with her.\n\nAnd yet I'm loath to tell the worker the bind I'm in because I know how it will sound. And so instead of coming clean, I root around in the pockets of my jeans, extracting the lining. I dive a hand deep into the depths of my bag searching for something I know isn't there. I pluck thirty dollars out of my wallet\u2014the cost of the birth certificate is only fifteen\u2014and try handing it to the woman. \"Keep the change, please,\" I say, bemoaning in a low voice how my license was in my bag just this morning. How it must have fallen from my wallet on the way in. How it was there, but now it's gone.\n\nI press the urn to my chest, hoping the woman's mercy will prevail and she'll pocket the extra fifteen bucks and get me what I need. She stares at the money for a minute and then asks whether I have any other form of ID. An insurance card or voter registration, but I shake my head and tell her no. I don't have either of these things. Mom had health insurance. A rock-bottom plan that helped pay for cancer treatment, though I'm still in the hole more than I care to think about. But Mom never added me to her insurance plan because she said it wasn't something I needed. I was young and healthy and the rare trip to the clinic could be paid for with cash. Those required school vaccines I got at the Department of Public Health because they were cheap.\n\n\"Got any mail with your name on it?\" the woman wants to know, but I shrug my shoulders and tell her no. She gives me a look. Disbelief, I think. I'm as much of a skeptic as the next guy; I know how this sounds.\n\n\"Please, ma'am,\" I beg. I'm tired and I don't know what else to do. My eyes feel heavy, threatening to close. There's the greatest desire to lie down on the floor and sleep. Except that it's only a tease, my body playing tricks on me. Even if I lie outstretched on the linoleum tiles, I still wouldn't sleep.\n\n\"I really need that birth certificate, ma'am,\" I say, shuffling in place, and it must be something about the way my voice cracks or the tears that well in my eyes that makes her lean forward and snatch the money from the countertop. She gathers the bills into her hand, counting them one at a time. Her eyes take a quick poll of the room to see if anyone else is watching, listening, before she whispers, \"How about this. How about I see if I can find anything first. Then we'll figure out what to do about the ID.\"\n\nI say okay.\n\nShe takes the form and begins typing information onto the rows of keys.\n\nMy heart pounds inside my chest. My hands sweat. In just a few short minutes, I'll know who my father is. I start thinking about his name. Whether he's still alive. And if he is, if he thinks about me the way I think about him.\n\nBy now, there are at least twenty people in line behind me. The room isn't large by any means. It's stodgy and drab, and everyone is looking at everyone else like they're a common criminal. Ladies clutch their purses to their sides. A kid in line screams that he has to pee. As he yells, I glance over my shoulder to see this poor kid, maybe four years old, hand pressed to his groin, eyes wide and ready to burst, his mother reading him the riot act for nature's call.\n\n\"There were no records found,\" the woman says to me then. Not at all the words I expected to hear. My face falls flat; my mouth parts. For a second I'm confused, unable to produce coherent thoughts or words.\n\nI fight to find my voice, asking, \"Are you sure you spelled it correctly?\" imagining her hunting and pecking for the letters, clipping the corner edge of some surplus letter by mistake, misspelling my name.\n\nBut her face remains motionless. She doesn't attempt another search, as I'd hoped she'd do. She doesn't glance down at the computer or check her work.\n\n\"I'm sure,\" she says, raising a hand into the air to beckon for the next customer.\n\n\"But wait,\" I say, stopping her. Not willing to give up just yet.\n\n\"There were no records found, miss,\" she tells me again, and I ask, feeling incredulous, \"What does that mean then, _no records found_?\" because what I'm suddenly realizing is that, instead of being _dead_ , the crux of the matter is that there is no birth record on file for me.\n\nI can't be dead because I haven't yet been born.\n\nThe Bureau of Vital Records doesn't even know I exist.\n\n\"Of course you must have found something,\" I argue, not waiting for a reply. My voice elevates. \"How can there be no birth certificate for me when clearly I'm alive?\"\n\nAnd then I pinch a fold of skin on my arm, watching as it swells and turns red before shriveling back down to size. I do it so that she and I can both see I'm alive.\n\n\"Ma'am,\" she says, and there's a shift in posture, her empathy quickly giving way to aggravation. I've become a pest. \"You left half this form blank,\" she says.\n\nI argue that she told me I could. That she was the one who said I didn't have to fill it all out. She ignores me, continues to speak. \"Who's to say you were even born in Illinois? Were you born in Illinois?\" she asks, challenging me, calling my bluff, and I realize that I don't know. I don't know where I was born. All my life, I only assumed. Because Mom never told me otherwise and I never thought to ask.\n\n\"No records found means that I couldn't locate a birth certificate based on the information you gave me. You want to find your birth certificate, you need to fill in the rest of these blanks,\" she tells me, slipping the request form back to me as I stare down helplessly at all the missing information, _name of father_ , _place of birth_ , wondering if what I filled in was even correct to begin with.\n\nWas Mom always a Sloane like me? That I'd also assumed. But if she was married when I was born, then maybe she had a different last name, one she ditched at some point over the last twenty years for some reason I don't know?\n\n\"And next time,\" the employee tacks on as I back dismally away, losing hope, running blindly into another woman in line, \"be sure and bring your ID.\"\n\nI make my way out the door, climbing back up to the first floor two steps at a time. The building's stairwell is industrial and dark, a flash of gray that comes at me quickly. It spirals upward in circles for thirty floors or more. When I arrive on the first floor, slipping through the stairwell door, crowds flood the lobby of the Daley Center. I'm grateful for this, for the anonymity of it all. I camouflage myself among the wayward teens who've been summoned here for court, those with purple-dyed hair and heads hidden beneath sweatshirt hoods. I make my way back outside, nowhere closer to finding my father or proving my identity.\n\nAs far as the world is concerned, I'm still dead.\neden\n\nSeptember 14, _1996  \n_ Egg Harbor\n\nThe town was mobbed with people today as it always is on Saturdays, vacationers trying hard to take advantage of the last few warm days before fall arrives. It's September now, days shy of the equinox, and as September eventually bleeds into October, the seas of people will finally leave. They come for the hundreds of miles of shorelines, the extensive gift shops, the food. But by December, this far north into Wisconsin, the temperatures will hurtle to twenty or thirty degrees, mounds of snow will obstruct the streets, and the skies will be endlessly gray. And then no one will want to be here, least of all me. Aaron and I will spend the Midwest winter as we always do, imagining the warm places in the world we hope to one day go, places where cold and snow don't exist. St. Lucia, Fiji, Belize.\n\nPlaces we will never go.\n\nI spent the day while Aaron was at work wandering the town's streets, simulating a tourist. I visited gift shops; I bought a T-shirt and ice cream, a book on sailing. I rode the Washington Island Ferry through Death's Door, spending the late-afternoon hours exploring the crystal clear waters and the polished white stones of Schoolhouse Beach, trying to skip rocks out over the lake, and like getting pregnant, failing at that too.\n\nBack in town I watched families wander from store to store, mothers with buggies, fathers with toddlers perched on their backs. I stared at them as afternoon blended into evening, seated on a bench at Beach View Park, watching as families laid out blankets, staking their claim to a patch of land for the night's sunset display.\n\nThe children were everywhere, and I started to wonder why something in so much abundance could ever be hard to achieve.\n\nOctober 8, 1996 _  \n_Egg Harbor\n\nEach time Miranda and her boys stop by, she has a new suggestion for me, some tip on how to hasten conception. No subject is too personal or too taboo to discuss, from the style of Aaron's underwear to various positions that supposedly aid in fertilization as she lounges on my back patio or living room sofa, weather depending, and cites for me the reasons she believes Aaron and I are not yet pregnant\u2014though never once did I ask.\n\nAs she talks, Jack and Paul loiter before us, performing for me a song they learned, a magic trick, how they can make their eyes go crossed. They stand before me as Miranda spells out the effects of tight underpants on the male genitalia, saying over and over again, \"Look at me, Miss Eden. Look what I can do,\" __ while folding their tongues in half, or trying to make them stretch clear to the ends of their noses, and, as Miranda talks louder to counter their escalating tones, it hits me how attention-starved they are, how they would give anything for her to watch them for a minute, to praise their talents. Every day, there is dirt wedged beneath their nails and some sort of food on their cheeks and chins. Their outfits are cobbled together with clothing that doesn't match and hardly fits.\n\nI clap my hands for Jack and Paul, but Miranda tells them to go away. To go play.\n\nEvery day.\n\nAs her baby bump swells more and more, I'm pestered by Miranda to _hurry up_ , to get knocked up, so that her baby and my baby can still go to school together as I've promised her they would.\n\nIf I wait much longer they'll be in different grades.\n\nThat's what Miranda has told me.\n\n\"September is the cutoff, don't you know?\"\n\nAccording to Miranda's timeline, I have until September of next year to have a baby. Twelve months, which leaves only three to get pregnant.\n\n\"It's not that we're not trying,\" I've tried to explain, and she counters with a flip of the hand and a slapdash \"I know, I know,\" and then it's back to the underwear we go. To help with Aaron's and my fertility issues, she suggested a pillow beneath my hips to help steer sperm in the right direction. \"It's all about gravity,\" she says.\n\nAt every visit I watch the size of her own baby bump swell, her maternity shirts no longer able to cover its overwhelming girth. I tell myself that her suggestions are only old wives' tales, not rooted in truth, but how am I to know if that's true?\n\nBut today when she lounged on my sofa, peering at me with that same expression on her face\u2014mouth parted, eyebrows raised\u2014and asked if I was keeping track of my ovulation, I realized how stupid it was of me, how naive.\n\nThis was Aaron's and my first foray into babymaking. I was sure it was something that just _happened_ , that there was no need to time or plan. In the moment, I told her yes, of course I was keeping track of my dates, because I couldn't bring myself to say otherwise, to admit to her that it never occurred to me to figure out when I was and when I wasn't ovulating. Aaron and I both come from large families, and the number of grandchildren our parents have been blessed with is in no way in short supply. It seemed a given that after ample time, after many months of waking up in the morning to Aaron's soft fingers tracing my bare skin, thumbs hooking through the lacy edges of my underpants, gliding them proficiently over my thighs, sooner or later we'd succeed. We'd make a baby as we intended to do.\n\nBut for the first time I've come to realize that this is going to take more than time.\n\nAfter Miranda left I drove to the library and sought out a guidebook on pregnancy and there, in the stacks of books, plotted out my approximate menstrual cycle. I figured out the first date of my last period. I counted backward; I did the math. It wouldn't be perfect, that I knew\u2014my periods had never been perfect\u2014but it would be close. And close to perfect was better than nothing for me.\n\nAnd now, knowing that in just two days' time I will be ovulating fills me with an abundant amount of hope. Aaron and I were doing it wrong all along, missing out on the best times to get pregnant, likely omitting my most fertile days, those negligible hours when conception can occur. On the way home I stopped at the market and picked up a pocket-size calendar and, at home, with a red pen, circled my most fertile days for the next three months, through the end of the year.\n\nThis time we'll get it right.\njessie\n\nI push my way through the turnstile doors and step outside, making my way across the plaza. Beside the Eternal Flame, I pause, overcome with the sudden urge to scale the fence and lie down beside the puny little fire in the fetal position. To fall to my side on the cold concrete, beside the memorial for fallen soldiers. To pull my knees up to my chest in the middle of all those pigeons who huddle around it, trying to keep warm. The land around the flame is thick with birds, the concrete white from their waste. That's where I want to lie. Because I'm so tired I can no longer stand upright.\n\nPeople breeze past me. No one bothers to look. A passing shoulder slams into mine. The man never apologizes and I wonder, _Can he see me? Am I here?_\n\nI head to the bike rack, finding Old Faithful ensnared beneath the pedals and handlebars of a dozen or more poorly placed bikes. I have to tug with all my might to get her out and still I can't do it. The frustration over my identity boils inside me until I feel myself begin to lose it. All this red tape preventing me from getting what I need, from proving who I am. I'm starting to question it myself. _Am I still me?_\n\nThe debilitating effects of insomnia return to me then, suddenly and without warning. General aches and pains plague every muscle in my body because I can't sleep. Because I haven't been sleeping. My feet hurt. My legs threaten to give. I shift my weight from one leg to the next, needing to sit. It's all I can think about for the next few seconds.\n\nSitting down.\n\nPins and needles stab my legs. I wrench on the bike, yanking as hard as I can, but still she doesn't budge. \"Need a hand?\" I hear, and though clearly I need a hand, there's a part of me feeling so suddenly indignant that I turn with every intent of telling the person that _I've_ _got it_. Words clipped. Expression flat.\n\nBut when I turn, I see a pair of blue eyes staring back at me. Royal blue eyes like the big round gum balls that drop down the chute of a gum-ball machine. And my words get lost inside my throat somewhere as I rub at my bleary eyes to be sure I'm seeing what I think I'm seeing. Because I know these eyes. Because I've seen these eyes before.\n\n\"It's you,\" I say, the surprise in my voice clear-cut.\n\n\"It's me,\" he says. And then he reaches over and hoists Old Faithful inches above the other bikes, those that have held her prisoner all this time. It's effortless to him, like nothing.\n\nHe looks different than the last time I saw him. Because the last time I saw him he was folded over the cafeteria table, drinking coffee in a sweatshirt and jeans. Now he's dressed to the nines in black slacks, a dress shirt and tie, and I know what it means. It means that his brother has died. His brother, who was hurt in a motorcycle accident after a car cut him off and he went flying off the bike, soaring headfirst through the air and into a utility pole without a helmet to protect his head.\n\nHe held vigil beside his brother's hospital bed while I held vigil beside Mom's. And now, six days later, his eyes still look tired and sad. When he smiles, it's strained and unconvincing. He's gotten a haircut. The dark, messy hair has been given a trim and though it's not prim or tidy\u2014not by a long shot\u2014it looks clean. Combed back. Much different than the hair I saw those days and nights in the hospital cafeteria, his head stuffed under the hood of a red sweatshirt. We only spoke the one night, him fussing about the coffee, telling me how he'd rather be anywhere but there. But still, there's the innate sense that I know him. That we shared something intimate. Something much more personal than coffee. That we're bound by a similar sense of loss, united by grief. Both collateral damage in his brother's and my mother's demise.\n\nHe sets Old Faithful down on the ground and passes the handlebar to me. I take her in my hand, seeing the way his nails are bitten to the quick, the skin torn along the edges. A row of rubber bands rests on his wrist, the last one tucked halfway beneath the cuff of the dress shirt. A single word is written on the back of the hand with blue ink. I can't read what it is.\n\nHe runs his hands through his hair and only then do I think what I must look like.\n\nIt can't be good.\n\n\"What are you doing here?\" I ask, as if I have any more right being here than him.\n\nHe speaks in incomplete sentences, and still I get the gist. \"The wake,\" he says. \"St. Peter's. I needed some air.\"\n\nHe points in the direction of some church just a couple of blocks from here, one that's too far to see from where we stand. Though still I look, seeing that the sun has slipped from the sky and is hidden now behind a cloud. While I was inside the building, the clouds rolled into the city, one by one. They changed the morning's blue sky to one that is plush and white, filling the sky like cotton balls, making the day ambiguous and gray.\n\nI don't ask when or how his brother died and he doesn't ask about Mom. He doesn't need to because he knows. He can see it in my eyes that she has died. Neither of us offer our condolences.\n\nHe rams his hands into the pockets of his slacks. \"You never told me your name,\" he says. If I was the kind of girl that felt comfortable in situations like these, I'd say something snarky like _Well, you never asked._\n\nBut I don't because it's not that type of conversation, and I'm not that girl.\n\n\"Jessie,\" I say, sticking my hand out by means of introduction. His handshake is firm, his hand warm as he presses it to mine.\n\n\"Liam,\" he says, eyes straying, and I take it as my cue to leave. Because there isn't anything more to say. The one and only conversation we had in the hospital, words were sparse, but unlike in the hospital we're no longer killing time, just waiting for people to die. That night, before the conversation drifted to quiet and we sat in silence for over an hour, sipping our coffees, we talked about private things, nonpublic things, things we weren't apt to tell the rest of the world. He told me about his brother beating him up when they were kids. About how he would lock him out of the house in the rain and shove his head in the toilet, giving him a swirly when their folks weren't home. _Such a bastard_ , __ he said, though I got the sense that that was then and this was now. That over the years, things changed. But he didn't say when or how.\n\nI told him about Mom's hair and fingernails, both of which she lost thanks to chemotherapy. Her eyelashes too. I told him about the clumps of hair that fell out, and how I watched on in horror as Mom held fistfuls of it in her hands. How there were whole clods of it on her pillowcase when she awoke in the morning, masses of it filling the shower drain. I said that Mom never cried, that only I cried. It grew back, after the cancer was in remission for the first time, soft fuzz that grew a little thicker than it was before chemotherapy. A little more brown. It never reached her shoulders before the cancer returned.\n\n\"You should get back to the wake,\" I tell him now as we stand there in the middle of Daley Plaza. But he only shrugs his shoulders and tells me that the wake is through. That everyone split.\n\n\"The funeral's tomorrow,\" he says as I wrap my fingers around Old Faithful's handlebars. I don't know what to say to that. There isn't anything to say to that.\n\nTurns out, I don't need to say anything. \"You never said what you're doing here,\" he says then, but as I'm about to explain I realize that there's no easy answer for it, because the reason I'm standing outside Daley Center is far more convoluted than his. And so instead of answering, I sigh and say, \"Long story,\" thinking that he'll just say okay and walk away because chances are good he didn't want to know in the first place. He was probably only being polite because I asked what he was doing here, and so he thought he should too, that he should reciprocate out of courtesy.\n\nBut as he shifts in place and tells me, \"I have time,\" I realize that he wants me to stay.\n\nThere's a sadness in his eyes, the likeness to mine uncanny.\n\nWe walk. Out of the plaza, down Washington and toward Clark Street, me towing Old Faithful by the handlebar. We walk in the street because it's illegal to ride a bike on the sidewalk in the city. I don't know what time it is, but what I can say is that the haste of rush hour is past, the clog of morning traffic like hair in a shower drain. Impossible to get through. It's gone, as if some plumber stopped by and dropped a gallon of Drano on the street, ameliorating the clog. People move slowly now. They take their time. Without the blockage we easily slip through, weaving in and out of pedestrians and cars.\n\n\"I stopped by vital records,\" I say. \"I needed to get my birth certificate. Except that didn't go as planned,\" I explain as we turn a right on Clark, which is a one-way street around here. All the cars come directly at us. They miss us by a hair's breadth at times because there are no bike lanes. Not that it matters because half of the time when there are, cars and trucks illegally park and I have to veer around them and into traffic. The number of bike-related deaths in the city is staggering; I just hope that one day one of them isn't me.\n\nLiam asks why getting my birth certificate didn't go as planned. He's a good ten inches taller than me, broad in the shoulders but narrow around the hips. At just over five feet, I've always been on the short side. My whole life, for as long as I can remember, I've been short. Kids in school used to make fun of me. They'd call me names like shrimp, peanut. Squirt.\n\nHe towers over me, his body slim but in the tailored clothes, he doesn't look too thin. I remember him in the hospital\u2014oversize sweatshirt and jeans, getting swallowed up by fabric. Then he looked thin.\n\nI start at the beginning and tell Liam the whole story. Otherwise it won't make sense. And even then it doesn't make much sense because I'm having a hard time wrapping my head around it myself. I tell him about applying to college, the phone call from the financial aid office. The woman's cheery voice on the other end of the line, laughing, telling me I'm dead. I tell him about the wasted time spent trying to find my social security card, the worthless trip to the Social Security Office. The one that led me here, to the Daley Center in search of my birth certificate, though that too was a waste.\n\n\"I don't have a birth certificate,\" I close with. \"At least not one in the state of Illinois. And without a birth certificate or a social security number, there's no way to prove who I am or that I even exist. But what freaks me out even more,\" I admit, \"is this implication that\u2014\"\n\nBut before I can get the words out, birds swarm around me, moving in from all directions. Pigeons with beady eyes and little bobbing heads, pecking at something on the street. They fight over it, their squawks loud and angry. I try to sidestep them, but their movements are arbitrary, aimless; there's no predicting where they'll go. I step on the tail feathers of one by chance and it scurries, wings slapping together to get away from me.\n\nAs I go to take another step, I see what the skirmish is all about. It's another pigeon, dead, lying on the street where my foot should go. The other birds move in on it, pecking at it, trying to eat it, and just like that, there's nowhere for me to put my foot. It throws off my stride, makes me lose balance. The dead pigeon lies on its back, spread-eagle-like. Its wings are fanned on either side of its body, white belly exposed, its neck turned too far in one direction, broken I think. I see only one beady eye, the other somehow missing. Its beak is tucked into the crook of a neck, and on the street beside it are flecks of blood.\n\nI nearly step on the carcass as my body lurches forward, stumbling, and I'm sure I'll fall. My heartbeat kicks up a notch or two, hands sweaty, and like that I'm at the mercy of the bird and the street.\n\nI let go of Old Faithful's handlebars by accident. I watch as she topples onto the street, certain I'm about to go with her. People turn to see what the racket is, the clang of the bike on the street, the sound of my scream. My hands reach for something to latch on to, coming up empty until Liam grabs me by the wrist, steadying me.\n\n\"Jessie?\" he asks, and I have to fight for a minute to catch my breath. I'm breathing hard, seeing only pigeons nipping the bloody flesh of a dead bird. And I'm thinking about that bird, wondering what happened to kill the bird. How did it die? If it was killed by a car or a bike, or a run-in with a building window maybe. Maybe it flew headfirst into the Thompson Center before sliding down, down, down to the ground.\n\n\"Jessie?\" Liam asks again because I still haven't replied. His eyes watch me, uneasy, as he makes sure I'm steady on my feet before leaning down to reclaim Old Faithful from the street.\n\n\"Are you okay?\" he asks, and, \"What happened?\" and I shake my head and say, \"The damn bird. Those pigeons.\"\n\n\"What bird?\" he asks. \"What pigeons?\"\n\nI turn to point them out to him. But when I look back on to the street behind me, there's no bird. No pigeons. The only thing there is a squandered hot dog that lies on the asphalt. Half-eaten, gravel stuck to what remains of it. Chunky green relish spilling from the bun, red ketchup splattered here and there like blood.\n\nThere's no dead bird.\n\nThere was never a dead bird.\n\nThe world loses balance all of a sudden, the street beneath my feet unpredictable and insecure. I think of sinkholes, when the earth suddenly decides to give, roadways collapsing like Play-Doh, sucking people in and swallowing them whole.\n\nI shake my head. \"Just tripped over my own feet,\" I say, but I can see in Liam's eyes: he doesn't believe me.\n\nWe move on.\n\nLiam waits for me to finish whatever it is I was saying before I saw the bird, but now my train of thought is gone. I can think only of the bird, the pigeons, the flecks of blood. And so he reminds me. And then I remember.\n\nWhat freaks me out the most, I tell him, is the implication that I'm already dead.\n\nHe asks about the death database. What it is and what it's called, and so I tell him what the woman from the financial aid office told me.\n\n\"The Death Master File,\" I say, which in and of itself sounds like something the grim reaper must carry along with him, a listing of all the souls he's sent to collect. Liam looks it up on his smartphone, and soon finds out that access to the file is restricted. That not just anybody can look at it. He tells me what I already know. That it's a listing of millions of people who have died, them and their social security numbers. It's used as a means to prevent fraud and identity theft. To stop living people from opening credit cards and getting mortgages in the name of someone who's already dead.\n\n\"So somehow I got placed on this list, and now my social security number is good for nothing until I clean up this mess. Because on paper, I'm dead. And I can't find my social security card or figure out how to get a new card because I don't have the other documentation I need to do it.\"\n\n\"Listen to this,\" Liam says as a disclaimer pops up on his phone and he reads, quoting verbatim, \"'In rare instances it is possible for the records of a person who is not deceased to be included erroneously in the DMF.'\"\n\nI ask him how that can happen. \"A clerical error,\" he says, meaning with the stroke of one wrong computer key someone who's alive and well is suddenly dead. Or not dead but undocumented, which is almost as good as being dead, I'm quickly learning.\n\nThe only reason my own death went unnoticed for all this time, I think, is because I haven't once been asked to give my social security number. But sooner or later it was bound to happen. When I sought out a driver's license, made an attempt to open a credit card. An attempt that would have been denied.\n\nAs we scoot onto the pedestrian side of the Clark Street Bridge and cross over the Chicago River, I think of people in the same situation as me, unable to access their own bank accounts and going broke. Those who don't have the money for food or shelter, though they _do_ have the money; it's just that it's tied up in some bank account they can't access because the bank is certain they're dead.\n\n\"People get locked out of their own lives, interrogated by police for suspected identity theft when the person whose identity they've supposedly stolen is themselves,\" Liam says as he drops his phone into the back pocket of his pants, and I utter under my breath, \"What a mess.\"\n\nI stare down below, beneath the metal grates of the bridge, where a tour is underway, tourists exploring the polluted grayish-green waters of the Chicago River. The tour guide steers passengers' attention to the bridge\u2014built in 1929, a bascule bridge, she says\u2014and all eyes move to Liam and me, taking photos, pointing upward some twenty feet or more to the bridge on which we stand.\n\n\"You really are Jessie, aren't you?\" His words are dry, meant to be funny, though they're not. His tone is deadpan, his face expressionless.\n\nAnd though I know it's in jest, it's a question that nags at me.\n\nI am Jessie, aren't I? Am I Jessica Sloane?\n\nWe continue to walk. Down Clark and left on Superior, my feet following Liam's lead. We're quiet. We don't speak much. He asks if I've been sleeping. He says that I look tired and I pause, looking at my own reflection in the glass facade of a building. I see what he sees. The sunken eyes surrounded by puffy red skin, the tip of my nose red.\n\nI make light of the insomnia. I say that sleep is a waste of time. That there are so many more productive things I could be doing instead of sleeping.\n\n\"It's not good for you, Jessie,\" he tells me. \"You need to sleep. The melatonin,\" he says, same as he did in the hospital when he slipped those pills into the palm of my hand. \"Give it a try.\" _I did give it a try_ , I think. I tried the melatonin\u2014that and the clonazepam\u2014and slept right on through Mom's death. Never again.\n\nI tell him that I will but I won't.\n\nAnd then he stops beside a mid-rise, saying, \"This is me. This is where I live.\"\n\nThis building beside us is five or six stories tall, flanked with floor-to-ceiling windows. A sign out front offers spacious open-plan lofts for sale. A doorman patrols the revolving front door and there's something very moneyed about it that makes me feel out of place and ill at ease. The Liam before me is suddenly at odds with the Liam I remember from the hospital, the one who was bedraggled, a bit dog-eared like me.\n\nA look of confusion must pass on my face. \"My brother and I lived here together,\" he explains. His voice is deep and there's no rise or fall to his intonation as he speaks, telling me, \"He was a software engineer.\"\n\nI fill in the missing pieces. His brother made the money. He paid for the condo. And now he's gone.\n\n\"You'll be okay?\" I ask, and his reply is detached.\n\n\"What's that they say?\" he asks, plucking at the row of rubber bands on his wrist so that I see now what it says on his hand in the blue ink. _Adam._ His brother's name, I think. \"About death and taxes?\"\n\nThat nothing is certain but death and taxes. That's what they say. But he's not looking for an answer. What he's saying is that he may or may not be okay, but there's no way to know right now. Same as me.\n\nWe say our goodbyes. I watch as he slips through the doors of the apartment building, disappearing behind a wall of glass.\neden\n\nNovember 14, _1996  \n_ Egg Harbor\n\nIt's November now.\n\nThe gray skies have descended, everything perpetually overcast and sad. The boats have been pulled from the bay, leaving it barren and empty, like my womb. The seasonal shops are closed. The tourists took their cue to leave.\n\nTwo weeks ago, on the first of November, Miranda had that baby of hers, a seven pound, three ounce beautiful baby boy who she and Joe named Carter. I visited in the hospital the day after he was born, her only visitor aside from Joe. I saw it in her eyes as soon as I entered the postpartum room, Miranda swaddling baby Carter with a look of arrant dissatisfaction on her face. Her lips were pursed, her eyebrows creased, crow's-feet forming around the eyes.\n\nAs I walked in\u2014swapping places with Joe, who went to the cafeteria for coffee\u2014her disillusioned eyes rose to mine and she confessed aloud so that baby Carter could hear, not bothering to lower her voice or to press her hands to his ears to muffle the rotten words, \"All I wanted was a baby girl. Is it too much to ask for one little girl? But instead it's another goddamn boy.\"\n\nHer words knocked the wind out of my lungs. They made it hard to breathe. They were so ugly and vile, and I saw a look in Miranda's eyes as she spoke of him, eyes dropping to his. A look that made my heart hurt. Only a day old and already she abhorred her baby boy.\n\nI asked if I could hold him and she said yes, handing him off with too much inclination, too much ease, as if grateful to be rid of him. I took baby Carter to a chair in the corner of the room and peered at his inappreciable wisps of blond hair and his heavy, tired eyes, thinking to myself, _What difference does it make if he's a boy or girl, only that he's happy and healthy?_\n\nAnd I felt angry for the first time at Miranda. Not just annoyed but truly angry. Angry that she had three beautiful baby boys and I had none. Angry that she didn't love her babies or value her babies, that she couldn't understand another woman\u2014one like me\u2014would give life and limb for a child.\n\nOh, what I wouldn't give for a child.\n\nI had a thought then.\n\nWould Miranda care if I rose to my feet and carried baby Carter from the room?\n\nWould she even notice?\n\nThe hospital experience, for Miranda, was a welcomed furlough from motherhood. From what I'd been told, Carter spent his time in the nursery, being cared for by nurses round the clock except when he needed to eat, and only then did nurses carry him, crying and discontent, to his mother, and she welcomed him grudgingly, embittered to her chest. And then as soon as he was full, his eyes drifting lazily to sleep, she asked the nurses to take him away so that she could sleep.\n\nSpread out on her hospital bed in an indiscreet polka-dot gown, I watched Miranda sleep. Or pretend to sleep at least, so that she didn't have to tend to her child. She was exhausted, yes, from many hours of labor and delivery, from the every-other-hour feedings, and yet I wondered if her eyes were merely closed so that she could remain insensible to her baby boy, who lay limply in my arms, head misshapen, skin wrinkly and pink as a newborn's should be. Miranda's hair was brushed away from her face, pulled taut into a ponytail that lay flat against the bed. Her arms and hands were stretched out by her sides. She breathed with her mouth open, nostrils flaring with each inhalation and exhalation of air.\n\nI whispered her name. There was no reply.\n\nIt was almost as if she was asking me, begging me, daring me to take her child.\n\nAnd so I did.\n\nI stood from the inflexible armchair, slowly, gradually, piecemeal-like so that the chair wouldn't make a sound. So that the floor wouldn't cheep. So that my own two feet wouldn't betray me. I flexed one muscle and then the next until I was standing upright, holding my breath.\n\nI crossed the room, creeping by degrees so my shoes wouldn't squeak on the floors. Miranda's eyes were closed, enjoying the peace and serenity of having someone else care for her child.\n\nIt didn't occur to her for one instant that someone might try and take her baby.\n\nI slipped into the hallway without a peep. Two left turns and there Carter and I were, standing before the nursery, staring through glass at a half dozen sleeping babies. They lay bundled like burritos in their pink and blue blankets, with knitted hats atop their near-bald heads. They were sleeping, every last one of them, completely tuckered out. The newborns slept in rolling bassinets all arranged on display so that grandmas and grandpas could see. If it wasn't for the slip of paper in each bassinet with the baby's name and date of birth in blue ink, there was no telling them apart aside from the obvious distinction of pink and blue.\n\nHow easy it would be for two to be swapped, or for one to up and disappear.\n\nOne nurse stood guard of them all, a shepherd in the pasture keeping watch over her sheep. What I wouldn't give to be that nurse, to be tasked with caring for the infinite number of newborn babies that rotated in and out of the nursery each day.\n\nI wondered if she ever had a weakness for any one of these babies. A fondness. Was there ever one colicky child who caught her eye, the runt of a litter of multiples she wanted to bring home as her own?\n\nFrom down the hallway a door opened and I saw the main hospital on the other side, areas other than the labor and delivery ward. A common hallway. The hospital's information booth. The doorway was twenty steps away at best, and there was nothing but two unlocked doors to prevent Carter and me from leaving. There was no alarm, at least none that I could see. There was no system to buzz people in and out. It was an open door, an invitation.\n\nHow easily Carter and I could just leave.\n\nI looked around; the nursery nurse had her back in my direction, attention now focused on one little baby who was trying to wake up. Behind me, there was only a single woman at the nurses' station, a middle-aged lady on the phone. Other than that, the ward was quiet and still, all patient doors pulled closed, mothers on the other side in the throes of labor or fast asleep.\n\nI peered to the doorway again, those swinging double doors just twenty steps away from where I stood. I didn't think about the rest, about what I would tell Aaron or what Miranda might do when she awoke and realized Carter was gone. My heart beat quickly as desire and instinct told me to do it and to do it quickly, to move with purpose, to not draw attention to myself. In my arms, I held the very thing Aaron and I had been trying for for months. A baby.\n\nMiranda didn't want him anyway. I was doing her a favor, I reasoned.\n\nHow easily this baby could be mine.\n\nI thought of only one thing in that moment as I stood frozen, staring through glass at the plentiful sleeping babies.\n\nHow easy it would be to just go.\n\nI didn't do it, of course, but it would be remiss to say the idea never crossed my mind.\njessie\n\nI pedal toward Roscoe Village. As I do, I stare over my shoulder, back into the Loop at the peaks of skyscrapers that rise into the sky like distant mountain summits. I watch as the urban streets become residential.\n\nOnce in Roscoe Village, I duck into a burger joint on Addison. My stomach is empty by now, the morning's sugar high having given way to a glucose crash, one which makes me irritable and edgy. I've had nothing to eat but a donut all day\u2014a donut and coffee\u2014though since Mom's death, my hip bones protrude from my waistline and the bones of my rib cage are startlingly transparent.\n\nI'm not _not_ eating on purpose. I've just had no desire to eat.\n\nI order a hamburger and take it to the counter to eat. There, I stare out the window at the world as it passes by without me. A bus goes by, the 152 heading east. A plastic bag floats through the air, surfing the airstream. Middle school kids amble by in private school uniforms\u2014starchy plaid split-neck jumpers; burgundy sweater-vests; pressed pants\u2014with backpacks so heavy they nearly tip over from the weight of them. An older woman stands beside the bus stop. The 152 gathers her up and goes, disappearing in a puff of smoke.\n\nI eat part of my burger, wrapping the rest up for the trash. As I'm about to go, a voice stops me. I turn to see a woman standing beside me in jeans and a cardigan, a pair of white gym shoes on her feet. Her graying hair is wound back into a bun.\n\n\"Jessie? Jessie Sloane? Is that you?\"\n\nBut before I can say one way or another if it's me, she decides for me. \"It _is_ you,\" she declares as she tells me that she remembers me when I was yea high, her hand pegged at about thirty-seven inches in the air. And then she embraces me, this strange woman wrapping her thickset arms around my neck and declaring again, \"It is you.\"\n\nExcept that I don't know who she is. Not until she tells me.\n\nAnd even then, I still don't know.\n\n\"It's me,\" she says. \"Mrs. Zulpo. Eleanor Zulpo. Your mother used to clean my home when you were a girl. In Lincoln Park,\" she tells me, tacking on details as if it might help me remember. \"Tree-lined street, beautiful box beam ceilings, rooms flooded with natural light,\" she says, though she and her husband don't live there anymore, not since the housing market crash when she had to give up her home. When they had to downsize. That's what she tells me.\n\nI draw a blank. I don't remember.\n\nLike me, Mom used to clean homes. Mostly upscale places that we could never afford. She taught me everything I know. My first foray into the family business came when I was about twelve years old and would get down on my hands and knees beside her and scrub floors.\n\nBut before that, when I was too young to clean, Mom would lug me along on assignments and there I'd spend my days playing pretend in strangers' homes. Cooking imaginary meals in their palatial kitchens, tucking my imaginary children into their mammoth beds before Mom scooched me out of the way so she could wash the sheets.\n\n\"You don't remember me,\" Eleanor Zulpo decides, realizing that it must have been sixteen or seventeen years ago or so, when I was three or four. \"Of course you don't remember,\" she says, loosening her hold on my neck, telling me that I look just the same as I did back then. \"It's those dimples,\" she says, pointing at them. \"Those adorable dimples. I'd know these dimples anywhere.\n\n\"I read about your mother in the paper,\" she says then, sitting beside me on her own stool, unwrapping a hot dog. The sight of it alone, that hot dog, lying out on a foil wrapper, slathered in ketchup and relish\u2014that and the smell\u2014reminds me of the dead bird. The pigeon. And instead of a hot dog, I suddenly see blood, guts, gore, and I gag, vomit inching its way up my esophagus. I reach for my drink and force it back down, gargling, trying to get the taste of vomit from my mouth.\n\nMrs. Zulpo\u2014 _Eleanor_ , she says to call her\u2014doesn't notice. She keeps going. \"I saw her obituary,\" she's saying. \"It was a great write-up, a lovely tribute for a lovely woman,\" she says. I tell her that it was.\n\nI submitted the death notice to the newspaper. I covered the cost of the obituary. I found an old photo of Mom to use, one that was a good six years old at least, taken back before she got sick.\n\nWe'd lived our entire lives in private, but for whatever reason I felt the whole world should know that she was dead.\n\n\"There have been other cleaning women since your mother. But never anyone as good as she was, as conscientious, as thorough. She was one of a kind, Jessie,\" she says, and I tell her I know. Eleanor tells me stories. Things I didn't know, or maybe I did. Memories that have been lost to time, erased clear from my brain's hard drive. About the time I helped myself to her Wedgwood china when Mom was cleaning. How I snatched it right from her hutch and set the dining room table to have a tea party with. \"Wedgwood china,\" she tells me, grinning. \"A single cup and saucer go for about a hundred dollars each. They had been my own mother's, given to me when she died. Heirlooms. Your poor mother,\" she laughs. \"She nearly had a heart attack when she found you. I told her it was fine, that it wasn't like anything had gotten hurt. And besides, it was nice to see the dishes being put to use for a change.\"\n\nAnd then she tells me that, at her suggestion, the three of us sat down at the dining room table and drank lemonade from the Wedgwood china.\n\nIt fills me with a sudden sense of nostalgia. A yearning for the past.\n\n\"What else do you remember?\" I ask, needing more. Needing someone to fill in the gaps for me, all those details I can no longer remember.\n\nEleanor tells me how her children were grown by the time I arrived, and so it was nice to have a child in the house again. She didn't work outside of the home. When Mom and I came, she was grateful for the company. She used to look forward to the days we'd come. Usually she'd play with me while Mom cleaned, hide-and-go-seek in her home, or build forts from the newly washed sheets.\n\n\"You were a funny girl, Jessie,\" she tells me. \"Silly and strong willed, a great sense of humor to boot,\" she says. \"A bit ornery too. But those dimples,\" she adds as she takes a bite of the hot dog, speaking through a full mouth, \"with those dimples you could get away with murder, Jessie.\" She laughs.\n\nShe says that anything Mom wanted done, she had to ask me twice. That the lunch Mom brought along for me, I refused to eat. That I was a far cry from shy, and would spend half of my days in her home creating a show to perform for her and Mom before we'd leave.\n\n\"You used to march around, insisting like the dickens that your name wasn't Jessie. Because you didn't like it back then, I think,\" she says then, saying I was adamant about it, insistent that my name wasn't Jessie. That my name was something else, but she doesn't remember what. \"You would pout your face and stomp your foot and insist that people stop calling you Jessie. _Stop calling me that_ , you'd cry, face turning red. Your mother would go along with it for a while, trying to ignore your antics. Because she knew you were doing it for attention and, if she didn't give in to you, sooner or later you'd quit. Though rarely did you quit,\" she smiles, telling me I was a headstrong little girl.\n\n\"You knew what you wanted,\" she says.\n\nEventually Mom would have enough of it, Eleanor tells me, and she'd get down to eye level and say, _That's enough, Jessie. We talked about this, remember?_\n\nBut I have no memory of this at all.\n\nWhy would I go around masquerading as something other than Jessie? I don't have time to come up with an answer because soon Eleanor is telling me how I used to carry an animal everywhere I went\u2014a stuffed dog or a bear or a rabbit\u2014but I couldn't care less about that because what I'm wondering is why in the world I would be so unrelenting about that name. About the name Jessie. Why I would insist it wasn't mine.\n\n\"And then there was your mother's name,\" Eleanor says before I have a chance to think it through, and I ask, \"What about it?\"\n\nHer eyebrows crease. She removes a pair of glasses and sets them on the countertop, rubbing at her eyes. \"It's just that most little girls call their mother _Mom_ or _Mommy_.\"\n\nShe leaves it at that and so I ask, \"And I didn't?\" thinking suddenly that Eleanor is mistaken. That she's wrong. Time has altered these memories of hers, or she's mistaken Mom and me for some other cleaning lady and child. Another child with dimples like mine. Because in all my life, she's only ever been one thing to me\u2014Mom\u2014or so I think.\n\nEleanor shakes her head and at the same time I see my hands before me, gripping the edges of the countertop, also shaking.\n\n\"You didn't,\" she says. \"You called her by her given name _._ \"\n\nEleanor tells me that Mom would put up with it to a certain extent but then every now and again she'd get down and whisper in my ear, _We've talked about this, Jessie. Remember?_ Same as she said about my own name. _You're to call me Mom._\n\n\"For a short while, you'd remember. You'd remember to call your mother _Mom._ But before too long, you'd forget and go back to calling her by her Christian name. Eden.\"\n\nI don't remember doing that.\neden\n\nJanuary 16, 1998   \nChicago\n\nI drove the speed limit the entire way, not wanting to draw attention to myself. It snowed much of the time and the roads were slick, though being a Midwesterner, I'm quite accustomed to driving on slick roads. This wasn't my first time with snow. And yet it was my first getaway, my first flight. My first vanishing act of what I hoped wouldn't be many, because I prayed that the world would let me disappear, that he would let me go.\n\nI found myself staring in the rearview mirror nearly the entire time, all along Highway 42 and to the interstate, knuckles turning white from their grip on the steering wheel, though I knew there was no logical way he knew where I was, or that he watched me leave. But still.\n\nHe might just be there.\n\nWhen I arrived, the first thing I did was find an apartment that I could afford, which wasn't easy considering I have so little in the way of money, nearly nothing at all, quite literally ten dollars more than was the rent payment, which means that for the immediate future, we'll be eating bread and cheese. I purchased a paper at a newsstand and, on a snow-covered park bench, scanned the for-rent ads, settling on a studio apartment in Hyde Park. The building is all wrapped up in a creamy yellow brick facade that's gone to rack and ruin; it looks abandoned, uncared for and unloved, like me. The ad trumpeted a French Renaissance charm but if it's there, I can't see it.\n\nOn the way into the building, I watched a drug deal transpire on the street. It happened right there, right before my eyes, two shadowy figures lurking beside the building, where the tall structure obstructs the sun's rays, making the men harder to see. They were men, of course, because I find it hard to believe that two women would stand on the street corner trading money for drugs, a wad of folded-up cash for the clear plastic bag of pills that passed from one hand to the next. I never saw their faces or their eyes, for their heads were cloaked in the hoods of sweatshirts like headscarves. And yet the men were tall, lanky, flat. Undeniably men.\n\nWe passed by quickly, my eyes tethered to the broken concrete of the street, feet kicking up pebbles as I went, certain I could feel their eyes on me. I inserted my key and ducked into the foyer of the apartment complex, grateful to be separated by a wall of glass.\n\nWe can't stay here forever. It isn't safe, I don't think.\n\nAnd now, inside the apartment, I bolt the door behind me and stare out the peephole for a minute or two, to be certain no one followed me in. Not the drug dealer or his buyer, and not anyone else. I move to the window next, parting the dusty, broken mini-blinds with my fingertips, peering out, my fingers turning gray with dust. I survey the street below to be sure we haven't been followed, that no one knows we are here.\n\nThe last tenant had been recently evicted, her belongings never reclaimed. Because of this, I've been endowed with a foul-smelling sofa, a banged-up table, a mattress with worrisome stains. That and a carton of eggs that expired last week. I don't think we'll eat them.\n\nI open the newspaper and again turn to the classified ads. But this time, instead of searching the apartment listing, I go to the wanted ads, searching for a job as a house cleaner because really, that's the extent of my qualifications, and after the stunt I pulled at the hospital, references are out of the question. _Must be courteous, conscientious, previous experience preferred_ , I read. _Must speak English. Have good communication skills, a great work ethic._ The wages are noted; I tally the number of hours I will need to work to pay another month's worth of rent in this shoddy complex. Sixty hours\u2014that's what I'd need to work. Though we also need to eat.\n\nThis is no longer just about me.\n\nI try to relax but she's kicking and upset now, thrashing about, and I find that I can't relax. I tell her it's okay, that she doesn't need to worry, that she's safe here with me, though even I don't know if that's true because I have yet to decipher if we're safe here, if I'm safe. I stroke her, running my hand along the flushed skin, and for a moment\u2014only a moment\u2014she stops fighting. She gives in.\n\nI try on a name for size.\n\n\"Jessie,\" I say, taking her stillness as consent.\n\nI'll call her Jessie.\n\nI'm not a bad person, I remind myself, though in that moment as I sit\u2014watching a roach as it scurries across the worn carpeting, reaching a wall, shimmying along the baseboards to where the rest of its family no doubt lies waiting\u2014reflecting on the last twenty-four hours of my life, the last twenty-four days and weeks, I'm not entirely certain that's true. All sorts of emotions get churned up inside me, everything from sadness to regret and shame, and I think of him standing unsuspectingly at the cottage, knocking on the door in vain.\n\n\"You're not a bad person,\" I incant, believing that if I think on it long enough, if I say it enough times, a thousand times over, it might somehow turn true.\n\nI didn't set out to do the things I did. There was never any willful intent, any malice, only a pining for something I didn't have, something I so desperately needed. You wouldn't condemn a famished child for stealing a loaf of bread, now would you? A homeowner for shooting an armed intruder to protect his family?\n\nI'm not a bad person, I decide, far more resolute this time.\n\nI only did what I had to do.\njessie\n\nWhen I finally make my way back to Cornelia Avenue, it's evening. The colors of the sky have begun to change. Shadows fall across the street. The sun is thinking about going down.\n\nI walk along Cornelia beside Old Faithful, staring at the million-dollar homes that fringe the street. They're mostly newly gutted homes with small tracts of grass. For each home lies a single tree on the road verge, fully grown. Its leaves form a canopy over the street where it joins with the tree on the other side. Conjoined twins.\n\nThe temperatures have fallen. It's no more than fifty-some degrees outside, a cold that creeps under my clothing, chilling me to the bone. The heat in the carriage home is stingy at best, when it runs. Though I toyed with the thermostat this morning, setting the temperature to seventy-two degrees, the furnace never kicked on before I left. When I arrive, it will be cold inside.\n\nAs I make my way along the street, the dread of nighttime creeps in. The fear of eight long hours of darkness with nothing to do and only morbid thoughts to keep me company.\n\nThe front door of the greystone is open as I approach, though Ms. Geissler is nowhere to be seen. I stop on the sidewalk, wondering if I should let her know or if I should keep going. What I want to do is keep going, but my conscience says otherwise.\n\nThere's a garden on the front lawn of the greystone, one I didn't notice before, but now I do. It's not huge because city living doesn't allow for things to be huge. But it's magical. A blanket of yellows and oranges and reds that warms the earth. Tiny white butterflies hover above the blooms, levitating midair.\n\nI blink once and they're gone because most likely they were never really there.\n\nI make my way down the walkway, climbing the steps toward the front door. The home is large; three stories tall with a garden apartment to boot, one that peeks at me from beneath street level, hidden behind a black metal fence.\n\nAs I knock on the door, it pushes open more than it was before. My eyes take in the foyer, a carpeted runner, an unlit chandelier that dangles from the ceiling. \"Hello?\" I call out into the empty space, but if my landlord is here, she doesn't hear me.\n\nMy fingers press the doorbell and I hear the chime of it from inside, but still, there's no reply. \"Hello?\" I call again, laying a hand flat against the door and pressing it the rest of the way open. My feet cross the threshold as I step into the home.\n\nI reach for a light switch and toggle it up and down, but nothing happens. The chandelier above me remains dark. It's not black in the home because the sun has yet to go all the way down. There's still some light outside, but it's fading fast. Soon it will be gone.\n\n\"Ms. Geissler?\" I call out, explaining who I am and why I'm here. \"It's Jessie,\" I say. \"Jessie Sloane. Your new tenant. I just moved in to the carriage home,\" I call out, and at first I think the worst, that she's here somewhere, but that she's hurt. That she's had a nasty fall. That she can't answer me because she's lying on the ground just waiting to be found. That she's dead.\n\nI don't think the obvious. That Ms. Geissler's in the shower and can't hear me. That she forgot to close the door on the way out rather than the way in. That she's not here.\n\n\"Ms. Geissler?\" I call again, with an urgency to my voice this time. \"Hello? Are you here?\"\n\nAnd it's only then that I hear the sound of a piano playing from upstairs. Classical music, I think. The kind you've heard before because it's famous. Mozart. Beethoven. I don't know which. The piano is quieted down from the distance, diluted, but still I hear it, the music staccato-like, sharp and disconnected.\n\nAnd I breathe a sigh of relief because she's here. Because she's fine.\n\nI could go home now.\n\nI _should_ go home now.\n\nI should pull the door fully closed behind me and leave.\n\nBut instead I find myself hesitating at the base of the stairs. My hand grips the baluster as I stare up the flight of stairs, into the dark, cavernous second floor of the home. Because now the classical music has turned into some sort of ballad, and I find that it's haunting and beautiful.\n\nThat it's calling me, summoning me up the stairs.\n\nBegging me to come and listen, to come and see.\n\nAnd instead of leaving, my feet carry me up the stairs before I can think this through. I hold my breath as I go, listening only to the sound of the piano. Climbing upward, one step at a time.\n\nThe house is large, each room sprawling and grand, though they're hard to see for the scarcity of light, which becomes even more dim with each minute that passes by. Upstairs, my legs carry me to the bedroom from which the music comes. The only room that, as far as I can see, boasts light. The door is pulled to and so there's only a sliver of it. Only a sliver of light peeking from beneath the door slab.\n\nI go to it.\n\nStanding before the closed door, I listen to the sound of the piano play. My hand drops to the door's handle and it's unintentional when I turn the knob. I can't help myself; it just happens. I press a hand flat against the door and push it open, so slowly so that it doesn't squeak. I see her there on the piano's bench, her back to me. Her fingers move nimbly over the piano keys, foot pressing against the pedal with obvious expertise. I find myself entranced by her song, by the rhythmic motion of her hands and feet.\n\nAnd then she stops playing.\n\nAnd it strikes me suddenly, an awareness.\n\nShe knows that I am here.\n\nI shouldn't be here.\n\nAll at once I feel like a trespasser. Like I've gone too far. This is not my home and I have no business being here.\n\nShe doesn't turn. \"Something I can help you with?\" she asks and I gasp first before I laugh. A nervous laugh. An exhausted laugh. One I can't make stop though I try. And only then does she turn and look at me as I press my hands to my mouth to smother the laugh.\n\nMs. Geissler looks to me to be about sixty years old. Her hair is short, a dyed blond that's feathered around the edges. She wears glasses, dark, plastic frames that sit on the bridge of her nose. There's a frailty about her, her body gaunt, cloaked in a cotton dress. She rises to her feet and only then do I see that she's petite. There are lines on her face, laugh lines, frown lines, crow's-feet. And yet they look more regal than old. She's a beautiful woman.\n\n\"Jessie, isn't it?\" she asks, and though it takes a minute to find my voice, I say that it is. She says that it's nice to meet me. She steps toward me, slipping her hand into mine. My hand shakes as it did this afternoon, a quiver that won't quit.\n\n\"I'm sorry,\" I stammer. \"I didn't mean to interrupt,\" I say. Though I've done far worse than interrupt. \"I rang the doorbell. I knocked. The front door was open,\" I explain, voice as doddery as my hands, just barely managing to scrape the memories together and remember why I'm here. \"You left your front door open,\" I say again, for lack of anything better to say.\n\n\"Oh,\" she says, chastising the door latch. How it's old. How it doesn't work properly. How she needs to get it fixed, as she needs to get many things in this old home fixed.\n\n\"How is everything with the carriage home?\" she asks instead, and I tell her fine. I say how much I like it. I compliment the hardwood floors because I can think of nothing else to say. I say that they are pretty. I thank her for letting me stay there. She says it's no bother.\n\nIt's awkward and uncomfortable, all the conversation forced. I think then that I should leave. I've overstayed my welcome because I was never welcome in the first place.\n\nBut just as I'm about to say my goodbyes and go, a noise comes from somewhere upstairs. From the third floor of the home. What it sounds like to me is the thud of a textbook falling. Something heavy and dense. I glance upward, finding a hatch there, a pulldown ladder that when folded up and stowed away becomes one with the ceiling, as it is now.\n\n\"What's that?\" I ask, but Ms. Geissler's face goes suddenly blank, and she shakes her head, asking, \"What's what?\"\n\n\"The noise,\" I say. \"Is someone there?\" as I point up toward the ceiling.\n\n\"I didn't hear anything,\" she replies.\n\nI hold my breath and listen in vain for more noises coming from up above. But they don't come. The house is silent, and I know then: I made it up.\n\nMy eyes burn. I rub at them, making them more red than they were before, still aware of my shaking hands.\n\n\"I must be mistaken,\" I say, holding my hands out before me so that I can see the way they tremble. They're cold. But that's not the reason for the trembling. It's something far worse than that, I think. Something neurological. I have my brain to thank for this. Because after all these nights without sleep, my brain functions are out of whack.\n\nI try and convince myself that the shaky hands aren't degenerative. That they aren't getting worse. And yet there's no denying the fact.\n\nMy hands are shaking far more than they were this afternoon.\n\nIt's as if she can read my mind.\n\n\"You've been having trouble sleeping,\" she says, more of a statement than a question. She's not asking me because she knows. Behind the glasses, her eyes are a soft gray, staring at me in pity. I wonder how it is she knows I haven't been sleeping. Does it have something to do with the dark circles under my eyes, the bags, the red pools of blood that flood my sclera?\n\n\"I saw your light on late last night,\" she says by means of explanation, and I think of myself last night. Hearing the strange pinging sound through the floor register, the voices, and turning the light on to investigate. It was nothing, of course, though still I spent the rest of the night lying in bed unable to sleep, forever indebted to the sun when it finally decided to rise and I headed off in search of caffeine, my magical potion, which becomes far less potent with each passing day that I don't sleep.\n\nWhat makes not sleeping even worse than the crippling fatigue is the boredom that infiltrates those nighttime hours. The misery. The morbid thoughts that keep me company all night long. Last night I found myself thinking about ashes and bone fragments. That's what remains after a body has been incinerated. When Mom came back to me from the crematorium, I expected something soft, like the ashes left behind in Mom's and my fireplace. On cold nights, she and I used to toss in a few logs, sit on the floor beneath the same blanket, trying to stay warm. When the fire burned out, the ashes that remained were soft. Delicate. I didn't know that Mom's ashes would be coarse like sand, like cat litter, and not soft like ashes. Or that there would be bone fragments.\n\nAfter Mom's 130 pounds were reduced to just 4, I didn't have the wherewithal to bring the urn to the crematorium with me so that they could place her inside. And so instead she came to me in a little baggie in a sturdy box. I was tasked with making the transfer to the rhubarb urn, this straight, canister-like contraption that's anything but the round body, narrow neck of your classic urn. You wouldn't even know it was an urn except for Mom's name impressed in the clay along with the years of her birth and death. Her stint on earth. Forty-nine years.\n\nI made the transfer at the kitchen table, the day after I brought her remains home from the crematorium. The same table where we used to eat. I used a funnel. Same funnel we used to use when transferring sugar cookie icing to the piping bags. When I was done, a fine mist of Mom covered the tabletop. I wiped her away with the palm of a hand. Then Mom was stuck to me, and it wasn't like I could just wash her off with soap and water. Because it was Mom. I couldn't just wash Mom down the kitchen sink.\n\nThese are the things you don't think about when someone has died. You don't want to think about them.\n\nAnd yet these are the thoughts that keep me up all night. A fine mist of Mom on the palm of my hand.\n\n\"I couldn't sleep,\" I say, leaving it at that, pretending it was a one-time thing, not letting on to the fact that I haven't slept in all these nights.\n\n\"Try a glass of warm milk,\" she offers. \"It always helps me sleep like a baby,\" she says, and I tell her I will. But I won't. I've tried that already and besides, I hate the taste of warm milk.\n\nBut then it comes again. The noise, one I'm certain I didn't imagine this time. Another dull thud.\n\nAnd it's unintentional when my shaky hand lifts up to tug down the ladder and see for myself what's inside.\n\n\"You don't want to do that,\" Ms. Geissler snaps, her words brusque.\n\nI freeze in place, insisting, \"There's something there,\" and only then does she reconsider.\n\n\"I didn't want to scare you,\" she explains, and I breathe a sigh of relief. I didn't imagine the noise. It was there.\n\n\"It's squirrels,\" she tells me. \"They've taken over the place,\" she laments. \"I haven't stepped foot up there for a while.\" She says that she's been working with a pest control service to have them removed, but she's quite sure the service is more adept at bringing squirrels into her home then getting them out. The space is uninhabitable for now, until the problem gets sorted out. She can't bring herself to go up there, not until the squirrels are gone and her contractor repairs the damage.\n\n\"The squirrels,\" she complains, \"have chewed holes in the walls. They've gnawed their way through electrical wires. They've ruined a perfectly good lamp. I've switched services, mind you. But getting rid of squirrels is no easy task. I need a roofer to come and replace the tiles and block the squirrels' way in, but the roofer won't come until all the squirrels are out. The darn things have it in for me,\" she says, sighing exasperatedly, and it doesn't once occur to me not to believe her.\n\nI say to her, \"Sounds like a mess.\"\n\nOut the window, I see that the sun has finished setting. Darkness has arrived, anchoring itself to the earth for the night. Ending another day.\n\n\"It's getting late,\" I say, excusing myself, saying my goodbyes, and leaving.\n\nI make my way around the periphery of the house, cutting across the patio and onto the lawn. There I pause midstride, hands on hips, and look skyward to see that the stars are lost somewhere behind the clouds. That there isn't a star in sight. The moon is there, but only a sliver of it. A crescent moon that doesn't do anything to light up the night. The fall air is cool; goose bumps appear on my arms. I rub at them, hoping the friction will make them go away. For now it does, though I'm dreading another night in the freezing cold carriage home.\n\nThe home is enveloped in blackness as I arrive. I have to fight to get the key into its hole. Twice I drop it, scrabbling around on the stoop to hunt it down.\n\nThe sound of a siren in the distance startles me. As I glance backward, over my shoulder to see the red and blue emergency lights whirling through the sky, I find Ms. Geissler standing in the back doorway of her home. She's illuminated by kitchen lights, easier to see than me, who stands in total darkness.\n\nAnd yet her eyes are unmistakably on mine as if she's been watching me the entire time.\n\nI find the keyhole and open the door. I hurry inside.\n\nAs I climb the lopsided steps, I feel the weight of fatigue bearing down on me. Fatigue from physical exertion and fatigue from lack of sleep. I lie down on the mattress, staring at my shaky hands before my eyes. There's an anemic quality to them. Blanched and mealy, the skin at their edges disappearing somehow, evanescing, like a loose thread being tugged from the hem of a shirt, the whole thing unraveling, coming apart at the seams. That's me. Coming apart at the seams. Little by little, I'm disappearing.\n\nI look again at my hands, and this time they are fine. Intact.\n\nBut still shaking.\n\nI close my eyes and even though sleep is there within reach and I stretch my hand out to grab it, it's unattainable. Elusive and shifty. It moves away, mocking me. Laughing in my face.\n\nFor as tired as I am, I still cannot sleep.\neden\n\nDecember 21, _1996  \n_ Egg Harbor\n\nFor months now, Aaron and I have become slaves to the red circles on my pocket-size calendar, our intimacies slated out in advance. During my most fertile days we make love two, sometimes three times a day, though there's something inorganic about it now, something mechanical and forced. Our entire world, it seems, has become about making a baby, and I struggle to remember what our lives were like before we made the decision to start a family.\n\nTwo nights ago I stayed up until he was home from work, in bed, reading my book. Twice I rose from bed to look outside, searching through the bare trees for signs of headlights in the distance\u2014a shock of blinding yellow against the blackness of night\u2014rambling down the long, winding drive. But there were none. The night was pitch-black, no moon anywhere, not a star to be seen. It seemed to take forever for him to be home.\n\nThe bedroom lamp was dimmed, a candle burning on the dresser for ambience, though when he finally did arrive, Aaron took one look at that candle and blew it out, thinking I'd gotten tired and plumb forgot about the burning candle. The small room filled with the noxious smell of smoke as he pulled his chef getup from his body, dropping it to the floor. He climbed into bed beside me, saying how he was so tired, how his feet hurt. His words were slurred with simple lethargy and fatigue. He didn't bother to turn off the light. I smelled the chophouse on him, the garlic, the Worcestershire sauce, the flesh of chops and steaks.\n\nAnd yet there it was, another red-circled date on the calendar.\n\nBeneath the blankets I wore a satin robe and beneath the robe nothing, though in it I didn't feel nearly as sexy as I'd thought I would, as I'd _hoped_ I would, a feeling that was only exacerbated when I untied the ribbon from around my waist, revealing myself to him, and in Aaron's eyes spied a moment of hesitation, an excuse ready to form on his lips.\n\n\"Remember?\" I asked, childlike hope in my eyes. \"I'm ovulating,\" I reminded him, and before he could speak, before he could tell me why it wasn't a good night, I lowered myself beneath the sheets and easily changed his mind.\n\nI don't think he minded that I did. In fact, I think he was quite pleased.\n\nWhen we were through, Aaron pulled away and moved to his side of the bed, leaving me and my elevated hips alone in the hopes that this time, gravity might work its magic.\n\nFor three days in a row now it's gone like this, though tonight Aaron did object and it was much harder to make him acquiesce, and even when he did there was little satisfaction in it, little pleasure, but rather the knowledge that he was doing this for me. Because I wanted him to. Because I was making him do it. There was resentment in it, disgruntlement in his every move. When we were done I offered a pitiful _thank you_ , which felt entirely wrong, as we each drifted to our own side of the bed, an ocean of space spread between us.\n\nIt's become apparent that these days we do it because we have to, not because either of us wants to have sex. We skip any sort of foreplay and get straight to the grunt work, finding sex as pleasurable as brushing teeth or washing dishes. Our movements have become as repetitive and predictable as cleaning laundry.\n\nJust like any other of our daily chores, we've begun to grudgingly make love for three days out of the month, finding the other twenty-seven to be a blissful reprieve.\n\nJanuary 9, 1997  \nEgg Harbor\n\nIt took nearly thirty minutes to get to the obstetrician appointment, and all along the way, all I could hear was the grinding of snow beneath the tires' tread. Outside it was cold, a frosty thirty-two degrees, and the plump clouds looked like they might burst apart at the seams at any moment, burying us with three more inches of snow. Aaron was torn, worried we wouldn't be home in time for his shift, but feeling the need to go too. To be at the appointment. He vacillated about it for a good five or ten minutes, standing in the open doorway, letting the cold air into our home.\n\nIn the end, he decided to go so that as we drove south on Highway 42, both of us quiet, I felt a great guilt about it, knowing that if he was late to work, it would be my fault.\n\nThe obstetrician was in Sturgeon Bay, a seventeen-mile drive. He was the closest I could find and also came with a recommendation, one from Miranda, the only woman in town I knew well enough to ask. Miranda, who drove her three boys to my home last week in her Dodge Caravan\u2014windshield caked with ice crystals still, so that it was near impossible to see through the rimy glass\u2014because it was far too cold to walk.\n\nMiranda, who sat sprawled on my sofa, feet raised to the coffee table, while I rocked her crying two-month-old baby, Carter, to sleep.\n\nMiranda, who was so overwhelmed with motherhood that she couldn't stand to be alone with her own three kids.\n\nMiranda, who revealed to me that she was pregnant again, that it was a mistake this time, that they hadn't been trying. \"Because who in their right mind tries for more when they already have three kids?\" she asked, staring at me sadly as if I should take pity on her for this obvious misfortune, but what I felt instead was infuriated and sick, anger and bile rising quickly inside me.\n\nMiranda confessed to me that though she and Joe had waited the recommended six weeks after Carter was born to fool around, sure enough, Joe managed to knock her up on the first try, and already the morning sickness had set in so that her boys were forced to watch even more TV than ever before because Miranda didn't have the stamina to entertain them all day, let alone feed them. \"The nerve of that bastard,\" __ she said of Joe and his evident virility, __ and then she asked what in the world was taking Aaron and me so long to conceive.\n\n\"You don't think,\" she asked, eyes wide, \"that you're infertile, do you? That that handsome husband of yours is shooting blanks?\"\n\nAs I sat beside him in the car, driving to the obstetrician appointment, listening to the pulverization of snow beneath the wheels, staring at the clouds, I couldn't help but wonder if that was the case. Was Aaron shooting blanks? Was Aaron _infertile_?\n\nAaron, who could do anything, who could fix anything, could not create a child?\n\nAaron, to whom everything came so easily, had difficulty making a baby?\n\nThe thought alone made me angry and annoyed. Angry at Aaron because why, for all the things he was so capable of doing, was he incapable of doing _this_?\n\nWhy couldn't he fix this? Why couldn't he make this right?\n\nAssigning fault seemed to be the name of the game these days, pointing fingers, attaching blame. Whose fault was it that we didn't yet have a baby?\n\nMarch 11, 1997 _  \n_Egg Harbor\n\nWhat I've come to learn after being referred to a fertility specialist is that even though I get my period each month with moderate regularity, my body isn't ovulating correctly, isn't always ovulating. _Anovulation_ , it's called, a word I've never heard of before but now think about at every waking hour and when I should be asleep. If I'm being honest, this comes as little surprise to me. My body is simply going through the motions, the preparations of the endometrium\u2014the lining of my uterus readying itself to welcome a fertilized egg\u2014and then sloughing off when no egg moves in. It's not that the egg wasn't fertilized by Aaron's sperm. It's that it simply wasn't there to begin with.\n\nToday I began my third cycle of Clomid. After months of this, I have no sense of humility left, no modesty. I've paraded my private parts for every doctor, nurse and technician in the fertility clinic to see, while all Aaron ever had to do was drop off a sperm sample and endure a simple blood draw. It hardly seems fair. The first month I didn't ovulate. Last month we upped the dosage and, though Dr. Landry spied two follicles when he performed his ultrasound\u2014forcing the transvaginal ultrasound probe between my legs so that I should rightfully have felt violated and ashamed, but no longer did, sending Aaron and me home with strict orders to have sex\u2014we didn't get pregnant.\n\nThe pills make me weepy all the time, for no apparent reason at all, though having seen the inventory of potential side effects, I consider it a blessing that the only one I'm doomed to endure is the predisposition for crying. I cry at the market; I cry in the car. I cry at home while mopping floors and folding laundry and standing in the doorway to one of the spare bedrooms, wondering if it will ever hold a child, steeling myself for another cycle of Clomid that will likely end again with my monthly flow.\n\nTo counter Aaron's low sperm motility, as it's called, he's switched to wearing loose-fitting underpants (I don't tell Miranda this), and is tasked with finding ways to reduce stress in his life, stress which neither of us knew he had. He now sleeps until after ten o'clock every morning so that we no longer share our day's coffee on the dock, which is fine anyway seeing as the eternal winter has trapped us indoors and there are no sailboats to be seen on the bay, none until spring. He takes herbal supplements and when the temperatures aren't too abysmal will go for a walk or a run, so that our days together are mere hours at best. This too is fine, seeing as we don't have much to talk about anymore, nothing that doesn't involve the many things the world is reluctant to let us have: strong, capable sperm; regular ovulation; a positive pregnancy test; a baby.\n\nIt isn't that Aaron doesn't have enough sperm\u2014he does\u2014it's that what he has doesn't swim properly and isn't able to travel the four inches or so to where my egg may or may not be waiting.\n\nIn short, we're both to blame, though there isn't a moment that I don't wonder which of us is to blame more and even though I think it's me, I _know_ it's me, there is a part of me aggrieved that I'm the only one forced to record my body temperature, to take ovulation tests, to cry in public for no sound reason at all, to travel to the fertility clinic again and again, to be probed so that some doctor or technician can gaze inside me and at my ovaries, while all Aaron has to do is take an herbal supplement from time to time and exercise on occasion.\n\nIt doesn't seem fair. It doesn't seem right.\n\nI've come to resent Aaron for this, as I've come to resent him for many things.\n\nMarch 13, 1997 _  \n_Egg Harbor\n\nI field questions nearly every day about when Aaron and I are going to have a baby, often from my stepmother or Aaron's mother, calling on the phone when he's at work, asking not-so-subtly for grandchildren.\n\nWhen can they expect them? When will there be good news to share?\n\nIt's not that grandchildren are in short supply because they aren't. Instead it's that Aaron and I have been married for over two years, and society doesn't take well to that: two nearly thirty-years-olds, married for over two years without kids, as if there's something unthinkable about it, something taboo.\n\nIs there something wrong with that?\n\nIt feels as if there is.\n\nA married woman of my age without a child is quite the anomaly these days.\n\nI can't bring myself to say aloud that we're trying, _trying and failing_ to make a baby because I don't want pity and I don't want advice. And so instead I tell Aaron's mother and my stepmother _soon_ , wishing that my own mother were still alive because hers is the only advice I want and need.\n\nI spend my days waiting. Waiting for Aaron to wake up, waiting for Aaron to leave, waiting for Aaron to get home so I can again close my eyes and sleep. Waiting for a new cycle of Clomid to begin, to ovulate, to make love to Aaron like robots would do, hasty and unfeeling, and then waiting for the negative pregnancy test results, the loyal, trusty blood.\n\nIt's the only thing I can depend on anymore. That sooner or later, my period will come.\n\nMarch 14, 1997  \nEgg Harbor\n\nSpring looms on the horizon.\n\nIt's weeks away still, but every now and then a day blooms before me, fifty or sixty degrees and full of sun, so that it's easier to get through than the endlessly gray winter days.\n\nThese rare springlike days I leave the cottage when Aaron is away and head into town. I've discovered a dance studio there, completely by chance\u2014I didn't seek it out\u2014a small single-story cottage on Church Street that tiny ballerinas move in and out of all day.\n\nThe first day I spotted the studio, I saw an empty park bench nearby, which was warm and welcoming, set directly in a shaft of sunlight so that even though it was no more than fifty-two degrees outside, I felt snug, my skin warm from the sun's generous beams.\n\nFor nearly an hour I watched the ballerinas, toddlers mainly in leotards with their hair pinned neatly back in buns. Their little voices were happy and high-pitched, like birds, as they clung to their mothers' hands, coming and going like clockwork, nearly every hour on the hour.\n\nThere was one group in particular that caught my eye. A group of sixteen\u2014eight mothers and their daughters\u2014who arrived en masse around noon, a whole bundle of giggly girls with women trailing behind, women who sipped lattes and gossiped while I sat alone on a park bench, feeling sorry for myself, isolated from society because I didn't fit in. Because I didn't have a child.\n\nThe women were beautiful, every last one of them, which for whatever reason made me feel dirty, self-conscious and ashamed. I smiled as they walked by, but not one looked at me and no one smiled in reply. They wore peasant tops and floaty skirts; cowboy boots; big, baggy sweaters; hobo bags; while me, on the other hand, I sat wrapped up in a sweatshirt of Aaron's that had faded and shrunk in the wash, feeling alone, bloated, desperate, wanting for a child.\n\nHow different I am from those mothers.\n\nI could never be one of them, one of those women who travel in a pack, whispering secrets about their husbands, their children's nighttime habits, which little ones still wet the bed. All because I didn't have a child. Because without a child, I had nothing to offer them.\n\nBecause I'm nothing, I easily reasoned then, if not a mother.\n\nThere's no other justification for my life.\n\nI watched them as they walked by, as they closed in on the dance studio. And then, after the women had passed and I assumed the parade was through, I noticed one little girl straggling behind, nearly stagnant on the sidewalk. Struggling to keep up. Too busy examining the buds on the trees. Smaller than the rest, which made me think of the piglet in Charlotte's Web. Wilbur, saved from slaughter by little Fern. I was captivated by her, holding my breath as she passed by, joining the others in the studio. Only when she was gone did I allow myself to breathe.\n\nAnd now twice, sometimes three times a week I find myself sitting there on that bench, watching the dancers come and go, wishing one of them, any single one of them\u2014but especially the littlest one, a head shorter than the rest, straw-colored hair and a collection of freckles, whose tiny feet always lag behind so that one day I worry she'll be forgotten\u2014was mine.\n\nI've become an addict really, and the only thing that eases the symptoms of withdrawal is seeing children, is being in the company of children. They are my fix, an antidote for the restlessness, the irritability, the tremor of my hands that is only exacerbated with each passing month that I don't get pregnant.\n\nThe little girl can't be more than three years old, pudgy arms, legs and cheeks still padded with baby fat that will one day wear away, no doubt, so that she'll look like any one of the ladies she tags along after, with their long limbs and their long hair and their coffee.\n\nI don't like the way I feel sitting there on that park bench, eyeing children who are not mine. But I have nothing better to do with my time, and I don't think I could stop if I tried.\n\nI suggested to Aaron that I look for a job, for some diversion from the long, lonely afternoons while he is away. Aaron isn't game. He'd rather I _not_ work, which makes no sense to me. The financial burden of fertility treatments is steep; we could use the additional income. We've begun to argue about things like the cost of ground beef, the cost of electricity.\n\nAaron and I are monitoring the Clomid cycles, which means for each failed attempt we are quite literally throwing away hundreds of dollars for the medication, blood work and ultrasounds to see whether my body is releasing eggs, and when. Insurance won't cover these costs because, of course, some high-and-mighty insurance company doesn't give a darn whether Aaron and I ever have a baby, and so the procedure is considered _elective_. We are electing to waste thousands of dollars to try and conceive a baby, while other parents, far less capable or worthy parents, are given one for free.\n\n\"You're under so much stress already,\" Aaron said when I suggested applying for a job, and \"Why not just focus on this?\" meaning making a baby, as if somehow I'd been unfocused, and as if that lack of focus was the reason we were still without a child. I'd been too cavalier about it, too casual, too devil-may-care. He didn't use those words, not a single one of them, and yet that's exactly what I heard when he came home from work after midnight that night and, though I lamented about being bored all day, about being alone, he suggested I not apply for a job, but rather focus on _this_ , with a sweeping gesture toward my vacuous womb.\n\nI screamed at him then. I slammed a door. I locked him out of the bedroom so he slept on the sofa for the night.\n\nNever before have I screamed at him. Never before have I raised my voice.\n\nHe didn't object to sleeping on the sofa. It was one in the morning. He was tired, he told me. \"Eden, that's enough,\" __ he said with a sigh while gathering his pillow from the head of our bed. \"I need to sleep.\"\n\nI sat there in the bedroom that night, in the dark, propped up against pillows and not lying down. My hands still shook even hours after my fit was through. A headache slunk up the base of my neck and consumed my skull so that every part of my head hurt. My eyes burned from crying and though I tried to blame the medication for this\u2014after all, mood swings and a propensity for crying were both common side effects of the Clomid\u2014I didn't know whether or not they were to blame this time.\n\nMaybe it was just me.\n\nI felt sorry come morning.\n\nBut I didn't apologize and neither did Aaron. Instead he left for work earlier than ever before and I returned to the dance studio, an addict in need of a fix.\n\nMarch 19, 1997 _  \n_Egg Harbor\n\nWhen Clomid alone failed to work, Dr. Landry suggested IUI. Intrauterine insemination. Placing Aaron's sluggish sperm directly into my uterus so that they don't have to paddle through those four inches of mucousy space all on their own, so that they will have an easier time finding and fertilizing my egg without getting lost, swimming in circles in my vaginal canal as they are apt to do. Each month, Aaron and I have quite literally thrown away money, frittered away follicles and eggs, doled out hundreds of dollars on medication and ultrasounds for nothing. My trips to see Dr. Landry have been a waste. It's time to try something new. Intrauterine insemination will add a couple hundred dollars to our monthly expenditure, but will also increase the likelihood of conception, especially in cases like ours where low sperm motility is to blame.\n\nThere it was again, that word: _blame._\n\nThere is also the added benefit that with IUI Aaron and I won't have to have sex, which is a blessing in and of itself. Aaron is capable of collecting his sample all on his own in the comfort of a private room at the clinic, complete with pornographic videos and magazines, where sexy, buxom women far more appealing to the eye than me will help us create a child. It mortified him to have to do this, and yet after months of invasive ultrasounds and repeated blood draws, after digesting medication that made me moody, that made me cry, after poking myself in the gut with shots of hormones for months on end, this seemed only fair. This seemed right. The nurse tendered my assistance, saying I could keep Aaron company if I'd like, but with a sideways glance, he went in without me and closed the door, and there was a spasm of jealousy, a shooting pain searing through my head as though someone had momentarily lodged an ice pick into my skull.\n\nI envisioned Aaron on the other side of that door, aroused by some strumpet on the television screen and not me.\n\nAnd then hours later, after the sperm had been collected and cleaned, it was my turn to be put to work, to lie on the exam table, completely undressed from the waist down with only a drape sheet to provide that false sense of privacy, while Dr. Landry placed first a catheter and then Aaron's sperm inside me.\n\nAnd then sent us home to wait.\n\nAaron, as always, went to work, leaving me alone and bored, and so I drove into town and sought out that small dance studio on Church Street and sat on the park bench, watching the little ballerinas come and go, searching for the smallest one with the straw-like hair and freckles, a head shorter than the rest, who always struggled to keep up with her mother and friends.\n\nI had to wait awhile, but eventually she came and my heart skipped a beat. My hands went numb. I held my breath.\n\nI saw her ambling first through the double blue doors of the studio, already lagging behind before she'd ever stepped foot outside, grappling with the weight of the door because there was no one around to hold it for her. Her tiny head barely surpassed the door's crash bar. The others were already a good five or ten paces ahead, moving down the concrete sidewalk in the direction of town, little girls gabbing merrily about an afternoon playdate while their mothers followed behind, paper cups of coffee in hand. Only once did a mother turn around to see where she was, calling out, \"Snap to it, Olivia, or you'll get left behind,\" __ and then she turned again, facing forward, never again checking on Olivia, who brought up the rear, the caboose on some sort of high-speed train that had somehow gotten off track.\n\nShe had a name now. _Olivia._\n\nBut Olivia's mother was fully immersed in a conversation with the ladies, listening to one of the other mothers complain about her husband's long hours and relentless travel schedule. He was in Tampa Bay this week on business, and that new admin assistant from the office had gone along too, the one her husband talked about at the dinner table, so that she couldn't help but be concerned.\n\n\"You don't think?\" asked one of the other ladies, and Olivia's mother piped in with \"Oh, you poor thing.\"\n\nAnd it was decided then. This woman's husband was having an affair.\n\nThrough all this, no one paid attention to Olivia, who had fallen even farther behind.\n\nI had no intent of rising from the park bench as she passed by. None at all. The thought didn't cross my mind until a single bobby pin fell from her hair, a silver sprung hairpin that dropped to the ground at such a frequency only I could hear. Little Olivia kept walking, leaving the hairpin behind. Her mother kept walking, now nearly twenty or thirty paces ahead. Only I paused to retrieve the hairpin, falling in line behind Olivia and the rest of her troupe, six steps behind and struck dumb.\n\nI couldn't speak.\n\nI could have called her by name; I could have tapped her on the shoulder and handed her the bobby pin. But I didn't. Instead I shadowed her by a mere three feet, eyes gaping at the lavender leotard and tutu, the sheer white tights, the hair done up in a bun, starting to lose its hold as strands of brown and yellow drifted through the springlike air. Beside our feet, the snow had melted, leaving puddles that returning birds paused to drink from. On the trees there were buds, tiny shamrock-green buds about to burst forth with leaves.\n\nI never once thought about taking her, about grabbing a hold of her with my hand pressed to her mouth so that she wouldn't scream. I didn't think of luring her away, bewitching her with the promise of a puppy or ice cream. I only wanted to watch for a while, to walk a breath behind and pretend for just this one moment in time that she was mine.\n\nAs I followed Olivia down the sidewalk, a conversation played out in my mind.\n\n_Slow down, baby girl_ , __ I thought to myself, whispering the words in my head. _Come hold Mama's hand_ , I urged, and in my imaginings I held out my hand as little Olivia slackened a bit, slowing down, turning to me so I could see the color of her eyes, the wealth of freckles she'd no doubt one day either outgrow or grow to hate. She slipped her hand inside mine and I squeezed tight, careful not to let go as we passed through an intersection while the traffic on either side paused to let us through. Olivia's hand was easy to hold. Her steps fell into sync with mine.\n\nIt was the raucous laughter of the other girls that broke my trance, bringing me back to the earth, back to my physical existence. To reality. They had all turned at once, calling Olivia a snail, a slowpoke, waiting for her to catch up so they could go get ice-cream cones, and even though I knew it was all in jest\u2014Olivia's piping laughter was proof of this, no?\u2014my heart ached for her for being called names, for being the poky little puppy, always lagging behind.\n\nAnd then my heart ached for me when she skipped off with her friends, leaving me behind, standing alone on the sidewalk with her bobby pin in my hand.\n\nI hoped that just once she would turn and see me and know that I was there.\n\nI kept Olivia's bobby pin as a token of luck.\n\nTen days later, my period arrived.\n\nAnd now another month has come and gone without a baby.\njessie\n\nMy nighttime thoughts can be grouped into four categories. They follow the same pattern, the same predictable rotation each night. Wash, rinse, dry, repeat.\n\nIt all begins with the morbid thoughts where I obsess over death and dying, of being dead, trapped inside an urn, unable to breathe. They settle in around twilight, when the sun sinks beneath the horizon, slipping away to play with kids on the other side of the world. It's then that I start to wonder how much time I have left on earth. I think about how and when I will die. Will it hurt when I die? Did it hurt when Mom died?\n\nThese morbid thoughts soon mutate into grieving, sinking ones where I miss Mom so much it hurts. By this point in the night, the world has turned black and I lie on my mattress in a black room, confined by blackness. A prisoner of the night. In all my life, it was always Mom and me, like Batman and Robin, Lucy and Ethel. Shaggy and Scooby-Doo. We were a team. Without her I don't know what to do. I spend half the night pleading for her back. Because I don't know who I am without her. Because, without her, I am nothing.\n\nI don't cry about it because my eyes are done crying. They've dried up. And so instead I think things like, if Mom isn't here, then I don't want to be here either. It's grim, and yet it's true.\n\nMy thoughts go on like this for what feels like hours because it probably is. Eventually they turn into a guilt trip, where I loathe myself for sleeping through Mom's death. For getting testy with her when she puked for the sixth time in a row, missing the toilet by a mile. For not speaking to her for weeks when she wouldn't come clean to me about my dad. For not holding her hand the time she chaperoned my fifth grade field trip to the planetarium, or bothering to thank her for the embroidery thread she got me in middle school\u2014a half dozen colors to make friendship bracelets with. I'd only huffed and stomped off to another room, thinking how stupid could she be. Didn't she know I didn't have any friends? These memories haunt me now.\n\nIn truth, Mom and I hardly fought. The only arguments she and I ever had were mostly over my father. Mom never wanted to talk about him\u2014she refused to talk about him\u2014and so I snuck around her back to try and learn more.\n\nI was six years old when I first realized I didn't have a father. Until then I was too oblivious to see that other kids did and I didn't. Mom and I lived alone. We kept to ourselves much of the time. I didn't go to preschool and I didn't have friends. I didn't know much of anything outside of my world with Mom, not until school began, and then my world grew exponentially larger, though still, in comparison to everyone else's, it was small.\n\nIt was my first day of kindergarten when I realized that all of the kids in the class, aside from me, had both a mom and a dad. I remember that day, organizing our belongings inside the bulky metal cubbyholes, while our moms hovered in the classroom, talking to the teacher, talking to other moms. Everyone except for my mom, because she stood there alone, talking to no one. This confused me. Why didn't Mom talk to the other women?\n\nBut what confused me even more was the huddle of men in the classroom. A whole busload of them. Not just moms, but moms _and_ men. Who were these men, and what were they doing here?\n\nI asked one little girl. I pointed at the giant of a man standing by her side. _Who is that?_ I asked, eyes wide, looking skyward. She said it was her dad, and though I'd heard that word before, it wasn't one that was readily in my vocabulary.\n\nI tallied up the men in the room, realizing that every single child had one but me.\n\nThe mention of my father didn't come up again until later in the school year, when some kid asked where he was. We'd had a music performance and, while everyone else had a mom and dad in tow\u2014grandma and grandpa too\u2014I only had Mom. And things like that, when you're six, are big news. How Jessie Sloane doesn't have a dad.\n\n_Where's your dad?_ the kid asked, all dressed up in a sweater-vest and pants.\n\n_I don't have one_ , __ I said, thinking that was the end of it. But he came back with some comment about how _everyone has a dad_ , and others started to laugh.\n\nI asked Mom about it that night at home. I had to know. _Where's my dad?_ I asked, standing in her bedroom doorway while she lay on the bed, bare feet crossed at the ankles, reading a book. Even at six years old, I could see that she was tired from a day spent cleaning someone else's home.\n\nI didn't wait for her reply. _Joey Malone said everyone has a dad_ , I told Mom as she uncrossed her ankles and set her bookmark between the pages of her book. _So where's mine?_ I asked, feeling aggrieved all of a sudden. As aggrieved as a little kid can be.\n\nMom was keeping something from me.\n\nMom had a secret that she wouldn't share with me.\n\nMom's face turned as red as hot coal. _Joey had no right to say that_ , __ she told me. _Not everyone has a dad. Not you._\n\nBut her answer came with no explanation.\n\nMaybe he was dead. Maybe they were divorced. Maybe they were never married in the first place. Or maybe I never really had a dad.\n\nStill, I started snooping around the house to be sure, in case there was something hiding there that I might find. Evidence. A clue.\n\nA few years later I became more tenacious about it, more annoying. I asked Mom again where my father was. What had happened to him. _Is he dead?_ I wanted to know. I said that word with the testiness of a preteen. The exasperation. _Dead._\n\nBut she wouldn't say. Time and again, she changed the subject; she pretended not to hear me ask. She had a brilliant way of mincing words, of making me forget what I had asked. Of clamming up and saying nothing.\n\nAnd yet, again and again, I asked. A hundred times after that. But never did she tell me.\n\nI became ruthless about it.\n\nWhen I was twelve I set a place at the dinner table for him. Whoever he might be. Just in case he decided to show. Mom swiped his silverware from the table post-haste. Flung it back in the drawer.\n\n_Let's not do this, Jessie_ , __ she said.\n\nI searched city streets for his face. Never sure what I was looking for, but always looking. I wondered if he had blondish hair and dimples like me. Or if he was a brunette, a redhead, maybe even some other ethnicity.\n\nMaybe we looked nothing alike.\n\nOr maybe we were the kind that could pass for twins.\n\nI learned that dimples are inherited. A dominant trait. Meaning only one parent would have to have them for me to have them. And seeing as Mom had none, I easily reasoned that they came from _him_. From _Dad_. That, barring some sort of genetic mutation, I'd inherited them from my father.\n\nWhat a dimple really is is a birth defect. A short facial muscle that pulls on your skin when you smile, causing indentations in the cheek. My father and I are, therefore, both defected.\n\nI made up names for him. Occupations. I sized up men with dimples at random, wondering if any of them were him.\n\nI imagined him with a different wife and kids. Me with half brothers and sisters, a family. In my delusion, every last one of them had dimples.\n\nBefore bed, I'd leave the porch light on, so that he could find our home if ever he came to visit. So that he'd know which one was ours. Which bungalow in a sea of bungalows belonged to me.\n\nWhen I was fourteen, I attempted a crop top for school. It wasn't my thing, bearing my belly button for all to see. But it was a camouflage T, soft and green, and I was fourteen. Feeling rebellious. Trying to fit in with the crowd but failing. Instead I stood out like a sore thumb, always light-years behind the latest fad.\n\nMom's mouth dropped. She shook her head. She said no to the crop top, told me to march upstairs and change. To _march_. I put up a fight, standing with my hands on my hips, pouting. Sputtering the nonsense of a fourteen-year-old girl.\n\nBut Mom would have none of it. It wasn't up for discussion, she told me, saying for a third time to _march_. Pointing at the stairs.\n\nMy words were brisk. _I bet that if my dad were here_ , _he'd let me_ , I said. She looked hurt, visibly wounded. I'd hurt her and I was glad I did.\n\n_Are you ever going to tell me about him?_ I asked. It was a fair question. I deserved to know, or so my fourteen-year-old self believed I did. I didn't once consider the reasons she kept him from me, or the ramifications of knowing who he was. But Mom did.\n\n_Qui vivra verra_ , Mom replied, holding her hands up in the air. Her favorite saying, one that rolled eloquently off her tongue. _Only time will tell_ is in essence what it means, but this time what it was was a way to be evasive. To avoid my question yet again.\n\nI stormed out of the room. Marched up the stairs and slammed a bedroom door. I put on a sweatshirt that covered every square inch of me.\n\nNot a year later, the cancer came.\n\nAnd then I started wishing I'd never asked about my dad.\n\nI dwell on those memories now, hating myself for what I put Mom through.\n\nBut every night around 3:00 a.m., when I've exhausted all the thoughts of death and grief and guilt for a single night, my imagination begins to take flight. My imagination or my memories, though some nights I have a hard time determining which is which. Tonight it's a memory, I think, one so far-flung that my brain has to cobble pieces of it together, adding to the gaps so that it makes sense. Filling in the blanks. I see that kindergarten classroom, a poster of the golden rule taped to the cinder block walls. A big bookcase, a rectangular rug with the alphabet depicted on it\u2014the alphabet plus simple pictures, an apple for _A_ , a bird for _B_ \u2014the American flag. A chalkboard with the teacher's name written on it in perfect penmanship. I see Mom standing there before the teacher, making an introduction, saying to the teacher that she is Eden and I am Jessie, and then the teacher squats just so and reaches out a hand to me and I shake it. Her smile is warm and sincere as she rises back up to Mom.\n\nMrs. Roberts stands with the clipboard in hand, making sure each child's paperwork is complete and that they've brought their supplies. Mom hovers self-consciously before her, hands behind her back, fingers laced. Mom and Mrs. Roberts talk and as they do, words reach my ears\u2014 _birth certificate_ , __ I think I hear\u2014and Mom stiffens at Mrs. Roberts's request.\n\n\"Pardon me?\" Mom asks, and Mrs. Roberts explains how there's a note from the school office that she's yet to provide a copy of my birth certificate with the other registration materials. A certified copy, with the raised seal.\n\nMom doesn't miss a beat. She says something about a house fire. \"We lost everything,\" she says, and Mrs. Roberts's face turns sad.\n\n\"How awful,\" Mrs. Roberts says consolingly as I, six-year-old me, asks unsuspectingly, \"What fire?\" Because there was never a fire. Not in our home. We didn't lose a thing.\n\nMom shushes me. Mrs. Roberts lays a hand on Mom's arm and says just as soon as she can get a replacement, that would be fine.\n\nBut then, like that, the memory disappears, and I have to wonder if it was a memory at all or only my imagination.\n\nTonight as I lie on the mattress in a misplaced belief that if I lie here long enough, eventually I will sleep, I think of a dead three-year-old Jessica Sloane, having to remind myself that it's a typographical error only, that she doesn't exist.\n\nThe room is quiet as I lie in bed wondering what she looked like. For three years old, I picture chubby wrists and knees, innocent eyes, an endless smile. I wonder if that's what she looked like. But then again I remember. There is no other Jessica Sloane. She _is_ me.\n\nA heavy silence flattens me in bed, filling every crevice in the room like a poisonous gas. I think that maybe it could kill me, that silence. Displacing all the oxygen in the room with a smothering quiet. The only thing I hear is the _tick, tock, tick, tock_ of the wall clock, keeping time.\n\nI rise to my knees and gaze out the window into the yard, seeing only the back of Ms. Geissler's home from here. It's tall and imposing, three floors of limestone and brick. Such a big home for one woman alone.\n\nThere's a balcony in the back of the home, a basic, rudimentary sort of thing. Wooden scaffolding that soars up three floors, a wooden slab to stand upon. It looks unsafe to me. Unsound. Not up to code.\n\nAs I kneel before the window, I rest my elbows on the sill. Foolishly believing that I blend into the blackness of the room, that no one can see me from here. The house is dark, except for a single light that's turned on. A yellow hue fills the margins of a window. The rest of it is blocked by a drawn window shade. I can't see into the room, just that frame of light around the window shade.\n\nMs. Geissler must have forgotten to turn the light off before she went to bed, I rationalize, because it's the middle of the night, and no one should be awake but me.\n\nBut as I stare, I see that the frame of light behind the window shade is moving, because the window shade inside it is also moving. It's a gentle back and forth motion, as if a person had been standing just seconds ago behind it, lifting the edges of the shade to peer out.\n\nI imagine her at the window, gazing out, seeing me lying on the mattress, pretending to sleep. I think of her own admission\u2014 _I saw your light on late last night_ \u2014and imagine that last night, like this, she stood at the window, staring at me. Me, who naively obliged, leaving the shades open wide, basic white roller shades that I didn't once think necessary to pull down.\n\nBut now suddenly I do.\n\nI watch the motion of the window shade as it slows and then stops.\n\nAnd then, like that, the light flicks off.\n\nThe yellow edges of the window disappear. The greystone is engulfed in total darkness. I'd think nothing of it, but then it occurs to me that the light was coming from the third floor of the home. The place with the squirrels. The place where Ms. Geissler doesn't go.\n\nShe was lying to me.\n\nWhy would she lie to me?\n\nI crawl back into bed. I throw the covers over my head.\n\nI make poor attempts to placate myself, to convince myself that the light is on a timer. That it's automated. That it goes on and off of its own free will. That a heat vent was spewing warm air directly at the window shade, making it move.\n\nBut it's not so easy to believe.\neden\n\nMay 14, 2001   \nChicago\n\nI watch as, beside me, Jessie sleeps. She's out for the count now\u2014finally, after a long, feverish night\u2014spread out on a blanket on the floor, arms splayed in opposite directions like the wings of a jetliner. Her pale face is placid and calm, unlike last night when it was a fiery red, the fever and the fury creeping up her neckline, inflaming her forehead and cheeks. She'd cried out all night in discomfort, wailing, unable to get a hold of her own breath. Her fever capped at 103 degrees and I was grateful for this, for the fact that it wasn't high enough to necessitate a visit to the emergency room. I don't know if I'd have had it in me to go to the hospital had we needed to. I find that the very notion of hospitals\u2014the antiseptic smells, the insipid hallways, the vigilant eyes\u2014still gets under my skin sometimes, like some form of PTSD, I think, because just thinking about being in one rattles my nerves, makes me dizzy, makes my chest hurt. I don't know that I could ever go back to one, not after what I've done. I'm certain they'd see clear through me, that\u2014even with all these miles spread between us\u2014they, the doctors, the nurses, the ladies at the reception desk, would know just exactly who I am, as if I have my own scarlet _A_ forever etched into my shirt as a reminder of my guilt.\n\nI stare at Jessie, sound asleep on the quilted blanket beside me. Her hair fans out around her face. Her arms, both of them, are thrust upward and over her head now like goalposts. There isn't a single line on her skin anywhere, and though I don't want to wake her, I stroke the back of a finger across her tranquil ivory cheek, grateful she still sleeps.\n\nIt takes my breath away sometimes, the way that she looks absolutely nothing like me, but is instead all blond hair and blue-eyed. And then there are those dimples\u2014those dimples!\u2014the most telling of all, so that I've tried sucking my cheeks in from time to time in the hopes of replicating them on my own skin. It doesn't work, of course, and instead of dimples I'm left with a fish face that makes Jessie laugh. There are times I find that I have to remind myself that I am a mother, that I am her mother, and I wonder if others see the hesitation in me, the doubt, or if it's only in my mind.\n\nYesterday as we were walking from the French bakery, the one with the luscious petits fours for which I had a sudden craving, the woman behind the counter wished me a happy Mother's Day, and there was something querying about it that I didn't like. It rubbed me the wrong way. What started as a polite greeting turned into a question instead, as if she doubted at that last moment\u2014words already out of her mouth, too late to pluck them back\u2014whether she should be wishing me a happy Mother's Day.\n\nWas I the child's mother? Was I a mother? After all, we looked nothing alike, and of course the lack of a wedding ring raised a red flag. Perhaps I was only the child's babysitter, her nanny, the au pair.\n\nAs I thanked the woman I saw her turn red with shame, believing she'd misspoken. But I grabbed for Jessie and said, \"Come along, my darling girl,\" as if that might validate it for both her and me. As if it might make my maternity more real.\n\nAll afternoon I found myself overthinking, wondering what exactly that woman saw that made her question whether I was Jessie's mother. Was it the manner in which I carried myself, the way I spoke, the lack of a physical resemblance? I thought about it all day and night, wanting to know, needing to know, so that whatever it was, I could next time disguise it better.\njessie\n\nThe day begins with a cleaning assignment, the first in two weeks. It's a good thing for more reasons than one. These days, cash is in short supply, and I need something to do with my time. Something better than to obsess over my social security number or lack thereof, Ms. Geissler staring out her window, watching me\u2014which, even by the light of day, still rattles me. So much so that before I leave, I eye the window shades in the carriage home, fully intent on pulling each and every one down so that no one can see inside while I'm gone.\n\nI slip out of the carriage home quietly, setting the door closed.\n\nI make my way down the alleyway in back, avoiding Ms. Geissler.\n\nAt 7:30 a.m., I arrive at the home on Paulina, a typical workers cottage. I have to ring the doorbell twice before Mrs. Pugh comes to the door and even then, when she draws it open, there's a deliberateness about it. It's not the breezy way she typically throws open the door and welcomes me in. Her voice is out of joint, uncharacteristic of her typical chirpiness. \"Jessie,\" she says at seeing me standing there. The word falls flat, her eyes dropping to the mop and bucket in my hands, the cleaning caddy stuffed up under my arm. It's far more than my two arms can carry, so that I feel clumsy though I haven't dropped anything. Not yet.\n\nAs the sun rises, it lands on the nape of my neck, making it warm, which is a relief from the near-hypothermic way I spent the night in the carriage home. Cold enough to freeze. My teeth chattered all night, body wrapped up in the one blanket I could find. Three pairs of socks on my feet.\n\nIt isn't so much a welcome. \"Jessie?\" is what Mrs. Pugh really means, a question more than anything, as if she's surprised to see me, as if she's asking why I'm here. She stands before me in a robe and slippers, shielded by the door. There's no workout attire as expected. No yoga mat and no gym shoes. She must not be feeling well, I guess, because at eight in the morning Mrs. Pugh has yoga, so that by seven thirty, she's always dressed, hair done up in a ponytail with strands that hang loose and frame her face. But not today.\n\n\"Am I early?\" I ask, looking at my watch, which tells me it's seven thirty. I'm not early because I'm right on time. I hear Mr. Pugh call from the distance, \"Who's there?\" he asks.\n\n\"It's Jessie,\" she says.\n\n\"Jessie?\" he asks, the tone of his voice equally confused. As if he doesn't know who I am, which of course he does. I've been cleaning their home for years. Every Tuesday.\n\n\"It's Wednesday,\" Mrs. Pugh tells me. \"You're not early, Jessie,\" she says. And I can't make out that expression on her face, but I can see that she's not happy. \"You're a day _late_. You were supposed to be here yesterday,\" she tells me, and it startles me, this sudden revelation that today is Wednesday. That it's not Tuesday after all, in which case my whole week's been mixed-up. I wonder what else I missed. I feel groundless all of a sudden, standing high on a ledge with nothing to hang on to.\n\nMy apology is effusive. \"I'm sorry,\" I sputter. \"I'm so, so sorry,\" as I try and make my way past Mrs. Pugh and into their home to clean it now, but she stands in my way and says not to bother. \"We had friends over last night, Jessie. Parents from the preschool. We needed the home cleaned,\" she says as she tugs tighter on the cord of the robe to keep whatever's inside concealed.\n\n\"I had to find someone else to clean it,\" she says as she stares at me, not into my eyes, but somewhere beneath. She raises a finger, points at my chest so that I look down but see nothing. She says, \"Jessie, your...\" but then her voice drifts off. She reconsiders. Puts her hand down and says instead, \"I tried calling you. You didn't answer.\"\n\n\"I'm so sorry,\" I say again. \"I could rake the leaves,\" I suggest, though the number of leaves on their lawn is negligible. It's too early in the season for many leaves to be falling. But I say it so that I'll have something, anything to do. \"Mow the lawn?\" I ask, hearing how desperate I sound, but she shakes her head and tells me, \"We have a service. They take care of the yard work.\"\n\n\"Of course,\" I say, feeling stupid. I back away, not bothering to turn and look where I'm going, missing the one concrete step that separates the front stoop from the walkway. One step, a ten-inch rise. I drop straight down, landing gracelessly somehow or other on the balls of my feet, whacking my teeth together in the process. I don't fall, but the mop slips from my hands, its clang echoing up and down the street.\n\nI turn to leave, tripping over the mop as I do, and only then does Mrs. Pugh take pity on me. \"Our company,\" she begins, \"last night. Six kids and twelve adults can make quite the mess.\"\n\nShe opens the door wider and invites me inside. My thanks is as over-the-top as my apology. It has nothing to do with money, but everything to do with time. Everything to do with keeping myself occupied.\n\nI wipe down the kitchen countertops and cabinets; I wash the floors. In the bathroom, I scrub like the devil, taking out all my anxiety on the subway tiles. It doesn't help.\n\nAs I move from the bathroom to Mr. and Mrs. Pugh's bedroom, I catch sight of a computer sitting on a writing desk and it gives me an idea. The desk is minimalist, as is the computer. A sleek silver laptop that prompts me for a password as I lift it open and press the return key, holding my breath to listen for the sound of footsteps sweeping down the hall. It doesn't take a genius to figure this one out. Taped there to the desk is the password, as well as the password for every one of Mr. and Mrs. Pugh's financial accounts. Their credit cards, their bank accounts. Their Vanguard funds. I type the code and easily get in. I could probably appropriate a few hundred thousand from them if I wanted to. But that's not what I'm here to do.\n\nMr. Pugh has gone off to work and so for now it's only Mrs. Pugh and me. Mrs. Pugh, who sat in the sunroom drinking her coffee and reading a book when I excused myself to clean. I pray she stays put, that she doesn't come wandering into her bedroom and catch me meddling with her things.\n\nI pull up a search engine and type my own name into it. Jessica Sloane. I'm not sure what I expect to find. Or rather what I expect to find is nothing. But instead I find an interior designer with my name, one that takes up the first two pages of results. Around page three I find a doctor named Jessica Sloane. Even farther down the page, a Pilates instructor. A Tumblr account for a fourth woman of the same name.\n\nBut me specifically, I'm nowhere there. Though it's not like I'd have a reason to be on the internet. I've done nothing noteworthy with my life; I don't have social media; I've never been on the news. For the last twenty years, Mom and I have lived as sequestered a life as we could. Like nuns, except that we didn't pray. We just kept to ourselves.\n\nI click on the tab for images. Hundreds of photographs load before my eyes. Hundreds of photographs of rooms the interior designer Jessica Sloane has designed. They're dramatic and fussy and not at all my style. There are photographs of her too. Her and Jessica Sloane, MD, all decked out in a white lab coat with a stethoscope slung around her neck, smiling. Trying hard to look empathetic and intelligent all at the same time. I click the news tab at the top of the page, finding articles about them too.\n\nI pause then, hands frozen above the keyboard, hearing a noise from down the hall. The house is long and narrow, each of the rooms small. I listen, hearing water streaming from the kitchen faucet, the coffee maker warming up to brew another pot. Mrs. Pugh is making herself more coffee.\n\nOnly when Mrs. Pugh's gentle footfalls drift away do I return to the screen.\n\nOn a whim, I insert my middle name, certain the search will come back empty. But instead it narrows the results down to a manageable thirty-two, which is not at all what I was expecting, and at first I think the computer is wrong.\n\nIt's the top hit on the page that catches my eye, a newspaper clipping dated seventeen years ago. The headline reads Hit-and-Run Driver Kills Girl, Age Three.\n\nIt takes my breath away. My eyes can't believe what they see. The words. The picture. The caption beneath the image that reads, in italics, Jessica Jane Sloane.\n\nThat's me.\n\nMy hands clutch the edge of the desktop, squeezing hard, white-knuckled from the grip.\n\nI go on to read an article that describes a child walking into traffic and being struck by a car. The car sped on, it says, leaving the girl for dead in the street. According to witness accounts, the car was going too fast, driving erratically. Assumptions were made that the driver was drunk, though no one got a good look at him or her, nor did anyone catch a glimpse of the license plate number. There were discrepancies as to the color of the car, which went to prove the unreliability of eyewitness accounts. They couldn't be trusted. The girl, Jessica Jane Sloane, was carted to the local hospital via ambulance, and there she died.\n\nI click back on the images tab and spy a photograph of little Jessica Sloane in a purple bathing suit. In it, she's happy. She's three years old.\n\nMy head spins. My fingers go numb. They lose feeling completely as I stare at the little girl's face and think, _Who is this girl and what's she got to do with me?_\neden\n\nMarch 29, _1997  \n_ Egg Harbor\n\nThey say that vodka has no smell to it, and yet it was clear as day to me, the smell of it on Aaron's breath as he dropped into bed beside me tonight, the clock trumpeting 1:13 in the morning. Over the last few weeks, I'd noticed a gradual shift in his work schedule, each night him coming home later than ever before.\n\nAt first he said nothing, just stared blankly at me when I asked if he'd had something to drink. He didn't say yes, but he didn't say no either, and it seemed reasonable enough to assume he _had_ been drinking, though he need not say one way or the other because I could smell it on his breath.\n\nIt just so happened that Aaron and a couple of coworkers had stuck around for a nightcap after their shift was through. It had been a bad night, _shitty_ was the word Aaron used, Aaron who didn't ever used to complain. Damien was a no-show and Aaron was in the weeds all night, struggling to keep up on the line.\n\n\"It was just one for the road,\" he said. \"It's not like I'm drunk, Eden. It was one drink. One stupid drink,\" __ he said as he set the pillow over his head.\n\nI didn't need to remind him of the effects of alcohol on male fertility. He knew. He knew because Dr. Landry had told us all those many months before when we discussed ways to better improve Aaron's low sperm motility.\n\nI didn't need to tell him how I had been alone all day, for eleven hours this time. Nearly twelve. He knew this too. He knew that I didn't like to go to sleep until he was here, in bed beside me. He knew that most days the boredom and loneliness consumed me, and what else was there to do for those eleven or twelve hours besides think about how much I craved a baby?\n\nI rolled over onto my side of the bed, taking the blanket with me.\n\n\"So now you're mad?\" __ Aaron asked as he sat there, exposed. I didn't say yes or no but I didn't need to say one way or the other because Aaron could see my posture, could sense me tense up in fury and rage. He tried to reach out for me, but I pulled away. He sighed. \"I needed to unwind for a bit. To have a little fun,\" __ he said by means of explanation, but it only made things worse, imagining him with coworkers, drinking vodka and having fun.\n\n\"What's so wrong with that?\" he asked. \"Do you have any idea how stressful it is for me at work?\" but I had a different thought then, one that went back to money. Not only was Aaron coming home later each night, drinking after work with friends\u2014 _female friends?_ I wanted to ask, but couldn't do it quite yet, too afraid to know the truth, that Aaron was throwing back shots of vodka with the pretty cooks and waitresses while I sat, a prisoner in my own home\u2014he was blowing our money on booze. Money that could otherwise be saved for fertility treatments. For a baby.\n\n\"You don't need to be wasting our money like that,\" I said. \"We hardly have enough as it is.\"\n\nAnd then I did ask him who he was drinking with __ and he rattled off names. Casey. Riley. Pat. Names that were all conveniently unisex. Names that kept me up half the night wondering if they were male or female.\n\n\"Who's Casey?\" I asked, censoriously, and when he didn't reply I created her in my mind's eye: tall and svelte with long butterscotch hair and pecan eyes. Flirtatious and tactile, predisposed to standing too close and touching so that I envisioned her, this make-believe woman, with her nimble hand on Aaron's arm.\n\nPerfect teeth.\n\nA flawless complexion.\n\nAn effortless laugh.\n\nI've gained ten pounds now due to the many months of fertility treatments. I'm bloated all the time, in addition to moody and upset. The water retention has made my fingers grow fat. Most days my wedding ring barely fits with the water weight and stays hidden at home in a dresser drawer.\n\nAnd then Aaron asked, \"What happened to you, Eden? You used to be so much fun,\" __ while pulling the blanket from me. His final hurrah.\n\nI lay there in the dark, completely exposed.\n\nThere was a part of me that remembered that Eden, the fun Eden, but in the moment she seemed so far gone, she was hard to remember anymore.\n\nApril 14, 1997 _  \n_Egg Harbor\n\nToday I watched a mourning dove in the gutter of our home get pelted with hail. She was female, a mother-to-be, beautiful with delicate beige plumage, perched on three oval eggs in the aluminum gutter. She'd spent days with her man friend, methodically assembling the nest of twigs and grass blades\u2014while I watched on from the second-story window as they scurried back and forth from tree to trough, collecting materials and sticking them flimsily together\u2014not thinking once of the rainwater that would soon stream past her shanty or the pellets of frozen ice that would one day take her life.\n\nIt was golf ball\u2013size hail, a fusillade of machine-gun fire streaming down from the pale green sky. I've never felt so helpless, watching as she sat there, hunkered down over her eggs, protecting them until the bitter end. It went on for six and a half calamitous minutes, and when it was through she lay there, unmoving, folded lifeless over the eggs like a hooded cloak and I didn't know what to do. There was no blood. I would have expected there to be blood, and yet the internal damage was no doubt worse than that which I could see from the outside, evidence of the great lengths some mothers will go to protect their children. She could have flown away, sought shelter beneath the elm or cottonwood trees that crowded the yard, diminishing our view of the lake.\n\nBut she didn't. She stayed.\n\nThe storm passed. The clouds drifted away and the sun began to shine. A rainbow appeared in the sky. The hail melted. Rainwater evaporated. The only sign of the storm was the dead bird.\n\nAaron watched on as I schlepped the old wooden ladder to the back of the cottage and began to climb. He asked what I was doing as I shimmied up those steps in bare feet, the shaky ladder teetering on the lawn. At the top rung I saw her, splayed sideways, head lolling over the edge of the gutter. I pressed a single finger to her chest, feeling for a heartbeat and, at finding none, removed her body from the trench. Beneath her corpse, the eggs were still intact.\n\nShe died a martyr.\n\nI buried her beneath the trellis, which the snowdrift clematis had overtaken at this time of year, white flowers powdering the wood.\n\nThey say that mourning doves mate for life. As far as I could tell, her man friend never returned to grieve his loss or to check on the eggs.\n\nSometimes this is the way it is with men.\n\nApril 24, 1997 _  \n_Egg Harbor\n\nI can't trust myself to stay at home all day anymore.\n\nAll too often, I drive into town and park outside the dance studio, watching the little ballerinas come and go. It rains many days now, this time of year, and so they come toting umbrellas, skipping over puddles, walking faster than ever before, though always, _always_ , does little Olivia lag behind, and on the most inclement of days, when no one else wants to be outside, I am sure that she will be forgotten. It makes me sick to my stomach to do so, to watch the ballerinas in their leotards and tutus and tights, a Peeping Tom by my own right; it isn't perverse, there's nothing depraved about the thoughts that run through my mind, and yet I know in my heart of hearts that it's unhealthy, pining this way for someone else's child.\n\nAnd so, against Aaron's will, I found a job. Some useful way to spend my days other than keeping vigil of the ballet studio, watching the ballerinas come and go.\n\nWe rarely talk these days anymore, other than that time spent in limbo each month, while they wash Aaron's sperm before injecting it inside me. Then we talk. About what, I don't know. About nothing. When I ask him questions, I'm astounded by the brevity of his replies, one-or two-word responses that leave no room for dialogue. He doesn't make eye contact. He asks me nothing. We kill time in the lobby of the fertility clinic before my name is called and only then am I granted amnesty, a pardon, a reason not to have to sit in the lobby and speak to my husband.\n\nThe job is at the hospital. The position is in billing as a medical coder, one I have ample experience in after all those years working for a pediatrician in Green Bay. And so now, I spend eight hours a day reading through patient files to figure out what they're to be billed for; I enter data; I submit to insurance companies; I mail invoices to patients. It feels good to be doing something with my days, to be earning an income.\n\nAnd yet the position comes with its fair share of downsides too.\n\nYesterday as I sifted through patient files, I came across a little girl, a toddler killed in an auto-ped accident. In other words, she wandered into the street when her mother wasn't looking and was hit by a passing car on the roadway, a four-lane highway that cut right through town. The little girl (and though I, myself, never laid eyes on her, I conjured her up in my mind anyway, her tiny, broken frame still clad in a pair of denim overalls with blood-stained pigtails in her light brown hair) was transported to our hospital by ambulance, and there, received a multitude of treatments, from a CT scan to assess brain damage, to an operation to control internal bleeding and swelling in the brain. A decompressive craniectomy, as was noted in the extensive patient chart. There were blood transfusions. She was on narcotics for pain. An anesthesiologist was called to deliver a local anesthetic to put her to sleep for the surgery. The surgery itself lasted six hours, and each of these items came with an exorbitant price tag, one the family's shoddy insurance company was loath to pay. For six tortuous hours while the little girl's mother, I can only imagine, sat on a chair in the waiting room, biting her nails to the quick, a neurosurgeon, along with a team of doctors and nurses and scrub techs, removed part of the girl's brain to allow room for the swelling inside.\n\nStill, she died.\n\nBy the time the paperwork made it to me in coding and billing, it had been days since the angels carried her away. Her mother no longer stood within the hospitals' walls, sobbing for her child. Her body had been removed from the morgue, transported to the funeral home, buried in the ground.\n\nAnd yet for me, it's a fresh wound. One that will stay with me for a long time to come. As I typed the billing codes into the system, I cried for a little girl I've never met, tears snaking down my eyes and onto the computer keys, knowing that she will never truly be anything more than a name and a social security number to me, but still it makes me cry, grieving for someone I don't know, consumed with the unwanted knowledge that healthy little girls\u2014like the sick and the elderly\u2014die too.\n\nBut there are perquisites to the job too.\n\nI wear a name badge that gains me access to every nook and cranny in the whole entire hospital, including the birthing center, where I can watch newborn babies being tended to in the nursery, lying immobile in their rolling bassinets, bundled like burritos with knitted hats on their perfectly pink and misshapen heads. I didn't seek them out\u2014in fact, I swore to myself that I would abstain from visiting the newborn babies\u2014but I saw them anyway when a pair of grandparents-to-be stopped me in the hall and asked the way to labor and delivery. I had no choice but to lead them there, to steer them through the mazelike hospital halls, through the double doors and into the unit where the newborn babies caught my eye.\n\nAnd now I stand there for what feels like hours, staring through glass, coming to terms with my fait accompli. No workday passes without at least one visit to the nursery room and as I sit at my desk coding patient files, it's all I can think about, seeing those babies. Getting my fix. I've come to know the nurses now\u2014thanks to the frequency of my visits. They address me by name, sometimes holding up the newest infants so I can see their puffy, half-closed eyes, their still-bowed legs from being cramped inside a warm, cozy uterus, their cone-shaped heads from being suctioned through their mother's vaginal canal and into the world.\n\nWhen they ask, I tell them I'm training to be a nursery room nurse myself, that I'm in the process of earning my associates' degree in nursing and, that as soon as I do, I'm going to apply for a job here, in our hospital's nursery room. I tell them I come to watch and learn, to see how the experts do it. I flatter the nurses so they don't think it odd that I spend every free second away from billing and coding staring at babies who are not mine. They smile and say how fantastic that is and sometimes, if I'm really lucky, they sneak me inside so I can stroke the soft skin of a tiny babe.\n\nThough that, of course, isn't the real reason I come.\njessie\n\nIt's not yet ten in the morning when I leave the Pughs' home. The day stretches out before me like the Sahara, massive and deserted and dry. And now I'm even more agitated than I was before, all nervous energy with nothing to do. Nowhere to go. No one to talk to.\n\nI carry with me in my bag a printout of the newspaper article I found on the Pughs' laptop, grateful when Mrs. Pugh called down the hall that she was stepping out for a bit, and I was able to send it to the printer without her hearing me. Because if I knew one thing, it was that I needed to take the article with me.\n\nI hop on Old Faithful and ride. I turn aimlessly, unplanned at each intersection, my head lost in the clouds. I move in circles so that three times I pass by the very same delicatessen without meaning to. I speak to Mom. I ask her questions about my lack of a birth certificate, my missing social security card, the girl in the article. Who is she, and what does she have to do with me? Does she have anything to do with me? _Tell me, Mom_ , I scream in my head. _Tell me!_\n\nIt isn't until a woman standing on a street corner stares at me like I'm crazy that I realize I've been speaking out loud.\n\nIn time I find my way into the Loop. It isn't intentional. I don't go there on purpose. It's something far more subliminal than that, that makes my legs pedal hard, steering me to the Art Institute on Michigan Avenue, where I park Old Faithful just steps from the bronze lions and walk.\n\nI don't go to the museum.\n\nRather I head to the south end of the building where, just off Michigan Avenue, I slip into this secret world of raised flower beds and a grove of hawthorn trees. I'd never have known what kind of tree they were, but Mom knew, Mom who found this spot by accident one day when I was young and we were exploring. It was fall and the trees were angular and uneven, a brassy shade of copper that peeked through the green of nearby trees as they do now.\n\n_Let's see what this place is_ , Mom had said that day, grabbing a hold of my hand and drawing me in. That first day, I didn't want to go. Rather I wanted to climb on the lions' backs and ride. But Mom had said no. The lions were to look at. They weren't for riding, though she let me pet them as we passed by.\n\nThe entrance to the garden is guarded by honey locust trees, which keep the rest of it hidden from the urban world on the other side. I slip in. I walk down a handful of steps that dip inches below street level. I move between the trees, lost in an enclave beneath an awning of leaves. Transported somewhere hundreds of miles from a city street.\n\nThere are people here. It's not as if I'm the only one who knows about this place. And yet those that are here are placid. Quiet. Drinking coffee and smoothies, reading books, staring into space. A woman picks at the edges of a muffin wrapper, offering scraps to a nearby bird.\n\nThis was one of Mom's favorite spots in the city. We'd come here and she'd spend hours sitting on the edge of the raised beds. She'd watch as I scaled them with my arms extended, imagining myself as a tightrope walker. They're large\u2014a good twenty feet by twenty feet or more\u2014so it was always quite the feat when I could get around without falling.\n\nMom let me do it for hours. She never got bored.\n\nThere was one place in the garden Mom liked more than the rest because it was secluded, set back from the street entrance, the water fountain and the pool. Even in the most secluded of places, she found the most covert place to hide.\n\nI make my way there now because I think that somehow I might feel closer to Mom if I sit there. That somehow we'll be able to commune.\n\nBut when I get there, that spot is already occupied. A man sits there, reading the newspaper. Truth be told, it makes me crabby, thinking what nerve he has to sit in Mom's favorite spot. And so I sit opposite him on another bed, twenty feet away or more, watching him, waiting for him to leave. I stare at him, thinking it'll make him uncomfortable and he'll go.\n\nBut he's not uncomfortable because he doesn't even see me staring. He's too preoccupied by the newspaper in his hands.\n\nI can't say one way or the other if he's tall or short because he's sitting. He's got his legs crossed, ankle to knee, and his clothes are all sorts of nondescript. Pants, shirt, shoes. Nothing noteworthy about them. They're clothes. The sleeves of his shirt are thrust to the elbows. There, on his left arm beneath the cuff of the shirt, is a scar. It peeks out from beneath the sleeve, a six-inch gash that's healed poorly. The skin around it is puckered and pink.\n\nHis face looks sad. That's the first thought I have. That the expression on his face\u2014that and his body language\u2014is one of sadness. The way his mouth pulls down at the corners, a slight tug there at the edges of his lips. The way his shoulders slouch. I should know because each time I look in the mirror, I see the very same thing. On his face is a patch of hair, a tight beard, trimmed and tidy. It gives off an aura of mystery and regality. His skin is tanned like the hide of a moose, stretched and dried in the sun before being smoked over a fire. Like he's spent too much time outside in the sun.\n\nHe isn't thumbing through the newspaper, but instead he's got his eyes peeled to some story on the top page, the paper folded so that he can hone in on it. Something bad has happened in the world, I think. Something bad always happens. I wonder what it is this time. Terrorist attack. Women and children being slaughtered by their own leaders. A shooting in an elementary school. Children murdered by their own moms and dads.\n\nI watch his eyes, the movement of them as he scans the story. Moving left to right. Dropping down to read the next line. But his eyes are lowered, gazing down on the newspaper and so I can't see much, none other than the lashes and the lids. He bites a lip. He bites hard so that the pain of the lip overrides whatever it is he's feeling on the inside. I do that too.\n\nHe reaches for a cup of coffee set on the marble edge of the raised bed. I read the corrugated sleeve on the cup. A coffee bar on Dearborn. I've never been there before, but I know the place. I've seen it before.\n\nAnd then he gets up to go, and I ready myself to make a move for his seat. He slips an orange baseball cap over the brown hair, though as he goes, he leaves his newspaper behind. Because he's sad. Because he's distracted.\n\nHe walks away and I notice a shoe is untied, the cuff of a pant leg stuck in the shoe's tongue. He leaves it there. For a second or two I watch him go.\n\nBut then, standing and making my way to the raised bed, I call to him, \"Sir,\" while grabbing the newspaper so that the wind doesn't have a chance to scatter it around the garden. \"Sir,\" I call again, \"your newspaper.\"\n\nBut he's walking away and before I can run to him, something leaps off the page at me. It grabs me by the throat so that I can't speak and I can't move. I'm frozen in place, a bronze statue like the lions who stand before the Art Institute, guarding its entrance.\n\nThere on the top of the newspaper is Mom's beautiful face. Her beautiful brown eyes and brown hair, both watered down by the black-and-white newsprint.\n\nHer obituary. The one I put in the paper because I needed the world to know she was dead. To solidify it. To make it real. Because only then, when it was written in print for all of the world to see, would I believe it.\n\nThis man. The sad man sitting in Mom's and my spot in the garden. He was reading her obituary.\n\nHe had the newspaper folded so that Mom's face was on top, and it was these words his eyes spanned as he bit his lip so that he wouldn't cry.\n\nMom's obituary is what made this man sad.\n\nI read over the words. Mom's death notice, which was brief because there wasn't a whole lot of information to provide. No memorial service. No one to send flowers to.\n\nThe final line reads \"Eden Sloane is survived by her daughter, Jessica.\"\n\nMy legs lose feeling. I go slack jawed. Because there's one word on the newsprint that the man has circled and it's my name. Jessica.\neden\n\nJuly 12, 2003   \nChicago\n\nThe park is named after some poet, I've come to learn. Though no one pays attention to things like that because, to most people, it's just a park where kids romp around on the playground and, on the other side of a chain-link fence, boys play basketball, the repeated _thump_ , _thump_ , _thump_ of the ball on concrete a steady refrain. They're older boys mostly, teenagers, and they spout from their mouths a flurry of curse words at regular intervals, and I feel grateful Jessie is still too young to know what any of it means, though she pauses from time to time to watch them. To just stand on the playground and stare.\n\nThere are baseball fields off in the distance, and on the other side of a bridge, a path that snakes along the river where she and I sometimes walk, but not today. Today she played on the playground, and for the first time ever, found a friend. Not the kind of friend we'd keep in touch with after today or invite over for a playdate. No, Jessie and I don't have those types of friends.\n\nRather she's the kind of friend who, for fifteen or twenty minutes at best, is a bosom body. A soul mate.\n\nI watched as Jessie and the little girl chased each other in dizzying circles, up the stairs and down the slide. Again and again and again. As far as I knew, they never exchanged names. Because that's the way it is with kids. Uncomplicated. Straightforward. Easy.\n\nThere was no one else on the playground but the two of them, and the only ones sitting on the periphery of it were the little girl's mother, pushing a newborn in an old-fashioned buggy, and me. It took some time, a few awkward glances my way, before she rose from her own park bench and came to mine, standing before me, offering a hello. I too said hello, staring down into the buggy at the infant sound asleep beneath a yellow blanket.\n\n\"How sweet,\" I said.\n\nThe baby, Piper, she told me, was twelve days old, born on the first of July. The woman moved guardedly, as if in pain, and I didn't ask before she told me, \"Piper was breech,\" telling me how her baby was fully intent on entering the world feetfirst. \"The doctors did everything they could to change that. But no such luck,\" she explained, sitting softly beside me on the park bench and describing in too much detail what a C-section is like. The incision. The surgical staples. The scar she'd no doubt have. She lifted the hem of her shirt then so that I could see it myself, and I blushed at the sight of her still-pregnant belly, at the bloated butterfly tattoo that sat just inches from the healing incision, at the canvas of fair skin. She was oversharing and I blamed the newness of childbirth for it, the fact that to her it was still fresh. The only thing these days that occupied her mind.\n\n\"With Amelia it was different,\" she admitted, and I made the easy assumption that Amelia was the older of the two, the little girl, maybe five years old, who Jessie made a train with at the top of the slide\u2014wrapping her skinny legs around the midsection of a girl she hardly knew\u2014and together they catapulted down to the wood chips below, landing on their rear ends, laughing. \"Twenty-some hours of labor, three hours of pushing,\" she said, going on far too long about the gush of water when her membranes ruptured, like the pop of a water balloon. Her, worried only that she might poop on the bed, as one of her girlfriends had done. The broken blood vessels left behind on her face from hours of pushing, thin, red veins that snaked this way and that across her skin. Some doctor she didn't know delivering her baby. Her breasts engorged, her unable to produce milk following childbirth. Having to relent to formula, which her mommy's groups abhorred.\n\nI felt uncomfortable, if I was being honest, about this sudden revelation of information from a woman I didn't know. But it dawned on me then that this is the type of thing women do, this is the type of thing mothers do: share their experiences, swap stories, foster camaraderie.\n\nShe looked at me expectantly, as if it was my turn to share. She was quiet, watching me, and when I didn't respond, she prompted, \"And your girl?\" and I knew then that I must tell her something, that I must offer up some version of the truth. I pictured those wide hospital halls, the glaring lights. \"She was a vaginal birth?\" she asked, that word alone\u2014 _vaginal_ \u2014making me turn redder than I was before. Because these were the kinds of conversations I didn't have. Intimate. Friendly.\n\nMost of my conversations ended at hello.\n\nI felt my head nod without my permission, and I knew I must say more, that a nod of the head alone wouldn't suffice.\n\nAnd so I told her about the hospital room. I told her about the huddle of people who gathered around me, the nurses clinging to either of my legs, encouraging me to push. Incanting it in my ear\u2014push, push\u2014as I gathered handfuls of bedding in my hands and bore down with all of my might. The epidural had worn off by then, or maybe it was never there to begin with. All I felt was pain, a pain so intense it was as if my insides were on fire, about to rupture. I was certain I would soon explode. A hand stroked the sweaty hair from my face, whispering words of encouragement into an ear as I screamed, this crude, ugly scream, but I didn't care how crude or ugly it was. The nurses wrenched on either of my legs, stretching me apart, making me wide. _Push_ , they said again and again, and I did, I pushed for dear life, watching as that flash of black spilled from inside of me and into the doctor's gentle hands.\n\nBut then I remembered.\n\nThat wasn't me.\njessie\n\nBefore I can tear my eyes from Mom's face on the newspaper's obituaries page, from my own circled name, the man has slipped from the garden and disappeared from sight. I attempt to run after him, barreling through the rows of hawthorn trees as quickly as my legs can carry me. But still, when I come rushing out onto Michigan Avenue, chest heaving, breathing hard, he's gone. The sidewalk is inundated with people, with kids, a middle school field trip to the Art Institute, and they're all lined up in two parallel rows before the museum's concrete steps. Clogging the sidewalk. I push past bubbly preteens who are incognizant of my desperation, who don't care. By the time I reach the other side, there's no sign of the man anywhere. The man with the sad eyes and the untied shoe.\n\nI stare up and down the street, completely aghast. A muscle in my eyelid twitches, a spasm. Something involuntary, something I can't make go away though I try. It's extremely annoying. The street is a wide six-lane divided street jam-packed with people and cars, a median strip in the center that's plugged with flowers and trees, making it even harder to see the other side. But still I walk, searching the streets for the man.\n\nI hurry down Michigan with a heavy, desperate tread. The wind is a wall by now, and I lean into it to walk. It's exhausting. All the while, my eyelid twitches. I turn left at Randolph, a temporary reprieve from the militant headwind, which now comes at me from the side so that I slope laterally, a perfect seventy-five-degree angle. At Clark, I turn right, not quite knowing where I'm going, but trying desperately to find the man. I climb northward, gazing into storefronts to see if he's there. I stare down alleyways, out of breath by the time I come to a six-story building on Superior Street, one that's flanked with floor-to-ceiling windows and looks oddly familiar to me.\n\nI spin in a circle, taking it in, the doorman in uniform, the sign outside that reads Spacious, Open-plan Lofts for Sale. Inquire Inside. __ I know where I am. I've been here before.\n\nJust like that, I'm standing at Liam's front door.\n\nI didn't know I was coming here. I didn't come here on purpose. But here I am, and now that I'm here, I make an attempt to scoot past the doorman and into the building. Because maybe Liam can help me think this through. The little girl in the car accident, the man in the garden. He'll make me see that there's nothing sordid going on. That it's only a coincidence.\n\nThe doorman stands on the curb, hailing a cab for a resident. \"Can I help you, miss?\" he asks, catching sight of me out of the corner of his eye, as he steers the resident into the back seat of the cab and closes the door for her.\n\nHe steps closer to me. \"I'm here to see Liam,\" I say.\n\nHis smile is mocking. Wary. \"Liam who?\" he asks, playing dumb, and I freeze, realizing only then that I don't have a last name. That to me he's just Liam. That until yesterday he wasn't even that, because before yesterday he didn't have a name. He was only the guy from the hospital, the one with the blue eyes.\n\nBut I also realize that the doorman knows fully well what Liam's last name is. He isn't curious. He's testing me, checking to see how well I know Liam before he lets me in.\n\n\"I don't know his last name,\" I admit, feeling uncomfortable as my feet shift in place. At first he's hesitant, not sure he wants to phone Liam or not. For all he knows, I'm someone Liam is avoiding, someone he doesn't want to see. And that's his job, to keep unwanted visitors at bay, unwanted visitors like me.\n\nHe sizes me up and down. He asks twice what my name is. Both times I say Jessie, though for the first time I start to doubt that it is. I feel disheveled, disoriented, and though I have no idea what I look like, I can see it in the doorman's eyes. It's not good. I run my hands through my hair; I rub at my twitching eyes.\n\n\"Is Liam expecting you?\" he asks, and I'm not quick enough on my feet to lie. I tell him no.\n\n\"Can you call him for me, please?\" I plead, the desperation in my voice palpable to both him and me.\n\nThe doorman reluctantly phones Liam for me, but Liam doesn't answer his call. \"He's not home,\" he tells me, setting the phone down. I feel the skeptic in me start to take hold. He's lying. He didn't call him. He only pretended he did, but he didn't. I think that maybe the number he dialed wasn't Liam at all, or maybe he didn't push enough digits for the call to go through. Or he hung up before Liam had a chance to answer.\n\nI'm about to get angry, but then I remember. The funeral. Liam's brother's funeral is today. He's at the funeral. He's not home.\n\nI excuse myself, walking from the building, feeling muddled. There's a convenient mart next door to the apartment building. I slip inside and buy a Coke, hoping the caffeine will make me feel less mixed-up. Or, at minimum, curtail the throbbing in my head from the day's lack of caffeine.\n\nBack outside, I drop down onto the curb to catch my breath. I need to think things through, but my mind can focus on only one thing. What if Jessica Sloane with my social security number did die when she was three? She wasn't erroneously classified as dead because she was really dead. Then I've been living with a mistaken social security number all this time, with a mistaken identity.\n\nIs it possible that the other Jessica Sloane and I have social security numbers so close they're off by only a single digit, or have two numbers that are interchanged? Maybe she died and someone unwittingly typed my social security number into the death database. The names matched, so they didn't think twice. An oversight only.\n\nDoubtful.\n\nAnd then my mind gravitates to the man in the garden. Who is he, and what was he doing there? What does he want with me?\n\n\"Jessie?\" I hear, and when I look up from the street, I see Liam making his way toward me. All dressed up in a black suit and tie. Looking undeniably sleek but also tired like me.\n\nI rise from the curb and bridge the gap, and, as we close in on one another, his face darkens. \"Your shirt,\" he says as he points to it, to my shirt, and tells me that I've got it on inside out. Which wouldn't be so obvious were it not for the label sitting smack-dab beneath my chin, a blaring thing. I pluck it from my skin for a better look.\n\nNot only do I have my shirt on inside out, but it's backward. And now that Liam has pointed it out for me, I feel the high neckline, the cotton taut in places it isn't meant to be taut. In that moment I have no memory of ever grabbing the shirt from the closet, slipping it from its hanger, of ever putting it on.\n\nIt's a blessing that I'm even dressed.\n\n\"Come inside,\" he says, his eyes hanging on a little longer than they ordinarily would. \"You can fix it there.\"\n\nBut I say, \"No,\" shooing him off, feeling suddenly asinine. \"It's just a stupid shirt anyway; it's not like anyone noticed.\" And then I sigh, feeling completely exasperated. Exasperated and exhausted. He hears it in my voice.\n\n\"Jessie,\" he says, his voice far more resolute this time. \"Come inside. Keep me company.\"\n\nWe step inside the building and wait for the elevator to come. \"Did you sleep last night?\" he asks. I don't say yes or no, but my silence gives it away. In my head, I tally the days up. I lose track at number four and have to start again, counting on my fingers this time, reaching seven.\n\nIt's been seven days since I've slept.\n\n\"I looked it up,\" Liam tells me as the elevator comes for us. Though it doesn't align with the lobby floor\u2014a fact that I realize all too late\u2014and so I trip on the way in, stumbling over that one-inch rise. Liam latches on to my arm, steadying me. He doesn't let go. Not until I draw my arm away, stepping closer to the wall so that I can use it for leverage if need be.\n\n\"Looked what up?\" I ask as the elevator sweeps us up to the sixth floor. I feel suddenly rocky on my feet. Nauseous.\n\n\"The longest a person has ever gone without sleep,\" Liam says.\n\nHe tells me how people die from lack of sleep. About lab rats who died from lack of sleep. \"How long?\" I ask.\n\n\"Eleven days,\" he says. \"Eleven, Jessie,\" he repeats to drive the point home, I think. \"You need to sleep.\"\n\n\"I will,\" I say, but chances are good that I won't.\n\nI ask how the funeral went because I don't want to talk about my lack of sleep or the fact that in four more days I'm liable to die because of it. The funeral, he says, went as well as to be expected for a funeral. His shoulders shrug and his expression is flat. He doesn't say more.\n\nThe elevator arrives at the sixth floor. He leads us to his apartment, walking a half step ahead of me. At the door, I stop a few feet back, waiting as he opens it. Inside, the space is big and roomy with ceilings that are extraordinarily high, track lighting, exposed brick. Sunlight pours in through floor-to-ceiling windows. \"You coming?\" he asks.\n\nI walk past him and into the apartment as behind me he closes the door.\n\nHe offers me something to drink. I say no because I have my Coke, which I uncap and take a swig of. But as I raise the bottle up to my lips, there's that tremor to my hand again, the one I can't make stop.\n\nLiam tugs the tie from his neck and slips the suit jacket off. Throws it over the arm of a chair. Unbuttons his shirt. Rolls the cuffs of it to his elbows. Finds himself a water in the refrigerator and sinks into a low-slung chair. He never asks what I'm doing here.\n\nI give the article to Liam, my hand still shaking as I do. I sit on a chair opposite him. I don't bother fixing my shirt.\n\n\"What's this?\" he asks, but it's one of those questions that isn't really a question because already he's reading the story of Jessica Sloane, who was killed by a hit-and-run driver at the age of three. When he comes to the end of it he tells me what I already know. He says that this is strange.\n\nI assert, \"I mean, it's just a coincidence, right? A mistake?\"\n\nHis face is impassive. He doesn't say an emphatic _yes_ as I'd hoped he would; he doesn't put my mind at ease. This time, there are too many holes that don't line up.\n\n\"I don't know,\" he admits, saying, \"It's just that it's strange, Jessie. I mean, yesterday it was a coincidence. Yesterday it was a mistake. Yesterday someone screwed up. But now it's like it isn't so much an accident as it is someone intentionally trying to keep you off the radar. You have no birth certificate, you can't find your social security card and the social security number you think is yours matches up with that of a dead girl. One who might just have the same name as you.\"\n\nThe expression on his face says it all. Something sordid is going on here. Something bad.\n\n\"It's just hard to believe that she's not you,\" he says while motioning to the photograph in the article, but when I look at the child's face, I see nothing but a stranger looking back at me. I've never seen this girl before.\n\n\"But it's not me,\" I argue, voice trembling. \"She doesn't look a thing like me. Look at the shape of her eyes, her nose. It's all different,\" I allege, rising to my feet. \"It's all wrong.\"\n\n\"I didn't mean that,\" he says, his voice gentle. \"That's not what I meant, Jessie. I just mean,\" he says. \"I just mean that I think it's possible there's something going on here, some sort of identity theft.\"\n\n\"What do you mean, identity theft?\" I ask, except I know what he means. What he's suggesting is not that my identity has been stolen, but that I've stolen the identity of someone else\u2014unpremeditated on my part, but still identity theft.\n\n\"Jessie,\" he starts, but I shake my head and he stops.\n\nAt first there's nothing but silence. I drop back down into my chair. I think it through. \"You think my mother changed my name, gave me a phony identity and passed me off as a dead girl?\" I ask, the words themselves unthinkable. Not something that could possibly be real. For a second I feel like I might vomit. The Coke gathers in my stomach, burning the lining of it. There's hardly any food inside me, which, when coupled with everything else, doesn't sit well. The pain starts somewhere around my navel and creeps up my chest. An agonizing lump that plunks itself behind the breastbone.\n\n\"But no,\" I say decisively, rising to my feet again and beginning to pace. Why would Mom do that? Why in the world would Mom steal the identity of a girl who had died and give it to me? \"Why?\" I ask out loud, though the answer slowly dawns on me, that if Mom went around passing me off as a dead girl, then no one would know she had stolen another child's identity. Because that child was dead.\n\nI watch as Liam grabs for a laptop on the coffee table and types quick, harried words into it. He moves from his chair and comes to me and together we stare at the words on the screen. There's a whole word for it, he tells me. \"Ghosting. Thieves open bank accounts and credit cards using a dead person's social security number,\" he says. \"They pore over obituaries to see who's died, and then rack up thousands of dollars of debt in some stiff's name.\"\n\n\"But why?\" I ask dumbly, though I'm not that dumb. I just can't wrap my head around it. People do this kind of thing for financial gain, but Mom and I were never rich. We weren't living a life of riches. We lived paycheck to paycheck.\n\nBesides, Mom would never do anything to harm someone; she would never steal.\n\nThere has to be more to it than that.\n\nIf\u2014and that's a big _if_ \u2014she took the identity of a dead child and gave it to me, then it was for some other reason than financial gain. But what? I can't even begin to guess.\n\nI swallow the last of my Coke. It's like rubbing salt in an open wound. The pain in my chest gets worse so that I cough and, as I do, all I can think of is corroded pipes, the lining of my esophagus plugged up and rusty.\n\nI let an idea dwell for a short time, and then quickly expunge it from my mind.\n\n_Find yourself_ , Mom told me. One of two wishes she had for me before she died.\n\nMaybe she didn't mean for me to apply to college. Maybe it was far less esoteric than that. Maybe it was quite literal.\n\n_Find yourself_ , she said, because Jessie Sloane isn't you.\neden\n\nMay 28, _1997  \n_ Egg Harbor\n\nAs spring ripens into summer, tourists reappear. The town comes to life with a certain vivacity that was missing during the dismal days of winter. Trees burgeon, flowers bloom.\n\nMiranda and her three boys appear like magic at my front door each day that I'm not working\u2014and often, I'd venture to guess, when I am\u2014toting blueberry loaves and apple pies.\n\nAs the boys play in the tree swing (that by now was meant to hold my own child, the two of us nestled snugly together, he or she on the seat of my lap, weightless and grinning as we lift off from the ground and take flight), Miranda and I sip lemonade. As always she sells short the joys of marriage and motherhood, while little Carter crawls on the lawn before us on all fours, eating dirt. She complains about everything from what a jerk her husband, Joe, can be\u2014coming home late from work, missing dinner, not helping with the boys' bedtime routine\u2014to the monotony of her days, to the amount of food three growing boys consume. She can never keep the cabinets fully stocked, she tells me, because the minute she buys it they eat it all, which leads into an onslaught on the difficulties of grocery shopping with three boys, and she describes it for me: the poking and the prodding of each other, the name-calling\u2014birdbrain, imbecile, idiot\u2014the running off headlong down the market's aisles, bumping into strangers, begging and crying for things that Miranda has already said no to, trying to sneak it past her and into the basket, screaming and calling her names when she snatches it out of their dirty hands and returns it to the store shelf.\n\n\"That must be so difficult,\" I say, trying my hardest to sound empathetic, but when Miranda replies with \"You have no idea, Eden. Can you even grasp how lucky you are, getting to grocery shop alone?\" it's all I can do not to scream.\n\nI would give life and limb to grocery shop with a child.\n\nMiranda doesn't bother asking how the fertility treatments are going, though just last night Aaron and I made the decision to give in vitro fertilization a try. Or rather, I should say, _I_ made the decision to give in vitro a try. The cost of it is extortionate, thousands of dollars for a single cycle, for Dr. Landry to go inside one time and pluck an egg or two from my ovaries to combine with Aaron's sperm, making an embryo, a _baby_ , in a culture dish. As one grows bacteria. It seems scientific, synthetic, and yet there isn't anything I wouldn't do for a child.\n\nI know this now.\n\nBut Aaron isn't so sure. As we stood in the kitchen last night, both of us speaking in acerbic tones, he calculated the costs we've paid over the year, all the pelvic ultrasounds and semen analyses, the Clomid cycles, the trigger shots, intrauterine insemination. The grand total tallied up to some ten thousand dollars already spent trying to create a child, an expenditure that will nearly double with one single cycle of IVF. Aaron and I don't have this kind of money. He reminds me of this relentlessly, as he reminds me how happy we used to be before we ever made the decision to start a family, and I have this vague recollection of a couple, a man and a woman\u2014as if in another life\u2014sitting on a dock, holding hands, watching sailboats float by on a bay.\n\n\"I think we should stop, Eden,\" he said, trying hard to reach out to me but I pulled away. \"I think we should be happy with what we have.\"\n\n\"And what's that?\" __ I asked, up in arms. What did we possibly have without a baby?\n\n\"Us,\" he said, looking sad. \"You and me. That's what we have.\"\n\nI wouldn't be deterred.\n\n\"We will do this,\" I told him of the in vitro fertilization. Hands on hips, my expression flat. An imperial fiat.\n\nI left the room so it couldn't be further discussed.\n\nI've taken out three credit cards in my own name, and charge each appointment with Dr. Landry to them in sequence. Never are we able to pay more than the minimum payment for each. The interest fees soar monthly as the cottage degenerates bit by bit. The furnace went out; we need to replace the plumbing throughout the entire home before the decades-old steel pipes wear out for good. The windows are drafty; they too need to be replaced before another winter comes or we'll spend an arm and a leg to heat the home, watching our money quite literally go out the window.\n\nBut each of these plays second fiddle to making a baby.\n\nAaron and I argue daily about money. The cost of groceries, the cost of clothes.\n\nWhat concessions can we make so that we can save more for a baby?\n\nDo we really need two cars, cable TV, a new pair of shoes?\n\n\"This is ridiculous,\" Aaron says as he holds up a shoe, the outsole flapping loose like a hangnail. \"I can't go to work like this.\" And yet I argue with him, claiming he's being extravagant by not making do with the shoe. \"Surely you can get another month out of those shoes,\" __ I say, suggesting he use some glue, though it isn't about the shoe, but rather what the hundred dollars for another pair of shoes will buy. An appointment with Dr. Landry, a hormone shot, a month's worth of Clomid.\n\nBut Aaron swears he needs the shoe, which inside makes me fume.\n\nHow selfish can he be? Where are his priorities?\n\nAt each unwelcome visit, when Miranda and her boys appear at my door without invitation, her belly continues to swell, another baby on the way, \"Hopefully a girl this time,\" __ she says, fingers laced together in the air.\n\nIf Aaron and I hurry up, she reminds me for the umpteenth time, joining me in the backyard for another glass of lemonade, her baby and my baby can one day go to school together. They can be friends.\n\nI smile.\n\nAnd though I don't say it aloud, I think to myself that I'd rather die than have my baby and Miranda's baby be friends.\n\nJune 13, 1997 _  \n_Egg Harbor\n\nThe hollyhocks are in bloom. Just the sight of them lined up defiantly against the weathered picket fence stabs me in the chest. They stand high above the rest of the flowers in the garden, six feet tall or more. Their bold bell-shaped flowers burn red against the greenery.\n\nIt's been a year then since Aaron and I planted the seedlings in the lawn against the fence where they'd be sheltered from the rain and the wind. And now here they are, exhibitionists in my flower bed, outshining the roses and lilies.\n\nReminding me of all the wasted time Aaron and I have spent trying to have a baby.\n\nWhen Aaron was at work, I took a pair of scissors to them, cutting hard through the thick stem. I seethed as I did it, crying, taking out a year's worth of rage on the flowers. I screamed like a maniac, grateful that, thanks to the deep rim of trees surrounding our yard, no one was around to see or hear my outburst. I grabbed handfuls of stems and tugged with all my might, wresting the roots from the ground where I stomped on them like a child. I tore the flowers from their stems, shredding them into a million pieces until my hands were yellow with pollen and I was out of breath from the outburst.\n\nWhen I was finished, I threw them away, beneath the garbage where all the negative pregnancy tests go.\n\nThe deer, I'll blame, when Aaron asks what happened to the flowers. I'll say that the deer have had their way with the hollyhocks, eating them to the quick.\n\nAnd he'll be more upset about this than he is our lack of a baby.\n\nAfter all our hard work.\n\n\"Such a shame,\" he'll say, before waging a war against the innocent deer.\njessie\n\nI take the Brown Line back to the carriage home, walking the last couple of blocks from the station at Paulina. I feel lost without my bike. I don't have my bike, Old Faithful, because I left it outside the Art Institute, tethered to some sort of loopy bike rack, when I walked to Liam's, chasing after the mystery man.\n\nIt's dark inside by the time I arrive, night falling quickly. I close the door behind me and jiggle the handle a couple of times to be sure it's closed tight. I'm in a trance, thinking about little else but the dead Jessica Sloane. The one who is three years old. The one who is me but not me all at the same time. Lines from the newspaper article run through my mind, committed to memory already.\n\n_A four-lane highway with a speed limit of just twenty-five._\n\n_The road twisted through the small seaside town._\n\n_The driver rounded a bend at nearly twice that speed._\n\nEvery time I close my eyes I see her face.\n\nI have only a vague recollection of riding the elevator downstairs; of pushing my way through the turnstile doors of Liam's apartment building; of walking to the Merchandise Mart to catch the train with him at my side. He'd offered to cover the cost of a cab for me but I said no.\n\nStill, he walked me there, to the Merchandise Mart, and paid to stand on the platform beside me, waiting until the Brown Line came. And now that I look, I see his jacket draped over me, keeping me warm. He must've put it there, but that I don't remember.\n\nI turn and walk up the carriage house's stairway, a rickety old thing with steps that are a bit concave, the edges worn away. The steps sink at their center. They squeak. The tread pitches downward from a century's worth of weight, and I cling to the railing so I don't fall.\n\nWhen I get to the top I have to fight for breath. The steepness of the steps isn't to blame, nor for once my overwhelming fatigue.\n\nWhat knocks the wind from my lungs is something else entirely.\n\nBecause as my feet hit the wooden floorboards and my eyes size up the open rooms, I see that the white window curtains I'd pulled shut before I left, so that no one could see inside while I was gone\u2014every single one of them is open wide.\n\nIt's instinctive, the way the blood coagulates inside me. It becomes thick and gooey so that I can't move.\n\nSomeone was here.\n\nMy gut feeling is to hide. There's a closet nearby, a catchall for coats and shoes. My eyes go to it. I could hide. I could bury myself in a dark nothingness and cower on the floor in fear. Because whoever opened the blinds might still be here. Inside the old home.\n\nI listen for strange noises. For calculated footfalls coming for me. For the sound of restrained breaths, slow, repressed and controlled unlike mine. I listen for the groan of floorboards, but the only sound I hear is that of my own heartbeat.\n\nI don't hide.\n\nI've never been a particularly courageous person. Mom always said to face my fears, to take matters into my own hands, to fight for what was mine. And so I make my way slowly through the home, searching for signs of life.\n\nMuch of the carriage house is easy to see from where I stand. But then there are those places I can't see. An upstairs closet, the bathroom, under the eaves of the pitched roof where shadows make it hard to see. All of that is up another set of stairs, on the third floor of the home.\n\nI ascend those steps on tiptoes, the arches of my feet beginning to burn. Convinced that if I walk on tiptoes, the intruder won't hear me, that he or she won't know that I am here.\n\nUpstairs, I see a figure hunkered down beneath the sloped ceiling and my breath leaves me. It's hidden to the side of the mattress, trying to hold still and yet moving in a gentle rhythm.\n\nWhat I see is a man on bent knee, crouched down, waiting to lunge at me as I reach the top of the staircase.\n\nI gasp aloud, attempting to brace for impact. But instead I lose balance, slipping backward on the top step and sliding downward the eight-or nine-inch rise to the step below. I catch myself there, gripping tightly to the stairwell banister before I plunge down an entire flight of stairs, head over heels over head. Breaking my neck.\n\nMy heart pounds hard.\n\nI cling to the banister and realize that no one has lunged at me.\n\nAnd this time, when I look again, there's no one there.\n\nIt's just the shadow of a tree streaming in through an open window. The leaves are hair, the branches arms and legs. The gentle rhythm, the movement of wind. No one is there.\n\nI turn to make my way to the bathroom. It's a small room, but as I come to it, I take note: the door isn't pressed flush against the wall as it should be. Behind the open door, there is enough space for a body to hide.\n\nI have to muster every ounce of courage I have to go on. It isn't easy. My feet don't want to move, but they do. It's slow, deliberate.\n\nWhen I reach the bathroom door, I don't step inside. I don't look behind the door.\n\nRather my movements are sudden and abrupt, an impulse. I kick the door as hard as I can, where it ricochets off the wall, the rubber stopper running headfirst into the baseboard, not bumping into a person first. Because there's nobody there to slow it down. There's nobody there at all.\n\nAs I make my way inside the bathroom, I find the shower curtain pulled tight, stretched from wall to wall. It billows slightly. Heat spews from a nearby vent, though that's not the reason for the movement. Instead what I envision is a figure standing on the other side of the curtain, the breath from his or her lungs making the curtain move.\n\nSomeone is there, hiding behind the shower curtain.\n\nI tread delicately. On tiptoes. Two steps, and then three.\n\nI reach out a hand, aware that the blood throughout my entire body has stopped flowing. That I'm holding my breath. That my heart has ceased beating.\n\nI feel the cotton of the shower curtain in my shaking hand, the plastic of its liner. I grab a fistful of it and pull hard, finding myself face-to-face with the white tiles of the shower wall.\n\nThere's no one there. It's only me.\n\nThe carriage home is empty. Whoever was here has gone for now.\n\nI do only one thing then, and that's check the fire safe box where I keep my money, to be sure someone hasn't swiped every last penny from me. Because why else would someone break into the carriage home except to steal from me? I keep the box in the closet these days, hidden in the corner beneath the hem of a long winter coat where, God willing, no one will ever find it. I open the closet door, drop to my knees and gather the box in my hands. The box is locked. When I slip the key inside, I find every dollar accounted for. Whoever was here didn't steal money from me.\n\nI try not to let my imagination get the best of me, but to force logic to prevail. I tell myself that I never closed the shades in the first place. That I only thought about doing it, but never did. I think long and hard, trying to remember the smooth, woven feel of the white roller shade in my hand as I drew it southward and let go, watching it hold.\n\nDid that happen, or did I only imagine it did?\n\nOr maybe whatever springlike mechanism that makes the shades open failed to keep them closed. The ratchet and pin that hold them in place didn't work. Simple human error or mechanical failure.\n\nOr maybe someone _was_ there, lifting the roller shades one by one so that when I returned, they could see me. I tell myself no. That the front door was locked. And that, as far as I know, only one person but me has a key. My landlord.\n\nI step from the closet and make my way to a nearby window where I stare out and toward Ms. Geissler's home. The room turns warm all of a sudden. Beneath my arms, I sweat.\n\nThere's no one there, no one that I can see.\n\nAnd yet, as it was last night, there's a light on in the third-story window of the greystone home. The window shades are lowered, but not pulled all the way down. They don't lie flush against the window sill. There's a gap. Albeit a small one, only a couple of inches at best.\n\nBut still, a gap.\n\nAnd as I stare at that gap for half the night, sometime around midnight I see a shadow pass by. Just a shadow, but nothing more.\neden\n\nJuly 2, _1997  \n_ Egg Harbor\n\nBe still my beating heart, it worked! We're going to have a baby!\n\nOne single cycle of IVF and, as I sat on the toilet today after Aaron had gone off to work, the all-familiar pregnancy test cradled between my fingers, I spied not one single line this time, but two. Two! Two pink lines running parallel on the display screen.\n\nMy heart hammered quickly inside my chest. It was all I could do not to scream.\n\nAnd still I had my doubts\u2014after months of seeing only one line, it was easy to convince myself that I was imagining the second one there, that I had quite simply fashioned it in my mind. The one line was bright pink like bubble gum, the same dependable line that greeted me each month, bringing stinging tears to my eyes.\n\nBut the other, this new line, was a light pink, the lightest of light, the mere suggestion of pink, a whisper that something might be there.\n\nI pray that it's not a deception of my mind.\n\nI went to the market wearing mismatching shoes. I drove above the speed limit with the window open, though outside it poured down rain. I ran into the store without an umbrella, saturating my hair. If anyone noticed my shoes, they didn't point it out.\n\nI purchased three additional pregnancy tests of assorted brands in case one had a tendency toward being inaccurate. I took them home and urinated on them all, every last one of them, and in the end, there were six lines.\n\nThree additional pregnancy tests.\n\nSix pink lines.\n\nAaron and I are going to have a baby.\n\nJuly 5, 1997 _  \n_Egg Harbor\n\nFor days we've been living in a constant state of euphoria.\n\nI walk around the home, floating on air. I dream up baby names for boys and girls. I go to the hardware store and get samples of paint for the nursery room walls.\n\nAt home alone, I find myself dancing. Spinning in graceful circles around the living room floors. In all my life, I've never danced before. But I can't help myself. I can't stop my feet from swaying, my arms wrapped around myself, holding on to the life within. Dancing with my unborn child. I find an old record and set it on the turntable. I carefully place the needle on it, and move in tune to the music as Gladys Knight sings a song for me.\n\nThe day I discovered the positive pregnancy tests, I phoned Aaron at work to deliver the news. He was euphoric in a way I'd rarely seen him before. He left work at once and came home earlier than ever before, pulling his car in to the drive minutes before eight o'clock.\n\nHe brought me ice cream in bed; he fed it to me with a bent-out-of-shape spoon. He lay in bed beside me and rubbed my back. He massaged my feet. He stroked my hair. He told me how amazing I was, how gorgeous, and how already I had that beautiful pregnancy glow.\n\nHe stared at me then, just stared, and inside my heart began to cantor, a kaleidoscope of butterflies flitting inside me. I knew what would come next and it was then that my body began to want him, to need him like it hadn't for so long before. I soughed at his touch, my skin breaking out in gooseflesh as he ran a hand across my arm, lacing his fingers through mine. As he stared, he said again that if our baby girl looked anything like me, that she would be the prettiest thing around. And then he tucked a strand of hair behind my ear and I knew that in that moment, I was the most beautiful girl in the world to him.\n\n_Our baby girl._\n\nHe held me tightly and kissed me like he hadn't in months, slowly and deeply at first, growing ravenous, a starved man who hadn't been fed in years, and it was then that I realized I too was empty and famished.\n\nMy breath quickened as he slid a steady hand up the skirt of my nightgown.\n\n\"You think it's okay?\" I gasped as Aaron withdrew my underpants and set them aside, though there was nothing more that I wanted in that moment than a fresh start for Aaron and me and our baby, to be able to erase all the animosity in a single moment, with a single deed.\n\n\"You think it's safe?\" I begged, and Aaron assured me that everything was okay, and, as we moved together there on the bed, I believed him. For the first time in a long time, I believed him.\n\nJuly 14, 1997 _  \n_Egg Harbor\n\nAn ultrasound with Dr. Landry confirmed the pregnancy, though there was no need for Dr. Landry's attestation because I, for one, already knew that it was true, that the manifold of pregnancy tests didn't lie. The battle with morning sickness had begun already, a misnomer if I'd ever heard one for it was morning, noon and night sickness. Not once did I complain, but rather welcomed the nausea and the fatigue as a gift.\n\nDr. Landry told Aaron and me that our tiny embryo is currently measuring one-half of a centimeter from crown to rump. As I lay on the examination table, feet in stirrups, for once not put off by the wand inside me, the complete invasion of privacy that I've come to accept as par for the course, Dr. Landry pointed out the gestational sac and the yolk sac, but I couldn't take my eyes off that pint-size nub that would one day be a baby.\n\nAaron held my hand the entire time. He stroked my hair. He kissed my lips when the image appeared, dark and grainy and impossible to see were it not for Dr. Landry's informative voice and thin finger telling us what was the gestational sac and what was the yolk sac, and where our baby was growing, and then, once I found it, the embryo\u2014a half centimeter long with paddle-like arms and legs and webbing between its toes and fingers, none of which I could see for myself though Dr. Landry told us were there\u2014the one thing in the world I loved more than anything else, I couldn't divert my eyes.\n\nThere was a heartbeat. We couldn't hear it yet, but we could _see_ it. It was there, the movements of it on the ultrasound screen. Our baby had a heart and a heartbeat, and blood that coursed through his or her tiny body. Its heart had chambers\u2014four of them Dr. Landry said!\u2014and beat like a racehorse, a heartbeat that easily trumped mine, though it too was going at a steady gallop.\n\nI'm six weeks along. And we have a due date now.\n\nBy May, Aaron and I will finally have a baby. We'll be parents!\n\nHow will I possibly be able to wait that long to hold my baby in my arms?\n\nJuly 16, 1997 _  \n_Egg Harbor\n\nI told my stepmother about the baby today. I didn't mean to; it just happened. We were on the phone when she asked\u2014as she had so many times in the past\u2014\"How much longer are we going to have to wait for you and Aaron to have a baby?\" __ and it wasn't so much that I told her, because I didn't, but it was the lack of a response that gave it away, the silence, because I was too busy beaming behind the handset, trying to no avail to manufacture a lie.\n\nIf Nora could have seen me, she would have noticed the way my skin turned pink; she, like Aaron, would have seen the way I glowed. She would have seen me run a delicate hand across the cotton of my blouse\u2014a link to the life inside\u2014and triumphantly smile.\n\nShe said nothing at first, nothing in response to my nothing.\n\n\"When were you thinking you'd tell us?\" she asked then with the slightest hint of malice\u2014Nora, of course, needs to be the first to know everything\u2014followed immediately by \"Does Aaron's mother already know?\" and there was jealousy and skepticism in her voice long before she offered her congratulations and said how happy she is for Aaron and me.\n\nI called Aaron's mother next before Nora had a chance to call for herself, boasting that she knew a whole thirty seconds before Aaron's mother did.\n\nIt was like a wildfire then, that instant burst of pregnancy news that caught quickly, spreading through the family from phone call to phone call like a raging inferno. By the end of the day, nearly everyone would know our news.\n\nMiranda arrived as Aaron's mother and I were saying goodbye, and catching a glimpse of my hand still situated on the cotton of my blouse, she said to me, \"It's about goddamn time, Eden.\"\n\nAnd then she hugged me, a quick, careless hug, sending her boys into the backyard to play alone so she could lie on my sofa and rest. Little Carter didn't want to go; he, himself, was still a baby, and so she picked him up and plopped him in Jack's arms and said again to _go_ and we stood there, watching them walk away, listening as Carter cried _._ She was massive again, still months away from giving birth to baby number four, and the evidence of it was everywhere: in her tired eyes, her unwashed hair, her inflated legs.\n\nPregnancy did not suit Miranda well.\n\nHer maternity shirts no longer fit correctly, leaving a fraction of her stomach exposed, ashy skin drawn tightly around her baby, a black, vertical line etched on her body from belly button down. Miranda herself didn't have a pregnancy glow, but rather was covered in blotchy brown spots all over her skin; the hormones were not working in her favor.\n\n\"Just wait until you're as fat as me,\" she said, seeing the way I watched her drop onto the sofa, a giraffe making an ungainly attempt to sit.\n\n\"Well I have news too,\" she said then, as if she couldn't stand me being happy, as if she couldn't take a back seat to my glad tidings for once. \"We're going to have a girl!\" she screeched, clapping her own hands, going on to say how\u2014though Joe didn't know it yet\u2014she'd had a peek at her medical file when the obstetrician was out of the room during her last appointment, and there, in the margins of the paperwork, saw the Venus symbol written with black ink.\n\n\"Finally,\" she said, frowning out the window at her three boys, fifty-pound Jack lugging twenty-pound Carter around, Carter who still cried. \"After everything I've been through,\" she said, and I wanted to be happy for her, I really did.\n\nBut I couldn't bring myself to be.\n\nShe didn't deserve another baby any more than a murderer deserves clemency.\n\nI was grateful when, an hour later, Paul wet himself and they had to leave.\n\nAaron had wanted to keep the news of our pregnancy a secret for a while longer, but I couldn't help myself. I wanted to shout it from the rooftops, to let everyone in the whole entire universe know that I was going to be a mother. \"Why wait?\" __ I asked later that afternoon as he prepared for work. I frowned at him, feeling punctured that he would want to keep our baby a secret. We'd spent a year trying to achieve this, watched our lives and our marriage flounder to make a baby, drained our savings and accrued mass amounts of debt on our credit cards.\n\nAnd yet I couldn't be happier. I couldn't be more thrilled.\n\nThis was the one thing that I wanted more than anything. More than _anything_.\n\nI wanted everyone to know about it.\n\n\"Just in case,\" Aaron replied when I asked why we should keep our baby a secret, why we should wait to share the news.\n\n\"In case what?\" I asked, provoking him, but he wouldn't say the words out loud. He was being cautiously optimistic, I knew, but what I wanted was for him to be jubilant like me. He stood before me in the kitchen, slipping his feet into a pair of new shoes, waterproof, slip-proof black loafers that cost us an arm and a leg. But none of that mattered now, not trivial things like the cost of groceries, the cost of shoes.\n\nWe were going to have a baby.\n\nHe stood and came to me, wrapping his arms around the small of my back, and I breathed him in, the scent of his aftershave and soap because Aaron, of course, didn't wear cologne. His hands were rough from years of hard work, the scrubbing of dishes, the scalding sauces that bubbled over onto his hands, burning them. The many near misses with a utility knife. The gashes and lacerations, healed now but always there. Aaron's hands were rough and worn, but also the softest things in the world to me as they slipped under the hemline of my blouse and stroked my bare skin.\n\nHe wouldn't say the words out loud, but he didn't have to.\n\nI knew exactly what he was thinking.\n\n\"We saw our baby,\" I told him, whispering the words into his ear. \"We saw the heartbeat. Everything is fine.\"\njessie\n\nI'm out the door early, hurrying to the side of the carriage home to collect my bike, but when I arrive, I see that she's gone. That she's not there. That the spot where I left her last night is completely empty.\n\nThere's a moment of panic.\n\nSomeone has stolen Old Faithful from me.\n\nMy heart picks up speed, my face warming with frustration and anger and fear. I look up and down the alleyway as my heart sinks. For a minute, tears well in my eyes. I could cry.\n\nBut then I remember leaving Old Faithful tethered to the bike rack outside the Art Institute. No one has taken her from me. I left her there.\n\nI take the Brown Line out to Albany Park, getting off at Kimball. From there it's a walk to Mom's and my old home, a classic Chicago-style bungalow that's boxy and brick with a low-pitched roof on a street where every single home is a replica of the next. The desperation has gotten under my skin now, a do-or-die need to find my birth certificate, to find my social security card, to figure out who the hell I am. I need to make a final sweep of the home to see if there's anything there, anything I may have missed. Because the estate sale will kick off soon, and then it will be too late. Everything that was once mine will be gone.\n\nI've only been gone a couple of days. But as I make my way down the sidewalk, I feel homesick. I miss Mom more than ever. I miss my home. The sight of the for-sale sign plunged into the green grass makes my stomach churn, my Realtor's pretty face smeared across the corner of it. I'd picked her, this Realtor, because I saw her face and name on a similar sign somewhere down the street. There was a number to call and so I called it. And like that, the house is on the market and soon, any evidence of my time with Mom will be gone.\n\nThe house looks different than it did before. The only thing still here are the ghosts we've left behind. Aside from our house, the rest of the block looks annoyingly the same, as if no one noticed that I'd left or that Mom died, which most likely they didn't. The only person I see outside is our neighbor Mr. Henderson from next door. There he stands on his own front porch, thinning hair standing vertical, a cigar in hand. Smoke billowing around his head. Mr. Henderson wears corduroy pants, slippers, a fisherman cardigan. Though as far as I know, he doesn't fish. Instead he teaches English lit at a local college and is pretentious as all get-out. Mr. Henderson couldn't be bothered to help after Mom's cancer spread to the bones, leaving her far more susceptible to fracture. She fell one morning when I was at school, shattering a hip, lying there on her back, calling out an open window for help.\n\nHe heard her cries as he sat there in his own front room, sucking away on his cigar. No doubt he heard her cries, though later, as the ambulance carried her away and he stood watching from his porch steps\u2014merely a snoop and not a Samaritan\u2014he claimed he did not.\n\nI pay extra attention to the sidewalk as I walk along, taking care not to step on the cracks. Not that it matters because Mom wouldn't feel it anyway if my footfalls broke her back. _Step on a crack, break your mother's back._\n\nI cross the street, refusing to say hello to Mr. Henderson, refusing to meet his curious eye. I dig into my bag for the keys, climbing up the stairs and to the front door.\n\nThis neighborhood has been around for near forever. Most of the homes are circa 1920-something, during some sort of housing boom when thousands of bungalows sprung up overnight, fulfilling dreams of homeownership for that exploding middle class. Because the homes were practical and affordable. And because there were a ton of them. Up and down the street, all I see is nothing but trees and brick, trees and brick. Trees and brick as far as the eye can see. I have no doubt Mom chose Albany Park to live because it's relatively affordable, a good place to raise kids. Money was a luxury Mom and I didn't have. Not that I can say I grew up poor, because I didn't. But Mom was frugal and we weren't rich.\n\nWe planned a big dinner out for when the cancer was finally in remission for the second time around. Gibsons Steakhouse. Mom was going to buy a new dress to wear because she never spent money on herself. Any time there was a little extra to spare, she spent it on me.\n\nNeedless to say, Gibsons Steakhouse never happened.\n\nThe day Mom found out the cancer was back, she was sitting outside on the front stoop when I got home from school. She'd been to the doctor for back pain, the kind that no amount of ibuprofen could fix. Pain she hadn't told me about until that afternoon. She thought it was a herniated disc, back strain, sciatica. Effects of the job.\n\nAs it turned out, it was the breast cancer, back for vengeance. Metastasized to the bones, the lungs.\n\nShe told me to sit. She held my hand, caressed each finger one at a time while I committed to memory the length and shape of her fingers, the asymmetry of the knucklebones, the blue rivers of veins that swept across the thinning skin.\n\nShe said to me that day on the stoop, _Jessie, I'm dying. I'm going to die._\n\nI cried. But she said it was all right. She wasn't afraid to die. She was stoic. _When?_ I asked, like some stupid child. Like Mom had any way of knowing exactly when it would happen.\n\nWhat she said was _Sooner or later we all die, Jessie_. _It's only a matter of time. And this is mine_.\n\nI unlock the door and step inside. I'm inundated with the smell almost immediately. The smell of Mom. Her hand lotion, Crabtree & Evelyn's Summer Hill. It nearly knocks me from my feet. It's diffused through the rooms and if I didn't know any better, I'd think that Mom was still here with me. Heart still ticking, not yet dead. I hear that death rattle, the saliva pooling there in the back of her throat. The nurses' gentle footfalls, close enough to touch. As if they're still there, still walking in orbits around me. Lathering lotion on Mom's hands and feet, turning her every few hours to keep bedsores from forming on her skin.\n\nThe smell of the lotion is overpowering. It binds to the millions of tiny little hairs in my nose, bringing me to my knees every time I breathe. _Mom_.\n\nAnd I find that I'm looking for her, half-certain that when I turn she'll be standing there in the arched doorway of the kitchen, sagging body leaning against the doorway because she doesn't have the energy to hold it upright anymore, a soft cotton hat covering her bald head. Asking how I got along at school today in that way that she does, teeth gritted through the pain that managed to breeze in and past the narcotics sometimes.\n\n_How'd you get along at school today, Jessie?_\n\nBut it's not real, I remind myself.\n\nThe nurses are not here.\n\nMom is dead.\n\nAnd only then am I aware of the silence. Of the earsplitting silence that now worms its way through the cracks of our home.\n\nI don't know where to begin. I searched the entire home already, but I look anyway, starting in my bedroom, planning to work my way down in search of the social security card. I pluck desk and dresser drawers from their tracks. I dig beneath clothes I've intentionally left in the dresser drawers, those I no longer need. I lift rugs from the floor and check beneath. I canvass my closet. No luck.\n\nI make my way to Mom's bedroom, where I see that the liquidator has begun to tag items for sale. Mom's clothes now hang from a rolling rack beside her bed. I run my hands over a knit cardigan, her favorite. If I'd had my wits about me at the time, I would have had Mom cremated in the cardigan so she could spend all of eternity in it. But instead she wore a hospital gown, white and wrinkled with snowflakes, a single tie on the otherwise open back. The funeral home gathered her body from the hospital within hours after she died. But there was a mandatory waiting period before the cremation could begin. Twenty-four hours, in case I changed my mind.\n\nI spent those twenty-four hours parked outside the funeral home's doors, sitting on the curb because they didn't have a bench. And because I couldn't bring myself to go home without Mom.\n\nThe liquidator will take some 40 percent of all sales, which is fine by me. Anything so that I don't have to be involved in the process, so that I don't have to watch our possessions walk out the door in the arms of someone new.\n\nI pull open the closet door to reveal a large walk-in. It's empty now, all of Mom's clothes moved to the rack beside the bed. Only hooks and a mirror remain\u2014a silver-framed oval mirror that Mom and I used to make silly faces in front of when I was a girl. I'd stand on a chair so that I could see inside, and there we'd stare at our reflections side by side in the glass.\n\nThe mirror hangs on the closet wall, an oversight only, for it won't be long before the liquidator pulls that too from the wall and sticks a price tag on it, snatching memories right along with it, memories of my crossed eyeballs, Mom's fish face.\n\nI run a hand along the glass, remembering how sometimes we didn't make silly faces at all. How sometimes I'd just sit on the floor beside her feet and watch as Mom stared at herself, her dark hair and eyes so unlike the dishwater-blond hair that sat on my head, the tufts of eyebrow hair that stuck straight up, same as they do now. Unlike me, Mom didn't have dimples. My dimples are much more than simple holes in the cheeks, but more like deep comma-shaped gorges. I didn't get those from Mom. There isn't one feature on my face that came from her.\n\nEven as a kid, I saw the way Mom looked when she stared at her reflection in the mirror. She looked sad. I wondered what she saw. For some reason I don't think it was the same pretty face that I saw.\n\nI'm about to leave when I spot something out of the corner of my eye, something I've never noticed in Mom's closet before. Something that would have otherwise been hidden behind the hems of clothes, except that now there are no clothes to taint the view.\n\nI have to look twice to be sure that it's there, that I'm not only imagining it's there. What it is is black, metal, covered in louvers. A door. A boxy little door that hovers less than a foot above the hardwood floors.\n\nI drop to my hands and knees and pull on the door's knob, finding a crawl space on the other side. A _crawl space_. I never knew we had a crawl space before.\n\nThe space is dark and dingy, the ceiling low. The floor is dirt, covered only by a thick sheet of plastic. I can't believe I never found this place before. How many times did I dig my way through Mom's closet for clues as to who my father could be? But as it so happened, I never dug far enough. Instead I gave up when I got to the clothes, taking for granted that there was nothing on the other side but a wall.\n\nOnly one time did Mom bring my father up all on her own, without my begging. I was twelve years old. Mom had had a glass of wine before bed. She said to me that night, seconds before she fell asleep, head draped over the rock-hard sofa arm, _A long time ago, I did something I'm not proud of, Jessie. Something that shames me. And that's how I got you._\n\nThe next thing I heard was the sound of her half-drunk snore, but by morning I couldn't bring myself to ask what she'd meant by it.\n\nI reach inside the crawl space and drag something out. What it is, I don't know. Not until I get it into the closet's light do I see that it's a plastic storage bin, and the adrenaline kicks in at the prospect of what I might find inside. My social security card, for one, but more likely, something having to do with my father, which suddenly, in this moment, takes precedence. Something Mom kept tucked away so that I wouldn't find it.\n\nI tear the lid off, finding photo albums inside. I find myself feeling hopeful, wondering what I'll find in them. Photos of Mom, photos of my father, photos of Mom and her own mom and dad.\n\nBut of course not. Instead it's me. All me.\n\nI set the album aside to take back to the carriage home with me.\n\nI crawl toward the crawl space, feeling blindly inside for another box. I can't reach far enough in to grab it, and so I have to crawl in through the door. Inside, the space is only about thirty-six-inches tall. I'm not fully in before claustrophobia settles in. The dirty floors and wooden beams close in around me. The darkness is smothering. The only light comes from behind. I find another storage bin and drag it out backward, through the access panel and onto the closet floor, grateful for a little elbow room.\n\nI open the lid and have a look, hoping that this is the mother lode I've been in search of. The answer to all the questions I have. But it's not. It's nothing, just a bunch of inconsequential items in a plastic storage bin, which makes me realize this isn't a secret crawl space at all, but just a _crawl space_. For storage. For stuff Mom had no other place to put.\n\nShe didn't intentionally keep this a secret from me. I just never knew it was here.\n\nI sigh, feeling uncomfortable and glum. I rise to my feet, stretching my hands above my head, arching my back. But my movements are quick and careless. The blood flees my brain as I stand up, leaving me light-headed and dizzy. All of these nights without sleep are taking their toll on me. I reach for the wall to steady myself, crashing into the mirror as I do. I watch on helplessly as the mirror loses its hold on the wall and I can't catch it in time. I'm too slow to stop it from falling.\n\nIt slips from its nail and slams to the ground, scratching the wall as it does, leaving a four-foot scrape in the paint. The entire mirror shatters before my eyes. Broken glass spreads like spiderwebs, chunks falling to the hardwood floors. And all I can think about is bad luck. Seven entire years of bad luck that await me now.\n\nI curse out loud, wondering if there's any hope of salvaging the mirror. I start to gather the biggest chunks in my hand, careful that I don't step on the tiny shards of glass.\n\nMy eyes are so caught up in what's happening on the floor that at first I don't see the small compartment dug into the wall. A little recess carved there into the drywall, hidden behind where the mirror should go. A hole that's been fitted with a sturdy box.\n\nAnd I think for a moment that my eyes deceive me. That I'm only imagining the compartment is there. Because why in the world would there be a secret storage compartment on Mom's closet wall? I rub at my eyes, certain it will disappear as I do. But sure enough, it's still here.\n\nFor at least twenty seconds I stare at that box without moving.\n\nMom had a stash of personal stuff she kept hidden from me.\n\nI think of all the times Mom and I looked together in the mirror when I was a girl. All I ever saw was a mirror\u2014our own silly expressions looking back at us through the glass\u2014but for Mom it was a portal to her private world, a gateway to the things she didn't want me to see.\n\nIt feels an enormous invasion of privacy for me to snoop but I can't help myself. I reach my hand inside Mom's secret box. There's only one item there. It's a scrap of glossy white paper pressed into the corner of the box. My chest clenches. I hold my breath.\n\nThis could be something.\n\nOr, like the plastic storage bins hidden in the crawl space, this could be nothing.\n\nI have to use a fingernail to emancipate the scrap. When I do, I turn it over in my hands to see. It's a photograph that some part of my memory reminds me I've seen before.\n\nBut with the memory of the photograph comes the memory of Mom's face. Openmouthed and afraid. She knew I'd seen it. But what happened next has been wiped clean from my brain's hard drive. Either that or entombed beneath a gazillion other memories, harder than others to dig up.\n\nMom hid this photograph from me.\n\nIt's the kind of photograph that looks a little dated, a little old. Not crazy old, like archaic. But older than me. The colors are faded, the blues a little less blue, and the greens a little less green than they used to be. It's a picture of a lake. A long seashore of blue. Tan sand, darker where the water hits it. White ripples of waves. Evergreen trees line the edges of the lake. There is a pier suspended over the water, one that looks unsound, unsafe. Like at any moment it could sink into the lake and get carried away with the waves. If I squint my eyes up tight, there's a boat out there on the water. A sailboat, just a simple sloop with a single white mast. That's what I see.\n\nMom knew a whole lot about sailboats, which she relayed to me when we used to walk past DuSable Harbor on occasion, hand in hand. _See that one over there, Jessie?_ she'd ask, slipping her hand away long enough to point at it. I'd pretend to look. Pretend to look because I didn't really care, her words falling on deaf ears. _That's a cutter_ , __ or, _that's a catamaran_ , she'd tell me. She had a book on them, a heavy coffee-table book called _Sailing_. Though as far as I knew, Mom had never once stepped foot on a sailboat in her entire life. At least not since I've been alive. I forget sometimes that Mom had a life that preceded mine.\n\nBut the lake and the sailboat are only an afterthought to the image I see, because there's also a man in the photograph, one with brown hair and a large stature. He's tall and husky with thick wrists exposed by a flannel shirt that's rolled up to the elbows. There's a watch on a right wrist, a hat in his hand. He stands with his back to the camera, blurred at the edges because he wasn't standing still when the shutter button was pressed. He's not centered on the photo paper, as if he was moving away when the picture was snapped.\n\nThe photograph wasn't meant to be of him.\n\nThe central object is the sailboat. The picture is of the boat. And the man only got in the way. By today's terms, a photobomb.\n\nThe man stands with his hands on his hips, left knee bent a bit. His head is pitched to the right. He has blue jeans on\u2014saggy ones, not formfitting. The ends are fraying, turning white. One of his gym shoes is untied. Strands of hair move in the wind.\n\nI wish that he would turn and look at me, so that I could see his eyes, the shape of his nose. Whether we look anything alike.\n\nIs this man my father?\n\nWhy did Mom hide this photo from me?\n\nWhy did she not want me to know anything about this lake or this boat or this man?\n\nI think of all those times I sat cross-legged on the closet floor beside her feet, watching as she stared sullenly at her own reflection in the mirror. What I thought was that she didn't like what she saw. A modest, unpretentious face, a bit earthy with dark hair and dark eyes.\n\nAnd then, years later when the cancer settled in, that same face became cadaverous. She lost more weight than she had to spare, face thinning, cheekbones hollowed out\u2014an image she despised. That's what I thought she was looking at when she stared in the mirror.\n\nBut now I think that maybe she wasn't looking at herself as much as she was looking through the glass, reflecting on the life she left behind, the one she kept hidden from me behind that mirror.\neden\n\nJuly 21, _1997  \n_ Egg Harbor\n\nWhat Aaron told the emergency room physician was that there was blood, \"Some blood,\" he said, \"spotting,\" which to me equated to a teaspoon or two, enough to dirty a single pad, but the amount of blood I saw was measured in liters and gallons.\n\nIt came gushing out of me, a deluge of blood pouring down from the sky, rivers and streams overflowing their banks, dousing the earth, sweeping homes from their foundations. Everywhere I looked there was blood.\n\nThe day was hot and I wore shorts, and the blood, it saturated my underpants first before snaking down the inside of my bare leg, a thin, red zigzag emblazoned against my fair white skin.\n\n\"I have my period,\" I told Aaron as we were there in the backyard\u2014he staring openmouthed at me, on his knees, installing chicken wire around the flower bed so the deer couldn't poach from us again, making off with our beautiful hollyhock blooms.\n\nIn retrospect there were warning signs, maybe: the suggestion of a cramp, some lower back pain, tokens of pregnancy as well as menstruation and miscarriage. The fact that the nausea had abated during the last twenty-four hours was, to me, a welcome blessing and not once a sign of catastrophe.\n\n\"You're pregnant,\" Aaron said lightly, rising to his feet and coming to me, but I couldn't process his words, couldn't make sense of what was happening. It was my period again, come to me like it does every month without fail. There was dirt on his forehead, and his hands were red, etched with the impression of chicken wire. \"You don't have your period, Eden,\" he said, dropping the wire cutters to the ground and taking my hands into his.\n\nHe wiped the blood from my leg with his own sweaty T-shirt.\n\nIn the car I sat on a kitchen towel.\n\nWe didn't speak on the way to the emergency room.\n\nHe told the attending physician that there was _some blood_ , that I was _spotting_.\n\nAn ultrasound was performed. This time, there was no heartbeat.\n\nThe baby's heartbeat had disappeared.\n\nBoth Aaron's and the doctor's eyes wandered to mine, though I wouldn't meet theirs, too busy staring at the black gestational sac on the monitor, at the stillness of the screen, the lack of movement. The absence of sound.\n\nAaron reached out a hand to mine but I couldn't feel it. I only saw that it was there.\n\nI was insensible. I was stone-cold.\n\n\"What now?\" Aaron asked the physician who'd been sent down from obstetrics to perform the ultrasound, a woman who would soon return to labor and delivery to deliver someone else's healthy newborn.\n\n\"We'll perform a dilation and curettage,\" she said, \"to eliminate any remaining tissue from the womb.\"\n\nTissue. As if only a few hours ago that tissue hadn't been a child.\n\nIn that moment, I couldn't speak. I couldn't even bring myself to cry.\n\nThey put me to sleep for the procedure.\n\nI prayed I'd never wake up again.\njessie\n\nI leave Albany Park, taking the train into the Loop, where I make my way to the Art Institute to collect my bike. From there, I pedal to the coffee bar on Dearborn, the one where the man at the garden had purchased his coffee, the name I'd read on the paper sleeve of his coffee cup. People are religious about their coffee and their routines, and so it seems logical enough to think that if he was here yesterday, he'll come again today. I need to find him. I need to ask him why he was in the garden\u2014 _Mom's and my_ special garden\u2014sitting there, reading her obituary. I need to hear why Mom's obituary made him sad. How does he know Mom?\n\nI bring the photograph of the man with me. I carry it in the front pocket of my bag.\n\nThe man who I think might be my father.\n\nAt random stoplights I slip my hand into the pocket of my bag and pull it out. I try to spot some nicety I haven't yet seen, some minor detail in the image I've managed to overlook, like the swollen clouds or the gangly-looking bird that perches on a rock at the water's edge.\n\nThe sleeves of the man's flannel shirt are shoved to his elbows in the picture. A raised red line bridges a lower arm. Scar tissue, I think, or maybe just an anomaly in the photograph, a streak of light or a reflection. I wonder what any of it means. If it means anything. If the clouds or the birds or the scar can provide details about the man or the land on which he stands.\n\nWhere was this picture taken?\n\nAnd more importantly, who is he?\n\nI search in vain for the smoking gun to tell me who he is. How I know him. What this man has to do with me. I wonder if the answer is there, staring me in the face, and I simply can't see it.\n\nAnd then the light turns green and I carefully shove the picture back into my bag and pedal on toward the coffee shop.\n\nWhen I arrive, I press in through the door, past people who are coming out. The coffee bar is eclectic, cluttered with mismatching tables and chairs. There are stacks of magazines and books.\n\nBetween the grinding and gurgling of the espresso machine, the roar of people talking, the coffee shop is loud. I order a coffee and carry it to the kiosk to douse it with sugar. A blue velvet sofa lines a wall, and I help myself to it, sinking into the wilted center, watching as caffeine-deprived customers come and go. The line grows long enough that the last person stands in the doorway because he doesn't clear the doorframe. Instead he props it open with his body, letting the fall air in. Napkins blow from a table and litter the floor.\n\nAs I sit there waiting for the man from the garden to magically appear, I pull the photograph from my pocket one more time, taking in the man's stature, the color of his hair. Imagining his eyes. In the image, they're looking out toward the sailboat, away from the camera lens, and so I can't see them. I can't see what they look like, but I can imagine.\n\nThey're blue like mine, and he has dimples too.\n\nI sip from my coffee, place the photograph back in my bag.\n\nMy mind drifts and I find myself thinking about the other Jessica Sloane. The one who is not me. And I know with a sudden translucence that I am not Jessica Sloane, but that I'm somebody else. That Jessica Sloane died when she was three and for whatever reason, Mom stole her social security number and gave it to me. This is no longer a hypothetical. I know.\n\nBut there are ways of finding out who you are, aside from a birth certificate, social security number or name. Because if I'm not Jessica Sloane, then I need to know who I really am. I think of forensic identification, stuff like fingerprints, DNA, handwriting analysis, dentistry. Ways to prove one's identity aside from birth certificates and social security numbers. Everyone in the whole wide world is supposedly unique, like the stripes of a zebra or the spots of a giraffe. Snowflakes. It's near mathematically and scientifically impossible that any two could be the same. Even the creases of our feet are distinct, which is one of the reasons babies' footprints are taken after birth. For identification purposes. Because no two footprints are alike. So hospitals know which baby is which if ever they get separated from their moms or dads. In case the ID bands slip from their ankles or wrists. I stare at my fingerprints, thinking the answer to who I am is sitting there, in all those miniscule lines that make me unique, a single snowflake, one in twenty trillion falling in a snowstorm, drifting aimlessly and alone.\n\nI don't know who I am, but I'm not Jessica Sloane.\n\nIt's hours later when I catch a smidge of orange pass by the storefront window, and I know right away: it's him. It's the orange baseball cap that he wore, slipping it over his hair before he left the garden. He's here, come and gone for coffee and somehow, in a daze, I all but missed him.\n\nI rise too quickly from the blue velvet sofa, spilling a lukewarm coffee, my third of the day, down the front of me, staining my shirt a translucent brown. I don't bother blotting it with napkins before I go scurrying for the door, knocking into a stanchion post along the way. I knock it over with a clang, leaving it on the floor as people stare. \"What's the hurry?\" I hear breathed through the air. \"What's her problem?\" followed by a giggle, a snort.\n\nI press my way out onto the city street, following the pinprick of orange in the distance, a beacon of light as it slaloms this way and that down the street. I run, pushing my way past people walking too slowly, trying desperately to bridge the gap from him to me.\n\nAs I narrow in on him, I reach out and tug on something, my hand bearing down hard. A little boy cries out, and, as they turn to me, I see. A little boy in a superhero costume. The Flash. He's perched on his father's shoulders, making him tall. The costume is red and yellow with a mask that covers his face. It's the type of mask that covers everything, leaving only slits for the mouth, nose and eyes. Like the costume, it's also red and yellow. Not orange, though my mind mutated them for me, mixing the red and yellow, turning them into orange.\n\nOnce again, my eyes have deceived me.\n\nHe isn't the man from the garden after all.\neden\n\nJune 17, 2005   \nChicago\n\nIt's been a couple of hours since it happened, and still I can't get my heart rhythm to slow. I feel off, a dull headache in the back of my neck that simply won't quit, my handwriting like chicken scratch from the shaking hands. Jessie is quiet now, tucked into bed with her lights turned off. I'd read her a story before bed, hoping it might help her forget. Hoping it might replace the photograph she saw with the fun of leading imaginary beasts on a wild rumpus around her bedroom. She was laughing by the time she went to bed, and I can only pray that she dreams tonight of Emile and Bernard, and not of Aaron.\n\nI, however, will dream only of Aaron.\n\nI think I covered my tracks quite well, but I won't ever know. There's no telling what goes on inside a little girl's mind, which details of our lives are committed to memory and which we forget.\n\nFor the first time tonight, past and present collided, and it made me realize one thing: that I have to be more wary of where I hide my things. Jessie is older now and more inquisitive. She's liable to have questions for me that I can't answer because I don't want to answer them. I have to be more careful if I'm going to keep my past from her.\n\nIt's not that I don't love her. It's that I do.\n\nWe'd just finished up dinner when it happened. I was in the kitchen, wiping down the countertops, and she'd disappeared down the hall to, presumably, go play. She was in her room, or so I thought at the time, quiet as a church mouse. That should have been my warning, because for as fiery and high-spirited as she is, Jessie is rarely quiet.\n\nI don't know how much time passed\u2014ten minutes, an hour while I was stupidly relishing in the quiet and didn't once think to check on her\u2014when she appeared there in the doorway to the kitchen with an item in her hand, asking of me, \"Who's this?\"\n\nHer eyes, when I turned to her, were doe-eyed, her hair falling into her forehead like it hadn't seen a brush in weeks. There were dust bunnies clinging to the fabric of her pants and I knew right away that she'd been somewhere she shouldn't have been, on her hands and knees, digging through things.\n\n\"Where'd you find this?\" I asked, taking it from her hand. I heard my voice crack as I said it, and though I couldn't see it, I was certain my face was masked in fear. My voice wasn't angry. It was scared.\n\nJessie had found it under my bed, of course, where she'd been snooping. The photograph had been stashed inside an envelope, inside a box, and under the bed, the kind of thing one didn't just happen to stumble upon. She went searching for it. Or rather she went searching for something and she found it, because up until a few minutes before, she didn't know this photograph existed, the photograph I'd snatched all those years ago in the yard of our cottage, a photograph of our glorious view\u2014the lake with a sailboat out at sea\u2014meant to be only of the lake and the sailboat, though Aaron stepped into the frame just as I took the picture. He'd apologized and later, after the pictures were developed, we'd laughed over it. Aaron thought he'd ruined my photograph, but what he'd done was the opposite of that. He'd made it perfect. He'd made it complete.\n\nUp until a few minutes ago, Jessie didn't know Aaron existed because those Who's my father? questions have only just begun to surface, and so far I've been able to quell them all with the suggestion of milk and cookies or ice cream.\n\n\"Who is it, Mommy?\" she asked again when I didn't respond.\n\n\"Just an old friend,\" I said, trying to settle my jittery voice as I opened a kitchen drawer\u2014the closest thing to me\u2014and slid it inside. I'd find a better hiding spot later after she'd gone to sleep. I could feel my cheeks inflame, my hands start to shake.\n\n\"Are you mad at me?\" Jessie asked then, eyes swelling with tears, mistaking what I was feeling for anger when what it was was sadness and regret and shame.\n\n\"No, baby,\" I said, dropping down to my knees and drawing her into me. \"Mommy could never be mad at you,\" I told her, and then I smiled as widely as I could and grabbed a hold of her hand. \"How about some ice cream before we get ready for bed?\" I proposed, and of course there was no hesitation, no wavering. Jessie screamed an easy yes! while jumping up and down, and so we carried bowls of chocolate ice cream onto the front porch to eat, watching as the sun made its final descent beneath the horizon. I helped her with a bath and we read about the wild rumpus. I tucked her into bed. She asked me to lie with her as she always does these days, and so I curled under the covers beside her, and she pressed her body into mine, a lean arm flung across my chest, pinning me down.\n\nThis was everything I ever wanted and more.\n\nI lay there until her breaths became flat and slow, and then I returned to my own room. There I sat on the bed, clutching the photograph of Aaron in my hand, still trying to catch my breath. This photograph had been hidden beneath the bed for years. I've known it was there, of course, but couldn't bear the idea of looking at it, not until it was forced quite literally into my hand. It was the only keepsake I kept of him, just the one single photograph\u2014not our wedding photographs, not my engagement ring\u2014because in it, he's looking away. He's not looking at me, and so I can't see that love and adoration in his eye.\n\nI can't see the anger.\n\nI stare at the photograph, wondering what Aaron must look like now. Is he graying slowly like me, or is his hair still a chestnut brown? Is he fuller around the middle, or maybe he's more slim? And then I start to wonder if he's eating okay, if he's sleeping okay, if some other woman now spends her nights beside him in bed. My mind gets stuck there, a skipping record. I can't unsee this image, imaginary as it may be, of a woman lying beside Aaron, peacefully asleep\u2014her head tucked into the crook of his arm, his hand on the small of her back\u2014where I used to be.\n\nI won't let myself dwell on the past.\n\nI move quickly, having to get rid of the evidence before Jessie wakes up and goes snooping again. I put the photograph where she'll never find it, and then, when it's done, I tiptoe back into Jessie's room and stand there at the edge of the bed, forcing the past to some locked chamber in my mind, the same spot where that woman's voice is buried, the high-pitched squeal as she chased me down on the street.\n\nGet your hands off my child.\n\nI slip back under the covers beside Jessie so that when she awakes in the morning, she'll never know I was gone. A simple sleight of hand.\njessie\n\nThat night, I climb into bed with my clothes still on. I don't bother changing them. I just want to get into bed, to be in bed. The bed used to be my safe place. But after all these nights not sleeping\u2014eight of them, eight days and nights without sleeping now\u2014the bed is my torture chamber too.\n\nI read once about a man who died because he couldn't sleep. Fatal familial insomnia, it was called. Within twelve months from the time symptoms appeared, he was dead.\n\nI think this is what's happening to me.\n\nIt started with a single bad night of sleep. For whatever reason, his mind wouldn't shut off. Wouldn't let him rest. One night turned into two, and before long he'd gone weeks without a decent night of sleep. _Relaxed wakefulness_ is what it was called, though it was anything but relaxed. He never made it past stage 1 of non-REM sleep, the stage between wakefulness and sleep. He never dreamed. It was a light sleep at best when he was lucky, lasting less than ten minutes at a time, the kind of sleep interrupted by a hypnic jerk, by an overwhelming sense of falling.\n\nI have it worse, I think. Because a light sleep, to me, would be a dream come true.\n\nHe walked the earth in a stupor, asleep but awake. Awake but asleep. He spent his days in a hallucination of sorts, not sure if he was alive or dead. He heard buzzing noises all the time. People calling out his name though no one was there. A voice whispering odd decrees on repeat. _Just do it already. Just jump._ A hand touching his arm and he'd whirl around, agitated and afraid, to find himself alone. The panic attacks were infinite. His brain was on overdrive all the time. There was no way to hit the switch and shut it down.\n\nAs a result, his brain's tasks were all out of whack. His muscles twitched. His heart raced. His blood pressure soared. Coordination was lost. He could no longer function properly. It went on like this until he died.\n\nThe most gruesome part? Though the body goes to pot, the mind does not. Thought processes remain relatively intact. They're clued in completely to their own demise.\n\nThe ill sweat profusely.\n\nThey stop eating, speaking.\n\nThey shrivel to nothing but a glassy-eyed stare, eyes shrunken to mere pinpricks, like mine. And then they die. Because, after those long, agonizing nights lying in bed, failing to truly sleep, fatal familial insomnia is nothing but a death sentence for them. The grim reaper coming to steal their life.\n\nI'm waiting for my time.\n\nI sit up in bed. I don't delude myself into lying down because I know I won't sleep. And so I sit, engulfed in blackness, legs pulled up to my chest. The blanket is kicked to the end of the bed because, though it's cold in the carriage home, I've begun to sweat. The sweat, it gathers under my arms and in my hairline. My palms are damp with it. The soles of my feet. The skin between my fingers and toes.\n\nMy heart beats rapidly.\n\nMy head spins.\n\nI stare into blackness, seeing things that I hope are not there. I go through the motions. The typical night, thinking the morbid thoughts, followed by the grieving ones where I miss Mom so much it hurts. It's a pain in my sternum this time, like heartburn or indigestion. Except that it's grief.\n\nAnd then when I'm done grieving, the self-loathing comes, where I despise myself for all of that which I _would've_ , _could've_ , _should've_ done differently. Said I love you while she could still hear me. Hugged her longer and with more frequency. Run a hand over the dark chocolate fuzz that had started to regrow on her scalp after her last round of chemo was through.\n\nI bullet point them all in my mind. All the things I should've done.\n\nThe silence and the blackness of the room become suddenly suffocating and I feel like I can't breathe. I'm drowning in silence. Being asphyxiated by it.\n\nI turn to my knees and peel the shade back, gazing outside. The world tonight is dark, a carbon gray. Not quite black, but close enough. Little by little, my eyes adapt to it, and though it's dark outside, I can see. Not perfectly, but I can see something. A halo of light from a streetlamp, a half a block away. Orion the hunter, brightening the sky. His shield is aimed at me as he hovers, light-years above the greystone, club hoisted above his head with a dog at his feet. For whatever reason, the light makes me feel less alone and less scared.\n\nAnd then, as the moonlight slips out from behind a cloud, it settles on the greystone. As my eyes adjust to it, the house begins to slowly take shape. My eyes rise up from ground level, grazing over the kitchen's sliding glass door, an enclosed porch, up the home's rear facade, and there they make out an amorphous shape standing in the open window of the third floor. The very same window, which, for the last two nights, radiated light.\n\nExcept that tonight it's dark. There is no light, but rather a pair of eyes.\n\nThe bile in my stomach begins to rise. I feel like I could be sick. I press a hand to my mouth to silence my own scream.\n\nThe moonlight reflects off the eyes, making them glint in the darkness of night. They're undeniable. They're _there_. I'm not just making them up.\n\nBut beyond the eyes I see little else. Just a formless, shadowy shape to let me know that someone is standing in the window, watching me.\n\nI let the shade go and it falls closed.\n\nI grab Mom and her urn from a bedside table and slip to the floor, thinking that I don't want to be here in this carriage home, that I want to leave. That I'd rather be anywhere else in the world but here. But also realizing that I have nowhere to go. I press Mom to me and hold her tight because with her in my arms, I feel less alone. I scoot to a wall and press my back to it, heart beating hard. I try to defuse my fears, to make myself feel better, by telling myself that it's only Ms. Geissler. That it's only Ms. Geissler watching me.\n\nAnd yet it doesn't make me feel better. Because Ms. Geissler is a stranger to me. We've hardly met. I don't know a single thing about her, other than she lied about the squirrels inside her home, but for what reason, I don't know.\n\nMy heart pounds. My hands are moist. They sweat and again I'm sure that I am dying. That the perspiration is a symptom of fatal familial insomnia, which has stolen my sleep from me and is now coming to take my life.\n\nI want to get out of here. I want to leave. And yet I paid nearly everything I have to be here. I can't get out of here, I can't leave. I have nowhere to go.\n\nI pull my knees into my chest. I drop my head to them and close my eyes. I pray to sleep, over and over I say it. _Please just let me sleep. Please just let me sleep. Please just let me sleep._ I beg for morning to come, for the sun to rise higher and higher in the sky, chasing the nighttime away.\n\nFor eight days now it's gone like this. Eight nights.\n\nHow many more days and nights can I go on without sleep?\n\nAnd then I hear something. Just a murmur, faint at first like the sound of a piano playing from some other room. A gentle melody. But, of course, that can't be because there's no piano in the next room, and no one here to play it but me. And I'm not playing a piano.\n\nMy ears stand at attention. My head tips. I listen, and though I want to stay, firmly anchored to the wall where I can see through the darkness to know what's coming for me, I lift my body from the floor, carrying Mom's ashes with me. It's unintentional when I press a single palm down on the ground to hoist myself up. The other clutches tightly to Mom, pressing her to my chest like a newborn baby. I stand to an almost-upright position, bent at the shoulders so I don't hit my head on the low ceiling. And still I do hit my head, crashing into a low-lying wooden beam, so hard that when I press my fingers to it I feel the undeniably sticky texture of blood.\n\nI tiptoe down the steps, one tread at a time, so slowly that it's almost as if I'm not moving at all. As I descend, voices surface. Not just one, but two or three or four. One lead and a host of background singers to accompany the piano. It makes me gasp for breath. My legs become weak, incapacitated; they start to give as I clutch the stair railing for support, squeezing so tightly the muscles of my hands cramp.\n\nI can't go on. I don't want to go on. But I do because I have to. Because there's nothing there, because there's some reasonable explanation for the sound. A car stereo playing outside the carriage home, maybe, the tune getting carried in through an open window.\n\nBut I won't know what it is unless I go see.\n\nI force myself to creep down the steps. I edge across the floorboards, willing myself forward, creeping, one step at a time. Following the sound, which comes from a wall and not the window at all because the window is closed tight.\n\nThe song isn't coming from the stereo of a car parked somewhere outside.\n\nIt's coming from inside the carriage home.\n\nI go after the sound, and it leads me to an old vintage pie safe pressed flush against a wall, a petite bookshelf with a couple of shelves and a door. It's one of the few pieces of furniture that came with the carriage home.\n\nI grab a hold of the knob and pull the door open swiftly, dropping to my knees. As I gaze inside, I find that it's empty, which makes no sense because the song is in there. It's coming from the pie safe. I feel blindly with my hands, moving them up and down the edges of the shelves, feeling for something, though what I don't know.\n\nAnd then a thought comes to me.\n\nWhat if the sound isn't coming from the pie safe? What if it's coming from somewhere behind?\n\nI don't think twice. I shove the pie safe out of the way. It isn't heavy, but it isn't light either. I press a shoulder into it. It takes some jostling as it skids across the floor.\n\nAnd there, on the wall behind where the pie safe was placed, I discover a cast-iron air return grille. One of those wall-mounted vents that leads into the duct system. It's an air return, one that sucks stale air from the room and cycles it back through the home's ductwork, leading, I have to assume, to the floor register upstairs where I heard the undeniable ping the other night. Ping, and then nothing. Ping, and then nothing.\n\nExcept that nothing is getting sucked up in here. Instead it's getting forced out.\n\nAnd it's not air at all, but music. Gladys Knight & the Pips, \"Midnight Train to Georgia.\"\n\nHow can this be?\n\nI press my whole body against the grille to listen to the song. Mom's favorite. One she used to play over and over again until I got sick of it. Until I pouted and told her to turn it off because it was old people music. Those were the words I used. _Old people music._\n\nI'm stricken with the most impossible of thoughts, one that makes the hairs on my arms stand on end.\n\nMom is there. Inside the home's ductwork.\n\nI set the urn down on the floor and, at first, try to jerk the whole thing off the wall with both of my hands. It won't budge. I grip the edges of the grille and pull, but I don't have a good grip on it and it slips easily from my grasp. I tumble backward, falling to the floor. The air return grille is wedged on too tight, held to the wall with four screws, one in each corner. I make an attempt to unscrew each with a bare hand, pinching and twisting the jagged screws until the skin splits, catching a sharp edge of it, one that's been whet over time. My finger starts to drip with blood.\n\nBut the screws don't move. Not even a little bit.\n\nI grit my teeth and pinch and twist harder, but still nothing. They don't budge the slightest bit.\n\nAnd so I wedge a fingernail into the slotted screw and turn. But all that happens is my nail breaks, getting ripped in two, leaving my nails in tatters. I curse out loud from the pain of it before hoisting myself from the floor and hurrying to the kitchen for a knife. I shuffle through a cutlery drawer\u2014tossing forks and spoons out of the way, spilling them one by one to the floor\u2014and find a butter knife.\n\nI run back to the air return. I fall again to my knees.\n\nI stick the knife into the screw head, turning counterclockwise as hard as I can. Bearing down on that knife with my whole body weight.\n\nThis time, it turns.\n\nI spin and I spin that knife, desperate, gasping, as if I might just find Mom inside the air return. Because for this moment that's exactly what I'm thinking. That that's where she is. Inside the air return. I don't know how or why, but she is. She's _there_. I'm just sure of it.\n\nI pluck one screw from the wall and move on to the next one. And the next one. And the next. All four screws tumble to the ground.\n\nThe grille loses its grip on the wall and falls. The sound is clamorous. I shove it out of the way and look inside. It's some sort of stainless steel box set there behind the air return grille, one that changes course about a foot of the way in. I can't see far enough inside to see where it goes and so I reach in a hand, grasping, sweating, but come up empty, thinking that behind that curve there are miles and miles of pipes and tubes which somehow or other lead to Mom. Mom is at the end of those tubes, listening to her music, speaking to me.\n\nI try going in headfirst and then feetfirst. But I don't fit and in time give up, because I don't know what else to do.\n\nI spend the rest of the night lying on the floor beside the air return in the fetal position, listening to Gladys Knight sing to me.\neden\n\nSeptember 26, 2010   \nChicago\n\nWe bought our first computer today, at Jessie's insistence. I'd been saving for some time for it, hoping to surprise her because, as Jessie says, we're the last two people in the world without a computer, which may or may not be true.\n\nWe had to take a cab to the store for it, so that we could tote the boxes home in the trunk of the cab, while the driver waited impatiently for us to load and unload, meter running the whole time, never once offering to help. And then, at home, after Jessie and I lugged the boxes to the office, we sat on the floor, methodically reading instructions and trying our best to decipher which cords went where. The directions might as well have been written in Japanese, the illustrations done up for a four-year-old.\n\nWhen all was said and done, I was shocked to find that, when we turned it on, the thing came to life, some sort of revolving image\u2014a screen saver, Jessie told me\u2014moving about on the screen.\n\nJessie went straight for the internet. \"Look yourself up,\" she encouraged me, and I asked what she meant by that, thinking she'd just use this computer to type up papers for school. I hadn't thought much of her fiddling around on the internet, but I saw quickly that it was the one thing on her mind, the reason she wanted this computer. To look stuff up on the internet.\n\n\"Go ahead,\" she said again with an enthusiastic nod of the head, dishwater hair falling into her eyes. \"Type in your name,\" she told me, \"and see what you find.\"\n\nBut I laughed only, telling her we wouldn't find anything, because certainly I'm not on the internet. That's the kind of thing reserved for celebrities and politicians. Not everyday, ordinary people like me. But Jessie was certain.\n\n\"The internet knows everything,\" she told me, emphasizing that word everything, and I filled instantly with dread, trying to assure myself that it was only the ramblings of an eager preteen, that certainly the internet couldn't know everything, like some sort of omniscient god.\n\nBut Jessie's hands breezed past mine, and with nimble fingers, she typed Eden Sloane onto the keyboard and pressed the return key.\n\nIt didn't happen right away.\n\nNo, there was a moment of naive disbelief while the computer did its thing. In that moment, I assured myself that we'd find nothing. Nothing at all. Of course the internet didn't know anything about me because why would it? What reason did I have to be on the internet?\n\nBut then an image popped onto the screen before us. And there was my name, highlighted any number of times. My stomach dropped at the sight of them, all these results the computer had gathered for Eden Sloane. Some of them, I saw\u2014as my eyes sailed past the results one at a time, trying to decipher which secrets of mine Jessie would soon find\u2014were not me. There was a split second of relief.\n\nIt's another Eden Sloane. It's not me.\n\nBut then one listing caught my eye, rattling me to the core. Because there, on the internet, for anyone to see, was my name and, beside it, the address of Jessie's and my home, our little bungalow on the northwest side of Chicago, where I thought no one could find us, where I stupidly believed there was no way to know where we were.\n\nI was wrong.\n\nBecause now I see that any and everyone is privy to that information, that anyone who's looking for me can find out just exactly where I am.\n\nIt was unconscious then, the way that I rose to my feet quickly and moved to the window, pulling the curtains closed post-haste. When Jessie gave me a look, I blamed the glare of sunlight on the computer screen\u2014a glare that wasn't ever there\u2014and she believed me.\n\nI haven't disappeared after all.\n\nAll this time, I've been out in the open, living right under everyone's noses.\n\nMy throat constricted and went dry. I choked on my own saliva. I coughed, a desperate, panicked cough, unable for a moment to breathe past the saliva that was lodged in my throat.\n\n\"You okay?\" Jessie asked, patting my back, and I nodded my head yes, though even I didn't know if that was true or not. Was I okay?\n\nWhen I could speak, I asked her to run down the hall and fetch me a glass of water.\n\nAs she did, I snatched the electrical cord from the socket, watching as the screen turned blissfully black. I started packing the computer back in its box the moment Jessie left and, that very same afternoon, planned to hail another cab and return it to the store.\n\nWhen Jessie returned with the glass of water and asked what I was doing\u2014as I sat there on my haunches, wrapping foam paper around the computer parts\u2014I told her that the computer was broken. That there was something wrong with it, which of course there was. There was something very wrong with it.\n\nI told her that it would have to go back. I avoided Jessie's eyes as, there in the doorway, her face quickly fell. \"Can we get a new one?\" she asked, and though I said yes, I didn't for one second mean it.\n\nBecause there would be something wrong with that computer too.\n\nAs Jessie and I stood on the drive, waiting for a cab to appear, I couldn't help but wonder, What other secrets of mine did the internet hold?\njessie\n\nThe music gets chased away with morning's first light, and now the house is silent and still. It startles me, the way the music suddenly stops, and now that it's gone, I have to wonder if it was ever really there. I sit up with a start, sticking to the wooden floor. I've been sweating. I say my own name aloud to be sure I can still speak, that fatal familial insomnia hasn't stolen my voice from me already. \"Jessica Sloane,\" I say, my words slurred.\n\nI find myself on my hands and knees searching for Mom's urn, knowing I left her here beside me last night. I comb through the planks of the hardwood floors, as if somehow or other she's slipped through the millimeter gap between boards.\n\nIt's a sinking feeling. A spreading, sinking feeling that comes to me at once.\n\nI've lost Mom.\n\nI don't know who I am anymore. I can't go on, I won't go on without her here. I hold my breath and refuse to breathe. And just when I think I'm about to die, I see her. Just two feet away, on the other side of me, right where I left her. My panic comes to a halt.\n\nMom is still here. She's not yet gone. I release my breath and, at the same time, somehow hear the labored sound of Mom breathing through the air return. Short, shallow breaths followed by no breaths at all.\n\nOnly in daylight do I give up my perch. I rise to my feet, arching my back from the stiff muscles that come with three or four hours of lying on the hardwood floors. I creep across the room slowly, deliberately, one step at a time, my legs half-asleep. And I'm jealous of them because at least some small part of me still knows how to sleep.\n\nIn the shower, I shampoo my hair. I reach for the conditioner and end up dumping another handful of shampoo on my scalp. I wash my body and then, because I can't remember if I did, I wash it again. Though later, when my skin starts to secrete a sour smell, I wonder if I washed at all.\n\nI head off for a cleaning assignment. As I scrub away on the homeowner's porcelain floors, I notice that my fingernails are still intact. Not a single nail is torn. There's no dried blood clinging to my fingertips because they haven't been bleeding. Even now I feel the sharp edges of the screw head burrowing into my fingertips, and I'm not sure if that happened or if I only imagined it did.\n\nI lock up before I leave. I load my paraphernalia onto the back end of Old Faithful. Mop, bucket, rubber gloves. The September day is sunny and warm. I ride in the street, on tapered one-way streets, which narrow with parked cars like the thickening of arteries with deposits of fat.\n\nI stop for coffee and a donut, taking them to go. \"Have a good one, Jenny,\" the owner of the bakery says to me as I leave, and I think maybe she doesn't have it wrong after all. Maybe she knows something I don't know. Maybe I really am a Jenny, since I'm no longer Jessica Sloane.\n\nI pedal past a police station. On the sidewalk before the brick building, I pause. I think about stepping inside, asking them to fingerprint me. Maybe they can look my prints up in their system and tell me who I am. But I'm not sure that's how it works. I'm sure they'd need a reason to fingerprint me, and I'm not sure I have one to give. Not a good one anyway. Not one that wouldn't raise red flags.\n\nBut then my mind drifts to the notion of DNA, one of those in-home kits that you mail away. Those that claim, with a simple swab of the cheek, to help you figure out your family tree, find distant relatives, discover unknown ethnicities. It's just what I need. To figure out who I am.\n\nI return to the coffee bar on Dearborn and sit there on the blue velvet sofa, waiting for the man from the garden. Hoping he'll come today. I see orange everywhere I look. On a shirt, a shoelace, a flyer taped to a store window, in a flower bed. But none are the man.\n\nI go to the garden, slipping back in between the honey locust trees and finding my way to Mom's favorite spot. It's empty, except for a bird, a little brown thing, a sparrow, pecking away in the dirt for food. I scare it away as I make my way to the edge of the raised bed, sitting on the marbled edge, my eyes circumspect but also tired. The twitch in my eye has yet to go away. If anything, it's gotten worse. It twitches incessantly, only stopping when I dig the heels of my hands into it and press hard.\n\nAfter an hour or two, I give up. I take the long way back to the carriage home because I'm in no hurry to return. I bike past the elementary school at the corner of Cornelia and Hoyne, a stately structure made almost entirely of red brick, four floors that are tall and thin and deep. Kids play outside, on a parking lot playground beside the school building. The flag is at half-staff; someone has died. The kids are rowdy, unruly, loud, like howler monkeys defending their territory. They scurry to the top of the jungle gym, laying claim to the swings and slides.\n\nI round the corner at Cornelia. A bell rings, calling the kids inside from play. They'll go home soon; it's midafternoon. Once they're gone, the world is suddenly silent. The trees stand tall and proud, the sun's light getting scattered at random through their leaves, dusting the sidewalk.\n\nAs I near in on the greystone, I watch as, across the street from it, a little boy schlepps a bucket, waddling down to the sidewalk with his mother on his heels. He flips the bucket upside down and a stack of chalk falls to the concrete. It makes a racket. A single blue piece nearly rolls into the street but he stops it in time, running awkwardly after it. His mother asks him what he's going to draw, waving her hand at me, calling out hello. He's going to draw a hippopotamus.\n\nMs. Geissler is also outside. She's bent at the waist, picking weeds from her flower bed, plucking and gathering them in her hands. She wears gaudy gardening gloves and, on her head, a wide-brimmed straw hat that keeps the sun from her skin.\n\nI see her and feel a rush of anger well inside me. A rush of anger and unease, among other things. I think of Ms. Geissler there in the third-story window watching me at night. The third story, which is overrun with squirrels. The third story, where she claims she hasn't been in months. I think of the eyes, of _her_ eyes, pressed to the window like the eyes of an owl, big enough and bright enough to catch prey on even the darkest of nights.\n\nBut it's more than that too, because I'm certain that someone has been in the carriage home when I wasn't there. Only two people should have a key to that home, and it's Ms. Geissler and me.\n\nThe carriage home is technically hers, but as far as I'm concerned, she shouldn't be allowed to come and go without reasonable notice. Without letting me know in advance, twenty-four hours in my opinion. It's one thing if the pipes had burst or sewage was overflowing from the toilet, but so far, that's not the case.\n\nI think of what Lily the apartment finder said about carriage homes not abiding by the same rules as prescribed in the city's landlord-tenant ordinance. Living here, I wouldn't be protected in the same way, she'd told me.\n\nDid she mean I'd have a complete lack of privacy? That Ms. Geissler could enter my home without permission? Open and close my window shades? Stare in through the glass at me?\n\nFor some reason, I don't think so.\n\nAt first I think I should keep going, that I should pedal right on by. But then I have second thoughts. I want to speak to her, because there's something nefarious going on here\u2014many nefarious things\u2014and I want to know what it is.\n\nI force down the kickstand of Old Faithful and stand, hands on my hips behind Ms. Geissler. As I do, words emerge. I don't think them through.\n\n\"Why have you been watching me?\" I ask.\n\nHer smile is warm. \"Jessie,\" she says kindly, as if she didn't hear my question or the tone of my voice at all. Instead she says that it's nice to see me today. \"How about this weather?\" she asks, hands elevated, praising the sun and the sky for this glorious day.\n\nAnd I'm thrown easily off track, thinking then only about the weather. Forgetting about the pair of eyes watching me at night. Forgetting the fear I felt at stepping inside the carriage home and finding the shades open wide.\n\nI snap to. \"Why have you been watching me?\" I ask, and her face clouds over in confusion. Her eyebrows crease.\n\n\"I don't know what you mean,\" she declares.\n\n\"I saw you,\" I assert, pointing a finger at the windows up above. The windows that are dark now, not a light on inside. They're obscure, shadows only. The only thing that I can see is the outside world getting cast back at me. A reflection. \"Standing up there,\" I say. \"Three nights in a row now,\" I say, though the truth is that I've lost count. It could be three. It could be four or more. \"You've been staring into the house, watching me. Spying on me. Why?\" I demand. \"Why are you watching me?\"\n\nThe smile slips from her face. Or rather gets replaced with one that's more pitying. Ruts form between her eyes, deep trenches in the skin. She pulls the hat from her head and a great big cluster of hair falls from her head, getting trapped in the straw brim. Like Mom's used to do before she bit the bullet and shaved it all off. I see her and me standing together in the shower basin. Starting with an electric shaver first, and then a cheap, plastic disposable razor. Rubbing gobs of aloe vera on it when we were through.\n\n\"Well, aren't you going to say something?\" I ask when Ms. Geissler doesn't say anything. I can't stand to see her looking at me piteously, saying nothing. \"You have no right,\" I say, my eyes lost on the clump of hair that has fallen out of her scalp. She grabs a hold of it, plucks it from the hat and releases it to the wind. \"No right,\" I tell her, \"to be spying on me.\"\n\n\"Jessie,\" Ms. Geissler says. Her voice bleeds of sympathy, empathy. Or darn good theatrics. I don't know which, but whatever it is, I don't like it one bit. \"Jessie, dear,\" she says again. \"You're still not sleeping, no?\" she asks. I feel my knees become liquid. They soften. I want to say no, that I haven't been sleeping. I want her to tell me to try warm milk. A spoonful of honey. To listen to music before I go to bed. Calming music. Lullabies. Not because I trust her; I don't. But because I want someone to tell me about the music and the voices that come to life in the ductwork at night. About Jessica Sloane.\n\nIn that moment I see her, Jessica Sloane, in her purple bathing suit, lying dead on the street. Pigeons circle around her, staring at her with their beady eyes.\n\nMs. Geissler stands before me, staring. \"Jessie, are you all right?\" she asks, and only then do I realize that she's been speaking to me. That she's been speaking to me and I didn't hear a word. \"You don't look all right,\" she decides, empathy in her eyes, but I won't let her divert me from my track. I look around, remembering where I am. Remembering what I was going to say.\n\n\"Slept like a baby,\" I lie.\n\nI look to the ground for the clump of hair that fell from Ms. Geissler's head, but it's not there. All there is is a cluster of leaves, a mixture of yellows and browns that shrivel on the lawn. As my eyes rise to Ms. Geissler, she replaces the hat on her head. And there I see it. A single wilted yellow leaf, folded like a moth in its cocoon, clinging to the straw of the hat.\n\nThere was never a clump of hair. I'd only imagined it was hair. It was just leaves. Leaves falling from a nearby tree, getting snagged on the hat as she hunched over the lawn, tending garden.\n\n\"I see you there in the window. Every single night. I know you see me. You were in my home,\" I snap, my tone turning vitriolic. \"That's trespassing, you know?\" I say. \"An invasion of privacy. I could call the police. I should call the police.\"\n\nShe's quiet at first. \"Jessie, honey,\" she says, the look on her face one of concern. Condolence. Shame. \"Oh, Jessie. Poor, poor, Jessie,\" she says instead, pitying me, ignoring my threat to call the police. She takes a step toward me, makes an attempt to stroke my arm with her gaudy gardening gloves. But I pull back. \"You must be mistaken, dear,\" she says. \"The third floor, I told you already,\" she says, making a sweeping gesture of the greystone behind her. \"I don't go up there anymore. I haven't been up there in months.\"\n\nIt's a lie. I know that's not true. I know because she was there.\n\n\"I saw the light on in the attic. I saw you standing there in the window looking out. Watching me.\"\n\n\"No,\" she says to me, shaking her head, looking concerned and confused. \"There are no lights up there in the attic. I'd had a lamp once, just an old floor lamp, nothing special, but the squirrels chewed their way right through the cord. Can you imagine?\" she says then, tsking her tongue and shaking her head. \"Pesky little things. It's a wonder they didn't electrocute themselves.\" And for the briefest of moments it sounds so genuine, so real, that I almost see the squirrels' overgrown teeth gnawing their way through the cord, cutting power to the floor lamp.\n\nBut not quite.\n\n\"I know what I saw,\" I insist.\n\nBut somewhere deep inside me, I also wonder if I do.\n\n\"You must be mistaken, Jessie,\" she says. \"Maybe it was a dream. You've lost your mother. Grief can be a terrible thing. The isolation, the desperation\u2014\" But I stop her before she can cite for me the stages of grief. Her eyes now are chock-full of condolence. Sorrow. They mock me. I know what she's doing. With her pitiful eyes and her compassion, she's trying to make me question my own sanity, to make me think I'm crazy. A by-product of the insomnia and the grief.\n\nBut I know what I saw. There was a light on in the third floor. There were eyes in the window, watching me.\n\n\"Then let me see,\" I insist. My words are assertive. I attempt to call her bluff. \"Let me go to the attic. Let me see for myself that there is no light there.\"\n\nHer lips curve upward. She grins. Not a happy smile, not a mocking smile, but an appeasing one. She's placating me. \"Oh, I don't think so. It's quite the mess, Jessie. I don't even think it's safe to go up there,\" she says. Not until she can get her contractor out to clean it up, which she says she really needs to do. It's been too long and the attic, for now, is just a waste of space. And then she says that she must go. Rain is on its way, she says, staring skyward. Until now I didn't notice the storm clouds rolling in. It was all blue sky and sun, but now it's not. Now there are clouds. \"The weeds are calling me,\" Ms. Geissler says, turning, stepping closer to the thistle and away from me.\n\nAnd then, in that moment, from up above, the clouds burst apart at the seams. Rainwater comes pouring down. Just like that, the sun-dappled sidewalks are gone, getting replaced with puddles. I take my eyes off Ms. Geissler, looking down, to see my feet submerged in a puddle of water. Across the street, the little boy's chalk hippopotamus gets washed away, rallying his tears. He begins to cry. But not before first throwing his chalk so that it breaks in two, screaming, \"It's ruined,\" and then stomping off and heading inside, hot on his mother's heels.\n\nI look back toward Ms. Geissler, but already she's gone.\n\nIn the distance, a screen door slams and there I am.\n\nHair matted down, wet clothes binding to me. All alone.\n\n* * *\n\nThe rain, only a cloudburst, is through. Over and done with. No sooner had I fled the lawn for the cover of indoors than it stopped. The sun forced its way through the clouds again like a baby chick breaking free from an eggshell. The world turned yellow, golden.\n\nDrop by drop the rain disappeared, going back up the way it came down. And then the sun set, turning the world to pink and then purple and then black, welcoming another sleepless night.\n\nI stare out the window and into the third story of Ms. Geissler's home. I stare until my eyes get tired from it, so tired that my retinas begin to burn, the lid continuing to twitch. And yet I can't bring myself to blink because in those milliseconds, I might miss something, a flicker of light, eyes in the window staring back at me. The house itself blurs, softening at the edges because I've been staring too long.\n\nBut still, I don't blink.\n\nThe shades on the third-story window are drawn. All three of them pulled taut. Like the world outside, the room is dark. For hours on end, there's no one there. Evidence that I'm mistaken. Evidence that I am wrong. That Ms. Geissler hasn't been standing in the window watching me at night, and that my imagination only made it up. It couldn't have been a dream because when you don't sleep you don't dream. And so instead it was my mind playing games with me.\n\nAll night long, the window remains empty and black. It's cold in the carriage home because I've turned the heat completely off in an effort to prevent noises from sneaking in through the ductwork. So far, it's working. There are no voices; there are no pings. No music. But as a result, the temperature in the carriage home hurtles to fifty degrees. My fingers and toes go numb.\n\nAs I lie there in bed listening to the _tick_ , _tock_ of the wall clock, it dawns on me. Mom is not my biological mom. It seems so transparent, so _glaring_ there in the witching hour. As if it's been staring me in the face all this time and I just failed to see. I look nothing like her, for starters, which doesn't necessarily matter because for all I know I'm a dead ringer of my dad. But still, it's cause for doubt.\n\nIf Mom is not my biological mother, then how did I come to be with her? How did I come to think of her as my mom?\n\nMaybe it was something innocuous, like she adopted me as a child. And in an effort to keep me from my birth mom\u2014who, for all she knew, would try and track me down in an attempt to regain custody\u2014she stole a dead child's identity and gave it to me so that I'd be impossible to find. Maybe my birth mother was abusive, neglectful. Or maybe she was thirteen years old, a victim of rape, not ready to be a mother. A teenager who'd gotten loaded at a party and went too far with some guy. Mom was saving me from a life of abuse and neglect at the hands of a reluctant mother.\n\nOr maybe it's not so innocuous after all.\n\nMaybe it's more toxic than that. Maybe I wasn't adopted, but rather taken. Kidnapped. It's a thought I go to only because it's the middle of the night, the time my imagination most often takes flight.\n\nDid Mom _kidnap_ me?\n\nI feel an overwhelming sense of guilt for thinking these things about Mom. That she took me. That I'm not hers. That she did something illicit, that she did something wrong. I think of myself, twelve years old, Mom, woozy on a glass of wine, confessing, _A long time ago, I did something I'm not proud of, Jessie. Something that shames me._\n\n_And that's how I got you._\n\nI know now what she means.\n\nEleanor Zulpo, the woman Mom used to work for when I was a girl, told me that as a child, I insisted my name was something other than Jessie. She remembered that I'd pout my face and stomp my foot and demand that Mom stop calling me Jessie.\n\nJessie isn't my real name. That much I already know. It's a name Mom forced on me, one I accepted with resistance, because even a three-or four-year-old knows their name and isn't quick to change it.\n\nBut not only did I call myself by a different name, but I called Mom _Eden_. Did I call her Eden because she wasn't my mom? Because she'd _kidnapped_ me? Because my mom was someone else, and if so, then who?\n\nIn the back of my mind I tell myself that if, _if_ , Mom kidnapped me\u2014that word itself lumbering through my brain, clumsy and awkward, finding it hard to travel from neuron to neuron because the very idea of it is so incompatible with _Mom_ , who was always so loving, so kind\u2014she had a good reason to do it. She wasn't just some run-of-the-mill child abductor.\n\nBut something doesn't add up, like a puzzle with interlocking pieces, the rounded tabs and the carved-out openings that are all supposed to connect. They don't.\n\nBecause there's the photograph of the man. The one I hold so tightly in my grip that the edges of it begin to disintegrate with sweat. I spend the night holding the photograph of the man with the lake and the trees, knowing he meant something to Mom, that she intentionally kept this photograph and this man from me.\n\nWho is he? I have to find him. I have to find him so that I can know who he is, if he's my father. Then I'll know how and when and why I came to be with Mom. Mom who is not my mom.\n\nI look hard for something, for some clue that I've failed to see. The cut of his hair, the color of the lake, the type of trees in the backdrop. The way he stands, the brand of his jeans\u2014the tag far too small to read, but still I try\u2014that sailboat in the distance. Is it really white like I believed it to be, or is it more of a pale yellow, or white with pale yellow stripes? None of which matter.\n\nAnd then I see it. It's a small thing, but significant enough to me. Because suddenly every detail is significant to me.\n\nThis man is left-handed. I know this, or convince myself I know it, because he's wearing his watch on his right wrist. It isn't one of those hard-and-fast rules, and yet it's common enough to be true. People tend to wear their watches on their nondominant wrists.\n\nI think that there are only a handful of lefties in the world, which narrows down my search exponentially, though still the field is huge. Instead of being one in seven billion, the odds that I'll find this man are now more like one in seven hundred million.\n\nAnd I know it then; I'll never find this man.\n\nHe could be anywhere. He could be anyone. Even if I found myself staring right at him, I'd never know it because I've never seen his face before.\n\nI set the photograph aside. I'm so cold that my skin turns mottled and gray. It's got a purplish tint to it and looks like it's covered in lace, a white overlay to the purple skin. I sit there on the floor, staring at my hands, my legs. All that exposed skin, which is as cold and as mottled as Mom's was before she died.\n\nAnd I come to one conclusion: like Mom, I'm also dying.\n\nAt first, everything around me is black. I can't bring myself to move. I'm too cold, too tired, too scared to move. I can't bring myself to throw the covers over my arms and legs. Night goes by with the speed of a sloth. Painfully slow.\n\nBut then it begins. Sunrise. Daylight. Morning comes. Out the window, I watch it happen.\n\nIt starts as a single pixel of light. The sun still tucked safely below the horizon, scattering its light into the atmosphere. A semidarkness. A soft glow of yellow and blue. The clouds thicken around it, getting drawn in, like nuts and bolts to a magnet. They flush at their edges, turning shades of pink and red. As if the clouds themselves are embarrassed.\n\nThe sun rises higher and higher into the sky.\n\nAnd just like that, day has arrived.\n\nThe air in the room starts to warm thanks to the sun's rays pouring in the open window. My mottled, purple skin disappears, getting replaced with a healthy pink. I'm not dying after all. I'm still very much alive, it seems. For now at least.\neden\n\nAugust 4, _1997  \n_ Egg Harbor\n\nIt's been two weeks since they took my baby from me.\n\nToday, Aaron and I sat in Dr. Landry's office.\n\n\"The good news,\" Dr. Landry said, face firm, undeviating, with no hint of a smile, \"is that we now know you can get pregnant. Your body is capable of that. But maintaining the pregnancy is proving to be another matter.\"\n\nWe had only been there a couple of minutes. Aaron and I sat beside each other on matching tufted armchairs, Dr. Landry on a swivel chair behind his desk. In my hand I clenched a tissue, dabbing at my cheeks as Dr. Landry stared at me.\n\nI asked him, \"How long until we can try again?\" meaning all of us, another round of IVF at the cost of another ten thousand dollars, money that Aaron and I most certainly didn't have because we didn't have it the first time around. I now had three credit cards in my name and each were nearly maxed out. The minimum payment alone was more than I could pay. I'd never been in debt before; I'd never been behind on payments; I'd never been in the red. I'd never been bankrupt. It made me anxious, and yet I easily reasoned that it was money well spent.\n\nI'd sell my own organs\u2014a spare kidney or the lobe of a lung\u2014before giving up on a baby.\n\nHe was dressed down today, no lab coat as usual, and, as Aaron attempted to cling to my hand, I pulled away, folding my hands in my lap. The numbness, the narcosis, it stuck around me like a cold that wouldn't quit. When I wasn't in bed crying, then I was numb. I felt nothing. I had only two modes these days: sad and numb.\n\nDr. Landry replied with \"There's really no definitive answer to that; we can try again whenever you're ready,\" but his words were blighted by Aaron's incredulous sigh because Aaron, as he'd already told me, didn't want to try again. He wanted to be through.\n\nThe reason was simple.\n\nThe reason was me.\n\nFor the last two weeks, I couldn't bring myself to get out of bed. Morning, noon and night, I cried for my lost child, wondering how it was possible to grieve for something that was never truly mine.\n\n_Man plans, and God laughs._ Isn't that what they say?\n\nAaron didn't want me to make another appointment with Dr. Landry. He had other suggestions for whom I should call instead: a therapist, a support group. Maybe all I needed was some time away, he foolishly believed. A trip by myself to one of those places I've forever longed to go. St. Lucia, Fiji, Belize. As if lying by the seashore and drinking a cocktail might help me forget the fact that I'd just lost a child, might annihilate that desire to ever have a child, so that when I returned I'd feel fresh, revived, happy.\n\n\"I don't want a goddamn vacation!\" I screamed at him then, lying in bed, blankets over my head, coming up from under the covers only for air. \"I want a baby. Why don't you get that, Aaron? What's so difficult to understand?\"\n\nAnd it was only then in broad daylight, when I dared to poke my head out of my own dark cavern, that I could see Aaron's eyes were red and swollen, his heart visibly broken like mine. His shirt was wrinkled, the buttons lined up incorrectly, his hair standing on end. His facial hair had grown threefold, proof to me that he, like me, wasn't leaving the house, that he too couldn't bring himself to go to work.\n\nBut I didn't acknowledge this.\n\n\"I know what you're feeling,\" he said quietly, compassionately, his voice losing control as he wiped at his eyes with the back of a shirtsleeve.\n\n\"Trust me,\" he said. \"I get it.\"\n\nA better person would have realized that Aaron had lost something too. A better person would have consoled him, would have let him console and be consoled. But not me.\n\nThis was my loss, not his.\n\n\"Go away,\" I barked then, and I heard it in my own voice, heard it and hated it but said it nonetheless. \"You have no idea what I'm feeling. Don't stand there and pretend you know what it's like to lose a child.\"\n\nI returned to my cave, throwing the blankets back over my head where I could scarcely breathe.\n\n\"This baby. This pregnancy. This need to get pregnant,\" Aaron lamented as he stood in the doorway, urging me to eat, to get out of bed, to go for a walk, to get some fresh air. \"They've gotten the best of you, Eden. They've turned you into someone I don't recognize anymore. Someone I don't know.\"\n\nAnd then he reminded me of who I was before that day we decided to start a family.\n\nFun loving. Benevolent and genuine. Carefree.\n\n\"I'd give anything to go back to being Aaron and Eden. Just us. Just you and me,\" he said _,_ and for a bat of an eye I remembered us on our wedding day, riding in on horseback on Aaron's family farm in a regal ball gown, exchanging nuptials beneath the nighttime sky. A celebration worthy of a fairy tale. I had found my everything. I had married my prince.\n\nBut suddenly my everything wasn't good enough.\n\nI needed more.\n\n\"I want to try right away. As soon as we can,\" I told Dr. Landry today as we sat in his office, and it was then that Aaron stood up from his tufted armchair and left the room.\n\nSeptember 8, 1997 _  \n_Egg Harbor\n\nAaron didn't show up at the fertility clinic for today's appointment.\n\nFor weeks I've gone through the whole rigmarole, the process of developing follicles, of returning to Dr. Landry's office every few days to have my blood drawn and an ultrasound performed to see if there were any viable candidates for the procedure. I've been injected with a legion of hormones, each which leave blood blisters along my skin and a gamut of side effects, from headaches to hot flashes to moodiness and pain.\n\nAlready, Dr. Landry has forewarned me that, should implantation occur, Aaron and I will need to administer shots of progesterone into my backside to not only make a baby this time, but to help maintain the pregnancy. We'll do it daily, for ten weeks or more. \"It's not for the faint of heart,\" he assured me, but I told him I'm ready. \"I'll do anything, _anything_ ,\" I swore to Dr. Landry as he listed the side effects of the progesterone shots\u2014the weight gain, the facial hair, the unbearable pain\u2014to have a baby.\n\nAnd then, when a mature follicle was ready, spotted on Dr. Landry's ultrasound monitor where not so long ago sat the image of a baby with a heartbeat and webbed hands and feet, we scheduled an appointment for the egg retrieval, where Dr. Landry planned to insert a needle deep inside me to remove the eggs from my womb.\n\nToday was that day.\n\nExcept it wasn't.\n\nI sat for hours waiting for Aaron to come and deliver his sperm.\n\nThree hours and fourteen minutes to be precise, watching as other couples\u2014six, eight, ten of them\u2014came and went through the glass doors.\n\nI read each of the magazines in the waiting room two times.\n\nI made an attempt to phone Aaron, but he didn't answer my call.\n\nI told the receptionist, who stared at me with shame and regret, that Aaron was only running late, that he would be here soon.\n\nThat he was caught up in traffic.\n\nAnd then, after another hour of waiting, I asked to speak to a nurse and one was fetched for me, and, standing closer to her than appropriate so that she had to take a half step back to regain her personal space, I wondered whether they had any of Aaron's sperm remaining from the analysis or our first round of IVF. Certainly they had some remaining in storage, a few drops even, a single sperm, half-dead, clinging to the edges of a petri dish.\n\nBut she shook her head remorsefully, apologized and said no.\n\n\"I'm sorry,\" she said. \"There is no sperm.\"\n\nThe nurse took a step away, but before she could go I laid a hand on her arm and asked whether my eggs could be removed now, if they could simply be stored somewhere, held on to until Aaron came to make his deposit. It seemed completely possible, like layaway, but I was reminded then of the little life span an egg has after ovulation. \"There isn't time,\" she said, and it was then and only then that I inquired about donor sperm, an idea that settled in my mind slowly, one morsel at a time, while I spent hours in the fertility reception room, reading pregnancy magazine after pregnancy magazine, waiting for Aaron to come.\n\n_Donor sperm_.\n\nTwo words I thought I'd never have to use in my entire life.\n\nThe desperation in my voice was tangible to every single person in the room, but none more than me. \"Can we use donor sperm?\" I begged, latching on to her arm now, fingernails leaving crescent-shape indentations in her skin.\n\n\"Where is your husband, Eden?\" the nurse asked, stepping away, pretending altogether that I had never uttered those words, _donor sperm_. She was speaking down to me, that I knew. That I could clearly hear, as she riffled through a patient file in her hands, another patient's file, obviously distracted and needing to be somewhere other than in the reception area with me.\n\n\"Where's Aaron?\"\n\nIt was then that I told her how he must have gotten caught up at work, except that was a lie because it was Monday, the one day in which Aaron never worked.\n\nI inquired again about donor sperm\u2014certainly they had vials and vials of male sperm stored somewhere in this facility that I could use\u2014but the nurse assured me that they would need consent from me and Aaron\u2014from the both of us\u2014to use someone else's sperm.\n\nIn other words, Aaron would need to be present to give his consent, he would need to be here, and Aaron wasn't here.\n\nAaron wasn't running late and he wouldn't be there soon.\n\nHe had no intention of coming at all.\n\nHe just didn't tell me.\n\nNot until I came home from the fertility clinic to find him at the kitchen table, drinking a beer. A beer! Suffice to say I lost it completely, feeling enraged. I screamed at him then like I'd never done before, uttering words I could never take back. Coming at him with fists raised, thinking for a minute that I could hit him. That I _would_ hit him. I'd never done a thing like that before, and my fists stopped just shy of him as I turned on myself instead, pulling my own hair, screaming like a maniac. Aaron didn't flinch. I'd scarcely ever raised my voice to Aaron before, and it left me feeling rattled long after he left the room, walking out on me midsentence. It was the medication doing it, I convinced myself as I stood there in the empty kitchen in silence, strands of my own hair in my hands, watching as outside the sun went down\u2014an arc of pinks and blues setting over our share of the bay, heaven on earth as we once so foolishly believed\u2014the myriad fertility drugs affecting my judgment. They were the reason why I screamed and yelled, and yet I had every reason in the world to be angry.\n\nAaron never showed for the appointment.\n\nHe never came to deliver his sperm.\n\nMy eggs were ready and waiting, but where was he?\n\n\"Where were you?\" I demanded as I lifted his empty beer bottle from the kitchen table and hurled it against the wall, longing and hoping for the release of a thousand minuscule shards of glass when all it was was two. Two large chunks of amber glass falling to the floor with a dull thud, leaving me far from satisfied. I reached for a collection of mail then, set there on the table's edge, and hurled that every which way too, bills and late notices drifting to the ground like fallen leaves.\n\n\"I told you,\" he said after I'd followed him into the living room\u2014voice remarkably composed because he had likely sat there half the day rehearsing what he was going to say\u2014\"that I was through. We've been at this for a year,\" he said. \"Over a year. We're broke, Eden. Everything we've worked for is gone. We have no more money to invest in this,\" he said, holding out a bill for me, one that arrived in today's mail, a credit card statement with a seventeen-thousand-dollar debt. __ \"Look what this has done to us. To our marriage. To you.\n\n\"I can't keep doing this,\" he told me.\n\n\"I meant it, Eden. I'm through.\n\n\"It's time for you to choose.\"\n\nAnd then he reached for a packed bag that sat on the hallway floor.\n\nThe front door opened and then closed again, and I wondered if that was it then.\n\nIf that was the last time I'd ever lay eyes on Aaron.\njessie\n\nI sit on the sofa beside Liam, in his apartment. His laptop is on my thighs. I find my way to the website for the National Center for Missing & Exploited Children, thinking that if Mom stole me, for whatever reason, if I had a family before her, then maybe someone once reported me as missing. Maybe my real family is missing me.\n\nOn the website, I discover countless babies stolen from their cribs. Kids who got on the bus, but never made it home from school. Pregnant women last seen on the gritty footage of parking lot surveillance cameras. Infant twins missing after a parent was found dead. Babies lifted from hospital nurseries.\n\nOne by one, I become absorbed in the sad stories. The stories of the missing. Thumbnail image by thumbnail image, I open them all. I read about a toddler who was last seen playing on his own front porch in some small town in Georgia, where he lived with his father and stepmother in Jeffersonville, Georgia. He was last seen at approximately ten fifteen on a Tuesday morning, way back in 1995. His hair is sandy and his eyes are green. An age-progressed photo shows what he might look like today, if he's even still alive. He was taken by his mother in a custody battle. There's a picture of her too. In it, she looks a little agrarian, a little unsophisticated, a little mean. Her hair is sparse and thin, her skin weathered and blotchy.\n\nI think that it's possible that, like the little boy from Jeffersonville, Georgia, I am a child stolen from my front porch or from my crib, a child that climbed aboard the school bus one morning and never came home that afternoon.\n\n\"What did you find?\" Liam asks as I scroll through the website, finding a search form.\n\n\"Nothing yet,\" I tell him. But I hope I will soon.\n\nI fill in as much as I think I know and leave the rest blank. I am child. I am female. These are things I know. I make up the dates I may or may not have gone missing. I fill them in, this three-year gap that stretches clear from the day Jessica Sloane was born until the day she died. Three years. Three momentary years. Shorter than a presidential term, than the span between Olympic Games, between leap years.\n\nI watch as over one hundred cases load. One hundred little girls missing in a three-plus-year time span. One hundred little girls missing in a three-year time span that now, seventeen years later, still haven't been found. It makes me sad. I think of their parents, of their real moms and dads.\n\nOne by one I click on the images and they tell me everything I need: when and where the child went missing; the color of their eyes, their hair, how old they would be. There are age-progressed photos, though how accurate they are, I don't know. Ivy Marsh went missing at the age of two. She was last seen in Lawton, Oklahoma, a little girl with blue eyes, blond hair, dimples like me. Kristin Tate went missing on her third birthday, last seen in Wimberly, Texas. She too has dimples.\n\nI scroll down the page and click the arrow, move to the second page, and then the third. The fourth. \"What are you looking for?\" Liam asks, glancing over my shoulder to see what I see.\n\n\"I don't know,\" I say, but then I take it back, telling him that what I'm looking for is me. I take in the age-progressed photos of children who went missing nearly twenty years ago, wondering if any of them might look like me. Though I tell myself that an age-progressed image of an infant wouldn't have the same accuracy as that of an older kid because of how the face changes over the years. All babies have big, round eyes, chubby cheeks. Enormous foreheads. Toothless grins. There's nothing distinguishable about them. They all look the same to me. So who's to say what a baby's face would look like in twenty years?\n\nAnd it doesn't matter anyway because scanning the missing children, not one bears a resemblance to me.\n\nI push the laptop away. I reach into my bag and for the first time show the photograph to Liam. He asks who it is and I say, \"Just some guy,\" though I feel in my gut that there's more to it than that. Because of some primal instinct to be close to this man, to know who he is. I tell Liam where I found the photograph, hidden in the cubbyhole behind the closet mirror.\n\n\"You think he's your father?\" he asks, both a question and a statement. I shrug. He takes the photograph into his hands, holds it closely to his eyes, examining it before he slips the photograph back into my hands. My hands still shake, the tremor that for all these days won't go away. The room goes quiet, all except for the steady beat of rain against the window. It's a drizzle only, not a complete washout, though the day outside is ugly and gray. The morning's beautiful sunrise has been clouded over now; it's long gone. The melody of rain on glass is calming. I find myself soothed by it, tuning out everything else but that sound, wanting to sing along with it somehow, like a song's refrain.\n\nAnd then it happens again. My eyelids close. They do it against my will. My head slumps forward, my neck no longer able to hold it up. It lasts a second. Only a second.\n\nFor one blissful second, I am asleep.\n\nBut then a jolt of electricity tears through me and my head snaps to. I'm awake.\n\n\"Jessie,\" I hear. I see Liam's hand fall to my knee. I turn to face him, his blue eyes so well-meaning. I'm overcome with a sense of belonging that I've rarely known before, only ever with Mom.\n\nHe touches my hair and for a single moment, something inside me feels warm.\n\nHe urges me to lie on his sofa. He offers up a pillow and a blanket, but I say no thanks. That I'm all right. \"Jessie,\" he argues, but I say it again. I'm all right, though we both know that's not true.\n\nI excuse myself, pushing my body from the sofa as if I weigh three hundred pounds. In the bathroom I splash cold water on my face. I stare at my reflection in the mirror. My skin has a grayish-green tint to it. I look sick, like I'm dying. My eyes sink into their sockets, deep bags formed beneath each. I press a finger to them, watching as they sink and then swell. Sink and then swell. My lips are dry, chapped around the edges, blistered, my cheeks concave.\n\nI count the days on my fingertips. The days since I've been asleep.\n\nThe longest anyone has gone without a drop of sleep is eleven days.\n\nI stare at my own sunken reflection, not able to make sense of what I see, but knowing that by this time tomorrow, I will be dead.\neden\n\nSeptember 23, _1997  \n_ Egg Harbor\n\nI tried not to let the desperation get the best of me, too afraid of what I might do if it did. I tried hard to keep busy, taking on extra shifts at the hospital, working overtime because being at home, alone, threw me easily off balance and I didn't like the feeling of being off balance, of being desperate, of feeling like I was losing control.\n\nMy home, Aaron's and my utopian cottage, quickly became a dystopia to me, a place where everything was undesirable and sad, and where I was in a constant state of dysphoria; I couldn't stand to be there and so I took to keeping myself out of the home all day, every day, doing everything imaginable to avoid the pine floors and whitewashed walls, the glorious tree swing that had once deceived me into believing this place was home.\n\nI spent ten hours a day reading through patient files, trying to decipher what they were to be billed for and entering it into the hospital's system. It was meaningless and mundane, and yet a wonderful way to waste time. I took odd jobs on occasion, answering ads for a temporary cleaning lady or a dog walker or a driver to take a sweet elderly woman for dialysis treatments, keeping her company for the four hours it took to eliminate waste from her blood three times each week. It kept me busy and more than anything, I needed to be busy.\n\nTime passed.\n\nLast week I came home to find a separation agreement in a manila envelope, set beside the front door. In it, Aaron left me the house and all of our assets, taking from me only the debt, as much as he could anyway, the credit cards that were in both of our names.\n\nEven in divorce he was protecting me.\n\nI signed the paperwork post-haste, knowing that the sooner I did, the sooner the divorce was complete, I could ask for donor sperm without Aaron's consent.\n\nIn the meantime, I did everything I could to keep busy, knowing it would take months, nearly six of them, until the divorce was finalized.\n\nCould I wait that long for a baby?\n\nOh, how I would try.\n\nBut as they say, the road to hell is paved with good intentions.\n\nBecause the minute the well ran dry and I found myself with nothing better to do, I drove by the quaint dance studio on Church Street and sat on the park bench, watching the little ballerinas come and go, and it was different now because I hadn't been there in months, since springtime, but everything was still the same. The bigger girls scurried out of the studio first, followed by their mothers, who carried coffee and talked.\n\nAnd then, just when I'd begun to think that was it, the end of the procession, there came little Olivia with her short legs lagging behind, waylaid by things like heavy doors and sidewalk cracks, struggling to keep up. Her hair had been cut short, no longer in a bun but pinned to the sides of her head with barrettes. She was still easily distracted, that I'd come to learn, sidetracked by things like birds and bugs and today a leaf, bright red on the white concrete, the first indication of fall.\n\nShe paused to poke and prod at it as if it were alive, examining the redness of the leaf, the shape of its lobes, while the others gravitated away at their own pace so that the distance between them grew exponentially, and this time, Olivia's mother was too caught up in her conversation that she didn't see her daughter on her haunches, examining the leaf with the concentration and single-mindedness of a microbiologist. The woman's feet hit the street and she crossed the intersection, unaware of the fact that she and her child were now separated by a highway, the very same highway that once took a little girl's life when her mother was also not watching.\n\nSome women were not meant to be mothers.\n\nAnd some who were, some who would make the very best mothers, were refused the right.\n\nIt didn't seem fair.\n\nOh, what a good mother I would be, if only the universe would let me.\n\nSuddenly Olivia's eyes peered up from the fallen leaf and, at seeing that she was alone, she began to cry. It was a process that went by degrees, a feeling of excitement first at finding the leaf, followed by frustration that there was no one around to show the leaf to, before sadness crept in, a great heartache that the others had left without her, leading to panic. Sheer panic. Olivia gasped first, choking unexpectedly on her own saliva, and then she began to cry, quiet tears, choked-up tears, while her little knees shook beneath their shiny white tights.\n\nI'd be remiss to say that a series of thoughts didn't move swiftly through my mind.\n\n_How would I hide her?_\n\n_Where would we go?_\n\n_What would I call her?_ Because surely if she was a missing child, she couldn't parade around town as Olivia still. She'd have to be something else.\n\nI leaned forward from the bench to lift the leaf from the concrete and asked if she ever collected leaves and pressed them between the pages of a heavy book. The sound of my voice, the sight of her leaf in my hand, gave her pause. Her eyes rose from the earth and landed on my smile, and for a moment there was a cessation of tears as I extended the leaf toward her and she took it from me with a shaky hand.\n\nI rationalized in my mind that it would be Aaron's fault if I took the child\u2014not mine, no, not ever mine\u2014because that was the name of the game these days: blame.\n\nIf only he had shown up at the fertility clinic...\n\nIf only he hadn't walked out of my life...\n\nHe and I still would have a chance at our own child.\n\nI wouldn't have had to take one that wasn't mine.\n\n\"Why are you crying?\" I asked, though of course I knew the reason why. I remained seated, not wanting to scare her by standing tall and towering over her small frame. Outside, the temperatures were dropping again, fall drawing near. Soon the tourists would leave. On her arms there were goose bumps as loose strands of dishwater hair clung to the puddles of tears.\n\n\"Where's Mommy?\" she asked, eyes searching the street. But only I heard it in the distance: the sound of girls' laughter over the sound of the wind. Olivia didn't hear.\n\nThrough the trees I could barely make out the red sleeve of a cardigan, the pink of a tutu, a length of brown hair.\n\n\"You lost your mommy?\" I asked and, extending my own hand to hers, said, \"Would you like for me to help you find her? Would you like for me to help you find your mother?\n\n\"It's okay,\" I said when she hesitated. \"I won't hurt you.\"\n\nIt would be a lie to say she took my hand with ease, that she didn't stare at it for a minute, overthinking, some disquisition about not talking to strangers coursing through her mind.\n\nBut then she did take my hand, slipping it inside. It was a great shock to my system to feel this small, soft hand within mine, and it was all I could do not to squeeze tight with instinct, knowing that might make her scream. I didn't want Olivia to scream. I didn't want to scare her, but more so, I didn't want to draw attention to ourselves. For all intents and purposes, this was how it should be. I was hers and she was mine.\n\nAnd then I began to lead her in the opposite direction of where her mother had disappeared. The direction of my car.\n\nOlivia stopped, peering the other way over her shoulder\u2014even a young girl could remember which way her mother had last been walking\u2014but I said to her not to worry, that if we took the car we might find her mother more quickly than if we walked.\n\nI pointed to my car in the distance. \"It's right there,\" I said.\n\nShe thought about this a moment, standing frozen on the pavement, hemming and hawing, eyes moving back and forth from me to the car. A band of clouds had rolled in, blocking the earth from the sun, and as it did, the wind picked up its speed, chasing the warm day away. Outside, the temperature dropped by as many as five degrees and the day turned gray.\n\nFall was coming; fall was here.\n\n\"Well, that's okay,\" I said then, letting go of her hand. \"If you don't want to find your mother, we don't need to,\" and it was reverse psychology, of course, making her believe that if she didn't get in the car with me, I might just leave her behind.\n\nI didn't want to scare her, and yet there was no other way.\n\nI was only doing what I needed to do.\n\nI reasoned that we would only drive to the next town and then stop for ice cream. That I'd have her just long enough to teach her mother a lesson. Then I'd return her. Certainly I wasn't planning to _steal_ the child, because that's not the type of person that I am. A kidnapper and a thief. I only wanted to borrow her for a while, like a library book on loan. To satisfy my craving for the time.\n\nI had taken no more than two steps away when I heard Olivia's tiny feet scurrying quickly on the concrete, running after me. It worked.\n\nHer hand reached up, and she grabbed a hold of mine, squeezing tightly, careful not to let go. I smiled at her and she smiled back, the tears evaporating quickly from her cheeks.\n\n\"Your mother must be here somewhere,\" I said then, and we walked that way, hand in hand, for a good ten feet or more. We moved slowly\u2014at Olivia's pace, though I wanted to tug on her hand and run\u2014and still, it took twenty seconds or less to traverse those ten feet. But in those twenty seconds I convinced myself that in some minute, negligible way, we looked alike, Olivia and me, though in reality we didn't. We looked nothing alike.\n\nI wondered if, once she and I were sitting across from one another at a local diner, eating strawberry sundaes with whipped cream on top, I'd ever be able to return her to her mother.\n\nAnd then a new thought crossed my mind. I could drive farther south, south of Sturgeon Bay, south of Sheboygan, south of Milwaukee. We could live somewhere else, far away from here, where people might believe that we were mother and child.\n\nThey would have no reason not to believe.\n\nI'd rename her. I'd call her something other than Olivia.\n\nAnd in time, she'd come to think of it as her given name.\n\n\"I don't have a booster seat,\" I said as we approached the car, \"but that's okay for now. The seat belt will do just fine.\" And as we closed in on the car I extended a hand toward the handle, reaching out to open the back door for Olivia to climb through. \"It will only be a short drive after all,\" I promised her. \"I'm sure your mother is here somewhere.\"\n\nIn a single moment, I thought this through. I made a plan and it went like this. Once Olivia was in the car I would speed off the opposite way, far from town, away from her mother, not stopping until we'd passed Sturgeon Bay. There I would stop only to buy Olivia ice cream, something to soothe her, to make her not be scared, to quiet her certain tears. Ice cream and a stuffed bear or a toy from a gas station store, something she could clutch to her chest to make her feel safe. We'd drive all night, as far as we could go. Far away from here.\n\nAnd that's when I heard it.\n\nOlivia's name screamed urgently, emphatically through the cold air.\n\nIt was a high-pitched screech, whiny like a whistle. A distressed sound. What followed were the footsteps of a stampede, thousands of wildebeests running down the street. That's what it sounded like anyway, and as I peered up, hand still six inches away from the door, I saw Olivia's mother and her herd hurrying toward me, eight ladies with seven little ballerinas in tow, shouting commands.\n\n\"Olivia, come here right now.\n\n\"What do you think you're doing?\n\n\"Get your hands off my child!\"\n\nMy hands grew slick. My heart beat quickly, more quickly than it was already beating. Under my arms there was wetness. Sweat. My head suddenly hurt. My brain thought quickly to manufacture a lie, as one of the ladies pointed at me and said, \"I've seen you around here before,\" __ and I ransacked my mind for words, any words, but the words wouldn't come. My mind was holding them captive, detaining my words from me, though what it did do was measure the distance\u2014computing the distance from the ladies to me, the distance from me to the car\u2014doing the math, figuring it out, whether I could get Olivia inside the car before her mother and the other ladies reached us.\n\nI _could_ , I decided. But there needed to be no hesitation.\n\nI needed to go.\n\n_Go!_\n\nI needed to go _now_.\n\nBut my feet wouldn't work properly, and my hand, slick with sweat, let go of Olivia's hand and suddenly she was running in the wrong direction, running toward her mother and away from me and away from the car.\n\n\"Who in the hell do you think you are?\" Olivia's mother asked pointedly as she gathered Olivia into her arms and hoisted her to her chest. \"What did you think you were doing with my child?\"\n\nAnd though I was completely tongue-tied, it was Olivia who did the speaking for me, who struggled in her mother's arms to be set free and there, once her feet were firmly planted back on the concrete while twirling her red leaf in her hand, she said, \"You forgot me, Mommy.\"\n\nAnd with that she took six tiny steps away from her mother's reach and extended her leaf to me. A parting gift.\n\nI took it in my hand. \"She was helping me find Mommy,\" Olivia crooned, smiling a toothless grin, but still, I could muster no words.\n\nAnd then Olivia's mother changed tack, and her tone softened. The lines of her face disappeared and instead of reprimanding me or calling the police, as one of the ladies in the backdrop suggested she do, she thanked me. _She thanked me_. She thanked me for helping Olivia. Her cheeks turned red and her eyes filled with tears, and in that moment she believed were it not for me, she may have lost her child.\n\n\"You should keep a better eye on your daughter,\" I threatened, my voice and hands shaking like the leaves in the trees, clinging to their branches for dear life.\neden\n\nMay 11, 2016   \nChicago\n\nI sit on the front stoop, hands pressed between my knees to curb their shaking. I stare expectantly down the street, searching for that first glimmer of yellow to come bobbing along, the school bus, with Jessie on it. I check the time on my watch, knowing down to the minute what time the school bus arrives, but not having the tenacity to wait another three, because if I have to wait much longer I might get cold feet.\n\nI need to get this over and done with. I need for this to be through.\n\nI've combed and curled my hair. I lathered blush onto my cheeks for color, not so that I'll look nice, but so that I look alive, my current pallid tone far more synonymous with death and dying than with vigor. If I look healthy and robust, then maybe Jessie won't be as concerned. I wear a nice shirt. I plaster a smile to my face, one that sours the longer I wait.\n\nI practice the words I'll soon say, saying them aloud so that I can get control of my cadence and rhythm, so that my voice doesn't shake the way it often does when I'm scared. Truth be told, I am scared, yes. I'm absolutely terrified. Though I won't dare say that to Jessie; for Jessie's sake, I'll put on a brave face.\n\nThe braking of the school bus sounds to me like the screech of a barn owl. I watch as Jessie clambers down the massive steps on the heels of her classmates, eyes lost on the ground as they often are these days. Her backpack is heavy; she slumps forward to counter the weight of it and I force back tears, knowing that my days of watching Jessie emerge from the school bus are coming quickly to an end.\n\nI smile and she knows, the moment she arrives, that something is wrong.\n\n\"What's happened?\" she asks, staring at me with the deadpan expression of a teenage girl, one that hides a legion of feelings behind that single blank stare. Sadness, confusion, fear. Her eyes\u2014oh, how blue they are! Even to this day, they shake me to the core\u2014are poker-faced. But not for long.\n\nAs I take her in, I realize that though she's wise beyond her years, she's still a child. A child who will be an orphan soon. I pat at the step beside me and tell her to sit down, cursing myself for trying too hard to look nice. I forget in that moment everything that I'd planned to say\u2014all the wise old adages on life and death that I prepared to quote\u2014and say outright to her instead, \"Jessie, I'm dying,\" my voice flat and even, just barely above a whisper, trying desperately to stay calm for her sake. \"I'm going to die,\" I say as that inexpressive demeanor cracks before me and tears rush to Jessie's blue eyes, flooding them instantly, a flash flood of tears.\n\nI stare at her stoically, trying not to cry as Jessie breaks down before me. But it's hard to do. Jessie rushes into me, throwing her arms around my shoulders and neck. She pulls me in tightly as I purr into her ear, \"Now, now. Don't cry. Everything will be all right,\" enveloping her in my arms, patting her back, stroking her hair.\n\n\"I'm not scared,\" I tell her, lying through my teeth because these are the words she needs to hear. \"Sooner or later we all die, Jessie. It's only a matter of time. And this is mine.\"\n\nTo say I'm not heartsick would be a lie. To say I don't feel ashamed would be too.\n\nBecause after everything I did to make Jessie a part of my world, I'm leaving her alone to fend for herself, and for this, I feel guilty as sin.\njessie\n\nI'm lying in bed when I hear a noise from outside. It makes me jump suddenly, makes me spring inches from the mattress and into the air.\n\nWhat I expect to see when I look outside is a garbage can lid getting hurled to the ground, one of those galvanized steel ones clanging to the concrete. Because that's the sound I hear, the din of metal on concrete, and I imagine a colony of hulking rats climbing on shoulders to scale the garbage can, working together to carry off whatever's inside.\n\nBut instead when I peel the shade back and gaze out, I see nothing.\n\nThe moon, the stars are nowhere to be seen tonight. It's pitch-black outside.\n\nFor hours on end I find myself staring into the black nothingness that is Ms. Geissler's home. My body shakes from the cold, though as always I sweat. And I think that I have a fever, because that's the way it feels to me. Icy cold on the inside, but sweating through layers of clothes, my skin damp with sweat. My clothes stick to me as my teeth chatter. I'm not sure I have it in me to survive another night. I wonder what a panic attack feels like, a breakdown. I think that's what's happening to me.\n\nMy eyes adjust to the darkness, making out shapes. The blackened windows, the balcony suspended three stories in the air on stilts, the flat roofline, the porch, the sliding glass door.\n\nAs I stare, I watch a squirrel leap from the branches of an oak tree and onto the rooftop. It vanishes into the eaves of the rooftop, as voices speak to me through the floor register again. _Peripheral cooling_ , they say this time, and _mottling of the skin_ , their voices weak and watered down, far away from here. But I can't be bothered this time to run and throw myself down over the metal grate because I know they won't hear me if I do. Even if I scream at them through the vent, they won't reply because they never do. Because they're only in my mind.\n\nI hear the sound of footsteps too, quiet, restrained footsteps that slink up through the floor register and into the room with me. A giggle.\n\n_Shh_ , __ someone says, voice suppressed. _Let her sleep._\n\nI can't turn away from the window. Like bugs drawn to a light at night, I can't bring myself to look away. The window is the color of ebony, of charcoal. It's jet-black, the window shade motionless, completely inert.\n\nI take in the rectangular shape of the glass itself\u2014narrow and tall\u2014the stagnancy of the shade. There is no one there. Behind the window and the shade, the room is empty and dark.\n\nUntil it's not.\n\nBecause, at three in the morning, the light flicks on.\n\nThere's an immediacy to it, a sudden unexpectedness. So much so that I almost fall from the edge of the bed. It happens all at once. A lamp turns on and the shades go up at the same time. The room becomes flooded with light.\n\nFor the first time I have a clear view of the room inside. What I see is a bedroom of sorts. An attic room, one space divvied up by three windows. Like a triptych, a painting where three canvas panels come together to create one scene.\n\nIn the first, a bed's headboard is pressed up against a wall paneled with a dated oak that stretches from floor to ceiling. The bed is unmade, a marshmallow-white comforter pulled down a foot from the head of the bed, pillows lay flat. There is a lamp on beside the mattress that lobs the soft yellow light across the room.\n\nIn the second canvas is the foot of the bed and the bottom two vertical columns of a four-poster bed frame. There is a wooden door on the back wall that leads to a hall. Or a closet. It's closed, so I don't know where it goes. A random cord dangles from the ceiling, belonging to seemingly nothing. At some point in its life, it might have been a fan or a light.\n\nIn the third canvas is the man.\n\nWhich makes me clutch a hand to my mouth, to keep myself from screaming.\n\nHe's leaned up against the window, the very same window where someone has been standing behind the shade watching me. His back is turned to me, as he sits on a ledge, pressing his back to the glass. He's dressed in brown, all of it, everything I can see, blending into the walls. Camouflage, a disguise. His hair is brown, pruned close to his head. I can't see his face or his eyes.\n\nI stare at him for minutes, unmoving, he and I both frozen in place.\n\nAnd then he rises. And as he does, I see that he is tall. He stretches in place, hands above his head, back arched. His stride is long and decisive. He crosses the room in three easy steps\u2014what might take me eight or ten\u2014all with his back in my direction, as if he knows I'm watching him. As if he knows, and he's toying with me. Playing a game of peekaboo. Of blind man's bluff.\n\nHis hands hang limply by his sides. I set my own hands on the window glass, as if reaching for the man on the other side of it.\n\nI can feel it beneath my skin, something I can't quite put my finger on. Something about this man strikes a chord with me. His stature, his posture, the color of his hair. I've seen him before. Like Michelangelo's statue of David. You'd know it by David's carriage even if you never saw his face. He stands with his hand on his hips, left knee bent just a bit. His head is pitched to the right, looking at something off in the distance, something only he can see. Not me.\n\nAs my eyes fall to his right arm, I notice a watch on his wrist. A watch on his right wrist, which means to me that he, like the man in the photograph, is left-handed. I think of the man standing there in Mom's photograph in the saggy blue jeans. An afterthought to the lake and the boat and the trees. An addendum tucked neatly away in parenthesis. Almost forgotten, but not quite. I race to my bag and withdraw the photograph, holding it to the window so that I can see.\n\nThe stature is the same. Not just similar, but the same.\n\nHe's the man from Mom's photograph.\n\nAnd then he turns, wheeling toward the window, quickly, in an instant. My hand slips unintentionally from my mouth as a scream slips out. I hold my breath, taking in his trim beard and his sun-tanned skin, knowing I've seen him before. I don't blink and I don't breathe. Because I know this man. It isn't just a hunch. Because there on his forearm is the very same scar, harder to see from the distance, but undeniably there. A six-inch gash, one that stretches clear from his wrist to beneath the cuff of a shirt, the skin around it puckered and pink.\n\nAnd only then do I remember that the man in the photograph also had a scar.\n\nAs did the man at the garden. The one who sat reading Mom's obituary and looking sad.\n\nThe scar is the smoking gun. The one I was looking for. The one I couldn't see.\n\nThey're not two men who I've been searching for, but rather one man. And though it feels unimaginable, impossible, outrageous and far-fetched, I know it's true.\n\nThis man is my father.\n\nHe knows that I am here. He knows that I am here and he's come for me.\n\nBecause why else would he be there?\n\nNow that I see them, his eyes are like hazelnuts, small and dark. He stares at me. Like me, he doesn't blink.\n\nAnd then he steps from the window and reaches for the lamp. It turns off and then on again. A distress signal. An SOS. Morse code. Three short, three long, three short flashes of light. Save me.\n\nHe's speaking to me. Communicating.\n\nI rise from the bed, sitting on the edge of it. I force my feet into a pair of gym shoes. The shoes resist. My feet have been sweating. They're tacky and they don't slip easily on. The laces of my shoes remain untied, trailing me as I go down the treacherous steps. I race out the front door, leaving it open wide, and across the dew-covered lawn.\n\nAt first I don't think. I just go.\n\nBlades of grass reach out to tickle my legs as I cross the yard. The grass is long, in need of a trim, and my legs are bare, wearing only a pair of shorts. The air is nippy and brisk, but still, somehow, I sweat. It comes streaming down my hairline, gathering like swimming pools beneath my arms.\n\nAnd then, ten or twenty feet from the carriage home, I start to question myself. What am I doing?\n\nSuddenly I'm scared.\n\nThree times I stop to get my bearings, looking around, in front of me and behind. Listening. A tree reaches out for me, brushes my arm, its leaves like the gentle caress of a human hand. I jerk back, startled and afraid.\n\nIt's dark outside. So dark that I can't see what's three feet before my eyes. I don't know what's there, if anything's there. My heart pounds inside me. \"Is anyone there?\" I call out, but no one replies.\n\nAbove me, I'm keenly aware that the blaze of light from the third-floor bedroom has gone dark. The house is black, no light anywhere. I think about going back, about turning around and going home. Of crawling onto the bed, of hiding beneath the sheets where I'll be safely on base.\n\nBut then I come to a spot that's halfway to the greystone and halfway back. I'm stuck in the middle, and the thought of going back seems as ominous as moving forward, especially since I left the door open wide. By now, who knows who's let themselves inside.\n\nI hear scavengers in the distance. Raccoons, crows, rats. A creature scampers away from me on the lawn. Ringed tail. Masked face. Footprints like human hands. And I imagine a contorted human crawling by on all fours, releasing a guttural growl at me. Running away.\n\nI make my way along the brick paver patio and toward the front door. There I climb the steps to the front door. Nearly ten of them, each tread precariously thin. At the top I pause to catch my breath. I breathe in, holding the air in my lungs. Absorbing it. Letting it fill my cells and bob through my bloodstream like a buoy at sea.\n\nTwo sidelights flank the solid mahogany door. I see my bedraggled reflection in each as I stand with my hand on my heart, gasping for air. My hair stands every which way; my skin is a bloodless white, deathly pale. There are purple bags beneath my eyes.\n\nI knock on the door. It's a knock that's uncertain at first, but one that becomes more certain with each second that passes by. Once, twice, three times I knock. There's no reply.\n\nBefore I know it, I've knocked twenty-three times, each knock progressively louder, so that by number twenty-three, the knock is a pound. I raise my arm again but before I can bang once more, the porch light switches on. It startles me, the abruptness of it. Though after all this time, it's anything but abrupt.\n\nSuddenly I'm no longer trapped in a black hole but instead doused with a bright white light that makes me go momentarily blind. For a whole six seconds after the front door opens, I see nothing. Just blotches, spots, dots. \"Who's there?\" I ask, voice still breathless, knowing it can be one of two people standing in the doorframe: the man in the attic window or Ms. Geissler.\n\n\"It's three in the morning, Jessie,\" she says to me. Her words are tired and annoyed. It's Ms. Geissler, who, unlike me, had apparently been sleeping, spared from a night of insomnia, unlike me.\n\nMy eyes focus to see her wrapping a red cotton robe around herself, tying the belt into a bow and patting down her hair. \"What's the problem, dear?\" she asks, her eyebrows scrunched up. \"Is everything all right?\"\n\n\"He's here,\" I say quickly, taking two small steps toward Ms. Geissler, bridging the gap from her to me. She takes a step in retreat.\n\n\"Who's here?\" she asks. And I say, \"Him. A man. Upstairs.\"\n\nAnd it's the blankness of her expression that gets me upset, that makes me snap. That and my overwhelming fatigue, my persistent irritability thanks to a lack of sleep. \"You know who I mean. You know exactly who I mean,\" I say roughly because I know he's here, in her home. She has to know that he's here. She has to _know_ him, because why else would he be here? \"The man in the window upstairs. The one who's been watching me. He's here.\"\n\nShe presses a hand to her heart. Gasps, \"There's a man here? In my home?\"\n\nHer face goes white. She makes an offhand effort to peer over her shoulder and into the vacuous foyer as I take another step forward, one that gets my toes just inside her home. But only my toes. She resists, grasping the door hard and putting a foot behind it. She nearly shuts the door in my face. I lose balance, stumbling back onto the concrete stoop.\n\n\"I saw him in the window,\" I tell her, pointing at the staircase behind her. \"Upstairs. A man in the third-story window,\" I say, and at this she relaxes visibly and smiles. She shakes her head and tells me that there's no one in the third-story bedroom, her voice so sure that for a second I believe it. She says again that no one's been in that room for months. Not since the squirrel incident, and then I think she's going to rehash it for me, the whole story about the squirrels inhabiting the third floor. I know now that it isn't true because there were never squirrels in that room but rather a man, my father, who she's been hiding from me all these days.\n\n\"The attic ladder,\" she tells me this time instead, \"it's a pulldown thing,\" at which, like a mime, she grabs for an imaginary string over her left shoulder and pulls. \"Broken for a couple of months. Wouldn't you know it,\" she says, \"the exterminator managed to break the darn thing. I just haven't gotten around to getting it fixed.\"\n\n\"But I saw him,\" I insist, and she says quite simply, \"You must be mistaken. There's no one there. Because how would anyone get up there, Jessie, without a ladder?\"\n\nIt seems so sensible, the way that she says it. And for a fraction of a second, I doubt myself as she hoped I would do. But then his image returns to me\u2014him standing there in the window, looking out at me\u2014and I know that she's lying. That she's keeping him from me. Hiding him from the world just as Mom has always done.\n\n\"Let me in,\" I insist, pushing the door against the weight of her, and she says to me, \"Now, now, Jessie. You had a bad dream, that's all,\" but of course this can't be true.\n\n\"You were dreaming there was a man in the room,\" she says to me. She reaches out a hand to mine but I pull briskly away. \"Just a bad dream, that's all. It will all be clearer come morning.\"\n\n\"I know what I saw,\" I tell her, voice cracking. But her face is suddenly so pacific, so kind, and she asks if I'd like for her to walk me back to the carriage home so I don't have to go alone. It's dark out, she says. Hard to navigate the way. \"But not to worry,\" she tells me. \"I know this yard like the back of my hand,\" and she reaches for my arm to lead the way home. She winks at me and says, \"And besides, I have a flashlight.\" And there it is, in the pocket of her robe. She flicks it on as if this conversation is over, as if she's put my worries to rest and now I can go home, feeling assured that there's no man in this home. No man watching me.\n\nBut I yank my arm away. \"Why are you hiding him from me?\" I ask. My voice becomes elevated, high-pitched, defensive. \"Why don't you want me to see him? Why don't you want me to know that he's there?\"\n\nAnd then I let slip the one thought that's put down roots in the back of my mind, that's replaced all logical thought.\n\n\"Why are you keeping my father from me?\" I scream.\n\nHer face falls flat and she goes white, even whiter than she was before. She shakes her head, presses a hand to her mouth but says nothing. Nothing at first, before she carefully breathes out, treading lightly, \"You're quite sure you saw a man in there?\"\n\nMy heart nearly sings in relief. She believes me. _She believes me_.\n\nI nod vigorously.\n\n\"Perhaps you're right then. Perhaps someone is there,\" she says with concern as she draws back the door and lets me in. \"Why don't you go see,\" she suggests.\n\nI think of my father, so close within reach. I soar past Ms. Geissler on the staircase, taking the steps two at a time up to the second floor. There I stand beneath that little hatch that leads up to the third floor. I listen for footsteps at first, hearing nothing, but remembering that I've stood here before and heard something.\n\nHe was here that night. Standing above me. Was he trying to contact me, to get my attention? To let me know that he was here?\n\nI reach for the cord and give it a tug. The ladder unfurls before me, unfolding into makeshift steps. Two of the steps are split. Another is missing, just as Ms. Geissler said.\n\nShe warns me, \"The steps, Jessie. They're not safe,\" though I go anyway, clutching the hand railing, which is unstable at best. \"Bring the flashlight with you,\" she says, attempting to hand it to me. But I don't take it.\n\n\"There's a light,\" I tell her. \"I saw the light. I don't need a flashlight,\" but she tells me to take it anyway, as she gives it a shake. I take it only to appease her, tucking it under the crook of an arm.\n\nI begin to climb. I move slowly, walking though I want to run. The fourth step gives on me, splintering, and I shriek.\n\n\"Jessie!\" Ms. Geissler yells, asking if I'm okay.\n\n\"I'm fine, I'm fine,\" I say, gripping the railing harder and pulling myself up and over the broken step.\n\nMs. Geissler makes no attempt to follow, but stands instead at the bottom of the stairs. She crosses her arms against her chest, watching as I go. She tells me to be careful. She tells me to go slow.\n\nI reach the top step and hoist myself into the attic. The room is murky. Out the open windows, the sun is lost somewhere beneath the horizon. It's still nighttime, and yet there's a flush to the sky. Morning will be here soon.\n\nI barely make out a lamp, the same lamp that for the past few nights radiated light. One of those old Tiffany-style lamps, with the stained-glass shade. But when I go to turn it on, nothing happens. The lamp is dead, the lightbulb burned out. I turn the knob around and around but still nothing happens. All I hear is the idle click that mimics my heartbeat.\n\nI orbit the room, looking for him. I trip over things that I can't see. I hold my breath and listen, but I hear nothing. \"Hello?\" I ask, more begging than inquisitive.\n\n\"Come out so I can see you,\" I whisper to the man. My father. I tell him I know that he's here. That I want to see him, to meet him. That I've been waiting my whole life. I take small steps around the room, using my hands as a guide. My heartbeat pounds in my ear as I hold my breath, listening for breath, for footsteps, for him. A game of Marco Polo.\n\n\"Marco,\" I chant aloud to myself, but there's no reply.\n\nI reach for the flashlight Ms. Geissler gave me. I turn it on. It casts a meager glow around the room, not much but enough. The light bounces on the wall from the tremor of my hands.\n\nWhat I find is a wall of cardboard bankers boxes\u2014dozens of them\u2014with holes chewed out. Rodent droppings and old building supplies. Gallons of paint, boards of hardwood, boxes of screws and nails.\n\nA makeshift nest\u2014clumps of twigs and leaves\u2014is nestled into the corner of the attic, and on it, there's some hairless and fetal-looking thing that looks like it's just climbed out of its mother's womb. A mother squirrel stands over her baby, scowling at me.\n\nWhat I don't find is a four-poster bed. A white comforter. A cord dangling from the ceiling. A man. None of those things are here. It's just a ratty and dilapidated attic inhabited by squirrels, just as Ms. Geissler has said.\n\nI feel like I can't breathe. The pain in my chest is immense, in my arm, my jaw, my abdomen. The room is empty, though as sure as I live and breathe, I saw a man here.\n\nI stand looking out the window and toward the carriage home. I don't know how long I stand there, staring, thinking that maybe he will appear. That somehow we'll have swapped places. But he never appears.\n\nI make my way back down the steps, where Ms. Geissler stands waiting for me. On her face is a complacent look. An _I told you so_ look.\n\n\"Find what you were looking for?\" she asks, though I can't speak. A lump forms in my throat, but I will not cry. I cannot cry.\n\n\"I told you, Jessie,\" she gloats, and I know then that she did this only to humor me. \"There is no man there. Squirrels. Only squirrels.\" And then she thrusts the ladder back up so that the squirrels can't take over the rest of her home.\n\nShe shows me the door, but before closing it on me, she first asks, \"Did you ever think, Jessie, that you're only seeing what you want to see? You need help.\" She all but pushes me out of her house and slams the door behind me. I hear the sound of a lock clicking shut.\n\nThe porch light goes off, and once again I am submerged in darkness.\n\nI set myself down on the top porch step, feeling exhausted. My body aches from the lack of sleep, from ten nights of my mind depriving me of sleep. It's an insidious way to die, I think, from lack of sleep because there is nothing gory about it, no blood, no guts, and yet the effects are just as gruesome. I know because I'm living it.\n\nAs the sun begins to rise on the eleventh day, it's only a matter of time until I die.\n\nThis is what it feels like knowing you're about to die.\n\nThis is what Mom must have felt like knowing she would die.\n\nI sit on the stoop and talk to myself, blathering about what's happening to me, hoping to make sense of it, but striking out. I can't make sense of it. I count to ten to make sure I can still do it, losing track at number six. I cry, a proper cry, shoulders heaving, the first in a long time. My heart, my head, everything hurts. I fold over sideways on the porch step, rolling up into the fetal position, pulling my knees into my chest, wondering if this is where I'll die.\n\n* * *\n\nAll at once I look up and have no clue where I am.\n\nBy now the sun is just barely beginning to rise. It turns the world from black to gray. One by one people appear on the street before me. Joggers, early-morning commuters.\n\nAs a hint of daylight fills the sky, I suddenly catch a glimpse of something on the other side of the street. It's a man in jeans and a jacket, bustling down the street with his hands in the pockets of his pants. His chin is tucked into the coat to keep warm, and there's a hat on his head, an orange baseball cap, and for this reason I know that it's him.\n\nBut how did he get here? How did he slip out of Ms. Geissler's home without me seeing him?\n\nAnd that's when the answer comes to me. The balcony. The one that leads from street level up to the third floor.\n\nHe climbed down the balcony before I had a chance to go up the stairs, sneaking out as I cut across Ms. Geissler's lawn. That's when the light in the window went black. It went black because he'd already left. As I examined the attic with a flashlight, he was at ground level, looking in through the windows, watching me.\n\nI rise quickly, calling for him, waving my hands to get his attention. I fall down the porch steps, all six or eight or ten of them. \"Excuse me!\" I scream, but if he sees, if he hears, he doesn't look and he doesn't wave back. He doesn't slow down. He never stops moving. He's in a hurry. He has somewhere to be.\n\nI run as fast as my legs will carry me, which isn't fast.\n\nThe twitch in my eye has gone from one eye to two, so that they both spasm and I can't get them to stop. My hands shake. My arms ache, my legs ache, my back aches and, as I move across the street, not looking either way before I cross, a passing car nearly runs into me. The driver slams on their brakes to keep from hitting me.\n\nI stand there in the street, three inches before the hood of the car, staring at the panicked driver, myself unfazed. Because I don't have it in me to be scared. The driver shoots me a dirty look. When I don't move, she douses the window with windshield wiper fluid, splashing me as she hoped to do. She screams out the window at me, and only then do I go.\n\nBy the time I turn away from the car, the man has advanced a quarter of a block or more. He's harder to see than he was before, farther away. Every now and then I see the orange cap bobbing and weaving down the street, but then it gets blocked by a low-hanging tree limb and I can't see him.\n\nI panic; I've lost him.\n\nBut then again it returns, and I follow along.\n\nI listen for the sound of footsteps, and though I'm a half a block away, I hear them. They're tenacious and quick, and for this reason, I know that they're his. I follow, having only the drum of footsteps to guide me, the drum of footsteps, steady like a beating heart.\n\nBut then, as I round the corner and pick up the pace, I hear something else too. They're words, breathed into my ear. _Earth to Jessie_ , I hear, and I spin suddenly on my heels, glaring at a man who follows from behind. He's dressed in a suit and tie, an overcoat draped over him, smoking a cigarette. In the other hand, a coffee cup.\n\n\"What are you looking at?\" he grills, tossing the cigarette to the ground. He grinds it into the concrete with the toe of a shoe and immediately reaches into his breast pocket for another. I turn away, saying nothing.\n\nAnd then another noise comes. It's so soft, so subtle, hardly more than a whoosh of air against my ear, as I come to a red light and stop. _Psst_ , __ says the noise, like the buzzing of a mosquito in my ear. I'm at a street corner, my eyes peeled to the walk signal, waiting for my turn to cross, hoping it was soon before I lose track of the orange cap. The street is congested, early-morning rush hour dissecting me from it.\n\n_Psst. Hey you,_ I hear, _hey, Jessie_ , __ and I jump, my eyes turning away from the street to see who it is and who's calling me. The man with his cigarette is gone now, around the corner and out of sight, leaving a wake of smoke trailing behind. Behind me stands a corner coffee shop, the first floor of a three-story light-colored brick building. There are people milling around outside, just a small handful of them, though their bodies are turned away from me.\n\n_Jessie_ , __ I hear again, and I snap to attention. Who said that? Who's calling me?\n\nThere's a sudden chill in the air. I shiver. I pull my sweatshirt tighter around me, eyeing the people outside the coffee shop and taking them all in. But there's no one here that I know.\n\nI turn away but still can't shake the feeling that someone is following. That someone is watching me. It's a gut feeling and there, at the fringes of my awareness, I feel it. Eyes on me though they're outside my field of view, burning a hole in my back.\n\nOn the other side of the intersection I pause, looking backward one last time, because I just can't shake that sense of being watched. And then I hear it again.\n\n_Psst. Hey. Hey, Jessie_ , __ and I turn suddenly, a spinning toy top on its tip. I almost lose balance; I almost fall to the ground. The world spins on its axis and I don't know what to blame for it, the lack of sleep or grief.\n\nA man and woman walk behind me now, holding hands. Midthirties, pushing forty. They look slick and sophisticated, she taller than him in high heel boots, though they're both pinched and slim. \"Did you call me?\" I ask, but they exchange a look and tell me no. They part ways, slipping around me, one on either side. Once they pass, they rejoin hands, looking into each other's eyes before gazing over their shoulders at me. They laugh. I hear words giggled between them. _Lunatic_ and _crazy._ They're talking about me.\n\nAnd then there's a hand on my back. A warm hand that touches my bare skin from behind. It caresses me as every single hair on my arms and legs goes erect and I can't help myself. I scream. I jerk away, spinning around to find no one there. There's no one standing on the sidewalk behind me, though I hear it again. I feel it again. Lips pressed to my ear, whispering, _Earth to Jessie_.\n\nI shake my head, willing it away, telling myself that it's nothing. That it's only the wind. I look up, coming to, realizing that I've lost track of the man I am sure now is my father. He's gone. I listen for the sound of his footsteps, searching the horizon for the orange baseball cap. I start to panic\u2014eyes desperately lurching this way and that, hoping to see that pinprick of orange way off in the distance. I stagger down the street like a drunk. I can no longer hold my body upright because it's begun to collapse on me. I try running but I can't run, and so it's a shamble at best, feet dragging.\n\nA hand latches onto my arm, a voice asks if I'm all right. I peer down at the hand on my arm, seeing a spindly hand, a bony hand. Rivers of blue veins roll across it. There's dirt wedged beneath the fingernails, lining the edges of the nail bed, and that's how I know. I know this hand; I'd know this hand anywhere. This is Mom's hand.\n\nMy eyes shoot up, taking in the woman draped all in white. She looks nothing like Mom. And yet, she says, \"Jessie.\"\n\nI'm so taken aback that I don't have it in me to respond. She stands before me, a halo of sunlight bearing down on her. She wears a wispy white blouse that billows in the early-morning breeze, the top button undone so I catch a hint of the pale skin beneath. On her bottom half is a skirt, a long one, stretching clear to her feet so that I can't be certain they're there. She looks fragile, delicate and, as she draws her hands through her hair, strands come with it. Clumps of hair fall from her scalp just like that, getting trapped between her thin fingers. Through the thin, floaty blouse I catch sight of her breasts. The breasts flat, nipples gone. Serrated suture marks crisscrossing her chest, the way Mom's used to be.\n\n\"Mom,\" I say. As impossible as it sounds, this woman standing before me is Mom.\n\n\"Mom,\" I beg this time, trembling as I reach for her, wanting nothing more than to draw her close, to wrap my arms around her shoulders and pull her in tight. I'm crying now, tears falling freely from my eyes. \"Mom!\" I plead, but before me she pulls suddenly back, sharply back, her eyebrows pleated. Her mouth drops open and she asks, \"Do you need me to call someone for you? An ambulance, maybe?\" as she stands a good three feet away and retreats a step for every step that I draw near. I grab for her again, but she tugs her arms out of reach from mine, setting them behind her back.\n\n\"I'm not your mom,\" she states. And it's so assertive, so firm, it gives me pause.\n\nMy eyes calibrate the image I see, the woman with the red hair and green eyes dressed in all white. Except that her hair is intact and what I saw as suture marks are instead lace.\n\nIt's not Mom.\n\nI drop my hands to my sides, as she asks again if I need help, if there's someone she can call for me. I bark out _no_ , though all at once I realize that I have no idea where I am, that the streets and the buildings are unrecognizable to me. That I've never seen them in my whole life.\n\nWhere am I?\n\nHow did I get here?\n\nA siren wails off in the distance.\n\nA car door squeaks open and then slams closed.\n\nPeople push past me on the sidewalk, in a hurry to get here or there as the woman disappears into the crowds.\n\nI cup my hands around my mouth, screaming up and down the street for my father.\n\nAnd then, when I think all hope is lost, I see him. Out of the corner of my eye, somewhere in my peripheral vision, I catch a glimpse of orange as it slips behind the glass door of an apartment building on the other side of the street. I go to it, tugging on the door handle to follow him in, but find the door locked.\n\nI press my face to the glass, staring inside. The lobby of the building is near empty. It's dated and retro with 1970s linoleum tile, the kind that seeps with asbestos. Where are we? Does he live here? Does he know someone who lives here? The tile is partly covered with some sort of commercial carpeting, bland and gray, to disguise the ugly tile. A postal worker separates mail into a million bins and though I knock on the glass for him to let me in, he ignores me. Either he can't hear or he doesn't care. He just goes about sorting the mail as if I'm not here, as if he can't see me, as if I'm invisible.\n\nAnd I wonder then if I am invisible, if I am already dead.\n\nI tug again on the tempered glass door. The hollow metal frame rattles in place. I smash the heel of my hand against the glass to no avail.\n\nI begin to make my way around the building, in search of another way in. A freight entrance, maybe. But before I've gone twenty feet, a tenant comes tearing out of the building, eyes set on an incoming bus. I race back to the door, managing to slip in a hand in time to prevent it from latching. I sail inside. Behind me, the door closes tight.\n\nMy eyes look to the left just as a flash of orange disappears behind a door. A black-and-white sign beside it reads Stairs _,_ the steps themselves explicated by a zigzag line. He's going upstairs. I follow along, racing toward the stairwell and after him.\n\nI press hard on the steel door's push bar, making my way into the stairwell. I run, scaling the steps two at a time, clinging to the banister with a sweaty hand, pulling myself up the concrete stairs. The air is stuffy, suffocating, hard to breathe. There's a notable lack of oxygen in here. It's unventilated; there's no access to fresh air. I choke on nothing and it takes a moment to regain my composure, to stop myself from choking on the musty air.\n\nThere are sixteen floors in the building. Above me, I hear footsteps as they climb upward at a better clip than me. He's going too fast. I can't catch up. I call to him, but if he hears, he doesn't let up.\n\nSomewhere between the eighth and ninth floors, my feet slip on the edge of the step, on some sort of tactile paving, yellow, rubbery lumps that are meant to have the opposite effect, to prevent people from falling. But not me. Rather my body keeps going, the momentum of the run thrusting me forward at a blistering pace. But, thanks to the tactile paving, my feet slow down, two things which are mutually exclusive because I can't stop and go at the same time. And so instead I trip, feet skidding beneath me. My body jerks, my hand latching on to the banister to keep me upright. Pain radiates down my arm, into my hands, seeping into the muscles of my rib cage, my neck, my back. But I keep going. He is right there, within reach. I can't lose him this time.\n\nI scurry up yet another flight of stairs. I keep running, up the steps. Though before I know it, we've reached the top floor. The highest floor in the entire building, the sixteenth floor. The end of the line, I think at first, but not quite. Because he's still climbing. Because there's still one more flight of stairs, different from all the rest. More industrial, more heavy-duty. Not meant for everyday pedestrian use. It's more of an elaborate stepladder than stairs. But I scale it nonetheless, ten feet behind him. Beside it, a sign reads Roof Access _._\n\nThere's a hatch at the top, a single slab of aluminum with a hinged lid. He pushes through it and I follow, mounting the last few steps of the stepladder and breaking free onto the rooftop of the apartment building.\n\nAt the top, the hatch door closes all on its own behind me. The wind forces it shut, the sound of it slamming closed, startling me.\n\nI reach for the handle to tug it open again, finding it suddenly locked.\n\nI'm trapped on the building's rooftop.\n\nThe city surrounds me. A panorama. With arms outstretched, I can't help but spin, taking it all in. Enjoying the view, knowing fully well this may be the last thing my eyes ever see.\n\nThe buildings and skyscrapers rise up like dominos around me and I stand on my own domino, waiting for my turn to fall. The lake is bluer than I've ever seen, a luminous blue that makes the blue of the sky inferior. An underling. Sunlight reflects off the glass of the buildings so that the whole world is suddenly aglow.\n\nI circulate the building, looking for him, for my father. Now that we're here, he's somehow disappeared. He's hiding from me. I call to him, but he doesn't reply. \"Hello!\" I scream. \"I know you're up here!\"\n\nThe roof itself is filled with all sorts of miscellany. An industrial cooling system. Exhaust vents. Access panels to this and that. It makes it hard to see. I search among various parts of the cooling system, looking for him. They're big, boxy things that make noise from time to time, like the whirring of a fan inside. I hold my breath. I refuse to breathe. Breathing makes noise and I don't want to make noise. I only want to listen.\n\nA hand strokes me again, whispering into my ear, _Earth to Jessie._\n\nI pull back, drawing sharply away from the strange caress.\n\nTo the west end of the building, there's a fire escape, one that runs from ground level clear to the top, a thing so basic, so rudimentary, it terrifies me. It's little more than a metallic swimming pool ladder, four treads that lift you from the rooftop to the other side of the building.\n\nThat's where my father stands. On the fire escape. Now that the hatch is closed, the only way out of here, aside from a free fall, is the fire escape.\n\nI go there, legs shaking. I call to him, voice more subdued now that I see him. Now that I've found him. Now that he's in reach. He's climbed over the roof wall, a three-foot thing, and onto the fire escape.\n\nMy hand reaches out for the ladder's handrail and I grab a hold of it and pull myself up. My hands are dripping and slippery. I go up one tread. It gives on me and I fall back down to the ground. I start again. One step and then two, watching on in horror as my father begins his descent without me, jogging down the steps at a steady clip, unfazed by the great height.\n\n\"Stop, please,\" I beg, hearing the anguish in my voice. \"Please, don't go.\"\n\nAs I near the top, there's a moment of calm that comes and goes so quickly I almost don't notice it. For one split second the world is still. I'm at peace. The sun moves higher and higher into the sky, yellow-orange glaring at me through the buildings, making me peaceful and warm. My hands rise beside me as a bird goes soaring by. As if my hands are wings, I think in that moment what it would be like to fly.\n\nAnd then it comes rushing back to me.\n\nI'm hopelessly alone. Everything hurts. I can no longer think straight; I can no longer see straight; I can no longer speak. I don't know who I am anymore. If I am anyone.\n\nAnd I know in that moment for certain: I am no one.\n\nI think what it would feel like to fall. The weightlessness of the plunge, of gravity taking over, of relinquishing control. Giving up, surrendering to the universe.\n\nThere's a flicker of movement beneath me. A flash of brown, and I know that if I wait any longer, it will be too late. The decision will no longer be mine. I cry out one more time. And then I go, legs convulsing as I swing one leg over the edge of the building and onto the fire escape on the other side. I have to force myself to do it. It takes everything I have. All that's on the other side is a measly shelf, an overhang, that hovers seventeen floors above land.\n\nI make my way toward him, but he's moving far too fast for me. And I'm scared, looking down where, beneath me, the earth tilts and sways. I'm overcome with vertigo. I feel nauseous; I feel like I could be sick. The steps of the fire escape are perforated to prevent snow and ice from forming, which does nothing for me now. I can see straight through them to the street beneath my feet. People like ants walk up and down the street, minding their own business, paying no attention to me. Cabs like matchbooks soar past.\n\nThe steps beneath me are corroded and weak. A handful are missing. In some spots, the fire escape pulls away from the building's masonry, bolts no longer holding tight. I take the steps two at a time, though they clatter each time my feet hit, the entire fire escape bucking beneath me. I have to take long strides over the missing steps.\n\nI make it down only half a flight of steps before my knees give.\n\nAs they do, I lurch forward, staggering. I fall down the second half flight of stairs. The railing at the end is corroded, as much of the fire escape is. It's the red-orange of rust. As my body goes hurtling into it, the spindles give and I slip straight through, with nothing there to prevent my fall.\n\nAs I tumble off the side of the fire escape, my head swims.\n\nI take one final look at the great distance to the ground, the distance I'll soon fall.\n\nAll at once, I'm falling. My legs follow the rest of me, feet making a last-ditch effort to cling to something, trying in vain to tether themselves to the steel of the fire escape. I try to grab it with a hand, but it slips straight through time and again, as I soar along beside it, unable to grab hold.\n\nMy arms and legs kick. They do the doggy paddle as I soar downward. I flail and kick, my body splayed as air rushes from beneath me, wrapping my hair around my face. I can see nearly nothing. Not that there's much to see anyway, other than the blue of the sky as I fall. There's no air resistance. The air does nothing to slow me down. My hands make a meek attempt to protect my head, some sort of Pavlovian response, as I thrust my feet downward, knowing my only chance of survival hinges on landing feetfirst. It doesn't work. I can't get them down. Another fire escape landing soars past but I can't get to it in time.\n\nMy insides scuttle to my center from the speed, from the velocity of the fall. A fall that feels like forever. Like I am forever falling. My face molded in fear.\n\nI open my mouth to scream, but nothing comes out.\neden\n\nOctober 3, _1997  \n_ Egg Harbor\n\nIt's become an itch that I can't reach. A hunger that no amount of food can satiate. A drought that a thousand rainfalls can't fix.\n\nThat unquenchable need to be a mother.\n\nI think about it morning, noon and night.\n\nAt night I lie awake not sleeping, wondering how I will ever be a mother.\n\nI don't know that I have it in me to wait until Aaron's and my divorce is complete.\n\nThere is adoption, of course, but as a single mother going through divorce proceedings and carrying an exorbitant amount of debt, I hardly think I'm a suitable candidate for adoption.\n\nAnd so I must find another way.\n\nI go to work early and I leave late, spending those extra few minutes staring at the babies through the nursery room glass. On my lunch break I eat quickly so that I have time to wander down to the labor and delivery unit and salivate over the newborns while the nursery room nurses tend to their every need, the bottles and clean diapers and the endless rocks in the rocking chairs.\n\nI don't want to feel the way I do.\n\nI'm not a bad person, not by any means, and yet it's an addiction to me. A disease. I'm unable to abstain from thinking about babies, from wanting a baby, from craving a baby as one does gambling or cocaine.\n\nI've lost control of my own behavior. I don't know what I'm capable of, what I might do, and that in itself terrifies me. Once I was very rule abiding; I always did as I was told.\n\nBut now my neurotransmitters are in disrepair and quite simply, I'm not the person I used to be. That Eden is gone, replaced with someone I scarcely recognize anymore, someone I don't know.\n\nOctober 7, 1997 _  \n_Egg Harbor\n\nSomething happened today.\n\nI had eaten my lunch\u2014roast beef on rye from the hospital's cafeteria\u2014sitting all alone at one of the smaller round tables, nibbling quickly, quietly and staring out the window at the visitors and outpatients who came and went through the revolving front doors, realizing how utterly alone I felt as the other tables spilled over with groups of four, five, six, all involved in conversations that didn't have a thing to do with me. Oh, how I felt so alone. When I was through eating, I set my tray beside the trash can and then went to visit the babies in the nursery.\n\nA drug addict needing her fix.\n\nAs I stood there, peering through glass at the newborns sound asleep in their knitted blankets with their knitted hats, one little one in particular caught my eye, the name in the bassinet reading Jade Cutter _._ It was the name that caught my attention, not necessarily the baby herself, though she was perfect in every way, from the roundness of her head to the redness of her cheeks. But more so, it was the slip of pink paper in the bassinet that caught my eye, the one that listed her name, date and time of birth, pediatrician, and the names of her parents.\n\nJoseph and Miranda Cutter _._\n\nJoe and Miranda.\n\nThey'd had their baby girl. They'd had their baby girl and they didn't tell me.\n\nShe was swaddled in a pink cotton blanket, eyes closed, mouth parted as she breathed in her sleep. A single hand had forced its way from the blanket, but Jade seemed unmindful of this, unlike other infants\u2014I'd come to learn from my time spent observing them in the nursery\u2014whose limbs needed to be controlled so that they could sleep. There was dark hair, a mound of it, that sneaked out from the edges of the pink hat and, though they were shut tight, I had to imagine her eyes too were dark like Joe's, though these, of course, were things that often changed over time.\n\nAnd in that moment little Jade's eyes parted and she gazed at me, it seemed, and I was stricken with a sudden, purposeful, persistent need to hold her in my arms.\n\nI stepped into the nursery, greeting the ladies there by name. There were two of them, one older and one younger, both of whom I knew fairly well. How many months had I been stopping by, telling the tale of how I was working hard to earn my degree so I could be one of them, nursery room nurses who tended to newborn babies? How many times had they let me into the nursery, allowing me to watch as they changed diapers and swaddled with the expertise of someone who'd done it a million times? How many times had they let me stroke an infant's cheek in his or her sleep, never once needing to remind me to wash my hands because I always remembered?\n\nBut not this time.\n\n\"You'll need to stay in the hall, Eden,\" the older of the two nurses said, a woman by the name of Kathy, and I felt a stabbing sensation in the chest as she pointed to the floor, to an imaginary line that dissected the nursery from the hallway tiles.\n\nThat's the line where I was to remain behind.\n\n\"But, Kathy,\" I attempted to argue, but she held up her hands and told me that there had been complaints that they'd been too lax of late, and hospital officials were cracking down on security protocols, and I wondered now if the babies were fitted with tiny security devices on their wee wrists or ankles to keep someone from walking out the front door with them. Someone like me.\n\n\"But it's just me,\" I reassured her, holding up my badge, reminding her that _I work here._\n\nExcept that by that time, she'd turned her back to me and was attending to an infant who'd begun to fuss; she wouldn't let me in, and I could feel my hands begin to shake in withdrawal. There was a tightness in my chest and my head suddenly hurt. For a minute or two, I couldn't breathe. My heart was palpitating, strong, irregular beats that left me light-headed, though neither of the nurses seemed to notice; no one noticed but me.\n\nAnd then Joe was there coming to my rescue, as he appeared at the nursery to come lay claim to his baby girl.\n\n\"Joe!\" I said too loudly, thrilled to see him, knowing that he was my key to that baby. Joe would get Jade out of the nursery; Joe would let me hold and coddle baby Jade as I needed to do.\n\nHe said hello to me, and what a nice surprise to see me, and at this, I felt a smile spread widely across my lips. My heartbeat slowed; the tension in my head and neck began to ease.\n\n\"Miranda called you to tell you the news?\" he asked, but I said no, that I was working, that I had just come to visit the babies in the nursery when I saw baby Jade.\n\n\"Congratulations!\" I said, offering an awkward embrace. Joe was not a man I knew well, and what I did know came from Miranda's own complaints about him. How he was a jerk, a lousy father. But I couldn't let this deter me now.\n\n\"How is Miranda?\" I asked, and Joe replied as expected: Miranda was tired, Miranda wanted nothing but to sleep and already I imagined her, sore over her infant's need to eat.\n\nKathy glided the rolling bassinet out the nursery door to Joe's waiting hands, and when I made an attempt to follow, to accompany Joe to Miranda's room where I would sit on the corner armchair with Jade in my arms as I'd once done with Carter, he said to me, \"Another time, Eden? My parents are here,\" meaning that they already had company, that Joe's mother and father were here to see baby Jade.\n\nThat I wasn't welcome in the hospital room with Joe's mother and father because as everyone knows, three's a crowd.\n\n\"Just for a minute?\" I pleaded, staring at Joe, who looked worse for wear in that moment, tired and jaded. I could see it in his eyes: four children was too much.\n\nI could help him.\n\nI could take a single one off his hands.\n\nJust one child.\n\n\"Just to offer my congratulations to Miranda and then I'll go?\" I begged\u2014and even I could hear the desperation in my voice\u2014but Joe shook his head, and I felt like a child then, like a five-year-old child who'd just been told no.\n\nJoe said that he'd pass my message along to Miranda and then he turned to go without me. He walked quickly on purpose, faster than my legs could go, and I felt the dismissal a thousandfold then. I was being brushed off, given the cold shoulder as if I carried a stigma on my sleeve.\n\nThe stigma of infertility, the stigma of miscarriage, the stigma of a woman whose husband was in the process of divorcing her.\n\nI blinked and Joe was gone, disappeared down the hall and around the corner where I couldn't see him anymore, and immediately the headache returned, the palpitations, the sweat. The hospital walls began closing in on me as in a room nearby a lady, deep in the throes of labor, screamed, and instead of feeling sympathy for her, I felt a surge of jealousy and spite.\n\nOh, how I wanted to be the one screaming in the throes of labor pain! How I wanted to feel a baby inside me, wedging itself headfirst to get out. How I wanted to feel that baby press between my legs, to feel it crown as doctors and nurses gathered around telling me to push. _Push!_\n\nMy feet crept toward her room with instinct, setting my hand on the doorknob and turning it, opening it just a sliver so that I could see in. There was far too much happening inside the room for anyone to hear the door squeak. I stood in the doorway, inching a foot back so no one would see. A Peeping Tom. The door wasn't open and yet it was ajar, not quite closed tight, and through the crack I saw her laid out on her back, gasping from pain. I saw her gather handfuls of blanket in her hands and squeeze, pushing to get that baby out. I heard her scream, this throaty, guttural scream, crude and uninhibited as a nurse on either side told her to push. \"Push!\" Her husband stroked her sweaty hair, brushing it out of her eyes. Between her legs was a shock of black and there I stared, wondering just where exactly she ended and the baby began as she pushed again, holding her breath\u2014as I, in turn, held mine, parting my legs ever so and pushing too\u2014bearing down, and this time, as she pushed, a baby came spilling out of her insides, covered in mucous, and the room was filled with a sudden rapturous bliss.\n\nThe door slammed shut in my face.\n\nSomeone had seen me.\n\nI ran away, out of labor and delivery.\n\nI was due back at my desk in just a moment. Soon the other women in billing would wonder where I had gone, and why I wasn't yet back from lunch. They would tell our manager. I would be given a scolding.\n\nBut I couldn't go back to billing at that moment.\n\nI needed to get away.\n\nI got behind the wheel of my car and I drove and drove.\n\nI drove to the chophouse, needing to see Aaron, desperate suddenly to see him, for him to hold me in his arms, to stroke my hair and tell me everything would be all right. If I'm being honest, I was scared of the person I was, scared of the person I'd become. I was quite terrified, if Aaron didn't put a stop to it, of what I might do. My thoughts were scattered, sown like seeds in my mind, and there was no telling which ideas would bloom, the sensible ones like going home and putting myself to bed, or the misguided ones where I return to the hospital and force myself into Joe and Miranda's room, screaming like a lunatic, demanding that they give me their child.\n\nI left the car parked haphazardly across parallel lines on the street outside, nearly a block from the chophouse. Parking in town was never easy to come by. I stepped from the car, my ankle giving on me as it sunk deep into a crater on the street. I shook it off, kept moving, feeling the ligaments beneath my shoes begin to ache and swell.\n\nIt had begun to rain outside, the sky darkening. The restaurants, the gift shops, the galleries that lined the street radiated light. They beamed from the inside out, while outside people scattered like roaches in daylight, hiding under canopies and slipping inside stores, seeking shelter, huddled in throngs beneath ample-size golf umbrellas, clutching one another, laughing.\n\nBut not me.\n\nI made my way to the chophouse alone, fully intent on going inside. On speaking to Aaron. On begging him to help me, on pleading with him to take me back. I was desperate. What else could I do? The rain came pouring down, permeating my skin, so that I could feel it inside my bones. I hurried past people tucked warmly, drily beneath their umbrellas, no one offering to share. The rib of a passing umbrella poked me in the shoulder, but no apology came thereafter, as if it was my fault, as if it was my shoulder's fault for getting in the way of this man's umbrella.\n\nI closed in on the chophouse, smelling that scent that always followed Aaron home and into bed with us, that coiled around us while we slept. Grease, Worcestershire sauce, the flesh of meat.\n\nBut before stepping inside, I caught a fleck of Aaron through the restaurant window, seeing his face through the small partition that separates the kitchen from the dining room. A flyspeck only, but in that flyspeck, there was a lightness about him, a nimbleness, a radiance to his skin. Rain streaked down the window, but I peered past it, watching as a smile danced on the edges of Aaron's face. In the very same fleck some other man made a wisecrack, I could only assume, because then Aaron was laughing, _laughing!_ , the edges of his lips reaching upward to the sky like he hadn't done in years. Aaron was laughing and it was beautiful to see, an openmouthed laugh, nothing curbed or restrained about it, and I saw in Aaron's eyes a felicity that I hadn't seen in quite some time. Never did he press his hand to his mouth to hide the smile, but rather chuckled with all of his might.\n\nAaron was happy. Aaron had found his happy place.\n\nUnlike me, his heart had healed and he was no longer broken. He was whole.\n\nOh, how I wanted to be there beside him, laughing too.\n\nBut I couldn't bring myself to do it, to shatter what had already been fixed. I'd ruin him, that I knew, if I stepped foot into the chophouse, as I imagined the laughter drawing to a sudden close if I walked in, that lovely smile vaporizing from his face at the sight of me.\n\nAnd so instead, when a hostess poked her head outside and asked if I'd like to take a peek at the menu, I shook my head, scurrying the other way like all the other roaches, seeking shelter indoors from the rain.\n\nIt was an upscale restaurant where I went, fine dining with a bar attached, the kind where one might have a glass of wine while waiting for their table to be set. This hostess offered a table, but I strutted straight past her and to the bar\u2014sopping wet, leaving a trail of rainwater behind me as I walked. I climbed onto one of the tall stools and ordered a chardonnay to drink. A chardonnay! The glass came to me full to the rim, a generous pour at the hand of a bartender with cavernous dimples and sparkling blue eyes, a man who must have been six years younger than me, barely old enough to be serving alcohol at an upscale establishment. And yet here he was, and in the moment I felt suddenly old, much older than my twenty-nine years, but that didn't matter. That was the least of my concerns.\n\nWith the wine he also brought a dish towel, which I used to towel dry the ends of my hair.\n\nThe first sip of wine tasted like battery acid to me.\n\nIt choked me on the way down, burning the lining of my esophagus so that the bartender raised an eyebrow at me and asked if I was all right. I pressed a hand to my mouth, nodding, but I wasn't sure that I was all right. The wine settled in the pit of my stomach, and the feeling was a mix of repulsion and nausea, along with a warmth and prickling that I quite liked.\n\nAnd so I had another sip, wanting the warmth and prickling to have its way with me, to help me forget about Aaron and the miscarriage, all those wasted months trying unavailingly to create a baby.\n\nHow stupid I'd been in believing that with Dr. Landry's help we could outsmart nature. Aaron and I were infertile; that was the nature of the beast. That couldn't be changed.\n\nThe universe was laughing at me for my arrogance and my vanity.\n\nI took another sip of wine and this time, I didn't choke.\n\nI thought of my baby, of my unborn baby. Of my dead baby. I wondered what she would have looked like had she had a chance to grow full-term. Would she have looked like Aaron, with dark hair and light eyes, or would she have looked like me?\n\nWould she have been a she, or would she have been a he?\n\nI still think about her all the time.\n\nHad she been a girl, I would have named her Sadie.\n\nI raised my glass to my lips and swallowed a mouthful, wondering if she ever crossed Aaron's mind.\n\nWondering if I ever crossed Aaron's mind.\n\nBy the second glass, the wine was no longer battery acid to me. It quenched that _hunger_ , that _thirst_ , like nothing else in the world was able to do. It spilled through my veins, anesthetizing my arms and legs, dulling my senses. I hadn't had a drop to drink in quite some time and so it didn't take much for the room to start to blur at the edges, for the stool to feel insecure beneath my seat.\n\nWith every sip thereafter I became a more youthful version of myself, someone more energetic, someone more carefree.\n\nWith every sip I became blissfully forgetful, forgetting at once that I was a soon-to-be divorc\u00e9e, a woman who would never have a baby.\n\nIt was a quiet night, a Tuesday night, and so the bartender happily filled his free time speaking to me\u2014about what, I hardly remember anymore\u2014and, after that second glass of chardonnay was poured, I plucked a credit card from my purse, one that wouldn't be denied, and the bartender started a tab for me, telling me his name was Josh.\n\n\"You have a beautiful smile,\" he said to me, and I blushed, grinning, and he pointed at it and said, \"Yup, that's the one,\" while smiling his own beautiful smile. For whatever reason I dug a tube of lipstick from my purse, a light shade of pink, and applied it to my lips, leaving light pink prints around the rim of my wineglass that he filled each time with a bountiful pour.\n\nI unbuttoned the top button of my blouse, leaning farther over the bar, fully aware of just how pathetic it was, me, a lonely, depressed woman hitting on a bartender in a near-abandoned bar.\n\nI had become a clich\u00e9.\n\n\"What's your name?\" he asked, setting a bowl of nuts before me, a single finger brushing against my skin as he did, and I told him that it was Eden. He equated it to the garden of Eden, in other words, _paradise_ , and I smiled and said I'd never heard that before, though of course I had, from each and every one of the lowlifes who came before Aaron when they were trying to pick me up in bars far less classy than this.\n\n\"What are you doing here all by yourself, Eden?\" he asked while swirling a dishrag in circles before me.\n\nI shrugged my shoulders and said that I didn't know.\n\nWhat was I doing here?\n\nI reached for my glass and downed the last few drops. At once it was refilled, and I downed that too, scarcely able to recall what came next.\n\nOnly bits and pieces stayed with me until morning, a montage of what may have occurred. Sliding from the barstool with the third glass of wine. Laughing at myself as strange hands helped me to my feet, refreshing my glass. A face far too close to mine. The deep groove of dimples. Words whispered in my ear.\n\n\"Wait for me,\" he said.\n\nStanding on the street corner in the dark autumn night, I leaned against a streetlamp that didn't give off an ounce of light. Getting absorbed by blackness until even I wasn't sure if I was still there. It was raining still, a fine mist in the air, one which seemed to levitate and not fall.\n\nAnd then suddenly there were lips on my neck, hands kneading my skin, though who they belonged to, I couldn't see. It was far too dark to see, but it didn't matter to me. I knew only that my extremities were numb from the alcohol, and it was cathartic to me, strange hands wandering along the landscape of my skin, exploring the valleys and hills with a certain vehemence I'd never felt before. A body pressed against mine, pinning me to the streetlamp, whispering breathless words into the lobe of an ear.\n\n\"Where's your car? I'll drive.\"\n\nI heard the sound of an engine gunning, the stars coming at me at a dizzying speed before the world turned black again, and then the scratch of facial hair on my cheek, a hand groping at my chest with the impatience of a sixteen-year-old boy. A hasty man pawed at me, tearing at my blouse. What buttons remained clung to the fabric by strings, as he pushed me into the back seat of the car, moving with the deftness and agility of someone who knew what they were doing, of someone who had a history of strange women in the back seats of cars.\n\nI felt the force of my skirt getting thrust clear up to my rib cage. The scratch of a fingernail as he tore at my panties, pushing them aside. The sound of a moan, my own forced moan tolling through the airless space because, even with the continuous thrust of his hips into me, I felt nothing and I wanted more than anything to feel something, to feel _anything_ , because feeling something was far better than feeling nothing, and in that moment all I felt was nothing. Nothing that mattered anyway.\n\nInstead, hot breath on the lobe of my ear. Handfuls of hair being clenched between hands, tugged consciously or unconsciously, I didn't know. Reggae music on a car stereo.\n\nHe panted out a name in rhythm, \"Anna, Anna.\" Did he think that that was my name, Anna, or was there another woman in his life, a woman named Anna, and he was only pretending that I was her? I replied with \"Yes, yes!\" deciding that I would be his Anna if that's who he wanted me to be. A seat belt buckle drilled a hole into the small of my back, plastic plunging itself into me with every thrust of his hips, leaving its mark, though still I felt nothing, nothing at all, not until finally a spasm tore through him like a lightning strike and he collapsed against me, and then there was the weight of him, no longer supported by his own hands.\n\nThe weight of him. That I felt.\n\nAnd then weightlessness.\n\nA car door opened and closed and then there was silence.\n\nHe was gone.\n\nI woke up in the morning in the back seat of my car, parked at the far edge of a public playground parking lot, beneath the shadow of a tree, my skirt still thrust clear up to my rib cage, the rest of me exposed, hidden only by the dewdrops that had settled on the windows overnight.\njessie\n\nMy heart beats inside me like a cheetah. I'm screaming.\n\n\" _Psst_. Hey you, hey, Jessie.\"\n\nThere's a hand at my shoulder, rattling me. It's gentle, but insistent. I jerk away from the hand, arms flailing. I'm no longer falling.\n\nA mouth presses closely to my ear, speaks in a breathy voice. A stage whisper. \"Earth to Jessie,\" she says, and it's a numbing voice. A hypnotic voice. The perfect opiate.\n\nI imagine where I am. On the grass. Body in bits on the ground, bleeding and broken, hardly able to move. In the distance, the sound of an ambulance's wailing siren as my father walks away from the scene unscathed.\n\nThe voice says _it's okay, it's okay_ , three times or more while stroking my hair. I can't open my eyes. And yet I see her, a woman hunched over me on the lawn, while others crowd around her. She's gawking, her eyes fixated on the most gruesome parts of my battered body. A leg that bends backward, organs that protrude from the skin.\n\nI know the voice. I've heard it before. But I can't place it.\n\nI'm swimming beneath water. Sounds are muffled above my head. The dropping of a needle onto an old vintage vinyl record. Voices talking. A measured, high-pitched ping. Ping, and then nothing. Ping, and then nothing. Ping, ping. Voices in the background. Talking. Saying things like _morphine_ and _slipper socks_ and _ice chips_.\n\nWhen I go to open my eyes, they're sealed shut. Taped down. Impossible to open.\n\nMy hands rise and I'm surprised to find that I can still move them, my arms and hands. That they're not broken after all. Not shattered into a million pieces across the concrete.\n\nI press the heels of my hands against my eyes and rub hard, wiping the crusty discharge. Inside, my heart pounds hard. A song begins to play. Quietly. Background music. __ It's a song I know well because it's Mom's favorite song.\n\nWhen I finally get my eyelids to lift, all I see is yellow. A blinding yellow light.\n\nAnd that's when I know that I'm dead. That's the first clue.\n\nThe yellow light charges my eyes. It stuns and overpowers them, making them close again because I can't stand to keep them open; it hurts too much. I blink repeatedly, trying to adapt to the light. To orient myself, to find a reference point, to figure out where I am.\n\nThe second clue that I have that I'm dead is Mom. Because Mom is also dead. And yet, as I open my eyes, she's here, sitting five or six feet from me. She sits upright, on some sort of reclining armchair with castors on its feet, her gaunt legs propped on the chair's footrest. She's dressed in a roomy gown that slips carelessly from a shoulder, the hair on her head merely fuzz, as it was the last time I laid eyes on her alive. Which is why I know this is some sort of afterlife we're stuck in. Mom and me.\n\nThe room around me is blue. Blue walls. Blue sheets. A comforting, pastel shade of blue. I'm not on the lawn after all. I'm not outside, lying in the shadow of the building from which I fell. Rather I'm in a room, on a bed.\n\nA woman stands beside Mom, lathering lotion onto her arms and hands, massaging the purplish, blotchy skin. I know who she is because I've seen her before, at the hospital before Mom died. She was Mom's nurse, one of them anyway. A woman named Carrie who was more religious than any about applying lotion to Mom's hands and feet, about turning her so she didn't get bedsores. Even when I begged for them to leave her alone so Mom could sleep.\n\nShe looks over at me and says, \"Well, it's about time,\" and that's when I know that we're all dead. Mom, the nurse and me. They've just been waiting for me to arrive.\n\nI know how Mom and I died, but I wonder, how did she?\n\n\"That stuff knocked you out cold,\" says the woman who squats on her haunches beside me, a second nurse. Her hand rests on my shoulder, the very same hand that only moments ago rattled me, mouth purring into my ear, _Psst. Hey you, hey, Jessie. Earth to Jessie._\n\n\"What stuff?\" I ask, feeling dazed and confused. Behind me, from a record player, Gladys Knight sings to me. There's the greatest sense that I'm still falling, though I'm well aware that it didn't hurt when I hit the ground. That when I crash-landed into the concrete beside the apartment building, I felt nothing. I don't even remember it happening. I must've been dead by then, I decide. A heart attack, a broken neck.\n\nThe room whirls around me. I push myself up so I sit, perpendicular, no longer lying down on a bed. There's a puddle of blankets on the floor, a pillow beneath my head. The second woman rises from the ground beside me and pulls the strings of a window shade so that they rise. I've seen her before. She's the same woman who kept me company the night before Mom died, and now she too is dead like me. How can that be?\n\nHow can we all be dead?\n\nMore blinding yellow infiltrates the room, making it hard to see much of anything clearly. But Mom. I see Mom. My eyes go back to Mom. To Mom sitting there. Mom, in the flesh. No longer listless. No longer bed bound. She looks sleepy still, her eyes glazed over, and yet on her face, a smile. \"How about some ice chips, Miss Eden?\" Nurse Carrie asks, offering a single piece of ice from the end of a spoon.\n\n\"The clonazepam,\" I hear, and it takes a minute to realize the nurse is talking to me, that I asked a question and she's answering it for me. \"The stuff doc gave you to sleep. He'll be happy to hear it worked. You needed a good night's sleep. You were dreaming,\" she says. \"Calling out, kicking in your sleep. Must've been one hell of a dream.\"\n\nAnd as I finally start to get my bearings, I realize where I am. I'm in Mom's hospital room. Mom. Who sits six feet from me, upright, sucking on a cube of ice. Not six feet under, but six feet from me. No longer ashes, but now whole.\n\nThe clonazepam. The melatonin. That I remember. My own bloody, inflamed eyes. The doctor, concerned, offering something to help me sleep. Watching a newsmagazine show on the TV, a story about identity theft, while waiting for the pills to kick in, the nurse tucking me into bed, telling me about her daughter, dead in a car accident at the age of three. The purple swimsuit, her daughter collecting shells from the sea. That I remember.\n\nBy the time I woke up Mom was dead, except that she wasn't dead.\n\nIt was all a dream.\n\nMy eyes adapt. The light becomes less painful, less blinding.\n\nAnd that's when I see a man in the room too, and I know straightaway that it's the man from the dream. And I wonder if I'm still dreaming. If this is like the purgatory of dreams and I'm trapped somewhere between sleep and awake, having to atone for my sins before I can fully wake up. His back is to me as it's almost always been because he's there on a chair before Mom. He sits, though I see it in the body posture, the carriage, and I know that it's him. I'm not chasing him anymore because now he's here.\n\nPing, I hear then. Ping. And I turn to watch the movement of lines across Mom's EKG, the spikes and dips of her heartbeat.\n\n\"Dad,\" I breathe, my voice gravelly and hoarse. My heart throbs. Because after chasing him for all those days and nights, after spending my entire life trying to find him, he's here.\n\nHe's been here all along, waiting for me to wake up.\n\nExcept that as the man turns to me, I see that he's different. His face is not the face from my dreams. There's no facial hair anywhere, and his eyes are a grayish-green like sage. They're not brown. His hair is streaked with gray and there are lines across his face, forehead lines mostly, deeply set. His arms are blotched with pale pink scars.\n\nIt dawns on me then, slowly. Of course he's different from the man in my dream. Because in real life I never saw his face. I only caught a glimpse of the back of him when I was a girl, before Mom snatched the photograph from my hand. Before we read a book, before we ate ice cream. Now I remember. I never saw that photograph again until it returned to me in a dream.\n\nThere's a book on his lap.\n\nHe leans forward, gathering Mom's hands into his. Hers are limp. He strokes her cheek, and I see in his eyes the look that he has for her. A look of adoration, a look of love. It makes me feel embarrassed, watching them. This moment of intimacy. It's not for me to see.\n\nIn all my life, no one has ever looked at me that way. I doubt anyone ever will.\n\nHis smile is deferential, kind. \"No, Jessie,\" he says as he lets go of Mom's hands and turns back to me. \"I'm not your father,\" he tells me, and at first I'm speechless because if not my father, then who? My eyes well with tears\u2014wanting, _needing_ him to be my father\u2014as I sputter, \"There was a photograph Mom had of you. I remember seeing it a long time ago. She took it away, she hid it, but it stayed with me. It was a picture of my father. It was _you_. You have to be him,\" I say, and he leaves her side to come to me.\n\nHe sits down beside me on the bed, a gap spread between us. He pats my hand, tells me his name is Aaron. \"I knew your mother a long time ago,\" he explains. \"We were married. She was my wife,\" but then he pauses, his own eyes red, and gathers himself. He won't cry in front of me. \"I don't have any children, Jessie,\" he says, as if that should make it clear, but it only makes me more confused. More angry and more confused. Because how could he be Mom's husband but not my father? Didn't he want me?\n\nMy tone is more scathing, more exasperated than I mean for it to be. \"Then why are you here?\" I ask, and I see the anguish in his eyes, the grief. I pull my hand from his, seeing then that he doesn't have a wedding band. He's not married and I wonder if, after he and Mom were married, he ever was. He divorced Mom, he left her, I think, and there's a groundless anger that swells up inside me.\n\nThis man hurt Mom.\n\n\"I loved your mother very much,\" he says, as if he can read my mind. But then he rethinks and alters it a bit. \"I _love_ your mother very much,\" he says, before holding up the book from his lap. \"Eden,\" he tells me, \"your mother, she sent this to me,\" and I look at it, a brown leather book with a stitched edge, and in his other hand a note, written in Mom's handwriting on a piece of stationery. Stationery with her own name engraved along the edge. \"It's her journal,\" he explains, though in all my life, I never once knew Mom kept a journal.\n\nHe hands the note to me. I skim Mom's words. In them, she tells him she's dying. She says that she wants him to have this journal so that he can finally have closure, so he can finally understand.\n\nThe last line reads, _With love, Eden._\n\n\"Understand what?\" I ask. And there it is again, that exasperation.\n\nBut his tone is compassionate and warm, his eyes soft. He rubs at his forehead, confesses, \"There were some loose ends, Jessie,\" he says. \"Some unfinished business between your mother and me.\" He asks, \"What did Eden tell you about your father?\" and I shake my head and admit, \"Nothing. She never told me a thing about him.\"\n\nHe passes the book to me, the journal. He says that he thinks I should read it, that it would help me understand.\n\n\"Everything she did,\" he tells me, voice cracking, \"she did for you. You should know that.\"\n\nAnd then he rises to his feet to leave, but not before first confessing, \"I wanted to be a father, Jessie. I would have loved to be a father. I would have loved to be _your_ father. But sometimes life doesn't go as planned.\"\n\nI don't know what he means by that. But I grip the journal in my hand; I press it to my heart, knowing I'll soon understand.\n\nHe says that he'll give Mom and me a few minutes. And then he leaves the room.\n\nMy eyes turn to Mom's. They're unfocused and disoriented, the top lid puffed up. She sees me but doesn't see me all at the same time. I wave; she waves back. But not right away, as if there's a broadcast delay. Her lips are a length of string, pilled and thin. They're dry, chapped, some sort of gunk collecting around the edges, which no one bothers to wipe. Her skin is a washed-out shade of gray blotched with purple and blue. A lack of oxygen. Poor circulation flow.\n\nAnd yet she's there. Sitting upright. Waving.\n\n\"You're alive,\" I breathe as I go to her one last time.\n\n* * *\n\nThe nurse leads the way as we drift into the hall. I take one look over my shoulder as we go, saying to her quietly, in a whisper, \"It's a miracle.\" Because I don't want Mom to know how close she came to dying. \"She's better. She's all better. Just like that. Overnight, and she's better,\" I say, a smile as wide as the Grand Canyon on my face. And suddenly nothing else matters. All that matters is Mom. I clutch the nurse's hand, wanting to celebrate the moment, to savor it. Relief consumes me, seeing that Mom has her strength back, some of it anyway. That she can sit up, that she can swallow. I'm thinking of next steps already. We'll begin chemo again. Maybe there is some clinical trial that Mom can participate in, some new medicine we can give a try.\n\n\"Oh, Jessie,\" the nurse says as we watch a family pass by in the hallway, flowers and balloons tethered to their hands. Her face drops. It gets overpowered with empathy, and for a minute or two she's speechless. The only smile she has to offer is a comforting one. Not a happy one. Not a celebratory one like mine.\n\nIt's a sympathy smile.\n\n\"Jessie,\" she says as she ushers me to a nearby bench and we sit, just across the hall from Mom's room so we can still see inside. \"Your mother,\" she says, hesitating. \"She doesn't have much time left.\"\n\n\"But\u2014\" I argue, thinking of Mom, sitting there in the room in a chair. Mom, more energetic than I've seen for weeks. Mom, making what looks to me like a speedy recovery. There's a spark to her eye, just a dot of light that wasn't there the last time I looked, days ago when she last opened her eyes. She'd been comatose for days like that; she couldn't swallow, she couldn't eat. The doctor said it wouldn't be long. And now here she is. Clearly he was wrong. Through the doorway I see Mom reach a hand out to nurse Carrie, rub at her throat. She can't speak. But she's asking for more ice chips, for a drink. \"Look at her. See for yourself. She looks _fine_.\"\n\n\"She looks better. But the cancer. The cancer is still there. This happens, Jessie. A death rally, we call it. She will relapse, honey, and likely soon. Maybe hours, maybe days. There's no way to know for sure, but her body is still deteriorating. The cancer isn't cured. It's metastasized to the lungs, the bones. It's getting worse.\"\n\nWhich I know. Of course I know. I've heard this all before, many times. But looking at Mom now, it can't be true. It's like she's had this surge of brain power and awareness. Like she's come back from the dead.\n\nAnd then I understand.\n\nTerminal lucidity. As imminent a sign of death as any. The final blessing I'd been hoping for. Five more lucid moments with Mom. That's all I asked for. And here they are.\n\n* * *\n\nThe nurse graciously turns off the machine as it flatlines. I wonder if she always does that, if her hand shoots there automatically the moment a patient dies so their beloved family members don't have to hear the damn thing scream. The ping from my dream has finally gone silent.\n\nMom's doctor presses the end of his stethoscope to her chest and we all look to him for guidance, for him to tell us she's dead, though we already know that she is. Her body lies peacefully on the bed, skin going white, blood draining from it. Already she's colder and more synthetic feeling than she was before. Her hands and toes unclench; her body goes lax. The doctor speaks. \"Time of death,\" he says, \"Two forty-two.\"\n\nAnd with it comes great relief.\n\nMom's battle with cancer is done.\n\nMom's death rally lasted a total of three hours and fourteen minutes. For some of it she sat up with me in the chair, while Aaron watched from the corner of the room. He thought he should leave, that Mom and I should have this time together, but I asked him to stay. I did most of the talking. Mom could talk once she warmed up to it, but talking didn't come with ease.\n\nI spent the time trying not to cry. But then, when I couldn't hold it in any longer, I sobbed, gulping down air and choking on it. Because there were things that needed to be said and I didn't have much time left. If I didn't say them, I'd regret it forever, for the rest of my life. \"I don't know who I am without you,\" I confessed. \"I'm no one without you.\" And though I didn't say it aloud, I thought to myself that I'm vapor without Mom around. I'm nothing. A nonentity. A rock, a clock, a can of baked beans.\n\nMom stroked my hand as she did the day she told me she was dying. Caressed my fingers one at a time, and forced a smile that was as sad as mine.\n\n\"You're _you_ ,\" she said. \"The one and only Jessie Sloane,\" as she stroked my arm with an anemic hand, the white flesh darkened here and there with bruise-like marks.\n\nI squeezed into the same chair beside her, as if I was still a little girl. I don't know how we fit, but somehow we did. We sat like that for a while. As we did, one of my earliest memories returned to me, one of the few that hadn't been lost to time. In it, I'm about five years old. It's the middle of the night when Mom comes to me in my room. I'm sound asleep when she kneels on the floor beside my bed, whispering into an ear, _Jessie, honey. Wake up._\n\nAnd I do.\n\nShe helps me get dressed. Not fully dressed, but instead we slip a sweater over my nightgown, a pair of leggings beneath its hem. Socks and shoes. I follow her out the front door and into the blackness of night, asking at least a gazillion times where we're going, though all she ever says is _You'll see._ We walk hand and hand down the street.\n\nThere's a rare giddiness about Mom in that moment. A frivolity. She isn't restrained as she often is, but instead is playful and bright. We only walk as far as the home next door, but for me it seems an incredible adventure, some sort of magical, midnight escapade. We have to walk to the far side of the Hendersons' home where the gate is, cutting through their lawn as we go. Mom stands on tiptoes\u2014her feet, I realize only then, are bare\u2014to unlatch the gate, pushing it slowly open so it doesn't squeak.\n\n_Where are we going?_ I ask, and she says, _You'll see._\n\nWe creep through the grass to a tree at the center of the backyard. A tall tree, sky-high, as high as my five-year-old eyes can see. Though it's too dark to see, I'm pretty sure the crown of the tree stretches clear into the clouds. There's a swing hanging there from one of the tree's branches, just a slab of wood with a thick rope that's looped through holes on each side. Mom tells me to hop on and at first I resist, thinking we can't possibly ride on the Hendersons' tree swing without asking. But Mom's face is radiant, her smile wide.\n\nShe sits down on the wood herself, pats the thighs of her pants. She tells me again to hop on, only this time she means on her lap. And I do.\n\nI scramble awkwardly on, Mom hitching an arm around my stomach to help hoist me up. I sit on her lap, leaning back and into her as she gets set to launch us from the ground. Mom holds onto the rope with a single hand, the other folded around my belly. She walks backward as far as her bare feet can reach and then all at once, she lifts her feet from the earth and, just like that, we fly.\n\n\"Do you remember,\" I asked, snuggled there beside her on the hospital chair, \"the time we broke into the Hendersons' backyard and snuck a ride on their tree swing?\"\n\nFor as long as I live, I'll never forget the smile that bloomed on her face right then. She closed her eyes, reveling in the moment. The memory of the two of us nestled together on that tree swing. \"It was the best night of my life,\" I said.\n\nShortly after the memory left, Mom got tired. The nurses and I helped her back to bed. Minutes later, Mom fell asleep. She drifted back into some sort of minimally conscious state and passed away two hours later with me there at her side.\n\n* * *\n\nIt's only after the funeral director comes to collect Mom's body that I finally rise from the chair. The room is remarkably quiet. No music playing, no familiar sound of the EKG.\n\nThe only sounds I hear now come from down the hall where other people lie dying.\n\nBefore he leaves, Aaron asks if I'll be all right and I tell him that I will. \"I may not be your father,\" he says, \"but it would mean the world to me if I could be your friend,\" and I tell him that I'd like that very much. He goes, and after he does, I see the nurse has already begun to strip the sheets from Mom's bed. Soon another patient will be here, another family surrounding them, watching as they die.\n\n\"Where are you going to go?\" she asks, and I shrug and say stupidly, \"Coffee,\" because nothing else comes to mind.\n\nBeyond that, I have no idea where to go, what to do with my life.\n\nBut there's a part of me that thinks I can figure this out in time.\n\nI try to reconstruct the dream. As I move down the bright, buzzing hospital halls, I try to piece it back together. But dreams have a way of fading fast, the mind a habit of deleting nonessential things. It's as if there's a fifty-piece puzzle before me and I'm missing all but five pieces. I've lost forty-five and only some of them connect. I remember only squirrels. Hot dogs. A hippopotamus. But I don't know what any of it means.\n\nIt's only as I cut through the hospital lobby, passing by the cafeteria, that I'm struck with the sudden sense that something is missing. Something that makes it harder to breathe. I come to a sudden stop and as I do, a body plows into me from behind, making my bag drop to the ground, contents spilling across the hospital floor. Mom's stuff\u2014her lotion, her ChapStick, her journal\u2014as well as mine. My driver's license, my credit card, dollar bills.\n\n\"My fault, my fault,\" I hear as I turn to see a man scramble to the ground to pick up my stuff. \"I didn't see you. I wasn't looking where I was going,\" he admits as he rises to his feet and holds out the bag for me, my things shoved indelicately back in.\n\nAs he does I catch a look at his face for the very first time, and only then do I remember. I gasp. It's him. \"Liam,\" I breathe, taking in that shaggy brown hair and the blue gum-ball eyes, knowing with certainty that he was there in my dream with me. There's the vaguest recollection of sitting on a sofa beside him, of his hand stroking my hair. It's a thought that makes me blush as I take a step closer to him. And though I don't know him, there's the greatest sense that I do. That we're already friends. \"Liam,\" I say again.\n\nBut his face only clouds over in confusion. He shakes his head, stares vacantly at me like I'm mistaken. He looks tired. Stubble has all but taken over his face, and his hair stands on end. His bloodshot eyes are even bloodier than they were before, rivers of red running through the white. He shakes his head. \"Jackson,\" he says. \"Jack.\" And I find that I'm thrown completely off, feeling out of sorts because he's not Liam. Of course he's not Liam. Because Liam was only a dream. This man is a different man completely, though our late-night confessions over coffee were real. That was real, I remind myself, finding it suddenly impossible to remember what's real and what's not.\n\n\"I'm sorry,\" I stammer. \"I thought,\" and I feel silly all of a sudden. \"I should go,\" I say, taking the bag abruptly from his hands, excusing myself, trying to sidestep him and leave. But he doesn't let me leave. Instead he steps in front of me, reaches out his hand and says, \"You never told me your name,\" and for a second there's the sense that he doesn't want me to leave. That he wants me to stay.\n\nHis handshake is warm and firm. He holds on a second longer than he needs to.\n\nI reply, \"Jessie,\" knowing for the first time in a long while that I am. I am Jessie Sloane.\n\n\"You're leaving, Jessie?\" he asks, and I say, \"No reason to stick around here any longer.\"\n\nI don't have to tell him that Mom is dead, because he already knows. He can see it in my eyes. \"Your brother?\" I ask, thinking of the motorcycle accident. His brother flying headfirst into a utility pole. \"Is he going to be all right?\" For a moment Jack\u2014Jackson\u2014is silent, but then he says, \"Bit the dust last night,\" and my heart breaks for the both of us.\n\nBut there's also a sense of relief because, though we lost the war, the battle is finally through.\n\n\"Where are you going?\" he asks, and I tell him I'm not sure. Anywhere. That I just have to get out of here, and he says he knows what I mean. His family waits upstairs in his brother's hospital room for the funeral home to arrive, to carry the body away. That's the last thing he needs to see. That's what he tells me. He shuffles from foot to foot, looking antsy and strung out, desperately in need of a good night's sleep.\n\nI ask him if he wants to go for coffee, and together we leave.\n\n* * *\n\nThat night, at home alone, I find the courage to open the journal. I caress its cover for a good fifteen minutes first, scared to death of what I might find inside. Maybe my father. Maybe not.\n\nI sit on the sofa in Mom's and my home in Albany Park. Because for now it's not yet on the market, though I know that soon it will be. I carefully pull the cover back. A flattened leaf slips from its inside and onto my lap\u2014red, with edges that fold up slightly at their edges\u2014as does a photograph, which falls facedown on my thighs. There's a name etched on the back. _Aaron_. I know what the picture is before I ever look. The photograph I found as a child. The one Mom hid away in this journal so that I couldn't find it again.\n\nMy heart breaks at the familiar sight of Mom's handwriting.\n\nMy eyes wade through the pages, tears blurring my vision. Making it hard to take in the words. But I do anyway, curled into a ball on the sofa, beneath a blanket Mom and I once shared, listening to her favorite records over and over again on repeat.\n\n_Aaron showed me the house today_ , it reads. _I'm in love with it already\u2014a cornflower blue cottage perched on a forty-five-foot cliff that overlooks the bay. Pine floors and whitewashed walls. A screened-in porch. A long wooden staircase that leads down to the dock at the water's edge where the Realtor promised majestic sunsets and fleets of sailboats floating by..._\neden\n\nNovember 10, _1997  \n_ Egg Harbor\n\nWhen I awoke this morning there was the most unpleasant sense in my stomach, as if I'd swallowed some sort of gastric acid in the middle of the night and there it sat, lost somewhere between my throat and my intestines, not sure which way to go. Up or down. There was an awful taste in my mouth, as if I'd drunk a vat of vinegar before bed, and when I hurried for a glass of water to wash it down, I wound up hurling the water and everything else inside my stomach into the kitchen sink and then stood, clutching the countertop, tasting vomit, trying hard to catch my breath. There was saliva on my chin and tears in my eyes.\n\nWhat did I eat last night?\n\nWhatever it was, it wasn't much. I haven't eaten much for weeks, having subjected myself to a life of seclusion since my brush with that bartender in the back seat of my car. I haven't left the house other than for the bare necessities, for fear of running into him on the street. My home is my prison. I've been too ashamed to go outside.\n\nAshamed for a whole slew of reasons, my promiscuity only being one of them.\n\nOvernight I had gone from being a respectable human being to a voyeur, a kidnapper, a misfit, a freak. The morning after my encounter with that bartender, I came home to find bruises on my neck from where he sucked my skin raw so that I couldn't leave the house until they healed, my skin returning to its usual shade of peach. Day and night I stared at those bruises, hating myself. What kind of person was I? What kind of person had I become?\n\nI remembered the feeling of little Olivia's hand in mine.\n\nHad that really happened, or was it only a dream?\n\nDid I nearly steal another woman's child?\n\nTwo women's children?\n\nThe bartender had taken off with my purse too, snatched it right from the front seat while I lay in the back in a daze, leaving the car door unlatched, the interior lights on so that by morning the battery was completely drained. I walked the three miles home with a swollen ankle, clutching the plackets of my shirt together since the buttons had snapped clear off at his hasty hand. I spent the morning after on the telephone with various credit card companies, reporting the cards stolen, despising myself for getting into this situation in the first place, for letting myself be a floozy and a victim. I avowed to pay off my debt and cut the new cards the credit card company would no doubt send me to shreds.\n\nI would never be a victim again.\n\nI'd never trust anyone again.\n\nI would never leave the house for fear I might try and pilfer someone else's child.\n\nAnd so I've become a recluse, plunged into a state of depression where I go unshowered for days at a time, oftentimes not getting out of bed from morning until night. I eat only when I need to, when the hunger pangs are more than I can bear. I've lost my job, no doubt, though no one told me as much, but one can't expect to stay employed when they haven't gone to work for thirty-odd days. I'm drowning in debt, I assume, though I haven't found the energy to drag myself to the mailbox to retrieve the bills, but I'm certain I must be because just last night when I flipped a light switch on, nothing happened. I jiggled the toggle up and down and when that failed, tried another light switch.\n\nIt appeared the electricity had been shut off for nonpayment.\n\nI went back to bed in the dark, planning to stay there for the rest of my life, which would be short as I swore off water and food too.\n\nBut then this morning the nausea wrenched me from bed, dragging me to the kitchen sink, where again and again I heaved, wondering what in the world was wrong with me.\n\nAnd it was a slow dawning then, daylight arriving at its own sweet time, one shaft of light at a time.\n\nFor thirty-odd days I had lain in bed since my encounter in the back seat of the car, and in those thirty-odd days, my period\u2014my ever-reliable period\u2014hadn't come.\n\nAnd now there was the nausea, the vomiting, and though every rational thought in my mind told me it wasn't true, it couldn't be true\u2014after all, I was infertile; there was no way I could get pregnant of my own accord, without Dr. Landry's menagerie of drugs and devices\u2014I knew instinctively that it was true.\n\nI was pregnant.\n\nTo say I was happy would be a lie.\n\nIt wasn't that I didn't savor the thought for a second or two, that I didn't relish the idea of carrying a child, of birthing a child, of being a mother. There was no greater desire in the whole entire world for me. It's all I wanted; it's the only thing that mattered in my life.\n\nBut deep inside I knew this child would never come to fruition. A fetus it was, but a baby it would never be. It would be as it was the last time with the heartbeat that was there and then not there, the gallons of blood. I would lose this baby as I had the last, and it would be my purgatory, my punishment, being forced to endure weeks, maybe a month, of pregnancy, knowing as always that it would end with blood.\n\nThat trusty, reliable blood.\n\nAnd so instead of being happy I stood there, back to the countertop, steeling myself for another miscarriage, to lose this baby like I had the last. Certainly the universe wouldn't let me keep this child. This, truly, was my penance, a gift that was given only to be taken away.\n\nJanuary 15, 1998 _  \n_Egg Harbor\n\nThe joke is on me it seems, for I've made it through the first trimester without a single drop of blood.\n\nThe baby has survived thirteen weeks in my wasteland of a womb.\n\nOnly by necessity have I left the house, taking a job at a local inn where I clean rooms once the guests leave. There's nothing glamorous about it. Just stripping beds of sheets and washing endless mountains of laundry, scrubbing someone else's excrement off a toilet seat. The perk of the job, however, is that I essentially speak to no one, working alone in an uninhabited guest room or the laundry room, dealing only with dust spores and mildew, as opposed to the human race.\n\nBut the work itself is backbreaking. And those first thirteen weeks of the pregnancy were anything but fun and fancy-free. The morning sickness, the lethargy nearly got the best of me until the empty hotels beds were hard to resist\u2014I envisioned myself sprawled out across them, wrapped up in one of the hotel's velour robes\u2014but, for as much as I wanted to, I didn't give in to the whim.\n\nOnly second to a baby, I needed this job more than anything.\n\nI haven't been to see Dr. Landry or another obstetrician, though there's a slight outgrowth to my midsection now, a bulge that makes my pants fit tightly so that I've taken to wearing sweatpants when I'm not stuffed into the uniform I wear for work, the polo shirt and the khaki pants, which I now leave unbuttoned so I don't flatten the baby.\n\nThe cottage is on the market again.\n\nI can no longer afford to pay for it. I haven't been able to for months so that I'm in debt to the bank and the foreclosure threats have begun to arrive. The sign went in today, stuck there\u2014forced into the nearly frozen ground\u2014by the very same Realtor who sold us the home.\n\nOh, what she must think, looking at me now. How I've changed.\n\nThe Realtor didn't look the least bit different to me, but I was changed, hardly the same woman I was when we first met, less than two years ago.\n\nAfter she left, I sat myself on the tree swing and swayed, moving back and forth through the nippy winter air. I did it until my fingers were numb and I could no longer feel the sturdy rope beneath my hands.\n\nThis was the closest my child would ever come to a ride on this swing.\n\nThe bay was empty now, not a boat anywhere, and snow flurries fell on the dock, collecting like powdered sugar. There were birds in the trees, winter birds, cardinals and chickadees, but everyone else was gone, sunning themselves on one of those tropical islands where I only dreamed I might one day go.\n\nThe greenhouse door was frozen shut.\n\nThe flowers in the flower bed were dead.\n\nI was still outside when I heard the doorbell ring, and thinking it was the Realtor\u2014that she had an offer _already!_ \u2014I left my post to see.\n\nBut it was not the Realtor.\n\nAaron stood before me, his chestnut hair getting peppered with soft powdery snow. His eyes had a forlorn look about them, sad. He wore a coat, his hands set in the pockets of it, and as I pulled the door to, he offered a simple smile.\n\n\"Aaron,\" I said.\n\n\"Eden.\"\n\nI couldn't bring myself to invite him inside, for the cottage was truly a mess, in a state of bedlam; I couldn't bring myself to show him what had become of our home. And so I stepped outside, onto the porch, my hair also getting peppered with snow. I pulled the door closed behind me. My feet were bare, covered only in socks, and against the concrete, they grew cold. Aaron, ever-obliging Aaron, ever-unselfish Aaron, ever-benevolent Aaron, shimmied at once out of his coat and wrapped it around my shoulders, saying to me, \"You'll catch your death out here,\" and beneath the weight of his hands\u2014which lingered there on my shoulders, gently liberating strands of hair that were trapped under the heavy coat, warm hands tucking them behind my ear, pausing there\u2014I softened like a stick of butter left on the table too long.\n\nWe said nothing.\n\nBut I could see in his eyes that I had been wrong. That Aaron wasn't healed as I'd believed him to be the day I saw him through the chophouse windows. That he was only taped back together that day, a skimpy job at best, for the tape had come undone, it had lost its stick, and Aaron was once again broken, standing before me now, mere fragments of himself.\n\nOh, what have I done?\n\nHe hunched to my height, bending his knees ever so. He cupped his hands around my face\u2014softly, delicately\u2014as if those hands cradled an heirloom crystal vase, and I could see in Aaron's eyes that what he held was, to him, something fragile, something magical, something irreplaceable and beyond compare.\n\nThat, to him, I was irreplaceable and beyond compare as I'd always been.\n\nThat, in all these months apart, that hadn't changed for him.\n\nHis lips felt warm as he pressed them to mine, and there was nothing rushed about it, nothing presumptuous or brusque. \"I want you back,\" he whispered into my ear.\n\n\"I need you back.\n\n\"I miss you, Eden.\n\n\"I am nothing,\" he said, \"without you.\"\n\nI am nothing.\n\nWas it just my imagination, or did the baby inside me kick?\n\nI stepped back from Aaron, tugging on the ends of my sweater to make sure that tiny bulge was concealed. Inside me the baby\u2014not Aaron's baby, but the baby of some man I would never know\u2014knew how to squint its eyes and to suck its thumb. Each day it grew bigger, arms and legs lengthening, organs and cells unfolding in my womb. It would come to be a person one day, a person perhaps with cavernous dimples and sparkling blue eyes, but never would I resent this child for the choices that I made.\n\n_Be careful what you wish for_ , __ the saying goes, but never would I harbor a grudge for all that I lost to have this baby. All that I will lose.\n\n_It might just come true._\n\nI would have done anything for a baby. This I know without a shred of doubt.\n\nThe lump in my throat was nearly impossible to speak past. Something inside my larynx had swollen to two times its size and my eyes burned with tears. As they began to fall, Aaron wiped them from my cheeks with the pad of a thumb and again pressed his lips to mine, saying that everything was okay, that everything would be fine. He held me close, stroking my hair, pressing my hands between his to keep them warm.\n\n\"Can I come home?\" he asked.\n\nAnd I thought what it would do to him if I told him about the baby.\n\nIt would take those broken pieces of Aaron that remained and sliver them completely. It would pulverize them so that all that was left of Aaron would be ashes and dust.\n\n\"Yes,\" I said, feigning a smile, forcing the word past that knot. \"Yes.\"\n\nAaron's knees nearly collapsed from the relief of it. He kissed me again, this time with passion and zest, then reached his hand toward the doorknob to let us both inside.\n\nBut I stopped him.\n\n\"Not yet,\" I said. \"The house is a mess,\" I said. \"Complete bedlam. Let me clean it first,\" I told him, and though Aaron tried to shoo it off, to tell me it didn't matter, that we'd clean it together, I said no.\n\nThat I wanted it to be just right.\n\nThat I wanted it to be perfect for him.\n\nThat _I_ wanted to be perfect for him.\n\nAnd at this he relented, and an agreement was made.\n\nThe following morning he would return with all of his belongings, and we'd start over with a clean slate. We'd be Aaron and Eden again. Just us. Just Aaron and Eden.\n\nHe kissed me goodbye\u2014lips lingering on mine for what only I knew would be the last time\u2014and then he was gone, his car pulling out backward down the long, winding drive, disappearing through dark tree bark. The leaves of the trees were gone, as soon I would be.\n\nLife is full of regrets and this is only one of them.\n\nIt didn't take long to pack a bag.\n\nBy the time it was dark outside, I was heading south, south of Sturgeon Bay, south of Sheboygan, south of Milwaukee. Soon I would be living far away from here. My baby and me.\n\n_Dear Aaron,_\n\n_I had a dream last night. In it, I was being chased. I ran in slapdash circles all night long, sweating and panicked as people tend to do in dreams, and for the longest time I couldn't see the angry face of the man who was chasing me. It wasn't until later, when I finally awoke, delirious and frightened, that I realized it was you, which puzzled me a great deal because after all of the grief and the heartbreak I've put you through, you have never been anything but selfless toward me. Compassionate and kind._\n\n_You, of all people, would never hurt me._\n\n_I remembered the way it is with dreams sometimes, how they have a habit of being less literal and more metaphoric, and I thought that sometimes with dreams like this, it's not about who's chasing you, but what you're running from._\n\n_I've spent the last twenty years running from the past, Aaron, from all the horrible things I put you through. And now I'm dying of cancer. I'm going to die. But I can't stand the idea of leaving this world without explaining things to you first so that you'll understand. It's only right that you have the closure you deserve. Every single day for the past twenty years I thought about calling you, asking you to meet. But I knew I'd never be able to verbalize all that I was feeling, that I could never put it into intelligible words, nor could I bear the thought of looking you in the eye and admitting what I'd done. And so for now, my journal will have to suffice._\n\n_I have a child, Aaron, a daughter, named Jessie, who means everything to me\u2014and more. A mistake is what some might call it, but to me, she's perfection. Jessie has spent her entire life searching for her father. It should have been you._\n\n_With love,_\n\n_Eden_\n\n* * * * *\nacknowledgments\n\nFirst and foremost, thank you to my smart, savvy and thoughtful editor, Erika Imranyi, for always having faith in me and my books, and for helping take my rough drafts and transform them into something that shines. I'm so proud of the work we do together and know with confidence that my novels are far better after you've left your mark on them.\n\nThank you to my wonderful and always encouraging friend and literary agent, Rachael Dillon Fried, for having my back, for knowing just what to say when I'm in need of reassurance and for always seeing the positive in everything I do.\n\nThank you to everyone at Sanford Greenburger Associates, Harper Collins and Park Row Books, including my publicity team of Emer Flounders and Shara Alexander; Reka Rubin for sharing my novels with the world; Erin Craig and Sean Kapitain for another gorgeous cover design; the copy editors, proofreaders, sales and marketing teams, and all those who play a role in getting my story into the hands of readers. I couldn't do what I do without any of you. I'm forever grateful for your diligence, enthusiasm and support.\n\nTo my wonderful friends who do not (I promise!) provide the inspiration for my less savory characters, but always supply an abundance of encouragement throughout the process, thank you! To the booksellers, librarians, book bloggers and the media who read and recommend my novels to your customers, patrons and fans, you are amazing! And to my readers worldwide, you're the reason I keep writing.\n\nLastly, thank you to my parents, Lee and Ellen; my sisters, Michelle and Sara; the entire Kubica, Kyrychenko, Shemanek and Kahlenberg families; to my husband, Pete, and our children, Addison and Aidan; and to Holly, who kept me company while I wrote\u2014there will always be a spot for you by my side.\nISBN-13: 9781488023576\n\nWhen the Lights Go Out\n\nCopyright \u00a9 2018 by Mary Kyrychenko\n\nAll rights reserved. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this ebook on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of the publisher, Harlequin Enterprises Limited, 22 Adelaide St. West, 40th Floor, Toronto, Ontario M5H 4E3, Canada.\n\nThis is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental. This edition published by arrangement with Harlequin Books S.A.\n\n\u00ae and \u2122 are trademarks of the publisher. Trademarks indicated with \u00ae are registered in the United States Patent and Trademark Office, the Canadian Intellectual Property Office and in other countries.\n\nwww.Harlequin.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## The Icarus Syndrome\n\nA History of American Hubris\n\n## Peter Beinart\n\nTo Ezra and Naomi\n\n## Contents\n\nIntroduction\n\nPart I: The Hubris of Reason\n\nChapter 1 A Scientific Peace\n\nChapter 2 The Frightening Dwarf\n\nChapter 3 Twice-Born\n\nChapter 4 I Didn't Say It Was Good\n\nPart II: The Hubris of Toughness\n\nChapter 5 The Murder of Sheep\n\nChapter 6 The Problem with Men\n\nChapter 7 Saving Sarkhan\n\nChapter 8 Things Are in the Saddle\n\nChapter 9 Liberation\n\nChapter 10 The Scold\n\nChapter 11 Fighting with Rabbits\n\nChapter 12 If There Is a Bear?\n\nPart III: The Hubris of Dominance\n\nChapter 13 Nothing Is Consummated\n\nChapter 14 Fukuyama's Escalator\n\nChapter 15 Fathers and Sons\n\nChapter 16 Small Ball\n\nChapter 17 The Opportunity\n\nChapter 18 The Romantic Bully\n\nChapter 19 I'm Delighted to See Mr. Bourne\n\nConclusion The Beautiful Lie\n\nNotes\n\nSearchable Terms\n\nAcknowledgments\n\nAbout the Author\n\nCredits\n\nCopyright\n\nAbout the Publisher\n\n## INTRODUCTION\n\n\"If you want war, nourish a doctrine. Doctrines are the most frightful tyrants to which men are ever subject.\"\n\n\u2014WILLIAM GRAHAM SUMNER\n\nA story: the reason I wrote this book. I was sitting at a restaurant on New York's Upper East Side in early 2006, squeezed into a corner table next to an elderly, elfin man sipping a martini. He was Arthur Schlesinger, Jr.; I was petrified. I'd hero-worshipped the man since high school, when I developed a bout of Kennedy mania not uncommon among young liberals in Reagan-era Boston. I'd consumed his memoir of the Kennedy White House, and his cold war liberal manifesto, The Vital Center, and stacks of his op-eds. I didn't really understand why he was having lunch with me, and as far as I could tell, none of my halting efforts at flattery and political insight were validating his decision. He mostly wanted to gossip, but even there, I was out of my depth.\n\nThen his tone changed: \"Why did your generation support this war?\" I began to stammer, something along the lines of \"Well, of course, not everyone did; they weren't all as dumb as me.\" No laughter. \"But yes, perhaps there was something in our experience that, on average...\" Then he spilled his martini, and for the first time looked frail. Waiters swarmed; I was off the hook. We never spoke again. Less than a year later, he died.\n\nBut as one often does after flubbing a pregnant moment, I kept going over the scene in my mind, trying to formulate an answer worthy of the question and the questioner, something that convinced him I wasn't a fool. His assumption was right: Young people had supported the Iraq invasion at higher rates than their elders. According to an October 2002 Pew Research Center poll, Democrats under the age of thirty were almost as pro-war as Republicans over the age of sixty-five. A survey of bloggers, pundits, and op-ed writers revealed the same generational skew. And the reason, the culprit...suddenly it hit me: One of the culprits was him!\n\nIt was Schlesinger, after all, who had written in January 1991 that the Gulf War \"will most likely be bloody and protracted,\" that it \"will be the most unnecessary war in American history, and it may well cause the gravest damage to the vital interests of the republic.\" He may not have remembered that op-ed, but I did. I was a sophomore in college then, a dove, a worshipper of Arthur Schlesinger. On the eve of the first major war of my adult life, I was looking for guidance, and everywhere I looked, from my professors on up, the guidance was: Don't do it. Remember Vietnam. Tremble when you contemplate war. And so I trembled, and war came, and it was neither protracted nor bloody, at least for our side. In fact, the Kuwaitis greeted us as liberators, and it turned out that Saddam Hussein had been hiding a crash nuclear program, which might have reached fruition had he not been thrashed\u2014all of which made me tremble, in retrospect, about what might have happened had we not gone to war.\n\nThen came Bosnia. In 1993, with Bill Clinton trying halfheartedly to convince America's European allies to support air strikes against the Serb army that was raping and murdering Bosnia's Muslims, Schlesinger warned that \"the arguments used today for intervention in Bosnia have disquieting echoes of the arguments used 30 years ago for intervention in Vietnam.\" Vietnam! The word made me shiver. But two years later, Clinton did launch air strikes, and then he sent peacekeepers, and there was no quagmire. On the contrary, the rapes and murders stopped. The barbed wire from the concentration camps came down.\n\nFinally, in November 2001, early in the Afghan War, my hero put pen to paper yet again, this time insisting that the Bush administration's effort to depose the Taliban was proving futile. \"Perhaps,\" he wrote, \"they should have reflected on Vietnam.\" But by now I was sick of reflecting on it. It had never been my war, after all. It was a bogeyman, conjured by well-meaning but patronizing elders\u2014in the way a grandparent who had witnessed the Depression might warn against investing too heavily in stocks. By 2003, when America invaded Iraq, my generation had witnessed its own history, and it was not a history that made one tremble at the prospect of war. I was tired of hearing people cry wolf. Vietnam\u2014just hearing it invoked in a foreign policy debate made me roll my eyes.\n\nSo I had my answer to Schlesinger's question. And I imagined\u2014in that puffed-up way you imagine when you're on a pedestal speaking to yourself\u2014that he would have been unsettled. And then I had another thought, which unsettled me: My answer wouldn't have surprised him a bit. He had gone through the same process himself. As a college student in the late 1930s, he too had heard his elders invoke a traumatic war\u2014World War I\u2014as a reason not to fight again. At first he believed them, clinging to isolationism until Nazi troops reached French soil. But then he saw his elders proved wrong: For America, World War II was not a disaster; it was a triumph. And after the war, the triumphs continued. If 1989 through 2003 were our golden years, 1945 through 1965 were his. He too had been a child of the age of victory. And then Vietnam came, and he tasted tragedy, and politically he was never the same.\n\nNow history's wheel had turned again. Another generation\u2014mine\u2014had seen so much go right that we had difficulty imagining anything going wrong, and so many of us grew more and more emboldened until a war did go hideously wrong. Maybe there was an argument here about cycles\u2014Schlesinger loved cycles\u2014cycles of success leading to hubris leading to tragedy, leading, perhaps, to wisdom. Hubris. The word stuck in my brain.\n\nIt comes from the ancient Greeks, who defined it as insolence toward the gods. In Aeschylus's play Agamemnon, the great warrior, fresh from his triumph at Troy, strides upon a lush purple robe, even though he knows that purple symbolizes divine power. Angered, the gods withdraw their protection, and Agamemnon's vengeful wife hacks him to death in the bath. In another Aeschylus drama, The Persians, a young king, Xerxes, inherits a mighty empire from his father, Darius. From the start, Xerxes is tormented by feelings of inferiority, constantly reminded that Darius's empire \"was won at spearpoint \/ while he [Xerxes] not half the man \/ secretly played toy spears at home \/ and added nothing \/ to inherited prosperity.\" Determined to surpass his father's accomplishments, and emboldened by Persia's military might, he traverses the holy Bosporus, a body of water humans are not supposed to cross. Once again the gods retract their favor, and Persia's enemies massacre Xerxes' forces. Upon learning the news, Darius's spirit rises from the grave to bemoan \"my son in his ignorance \/ his reckless youth.... Mere man that he is \/ he thought, but not on good advice \/ he'd overrule all gods.\"\n\nIn Greek literature, people sometimes create their own hubris: Like Agamemnon, they win epic triumphs and thus decide they are more than human. And sometimes, like Xerxes, they inherit their hubris from the triumphs of generations past. Take the legend of Icarus. Father and son are trapped on the island of Crete. But Daedalus, a resourceful man, builds wings of feathers and wax. Don't fly too low, he cautions his son, a lighthearted youth named Icarus, or too high: \"Keep to a middle range if you can, and don't try to show off.\" At first, Icarus\u2014heeding his father's warning\u2014flies cautiously. But in his exhilaration, he gradually forgets Daedalus's words. As the Roman poet Ovid recounts the story, \"the youngster's initial fears have been mostly calmed. His confidence now has developed. He wonders what he can do with this splendid toy, what limits there are to his father's invention. He flaps his wings and rises higher\u2014but nothing bad happens. He figures he still has plenty of margin and rises higher still.\" Watching from earth, observers assume that this winged creature must be a god. \"It's exciting, wonderful fun, as he soars and wheels, but he doesn't notice the wax of his wings is melting and feathers are falling out.\" He has flown too near the sun. The wings crumble, revealing him to be mortal after all. And he plunges to his death, into the sea.\n\nThere's nothing intrinsically American about hubris. As Aeschylus and Ovid testify, it's a vice that long predates us. But since it's an affliction born from success, we've been especially prone. Over the last one hundred years, no world power has come close to matching our run of good fortune. When your cities are bombed and your lands are plundered and your government is toppled and your empire dissolves\u2014which is what happened, in varying degrees, to America's major competitors during the twentieth century\u2014you have lots of problems, but hubris is no longer one of them. We, on the other hand, who did much of the bombing and toppling and dissolving, have as a result sometimes been tempted to believe\u2014to paraphrase the campaign slogan of a certain Texas governor turned president\u2014that whatever Americans can dream, Americans can do. A rousing sentiment, but dangerous, the kind of thing that can get you into trouble with the gods.\n\nThis book is about American hubris, American tragedy, and the search for American wisdom. It's about three moments in the last century when a group of leaders and thinkers found themselves in possession of wings. In each case, the Icarus generations flapped gently at first, unsure how much weight the contraptions could bear. But the contraptions worked marvelously, and so people gradually forgot that they were mere human creations, finite and fragile. Politicians and intellectuals took ideas that had proved successful in certain, limited circumstances and expanded them into grand doctrines, applicable always and everywhere. They took military, economic, and ideological resources that had proved remarkably potent, and imagined that they made America omnipotent.\n\nIn critical ways, each generation was different. Woodrow Wilson and the pro-war progressives were more like Agamemnon; they mostly generated rather than inherited their hubris. Lyndon Johnson and the Camelot intellectuals of the 1960s, by contrast, and George W. Bush and the post\u2013cold war conservatives, were more like Xerxes and Icarus: eager to surpass the accomplishments of the Daedalus generations that came before. The ideas were different, too. Wilson succumbed to the hubris of reason: the belief that America was wise and powerful enough to turn the jungle of international affairs into a garden. Johnson and the Camelot intellectuals (including, for a time, Arthur Schlesinger) fell in love with toughness: the idea that through unyielding force America could halt communism's march anywhere in the world. George W. Bush and his conservative supporters (with an assist from liberal hawks like me) grew entranced by the idea of dominance: the belief that America could make itself master of every important region on earth.\n\nIn each case, politicians and intellectuals built on what came before, pushing ideas further and further, until, like a swelled balloon, they burst. The problem wasn't the ideas themselves; in limited form, they had often served the nation well. In fact, ironically enough, they had emerged partly as correctives to the hubris of a prior age, before inflating themselves. It is tempting, in the wake of foreign policy disaster, to damn an entire intellectual tradition for what it ultimately wrought. Thus, since Iraq, some liberals have projected back into history a unified conservative foreign policy lineage, always pregnant with the sins that the Bush administration would bring forth into the world. But I've grown distrustful of this way of thinking, which imagines that there are stained bloodlines and pure ones, and grown more partial to Reinhold Niebuhr's suggestion that in foreign policy, as in life, evil often grows from unchecked good. Harry Truman's efforts to contain communism in Europe were wise, even though they helped plow the ideological soil for Vietnam. Bush the father was right to fight the Gulf War, even though it helped breed the confidence that led his son to overthrow Saddam. Hubris is not the possession of any one party or intellectual tradition; it is any intellectual tradition taken too far. And foreign policy wisdom sometimes consists of understanding that the very conceptual seedlings you must plant now can, if allowed to grow wild, ravage the garden.\n\nBut how do you know when the former sapling, which once bore tasty fruit, is starting to strangle the other plants? In Greek literature, the gods don't tell you. Humans are simply expected to know. And to make matters even trickier, excessive ambition is not the only human pitfall; insufficient ambition can be deadly, too. Daedalus didn't just warn Icarus not to fly too high; he also warned him not to fly too low. In fact, Daedalus was, in his way, quite a risk-taker himself. He believed that he and his son could fly, which in itself might have been considered a transgression against the divine. The Greeks could have spun a legend that culminated with his death as well, thus implying that he should have remained earthbound and imprisoned. But they didn't, even though they had no evidence that humans could fly. They imagined it, and because of human ambition, now we can.\n\nSo where does ambition end and hubris begin? There's no formula for answering that. In fact, the belief that you've discovered a formula that works in all situations is itself a sign that you've crossed the line. To some degree, foreign policy is all about deciding in which direction you'd rather be wrong. Are you so intent on making sure America doesn't fly too high that you oppose not only invading Iraq, but saving Kuwait? Are you so determined to avoid flying too low that you support not merely World War II but Vietnam? Barely anyone will be right every time, because the gods don't speak to us. Or, as Warren Buffett has said about investing in a bull market, it's like Cinderella at the ball. She knows that if she stays too late her chariot will turn into a pumpkin and her gown will turn to rags. But she doesn't want to leave too early and miss meeting Prince Charming. The problem is that there are no clocks on the wall.\n\nNo clocks, but there are warning signs, the kind that someone with lots of experience at balls might notice. One warning sign is overconfidence: a political climate in which influential people assume that the war's outcome is preordained; that because of America's military prowess or economic resources or ideological appeal, we cannot possibly lose. When influential Americans talk that way, it's usually because America has not lost in a long time. We're all prisoners of analogy, and people tend to imagine that the future will be like the immediate past: that Vietnam will be like Korea or World War II; that Iraq will be like the Gulf War or the initial phase of Afghanistan. But that reliance on analogy often blinds us to the ways in which, rather than replicating the successes of the past, we are reaching beyond them, taking on more risk. We think we are running on a treadmill when we are actually ascending a ladder.\n\nIf overconfidence is one danger sign, unilateralism is another. France, Canada, and Britain, which had fought alongside the United States in World War II and Korea, refused to join us in Vietnam. France, Canada, and Germany, which had supported the United States in the Gulf War, the Balkans, and Afghanistan, opposed invading Iraq. The problem with failing to convince other countries\u2014and particularly the Western democracies that broadly share our values and interests\u2014to back us in war is not necessarily that we need their help to win. (These days, given the vast gap between our military capacity and theirs, they may be as much a burden as a help.) The problem is that their unwillingness to back us may be evidence that winning is impossible. What we need, in other words, is not our allies' tanks but their judgment. It's like a patient contemplating a high-risk medical procedure: You may not need several doctors to perform the operation, but you want several doctors to confirm that the operation can be successfully performed at all.\n\nThe sober judgment of allies is especially important for a nation intoxicated with success. (As the old saying goes, when three friends say you're drunk, lie down.) Americans sometimes dismiss our European allies as defeatist, so burdened by their tragic histories and so enfeebled by their military weakness that they instinctively choose retreat over confrontation. But it is precisely because they have been battered by history that they may be able to spot hubris when we, because of our more triumphant experience, cannot. Before Vietnam, and again before Iraq, French leaders urged the United States to learn from France's imperial misfortunes in Southeast Asia and the Arab world. But Lyndon Johnson and George W. Bush scoffed at the idea that we had anything to learn. The French, after all, were history's losers. We were its winners. They were mere mortals; we were America.\n\nA third flashing light is excessive fear. Even when America's leaders fly nearest the sun, they generally insist that they are merely taking defensive measures against grave threats. Hubris rarely speaks its name. And to some extent, those leaders are right: Foreign threats usually have something to do with America's decision to launch a war. Woodrow Wilson would not have taken America into World War I just because he believed he could reorder world politics; German subs were sinking our ships. The United States would not have fought the Vietnam War had Ho Chi Minh not been a communist, no matter how confident we were of our military might. But the problem with explaining America's wars solely as a response to threats is that our threat perceptions vary wildly over time. Things we take in stride at one moment terrify us at another. In 1939, few American politicians believed that a Nazi takeover of Warsaw constituted a grave danger to the United States. By 1965, many believed we couldn't live with a North Vietnamese takeover of Saigon. In the 1980s, Americans lived peacefully, albeit anxiously, with thousands of Soviet nuclear warheads pointed our way. By 2003, many Washington commentators claimed that even Iraqi biological or chemical weapons put us in mortal peril. How threatened American policymakers feel is often a function of how much power they have. The more confident our leaders and thinkers become about the hammer of American force, the more likely they are to find nails.\n\nThat's the problem with explaining Iraq merely as a response to the terrorist attacks of September 11, 2001, as the Bush administration did. Obviously, 9\/11 mattered: Without it, we would not have sent more than one hundred thousand U.S. troops into Iraq. But had 9\/11 occurred in 1979 or 1985 or 1993, moments when America was less confident that its military was unstoppable, its economic resources plentiful and its ideals universal, it is unlikely that we would have responded by launching two distant wars in short succession, something the United States had never done in its entire history. In fact, even in 2002, had the war in Afghanistan not initially gone well, thus further buoying the Bush administration's self-confidence, America probably would not have invaded Iraq. And had Bush officials doubted that they could successfully invade and remake Iraq, they probably would not have declared Saddam an urgent and intolerable threat. America's leaders tend not to tell us we are in grave danger unless they think they can do something about it.\n\nThis is not to say it's always a mistake for the United States to wage war in places it once considered irrelevant. Over the course of the last century, as America has risen to global power, we have naturally come to worry about regions of the world to which we previously gave little thought. But when we allow our fears to swell so dramatically that quelling them would require virtually unlimited quantities of money and blood, something has gone wrong. Once Lyndon Johnson declared a communist takeover in backward and remote Vietnam to be a grave threat to the United States, then a communist takeover in virtually any country constituted a grave threat. Once the Bush administration said America was in mortal danger because Saddam was supporting terrorism (not even terrorism against the United States, just terrorism) and seeking weapons of mass destruction, America was suddenly in mortal danger from Iran, North Korea, Syria, and perhaps Pakistan, too.\n\nA wise foreign policy starts with the recognition that since America's power is limited, we must limit our enemies. That's why Franklin Roosevelt hugged Stalin close until the Soviet Union had helped us defeat Nazi Germany, and why Richard Nixon opened relations with China, so the United States wasn't taking on Moscow and Beijing at the same time. By contrast, when America's leaders outline doctrines that require us to confront long lists of movements and regimes simultaneously, it's a sign that we've lost the capacity to prioritize. And that, in turn, is a sign that we think we are so powerful that we don't need to prioritize. And that, in turn, is a sign that we're flying too high.\n\nThese warning signs are a starting point. But in and of themselves, they are too crude. Because people are not particles, history is not physics. History does contain cycles\u2014attitudes, behaviors, and outcomes that rhyme\u2014but every moment is also unique and every event has multiple causes. Telling the story of the last century of American foreign policy as cycles of success leading to hubris leading to tragedy leading to wisdom, and then leading to more success, is like looking at a page covered with tiny, fuzzy dots and graphing the places they rise and fall. Reality is far messier than your smooth lines suggest. Others can map the evidence a different way.\n\nFor policymakers, the challenge is even tougher: to predict, on the basis of the dots you've already seen, where the next ones will fall. To do that well requires a trained eye. Just as a skilled radiologist can see patterns on an X-ray that may elude a medical student, the people who make foreign policy must draw meaning from a hazy landscape. In this effort, two kinds of knowledge can help. The first is a broad knowledge of history: not just American history, but the history of other great powers that in their day felt history's wind at their backs. This knowledge can provide a wider menu of analogies than the ones offered by America's recent past. Maybe Vietnam had more in common with the Spanish-American War than with Korea? Maybe our invasion of Iraq was more like Napoleon's invasion of Egypt than like the Gulf War? That American hawks relentlessly compare our adversaries to Hitler, and American doves relentlessly compare our wars to Vietnam, suggests that for much of our political class, the menu of available historical analogies is depressingly small.\n\nIf a broad sense of history can help policymakers navigate the fog, so can an intimate feel for one specific place. It is ironic that in recent years many American foreign policy commentators and practitioners, hoping to be the next George Kennan, have tried to formulate a universal foreign policy doctrine for the post\u2013cold war, or post-9\/11, age. In reality, Kennan, like William Graham Sumner, hated and feared universal doctrines. And after he saw containment swelling into one, starting in the late 1940s, he spent the next four decades screaming himself hoarse in opposition. What Kennan believed in, and what his writing on the Soviet Union exemplified, was the value of deep, local knowledge. By the time he wrote his famous \"Long Telegram\" from Moscow in February 1946, Kennan was on his third Russian posting and had also served in Riga and Tallinn, on Russia's northwest border. He spoke formal Russian better than Stalin (whose native language was Georgian), was writing a biography of the Russian writer Anton Chekhov, and would later author a prizewinning account of American foreign policy during the Russian Revolution. It was his profound understanding not only of Russia's politics but of its history and culture that allowed Kennan to conceive of containment, a strategy he developed not for dealing with \"communism\"\u2014an abstraction he thought had little foreign policy meaning in and of itself\u2014but for dealing with one communist regime at one moment in time.\n\nIn post\u2013World War II America, where many saw the Soviet Union through the prism of America's last totalitarian foe, Nazi Germany, Kennan showed why the present was unlike the past; why Stalin, unlike Hitler, could be stopped by measures short of war. Like Daedalus, he crafted a set of conceptual wings appropriate to the specific problem at hand. And because he had built the wings, he could see when they were starting to melt.\n\nIt is no coincidence that a generation later, when the Johnson administration waded into its Indochinese bog, there were no high-level policymakers whose knowledge of Vietnam remotely rivaled Kennan's knowledge of Russia. On the contrary, America's best Asia specialists\u2014people like Kennan's friend John Paton Davies\u2014had been hounded from government by Joseph McCarthy, in part for daring to suggest that Asia's communist parties might not be clones of their Soviet counterparts. Similarly, there were no high-level policymakers in the Bush administration whose knowledge of Iraq approached the Kennan standard, either. Partly as a result, after 9\/11, Bush and his supporters implied that all Middle Eastern terrorist movements and terror-supporting regimes were basically the same. They lumped together Iraq, Iran, Syria, Al Qaeda, Hezbollah, Hamas, and the Taliban into a single ideological construct called \"jihadist terrorism\" or \"Islamofascism\" and then outlined a set of responses\u2014preventive war and coercive regime change\u2014that they proudly equated with containment. The irony was that the Bush administration's one-size-fits-all approach to fighting terrorism was like containment: not Kennan's narrowly tailored strategy for containing Stalin's U.S.S.R., but Johnson's hubristic effort to contain communism everywhere in the world.\n\nHaving implied that national differences were irrelevant, the Bush administration then proceeded to appoint men too ignorant to spot them anyway. In May 2003, in the crucial first months after the overthrow of Saddam, Bush sent a man to run Iraq, L. Paul Bremer, who had never before been posted to the Arab world. To grasp the intellectual chasm between American foreign policy toward the U.S.S.R. in 1946 and American foreign policy toward Iraq in 2003, one need only try to envision Bremer writing a biography of an Iraqi writer, or, for that matter, being able to name one.\n\nIn important ways, America's wars in Afghanistan and Iraq echo the Icarus tale. But they differ in one crucial respect: Icarus dies from his hubris; America does not. In this sense, we are more like Xerxes in The Persians, who survives the consequences of his arrogance and returns home in rags. Tormented by the chorus, which demands an accounting for all the great soldiers who now lie dead, Xerxes at first wallows in self-pity. He blames the gods; he declares himself cursed; he says he wishes he too had died. But then he puts on a new robe, and rather than suffering the chorus' abuse, he begins to direct its actions. \"We have been shattered by such a fate,\" he declares, but then tells the wailers to go home and prepare for the future.\n\nAs the play ends, there is at least the hint that although Xerxes has been transformed by tragedy, he has not in fact been shattered by it. He no longer feels sorry for himself. He can still lead his people. He turns their attention to the possibilities of a new day. Robert Kennedy, who like Arthur Schlesinger was transformed by Vietnam, and who late in life developed a fascination with the Greeks, once said that \"tragedy is a tool for the living to gain wisdom, not a guide by which to live.\" If the last few years have acquainted a new American generation with tragedy and made us less inclined to believe that the course of history\u2014or the power of the gods\u2014is on America's side, we should see that as a cause not for despair, but for hope. For it is only when we abandon our false innocence and unearned pride that we can look around at the bits of feathers and wax that still surround us, and begin, carefully, to build wings.\n\n## PART I\n\n## THE HUBRIS OF REASON\n\n## CHAPTER ONE\n\n## A SCIENTIFIC PEACE\n\nWoodrow Wilson had very few friends, and that bothered him. People considered him \"cold and removed,\" he groused. He wished journalists would write about his lighter side\u2014his love of baseball, his gift for mimicry, his flair for limericks\u2014but instead they depicted him as a bloodless \"thinking machine.\" He longed for a nickname. Perhaps if he had kept his birth name, Thomas, he mused, people would call him \"Tommy\" and thus find him more approachable. Theodore Roosevelt was known as \"Teddy,\" and no one ever called him cold and removed.\n\nThe problem was that while Wilson loved the idea of friends, he wasn't very friendly up close. One reporter compared his handshake to \"a ten-cent mackerel in brown paper.\" A Baltimore ward leader said the president \"gives me the creeps. The time I met him, he said something to me, and I didn't know whether God or him was talking.\" Wilson knew his reputation as aloof was partly his own fault. \"I have a sense of power in dealing with men collectively which I do not feel always in dealing with them singly,\" he acknowledged, which may have had something to do with the fact that when he dealt with them singly, they often talked back.\n\nThere was one big exception, a genuine, honest-to-goodness presidential friend: Colonel Edward House. The two men took drives together, gossiped in their nightgowns, read poetry aloud to one another, shared secrets of their love lives. House, remarked Wilson, \"is my second personality. He is my independent self. His thoughts and mine are one.\"\n\nFor Wilson, House had three endearing characteristics. First, he shared the president's background, having been raised in the genteel South. Second, he was a world-class sycophant. Third, he did the things that Wilson needed done but didn't like doing himself. The \"Colonel\" was not actually a military man at all; he was a fixer, highly skilled in the dark arts of patronage, conspiracy, and intimidation. \"He could walk on dry leaves,\" remarked Oklahoma Senator Thomas Gore, \"and make no more noise than a tiger.\" He always made you feel \"intimate,\" noted another observer, \"even when he was cutting your throat.\"\n\nWoodrow Wilson, a politician with little patience for the more corporeal elements of his craft, needed someone like that, and he rewarded House with both affection and power. For a long spell in the middle of Wilson's presidency, House wielded more influence over American foreign policy than the secretary of state and the secretary of war combined. He was a kind of ambassador without portfolio: Wilson's emissary to the actually existing human race.\n\nWalter Lippmann also seduced the powerful, but he did it through the front door. Disappointed by his own father, he acquired others, becoming the brilliant son that great men felt they deserved. At nineteen, as a Harvard junior, he took weekly tea with one world-renowned philosopher, William James, and gossiped over dinner with another, George Santayana. After graduating, he apprenticed for the famed muckraking journalist Lincoln Steffens. His first book, written at age twenty-three, drew praise from Sigmund Freud. After his second, published the following year, Theodore Roosevelt declared him \"the most brilliant young man of his age in the United States.\" By 1916, at age twenty-seven, he was dining at the White House, and his editorials for a new progressive journal, the New Republic, were finding their way into Wilson's speeches. In the spring of 1917, while his contemporaries were being shipped across the Atlantic to the charnel house of World War I, he procured a draft exemption from Secretary of War Newton Baker, who was accumulating whiz kid assistants. Then, a few months later, as a damp Washington summer gave way to a frigid fall, Baker called Lippmann into his office. Colonel House wanted to meet him outside.\n\nThe fixer and the wunderkind began to walk, past Pennsylvania Avenue and the White House, and down Seventeenth Street toward the Potomac. In the distance spread the marshland that would become the National Mall, and beyond that the partially completed Lincoln Memorial, its white columns already standing, but empty of the seated figure who would reside within. As they walked, House described a project of the most fearsome secrecy. America had been at war for only six months, but already Wilson was designing a peace to redeem the slaughter. Three million Russians and five million Germans were either wounded or dead. In France, four thousand villages had simply ceased to exist. The European state system, long anchored by four vast overland empires, based in Berlin, Moscow, Vienna, and Istanbul, was imploding. Like most Americans, Wilson had hoped against hope that the United States could escape the widening gyre. He had resisted fierce pressure to join the fray, even after hundreds of U.S. civilians were drowned by German subs. On the night he finally asked Congress to declare war, he returned to the White House and wept. But now that America had joined the battle, Wilson was determined to build a new world on the ruins of the old, to ensure that barbarism never overthrew civilization again.\n\nWhen the slaughter stopped, the world would need a new map, which gave every nation the territory it deserved and not an inch more. And it would need laws, so that international affairs no longer followed the law of the jungle. For Wilson, it fell to America\u2014which he considered the sole disinterested combatant, the only nation that wanted nothing out of the war save that it not happen again\u2014to draw up the plans. And House was tapping the country's best minds for the task. He told Lippmann to join a group of experts working from a secret office in the New York Public Library. To disguise the project's import, they would call it simply \"the Inquiry.\" From the ashes of war they would construct what House called a \"scientific peace.\"\n\nA scientific peace. The concept came straight from the belly of the progressive movement. Progressivism, historians insist, was not one thing. It was a swarm of impulses and interests, often colliding with each other. But if progressivism was an ideological cacophony, one note cut through the din: faith in human reason. This faith predated World War I. Indeed, it grew from progressivism's success in remaking the United States. Surveying their country at the dawn of the twentieth century, many progressives had seen a society hurtling toward a second civil war. Industrialization, urbanization, and immigration, they believed, were dividing the nation against itself, making it a battlefield of hostile tribes\u2014business versus labor, urban versus rural, immigrant versus native-born\u2014each trying to smash each other into submission, each trampling democracy along the way. It was up to government to impose order, not through brute force\u2014then it would be just another selfish tribe\u2014but through the force of reason. Government would provide answers to society's conflicts: answers so disinterested, so rational, so self-evidently fair that the tribes would lay down their arms. It was no accident that many progressives saw a scientific peace as the answer to Europe's war. For close to two decades, they had been fashioning a scientific peace inside the United States.\n\nRational, disinterested policies required rational, disinterested policymakers: experts. And experts occupied a hallowed place in the progressive mind. Lippmann's boss at the New Republic, Herbert Croly, a homely, taciturn man\u2014alternately likened to a yogi and a crab\u2014suggested entrusting them with a fourth branch of government. The eccentric Norwegian-American economist Thorstein Veblen proposed handing them control over the economy. And no one gave the concept grander expression than House himself. Five years earlier he had anonymously published a strange little novel, in which a young \"mastermind\" named Philip Dru, watching the United States sink into civil war, seizes power and declares himself \"Administrator of the Republic.\" He replaces Congress with a commission of five expert lawyers who decree all the high-minded reforms that selfish interests have long stymied. With order and reason thus enthroned in the United States, Dru sails off into the Pacific with his bride, determined to bring a scientific peace to the world beyond America's shores.\n\nPhilip Dru: Administrator was a lousy book, but a revealing one. Dru is not merely an expert. He is also an inspirational leader, able to rally the public to his cause. And as a great leader, he is largely an educator. While they revered experts, the progressives knew it took leaders to convince\u2014to educate\u2014the public to accept their conclusions. \"The nation,\" wrote Croly, \"like the individual, must go to school.\" And if the nation was a schoolhouse, the president was its schoolmaster. It was no coincidence that John Dewey, progressivism's most influential democratic theorist, was also a philosopher of education. To make democracy work, leaders had to educate Americans to embrace\u2014indeed, to demand\u2014the rational answers formulated on high.\n\nThis, then, was the essential progressive equation: Objective experts plus inspiring leaders plus educated citizens equal a society governed by reason, not force. And underlying it all was the most basic faith of all: in humanity itself. Against the late-nineteenth-century Social Darwinists who declared people fundamentally selfish, progressive thinkers like Dewey, Veblen, and the historian Charles Beard insisted that they were naturally generous and cooperative; it was the evil of anarchic capitalism that made them act like beasts. Against the theologians who saw humanity forever maimed by original sin, Walter Rauschenbusch and the champions of Social Gospel Christianity insisted that evil resided in the world, not in man.\n\nProgressivism's great discovery, declared the historian Christopher Lasch, was the \"lost innocence of the race.\" Progress was possible because deep inside, people were better\u2014much better\u2014than the world in which they lived. It was a stirring faith, nurtured by the progressive movement's success in bringing reason and order to the United States. And at a fevered moment at the height of World War I, after the planet itself had erupted in civil war, Woodrow Wilson decided to export it\u2014via the sword\u2014to the entire world. For a glittering constellation of American intellectuals, it was the crusade of a lifetime, a high-water mark of American optimism that would not be reached again until our own time. And it was shot through with hubris, hubris that America could make politics between nations resemble politics between Americans, hubris that the progressives could build a world governed by reason when, as it turned out, they weren't always that reasonable themselves. It was a hubris from which America would not fully recover until Wilson and House were dead, Lippmann was a weary, humbled man, and the world had endured more catastrophe than their innocent minds could even imagine in the brisk and intoxicating fall of 1917.\n\nBy the time House dispatched Lippmann to help draft a progressive charter for the postwar world, progressivism had already transformed public life in the United States. And it was because of their success rationalizing government at home that reformers dared imagine they could rationalize the entire globe. Until 1900, the tribes\u2014especially the mega-corporations, or \"trusts,\" which had amassed vast wealth since the Civil War\u2014had dominated the impotent, corrupt federal government. But as the new century dawned, impartial experts and heroic leaders began to seize command.\n\nThey didn't come much more heroic than Theodore Roosevelt. A war hero, a boxer, a ferocious reader with a photographic memory, and a big-game hunter who did his own taxidermy, America's twenty-sixth president was also the author of a four-volume history of the American West; biographies of Gouverneur Morris, Thomas Hart Benton, and Oliver Cromwell; the standard work on the Naval War of 1812; a book on child rearing; and several on the birds of New York state. As deputy sheriff of Medora, North Dakota, in the 1880s, he had captured three men who tried to steal his boat, and guarded them single-handedly for forty hours, keeping himself awake by reading Tolstoy.\n\nRoosevelt inaugurated the progressive age. In May 1902, 150,000 soot-covered men emerged from deep inside the earth in eastern Pennsylvania. They were anthracite coal miners, men at constant risk of suffocation, asphyxiation, or explosion, and they were on strike in pursuit of an eight-hour workday, a 20 percent wage hike, and recognition of their union. Past strikes had usually been resolved with buckshot, as the trusts and their government lackeys bloodied workers into submission. But Roosevelt didn't want a class war; he wanted a scientific peace, a settlement based on reason, not force. So he created what would become the quintessential progressive entity: an expert commission. Two engineers, a business specialist, a union leader, a judge, a priest, and the federal commissioner of labor heard testimony from 558 witnesses. They compiled statistics on the frequency of mine accidents, the quality of company-sponsored medical care, and the number of children working underground. In the end they gave the miners almost exactly half of what they asked for, along with a permanent commission to settle future disputes\u2014all without anyone getting shot.\n\nUsing the coal settlement as a model, Roosevelt then proposed a Bureau of Corporations to gather data on business practices, and a Bureau of Labor to do the same for working conditions. When conservative senators balked, he took his case to the people, telling cheering crowds that selfish corporations must submit to rational control. In 1906, he signed the Hepburn Act, which empowered another group of experts\u2014the Interstate Commerce Commission\u2014to hold hearings on railroad pricing, and in case of abuse, set rates itself. Meanwhile, Upton Sinclair's bestseller about the revolting conditions in Chicago packing houses, The Jungle, was sparking a furor over the safety of American meat. Roosevelt invited Sinclair to the White House, then sent government investigators to confirm his findings. When they did, Roosevelt muscled through the Meat Inspection Act, which authorized Agriculture Department experts to ensure that America's chickens, hogs, and cows were not poisoning America's people, and to shut down slaughterhouses if they did.\n\nFrom coal to railroads to beef, this was the ethic of reason sprung to life. Experts and muckraking journalists gathered data about the irrationality and injustice tearing America apart. A charismatic leader used it to educate the people, building a mighty tide of public opinion that overwhelmed the selfish tribes. And thus empowered by an active and reasoned populace, the leader built a permanent machinery of investigation, creating a virtuous cycle of objective information, public education, and scientific peace.\n\nBut these achievements were a mere warm-up for what progressives would accomplish under their greatest White House champion, the man with few friends but epic dreams, Thomas Woodrow Wilson. Even more than other progressives, Wilson feared a second civil war, mostly because he remembered the first one. His earliest memory was of standing outside his family's home in Augusta, Georgia, at age four and hearing people yell that Abraham Lincoln had been elected and there would be war. When war came, he watched Confederate soldiers moaning on gurneys inside his father's church, which had been turned into a makeshift hospital, and he peered at the Union soldiers milling in the courtyard outside, which had been turned into a makeshift jail. The war split the Wilson family asunder; its northern and southern branches never spoke again. \"A boy never gets over his boyhood,\" Wilson later declared, and his left him with a gnawing anxiety that order might break down once more.\n\nWilson never truly identified with either Blue or Gray. In his mind, they were both selfish tribes. Even as a boy, he tried to stand outside the conflict, to find an impartial, rational vantage point. When he played soldier, his imaginary armies flew neither the Stars and Stripes nor the Stars and Bars, but rather the Union Jack. As he grew older, Wilson came to idolize Lincoln, not because Lincoln had led the North to victory, but because, in Wilson's mind, he had stood above the parochial interests of both sides and thus restored unity to a divided nation. Lincoln \"was detached from every point of view and therefore superior,\" Wilson said of his political hero. \"You must have a man of this detachable sort.\"\n\nIn his own mind, Wilson was such a man. He considered himself detached from narrow regional interests because although born in the South, he had spent his adulthood in the North and so understood both worlds. And he considered himself detached from America's looming second civil war\u2014between rich and poor\u2014because he was neither a worker nor a capitalist; he was a scholar. He believed government could be \"reduced to science,\" a science in which he had particular expertise, since his own writings had helped establish public administration as an academic discipline in the United States. In 1910, when he ran for governor of New Jersey, his first foray into electoral politics, Wilson boasted that as the president of Princeton University he had run an institution dedicated to the production of unbiased expertise.\n\nThe job of impartial experts, in Wilson's view, was to write impartial rules. And in their grandest form those rules took the form of constitutions. Constitutions were precious to Wilson. The son and grandson of Presbyterian ministers, he was reared in a religious tradition that placed special emphasis on the covenant between God and his people. For Wilson it was this basic compact, writes one biographer, that \"imposed a comprehensible pattern\u2014orderly, predictable, and permanent\u2014upon the transient character of human affairs.\" Wilson saw the Bible as the greatest constitution of all, \"the 'Magna Charta' of the human soul.\"\n\nIt was fitting, in Wilson's view, that the Civil War had ended in a rewritten American constitution, a new, fairer set of rules under which the entire country could peaceably live. And in his own life, Wilson drafted constitutions wherever he went. As a kid, he wrote one for his local baseball league. When he turned seventeen, he founded an imaginary yacht club and wrote a charter for it as well, complete with bylaws, regulations, and penalties for those who disobeyed. As an undergraduate at Davidson College, he transcribed the debating society's constitution by hand. At Princeton, where he transferred, he tinkered with the constitution of not one debating society, but two. From the University of Virginia, where he studied law, to Johns Hopkins, where he got his Ph.D., to Wesleyan, where he taught, it was more of the same. Like a legalistic Johnny Apple-seed, Wilson traveled from place to place replacing disorder with order, unfairness with fairness, anarchy with law. He even recommended to his wife that they draft a constitution for their marriage. Let's write down the basic principles, he suggested; \"then we can make bylaws at our leisure as they become necessary.\" It was an early warning sign, a hint that perhaps the earnest young rationalizer did not understand that there were spheres where abstract principles didn't get you very far, where reason could never be king.\n\nIf Wilson's devotion to order and reason was classically progressive, so was his faith that leaders could educate people to want them. It was no coincidence that so many of his constitutions were written for debate societies. \"Statesmen,\" Wilson declared, \"must possess an orator's soul,\" and he certainly did: He was among the greatest orators in the history of American politics. If Wilson believed he was impartial, he believed just as strongly that his words could make ordinary people impartial, too. He possessed, his press secretary explained, \"an almost mystical faith that the people would follow him if he could speak to enough of them.\" He knew that people sometimes believed irrational, selfish things. He just didn't think they would continue to believe them after listening to him.\n\nNothing in Wilson's early years as president shook that faith. On the contrary, his already considerable self-confidence swelled as he compiled one of the most dazzling first terms in American history. A month after taking office, he put his oratorical gifts to use, becoming the first president since John Adams to address Congress in person. His topic was the tariff on imports, which had more than doubled since the Civil War. In Wilson's view, the tariff was irrational: Many of the sheltered industries were perfectly capable of competing with their foreign competitors. They enjoyed government protection not because it served the public interest, but because they had members of Congress in their pockets. Outraged by Wilson's proposal, the selfish tribes descended on Washington to preserve their irrational advantage. But Wilson met them head-on. In a dramatic appeal to the American people, he denounced the \"insidious\" lobbyists seeking \"to overcome the interest of the public for their private profit.\" Progressive senator Robert La Follette then shrewdly launched an investigation into whether the senators who opposed reform were benefiting personally from particular tariffs. Humiliated by what La Follette's investigation uncovered, several flipped their vote and the tariff bill passed. \"I did not much think we should live to see these things,\" exclaimed Wilson's ecstatic secretary of agriculture. Three years later tariff rates were entrusted to an expert commission, ostensibly shutting out the selfish tribes forever.\n\nFrom there, Wilson turned to the banks, which, according to a muckraking investigation by progressive House Democrats, used political rather than scientific criteria to make loans. The result was a Federal Reserve Board staffed with presidentially appointed expert economists to oversee America's regional banks. In the space of eight months, Wilson had passed the first tariff reduction in almost twenty years and the first banking reorganization in almost fifty.\n\nEleven months later, he passed the third jewel of his progressive agenda: a Federal Trade Commission empowered to investigate business fraud and abuse, ban unfair practices, and force the offenders to compensate their victims. And when Congress reconvened in December 1915, it passed seven smaller pieces of progressive legislation, including the creation of a Shipping Board to ensure safe working conditions for sailors, government loans to struggling farmers (the recommendation of a Wilson-appointed expert commission), a ban on child labor, workers' compensation for government employees, and an eight-hour day for railroad workers, which averted a potentially bloody nationwide strike.\n\nYears later, after the war had made progressive an epithet\u2014when Wilson was a pariah and reform lay flat on its back\u2014critics would look back at these initiatives and see not impartial reason, but selfish interest in disguise. Entrenched industries learned how to turn the bodies designed to oversee them to their own advantage, often by helping to write regulations with which their smaller competitors could not afford to comply. No matter how tightly the progressives tried to lock the door, the inequities of power usually found a way in.\n\nAnd the corruption did not only come from outside; the progressives themselves were not as disinterested as they liked to believe. Many of the experts appointed to the Tariff Commission, the Federal Trade Commission, and the Federal Reserve turned out to have their own biases and interests. Even Wilson himself, though convinced that he, like Lincoln, was detached from irrational prejudices, actually held quite a few of them, including a virulent antipathy toward blacks, an instinct that women shouldn't vote, periodic suspicions about the loyalty of America's newer immigrants, and an unshakable faith in the number 13. In expecting the public to be motivated by logic alone, Wilson was holding it to a standard that he could not meet himself.\n\nThis hubris of reason would become clearer when Wilson exported his progressive assumptions overseas. But in his triumphant first term, it was easy to be bullish about human progress. America had not experienced a traumatic war in fifty years, or a serious economic downturn in twenty-five. The country had been booming for so long that the influential University of Pennsylvania economist Simon Patten declared that America had transitioned from a \"Pain Economy\" to a \"Pleasure Economy,\" in which scarcity would soon be a thing of the past. All this economic dynamism had produced class conflict, to be sure, but after more than a decade of progressive reform the government seemed to have that well in hand. Experts were steadily expanding their writ over the American economy, replacing selfishness with science and anarchy with order, and the American people\u2014judging by their enthusiasm for first Roosevelt and now Wilson\u2014were cheering them on. In the early years of a thrilling new century, America was fulfilling its destiny: It was becoming a republic of reason. \"It was a happy time, those last few years before the First World War,\" Lippmann would later reflect. \"The air was soft, and it was easy for a young man to believe in the inevitability of progress, in the perfectibility of man.\"\n\nIn 1910, the progressive-minded steel magnate Andrew Carnegie wrote a letter to the trustees of his newly formed Carnegie Endowment for International Peace. Their first project, he instructed, was to abolish war. Once that was accomplished, they should reconvene to \"consider what is the next most degrading evil or evils whose banishment...would most advance the progress, elevation and happiness of man.\" He envisioned the process continuing on like this, with the trustees periodically checking off problems and thus assisting \"man in his upward march to higher and higher stages of development unceasingly; for now we know that man was created, not with an instinct for his own degradation, but imbued with the desire and the power for improvement to which, perchance, there may be no limit short of perfection even here in this life upon earth.\" It was the kind of thing that, looked back upon from the other side of the abyss, could make you either laugh or cry.\n\nFor some Americans, the outbreak of World War I challenged this soaring optimism. Progressivism's success \"was beginning to shake me in my very firm belief in original sin,\" wrote a friend of Lippmann's. \"This war has restored it triumphantly.... How is Walter going to quench this fundamental and illogical passion in us all?\" It was the question: If humans were rational creatures and progress was history's natural course, why were the most advanced nations on earth sending their young men to cower in trenches and gag on poison gas? But most progressives parried it in classic American fashion: by drawing a bright line between the enlightened new world and the degenerate old. Europe was the past; America was the future. Europe was the disease; America was the cure. \"Evil and suffering did not of itself invalidate progressivism,\" notes the historian John Thompson; \"on the contrary, its existence had always been the spur\u2014so long as belief in the possibility of amelioration was retained.\" For Wilson, progressivism's success in fostering a scientific peace at home proved that amelioration was not just possible, but likely. \"Europe is still governed by the same reactionary forces which controlled this country,\" he explained, but that \"old order is dead...[and] the new order, which shall have its foundation on human liberty and human rights, shall prevail.\"\n\nBy \"old order,\" Wilson meant, above all, \"that unstable thing which we used to call the 'balance of power.'\" For centuries, it had been the core principle of European statecraft. Nations looked out for their self-interest, and only their self-interest. They allied, schemed, armed, even warred, all to ensure that no adversary\u2014or combination of adversaries\u2014grew powerful enough to threaten their very existence. The balance of power was disorderly, unfair, and unscientific, everything Wilson disdained. At best, each country's efforts at self-preservation and self-aggrandizement produced an equilibrium in which no nation, or group of nations, grew so powerful as to launch an attack, and none grew so weak as to invite one. The most anyone could hope for was a peace of drawn swords. The logic behind the balance of power mirrored the logic behind the unregulated free market. Given that nations, like people, were inherently selfish, a system based upon that selfishness was wiser than a system based upon the illusion that anyone gave a damn about anyone else.\n\nFor many European statesmen, who had learned from harsh experience to distrust their carnivorous neighbors, the balance of power was like gravity. You might not love it; but you defied it at your peril. \"We too came into the world with the noble instincts and the lofty aspirations which you express so often and so eloquently,\" explained grizzled French premier Georges Clemenceau to Wilson. \"We have become what we are because we have been shaped by the rough hand of the world in which we have to live and we have survived only because we are a tough bunch.\"\n\nBut to progressives such as Wilson, who had witnessed less tragedy than their European counterparts, and more triumph, the balance of power looked both immoral and archaic, the global equivalent of America's selfish tribes. Once upon a time, argued Lippmann, American politics had also been like Europe's, with the North and South \"each trying to upset the balance of power in its own favor.\" But Lincoln had remedied that. The progressives also compared Europe's balance of power to the savage, anarchic capitalism of late-nineteenth-century America, which Social Darwinists had defended on the grounds that selfishness was the way of the world. Now, as the result of Roosevelt and Wilson's reforms, that too was being tamed.\n\nFinally, many progressives pointed to the Western Hemisphere as a third model that proved that the balance of power could be overcome, that politics between nations, like politics between individuals, could be governed by impartial reason, not brute force. America's policy toward its southern neighbors, Wilson declared, \"is known not to be [shaped by] a selfish purpose. It is known to have in it no thought of taking advantage of any government in this hemisphere or playing its political fortunes for our own benefit.\"\n\nWhen Wilson said the United States had no selfish motives in Latin America, he believed it. Just as he considered himself a neutral arbiter of race, class, and regional interests in the United States\u2014despite being a racist\u2014he cast himself in the same role when it came to America's relations with its neighbors to the south. He proposed that the countries of the Western Hemisphere agree to a Pan-American collective security treaty under which they would protect each other from external aggression and internal disorder. On its face, it looked as objective as the Tariff Commission. But in reality, it was the United States that had the power to intervene to prevent disorder in its weaker southern neighbors, not the other way around. And while Wilson genuinely believed that such interventions were unselfish, it was no coincidence that where the United States did intervene\u2014in Haiti, in the Dominican Republic, and in Mexico (twice)\u2014it had the non-altruistic effect of securing U.S. investments and preventing meddling by European powers.\n\nFor Wilson, it was an article of faith that he wanted for the Latins only what they wanted for themselves. Unlike more naked U.S. imperialists\u2014who would have been happy installing dictatorships in the countries to their south\u2014Wilson was enough of an idealist to believe that the people of Latin America could achieve what the United States had achieved: democratic capitalist governments where property was respected and change occurred only via the law. But his idealism was marred by parochialism: He didn't understand Latin America well enough to realize that given the historical experience and economic realities of many of its countries, even necessary change would be far more chaotic than in the United States. Wilson wanted America's neighbors to be democratic, but only if they elected people like him\u2014progressive capitalists\u2014not radicals who might foment disorder or interfere with American business. Ever the believer in political education, he famously declared that \"I am going to teach the South American republics to elect good men!\"\n\nWhen they did not, he convinced himself that the leaders he disliked didn't represent their people as well as he did. For Wilson, it was nearly axiomatic that selfish, irrational nationalism was not real nationalism at all. And so he sent U.S. troops to depose an anti-American dictator in Mexico, confident that the Mexicans would welcome the arriving U.S. soldiers as liberators. Instead they rose up in fury against the gringo invaders, and America quickly withdrew. Wilson was bothered by the experience, but not too bothered. He still saw his Latin America policy as a success. After all, U.S. economic and political influence in the hemisphere expanded dramatically during his presidency, largely because World War I cut off South American banks and businesses from Europe. Wilson's Pan-American treaty was never ratified, but it remained his model for the world.\n\nDuring World War I, Wilson, who had never visited continental Europe before the war and spoke none of its languages, repeatedly recommended both U.S. domestic politics and U.S. relations with its Latin neighbors as templates for a world governed by reason, not force. His basic error was in not recognizing what was blindingly obvious to most African-Americans and Mexicans: that politics within the United States, and within the Western Hemisphere, was absolutely dependent on force. In the United States, it was the government's monopoly on military power that ultimately made possible the progressive agenda. No matter how rational Wilson's commissions, and how inspiring his rhetoric, they would not have mattered had Washington not been able to enforce its edicts via the sword. Similarly, in the Western Hemisphere, it was Washington's near monopoly on military power that undergirded Wilson's Pan-American visions. In the Americas, and in the United States, it was because Wilson had all the guns on his side that he could imagine that his authority did not really depend on them.\n\nAs Wilson saw it, the United States, given its lack of selfish motives, stood apart from Europe's balance-of-power clash in the same way government experts had stood apart from the clash between industry and labor in Pennsylvania's anthracite coal mines. America was not a player in Europe's game, but it could be an umpire, since its own domestic accomplishments suggested a better way. The United States, Wilson declared, must play \"a part of impartial mediation.\" The war's \"causes cannot touch us,\" he added, but its \"very existence affords us opportunities for friendship and disinterested service.\"\n\nBut as was so often the case, Wilson was not as disinterested as he liked to believe. The supposedly peaceful, rational politics of the new world and the brutal, benighted politics of the old were deeply intertwined. The Monroe Doctrine, which for close to a century had kept the Americas free from European power politics, had been enforced for most of that time not by the United States, whose navy was comparatively puny, but by Britain, the greatest naval power on earth. For its own reasons, London had decided to allow the United States a safe haven in the Western Hemisphere, but its ability to do so depended on a balance of power in Europe. For centuries, the central goal of British foreign policy had been to prevent any one nation from dominating the European continent, since that nation could then threaten the royal fleet that protected the British Isles. Without a European balance of power, Britain could not control the high seas. And without British control of the high seas, the United States could not be sole master of its hemispheric domain. Wilson saw the Americas as an Eden free of balance-of-power politics, but he could only indulge that conceit by ignoring Beelzebub guarding the gates outside.\n\nBritain entered World War I because it feared that if it did not, Germany might vanquish France and Russia and make itself overlord of all Europe. Since 1871, when Bismarck welded the German states into one, Germany had been casting an ever-longer shadow over the continent. In 1890, its population had been 11 million higher than France's; by 1913, the gap was 30 million. In 1890, it had produced about 2 million more tons of steel annually; by 1913, the gap was 13 million. In 1880, the British fleet had boasted more than seven times Germany's tonnage; by 1914, the ratio was less than two to one.\n\nThere were influential Americans who looked upon Germany's growing might the same way London did: with dread. Many of them were conservatives like Massachusetts Senator Henry Cabot Lodge, a dour, haughty, and cerebral Boston Brahmin, the first person ever to receive a Harvard history Ph.D. Lodge believed in the balance of power for the same reason he opposed most progressive reforms at home: He thought that trying to eradicate selfishness was a fool's errand, and a dangerous one. \"We must deal with human nature as it is,\" he declared, \"and not as it ought to be.\" But in his view of Europe's war, Lodge was joined by Theodore Roosevelt, who broadly shared Wilson's domestic views but, crucially, did not see them as a template for American policy overseas. Unlike Wilson, who had little experience abroad, Roosevelt had long been fascinated by America's role in the world and had for years maintained a lively correspondence with friends and associates in Europe. As early as 1910 he had traveled through Europe after a post-presidential African safari, and returned home worried that the Huns were growing too strong.\n\nTemperamentally, Roosevelt also liked power politics. He liked\u2014he was blunt about it\u2014war. For Wilson, a domesticated world order, where decisions were made via education and law, not force, was heaven. For Roosevelt, who thought war made men virile and life exciting, a domesticated world order was hell. Roosevelt's bloodthirsty side was often ugly; in fact, it had produced a kind of hubris of its own in the Spanish-American War a decade and a half earlier. But his grasp of the fundamental difference between domestic and international politics made him more alive than Wilson to the role that force played\u2014and would always play\u2014in world affairs. And in Europe, at least, that understanding proved useful, since it helped him see the potential threat that Germany posed. \"Do you not believe that if Germany won this war, smashed the English Fleet and destroyed the British Empire, within a year or two she would insist upon taking the dominant position in South and Central America?\" he asked in 1915. Within months of the war's outbreak, Roosevelt and Lodge were demanding that America arm itself, if not to enter the war on Britain's side (something most Americans would not yet consider), then at least to ensure that if the Royal Navy did fall, America could defend its hemisphere if Germany went after that next.\n\nSpurning Roosevelt and Lodge's demands, Wilson initially resisted building up American arms. He saw the United States as an impartial mediator\u2014helping the Europeans evolve beyond power politics\u2014not another selfish tribe. Impartiality, however, was harder than it looked. When Wilson declared America neutral, he demanded that the combatants respect its right to trade with both sides under international law. But enforcing that law against the world's two greatest military powers was not like enforcing it against the trusts, especially when America's military was so feeble. The British and Germans thumbed their noses at Wilson's request. London imposed a blockade on German ports, and Berlin countered with a blockade of its own, aimed at cutting the British Isles off from crucial transatlantic supplies. There was, however, a crucial difference. Lacking His Majesty's mighty fleet, Berlin employed a newer technology: submarines. Britain enforced its blockade by boarding and searching U.S. ships. Submarines, by contrast, relied on stealth. If they surfaced to inspect passing ships they would be easy prey for an armed merchant vessel. Subs could only enforce a blockade one way: by sinking whatever tried to pass.\n\nIn May 1915, a German U-boat blew a hole in the British ocean liner R.M.S. Lusitania, drowning 1,198 people, including 128 American civilians, in the icy waters off the Irish coast. To Americans, who had seen the war as a remote affair, the shock was seismic. A decade later, many still remembered where they were when they heard the news. For Roosevelt and Lodge, it just underscored the German threat. They knew the American people weren't ready for war, but in their hearts, Lodge and TR were already there. For Wilson, however, the problem wasn't German power; it was international law. \"The rights of neutrals in time of war are based upon principle, not upon expediency,\" he wrote in a diplomatic note to Berlin, \"and the principles are immutable.\"\n\nIt was a fateful choice. By insisting that Americans had a sacred right to travel the high seas, Wilson was putting the United States on a path to war with Berlin. But he was justifying it on legal and moral grounds, not geopolitical ones. As Wilson described it, Germany was dangerous not because it threatened to dominate the European continent and imperil the Western Hemisphere, but because it was violating America's neutrality rights. This rationale preserved the fa\u00e7ade of objectivity that he cared about so much, the sense that America was acting from impartial principle, not selfish interest. But there was a cost: Wilson never used Germany's submarine warfare, as Lodge and Roosevelt urged him to, to tell Americans that they had a stake in a particular distribution of power on the European continent, that Germany was a danger not because it was violating international law but because it could threaten America. The United States was inching toward war, but with the illusion that it was still an umpire in the game of world politics, not a player on the field. It was an illusion that would prove extremely costly in the years to come.\n\nThe sinking of the Lusitania was like the starting of a stopwatch. Given Wilson's definition of neutrality rights, and Germany's need to blockade British ports, it was just a matter of time until U-boats drowned enough Americans to draw the United States into war. Well aware of this, Wilson flung himself into an effort to end the conflict before it engulfed his country. Colonel House shuttled between London, Paris, and Berlin trying to convince the belligerents to agree to a peace deal, any peace deal. Don't focus on \"local settlements [such as] territorial questions, indemnities, and the like,\" Wilson instructed House. But for Europeans, those \"local settlements\" were what the war was all about. Germany wanted enough money and land to ensure that in tandem with its lackey, Austria-Hungary, it could dominate Europe; Britain, France, and Russia (and later Italy) wanted to thwart Germany's plans and aggrandize themselves at Berlin's and Vienna's expense.\n\nIn May 1916, Wilson tried to cut the Gordian knot. He announced that if the combatants laid down their arms, America would join a League to Enforce Peace, a global version of the Pan-American pact he had been pushing in the Western Hemisphere. Together, the league's members would guarantee freedom of the seas and repel aggression by any country against another. This system of collective security, in which countries respected and enforced rules of civilized behavior, would replace the balance-of-power system, in which countries\u2014or alliances of countries\u2014looked out only for themselves. It would ensure that international disputes were resolved based upon reason, not force. The implication was that Europe's belligerents need not worry where they drew the armistice lines; the balance of power between adversaries would no longer matter.\n\nAmong American progressives, many of whom had been championing such a league for years, Wilson's speech was a triumph. Lippmann declared it \"one of the greatest utterances since the Monroe Doctrine.\" But Europe's leaders were less enchanted. It was as if America were demanding that they call a halt to the game and insisting that the score no longer mattered. For statesmen who genuinely believed that they were fighting for survival and who, in pursuit of that belief, had sent millions of young men to be maimed and killed, there was something infuriating about Wilson's insistence that their death struggle for security was a triviality and a superstition. Wilson's vision of a world without power politics, declared French author Anatole France, was like \"a town without a brothel.\" It was too innocent, too antiseptic, too inhuman. Wilson was casting off America's historic isolation and committing it to an active role in European affairs, but only conditionally: only if the old world became something radically different from what it was. The United States, Wilson declared, \"can in no circumstances consent to live in a world governed by intrigue and force. We believe that our own desire for a new international order under which reason and justice and the common interests of mankind shall prevail is the desire of enlightened men everywhere.\" But what if it wasn't? Lurking behind Wilson's words was an implicit threat, which would come back to haunt him: If the world did not live up to its standards, America would pick up its marbles and go home.\n\nSo Wilson's mediation efforts failed, and slowly but surely, the war reeled America in. In March 1916, when a U-boat sank the French passenger ferry Sussex, injuring several Americans, Wilson issued an ultimatum: Unless Germany ceased attacking civilian ships, the United States would break off diplomatic relations. After a fierce internal debate, the kaiser's government backed down, partly because it was running low on subs anyway. But Wilson had placed himself in a diplomatic straitjacket: The moment Berlin resumed its submarine blockade, his own words would push him to the brink of war.\n\nOn January 31, 1917, Germany's ambassador informed Washington that his government was doing just that. The Germans now believed they had enough subs to enforce the embargo, and, with Russia crumbling on their eastern front, they saw a chance to rapidly win the war. \"England will lie on the ground in six months,\" predicted German naval authorities, \"before a single American has set foot on the continent.\"\n\nThree days later, Wilson broke off diplomatic relations with Berlin. He still hoped for a miracle, but on March 18, Germany sank three U.S. merchant ships, killing fifteen Americans. Wilson's entire cabinet was now demanding war. Adding to the fever, the British had intercepted a telegram, sent from German foreign secretary Arthur Zimmermann to the government of Mexico, proposing a military alliance. If the United States declared war on Germany, and Mexico came to Germany's aid, Berlin would help it regain Texas, New Mexico, and Arizona. The telegram confirmed Roosevelt and Lodge's warning that if Germany won, its dark shadow would soon cross the Atlantic. When Wilson released Zimmermann's message to the Associated Press, it sparked a wave of public support for war.\n\nIf the Zimmermann Telegram demonstrated the dangers of a German victory, a very different event suggested the possibilities of a German defeat. On March 8, food riots broke out in Russia's imperial capital, Petrograd, and quickly grew into a general strike. When Czar Nicholas II tried to return to the city to reestablish control, soldiers refused to let his train pass. Seven days later, he abdicated. A dynasty that had ruled for three centuries had been toppled in a single week. And for the first time in its history, Russia was a republic, led by the moderate, pro-Western Alexander Kerensky.\n\nFor Walter Lippmann and John Dewey, Russia's liberal revolution was like a sign from God. Although aware that the carnage on the seas might force America into war, they yearned to enter it on behalf of something grander than the safety of America's merchant marine. If the United States had to descend into the valley of death, they wanted to believe it could lead Europe toward the light. Now, suddenly, the people of Russia, the most brutal, backward country on the continent, were breaking their chains, showing that the old world could indeed follow the path of the new. America could never join the war with its \"full heart and soul,\" Dewey had written, until \"the almost impossible happens...until the Allies are fighting on our terms for our democracy and civilization.\" Now the almost impossible had occurred: The most autocratic Allied power was embracing democracy. It is now \"as certain as anything human can be,\" wrote Lippmann, \"that the war which started as a clash of empires in the Balkans will dissolve into democratic revolution the world over.\"\n\nAmong pro-war progressives, the czar's overthrow was like the popping of a cork. In a great rush, the optimism that had been welling up during their years of domestic triumph bubbled over. Ambivalence, hesitation, even dread were replaced by giddy expectation. \"We have been, as it were, a laboratory set aside from the rest of the world in which to make, for its benefit, a great social experiment,\" Dewey declared. \"The war, the removal of the curtain of isolation, means that this period of experimentation is over. We are now called to declare to all the world the nature and fruits of this experiment.\" No longer was the war a mistake, or even a grim necessity; it was a crusade.\n\nOn the evening of April 2, 1917, as a light rain fell, mounted cavalry escorted Wilson's car from the White House to the Capitol. Soldiers saluted as he passed; thousands of well-wishers lined the route. Once he arrived, Wilson was taken to a congressional antechamber. Alone there for a moment, he began to shake uncontrollably, and had to stare hard into a mirror to regain his poise. At 8:32 P.M., the doors swung open and he walked into the House chamber, where he was greeted by a full two minutes of applause. With the exception of a tiny handful of war opponents, everyone in the audience wore or held an American flag.\n\n\"We are,\" Wilson declared, \"at the beginning of an age in which it will be insisted that the same standards of conduct and of responsibility for wrong done shall be observed among nations and their governments that are observed among the individual citizens of civilized states.\" This was the crusade in a nutshell: America would make politics between nations resemble politics between Americans. Wilson went on to salute the Russian people, who \"have been always in fact democratic at heart,\" for proving that America's democratic principles were becoming Europe's as well. And with the chief justice of the United States leading the crowd in raucous cheers, he ended by declaring that America \"shall fight for the things which we have always carried nearest [to] our hearts\u2014for democracy...for the rights and liberties of small nations, for a universal dominion of right by such a concert of free peoples as shall bring peace and safety to all nations and make the world itself at last free.\" As the crowd rose to its feet, members of Congress ripped the flags off their sleeves and lapels and waved them wildly. Five days later, America was at war.\n\nTo emphasize the conditional nature of America's commitment, Wilson declared that America would fight as an associate of Britain, Russia, France, and Italy, not an ally. Just as he believed Lincoln had done during the Civil War, he would fight and yet remain detached. Knowing (or at least suspecting) that London, Paris, Moscow, and Rome had made secret arrangements to feed on Germany's and Austria-Hungary's carcasses if they won the war, Wilson decided that America must establish a rational, disinterested process for designing the postwar world. To do that, in the fall of 1917, \"the Inquiry\" was born.\n\nLippmann was only one of the progressive luminaries whom House enlisted in the cause. Thorstein Veblen served as an informal adviser. So did the New Republic's Herbert Croly, Harvard president A. Lawrence Lowell, and the famed historian of the vanishing frontier, Frederick Jackson Turner. Future Supreme Court justice Felix Frankfurter requested a role, as did Judge Learned Hand. Dewey considered heading the Inquiry's Russia division. The effort began in secrecy, at the New York Public Library on Forty-second Street, where only the head librarian and one aide knew the true purpose of the men working furiously in back. From there it moved to the American Geographical Society on Broadway and 155th, largely because of the society's vast storehouse of maps. Finally, a reporter for Philadelphia's Public Ledger broke the story, and the Inquiry was deluged with job applicants. Satellite offices were established at Harvard, Princeton, and Yale. Guards began patrolling the Inquiry's headquarters night and day.\n\nThe Inquiry was modeled on those great temples of progressive reform: the Tariff Commission, the Federal Trade Commission, the Interstate Commerce Commission, and the Federal Reserve. Just as they had established scientific principles for American industry and finance, the Inquiry would now establish them for entire nations\u2014determining which countries should exist and where. In their effort to draw new European borders, Lippmann and his colleagues produced four types of studies. Historians determined which ethnic, religious, and linguistic groups had the best ancestral claim to a given patch of land. Geographers and sociologists searched for the right way of classifying the current residents. Economists allocated natural resources and industrial infrastructure so nations would be economically viable. And a final group\u2014the smallest\u2014handled politics.\n\nMuch of the work in this fourth category concerned domestic arrangements like federalism. Relations between states, by contrast, received little attention. How to put the genie of German dominance back in its bottle and rebuild Europe's shattered balance of power were questions that had no scientific answers, and were thus largely ignored. The whole idea behind the Inquiry, after all, was to replace borders imposed by force with borders conceived through reason. As Wilson's secretary of state, Robert Lansing, explained, \"the fixing of frontier lines with a view to their military strength and in contemplation of war was directly contrary to...the policy of the United States.\"\n\nThe Inquiry produced more than two thousand reports, many filled with statistical tables and charts, often with no interpretation whatsoever\u2014the facts were meant to speak for themselves. But like Wilson himself, Lippmann and his colleagues were not as objective as they liked to believe. \"In some respects the Koords [sic] remind one of the North American Indians,\" read one report on the Middle East. \"Their temper is passionate, resentful, revengeful, intriguing, and treacherous.\" And it wasn't only racism that infected the Inquiry's efforts; power politics did as well. Just as the trusts had wormed their way into the tariff and trade commissions, the British now influenced the Inquiry\u2014plying the Americans with data that buttressed London's favored outcomes. So deeply enmeshed were they in the Inquiry's activities that British diplomats in Washington knew more about its work than did most members of Congress.\n\nThat, in and of itself, was not such a terrible thing. Had Wilson told Americans that Germany's power and ambition made it a potential threat to the United States, and that America was entering the war because Britain, France, and Russia could no longer contain that threat alone, no one would have expected the Inquiry to be impartial. Had he forthrightly acknowledged that America was committing itself to a world it could not perfect, had he defined America's war aims as something less grandiose than the eradication of power politics, perhaps the postwar peace would not have proved such a crushing disappointment. Had he not allowed the hubris of reason to swell so dramatically, perhaps it would not have so violently popped.\n\nIn October, Russia's liberal experiment collapsed as Kerensky's government fell to the Bolsheviks of Lenin. Determined to discredit the capitalist powers and withdraw Russia from the war, Lenin published the Allies' secret treaties dividing up the spoils of war. Suddenly, for all his grand rhetoric, Wilson looked like an accomplice to a land grab. In mid-December, House summoned Lippmann to his Fifty-seventh Street apartment and told him that the Inquiry must show the world that America was still fighting for high principle, and fast. For the rest of the month, Lippmann and his fellow experts worked around the clock, often not even going home to sleep. We need \"genius,\" Lippmann told his colleagues, \"sheer, startling genius, and nothing else will do.\" Finally, on January 2, 1918, Lippmann gave House the Inquiry's final report. Three days later the president and his best friend met to mold it into a series of public recommendations. \"We actually got down to work at half past ten,\" House noted in his diary, \"and finished remaking the map of the world, as we would have it, at half past twelve.\"\n\nThe result, unveiled on January 8 before a joint session of Congress, was perhaps the most celebrated diplomatic initiative in American history: the Fourteen Points. It included free trade, freedom of the seas, disarmament, rights for colonial peoples, an end to secret treaties, and a League to Enforce Peace (increasingly called the League of Nations). When it came to borders, Wilson obscured some of the Inquiry's findings, employing vague language to avoid firm commitments, but the plan's eight territorial planks still veered toward national self-determination and the principle that borders should be determined by reason, not force. The boy who had once drafted constitutions for his baseball league and his imaginary yacht club had finally done it: He had drafted a constitution for the entire world.\n\nWilson knew that his wartime allies, especially the French, wanted a harsher peace. But he trusted in his capacity to teach. The experts had told him what was right, and now he would educate the people of Europe, rallying them against their own selfish regimes. That included the people of Germany. American planes dropped copies of the Fourteen Points behind enemy lines, and Lippmann joined a newly created Propaganda Commission, from which he planned \"a frank campaign of education addressed to the German and Austrian troops, explaining as simply and persuasively as possible the unselfish character of the war, the generosity of our aims, and the great hope of mankind which we are trying to realize.\"\n\nFinally, that August, after years of blood-soaked stalemate, the German military crumbled. With American troops now flooding into Europe, the Allies broke through enemy lines, destroying sixteen divisions in a matter of days. On the German home front, sailors mutinied; workers struck; starving women marched through Berlin holding their empty pots and pans. One of Germany's top generals, Erich Ludendorff, sank into depression; the other, Paul von Hindenburg, put on dark glasses and a fake mustache and sneaked across the border into Sweden. In November, the kaiser abdicated. A liberal government took power in Berlin and sued for peace on the basis of the Fourteen Points. The German people, it appeared, had heard Wilson's appeal. Now they too were demanding a scientific peace.\n\nThe stage was now set. No sitting U.S. president had ever left the Western Hemisphere. (According to some interpretations, doing so was actually illegal.) But overruling his top advisers, Wilson decided to oversee the peace negotiations himself, so no subordinate would steal the glory. On December 4, 1918, as gunships saluted, handkerchiefs waved, and pigeons were released into the sky, the passenger liner George Washington left the Hoboken, New Jersey, harbor, carrying the president and virtually everyone else of importance in the executive branch, a delegation 1,300 strong. It was flanked on its left and right by ten destroyers; a battleship sped ahead to smooth the waves. Nine days later, at just past 1:30 P.M. on December 13\u2014a time and date chosen because the president considered it lucky (his name contained thirteen letters)\u2014the armada, now sixty vessels strong, reached Brest, France, on the far side of the Atlantic. On the morning it arrived, the sun, which had been absent for days, suddenly cut through the gray sky.\n\nWhen Wilson disembarked, Europe's battered masses gave him a greeting that one journalist called \"inhuman\u2014or superhuman.\" At 3 A.M. that night, on the train carrying the American delegation to Paris, Wilson's doctor looked outside and saw men, women, and children lining the tracks as far as the eye could see. When the Americans reached the French capital, two million admirers jammed the streets, the largest crowd in French history. In Rome, the mayor likened Wilson's visit to the Second Coming. In Milan, banners compared him to Moses. Italian soldiers knelt before his picture; families placed his photograph on their windowsills, surrounded by sacred candles. \"For a brief interval,\" wrote H. G. Wells, \"Wilson stood alone for mankind.... He was transfigured in the eyes of men. He ceased to be a common statesman; he became a Messiah.\"\n\nThe lonely preacher's son had wagered an epic gamble: that through force he could build a world governed by reason. And now, with Germany vanquished and Europe's masses screaming his name, he seemed on the brink of success. It was the progressive dream, unfolding on a global scale. \"It might turn out well,\" wrote House in his diary, \"and yet again it might be a tragedy.\"\n\n## CHAPTER TWO\n\n## THE FRIGHTENING DWARF\n\nRandolph Bourne made people uncomfortable. Partly, it was his body. At birth, a brutal doctor, making promiscuous use of forceps, elongated his head and disfigured his face. At age four he developed a double curvature of the spine, which left him a hunchback. As an adult, he wore a long black cape, like something out of Phantom of the Opera. John Dos Passos called him a \"tiny twisted bit of flesh.\" Theodore Dreiser called him \"that frightening dwarf.\"\n\nBourne's mind was frightening, too. He began reading grocery labels as a toddler and the Bible by kindergarten. By the time he reached college, he had become a withering polemicist, perhaps Lippmann's only generational equal. But if Lippmann used his pen to impress and ingratiate, Bourne used his to cut\u2014a harsher style engendered by a harsher life. A self-described \"homesick wanderer,\" he felt little affection for his aristocratic but downwardly mobile mother, barely knew his alcoholic, absentee father, couldn't stand his teachers and his hometown, and spent his twenties flitting from one mindless, dead-end job to another. Then, in 1909, he won a scholarship to Columbia University, where he met Charles Beard and John Dewey, the men who changed his life.\n\nBeard, the charismatic son of an Indiana gentleman farmer, was progressivism's greatest historian. Dewey, a shy mumbler from Burlington, Vermont, was its greatest philosopher. On the eve of World War I, they were high priests of the church of reason, and their effect on Bourne was almost spiritual. From Dewey, whom he likened to a prophet, Bourne learned that schools could unlock the human instinct for cooperation and thus breed citizens who transcended their selfish desires. From Beard he learned to see American history as an upward march from plutocracy to democracy and anarchy to organization. By the time war struck, Bourne was earnestly spreading the gospel of progress. It was young people like himself, he argued, in whom selfishness had not yet fully taken root, who could serve as \"the incarnation of reason.\" In 1914, Beard got his young disciple a job at the fledgling New Republic. Not yet thirty, the brilliant hunchback was becoming one of progressivism's sharpest voices. And in those heady, innocent days before the war, his future\u2014like his movement's\u2014appeared bright.\n\nAt first the carnage across the Atlantic changed little. Like his New Republic colleagues, and like Woodrow Wilson, Bourne wanted no part of Europe's war. But as 1916 turned to 1917, he began to feel the ground shift. Lippmann and Croly were now visiting Colonel House's New York City apartment every week, and their editorials were speculating with mounting fervor about how America's entrance into war might transform the country and the world. For his part, Bourne was moving hard the other way\u2014his antiwar militancy growing ever stronger as America crept toward war. Not coincidentally, fewer and fewer of his articles made it into print, and by April 1917, when America finally joined the fray, the New Republic was no longer publishing his political writing at all.\n\nBut Bourne's bitterest break came not with his employer, but with his mentors, Dewey and Beard, who became two of the war's loudest academic cheerleaders. Wilson's vision of a rationalized postwar world, wrote Beard, reflected the \"slowly maturing opinion of the masses of the people everywhere in the earth.\" It heralded \"a new epoch in the rise of government by the people and in the growth of a concert among the nations.\" Dewey rhapsodized about the war's potential impact at home. Wartime mobilization, he predicted, would shift power from private to public hands, from selfish tribes to unbiased experts\u2014exactly the shift for which progressives had been striving since the century turned. It would usher in \"a more integrated, less anarchic [political] system\" marked by the \"systematic utilization of the scientific expert.\" For decades, Dewey had seen education as the key to overcoming American selfishness and disorder. Now he had a new laboratory: war.\n\nFor Bourne, the analogy was not just wrong, it was monstrous. Schools were \"rational entities,\" safe laboratories for tinkering with human behavior. Nations at war were entirely different. Far from being a mechanism for reasoned planning, he argued, \"war is just that absolute situation...which speedily outstrips the power of intelligent and creative control.\" War \"determines its own end, victory, and government crushes out automatically all forces that deflect, or threaten to deflect, energy from the path of organization to that end.\" For Bourne, Dewey was like a man who, having called a tiger a horse, thought he could restrict its diet to grass.\n\nWar progressivism, Bourne was coming to believe, rested upon a lie: Force, once unleashed, could not remain the servant of reason. If the Inquiry tested that proposition overseas, it found its domestic counterpart in something called the Committee on Public Information, which was tasked with building support for the war at home. Led by George Creel, a former urban reformer and muckraking journalist, it embodied progressivism's love affair with the fact. Among the men and women who stocked the committee\u2014many of whom had spent the prewar years exposing the abuses of the trusts\u2014it was a point of pride not to coerce Americans into supporting the war, nor even to play upon their emotions. \"A free people cannot be told what to think,\" declared Creel. \"They must be given the facts and permitted to do their own thinking.\"\n\nBourne's mentors nodded approvingly. Beard went to work for Creel's committee, editing some of its pamphlets on the causes and purposes of the war. Dewey applauded the committee's calm, deliberative spirit, which he considered appropriate for a country in which \"very large numbers of our citizens...have systematically taught themselves to discount all of the more violent appeals to passion.\"\n\nProgressives had long placed their faith in the rationality of the common woman and man. And Dewey had predicted that ordinary Americans would act rationally even\u2014and perhaps especially\u2014during war. But by 1917, that claim sat uncomfortably alongside the wave of anti-German frenzy spreading from Washington, D.C., across the American hinterland. After America entered the war, Congress passed espionage and sedition acts that outlawed \"disloyal, profane, scurrilous, or abusive language\" against the Constitution, the government, the military, and the flag. Cincinnati outlawed the sale of pretzels; Iowa's governor made publicly speaking German a crime. When a Wyoming man was overheard saying, \"Hoch der Kaiser\" (\"Up with the kaiser\"), a group of townspeople hanged him, cut him down while still alive, and made him kneel and kiss the American flag. In April 1918, a St. Louis mob abducted a young German-American, stripped him, dragged him through the streets, and then lynched him, while a crowd of five hundred cheered. At trial, the defense attorney called the murder patriotic, and it took a jury twenty-five minutes to acquit.\n\n\"The war,\" wrote Bourne's friend, Waldo Frank, \"which drove all the world, including Dewey mad, drove Bourne sane.\" In a series of scalding essays, Bourne put his mentors on intellectual trial. It wasn't just America's bloodthirsty yokels who proved that rationality was impossible in a time of war, Bourne argued; it was the war progressives themselves. Though Dewey and Beard saw themselves as apostles of reason, Bourne insisted that their motives for supporting the war were as primal as everyone else's. They were bored. Their domestic campaigns, having brought so much success, had lost the capacity to thrill. Having come of age at a time of swelling optimism, and possessing no primary experience of political tragedy, they were conditioned to see their people, their leaders, and themselves as capable of boundless achievement. And as a result, when Russia's liberal revolution offered a glimpse of a remade world, they could not resist the psychic pleasure of a crusade. \"Hesitations, ironies, consciences, considerations\u2014all were drowned in the elemental blare of doing something aggressive, colossal,\" wrote Bourne. \"There seems to have been a peculiar congeniality between the war and these men. It is as if the war and they had been waiting for each other.\"\n\nBetrayed by his prophets, Bourne cast about for a new creed. At times he veered toward isolationism. The horrors on the home front, he argued, showed that Dewey was wrong: American progressivism was not a successful experiment ready to be unveiled to the world. Americans should instead embark upon a \"stern and intensive cultivation of our garden...a turning within in order that we may have something to give without.\" At other moments, Bourne lurched toward pacifism. Wilson, he argued, had been right to seek to replace the balance of power with a world of reason and law. Where he had gone horribly wrong was in believing that this new world could be enthroned\u2014and maintained\u2014via the sword. There is no such thing as a \"democratic and antiseptic war,\" Bourne insisted. \"The pacifists opposed the war because they knew this was an illusion.\"\n\nHad Bourne lived longer, he might have reconciled these competing strains. But he did not live long enough to try. Cast out from the New Republic in 1917, he began writing for a small literary journal called Seven Arts. But it soon folded, largely because of the furor over his antiwar polemics. He briefly served as a contributing editor at another minor publication, Dial. But the journal's benefactor was also wooing John Dewey, the man Bourne was savaging in print. Dewey agreed to join the staff on one condition: that his nemesis be fired.\n\nUnder investigation by the Justice Department, increasingly unable to publish, and virtually homeless\u2014living in the apartments of friends\u2014Bourne died of influenza just weeks after the war's end. A legend soon grew that he had starved because no journal would print his work. That was a fable, but its appeal testified to Bourne's hold over the postwar imagination. In his final, fevered days, his assault on the hubris of reason offered a template for many of America's leading intellectuals and politicians in the two decades to come. Isolationists in the 1920s and '30s demanded that America never again bind its fate to the alien civilization across the Atlantic. Pacifists kept alive the dream of a rationalized world but insisted that achieving it required America not to fight wars, but to abolish them. And leading these two intellectual currents, oddly enough, were Charles Beard and John Dewey. In death, the prot\u00e9g\u00e9 became mentor, and the fathers carried on the work of the prodigal son. Had Bourne's ravaged body endured into the postwar age, the homesick wanderer might have found himself, finally, at home.\n\nIf the hubris of reason was cracking domestically, it would soon crack overseas as well. On January 18, 1919, five weeks after Wilson's triumphant arrival in Europe, the Paris Peace Conference began. The date was no accident. It had been on that same day in 1871 that Wilhelm I was coronated kaiser of a newly unified Germany. French premier Georges Clemenceau, who orchestrated the timing, remembered 1871. Back then, as a young man, he had helped barricade his neighborhood against the German siege of Paris. Now, in his seventies\u2014suffering from diabetes, insomnia, and eczema on his hands so severe that he was forced to wear gloves\u2014he had seen the Germans invade again. Germany, he told a journalist, was his \"life hatred.\" His one overriding goal before he died\u2014for which he was willing to beg, lie, cheat, steal, and kill\u2014was to ensure that France was never raped again. According to legend, he had asked to be buried upright, facing Germany, so he could keep vigil even from the grave.\n\nClemenceau wanted Wilson to see Germany as he saw it, to understand life in the lion's den. In January 1919, France was a ravaged nation. The war had taken one quarter of its men between the ages of eighteen and thirty. Double that number had been wounded; they were everywhere, pitiful, limbless creatures begging in the streets. On paper, France had won the war. But Clemenceau knew that it was less a victory than a stay of execution. France had lost a higher percentage of men than Germany, and since the fighting had occurred largely on its soil, it had suffered more damage to its infrastructure. By sparking the Bolshevik Revolution in Russia, the war had also removed France's most powerful ally\u2014and Germany's most powerful foe\u2014in Eastern Europe. Structurally, therefore, France finished the war even more vulnerable to its bigger, more productive neighbor than it had been at the start. Even the souvenir penknives that French shopkeepers sold to commemorate victory were made in German factories.\n\nClemenceau urged Wilson to see the destruction firsthand, the six thousand miles of French territory that the war had laid waste. \"There are hundreds of villages into which no one has yet been able to return,\" explained one French minister. \"Please understand: it is a desert, it is desolation, it is death.\" But Wilson resisted French invitations to tour the battlefields; he did not want to be swayed by emotion. He hadn't even wanted to hold the peace conference in Paris, preferring Geneva, where there would be fewer \"hysterical\" French. As always, Wilson saw himself and his nation as impartial, detached. Americans, he explained, were \"the only disinterested people at the Peace Conference.\" Asked by foreign correspondents what he would say to his fellow leaders once the conference began, he replied, \"[W]e come here asking nothing of ourselves and we are here to see you get nothing.\"\n\nClemenceau found Wilson's detachment maddening. He grumbled about the American president's \"cold reason\" and \"mathematical justice.\" Wilson, he moaned, \"believed you could do everything with formulas and his fourteen points. God himself was content with [only] ten commandments.\" For Clemenceau, there could be no objectivity between the lion and the lamb. In his mind, the purpose of the peace conference was not to repeal the balance of power\u2014a system he pointedly endorsed just before the negotiations began\u2014but to rectify it. France must be made stronger; Germany weaker. There must be a redistribution of force. \"I have come to the conclusion that force is right,\" he told a dinner guest. \"Why is this chicken here? Because it was not strong enough to resist those who wanted to kill it. And a very good thing too!\"\n\nFrance needed security; in that pursuit Clemenceau was unyielding. But he was flexible about how to attain it. The most obvious way was to ravage Germany as it had ravaged France: to strip it of as much land and money as possible. In addition to Alsace-Lorraine, whose return to France everyone took for granted, Clemenceau suggested annexing the Rhineland, a strategically vital, mineral-rich slice of land that hugged Germany's borders with Holland, Belgium, and France, or establishing a puppet state there. That way, if Germany struck again, at least France would have a territorial buffer to absorb the blow. He also proposed reparations so severe that they would not only help France rebuild but shift the economic balance of power with Berlin.\n\nBut France needed more than money and land. It needed allies, since without them it couldn't enforce a peace agreement anyway. Clemenceau was prepared to go softer on Germany in return for ironclad guarantees that America and Britain would come to France's aid if the Huns stirred again. But here his desires and Wilson's progressive vision clashed. Clemenceau wanted that most old-fashioned of things: an alliance against a potential foe. Wilson wanted a world without alliances, where nations no longer banded together in selfish blocs but instead championed the common interests of all humankind.\n\nFor five months, Wilson and Clemenceau bickered, with British prime minister David Lloyd George usually somewhere in between. Clemenceau embraced Wilson's call for a League of Nations but tried to turn it into an anti-German alliance. The League, he proposed, should exclude Germany and establish a standing army dominated by America, Britain, and France. When the Americans and British refused to create a League army, Clemenceau suggested a common military planning staff instead, so the League could organize quickly in response to German aggression. But Wilson rejected that, too. For Clemenceau, who assumed that Germany would resist whatever peace settlement the Allies imposed, the League was only useful as a tool of Allied enforcement. For Wilson, by contrast, the whole purpose of the peace settlement and the League was to establish principles so rational and fair that they would be accepted by all countries, Germany included. Wilson was no pacifist; he conceded that in theory the League might have to enforce its rulings at gunpoint. But he insisted that military action would be rare. As in his crusades for progressive reform at home, and for collective security in the Western Hemisphere, he claimed that it was education, not force, that would convince people to accept the scientific principles formulated on high. In the new world being born, he insisted, stopping aggression would depend \"primarily and chiefly upon one great force, and that is the moral force of the public opinion of the world.\"\n\nIn the end, the French and Americans struck a compromise: The Rhineland would stay in German hands but the Allies could occupy it for fifteen years. Washington and London also offered Paris security guarantees: They would come to its aid if the Germans attacked again. On paper Clemenceau had what he wanted. But the alliance was built on sand. Wilson downplayed it, saying it did not commit the United States to do anything it hadn't already pledged to do under the League. That made Clemenceau shiver. The League, in his mind, was utopian precisely because it bound its members to defend the borders of all nations\u2014whether those borders mattered to their security or not. In the real world, Clemenceau believed, that made America's pledges under the League virtually meaningless\u2014a point underscored by Wilson's insistence that upholding those pledges would rarely require force. Clemenceau wanted a French alliance with the United States to be everything the League was not: a binding, narrowly tailored commitment made for reasons of national security, not universal principle. He wanted Wilson to acknowledge that aggression against France mattered to America in a way that aggression against Latvia did not, to admit that America had entered the game of global politics not to protect the rights of all nations, but to protect the rights of those crucial few\u2014like France\u2014upon which America's own security relied.\n\nBut that is not how Wilson had sold the war back home. He had told Americans that the United States was an associate of Britain and France, not an ally\u2014that it was fighting not on their behalf, but on behalf of all humankind. The clear implication was that America's goal at the peace conference was not to buttress any one group of nations but to ensure that all nations, the victors as well as the vanquished, were treated rationally and fairly.\n\nIn Wilson's mind, his rapturous welcome across the continent proved that ordinary Europeans wanted this, too. They wanted to transcend their selfish, tribal desires; they just needed a great educator like himself to show them the way. \"National purposes have fallen more and more into the background,\" he declared, \"and the common purpose of enlightened mankind has taken their place.\" Mastodons like Clemenceau didn't understand that; they still hungered for alliances, reparations, and land. But Wilson told aides that Europe's leaders did not represent their people. \"If necessary,\" he insisted, \"I can reach the peoples of Europe over the heads of their rulers.\"\n\nBut the peoples of Europe let Wilson down. Time and again during the Paris negotiations, when Europe's leaders resisted his designs, Wilson threatened to take his case to their people, in a campaign of public education. And time and again, ordinary Europeans refused to be educated. When Wilson said he would appeal to the British to overturn their government's opposition to freedom of the seas, one of the Fourteen Points, Lloyd George urged him on, no doubt amused at the prospect of an American telling Britons to place their trust in his abstract principles rather than the Royal Navy. When Italian prime minister Vittorio Orlando walked out of the peace conference after his nation was denied the Adriatic city of Fiume, Wilson promised to explain the decision to the Italian people himself. Unmollified, the Italians, who had erected plaques to Wilson only weeks before, covered them over in disgust. If ordinary Americans weren't as rational in wartime as Wilson and his fellow war progressives had assumed, ordinary Europeans, it turned out, weren't, either.\n\nWilson was hoisted on his own petard. Against Roosevelt and Lodge's advice, he had denied that America was entering the war because its own security required preventing Germany from overthrowing the European balance of power. Instead he had told Americans that they were entering the war to abolish the European balance of power, to build a world in which reason governed force. The implication was that America's participation was conditional: If the old world refused to change, America could always retreat to its own hemisphere, since it could \"in no circumstances consent to live in a world governed by intrigue and force.\" Now, as the terms of the Paris Peace Conference trickled out, it became clear that the postwar world would indeed be governed by intrigue and force. The Treaty of Versailles was not as punitive as legend suggests. It was considerably less onerous, for instance, than the treaties that Germany had imposed upon Romania and Russia when they exited the war in May 1918. But it was far from a scientific peace. German territory was parceled out to France, Belgium, Denmark, Czechoslovakia, and Poland, sometimes in flagrant violation of the principles of national self-determination laid out by the Inquiry and the Fourteen Points. Italy seized several choice parts of the former Austro-Hungarian Empire; Japan grabbed German islands in the Pacific and a slice of China; Britain and France snatched Berlin's colonies in Africa and the Middle East. The settlement reflected the distribution of postwar power, not objective standards of right and wrong. The peace treaty failed that classic progressive test: It was not seen as impartial by all sides.\n\nWilson did his best to sugarcoat the results. He had convinced the Europeans to join a League of Nations, he explained, and over time the League would smooth out the treaty's rougher edges. But many of Wilson's progressive allies refused to be placated. Outraged that his work for the Inquiry had been in vain, Lippmann warned the Europeans that \"if you make it a peace that can be maintained only by the bayonet we shall leave you to the consequences and find our own security in this hemisphere.\" Wilson had made America's entry into the war sound altruistic: The United States had joined the fray to lift up the benighted Europeans, not because its security was intertwined with theirs. Now the Europeans had refused to be lifted up. And so for many disillusioned progressives\u2014and many Americans more generally\u2014there was only one thing to do: return home.\n\nOn July 10, 1919, Wilson submitted the Versailles Treaty\u2014which included American membership in the League\u2014to the Senate. There he faced three rough blocs. First were the forty-seven Senate Democrats, who as loyal partisans nearly all backed U.S. entrance into the League. But by themselves they lacked the two-thirds votes necessary for ratification. Next were a group of Republicans called \"Irreconcilables,\" led by Idaho's Senator William Borah, who for various reasons were certain to vote no. They numbered only about fourteen, but their voice was amplified by disillusioned war progressives like Lippmann, who now opposed the League with the fury of lovers scorned.\n\nThe opposition of the Irreconcilables meant that to win Senate ratification the treaty required votes from the third bloc: a group of roughly thirty-five Republicans called \"Reservationists.\" The Reservationists were prepared to swallow America's entrance into the League, but only if certain \"reservations\" were added to the treaty, the most important of which was that only a vote of Congress\u2014not a vote of the League\u2014could take America to war. Their patron saint was Theodore Roosevelt, who six months earlier had dropped dead of a heart attack. Their leader was Roosevelt's soulmate, Henry Cabot Lodge, who saw his battle with Wilson as a tribute to his dead friend.\n\nAt first glance, Lodge stood in the ideological middle, with the militantly pro-League Wilson and the militantly anti-League Borah occupying the two poles. But in a deeper sense he stood apart from both. Wilson and Borah were, in the political scientist Hans Morgenthau's words, \"brothers under the skin.\" Wilson claimed that the League would rationalize international affairs and therefore must be embraced. Borah insisted it would not and thus must be spurned. Wilson said America should engage with the world in order to perfect it; Borah said the world could not be perfected and so should not be engaged. Only Lodge articulated a realistic internationalism: He acknowledged that the world could not be perfected\u2014that reason would never fully govern force\u2014but urged that America bind itself to it anyway.\n\nLodge did not believe in detachment. He was a fierce partisan of his city, Boston; his political party, the Republicans; and his intellectual tradition, which he traced to Alexander Hamilton and the Federalists. Like Wilson, he revered Lincoln\u2014but Lincoln the partisan and Lincoln the conqueror, not Lincoln the impartial reconciler. When World War I broke out, Lodge became a partisan of Britain and France. The United States, he insisted, \"should act, not as an umpire between our allies and our enemies, but as one of the allies.\" Now, with the war over, his goal mirrored Clemenceau's: to put Germany in a box from which \"it will be physically impossible for her to break out again.\"\n\nFor Lodge, the League\u2014as Wilson described it\u2014was a dangerous delusion. America, in his view, would never go to war for Latvia, and he did not like making promises that his nation could and should not keep. He was willing to support U.S. membership, but only once Congress made clear that the League was a mere debating society, which would change nothing fundamental about international affairs. Lodge was not sentimental; he did not believe that international politics would ever be much more than a jungle. He urged Wilson to eschew \"efforts to reach the millennium of universal and eternal peace,\" to stop trying to make \"mankind suddenly virtuous by a statute or a written constitution.\" Better to seek a balance of power against Germany, anchored by an alliance with France. Better to fly low and steadily than be seduced by the clouds above.\n\nIn theory, Wilson and Lodge should have been able to come to terms. Wilson desperately wanted the League; Lodge desperately wanted the treaty with France. Each man had the power to grant the other's wish. But to convince Lodge to support U.S. membership in the League, Wilson had to admit that the emperor had no clothes: that America would never go to war at the League's behest. Even more humiliatingly, he had to separate the French security treaty from the League itself, to show that the former, unlike the latter, constituted a binding commitment. He had to acknowledge, in other words, that his dream of a rationalized world would not come true anytime soon.\n\nThis Wilson would not do. Instructing his Democratic Senate allies not to compromise, he instead set out to speak to the people, to rally them against selfishness and call them to reason in one last, great educational campaign. His longtime doctor, Cary Grayson, begged him not to go. Wilson had a history of strokes, and that April in Paris, while running a high fever, one side of his face had gone numb and his eye had twitched uncontrollably. Since the president's return to the United States, Grayson noticed, his memory had deteriorated; he seemed confused by simple things. But Wilson brushed off the warning. \"I know I am at the end of my tether,\" he declared, \"but...in the presence of the great tragedy which now faces the world, no decent man can count his personal fortunes in the reckoning.\"\n\nOn September 3, 1919, he left Washington in a custom-designed train. From Ohio to Nebraska to California and back through the interior West, he delivered thirty-seven speeches in twenty-two days over nearly ten thousand miles. The crowds often swelled into the thousands, and, in every speech save one, Wilson addressed them without a microphone. Finally he began to buckle under the strain. On September 25, during a speech in Pueblo, Colorado, he lost his bearings, and then, upon recalling America's fallen soldiers, burst into tears. That night he coughed so violently that he had to sleep upright to breathe and suffered a headache so painful he could barely see. His train raced back to Washington, but, on the car ride from Union Station to the White House, as his motorcade drove past deserted sidewalks, he began doffing his hat to imaginary crowds, fueling rumors that he had lost his mind. Within a week the left side of his body was paralyzed. He spent most of the rest of his presidency in seclusion, tended to by his wife, Edith. Neither his vice president nor any member of his cabinet would see him for the next year and a half, until he emerged to watch his successor sworn in.\n\nThe people had failed Wilson yet again. Even Democrats detected little shift in public opinion as the result of his trip; if anything, anti-League sentiment was building as more and more Americans came to see Wilson's European crusade as one big waste. Colonel House\u2014long Wilson's anchor to the messy, real world\u2014sent him two letters urging him to accept Lodge's changes. \"Your willingness to accept reservations rather than have the treaty killed,\" he implored, employing the flattery that had long greased their relationship, \"will be regarded as the act of a great man.\" But even before his stroke, Wilson had turned on his old friend, convinced that House had made concessions in Paris aimed at sabotaging his dream. Edith Wilson, who had long resented House for competing with her for the president's meager affections, refused the colonel's pleas for a meeting. So Wilson spent the disastrous fall of 1919 walled off from reality, barricaded in what one observer called \"the lonely citadel of his soul.\"\n\nWith Wilson a virtual cadaver, and the politically primitive Edith in firm control, the vultures circled. In November, the Irreconcilables and Reservationists joined forces to defeat the unamended treaty. Then, when Lodge's reservations were added, the treaty was voted down again, this time by a coalition of Irreconcilables and Senate Democrats, whom Wilson had instructed to oppose anything but a clean draft. The following March, when the treaty came up for one final vote, twenty-one Senate Democrats defied the president and backed the amended draft. But they still fell seven votes short of the two-thirds needed for ratification. In retaliation, Wilson refused to urge Senate Democrats to support the treaty with France, and so it died as well.\n\nThe tragedy was not, as Wilson's supporters claimed, that Lodge and the selfish Republicans had killed his dream of a rationalized world. That dream was always a fantasy, born of a fundamentally flawed analogy between politics in the United States\u2014where leaders possessed a near monopoly on force and thus could imagine that force had given way to reason\u2014and politics in the world at large, where no such monopoly existed and force stood naked as the arbiter of events. The real tragedy was that Wilson could not himself abandon his dream, even after Paris, and thus help bring Americans to the painful, crucial realization that they must commit themselves to a world they could not perfect. As a result, in the years after World War I, the hubris of reason gave way not to a sober and dogged realism, but to isolationism and pacifism\u2014new flights of fantasy that evaded the central truth that Wilson would not help his country learn: that America must bind itself to the world, even though the world would always let America down.\n\nFor John Dewey and Charles Beard, it had been a very bad war. Before America joined the fighting, Dewey had predicted that wartime Americans would be models of rationality. We will be \"lenient and amiable in our judgments,\" he prophesized, and \"discount all the more violent appeals to passion.\" But even on Dewey's own campus, violent passions stirred. Two months after America entered the fray, Columbia University president Nicholas Murray Butler told a commencement audience that \"what had been tolerated before becomes intolerable now. What had been wrongheadedness was now sedition. What had been folly was now treason.\" Four months later, Butler fired psychology professor James McKeen Cattell, the man who had recruited Dewey to Columbia, for writing a letter on university stationery claiming that draftees had the right to conscientiously object. Dewey protested, but most of his colleagues considered the firing entirely justified. Soon Dewey was being investigated by the Justice Department for \"pro-German\" leanings himself.\n\nBeard's experience was no better. In the spring of 1916, after publicly defending a man's right to shout \"To hell with the flag\" in a New York City public school, he was summoned by the Columbia trustees. For thirty minutes they questioned him about his allegedly subversive political and historical views before sending him away with a warning not to continue to \"inculcate disrespect for American institutions.\" Cattell's firing the following year was the final straw. On the afternoon of October 9, 1917, Beard finished his afternoon lecture, then told his students, \"This is my last lecture in Columbia University. I have handed in my resignation this afternoon to take effect at 9 A.M.\" Butler and the trustees, Beard later explained, were using \"the state of war to drive out, or humiliate, or terrorize every man who held progressive, liberal, or unconventional views in political matters.\" After a moment of stunned silence, Beard's students rose and applauded for twenty-five minutes straight, while their teacher wept.\n\nSoon Beard himself\u2014like Dewey\u2014was under investigation, in his case by the Senate Judiciary Committee for associating with groups deemed unfriendly to the war effort. The army banned some of his books from its training camps.\n\nPolitically, neither man was ever the same. The war, Dewey admitted, had fostered a \"cult of irrationality.\" The \"small voice of reason\" had been utterly silenced \"amid howling gales of passion.\" Equally disillusioned by the war's fruits overseas, he joined Lippmann and the Irreconcilables in urging the Senate to reject the Versailles Treaty. \"Were not those right who held that it was self-contradictory to try to further the permanent ideals of peace by recourse to war?\" he wrote in early 1919. It was a veiled apology to Randolph Bourne, the disciple he had betrayed, who had died just weeks before.\n\nIn what one friend called \"a state of tension that in most people would have been an illness,\" Dewey left for Asia, where he spent the next two years. He returned to the United States a devout pacifist, insisting, as Bourne had in his final days, that instead of using force to build a world of reason, America should use reason to build a world free of force. He would never support another war.\n\nFor his part, Beard carried on the other half of Bourne's revolt against the hubris of reason: isolationism. In 1922, he called for a foreign policy of \"continentalism,\" under which the United States would divest itself of all overseas territories and commitments, refuse all military and diplomatic efforts to protect U.S. investments abroad, and build a military solely devoted to protecting the homeland from foreign attack. Henceforth, Beard pledged, he would \"center my efforts on the promise of America rather than upon the fifty century-old quarrels of Europe.\" The United States, he declared, should \"set its own house in order under the stress of its own necessities and experiences.\"\n\nOn paper, Dewey's pacifism and Beard's isolationism were worlds apart. Dewey's vision remained every bit as hubristic as Wilson's; he was just no longer willing to kill to achieve it. Beard, by contrast, now rejected not merely Wilson's means, but his ends: He no longer wanted to rationalize the world. But despite this theoretical gulf, Dewey and Beard shared one core belief, which would define American foreign policy in the interwar years: that America should never again make any overseas commitment that would require enforcement via the sword.\n\nFor the French, who desperately needed America's drawn sword to keep the Germans at bay, it was a slow-motion tragedy. In August 1921, Wilson's successor, Warren Harding, signed a separate peace with Germany that freed the United States of all obligations to enforce the Treaty of Versailles. Since Britain had conditioned its own security alliance with France on America's, Paris was now on its own.\n\nA year later, Germany signed the Treaty of Rapallo, normalizing relations with the Soviet Union. For France, the rapprochement was ominous. Paris, which had long pursued close ties with the nations to Germany's east so that in the event of war Berlin would have to fight on two fronts, was now banking on alliances with the newly created nations of Eastern Europe, some of which had been carved from German or Russian soil. Rapallo was the first step in Germany and Russia's efforts to take that land back. Even worse, the treaty contained a secret clause allowing Germany to train its forces on Soviet soil\u2014a direct violation of Versailles, which required the Germans to substantially disarm. Less than three years after the peace treaty, Berlin was already breaking free.\n\nBy 1923, the Germans were no longer paying their war reparations, either. Ever since 1921, when an Allied commission had set the reparations bill at 132 billion marks, Berlin had been screaming about the insane, intolerable burden under which it labored. But in reality the burden wasn't as insane and intolerable as it appeared. Given the way the reparations were structured, the bulk of the 132 billion would never have to be paid; the real figure, by some estimates, was only 25 billion marks (about $550 million). Before the Allies presented the bill, the Germans had actually proposed paying more than that themselves.\n\nThe real problem was not the amount of reparations Germany had to pay; it was the fact that most Germans did not believe they should pay any reparations at all, just as they did not believe they should forfeit territory, armaments, or their right to immediately join the League. The Versailles peace, which the Germans called a virtual crime against humanity, was in many ways milder than the one the Allies would impose in 1945, and which Germans swallowed with barely a whimper. The difference was that after World War I\u2014unlike after World War II\u2014the Germans did not feel defeated. They had never seen foreign troops marching through their razed cities. Since the Allies, in a terrible blunder, had made Germany's postwar civilian leaders sign the hated Versailles Treaty, rather than forcing the generals who had launched the war to take ownership of the peace, German militarists and hypernationalists found it all too easy to claim that the war had not been lost on the battlefield. It had been lost because liberals, socialists, and Jews stabbed Germany in the back.\n\nClemenceau was right. The Germans had not abandoned their dreams of European hegemony. And his nightmare was coming true: France was being left to face them alone. It was a vicious cycle. The more abandoned France felt, the more it lashed out against Berlin in a desperate bid to strangle Germany's growing postwar power in the crib. And the more France lashed out, the more it alienated public opinion in America, where isolationists and pacifists alike insisted that the selfish, irrational French were as bad as their German foes, if not worse.\n\nIn January 1923, after Germany, which was paying some of its reparations in coal, defaulted on its thirty-fourth shipment in thirty-six months, France and Belgium sent troops into the Ruhr Valley to seize German mines and steel plants as collateral. German inflation had already been running dangerously high as the new Weimar-based government printed currency to fill the gap between what it received in taxes and customs duties (which had fallen sharply) and what it spent (which remained high). The Ruhr occupation tipped that problem into a catastrophe. Furious that French troops were squatting on German soil, Weimar encouraged passive resistance: paying miners and factory workers not to work. To do that, Weimar printed even more money. By fall, it required thirty paper mills and 150 presses all operating twenty-four hours a day to generate all the marks that Germany's government was sending into circulation. To purchase the goods that one mark would have bought in 1914 now required one trillion. Its savings rendered virtually worthless by hyperinflation, the German middle class was now easy prey for a Nazi Party whose popularity was beginning to build.\n\nAmericans mostly responded by blaming France. The Ruhr invasion, declared William Borah, was \"utterly brutal and insane.\" Oswald Garrison Villard, editor of the Nation, called it every bit as despicable as Germany's invasion of Belgium during World War I. Thousands assembled at New York's Madison Square Garden, where a litany of senators denounced French militarism. To register his disapproval, President Harding withdrew all remaining U.S. troops from the Rhineland, abandoning the final element of American enforcement of Versailles. Four years earlier, when they signed their names at Versailles, French leaders believed they had an American commitment to defend their borders and to help keep Germany from arming in the Rhineland. By 1923, they had neither.\n\nThe Ruhr occupation did not just offend American sensibilities; it hurt America's bottom line. World War I had turned the United States into the world's largest creditor, and U.S. investors were keen to sink their money into postwar Germany, whose reviving industries they believed would offer fat rates of return. But as long as the French occupied the Ruhr, Germany's economy could not revive. So in 1924, President Calvin Coolidge sent banker Charles Dawes to convince Berlin to start paying a reduced reparations bill, and to convince Paris to withdraw its troops. Dawes succeeded. The Germans began paying, France pulled back, and Americans made money. But for France, it was yet another defeat. By agreeing to the Dawes Plan, the French effectively conceded that it would be Washington\u2014not Paris\u2014that determined whether Germany was paying sufficient reparations. And since Dawes and Coolidge didn't much care whether Berlin paid its reparations to France, so long as it repaid its loans to America, Weimar could now default with virtual impunity. Just as significant was the way Dawes approached the two sides: as an impartial mediator. As Coolidge put it, \"We are independent, detached and can and do take a disinterested position in relation to international affairs.\" Wilson had at least made America an \"associate\" of the French cause. Coolidge went further: America was now strictly neutral between the nation it had fought alongside and the nation it had fought against, between France, which had no capacity to dominate Europe, and Germany, which still did.\n\nSoon the impartiality was enshrined in international law. In 1925, at the Swiss resort of Locarno, Germany, France, and Belgium pledged to respect each other's borders, with Britain and Italy serving as guarantors. It all seemed wonderfully hopeful: The wartime belligerents were finally starting to reconcile. Europeans began speaking of a \"spirit of Locarno\" breaking out across the continent; the British, German, and French negotiators all won the Nobel Prize. But there were darker forces at play. At Locarno, the final, slim prospect of a British alliance with France died. Instead of pledging to protect France against German attack, Britain was now pledging to protect Germany against French attack as well, which presumably meant that if the French again sent troops into the Ruhr to enforce Versailles, then London and Paris would be at war. At Locarno, the Germans also conspicuously refused to pledge to respect their borders with Austria, Czechoslovakia, and Poland. It was yet another sign that Berlin intended to break out of its Versailles straitjacket in the east, a move that would in turn imperil the two-front strategy on which French security relied.\n\nSlowly and artfully, using progressivism's language of impartiality as their pretext, the Germans were wriggling free. In 1926, they were admitted to the League and even given a seat on its council. Now any French bid to use the League to enforce Versailles was subject to Berlin's veto, and any League-sponsored effort at disarmament would have to apply equally to both Germany and France. Since France, given its visceral fear of its eastern neighbor, was unwilling to disarm, Germany no longer had to, either. In January 1927, the Inter-Allied Military Control Commission, which had been (halfheartedly) monitoring Berlin's compliance with its disarmament obligations, left German soil. In theory, under the Versailles Treaty, Germany still had only one hundred thousand troops and no tanks, heavy weaponry, or air force. Meanwhile, France began building a chain of military defenses across the length of its border with Germany: the Maginot Line.\n\nFor the pacifists and isolationists who shaped American foreign policy in the 1920s, it was France\u2014more than Germany\u2014that exemplified the selfish, irrational tendencies they disdained. The Germans, with their calls for international equality, mutual disarmament, and self-determination, spoke America's language. The French, by contrast, kept going on about fortifications, punishments, alliances, and the balance of power\u2014all grating to the American ear. France, wrote Dewey, \"affords a striking example of the fact that\" in international affairs, \"the old policies and the old type of politicians are still absolutely in control.\" When it came to security, the French were like a man desperate for food, and every time they lunged for it, Americans reprimanded them for their poor table manners.\n\nWhile France grimly prepared for another war, Americans in the 1920s fell madly in love with the idea of peace. From Ernest Hemingway to John Dos Passos to William Faulkner to H. L. Mencken, American novelists and critics savaged the idea that there was nobility in war. Harding and Coolidge's influential secretary of state, Charles Evans Hughes, began calling the State Department \"the Department of Peace.\" In 1922, publisher Edward Bok offered one hundred thousand dollars to anyone who could outline a scheme for world peace in less than five thousand words, and a staggering 250,000 Americans inquired about taking part. The Nation demanded \"a burning, an unconquerable, an undeviating hatred of war, any war for whatever reason.\" On Armistice Day 1932, Nicholas Murray Butler, the Columbia University president whose wartime jingoism had so traumatized Dewey and Beard, called on the nations of the world to destroy their weapons, but leave a few to put in museums so future generations could see how barbaric human beings had once been.\n\nInfluential Americans actually said such things in the 1920s, in full sincerity. What's more, they tried to turn their words into action. In 1920, Borah introduced legislation proposing a conference with Britain and Japan aimed at cutting each nation's naval expenditures in half. It passed the Senate, 74\u20130. The following fall, Secretary of State Hughes convened the Washington Naval Conference, which produced an agreement between America, Britain, Japan, France, and Italy on a ratio limiting their battleship fleets. Hughes, one British observer quipped, had destroyed more royal vessels than all the admirals in history. Scattered voices in the U.S. departments of Navy and War muttered that the ratios did not allow America to keep pace with Japan in the Pacific, but they were missing the larger point: Congress wouldn't have appropriated enough money to keep pace even without the treaty. Between 1922 and 1926, American naval spending dropped by one-third.\n\nBy 1927, the Coolidge administration was itching for another disarmament conference. But the French, who had been dragged unhappily into the first one, refused to play along. Instead, French foreign minister Aristide Briand\u2014a man renowned for his wit and great mop of disheveled gray hair\u2014proposed that the United States and France pledge never to go to war. It was a sly move. Since Versailles, Paris had been busily encircling Germany with alliances: with Belgium in 1920, Poland in 1921, Czechoslovakia in 1924, and Romania and Yugoslavia in 1926. However, in the rational, unselfish new world supposedly born from World War I, alliance was a dirty word. So the French didn't call their treaties \"alliances\" they called them pledges never to go to war (sometimes with secret addenda promising military aid in case either signatory went to war with anyone else). Now Paris was proposing such a pledge with the United States.\n\nHughes's replacement as secretary of state\u2014an elderly, profane, former corporate lawyer named Frank Kellogg\u2014was not fooled. He knew that if America and France pledged never to go to war, America would find it much harder not to go to war with whomever France was fighting against. With a U.S. nonaggression pact in its back pocket, France could infringe upon U.S. neutrality rights\u2014as the British had done during World War I\u2014secure in the knowledge that America could not retaliate. This would make the United States a de facto belligerent on France's side.\n\nThe American peace movement, however, was enthralled by Briand's idea, and so Kellogg needed a counterproposal. In that effort, he turned to the work of John Dewey. Upon his return from self-imposed exile in Asia, Dewey had thrown himself into a movement called the Outlawry of War. War, Dewey noted, was not currently illegal under international law. There were laws of war\u2014rules about treating prisoners, and the like\u2014but no laws against war. This, Dewey declared, made international politics barbaric and archaic. Once upon a time, he claimed, societies had not outlawed murder, either; they had just regulated it. Dueling was once a legal method of settling disputes within nations. But now humanity had progressed. Just as dueling had been outlawed between individuals, war should now be outlawed between nations.\n\nIt was a deeply progressive argument: History was marching upward from barbarism to civilization, and, in that effort, the triumph of reason and law within nations should serve as a template for the triumph of reason and law between nations. It was similar in spirit to what Dewey had believed in 1917, except that now Dewey insisted that this upward march must proceed not via war, but via its abolition. As always, the key was public opinion. Dewey, like Wilson during the Paris Peace Conference, insisted that ordinary people were more morally advanced than their leaders. Across the world, he insisted, a \"community of moral feeling\" against war already existed. By helping to outlaw war, America would nurture that moral feeling until it became a stigma so profound\u2014like the stigma against slavery or incest\u2014that war became almost unthinkable. Dewey's pacifism was the ethic of reason turned inside out, a kind of hubris of the impotent. America should insist on an utterly remade world, governed by reason, not force. But the means should be as pure as the end: To avoid sullying the dream of a rationalized world, America should shed no blood in its pursuit.\n\nIn 1921, Dewey had written to Borah, the flamboyant leader of the Senate Irreconcilables, urging him to draft legislation committing the United States to the abolition of war. Two years later, Borah did just that, and in 1927, when Kellogg needed an answer to Briand, Borah's legislation was still sitting there like an unlit match. When Kellogg struck the fuse and proposed a treaty banning war not just between the United States and France but between all nations, the peace movement erupted in even greater exhilaration. Carrie Chapman Catt's Committee on the Cause and Cure of War convened ten thousand town meetings to pass pro-Outlawry resolutions; side by side the resolutions stretched for two miles. Letters endorsing Kellogg's proposal flooded the State Department at the rate of six hundred per day.\n\nThe French were furious. Once again they had tried to drag Americans down into the harsh, real world only to see them float back up into the haze of dazzling abstraction. There was nothing left to do but smile grimly and play along. At the signing ceremony, where sixty-four nations pledged to forever abstain from war, Briand\u2014a glorious bullshitter when necessity required\u2014solemnly dedicated the treaty to the World War I dead. President Coolidge, with greater sincerity, declared that the Kellogg-Briand Treaty held \"greater hope for peaceful relations than was ever before given to the world.\" The Senate passed the treaty, 85\u20131. Dewey called it one of the most important events in the history of international affairs. Kellogg won the Nobel Prize.\n\nSome of the senators who voted for the treaty were isolationists. They didn't believe for a minute that the blood-soaked Europeans would refrain from war. But the treaty was a harmless sop to their pacifist constituents, a meaningless commitment that substituted for a real one. Pacifists, by contrast, supported the treaty in deadly earnest; they considered Kellogg-Briand the dawn of a virtual messianic age. Between them, the two factions embodied the same dual impulses that had animated Randolph Bourne in his final days: to avoid war by fleeing it and to avoid war by banning it. Wilson, for all his self-delusion, had actually risked something for his vision of a rationalized world. Beard and the isolationists abandoned that vision; Dewey and the pacifists thought they could achieve it by proclamations alone. What they all shared\u2014Wilson and those who rebelled against him in the giddy, unreal 1920s\u2014was a refusal to meet the world on its own terms, to accept that politics between nations would never resemble politics between Americans, yet must be embraced nonetheless. The real answer to the hubris of reason was still waiting to be born.\n\n## CHAPTER THREE\n\n## TWICE-BORN\n\nOn a fall day in 1932, John Dewey ascended a Manhattan podium to endorse a candidate for Congress: a theologian thirty years his junior named Reinhold Niebuhr, who was running on the Socialist ticket in New York state's 19th District. It was a decision he would regret. Unbeknownst to Dewey, Niebuhr had just completed a book that attacked him by name. Dewey, Niebuhr charged, placed too much faith in reason, forgetting that \"reason is always, to some degree, the servant of interest.\" And his pacifism was na\u00efve, since \"conflict is inevitable, and in this conflict power must be challenged by power.\" Niebuhr's book, Moral Man and Immoral Society, was a broadside against the faith to which Dewey had clung in both war and peace: that through education humanity could be fundamentally improved. For the aging philosopher, it would prove little consolation that the young cleric won only 4 percent of the vote.\n\nLike Randolph Bourne, Niebuhr had once been a fervent admirer of Dewey, and like Bourne, he had scores to settle from World War I. Niebuhr had been twenty-four years old when America plunged into the Great War, and his war progressivism had an almost desperate tinge. Niebuhr, after all, was German, the son of an immigrant from the province of Lippe-Detmold. When the war broke out, he was a student at Yale Divinity School, struggling to perfect his English. And when the United States entered the fray, he was a minister at a German-language church in northwest Detroit, having only recently convinced the board to allow English-language services every week. As war hysteria gripped the nation\u2014and publicly speaking German became in some places a crime and in others an invitation to mob attack\u2014Niebuhr refused to defend his besieged flock; to the contrary, he added to their torment. Eager for literary fame, he broke through with a cover story in the Atlantic implying that when it came to German-Americans, the public's \"suspicions of disloyalty\" were wholly justified. He demanded removal of the word German from his denomination's letterhead, and when a church newspaper gently criticized wartime censorship, Niebuhr chastised its editor, insisting that the paper express \"out-and-out loyalty.\"\n\nThe ambitious young preacher justified all this by declaring World War I a noble crusade: \"the inspiring spectacle of a nation making every sacrifice of blood and treasure for aims which do not include territorial ambitions or plans for imperial aggrandizement.\" He was gambling that Wilson and Dewey were right: that the war would midwife a nation and a world governed more by reason than force. When that hope died at Versailles, Niebuhr announced\u2014like Randolph Bourne and Walter Lippmann\u2014that he had been betrayed. He called Wilson a dupe. He denounced the wartime pastors so \"anxious to prove that they liked the smell of blood.\" He fretted that \"no one will ever know the tragedy in the millions of lives of German immigrants\" treated \"as potential criminals\" by their adopted nation. Niebuhr understood the tragedy all too well. He had seen it up close, not as a victim, but as a victimizer. Like Lippmann, he responded to the collapse of his war progressive dreams with rage when the more honest reaction would have been shame.\n\nFor a time after the war, Niebuhr called himself a pacifist. But his heart was never in it. Even when he supported outlawing war, he kept rebuking his fellow outlawers for their excessive idealism, for their failure to see that war was a product not of human ignorance, which could be eradicated, but of human sinfulness, which could not. World War I, he declared, \"made me a child of the age of disillusionment.\" And that disillusionment applied not merely to the belief that the world could be rationalized through war, but to the belief that it could be rationalized through peace as well.\n\nWhat exactly all this disillusionment meant for Niebuhr's foreign policy views was not yet clear, since in the 1920s foreign policy was not his major concern. But when it came to domestic affairs, he kept pointing out the limits of reason and the centrality of power, taking aim at core principles of the progressive creed. One of his favorite examples was Henry Ford, the auto titan who dominated Niebuhr's adopted city of Detroit. Ford made a great show of his Christian piety, thus suckering some progressive-minded ministers into believing that by citing the ethical teachings of the gospel they could convince him to treat his workers more fairly. Niebuhr called that a delusion. Ford didn't exploit his workers because he was ignorant of the Bible; he exploited them because it made him rich. The problem was not inadequate education; it was naked self-interest. And the answer, in Niebuhr's view, was to confront Ford with the organized self-interest of his workers, who should fashion themselves into a labor union and refuse to work at starvation wages. Unlike the progressives, Niebuhr didn't think the clash between Ford and his workers had an objective solution. He wasn't seeking a scientific peace. He was seeking a new industrial balance of power, in which the rich\u2014acting less from reason or love than from fear\u2014satisfied the selfish desires of the empowered poor. In 1929, Niebuhr made it official, joining the Socialist Party.\n\nThen the Depression hit. By 1933, America's gross national product was less than one-third its level in 1929. Food riots grew common; families foraged in garbage dumps for something to eat. In this bleak and savage new America, Niebuhr lurched further left. Once again he took aim at progressives, whom he accused of trying to persuade the rich to treat the poor more justly. The old progressive faith in education, and in the ultimate harmony of society, he argued, was na\u00efve. The class war had begun, and it was a contest not of reason, but of might. Dewey, Niebuhr claimed, would not stare this harsh reality in the face. He could not see that when oppression forces you to the barricades, the time for pedagogy is done.\n\nIn his attack on Dewey, Niebuhr reflected his times. The Depression politicized intellectuals and radicalized them, leaving many contemptuous of the claim that the struggle between business and labor could culminate in a scientific peace. In 1931, Lippmann's first boss, the famed muckraker Lincoln Steffens, published an autobiography mocking his earlier belief that by exposing society's crimes, journalists could educate those in power to remedy them. In his influential 1932 polemic, Farewell to Reform, John Chamberlain took stock of three decades of progressive good intentions and declared them a failure; revolution was the only option left. That same year, a host of prominent writers, including the novelist John Dos Passos, the philosopher Sidney Hook, and the literary critics Malcolm Cowley and Edmund Wilson, announced that they were voting communist for president.\n\nLooking at the election results, one might have wondered whether any of this mattered. American politics did move left in the 1930s, but the result was not revolution, but the New Deal\u2014not a farewell to reform, but its spirited revival. At the very moment leftist intellectuals were burying progressivism, Franklin Roosevelt seemed to be dredging it up.\n\nBut the intellectual rebellion that Niebuhr led\u2014against Dewey's faith in the inevitability of progress, the efficacy of education, and the inherent goodness of man\u2014would by the early 1940s profoundly affect American foreign policy, as the United States climbed out of the Depression and found itself standing at the precipice of a second world war. As in 1917, a vision of America would become the template for a vision of the entire globe. And a generation less intoxicated by success than its predecessor, and more toughened by tragedy, would do what the pacifists and isolationists of the 1920s and 1930s could not: bury the hubris of reason and reconcile America to a fallen world.\n\nAll this, however, was still a long way off when Dewey endorsed Niebuhr on that fall day in 1932. America's president was Herbert Hoover, a man who neatly blended the pacifism and isolationism of the interwar years. Hoover believed in reason, not force, and understandably so: He had seen the former transform his life and the latter devastate the world. His biography would have made Horatio Alger blush. Orphaned at age nine, he was sent to live with his uncle in Oregon, who forced young Herbert to labor such long hours in the family real estate office that he was unable to attend high school. Undaunted, the young striver took classes at night, where he mastered bookkeeping, typing, and math and so impressed his teachers that he gained admission to Stanford University's inaugural class. There he trained as a mining engineer, and by age forty his genius at exploiting inaccessible deposits had made him a multimillionaire. (To this day, Hoover's entry in the Australian Prospectors & Miners Hall of Fame does not even mention that he became president.)\n\nIn early-twentieth-century America, engineering was a quintessentially progressive profession, and Hoover was, in one colleague's words, the \"engineering profession personified.\" Engineers saw themselves as scientifically trained experts standing above the selfish clash between business and labor. They occupied, in Hoover's words, a \"position of disinterested service,\" \"want[ing] nothing...from Congress [except] efficiency.\" And this, in Hoover's mind, made them just the kind of people America needed to run its government.\n\nWorld War I provided Hoover his chance. In London on business, he was asked by the U.S. Embassy to help find food, lodging, cash, and safe passage back to the United States for the thousands of Americans marooned in Europe. This he did with dazzling efficiency: Of the $1.5 million in loans he arranged for roughly 120,000 stranded Americans, all but $300 was paid back. When the job was done, the U.S. ambassador called Hoover \"a simple, modest, energetic man who began his career in California and will end it in heaven.\"\n\nAfter that, Hoover was charged with aiding the people of Belgium, who were suffering both from German occupation and a British blockade. This time he oversaw the shipment of more than $5 billion in food and supplies while spending less than half a percent on overhead. By war's end, close to four million Europeans had written letters or signed petitions thanking him for his work. The people of Finland invented a new verb: hoover, meaning \"to help.\" Now something of an international celebrity, he was tapped by Wilson to head the American Food Administration, where he ran a vast public relations campaign aimed at convincing Americans to ration food and other staples so there would be enough for the troops overseas. Soon, the word Hoover entered the American lexicon as well, as a verb meaning \"to conserve.\" \"I can Hooverize on dinners and on lights and fuel too,\" read a popular Valentine's Day card in 1918, \"but I'll never learn to Hooverize when it comes to loving you!\"\n\nHoover's wartime experience left him with foreign policy views that blended the pacifism of John Dewey and the isolationism of Charles Beard. Already predisposed to nonviolence by his Quaker upbringing, Hoover now contrasted his accomplishments as an international engineer and relief worker\u2014using science, commerce, and moral appeals to improve humanity's lot\u2014with the savagery and waste of Europe's war. But if Hoover, like Dewey, believed that America should try to build a world free of force, his wartime experience also inclined him toward Beard's pessimism that Europeans would ever shed their murderous ways. The result was a foreign policy ideology best described as \"pacifism if possible, isolationism if necessary.\" The U.S. government, in Hoover's view, should appeal to other nations to cut armaments and renounce aggression, just as he had appealed to Europeans to aid Belgium and to Americans to stop eating meat. And the United States should spread capitalism across the globe, just as he had in his mining days. But America should always retain the freedom to retreat across the oceans if its efforts were rebuffed. The United States, Hoover believed, was militarily and economically self-sufficient. For him, as for Harding, Coolidge, and even Wilson, America's international engagement was conditional: If the world lived up to our standards, great. If not, we could survive perfectly well on our own.\n\nThis was exactly the attitude that had been making French leaders twitch ever since Versailles. In 1929, soon after entering the White House, Hoover, along with British prime minister Ramsay MacDonald, hatched plans for another naval disarmament conference, to extend to smaller ships the restrictions put in place in 1921. Thrilled, peace activists sent the State Department so many letters that clerks complained they could not process all the mail.\n\nIn London the following spring, the United States, Britain, and Japan agreed to limit their fleets of destroyers, cruisers, and submarines. But the French\u2014confirming their reputation as the troglodytes of Europe\u2014would not join in. Pressed to accept naval parity with Benito Mussolini's Italy, Paris responded with that most tedious and hoary of demands: a security alliance with America and Britain. When Hoover refused, the French walked out. France, Hoover commented, was \"rich, militaristic, and cocky; and nobody can get on with her until she has to be thrashed again.\"\n\nThe Germans could not have agreed more. Berlin's bid to free itself from the fetters of Versailles, which it had once pursued with stealth and good manners, was becoming more blatant. A big reason was the Depression, which further devastated a German middle class still reeling from the hyperinflation of 1923. In September 1930, with unemployment nearing 15 percent, the Nazis\u2014who had won roughly 800,000 votes two years earlier\u2014garnered more than 6 million, making them the Reichstag's second-largest party. Six months later, Berlin announced a customs union with Austria, the first step toward unifying the two German-speaking nations, which Versailles explicitly banned. The French were apoplectic and ultimately squashed the move, but Hoover, who saw the customs union as economically rational, not geopolitically threatening, gave his blessing. \"Evidently you don't approve of the Versailles Treaty,\" remarked Secretary of State Henry Stimson. \"Of course I don't,\" Hoover replied. \"I never did.\"\n\n\"In the realm of reason,\" observed Walter Lippmann, Hoover \"is an unusually bold man.\" But \"in the realm of unreason he is...easily bewildered.\" Unfortunately for the Great Engineer, unreason was on the march in the Depression years, and not just in Germany. In September 1931, a cadre of younger Japanese army officers swearing fealty to bushido, the ancient Japanese warrior code, engineered a massive invasion of the northern Chinese province of Manchuria. Hoover was stunned: He had thought American morality and commerce were taming the Japanese. Just a year earlier, Tokyo had signed the London Naval Treaty, and Japan's economy was deeply enmeshed with America's own. But the Japanese prime minister who signed that treaty was now dead, shot in the abdomen by a young hypernationalist. And during the Depression, America's own protectionist policies changed Tokyo's economic calculus. In an increasingly autarkic world, Japan could no longer depend on financial and commercial links across the oceans; it needed an economic bloc of its own. That required securing Manchuria, its primary source of iron and coal, and the recipient of 90 percent of its foreign investment. By year's end, Japanese troops had pushed several hundred miles into China. In Tokyo, yet another civilian prime minister lay dead, and the army and navy were firmly in control.\n\nIn response, Hoover turned to the methods that, in his own life, had served him so well. He issued a moral appeal, informing Tokyo that since it had violated the Kellogg-Briand Treaty banning war, America would not recognize the fruits of its aggression. But Japan's war-frenzied regime paid little heed. Hoover hoped the strain of conquest might spark Japan's economic collapse, but instead its economy improved. He backed a League of Nations investigation into the incursion, but when the world body issued its condemnation, Japan walked out. Finally, Secretary of State Stimson\u2014a blue-blooded Theodore Roosevelt prot\u00e9g\u00e9 who feared war less than his boss, and understood power politics more\u2014suggested threatening sanctions. But Hoover refused; his experience in Europe had bred in him a deep aversion to blockades. \"The United States,\" he declared, \"has never set out to preserve peace among other nations by force.\"\n\nIn the 1920s, America's refusal to help uphold the European or Asian balance of power had been more understandable. Given the lack of obvious foreign threats, and the American public's hostility to overseas obligations, it would have taken an unusual leader to see that Germany's potential strength and undaunted ambition necessitated an American commitment to France. Theodore Roosevelt and Henry Cabot Lodge might have been such leaders, but by the mid-1920s both were dead.\n\nHarder to defend was Hoover's unwillingness to build up American strength in the early '30s, with the Japanese rampaging through China, and Germany growing increasingly aggressive in its territorial demands. Even after the Manchuria invasion, Hoover still did not commission a single new warship. To the contrary, he began planning the grandest disarmament initiative yet. For years peace groups had been agitating for a meeting to address not merely limitations on a few classes of ships among a few powers, but the disarmament of the entire globe. And in 1932 they got their wish, when representatives from fifty-nine nations arrived in Geneva for the World Disarmament Conference. U.S. officials quickly electrified the delegates by proposing that all \"offensive\" weapons be banned, and all other armaments be cut by at least a quarter. A peace convoy of more than two hundred cars, carrying a petition with 150,000 signatures in support of Hoover's efforts, drove from Los Angeles to Washington, where it was taken to the White House by police escort.\n\nPredictably, the French were hostile. Naval disarmament was bad enough, but slashing the size of their army\u2014when intelligence reports showed that Germany was rapidly rearming\u2014was downright terrifying. Yet again Paris demanded security guarantees from Washington and London, and yet again Paris was rebuffed. \"When would the American government learn,\" exclaimed French foreign minister Andr\u00e9 Tardieu, \"that it could not brutalize France into concessions which it was not prepared to make?\" In July, the disarmament conference temporarily adjourned. Four months later, Germany's parliamentary elections produced an upset: the Nazis had won a plurality of the vote.\n\nIf Hoover clung to the foreign policy principles of a dying age, his successor, Franklin Roosevelt, was widely thought to lack any firm principles at all. Even as a child, his cousins, Theodore's children, considered him a toady, eager to tell adults whatever they wanted to hear. At Harvard, classmates called him a \"false smiler\" and \"two-faced.\" When FDR ran for president, H. L. Mencken quipped that if he \"became convinced tomorrow that cannibalism would get him the votes he so sorely needs, he would begin fattening up a missionary in the White House backyard.\" The charges of spinelessness and opportunism would never entirely stop. \"Every great leader had his typical gesture,\" declared the writer and Republican congresswoman Clare Boothe Luce years later. \"Hitler the upraised arm, Churchill the V sign. Roosevelt?\" She licked her index finger and raised it to the wind.\n\nBut FDR did have foreign policy principles, or at least foreign policy instincts, instincts that would eventually help him overcome the hubris of reason. Those instincts were just difficult to see early on because one of them was that a politician should conceal his views when they clashed with the public's, and for most of the 1930s, FDR's did. Had he been more forthright\u2014less two-faced and opportunistic\u2014it is unlikely he would have been elected, or reelected, president of the United States.\n\nFDR's foreign policy instincts were shaped by two mutually reinforcing passions: for Teddy Roosevelt and for the sea. As a boy, the young Franklin hero-worshipped his fifth cousin, whom he called the greatest man he had ever known. At fourteen, he bought a pair of TR's trademark pince-nez eyeglasses. At twenty-three, he married TR's favorite niece, Eleanor. He slavishly patterned his political career on his famous relative's: winning election to the New York state assembly, getting himself appointed assistant secretary of the navy (and trying to leave that job to raise a regiment to fight in Mexico, just as TR had done in Cuba during the Spanish-American War), and then becoming governor of New York.\n\nFDR mimicked his cousin ideologically as well. Although he was working in Wilson's Navy Department when World War I broke out, FDR's views more closely resembled those of TR and Lodge. Like them, he made little pretense of objectivity; he wanted Britain and France to win. And like them, and in defiance of his superiors in the administration, he urged as early as 1915 that the United States build up its military to deal with a potential German threat. FDR's wartime statements bristled with TR's belief that war between nations was inevitable, necessary, even fun, and that it would do America good to get into the fight. He mocked his boss at the Navy Department, a pacifist named Josephus Daniels, whose \"faith in human nature and civilization and similar idealistic nonsense was receiving such a rude shock\" from the war. And he derided the \"soft mush about everlasting peace which so many statesmen are handing out to a gullible public.\" When the war ended, Roosevelt supported American membership in the League of Nations but added that \"I don't care how many restrictions or qualifications are put on [it].\" The League, he explained, was \"merely a beautiful dream, a Utopia.\" In the harsh and exhilarating real world, the struggle for power would rage on.\n\nIf FDR's muscular foreign policy views echoed TR's, they also stemmed from his almost mystical connection to the sea. Growing up on an estate alongside the Hudson River, he built toy boats before he could read, sailed his first real one at age nine, and owned a vessel by age sixteen. At his prep school, Groton, he filled his room with prints of sailing ships and tried to run away and join the navy during the Spanish-American War. At Harvard he began collecting manuscripts about naval history and developed a lifelong fascination with geography. He also read the works of the famed Admiral Alfred T. Mahan, who argued that a strong navy, capable of protecting commercial shipping, securing colonies, and protecting home waters, was essential to a nation's power.\n\nFDR's love of the sea reinforced his cosmopolitanism: As a child, he spent so much time learning German and French from foreign-born governesses that when he entered Groton he spoke with a slight European accent. He traveled to Europe for the first time at age three, and then every summer from ages seven to fifteen. Unlike Wilson, whose image of the world derived largely from theories developed in the United States, FDR was used to seeing events through foreign eyes. And unlike Hoover, whose international mining and relief work inclined him to see foreign nations as consumers or charity cases, FDR, because of his naval obsession, could more easily see them as competitors. Expert sailor that he was, FDR knew that the Atlantic and Pacific were no longer America's moats\u2014that they grew smaller with every naval advance. Even before World War I, he urged Daniels and Wilson to build a fleet capable of patrolling one thousand miles off American shores. If \"an enemy of the United States obtains control of the seas,\" he warned, the Western Hemisphere would offer no refuge.\n\nFinally, it was the sea that introduced Roosevelt, just before his fortieth birthday, to tragedy and forced him to summon a previously untapped, inner strength. As a young man, he had indeed been\u2014as his detractors whispered\u2014pliable, smug, and light. For all his mimicry of his famous relative, the contrasts spoke louder than the similarities. While an undergraduate at Harvard, TR had begun writing his authoritative, five-hundred-page history of the Naval War of 1812. FDR, by contrast, wrote editorials for the Harvard Crimson urging louder cheering at football games. TR had actually waged war; FDR just pretended to. In July 1918, after much pestering, Daniels let him travel to wartime Europe to inspect the U.S. fleet. Although he later giddily recounted his voyage through (allegedly) sub-infested waters and described the artillery fire he heard on the battlefields of northern France, FDR never encountered any real hardship or danger. To the contrary, traveling with his old Harvard friend and golfing partner Livingston Davis, he stayed in the fanciest hotels, dined with foreign dignitaries, and drank himself sick. (After one particularly wild night of carousing in Scotland, he awoke to find himself sharing a bed with a fox.) Although already sporting a fever, he stayed out until 4:30 A.M. on his final night in Europe, and then promptly contracted double pneumonia on the voyage home. When the boat docked in New York City, he was carried by stretcher to his mother's house, where Eleanor, while unpacking his clothes, discovered his love letters to her former assistant, Lucy Mercer. Eleanor offered him a divorce, but when his mother threatened him with disinheritance, FDR declined. For Roosevelt's critics, the trip just illustrated his true nature: boastful, spoiled, irresponsible, dilettantish, and weak\u2014\"the featherduster,\" as college classmates called him behind his back.\n\nBut something changed in his thirty-ninth year. After a long and exhausting deep-sea fishing trip near the Roosevelt vacation compound off the New Brunswick coast, FDR fell into icy waters and soon began to shake uncontrollably. Within days, he was paralyzed from the waist down by polio. As his muscles atrophied, his shriveled legs began to bend backward behind his knees. To keep them straight, a doctor pressed them into heavy plaster casts, where they were forced upright by iron rods, in a procedure that one biographer compared to \"being stretched on a rack in a medieval torture chamber.\" Years later, FDR told a friend that in those days of agony, for the first time in his life he lost his faith in God, and then, somehow, found it again.\n\nOutwardly, FDR remained much the same. He rarely complained, and he retained his usual breezy demeanor. But those close to him noticed a profound change. Polio, noted Eleanor, was the \"turning point\" in her husband's life. It \"gave him strength and courage he had not had before.\" Roosevelt's close friend and future labor secretary Frances Perkins described \"a spiritual transformation\" and a new \"humility of spirit.\" FDR's uncle called him a \"twice-born man.\"\n\nThis can be taken too far. In important ways, the foreign policy realism that FDR smuggled into the White House antedated his disease. It had its roots in his adoration of TR, his reading of Mahan, his travels in Europe, and his image of the sea. But in some inchoate way, polio reinforced his view of the world as an irrational, unknowable, unyielding place. By 1932, FDR understood, in a way Wilson and Hoover never did, that politics, like life, did not permit tidy constructs or linear progress. His personal story\u2014brash optimism and easy success followed by sudden calamity and a struggle against the limits it imposed\u2014was a kind of metaphor for his entire generation. Like Niebuhr, he was also, in his way, a \"child of the age of disillusionment.\" As Roosevelt campaigned for president on crutches, rocking his stiff, steel-encased legs and hips forward in order to move a few excruciating feet, his old drinking buddy, Livingston Davis, shot himself. It was yet another sign of an end to childlike things.\n\nAt first, Charles Beard was enthusiastic about Roosevelt's presidency. Tacking as he so often did with the political wind, FDR had spent the 1920s downplaying his internationalist foreign policy views. At the 1932 Democratic National Convention, when the isolationist publisher and power broker William Randolph Hearst demanded that he renounce his support for the League, Roosevelt quickly complied, thus helping secure his party's nomination. In his first year in office, he torpedoed efforts to coordinate a global response to the Depression and his delegate to the London Disarmament Conference announced that America would make \"no commitment whatever to use its armed forces for the settlement of any dispute anywhere.\" FDR did triple the budget for new ship construction, the biggest increase since 1916, but even this he justified on isolationist grounds: as a way to fight unemployment and defend America's home waters. Impressed, Beard applauded Roosevelt's renunciation of \"any duty owed by the United States to benighted peoples.\" By the fall of 1933, Beard was dining at the White House and reporters were calling him \"one of the intellectual parents of the New Deal.\"\n\nHitler was now in power. Four days after being appointed chancellor, he ordered a massive rearmament effort. A few months later, he pulled Germany out of the London Disarmament Conference and the League. In early 1934, he made his rearmament efforts public, in bald defiance of Versailles. Seeing no response from Britain or France, he introduced conscription the following year. Then, on the morning of Sunday, March 7, 1936, German troops marched into the Rhineland. France asked if it could expect Britain's support if it met Hitler's move with force, support that Britain had twice pledged to provide, at Versailles and Locarno. But His Majesty's government refused, with its secretary of state for war telling Hitler's ambassador that the British public \"did not care 'two hoots' about the Germans reoccupying their own territory.\" For his part, FDR's secretary of state, Cordell Hull, declared that since the United States had signed neither Versailles nor Locarno, Hitler's actions were none of its concern. Only later would historians learn that Hitler, whose military buildup was still in its early stages, had instructed his generals to retreat at the first sign of foreign resistance.\n\nPrivately, FDR called Hitler a madman and mused about offering the Fuhrer a disarmament deal, and imposing a naval blockade if he refused. But publicly he said nothing of the sort. An astute student\u2014and often slavish follower\u2014of popular opinion, Roosevelt knew exactly how his fellow citizens felt about German aggression: Far from undermining American isolationism, Hitler's moves were bolstering it. During the 1920s, when the prospect of another European war seemed remote, pacifism had been as strong a public force as isolationism, although the line between the two sometimes blurred. But by the mid-1930s, with militarism rising in Germany, Italy, and Japan, Americans found it harder to imagine that through commerce, disarmament deals, and nonaggression pacts they could build a less violent, more rational world. And so pacifism, with its optimistic, outward-looking spirit, receded before isolationism: the sense that with storms brewing on the horizon, America should shut its windows and lock its doors. \"Let us turn our eyes inward,\" declared Pennsylvania governor George Earle in 1935. \"If the world is to become a wilderness of waste, hatred, and bitterness, let us all the more earnestly protect and preserve our own oasis of liberty.\"\n\nAmerican efforts to do that were shaped, to a striking degree, by the memory of World War I. From 1934 to 1936, North Dakota's Gerald Nye oversaw a vast Senate investigation, involving ninety-three hearings and more than two hundred witnesses, into whether bankers and arms dealers had lured America into the Great War. Bestselling books such as H. C. Englebrecht and F. C. Hanighen's Merchants of Death, Walter Millis's Road to War, and Beard's The Open Door at Home and The Idea of the National Interest hurled essentially the same charge. On April 6, 1935, the eighteenth anniversary of America's entry into the war, fifty thousand veterans marched for peace in Washington, laying wreaths at the graves of three congressmen who had voted no. A January 1937 poll revealed that 71 percent of Americans considered U.S. intervention in World War I a mistake.\n\nThe result of all this historical remorse was the Neutrality Acts, a series of laws that between 1935 and 1937 prohibited the sale of weapons or the lending of money to countries at war, and warned Americans that if they traveled on belligerents' ships, they did so at their own risk. This time reckless tourists would not be permitted to get themselves drowned, and weapons manufacturers and Wall Street speculators would not be allowed to sign contracts that gave them a financial incentive in the spilling of American blood.\n\nThe Neutrality Acts represented the final, tragic, unintended consequence of Wilson's justification for U.S. entry into World War I. Wilson had justified that entry by invoking America's sacred rights under international law. Now the historical revisionists were insisting that those rights hadn't really been sacred at all; they had masked the pecuniary interests of arms dealers and Wall Street fat cats\u2014fat cats who were none too popular with American capitalism on its knees. The war progressives had offered an idealistic rationale for war, not a geopolitical one, and so once Americans decided that this idealism was a fraud, no compelling argument for international engagement remained. In this sense, the isolationism of Charles Beard was war progressivism's bastard child.\n\nFranklin Roosevelt had World War I on the brain as well. \"From 1913 to 1921, I personally was fairly close to world events,\" he told the nation in a radio address, \"and in that period, while I learned much of what to do, I also learned much of what not to do.\" FDR didn't say so out loud, but unlike 71 percent of his fellow citizens, he still believed the war had been worth fighting. In his mind, Wilson's mistake was in how he justified it. Wilson had portrayed it as a war for high principle when it was actually a war to keep America safe.\n\nThrough the end of 1936, FDR kept his head down, signing the Neutrality Acts and promising to \"isolate America from war,\" thus helping ensure a reelection landslide. But in October 1937, when Tokyo resumed its attack on China, he began to do what Wilson had not: argue that the shifting balance of power on far-off continents endangered the United States. Condemning \"terror and international lawlessness,\" FDR warned that \"if those things come to pass in other parts of the world let no one imagine that America will escape, that it may expect mercy, that this Western Hemisphere will not be attacked.\" Roosevelt wasn't arguing that Japanese expansionism merely threatened principles like self-determination that Americans held dear. He was arguing that Japanese expansionism threatened the United States itself, that stopping it was a matter of necessity, not choice.\n\nFDR didn't say how the United States should stop Japan, except to speak vaguely about an international \"quarantine.\" Still, isolationists got the point. He was taking aim at their central argument: that America could live apart. Members of Congress threatened impeachment. \"Stop Foreign Meddling; America Wants Peace,\" screamed the Wall Street Journal. The quarantine speech, FDR later claimed, \"fell upon deaf ears, even hostile and resentful ears.\" The following January, after Japanese planes sank an American gunboat on the Yangtze River, the House of Representatives came within twenty-one votes of passing the Ludlow Amendment, which would have required a national referendum on taking the country to war except in cases of foreign attack. For the moment, FDR's bid to convince his fellow citizens that they had a stake in the global balance of power had failed. \"It's a terrible thing,\" he told an aide, \"to look over your shoulder when you are trying to lead and to find no one there.\"\n\nIn March 1938, Hitler swallowed Austria, with Washington offering only the mildest of protests. Then in September, in his most brazen move yet, he demanded the Sudetenland, the German-speaking chunk of Czechoslovakia. The Czechs asked the French, with whom they had a security treaty, if they could expect support in the event of war. The French in turn asked the British, and were told\u2014as Neville Chamberlain later famously put it\u2014that the Sudetenland crisis was \"a quarrel in a faraway country between people of whom we know nothing.\" Not only would Britain not fight to defend Czechoslovakia's border with Germany, it would not even commit to defending France's border with Germany (despite having pledged to do so at Locarno), if in trying to defend its imperiled Eastern European ally, Paris incurred Hitler's wrath. On paper, the U.S.S.R. was also committed to France and Czechoslovakia's defense. But with Britain fleeing its commitments, Moscow would not act alone.\n\nThe French were actually relieved. By the late 1930s, Hoover had gotten his wish: French leaders were no longer militaristic and cocky. To the contrary, they were almost morbidly fatalistic, like a man who knows he will soon face the gallows and has spent so many hours shouting himself hoarse from his prison cell, with no response, that he can no longer muster the breath. The French knew their best chance at stopping Hitler had been to act early, before he converted Germany's massive industrial capacity into military might. By 1938, it was too late: France was producing forty-five warplanes per month; Germany was producing 450. When the U.S. ambassador in Paris asked French premier Edouard Daladier if France would go to war for Czechoslovakia, Daladier replied, \"With what?\"\n\nOn September 15, Chamberlain flew to Hitler's mountain castle in Berchtesgaden\u2014a remote location that the Fuhrer had chosen largely to humiliate the British leader\u2014and told him that London and Paris would back a plebiscite in those parts of Czechoslovakia where ethnic Germans constituted a majority. But Hitler soon upped the ante. The Czechs, he demanded, must evacuate the Sudetenland by October 1 and hand the territory\u2014and all its military installations\u2014over to the German army, which would then supervise the vote. Hitler also insisted that Prague surrender its Polish-and Hungarian-speaking provinces to those neighbors, just to ensure that the rump Czech state would be utterly defenseless. At first Chamberlain refused. But his generals, long starved of resources by governments committed to disarmament, said they lacked the planes to defend London. So on September 28, in a last-ditch bid for peace, Chamberlain flew to meet Hitler again, this time in Munich. For thirteen hours, the leaders of Britain, Germany, Italy, and France negotiated, while Czech representatives waited at a nearby hotel. In the end, Hitler got the Sudetenland, and Chamberlain got a pledge that Germany and Britain would remain forever at peace. The British leader returned home triumphant, carrying his trademark umbrella, and telling a cheering crowd that he had achieved \"peace with honor...peace for our time.\" Privately, FDR compared Britain's treatment of the Czechs to Judas Iscariot's betrayal of Jesus. But he didn't say so publicly. To the contrary, he sent Chamberlain a two-word cable: \"Good man.\"\n\nAs 1938 turned to 1939, FDR kept telling Americans that their fate and Europe's were intertwined. But as his actions in the Munich crisis showed, he held out little hope of translating those words into actions anytime soon. There was something ironic in the fact that this master communicator, whose radio addresses had rallied the nation during the depths of the Depression, did not believe his oratory could rouse isolationist America from its slumber. But this was part of the lesson he had drawn from World War I. Wilson's failure during the treaty fight had left Roosevelt dubious about the degree to which a leader, even an articulate one with the facts on his side, could sway the public on questions of war and peace. In this way, FDR's cautious response to the darkening international scene wasn't mere timidity; it reflected a retreat from the progressive belief that citizens were essentially pliable. In FDR's mind, political education could only do so much. Events would have to speak for themselves.\n\nOn September 1, 1939, international dignitaries gathered in Geneva, home to the League of Nations, to unveil a long-planned statue of Woodrow Wilson. On the same day, hundreds of miles to the northeast, German soldiers poured across the Polish border. At last, Britain and France decided to fight. At 2:50 A.M. Washington time, Roosevelt was awakened by a call from his ambassador in Paris, William Bullitt, informing him that World War II had begun. The call, FDR told his cabinet later that week, provoked \"a strange feeling of familiarity, a feeling that I had been through it all before.\" The two decades since World War I suddenly seemed like a long intermission. Now he was \"picking up again an interrupted routine.\"\n\nAddressing the nation on September 3, FDR assured Americans that he had no intention of taking them to war. But neither did he ask them to be detached observers in the struggle between Nazism and its foes. \"I cannot ask that every American remain neutral in thought,\" he declared, in an oblique reference to Wilson's call for impartiality in the early days of World War I. \"There is a vast difference,\" he added in his State of the Union address the following January, \"between keeping out of war and pretending that war is none of our business.... For it becomes clearer and clearer that the future world will be a shabby and dangerous place to live in\u2014yes, even for Americans to live in\u2014if it is ruled by force in the hands of a few.\" With Europe at war, Congress finally amended the Neutrality Acts to allow Britain and France to buy U.S. arms, so long as they paid cash and carried them away in their own ships.\n\nAs 1939 turned to 1940, Hitler's Germany and Stalin's U.S.S.R.\u2014which had signed a nonaggression pact the previous summer\u2014carved up Eastern Europe. Then, in April, Germany conquered Denmark and Norway, and then Holland and Belgium in May. Keeping up his rhetorical campaign, FDR warned that if Hitler gained control of Europe, Americans would be a \"a people lodged in prison, handcuffed, hungry, and fed through the bars from day to day by the contemptuous, unpitying masters of other continents.\" But these were just words. \"If you cannot give France in the coming days a positive assurance that the United States will come into the struggle within a short space of time,\" wrote French premier Paul Reynaud on June 14, \"you will then see France go under like a drowning person after having thrown a last look towards the land of liberty from where she was expecting salvation.\" But with more than 80 percent of Americans still against entering the war, FDR let France drown. Only the sight of it, he reasoned cruelly but correctly, would shift the public mood. Eight days after Reynaud's plea, the Nazis accepted France's surrender on the same spot, in the forests of Compi\u00e8gne north of Paris, where Germany had surrendered twenty-two years before. Mercifully, Georges Clemenceau was not alive to see his nightmare come true.\n\nFrance's fall jolted American opinion like no presidential speech ever could, and FDR quickly capitalized. In August, with Winston Churchill now Britain's prime minister, and bracing his nation for a German assault, FDR sent the United Kingdom fifty destroyers in exchange for military bases in the Western Hemisphere. The following month, he convinced Congress to institute the draft, although mail to the White House ran ten to one against. And in December, with the British no longer able to pay for American arms, he pushed through \"lend-lease,\" which essentially provided them for free.\n\nAs he edged closer to war, FDR began laying the foundations that would undergird American foreign policy for the next quarter century. First, he convinced former Republican secretary of state Henry Stimson to become his secretary of war, and Frank Knox, who had been the GOP's vice presidential nominee in 1936, to serve as secretary of the navy. Stimson's former boss, Herbert Hoover, was now a rabid isolationist. But Stimson, who had quarreled with Hoover almost a decade earlier on Manchuria, now broke with him decisively and became the anchor for a revived Republican internationalism, which helped FDR push lend-lease and the draft through Congress. In choosing Stimson, FDR\u2014the country squire whose New Deal domestic policies had made him a hated figure in conservative East Coast financial circles\u2014was also reconciling with his class. American foreign policy would henceforth be led by a bipartisan, hard-nosed elite entirely comfortable with power politics. In America's hour of peril, the children of Theodore Roosevelt\u2014who espoused neither Woodrow Wilson's ethic of reason nor the pacifism of John Dewey nor the isolationism of Charles Beard\u2014were closing ranks. They would guide American foreign policy for a generation, until discredited by hubris of their own, in Vietnam.\n\nAt the same moment he embraced bipartisan internationalism, FDR also began dramatically expanding presidential power. The deal to provide Britain fifty destroyers violated the Neutrality Acts and would have met fierce resistance on Capitol Hill. But the White House, advised by a corporate lawyer named Dean Acheson, decided it had the legal authority to transfer them without congressional approval. Then, in May 1941, FDR declared an \"unlimited national emergency,\" which gave him the power to take any action he deemed necessary to protect the Western Hemisphere. In July, he used it to send four thousand Marines, without congressional approval, to Greenland and Iceland, to prevent them from falling into Nazi hands. It was an historic shift. Ever since Versailles, Congress had held foreign policy tightly in its grip, giving the president little scope for overseas adventures. Now executive power\u2014nurtured by America's bipartisan internationalist elite\u2014was starting to swell. It would swell unabated for a generation, until it too collapsed in the jungles of Vietnam.\n\nIn May 1941, FDR had a dream that Hitler's Luftwaffe was bombing New York. Privately, some historians believe, he had already decided to enter the war. By fall, U.S. warships were escorting British merchant vessels across the Atlantic, ranging hundreds of miles in pursuit of German subs, doing everything possible to provoke an incident. They soon got one. In September, a German U-boat fired torpedoes at the U.S.S. Greer near Iceland. In October the U.S.S. Kearny was hit, and later the U.S.S. Reuben James. Public opinion swung toward war, particularly since FDR claimed that the ships had been attacked without provocation, which was a lie. But still, Roosevelt waited. Calling for war now, he told Churchill, would trigger a bitter three-month congressional debate. And if events on the battlefield or at the subsequent peace conference turned sour, that initial disunity would return to haunt the nation.\n\nThe basis upon which America entered the war, FDR believed, would make all the difference once the shooting stopped. When subs attacked the Greer, he did not say\u2014as Wilson had\u2014that Germany had violated America's neutrality rights. Instead he warned that Hitler was seeking \"absolute control and domination of the seas.\" Roosevelt was groping for a way to convince the public that this time Germany threatened not just international law, but American security. Only then\u2014once Americans saw international engagement as survival, not social work\u2014would they sustain it even when the world let them down. \"Our policy is not based primarily on a desire to preserve democracy for the rest of the world,\" FDR added. \"It is based primarily on a desire to protect the United States and the Western Hemisphere from the effects of a Nazi victory.\" This, he believed, was what Wilson should have said in 1917.\n\nIt was easier in 1941, of course, because Hitler was a more terrifying figure than the kaiser, with more terrifying weapons and more grandiose plans. But FDR still needed an event to dramatize his words. It came, as it happened, not in the Atlantic but in the Pacific. When France fell to Germany in 1940, Japan conquered its colonies in northern Indochina, leading the United States to halt sales of scrap iron and steel in protest. The following summer, Japan took the rest of France's Asian empire, prompting FDR to take the much graver step of cutting off Tokyo's supply of oil. That set the clock ticking. With no fuel supply of its own, Japan faced a choice: ignominious withdrawal or a push farther south into the oil-rich Dutch East Indies. By the first week of December, Roosevelt knew that Tokyo was planning the latter course, which he feared would involve an attack on U.S. forces in the Philippines. His close aide Harry Hopkins suggested that America strike first and thus gain the element of surprise, but FDR refused, partly because he believed that only a Japanese attack would unify the country behind war.\n\nAt 7:55 A.M. Hawaii time on December 7, carrier-based Japanese aircraft began appearing in the Oahu sky. For two hours, 360 planes turned Pearl Harbor into an inferno, killing more than two thousand Americans and destroying most of America's Pacific fleet. The decision for war, FDR told Hopkins, was now out of his hands. Events, not mere words, had shown that America could not live apart. \"Franklin,\" commented Eleanor, \"was in a way more serene than he had appeared in a long time.\"\n\nOn May 22, 1940, Reinhold Niebuhr received a letter from the executive secretary of the American Socialist Party. The party had just held its annual convention, which reaffirmed its devotion to class struggle and its indifference to the struggle between nations currently bloodying much of the world. World War II, according to party doctrine, was a contest between capitalist powers in which Socialists should have no allegiance. This placed the party in direct conflict with groups like the Committee to Defend America by Aiding the Allies, which insisted that the United States had a strategic and moral interest in fascism's defeat. Niebuhr, the secretary noted, was a prominent member of both organizations. \"I am interested,\" he wrote, \"in getting your reaction on this matter.\"\n\nTwo days later, Niebuhr resigned his party membership. \"The Socialists have a dogma that this war is a clash of rival imperialisms,\" he wrote in the Nation. \"Of course they are right. So is a clash between myself and a gangster.\" The party was so wedded to \"purely ideal perspectives\"\u2014it viewed events from such Olympian heights\u2014that it could see no important difference between the capitalism of Winston Churchill and the capitalism of Adolf Hitler. That, Niebuhr argued, was \"utopianism.\" For him, few epithets were worse.\n\nNiebuhr's break with the left had been several years in coming. By the mid-1930s, with the New Deal in full swing, his fury over economic oppression had begun to cool. He still believed that ultimately the proletariat would rule. But FDR's \"amiable opportunism,\" he admitted, was \"arresting the decay of capitalism as effectively as that can be done.\" His sympathy for Marxism also faded as he learned more about its Soviet incarnation. In 1939, when Nazi Germany and Soviet Russia became allies, he began to stress their common totalitarian features. If the Socialists saw little difference between Churchill and Hitler, Niebuhr saw less and less difference between Hitler and Stalin.\n\nBut there was something odd about Niebuhr's ideological transformation. Despite moving from left to center, he kept sounding many of the same themes. In 1932, he had attacked progressives like Dewey as na\u00efve for thinking that through reason they could achieve justice without conflict. Now he accused the communists of na\u00efvet\u00e9 as well. They too, he argued, championed a supposedly scientific vision of inexorable progress. They too believed that if you overhauled society's institutions you could eradicate evil. Because they would not acknowledge that human nature was inherently sinful, they could not see that it wreaked havoc upon all efforts to rationalize society, including their own. Whereas Niebuhr had once applauded communism for its hard-boiled embrace of class conflict, he now lumped it with progressivism as yet another utopian creed.\n\nOnce again Niebuhr reflected his times. By the mid-1930s, many American intellectuals were shifting anxieties, worrying less about the injustice of Depression-era capitalism and more about the menace of fascist aggression. At first communism seemed a sturdy bulwark against this rising threat, and to underscore the point, in 1935 the American Communist Party ceased its revolutionary agitation and joined liberals in an antifascist Popular Front. But between 1936 and 1938, Stalin's show trials, which culminated in the murder of dozens of his political rivals, prompted some American intellectuals to reconsider their communist sympathies. And when Stalin and Hitler joined forces the following year, the trickle of anticommunist disillusionment became a flood. One after another, intellectuals who had embraced some species of communism during the Depression left the Marxist faith. The disillusioned communists\u2014a remarkable group that included the philosophers Sidney Hook and James Burnham; the journalists Max Lerner, Louis Fischer, Irving Kristol, and Max Eastman; the critics V. F. Calverton and Irving Howe; and Philip Rahv and William Phillips, editors of the new, fiercely anti-Stalinist journal Partisan Review\u2014did not all end up in the same place politically. But they shared a sensibility. Like Niebuhr, they had lost faith in grand schemes for remaking the world because they had lost faith in grand schemes for remaking human beings. The 1930s taught that evil did not stem merely from failures of education and expertise, or from one economic system. Its roots lay deeper, inside humanity itself. A whole generation of intellectuals carried this tragic insight with them as they journeyed back to the democratic fold. They approached American democracy not as virgin enthusiasts, like the progressives, but as weary travelers. Noting that they had gained wisdom through pain, the sociologist Daniel Bell called them \"twice-born,\" which, as it happens, was the same phrase FDR's uncle had applied to him.\n\nIt was these \"twice-born\" intellectuals who in the late 1930s provided intellectual ballast for Roosevelt's moves toward war. Unlike the war progressives, they held out little hope that through military action America could eradicate power politics. They conceded that Americans would die and kill for racist, colonial powers like Britain and France and for an immoral economic system at home. But they were comfortable fighting for lesser evils. In their minds, that was what politics was all about. \"The civilization we are called on to defend,\" wrote Niebuhr, \"is full of capitalistic and imperialistic injustice...[but] it is still a civilization.\"\n\nIn their argument for war, the ex-communists were joined by another child of the age of disillusionment: Walter Lippmann. Lippmann had walked his own tortured intellectual path since Versailles. Obsessed with the public irrationality that World War I had exposed, in the 1920s he had questioned whether ordinary people were competent to govern themselves at all, whether democracy could ever really work. But like the ex-Marxists, he became a partisan of liberal democracy in the late 1930s, not because he had regained his youthful optimism about what it could accomplish but because the alternatives were so horrifying.\n\nEarly in FDR's presidency, fearful of repeating the mistakes of 1917, Lippmann had urged America to steer clear of Europe's troubles. But as the Nazi menace grew, he came to terms with the war that as a young man he had championed and then spurned. He did so by replacing the idealistic arguments of his youth with realist ones. America had not entered World War I to rationalize the world, he now argued. It had fought to ensure \"the safety of the Atlantic highway\": to prevent Germany from smashing the British fleet that guaranteed American security and prosperity. This, he argued, \"is something for which America should [again] fight.\" He had, it turned out, discovered the same rationale for war as FDR, the man he once called an \"amiable boy scout\" who \"just doesn't happen to have a very good mind.\"\n\nBy the early 1940s, John Dewey and Charles Beard were old men. They could see war coming and had long ago resolved to resist it with everything they had so as never to repeat the sins of World War I. In 1940, when the City University of New York revoked a job offer to the British philosopher Bertrand Russell because he was a pacifist, Dewey felt he was reliving a nightmare. Beard, who had long since stopped receiving invitations to the White House, grew convinced that FDR's power grab was leading to dictatorship. He even alleged that Roosevelt had known the Japanese might bomb Pearl Harbor but did nothing because he needed a pretext for war. Both Dewey and Beard felt in their bones that this war would be worse than the last, that it might destroy American democracy once and for all. \"Day and night I wonder and tremble for the future of my country and mankind,\" Beard wrote to a friend. \"An epoch has come to an end,\" Dewey declared, \"but what is beginning is too much for me.\"\n\nIn their opposition to American entry into World War II, Dewey and Beard were fulfilling their silent oath to their martyred student, Randolph Bourne. But ironically, they now came under assault from a new group of younger intellectuals, for whom the Bourne legend held no appeal. To these sad-eyed realists, who in the 1930s had grown old beyond their years, Dewey and Beard were moral perfectionists, men who would allow the greatest of evils to triumph for fear of implicating themselves in the lesser evil of war. Dewey, Niebuhr charged, would not \"defend democracy because it is not pure enough.\" The historian Lewis Mumford, once a Beard disciple, now called him \"a passive\u2014no, active\u2014abetter of tyranny, sadism, and human defilement.\"\n\nIn 1941, the American Communist Party, now in bed with Nazism via the Hitler-Stalin pact, created a Randolph Bourne Award to honor an intellectual who opposed America's march to war. They chose the novelist Theodore Dreiser, a raving anti-Semite who considered FDR's foreign policy a Jewish plot. Pacifism and isolationism, which sought to keep America free from the barbarism of the old world, had become complicit in its greatest barbarism ever. The age of Randolph Bourne\u2014which had begun, oddly enough, when influenza took his life\u2014was now dead. In its wake, the real answer to the hubris of reason was being born.\n\n## CHAPTER FOUR\n\n## I DIDN'T SAY IT WAS GOOD\n\nOn Friday, September 15, 1944, Franklin Roosevelt went to the movies. The film was Wilson, a lavish, syrupy, star-studded account of the twenty-eighth's president's life that cost $5 million to produce, making it the most expensive film yet made. A decade earlier, when World War I revisionism ran rampant and Wilson was a figure of public scorn, such a project would have been unimaginable. But World War II rescued Wilson's reputation. In 1941, Otto Preminger directed In Time to Come, a Broadway play about the Paris Peace Conference, which painted Clemenceau and Lodge as villains and Wilson as a man too good for his time. Three years later, the magazine Look published a glossy pictorial biography of Wilson subtitled \"The Unforgettable Figure Who Has Returned to Haunt Us.\" At their 1944 convention, Democrats voted a resolution of tribute to the former president. And by the time Wilson hit the theaters, the nation was in the midst of a full-blown Wilson revival. At the Roxy in New York, twenty thousand people stood in line for a ticket; one million saw the film in its first five weeks alone. The movie, which noted Wilson's love of football, golf, and vaudeville, even made him seem friendly. But FDR was less charmed by the experience than stressed. While he was watching it, his blood pressure spiked to 240 over 130.\n\nFor Roosevelt, Wilson's failure was a near obsession. When writing speeches in the Cabinet Room of the White House, he often glanced up at Wilson's portrait, which he had installed above the mantelpiece. Versailles, wrote FDR's speechwriter Robert Sherwood, was \"always somewhere within the rim of his consciousness.\" But unlike the creators of Wilson, FDR did not exactly consider his old boss a role model; more like a cautionary tale. Wilson, he believed, had dug his own political grave by imagining that other countries and his own citizens were more malleable than they really were. In response, in the years before Pearl Harbor, FDR had labored mightily to convince Americans by word and deed that saving Europe and Asia from fascism was self-defense, not philanthropy. And now, with America in the fight, he had two big and rather un-Wilsonian ideas about the postwar world. First, it should be built on power, not reason, because only a coalition of the powerful could keep the peace. Second, this must be concealed\u2014as much as possible\u2014from the American people, who, if confronted with the ugly truths of international affairs, might scurry back to their shores, in which case the world would again go to hell. It was in his messy, often unsentimental, sometimes duplicitous efforts to square these dual imperatives that FDR, in his final years on earth, began formulating an answer to the hubris of reason. That answer\u2014which his successor, Harry Truman, amended and completed\u2014was far from perfect. In some ways, it was downright tragic. But it laid the foundation for a fundamentally better world.\n\nIf FDR had an historical inspiration for his postwar vision, it was not America's twenty-eighth president; it was the twenty-sixth. Theodore Roosevelt had also envisioned an international order built on power. As early as 1910, he had offered his own conception of a League of Nations. What distinguished it from Wilson's was that TR wanted \"to create the beginnings of international order out of the world of nations as those nations actually exist.\" In other words, power in the League should reflect power in the world: TR had little patience for an international body based upon the principle that strong and weak nations were equal in the eyes of international law, since for him law was only as good as the gun that enforced it. When the United States entered World War I, he saw the Allies as the core of this new body. He wanted a league of victors, built upon security treaties between America, Britain, and France. The victorious powers would be allowed their spheres of influence, no matter how unscientific and unfair, as long as they stuck together and thus kept the defeated Germans from again disturbing the peace.\n\nFDR's vision was strikingly similar. In August 1941, six months before America entered the war, he met Churchill off the coast of Newfoundland. They agreed to formulate an \"Atlantic Charter\" that outlined their hopes for the postwar world. But when the British prime minister proposed that this vision include an \"effective international organization,\" FDR balked, saying the League had been tried and had failed. Instead, he suggested, Britain and America should police the world themselves. \"The time had come,\" he declared, \"to be realistic.\"\n\nThe following May, Roosevelt broadened the idea, telling Soviet foreign minister Vyacheslav Molotov (whose country was now at war with Hitler) that the postwar world should include \"Four Policemen\": the United States, Britain, the U.S.S.R., and China, the four nations that would emerge as most powerful once Germany and Japan were crushed. Each great power would walk a particular beat: Britain would control Western Europe; the Soviet Union would control Eastern Europe; the United States would guard the Western Hemisphere and the Pacific; and China (with America's help) would control the Asian mainland. Together, these Four Policemen would keep down Germany and Japan, so that they couldn't stir up trouble as they had after World War I. Such an arrangement, FDR told the \"cannonball\"-headed, \"slab\"-faced Soviet foreign minister, was far preferable to another League, which would include \"too many nations to satisfy.\" As the Saturday Evening Post's Forrest Davis reported the following April, \"Mr. Roosevelt primarily is concerned not with aspirations toward a better world such as he articulated in the Atlantic Charter, but with the cold, realistic techniques, or instruments, needed to make those aspirations work. This means that he is concentrating on power.... The problem of security rests with the powers who have the military force to uphold it.\"\n\nThis time there would be no Inquiry poring over maps in the New York Public Library, scientifically determining which nations should exist, and where. Unlike Wilson, FDR clearly envisioned spheres of influence, in which great-power \"policemen\" made the rules and small nations did as they were told. And unlike Wilson, who had imagined a peace settlement so impartial that it was embraced by victor and vanquished alike and thus required little enforcement, FDR wanted the victors to impose peace and sustain it by confronting Germany and Japan with overwhelming force. His greatest desire was that the wartime allies stick together to uphold the peace. (Unlike Wilson, FDR always called America an ally of Britain, France, and the U.S.S.R., never a mere \"associate.\") Ideally, in Roosevelt's view, the Germans and Japanese would see the justice of this new postwar order. But he wasn't relying on public opinion to keep the postwar peace; he was relying on a powerful alliance. All this would have brought a smile not only to the faces of Teddy Roosevelt and Henry Cabot Lodge, but to that other mustachioed devotee of power politics: Georges Clemenceau.\n\nSecretary of State Cordell Hull and his influential deputy, Sumner Welles\u2014both unreconstructed Wilsonians\u2014were appalled. \"During the spring of 1943,\" wrote Hull in his memoirs, \"I found there was a basic cleavage between him [FDR] and me on the very nature of the postwar organization.\" Hull and Welles tried and failed to convince FDR that opposing another League of Nations was wrong on the merits. But fortunately for them, Harry Hopkins, the president's most influential adviser, convinced him he was wrong on the domestic politics. Roosevelt had assumed that the American people did not want another League; he had told Churchill as much at Newfoundland. Since America entered the war, however, public sentiment had shifted, as the reaction to the movie Wilson illustrated. In 1941, only 38 percent of Americans had backed U.S. entrance into a new international organization; by 1944, 72 percent did. This was, in a sense, a testament to FDR's own strategy: By waiting until Pearl Harbor to enter the war, he had convinced most Americans that isolationism was impossible. But now, having decided to participate in the world, Americans were again determined to bend it toward their ideals. For FDR, that was a recipe for disillusionment, since he suspected that the world would not bend very quickly or very much. Hopkins, however, convinced him to conceal those thoughts from public view. Henceforth FDR would continue crafting a postwar settlement based on power politics, but he would give it a Wilsonian face, especially when addressing audiences at home. And he would try, in this way, to do what Wilson and the war progressives could not: reconcile America's missionary impulse with the realities of an imperfect world.\n\nIn keeping with this political strategy, FDR in the spring of 1943 agreed to encase his Four Policemen idea within a Wilsonian shell. In a meeting with British foreign secretary Anthony Eden in March, he suggested a three-tiered international body including an executive committee, comprised of the Four Policemen, which made all important decisions, an advisory council including six or eight other nations, which handled smaller, nonmilitary matters; and a general assembly with universal membership that met once a year so that small countries could, in Eden's words, \"blow off steam.\"\n\nIn November, Roosevelt presented his vision to Churchill and Stalin. To meet them, he flew to Tehran, a trip of more than seventeen thousand miles, undertaken in an age when aircraft cabins often lost pressurization, by a man suffering serious respiratory problems and in chronic pain. His fellow leaders embraced the Four Policemen concept, but they were amused by FDR's insistence that they make no public reference to it and his claim that the American people would never accept great-power spheres of influence. It was as if, having drawn them a picture of a wolf, he was demanding that they call it a sheep.\n\nThe contradiction went to the heart of Roosevelt's political dilemma. He desperately wanted good postwar relations with Stalin. Otherwise, he feared, the Germans would exploit divisions between the wartime victors, as they had after World War I. And he knew Moscow's price for such a partnership: a sphere of influence in Eastern Europe, which could serve as a buffer in case Germany ever invaded again.\n\nOn the battlefield, that sphere was already being demarcated. By the time FDR went to Tehran, the U.S.S.R.\u2014having turned the tide against Hitler's forces at Stalingrad and Kursk, two of the bloodiest battles in the history of war\u2014was already emerging as the most powerful player in Eastern Europe. That was no accident. From the beginning, FDR had adamantly opposed spilling American blood in the lands between the Baltic and the Adriatic seas. When his adviser William Bullitt, and then Churchill, suggested that Washington and London alter their European invasion plans, choosing a route that took longer to defeat Hitler but put American and British troops on the ground in Eastern Europe to contain Soviet influence, FDR refused. Fighting in Eastern Europe, he knew, would require mobilizing far more American troops, which could disrupt the economy back home. The military warned that it would prove a logistical nightmare, since the region lacked good supply routes from the sea. And finally, Roosevelt feared that Americans would revolt against the additional loss of life. He had sold them not a war for the rights of small nations, but what he called \"The Survival War\"\u2014a war to defeat Hitler, Mussolini, and Tojo. He knew Americans didn't want Eastern Europe to fall under Stalin's thumb, but his gut told him that they were even more opposed to sacrificing their sons to prevent it. There was a gap, he believed, between the world that Americans wanted to create and the blood-price they were willing to pay to create it. And when that gap was exposed, he feared, a foul, isolationist wind might sweep across the American political landscape. His strategy, therefore, was to hide the discrepancy between ideals and reality as much as possible until America was firmly ensconced in a system of postwar security. Then, no matter how disillusioned Americans became, they would be unable to again retreat from the world.\n\nFor his strategy to work, FDR needed his fellow leaders to lie. That was a major reason he had traveled those forty excruciating hours on a primitive plane\u2014not merely to get Stalin and Churchill to agree to his scheme for postwar cooperation, but to enlist their aid in convincing his own people. In Tehran, FDR told Stalin he \"fully realized\" that Russia planned to reoccupy the Baltic countries, which it had ruled for roughly a century until they gained independence at Versailles. But since \"the question of referendum and self-determination\" would constitute a \"big issue\" in America, it would be helpful to him \"personally\" if Stalin gave Lithuania, Latvia, and Estonia some say over their fate. On Poland, Stalin informed Roosevelt and Churchill that he intended to annex an eastern slice of the country while compensating the Poles with a slice of eastern Germany. FDR said that this was fine but that he \"could not publicly take part in any such arrangement,\" since he needed the votes of \"six to seven million Americans of Polish extraction\" in his upcoming reelection bid.\n\nUpon returning to the United States, Roosevelt delivered a Christmas Eve radio address about his trip. \"The rights of every nation, large or small, must be respected and guarded as jealously as the rights of every individual within our own republic,\" he announced. \"The doctrine that the strong shall dominate the weak is the doctrine of our enemies\u2014and we reject it.\" Just back from a whorehouse, he was insisting that America remained chaste. One day, he knew, reality would become too graphic to ignore.\n\nAs 1943 turned to 1944, the particulars of FDR's vision grew clearer. By March, the State Department had hammered out an outline of the new international organization to be born after the war. Its very name\u2014the \"United Nations Organization\" (later referred to as simply the \"United Nations\")\u2014hinted at its differences with the League. \"United Nations\" was the appellation that America, Britain, the Soviet Union, and their smaller partners had given their alliance against Germany, Italy, and Japan. It suggested a continuation of the wartime partnership, an organization comprising (or at least dominated by) the alliance that had won the war.\n\nIn crucial ways, the UN was also structured differently from the League. The League had also contained a council composed of big countries, but it could merely offer recommendations to the General Assembly, which made the final decision. What's more, every member of that General Assembly could exercise a veto, which created a significant equality of power between members. At the UN, by contrast, the General Assembly would make recommendations to the Security Council, not the other way around. In fact, the Security Council would not even need a General Assembly recommendation; it could simply act on its own authority. Moreover, only the Security Council's permanent members\u2014the Four Policemen (with France later added as a fifth)\u2014enjoyed veto power. The result, observed British historian Evan Luard, was \"a revolutionary transformation of the existing international system in favour of the great.\"\n\nWith a draft of the prospective United Nations in hand, Secretary of State Hull began corralling congressional support, taking particular pains to woo prominent Senate Republicans, which Wilson had failed to effectively do while negotiating the League. Key Republicans such as Arthur Vandenberg, the GOP's ranking member on the Senate Foreign Relations Committee, were pleased by the Security Council veto and the absence of an international police force, both of which allayed their fears that the nascent UN would infringe upon American sovereignty. They were happier with the organization's structure, in fact, than some Wilsonians, who objected to the outsize power that the body afforded large nations. But politically, the UN's real vulnerability wasn't its structure but the territorial settlement it would implicitly ratify, since both Wilsonians and anticommunist conservatives strongly opposed granting the Soviets a sphere of influence in Eastern Europe. In May 1944, when Forrest Davis penned an article titled \"What Really Happened at Teheran,\" which claimed that, in his dealings with Stalin, FDR was \"willing to settle for the lowest common denominator...peace among the great powers,\" prominent senators began to suspect that the president was cutting secret deals on Eastern Europe. \"We all believe in Hull,\" wrote Vandenberg in his diary, \"but none of us is sure that Hull knows the whole story.\"\n\nThey were right; he didn't. That fall, Churchill traveled to Moscow without FDR, who authorized the U.S. ambassador to the U.S.S.R., Averell Harriman, to represent him. At 10 P.M. on October 9, at a meeting to which Harriman was not invited, Churchill took out half a piece of paper, wrote down percentages alongside the names of various countries, and handed it to Stalin, who made a check mark in blue pencil. It was a parody of sinister old-world power politics: The two leaders had just divided up southeastern Europe. Britain's sphere of influence would include 90 percent control in Greece, while Russia's would give it 90 percent in Romania and 75 percent in Bulgaria and Hungary. They would split Yugoslavia 50-50. Churchill told Stalin that this \"naughty document\" should be kept secret \"because the Americans might be shocked.\" That was true. Many Americans, including Vandenberg and probably also Hull, would indeed have been shocked. But FDR was not one of them. At Tehran, Roosevelt later confided, he had \"already told the Russians they could take over and control [Eastern Europe] completely as their sphere.\" The president, commented Harriman several weeks after the Moscow summit, \"consistently shows very little interest in Eastern European matters except as they affect sentiment in America.\"\n\nFDR spent 1944 in a race against time: hoping to win reelection to a fourth term before voters learned of his compromises on Eastern Europe. In November, he successfully crossed the finish line, after a campaign in which his Republican opponent, Thomas Dewey, rarely challenged his postwar efforts. But the 1944 campaign was just one leg of a longer contest, in which Roosevelt struggled to entrench America in the UN before the tragedy in Eastern Europe made Americans recoil. And within that competition lay another, more intimate one: against the limitations of FDR's own body. Since Tehran, Roosevelt's health had deteriorated. A medical examination in early 1944 revealed hypertension, an enlarged heart, and signs of cardiac failure. Over the course of the year, he lost twenty pounds, and a picture published in July, which captured his withered, dazed visage, briefly threatened his reelection bid. In September, when a former aide saw Roosevelt for the first time in eight months, he was appalled by \"the almost ravaged appearance of his face.\"\n\nNevertheless, a month later, FDR flew to Yalta to meet Churchill and Stalin for what would be the last time. The flight was again arduous, and the accommodations were worse. The leaders stayed in Livadia Palace, a fifty-room czarist castle perched on a cliff above the Black Sea. But the building, which like most things in Crimea had been ravaged by Hitler's armies, contained barely any working bathrooms and teemed with typhoid and lice. Churchill muttered that it must have taken the Russians ten years to find a site so ghastly, and endured the stay by fortifying himself regularly from a private supply of whiskey. FDR\u2014his face pallid, his lips blue, his hands trembling\u2014wrapped himself in a large cape to stay warm.\n\nStill, Roosevelt pressed his agenda: to secure Moscow's pledge to enter the war against Japan, to decide how to govern postwar Germany, to nail down the structure of the new United Nations, and to again fudge the question of Eastern Europe so anger over Soviet influence there didn't turn Americans against the UN. On the first issue, he succeeded: Stalin agreed to enter the Asian theater three months after the European war was won. (In return the U.S.S.R. expected to replace Tokyo as the dominant power in Manchuria and northern China.) The UN's organizational structure wasn't much of a problem, either. For six months Stalin had been demanding that the sixteen Soviet republics each have its own General Assembly vote, presumably to counter the pro-Western Latin American and British Empire blocs. At Yalta, he agreed that between them the republic would have two or three votes, and FDR said he would demand the same number for the United States if the Senate raised a fuss. But the issue, Roosevelt insisted, \"is not really of any great importance\" the UN was not going to make decisions by majority vote.\n\nThe tough nut was Poland. By the time of the Yalta Conference, Soviet troops were already occupying Warsaw, Krakow, Lodz, and Posen, while their American and British counterparts remained one thousand miles away. FDR had already agreed to shift the Polish border west, and Churchill had traded away Romania, Bulgaria, and Hungary on a single piece of paper. But the composition of Poland's postwar government was harder. The issue had special meaning for Churchill, since the destruction of Polish independence had been Britain's trigger for war. Roosevelt was less personally invested, but he knew that of all the Eastern European nations, Poland's fate was the most politically explosive back home. Weeks earlier, Moscow had installed a puppet regime based in Lublin. The United States and Britain, by contrast, still recognized the exiled prewar Polish leadership in London. At Yalta, Roosevelt and Churchill pushed Stalin to bring the London Poles into a broad-based government. In return they got words. In a Declaration on Liberated Europe, all three leaders called for \"broadly based\" governments and \"free elections\" in the lands liberated from Hitler. It was a lovely vision: a \"Crimean Charter\" that echoed the Atlantic one penned by FDR and Churchill in 1941. But with the Red Army squatting on Polish soil, Roosevelt knew he had no way of making the rhetoric real. All he could do was plead. \"I want this election in Poland to be the first one beyond question. It should be like Caesar's wife. I did not know her but they said she was pure,\" he told Stalin. \"They said that about her,\" replied the Soviet leader, his smallpox-scarred face widening into a grin, thus revealing his yellowish-brown teeth, \"but in fact she had her sins.\"\n\nOn March 1, Roosevelt went before Congress to report on what he had achieved. One aide said he looked \"ghastly, sort of dead and dug up.\" But his words stirred the chamber. At Yalta, he declared, he and his fellow leaders had put an end to \"the spheres of influence, the balances of power...that have been tried for centuries\u2014and have always failed.\"\n\nWithin weeks, events in Poland were mocking his words. Not only were the Soviets exercising a sphere of influence, but they were enforcing it with characteristic brutality: systematically silencing, deporting, or murdering Poles who wanted a government free from Moscow's grip. Furious, Churchill sent FDR a stream of missives proposing that they make a joint protest, thus exposing Stalin's violation of the principles to which he had signed his name. But Roosevelt did not want a public row. The creation of the United Nations, its ratification by the Senate, and the exorcism of the ghost of Versailles were within reach. He knew the realities in Eastern Europe would be nastier than the proclamations in the Atlantic and Crimean charters implied, but he believed that he had laid down a moral marker, a standard by which the world could judge. Over time, he hoped, things would improve, as long as America did not pull away from the world in disgust. \"Perfectionism,\" he declared in his 1945 State of the Union address, perhaps with an eye toward the realities Americans would soon discover, \"no less than isolationism or imperialism or power politics may obstruct the paths to international peace. Let us not forget that the retreat to isolationism a quarter of a century ago was started not by a direct attack against international cooperation but against the alleged imperfections of the peace.... We gave up the hope of gradually achieving a better peace because we had not the courage to fulfill our responsibilities in an admittedly imperfect world.\" For FDR, this was the core lesson of World War I, the true answer to the hubris of reason. America must dirty its hands. To preserve its freedom, it must practice power politics, even though power politics would threaten American freedom in other ways. And it must do so while recognizing that despite its efforts, the world would remain an ugly place, just not quite as ugly as if America tried to remain pure.\n\nThose close to the president knew the end was near. \"He is slipping away from us,\" said his scheduler, William Hassett, \"and no earthly power can keep him here.\" On April 12, as he sat for a portrait, FDR suddenly complained of a \"terrific headache,\" then slumped over unconscious in his chair. At 3:35 in the afternoon he was pronounced dead of a cerebral hemorrhage. By his side was Lucy Mercer, the mistress he had told Eleanor he would jettison more than twenty-six years earlier.\n\nTwelve days later, a U.S. platoon made contact with Russian troops in eastern Germany, near the city of Torgau, on the Elbe. The Americans followed the Russians back to the farmhouse where they were encamped and drank toasts to the Allied cause, its triumph over the Nazis now virtually assured. A few hours later, half a world away, 282 delegates entered the San Francisco Opera House for the opening ceremony of the United Nations. By late June they had approved a structure that closely resembled the one envisioned at Yalta. It would comprise a General Assembly, where all nations talked, and a Security Council, where the great powers\u2014vetoes in hand\u2014acted. The UN's Universal Declaration of Human Rights spoke about \"the equal and inalienable rights of all members of the human family,\" but once again reality could not match the soaring words. Despite some vague language about trusteeship, Britain and France kept their colonial possessions. On Poland, delegates deferred seating Russia's puppet government. But the ultimate outcome was already clear: With Soviet troops in Eastern Europe, and a Soviet veto on the Security Council, Poland's awful fate was sealed.\n\nA month later, the Senate took up American membership in the UN. Especially among Democrats, Wilson's ghost was summoned again and again. It is rare for a man \"to be given a second chance in his lifetime to correct a great mistake,\" declared Connecticut's Brien McMahon. \"It is even more seldom that that chance comes to a nation.\" Senator Charles Andrews informed his colleagues that he had recently visited Wilson's grave for inspiration. Florida's other senator, Claude Pepper, said that specters hovered over the proceedings, watching through the \"veil which their immortal eyes can always penetrate.... One of them is Woodrow Wilson, and the other is Franklin D. Roosevelt.\"\n\nIn the end, the vote wasn't close. Pearl Harbor had killed American isolationism, and the realities in Eastern Europe\u2014which FDR had helped mask\u2014hadn't yet revived it. On Saturday, July 28, with only two die-hard isolationists voting no, the Senate overwhelmingly ratified the Charter of the United Nations. Upon hearing the news, one aging League supporter said he \"almost shouted and cried with joy.\" Another wrote, \"The faith of Woodrow Wilson has been vindicated. The record of the United States of 1920 has been expunged.\" But in the Senate chamber, the mood was oddly muted. Spectators packed the galleries, but the vote sparked no cheering or applause. In truth, Wilson's spirit had not been entirely revived; his faith had not been fully vindicated. The world had been saved, but not exactly transformed. As average Americans would gradually come to realize, Eastern Europe was slipping away. And there were other horrors. As U.S. troops occupied Germany, they discovered a vast machinery of death, whole cities devoted to race murder. In August, Americans\u2014who a few years earlier had been shocked by the Spanish bombing at Guernica, which killed roughly two hundred people\u2014watched their own nation drop an atomic bomb on Hiroshima, killing eighty thousand in a single night. Meanwhile, from Arkansas to Arizona, one hundred thousand American citizens whose only crime was their Japanese ancestry were slowly exiting the camps where they had been imprisoned for most of the war. In such a world, who could declare, even in victory, that humanity was marching toward a bright dawn? A few idealists attacked the UN as a betrayal of the Wilsonian dream, \"a mere camouflage,\" in the words of one church group \"for the...exercise of arbitrary power by the Big Three for the domination of other nations.\" But many Americans seemed to intuitively grasp the lesson that FDR believed the war progressives and their pacifist and isolationist successors had never learned: that the world was a tragic place, which America could neither escape nor fully redeem. What the Senate had ratified, wrote Time magazine, was \"a charter written for a world of power, tempered by a little reason.\" Or as FDR exclaimed about the Yalta deal, in words that could have summed up his entire foreign policy, \"I didn't say the result was good. I said it was the best I could do.\"\n\nFDR wasn't much of an intellectual; he relaxed by building model ships, not reading highbrow journals. But the sentiment he expressed\u2014that people must do their best in the world while not expecting too much of it\u2014captured the intellectual spirit of the age. Even before World War II, the Great War, the Depression, and Stalin's crimes had darkened the mood of many American writers and thinkers. And now, the Holocaust, Hiroshima, and the Iron Curtain joined the register of human horror. Looking back, the intellectuals of the 1940s tended to lump together the political faiths that had come before\u2014war progressivism in the 1910s, pacifism in the '20s, Marxism in the '30s\u2014and call them the age of optimism and reason. Their own challenge, as they saw it, was to practice politics in a world that had reduced all utopian visions to rubble. \"Frustration is increasingly the hallmark of this century\u2014the frustration of triumphant science and rampant technology, the frustration of the most generous hopes and of the most splendid dreams,\" wrote a precocious Niebuhr disciple named Arthur Schlesinger, Jr., in his 1949 book, The Vital Center. \"Nineteen hundred looked forward to the irresistible expansion of freedom, democracy and abundance; 1950 will look back to totalitarianism, to concentration camps, to mass starvation, to atomic war.\"\n\nFor many intellectuals, coming to terms with these frustrations meant acknowledging that their roots lay not in any economic or political system, but in human nature itself. \"Our modern civilization...was ushered in on a wave of boundless social optimism,\" wrote Niebuhr in a series of 1944 lectures that became the book Children of Light, Children of Darkness. \"Modern secularism is divided into many schools. But all the various schools agreed in rejecting the Christian doctrine of original sin.\" This was a little overstated, as Niebuhr's generalizations often were. But it was a feature of intellectual life in the 1940s to see the past in prelapsarian terms.\n\nNiebuhr's vision of man as a fallen creature was rooted in his Christian theology. But his views penetrated far beyond the community of faith. Lionel Trilling, a secular Jew and America's most influential literary critic, was drawn to writers like Nathaniel Hawthorne and T. S. Eliot, whose work evoked a sense of tragedy about the human condition. In The Vital Center, Schlesinger attacked the progressives for their \"soft and shallow concept of human nature\" and their \"unwarranted optimism about man.\" There was a self-consciously jaded quality to many wartime intellectuals. They were like Rick, the hard-boiled nightclub owner in the 1942 film Casablanca, who seems utterly cynical about the world yet risks his life to resist the Nazis nonetheless. Niebuhr and Schlesinger loved calling their adversaries\u2014be they communists or old-style progressives\u2014idealistic and innocent. For their part, they prided themselves on seeing the world in all its awfulness without succumbing to despair.\n\nIn addition to assaulting the progressives for their overly sunny views of human nature, intellectuals in the 1940s attacked the claim, made by progressives and Marxists alike, that politics could be scientific. The wartime intellectuals were not hostile to science and technology per se. How could they be, since without radar and penicillin and the factories that churned out ships and planes faster than the Nazis, America might not have won the war? But they denied that the scientific method\u2014the gathering and testing of objective facts\u2014could underpin government, since people, unlike machines, did not always act in rational ways, and since politics, unlike engineering, was fundamentally a moral pursuit. Once again, John Dewey served as whipping boy. The University of Chicago philosopher Mortimer Adler, yet another student turned assailant, accused him of creating an educational philosophy based on the principle that \"science...would eventually supply all the guidance necessary for human conduct.\" That made a Deweyian education ultimately amoral, Adler alleged, since science \"cannot make a single judgment about good and bad, right and wrong.\"\n\nIf Adler accused Dewey of thinking America should brew its politics in a beaker, no one applied the critique more systematically to foreign policy than Hans Morgenthau, who took deadly aim at Colonel House's old idea of a scientific peace. A Bavarian-born Jew, Morgenthau spent his youth at the knee of scholars steeped in the tradition of Bismarckian realpolitik. He admired that conservative, Germanic tradition, with its reverence for the national interest and the balance of power, because he believed it respected limits, limits of both human reason and national power. He mourned its replacement by Nazism, and when Nazism chased him to America's shores, he used the realpolitik tradition of his youth to challenge the foreign policy culture of his adopted land, in hopes of better preparing it for its confrontations with the old world.\n\nFor too long, Morgenthau argued in his 1946 book, Scientific Man vs. Power Politics, Americans had succumbed to the \"misconception of international affairs as something essentially rational, where politics plays the role of a disease to be cured by means of reason.\" This, in Morgenthau's view, was a profound mistake; human behavior could not be rationalized, because human beings possessed an irrational and incurable lust for power. That was particularly true in international affairs. In the domestic arena, Morgenthau observed, liberal democracy camouflaged the harsh realities that underlay all politics, redirecting the struggle for power from the bullet to the ballot. As a result, people who took democracy for granted\u2014and no people did so more than Americans\u2014tended to believe that international politics could be tamed in the same way: by creating global parliaments that put international disputes to a vote or global courts that ruled aggression illegal. But the domestic camouflage of power, Morgenthau insisted, depends upon the fact that one sovereign possesses a monopoly of it. It is precisely because the state's power is so uncontested that it can sheath its sword, thus creating the illusion that its authority doesn't ultimately rest on force. In the international arena, by contrast, where there is no sovereign, the contest is naked. In a neighborhood with no policeman, no one can pretend that safety does not rely on the barrel of a gun. And thus, Morgenthau posited, it was in international affairs that the illusion that politics could be made rational and scientific\u2014made into something other than the self-interested search for power\u2014was most dangerous. Both the war progressives and their pacifist successors had relied heavily on analogies between domestic and international affairs. Those analogies, Morgenthau insisted, were dead wrong.\n\nMorgenthau urged Americans to abandon the progressive illusion that through scientific knowledge and charismatic leadership they could educate other nations to desire something besides power. And he urged them to abandon that illusion about themselves as well, recognizing what other nations already knew: that beneath America's moralistic rhetoric lay its own self-interest. Instead of trying to create a rational system to govern the world, he argued\u2014be it Wilson's collective security or Dewey's pacifism\u2014Americans should see themselves as guardians not of universal morality, but of their own national interest. For Morgenthau, a foreign policy based on national interest was not amoral; to the contrary, it was the most ethical response to an anarchic world. After all, in a town with no policeman, what better keeps the peace: the household that in the name of nonviolence disarms itself, and thus tempts others to attack; the household that equates its own interests with everyone else's, and thus seeks to deny others the right to self-defense; or the household that carefully delineates its domain and maintains sufficient strength to defend it? In foreign policy, Morgenthau argued, where precepts drawn from domestic affairs hold little sway, a concern for the national interest was most likely to establish a balance of power, an equilibrium that confronted potential aggressors with countervailing force. And a balance of power could effectively mitigate international conflict because rather than requiring nations to transcend their self-interest, it relied upon it. It appealed to something more solid than reason: fear.\n\nIn the 1940s, Morgenthau was not alone. He was part of a remarkable crop of foreign policy intellectuals determined to slay the hubris of reason once and for all. One of those intellectuals, ironically enough, was the former boy wonder of war progressivism, Walter Lippmann. Having already recast World War I as a war of self-defense, Lippmann, now America's most famous columnist, argued that after World War II America should not again try to create an institution like the League of Nations based upon \"a moral code which was beyond human capacity to observe and which the United States had been unable to honor.\" Instead, he argued, sounding a lot like FDR, that the United States should focus on maintaining cooperation among the big powers that had vanquished Germany and Japan. And if that required granting Moscow a sphere of influence in Eastern Europe so it would feel less threatened, so be it. After all, didn't America have its own sphere in Latin America? Here Lippmann picked up a theme that both Niebuhr and Morgenthau stressed: that America must acknowledge the self-interest in its own actions rather than demanding that other nations comply with moral norms it did not uphold itself.\n\nBy the end of World War II, Lippmann was no longer a wunderkind. Nearing sixty, he had enjoyed dazzling professional success. But he had also experienced not only national tragedy, but personal tragedy as well, having seen his marriage collapse in a blaze of public humiliation after an affair with the wife of one of his best friends. He was no longer the optimist and rationalist he had been when he met Colonel House for that autumn 1917 stroll. His views, he wrote in 1943, had evolved \"slowly over thirty years, and as a result of many false starts, mistaken judgments and serious disappointments. They represent what I now think I have learned, not at all what I always knew.\" It was a measure of the age that when it came to foreign policy, even a man as haughty as Walter Lippmann was in a chastened mood.\n\nIf Niebuhr was realism's theologian, Morgenthau its political scientist, and Lippmann its journalist, its greatest postwar practitioner was an introverted hypochondriac from Milwaukee named George Frost Kennan.\n\nKennan was too young to be a disillusioned progressive and too conservative to have ever been a communist, but he didn't need grand public embitterments to decide that the world was a grim place. Private embitterments more than did the job. He was born in a house built backward, so it received little natural light. His mother died when he was an infant, leaving him to be raised by a repressed, distant father and a stepmother who favored her own boy over him. At twelve they dispatched him to military school, where he was tormented by bigger boys and his class yearbook read, \"Pet peeve\u2014the universe.\" Fascinated with the American elite ever since reading F. Scott Fitzgerald, he somehow made it to Princeton, where he was even more miserable than he had been in high school. A poor, unsophisticated son of the Midwest with weak social skills and something of a persecution complex, he entered the Ivy League expecting to be looked down upon, and was not disappointed. In his first week, he asked a fellow student for the time and had smoke blown in his face. On a campus dominated by eating clubs, he had neither the money nor the cachet for sustained membership, and ate most of his meals alone. After graduating, he joined the foreign service, a profession he loved but tried to resign from eight times in response to imagined slights. In his time off he often checked himself into sanitariums to cure an array of real and perceived maladies. He enjoyed the company of nurses, he later admitted, because they conjured the mother he had never had.\n\nKennan once wrote that he felt like \"a guest of one's own time and not a member of its household.\" This alienation from the modern world was sometimes quaint: Kennan loathed cars and proposed that candidates for the foreign service be required to till the soil. And it was sometimes appalling: Kennan's love affair with the eighteenth century often left him unsympathetic to blacks, Jews, women, and immigrants who demanded liberation from ancient hierarchies. Kennan's affection for hierarchy, in fact, sometimes left him dubious about democracy itself and led him to muse about the benefits of a government run by unelected wise men like, well, himself.\n\nAs an observer of his own country, Kennan was often unfair and unreliable. But it wasn't his analysis of the United States that in the 1940s made him famous: It was his analysis of the U.S.S.R. Upon joining the foreign service, he was sent to study Russian at the Oriental Seminary in Berlin, a citadel of realism founded by Otto von Bismarck, and then to the Baltic capitals of Riga and Tallinn, where he was introduced to Russia by its czarist \u00e9migr\u00e9s, who bred in him an abiding love for its culture and a fierce hostility to its communist rulers. As the years passed, and his foreign policy views jelled, Kennan's realism and his anticommunism came together in a critique of what he called \"universalism,\" a pathology he believed that American progressives like Wilson and Soviet communists like Lenin shared. Both groups, Kennan argued, wanted to remake the world according to supposedly objective principles and promised that once they had, \"the ugly realities [of world politics]\u2014the power of aspirations, the national prejudices, the irrational hatreds and jealousies\u2014would be forced to recede\" and nations would live in harmony. The nature of those principles differed, of course. Wilson wanted a world of reason and law, a world that looked like America in the progressive age; Lenin wanted a global dictatorship of the proletariat, a world that looked like the U.S.S.R. But the fundamental problem with both visions, Kennan insisted, was that they imposed an abstract, universal vision on a messy, diverse world. For Kennan, like Morgenthau, foreign policy was not a science. Practitioners should not build logical models; they should tend to organic matter. They should be not mechanics, but gardeners, cultivating each plant or flower based on its own circumstances and needs. In foreign policy, as in marriage, Kennan argued, general principles of conduct are far less useful than an intimate understanding of the participants. Had he known that Wilson proposed drafting a constitution to govern his union, he would have laughed and laughed.\n\nWhen it came to Russia policy, Kennan saw two imperatives: to prevent Soviet universalism from threatening America and to prevent America from threatening itself by lapsing back into a Wilsonian universalism of its own. Kennan didn't begrudge FDR's willingness to grant the Soviets a sphere of influence, since preventing it was beyond America's power. But he considered Roosevelt na\u00efve for thinking that granting the U.S.S.R. such a sphere would make it an ally in policing the postwar world. Russian history, Kennan argued, inculcated a suspicion of outsiders, and Marxist-Leninism, with its ideological hostility to capitalism, confirmed the tendency. Given the Kremlin's revolutionary aspirations, he believed, FDR's efforts to enlist it as a partner in upholding the postwar status quo were doomed. Even worse, America's efforts at cooperation would likely be taken in Moscow as signs of weakness, thus increasing rather than reducing the chances of war.\n\nKennan did not consider Stalin another Hitler; he thought the Soviet leader was basically cautious. But if America, in its zeal to maintain the wartime alliance, left the door to expansion open, Stalin would walk through. Russia, wrote Kennan, is like a \"toy automobile wound up and headed in a given direction, stopping only when it meets with some unanswerable force.\" America needed to carefully choose which chunks of the globe were essential to its security, and in each one establish a balance of power sufficient to deter Soviet influence. Such balances would not require governments that resembled America's: that would be Wilsonian universalism, Kennan's other b\u00eate noire. They could be democratic or dictatorial, capitalist, socialist, or even communist. All America should ask of its allies was the will to resist Soviet domination. With this strategy, Kennan modified FDR's response to the hubris of reason. The final answer would not be called the Four Policemen; it would be called containment.\n\nBy 1946, Washington was finally ready to hear what Kennan had to say. During the war, given FDR's obsession with maintaining the anti-German alliance, Kennan and the other anti-Soviet hard-liners in the State Department's Eastern European division had been mostly ignored. But in his final days, even Roosevelt, by some accounts, was losing patience with Moscow. While he sympathized with Stalin's desire for friendly governments on Russia's border, he had grown increasingly annoyed by the Soviet leader's inability to orchestrate them with a light and subtle hand so as not to embarrass him at home. Roosevelt may have genuinely believed that the nations of Eastern Europe could elect governments that were both democratic and deferential to Moscow\u2014a wildly unrealistic hope in a country like Poland, which seethed with hatred toward its large and brutal neighbor. Or he may simply have wanted to delay any conflict with Stalin until Germany and Japan were defeated and America was ensconced in the UN. Either way, when Harry Truman succeeded him, relations between Washington and Moscow began to chill. Truman\u2014a failed businessman, a machine politician, the vice president largely because special interests had blocked more qualified contenders, and now the first president in a half century without a college degree\u2014was staggeringly unprepared for the challenges he would face. FDR hadn't even bothered to tell him that America was building an atomic bomb. On April 12, 1945, when Eleanor Roosevelt broke the news that the president was dead, Truman was silent for a long time; then asked, \"Is there anything I can do for you?\" To which she replied, \"Is there anything we can do for you? For you are the one in trouble now.\"\n\nOne consequence of Truman's ignorance was that he viewed U.S.-Soviet relations from the perspective of an outsider. Roosevelt had felt grateful to the U.S.S.R., which had lost ninety times as many lives in the war as had the United States. He knew that by delaying America's invasion of France until 1944 he had purchased victory cheap, and that he had been able to do so because for almost three years Soviet ground troops had faced the Nazi meat grinder largely alone. At Tehran and Yalta, FDR had also given Stalin the winks and nods that undercut Allied declarations about democracy and self-determination. Truman hadn't been in those back rooms, nor even told what transpired there, and so he saw the Allied victory the way most of his countrymen did: as the story of America saving Europe, and especially Russia, which Truman assumed had only survived Hitler's onslaught because of American aid. It was Stalin, in his view, who should be grateful, and should express that gratitude by upholding his public pledges to safeguard democracy in the nations he controlled. Over the course of 1945, as it became clear that Stalin was doing no such thing, Truman began to get tough.\n\nConvinced that FDR's diplomacy had involved too much carrot and too little stick, Truman spent much of 1945 trying to muscle Moscow into cooperating on America's terms. He brought Soviet foreign minister Molotov into his office and gave him \"the straight one-two to the jaw.\" He halted wartime aid and made it clear there would be none in the future unless the Kremlin cleaned up its act. He threatened to publicly denounce Soviet behavior in Eastern Europe and he brandished America's new atomic bomb, thinking it would scare Stalin into complying with U.S. demands. But all these efforts failed: The bomb was too crude an instrument to be wielded for geopolitical effect, and while the Soviets didn't relish the cutoff of U.S. aid, having pliant governments on their border mattered more. As for Molotov, it wasn't easy to intimidate a man who slept with a hunk of brown bread and a pistol under his pillow. So Truman and his advisers began a more fundamental reassessment. Perhaps the problem wasn't that FDR's efforts at cooperation had involved too many carrots. Perhaps cooperation wasn't possible at all.\n\nThat's where Kennan came in. In February 1946, in his capacity as deputy head of the U.S. mission in Moscow, he was asked by the State and Treasury departments to explain Stalin's increasingly anti-American rhetoric and his refusal to join the newly created International Monetary Fund and World Bank. Kennan read the request and groaned. He had explained the Kremlin's behavior to his superiors in Washington many times before, and received no response; it was like \"talking to a stone.\" What's more, the \"sunless, vitaminless\" environment of Stalinist Moscow was affecting his health. Suffering from a fever, cold, and ulcers, he retired to his sickbed and dictated an eight-thousand-word telegram, the longest ever sent from the embassy in Moscow. He expected that this time too it would disappear into the bowels of the Washington bureaucracy with barely a trace.\n\nIn his missive, Kennan took aim at the key assumption underlying both Roosevelt's and Truman's policies: that the United States\u2014either through carrots or sticks\u2014could maintain its alliance with the Soviet Union now that the war was won. Moscow, argued Kennan, was not eschewing cooperation because American policy was too tough or too soft; American policy was irrelevant. Stalin was not Georges Clemenceau, a prickly character eager for a postwar partnership with the United States. Stalin wanted conflict with the capitalist world because he ruled by repression, and to justify that repression he needed an external threat. With Hitler vanquished, and America the most powerful capitalist country in the world, we were now it.\n\nKennan's answer was containment. If Roosevelt had transgressed the Wilsonian faith by accepting spheres of influence, Kennan now committed an even greater apostasy: He blessed the balance of power. Even FDR, Kennan argued, had been dewy-eyed. The Soviets would not help America police the world. Rather, they would pocket their Eastern European spoils and seek new ones, foraging in places such as Western Europe, Japan, and the Middle East, which America must not allow a hostile power to control. To prevent this, Kennan argued, the United States would have to seek regional balances of power, alliances that met the U.S.S.R. and its clients with countervailing force.\n\nThis time, Kennan's telegram did not reach Washington and disappear. To the contrary, it detonated with near-atomic force. Kennan's boss, Ambassador to Russia Averell Harriman, gave it to Secretary of the Navy James Forrestal, who sent it to Truman, the entire cabinet, top military officers, key members of Congress, and influential columnists and publishers. The State Department distributed it to every U.S. Embassy in the world. Kennan had provided a rationale for the policy Washington was already groping toward. The central reality of the coming age, it was now clear, would be not great-power cooperation, as FDR had hoped, but great-power conflict. As Kennan's fellow Soviet expert Charles Bohlen noted, there would be not one postwar world, but two.\n\nAs containment became the prism through which Washington saw the world, other aspects of the postwar settlement fell into place. Originally, FDR had wanted Germany crushed. The more emasculated it was, he assumed, the less it could challenge the Four Policemen who patrolled the postwar world. In 1944, he endorsed Secretary of the Treasury Henry Morgenthau's plan to exile Germany to the nineteenth century, dismantling its industry and carving it into small agricultural states. When FDR talked about how to treat Hitler's vanquished nation, his favored image was castration, and it wasn't always clear that he was speaking entirely metaphorically.\n\nBy 1945, Roosevelt had ditched Morgenthau's plan, in part because of opposition from Secretary of State Hull, who feared inflaming the Germans. But what sealed the decision was containment. Preventing Soviet expansion, Kennan argued, required building up the capacity of key regions to resist. And unless Germany's economy revived (at least in that part of Germany under Western control), neither would Western Europe's as a whole, leaving it ripe for Soviet subversion. Ironically, then, it was the recognition that great-power cooperation was impossible\u2014that a balance of power was the best America could achieve\u2014that helped convince the United States to treat the defeated Germans more generously. The same principle underlay the Marshall Plan, America's initiative to alleviate the economic disorder and despair stalking postwar Western Europe. That noble effort would never have passed Congress had Americans not feared that Moscow would turn Europe's suffering to its advantage. Such ironies fit the intellectual ethos of the age: By lowering America's moral sights and substituting a more realistic vision (containment) for a more optimistic one (U.S.-Soviet cooperation), America was actually doing more good.\n\nThe contrast was clear. After World War I, the absence of foreign threats had contributed to America's withdrawal. After World War II, by contrast, the specter of Soviet power drew America further and further into European affairs. Soon the United States wasn't only paying to rebuild Western Europe economically; it was promising to defend it militarily. If in 1940, when France fell, Clemenceau's nightmare had come true, in 1949 Washington and Paris realized his dream, a binding security guarantee, the NATO pact.\n\nWhat held the postwar settlement together, then, was restraint on American universalism and fear of the universalism of the Soviet Union. FDR granted Moscow its sphere of influence; Truman contained it. Together they acknowledged the limits of America's capacity to remake the world yet convinced Americans to sacrifice so that Hitler and Stalin could not remake it, either. Roosevelt and Truman set their ideological ambitions lower than Wilson and paid a higher price to achieve them; they deployed greater power in pursuit of narrower goals. By eschewing a scientific peace, they forged one better able to endure in an imperfect world.\n\nAnd yet, while America's leaders often acted like realists, they didn't sound like realists. They outfitted their Lodgean peace with Wilsonian clothes. From Newfoundland to Yalta to San Francisco, FDR and Truman often spoke in universalistic, missionary terms. Partly it was politics. Kennan, Morgenthau, and Lippmann wanted to keep the citizenry out of foreign policy, since they believed that public participation would hamper the subtle and ruthless adjustments that an effective balance of power required. FDR and Truman, however, could not afford such an elitist view. They had to speak to Americans in their own language; they had to get legislation through Congress. They had to worry about the balance of power at home as well.\n\nBut there was more to it than that. The words were window dressing, but they were not only window dressing. They reflected a discomfort with brute, unreasoning power even as America accommodated itself to it. If FDR and Truman lacked Wilson's faith in a world governed by reason, neither were they fully reconciled to its opposite. Had they not at least partially believed their own words, they could not have spoken them so convincingly. In this sense, they were less like Kennan, Morgenthau, and Lippmann than like Niebuhr, who refused the realist label because while he acknowledged that self-interest powered human affairs, he did not want to sanctify it. He believed that while both nations and individuals were inherently sinful, they could still be redeemed through the mystery of God's grace. And it was this faith that ordinary individuals could somehow be redeemed that kept Niebuhr from the dark and sometimes ugly skepticism of democracy to which Kennan fell prey. \"Realities are always defeating ideals,\" he wrote, \"but ideals have a way of taking vengeance upon the facts which momentarily imprison them.\"\n\nThis is where Niebuhr and Roosevelt met. Perhaps it is no surprise that a man who for decades insisted, in the face of all reality, that he would cast off his own momentary imprisonment and walk again would insist on seemingly fanciful declarations about what was possible in the world. With the UN charter, as David Fromkin puts it, FDR created a sleeping beauty. Like the Declaration of Independence, its words rebuked their authors and offered the prospect, however distant, that justice and reason would one day rise from their slumber.\n\nIt was a treacherous balance. If the ideals grew too intoxicating they could blind Americans not only to the limits of what they could expect from the world, but of what they could expect from themselves. Missionaries unable to recognize their own sinfulness, Niebuhr noted again and again, were worse than merely na\u00efve; they were often monstrous. Yet if the ideals disappeared altogether, as the realists seemed to desire, the result might be utter cynicism, a world grown numb to evil.\n\nThere was no abstract formula for weighing these competing imperatives. Abstract formulas, in fact, had been part of the problem. Much of the time, FDR and Truman were simply feeling their way. But overall, their generation\u2014which was more aware than the war progressives and pacifists of the tragic nature of world affairs and more aware than the isolationists that America must struggle to improve it nonetheless\u2014found a balance that served their nation astonishingly well. For an extraordinary period during and immediately after World War II, they not only built a stable balance of power but swathed it in the garb of reason. They saved Asia from Japanese imperialism, and Western Europe from not one species of totalitarianism but two. They took a nation that wanted no commitments in the world, especially if they involved carrying a gun, and made it the leader of the mightiest wartime and peacetime alliances in human history. They did all this while strengthening America's economy and preserving its democracy, in the face of Beard's and Dewey's predictions that another world war would destroy both.\n\nAnd then, as the years passed, and Americans increasingly took the fruits of their labor for granted, a younger generation of thinkers and leaders began to suspect that they had set their sights too low.\n\n## PART II\n\n## THE HUBRIS OF TOUGHNESS\n\n## CHAPTER FIVE\n\n## THE MURDER OF SHEEP\n\nFor all his misanthropy, George Kennan had to admit that the world was treating him pretty well. Before his February 1946 \"Long Telegram\" from Moscow, he had been an obscure, miserable functionary; perhaps two hundred people in the entire government had known his name. Now he was the State Department's resident genius. In April he was recalled to Washington and made deputy commandant of the National War College, where he taught officers and diplomats about the U.S.S.R. A year later he became the State Department's first-ever director of Policy Planning, which put him in charge of foreign policy big-think and gave him direct access to Secretary of State George Marshall. Even his health began to improve.\n\nThe key figure in Kennan's ascent was Navy secretary James Forrestal. Like Kennan himself, Forrestal was an outsider in the patrician, East Coast, WASP-dominated foreign policy elite. Born to a working-class Catholic family, he had twice broken his nose boxing, which added an air of menace to his rugged good looks. Some people said he resembled the actor James Cagney, who was famous for playing gangsters. Like Kennan, Forrestal had made it\u2014against the odds\u2014to Princeton. But unlike Kennan, who was miserable there yet stayed true to himself, Forrestal used the Ivy League to don a new identity as an upper-class dandy. He shed his Catholicism, cut ties to his parents, made millions on Wall Street, and romped with the Jazz Age Long Island set like a character from a Fitzgerald novel. Yet he never seemed entirely at ease with his new self. His many business associates, his many drinking buddies, his many lovers, even his ill-fated wife, all admitted that beneath the high-achieving, high-living exterior lurked a man they didn't really know at all.\n\nPolitically, Forrestal was a realist of sorts. He certainly saw the world as a tough and irrational place. But for Kennan, Niebuhr, Morgenthau, and Lippmann, the recognition that the world was tough and irrational represented a reaction against Wilson's messianic fervor, a kind of anti-ideology. Forrestal, by contrast, invested toughness with an ideological fervor of its own. He saw the Soviets as Nazis, ruthless fanatics determined to snuff life from the free world. And he believed that only brute, unyielding force could keep them at bay. In his use of Nazism as a template for understanding international affairs\u2014his belief that evil was a permanent, pervasive force in the world and that any concession to it would unleash the same horror as Chamberlain's had at Munich\u2014Forrestal was, in 1946, a man ahead of his time, a harbinger of a new hubris gradually taking root in a new age.\n\nForrestal was morbidly fascinated by the writings of Marx and Lenin, which he saw as the postwar equivalent of Mein Kampf: a Rosetta stone for understanding America's new totalitarian foe. In late 1945, he asked an economist on his staff named Edward Willett to do a study of \"dialectical materialism,\" a term Willett had never heard before. Willett, a man whose intellectual reach often exceeded his grasp, read two books on Marxism and then produced a paper titled \"Random Thoughts on Dialectical Materialism and Russian Objectives,\" which lived up to its name. Forrestal distributed the paper to some Russia experts for review, but the general response was that Willett didn't know what he was talking about, and that in any case, Soviet foreign policy was motivated more by national interest than by Marxist theory. Forrestal was crestfallen. He had commissioned the paper precisely because he believed the Kremlin was motivated by Marxist theory and thus was hell-bent on global revolution. \"I realize it is easy to ridicule the need for such a study,\" he wrote to one associate, \"but I think in the middle of that laughter we always should remember that we also laughed at Hitler.\"\n\nJust then, like an answered prayer, Kennan's Long Telegram landed on his desk. It was well reasoned, well informed, articulate\u2014everything Willett's paper was not. Kennan, like most other Russia experts, believed that Stalin's foreign policy was motivated more by Russia's historic insecurity and animosity toward the West than by the writings of Lenin and Marx. But in his effort to puncture the still-prevalent view that America could charm Moscow into postwar cooperation, Kennan had described the Kremlin in harsh, pungent terms. And he did acknowledge that communism exacerbated Russian suspicions of the outside world. This was good enough for Forrestal, who now had a new horse to ride.\n\nAfter summoning Kennan back to Washington to teach at the National War College, which he had helped create, Forrestal asked him to revise Willett's article, hoping to invest its argument with greater intellectual heft. Kennan demurred, admitting that he didn't share Willett's belief that Stalin, like Hitler, wanted a world war. But Forrestal countered with a new request: Kennan should write a paper of his own. This put Kennan in a bind. He wanted to keep Forrestal, his most powerful patron, happy, yet he didn't really share his views. He tried to square the circle by drafting an essay that claimed Moscow's actions were motivated by both nationalist and ideological impulses, without suggesting which was more important. That was good enough for Forrestal, who distributed Kennan's essay with a cover note stating that \"nothing about Russia can be understood without also understanding the implacable and unchanging direction of Lenin's religion-philosophy.\" For Kennan, who later admitted that he had written the essay to serve \"what I felt to be Mr. Forrestal's needs at the time,\" it was a small, strategic breach of intellectual honesty. And it worked. Three months later, with a push from Forrestal, he got his dream job: head of the State Department's Policy Planning staff.\n\nBut Kennan's breach came back to haunt him. In January 1947, he gave a talk on Russia at the Council on Foreign Relations, which the editor of the Council's journal, Foreign Affairs, asked him to turn into an article. Kennan didn't have time to write something new, so he submitted the essay he had penned for Forrestal. The State Department gave him permission to publish it as long as it did not bear his name. Under the article's title, the byline read simply \"By X.\"\n\nOn the surface, the X article brought Kennan another burst of good fortune. It was excerpted in Reader's Digest and Life, and quoted in Newsweek. After Forrestal tipped off New York Times columnist Arthur Krock about the author's identity, Kennan was named one of U.S. News & World Report's people of the week. Suddenly he was deluged with speaking requests. At Radcliffe, where his daughter attended college, friends began calling her \"Miss X.\"\n\nBut beneath all the adulation lay a problem, one that would shadow Kennan until he was an old man: the problem of what containment actually was. For Kennan, it was something modest, something significantly less than a global doctrine. First of all, it was a strategy against a country, the Soviet Union, not an ideology, communism. If communist movements were pawns of Moscow, Kennan conceded, then their ascent to power in strategically important countries should be vigorously opposed. (He even supported covert action to undermine Soviet control in Eastern Europe.) But unlike Forrestal, he suspected that some communist regimes might refuse orders from the Kremlin and perhaps even grow openly hostile, in which case they themselves might become vehicles for containing Soviet power. For Kennan, it was critical that while the Soviet Union pursued \"universalism\"\u2014a world remade in its ideological image\u2014the United States should not. That, in his view, had been Wilson's mistake. The United States might be partial to democracy, but it should condone ideological diversity. It should tolerate governments of any type as long as they were not agents of a threatening power.\n\nSecondly, for Kennan, containment was primarily a political strategy, not a military one. He had seen firsthand the way World War II shattered Soviet society, and he believed the Kremlin had little appetite or capacity for another world war. Unlike Forrestal and Willett, therefore, he put little stock in the Hitler analogy. The real danger, in his mind, was not that the Red Army would sweep across Western Europe. It was that the people of Western Europe, made desperate by the continent's postwar misery and chaos, would elect Kremlin-controlled communist parties and thus unlock the door to Soviet domination from the inside. That's why for Kennan, containment's crown jewel was the massive program of American economic assistance dubbed the Marshall Plan. NATO, by contrast, which committed the United States to Western Europe's military defense, struck him as unnecessary and probably counterproductive.\n\nFinally, Kennan did not believe that containment applied everywhere. Even the spread of Soviet influence did not bother him if the spread was to countries that lacked the industrial capacity, natural resources, or strategic location to tip the geopolitical balance in Moscow's favor. If the Kremlin wanted to waste its money and blood in irrelevant backwaters, he believed, that was no reason for America to follow suit. Kennan was not always precise or consistent in defining which chunks of the globe he considered strategically important and which he did not, but he was adamant that containment have geographic limits. The central point, in his mind, was that America must have a prior conception of its interests, which it defended against Soviet threats. To oppose Soviet power everywhere would flip that on its head: It would mean deciding that anywhere the U.S.S.R. probed suddenly became in the interest of the United States to defend, in which case Moscow, not America itself, would determine which areas of the globe mattered to America.\n\nBy the summer of 1947, Kennan had articulated these principles many times. What he had not done\u2014on purpose\u2014was clearly articulate them in his paper for Forrestal, the paper that, to his surprise, made containment a household world.\n\nIn his summer home, surrounded by pine trees on an island off the coast of Maine, Walter Lippmann read the X article and decided that containment was a dangerous idea and Kennan was a dangerous man. He proceeded to write fourteen straight columns attacking the concept, which he packaged into a short book. He took its title from a French phrase describing the standoff between Berlin and Paris before World War II: la guerre froide, the cold war.\n\nFor Lippmann, Mr. X's containment doctrine had two big and interrelated problems. First, it assumed that the Soviets were motivated by ideology, not nationalism. As a result, it interpreted the Kremlin's insistence on an Eastern European sphere of influence not as the fulfillment of an historic Russian desire for a buffer against attack from the West (a desire that Hitler's invasion had graphically underscored), but as the first step in a bid to take over the world. Second, because Mr. X had defined Soviet ambitions as global, he was insisting that America's response be global, too. Containing Moscow everywhere, Lippmann warned, would require the United States to commit itself to a motley array of \"satellites, puppets, clients, agents about whom we can know very little.\" It would require \"a drawing account of blank checks both in money and military power.\"\n\nThe irony was exquisite. To please Forrestal, Kennan had described Soviet policy as more ideological than he thought it really was, and containment as more global than he believed it needed to be. To make a polemical point, Lippmann had put the most extreme gloss on Kennan's purposefully ambiguous pronouncements. The result was a grand, public feud between two men who basically agreed. Kennan called it a \"misunderstanding almost tragic in its dimensions.\" He checked into Bethesda Naval Hospital complaining of ulcers and began composing a letter to Lippmann expressing his \"bewilderment and frustration\" that the columnist had \"held against me so many views with which I profoundly agreed.\" Kennan considered publishing the letter, but Secretary of State Marshall, who believed his planning chief had gotten too much publicity already, refused. And Kennan, for reasons that remain obscure but probably stemmed from his generally neurotic personality, never sent it, either. Two years later, finding himself on a train with Lippmann from New York to Washington, he launched into an impassioned explanation of why the columnist had gotten his views all wrong. But by then it was too late.\n\nIt was too late because between 1946 and 1950, containment swelled from the limited doctrine Kennan espoused into the hubristic one Lippmann condemned. And the vehicle for that expansion, as usual, was success.\n\nWhat expanded was not merely a foreign policy doctrine, containment, but a foreign policy mind-set, toughness, which had its roots in World War II. For virtually everyone in the Truman administration, Munich had been a totemic trauma, a searing, life-altering event. And to most policymakers its lessons were clear: The world contained evil people and evil regimes, which were impervious to reason. If their aggression was not met, early on, by force, the result might be death, destruction, and dishonor on an epic scale.\n\nBut despite this, America in the early postwar years did not go searching the world for Hitlers to destroy. In Western Europe and Japan, to be sure, Truman administration officials generally agreed that America must contain Soviet power; World War II had shown we had vital interests there. There were other chunks of the world, however, where the United States had never before projected much military power and whose political fate had never seemed tightly bound up with its own. The United States in the late 1940s did not have global military commitments, and most politicians, on both sides of the aisle, wanted to keep it that way. In 1948, with the Truman administration negotiating the treaty that created NATO, thus obligating America to defend Western Europe, Arthur Vandenberg and John Foster Dulles, two of the Republican Party's most influential spokesmen on foreign affairs, both demanded Secretary of State Marshall's assurance that America not go around making similar pledges in other corners of the globe.\n\nOne reason they feared such pledges was financial. With memories of the Depression still fresh, many politicians and policymakers considered the U.S. economy a fragile organism, vulnerable to sudden collapse. Containing communism across the entire globe would require vast sums for defense, which would likely lead to budget deficits and their ugly cousin, inflation, a prospect almost as frightening to government officials in the early Truman years as the Red Army.\n\nAs a result, to Forrestal's dismay, Truman put sharp restrictions on the amount he would spend containing the Soviets, restrictions that buttressed Kennan's insistence on containment's geographic limits. Since \"our nation's economy under existing conditions can afford only a limited amount for defense,\" announced Omar Bradley, chairman of the Joint Chiefs of Staff, in October 1949, \"we must look forward to diminishing appropriations for the armed services.\" That same month, Truman unveiled his budget, which slashed military spending by 15 percent. In the late 1940s, America's leaders did not know how much military spending the U.S. economy could bear, and they were not eager to find out.\n\nBut success turned Kennan's version of containment, with its keen appreciation of America's financial and geographic limits, into something more like Forrestal's. It began in the Near East, where Britain, which had long checked Russian ambitions, was now too bankrupt and exhausted to continue the job. In 1946, Iran's shah complained of bullying from Moscow, which had stationed troops on his nation's soil during the war and was refusing to remove them until it got a share of Persian oil. Truman demanded that the Red Army leave and dispatched $10 million in military aid, which Tehran used to fight a separatist uprising that the Kremlin had stoked in the country's north. Within a year, the Soviet troops were gone; the separatists had been crushed; and the United States, not the U.S.S.R., had won lucrative new rights to drill for Iranian crude. It was Munich in reverse: by standing tough\u2014and not abandoning Iran, as Chamberlain had abandoned Czechoslovakia\u2014America had nipped aggression in the bud. Iran had been a \"test case,\" declared the State Department, of a \"small state victim of large state aggression.\" And this time, America passed.\n\nIt was, it turned out, just a warm-up. A year later, in February 1947, a British embassy official drove to the State Department to inform U.S. officials that His Majesty's government, now liquidating its once-mighty empire in a kind of geopolitical fire sale, could no longer afford to defend Greece or Turkey, either. In Greece, a right-wing government was fighting for its life against communist rebels. In Turkey, the Soviets were demanding bases near the Dardanelles, from which they could project power into the eastern Mediterranean and the Middle East.\n\nThat night, Kennan and other top State Department officials met to consider whether America should come to Greece and Turkey's aid. It was a potentially historic shift. The eastern Mediterranean had been a traditional zone of British\u2014not American\u2014influence. (In their 1944 meeting in Moscow, where Churchill and Stalin agreed that the United Kingdom would enjoy 90 percent control in postwar Greece, no American had even been in the room.) Moreover, defending Athens and Istanbul would cost far more than defending Tehran. Congress was in no hurry to send hundreds of millions of dollars to nondemocratic regimes whose fate had never troubled it before, and even some in Truman's own cabinet were dubious.\n\nThe State Department officials agreed that America should send the money. But Kennan shuddered when he heard the way Truman intended to justify it. \"It must be the policy of the United States to support free peoples who are resisting attempted subjugation by armed minorities or by outside pressures,\" read the speech Truman was planning to give before Congress. This was just what Lippmann had warned against: a pledge to contain communism everywhere it reared its head. The Truman Doctrine \"should not be a blank check to give economic and military aid to any area in the world where the communists show signs of being successful,\" Kennan pleaded in a memo to his superiors. The American people must be told that containment had limits. But Truman's political advisers said the universal rhetoric would help sway a skeptical Congress. Like the X article, it was a small, strategic breach of intellectual honesty, with consequences that turned out not to be small at all.\n\nOnce again containment worked. Congress appropriated the money, and by late 1949 the Greek communists (whom Stalin had never much cared for anyway) had laid down their arms and the Dardanelles was safe from Soviet threat. Truman's hard line boosted his approval ratings, as Democrats who had been pilloried as soft on communism in the disastrous midterm elections of 1946 regained their foreign policy edge. Things got even better in 1948, when the Soviets blocked land access to West Berlin and Truman responded with a massive airlift that forced them to back down. It was another step up the toughness ladder: Now the Americans weren't merely sending money to contain the U.S.S.R.; we were flying planes. And this brought yet more success. Abandoning Berlin to the Russians, warned one U.S. official, would have been \"the Munich of 1948.\" Instead the Soviets caved, and Truman's reputation for firmly resisting communist aggression helped power him to an upset reelection victory that fall.\n\nContainment was working beautifully; the Truman administration was flying high; and yet George Kennan\u2014who had built the conceptual wings\u2014was worried. \"If I thought for a moment that the precedent of Greece and Turkey obliged us to try to do the same thing in China,\" he declared, \"I would throw up my hands and say we had better have a whole new approach to the affairs of the world.\"\n\nAlmost everything Kennan knew about China he had learned from John Paton Davies, a man rather similar to himself. Although jollier than Kennan, Davies shared his talent for combining intimate knowledge of a foreign land with detached, almost clinical, judgments about its politics. Born to Baptist missionaries in the province of Sichuan, Davies had attended Yenching University alongside many of the people who would later populate China's political elite. After finishing his studies in the United States, he had returned to China as a Foreign Service officer, serving four diplomatic tours there over the course of twelve years. When Kennan became head of the Policy Planning staff, Davies was the first person he hired.\n\nAs Davies looked at China in 1948, he saw good news and bad news. The bad news was that there was no way on God's green earth that America could prevent Mao Zedong's Communist Party from taking power, since America's allies, the Nationalists of Chiang Kai-shek, whose corruption and incompetence Davies had witnessed up close, were in a state of terminal collapse. The good news was that Mao, who unlike the communist leaders in most of Eastern Europe had his own power base, was unlikely, in Davies' view, to remain Moscow's stooge for very long. If America played its cards right, in fact, Mao might even become America's ally in containing Soviet power. \"A united communist China,\" suggested Kennan, reflecting Davies, \"might threaten Russian security and Russian control of the communist movement.\"\n\nDavies and Kennan proposed that America limit aid to Chiang in hopes of establishing decent relations with Mao once he took power. Forrestal, who had ascended to the newly created position of secretary of defense, objected. But Truman overruled him, declaring, with typical pool-hall eloquence, that sending money to Chiang was like \"pouring sand in a rat hole.\" It was clear evidence that within the corridors of power, Kennan's modest version of containment remained alive and well. Forrestal notwithstanding, most Truman officials recognized that not every victory for communism was a victory for the U.S.S.R.; that containment applied in some parts of the world but not in others; and that given the limits of American money and power, sometimes the best thing the United States could do\u2014even in the face of communist victories\u2014was to leave well enough alone.\n\nBut that is not what they had told the public. From the X article to Truman's Greece and Turkey speech, the rhetoric of containment had soared higher than the reality, and Chiang's American supporters now demanded that the latter be brought in line. China, whose vast bounty of consumers and converts had long stirred the dreams of American business and the American church, occupied a special place in the national imagination. And as 1948 turned to 1949, a powerful China lobby, championed by Time magazine's founder, Henry Luce\u2014the son of a Presbyterian missionary who had harvested souls in the city of Dengzhou\u2014accused the administration of letting the communists win. Over administration objections, Congress appropriated billions in economic and military aid to Chiang.\n\nKennan urged his superiors to confront the \"confusion and bewilderment in the public mind regarding our China policy,\" to say publicly what they were saying behind closed doors: that Chiang was a lost cause and Mao did not pose a grave threat. But Truman, like FDR before him, was more comfortable acknowledging the limits of American power privately than in public. His administration did not challenge the China lobby until it was too late, and by October 1949, when Mao hoisted a red flag in Tiananmen Square, the American China debate had turned rancid. The timing of Chiang's defeat was terrible: It came soon after news that Moscow had tested an atomic bomb, and amid of a wave of high-profile espionage trials, which fueled suspicions that Truman had let Mao take power because deep within the U.S. government, traitors were helping the communists win.\n\nOn February 9, 1950, with these flammable ingredients already in place, an obscure Wisconsin senator named Joseph McCarthy lit the match. He had been looking for an issue that would make his name. Friends suggested the St. Lawrence Seaway; then a national pension plan; then it hit him: treason! In a speech to the Women's Republican Club in Wheeling, West Virginia, he brandished a piece of paper allegedly listing 205 communists who worked for the State Department. The intersection of China's fall and McCarthy's lies, which he would hurl without scruple or shame for the next four years, played a crucial role in the growth of the hubris of toughness. China, after all, should have been a shining example of the limits of the Munich analogy. It should have convinced Americans of what Kennan and Davies knew: that contrary to the X article and contrary to what Truman had said in his Greece and Turkey speech, there was no earthly way America could man every anticommunist barricade across the globe. Had America genuinely tried to keep Mao from power\u2014which would have required not merely epic sums of money, but U.S. casualties in numbers that would have made Okinawa look like afternoon tea\u2014those limits would have become brutally clear. A serious effort at actually applying global containment in China would likely have brought the whole doctrine crashing down. But precisely because Truman hadn't done that, McCarthy and his allies could insist that holding China would have been easy: It would have required just a bit more aid, a few military advisers, and perhaps a little airpower. Fueled by McCarthyism, the legend grew that America had \"lost\" China not because it lacked the power to keep it, but because it refused to try.\n\nIntellectually, the McCarthyite argument on China was preposterous. But its impact on American politics was immediate and profound. Within months of McCarthy's emergence and China's fall, Truman's job approval rating, which had been nearly 70 percent in January 1949, had been cut in half. Prominent Republicans were calling Secretary of State Dean Acheson a coward and his predecessor, George Marshall\u2014the five-star general who had masterminded America's victory in World War II\u2014a traitor. And the political claim that Truman officials were weak in their opposition to communism was often linked to a cultural claim, which had particular resonance in the toughness years: that they were weak as men. Acheson, sneered McCarthy, pranced around \"with a lace handkerchief, a silk glove, and with a Harvard accent.\" GOP senator Everett Dirksen vowed to expose the administration's \"lavender lads,\" and by 1953 more than four hundred State Department employees had been either fired or forced to resign for alleged homosexuality. Decades later, Lyndon Johnson, then a freshman senator from Texas, would remember that \"Harry Truman and Dean Acheson had lost their effectiveness from the day that the Communists took over in China.\" When it was his turn in power, he would make sure nothing like that ever happened to him.\n\nIt was in this toxic political climate, on June 25, 1950, that the White House learned that communist North Korea had invaded its southern twin. Truman administration officials, consistent with Kennan's view that containment did not apply on the Asian mainland, had repeatedly implied, both publicly and privately, that they would not defend Seoul. But when war came, all that went out the window. Truman was in Independence, Missouri, when he learned of the invasion, and he quickly rushed back to Washington. \"I had time to think aboard the plane,\" he later wrote. \"In my generation, this was not the first occasion when the strong had attacked the weak.... Communism was acting in Korea just as Hitler, Mussolini, and the Japanese had acted ten, fifteen, and twenty years earlier.\" Later he added, \"If...the threat to South Korea was met firmly and successfully, it would add to our successes in Iran, Berlin and Greece.\" Those were the choices: Munich or anti-Munich. Berlin and Greece had boosted Truman's approval ratings, while his failure to prevent Mao's takeover had sent them tumbling and fueled McCarthy's rise. The November midterm elections were only months away, and Republicans would surely cite another communist triumph as evidence that he was guilty of appeasement, if not treason. By 1950, Truman believed that acknowledging containment's limits was more than perilous. It was the fast lane to political death.\n\nWithin days, the United States was rushing troops to the battle. Inside the administration, the debate was no longer about whether America should fight; it was about what America was fighting for: the preservation of South Korea or the liberation of the communist North as well. When Kennan, who had been on leave at Princeton, returned to the State Department, Davies warned him that if U.S. forces tried to reunify the entire peninsula, China might enter the war. In response, Kennan proposed that Truman publicly pledge not to cross the 38th parallel, which divides the two Koreas. But in good toughness fashion, John Allison, director of the State Department's Office of Northeast Asian Affairs, accused him of espousing a \"timid, half-hearted policy\" of \"appeasement.\" As the summer dragged on, Kennan felt the momentum swinging toward reunification. \"The course upon which we are today moving,\" he wrote to Acheson in late August, \"is one, as I see it, so little promising and so fraught with danger that I could not honestly urge you to continue to take responsibility for it.\"\n\nBut Acheson did take responsibility for it. In mid-September, General Douglas MacArthur pulled off a daring amphibious landing at Inchon, two hundred miles behind enemy lines, and began racing toward the 38th parallel. Halting him there would have required a heroic act of political will, since dealing with the egomaniacal general was, in Kennan's words, like \"arranging the establishment of diplomatic relations with a hostile and suspicious foreign government.\" Besides, with the fall elections approaching, Truman needed a win, something that would silence the McCarthyite hordes. So far, every time he had taken another step up the toughness ladder, it had paid off.\n\nSo MacArthur crossed the 38th parallel into North Korea, and like Xerxes crossing the Bosporus, the gods took their revenge. In this case, the instruments of their vengeance were Chinese. In early October, with MacArthur's troops nearing the Yalu River, which separates North Korea from China, Stalin told Mao that if he threw five to seven divisions into the fight he could not only keep the Americans off his border, but deal them a crushing blow. After several sleepless nights, Mao decided\u2014as Davies and Kennan had feared\u2014to do just that. Within weeks, hundreds of thousands of Chinese troops were sneaking across the Yalu, carrying precooked meals so they wouldn't need to light fires visible to American planes. On October 26, U.S. forces took their first Chinese prisoner. Still, MacArthur pushed north, boasting that his men would be home by Christmas.\n\nIn late November, the Chinese launched an all-out attack, forcing U.S. troops into a panicked retreat. By spring, after months of ghastly fighting, American forces stabilized their positions roughly where the whole conflict began, at the 38th parallel, and State and Defense department officials began exploring negotiations to end the war. But MacArthur resisted, defying Truman's orders and demanding that America expand the conflict by blockading China's ports, bombing its military installations, and encouraging Chiang to launch an invasion from Taiwan. In April, after MacArthur's brazen insubordination became too much to bear, Truman fired him. But this just liberated the general to loudly accuse his former boss of being \"blind to history's clear lesson\" and practicing \"a form of appeasement on the lines of the Munich conference in 1938.\" For his part, McCarthy called Truman's refusal to pursue total victory a \"super-Munich,\" adding, \"The son of a bitch ought to be impeached.\" The young senator from Texas, Lyndon Johnson, saw his office flooded with pro-MacArthur mail.\n\nAs MacArthur's supporters saw it, Truman\u2014like Neville Chamberlain in 1938\u2014was showing dangerous weakness in the face of aggression. But the charge just showed how unreal their definition of weakness had become. In the late 1930s, many Americans had denied that their nation's safety required the survival of France. Before 1947, few had believed American security had much to do with what happened in Turkey and Greece. In 1949, most observers assumed that the United States would abandon South Korea to its fate. Yet by 1951, Truman's opponents were handing him umbrellas because he refused to launch a war inside China to liberate all of Korea from communism. America's mounting global strength, as it filled the gap left by Europe's evaporating empires, combined with the terror of weakness inherited from World War II, had turned every foe into Hitler and every compromise into Munich. Politicians who once considered it no shame to let America's enemies have Paris now cried treason because we were letting them have Pyongyang.\n\nMacArthur and his allies genuinely believed that the only reason containment kept expanding was that the communists kept confronting the United States with graver and graver threats. What they didn't grasp was that their definition of what constituted a threat had grown in parallel with America's growing confidence in its power. One big reason for that growing confidence was the U.S. economy. In the late 1940s, many Truman officials had seen it as fragile and thus unable to sustain too much defense spending. Their belief that the United States didn't need to stop communism everywhere in the world had been deeply bound up with their belief that America couldn't afford to. But by 1950, official views were beginning to change. In 1949, the new head of the Council of Economic Advisers, Leon Keyserling, began denying that deficits were as dangerous as most people assumed. Following the famed British economist John Maynard Keynes, he argued that the government spending that created deficits could also rev up the economy, thus producing higher economic growth\u2014and higher tax revenues\u2014and eventually bringing the budget back into balance. Keyserling cited World War II. After all, the massive military buildup\u2014and resulting deficits\u2014needed to fight the Nazis hadn't wrecked America's economy. To the contrary, they had ended the Depression. And in the years since, continued economic growth had helped balance the budget again. Recent experience, in other words, showed that the U.S. economy wasn't fragile at all. It could bear, and even benefit from, far more government spending than policymakers had previously imagined. The only real limits were in America's leaders' minds. As former undersecretary of state Robert Lovett told a committee tasked with putting America's containment strategy into written form, \"there was practically nothing that the country could not do if it wanted to do it.\" His words might have served as an epigraph to the mounting hubris of the age.\n\nIn April 1950, that committee\u2014led by Kennan's successor as head of Policy Planning, Paul Nitze\u2014produced a sixty-six-page top-secret document called National Security Council Report 68, which declared that America could, and must, dramatically boost defense spending. NSC 68 was partly a product of Keyserling's economic optimism, an optimism nurtured by America's burgeoning postwar boom. By 1955, Keyserling predicted, gross national product would grow to almost $300 billion, meaning that even at dramatically higher levels, military spending would not rise significantly as a percentage of GNP. Gradually, the Depression-induced caution of the 1940s was melting away. Truman officials were coming to see economic limits as a thing of the past.\n\nIn 1950, with NSC 68 claiming that the United States could afford to spend far more on defense, and Korea apparently proving that we needed to, a flurry of supplemental financing boosted military spending to more than $48 billion, an astounding increase over the roughly $13 billion in Truman's initial budget. Containment, which Kennan had conceived as primarily a political strategy, was now an unmistakably military one. And if NSC 68 eviscerated containment's financial restraints, it eviscerated its geographic ones as well. The report's authors admitted that not every communist victory represented a threat to the United States, but they insisted that America oppose communism's advance everywhere anyway because no matter how strategically insignificant a country might be, its loss would prove psychologically damaging. The perception of weakness, NSC 68 argued, would breed \"doubt and recrimination,\" both among America's allies and within the United States itself. And with McCarthy becoming a national plague, no one needed to spell out just how debilitating that domestic \"recrimination\" could be.\n\nNSC 68 highlighted a concept that would sit near the core of American foreign policy during the toughness years: \"credibility,\" the belief that showing weakness in unimportant places would unnerve our allies and embolden the communists in important ones\u2014which meant that unimportant places were important after all. America's challenge, therefore, was not merely to be strong, but to appear strong\u2014too look \"credible\"\u2014both to others and to ourselves. And in the funhouse mirror that was McCarthy-era Washington, where mighty America looked perpetually weak, creating the perception of strength required expanding containment to the far reaches of the earth.\n\nBeneath the policy differences lay a deeper, philosophical difference between Kennan's version of containment and NSC 68's version. Kennan, like Lippmann, Morgenthau, and Niebuhr, believed that Wilson and the progressives had been na\u00efve about human nature: There was darkness in the soul that made reason an insufficient guide to world affairs. \"The fact of the matter,\" Kennan told students at the National War College, \"is that there is a little bit of the totalitarian buried somewhere, way down deep, in each and every one of us.\" But for Kennan, that last clause was crucial: Every one of us was capable of evil, Americans included. That's why, although he believed America must sometimes use force to limit the evil done by its enemies, he also insisted that it limit itself, because it could do evil, too. For Kennan, containment was a partial doctrine because anticommunism was a partial truth\u2014valuable in moderation, but prone, if stretched too far, to resemble the very fanaticism it was pitted against. NSC 68, by contrast, threw around words like evil, irrational, and fanatical liberally, but applied them only to the other side. Americans were uniformly described as moral, decent, and rational\u2014so moral, decent, and rational, in fact, that they could take virtually unrestrained action to combat communism without ever imperiling their souls.\n\nBut by 1950, Kennan did believe that frenzied anticommunism was imperiling the American soul. There was madness in the air. Kennan's old patron, James Forrestal, whom Truman had recently replaced as secretary of defense, began claiming he had been fired because Soviet agents wanted to silence his warnings about Moscow's impending all-out attack. He also brooded endlessly over a Washington columnist's claim that he had shown cowardice when thieves came to steal his wife's jewelry. Within weeks of his dismissal, Forrestal was refusing to leave his house, talking in hushed tones for fear he was being bugged, keeping the blinds permanently drawn to guard against snipers. Friends convinced him to check into Bethesda Naval Hospital, where he was treated with sedatives, tranquilizers, and shots of insulin.\n\nAt first Forrestal's health seemed to improve. But a few weeks into his stay, he rose from his hospital bed in the middle of the night and began copying passages from Sophocles' poem \"The Chorus from Ajax.\" The poem tells the story of the legendary Greek warrior Ajax, a man so confident in his martial skill that he spurns the goddess Athena when she offers her assistance on the battlefield of Troy. In retaliation for this and other acts of mortal hubris, she afflicts him with madness. Believing himself under attack by his enemies, Ajax pulls out his sword and slaughters his foes, only to find that he has murdered a herd of sheep. Shamed, he turns the sword on himself.\n\nAfter finishing his transcription, Forrestal put the sheets of paper on the table beside his bed and walked across the corridor to an adjacent kitchen. There he opened a window and jumped. He landed facedown with a thud, thirteen floors below.\n\nThere was another man, even closer to Kennan, who by the early 1950s was checking for hidden microphones when he entered a room. And he, tragically, had reason to. In the summer of 1951, the Senate Subcommittee on Internal Security launched an investigation of John Paton Davies, charging that he had sabotaged Chiang, assisted Mao, and tried to infiltrate communists into the U.S. government. Kennan testified on Davies' behalf and even threatened to resign in protest. But given his opposition to global containment, Kennan was now a marginal, even suspect figure himself, and his protests had little effect. In 1953, Davies was removed from his post as assistant to the U.S. high commissioner in Germany and sent to stamp passports in Peru. When even that demotion didn't satiate McCarthy, the State Department launched yet another investigation into Davies' character, the ninth in five years\u2014and then fired him outright. Rather than return to the United States, Davies remained in Peru. For the next decade, the American government's foremost China expert lived at the foot of the Andes, sustaining his family by building furniture.\n\nIn one of his last Davies-inspired memos, written in August 1950, just months before his friend was consumed by McCarthyite attack, Kennan focused on a little-noticed consequence of NSC 68: the decision to send military advisers to battle a communist-led revolt in the obscure French colony of Vietnam. \"In Indochina, we are getting ourselves into the position of guaranteeing the French in an undertaking which neither they nor we, nor both of us together, can win,\" Kennan warned Acheson. But in a cover note he acknowledged that \"like many of my thoughts, [these] will be too remote from general thinking in the Government to be of much practical use to you.\" He was right: his memo received no response at all.\n\n## CHAPTER SIX\n\n## THE PROBLEM WITH MEN\n\nIn an age of hysteria, Dwight Eisenhower was the political equivalent of Valium. He tacked to the soothing center; he spoke in apple-pie platitudes; he practiced his chip shot in the Oval Office. As a result, critics often accused him of passivity. Behold the Eisenhower doll, they quipped: Wind it up and it does nothing for eight years.\n\nBut an ice cube dropped into boiling water expends tremendous energy, even if the resulting liquid is merely lukewarm. In reality, Ike was not passive at all. Behind closed doors, he worked strategically, forcefully, even ruthlessly, to hold the price of containment down, in both dollars and American lives. And because he succeeded so well, the hubris of toughness, which had swelled massively between 1946 and 1950, deflated somewhat during his eight years in office. It was only once he exited the political scene that a new generation of leaders\u2014emboldened by America's postwar success and lacking his hard-won knowledge of the limits of economic abundance and military might\u2014flew the wings of containment into the sun.\n\nWhen Eisenhower sought the presidency in 1952, some liberals feared that, because he had spent his life in the military, he might prove a warmonger. They could not have been more wrong. As a general, Ike had been the anti-MacArthur, a man known for his aw-shucks modesty, meticulous planning, and extreme caution. His mother was a pacifist, a philosophy he didn't share but never mocked. And his experience sending young men to their deaths bred in him an intimate hostility to war. Discussing dovish civilians in a letter to his brother in 1943, he wrote that \"I doubt whether any of these people, with their academic or dogmatic hatred of war, detest it as much as I do.... [They] probably have not seen bodies rotting on the ground and smelled the stench of decaying human flesh.\" As the commander of Allied forces in March 1945, with Churchill pressuring him to race to Berlin before the Soviets got there, Ike refused; he would not take large casualties for a prize of little military value. Seven years later, he lashed out at those who criticized his decision, declaring that \"none of these brave men of 1952 have yet offered to go out and pick the ten thousand American mothers whose sons would have made the sacrifice to capture a worthless objective.\"\n\nIf Ike learned wartime prudence the hard way, Americans in the early 1950s were learning it as well, in Korea. When the war began, only 14 percent of the public expected it to last more than a year. But by the time Eisenhower was elected, it had dragged on for thirty months, with no end in sight. For the last eighteen of those, little territory changed hands. It was World War I\u2013style trench warfare: endless artillery barrages, impenetrable defensive lines, ghoulish conditions, dead boys piling up in the snow. To hold Pork Chop Hill, a place of no particular significance except that the military brass wanted it held, one army company lost 121 of 135 men. By the fall of 1952, a plurality of Americans considered the war a mistake. And with more than one hundred thousand Americans dead, wounded, or missing, the McCarthyite illusion that global containment was easy\u2014that it required little more than cleansing Foggy Bottom of a few latter-day Benedict Arnolds\u2014had become the least lamentable casualty of war.\n\nEisenhower's chief opponent for the Republican presidential nomination, Robert Taft, had proposed following MacArthur's advice and doubling down in Korea: taking the fight to Mao in an effort to quickly win the war. And once Ike got the nomination, some in his party pressured him to follow suit. But Eisenhower knew enough about war\u2014and about MacArthur\u2014to suspect that escalation was folly. And he had seen enough as a candidate to realize that the American people wanted peace more than they wanted victory. His running mate, a young, red-baiting California senator named Richard Nixon, and his chief foreign policy adviser, the starchy, self-righteous John Foster Dulles, excoriated the Democrats for appeasement and pledged to roll back communism in Korea and across the globe. But Ike, for the most part, promised not to win the war, but to end it. In October, as a new round of fighting sent U.S. casualties above one thousand per week, he called the war a \"tragedy.\" American troops, he suggested, should cede much of the combat to South Koreans. And just eleven days before the election, in a nationally televised speech at Detroit's Masonic Temple, he promised that, if elected, he would go to Korea and make peace. \"For all practical purposes,\" declared a political writer for the Associated Press, \"the contest ended that night.\"\n\nWhen Eisenhower arrived in Seoul in late November 1952, he found the South Korean president Syngman Rhee and America's top military commander, Mark Clark, itching for a major offensive aimed at unifying the peninsula once and for all. But Eisenhower politely ignored them, wrapping himself in a thick jacket, thermal boots, and a fur hat and trudging out to inspect the situation firsthand. After interrogating front-line officers, eating with troops, and scouting the terrain from the air, he concluded that his intuition had been right: Militarily, unification was a pipe dream. Instead, he passed word to the Soviets and Chinese that, if a truce wasn't reached soon, he might expand the war, and perhaps even use nuclear weapons. Having stoked his adversaries' fears of war, he then moved aggressively to make peace. He was aided in this effort by Stalin's death, since it was Stalin\u2014more than Mao\u2014who had been keen to prolong the fighting. (After all, it was Chinese boys, not Russian ones, who were getting killed.) By summer, South Korean, North Korean, American and Chinese diplomats had negotiated an armistice, which left the border between North and South Korea almost exactly where it had been when the war began. Most Americans breathed a sigh of relief and vowed never to do anything like that again. For a time, Korea's bitter memory proved Ike's ally in keeping hubris in check\u2014until its chastening influence was gradually buried under a new wave of success.\n\nEisenhower's behavior in Korea set the tone: no more endless expenditures of money and blood. The first constraint was almost as important as the second. Nothing in Ike's experience growing up in hardscrabble nineteenth-century Kansas predisposed him to believe that money was infinite or debt a good thing. His economic views were shaped in an earlier, harsher era, when scarcity was the norm and free spending was a ticket to ruin. Since leaving the military, he had also become golfing buddies with a slew of prominent businessmen, who reinforced his antipathy to big government. Succeeding Truman, declared a congressman who shared Ike's economic views, was like \"taking over a hussy who had spent all her husband's money and run up a lot of bills at the local department store.\"\n\nAided by the armistice in Korea, Eisenhower slashed military spending from 13 percent of GDP in 1954 to barely 9 percent when he left office. If America's generals had assumed that having one of their own in the White House would be good for business, they were bitterly disappointed. During Ike's presidency, two army chiefs of staff resigned and a third, Maxwell Taylor, retired to write a book attacking Eisenhower's policies on national defense.\n\nIf Eisenhower refused to pay for containment, he also refused to fight for it with American troops. There would be no more Koreas on his watch. This posed a problem, because, like the authors of NSC 68, he insisted that containment be global. He still wanted to contain communism everywhere; he just wanted to do so cheaply and bloodlessly (or, at least, cheaply and bloodlessly for the United States). And in that effort, two methods proved particularly useful. The first was covert action. The CIA\u2014headed by John Foster Dulles's brother Allen\u2014was busy during the Eisenhower years. It toppled leaders in Iran and Guatemala; attempted coups in Indonesia, the Dominican Republic, and the Congo; and discussed putting radioactive dust in Fidel Castro's shoes to make his hair fall out and rob him of the virile image that supposedly kept Latin leaders in power. Containing communism via assassinations and coups usually proved disastrous for the targeted nations, and for their long-term relations with the United States. But financially, if not morally, it was cheap. And few Americans got killed.\n\nEisenhower's second major method of containment didn't get Americans killed, either, although it did scare some of them half to death. That method was nuclear weapons. During his presidency, Ike wielded nukes to scare the Soviets and Chinese into settling the Korean War, to deter Moscow from invading West Berlin, and, in a kind of reductio ad absurdum, to stop the Chinese from taking Quemoy and Matsu, islands that weren't even vital to Taiwan's defense, let alone America's. If Moscow stirs up trouble, Ike informed congressional leaders in late 1954, the United States will \"blow the hell out of them.\"\n\nEisenhower's discussion of the bomb\u2014which he said America would employ \"just exactly as you would use a bullet or anything else\"\u2014was chillingly nonchalant. And critics depicted him as a genial, vaguely senile, grandfatherly figure, mouthing banalities and cutting ribbons while subordinates prepared to blow the world to kingdom come. But as with many things Ike-related, appearances were deceiving. Eisenhower wasn't actually cavalier about nuclear weapons at all. To the contrary, he had seen enough of war to know that once unleashed, it could rarely be controlled. And as a result, he feared that if conventional fighting broke out between the superpowers or their proxies it might end in mushroom clouds. (One reason he favored covert action was its deniability, which allowed each side to pull back and save face before things got out of hand.) And so, in an audacious gambit, he loudly declared that, if conventional warfare broke out, he would initiate what he most feared: nuclear war. In so doing, he hoped not only to disabuse the Soviets of the belief that they could keep small wars small, but also to disabuse those in his own government. By insisting that any superpower conflict would descend the slippery slope to Armageddon, he hoped to prevent both sides from taking the first step.\n\nSo while Eisenhower's nuclear rhetoric was terrifying, his behavior was generally cautious. In 1954, with French troops besieged in the remote Vietnamese garrison of Dien Bien Phu, and Paris facing defeat at the hands of the communist Vietminh, Ike's air force and navy chiefs of staff, along with Vice President Nixon, urged American air strikes. But Ike, convinced that air strikes now would lead to ground troops or worse later, refused. On Quemoy and Matsu, although he threatened nuclear war, he resisted calls to initiate it, even after China shelled the islands. And in July 1958, although he sent troops to seize the Beirut airport in a show of support for Lebanon's beleaguered pro-Western regime, he overruled the joint chiefs, who wanted to occupy the entire country. By October, U.S. forces were gone.\n\nIke held down global containment's cost. Unfortunately, he never challenged the concept itself. He still hewed to the idea\u2014enshrined in NSC 68\u2014that if the United States permitted communism's advance anywhere, it would damage American credibility, thus emboldening communists everywhere. This idea, which eviscerated Kennan's effort to make distinctions between different regions of the globe and to see U.S. interests as something other than a function of communist threats, remained part of Eisenhower's conceptual arsenal. It was still lying there when he left office for his less skilled successors to pick up, like an unexploded grenade.\n\nIn this sense, for all his savvy, Ike paved the way for the hubris that followed. Because he reduced America's military resources without reducing the interests they were supposed to defend, he made it easy for younger men to attack him as passive. That was unfair: Eisenhower's strategy was not passive; it was downright daring. And for eight years at the height of the superpower standoff, it prevented war. \"We kept the peace,\" he told a journalist. \"People ask how it happened\u2014by God, it didn't just happen.\" It was a remarkable achievement, but by the time Eisenhower left office, it no longer felt so remarkable. The Korean trauma was fading: Between the fall of 1952 and the fall of 1956 the percentage of Americans who considered the war a mistake fell by 15 points. Precisely because of Ike's success\u2014because the communists made few significant gains and America avoided war\u2014average Americans, and especially the foreign policy elite, came to believe that Korea had worked. It had been hard but necessary, a win. The conflict was being absorbed into the toughness catechism, the latest in a valiant litany of anti-Munichs that ran from Iran to Berlin to Greece. And increasingly a new generation of American leaders and thinkers was eager for more. They thirsted for new challenges\u2014challenges that required danger, sacrifice, and strain. Eisenhower, who had seen enough danger and sacrifice to appreciate their absence, never grasped this yearning. He knew war too well to romanticize it, and he had nothing to prove. But he now belonged to a passing age.\n\nIn 1960, Ike's final year in office, a young intellectual named Daniel Bell published a book called The End of Ideology, in which he argued, among other things, that young intellectuals like himself were bored. They were bored because in America there were no more political crusades. In the Wilson years, intellectuals like Dewey and Beard had thrilled to progressivism's vision of a harmonious, rational world. In the 1930s, writers had flocked to communism's equally heady vision of inexorable scientific progress. But by the '40s, America's leading intellectuals\u2014men like Niebuhr and the literary critic Lionel Trilling\u2014had buried those grand visions and embraced tragic realism, not just in foreign policy but as a perspective on life.\n\nIt was a weary creed, befitting people who, as Bell noted, had seen horrors that made them old before their time. By the 1950s, few American intellectuals still believed\u2014as had Dewey and the social gospel theologian Walter Rauschenbusch\u2014that if government provided the right kind of education it could unlock the reason and love lodged in ordinary people's souls. Far from valorizing the common man, the intellectuals of the '50s breathed a sigh of relief that ordinary Americans didn't expect very much from politics. America's shopkeepers didn't harbor fascist dreams of national purity; they tried to keep taxes low. America's labor leaders didn't march for revolution; they tried to get their members a better dental plan. If the progressives had yearned to transcend humanity's petty, selfish concerns, the postwar realists embraced them. In cold war America, argued Bell, politics was about the \"unheroic, day-to-day routine of living.\" After \"living dangerously in the exciting land of either-or,\" declared Arthur Schlesinger, Jr., Americans had entered \"the unromantic realm of more-or-less.\"\n\nFor older intellectuals like Niebuhr, who had seen the catastrophe that political crusades could bring, this realism represented an intellectual life raft. But for many of the younger thinkers who followed, it was more like a straitjacket. Since World War II generally marked the beginning of their political consciousness, not the end, they tended to take America's postwar success for granted. With the exception of Korea (and even that became a happier story in retrospect), they had seen no costly overseas traumas. At home they had experienced almost ceaseless boom. Yet instead of contentment, many experienced what the sociologist David Riesman called the \"malaise of the privileged.\" There was something enervating about being told that 1950s America, with its private comfort and lack of public drama, was the best humanity could do. And the species of American that all this comfort produced\u2014the conformist, sheeplike \"Organization Man\" that William Whyte portrayed in his widely read 1956 book\u2014filled younger critics with disgust. \"The young intellectual is unhappy because the 'middle way' is for the middle-aged, not for him,\" wrote Bell. \"[There is] an underlying restlessness, a feeling of being cheated out of adventure, and a search for passion.\"\n\nAmong those who felt this restlessness was Schlesinger, a short, slight, balding, bespectacled, and bow-tied historian two years Bell's senior. Schlesinger was particularly worried about the impact that this lack of adventure was having on America's men. In 1958, in an essay for Esquire titled \"The Crisis of American Masculinity,\" he warned that women were on the march, \"an expanding, aggressive force, seizing new domains like a conquering army, while men, more and more on the defensive, are hardly able to hold their own and gratefully accept assignments from their new rulers.\" The signs of emasculation were everywhere: in the media's fascination with sex-change operations, in the rage for homosexuality, even in the movies. Why was it, Schlesinger asked, that Gary Cooper, Cary Grant, Clark Gable, and Spencer Tracy\u2014each well into his fifties\u2014kept playing romantic leads \"opposite girls young enough to be their daughters?\" Because none of the younger male stars looked man enough to pull it off.\n\nThe problem, he argued, stemmed from conformity and ease. Men drove in identical cars; worked in identical, air-conditioned offices; slept in identical, manicured suburbs. Nothing in their lives was unconventional; nothing involved danger. Postwar America had become homogenized, and it had gone soft. It no longer produced heroes; it produced drones. And a man without heroic dreams was not fully a man.\n\nSatire, Schlesinger suggested, might be one antidote to this cultural malaise. Inspiring art was another. And a third tonic was politics. If a leader confronted the nation's affairs in a sharp and daring way, pushing Americans beyond their comfort zone, demanding they meet epic challenges without hesitation or fear, he could create a space in which men might become heroes. He could make public life \"virile\" again. Although Schlesinger didn't say so, he had a candidate in mind: the junior senator from Massachusetts, John Fitzgerald Kennedy.\n\nIt was an apt choice, since for Jack Kennedy, virility was virtually a philosophy of life. Like Theodore Roosevelt, who was in crucial ways his cultural and ideological ancestor, Kennedy's identity had been forged, in boyhood, by a struggle against physical weakness. Born asthmatic and with a deformed back, he spent much of his youth wheezing, on crutches, and wearing a brace. His mother described him as \"elfin\" his grandfather offered him a dollar for every pound he gained. At two, he came so close to dying of scarlet fever that his father, Joseph Sr., promised God that he would donate half his money to charity if the boy lived. (God fulfilled his side of the bargain; Joseph Sr., it appears, did not.) At age fourteen, Jack suffered an attack of appendicitis so severe that a priest administered last rites. \"At least one half of the days that he spent on this earth,\" said his younger brother Robert, \"were days of intense physical pain.\"\n\nIt was not an easy family in which to be frail. Jack's father, who had clawed his way to great wealth only to find himself snubbed by Boston Brahmins who would not accept an Irish Catholic at any price, loathed weakness. And Jack's older brother, Joe Jr., a bigger, stronger boy with a brutal streak, punished it. At night, younger siblings lay awake listening to Joe smashing Jack's head against the wall; one particularly nasty confrontation landed Jack in the hospital with twenty-eight stitches. \"Jack always had something to prove, physically,\" remarked his prep school friend Lem Billings, \"to overcompensate and prove he was fit when he really wasn't.\"\n\nIt was hardly a surprise, therefore, that Jack Kennedy filled his adolescence with trials signifying manhood. Despite his size, he played football at Harvard. He racked up sexual conquests, first with prostitutes procured by his father and later with women he met on his own. (This particular show of physical vigor had the paradoxical effect of saddling him with yet another malady: the clap.) And he went to war. Rejected in 1941 by the army because of his bad back, he used his father's connections to find a spot in the navy, and then used them again to transfer from desk work into command of a PT boat in the South Pacific, one of the most glamorous jobs at sea. In July 1943, near the Solomon Islands, a Japanese destroyer sliced his boat in half. Two of his men were killed instantly; the other eleven were tossed into the ocean. With most of the surrounding territory controlled by the Japanese, Kennedy led his surviving men on a five-hour swim through crocodile-and shark-infested waters to a tiny deserted island. He towed one badly burned sailor by clenching a life-jacket strap in his mouth. Avoiding Japanese patrols, and despite repeatedly falling unconscious from exhaustion, he then led them on a second swim to a larger island two and a half miles away, where they survived on coconuts and water. There they were discovered by two Pacific islanders in a canoe, to whom Kennedy gave a coconut shell on which he scribbled a plea for help. The story of PT-109 would become legend. As president, Kennedy kept the coconut shell in a case on his desk.\n\nHad Kennedy's quest to prove his manhood been a merely personal drama, it would hardly have mattered. But by World War II, his journey had taken on a distinctly ideological cast, one that mirrored the nation's as a whole. Joseph Sr., after all, was an appeaser. Like many in his generation, his views of foreign policy had been shaped by World War I, a war whose draft he had unapologetically dodged. Military service, he insisted, was a \"sucker's game.\" Like Herbert Hoover, he considered war an inexcusable distraction from the real purpose of world affairs: making money.\n\nIn December 1937, FDR rewarded Joseph Sr. for the vast sums he had donated to his presidential campaigns by naming him ambassador to the Court of St. James (a position Kennedy wanted largely so he could lord it over his WASP tormenters back home). Arriving in London as Neville Chamberlain's appeasement policy was reaching its crescendo, Joe Sr. took an instant liking to the former Birmingham industrialist, a man who, like himself, valued commerce and loathed war. And he developed a sharp aversion to Chamberlain's rival, Winston Churchill, who struck Kennedy as not only militaristic, but perpetually drunk. When Chamberlain returned from Munich in September 1938, having purchased peace with Hitler at Czechoslovakia's expense, Joe Sr. called him the greatest statesmen of the age. A year later, with Hitler demanding part of Poland, Kennedy urged that London cede it to the Nazis as well. He even met secretly with the German ambassador, telling him that Britain lacked the stomach for war and accusing his own boss, Franklin Roosevelt, of succumbing to pressure from Jews.\n\nBy 1939, Joseph Kennedy, Sr., was a very controversial man. His declaration that \"democracy in England is finished\" and that fascism might prove a better alternative had evoked fury in both the British and American press, as had his prediction that if Britain did fight the Nazis, it would lose. The New Statesmen called him a \"frightened rich man who thinks only in terms of money.\" When war broke out, Fleet Street pointedly noted that Kennedy had followed British women and children to the countryside, where fewer bombs fell, rather than stay in London with the nation's men.\n\nNone of this was lost on his college-age sons, Joe Jr. and Jack, who in the late 1930s split their time between London and Harvard. Joe Jr. eagerly embraced his father's isolationist line. In 1938, when Walter Lippmann criticized one of the ambassador's speeches, Joe Jr. accused the columnist of exhibiting \"the natural Jewish reaction.\" The following year he toured civil war Spain, penning letters back home that championed the Fascist cause. Joe Sr., who thought becoming an author could boost a young man's career, hired an editor to turn his son's letters into a book. And when Joe Jr. returned to Cambridge in the fall of 1940, he helped found Harvard's Committee Against Military Intervention in Europe.\n\nBy 1940, however, with Paris under Nazi control and American opinion swinging hard against appeasement, publishers showed little appetite for a book praising Fascist Spain. Instead it was Jack who made his literary mark. In the fall of 1939, he had decided to write his Harvard thesis on British policy toward Hitler. Submitted in April 1940, \"Appeasement at Munich\" did not condemn Chamberlain's actions. To the contrary, it argued that he had faced little choice, since in a democracy like Britain's, public opinion naturally tended toward pacifism. Far from contradicting his father's view, Jack's paper subtly echoed it, implying that in a dangerous world, totalitarianism was the form of government most likely to survive.\n\nWhen Joe Sr. suggested that his younger son now seek publication, Jack jumped at the chance to turn his thesis into a book. But he still didn't envision it as a brief against appeasement, telling his father that \"I should like to get something in the conclusion about the best policy for America as learn't [sic] from a study of Britain's experience but of course don't want to take sides too much.\"\n\nWhen Jack sent the manuscript to New York Times columnist and family friend Arthur Krock, however, that ambivalence quickly disappeared. Krock urged Kennedy to shift the blame from British democracy to Chamberlain himself. And he urged him to come down clearly in favor of U.S. intervention, to use Britain's example to warn Americans of the perils of lethargy in the face of the Nazi threat. Krock even suggested a title. Churchill had recently named a collection of his anti-appeasement speeches While England Slept. Kennedy should play off it, calling his book Why England Slept.\n\nSo it was that in July 1940, with his father and older brother still sympathetic to fascism, hostile to Jews, and inclined toward appeasement, John F. Kennedy published an interventionist manifesto, which lionized Churchill and demanded that America institute a draft. Hitting shelves only weeks after Paris fell, Jack's book became an instant bestseller (it didn't hurt that Joe Sr. bought copies in bulk) and identified its author with arguments about the inescapability of evil, the danger of aggression, and the necessity of force that would define his generation. By war's end, Joe Jr. was dead, having volunteered for a near-suicide mission over Germany that some believe he took in a desperate bid to clear his and his father's names. Joe Sr., having resigned his ambassadorship, was a political pariah, a symbol of policies that many Americans now associated with cowardice, if not treason. Jack, by contrast, was a famed interventionist and a war hero. It would be excellent training for a cold war climate in which toughness, both ideological and personal, became a national obsession. Schlesinger was right: Jack Kennedy had a gift for the politics of manliness. His mastery of it helped propel him to the White House. And his fidelity to it helped propel America into Vietnam.\n\nWhat distinguished men like Kennedy and Schlesinger from their elders wasn't merely their fixation with Munich; it was their tendency to refract that experience through the lens of America's postwar success. As a result, they not only believed that communist expansion must be stopped (which Ike believed, too), but that America's capacity to stop it was nearly infinite (which Ike did not). For Kennedy and his supporters, in fact, it was precisely Eisenhower's unwillingness to confront communism with the full force of American ingenuity and might\u2014even though he had acknowledged the danger\u2014that made his presidency so dispiriting. The answer was to strip away Ike's self-imposed limits, to ask more of Americans, to get them fully into the game.\n\nHad Sputnik not arrived, America's restless young intellectuals might have needed to invent it. On October 4, 1957, from out of the sky, came evidence that while Americans were sitting fat and happy on the couch, the other side was racing ahead. Sputnik was a beeping, two-foot-wide, 184-pound aluminum sphere, the first man-made satellite in space, and it was the Soviets who had launched it, not the United States. Two months earlier, in another shock, Moscow had launched the first intercontinental ballistic missile (ICBM). Suddenly America seemed to be falling behind in everything from missiles to science labs to math teachers.\n\nPredictably, Ike greeted the furor with a yawn. When a committee led by Ford Foundation chairman H. Rowan Gaither used Sputnik to demand another round of NSC-68-style increases in defense spending, Eisenhower largely rejected the proposed buildup, blandly noting that it did not matter whether America had as many missiles as Moscow; we had enough to deter attack. (He also knew from secret U-2 spy planes that the Soviets had nowhere near as many ICBMs as panicked Americans assumed.) But this time, rather than tamping down national alarm, Eisenhower's low-key response inflamed it, further convincing his critics that he was lethargic and weak, the embodiment of what Schlesinger called \"the politics of fatigue.\" One prominent journalist compared Eisenhower to Stanley Baldwin, the appeasement-minded British prime minister who had preceded Chamberlain. \"Thank you, Mr. Sputnik,\" declared a commentator on the Mutual Broadcasting System. \"You gave us a shock which hit many people as hard as Pearl Harbor.... You woke us up out of a long sleep. You made us realize a...nation, like a man, can grow soft and complacent.\" It was time for Jack Kennedy to make his move.\n\nIn his 1960 presidential bid, Kennedy made the post-Sputnik \"missile gap\" the symbol of an America that, like Britain in the 1930s, lay asleep as threats gathered. His opponent, Vice President Richard Nixon, was McCarthy in a better suit, a master at painting Democrats as pink. Running with Ike in 1952, he had labeled Democratic nominee Adlai Stevenson a \"PhD grad of Dean Acheson's cowardly college of communist containment.\" But Kennedy was not Stevenson. In fact, he and his advisers saw the high-minded, self-consciously cerebral Illinois governor in much the same way Nixon did: as effete and soft. In Congress, Kennedy had joined McCarthy and Nixon in blasting Truman for losing China. Columnist Joseph Alsop dubbed Kennedy \"Stevenson with balls,\" and Jack relished the line.\n\nKennedy beat Nixon at his own game. Picking up Schlesinger's theme, he linked Nixon to the soft, materialistic ethos that had supposedly enfeebled the nation. Referring to Nixon's famous \"kitchen debate\" with Soviet premier Nikita Khrushchev in a model house at the 1959 American National Exhibition in Moscow, Kennedy accused his opponent of having told the Soviet leader \"that while we might be behind in space, we were certainly ahead in color television.\" The vice president \"may be very experienced in kitchen debates,\" Kennedy jibed; \"so are a great many other married men I know.\" But \"I would rather take my television black and white and have the largest rockets in the world.\"\n\nThe virtues of a hard and strenuous life, and their importance to America's cold war struggle, were a central Kennedy theme. In December 1960, he penned an article for Sports Illustrated titled \"The Soft American,\" in which he pointed to a recent study revealing that young Americans trailed Europeans in tests of flexibility and strength and warned that \"our increasing lack of physical fitness is a menace to our security.\" In an era of \"pre-cooked meals and prefab\" houses, Kennedy wrote that same year, Americans were losing \"that old fashioned Spartan devotion.... We stick to the orthodox, to the easy way and the organization man.\" Some wondered, he told an audience at Rice University, why America must beat the Soviets to the moon. \"Why climb the highest mountain?\" he responded, \"[W]hy does Rice play Texas?...because that goal will serve to organize and measure the best of our energies and abilities.\"\n\nKennedy's mania for personal and ideological toughness represented, in a sense, Theodore Roosevelt's final, posthumous victory over Woodrow Wilson. In 1917, when Wilson launched his crusade to tame world affairs, Roosevelt's dissenting view\u2014that seeing the world as a jungle would improve national character\u2014had been a minority strain. But during World War II, cousin Franklin had embraced much of TR's vision, even as he spoke Wilsonian words. And now JFK, in his obsession with vigor, was echoing TR with uncanny precision. Roosevelt had ordered that all U.S. Marines be able to march fifty miles in under twenty hours; when Kennedy learned of this, he renewed the order and publicly extended it to the members of his cabinet. Roosevelt encouraged reporters to cover him hunting and hiking but refused to let them watch him play tennis, which he considered effete. For his part, Kennedy encouraged journalists to report that he played touch football (so long as they explained that in the Kennedy clan, touch football was quite savage), but he avoided being photographed playing golf, which was Ike's soft, grandfatherly game. On one occasion, Jack even screamed at a New York Herald Tribune reporter who had revealed that the reason he campaigned without an overcoat in New Hampshire and Wisconsin in winter was not that he possessed the kind of frontier grit that made him impervious to cold, but, rather, that he wore thermal underwear.\n\nVigor, that quintessentially Kennedy word, connoted not only toughness but youth. And when Kennedy spoke in generational terms, which was often, he emphasized that while he and his contemporaries were idealistic, they were not innocent. They were, he noted in his inaugural address, \"tempered by war, disciplined by a hard and bitter peace.\" Unlike Wilson and the progressives, they knew that in the world, as in the heart, evil never died.\n\nThe Kennedy men, like the intellectuals and statesmen of the 1940s, saw themselves as coming of age in a world where innocence was no longer possible. (Kennedy's national security advisor, McGeorge Bundy, titled a 1963 foreign policy essay \"From Innocence to Engagement.\") In that way, toughness was central to the self-conceptions of both groups. But what they meant by it was different. In the '40s, moral toughness had meant recognizing that the world was inescapably tragic yet could not be escaped. It meant lowering one's expectations about what reason could accomplish without succumbing to despair. The ethos was defensive: The world must be engaged not so the children of light could transform it in their image, but to ensure that the children of darkness did not, either. For Kennedy, Bundy, and Schlesinger in the early '60s, by contrast, toughness connoted something more aggressive, something more like a crusade. It meant demanding more of oneself and one's nation, paying any price, bearing any burden, meeting any hardship, and thus achieving the impossible. As Schlesinger wrote about the early days of the Kennedy administration, \"We thought for a moment that the world was plastic and the future unlimited.\"\n\nFor the older men, World War II was the last in a sequence of brutal blows. For the younger generation, it was the beginning of the nation's astonishing rise. In the '40s, toughness meant living with limits; by the '60s it meant transcending them. Kennan, Morgenthau, Lippmann, and Niebuhr were realists. The Kennedy men, by contrast, were what the radical sociologist C. Wright Mills called \"crackpot realists\" and David Halberstam called \"ultra-realists.\" They started with the reasonable insight that evil could never be eradicated and that the currency of world affairs would always be power\u2014and forgot that there were limits to these truths as well.\n\nThe men of Camelot were fond of comparing the 1950s to the '30s, with the implication that they too had come to power after a period of innocence, inaction, and mounting danger. Kennedy was obsessed with Churchill: He bestowed honorary American citizenship on the former British prime minister, claimed to have read all his books, and asked to hang one of his paintings in the White House. But Churchill and FDR had roused their nations for defensive action in a moment of genuine and frightening weakness. America in the '50s, on the other hand, was the opposite of weak. Even with Eisenhower's cuts, the NSC 68 buildup had left the Soviets\u2014with their much smaller and less efficient economy\u2014far behind. What the Kennedy men called appeasement was simply the cautious use of awesome strength. And what they called toughness in the face of aggression was something close to aggression itself.\n\n\"In the long history of the world,\" Kennedy declared in his inaugural address, \"only a few generations have been granted the role of defending freedom in its hour of maximum danger. I do not shrink from this responsibility\u2014I welcome it.\" But 1961, unlike 1938, was not freedom's hour of maximum danger. And claiming that it was smacked of generational envy. The Kennedy men, who had found the '50s boring and vaguely effeminate, wanted epic challenges. That's why they took fifty-mile hikes; it's why they wanted to go to the moon; it's why Rice played Texas. They thought such struggles would have a salutary effect on the national character, and the cold war, of course, was the greatest struggle of all. But Churchill and FDR had not waged war against Hitler because they feared their people were succumbing to a life of ease. It was ease that their Depression-wearied people desperately wanted, but came to accept that they could not have. That was what made the wartime generation's struggle both tragic and heroic. Kennedy's, by contrast, was a blend of tragedy and farce.\n\n## CHAPTER SEVEN\n\n## SAVING SARKHAN\n\nIn 1958, a novel about American foreign policy stunned the publishing world. It sat on bestseller lists for seventy-eight weeks and sold almost five million copies. Hollywood turned it into a movie starring Marlon Brando. Critics compared its impact to Uncle Tom's Cabin and The Jungle. By one estimate, it inspired twenty-one bills in Congress.\n\nThe novel was The Ugly American. It is set in the fictional Southeast Asian nation of Sarkhan, which American diplomats are trying to save from falling into communist hands. Unfortunately, most of the Americans in Sarkhan are \"fat,\" \"ignorant\" organization men, more interested in gossiping at embassy cocktail parties than wading through remote swamps to battle an elusive and deadly foe. Sarkhan's only hope lies with the different breed of American embodied by Tex Wolchek, a \"tall and muscular\" army paratrooper intimately familiar with writings of Mao Zedong, and Colonel Edwin Hillandale, a charismatic, harmonica-playing expert on Southeast Asian languages and cultures who understands that the key to winning hearts and minds in Sarkhan is understanding that the natives are obsessed with astrology and palm-reading.\n\nJohn F. Kennedy loved it. In 1959, he sent the novel to all his colleagues in the Senate. As president, he encouraged its co-author, William Lederer, to send him memos on counterinsurgency. He tried to appoint Major General Edward Lansdale\u2014the man on whom Hillandale was based\u2014as his ambassador to South Vietnam. When the State Department resisted, he made him an adviser on Cuba instead.\n\nWhat appealed to Kennedy about The Ugly American was not just its theme\u2014the intersection of personal toughness and anticommunism\u2014but its setting: the third world. In 1951, as a thirty-four-year-old congressman, he had traveled with his brother Robert and sister Pat for seven weeks through India, Pakistan, Israel, Iran, Singapore, South Korea, Japan, Thailand, and Vietnam. He came back moved by the poverty he saw, and by the depth of nationalist passion in countries long bridled by colonialism. \"If one thing was bored into me as a result of my experience in the Middle as well as the Far East,\" he declared upon his return, \"it is that Communism cannot be met effectively by merely the force of arms. The central core of our Middle Eastern policy is (or should be) not the export of arms or the show of armed might but the export of ideas, of techniques, and the rebirth of our traditional sympathy for and understanding of the desires of men to be free.\" Kennedy returned to this anti-imperialist theme repeatedly in his congressional career. In 1957, he gave a speech so critical of France's colonial war in Algeria that it elicited an angry visit from the French ambassador. But Kennedy stood his ground, keeping the ambassador waiting and purposely serving him an awful lunch.\n\nThe trip, and what he learned from it, spoke well of Kennedy. Like Eisenhower in 1952 trudging through the Korean snow, he had gone to see for himself. And in so doing he had partially freed himself from the intellectual shackles of global containment. At ground level, he had sensed the absurdity of trying to force Asia's left-wing, anticolonial movements onto Nazism's Procrustean bed. He had learned what Kennan learned in Moscow and Davies learned in Chongqing: that up close, a grand abstraction like \"communism\" explains far less than it appears to in Washington, D.C.\n\nBut Kennedy was also a child of Munich and McCarthyism. In his formative years, he had seen appeasement fail, containment succeed, and men destroyed for acknowledging that it couldn't succeed everywhere. So he tried to keep his heresies in check, to reconcile what he had seen in Asia with what his constituents read about in Sarkhan. In this regard, The Ugly American proved inspiring. Hillandale, after all, is no armchair anticommunist; he speaks the local language and cares about the local people. But all this knowledge and empathy is also a means to an end: crushing the communists. For Kennedy, Hillandale squared the circle between reality and ideology, between the need to understand and the need to be tough. Unfortunately, Hillandale wasn't a real person and Sarkhan wasn't a real place. Vietnam was.\n\nIn the late 1950s, with scores of Asian and African nations on the cusp of independence, JFK wasn't alone in his fascination with the developing world. It was widely assumed, among the restless toughness intellectuals, that their generation's greatest test\u2014their World War II\u2014would come in what Kennedy called, with a touch of romance, \"the lands of the rising people.\" Politics may have grown boring in the West, argued Daniel Bell in The End of Ideology, but \"the trajectory of enthusiasm has curved East, where, in the new ecstasies for economic utopia, the 'future' is all that counts.\" In economics and political science departments across the United States, modernization theorists began studying how to export capitalism to the poor world so the masses of Africa, Asia, and Latin America would see that Marxism was not the true path to prosperity. \"The Communist bid to win Asia by demonstrating rapid industrialization is already launched,\" announced one such theorist, an ambitious MIT professor named Walt Rostow, in 1957. \"It behooves the Free World and especially the United States to decide promptly whether it is to observe or participate in this struggle on which so much of our destiny hinges.\" Observing, of course, was what Ike did. Participating, shaping, mastering\u2014that was the Kennedy way. When he became president, Kennedy made Rostow his deputy national security advisor.\n\nAs if to underscore Rostow's point, two weeks before Kennedy took office, Nikita Khrushchev gave a speech pledging Soviet support for \"wars of liberation\" across the third world. Kennedy opened his first National Security Council meeting by quoting the Soviet premier's words. The speech, he told a journalist, was \"Khrushchev's Mein Kampf. He tells us what he's going to do.\" It was as if the Soviet leader had sounded an opening bell.\n\nAmerica's diplomatic corps and its regular army, Kennedy feared, were too slow, timid, and dumb for this epic contest; he needed Hillandales. He established mandatory counterinsurgency classes at the army war colleges and the State Department. High-level diplomats were summoned to a special \"Interdepartmental Seminar\" taught by Lansdale, Rostow, and others, where they read Mao and Che Guevara. Kennedy himself wrote the introduction to a book on counterinsurgency. And even volunteers for the newly created Peace Corps, whose work digging wells and building schools across the third world initially struck Kennedy as a bit soft, were imbued with the Hillandale spirit. Recruits were sent to jungle boot camps, where they awoke at 5 A.M. for calisthenics and a day of obstacle courses and grueling runs. At one Peace Corps camp in Puerto Rico, young do-gooders were taught to rappel down the face of a dam, and thrown, hands and feet bound, into a river to learn how not to drown.\n\nKennedy approved, but his real favorites\u2014the men whom the Peace Corps civilians were trying to imitate\u2014were the Special Forces. Fluent in obscure languages, familiar with the work of both Mao and Rostow, able to descend deep into the jungle for weeks at a time with neither food nor maps, they terrorized the regular army in mock battles. Generals would awake in the morning to find their bedsheets scrawled with the words \"You are dead,\" and stickers notifying them that their jeeps had been blown up. Kennedy upped the Special Forces' budget fivefold and personally selected their equipment, which included sneakers with steel soles to protect against jungle booby traps. They were the only members of the U.S. military permitted to wear berets. In October 1961, Kennedy took the White House press corps on a field trip to Fort Bragg, North Carolina, where Special Forces troops staged ambushes and ate snake meat. Skydivers jumped from planes; frogmen swam from submarines. One soldier hoisted a rocket on his back, zoomed over a lake, and landed at the proud president's feet.\n\nFor Kennedy, the first round in this new cold war struggle came in Cuba, where on January 8, 1959, a thirty-two-year-old ex-pitcher named Fidel Castro\u2014having turned to revolution after failing to make the Pittsburgh Pirates\u2014had marched his guerrilla army into Havana, overthrowing the American-backed dictatorship of Fulgencio Batista. Initially, Castro's anti-imperialist rhetoric elicited Kennedy's sympathy. But when the Cuban leader's leftist proclivities became clear, that sympathy quickly dissipated. The NSC 68 rules\u2014that America must stop any new scrap of territory from being painted red (especially one within spitting distance of Miami)\u2014had been transgressed. And Kennedy, who had spent the campaign promising that in the fight against third-world communism he would get America off the couch and into the game, could not sit idly by.\n\nThe CIA, as it happened, had a plan to set things right. Buoyed by its success overthrowing leftists in Iran and Guatemala, the Agency had won Ike's permission to organize something similar in Cuba. Soon after Kennedy's inauguration, CIA deputy director Richard Bissell informed him that the Agency was training anti-Castro exiles in Guatemala for an invasion of Cuba's southern coast. Bissell and his boss, CIA director Allen Dulles, claimed that Castro's support was soft, both among the Cuban public and within the Cuban military. (The Agency's own Cuba analysts disagreed, but were ignored.) If Kennedy moved quickly, Dulles and Bissell promised, he could spark an uprising that, at worst, would enmesh Castro in a civil war. If he did nothing, by contrast, Moscow would help Castro consolidate control and foment Marxist revolution across the Americas.\n\nThe proposal pitted Kennedy's anti-imperialist instincts against his anticommunist ones. He was eager to launch the Alliance for Progress, an aid program aimed at putting the United States on the side of nationalist aspiration in the Americas, and he feared that invading Cuba would sabotage that effort. But while he had misgivings about the operation, he believed in the CIA. Smart, unconventional, tough\u2014they were his kind of people. He particularly liked Bissell, a forceful, brainy alumnus of Groton and Yale who at a well-lubricated, late-night CIA dinner at Washington's Alibi Club had delighted the president by declaring himself a \"man-eating shark.\" Bissell might not be Hillandale, but he reminded Kennedy of another favorite fictional character, James Bond, whose daring, elegance, anticommunism, and cost-free sexual promiscuity offered the president an idealized image of himself. (At a Georgetown dinner in 1960, Kennedy had asked Bond's creator, the British spy novelist Ian Fleming, how to handle Castro. Fleming suggested spreading rumors that Castro was impotent, which faintly echoed reality, since Schlesinger dubbed the CIA's invasion plan \"Operation Castration.\")\n\nSchlesinger, to his credit, opposed the Cuba operation, as did some in the State Department; they feared the public relations fallout and didn't understand the rush. But Kennedy looked down on the bureaucrats at Foggy Bottom; they were organization men, cautious, unimaginative, and slow. \"They're not queer at State,\" he told a friend, \"but, well, they're sort of like Adlai [Stevenson]...[I]f I need material fast or an idea fast, CIA is the place I have to go.\" The difference between Eisenhower's foreign policy and Kennedy's, Rostow had explained, would be \"a shift from defensive reaction to initiative.\" Now Kennedy had to choose between an audacious scheme run by an elite band or a policy of paper-pushing and wait-and-see. It was no contest.\n\nThe irony was that it would have required greater bureaucratic initiative, and greater political courage (a very different thing than cold war machismo), to rein containment in, as Eisenhower had done at Dien Bien Phu. But Kennedy was in a more vulnerable position than Ike: He didn't have his predecessor's experience running large, treacherous bureaucracies, and he didn't have the word General in front of his name. Near the end of a briefing on the invasion plan, Bissell looked at him and said, \"Don't forget one thing.... If we take these men out of Guatemala, we will have to transfer them to the United States, and we can't have them wandering around the country telling everyone what they have been doing.\" The subtext was clear: Once Republicans learned that Kennedy had quashed a plan to overthrow Castro, a plan Eisenhower had put into motion, he would face a firestorm on his right. (And the CIA, angry that its operation had been scrapped, and led by the brother of Ike's secretary of state, would discreetly fan the flames.) Dropping the invasion, Kennedy aide Kenny O'Donnell later explained, would have made the president look like an \"appeaser of Castro.\"\n\nThat fear haunted Kennedy's advisers as well. At the State Department, Undersecretary Chester Bowles warned that the department's Latin America experts put the invasion's chance of success at only one in three, and he urged his boss, Secretary of State Dean Rusk, to try to dissuade Kennedy. But Rusk had spent his entire life ensuring that he was never caught on the wrong side of the appeasement charge. So poor as a child that he sometimes attended school without shoes, and so patriotic that he enlisted in ROTC not at the beginning of college, but at the beginning of high school, he had\u2014rather miraculously\u2014become a Rhodes scholar. And as a result, he had been in the audience in 1933 when the fashionable aristocrats of the Oxford Union declared that if war came to Europe again they would not fight for King and Country. It was an act that Rusk never forgot nor forgave. As Truman's assistant secretary of state for Far Eastern affairs, he had urged going north of the 38th parallel and helped organize the first American military aid to Vietnam. As a result, during the height of the red scare, he had come out of the Asia bureau unscathed, even as experts like John Paton Davies were destroyed. The reason was that Rusk wasn't really an Asia expert at all; he was a Munich expert.\n\nRusk's whole life, in other words, had taught him not to do what Bowles was urging. Privately, he did fear that the invasion was ill conceived. But he kept those worries to himself. And so did almost everyone else. \"Nobody in the White House wanted to be soft,\" one aide later recalled. \"Everybody wanted to show they were just as daring and bold as everybody else.\"\n\nSo Kennedy told the CIA to go ahead with the plan, so long as it kept America's fingerprints off it\u2014wishful thinking, given that news of the exile training program had already shown up in the New York Times. On the evening of Saturday, April 14, fifteen hundred Cuban exiles loaded onto five freighters at a secret U.S. base in Puerto Cabezas, Nicaragua, bound for Cuba's southern coast. It would be, the historian Theodore Draper later wrote, \"one of those rare events in history\u2014a perfect failure.\" Castro, who knew an invasion was in the works, had arrested many of the people who might have supported one. And even if he hadn't, few Cubans were interested in joining an exile force composed of loyalists from the old regime. To make matters worse, the area the CIA had chosen for the exiles' landing, the Bay of Pigs, was a Castro stronghold, a place he often went to fish.\n\nA preparatory strike aimed at disabling Cuba's small air force failed, so Castro's pilots sunk the boats carrying most of the exiles' ammunition and communications equipment before they even reached the shore. Once they did, the Cuban army was there in force to greet them. Bissell had assured Kennedy that even in the worst case scenario, the exiles could disappear into the nearby Escambray Mountains and launch a guerrilla war from there. But he had neglected to mention that between the beaches and the hills lay eighty miles of swamp.\n\nFor a day and a half, the rebels radioed frantically to their CIA trainers for air support. And in turn, the CIA and the military beseeched the president. But Kennedy\u2014frightened of the international condemnation an all-out U.S. attack would bring\u2014refused. He had, he was beginning to realize, been set up. Dulles and Bissell had never really believed that the Cubans could defeat Castro alone. They had simply wanted to get him on the hook. Once the fighting began, they assumed, he would have no choice but to invade.\n\nThat is certainly what the exiles believed. \"Do not see any friendly air cover as you promised. Need jet support immediately,\" wired their commander at 1:45 P.M. on Tuesday, with his men facing imprisonment or death. And then, after his final plea was denied, this message: \"And you, sir, are a son of a bitch.\" Back in Washington, Kennedy was sitting alone in his bedroom, in tears.\n\nThe Bay of Pigs had a paradoxical impact on Kennedy. Privately, he grew more wary of military action. He now understood the reason for the habitual optimism of the Joint Chiefs and the CIA: If things went awry, they would simply push him to escalate. He fired Dulles and Bissell and began relying more heavily on longtime aides, especially his brother Bobby. But publicly, he kept these anxieties to himself. Precisely because, like Truman on China, he had halted U.S. intervention before it cost significant American lives, he had not forced the foreign policy elite to acknowledge the limits of American power. Hawks could still say, as they did in 1949, that with airpower, military advisers, and guts, America could contain communism everywhere at modest cost.\n\nHerein lay the tragedy of the toughness ethic: Even when presidents realized that global containment was impossible, they feared saying so publicly, and as a result, they perpetuated the political dynamic that held them captive. \"Defeats\" like Mao's takeover in China and the Bay of Pigs did not slow the climb up the ladder toward hubris, because politically they were processed not as battles America had lost but as battles America had chosen not to win. Since World War II, the only thing that had deflated the growing myth of American omnipotence and omniscience was agony, the agony of all those frostbitten American boys suffering and dying in Korea. But by 1961, that memory no longer stung.\n\nSo while Kennedy was privately chastened by the Cuban disaster, his political problem, as he perceived it, was not that he had been too much of a hawk, but too much of a dove. Within hours of the invasion's collapse, Bobby was demanding that the White House take steps to ensure that his brother did not look like a \"paper tiger.\" Two days later, JFK himself warned that \"our restraint is not inexhaustible,\" thus reassuring Americans that although he had quashed an all-out invasion, he might not be so sane the next time. Dulles and Bissell were replaced by men as hard-line as themselves. And the only other official to lose his job over the debacle was none other than Chester Bowles. At the White House, his warnings were seen not as evidence of good judgment but as confirmation of what everyone already suspected: that he was soft. Rusk, by contrast, soldiered on.\n\nIf Cuba was the first round, the second came in a country most Americans had never even heard of: the tiny, sleepy, landlocked nation of Laos. In early 1961, Vietnam was still an afterthought. (When he gave Kennedy a post-election foreign policy briefing, Eisenhower never even brought it up.) It was Laos, in Ike's words, that constituted the \"cork in the bottle\" in Southeast Asia. If it fell to communism, it might take the whole region down with it.\n\nBut for a country of such grave geopolitical import, the civil war in Laos had a distinctly Potemkin feel. The State Department wrote the speeches of the Laotian king. The Royal Lao Army, whose entire budget came from the Pentagon and CIA, showed little interest in fighting. Sometimes it canceled military maneuvers to pick flowers or swim. On one occasion, government and rebel troops skipped a planned battle and together participated in a local water festival. \"My people,\" the king explained apologetically, \"only know how to sing and make love.\" As a \"military ally,\" quipped Kennedy's ambassador to India, John Kenneth Galbraith, \"the entire Laos nation is clearly inferior to a battalion of conscientious objectors from World War I.\"\n\nDespite these realities, or perhaps because of them, Rostow and the Joint Chiefs of Staff wanted to flood Laos with U.S. troops. When Kennedy asked Joint Chiefs chairman Lyman Lemnitzer for a backup plan in case the troops failed to stamp out communism, Lemnitzer exclaimed, \"You start using atomic weapons!\"\n\nKennedy was not going to be sent down a rabbit hole again. Instead he asked his European allies for advice. British prime minister Harold Macmillan and French president Charles de Gaulle suggested that he try cutting a deal with the Soviets. Moscow, it turned out, didn't particularly relish a communist victory in Laos, either, since the rebels were mostly allied with Beijing. So, in May 1961, U.S. Ambassador at Large Averell Harriman and Soviet foreign affairs minister Andrei Gromyko began negotiations that culminated in an agreement guaranteeing Laos's cold war neutrality; it could have military alliances with neither east nor west. With that, Laos quickly fell out of the news, returning to what Galbraith called \"the obscurity that it richly deserves.\"\n\nOnce again Kennedy had acted wisely. And once again wisdom was politically perilous. Rostow was appalled by Kennedy's show of weakness. \"The cease-fire in Laos came as a cold war defeat for the U.S.,\" declared Henry Luce's Time magazine. Because Kennedy could not say publicly what Kennan had\u2014that a country like Laos did America little good as an ally and little harm as a foe\u2014his good judgment there actually made it harder to show the same good judgment elsewhere. He was still climbing up the hubris ladder. Khrushchev, he told an aide, \"must not misunderstand Laos and Cuba as an indication that the United States is in a yielding mood.\"\n\nUnfortunately, that is exactly what Khrushchev seemed to believe. In June, Kennedy went to meet the Soviet premier in Vienna. Confident that he could win over fellow politicians with his intelligence and charm, Kennedy hoped to convince his Russian counterpart to avoid aggressive moves, especially in the third world. But if Kennedy wielded power like a stiletto, Khrushchev wielded it like a club. So uneducated that he could barely sign his name, Khrushchev had a primitive understanding of American politics. (He assumed that since New York governor Nelson Rockefeller hailed from a family of famous capitalists, he must be the real power in Washington.) Already inclined to suspect that a young, inexperienced man like Kennedy was too weak to control the plutocrats who really ran American foreign policy, Khrushchev had felt his suspicions vindicated by the Bay of Pigs. By bullying the young leader, he hoped to win concessions, especially on Germany, which he cared about most.\n\nSo every time Kennedy said something gracious, Khrushchev said something brutal. Hour after hour, the Soviet leader lectured, hectored, and belittled, taking every Kennedy courtesy as an invitation for another verbal assault. Kennedy, who had expected a parlor discussion, found himself\u2014unprepared\u2014in a barroom brawl. And if the insults and bravado weren't enough, Khrushchev closed the summit with a threat: He would shut off the West's access to Berlin, the divided city located deep inside communist East Germany. If the United States resisted, there would be war.\n\nIn reality, Khrushchev's bluster stemmed more from weakness than strength. East Germany, the most economically advanced of the Soviet client states, supposedly a shining model of socialist development, was being slowly depopulated. In the summer of 1961 alone, more than twenty thousand East Berliners fled to the capitalist West, where per capita income was more than twice as high and consumers didn't have to wait two years to buy a refrigerator. If he did nothing, the Soviet leader feared, East Germany might collapse, thus creating the ultimate Russian nightmare: another unified, militarized Germany, this time backed by the United States. But the Munich paradigm predisposed America's leaders to see the Kremlin as strong, not weak, as zealously seeking new conquests rather than nervously trying to hold on to what it already had. And so for Kennedy, the obvious explanation for Khrushchev's belligerence was that after Cuba and Laos, the Soviet leader thought he was soft.\n\nAfter their final meeting, Kennedy slumped on a couch in a darkened hotel room, a hat pulled over his eyes. \"He savaged me,\" he told James Reston of the New York Times, who had scored an exclusive interview. \"He thinks because of the Bay of Pigs that I'm inexperienced...thinks I'm stupid. Maybe most important, he thinks that I had no guts.\" Reston thought the president was in shock. In London the next day, en route back to the United States, an emotional Kennedy told the hawkish columnist Joseph Alsop, \"I just want you to know, Joe, I don't care what happens. I won't give way. I won't give up, and I'll do whatever's necessary. I will never back down, never, never, never.\" Back in Washington, Rostow was composing a memo to the president. The spring and summer of 1961, he wrote, were analogous to 1942, when the Allies suffered setback after setback. But just as Churchill and Roosevelt had turned the tide, so would they. The place to start was Vietnam.\n\nKennedy had some experience with Vietnam. He had gone there in 1951, as part of his trek with Bobby and Pat across Asia. And he had found a country at war. France, which had lost its Southeast Asian colonies to Japan during World War II, had managed to regain them afterward, thus boosting its war-wounded pride. But French control was more apparent than real. Under Japanese occupation, Ho Chi Minh's Vietminh had developed into a powerful force, and for a brief period after Tokyo's defeat even governed the country. When Paris tried to reassert colonial rule, Ho's troops launched a guerrilla war. By the time Kennedy arrived, they controlled two-thirds of the countryside. With France buckling under the war's economic strain, the United States\u2014which saw the conflict less as an anticolonial struggle than as an anticommunist one\u2014was increasingly paying the bills.\n\nFrench colonialism left a poor impression on the young congressman. He found French officials obstinate and crudely racist, far inferior, from his Anglophile perspective, to their British counterparts in India, who had granted independence before the real ugliness began. In keeping with his generally anticolonial bent, Kennedy in 1953 introduced Senate legislation conditioning U.S. aid for the French effort on Paris's willingness to grant its Southeast Asian colonies independence. Unless Vietnam developed a sovereign, anticommunist \"native army\" free from French control, he insisted, Indochina was lost.\n\nThen, in 1954, one seemed to magically materialize. Rocked by their devastating loss to the Vietminh at Dien Bien Phu, the French agreed to grant Vietnam independence. But to deny Ho the full fruits of victory, Vietnam was to be temporarily partitioned. North of the 17th parallel, where the Vietminh enjoyed overwhelming control, would be a communist state. But in the South, where the French enjoyed greater influence, a pro-Western government would be established. In 1956, internationally supervised elections would unite the country once again.\n\nFor the young John Kennedy, Vietnam's partition was an intellectual trap. He badly wanted to believe that America could fight Vietnamese communism without becoming colonialism's heir. And on paper, the newly independent nation of South Vietnam enabled him to believe exactly that. It had an army, a flag, and a devoutly Catholic, avowedly anticommunist leader named Ngo Dinh Diem. Kennedy liked what he saw and in 1956 helped found the American Friends of Vietnam. South Vietnam, he declared, summoning all his best cold war metaphors, \"represents the cornerstone of the Free World in Southeast Asia, the keystone in the arch, the finger in the dike.\"\n\nBut it was a mirage: There was no South Vietnam. The Vietminh, by far the most powerful force in the country, had only accepted temporary partition because they were promised elections to reunify the entire nation, elections they (and everyone else) assumed Ho would win. South Vietnam, by contrast, was led by men who had either sat out their country's anti-imperial struggle or opposed it. In Washington, their titles and crisply pressed uniforms appeared impressive. But back home they were just the same colonial flunkies serving a new foreign master. \"Like water turning into ice,\" David Halberstam would later write, \"the illusion crystallized and became a reality, not because that which existed in South Vietnam was real, but because it became real in powerful men's minds.\" And John F. Kennedy, who should have known better, was one of them.\n\nIn the fall of 1961, with the Kennedy administration hungry for a win after Cuba, Laos, and Vienna, Vietnam forced its way onto the White House agenda. Two years earlier, having abandoned hope that national elections would ever be held, and after watching Diem ruthlessly suppress Vietminh supporters in the South, Ho's government in Hanoi began aiding the Vietcong, a southern insurgency dedicated to reunifying the country by force. The southern countryside soon erupted in violence, and Diem responded by forcing peasants from their guerrilla-infested villages, which alienated them further. By September 1961, the Vietcong were launching ever more brazen attacks, even briefly seizing a provincial capital fifty-five miles from the southern capital, Saigon. Alarmed, the Joint Chiefs proposed dispatching a large contingent of U.S. troops. But Kennedy distrusted their advice. So he dispatched Rostow and his special military advisor, General Maxwell Taylor, to travel to South Vietnam to investigate.\n\nThe choice of fact-finders predetermined the facts they found. If Kennedy, Rusk, and National Security Advisor McGeorge Bundy thought victory in Vietnam was important, Rostow went further: He believed it was virtually preordained by history itself. Rostow, the son of left-wing Jewish immigrants who in homage to their adopted land had named their three sons Walt Whitman, Ralph Waldo, and Eugene Victor (for Socialist leader Eugene Victor Debs), was socially insecure; colleagues in Cambridge and Washington often thought he was trying just a little too hard to fit in. (This was easy for people who graduated from Groton to say about someone who graduated from New Haven's Hillhouse High.) But what Rostow lacked in social self-confidence he more than made up for in intellectual self-confidence. He had entered Yale at fifteen, had won a Rhodes scholarship four years later, and by twenty-nine had become the youngest person ever offered a full professorship at Harvard. In his masterwork, The Stages of Economic Growth: A Non-Communist Manifesto, he set out to disprove Marx. The great philosopher was correct to see history as moving inexorably in one direction, Rostow conceded, but he had gotten the direction wrong. It was actually moving toward American-style capitalism. For older intellectuals like Niebuhr, accepting that capitalist America was the best humanity could do had been a sobering, even tragic realization, since it fell so far short of their earlier utopian dreams. But for Rostow, America was utopia. We had solved the big problems\u2014we had reached the end of ideology\u2014and our answers were universal. Since American society was \"now within sight of solutions to the range of issues which have dominated its political life since 1865,\" explained Rostow, domestic policy was \"becoming a bore.\" But helping third-world countries stuck far down on history's ladder was exciting as hell. Like Walter Lippmann and John Dewey in 1917, Rostow believed that he had cracked history's code, which made him serenely confident that\u2014with his guidance\u2014America could win in Vietnam.\n\nRostow's approach to foreign policy was the mirror image of Kennan's. Kennan had started with an intimate knowledge of a particular country at a particular time and then crafted a theory around it. Rostow, by contrast, started with a theory that was applicable always and everywhere and then grafted it onto a specific time and place. As a result, he was better at seeing forests than trees. Returning once from a trip to Latin America, he astonished colleagues by declaring that the key to understanding Hispanics was to realize that they were really Asians. \"Oh, for God's sake, Walt,\" someone replied, \"why are you talking about something you know nothing about?\"\n\nMaxwell Taylor also had too much invested in South Vietnam to see it for what it really was (or was not). Even by Kennedy administration standards, he cut an imposing figure. Movie-star handsome, he spoke French, Spanish, Chinese, German, and Japanese, and was given to quoting Thucydides in the original Greek. The New York Times called him a cross \"between Virgil and Clausewitz.\" During World War II, he had sneaked behind enemy lines to scout out locations for a potential American attack. Since the war, he had run the U.S. Military Academy at West Point and then, in a dazzling cultural shift, the Lincoln Center for the Performing Arts.\n\nHe had come to Kennedy's attention after leaving the Eisenhower administration and then slamming its overdependence on nuclear weapons, in a book called The Uncertain Trumpet. Ike had threatened nuclear war because he believed that small wars, once unleashed, could not be effectively controlled. But Taylor believed they could: The Uncertain Trumpet was a primer on how to fight them, and win. \"We can trigger near total destruction,\" declared its book jacket. \"But can we defend Berlin-South Korea-Vietnam-Iran-Thailand-America?\" Now Taylor was Kennedy's Special Military Advisor, and his job as he saw it was to answer that question yes.\n\nIn their report, Rostow and Taylor conceded that South Vietnam was suffering a \"deep and pervasive crisis of confidence.\" But rather than linking that crisis to the artificiality of the state itself, they attributed it to the Vietcong's recent gains and to America's neutrality deal in Laos. The implication was clear: Kennedy's soft line in Laos had destabilized the region, and only a hard line next door could stanch the bleeding. Rostow and Taylor recommended sending eight thousand U.S. combat troops. Officially, their mission would be to help repair the devastation from a recent flood in the Mekong Delta. But in reality they would be there to buck up the South Vietnamese and scare the North into stopping the war.\n\nKennedy hesitated. He suspected that if he sent the troops and the North's aggression did not cease, he would soon face pressure to send more. \"It's like taking a drink,\" he told Schlesinger. \"The effect wears off, and you have to take another.\" But he was determined to hold South Vietnam. It had become, said Bundy, \"a sort of touchstone of our will.\" So as he had on Cuba, Kennedy split the difference, refusing to send combat troops but substantially boosting the number of U.S. advisers and the amount of aid. As a model, he cited Truman's aid to Greece, which had halted communism's advance with U.S. money but without U.S. soldiers. It was a bad analogy: Greece was a real country; South Vietnam was not. But it was one of a number of comforting analogies that the Kennedy administration had inherited from almost two decades of foreign policy success. And neither Rostow nor Taylor nor Kennedy himself knew enough about Vietnam to grasp how poorly it applied.\n\nBy 1962, Kennedy was growing more confident. On Berlin, Khrushchev had backed down, abandoning his threat to block Western access to the city, and instead building a wall halfway through it, to keep the East Germans in. And if Kennedy's Berlin victory helped erase the humiliation at Vienna, he soon won an even bigger victory, which helped erase the humiliation of the Bay of Pigs.\n\nThat aborted invasion hadn't spelled the end of U.S. efforts to topple Castro. Kennedy, who had himself attacked Nixon for \"losing\" Cuba to the communists, knew that if it stayed lost he could be vulnerable, too. So in late 1961 he put Edward Lansdale, of Ugly American fame, in charge of a secret effort to overthrow the Cuban leader. The effort was dubbed Operation Mongoose, since Lansdale, who owned two of the catlike carnivores, insisted that they excelled at killing dangerous snakes.\n\nOnce again Castro was a step ahead. He asked the Soviets for help deterring another U.S. attack, and Khrushchev, angry that America had installed nuclear missiles near his border in Turkey (and worried that Castro might shift his allegiance to Beijing), decided to return the favor by putting nukes near America's shores. At first, evidence of the missile deployments in Cuba was murky, and Kennedy downplayed the rumored buildup. But as the summer of 1962 turned to fall, his GOP opponents\u2014led by New York Senator Kenneth Keating\u2014seized on the rumors and denounced the White House as weak for standing idly by. Then, at 8:45 A.M. on Tuesday, October 16, McGeorge Bundy walked into the president's bedroom and handed the bathrobe-clad commander in chief photos proving that the buildup was real. \"Ken Keating,\" Kennedy muttered, \"will probably be the next President of the United States.\"\n\nAt first Kennedy was not sure why the Cuban missiles posed such a grave threat. (\"What difference does it make?\" he mused. \"They've got enough to blow us up now anyway.\") But when he met with his advisers, a consensus quickly formed that the missiles were intolerable. Whether or not they actually threatened America, Kennedy later explained, they \"would have appeared to, and appearances contribute to reality.\" Whether this hoary old chestnut, enshrined by NSC 68, was really true\u2014whether concessions in one part of the world really did embolden the communists elsewhere\u2014was far from obvious. (In fact, a mountain of political science literature would later cast doubt on the claim.) But to men who had seen Hitler exploit every British and French concession, it made intuitive sense. And besides, even if weakness didn't embolden the communists; it would certainly embolden the GOP. Allowing Castro not merely to survive but to obtain missiles able to incinerate much of America, Bobby declared, would get his brother impeached.\n\nSeveral advisers proposed a sneak attack while America still enjoyed the element of surprise. But Kennedy, rightly terrified that an attack might spark World War III, decided to start with an embargo instead. \"The 1930s taught us a clear lesson,\" he told the nation on Monday, October 22: \"aggressive conduct, if allowed to go unchecked and unchallenged, ultimately leads to war.\" The embargo worked. Ships carrying Soviet arms were met before reaching Cuba by American vessels (including a 2,200-ton destroyer named for Joe Kennedy, Jr.), and rather than risk Armageddon, the Soviet ships turned around. But on the island, construction continued. For all Kennedy knew, the Soviets already had everything they needed to install and arm the missiles, even if another ship never got through.\n\nBy Sunday, October 28, the Kennedy administration was thinking seriously again about air strikes, to be launched before dawn on Tuesday morning. White House officials were handed envelopes marked \"To Be Opened in Emergency,\" which explained that in the event of nuclear war they would be ferried by helicopter to shelters in Maryland's Catoctin Mountains. (Each official was allowed to save one secretary.) Privately, Kennedy put the chance of war at \"somewhere between one and three and even.\" Then, that morning, Khrushchev cut a deal, agreeing to dismantle the missiles in return for a U.S. promise not to invade Cuba and a secret pledge to withdraw the missiles in Turkey.\n\nOn the surface, the outcome confirmed the toughness paradigm. \"If we have learned anything from this experience,\" declared the minutes of an October 29 National Security Council meeting, \"it is that weakness, even only apparent weakness, invites Soviet transgression. At the same time, firmness in the last analysis will force the Soviets to back away from rash initiatives.\" As many of Kennedy's advisers saw it, the missile crisis was his equivalent of Truman's aid to Greece or the airlift in Berlin, his very own anti-Munich. And like Truman, he reaped the rewards. By the end of October, Kennedy's approval rating was eleven points higher than it had been in August, when Republicans were slamming him as weak. That fall, Democrats broke the historic pattern for a president's first midterm elections and gained seats in the Senate.\n\nBut Kennedy knew\u2014even more vividly than before\u2014that the toughness ethic that he publicly upheld was at odds with reality. In truth, it wasn't only Khrushchev who had blinked. By promising not to invade Cuba and to withdraw U.S. missiles from Turkey, Kennedy had made two key concessions of his own. He just hadn't forthrightly explained them to the American people. Soon after the crisis, Kennedy brought in Schlesinger, who served as the White House's court historian, and told him that commentators were drawing dangerous conclusions from the missile crisis. Khrushchev, the president insisted, had backed down partly because the United States had met him halfway and partly because he didn't consider the Cuban nukes crucial to Soviet survival. \"It was because of these factors that our policy worked,\" wrote Schlesinger in his notes, \"not just because we were tough.... He [Kennedy] worried that people would take the wrong lessons away from the crisis.\"\n\nHad he known the truth, which was that Castro furiously opposed Khrushchev's concession because he believed the nukes were crucial to his survival, Kennedy might have taken his private heresies even further. It was one thing to expect Moscow to back down in showdowns far from home. It was another to expect indigenous communist movements to do so when they were fighting on their own soil. A generation earlier, Kennan had insisted on exactly this distinction\u2014between Soviet interests and the interests of local communist movements\u2014but by the early 1960s he had been in intellectual exile for a decade. And so when Kennedy began venturing, tentatively, away from the logic of global containment, he found few influential guides and few well-lit conceptual paths. When it came to the cold war, his top advisers, for all their surface brilliance, were intellectually conventional and timid. And they knew very little about the third world. Secretary of Defense Robert McNamara, in fact, was already thinking about how to apply the lessons of the Cuban Missile Crisis\u2014which were, of course, the same old lessons of Munich\u2014to another place where he believed communist aggression needed a firm American reply: Vietnam.\n\nConditions there were going from bad to worse. The new American aid, which Kennedy dispatched after the Rostow-Taylor mission, was supposed to come with strings. U.S. officials were unhappy with Diem, a short, squat man who liked to hear himself talk\u2014he could answer a single question with an hour-long reply\u2014and couldn't be bothered to listen. His government either neglected or abused South Vietnam's rural villagers, the overwhelming bulk of the population. He closed down newspapers and sent political opponents to \"reeducation centers.\" And his main criterion for choosing political advisers was blood. At one point the president's three brothers comprised half the South Vietnamese cabinet.\n\nBut when American officials tried to pressure Diem to make reforms, he either ignored them or flew into a rage, and Washington\u2014unable to credibly threaten to abandon Saigon\u2014backed down. Rather than grow more representative, Diem's regime grew more insular, with power increasingly confined to the president; his drug-addict brother, Ngo Dinh Nhu; and his poisonous wife, Madame Nhu, popularly nicknamed the Dragon Lady. Censorship increased, and the regime began requiring official permission for all public gatherings, including weddings and funerals.\n\nSoon this repression blew up in the government's face. On May 8, 1963, in violation of government orders, revelers in the city of Hue flew religious flags in a mass celebration of the Buddha's birth. Diem's troops responded by firing into the crowd, killing two adults and six children. In an overwhelmingly Buddhist country ruled by a small, repressive Catholic clique, that was like throwing a lit match on dry grass. Buddhist priests launched hunger strikes and mass protests. Then, on June 11, as cameras rolled, an elderly monk burned himself alive.\n\nMadame Nhu called the immolation a \"barbecue\" and offered gasoline and matches to anyone who wanted to follow suit. For his part, Diem sent troops to arrest monks by the thousands and ransack their pagodas. He had declared war on the most revered figures in Vietnamese society, and soon, in Halberstam's words, \"a form of madness seemed to take over in Saigon.\" University students walked out of their classes in protest, followed by high school students, and then elementary school students. Back in Washington, Kennedy's whiz-kid advisers strained to understand the crisis. They kept referring to the leader of the Buddhist rebellion as Mr. Bonze, not realizing that bonze was not his actual name; it was the Vietnamese word for \"monk.\"\n\nIn August, South Vietnamese generals asked U.S. ambassador Henry Cabot Lodge, Jr., grandson of the man who had battled Woodrow Wilson over the League, whether Washington would oppose a coup. At first the Kennedy administration was noncommittal. But spooked by rumors that Diem was secretly negotiating a deal with Hanoi to reunify Vietnam and expel the Americans, the White House eventually encouraged the plotters. That even America's handpicked ally was prepared to liquidate South Vietnam should have been a sign that it was not the place to draw an anticommunist line in the sand. But instead, Diem's overtures helped convince U.S. officials that he needed to go.\n\nIn the early afternoon of November 1, renegade troops occupied military bases and communications centers across Saigon and demanded that Diem resign. Within hours they had virtually encircled the government palace. By morning, Diem and his brother had fled through a secret underground tunnel and taken refuge in a Catholic church in the district of Cholon. There they gave confession and took communion before making a deal with the coup leaders for safe passage out of the country. But the deal didn't hold. After being picked up by an armored personnel carrier, they were shot and hacked to death by rebel troops, their bodies deposited in an unmarked grave. Informed of the news during a White House meeting, Kennedy turned pale and fled silently from the room.\n\nBy 1963, John F. Kennedy had become a mildly seditious man. The missile crisis had shaken him, and his apparent victory over Khrushchev had made him less vulnerable to attack from his right. In May, he asked a few White House aides to begin drafting what they called \"the peace speech,\" and not to show it to anyone in Rusk's State Department or McNamara's Pentagon until the last minute.\n\nOn June 10, Kennedy took the podium on commencement day at American University, and gently challenged some of the key assumptions that had governed U.S. foreign policy since NSC 68. First, he described the U.S.S.R. less as bloodthirsty and aggressive than as haunted and afraid. \"No nation in the history of battle ever suffered more than the Soviet Union in the Second World War,\" he declared. \"A third of the nation's territory, including two thirds of its industrial base, was turned into a wasteland.\" This was a far cry from equating the Soviet Union with Nazi Germany in 1938. Kennedy was finally describing Soviet leaders as Kennan had\u2014as leaders of a particular country, guided in large measure by their own historical experiences\u2014rather than as mere vessels for a demonic ideology. Yes, Kennedy acknowledged, Soviet communism sometimes appeared crazed and frightening, but Americans should take care \"not to fall into the same trap...not to see only a distorted and desperate view of the other side.\" Anticommunism, in other words, was only a partial truth; toughness could be taken too far. It may have been only a coincidence, but in the months before the speech, Kennedy had grown close to a new member of the National Security Council staff, thirty-five-year-old Michael Forrestal, whose father's own distorted and desperate anticommunism had taken his life fourteen years before.\n\nIn August, Kennedy and Khrushchev signed a treaty partially banning nuclear tests; it was the first arms control deal of the cold war. And while some Republicans denounced the treaty as appeasement, it easily passed the Senate. When Kennedy mentioned the agreement on a western tour in late September, in fact, he was surprised to find that it elicited loud applause, even in Billings, Montana, and Salt Lake City, Utah. Perhaps the country was changing; perhaps ordinary Americans were no longer stuck in 1950. Perhaps after a big win in 1964, Kennedy might voice even greater heresies.\n\nPerhaps America didn't need to be in Vietnam. \"If I tried to pull out completely now from Vietnam, we would have another Joe McCarthy red scare on our hands, but I can do it after I'm reelected,\" Kennedy told Senate Majority Leader Mike Mansfield. \"We don't have a prayer of staying in Vietnam,\" he insisted to his old friend Charlie Bartlett over drinks. \"Those people hate us. They are going to throw our asses out of there at almost any point. But I can't give up a piece of territory like that to the Communists and then get the American people to reelect me.\"\n\nWas this just talk? It was Kennedy's gift, and perhaps also his curse, to see better than his advisers that toughness had become a conceptual prison. But he was smarter than he was brave. Forced to choose between the risk of losing in Vietnam and the risk of really trying to win, his first choice, as it had been all along, was not to choose. In September, he once again suggested that they send Lansdale to Saigon; maybe he could square the circle. A few weeks later, as the president's motorcade approached the Texas School Book Depository in Dallas, he was shot and killed. The nation grieved, and the hubris of toughness continued to swell.\n\n## CHAPTER EIGHT\n\n## THINGS ARE IN THE SADDLE\n\nJohn F. Kennedy, said French president Charles de Gaulle, was America's mask. Lyndon Johnson was its face. With Kennedy, power was rarely crude. He hid it with elegance, irony, wit. Throughout his political career, he had left the unsightly work to others. Joe Sr. handled the bribes; Bobby leveled the threats. Jack was seldom in the room.\n\nRaw politics\u2014horse-trading, patronage, intimidation\u2014bored and embarrassed him. His grandfathers had played that game on the ghetto streets of Irish Boston. He had been sent to Choate and Harvard to transcend it. And although he had seen more combat than advisers like Bundy, McNamara, and Rostow, he was squeamish about raw force as well. When he thought of those Cuban exiles lying dead on the beach, or of Diem's mutilated body, he became almost physically ill. The icon was Bond: force veiled by style. The bad guys were dispatched in dramatic fashion, but you never actually saw them die.\n\nWith Lyndon Johnson, by contrast, power was naked, corporeal, animal. \"You really felt as if a St. Bernard had licked your face for an hour, had pawed you all over,\" commented the Washington Post's Ben Bradlee. \"He never just shook hands with you. One hand was shaking your hand; the other hand was always someplace else, exploring you, examining you.\" As a twenty-three-year-old congressional assistant, Johnson began his lifelong habit of demanding that subordinates accompany him not only into the bathroom, but into the stall. By the time he reached the Senate, he was urinating openly in his office washbasin during meetings. In mixed company, sometimes he pulled down his pants to scratch his rear end. Fellow members of Congress described encountering him in bathrooms and watching with discomfort as he shook his penis in their direction while commenting on its size. In the words of Robert Caro, \"None of the body parts customarily referred to as private were private when the parts were Lyndon Johnson's.\"\n\nUnlike Kennedy, he had never had the luxury of being above the fray. Lacking money and family connections, he had early on developed an ability to make men serve his interests through sheer force of will. With those more powerful than himself, he employed brazen, epic flattery. With those less powerful, he relied on humiliation. Kennedy outcompeted the people around him; Johnson emasculated them. He would brutalize an aide, berate him in public, reduce him to a whimpering wreck, and then turn around and give him a car or some other lavish gift. As president, he threatened cabinet members that if they resigned he would sic the FBI and IRS on them. Kennedy did not enjoy seeing men abase themselves; it made him uncomfortable. For Johnson, by contrast, it was the beginning of a good working relationship. \"How loyal is that man?\" he asked a White House aide about a potential hire. \"Well, he seems quite loyal, Mr. President,\" the aide replied. To which Johnson erupted, \"I don't want loyalty. I want loyalty. I want him to kiss my ass in Macy's window at high noon and tell me it smells like roses. I want his pecker in my pocket.\"\n\nThere was nothing subtle about the way Lyndon Johnson practiced the politics of toughness. He saw doves as little boys or, worse, as women. Informed that one official was growing squeamish about Vietnam, he shot back, \"Hell, he has to squat to piss.\" Munich, of course, was at the heart of it. \"Lyndon's ideas were set in concrete by World War II,\" wrote a former Senate colleague after he became president. \"Every big action he takes will be determined primarily on the basis [of] wether [sic] he thinks any other action will look like a Munich appeasement.\" In the summer of 1941, he had been a thirty-two-year-old congressmen, deputized by FDR, and by his mentor, Speaker of the House Sam Rayburn, to make sure that legislation instituting a military draft passed. In a country still unreconciled to war, resistance had been fierce. One congressional aide said that \"in forty years on the Hill he had never seen such fear of a bill.\" When the tally hit 203\u2013202 in favor, Rayburn halted the voting, in blatant violation of House rules. For his young prot\u00e9g\u00e9, the spectacle was not disturbing; it was inspiring. This was how real men wielded power, how evil was countered, and how democracy survived.\n\nIn the late 1940s, the lessons Johnson learned in the anti-Nazi struggle transferred seamlessly to the cold war. \"We have fought two world wars because of our failure to take a position in time,\" he declared in 1947, in endorsing Truman's aid to Turkey and Greece. \"We must apply to Dictator Stalin the same doctrine that we applied to the Kaiser and to Hitler.\" In anticommunism, Johnson's instincts and his interests aligned. As military budgets skyrocketed, defense became a big industry, especially in Texas. And Johnson, who secured a seat on the Senate Armed Services Committee just in time to support the NSC 68 buildup, became the industry's champion, and its beneficiary.\n\nIf historical analogy and home-state pork weren't reason enough to back global containment, there was, of course, political fear. Like Kennedy, Johnson saw McCarthyism from both ends: as hunter and prey. In 1948, he shamelessly linked his Democratic Senate opponent, Coke Stevenson, to labor unions allegedly under communist control. But soon after winning, he watched his support for Truman become a frightening liability in a state going wild for MacArthur. Johnson never fully got over the McCarthy years. As president, he sometimes told younger aides that they didn't know what it was like, didn't realize the value Congress placed on Asia, didn't understand the hell that would break loose if a Democrat let the communists paint another Asian ally red. That fear lodged deep within him, where it merged with terrors of an even more visceral sort. \"Every night when I fell asleep,\" he told a biographer about his years as president, \"I would see myself tied to the ground in the middle of a long, open space. In the distance, I could hear the voices of thousands of people. They were all shouting at me and running toward me: 'Coward! Traitor! Weakling.' They kept coming closer. They began throwing stones. At exactly that moment I would generally wake up...terribly shaken.\"\n\nTaking the helm of an anguished nation in late November 1963, Johnson decided that the best way to win its affection was to be more faithful to Kennedy's legacy than Kennedy would have been himself. He tried to retain Kennedy's foreign policy team, since they represented continuity. (Insecure around all those eggheads, he nonetheless tried to create a little camaraderie. A few weeks after the assassination, he invited several staffers to join him for a swim in the White House pool: naked, of course. But one egghead was so nervous that he dove in without taking off his glasses. And they spent much of the rest of the time trying to recover them.) The key Kennedy holdovers\u2014Bundy, McNamara, Rusk\u2014assured LBJ that continuity meant holding South Vietnam. It was the awful irony of the Kennedy-Johnson succession. In his final months, Kennedy had sensed that the path he was following in Vietnam was leading to disaster. (Whether he would have gotten off it is another question.) But Johnson insisted on following that path, at least initially, because it had been Jack's.\n\nFor most of 1964, Johnson tried to keep Vietnam out of the newspapers. He told the eggheads to manage the problem quietly so he could focus on winning the presidential election that fall\u2014the election that would make him a real president, not just a surrogate. But subtly, American policy was beginning to shift. Kennedy's decision to send advisers in 1961 had been meant to deter the North from supporting the Vietcong. His 1963 decision to ditch Diem had been meant to create a government in the South capable of rallying the country. By 1964, however, both efforts had failed. Diem's overthrow produced not a legitimate government, but virtually no government, as a dizzying array of factions jockeyed for control. In January, a group of young officers led by General Nguyen Khanh took control. But Saigon remained consumed by demonstrations, strikes, and rumors of additional coups. Khanh took up residence in a houseboat on the Saigon River in case he needed to make a fast getaway. Meanwhile, the South Vietnamese public was becoming increasingly anti-American and increasingly pro-reunification, and the flow of weapons and fighters heading south from Hanoi grew and grew.\n\nWatching with mounting frustration, the Johnson administration decided to make the North pay for its meddling. The real problem, of course, was the South, where the government and indeed the entire state enjoyed little popular allegiance. But the Munich and Korea analogies distorted Washington's view, inclining U.S. officials to see Vietnam as yet another case of cross-border aggression. And no one knew how to solve the South's internal problems anyway. So in a classic example of policymakers having a hammer and deciding that their problem was a nail, Johnson's advisers decided to take the fight to Hanoi.\n\nIt was this decision that led, in the summer of 1964, to something called OPLAN 34A: South Vietnamese commando raids against North Vietnam. On August 2, soon after one of those raids, the destroyer U.S.S. Maddox journeyed north of the border to conduct electronic espionage off the North Vietnamese coast, in a place called the Tonkin Gulf. Perhaps thinking that the Maddox had been part of the commando raids, North Vietnamese torpedo boats opened fire, and after a brief skirmish, were warded off. Johnson was furious. Moscow and Beijing think \"we're yellow and don't mean what we say,\" he told an aide. So he ordered the Maddox back into North Vietnamese waters, where on the night of August 4, sixty miles off the coast, it and another ship, the Turner Joy, radioed that they were under attack. In response, Johnson ordered air strikes against North Vietnamese ships and oil storage facilities. More important, he asked Congress for a resolution authorizing him to take \"all necessary measures to repel any armed attacks against the forces of the United States.\"\n\nIt was a blank check for war. (Johnson compared the resolution to \"grandma's nightshirt...it covered everything.\") Yet Congress quickly complied. Presidential power had grown massively in the quarter century since Franklin Roosevelt repealed the Neutrality Acts. Truman had waged war in Korea without asking Congress first. In 1955, Ike had demanded\u2014and received\u2014authority for military action over Quemoy and Matsu. Kennedy had kept Congress in the dark about the Bay of Pigs. For many in the bipartisan foreign policy elite born under FDR, the lesson of the 1930s was that average Americans were too parochial to respond proactively to foreign threats. And the lesson of the McCarthyite 1950s was that Congress was filled with wild men who, if allowed too much oversight of the executive branch, would wreak havoc with their demagogic crusades. Occasionally lying to Congress and the public, in other words, was part of protecting the country. And who could argue with the results? As the \"imperial presidency\" grew, so did American prosperity and power. Sometimes, it seemed, the American people and their representatives in Congress didn't even want to know all the things being done to keep them safe. In 1958, according to a University of Michigan study, 71 percent of Americans trusted their leaders to do the right thing all or most of the time. As late as 1966, asked why Congress didn't more carefully scrutinize the activities of the CIA, Massachusetts Senator Leverett Saltonstall explained that there was \"information and knowledge on subjects which I personally, as a Member of Congress and as a citizen, would rather not have.\"\n\nLyndon Johnson, who had been manipulating other officeholders since he stole the student council elections at Southwest Texas State Teachers' College, lustily embraced this tradition of \"benign\" duplicity. (\"How do you know when Lyndon Johnson is telling the truth?\" went a joke of the time. \"When he strokes his chin, pulls his earlobe, he's telling the truth. When he begins to move his lips, you know he's lying.\") Administration officials never mentioned the South Vietnamese raids that had preceded the August 2 attack, thus leaving the impression that it was entirely unprovoked. And they never mentioned their suspicions that the August 4 attack had not actually occurred at all. The Senate debated the Tonkin Gulf Resolution for less than ten hours, during which the chamber was mostly empty; then passed it, 88\u20132. The House took forty minutes, and its vote was unanimous.\n\nFor Johnson, it was a masterstroke. He invoked the now-familiar litany, comparing his actions in the Tonkin Gulf to Truman's in Greece, Turkey, and Berlin, and to Kennedy's in the Cuban Missile Crisis. And as it had for them, toughness paid political dividends, with the crisis boosting his popularity thirty points almost overnight. His Republican presidential opponent, Barry Goldwater, a man who seemed altogether too comfortable contemplating nuclear war, was marginalized on the far right. And having kept both the communists and the Republicans at bay, and with a big lead in the polls, Johnson seemed to have the country in the palm of his hand. \"I never had it so good,\" he exclaimed at a rally in Detroit, where he was endorsed by both auto titan Henry Ford and the head of the United Auto Workers, Walter Reuther. It was the perfect end-of-ideology image: the inclusive, centrist leader flanked by the enlightened capitalist and the anti-revolutionary worker. \"The times were good now,\" wrote Halberstam about that fall of 1964, \"and there were better ones, golden ones, ahead.\"\n\nThe country was booming: Unemployment was 4.5 percent; inflation was almost nonexistent; there hadn't been a recession in nearly five years. Militarily, Johnson declared, the United States was stronger \"than at any other time in our peacetime history.\" African-Americans, having watched Johnson muscle through the Civil Rights Act, seemed grateful. (\"Those Negroes,\" LBJ declared, \"cling to my hands like I was Jesus Christ.\") A cover story in Time commented on the wholesome, deferential quality of America's youth, noting that \"the classic conflict between parents and children is letting up.\" The country appeared almost devoid of worry. Its capacity to achieve seemed limited only by its capacity to dream\u2014and in Lyndon Johnson it had a president who dreamed big. \"I'm sick of all the people who talk about the things we can't do,\" he boomed. \"Hell, we're the richest country in the world, the most powerful. We can do it all.\"\n\nHis election victory was massive: 61 percent, the largest in American history. In the Senate, Democrats now claimed sixty-eight seats. In his inaugural address, Johnson called America a nation of \"miracles\" and announced that he would do FDR one better: He would end poverty for good. Aides thought he had his eye on Mount Rushmore, or higher. \"I understand you were born in a log cabin,\" commented West German chancellor Ludwig Erhard on a visit to the LBJ Ranch. \"No, Mr. Chancellor,\" Johnson replied. \"You have me confused with Abe Lincoln. I was born in a manger.\"\n\nJohnson was on top of the world, and to stay there all he had to do was defeat one \"raggedy-assed little fourth-rate country.\" In November, administration officials drew up bombing plans aimed\u2014depending on whom you asked\u2014at either reducing Hanoi's support for the Vietcong, halting it altogether, or buoying morale in the South. But Johnson hesitated. It was his war now, and although less attuned than Kennedy to the power of anti-imperial aspiration, he knew enough to know that Vietnam was a bad place to pick a fight. \"What the hell is Vietnam worth to me?\" he screamed at McGeorge Bundy. \"I don't think it's worth fighting for.\" Standing on the precipice, like Agamemnon before the purple robe, he paused, even trembled, searching for a way to avoid the fateful step. But a trembling man can still commit hubris. Johnson raged against the prison of his own assumptions\u2014assumptions both about America's enemies and about America's people\u2014but they imprisoned him nonetheless. \"If you start running from the Communists they just chase you right into the kitchen,\" he told Bundy, moments after declaring that Vietnam was not worth fighting for. And then, in a conversation the same day with his old Senate mentor, Richard Russell: \"They'd impeach a President who'd run out, wouldn't they?\"\n\nSeeking an alternative to both escalation and defeat, or perhaps just seeking to delay the reckoning, Johnson sent Bundy to South Vietnam to investigate. But Bundy was as much a prisoner of the toughness ethic as he. A legend at Yale, where he had arrived as the first student in school history with three perfect scores on his college entrance exams, Bundy in 1940 had broken with the prevailing isolationism on campus. Just weeks after Jack Kennedy published Why England Slept, he had issued his own interventionist manifesto in a book of student essays titled Zero Hour: Summons to the Free. When he returned to academia after the war as a preternaturally young government professor at Harvard, his Munich lecture became famous; students crowded into the lecture hall to hear his voice crack as he described the betrayal of little Czechoslovakia. \"It is certainly a peculiar statistical coincidence,\" remarked Bundy's brother William, who himself served as Johnson's assistant secretary of state for the Far East, \"that of the decision-making group [on Vietnam], Johnson, Rusk, McNamara, my brother, Walt Rostow, myself, all of those of us who were of age in 1940\u201341 were interventionists in the isolationist-interventionist debate.\" In making Bundy his envoy, Johnson was sending yet another American who knew too much about Central Europe in 1940 and too little about South Vietnam in 1965.\n\nDuring Bundy's trip, the Vietcong attacked a U.S. Army barracks in the town of Pleiku, killing eight and wounding 126. Within hours, Bundy sent word back to Washington: The bombing must begin immediately; America must stand firm. The next day, Johnson spoke from the East Room of the White House in his best Kennedyese: \"We love peace,\" he declared. \"We shall do all we can in order to preserve it for ourselves and all mankind. But we love liberty the more and we shall take up any challenge, we shall answer any threat. We shall pay any price to make certain that freedom shall not perish from this earth.\" Letting South Vietnam go communist was now the equivalent of letting freedom perish from the earth. And in this climate of rhetorical absurdity, in which America's leaders could not distinguish marginal threats from mortal ones, the air war against North Vietnam began.\n\nEleven years earlier, when the leaders of the navy and air force proposed air strikes to save the French at Dien Bien Phu, U.S. Army Chief of Staff Matthew Ridgway had warned them not to be na\u00efve: The air strikes would fail, and American ground troops would surely follow. Back then, Korea was still an open wound; Lyndon Johnson himself had counseled caution. By 1965, however, the only caution that Korea induced was tactical: If America went to war again in Asia, it must not draw China in. Memories of Korea had brightened over the intervening decade. Polls showed that almost two-thirds of Americans now considered the war worthwhile, close to twice the number in 1952. Korea was no longer considered a bloody stalemate; it was considered a win. In 1954, having fought one world war and one large regional war in less than a decade, the public was tired. But by 1965, success had made Americans malleable. They weren't crying out for war in Vietnam, but where their leaders pointed, they dutifully followed. It had worked so far.\n\nSo the bombing commenced and ground troops quickly followed. In late February, General William Westmoreland, head of the Military Assistance Command in Saigon, requested two battalions\u2014roughly 3,500 troops\u2014to protect the U.S. air base at Da Nang. To fight an air war, after all, you needed ground troops to guard the planes.\n\nBy March, it was clear the bombing wasn't working. (The CIA's own reports had predicted it would not work, but Johnson rarely read even the summaries.) In South Vietnam, which had just endured yet another coup, morale was as low as ever, and the North was gearing up for a major new offensive. Westmoreland now requested two entire army divisions, not merely to guard bases but to fight the enemy on the ground. Johnson agreed to send forty thousand, but limited their mission: They were only to fight in \"enclaves\" within fifty miles of a U.S. base. And they would stay only long enough for the South Vietnamese to build up their forces and for the bombing to take its toll. Johnson still believed he was standing on the precipice, that he could pull back whenever he wanted. But it was an illusion; he was already over the edge.\n\nThings were moving quickly now. By summer, the South Vietnamese army was visibly crumbling; desertion rates among draftees approached 50 percent. The North was no longer just sending supplies and advisers to the Vietcong; it was sending full regiments, who decimated the South Vietnamese in battle. Westmoreland asked for 150,000 troops and more intense bombing. Bundy, McNamara, and Rusk all concurred.\n\nBut one top official, Undersecretary of State George Ball, began to furiously resist. He was a bit older than Bundy, McNamara, and Rostow, and he wasn't awed by them. For all their supposed brilliance, they struck him as parochial. Unlike them, he knew the French experience in Vietnam well, having been the French government's attorney in the United States for much of the 1950s. While other administration officials invoked America's history in World War II, Greece, Berlin, Korea, and Cuba, the familiar histories of toughness and triumph, only Ball talked about France's history in Vietnam, a history of defeat. French president Charles de Gaulle, Ball noted, believed America would suffer the same agony in Vietnam as his country had. And that view was shared by most of America's European and Asian allies, as well as by the United Nations itself, which had endorsed the war in Korea but refused to endorse the war in Vietnam. Urged to heed the warnings of America's allies, Johnson replied, according to notes, \"that he did not pay the foreigners at the UN to advise him on foreign policy, but that he did pay [Dean] Rusk.\"\n\nAs spring turned to summer in 1965, four graying intellectuals echoed Ball's fears: Walter Lippmann, George Kennan, Reinhold Niebuhr, and Hans Morgenthau, the very men who had done so much to define the toughness creed a generation before. In February, Lippmann called the administration's escalation \"supreme folly. While the war hawks would rejoice when it began, the people would weep before it ended.\" Johnson and Bundy responded by repeatedly inviting Lippmann to the White House, where they lathered him with flattery and tried to deceive him into thinking they shared his views. But Lippmann\u2014who had been schmoozing with presidents for half a century\u2014was neither dazzled nor intimidated, and kept pounding away in print. So the White House decided to play rough. Two teams of administration aides launched the \"Lippmann Project,\" an investigation of everything he had written since the 1920s, aimed at exposing him as a fool. Johnson began telling anyone who would listen that Lippmann, now in his late seventies, was cowardly or senile or both. The campaign took a toll: Some of Lippmann's longtime friends began to snub him; government officials stopped answering his calls. But the old man did not buckle. \"The root of [Johnson's] troubles has been his pride, a stubborn refusal to recognize the country's limitations or his own,\" wrote Lippmann in one particularly acid column. \"Such pride goeth before destruction and an haughty spirit before a fall.\"\n\n\"I am dreadfully worried about what our people are doing in Southeast Asia,\" George Kennan confided to a friend in March. He tried to speak to Bundy but got nowhere. Finally he agreed to testify before the Senate. \"I think that no episode, perhaps, in modern history has been more misleading than that of the Munich conference,\" he declared. \"It has given to many people the idea that never must one attempt to make any sort of political accommodation in any circumstances. This is, of course, a fatally unfortunate conclusion.\"\n\n\"The policy of restraining Asian communism by sheer military might is fantastic,\" added Reinhold Niebuhr in September. When Hubert Humphrey journeyed to New York to give the keynote address at a dinner in his honor, the seventy-three-year-old theologian was appalled to find the vice president \"claiming my anti-Nazi stance of the thirties [as justification] for the present war.\"\n\nIn May, Hans Morgenthau took the stage at the first National Teach-In against the war. Yes, he conceded, South Vietnam would likely go communist if U.S. forces withdrew. But to claim, as the Johnson administration did, that it would therefore become a client of Beijing was just plain ignorant, given the historic animosity between the two nations. (In fact, communist Vietnam would go to war with communist China four years after the United States withdrew.) In June, Morgenthau accepted a CBS News invitation to debate Bundy on the war. But the younger man out-maneuvered him, selectively quoting from Morgenthau's articles to make it appear as if the aging professor were contradicting himself. Lacking Bundy's debating skills, or his instinct for the jugular, Morgenthau left the exchange embittered, later admitting to a friend that he felt a \"general discouragement, sometimes bordering on despair.\"\n\nOutside government, the Daedaluses of the toughness ethic\u2014the men who had helped craft the conceptual wings that Johnson was now flying into the sun\u2014saw disaster ahead. And inside the government, George Ball saw it, too. On June 18, 1965, he sent Johnson a memo that began with a poem by Emerson:\n\nThe horseman serves the horse,\n\nThe neat-herd serves the neat,\n\nThe merchant serves the purse,\n\nThe eater serves his meat;\n\n'Tis the day of the chattel,\n\nWeb to weave, and corn to grind,\n\nThings are in the saddle,\n\nAnd ride mankind.\n\nThings\u2014or as the Greeks would have put it, gods\u2014were in the saddle, not men. Vietnam was not a place that the United States, for all its might, could control. America's leaders needed to lower their sights, appreciate their limits, husband their strength, maybe even save their souls. None of the men who mattered agreed, or even really understood. And Lyndon Johnson sent another hundred thousand troops to South Vietnam.\n\n## CHAPTER NINE\n\n## LIBERATION\n\nUnderstanding how the hubris of toughness fell\u2014and what replaced it\u2014requires a half step back in time.\n\nOn Wednesday, April 19, 1961, McGeorge Bundy slipped out of his office on the third floor of the Old Executive Office Building and into a conference room across the hall, where the National Security Council staff sat waiting. Two days earlier, Cuban exiles had waded ashore at the Bay of Pigs. By Wednesday, those who remained alive were being hunted down by Castro's men in the Zapata swamp. The Kennedy administration's first major foreign policy initiative had proved a catastrophe. Yet Bundy's tone was self-assured, even cocky. The former Harvard dean compared the Cuban exiles, with their blind faith that Washington would come to their rescue, to assistant professors who na\u00efvely insisted they would gain tenure, until the fateful day arrived. Some of the men in the audience\u2014which included Walt Rostow, Arthur Schlesinger, and a few others\u2014chuckled. Others picked at a large bowl of fruit in the center of the table. But near the back of the ornate conference room, one junior staff member silently seethed. The gilded surroundings, the haughty repartee, the fruit juice dripping from powerful men's lips as they joked about weaker men's fate\u2014it reminded him of ancient Rome. \"I guess [Castro's ally] Che [Guevara] learned more from Guatemala than we did,\" quipped Bundy, referring to the 1954 coup on which the Cuba invasion was partly modeled. For Marcus Raskin, that was the final straw. \"It's interesting that Che learned from Guatemala,\" he interjected, \"but what have we learned?\" Bundy didn't respond; he just glared. Someone else scolded Raskin for indulging in recriminations. That afternoon, the twenty-six-year-old received a call from Bundy's assistant. His presence at National Security Council meetings would no longer be required.\n\nIn an administration of self-styled Hillandales, Raskin was that oddest of creatures: a man uninterested in appearing tough. The son of a Jewish immigrant plumber from Milwaukee, he had been, in his youth, a musical prodigy. He mastered the first-year piano instruction manual in a week; he was playing concertos by age twelve. But at his core, the disheveled young pianist was less a musician than a moralist. As a teenager, he walked with a classmate named Jerry Silberman through the hardscrabble neighborhood near his childhood home, burning with outrage at the squalor. Silberman later moved to Hollywood and changed his name to Gene Wilder. Raskin moved to Washington, where he began causing trouble.\n\nSeveral left-leaning Democratic congressmen had banded together to form something called the Liberal Project, and in 1959 they hired Raskin as the group's secretary. Several days into the job, he sent his bosses a memo. \"Americans are bored. They are apathetic about politics. They are afraid,\" he wrote. \"They have withdrawn from the awesome complexity and almost hopeless dread, which is the general social and political scene.\" On the surface, it was a familiar attack on the malaise of the late Eisenhower years, not much different from what Schlesinger and Bell were writing at the time. But beneath the surface lay the seeds of an ideological revolt. As Schlesinger saw it, Americans were bored because their government was not fighting the cold war with vigor and nerve. Once it did, public life would become exciting, and men could again become men. For Raskin, by contrast, the problem was not that America's leaders were waging the cold war lethargically; it was that they were waging the cold war at all. The militarization of America's foreign policy, he argued, was sucking the life out of America's people. Their creative, hopeful instincts were being suffocated by fear. It was a critique that, propelled by Vietnam, would move from the margins of American politics to the center during the 1960s, shattering the hubris of toughness and wrecking the careers of many of the smug young men who sat alongside Marcus Raskin that April day, eating fruit.\n\nBundy hired Raskin knowing that he was, by Kennedy standards, on the extreme left. But Bundy liked to be intellectually challenged. Silencing dissenting views was too easy; it was more satisfying to allow radicals a hearing, and then beat them in a fair fight. Bundy, a champion debater from his days at Groton, envisioned Raskin as a kind of sparring partner, someone who would keep him sharp and then politely retire once the decisions were made.\n\nBut Raskin did not play by the rules. Not content with mere intellectual banter, he crossed into open moralizing\u2014something the hard-nosed Camelot men could not abide. A few months after the Cuba meeting, Bundy asked an aide to draw up plans for a nuclear first strike if the Soviets invaded West Berlin. When Raskin found out, he said something indelicate: He called his colleagues war criminals. \"How,\" he asked, \"does this make us any better than those who measured the gas ovens or the engineers who built the tracks for the death trains in Nazi Germany?\" It was the ultimate taboo, turning the Munich analogy on its head and seeing America's leaders, not the communists, as Hitler's heirs. For hours Raskin and the aide screamed at each other, until both broke down in tears. Things were getting out of hand. \"I can't take Marc anywhere,\" Bundy muttered, \"without worrying that he won't pee on the floor.\"\n\nIn the spring of 1962, Raskin's former bosses in Congress published a book, The Liberal Papers, which thanked him by name. American foreign policy, declared its opening essay, was warped by \"fear about masculinity...distorted into a need to feel tough.\" Later chapters proposed that America cut its nuclear stockpiles, slash military aid to anticommunist dictatorships, and support China's admission into the UN\u2014in short, that America try to end the cold war. Republicans quickly pounced. Senate Minority Leader Everett Dirksen suggested that the book be renamed \"Our American Munich.\" With the GOP vowing to make Raskin an issue in the fall midterm elections, he was reassigned to domestic affairs. By year's end he was gone.\n\nIn his rage at the cold war, however, and at the tough-guy politics that sustained it, Raskin was not alone. He represented the tip of a large and growing iceberg floating beneath the surface of American public life. In 1964, Stanley Kubrick produced Dr. Strangelove, which depicted America's generals as nuclear-obsessed madmen convinced that communists were trying to contaminate their bodily fluids. The New York Times' fifty-nine-year-old film reviewer condemned the movie as disrespectful, since it depicted \"virtually everybody\" in power as \"stupid or insane\u2014or what is worse, psychopathic.\" But a younger critic wrote in to object, praising the film for exposing how \"after some 20 years of living under the constant threat of the great bomb we are intellectually and emotionally spastic.\" In 1959, a naval officer turned historian named William Appleman Williams published The Tragedy of American Diplomacy, which argued that it was America's insatiable search for overseas markets, not Soviet aggression, that had sparked the cold war. Williams was soon hauled before the House Un-American Activities Committee. But over the course of the 1960s, his academic disciples began hacking away at the idea that the Soviets represented evil and America embodied good. And in June 1962, fifty-nine students gathered at a United Auto Workers camp in Port Huron, Michigan, where they formed Students for a Democratic Society and issued a generational manifesto that would sell sixty thousand copies. Its message was similar to Raskin's: that the cold war was not a geopolitical necessity, forced upon America by a slave superpower bent on world domination. It was an elite strategy, used to keep Americans in a state of emotional terror and political lockdown so they would stand mute while their leaders pursued policies of violence and exploitation from Harlem and Mississippi to the far corners of the earth.\n\nIn the mainstream press, these cracks in the political ice went largely unnoticed. But they were signs of a deeper change. The cold war was beginning to thaw. By the late 1950s, Joseph McCarthy had drunk himself to death, and with anticommunist hysteria no longer at fever pitch, peace groups began emerging from the forced hibernation of the red scare years to demand a halt to the testing\u2014and ultimately the production\u2014of nuclear bombs. In Moscow, Stalin was dead, too, replaced by less fearsome men. In Beijing, Mao was growing increasingly anti-Soviet, giving lie to the specter of a unified communist menace. And, most important of all, in the United States, a generation with no memory of Munich was coming of age. They flooded America's universities: College enrollment nearly quadrupled between 1946 and 1970. Most of the students were apolitical, but a vocal few, like those who gathered at Port Huron, began contrasting the catechisms of the cold war with the new realities of their time. The younger generation, Lyndon Johnson later remarked in disgust, wouldn't \"know a Communist if they tripped over one.\" But that was precisely the point. For the twenty-six-year-old Raskin, and the even younger activists who trailed behind, Munich was not a template; it was a clich\u00e9. Soviet communism\u2014those tired old bureaucrats in Moscow mouthing slogans even they seemed to no longer believe\u2014did not strike them as particularly menacing. What was menacing was anticommunism. America's \"democratic institutions and habits have shriveled in almost direct proportion to the growth of her armaments,\" declared the Port Huron Statement. For a whole generation of younger activists, the cold war was not a struggle for democracy; it was the antithesis of democracy. It was not something to wage, but something to overcome.\n\nThe struggle against the cold war was, at its core, a struggle against the ethic of toughness that marked the Camelot age. It was a revolt, in the eyes of those who instigated it, against the violence and oppression that characterized American policies at home and abroad. And that revolt linked the anti\u2013cold warriors to another group of Americans struggling against violence and oppression: the activists of the burgeoning civil rights movement. How intimately the two movements were intertwined can be glimpsed in the unusual career of one Albert Bigelow. Bigelow had been a naval captain in the Pacific during World War II. But ten years after the war, his life changed forever when two horribly disfigured victims of Hiroshima who had come to the United States for plastic surgery stayed at his house. (Again the World War II analogy was being flipped on its head: For the anti\u2013cold warriors the good war wasn't quite so good.) Haunted by what his government had done, Bigelow decided that the next time America detonated a nuclear weapon, he would be there. So in March 1958 he and three other pacifists set sail in a thirty-foot sailboat, The Golden Rule, for the Marshall Islands, where the United States was set to test a hydrogen bomb. Arrested near Honolulu, Bigelow reluctantly concluded that placing himself in the path of nuclear explosions was impractical. But he still wanted to lay down his body for justice and peace. So in May 1961 he boarded a bus in Washington along with twelve other members of the Congress of Racial Equality (CORE), determined to integrate interstate transport across the South.\n\nCORE's founder, James Farmer, had cut his teeth working for the pacifist Fellowship for Reconciliation, which was formed in 1915 to oppose World War I. Bayard Rustin, the Fellowship's other African-American staffer, went on to organize the 1963 March on Washington. And the other two major civil rights organizations\u2014the Southern Christian Leadership Conference and the Student Nonviolent Coordinating Committee (SNCC)\u2014were founded on pacifist principles as well. So the movement to end the cold war helped to incubate the movement for civil rights\u2014and the pollination also went the other way. It was a summer spent working for SNCC in 1961 that helped convince the University of Michigan's Tom Hayden to cofound Students for a Democratic Society the following year.\n\n\"I am going,\" wrote Bigelow before setting sail for the Marshall Islands, \"because however mistaken, unrighteous, and unrepentant governments may seem, I still believe all men are really good at heart.\" Philosophically, this was the great divide, separating the practitioners of cold war toughness from the anti\u2013cold war activists who revolted against them. For Hans Morgenthau, Reinhold Niebuhr, and the realist intellectuals of the 1940s, the beginning of wisdom was recognizing that people were not necessarily good at heart, that evil could never be eradicated from the world because it could never be eradicated from the soul. This premise had profound implications for foreign policy: It meant that conflict was endemic, that evil means might be required to ensure that greater evil did not prevail. That had been the lesson of Munich: that sometimes the horror of war was the only way to prevent greater horror. And it was this lesson that \"ultra-realists\" like Bundy, Rusk, and Lyndon Johnson not only learned but overlearned, and that helped lead America into Vietnam.\n\nBigelow didn't see the world that way, because he didn't see human beings that way. And neither did many others on the New Left. It is the \"theological gospel according to Reinhold Niebuhr,\" wrote Marcus Raskin, \"namely, that we live in an immoral world that offers the statesmen no option but to choose evil, albeit a lesser one,\" that has \"made sonsofbitches of us all.\" From the struggle to end the cold war to the struggle for civil rights, the activists of the early 1960s were generally more hopeful; they refused to believe that when you stripped humanity naked you found the snake of original sin. (Even Martin Luther King, Jr., who admired Niebuhr, found his view of human nature too bleak.) It was no accident that America's leading pacifist journal, which published white anti\u2013cold warriors along with civil rights leaders like King and Rustin, was called Liberation. The assumption was that while society was filled with violence, fear, and hate, people were not born that way. If you liberated them from oppressive institutions like segregation and militarism, their inner goodness could break free.\n\nIf this optimism about human nature underpinned the struggle for racial justice and world peace, it was even clearer in the third leg of the 1960s protest triad: the counterculture. The closest thing hippies had to a philosopher was a rather square classics professor named Norman O. Brown. Brown was a sunny Freudian, which Freud himself might have considered an oxymoron. In Freud's view, the human subconscious was a dark and dangerous place. People had an instinct for sex, which if not constrained by cultural taboo would destroy civilization. And they had an instinct for death, which could easily explode into violence. So repression was tragic because it denied people their deepest desires, but it was also necessary because those desires, if unleashed, could spawn barbarism and chaos. Brown disagreed. He denied that people had an instinct for death, and he denied they should repress their instinct for sex. In his view, people's truest selves were not frightening; they were glorious. Brown urged Americans to unlock their inner Dionysus, the Greek god of wine and ecstasy, which \"breaks down the boundaries; releases the prisoners; abolishes repression.\" Liberate people from social constraint, he insisted, and they will become virtual geysers of love.\n\nBy the late 1960s, thousands of young hippies\u2014clad in tie-dye and tripping on LSD\u2014were attempting exactly that. Like the overall-wearing civil rights activists of SNCC and the coffeehouse peace activists of SDS, they believed they were championing love against hate, nonviolence against violence, liberation against repression. All three movements believed that the men who ran America, and kept the nation in a state of conformity and fear, were morally bankrupt and needed to be peaceably overthrown. All three believed that once they were, the love and compassion locked inside the American heart would break free. All three believed that ultimately the ethic of toughness was an ethic of death. And all three were radicalized and empowered when that ethic blew itself to bits in Vietnam.\n\nIn the summer of 1965, George Ball had warned Lyndon Johnson that once he began escalating in Vietnam, he would not be able to stop: The number of U.S. troops might eventually reach half a million. Robert McNamara called that prediction \"outrageous.\" But within eighteen months it had come true. The problem was simple: Every time America upped the ante, so did Hanoi. In July, when Johnson made the fateful decision to throw a hundred thousand U.S. troops into the war, there were two North Vietnamese regiments fighting in the South. By November, there were as many as nine. America escalated in the air as well. By 1967, the United States had dropped more bombs on Vietnam than it did in all of World War II. But Hanoi built tunnels, thirty thousand miles of them. And China and the Soviet Union, which had once supported a negotiated settlement, began vying to see who could aid Hanoi more. Between 1965 and 1968, by one estimate, the bombing cost America almost ten dollars for every one dollar of damage it did to North Vietnam.\n\nJohnson's actions didn't only spark a counter-escalation in Vietnam; they also sparked a counter-escalation at home. In the early 1960s, anti\u2013cold war activists had mostly focused on nuclear disarmament, not Southeast Asia. In five thousand words, the Port Huron Statement only mentioned Vietnam once. But with American planes and then American GIs pouring into Vietnam, the New Left's priorities began to shift. On Easter Sunday 1965, SDS held the first major rally against the war. In May, the University of Michigan held the first antiwar \"teach-in,\" and by summer this exercise in guerrilla education had spread to 120 campuses. In November, thirty thousand people protested in Washington; the following March, fifty thousand turned out in New York. Over the course of 1965, the membership of SDS quadrupled.\n\nWith large public majorities backing Johnson on the war, it was easy to dismiss the protesters as a lunatic fringe. But soon mainstream figures began rebelling as well. In March 1967, with roughly forty thousand Americans wounded or dead in Vietnam, Robert Kennedy called the war a \"horror\" and urged Johnson to halt the bombing. A month later Martin Luther King publicly denounced it, too, describing America as \"the greatest purveyor of violence in the world today.\" And with the antiwar movement growing bigger and angrier, and key members of the political establishment defecting to its side, the Johnson administration began to feel like an institution under siege. In November, when Dean Rusk traveled to New York for a speech, five thousand people turned out to protest, and members of SDS threw bottles and bags of blood. When Bundy went to defend the war at Harvard, a campus that a few years earlier had held him in awe, demonstrators carried signs asking, \"When Will Bundy Pay for His War Crimes?\" His old academic colleagues increasingly shunned him; his brother William's office at MIT was firebombed. Stephen Bundy, a Harvard undergraduate, walked into the offices of the Harvard Crimson newspaper one day to find its editors throwing darts at a picture of his father mounted on the wall.\n\nMcNamara, whose own son was becoming a radical peace activist, began to break under the strain. In November 1965, a thirty-two-year-old Quaker man sat down at the river entrance to the Pentagon, a mere fifty yards from McNamara's office, poured kerosene over his body, and set himself and his eighteen-month-old daughter on fire. (The baby lived; her father did not.) It was not an easy image to forget. At a New Year's Eve party a few weeks later, McNamara began talking with a woman who opposed the war, and then suddenly began to sob. Over the next two years the scene repeated itself several times. The man with the slicked-back hair and the clipped, machine-gun voice, who once epitomized the ultratough Camelot style, now seemed hard-pressed, when talking about the war he had helped launch, to keep from bursting into tears.\n\nThe heady optimism of 1964, when America seemed invincible and Lyndon Johnson held it in the palm of his hand, was a distant memory now. In 1966, George Ball quietly resigned. When McNamara suggested that the war could not be won, Johnson accused him of disloyalty and froze him out. He turned on Bundy, too, convinced that he was secretly colluding against him with Bobby Kennedy. By the end of 1967, Bundy and McNamara were both gone.\n\nReplacing Bundy as national security advisor was Rostow, whom one aide described as \"Rasputin to a tsar under siege.\" Rostow was a jovial figure, far gentler than the caustic Bundy. But on the subject of Vietnam, his historical determinism made him the truest of true believers, \"a fanatic in sheep's clothing,\" in one detractor's words. He carefully edited intelligence reports before Johnson saw them, weeding out bad news and trumpeting signs of progress, however trivial. To more pessimistic colleagues, his insistence that America was winning in Vietnam seemed increasingly divorced from reality. After one dispiriting report on the war in 1967, he turned to the briefer in a slightly desperate tone and demanded, \"You do admit that it'll all be over in six months?\" \"Oh,\" the man replied with a smirk. \"I think we can hold out longer than that.\"\n\nBut Rostow possessed the quality that his boss prized most: absolute loyalty. As former aides and allies turned against him on the war, Johnson, an insecure, unlovable man in the best of times, crawled into a dark space inside himself. He was, he insisted, the victim of a conspiracy. Communists had infiltrated the \"highest counsels\" of government; antiwar senators were taking orders from the Kremlin. No longer able to travel without being besieged by demonstrators, he became a virtual prisoner inside the White House, which was itself ringed by protesters chanting, \"LBJ, how many kids have you killed today?\" and \"Lee Harvey Oswald, where are you now?\" Increasingly, Johnson found it difficult to sleep; his doctor worried that his health might give out. Robert Kennedy privately described the president as \"very unstable.\" At a private meeting in 1967, when reporters repeatedly badgered him about why America was in Vietnam, Johnson finally unzipped his pants, pulled out his penis, and screamed, \"This is why!\"\n\n\"The only difference between the Kennedy assassination and mine,\" Johnson moaned, \"is that I am alive and it has been more torturous.\" The awful irony was that he didn't care very much about Vietnam. His real love, he insisted, was the Great Society, his massive effort to alleviate poverty, not \"that bitch of a war.\" But he remained captive to the toughness ethic, convinced, as he later told a biographer, that if he let South Vietnam fall, people would say \"I was a coward. An unmanly man. A man without a spine.\" So in November 1967 he summoned the commander of U.S. forces in Vietnam, William Westmoreland, to Washington to tell Americans that victory was near. \"We are making real progress,\" the square-jawed general told the National Press Club. \"We have reached an important point where the end begins to come into view.\"\n\nWestmoreland's timing was very bad. Even as he spoke, North Vietnamese and Vietcong units were quietly slipping into cities and towns across the South. A diversionary attack on a Marine base near the border with Laos drew U.S. forces out of urban areas. Many South Vietnamese troops had left their bases as well, having returned to their home villages to celebrate the lunar New Year, which the Vietnamese called Tet. And then, on January 30, 1968, all hell broke loose. At 2:45 A.M., nineteen Vietcong troops blew a hole in the wall surrounding the American embassy in Saigon. For six hours they crouched behind flowerpots in the embassy garden and traded fire with U.S. military police, until they were finally subdued. Over the next twenty-four hours, Hanoi attacked thirty-six of South Vietnam's forty-four provincial capitals, as well as the airport, the South Vietnamese military headquarters, and the presidential palace in Saigon. Communist forces seized control of the picturesque city of Hue, ancient seat of the Vietnamese empire. By the time U.S. Marines retook it a month later, it was \"a shattered, stinking hulk, its streets choked with rubble and rotting bodies.\"\n\nBack home, commentators reacted with shock. \"What the hell is going on?\" demanded CBS newscaster Walter Cronkite, who according to polls was the most trusted man in the nation. \"I thought we were winning the war!\" In purely military terms, Tet was actually an American victory. The Vietcong, who excelled at guerrilla warfare, were no match for U.S. forces in a conventional fight, and they sustained massive losses in the attack. But for ordinary Americans, who had repeatedly been told that victory was near, that was cold comfort. If the enemy could make it to the gates of the U.S. Embassy itself, the war was clearly far from over. In February, Washington Post columnist Art Buchwald conducted a mock interview with General Custer at the Battle of Little Big Horn. \"We have the Sioux on the run,\" Custer insisted, \"the Redskins are hurting badly and it will only be a matter of time before they give in.\"\n\nWestmoreland proposed upping the ante yet again: mobilizing the reserves and calling up another two hundred thousand troops. But politically, something had snapped. To replace McNamara, Johnson had chosen establishment superlawyer Clark Clifford, a man with doubts about the war and an ego so large that not even Lyndon Johnson could subordinate it to his own. (Clifford-watchers often told the story of the company president who called to ask him how to handle a legal problem. Clifford told the man to say and do nothing, and charged him ten thousand dollars. When the man called back to complain about the bill and to ask why he had to remain mum, Clifford replied, \"Because I told you to,\" and billed him another five thousand dollars.) Since his days teaching elementary school in Texas, Johnson had always made sure to establish dominion over his aides, to make it brutally clear who was beholden to whom. But he was weak now; he needed Clifford more than Clifford needed him. The new defense secretary quashed Westmoreland's request for massive reinforcements. And he gathered a group of seasoned cold warriors\u2014men like Dean Acheson and Council on Foreign Relations president John McCloy\u2014to tell Johnson the ugly truth. There could be no more escalations. The question was no longer how to win; it was how to get out.\n\nIn important ways, America had become a different country. Underpinning the hubris of toughness had been several assumptions, all born from the nation's extraordinary post\u2013World War II success. The first assumption was that when America put its shoulder to the wheel, no military force on earth could long stand in its way. That faith had shattered at Tet. The second assumption was that the American people would bear whatever burdens and pay whatever price their leaders demanded to stop communism. By the late winter of 1968, that too looked dubious. Tet had sent public opinion reeling. Peace protests were drawing hundreds of thousands into the streets, and in March, for the first time, Gallup showed that a plurality of Americans opposed the war.\n\nThirdly, the hubris of toughness had presumed boundless resources. Kennedy and Johnson had basically embraced NSC 68's assumption that when it came to foreign policy, the government had a blank check. But by 1968, resources no longer looked so boundless. Johnson's spending on the war and the Great Society\u2014and his refusal to raise taxes to pay for them until it was too late\u2014had sparked inflation and a gaping budget deficit. With the U.S. economy suddenly weak, foreign investors began to convert their dollars into gold, shaking the world's faith in the greenback. Johnson arranged a Band-Aid fix and the momentary crisis passed, but as Acheson acknowledged, \"The gold crisis has dampened expansionist ideas.\" Just a few years earlier, men like Rusk, Bundy, and McNamara had taken it as a virtual given that America had the money, the firepower, and the will to stop communist expansion anywhere on earth. By the spring of 1968, that confidence seemed like the fantasy of a bygone age.\n\nOn March 26, Acheson, McCloy, and the other establishment gray-beards went to the White House. Unlike Kennan, Lippmann, Morgenthau, and Niebuhr, most of them had nodded approvingly as the Camelot generation stretched containment far beyond its original size and shape. They had found it flattering that younger men wished to replicate in the third world what they had achieved in Europe. And they hadn't known enough about the world's hotter, darker regions to see that the analogy was horribly flawed. But by 1968, these foreign policy elders were less afraid of Ho Chi Minh than of their own grandchildren, who, radicalized by the war, seemed to be turning against them and everything they had built. Losing in Vietnam, they had come to believe, would endanger the United States less than staying there. Johnson, who had spent his entire career looking over his right shoulder, making sure that men like these never saw him as weak, was aghast. \"The establishment bastards,\" he fumed, \"have bailed out.\"\n\nFive days later, Johnson did, too. Politically he had become radioactive. On March 12, an obscure antiwar senator named Eugene McCarthy had stunned the political world by coming within four thousand votes of beating him in the New Hampshire primary. Polls suggested that the outcome in Wisconsin, the next state to vote, would be even worse. Johnson, interestingly enough, had been thinking a lot about Woodrow Wilson, the last president destroyed by hubris in war. As LBJ later told a biographer, he was haunted by images of Wilson's final eighteen months, when the twenty-eighth president was \"stretched out upstairs in the White House, powerless to move, with the machinery of the American government in disarray around him.\" Johnson would not let that happen to him: He would end the war and leave office, thus saving his political reputation and perhaps his life itself. And so at 9 P.M. on March 31, he went on television to announce that America was prepared to curb the bombing and negotiate a peace agreement with Hanoi\u2014and that he would not seek reelection as president.\n\nIn a sense, LBJ's gambit worked. He returned to Texas, grew his hair long, ate like a pig, smoked like a chimney, terrorized the workers on his ranch, and lived for another five years. But he had only saved his body. Close observers suspected that his soul was already dead. \"My daddy committed political suicide for that war in Vietnam,\" said Johnson's daughter Luci. \"And since politics was his life, it was like committing actual suicide.\" The war, and the hatreds it spawned, haunted Johnson to the grave. He was dissuaded from attending the 1968 Democratic convention for fear that his presence might fuel violence. He wasn't even invited to the Democratic convention in 1972, and his portrait did not hang alongside past presidents and current candidates in the party's gallery of honor. When he died, Johnson's New York Times obituary declared that \"his vision of the Great Society [had] dissolved in the morass of war in Vietnam.\" The first southerner to occupy the White House since Wilson, he too had seen war turn rancid, and died with its taste on his lips.\n\nJohnson wasn't the only one thinking about World War I. In myriad ways, the New Left also reached back in time, to the days before American foreign policy was swallowed by the Munich analogy, to the peace crusaders of interwar years. Randolph Bourne, who had languished in obscurity since World War II, suddenly became hot. In 1964, Harper & Row published an anthology of his essays; in 1966, E. P. Dutton put out another. The New Left historian Christopher Lasch devoted an admiring chapter to him in a book that took swipes at Niebuhr. In 1967, a radical young linguist named Noam Chomsky quoted Bourne at length in an essay on intellectuals and war.\n\nIf Vietnam revived Bourne's reputation, it resuscitated his two mentors as well: John Dewey and Charles Beard. Marcus Raskin called Dewey's bid to outlaw war a model for post-Vietnam foreign policy. And while writing the Port Huron Statement, Tom Hayden was influenced by Dewey's 1927 book, The Public and Its Problems, in which the philosopher argued, contra Walter Lippmann, that with the right education, ordinary people could effectively govern themselves. For Hayden and his intellectual mentor, C. Wright Mills\u2014who wanted a \"participatory democracy\" and loathed the postwar belief that elites should keep the irrational masses far from the levers of power\u2014Dewey was an inspiration. They even set out to reclaim the word utopian, which in the toughness years had rarely been uttered without a sneer.\n\nWhile New Left sociologists and philosophers were resurrecting Dewey, New Left historians\u2014led by William Appleman Williams\u2014were resurrecting Beard, hailing his warnings about presidential power and the economic motives for war. \"The war in Indochina...has led many liberals and radicals to look again at the criticism offered by such men as Charles Beard,\" wrote a young, militant historian named Ronald Radosh. Even Herbert Hoover's foreign policy received a second, more positive look.\n\nInvoking Beard, Dewey, and Bourne was a way of summoning an earlier time, before American soldiers patrolled the global beat, before American children cowered under their desks during nuclear drills, before America's leaders lathered themselves with machismo to cover their fear. Not everyone in the anti-Vietnam movement was a Deweyian pacifist, of course. Most protesters didn't oppose all wars, just this one. Still, as the fighting ground on, killing sixteen thousand Americans and hundreds of thousands of Vietnamese by the beginning of 1968, even many nonpacifists began to feel, as Raskin and SDS and the civil rights marchers had from the beginning, that Vietnam was a symptom of a larger disease: a penchant for violence that had become endemic to American foreign policy and American life.\n\nIf intellectuals and activists were creating post-toughness politics\u2014a politics aimed at liberating American from the Cold War\u2014in the late 1950s that political ethic moved from the streets into the Democratic Party. The first presidential candidate to give it voice was Eugene McCarthy, the man whose shocking success in the 1968 New Hampshire primary had chased Lyndon Johnson back to Texas. McCarthy was an unlikely trailblazer. A conventional cold warrior for most of his career, he had spent the late 1950s slamming Eisenhower as an appeaser. In 1964, he told Johnson that he would make a better running mate than Hubert Humphrey because Humphrey was apt to go soft on Vietnam. As late as 1965, with U.S. troops pouring into Indochina, McCarthy still held his tongue, hoping that LBJ might appoint him ambassador to the UN.\n\nBut by year's end, as it became clear that he wouldn't get a glamorous new job, McCarthy grew bitter and bored. An introvert who as a young man had toyed with becoming a Catholic monk, he didn't really like the Senate. He often skipped committee meetings to read theology or write poetry. And as time passed, his personal alienation morphed into a broader ideological discontent. For one thing, administration officials kept lying to him. One evening in February 1965, he went with a group of senators to the White House, where Johnson lavished praise on South Vietnamese leader Nguyen Khanh, insisting that he was popular and in control. The next day Khanh was ousted in a coup. The following year, in a Senate Foreign Relations Committee hearing, McCarthy asked Dean Rusk whether he believed MacArthur's old adage that America could not win a land war in Asia. Rusk replied with a blizzard of obfuscation, to which McCarthy responded, \"I don't think that quite answers my question.\" Rusk looked at him smugly and declared, \"I know it didn't, sir.\"\n\nBy 1967, McCarthy had become that most dangerous of Washington creatures: a politician with little to lose. He had just turned fifty; his job annoyed him, his marriage had turned sour, and he had the gnawing sense that life was passing him by. Young peace activists were scouring the Senate, looking for someone to run against Johnson in the primaries. McCarthy's children, two of them peace activists themselves, urged him to do it. An old monk told him to seize the day. He saw the chance to inject some excitement and purpose into his life, maybe even to become a hero.\n\nAs it turned out, he was not a very good hero. Although honest and intelligent, he was also lazy, arrogant, mean-spirited, reclusive, and opaque. Instead of prepping for an important debate, he passed the time singing Irish ballads. He made an influential New York Times columnist wait outside his office while he finished writing a poem about wolverines. One afternoon in Los Angeles, arriving early for an event, he simply walked to the podium and gave his prepared speech, even though the crowd had not yet arrived.\n\nAfter New Hampshire, he was joined in the race by Robert Kennedy, who immediately seized the hero's mantle. In this competition with Camelot, with the fairy-tale story of the shy, tortured younger sibling who rises to avenge his brother's death and save the nation in its moment of peril, McCarthy's deficiencies became downright painful. At his events, listeners nodded politely, or nodded off. At Kennedy's, girls screamed; fans ripped at his clothes; grown men cried. Emotionally McCarthy was a dud. But ideologically he was the truly momentous figure: the vehicle through which post-toughness politics entered the political mainstream.\n\nRobert Kennedy was too wedded to the past. Earlier in his career, he had taken the Kennedy cult of toughness to truly lunatic extremes. As a Senate investigator, he once challenged Teamster leader Jimmy Hoffa to a push-up contest. On a trip down the Amazon in 1965, he jumped into a river filled with piranhas and dared his boatmates to join him. The following year, on safari in Africa, he dismounted from a jeep, walked to within fifteen feet of a rhinoceros, and stared at the beast until it trotted away.\n\nOn foreign policy, he had been a rigid, brutal, hawk. In the 1950s, he hadn't merely worked for Joseph McCarthy; he had made the drunken demagogue godfather to his first child. A maniac for Edwin Hillandale and James Bond, he had kept a green beret on his Senate desk and organized such fierce mock counterinsurgency battles at the Kennedy vacation compound that relatives warned he was scaring the kids. In his brother's administration he was known for advocating the hardest of hard-line stances, and then calling his colleagues wimps when they pointed out the risks.\n\nBy 1968, his views had evolved, and he called for deescalating in Vietnam. But he never confronted the underlying assumptions that made the war possible. Attacking global containment itself\u2014linking the horrors of the late 1960s to the hubris of the early 1960s\u2014would have meant defiling his brother's memory, the one thing that he would not, under any circumstances, do. And so his critique of the war was passionate yet incoherent. No longer a cold warrior, and yet not an anti\u2013cold warrior, his true foreign policy ideology, if it can be called that, was simply Kennedyism: the belief that through the alchemy of Camelot, America could be both hawkish and dovish, tough and humane, crusading and restrained, that it could bear any burden in its long twilight struggle against communism yet escape burdens like Vietnam.\n\nMcCarthy was more lucid and more radical. Vietnam, he declared, \"originated in the containment doctrines of the 1950s.\" It was the bitter fruit of NSC 68, of the belief that every time communist guerrillas advanced across some godforsaken jungle outpost it was 1938 all over again. For McCarthy, enough was enough. He called not merely for deescalating in Vietnam but for recognizing Cuba, ending weapons sales to the third world, and slowing the nuclear arms race. It was the same agenda that Raskin had been fired for supporting in 1962. And McCarthy's philosophical assumptions also mirrored those of the New Left. He accused America's leaders of being too quick to see evil lurking around every corner, too unwilling to believe that people would respond to love, not force. We must place our \"hope and trust in our fellow men,\" McCarthy told antiwar activists. \"We do not have...to be afraid.\"\n\nIt was no surprise that many of those activists joined his campaign, believing\u2014for the first time in their lives\u2014that a mainstream presidential candidate saw the world as they did. In the late 1960s, the hubris of toughness, which had been building like a wave for two decades, finally broke. In the '70s, post-toughness politics would take over first the Democratic Party and then the government itself.\n\n## CHAPTER TEN\n\n## THE SCOLD\n\n\"The tragedy of Vietnam is the tragedy of the catastrophic overextension and misapplication of valid principles.\" With those words, published in 1967, Arthur Schlesinger traced not only the rise of the hubris of toughness, but the arc of his own career. In the 1940s he had been the junior member of a group of realist thinkers\u2014led by his mentor Reinhold Niebuhr\u2014who told Americans that the struggle against evil was forever, that reason would never govern the world. In the early 1960s he had watched contemporaries like Bundy and Rostow stretch that toughness ethic past the breaking point, turning the struggle against evil into a global crusade. And by decade's end he had become a senior member of a new group of intellectuals, many of them too young to remember Munich, who rejected toughness politics and began imagining a post\u2013cold war world.\n\nEugene McCarthy lost the Democratic nomination in 1968, first to Robert Kennedy\u2014who had taken a decisive lead by the time he was gunned down\u2014and then to his old rival Hubert Humphrey, who was even less inclined to challenge cold war orthodoxy. But McCarthy's ideas won. After 1968, the antiwar movement began taking over the Democratic Party. That year, for the first time since the cold war began, northern Democrats in Congress voted en masse against a major weapons system: the Sentinel antiballistic missile. In 1970, peace activists in Washington State rewrote the state party platform to demand a moratorium on the production of nuclear weapons. In 1971, Senate Majority Leader Mike Mansfield introduced legislation to cut the number of American troops in Europe in half.\n\nUnderlying these heresies was the belief\u2014once confined to a scruffy New Left fringe\u2014that, as Schlesinger put it, \"the military hang-up of our foreign policy is an exercise in futility.\" In Vietnam, America was using more firepower than any nation in history. (The United States would ultimately drop 14 million tons of explosives, more than five times as many as it dropped during World War II.) Yet it was losing to the Vietcong, a ragtag guerrilla force, backed by North Vietnam, a country that used one-fifth as much electricity as was generated annually by the Arlington, Virginia, branch of the Potomac Electric Power Company. \"Perhaps the principal lesson of the past decade,\" wrote Leslie Gelb and Paul Warnke, two rising stars in the post-Vietnam foreign policy establishment, \"is that military force is a singularly inept instrument of foreign policy.\"\n\nAmerican foreign policy, argued the post\u2013cold war thinkers of the early 1970s\u2014again echoing the New Left radicals of the early 1960s\u2014had been too governed by fear. The Soviet Union, they insisted, was not Nazi Germany. It was a tired, defensive gerontocracy, eager, above all, to be left alone. To be sure, communist movements might be gaining steam in certain pockets of the third world. But they were no more controlled by Moscow than they could be defeated by the United States. The key was to stop worrying about these obscure conflicts, to see them as local affairs that could not harm the United States unless the United States used them as an occasion to harm itself. What truly threatened America, in other words, was not the Soviet Union, but America's excessive fear of the Soviet Union, which led it to do self-destructive things. \"If we find ourselves in a hostile world,\" declared Warnke, \"it's going to mean our foreign policy has been idiotic.\"\n\nCold warriors often accused the new intellectuals of isolationism. But that was mostly wrong. The post-toughness Democrats were well aware that America could not escape the world. In fact, they popularized the term interdependence to describe the myriad new ways in which America was entangled in it. But it was precisely this entanglement, they argued, that required that America change the nature of its international involvement from conflict to cooperation. \"Perhaps the greatest challenge to American foreign policy makers in the next generation will be to find constructive ways in which to cooperate with other nations in 'managing interdependence,'\" wrote Anthony Lake, a talented post\u2013cold war thinker who, like Gelb, was born too late to remember Munich. The language was calmer and dryer than that of the New Left. It was the language of the seminar room, or even the boardroom, not the language of the soapbox and the street, but many of the same ideas were there: If you liberated people from their fear you could unlock their decency; if America's leaders got over their obsession with appearing tough, they could build a more cooperative, more humane world.\n\nAlthough their criticism of military force and their talk of global community sometimes made them sound like John Dewey, the post\u2013cold war thinkers were neither pacifists nor utopians. Dewey had opposed all U.S. military commitments, even after Hitler took power. The post\u2013cold war thinkers of the 1970s, by contrast, only suggested that America limit the scope of its military commitments and stop viewing communism as a unified menace\u2014which made sense, given that Beijing and Moscow threatened each other more than either threatened the United States. In the quarter century between Pearl Harbor and Pleiku, America's leaders had gradually forgotten that there were limits to American power, and as a result, that there must be limits to American fear. Now Vietnam had shown that America was not omnipotent, and so men like Lake, Warnke, and Gelb usefully pointed out that America did not need to be.\n\nWith their greater optimism about international cooperation, and about human nature itself, the post\u2013cold war intellectuals were temperamentally different from realists like Kennan and Morgenthau. But when it came to global containment, the views of the two groups tended to coincide. The reason was that the post\u2013cold war intellectuals were not rebelling against Kennan's and Morgenthau's ideas; they were rebelling against \"catastrophic overextension and misapplication\" of Kennan and Morgenthau's ideas. They were not rebelling against the realism of the 1940s; they were rebelling against ultra-realism of the 1960s. In suggesting that Lyndon Johnson and Dean Rusk's worldview was overly dark, men like Gelb, Warnke, and Lake were actually doing something similar to what Kennan and Morgenthau had done when they called John Dewey's worldview overly rosy: They were deflating an ideological tradition that had swelled too far. In insisting that the truths of realism, like the truths of progressivism, had limits, the post\u2013cold war thinkers were offering the beginnings of an answer to the hubris of toughness. All they needed was a leader who could make Americans listen.\n\nThat was the hard part, as Democrats learned in 1972, when they nominated George McGovern for president. Unlike Eugene McCarthy, McGovern was no latecomer to peace politics; it was in his blood. He hailed from South Dakota, two of whose three congressmen had opposed World War I. As a young man in the 1940s, he had joined the United World Federalists, which denounced the UN as too weak. In the presidential election of 1948, he had supported Henry Wallace over Harry Truman, and thus cast his lot with the last prominent Democrat to oppose the nascent cold war. McGovern never repudiated those early views. To the contrary, in his autobiography he insisted that \"the peace of the world would have been better served by the hopeful and compassionate views of Wallace than the get tough policy of the Truman administration.\" If Marcus Raskin and others in the antiwar movement yearned to revive the spirit of peace progressivism, they finally had their man.\n\nWhen McGovern called Wallace's worldview \"hopeful,\" he was referring to something specific: the hope that the United States and the Soviet Union, having cooperated to vanquish fascism, could make their continued cooperation the foundation of a peaceful world. McGovern never abandoned that hope. Throughout his career, he worked to demilitarize, and ultimately end, the cold war. At the height of McCarthyism, when many politicians were baying for war not merely with Pyongyang, but with Beijing as well, he opposed U.S. intervention in Korea. A decade later, in his first Senate speech, he proposed recognizing Castro's Cuba. By the time he ran for president in 1972, he was demanding that the Pentagon's budget be slashed by more than a third.\n\nEven George McGovern was no John Dewey or Herbert Hoover; he wasn't calling for an end to America's core military commitments in Western Europe and Japan. But he was the most radical critic of global containment ever nominated by a major party. \"The war against communism is over,\" he told the journalist Theodore White. \"Somehow we have to settle down and live with them.\"\n\nLike Warnke, who became his campaign's top foreign policy adviser, McGovern did not believe that America was fated to live in a hostile world. In his acceptance speech at the Democratic convention, he urged Americans to stop being \"so absorbed with fear and danger from abroad\" on the campaign trail, he conjured \"a future that is not based on outdated stereotypes of military confrontation and power politics.\" It was a revealing critique. Ever since the 1940s, America's most influential foreign policy thinkers had insisted that power politics was not a choice but a fact. It was the inevitable expression of the lust for power that resided in the human heart. But McGovern, like the New Left, rejected that tragic view. McGovern's Christianity was not Niebuhr's. In fact, his hero was the progressive theologian whom Niebuhr had rebelled against: Walter Rauschenbusch, father of the Social Gospel. Christianity's task, wrote Rauschenbusch, is \"transforming the life on earth into the harmony of heaven.\" For Niebuhr this was utopian nonsense. But Rauschenbusch's words so inspired McGovern that decades after reading them in college, he quoted them in his autobiography.\n\nParadoxically, it was this more hopeful view of human nature, and of foreign policy, that made George McGovern such an angry man. What Raskin introduced into the Kennedy White House, McGovern introduced into the 1972 presidential campaign: the language of moral fury. For Bundy, who considered himself a student of Niebuhr, who believed that evil was forever present in world affairs, and that \"grey is the color of truth,\" sermonizing about foreign policy was like \"pee[ing] on the floor.\" But McGovern, like Raskin, believed that foreign policy was not a choice between lesser evils but between evil and good. So he said what he believed, which was that Vietnam was more than a mistake. It was a crime.\n\n\"This chamber,\" McGovern told his Senate colleagues in September 1970, in perhaps the harshest attack on the toughness ethic ever by a national politician, \"reeks of blood. Every senator here is partly responsible for that human wreckage at Walter Reed and Bethesda Naval and all across our land\u2014young men without legs, or arms, or genitals, or faces, or hopes. There are not very many of these blasted and broken boys who think this war is a glorious adventure. Do not talk to them about bugging out, or national honor, or courage. It does not take any courage at all for a congressman, or a senator, or a president to wrap himself in the flag and say we are staying in Vietnam, because it is not our blood that is being shed. But we are responsible for those young men and their lives and their hopes. And if we do not end this damnable war, those young men will some day curse us.\"\n\nGeorge McGovern knew something about shedding blood. In early 1943 he had been a twenty-two-year-old college student with a pilot's license, a pregnant wife, and an absolute dread of flying a plane. Nonetheless, he piloted thirty-five bombing raids over Germany, sometimes amid enemy fire that forced him into terrifying emergency landings. On one mission his right tire blew out upon takeoff, meaning that his plane would likely crash when it tried to land. Rather than bailing out, McGovern and his crew completed their nine-hour bombing run, all the while knowing that they might die upon their return to base. On another flight, McGovern's plane was struck by more than a hundred pieces of shrapnel during withering German antiaircraft fire. One of his men was hit in the leg. The plane lost oxygen, and the heated suits that insulated the crew from the subzero weather gave out. Still, McGovern found a makeshift airstrip and successfully landed the plane.\n\nOne of McGovern's men, a streetwise, hard-drinking engineer from Bridgeport, Connecticut, was shattered by these harrowing ordeals. After the war he entered a mental asylum, and then stepped in front of a truck, in what may have been a suicide. McGovern's navigator, a somber Milwaukeean with dreams of entering the ministry, was flying with another pilot when he was forced to bail out over German soil just before his plane blew up. McGovern spent the remainder of the war sleeping next to the man's empty bunk, gazing at his clothes and photos of his family, praying that he would return. He never did. One of the deep ironies of the 1972 campaign, a campaign in which many Americans came to see McGovern as a pacifist, if not a coward, was that he was one of the greatest war heroes ever to run for president. He had been awarded the Distinguished Flying Cross for Bravery four times. But in his speech accepting the Democratic nomination, he excised mentions of his war record. \"It seemed hypocritical,\" said one of his aides, to boast that \"'I killed more people than any of you guys.'\"\n\nGeorge McGovern would not play the politics of toughness. He attacked the idea that \"tough talk and big Pentagon budgets are somehow synonymous with national manhood\" and said he would, if necessary, go on his hands and knees to Hanoi to retrieve American prisoners of war. He refused to boast about his wartime heroism, perhaps because he feared dishonoring the dead. He was a very unusual cold war politician: one with no need, or desire, to prove himself a man.\n\nIt was all quite admirable. There was just one problem: He lost forty-nine states.\n\nPolitically, rejecting the hubris of toughness was treacherous: Americans didn't want to go on bended knee to Hanoi. But sustaining it was also treacherous, since Americans didn't want to keep sending their sons to fight Hanoi, either. In 1968 and 1972, as the Democratic Party was abandoning the politics of machismo, America elected and reelected a man who said \"combat is the essence of politics\" and boasted about eating sheep's testicles. No one worked harder than Richard Nixon to maintain America's image of ferocity, and his own. Yet Nixon's bravado couldn't conceal the fact that fewer and fewer of his citizens were willing to pay the price to contain communism around the world. And so his foreign policy often resembled a man who hangs a \"beware of dog\" sign on his house while trying to rouse an exhausted and indifferent mutt. Nixon repeatedly bluffed foreigners about what the United States might do and lied to Americans about the commitments he had made in their name. Those efforts were sometimes ingenious but they involved too much secrecy, dishonesty, and immorality for a public that no longer gave its presidents the benefit of the doubt. Ultimately Nixon's efforts to practice the politics of toughness in an era of diminished power failed. And in 1977, Jimmy Carter did what McGovern could not: smuggle post-toughness foreign policy into the White House.\n\nIt was hardly surprising that Richard Nixon considered fear a more powerful force than love. His own life proved it. At an early age he discovered that people didn't much like him. In college his brooding manner and harsh tone repelled girls. After law school he was rejected by the fancy Manhattan firms. But he overcame this personal rejection through brass-knuckle determination. In his first political campaign, he defeated an entrenched congressman by insinuating that he espoused the \"communist line.\" Once in Congress, Nixon exposed establishment golden boy Alger Hiss as a spy, thus making himself a hated figure among right-thinking liberals but helping win himself a spot as Dwight Eisenhower's running mate. Even that triumph, however, didn't end the social rejection. In eight years as president, Ike never once asked his grasping understudy to join him in the White House private quarters. \"What starts the process really are laughs and slights and snubs when you are a kid,\" Nixon once told a friend. \"But if you are reasonably intelligent and if your anger is deep enough and strong enough you learn that you can change those attitudes by excellence, [and] personal gut performance, while those who have everything are sitting on their fat butts.\"\n\nIf Nixon saw life as a pitiless struggle in which grit\u2014not charm\u2014usually won out, he reinforced that worldview with his choice of national security advisor. Growing up in the Bavarian town of Furth in the 1930s, young Heinz Kissinger watched his meek, kindly father destroyed by repeated experiences of degradation and defeat. He developed such a terror of non-Jews that even after he immigrated to the United States as a teenager and changed his name to Henry, he still instinctively crossed the street when gentile boys approached. He abandoned his own, deep religious faith after the Holocaust claimed thirteen of his relatives. And in its place he substituted a view of life that would have made Hobbes shiver. Explaining the Holocaust to an American acquaintance after the war, Kissinger wrote that \"the intellectuals, the idealists, the men of high morals had no chance.... Having once made up one's mind to survive, it was a necessity to follow through with a singleness of purpose inconceivable to you sheltered people in the States. Such singleness of purpose broached no stopping in front of accepted sets of values; it had to disregard ordinary standards of morality. One could only survive through lies, tricks and by somehow acquiring food to fill one's belly. The weak, the old had no chance.\"\n\nAs an adult, Kissinger spoke little about the harrowing anti-Semitism of his youth. (\"Any people who have been persecuted for two thousand years,\" he joked, \"must be doing something wrong.\") But its impact was palpable. \"Life is suffering. Birth involves death,\" he wrote in his undergraduate thesis, a 388-page behemoth titled \"The Meaning of History\"\u2014the longest in Harvard's history\u2014which spurred the university to limit the length of all future submissions. If McGeorge Bundy and Dean Rusk had in their youth witnessed the Western democracies' failure to resist Hitler, Kissinger had witnessed something even more chilling: the failure of Germany's own citizens\u2014his family's Christian neighbors\u2014to resist the destruction of their fragile democracy from within. As a result, even more than native-born Americans of his generation, he tended to see democracies as weak and ordinary citizens as irresponsible and apathetic. Only strong leaders, acting independently of their people's whims, could restrain the chaos and evil lurking in international affairs, and within the hearts of women and men.\n\nKissinger, his colleagues were astonished to learn, feared that he might become the victim of a vicious anti-Semitic backlash if he presided over defeat in Vietnam. (It didn't help that Nixon himself often baited him with anti-Semitic remarks.) Asked once about White House chief of staff H. R. Haldeman, Kissinger replied that \"Haldeman was the kind of guy who would send you to the showers.\" It was hard to imagine a worldview more at odds with the one emerging inside the Democratic Party, which proposed that by ending the cold war, America's leaders could foster international harmony and unlock the decency buried inside the nation's soul.\n\nGiven their dread of weakness, it was no surprise that Nixon and Kissinger considered post-Vietnam America\u2014and especially the post-Vietnam foreign policy elite\u2014to be dangerously soft. And they saw their challenge as ensuring that the United States did not pay dearly for the infirmity of its ruling class. In part that meant giving America's enemies a reason not to exploit its vulnerability. Nixon opened diplomatic relations with China and he pursued d\u00e9tente with the Soviet Union. And by playing the two communist giants off against each other, and holding out the prospect of economic and diplomatic favors if they eschewed aggression, he tried to achieve the goals of containment without its price. Instead of containing America's foes, he gave them incentives to contain themselves.\n\nNixon's overtures to Moscow and Beijing helped slow the arms race and reduce the threat of nuclear war (at least to the United States; in the late 1960s, Moscow and Beijing came alarmingly close to war with each other). Thawing the cold war also promoted democratic change in the Soviet bloc, since the 1975 Helsinki Accords, a centerpiece of d\u00e9tente, helped spur dissident movements in Eastern Europe. That was the good news. The bad news, from Nixon and Kissinger's perspective, was that Moscow would not\u2014or could not\u2014contain communism's spread in Vietnam and other corners of the third world. For post-toughness liberals like Eugene McCarthy and George McGovern, that wasn't a big problem. After all, they argued, America's rapprochement with China proved Kennan's old argument that not every communist regime was America's enemy or Moscow's friend. For McCarthy and McGovern, it hardly mattered what form of government third-world countries chose: They couldn't bloody the United States unless we used them as an excuse to bloody ourselves. If America reached out its hand in friendship, in fact, it could cooperate with left-wing regimes to build a more just and peaceful world.\n\nBut Nixon and Kissinger had little interest in a foreign policy of kumbaya. What McCarthy and McGovern saw as the hand of friendship looked, through Nixon and Kissinger's darker lenses, like the white flag of surrender. Individually, third-world communist regimes might not threaten the United States, they conceded. But the perception of weakness did. The more blood there was in the water, the more ferocious the sharks would grow. For Nixon and Kissinger, it was easy to imagine just how merciless America's enemies would become once they realized Uncle Sam could be pushed around. All Nixon and Kissinger had to do was envision how merciless they would be. \"We'll get them on the ground where we want them. And we'll stick our heels in, step on them hard, and twist,\" Nixon told aides one day in 1971, in explaining how to handle a wounded opponent. \"Get them on the floor and step on them, crush them, show them no mercy.\" No wonder he didn't want to be the one lying on the ground.\n\nIf McCarthy and McGovern took America's post-Vietnam weakness as a license to stop viewing foreign policy as a contest of military strength, for Nixon and Kissinger the lesson was the opposite. The weaker America grew, the more crucial the old NSC-68 idea of \"credibility\"\u2014the belief that America would stand up for its allies and against its adversaries\u2014became. Nixon called it \"the madman theory.\" The more irrational it became for America to use massive force, the more America's enemies needed to believe that its leaders were just crazy enough to do it.\n\nIn their efforts to appear insane, Nixon and Kissinger occasionally convinced each other. Without his calming presence, Kissinger told aides, \"that drunken lunatic\" would \"blow up the world.\" He later informed them that \"our peerless leader has flipped out.\" For his part, Nixon mused that Kissinger required psychiatric care. But the Dr. Strangelove act made less of an impact on its intended audience: America's third-world foes. Soon after taking office in 1969, Nixon tried to bolster his reputation for bloodcurdling toughness by secretly bombing North Vietnamese sanctuaries in Cambodia, something Johnson had refused to do. But Hanoi was not intimidated and kept trying to topple the government of South Vietnam. The following year\u2014furious that Vice President Spiro Agnew had accused him of \"pussyfooting,\" and after considerable drinking and several screenings of Patton, a movie that begins with the legendary World War II general standing in front of a vast American flag and declaring, \"We're not just going to shoot the bastards, we're going to cut out their living guts and use them to grease the treads of our tanks\"\u2014Nixon startled aides by ordering that U.S. troops invade Cambodia outright. \"If, when the chips are down, the world's most powerful nation, the United States of America, acts like a pitiful, helpless giant,\" Nixon told the nation, \"the forces of totalitarianism and anarchy will threaten free nations and free institutions throughout the world.\" Two years later, the president\u2014now sporting a pipe in homage to his new favorite tough guy, General Douglas MacArthur\u2014responded to a communist offensive by blockading North Vietnam's ports and mining its harbors. \"The bastards,\" he vowed, \"have never been bombed like they're going to be bombed this time.\"\n\nIn opening relations with China and initiating d\u00e9tente with the U.S.S.R., Nixon and Kissinger broke from cold war orthodoxy. But in their obsession with \"credibility\" they were still ultra-realists, guided by the axioms of NSC 68. For realists like Kennan and Morgenthau, credibility had meant convincingly pledging to defend America's core interests. Such pledges should be made sparingly, they believed, because making them credible required convincing the American people that the country in question was worth incurring huge costs to defend. Nixon and Kissinger, by contrast, suggested that American credibility was on the line wherever communists and anticommunists squared off. Like his old boss, Dwight Eisenhower, Nixon was trying to limit containment's price without curtailing its scope, which created a mismatch between means and ends. Eisenhower had found that mismatch easier to conceal, because with much of the third world still under colonial rule, global containment wasn't yet truly global, and because he could deploy the CIA without Congress looking over his shoulder. For Nixon and Kissinger, however, the discrepancy was harder to mask. No amount of bombing or bluster could obscure the fact that the American people were no longer willing to sacrifice their sons for Saigon.\n\nIn January 1973, the United States and North Vietnam signed a peace deal that on paper kept South Vietnam alive. Afraid that American credibility would be damaged by Saigon's rapid military defeat, Nixon secretly promised its leaders that the United States would \"respond with full force should the settlement be violated by North Vietnam.\" But it was a bluff. Within months Congress passed legislation banning any further U.S. military operations in Vietnam. By April 1975, with Saigon on the verge of defeat, Gerald Ford, who had assumed the presidency after Nixon's resignation, made a last-ditch appeal to Congress for aid to save the crumbling regime: \"U.S. unwillingness to provide adequate assistance to our allies fighting for their lives would seriously affect our credibility throughout the world,\" Ford told a joint session of Congress in a speech written by Kissinger. Not a single member of Congress clapped, and two Democrats walked out. Congress rejected the aid and South Vietnam fell.\n\nVietnam was only the most graphic evidence that global containment was becoming the foreign policy equivalent of a bounced check. Nixon's resignation prompted a Democratic landslide in the midterm elections of 1974, which pushed Congress firmly into the post-toughness camp. \"Scratch a new House member,\" went the expression, \"and you're likely to find an old peace activist.\" Despite Nixon and Ford's efforts, the percentage of GDP spent on defense dropped from more than 8 percent in 1970 to less than 5 percent by 1977. Nixon and Kissinger helped get Chile's leftist president overthrown in 1973, just as Eisenhower had done in Guatemala in 1954. But Congress no longer tolerated that sort of thing. And when news of America's role in the coup became public, the Senate held investigations so grueling that they led to the conviction of Nixon's former CIA director for perjury. When civil war broke out in Angola, Ford funneled arms to anticommunist forces and Kissinger warned that \"if Moscow gets away with this one, it will try again soon in some other area.\" But Congress soon banned the aid and Ford meekly abandoned the policy, leading Kissinger to denounce his boss to reporters.\n\nWhat only a few New Left radicals had argued in the early 1960s\u2014that American democracy was threatened less by communists overseas than by what America's government did to fight them\u2014was becoming conventional wisdom. In 1974, Nixon resigned as a result of Watergate, a vast exercise in White House\u2013authorized criminality set in motion by his fury over newspaper leaks about Vietnam. And Watergate was only the most famous of the crimes that the U.S. government was found to have committed in the cold war's name. A few months after Nixon left office, the New York Times reported that the CIA had conducted a secret audit of its illegal behavior over the preceding decades, which insiders dubbed \"the family jewels.\" Pressed by Congress, Director of Central Intelligence William Colby tacitly admitted that the CIA had conspired to kill a bevy of foreign leaders. It had secretly opened and photographed hundreds of thousands of pieces of domestic mail. And it had administered LSD and other narcotics to unsuspecting Americans in an effort to develop tactics for mind control. In New York and San Francisco, the CIA had actually run brothels in which government-employed prostitutes lured men to secretly monitored rooms and then plied them with drugs to see how they would react. The Agency called it Operation Midnight Climax.\n\nAmerican foreign policy had passed through the looking glass. John Paton Davies was now back from Peru, with his government clearance restored. The policy that he had been persecuted for espousing\u2014diplomatic relations with communist China\u2014was becoming a reality. A New York Times columnist proposed that he be named America's first ambassador to the People's Republic. In 1971, the Senate Foreign Relations Committee\u2014once a symbol of congressional timidity, now a symbol of congressional power\u2014invited him to a kind of coming-out party for global containment's early foes. As cameras flashed, Missouri's Stuart Symington called Davies a \"great American.\" Arkansas' William Fulbright added, \"It is a very strange turn of fate that you gentlemen who reported honestly about conditions [in China] were so persecuted because you were honest about it. This is a strange thing to occur in what is called a civilized country.\"\n\nMeanwhile, in 1975, the Senate committee created to investigate the CIA hauled in Edward Lansdale, the inspiration for Edwin Hillandale of Ugly American fame, to answer questions about U.S. efforts to assassinate Fidel Castro. In incredulity and disgust, senators asked whether it was really true that Lansdale had proposed spreading a rumor among Cubans that Christ was about to return to take vengeance upon the godless Castro. (A U.S. submarine was then supposed to shoot flares into the night sky to convince Cubans that the Second Coming had begun.) Exploits that once seemed exciting and noble now looked immoral and a tad pathetic. The hubris of toughness, which Nixon and Kissinger had been keeping on life support, appeared dead. And in 1976, America elected the man who many assumed would be its undertaker.\n\nUnlike George McGovern, James Earl Carter had not spent his life opposing the cold war. In fact he hadn't thought about it much at all. His campaign autobiography, published in 1975, barely mentioned foreign policy, and he almost canceled a trip to Japan that same year when it turned out that none of his aides owned passports.\n\nBut the revelations of America's overseas crimes touched something deep within him. Jimmy Carter was, at his core, a scold. Growing up in a family marred by alcoholism, infidelity, and depression, he had been the straight arrow. When his father sent him to gather boll weevils that had fallen off the cotton shrubs near their Georgia home, young Jimmy rebuked other children for picking them directly from the plants. For his presidential inaugural address, he proposed a quote from Chronicles\u2014\"If my people...turn from their wicked ways; then I will hear from heaven, and will forgive their sin, and will heal their land\"\u2014until staffers convinced him it was too self-righteous. Well into his presidency, he still corrected aides' grammatical mistakes.\n\nCarter's moralism was also stoked by his experience with race. While only recently alerted to America's sins abroad, he was intimately familiar with the sins for which it needed to repent at home. And as a white Georgian who supported civil rights, he saw himself as something of an agent for that repentance. Carter's father was a staunch segregationist; he may even have taken part in a lynching. But Jimmy, the dutiful son in every other respect, bucked his father's views\u2014once even taking a date to a black church. As an adult, he courageously refused to join the racist White Citizens' Council, even though some members boycotted his business in response. In his inaugural address as Georgia's governor in 1971, he dramatically denounced racism in the state, thus gaining national attention. So while Carter entered presidential politics without firm foreign policy views, he was by temperament and experience conditioned to see world affairs less as a contest for power than as a quest for justice, less as a question of us versus them than right versus wrong. If the civil rights movement had overthrown a system of violence, fear, and hate, thus liberating the decency buried within his beloved South, Carter hoped to extend that struggle to America's role in the world. At home, building Martin Luther King's \"beloved community\" had required overcoming segregation. Around the world, building a \"global community\"\u2014one of Carter's favorite phrases\u2014would require overcoming the cold war.\n\nIn 1976, human rights were also shrewd politics. Liberals were outraged by the abuses of the CIA and by America's support for anticommunist tyrants overseas. Conservatives were furious over d\u00e9tente, which had led Nixon and Ford to downplay repression inside the U.S.S.R. During the campaign, Carter skillfully courted both sides, using human rights to attack Ford from both left and right. Unlike McGovern, Carter didn't challenge the toughness ethic head-on. In fact, his attacks on d\u00e9tente won him the support of many of the unreconstructed cold war Democrats, or \"neoconservatives,\" who had abandoned McGovern four years before.\n\nBut when Carter entered the White House, his balancing act collapsed. The reason was that when liberals and conservatives talked about human rights, they meant fundamentally different things. For conservatives, the essence of world affairs remained conflict. America was locked in an existential battle against an implacable foe, and human rights were a useful way of justifying and waging that battle. Since the cold war was a struggle between moral and immoral ideologies, anything that helped America win it was, in the long run, moral. For post-toughness liberals, by contrast\u2014as for the early New Left\u2014human rights were not a weapon in the cold war, but its antithesis. A moral foreign policy could only be built on love, not fear. And that meant seeing the world not as warring camps, but as a moral community, where all nations, regardless of ideology, cooperated to improve humanity's common estate.\n\nIf Carter flirted with both views during the campaign, as president he tilted clearly to the liberal side. His administration was not devoid of cold warriors: National Security Advisor Zbigniew Brzezinski was a hawk, although, sensing Carter's proclivities, he did not fully bare his talons at first. But the bulk of Carter's appointments were more dovish. Key post\u2013cold war intellectuals such as Gelb, Warnke, and Lake all got important jobs. Carter's ambassador to the United Nations, Andrew Young, and his assistant secretary of state for human rights, Patricia Derian, both came out of the civil rights movement. His first choice to head the CIA, Theodore Sorensen\u2014an outspoken liberal with no experience in intelligence work\u2014so frightened America's spooks that they derailed his nomination.\n\nCarter criticized human rights abuses in the Soviet Union but emphatically rejected the right's effort to use them to scuttle d\u00e9tente. To the contrary, he argued that it was obsessive anticommunism that had damaged human rights around the world. \"We are now free of that inordinate fear of communism which once led us to embrace any dictator who joined us in that fear,\" he declared a few months after taking office. The implication was clear: The world was less dangerous than the cold warriors had claimed. And if Americans stopped projecting their nightmares onto it, they could make it a more decent place.\n\nIn part that meant acknowledging the diminished significance of military force. Carter's first budget slashed defense spending by 5 percent, and he halted production of big-ticket weapons systems like the B-1 bomber and the neutron bomb. And he wasn't only less willing to build up America's armaments; he was less willing to deploy them. Six days into office, Carter pledged to withdraw U.S. troops from South Korea, whose authoritarian regime he had repeatedly condemned. \"We have an aversion to military involvement in foreign countries,\" he explained. \"We are suffering, or benefiting, from the experience that we had in Vietnam.\" And there was more. Carter not only abandoned Lyndon Johnson's policy of sending U.S. troops to fight third-world communism; he also moved away from Richard Nixon's policy of selling weapons to help third-world allies resist communism themselves. If Nixon and Kissinger had tried to practice global containment on the cheap, Carter moved away from the doctrine altogether. He announced sharp cuts in U.S. arms sales to the third world and he halted, or slashed, aid to a throng of anticommunist dictatorships. Carter was not always consistent: Some pro-American tyrannies continued to get aid. But compared to every president since NSC 68, his policies marked a genuine shift. \"We have witnessed perhaps, the end of a phase in our own foreign policy, shaped largely since 1945,\" declared Brzezinski. \"Preoccupation particularly with the cold war as a dominant [concern] of U.S. foreign policy, no longer seems warranted.\"\n\nIf post-toughness foreign policy meant reducing military support for pro-Western tyrants, it also meant reaching out to the kind of leftist regimes that the United States had once tried to overthrow. In 1977, Carter began to do what Raskin had proposed a decade and a half earlier: normalize relations with Cuba. He removed U.S. travel restrictions to the island, and the two nations established low-level diplomatic ties. In an interview with ABC, Castro told Barbara Walters that Carter was the first American president he actually liked. Carter also moved to normalize relations with communist governments in Angola and a newly unified Vietnam.\n\nThe Carter administration pointed to its Angola policy with particular pride. When Marxists seized power there, and Cuban troops began patrolling the country, Kissinger and others had predicted disaster unless the United States intervened. Instead Congress cut off aid to anticommunist forces, and Carter refused to reinstate it. And by 1977, the regime in Luanda was seeking closer ties to Washington and the Cuban troops were guarding the refineries where American companies drilled for oil. For Carter officials, the lesson was clear: \"If we maintain calm, keep our doors open, we are likely, in most cases, to find common interests with new governments,\" explained Undersecretary of State David Newsom.\n\nThat was too rosy. Not every leftist regime would abandon its anti-Americanism for a price. Carter sometimes seemed to forget that if nations had interests in common, they also had interests in conflict. But if Carter failed to establish a global community that cut across the cold war divide, neither did his pullback from global containment imperil American security. Some third-world countries went communist on his watch, but given their lack of military and economic power, they added little, if anything, to Soviet strength. Communism brought misery to people in countries like Angola, but the anticommunist alternatives were usually little better, and by refusing to arm regimes and rebel movements that brutalized their people, Carter won the United States some modest goodwill. By the middle of his term, the burying of the hubris of toughness appeared well under way. \"In the early period of the Carter administration,\" wrote Marcus Raskin, \"attempts were made...to organize a foreign policy that was not totally dependent on military power.\"\n\nAnd then, in the blink of an eye, events intruded, and the post-toughness moment was gone.\n\n## CHAPTER ELEVEN\n\n## FIGHTING WITH RABBITS\n\nOn April 20, 1979, Jimmy Carter got into a fight with a rabbit. He was fishing on a pond at his Georgia farm, and had just cast his rod into the water when he noticed the creature paddling toward him. It seemed angry. Its nostrils were flaring, its teeth were bared, and it was hissing in a menacing way. When it tried to board his boat, Carter clubbed it with his oar.\n\nUpon returning to the White House, he recounted the story over lemonade to several aides. They were dubious. Rabbits, they pointed out, are not known either for swimming or for attacking people. And if the animal was so threatening, why had the Secret Service agents onshore not come to the president's aid? Carter did not take kindly to having his honesty questioned. He had, after all, campaigned for president by pledging never to tell a lie. And so, propelled by two of his signature personality traits\u2014self-righteousness and micromanagement\u2014he went searching for a photo to corroborate his account. Eventually he found one: a White House photographer had captured the furry predator speeding away from his boat. Triumphant, the president showed the evidence to his staff.\n\nThat should have been the end of it. But in August, Press Secretary Jody Powell, perhaps looking to humanize his boss, passed on the anecdote to the Associated Press. Big mistake. The media, desperate for copy in a slow news month, leaped on the story. \"Bunny Goes Bugs: Rabbit Attacks President,\" screamed a front-page article in the Washington Post. In homage to the 1975 blockbuster about a killer shark, the paper dubbed Carter's assailant \"Paws.\"\n\nAll three nightly news programs featured the incident. Carter was questioned about it for a week, and when journalists learned that there was a photo floating around the White House, they filed a Freedom of Information Act request for its release. Conservative columnists Robert Novak and George Will alleged that Carter's weak response to his amphibious attacker mirrored his weak response to the Soviet threat. (It didn't help that rather than admitting that he had clubbed the rabbit, Carter, fearful of offending animal rights activists and in violation of his no-lying pledge, told the press that he had merely splashed water on it with his oar.) Evangelical leader Jerry Falwell explained that in the book of Revelation, swamp rabbits are associated with Satan. Carter, he declared, should have throttled the beast.\n\nJournalists loved the rabbit story for the same reason they loved stories about Gerald Ford falling down stairs: It confirmed their stereotypes. By the summer of 1979, the press no longer viewed Carter as an idealistic outsider bringing down-home decency to Washington. They viewed him as a wimp. Post-toughness foreign policy, with its vision of a world defined more by cooperation than conflict, more by love than fear, had appeared politically ascendant in the mid-1970s, with the public sick of war and comforted by d\u00e9tente. But by the time rabbit-gate hit the papers, the world looked nastier, and for Jimmy Carter, Washington was getting nastier, too.\n\nCarter had been blunt: There would be no more Vietnams. In 1978, when Soviet arms and Cuban troops poured into Ethiopia to help its Marxist president retake a stretch of desert called the Ogaden from neighboring Somalia, Carter rebuffed Brzezinski's suggestion that America send a naval task force to the Indian Ocean in response. His caution made sense, given that Somalia had committed the initial aggression, and that the Soviets and Cubans were actually restraining Ethiopia from launching a wider war. Still, among hawkish foreign policy watchers, the events in Africa's horn rankled. D\u00e9tente had been sold to Americans, in part, as a way of curbing Soviet meddling in the third world. (It was one of Kissinger's creative untruths: Moscow had never promised to do any such thing.) Yet while U.S. diplomats were backslapping their Russian counterparts at chummy superpower summits, communists had taken Vietnam, then Angola, and now Ethiopia. Brzezinski warned that the mood in Washington was shifting. American losses in the third world, he told his boss, \"are creating the conditions for domestic reaction.\"\n\nHe didn't know how right he was. The Ogaden may not have mattered very much to the United States, but Iran did. For a quarter century it had been a prize ally. In 1973, with the U.S. economy reeling from an Arab oil embargo, Iranian crude had been an economic lifeline. And in Vietnam's wake, with the United States no longer willing to contain third-world communism with American boys, the Nixon administration had turned the Shah's kingdom into a virtual U.S. armory, selling it some $20 billion of weapons designed to keep the Soviets from the oil fields of the Persian Gulf. By 1976, there were seventy thousand Americans living in Iran, mostly tending to its oil industry and burgeoning military machine. And while Carter privately chided the Shah for his dismal human rights record, even he considered Tehran too valuable to alienate because of moral qualms. On December 31, 1977, the Carters went to celebrate New Year's Eve at the Shah's royal palace, which one journalist compared to Versailles in the days of Louis XIV. As the minutes counted down to 1978, and waiters distributed caviar, Dom Perignon, flaming ice cream, and diced partridge, the president called Iran \"an island of stability in one of the most troubled areas of the world.\" He noted the \"respect and the admiration and the love\" that Iranians felt for their leader. And he said he felt the same way: \"There is no leader with whom I have a deeper sense of personal gratitude and personal friendship.\"\n\nEight months later, the island of stability was underwater. Riots gripped the country; the Shah spent the following New Year's Eve under siege; by mid-January 1979 he was in exile, replaced by the Ayatollah Ruhollah Khomeini, whose relationship to the United States was not exactly defined by gratitude and friendship. He called America the \"Great Satan.\"\n\nThen, in June, Nicaragua's pro-American dictator, Anastasio Somoza, fell to the leftist Sandinistas, giving Moscow its first new potential beachhead in the Western Hemisphere since the coup in Chile in 1973. In reality, these setbacks weren't as disastrous as Carter's critics claimed. Iran's new theocratic regime, while bad news for the United States, was bad news for the U.S.S.R., too, since it helped inspire Muslim opposition to communist rule across the border in Afghanistan and within the Soviet Union itself. Communist governments in Angola and Ethiopia offered the Soviets few benefits and lots of headaches. By decade's end, Moscow's new African clients were costing so much money and proving such poor Marxists that some Kremlin officials suggested pulling the plug. And, most important, even as communism made gains in the developing world, it was starting to rot inside the U.S.S.R. In 1975, a massive agricultural shortfall forced the Soviet Union to begin importing grain, something it would do every subsequent year until it ceased to exist. Despite high oil prices, which kept the Soviet economy afloat, Russians by 1979 were loudly complaining about shortages of medicine, soap, toothpaste, and thread. The Soviet Union was like a boxer who kept landing jabs but whose coronary arteries were contracting, slowly shutting off blood to the heart.\n\nBut in the late Carter years, Americans weren't particularly focused on the Soviet Union's weakness; they were too preoccupied with their own. A president skilled in the politics of manhood might have found ways of assuaging those fears at limited cost. But as a Democrat with little foreign policy experience, Carter was not well positioned to do so. And he didn't even fully grasp the problem. By 1979, the ethic of toughness\u2014which had appeared dead\u2014was rising from the grave. It would not be laid to rest until Carter was out of office and Ronald Reagan had taken his place.\n\nThe story of the resurgence of the toughness ethic in the 1970s is also the story of neoconservatism. And the story of neoconservatism is, to a large degree, the story of an impish, pugilistic garment worker's son named Irving Kristol. Even as a child, Kristol had the instincts of a conservative, which is to say, he revered authority and tradition. He experienced Orthodox Judaism in its crudest and dumbest form: The rabbi at his Brooklyn yeshiva told him to spit when he passed a church, slapped him in the mouth when he misbehaved, and made him deliver his bar mitzvah speech in Yiddish, a language Kristol did not understand. Yet unlike many of his contemporaries, Kristol venerated traditional Judaism nonetheless, even if he could never quite explain why.\n\nIn the late 1930s, he enrolled at City College of New York and became a Trotskyist, ostensibly committed to world revolution. Yet underneath he remained a conservative. He began reading Reinhold Niebuhr and other Christian theologians, which was an odd thing for a Jewish Trotskyist to do, and found himself attracted to their emphasis on original sin. He venerated Lionel Trilling, the great Columbia literary critic, who, like Niebuhr, emphasized the tragic element of the human condition. Then, after graduating, Kristol was drafted into World War II, where he fought in a unit \"heavily populated by thugs or near-thugs from places like Cicero, [Illinois] (Al Capone's home base).\" The experience left him even more skeptical of human nature. His fellow enlisted men, Kristol discovered, were \"inclined to loot, to rape, and to shoot prisoners of war. Only army vigilance kept them in check.\" And so for Kristol, the vigilance of people in power became very important. \"My wartime experience,\" he later wrote, \"did have the effect of dispelling any remnants of antiauthority sentiments (always weak, I now think) that were cluttering up my mind.\"\n\nBy the 1950s, Kristol was no longer a Troskyist; he was a cold war liberal. Calling himself a conservative barely occurred to him, since, as far as he could tell, there were no conservative intellectuals in 1950s America, and if there were, they hated the New Deal, which he did not. But it was the conservative elements of '50s liberalism\u2014its anticommunism, its anti-utopianism, its belief that ordinary people were no better than the societies in which they lived, and quite possibly worse\u2014that Kristol liked best. In the '50s, when you could be a liberal and a believer in the toughness ethic at the same time, Irving Kristol was both.\n\nBut by the early 1970s, that was no longer possible. The Democratic Party was repudiating the toughness ethic, and so Irving Kristol decided that to remain an old-fashioned toughness liberal, he had to become, oddly enough, a \"conservative.\" Many other \"neoconservatives\" clung to the term liberal. They accused the McGovern Democrats of linguistic theft. But for his part, Kristol embraced the new term. He explained that when you're named Irving, you learn not to worry about labels.\n\nWhat made Irving Kristol a neoconservative was not his support for Vietnam. In fact, like many neocons in the late 1960s and early '70s, he was ambivalent about the war. What Kristol and his allies really cared about was not the war but the movement against it: the New Left. And what they hated about the New Left was its revolt against authority, its belief that by liberating Americans from their existing institutions it could build far better ones in their place. For Kristol, who believed that people liberated from traditional constraints were little more than animals, this was utopianism: \"the main source of all evil in the world.\" And if liberals were no longer willing to do what the army had done to those thugs from Cicero, Illinois\u2014put down their convulsions by force\u2014then he would ally himself with whoever would.\n\nThe first institutions that the neoconservatives set out to defend against the utopian threat were universities, which held a special place in their heart. At City College in the 1930s, Kristol and his friends had been too poor to buy a sandwich. Yet they used that ramshackle institution as a springboard from the ghettos of Brooklyn and the Bronx into a world of intellectual exploration and middle-class comfort. It was quite a shock, therefore, to watch the \"rich college fucks\" (in Daniel Patrick Moynihan's words) of SDS denounce America's great universities as tools of cold war repression. Amazement turned to horror as the pampered radicals employed rhetorical and sometimes actual violence to achieve their demands. And horror turned to fury as campus administrators buckled before the student mob.\n\nBy the late 1960s, SDS, radicalized by Vietnam and by the brutality with which the police often greeted antiwar protest, had abandoned non-violence in favor of what some called \"armed love.\" (\"We must learn to fight as well as seek love,\" explained a New York\u2013based SDS faction called the Motherfuckers, which blended hippie values with antiwar fury. \"We must take up the gun as well as the joint.\") Kristol and the neoconservatives were not surprised: In their mind, liberation from existing authority usually ended in blood. In 1968, the SDS leader at Columbia, a suburban Jewish kid from New Jersey named Mark Rudd\u2014whose mother once brought him a home-cooked meal in the middle of a sit-in\u2014sent a letter to the university's president. \"There is only one thing left to say,\" he wrote. \"It may sound nihilistic to you, since it is the opening shot in a war of liberation.... 'Up against the wall, motherfucker, this is a stickup.'\" Two years later, a garment worker's son turned sociologist named Nathan Glazer, who had graduated from City College in 1944, announced that he had seen enough. \"How does a radical\" like himself, he asked, \"end up by early 1970s a conservative?\" His answer: by watching nihilists like Rudd, who sought the \"destruction of authority\u2014any authority.\" He had \"learned,\" Glazer explained, \"in quite strictly conservative fashion, to develop a certain respect for what was.\"\n\nIf that respect applied to universities, it also applied to cities. Neoconservatives loved cities; urban America was the only America most of them knew. And in the late 1960s, they came to believe that the New Left and its soft liberal enablers were destroying city life. In 1965, Kristol, Glazer, and Daniel Bell helped found the Public Interest, which cast a skeptical eye on liberal efforts to radically improve the estate of the urban poor. The writers for the Public Interest had no ideological problem with government; unlike traditional conservatives, they did not worship the free market. But they believed in authority. And they decided that in its efforts to liberate the poor from the conditions that oppressed them, the New Left was undermining the traditional forms of authority\u2014be they family, police, ward boss, church, or neighborhood school\u2014that made ghetto life tolerable at all. When reformers bused kids across town, or increased welfare payments to single mothers, or tore down old neighborhoods to build new ones in their place, they may have believed they were merely removing the obstacles that denied people opportunity and happiness. But they were actually meddling with entrenched patterns of life, patterns that reflected human nature as it really was, not as the New Left utopians wished it to be. Trying to change those patterns, warned Kristol and his colleagues, would create more misery and violence, not less. The more dramatic the effort at progress, the more dramatic the failure that would result.\n\nWhat was true for universities and cities, the neocons argued, was also true for families. Patriarchal authority might be arbitrary and unjust; it might involve force. But it was rooted in human nature. Efforts to liberate women and children from the traditional family, insisted Commentary editor Norman Podhoretz\u2014the second most important neoconservative polemicist, after Kristol\u2014would only make things worse.\n\nNeocons made a similar argument about the Democratic Party, an institution to which, in the 1970s, many of them still felt attached. Between 1968 and 1972, a commission headed by\u2014who else?\u2014George McGovern had instituted a flurry of reforms aimed at fumigating the party's smoke-filled rooms. But a little-known neoconservative political scientist named Jeane Kirkpatrick insisted that the reforms had backfired. Rather than making the party more democratic, they had merely shifted power to a new class of political consultants who were even more selfish and unrepresentative than the party bosses they replaced. It was no surprise that Kirkpatrick, like Kristol, admired Niebuhr. Like Niebuhr, she insisted that the language of reason often concealed the reality of interest, and that the pursuit of political perfection was a dangerous game. \"In party reform as in life,\" Kirkpatrick wrote, \"good intentions are not enough, and wishing does not make it so.\"\n\nIf neoconservatives loved universities, cities, patriarchy, and the Democratic Party, they also loved American power. Since many of them were second-generation American Jews, who had entered the middle class just a few years after their European brethren entered Dachau and Buchenwald, they felt with particular intensity that only American strength kept barbarism at bay. And since most of them knew little about Russia, but a lot about communism, having witnessed Stalinist thuggishness and duplicity from the other side of the City College cafeteria, they generally interpreted Soviet actions as the result of a fanatical ideological drive. Stalinism and Nazism had taught them that evil was real and that America\u2014even in its worst moments\u2014was not it. In the late 1960s they watched in disgust as the New Left compared Lyndon Johnson to Adolf Hitler and Fidel Castro to Jesus Christ. And in the 1970s that disgust only grew as they saw America retreat, its enemies swarm, and post-toughness liberals declare that harmony was breaking out across the globe.\n\nThe neoconservatives generally insisted that when it came to foreign policy they held the same beliefs as John F. Kennedy and Lyndon Johnson; it was the post-toughness liberals who had changed. \"In order to become a neoconservative,\" Kristol quipped, \"all you had to do was stand in place.\" There was a lot of truth in that. Like Kennedy and Johnson, the neocons saw Hitlers everywhere, and like Kennedy and Johnson, they saw American power as constrained not by finite resources but merely by an irresolute will. What the neocons did not acknowledge was that in carrying on the ultra-realist tradition of the Best and the Brightest they were actually abandoning the older realist tradition of Niebuhr, Morgenthau, Lippmann, and Kennan, which had recognized that American power and wisdom were limited, and had seen anticommunism as a vital but partial truth. While Kristol and Kirkpatrick paid homage to the tragic toughness of the 1940s, they were actually heirs to the crusading toughness of the 1960s. In their obsession with anti-utopianism, they forgot that anti-utopianism, when pushed too far, can become hubristic, too.\n\nDuring the 1976 campaign, the neoconservatives had held out some hope for Jimmy Carter. They were still mostly Democrats, and he had attacked Ford from the right as well as the left. But when Carter stocked his administration with post-toughness liberals like Warnke, Lake, Gelb, and Young, the neocons were enraged. Carter offered only one of the neocons a job: as special envoy to Micronesia. (\"Not for Polynesia. Not for Macronesia,\" joked Podhoretz's stepson-in-law, Elliott Abrams. \"But Micronesia.\") So they decided to tear him apart.\n\nIf the Camelot gang had often compared the 1950s to the '30s, the neoconservatives now called the '70s America's decade of drift and peril. \"The parallels with England in 1937 are here, and this revival of the culture of appeasement ought to be troubling our sleep,\" wrote Podhoretz in 1977. (If that was too subtle, the essay was accompanied by a drawing of Carter carrying an umbrella.) Kennedy had cited the Gaither Commission as evidence that Moscow was pulling ahead while America slept. Now the neocons pointed to Team B, a group of cold warriors\u2014led by Paul Nitze, the crusty author of NSC 68; Richard Pipes, a right-leaning historian of Russia; and a young arms control wonk named Paul Wolfowitz\u2014who claimed that while America was slashing its defense budget Moscow had taken a decisive edge. And like the men of Camelot, the neocons often claimed that America's \"culture of appeasement\" was a culture of effeminacy; the missile gap stemmed, once again, from a manliness gap. Podhoretz, echoing Schlesinger in 1958, announced that homosexuality was sapping America's martial spirit, just as it had sapped Britain's after World War I. Carter, a writer for the Wall Street Journal declared, \"wouldn't twist arms. He didn't like to threaten or rebuke.\" He donned sweaters. \"He even kissed Brezhnev!\" He was America's first female president.\n\nThe neocon critique of Carter was even harsher than Kennedy's and Schlesinger's attack on Ike. For the restless cold war intellectuals of the late 1950s, Eisenhower was simply tired and complacent. For the neocons, by contrast, Carter's problem went deeper. He did not merely lack the energy to wield force; he inhabited a fantasy world in which force no longer mattered. He was like the university presidents who spoke of understanding student radicals because they lacked the guts to expel them or the mayors who talked about the root causes of crime because they feared locking up rioters. Like other authority figures unwilling to defend their domains, Carter was using the rhetoric of love to mask a failure of will.\n\nThe result, from the neocon perspective, was predictable. Because Carter would not use force to maintain order, the world's \"juvenile delinquents\"\u2014a phrase Kristol applied to anti-American regimes in the third world\u2014were running wild. And as in America's cities and college campuses, the result was not harmony and love, but barbarism and chaos. It was true, Kristol conceded, that American \"power may indeed corrupt.\" But \"we are now learning that, in the world of nations as it exists, [American] powerlessness can be even more corrupting and demoralizing.\"\n\nFor Jeane Kirkpatrick, the Oklahoma-born Baptist and mother of three who would become an unlikely icon in a movement dominated by Jewish men, it was d\u00e9j\u00e0 vu. The same post-toughness na\u00efvet\u00e9 that had undermined traditional authority in the Democratic Party was now undermining American authority around the world. Once again a romantic view of human nature, and a lack of respect for institutions as they actually were, was producing unintended consequences of the ugliest kind. Kirkpatrick was very upset about Nicaragua and Iran. In her view, Somoza and the Shah were a bit like the bosses who had run the old Democratic Party. Their authority did not stem from democratic principle; it stemmed from tradition, and it relied on force. And so when antigovernment militants in Nicaragua and Iran rebelled, demanding liberation from authoritarian rule, Carter sympathized. He told Somoza and the Shah that America, with its new fidelity to human rights, would not support a military crack-down. Instead America's longtime clients would have to find a more democratic basis on which to continue their rule. In Kirkpatrick's view, this was na\u00efvet\u00e9 on stilts. Authoritarian regimes like Nicaragua's and Iran's were not thin crusts, which could be torn off to reveal a glimmering democracy underneath. They reflected the culture of their societies, societies no more capable of sustaining liberal democracy than of flying a man to the moon. In a 1979 Commentary essay titled \"Dictatorships and Double Standards,\" Kirkpatrick wrote that \"democratic governments have come into being slowly, after extended prior experience with more limited forms of participation.\" The process usually took \"decades, if not centuries.\" Believing Iran and Nicaragua could democratize overnight was like imagining that you could eliminate poverty by bulldozing a slum. Carter, she argued, \"is, par excellence, the kind of liberal most likely to confound revolution with idealism, change with progress, optimism with virtue.\"\n\nFor Kirkpatrick, it was the same, awful story: Believing people were fundamentally better than the institutions that governed their lives, and would create a freer, more loving society if liberated from external constraint, was as na\u00efve in Iran and Nicaragua as in Harlem. It was, she believed, high time that Americans accepted the world for what it was: a nasty place in which people were generally no better than their governments, professions of goodwill usually weren't worth the paper they were written on, and moral progress\u2014when it came at all\u2014was grindingly slow. Foreign policy was a long, hard, often ugly battle between a lesser evil called the United States and a network of greater evils headquartered in Red Square. To think it was anything else, or could become anything else anytime soon, would only make things worse. \"We are daily surrounded by assertions that force plays no role in the world,\" she declared. \"Unfortunately it does.\"\n\nIn the final months of 1979, those assertions suddenly stopped. On November 4, Iranian militants scaled the walls of the U.S. Embassy in Tehran; then flung open the gates for their armed colleagues as local police stood by. By nightfall, seventy-six Americans sat blindfolded, tied to chairs or beds, in the ambassador's house. Nothing could have more graphically illustrated the neocons' claim that America was weak in a dangerous world.\n\nThe hostage crisis transfixed the nation. American television reporters set up shop outside the embassy gate, where they filmed shackled and blindfolded hostages being paraded before jeering crowds. Local news stations hunted down hostage relatives from their city or state, staking out their houses and broadcasting their anguished pleas. On the CBS Evening News, Walter Cronkite signed off every night by announcing the number of days the Americans had been held captive. If more than a decade earlier his on-air revulsion during the Tet offensive had marked the end of the toughness era, now his on-air revulsion at the hostage crisis seemed to mark the end of the post-toughness era as well.\n\nCarter tried diplomacy and imposed sanctions. But the big question\u2014the neocon question\u2014was force. Secretary of State Cyrus Vance argued no. He had been a junior member of the cold war establishment: Yale and Yale Law, a gunnery officer during World War II, a distinguished corporate lawyer, McNamara's deputy secretary of defense. But Vietnam had changed him; he was now a senior member of the post-toughness foreign policy class, a patron to men like Gelb, Warnke, and Lake. He was less of a moralist than McGovern and Carter but he shared their belief that military force was usually useless or worse. Diplomacy and accommodation, while hardly glamorous, were the best tools America had. \"Most Americans,\" he declared after the hostages were seized, \"now recognize that we alone cannot dictate events. This recognition is not a sign of America's decline. It is a sign of growing American maturity in a complex world.\"\n\nHis chief bureaucratic rival, National Security Advisor Zbigniew Brzezinski, did not believe that for a minute. Born in Poland and educated in Canada and at Harvard, Brzezinski was naturally aggressive. His relentless style and angular features led Carter's chief of staff to dub him \"Woody Woodpecker.\" Playing soccer once at a White House picnic, he charged the ball with such ferocity that colleagues ran for cover.\n\nIn his early career, he had been an unabashed hawk. During the Cuban Missile Crisis, he called on the Kennedy administration to bomb. In 1965, he debated alongside McGeorge Bundy, and against Hans Morgenthau, in favor of American intervention in Vietnam. As late as 1968, he was still urging America to stay there, if necessary for thirty years. We must show \"we have the staying power,\" he pleaded. America must not \"chicken out.\"\n\nBut by the 1970s, Brzezinski could see which way the wind was blowing. And he was not one to tilt at windmills. So at Columbia, where he taught, he changed the name of his center from the Research Institute on Communist Affairs to the Research Institute on International Change and shifted his focus from anticommunism to interdependence. By the time he entered the Carter administration he was, like his boss, heralding the emergence of a \"global community\" where \"elements of cooperation prevail over competition.\"\n\nCritics suspected that this was all an act, that in his heart, Zbig still considered the world a harsh and unforgiving place. As early as 1978, he had urged a show of force in response to the Soviet intervention in the Horn of Africa. And in 1979, when Somoza and the Shah fell, he sloughed off his prior talk of global cooperation like an outer skin. On Iran he urged military action, even though it might endanger the hostages, so America's enemies would learn that molesting the United States still carried a price. He mocked Vance as \"the last Vietnam casualty,\" declaring that his colleague was \"so worried after the disastrous use of force in Vietnam that he lacks the will to use force again.\" When Carter fretted about the loss of life that an attack on Iran might bring, Brzezinski bluntly informed him that international affairs were \"not a kindergarten.\"\n\nIt was a sign of how far the pendulum had swung\u2014of how eager Carter now was to use force\u2014that he authorized a rescue effort that stood little chance of success. (The plan's proponents actually called that one of its virtues: The mission was so implausible, they argued, that it would take the Iranians by surprise.) On April 24, 1980, eight U.S. helicopters flew to a secret location in the Iranian desert. There they were meant to rendezvous with cargo planes that would help them refuel before flying hundreds more miles to another secret base, nearer Tehran. From there the rescuers would transfer to trucks, drive to the U.S. Embassy, overwhelm the captors, and bundle the hostages onto helicopters, which would fly to another secret location, where planes would whisk them out of Iran. That was the plan. In reality, as the helicopters reached their first desert outpost, massive clouds of dust caused several of them to malfunction, leading Carter to abort the mission. To make matters worse, a helicopter and a plane crashed into one another as they were trying to leave, killing eight American troops. At 1 A.M. on April 25, Carter broke the news to the American people. It was yet another humiliation. Vance had already resigned in disgust.\n\nAnalytically, it made little sense to blame the disasters in Iran on post-toughness foreign policy. The Islamic revolution, after all, was the bitter fruit of America's decades-long support for that country's tyranny. The real presidential villains were not Carter\u2014who just happened to be in office when the dam broke\u2014but Eisenhower, who had overthrown a democratically elected Iranian prime minister in 1953, and Nixon, whose endless weapons sales had made the Shah think he was invincible. The problem wasn't that America hadn't seen Iran through a cold war prism; it was that America had only seen Iran through a cold war prism. And had Carter followed Kirkpatrick's advice and taken forceful military action to save the Shah, he just would have made Iranians hate the United States, and its puppet regime, even more. As for the hostages, Carter did try military intervention. The failure of his rescue effort actually illustrated one of post-toughness foreign policy's main axioms: Military force often doesn't work.\n\nBut for Carter politically, the events in Iran were a catastrophe. And, incredibly, things got worse. In the middle of the hostage nightmare, eighty-five thousand Soviet troops entered Afghanistan to prop up a communist regime threatened by Islamist rebels (rebels who, ironically, drew inspiration from Iran). Never before had Moscow invaded a country it had not conquered during World War II. The neocons brandished their Munich analogies. Brzezinski warned that Afghanistan was the first step in a Soviet bid for the oil fields of the Persian Gulf, an area he called \"the arc of crisis.\"\n\nIn truth, as Soviet archives would later reveal, Kremlin officials had no such designs; they merely hoped to avoid a fundamentalist Afghan regime that might stir up rebellion among Muslims inside the U.S.S.R. At one point, Soviet premier Leonid Brezhnev even asked an aide where this \"arc of crisis\" that the Americans kept talking about actually was. Far from being a launching pad for further conquests, the Afghan invasion left the Soviet Union dangerously overstretched. To hear the neocons tell it, the U.S.S.R. in the 1970s was like Germany in the 1930s, growing ever more powerful as the Western democracies buried their heads in the sand. But, geopolitically, the 1970s actually proved a net loss for the Soviets, since both Egypt, once Moscow's most important ally in the Middle East, and China, once Moscow's most important ally in Asia, shifted to America's side. The Afghan invasion proved the biggest net loss of all, as Moscow threw vast sums of money and vast numbers of lives into a savage, unnecessary, unwinnable war that hastened the demise of the Soviet Union itself.\n\nBut to the Carter administration, already reeling from a string of setbacks and humiliations, the Afghan invasion looked downright apocalyptic. Aides offered Carter a range of retaliatory measures. He chose them all. He embargoed grain exports to the U.S.S.R., expelled Soviet diplomats, cut cultural ties, restricted Aeroflot flights into the United States, and announced that America would boycott the 1980 Olympic Games in Moscow. And that was only the beginning. If he had backed away from post-toughness foreign policy when the hostages were seized, he now ripped it to shreds. Human rights became an afterthought, as America rushed aid to autocratic allies like Pakistan. The CIA was unleashed. (\"We need to remove unwarranted restraints on America's ability to collect intelligence,\" declared Carter, failing to mention that he was one of the people who had imposed those restraints in the first place.) Arms control was now dead, as Carter told the Senate not even to bother voting on the SALT II nuclear agreement, which his administration had painstakingly negotiated. U.S. forces flooded into the Middle East as the Pentagon scrambled to build military installations in the dictatorships of Kenya, Somalia, and Oman. Americans, Carter declared, must abandon their dream of a \"simple world\" of \"universal goodwill\"\u2014who could have encouraged that?\u2014and face the \"sometimes dangerous world that really exists.\"\n\nIt was like a time warp; all the old analogies came flooding back. Brzezinski compared Carter's response to the Afghan invasion to Truman's response to communist aggression in Turkey and Greece. The White House unveiled a \"Carter Doctrine,\" deliberately meant to echo Truman's in 1947, under which America would forcibly repel any threat to the Persian Gulf. \"We must pay whatever price\"\u2014where had Americans heard that before?\u2014\"to remain the strongest nation in the world,\" Carter declared. \"Aggression unopposed,\" he warned, \"becomes a contagious disease.\"\n\nThat spring, Vice President Walter Mondale addressed the U.S. Olympic Committee, urging them to comply with Carter's call for a boycott of the 1980 games. Going to Moscow, he declared, would be like going to the Berlin Olympics of 1936. \"The story of Hitler's rise,\" he declared, was \"a chronicle of the free world's failure: of opportunities not seized, aggression not opposed, appeasement not condemned.\" Now America faced the same choice and \"history holds its breath.\"\n\nIt was a resurrection story, or so it seemed. Born from the shame of Munich, toughness foreign policy had grown to excess in the golden decades of the early cold war, died in the rice paddies of Vietnam, and was now being reborn in the snowcapped peaks of the Hindu Kush. Except that it was not; it was all a mirage. The hubris of toughness was not being revived, any more than Franklin Roosevelt had revived Woodrow Wilson's hubris of reason during World War II. It had simply never received a proper burial. No post-Vietnam president had successfully reconciled the politics of manhood with the reality of limits. Until Ronald Reagan, a man who considered himself FDR's heir, and in a strange sort of way, actually was.\n\n## CHAPTER TWELVE\n\n## IF THERE IS A BEAR?\n\nOne summer day, sometime in the 1980s, an army colonel named Samuel Trautman went to see a Green Beret who had served under his command in Vietnam. The man had fallen on hard times. Upon his return to the United States, hippies had thrown garbage on him and called him \"baby killer.\" Brutal flashbacks from his days as a prisoner of war had left him unable to hold down a job. Now he was a prisoner again, arrested for rampaging through a northwestern town where he had gone to find the last remaining member of his unit, only to learn that the man had died from Agent Orange.\n\nTrautman came with an offer. The Green Beret could go free on one condition: He must return to Vietnam, on a top-secret mission to find the POWs still under Hanoi's control. In the courtyard of a prison labor camp, as inmates broke rocks under a sweltering sun, John Rambo looked at his former commander and asked, \"Do we get to win this time?\"\n\nIn 1985, Rambo became the smash hit of the summer by answering that question with an earsplitting, kick-ass \"Yes!\" In the film, Rambo returns to Vietnam, kills communists by the dozen, frees his imprisoned comrades, and is awarded the Medal of Honor by America's patriotic new president, Ronald Reagan.\n\nArt, if you can call it that, was imitating life. In the 1980s, Reagan\u2014a career Hollywood actor whose wife sometimes greeted him when he left the Oval Office by yelling \"cut\"\u2014often seemed to be starring in a movie of his own, which followed the Rambo script. Less than a month after taking office, he publicly awarded the Medal of Honor to a Green Beret who had saved the lives of eight men while enduring thirty-seven bullet and bayonet wounds in eastern Laos, but had been denied the award because of a technicality. America's Vietnam veterans, Reagan told a Pentagon crowd, \"came home without victory not because they'd been defeated but because they'd been denied permission to win.\"\n\nIn Reagan: The Movie, America, after a long string of indignities and defeats, finally remembers how to win. It returns to Vietnam, slays its demons, and recaptures the confidence of a bygone age. It was as if the country had turned back the clock to 1964, relegating the entire post-toughness era to the status of a bad dream. (The other big movie of 1985 was actually titled Back to the Future.) In 1982, Hasbro Toys resumed producing the G.I. Joe action figure, which it had discontinued the year after Saigon fell. By 1985, it was America's bestselling toy. That same year in New York, Vietnam vets got the ticker-tape parade they had long been denied. \"This country has really needed to flex its muscles,\" declared the man who played Rambo, Sylvester Stallone. \"The other little nations were pulling at us, saying, 'You're bullying. Don't tread on us.' So we pulled back.... And what happened, as usual, is people took kindness for weakness, and America lost its esteem. Right now, it's just flexing. You might say America has gone back to the gym.\"\n\nReagan: The Movie delighted audiences. But it was a fantasy. When it came to foreign policy, the real Reagan didn't turn back the clock to 1964. He never seriously considered enforcing global containment with U.S. troops. Instead of refighting Vietnam, he created Potemkin Vietnams where America won because it could not possibly lose. He served up victories on the cheap, triumphs without risk. Reagan's critics often accused him of reviving the chest-thumping spirit that had led to Vietnam. But they missed the point. For Reagan, chest-thumping was in large measure a substitute for Vietnams, a way of accommodating to the new restraints on U.S. power while still helping Americans feel strong and proud. Reagan didn't revive the hubris of toughness. He did what Carter had tried but failed to do: He performed the last rites.\n\nThe key was his relationship to evil. Or, more precisely, his lack of relationship. For a half century, evil had lain at the heart of the toughness ethic. American intellectuals had discovered it in the 1930s and '40s as the Depression and Stalinism and Nazism shattered the progressive dream of reason. By the mid-1960s, when America entered Vietnam, the discovery had become an obsession. Since evil was terrifying, so were the Vietcong, and since evil was permanent, it was always 1938. The New Left, and for a time, Jimmy Carter, had rebelled against that, insisting that despite their differences, nations\u2014like individuals\u2014were capable of cooperation, harmony, even love. But Carter's timing and his political instincts were both bad, and in 1979 post-toughness foreign policy collapsed.\n\nFor neoconservatives like Kristol and Kirkpatrick, Carter's failures reaffirmed the toughness ethic. They showed that in foreign policy, as in urban policy, dramatic progress was usually a mirage; people just weren't built that way. The key was to embrace the lesser evil and defend it with force. When Reagan's ideological allies talked about overcoming the Vietnam syndrome and getting back to fighting the cold war, that's what they meant. It's what they believed Ronald Reagan had been elected to do. But the joke was on them because Reagan didn't believe in evil, at least not as a powerful or permanent thing. He believed what the New Left believed: that beneath all bad authority lay good people, waiting to be set free. Communism, the nuclear standoff, the cold war\u2014he saw them as ugly crusts concealing the glorious humanity concealed beneath. \"We have it in our power to begin the world again,\" Reagan liked to say, quoting Thomas Paine. It was, the columnist George Will complained, \"the least-conservative sentiment conceivable,\" a hint of the transgressions to come.\n\nReagan didn't fret about evil because he didn't see misfortune. He was America's Mr. Magoo: No matter how many buildings collapsed around him, he kept marching to the merry beat of his internal drummer. (In this regard, he was the temperamental opposite of Richard Nixon, who seethed with bitterness no matter how high he climbed.) Reagan's childhood can fairly be described as harrowing\u2014yet he refused to be harrowed. His father was a drunk, a man who kept destroying the family's screen door because rather than opening it he would walk right on through. One day when Ron was eleven, he found his old man passed out, spread-eagle in the snow, and had to drag him by the armpits to bed. Not surprisingly, the family was poor. No longer able to afford chicken on Sunday, they began eating liver, which the local butcher usually sold as pet food. Yet in his autobiography, Reagan called his hometown \"heaven\" and his childhood \"one of those rare Huck Finn\u2013Tom Sawyer idylls.\" About the Depression, which hit just as he went looking for his first full-time job, and which sent armies of vagrants scavenging for food, Reagan remembered \"that there was a spirit of helpfulness and yes kindliness abroad in the land.\" Among friends, his optimism became a running joke. In Hollywood, when rain threatened to cancel a shoot, he would tell the director, \"Don't worry. It's going to clear up any time.\" He couldn't play villains because he couldn't get in touch with his dark side. And he hated attending funerals. During the recession of 1982, economic advisers showed him a litany of grim statistics in a bid to convince him to change his policies. To their amazement, Reagan focused on the only positive one.\n\nThis almost surreal optimism may have grown out of Reagan's childhood efforts to cope with his father's alcoholism. The children of alcoholics sometimes take refuge in dream worlds, and Reagan, remarked an early girlfriend, \"had an inability to distinguish between fact and fantasy.\" He also inherited his cheeriness from his mother, a deeply religious woman who raised him in the Church of Disciples, a spinoff from Unitarianism that downplayed the idea of sin. Reagan spent his youth marinating in this theological tradition. He spent every Wednesday night and virtually all of Sunday in church, dated the local minister's daughter, and attended Eureka College, a Disciples' institution. And from the Disciples he acquired a view of human nature right out of the Social Gospel. \"My personal belief,\" Reagan wrote to a friend in 1951, \"is that God couldn't have created evil so the desires he planted in us are good.\" In a 1981 speech, Kirkpatrick, whom Reagan appointed ambassador to the United Nations, called her boss's election a reproach to \"an age and to a society like ours, which regularly deny the existence of evil.\" But she had it all wrong. In a fundamental sense, Reagan did deny the existence of evil: He saw it as merely the transitory and artificial absence of good.\n\nTo be sure, Reagan loathed communism. His animosity was ancient and intimate; it dated from his fights with communists in Hollywood in the 1940s. But while Reagan shared the right's hatred of communism, he didn't share its fear. Communism didn't scare him because he saw it as an aberration, not as the manifestation of something terrible lodged in the soul. So while his passionate anticommunism set him apart from SDS, and from post-toughness liberals such as McGovern and Carter, his confidence that communism could be transcended, thus unlocking a harmonious world, created a strange ideological kinship with the New Left. The New Left believed it could end the cold war, and so did Reagan. He thought he could end it by overcoming communism; they thought they could end it by overcoming anticommunism. But intellectually, the connective tissue was a belief that evil\u2014however defined\u2014could be cast off, revealing something far better underneath.\n\nIf Carter's optimism about cooperating with the Soviets led him to initially ease up on the cold war, Reagan's optimism about defeating the Soviets led him to initially escalate it, in the hope that Moscow's empire might crack. To this end he dramatically boosted defense spending and aided anticommunist regimes and rebels in the third world. But even in those early years, when his cold war policies were most aggressive, Reagan still placed careful limits on the force he was willing to employ and the risks he was willing to take. In 1981, when Poland's communist government cracked down on the independent labor movement, Solidarity, Jeane Kirkpatrick, Irving Kristol, Norman Podhoretz, and George Will all urged Reagan to declare Warsaw in default of its overseas loans and thereby economically destabilize the Polish regime. But Reagan deferred to America's Western European allies, who feared that calling in the loans would destabilize the entire European banking system. Reagan, noted George Shultz, who in 1982 took over as his secretary of state, was \"a cautious though decisive man, sensitive about being viewed as too pugnacious.\" He wanted to halt, and even roll back, Moscow's advance, but not at too high a cost. He didn't ask Americans to \"pay any price, bear any burden\" to keep countries from going communist; he didn't think he had to, since time was on our side. He didn't see America's back as perpetually against the wall.\n\nIn part that is because his back wasn't perpetually against the wall. Since, unlike Truman, Kennedy, Johnson, and Carter, he hailed from the political right, Reagan didn't have to worry as much about looking soft. In fact, he recognized, in a way his conservative supporters rarely did, that after Vietnam, looking like too much of a hawk was almost as bad as looking like too much of a dove. Aided by shrewd political advisers like James Baker, Michael Deaver, and Richard Wirthlin, Reagan grasped the public's curious relationship to Vietnam. On the one hand, they wanted to avenge it, to show the bad guys once and for all who was boss. When Carter rubbed salt into the wound by allowing America to be humiliated in the third world yet again, they turned on him. On the other hand, most Americans had internalized Vietnam's basic lesson: America should never again put its boys on the line trying to keep communists from power in the third world. The public loved Rambo because in the movies, America won and no Americans died. Reagan knew that if he could gin up a war like that, Americans would cheer. If he couldn't, it was better to let the bad guys win.\n\nAlthough neoconservatives sometimes called Reagan Kennedy's heir, the two presidents reflected their very different times. In the early 1960s, sacrifice had been a political winner. Almost two decades of nearly unbroken success had convinced the country, and particularly its foreign policy elite, that it was capable of anything, as long as it got off the couch. In the early '80s, by contrast, Americans didn't want to be told to sacrifice; between Vietnam and stagflation they had been sacrificing more than enough. (In 1984, when Walter Mondale pledged that if elected he would raise taxes, that call for sacrifice helped ensure his landslide defeat. For his part, Reagan told Americans that they could have it all: a big defense increase, popular domestic programs, and lower taxes.) Americans no longer wanted their president to find mountains for them to climb, because they were no longer sure mountains could be climbed. They wanted him to find a molehill and call it Mount Everest. It was like the 1984 Olympics, which the Soviets boycotted. Did Americans cheer any less loudly because their best opponents weren't on the field? Hell no. These weren't the Camelot years, when Kennedy compared the cold war to a game between Texas and Rice. Playing the best, he had said back then, brings out your best. By the '80s, America was happy to play Grenada. We needed the win.\n\nMost of Reagan's intellectual allies didn't understand this. They were still ultra-realists, too afraid of Moscow to endorse post-Vietnam limits on America's cold war fight. And since they didn't have to get elected themselves, they could afford to ignore the public's post-Vietnam terror of war. The result was a series of skirmishes, which pitted Reagan's conservative supporters and a few hard-line aides against his more pragmatic advisers, and most importantly, against Reagan himself. Time and again, the result was the same. When looking tough was risk-free, Reagan played the part for all it was worth. But when the ante went up, he cashed in his chips.\n\nTake Central America. To Reagan's liberal critics, it was Vietnam all over again. They watched in horror as he invoked the dreaded domino theory, warning that if the communists in Nicaragua consolidated their rule, and the pro-American regime in El Salvador fell, Mexico might be next. \"Here we go again,\" warned a dovish senator. \"El Salvador,\" read a bumper sticker of the time, \"is Spanish for Vietnam.\"\n\nHad Reagan wanted to refight Vietnam, El Salvador would have been a logical place. Its smaller size and greater proximity to the United States would have made the logistics easier. The American military had a better grasp of its language, culture, and terrain. Its leftist rebels were less unified and battle-hardened than the Vietcong. And if you saw those rebels as agents of Moscow, as Reagan did, the Monroe Doctrine offered a rationale for intervention that dated back to the nineteenth century.\n\nBut Reagan never even considered it. At a meeting early in 1981, his first secretary of state, Alexander Haig\u2014the senior official most interested in picking a direct fight south of the border\u2014told his colleagues that if Cuba didn't stop arming El Salvador's communist rebels, \"I'll make that island a fucking parking lot.\" Among conservative intellectuals, Haig's vision of a 1962-style blockade to prevent Castro from exporting arms caught on. William F. Buckley, the longtime editor of National Review, endorsed it explicitly, adding that it should be accompanied by declarations of war against both Cuba and Nicaragua. Norman Podhoretz wrote that \"the United States must therefore do whatever may be required, up to and including the dispatch of American troops, to stop and then to reverse the totalitarian drift in Central America.\" Irving Kristol added that, \"The world should know, and the American people must be told, that under no circumstances will we permit El Salvador to go the way of Cuba and Nicaragua, and that we will use all means to prevent this.\" But among Reagan's key advisers, Haig's talk of direct military action was greeted with a chorus of anxious references to Vietnam. Assured that Haig had been merely speaking for effect, Michael Deaver, the White House aide closest to the president, replied, \"It certainly had a good effect on me. It scared the shit out of me.\" From that meeting until Haig resigned more than a year later, Cuba wasn't blockaded; Haig was. Deaver ensured that Haig never saw the president alone. Chief of Staff James Baker kept him off television. And they were almost certainly reflecting the wishes of their boss. As Deaver later noted, Haig's comments \"scared the shit out of Ronald Reagan,\" too.\n\nHaig's problem was that his tough-guy act wasn't an act. He couldn't turn off the projector. \"You get a band of brothers from CIA, Defense, and the White House and you put together a strategy for toppling Castro. And in the process we're going to eliminate this lodgment in Nicaragua from the mainland,\" he instructed his assistant Robert McFarlane. But when McFarlane went looking for that band, he found the Pentagon in a distinctly unfraternal mood. Reagan's defense secretary, Caspar Weinberger, was a hoarder: He loved buying weapons; he just didn't like using them. He considered Vietnam \"the crime of the century\" because America's leaders had tried to fight it without enduring popular support. (In 1984, in a speech that left a lasting impression on his young military aide, Colin Powell, he would lay out a series of tests for military intervention\u2014including sustained public backing and clearly defined goals\u2014that El Salvador, like Vietnam, plainly flunked.) When McFarlane reported that the Defense Department had little interest in another Bay of Pigs, Haig flew into a rage, denouncing the Pentagon's caution as \"limp-wristed, traditional cookie-pushing bullshit.\" But Weinberger was living in reality, as manifested not only by opinion within the military, but in the nation at large. Three-quarters of Americans, according to a 1985 survey, opposed invading Nicaragua. A 1983 ABC poll found that Americans opposed sending troops to El Salvador by a margin of almost six to one, even if that meant allowing a communist takeover. That same year, when Reagan reassured a joint session of Congress that \"to those who invoke the memory of Vietnam: there is no thought of sending American combat troops to Central America,\" members of both parties erupted in cheers.\n\nCongress was a big reason for Reagan's restraint. An essential condition of the hubris of toughness had been the explosion of executive authority in the years between FDR and LBJ. In the 1980s, however, Reagan enjoyed no such free rein. Americans generally liked him, but after Vietnam they no longer associated unchecked presidential power with national success. Even his efforts to aid the Nicaraguan Contras provoked a brawl on Capitol Hill. In the 1950s and '60s, the CIA had overthrown third-world governments whenever presidents felt like it; Congress read about it in the newspapers, if at all. But in 1984, when the Wall Street Journal revealed that the Agency had been mining Nicaragua's harbors, the Senate denounced the move, 84\u201312. That same year, Congress outlawed covert aid to the Contras, and when news broke that the Reagan administration had been aiding them anyway\u2014selling arms to Tehran and funneling the profits to the Contras in what was dubbed the Iran-Contra Affair\u2014the political fallout was devastating. Reagan's chief of staff was forced to resign. Nine administration officials were indicted, and six convicted. Reagan's advisers feared he might be impeached.\n\nThe president's conservative backers were forever chanting \"let Reagan be Reagan,\" implying that a pantywaist State Department, a gun-shy military, an insolent Congress, and a slothful public were squelching his hawkish convictions. But for Reagan, convictions were often a malleable thing; he molded them to political reality without ever believing he had changed his mind. He certainly wanted to keep Central America from falling to communism, and his efforts in that regard helped get a vast number of Nicaraguans and Salvadorans killed. (As a percentage of its population, Nicaragua lost more people in the Contra-Sandinista war than the United States did in the Civil War, World War I, World War II, Korea, and Vietnam combined.) But Reagan knew he couldn't get any of his own citizens killed. So when hawks proposed sending U.S. troops to Central America, he pointed out that Latins didn't like \"the colossus of the north sending in the Marines.\" As hawks like Buckley, Podhoretz, and Kristol recognized, this anti-interventionism undermined Reagan's anticommunism, since the Contras were unlikely to topple Nicaragua's government by themselves. But Reagan refused to see the contradiction, because, like the public, he wanted to fight the cold war without sacrifice or risk. After Iran-Contra, when it became clear that Congress would no longer permit military aid to the Contras, Reagan did not dwell on his failure to overthrow a Soviet ally less than a thousand miles from the Rio Grande. He saw the bright side: He had prevented another Vietnam. Whereas he had once vowed to stare down Moscow and Havana, he now congratulated himself for staring down his own right-wing base. \"Those sons of bitches won't be happy until we have 25,000 troops in Managua,\" he told his chief of staff triumphantly in 1988, \"and I'm not going to do it.\"\n\nReagan knew how to do what Truman and Eisenhower had done in Korea but Kennedy and Johnson never managed in Vietnam: cut his losses. The best example came late in his first term, as he was turning his eye toward reelection. Over Weinberger's objection, and without thinking through the details (which for Reagan was not uncommon), he had approved the deployment of U.S. peacekeepers to Lebanon as part of a deal to end fighting between Israel and its Palestinian and Syrian foes. Originally the deployment was scheduled to last only a month. But U.S. Marines were sucked into the quicksand of Lebanese politics and eventually took up arms to assist a Christian, pro-Western government at war with segments of its own society. In the early morning of October 23, 1983, those segments took revenge. A terrorist with a thick mustache and a wide grin drove a Mercedes truck past Marine guards and around a series of concrete and barbed-wire obstacles before detonating twelve thousand pounds of TNT at the mouth of the four-story building where the 1st Battalion, 8th Marine Regiment slept. The building was shoved off its foundation and collapsed, killing 241 Marines. For more than six hours, rescuers waded through the rubble searching for wounded, guided by screams. Reagan called it the \"saddest day of my presidency, perhaps the saddest day of my life.\"\n\nDemocratic House Speaker Tip O'Neill demanded a U.S. withdrawal. But Reagan refused and accused him of appeasement. \"He may be ready to surrender,\" Reagan declared, doing his best Rambo imitation, \"but I'm not.\" Unfortunately for Reagan, the American people were. Immediately his approval ratings began to sink. Nearly 60 percent of the public disapproved of his policy. Congressional Republicans were growing restless. And so five days after calling O'Neill Neville Chamberlain for wanting to pull U.S. troops out of Beirut, Reagan decided to do exactly that. Midge Decter, Podhoretz's wife and a noted neoconservative in her own right, declared herself \"disgusted.\" Baker and Deaver were thrilled: Their boss's poll numbers soon ticked back up.\n\nBut the Lebanon retreat only illustrated part of Reagan's political brilliance. He buried the hubris of toughness not only by avoiding unpopular wars (after all, even Jimmy Carter did that), but by staging popular ones and thus expunging the stench of defeat. The day the Marine barracks were destroyed, Reagan was confronted by a different foreign crisis. In the tiny Caribbean island of Grenada, known for its exports of nutmeg (if for anything at all), coup plotters had murdered the prime minister. Anarchy reigned and U.S. officials claimed that the eight hundred American students at the St. George's Medical School might be in danger. Reagan also feared that the coup would help Castro turn Grenada into a communist base. As with many of Reagan's pronouncements, the conviction was sincerely held but not entirely logical, since Castro's ties had been to the murdered prime minister, and the Cuban leader denounced the coup. Reagan's insistence that Cuba was building an airfield in Grenada for Soviet planes did not pass muster, either, since the planned runway was too small.\n\nBut Reagan was not a stickler for detail. (He was the same man, after all, who the year before had toasted the people of Bolivia at a state dinner in Brazil, and then explained away the mistake by claiming he was going to Bolivia next, which bewildered the government of Colombia, where he was due the following day.) So at 5:36 A.M. on October 25, just two days after the Lebanon attack, four hundred Marines landed on Grenada's western shore. A little after 6 A.M., U.S. Army Rangers parachuted into the southern part of the island. Thirty hours later, the war was won and American medical students were kissing the ground in Charleston, South Carolina. More than 70 percent of Americans backed the Grenada invasion. Mail to the White House ran more than ten to one in favor.\n\nAccording to his foremost biographer, Lou Cannon, Reagan didn't hatch the invasion to divert attention from Lebanon. But he was \"shameless\" in using it to turn the public mood from humiliation to pride. The Grenada invasion spawned more medals per soldier than any war in U.S. history. Reagan hosted the St. George's medical students at the White House, where they waved little American flags. Three years later, he staged another Potemkin war, dropping more than ninety 2,000-pound bombs on Libya in retaliation for its involvement in a Berlin terrorist attack that killed a U.S. soldier. Like Grenada, the Libya bombing garnered more than 70 percent support at home. \"Every nickel-and-dime fanatic and dictator knows that if he chooses to tangle with the United States of America, he'll have to pay a price,\" declared Reagan, morphing into Rambo again. The last phrase was key: It was our enemies who would pay the price, not us. Americans could thump their chests, swell with pride, and never get up from the couch.\n\n\"There's a bear in the woods.\" Viewers heard a menacing, pounding beat. Against a bleak landscape, the mammoth creature prowled across America's television screens in the fall of 1984. \"For some people, the bear is easy to see. Others don't see it at all. Some people say the bear is tame. Others say it's vicious and dangerous. Since no one can be really sure who's right, isn't it smart to be as strong as bear? If there is a bear?\" The screen went blank and then Reagan's picture appeared, above the words PREPARED FOR PEACE.\n\nAt first glance, it was everything the neoconservatives believed, in visual form. The world was a wilderness, devoid of civilization's rules. America shared it with a predator, deaf to morality and reason. And America was alone. In the commercial's final frame, you see the grainy outline of a man, standing only a few paces from the beast on a barren hillside. Does he have a gun? You sure as hell hope so.\n\nBut what about the last line: \"If there is a bear?\" You could almost see Jeane Kirkpatrick and Irving Kristol furrowing their brows. Of course there is. The bear is eternal. You can call him Hitler or Brezhnev or Beelzebub, but he's always there. And you had better not let down your guard.\n\nThe last line, however, was classic Reagan. For the moment, sure, there's a bear. But if you show you're strong he may turn into a teddy, in which case you can throw away your gun and have a picnic.\n\nKirkpatrick had made her reputation explaining why that couldn't happen. In her famous essay, \"Dictatorships and Double Standards,\" she argued that authoritarian regimes like those of Somoza and the Shah took decades or centuries to democratize, and that totalitarian regimes like Moscow's would never democratize at all. Her neoconservative allies generally agreed. There is not \"the slightest possibility that even the most minimal degree of civil or political liberty will ever be allowed under Communism,\" declared Norman Podhoretz in 1976. \"All hopes for 'liberalization' in the Soviet Union are so chimerical,\" added Irving Kristol in 1984. \"There may be occasional spasms in this direction, but they do not and cannot last.\"\n\nThat's what Kirkpatrick thought her boss believed as well. \"The Reagan administration...is less optimistic than recent American governments about the evolution of Soviet society,\" she explained in 1981. But that was wrong: Reagan was actually more optimistic about the evolution of Soviet society than his predecessors. Although his right-wing allies generally described the U.S.S.R. as a fearsome juggernaut, he thought\u2014correctly\u2014that it was running out of gas. In part that stemmed from his deep conviction that communism defied the laws of economics. But more fundamentally it flowed from his belief that evil systems could not forever keep good people down. Communism, he wrote in 1975, \"is a form of insanity\u2014a temporary aberration which will one day disappear from the earth because it is contrary to human nature.\" Seven years later, in a speech to the British parliament, he said the Soviet Union was experiencing a \"revolutionary crisis\" because it \"runs against the tide of history by denying human freedom and human dignity to its citizens.\"\n\nThis was a far cry from the toughness ethic. For intellectuals like Niebuhr and Schlesinger, who in the 1930s had seen democracy fall in the most advanced societies on earth, and had watched the best minds of their generation enlist willingly in totalitarianism's armies of the night, the idea that an inexorable \"tide of history\" lifted humanity toward freedom was a progressive fantasy. Kirkpatrick and Kristol, as children of the toughness age, were heirs to that harsh view. \"Americans,\" demanded Kristol in 1982, should stop \"thinking that time is on our side.... For the Soviet leaders are utterly convinced that time\u2014i.e. history\u2014is on their side. And so far there is precious little evidence to prove them wrong.\"\n\nThis skepticism about progress was intimately related to skepticism about human nature. Among the cold war liberals of the 1950s, it had been widely accepted that people were generally no better than the political systems that governed them, and that when they sought revolutionary change, things usually got worse. For Kristol and Kirkpatrick, both former cold war liberals, the actions of the New Left graphically confirmed this view. And their new allies on the right, traditional conservatives like Buckley and George Will, took an ever darker view of humanity's propensity to evil. In a gurgling infant, they saw original sin. In a Siberian gulag, Reagan saw the untapped goodness of man.\n\nIf cold war conservatives and neoconservatives took a sober view of humanity in general, many took a particularly dim view of Russians. For Kirkpatrick, culture mattered; it was the soil in which abstractions like democracy either sprouted or withered. And Russia, she suspected, was arctic tundra. Like Kennan, she saw Soviet communism not merely as an ideological veneer but also as the latest expression of a tradition of \"oriental despotism\" that stretched back to the czars. The historian Richard Pipes, who had helped paint a menacing picture of Soviet strength on Team B and served as the first Soviet expert on Reagan's National Security Council, called Russian communism the triumph of the \"muzhik, the Russian peasant. And the muzhik had been taught by long historical experience that cunning and coercion alone ensured survival.... Marxism...has merely served to reinforce these ingrained convictions.\"\n\nReagan didn't know from muzhiks. When he thought about Russians, he imagined \"Ivan and Anya,\" decent folks who, but for an accident of birth, could have appeared on Father Knows Best. \"Just suppose with me for a moment that an Ivan and an Anya could find themselves, say, in a waiting room, or sharing a shelter from the rain or a storm with a Jim and Sally, and that there was no language barrier to keep them from getting acquainted,\" he told the nation in 1984, in a passage he wrote by hand. \"Before they parted company they would probably have touched on ambitions and hobbies and what they wanted for their children and the problems of making ends meet. And as they went their separate ways, maybe Anya would say to Ivan, 'wasn't she nice, she also teaches music.' Maybe Jim would be telling Sally what Ivan did or didn't like about his boss. They might even have decided that they were all going to get together for dinner some evening soon. Above all, they would have proven that people don't make wars.\"\n\nThere was a parallel with Reagan's view of the United States. Commentators often noted that he saw America as one giant Hallmark card, obscured by an overgrown and dastardly federal government. But because of his reputation as a screeching hawk, they mostly failed to notice that he saw the U.S.S.R. the same way: as a land of good-hearted Ivans and Anyas laboring under an oppressive but artificial regime. In particular, critics were misled by his phrase \"evil empire,\" a term that drove the Kremlin and many liberals batty but that Reagan used exactly once, in 1983. It was the handicraft of a speechwriter named Anthony Dolan, a Buckley prot\u00e9g\u00e9. And the original text read: \"Now and forever, the Soviet Union is an evil empire.\" Reagan cut out the first three words. Asked four years later, when he and Mikhail Gorbachev were fast friends, why he no longer used the phrase, Reagan replied, \"I was talking about another time and another era.\" The bear was no longer a bear.\n\nBecause he believed the Soviet Union could radically change, Reagan also believed something else alien to the toughness tradition: that America could eventually disarm. In the mid-1940s, before he became a conservative, he, like McGovern, had been a member of the United World Federalists, a group that sought to outlaw nuclear weapons. And even as he moved right, Reagan clung to his dreams of nuclear abolition, although both his allies and his adversaries largely overlooked them, since they didn't fit his image as a cold war hawk. As was often the case with Reagan, movies played a big role. Reagan loved science fiction. And Colin Powell, who eventually became Reagan's national security advisor, believed his boss had been particularly affected by a 1951 film called The Day the Earth Stood Still. In it, earth is visited by an ambassador from a higher civilization, which has abolished violence by handing over military power to an army of intergalactic robots, who are programmed to destroy all planets that still resort to war. The aliens have been watching events on earth with dismay. They fear the earthlings will learn how to travel through space and thus infect the solar system with their bloodlust. Before departing in his flying saucer, the alien visitor warns that unless the people of earth abandon war, the robots will descend and kill them all.\n\nFor Reagan, this was powerful stuff. Again and again during his presidency, he cited a potential alien invasion as one reason America and the Soviets must overcome their differences. In 1985, he mentioned the idea to Gorbachev, who gave him a strange look and changed the subject. He frequently discussed the alien threat with aides, who worked feverishly to keep it out of his speeches. Whenever Powell got word that Reagan was ruminating about an attack from outer space, he would roll his eyes and say, \"Here come the little green men again.\"\n\nWhen it came to ensuring peace and harmony between the superpowers, Reagan saw nuclear abolition as the long-term goal. His short-term strategy was \"Star Wars,\" a missile shield that would protect America until the Soviets disarmed (and presumably protect against space aliens as well). To most scientific experts, seeing Star Wars as a quick fix was downright bizarre, since they didn't believe it would work ever, let alone anytime soon. And even the hawks who favored Star Wars generally did so because they considered it an alternative to disarmament treaties, a way of reducing the nuclear threat without having to rely on the Soviets to keep their word. But in the Disney movie that played inside Reagan's head, Star Wars and disarmament complemented each other. America would build a missile shield and share it with Moscow, thus rendering both sides invulnerable to attack. Then they would eliminate their nuclear weapons, eventually leaving them with no swords, only massive shields. Reagan\u2014who believed in reincarnation\u2014told Gorbachev that in a prior life he had probably been the inventor of the shield.\n\nStar Wars sparked excitement among Reagan's backers and ridicule from his critics. But during his first term, his desire to abolish nuclear weapons attracted much less notice. One reason is that Reagan, like many conservatives, believed that in the 1970s Moscow had taken the nuclear lead, and he didn't want to negotiate seriously on arms control until America had caught up. For many of those conservatives, however, the belief that America was too vulnerable to let down its guard was perpetual. It did not reflect their assessment of the superpower balance at one moment in time; it reflected their eternal view of world affairs. In 1984, the hawkish Committee on the Present Danger warned that the \"the gap between U.S. and Soviet military capabilities continues to grow.\" But Reagan disagreed. By the end of his first term, after several years of massive military expenditure, he believed that America had caught up.\n\nIn 1983, several events also triggered Reagan's latent terror of nuclear war. It began, predictably, with movies. In June, he took in WarGames, which starred Matthew Broderick as a hacker who breaks into the North American Aerospace Defense Command (NORAD) and almost starts a nuclear war. Reagan\u2014who, as his view of Star Wars revealed, tended to believe that pretty much anything was technologically possible\u2014found the film disturbing. Two days later, at a meeting with members of Congress, he asked if anyone had seen the movie. None had, so Reagan began describing the premise. \"I don't understand these computers very well,\" he explained, \"but this young man obviously did. He had tied into NORAD!\" A few months later, the president was even more deeply affected by The Day After, a made-for-TV movie that depicts Lawrence, Kansas, in the aftermath of a nuclear war. \"It is very effective and left me greatly depressed,\" he wrote in his diary. \"[W]e have to do all we can...to see that there is never a nuclear war.\"\n\nReagan didn't draw a sharp contrast between reality and celluloid. (In the most bizarre example, he told Israeli prime minister Yitzhak Shamir that he had visited Nazi concentration camps at the end of World War II, an odd claim, given that he had spent the entire war inside the United States. In truth, he had seen footage of the camps in a film.) And so just a few days after viewing The Day After, when he attended a briefing on U.S. military procedure in the event of a Soviet nuclear attack, Reagan saw the movies he had watched playing out before him in real life. His dread only grew when he learned that his nuclear buildup and anti-Soviet rhetoric had so terrified the Kremlin that they interpreted a NATO \"war game\" (there was that phrase again) called Able Archer as preparation for a real attack, and put their military on high alert.\n\nIn his mounting fear of war, Reagan once again reflected the public mood. After several years of epic military spending, cold war vitriol, and no superpower summits, the percentage of Americans favoring arms control had shot through the roof. In 1983, the House even passed a resolution endorsing a mutual freeze on the production of nuclear arms. Reagan's political advisers urged him to make an overture to the Kremlin so Democrats could not portray him as a warmonger in his 1984 reelection campaign.\n\nShultz, a beefy ex-academic with a tattoo of a Princeton tiger on his behind, was also eager to get disarmament talks going. And so in January 1984, more than a year before Gorbachev took power, Reagan began a dramatic rhetorical shift. Declaring that \"nuclear arsenals are far too high,\" he told the nation that \"my dream is to see the day when nuclear weapons will be banished from the face of the earth.\" Then, in September at the UN, he quoted Gandhi as saying that \"I have been convinced more than ever that human nature is much the same, no matter under what clime it flourishes, and that if you approached people with trust and affection, you would have tenfold trust and thousand-fold affection returned to you.\" It was the kind of thing that could make a neocon's hair stand on end.\n\nEven after Reagan won reelection, the antinuclear talk continued. \"I just happen to believe that we cannot go into another generation of the world living under the threat of those weapons,\" he said in November. Irving Kristol was starting to get anxious. \"To be conciliatory on the arms-control issues during the election campaign might have been politically expedient,\" he wrote in December. \"But why take it so seriously now? Nothing of any significance is going to happen, and it would be the height of naivete to think otherwise.\" By toughness standards, however, Reagan was indeed na\u00efve, and growing more so. Nothing in the Soviet Union had yet changed, but he was insisting, with mounting vehemence, that cooperation, not conflict, must define relations between the United States and the U.S.S.R. He was increasingly repudiating the toughness ethic. For conservatives, he was becoming a dangerous man.\n\nOf course, none of this would have mattered nearly as much had the Politburo in March 1985 not chosen Mikhail Sergeyevich Gorbachev as general secretary of the Soviet Communist Party. The fifty-four-year-old Gorbachev represented a younger generation that had come of age during Nikita Khrushchev's post-Stalin thaw, and had spent the subsequent decades quietly seething at the geriatric, mulish, often inebriated commissars who were driving Russia's economy into the ground. Gorbachev didn't want to abandon communism; he wanted to save it. But to revitalize Soviet industry and agriculture he had to slash defense spending, which by the mid-1980s was consuming a jaw-dropping 40 percent of his government's budget. And he knew that the Politburo wouldn't permit those cuts unless superpower relations improved and America looked like less of a threat.\n\nReagan made that possible. Within hours of Gorbachev's selection, and without knowing anything about the radical reforms upon which the Soviet leader would later embark, Reagan invited him to a summit meeting, without preconditions. Two days later, Gorbachev met with Vice President George H. W. Bush, who was in Moscow to attend the funeral of Gorbachev's predecessor. Bush's talking points would have made Jeane Kirkpatrick twitch. (As it happened, she had left her post several months earlier, declaring her \"disappointment\" that so \"many of the people in the administration\" did not share her \"foreign policy objectives.\") \"We should strive to eliminate nuclear weapons from the face of the earth,\" read Bush's notes. \"We should seek to rid the world of the threat or use of force in international relations.\"\n\nBy now Reagan was hungry to meet Gorbachev and begin dismantling the nuclear weapons he had spent so much money to build. Although for years he had denounced the never-ratified SALT II treaty on the grounds that it enshrined America's nuclear inferiority, in 1985 he overruled administration hard-liners and quietly scrapped some older submarines so that America would not exceed the treaty's limits and upset the Kremlin. In March, when Soviet troops in East Germany killed a U.S. soldier, giving Reagan a perfect excuse to avoid scheduling a superpower summit, he instead told journalists that such incidents just made him want to meet his Soviet counterpart more. And once the summit was set for November, he did something that astonished his aides: study. (This was, after all, the same man who in 1983 had refused to prepare for a meeting with world leaders because, as he told James Baker, \"the Sound of Music was on last night.\") Whereas in the past Reagan's advisers had been forced to prepare one-page \"mini-memos\"\u2014or actually commission short documentary films\u2014to get him to digest information, he now devoured twenty-five briefing papers on Soviet politics and culture. It was like that other classic Stallone character, Rocky, rousing himself from his lethargy before the big fight. At their meeting in Geneva, Reagan and Gorbachev were scheduled to meet one-on-one for only fifteen minutes. Instead they talked for almost five hours. At one point Reagan leaned over to his Soviet counterpart and whispered, \"I bet the hard-liners in both our countries are bleeding when we shake hands.\"\n\nThey were bleeding soon enough. No treaties came out of Geneva, but the atmosphere was chummy and the two leaders agreed to meet again the following year, with both sides confident an arms control deal was just a matter of time. Podhoretz was beside himself. Gorbachev, he insisted, was the shrewdest, most sinister Soviet leader yet. He was duping the West into believing that Russia no longer posed a threat, and thus getting the United States to pay to revitalize the collapsing Soviet economy. But since the arms control deals that Washington and Moscow were contemplating were (of course) ridiculously skewed in the Kremlin's favor, Gorbachev was dismantling none of his massive military machine. With the West's guard down, and the military balance shifted further to Moscow's advantage, the Soviet leader would then follow up his charm offensive with a military one, ripping off his smiley mask and revealing himself as the Hitler-Stalin that he really was. As Reagan and Gorbachev edged closer to a disarmament treaty, Podhoretz published a stream of articles with titles like \"How Reagan Succeeds as a Carter Clone\" and \"What If Reagan Were President?\" In 1986, when Reagan refused to use Moscow's imprisonment of an American journalist as pretext to cancel his second summit with Gorbachev, Podhoretz accused him of having \"shamed himself and the country\" in his \"craven eagerness\" to give away the nuclear store. George Will said the administration had crumpled \"like a punctured balloon.\" What particularly galled Podhoretz and Will was that Reagan, because of his tough-guy reputation, was burying the toughness ethic more successfully than Carter could have dreamed of. If Gorbachev was a wolf dressed as a sheep, Reagan was a sheep dressed as a wolf, and as a result, America was being eaten alive.\n\nAt first most conservatives avoided attacking Reagan personally. Throughout 1985 and 1986, they largely blamed his advisers for sabotaging his allegedly hawkish views. But the truth was closer to the reverse. In January 1986, Gorbachev suggested eliminating all nuclear weapons by the year 2000. Administration hard-liners were appalled, but Reagan was intrigued, reminding aides that \"I have a dream of a world without nuclear weapons.\" That fall, when the two leaders met in Reykjavik, Iceland, Gorbachev again proposed complete abolition, this time by 1996. It was a mind-bending proposal. Nuclear weapons were crucial to America's alliance with Western Europe, not to mention its power in the world. America's European allies were shocked, but Reagan was typically uninterested in the details. This was the happy ending that he believed every drama should have. He told Gorbachev that they should return to Iceland in ten years, each carrying their country's last nuclear missile. Then they would publicly demolish them and \"give a tremendous party for the whole world.\"\n\nIt never happened, since Gorbachev conditioned his offer on a U.S. promise not to deploy Star Wars, a promise Reagan would not make. Still, the following December they agreed to eliminate all Intermediate-Range Nuclear Forces (INF). This went far beyond anything Nixon or Carter had done. SALT I and II had merely limited how many new nuclear weapons the superpowers could build. Under Reagan, for the first time, the two sides began dismantling them.\n\nThe INF deal was sharply slanted in America's favor. Desperate to cut military spending, Gorbachev had given in on almost everything: agreeing to dismantle Soviet missiles in Asia as well as Europe, long a Kremlin red line, while allowing Britain and France to keep theirs. But for many on the American right, details weren't the point. Reagan was accepting that the Soviet Union was no longer really evil, that America no longer lived in a 1930s world. It was as if Churchill had invited Hitler over for tea and begun dismantling the Royal Air Force.\n\nNo longer could conservatives pretend that the president had been hoodwinked by his liberal advisers. In his final year in office, with Reagan himself almost giddy about his relationship with Gorbachev, the ultra-realists attacked him in exactly the same way they had long attacked the post-toughness left. Will accused Reagan of \"elevating wishful thinking to the status of political philosophy.\" The Wall Street Journal called him \"a utopian.\" Kirkpatrick insisted that Gorbachev \"is not the architect of Soviet retreat.\" A Republican congressman named Dick Cheney called glasnost a fraud.\n\nTo fight the INF deal, Podhoretz called for reviving the Committee on the Present Danger, originally founded in 1950 to lobby for NSC 68 and then reestablished in 1976 to oppose d\u00e9tente. It had grown moribund after Reagan's election, but now that Reagan had proved even worse than Carter, Podhoretz proposed reconstituting it. At a Ramada Inn in Tysons Corner, Virginia, another group of conservatives formed the Anti-Appeasement Alliance, which aimed to defeat the treaty in the Senate. In conservative newspapers across the country, they ran full-page ads declaring, \"Appeasement is as unwise in 1988 as in 1938.\" Underneath the text were pictures of Ronald Reagan, Mikhail Gorbachev, Neville Chamberlain, and Adolf Hitler. It was the reductio ad absurdum, the comic ending to a story that was turning out magically well. If Munich had been a genuine tragedy, and in Vietnam the Munich paradigm had produced a blend of tragedy and farce, this was pure farce. Invited to a wedding, a few die-hards insisted that it was really a funeral and donned sackcloth and ashes, which made the whole affair merrier still.\n\nWith only months left in his presidency, Reagan traveled to Moscow, where Ivans and Anyas mobbed him everywhere he went. \"Systems may be brutish, bureaucrats may fail. But men can sometimes transcend all that,\" he told students at Moscow State University, sounding a little like the Port Huron Statement. \"The accumulated spiritual energies of a long silence yearn to break free.\"\n\nA year later, Eastern Europe did break free, and then the captive nations of the soon-to-be-former U.S.S.R. Reagan's vision of a world liberated from the cold war and of a Soviet Union liberated from communism had come true. In future years, many on the right would try to reverse the causality of these earthshaking events, arguing that it was Reagan's merciless pressure that brought the Soviet regime to its knees. Had that truly been the case, there would have been no reason for their anguish, no reason to call him a dupe and a fool. Conservatives called Reagan those things for good reason: because he had betrayed them, and the toughness tradition itself. He had refused to spill American blood fighting communism in the third world. And he had turned away from nuclear confrontation and toward nuclear peace before Gorbachev even took power, let alone deposited communism in history's dustbin. \"If Reagan had stuck to his hard-line policies in 1985 and 1986,\" wrote Anatoly Dobrynin, longtime Soviet ambassador to the United States, \"Gorbachev would have been accused by the rest of the Politburo of giving everything away to a fellow who does not want to negotiate. We would have been forced to tighten our belts and spend even more on defense.\" The cold war ended, and Soviet communism collapsed, not because Reagan made America more frightening but because he made it less so; not because he learned the lessons of Munich but because he unlearned them, and helped the American people unlearn them, without surrendering their pride.\n\nThe claim that America had overthrown the Soviet empire by threat of force was a toxic seed that would grow in future decades into a hubris of its own, nourished by the rich soil of post\u2013cold war success. But no one could foresee that in the years of wonder that brought the cold war, and the hubris of toughness, to a close. George Kennan was an old man now, the sole surviving member of that extraordinary realist circle\u2014including Niebuhr, Morgenthau, and Lippmann\u2014that had helped bury the hubris of reason a half century before. The others had all died in political despair. Lippmann had paid dearly for opposing Vietnam, eventually leaving Washington, where Lyndon Johnson and his henchmen had made him a figure of ridicule. It is a \"much less pleasant world to live in,\" he told an interviewer in his final days. \"I think it's going to be a minor Dark Age.\" Niebuhr also found Vietnam a brutal blow. \"I am scared by my own lack of patriotism,\" he confided near the end. \"For the first time I fear I am ashamed of our beloved nation.\" Morgenthau admitted to \"a general discouragement, sometimes bordering on despair\" in his declining years. \"I am no longer 'my usual combative self,'\" he told a friend in 1972. \"I have become convinced, however reluctantly, of the hopelessness of fighting.... Perhaps to save one's soul is all that is left.\"\n\nBy temperament, Kennan was the darkest of them all. Like the others, he had preached the gospel of toughness in the 1940s, when it was still an insurgent idea. He had told Americans to see world politics as an arena of struggle, not accord, a sphere governed by power, not reason or love, where progress was often a delusion and evil never died. But his ideas, so precious in moderation, had grown monstrous in the hands of the ultra-realists. He was containment's Dr. Frankenstein, and by the 1980s he had spent so many years raging at his creation to no avail that he too fell prey to national self-loathing and moral despair. \"Poor old West,\" he wrote, \"succumbing feebly, day by day, to its own decadence.\" Maybe it wasn't even worth defending. Maybe the greatest evil was within.\n\nReagan, he initially assumed, was rock bottom. His administration was \"piling weapon upon weapon, missile upon missile...like the victims of some sort of hypnosis, like men in a dream, like lemmings headed for the sea.\" Nearing eighty, Kennan turned back to the disaster that in his view had started it all: World War I. He embarked on a three-volume history of the origins of the conflict, all the while suspecting that it was mere prelude to the far greater catastrophe toward which America now blindly marched: a nuclear World War III \"from which there can be no recovery and no return.\" Perhaps, he mused, it was just recompense for Hiroshima and Nagasaki. The chickens were coming home to roost.\n\nAll this cultural gloom made him an easy mark. The New Republic declared him senile and suggested that he be \"put out to pasture.\" \"The great container,\" it jeered, \"springs a leak.\"\n\nHis opinions marginalized, and his faculties derided, Kennan in 1981 suggested scrapping SALT II as insufficiently bold and instead cutting all nuclear stockpiles in half, en route to the eventual abolition of all nuclear weapons. It was a bizarre turn of events: the architect of the toughness ethic trying to stop his own runaway tradition by embracing the most utopian of post-toughness ideas. He never thought his proposal would gain a serious hearing. It was a cry in the dark, the desperate offering of a sad old man to a president he assumed held his views in contempt.\n\nInstead, five years later, Reagan offered the same proposal himself. For Kennan it was something of a mystery. He was suspicious of Reagan. He distrusted people who lacked an appreciation for tragedy, who insisted that history was forever moving in the right direction. But it was Reagan's peculiar sunniness, his insistence that the world need not be a dark and dangerous place, that helped tame Kennan's Frankenstein monster. The ethic of toughness, which had begun with the recognition that the optimism of the progressive era had its limits, was now ending with the recognition that the pessimism of the cold war had its limits, too. It was the unlikeliest of outcomes, led by the unlikeliest of peacemakers. But it was glorious nonetheless. The end of the cold war, admitted Kennan, \"is a fit occasion for satisfaction.\" For him, that was a scream of joy.\n\n## PART III\n\n## THE HUBRIS OF DOMINANCE\n\n## CHAPTER THIRTEEN\n\n## NOTHING IS CONSUMMATED\n\nIn April 1989, with the Soviet empire teetering on the brink of collapse, the Senate Foreign Relations Committee called eighty-five-year-old George Kennan to explain what was happening to the world. It was a very unusual hearing. For one thing, Kennan was the sole witness. For two hours, he alone held the floor, his voice occasionally quivering, and dropping to little more than a whisper. For another, the senators didn't bloviate, at least not as much as usual. They listened like attentive schoolchildren, prefacing their questions with little tributes, telling him what a privilege this was, offering the thanks of a grateful nation. Since the United States had won the cold war as much with ideas as with armies\u2014and since containment was the most famous idea of all\u2014the little old man across the dais was the closest thing America had to a conquering general. (If the senators knew that Kennan had renounced containment almost four decades earlier, and watched in horror as it grew into a global crusade, they didn't let on.) When the hearing closed, everyone rose to applaud: the senators, their staff, even the committee stenographer. Of course she did, wrote the Washington Post's Mary McGrory, \"Considering the guff she usually has to take, the self-serving, pettifogging, jargon-ridden, sometimes semiliterate chest-pounding, Kennan's profound and luminous comments must have sounded like a Mozart oboe concerto.\"\n\nThere was only one discordant note. It came from Senator Daniel Patrick Moynihan, no intellectual piker himself, a former Harvard government professor and ambassador to the UN and India, a man, quipped George Will, who had written more books than most of his colleagues had read. Moynihan had a score to settle on behalf of Woodrow Wilson. For half a century, the melancholy Kennan had been shadowboxing with America's twenty-eighth president, landing blow after blow against Wilson's righteous optimism, his faith that reason and democracy would eventually rule the world. For a long time now, Kennan had had his fellow Princetonian on the ropes. The Somme, Auschwitz, Hiroshima, the Gulag, Cambodia, Iran: History itself had sent Wilson down for the count. In the battle between the rationalizer and the misanthrope, Kennan had a mountain of skulls on his side.\n\nBut now, in the virtual blink of an eye, history had turned. From Asia to Latin America to Eastern Europe, from the mud villages of the third world to the citadels of totalitarian power, freedom was washing across the world like a mighty wave. It was time, Moynihan believed, to give Wilson his due.\n\n\"I know you have been skeptical about Wilson,\" Moynihan ventured, tiptoeing toward a rebuke. But has not the world \"accommodated [itself] to Wilson's visions?\" And there, in front of everyone, after a lifetime of intellectual combat, the conquering general raised the white flag. \"You are right. I was long skeptical about Wilson's vision,\" Kennan replied. \"But I begin today in the light of just what has happened in the last few years to think that Wilson was way ahead of his time.\"\n\nNineteen eighty-nine was a year that blew men's minds. Six months after Kennan's testimony, Yoshihiro Francis Fukuyama, deputy director of the State Department's Office of Policy Planning, traveled to the south of France. Like Jeane Kirkpatrick and Richard Pipes, he was a Sovietologist and a conservative, not a combination that traditionally inclined one toward optimism about world affairs. But he had been watching in wonder as Mikhail Gorbachev and his top advisers abandoned the language of Marxism and began speaking a tongue that sounded remarkably like America's own. Pipes was still comparing Gorbachev to Brezhnev. Norman Podhoretz was still comparing him to Lenin. But Fukuyama believed that something big was happening; he could feel the earth moving beneath his feet.\n\nWhen he arrived at the French Riviera for a NATO meeting, his European counterparts told him to calm down. Forget about it, declared the West German representative: Germany won't be unified in my lifetime. When Fukuyama went to Germany to see for himself, he heard more of the same. In East Berlin he snapped photos of demonstrators in the streets. Forget about it, said U.S. Embassy officials: East German communism is here to stay.\n\nTwo weeks later, the Berlin Wall fell.\n\nAn old mentor of Fukuyama's, a University of Chicago professor and pied piper of the intellectual right named Allan Bloom, was hosting a lecture series titled \"Decline of the West?\" It was the same old conservative gloom: America and its fellow democracies were culturally weak, morally weak, ideologically weak, militarily weak. It was still 1938. But Fukuyama had said good-bye to all that, and when Bloom asked him to take part, he gave a talk that was not gloomy at all. It was, in fact, the most breathtakingly optimistic statement by an American intellectual since Walter Lippmann and John Dewey glimpsed utopia in the rubble of World War I. Fukuyama's title: \"Are We Approaching the End of History?\"\n\n\"The twentieth century,\" he explained, \"has made us all into deep historical pessimists.\" He recounted the litany of horrors: \"two destructive world wars, the rise of totalitarian ideologies, and the turning of science against man in the form of nuclear weapons and environmental damage.\" Thoughtful people like Richard Pipes and Allan Bloom had witnessed so much despair over such a long time that they had come to believe it was the normal order of things. They were so used to tragedy that they could not recognize triumph, even when it stared them in the face.\n\nBut there it was, the triumph of democracy\u2014of Wilson's dream\u2014bursting out across the globe. In Poland, Lech Walesa was quoting Jefferson; in China, students hoisted a Goddess of Democracy in Tiananmen Square. Things had come full circle. In 1918, with liberal revolution seemingly cascading across Europe, the New Republic had declared, in a fit of Wilsonian ecstasy, that \"at this instant of history, democracy is supreme.\" Now, more than seventy years later, after decades upon decades of blood and tears, Fukuyama was saying so again. History with a capital H, history as the epic struggle over how humanity should be ruled, was coming to an end. The fascist idea had died; the communist idea was dying; the democratic idea stood alone. Democracy was history's ultimate victor, the final ideological destination of man.\n\nThe battle of ideas was over. Intellectually, there was nothing left to contain. At the core of the toughness creed had been the belief that America's enemies were fearsome not merely militarily, but ideologically as well. The fascists and communists were terrifying because they tapped into something deep in human nature. They were part of us, the dark side of the soul. Ronald Reagan had never believed that: For him, there was no dark side of the soul. And now Fukuyama was turning Reagan's insurgent optimism into a new creed, a Wilsonianism for a one-superpower age. The ethic of toughness, which had emerged to explain and combat the great totalitarian ideologies of the twentieth century, was being laid to rest alongside them. In Social Sciences 122, a lecture hall at the University of Chicago, the ethic of dominance was being born.\n\nFukuyama's lecture became an article, and then a book, and made him a star. The end of history became a synonym for intellectual audacity, for the biggest of big-think. But as a foreign policy doctrine, the ethic of dominance actually began life small. Like the ethic of reason and the ethic of toughness before it, it started off as a limited creed and then grew with success. Fukuyama in 1989, like Kennan in 1946, was Daedalus. He built wings, then watched his intellectual progeny fly them into the sun.\n\nFor Fukuyama it was democracy that was dominant, not necessarily the United States. America, in fact, was ending the cold war a bit like it ended World War II: triumphant, but exhausted and insecure. As one top U.S. official put it, we \"crossed the finish line out of breath.\"\n\nAmerica's totalitarian competitor may have been crumbling, but the United States wasn't the only democracy on the world stage, and in the late 1980s it wasn't the most vigorous. America's share of world GDP had been dropping for as long as anyone could remember. Its economic growth was sluggish; its education and health-care systems were embarrassing; its inner cities were terrifying; and everyone in the country seemed to be in debt, the government included. During the Reagan years, America had gone from the world's largest creditor to its largest debtor. And on October 19, 1987, roughly two years before the Soviet empire fell, the United States had suffered a day of reckoning of its own. The stock market plunged 22 percent, more than the worst day of 1929.\n\nSo while Fukuyama was dubbing the end of the cold war \"the end of history,\" he shared intellectual center stage with another academic super-star, Yale historian Paul Kennedy, who gave it a less triumphal name: \"the end of the American era.\" The United States and the Soviet Union, Kennedy claimed, were partners in decline, twin victims of a historical disease called \"imperial overstretch,\" in which great powers assume overseas obligations they cannot afford. In winning the cold war, in other words, America had actually lost. Germany and Japan, nations that could build cars and VCRs as opposed to missiles\u2014products people actually wanted to buy\u2014were eating our lunch. In a miracle of good timing, Random House published Kennedy's book, The Rise and Fall of the Great Powers, only months after the stock market crash. Initially it printed nine thousand copies; within a year it had sold 225,000. For thirty-four weeks, Kennedy's book graced the New York Times' bestseller list. He had delivered his message, and the intellectual class heard him loud and clear: Before getting too self-congratulatory about the demise of our slow, dumb, overmuscled foe on the other side of the earth, Americans should take a hard look in the mirror. If the Soviets were geopolitical mastodons, we were Tyrannosaurus rex.\n\nSo 1989 was a heady time, but not a hubristic one. The American idea\u2014democracy\u2014was triumphant, but America itself was not. Economically the nation was anxious, and just as in the early days of containment, economic anxieties made America's leaders wary of projecting military power. One of the most popular phrases in early post\u2013cold war Washington was peace dividend. With no threat on the horizon, and America's military expenditures bleeding it dry, politicians looked forward to cutting back foreign deployments and cutting the Pentagon down to size. By scaling back America's ambitions overseas, they hoped to bind its festering wounds at home.\n\nThere was another inhibition as well. It wasn't just that America lacked the money to run its vast military machine; many influential Americans doubted whether that military machine was much use anyway. The conventional wisdom, as the cold war wound to a close, was that big armies were like Bentleys: nice to look at but unlikely to take you very far. In Afghanistan the Soviets had driven theirs into a ditch, and the Japanese and Germans were zipping ahead on civilian power alone. For its part, America hadn't fought a real war since it left Saigon with its tail between its legs in 1975, and Reagan, for all his Rambo bluster, had never dared send American troops in large numbers into harm's way. If the American economy stood in the shadow of a stock market crash and a prophecy of decline as the cold war closed, the American military, Reagan notwithstanding, still stood in the long shadow of Vietnam.\n\nBetween 1989, when the cold war ended, and 2003, when America invaded Iraq, all this changed. Militarily America won and won, until America's leaders found it hard to imagine America could lose. Economic decline turned to economic mastery, and soon resources seemed almost infinite. Success was like helium, and the ethic of dominance grew. Fukuyama's belief that history was moving democracy's way became George W. Bush's belief that within every dictatorship lay a democracy ready to burst free. And Fukuyama's belief that democratic ideals were supreme became Bush's belief that American power was supreme, that the United States could dominate\u2014ideologically, economically, and militarily\u2014every important region on earth. If the cold warriors had inflated the lessons of 1938 until they believed that virtually every foe could and must be stopped, the post\u2013cold warriors inflated the lessons of 1989 until they believed that virtually every foe could and must be smashed. By 2003 the ethic of dominance had become the equivalent of the ethic of reason in 1917 and the ethic of toughness in 1965: a good idea pushed too far, a theory that explained too much, a species of hubris, a very dangerous thing.\n\nIt started with a war Americans barely even remember, and a hidden struggle that presaged much that was to come. In 1988, with Ronald Reagan on his way out the door, a bureaucratic fight broke out within the bowels of his administration. It pitted two men who were almost caricatures of their breed. On one side was Admiral William Crowe, chairman of the Joint Chiefs of Staff, a big, bald, sixty-three-year-old good ol' boy whose lazy Oklahoma drawl made Ivy League civilians forget that he had a master's from Stanford and a doctorate from Princeton. On the other was Assistant Secretary of State Elliott Abrams, the quintessential neocon: a Jewish, New York\u2013born ex-liberal who had attended high school with the children of Julius and Ethel Rosenberg, lived in Nathan Glazer's attic during law school, and married Norman Podhoretz's stepdaughter. The subject was Panama, where the United States and its longtime client, Manuel Noriega, were in the midst of a nasty divorce.\n\nGiven his pedigree, observers often assumed that Abrams was an ideological clone of Jeane Kirkpatrick, the Reagan administration's most famous neocon. But there was a generational difference. Kirkpatrick was fifty-five when Reagan took the oath of office; Abrams was only thirty-two. As a result, Abrams, like his generational contemporary Francis Fukuyama, was more deeply influenced by the wave of democratization that began sweeping the world in the 1980s. In Reagan's first term he served as assistant secretary of state for human rights and humanitarian affairs, a job in which, critics assumed, he would follow the Kirkpatrick line: soft on pro-American dictators. But that line was premised on pessimism that third-world autocracies could democratize anytime soon (and a corresponding belief that if America pushed them hard to do so, the result would be not democracy, but communism). In 1983, Irving Kristol was still writing that \"the traditions\u2014political, religious, cultural\u2014that shape Latin American thinking and behavior are such as to make it exceedingly difficult for the countries of Southern America to proceed along the [democratic] lines followed by Northern America and Western Europe.\" And yet, between 1980 and 1986, ten former autocracies embraced democracy in Latin America alone. So Abrams grew more optimistic about democracy's chances, more willing to push for human rights even in places like Chile, Paraguay, South Korea, and the Philippines, where the thugs were on our side.\n\nIn the summer of 1985, Abrams took over the State Department's Latin America bureau. A few months later, Noriega\u2014then receiving a $200,000 annual salary from the CIA (the same amount the U.S. government paid Ronald Reagan)\u2014ordered his henchmen to abduct Hugo Spadafora, a prominent critic of the regime. Over the course of seven hours, Noriega's men repeatedly rammed a stick up Spadafora's rectum, pounded his genitals until they were grotesquely swollen, pushed sharp objects under his fingernails, slashed the muscles in his thighs, broke several of his ribs, and then beheaded him. After obtaining photos of his brother's corpse, Winston Spadafora traveled to Washington, where he showed the pictures to anyone with the stomach to look at them. Even Jesse Helms was appalled.\n\nQuietly, Abrams began pushing for a tougher line against Noriega. By 1987 he was denouncing him publicly. And by 1988 he was proposing military force. It helped that with the superpower standoff winding down, Noriega's role as a conduit for aid to the Nicaraguan Contras no longer seemed as vital. During the cold war, the ethic of toughness had required embracing lesser evils like Noriega to contain the greater evil, communism. Now, in a world where America was becoming ideologically dominant, America could set its sights higher. Rather than merely containing evil, it could impose good.\n\nAbrams's desire to topple Noriega wasn't purely humanitarian, of course. In the late 1980s, Americans were in a frenzy over drugs. According to a 1988 Gallup poll, they rated drugs their number-one concern. Noriega was up to his eyeballs in the stuff, and in February of that year two Florida grand juries indicted him for helping to smuggle marijuana and cocaine into the United States. (In the past, U.S. officials had leaned on prosecutors not to charge Noriega, because he was providing useful cold war intelligence. But by 1988 he was no longer important enough to protect.) So Abrams's argument for armed regime change wasn't premised solely on spreading democracy. But in his mind, as in Woodrow Wilson's, spreading democracy and safeguarding American security were interlinked: Regimes that did bad things at home generally did bad things abroad. It was a crucial intellectual shift. Cold war conservatives like Jeane Kirkpatrick had generally warned against trying to rapidly export democracy, both because they believed it wouldn't grow in barren third-world soil and because they feared the effort would imperil American security, which they believed pro-American thugs were serving rather well. Abrams, by contrast\u2014like many other young, post\u2013cold war conservatives\u2014believed that democracy could quickly take root in foreign lands; they had seen it happen before their eyes. And they feared that if it didn't, continued dictatorship might put American security at risk. For Kirkpatrick, toughness meant recognizing that in a world where evil never dies, trying to impose American values can imperil American interests. For Abrams, dominance meant believing that evil could be vanquished, and that by imposing its values, America could further its interests at the same time.\n\nUnlike Abrams, William Crowe wasn't particularly ideological. And that was precisely the point. In war games, he noticed that civilians were generally quicker than military men to resort to force, and he suspected that it was because for them, war was a kind of game. They lived in the world of theory; war didn't turn their stomach because they had never smelled it up close. Abrams, Crowe often noted, had procured an educational deferment to avoid Vietnam. His views \"were both na\u00efve in their formulation and reckless in their casual commitment of our military men and women. This latter phenomenon is not unknown among young political appointees who have never served in uniform. However, Mr. Abrams raised it to an art form.\"\n\nThroughout 1988, Abrams and Crowe battled. Abrams wanted to establish a democratic government in exile on Panamanian soil to pressure Noriega; Crowe didn't consider Panama's opposition very effective or very democratic. Abrams said the United States could overwhelm Noriega's Panamanian Defense Forces with a small force; Crowe made invading Panama sound like D-day. Abrams said the Panamanians would welcome U.S. troops as liberators; Crowe asked why, if the Panamanians loved democracy so much, they weren't fighting for it themselves. Abrams believed that Crowe couldn't get over Vietnam; Crowe believed that if Abrams had managed to get himself over to Vietnam in the 1960s, he wouldn't be demanding a repeat performance south of the border now.\n\nIn 1988, the Crowe-Abrams feud was a fairly one-sided affair. Ronald Reagan\u2014that dove in hawk's feathers\u2014had sent ground troops into battle only twice: in Grenada, against an opposing army of six hundred men, and in Lebanon, a disaster that still haunted his sleep. (According to his press secretary, Reagan's final words in the Oval Office were \"The only regret I have after eight years is sending those troops to Lebanon.\") Ideologically, Reagan was a transitional figure: more convinced than most of his conservative contemporaries that democracy would triumph across the globe, but more skeptical than many younger conservatives that U.S. troops could help bring that triumph about. So to Abrams's dismay, the president sided with Crowe, opposing any action that might require \"counting up the bodies.\" Instead Reagan tried unsuccessfully to cut a deal with Noriega: Prosecutors would drop the drug charges if he gave up power. America, insisted Deputy Secretary of State Lawrence Eagleburger after Reagan's decision, \"will never invade Panama.\" The Pentagon gleefully sent around a mock invitation to Noriega's farewell party as Panamanian leader. Across the front it read \"Due to circumstances beyond Elliott Abrams's control the farewell party has been indefinitely postponed.\"\n\nBut Abrams had the last laugh. When Reagan's vice president, George H. W. Bush, ran to succeed him in 1988, Reagan's offer to drop the drug charges became a millstone around his neck. It didn't help that as director of the CIA in the 1970s, Bush had met Noriega and kept him on the U.S. payroll. Democratic Party bumper stickers read \"Bush-Noriega '88\u2014You know they can work together.\" And in response, Bush was forced to promise that if he won the White House, there would be no deals.\n\nOnce elected, Bush tried fomenting a coup in Panama. But the effort ended in disaster as Noriega's men foiled the plot and tortured its ringleader to death. In May, Panamanians went to the polls and overwhelmingly voted Noriega's allies out of office. But the dictator refused to cede power. Outraged, demonstrators flooded into the streets chanting, \"Down with the pineapple\" (a reference to Noriega's acne-covered face). They hung pineapples from telephone polls in disgust. (The fruit-based protest may also have been a subtle reference to Noriega's flexible sexuality, since it was an open secret that he was a bisexual who periodically dressed in hot yellow, lathered himself in perfume, and cavorted with a pilot boyfriend.) But Noriega foiled the protests by buying up the entire country's pineapple supply. And when Guillermo Endara, the man who had won Panama's presidential election, led a protest rally, government goons smashed him in the face with an iron bar.\n\nFinally, with the United States demanding that Noriega step down, Panamanian forces detained a U.S. Marine at a roadblock and then shot him as he tried to drive away. An American navy lieutenant who had the misfortune to witness the murder was blindfolded and repeatedly kicked in the head and groin. His wife was threatened with rape and forced to stand with her hands above her head until she collapsed.\n\nReagan might still have resisted military action; no one would have dared call him a wimp. But George H. W. Bush had a high-pitched, squeaky voice, goofy-preppy diction, and people did call him a wimp, often. In his native milieu, the fading world of Brahmin New England, where bravado was for the nouveau riche and the first-person singular was a dirty word, Poppy Bush had actually been something of a stud. He'd been tapped for his first secret society at fourteen, joined the navy as its youngest flier four years later, captained the baseball team at Yale, and made his first million by thirty. But the media did not care that he made his motorcade stop at red lights so as not to inconvenience other drivers or that he played a mean game of horseshoes. He might have been a masterful president before television, or perhaps radio. But he proved dismally unable to cultivate his public image in the CNN age. And because, unlike Reagan, he did not look like a wartime leader, he had to become one.\n\nIn late December 1989, wearing a pair of bright red socks\u2014one reading \"Merry\" and the other reading \"Christmas\"\u2014Bush authorized U.S. troops to invade Panama, taking the first, small step on the ladder from inhibition to confidence to hubris that would culminate with his son's invasion of Iraq. As Americans were just beginning to learn, war was easier in the post\u2013cold war age: Noriega, unlike Ho Chi Minh, had no rival superpowers to funnel him arms. Militarily the invasion was a rout. Noriega, who was visiting a brothel when the U.S. attack began, took refuge at the house of the papal nuncio, the Vatican's man in Panama. For two days and nights, U.S. troops surrounding the building blasted rock music (which Noriega loathed), including such topical choices as \"You're No Good,\" \"I Fought the Law,\" and \"Voodoo Child,\" a reference to the discovery in Noriega's apartment of tamales in which he had written the names of his political enemies. Eventually Noriega gave himself up and was flown to a Florida jail. In the entire operation, twenty-three U.S. soldiers died.\n\nBefore the war, according to polls, only a third of Americans had wanted to invade. But once the invasion proved a success, support jumped to 80 percent, and Bush's approval ratings hit 76 percent, the highest for any U.S. president since Vietnam. It was the first sign that in the post\u2013cold war age, starting a war could prove a political boon.\n\nFor a small war, the Panama invasion boasted an astounding number of code names: Nimrod Dancer, Sand Fleas, Ma Bell, Post Time, Klondike Key, Stumbling Block, Lima Bean, Purple Storm, Blind Logic, Prayer Book, Gabel Adder, Silver Bullet, Acid Gambit, Nifty Package, Robin Quart, Pole Tax, Krystal Ball, Just Cause, and Promote Liberty. But the last one proved the most apt. In the run-up to war, critics had called the idea of invading to impose democracy absurd. \"Surely it is a contradiction in terms and a violation of America's best ideals to impose democracy by the barrel of a gun,\" declared Massachusetts Senator Edward Kennedy. Jimmy Carter warned that \"any sort of military involvement there would immediately alienate the Panamanian people.\"\n\nBut on the ground, to the astonishment of many foreign observers, U.S. troops were in fact greeted as liberators. A CBS poll found that the war was even more popular in Panama than in the United States. When Vice President Dan Quayle arrived in Panama City soon after the invasion, he was greeted with signs reading \"Viva Quayle\" and \"Gringos Don't Go Home. Clean Panama First.\" (Only later, when the Bush administration reneged on its promise of $1 billion to \"clean up\" post-Noriega Panama, did the pro-gringo sentiment fade.) And confounding pessimists like Jeane Kirkpatrick and Irving Kristol, Panama didn't revert to dictatorship. Endara was sworn in as president, and then peacefully voted out four years later as Panama joined the swelling ranks of Latin American democracies.\n\nHow was all this possible in a country with weak democratic traditions and a history of nationalist resentment toward the United States? In a sermon several weeks after the war, a Panamanian priest offered an answer. The real Panama, he insisted, had not been invaded by the United States. The real Panama\u2014which offered its people freedom and a better life\u2014had actually been invaded years earlier by Noriega and his goons. It was this imposter nation, this tyrannical \"anti-country,\" whose sovereignty the United States had trampled, not the real Panama's. And because the United States had done so, \"the true country now starts to take its first difficult and costly steps toward its reconquest: a free country with justice and liberty.\" Authoritarianism, the priest was saying\u2014repudiating Kirkpatrick's argument in \"Dictatorships and Double Standards\"\u2014was not organic; it was artificial, unreal. When America peeled away Panama's ugly, tyrannical veneer, it found a democracy trapped underneath. Fukuyama had said history was on democracy's side; now America was learning that through war it could speed history up. Years later, Bush's defense secretary Dick Cheney would call the invasion \"good practice,\" \"a trial run.\"\n\nOn November 27, 1990, a little less than a year after Noriega fell, William Crowe, now retired, went to visit the man holding his old job, Chairman of the Joint Chiefs Colin Powell. That August, Saddam Hussein had invaded Kuwait. He had done so for simple, primal reasons, the same reasons a hungry predator devours farm-fattened prey. Iraq was a big, powerful country and it was starving. Its eight-year war with Iran had left it bankrupt, and without money to lubricate his rule, Saddam feared losing power. His regime owed $30 billion to its smaller, richer neighbor to the south, and the Kuwaitis\u2014disregarding Saddam's suggestion that they forgive his debts because he had defended the Arab world against Persian marauders\u2014were demanding that he pay up. The Kuwaitis also refused to cut oil production and thus hike prices, which Saddam hoped would raise some much-needed cash. And they refused to stop the \"slant drilling\" that Saddam claimed was stealing Iraqi oil. In principle, Kuwait's ruling Al-Sabah family had every right to spurn Saddam's pleas: Iraq's desperation was his own fault. But there was something brazen about their unwillingness to strike a deal. It was like a rich man walking past a street tough and waving his gold watch.\n\nThe CIA thought Saddam would merely dip his toe across the border and snatch a few oil wells. Instead he swallowed Kuwait whole. Suddenly, Saddam controlled 20 percent of the world's proven oil reserves, and within the Bush administration, there was unanimity on one thing: He could not have Saudi Arabia, which would give him more than 45 percent. Within days, America was dispatching troops to Saudi soil.\n\nBut when it came to Kuwait itself, the Panama divisions broke open again. In theory, everyone was for sanctions, which were meant to push Saddam back across the border nonviolently. In reality, however, some people were more for them than others. Many of Crowe's military buddies wanted to wait a year or more for the embargo to take its toll, in hopes of avoiding war. Top Pentagon civilians, by contrast, led by Cheney and his undersecretary for policy, Paul Wolfowitz, believed the United States should wait only a few months, and then draw its sword. On the surface, it was merely a debate about means: would economic pressure or military force best restore Kuwait? But under the surface, it was also emphatically a debate about ends. The doves wanted to restore the status quo ante; the hawks wanted to change it. The doves just wanted Saddam out of Kuwait. The hawks wanted him not only out of Kuwait, but so emasculated that he could never again threaten his neighbors and hopefully, so emasculated that he lost power.\n\nDuring the cold war, the hawkish option would probably not even have been on the table. From the late 1950s until the collapse of Soviet power, Iraq had been Moscow's ally, and the Kremlin would not have permitted the United States to beat Iraq to a pulp. Had Saddam invaded in 1980 instead of 1990, America would have been lucky to get him out of Kuwait at all, thus restoring the prewar balance of power. But now, in a one-superpower world, Cheney and Wolfowitz realized that they could do more than hold the line; they could extend it. They could do more than contain Saddam; they could cripple him. Still squinting in the light of a changed world, they were coming to see that with the cold war over, no great power could prevent America from expanding its dominance. In the Persian Gulf, the door to U.S. hegemony was wide open. The American military simply needed to walk through.\n\nCrowe, like many in uniform, dreaded the thought. He worried that Cheney and Wolfowitz\u2014who like Abrams had not served in Vietnam\u2014were unlearning its lessons, and that they didn't have blood in the fight this time, either. (He did. His son Blake was a Marine captain sweltering in the Saudi desert.) \"War is not neat, it's not tidy. It's a mess,\" Crowe told a Senate panel; the Pentagon had already ordered sixteen thousand body bags. For him the cold war was still the template, and the cold war proved the virtue of patience. Attacking Iraq before sanctions had time to work, he argued, would be like trying to militarily roll back the U.S.S.R.\n\nSitting in his old Pentagon office, now redecorated by Powell, Crowe tried to nudge his successor into doing what he wished his predecessors had done in Vietnam: lay his career on the line to prevent an ill-conceived war. The country isn't eager to fight, Crowe insisted; Powell nodded in agreement. War can be legitimate, Crowe went on, but only after you've been attacked. Powell nodded again. They were sitting at an antique table, as a steward in a bright yellow jacket served lunch. Finally, Crowe asked Powell straight out: \"Where are you on the Gulf deployment?\" Powell gestured toward the window, which offered a gorgeous view of the Potomac River, and Washington, D.C., on the other side. \"I've been for a containment strategy,\" he replied, \"but it hasn't been selling around here or over there.\"\n\nFor Colin Luther Powell, the army was more than a career; it was his way of claiming ownership of his country. His Jamaican immigrant parents had not felt fully American; they lived in a West Indian cultural bubble, dancing to calypso and revering the queen. His own generation knew America better, but for young Caribbean-Americans growing up in the shadow of Jim Crow, to know America was not necessarily to love it. Colin Powell was not the only son of West Indian immigrants to figure prominently in the politics of late-twentieth-century America, after all. The other three were Stokely Carmichael, Louis Farrakhan, and Malcolm X.\n\nAs a young man, Powell had seen the army as both a refuge from the actually existing United States and a vision of what it might one day be. By the time he entered the service in the late 1950s, the military was fully integrated, but America itself was not. As a young officer stationed in the South, leaving base meant searching in vain for a restaurant that would serve him a hamburger or let him use the bathroom. When he and his young wife went looking for housing near Fort Bragg, North Carolina, in 1962, a real estate agent showed them an empty shack in the middle of a garbage dump. The army, by contrast, offered camaraderie, shared risk, and common purpose, across the color line. While hardly devoid of racism, it rewarded his talent and his willingness to play the game. \"I don't shove it in their face,\" he said years later, in explaining his success. \"I don't bring any stereotypes or threatening visage to their presence,\" and \"I perform well.\" That he did, in both senses of the word.\n\nHe had risen within the army by fiercely protecting its interests and risen in Washington by skillfully serving the interests of the civilians for whom he worked. But now, with Iraq's invasion of Kuwait, those imperatives collided. Crowe had done only one tour in Vietnam; Powell had done two and thus the war was even more seared into his consciousness. He had seen it corrupt and degrade his beloved army; he had watched \"recently healthy young American boys, now stacked like cordwood\" and sat on helicopter rides alongside the bodies as they began the long journey home. Lebanon had left its mark, too. As Caspar Weinberger's military aide in 1983, he had taken the late-night call from Beirut informing him that hundreds of Marines had been blown to bits. Weinberger had responded by setting the bar for military action so high that it virtually guaranteed Reagan would not use force again, and that suited Powell just fine.\n\nWhen Bush administration officials discussed how to respond to Iraq's invasion of Kuwait, Powell felt the ghosts of Vietnam and Lebanon in the room. Civilians kept asking him about military options for liberating Kuwait; he kept asking whether Kuwait was worth dying for. Claims that Iraq could be defeated from the air terrified him; that's what Rostow and company had claimed in Vietnam. Bush said Iraq didn't look that tough; it hadn't even managed to beat Iran. \"But Iran paid in manpower,\" Powell replied. It lost close to half a million men.\n\nPowell's objections didn't help his relationship with Secretary of Defense Dick Cheney. Powell wasn't crazy about his boss. Only half in jest, he called Cheney a \"right wing nut,\" and as Crowe had done with Abrams, Powell often noted Cheney's deferments during Vietnam. It was harder to know what Cheney thought about Powell, or about most things, for that matter. That's why people called him \"the Sphinx.\" But he had a reputation for humiliating generals. \"You're chairman of the Joint Chiefs,\" he bellowed after a meeting where Powell raised questions about the wisdom of war. \"You're not secretary of state. You're not the national security advisor. And you're not secretary of defense. So stick to military matters.\"\n\nPowell didn't give up. If he was \"the skunk at the picnic,\" he said later, they could all \"take a deep smell.\" That fall, with U.S. troops streaming into the Gulf and war drawing nearer, he mounted another protest. First he tried to convince Cheney that they should give sanctions more time to work, but Cheney replied that Bush had already made up his mind. Then he went to Secretary of State James Baker, who he suspected wanted a diplomatic way out, and received mild encouragement. Then he talked to Brent Scowcroft, Bush's national security advisor, who seemed annoyed that Powell wouldn't let the matter lie. Finally he got his meeting with Bush himself. It was a Friday afternoon in early October, and the president was in one of his light and goofy moods, which made Powell even more nervous. Powell wanted to make a stand, to do what he\u2014like Crowe\u2014wished his predecessors had done in Vietnam. He tried to make the case for containment, which he called \"strangulation\" because it sounded more aggressive. He acknowledged that his strategy could take as many as two years, but said he was convinced that the embargo would eventually work, at least in restoring the prewar status quo, which was all he thought America needed to do. What he didn't say\u2014it was just too difficult in that atmosphere; it might have elicited even harsher rebukes\u2014was that two years was sure as hell an acceptable wait if it meant sparing the thousands of young Americans about to be sent into the meat grinder, young Americans who were a lot more real to him than to his country-club bosses. But Powell didn't say that. He had built his career on giving people in power what they wanted and he knew even better than they did what they didn't want: an angry, self-righteous black man \"peeing on the floor.\" He could live, he declared, with whatever decision Bush made. And Bush\u2014who liked Powell, liked what he represented about America, sometimes almost forgot he was black\u2014seemed pleased.\n\nA decade later, when America's military confidence had swelled, and the Gulf War looked in retrospect like an easy and obvious thing, people often forgot how close the United States came to not fighting the war at all. The military wanted to give sanctions more time to work. Baker was itching to cut a diplomatic deal. Scowcroft was trying to anticipate Bush, and Bush\u2014although he grew more certain over time\u2014was tentative early on.\n\nFor its part, the public was hardly gung ho. Polls showed that Americans supported military action, but only when told it would be cheap and quick and that the blood would be spilled by the other side\u2014exactly what retired military leaders were telling them not to think. In Congress, virtually the entire Democratic leadership, and most of the rank and file, voted no; it was the closest vote on authorizing military force since 1812. Zbigniew Brzezinski warned that the war could produce twenty thousand U.S. casualties and \"potentially devastating economic consequences.\" Aging veterans of Camelot such as Robert McNamara and George Ball came out of the woodwork in opposition. In testimony opposing the war, Arthur Schlesinger urged senators to \"take a few quiet moments to revisit the Vietnam monument\" before casting their votes. \"Vietnam hangs in the collective subconscious like a bad dream, a psychic wound,\" observed Newsweek, \"It hovers over politicians and policymakers, the past that will not die.\"\n\nBut the past did begin to die. Analogies don't live forever, and just as the \"lessons\" of Munich had superseded the \"lessons\" of World War I, America's victories in the cold war and the Gulf began to undermine the \"lessons\" of Vietnam. Before the war, Saddam had warned that \"the United States relies on the Air Force and the Air Force has never been the decisive factor in a battle in the history of wars,\" which was pretty much Colin Powell's view. But they were both proved wrong. During the post-Vietnam years, while army men like Powell had been cursing airpower and the mirage-like promise that it would make warfare cheap and clean, airpower had undergone a technological revolution. During Nixon's Christmas bombings of Hanoi in 1972, the average B-52 missed its target by 2,700 feet. In the Gulf War, by contrast, planes using laser-guided smart bombs regularly came within a foot or two. The air force could now choose not merely which building to hit, but which floor. At one point a stealth fighter guided a bomb through an air shaft in the roof of the Iraqi air command.\n\nThe desert, it turned out, was an easier battlefield than the jungle; and Iraq, a more urbanized, industrialized country than North Vietnam, was more vulnerable to attacks from the air. War, Crowe had insisted, is never neat and tidy. But for America in the Gulf, the air war came close. Leslie Gelb called it \"immaculate destruction.\"\n\nBy the time American troops went in on the ground, the Iraqi military had been decapitated. Its various units were like limbs without a brain, unable to communicate with one another or do much of anything except flail. In one massive battle, Iraq lost eight hundred tanks; America didn't lose a single one. One hundred hours after the ground fighting began, it was over. Terrified Iraqi soldiers were streaming back along Highway 80 into Iraq. It was a terrible, savage war, but only for one side. Perhaps one hundred thousand Iraqis died. For the United States, the figure was 146.\n\nWar, American policymakers were coming to believe, was easier in a one-superpower world. Saddam, like Noriega, had no geopolitical big brother to come to his aid. The flip side was that because America didn't have to worry about holding off communists in other parts of the globe, it could focus its firepower on the Gulf. And because communism was dead, and the Bush administration had rallied most of the world to America's side, other countries paid the bills. Germany and Japan both ponied up; the Saudis alone pledged $15 billion. The Gulf War didn't land the U.S. government deeper in debt, as Brzezinski and others feared. By some accounts, America actually turned a profit on the war.\n\n\"By God,\" exclaimed President Bush, \"we've kicked the Vietnam syndrome once and for all.\" In New York City, Colin Powell rode in a white Buick convertible, with Cheney in the car behind, while a million yellow ribbons fluttered, six tons of confetti fell, and close to five million people cheered. A radio station in Raleigh, North Carolina, began opening its programs with the words \"broadcasting from the most powerful nation in the world.\" House Minority Whip Newt Gingrich and spy novelist Tom Clancy consulted with an expert in Latin and coined the phrase Venimus, Vidimus, Icimos Gluteos: \"We came, We saw, We kicked ass.\"\n\nBut for all the chest-thumping, the victory felt incomplete. It \"is like coitus interruptus,\" declared Norman Podhoretz. \"Nothing is consummated.\" Precisely because America's military had proved so awesome, critics began to wonder why it hadn't removed Saddam from power. As in Korea forty years earlier, after MacArthur's stunning maneuver at Inchon, the thrill of success bred the temptation to reach for more. Polling showed that while only a third of Americans supported overthrowing Saddam before the war began, by a week into the air campaign the figure had more than doubled.\n\nBush also wanted Saddam gone. He had assumed that when Saddam's military rivals saw him thrashed they would oust him from power. The problem was that Saddam had no military rivals; he had murdered everyone with talent or an independent power base. As the war drew to a close, Bush encouraged Iraqis to \"take matters into their own hands, to force Saddam the dictator to step aside,\" but Bush's intended audience\u2014Saddam's fellow generals\u2014didn't respond. Instead, it was Iraq's Kurds and Shia, whom Saddam and his Sunni Arab clique had brutally repressed, who rose up against his rule.\n\nNo senior Bush administration official argued that America should march to Baghdad and depose Saddam itself. That idea, which many hawks would come to support in retrospect, was still virtually unthinkable in 1991. Assumptions about what the U.S. military could achieve, how much the U.S. economy could afford, and how much the American people would bear remained far too low, even after America's success in expelling Saddam from Kuwait. The hubris bubble had not yet fully swelled. The Vietnam syndrome was dying but not yet dead.\n\nThere was a narrower debate, however, which was revealing in its own right. Bush did not want to help the Kurds or Shia at all. On Powell's recommendation, he would not even ground Saddam's helicopters, which were slaughtering the Shia rebels who had risen up on what they thought was Bush's command. Partly that was because he feared getting sucked into an Iraqi civil war. But his reluctance ran deeper than that. Bush worried that if he helped the Shia, the Sunni Arab regimes that had backed the war, but deeply feared a Shia, pro-Iranian regime in Baghdad, would denounce him, shattering the international coalition that had prosecuted the war. That mattered because in 1991\u2014as opposed to 2003\u2014America's leaders didn't believe they could fight Saddam on their own dime. Poverty bred multilateralism and multilateralism induced restraint. In an era of economic scarcity, America needed rich allies like Saudi Arabia to foot the bill.\n\nWhat's more, Bush agreed with his Sunni Arab allies. Like them he saw the Shia rebellion not as a democratic uprising but as a sectarian one, likely to replace one species of tyranny with another, this one quite possibly fundamentalist. For Bush, the Shia rebellion in Iraq's south evoked less Poland in 1989 than Iran in 1979, and in this respect he was closer to Jeane Kirkpatrick than to Francis Fukuyama, closer to William Crowe than to Elliott Abrams. He didn't believe that within the cocoon of Saddam's dictatorship lay an Iraqi democracy ready to sprout wings. He had hoped for a coup that would produce a milder, pro-American version of Saddam, a Sunni general who would be our bastard, not his own. But when he saw that this was not possible, he stuck with Saddam as the lesser evil.\n\nFor Bush, lesser evils were about the best one could expect. The last American president to serve in World War II, he had come of age when democracy remained a rather exotic and beleaguered form of government. And his realist instincts were reinforced by Scowcroft, a Kissinger disciple with a gentle disposition but an extremely harsh view of the world. Bush simply did not believe that trying to promote American ideals would promote American interests; like Irving Kristol, he feared the unintended consequences of good intentions. He had not been particularly happy about the breakup of the Soviet Union, preferring autocratic stability to democratic chaos, preferring the devil he knew to the one not yet born. For him the Panama invasion had been about drugs, domestic politics, and American honor. Democracy hadn't played much of a role. And now he would not lift a finger to give democracy a chance in Iraq. His signature word\u2014it became a running joke on Saturday Night Live\u2014was prudence, which Merriam-Webster defines as \"caution or circumspection as to danger or risk.\" He was like Eisenhower in this way. His foreign policy was Hippocratic: First, do no harm.\n\nCheney and Wolfowitz were different: more optimistic and more aggressive. Cheney had been slow to recognize the changes in the U.S.S.R., but once he did, he wanted to dismember America's old foe. He wasn't trying to promote democracy necessarily; he wasn't a terribly idealistic sort. But he wanted to expand American power. To be truly safe, he believed, America must extend its dominance, and in that way he exemplified the spirit of the emerging post\u2013cold war age. Bush was comfortable limiting America's power and his own; limits protected you against bad things. He didn't want to restore the imperial presidency; he believed Johnson had only hurt himself by hoodwinking Congress on Vietnam. So he let Congress vote on the Gulf War, thus giving himself cover if it went bad. Cheney had opposed that. For him, letting Congress in the game wasn't cover; it was a straitjacket. Cheney hadn't wanted to run the war through the UN, either. Why give anyone else a say? And when the war ended, he was less inclined than Bush to defer to America's Arab allies. He didn't want to get drawn into an Iraqi civil war any more than Bush did, but he supported shooting down Saddam's helicopters. He didn't necessarily believe that a Shia government would be democratic, but he did think it might be more pliant. He was hungrier than Bush to extend America's reach and more willing to take risks to bring that about.\n\nLike Cheney, Wolfowitz also believed in extending American dominance. But he saw the spread of democracy and the spread of American power as more closely intertwined. Like Abrams and Fukuyama, who was his close friend, he had been in his thirties in the early Reagan years. And like them he had been shaped by the great tide of democratization that washed across the developing world. If Abrams's crucible had been Panama, Wolfowitz's had been the Philippines. At first, the Reagan administration\u2014in keeping with the Kirkpatrick line\u2014had embraced Manila's pro-American dictator, Ferdinand Marcos. But as Marcos weakened and a communist insurgency grew, Wolfowitz, as assistant secretary of state for East Asia, began pushing for reform. By late 1985, Kirkpatrick, now out of government, was accusing her former colleagues of the same democratic \"purism\" that had led to the downfalls of Somoza and the Shah. \"Anyone who thinks it [the Marcos regime] will be succeeded by an authentic parliamentary government with sensible economic policies,\" added Irving Kristol, \"has been swallowing too many happiness pills.\" But Wolfowitz kept swallowing them. In 1986, when Marcos rigged an election and Filipinos poured into the streets in protest, Wolfowitz helped convince Secretary of State George Shultz that America must force Marcos out, and Shultz in turn convinced Reagan. A democratic, pro-American government (with sensible economic policies) took power, and the communist insurgency faded. Wolfowitz later called it \"the high point of my career.\"\n\nNot surprisingly, then, Wolfowitz saw the Shia rebels differently than Bush and Scowcroft did: not as fundamentalists, but as democrats. Of all the top Bush officials, he was the most adamant that America come to their aid. He kept urging that Saddam's helicopters be shot down, even after the bureaucratic fight was lost.\n\nIn so doing, he slammed up against Powell. Like Abrams and Crowe before them, Wolfowitz and Powell were oil and water. Powell loathed disorganization; in Wolfowitz's office, the papers were stacked up so high you could barely see. Wolfowitz was a nerd: He spoke six languages and kept ten ballpoint pens inside his suit pocket. Powell distrusted intellectuals. \"There are some people at that school you hang out at,\" he later told a Harvard professor, \"who contribute nothing else to the national GDP except to create these great schemes.\"\n\nIn late March, with Wolfowitz still kicking up a fuss about the helicopters, Powell screamed at him that the question was closed. Inside the administration, cautious realism still held the upper hand. But things were changing. Before the war, critics had mostly called Bush reckless for going to war too early. It was a sign of the shifting political mood that they now called him timid for ending it too soon.\n\nBy any reasonable standard, the Gulf War was a success. Facing a public, a Congress, and a military still deeply scarred by Vietnam, Bush had taken America to war and won it at a price\u2014in money and lives\u2014that Americans were willing to pay. Yes, Saddam was still in power, but he was weaker. His military had been ravaged and as the price of defeat he was forced to accept a humiliating regime of sanctions and inspections designed to ensure that it was not rebuilt. The Kurds and Shia paid a terrible price for America's decision to end the war early, but the Bush administration eventually established a safe haven in Iraq's north that gave many Kurds some respite from Saddam's horrors. And in both north and south, American planes patrolled the skies to make sure Saddam did not menace his neighbors again. The hawks who had wanted not merely to liberate Kuwait but to cripple Iraq had achieved their goal.\n\nThe war also dramatically expanded American dominance of the Middle East. When Reagan took office, Central Command (CENTCOM), which oversaw U.S. military operations from Kenya to Kazakhstan, and ran the war in the Gulf, had not even existed. Back then, America had jostled for influence in the Middle East with the U.S.S.R. Now Syria, one of Moscow's former clients, had joined a U.S.-led war against one of its other former clients, Iraq, while the Soviets watched from the sidelines. In the 1980s, America's military presence on Middle Eastern soil had been limited to a navy supply base in tiny Bahrain and a few hundred peacekeepers in the Sinai desert. Now, after the Gulf War, U.S. ground troops occupied six Middle Eastern countries and U.S. planes patrolled the skies above much of Iraq. In 1985 the United States had accounted for 15 percent of arms sales to the Middle East. A decade later it was 72 percent.\n\nYet despite all this, as George Bush began his 1992 reelection campaign he found himself attacked by both conservatives and leading Democrats for having left Saddam in power. In April 1991, Tennessee Senator Al Gore, who only months earlier had agonized about whether to support the war at all, demanded that Bush develop a strategy of democratic regime change so that \"we advance the day not just when Saddam Hussein is out of power, but when there exists in Iraq a government that grants reasonable consideration to all its peoples.\" At the 1992 Democratic convention, Georgia governor Zell Miller said that President Bush \"talks like Dirty Harry but acts like Barney Fife.\" It was the same fate that had befallen Harry Truman when he pulled back from MacArthur's bid to liberate the Korean peninsula from communism. American power had expanded dramatically, just not as dramatically as American confidence. It was a telltale sign that hubris was starting to build. The higher America flew, the more fervently critics demanded that it fly higher still.\n\n## CHAPTER FOURTEEN\n\n## FUKUYAMA'S ESCALATOR\n\nFor Colin Powell, Bill Clinton's presidency constituted an irony, maybe even a cruel joke. He had been sad to see the Bushies go. Sure, Cheney was a bit icy. Wolfowitz's ideas were a little grand. But Powell liked Republicans. Like him, they believed in authority, discipline, order. With Republicans, meetings started on time; decisions got made; you knew who was in charge. He particularly liked Reagan and Bush. They were courtly and dignified, traditional in a way he appreciated. They looked the way he thought presidents should look. Powell considered many in his own age cohort, and among the slightly younger baby boomers, to be self-indulgent, whiny, unkempt. The military was a museum of old-fashioned virtues, and when it came to civilians, Powell found them best expressed in men and women a generation older than him.\n\nIt was a challenge, therefore, to serve a commander in chief who, on at least one occasion, sat in the Oval Office in his sweatpants, munching a banana smeared with peanut butter. Powell, who still had eight months left as chairman of the Joint Chiefs when Clinton took office, was a fanatic about punctuality: At his morning staff meetings, everyone took their seats at 8:30; he walked in at the stroke of 8:31. The new White House, by contrast, ran on what reporters called \"Clinton Standard Time\": anywhere from a few minutes to a few hours late. (It was an irony of the Clinton-Powell relationship. Powell had developed his mania for promptness in part to combat racist stereotypes of blacks. Clinton, a son of the redneck South, conformed to those stereotypes and partly as a result was dubbed \"the first black President.\") Powell rose early; Clinton stayed up late, often working\u2014and expecting his staff to work\u2014until 2 A.M. Powell was stoic: When injured for the second time in Vietnam, he asked the army not to inform his wife. Clinton loved to emote. At a touchy-feely, get-to-know-you session for cabinet members and White House staff, he described how painful it was as a young boy to be fat.\n\nFinally, Powell prized proper appearance. When meeting someone for the first time, he instinctively looked to see if their shoes were shined. Clinton, by contrast, often took reporters' questions in his tracksuit, while sweating profusely from his morning run. He was also a binge eater, whose intake increased with stress. And some of his advisers were worse. Powell's new boss at the Pentagon was Les Aspin, a disheveled, doughy man known to lunch on potato chips doused in mayonnaise. At a meeting with Jordan's King Hussein, Powell watched in horror as Aspin ate thirteen hors d'oeuvres.\n\nThere should have been at least one note of consolation: At least these graying beatniks wouldn't start many wars. Powell distrusted foreign policy makers who hadn't served in uniform, but he actually preferred those who had avoided Vietnam because they opposed it to couch-potato militarists like Abrams, Wolfowitz, and Cheney. In the culture war, he and the Clintonites were often on opposite sides, but when it came to real war, Powell hoped that the children of Woodstock and the children of Khe Sanh would find common ground in their resolve to never do anything like Vietnam again.\n\nThat's why the Clinton years were a cruel joke, because the hippies had changed. Vietnam no longer traumatized them; they were starting to appreciate war. It wasn't apparent right away, but by the mid-1990s, after some humiliating false starts, the Clintonites began learning many of the same lessons about the post\u2013cold war age that the Bushies had learned: that military force worked, that America's political and economic model was universal, and that when America pushed hard in one direction, its allies fell in line. There were partisan differences, to be sure, differences that would become clearer when Cheney, Wolfowitz, and Powell returned to office under a new President Bush. But by Clinton's second term, the hubris bubble was again starting to swell. The ethic of dominance, it turned out, was ideologically ambidextrous. In the Clinton years, American hegemony learned to hit from the left side.\n\nThe Clintonites, to be fair, had never really been hippies. During Vietnam they had formed the right edge of the antiwar movement, the students who wanted to save the system, not smash it, and the left edge of the establishment, the people nudging it to adapt to changing times. Many of them\u2014like Sandy Berger and Clinton himself\u2014had worked in the McGovern campaign. Some\u2014like Anthony Lake and Richard Holbrooke\u2014had served as discontented young diplomats in Vietnam. Virtually none had joined SDS.\n\nBut Powell was not wrong to assume that he could dissuade the Clintonites from using force. Compared to Cheney, who ate generals for breakfast, the Clintonites were more deferential to the military, or maybe just more afraid of it. And the military certainly wasn't very deferential to them. Two months into his presidency, when Clinton visited the U.S.S. Theodore Roosevelt, its crew mocked him behind his back. \"What did Clinton do when a protester threw a beer at him?\" went one joke making the rounds of the ship. \"It was a draft beer so he dodged it.\" Two months after that, an air force general accused Clinton of \"pot-smoking, draft-dodging [and] womanizing.\" Clinton had taken office promising to lift the ban on gays in the military, but after encountering fierce resistance from Powell and the other Joint Chiefs, he backed down. After that, many in the military considered him an easy mark. Presumably he would back down even faster if they opposed going to war.\n\nClinton had talked tough on foreign policy during the campaign, slamming Bush for leaving Saddam in power, tolerating \"ethnic cleansing\" in the former Yugoslavia, and coddling the dictators in Beijing. But less important than what he said was how little he said it. In his four-thousand-word convention acceptance speech, foreign policy received only 141 words. He had been elected, Clinton told ABC's Ted Koppel, to \"focus like a laser beam on the economy.\" Foreign policy aides\u2014except for those who dealt with international economic issues like trade\u2014were warned not to take too much of his time. When a deranged man crashed a small plane onto the White House lawn in 1994, insiders joked that it was Clinton's CIA director trying to get a meeting.\n\nPowell suspected that as long as Clinton didn't devote much personal attention to foreign policy, his foreign policy wouldn't be particularly aggressive. And he was right. In Clinton's first two years as president, his personal hesitancy on international affairs\u2014combined with America's ongoing economic troubles and the fading, but not yet dead, legacy of Vietnam\u2014left him hesitant to use military force. The early Clinton years were to the rising ethic of dominance what the Eisenhower years had been to the rising ethic of toughness: an interregnum, a pause before the march toward hubris resumed.\n\nA big reason for that pause was the U.S. economy, which in 1993 still looked every bit as fragile as it had five years earlier when Paul Kennedy predicted America's decline. Echoing Kennedy, top Clinton officials privately warned that America couldn't afford costly new foreign commitments. \"We simply don't have the leverage, we don't have the influence, [or] the inclination to use military force,\" conceded Undersecretary of State Peter Tarnoff in May. \"We don't have the money to bring positive results any time soon.\"\n\nWestern Europe, by contrast, was feeling its geopolitical oats. No longer dependent on America for protection against the Soviet Union, many Europeans believed they were finally becoming masters of their own house. And in 1993 they proudly inaugurated the European Union, which was meant to help the continent speak with one loud voice on the world stage. The Gulf War may have left America dominant in the Middle East, but in Europe, American power appeared to be in retreat. As EU chairman Jacques Poos exulted, \"The Age of Europe has dawned.\"\n\nIf European problems were now meant to have European solutions, the test case was Yugoslavia, which in the early 1990s had ruptured in a gush of blood. From the beginning the Europeans called the tune. In 1991, when first Slovenia and then Croatia seceded from the Serb-dominated Yugoslav federation, the Bush administration warned against recognizing their independence. But Germany\u2014recently unified, historically allied with Croatia, and basking in its newfound geopolitical power\u2014brushed Washington aside. Bonn recognized the two breakaway nations; the rest of Western Europe quickly followed suit, and American officials watched with a combination of passivity and indifference. As Secretary of State James Baker liked to say, \"We don't have a dog in this fight.\"\n\nSoon all hell broke loose. Slovenia contained barely any Serbs and thus made a fairly clean getaway. But in Croatia, Serbs comprised more than 10 percent of the population, and many of them wanted no part of an independent Croatian state led by ultranationalist, anti-Serb bigots. In May 1991, Serbs in the small eastern Croatian town of Borovo Selo raised the old Yugoslav flag. Croatian policemen arrived to take it down. Suddenly the Serbs began shooting. They not only killed a dozen policemen but also gouged their eyes out. It was a small taste of things to come.\n\nA year later, in the spring of 1992, heavily Muslim Bosnia seceded as well. But with Orthodox Christian Serbs comprising roughly a third of its population, and Catholic Croats another fifth, it was the most vulnerable of all. On April 6, the Europeans recognized Bosnia's independence; the next day America did, too. Two weeks later, local Serbs, backed by the Yugoslav army, laid siege to the Bosnian capital, Sarajevo, shelling it night after night. By midsummer, the city that had hosted the 1984 Winter Olympics was a horror show. With its cemeteries overflowing, people began burying bodies in public gardens, even backyards. There was barely any water, food, or fuel. \"What is the difference between Sarajevo and Auschwitz?\" went a local joke. \"In Auschwitz at least they had gas.\"\n\nIt got worse. In the Bosnian hinterland, the Serbs were building concentration camps. In the town of Banja Luka, they dispossessed Bosnian Muslims of their homes and possessions, shaved their heads, herded them onto cattle trains, and dumped them at a place called Omarska, an old iron mine now encased by barbed wire. Inside, inmates were made to endure\u2014and commit\u2014unspeakable crimes. Men were forced at gunpoint to castrate their sons and rape their daughters. Lacking adequate supplies of food and water, some inmates were reduced to drinking their own urine; others tried to survive by eating grass. The goal was to \"cleanse\" Bosnia of Muslims: Some would be killed, others deported, and the rest would be too psychologically shattered to ever assert their independence again.\n\nEurope's response was to push through a UN arms embargo, which, given that Belgrade controlled virtually all of the Yugoslav army's heavy weaponry, merely prevented the Bosnians from defending themselves. Britain, France, and several other European nations also sent troops as part of a UN \"peacekeeping\" force. But in Bosnia there was no peace to keep. Allowed under UN guidelines to defend their aid convoys, but not allowed to defend Bosnians, the peacekeepers proved worse than bystanders. They became potential hostages, whom the Serbs could seize if the West threatened military action. Useless at preventing Bosnia's destruction, the UN troops proved effective only at preventing the world from doing anything about it.\n\nThe Clintonites were appalled. Vice President Al Gore, National Security Advisor Anthony Lake, Ambassador to the United Nations Madeleine Albright, and Ambassador to Germany Richard Holbrooke began pushing a policy called \"lift and strike,\" aimed at saving Croatia and Bosnia by lifting the UN arms embargo and striking Serb positions from the air. But implementing \"lift and strike\" required overruling the Europeans, who still saw the former Yugoslavia as their show. And before the Clintonites could overrule the Europeans, they first had to overrule Colin Powell.\n\nThat proved impossible. On national security, Powell was Clinton's most powerful adviser. He had more foreign policy experience than the new president and many of his top aides, and was vastly more popular on Capitol Hill. Within the military, he sometimes seemed to be the only person preventing outright insubordination. Clinton was also terrified that Powell would resign and run against him for reelection in 1996. In their first clash, on gays in the military, it was the president\u2014not the general\u2014who had caved. \"Powell simply overwhelmed the administration,\" noted Holbrooke. \"He regarded the new team as children. And the new team in turn regarded him with awe.\"\n\nThe Gulf War had made Powell a national hero, but it hadn't altered his reluctance to use force. Bosnia struck him as a cross between Lebanon and Vietnam. As in Lebanon, America would be putting itself in the middle of a Byzantine, brutal civil war that had little direct bearing on its security. And as in Vietnam, the former Yugoslavia's terrain was a forbidding patchwork of forests and mountains, nothing like the open desert of Kuwait. Civilians sometimes called the Serbs a lackluster force, undisciplined and frequently drunk, but Powell likened them to the Vietcong: tenacious defenders of what they considered their home soil. Powell grew particularly agitated when Clintonites suggested that America could defeat the Serbs from the air. \"When I hear someone tell me what airpower can do,\" he declared, \"I head for the bunker.\" When administration officials asked him what it would take to save Bosnia, Powell replied two hundred thousand ground troops. That was meant to be a conversation-stopper, and usually was.\n\nDespite Powell's opposition, by May 1993 Clinton officials thought they had reached a consensus on \"lift and strike.\" Secretary of State Warren Christopher was dispatched to consult with the Europeans. But the key word was consult. The Christopher trip was multilateralism in its truest form. America was not presenting its allies with a fait accompli; it was genuinely seeking their opinion. It was a sign that when it came to Europe, the balance of power had changed: America would ask, not tell.\n\nThe Europeans, in turn, told Christopher to go to hell. British and French leaders refused to endanger their peacekeepers by taking steps sure to enrage the Serbs. \"We do not interfere in American affairs,\" declared one top European official. \"We hope they will have enough respect not to interfere in ours.\"\n\nWhen Christopher reached Bonn, he got a call from Les Aspin telling him he might as well come home. Clinton had changed his mind; he no longer supported \"lift and strike.\" Powell had given him a copy of Robert Kaplan's travelogue, Balkan Ghosts, which Clinton read as suggesting that the peoples of the region had been slaughtering each other for five hundred years. Soon Christopher was saying the same thing. In testimony before Congress soon after his return, he emphasized that the killing went both ways. State Department underlings were dispatched to find examples of atrocities committed by Bosnians against Serbs.\n\nChristopher's trip had been \"an exchange all right,\" snarled former Reagan official Richard Perle. \"Warren Christopher went to Europe with an American policy and he came back with a European one.\" But it was more complicated than that. The Europeans had rolled Christopher in large measure because Powell had again rolled Clinton. As usual there was a close association between presidential power and American assertiveness. It was no surprise that at the same moment Time magazine dubbed Clinton \"The Incredible Shrinking President,\" many in Europe were talking about the incredible shrinking United States.\n\nEventually Clinton's fortunes would turn around, and America would resume the upward climb that had begun with Panama and would end with Iraq. But in 1993 that was still several years away. Before the Clinton administration could ascend the ladder of dominance, it would taste even more humiliation first.\n\nIf the administration's inability to end the war in Bosnia was embarrassing, equally embarrassing was its inability to win the war three thousand miles to the south, in Somalia. In Bush's final months, in a move that struck some observers as exceedingly noble and others as exceedingly cynical, America had sent thirty thousand troops to Somalia, a cold war pawn turned failed state whose people were starving because warlords prevented the distribution of food. Throughout 1992, top Bush officials\u2014and particularly Powell\u2014had opposed military intervention, before quickly reversing themselves after Clinton's election in November. In trying to explain the about-face, some observers cited the mounting horror of the situation, captured on CNN: the skeletal children too weak to swat away flies. Cynics, who included some inside the Bush administration itself, offered a harsher explanation: Bush and Powell were going to war in East Africa so Clinton wouldn't be able to in the Balkans. In Somalia, admitted Scowcroft, it was \"a lot cheaper\" than in Bosnia \"to demonstrate that we had a heart.\"\n\nAmerican troops were supposed to be gone by the time Bush left office, but the deployment lingered on in a fit of absentmindedness. In the early Clinton administration, where foreign policy was often an afterthought, Africa was less than an afterthought, and anything closely associated with George Bush was suspect, Somalia received a fraction of the high-level attention devoted to the controversy at the White House travel office. No top Clinton official even visited Somalia. Aspin had wanted to go, but his less than robust constitution was so unsettled by the required inoculations that he canceled his trip.\n\nBut someone was paying attention: Boutros Boutros-Ghali, the imperious secretary-general of the United Nations. As an Egyptian, he had strong feelings about the hellhole nearby, and one of those feelings was intense hostility toward its strongest warlord, Mohammed Farrah Aidid. Over the course of 1993, Boutros-Ghali stood at the intersection of two dangerous trends. Militarily the peacekeeping force in Somalia was growing weaker as third-world troops gradually replaced their more intimidating U.S. counterparts. But politically the operation was growing more ambitious as peacekeepers expanded their mandate from merely safeguarding the distribution of food to building a functional Somali government, which required sidelining Aidid.\n\nOne day in June, Pakistani peacekeepers announced that they were planning to confiscate some of Aidid's weapons. Aidid's men responded by killing twenty-four of them and mutilating their bodies. The American general in charge of UN forces then upped the ante by putting a bounty on Aidid's head.\n\nBy now, Powell wanted U.S. forces removed. So did Aspin, but he was shouting into a void. (During his yearlong tenure as defense secretary, he had a grand total of two meetings with the president.) Powell might have gotten Clinton's attention, but he was weeks away from retirement and had largely checked out. So the White House sleepwalked toward a showdown with a warlord it knew little about in a war it thought it could ignore.\n\nOn October 3, U.S. Special Forces tried to capture two of Aidid's top lieutenants at a hotel in downtown Mogadishu. Suddenly one of their Black Hawk helicopters was shot down, leaving wounded American soldiers trapped in the middle of a Somali slum. Aidid's men burned tires to alert their comrades, and throngs of AK-47-wielding fighters descended on the besieged Americans. By day's end, eighteen Americans\u2014along with roughly one thousand Somalis\u2014were dead. CNN showed a Somali mob dragging one U.S. soldier's disfigured corpse through the streets.\n\nHolbrooke called it \"Vietmalia\": Vietnam plus Somalia. Powell had retired, but the Powell Doctrine\u2014that America should go to war in only the most extreme and auspicious of situations (and then with overwhelming force)\u2014was now more alive than ever. For one thing, after Mogadishu, presidential power over foreign policy declined even further. Four days after the massacre, West Virginia Senator Robert Byrd, chairman of the powerful appropriations committee, literally screamed at Clinton that unless he withdrew U.S. troops immediately, Congress would cut off funding. A few months later, they were gone. When Clinton awarded a posthumous Medal of Honor to one of the soldiers killed in Somalia, the man's father told him he was unfit to be commander in chief.\n\nIf Bosnia had proved America weak in Europe, suddenly it looked weak almost everywhere. In Haiti, the UN had brokered a deal with the junta that overthrew elected president Jean-Bertrand Aristide. Two hundred U.S. troops were to begin training the Haitian military as part of a plan to return the island to democracy. But when the U.S.S. Harlan County steamed into Port-au-Prince a week after the killings in Mogadishu, it encountered a mob on the dock chanting \"Somalia! Somalia!\" For a day the ship idled offshore as Clinton officials tried to decide what to do. Then it turned around and headed back home. Critics called it Somalia Two.\n\nRock bottom came the following year in the tiny central African nation of Rwanda. In January 1994, a Canadian major general named Rom\u00e9o Dallaire, who was heading the small UN peacekeeping mission there, received chilling news. An informant told him that Hutu militia-men, trained by Rwanda's overwhelmingly Hutu military, were planning to exterminate the country's Tutsi minority. Everything was in place. They were simply awaiting the signal to begin.\n\nDallaire faxed UN headquarters in New York. The informant was willing to show him where the militias stashed their weapons, and Dallaire wanted permission to seize them. His UN superiors refused. He was expanding the scope of his mission. It sounded like Somalia. The Clinton administration would never go along.\n\nThree months later, a plane carrying Rwanda's president was shot down: It was the signal to begin. Teams of Hutu extremists began hunting down Tutsis and moderate Hutus village by village, neighborhood by neighborhood, house by house. In the city they used guns; in the countryside, where guns were scarcer, they used machetes, clubs studded with nails, hammers, screwdrivers, even the handlebars of bicycles. At a school where two thousand Tutsis had taken refuge among a few peacekeepers, the Tutsis pleaded with the UN troops to shoot them rather than leave. Death by machine gun was preferable to being hacked to death.\n\nDallaire appealed for reinforcements, but the Pentagon resisted fiercely. Even though the peacekeeping mission didn't include Americans, Clinton officials feared that the United States might get dragged in if the peacekeepers got in trouble. The best way to ensure that Rwanda didn't become Somalia was to withdraw the UN altogether. For its part, the U.S. Embassy staff was already on its way out. The American ambassador's chief steward, a Tutsi, begged to be taken along. \"We're in terrible danger,\" he pleaded. \"Please come and get us.\" The ambassador refused. The steward's family and more than thirty other Rwandans employed by the United States were subsequently killed.\n\nSpooked by Somalia, the Clinton administration did not hold a single high-level meeting to discuss intervention. Government spokesmen tied themselves in rhetorical knots to avoid calling the killings \"genocide,\" for fear that if they did, America would be obligated to try to stop it. By the time the carnage was over, one hundred days later, eight hundred thousand Rwandans were dead in what one journalist called \"the most efficient killing since the atomic bombing of Hiroshima and Nagasaki.\" Back home in his native Quebec, Dallaire swallowed a fistful of pills, washed them down with a bottle of Scotch, lay down on a park bench, and tried to die.\n\nFour years later, on a tour through Africa, Bill Clinton arrived in Rwanda's capital and apologized. He met survivors and handed the country's new president a plaque commemorating the dead. Back home, critics were scathing: Clinton apologized too often and too well. He had done so twice already on the same trip.\n\nBut the apology mattered. It mattered that when asked to name the greatest regret of his presidency, Clinton did not mention Somalia, as Reagan had mentioned Lebanon, an intervention gone bad. He mentioned Rwanda, an intervention never carried out. It mattered because in death the Rwandans came to matter more to American foreign policy than they had in life. Rwanda became the anti-Vietnam: a parable about the horror of not going to war. Along with Bosnia, it became the occasion for an historic reconciliation between the American left and American power.\n\nUnderstanding this reconciliation requires understanding that intellectually, Elliott Abrams and Paul Wolfowitz were not the only people who embraced Fukuyama's vision. In the 1990s, many liberals also came to believe that they were living at history's end. Among liberals, the idea that all people hungered for democracy and human rights was not new. Carter had said as much in the 1970s when he made human rights the rhetorical centerpiece of his foreign policy. But during the cold war, there had been a problem: If all people wanted human rights, why did so few governments\u2014especially in the third world\u2014respect them? Mainstream post-toughness liberals (unlike some on the New Left) did not admire dictators like Fidel Castro and Ho Chi Minh. Yet many liberals suspected that for all their nastiness, Castro and Ho were giving their people things that they genuinely wanted: the dignity that comes from casting off imperial rule, the rough-hewn social justice that comes from breaking up feudal estates and giving long-suffering peasants some land. Why else had the Vietnamese been more willing to fight for Hanoi than for Saigon and had the Cubans refused to rise up against Castro, despite the CIA's best efforts? Most liberals in the '70s still considered political freedom a universal desire. But many accepted that, in some places at least, economic development and economic justice might have to come first. In a nod to third-world autocracies, Carter even defined human rights as comprising not just freedom of speech and the secret ballot but also the right to food, shelter, health care, and education. \"A lot of third world countries...insisted that 'human rights begins at breakfast, and you cannot expect us to worry about frills like civil and political liberties until we can feed our people,'\" noted Jessica Tuchman, the human rights expert on Carter's National Security Council, \"and there was a lot of sensitivity to that in the State Department.\"\n\nIt was the liberal equivalent of Jeane Kirkpatrick's argument for why America should tolerate anticommunist third-world thugs. Kirkpatrick and other cold war conservatives said dictators like Somoza and Marcos were necessary evils. We should support them because at least they were better than communism. Many post-Vietnam liberals, by contrast, saw communists like Castro, Ho, and the Sandinistas as the necessary evils. We should tolerate them because at least they were better than imperialism. In the long, long term, both liberals and conservatives thought greater freedom was possible. Kirkpatrick believed that over the course of decades, or perhaps centuries, pro-American autocracies could evolve toward democracy. Liberals hoped that after fulfilling their people's basic needs, leftist tyrants would loosen up. But in the near term, both sides accepted that oppression was sometimes inevitable. Demanding immediate change might just make things worse.\n\nThe democratic revolutions of the 1980s\u2014culminating in the euphoria of 1989\u2014changed that. They not only shattered the right's cultural pessimism; they shattered the left's cultural relativism as well. From Asia to Africa to Latin America to Eastern Europe, dozens of poor countries embraced democracy, giving the lie to the claim that freedom must wait until everyone ate a good breakfast. What's more, it became painfully obvious that left-wing tyrannies hadn't been providing good breakfasts anyway; in former communist bastions like China and Vietnam, ex-revolutionaries began scrambling to lure investment from the capitalist West.\n\nThe impact on liberals was profound. The more human rights spread in the poor world, the easier it was to unequivocally condemn those regimes that still denied them. If South Africa was embracing liberal democracy, what was Rwanda's excuse? If democracy was breaking out in Budapest and Bucharest, what justification could there be for Belgrade's crimes? Communism's collapse also meant that where dictatorships remained, they often lost their leftist coloration. With the end of the cold war it became easy to see that Serbia's murderous leader, Slobodan Milosevic, was less a Marxist than a racist hypernationalist. The Bosnian genocide, many liberals insisted, was being perpetrated not by the far left, but by the far right: by \"fascists.\" Among liberals, fighting European fascism\u2014as George Orwell had done during the Spanish Civil War\u2014had a very good pedigree. And the fact that the Serbs were ravaging Bosnia, which was diverse, secular, and tolerant\u2014a liberal dream\u2014made the cause particularly compelling. \"Bosnia isn't Vietnam,\" declared Todd Gitlin, a former SDS president who shed his pacifism during the Balkan wars, \"it's Spain.\"\n\nLiberals didn't always acknowledge that they were viewing Bosnia and Rwanda through Fukuyama's prism. In fact, some suggested that these spasms of evil proved that he had been too optimistic about history's course. But Fukuyama had not claimed that political evil would disappear with history's end; he had claimed that political evil would no longer be intellectually respectable. Bad leaders in Serbia and Rwanda might still do bad things, but unlike the fascists or communists of old, their rationales would no longer capture the imagination of people across the world. This was a big change. In 1949, in the Vital Center, Schlesinger had argued that what made totalitarianism so frightening was its genuine appeal, even to the people fighting it. It had seduced some of the greatest minds in the Western world. It wasn't just an awesome foe militarily; it was an awesome foe intellectually, even spiritually.\n\nThe Serbs and Hutus, by contrast, didn't seduce anyone. They were not just militarily puny; they were intellectually puny. They offered no sweeping alternative vision for how to organize society. They had no sympathetic intellectuals at Berkeley and the Sorbonne. Unlike Nazi and Soviet totalitarianism, which had provoked deep philosophical inquiry, the ideologies of Milosevic and the Hutu genocidaires were interesting only as anthropology. They represented not a competing vision of humanity's future, but an ugly vestige of its tribal past. When liberals remarked in horror that genocide was occurring in 1994, they were assuming that 1994 should be better than 1944. They were buying into Fukuyama's belief that history climbs upward, that progress is the normal course of things. And they were doing so because with the cold war's end they had seen so much progress themselves. In this regard, they were, like Fukuyama himself, heirs of Woodrow Wilson and the optimistic early-twentieth-century progressives. And they were unlike the chastened toughness intellectuals who emerged in their wake. Arthur Schlesinger and Reinhold Niebuhr had not assumed that 1944 would naturally be better than 1904. They had not assumed that history only marched one way.\n\nSo intellectually, Clinton-era liberals were more confident than their cold war predecessors that human rights were achievable everywhere, soon. And militarily, they were more confident that America could defend those rights at the point of a gun. Comparing the liberal reaction to Bosnia and Rwanda to the liberal reaction to another genocide\u2014two decades earlier\u2014illustrates the point. In 1975, with the Khmer Rouge on the verge of taking power in Cambodia, President Gerald Ford had warned that their victory would spark a \"bloodbath,\" and urged continued military aid to prop up their anticommunist opponents. His warning was prophetic: In their three and a half years in power, the Khmer Rouge would kill or starve two million of Cambodia's seven million people. But in the mid-1970s, Americans were far too exhausted by Vietnam to seriously contemplate military action. And liberals, in particular, found it almost impossible to imagine\u2014after all the destruction the United States had wreaked in Southeast Asia\u2014that further American military action could serve the cause of human rights. \"It is argued that we must give military aid because if we do not there will be a bloodbath,\" noted New York Democratic congresswoman Bella Abzug, but \"there is no greater bloodbath than that which is taking place presently and can only take place with our military assistance.\"\n\nIf liberals in the 1970s found it hard to believe that an American military intervention could have humanitarian effects, they found it equally hard to believe that such an intervention could have a humanitarian intent. Ford's moral pleas, many assumed, were bogus; he just did not want another cold war domino to fall. But by the 1990s, when genocide broke out in Bosnia and Rwanda, the world was no longer a superpower chess-board, and liberals had to squint to see sinister geopolitical motivations behind the call for humanitarian war. To the contrary, what struck many as sinister was America's palpable indifference to atrocities in countries where it had few economic interests. If America could save Kuwait's oil, many liberals suggested, surely it should save Bosnia's people. Leslie Gelb, the young post-toughness intellectual who in 1971 had written that \"military force is a singularly inept instrument of foreign policy,\" was by 1993 a columnist for the New York Times. \"Diplomacy without force is farce,\" he thundered in a column on Bosnia; NATO should begin bombing immediately.\n\nBy the time Clinton took office, liberals were also shedding another inhibition against American military force: They were losing faith in the United Nations. In the Bush years, with the cold war over, the UN had gotten a new lease on life. Since Washington and Moscow no longer lined up on opposite sides of every third-world squabble, the Security Council finally seemed able to resolve them. It authorized the Gulf War, and in the early 1990s blue-helmeted UN peacekeepers fanned out across the world, including to Somalia, Bosnia, and Rwanda. During the 1992 campaign, Clinton even mused about a standing UN army. Perhaps, many liberals hoped, it would be UN\u2014not U.S.\u2014muscle that saved poor Bosnia from destruction.\n\nBut by 1994, the UN dream was dying. In Somalia, even congressional Democrats were furious that Boutros-Ghali had led America by the nose into the Mogadishu massacre. In Rwanda, Dallaire's peacekeeping force had proved impotent. In truth, these disasters were as much America's fault as the UN's: In both cases the organization was largely doing America's bidding. But the UN, and especially its prickly secretary-general, were convenient scapegoats. Even among liberals, the vision of a UN-led order was losing its luster.\n\nBosnia killed it. The Europeans had run the show there from the beginning, and they had operated through the UN, which in theory was what multilateral-minded American liberals wanted. The problem was that in practice the UN operation was a gruesome farce. As Blue Helmets watched, the Serbs swallowed much of Bosnia. By the fall of 1994, those Bosnian Muslims who were not exiled or dead had mostly clustered in a few \"safe havens,\" supposedly protected by UN troops. Except that they weren't really protected at all. In October the Serbs attacked the \"safe haven\" of Bihac, shelling it with cluster bombs and napalm. (It was a symbol of how the moral symbolism had flipped. In Vietnam we used napalm. In Bosnia our enemies did.) When NATO warplanes bombed Serb positions in retaliation, the Serbs took UN peacekeepers hostage and threatened to kill them unless NATO stopped. So NATO stopped and the rape of Bihac went on.\n\nBy 1995, the UN in Bosnia had become a symbol of moral equivalence between murderers and the murdered, of pitiful weakness in the face of genocide. The post\u2013cold war liberal faith in international institutions had collided with the post\u2013cold war liberal faith in human rights. Eight years later, many liberal hawks would see the Iraq War in similar terms: as a choice between form and substance, process and outcome; between solidarity with a set of procedures and solidarity with flesh-and-blood human beings crying out in muffled agony from across the globe. For a whole generation of liberal intellectuals and policymakers, Bosnia preconditioned the answer. After Bosnia, full-throated multilateralism never regained its allure.\n\nThe disillusionment with the UN also reflected disillusionment with Europe. On the Security Council, the Russians protected their fellow Orthodox Christians, the Serbs. The French, who had been Belgrade's allies between World Wars I and II, sympathized with the Serbs as well, especially under French president Fran\u00e7ois Mitterrand. The Germans looked out for their World War II allies, the Croats. Few European governments seemed to care much about the Bosnian Muslims, who they sometimes implied were not fully European because they were not Christian.\n\nFor American liberals, especially Jewish-American liberals, who were thickly represented among the Bosnia hawks, it all sounded hideously familiar. \"Fifty years ago, there was no room for Jews in Europe. Now there is no room for Muslims in Europe,\" wrote the New Republic's Leon Wieseltier, whose parents both survived the Holocaust. \"There is no such place as 'the West.' There is Europe and there is America and they are distinct chapters in the history of decency.\" The implication was clear. The world's natural champion of human rights, the true executor of Fukuyama's vision of a postdictatorial world, was not the UN or even \"the West\" it was the United States. With their post\u2013cold war confidence that human rights were universal, their post\u2013Gulf War confidence in military force, and their Bosnia-induced disgust with Europe and the UN, many American liberals were turning, without apology, to American power.\n\nIn Washington, this intellectual ferment mattered. Bosnia didn't stir passions in Peoria, but Bill Clinton closely followed elite, as well as mass, opinion. (He was so frequently influenced by op-eds that Vice President Gore began writing rebuttals to ones he felt were leading the president astray.) And by 1995 the anger in elite circles over Bosnia was boiling over. The New York Times, which some suspected was trying to expiate its guilt for its inadequate coverage of the Holocaust, was relentless. (\"What the fuck would they have me do?\" Clinton exploded after reading one Times column.) Low-level State Department officials resigned in protest. Twenty-seven liberal human rights, religious, and professional groups\u2014many with histories of near pacifism\u2014jointly demanded military action. (In Bosnia, even the Quakers were for war.) At the dedication of the newly opened Holocaust Memorial Museum on the National Mall, famed Holocaust survivor Elie Wiesel demanded that \"something, anything, must be done.\" As Clinton shifted uncomfortably in his seat, the audience broke into applause.\n\nIn June 1995, the House of Representatives voted to unilaterally defy the UN arms embargo. It was a declaration of independence: America would no longer practice multilateralism at Bosnia's expense. The Europeans were furious; if America stopped enforcing the embargo, the UN's Bosnia mission would collapse. But it was collapsing anyway. In July, the Serbs went after the \"safe haven\" of Srebrenica, where forty thousand Bosnian Muslims had taken refuge. First the Serbs captured thirty of the Dutch peacekeepers supposedly keeping Srebrenica safe, dispossessing them of their weapons, vehicles, and berets. Since the UN had to approve any NATO military response, and such a response might get the Dutch hostages killed, that gave the Serbs an open door. While UN peacekeepers watched, the Serbs put Srebrenica's women, children, and elderly on buses, which eventually dumped them at another \"safe haven\" a few hours away. Along the route, Serb soldiers periodically stopped the buses to select attractive women for roadside rape. Srebrenica's adult men were blindfolded, stripped to their underwear, taken into the countryside, and, by the thousands, shot. For their part, the Dutch peacekeepers left Srebrenica for the Croatian capital of Zagreb, where they celebrated their freedom by drinking and dancing into the night. The Dutch commander informed reporters that \"the parties in Bosnia cannot be divided into 'the good guys' and 'the bad guys.'\"\n\nAfter Srebrenica, the UN in Bosnia was effectively done. Congress, led by Senate Majority Leader Robert Dole, who was preparing to run against Clinton in 1996, voted again to lift the arms embargo. Britain and France reminded the White House that since their peacekeepers were members of NATO, the United States would be obligated to help airlift them out. It was the worst of both worlds. America would have to intervene militarily and Bosnia would still die.\n\nFor Bill Clinton, Srebrenica culminated a miserable first two years in office. The public had massively repudiated him in the 1994 midterm elections, and he was so overshadowed by the new Republican Congress that he was reduced to reminding reporters in April 1995 that \"the president is relevant.\" His foreign policy team was a shambles. Secretary of Defense Les Aspin had been fired in 1993. Secretary of State Warren Christopher had tried to resign in late 1994. National Security Advisor Anthony Lake had seriously considered doing the same. CIA director James Woolsey had left in early 1995 and would soon endorse Dole's presidential bid. After a trip to Washington, French president Jacques Chirac said the position of leader of the free world was \"vacant.\"\n\nFor liberal Bosnia hawks, Clinton was a wretched disappointment. At the very moment that American liberals were reconciling themselves to American power, a liberal president seemed unable to wield it. But Bill Clinton was a bad person to count out, because his entire career was a story of improbable last-second triumphs, of victories snatched from the jaws of defeat. He had lost his first race for reelection as Arkansas' governor before coming to politically dominate the state. Rumors of adultery had almost blown up his presidential candidacy early in the primaries, leading him to dub himself \"the comeback kid\" when a strong showing in New Hampshire saved his campaign. He had begun the general election in third place, behind Bush and Ross Perot. At 4:30 A.M. on inauguration morning, he was still writing his inaugural address.\n\nIf Clinton had a rendezvous with destiny, he usually arrived late. He often seemed unable to focus until his own indecisiveness or irresponsibility had brought events to the brink of crisis. But the worse things got, the better he performed. \"Do you know who I am?\" he once told Republican House leader Newt Gingrich. \"I'm the big rubber clown doll you had as a kid.... [T]he harder you hit me, the faster I come back up.\"\n\nNow, after Srebrenica's fall, it was five minutes to midnight. Congress and the Europeans were forcing Clinton's hand, and Bosnia was even starting to hurt politically. Clinton's political Svengali, Dick Morris, who had previously opposed military intervention, now decided that America's failure in Bosnia was reinforcing the public's perception of his boss as weak. All of a sudden, the famously tentative Clinton administration grew focused, aggressive, even domineering. Lake was dispatched to inform the Europeans that the next time the Serbs did something awful, America would bomb the hell out of them. The contrast with Christopher's 1993 trip was clear: America was no longer asking; it was telling. To make such bombing possible, the Clintonites bluntly informed Boutros-Ghali that NATO would no longer ask his permission to use military force. On Bosnia, the UN was being cut out of the game. Inside the administration, the military was also being shunted aside. Powell's successor as chairman of the Joint Chiefs, John Shalikashvili, was a capable, well-liked soldier with a made-for-Hollywood life story. Born in Poland to refugees from Soviet Georgia, he had come to America at age sixteen and learned English from John Wayne movies. But he was not a national icon able to awe the Clintonites into submission, and unlike Powell, he saw himself as part of the Clinton team. He deferred to them rather than the other way around.\n\nAlmost as important to the military's new willingness to use force was Shalikashvili's director of strategic policy and plans, a cerebral, tightly coiled lieutenant general named Wesley Clark. Within the military, Clark wasn't particularly popular. He didn't backslap, and he liked showing people how smart he was. He was the sort of person, one associate remarked, who would finish a three-hour exam in two hours and then tell you how easy it was.\n\nSome of Clark's uniformed colleagues considered him an apostate, a guy who cared more about helping the White House than sticking up for the military. The fact that he, like Clinton, was a Rhodes scholar from Arkansas fueled suspicions that the two men were close. That wasn't true, but the anti-interventionists in the Pentagon had reason to worry nonetheless. The reason was that Clark\u2014unlike Crowe on Panama or Powell on the Gulf or the Balkans\u2014was becoming emotionally invested in Bosnia's fate. On the morning of August 19, 1995, he was driving with a group of U.S. officials toward Sarajevo along a notoriously dangerous mountain road that they had been forced to traverse because Milosevic refused to offer them a safer route through Serb territory. Suddenly one of the American vehicles veered off the cliff and hurtled down the side of the mountain. Clark rushed to where it had landed and found the bodies of two of his colleagues. From then on, observers noted, he spoke about Milosevic with a particular intensity. Powell's voice had only welled up with emotion when discussing the army grunts in Vietnam or the dead Marines in Beirut. Clark began speaking that way about the people Milosevic killed.\n\nOn August 28, the Serbs did something awful: They shelled a Sarajevo market, killing thirty-seven Bosnian civilians. NATO responded with a massive bombing campaign. Planes from the U.S.S. Theodore Roosevelt, the same aircraft carrier whose sailors had ridiculed Clinton two years earlier, blasted Serb positions. The bombing coincided with a ground push by the Croats, who had evaded the arms embargo with clandestine American help. Together these attacks forced the Serbs\u2014for the first time since the fighting began\u2014into a headlong retreat. And with the war suddenly turning against them, they showed a new willingness to discuss peace.\n\nThe man charged with discussing peace with them was Richard Holbrooke, who came to symbolize the newly aggressive Clinton style. On Bosnia, Holbrooke was white-hot. He had traveled there before Clinton even took office, meeting with concentration camp survivors and staying in a Sarajevo Holiday Inn whose rooms were stained with blood. He had also been on that road with Clark when their colleagues plunged to their death. But other U.S. officials were passionate, too. What distinguished Holbrooke was his combination of moral passion and comfort with raw power. He was a steamroller, far more interested in results than in procedural niceties. When you gave Holbrooke a mission, heads butted, china broke, feathers were often deeply ruffled, but the job got done. In this sense he was the temperamental opposite of Warren Christopher, whose 1993 trip to Europe had been characterized by excellent form, unfailing courtesy, and utter failure. Holbrooke embodied the new liberalism that in 1995 was being born: idealistic about ends, but somewhat brutal about means, committed not merely to containing evil but to vanquishing it, and confident that vanquishing evil and extending American dominance were usually one and the same.\n\nHolbrooke summoned the Balkan leaders to an air force base in Dayton, Ohio. It was a symbol that America\u2014not Europe\u2014was now in control. And after twenty days of extremely muscular diplomacy, he brokered a deal, which U.S. peacekeepers would help enforce. For Clinton, Dayton was part of a larger political comeback. That winter he bested the Gingrich Republicans in a standoff over the budget and the following year cruised to reelection. And for American foreign policy, Dayton was part of a comeback story as well: After two years of hesitation and failure, American confidence was again starting to build. A year earlier, the Clinton administration had finally forced out the Haitian junta that humiliated the U.S.S. Harlan County. If the Bushies had overthrown Noriega, the Clintonites had now also deposed a dictatorship that posed no military threat to the United States, and installed an elected leader in its place.\n\nIf the Gulf War had shown liberals that not every war need be Vietnam, Bosnia underscored the point. In the entire NATO bombing campaign, not a single American died. And for a whole generation of liberal foreign policy makers and commentators, Bosnia taught lessons directly at odds with the lessons of Vietnam: First, that military force could serve moral ends. Second, that the Pentagon's reluctant warriors did not always know best; their prophecies of disaster should be taken with spoonfuls of salt. Third, that the UN should be relegated to the backseat because it lacked the moral clarity and military muscle for humanitarian war. And fourth, that the Europeans were useful allies, but not equal partners, good to have along for the ride but utterly unable to steer the car. The Clintonites were still more multilateral than the emerging post\u2013cold war conservatives. They weren't ideologically hostile to international institutions. But they did want America firmly in command. As Clark exclaimed during Anthony Lake's trip to Europe, after U.S. officials told America's allies that Bosnia policy was about to change, \"The big dog barked today.\" It would bark even more loudly before the Clinton administration was through.\n\nIn 1996, the White House helped dump Boutros Boutros-Ghali as head of the UN, making him the first secretary-general ever denied a second term. The message was clear: Secretaries-general shouldn't get too haughty or independent. In managing the post\u2013cold war world, Turtle Bay would be subordinate to Pennsylvania Avenue, not the other way around.\n\nIf the age of the UN was now over, so was the age of Europe. In the early 1990s, many on both sides of the Atlantic had predicted that with no Soviet Union to contain, NATO would wither, and with it America's political and military hold over Western Europe. Instead the European Union would assume control, in partnership with the UN and perhaps the reconstituted Organization for Security and Co-operation in Europe. The former Yugoslavia was to be the showcase for this new institutional order.\n\nBut NATO saved Bosnia and Bosnia in turn saved NATO, extending it\u2014and American power\u2014deep into the former Soviet bloc. Holbrooke, whose position as assistant secretary of state for European affairs put him in charge of NATO expansion, steamrolled a skeptical Pentagon, and America steamrolled Russia, which was desperately opposed. Even America's NATO allies, some of whom opposed expanding the alliance, and some of whom wanted to admit a different slate of members, were shunted aside. \"Washington was riding roughshod over its allies,\" lamented Germany's representative to NATO. \"When exactly did the Americans go from leadership to hegemony?\" wailed a top French official. The Russians ranted and raved but were plainly impotent. In the end they merely pleaded for NATO to stay off former Soviet soil.\n\nAmerica's appetite, however, grew with the eating. The success of NATO in incorporating countries like Poland, Hungary, and the Czech Republic seemed to illustrate what academics called \"democratic peace\" theory\u2014the theory that democracies did not war with one another. Fukuyama had argued that all nations were moving toward democracy; now democratic peace theory suggested that the move toward democracy would also be a move toward peace. So NATO did eventually push onto former Soviet soil. There was nothing for Moscow to do but scream.\n\nThe Bosnia intervention was to Eastern Europe what the Gulf War had been to the Middle East: the gateway for a massive extension of American power. And Kosovo was to Bosnia what the Korean War had been to Truman's 1947 decision to aid Greece and Turkey: the same principle, stretched further. If the ethic of toughness had grown in the late Truman years because it brought success, the ethic of dominance grew in the late Clinton years for the same basic reason.\n\nUnlike Bosnia, Kosovo was not an independent country. It was a province of Serbia whose overwhelmingly Albanian Muslim population had enjoyed autonomy until Milosevic withdrew it as part of his effort to Serbianize everything. As the regime in Belgrade grew more brutal, a Kosovar guerrilla movement emerged, and its presence just made the Serb authorities crack down harder. Finally, in January 1999, the Serbs executed forty-five Kosovars\u2014men, women, and children\u2014and left their bodies in the snow.\n\nIn Bosnia, it had taken years of this sort of thing, and worse, to overcome the Clinton administration's inhibition about military force. But the Clintonites were not inhibited anymore. For one thing, a different foreign policy team was now in charge. Wesley Clark had become Supreme Allied Commander in Europe, and while others in the Defense Department still feared military action, his position as NATO's top military man meant that the Pentagon's efforts at obstructing military action were virtually doomed from the start.\n\nRunning things on the civilian side was Clinton's second-term secretary of state, Madeleine Albright, who in her own way embodied the administration's newly muscular style as much as Holbrooke did. If Holbrooke's comfort with American power was a function of personality, Albright's was largely a function of genealogy. Her father had been a high-ranking Czech diplomat who fled the Nazis and returned home after World War II only to flee again when Moscow took Czechoslovakia in its grip. Czechoslovakia's betrayal at Munich, Albright often noted, was her formative foreign policy experience, even though it happened when she was only one year old. She experienced it through her parents, and as frequently happens in such circumstances, personal and political history fused, giving the latter a special intensity. Albright's relationship to Vietnam, by contrast, was unusually distant. She had spent the Vietnam years as a wife, mother, and student, not a frustrated young diplomat like Anthony Lake or an earnest young protester like Clinton or Sandy Berger. Because her life had been touched by the crimes of America's enemies but not by America's own, and because America was her refuge, not merely her home, she combined a deep idealism about the United States with a very personal hostility to its dictatorial foes. And in combination, that made her the top Clinton official with the fewest moral or practical qualms about American force.\n\nWith Albright in charge, America no longer negotiated with Milosevic; it gave him orders at gunpoint. In February 1999, at a ch\u00e2teau in Rambouillet, France, he was handed a take-it-or-leave-it offer: Grant Kosovo autonomy, withdraw most Serbian troops from the province, and see them replaced with NATO peacekeepers, or else be bombed. It was an audacious proposal. Kosovo, unlike Bosnia, was not an independent country. America was now dictating internal Serbian affairs, telling Belgrade to remove soldiers from part of its own country.\n\nMilosevic tried stalling for time. At Rambouillet, the Serb delegates did little serious negotiating and much serious drinking, downing almost four hundred bottles of wine in the first five days alone. (This in turn spawned a great deal of singing, as woozy Serbs crooned patriotic tunes late into the night.) The Kosovars, by contrast, stayed sober and took America's deal. As a result, six days after the conference closed, the bombing began.\n\nThe air war started slowly and inefficiently, as the Clinton administration's fear of U.S. casualties limited its scope. But gradually, NATO airpower (which was to say, American airpower, since U.S. planes flew 80 percent of the sorties) proved ferociously effective\u2014more effective, even, than during the Gulf War. In 1991, only 9 percent of the bombs dropped on Iraq had been precision-guided; by 1999 in Kosovo, it was over 60 percent. Bombarded by NATO from the air and by Kosovar rebels on the ground, Serbian troops began to desert in mounting numbers. NATO also targeted the houses and businesses of Serbia's notoriously corrupt political elite, making them pay a highly personal price for keeping Kosovo in their clutches. Finally, Milosevic caved and Serbian forces withdrew. The United States and its allies had threatened to send in ground troops, but unlike in the Gulf, they never needed to. As the military historian John Keegan declared, \"There are certain dates in the history of warfare that mark real turning points.... Now there is a new turning point to fix on the calendar: June 3, 1999, when the capitulation of President Milosevic proved that a war can be won by airpower alone.\" In thirty-four thousand sorties, only two NATO planes had been shot down. Not a single American died in combat.\n\nIf the 1990s were a story of rising American military and ideological dominance, Kosovo was the apex. First, it represented yet another case of war made easy. From Panama to the Gulf to Bosnia and now to Kosovo, almost everything the U.S. military touched had turned to gold. The sole exception was Somalia, and even there the losses had been comparatively small: only eighteen dead, nothing like Vietnam, Korea, or even Beirut. By 1999, Somalia's cautionary tale had been forgotten amid the broader narrative of Clinton-era military success.\n\nSecond, Kosovo illustrated the increasing marginalization of the United Nations. Knowing that Russia would veto military action at the Security Council, America had circumvented it, using NATO instead. But once Milosevic was vanquished, the UN retroactively blessed the war and took over Kosovo's political administration. It was a lesson many liberal hawks would remember four years later during the debate over Iraq: Not only could the United States win wars without the help of Turtle Bay, but once it had, the UN would help tidy up the mess.\n\nThird, Kosovo underscored America's growing willingness to violate its enemies' sovereignty. In Panama, in the \"no-fly\" zones in northern and southern Iraq, in Haiti, and now in Kosovo, America had used force to prevent regimes from committing abuses on their own soil. In each case, U.S. officials had claimed those abuses threatened the United States or its allies, but that was a stretch; the threats were hardly imminent or grave. In reality, as America expanded its power, it was expanding its definition of what constituted a threat, just as it had during the early cold war, when the last hubris bubble grew. The more confident America's leaders became in the hammer of military force, the more closely they looked for nails.\n\nFourth, presidential power over foreign policy was again on the rise. Even as Clinton battled for his political life against Republican efforts at impeachment over the Monica Lewinsky affair, he waged war in Kosovo and bombed Iraq without congressional authorization. In late 1994, in a move of dubious legality, he also bypassed Congress to bail out Mexico, which was on the verge of financial default. In the 1970s and '80s, the left had championed restraints on presidential power. But now many liberals were beginning to feel about executive authority the way they had felt during the McCarthy years. The congressional Republicans of the late 1990s, like the congressional Republicans of the early 1950s, seemed so primitive and reckless that many liberals hoped the presidency would become a bit more imperial. When it came to foreign policy, they didn't want Congress in the game.\n\nFinally, there was something else that made Kosovo a harbinger of things to come: It was a \"preemptive\" war. In Kosovo, the United States did not merely respond to ethnic cleansing faster than it had in Bosnia; it responded before most of the ethnic cleansing even began. When NATO started bombing, the Serbs had not yet forced large numbers of Kosovars out of their homeland (although significant numbers had been internally displaced from their villages). In other words, the war was justified less by what Milosevic had already done in Kosovo than by what Americans believed he would do there in the future, judging by his behavior elsewhere. For many human rights\u2013minded liberals, this was what made Kosovo so exciting. It suggested that the United States and its allies could act before the slaughter and expulsions truly began. In Kosovo, the argument for preemption was strong: Given Milosevic's record in Bosnia, and what was already starting to happen on the ground, there was good reason to believe that terrible crimes were about to occur. But in America, where leaders did not usually justify war on preemptive grounds, Kosovo nudged open an intellectual door, a door George W. Bush would fling wide open four years later, when he cited \"preemption\" to justify his invasion of Iraq.\n\nIt was a giddy time, in some ways even more intoxicating than 1989. As the new millennium dawned, America dominated the world not only ideologically and militarily, but now economically as well. Germany and Japan, which a decade earlier had seemed poised to leave America in the dust, were by Clinton's second term growing at less than half the rate of the United States. Between 1992 and 1996, the U.S. stock market doubled in value. By 1997, unemployment was at its lowest rate in twenty-four years and economic growth was rising at a phenomenal 8 percent per year. In Washington, it began raining revenue, and by 1998 the budget deficit that Paul Kennedy had cited as evidence of America's impending economic doom was virtually gone. Entrenched cultural maladies like crime, welfare dependency, and teen pregnancy, which had vexed intellectuals and policymakers for decades, also began to improve dramatically.\n\nThe New York Times called the erasing of the deficit \"the fiscal equivalent of the fall of the Berlin Wall,\" and the psychological effect was comparable. By 1999, Americans had seen their country repeatedly conquer enemies and problems that supposedly could not be conquered. Saying America couldn't accomplish something looked like a fool's bet. Ideologically, the spread of democracy had erased the cultural pessimism and relativism that flourished on both right and left in the 1970s and '80s. Militarily, America's triumphs in Panama, the Gulf, and the Balkans had largely erased the specter of Vietnam. And economically, these wars hadn't bankrupted the United States. To the contrary, the further America extended its military power, the richer it grew.\n\nPaul Kennedy offered a mea culpa. \"Nothing has ever existed like this disparity of power, nothing,\" he wrote in evident wonder. America was spending more on defense than the next nine nations combined, yet fiscally it wasn't even breaking a sweat. \"Being Number One at great cost is one thing,\" he observed. \"Being the world's single superpower on the cheap is astonishing.\" A decade earlier he had insisted that the laws of geopolitical gravity would force America down. Yet higher and higher it flew.\n\nIn myriad ways, this triple dominance\u2014ideological, military, and economic\u2014was self-reinforcing. America's \"end of history\" confidence that every country could sustain democracy and respect human rights boosted the arguments for military action in Panama, Haiti, Bosnia, and Kosovo. And when military action worked\u2014sparking jubilation among the Panamanians, Haitians (at least initially), Bosnians, and Kosovars\u2014that ideological confidence only grew. The post\u2013cold war economic boom, like the post\u2013World War II economic boom, created a new abundance of resources, which quieted those who had warned that America couldn't afford far-flung military commitments. And the boom also subtly altered Americans' understanding of what had triumphed at history's end. In 1989, Fukuyama had merely heralded the ideological triumph of free-market democracy. Now, by 1999, with the European and Japanese versions seemingly discredited, many in Washington believed that American-style free-market democracy\u2014with its signature antipathy to high taxes and government regulation\u2014was history's final destination. By Bill Clinton's second term, what Americans called \"globalization\" was in the rest of the world increasingly being called \"Americanization.\" It meant more than the spread of capitalism and democracy. It meant the spread of a particularly American brand of democracy, symbolized by Clinton's political consultants, who introduced U.S.-style campaigning across the globe; and it meant a particularly American brand of capitalism, spread by American management consultants, economics professors, and investment bankers, with help from the U.S.-dominated IMF and World Bank. By 1999, Fukuyama's democratic triumphalism had morphed into American triumphalism. The \"end of history\" was not just an idea anymore; it was a place.\n\nSome countries had trouble getting with the program, of course. There were political laggards like Milosevic and the Hutu genocidaires who threatened to destabilize whole regions. There were also economic laggards in Latin America and East Asia whose late-1990s meltdowns almost took down the whole global economy. For many non-Americans, these economic crises suggested basic problems with the ultracapitalism America was spreading. But in turn-of-the-millennium Washington, the general assumption was that the Latin Americans and Asians had gotten into trouble not because they had followed the American gospel but because they had not followed it devoutly enough. In any case, the debate soon subsided because American-led economic intervention set things right (at least it looked that way from Washington) just as American-led military intervention had in Bosnia and Kosovo. After the East Asian crisis eased, Time pictured Secretary of the Treasury Robert Rubin, his deputy Lawrence Summers, and Federal Reserve chairman Alan Greenspan above the words \"The Committee to Save the World.\"\n\nDuring the cold war, many Americans had seen the world as defined by two huge, opposing magnets: America and the Soviet Union, democracy and communism, good and evil. Each was a powerful attractive force, pulling nations and even individuals its way. No one could be sure which way the smaller magnets would go.\n\nThe cold war's end, however, had eliminated the evil magnet, and with it the grand historical uncertainty. Americans no longer related quite as easily to the movie Star Wars, with its story of two opposing armies: the light and dark sides of the force. To read the most influential 1990s commentators\u2014columnist Thomas Friedman, for instance, or the Economist magazine\u2014was to see the world not as dueling magnets, but as an escalator, with democratic capitalism at the top. Some nations were further along than others, of course. The journey was occasionally bumpy, and some nations were even trying to climb down. But the futility of the effort would force them to eventually turn around and begin trudging upward, in the same direction as everyone else.\n\nThe 1990s had shown that America\u2014and not any rival democracy or international body\u2014stood on the escalator's highest step. And they had shown that America could reach down and corral countries going in the wrong direction before they did too much damage to themselves and others. America did not have to wait for history's escalator to gradually move nations in the right direction; it could speed up the process, if need be by force.\n\nThis was not Fukuyama-ism; it was ultra-Fukuyama-ism. The escalator metaphor fit his theory, but he had never said that America could easily crank up the gears. To him, the 1990s hadn't proved that military intervention was easy; they had proved that it was hard. After all, to secure its military victories, the United States had been forced to station troops indefinitely in Bosnia, Kosovo, Haiti, and Iraq, and there was every reason to fear that once they left, things would spin back out of control. When it came to promoting democracy, Fukuyama argued, \"there are very sharp limits to American resources and patience.\" In fact, \"given that the United States can build democracy in a poor, agrarian country only through massive intervention over the course of a couple of generations, it is probably better in most cases not to try.\" The better strategy was simply to trade with authoritarian countries and thus bring them into the world economy, which might gradually produce a middle class that demanded political reform. Countries needed to evolve toward democracy at their own pace, he argued. The world wasn't as malleable as it seemed.\n\nBut Fukuyama's anxieties had little resonance in an era overflowing with optimism. \"We are fortunate to be alive at this moment in history,\" declared Bill Clinton in 2000, in his final State of the Union address. \"Never before has our nation enjoyed, at once, so much prosperity and social progress with so little internal crisis and so few external threats.\" A poll taken by the Pew Research Center on the cusp of the millennium found that \"Americans are near unanimous in their confidence that life will get better for themselves, their families and the country as a whole.\" Majorities predicted that the twenty-first century would bring cures for cancer and AIDS, a cleaner environment, and people living in space. A slightly smaller number, 44 percent, predicted Christ's return to earth.\n\nClinton compared the mood to his youth in the early 1960s, before Vietnam, the last time anything seemed possible. The political writer David Brooks reached back further, comparing young Americans at the twentieth century's end to young Americans at the twentieth century's dawn, before World War I, the last time a generation truly believed in the inevitability of progress and the goodness of man. Neither mentioned the historical corollary: that both times before, it had all ended in tears.\n\n## CHAPTER FIFTEEN\n\n## FATHERS AND SONS\n\nIn 1995, Irving Kristol published his fourth autobiographical essay. (That was nothing: Norman Podhoretz would soon publish his fourth autobiographical book.) In general, the essay's tone was jolly, even triumphant. America had won the cold war; American capitalism was booming; and even with a Democrat in the White House, right-wing ideas shaped the Washington debate. \"I have much to be cheerful about,\" declared Kristol, now seventy-five years old. \"I deem the neoconservative enterprise to have been a success.\"\n\nBut in the final paragraph, Kristol's tone darkened, and he offered a kind of warning. \"I am well aware that the unanticipated consequences of ideas and acts are often very different from what was originally intended,\" he noted, somewhat cryptically. \"That, I would say, is the basic conservative axiom, and it applies to conservatives as well as liberals and radicals.\"\n\nIf Kristol feared any particular unanticipated consequence of neoconservatism's success, he didn't say. Yet even as he wrote, his prophecy was coming true. Neoconservatism's progeny, buoyed by the very success Kristol was celebrating, were fashioning a foreign policy vision sharply at odds with his own. In retrospect, 1995 was a hinge year on the intellectual right, a year in which one generation's voice began to fade and another's began to rise. In the early 1990s it had been mostly the men and women of Kristol's generation who spoke, and what they advocated, for the most part, were limits on American power. In the late '90s their ideological offspring replied, and what they denounced, for the most part, were limits on American power. There was an odd time lag to the discussion. It was as if a group of parents had written their children a letter about the kind of nation America should be after the cold war, and waited for them to grow old enough to respond. And when the children did grow old enough, they read the letter and tore it to shreds.\n\nOf the older generation, the most important foreign policy thinkers were Kristol and Jeane Kirkpatrick, two tough old birds. Intellectually, their defining feature was their hostility to grand efforts to remake the world. At formative stages in their youth, each had bumped up against totalitarianism, and the collisions had left permanent scars. For Kristol, it had occurred at City College in the late 1930s, where he and a few Trotsykist dissenters squared off against hordes of thuggish, brain-dead Stalinoids. For Kirkpatrick, it had come in the early 1950s, when she worked at the State Department interviewing refugees who brought with them harrowing tales from Stalinist Eastern Europe and Maoist China. From these experiences, Kristol and Kirkpatrick drew a similar lesson: Totalitarianism was monstrous because it was unrealistic. Stalin and Mao wanted to create perfect societies; but perfect societies required perfect people, and since people were by nature imperfect, the communists took it upon themselves to perfect them by force, which required crushing the spirit and cracking the skull.\n\nWhen the New Left came along in the 1960s, Kristol and Kirkpatrick opposed it for the same reason: because they believed it was trying to remake the United States along utopian lines, a project they were sure would end in chaos and blood. They defended America's existing institutions and practices\u2014its universities, cities, families, political system, and cold war foreign policy\u2014not because they believed those institutions and practices were flawless, but because they believed they were pretty darn good considering what human beings were really like. It was no surprise that they both admired Reinhold Niebuhr; their foreign policy realism stemmed from their moral realism. Once, late in life, Kirkpatrick spent an afternoon with a friend and her two small children. Over hot chocolate, the children suggested a game of Sorry! Kirkpatrick agreed, and then trounced them. Asked afterward by their mother why she hadn't let the kids win, Kirkpatrick replied, \"I'm merciless when it comes to Sorry! It's the real world.\"\n\nLike other ultra-realists, Kirkpatrick and Kristol often read Niebuhr with one eye closed, ignoring his warning that if communism was messianic, self-righteous anticommunism could be messianic, too. But in their minds, global containment was not messianic; it was defensive. The communists were the utopians, trying to forcibly remake the world in their image. America was simply holding the line. \"Global political instability,\" Kirkpatrick insisted, \"stems from the fact that the Soviet Union is frankly, proudly, a revolutionary power.\" The United States, by contrast, employs a \"minimal use of force for limited, defensive purposes.\"\n\nWhether or not this defensive, anti-utopian interpretation of the cold war was correct, it proved very important when the cold war ended. When the Soviet empire crashed, Kirkpatrick and Kristol let out a sigh of relief and declared that it was time for America to become, in Kirkpatrick's words, \"a normal country in a normal time.\" The cold war, they argued, had been necessary, and even heroic, but also costly and unnatural. The presence of a revolutionary global superpower had forced the United States to become a defensive global superpower, expending blood and treasure in every continent on earth. Now, with its survival no longer threatened, America could climb down from the barricades. In Kirkpatrick's mind, the United States was like a man forced to defend his village against an invading army. The battle had gone on for a long time, and summoned within him great reservoirs of resourcefulness and courage. But it had also forced him to neglect hearth and home. His crops lay untended; his house was in disrepair; he barely knew his family anymore. Now the marauders had laid down their arms. Waking up the next morning, he should not strap on his armor and go looking for new armies to slay.\n\nKristol and Kirkpatrick saw no reason whatsoever for America to try to dominate the post\u2013cold war world. First, they didn't think America had the money. They had been willing to pay any price to contain Soviet power, but like Paul Kennedy, they believed the price had been high. When it came to foreign policy, Kirkpatrick argued, there was no longer any need for \"expansive, expensive, global purposes,\" which was a good thing, because the United States lacked \"boundless resources\" and could not \"continue to live beyond our means.\"\n\nIf America lacked the money for world dominance, it also lacked the will. Kirkpatrick and Kristol were convinced that if American elites tried to keep sending the nation's cash\u2014not to mention its sons and daughters\u2014around the world in the absence of a mortal threat, the America public would rise up. They urged disbanding NATO, withdrawing most U.S. troops from Europe (and perhaps Asia as well), cutting the defense budget, and preparing for a multipolar world. \"There are theorists who would happily burden us with the mission of monitoring and maintaining a 'balance of power' among other nations, large and small, in Europe, the Middle East, Asia, etc.,\" wrote Kristol. \"We are just not going to be that kind of imperial power.... [T]he American people violently reject any such scenario.\"\n\nThirdly, if America lacked the money and will for global dominance, it also lacked the wisdom. Kirkpatrick and Kristol did not want the United States to try to convert the world to its political ideology; that had been the Kremlin's ugly game. The democratic transformations of the 1980s had convinced younger conservatives like Elliott Abrams and Paul Wolfowitz that people around the world wanted democracy, and that America could help them get it. But Kirkpatrick and Kristol were both in their sixties when the cold war ended, and the new optimism about global democracy did not penetrate their hardened intellectual shells. They weren't at all sure that the desire for freedom was universal, and even if it was, they were adamant that trying to turn that desire into reality was beyond America's capacity. In 1990, at a conference of former cold warriors, Kirkpatrick congratulated the Bush administration for standing by while Moscow tried to crush Lithuania's fledgling democracy. \"Americans do not know at this stage what is best for the Soviet people,\" she explained. \"Any notion that the United States can manage the changes in that huge, multinational, developing society is grandiose. It is precisely the kind of thinking about foreign policy which Americans need to unlearn.\" On a later panel, Kristol denounced efforts to promote democracy in Ukraine. \"The function of the United States is not to spread democracy all over the world,\" he insisted, adding that historically such efforts had mostly failed. When someone suggested that the Reagan era offered grounds for optimism, Kristol called the supposed \"worldwide sweep of democracy\" a mirage. \"It is just too good to be true,\" he insisted. \"The world is not like that. It is not going to happen. Something will screw it up.\"\n\nWhen Saddam Hussein invaded Kuwait, Kirkpatrick's first instinct was that it was not America's problem. Like George Kennan at the dawn of the cold war, before containment was inflated by success, she defined America's interests narrowly because she had grave reservations about its resources. It was unfair, she argued, for the world to expect America to fight Saddam when \"it is believed almost everywhere outside the US that the US is a world leader in serious economic decline.\"\n\nFor a while, she urged giving sanctions time to work. By November, when they still had not, and President Bush had staked his credibility on restoring Kuwait, she reconciled herself to war. But she still opposed sending U.S. ground troops, arguing that America should leave the land fighting to Arab nations and merely bomb from the air. If Saddam fell, she suggested, perhaps the Arabs could manage Iraq's reconstruction. The implication was clear: If America took the lead in fighting Saddam, it would find the price too high, and if America tried to dominate the Middle East, it would be drinking from a poisoned cup.\n\nKristol was less ambivalent: America needed Gulf oil, and so he supported an American war. But once the United States pushed Saddam out of Kuwait, he heaped scorn on those who advocated democratic regime change in Iraq itself. Unlike Wolfowitz, who believed the Shia and Kurdish rebels were fighting for freedom, Kristol saw them the way George H. W. Bush and Colin Powell did: as secessionists and fanatics fighting for nothing nobler than the chance to visit upon Iraq's Sunni Arabs what the Sunni Arabs had visited upon them. Kristol's view of the Arab world was suffused with old-fashioned conservative pessimism about the capacity of people in exotic lands to build democracy. Fukuyama-style universalists, he declared, find \"it close to impossible to understand that the world is populated by other people who are really different from us.\"\n\nWhen Bill Clinton took office, Kirkpatrick and Kristol took a generally dim view of his humanitarian wars. \"Military intervention in the internal affairs of another state is undesirable,\" wrote Kirkpatrick in 1995, in denouncing the U.S. occupation of Haiti. \"Overturning another government by force or threat of force is still more objectionable\u2014except if there are truly vital American interests and lives at stake.\" On the former Yugoslavia, Kristol made skeptical noises about military intervention early on, and then ignored the subject. Kirkpatrick supported lifting the arms embargo and even bombing the Serbs, but remained extremely wary of putting U.S. peacekeepers on the ground.\n\nNeither Kirkpatrick nor Kristol was an isolationist. But they were comfortable with a world where other powers\u2014be they European, Asian, or Arab\u2014supervised their corners of the globe, so long as those powers did not pursue a global, revolutionary mission like the Soviet Union's. And they had little interest in America telling other countries how to govern themselves, especially at gunpoint. Their belief that efforts to make society conform to an abstract ideal usually failed, whether in Mao's China or John Lindsay's New York, infused them with the same unsentimental skepticism about America's ability to reshape the world that Colin Powell had learned on his tours in Vietnam. Kirkpatrick and Kristol saw neither American interests nor American ideals as universal. They believed that the United States would be better off, most of the time, if it let other nations be.\n\nIn the early 1990s, Kirkpatrick's and Kristol's foreign policy vision held sway on the intellectual right. It was shared not only by older neo-conservatives like Nathan Glazer, Owen Harries, and James Q. Wilson, but by traditional conservatives like National Review's William F. Buckley and the columnist George Will. Patrick Buchanan, then still a respectable figure on the mainstream right, was even more hostile to a global American role. Not every older conservative foreign policy intellectual opposed American dominance, to be sure. Norman Podhoretz, for instance, was more bellicose and moralistic. But in 1996 even Podhoretz admitted that \"only a tiny\" number of neoconservatives supported \"expansive Wilsonian interventionism.\" A study of right-wing intellectuals published the same year noted their \"strengthening commitment to realism,\" \"narrowed view of American vital interests,\" and \"general reluctance to crusade.\"\n\n\"There are no normal times.\" With those words, written in 1991 and aimed straight at Jeane Kirkpatrick, the younger conservative generation fired its first shot.\n\nThe marksman was columnist Charles Krauthammer, an acid-tongued ex-psychiatrist from Montreal, and a man young enough to be Kirkpatrick's son. What Kirkpatrick meant by \"normal times\" was that since America no longer faced a superpower with messianic ambitions and the muscle to carry them out, it no longer needed to man barricades all over the globe. What Krauthammer meant by \"There are no normal times\" was that if America climbed down from those barricades the world would go to hell in a handbasket and from out of the chaos new and equally dangerous threats would arise. If Kirkpatrick saw America as a man who could abandon his patrol because invaders no longer threatened his village, Krauthammer warned that once he did, new marauders, seeing that the village was defenseless, would soon begin their attack.\n\nWho were these new marauders? Krauthammer gave two answers. Answer number one was what he called \"weapons states\": small, tyrannical third-world countries like Iraq and North Korea that through the miracle of high technology could build chemical, biological, or nuclear weapons and thus threaten America's allies or America itself. Answer number two was essentially: \"Who knows and it doesn't really matter.\" By patrolling the world, Krauthammer argued, America preserved global stability. If it stopped, the result would be global chaos, and from that chaos all manner of currently imperceptible dangers would emerge. You didn't have to know who would attack the village, only that if you stopped guarding it sooner or later someone would.\n\nAt first glance, Krauthammer was merely proposing that America keep doing what it had done during the cold war. Kirkpatrick wanted to come home; he wanted to stay on patrol. But in truth he was arguing for something far more aggressive than that. During the cold war, after all, America had tolerated \"weapons states\": not small ones with one or two warheads, but giant ones like the Soviet Union and China with hundreds or thousands of them. We had stood by as hostile powers developed huge nuclear arsenals, and we had responded with a policy of deterrence: warning that if they used their nukes, we would, too. Had Krauthammer simply wanted America to maintain the same level of security it enjoyed during the cold war, he would have proposed that the United States do the same thing now: deter any new \"weapons state\" with the threat of nuclear retaliation.\n\nBut Krauthammer was not proposing that. America, he argued, must \"confront\" and \"if necessary, disarm\" the weapons states. In other words, it must use force\u2014or the threat of force\u2014to make sure the arsenals were never built (and for Krauthammer, that meant not only nuclear arsenals, but even chemical and biological ones). This required doing a lot more than simply standing outside the village deterring invading armies. It required going around to neighboring villages looking for armies about to form, and attacking (or threatening to attack) them before they did. What Krauthammer was proposing was a doctrine of preventive (or what George W. Bush would wrongly call \"preemptive\") war. And since he wasn't merely worried about \"weapons states\" but also about the more amorphous threat of global instability\u2014which could break out in any strategically significant region not dominated by the United States\u2014Krauthammer was essentially suggesting that America encamp outside any significant village, even if there was no evidence that it was raising an army, just to make sure no one got any big ideas. During the cold war, America had patrolled large swaths of the globe, but it had still acknowledged that there were big, important regions where Moscow and its allies held sway. For the most part, America had been content to live in a bipolar world. Now, Krauthammer was arguing, America must dominate every important region on earth. \"The alternative to unipolarity,\" he declared, \"is chaos.\"\n\nIt was no coincidence that Krauthammer published his attack on Kirkpatrick soon after the Gulf War. As usual in the development of hubris bubbles, it was only once things that formerly looked hard\u2014like liberating Kuwait\u2014had been made to look easy that people set their sights higher. Had America proved militarily unable to keep Saddam from gobbling his neighbor, Krauthammer could not have seriously proposed launching a new war, inside Iraq itself, to rid him of his unconventional weapons.\n\nThe Gulf War gave Krauthammer confidence that America had the money and muscle for preventive wars. But he worried about public will. Kristol and Kirkpatrick insisted that average Americans would never permit the United States to play global policeman in the absence of a grave foreign threat, and Krauthammer half agreed. Yes, he conceded, Americans inclined toward isolationism. They could be roused from that isolationism, however, by the chance to accomplish great and moral deeds. \"Americans will venture abroad to do right things,\" he argued, \"but only to do right things. Otherwise they would rather stay home.\" This was not how the older generation saw it. In Kirkpatrick and Kristol's view, Americans had fought the cold war not because they wanted to remake the world but because they considered themselves in danger from communism. Krauthammer turned that argument on its head. Because Americans were naturally isolationist, he suggested, they would bury their heads in the sand even when they were in danger. The only way to get them to fight in their own defense was to tell them they were remaking the world.\n\nKrauthammer was making a subtle point. He wasn't arguing that America should fight wars to make the world a better place. He was arguing that unless Americans believed their wars were making the world a better place they would not fight in their own self-protection. And for him, self-protection required American hegemony. Dominance was Krauthammer's goal; idealism was the means of convincing Americans to achieve it.\n\nHerein lay the key difference between the younger, dominance-oriented conservatives and the liberal hawks. For the liberal hawks, spreading American power was worthwhile if it spread democracy and human rights. For the younger conservatives, by contrast, spreading human rights and democracy was worthwhile if they spread American power. Both groups were more confident than their ideological forefathers that democratic values and American might generally went hand in hand. But their criteria for war differed. Liberal hawks supported humanitarian interventions in strategically unimportant places like Haiti and Rwanda, though they might cloak those interventions in the language of national interest to win public support. Dominance conservatives supported cold-blooded, national-interest interventions like the Gulf War, though they might cloak them in moral garb to win public support. Such garb was crucial, Krauthammer argued, because otherwise the American people might not play along. It was an argument reminiscent of the pre-Vietnam years, when foreign policy elites often justified presidential duplicity as a way of protecting Americans from the consequences of their isolationism. Now, as memories of Vietnam faded, and another hubris bubble grew, Krauthammer was making that argument again.\n\nIf Krauthammer's attack on Kirkpatrick marked the first big sign that younger conservative intellectuals saw the world differently than their elders, the second came from within the Bush administration itself. Every two years, the Pentagon published something called the Defense Planning Guidance (DPG), a statement of America's foreign policy goals aimed at helping military planners decide where to spend money and deploy forces. In 1991, the document was unusually important, since with the cold war over, no one was quite sure what America's new foreign policy goals actually were.\n\nSecretary of Defense Dick Cheney delegated the job to his third in command, Paul Wolfowitz, who delegated it to an aide named I. Lewis \"Scooter\" Libby, who delegated it to a guy on his staff named Zalmay Khalilzad, who consulted with Richard Perle and a few others and then produced a draft. The goal of American foreign policy, Khalilzad declared, should be to \"prevent any hostile power from dominating a region whose resources would, under consolidated control, be sufficient to generate global power.\" At first blush\u2014like Krauthammer's essay\u2014this sounded rather defensive. After all, preventing a \"hostile power\" (the Soviet Union) \"from dominating a region whose resources would, under consolidated control, be sufficient to generate global power\" (Europe, East Asia, or the Middle East) had been U.S. policy in the cold war. And just six months before Khalilzad wrote his draft, the United States had shown how this principle might work in the post\u2013cold war age: Saddam had invaded Kuwait, thus potentially dominating the Persian Gulf, and the United States had rallied a coalition to repel him. The goal seemed familiar; it even seemed affordable, since other countries would presumably share America's interest in preventing one power from dominating a key region of the world and thus help pick up the tab, as they had in the Gulf.\n\nBut Khalilzad's draft also said this: \"The U.S. may be faced with the question of whether to take military steps to prevent the deployment or use of weapons of mass destruction\" (WMD). Preventive war to stop countries from acquiring WMD was not a continuation of U.S. strategy in the cold war; it was a sharp break. And it offered a window into the very peculiar way in which Khalilzad's paper defined foreign \"domination\" of a key region. For Khalilzad, hostile nations could \"dominate\" key regions not just by conquering their neighbors but also by taking an \"aggressive posture\" internally\u2014in other words, by developing weapons of mass destruction. By that standard, preventing Soviet \"domination\" of Europe in the late 1940s would have required more than merely defending countries like Italy, Britain, and France; it would have required preventive war to make sure Moscow never acquired a nuclear bomb. Khalilzad's argument was a lot like Krauthammer's: Superficially, he was merely outlining a strategy to protect America against foreign threats. But the definition of what constituted a threat had been so radically lowered that protecting America now required crippling\u2014if not toppling\u2014any hostile regime in any important corner of the globe. Why America's allies would help pay for that, Khalilzad didn't say.\n\nWhen the White House learned of the draft through a leak to the New York Times, it flipped out. President Bush, who was in his late sixties, two decades older than Cheney, Wolfowitz, Libby, and Khalilzad, was comfortable talking about resistance to aggression, as he had during the Gulf War. But like Kirkpatrick and Kristol, he was wary of proactive moves to extend the frontiers of American power, as evidenced by his reluctance to dismember the Soviet Union, intervene in the former Yugoslavia, or help depose Saddam. He pleaded with reporters not to \"put too much emphasis on leaked reports, particularly ones that I haven't seen.\"\n\nWolfowitz and Khalilzad feared that their careers were over. But when Cheney finally read the draft, he loved it. It was the clearest sign yet that he and the rest of the Bush administration didn't see the world the same way. Unlike Bush, he had wanted to pry the Soviet Union apart and opposed running the Gulf War through the UN. Unlike Powell, he had opposed any significant post\u2013cold war cuts in defense spending and supported shooting down Saddam's helicopters. Now he wanted America to dominate every important region on earth. He told Libby to rewrite the document in subtler language but preserve its basic argument. Congratulations, he told Khalilzad. \"You've discovered a new rationale for our role in the world.\"\n\nKrauthammer loved the Defense Planning Guidance. So did two former Reagan administration officials, both even younger than him: Robert Kagan and William Kristol. Kagan was son of a conservative Yale classicist. Kristol was the son of Irving Kristol. With their rise to intellectual stardom, the intraconservative debate became an argument not only between generations, but between a father and son, except that increasingly, only one side spoke. In the late 1990s, Irving Kristol fell mostly silent as his son eviscerated his vision of America's role in the world.\n\nIf the first two major statements of dominance conservatism\u2014Krauthammer's 1991 essay and the 1992 Defense Planning Guidance\u2014came in the wake of America's victory in the Gulf, Kristol and Kagan's came in 1996, soon after America's triumph in Bosnia. Their basic argument was similar: that Americans should not be lulled into complacency by the absence of foreign threats, because unless the United States dominated every important region of the world, such threats would arise soon enough. But Kristol and Kagan added two important wrinkles. First, they embraced humanitarian war. The Defense Planning Guidance had largely ignored the topic, and Krauthammer was openly hostile. While he wanted to cloak hardheaded, security-based interventions in moral language to win public support, he opposed going to war on humanitarian grounds alone. \"In a country with strong isolationist tendencies,\" he wrote, \"you do not squander blood and treasure on teacup wars.\" Kristol and Kagan, by contrast, incorporated Bosnia into their argument for American dominance. Humanitarian wars, they argued, combated public isolationism by showing Americans the good their power could do. As a result, they primed the pump for the more strategically important battles to come.\n\nThe second innovative feature of Kristol and Kagan's call for American dominance was the person in whose name they issued it: Ronald Reagan. If foreign policy hubris consists of thinking that you are merely applying the lessons of the past while actually expanding them as the result of success, Kristol and Kagan's doctrine of \"neo-Reaganism\" was the great intellectual example of the post\u2013cold war age.\n\nThat Kristol and Kagan invoked Reagan is no surprise. Politically, he was their god. For them, as for Elliott Abrams, Paul Wolfowitz, and Francis Fukuyama, the Reagan presidency was a formative life experience. Kristol, who had joined the Reagan administration at age thirty-two and worked as chief of staff to Secretary of Education William Bennett (before staying into the Bush administration to do the same job for Vice President Dan Quayle), was particularly influenced by Reagan's political success. Kristol had no background in foreign affairs. He was an operator and a partisan, mostly interested in shaping media debates and getting Republicans elected. He was, noted one administration colleague, \"a man of action, not introspection,\" a man who \"understood it was all a game.\" The lesson of the Reagan years, he told a friend during the 1990s, was that when Americans focused on foreign policy, Republicans won.\n\nIn the Kristol-Kagan duo, Kristol was Art Garfunkel: He played little role in crafting the lyrics but helped to amplify the sound. Kagan was less partisan and more substantive: He was, in fact, the most gifted conservative foreign policy thinker of the post\u2013cold war age.\n\nKagan's career and Jeane Kirkpatrick's were in important ways mirror-images. He had entered the Reagan administration just as she was leaving, when she was fifty-eight and he was less than half that. Abrams, who had been her prot\u00e9g\u00e9, became his mentor. Nicaragua, one of her passions, became his as well. And it was on Nicaragua that Kagan's intellectual assault on Kirkpatrick began.\n\nShe had argued famously in \"Dictatorships and Double Standards\" that Carter's refusal to support Somoza as the necessary evil\u2014his utopian faith that democracy was possible everywhere, right away\u2014had ushered the Sandinistas to power. Kagan disagreed. After leaving Reagan's State Department, in fact, he wrote a nine-hundred-page book making exactly the opposite claim. Carter's real failure, Kagan insisted, had been to not dump Somoza earlier, when there was still a nonviolent, non-Marxist alternative. The implication of Kirkpatrick's argument was that Reagan should stick by pro-American tyrants like Ferdinand Marcos and Chile's Augusto Pinochet, which at first he did. The implication of Kagan's argument, by contrast, was that Reagan's great accomplishment had been switching course and helping to push those dictators aside. For Kagan, Reagan's greatness lay in his rejection of Kirkpatrick's moral pessimism\u2014his faith that democracy could indeed take root everywhere soon. Reagan, wrote Kagan, \"completely reversed the grim pronouncements of...traditional conservative Republicans. It was a message of optimism, rather than of despair; it pointed to opportunities rather than dangers. Reagan did not have to tell Americans that support for brutal dictatorships was a necessarily evil, nor that the struggle with communism was endless.\"\n\nIn 1997 Kagan made his critique of Kirkpatrick even more explicit. He took to the pages of Commentary, where in 1979 she had written \"Dictatorships and Double Standards,\" to publish an essay titled \"Democracies and Double Standards.\" Kirkpatrick, he noted, had denied that democracy could emerge \"anytime, anywhere, under any circumstances.\" But she was wrong; the intervening two decades had proved that it could.\n\nFor Kagan and Kristol, being a \"neo-Reaganite\" meant believing that democracy could triumph anytime, anywhere, and that its triumph would likely extend American power. But it also meant believing that America could intervene aggressively to hasten those triumphs. As one critic later put it, Fukuyama was a Marxist: He believed history was moving inexorably toward America's democratic ideals. Kagan and Kristol were Leninists: They believed in speeding history up.\n\nIn Kagan and Kristol's rendition, Reagan was a Leninist, too. He had, they wrote, \"refused to accept the limits on American power imposed by the domestic political realities that others assumed were fixed.\" There was some truth to that: Reagan did spend far more on defense, and do far more to support anticommunist rebels, than any predecessor. But in other ways\u2014the most important ways, in fact\u2014he adhered scrupulously to the limits imposed by Vietnam. He never sent U.S. forces to El Salvador, Nicaragua, or Cuba, as many conservative commentators proposed, or even to Panama, despite Abrams's pleas. In Lebanon, his one nation-building campaign, he responded to a terrorist attack by quickly pulling the plug. When it came to direct military action to promote American ideals and power, Reagan wasn't a Leninist at all. He was, in fact, more cautious than either George H. W. Bush or Bill Clinton, both of whom Kagan and Kristol considered wimps.\n\nKagan also claimed it was Reagan's \"ideological confrontation\" that had helped to bring down the Soviet empire. And at times, Reagan certainly had been ideologically confrontational. But in his final years in office, it was Reagan's lack of ideological confrontation\u2014his willingness to abandon his evil-empire rhetoric and negotiate the deepest arms reductions of the cold war, long before Gorbachev allowed freedom for Eastern Europe\u2014that led prominent conservatives to compare him to Neville Chamberlain. In fact, as Norman Podhoretz noted bitterly at the time, Reagan began moving away from ideological confrontation in 1984, before Gorbachev even took power. And it was that very lack of ideological confrontation\u2014not Reagan's initial defense buildup and rhetorical belligerence\u2014that helped Gorbachev convince Kremlin hard-liners that the U.S.S.R. could abandon its Eastern European safety vest without fearing U.S. aggression.\n\nReagan had indeed dreamed of regime change in the Soviet Union, of a world where America was ideologically and geopolitically dominant. But his efforts to bring it about had been highly constrained, both by the trauma of Vietnam, which made large-scale military action politically treacherous, and by his own very personal terror of war. Kristol and Kagan, by contrast, were writing after Panama, the Gulf War, and Bosnia had largely buried the Vietnam syndrome and made war less politically frightening. They also lived in a post\u2013cold war world in which America was squaring off not against another superpower, but against a series of second-and third-rate powers, which seemed like easy prey for America's awesome military machine. War just didn't scare them as much. In truth, Kristol and Kagan were less \"neo-Reaganites\" than \"ultra-Reaganites.\" If Lyndon Johnson had taken George Kennan's ethic of toughness and stretched it to the point of elephantiasis, now Kristol and Kagan were taking Reagan's ethic of dominance and doing the same. They shared Reagan's basic goals; they just abandoned the limits that had constrained his efforts to achieve them.\n\nLike the Camelot intellectuals of the late 1950s and '60s, Kristol and Kagan suffered from generational envy. Too young to have played major roles in the great anticommunist struggle, they were to the cold war what John F. Kennedy's generation had been to World War II: the junior officers, often relegated to watching events from belowdeck. Irving Kristol and Jeane Kirkpatrick were not nostalgic for the cold war. They were a bit like Dwight Eisenhower in the 1950s: They had seen enough of heroism and its awful price. Normalcy struck them as a pretty attractive thing. But the younger conservative intellectuals, who feared normalcy and its corresponding lack of potential for heroism, were intensely nostalgic. If Kennedy had tried to hang Churchill's paintings in the White House, Kristol and Kagan were constantly lighting votive candles to Reagan. Four years after their call for a \"Neo-Reaganite\" foreign policy, they edited a book called Present Dangers\u2014the title being a play on Podhoretz's 1980 polemic against d\u00e9tente, Present Danger, which had influenced Reagan. \"Surely we are entitled to hope for another Reagan,\" wrote Kristol in 1997. \"Who will be this era's Ronald Reagan?\" asked his new magazine, the Weekly Standard, in 1998. The answer was implicit: whoever listened to them.\n\nCamelot intellectuals like Bundy and Rostow had often compared the late 1950s to the late '30s: a moment of drift and peril requiring a heroic new generation to save freedom in what Kennedy called its \"hour of maximum danger.\" Now Kristol and Kagan announced that \"the current situation is reminiscent of the mid-1970's.\" If Eisenhower had been Neville Chamberlain, Bill Clinton was now Jimmy Carter. But if comparing the 1950s to the '30s was bad, comparing the 1990s to the '70s was even worse. Kagan and Kristol were sure the Clinton years were a period of grave and mounting danger, but they couldn't even quite identify what the mounting danger was. Like Krauthammer, they mentioned \"weapons states\" with WMD, threw in a resurgent Russia and a potentially expansionist China, and suggested that taken together, these threats were as fearsome as the U.S.S.R. (leading Irving Kristol's close friend Owen Harries to reply that \"a bad head cold plus a skin rash plus a slipped disk does not add up to a brain tumor\"). But Kristol and Kagan's fallback answer was that the danger was no particular enemy at all; it was America's unwillingness to recognize that it was in danger. \"The ubiquitous post\u2013Cold War question\u2014where is the threat?\u2014is thus misconceived,\" Kristol and Kagan explained. \"In a world in which peace and American security depend on American power and the will to use it, the main threat the United States faces now is its own weakness.\"\n\nIt was a remarkable act of intellectual jujitsu. What Kristol and Kagan were proposing was brazenly aggressive: They didn't want merely to contain anti-American regimes; they wanted to overthrow them, and not through the slow and subtle infiltration of global capitalism, but through ideological warfare and quite possibly outright force. As they themselves declared, the United States should pursue \"benevolent global hegemony.\" Yet global hegemony, they insisted\u2014echoing Krauthammer and the Defense Planning Guidance\u2014was merely defensive; it was ultimately no different than repelling the marauders at the gates because any region that America did not dominate would ultimately represent a mortal threat. Essentially, their slogan was \"Give me dominance or give me death.\"\n\nIt took a special kind of person to gaze out at America in the Clinton years\u2014a country with a defense budget bigger than all its competitors' combined\u2014and see weakness, but Kristol and Kagan were not comparing post\u2013cold war America to any other country at any other time; they were comparing it to what they considered possible. Like the proponents of NSC 68, they assumed America's economic resources were nearly infinite. (Of the fifteen essays in Present Dangers, not one discussed how to pay for the massive increase in military spending that the book proposed.) Because they had come of age during a great wave of global democratization, they tended to believe that democracy could arise quickly anywhere on earth. And because they had seen America win wars so easily, the prospect of military confrontation did not fill them with dread. Kristol and Kagan wanted to replicate the triumphs of their elders: to conquer new Soviet Unions and usher in their very own 1989, and they planned to do it the way they imagined that Reagan had: by confrontation and force. They were the very thing that Irving Kristol and Jeane Kirkpatrick feared most: intellectuals with a messianic vision to remake the world and little sense that national or human fallibility might stand in their way. The whole thing was drenched in irony. Irving Kristol had always associated utopianism with the left; it was the monster he had spent his career trying to slay. Now, in the twilight of his life, it had returned, on the right, championed by his own son. Unanticipated consequences indeed.\n\nBy decade's end, the dominance conservatives had congealed into a potent network, which brought policymakers and pundits together in common cause. Abrams, who now also called himself a \"neo-Reaganite,\" penned an essay for Present Dangers, as did Wolfowitz. Both men also wrote for the Weekly Standard and helped Kristol and Kagan establish a think tank, the Project for the New American Century (PNAC), whose other founding associates included Scooter Libby, Zalmay Khalilzad, Donald Rumsfeld, and Dick Cheney. (Just as significant were the people who did not join: Irving Kristol, Jeane Kirkpatrick, Owen Harries, William F. Buckley, George Will, and Colin Powell.) In the late 1990s, PNAC called for overthrowing Slobodan Milosevic, signaling that for the dominance conservatives, going to war to stop genocide in Bosnia and Kosovo was not enough; without democratic regime change there could be no \"full and complete victory\" in the Balkans. The bar for going to war kept going down, and the bar for achieving victory kept going up. But in 2000, complete victory was achieved: A democracy movement forced Milosevic from power. That left PNAC free to focus on overthrowing Saddam Hussein.\n\nThe intellectual infrastructure for global dominance was now in place. With the U.S. economy booming, money no longer appeared to be a problem, and no one in the PNAC circle worried that America lacked the wisdom. The final obstacle was the one that Irving Kristol and Jeane Kirkpatrick had seen as their ultimate safeguard: the American people. Did they have the will?\n\nThe dominance conservatives feared the answer was no. The Republican Congress, after all, was rather isolationist. Clinton had managed to take the country to war in the Balkans nonetheless, but his administration seemed to assume that in an era devoid of grave threats, the public would only support wars that got no Americans killed. For the PNAC crowd, this was evidence of a severe cultural rot. Sounding a lot like Arthur Schlesinger in 1958, they declared that Americans had grown soft. \"We have become a nation obsessed with risk avoidance and safety,\" wrote Kristol's Weekly Standard colleague David Brooks, who delved most deeply into this alleged malaise. Americans, Brooks claimed, had \"replaced high public aspiration with the narrower concerns of private life.\" It was up to government to call them to a higher purpose, one that required sacrifice and strain. Four decades earlier, Schlesinger had denounced America's \"vacation from public responsibilities\" Krauthammer now spoke of a \"holiday from history.\" Schlesinger had lamented the \"decline of greatness\" Brooks decried \"post-greatness America.\" To remedy the problem, he and Kristol offered a new vision: \"National Greatness Conservatism.\" The phrase was cribbed from Theodore Roosevelt, the scourge of softness who had inspired JFK's fifty-mile hikes.\n\nWhen it came to domestic policy, national-greatness conservatism was paper thin. Like Schlesinger, both Kristol and Brooks wanted government to lead heroic missions at home as well as abroad. But unlike Schlesinger, they were conservatives who believed that, domestically at least, government usually screwed things up. They disliked the extreme hostility to government that had taken hold on the post\u2013cold war right, which they believed offered Americans no goal nobler than being left alone, but they struggled to explain what large domestic initiatives they actually trusted government to undertake. Brooks suggested constructing grand public buildings like the Library of Congress, a project unlikely to rouse Americans from their e-trades, or privatizing Social Security, Medicare, and education, which hardly seemed like a grand public mission at all. Krauthammer added that America should journey to Mars. But as the conservative columnist Jonah Goldberg noted, all this \"was very hard to understand except as public relations.\" Brooks and Kristol had diagnosed what they believed was a cultural malady, and a political one, too, since the Gingrich Congress's strident hostility to government was proving unpopular. But their bid to solve it through inspiring government-led projects kept running into a fundamental problem: As conservatives, they still basically believed that when government messed around in the free market, it usually made things worse.\n\nThe only domestic character-building project that gained any traction was the campaign to impeach Bill Clinton. For the writers at the Weekly Standard, Clinton embodied the cultural degradation that was preventing America from achieving true greatness. \"He is supremely self-seeking, in every conceivable sense,\" wrote one of Kristol's Standard colleagues about the president. \"His politics, too, are based on self-interest; the stroking and stoking of many small appetites. He does not lead citizens but rapacious consumers, whose sense of grievance he tries to exaggerate. His mode is to focus on the small glitch and call it a crisis, so that he can step in to cushion still further the already soft edges of boomer life.\"\n\nThe impeachment crusade, however, ended in disaster. The American people, who generally liked Clinton's small-bore, low-strain policies and wanted no part of a jihad against the era in which they lived, punished Republicans at the polls in the midterm elections of 1998. Rather than striking a blow against America's cultural decay, the impeachment struggle became, in Kristol's view, further evidence of it. Rather than serving as the great public cause for which the national-greatness conservatives hungered, it simply reaffirmed how desperately Americans needed such a cause, how badly America needed to become something other than what it was: a pampered, indulgent, dull place.\n\nFukuyama had seen all this coming. \"The end of history will be a very sad time,\" he had written back in 1989. \"The willingness to risk one's life for a purely abstract goal, the worldwide ideological struggle that called forth daring, courage, imagination, and idealism will be replaced by economic calculation, the endless solving of technical problems, environmental concerns, and the satisfaction of sophisticated consumer demands.\" After returning home with the battle won, the warrior might spend some happy time cultivating his crops, fixing his house, loving his wife, and enjoying the soft comforts of home. If he had grown old fighting, and years of hardship had sapped the animal energies of youth, he might even be content to pass his remaining days in this state of repose. But what about his son? Would he not feel the call to struggle, to danger, to greatness?\n\nPerhaps most Americans did not feel this pull. For them, safety and prosperity were not yet boring. They still had roofs to repair and fields to sow. But Kristol and his colleagues found all this tinkering tedious, even degrading; unlike Clinton, they took no joy in helping Americans with such mundane work. And they searched for signs that average Americans might be growing tired of it as well. How else, they asked, to explain the wave of World War II nostalgia that hit America in the late 1990s, culminating with Tom Brokaw's bestseller, The Greatest Generation, and the movie Saving Private Ryan\u2014the palpable envy with which younger Americans discussed the savage trials of their parents and grandparents? The dominance conservatives were sure that something was stirring, if only it could be harnessed.\n\nAfter all, there were still places that history had not yet tamed. Those places could be depicted as a menace to everything Americans held dear, perhaps as great a menace as any that had come before. And even better, because the grand ideological contest was over, and history's escalator was inexorably moving America's way, the outcome of this new battle was preordained. It was the best kind of battle, the kind you could not lose. In the summer of 2000, Jonah Goldberg, the young conservative who had derided Brooks's and Kristol's national-greatness vision as mere public relations, hit upon a mission of his own: Invade Africa. \"I mean going in\u2014guns blazing if necessary\u2014for truth and justice,\" he explained. \"I am quite serious about this.... We should spend billions upon billions doing it. We should put American troops in harm's way. We should not be surprised that Americans will die doing the right thing.\" It was a curious pronouncement, a kind of eruption of the dominance conservative id. But there was an honesty to it, a recognition that if the problem was deep boredom born of profound success, the solution did not lie in building big libraries and tinkering with Social Security. It would take something grander, and bloodier, too.\n\n## CHAPTER SIXTEEN\n\n## SMALL BALL\n\nNational-greatness conservatives had a presidential candidate in 2000, and his name wasn't George W. Bush. It was John S. McCain, III, a man who had been surfing America's waves of hubris and tragedy since he was in his teens. They were, in a sense, the story of his life.\n\nMcCain had graduated from the U.S. Naval Academy in 1958, the last time America seemed invincible. He was a navy flier, restless and brash, a cocky kid at a cocky time. The Camelot gang, reared on the lessons of Munich, was itching for its own showdown with totalitarianism in the jungles of the developing world. And like them, McCain\u2014the son and grandson of admirals who served in World War II\u2014yearned to fight the dark-skinned Hitlers of his age. It never occurred to him that the Hitlers might win. \"I believed,\" he later explained, \"that militarily we could prevail in whatever conflict we were involved in.\"\n\nVietnam changed that. Five years in a Hanoi prison turned McCain's hair white. By the time he returned home, he could no longer raise his arms above his head, and he no longer believed America could vanquish every foe. \"The American people and Congress now appreciate that we are neither omniscient nor omnipotent,\" he told the Los Angeles Times. \"If we do become involved in combat, that involvement must be of relatively short duration and must be readily explained to the man in the street in one or two sentences.\" Colin Powell couldn't have said it better himself.\n\nIn the 1980s, McCain won a seat in Congress, where he proved a Reaganite in the true sense of the word: slow on the trigger when it came to military force. Like Powell he was appalled by the 1982 intervention in Lebanon, which he saw as Vietnam writ small. In 1987 he opposed putting U.S. flags on Kuwaiti oil tankers for fear the United States would be drawn into war with Iran. When Iraq invaded Kuwait three years later, his first instinct was anything but bellicose. \"If you get involved in a major ground war in the Saudi desert, I think [public] support will erode significantly,\" he warned. \"Nor should it be supported. We cannot even contemplate, in my view, trading American blood for Iraqi blood.\"\n\nOn the Gulf War, McCain\u2014like Jeane Kirkpatrick\u2014eventually came around. But he still opposed marching to Baghdad, and every other war of the early 1990s. He railed against Clinton's intervention in Haiti and sponsored legislation to pull U.S. troops from Somalia. And \"on Bosnia,\" wrote a hostile Robert Kagan in 1995, \"Senator John McCain led the Republican attack, warning that any use of military power there would result in another failure like Vietnam or Lebanon.\"\n\nMcCain would later say that Srebrenica changed him: All those Bosnians herded to the rape chamber and the slaughterhouse while UN peacekeepers skulked away. But in fact, days after the massacre, he still denied that \"our security is so gravely threatened in Bosnia that it requires the sacrifice, in great numbers, of our sons and daughters.\" What changed McCain wasn't Serb barbarism; it was American success. It was only once NATO warplanes had pounded the Serbs into submission\u2014once it became clear that Bosnia wasn't Vietnam\u2014that he grew talons and became a hawk.\n\nBy the 2000 presidential primaries he had come full circle, back to the jaunty confidence of his youth. On Kosovo, while many Republicans wrung their hands, McCain called the Clinton administration's air war timid and urged invading on the ground. He hooked up with his former detractors at the Weekly Standard and, at their urging, proposed what he called a \"21st century...Reagan Doctrine.\" Just as President Reagan had allegedly overthrown the evil empire by force of arms, President McCain would now overthrow its pipsqueak heirs: Iraq, North Korea, and Iran. Once a Reaganite, McCain was now an ultra-Reaganite, which meant he was reenacting the Gipper's foreign policy\u2014minus the dread of war.\n\nEqually exciting, in Kristol, Brooks, and Kagan's eyes, McCain wrapped his policy agenda in the rhetoric of national greatness. He railed against the tranquilizer of affluence, which had left Americans snoozing lazily on the couch, too fat, happy, and selfish to do anything big, like fight another cold war. The Standard compared him to Reagan, Teddy Roosevelt, and, surprisingly for a conservative magazine, JFK. It was 1958 all over again, and after decades of Vietnam-induced caution, John McCain had rediscovered a lost part of himself.\n\nMcCain's chief rival for the Republican nomination, Texas governor George W. Bush, didn't have such strong foreign policy views, in large part because foreign policy hadn't directly influenced his life. While McCain had been rethinking American omnipotence between torture sessions at the Hanoi Hilton, Bush had been sipping banana daiquiris and playing midnight water polo at a swanky Houston singles complex called Chateau Dijon. He too had been a brash, restless kid, eager to match the heroism of his greatest-generation dad. But unlike McCain, he had discovered no grand cause into which to channel his swagger. He never found his own World War II, and even when he tried to walk his father's path in smaller ways, he usually stumbled and fell.\n\nThe generational contrast was downright painful. George H. W. Bush had excelled at Andover; George W. almost flunked out. H.W., even as a teenager, had been the family star, dutiful and upright; the teenage W., by contrast, had a mortifying habit of walking up to older women at church and asking whether they still enjoyed sex. (He also relished a good fart joke in mixed company, a predilection that continued into the White House.) H.W. was an accomplished athlete, captaining the baseball team at Yale, as his father had before him; W. was a benchwarmer on the junior varsity, and commissioner of an intramural stickball league. H.W. drank in moderation; W. drank to excess, repeatedly driving drunk with his younger siblings in the car. In his early twenties, H.W. married the daughter of a prominent local family; at the same age, W. tried to do the same thing, but his fianc\u00e9e broke off the engagement. H.W.'s exploits as a fighter pilot in World War II were the stuff of family legend; during Vietnam, W. hoped to be a pilot as well, but never made it off American soil, serving instead with other trust funders in the \"champagne unit\" of the Texas National Guard. H.W. started an oil company with a Spanish name (Zapata) and made a million by the time he was thirty; W. started one, too (Arbusto), but it went belly-up.\n\nBy the 1970s, Bush's life was a shambles. Well into his thirties, he still didn't have a steady job. He was drinking almost every night and had downsized from Chateau Dijon to a dingy apartment above a garage. After one particularly ignominious professional failure, his father confessed himself \"disappointed.\" At a state dinner, W. introduced himself to Britain's Queen Elizabeth II as the family's black sheep.\n\nEventually, with middle age fast approaching, Bush found himself\u2014which meant, in large part, no longer trying to be his dad. In 1977 he ran for Congress, a decision that violated his father's code: that no Bush should enter politics before ensuring his family's financial security first. But for W. the experience was empowering. Even though he lost, he discovered a talent for good-ol'-boy glad-handing that his father lacked. In 1985 he quit drinking and embraced evangelical Christianity, rejecting his father's high WASP Episcopalianism in favor of a more emotional, populist faith, which helped him forge ties to social conservatives who distrusted his dad. (Bush's newfound religious fervor also helped him turn the tables on his judgmental parents: After his conversion, W. told his mother that unless she became born-again, too, she wouldn't get into heaven.) Then, in 1989, he became managing partner of the Texas Rangers baseball team, finally finding a job in which\u2014at least in his own mind\u2014he excelled.\n\nBy the early 1990s, Bush had developed a signature personal and professional style. Instead of biding his time, playing by the rules, and carefully preparing the ground, as his father had done, he would seize the initiative, aim high, and follow his gut. At a time when complex statistical analysis was transforming the way front offices analyzed baseball, Bush disregarded the advice of his scouts and judged players less on the numbers than on their character. Lacking his father's penchant for dispassionate analysis, he fell back on the emotional intelligence that he considered his strength. He sized up players quickly and made bold, unorthodox decisions, like trading Sammy Sosa to the Chicago White Sox for Harold Baines. In the pickup games he organized for team owners, Bush played the same way. \"I remember him striking out a lot,\" one observer remembered. He took \"wild swings with lots of muscle. But he was swinging so hard, trying so hard, he didn't take the chance to watch the ball.\"\n\nSosa went on to hit 608 home runs after Bush traded him, while Baines only hit another 195, and in Bush's five years as managing partner, the Rangers hugged the bottom of their division. But Bush considered his baseball years a success. He sold his stake in the Rangers for $18 million, finally making himself a man of means. And the job raised his public profile, setting the stage for another political bid.\n\nIf Bush was growing more confident in his aggressive, intuitive style, he was also emboldened by seeing his father's more cautious approach fail when it mattered most. In W.'s mind, his dad lost his 1992 reelection bid because he had grown preoccupied with the minutiae of governance and offered the country no rousing agenda for a second term. \"The vision thing matters,\" W. later told Bob Woodward, in explaining what he had learned from his father's defeat. In politics, the younger Bush decided, victory goes to the bold.\n\nIn 1994, Bush told his parents that he wanted to run for governor of Texas. \"You can't win,\" barked his mother, worried that he would siphon off contributions from his more studious brother Jeb, who was seeking Florida's top job. But W. turned out to be the best campaigner in the family, and his grand, if vague, campaign slogan, \"What Texans Can Dream, Texans Can Do,\" appealed to the state's outsize image of itself. By 2000, the family screw-up was the front-runner for the Republican nomination for president. His chief competitor was John McCain.\n\nAt first glance, the Bush and McCain campaigns were utterly different. McCain attacked corporate interests; Bush seemed at times to be a virtual front man for them. Bush was most comfortable discussing \"compassionate conservatism,\" which mostly meant standardized tests in schools and government money for religious charities, topics in which McCain showed little interest. McCain spoke in considerable detail, and with great passion, about foreign policy. Bush couldn't publicly name the leaders of India or Pakistan, and in private didn't know that Germany was a member of NATO. His foreign policy statements often seemed guided by no coherent principle at all except the desire to be against whatever Bill Clinton was for.\n\nBut rhetorically, the Bush campaign actually shared a theme with McCain, and with JFK as well: the need for Americans to find a cause larger than the pursuit of mere material things. Bush's slogan was \"Prosperity with a Purpose.\" As he told the convention that nominated him, \"Prosperity can be a tool in our hands used to build and better our country, or it can be a drug in our system dulling our sense of urgency, of empathy, of duty. Our opportunities are too great, our lives too short, to waste this moment. So tonight, we vow to our nation we will seize this moment of American promise. We will use these good times for great goals.\" This was a slap at Clinton, of course, who to conservatives embodied purposeless prosperity and degraded, hedonistic ease. But it was also, more subtly, a slap at Bush's father, who Bush believed had wasted his presidency pushing paper. If elected president, Bush vowed, he would not play \"small ball.\" (The Gulf War, which had once turned the knuckles of America's leaders white, was now \"small ball.\" It showed how much national confidence had swelled in the decade since the end of the cold war.) Bush's own term in office would be, in his baseball-heavy vocabulary, a \"game changer.\"\n\nSo if McCain was speaking the language of national greatness, Bush was, too. It was just harder to know what he thought greatness meant. For a Republican, he talked a lot about poverty\u2014about kids betrayed by bad schools and lives disfigured by drugs. But like the national-greatness intellectuals, he was handicapped in proposing bold solutions by his distrust of government. On foreign policy, his answers mostly left Kristol and Kagan cold. While they believed Clinton had been timid in his use of military force, Bush seemed to score him for being too aggressive. He denounced the intervention in Haiti, talked about removing U.S. peacekeepers from the Balkans, and warned that American forces were being spread too thin.\n\nBut Bush also struck another chord, which echoed, if not Kristol and Kagan, then at least Charles Krauthammer. He opposed the Clinton interventions, he explained, because they were small ball, a distraction from the main event. \"We must be selective in the use of our military precisely because America has other great responsibilities that cannot be slighted or compromised,\" he declared in his first foreign policy address of the campaign. What were those great responsibilities, which might require America to unsheathe its sword? Paul Wolfowitz, one of Bush's foreign policy advisers, spoke\u2014as Krauthammer and McCain did\u2014about toppling dictatorships that were developing WMD. But other advisers said nothing of the sort. It was impossible to know exactly what Bush had in mind, or whether he had anything in mind at all. Unlike McCain, he did not have a bold foreign policy agenda. He had something more inchoate: a taste for boldness, a disdain for merely managing problems, a sense that his political and personal calling lay in epic, world-altering things, if only he could figure out what they were. In 2000, it was only an instinct, and on foreign policy it had little intellectual scaffolding. But it went to the core of who George W. Bush was, who he had decided to be. And in ways that only became clear later, it fit the moment, since a whole cadre of younger conservative intellectuals, emboldened by more than a decade of American military, ideological, and economic triumph, were growing bolder, too. The observers who assumed that W. would be a dumber, looser version of his father, content to \"amble into history\" with a cigar in one hand and an O'Doul's in the other, didn't understand the man\u2014and the moment\u2014at all.\n\nAmong those who didn't understand the new president was his secretary of state, Colin Powell. The years out of government had been good to Powell. His memoir had sold more than a million copies, and polls rated him among the most admired people on earth. Seeking to bask in his reflected glow, both Bush and McCain had promised during the campaign to appoint him to high office. And when Bush eventually did, observers mused that he might become the most powerful secretary of state ever. Watching the press conference where his selection was unveiled, in which Powell answered question after question with effortless command while his new boss fidgeted in the background, the Washington Post noted that \"Powell seemed to dominate the President-elect.\" In the New York Times, Thomas Friedman added, \"I sure hope Colin Powell is always right in his advice to Mr. Bush because he so towered over the President-elect...that it was impossible to imagine Mr. Bush ever challenging or over-ruling Mr. Powell on any issue.\"\n\nPowell had not been a close adviser during the campaign, and he didn't know Bush well. But he knew his father\u2014knew and admired him\u2014and assumed that the apple hadn't fallen far from the tree. Certainly W.'s vague and vaguely realist campaign pronouncements didn't suggest a radical change. Bush had, to be sure, distanced himself from his dad during the campaign, and invoked Reagan's name instead, but that didn't worry Powell. After all, he knew the real Reagan, the guy who wouldn't even invade Panama. All in all, Powell's return to government looked like the capstone to a glittering career. Facing the cameras as he stepped back onto the public stage, he had every reason to believe that when it came to foreign policy, he would be regent to a dutiful boy king.\n\nThe Bush administration's first National Security Council meeting began on January 30, 2001, at 3:35 P.M. exactly. A good omen, Powell thought. The new president, it turned out, was as fanatical about punctuality as he was. He was also an early riser and a stickler for proper dress, never entering the Oval Office except in suit and tie. The NSC meeting was carefully choreographed, each official in his or her assigned seat, each speaking in turn, by order of rank. Powell was impressed. It was good to have Republicans back in charge.\n\nBut there were other, darker omens, whose significance Powell did not yet fully grasp. To Bush's right sat old frosty himself, Dick Cheney, now vice president. In the half century since the NSC was created, vice presidents had rarely participated. Cheney, however, would attend almost every session, even taking part by videoconference when he was on the road. He also sat in on the president's morning CIA briefing; that was unprecedented, too. (In fact, the CIA first briefed Cheney alone, which allowed him to instruct the briefers to emphasize certain points with the president and downplay others\u2014in other words, to stack the deck.) Cheney had also headed Bush's transition team\u2014again, highly unusual. For top cabinet posts, he had chosen the candidates Bush interviewed, and while Bush made the final decision, he ratified Cheney's recommendations every time. The result was an administration teeming with vice presidential Mini-Me's. National Security Advisor Condoleezza Rice was Bush's close friend. But she was sandwiched between Cheney's own powerful adviser on national security, Scooter Libby, who in the first Bush administration had outranked her, and her deputy, Stephen Hadley, who had worked for Cheney at the Pentagon and whom Powell's close ally Richard Armitage called the vice president's \"mole.\" The new treasury secretary was Paul O'Neill, a longtime Cheney associate. At Defense was Cheney's mentor, Donald Rumsfeld, a renowned bureaucratic shark. (Powell had wanted Pennsylvania governor Tom Ridge, a less formidable figure, but Cheney pushed hard for Rummy, his old boss from the Nixon and Ford administrations, a man Henry Kissinger\u2014Henry Kissinger!\u2014once called the \"most ruthless\" government official he had ever met.) Rumsfeld's Pentagon deputy was Wolfowitz, another former Cheney subordinate, and a former boss and mentor to both Hadley and Libby. Cheney even stocked Powell's own State Department with moles, including Undersecretary John Bolton, who would spend Bush's first term spreading anti-Powell gossip, and Cheney's own thirty-six-year-old daughter, Elizabeth.\n\nAt the January 30 meeting, Powell spoke first, warning that violence between Israel and the Palestinians was spiraling out of control, and urging that the administration push the parties back to the negotiating table. Nothing doing: Cheney and Rumsfeld declared the peace process a waste of time and shifted the conversation to Iraq. Powell was ready: He suggested refining the sanctions that the UN had imposed after the Gulf War, making them more enforceable and less onerous for the Iraqi people. But Rumsfeld said sanctions would never stop Iraq from acquiring WMD and proposed using U.S.-patrolled no-fly zones to bomb Iraq instead. The conversation then turned to possible covert action to topple Saddam, and eventually adjourned. The Bush administration was only days old but it already had two Iraq policies: Powell was working to contain Saddam; Rumsfeld\u2014with Cheney's patronage\u2014was working to overthrow him.\n\nAt a \"Principals\" meeting (an NSC meeting the president does not attend) ten days later, Rumsfeld went further. It would be convenient, he mused, if the Iraqis shot down a plane enforcing the no-fly zone; that would justify a major U.S. air strike. (It would be a kind of aerial Tonkin Gulf.) Wolfowitz suggested dramatically expanding support for Iraqi opposition forces, perhaps even helping them establish a provisional government in Iraq's Shia south. \"Imagine what the region would look like without Saddam and with a regime that's aligned with U.S. interests,\" Rumsfeld declared. \"It would change everything in the region and beyond it.\"\n\nChange everything: That's what Cheney, Rumsfeld, and Wolfowitz were back in government to do. Even in the first Bush administration, Cheney and Wolfowitz had been the house gamblers, eager to dismember the U.S.S.R. rather than stabilize it, and to help the Shia topple Saddam, even at the risk of Iranian influence or civil war. On their way out the door, they had sketched a blueprint for global dominance, which contemplated preventive war to ensure American hegemony over every important region of the world. And after eight years on the sidelines, watching even the feckless Clintonites win war after war, they were even more confident of what American power could do. If Bush had an instinct for boldness, Cheney and company had at least the beginnings of a plan.\n\nIn imagining a transformed Iraq, flipped from adversary to client, which would further America's dominance of the Middle East, and the world, Cheney, Rumsfeld, and Wolfowitz believed they were walking in Reagan's footsteps. Like John McCain, William Kristol, and Robert Kagan, Wolfowitz had spent the late 1990s advocating a new \"Reagan doctrine.\" In Afghanistan, Nicaragua, and Angola, he argued, Reagan's aid to anticommunist rebels had helped bring the Soviet Union to its knees. In the Balkans, Bill Clinton had gone even further, not only aiding the Croats, the Bosnians, and later the Kosovars but also bombing the Serbs on their behalf. (The dominance conservatives had an odd relationship to Clinton: They bashed him mercilessly, but when it came to foreign policy, they stood on his shoulders.) The lesson seemed clear: By offering money, weapons, and air support to the opponents of Saddam, the United States could turn Iraq from a hostile tyranny to a democratic ally.\n\nCheney and Rumsfeld were less concerned about the democratic part, but for Wolfowitz, Islamic democracy had become something of a passion. After his success in helping to oust Ferdinand Marcos, he had become Reagan's ambassador to Indonesia, where, in his final days in his post, he had given a controversial speech urging Jakarta to embrace democracy. At the time it seemed quixotic, but in 1999, Indonesia\u2014the world's largest Muslim country\u2014held its first free presidential election.\n\nIt underscored what Wolfowitz called \"a remarkable phenomenon of our time\u2014the triumph of democracy in country after country, including some with no previous history of democratic rule.\" And this ideological optimism left him susceptible to the charms of one Ahmed Chalabi, the leader of the opposition group the Iraqi National Congress (INC), who spent the 1990s telling Wolfowitz, Cheney, and anyone else who would listen that pro-American democracy could triumph in Iraq, too. Chalabi, the scion of one of Iraq's wealthiest families, an elegant man with a University of Chicago mathematics Ph.D., knew how to market his movement to Americans. Its very name, the INC, evoked Nelson Mandela's ANC, which had brought democracy to South Africa. And Chalabi's most important intellectual ally, a rumpled Brandeis professor named Kanan Makiya, had authored a democratic charter for Iraq that was explicitly modeled on the work of Eastern European dissidents like Vaclav Havel. Wolfowitz was a smart man: He didn't believe Chalabi and Makiya could turn Iraq into the Czech Republic overnight. But from the Philippines to Panama to Indonesia to Serbia, he had spent the previous decade and a half watching Fukuyama's prophecy come true. So maybe Iraq wouldn't be the Czech Republic. Surely, however, it could be Romania, the Eastern European country with the nastiest dictator and the bumpiest path from communism to freedom. Romania, that was Wolfowitz's preferred analogy\u2014not perfection, but one hell of a game changer nonetheless.\n\nWhat Wolfowitz and his \"neo-Reaganite\" allies seemed not to understand was that in imagining that military aid and air support for the INC could usher in a pro-American, democratic Iraq, they were going well beyond what Reagan (or Clinton, for that matter) had actually done. As frequently happens during hubris bubbles, they were invoking the successes of the past to justify something far more ambitious in the present. Of the three countries where the \"Reagan Doctrine\" had been tried\u2014Afghanistan, Nicaragua, and Angola\u2014America's proxies had won military victory in only the first. And in Afghanistan, that had happened largely because the occupying Soviet army withdrew\u2014a precedent with little relevance to Saddam's Iraq, where there was no foreign occupier. (Nor was Afghanistan exactly a victory for democracy, since America's allies, the mujahideen, were mostly theocrats.) The Clinton analogies were equally misleading, since the Bosnia war was aimed at preserving a regime (and its people), not overthrowing one. It was true that in 2000, fifteen months after the Kosovo war, Slobodan Milosevic did fall, but he was toppled by peaceful revolution, not U.S. bombs or Kosovar troops.\n\nWolfowitz had every reason to be optimistic about the long-term spread of global democracy. Many new countries had indeed embraced representative government in the 1980s and '90s, but with the exception of Panama, none of those democratic revolutions had come primarily through the barrel of an American gun. (And even Panama had held a democratic election seven months before the United States invaded. After deposing Noriega, the United States simply installed the man Panamanians had already elected.) What Wolfowitz was proposing for Iraq\u2014an American-proxy war to produce a pro-American democracy\u2014was not the \"Reagan Doctrine.\" It was the \"Reagan Doctrine\" on steroids.\n\nIt all made Colin Powell extremely queasy. When he heard people talk about forcibly remaking whole regions of the world, his mind sometimes drifted back to his days as a second lieutenant, stationed in West Germany in the late 1950s. He had been drinking with some other up-and-coming young officers, and they were all feeling pretty good about themselves when their captain decided to take them down a notch. \"You guys think you're really hot stuff,\" he bellowed. \"You've got all your little troops in bed and you're sitting here patting yourselves on the back about how you solved all the world's problems today. And you're going to go home and lie down with your little wives and sleep well. But you know what? While you're sleeping, the world is going to get fucked up all over again and you're going to have to get up in the morning and start at the beginning.\" Powell didn't believe in game changers. For him, foreign policy leadership, like military leadership, mostly consisted of holding things together, keeping all hell from breaking loose, making incremental change. He had no problem with small ball. It was small ball\u2014realistic expectations, meticulous attention to detail\u2014that kept your men alive. If Bush's sports vocabulary was baseball, Powell's was football, a rougher game, and more complex, too. \"Success doesn't always come with a deep pass,\" he liked to say. \"Sometimes it's a ground game...building up slowly but surely, layer by layer.\"\n\nPowell didn't have much patience for the Iraqi exiles. If Wolfowitz likened them to the Nicaraguan Contras, the Bosnians, and the Afghan mujahideen, Powell viewed them through the lens of South Vietnam, where America's local allies had promised much, delivered little, and gotten a lot of his buddies killed. He viewed Wolfowitz's proposal for a liberated enclave in the Iraqi south the same way William Crowe had viewed Elliott Abrams's proposal for a liberated enclave in Panama: as the kind of militarily ludicrous suggestion you got from people who had spent their twenties in think tanks, not foxholes. For Powell, Iraq was a problem\u2014not an existential crisis, not a golden opportunity, just a problem to be managed. At his confirmation hearings, he had called Iraq a \"broken, weak country.\" In February he declared that containment was working, and that Saddam posed no threat to his neighbors. It was an extraordinary statement considering that Cheney, Wolfowitz, and their intellectual allies were declaring containment dead, and Iraq as grave a danger as the U.S.S.R. But Powell downplayed the threat from Saddam in part because he feared the costs of trying to eliminate it. Less confident that America had the power, wisdom, and money to midwife a democratic, pro-American Iraq, he was content to muddle through. And muddling through was exactly what Cheney, Rumsfeld, and Wolfowitz were determined not to do.\n\nIt was same argument that had pitted Crowe against Abrams more than a decade earlier: whether to tolerate adversaries or topple them. And in the Bush administration's early months, Iraq was only one of the places it played itself out. In March, with the South Korean president about to arrive in town, Powell told the Washington Post that the Bush administration would continue Clinton's position of negotiating with the Stalinist North. When the Post story came out the next day, he got an irate call from Rice: He had misstated the administration's position. The new policy was not negotiation; it was regime change.\n\nPowell was stunned. The cabinet had never met on North Korea; if coercive regime change was the new line, no one had asked his opinion. To Powell's staff, it smelled like a Cheney power play: a policy hatched in secret and then wielded like a club to show that Powell was not as powerful as everyone thought. It was classic Cheney. Whenever he started a new job, he liked to take down some big game just to prove who was in charge. At the Pentagon in the early 1990s, he had taken down generals. Now that Cheney was vice president, the big game was Colin Powell.\n\nIt was ironic. During the Clinton years, Powell had moaned about the length of meetings, about Clinton's inability to make up his mind, about the fact that even junior officials demanded their say. Now information was so tightly held, and decisions so briskly made, that even he was not always in the room. After Rice's call, he sheepishly told the press that he had misstated administration policy. \"Sometimes,\" he explained, \"you get a little far forward on your skis.\"\n\nEven the fights Powell won were painful. A few weeks after the Korea humiliation, a U.S. spy plane collided with a Chinese jet off the coast of China and made an emergency landing on Chinese soil. Before returning the crew, Beijing demanded an apology and a halt to U.S. spying over what it claimed were its territorial waters. For a week Powell negotiated for the crew's release, finally securing it with a classic diplomatic fudge, a letter that in English offered no apology but did so in the Chinese translation. Powell was ecstatic; this was high-level, cool-headed problem-solving, just what he did best. But not everyone was pleased. The White House refused to let him go on television to trumpet his achievement. Kristol and Kagan, whose anti-Chinese rhetoric had been growing louder and louder, called the non-apology apology a \"national humiliation\" and implied that Cheney and Rumsfeld disapproved. Then, a few weeks later, Bush told ABC News that America would defend Taiwan in the event of a Chinese attack, junking several decades of carefully crafted diplomatic ambiguity aimed at keeping the United States out of a Pacific war. Wolfowitz and PNAC had called for just such a declaration in 1999, and Powell aides even suspected that top Pentagon officials were pushing Taiwan to declare independence, a move that might well prompt Beijing to strike. Was this Bush's effort at a game changer, a bold, impulsive move aimed at punishing the Chinese for their temerity and showing once and for all that in East Asia, the United States was in charge? The White House walked Bush's statement back, but the Standard said Bush had known exactly what he was saying, and Cheney seemed to go out of his way to reaffirm Bush's original remark. That same month, the Pentagon offered Taiwan its largest arms sale in a decade.\n\nHad Powell been kneecapped again? It was hard to be sure, since Cheney\u2014a man known for driving around his home state of Wyoming and only speaking to fellow passengers once or twice per hour\u2014said little, at least when Powell was around. Rumsfeld was opaque, too, often speaking in what Powell called the \"third-person passive once removed\"\u2014with statements like \"one would think\" and \"one might expect.\" Powell's deputies were equally baffled. When they went to meetings with Hadley, Libby, Wolfowitz, and Undersecretary of Defense for Policy Douglas Feith, they often suspected that their Pentagon and White House colleagues knew things they didn't, that the real decisions were being made someplace else. (Only later would they learn that Cheney's office sometimes used the National Security Agency to spy on State Department officials when they traveled abroad, and that when Powell's aides received e-mails, blind copies were frequently sent to the vice president's staff.)\n\nWhat was clear, by the summer of 2001, was that unnamed colleagues were gunning for Powell in the press. One article declared that he was \"on a short leash...nothing more than a silent symbol or a messenger boy.\" In anonymous quotes, administration officials suggested that he had lost a step. \"Powell's megastar wattage looks curiously dimmed, as if someone has turned his light way down,\" declared a September 10 cover story in Time. For Kristol and Kagan, who were unhappy that the Bush administration was not hewing consistently to a \"neo-Reaganite line,\" Powell had become enemy number one.\n\nIn ways Powell hadn't fully appreciated, Washington had changed since the end of the cold war. In Congress, moderate Republicans like himself were now rare. The right had also built a much stronger intellectual and journalistic network, symbolized by magazines like the Standard, think tanks like PNAC, nationally syndicated radio talk show hosts like Rush Limbaugh, and talking-head TV channels like Fox News, which had not even existed in the Reagan years. They provided Cheney, Rumsfeld, and Wolfowitz an echo chamber that Powell lacked. That Powell was having problems with the media was surprising; he had long been considered a master of the press. But his appeal was personal, not ideological. As a pragmatist, distrustful of theory, his perspective lacked appeal for the conservative journals, think tanks, and chat shows that specialized in big, aggressive, Manichean visions. In his years out of power, American dominance had become, for a whole generation of conservative commentators, an ideology. And Powell, for all his political savvy, was less formidable in this more ideological age.\n\nAs the summer of 2001 faded into fall, Powell was growing weary; Bush II was not going as planned. But he still had one ace in the hole: the American people. It was the very thing that drove the national-greatness conservatives mad. In an era without obvious threats, the public would only support wars in which barely any Americans got killed. Inside the Beltway, the dominance conservatives were winning the bureaucratic battle, but in the country at large, Powell's low-key, low-cost, low-blood foreign policy still enjoyed the upper hand.\n\nIf Powell took solace from his public standing, he also retained hopes for Bush himself. In his more optimistic moments, he considered the president a work in progress, with potential to grow in the job. Perhaps as Bush grew more confident, Powell thought, he would bring the Cheney-Rumsfeld-Wolfowitz nexus under control. In a sense, Powell was right: Bush hadn't fully settled into the presidency. He had pushed through new policies on education, taxes, stem cells, and funding for religious charities, but his agenda, in his first eight months, was hardly epic or stirring; it just wasn't an epic or stirring time. Even Bush's \"successes,\" chief speechwriter Michael Gerson later acknowledged, \"had a quality of randomness, disconnected from larger purposes.\" At times Bush seemed to be coasting, spending long stretches at his Texas ranch. Powell hoped that when the president found his feet, and his true calling, he would become a wiser, more diligent, more responsible steward of the nation's fortunes, more his father's son. After eight months, he still didn't really know the man at all.\n\n## CHAPTER SEVENTEEN\n\n## THE OPPORTUNITY\n\n\"We have to think of this as an opportunity.\" It was an odd thing for President Bush to say at 9:30 P.M. on September 11, 2001, with thousands of New Yorkers and Virginians freshly dead beneath mountains of twisted concrete. But he kept saying it. \"This is a new world.... This is an opportunity,\" he told a National Security Council meeting two days later. And then again, in an interview: \"I will seize the opportunity.\"\n\nOpportunity for what? When it came to the Muslim world, Bush did not yet know. South Asia and the Middle East were not regions with which he had deep experience, to put it mildly. Before entering the White House, he had been to the Middle East exactly once, in 1998, on a brief swing designed to burnish his foreign policy credentials, and he had never traveled to the subcontinent at all. Asked about the Taliban by Glamour magazine in 2000, he had responded with a blank stare, and then, after being reminded that they ruled Afghanistan, chuckled, \"I thought you said some band.\" As late as January 2003, according to visitors to the Oval Office, he seemed unaware that Iraq was divided between Sunni and Shia.\n\nBut it was precisely because Bush knew so little about the greater Middle East that he so easily made it a mirror of his views about the United States. The terrorists, he decided, had attacked America because they considered it weak, and they considered it weak because Americans lacked moral purpose. \"[T]here is the image of America out there that we are so materialistic, that we're almost hedonistic, that we don't have values, and that when struck, we wouldn't fight back,\" he explained. Al Qaeda had created \"a mythology of American weakness and decadence,\" explained Gerson. \"President Bush set out radically to change this course of events and smash this myth.\"\n\nIt was a remarkable act of projection. To be sure, Osama bin Laden disdained Western culture, and had once called America the \"weak horse\" as a way to rally his troops. But he hadn't toppled the Twin Towers because he saw America as weak; he had toppled them because America had grown so ferociously strong, because its victories in the cold war and Gulf War had made it the undisputed master of the Middle East, patron to the Arab and Israeli leaders whom bin Laden loathed, and military defiler of Saudi Arabia's sacred soil. If bin Laden had been truly emboldened by weakness, he would have attacked his old foes in Russia on 9\/11. They were the ones who had shown real decadence in the 1990s, watching in a vodka-induced stupor as their entire empire collapsed. Instead, bin Laden had attacked the Soviets after they invaded Afghanistan in the early 1980s, when their empire pushed deep into the Muslim world. Now he was attacking America, the world's new behemoth, because it held Muslim lands in its grasp.\n\nIt was not bin Laden who had spent the Clinton years bemoaning American weakness and decadence; it was the national-greatness conservatives. And it was not bin Laden who had railed against Clinton's prosperity without purpose, his blend of 1960s depravity and political small ball; it was George W. Bush. Now, after 9\/11, Bush turned that critique into a deeply solipsistic explanation for why three thousand Americans lay dead. Clinton, he declared, had invited attack by responding to past provocations with \"pinprick\" strikes that merely \"pounded sand.\" America had emboldened bin Laden with a policy of what Rumsfeld called \"reflexive pullback.\" Now that would change: Bush's anti-terror policies would be bold, consequential, decisive. As he told members of Congress soon after 9\/11, \"I'm not going to fire a two-million-dollar missile at a ten-dollar empty tent and hit a camel in the butt.\"\n\nAs a description of Clinton's actual foreign policy, this was sheer fantasy. Clinton's \"reflexive pullback\" had extended NATO to the borders of the former U.S.S.R., produced a military budget larger than all of America's competitors combined, and left awed foreigners comparing the United States to ancient Rome. Among Clinton's \"pinprick\" strikes had been Operation Desert Fox in 1998, a four-day campaign conducted without UN approval, in which the United States launched more cruise missiles against Iraq than it had during the entire Gulf War, killed 1,400 Iraqi troops, and\u2014U.S. weapons inspectors later learned\u2014frightened Saddam into aborting his efforts at producing WMD. In reality, American power had expanded mightily in the Clinton years; it just hadn't expanded as much as the national-greatness conservatives wanted, since, engorged by more than a decade of military, ideological, and economic triumph, they believed America could, and should, vanquish every important adversary on earth.\n\nIf Bush's interpretation of 9\/11 fit the mounting hubris of the time, it also fit his personal history. The parable of big goals replacing small ball, and moral responsibility replacing self-indulgence, was not merely his narrative for the \"war on terror\" it was his narrative for his entire life. As a man, he had meandered aimlessly and self-destructively, then found a higher purpose by overcoming the evil of alcoholism. And now, as president, the story was playing itself out again. In his first eight months in office, he had searched in vain for his presidential calling; now evil had arrived and made that calling clear. He told aides that 9\/11 gave his fellow baby boomers a chance to invest their lives with deeper meaning, to show the kind of spirit that their parents had shown in World War II. \"In our grief and anger,\" he told a joint session of Congress nine days after the attacks, \"we have found our mission and our moment.\" At the White House, amid the anguish and fear, there was exhilaration. \"No more stilted generational summonses, no more made-up 'callings,'\" declared White House speechwriter Matthew Scully after Bush's speech. \"Here, finally, was the real thing\u2014a real calling with real heroism.\" That's what Bush meant, at least initially, by 9\/11's \"opportunity.\" It was the opportunity to infuse an aimless presidency, and a pampered age, with heroism, purpose, greatness.\n\nAs usual, Powell was the odd man out. When the towers fell, he was in Lima for a meeting of the Organization of American States, listening to Peru's president plead for a reduction in U.S. cotton tariffs. As a result, he was not in Washington that evening to help shape Bush's address to the nation, in which the president declared that in the new \"war on terror,\" the enemy would be not merely the terrorists who had struck Virginia and New York, but the regimes that harbored them.\n\nPowell disliked the formulation. At a cabinet meeting the next morning, he tried to narrow the focus, declaring that America should concentrate \"first on the organization that acted yesterday.\" But Cheney pushed back. \"To the extent we define our task broadly,\" he replied, \"including those who support terrorism, then we get at states.\" On September 13, the dispute spilled into public view. \"It's not just simply a matter of capturing people and holding them accountable, but removing the sanctuaries, removing the support systems, ending states who sponsor terrorism,\" Wolfowitz declared at a Pentagon briefing. Once again Powell tried to dial things back. \"We're after ending terrorism,\" he told reporters, \"and if there are states and regimes\u2014nations\u2014that support terrorism, we hope to persuade them that it is in their interest to stop doing that. But I think 'ending terrorism' is where I would leave it and let Mr. Wolfowitz speak for himself.\"\n\nBehind the newfangled war-on-terror lingo, it was the same debate that had been playing itself out since Panama: negotiation or regime change. Wolfowitz wanted to \"end\" terror-supporting regimes; Powell wanted to persuade them to change their ways and thus restrict the war to Al Qaeda itself. Wolfowitz's strategy would extend the frontiers of American dominance, creating new clients in countries where the United States did not currently hold sway. Powell's was cheaper in both money and blood.\n\nIn the days immediately following 9\/11, the debate was about two countries: Afghanistan and Iraq. For Cheney, Wolfowitz, and Rumsfeld, it was Iraq that really mattered. For starters, they genuinely believed that Saddam had had a hand in the attack, and they didn't much care that the administration's terrorism experts thought they were nuts. (On the subject of Iraq and terrorism, Wolfowitz was used to people thinking him nuts. He believed Saddam had masterminded the 1993 World Trade Center bombing, and for a time even suspected him of orchestrating the 1995 bombing in Oklahoma City. Wolfowitz-watchers considered it an intellectual eccentricity, like an otherwise rational man who decides carrot juice can cure cancer.) Cheney, Rumsfeld, and Wolfowitz also considered Afghanistan a bad place to fight a war: the mountainous terrain made ground operations difficult, and the country was so primitive that there was barely anything worthwhile to bomb. Finally, and most important, Afghanistan wasn't a game changer. A geopolitical backwater, it lacked strategic position or critical resources (a polite way of saying oil). \"If the war [on terror] does not significantly change the world's political map,\" Rumsfeld wrote in a memo to Bush, \"the U.S. will not achieve its aim\"\u2014and attacking Afghanistan didn't do that. It was like invading Haiti: a lot of work for little reward.\n\nPowell thought they were in cloud-cuckoo-land. \"What the hell, what are these guys thinking about? Can't you get these guys back in the box?\" he moaned to Joint Chiefs of Staff chairman Henry Shelton. Cheney, Rumsfeld, and Wolfowitz wanted to go after Iraq because doing so might reshape the Middle East. Powell wanted to go after Afghanistan because that's where the terrorists were. He wanted to fight Al Qaeda; they believed that to effectively fight Al Qaeda, America must expand its hegemony in the Arab world. In his mind, it was a bait and switch.\n\nInitially, Bush split the difference. He too believed Saddam was involved in 9\/11, but he knew he couldn't prove it. He loved the idea of doing something big. As Gerson put it, \"All of his instincts tended toward a single ambition: a desire to reshape the security environment we found in the world, rather than endlessly responding to escalating dangers and attacks\"\u2014and he feared that merely getting bin Laden and his cronies was not big enough. But still, he could see Powell's point. Americans wanted bin Laden's head on a stick; Afghanistan had to be first. Anything else risked a public revolt. \"Start with bin Laden, which Americans expect,\" he told his advisers. \"And then if we succeed, we've struck a huge blow and can move forward.\" It was a shrewd insight into the politics of hubris. If America won easily in Afghanistan, then the public would grow more malleable on Iraq. After overthrowing the Taliban, overthrowing Saddam would no longer seem so hard.\n\nBut in the days after 9\/11, even overthrowing the Taliban seemed hard. In 2002, once American confidence had swelled further, toppling the regime would be remembered as easy and obvious, a mere appetizer before the Mesopotamian main course. But it didn't look that way at the time. Initially the military wanted no part of Afghan regime change, which it assumed would require tens of thousands of troops. In fact, when Shelton addressed Bush's war cabinet on September 15, two of the three military options he proposed involved no U.S. ground forces at all. Afghanistan was a \"graveyard for the interests of great powers,\" cautioned the New York Times, \"a general's nightmare and guerrilla commander's fantasy.\" U.S. allies were even more squeamish. French foreign minister Hubert V\u00e9drine warned that bin Laden had set a \"diabolical trap\" for the United States, luring it into a war it could not win. His German counterpart worried that invading Afghanistan might \"create more instability.\" When administration representatives traveled to Russia, a country that knew something about Afghan wars, a Kremlin official told them, \"With regret, I have to say you're really going to get the hell kicked out of you.\"\n\nPowell feared the Russians were right. Like his old colleagues in the military, and like the CIA, he wanted to blow up some Al Qaeda bases and lure the Taliban from their alliance with bin Laden, not overthrow them. \"It is not the goal at the outset to change the regime,\" he told a Principals meeting on September 23, \"but to get the regime to do the right thing.\" It was a lot like his debate with Wolfowitz at the close of the Gulf War. Once again Powell preferred sticking with the barbaric regime he knew to deposing it and potentially getting mired in civil war.\n\nFor the dominance conservatives, Powell was on the wrong side of history yet again. \"It is deeply troubling to see the secretary of state begin to wobble,\" wrote Charles Krauthammer in late September. \"Eleven years ago, then-President Bush overrode Powell's resistance to fighting Saddam,\" wrote William Kristol. \"Bush was vindicated in doing so. Will the current President Bush follow Powell's lead? Or will Bush lead and demand that Powell follow?\" Like Wolfowitz, both Krauthammer and Kristol would have been happy for America to go after Saddam right away, but they understood Bush's logic. If America didn't successfully overthrow the Taliban first, there would be no war in Iraq.\n\nOn Afghanistan, Powell lost. To Bush, merely lobbing missiles meant pounding sand, killing camels, Clinton stuff. \"We'll attack with missiles, bombers and boots on the ground,\" he told the NSC on September 17. \"Let's hit them hard. We want to signal this is a change from the past.\"\n\nIronically, while Bush believed he was breaking with Clinton, he was actually building on Clinton's example. America's Afghan war strategy, devised largely by the CIA, involved massive U.S. airpower, and a small number of U.S. trainers, arming and advising the Northern Alliance so they could defeat the Taliban on the ground. It was a lot like what the Clinton administration had done in the Balkans, where the United States had assisted first Croat, and then Kosovar, ground forces while NATO pounded Serb forces from the air. And it was because this \"Clinton Doctrine\" (or \"Reagan Doctrine,\" in Wolfowitz's terminology, since the Republican bloodlines had to be kept pure) had worked in Bosnia and Kosovo that Bush pushed it further in the Hindu Kush. It was another example of how success begets ambition. Commentators would later say that the 9\/11 attacks made overthrowing the Taliban inevitable, that doing less was inconceivable after the murder of three thousand on American soil. But had Kosovo turned into Vietnam, or even Somalia, doing less would have been highly conceivable. September 11 was crucial to America's decision to invade Afghanistan, of course; without the attacks, invading would never have been seriously discussed. But without the self-confidence engendered by a decade of military, economic, and ideological triumph, policymakers might well have chosen the safer and more limited military response that Powell and many in the military and intelligence community preferred. When it came to the Afghan War, September 11 was the match. American self-confidence was the dry tinder that burst into flame.\n\nAt first, everything went wrong. Starting in early October, America bombed Afghanistan for days, and then weeks, but the Taliban didn't buckle. Efforts to foment a rebellion among Pashtuns in the Afghan south failed miserably, as a key anti-Taliban exile was captured just hours after crossing with his fighters onto Afghan soil. \"Like an unwelcome specter from an unhappy past, the ominous word 'quagmire' has begun to haunt conversations among government officials and students of foreign policy, both here and abroad. Could Afghanistan become another Vietnam?\" mused a late October news analysis in the New York Times. With the Northern Alliance looking amateurish and ineffectual, and American air strikes having little apparent effect, the military began making plans for the dreaded land invasion\u2014fifty thousand troops, for starters. To Powell, it just made the Vietnam analogy all the more apt: When bombing and local allies failed, the next step was always U.S. troops on the ground. He began to protest, telling a Principals meeting that \"I'd rule out the United States going after the Afghans [on the ground], who have been there for 5,000 years.\" It was one of Powell's verbal tics. When he wanted America to stay out of someone else's war\u2014whether in Lebanon, Bosnia, or now Afghanistan\u2014he claimed the people there had been fighting for hundreds or thousands of years. It was a window into his view of the world, his sense that problems were often incorrigible and that at least in some places, not only was history not ending, but it didn't progress much at all.\n\nIt was a crucial moment, those days just before and after Halloween 2001, and it set the stage for much that was to come. The Europeans were skittish; the New York Times was talking quagmire; Powell seemed to want to cut America's losses. Bush hated quitters; his natural instinct, when challenged, was to double down. He began privately cursing the media, suggesting that they wanted him to fail. The Weekly Standard attacked Powell as defeatist and demanded that Bush send in the army. At one point Bush took Cheney aside and asked him if he still believed they would win. Cheney said he had absolute faith.\n\nThen, suddenly, the tide turned. On November 2, an American air strike destroyed a famed transmission tower that the Soviets had tried for years to hit, but never could. The smart bombs that had worked such wonders in the 1990s were finally working their magic again. Outside the town of Mazar-i-Sharif, a U.S. Special Forces soldier on horseback spotted a clump of Taliban fighters, tapped their coordinates into his laptop, and summoned an armed pilotless drone\u2014the newest gadget in America's high-tech arsenal\u2014along with a B-52 bomber. Nineteen minutes later, the Taliban fighters were dead.\n\nQuickly, the Taliban began to crumble. Between November 9 and November 12, the Northern Alliance increased its share of the country from 15 percent to 50 percent. On November 13, Afghanistan's capital, Kabul, fell. Kites, an Afghan passion banned by the Taliban as unIslamic, started to dot the sky. Indian pop music, also formerly banned, soon blared from taxi radios and boom boxes. In Kabul's main stadium, which the Taliban had used to publicly execute those who defied religious law, soccer games broke out. Women began removing their burkas; men shaved their beards. Girls, doomed under the Taliban to a life of illiterate, beastlike submission, began\u2014often for the first time in their lives\u2014to attend school. By Christmas, Afghanistan had sworn in a new leader, Hamid Karzai, not a Northern Alliance warlord, but an apparently decent and modern man with brothers who ran restaurants in Chicago, San Francisco, Baltimore, and Boston, a democrat. His new government did not merely replace tribe with tribe; it represented Afghan's ethnic diversity. Karzai himself was a Pashtun. In January, at Bush's State of the Union address, he sat next to first lady Laura Bush, along with a female member of the new Afghan cabinet.\n\nIt was, in a sense, the culmination of all the success America had achieved since 1989. Militarily, the accomplishment was breathtaking, exceeding even what the United States had done in Panama, the Gulf, Bosnia, and Kosovo. From a standing start, America had gone to war in a forbidding, landlocked country half a world away, a legendary graveyard of empire that had brought the mighty U.S.S.R to its knees. It had done so with a ground force of 316 Special Forces troops and 110 CIA agents. And in just over a month, it had brought down the enemy regime at a total cost of $1 billion to $2 billion, less than a single B-2 bomber. In the wake of Afghanistan, suggested an article in the conservative magazine National Review, \"we face the prospect of being able to dominate the world at the touch of a button\u2014and at virtually no cost in casualties.\" It was a dream come true.\n\nIdeologically, the victory was just as heady. Even more than Panama or the Balkans, Afghanistan was a place that by history, culture, location, and economics seemed condemned to despotism, yet its people were embracing democracy with tears of joy. Afghan's new leaders rushed to tell Americans the same thing that the Panamanian priest had told them more than a decade before: that their real country was a free, decent, modern place, concealed from the world by a small, monstrous faction. The Taliban were not Afghanistan's true face; they were its mask. And America was not Afghanistan's occupier, bringing alien, unwelcome ideas, but its liberator, allowing it to become, once again, its truest self. A country that only weeks before had been marching backward toward the seventh century now seemed like just another young democracy, hopping onto Fukuyama's escalator, wobbling a bit, but peering up toward the light.\n\nGeopolitically, the war extended American power deep into Central Asia, just as the Gulf War had in the Middle East and Bosnia had in Eastern Europe. By 2002, not only was the U.S. Army deployed across Afghanistan, but American troops and planes were nestled at bases in Kazakhstan, Uzbekistan, Tajikistan, and Kyrgyzstan, in the former U.S.S.R.\n\nFor the Bush administration and its intellectual allies, who only days before had been accused of leading the nation into a second Vietnam, vindication was sweet and reinforced one of the central tenets of dominance conservatism, not to mention George W. Bush's life: that glory goes to those with ambition and faith, not the faint of heart. \"Critics of the U.S. war in Afghanistan have been wrong about virtually everything,\" crowed an article in the Weekly Standard. At a press conference in late November, Rumsfeld taunted the press, declaring that from the beginning the Afghan War had gone as planned, but that thanks to negative reporting, \"It looked like we were in a\u2014all together now\u2014QUAGMIRE!\" In the wake of Bush's success, the press grew more quiescent; a Saturday Night Live skit showed reporters so intimidated by Rumsfeld that they would not ask any questions at all. In December, People named the sixty-nine-year-old defense secretary one of the sexiest men of the year, after which Bush dubbed him \"Rumstud.\"\n\nPolitically, Bush looked like quite a stud himself. September 11 had sent his approval rating into the stratosphere, as Americans rallied around the flag, but it was the Afghan victory that kept it there. By the fall of Kabul, according to Gallup, he enjoyed the support of 87 percent of Americans. On September 10, he had been an accidental president, widely considered out of his depth. Now he dominated Washington as few presidents ever had.\n\nAs Bush grew stronger, so did the presidency itself. As Gerald Ford's thirty-four-year-old chief of staff in the dying days of Vietnam, Dick Cheney had run a White House under siege, as the hubris of toughness collapsed amid an orgy of congressional and journalistic recrimination. Deeply scarred by the ordeal, Cheney had devoted his career to the restoration of the imperial presidency of the early cold war. Now 9\/11 and Afghanistan gave him the power to make that dream come true. In January 2002, with no input from Congress, he steamrolled Powell and convinced Bush that the combatants the United States had picked up on the Afghan battlefield should not enjoy the protections of the Geneva Convention. In February, at Cheney's insistence, the White House went to federal court to deny the General Accounting Office (GAO) access to records of an energy task force he had led. The Bush administration, wrote the GAO's head, \"seeks to work a revolution in separation of powers principles, one that would drastically interfere with Congress's essential power to oversee the activities of the executive branch.\" Bush and Cheney won the case.\n\nBy every measure, it looked like an awesome performance. The war in Afghanistan was not over, to be sure. In a rugged mountainous region called Tora Bora, roughly fifteen hundred Al Qaeda fighters had escaped across the border into Pakistan. They would need to be mopped up. But few worried about that in the thrilling winter of 2001. \"The initial reaction to 9\/11,\" Gerson later remembered, \"had been successful beyond expectation, and perhaps beyond precedent.\" And Bush was determined not to rest on his laurels, as his father had done after the Gulf War. There was even greater glory to come. \"They won in Afghanistan when everybody said it wouldn't work, and it's got them in a euphoric mood of cockiness,\" trembled one Powell ally, \"and anyone who now preaches any approach of solving problems with diplomacy is scoffed at. They're on a roll.\"\n\n## CHAPTER EIGHTEEN\n\n## THE ROMANTIC BULLY\n\nWhy did America invade Iraq? The Bush administration's answer went something like this: Before September 11, Saddam Hussein was a problem. He had attacked his neighbors; he was building weapons of mass destruction (or so it seemed); he was defying UN resolutions; he cavorted with terrorists. Then came 9\/11, and America's leaders peered into the abyss. If terrorists could kill three thousand Americans with boarding passes and box cutters, imagine what they could do with weapons of mass destruction. The anthrax attacks a month later underscored the danger. So the Bushies asked the obvious question: Who might supply terrorists with WMD? Who had the motive and the means? Saddam!\n\nAmerica invaded, in other words, out of fear. September 11 showed that the terrorist threat was graver than previously understood, and so America took graver steps to confront it. It's not hard to see why Bush officials favored this argument: It made the Iraq War sound defensive. But it ignored something crucial: Fears don't exist in isolation. They tend to rise and fall depending on what people think they can do about them. In Robert Kagan's words, \"A man armed only with a knife may decide that a bear prowling the forest is a tolerable danger.... The same man armed with a rifle, however, will likely make a different calculation.\"\n\nHad September 11 happened in 1977, or 1983, or 1989, America would have been more like the man with the knife. Back then, Vietnam still haunted discussions of military force; America's post\u2013cold war triumphs had not yet exorcised the ghost. In the 1970s and '80s, nothing in America's recent experience suggested that America could successfully invade and occupy a large, distant country like Iraq (especially after it had just invaded and occupied another large, distant country: Afghanistan). After all, the United States hadn't even managed to rescue the hostages in Iran in 1980 or protect its Marines in Beirut in 1983. And if America's leaders doubted their capacity to rid the world of Saddam, they would have likely hesitated before describing him as a mortal threat. Why stoke public fear unless you can put it to rest?\n\nBush, by contrast, stoked public fear relentlessly. In the run-up to war with Iraq, he claimed that while deterrence had worked against the Soviet Union and China, it couldn't work against dictators like Saddam. As an argument about threats, that made little sense: There was little in Saddam's record to suggest that he was less deterrable than Stalin or Mao. But Bush encouraged Americans to believe that the threat was greater because he thought he could eliminate it. During the cold war, America had chosen deterrence because preventive war against the Soviet Union or China was too frightening to seriously contemplate. Preventive war against Iraq, however, was not so frightening, especially given America's recent run of military success. Top Bush officials thought America could invade and occupy Iraq with relative ease. And so they depicted Saddam, and his presumed weapons, as an intolerable, undeterrable threat.\n\nThe timing of Bush's decision to go to war illustrates the point. In October 2001, when Afghanistan looked like a quagmire, there had been little talk inside the White House about a second war. When Rumsfeld raised the issue at an NSC meeting on October 9, he was rebuffed. But when the Taliban fell in mid-November, the mood quickly changed. Eight days after the capture of Kabul, Bush told Rumsfeld to develop a strategy for toppling Saddam. By February 2002, the U.S. military was shifting forces from Afghanistan to the Persian Gulf. The Iraqi threat had not grown as a result of America's apparent victory over the Taliban, but the Bush administration's self-confidence had.\n\nThat confidence came in three parts, the first\u2014and least appreciated\u2014of which was economic. In 2002, as the White House prepared for a second war, the budget surplus of the 1990s turned to deficit. But the dominance conservatives, like the authors of NSC 68, did not fear deficits. After all, Reagan had racked up huge ones fighting the cold war, and although Paul Kennedy and others had predicted catastrophe, those deficits had disappeared by century's end. For the Bush administration's \"neo-Reaganites,\" the moral of the story was that deficits were a small and temporary price to pay for vanquishing America's foes. At a meeting in November 2002, when Secretary of the Treasury Paul O'Neill warned that America was \"moving toward a fiscal crisis,\" Cheney interrupted. \"Reagan,\" he declared, \"proved deficits don't matter.\"\n\nRather than raising taxes to reduce the deficit and pay for America's two wars, the Bush administration cut them in 2002, and then again in 2003. The tax cuts, it insisted, would largely pay for themselves as the economy boomed, and so would Iraq, whose vast oil wealth would fund its own postwar reconstruction. When White House economic adviser Lawrence Lindsey suggested that toppling Saddam might cost $200 billion, he was reprimanded, then fired. Top administration officials derided his suggestion as ludicrous. After all, toppling the Taliban had cost less than 1 percent of that.\n\nIf the Bushies were brimming with economic confidence, their military confidence was equally high. It wasn't just that America had spent the last decade winning wars\u2014since Bosnia, it had been winning wars with barely any U.S. ground troops. From this, Donald Rumsfeld drew a striking conclusion: The era of massive land armies was over. The future of war belonged to the lean, fast, and ultrahigh-tech. In his mind, Afghanistan proved the point. Before the war, General Tommy Franks, head of Central Command, had told him that overthrowing the Taliban would require tens of thousands or even hundreds of thousands of GIs, a hulking force that would take nine months to fully deploy. Instead the CIA had parachuted in one hundred or so agents carrying laptops and suitcases of cash, and, with help from the U.S. Air Force, the Northern Alliance, and a few hundred Special Forces troops on horseback, they had taken down the Taliban in little more than a month.\n\nIt was more than the hypercompetitive Rumsfeld could bear. His doctrine had apparently been vindicated, but his CIA rivals were getting the glory. A haughty man in the best of times, he developed an epic disdain for the army's top brass, whom he considered timid, conformist, and flat-out dumb. He felt special contempt for Franks, no one's idea of an intellectual, a man whose favorite movie was The Nutty Professor. As a result, in early December, when Franks said invading Iraq would require close to four hundred thousand troops, Rumsfeld shot him down, remarking that \"I'm not sure that that much force is needed given what we've learned coming out of Afghanistan.\" Over the next fifteen months, he forced the troop number down by almost a third.\n\nWhen William Crowe or Colin Powell ran the Joint Chiefs of Staff, America's generals had not been so easily bullied. But by 2002, after more than a decade of successful wars, the civilian-military balance of power had shifted. The military's warnings no longer made civilians' teeth chatter. \"America's senior generals have opposed nearly every intervention that the United States has undertaken in the post\u2013cold war era,\" wrote William Kristol and the New Republic's Lawrence Kaplan. (The implication was too obvious to require spelling out: Every time, the generals had been wrong.) In the summer of 2002, Bush was seen toting a book called Supreme Command, which argued that the greatest wartime leaders regularly overruled their generals. And with his backing, Rumsfeld developed a relationship with the uniformed military that approximated the relationship between a batterer and his spouse. \"Shut up. I don't want any excuses. You are through and you'll not have time to clean out your desk if this is not taken care of,\" he barked at one admiral. Asked once by the president for his opinion, Franks replied jokingly, \"I think exactly what my secretary thinks, what he's ever thought, what he will ever think, or whatever he thought he might think.\" In early 2002, when Bob Woodward asked Chairman of the Joint Chiefs Richard Myers to describe Rumsfeld's personality, the highest-ranking general in the U.S. military simply buried his head in his arms.\n\nFor Rumsfeld and Cheney, economic and military confidence was enough. They did not much care whether Iraq became a democracy after Saddam fell, as long as it served America's will. Their foreign policy strategy, like their management style, was directed less at hearts and minds than at the lower regions of the anatomy, in the belief that if intense pressure was applied there, heart and mind would surely follow. Where \"there was no room for idealism or sentimentality,\" remarked one White House lawyer, \"you'd find the vice president there.\"\n\nBut for Bush, idealism did play a role. For him, ideological confidence\u2014his belief that Iraq could become a democracy and a model for the region, a shining Arab city on a hill\u2014was as important as military and economic confidence. In this regard, the president was a more contradictory figure than his vice president and secretary of defense. On the one hand, he believed as they did in establishing dominance and inspiring fear. Bush may have become a committed Christian in middle age, but he had been a bully all his life. You could see it in his habit of assigning nicknames, which established a subtle supremacy over those around him. (Bush's subordinates were never encouraged to give him nicknames in return.) It was evident during his dad's presidential campaigns, in which he played the role of enforcer. And even family members quietly testified to his thuggery. Once during an intramural basketball game at Harvard Business School, W. had brazenly elbowed the opposing captain in the mouth, nearly sparking a brawl. Many years later, the recipient of that blow met Jeb and recounted the incident. George, his younger brother explained, \"truly enjoys getting people to knuckle under.\"\n\nSo it was little surprise that Bush attributed 9\/11 to the fact that America wasn't sufficiently frightening. But that was only half the story. If the president was part bully, he was part romantic, too. Cheney and Rumsfeld were not huggers, and they did not openly weep. But Bush did both, often. As public policy, compassionate conservatism may have been thin gruel, but for Bush emotionally, it was real. Running for reelection as Texas governor, he had encountered a skinny fifteen-year-old black kid named Johnny Baulkmon in a juvenile prison. \"What do you think about us?\" Baulkmon asked, catching the governor off guard. \"I think you can succeed,\" Bush responded. \"The state of Texas still loves you all. We haven't given up on you,\" and as he spoke, he almost began to cry. For Bush, the author Robert Draper notes, the incident produced \"a strange euphoria.\" For weeks he spoke about it to anyone who would listen, not only in public\u2014where the encounter became part of his stump speech\u2014but also in private. His imagined communion with the kid became, in his own mind, the raison d'\u00eatre of his campaign. (For the kid, the experience was less revelatory. Asked about the experience years later, Baulkmon\u2014now an adult criminal\u2014said Bush \"doesn't care about anything but himself. He's complete trash, a horrible evil person.\")\n\nAfter 9\/11, Bush began to feel a bond with people suffering oppression in the Muslim world similar to the bond he had felt with Johnny Baulkmon. It started with the heartrending scenes from liberated Kabul. After that, Bush started seeking out Iraqis who had endured Saddam's horrors, and upon hearing their stories, he often broke down. Bush's critics would later call his freedom rhetoric cynical, a pretext hatched once it became clear that Saddam had no WMD. But many of those close to the president saw it the other way around: For Bush personally, they believed, WMD was the pretext for war and democratic transformation the real cause. \"For Bush,\" wrote Press Secretary Scott McClellan, \"removing the 'grave and gathering danger' that Iraq supposedly posed was primarily a means for achieving the far more grandiose objective of reshaping the Middle East as a region of peaceful democracies.\" Cheney told confidants in March 2003, \"Democracy in the Middle East is just a big deal for him [Bush]. It's what's driving him.\" And a White House aide told the Standard in February that Iraqi democracy is \"what animates him. It's on his heart, his mind, his agenda. This is what he wants to talk about.\" It is certainly true that Bush didn't stress democracy as a public justification for war until after Saddam was gone, but that may be more because he believed it wouldn't convince the American people than because it didn't convince him.\n\nThat Bush believed Iraq capable of dramatic transformation was not surprising; dramatic transformation was the story of his life. His victory over alcohol had not been slow, messy, and complex; he had freed himself in one epic act of will. That had been the template for compassionate conservatism. When it came to poverty, Bush showed scant interest in structural conditions and incremental improvements. He believed people could radically improve their lives if they changed their hearts. He believed the vehicles for that change would be religious charities, the same \"armies of compassion\" that had changed his life. And now, after 9\/11, he envisioned the U.S. military as the greatest army of compassion of all. In 2002, Bush exhibited no more curiosity about the cultural and historical roots of Iraqi tyranny than he had exhibited about the cultural and historical roots of American poverty. Rather, he identified with ordinary Iraqis suffering under Saddam and believed them capable of transforming their lives, just like Johnny Baulkmon\u2014and like Bush himself\u2014if only someone gave them a chance.\n\nFor a conservative, Michael Gerson noted, Bush had a view of human nature that was uncommonly bright, and his brand of Christianity striking untroubled by original sin. Like the New Left activists who rebelled against the ethic of toughness in the 1960s, and like his hero, Ronald Reagan, Bush believed that people were generally better than their governments. And he believed this was true of people of all nations and creeds, from the inner-city criminal to the Iraqi peasant. Bush's critics sometimes called him a Christian crusader, leading a \"clash of civilizations\" against Islam. But in reality, Bush's faith made him a universalist, a fervent believer that all God's children were basically the same, and basically good. \"The human heart desires the same good things, everywhere on Earth,\" he declared in February 2003. \"In our desire to be safe from brutal and bullying oppression, human beings are the same. In our desire to care for our children and give them a better life, we are the same. For these fundamental reasons, freedom and democracy will always and everywhere have greater appeal than the slogans of hatred and the tactics of terror.\"\n\n\"Always and everywhere\"? In the age of Hitler and Stalin, when democracy looked like a fading force and men like Reinhold Niebuhr and Arthur Schlesinger understood that totalitarianism also had a claim on the human heart, declaring that freedom was humanity's inevitable choice would have sounded bizarre. But Bush's personal optimism mingled with the ideological optimism of an era in which country after country had overthrown tyranny. As he prepared for war with Iraq, Bush peppered his speeches with the language of democratic peace theory and of Fukuyama's end of history. Just as democracies did not war with each other, he argued, democracies would not incubate terrorism. And democracy was possible in the Middle East, because democracy was the universal yearning of all humanity. \"The 20th century ended with a single surviving model of human progress,\" Bush told cadets at West Point. \"For most of the twentieth century, the world was divided by a great struggle over ideas: destructive totalitarian visions versus freedom and equality,\" declared his administration's National Security Strategy. \"That great struggle is over.\"\n\nThe war with Iraq had not yet begun, but ideologically the great global struggle was already over. The implication was clear. Ultimately America could not fail in Iraq, because American ideals were the ultimate destination of all humankind. History itself was moving America's way; George W. Bush was simply giving it a push.\n\nHad Bush been a more introspective man, he might have pondered the tension between the two sides of his nature: the bully who wanted to frighten Arabs and Muslims into submission and the romantic who wanted to liberate them from bondage. Had he known more about the Middle East, he might have recognized that to people with a history of imperial subjugation, whose parents and grandparents had been subjugated by idealists carrying Maxim guns, this twin mission would look both familiar and ominous.\n\nBut there is no evidence that he did. In Bush's own life, after all, virtue and power had gone hand in hand; when he quit drinking he became a better person and a more successful one, too. He saw recent American history the same way. Under Reagan, he believed, America had done well by doing good. From Latin America to East Asia to Eastern Europe, the United States had pursued dominance and democracy at the same time; now America just needed to apply that template to the Middle East. The older, darker conservative view, born in the totalitarian age and embodied by Jeane Kirkpatrick and Irving Kristol\u2014which stressed the limits of good intentions and the incorrigibility and inscrutability of far-off lands\u2014had less resonance in an era that had witnessed so much moral and geopolitical triumph. And it was particularly alien to the sunny, unreflective mind of George W. Bush.\n\nThat's why Chalabi and his Iraqi National Congress proved so useful. They told the Bushies that post-Saddam Iraq would be pro-American, even pro-Israeli, and yet democratic as well. They squared the circle that Bush needed squared: between American dominance and Iraqi freedom. By depicting himself as the heir of Lech Walesa and Vaclav Havel, Chalabi allowed the \"neo-Reaganites\" to imagine Iraq as the reincarnation of Poland, Czechoslovakia, or at least Romania after the cold war: countries where democratic transformation and pro-American transformation had occurred in tandem. He allowed them to disregard warnings that in the Arab world, a region with fewer democratic traditions than Eastern Europe and much fiercer hostility to the United States, making Iraq an American client would be hard, making it a stable democracy would be harder, and making it both would be hardest of all.\n\nThe Iraqi exiles\u2014who had been in Washington long enough to know what the Bushies wanted to hear\u2014insisted not only that all this was possible, but that it would be cheap. In the run-up to war, the administration and its supporters sometimes cited the examples of Germany and Japan, where America had toppled hostile dictatorships and built pro-American democracies in their place. But those analogies, if taken seriously, would have implied a long and costly postwar occupation\u2014exactly what the Bush administration was insisting Iraq would not require. The more influential, and more comforting analogy, was to the Reagan Doctrine, which Wolfowitz had written about so much in the 1990s. In this model, America didn't need to occupy and govern Iraq. It just needed to install its allies, who would do the occupying and governing themselves. In the 1980s, those allies had been the Contras and the Afghan mujahideen. In the 1990s, they had been the Bosnians and the Kosovars. And after 9\/11, they had been Hamid Karzai and the Northern Alliance. As Bush himself told a reporter, America would do in Iraq \"the same [thing] we did in Afghanistan\u2014it's a blueprint, a model.\"\n\nThe Afghan example was particularly reassuring because America seemed to have birthed a fledgling democracy with very few troops, thus reconciling Bush's dream of democratic transformation with Rumsfeld's hatred of nation building. And with Chalabi's help, top Bush officials hoped to do the same in Iraq. In February 2003, when General Eric Shinseki famously warned that occupying Iraq would require several hundred thousand U.S. troops, Wolfowitz said he was wrong because \"we are training free Iraqi forces to perform functions of that kind,\" by which he meant the Iraqi National Congress. Wolfowitz and other top Pentagon civilians hoped to establish the INC as a provisional government-in-exile even before the war began, with its own U.S.-trained army, and then hand it the reins soon after Saddam fell. \"[W]e had in mind our recent experience in Afghanistan,\" explained Douglas Feith, \"where the United States immediately recognized an interim government of Afghans and never became an occupying power.\"\n\nFor Rumsfeld and Cheney, the INC's democratic bona fides were secondary. It mattered less that they were democrats than that they were ours. But for Wolfowitz, who had become captivated by the idea of Islamic democracy during his time as ambassador to Indonesia and after watching Iraq's Shia rise up against Saddam in 1991, those bona fides mattered a lot. More than Rumsfeld or Cheney, he shared Bush's passion for spreading freedom, and because he did, he gained influence beyond his station. Wolfowitz, noted the Weekly Standard's well-connected White House reporter Fred Barnes, did more to shape Bush's post-9\/11 thinking than any other adviser. They were an odd pair: the bookish, Jewish defense intellectual and the swaggering, faith-based president. But they had both come to see dictatorship as unnatural, alien to the basic yearnings of humankind. In his discussion of Iraq, Wolfowitz resisted the terms occupation and nation building, preferring liberation instead. It was a revealing choice. Nation building implies constructing something new; liberation implies freeing something that already exists. \"'Export of democracy' isn't really a good phrase,\" Wolfowitz told an interviewer. \"We're trying to remove the shackles on democracy.\" Wolfowitz's faith in liberation\u2014his belief that if repressive structures were torn down, humanity's innate goodness would break free\u2014put him philosophically closer to the New Left of the 1960s, which also believed that human beings were fundamentally superior to the societies in which they lived, than to older conservatives like Kirkpatrick and Irving Kristol. For Wolfowitz, it was as if democracy already existed in the hearts and minds of ordinary Iraqis. This nascent democracy sat captive within the walls of Saddam's prison-state. All America had to do was turn the key.\n\nFor Colin Powell, by contrast, Saddam's Iraq was not a prison; it was a vase. It would not unlock, releasing the democracy inside; it would shatter, bloodying America's hands. When discussing Iraq, Powell liked to cite the \"Pottery Barn rule\": You break it, you own it. (In fact, Pottery Barn had no such rule and accused Powell of slandering the company and driving away business.) To Wolfowitz and Bush, Saddam was Iraq's jailer; to Powell, he was its glue.\n\nBy the summer of 2002, Powell was no longer the commanding figure he had been when the administration began. He was still popular among his foreign counterparts, but admiration was now tinged with pity. To get something done with the Americans, diplomats whispered, you had to deal with the barbarians at the White House. That's where the power lay.\n\nPowell doubted that Rice, whom he considered far too eager to please, was telling Bush about the dangers of war, and he was sure that Cheney wasn't. In the conservative press, the mantra repeated again and again was that it was \"hard to imagine\" how toppling Saddam could leave America and Iraq worse off. (It was another revealing turn of phrase. People who had grown accustomed to seeing dictatorship give way to democracy, not chaos, and had little memory of an American war gone horrifically wrong, found such things \"hard to imagine.\") But Powell could imagine it; Vietnam and Lebanon were like shrapnel lodged in his brain. In early August 2002, on a flight back from Asia, he resolved to tell Bush everything that might go wrong.\n\nOn the evening of August 5, he dined with Bush and Rice at the White House. \"When you hit this thing,\" he warned, \"it's like crystal glass.... [I]t's going to shatter. There will be no government. There will be civil disorder.\" He cited Iraq's lack of democratic traditions, the war's potentially massive economic costs, and its destabilizing impact on America's Arab allies. It was a grim litany, a clear call for Bush to reconsider his path. And then Powell reached the punch line, the advice he had been waiting months to give: Ask for UN support. Huh? It didn't make sense. Many of the horrors Powell was outlining were intrinsic to any invasion of Iraq; they were likely even if the UN did give its blessing. Perhaps Powell believed the Security Council would resist the push for war and make Bush back down\u2014in other words, that the French, Russians, and Chinese would do his work for him. But by making an argument about how to go to war, not whether to go, Powell implicitly conceded the central point.\n\nIt was eerily similar to his encounter with Bush's father before the Gulf War. Once again he had summoned his courage, built up a head of steam, and then swerved at the last minute to avoid a collision. Once again he had refused to go baldly into opposition, to put his reputation, and his job, on the line. Those close to him believed it just wasn't in his nature. His great strength\u2014which set him apart from Cheney, Wolfowitz, and Bush\u2014was that he lacked certainty. He was not an ideologue; he distrusted abstractions; he saw shades of gray. But precisely because he did lack certainty, he could not say with total conviction that the war was a mistake. It must have crossed his mind that he had warned of disaster three times before\u2014before the Gulf War, Bosnia, and Afghanistan\u2014and each time disaster had not struck. Three times presidents had ignored his advice, and three times everything had worked out fine, both for America and for him. If a decade of military triumph had made the hawks too willing to disregard his warnings, perhaps it had made him too comfortable seeing them disregarded.\n\nIf Powell did not go baldly into opposition, neither did the leaders of the Democratic Party. Before the Gulf War, the Democratic leadership had lined up in loud and virtually unanimous opposition. But by 2002, after a decade of military triumph, opposing an American war seemed like an excellent way to kill your political career. Everyone knew the parable of Georgia Senator Sam Nunn, who by forcefully opposing the Gulf War had sunk his chances of reaching the White House. Nor did it escape notice that of the ten Senate Democrats who had backed the Gulf War, two of them\u2014Joe Lieberman and Al Gore\u2014had ended up on the 2000 Democratic presidential ticket. When the resolution authorizing the Iraq War came before Congress in October 2002, Democrats with safe seats and without national ambitions generally voted no. But most of the party hierarchy\u2014and almost every Democrat thinking seriously of running for president, from John Kerry to John Edwards to Joseph Biden to Hillary Clinton\u2014voted yes.\n\nDuring the debate over the Gulf War, congressional Democrats had endlessly invoked Vietnam. But this time, even though America was contemplating a far more ambitious mission, Vietnam analogies were scarce. In his 1991 speech opposing the Gulf War, Massachusetts Senator Kerry, who had launched his political career protesting Vietnam, mentioned it ten times. In his 2002 speech supporting war against Iraq, he mentioned Vietnam only once.\n\nFor many Washington Democrats in 2002, there was something pass\u00e9 about citing Vietnam. It was the foreign policy equivalent of wearing tie-dye. It pegged you as a relic of a bygone age, too traumatized by the past to see that military force now worked. This was particularly true among younger Democrats. Fifty-eight percent of House Democrats ages forty-five or younger voted to authorize the Iraq War, compared to only 35 percent over forty-five. (In the Senate, there were only two Democrats younger than forty-five, and both voted yes, compared to 56 percent of older Senate Democrats.) Age forty-five was a useful dividing line because someone who was under forty-five in 2002 would have been under fifteen in 1972, meaning they would likely have been too young to be directly shaped by Vietnam. Even among older Democrats, who did remember that war, its impact had been dulled by America's post\u2013cold war military success. But among younger Democrats, who had seen nothing but military success, Vietnam's impact was fainter still.\n\nAs a result of their experience, younger Democrats were also particularly inclined to believe that military force and liberal values could go hand in hand. That faith had begun during Panama and the Gulf War, when liberals learned that the American military could shoot straight, that it could do more than napalm villages; it could win wars. It had grown during Bosnia and Kosovo, when liberals saw that not only could America win, but it could win in liberalism's cause, that the fight for human rights truly could be waged through the barrel of a gun.\n\nFor many liberal intellectuals in the 1990s, Sarajevo had been the perfect city: cosmopolitan, multi-ethnic, secular, and fighting for its life. And Milosevic's Serbs had been the perfect enemy: racist, sexist, bloodthirsty. Fascist! The word gained currency on the Clinton-era left; it evoked a spirit of World War II\u2013era leftist hawkishness long thought dead. Then came 9\/11. Now another multi-ethnic, cosmopolitan, culturally left-wing city was under attack, and it was the very city where many of the liberal hawks lived: New York. Like Sarajevo, it had been gouged by people in love with purity, willing to kill to prevent the mongrelization of races and religions, the very things liberals loved. If liberal hawks had seen Milosevic as the face of a resurrected European fascism, climbing out of history's grave, they coined a parallel term to describe Al Qaeda and the Taliban: Islamofascism. Although later associated with the neocons, the term was actually coined by Paul Berman and Christopher Hitchens, two of the most eloquent left hawks. Its function was to explain that America's new terrorist enemy\u2014like its Balkan enemy a few years earlier\u2014resided on the ideological right, and that fighting it was thus more than kosher; it was the great liberal duty of the age.\n\nWhen liberal hawks said that America was fighting fascism, they meant it in part as a rebuke to Fukuyama. Berman attacked \"the deluded, triumphalist atmosphere of 1989,\" when Fukuyama declared that democracy had won. Totalitarianism, Berman insisted, had never really died; Americans had just failed to recognize it in its new, Islamic garb. But Berman and the other liberal hawks owed more to Fukuyama than they cared to admit. Al Qaeda and the Taliban may indeed have been totalitarian, but unlike the totalitarianisms of the mid-twentieth century, they posed no serious ideological challenge to democracy. The Taliban was a mud-hut pariah regime, and while bin Laden enjoyed some prestige in the Muslim world for having bloodied the American Goliath, his vision of society stirred few Middle Eastern hearts and minds. Barely anyone believed that Al Qaeda and the Taliban offered an economic model that could outperform democratic capitalism, as many had believed about fascism and communism during the Depression. The jihadists were not only militarily weak, they were ideologically weak. Like Milosevic, they were totalitarians in a post-totalitarian age\n\nSome liberal hawks tacitly acknowledged this. As Berman himself wrote, in \"the revolutions of 1989...the notion that one or another race or culture or religion is hopelessly allergic to liberal ideas\u2014this notion did pretty much explode.\" In other words, America's new anti-totalitarian struggle differed fundamentally from the struggles against Nazism and Stalinism because 1989 had shown that liberal democracy could penetrate every society on earth. Mid-century intellectuals like Niebuhr and Morgenthau had not been nearly so confident that democracy was the world's universal creed. They had hoped merely to contain Soviet totalitarianism and perhaps soften it, but held out little hope that it could be wiped from the earth. Berman and the liberal hawks, by contrast\u2014like Bush and Wolfowitz\u2014believed in America's ideological dominance; they saw their anti-totalitarian struggle as a kind of ideological mopping-up operation, with the ultimate outcome preordained by the trajectory of history itself. The war on terror was less a sequel to the anti-totalitarian struggles of the twentieth century than an epilogue. There were a few more pages to write, some new characters and plot twists, but everyone already knew how the story would end.\n\nFor the liberal hawks, like the dominance conservatives, Afghanistan added to the ideological swagger. Before the war, some left-wing doves had warned that intervening would bring military disaster and moral horror. Noam Chomsky, the anarchist-linguist who decades earlier had invoked Randolph Bourne to oppose Vietnam, warned that if America attacked the Taliban millions might die. But America attacked, and instead of mass murder there was celebration, and an embryonic democracy. As suspected, the new Islamic totalitarianism was not very strong at all.\n\nThus emboldened, many liberal hawks urged war against Saddam. As in Bosnia, the UN\u2014whose weapons inspectors had left Iraq in 1998\u2014seemed feckless. So once again liberal hawks turned to American power. \"Multilateral solutions to the world's problems are all very well, but they have no teeth unless America bares its fangs,\" wrote Michael Ignatieff, another influential liberal hawk. Or as one younger liberal foreign policy hand put it, \"I can't say with a straight face that it's fine to go around the UN for Kosovo and not do it in Iraq.\"\n\nBut Iraq was not Kosovo, or even Afghanistan. In the Balkans, the casus belli had been genocide, an unfolding moral emergency. In Afghanistan, whatever the human rights side benefits, the casus belli had been self-defense. In Iraq, by contrast, not only was there no imminent threat; there was no genocide, either. Some liberal hawks argued that Saddam's rule constituted a moral emergency in and of itself. But by that standard so did Burma, North Korea, Zimbabwe, and a host of other nasty dictatorships. Intellectually, the liberal hawks who went from backing war in the Balkans and Afghanistan to backing war in Iraq were not walking a treadmill; they were climbing a ladder.\n\nSo it was that a significant portion of the left, whether because of political ambition or moral ambition or both, waved the Bush administration on as it sped toward war. And so did many on the older, anti-dominance right. Jeane Kirkpatrick, now seventy-six years old, privately worried that trying to violently remake Iraq was utopian, and would bring not freedom, but chaos. But she never said so publicly, where it might have made a difference. She was personally close to the Cheneys and to many other leading hawks. Perhaps she thought public dissent would give aid and comfort to the enemies of her friends, or perhaps, after a decade of victorious wars, she, like Powell, had lost the confidence of her convictions. For whatever reason, she not only failed to publicly object; she flew\u2014at the Bush administration's request\u2014to Geneva to defend the war before the UN Human Rights Commission. Later she insisted that she had not defended Bush's decision to invade Iraq, only his right to do so under international law. It was a subtle distinction, and utterly lost in the clamor for war.\n\nIrving Kristol, now eighty-two, said even less. With his son leading an assault on the foreign policy axioms that he had espoused for much of his life, the man sometimes called the godfather of neoconservatism wrote barely a single word about what many were calling the neocon war. One writer heard him mutter that his son was trying to turn George W. Bush into Napoleon. Others heard rumors that he was privately critical of what the Standard wrote. But publicly, he said virtually nothing at all.\n\nAnother father also held his tongue. In August 2002, George H. W. Bush's former national security advisor and close confidant, Brent Scowcroft, publicly denounced the impending war. Scowcroft would later claim that the elder Bush shared his views, but the elder Bush himself stayed silent. In January 2003, the former first lady, Barbara Bush, approached ex-Democratic senator David Boren at a dinner and asked, \"Are we right to be worried about this Iraq thing?\" Boren answered yes. \"Well, his father is certainly worried and is losing sleep over it,\" she replied. \"He's up at night worried.\" Boren suggested that the former president talk to his son, but she answered, \"He doesn't think he should unless he's asked.\" For his part, W. told Fox News that he didn't frequently consult his father on foreign policy, because his dad didn't have the latest intelligence.\n\nOne conservative who did publicly object was Francis Fukuyama, the Daedalus of the dominance age, inventor of a doctrine that was soaring beyond his control. In 1998, he had signed a letter\u2014along with William Kristol, Robert Kagan, Elliott Abrams, and Paul Wolfowitz\u2014urging that America pursue regime change in Iraq. But in 2002, when they went a step further\u2014urging that America pursue regime change with U.S. troops\u2014he grew skittish. The Iraq War, Fukuyama warned in December 2002, is \"an immensely ambitious exercise in the political re-engineering of a hostile part of the world,\" exactly the kind of exercise that once made conservatives tremble. \"The United States is not good at either implementing or sticking to such projects over the long run,\" he added, picking up one of Irving Kristol's favorite themes. And \"it is not at all clear that the American public understands it is getting into an imperial project.\" Eventually, Fukuyama argued, liberal democracy could sprout in the Middle East, as it had in other formerly authoritarian parts of the world. And when that happened, America would be safer. But that growth would have to be organic. Spilling Iraqi blood in a preventive war, he insisted, would not fertilize the soil.\n\nFukuyama did not just publish these arguments; he said them to Paul Wolfowitz's face. The two men had been friends for thirty-five years, since Fukuyama's days as an undergraduate at Cornell. Wolfowitz had given Fukuyama his first government job in the 1970s, then hired him again during the Reagan years, and later brought him to teach at Johns Hopkins's School of Advanced International Studies, where Wolfowitz was dean. After 9\/11, Wolfowitz convened three study groups to make suggestions for the broad direction of the war on terror, and asked Fukuyama to head one. One day in January 2003, Fukuyama went to an office building in Arlington, Virginia, to present his findings. Jihadist terrorism, he argued, was not another great totalitarian foe like communism or fascism, and America should not treat it as such. The important thing was not to overreact, not to take military action that alienated Muslims, thus strengthening Al Qaeda's inherently weak hand. Instead America should rely on diplomacy, intelligence gathering, law enforcement, and patience. History was on its side, but history could not be rushed.\n\nFukuyama was arguing against making the war on terror a grand, heroic project, a successor to the cold war; he was urging small ball. It was an assault on the basic premises underlying Bush and Wolfowitz's war on terror, phrased in the language of a more cautious conservatism, forged in a more humble age. More than many conservatives, who only mumbled their fears, Fukuyama was showing intellectual courage. His was the only presentation Wolfowitz attended. The deputy defense secretary listened and then exited the room without comment. The two never discussed Fukuyama's presentation again.\n\nFrom liberals to conservatives to economists to generals, by 2003 many of the people who in the past might have resisted war had either fallen silent or been shoved aside. Without them, Colin Powell was a very vulnerable man. \"Someday when you're retired and I'm retired,\" he told Democratic senator Joseph Biden, \"I'll tell you about all the pressure I've been put under over here.\"\n\nOn January 13, Bush called Powell into the Oval Office, sat down in front of the fireplace, and announced, \"I really think I'm going to have to do this.\" Powell asked if he was sure. It was a silly question: Bush was always sure. \"Are you with me on this?\" Bush asked. \"I'm with you, Mr. President,\" came the reply. The entire conversation lasted twelve minutes. Bush had still never asked Powell whether he thought war was wise. \"I didn't need his permission,\" he later explained.\n\nPowell was on the hook, which in the Bush administration was a dangerous place to be. On January 25, Bush asked him to publicly make the case that Saddam was hiding WMD. \"You have the credibility to do it,\" Bush explained. He was right: According to one poll, Americans trusted Powell over the president on Iraq by a margin of almost three to one. Cheney put it more aggressively. \"You've got high poll ratings,\" he said, jabbing his finger into Powell's chest; \"you can afford to lose a few points.\" This, in Bush and Cheney's mind, was what Powell was there for: not to make Iraq policy but to sell it. He was\u2014there was no gentler way to say it\u2014being used.\n\nThree days later, the White House sent Powell a forty-eight-page, single-spaced memo drafted, Powell suspected, in Cheney's office. It was filled with falsehoods, like the claim that 9\/11 ringleader Mohammed Atta had met Iraqi intelligence officials in Prague. Powell tossed the document aside and on Saturday, February 1, parked himself\u2014along with key aides\u2014at the CIA, where he began looking for the truly convincing evidence on Iraqi WMD, the stuff that you didn't have to be a fanatic to believe. Tensions ran high, as Cheney's staff\u2014whom Armitage privately called \"the Gestapo\"\u2014kept pushing to restore allegations that officials from State and the CIA considered garbage. As Powell's chief of staff, a no-nonsense retired colonel named Lawrence Wilkerson, later admitted, \"We were beginning to get leery of our own presentation.\" But Powell could not afford to get too leery. By telling Bush he would support him on the war, he had bolted the door behind him. So he did what he always did: He managed the situation, making the best of the circumstances he was in. After a while, he grew more comfortable with the presentation. He had never bailed out before, and things had always turned out okay.\n\nJust before 10:30 A.M. on a frigid Wednesday, February 5, Powell strode into the UN Security Council chamber, dressed in a dark suit and red tie, and made the case for preventive war. \"Every statement I make today,\" he declared, \"is backed up by sources, solid sources. These are not assertions. What we are giving you are facts and conclusions based on solid intelligence.\" Even members of Powell's own staff doubted that was true.\n\nIn the United States, commentators pronounced the speech a triumph. The Washington Post called it \"irrefutable\" (though, in fact, UN weapons inspectors began refuting it almost instantly). The New York Times called it \"the most powerful case to date.\" A Newsweek poll found that the percentage of Americans supporting war jumped almost instantly. The White House was overjoyed: Good old Colin Powell had finally shown he was on the team. But Powell's wife, who had traveled with him to New York, was overcome with apprehension, a premonition of bad things ahead. His daughter, who heard the presentation on the radio, thought her father did not sound like himself. When Powell's talk was over, Wilkerson left the UN for his Manhattan hotel room, where he fell into a restless sleep and awoke in despair. It was, he would later say, \"the lowest moment of my life.\" When he returned to Washington, he ordered plaques for everyone who had worked on the speech. But when Powell handed them out, he noticed that his chief of staff hadn't ordered one for himself. I don't want one, Wilkerson explained.\n\nAt 10:15 P.M., on Wednesday, March 19, 2003, President Bush addressed the nation from the Oval Office. The invasion of Iraq had begun. In the days before the Gulf War, his father had found it difficult to sleep; he had lost weight and occasionally struggled to breathe. The decision to send young Americans to their death, he wrote to his children, \"lingers and plagues the heart.\" Before the Iraq War, by contrast, his son let those around him know that he was sleeping well. \"There is no doubt in my mind we're doing the right thing,\" he insisted. \"Not one doubt.\" Just before the cameras rolled, as he prepared to tell the nation it was at war, Bush pumped his fist and said, \"Feels good.\"\n\nThe beginning of the Iraq War resembled the beginning of the Afghan War: Progress was alarmingly slow. While Saddam's regular troops largely melted away, the United States met fierce resistance from Iraqi paramilitaries in civilian clothes who hid among the local population and launched raids on America's supply lines, which U.S. forces lacked sufficient numbers to defend. On television, retired generals began saying that the Bush administration had attacked with too few troops, and that as a result, things were going dangerously wrong.\n\nThen Rumsfeld's light but ultrafast force reached Baghdad, and everything changed. On April 9, only days after the TV generals had warned of a quagmire, Saddam's regime collapsed. In the city's main square, Marine Corporal Edward Chin attached a hook from his tank to a statue of the tyrant and hauled it down, as locals cheered. The next day, a group of distraught Iraqis led Lieutenant Colonel Frank Padilla and his battalion to a forbidding, walled compound. When the Marines broke through its outside gate, 150 disheveled, beaten, and malnourished children\u2014some as young as seven\u2014rushed into the arms of the Iraqis waiting outside. It was a children's prison where Saddam kept the sons and daughters of families he considered disloyal. The Iraqi parents showered the Marines with kisses. If this wasn't liberation, nothing was.\n\nIn an age of intoxicating military victories, it was the most intoxicating of all. Toppling Saddam had taken three weeks and left 108 U.S. soldiers dead from hostile fire, fewer than in the Gulf War. None of the parade of horribles that Powell, Fukuyama, Scowcroft, and others predicted had come true. The Arab street had not risen up; Saddam had not torched his oil fields or attacked Israel; Iraq's neighbors had not intervened. There was looting, to be sure, but top Bush officials and their intellectual allies generally shrugged it off. \"It is hard to be overly troubled by the sight of Iraqis looting the homes and offices of leading Baathists,\" wrote Max Boot in the Weekly Standard. \"Why shouldn't the people take back a few of the regime's ill-gotten gains?\"\n\nCommentators struggled to comprehend the magnitude of America's military achievement. \"The stunning success of the 'combat portion' of Operation Iraqi Freedom challenges any understanding based upon previous military history,\" declared one article in the Standard. \"One gets the impression that U.S. military dominance is now so overwhelming that the rules of conflict are being rewritten,\" added David Brooks. \"We could be entering the age of decapitating wars, in which the United States can change evil regimes without widespread loss of life.\" General Tommy Franks declared that America's ultrasophisticated technology meant that it could now see the battlefield with the \"kind of Olympian perspective that Homer had given his gods.\"\n\nIdeologically, American dominance seemed equally profound. \"In the images of celebrating Iraqis,\" Bush declared on May 1, from the deck of the aircraft carrier Abraham Lincoln, \"we have also seen the ageless appeal of human freedom.... Men and women in every culture need liberty like they need food and water and air.\" For Brooks, this was what national greatness was all about: a heroic leader leading a heroic people in heroic deeds. The post\u2013cold war generation had found its calling. The Iraq War, Brooks wrote, is \"what the United States is on earth to achieve.\"\n\nThe dominance conservatives began imagining a new global order, redesigned to reflect America's epic might. Kristol said it was time to consider withdrawing from the UN; America would have more power, and more legitimacy, without it. Another article in the Standard suggested pulling out of the G-8, the world's club of major industrial democracies, because it gave antiwar countries like France and Germany too much say. Krauthammer went furthest of all, arguing that even NATO, by refusing to endorse the war, had made itself worthless. If the post\u2013cold war era had begun with visions of a new, Wilsonian world of international cooperation and international law, the opposite had now come to pass. America be-strode the world unfettered, Gulliver freed from the Lilliputians' chains.\n\nBush's approval rating, which had drooped into the mid-50s in March, spiked back up to 77 percent. Donald Rumsfeld basked in the adulation as well. On April 14, Michael Jordan's final home game in the NBA, Rumsfeld presented Jordan with an American flag at Washington's MCI Center and got a standing ovation even more thunderous than the greatest basketball player of all time. Tommy Franks and CIA director George Tenet did well from the war, too. After retiring from the army, Franks signed a multimillion-dollar book contract, and pocketed a million more in speaking fees. In 2004, Bush awarded both him and Tenet the Presidential Medal of Freedom.\n\nThe only top administration official who didn't prosper from Saddam's fall was Colin Powell. Before the war, he had been merely vulnerable. Now he was expendable, and the jackals circled. On April 22, former House Speaker Newt Gingrich, a longtime ally of Cheney's, and a member of Rumsfeld's Defense Policy Board, charged Powell's State Department with a \"deliberate and systematic effort to undermine the President's policies.\" Asked by reporters whether he agreed with Gingrich's charge, Rumsfeld dodged the question. Bush also declined to defend his secretary of state.\n\nIn mid-April, Cheney invited Wolfowitz, Scooter Libby, and an old Ford administration colleague and war booster named Kenneth Adelman to a celebratory dinner at his house with their wives. They talked about what a mistake it had been to halt the Gulf War in 1991 and marveled at how easy taking Baghdad had ultimately proved. They traded affectionate stories about their friend and ally Rumsfeld, and they toasted the president. Then someone mentioned Powell, and they all laughed.\n\n## CHAPTER NINETEEN\n\n## I'M DELIGHTED TO SEE MR. BOURNE\n\n\"The scenes we've witnessed in Baghdad and other free Iraqi cities belie the widespread early commentary suggesting that Iraqis were ambivalent or even opposed to the coalition's arrival in their country.\" Donald Rumsfeld was lecturing the press, two days after Saddam's fall. Behind him, photos flashed across a Pentagon screen. \"Iraqis share laugh with a U.S. Army soldier,\" read the caption beneath one. \"Jubilant Iraqis cheer U.S. Army soldiers\" read another. \"Happy Iraqis pose with a U.S. Army soldier,\" read a third.\n\nA reporter piped up. \"Mr. Secretary, you spoke of the television pictures that went around the world earlier of Iraqis welcoming US forces with open arms. But now television pictures are showing looting and other signs of lawlessness....\"\n\nRumsfeld responded philosophically. \"I think the way to think about that is that if you go from a repressive regime...and then you go to something other than that\u2014a liberated Iraq\u2014that you go through a transition period. And in every country, in my adult lifetime, that's had the wonderful opportunity to do that, to move from a repressed dictatorial regime to something that's freer, we've seen in that transition period there is untidiness.\"\n\nThe reporter tried again: \"Do you think that the words 'anarchy' and 'lawlessness' are ill-chosen....\"\n\nRumsfeld cut him off. \"Absolutely. I picked up the newspaper today and I couldn't believe it. I read eight headlines that talked about chaos, violence, unrest. And it was just Henny Penny\u2014'the sky is falling.' I've never seen anything like it! And here is a country that is being liberated, here are people who are going from being repressed and held under the thumb of a vicious dictator, and they're free!\"\n\nSixty-two hundred miles away, in Baghdad, newly liberated Iraqis were liberating their government of refrigerators, desks, chairs, mattresses, even a plastic Santa Claus. From the house of Saddam's son Uday, vandals snatched liquor, guns, lewd paintings, and white Arabian horses. From the house of one of Saddam's first cousins, they seized a battery-powered model Ferrari, a parachute, and a motorized water scooter. One enterprising burglar managed to haul away a boat. All in all, looters gutted seventeen of Iraq's twenty-three government ministries, burning many to the ground. The total cost of the rampage: $12 billion.\n\nIt seemed like a paradox. In Afghanistan and now Iraq, the American military had ventured halfway across the globe and toppled hostile governments at low cost and lightning speed, awing the world. Yet this global Goliath, which used Google-age technology to ensure that even as Iraq's government fell its public infrastructure remained largely unscathed, could not stop unarmed Iraqi civilians from plundering that infrastructure like the armies of Genghis Khan.\n\nPart of the answer was that America had not really tried to stop them. To Powell's chagrin, Rumsfeld had kept the invading force ultra-lean, fast enough to sprint to Baghdad in record time but too small to squat there and police a city of more than six million people, where American troops were the only law. What's more, those troops were given little guidance about what to do once Saddam fell, largely because Rumsfeld and General Tommy Franks didn't want them to do much of anything except turn around and go home. When an anguished colonel pleaded with Lieutenant General David McKiernan, commander of U.S. ground forces, that \"You have got to stop this.... [E]verything's being destroyed,\" McKiernan replied in cold fury: \"I don't ever want to hear that from your lips again. This is not my job.\"\n\nIt was the job of a breezy retired general named Jay Garner, head of the Pentagon's newly created Office for Reconstruction and Humanitarian Assistance, which held its first major planning meeting less than a month before the war began, and arrived in Baghdad only after looters had already picked Iraq's government clean. Garner's office, confided Britain's top diplomat in postwar Iraq, \"is an unbelievable mess.\" Lawrence Diamond, a Stanford sociologist dispatched to Iraq to help build democracy, found the Pentagon's postwar efforts so appalling that he rifled through the California penal code searching for an appropriate indictment. He settled on \"criminal negligence.\"\n\nSo America's failure to control postwar Baghdad was partly a matter of choice. But it was a matter of ability, too. The American military was built for fast wars and bloodless occupations, occupations of stable, peaceful countries like Germany, South Korea, and Japan, and small ones like Bosnia and Kosovo, where America's allies contributed most of the troops. The GIs who found themselves running (or not running) Baghdad in the weeks after Saddam's fall were mostly trained to kill enemy soldiers, not stop looters, restock ransacked hospitals, and direct traffic. The 1.3 million-person active-duty U.S. Army contained only 16,000 military police and civil affairs officers (with another 47,000 in the National Guard and Reserves). Garner's office, which was supposed to run the occupation's civilian side, boasted barely any Arabic-speakers, partly because Rumsfeld and his top aides distrusted regional experts, whom they suspected of ideological disloyalty, but mostly because there were only fifty or sixty fluent Arabic-speakers in the entire U.S. diplomatic corps. Garner himself was so ignorant of the country he was supposed to rebuild that when someone mentioned Ayatollah Ali al-Sistani, Iraq's most powerful man, he responded with a blank stare. Successfully occupying Iraq, as opposed to merely invading it, required lots of deployable bodies, lots of local knowledge, and lots of patience, traits that had not been required during America's flurry of post\u2013cold war wins, and which the American military\u2014and indeed, American society\u2014did not possess.\n\nIn truth, the scene was not as paradoxical as it appeared. The American military was extraordinarily good at some things and not very good at others\u2014and those others proved crucial when Saddam fell. It was like watching Michael Jordan play baseball. In the summer of 2003, as American soldiers in full body armor cursed and sweltered under Baghdad's 110-degree sun, America's military confidence, which had been growing like a wave for a decade and a half, began to break. America's ideological confidence would soon follow, and then its economic confidence after that. The hubris of dominance was beginning to come apart.\n\nIf postwar Iraq confounded the Bush administration's military expectations, it confounded its ideological expectations as well, and the two were intertwined. For his part, Rumsfeld didn't much care how Iraqis governed themselves with Saddam gone. The invasion had toppled one enemy, scared others, and vindicated his vision of a light, agile, high-tech military\u2014or, at least, vindicated it if you considered the postwar irrelevant, which Rumsfeld basically did. For Bush and Wolfowitz, however, what happened in post-Saddam Iraq mattered a great deal. They opposed nation building not out of indifference but out of faith, not because they didn't care whether democracy arose in Iraq but because they believed it would arise spontaneously, without lots of U.S. money or troops. For Wolfowitz, in particular, Chalabi's exiles were crucial. In late 2002, he proposed training them to fight alongside U.S. troops, thus perpetuating the idea that, as in Nicaragua, the Balkans, and Afghanistan, America's local allies were actually liberating themselves, with the United States playing only a supporting role. Chalabi's fans in the media often talked about \"Free Iraqi\" forces, an allusion to the \"Free French\" who had fought alongside America and Britain in World War II, and a memo by Assistant Secretary of Defense Peter Rodman, which Rumsfeld forwarded to Bush's entire war cabinet, made the analogy explicit. \"Had FDR and Churchill actually imposed an occupation government\" in France, Rodman argued, \"the Gaullists would have been neutered.\" The implication was clear: Chalabi was Iraq's de Gaulle, and with him and his army in charge, the United States could quickly draw down its forces and watch an indigenous, pro-American Iraqi democracy blossom.\n\nChalabi certainly had de Gaulle's ego, but in most other respects he and his men proved dismally unable to play the role the Bushies had assigned them. As often happens in a hubris bubble, policymakers who thought they were merely learning the lessons of the past actually stretched and distorted them. In Afghanistan in the 1980s, Bosnia and Kosovo in the '90s, and Afghanistan again after 9\/11, a nationalist fighting force had already existed. In Iraq, by contrast, Wolfowitz and his allies simply assumed that one should exist because it fit a historical pattern that they believed had brought great success. (Iraq did have one indigenous rebel force, the Kurdish Peshmerga, but they fought for Kurdistan, not Iraq, which was exactly the problem.) At Wolfowitz's insistence, in the months before the war, the Pentagon began hastily raising a prefabricated \"Free Iraqi\" army. Chalabi promised 10,000 men; the U.S. military planned for 6,000. Ultimately, after crippling problems with recruitment, screening, and instruction, the force dispatched to Iraq totaled 73.\n\nIn mid-April, the United States flew Chalabi and some supporters to the outskirts of the southern Iraqi city of Nasariyah, where they took part in a political assembly designed to form the nucleus of a post-Saddam regime. But to the surprise and dismay of his American backers, Chalabi turned out to have virtually no local following. He hadn't lived in Iraq since 1958, during which time Islamists had largely supplanted the secular elite from which he hailed. Moreover, he didn't exactly endear himself to his countrymen once he hit the ground. When Baghdad fell, Chalabi's men quickly claimed a series of mansions favored by one of Saddam's sons. Then they appropriated some of Saddam's SUVs, which they allegedly sold overseas for a fat profit. Then fellow Iraqis began accusing them of stealing reconstruction funds. In 2004, Iraqi and American forces raided Chalabi's offices on charges of embezzlement, theft, kidnapping, and passing classified U.S. intelligence to Iran. Polls showed him with the lowest approval ratings of any Iraqi politician, including Saddam.\n\nIn Iraq, it turned out, Chalabi was not the man he had appeared to be in Washington. To his boosters at the American Enterprise Institute, the Pentagon, and the Weekly Standard, he had seemed like the archetypal end-of-history figure: a man dedicated to putting his nation on the escalator to democracy, an Arabic version of the leaders who had toppled dictators from Poland to Panama and South Africa to South Korea. But back in Iraq, he became ideologically inscrutable, a man of shadowy dealings, hidden loyalties, and dark misdeeds. And in this sense, he typified the Bush administration's experience with Iraq more generally, a place that\u2014like Vietnam and like Europe during World War I\u2014grew more alien the deeper America waded in.\n\nIn May, the White House replaced Garner with Paul Bremer, a man even more culturally ignorant and even more ideologically self-assured. Bremer, who had never before served in the Arab world, saw Saddam's regime the way Bush and Wolfowitz did: as an ugly skin concealing the pro-American democracy trapped inside. So being a can-do guy (critics preferred the phrase \"control freak\"), Bremer boldly tore off the remaining scab. Four days after arriving in Iraq, in his first public act as head of the newly formed Coalition Provisional Authority (CPA), he banned thirty thousand members of Saddam's Baath Party from holding jobs in Iraq's new government. Assuming\u2014wrongly\u2014that Baathists dominated Saddam's armed forces, he followed up a week later by dissolving the Iraqi military, interior ministry, and presidential guards. Then he abolished all restrictions on foreign imports and began rapidly privatizing the two hundred state-owned companies that had formed the backbone of the Iraqi economy. Soon, more than a million government employees were out of work. \"We must make it clear to everyone that we mean business: that Saddam and the Baathists are finished,\" Bremer wrote to President Bush. \"The dissolution of his chosen instrument of political domination, the Baath Party, has been very well received.\"\n\nMeanwhile, throngs of angry former soldiers and government workers began to congregate at the CPA's gates. \"We will start ambushes, bombings and even suicide bombings. We will not let the Americans rule us in such a humiliating way,\" declared one fired Iraqi officer. \"The only thing left for me is to blow myself up in the face of tyrants,\" said another. Bremer vowed that he would not be blackmailed by terrorists. Around the same time, the first roadside bomb hit an American Humvee driving from the Baghdad airport. Later, those bombs would gain a name: improvised explosive devices (IEDs).\n\nOnce again Iraq was deviating from the ideological script. Ever since Panama, a central premise of dominance foreign policy had been that anti-American dictators were alien to the societies they ruled and that pro-American rebels represented the true popular will. But in Iraq it was Chalabi who lacked deep roots in the political soil, and Saddam's army and political party that proved far more entrenched than Bremer and his fellow ideologues had imagined. If the Iraqi military and Baath Party were Iraq's ugly outer skin, they were also\u2014as Powell had warned\u2014its glue. Many Sunnis, in particular, had become Baathists as a requirement for employment. And even more had joined the army, which in Sunni communities was a key source of status and jobs. With the army dissolved, large numbers of Sunnis suddenly found themselves unemployed, humiliated, and heavily armed. When Bremer ripped off Iraq's governing skin, he gashed a large segment of Iraqi society in the process, and opened a sore that soon began gushing blood.\n\nSoon Americans and Iraqis were engaged in a darkly comic, savagely violent, dialogue of the deaf. In the center of Baghdad, behind seventeen-foot-high concrete walls topped with razor wire, the CPA created the Green Zone, an Epcot America where the televisions played Fox, the radios blared classic rock, the recreation officers taught yoga and salsa dancing, and the stores sold Cheetos, Dr. Pepper, booze, protein powder, and T-shirts reading \"Who's Your Baghdaddy?\" At the cafeteria in Saddam's former Republican Palace, well-mannered South Asian workers served cheeseburgers, hot dogs, grits, fried chicken, and freedom fries. The menu, which had a distinctly southern flavor, included large quantities of pork, which Iraqi Muslims might have found offensive to prepare. But that wasn't an issue, since Iraqis weren't permitted to work in the dining hall for fear they would poison the food.\n\nWhen Americans ventured into \"the red zone\"\u2014otherwise known as Iraq\u2014they passed through the looking glass, into a parched, dust-brown riddle of a country where American logic often seemed turned upside down. When the Americans set about building an army to replace the one they had disbanded, they dubbed it the New Iraqi Corps, or NIC, only to later learn that in Iraqi Arabic nic resembles the word for \"fuck.\" When a visiting administration official toured Iraq's streets, he was pleased to see kids flash him the thumbs-up sign, only to be told that in Iraq a raised thumb was the equivalent of a raised middle finger. U.S. officials talked incessantly about freedom. (At one CPA briefing, an Iraqi journalist asked an American general why U.S. helicopters flew so low to the ground, scaring local children. \"What we would tell the children of Iraq,\" the general replied, \"is that the noise they hear is the sound of freedom.\") But Arabic-speakers noticed that when Iraqis spoke back, they talked less about \"freedom\" (hurriya)\u2014which according to George W. Bush all people desired like food and water\u2014than \"justice\" (adil ), which had a more confrontational ring. \"Marines are from Mars, Iraqis are from Venus,\" wrote one young major in an e-mail to friends. \"I started to realize,\" noted the Washington Post's Pulitzer Prize\u2013winning correspondent, Anthony Shadid, \"how little any of us\u2014journalists, policy makers, citizens\u2014really understood about Iraq.\"\n\nThe assumptions of ideological dominance were proving wrong, with consequences that undermined America's military dominance as well. Between the spring and fall of 2003, the number of insurgent attacks against occupying forces and Iraqi sympathizers tripled, to roughly one thousand per month. Mortar attacks on the Green Zone became a near-daily occurrence, and Bremer's staff began carrying guns to their offices. In September, 94 percent of Iraqis said Baghdad was more dangerous than it had been under Saddam.\n\nAs Iraq grew more savage, the U.S. military grew more isolated. Troops hunkered down in heavily fortified mini Green Zones across the country, periodically venturing out in full body armor, accompanied by Abrams tanks, Bradley fighting vehicles, and Apache attack helicopters for raids against suspected insurgents. But the more raids they conducted, the bloodier the insurgency got. Given America's ignorance of Iraq, its military strategy was the equivalent of putting a blindfolded man in a room filled with fragile objects, handing him a flyswatter, and telling him to smash the buzzing sound. For every insurgent that U.S. forces captured or killed when they stormed through Iraqi neighborhoods, they enraged dozens of ordinary Iraqis whose homes were violated, damaged, or destroyed, and whose relatives were arrested, injured, or killed. Realizing that they lacked the information to hunt insurgents effectively, America's military leaders began a frantic search for better intelligence. U.S. troops rounded up tens of thousands of detainees and dispatched them for interrogation. But here again, America's vaunted military was not up to the job. The endless shipments of detainees overwhelmed the small number of military police assigned to handle them, turning the military prison system into a black hole into which innocent Iraqis disappeared for weeks, months, or even years. Worse, the stress of handling far too many detainees under dangerous and chaotic conditions, while being pressured to gather more and more intelligence, led to a grotesque moral breakdown, as ill-trained and traumatized prison guards brutalized their wards in ways that turned the stomach of the world.\n\nBy spring 2004, when this breakdown made Abu Ghraib prison a household name, the battle for Iraqi hearts and minds had turned into a rout. A Gallup poll found that only 5 percent of Iraqis believed the United States was in Iraq \"to assist the Iraqi people\" and only 1 percent thought it was there to establish democracy. By contrast, 50 percent believed America's primary mission was \"to rob Iraq's oil.\" In March, four U.S. contractors were kidnapped and murdered in the Sunni stronghold of Fallujah; their bodies were mutilated and hanged from the city's main bridge, as locals cheered. Days later, a rebel Shia militia\u2014led by the virulently anti-American and exceedingly popular cleric Moqtada al-Sadr\u2014rose up as well, seizing control of the vast Baghdad slum known as Sadr City. Suddenly a large chunk of Sunni Iraq and a large chunk of Shia Baghdad were in insurgent hands. Between the fall of 2003 and the fall of 2004, insurgent attacks tripled again, to roughly three thousand a month. U.S. troops began writing their blood type on the inside of their helmets before going out on patrol.\n\nUnderlying the ethic of dominance had been an assumption of mutual innocence, the belief that beneath every tyrant lay a decent, democratic people, and that America could be the agent of their liberation because its own government was so committed to decency and democracy around the world. But by 2004, after the mutilations in Fallujah and the molestations at Abu Ghraib, Americans and Iraqis looked at each other with cold, un-innocent eyes. Average Americans increasingly saw Iraqis as ungrateful, bloody-minded savages, and average Iraqis saw Americans as haughty, rapacious occupiers. What's more, some Americans began to suspect that the Iraqis were right: that the experience of occupation was debasing America's military, and its entire government. The Iraqis believed America was defiling Iraq, and growing numbers of Americans believed Iraq was corrupting America's soul.\n\nDuring Vietnam, Lyndon Johnson's brave public face had concealed private agony. He had raged against his predicament: a war he didn't want to fight but thought he couldn't afford to lose; an ethic of foreign policy toughness that was supposed to protect his beloved domestic agenda but was swallowing it instead. George W. Bush was different: He didn't agonize; he believed. In late 2003 and early 2004, as Iraq descended into hell, he gave Americans platitude-laden pep talks, which boiled down to \"Buck up, we're going to win.\" And in private, to the amazement of White House visitors, he said essentially the same thing. \"It takes an optimistic person to lead. Who's going to follow a leader who says, 'Follow me, things are going to get worse?'\" he told aides. \"I'm the calcium in the backbone.\" Calcium in the backbone\u2014Bush loved the phrase and repeated it often. The implication was clear: Doubt, even private doubt, perhaps especially private doubt, makes you spineless. \"If you're weak internally,\" he insisted, \"this job will run you all over town.\"\n\nBush's harshest critics often assumed he didn't believe his own words, that he was simply lying, which was understandable given his tendency to say things like \"We found the weapons of mass destruction. We found biological laboratories.... [F]or those who say we haven't found the banned manufacturing devices or banned weapons, they're wrong. We found them\"\u2014as he did in late May 2003. But the accusations of duplicity partly missed the point. According to Chief of Staff Andrew Card, Bush genuinely believed that Saddam had possessed WMD, not just in 2003 and 2004, but as late as 2006. If Bush was lying, he was first and foremost lying to himself.\n\nFar from admitting error, in fact, Bush was doubling down. Soon after Saddam's fall, he created the Greater Middle East Initiative, a White House office meant to promote freedom across the Muslim world. It was led by Elliott Abrams, the man who a decade and a half earlier had helped to inaugurate dominance foreign policy by pushing to invade Panama. Then, in January 2005, Bush delivered an inaugural address that employed variations of \"freedom\" and \"liberty\" forty-four times, including nine in the last two paragraphs alone. \"History,\" he declared, \"has a visible direction.\" And America would help push it forward, \"with the ultimate goal of ending tyranny in our world.\"\n\nFor a moment, events actually seemed to be moving that way. In January, an impressive 59 percent of Iraqis went to the polls to elect a National Assembly. In February, the Cedar Revolution brought an end to Syria's three-decade occupation of Lebanon. In March, the Tulip Revolution brought democracy to Kyrgyzstan. Palestinians held relatively free elections in the Gaza Strip and West Bank. Like the toppling of Saddam Hussein in spring 2003, the democratic breakthroughs of spring 2005 offered a tantalizing glimpse of a transformed world. But once again it proved a mirage. In the Gaza Strip, Palestinians elected the theocratic, virulently anti-Israeli militia Hamas\u2014which led the Bush administration, in flagrant violation of its democratic sermonizing, to privately urge its moderate Palestinian allies to try to overturn the election results by force. In Lebanon, the coalition that had risen up against Syrian rule fractured along largely religious lines, with the theocratic, virulently anti-Israeli militia Hezbollah emerging as the nation's most powerful force. In Iraq, Sunnis largely boycotted the vote and multi-ethnic, secular parties were crushed. The insurgency raged on.\n\nThe democracy escalator, it turned out, was not progressing ever upward. Not in the Arab world, where people kept electing medievalists, sectarians, and thugs. Not in theocratic Iran, which was exploiting Iraq's weakness to challenge American dominance of the Persian Gulf. Not in Latin America, where Hugo Ch\u00e1vez's illiberal, anti-American populism was metastasizing across the continent. Not in Russia, which under Vladimir Putin was marching brazenly down the escalator toward autocracy while ordinary Russians cheered. And, most important, not in China, whose dictatorship Americans had once assumed was living on borrowed time. Now Chinese tyranny seemed stable, prosperous, and confident\u2014an alternative model for the world. Beijing \"no longer...tries to cultivate influence in distant African countries as it did in the 1960s,\" Fukuyama had written in 1989, in explaining that America no longer had global ideological competitors. But by Bush's second term, Beijing was cultivating widespread influence in Africa, and across the globe. One academic declared it \"the end of the End of History.\"\n\nIn 2006, Jeane Kirkpatrick died. Her final years had been rough. She had struggled to finish a book on post\u2013cold war foreign policy, a task that grew harder as her faculties diminished and her strength ebbed. Among top officials at the American Enterprise Institute (AEI), where she had once been a towering force, she became a figure of ridicule. When discussing which scholar to recommend for a prestigious commission in 2005, one AEI administrator joked, \"Maybe we'll send Jeane,\" prompting snickers among colleagues. Eventually, after almost three decades, AEI booted her out the door without so much as a farewell party.\n\nBut the book did finally appear, posthumously, with the help of a friend. From the grave, Kirkpatrick lashed out against the Iraq War and planted a final flag for the older, grimmer conservatism that institutions such as AEI had abandoned. \"Democracy requires security as a prerequisite,\" she declared. \"That is why, throughout history, if the single force of political stability in a region is removed without critical institutions in place to fill the resulting vacuum of power, the security of societies and their budding institutions will be precarious at best.\" It was \"Dictatorships and Double Standards\" all over again, an attack on the assumption that beneath every tyrant was a democracy waiting to flower. Kirkpatrick, noted her co-author, had learned her politics in the shadow of the Holocaust. Unlike George W. Bush, she believed \"that we are born not as angels, but more like animals,\" pulled as powerfully toward darkness as toward light.\n\nThe book itself garnered only modest attention. But by 2006, the year of Kirkpatrick's death, her ideas were gaining new life, even among some of the younger conservatives who had once scorned them. \"I used to see the world as a landscape of rolling hills,\" wrote David Brooks. \"People everywhere seemed to want the same things: to live in normal societies, to be free, to give their children better lives. Now it seems that was an oversimplified view of human nature.\" Conservatism, declared George Will, had become infected by a missionary, crusading spirit, and it was time for an exorcism. \"Three years ago,\" he wrote, \"the [Bush] administration had a theory: Democratic institutions do not just spring from a hospitable culture, they can also create such a culture. That theory has been a casualty of the war that began three years ago.\"\n\nIn March, Fukuyama joined the counterattack, publishing America at the Crossroads, a kind of eulogy for Kirkpatrick and Irving Kristol's older, more cautious neoconservatism. The dominance conservatives, he argued, had betrayed their ideological parents and left a once-proud intellectual tradition in ruins. Kagan responded with a book called The Return of History and the End of Dreams, which claimed that it was actually Fukuyama who had been too quick to count dictatorship out. \"The autocratic tradition,\" Kagan insisted in a companion article, \"has a long and distinguished past, and it is not as obvious as it once seemed that is has no future.\" All of a sudden, the most important intellectuals on the foreign policy right were loudly denying that democracy was on the march, and attacking each other for ever having been so na\u00efve as to believe such a thing.\n\n\"I'm delighted to see Mr. Bourne with us today.\" Casey Blake, a Columbia University history professor, was welcoming Randolph Bourne back to his alma mater, which was a little odd, given that Bourne had been dead for more than eighty years. But surveying the crowd of antiwar students in the fall of 2004, Blake detected Bourne's presence. The tiny, bent figure with the acid pen had been rescued from oblivion once before, during Vietnam, when anti-toughness intellectuals like Marcus Raskin and Noam Chomsky needed a muse. Now, Blake was suggesting, he had returned again. In 2001, activists opposed to the war on terror had founded the Randolph Bourne Institute. Earlier in 2004, the Atlantic's James Fallows had invoked Bourne to protest the Bush administration's wartime propaganda. Now Columbia was sponsoring a symposium on Bourne's new relevance. And Bourne wasn't the only ghost of hubris past to haunt American foreign policy during the Iraq years. At around the same time, a political scientist and former army officer named Andrew Bacevich\u2014who would become one of the war's most influential critics\u2014set out to revive the memory of Bourne's old professor, Charles Beard.\n\n\"If Bourne's ghost hovers around us,\" declared the historian Robert Westbrook, one of the speakers at the Columbia symposium, \"it is less William Kristol, Richard Perle and Paul Wolfowitz who would interest him than the likes of Paul Berman, Christopher Hitchens, and Michael Ignatieff, left-wing intellectuals who echoed the arguments of [John] Dewey and other pro-war progressives as they rushed to support the war in Iraq as above all a war for human rights.\" Bourne would have known what to do with such people, Westbrook suggested. He would have gutted the liberal hawks as mercilessly as he gutted their intellectual forefathers during World War I.\n\nBy 2004, the gutting was well under way on the intellectual left. Before Iraq, what had united liberal hawks and dominance conservatives was the belief that the spread of human rights and the spread of American military power generally went hand in hand. That had been the lesson of the Balkans, where American bombers helped stop genocide, and of Afghanistan, where GIs escorted little girls to school. But in Iraq, many liberals again learned the lesson of Bourne and the New Left: that war was not an instrument of democracy, but a menace to it. The \"war-technique,\" wrote Westbrook, quoting Bourne, \"determines its own end\u2014victory, and government crushes out automatically all forces that deflect, or threaten to deflect, energy from the path of organization to that end.\" In the opinion of many liberals, that was exactly what America's government had done in Iraq. To justify the war it had hyped the threat of WMD, and to prosecute the war it had sodomized innocents at Abu Ghraib. And since the \"war on terror\" threatened to go on forever, the \"war-technique\" threatened to eat away at American democracy until it was an empty shell. If influential conservatives were turning against dominance foreign policy because they doubted war could bring democracy to Iraq, many liberals were turning against it because they believed the war in Iraq was corroding democracy in the United States.\n\nThe hubris of dominance, like the hubris of reason and the hubris of toughness before it, had relied on faith in political authority. Public trust and public malleability had permitted elites to march the nation to war. But in the late Bush years, Iraq generated a distrust of political authority unseen since Vietnam. Hollywood began pumping out TV shows and movies that pictured Islamic terrorists as mere pawns of the world's true villains, who resided at the White House, Pentagon, and CIA. In Syriana (2005), the CIA assassinates a reformist Arab prince trying to free his nation from America's imperial death grip. In the Bourne trilogy (no relation to Randolph), a latter-day James Bond uses quick wits, martial arts, and high-tech gadgets to defeat the bad guys and get the girl\u2014except that in this case, the bad guys are his own former bosses at the CIA. In Mission: Impossible III (2006), another supertough, supercool intelligence agent battles a dastardly international arms smuggler, only to learn that the smuggler is a front man for neoconservatives eager to provoke war in the Middle East. Even the television series 24, which initially depicted the war on terror as a brutal but necessary struggle, in 2006 swerved into antigovernment paranoia. The show's first president, the wise and honest David Palmer, is succeeded by Charles Logan, who tries to provoke a terrorist attack against America so he can seize Central Asia's oil.\n\nIf, in Iraq's wake, filmmakers saw Washington as a treacherous place, so did a new generation of liberal activists dedicated to exposing the fiction that America's governing elite was either wise or benign. Just as Vietnam had helped create the adversary journalism of the 1970s\u2014unwilling to take official statements at face value, always on the prowl for corruption, disdainful of the deference previously accorded people in power\u2014Iraq also created a brazen, in-your-face journalistic culture, this time on the Internet. On the \"netroots\" blogs of the late Bush years, profanity was common, insults were rampant, and the guiding assumption was that the \"Washington establishment\" (a term that covered pundits as well as politicians) was stupid, pompous, predatory, and corrupt. Iraq, of course, was Exhibit A. The netroots, explained Matt Stoller, one of its stars, \"is a group of people who came to the web because we felt betrayed by a system we formerly trusted.\" The most successful liberal bloggers, he noted, win their audiences by exposing the fact \"that power and authority was built on silly illusions.\"\n\nAs in the 1920s and the 1970s, this revolt against political authority was, above all, a revolt against presidential authority. In 2007, in a sign that the Internet's adversarial culture was permeating the journalistic establishment, the New York Times announced that it would no longer attend the White House Correspondents' Association Dinner, which had become a symbol of the media's incestuous relationship with the executive branch. In 2005 and 2006, Congress placed the first real restrictions on Bush's ability to detain and torture \"war on terror\" suspects. It also squashed his domestic agenda, rebuffing White House efforts to alter immigration law, partially privatize Social Security, and extend tax cuts. \"What you have seen,\" said Republican senator Lindsey Graham, \"is a Congress, which has been AWOL through intimidation or lack of unity, get off the sidelines and jump in with both feet.\"\n\nThe judiciary also awoke from its slumber. In 2004, after three years of docility, the Supreme Court ruled that judges could decide whether non-Americans were wrongfully imprisoned at the U.S. naval base at Guant\u00e1namo Bay, Cuba. Two years later, it deemed the military tribunals that the Bush administration had established there unconstitutional. In 2007, the U.S. Court of Appeals for the District of Columbia Circuit ordered the Bush administration to release information about Gitmo detainees. And in 2008, in a fourth straight legal slap, the Supreme Court ruled that detainees could challenge their detention in federal court. \"The irony,\" noted former Reagan administration lawyer Bruce Fein, \"is that the president has now ended up with lesser powers than he would have had if they had made less extravagant, monarchial claims.\"\n\nFinally, the public itself delivered the most crushing blow to Bush's once-imperial presidency, handing the Democrats control of both houses of Congress in the landslide midterm elections of 2006. Rumsfeld was fired. On his last night as defense secretary, the man who in 2003 had received a louder ovation than Michael Jordan took his family to an upscale Washington restaurant. It was not exactly a triumphant scene. \"I'm not serving a war criminal,\" shouted the chef, who only relented after another cook agreed to prepare Rumsfeld's meal.\n\nFor his part, Cheney now registered lower public approval ratings than Michael Jackson after he was accused of pedophilia and O. J. Simpson after he was tried for murder. In Bush's second term, his network of allies and moles\u2014Rumsfeld, Wolfowitz, John Bolton, and, most important, Chief of Staff Scooter Libby, who was sentenced to thirty months in prison for lying under oath\u2014largely came apart, substantially reducing the vice president's influence. Thirty pounds overweight, suffering from gout, and the victim of no less than eight cardiac episodes during his time in the White House, the most powerful vice president in history was now less fearsome than pitiable.\n\nBush, who after 9\/11 had boasted the highest presidential approval rating in American history, was now the most disliked president on record. Pollsters found that attaching his name to even popular proposals turned the public against them. Even within the military, a Republican bastion, he could not escape public fury over the war. \"See? It's not worth it,\" yelled a family member at Brooke Army Medical Center in San Antonio during Bush's visit, pointing to their disfigured relative. At the National Naval Medical Center in Bethesda, Maryland, the mother of a dying soldier burst into tears, then screamed at the president, \"This is not your daughter!\" In the summer of 2008, Bush became the first sitting president since Lyndon Johnson not to attend his party's presidential convention.\n\nAt first, Bush's descent and Iraq's went hand in hand. In February 2006, a Sunni attack on a Shia shrine in the town of Samarra triggered an orgy of sectarian slaughter horrifying even by Iraqi standards. Cities, neighborhoods, even streets were \"ethnically cleansed.\" People on the wrong side of the sectarian divide ended up with limbs hacked off or drill holes in their head.\n\nIn May, with domestic opinion turning hard against the war, Congress tried to seize control. It created an outside panel, the Iraq Study Group, led by George H. W. Bush's former secretary of state, James Baker, and stocked with former Bush I officials. It was the foreign policy equivalent, one panelist quipped, of a family intervention. The Study Group proposed withdrawing tens of thousands of U.S. combat troops from Iraq and initiating negotiations with the anti-American dictatorships in neighboring Syria and Iran. Militarily, the group was suggesting, America could not impose its will on Iraq, and ideologically it could not impose its will on the Middle East. For the first time in the dominance era, a prominent, bipartisan group of foreign policy elites was demanding the retrenchment of American power.\n\nIn 1968, when the titans of the foreign policy establishment told Lyndon Johnson that the jig was up in Vietnam, he cursed and moaned, but ultimately conceded defeat. But in 2006, when they delivered the same news to George W. Bush, he told his father's consiglieres to go to hell. He made no bold diplomatic overtures to Damascus and Tehran, and instead of withdrawing troops from Iraq, he added them, in what became known as \"the surge.\" He would keep the troops in Iraq, Bush vowed, even if his only supporters were his wife, Laura, and Barney, the dog.\n\nMost foreign policy experts, most politicians, and most ordinary Americans reacted with a mixture of disbelief and rage. Soon, however, something extraordinary happened: The situation in Iraq began to improve. By the time additional U.S. troops arrived in 2007, Iraq's Sunnis had come to a strange and painful realization: America was their last, best hope. Ever since the Iraq War began, Sunni leaders had deluded themselves into believing that if they drove out the U.S. occupiers they could reestablish their historic dominance over Iraq's Shia majority. In the civil war of 2006, however, that dream died as Shia militias expelled Sunnis from much of Baghdad. Suddenly, the 160,000 U.S. troops in Iraq no longer looked like the barrier standing between Sunnis and supremacy; they looked like the barrier standing between Sunnis and extermination. As a weapon against the Americans, many Sunni leaders had found Al Qaeda useful. But now those leaders wanted U.S. protection, which required casting the jihadists aside. This they did with little remorse, since the jihadists had alienated even conservative Sunnis with their nasty habit of maiming and killing Iraqis for alleged violations of Islamic law.\n\nThe timing was most fortuitous. At the very moment Sunnis realized they needed American protection, American troops started flooding their cities and towns. The United States not only had more forces in Iraq as a result of the surge but also began deploying them differently, sending troops out of their isolated bases to live among ordinary Iraqis, as dictated by the counterinsurgency doctrine embraced by General David Petraeus. By late 2007, Sunni tribesmen were turning en masse against Al Qaeda and toward the United States.\n\nThis, in turn, changed Shia politics. When Sunni violence was rampant, many Shia had turned to Moqtada al-Sadr's Mahdi Army for protection. But over time, the Mahdi Army had degenerated into a criminal gang, offering protection of a Soprano-esque sort while muscling in on local businesses. When Sunni violence declined, the Mahdi Army's services became less appreciated, and when a resurgent Iraqi army\u2014backed by more U.S. troops\u2014began challenging Sadr's men for control of the Shia streets, the Mahdi Army lost public support. Realizing that he could summon neither the military firepower nor the popular enthusiasm necessary to defeat the Americans, Sadr called a cease-fire and then dropped out of sight.\n\nIt was a stunning turnaround: Attacks on U.S. soldiers, which had totaled roughly 1,500 a week when the surge began, dropped to around 150 a week by spring 2009. But the larger truth remained: Dominance foreign policy was dead. The ethic of dominance had been premised on the belief that the United States could win wars fast, and with minimal outlays of American cash or blood. But by 2009, an occupation that was supposed to last six months was entering its sixth year, and \"the quiet consensus emerging among many people who have served in Iraq,\" noted military expert Thomas Ricks, was that the United States would still be fighting there in another six. In 2002, White House economic adviser Lawrence Lindsey had been fired for suggesting that the war might cost $200 billion; by 2009, according to one estimate, it had cost America $3 trillion. On September 11, 2001, Al Qaeda had taken three thousand American lives; by 2009, the war in Iraq had taken more than four thousand.\n\nPetraeus's counterinsurgency strategy had proved brilliant, but it was also slow, costly, painful, and morally ambiguous\u2014exactly what the Bush administration had promised the Iraq War would not be. The United States had not vanquished the Sunni insurgents; it had cut deals with them. Iraqis who killed Americans had not received retribution; they had received paychecks. The American military had shown that it could fight an insurgency, but the Iraqi insurgents had offered the world an instruction manual for grounding the American juggernaut: Bog the Americans down in a protracted, gruesome guerrilla war and bleed them until their ADD-afflicted public demands that their leaders turn off the TV. Even if you don't force the Americans out, you will force them to the bargaining table, where they will shower you with money, which you can use for the power struggle that begins in earnest when they leave.\n\nThe Taliban got the message loud and clear. In 2008 and 2009, even as Americans grew more optimistic about Iraq, they grew increasingly despondent about Afghanistan. Exploiting the Bush administration's refusal to launch a serious nation-building effort in 2002 and 2003, the Taliban and Al Qaeda had come roaring back, seizing large chunks of the Afghan south and sizable portions of neighboring Pakistan. In the United States, views of Afghanistan were coming full circle. Right after 9\/11, many foreign policy commentators had warned that building a functioning, liberal, pro-American regime there was impossible: the country was too backward, too savage, too xenophobic. By 2002, that pessimism had been largely forgotten as Americans celebrated their rapid overthrow of the Taliban. By 2009, however, all the old clich\u00e9s\u2014\"quagmire,\" \"graveyard of empires\"\u2014were back. Vietnam analogies, which had been so out of style when America went to war in Iraq, were suddenly everywhere. Commentators who had once noted fondly that Hamid Karzai had brothers who were restaurateurs in Boston and Baltimore now discovered that another brother was one of Afghanistan's biggest drug kingpins. Americans were again speaking about Afghanistan in the language of despair.\n\nBy 2009, even policymakers who believed victory in Afghanistan was possible were defining it far more modestly than they had in the hubris-swollen days of 2002 and 2003. For America's top military leaders, the lesson of Iraq was that the United States would have to pay a far higher price for far humbler goals. On Iraq, Petraeus called himself a \"minimalist.\" The United States, explained his strategic adviser, Major General David Fastabend, will have \"to settle for far less than the vision that drove it to Baghdad.\" And when Petraeus was named head of Central Command, which gave him authority over the Afghan War, he encouraged this minimalism there, too. Even as the new administration of President Barack Obama sent more troops to Afghanistan, it signaled a willingness to cut deals that allowed the brutal, misogynistic Taliban a share in power. The Taliban, Obama officials bluntly conceded, would never be vanquished.\n\nIn both Mesopotamia and the Hindu Kush, the dream of ideological dominance was fading. After a massively rigged Afghan election in 2009, few Americans still saw Karzai as a committed democrat. And in Iraq, noted the Brookings Institution's Kenneth Pollack, \"Many Iraqis (and Americans) believe [Prime Minister Nuri al-] Maliki intends to make himself a new dictator.\" American policymakers still hoped that democracy would survive in Afghanistan and Iraq, of course, but the bald truth was that ensuring its survival was no longer part of America's mission. In both countries, America now defined \"victory\" not as the creation of governments that served as ideological models for the Muslim world, or even governments that served as bases for U.S. power, but merely governments that could prevent their territory from incubating attacks against the United States. In defining victory down, U.S. officials backhandedly accepted Kirkpatrick's old logic: that in some parts of the world, the lesser evil was the best America had a right to expect.\n\nIf America's military and ideological dominance was ebbing, its economic dominance soon followed. In the late 1990s, when Governor George W. Bush was pondering a presidential run, Lawrence Lindsey had warned him that a bubble was developing on Wall Street as the result of inflated high-tech stocks. \"There's a good chance it'll burst while you're president,\" Lindsey cautioned. Bush took little heed. In fact, as president he occasionally needled Lindsey on the subject. \"How's that bubble doing, Lindsey?\" he joked, as mortgage-backed securities replaced Internet stocks as the darling of Wall Street and the stock market soared ever higher. \"Gonna burst any day now?\"\n\nIn the fall of 2008, it did. Even before the meltdown, commentators had warned that U.S. economic power was waning. From 2001 to 2008, America's share of world GDP declined every year. America remained the world's largest consumer, the primary place where Chinese factories disgorged their mountain of manufactured goods, and the largest recipient of foreign investment, the primary place where Chinese and Middle Eastern investment funds parked their vast savings. All this investment subsidized the profligate habits of America's government and people, both of whom lived beyond their means. Even before the crash, however, experts warned that all this borrowing was sapping America's geopolitical muscle. As the Council on Foreign Relations' Bradley Setser noted in a 2008 paper, \"a debtor's capacity to project military power hinges on the support of its creditors.\" One of dominance foreign policy's critical, if generally unspoken, assumptions was that the United States could economically sustain its aggressive military posture. But that aggressive posture, according to Setser, now required the forbearance of China, which served as America's bank.\n\nThen came the financial deluge. In the short term, investment in the United States actually rose, since in a time of panic America seemed a safer bet than most other places. But to forestall an economic depression, Washington began printing money wildly\u2014and plunged massively into debt. In financial circles, observers grew increasingly pessimistic about the long-term value of the dollar, sparking fear that Asian and Middle Eastern investors might stop buying as many greenbacks, which might force Americans to begin living within their means\u2014a frightening prospect for a populace and a government that have not done so in a very long time.\n\nAnd if the financial crisis threatened to drain America of cash, it also drained it of prestige. In the late 1990s, America's ideological dominance had rested not only on the belief that history's escalator was carrying the world toward democracy, but that it was carrying the world toward American-style capitalism as well. But that trajectory looked far less inevitable after the financial crash, when America\u2014which had spent the past two decades telling the world to deregulate its financial markets\u2014began struggling to reregulate its own. \"The teachers now have some problems,\" noted Chinese vice premier Wang Qishan drolly. French president Nicolas Sarkozy was blunter: \"Le laisser-faire, c'est fini.\"\n\nIt was a good moment to be Paul Kennedy. In the 1990s he had been a figure of intellectual sport, the poor sap who bet against the United States just before it left the world in its dust. But the critics weren't laughing anymore. In 2006, China published a new edition of Rise and Fall of the Great Powers, accompanied by an eight-part television series, largely cribbed from its pages. In 2008, the American Political Science Association hosted a panel to mark the twentieth anniversary of Kennedy's book. Its title: \"Is the United States in Decline Again?\"\n\nWith a roar, American intellectuals shouted yes. In January 2008, in a New York Times Magazine cover story, the New America Foundation's Parag Khanna heralded the birth of the \"Non-American world.\" Four months later, Newsweek's Fareed Zakaria published The Post-American World. And in 2009, National Journal's Paul Starobin came out with After America, which lionized Paul Kennedy for being ahead of his time. \"Kennedy was savaged in certain quarters,\" noted Starobin in a companion article. \"And then he was ridiculed.... [But] these days, Kennedy is looking less like a heretic and more like a prophet.\"\n\n\"American civilization,\" wrote Starobin, \"has reached the end of its long ascendancy in the world.\" And many ordinary Americans seemed to agree. A 2009 Pew Research Center poll found that only 36 percent of Americans believed their children would have a better life than them, down 19 points from 1999. Among Indians, by contrast, the figure was 78 percent, and among Chinese, 89 percent.\n\nAmong the American pessimists was George Kennan. His great realist contemporaries\u2014Morgenthau, Lippmann, and Niebuhr\u2014had all died during the Vietnam War or its aftermath, and spent their final years in political despair. Kennan, however, had lived to see the miracle of the cold war's end, and having seen it, he told family and friends that he was now content to die. But he didn't die. He lived on, past the fat, carefree '90s, past 9\/11, and all the way to Iraq, a war that filled him with dread. \"I take an extremely dark view of all this\u2014see it, in fact, [as] the beginning of the end of anything like a normal life for all the rest of us,\" he wrote to his nephew on the eve of the war. \"What is being done to our country today is surely something from which we will never be able to restore the sort of country you and I have known.\" He asked that the letter be destroyed, but it never was. Having lived to see his greatest hopes realized, he had now lived to see them dashed again. A decade and a half after tipping his hat to Wilson, his pessimism had been vindicated after all. In 2005 he died, like the others, in political despair.\n\n## CONCLUSION\n\nTHE BEAUTIFUL LIE\n\n\"We need some great failures...we ever-successful Americans\u2014conscious, intelligent, illuminating failures.\"\n\n\u2014LINCOLN STEFFENS\n\nEasy for him to say. It's hard to be illuminated by failure in a country where success is a national religion. It's hard to say \"no, we can't\" in a country that spent its first century conquering a continent and its second conquering much of the world. The American character, wrote Arthur Schlesinger, Sr., \"is bottomed upon the profound conviction that nothing in the world is beyond its power to accomplish.\"\n\nBut that conviction is a lie. Much is beyond our power to accomplish, especially when it comes to the world beyond our shores. When America's leaders fall in love with the lie, when success convinces them it might actually be true, when they forget we fly on wings of wax, the gods take their revenge.\n\nYet if they denounce the lie, expose its absurdity, declare themselves deaf to its infantile charms, they exile themselves from our political tradition. Intellectuals may warn that we are too ignorant and overstretched to remake the world; ordinary Americans may quietly acknowledge that we will never fully vanquish our foes\u2014in the same way they quietly acknowledge that politicians will always have affairs. But woe to the leader who speaks those heresies out loud. \"My fellow Americans, we must lower our sights, reconcile ourselves to evils that have no remedy, admit that the world will not bend to our will\"\u2014the words cannot be spoken. You cannot stand up in church and denounce the virgin birth. Privately, we speak in different tongues, but officially, can-do-ism is the only vocabulary we have.\n\nSo the lie is politically essential. But more than that: It's morally essential. In bad times, it helps us to look beyond the dismal present, to find hope in the unseen. Arthur Schlesinger, Sr.'s, analysis of the American character, after all, was delivered during the winter of 1942, when Nazi flags flew from the Atlantic to the Volga and the Arctic to the Sahara. His optimism was less an invitation to overreach than a counsel against despair. For the French and British in the 1940s, or the Bosnians and Kosovars in the 1990s, American can-do-ism was a very welcome thing.\n\nWhat America needs today is a jubilant undertaker, someone like Franklin Roosevelt and Ronald Reagan who can bury the hubris of the past while convincing Americans that we are witnessing a wedding, not a funeral. The hubris of dominance, like the hubris of reason and the hubris of toughness before it, has crashed against reality's shoals. Woodrow Wilson could not make politics between nations resemble politics between Americans. Lyndon Johnson could not halt every communist advance. And we cannot make ourselves master of every important region on earth. We have learned that there are prices we cannot pay and burdens we cannot bear, and our adversaries have learned it, too. We must ruthlessly accommodate ourselves to a world that has shown, once again, that it is not putty in our hands.\n\nFor starters, that means remembering that we did not always believe we needed to dominate the world in order to live safely, profitably, and ethically in it. In the decade and a half after the Soviet empire fell, dominance came so easily that we began to see it as the normal order of things. We expanded NATO into East Germany, then into Eastern Europe, then onto former Soviet soil, while at the same time encircling Russia with military bases in a host of Central Asian countries that once flew the hammer and sickle. We established a virtual Monroe Doctrine in the Middle East, shutting out all outside military powers, and the Bush administration set about enforcing a Roosevelt Corollary, too, granting itself the right to take down unfriendly local regimes. In East Asia we waited expectantly for China to democratize or implode and thus follow Russia down the path to ideological and strategic submission. And we stopped thinking about Latin America much at all, since we took it as a virtual fact of nature that no foreign power would ever again challenge us in our backyard.\n\nWe were like the warrior guarding his village who suddenly finds that the enemy has abandoned the battlefield, leaving vast tracts of territory undefended, and so takes them for his own, since the acquisition apparently involves little risk and cost. And once those lands have been incorporated, he sees that even more is available: The inhabitants offer little resistance and even appear pleased to join the realm. And as his domain extends outward, the warrior begins to see its new size less as a choice than a necessity: the bare minimum necessary to keep his family safe. The old borders, which he once deemed sufficient, now strike him as frighteningly exposed. In fact, he comes to suspect that even his current territory is inadequate; he has grown so used to expansion that mere stasis strikes him as a form of retreat. And meanwhile, the lands just beyond his domain are no longer so welcoming or unguarded, and mutinies have broken out in some of his recent acquisitions. Fulfilling his obligations is no longer so effortless and the resources at his disposal are no longer so plentiful. His challenge is to step back from the border skirmishes that now consume his time and try to recover the more disciplined habits of mind that guided him in the days before the recent windfall, because the days of windfall are now clearly gone.\n\nIf the men and women who shape American foreign policy conduct this intellectual audit they will discover a sharp discontinuity between some of today's widely held assumptions and the assumptions of successful American policymakers in eras past. After 9\/11, in the name of fighting terror, the Bush administration declared war or cold war on Iraq, Iran, Syria, the Taliban, Hezbollah, and Hamas, virtually every significant regime and militia in the greater Middle East that did not kiss our ring. And in its pursuit of regional dominance, it claimed that it was merely doing in the Muslim world what past generations had done in Europe and Asia. But that's not right. Franklin Roosevelt did not wage World War II so America could be the world's sole superpower, or even Europe's. He wanted Four Policemen; unipolarity was Hitler's goal. And FDR did not wage war against all the enemies of freedom. He allied with Stalin to defeat Hitler and Tojo. Similarly, during the cold war America did not take on the entire communist world, except for a period of hubristic intoxication that began with McCarthyism and culminated in Vietnam. In the late 1940s we made common cause with the communists in Belgrade, and in the 1970s and '80s we made common cause with the communists in Beijing, all to contain the communists we feared most, who resided in Moscow. George Kennan saw the purpose of containment as ensuring that no single power controlled the world's centers of economic and military might, not ensuring that that single power was the United States.\n\nHow could our forefathers have been so cowardly and immoral? Stalin was a monster; so was Mao. And they both had nuclear weapons aimed at us. Why did we live with that sword of Damocles? Why did we accept their dominion over billions of souls? Once upon a time, the answer was obvious: because we lacked the power not to. Franklin Roosevelt knew the American people would not sacrifice their sons by the thousands to keep Eastern Europe from Soviet hands. During Korea, Harry Truman blundered into war with Beijing and realized that in Asia too the price of denying America's communist foes a sphere of influence was far too high. Even Ronald Reagan proved so reluctant to challenge Soviet control over Poland in the early 1980s that conservative commentators cried betrayal. In different ways, all these presidents understood that in foreign policy, as in life, there are things you may fervently desire but cannot afford. And in foreign policy, that recognition is even more important, since you are not merely spending other people's money; you are spilling other people's blood.\n\nIn our time, these tragic choices have been largely airbrushed from public memory. World War II has become the story of America single-handedly saving Europe from Nazi totalitarianism, even though U.S. troops didn't hit the beaches of Normandy until our Soviet totalitarian allies\u2014who lost ninety men for every one of ours\u2014had already turned the tide. The cold war has become the story of America's triumph over communism, even though we played off one communist giant against the other. As a result, many policymakers and pundits have come to see prioritizing among adversaries as immoral and un-American, a view exemplified by the Bush administration, which spurned Iran's offers of help against Al Qaeda, did little to turn Damascus against Tehran, waited until 2007 to exploit divisions between nationalist and jihadist insurgents in Iraq, and made no meaningful distinction between Al Qaeda and the Taliban in Afghanistan. The assumption was that a nation as powerful as ours did not need to choose.\n\nNow the days of reckoning have arrived. Our commitments have grown massively since the cold war's end, but our resources\u2014economic, military, and ideological\u2014are not what they once appeared. In Walter Lippmann's phrase, American foreign policy is \"insolvent.\" Our obligations exceed our power. Yet like a homeowner unwilling to sell his house for $500,000 because he once believed it worth $1 million, we cling to illusions born in easier times. We lack even the vocabulary for hard choices. In public debates over foreign policy, commentators talk endlessly about combating foreign threats and promoting American values, but much less about safeguarding American interests. Without a prior conception of interests, however\u2014a conception of what the world must look like for America to be safe, prosperous, and free\u2014intelligently deciding what constitutes a threat is impossible. And without a prior conception of interests, promoting American values is an infinite project. Talking about threats makes us feel tough and talking about values make us feel virtuous, but only talking about interests forces us to acknowledge the limits of our ability to be either tough or virtuous. This discomfort with the language of interest is a symptom of America's post\u2013cold war inability to prioritize. We remain in thrall to a series of assumptions about American omnipotence that, if not challenged, threatens to drive our foreign policy deeper into the red.\n\nThe first assumption concerns terrorism. After September 11, America's leaders warned that Al Qaeda would likely strike again on American soil and with even greater force. Partly this was a natural response to the agony of the moment. But the decision to describe terrorism in apocalyptic terms\u2014to see 9\/11 not as the limit of Al Qaeda's capacities but as a mere warm-up for the real horror to come\u2014also stemmed from America's extraordinary post\u2013cold war confidence. We defined the terrorist threat as virtually unlimited because we believed we had virtually unlimited resources to fight it. Today, by contrast, our resources are gravely strained, and yet in Washington, official rhetoric about terrorism remains almost as apocalyptic as it was eight years ago, even though Al Qaeda has not struck the United States\u2014or any other country\u2014on nearly the scale of 9\/11. Obviously, jihadist terrorism remains a threat. Yet our refusal to reevaluate the severity of that threat in the face of new information carries a cost. Since, officially, the terrorist threat has not been downgraded, homeland security and defense spending remain politically sacrosanct, despite a budget deficit that has swelled dramatically. And since, officially, the terrorist threat has not been downgraded, it remains politically treacherous to eschew the most aggressive counterterrorist actions\u2014drone attacks in Pakistan, for instance, which kill large numbers of civilians and spark hatred of the United States\u2014even if those actions harm American security more than they help. In their unwillingness to revise official rhetoric about terrorism in the face of eight years' worth of evidence that Al Qaeda is not\u2014thank God\u2014the danger Americans once imagined, American politicians resemble Lyndon Johnson and Dean Rusk, who in the mid-1960s spoke about the communist threat as if Moscow and Beijing were still allies and Stalin was still alive.\n\nThe second assumption concerns nuclear deterrence. In 2002, George W. Bush declared the concept dead. In the post-9\/11 world, he insisted, threats of nuclear retaliation were not only useless against Al Qaeda; they were also useless against anti-American dictatorships with WMD. Analytically the argument made little sense. Why, exactly, would deterrence not work against Iraq, Iran, and North Korea? Because they were led by ruthless dictators who espoused fanatical ideologies, aided terrorists, and butchered their own people\u2014unlike, say, Joseph Stalin or Mao Zedong, whom America successfully deterred during the cold war? The logic behind deterrence does not rely on a regime's decency; it relies on a regime's desire to stay in power, something a nuclear exchange would put at risk. In fact, even as the Bush administration repudiated deterrence, it began quietly deterring North Korea, which it had permitted to produce several nuclear bombs.\n\nThere is no guarantee that deterrence will work always and everywhere, of course. Although Tehran's theocratic rulers have no history of suicidal behavior (they have never given their terrorist allies biological or chemical weapons, for instance), America and the world would certainly be better off if they didn't get the bomb. But America and the world would have been better off had Stalin, Mao, and Kim Jong Il never gotten the bomb, too; we relied on nuclear deterrence not because it was ideal but because the military alternatives were ghastly. Now the military alternatives are ghastly again. An American or Israeli strike would have grave consequences for the two wars America is already fighting on Iran's border: in Afghanistan and Iraq. If commentators who urge military action against Tehran are willing to imperil success in those conflicts to prevent an Iranian bomb, so be it. But pro-war commentators rarely make that argument, because it would require painful choices among competing priorities. And in Washington today, such choices are rarely publicly acknowledged. Seven years after the invasion of Iraq, the illusions of omnipotence remain: We are so powerful that we do not need to choose.\n\nThe third area in which these illusions warp our foreign policy debate concerns Russia. Officially, the United States remains committed to admitting Georgia and Ukraine into NATO, thus continuing the eastward expansion that began when the Soviet Union fell. The logic behind this expansion was that Washington and its allies were so strong, and Moscow was so weak, that we could push closer and closer to Russia's border without ever paying a price. But in 2008, when it invaded Georgia, Russia showed that it could indeed exact a price. Since NATO is a mutual defense pact, admitting Georgia would mean that if Russia attacked again, the United States would be obligated to send young men and women from Cleveland to fight and die for Gori, something the American people would never tolerate.\n\nThe United States has been in this kind of situation before. During the cold war, Washington and Moscow struck a de facto bargain over Finland, the U.S.S.R.'s neighbor to the northwest. The United States tacitly accepted Soviet influence over Finnish foreign policy while the Soviets allowed Finland to remain democratic and self-governing in its domestic affairs. The arrangement was ugly and unfair. It required acknowledging that Finland, because of its size and location, would not enjoy the full privileges of sovereignty. But America's leaders knew that they would be doing the Finns no favors by encouraging them to poke the Kremlin in the eye, since America would not protect them from the Soviet response. Today a similar arrangement is likely the best America can expect when it comes to Georgia and Ukraine. But acknowledging that is unpleasant and so most politicians do not. Instead they perpetuate the fiction that we can keep pushing the frontiers of American hegemony outward, even though it requires writing the foreign policy equivalent of bounced checks.\n\nThere is a reason politicians do this: In our political culture, publicly acknowledging that something is beyond America's power is perilous. The president who presides over a nuclear Iran or concedes a Russian sphere of influence in parts of the former U.S.S.R. or admits that we will never fully vanquish jihadist terrorism will be, politically, a marked man. He will be accused of betraying Americanism itself, of having desecrated the church of optimism. The aftershocks may not be as great as the ones that upended American politics after we conceded China in 1949 or fled Vietnam in 1975, but they will be tumultuous nonetheless. Ideologues and opportunists alike will insist that the talk of limits was an illusion, that the only limits that mattered were the limits of presidential will. They will insist that everything would have been possible had only real Americans roamed the corridors of power. If only the men and women in the White House had truly believed.\n\nTo survive this onslaught\u2014to truly lay the hubris of dominance to rest\u2014Barack Obama will need to redefine our national faith, to decouple American optimism from the project of American global mastery. He will need to find new, more manageable missions that draw upon that beautiful American lie: that nothing is beyond our reach.\n\nFirst, that means redirecting American can-do-ism inward. During the Bush years, a strange inversion occurred: America's government grew more ambitious overseas even as it grew more complacent at home. When it came to protecting the environment, improving health care, or overseeing Wall Street, President Bush oozed fatalism even as he invaded two countries and vowed to end tyranny on earth. This gets things backward. In politics, as in life, we should be most ambitious in those spheres where we have the most power. There are limits to the federal government's capacities at home as well, of course: limits of both money and knowledge. But we know better how to rebuild New Orleans than how to rebuild Afghanistan, more about how to regulate the U.S. financial system than how to establish one in Iraq. The main purpose of foreign policy is to serve, in Lippmann's words, as the \"shield of the republic\": to protect our democracy so it can thrive. If American power swells overseas but the quality of life for Americans deteriorates at home, then American foreign policy has failed.\n\nThe Bush administration's combination of domestic complacency and overseas grandeur stemmed partly from ideology. But it also stemmed from a particular allocation of American resources. Defense now accounts for well over half of the federal government's discretionary budget, close to five times as much as discretionary spending on health care and education combined. All that money is both a cause and an effect of the fact that Americans think more highly of the military than of other institutions of government. The U.S. military is, to be sure, a remarkable institution, more than worthy of pride. But practically and psychologically, we rely on it too much. As our diplomatic capacity has withered, the military\u2014whose budget is ten times that of the State Department\u2014has become our primary instrument for interacting with the world. As our welfare state has withered, the military has become one of our few effective instruments for economic security and upward mobility at home. It has also become an island of discipline, self-sacrifice, and public-spiritedness in a culture that is materialistic, hedonistic, and guilty about it. We are like a high school that cheers lustily every Friday night for its champion football team, thus distracting itself from its overcrowded classrooms, mediocre test scores, and dismal chess club. And when we do acknowledge that the school is in disrepair, our solution is to ask the football coaches to start purchasing computers and establishing after-school math programs because they have all the money and are the only ones we trust to get anything done.\n\nNow the days of easy gridiron triumph are over, and we must find new sources of school spirit. We must focus more attention on\u2014and devote more money to\u2014the other ways in which we compete with the world: from educational standards to environmental quality to scientific achievement to public health. We must remember that the military and geopolitical triumphs we commemorate were, at their core, economic triumphs. We won World War II as much on the factory floor\u2014where American workers churned out more and better planes and tanks than their German and Japanese counterparts\u2014as on the battlefield. And we won the cold war because America and its democratic allies made capitalism stable and humane while the Soviets never made communism creative and dynamic. We must retell our narrative of national success as a narrative of economic and technological success. We must remember the things we admired about our country when militarism and Americanism were not so deeply intertwined.\n\nBut redirecting American ambition inward will not, by itself, reconcile Americans to the new limits on our power. Americans want their government to project optimism and energy at home, but they also want to be strong overseas. Obama's challenge is to show that strength and dominance are not the same thing. In this regard, he should learn from Ronald Reagan, who scrupulously avoided Vietnam-type military interventions yet found symbolic ways\u2014like awarding the Medal of Honor to overlooked Vietnam hero Roy Benavidez\u2014to make Americans feel proud and strong. As Reagan understood, foreign policy debates are often cultural debates in disguise. Whether it was the Swift Boat attack on John Kerry in 2004 or the attacks on Obama himself for not wearing a flag lapel pin, liberals are often vulnerable during such debates because conservatives cast their criticisms of U.S. policy as evidence of lack of patriotism. Politically, the answer is not for liberals to abandon such criticism, which itself represents a form of patriotism, but rather to leaven patriotic criticism with patriotic affirmation. That means more than ritual incantations about flag and country; it means aggressively challenging those who unfairly deride the United States and its institutions. From a purely foreign policy perspective, publicly confronting Mahmoud Ahmadinejad or Hugo Ch\u00e1vez when they malign the United States may seem useless, or even counterproductive. Likewise, there may be no purely foreign policy rationale for publicly confronting universities that ban military recruiters from campus. But there is no such thing as pure foreign policy; foreign policy requires domestic support. If Barack Obama does not want to be Jimmy Carter, if he does not want Americans to equate his restraint with their humiliation, he must be as aggressive as Reagan in finding symbolic balm for America's wounded pride.\n\nEven this, however, is not enough. Great presidents reconcile a respect for limits not merely with an affirmation of American pride, but with an affirmation of American ideals. Despite everything, we remain a missionary nation with an enduring desire to repair and redeem the world. That missionary zeal, which has at times produced delusion and catastrophe, can be a force for extraordinary progress if it is tempered by two kinds of humility.\n\nThe first is the humility of time. What distinguished George W. Bush in 2003\u2014and Woodrow Wilson in 1917\u2014from more successful presidents was not their belief that liberty could spread around the world; it was their millenarianism, their belief that through the wars they launched it could spread in one big bang. We need more patience. In general, America cannot bludgeon dictatorships into democratizing. In fact, we cannot even deny large authoritarian regimes like those in Russia and China influence over their smaller neighbors. In the short term, our strategy should resemble the vision of great-power cooperation across ideological lines that animated Franklin Roosevelt during World War II. In a world threatened by global warming, financial panic, and swine flu, in fact, the need for such cooperation is even greater now than it was in FDR's day. And because none of the current great powers espouses a revolutionary ideology aimed at overturning the current international order, as Nazi Germany and the Soviet Union did, the opportunity for cooperation is greater, too.\n\nBut if cooperation with authoritarian powers requires realism, we should be short-term realists with long-term, nonrealist dreams. Like Roosevelt and Harry Truman, we should seek to enmesh undemocratic regimes in an open, rule-based, global trading system, because capitalism can slowly and subtly create pressures for the rule of law. It is capitalism, after all, that has turned China from a totalitarian state into an authoritarian one, thus allowing over one billion people a degree of personal freedom unimaginable in Mao's day. In addition, like Roosevelt and Truman, we should try to convince governments to sign declarations\u2014about freedom of speech, freedom of religion, due process, and free elections\u2014that rebuke their own behavior. Critics will dismiss these declarations as worthless because the words will fall far short of current reality, and the organizations created to monitor them will be weak, and America will continue to do business with some of the worst offenders. But forcing regimes into hypocrisy will be precisely the point. In a world where technology gives citizens the power to organize as never before, ordinary people may be able to wield these charters as weapons against an unjust status quo, as dissidents in the Soviet bloc did after the Helsinki Accords, and as Americans themselves did when they protested the Bush administration's violations of the international ban on torture. Activists may even band together across borders in ways scarcely imaginable in the pre-Internet age. \"Ideals,\" wrote Reinhold Niebuhr, \"have a way of taking vengeance upon the facts which momentarily imprison them.\" When it comes to ugliness in far-off places, America usually lacks the power to vindicate those ideals in the here and now. But we can help plant seeds that others may one day reap.\n\nFor that to happen, however, the declarations must bind us as well. Precisely because freedom and dignity are universal values, they are not ours alone to define. George W. Bush not only lacked the humility of time; he lacked the humility of place. His universalism was oddly parochial. The freedom agenda, as the Bush administration defined it, meant pressure for democratic elections, but only if they produced the outcomes we wanted; pressure for human rights, but not in war-on-terror allies like Israel, Saudi Arabia, and Pakistan; and a refusal to even contemplate the possibility that America's own anti-terror policies might violate the principles we preach. What non-Americans know\u2014and Americans should admit\u2014is that our national interests and prejudices will always taint our pursuit of universal ideals. We are not some deracinated global umpire, sitting in judgment over humankind; we are one of the players on the field.\n\nIt is this recognition that our idealism is tainted by self-interest that should make us pause and pause and pause again before unilaterally invading tyrannical nations on the assumption that their people will thank us for it. Even if we genuinely believe that we are acting from altruistic motives, the people whose country we invade will generally be more suspicious, especially if they have been on the receiving end of armed Western altruism before, and especially once an American soldier shoots their cousin or breaks down their door. And even if we are initially welcomed as liberators, countries with awful governments usually have awful political cultures, which do not change easily or quickly. Unless we are willing to stay long enough to change those political cultures\u2014and Americans are not known for our patience, especially when it comes to costly endeavors in far-off lands\u2014we will be building castles in the sand.\n\nGeorge W. Bush was right that people want freedom and dignity for themselves, but they also want freedom and dignity for their nation, and they don't always draw a bright line between the two. This is particularly true for people whose nations have suffered the indignity of imperial rule, nations that wield far more global influence today than they did a generation or two ago. The shift in power from countries that were once colonizers to countries that were once colonized constitutes one of the most profound trends in world politics, and any project that looks like idealism to us but imperialism to them will collide disastrously with the nationalism of peoples whose historical memory has left them with a chip on their shoulders about Western governments telling them what to do.\n\nOften, our ignorance leaves us unable to even understand why the chip is there. Listening to the foreign policy debate in Washington, one would not know that the United States was once a quasi-imperial power in China, or that we helped overthrow an elected prime minister of Iran. As a result, Americans are often genuinely confused when Iranians and Chinese\u2014including Iranians and Chinese deeply committed to human rights\u2014bristle at our lectures. For such a lustily nationalistic people, we remain curiously blind to the nationalism of others.\n\nThe challenge is to convince postcolonial regimes that their quest for greater dignity and power ultimately rests on providing their people greater freedom, and that America can be their partner in that effort. In this regard, we should keep pushing to reshape global bodies like the UN Security Council, World Bank, and International Monetary Fund so their composition no longer reflects the imperial age. We should stop demanding that third-world nations halt their pursuit of nuclear weapons without simultaneously acknowledging our own responsibility to curb our arsenals. We should accept that as a rich nation, which industrialized with little heed to its effect on the environment, we bear more of the initial burden of curbing global warming. And we should acknowledge that the struggle against global poverty\u2014a struggle that looms far larger in countries where millions go hungry\u2014is also a struggle for human rights, and that in this struggle, America can do far more.\n\nAll these efforts differ from the hubris of dominance in one fundamental way: Rather than merely requiring that other nations change, they require that we change as well. They combine greater modesty in our demands of others with more strenuous demands of ourselves. They require that instead of imagining that we stand, self-satisfied, on history's highest rung, we see ourselves as fellow travelers on a path whose course we alone do not set, and along which we have far to climb.\n\nLurking behind many of America's current anxieties is the fear that what will follow hubris is decay; that we are in the autumn of our power; that other nations will soon occupy history's center stage. But we have been through this before. In 1969, with America neck-deep in Vietnam, the historian C. Vann Woodward wrote that \"history has begun to catch up with Americans, the fabled immunity from frustration and defeat has faltered.... America has at last encountered problems that are difficult to reconcile with traditional myths of indomitable optimism.\" Four decades later, we can see that although communism won the war for Vietnam, capitalism won the war for East Asia, and democracy has won stirring battles there, too. The trauma of our defeat gradually faded, yet American optimism endured, not because we clung to the impossible dream of global containment but because we abandoned it and created space for other, better dreams to take its place. Now another generation of Americans must jettison our visions of invincibility. We should do so joyfully, for the recognition that no collection of mortals can impose its will on an unruly globe is not a sign of decay, but of wisdom. And tempered by wisdom, American optimism is\u2014and always will be\u2014one of the great wonders of the world.\n\n## NOTES\n\nIntroduction\n\n\"tyrants to which men are ever subject\": William Graham Sumner, War and Other Essays, ed. Albert Galloway Keller (New Haven, CT: Yale University Press, 1911), 36.\n\nRepublicans over the age of sixty-five: According to Pew, 94 percent of Republicans under the age of thirty backed the Iraq War, compared to only 65 percent of Republicans over sixty-five. By contrast, 61 percent of Democrats under thirty backed the war compared to only 33 percent over sixty-five. \"Generations Divide over Military Action in Iraq,\" Pew Research Center for the People & the Press, October 17, 2002, http:\/\/people-press.org\/commentary\/?analysisid=57. A later Pew poll, in February 2003, once again found that Americans under thirty were more pro-war than Americans over sixty-five, although this time the gap was only eight points. \"Public Favors Force in Iraq, but U.S. Needs More International Backing,\" Pew Research Center for the People & the Press, February 20, 2003, http:\/\/people-press.org\/reports\/pdf\/173.pdf.\n\nrevealed the same generational skew: At my request, two enterprising Yale undergraduates, Chris Chen and Adam Stempel, surveyed the views of influential bloggers, television and radio pundits, and print columnists on the Iraq War between September 1, 2002, and March 15, 2003. To gather their cohort of influential bloggers they used a list published in June 2003 by the Online Journalism Review (http:\/\/www.ojr.org\/ojr\/glaser\/1056050270.php), disregarding those bloggers whose views on Iraq could not be discerned. For the print columnists, they surveyed all the signed op-eds written about the war in the New York Times, Washington Post, and Wall Street Journal between September 1, 2002, and March 15, 2003. (If a columnist's opinion was ambiguous, or the columnist was a government official or member of Congress, he or she was not included.) For television and radio pundits, they surveyed the views of the hosts of the twenty-five most-watched television talk shows and most-listened-to radio talk shows, according to the Nielsen television ratings and Arbitron radio ratings closest to October 2002 (http:\/\/www.mediabistro.com\/tvnewser\/original\/0409_q3ratings.pdf; http:\/\/www.talkers.com\/talkhosts.htm). (If a television or radio host did not offer explicit views on the war, he or she was discarded.) They also surveyed the views of a list of influential Iraq-specific pundits compiled by Slate in 2003 (http:\/\/slate.msn.com\/id\/2080099\/). Where a search of Wikipedia pages, personal profiles, Library of Congress listings, and dates of college graduation did not reveal a commentator's age, he or she was discarded. Ultimately, Chen and Stempel found that 72 percent of the bloggers, op-ed writers, and television and radio pundits under the age of forty-five supported the war, compared to only 60 percent of those over forty-five. Forty-five was chosen as the cutoff date because someone who was forty-five in 2003 would have been only fifteen when the final U.S. ground troops left Vietnam in 1973, and thus, likely too young to have been heavily influenced by that war. The raw data that Chen and Stempel used is available at http:\/\/sites.google.com\/site\/icarussyndromebook.\n\n\"vital interests of the republic\": Arthur Schlesinger, Jr., \"White Slaves in the Persian Gulf,\" Wall Street Journal, January 7, 1991, A14.\n\n\"30 years ago for intervention in Vietnam\": Arthur Schlesinger, Jr., \"How to Think About Bosnia,\" Wall Street Journal, May 3, 1993, A16.\n\n\"they should have reflected on Vietnam\": Arthur Schlesinger, Jr., \"Are We Trapped in Another Vietnam?\" The Independent, November 2, 2001, 5.\n\nuntil Nazi troops reached French soil: Arthur Schlesinger, Jr., A Life in the Twentieth Century: Innocent Beginnings, 1917\u20131950 (New York: Houghton Mifflin, 2000), 231\u201332.\n\nleading, perhaps, to wisdom: Schlesinger's famous book on cycles of reform and reaction in American politics is The Cycles of American History (New York: Mariner, 1999).\n\ndefined it as insolence toward the gods: An alternative theory, associated with the classicist N.R.E. Fisher, argues that while the Greeks were certainly interested in godly retribution against humans who displayed excessive pride, the word hubris or hybris actually meant something different: an act of aggression by one person against another designed to inflict shame or assert superiority. See N.R.E. Fisher, Hybris: A Study in the Values of Honour and Shame in Ancient Greece (Warminster, England: Aris & Phillips, 1992).\n\nhacks him to death in the bath: Aeschylus, 1: The Oresteia, trans. David R. Slavitt (Philadelphia: University of Pennsylvania Press, 1998).\n\n\"but not on good advice \/ he'd overrule all gods\": Aeschylus, Persians, trans. Janet Lembke and C. J. Herington (New York: Oxford University Press, 1981), 76\u201377.\n\nplunges to his death, into the sea: Ovid, Metamorphoses, trans. David R. Slavitt (Baltimore: Johns Hopkins University Press, 1994), 154\u201355.\n\nwould bring forth into the world: My own book The Good Fight: Why Liberals\u2014and Only Liberals\u2014Can Win the War on Terror and Make America Great Again (New York: HarperCollins, 2006) is somewhat guilty of this.\n\nrefused to join us in Vietnam: Sylvia Ellis, Britain, America and the Vietnam War (Westport, CT: Praeger, 2004), 116; Fredrik Logevall, \"De Gaulle, Neutralization, and American Involvement in Vietnam, 1963\u20131964,\" Pacific Historical Review 61, no. 1 (1992): 91\u201393; Fredrik Logevall, Choosing War: The Lost Chance for Peace and the Escalation of War in Vietnam (Berkeley: University of California Press, 1998), 156\u201360; David Kaiser, American Tragedy: Kennedy, Johnson, and the Origins of the Vietnam War (Cambridge, MA: Belknap, 2000), 315.\n\nhad never done in its entire history: Jennifer K. Elsea and Richard F. Grimmett, \"Declarations of War and Authorizations for the Use of Military Force: Historical Background and Legal Implications,\" Congressional Research Service, March 8, 2007, http:\/\/fas.org\/sgp\/crs\/natsec\/RL31133.pdf.\n\noffered by America's recent past: For a detailed discussion of how policymakers can effectively use historical analogies, see Richard E. Neustadt and Ernest R. May, Thinking in Time: The Uses of History for Decision Makers (New York: Free Press, 1986).\n\nhated and feared universal doctrines: \"There are very few general observations which can be made about the conduct of states which have any absolute validity at all times and in all cases,\" Kennan wrote. \"The few that might have such validity are almost invariably to be found in the realm of platitude. If this absolute validity is lacking, the chances are that the utterance in question will some day rise to haunt us in a context where it is no longer fully applicable. If, on the other hand, the utterance remains in the realm of platitude, then there is all the more reason why we should not associate ourselves with it.\" John Lewis Gaddis, Strategies of Containment, rev. ed. (New York: Oxford University Press, 2005), 51\u201352.\n\non Russia's northwest border: For a list of Kennan's diplomatic postings, see David Mayers, George Kennan and the Dilemmas of US Foreign Policy (New York: Oxford University Press, 1988), xiii\u2013xiv.\n\nAmerican foreign policy during the Russian Revolution: Nicholas Thompson, The Hawk and The Dove (New York: Henry Holt, 2009), 9, 158, 166; Mayers, George Kennan and the Dilemmas of US Foreign Policy, 31.\n\nbe clones of their Soviet counterparts: For a discussion of the destruction of the \"China hands\" and its effect on Vietnam policy, see David Halberstam, The Best and the Brightest (New York: Ballantine, 1992), 103\u20134, 188\u201391; and E. J. Kahn, Jr., The China Hands: America's Foreign Service Officers and What Befell Them (New York: Penguin, 1972).\n\nnever before been posted to the Arab world: Michael R. Gordon and Bernard E. Trainor, Cobra II: The Inside Story of the Invasion and Occupation of Iraq (New York: Vintage, 2007), 545.\n\ngo home and prepare for the future: For an argument about the shifting relationship between Xerxes and the chorus, see Harry C. Avery, \"Dramatic Devices in Aeschylus' Persians,\" American Journal of Philology 85, no. 2 (1964): 180\u201383.\n\n\"not a guide by which to live\": Evan Thomas, Robert Kennedy (New York: Simon & Schuster, 2000), 286\u201387; \"The Politics of Restoration,\" Time, May 24, 1968, http:\/\/www.time.com\/ time\/magazine\/article\/0,9171,844434,00.html.\n\nPART I: THE HUBRIS OF REASON\n\nChapter 1: A Scientific Peace\n\nno one ever called him cold and removed: John Milton Cooper, Jr., The Warrior and the Priest: Woodrow Wilson and Theodore Roosevelt (Cambridge, MA: Harvard University Press, 1983), 246\u201347; Alexander L. George and Juliette L. George, Woodrow Wilson and Colonel House: A Personality Study (New York: John Day, 1956), 16.\n\n\"a ten-cent mackerel in brown paper\": Walter LaFeber, The American Age, 2nd ed. (New York: Norton, 1994), 270.\n\n\"I didn't know whether God or him was talking\": Eric F. Goldman, Rendezvous with Destiny: A History of Modern American Reform (New York: Ivan R. Dee, 2001), 213.\n\nthey often talked back: John M. Mulder, Woodrow Wilson: The Years of Preparation (Princeton, NJ: Princeton University Press, 1978), 273.\n\n\"His thoughts and mine are one\": Godfrey Hodgson, Woodrow Wilson's Right Hand: The Life of Colonel Edward M. House (New Haven, CT: Yale University Press, 2006), 10.\n\n\"even when he was cutting your throat\": Hodgson, Woodrow Wilson's Right Hand, 2; Margaret MacMillan, Paris 1919: Six Months That Changed the World (New York: Random House, 2003), 17.\n\nColonel House wanted to meet him outside: Ronald Steel, Walter Lippmann and the American Century (New York: Transaction, 1999), 17, 20, 107, 109, 117, 129; Charles Forcey, The Crossroads of Liberalism: Croly, Weyl, Lippmann and the Progressive Era, 1900\u20131925 (New York: Oxford University Press, 1961), 88\u201389, 109.\n\na project of the most fearsome secrecy: Steel, Walter Lippmann, 127; Lawrence E. Gelfand, The Inquiry: American Preparations for Peace, 1917\u20131919 (New Haven, CT: Yale University Press, 1961), 26.\n\nfive million Germans were either wounded or dead: James McRandle and James Quirk, \"The Blood Test Revisited: A New Look at German Casualty Counts in World War I,\" Journal of Military History 70 (2006): 688; Thomas J. Knock, To End All Wars: Woodrow Wilson and the Quest for a New World Order (New York: Oxford University Press, 1992), 32.\n\nfour thousand villages had simply ceased to exist: LaFeber, The American Age, 316.\n\nhe returned to the White House and wept: Arthur Link, Woodrow Wilson and the Progressive Era 1910\u20131917 (New York: Harper, 1954), 282.\n\nthey would call it simply \"the Inquiry\": Gelfand, the Inquiry, 41.\n\nwhat House called a \"scientific peace\": Lloyd Ambrosius, Woodrow Wilson and the American Diplomatic Tradition: The Treaty Fight in Perspective (New York: Cambridge University Press, 1987), 32.\n\nhistorians insist, was not one thing: See, for instance, Daniel T. Rodgers, \"In Search of Progressivism,\" Reviews in American History 10 (1982): 113\u201332.\n\na fourth branch of government: Forcey, The Crossroads of Liberalism, 6\u20138; Goldman, Rendezvous with Destiny, 189; Sidney Kaplan, \"Social Engineers as Saviors: Effects of World War I on Some American Liberals,\" Journal of the History of Ideas 17 (1956): 355.\n\nhanding them control over the economy: John Patrick Diggins, Thorstein Veblen: Theorist of the Leisure Class (Princeton, NJ: Princeton University Press, 1999), 20\u201324, 194\u201396.\n\nto the world beyond America's shores: Edward M. House, Philip Dru: Administrator, A Story of Tomorrow 1920\u20131935 (New York: B. W. Huebsch, 1912), http:\/\/www.gutenberg.org\/dirs\/ etext04\/8phlp10h.htm.\n\n\"like the individual, must go to school\": Forcey, The Crossroads of Liberalism, 43\u201344.\n\nthat made them act like beasts: For Veblen, for example, prehistoric society was \"characterized by considerable group solidarity.... The prime requisite for survival under the conditions would be a propensity unselfishly and impersonally to make the most of all the material means at hand and a penchant for turning all resources of knowledge and materials to account to sustain the life of the group.\" Quoted in David W. Noble, The Progressive Mind, 1890\u20131917 (Minneapolis: Burgess, 1981), 50. On Beard's belief that absent the distorting effect of unregulated capitalism people would incline toward cooperation, see Richard Hofstadter, The Progressive Historians: Turner, Beard, and Parrington (New York: Knopf, 1968), 175\u201377.\n\nevil resided in the world, not in man: Noble, The Progressive Mind, 76.\n\n\"lost innocence of the race\": Christopher Lasch, The New Radicalism in America, 1890\u20131963: The Intellectual as a Social Type (New York: Knopf, 1965), 143\u20134.\n\nkeeping himself awake by reading Tolstoy: Cooper, The Warrior and the Priest, 27\u201335; William Henry Harbaugh, Power and Responsibility: The Life and Times of Theodore Roosevelt (New York: Farrar, Straus & Cudahy, 1961), 47\u201356; Edmund Morris, The Rise of Theodore Roosevelt (New York: Ballantine, 1980), 321\u201325.\n\nrecognition of their union: For conditions in the anthracite mines, see Harbaugh, Power and Responsibility, 169; James Abrams, \"Anthracite Mining Unionism and the UMW: An Oral History,\" Pennsylvania History 58 (1991): 330\u201337; Perry K. Blatz, \"Local Leadership and Local Militancy: The Nanticoke Strike of 1899 and the Roots of Unionization in the Northern Anthracite Field,\" Pennsylvania History 58 (1991): 278\u201397.\n\nlackeys bloodied workers into submission: The historian John Blum calls Roosevelt's actions during the coal strike \"an extraordinary departure from the protection that presidents had long afforded to management.\" John M. Blum, The Progressive Presidents: Roosevelt, Wilson, Roosevelt, Johnson (New York: Norton, 1989), 40.\n\nwithout anyone's getting shot: Robert H. Wiebe, \"The Anthracite Strike of 1902: A Record of Confusion,\" Mississippi Valley Historical Review 48 (1961): 229\u201351; Jonathan Grossman, \"The Coal Strike of 1902\u2014Turning Point in U.S. Policy,\" Monthly Labor Review, October 1975, http:\/\/www.dol.gov\/oasam\/programs\/history\/coalstrike.htm; Forcey, The Crossroads of Liberalism, 67.\n\nand to shut down slaughterhouses if they did: Blum, The Progressive Presidents, 39\u201344; Michael McGerr, A Fierce Discontent: The Rise and Fall of the Progressive Movement in America, 1870\u20131920 (New York: Free Press, 2003), 161\u201363.\n\norder might break down once more: \"Behind the apparent optimism of the belief in the inevitability of progress,\" writes one analyst of Wilson's thought, \"there was the perception of a demand that society be driven forward lest it should again split apart into warring camps.\" Niels Aage Thorsen, The Political Thought of Woodrow Wilson (Princeton, NJ: Princeton University Press, 1988), 15. See also Mulder, The Years of Preparation, 12, 29; Knock, To End All Wars, 3; Anthony Gaughan, \"Woodrow Wilson and the Legacy of the Civil War,\" Civil War History 43 (1997): 225\u201343.\n\nthe production of unbiased expertise: Gaughan, \"Woodrow Wilson and the Legacy of the Civil War,\" 225\u201343; Thorsen, The Political Thought of Woodrow Wilson, 12, 126\u201329, 192.\n\n\"at our leisure as they become necessary\": Mulder, The Years of Preparation, 34, 38\u201339, 41\u201342, 51\u201352, 56, 66, 82\u201383, 270; Arthur S. Link, \"His Presbyterian Inheritance,\" in The Higher Realism of Woodrow Wilson, ed. Arthur S. Link (Nashville, TN: Vanderbilt University Press, 1971), 16; Knock, To End All Wars, 4.\n\nin the history of American politics: Thorsen, The Political Thought of Woodrow Wilson, 28.\n\n\"if he could speak to enough of them\": Ray Stannard Baker, Woodrow Wilson: Life and Letters (New York: Greenwood, 1968), iii, 173.\n\nthe most dazzling first terms in American history: \"During his first term,\" writes John Milton Cooper, \"Wilson compiled a spectacular, possibly unmatched, record of legislative and party leadership.\" Cooper, The Warrior and the Priest, 229.\n\ndoubled since the Civil War: Link, Woodrow Wilson and the Progressive Era, 35; Karen E. Schnietz, \"The 1916 Tariff Commission: Democrats' Use of Expert Information to Constrain Republican Tariff Protection,\" Business and Economic History 23 (1994): 179.\n\npolitical rather than scientific criteria to make loans: Allan H. Meltzer, A History of the Federal Reserve (Chicago: University of Chicago Press, 2003), 69\u201371; Blum, The Progressive Presidents, 69\u201370; Link, Woodrow Wilson and the Progressive Era, 45, 48\u201349; Bruce Champ, \"The National Banking System: Empirical Observations,\" Working Paper 07-19, Federal Reserve Bank of Cleveland, December 2007, 26\u201329, http:\/\/www.clevelandfed.org\/research\/work paper\/2007\/wp0719.pdf.\n\nthe first banking reorganization in almost fifty: Schnietz, \"The 1916 Tariff Commission,\" 178\u201379; Bruce Champ, \"The National Banking System: A Brief History,\" Working Paper 07-23, Federal Reserve Bank of Cleveland, December 2007, http:\/\/www.clevelandfed.org\/ research\/workpaper\/2007\/wp0723.pdf.\n\naverted a potentially bloody nationwide strike: Link, Woodrow Wilson and the Progressive Era, 59, 61\u201363, 67\u201374; Blum, The Progressive Presidents, 72\u201374, 78\u201379; Cooper, The Warrior and the Priest, 251.\n\nan unshakable faith in the number 13: Hodgson, Woodrow Wilson's Right Hand, 211; Link, Woodrow Wilson and the Progressive Era, 59\u201361. For Wilson's appalling views of blacks, see Lloyd E. Ambrosius, \"Woodrow Wilson and The Birth of a Nation: American Democracy and International Relations,\" Diplomacy and Statecraft 18 (2007), 689\u2013718.\n\na serious economic downturn in twenty-five: Some might call the Spanish-American War traumatic, but it cost the United States a mere 385 combat deaths, with another 1,662 Americans wounded. U.S. troops kept fighting off and on in the Philippines until the 1930s, but after the election of 1900 those skirmishes were never a major domestic political issue. Allan R. Millett and Peter Maslowski, For the Common Defense: A Military History of the United States of America (New York: Free Press, 1994), 284\u2013301; E. Berkeley Tompkins, \"Scylla and Charybdis: The Anti-Imperialist Dilemma in the Election of 1900,\" Pacific Historical Review 36 (1967): 143\u201361; Fred H. Harrington, \"The Anti-Imperialist Movement in the United States, 1898\u20131900,\" Mississippi Valley Historical Review 22 (1935): 211\u201330.\n\nsoon be a thing of the past: Robert H. Wiebe, The Search for Order 1877\u20131920 (New York: Hill & Wang, 1967), 141; McGerr, A Fierce Discontent, 116.\n\n\"the inevitability of progress, in the perfectibility of man\": Steel, Walter Lippmann, 45.\n\n\"even here in this life upon earth\": William T. R. Fox, \"Interwar International Relations Research: The American Experience,\" World Politics 2 (1949): 68.\n\n\"illogical passion in us all?\": Forcey, The Crossroads of Liberalism, 224\u201325.\n\n\"possibility of amelioration was retained\": J. A. Thompson, \"American Progressive Publicists and the First World War, 1914\u20131917,\" Journal of American History 58 (1971): 369.\n\n\"and human rights, shall prevail\": Noble, The Progressive Mind, 184.\n\nwe used to call the 'balance of power'\": Inis L. Claude, Power and International Relations (New York: Random House, 1988), 82.\n\n\"because we are a tough bunch\": MacMillan, Paris 1919, 23.\n\n\"the balance of power in its own favor\": David Steigerwald, Wilsonian Idealism in America (Ithaca, NY: Cornell University Press, 1994), 55.\n\n\"political fortunes for our own benefit\": Lloyd E. Ambrosius, Wilsonianism: Woodrow Wilson and His Legacy in American Foreign Relations (New York: Palgrave Macmillan, 2002), 37.\n\nexternal aggression and internal disorder: Mark T. Gilderhus, Pan American Visions: Woodrow Wilson in the Western Hemisphere 1913\u20131921 (Tucson: University of Arizona Press, 1986), 49.\n\npreventing meddling by European powers: Walter H. Posner, \"American Marines in Haiti, 1915\u20131922,\" Americas 20 (1964): 231\u201366; Bruce J. Calder, \"Caudillos and Gavilleros versus the United States Marines: Guerrilla Insurgency during the Dominican Intervention, 1916\u20131924,\" Hispanic American Historical Review 58 (1978): 649\u201375.\n\n\"South American republics to elect good men!\": Wiebe, The Search for Order, 247.\n\nSouth American banks and businesses from Europe: Gilderhus, Pan American Visions, 37.\n\nworld governed by reason, not force: Hodgson, Woodrow Wilson's Right Hand, 231.\n\n\"a part of impartial mediation\": Blum, The Progressive Presidents, 86.\n\n\" friendship and disinterested service\": Cooper, The Warrior and the Priest, 273.\n\nless than two to one: Paul Kennedy, The Rise and Fall of the Great Powers: Economic Change and Military Conflict from 1500 to 2000 (New York: Vintage, 1989), 199, 200, 203.\n\n\"and not as it ought to be\": Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 85.\n\nHuns were growing too strong: Cooper, The Warrior and the Priest, 154.\n\n\"Central America?\" he asked in 1915: Henry A. Kissinger, Diplomacy (New York: Simon & Schuster, 1994), 42.\n\nif Germany went after that next: Cooper, The Warrior and the Priest, 277\u201378.\n\nby sinking whatever tried to pass: David Stevenson, Cataclysm: The First World War as Political Tragedy (New York: Basic Books, 2004), 199\u2013214.\n\nwhen they heard the news: Cooper, The Warrior and the Priest, 288.\n\nLodge and TR were already there: William C. Widenor, Henry Cabot Lodge and the Search for an American Foreign Policy (Los Angeles: University of California Press, 1980), 185\u2013198; Harbaugh, Power and Responsibility, 474\u2013476.\n\n\"and the like,\" Wilson instructed House: Lloyd E. Ambrosius, Wilsonian Statecraft: Theory and Practice of Liberal Internationalism During World War I (Wilmington, DE: SR Books, 1991), 40, 49.\n\n\"utterances since the Monroe Doctrine\": Steel, Walter Lippmann, 96.\n\n\"a town without a brothel\": LaFeber, The American Age, 294\u201395.\n\n\"prevail is the desire of enlightened men everywhere\": Ambrosius, Wilsonian Statecraft, 113\u201314.\n\nrunning low on subs anyway: Paul Birdsall, \"Why We Went to War,\" in The Shaping of Twentieth Century America, ed. Richard Abrams and Lawrence Levine (Boston: Little, Brown, 1965), 317; Blum, The Progressive Presidents, 91\u201392; Stevenson, Cataclysm, 210\u201312, 254\u201356.\n\nhis government was doing just that: David Kennedy, Over Here: The First World War and American Society (Oxford: Oxford University Press, 2004), 5.\n\n\"American has set foot on the continent\": LaFeber, The American Age, 294; Stevenson, Cataclysm, 264\u201366.\n\nbroke off diplomatic relations with Berlin: Link, Woodrow Wilson and the Progressive Era, 268.\n\nkilling fifteen Americans: \"Thrown from a Small Boat,\" New York Times, March 20, 1917, 1.\n\na wave of public support for war: Link, Woodrow Wilson and the Progressive Era, 271\u201373.\n\nthe moderate, pro-Western Alexander Kerensky: Stevenson, Cataclysm, 247\u201354.\n\n\"our terms for our democracy and civilization\": John Dewey, \"In a Time of National Hesitation,\" in John Dewey: The Middle Works, vol. 10, 1899\u20131924, ed. Jo Ann Boydston (Carbondale: Southern Illinois University Press, 1980), 259.\n\n\"democratic revolution the world over\": Steel, Walter Lippmann and the American Century, 113.\n\n\" fruits of this experiment\": John Dewey, \"America in the World,\" in John Dewey: The Middle Works, vol. 11, 72.\n\nAmerica was at war: Woodrow Wilson, \"War Message,\" speech to Joint Session of Congress, Washington, D.C., April 2, 1917, http:\/\/wwi.lib.byu.edu\/index.php\/Wilson%27s_War_Message_to_Congress; \"Must Exert All Our Power,\" New York Times, April 3, 1917, 1; Robert H. Ferrell, Woodrow Wilson and World War I, 1917\u20131921 (New York: HarperCollins, 1985), 1; Link, Woodrow Wilson and the Progressive Era, 282.\n\ndesigning the postwar world: At the Paris Peace Conference, Wilson claimed that he had never seen the secret treaties, a claim that his press secretary, Ray Stannard Baker, reasserted in his early biography of Wilson. But this is probably untrue. After America entered the war, British foreign secretary Arthur Balfour traveled to Washington to confer. During a series of meetings with Colonel House, Balfour explained the content of the secret agreements concluded by the French, British, Italians, and Russians. Balfour also promised to provide copies of these agreements to Wilson and House. Godfrey Hodgson notes that House and Wilson corresponded about the secret agreements well before Lenin's government published them in late 1917. Hodgson, Woodrow Wilson's Right Hand, 144\u201347.\n\nthe Inquiry's headquarters night and day: Gelfand, The Inquiry, 37, 39, 41\u201342, 45, 103, 121; Steel, Walter Lippmann, 129.\n\nwere thus largely ignored: Gelfand, The Inquiry, 33, 187\u201390.\n\n\"the policy of the United States\": Ambrosius, Wilsonianism, 101.\n\nthan did most members of Congress: Gelfand, The Inquiry, 120\u201321, 243, 317, 330\u201331.\n\nsecret treaties dividing up the spoils of war: Arno J. Mayer, Wilson vs. Lenin: Political Origins of the New Diplomacy 1917\u20131918 (New York: Meridian, 1969), 279.\n\n\"and nothing else will do\": Steel, Walter Lippmann, 129, 133.\n\n\"at half past twelve\": Knock, To End All Wars, 142.\n\ndetermined by reason, not force: Mayer, Wilson vs. Lenin, 267\u201368; Knock, To End All Wars, 143\u201344; LaFeber, The American Age, 309\u201310.\n\n\"great hope of mankind which we are trying to realize\": Steel, Walter Lippmann, 141\u201343.\n\nsneaked across the border into Sweden: MacMillan, Paris 1919, 157\u201358; E. Eyck, \"The Generals and the Downfall of the German Monarchy 1917\u20131918,\" Transactions of the Royal Historical Society 2 (1952): 47\u201367; D. K. Buse, \"Ebert and the German Crisis, 1917\u20131920,\" Central European History 5 (1972): 234\u201355; Klaus Epstein, \"Wrong Man in a Maelstrom: The Government of Max of Baden,\" Review of Politics 26 (1964): 215\u201343.\n\non the basis of the Fourteen Points: Steel, Walter Lippmann, 149\u201350; David Fromkin, In the Time of the Americans: The Generation That Changed America's Role in the World (New York: Vintage, 1996), 260.\n\nno subordinate would steal the glory: Fromkin, In the Time of the Americans, 267\u201369; MacMillan, Paris 1919, 3.\n\ncut through the gray sky: Ferrell, Woodrow Wilson and World War I, 138; Steel, Walter Lippmann, 151; Knock, To End All Wars, 194; MacMillan, Paris 1919, 3.\n\nthe largest crowd in French history: Fromkin, In the Time of the Americans, 284\u201385; Knock, To End All Wars, 194\u201395; MacMillan, Paris 1919, 15\u201316.\n\nsurrounded by sacred candles: MacMillan, Paris 1919, 15; Ferrell, Woodrow Wilson and World War I, 139\u201340.\n\n\"he became a Messiah\": Kennedy, Over Here, 356; Goldman, Rendezvous with Destiny, 262; John B. Judis, The Folly of Empire: What George W. Bush Could Learn from Theodore Roosevelt and Woodrow Wilson (New York: Scribner, 2004), 95\u201396; Knock, To End All Wars, epigraph.\n\n\"and yet again it might be a tragedy\": MacMillan, Paris 1919, 27.\n\nChapter 2: The Frightening Dwarf\n\nand the Bible by kindergarten: Carl Resek, \"Introduction,\" in Randolph Bourne, War and the Intellectuals: Collected Essays, 1915\u20131919, ed. Carl Resek (Indianapolis: Harper & Row, 1999), viii\u2013ix; Eric J. Sandeen, \"Bourne Again,\" American Literary History 1 (1989): 491\u201392; Bruce Clayton, Forgotten Prophet: The Life of Randolph Bourne (Baton Rouge: Louisiana State University Press), 1, 8.\n\nthe men who changed his life: Resek, \"Introduction,\" viii, ix\u2013x.; Roderick Nash, The Nervous Generation: American Thought, 1917\u20131930 (Chicago: Elephant Paperbacks, 1970), 36\u201338 Lasch, The New Radicalism in America, 76\u201379.\n\nprogressivism's greatest historian: Ellen Nore, Charles A. Beard: An Intellectual Biography (Carbondale: Southern Illinois University Press, 1983), 4\u201312.\n\nits greatest philosopher: Charles F. Howlett, Troubled Philosopher (Port Washington, NY: National University Publications, 1977), 59\u201360; Robert Westbrook, John Dewey and American Democracy (Ithaca, NY: Cornell University Press, 1991), 1; Goldman, Rendezvous with Destiny, 156\u201358.\n\nwho transcended their selfish desires: Randolph Bourne, \"Twilight of Idols,\" in Bourne, War and the Intellectuals, 60; Westbrook, John Dewey and American Democracy, 195.\n\ndemocracy and anarchy to organization: Noble, The Progressive Mind, 28.\n\n\"the incarnation of reason\": Kaplan, \"Social Engineers as Saviors,\" 350\u201351.\n\njob at the fledgling New Republic: Resek, \"Introduction,\" xi.\n\nmight transform the country and the world: Steel, Walter Lippmann, 109.\n\npublishing his political writing at all: Forcey, The Crossroads of Liberalism, 249, 283.\n\n\"a concert among the nations\": Frederic A. Ogg and Charles A. Beard, National Governments and the World War (New York: Macmillan, 1919), 570.\n\n\"systematic utilization of the scientific expert\": John Dewey, \"What Are We Fighting For?\" in John Dewey: The Middle Works, vol. 11, 99, 104.\n\nlaboratories for tinkering with human behavior: Bourne, \"Twilight of Idols,\" 56.\n\n\" from the path of organization to that end\": Westbrook, John Dewey and American Democracy, 208; Randolph Bourne, \"A War Diary,\" in Bourne, War and the Intellectuals, ed. Resek, 41.\n\n\"permitted to do their own thinking\": Kennedy, Over Here, 59\u201360; McGerr, A Fierce Discontent, 290.\n\nthe causes and purposes of the war: Thomas C. Kennedy, Charles Beard and American Foreign Policy (Gainesville: University of Florida Press, 1975), 32\u201333; Nore, Charles A. Beard, 74.\n\n\"the more violent appeals to passion\": John Dewey, \"What America Will Fight For,\" in John Dewey: The Middle Works, vol. 10, 273; Clayton, Forgotten Prophet, 223.\n\nit took a jury twenty-five minutes to acquit: Walter I. Trattner, \"Progressivism and World War I: A Reappraisal,\" Mid-America 44 (1962): 137; Kennedy, Over Here, 68, 80; Knock, To End All Wars, 133.\n\n\"including Dewey mad, drove Bourne sane\": Clayton, Forgotten Prophet, 4.\n\n\"they had been waiting for each other\": Randolph Bourne, \"The War and the Intellectuals,\" in Bourne, War and the Intellectuals, ed. Resek, 11; Bourne, \"Twilight of Idols,\" 60.\n\n\"because they knew this was an illusion\": Bourne, \"A War Diary,\" 41, 45.\n\nthat his nemesis be fired: Louis Filler, Randolph Bourne (Washington, DC: American Council on Public Affairs, 1943), 113; Resek, \"Introduction,\" xiii; Clayton, Forgotten Prophet, 229, 232\u201335; Westbrook, John Dewey and American Democracy, 212, 233\u201335.\n\ninfluenza just weeks after the war's end: Sandeen, \"Bourne Again,\" 504; Resek, \"Introduction,\" xiii; Clayton, Forgotten Prophet, 255\u201356.\n\nno journal would print his work: Lasch, The New Radicalism in America, 74.\n\na newly unified Germany: MacMillan, Paris 1919, 63.\n\nkeep vigil even from the grave: Gregor Dallas, At the Heart of a Tiger: Clemenceau and His World 1841\u20131929 (New York: Carroll & Graf, 1993), 111\u201312; MacMillan, Paris 1919, 27\u201328, 33.\n\nwere made in German factories: MacMillan, Paris 1919, 26, 28, 31\u201332; Gerhard Weinberg, \"The Defeat of Germany in 1918 and the European Balance of Power,\" Central European History 2 (1969): 256\u201357.\n\n\"it is desolation, it is death\": MacMillan, Paris 1919, 22, 28, 185.\n\ndid not want to be swayed by emotion: Eventually, in March, Wilson did make a brief battlefield tour. But it did not mollify the insulted French. MacMillan, Paris 1919, 22.\n\nfewer \"hysterical\" French: MacMillan, Paris 1919, 27, 168.\n\n\"only disinterested people at the Peace Conference\": MacMillan, Paris 1919, 9.\n\n\"we are here to see you get nothing\": Ferrell, Woodrow Wilson and World War I, 135.\n\n\"And a very good thing too!\": Steigerwald, Wilsonian Idealism in America, 77; Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 54; MacMillan, Paris 1919, 23\u201324, 33.\n\nthe economic balance of power with Berlin: MacMillan, Paris 1919, 170\u201373, 191; Sally Marks, \"The Myths of Reparations,\" Central European History 11 (1978): 254\u201355.\n\nBut Wilson rejected that, too: Ambrosius, Wilsonianism, 68\u201369; Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 75\u201377.\n\n\"of the public opinion of the world\": Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 78.\n\nthe Allies could occupy it for fifteen years: Ambrosius, Wilsonianism, 72.\n\nalready pledged to do under the League: Ambrosius, Wilsonianism, 71.\n\n\"over the heads of their rulers\": Ray Stannard Baker and William E. Dodd, eds., The Public Papers of Woodrow Wilson: War and Peace, vol. 5 (New York: Harper & Brothers, 1927), 259; E. H. Carr, The Twenty Years' Crisis, 1919\u20131939: An Introduction to the Study of International Relations (New York: Palgrave, 2001), 32\u201334; Knock, To End All Wars, 162.\n\nrather than the Royal Navy: LaFeber, The American Age, 315.\n\ncovered them over in disgust: Kennedy, Over Here, 357. Wilson also threatened to appeal to the people of France against their government's hard line on Germany, predicting that if he did, Clemenceau's government would fall. Fromkin, In the Time of the Americans, 322.\n\n\"a world governed by intrigue and force\": Ambrosius, Wilsonian Statecraft, 113\u2013114.\n\nnot as punitive as legend suggests: As Margaret McMillan writes, \"When historians look, as they have increasingly been doing, at the other details, the picture of a Germany crushed by a vindictive peace cannot be sustained. Germany did lose territory; that was an inevitable consequence of losing the war. If it had won, we should remember, it would have certainly taken Belgium, Luxembourg, parts of the north of France and much of the Netherlands. The Treaty of Brest-Litovsk showed the intentions of the German supreme command for the eastern frontiers. Despite its losses Germany remained the largest country in Europe west of the Soviet Union between the wars. Its strategic position was significantly better than it had been before 1914. With the reemergence of Poland, there was now a barrier between it and the old Russian menace. In place of Austria-Hungary, Germany had only a series of weaker and quarreling states on its eastern frontier. As the 1930s showed, Germany was well placed to extend its economic and political sway among them.\" MacMillan, Paris 1919, 481\u201382.\n\nwhen they exited the war in May 1918: McMillan, Paris 1919, 161, 481\u201382; Weinberg, \"The Defeat of Germany in 1918 and the European Balance of Power,\" 260; Stevenson, Cataclysm, 323\u201324.\n\n\"our own security in this hemisphere\": \"The Political Scene,\" New Republic, March 22, 1919, 14.\n\nnearly all backed U.S. entrance into the League: Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 91, 146.\n\nwere certain to vote no: The Irreconcilables were themselves divided into three categories: a group of isolationists who opposed all binding U.S. commitments overseas, a group of idealists or pacifists who thought the League didn't go far enough toward disarmament and decolonization, and a group of realists who considered the League utopian. Ralph A. Stone, \"The Irreconcilables' Alternatives to the League of Nations,\" Mid-America 49 (1967): 165\u201371; Robert J. Maddox, William E. Borah and American Foreign Policy (Baton Rouge: Louisiana State University Press, 1969), 62, 64.\n\nwith the fury of lovers scorned: Steel, Walter Lippmann, 163.\n\ncould take America to war: Historians have long debated whether the Senate's \"strong Reservationists,\" led by Henry Cabot Lodge, were sincere in their reservations, or were simply using them as cover to defeat the League. David Mervin portrays Lodge as deeply opposed to U.S. membership in the League but unwilling to say so publicly for political reasons. But Lloyd Ambrosius and William Widenor make a convincing case that Lodge would have swallowed U.S. membership, with reservations, in return for America's ratification of the French security treaty, about which he cared passionately. They cite numerous examples of Lodge declaring, in private, that he could support U.S. entry into the League if, as he told the novelist Edith Wharton in early 1919, it is \"so constituted that it will...not endanger or imperil the United States.\" David Mervin, \"Henry Cabot Lodge and the League of Nations,\" Journal of American Studies 4 (1971): 210; Widenor, Henry Cabot Lodge and the Search for an American Foreign Policy, 308\u201311; Ambrosius, Woodrow Wilson and the American Diplomatic Tradition,137.\n\ndropped dead of a heart attack: Harbaugh, Power and Responsibility, 518\u201319.\n\n\"brothers under the skin\": Hans Morgenthau, In Defense of the National Interest (New York: Knopf, 1951), 29.\n\nnot Lincoln the impartial reconciler: Widenor, Henry Cabot Lodge and the Search for an American Foreign Policy, 19\u201321, 216, 217, 284.\n\n\"but as one of the allies\": Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 86.\n\n\"impossible for her to break out again\": Widenor, Henry Cabot Lodge and the Search for an American Foreign Policy, 298.\n\n\"by a statute or a written constitution\": Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 86; Widenor, Henry Cabot Lodge and the Search for an American Foreign Policy, 351.\n\n\"count his personal fortunes in the reckoning\": Knock, To End All Wars, 251, 259; MacMillan, Paris 1919, 490; Kenneth S. Davis, FDR: The Beckoning of Destiny, 1882\u20131928 (New York: G. P. Putnam's Sons, 1972), 565, 583.\n\nemerged to watch his successor sworn in: Fromkin, In the Time of the Americans, 348, 350; Kennedy, Over Here, 361; Ferrell, Woodrow Wilson and World War I, 169; Knock, To End All Wars, 262\u201363.\n\nWilson's European crusade as one big waste: Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 187, 252.\n\n\"the lonely citadel of his soul\": Hodgson, Woodrow Wilson's Right Hand, 215\u201356; George and George, Woodrow Wilson and Colonel House, xv.\n\nthe two-thirds needed for ratification: Knock, To End All Wars, 263\u201364.\n\nwith France, and so it died as well: Ambrosius, Woodrow Wilson and the American Diplomatic Tradition, 212\u201314; Ambrosius, Wilsonianism, 141\u201342.\n\n\"discount all the more violent appeals to passion\": John Dewey, \"What America Will Fight For,\" in John Dewey: The Middle Works, vol. 10, 273.\n\nfor \"pro-German\" leanings himself: Carol Signer Gruber, \"Academic Freedom at Columbia University, 1917\u20131918: The Case of James McKeen Cattell,\" AAUP Bulletin 58 (1972): 297, 301\u20133; Westbrook, John Dewey and American Democracy, 117, 222.\n\n\"inculcate disrespect for American institutions\": Carol S. Gruber, Mars and Minerva: World War I and the Uses of Higher Learning in America (Baton Rouge: Louisiana State University Press, 1975), 189; Nore, Charles A. Beard, 78\u201379.\n\n\"unconventional views in political matters\": Goldman, Rendezvous with Destiny, 259\u201360.\n\nwhile their teacher wept: Nore, Charles A. Beard, 81; Goldman, Rendezvous with Destiny, 259\u201360; William Summerscales, Affirmation and Dissent: Columbia's Response to the Crisis of World War I (New York: Teachers College Press, 1970), 94\u201396.\n\nbanned some of his books from its training camps: Kennedy, Charles Beard and American Foreign Policy, 38; Nore, Charles A. Beard, 84.\n\n\"amid howling gales of passion\": Westbrook, John Dewey and American Democracy, 210.\n\nwho had died just weeks before: Kennedy, Over Here, 90; John Dewey, \"The Cult of Irrationality,\" in Characters and Events: Essays in Social and Political History, ed. Joseph Ratner (New York: Henry Holt, 1929), 587\u201391; Westbrook, John Dewey and American Democracy, 210, 239.\n\nwhere he spent the next two years: Howlett, Troubled Philosopher, 39; Westbrook, John Dewey and American Democracy, 239\u201340, 256.\n\nto build a world free of force: Westbrook, John Dewey and American Democracy, 260.\n\n\"its own necessities and experiences\": Nore, Charles A. Beard, 99\u2013100; Kennedy, Charles Beard and American Foreign Policy, 45, 49\u201350.\n\nParis was now on its own: Ambrosius, Wilsonianism, 93, 104.\n\nrequired the Germans to substantially disarm: Hans W. Gatzke, \"Russo-German Military Collaboration During the Weimar Republic,\" American Historical Review 63 (1958): 565\u201397.\n\npaying more than that themselves: Sally Marks, \"Reparations Reconsidered: A Reminder,\" Central European History 2 (1969): 359\u201361, 365; Gustav Stolper, Karl Hauser, and Knut Borchardt, The German Economy: 1870 to the Present (New York: Harcourt, 1967), 83; Marks, \"The Myths of Reparations,\" 237.\n\nwhich Germans swallowed with barely a whimper: Although the Treaty of Versailles has gone down in historical memory as savagely harsh, and the Allies' treatment of Germany after World War II is, by contrast, remembered as generous and enlightened, the truth is in some ways the reverse. After World War II, Germany, along with Berlin, was divided and occupied by the Allied powers, with full civil authority only being restored in 1949 (and reunification not coming until 1990). While the reparations imposed on Germany after World War I have become synonymous with sadistic excess, they were at least restricted to a particular figure (with the real figure, as noted above, far below that). After World War II, by contrast, the Potsdam Agreement set no reparations limit; it simply allowed each occupying power to expropriate from its occupation zone as it saw fit. The Soviets and French responded by treating their occupation zones as cash cows, stripping them of vast amounts of industrial machinery, supplies, and other materials, which they shipped back home. Finally, it is true that the victors dispossessed Germany of slightly less land in 1945 than in 1919 (10 percent of Germany's prewar total compared to 13 percent), but much of the post\u2013World War I territory was populated by non-German speakers, while after World War II German speakers comprised the vast bulk of people living on land parceled out to Germany's neighbors. MacMillan, Paris 1919, 465; Joseph B. Schechtman, \"Postwar Population Transfers in Europe: A Survey,\" Review of Politics 15 (1953): 151.\n\ntheir dreams of European hegemony: Historians once saw World War I as a mere accident, stumbled into by powers that did not truly want war. But that view has been significantly undermined in recent decades by the German historian Fritz Fischer, who argued in his 1961 book, Germany's Aims in the First World War, that a wide swath of the German political elite had since the early 1900s been eager for war with Russia, which they believed would give Germany hegemony over the European continent. Germany, Fischer argues, was not dragged into World War I by its weak ally, Austria-Hungary, but actually pushed a hesitant Hapsburg empire to make extreme demands of the Serbs following the assassination of Archduke Ferdinand, demands that Berlin knew would likely lead to war. In recent years, historians have gone even further, suggesting that German leaders desired war even though they knew Britain would probably enter the fray, and even though they knew the conflict might be long and painful. See, for instance, Keir A. Lieber, \"The New History of World War I and What It Means for International Relations Theory,\" International Security 32, no. 2 (2007): 155\u201391; Frederick A. Hale, \"Fritz Fischer and the Historiography of World War One,\" History Teacher 9 (1976): 258\u201379.\n\nto seize German mines and steel plants as collateral: Marks, \"The Myths of Reparations,\" 241.\n\nwhose popularity was beginning to build: Stolper, Hauser, and Borchardt, The German Economy, 74\u201389.\n\n\"utterly brutal and insane\": Warren Cohen, Empire Without Tears (Philadelphia: Temple University Press, 1987), 92.\n\na litany of senators denounced French militarism: Royal J. Schmidt, Versailles and the Ruhr: Seedbed of World War II (The Hague: Martinus Nijhoff, 1968), 210\u201312.\n\nAmerican enforcement of Versailles: Betty Glad, Charles Evans Hughes and the Illusions of Innocence: A Study in American Diplomacy (Urbana: University of Illinois Press, 1966), 223; Melvyn P. Leffler, The Elusive Quest: America's Pursuit of European Stability and French Security, 1919\u20131933 (Chapel Hill: University of North Carolina Press, 1979), 83; Jolyon P. Girad, \"Congress and Presidential Military Policy: The Occupation of Germany, 1919\u20131923,\" Mid-America 56 (1974): 211\u201319.\n\ncould now default with virtual impunity: Frank Costigliola, \"The United States and the Reconstruction of Germany in the 1920s,\" Business History Review 50 (1976): 493; Marks, \"The Myths of Reparations,\" 249.\n\n\"a disinterested position in relation to international affairs\": Robert H. Ferrell, Peace in Their Time: The Origins of the Kellogg-Briand Pact (New Haven, CT: Yale University Press, 1952), 131.\n\nall won the Nobel Prize: Kissinger, Diplomacy, 272\u201376.\n\nthe two-front strategy on which French security relied: Ferrell, Peace in Their Time, 65; Ambrosius, Wilsonianism, 106\u20137.\n\nthe Maginot Line: Ferrell, Peace in Their Time, 49\u201350; Ambrosius, Wilsonianism, 107; MacMillan, Paris 1919, 176, 481; Kissinger, Diplomacy, 280.\n\n\"are still absolutely in control\": John C. Farrell, \"John Dewey and World War I: Armageddon Tests a Liberal's Faith,\" Perspectives in American History 9 (1975): 329.\n\nthat there was nobility in war: John Dos Passos, \"The Scene of Battle,\" in The Culture of the Twenties, ed. Loren Baritz (Indianapolis: Bobbs-Merrill, 1970), 5\u201311; Henry May, The Discontent of the Intellectuals: A Problem of the Twenties (Chicago: Rand McNally, 1963), 21\u201322; Cohen, Empire Without Tears, 45; Robert Osgood, Ideals and Self-Interest in America's Foreign Relations (Chicago: University of Chicago Press, 1953), 330.\n\n\"the Department of Peace\": J. Chalmers Vinson, \"Military Force and American Policy, 1919\u20131939,\" in Isolation and Security: Ideas and Interests in Twentieth Century American Foreign Policy, ed. Alexander DeConde (Durham, NC: Duke University Press, 1957), 57.\n\n250,000 Americans inquired about taking part: Ferrell, Peace in Their Time, 25; Charles De Benedetti, \"The $100,000 American Peace Award of 1924,\" Pennsylvania Magazine of History and Biography 98 (1974): 224\u201349.\n\nhow barbaric human beings had once been: Osgood, Ideals and Self-Interest in America's Foreign Relations, 324\u201325.\n\nIt passed the Senate, 74\u20130: Cohen, Empire Without Tears, 46\u201347; Maddox, William E. Borah and American Foreign Policy, 86\u201387.\n\nnaval spending dropped by one-third: Glad, Charles Evans Hughes, 270; Phillips Payson O'Brien, British and American Naval Power: Politics and Policy, 1900\u20131936 (Westport, CT: Praeger, 1998), 149\u201376, 180\u201381, 230\u201336.\n\na de facto belligerent on France's side: Ferrell, Peace in Their Time, 62, 64, 73, 80.\n\nthat war became almost unthinkable: Westbrook, John Dewey and American Democracy, 260\u201371.\n\nat the rate of six hundred per day: Howlett, Troubled Philosopher, 87\u201388; Ferrell, Peace in Their Time, 33\u201335, 232\u201333, 239.\n\nin the history of international affairs: Ferrell, Peace in Their Time, 208, 218; Osgood, Ideals and Self-Interest in America's Foreign Relations, 348; Farrell, \"John Dewey and World War I,\" 331.\n\nChapter 3: Twice-Born\n\nwon only 4 percent of the vote: Richard Wightman Fox, Reinhold Niebuhr: A Biography (Ithaca, NY: Cornell University Press, 1996), 135\u201336; Reinhold Niebuhr, Moral Man and Immoral Society: A Study in Ethics and Politics (Westminster, MO: John Knox, 2002), xxviii.\n\nscores to settle from World War I: Daniel F. Rice, Reinhold Niebuhr and John Dewey: An American Odyssey (Albany: State University of New York Press, 1993), 3\u20134; Martin Halliwell, The Constant Dialogue: Reinhold Niebuhr and American Culture (New York: Rowman & Littlefield, 2005), 60.\n\n\"out-and-out loyalty\": Fox, Reinhold Niebuhr, 3\u201321, 28\u201329, 42, 45, 52\u201356; William G. Crystal, \"'A Man of the Hour and of the Time': The Legacy of Gustav Niebuhr,\" Church History 49 (1980): 417.\n\nHe called Wilson a dupe: Fox, Reinhold Niebuhr, 53, 58\u201359.\n\n\"as potential criminals\" by their adopted nation: Reinhold Niebuhr, \"What the War Did to My Mind,\" Christian Century, September 27, 1928, 1161\u201362.\n\n\"a child of the age of disillusionment\": Fox, Reinhold Niebuhr, 72, 74, 79, 83, 99; Niebuhr, \"What the War Did to My Mind,\" 1161.\n\njoining the Socialist Party: Fox, Reinhold Niebuhr, 94\u2013100, 109\u201310.\n\nless than one-third its level in 1929: Lynn Dumenil, The Modern Temper: American Culture and Society in the 1920s (New York: Hill & Wang, 1995), 305.\n\ngarbage dumps for something to eat: Goldman, Rendezvous with Destiny, 321.\n\nthe time for pedagogy is done: Dewey's foremost biographer, Robert Westbrook, argues that Niebuhr's attack was unfair. \"Although at an earlier stage in Dewey's career Niebuhr's contention that he hoped to persuade the powerful to admit the injustice of their rule and to relinquish their power might have had some force, by the early thirties it was misplaced. Dewey's call for an 'intelligent' politics was not a plea to the oppressed to abandon the effort to match power with power in favor of reasoning with the 'dominant economic interests' but rather an appeal to them to wage their struggles intelligently.\" Westbrook, John Dewey and American Democracy, 526\u201327. Niebuhr's best biographer, Richard Wightman Fox, concurs, writing that \"Niebuhr was not making a close study of his opponent's thought; he was constructing an ideal-type opponent who was easy to take down.\" Fox, Reinhold Niebuhr, 136\u201337.\n\neducate those in power to remedy them: Lincoln Steffens, The Autobiography of Lincoln Steffens (Berkeley, CA: Heyday Books, 2005), 374\u201375.\n\nrevolution was the only option left: John Chamberlain, Farewell to Reform: The Rise, Life and Decay of the Progressive Mind in America (Chicago: Quadrangle, 1932), 306\u201310.\n\nthey were voting communist for president: David M. Kennedy, Freedom from Fear: The American People in Depression and War, 1929\u20131945 (New York: Oxford University Press, 1999), 222; Arthur M. Schlesinger, Jr., The Crisis of the Old Order, 1919\u20131933 (New York: Houghton Mifflin, 2003), 436\u201337.\n\nhad made him a multimillionaire: Joan Hoff Wilson, Herbert Hoover: Forgotten Progressive (Boston: Little, Brown, 1975), 4\u201317.\n\nmention that he became president: Australian Prospectors and Miners Hall of Fame, \"Herbert Hoover,\" http:\/\/mininghall.com\/MiningHallOfFame\/HallOfFameDatabase\/Inductee.php?InducteeID=1164.\n\n\"want[ing] nothing...from Congress [except] efficiency\": Hoff Wilson, Herbert Hoover, 37, 43.\n\n\"began his career in California and will end it in heaven\": Hoff Wilson, Herbert Hoover, 44; Richard Norton Smith, An Uncommon Man: The Triumph of Herbert Hoover (New York: Simon & Schuster, 1984), 80.\n\nhoover, meaning \"to help\": Hoff Wilson, Herbert Hoover, 46\u201347.\n\n\"Hooverize when it comes to loving you!\": Hoff Wilson, Herbert Hoover, 59\u201360.\n\nwould ever shed their murderous ways: Selig Adler, The Uncertain Giant: 1921\u20131941; American Foreign Policy Between the Wars (New York: Macmillan, 1965), 126.\n\nthey could not process all the mail: Charles Chatfield, For Peace and Justice (Knoxville: University of Tennessee Press, 1971), 160.\n\ndestroyers, cruisers, and submarines: O'Brien, British and American Naval Power, 212\u201314.\n\n\"until she has to be thrashed again\": Adler, Uncertain Giant, 128\u201329; Cohen, Empire Without Tears, 104\u20135; Zara Steiner, The Lights That Failed: European International History, 1919\u20131933 (Oxford: Oxford University Press, 2005), 589\u201391; Henry Blumenthal, Illusion and Reality in Franco-American Diplomacy 1914\u20131945 (Baton Rouge: Louisiana State University Press, 1986), 158\u201359.\n\nthe Reichstag's second-largest party: Steiner, Lights That Failed, 642; E.J. Feuchtwanger, From Weimar to Hitler: Germany, 1918\u20131933 (New York: St. Martin's Press, 1993), 337.\n\nwhich Versailles explicitly banned: Frank Costigliola, Awkward Dominion: American Political, Economic, and Cultural Relations with Europe, 1919\u20131933 (Ithaca, NY: Cornell University Press, 1984), 235.\n\n\"I never did\": Ambrosius, Wilsonianism, 108.\n\nby a young hypernationalist: Steel, Walter Lippmann, 288; Adler, Uncertain Giant, 140\u201341; Wilfrid Fleisher, \"Premier Is Shot by Tokyo Fanatic; Feared to Be Dying,\" New York Times, November 14, 1930, 1; James B. Crowley, Japan's Quest for Autonomy: National Security and Foreign Policy, 1930\u20131938 (Princeton, NJ: Princeton University Press, 1966), 78.\n\narmy and navy were firmly in control: Crowley, Japan's Quest for Autonomy, 168\u201375.\n\nJapan walked out: Myung Soo Cha, \"Did Takahashi Korekiyo Rescue Japan from the Great Depression?\" Journal of Economic History 63 (2003): 127\u201344; Adler, Uncertain Giant, 146.\n\n\"peace among other nations by force\": Hoff Wilson, Herbert Hoover, 205; Elting E. Morrison, Turmoil and Tradition: A Study of the Life and Times of Henry L. Stimson (Boston: Houghton Mifflin, 1960), 382; Henry L. Stimson and McGeorge Bundy, On Active Service in Peace and War (New York: Harper & Row, 1948), 244; J. Chalmers Vinson, \"Military Force and American Policy, 1919\u20131939,\" 57.\n\ndid not commission a single new warship: Cohen, Empire Without Tears, 106.\n\narmaments be cut by at least a quarter: Steiner, Lights That Failed, 781.\n\ntaken to the White House by police escort: Associated Press, \"Says Hoover Plan Saved Arms Parley,\" New York Times, November 17, 1932, 8; Chatfield, For Peace and Justice, 161.\n\nwas downright terrifying: Steiner, Lights That Failed, 767.\n\n\"which it was not prepared to make?\": Steiner, Lights That Failed, 777; Blumenthal, Illusion and Reality, 165, 181.\n\na plurality of the vote: Cohen, Empire Without Tears, 121\u201322; Leffler, Elusive Quest, 274\u2013301; H. Arthur Steiner, \"The Geneva Disarmament Conference of 1932,\" Annals of the American Academy of Political and Social Science 168 (1933): 214\u201317; Marlies Ter Borg, \"Reducing Offensive Capabilities: The Attempt of 1932,\" Journal of Peace Research 29 (1992): 147\u201353; Blumenthal, Illusion and Reality, 180\u201387.\n\nand raised it to the wind: Robert Dallek, Franklin D. Roosevelt and American Foreign Policy, 1932\u20131945 (New York: Oxford University Press, 1995), 336; Fromkin, In the Time of the Americans, 41\u201342; Davis, The Beckoning of Destiny, 152; Steven Casey, Cautious Crusade: Franklin D. Roosevelt, American Public Opinion and the War Against Nazi Germany (New York: Oxford University Press, 2001), xxiii.\n\nthen becoming governor of New York: Blum, Progressive Presidents, 108\u20139; Fromkin, In the Time of the Americans, 40, 42; Dallek, Franklin D. Roosevelt and American Foreign Policy, 8.\n\na potential German threat: Robert A. Divine, Roosevelt and World War II (Baltimore: Johns Hopkins University Press, 1969), 50; Willard Range, Franklin D. Roosevelt's World Order (Athens: University of Georgia Press, 1959), 20; Davis, The Beckoning of Destiny, 383\u201394; Dallek, Franklin D. Roosevelt and American Foreign Policy, 9\u201310.\n\n\"handing out to a gullible public\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 9\u201310.\n\n\"merely a beautiful dream, a Utopia\": Range, Franklin D. Roosevelt's World Order, 29, 31, 168.\n\nessential to a nation's power: Davis, The Beckoning of Destiny, 71\u201372, 123; Dallek, Franklin D. Roosevelt and American Foreign Policy, 7\u20138; Robert Osgood, Ideals and Self-Interest in America's Foreign Relations, 411; Alfred T. Mahan, The Influence of Seapower upon History (Boston: Little, Brown, 1890).\n\nevery summer from ages seven to fifteen: Davis, The Beckoning of Destiny, 102; Dallek, Franklin D. Roosevelt and American Foreign Policy, 3.\n\nthe Western Hemisphere would offer no refuge: Davis, The Beckoning of Destiny, 389.\n\nurging louder cheering at football games: Davis, The Beckoning of Destiny, 164\u201365.\n\nher former assistant, Lucy Mercer: Fromkin, In the Time of the Americans, 291\u201393; James MacGregor Burns and Susan Dunn, The Three Roosevelts: Patrician Leaders Who Transformed America (New York: Atlantic Monthly Press, 2001), 84, 155; Jean Edward Smith, FDR (New York: Random House, 2007), 153, 159; Conrad Black, Franklin Delano Roosevelt: Champion of Freedom (New York: PublicAffairs, 2003), 96; \"Livingston Davis Commits Suicide,\" New York Times, January 12, 1932, 18.\n\nclassmates called him behind his back: Davis, The Beckoning of Destiny, 152.\n\nparalyzed from the waist down by polio: Davis, The Beckoning of Destiny, 648\u201350. In 2003, in the Journal of Medical Biography, several medical experts speculated that, in fact, FDR's disease may have been not polio, but Guillain-Barr\u00e9 syndrome. Armond S. Goldman, Elisabeth J. Schmalstieg, Daniel H. Freeman, Jr., Daniel A Goldman, and Frank C Schmalstieg, Jr., \"What Was the Cause of Franklin Delano Roosevelt's Paralytic Illness?\" Journal of Medical Biography 11 (2003): 232\u201340.\n\n\"a medieval torture chamber\": Davis, The Beckoning of Destiny, 668\u201369.\n\nthen, somehow, found it again: Davis, The Beckoning of Destiny, 655.\n\n\"twice-born man\": Davis, The Beckoning of Destiny, 669, 677\u201378, 680.\n\nLivingston Davis, shot himself: \"Livingston Davis Commits Suicide,\" 18.\n\ndefend America's home waters: Range, Franklin D. Roosevelt's World Order, 95; Divine, Roosevelt and World War II, 54\u201355; Robert A. Divine, Second Chance: The Triumph of Internationalism in America During World War II (New York: Atheneum, 1967), 19; Adler, Uncertain Giant, 152\u201353, 157; Dallek, Franklin D. Roosevelt and American Foreign Policy, 15\u201316, 54, 75.\n\n\"intellectual parents of the New Deal\": Warren I. Cohen, The American Revisionists (Chicago: University of Chicago Press, 1967), 133; Kennedy, Charles A. Beard and American Foreign Policy, 73.\n\nretreat at the first sign of foreign resistance: Blumenthal, Illusion and Reality, 183, 223; Kissinger, Diplomacy, 292\u201393, 302\u20135, 378.\n\na naval blockade if he refused: Fromkin, In The Time of the Americans, 335\u201336; Dallek, Franklin D. Roosevelt and American Foreign Policy, 102\u20133.\n\n\"preserve our own oasis of liberty\": Kennedy, Freedom from Fear, 386.\n\nhurled essentially the same charge: Kennedy, Freedom from Fear, 388; Adler, Uncertain Giant, 160\u201366; Richard W. Leopold, \"The Problem with American Intervention, 1917: An Historical Retrospect,\" World Politics 2 (1950): 412\u201313, 418\u201320; Dallek, Franklin D. Roosevelt and American Foreign Policy, 102; Goldman, Rendezvous with Destiny, 377.\n\nUS intervention in World War I a mistake: Kennedy, Freedom from Fear, 394; Hadley Cantril, Public Opinion, 1935\u20131946 (Princeton, NJ: Princeton University Press, 1951), 201.\n\nthey did so at their own risk: Adler, The Uncertain Giant, 173\u201375, 182.\n\n\"I also learned much of what not to do\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 152.\n\nensure a reelection landslide: Jane Karoline Vieth, \"The Diplomacy of the Depression,\" in Modern American Diplomacy, ed. John M. Carroll and George C. Herring (Wilmington, DE: Scholarly Resources, 1996), 88.\n\n\"Western Hemisphere will not be attacked\": Divine, Roosevelt and World War II, 16.\n\nabout an international \"quarantine\": Divine, Roosevelt and World War II, 17\u201318; Dallek, Franklin D. Roosevelt and American Foreign Policy, 148.\n\nCongress threatened impeachment: Adler, Uncertain Giant, 190; Vieth, \"The Diplomacy of the Depression,\" 91.\n\nscreamed the Wall Street Journal: Travis Beal Jacobs has argued that, contrary to early historical interpretations, popular reaction to the quarantine speech was not particularly negative. But there is no doubt that the speech stirred isolationist anger, and rightly or wrongly, FDR interpreted that isolationism as the dominant public mood. Travis Beal Jacobs, \"Roosevelt's 'Quarantine Speech,'\" Historian 24 (1962): 488. See also Dorothy Borg, \"Notes on Roosevelt's 'Quarantine' Speech,\" Political Science Quarterly 72 (1957): 405\u201333.\n\n\"even hostile and resentful ears\": Kissinger, Diplomacy, 379.\n\nexcept in cases of foreign attack: Adler, Uncertain Giant, 195\u201396; Chatfield, For Peace and Justice, 283\u201385; Vinson, \"Military Force and American Policy, 1919\u20131939,\" 80.\n\n\"trying to lead and to find no one there\": Vieth, \"The Diplomacy of the Depression,\" 91\u201392; Burns and Dunn, The Three Roosevelts, 357.\n\noffering only the mildest of protests: Vieth, \"The Diplomacy of the Depression,\" 92.\n\n\"of whom we know nothing\": Telford Taylor, Munich: The Price of Peace (New York: Doubleday, 1979), 884.\n\nMoscow would not act alone: Blumenthal, Illusion and Reality, 242\u201343; Taylor, Munich, 99\u2013102.\n\nDaladier replied, \"With what?\": Kissinger, Diplomacy, 293, 302\u20133, 308\u20139; Blumenthal, Illusion and Reality, 243\u201344.\n\nlacked the planes to defend London: Taylor, Munich, 801\u20132, 807, 835\u201338; Kissinger, Diplomacy, 312\u201314.\n\na two-word cable: \"Good man\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 161\u201366; Kissinger, Diplomacy, 314; Divine, Roosevelt and World War II, 21; Vieth, \"The Diplomacy of the Depression,\" 94.\n\ntheir fate and Europe's were intertwined: Divine, Roosevelt and World War II, 25; Kissinger, Diplomacy, 383.\n\nEvents would have to speak for themselves: For FDR, writes the historian Robert Dallek, America \"would have to enter the fighting with a minimum of doubt and dissent, and the way to achieve this was not through educational talks to the public or strong Executive action, but through developments abroad which aroused the country to fight.\" Dallek, Franklin D. Roosevelt and American Foreign Policy, 267. John Milton Cooper adds that \"the chief lesson Roosevelt drew from Wilson was not to pitch his appeals to people too high or to expect too much in the way of educating the public. The deviousness and self-deception in his policies toward the warring powers from 1939 through 1941 reflected that lesson.\" Cooper, The Warrior and the Priest, 360\u201361.\n\nBritain and France decided to fight: Divine, Second Chance, 29.\n\n\"picking up again an interrupted routine\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 197\u201398; Fromkin, In the Time of the Americans, 484\u201388.\n\n\"ruled by force in the hands of a few\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 199, 215; Fromkin, In the Time of the Americans, 489.\n\ncarried them away in their own ships: Divine, Roosevelt and World War II, 29.\n\n\"unpitying masters of other continents\": Divine, Roosevelt and World War II, 31.\n\n\"she was expecting salvation\": Blumenthal, Illusion and Reality, 260\u201361.\n\nhad surrendered twenty-two years before: Cantril, Public Opinion, 1935\u20131946, 971\u201372; Adler, Uncertain Giant, 229.\n\nmilitary bases in the Western Hemisphere: Divine, Roosevelt and World War II, 33\u201335.\n\nwhich essentially provided them for free: Dallek, Franklin D. Roosevelt and American Foreign Policy, 248\u201349, 255\u201360.\n\nwas now a rabid isolationist: Osgood, Ideals and Self-Interest in America's Foreign Relations, 379.\n\nprevent them from falling into Nazi hands: Robert L. Beisner, Dean Acheson: A Life in the Cold War (New York: Oxford University Press, 2006), 14; Fromkin, In the Time of the Americans, 527; Dallek, Franklin D. Roosevelt and American Foreign Policy, 245; Blum, The Progressive Presidents, 149; Divine, Roosevelt and World War II, 43; Kissinger, Diplomacy, 389\u201390.\n\nhe had already decided to enter the war: Dallek, Franklin D. Roosevelt and American Foreign Policy, 530; Casey, Cautious Crusade, 11, 14\u201315.\n\nwould return to haunt the nation: Adler, Uncertain Giant, 259; Divine, Roosevelt and World War II, 48\u201349; Cantril, Public Opinion, 1935\u20131946, 977; Dallek, Franklin D. Roosevelt and American Foreign Policy, 285\u201386.\n\n\"control and domination of the seas\": Divine, Roosevelt and World War II, 44.\n\n\" from the effects of a Nazi victory\": Osgood, Ideals and Self-Interest in America's Foreign Relations, 425.\n\ncutting off Tokyo's supply of oil: Walter LaFeber, The Clash: A History of U.S.-Japan Relations (New York: Norton, 1997), 192\u2013200.\n\nwould unify the country behind war: Dallek, Franklin D. Roosevelt and American Foreign Policy, 303\u201310.\n\n\"than he had appeared in a long time\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 311.\n\nfew epithets were worse: Fox, Reinhold Niebuhr, 193\u201394.\n\n\"as effectively as that can be done\": Fox, Reinhold Niebuhr, 149.\n\nend up in the same place politically: Howard Brick, Daniel Bell and the Decline of Intellectual Radicalism (Madison: University of Wisconsin Press, 1986), 35\u201336, 55\u201356.\n\nthe same phrase FDR's uncle had applied to him: Daniel Bell, The End of Ideology: On the Exhaustion of Political Ideas in the Fifties (New York: Free Press, 1962), 300.\n\n\" it is still a civilization\": Justus D. Doenecke, \"Reinhold Niebuhr and His Critics: The Interventionist Controversy in World War II,\" Anglican and Episcopal History 64 (1995): 461.\n\nwhether democracy could ever really work: See, for instance, Walter Lippmann, Public Opinion! (New York: Harcourt Brace, 1922); Walter Lippmann, The Phantom Public (New Brunswick, NJ: Transaction, 1993); and Walter Lippmann, Liberty and the News (Princeton, NJ: Princeton University Press, 2008).\n\n\"have a very good mind\": Steel, Walter Lippmann, 291, 382.\n\nDewey felt he was reliving a nightmare: Westbrook, John Dewey and American Democracy, 512; Howlett, Troubled Philosopher, 145.\n\nBeard wrote to a friend: Kennedy, Charles Beard and American Foreign Policy, 93, 105, 120\u201321, 126\u201327, 152.\n\n\"what is beginning is too much for me\": Westbrook, John Dewey and American Democracy, 512.\n\n\"because it is not pure enough\": Richard W. Fox, \"Reinhold Niebuhr and the Emergence of the Liberal Realist Faith, 1930\u20131945,\" Review of Politics 38 (1976): 257\u201358.\n\n\"tyranny, sadism, and human defilement\": Nore, Charles A. Beard, 183, 200.\n\nFDR's foreign policy a Jewish plot: Clayton, Forgotten Prophet, 265; W. A. Swanberg, Dreiser (New York: Scribner, 1965), 461\u201365.\n\nChapter 4: I Didn't Say It Was Good\n\nthe most expensive film yet made: Thomas J. Knock, \"'History with Lightning': The Forgotten Film Wilson,\" American Quarterly 29 (1976): 529.\n\ntribute to the former president: Divine, Second Chance, 57, 168, 213.\n\neven made him seem friendly: Knock, \"'History with Lightning,'\" 535, 538.\n\nhis blood pressure spiked to 240 over 130: FDR saw the movie while meeting with Churchill in Quebec. Divine, Second Chance, 169\u201370; Knock, To End All Wars, 272; John Morton Blum, V Was for Victory: Politics and American Culture During World War II (New York: Harcourt Brace, 1976), 8; United States Department of State, Foreign Relations of the United States: Conference at Quebec, 1944 (1944), http:\/\/digicoll.library.wisc.edu\/cgi-bin\/FRUS\/FRUS-idx?type=goto&id=FRUS.FRUS1944&isize=M&submit=Go+to+page&page=292; Howard G. Bruenn, \"Clinical Notes on the Illness and Death of President Franklin D. Roosevelt,\" Annals of Internal Medicine 72 (1970): 587.\n\n\"somewhere within the rim of his consciousness\": Robert E. Sherwood, Roosevelt and Hopkins: An Intimate History (New York: Harper, 1948), 227.\n\nit was the twenty-sixth: Forrest Davis, a journalist FDR repeatedly took into his confidence, says this outright: \"In his management of foreign affairs, Franklin Roosevelt follows more closely the path of his distant cousin, Theodore, than of Woodrow Wilson.\" Forrest Davis, \"Roosevelt's World Blueprint,\" Saturday Evening Post, April 10, 1943, 110.\n\nbetween America, Britain, and France: William Clinton Olson, \"Theodore Roosevelt's Conception of an International League,\" World Affairs Quarterly 29 (1959): 346; Cooper, The Warrior and the Priest, 333.\n\n\"to be realistic\": Divine, Second Chance, 43\u201344; Sherwood, Roosevelt and Hopkins, 359\u201360.\n\nwould control the Asian mainland: Warren F. Kimball, The Juggler: Franklin Roosevelt as Wartime Statesmen (Princeton, NJ: Princeton University Press, 1991), 86; Range, Franklin D. Roosevelt's World Order, 178; Robert L. Messer, \"World War II and the Coming of the Cold War,\" in Modern American Diplomacy, ed. Carroll and Herring, 125\u2013126.\n\n\"too many nations to satisfy\": George Schild, \"The Roosevelt Administration and the United Nations: Re-Creation or Rejection of the League Experience?\" World Affairs 158 (1995): 29. The descriptions of Molotov are Churchill's. Raymond H. Anderson, \"A Lifetime of History: Vyacheslav M. Molotov, Close Associate of Stalin, Is Dead in Moscow at 96,\" New York Times, November 11, 1986, B1, B7.\n\n\"the military force to uphold it\": Davis, \"Roosevelt's World Blueprint,\" 20, 110.\n\nwhich nations should exist, and where: The State Department, at the urging of Undersecretary Sumner Welles (one of the strongest Wilsonians in the Roosevelt administration), did launch a postwar planning effort involving regional experts inside and outside government, but it had far less impact given Roosevelt's realist proclivities.\n\nas much at Newfoundland: Schild, \"The Roosevelt Administration and the United Nations,\" 30; Divine, Second Chance, 43\u201344; Sherwood, Roosevelt and Hopkins, 359\u201360.\n\nby 1944, 72 percent did: Divine, Second Chance, 68; Knock, \"'History with Lightning,'\" 540.\n\n\"blow off steam\": Divine, Second Chance, 158; Kimball, The Juggler, 86.\n\nrespiratory problems and in chronic pain: Kimball, The Juggler, 88, 238.\n\ndemanding that they call it a sheep: Fromkin, In the Time of the Americans, 603.\n\nmost powerful player in Eastern Europe: Messer, \"World War II and the Coming of the Cold War,\" 118, 124\u201325.\n\nrevolt against the additional loss of life: Gaddis, Strategies of Containment, 6\u20137; Messer, \"World War II and the Coming of the Cold War,\" 111; Dallek, Franklin D. Roosevelt and American Foreign Policy, 433\u201334.\n\nto defeat Hitler, Mussolini, and Tojo: Goldman, Rendezvous with Destiny, 390.\n\nin his upcoming reelection bid: Dallek, Franklin D. Roosevelt and American Foreign Policy, 436; Divine, Roosevelt and World War II, 92.\n\n\"and we reject it\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 439.\n\n\"in favour of the great\": Schild, \"The Roosevelt Administration and the United Nations,\" 31; Leland M. Goodrich, \"From League of Nations to United Nations,\" International Organization 1 (1947), 8\u201310; Divine, Roosevelt and World War II, 66; Evan Luard, A History of The United Nations, Volume 1: The Years of Western Domination, 1945\u20131955 (New York: St. Martin's, 1982), 44.\n\nthe body afforded large nations: Divine, Second Chance, 196\u201399.\n\n\"Hull knows the whole story\": Forrest Davis, \"What Really Happened at Teheran,\" 46; Divine, Second Chance, 201.\n\n\"Americans might be shocked\": Daniel Yergin, Shattered Peace (Boston: Houghton Mifflin, 1977), 59; Dallek, Franklin D. Roosevelt and American Foreign Policy, 479; Joseph M. Siracusa, \"The Night Stalin and Churchill Divided Europe: The View from Washington,\" Review of Politics 43 (1981): 381\u2013409; Kimball, The Juggler, 162.\n\n\"as they affect sentiment in America\": Fromkin, In the Time of the Americans, 599; Dallek, Franklin D. Roosevelt and American Foreign Policy, 503; Wilson D. Miscamble, From Roosevelt to Truman: Potsdam, Hiroshima, and the Cold War (New York: Cambridge University Press, 2007), 54\u201357.\n\nrarely challenged his postwar efforts: Dallek, Franklin D. Roosevelt and American Foreign Policy, 483\u201384; James MacGregor Burns, Roosevelt: The Soldier of Freedom (New York: Harcourt Brace, 1971), 516, 528\u201329; Divine, Second Chance, 219; John Foster Dulles, War or Peace (New York: Macmillan, 1950), 124\u201325.\n\n\"the almost ravaged appearance of his face\": Dallek, Franklin D. Roosevelt and American Foreign Policy, 442, 481.\n\nwrapped himself in a large cape to stay warm: Dallek, Franklin D. Roosevelt and American Foreign Policy, 508; Burns, Roosevelt, 564.\n\nManchuria and northern China: Messer, \"World War II and the Coming of the Cold War,\" 133.\n\nmake decisions by majority vote: Dallek, Franklin D. Roosevelt and American Foreign Policy, 466\u201367; Kimball, The Juggler, 98; Divine, Second Chance, 276.\n\nexiled prewar Polish leadership in London: Melvyn P. Leffler, \"Adherence to Agreements: Yalta and the Experiences of the Early Cold War,\" International Security 11 (1986): 94\u201395; Jaime Reynolds, \"Communists, Socialists and Workers: Poland 1944\u20131948,\" Soviet Studies 30 (1978): 517.\n\n\"but in fact she had her sins\": LaFeber, The American Age, 439\u201340; Dallek, Franklin D. Roosevelt and American Foreign Policy, 516; Vladislav Zubok and Constantine Pleshakov, Inside the Kremlin's Cold War: From Stalin to Khrushchev (Cambridge, MA: Harvard University Press, 1996), 10.\n\n\"and have always failed\": Fromkin, In the Time of the Americans, 618; Franklin D. Roosevelt, \"Address to Congress on Yalta,\" March 1, 1945, http:\/\/millercenter.org\/scripps\/archive\/speeches\/detail\/3338.\n\nRoosevelt did not want a public row: Richard F. Staar, \"Elections in Communist Poland,\" Midwest Journal of Political Science 2 (1958): 202\u20133; Reynolds, \"Communists, Socialists and Workers,\" 519\u201322; Dallek, Franklin D. Roosevelt and American Foreign Policy, 524\u201325; Kimball, The Juggler, 176\u201378.\n\n\"in an admittedly imperfect world\": Franklin D. Roosevelt, \"State of the Union Address,\" January 6, 1945, http:\/\/www.presidency.ucsb.edu\/ws\/index.php?pid=16595.\n\nmore than twenty-six years earlier: Divine, Second Chance, 277; Dallek, Franklin D. Roosevelt and American Foreign Policy, 528; Burns, Roosevelt, 600.\n\nopening ceremony of the United Nations: Divine, Second Chance, 287; John Lewis Gaddis, The Cold War: A New History (New York: Penguin, 2005), 5.\n\nkept their colonial possessions: Divine, Second Chance, 297; UN Universal Declaration of Human Rights, December 10, 1948, http:\/\/www.un.org\/Overview\/rights.html.\n\ndeferred seating Russia's puppet government: Divine, Second Chance, 290.\n\n\"other is Franklin D. Roosevelt\": Divine, Second Chance, 311.\n\nsparked no cheering or applause: Divine, Second Chance, 290, 311\u201314.\n\nkilling eighty thousand in a single night: Arnold Offner, Another Such Victory: President Truman and the Cold War, 1945\u20131953 (Stanford, CA: Stanford University Press, 2002), 92\u201393.\n\nimprisoned for most of the war: LaFeber, The American Age, 406; Greg Robinson, By Order of the President: FDR and the Internment of Japanese-Americans (Cambridge, MA: Harvard University Press, 2001), 250.\n\n\"I said it was the best I could do\": Divine, Second Chance, 285, 297; Dallek, Franklin D. Roosevelt and American Foreign Policy, 521.\n\nnot reading highbrow journals: Richard Hofstadter, The American Political Tradition and the Men Who Made It (New York: Knopf, 1948), 319\u201320.\n\n\"to atomic war\": Arthur M. Schlesinger, Jr., The Vital Center: The Politics of Freedom (Boston: Houghton Mifflin, 1949), 1\u20132.\n\n\"the Christian doctrine of original sin\": Reinhold Niebuhr, The Children of Light and the Children of Darkness: A Vindication of Democracy and a Critique of Its Traditional Defense (New York: C. Scribner's Sons, 1944), 16.\n\nto see the past in prelapsarian terms: It was during the 1920s, after all, that Sigmund Freud made his American splash. Among the many U.S. intellectuals he influenced during that decade was Walter Lippmann, whose extremely influential books about public irrationality hardly brimmed with optimism about the human condition. See Steel, Walter Lippmann, 173, 262; and Dumenil, The Modern Temper, 146\u201347, on Freud's influence on 1920s intellectual life.\n\ntragedy about the human condition: Richard H. Pells, Radical Visions and American Dreams: Culture and Social Thought in the Depression Years (New York: Harper & Row, 1977), 358; Kevin Mattson, When America Was Great: The Fighting Faith of Postwar Liberalism (New York: Routledge, 2004), 34\u201335.\n\n\"unwarranted optimism about man\": Schlesinger, The Vital Center, 40.\n\nresist the Nazis nonetheless: I am grateful to Professor Casey Blake for the analogy. Norman Graebner makes a similar point in The Age of Doubt: American Thought and Culture in the 1940s (Boston: Twayne, 1991), 4.\n\nmight not have won the war?: Kennedy, Freedom from Fear, 653\u201358, 666; Graebner, The Age of Doubt, 51.\n\n\"necessary for human conduct\": Mortimer J. Adler, \"This Pre-War Generation,\" Harper's, October 1940, http:\/\/radicalacademy.com\/adlerprewargeneration.htm.\n\n\"good and bad, right and wrong\": Mortimer J. Adler, \"God and the Professors,\" Conference on Science, Philosophy and Religion (1941), http:\/\/radicalacademy.com\/adlergodprofessors.htm; Westbrook, John Dewey and American Democracy, 518.\n\nits confrontations with the old world: Joel H. Rosenthal, Righteous Realists: Political Realism, Responsible Power, and American Culture in the Nuclear Age (Baton Rouge: Louisiana State University Press, 1991), 2\u20137; Greg Russell, Hans J. Morgenthau and the Ethics of American Statecraft (Baton Rouge: Louisiana State University Press, 1990), 62, 66; Lewis A. Coser, Refugee Scholars in America: Their Impact and Their Experiences (New Haven, CT: Yale University Press, 1984), 219\u201320; William E. Scheuerman, \"Carl Schmitt and Hans Morgenthau: Realism and Beyond,\" in Realism Reconsidered: The Legacy of Hans Morgenthau in International Relations, ed. Michael C. Williams (New York: Oxford University Press, 2007), 62\u201377; Richard Little, \"The Balance of Power in Politics Among Nations,\" in Realism Reconsidered, ed. Williams, 153\u201354.\n\n\"cured by means of reason\": Hans J. Morgenthau, Scientific Man vs. Power Politics (Chicago: University of Chicago Press, 1946), 71.\n\nincurable lust for power: Ghazi A. R. Algosaibi, \"The Theory of International Relations: Hans Morgenthau and His Critics,\" in Realism Reconsidered, ed. Williams, 229, 240.\n\ncourts that ruled aggression illegal: Morgenthau, Scientific Man, 50\u201351.\n\nmoral norms it did not uphold itself: Divine, Second Chance, 180\u201381; Steel, Walter Lippmann, 405, 426.\n\n\"not at all what I always knew\": Walter Lippmann, US Foreign Policy: Shield of the Republic (New York: Pocket, 1943), viii.\n\nate most of his meals alone: Walter Isaacson and Evan Thomas, The Wise Men: Six Friends and the World They Made (New York: Touchstone, 1986), 76\u201378, 228; Thompson, The Hawk and the Dove, 24\u201326.\n\nthey conjured the mother he had never had: Isaacson and Thomas, The Wise Men, 140, 145, 147, 153, 165, 227; Mayers, George Kennan and the Dilemmas of US Foreign Policy, 101.\n\n\"not a member of its household\": Isaacson and Thomas, The Wise Men, 74.\n\nunelected wise men like, well, himself: On Kennan's prejudices, see Joshua Botts, \"'Nothing to Seek and...Nothing to Defend': George F. Kennan's Core Values and American Foreign Policy, 1938\u20131993,\" Diplomatic History 30 (2006): 839\u201366.\n\nits own circumstances and needs: Isaacson and Thomas, The Wise Men, 147\u201351; Rosenthal, Righteous Realists, 25\u201326.\n\nan intimate understanding of the participants: Michael Joseph Smith, Realist Thought from Weber to Kissinger (Baton Rouge: Louisiana State University Press, 1986), 170.\n\nwhen it meets with some unanswerable force\": X, \"The Sources of Soviet Conduct,\" Foreign Affairs 25 (1947): 574.\n\nwas losing patience with Moscow: See Dallek, Franklin D. Roosevelt and American Foreign Policy, 550\u2013551; Gaddis, Strategies of Containment, 14\u201315; Arthur M. Schlesinger, Jr., \"Origins of the Cold War,\" Foreign Affairs 46 (1967): 24. For an opposing view, see Kimball, The Juggler, 180; Miscamble, From Roosevelt to Truman, 73\u201379.\n\nlarge and brutal neighbor: John Lewis Gaddis, \"Was the Truman Doctrine a Real Turning Point?\" Foreign Affairs 52 (1974): 388.\n\nthe challenges he would face: Offner, Another Such Victory, 22\u201325.\n\n\"are the one in trouble now\": Miscamble, From Roosevelt to Truman, ix.\n\nin the war as had the United States: Gaddis, The Cold War, 9.\n\nbegan to get tough: Messer, \"World War II and the Coming of the Cold War,\" 128\u201329.\n\n\"the straight one-two to the jaw\": Miscamble, From Roosevelt to Truman, 137. In recent years, some historians have suggested that in his memoirs Truman exaggerated how tough he actually was on Molotov during their April 1945 meeting. See Geoffrey Roberts, \"Sexing Up the Cold War: New Evidence on the Molotov-Truman Talks of April 1945,\" Cold War History 4 (2004): 105\u201325. More generally, Wilson Miscamble has argued that Truman's hard line evolved more slowly and fitfully than traditional accounts suggest. From Roosevelt to Truman, 87\u2013332. But regardless of the speed at which the change occurred, there is no question that U.S. policy toward the U.S.S.R. grew tougher over the course of 1945.\n\nbread and a pistol under his pillow: \"Gun No. 242332,\" Time, September 12, 1955, http:\/\/www.time.com\/time\/magazine\/article\/0,9171,893108,00.html.\n\nthe longest ever sent from the embassy in Moscow: Mayers, George Kennan and the Dilemmas of US Foreign Policy, 99; Walter L. Hixson, George Kennan: Cold War Iconoclast (New York: Columbia University Press, 1989), 32; Miscamble, From Roosevelt to Truman, 279.\n\nwe were now it: Gaddis, Strategies of Containment, 18\u201320, 380.\n\nnot one postwar world, but two: Mayers, George Kennan and the Dilemmas of US Foreign Policy, 100; Townsend Hoopes and Douglas Brinkley, Driven Patriot: The Life and Times of James Forrestal (Annapolis, MD: Naval Institute Press, 1992), 272; Gaddis, Strategies of Containment, 56.\n\nhe was speaking entirely metaphorically: Dallek, Franklin D. Roosevelt and American Foreign Policy, 472; Miscamble, From Roosevelt to Truman, 70\u201371.\n\ninflaming the Germans: Mary N. Hampton, \"NATO at the Creation: U.S. Foreign Policy, West Germany and the Wilsonian Impulse,\" Security Studies 4 (1995): 619\u201320.\n\nEurope's suffering to its advantage: Beisner, Dean Acheson, 69\u201371, 76\u201389.\n\n\" facts which momentarily imprison them\": Mary S. McAuliffe, Crisis on the Left: Cold War Politics and American Liberals (Amherst: University of Massachusetts Press, 1978), 65, 160; Smith, Realist Thought, 113\u201314, 130; Fox, Reinhold Niebuhr, 59.\n\nFDR created a sleeping beauty: Fromkin, In the Time of the Americans, 647.\n\nPART II: THE HUBRIS OF TOUGHNESS\n\nChapter 5: The Murder of Sheep\n\nSecretary of State George Marshall: Thompson, The Hawk and the Dove, 61; Hoopes and Brinkley, Driven Patriot, 272; Wilson D. Miscamble, George F. Kennan and the Making of American Foreign Policy, 1947\u20131950 (Princeton, NJ: Princeton University Press, 1992), 218.\n\na man they didn't really know at all: Hoopes and Brinkley, Driven Patriot, 15, 36\u201347.\n\n\"we also laughed at Hitler\": Hoopes and Brinkley, Driven Patriot, 266\u201369.\n\nthat Stalin, like Hitler, wanted a world war: Hoopes and Brinkley, Driven Patriot, 275; Mayers, George Kennan and the Dilemmas of US Foreign Policy, 109\u201310.\n\n\"Lenin's religion-philosophy\": Mayers, George Kennan and the Dilemmas of US Foreign Policy, 110\u201311; Hoopes and Brinkley, Driven Patriot, 280.\n\nState Department's Policy Planning staff: Hoopes and Brinkley, Driven Patriot, 278; Miscamble, George F. Kennan and the Making of American Foreign Policy, 10.\n\nthe byline read simply \"By X\": Mayers, George Kennan and the Dilemmas of US Foreign Policy, 113; X, \"The Sources of Soviet Conduct,\" 566\u201382; George Kennan, Memoirs 1925\u201350 (Boston: Atlantic Monthly Press, 1967), 354\u201358.\n\nfriends began calling her \"Miss X\": Hoopes and Brinkley, Driven Patriot, 276; Miscamble, George F. Kennan and the Making of American Foreign Policy, 64, 66; Thompson, The Hawk and the Dove, 75\u201376.\n\nto undermine Soviet control in Eastern Europe: Gregory Mitrovitch, Undermining the Kremlin: America's Strategy to Subvert the Soviet Bloc, 1947\u20131956 (Ithaca, NY: Cornell University Press, 2000), 5\u201346.\n\nunnecessary and probably counterproductive: Mayers, George Kennan and the Dilemmas of US Foreign Policy, 153\u201355.\n\nla guerre froide, the cold war: Steel, Walter Lippmann, 444\u201345.\n\n\"both in money and military power\": Walter Lippmann, The Cold War (New York: Harper & Brothers, 1947), 22\u201323.\n\n\"many views with which I profoundly agreed\": Steel, Walter Lippmann, 445; Thompson, The Hawk and the Dove, 78.\n\ntoo much publicity already, refused: Miscamble, George F. Kennan and the Making of American Foreign Policy, 68.\n\nthe columnist had gotten his views all wrong: Thompson, The Hawk and the Dove, 79; Miscamble, George F. Kennan and the Making of American Foreign Policy, 67; Kennan, Memoirs 1925\u20131950, 359\u201362.\n\nsimilar pledges in other corners of the globe: Robert Jervis, \"The Impact of the Korean War on the Cold War,\" Journal of Conflict Resolution 24 (1980): 574.\n\nKennan's insistence on containment's geographic limits: Melvyn P. Leffler, Preponderance of Power: National Security, the Truman Administration, and the Cold War (Stanford, CA: Stanford University Press, 1992), 304\u201311.\n\nslashed military spending by 15 percent: Gaddis, Strategies of Containment, 91; Jervis, \"The Impact of the Korean War on the Cold War,\" 568.\n\nAnd this time, America passed: Offner, Another Such Victory, 138\u201343; Bruce R. Kuniholm, The Origins of the Cold War in the Near East: Great Power Conflict and Diplomacy in Iran, Turkey, and Greece (Princeton, NJ: Princeton University Press, 1980), 321\u201334.\n\nto defend Greece or Turkey, either: Beisner, Dean Acheson, 54.\n\nthe Eastern Mediterranean and the Middle East: Leffler, Preponderance of Power, 123\u2013124, 143; Beisner, Dean Acheson, 39.\n\ncome to Greece and Turkey's aid: Thompson, The Hawk and the Dove, 71.\n\nKennan pleaded in a memo to his superiors: Mayers, George Kennan and the Dilemmas of US Foreign Policy, 349.\n\nregained their foreign policy edge: Offner, Another Such Victory, 201\u20139; Zubok and Pleshakov, Inside the Kremlin's Cold War, 46, 57, 127.\n\nan upset reelection victory that fall: Offner, Another Such Victory, 254\u201366.\n\n\"a whole new approach to the affairs of the world\": Miscamble, George F. Kennan and the Making of American Foreign Policy, 217.\n\nDavies, a man rather similar to himself: Kennan openly admitted this, saying that Davies was responsible for \"whatever insight I was able to muster in those years into the nature of Soviet policies toward the Far East.\" Miscamble, George F. Kennan and the Making of American Foreign Policy, 215\u201316.\n\nDavies was the first person he hired: Miscamble, George F. Kennan and the Making of American Foreign Policy, 213, 218; Halberstam, The Best and the Brightest, 384; Kahn, The China Hands, 54.\n\n\"control of the communist movement\": Mayers, George Kennan and the Dilemmas of US Foreign Policy, 173. In Moscow, as Soviet archives would later reveal, Stalin was already worrying about exactly that. Zubok and Pleshakov, Inside the Kremlin's Cold War, 56.\n\n\"pouring sand in a rat hole\": Hoopes and Brinkley, Driven Patriot, 307\u20138; Offner, Another Such Victory, 200.\n\ndemanded that the latter be brought in line: See, for instance, Richard Freeland's account of a congressional hearing on March 20, 1947, in which Representative James Fulton of Pennsylvania repeatedly tried to get Acting Secretary of State Dean Acheson to promise that American policy in China would abide by the principles laid out in Truman's Greece and Turkey speech. Richard M. Freeland, The Truman Doctrine and the Origins of McCarthyism: Foreign Policy, Domestic Politics and Internal Security, 1946\u20131948 (New York: Knopf, 1972), 112\u201313.\n\nbillions in economic and military aid to Chiang: Mayers, George Kennan and the Dilemmas of US Foreign Policy, 179.\n\nMao did not pose a grave threat: Miscamble, George F. Kennan and the Making of American Foreign Policy, 227.\n\nthen it hit him: treason!: Halberstam, The Best and the Brightest, 117.\n\nhad been cut in half: Ernest R. May, \"Lessons\" of the Past: The Use and Misuse of History in American Foreign Policy (New York: Oxford University Press, 1973), 78\u201379.\n\n\"with a Harvard accent\": Beisner, Dean Acheson, 307.\n\n\"the Communists took over in China\": Geoffrey S. Smith, \"National Security and Personal Isolation: Sex, Gender and Disease in the Cold War United States,\" International History Review 14 (1992): 319; Robert D. Dean, Imperial Brotherhood: Gender and the Making of Cold War Foreign Policy (Amherst: University of Massachusetts Press, 2001), 66; Gaddis, Strategies of Containment, 240.\n\nthat they would not defend Seoul: Beisner, Dean Acheson, 326\u201328; Bruce Cumings, Origins of the Korean War, Volume II: The Roaring of the Cataract, 1947\u20131950 (Princeton, NJ: Princeton University Press, 1990), 420\u201327; William Stueck, The Korean War: An International History (Princeton, NJ: Princeton University Press, 1995), 30.\n\n\"Iran, Berlin and Greece\": May, \"Lessons\" of the Past, 77, 81\u201382.\n\n\"continue to take responsibility for it\": Miscamble, George F. Kennan and the Making of American Foreign Policy, 324; Stueck, The Korean War, 61, 63; Thompson, The Hawk and the Dove, 119.\n\n\"suspicious foreign government\": Miscamble, George F. Kennan and the Making of American Foreign Policy, 326; Thompson, The Hawk and the Dove, 123.\n\nto do just that: Chen Jia, \"The Sino-Soviet Alliance and China's Entry into the Korean War,\" working paper of the Cold War History Project, Woodrow Wilson International Center, Washington, D.C., 1991; Michael Hunt, \"Beijing and the Korea Crisis,\" Political Science Quarterly (1992): 453\u201378.\n\nhis men would be home by Christmas: Stueck, The Korean War, 93\u201394, 97\u2013103; Thompson, The Hawk and the Dove, 123; Walter LaFeber, America, Russia, and the Cold War, 1945\u20131975 (New York: Wiley, 1976), 121\u201322; Miscamble, George F. Kennan and the Making of American Foreign Policy, 327.\n\nTruman fired him: Stueck, The Korean War, 167\u201368; David Halberstam, The Coldest Winter: America and the Korean War (New York: Hyperion, 2007), 597\u2013606.\n\n\"the Munich conference in 1938\": Goran Rystad, Prisoners of the Past? The Munich Syndrome and Makers of American Foreign Policy in the Cold War Era (Lund, Sweden: CWK Gleerup, 1982), 37; David McLellan, \"Dean Acheson and the Korean War,\" Political Science Quarterly 83 (1969): 32.\n\nflooded with pro-MacArthur mail: LaFeber, America, Russia and the Cold War, 128; Robert Dallek, Lone Star Rising: Lyndon Johnson and His Times, 1908\u20131960 (New York: Oxford University Press, 1991), 398\u2013399.\n\nhadn't wrecked America's economy: Alan Brinkley, Unfinished Nation: A Concise History of the American People (New York: Knopf, 1993), 817.\n\n\"not do if it wanted to do it\": Gaddis, Strategies of Containment, 92.\n\nwould not rise significantly as a percentage of GNP: Leffler, Preponderance of Power, 355\u201360; Gaddis, Strategies of Containment, 88, 91.\n\nwithin the United States itself: Gaddis, Strategies of Containment, 95, 111.\n\nboth to others and to ourselves: On the idea of credibility, and its importance to American foreign policymakers, see Robert J. McMahon, \"Credibility and World Power: Exploring the Psychological Dimension in Postwar American Diplomacy,\" Stuart L. Bernath Memorial Lecture, Diplomatic History 15 (1991), 455\u201371; Daryl G. Press, Calculating Credibility: How Leaders Assess Military Threats (Ithaca, NY: Cornell University Press, 2005); and Ted Hopf, Peripheral Visions: Deterrence Theory and American Foreign Policy in the Third World, 1965\u20131990 (Ann Arbor: University of Michigan Press, 1994).\n\n\"in each and every one of us\": Gaddis, The Cold War, 46.\n\napplied them only to the other side: National Security Council, \"NSC\u201368: United States Objectives and Programs for National Security,\" April 14, 1950, http:\/\/www.fas.org\/irp\/offdocs\/nsc-hst\/nsc\u201368.htm.\n\nsedatives, tranquilizers, and shots of insulin: Hoopes and Brinkley, Driven Patriot, 440, 450\u201351, 460.\n\nhe turns the sword on himself: Sophocles, Ajax, trans. John Tipton (Chicago: Flood, 2008).\n\nwith a thud, thirteen floors below: Hoopes and Brinkley, Driven Patriot, 465.\n\nwhen he entered a room: Halberstam, The Best and the Brightest, 381.\n\ninto the U.S. government: Hixson, George Kennan, 164\u201365; Mayers, George Kennan and the Dilemmas of US Foreign Policy, 202, 205; Kahn, The China Hands, 244\u201346; Michael T. Kaufman, \"John Paton Davies, Diplomat Who Ran Afoul of McCarthy over China, Dies at 91,\" New York Times, December 24, 1999, http:\/\/www.nytimes.com\/1999\/12\/24\/world\/john-paton-davies-diplomat-who-ran-afoul-of-mccarthy-over-china-dies-at\u201391.html.\n\nsustaining his family by building furniture: Kahn, The China Hands, 258, 261\u201362; Hixson, George Kennan, 167; Halberstam, The Best and the Brightest, 381.\n\n\"to be of much practical use to you\": Miscamble, George F. Kennan and the Making of American Foreign Policy, 277\u201378.\n\nChapter 6: The Problem with Men\n\nit does nothing for eight years: Herbert S. Parmet, Eisenhower and the American Crusades (New Brunswick, NJ: Transaction, 1999), xiii, xvii.\n\n\"the sacrifice to capture a worthless objective\": Stephen Ambrose, Eisenhower: Soldier and President (New York: Simon & Schuster, 1990), 87\u201388, 96\u201397, 181, 189\u201392, 270.\n\nthe least lamentable casualty of war: Halberstam, The Coldest Winter, 629\u201330; John E. Mueller, \"Trends in Popular Support for the Wars in Korea and Vietnam,\" American Political Science Review 65, no. 2 (1971): 360\u201361; William Stueck, Rethinking the Korean War: A New Diplomatic and Strategic History (Princeton, NJ: Princeton University Press, 2002), 157.\n\nshould cede much of the combat to South Koreans: Robert A. Divine, Foreign Policy and U.S. Presidential Elections, 1952\u20131960 (New York: Franklin Watts, 1974), 10, 45\u201346, 53\u201354, 71, 74, 80.\n\n\"the contest ended that night\": Elie Abel, \"General in Pledge, First Task Would Be Early and Honorable End of the War,\" New York Times, October 25, 1952, A1; Parmet, Eisenhower and the American Crusades, 143; Robert A. Divine, Eisenhower and the Cold War (New York: Oxford University Press, 1981), 18\u201319; Halberstam, The Coldest Winter, 626.\n\nwhere it had been when the war began: Roger Dingman, \"Atomic Diplomacy During the Korean War,\" International Security 13, no. 3 (1988\/1989): 85\u201388; Ambrose, Eisenhower, 294\u2013295, 327\u2013331; Stueck, Rethinking the Korean War, 171\u2013181; Zubok and Pleshakov, Inside the Kremlin's Cold War, 71, 155; Stueck, The Korean War, 224, 302; Divine, Eisenhower and the Cold War, 27\u201331; Philip West, \"Review: Interpreting the Korean War,\" American Historical Review 94, no. 1 (1989): 80.\n\n\"bills at the local department store\": Gaddis, Strategies of Containment, 132.\n\nEisenhower's policies on national defense: Gaddis, Strategies of Containment, 162, 169; Stephen E. Ambrose and Douglas G. Brinkley, Rise to Globalism: American Foreign Policy Since 1938 (New York: Penguin, 1997), 131; Halberstam, The Best and the Brightest, 40.\n\nsupposedly kept Latin leaders in power: John Prados, Safe for Democracy: The Secret Wars of the CIA (Chicago: Irvin R. Dee, 2006), 99\u2013123, 166\u201383, 215, 275\u201380, 432; Tim Weiner, Legacy of Ashes: The History of the CIA (New York: Doubleday, 2007), 94, 171\u201372; Stephen Kinzer, Overthrow: America's Century of Regime Change from Hawaii to Iraq (New York: Times Books, 2006), 136\u201338.\n\n\"blow the hell out of them\": Chen Jian, Mao's China and the Cold War (Chapel Hill: University of North Carolina Press, 2001), 167\u201370, 199\u2013202; Campbell Craig, Destroying the Village: Eisenhower and Thermonuclear War (New York: Columbia University Press, 1998), 52, 78\u2013107; Gaddis, Strategies of Containment, 147, 167, 169.\n\nprevent both sides from taking the first step: Craig, Destroying the Village, 52\u2013118.\n\nor worse later, refused: George C. Herring and Richard H. Immerman, \"Eisenhower, Dulles, and Dienbienphu: 'The Day We Didn't Go to War' Revisited,\" Journal of American History 71 (1984): 343\u201363.\n\nU.S. forces were gone: Gordon H. Chang, \"To the Nuclear Brink: Eisenhower, Dulles, and the Quemoy-Matsu Crisis,\" International Security 12, no. 4 (1988): 96\u2013123; H. W. Brands, Jr., \"Testing Massive Retaliation: Credibility and Crisis Management in the Taiwan Strait,\" International Security 12, no. 4 (1988): 124\u201351; Leonard H. D. Gordon, \"United States Opposition to the Use of Force in the Taiwan Strait, 1954\u20131962,\" Journal of American History 72, no. 3 (1985): 637\u201360; Divine, Eisenhower and the Cold War, 55\u201366, 97\u2013104; Ambrose, Eisenhower, 469.\n\nthe war a mistake fell by 15 points: Stephen G. Rabe, \"Eisenhower Revisionism,\" Diplomatic History 17, no. 1 (1993): 100; Yuen Foong Khong, Analogies at War: Korea, Munich, Dien Bien Phu, and the Vietnam Decisions of 1965 (Princeton, NJ: Princeton University Press, 1992), 114.\n\nhorrors that made them old before their time: Bell, The End of Ideology.\n\n\"the unromantic realm of more-or-less\": Richard H. Pells, The Liberal Mind in a Conservative Age: American Intellectuals in the 1940s and 1950s (Middletown, CT: Wesleyan University Press, 1989), 138, 140.\n\nthey had experienced almost ceaseless boom: John Patrick Diggins, The Proud Decades: America in War and in Peace, 1941\u20131960 (New York: Norton, 1988), 345; Diane Kunz, Butter and Guns: America's Cold War Economic Diplomacy (New York: Free Press, 1997), 66; The Misery Index, \"The U.S. Inflation Rate, 1950\u20131960,\" http:\/\/www.miseryindex.us\/iRbyyear.asp?StartYear=1950&EndYear=1960.\n\n\"malaise of the privileged\": Pells, The Liberal Mind in a Conservative Age, 240.\n\n\"and a search for passion\": Bell, The End of Ideology, 404, 300.\n\njunior senator from Massachusetts, John Fitzgerald Kennedy: Arthur Schlesinger, Jr., \"The Crisis of American Masculinity,\" in The Politics of Hope and The Bitter Heritage (Princeton, NJ: Princeton University Press, 2008), 292\u201393, 295, 303.\n\n\"were days of intense physical pain\": Edward J. Renehan, Jr., The Kennedys at War, 1937\u20131945 (New York: Doubleday, 2002), 7\u20138; John Hellmann, The Kennedy Obsession (New York: Columbia University Press, 1997), 9; Doris Kearns Goodwin, The Fitzgeralds and the Kennedys (New York: Simon & Schuster, 1987), 309\u201312. On Kennedy's poor health, see Robert Dallek, An Unfinished Life: John F. Kennedy, 1917\u20131963 (Boston: Little, Brown, 2003), 85\u201388, 100\u20135, 397\u201399, 704\u20136.\n\n\"when he really wasn't\": Kearns Goodwin, The Fitzgeralds and the Kennedys, 354; Hellmann, The Kennedy Obsession, 9; Thomas, Robert Kennedy, 39; Renehan, The Kennedys at War, 7, 220.\n\nAnd he went to war: Renehan, The Kennedys at War, 162.\n\nthe coconut shell in a case on his desk: Dallek, An Unfinished Life, 82\u201388, 95\u201398; Renehan, The Kennedys at War, 223, 256\u201371.\n\nthe real purpose of world affairs: making money: Dallek, An Unfinished Life, 21; Renehan, The Kennedys at War, 34, 63, 142.\n\nof succumbing to pressure from Jews: Dallek, An Unfinished Life, 54; Renehan, The Kennedys at War, 16, 24, 45, 71, 105, 142.\n\nstay in London with the nation's men: Renehan, The Kennedys at War, 80, 117\u201318, 160, 175.\n\nCommittee Against Military Intervention in Europe: Renehan, The Kennedys at War, 73, 74, 91\u201392, 182.\n\nthe form of government most likely to survive: Renehan, The Kennedys at War, 92, 138.\n\n\"don't want to take sides too much\": Renehan, The Kennedys at War, 139.\n\ncalling his book Why England Slept: Renehan, The Kennedys at War, 139, 153\u201354; Dallek, An Unfinished Life, 65; John D. Fair, \"The Intellectual JFK: Lessons in Statesmanship from British History,\" Diplomatic History 30, no. 1 (2006), 129\u201330.\n\nclear his and his father's names: Hellmann, The Kennedy Obsession, 21. Arthur Krock claimed that Joe Jr. \"was seeking to prove by its [his mission's] very danger that the Kennedys were not yellow. That's what killed that boy.... And his father realized it.\" Ronald Steel, In Love with Night: The American Romance with Robert Kennedy (New York: Simon & Schuster, 2000), 33.\n\ncowardice, if not treason: Renehan, The Kennedys at War, 254.\n\nnot the United States: John Noble Wilford, \"With Fear and Wonder in Its Wake, Sputnik Lifted Us into the Future,\" New York Times, September 25, 2007, http:\/\/www.nytimes.com\/2007\/09\/25\/science\/space\/25sput.html.\n\nscience labs to math teachers: Brinkley, Unfinished Nation, 885\u201386; LaFeber, America, Russia and the Cold War, 203.\n\nnowhere near as many ICBMs: Gaddis, Strategies of Containment, 182\u201386; David L. Snead, The Gaither Committee, Eisenhower and the Cold War (Columbus: Ohio University Press, 1999), 154\u201356, 180\u201381; Peter J. Roman, Eisenhower and the Missile Gap (Ithaca, NY: Cornell University Press, 1995), 118\u201340.\n\n\"the politics of fatigue\": Schlesinger, \"The New Mood in Politics,\" in The Politics of Hope and The Bitter Heritage, 109.\n\nprime minister who had preceded Chamberlain: William V. Shannon, \"Eisenhower as President,\" Commentary 26, no. 5 (1958): 398.\n\n\"like a man, can grow soft and complacent\": Paul Dickson, Sputnik: The Shock of the Century (New York: Walker, 2001), 223.\n\nJack relished the line: Elie Abel, \"Stevenson Called Appeaser by Nixon,\" New York Times, October 17, 1952, 19; Thomas G. Paterson, \"Bearing the Burden: A Critical Look at JFK's Foreign Policy,\" Virginia Quarterly Review 54 (1978): 196; Dallek, An Unfinished Life, 160, 259.\n\n\"have the largest rockets in the world\": Robert Dean, \"Masculinity as Ideology: John F. Kennedy and the Domestic Politics of Foreign Policy,\" Diplomatic History 22, no. 1 (1998): 45\u201346.\n\n\"lack of physical fitness is a menace to our security\": John F. Kennedy, \"The Soft American,\" Sports Illustrated, December 26, 1960, www.ihpra.org\/soft_american.htm.\n\n\"the best of our energies and abilities\": John F. Kennedy, The Strategy of Peace (New York: Harper & Brothers, 1960), 201, quoted in Dean, \"Masculinity as Ideology,\" 29; Gaddis, Strategies of Containment, 232.\n\nIke's soft, grandfatherly game: Cooper, The Warrior and the Priest, 70; Gaddis, Strategies of Containment, 198; Dean, \"Masculinity as Ideology,\" 47; Dean, Imperial Brotherhood, 183; Richard Reeves, President Kennedy: Profile of Power (New York: Simon & Schuster, 1993), 471\u201372; Dallek, An Unfinished Life, 472.\n\nhe wore thermal underwear: Halberstam, The Best and the Brightest, 241.\n\n\"by a hard and bitter peace\": John F. Kennedy, \"Inaugural Address,\" January 20, 1961, http:\/\/www.jfklibrary.org\/Historical+Resources\/Archives\/Reference+Desk\/Speeches\/JFK\/003POF03Inaugural01201961.htm.\n\n\"From Innocence to Engagement\": McGeorge Bundy, \"Foreign Policy: From Innocence to Engagement,\" in Paths of American Thought, ed. Arthur M. Schlesinger, Jr., and Morton White (Boston: Houghton Mifflin, 1963), 293\u2013308.\n\n\"the world was plastic and the future unlimited\": Paterson, \"Bearing the Burden,\" 203.\n\nDavid Halberstam called \"ultra-realists\": Ambrosius, Wilsonianism, 146; C. Wright Mills, The Power Elite (New York: Oxford University Press, 1956), 313.\n\ninnocence, inaction, and mounting danger: For an example of Kennedy comparing the 1950s to the '30s, see his November 13, 1959, speech to the annual convention of the Democratic Party of Wisconsin, printed in Kennedy, The Strategy of Peace, 193, and quoted in Dean, \"Masculinity as Ideology,\" 45.\n\none of his paintings in the White House: Fair, \"The Intellectual JFK,\" 133, 141.\n\nfar behind: On America's military advantage over the Soviet Union when Kennedy took office, see Gareth Porter, The Perils of Dominance (Los Angeles: University of California Press, 2006), 8\u20139, 15; David Rosenberg, \"The Origins of Overkill: Nuclear Weapons and American Strategy, 1945\u20131960,\" International Security 7, no. 4 (1983): 27\u201371; James G. Hershberg, \"The Cuban Missile Crisis,\" in The Cambridge History of the Cold War, Volume 2: Crises and D\u00e9tente, ed. Melvyn P. Leffler and Odd Arne Westad (New York: Cambridge University Press, 2009), 66.\n\nChapter 7: Saving Sarkhan\n\nare obsessed with astrology and palm-reading: William J. Lederer and Eugene Burdick, The Ugly American (New York: Norton, 1958), 14\u201315, 69, 84\u201385, 91, 109\u201312, 116, 127, 145, 174\u201381, 187.\n\nan adviser on Cuba instead: Jonathan Nashel, Edward Lansdale's Cold War (Amherst: University of Massachusetts Press, 2005), 71, 191; Dean, \"Masculinity as Ideology,\" 36\u201337, 43, 49; Dean, Imperial Brotherhood, 172\u201379; John Hellmann, American Myth and the Legacy of Vietnam (New York: Columbia University Press, 1986), 17.\n\npurposely serving him an awful lunch: Dallek, An Unfinished Life, 165, 167; Halberstam, The Best and the Brightest, 95.\n\n\"the lands of the rising people\": Ambrose and Brinkley, Rise to Globalism, 173.\n\n\"the ' future' is all that counts\": Bell, The End of Ideology, 405.\n\n\"on which so much of our destiny hinges\": Nils Gilman, Mandarins of the Future: Modernization Theory in Cold War America (Baltimore: Johns Hopkins University Press, 2003), 45.\n\n\"He tells us what he's going to do\": Melvyn Leffler, For the Soul of Mankind (New York: Hill & Wang, 2007), 176; Dallek, An Unfinished Life, 350; David Milne, America's Rasputin: Walt Rostow and the Vietnam War (New York: Hill & Wang, 2008), 75; Jack Anderson with Daryl Gibson, Peace, War and Politics: An Eyewitness Account (New York: Forge, 1999), 103.\n\nto learn how not to drown: Gaddis, Strategies of Containment, 216; George J. W. Goodman, \"The Unconventional Warriors,\" Esquire, November 1961, 130; Dean, \"Masculinity as Ideology,\" 51, 56; Dean, Imperial Brotherhood, 187\u201397.\n\nlanded at the proud president's feet: Goodman, \"The Unconventional Warriors,\" 129; Dean, \"Masculinity as Ideology,\" 50; Joseph Kraft, \"Hot Weapon in the Cold War,\" Saturday Evening Post, April 28, 1961, 88; Halberstam, The Best and the Brightest, 123\u201324; Reeves, President Kennedy, 284.\n\nAmerican-backed dictatorship of Fulgencio Batista: J. David Truby, \"Castro's Curveball,\" Harper's, May 1989, 32\u201334.\n\nfoment Marxist revolution across the Americas: Evan Thomas, The Very Best Men: The Daring Early Years of the CIA (New York: Simon & Schuster, 1995), 241, 249; Prados, Safe for Democracy, 236\u201342, 265; Lawrence Freedman, Kennedy's Wars: Berlin, Cuba, Laos, and Vietnam (Oxford: Oxford University Press, 2000), 124\u201327; Reeves, President Kennedy, 69\u201370.\n\ninvading Cuba would sabotage that effort: Freedman, Kennedy's Wars, 124.\n\n\"Operation Castration\": Thomas, The Very Best Men, 237; Dallek, An Unfinished Life, 362; Skip Willman, \"The Kennedys, Fleming, and Cuba,\" in Ian Fleming and James Bond: The Cultural Politics of 007, ed. Edward P. Comentale, Stephen Watt, and Skip Willman (Indianapolis: Indiana University Press, 2005), 178, 193; Dean, \"Masculinity as Ideology,\" 48.\n\n\"a shift from defensive reaction to initiative\": Thomas, The Very Best Men, 251; Reeves, President Kennedy, 72; Paterson, \"Bearing the Burden,\" 195\u201396.\n\nhe would face a firestorm on his right: Reeves, President Kennedy, 71. Historians are divided over whether Eisenhower, had he still been in office, would have approved the Bay of Pigs operation. Several of his comments to Kennedy before and after the invasion were quite hawkish. Also, having sent U.S. Marines into Lebanon in 1958, Eisenhower was not opposed to military interventions in the third world. Yet he also demonstrated caution during the Dien Bien Phu and Quemoy-Matsu crises. Further, after the Bay of Pigs invasion, he reportedly cautioned Kennedy against future military intervention. Ambrose, Eisenhower, 385, 553\u201354; Divine, Eisenhower and the Cold War, 65; Prados, Safe for Democracy, 264.\n\n\"appeaser of Castro\": Dallek, An Unfinished Life, 358.\n\nto try to dissuade Kennedy: Freedman, Kennedy's Wars, 148; Halberstam, The Best and the Brightest, 67.\n\nfirst American military aid to Vietnam: Thomas J. Schoenbaum, Waging Peace and War: Dean Rusk in the Truman, Kennedy, and Johnson Years (New York: Simon & Schuster, 1988), 49\u201350, 233\u201334; Robert Mann, A Grand Delusion: America's Descent into Vietnam (New York: Basic Books, 2001), 73, 78\u201379.\n\nso did almost everyone else: Schoenbaum, Waging Peace and War, 291\u2013301, 303\u20134. Senator William Fulbright, invited to a meeting at the White House to discuss the proposed invasion, did protest on moral grounds, but was dismissed as na\u00efve. Freedman, Kennedy's Wars, 132\u201333; Thomas, The Very Best Men, 251\u201352.\n\n\"bold as everybody else\": Paterson, \"Bearing the Burden,\" 202.\n\na place he often went to fish: Reeves, President Kennedy, 83; Dallek, An Unfinished Life, 363; Thomas, The Very Best Men, 243, 260\u201361; Aleksandr Fursenko and Timothy Naftali, \"One Hell of a Gamble\": The Secret History of the Cuban Missile Crisis (New York: Norton, 1997), 99\u2013100.\n\nU.S. attack would bring\u2014refused: Freedman, Kennedy's Wars, 141\u201344.\n\nwould have no choice but to invade: As Dulles acknowledged in private notes, \"We [the leadership of the CIA] felt that when the chips were down, when the crisis arose in reality, any action required for success would be authorized rather than permit the enterprise to fail.\" Thomas, The Very Best Men, 247.\n\nalone in his bedroom, in tears: Reeves, President Kennedy, 93, 95; Thomas, The Very Best Men, 263; Prados, Safe for Democracy, 263.\n\nthe limits of American power: Only four American airmen died in the Bay of Pigs invasion. Prados, Safe for Democracy, 263.\n\nthat he was soft: Reeves, President Kennedy, 95; John F. Kennedy, \"Address Before the American Society of Newspaper Editors,\" April 20, 1961, http:\/\/www.presidency.ucsb.edu\/ws\/index.php?pid=8076&st=&st1; Freedman, Kennedy's Wars, 337.\n\nmight take the whole region down with it: Reeves, President Kennedy, 74; George Herring, America's Longest War: The United States and Vietnam, 1950\u20131975 (New York: McGrawHill, 2002), 92; Freedman, Kennedy's Wars, 294\u201395.\n\n\"conscientious objectors from World War I\": Reeves, President Kennedy, 74; Freedman, Kennedy's Wars, 295\u201396.\n\n\"You start using atomic weapons!\": Logevall, Choosing War, 24\u201325; Reeves, President Kennedy, 111; Dallek, An Unfinished Life, 353; Michael Hunt, Lyndon Johnson's War: America's Cold War Crusade in Vietnam, 1945\u20131968 (New York: Hill & Wang, 1996), 55.\n\nmilitary alliances with neither east nor west: Reeves, President Kennedy, 75, 115\u201316; Thomas G. Paterson, J. Garry Clifford, and Kenneth J. Hagan, \"JFK: A Can-Do President,\" in To Reason Why: The Debate About the Causes of U.S. Involvement in the Vietnam War, ed. Jeffrey P. Kimball (Philadelphia: Temple University Press, 1994), 183.\n\n\"the obscurity that it richly deserves\": Galbraith's comment actually referred to all of Southeast Asia. \"The Importance of Obscurity,\" Time, May 6, 1966, http:\/\/www.time.com\/time\/magazine\/article\/0,9171,901831,00.html. In truth, Laos did not return to complete obscurity after the U.S.-Soviet deal. North Vietnamese forces remained there and steadily expanded sections of the Ho Chi Minh trail, thus helping supply communist forces in South Vietnam. In 1964, fighting resumed between the Pathet Lao and the royal Laotian military. As U.S. involvement in Vietnam deepened, America began massive air strikes against North Vietnamese forces in Laos. The royal government slowly lost control of the country to the better-equipped and better-trained Pathet Lao, and in 1975 the Pathet Lao seized complete control of the country. Herring, America's Longest War, 340\u201341; Stanley Karnow, Vietnam: A History (New York: Penguin, 1997), 346\u201347.\n\n\"the United States is in a yielding mood\": Milne, America's Rasputin, 123; Reeves, President Kennedy, 157.\n\nthere would be war: Reeves, President Kennedy, 160\u201372; Dallek, An Unfinished Life, 407; Zubok and Pleshakov, Inside the Kremlin's Cold War, 180, 236, 243\u201353.\n\nbacked by the United States: Reeves, President Kennedy, 185, 187, 203; Zubok and Pleshakov, Inside the Kremlin's Cold War, 195; Hope Harrison, Driving the Soviets up the Wall: Soviet\u2013East German Relations, 1953\u20131961 (Princeton, NJ: Princeton University Press, 2003), 227.\n\nThe place to start was Vietnam: Reeves, President Kennedy, 173; Herring, America's Longest War, 96.\n\nwas increasingly paying the bills: Herring, America's Longest War, 6\u201329; Khong, Analogies at War, 237.\n\nIndochina was lost: Halberstam, The Best and the Brightest, 94; Dallek, An Unfinished Life, 167, 185\u201387.\n\nwould unite the country once again: Herring, America's Longest War, 49.\n\nAmerican Friends of Vietnam: Diane Kunz, \"Camelot Continued: What If John F. Kennedy Had Lived?\" in Virtual History: Alternatives and Counterfactuals, ed. Niall Ferguson (London: Papermac, 1998), 378.\n\n\"the finger in the dike\": Mann, A Grand Delusion, 200.\n\n\"real in powerful men's minds\": Halberstam, The Best and the Brightest, 148.\n\ntravel to South Vietnam to investigate: Herring, America's Longest War, 80\u201383; Freedman, Kennedy's Wars, 322\u201323.\n\nAmerica was utopia: Milne, America's Rasputin, 21, 23, 26, 35, 60\u201363; Gilman, Mandarins of the Future, 202.\n\n\"becoming a bore\": Milne, America's Rasputin, 63.\n\n\"something you know nothing about?\": Halberstam, The Best and the Brightest, 159.\n\nLincoln Center for the Performing Arts: Freedman, Kennedy's Wars, 18\u201319; Halberstam, The Best and the Brightest, 162\u201363, 171\u201372, 468\u201370.\n\nit was to answer that question yes: Maxwell D. Taylor, The Uncertain Trumpet (New York: Harper & Brothers, 1959); Halberstam, The Best and the Brightest, 162\u201363.\n\nscare the North into stopping the war: Halberstam, The Best and the Brightest, 169\u201370; Herring, America's Longest War, 97; Freedman, Kennedy's Wars, 323\u201327.\n\nwithout U.S. soldiers: Herring, America's Longest War, 100; Kai Bird, The Color of Truth: McGeorge Bundy and William Bundy, Brothers in Arms (New York: Touchstone, 1998), 222; Freedman, Kennedy's Wars, 330\u201334; Khong, Analogies at War, 96.\n\nthey excelled at killing dangerous snakes: Reeves, President Kennedy, 335\u201336, 343; Nashel, Edward Lansdale's Cold War, 71, 239; Freedman, Kennedy's Wars, 153\u201359; Prados, Safe for Democracy, 298\u2013306.\n\n\"the next President of the United States\": Hershberg, \"The Cuban Missile Crisis,\" 67\u201370; Fursenko and Naftali, \"One Hell of a Gamble,\" 92, 97\u201398, 139\u201340; Reeves, President Kennedy, 339, 343\u201347, 350, 368\u201370, 372.\n\n\"and appearances contribute to reality\": Hershberg, \"The Cuban Missile Crisis,\" 72; Leffler, For the Soul of Mankind, 175.\n\nwould later cast doubt on the claim: See, for instance, Press, Calculating Credibility; Hopf, Peripheral Visions; Stephen M. Walt, The Origins of Alliances (Ithaca, NY: Cornell University Press, 1987); Jonathan Mercer, Reputation and International Politics (Ithaca, NY: Cornell University Press, 1996).\n\nwould get his brother impeached: Reeves, President Kennedy, 401.\n\n\"ultimately leads to war\": Hershberg, \"The Cuban Missile Crisis,\" 71\u201373; Reeves, President Kennedy, 395\u201396; John F. Kennedy, \"Report to the American People on the Soviet Arms Buildup in Cuba,\" October 22, 1962, http:\/\/www.jfklibrary.org\/Historical+Resources\/Archives\/Reference+Desk\/Speeches\/JFK\/003POF03CubaCrisis10221962.htm.\n\nto withdraw the missiles in Turkey: Michael Dobbs, One Minute to Midnight: Kennedy, Khrushchev, and Castro on the Brink of Nuclear War (New York: Knopf, 2008), 310\u201311; Reeves, President Kennedy, 400\u201324; Hershberg, \"The Cuban Missile Crisis,\" 80\u201383; Fursenko and Naftali, \"One Hell of a Gamble,\" 285\u201387, 321.\n\n\"back away from rash initiatives\": Reeves, President Kennedy, 428.\n\ngained seats in the Senate: Reeves, President Kennedy, 426, 429.\n\n\"take the wrong lessons away from the crisis\": Reeves, President Kennedy, 427.\n\ntaken his private heresies even further: Fursenko and Naftali, \"One Hell of a Gamble,\" 291\u201393; Dobbs, One Minute to Midnight, 45\u201346; Hershberg, \"The Cuban Missile Crisis,\" 81.\n\nhalf the South Vietnamese cabinet: Freedman, Kennedy's Wars, 325\u201326; Herring, America's Longest War, 59\u201360, 77\u201378.\n\nincluding weddings and funerals: Halberstam, The Best and the Brightest, 181\u201382; Herring, America's Longest War, 101, 108\u20139.\n\nkilling two adults and six children: Mann, A Grand Delusion, 284.\n\nit was the Vietnamese word for \"monk\": Mann, A Grand Delusion, 284\u201386; Reeves, President Kennedy, 491.\n\nhelped convince U.S. officials that he needed to go: Freedman, Kennedy's Wars, 368; Logevall, Choosing War, 45\u201350.\n\nturned pale and fled silently from the room: Freedman, Kennedy's Wars, 390\u201395; Reeves, President Kennedy, 649.\n\nMcNamara's Pentagon until the last minute: Reeves, President Kennedy, 507; Dallek, An Unfinished Life, 619.\n\nhad taken his life fourteen years before: John F. Kennedy, \"Commencement Address at American University,\" June 10, 1963, http:\/\/www.jfklibrary.org\/Historical+Resources\/Archives\/Reference+Desk\/Speeches\/JFK\/003POF03AmericanUniversity06101963.htm; Reeves, President Kennedy, 447; Bird, The Color of Truth, 251\u201352.\n\nBillings, Montana, and Salt Lake City, Utah: Reeves, President Kennedy, 606\u20137; Halberstam, The Best and the Brightest, 295\u201396.\n\n\"get the American people to reelect me\": Reeves, President Kennedy, 444; Logevall, Choosing War, 38\u201349.\n\nmaybe he could square the circle: Logevall, Choosing War, 73; Reeves, President Kennedy, 600\u20131; Nashal, Edward Lansdale's Cold War, 57.\n\nChapter 8: Things Are in the Saddle\n\nLyndon Johnson was its face: Dallek, Lone Star Rising, 7.\n\n\"when the parts were Lyndon Johnson's\": Robert Dallek, \"Presidential Address: Lyndon Johnson and Vietnam: The Making of a Tragedy\" Diplomatic History 20, no. 2 (1996): 155; Robert Caro, Master of the Senate (New York: Knopf, 2002), 121\u201322.\n\n\"I want his pecker in my pocket\": Halberstam, The Best and the Brightest, 434, 436\u201337.\n\nhow democracy survived: Halberstam, The Best and the Brightest, 444; Rystad, Prisoners of the Past, 55; D. B. Hardeman and Donald C. Bacon, Rayburn: A Biography (Austin: Texas Monthly Press, 1987), 265\u201370; Dallek, Lone Star Rising, 226\u201327.\n\nbecame the industry's champion, and its beneficiary: Dallek, Lone Star Rising, 292, 294, 383.\n\n\"terribly shaken\": Dallek, Lone Star Rising, 323\u201324, 399; Halberstam, The Best and the Brightest, 590\u201391; Lyndon B. Johnson, \"Nightmares of Crucifixion,\" in To Reason Why, ed. Kimball, 45\u201346.\n\ntime trying to recover them: Halberstam, The Best and the Brightest, 298.\n\ngrew and grew: Logevall, Choosing War, 108, 321; Kaiser, American Tragedy, 304\u20135; Herring, America's Longest War, 133\u201339.\n\ndecided to take the fight to Hanoi: Herring, America's Longest War, 139; Kaiser, American Tragedy, 331.\n\n\"repel any armed attacks against the forces of the United States\": Herring, America's Longest War, 142\u201344; Logevall, Choosing War, 196\u2013205; Hunt, Lyndon Johnson's War, 79.\n\n\"would rather not have\": Logevall, Choosing War, 205; Kaiser, American Tragedy, 336\u201338; Ambrose and Brinkley, Rise to Globalism, 142; Paterson, \"Bearing the Burden,\" 211; David Frum, How We Got Here: The 70s (New York: Basic Books, 2000), 4, 39.\n\nits vote was unanimous: Dallek, Lone Star Rising, 84\u201385; Dallek, \"Presidential Address: Lyndon Johnson and Vietnam,\" 151; Herring, America's Longest War, 144; Logevall, Choosing War, 198; Kaiser, American Tragedy, 336\u201338.\n\n\"there were better ones, golden ones, ahead\": Rystad, Prisoners of the Past, 56; Logevall, Choosing War, 205; Herring, America's Longest War, 145; Halberstam, The Best and the Brightest,423.\n\n\"conflict between parents and children is letting up\": Kaiser, American Tragedy, 483; Leffler, For the Soul of Mankind, 221; Halberstam, The Best and the Brightest, 423; \"Students: On the Fringe of a Golden Era,\" Time, January 29, 1965, http:\/\/www.time.com\/time\/printout\/0,8816,839175,00.html; \"Lyndon B. Attitudes,\" Time, September 10, 1965, http:\/\/www.time.com\/time\/printout\/0,8816,834259,00.html.\n\n\"We can do it all\": Rick Perlstein, Nixonland: The Rise of a President and the Fracturing of America (New York: Scribner, 2008), 5.\n\nend poverty for good: Robert Dallek, Flawed Giant: Lyndon Johnson and His Times, 1961\u20131973 (New York: Oxford University Press, 1998), 184; Lyndon B. Johnson, \"Inaugural Address,\" January 20, 1965, http:\/\/www.bartelby.org\/124\/pres57.html.\n\n\"I was born in a manger\": Halberstam, The Best and the Brightest, 431; \"Lyndon B. Attitudes.\"\n\n\"raggedy-assed little fourth-rate country\": Halberstam, The Best and the Brightest, 512.\n\nbuoying morale in the South: For a discussion of the Johnson administration's planning for military action in November 1964, see Kaiser, American Tragedy, 355\u201374.\n\n\"They'd impeach a President who'd run out, wouldn't they?\": Logevall, Choosing War, 145; Kaiser, American Tragedy, 319\u201320.\n\nZero Hour: Summons to the Free: McGeorge Bundy, \"They Say in the Colleges...,\" in Zero Hour: A Summons of the Free, ed. Stephen Vincent Ben\u00e9t (New York: Farrar & Rinehart, 1940), 79\u2013116; Bird, The Color of Truth, 66\u201367; Leonard Lyons, \"The New Yorker,\" Washington Post, July 20, 1940, 4; Thomas C. Lyons, \"Books of the Times,\" New York Times, August 3, 1940, 17.\n\n\"the isolationist-interventionist debate\": Bird, The Color of Truth, 107; Khong, Analogies at War, 197.\n\n\"shall not perish from this earth\": Logevall, Choosing War, 324\u201327; Bird, The Color of Truth, 306\u20137; Halberstam, The Best and the Brightest, 523.\n\nJohnson himself had counseled caution: Herring and Immerman, \"Eisenhower, Dulles, and Dienbienphu,\" 353\u201354; Hunt, Lyndon Johnson's War, 76, 101.\n\nclose to twice the number in 1952: Mueller, \"Trends in Popular Support for the Wars in Korea and Vietnam,\" 360\u201361; Khong, Analogies at War, 102.\n\nIt had worked so far: Logevall, Choosing War, 360\u201361.\n\nhe was already over the edge: Logevall, Choosing War, 330, 370; Herring, America's Longest War, 156\u201358, 162.\n\n150,000 troops and more intense bombing: Logevall, Choosing War, 367; Herring, America's Longest War, 162\u201363.\n\n\"but that he did pay [Dean] Rusk\": Daniel DiLeo, George Ball, Vietnam and the Rethinking of Containment (Chapel Hill: University of North Carolina Press, 1991), 25\u201329; George W. Ball, The Past Has Another Pattern (New York: Norton, 1982), 152\u201353, 365; Kaiser, American Tragedy, 350\u201351, 362, 395\u201396, 402, 405\u20136; Logevall, Choosing War, 361\u201362.\n\n\"an haughty spirit before a fall\": Steel, Walter Lippmann and the American Century, 558\u201380; Fredrik Logevall, \"First Among Critics: Walter Lippmann and the Vietnam War,\" Journal of American-East Asian Relations 4, no. 4 (1995), 351\u201354, 364\u201372.\n\n\"a fatally unfortunate conclusion\": Mayers, George Kennan and the Dilemmas of US Foreign Policy, 277\u201379; Hixson, George Kennan, 227; Khong, Analogies at War, 174.\n\n\"[as justification] for the present war\": Fox, Reinhold Niebuhr, 284\u201385.\n\n\"general discouragement, sometimes bordering on despair\": Jennifer W. See, \"A Prophet Without Honor: Hans Morgenthau and the War in Vietnam, 1955\u20131965,\" Pacific Historical Review 70, no. 3 (2001), 437, 439\u201340; Bird, The Color of Truth, 321; CBS News, \"Dean Bundy Meets Professor Morgenthau,\" in Louis Menashe and Ronald Radosh, eds., Teach-Ins: USA Reports, Opinions, Documents (Washington, DC: Praeger, 1967), 198\u2013206; Christoph Frei, Hans J. Morgenthau: An Intellectual Biography (Baton Rouge: Louisiana State University Press, 2001), 221.\n\nanother hundred thousand troops to South Vietnam: Khong, Analogies at War, 151; Herring, America's Longest War, 164\u201365.\n\nChapter 9: Liberation\n\nuntil the fateful day arrived: Freedman, Kennedy's Wars, 142, 145; Prados, Safe for Democracy, 262\u201363; Halberstam, The Best and the Brightest, 68.\n\nwould no longer be required: Bird, The Color of Truth, 200; Marcus Raskin and Robert Spero, \"Ahead of History: Marcus Raskin and the Institute for Policy Studies,\" in The Four Freedoms Under Siege: The Clear and Present Danger of Our National Security State, ed. Marcus Raskin and Robert Spero (Westport, CT: Praeger, 2007), 276; Marcus Raskin, interview with author, February 22, 2008.\n\nwhere he began causing trouble: Raskin and Spero, \"Ahead of History,\" 273\u201374.\n\nretire once the decisions were made: Bird, The Color of Truth, 187, 219; Marcus Raskin, interview with author, February 22, 2008.\n\n\"that he won't pee on the floor\": Bird, The Color of Truth, 200, 206\u20137; Marcus Raskin, interview with author, February 22, 2008.\n\nthat America try to end the cold war: James Roosevelt, ed., The Liberal Papers (New York: Doubleday, 1962), 17, 27, 58\u201359, 146, 175, 251\u201352, 267, 279, 321, 326.\n\nBy year's end he was gone: Bird, The Color of Truth, 218\u201319; Raskin and Spero, \"Ahead of History,\" 277.\n\n\"we are intellectually and emotionally spastic\": Halberstam, The Best and the Brightest, 426\u201327; Gilbert Seldes, \"'Patient' Diagnosis of 'Dr. Strangelove,'\" New York Times, April 5, 1964, X7.\n\nHouse Un-American Activities Committee: Paul Buhle and William Rice-Maximin, William Appleman Williams: The Tragedy of Empire (New York: Routledge, 1995), 115\u201316; Jonathan M. Wiener, \"Radical Historians and the Crisis in American History, 1959\u20131980,\" Journal of American History 76 (1989): 407.\n\na generational manifesto that would sell sixty thousand copies: Allen J. Matusow, The Unraveling of America: A History of Liberalism in the 1960s (New York: Harper Torchbooks, 1986), 313; James Miller, Democracy Is in the Streets: From Port Huron to the Siege of Chicago (New York: Simon & Schuster, 1987), 13; Todd Gitlin, The Sixties: Years of Hope, Days of Rage (New York: Bantam, 1993), 111.\n\nof nuclear bombs: Howard Brick, Age of Contradiction: American Thought and Culture in the 1960s (New York: Twayne, 1998), 149\u201350; Charles DeBenedetti, An American Ordeal: The Antiwar Movement of the Vietnam Era (Syracuse, NY: Syracuse University Press, 1990), 389; Maurice Isserman, If I Had a Hammer: The Death of the Old Left and the Birth of the New Left (Urbana: University of Illinois Press, 1993), 149.\n\nCollege enrollment nearly quadrupled between 1946 and 1970: Matusow, The Unraveling of America, 308.\n\n\"know a Communist if they tripped over one\": Herring, America's Longest War, 219.\n\ndeclared the Port Huron Statement: Students for a Democratic Society, \"The Port Huron Statement,\" June 15, 1962, http:\/\/history.hanover.edu\/courses\/excerpts\/111hur.html.\n\nlay down his body for justice and peace: Isserman, If I Had a Hammer, 151\u201355.\n\nintegrate interstate transport across the South: Raymond Arsenault, Freedom Riders: 1961 and the Struggle for Racial Justice (New York: Oxford University Press, 2006), 99\u2013101.\n\nthe 1963 March on Washington: John D'Emilio, Lost Prophet: The Life and Times of Bayard Rustin (New York: Free Press, 2003), 37\u201350, 327\u201360.\n\nfounded on pacifist principles as well: Akinyele O. Omoja, \"The Ballot and the Bullet: A Comparative Analysis of Armed Resistance in the Civil Rights Movement,\" Journal of Black Studies 29 (1999): 560\u201361.\n\nthe following year: Eric Stoper, \"The Student Nonviolent Coordinating Committee: Rise and Fall of a Redemptive Organization,\" Journal of Black Studies 8 (1977): 31; Arsenault, Freedom Riders, 451; Tom Hayden, Reunion: A Memoir (New York: Random House, 1988), 53\u201355.\n\n\"all men are really good at heart\": Isserman, If I Had a Hammer, 127.\n\n\"made sonsofbitches of us all\": Marcus Raskin, \"A National Security Manager Tries to Explain,\" in Essays of a Citizen: From National Security State to Democracy, ed. Marcus Raskin (Armonk, NY: M. E. Sharpe, 1991), 104.\n\nfound his view of human nature too bleak: Halliwell, The Constant Dialogue, 230\u201336; David J. Garrow, Bearing the Cross: Martin Luther King, Jr., and the Southern Christian Leadership Conference (New York: William Morrow, 1986), 45.\n\n\"releases the prisoners; abolishes repression\": Douglas Martin, \"Norman O. Brown Dies, Playful Philosopher was 89,\" New York Times, October 4, 2002; Matusow, The Unraveling of America, 277\u201379; Brick, Age of Contradiction, 134\u201335.\n\nevery one dollar of damage it did to North Vietnam: Halberstam, The Best and the Brightest, 241; Herring, America's Longest War, 168, 172\u201377, 179; Logevall, Choosing War, 409\u201310.\n\nthe membership of the SDS quadrupled: DeBenedetti, An American Ordeal, 107\u20138, 111\u201312, 132, 150; Matusow, The Unraveling of America, 320.\n\nurged Johnson to halt the bombing: Mueller, \"Trends in Popular Support for the Wars in Korea and Vietnam,\" 363; Thomas, Robert Kennedy, 334\u201335; Herring, America's Longest War,267.\n\n\"the greatest purveyor of violence in the world today\": DeBenedetti, An American Ordeal, 172.\n\nthrew bottles and bags of blood: Gitlin, The Sixties, 254.\n\na picture of his father mounted on the wall: Bird, The Color of Truth, 323, 372, 397\u201398, 401.\n\nbegan to break under the strain: Deborah Shapley, Power and Promise: The Life and Times of Robert McNamara (Boston: Little, Brown, 1993), 408, 483.\n\nher father did not: DeBenedetti, An American Ordeal, 129; Shapley, Power and Promise, 354\u201355.\n\nbursting into tears: Bird, The Color of Truth, 345.\n\nBundy and McNamara were both gone: Halberstam, The Best and the Brightest, 629; Bird, The Color of Truth, 342, 347.\n\n\"a fanatic in sheep's clothing,\" in one detractor's words: Halberstam, The Best and the Brightest, 628; Milne, America's Rasputin, 7.\n\n\"I think we can hold out longer than that\": Halberstam, The Best and the Brightest, 637\u201338.\n\nantiwar senators were taking orders from the Kremlin: Dallek, \"Presidential Address: Lyndon Johnson and Vietnam,\" 467.\n\npulled out his penis and screamed, \"This is why!\": Dallek, Flawed Giant, 448, 452, 491, 526.\n\n\"I am alive and it has been more torturous\": Halberstam, The Best and the Brightest, 640.\n\n\"bitch of a war\": Gaddis, Strategies of Containment, 268.\n\n\"A man without a spine\": Doris Kearns, Lyndon Johnson and the American Dream (New York: Harper & Row, 1976), 253.\n\n\"the end begins to come into view\": Herring, America's Longest War, 221.\n\n\"choked with rubble and rotting bodies\": Herring, America's Longest War, 225\u201331; Karnow, Vietnam, 538\u201340.\n\n\"I thought we were winning the war!\": Herring, America's Longest War, 232; Dallek, Flawed Giant, 506.\n\n\"it will only be a matter of time before they give in\": Herring, America's Longest War, 229\u201331; Halberstam, The Best and the Brightest, 648.\n\nsubordinate it to his own: Gaddis, Strategies of Containment, 256; Herring, America's Longest War, 163; Halberstam, The Best and the Brightest, 597.\n\nand billed him another five thousand dollars: Halberstam, The Best and the Brightest, 650\u201351.\n\nwho was beholden to whom: Dallek, Lone Star Rising, 39\u201340.\n\na plurality of Americans opposed the war: Beisner, Dean Acheson, 633; Matusow, The Unraveling of America, 391; Gitlin, The Sixties, 293.\n\n\"has dampened expansionist ideas\": Herring, America's Longest War, 245\u201357; Kunz, Butter and Guns, 114\u201319; Walter LaFeber, Deadly Bet: LBJ, Vietnam, and the 1968 Elections (Lanham, MD: Rowman & Littlefield, 2005), 55\u201357; Robert M. Collins, \"The Economic Crisis of 1968 and the Waning of the 'American Century,'\" American Historical Review 101 (1996): 396\u2013422.\n\n\"have bailed out\": Herring, America's Longest War, 250. Although a majority of the \"wise men\" argued that it was time to disengage from Vietnam, a few hawks\u2014such as Maxwell Taylor and Supreme Court justice (and crony of Johnson) Abe Fortas\u2014argued for seeing the war through. Isaacson and Thomas, The Wise Men, 702\u20133.\n\n\"in disarray around him\": Theodore White, The Making of the President, 1968 (New York: Atheneum, 1969), 102; Gitlin, The Sixties, 304; Dallek, Flawed Giant, 528.\n\nand perhaps his life itself: Dallek, Flawed Giant, 522\u201323.\n\n\"it was like committing actual suicide\": Dallek, Flawed Giant, 601\u20135, 613\u201314.\n\nin the party's gallery of honor: Halberstam, The Best and the Brightest, 658; Dallek, Flawed Giant, 573, 617.\n\n\"dissolved in the morass of war in Vietnam\": \"Lyndon Johnson, 36th President, Is Dead,\" New York Times, January 23, 1973, 81.\n\nin an essay on intellectuals and war: Bourne, War and the Intellectuals, ed. Resek; Lillian Schlissel, ed., The World of Randolph Bourne: An Anthology (New York: Dutton, 1966); Lasch, The New Radicalism in America, 69\u2013103, 300\u20131; Noam Chomsky, American Power and the New Mandarins (New York: Pantheon, 1969), 4\u20138.\n\na model for post-Vietnam foreign policy: Marcus Raskin, \"The Erosion of Congressional Power,\" in Washington Plans an Aggressive War, ed. Stavins, Barnett, and Raskin, 255; Marcus Raskin, interview with author, February 22, 2008.\n\nDewey was an inspiration: Miller, Democracy Is in the Streets, 16, 149; Westbrook, John Dewey and American Democracy, 547\u201349.\n\nhad rarely been uttered without a sneer: For instance, \"If there is to be a politics of the New Left...our work is necessarily structural\u2014and so, for us, just now\u2014utopian.\" C. Wright Mills, \"The New Left,\" in Power, Politics and People: The Collected Essays of C. Wright Mills, ed. Irving Louis Horowitz (New York: Oxford University Press, 1963), 254. And \"the masses of hungry, aspiring, utopian peoples [are] intervening in history for the first time.\" Tom Hayden, quoted in Miller, Democracy Is in the Streets, 101.\n\nthe economic motives for war: Kennedy, Charles Beard and American Foreign Policy, 160, 165\u201367; H. W. Brands, What America Owes the World (New York: Cambridge University Press, 1998), 242\u201343.\n\na young, militant historian named Ronald Radosh: Ronald Radosh, Prophets on the Right: Profiles of Conservative Critics of American Globalism (New York: Simon & Schuster, 1975), 51.\n\na second, more positive look: See, for instance, Wilson, Herbert Hoover.\n\nendemic to American foreign policy and American life: Herring, America's Longest War, xiv, 267.\n\nhoping that LBJ might appoint him ambassador to the UN: Dominic Sandbrook, Eugene McCarthy: The Rise and Fall of Postwar American Liberalism (New York: Knopf, 2004), 120\u201321, 127, 133.\n\n\"I know it didn't, sir\": Sandbrook, Eugene McCarthy, 17\u201318, 132\u201333, 139, 145, 155.\n\nAn old monk told him to seize the day: Sandbrook, Eugene McCarthy, 171.\n\neven though the crowd had not yet arrived: Sandbrook, Eugene McCarthy, 190\u201391.\n\nstared at the beast until it trotted away: Steel, In Love with Night, 39, 125.\n\nwimps when they pointed out the risks: Halberstam, The Best and the Brightest, 70; Steel, In Love with Night, 25\u201326, 49, 76\u201378.\n\n\"We do not have...to be afraid\": Sandbrook, Eugene McCarthy, 150, 195; Matusow, The Unraveling of America, 421.\n\nChapter 10: The Scold\n\nthe arc of his own career: Schlesinger, \"Vietnam: Lessons of the Tragedy,\" in The Politics of Hope and The Bitter Heritage, 514.\n\na moratorium on the production of nuclear weapons: Robert David Johnson, Congress and the Cold War (New York: Cambridge University Press, 2006), 146; Benjamin O. Fordham, \"The Evolution of Republican and Democratic Positions on Cold War Military Spending,\" Social Science History 31 (2007): 609, 627; Theodore White, The Making of the President 1972 (New York: Atheneum, 1973), 46.\n\ncut the number of American troops in Europe in half: Gaddis, Strategies of Containment, 318.\n\n\"an exercise in futility\": Schlesinger, \"Vietnam: Lessons of the Tragedy,\" in The Politics of Hope and The Bitter Heritage, 522.\n\nPotomac Electric Power Company: Bird, Color of Truth, 356; Karnow, Vietnam, 471.\n\n\"a singularly inept instrument of foreign policy\": Paul C. Warnke and Leslie H. Gelb, \"Security or Confrontation: The Case for a Defense Policy,\" Foreign Policy 1 (1970\/1971): 15.\n\n\"it's going to mean our foreign policy has been idiotic\": \"The Real Paul Warnke,\" New Republic, March 26, 1977, 22\u201325.\n\naccused the new intellectuals of isolationism: Norman Podhoretz, for instance, wrote that \"the new isolationism, then, was what happened to the [New Left] Movement's ideas about the American role in world affairs when they were cleaned up to 'work within the system.'\" Norman Podhoretz, Breaking Ranks: A Political Memoir (New York: Harper & Row, 1979), 291\u201392.\n\nborn too late to remember Munich: Anthony Lake, \"Introduction,\" in The Vietnam Legacy: The War, American Society and the Future of American Foreign Policy, ed. Anthony Lake (New York: New York University Press, 1976), xxviii.\n\ntwo of whose three congressmen had opposed World War I: \"Detailed Vote in the House of Representatives on the Passage of the Resolution Declaring War,\" New York Times, April 7, 1917, 4.\n\nwhich denounced the UN as too weak: Bruce Miroff, The Liberals' Moment: The McGovern Insurgency and the Identity Crisis of the Democratic Party (Lawrence: University Press of Kansas, 2007), 30; Bernard Hennessy, \"A Case Study of Intra-Pressure Group Conflicts: The United World Federalists,\" Journal of Politics 16 (1954): 76\u201395; \"Group Asks Drive for World Rule,\" New York Times, November 2, 1947, 50.\n\n\"the get tough policy of the Truman administration\": George McGovern, Grassroots: The Autobiography of George McGovern (New York: Random House, 1977), 45; Miroff, The Liberals' Moment, 123.\n\n\"Somehow we have to settle down and live with them\": McGovern, Grassroots, 49; Miroff, The Liberals' Moment, 36, 132; White, The Making of the President 1972, 116\u201317.\n\nfated to live in a hostile world: Michael T. Kaufman, \"Paul Warnke, 81, a Leading Dove in Vietnam Era, Dies,\" New York Times, November 1, 2001.\n\n\"outdated stereotypes of military confrontation and power politics\": George McGovern, \"Acceptance Speech Before Democratic National Convention,\" July 14, 1972, http:\/\/www.4president.org\/speeches\/mcgovern1972acceptance.htm; John Ehrman, The Rise of Neoconservatism: Intellectuals and Foreign Affairs, 1945\u20131994 (New Haven, CT: Yale University Press, 1995), 59.\n\nhe quoted them in his autobiography: McGovern, Grassroots, 35.\n\nlike \"pee[ing] on the floor\": Bird, The Color of Truth, 200.\n\n\"those young men will some day curse us\": Miroff, The Liberals' Moment, 38.\n\nan absolute dread of flying a plane: McGovern, Grassroots, 20\u201323.\n\n\"'I killed more people than any of you guys'\": McGovern, Grassroots, 25\u201328; Miroff, The Liberals' Moment, 29\u201330.\n\nto retreive American prisoners of war: \"Senator McGovern Presenting His Views on Foreign Policy,\" New York Times, October 6, 1972, 26; Frum, How We Got Here, 308; Jonathan Rieder, \"Rise of the 'Silent Majority,'\" in Rise and Fall of the New Deal Order, ed. Steve Fraser and Gary Gerstle (Princeton, NJ: Princeton University Press, 1989), 262.\n\nboasted about eating sheep's testicles: Bruce J. Schulman, The Seventies: The Great Shift in American Culture, Society, and Politics (New York: Free Press, 2001), 23; DeBenedetti, An American Ordeal, 239; Gaddis, Strategies of Containment, 332; Walter Isaacson, Kissinger: A Biography (New York: Simon & Schuster, 1992), 259\u201367; Herring, The Longest War,291.\n\nrejected by the fancy Manhattan firms: Tom Wicker, One of Us: Richard Nixon and the American Dream (New York: Random House, 1991), 8, 10; Perlstein, Nixonland, 20\u201324.\n\ninsinuating that he espoused the \"communist line\": Irwin F. Gellman, The Contender: Richard Nixon, The Congress Years 1946\u20131952 (New York: Free Press, 1999), 65, 84\u201385; Stephen E. Ambrose, Nixon, Vol. 1: The Education of a Politician, 1913\u20131962 (New York: Simon & Schuster, 1987), 117\u201340; Wicker, One of Us, 39\u201350.\n\n\"sitting on their fat butts\": Ambrose, Eisenhower, 272; Wicker, One of Us, 9, 10.\n\nwhen gentile boys approached: Jeremy Suri, Henry Kissinger and the American Century (Cambridge, MA: Belknap, 2007), 39\u201341, 45, 221; Isaacson, Kissinger, 26, 30, 33.\n\n\"The weak, the old had no chance\": Jussi Hanhim\u00e4ki, The Flawed Architect: Henry Kissinger and American Foreign Policy (New York: Oxford University Press, 2004), 3\u20134; Robert Dallek, Nixon and Kissinger: Partners in Power (New York: HarperCollins, 2007), 34\u201335; Isaacson, Kissinger, 28, 30, 52.\n\nlimit the length of all future submissions: Suri, Henry Kissinger and the American Century, 29; Isaacson, Kissinger, 65, 561, 697.\n\nwithin the hearts of women and men: This is a major theme of Suri's Henry Kissinger and the American Century.\n\n\"the kind of guy who would send you to the showers\": Dallek, Nixon and Kissinger, 170\u201371; Suri, Henry Kissinger and the American Century, 207\u201310; Isaacson, Kissinger, 279\u201380, 393.\n\nto be dangerously soft: Shulman, The Seventies, 24\u201325; Isaacson, Kissinger, 120.\n\nhelped spur dissident movements in Eastern Europe: Hanhim\u00e4ki, The Flawed Architect, 433\u201337; Raymond L. Gartoff, D\u00e9tente and Confrontation: American-Soviet Relations from Nixon to Reagan (Washington, DC: Brookings Institution Press, 1994), 527\u201337.\n\n\"step on them, crush them, show them no mercy\": Isaacson, Kissinger, 328.\n\nNixon called it \"the madman theory\": Dallek, Nixon and Kissinger, 106\u20137.\n\nmused that Kissinger required psychiatric care: Isaacson, Kissinger, 262\u201363, 391.\n\nkept trying to topple the government of South Vietnam: Hanhim\u00e4ki, The Flawed Architect, 43\u201346.\n\nordering that U.S. troops invade Cambodia outright: Dallek, Nixon and Kissinger, 194\u2013200; Perlstein, Nixonland, 472.\n\n\" free nations and free institutions throughout the world\": McMahon, \"Credibility and World Power,\" 467.\n\n\"like they're going to be bombed this time\": Dallek, Nixon and Kissinger, 385; Isaacson, Kissinger, 419.\n\n\"should the settlement be violated by North Vietnam\": Hanhim\u00e4ki, The Flawed Architect, 256\u201357, 267\u201368.\n\nCongress rejected the aid and South Vietnam fell: Dallek, Nixon and Kissinger, 472\u201373; Isaacson, Kissinger, 642\u201343; Hanhim\u00e4ki, The Flawed Architect, 393; Gerald Ford, \"Reporting on the United States Foreign Policy,\" April 10, 1975, http:\/\/www.presidency.ucsb.edu\/ws\/?pid=4826.\n\n\"and you're likely to find an old peace activist\": DeBenedetti, An American Ordeal, 304.\n\nless than 5 percent by 1977: Gaddis, Strategies of Containment, 319.\n\ndenounce his boss to reporters: Prados, Safe for Democracy, 425\u201326, 429\u201330; Garthoff, D\u00e9tente and Confrontation, 564\u201365; Gaddis, Strategies of Containment, 329; Isaacson, Kissinger, 683\u201384; Odd Arne Westad, The Global Cold War (New York: Cambridge University Press, 2009), 218\u201341.\n\nOperation Midnight Climax: Prados, Safe For Democracy, 437; Martin A. Lee and Bruce Shlain, Acid Dreams: The Complete Social History of LSD (New York: Grove, 1985), 32.\n\nAmerica's first ambassador to the People's Republic: C. L. Sulzberger, \"Our Next Man in Peking,\" New York Times, June 27, 1971, E15.\n\n\"what is called a civilized country\": Kahn, The China Hands, 283\u201384, 290; U.S. Congress, Senate, Hearing before the Committee on Foreign Relations, The Evolution of U.S. Policy Toward Mainland China, 92nd Cong., 1st Sess., Report No. 67-891 (Washington, DC: U.S. Government Printing Office, 1971), 7.\n\nimmoral and a tad pathetic: Cecil Currey, Edward Lansdale: The Unquiet American (Boston: Houghton Mifflin, 1988), 339; U.S. Congress, Senate, Select Committee to Study Governmental Operations with Respect to Intelligence Activities, Alleged Assassination Plots Involving Foreign Leaders: An Interim Report, 94th Cong., 1st Sess., Report No. 94-465 (Washington, DC: U.S. Government Printing Office, 1975), 142.\n\nnone of his aides owned passports: Gaddis Smith, Morality, Reason and Power: American Diplomacy in the Carter Years (New York: Hill & Wang, 1986), 27; Westad, The Global Cold War, 195.\n\nhe still corrected aides' grammatical mistakes: Kenneth E. Morris, Jimmy Carter: American Moralist (Athens: University of Georgia Press, 1996), 5\u20136, 30\u201331, 47\u201348, 60, 67, 141.\n\ndenounced racism in the state: Morris, Jimmy Carter, 30\u201332, 72, 118, 152.\n\nattack Ford from both left and right: Joshua Muravchik, The Uncertain Crusade: Jimmy Carter and the Dilemmas of Human Rights Policy (Washington, DC: Hamilton, 1986), 2\u20137.\n\nderailed his nomination: Smith, Morality, Reason and Power, 51; Muravchik, The Uncertain Crusade, 8\u20139; Robert A. Strong, Working in the World: Jimmy Carter and the Making of American Foreign Policy (Baton Rouge: Louisiana State University Press, 2000), 12\u201313.\n\nhe declared a few months after taking office: Jimmy Carter, \"University of Notre Dame Commencement,\" May 22, 1977, http:\/\/millercenter.org\/scripps\/digitalarchive\/speeches\/spe_1977_0522_carter.\n\nthe B-1 bomber and the neutron bomb: Jerel A. Rosati, The Carter Administration's Quest for Global Community: Beliefs and Their Impact on Behavior (Columbia: University of South Carolina Press, 1987), 122; David Skidmore, Reversing Course: Carter's Foreign Policy, Domestic Politics, and the Failure of Reform (Nashville, TN: Vanderbilt University Press, 1996), 45; Smith, Morality, Reason and Power, 79\u201381; Morris, Jimmy Carter, 272.\n\naid to a throng of anticommunist dictatorships: Rosati, Quest for Global Community, 120\u201322, 129\u201330; Smith, Morality, Reason and Power, 54, 103\u20134, 148.\n\nhis policies marked a genuine shift: Rosati, Quest for Global Community, 154.\n\n\"no longer seems warranted\": Rosati, Quest for Global Community, 41.\n\nin Angola and a newly unified Vietnam: Smith, Morality, Reason and Power, 116\u201317; Rosati, Quest for Global Community, 119; Skidmore, Reversing Course, 30.\n\nexplained Undersecretary of State David Newsom: Skidmore, Reversing Course, 41\u201342.\n\n\"not totally dependent on military power\": Raskin, Essays of a Citizen, 82.\nChapter 11: Fighting with Rabbits\n\nshould have throttled the beast: Jody Powell, The Other Side of the Story (New York: William Morrow, 1984), 104\u20138; Brooks Jackson, Associated Press, \"Bunny Goes Bugs: Rabbit Attacks President,\" Washington Post, August 30, 1979, A1; \"The Bunny Encounter: Carter Says He Splashed 'Nice Rabbit' but Denies There Was an Attack by Either,\" Washington Post, September 1, 1979, A2; Art Buchwald, \"Carter Staff Pondering Harey Question: Why Not the Beast?\" Washington Post, September 11, 1979, E1; Morris, Jimmy Carter, 274\u201375.\n\nThey viewed him as a wimp: On press coverage of Carter, see John M. Orman, Comparing Presidential Behavior: Carter, Reagan, and the Macho Presidential Style (New York: Greenwood, 1987), 158, 164\u201365.\n\n\"creating the conditions for domestic reaction\": Westad, The Global Cold War, 195, 282\u201383; Leffler, For the Soul of Mankind, 282\u201385; Elizabeth Drew, \"A Reporter at Large: Brzezinski,\" New Yorker, May 1, 1978, 113\u201316; Smith, Morality, Reason and Power, 153\u201356; Gartoff, D\u00e9tente and Confrontation, 698\u2013705, 713\u201314.\n\nIranian crude had been an economic lifeline: Kunz, Butter and Guns, 223\u201341.\n\noil fields of the Persian Gulf: Jerry W. Sanders, Peddlers of Crisis: The Committee on the Present Danger and the Politics of Containment (Boston: South End, 1983), 239.\n\ncalled America the \"Great Satan\": Smith, Morality, Reason and Power, 181\u201387; Strong, Working in the World, 59\u201361.\n\nshortages of medicine, soap, toothpaste, and thread: Leffler, For the Soul of Mankind, 329, 333; Westad, The Global Cold War, 242, 284\u201385, 307\u20138, 315, 323. For a discussion of the stagnation of the Soviet economy, beginning in the 1970s, see Philip Hanson, The Rise and Fall of the Soviet Union: An Economic History of the USSR from 1945 (London: Pearson Education, 2003), 128\u201363.\n\neven if he could never quite explain why: Irving Kristol, \"An Autobiographical Memoir,\" in Neoconservatism: The Autobiography of an Idea, ed. Irving Kristol (New York: Free Press, 1995), 4.\n\nostensibly committed to world revolution: The more common term, Trotskyite, Kristol insisted, was invented by their Stalinist foes, and never used by Trotskyists themselves. Irving Kristol, \"Memoirs of a Trotskyist,\" New York Times, January 23, 1977, 43.\n\n\"that were cluttering up my mind\": Kristol, \"An Autobiographical Memoir,\" 6\u20137, 13; Kristol, \"My Cold War,\" in Neoconservatism, ed. Kristol, 483\u201384.\n\nthey hated the New Deal, which he did not: Murray Friedman, The Neoconservative Revolution (New York: Cambridge University Press, 2005), 78\u201379; Irving Kristol, \"American Conservatism, 1945\u20131995,\" Public Interest 121 (1995): 82\u201384; Kristol, \"An Autobiographical Memoir,\" 32.\n\naccused the McGovern Democrats of linguistic theft: Ehrman, The Rise of Neoconservatism, 46.\n\nnot to worry about labels: Irving Kristol, \"Confessions of a True, Self-Confessed\u2014Perhaps the Only\u2014Neoconservative,\" in Reflections of a Neoconservative, ed. Irving Kristol (New York: Basic Books, 1983), 74.\n\nhe was ambivalent about the war: Robert R. Tomes, Apocalypse Then: American Intellectuals and the Vietnam War (New York: New York University Press, 1998), 194, 219; Gary Dorrien, The Neoconservative Mind (Philadelphia: Temple University Press, 1993), 91\u201392; Irving Kristol, \"American Intellectuals and Foreign Policy,\" in Neoconservatism, ed. Kristol, 76.\n\n\"the main source of all evil in the world\": Irving Kristol, \"The Twisted Vocabulary of Superpower Symmetry,\" in Scorpions in a Bottle: Dangerous Ideas About the United States and the Soviet Union, ed. Lissa Roche (Hillsdale, MI: Hillsdale College Press, 1986), 20.\n\ntoo poor to buy a sandwich: Kristol, \"Memoirs of a Trotskyist,\" 43.\n\ntools of cold war repression: Mark Gerson, The Neoconservative Vision: From the Cold War to the Culture Wars (New York: Madison, 1996), 109.\n\n\"the gun as well as the joint\": David Farber, \"The Counterculture and the Antiwar Movement,\" in Give Peace a Chance: Exploring the Vietnam Antiwar Movement, ed. Melvin Small and William D. Hoover (Syracuse, NY: Syracuse University Press, 1992), 19.\n\nsent a letter to the university's president: Matusow, The Unraveling of America, 332\u201333.\n\n\"'Up against the wall, motherfucker, this is a stickup'\": Daniel Bell, \"Columbia and the New Left,\" Public Interest 13 (1968): 66.\n\n\"develop a certain respect for what was\": Nathan Glazer, \"On Being Deradicalized,\" Commentary 50, no. 4 (1970): 74\u201376; James Traub, \"Nathan Glazer Changes His Mind, Again,\" New York Times Magazine, June 28, 1998, 23.\n\nradically improve the estate of the urban poor: Friedman, The Neoconservative Revolution, 118\u201319.\n\nwould only make things worse: Podhoretz, Breaking Ranks, 363\u201364.\n\nKirkpatrick, like Kristol, admired Niebuhr: Jeane Kirkpatrick, \"Personal Virtues, Public Vices,\" in The Reagan Phenomenon, and Other Speeches on Foreign Policy, ed. Jeane Kirkpatrick (Washington, DC: American Enterprise Institute, 1983), 214.\n\n\"wishing does not make it so\": Ehrman, The Rise of Neoconservatism, 119.\n\n\"all you had to do was stand in place\": J. David Hoeveler, Jr., Watch on the Right: Conservative Intellectuals in the Reagan Era (Madison: University of Wisconsin Press, 1991), 85.\n\n\"But Micronesia\": Sidney Blumenthal, The Rise of the Counter-Establishment: From Conservative Ideology to Political Power (New York: Times Books, 1986), 128.\n\n\"troubling our sleep\": Norman Podhoretz, \"The Culture of Appeasement,\" Harper's, October 1977, 32.\n\nMoscow had taken a decisive edge: Sanders, Peddlers of Crisis, 197\u2013204.\n\nsapped Britain's after World War I: Podhoretz, \"The Culture of Appeasement,\" 29\u201331.\n\nAmerica's first female president: John H. Mihalec, \"Hair on the President's Chest,\" Wall Street Journal, May 11, 1984, 30.\n\nregimes in the third world\u2014were running wild: Kristol, \"We Can't Resign as 'Policeman of the World,'\" New York Times, May 12, 1968, 109; Irving Kristol, \"Our Foreign Policy Illusions,\" Wall Street Journal, February 4, 1980, 26.\n\n\"even more corrupting and demoralizing\": Irving Kristol, \"Foreign Policy: The End of an Era,\" Wall Street Journal, January 18, 1979, 16.\n\nit was d\u00e9j\u00e0 vu: Hoeveler, Watch on the Right, 153; Michael Novak, \"In Praise of Jeane Kirkpatrick,\" December 8, 2006, http:\/\/www.aei.org\/publications\/filter.all,pubID.25268\/pub_detail.asp.\n\n\"change with progress, optimism with virtue\": Jeane Kirkpatrick, \"Dictatorships and Double Standards,\" Commentary 68, no. 5 (1979): 5, 13.\n\n\"Unfortunately it does\": Sanders, Peddlers of Crisis, 162.\n\ntied to chairs or beds, in the ambassador's house: Mark Bowden, Guests of the Ayatollah: The First Battle in America's War with Militant Islam (New York: Atlantic Monthly Press, 2006), 8\u201367, 127\u201329.\n\nnumber of days the Americans had been held captive: Ellen Goodman, \"And That's the Way It Is\u2014Or Is It?\" Washington Post, June 17, 1980.\n\nend of the post-toughness era as well: Bowden, Guests of the Ayatollah, 189\u201391.\n\nCarter tried diplomacy and imposed sanctions: Smith, Morality, Reason and Power, 199; Rosati, The Carter Administration's Quest for Global Community, 139.\n\nMcNamara's deputy secretary of defense: Smith, Morality, Reason and Power, 40.\n\na patron to men like Gelb, Warnke, and Lake: Rosati, The Carter Administration's Quest for Global Community, 111\u201312; I. M. Destler, Leslie Gelb, and Anthony Lake, Our Own Worst Enemy: The Unmaking of American Foreign Policy (New York: Simon & Schuster, 1984), 95\u201398; Marilyn Berger, \"Cyrus R. Vance, A Confidant of Presidents, Is Dead at 84,\" New York Times, January 13, 2002.\n\n\"growing American maturity in a complex world\": Frum, How We Got Here, 343.\n\n\"Woody Woodpecker\": Drew, \"A Reporter at Large: Brzezinski,\" 106.\n\ncolleagues ran for cover: Strong, Working in the World, 11.\n\ncalled on the Kennedy administration to bomb: Smith, Morality, Reason and Power, 36.\n\nin favor of American intervention in Vietnam: CBS News, \"Dean Bundy Meets Professor Morgenthau,\" 198\u2013206.\n\nAmerica must not \"chicken out\": Rosati, The Carter Administration's Quest for Global Community, 110; Simon Serfaty, \"Brzezinski: Play It Again, Zbig,\" Foreign Policy 32 (Fall 1978): 6.\n\nfrom anticommunism to interdependence: Gerry Argyris Andrianopoulos, Kissinger and Brzezinski: The NSC and the Struggle for Control of US National Security Policy (New York: St. Martin's, 1991), 40\u201342; Rosati, The Carter Administration's Quest for Global Community, 115, n. 9.\n\n\"elements of cooperation prevail over competition\": Rosati, The Carter Administration's Quest for Global Community, 41, 56; Zbigniew Brzezinski, \"American Policy and Global Change,\" address before the Trilateral Commission, Bonn, West Germany, October 30, 1977.\n\ninternational affairs were \"not a kindergarten\": Smith, Morality, Reason and Power, 37, 199; Rosati, The Carter Administration's Quest for Global Community, 115, n. 14.\n\nCarter broke the news to the American people: Bowden, Guests of the Ayatollah, 226\u201333, 445\u201368.\n\nVance had already resigned in disgust: Smith, Morality, Reason and Power, 203\u20134; Natasha Zaretsky, No Direction Home: The American Family and the Fear of National Decline (Chapel Hill: University of North Carolina Press, 2007), 229.\n\nmade the Shah think he was invincible: Gary Sick, All Fall Down: America's Fateful Encounter with Iran (London: I. B. Tauris, 1985), 13\u201314, 173.\n\na country it had not conquered during World War II: Ambrose and Brinkley, Rise to Globalism, 293\u201395.\n\nthat the Americans kept talking about actually was: Leffler, For the Soul of Mankind, 336; Westad, The Global Cold War, 322.\n\nripped it to shreds: Rosati, The Carter Administration's Quest for Global Community, 142\u201344; Smith, Morality, Reason and Power, 224\u201328, 230.\n\nimposed those restraints in the first place: Jimmy Carter, \"State of the Union Address,\" address to Joint Session of Congress, January 23, 1980.\n\nhis administration had painstakingly negotiated: Smith, Morality, Reason and Power, 243.\n\ndictatorships of Kenya, Somalia, and Oman: Rosati, The Carter Administration's Quest for Global Community, 144\u201345.\n\n\"sometimes dangerous world that really exists\": Rosati, The Carter Administration's Quest for Global Community, 81.\n\n\"to remain the strongest nation in the world\": Rosati, The Carter Administration's Quest for Global Community, 88; Smith, Morality, Reason and Power, 230.\n\n\"becomes a contagious disease\": Ambrose and Brinkley, Rise to Globalism, 288.\n\n\"history holds its breath\": Smith, Morality, Reason and Power, 227\u201328; Steven R. Weisman, \"Mondale Expects Support,\" New York Times, April 12, 1980, 5.\n\nChapter 12: If There Is a Bear?\n\n\"Do we get to win this time?\": Susan Jeffords, The Remasculinization of America: Gender and the Vietnam War (Bloomington: Indiana University Press, 1989), 127\u201329.\n\nan earsplitting, kick-ass \"Yes!\": Box Office Mojo, \"Rambo First Blood: Part II,\" http:\/\/www. boxofficemojo.com\/movies\/?page=main&id=rambo2.htm.\n\nAmerica's patriotic new president, Ronald Reagan: Jeffords, The Remasculinization of America, 129, 142.\n\na movie of his own, which followed the Rambo script: Lou Cannon, President Reagan: The Role of a Lifetime (New York: Touchstone, 1991), 186.\n\ndenied the award because of a technicality: Initially it was believed that there had been only one witness to his heroism, not the required two. Gerry J. Gilmore, \"USNS Benavidez Honors Army Medal of Honor Hero,\" American Forces Press Service, http:\/\/www.defenselink.mil\/news\/newsarticle.aspx?id=45442; Albin Krebs and Robert McG. Thomas, Jr., \"Notes on People: A Green Beret's Bravery Gains Additional Recognition,\" New York Times, February 20, 1981, C24.\n\n\"because they'd been denied permission to win\": Ronald Reagan, \"Remarks on Presenting the Medal of Honor to Master Sergeant Roy P. Benavidez,\" Public Papers of the President, 155, February 24, 1981.\n\nBy 1985, it was America's bestselling toy: Gil Troy, Morning in America: How Reagan Invented the 80s (Princeton, NJ: Princeton University Press, 2005), 241; ICON Group International, Inc., Hobbies: Webster's Quotations, Facts and Phrases (San Diego: ICON Group International, 2008), 36.\n\nthe ticker-tape parade they had long been denied: Jeffords, The Remasculinization of America, 2.\n\n\"You might say America has gone back to the gym\": Gerald J. DeGroot, Noble Cause: America and the Vietnam War (Harlow, England: Longman, 2000), 357.\n\na hint of the transgressions to come: George F. Will, \"'Fresh Starts' and Other Fictions,\" in The Morning After, ed. George F. Will (New York: Free Press, 1986), 343.\n\nhad to drag him by the armpits to bed: Cannon, President Reagan, 207\u20138.\n\nwhich the local butcher usually sold as pet food: John Patrick Diggins, Ronald Reagan: Fate, Freedom, and the Making of History (New York: Norton, 2007), 58.\n\n\"one of those rare Huck Finn\u2013Tom Sawyer idylls\": Blumenthal, The Rise of the Counter-Establishment, 244; Leffler For the Soul of Mankind, 342.\n\nReagan focused on the only positive one: Cannon, President Reagan, 25, 179, 746; Diggins, Ronald Reagan, 67, 73.\n\n\"inability to distinguish between fact and fantasy\": Sean Wilentz, The Age of Reagan (New York: HarperCollins, 2008), 130.\n\nspinoff from Unitarianism that downplayed the idea of sin: Diggins, Ronald Reagan, 25.\n\nEureka College, a Disciples' institution: Stephen Vaughn, \"The Moral Inheritance of a President: Reagan and the Dixon Disciples of Christ,\" Presidential Studies Quarterly 25, no. 1 (1995): 109\u201327; Beth A. Fischer, The Reagan Reversal: Foreign Policy and the End of the Cold War (Columbia: University of Missouri Press, 1997), 106; Paul Lettow, Ronald Reagan and His Quest to Abolish Nuclear Weapons (New York: Random House, 2005), 7\u20138; Wilentz, The Age of Reagan, 130.\n\n\"the desires he planted in us are good\": Diggins, Ronald Reagan, 30.\n\n\"which regularly deny the existence of evil\": Jeane Kirkpatrick, \"The Reagan Reassertion of Western Values,\" in The Reagan Phenomenon, ed. Kirkpatrick, 34.\n\nfights with the communists in Hollywood in the 1940s: Cannon, President Reagan, 284\u201387; Diggins, Ronald Reagan, 99\u2013101.\n\nhe didn't share its fear: Leffler, For the Soul of Mankind, 340.\n\ndestabilize the entire European banking system: Dan Morgan and Robert G. Kaiser, \"Group of Aides Sought Tougher Stand on Poland,\" Washington Post, January 15, 1982, A1; Dan Morgan, \"West Faces Dilemma on Polish Debt,\" Washington Post, January 11, 1982, A15; George F. Will, \"Reagan's Dim Candle,\" Newsweek, January 18, 1982, 100; Norman Podhoretz, \"The Neo-Conservative Anguish over Reagan's Foreign Policy,\" New York Times, May 2, 1982, 30.\n\n\"sensitive about being viewed as too pugnacious\": Robert Kagan, A Twilight Struggle: American Power and Nicaragua, 1977\u20131990 (New York: Free Press, 1996), 270.\n\n\"is Spanish for Vietnam\": Thomas Paterson, \"Historical Memory and Illusive Victories: Vietnam and Central America,\" Diplomatic History 12, no. 1 (1988): 2\u20133.\n\ndated back to the nineteenth century: Paterson, \"Historical Memory and Illusive Victories,\" 3, 18.\n\n\"a fucking parking lot\": Cannon, President Reagan, 196.\n\ndeclarations of war against both Cuba and Nicaragua: William F. Buckley, \"Who Was Right?\" in Right Reason, ed. Richard Brookhiser (Boston: Little, Brown, 1985), 123\u201325; Buckley, \"Next in Central America?\" in Right Reason, ed. Brookhiser, 257.\n\n\"reverse the totalitarian drift in Central America\": Norman Podhoretz, \"Appeasement by Any Other Name,\" Commentary 76, no. 1 (1983): 38.\n\n\"we will use all means to prevent this\": Irving Kristol, \"The Muddle in Foreign Policy,\" Wall Street Journal, April 29, 1981, 28.\n\na chorus of anxious references to Vietnam: Kagan, A Twilight Struggle, 175.\n\n\"scared the shit out of Ronald Reagan\": Cannon, President Reagan, 196\u201397.\n\n\"limp-wristed, traditional cookie-pushing bullshit\": Cannon, President Reagan, 345, 348.\n\nopposed invading Nicaragua: William Schneider, \"'Rambo' and Reality: Having It Both Ways,\" in Eagle Resurgent? The Reagan Era in American Foreign Policy, ed. Kenneth A. Oye, Robert J. Lieber, and Donald Rothchild (Boston: Little, Brown, 1983), 60.\n\nallowing a communist takeover: Barry Sussman, \"Poll Finds a Majority Fears Entanglement in Central America,\" Washington Post, May 25, 1983, A1.\n\nmembers of both parties erupted in cheers: Podhoretz, \"Appeasement by Any Other Name,\" 27; Steven R. Weisman, \"President Appeals Before Congress for Aid to Latins,\" New York Times, April 28, 1983, A1; Gerald F. Seib, \"Central America Leftists Called Threat by Reagan,\" Wall Street Journal, April 28, 1983, 3.\n\nReagan's advisers feared he might be impeached: Cannon, President Reagan, 380, 718; Troy, Morning in America, 244; Wilentz, The Age of Reagan, 232, 242.\n\nWorld War II, Korea, and Vietnam combined: Westad, The Global Cold War, 347.\n\n\"and I'm not going to do it\": Cannon, President Reagan, 336, 337.\n\n\"perhaps the saddest day of my life\": Cannon, President Reagan, 389\u2013457, 442.\n\nReagan decided to do exactly that: Cannon, President Reagan, 445, 454\u201357, 510.\n\ndeclared herself \"disgusted\": Bernard Weinraub, \"On the Right: Long Wait for Foreign Policy Hero,\" New York Times, July 12, 1985, A12.\n\nTheir boss's poll numbers soon ticked back up: Dan Balz and Thomas B. Edsall, \"The Invasion of Grenada; GOP Rallies Around Reagan,\" Washington Post, October 26, 1983, A8; David Shribman, \"Poll Finds a Lack of Support for Latin Policy,\" New York Times, April 29, 1984, A1; Cannon, President Reagan, 510\u201311.\n\nsince the planned runway was too small: Cannon, President Reagan, 445\u201349.\n\nMore than 70 percent of Americans backed the Grenada invasion: Cannon, President Reagan, 448, 462; Schneider, \"'Rambo' and Reality,\" 59.\n\nmore than ten to one in favor: Diggins, Ronald Reagan, 249.\n\nfrom humiliation to pride: Cannon, President Reagan, 448\u201349.\n\nwhere they waved little American flags: Jon Western, Selling Intervention and War: The Presidency, the Media and the American Public (Baltimore: Johns Hopkins University Press, 2005), 95; Francis X. Clines, \"Medical Students Cheer Reagan at a White House Ceremony,\" New York Times, November 8, 1983, A10.\n\nmore than 70 percent support at home: Cannon, President Reagan, 653\u201354; Troy, Morning in America, 246.\n\nmorphing into Rambo again: Cannon, President Reagan, 663.\n\ndeclared Norman Podhoretz in 1976: Dorrien, Neoconservative Mind, 178.\n\n\"they do not and cannot last\": Irving Kristol, \"An Automatic-Pilot Administration,\" Wall Street Journal, December 14, 1984, 26.\n\nshe explained in 1981: Jeane Kirkpatrick, \"The Reagan Phenomenon and the Liberal Tradition,\" in The Reagan Phenomenon, ed. Kirkpatrick, 15.\n\n\"human freedom and human dignity to its citizens\": Leffler, For the Soul of Mankind, 339\u201340.\n\n\"precious little evidence to prove them wrong\": Irving Kristol, \"The Succession: Understanding the Soviet Mafia,\" Wall Street Journal, November 18, 1982, 30.\n\n\"oriental despotism\" that stretched back to the czars: Adam Wolfson, \"The World According to Kirkpatrick: Is Ronald Reagan Listening?\" Policy Review 31 (1985): 70.\n\n\"served to reinforce these ingrained convictions\": Sanders, Peddlers of Crisis, 149.\n\n\"people don't make wars\": Gaddis, Strategies of Containment, 360\u201361; James Mann, The Rebellion of Ronald Reagan: A History of the End of the Cold War (New York: Viking, 2009), 80.\n\n\"another time and another era\": Diggins, Ronald Reagan, 3, 8, 29; Mann, The Rebellion of Ronald Reagan, 304.\n\na group that sought to outlaw nuclear weapons: Cannon, President Reagan, 62.\n\ndidn't fit his image as a cold war hawk: On Reagan's nuclear abolitionism, see Lettow, Ronald Reagan and His Quest to Abolish Nuclear Weapons.\n\nthe robots will descend and kill them all: Cannon, President Reagan, 61\u201362.\n\n\"Here come the little green men again\": Cannon, President Reagan, 62\u201363.\n\nthe inventor of the shield: Thompson, The Hawk and the Dove, 300.\n\nuntil the Americans had caught up: Beth A. Fischer, \"Toeing the Hardline? The Reagan Administration and the Ending of the Cold War,\" Political Science Quarterly 112, no. 3 (1997): 483.\n\nit reflected their eternal view of world affairs: In 1982, for instance, Podhoretz attacked Reagan for saying that once America caught up with the Soviets he would support a mutual freeze on the production of new nuclear weapons. Podhoretz, \"The Neo-conservative Anguish,\" 30.\n\n\"the gap between U.S. and Soviet military capabilities continues to grow\": Walter Pincus and Don Oberdorfer, \"A-Arms Balance Holds,\" Washington Post, December 16, 1984, A1.\n\n\"He had tied into NORAD!\": Cannon, President Reagan, 58, 85.\n\n\"to see that there is never a nuclear war\": Fischer, The Reagan Reversal, 120.\n\nhe had seen footage of the camps in a film: Cannon, President Reagan, 486\u201390.\n\nplaying out before him in real life: \"In several ways,\" Reagan noted, \"the sequence of events described in the briefings paralleled those in the ABC movie.\" Barbara Farnham, \"Reagan and the Gorbachev Revolution: Perceiving the End of Threat,\" Political Science Quarterly 116, no. 2 (2001): 232. See also Leffler, For the Soul of Mankind, 359; and Fischer, The Reagan Reversal, 120\u201321.\n\nput their military on high alert: Fischer, The Reagan Reversal, 122\u201336; Don Oberdorfer, From the Cold War to a New Era: The United States and the Soviet Union, 1983\u20131991 (Baltimore: Johns Hopkins University Press, 1998), 65\u201367.\n\npercentage of Americans favoring arms control had shot through the roof: Schneider, \"'Rambo' and Reality,\" 43; John Rielly, \"American Opinion: Continuity, Not Reaganism,\" Foreign Policy 50 (1983): 96.\n\na mutual freeze on the production of nuclear arms: Troy, Morning in America, 138.\n\na warmonger in his 1984 reelection campaign: Cannon, President Reagan, 508\u201311, 740; Jeremi Suri, \"Explaining the End of the Cold War: A New Historical Consensus?\" Journal of Cold War Studies 4, no. 4 (2002): 92; Fischer, \"Toeing the Hardline?\" 496; Farnham, \"Reagan and the Gorbachev Revolution,\" 233\u201334.\n\nalso eager to get disarmament talks going: Cannon, President Reagan, 309; Barbara Farnham, \"Reagan and the Gorbachev Revolution,\" 230; Leffler, For the Soul of Mankind, 362.\n\n\"nuclear weapons will be banished from the face of the earth\": Ronald Reagan, \"Address to the Nation and Other Countries on United States-Soviet Relations,\" January 16, 1984, http:\/\/www.reagan.utexas.edu\/archives\/speeches\/1984\/11684a.htm.\n\n\"thousand-fold affection returned to you\": Fischer, The Reagan Reversal, 40\u201341.\n\n\"under the threat of those weapons\": Fischer, The Reagan Reversal, 42.\n\n\"it would be the height of naivete to think otherwise\": Kristol, \"An Automatic-Pilot Administration,\" 26.\n\na jaw-dropping 40 percent of his government's budget: Zubok, Failed Empire, 277.\n\nAmerica looked like less of a threat: Suri, \"Explaining the End of the Cold War,\" 70\u201378.\n\na summit meeting, without preconditions: Fischer, \"Toeing the Hardline?\" 494.\n\nin Moscow to attend the funeral of Gorbache's predecessor: Bernard Weinraub, \"Bush Sent to Rites,\" New York Times, March 12, 1985, A1.\n\ndid not share her \" foreign policy objectives\": Bernard Weinraub, \"Reagan Is Told by Kirkpatrick She Will Leave,\" New York Times, January 31, 1985, A6.\n\n\"the threat or use of force in international relations\": Leffler, For the Soul of Mankind, 364\u201365.\n\nbriefing papers on Soviet politics and culture: Cannon, President Reagan, 57, 155, 157, 746\u201349.\n\ntalked for almost five hours: Fischer, The Reagan Reversal, 48.\n\n\"bleeding when we shake hands\": Cannon, President Reagan, 754.\n\n\"craven eagerness\" to give away the nuclear store: Dorrien, The Neoconservative Mind, 197\u2013200.\n\n\"like a punctured balloon\": George Will, \"Reagan Botched the Daniloff Affair,\" Washington Post, September 18, 1986, A25.\n\n\"a dream of a world without nuclear weapons\": Leffler, For the Soul of Mankind, 387\u201388, 392.\n\n\"give a tremendous party for the whole world\": Gaddis, Strategies of Containment, 366; Richard Rhodes, Arsenals of Folly: The Making of the Nuclear Arms Race (New York: Knopf, 2007), 230; Oberdorfer, From the Cold War to a New Era, 169\u201374, 207\u20138; Jack F. Matlock, Jr., Autopsy of an Empire: The American Ambassador's Account of the Collapse of the Soviet Union (New York: Random House, 1995), 93\u201398; Lettow, Ronald Reagan and His Quest to Abolish Nuclear Weapons, 217\u201326.\n\nthe two sides began dismantling them: Leffler, For the Soul of Mankind, 394, 401.\n\nwhile allowing Britain and France to keep theirs: Leffler, For the Soul of Mankind, 400.\n\n\"elevating wishful thinking to the status of political philosophy\": Diggins, President Reagan, 384.\n\n\"a utopian\": \"Mad Momentum,\" Wall Street Journal, April 13, 1988, 28.\n\n\"is not the architect of Soviet retreat\": Jeane Kirkpatrick, The Withering Away of the Totalitarian State and Other Surprises (Washington, DC: AEI Press, 1990), 52; Hedrick Smith, \"The Right Against Reagan,\" New York Times Magazine, January 17, section 6, 36.\n\nDick Cheney called glasnost a fraud: Diggins, Ronald Reagan, 384.\n\nPodhoretz proposed reconstituting it: Dorrien, The Neoconservative Mind, 199; Sanders, Peddlers of Crisis, 152\u201354.\n\nNeville Chamberlain, and Adolf Hitler: Smith, \"The Right Against Reagan,\" 36; John Hanrahan, United Press International, \"Conservatives Escalate Anti-Treaty Campaign,\" January 20, 1988.\n\nsounding a little like the Port Huron Statement: Diggins, Ronald Reagan, 392.\n\n\"a long silence yearn to break free\": Cannon, President Reagan, 787.\n\n\"tighten our belts and spend even more on defense\": Barbara Farnham, \"Reagan and the Gorbachev Revolution,\" 250. The historian Thomas Risse-Kappen adds that there is \"not a shred of substantiation for the claim that there is any connection between the US [military] buildup and the Soviet turnabout.\" Diggins, Ronald Reagan, 405. Walter Uhler has debunked the claim that it was fear of SDI that led Gorbachev to seek to end the cold war. Walter C. Uhler, \"Misreading the Soviet Threat,\" Journal of Slavic Military Studies, March 2001, www.walter-c-uhler.com\/Reviews\/Misreading.html. Reagan himself wrote that \"I might have helped him see that the Soviet Union had less to fear from the West than he thought, and that the Soviet empire in Eastern Europe wasn't needed for the security of the Soviet Union.\" Gaddis, Strategies of Containment, 374.\n\n\"a minor Dark Age\": Steel, Walter Lippmann, 592.\n\n\"I am ashamed of our beloved nation\": Fox, Reinhold Niebuhr, 285.\n\n\"Perhaps to save one's soul is all that is left\": Frei, Hans J. Morgenthau, 221\u201322.\n\n\"succumbing feebly, day by day, to its own decadence\": Mayers, George Kennan and the Dilemmas of US Foreign Policy, 299.\n\n\"no recovery and no return\": Hixson, George Kennan, 283\u201385.\n\nchickens were coming home to roost: Mayers, George Kennan and the Dilemmas of US Foreign Policy, 314.\n\nsuggested that he be \"put out to pasture\": Henry Fairlie, \"Mr. X: The Special Senility of the Diplomat,\" New Republic, December 24, 1977, 9\u201311.\n\n\"springs a leak\": Paul Seabury, \"George Kennan vs. Mr. 'X': The Great Container Springs a Leak,\" New Republic, December 16, 1981, 17\u201320.\n\nthe eventual abolition of all nuclear weapons: Hixson, George Kennan, 282.\n\n\"is a fit occasion for satisfaction\": George Kennan, \"Republicans Won the Cold War?\" New York Times, October 28, 1992, in George Kennan, At a Century's Ending: Reflections, 1982\u20131995 (New York: Norton, 1995), 187.\n\nPART III: THE HUBRIS OF DOMINANCE\n\nChapter 13: Nothing Is Consummated\n\n\"like a Mozart oboe concerto\": Mary McGrory, \"Kennan\u2014A Prophet Honored,\" Washington Post, April 9, 1989, B1; Don Oberdorfer, \"Revolutionary Epoch Ending in Russia, Kennan Declared,\" Washington Post, April 9, 1989, A22.\n\n\"Wilson was way ahead of his time\": Senate Committee on Foreign Relations, The Future of U.S.-Soviet Relations: Hearing Before the Committee on Foreign Relations, 101st Cong., 1st sess., 1989, 27\u201328; Ray Moseley, \"Expert on USSR Says Communism Is Dying,\" Chicago Tribune, April 5, 1989, 1.\n\nstill comparing Gorbachev to Brezhnev: Richard Pipes, \"The Russians Are Still Coming,\" New York Times, October 9, 1989, A17.\n\nstill comparing him to Lenin: Dorrien, The Neoconservative Mind, 199.\n\nEast German communism is here to stay: Francis Fukuyama, interviews with the author, July 15 and 21, 2008.\n\n\"nuclear weapons and environmental damage\": Francis Fukuyama, The End of History and the Last Man (New York: Free Press, 1992), xiii, 3.\n\nGoddess of Democracy in Tiananmen Square: Peter LaBarbera and Paul Bedard, \"Walesa Thanks America for Aiding Poland,\" Washington Post, November 16, 1989, A1.\n\n\"democracy is supreme\": \"The Pivot of History,\" New Republic, November 16, 1918, 58.\n\n\"crossed the finish line out of breath\": Robert Kagan, \"A Retreat from Power?\" Commentary 100, no. 5 (1995): 22.\n\nworld's largest creditor to its largest debtor: Samuel Huntington, \"The US\u2014Decline or Renewal?\" Foreign Affairs 67, no.2 (1988): 78.\n\nthe worst day of 1929: E. S. Browning, \"Exorcising Ghosts of Octobers Past: Despite Housing Slump, Crashes Such as in 1987 Likely to Stay Memories,\" Wall Street Journal, October 15, 2007, C1.\n\nit had sold 225,000: James Mann, Rise of the Vulcans: The History of Bush's War Cabinet (New York: Viking Penguin, 2004), 161.\n\nthe New York Times' bestseller list: Derek Chollet and James Goldgeier, America Between the Wars (New York: PublicAffairs, 2008), xiii; Hawes Publications, \"New York Times Bestseller List,\" http:\/\/www.hawes.com\/1988\/1988.htm.\n\nAdmiral William Crowe, chairman: Frederick Kempe, Divorcing the Dictator: America's Bungled Affair with Noriega (London: I. B. Tauris, 1990), 294; Dennis Hevesi, \"Adm. William Crowe Dies at 82; Led Joint Chiefs,\" New York Times, October 19, 2007; Stanley Karnow, In Our Image: America's Empire in the Philippines (New York: Random House, 1989), 407.\n\nNorman Podhoretz's stepdaughter: Eric Alterman, \"Elliott Abrams: The Teflon Assistant Secretary,\" Washington Monthly, May 1987; Michael Crowley, \"Assessment: Elliott Abrams,\" Slate, February 17, 2005, http:\/\/www.slate.com\/id\/2113690\/; Jay Winik, On the Brink: The Dramatic, Behind-the-Scenes Saga of the Reagan Era and the Men and Women Who Won the Cold War (New York: Simon & Schuster, 1996), 425\u201332.\n\nAbrams was only thirty-two: Winik, On the Brink, 433.\n\n\"Northern America and Western Europe\": Irving Kristol, \"What Choice Is There in Salvador?\" Wall Street Journal, April 4, 1983, 16.\n\nthe thugs were on our side: Pamela Constable and Arturo Valenzuela, \"Is Chile Next?\" Foreign Policy, 63 (1986): 58\u201359, 71\u201373; David P. Forsythe, \"Human Rights in U.S. Foreign Policy: Retrospect and Prospect,\" Political Science Quarterly 105, no. 3 (1990): 435\u201354; Christopher Madison, \"Abrams Policy Skills to Be Put to Test in Central America Hot Seat at State,\" National Journal 17, no. 28 (1985): 1619\u201322; George Shultz, Turmoil and Triumph (New York: Scribner, 1993), 972, 974.\n\na prominent critic of the regime: Kempe, Divorcing the Dictator, 26; Emily Yoffe, \"A Presidential Salary FAQ,\" Slate, January 3, 2001, http:\/\/www.slate.com\/id\/1006798\/.\n\nJesse Helms was appalled: Kevin Buckley, Panama: The Whole Story (New York: Simon & Schuster, 1991), 27\u201328, 41\u201342; Kempe, Divorcing the Dictator, 176\u201377.\n\nhe was denouncing him publicly: Buckley, Panama, 90; Kempe, Divorcing the Dictator, 169, 294.\n\nimportant enough to protect: Eytan Gilboa, \"The Panama Invasion Revisited: Lessons for the Use of Force in the Post Cold War Era,\" Political Science Quarterly 110, no. 4 (1995): 541, 546\u201347.\n\nnever smelled it up close: Elaine Sciolino, \"Washington at Work: Crowe v. Abrams: A Private Feud over Handling Panama Becomes Public,\" New York Times, October 23, 1989, A14.\n\ndeferment to avoid Vietnam: Kempe, Divorcing the Dictator, 299.\n\n\"raised it to an art form\": William Crowe, letter to the editor, \"Elliott Abrams Remains Reckless on Panama,\" New York Times, October 16, 1989.\n\nweren't fighting for it themselves: Kempe, Divorcing the Dictator, 297, 303\u20134; Buckley, Panama, 138.\n\nsouth of the border now: Gilboa, \"The Panama Invasion Revisited,\" 548\u201349.\n\nstill haunted his sleep: Stefan Halper and Jonathan Clarke, America Alone: The Neo-Conservatives and the Global Order (New York: Cambridge University Press, 2004), 171.\n\n\"sending those troops to Lebanon\": Marlin Fitzwater, phone interview with the author, August 7, 2008.\n\nif he gave up power: Kempe, Divorcing the Dictator, 311; Buckley, Panama, 143\u201345.\n\n\"will never invade Panama\": Bob Woodward, The Commanders (New York: Simon & Schuster, 1991), 116.\n\n\"the farewell party has been indefinitely postponed\": Buckley, Panama, 166.\n\nthere would be no deals: Buckley, Panama, 147\u201348, 169; Kempe, Divorcing the Dictator, 11, 28, 313, 347.\n\nsmashed him in the face with an iron bar: Kempe, Divorcing the Dictator, 83, 225; Buckley, Panama, 152\u201356, 208; Woodward, The Commanders, 84.\n\nabove her head until she collapsed: Woodward, The Commanders, 84, 157\u201358; Buckley, Panama, 227.\n\npeople did call him a wimp, often: See, for instance, Margaret Warner, \"Bush Battles the 'Wimp Factor,'\" Newsweek, October 19, 1987, 28.\n\nplayed a mean game of horseshoes: Richard Ben Cramer, What It Takes (New York: Random House, 1992), 9, 116\u201317; Jacob Weisberg, The Bush Tragedy (New York: Random House, 2008), 23, 31\u201332, 34.\n\nthe names of his political enemies: Woodward, The Commanders, 168, 191\u201393; Buckley, Panama, 247.\n\nflown to a Florida jail: Andrew Rosenthal, \"Noriega's Surrender: Overview; Noriega Gives Himself Up to U.S. Military; Is Flown to Florida to Face Drug Charges,\" New York Times, January 4, 1990, A1; Adam Pertman, \"Closed Door Arguments Continue over Relocating Noriega,\" Boston Globe, January 13, 1990, A4.\n\ntwenty-three U.S. soldiers died: Woodward, The Commanders, 195.\n\nany U.S. president since Vietnam: Bruce Jentleson, \"The Pretty Prudent Public,\" International Studies Quarterly 36, no. 1 (1992): 55; Russell Crandall, Gunboat Democracy: US Interventions in the Dominican Republic, Grenada and Panama (New York: Rowman & Littlefield, 2006), 209.\n\n\"alienate the Panamanian people\": Crandall, Gunboat Democracy, 25, 197, 204, 215; Woodward, The Commanders, 85, 90, 136, 138, 144, 173, 189; Karin von Hippel, Democracy by Force: US Military Intervention in the Post\u2013Cold War World (Cambridge, England: Cambridge University Press, 2000), 35; Buckley, Panama, 179.\n\nthe swelling ranks of Latin American democracies: Crandall, Gunboat Democracy, 209\u201311, 216; Buckley, Panama, 179, 259.\n\n\"a free country with justice and liberty\": Crandall, Gunboat Democracy, 210.\n\n\"a trial run\": Stephen Hayes, Cheney: The Untold Story of America's Most Powerful and Controversial Vice President (New York: HarperCollins, 2007), 224.\n\nwaving his gold watch: Christian Alfonsi, Circle in the Sand: Why We Went Back to Iraq (New York: Doubleday, 2006), 21, 26, 44, 56; Thomas C. Hayes, \"The Oilfield Lying Below the Iraq-Kuwait Dispute,\" New York Times, September 3, 1990.\n\nmore than 45 percent: Energy Information Administration, \"International Petroleum (Oil) Reserves and Resources Data,\" http:\/\/www.eia.doe.gov\/pub\/international\/iealf\/crudeoilre serves.xls.\n\ndispatching troops to Saudi soil: Alfonsi, Circle in the Sand, 89.\n\nand then draw its sword: Steve A. Yetiv, Explaining Foreign Policy: U.S. Decision-Making and the Persian Gulf War (Baltimore: Johns Hopkins University Press, 2004), 127\u201329; Mann, Rise of the Vulcans, 185\u201386; Richard N. Haass, War of Necessity, War of Choice: A Memoir of Two Iraq Wars (New York: Simon & Schuster, 2009), 94.\n\nthe United States to beat Iraq to a pulp: Tareq Y. Ismael and Andrej Kreutz, \"Russian-Iraqi Relations: A Historical and Political Analysis,\" Arab Studies Quarterly 23, no. 4 (2001): 89\u201390.\n\nsweltering in the Saudi desert: Woodward, The Commanders, 37.\n\n\"been selling around here or over there\": Woodward, The Commanders, 35\u201339.\n\ndancing to calypso and revering the queen: Karen DeYoung, Soldier: The Life of Colin Powell (New York: Vintage, 2007), 19, 23.\n\nin the middle of a garbage dump: DeYoung, Soldier, 49, 69\u201370.\n\n\"I perform well\": Henry Louis Gates, Jr., \"Powell and the Black Elite,\" New Yorker, September 25, 1995, 73.\n\nseared into his consciousness: Patricia Sullivan, \"William Crowe, Jr.; Joint Chiefs Leader Had Diplomat's Touch,\" Washington Post, October 19, 2007, B6; DeYoung, Soldier, 51\u201368, 74\u201391.\n\nthey began the long journey home: DeYoung, Soldier, 76.\n\nhad been blown to bits: Mann, Rise of the Vulcans, 118.\n\nlost close to half a million men: Woodward, The Commanders, 342; DeYoung, Soldier, 193; Alfonsi, Circle in the Sand, 67; Michael R. Gordon and Bernard E. Trainor, The Generals' War: The Inside Story of the Conflict in the Gulf (Boston: Little, Brown, 1995), 33, 130\u201333; James A. Bill, \"Why Tehran Finally Wants a Gulf Peace; Iraq's Missile Blitz Broke Their Will to Keep Fighting,\" Washington Post, August 28, 1988, B1; Patrick E. Tyler, \"Gulf War Cease-Fire Begins,\" Washington Post, August 20, 1988, A1; Lawrence Freedman and Ephraim Karsh, The Gulf Conflict 1990\u20131991 (Princeton, NJ: Princeton University Press, 1995), 74, 203.\n\nCheney's deferments during Vietnam: Colin Powell, My American Journey (New York Ballantine, 1995), 393; Gary Dorrien, Imperial Designs: Neoconservatism and the New Pax Americana (New York: Routledge, 2004), 39.\n\npeople called him \"the Sphinx\": Dorrien, Imperial Designs, 38.\n\na reputation for humiliating generals: Hayes, Cheney, 216, 234\u201338; Gordon and Trainor, The Generals' War, 100\u20131.\n\n\"So stick to military matters\": Powell, My American Journey, 451\u201352.\n\nalmost forgot he was black\u2014seemed pleased: DeYoung, Soldier, 195; Woodward, The Commanders, 41\u201342, 299\u2013301.\n\nitching to cut a diplomatic deal: Yetiv, Explaining Foreign Policy, 117, 128; Haass, War of Necessity, War of Choice, 94, 107.\n\nwas tentative early on: Alfonsi, Circle in the Sand, 53; George [H. W.] Bush and Brent Scowcroft, A World Transformed (New York: Knopf, 1998), 315\u201318.\n\ntelling them not to think: Richard Morin and E. J. Dionne, \"Vox Populi: Winds of War and Shifts of Opinion,\" Washington Post, December 23, 1990, C1; Donna Cassata, Associated Press, \"Former Military Leaders Caution Against Rushing to War,\" November 29, 1990; Freedman and Karsh, The Gulf Conflict, 285.\n\nauthorizing military force since 1812: Adam Wolfson, \"Humanitarian Hawks? Why Kosovo but Not Kuwait,\" Policy Review 98 (2000): 30; Jeane Kirkpatrick, Making War to Keep Peace (New York: Harper, 2007), 25; Max Elbaum, \"The Storm at Home,\" Crossroads, April 1991, http:\/\/www.revolutionintheair.com\/histstrategy\/gulf1.html.\n\n\"potentially devastating economic consequences\": Zbigniew Brzezinski, interview with Paul Begala and Tucker Carlson, Crossfire, CNN, September 21, 1990; Zbigniew Brzezinski, \"Patience in the Persian Gulf, Not War,\" New York Times, October 7, 1990, 19.\n\ncame out of the woodwork in opposition: Senate Committee on Armed Services, Crisis in the Persian Gulf Region: US Policy Options and Implications, 101st Cong., 2nd sess., December 3, 1990; House Committee on Armed Services, Crisis in the Persian Gulf: Sanctions, Diplomacy and War, 101st Cong., 2nd sess., December 20, 1990.\n\nbefore casting their votes: Senate Committee on Armed Services, Crisis in the Persian Gulf Region: US Policy Options and Implications, 101st Cong., 2nd sess., December 3, 1990.\n\n\"the past that will not die\": Evan Thomas, \"No Vietnam,\" Newsweek, December 10, 1990, 24.\n\nthe average B-52 missed its target by 2,700 feet: Marshall L. Michel, III, Eleven Days of Christmas: America's Last Vietnam Battle (New York: Encounter, 2001), 223.\n\nin the roof of the Iraqi air command: Mark Clodfelter, \"Of Demons, Storms, and Thunder,\" Airpower Journal (1991), http:\/\/www.airpower.maxwell.af.mil\/airchronicles\/apj\/apj91\/win91\/clod.htm.\n\nvulnerable to attacks from the air: Clodfelter, \"Of Demons, Storms, and Thunder.\"\n\n\"immaculate destruction\": David Halberstam, War in the Time of Peace: Bush, Clinton and the Generals (New York: Simon & Schuster, 2001), 56. Michael Gordon and Bernard Trainor call the Gulf War \"the first war in history in which airpower not ground forces played the dominant role.\" Gordon and Trainor, The Generals' War, x.\n\nAmerica didn't lose a single one: Alfonsi, Circle in the Sand, 171.\n\nthe figure was 146: Robert W. Tucker and David C. Hendrickson, The Imperial Temptation: The New World Order and America's Purpose (New York: Council on Foreign Relations Press, 1992), 74; Gordon and Trainor, The Generals' War, 457.\n\nAmerica actually turned a profit on the war: Tucker and Hendrickson, The Imperial Temptation, 74.\n\n\"the Vietnam syndrome once and for all\": Tucker and Hendrickson, The Imperial Temptation, 152.\n\nclose to five million people cheered: DeYoung, Soldier, 209.\n\n\"the most powerful nation in the world\": Peter Applebome, \"After the War: National Mood; War Heals Wounds at Home, but Not All,\" New York Times, March 4, 1991, A1.\n\n\"We kicked ass\": Alfonsi, Circle in the Sand, 177.\n\n\"Nothing is consummated\": Chollet and Goldgeier, America Between the Wars, 34.\n\nthe figure had more than doubled: Jentleson, \"The Pretty Prudent Public,\" 67\u201369.\n\nindependent power base: Alfonsi, Circle in the Sand, 192\u201393.\n\nand depose Saddam itself: Freedman and Karsh, The Gulf Conflict, 403.\n\nwhat they thought was Bush's command: Alfonsi, Circle in the Sand, 215; Mann, Rise of the Vulcans, 193; Freedman and Karsh, The Gulf Conflict, 420.\n\n\"caution or circumspection as to danger or risk\": \"Prudence,\" Merriam-Webster Online Dictionary, http:\/\/www.merriam-webster.com\/dictionary\/prudence.\n\nto dismember America's old foe: Bush and Scowcroft, A World Transformed, 44, 208, 541.\n\nto run the war through the UN, either: Alfonsi, Circle in the Sand, 101, 152\u201353.\n\nshooting down Saddam's helicopters: Gordon and Trainor, The Generals' War, 455; Lewis D. Solomon, Paul D. Wolfowitz: Visionary Intellectual, Policymaker and Strategist (Westport, CT: Praeger Security International, 2007), 68.\n\nthe downfalls of Somoza and the Shah: Jeane Kirkpatrick, \"Our Interests in the Philippines,\" Chicago Tribune, December 15, 1985, 23.\n\n\"swallowing too many happiness pills\": Irving Kristol, \"Now What for US Client States?\" Wall Street Journal, March 3, 1986, 18.\n\nShultz in turn convinced Reagan: Raymond Bonner, Waltzing with a Dictator: The Marcoses and the Making of American Policy (New York: Times Books, 1987), 432; Solomon, Paul D. Wolfowitz, 30\u201337.\n\n\"the high point of my career\": Michael Dobbs, \"For Wolfowitz, a Vision May Be Realized,\" Washington Post, April 7, 2003, A17.\n\nthe most adamant that America come to their aid: Gordon and Trainor, The Generals' War, 451, 455; Mann, Rise of the Vulcans, 193; Solomon, Paul D. Wolfowitz, 68.\n\nso high you could barely see: Mann, Rise of the Vulcans, 263.\n\ninside his suit pocket: Dorrien, Imperial Designs, 57; Bill Keller, \"The Sunshine Warrior,\" New York Times Magazine, September 22, 2002, 48.\n\n\"to create these great schemes\": Gates, \"Powell and the Black Elite,\" 73.\n\nthat the question was closed: Gordon and Trainor, The Generals' War, 456.\n\nin the Gulf, had not even existed: Mann, Rise of the Vulcans, 89.\n\nabove much of Iraq: Joe Stork and Martha Wenger, \"The US in the Persian Gulf: From Rapid Deployment to Massive Deployment,\" Middle East Report 168 (1991): 25; Kenneth Katzman, \"The Persian Gulf: Issues for U.S. Policy, 2000,\" CRS Report for Congress, November 3, 2000, 18\u201324; Multinational Force and Observers, \"Assembling the Force,\" http:\/\/www.mfo.org\/1\/4\/22\/25\/base2.asp, MFO.\n\nA decade later it was 72 percent: Stockholm International Peace Research Institute, \"SIPRI Arms Transfers Database,\" http:\/\/armstrade.sipri.org.\n\n\"talks like Dirty Harry but acts like Barney Fife\": Alfonsi, Circle in the Sand, 208, 214.\n\nChapter 14: Fukuyama's Escalator\n\na banana smeared with peanut butter: John F. Harris, The Survivor: Bill Clinton in the White House (New York: Random House, 2005), 16.\n\nwalked in at the stroke of 8:31: DeYoung, Soldier, 186.\n\nanywhere from a few minutes to a few hours late: Harris, The Survivor, 54.\n\n\"the first black President\": Toni Morrison, \"The Talk of the Town,\" New Yorker, October 5, 1998, 31\u201332.\n\nexpecting his staff to work\u2014until 2 A.M.: Harris, The Survivor, 53.\n\nhe asked the army not to inform his wife: DeYoung, Soldier, 81.\n\nas a young boy to be fat: Harris, The Survivor, 35.\n\nif their shoes were shined: Gates, \"Powell and the Black Elite,\" 64.\n\nwhose intake increased with stress: Harris, The Survivor, xxiii, 56\u201357.\n\nas Aspin ate thirteen hors d'oeuvres: Bob Woodward, \"The Secretary of Analysis,\" Washington Post, February 21, 1993, W8; Halberstam, War in a Time of Peace, 245\u201346.\n\nmilitarists like Abrams, Wolfowitz, and Cheney: DeYoung, Soldier, 227.\n\nhad worked in the McGovern campaign: David Maraniss, First in His Class: A Biography of Bill Clinton (New York: Simon & Schuster, 1996), 264\u201386; Halberstam, War in a Time of Peace,20.\n\ndiscontented young diplomats in Vietnam: Halberstam, War in a Time of Peace, 180\u201381.\n\nVirtually none had joined SDS: Former SDS president Todd Gitlin, e-mail message to author, June 30, 2008.\n\n\"It was a draft beer so he dodged it\": Harris, The Survivor, 51.\n\n\"pot-smoking, draft-dodging [and] womanizing\": John Lancaster, \"Accused of Ridiculing Clinton, General Faces Air Force Probe,\" Washington Post, June 8, 1993, A1.\n\nhe backed down: Harris, The Survivor, 16\u201318; DeYoung, Soldier, 230\u201333.\n\nforeign policy received only 141 words: Halberstam, War in a Time of Peace, 193.\n\n\" focus like a laser beam on the economy\": Steven Mufson, \"Clinton to Send Message with Economic Choices; Appointments Will Show Administration's Direction,\" Washington Post, November 8, 1992, A33; William Clinton, interview with Ted Koppel, Nightline, ABC News, November 4, 1992.\n\nnot to take too much of his time: William G. Hyland, Clinton's World: Remaking American Foreign Policy (New York: Praeger, 1999), 18; Halberstam, War in a Time of Peace, 242.\n\nClinton's CIA director trying to get a meeting: Halberstam, War in a Time of Peace, 244; \"White House Has Not Been Impenetrable; Security Breached on Many Occasions,\" Washington Post, May 21, 1995, A14.\n\n\"bring positive results any time soon\": Chollet and Goldgeier, America Between the Wars, 64. For more on the controversy regarding Tarnoff's statements, see Jim Anderson, \"The Tarnoff Affair,\" American Journalism Review, March 1994, http:\/\/ajr.org\/Article.asp?id=1255.\n\n\"The Age of Europe has dawned\": Halberstam, War in a Time of Peace, 85.\n\n\"We don't have a dog in this fight\": Laura Silber and Allan Little, Yugoslavia: Death of a Nation (New York: Penguin USA, 1996), 198\u2013201; David C. Gompert, \"The United States and Yugoslavia's Wars,\" in The World and Yugoslavia's Wars, ed. Richard H. Ullman (New York: Council on Foreign Relations, 1996), 127\u201328.\n\nmade a fairly clean getaway: Halberstam, War in a Time of Peace, 31.\n\nultranationalist, anti-Serb bigots: Silber and Little, Yugoslavia, 92\u201398.\n\na small taste of things to come: Halberstam, War in a Time of Peace, 96\u201397.\n\nit was the most vulnerable of all: Halberstam, War in a Time of Peace, 121.\n\nshelling it night after night: Silber and Little, Yugoslavia, 222\u201328.\n\nburying bodies in public gardens, even backyards: Ian Traynor, \"The Slow but Sure Sacking of Sarajevo,\" Guardian, May 11, 1992, 22; Christine Bertelson, \"Escape from Sarajevo, and Death,\" St. Louis Post-Dispatch, January 19, 1993, A1; John Pomfret, \"The Hidden Agony of Mostar's Muslims,\" Washington Post, August 22, 1993, A1.\n\n\"In Auschwitz at least they had gas\": Halberstam, War in a Time of Peace, 123.\n\nto ever assert their independence again: Halberstam, War in a Time of Peace, 129\u201330; Samantha Power, A Problem from Hell: America and the Age of Genocide (New York: Harper Perennial, 2007), 251, 269.\n\nas part of a UN \"peacekeeping\" force: UN Department of Public Information, \"Background: United Nations Protection Force,\" September 1996, http:\/\/www.un.org\/Depts\/dpko\/dpko\/co_mission\/unprof_b.htm; Power, A Problem from Hell, 248\u201349; Silber and Little, Yugoslavia, 198\u201399.\n\npreventing the world from doing anything about it: Halberstam, War in a Time of Peace, 125\u201327; Thomas Weiss, \"Collective Spinelessness: U.N. Actions in the Former Yugoslavia,\" in Ullman, ed., The World and Yugoslavia's Wars, 59\u201396.\n\nstriking Serb positions from the air: Halberstam, War in a Time of Peace, 158, 196\u201397; Hyland, Clinton's World, 34.\n\n\"the new team in turn regarded him with awe\": Harris, The Survivor, 49.\n\nlittle direct bearing on its security: Mann, Rise of the Vulcans, 118.\n\nPowell replied two hundred thousand ground troops: Halberstam, War in a Time of Peace, 42; Michael R. Gordon, \"Powell Delivers a Resounding No on Using Limited Force in Bosnia,\" New York Times, September 28, 1992, A1; Colin L. Powell, \"Why Generals Get Nervous,\" New York Times, October 8, 1992, A35; Nancy Soderberg, The Superpower Myth: The Uses and Misuses of American Power (New York: Wiley, 2006), 23\u201327.\n\ntaking steps sure to enrage the Serbs: Halberstam, War in a Time of Peace, 227.\n\n\"enough respect not to interfere in ours\": Owen Harries, \"The Collapse of the West,\" Foreign Affairs 72 (September\/October 1993), http:\/\/www.foreignaffairs.org\/19930901faessay8562\/owen-harries\/the-collapse-of-the-west.html.\n\nhe no longer supported \"lift and strike\": Halberstam, War in a Time of Peace, 228.\n\nslaughtering each other for five hundred years: In fact, Kaplan's book was not mostly about Bosnia at all, but about Romania, Bulgaria, Greece, and other parts of the former Yugoslavia. And despite depicting Balkan hatreds as ancient, Kaplan actually supported U.S. military intervention in Bosnia. Robert D. Kaplan, Balkan Ghosts: A Journey Through History (New York: Vintage, 1996), ix\u2013xii; Halberstam, War in a Time of Peace, 228; Elizabeth Drew, On the Edge: The Clinton Presidency (New York: Touchstone, 1995), 157; Paul Starobin, \"The Liberal Hawk Soars,\" National Journal, May 15, 1999, 1310.\n\natrocities committed by Bosnians against Serbs: Halberstam, War in a Time of Peace, 230; Power, A Problem from Hell, 308.\n\n\"he came back with a European one\": Halberstam, War in a Time of Peace, 229.\n\nthe incredible shrinking United States: Harris, The Survivor, 62; Michael Duffy, \"The Incredible Shrinking President,\" Time, June 29, 1992, http:\/\/www.time.com\/time\/magazine\/article\/0,9171,975890,00.html.\n\n\"to demonstrate that we had a heart\": Western, Selling Intervention and War, 133, 135\u201337, 155, 163, 172; Halberstam, War in a Time of Peace, 248\u201352; Maryann K. Cusimano, \"Operation Restore Hope: The Bush Administration's Decision to Intervene in Somalia,\" Case Study No. 463, Institute for the Study of Diplomacy, 1995, 4\u20136; Ken Menkhaus and Louis Ortmayer, \"Key Decisions in the Somalia Intervention,\" Case Study No. 464, Institute for the Study of Diplomacy, 1995, 2\u20138.\n\ninoculations that he canceled his trip: Halberstam, War in a Time of Peace, 252, 254.\n\nwhich required sidelining Aidid: Halberstam, War in a Time of Peace, 254\u201357; Menkhaus and Ortmayer, \"Key Decisions in the Somalia Intervention,\" 13\u201323; Boutros Boutros-Ghali, Unvanquished: A U.S.-U.N. Saga (New York: Random House, 1999), 93, 110.\n\nputting a bounty on Aidid's head: Halberstam, War in a Time of Peace, 257\u201358.\n\ntwo meetings with the president: Halberstam, War in a Time of Peace, 244.\n\nat a hotel in downtown Mogadishu: Mark Bowden, Black Hawk Down: A Story of Modern War (New York: Atlantic Monthly Press, 1999), 3, 8.\n\nU.S. soldier's disfigured corpse through the streets: Halberstam, War in a Time of Peace, 261\u201362; Bowden, Black Hawk Down, 108\u201325.\n\n\"Vietmalia\": Richard Holbrooke, To End a War (New York: Modern Library, 1998), 217.\n\nCongress would cut off funding: Harris, The Survivor, 122; Michael Kranish, \"Clinton Builds Force, Sets Pullout; Says Somalia Withdrawal Can't Be Instant,\" Boston Globe, October 8, 1993, A1.\n\nhe was unfit to be commander in chief: Chollet and Goldgeier, America Between the Wars, 79.\n\nCritics called it Somalia Two: Halberstam, War in a Time of Peace, 270\u201371; Harris, The Survivor, 123.\n\nlay down on a park bench, and tried to die: Power, A Problem from Hell, 273\u201377, 388\u201389.\n\ntwice already on the same trip: Halberstam, War in a Time of Peace, 277; James Bennet, \"Clinton in Africa: The Overview; Clinton Declares U.S., with World, Failed Rwandans,\" New York Times, March 26, 1998, A1; John F. Harris, \"Clinton Tells Rwandans: World Too Slow to Act; Genocide Survivors Give Accounts of '94 Horror,\" Washington Post, March 26, 1998, A1.\n\nan intervention never carried out: Virginia Heffernan, \"Looking Back Across a Decade, with Bloody Regret,\" New York Times, April 1, 2004, E1.\n\nshelter, health care, and education: Richard A. Falk, \"Ideological Patterns in the United States Human Rights Debate: 1945\u20131978,\" in The Dynamics of Human Rights in U.S. Foreign Policy, ed. Natalie Kaufman Hevener (New Brunswick, NJ: Transaction, 1981), 43.\n\nsensitivity to that in the State Department: Muravchik, The Uncertain Crusade, 92.\n\n\"it's Spain\": Todd Gitlin, \"Bosnia Isn't Vietnam, It's Spain,\" Los Angeles Times, September 14, 1993, F7.\n\nCambodia's seven million people: Power, A Problem from Hell, 143.\n\nthe cause of human rights: A surprising and honorable exception was George McGovern. Power, A Problem from Hell, 132\u201336.\n\n\"with our military assistance\": Power, A Problem from Hell, 103.\n\nNATO should begin bombing immediately: Leslie Gelb, \"Balkan Strategy, Part II,\" New York Times, February 28, 1993, 15.\n\nincluding to Somalia, Bosnia, and Rwanda: In the four decades between the UN's creation and 1991, the organization authorized eighteen peacekeeping missions. Between 1991 and 1995, it authorized twenty-two. In 1992 alone, the number of UN peacekeepers worldwide more than quadrupled, from 11,000 to 52,000. United Nations, \"List of Operations, 1948\u20132009,\" http:\/\/www.un.org\/Depts\/dpko\/dpko\/list.shtml; Marrack Goulding, \"The Evolution of United Nations Peacekeeping,\" International Affairs 69, no. 3 (1993): 451.\n\nabout a standing UN army: Harris, The Survivor, 125.\n\nrape of Bihac went on: Halberstam, War in a Time of Peace, 284\u201385.\n\nbecause they were not Christian: Halberstam, War in a Time of Peace, 90\u201391, 304\u20135.\n\n\"distinct chapters in the history of decency\": Leon Wieseltier, \"Curses,\" New Republic, October 25, 1993, 46.\n\nwere leading the president astray: Drew, On the Edge, 158.\n\ncoverage of the Holocaust, was relentless: Starobin, \"The Liberal Hawk Soars,\" 1314.\n\nafter reading one Times column: George Stephanopoulos, All Too Human (Boston: Back Bay, 2000), 216.\n\neven the Quakers were for war: Power, A Problem from Hell, 428, 435.\n\nbroke into applause: Harris, The Survivor, 42\u201343.\n\nthe UN's Bosnia mission would collapse: Michael Dobbs, \"Embargo Vote Is Bipartisan Slap at Clinton; Dismay over Bosnia Reflects Setbacks,\" Washington Post, June 9, 1995, A22.\n\n\"'the good guys' and 'the bad guys'\": Power, A Problem from Hell, 394, 417; David Rohde, Endgame: The Betrayal and Fall of Srebrenica, Europe's Worst Massacre Since World War II (Boulder, CO: Westview, 1998), 326.\n\n\"the president is relevant\": Harris, The Survivor, 178.\n\nwould soon endorse Dole's presidential bid: Halberstam, War in a Time of Peace, 285\u201386, 299.\n\nleader of the free world was \"vacant\": Hyland, Clinton's World, 40.\n\nstill writing his inaugural address: Harris, The Survivor, xxiv, 11.\n\n\"the faster I come back up\": Harris, The Survivor, 334.\n\nbomb the hell out of them: Power, A Problem from Hell, 438; Halberstam, War in a Time of Peace, 340; Harris, The Survivor, 194, 196, 201.\n\nas part of the Clinton team: Halberstam, War in a Time of Peace, 319, 324\u201328; Harris, The Survivor, 197.\n\ntell you how easy it was: Halberstam, War in a Time of Peace, 391, 431\u201332.\n\nabout the people Milosevic killed: Halberstam, War in a Time of Peace, 395.\n\na new willingness to discuss peace: Halberstam, War in a Time of Peace, 339.\n\ntheir colleagues plunged to their death: Power, A Problem from Hell, 294; Halberstam, War in a Time of Peace, 395; Holbrooke, To End a War, 6\u201313, 231\u2013312.\n\nU.S. peacekeepers would help enforce: Holbrooke, To End a War, 308\u20139.\n\nnot a single American died: Chollet and Goldgeier, America Between the Wars, 131.\n\n\"The big dog barked today\": Halberstam, War in a Time of Peace, 340.\n\nthe first secretary-general ever denied a second term: United Nations, \"Former Secretaries-General,\" http:\/\/www.un.org\/sg\/formersgs.shtml.\n\nthis new institutional order: James M. Goldgeier, Not Whether but When: The U.S. Decision to Enlarge NATO (Washington, DC: Brookings Institution Press, 1999), 3, 17.\n\nfor NATO to stay off former Soviet soil: Jonathan Eyal, \"NATO's Enlargement: Anatomy of a Decision,\" International Affairs 73, no. 4 (1997): 709; Mark Kramer, \"The Myth of a No-NATO-Enlargement Pledge to Russia,\" Washington Quarterly 32, no. 2 (2009): 39\u201361; Goldgeier, Not Whether but When, 73, 110\u201317, 120\u201321, 135\u201338, 140\u201343; Strobe Talbott, interview with author, July 10, 2008.\n\nleft their bodies in the snow: R. Jeffery Smith, \"This Time, Walker Wasn't Speechless,\" Washington Post, January 22, 1999, A15.\n\nMoscow took Czechoslovakia in its grip: Halberstam, War in a Time of Peace, 378\u201379.\n\nwhen she was only one year old: Chollet and Goldgeier, America Between the Wars, 146.\n\nafter the conference closed, the bombing began: Ivo H. Daalder and Michael E. O'Hanlon, Winning Ugly: NATO's War to Save Kosovo (Washington, DC: Brookings Institution Press, 2000), 77\u201384; Power, A Problem from Hell, 447\u201348; Madeleine Albright, Madam Secretary (New York: Miramax, 2005), 505\u201318.\n\nfear of U.S. casualties limited its scope: Daalder and O'Hanlon, Winning Ugly, 106, 122.\n\nthan during the Gulf War: Stuart Croft, \"Guaranteeing Europe's Security? Enlarging NATO Again,\" International Affairs 78, no. 1 (2002): 98.\n\n\"war can be won by airpower alone\": Halberstam, War in a Time of Peace, 471\u201374, 478; Power, A Problem from Hell, 456; Daalder and O'Hanlon, Winning Ugly, 200\u20132.\n\ntwo NATO planes had been shot down: Power, A Problem from Hell, 459.\n\nNot a single American died in combat: Chollet and Goldgeier, America Between the Wars, 214.\n\ntook over Kosovo's political administration: North Atlantic Treaty Organization, \"Resolution 1244 (1999),\" adopted by UN Security Council, June 10, 1999, http:\/\/www.nato.int\/Kosovo\/docu\/u990610a.htm.\n\nthe verge of financial default: Chollet and Goldgeier, America Between the Wars, 166, 229.\n\ninternally displaced from their villages: Daalder and O'Hanlon, Winning Ugly, 40\u201341, 108\u20139.\n\nhalf the rate of the United States: World Bank Development Data, http:\/\/devdata.worldbank.org\/data-query.\n\nat a phenomenal 8 percent per year: Harris, The Survivor, 263; President William J. Clinton, \"President Clinton Addresses the Citizens of Colorado on the Global Economy,\" Littleton, Colorado, June 19, 1997.\n\nbegan to improve dramatically: Henry J. Kaiser Family Foundation, \"U.S. Teen Sexual Activity Factsheet,\" January 2005, http:\/\/www.kff.org\/youthhivstds\/upload\/U-S-Teen-Sexual-Activity-Fact-Sheet.pdf; Jeffrey Grogger, Stephen J. Haider, and Jacob Klerman, \"Why Did the Welfare Rolls Fall During the 1990s?\" Labor and Population Program Working Paper Series 03\u201307, Santa Monica, California, Rand Corporation, March 2003, 1; U.S. Department of Justice, \"Crime Trends,\" Bureau of Justice Statistics, http:\/\/bjsdata.ojp.usdoj.gov\/dataonline\/Search\/Crime\/Crime.cfm.\n\nthe psychological effect was comparable: Harris, The Survivor, 263.\n\n\"single superpower on the cheap is astonishing\": Paul Kennedy, \"The Eagle Has Landed,\" Financial Times, February 2, 2002.\n\n\"The Committee to Save the World\": Joshua Cooper Ramo, \"The Three Marketeers,\" Time, September 15, 1999, http:\/\/www.time.com\/time\/magazine\/article\/0,9171,990206,00.html.\n\n\"sharp limits to American resources and patience\": Francis Fukuyama, \"The Beginning of Foreign Policy,\" New Republic, August 17, 1992, 24\u201332.\n\nthat demanded political reform: Francis Fukuyama, \"The Global Optimists,\" New Republic, February 6, 1995, 40\u201342.\n\n\"so few external threats\": Chollet and Goldgeier, America Between the Wars, 278.\n\npredicted Christ's return to earth: Pew Research Center for the People & the Press, \"Optimism Reigns, Technology Plays a Key Role,\" October 24, 1999, http:\/\/people-press.org\/report\/51\/optimism-reigns-technology-plays-key-role.\n\nthe last time anything seemed possible: Harris, The Survivor, 393.\n\nprogress and the goodness of man: David Brooks, \"The Organization Kid,\" Atlantic, April 2001, http:\/\/www.theatlantic.com\/doc\/print\/200104\/brooks.\n\nChapter 15: Fathers and Sons\n\nhis fourth autobiographical essay: Kristol, \"An Autobiographical Memoir.\" The first three were \"Memoirs of a Trotskyist\" \"Memoirs of a Cold Warrior,\" New York Times Magazine, February 11, 1968; and \"My Cold War,\" National Interest 31 (1993): 141.\n\npublish his fourth autobiographical book: Podhoretz's four autobiographies are Making It (New York: Random House, 1967), Breaking Ranks (New York: Harper & Row, 1979), Ex-Friends (New York: Free Press 1999), and My Love Affair with America (New York: Free Press, 2001).\n\n\"as well as liberals and radicals\": Kristol, \"An Autobiographical Memoir,\" 40.\n\nStalinist Eastern Europe and Maoist China: Hoeveler, Watch on the Right, 82\u201383, 153.\n\n\"Sorry! It's the real world\": Jacquelyn Hardy, interview with the author, July 10, 2008.\n\n\" frankly, proudly, a revolutionary power\": Jeane Kirkpatrick, \"The Search for a Stable World Order,\" in Legitimacy and Force: Political and Moral Dimensions, ed. Jeane Kirkpatrick, vol. 1 (New Brunswick, NJ: Transaction, 1988), 367.\n\n\" for limited, defensive purposes\": Jeane Kirkpatrick, \"Moral Equivalence,\" in Legitimacy and Force, ed. Kirkpatrick, 70.\n\n\"a normal country in a normal time\": Jeane Kirkpatrick, \"A Normal Country in a Normal Time,\" National Interest 21 (1990): 40\u201345.\n\n\"continue to live beyond our means\": Kirkpatrick, \"A Normal Country in a Normal Time,\" 40\u201345; Jeane Kirkpatrick, \"Counseling the President,\" National Review, February 10, 1989, 28.\n\npreparing for a multipolar world: Kirkpatrick, The Withering Away of the Totalitarian State, 165; Kirkpatrick, \"A Normal Country in a Normal Time,\" 40\u201343; Jeane Kirkpatrick, \"Marriage of Convenience for the New Europe,\" Financial Post, July 10, 1990, 9; Jeane Kirkpatrick, \"An 'Alice in Wonderland' Defense Budget,\" Washington Post, April 5, 1993, A21; Irving Kristol, \"Defining Our National Interest,\" National Interest 21 (1990): 23.\n\n\"people violently reject any such scenario\": Kristol, \"Defining Our National Interest,\" 23.\n\nLithuania's fledgling democracy: Does \"the West\" Still Exist? A Conference of the Committee for the Free World (New York: Orwell, 1990), 39.\n\n\"which Americans need to unlearn\": Kirkpatrick, \"A Normal Country in a Normal Time,\" 43; Ralph Z. Hallow, \"Neoconservatives Meet in Search of Common Ground,\" Washington Times, April 24, 1990, A3.\n\n\"Something will screw it up\": Does \"the West\" Still Exist, 109, 117; Kristol, \"In Search of Our National Interest,\" Wall Street Journal, June 7, 1990, A14.\n\n\"serious economic decline\": Jeane Kirkpatrick, \"Ultimate Responsibility Falls on Arab Shoulders,\" Financial Post, August 16, 1990, 11.\n\ngiving sanctions time to work: Jeane Kirkpatrick, \"Gulf Crisis Proves That the World Needs a Policeman,\" Financial Post, August 28, 1990, 9.\n\nmerely bomb from the air: Jeane Kirkpatrick, \"Will We Liberate Kuwait...,\" Washington Post, November 12, 1990, A19.\n\nhe supported an American war: Irving Kristol, \"The Gulf: Born-Again Isolationists,\" Washington Post, August 22, 1990, A21.\n\n\"people who are really different from us\": Irving Kristol, \"After the War, What?\" Wall Street Journal, February 22, 1991, A10; Irving Kristol, \"Tongue-Tied in Washington,\" Wall Street Journal Europe, April 15, 1991, A14.\n\n\"vital American interests and lives at stake\": Jeane Kirkpatrick, \"Haiti's Looking Like Another Clinton Mistake,\" Post and Courier, March 14, 1995, A11.\n\nthen ignored the subject: Kristol, \"Defining our National Interest,\" 21; Ehrman, The Rise of Neoconservatism, 184; Halper and Clarke, America Alone, 99.\n\nputting U.S. peacekeepers on the ground: Jeane Kirkpatrick, \"U.S. Must Retaliate Strongly Against Serbs,\" Post and Courier, June 11, 1995, A21; Jeane Kirkpatrick, \"U.S. Needs Clear Goals in Bosnia,\" Post and Courier, November 19, 1995, A23.\n\nand the columnist George Will: Buckley defined his foreign policy vision as \"measured internationalism\" or \"prudent isolationism.\" William F. Buckley, \"American Power\u2014For What? A Symposium,\" Commentary 109, no. 1 (2000): 23\u201324. In 1991, Will wrote that \"concerning the question of U.S. military intervention in Iraq's civil war, President Bush has the traditional conservative's wariness about uncertain undertakings, a prudent skepticism about the promiscuous minting of abstract rights and duties, and an inclination to anchor U.S. policy in the rock of U.S. national interests. Today's imperial conservatives consider such thinking crabbed, mean-spirited and (adopting the language of liberal sensitivity-mongers) 'insensitive.' They want America to do for the world what Lyndon Johnson's Great Society was supposed to for America: fix it.\" George Will, \"Conservative Factions Debate What's Best for Kurds,\" Seattle Post-Intelligencer, April 18, 1991, A13. Will's own writing left little doubt that he was on Bush's side.\n\n\"expansive Wilsonian interventionism\": Norman Podhoretz, \"Neoconservatism: A Eulogy,\" Commentary 101, no. 3 (1996): 24.\n\n\"general reluctance to crusade\": Ehrman, The Rise of Neoconservatism, 184, 206.\n\nthe younger conservative generation fired its first shot: Charles Krauthammer, \"The Unipolar Moment,\" Foreign Affairs 70, no. 1 (1990): 29.\n\nyoung enough to be Kirkpatrick's son: Dorrien, Imperial Designs, 81; Winik, On the Brink, 457.\n\n\"if necessary, disarm\" the weapons states: Charles Krauthammer, \"The Lonely Superpower,\" New Republic, July 29, 1991, 27; Krauthammer, \"The Unipolar Moment,\" 29\u201333.\n\n\"is chaos\": Krauthammer, \"The Unipolar Moment,\" 32.\n\n\"Otherwise they would rather stay home\": Krauthammer, \"The Lonely Superpower,\" 27.\n\nrunning the Gulf War through the UN: Mann, Rise of the Vulcans, 201\u20132; Woodward, The Commanders, 106; Dorrien, Imperial Designs, 28; Alfonsi, Circle in the Sand, 101.\n\nshooting down Saddam's helicopters: Dorrien, Imperial Designs, 32; Gordon and Trainor, The Generals' War, 455.\n\n\"our role in the world\": Mann, Rise of the Vulcans, 208\u201313; Dorrien, Imperial Designs, 38\u201343; Patrick E. Tyler, \"US Strategy Calls for Insuring No Rivals Develop,\" New York Times, March 8, 1992, A1; Barton Gellman, \"Keeping US First: Pentagon Would Preclude a Rival Superpower,\" Washington Post, March 11, 1992, A1; Patrick E. Tyler, \"Senior US Officials Assail Lone-Superpower Policy,\" New York Times, March 11, 1992, A6; Barton Gellman, \"Aim of Defense Plan Supported by Bush: But President Says He Has Not Read Memo,\" Washington Post, March 12, 1992, A18.\n\nthe Defense Planning Guidance: Charles Krauthammer, \"What's Wrong with the 'Pentagon Paper'?\" Washington Post, March 13, 1992, A25.\n\nRobert Kagan and William Kristol: Dorrien, Imperial Designs, 139\u201340; Robert Kagan, \"American Power\u2014A Guide for the Perplexed,\" Commentary 101, no. 4 (1996): 30.\n\nafter America's triumph in Bosnia: Kagan did write two essays in Commentary in 1994 and 1995 (\"The Case for Global Activism,\" September 1994, 40\u201344; and \"A Retreat from Power,\" 19\u201325) that previewed his globalist worldview, but neither got nearly as much attention as his Foreign Affairs essay with William Kristol in the summer of 1996, \"Toward a Neo-Reaganite Foreign Policy,\" Foreign Affairs 75, no. 4 (1996), http:\/\/www.carnegieendowment.org\/publications\/index.cfm?fa=view&id=276.\n\n\"blood and treasure on teacup wars\": Charles Krauthammer, \"American Power\u2014For What? A Symposium,\" Commentary, January 2000, 34.\n\nstrategically important battles to come: Robert Kagan, \"America, Bosnia, Europe: A Compelling Interest,\" Weekly Standard, November 6, 1995, 27.\n\n\"understood it was all a game\": John Podhoretz, Hell of a Ride: Backstage at the White House Follies, 1989\u20131993 (New York: Simon & Schuster, 1993), 114.\n\non foreign policy, Republicans won: Unnamed Washington writer, interview with the author, August 7, 2009.\n\nher prot\u00e9g\u00e9, became his mentor: Kagan, A Twilight Struggle, x.\n\n\"the struggle with communism was endless\": Kagan, A Twilight Struggle, 55\u201377, 212.\n\ndecades had proved that it could: Robert Kagan, \"Democracy and Double Standards,\" Commentary 104, no. 2 (1997): 19\u201326.\n\nin speeding history up: Ken Jowitt, \"Rage, Hubris, and Regime Change,\" Policy Review 118 (2003): 33\u201342.\n\n\"others assumed were fixed\": Kagan and Kristol, \"Toward a Neo-Reaganite Foreign Policy.\"\n\nbring down the Soviet empire: Kagan, \"American Power\u2014A Guide for the Perplexed,\" 26.\n\nbefore Gorbachev even took power: Norman Podhoretz, \"The First Term: The Reagan Road to D\u00e9tente,\" Foreign Affairs 63, no. 3 (1984): 447\u201364.\n\nwithout fearing U.S. aggression: See Farnham, \"Reagan and the Gorbachev Revolution,\" 225\u201352; Leffler, For the Soul of Mankind, 436; Suri, \"Explaining the End of the Cold War,\" 60\u201392.\n\nPresent Danger, which had influenced Reagan: Norman Podhoretz, The Present Danger: Do We Have the Will to Reverse the Decline of American Power? (New York: Simon & Schuster, 1980); William Kristol and Robert Kagan, eds., Present Dangers: Crisis and Opportunity in American Foreign and Defense Policy (San Francisco: Encounter, 2000).\n\n\"another Reagan\": William Kristol, \"Reagan's Greatness,\" Weekly Standard, November 10, 1997, 31.\n\nthe Weekly Standard, in 1998: William Kristol, \"Clinton's Feckless Foreign Policy,\" Weekly Standard, May 25, 1991, 11.\n\n\"is reminiscent of the mid-1970's\": Kagan and Kristol, \"Toward a Neo-Reaganite Foreign Policy.\"\n\n\"does not add up to a brain tumor\": Owen Harries, \"American Power\u2014For What? A Symposium,\" Commentary 109, no. 1 (2000): 28\u201329.\n\n\" faces now is its own weakness\": Kagan and Kristol, \"Toward a Neo-Reaganite Foreign Policy.\"\n\n\"benevolent global hegemony\": Kagan and Kristol, \"Toward a Neo-Reaganite Foreign Policy.\"\n\nwhat they considered possible: Mann, Rise of the Vulcans, 215.\n\nPresent Dangers, as did Wolfowitz: Elliott Abrams wrote a chapter titled \"Israel and the 'Peace Process.\" Wolfowitz's was called \"Statesmanship in the New Century.\" Abrams called himself a \"neo-Reaganite\" in his contribution to \"American Power\u2014For What? A Symposium,\" Commentary 109, no. 1 (2000): 21.\n\nDonald Rumsfeld, and Dick Cheney: Dorrien, Imperial Designs, 130.\n\n\"complete victory\" in the Balkans: \"Letter to President Clinton,\" Project for the New American Century, September 11, 1998, http:\/\/www.newamericancentury.org\/kosovomilosevicsep98.htm; William Kristol and Robert Kagan, \"Victory,\" Weekly Standard, June 14, 1999.\n\nthis alleged malaise: David Brooks, \"Politics and Patriotism: From Teddy Roosevelt to John McCain,\" Weekly Standard, April 26, 1999, http:\/\/www.weeklystandard.com\/check.asp?idArticle=10411&r=onujh.\n\n\"the narrower concerns of private life\": David Brooks, \"A Return to National Greatness: A Manifesto for a Lost Creed,\" Weekly Standard, March 3, 1997, 16.\n\n\"holiday from history\": Schlesinger, \"The New Mood in Politics,\" in The Politics of Hope and the Bitter Heritage, 109; Charles Krauthammer, \"Holiday from History,\" Washington Post, February 14, 2003, A31.\n\n\"post-greatness America\": Schlesinger, \"The Decline of Greatness,\" in The Politics of Hope and the Bitter Heritage, 37; Brooks, \"A Return to National Greatness,\" 16.\n\ninspired JFK's fifty-mile hikes: Roosevelt used the phrase \"national greatness\" in an essay titled \"National Life and Character,\" which was published in the collection American Ideals (New York: G. P. Putnam's Sons, 1920), 268.\n\ngrand public mission at all: Brooks, \"A Return to National Greatness,\" 16.\n\nAmerica should journey to Mars: Charles Krauthammer, \"On to Mars: America Has Been Lost in Space; It's Time to Find Our Nerve Again,\" Weekly Standard, January 31, 2000, 23.\n\n\"except as public relations\": Jonah Goldberg, \"Grading Greatness,\" National Review Online, May 21, 2001, http:\/\/www.nationalreview.com\/goldberg\/goldbergprint052101.html.\n\n\"the already soft edges of boomer life\": Noemie Emery, \"Ask Not: John McCain Belongs to Our Oldest Political Party\u2014the Party of Teddy Roosevelt and FDR, JFK and Ronald Reagan\u2014the Patriot Party,\" Weekly Standard, February 21, 2000, http:\/\/www.weeklystandard.com\/check.asp?idArticle=10542&r=ivvnb.\n\n\"sophisticated consumer demands\": Francis Fukuyama, \"The End of History?\" National Interest 16 (1989): 18.\n\n\"Americans will die doing the right thing\": Jonah Goldberg, \"A Continent Bleeds: Taking Africa\u2014and Our Responsibilities\u2014Seriously,\" National Review Online (May 3, 2000), http:\/\/article.nationalreview.com\/?q=YmMzMDA0MWVhN2JjY2RhN2E4NmVkNTU1ZjJjMDU0MzQ.\n\nChapter 16: Small Ball\n\n\"in one or two sentences\": Norman Kempster, \"Vietnam War Leaves Legacy of Anguish; Still Overshadows Lives, U.S. Policies,\" Los Angeles Times, April 28, 1985, 1.\n\nbe drawn into war with Iran: Matt Bai, \"The McCain Doctrines,\" New York Times Magazine, May 18, 2008, http:\/\/www.nytimes.com\/2008\/05\/18\/magazine\/18mccain-t.html?ref=magazine.\n\n\"trading American blood for Iraqi blood\": Jacob Weisberg, \"Gulfballs,\" New Republic, March 25, 1991, 19.\n\n\"another failure like Vietnam or Lebanon\": Kagan, \"A Retreat from Power,\" 20.\n\nhe grew talons and became a hawk: John B. Judis, \"Neo-McCain,\" New Republic, October 16, 2006, http:\/\/www.tnr.com\/article\/politics\/neo-mccain.\n\n\"21st century...Reagan Doctrine\": Brooks, \"Politics and Patriotism\" David D. Kirkpatrick, \"Response to 9\/11 Offers Outline of McCain Doctrine,\" New York Times, August 17, 2008, http:\/\/www.nytimes.com\/2008\/08\/17\/us\/politics\/17mccain.html.\n\na conservative magazine, JFK: David Brooks and William Kristol, \"The McCain Insurrection; The Republican Establishment and the Conservative Movement Rallied to George W. Bush. The Voters Went for the Insurgent,\" Weekly Standard, February 14, 2000, http:\/\/www.weeklystandard.com\/check.asp?idArticle=10561&r=jyqrp; Noemie Emery, \"Ask Not\" David Brooks, \"The Anti-Boomer Candidate,\" Weekly Standard, February 21, 2000, http:\/\/www.weeklystandard.com\/check.asp?idArticle=10534&r=ghwet.\n\nsingles complex called Chateau Dijon: Jo Thomas, \"Governor Bush's Journey; After Yale, Bush Ambled Amiably into His Future,\" New York Times, July 22, 2000, A1; Weisberg, The Bush Tragedy, 46.\n\nheroism of his greatest-generation dad: David Greenberg, \"Fathers and Sons: George W. Bush and His Forebears,\" New Yorker, July 12, 2004, http:\/\/www.newyorker.com\/archive\/2004\/07\/12\/040712crbo_books.\n\nalmost flunked out: Nicholas Lemann, \"The Redemption,\" New Yorker, January 31, 2000; Weisberg, The Bush Tragedy, 38\u201340.\n\nwhether they still enjoyed sex: Todd S. Purdum, \"43+41=84,\" Vanity Fair, September 2006, http:\/\/www.vanityfair.com\/politics\/features\/2006\/09\/bushes200609.\n\nan intramural stickball league: Lou Cannon and Carl M. Cannon, Reagan's Disciple: George W. Bush's Troubled Quest for a Presidential Legacy (New York: PublicAffairs, 2008), 4, 5, 19; Peter Schweizer and Rochelle Schweizer, The Bushes: Portrait of a Dynasty (New York: Anchor, 2005), 153.\n\nhis younger siblings in the car: Weisberg, The Bush Tragedy, 46\u201347.\n\nhis fianc\u00e9e broke off the engagement: Kevin Phillips, American Dynasty: Aristocracy, Fortune and the Politics of Deceit in the House of Bush (New York: Viking, 2004), 44.\n\nbut it went belly-up: Weisberg, The Bush Tragedy, xvi, 32, 34, 49; Schweizer and Schweizer, The Bushes, 140.\n\nhis father confessed himself \"disappointed\": Weisberg, The Bush Tragedy, 47, 49.\n\nas the family's black sheep: Gail Sheehy, \"The Accidental Candidate,\" Vanity Fair, October 2000, http:\/\/www.vanityfair.com\/politics\/features\/2000\/10\/bush200010.\n\nshe wouldn't get into heaven: Weisberg, The Bush Tragedy, 50\u201352, 79\u201380, 83\u201384.\n\nin his own mind\u2014he excelled: Schweizer and Schweizer, The Bushes, 388.\n\n\"the chance to watch the ball\": Schweizer and Schweizer, The Bushes, 133, 389.\n\nhugged the bottom of their division: \"Harold Baines,\" http:\/\/www.baseball-reference.com\/b\/baineha01.shtml; \"Sammy Sosa,\" http:\/\/www.baseball-reference.com\/s\/sosasa01.shtml; \"Rangers Year-by-Year Results,\" http:\/\/texas.rangers.mlb.com\/tex\/history\/year_by_year_results.jsp.\n\nno rousing agenda for a second term: Schweizer and Schweizer, The Bushes, 458, 504.\n\nlearned from his father's defeat: Bob Woodward, Bush at War (New York: Simon & Schuster, 2003), 341.\n\nwas seeking Florida's top job: Schweizer and Schweizer, The Bushes, 413; Weisberg, The Bush Tragedy, 62.\n\nstate's outsize image of itself: Sam Howe Verhovek, \"Texas Governor Succeeds, Without the Flash,\" New York Times, June 14, 1995.\n\nGermany was a member of NATO: Mann, Rise of the Vulcans, 255; Weisberg, The Bush Tragedy, 146.\n\n\"Prosperity with a Purpose\": Lemann, \"The Redemption.\"\n\n\"good times for great goals\": \"GOP Platform 2: Prosperity with a Purpose,\" ABC News, July 31, 2000, http:\/\/abcnews.go.com\/Politics\/story?id=123292&page=1&page=1.\n\nwasted his presidency pushing paper: Weisberg, The Bush Tragedy, 65, 105.\n\nnot play \"small ball\": Michael Gerson, Heroic Conservatism: Why Conservatives Need to Embrace America's Ideals (and Why They Deserve to Fail If They Don't) (New York: HarperCollins, 2007), 42\u201343.\n\n\"game changer\": Ron Suskind, The Price of Loyalty: George W. Bush, the White House, and the Education of Paul O'Neill (New York: Simon & Schuster, 2003), 299.\n\nfirst foreign policy address of the campaign: George W. Bush, \"A Period of Consequence,\" address given at the Citadel, South Carolina, September 23, 1999, http:\/\/www.citadel.edu\/r3\/pao\/addresses\/pres_bush.html; Terry M. Neal, \"Bush Outlines Defense Plan in Address at Citadel,\" Washington Post, September 24, 1999, A3.\n\nand the moment\u2014at all: Frank Bruni, Ambling into History: The Unlikely Odyssey of George W. Bush (New York: HarperCollins, 2002); Gerson, Heroic Conservatism, 80.\n\n\"over-ruling Mr. Powell on any issue\": DeYoung, Soldier, 282, 287\u201388, 295, 297\u201398; Halberstam, War in a Time of Peace, 232, 238.\n\nregent to a dutiful boy king: DeYoung, Soldier, 286\u201388, 301\u20132; Mann, Rise of the Vulcans, 160.\n\nexcept in suit and tie: Ron Suskind, The One Percent Doctrine: Deep Inside America's Pursuit of Its Enemies Since 9\/11 (New York: Simon & Schuster, 2006), 151; Purdum, \"43+41=84\" Schweizer and Schweizer, The Bushes, 501.\n\nRepublicans back in charge: DeYoung, Soldier, 314.\n\nwhen he was on the road: Barton Gellman, The Angler: The Cheney Vice Presidency (New York: Penguin, 2008), 54; Bob Woodward, Plan of Attack (New York: Simon & Schuster, 2004), 176.\n\nin other words, to stack the deck: Gellman, The Angler, 244\u201345; Walter Pincus, \"Under Bush, the Briefing Gets Briefer,\" Washington Post, May 24, 2002, A33.\n\ncalled the vice president's \"mole\": Gellman, The Angler, 32\u201333, 36, 38, 189. In the first Bush administration, Libby was the Pentagon's principal deputy undersecretary for strategy and resources and later deputy undersecretary of defense for policy, positions that ranked higher in the interagency process than director of the National Security Council, which was Rice's job. Gellman, The Angler, 43\u201344.\n\na longtime Cheney associate: Hayes, Cheney, 301.\n\n\"most ruthless\" government official he had ever met: Roger Morris, \"The Undertaker's Tally: Sharp Elbows,\" Tom Dispatch, February 13, 2007, http:\/\/www.tomdispatch.com\/indexprint.mhtml?pid=165669; Bradley Graham, By His Own Rules: The Ambitions, Successes, and Ultimate Failures of Donald Rumsfeld (New York: PublicAffairs, 2009), 201.\n\nmentor to both Hadley and Libby: Gordon and Trainor, Cobra II, 169; Gellman, The Angler, 42.\n\nthirty-six-year-old daughter, Elizabeth: DeYoung, Soldier, 320; Gellman, The Angler, 37.\n\nwas working to overthrow him: DeYoung, Soldier, 314\u201317.\n\n\"everything in the region and beyond it\": Suskind, The Price of Loyalty, 85, 96\u201397; DeYoung, Soldier, 345; Gordon and Trainor, Cobra II, 16\u201317.\n\na new \"Reagan doctrine\": Paul Wolfowitz, \"Historical Memory: Setting History Straight,\" Current, June 2000, 22; Paul Wolfowitz, \"American Power\u2014For What? A Symposium,\" Commentary 109, no. 1 (2000): 47; Paul Wolfowitz, \"Statesmanship in the New Century,\" in Present Dangers, ed. Kagan and Kristol, 323.\n\nheld its first free presidential election: Solomon, Paul D. Wolfowitz, 39.\n\n\"history of democratic rule\": Paul Wolfowitz, \"Is the Atlantic Community Obsolete?\" in The Congress of Phoenix: Rethinking Atlantic Security and Economics, ed. Gerald Frost (Washington, DC: AEI Press, 1998), http:\/\/www.aei.org\/docLib\/20040217_book35.pdf.\n\ndissidents like Vaclav Havel: Jane Mayer, \"The Manipulator,\" New Yorker, June 7, 2004, http:\/\/www.newyorker.com\/archive\/2004\/06\/07\/040607fa_fact1; George Packer, The Assassin's Gate: America in Iraq (New York: Farrar, Straus & Giroux, 2006), 12.\n\na game changer nonetheless: Michael Dobbs, \"For Wolfowitz, a Vision May Be Realized,\" Washington Post, April 7, 2003, A17; Peter J. Boyer, \"The Believer: Paul Wolfowitz Defends His War,\" New Yorker, November 1, 2004, http:\/\/www.newyorker.com\/archive\/2004\/11\/01\/041101fa_fact.\n\nthrough the barrel of an American gun: While the Contras may have played some role in the decision by Nicaragua's Sandinista regime to hold the free elections that led to their losing power in 1990, the much larger factor was the Soviet Union's collapse, which left the Sandinistas\u2014like so many other third-world communist regimes\u2014orphaned.\n\n\"start at the beginning\": DeYoung, Soldier, 426.\n\nmaking incremental change: As Douglas Feith notes, \"Powell presented himself as the practical man of affairs, taking the world as he found it, focused on the here and now, intent on getting organized for the next day's set of meetings on whatever crisis was at hand.\" Douglas J. Feith, War and Decision: Inside the Pentagon at the Dawn of the War on Terrorism (New York: Harper, 2008), 60.\n\n\"slowly but surely, layer by layer\": DeYoung, Soldier, 487.\n\n\"broken, weak country\": DeYoung, Soldier, 305, 345; Woodward, Plan of Attack, 22.\n\nno threat to his neighbors: Thomas E. Ricks, Fiasco: The American Military Adventure in Iraq (New York: Penguin, 2006), 27.\n\nthe big game was Colin Powell: DeYoung, Soldier, 323\u201325.\n\nhe was not always in the room: DeYoung, Soldier, 329.\n\n\"little far forward on your skis\": Mann, Rise of the Vulcans, 278.\n\nto trumpet his achievement: Woodward, Bush at War, 13.\n\nCheney and Rumsfeld disapproved: Robert Kagan and William Kristol, \"A National Humiliation,\" Weekly Standard, April 16, 2001, http:\/\/www.weeklystandard.com\/check.asp?idArticle=11457&r=elzif; Robert Kagan and William Kristol, \"The 'Adults' Make a Mess,\" Weekly Standard, May 14, 2001, http:\/\/www.weeklystandard.com\/check.asp?idArticle=583&r=oglxj.\n\nkeeping the United States out of a Pacific war: Bob Woodward, State of Denial (New York: Simon & Schuster, 2007), 33.\n\nmight well prompt Beijing to strike: Solomon, Paul D. Wolfowitz, 48; Graham, By His Own Rules, 679; Andrew Cockburn, Rumsfeld: His Rise, Fall and Catastrophic Legacy (New York: Scribner, 2007), 140.\n\nreaffirm Bush's original remark: Kagan and Kristol, \"The 'Adults' Make a Mess\" \"Interview with Dick Cheney,\" Fox News Sunday, Fox News Network, April 29, 2001.\n\nlargest arms sale in a decade: Solomon, Paul D. Wolfowitz, 49.\n\nwhen Powell was around: Dorrien, Imperial Designs, 38; Todd S. Purdum, \"A Face Only a President Could Love,\" Vanity Fair, June 2006, http:\/\/www.vanityfair.com\/politics\/fea tures\/2006\/06\/cheney200606; Hayes, Cheney, 6.\n\n\"one might expect\": Woodward, Plan of Attack, 182\u201383.\n\nwere being made someplace else: DeYoung, Soldier, 335; Haass, War of Necessity, War of Choice, 213.\n\nsent to the vice president's staff: Gellman, The Angler, 242\u201343, 376\u201377; Cullen Murphy and Todd Purdum, \"Farewell to All That: An Oral History of the Bush White House,\" Vanity Fair, February 2009, http:\/\/www.vanityfair.com\/politics\/features\/2009\/02\/bush-oral-history200909; Haass, War of Necessity, War of Choice, 220.\n\nsuggested that he had lost a step: DeYoung, Soldier, 335\u201336.\n\nSeptember 10 cover story in Time: Johanna McGeary, \"Odd Man Out,\" Time, September 10, 2001, http:\/\/www.time.com\/time\/magazine\/article\/0,9171,1101010910\u2013173441,00.html.\n\n\"disconnected from larger purposes\": Gerson, Heroic Conservatism, 73. The author Robert Draper, who interviewed Bush at length, concurs with Gerson, noting that Bush \"was at root a man who craved purpose\u2014a sense of movement, of consequence. And things did not seem especially consequential in the summer of 2001.\" Robert Draper, Dead Certain: The Presidency of George W. Bush (New York: Free Press, 2008), 133.\n\nlong stretches at his Texas ranch: Fred Kaplan, Daydream Believers: How a Few Grand Ideas Wrecked American Power (Hoboken, NJ: Wiley, 2008), 114.\n\nChapter 17: The Opportunity\n\n\"I will seize the opportunity\": Woodward, Bush at War, 32, 62, 282.\n\ntraveled to the subcontinent at all: \"George W. Bush, Travelling Man?\" Washington Post, November 28, 2007, A10.\n\n\"I thought you said some band\": Mann, Rise of the Vulcans, 255.\n\ndivided between Sunni and Shia: Kaplan, Daydream Believers, 162; Weisberg, The Bush Tragedy, 207.\n\n\"we wouldn't fight back\": Woodward, Bush at War, 38\u201339.\n\n\"smash this myth\": Gerson, Heroic Conservatism, 82\u201383.\n\na way to rally his troops: In mid-November 2001, in a videotaped speech from Kandahar, Afghanistan, bin Laden used the \"weak horse\" line; http:\/\/www.npr.org\/news\/specials\/re sponse\/investigation\/011213.binladen.transcript.html.\n\n\"pounded sand\": Weisberg, The Bush Tragedy, 188; Feith, War and Decision, 12.\n\n\"reflexive pullback\": Woodward, Bush at War, 20; Cockburn, Rumsfeld, 151.\n\n\"hit a camel in the butt\": Cannon and Cannon, Reagan's Disciple, 184.\n\ncomparing the United States to ancient Rome: Mann, Rise of the Vulcans, 215.\n\nhis efforts at producing WMD: Ricks, Fiasco, 19, 21; Gordon and Trainor, Cobra II, 14.\n\nhad shown in World War II: Judy Keen, \"Same President, Different Man in Oval Office,\" USA Today, October 29, 2001, http:\/\/www.usatoday.com\/news\/sept11\/2001\/10\/29\/bushmood-usat.htm; Woodward, Bush at War, 205.\n\n\"our mission and our moment\": George W. Bush, \"Address to a Joint Session of Congress and the American People,\" September 20, 2001, http:\/\/georgewbush-whitehouse.archives.gov\/news\/releases\/2001\/09\/20010920\u20138.html.\n\n\"real calling with real heroism\": Matthew Scully, \"Present at the Creation,\" Atlantic, September 2007, http:\/\/www.theatlantic.com\/doc\/200709\/michael-gerson.\n\nthe regimes that harbored them: DeYoung, Soldier, 338, 346\u201347; Woodward, Bush at War, 31\u201332.\n\n\"let Mr. Wolfowitz speak for himself\": Woodward, Bush at War, 43, 60\u201361; Mann, Rise of the Vulcans, 302.\n\nthe 1995 bombing in Oklahoma City: Sam Tanenhaus, \"Bush's Brain Trust,\" Vanity Fair, July 2003, 114.\n\nbarely anything worthwhile to bomb: Woodward, Bush at War, 33, 83; Richard Clarke, Against All Enemies: Inside America's War on Terror (New York: Free Press, 2004), 31; Suskind, The Price of Loyalty, 187; Haass, War of Necessity, War of Choice, 196, 235.\n\nattacking Afghanistan didn't do that: Feith, War and Decision, 82.\n\nChiefs of Staff chairman Henry Shelton: Woodward, Bush at War, 61.\n\nbut he knew he couldn't prove it: Woodward, Bush at War,99.\n\ncronies was not big enough: Gerson, Heroic Conservatism, 122.\n\n\"and can move forward\": Woodward, Bush at War,43.\n\nno U.S. ground forces at all: Woodward, Bush at War, 79\u201380.\n\n\"guerrilla commander's fantasy\": \"Rendezvous with Afghanistan,\" New York Times, September 14, 2001, A26; \"War Without Illusions,\" New York Times, September 15, 2001, A22.\n\n\"create more instability\": Charles M. Sennott, \"Allies Caution Bush Against Military 'Trap,'\" Boston Globe, September 18, 2001, A21.\n\n\"get the hell kicked out of you\": Woodward, Bush at War, 103.\n\nbin Laden, not overthrow them: Mike Allen and Alan Sipress, \"Attacks Refocus the White House on How to Fight Terrorism,\" Washington Post, September 26, 2001, A03; Feith, War and Decision, 78\u201380.\n\ngetting mired in civil war: Woodward, Bush at War, 123, 124.\n\nwrote Charles Krauthammer in late September: Charles Krauthammer, \"The War: A Road Map,\" Washington Post, September 28, 2001, A39.\n\n\"and demand that Powell follow?\": William Kristol, \"Bush vs. Powell,\" Washington Post, September 25, 2001, A23.\n\n\"this is a change from the past\": Woodward, Bush at War, 98.\n\ndefeat the Taliban on the ground: Woodward, Bush at War, 314; Suskind, The One Percent Doctrine, 79.\n\nfighters onto Afghan soil: Molly Moore and Kamran Khan, \"Slain Rebel's Brief but Disastrous Foray,\" Washington Post, October 28, 2001, A20.\n\nnews analysis in the New York Times: R. W. Apple, Jr., \"A Military Quagmire Remembered: Afghanistan as Vietnam,\" New York Times, October 31, 2001.\n\n\"who have been there for 5,000 years\": Woodward, Bush at War, 275, 291.\n\nsuggesting that they wanted him to fail: Woodward, Bush at War, 262.\n\ndemanded that Bush send in the army: Fred Barnes, \"Bush Only Needs to Do One Thing; Win the War,\" Weekly Standard, November 5, 2001, 12.\n\nCheney said he had absolute faith: Hayes, Cheney, 360.\n\ntried for years to hit, but never could: Woodward, Bush at War, 312.\n\nthe Taliban fighters were dead: Kaplan, Daydream Believers, 35\u201336.\n\nits share of the country from 15 percent to 50 percent: Woodward, Bush at War, 312.\n\nto attend school: Ilana Ozernoy, \"Liberation Day,\" U.S. News & World Report, November 26, 2001, 30; Nancy Gibbs, \"Blood and Joy,\" Time, November 26, 2001, 28; Anthony Davis, \"Eyewitness to a Sudden and Bloody Liberation,\" Time, November 26, 2001, 58.\n\nBaltimore, and Boston, a democrat: Christiane Amanpour and Andrea Koppel, \"Hamid Karzai No Stranger to Leadership,\" October 10, 2002, http:\/\/archives.cnn.com\/2001\/WORLD\/asiapcf\/central\/12\/21\/ret.karzai.profile\/.\n\nfemale member of the new Afghan cabinet: George W. Bush, \"President Bush's State of the Union Address to Congress and the Nation,\" New York Times, January 30, 2002, http:\/\/www.nytimes.com\/2002\/01\/30\/us\/state-union-president-bush-s-state-union-address-con gress-nation.html.\n\nSpecial Forces troops and 110 CIA agents: Woodward, Bush at War, 314.\n\nless than a single B-2 bomber: Steven M. Kosiak, \"Estimated Cost of Operation Enduring Freedom: The First Two Months,\" Center for Strategic and Budgetary Assessments, December 7, 2001, available at http:\/\/www.csbaonline.org; Federation of American Scientists, \"B-2 Spirit,\" Nuclear Information Project, November 30, 1999, http:\/\/www.fas.org\/nuke\/guide\/usa\/bomber\/b\u20132.htm.\n\n\"virtually no cost in casualties\": Stanley Kurtz, \"Push-Button Warriors,\" National Review Online, November 29, 2001, http:\/\/www.nationalreview.com\/contributors\/kurtz112901.shtml.\n\nin the former U.S.S.R.: Andrew Bacevich, The Limits of Power: The End of American Exceptionalism (New York: Henry Holt, 2008), 47.\n\n\"wrong about virtually everything\": Joe Loconte, \"Rumsfeld's Just War; Generals Meet Theologians at the Pentagon,\" Weekly Standard, December 24, 2001, 13.\n\n\"all together now\u2014QUAGMIRE!\": Woodward, Plan of Attack, 37.\n\nafter which Bush dubbed him \"Rumstud\": Mann, Rise of the Vulcans,307.\n\n87 percent of Americans: Gallup Organization, \"Presidential Job Approval in Depth,\" http:\/\/www.gallup.com\/poll\/116500\/Presidential-Approval-Ratings-George-Bush.aspx.\n\ncongressional and journalistic recrimination: Mann, Rise of the Vulcans, 72.\n\nshould not enjoy the protections of the Geneva Convention: DeYoung, Soldier, 365\u201372; Gellman, The Angler, 170.\n\n\"activities of the executive branch\": Gellman, The Angler, 106.\n\nacross the border into Pakistan: Mann, Rise of the Vulcans,308.\n\n\"perhaps beyond precedent\": Gerson, Heroic Conservatism, 122.\n\n\"They're on a roll\": Seymour M. Hersh, \"The Iraq Hawks: Can Their War Plan Work?\" New Yorker, December 24, 2001, 58.\n\nChapter 18: The Romantic Bully\n\nmotive and the means? Saddam!: For an example of this reasoning, see Feith, War and Decision, 504.\n\n\"likely make a different calculation\": Robert Kagan, \"Power and Weakness,\" Policy Review 113 (June\u2013July 2002), http:\/\/www.hoover.org\/publications\/policyreview\/3460246.html.\n\non October 9, he was rebuffed: Woodward, Bush at War, 216.\n\nstrategy for toppling Saddam: Woodward, Plan of Attack, 1\u20134.\n\nAfghanistan to the Persian Gulf: Packer, The Assassin's Gate, 45; Barton Gellman and Dafna Linzer, \"Afghanistan, Iraq: Two Wars Collide,\" Washington Post, October 22, 2004, A1.\n\nturned to deficit: U.S. Office of Management and Budget, FY2008 Budget Request, http:\/\/www.whitehouse.gov\/omb\/budget\/fy2008\/pdf\/hist.pdf.\n\n\"proved deficits don't matter\": Suskind, The Price of Loyalty, 291.\n\nand then again in 2003: Glenn Kessler and Juliet Eilperin, \"In House, a Magic Balancing Act,\" Washington Post, March 14, 2002, A7; Jonathan Weisman, \"Late Deals Got Tax Cut Done,\" Washington Post, May 30, 2003, A5.\n\nhe was reprimanded; then fired: Packer, The Assassin's Gate, 116.\n\nless than 1 percent of that: Kosiak, \"Estimated Cost of Operation Enduring Freedom\" Murphy and Purdum, \"Farewell to All That.\"\n\nnine months to fully deploy: Woodward, Bush at War,43; Packer, The Assassin's Gate, 118; Kaplan, Daydream Believers, 39\u201340.\n\nconformist, and flat-out dumb: Cockburn, Rumsfeld, 153.\n\nThe Nutty Professor: Kaplan, Daydream Believers, 40\u201341; Gordon and Trainor, Cobra II, 27\u201328.\n\n\"learned coming out of Afghanistan\": Gordon and Trainor, Cobra II, 32; Woodward, Plan of Attack, 41.\n\ntroop number down by almost a third: Gordon and Trainor, Cobra II, 54\u201355.\n\n\"has undertaken in the post\u2013cold war era\": Lawrence F. Kaplan and William Kristol, The War over Iraq: Saddam's Tyranny and America's Mission (San Francisco: Encounter, 2003), 117.\n\nrelationship between a batterer and his spouse: William Kristol, \"From Truth to Deception,\" Washington Post, October 12, 2002, A31.\n\nsimply buried his head in his arms: Woodward, State of Denial, 19, 72; Woodward, Bush at War, 23, 251.\n\n\"you'd find the vice president there\": Gellman, The Angler, 161\u201362. In his memoir, Douglas Feith calls Rumsfeld \"thoroughly antisentimental.\" Feith, War and Decision, 75.\n\n\"truly enjoys getting people to knuckle under\": Suskind, The One Percent Doctrine, 215. Robert Draper recounts a similar story of bullying, this one from Bush's time as Texas governor. In a meeting of economic advisers, Karl Rove had been speaking at great length, annoying Bush with his self-importance. \"Karl,\" Bush suddenly yelled. \"Hang up my jacket.\" The room fell silent, and Rove did as he was told. Draper, Dead Certain, 102.\n\nthey did not openly weep: Gerson, Heroic Conservatism, 204.\n\nBut Bush did both, often: Robert Draper, \"The Prez & I,\" GQ Online, http:\/\/men.style.com\/gq\/features\/full?id=content_7778.\n\n\"complete trash, a horrible evil person\": Draper, Dead Certain, 51.\n\noften broke down: Cannon and Cannon, Reagan's Disciple, 195; Woodward, State of Denial, 270.\n\n\"region of peaceful democracies\": Scott McLellan, What Happened: Inside the Bush White House and Washington's Culture of Deception (New York: PublicAffairs, 2008), xii\u2013xiii.\n\n\"It's what's driving him\": Woodward, Plan of Attack, 412.\n\n\"he wants to talk about\": Fred Barnes, \"Bush Zeroes In,\" Weekly Standard, February 10, 2003.\n\nwas the story of his life: Gerson makes this point explicitly, writing that Bush was \"convinced that societies are capable of hopeful change because individuals are capable of hopeful change, based on his own experience.\" Gerson, Heroic Conservatism, 51.\n\nuntroubled by original sin: Gerson, Heroic Conservatism, 51, 99.\n\n\"hatred and the tactics of terror\": George W. Bush, \"President Discusses the Future of Iraq,\" February 26, 2003, http:\/\/georgewbush-whitehouse.archives.gov\/news\/releases\/2003\/02\/print\/20030226\u201311.html.\n\nBush told cadets at West Point: George W. Bush, \"President Bush Delivers Graduation Speech at West Point,\" June 1, 2002, http:\/\/georgewbush-whitehouse.archives.gov\/news\/re leases\/2002\/06\/print\/20020601\u20133.html.\n\n\"That great struggle is over\": White House, \"The National Security Strategy of the United States,\" September 2002, 1, http:\/\/georgewbush-whitehouse.archives.gov\/nsc\/nss\/2002\/nss1.html.\n\n\"it's a blueprint, a model\": Draper, Dead Certain, 188.\n\nby which he meant the Iraqi National Congress: House Budget Committee, Hearing on FY2004 Defense Budget Request, 108th Cong., 1st sess., 2003.\n\nsoon after Saddam fell: Gordon and Trainor, Cobra II, 121\u201322.\n\n\"became an occupying power\": Feith, War and Decision, 403.\n\nBush's post-9\/11 thinking than any other adviser: Fred Barnes, Rebel-in-Chief: Inside the Bold and Controversial Presidency of George W. Bush (New York: Crown Forum, 2006), 48, 61.\n\n\"remove the shackles on democracy\": Mark Bowden, \"Wolfowitz: The Exit Interviews,\" Atlantic, July\/August 2005, http:\/\/www.theatlantic.com\/doc\/200507\/bowden.\n\nYou break it, you own it: Cannon and Cannon, Reagan's Disciple, 196; DeYoung, Soldier, 401\u20132.\n\nand driving away business: Helen Huntley, \"Rule That Isn't Upsets Pottery Barn,\" St. Petersburg Times, April 20, 2004, http:\/\/www.sptimes.com\/2004\/04\/20\/Business\/Rule_that_isn_t_its_r.shtml.\n\nThat's where the power lay: DeYoung, Soldier, 392.\n\nhe was sure that Cheney wasn't: DeYoung, Soldier, 417.\n\nleave America and Iraq worse off: In January 2002, Kristol and Kagan wrote that \"it is almost impossible to imagine any outcome for the world both plausible and worse than the disease of Saddam with weapons of mass destruction.\" Robert Kagan and William Kristol, \"What to Do About Iraq,\" Weekly Standard, January 21, 2002, 23. \"That things might be worse without him [Saddam] is of course a possibility,\" wrote Kristol and Lawrence Kaplan in their 2003 book advocating war. \"But given the status quo in Iraq, it is difficult to imagine how.\" Kristol and Kaplan, The War over Iraq, 96. \"It's hard to imagine how the alternative [to Saddam] could possibly be worse,\" wrote Max Boot in the Standard. Max Boot, \"The False Allure of 'Stability,'\" Weekly Standard, December 9, 2002. And in slapping down Shinseki in February 2003, Wolfowitz declared, \"It is hard to conceive that it would take more forces to provide stability in post-Saddam Iraq than it would take to conduct the war itself and to secure the surrender of Saddam's security forces and his army\u2014hard to imagine.\" Ricks, Fiasco, 97\u201398; House Budget Committee, Hearing on FY2004 Defense Budget Request, 108th Cong., 1st sess., 2003.\n\neverything that might go wrong: Woodward, Bush at War, 334.\n\n\"There will be civil disorder\": DeYoung, Soldier, 401\u20132.\n\nimpact on America's Arab allies: Woodward, Plan of Attack, 149\u201351; Gordon and Trainor, Cobra II, 81.\n\nit just wasn't in his nature: DeYoung, Soldier, 509.\n\nvoted yes: Cannon and Cannon, Reagan's Disciple, 93.\n\nhe mentioned Vietnam only once: Authorization of the Use of United States Armed Forces Against Iraq, 107th Cong., 2nd sess., Congressional Record 148 (October 9, 2002), S10170\u2013S10175.\n\n56 percent of older Senate Democrats: \"Final Vote Results for Roll Call 455,\" H.J. Res. 114, October 10, 2002, http:\/\/clerk.house.gov\/evs\/2002\/roll455.xml; \"U.S. Senate Roll Call Votes 107th Congress\u20142nd Session,\" H.J. Res. 114, October 11, 2002; Biographical Directory of the United States Congress, http:\/\/bioguide.congress.gov\/biosearch\/biosearch.asp.\n\nthe very things liberals loved: For a good example of this argument, see David Talbot, \"The Making of a Hawk,\" Salon.com, January 3, 2002, http:\/\/www.salon.com\/books\/fea ture\/2002\/01\/03\/hawk\/print.html.\n\nit was the great liberal duty of the age: Peter Beinart, \"Ism Schism,\" New Republic, September 25, 2006, 6.\n\nwhen Fukuyama declared that democracy had won: Paul Berman, Terror and Liberalism (New York: Norton, 2004), 176.\n\n\"this notion did pretty much explode\": Berman, Terror and Liberalism, 190.\n\nthe Taliban millions might die: Noam Chomsky, interview with Svetlana Vukovic and Svetlana Lukic, Radio B92, Belgrade, September 19, 2001, http:\/\/www.b92.net\/intervju\/eng\/2001\/0919-chomsky.phtml.\n\nanother influential liberal hawk: Michael Ignatieff, \"The American Empire; The Burden,\" New York Times Magazine, January 5, 2003, http:\/\/www.nytimes.com\/2003\/01\/05\/magazine\/the-american-empire-the-burden.html.\n\n\"Kosovo and not do it in Iraq\": Former Clinton administration official, phone interview with the author, June 27, 2008.\n\nwould bring not freedom, but chaos: Kirkpatrick, Making War to Keep Peace, 279.\n\nhad lost the confidence of her convictions: Jacquelyn Hardy, phone interview with the author, July 10, 2008.\n\nunder international law: Kirkpatrick, Making War to Keep Peace, 281.\n\nturn George W. Bush into Napoleon: Washington, D.C., writer, interview with the author, June 9, 2008.\n\ncritical of what the Standard wrote: Washington, D.C., writer (different from above), interview with the author, July 15, 2008.\n\ndenounced the impending war: Brent Scowcroft, \"Don't Attack Saddam,\" Wall Street Journal, August 15, 2002, http:\/\/www.opinionjournal.com\/editorial\/feature.html?id=110002133; Suskind, The One Percent Doctrine, 167.\n\nBush himself stayed silent: Haass, War of Necessity, War of Choice, 217.\n\n\"He doesn't think he should unless he's asked\": Woodward, State of Denial, 114.\n\ndidn't have the latest intelligence: \"Text of Bush Interview,\" Fox News, September 22, 2003, http:\/\/www.foxnews.com\/story\/0,2933,98006,00.html.\n\nAmerica pursue regime change in Iraq: \"Letter to President Clinton on Iraq,\" Project for a New American Century, January 26, 1998, http:\/\/www.newamericancentury.org\/iraqclintonlet ter.htm.\n\nwould not fertilize the soil: Francis Fukuyama, \"American Conservatism: Beyond Our Shores,\" Wall Street Journal, December 24, 2002, A10.\n\nbut history could not be rushed: Robert S. Boynton, \"The Neocon Who Isn't,\" American Prospect Online, October 5, 2005, http:\/\/www.prospect.org\/cs\/articles?articleId=10304.\n\nFukuyama's presentation again: Francis Fukuyama, interview with the author, July 15, 2008.\n\n\"I've been put under over here\": Draper, Dead Certain, 186.\n\n\"his permission\": Woodward, Plan of Attack, 269\u201374.\n\n\"the credibility to do it\": Woodward, Plan of Attack, 291.\n\n\"you can afford to lose a few points\": DeYoung, Soldier, 441, 448.\n\nIraqi intelligence officials in Prague: Woodward, Plan of Attack, 289\u201391; DeYoung, Soldier, 436\u201341.\n\n\"leery of our own presentation\": DeYoung, Soldier, 444\u201345; Murphy and Purdum, \"Farewell to All That.\"\n\n\"conclusions based on solid intelligence\": DeYoung, Soldier, 449.\n\nUN weapons inspectors began refuting it almost instantly: Ian Fisher, \"Reporters on Ground Get Iraqi Rebuttal to Satellite Photos,\" New York Times, February 8, 2003, A8; Ian Fisher, \"U.N.'s Chief Inspectors Will Press Iraq for Hard Data That Could Avert War,\" New York Times, February 9, 2003, A14; Peter Slevin and Colum Lynch, \"U.S. Meets New Resistance at U.N.,\" Washington Post, A1; Glenn Kessler and Colum Lynch, \"Blix's Report Deepens UN Rift over Iraq,\" Washington Post, March 9, 2003, A1.\n\nI don't want one, Wilkerson explained: DeYoung, Soldier, 450\u201352.\n\nThe invasion of Iraq had begun: Woodward, Bush at War,356.\n\n\"lingers and plagues the heart\": Schweizer and Schweizer, The Bushes, 398\u201399.\n\nhe was sleeping well: Woodward, State of Denial, 155.\n\n\"Not one doubt\": Woodward, Bush at War,256.\n\n\"Feels good\": Weisberg, The Bush Tragedy, 70.\n\nthings were going dangerously wrong: Gordon and Trainor, Cobra II, 273\u201374; 346\u201361.\n\nIf this wasn't liberation, nothing was: Gordon and Trainor, Cobra II, 490; Alfonsi, Circle in the Sand, 401.\n\nfewer than in the Gulf War: \"Iraq Coalition Casualty Count\u2014Deaths by Year and Month,\" March-April 2003, http:\/\/www.icasualties.org\/Iraq\/ByMonth.aspx; Department of Veterans Affairs, \"Fact Sheet: America's Wars,\" May 2008, http:\/\/www1.va.gov\/opa\/fact\/amwars.asp.\n\n\"regime's ill-gotten gains?\": Max Boot, \"Good News, Operation Iraqi Freedom Went About as Well as Anyone Could Have Hoped. Why Is the Media So Glum?\" Weekly Standard, April 15, 2003, http:\/\/www.weeklystandard.com\/Content\/Public\/Articles\/000\/000\/002\/551wqxuo.asp.\n\ndeclared one article in the Standard: Thomas Donnelly, \"Lessons of a Three Week War,\" Weekly Standard, April 28, 2003, http:\/\/www.aei.org\/article\/17006.\n\n\"without widespread loss of life\": David Brooks, \"The Phony Debate,\" Weekly Standard, March 31, 2008.\n\n\"Homer had given his gods\": Andrew J. Bacevich, The New American Militarism: How Americans Are Seduced by War (Oxford: Oxford University Press, 2005), 22.\n\n\" food and water and air\": \"President Bush Announces Major Combat Operations in Iraq Have Ended,\" Remarks by the President from the U.S.S. Abraham Lincoln, Released by the Office of the Press Secretary, May 1, 2003, http:\/\/georgewbush-whitehouse.archives.gov\/news\/releases\/2003\/05\/20030501\u201315.html.\n\n\"United States is on earth to achieve\": David Brooks, \"Today's Progressive Spirit: The Scenes in Baghdad Flow from Understandings Realized at the American Founding,\" Weekly Standard, April 9, 2003, http:\/\/www.weeklystandard.com\/Content\/Public\/Articles\/000\/000\/002\/519jigox.asp.\n\nand more legitimacy, without it: Wes Vernon, \"Kristol: U.N. Has Gone from 'Useless' to 'Harmful,'\" March 11, 2003, http:\/\/archive.newsmax.com\/archives\/articles\/2003\/3\/11\/110420.shtml.\n\nFrance and Germany too much say: Fred Barnes, \"The Tempting of the President,\" Weekly Standard, April 21, 2003, http:\/\/www.weeklystandard.com\/Content\/Public\/Articles\/000\/000\/002\/543ayyjy.asp.\n\nhad made itself worthless: Charles Krauthammer, \"A Costly Charade at the U.N.,\" Washington Post, February 28, 2003, A23.\n\nspiked back up to 77 percent: Ricks, Fiasco, 134.\n\ngreatest basketball player of all time: Linton Weeks, \"23 Skiddoo; Michael Jordan Flies Down the Court for the Last Time at MCI Center,\" Washington Post, April 15, 2003, C1.\n\nmillion more in speaking fees: Woodward, Plan of Attack, 413.\n\n\"to undermine the President's policies\": Newt Gingrich, \"Transforming the State Department: The Next Challenge for the Bush Administration,\" American Enterprise Institute, Washington, D.C., April 22, 2003, http:\/\/www.aei.org\/publications\/filter.all,pubID.16992\/pub_detail.asp; Woodward, State of Denial, 251.\n\ndeclined to defend his secretary of state: DeYoung, Soldier, 466\u201368.\n\nsomeone mentioned Powell, and they all laughed: Woodward, Plan of Attack, 409\u201312.\n\nChapter 19: I'm Delighted to See Mr. Bourne\n\n\"vicious dictator, and they're free!\": Department of Defense, \"DoD News Briefing by Secretary Rumsfeld and General Myers,\" April 11, 2003, http:\/\/www.defenselink.mil\/transcripts\/transcript.aspx?transcriptid=2367; Woodward, State of Denial, 164.\n\ncost of the rampage: $12 billion: John Kifner and John F. Burns, \"As Tanks Move In, Young Iraqis Trek Out and Take Anything Not Fastened Down,\" New York Times, April 10, 2003; John F. Burns, \"Looting and a Suicide Attack as Chaos Grows in Baghdad,\" New York Times, April 11, 2003; Larry Diamond, Squandered Victory: The American Occupation and the Bungled Effort to Bring Democracy to Iraq (New York: Times Books, 2005), 282; Woodward, State of Denial, 179, 183; Rajiv Chandrasekaran, Imperial Life in the Emerald City: Inside Iraq's Green Zone (New York: Random House, 2006), 40.\n\n\"This is not my job\": Woodward, State of Denial, 179.\n\npicked Iraq's government clean: James Fallows, \"Blind into Baghdad,\" Atlantic, January\/February 2004, http:\/\/www.theatlantic.com\/doc\/200401\/fallows.\n\n\"is an unbelievable mess\": Gordon and Trainor, Cobra II, 541.\n\nsettled on \"criminal negligence\": Diamond, Squandered Victory, 292.\n\nthe National Guard and Reserves: Department of Defense, \"Defense Manpower Requirements Report,\" June 1999, http:\/\/www.defenselink.mil\/prhome\/docs\/fy2000.pdf; Lawrence Kapp, \"Reserve Component Personnel Issues,\" CRS Report for Congress, January 18, 2006, 4; Stephen Daggett and Amy Belasco, \"Defense Budget for FY2003: Data Summary,\" CRS Report for Congress, March 29, 2002, 15.\n\nthe entire U.S. diplomatic corps: Cockburn, Rumsfeld, 106\u20137; Gordon and Trainor, Cobra II, 545\u201347; Linda Robinson, Tell Me How This Ends: General David Petraeus and the Search for a Way Out of Iraq (New York: PublicAffairs, 2008), 4; \"Highlighting Public Diplomacy Needs,\" American Academy of Diplomacy Newsletter no. 59 (October 2003), http:\/\/www.academyofdiplomacy.org\/publications\/newsletter_archive\/ newsletter_issue_59.htm.\n\nresponded with a blank stare: Diamond, Squandered Victory, 35.\n\nUnited States playing only a supporting role: Gordon and Trainor, Cobra II, 121\u201322.\n\n\"the Gaullists would have been neutered\": Ricks, Fiasco, 124; Bradley Graham, \"US Airlifts Iraqi Exile Force for Duties near Nasiriyah,\" Washington Post, April 7, 2003, A1; Linda Robinson and Kevin Whitelaw, \"Deploying the 'Free Iraqi Forces,'\" U.S. News & World Report, April 7, 2003, http:\/\/www.usnews.com\/usnews\/news\/iraq\/articles\/fiff030407.htm; Feith, War and Decision, 253.\n\nfighting force had already existed: Military analyst Stephen Biddle made exactly that point in an analysis of America's overthrow of the Taliban. Noting the crucial role that the Northern Alliance played, he wrote that \"we should be wary of suggestions that precision weapons, with or without special operations forces to direct them, have so revolutionized warfare that traditional ground forces are now superseded.\" Stephen Biddle, Afghanistan and the Future of Warfare: Implications for Army and Defense Policy (Carlisle, PA: Strategic Studies Institute, 2002), ix.\n\nforce dispatched to Iraq totaled 73: Gordon and Trainor, Cobra II, 122\u201323, 360; Bob Woodward, The War Within: A Secret White House History (New York: Simon & Schuster, 2008), 49; Graham, \"US Airlifts Iraqi Exile Force for Duties near Nasiriyah\" Robinson and Whitelaw, \"Deploying the 'Free Iraqi Forces.'\"\n\nthe nucleus of a post-Saddam regime: Graham, \"U.S. Airlifts Iraqi Exile Force for Duties Near Nasariyah\" Marc Santora with Patrick E. Tyler, \"Pledge Made to Democracy by Exiles, Sheiks and Clerics,\" New York Times, April 16, 2003, A1.\n\nsecular elite from which he hailed: Dexter Filkins, \"Where Plan A Left Ahmad Chalabi,\" New York Times Magazine, November 5, 2006, http:\/\/www.nytimes.com\/2006\/11\/05\/magazine\/05CHALABI.html.\n\nmansions favored by one of Saddam's sons: Gellman, The Angler, 247.\n\nsold overseas for a fat profit: Jane Mayer, \"The Manipulator,\" New Yorker, June 7, 2004, http:\/\/www.newyorker.com\/archive\/2004\/06\/07\/040607fa_fact1.\n\nstealing reconstruction funds: Mayer, \"The Manipulator\" Ricks, Fiasco, 124.\n\nclassified U.S. intelligence to Iran: Scott Wilson, \"U.S. Aids Raid on Home of Chalabi,\" Washington Post, May 21, 2004, A1.\n\nany Iraqi politician, including Saddam: ABC News, \"Poll: Iraq\u2014Where Things Stand,\" March 15, 2004, http:\/\/abcnews.go.com\/images\/pdf\/949a1IraqPoll.pdf.\n\neven more ideologically self-assured: Cannon and Cannon, Reagan's Disciple, 199.\n\npro-American democracy trapped inside: Gordon and Trainor, Cobra II, 545.\n\nboldly tore off the remaining scab: Diamond, Squandered Victory, 349, n. 38; Chandrasekaran, Imperial Life in the Emerald City, 63.\n\nholding jobs in Iraq's new government: Sharon Otterman, \"Iraq: Debaathification,\" CFR Backgrounder, April 7, 2005, http:\/\/www.cfr.org\/publication\/7853\/iraq.html; Gordon, Cobra II, 549; Ricks, Fiasco, 158\u201359.\n\ninterior ministry, and presidential guards: In fact, of Iraq's 80,000 commissioned officers, only 8,000 were members of the Baath Party. Walter Slocombe, \"Status of Rebuilding and Training the Iraqi Army,\" press conference, State Department, Washington, D.C., September 17, 2003, http:\/\/2002\u20132009-fpc.state.gov\/24230.htm.\n\nbackbone of the Iraqi economy: Naomi Klein, \"Baghdad: Year Zero,\" Harper's, September 2004, http:\/\/www.harpers.org\/archive\/2004\/09\/0080197; Roger Burbach and Jim Tarbell, Imperial Overstretch: George W. Bush and the Hubris of Empire (London: Zed, 2004), 186.\n\nimprovised explosive devices (IEDs): Woodward, State of Denial, 202, 205\u20136, 211; Ricks, Fiasco, 164.\n\n\"Who's Your Baghdaddy?\": Chandrasekaran, Imperial Life in the Emerald City, 4, 14\u201315, 18, 130.\n\nfear they would poison the food: Chandrasekaran, Imperial Life in the Emerald City, 9\u201310.\n\nseemed turned upside down: Ricks, Fiasco, 206.\n\nresembles the word for \" fuck\": Gordon and Trainor, Cobra II, 554; Feith, War and Decision, 434.\n\nequivalent of a raised middle finger: Woodward, State of Denial, 290.\n\n\"is the sound of freedom\": Chandrasekaran, Imperial Life in the Emerald City, 127.\n\na more confrontational ring: Anthony Shadid, Night Draws Near: Iraq's People in the Shadow of America's War (New York: Picador, 2006), 15.\n\nin an e-mail to friends: Draper, Dead Certain, 204.\n\n\"really understood about Iraq\": Shadid, Night Draws Near, 8.\n\nto roughly one thousand per month: Hayes, Cheney, 426; Woodward, State of Denial, 261.\n\nbegan carrying guns to their offices: Chandrasekaran, Imperial Life in the Emerald City, 181\u201382.\n\nmore dangerous than under Saddam: Diamond, Squandered Victory, 26.\n\nraids against suspected insurgents: Ricks, Fiasco, 234, 236, 301\u20132.\n\narrested, injured, or killed: The International Red Cross report described U.S. raids this way: \"Arresting authorities entered houses usually after dark, breaking down doors, waking up residents roughly, yelling orders, forcing family members into one room under military guard while searching the rest of the house and further breaking doors, cabinets and other property. They arrested suspects, tying their hands in the back with flexicuffs, hooding them, and taking them away. Sometimes they arrested all adult males present in a house, including elderly, handicapped and sick people. Treatment often included pushing people around, insulting, taking aim with rifles, punching and kicking and sticking with rifles. Individuals were often led away in whatever they happened to be wearing at the time of arrest\u2014sometimes in pyjamas [sic] or underwear\u2014and were denied the opportunity to gather a few essential belongings, such as clothing, hygiene items, medicine or eyeglasses.\" Quoted in Ricks, Fiasco, 235.\n\nturned the stomach of the world: Ricks, Fiasco, 195\u2013200, 238\u201340, 283\u201392.\n\n\"to rob Iraq's oil\": Diamond, Squandered Victory, 25\u201326.\n\nslum known as Sadr City: Ricks, Fiasco, 335\u201338.\n\nroughly three thousand a month: Woodward, State of Denial, 368.\n\nbefore going out on patrol: Ricks, Fiasco, 350.\n\n\"this job will run you all over town\": Draper, Dead Certain, x, 357; Woodward, State of Denial, 490.\n\nas he did in late May 2003: Woodward, State of Denial, 209.\n\nbut as late as 2006: Weisberg, The Bush Tragedy, 209.\n\npushing to invade Panama: Kaplan, Daydream Believers, 134.\n\nthe last two paragraphs alone: Woodward, State of Denial, 378.\n\n\"ending tyranny in our world\": George W. Bush, \"Second Inaugural Address,\" January 20, 2005, http:\/\/georgewbush-whitehouse.archives.gov\/news\/releases\/2005\/01\/20050120\u20131.html.\n\nto elect a National Assembly: Angus Reid Global Monitor, \"Iraq: Election Tracker,\" Angus Reid, http:\/\/www.angus-reid.com\/tracker\/view\/5143.\n\nbrought democracy to Kyrgyzstan: Karl Vick, \"New Leadership Established in Kyrgyzstan,\" Washington Post, March 26, 2005, A8.\n\noverturn the election results by force: David Rose, \"The Gaza Bombshell,\" Vanity Fair, April 2008, http:\/\/www.vanityfair.com\/politics\/features\/2008\/04\/gaza200804.\n\nin Africa, and across the globe: Gary J. Bass, \"Humanitarian Intervention in the 21st Century,\" Tocqueville Review\/La revue Tocqueville 30, no. 1 (2009): 25; Fukuyama, \"The End of History?\" 17.\n\n\"the end of the End of History\": Azar Gat, \"The Return of Authoritarian Great Powers,\" Foreign Affairs 86, no. 4 (July\/August 2007), available at http:\/\/www.foreignaffairs.org.\n\nwithout so much as a farewell party: Joshua Muravchik, interview with the author, September 4, 2008.\n\n\"institutions will be precarious at best\": Kirkpatrick, Making War to Keep Peace, 279.\n\ntoward darkness as toward light: Allan Gerson, \"Postscript,\" in Kirkpatrick, Making War to Keep Peace, 308.\n\n\"oversimplified view of human nature\": David Brooks, \"The Jagged World,\" New York Times, September 3, 2006, http:\/\/select.nytimes.com\/2006\/09\/03\/opinion\/03brooks.html.\n\n\"war that began three years ago\": George F. Will, \"Can We Make Iraq Democratic?\" City Journal, Winter 2004, http:\/\/www.city-journal.org\/html\/14_1_can_we_make_iraq.html; George Will, \"How Bush Can Try to Recover from Errors in Iraq War,\" Chicago Sun-Times, March 19, 2006, B7.\n\nquick to count dictatorship out: Robert Kagan, The Return of History and the End of Dreams (New York: Knopf, 2008).\n\n\"seemed that it has no future\": Robert Kagan, \"End of Dreams, Return of History,\" Policy Review 144 (August\u2013September 2007), http:\/\/www.hoover.org\/publications\/policyre view\/8552512.html.\n\ndead for more than eighty years: Casey Blake, \"Randolph Bourne's America,\" lecture, Columbia University, October 11, 2004, http:\/\/www.randolphbourne.columbia.edu\/panel_1.pdf.\n\nfounded the Randolph Bourne Institute: See http:\/\/randolphbourne.org\/.\n\nadministration's wartime propaganda: James Fallows, \"We've Been Here Before,\" January 21, 2004, http:\/\/thebigstory.org\/fallows.htm.\n\nBourne's old professor, Charles Beard: Andrew J. Bacevich, American Empire: The Realities and Consequences of U.S. Diplomacy (Cambridge, MA: Harvard University Press, 2002), 11\u201323.\n\n\"path of organization to that end\": Robert Westbrook, \"Randolph Bourne's America,\" lecture, Columbia University, October 11, 2004, http:\/\/www.randolphbourne.columbia.edu\/panel_1.pdf.\n\nso he can seize Central Asia's oil: Logan is actually the show's third president. Its second, Jack Keeler, decides not to seek a second term after being wounded in a terrorist attack. For a thoughtful discussion of Iraq's impact on Hollywood, see Ross Douthat, \"The Return of the Paranoid Style,\" Atlantic, April 2008, http:\/\/www.theatlantic.com\/doc\/200804\/iraq-movies.\n\n\"authority was built on silly illusions\": Matt Stoller, \"PDF2007: The Rise of the Netroots,\" Personal Democracy Forum, May 18, 2007, http:\/\/personaldemocracy.com\/node\/1432.\n\nincestuous relationship with the executive branch: Amy Argetsinger and Roxanne Roberts, \"White House Press Dinner: De-Wonked!\" Washington Post, April 21, 2008, C3.\n\ntorture \"war on terror\" suspects: Eric Schmitt, \"Senate Moves to Protect Military Prisoners Despite Veto Threat,\" Washington Post, October 6, 2005, A22; Gellman, The Angler, 357.\n\n\" jump in with both feet\": Jim VandeHei and Charles Babington, \"Newly Emboldened Congress Has Dogged Bush This Year,\" Washington Post, December 23, 2005, A5.\n\nat Guant\u00e1namo Bay, Cuba: Charles Lane, \"Justices Back Detainee Access to U.S. Courts; President's Powers Are Limited,\" Washington Post, June 29, 2004, A1.\n\nrelease information about Gitmo detainees: Josh White, \"Government Must Share All Evidence on Detainees,\" Washington Post, July 21, 2007, A2.\n\nchallenge their detention in federal court: Robert Barnes, \"Justices Say Detainees Can Seek Release,\" Washington Post, June 13, 2008, A1.\n\n\"extravagant, monarchial claims\": Gellman, The Angler, 355; Jonathan Mahler, \"After the Imperial Presidency,\" New York Times Magazine, November 9, 2008, http:\/\/www.nytimes.com\/2008\/11\/09\/magazine\/09power-t.html.\n\ncook agreed to prepare Rumsfeld's meal: Thomas E. Ricks, The Gamble: General David Petraeus and the American Military Adventure in Iraq, 2006\u20132008 (New York: Penguin, 2009), 77.\n\nafter he was tried for murder: Hayes, Cheney, 504; Richard Morin, \"18%? Just How Low Is,\" Washington Post, March 5, 2006, B3.\n\nsubstantially reducing his influence: Gellman, The Angler, 364.\n\nnow less fearsome than pitiable: Todd S. Purdum, \"A Face Only a President Could Love,\" Vanity Fair, June 2006, http:\/\/www.vanityfair.com\/politics\/features\/2006\/06\/cheney200606; Gellman, The Angler, 389.\n\nmost disliked president on record: See Kathy Frankovic, \"Bush's Popularity Reaches Historic Lows,\" CBS News, January 15, 2009, http:\/\/www.cbsnews.com\/stories\/2009\/01\/15\/opinion\/ pollpositions\/main4724068.shtml; Frank Newport, \"Bush Job Approval at 28%, Lowest of Administration,\" Gallup News, April 11, 2008, http:\/\/www.gallup.com\/poll\/106426\/Bush-Job-Approval\u201328-Lowest-Administration.aspx.\n\nturned the public against them: Cannon and Cannon, Reagan's Disciple, 292.\n\npointing to their disfigured relative: Woodward, State of Denial, 437\u201338.\n\n\"This is not your daughter!\": Draper, Dead Certain, 350.\n\nattend his party's presidential convention: Murphy and Purdum, \"Farewell to All That.\"\n\neven by Iraqi standards: Ellen Knickmeyer and K. I. Ibrahim, \"Bombing Shatters Mosque in Iraq,\" Washington Post, February 23, 2006, A1.\n\ndrill holes in their heads: Robinson, Tell Me How This Ends, 163.\n\na family intervention: Weisberg, The Bush Tragedy, 218.\n\nLaura, and Barney, the dog: Todd S. Purdum, \"Inside Bush's Bunker,\" Vanity Fair, October 2007, http:\/\/www.vanityfair.com\/politics\/features\/2007\/10\/purdum200710.\n\nthen dropped out of sight: Sabrina Tavernise, \"A Shiite Militia in Baghdad Sees Its Power Wane,\" New York Times, July 27, 2008.\n\n150 a week by spring 2009: Michael E. O'Hanlon and Jason H. Campbell, \"Iraq Index: Tracking Variables of Reconstruction and Security in Post-Saddam Iraq,\" Brookings Institution, August 20, 2009, http:\/\/www.brookings.edu\/saban\/~\/media\/Files\/Centers\/Saban\/Iraq%20Index\/index20090820.pdf.\n\nwould still be fighting there in another six: Ricks, The Gamble, 325.\n\nit had cost America $3 trillion: Joseph E. Stiglitz and Linda J. Bilmes, \"The $3 Trillion War,\" Vanity Fair, April 2008, http:\/\/www.vanityfair.com\/politics\/features\/2008\/04\/stiglitz200804.\n\nIraq had taken more than 4,000: Iraq Coalition Casualty Count, http:\/\/www.icasualties.org\/Iraq\/index.aspx.\n\nsizable portions of neighboring Pakistan: Kenneth Katzman, \"Afghanistan: Post-War Governance, Security, and U.S. Policy,\" CRS Report for Congress, November 26, 2008, 22\u201324.\n\nwere suddenly everywhere: See, for instance, Peter Baker, \"Could Afghanistan Become Obama's Vietnam?\" New York Times, August 22, 2009, WK1; and Frank Rich, \"Obama at the Precipice,\" New York Times, September 26, 2009, WK12.\n\nAfghanistan in the language of despair: For instance, see Helene Cooper, \"Obama's War: Fearing Another Quagmire in Afghanistan,\" New York Times, January 25, 2009, http:\/\/www. nytimes.com\/2009\/01\/25\/weekinreview\/25cooper.html; Michael Crowley, \"Obama vs. Osama: Has He Picked the Right War?\" New Republic, December 24, 2008, http:\/\/www.tnr.com\/article\/obama-vs-osama.\n\nPetraeus called himself a \"minimalist\": Ricks, The Gamble, 287.\n\n\"less than the vision that drove it to Baghdad\": Ricks, The Gamble, 156.\n\nwould never be vanquished: As Robert Gates has said, \"there has to be ultimately...reconciliation\" with the Taliban. Kristin Roberts, \"Pentagon Sees Reconciliation with Taliban, Not Qaeda,\" Reuters, October 9, 2009, http:\/\/www.reuters.com\/article\/idUSTRE4987PH20081009.\n\n\"Maliki intends to make himself a new dictator\": Kenneth M. Pollack, \"The Battle for Baghdad,\" National Interest, September\/October 2009, http:\/\/www.nationalinterest.org\/Article.aspx?id=22018.\n\n\"Gonna burst any day now?\": Draper, Dead Certain, 119.\n\nshare of world GDP declined every year: Roger Altman, \"The Great Crash, 2008,\" Foreign Affairs (January\/February 2009): 11.\n\n\"hinges on the support of its creditors\": Brad W. Setser, \"Sovereign Wealth and Sovereign Power,\" Council on Foreign Relations, Council Special Report No. 37, September 2008, 3\u20134.\n\n\"Le laisser-faire, c'est fini\": Altman, \"The Great Crash, 2008,\" 11.\n\n\"Non-American world\": Parag Khanna, \"Waving Goodbye to Hegemony,\" New York Times Magazine, January 27, 2008, http:\/\/www.nytimes.com\/2008\/01\/27\/magazine\/27world-t.html.\n\n\"more like a prophet\": Paul Starobin, \"Beyond Hegemony,\" National Journal, December 1, 2006, http:\/\/www.nationaljournal.com\/about\/njweekly\/stories\/2006\/1201nj1.htm.\n\n\"ascendancy in the world\": Paul Starobin, After America: Narratives for the Next Global Age (New York: Viking, 2009), 6.\n\namong Chinese, 89 percent: Pew Global Attitudes Project, \"Confidence in Obama Lifts U.S. Image Around the World,\" July 23, 2009, http:\/\/pewglobal.org\/reports\/pdf\/264.pdf; Paul Taylor, Cary Funk, and Peyton Craighill, \"Once Again, the Future Ain't What It Used to Be,\" Pew Research Center, May 2, 2006, http:\/\/pewresearch.org\/pubs\/311\/once-again-the-future-aint-what-it-used-to-be; Starobin, After America, 94.\n\nhe was now content to die: John Lukacs, George Kennan: A Study in Character (New Haven, CT: Yale University Press, 2007), 177.\n\ndestroyed, but it never was: Lukacs, George Kennan, 187.\n\nConclusion: The Beautiful Lie\n\n\"intelligent, illuminating failures\": Steffens, The Autobiography of Lincoln Steffens, 788.\n\n\"beyond its power to accomplish\": Arthur M. Schlesinger, Sr., \"What Then Is the American, This New Man?\" American Historical Review 48, no. 2 (1943): 244. Walter Russell Mead echoes the point in his excellent book, God and Gold: \"Optimism is the default mode of Anglo-American historical thought. How could it not be? The Whig narrative teaches us plainly that God is on our side, and centuries of victorious experience and economic progress confirm that the message is right.\" Walter Russell Mead, God and Gold: Britain, America and the Making of the Modern World (New York: Knopf, 2007), 314.\n\nOur obligations exceed our power: Lippmann, U.S. Foreign Policy, 5.\n\nharm American security more than they help: See David Kilcullen and Andrew McDonald Exum, \"Death from Above, Outrage from Below,\" New York Times, May 16, 2009.\n\nif they didn't get the bomb: For a history of Iranian foreign policy since the Islamic revolution, which makes this point, see Ray Takeyh, Guardians of the Revolution: Iran and the World in the Age of the Ayatollahs (New York: Oxford University Press, 2009).\n\ndemocratic and self-governing in its domestic affairs: On the superpowers and Finland, see Jussi Hanhim\u00e4ki, Containing Coexistence: America, Russia, and the \"Finnish Solution\" (Kent, OH: Kent State University Press, 1997); and Jussi Hanhim\u00e4ki, \"Self-Restraint as Containment: United States' Economic Policy, Finland, and the Soviet Union, 1945\u20131953,\" International History Review 17, no. 2 (May 1995): 287\u2013305.\n\nto protect our democracy so it can thrive: Lippmann, U.S. Foreign Policy, iii.\n\nhealth care and education combined: Anup Shaw, \"World Military Spending,\" Global Issues, http:\/\/www.globalissues.org\/article\/75\/world-military-spending#USMilitarySpending.\n\nthan of other institutions of government: Thomas Sander, \"Trust Declining in All Institutions Other Than the Military,\" Social Capital Blog, http:\/\/socialcapital.wordpress.com\/2009\/06\/08\/trust-declining-in-all-institutions-other-than-the-military\/.\n\nprimary instrument for interacting with the world: Bureau of Public Affairs, \"International Affairs\u2014FY 2010 Budget,\" U.S. Department of State, http:\/\/www.state.gov\/r\/pa\/prs\/ps\/2009\/05\/123160.htm; Department of Defense, \"A New Era of Responsibility,\" http:\/\/www.whitehouse.gov\/omb\/assets\/ fy2010_new_era\/Department_of_Defense.pdf.\n\n\" facts which momentarily imprison them\": Fox, Reinhold Niebuhr, 59.\n\n\"traditional myths of indomitable optimism\": C. Vann Woodward, \"A Second Look at the Theme of Irony,\" in The Burden of Southern History, ed. C. Vann Woodward (Baton Rouge: Louisiana State University Press, 1993), 214, 216.\n\n## SEARCHABLE TERMS\n\nThe pagination of this electronic edition does not match the edition from which it was created. To locate a specific passage, please use the search feature of your e-book reader.\n\nAble Archer, 233\n\nAbraham Lincoln, U.S.S., 355\n\nAbrams, Elliott, 211, 248\u201351, 274, 304, 365\n\n\"neo-Reaganite,\" 308, 450n\n\nPanama and, 248\u201351, 262, 305, 322\u201325\n\nAbu Ghraib, 364, 369\n\nAbzug, Bella, 277\n\nAcheson, Dean, 77, 119, 120, 125, 137, 182\u201383, 418n\n\nAdelman, Kenneth, 356\n\nAdler, Mortimer, 95\n\nAeschylus, 3\u20134, 11, 392\n\nAfghanistan, 327\u201339, 383\n\nbeginning of the war, 332\u201336\n\nliberal hawks and, 349\u201350\n\n9\/11 and, 8, 327\u201329, 330\u201332\n\n\"quagmire in,\" 2, 338\u201339, 373\u201374\n\nSoviet Union and, 206, 216\u201317, 331\n\nAgency, the. See CIA\n\nAgnew, Spiro, 197\n\nAjax (Sophocles), 124\n\nAlbright, Madeleine, 269, 286\u201387\n\nAlgeria, 142\n\nAlliance for Progress, 145\n\n\"alliances,\" 57\u201358\n\nAllison, John, 120\n\nal-Maliki, Nuri, 374\n\nAl Qaeda, 327\u201328, 330, 331\u201332, 336, 348\u201349, 352, 372, 373\u201374, 381, 382\n\nAlsace-Lorraine, 44\n\nal-Sadr, Moqtada, 364, 372\u201373\n\nal-Sistani, Sayyid Ali, 359\n\nAlsop, Joseph, 137, 150\n\nambition and hubris, 3\u20136\n\nAmbrosius, Lloyd, 405n\n\nAmerica at the Crossroads (Fukuyama), 367\u201368\n\nAmerican Civil War, 21\u201322\n\nAmerican Communist Party, 80, 82\n\nAmerican Enterprise Institute (AEI), 361, 366\u201367\n\nAmerican Food Administration, 64\n\nAmerican Friends of Vietnam, 152\n\nAmerican Geographical Society, 35\n\n\"Americanization,\" 290\n\nAmerican Political Science Association, 376\n\nAmerican Socialist Party, 62, 79\n\nAmerican University, 159\n\nAndrews, Charles, 93\n\nAngola, 198\u201399, 203, 206\n\nanthracite mining, 19\u201320, 28, 400n\n\nanthrax attacks, 337\n\nanti-Americanism, 102, 164, 203, 362, 366\n\nAnti-Appeasement Alliance, 236\u201337\n\nanticolonialism, 142, 151\n\nanticommunism. See Communism anti-imperialism, and Kennedy, 142, 145, 167\n\nanti-Semitism, 82, 194\u201395\n\nanti-totalitarianism. See totalitarianism antiwar movement, 176\u201380, 184\u201385, 188\u201389, 208\u20139, 266\u201367, 368\n\nappeasement, 120, 127, 134, 135, 140, 159, 211, 236\u201337\n\nAristide, Jean-Bertrand, 273\n\nArmistice Day (1932), 57\n\nArmitage, Richard, 319, 353\n\narms control, 37, 56\u201358, 57, 66\u201367, 217, 231\u201337\n\narms embargoes, 269, 280, 281, 283, 297\n\narms sales, 202, 264, 324\n\nAsian financial crisis of 1997, 290\n\nAspin, Les, 266, 270\u201371, 272, 281\n\nAssociated Press (AP), 33\n\nAtlantic, 60\u201361\n\n\"Atlantic Charter,\" 84\u201385\n\natomic bombs. See nuclear weapons\n\nAtta, Mohammed, 353\n\nBaath Party, 361\u201362, 461n\n\nBacevich, Andrew, 368\n\nBack to the Future (movie), 219\n\nBaines, Harold, 315\n\nBaker, James, 222, 224, 227, 234, 257\u201358, 268, 371\u201372\n\nBaker, Newton, 16\n\nBaker, Ray Stannard, 402\u20133n\n\nbalance of power, 25\u201326, 28\u201329, 42, 47, 97, 102\u20133\n\nBaldwin, Stanley, 137\n\nBalfour, Arthur, 402\u20133n\n\nBalkan Ghosts (Kaplan), 270\u201371, 445n\n\nBall, George, 169, 171, 179\u201380, 258\n\nBarnes, Fred, 345\n\nBartlett, Charles, 160\n\nBatista, Fulgencio, 144\n\nBaulkmon, Johnny, 341\n\nBay of Pigs, 144\u201348, 165, 172, 423n\n\nBeard, Charles, 18, 39\u201340, 51\u201353, 64, 70\u201371, 72, 81\u201382, 184\u201385, 368\n\nbeautiful lie, 378\u201390\n\nBell, Daniel, 80, 131\u201332, 143, 209\n\nBenavidez, Roy, 386\n\nBennett, William, 304\n\nBenton, Thomas Hart, 19\n\nBerchtesgaden, 74\n\nBerger, Sandy, 266\u201367\n\nBerlin Wall, 244\u201345\n\nBerman, Paul, 348\u201349\n\nBethesda Naval Hospital, 124\n\nBiddle, Stephen, 461n\n\nBiden, Joseph, 347, 352\n\nBigelow, Albert, 176\u201377\n\nbin Laden, Osama, 327\u201328, 331, 349\n\nBismarck, Otto von, 29, 99\n\nBissell, Richard, 144\u201347\n\nBlack Monday (1987), 246\n\nBlake, Casey, 368\n\nBloom, Allan, 245\n\nBlum, John, 400n\n\nBohlen, Charles, 103\n\nBok, Edward, 57\n\nBolivia, 227\n\nBolshevik Revolution. See Russian Revolution\n\nBolton, John, 319, 371\n\nBoot, Max, 355, 458n\n\nBorah, William, 48, 55, 57, 59\n\nBoren, David, 351\n\nBosnia, 2, 268\u201371, 276, 280\u201385, 308, 313, 332, 348\n\nBourne, Randolph, 39\u201343, 52\u201353, 60, 82, 184\u201385, 349, 368\u201369\n\nBoutros-Ghali, Boutros, 272, 278, 282, 284\n\nBowles, Chester, 146, 148\n\nBradlee, Ben, 161\n\nBradley, Omar, 115\n\nBrando, Marlon, 141\n\nBremer, Paul, 11, 361\u201362, 363\n\nBrest-Litovsk Treaty (1918), 405n\n\nBrezhnev, Leonid, 216\n\nBriand, Aristide, 57\u201359\n\nBroderick, Matthew, 232\n\nBrokaw, Tom, 311\n\nBrooks, David, 292, 309, 311, 355, 367\n\nBrown, Norman O., 177\u201378\n\nBrzezinski, Zbigniew, 201, 202, 205, 214\u201317, 258, 259\n\nBuchanan, Patrick, 298\n\nBuchwald, Art, 181\n\nBuckley, William F., 224, 229, 298, 449n\n\nbudget deficits, 114, 122, 182, 289, 338\u201339.\n\nSee also defense spending\n\nBuffett, Warren, 6\n\nBullitt, William, 75, 87\n\nBundy, McGeorge, 139, 152\u201355, 167\u201374, 179\u201380, 192, 195, 214\n\nBundy, Stephen, 179\n\nBundy, William, 167, 179\n\nBureau of Corporations, 20\n\nBureau of Labor, 20\n\nBurnham, James, 80\n\nBush, Barbara, 351\n\nBush, George H. W., 251\u201364\n\nBush, George W., and, 314\u201316, 318\n\nDPG statement and, 301, 302\u20133\n\nGulf War and, 5, 254\u201363, 296\u201397\n\nIraq War and, 351\n\nPanama and, 251\u201354\n\nreelection campaign of 1992, 264, 267\n\nSomalia and, 271\u201372\n\nSoviet Union and Gorbachev and, 234\n\nBush, George W., 314\u201348, 382\u201385\n\nAfghanistan and, 2, 8, 330\u201336, 338\u201339, 344\n\ndominance ethic and, 5, 247\u201348, 369\n\nfreedom agenda, 297, 341\u201343, 388\u201389\n\nidealism of, 340\u201343\n\nIraq War and, 10\u201311, 319\u201323, 337\u201348, 354\u201356, 359\u201360, 365\u201366\n\nmidterm elections of 2006, 370\u201371\n\nmillenarianism of, 387\n\n9\/11 and, 327\u201329, 337, 341\n\nPowell and, 317\u201320, 323\u201326, 329\u201333, 335\u201336, 345\u201347\n\npresidential campaign of 2000, 313, 316\u201317\n\nReagan and, 318, 320, 322, 325, 338, 342, 343, 344\n\nwar on terror, 329, 349, 351\u201352, 370, 380\n\nBush, John Ellis \"Jeb,\" 316\n\nbushido, 65\n\nButler, Nicholas Murray, 51, 52, 57\n\nByrd, Robert, 273\n\nCalverton, V. F., 80\n\nCambodia, 197, 277\u201378\n\nCannon, Lou, 227, 435n\n\ncapitalism, 64, 79, 143, 387\n\nCard, Andrew, 365\n\nCarmichael, Stokely, 256\n\nCarnegie, Andrew, 24\u201325\n\nCarnegie Endowment for International Peace, 24\u201325\n\nCarter, Jimmy, 200\u2013207, 211\u201317\n\nAfghanistan and, 216\u201317\n\ncivil rights and, 200\u2013201\n\nhuman rights and, 201\u20132, 206, 212, 216\u201317, 274\u201375\n\nIran and, 206, 213\u201316\n\nneocons and, 204\u20135, 207\u20138, 211\u201313, 219\u201320, 304\n\npost-toughness foreign policy, 194, 201\u20133, 207, 211\u201312\n\nrabbit story, 204\u20136\n\nSoviet Union and, 202, 205\u20137, 214\u201315, 221\u201322\n\nCasablanca (movie), 95\n\nCastro, Fidel\n\nCarter and, 202\u20133\n\nCIA plots against, 129, 199\u2013200, 275\n\nCuban Missile Crisis and, 155\u201357\n\nMcGovern and, 191\n\nReagan and, 223\u201324, 227\n\nCatt, Carrie Chapman, 59\n\nCattell, James McKeen, 51\u201352\n\nCedar Revolution, 366\n\nCentral American policy. See Latin America policy\n\nCentral Intelligence Agency. See CIA\n\nChalabi, Ahmed, 321, 344\u201345, 360\u201362\n\nChamberlain, John, 62\n\nChamberlain, Neville, 74\u201375, 134\u201335, 137, 227, 305\n\nCh\u00e1vez, Hugo, 366, 386\n\nChekhov, Anton, 10\n\nChen, Chris, 397\u201398n\n\nCheney, Dick\n\nAfghanistan and, 329\u201330, 335\u201336\n\nbudget deficits and, 338\u201339\n\nDPG statement and, 301\u20132\n\nGulf War and, 254\u201355, 257, 261\u201362\n\nIraq and, 318\u201320\n\nIraq War and, 341\u201342, 345, 346, 352\u201353, 356, 371\n\nPowell and, 319\u201320, 324\u201325, 329\u201330, 352\u201353\n\nSoviet Union and, 236\n\nCheney, Elizabeth, 319\n\nChiang Kai-shek, 117\u201318, 121, 125\n\nChicago White Sox, 315\n\nChildren of Light, Children of Darkness (Niebuhr), 94\u201395\n\nChile, 198, 206, 304\n\nChin, Edward, 354\n\nChina, 116\u201322, 125, 294\u201395, 375, 387\n\nHainan Island incident (2001), 323\u201324\n\nJapanese invasion of Manchuria, 65\u201366\n\nNixon and, 9, 195\u201396\n\nTiananmen Square protests of 1989, 245\n\nChirac, Jacques, 281\n\nChomsky, Noam, 184, 349, 368\n\nChristopher, Warren, 270\u201371, 281\u201383\n\nChurchill, Winston\n\n\"Atlantic Charter,\" 84\n\nFour Policemen idea, 86\u201388, 89\n\nKennedy and, 134, 139\u201340\n\nWorld War II and, 76, 78, 79, 90\u201391, 92, 126\u201327, 134\n\nYalta Conference, 90\u201392, 101\n\nChurch of Disciples, 221\n\nCIA (Central Intelligence Agency)\n\nAfghanistan and, 332, 339\n\nCheney and, 318\u201319\n\nContras and, 225\u201326, 249, 322\n\nCuba and, 129, 144\u201347, 199\u2013200, 275\n\nHollywood images of, 369\n\nIraq and, 254, 353\n\nPakistan and, 216\u201317\n\nPanama and, 251\u201354\n\nVietnam War and, 168\u201369\n\nCity College of New York, 81, 207\u201310, 294\n\nCivil Rights Act of 1964, 166\n\ncivil rights movement, 176\u201377, 200\u2013201\n\nCivil War, 21\u201322\n\nClancy, Tom, 259\u201360\n\nClark, Mark, 128\n\nClark, Wesley, 282\u201384, 285\u201386\n\nClemenceau, Georges, 26, 43\u201347, 49, 54, 83, 85\n\nClifford, Clark, 182\n\nClinton, Bill, 265\u201374, 281\u201392, 320\n\nBalkans and, 2, 268\u201371, 280\u201385, 297, 320\n\nHaiti and, 273, 297\n\nimpeachment of, 288, 310\n\nKosovo and, 285\u201388\n\nneocons and, 297, 305, 307, 309\u201310\n\n9\/11 and, 328\n\nRwanda and, 273\u201374\n\nSomalia and, 271\u201373\n\nVietnam and, 266\u201367\n\nClinton, Hillary, 347\n\n\"Clinton Standard Time,\" 265\n\nCoalition Provisional Authority (CPA), 361\u201363\n\nColby, William, 199\n\ncold war, 172\u201387, 290\u201391, 294\u201395\n\nReagan and, 221\u201323, 227\u201337\n\nColumbia University, 39, 51\u201352, 209, 368\n\nCommentary, 210, 213, 305\n\nCommittee Against Military Intervention in Europe, 135\n\nCommittee on Public Information, 41\n\nCommittee on the Present Danger (CPD), 232\n\nCommittee to Defend America by Aiding the Allies (CDAAA), 79\n\nCommunism. See also China; Soviet Union\n\nEisenhower and, 126\u201331\n\nJohnson and, 180\u201383, 202\n\nKennedy and, 136, 141\u201345, 147\u201352, 154\u201355, 159\n\nNiebuhr and, 79\u201380\n\nNixon and, 193\u201399\n\nReagan and, 221\u201323, 227\u201337\n\nTruman and, 116\u201322\n\n\"compassionate conservatism,\" 316, 341, 342\n\nComprehensive Nuclear-Test-Ban Treaty, 159\u201360\n\nCongress of Racial Equality (CORE), 176\u201377\n\nconservatism. See dominance conservatism; national-greatness conservatives; neoconservatism\n\ncontainment\n\nCarter and, 202\u20133\n\nEisenhower and, 126\u201331\n\nKennan and, 10\u201311, 100, 102\u20134, 109\u201319, 123\u201324, 190, 380\u201381\n\nKennedy and, 142\n\nMcGovern and, 190\u201391\n\nPowell and, 257\u201358\n\nTruman and, 102\u20134, 114\u201319\n\ncontinentalism, 52\u201353\n\nContras, 225\u201326, 249, 322, 453n\n\nCoolidge, Calvin, 55, 59\n\nCooper, John Milton, 412n\n\ncosmopolitanism, 68\u201369\n\nCouncil on Foreign Relations, 111\n\ncounterculture, 177\u201380, 208\u20139, 266\u201367, 368\n\ncounterinsurgency, 141, 143, 372\u201373\n\nCowley, Malcolm, 62\n\ncredibility, 123, 196\u201398, 419n\n\nCreel, George, 41\n\nCroatia, 268\u201371, 276\u201380\n\nCroly, Herbert, 18, 35, 40\n\nCromwell, Oliver, 19\n\nCronkite, Walter, 181, 213\u201314\n\nCrowe, Blake, 255\n\nCrowe, William, 248, 250\u201351, 254\u201357, 259, 322\u201323\n\nCuba\n\nBay of Pigs, 144\u201348, 165, 172, 423n\n\nCarter and, 202\u20133, 205\n\nCIA plots against, 129, 199\u2013200, 275\n\nKennedy and, 144\u201348, 155\u201357, 172, 214\n\nMcCarthy and, 187\n\nMcGovern and, 191\n\nReagan and, 223\u201324, 227, 305\n\nCuban Missile Crisis, 155\u201357, 214\n\nCuster, George Armstrong, 181\n\nCzechoslovakia, 47, 56, 57, 74\u201375, 286\n\nDaedalus, 3\u20136\n\nDaladier, Edouard, 74\n\nDallaire, Rom\u00e9o, 273\u201374, 278\n\nDallek, Robert, 412n\n\nDa Nang, 168\n\nDaniels, Josephus, 68, 69\u201370\n\nDardanelles, 84, 115, 116\n\nDavies, John Paton, 117\u201318, 120, 125, 146, 199, 418n\n\nDavis, Forrest, 85, 89, 413n\n\nDavis, Livingston, 69, 70\n\nDawes, Charles, 55\n\nDay After, The (movie), 232\u201333, 438n\n\nDay the Earth Stood Still, The (movie), 231\n\nDayton Accords (1995), 283\u201384\n\ndeath instinct, 177\u201378\n\nDeaver, Michael, 222, 224, 227\n\nDeclaration on Liberated Europe, 91\n\nDecter, Midge, 227\n\nDefense Planning Guidance (DPG), 301\u20133\n\ndefense spending, 114\u201315, 122\u201324, 128\u201329, 137, 163, 182, 198, 202, 305, 307\u20138, 385\n\ndeficits. See budget deficits; defense spending de Gaulle, Charles, 149, 161, 169\n\n\"Democracies and Double Standards\" (Kagan), 305\n\ndemocracy (democratization), 246\u201348, 274\u201377, 300\u2013301\n\nIraq and, 341\u201345\n\nDemocratic National Convention\n\n1932, 70\u201371\n\n1968, 184, 188, 191\n\n1972, 184\n\n1992, 264, 267\n\nDemocratic Party\n\nGulf War and, 347\n\nIraq War and, 347\u201348\n\nneocons and, 208, 210\u201311, 212\n\n\"democratic peace\" theory, 285, 343\n\nDepression, the, 62\u201363, 65\u201366, 122, 220\n\nDerian, Patricia, 201\n\nd\u00e9tente, 195\u201396, 201\u20132\n\nDewey, John, 39\u201343, 184\u201385, 190\n\nBourne and, 39, 42\u201343\n\neducation and, 18, 40, 95\n\nFrance and, 56\n\nHoover and, 63\u201364\n\nNiebuhr and, 60\u201363, 81\u201382, 408\u20139n\n\nOutlawry of War, 58\u201359, 184\n\nRussian Revolution and, 33\u201334\n\nWorld War I and, 40\u201342, 51, 52, 53, 60\u201361\n\nWorld War II and, 81\u201382\n\nDewey, Thomas, 90\n\nDial, 42\n\ndialectical materialism, 110\u201311\n\nDiamond, Lawrence, 358\n\n\"Dictatorships and Double Standards\" (Kirkpatrick), 213, 228, 253, 304, 305, 367\n\nDiem, Ngo Dinh, 151\u201352, 157\u201359, 164\n\nDien Bien Phu, 130, 145, 151, 168, 423n\n\nDirksen, Everett, 119, 174\n\ndisarmament, 37, 56\u201358, 66\u201367, 217, 231\u201337\n\nDisarmament Conferences, 67, 71\n\nDobrynin, Anatoly, 237\n\nDr. Strangelove (movie), 174\n\nDolan, Anthony, 230\n\nDole, Robert, 281\n\ndominance, 261\u201362\n\ndominance conservatism (conservatives), 301, 303, 308\u201311, 320, 325\u201326, 331\u201332, 335, 355, 367\u201369\n\ndominance ethic, 246\u201348, 267\u201368, 289\u201390, 295\u201396, 298\u2013311, 338, 369, 373, 379\n\nDos Passos, John, 39, 57, 62\n\nDraper, Robert, 341, 454n\n\nDraper, Theodore, 146\u201347\n\nDreiser, Theodore, 39, 82\n\nDru, Philip, 18\n\ndrug war, 249\u201350\n\nDulles, Allen, 129, 144\u201345, 147\u201348\n\nDulles, John Foster, 114, 127, 129\n\nEagleburger, Lawrence, 251\n\nEarle, George, 72\n\nEast Asian crisis of 1997, 290\n\nEast Berlin, 150, 244\n\nEastern Europe, and Soviet Union, 87, 89\u201393, 101\u20132, 111\u201312\n\nEastman, Max, 80\n\neconomic crisis of 2008, 375\u201376\n\nEconomist, 291\n\nEden, Anthony, 86\n\neducation, 18, 40, 95, 131, 184\n\nEdwards, John, 347\n\nEisenhower, Dwight, 126\u201331\n\ncontainment policy, 126\u201331, 198\n\nCuba and, 129, 145\u201346\n\nKorean War and, 127\u201328, 130\u201331\n\nNixon and, 194, 198\n\nSoviet Union and, 128\u201330, 136\u201337\n\nEliot, T. S., 95\n\nElizabeth II of England, 314\n\nEl Salvador, 223\u201325, 305\n\nemasculation, 132\u201333, 137\u201338, 309\n\nEmerson, Ralph Waldo, 171\n\nEndara, Guillermo, 251, 253\n\n\"end of history,\" 245\u201346, 274\u201377, 289\u201390, 310, 343, 366\n\nEnd of Ideology, The (Bell), 131\u201332, 143\n\nengineering (engineers), 63\n\nEnglebrecht, H. C., 72\n\nErhard, Ludwig, 166\n\nEsquire, 132\n\nEthiopia, 205, 206\n\nEureka College, 221\n\nEuropean Union (EU), 268\n\nevil, 5, 18, 80, 114\n\nReagan and, 219\u201320, 221\n\nexcessive fear, as warning sign, 7\u20138\n\nexecutive powers, 77, 165, 225, 288\n\nFallows, James, 368\n\nFalwell, Jerry, 205\n\nFarewell to Reform (Chamberlain), 62\n\nFarmer, James, 176\n\nFarrakhan, Louis, 256\n\nFastabend, David, 374\n\nFaulkner, William, 57\n\nfederalism, 35\n\nFederal Reserve, 23, 24, 35\n\nFederal Trade Commission (FTC), 23, 24, 35\n\nFein, Bruce, 370\n\nFeith, Douglas, 324, 345, 453n\n\nFellowship for Reconciliation, 176\n\nfinancial crisis of 2008, 375\u201376\n\nFinland, 64, 384\n\nFischer, Fritz, 407n\n\nFischer, Louis, 80\n\nFisher, N.R.E., 398n\n\nFleming, Ian, 145\n\nFord, Gerald, 198\u201399, 277\u201378\n\nFord, Henry, 61\u201362, 166\n\nForeign Affairs, 111\n\nForrestal, James, 103, 109\u201315, 117, 124\n\nForrestal, Michael, 159\n\nFortas, Abe, 429n\n\nFour Policemen idea, 85\u201390, 100, 103\n\nFourteen Points, 37, 46\n\nFox, Richard Wightman, 409n\n\nFox News, 325, 351\n\nFrank, Waldo, 41\u201342\n\nFrankfurter, Felix, 35\n\nFranks, Tommy, 339\u201340, 355, 356, 358\n\nfreedom, 275\u201376\n\nBush and, 297, 341\u201343, 388\u201389\n\nfreedom of the seas, 31, 37, 46\n\nFreeland, Richard, 418n\n\nfree markets, 26, 209, 309\n\nFreud, Sigmund, 16, 177\u201378, 415n\n\nFriedman, Thomas, 291, 318\n\nFromkin, David, 105\n\nFukuyama, Francis, 244\u201346, 291\u201392\n\n\"end of history,\" 245\u201346, 274\u201377, 310\n\nIraq War and, 348\u201349, 351\u201352, 367\u201368\n\nFulbright, William, 199, 423n\n\nFulton, James, 418n\n\nGaither, H. Rowan, 137, 211\n\nGalbraith, John Kenneth, 148\u201349\n\nGarner, Jay, 358\u201359, 361\n\nGates, Robert, 464n\n\ngays in the military ban, 267, 270\n\nGaza Strip, 366\n\nGelb, Leslie, 189, 190, 201, 259, 278\n\nGeneral Accounting Office (GAO), 336\n\nGeneva Convention, 336\n\ngenocide, 274, 276\u201380, 308\n\nGeorge Washington, 38\n\nGeorgia, 383\u201384\n\nGerson, Michael, 326, 327, 331, 336, 342, 457n\n\nG.I. Joe, 219\n\nGingrich, Newt, 259\u201360, 281, 283, 309, 356\n\nGlazer, Nathan, 209, 248, 298\n\n\"globalization,\" 290\n\nGoldberg, Jonah, 309, 311\n\nGold Crisis of 1968, 182\u201383\n\nGolden Rule, 176\n\nGoldwater, Barry, 166\n\nGorbachev, Mikhail, 230\u201337, 244, 306, 439n\n\nGordon, Michael, 443n\n\nGore, Albert, Jr., 264, 269\n\nGore, Thomas, 16\n\nGraham, Lindsey, 370\n\nGrayson, Cary, 49\u201350\n\nGreat Depression, 62\u201363, 65\u201366, 122, 220\n\nGreater Middle East Initiative, 365\n\ngreatness. See national-greatness conservatives\n\nGreat Society, 180\u201381, 182, 184\n\nGreece, 89, 115\u201318, 121, 154\n\nGreek literature, 3\u20134. 6, 11, 398n\n\nGreenspan, Alan, 290\n\nGreen Zone (Iraq), 362\u201363\n\nGreer, U.S.S., 77\u201378\n\nGrenada, 227\u201328, 250\u201351\n\nGromyko, Andrei, 149\n\ngross domestic product (GDP), 128, 198, 375\n\ngross national product (GNP), 123\n\nGroton School, 68, 69, 173\n\nGuant\u00e1namo Bay, Cuba, 370\n\nGuatemala, 129, 172, 198\n\nGuevara, Ernesto \"Che,\" 172\n\nGulf of Tonkin, 164\u201366\n\nGulf War, 2, 5, 254\u201363, 296\u201397, 300, 312\u201313, 347\n\nHadley, Stephen, 319\n\nHaig, Alexander, 223\u201325\n\nHainan Island incident (2001), 323\u201324\n\nHaiti, 273, 284, 297, 313\n\nHalberstam, David, 139, 152, 158, 166\n\nHaldeman, H. R., 195\n\nHamas, 10, 366, 380\n\nHand, Learned, 35\n\nHanighen, F. C., 72\n\nHarding, Warren, 53, 55, 57\n\nHarlan County, U.S.S., 273, 284\n\nHarries, Owen, 298, 307\n\nHarriman, Averell, 89\u201390, 103, 149\n\nHarvard Crimson, 69, 179\n\nHarvard University, 69, 133, 135, 161\n\nHassett, William, 92\n\nHavel, Vaclav, 321, 344\n\nHawthorne, Nathaniel, 95\n\nHayden, Tom, 176, 184\n\nHearst, William Randolph, 71\n\nHelms, Jesse, 249\n\nHelsinki Accords (1975), 196\n\nHemingway, Ernest, 57\n\nHepburn Act of 1906, 20\n\nHindenburg, Paul von, 37\n\nHiroshima, 93\u201394, 176\n\nHiss, Alger, 194\n\nHitchens, Christopher, 348, 368\n\nHitler, Adolf, 10, 71\u201372, 74\u201380, 91, 112, 134, 135, 174, 210\n\nHobbes, Thomas, 194\u201395\n\nHo Chi Minh, 7, 151\u201352, 183, 275\n\nHoffa, Jimmy, 186\n\nHolbrooke, Richard, 267, 269, 270, 272, 283\u201386\n\nHolocaust, 194\u201395, 279\u201380\n\nHolocaust Memorial Museum (Washington, D.C.), 280\n\nhomosexuality, 119, 132, 211\n\nHook, Sidney, 62, 80\n\nhoover, 64\n\nHoover, Herbert, 63\u201367, 77\n\nHopkins, Harry, 78\u201379, 86\n\nHouse, Edward M., 15\u201319, 31, 36\u201340, 50, 96, 402\u20133n\n\nHouse Un-American Activities Committee (HUAC), 174\u201375\n\nHowe, Irving, 80\n\nhubris, 4, 398 n\n\nHughes, Charles Evans, 57, 58\n\nHull, Cordell, 71, 85\u201386, 89, 103\n\nhumanitarian wars, 277\u201378, 284, 297, 300\u2013301, 303\n\nhuman nature, 29, 80, 95, 123\u201324, 176\u201378, 191\u201392, 229, 245\u201346\n\nhuman rights, 93, 201\u20132, 216\u201317, 274\u201375, 300\u2013301\n\nHumphrey, Hubert, 170, 185, 188\n\nHussein, Saddam\n\nGulf War and, 2, 254\u201363, 296\u201397, 300, 312\u201313\n\nIraq War and, 345\u201346, 352\u201355, 457\u201358n\n\n9\/11 and, 330\u201331, 341\u201342\n\nHussein, Uday, 358\n\nHutus, 273\u201374, 276\u201377\n\nIcarus, 3\u20136, 11\n\nidealism, 388\n\nIdea of the National Interest (Beard), 72\n\nIgnatieff, Michael, 350, 368\n\nimprovised explosive devices (IEDs), 362\n\nInchon, 120, 260\n\nIndochina, 125, 185\n\nIndonesia, 320\n\n\"Inquiry, the,\" 17\u201318, 35\u201337\n\nInter-Allied Military Control Commission, 56\n\nintercontinental ballistic missiles (ICBMs), 137\n\ninterdependence, 189\u201390, 214\u201315\n\nIntermediate-Range Nuclear Forces (INF), 236\n\ninternationalism, 48\u201349, 77\n\nInternational Monetary Fund (IMF), 102, 290, 389\n\nInterstate Commerce Commission, 20, 35\n\nIn Time to Come (play), 83\n\nIran, 212\u201316, 366, 381, 383\n\nhostage crisis, 213\u201316\n\nShah and, 205\u20136, 212\u201313\n\nSoviet Union and, 115\n\nIran-Contra Affair, 225\u201326\n\nIraq, 319\u201323\n\nGulf War, 254\u201363, 296\u201397, 312\u201313\n\n9\/11 and, 330\u201336\n\nno-fly zone, 319\u201320\n\nOperation Desert Fox, 328\n\nregime change, 264, 323\u201325\n\nsanctions, 254\u201355, 257, 296\u201397, 319\n\nwar. See Iraq War\n\nIraqi National Congress (INC), 321, 344\u201345, 360\u201361\n\nIraq Study Group (ISG), 371\u201372\n\nIraq War, 7\u20138, 10\u201311, 337\u201365\n\nDemocratic Party and, 347\u201348\n\ndemocratization rationale for, 341\u201345\n\nfailure to control postwar Baghdad, 357\u201362\n\ninvasion begins, 354\u201356\n\nliberal hawks and, 348\u201350, 368\u201369\n\nneocons and, 350\u201352, 457\u201358n\n\npublic support, 1\u20132, 397n\n\nrun-up to, 338\u201345\n\nsurge strategy, 371\u201375\n\ntiming of Bush's decision, 338\u201340\n\nIrreconcilables. See Senate \"Irreconcilables\"\n\nIslamic terrorism. See Al Qaeda; jihadist terrorism\n\nIslamofascism, 10\u201311, 348\u201349\n\nisolationism, 32, 42\u201343, 51, 52\u201353, 59, 64, 71\u201373, 82, 92, 189, 300\n\nIsrael, 226, 319, 328, 366\n\nJackson, Michael, 371\n\nJacobs, Travis Beal, 411n\n\nJames, William, 16\n\nJapan, 57, 65\u201366, 73, 78, 85, 133\u201334, 151, 289\n\njihadist terrorism, 10\u201311, 348\u201349, 352, 382, 384. See also Al Qaeda\n\nJohnson, Luci Baines, 183\n\nJohnson, Lyndon B., 161\u201371\n\nCommunism and, 119, 162\u201363, 167, 175, 379\n\nGreat Society, 180\u201381, 182, 184\n\nneocons and, 210\u201311\n\ntoughness ethic, 5, 162, 166, 180\u201383, 210\u201311\n\nVietnam War and, 8, 10, 162, 163\u201371, 178\u201384, 238\n\nJordan, Michael, 355\u201356\n\nJungle, The (Sinclair), 20\n\nKagan, Robert, 303\u201310, 313, 337, 367\u201368, 450n, 457\u201358n\n\nKaplan, Lawrence, 340\n\nKaplan, Robert, 270\u201371, 445n\n\nKarzai, Hamid, 334, 344, 374\n\nKearny, U.S.S., 77\u201378\n\nKeating, Kenneth, 155\n\nKeegan, John, 287\n\nKeeler, Jack, 463n\n\nKellogg, Frank, 58\u201359, 66\n\nKellogg-Briand Pact (1928), 59, 66\n\nKennan, George Frost, 98\u2013100, 376\u201377\n\ncontainment strategy, 10\u201311, 100, 102\u20133, 109\u201319, 123\u201324, 190, 380\u201381\n\nKorean War and, 120\n\nReagan and, 238\u201339\n\nSoviet Union and, 10, 99\u2013100, 102\u20133, 157, 238\u201339, 243\u201344\n\nVietnam War and, 125, 169\u201370\n\nKennedy, Edward, 253\n\nKennedy, John F., 133\u201362\n\nassassination of, 160, 163\n\nCommunism and, 136, 141\u201345, 147\u201352, 154\u201355, 159\n\nCuba and Bay of Pigs, 144\u201348, 172\n\nCuban Missile Crisis, 155\u201357, 214\n\ndeveloping world and, 141\u201343\n\nJohnson compared with, 161\u201362\n\nLaos and, 148\u201349\n\nneocons and, 210\u201311\n\n\"peace speech\" (1963), 159\n\npresidential campaign of 1960, 137\u201338\n\nSoviet Union and, 136\u201338, 143, 149\u201351, 154\u201357, 159\u201360\n\ntoughness ethic, 132\u201333, 137\u201340, 148, 156\u201357, 162\n\nThe Ugly American and, 141\u201342, 155\n\nVietnam War and, 151\u201354, 157\u201359, 163\u201364\n\nWorld War II and, 133\u201336\n\nKennedy, Joseph, Sr., 133, 134\u201336, 161\n\nKennedy, Paul, 246\u201347, 267\u201368, 289, 376\n\nKennedy, Robert, 11, 133, 141\u201342, 186\u201387, 188\n\nCuba and, 147, 148, 155\n\nVietnam War and, 11, 179, 180, 186\u201387\n\nKerry, John, 347, 386\n\nKeynes, John Maynard, 122\n\nKeyserling, Leon, 122\u201323\n\nKhalilzad, Zalmay, 301\u20133, 308\n\nKhanh, Nguyen, 164, 185\n\nKhanna, Parag, 376\n\nKhmer Rouge, 277\u201378\n\nKhomeini, Ruhollah, 206\n\nKhrushchev, Nikita, 137\u201338, 143, 149\u201351, 154\u201357, 159\u201360\n\nKing, Martin Luther, Jr., 177, 179\n\nKirkpatrick, Jeane, 210, 212\u201313, 294\u2013301, 306, 366\u201367\n\nCarter and, 211\u201313, 215\n\nClinton and, 297\n\ndemocratization and, 212\u201313, 275, 367\n\n\"Dictatorships and Double Standards,\" 213, 228, 253, 304, 305, 367\n\nGulf War and Bush, Sr., 296\u201397, 301\u20132\n\nIraq War and Bush, Jr., 350, 366\u201367\n\nKagan and, 304\u20136, 308\n\nKrauthammer and, 298\u2013301\n\nPoland Solidarity and, 222\n\nReagan and, 221\u201322, 228\u201330, 234, 236, 248\u201350, 262, 305\u20136\n\nKissinger, Henry, 194\u201399\n\nKnox, Frank, 77\n\nKoppel, Ted, 267\n\nKorean War, 119\u201322, 127\u201328, 130\u201331, 168\n\nKosovo, 276\u201377, 285\u201388, 308, 313, 332, 348\n\nKrauthammer, Charles, 298\u2013301, 309, 317, 332, 355\n\nKristol, Irving, 207\u201311, 212, 293\u201398, 303, 306, 308\n\nCarter and, 211\u201312\n\nGulf War and Bush, Sr., 211\u201312\n\nIraq War and, 350\u201351\n\nNicaragua and, 224\n\nPoland Solidarity and, 222\n\nReagan and, 222, 224, 228\u201329, 233, 248\u201349, 262\n\nVietnam War and, 208\u20139\n\nKristol, William, 303\u201310, 332, 340, 355, 457\u201358n\n\nKrock, Arthur, 111, 135\u201336, 421n\n\nKubrick, Stanley, 174\n\nKurds, 260\u201362, 263, 297, 360\n\nKuwait, Gulf War, 254\u201363, 296\u201397, 300, 312\u201313\n\nKyrgyzstan, 335, 366\n\nlabor strikes, 19\u201320\n\nlabor unions, 62, 166\n\nLa Follette, Robert, 23\n\nLake, Anthony, 189, 190, 201, 267, 269, 281\u201382, 284, 286\n\nLansdale, Edward, 141, 155, 160, 199\u2013200\n\nLansing, Robert, 35\u201336\n\nLaos, 148\u201349, 154, 181, 218, 424n\n\nLasch, Christopher, 18\u201319, 184\n\nLatin America policy. See also specific countries\n\nReagan and, 223\u201326, 248\u201352\n\nWilson and, 26\u201329\n\nLatvia, 46, 49, 88\n\nLeague of Nations, 37, 45\u201351, 56, 68, 71, 75, 84, 86, 88\n\nLeague to Enforce Peace, 31\u201332, 37\n\nLebanon, 226\u201327, 250\u201351, 256, 270, 305\u20136, 366, 423n\n\nLederer, William, 141\u201342\n\nLemnitzer, Lyman, 149\n\nLenin, 36, 99, 110\n\nLerner, Max, 80\n\nLewinsky, Monica, 288\n\nLibby, I. Lewis \"Scooter,\" 301\u20133, 308, 319, 356, 371, 453n\n\nliberal hawks, 279, 287, 300\u2013301, 348\u201350, 368\u201369\n\nliberalism. See Democratic Party; New Left; progressivism\n\nLiberal Papers, 174\n\nLiberal Project, 173\n\nLiberation, 177\n\nLibya, 227\u201328\n\n\"lift and strike,\" 269, 270\n\nLincoln, Abraham, 21, 49\n\nLindsey, Lawrence, 339, 373, 375\n\nLippmann, Walter, 16\u201319, 24, 26, 104\u20135, 381\n\ncontainment strategy, 113\n\nFreud and, 415n\n\nHoover and, 65\n\nKennedy and, 135\n\n\"the Inquiry,\" 35\u201337, 47\n\nVersailles Treaty and, 47, 48, 52\n\nVietnam War and, 169\u201370, 238\n\nWorld War I and, 16\u201317, 32\u201340, 47\u201348, 81\n\nWorld War II and, 81, 97\u201398\n\nLithuania, 88, 296\n\nLloyd George, David, 45, 46\u201347\n\nLocarno Treaty (1925), 55\u201356\n\nLodge, Henry Cabot, 29\u201331, 33, 48\u201351, 83, 405n\n\nLodge, Henry Cabot, Jr., 158\n\nLondon Naval Treaty (1930), 65\u201366\n\n\"Long Telegram\" (Kennan), 10, 109, 110\n\nLook, 83\n\nLos Angeles Times, 312\n\nLovett, Robert, 122\n\nLowell, Abbott Lawrence, 35\n\nLSD, 178, 199\n\nLuard, Evan, 89\n\nLuce, Clare Boothe, 67\n\nLuce, Henry, 118, 149\n\nLudendorff, Erich, 37\n\nLudlow Amendment, 73\n\nLusitania, 30\u201331\n\nMacArthur, Douglas, 120\u201321, 122, 127, 163, 197, 260, 264\n\nMcCain, John S., III, 312\u201314, 316\u201317\n\nMcCarthy, Eugene, 183, 185\u201387, 188, 196\n\nMcCarthy, Joseph, 10, 118\u201319, 121, 125, 160, 163, 175, 186\u201387, 191\n\nMcClellan, Scott, 341\n\nMcCloy, John, 182, 183\n\nMacDonald, Ramsay, 65\n\nMcFarlane, Robert, 224\n\nMcGovern, George, 190\u201393, 196\n\nMcGrory, Mary, 243\n\nMcKiernan, David, 358\n\nMcMahon, Brien, 93\n\nMacmillan, Harold, 149\n\nMcMillan, Margaret, 405 n\n\nMcNamara, Robert, 157, 178, 179\u201380, 258\n\nMaddox, U.S.S., 164\u201365\n\nMaginot Line, 56\n\nMahan, Alfred T., 68, 70\n\nMahdi Army, 364, 372\u201373\n\nMakiya, Kanan, 321\n\nMalcolm X (Malcolm Little), 256\n\nManchuria, 65, 66, 77\n\nManila, 262\n\nMansfield, Mike, 160, 188\n\nMao Zedong, 117\u201319, 121, 125, 127, 175, 380\u201381\n\nMarch on Washington (1963), 176\n\nMarcos, Ferdinand, 262, 304, 320\n\nMarshall, George, 109, 113, 119\n\nMarshall Islands, 176\u201377\n\nMarshall Plan, 103\u20134, 112\n\nMarx, Karl (Marxism), 79, 110, 153, 244\n\nMatsu, 129, 130, 165, 423n\n\nMead, Walter Russell, 465n\n\nMeat Inspection Act of 1914, 20\n\nMencken, H. L., 57, 67\n\nMercer, Lucy, 69, 92\n\nMerchants of Death (Englebrecht and Hanighen), 72\n\nMervin, David, 405n\n\nMexico, 27, 33\n\nMicronesia, 211\n\nmilitary tribunals, 370\n\nmillenarianism, 387\n\nMiller, Zell, 264\n\nMillis, Walter, 72\n\nMills, C. Wright, 139, 184\n\nMilo\u0161evic, Slobodan, 276, 277, 282\u201383, 285\u201388, 290, 308, 321, 348\n\nMiscamble, Wilson, 416n\n\nMission: Impossible III (movie), 369\n\nMitterrand, Fran\u00e7ois, 279\n\nMogadishu, 272, 273, 278\n\nMolotov, Vyacheslav, 85, 101\u20132, 416n\n\nMondale, Walter, 217, 223\n\nMonroe Doctrine, 28, 32, 233, 379\n\nMoral Man and Immoral Society (Niebuhr), 60\n\nMorgenthau, Hans, 48, 96\u201399, 169\u201371, 190, 238, 349\n\nMorgenthau, Henry, 103, 104\u20135\n\nMorris, Dick, 282\n\nMorris, Gouverneur, 19\n\nMoscow State University, 237\n\nMotherfuckers, 209\n\nMoynihan, Daniel Patrick, 208, 243\u201344\n\nmultilateralism, 261, 270, 279, 280, 350\n\nMumford, Lewis, 82\n\nMussolini, Benito, 65\n\nNation, The, 55, 57, 79\n\nNational Committee on the Cause and Cure of War (NCCCW), 59\n\nnational-greatness conservatives, 309, 311\u201314, 316\u201317, 328\n\nnational interest, 97, 301, 381\u201382\n\nNational Naval Medical Center (Bethesda), 371\n\nNational Press Club, 181\n\nNational Review, 224, 298, 334\n\nNational Security Council (NSC), 143, 172\u201373, 318\u201319\n\nReport 68, 122\u201324, 130, 140, 144, 182, 236\n\nNational Security Strategy, 343\n\nNational War College, 109, 110\u201311, 123\u201324\n\nnation building, 345, 359\u201361\n\nNATO (North Atlantic Treaty Organization), 104, 112, 114, 278\u201389, 328, 355, 379, 383\u201384\n\nNaval War of 1812, 69\n\nNazi Party (Nazism), 54\u201355, 65, 75\u201377, 81, 82, 96, 110, 184\n\nneoconservatism (neocons), 207\u201317, 293\u2013311\n\nAfghanistan and, 216\u201317\n\nCarter and, 204\u20135, 207\u20138, 211\u201316, 219\u201320\n\nCommunism and, 221\u201323, 227\u201330, 236\u201337, 305\u20136\n\ndominance ethic and, 295\u201396, 298\u2013311\n\nDPG statement and, 301\u20133\n\nIran hostage crisis and, 213\u201316\n\nIraq War and, 350\u201352, 366\u201368\n\nReagan and, 221\u201323, 227\u201330, 236\u201337, 248\u201350, 303\u20136\n\n\"neo-Reaganism,\" 303\u20136, 308\n\nnetroots, 370\n\nNeutrality Acts, 72\u201373, 76, 77\n\nNew America Foundation, 376\n\nNew Deal, 62, 79\n\nNew Left, 173\u201380, 184\u201387, 208\u201311, 219\u201320, 294\n\nNew Republic, 16, 18, 35, 40, 42, 239, 245, 279, 340\n\nNewsom, David, 203\n\nNew Statesmen, 135\n\nNewsweek, 111, 258, 353, 376\n\nNew York Herald Tribune, 138\n\nNew York Public Library, 17, 35, 85\n\nNew York Times, 111, 135\u201336, 146, 150, 153, 174, 184, 186, 199, 246, 278, 280, 289, 302, 318, 331, 333, 353, 370, 376\n\nNgo Dinh Diem, 151\u201352, 157\u201359, 164\n\nNgo Dinh Nhu, 158\n\nNicaragua, 206, 212\u201313, 223\u201326, 304, 305, 453n\n\nNicaraguan Contras, 225\u201326, 249, 322, 453n\n\nNicholas II of Russia, 33\u201334\n\nNiebuhr, Reinhold, 60\u201363, 79\u201381, 82, 277, 388\n\nDewey and, 60\u201363, 81\u201382, 408\u20139n\n\nhuman nature and evil, 5, 94\u201395, 177, 191\n\nKristol and Kirkpatrick and, 207, 210, 294\u201395\n\nVietnam War and, 169\u201370, 238\n\nWorld War I, 60\u201361\n\nWorld War II and, 79\u201381, 82\n\n9\/11 terrorist attacks (2001), 8, 327\u201332, 348, 382\n\nAfghanistan and, 330\u201336\n\nIraq War and, 337\n\nNitze, Paul, 122\u201323, 211\n\nNixon, Richard, 9, 127, 137\u201338, 193\u2013202, 205\u20136\n\nNORAD (North American Aerospace Defense Command), 232\u201333\n\nNoriega, Manuel, 248\u201354\n\nNorthern Alliance, 332\u201334, 339, 344, 461n\n\nNorth Korea, 119\u201322, 127\u201328, 130\u201331, 323\u201324, 383\n\nNovak, Robert, 204\u20135\n\nNSC. See National Security Council\n\nNSC Report 68, 122\u201324, 130, 140, 144, 182, 236\n\nnuclear deterrence, 382\u201383\n\nNuclear Test Ban Treaty (1963), 159\u201360\n\nnuclear weapons, 66\u201367, 93\u201394, 101\u20132, 129\u201330, 155, 174, 176\u201377, 231\u201337. See also weapons of mass destruction\n\nNunn, Sam, 347\n\nNye, Gerald, 72\n\nObama, Barack, 374, 384\u201385, 386\n\nO'Donnell, Kenneth \"Kenny,\" 146\n\nOffice for Reconstruction and Humanitarian Assistance (ORHA), 358\u201359\n\nOklahoma City bombing (1995), 330\n\nOlympic Games\n\n1936, 217\n\n1980, 216, 217\n\n1984, 223, 269\n\nO'Neill, Paul, 319, 338\n\nO'Neill, Thomas Phillip \"Tip,\" 226\u201327\n\nOpen Door at Home (Beard), 72\n\nOperation Desert Fox, 328\n\nOperation Iraqi Freedom, 355\n\nOperation Midnight Climax, 199\n\nOperation Mongoose, 155\u201356\n\nOPLAN 34A, 164\u201365\n\nOriental Seminary, 99\n\nOrlando, Vittorio Emanuele, 47\n\nOutlawry of War, 58\u201359\n\noverconfidence, as warning sign, 6\u20137\n\nOvid, 4\n\npacifism, 42, 43, 51, 59, 64, 71\u201372, 82\n\nPadilla, Frank, 354\n\nPaine, Thomas, 220\n\nPakistan, 216\u201317, 272, 336, 382\n\nPalestinians, 319, 366\n\nPanama, 248\u201354, 305, 322\u201323\n\nParis Peace Conference (1919), 43\u201348, 83, 402\u20133n\n\nparochialism, 21, 27, 388\n\nPartial Test Ban Treaty (PTBT), 159\u201360\n\nPartisan Review, 80\n\nPathet Lao, 424n\n\npatriarchal authority, 210\n\npatriotism, 386, 394\n\nPatten, Simon, 24\n\nPatton (movie), 197\n\nPeace Corps, 143\n\npeace dividend, 247\n\nPearl Harbor, 78\u201379, 93\n\nPepper, Claude, 93\n\nPerkins, Frances, 70\n\nPerle, Richard, 271, 301\u20133\n\nPerot, Ross, 281\n\nPersians, The (Aeschylus), 3, 11\n\nPetraeus, David, 372\u201374\n\nPew Research Center, 1\u20132, 292, 376\n\nPhiladelphia Public Ledger, 35\n\nPhillips, William, 80\n\nPinochet, Augusto, 304\n\nPipes, Richard, 211, 230\n\nPodhoretz, Norman, 293, 298\n\nCarter and, 211\n\n\"new isolationism\" and, 430n\n\nNicaragua and, 224\n\nPoland Solidarity and, 222\n\nReagan and, 222, 224, 228, 235\u201336, 305\u20136, 438n\n\nSoviet Union and, 235\u201336, 244, 305\u20136, 438n\n\ntraditional family values and, 210\n\nVietnam War and, 260\n\nPoland, 47, 56, 57, 88, 91\u201392, 93, 222, 245, 381\n\npolio, 70\n\nPollack, Kenneth, 374\n\nPoos, Jacques, 268\n\nPopular Front, 80\n\nPork Chop Hill, 127\n\nPort Huron Statement, 175, 178, 184, 428n\n\nPotomac Electric Power Company, 189\n\nPotsdam Agreement, 407n\n\n\"Pottery Barn rule,\" 345\n\nPowell, Colin, 317\u201320, 322\u201326\n\nAfghanistan and, 329\u201333\n\nBush, Jr., and, 317\u201320, 323\u201326, 329\u201333, 335\u201336, 345\u201347\n\nClark and, 282\u201383\n\nClinton and, 265\u201367, 269\u201372, 323\n\nGulf War and, 254\u201360, 297\n\nIraq War and, 345\u201347, 352\u201354, 356, 358\n\n9\/11 and, 329\u201332\n\nReagan and, 231\n\ntests for military intervention, 224, 272\u201373, 312\n\nWolfowitz and, 263, 322\u201323\n\nPowell, Jody, 204\n\nPowell Doctrine, 272\u201373\n\nPreminger, Otto, 83\n\nPresent Dangers (Kristol and Kagan), 306, 307\u20138\n\npresidential powers, 77, 165, 225, 288\n\npreventive (\"preemptive\") war, 10\u201311, 288, 299\u2013303, 307\u20138, 338\n\nPrinceton University, 22, 98, 109\n\nprogressivism, 17\u201329, 32, 39\u201342, 58, 62, 96\u201397\n\n\"the Inquiry,\" 17\u201318, 35\u201337\n\nProject for the New American Century (PNAC), 308\u20139, 324, 325\n\nPromote Liberty, 253\n\nPropaganda Commission, 37\n\nPT-109, 133\u201334\n\nPublic and Its Problems, The (Dewey), 184\n\nPublic Interest, 209\n\nPutin, Vladimir, 366\n\nQaeda, al. See Al Qaeda\n\nQuayle, Dan, 253, 304\n\nQuemoy, 129, 130, 165, 423n\n\nRadosh, Ronald, 184\u201385\n\nRahv, Philip, 80\n\nRambo (movie), 218, 219, 222\n\nRandolph Bourne Award, 82\n\nRandolph Bourne Institute, 368\n\nRapallo Treaty (1922), 53\n\nRaskin, Marcus, 172\u201374, 175, 177, 184, 191, 203, 368\n\nRauschenbusch, Walter, 18, 131, 191\u201392\n\nRayburn, Sam, 162\n\nReagan, Ronald, 218\u201339\n\nCentral American policy, 223\u201326, 248\u201352\n\nCommunism and, 221\u201323, 227\u201337\n\nevil and, 219\u201320, 221\n\nGrenada and, 227, 250\u201351\n\nKennan and, 238\u201339\n\nLebanon and, 226\u201327, 250\u201351, 305\u20136\n\nLibya and, 227\u201328\n\n\"neo-Reaganism,\" 303\u20136, 308\n\noptimism of, 220\u201321, 245\u201346\n\nPanama and, 248\u201354\n\nSoviet Union and, 228\u201337, 305\u20136, 381\n\nReagan Doctrine, 320\u201322, 332, 344\n\nReagan: The Movie, 219\n\nrealism, 70, 99, 131\u201332, 387\n\nreason, 17\u201318, 58\n\nregime change, 264, 306, 323\u201325\n\nreparations, 53\u201355, 406\u20137n\n\nResearch Institute on International Change, 214\n\nReservationists. See Senate \"Reservationists\"\n\nReston, James, 150\n\nReturn of History and the End of Dreams (Kagan), 367\u201368\n\nReuben James, U.S.S., 77\u201378\n\nReuther, Walter, 166\n\nReynaud, Paul, 76\n\nRhee, Syngman, 128\n\nRhineland, 44\u201345\n\nRice, Condoleezza, 319, 323, 346\n\nRice University, 138\n\nRicks, Thomas, 373\n\nRidge, Tom, 319\n\nRidgway, Matthew, 168\n\nRiesman, David, 132\n\nRise and Fall of the Great Powers (Kennedy), 246\u201347, 376\n\nRisse-Kappen, Thomas, 439n\n\nRoad to War (Millis), 72\n\nRockefeller, Nelson, 149\n\nRodman, Peter, 360\n\nRomania, 47, 57, 89, 321, 344\n\nRoosevelt, Eleanor, 68, 69\u201370, 79, 101\n\nRoosevelt, Franklin Delano, 67\u201379, 83\u201394, 104\u20135\n\nBeard and, 70\u201371, 72\n\ndeath of, 92\n\nFour Policemen idea, 85\u201390, 100, 103\n\nKennedy, Sr., and, 134\n\nNew Deal and, 62, 79\n\npostwar vision, 83\u201390\n\nquarantine speech, 73, 411n\n\nreelection campaign of 1944, 90\n\nSoviet Union and, 74, 76, 79, 86\u201393, 100\u2013101, 104\u20135, 381\n\nWilson and, 69, 72\u201373, 78, 83\u201384, 85, 138\n\nWorld War I and, 68, 73, 84\n\nWorld War II and, 9, 73\u201379, 81\u201382, 90, 100\u2013101, 103, 380\n\nYalta Conference, 90\u201392, 101\n\nRoosevelt, Theodore \"Teddy,\" 19\u201320\n\nFDR and, 67\u201368, 69, 84\n\nKennedy and, 133, 138\n\nLippman and, 16\n\nSenate \"Reservationists\" and, 48\n\nWilson and, 15\n\nWorld War I and, 29\u201331, 33\n\nRoosevelt Corollary, 379\n\nRosenberg, Ethel, 248\n\nRosenberg, Julius, 248\n\nRostow, Walt, 143\u201345, 149, 150, 152\u201354, 167, 180\n\nRoyal Lao Army, 148\n\nRubin, Robert, 290\n\nRudd, Mark, 209\n\nRuhr Valley occupation, 54\u201355, 56\n\nRumsfeld, Donald, 308\n\nAfghanistan and, 328, 330, 335\n\ndismissal of, 370\u201371\n\nIraq no-fly zone and, 319\u201320\n\nIraq regime change and, 323\u201325, 324\u201325\n\nIraq War and, 338\u201341, 344, 345, 354\u201360\n\nRusk, Dean, 146, 148, 159, 169, 179, 185, 195\n\nRussell, Bertrand, 81\n\nRussell, Richard, 167\n\nRussia, and NATO, 383\u201384\n\nRussian Revolution, 10, 33\u201334, 36\u201337, 43\u201344.\n\nSee also Soviet Union\n\nRwanda, 273\u201374, 276\u201377\n\nsacrifice, 104, 222\u201323, 309, 381\n\nSadr, Moqtada al-, 364, 372\u201373\n\nSt. George's Medical School (Grenada), 227\u201328\n\nSt. Lawrence Seaway, 118\n\nSALT II treaty, 217, 234\u201337, 239\n\nSaltonstall, Leverett, 165\n\nsanctions, 66, 214\n\nIraq and, 254\u201355, 257, 296\u201397, 319\n\nSandinistas, 206, 304, 453n\n\nSantayana, George, 16\n\nSarajevo, 268\u201369, 282, 283, 348\n\nSarkozy, Nicolas, 376\n\nSaturday Evening Post, 85\n\nSaturday Night Live (TV show), 261, 335\n\nSaudi Arabia, 254, 261, 328\n\nSaving Private Ryan (movie), 311\n\nSchlesinger, Arthur, Jr., 1\u20133, 131, 132\u201333, 137, 173, 309\n\nCuba and Bay of Pigs, 145\n\nGulf War and, 2, 258\n\nKennedy and, 145, 154, 157\n\nVietnam War and, 154, 157, 188\u201389\n\nThe Vital Center, 1\u20133, 94, 95, 276\n\nSchlesinger, Arthur, Sr., 378, 379\n\nScientific Man vs. Power Politics (Morgenthau), 96\n\nscientific peace, 17\u201318, 37\n\nScowcroft, Brent, 257, 258, 261, 271, 351\n\nScully, Matthew, 329\n\nself-interest, 18, 25\u201327, 62, 97, 105, 388\n\nSenate Armed Services Committee, 163\n\nSenate Foreign Relations Committee, 89, 185, 199, 243\n\nSenate \"Irreconcilables,\" 48, 50\u201352, 59, 405n\n\nSenate Judiciary Committee, 52\n\nSenate \"Reservationists,\" 48, 50\u201352, 405n\n\nSentinel anti-ballistic missiles, 188\n\nSeptember 11th terrorist attacks (2001), 8, 327\u201332, 348, 382\n\nAfghanistan and, 330\u201336\n\nIraq War and, 337\n\nSerbs (Serbia), 268\u201371, 276\u201388, 297, 313, 348\n\nSetser, Bradley, 375\n\nSeven Arts, 42\n\nsex instinct, 177\u201378\n\nShadid, Anthony, 363\n\nShah of Iran, Mohammad Reza Pahlavi, 115, 205\u20136, 212, 215\u201316\n\nShalikashvili, John, 282\n\nShelton, Henry, 330, 331\n\nSherwood, Robert, 83\n\nShia Muslims, 260\u201363, 297, 320, 327, 345, 364, 371\u201373\n\nShinseki, Eric, 344\u201345\n\nShipping Board, 23\n\nShultz, George, 222, 233, 262\n\nSimpson, O. J., 371\n\nSinclair, Upton, 20\n\nSistani, Sayyid Ali al-, 359\n\nSlovenia, 268\u201371\n\nSocial Darwinism, 18, 26\n\nSocial Gospel, 18, 131, 191\u201392, 221\n\nSolidarity (Polish trade union), 222\n\nSomalia, 205, 271\u201373, 278\n\nSomoza, Anastasio, 206, 212, 215, 304\n\nSophocles, 124\n\nSorensen, Theodore, 201\n\nSosa, Sammy, 315\n\nSouthern Christian Leadership Conference, 176\n\nSouth Korea, 119\u201322, 127\u201328, 130\u201331, 202\n\nSouthwest Texas State Teachers' College, 165\n\nSoviet Union\n\nAfghanistan and, 206, 216\u201317, 331\n\nCarter and, 202, 205\u20137, 214\u201315, 221\u201322\n\nEastern Europe and, 87, 89\u201393, 101\u20132, 111\u201312\n\nfall of Berlin Wall, 244\u201345\n\nFour Policemen idea, 85\u201390, 100, 103\n\nFukuyama and, 244\u201345\n\nIraq and, 255\n\nKennan and, 10, 99\u2013100, 102\u20133, 109\u201317, 157, 238\u201339, 243\u201344\n\nKennedy and, 136\u201338, 143, 149\u201351, 154\u201357, 159\u201360\n\nKorean War and, 128\n\nneoconservatism and, 221\u201323, 227\u201330, 236\u201337, 305\u20136\n\nNew Left and, 174\u201375, 189, 210\n\nNixon and, 137\u201338, 195\u201396, 202\n\nReagan and, 221\u201323, 228\u201337, 305\u20136\n\nspace race, 136\u201337\n\nTruman and, 101\u20133, 114\u201316\n\nWorld War II and, 74, 76, 79\n\nYalta Conference, 90\u201392, 101\n\nspace race, 136\u201337\n\nSpadafora, Hugo, 249\n\nSpanish-American War, 9, 68, 401n\n\nSpanish Civil War, 276\n\nSpecial Forces, U.S., 143\u201344\n\nSports Illustrated, 138\n\nSputnik, 136\u201337\n\nSrebrenica, 280\u201382, 313\n\nStages of Economic Growth (Rostow), 153\n\nStalin, Joseph, 80, 82, 86\u201391, 110\u201311, 210\n\nFour Policemen idea, 86\u201390\n\nKorean War and, 121, 128\n\nYalta Conference, 90\u201391, 101\n\nStallone, Sylvester, 219, 235\n\nStanford University, 63\n\nStarobin, Paul, 376\n\n\"Star Wars,\" 231\u201333, 236\n\nState Department, U.S., 57, 88, 109, 120, 143, 145\n\nSteffens, Lincoln, 16, 62, 378\n\nStempel, Adam, 397n\n\nStevenson, Adlai, 137\n\nStevenson, Coke R., 163\n\nStimson, Henry, 65, 66, 77\n\nStoller, Matt, 370\n\nStudent Nonviolent Coordinating Committee (SNCC), 176, 178\n\nStudents for a Democratic Society (SDS), 175, 176, 178\u201389, 208\u20139\n\nsubmarines, 30, 31, 32, 234\n\nSudetenland, 74\u201375\n\nSummers, Lawrence, 290\n\nSumner, William Graham, 1, 10\n\nSussex, 32\n\nSymington, Stuart, 199\n\nSyria, 10, 226, 263\u201364, 366\n\nSyriana (movie), 369\n\nTaft, Robert, 127\n\nTaliban, 327\u201336, 330\u201335, 338\u201339, 348\u201349, 373\u201374, 461n\n\nTardieu, Andr\u00e9, 67\n\nTariff Commission, 24, 27, 35\n\ntariffs, 23, 36, 329\n\nTarnoff, Peter, 268\n\nTaylor, Maxwell, 128, 152\u201354, 429n\n\nTenet, George, 356\n\nterrorism. See Al Qaeda; jihadist terrorism; war on terror\n\nTet offensive, 181, 182, 213\u201314\n\nTexas National Guard, 314\n\nTexas Rangers, 315\n\nTheodore Roosevelt, U.S.S., 267, 283\n\nThompson, John, 25\n\nTiananmen Square protests of 1989, 245\n\nTime, 94, 118, 149, 166, 271, 290\n\nTonkin Gulf, 164\u201366\n\nTonkin Gulf Resolution (1964), 165\n\nTora Bora, 336\n\ntorture, 336, 370\n\ntotalitarianism, 381\n\nBerman and, 348\u201349\n\nBush and, 342\u201343\n\nKennedy and, 135\n\nKristol and Kirkpatrick and, 294\n\nNiebuhr and, 79\n\nNixon and, 197\n\nReagan and, 245\u201346\n\ntoughness ethic, 175\u201376, 229, 245\u201346\n\nJohnson and, 5, 162, 166, 180\u201383, 210\u201311\n\nKennedy, John F., and, 132\u201333, 137\u201340, 148, 156\u201357, 162, 210\u201311\n\nKennedy, Robert, and, 186\n\nneoconservatism and, 207, 210\u201311, 220, 229\n\nNixon and, 193\u201394\n\nNSC Report 68, 122\u201324\n\nTragedy of American Diplomacy (Williams), 174\u201375\n\nTrainor, Bernard, 443n\n\nTrautman, Samuel, 218\n\nTreaty of Versailles. See Versailles Treaty\n\nTrilling, Lionel, 95, 207\n\nTrotskyists, 207, 294, 433n\n\nTruman, Harry, 101\u20134, 114\u201324\n\nChina and, 116\u201322\n\ncontainment policy, 102\u20134, 114\u201319, 123\u201324\n\nGreece and Turkey speech, 115\u201319, 154\n\nKorean War and, 119\u201322, 123, 381\n\nSoviet Union and, 101\u20133, 114\u201316\n\nTuchman, Jessica, 275\n\nTulip Revolution, 366\n\nTurkey, 115\u201318, 121, 156\u201357\n\nTurner, Frederick Jackson, 35\n\nTurner Joy, U.S.S., 164\u201365\n\nTutsi, 273\u201374, 276\u20137724 (TV series), 369\n\nUgly American, The (Burdick and Lederer), 141\u201342, 199\n\nUkraine, 296, 383\u201384\n\nultra-realism, 190, 197\u201398\n\nUncertain Trumpet, The (Taylor), 154\n\nunilateralism, as warning sign, 7\n\nUnitarianism, 221\n\nUnited Auto Workers (UAW), 166, 175\n\nUnited Nations (UN), 88\u201394\n\narms embargos, 269, 280, 281, 283, 297\n\nBalkans and, 278\u201382, 284\u201385, 287\n\nGeneral Assembly, 88\u201389, 91, 93\n\nHuman Rights Commission, 350\n\nIraq War and, 346, 350, 353\u201354\n\nLeague of Nations compared with, 88\n\nopening ceremony, 92\u201393\n\npeacekeeping missions, 446n\n\nSecurity Council, 88\u201389, 93, 278\u201379, 287, 346, 353\u201354, 389\n\nUniversal Declaration of Human Rights, 93\n\nVietnam War and, 169\n\nUnited World Federalists, 190\u201391, 231\n\nUniversal Declaration of Human Rights, 93\n\nuniversal foreign policy doctrine, 10\u201311\n\nuniversalism, 99\u2013100, 104, 112\n\nUniversity of Virginia, 22\n\nU.S. News & World Report, 111\n\nU.S.S.R. See Soviet Union utopianism, 79\u201380, 184, 208\u20139, 211, 294\u201395, 308\n\nVance, Cyrus, 214, 215\n\nVandenberg, Arthur, 89, 114\n\nVeblen, Thorstein, 18, 35, 400n\n\nV\u00e9drine, Hubert, 331\n\nVersailles Treaty (1919), 47, 48, 50\u201354, 56, 64\u201365, 83, 406\u20137n\n\nVietnam War\n\nBosnia compared with, 2, 270\n\nClinton and, 266\u201367\n\nGulf War compared with, 258\u201359\n\nJohnson and, 8, 10, 162, 163\u201371, 178\u201384, 238\n\nKennedy, John F., and, 151\u201354, 157\u201359, 163\u201364\n\nKennedy, Robert, and, 11, 179, 180, 186\u201387\n\nKristol, Irving, and, 208\u20139\n\nMcCain and, 312, 314\n\nNew Left and, 184\u201385, 188\u201389\n\nNixon and, 197\u201398\n\nReagan and, 218\u201319, 222\n\nvigor (virility), 132\u201333, 137\u201340\n\nVillard, Oswald Garrison, 55\n\nVital Center, The (Schlesinger), 1\u20133, 94, 95, 276\n\nWalesa, Lech, 245, 344\n\nWallace, Henry, 190\u201391\n\nWall Street, 375\u201376\n\nWall Street Journal, 73, 211, 225, 236\n\nWalters, Barbara, 202\u20133\n\nWang Qishan, 376\n\nWar Games (movie), 232\n\nwarning signs, 6\u20139\n\nWarnke, Paul, 189\u201391, 201\n\nwar on terror, 329, 349, 351\u201352, 370, 380\n\n\"war-technique,\" 368\u201369\n\nWashington Naval Conference, 57\n\nWashington Post, 161, 181, 204, 243, 318, 323, 353, 363\n\nWatergate scandal, 199\n\nweapons of mass destruction (WMD), 301\u20133, 307, 319, 328, 337, 341, 352\u201353, 365, 369, 457\u201358n\n\n\"weapons states,\" 298\u2013300, 307\n\nWeekly Standard, 306, 308, 309, 310, 313\u201314, 325, 333, 335, 345, 355, 361\n\nWeinberger, Caspar, 224\u201325, 226, 256\n\nWelles, Sumner, 85\u201386, 413n\n\nWells, H. G., 38\n\nWest Berlin, 116, 150, 154\u201355, 174, 244\u201345\n\nWestbrook, Robert, 368\u201369, 408\u20139n\n\nWestmoreland, William, 168\u201369, 181\u201382\n\nWhile England Slept (Kennedy), 135\u201336\n\nWhite, Theodore, 191\n\nWhite Citizens' Council (WCC), 200\n\nWhite House Correspondents' Association Dinner, 370\n\nWhyte, William, 132\n\nWidenor, William, 405n\n\nWiesel, Elie, 280\n\nWieseltier, Leon, 279\n\nWilder, Gene (Jerry Silberman), 173\n\nWilhelm I (the Great), 43\n\nWilkerson, Lawrence, 353\u201354\n\nWill, George, 204\u20135, 220, 222, 235, 236, 243, 298, 367, 449n\n\nWillett, Edward, 110\u201311\n\nWilliams, William Appleman, 174\u201375, 184\n\nWilson (movie), 83, 86\n\nWilson, Edith, 50\u201351\n\nWilson, Edmund, 62\n\nWilson, James Q., 298\n\nWilson, Woodrow, 15\u201317\n\nJohnson and, 183\n\nKennan and, 99, 243\u201344\n\nLatin America policy, 26\u201329\n\nLeague of Nations and, 45\u201346, 47\u201351\n\nParis Peace Conference, 43\u201344\n\nprogressivism of, 20\u201325, 35\u201337\n\nWorld War I and, 7, 16\u201317, 19, 25\u201338, 61, 72\u201373, 83\n\nWirthlin, Richard, 222\n\nWMD. See weapons of mass destruction\n\nWolfowitz, Paul, 211, 308, 317\n\nDPG statement and, 301\u20133\n\nFukuyama and, 351\u201352\n\nGulf War and, 254\u201355, 261\u201363\n\nIraq and, 319\u201322, 324\u201325\n\nIraq War and, 344\u201345, 356, 359\u201360, 458n\n\n9\/11 and, 329\u201330, 351\u201352\n\nWoodward, Bob, 316, 340\n\nWoodward, C. Vann, 390\n\nWoolsey, James, 281\n\nWorld Bank, 290, 389\n\nWorld Trade Center attack (2001), 8, 327\u201332, 348, 382\n\nAfghanistan and, 330\u201336\n\nIraq War and, 337\n\nWorld Trade Center bombing (1993), 330\n\nWorld War I\n\nBeard and, 40\u201343, 51\u201353\n\nBourne and, 39\u201342\n\nDewey and, 40\u201341, 51\n\nFDR and, 68, 73, 84\n\nHoover and, 63\u201364\n\nKennan and, 238\u201339\n\nNiebuhr and, 60\u201361\n\nreparations, 54, 406\u20137n\n\nVersailles Treaty, 47, 48, 50\u201354, 56, 64\u201365, 83\n\nWilson and, 7, 16\u201317, 19, 25\u201338, 47\u201348, 61, 72\u201373, 83\n\nWorld War II, 73\u201382, 381\n\nChurchill and, 76, 78, 79, 90\u201391, 92, 126\u201327, 134\n\nEisenhower and, 126\u201327\n\nJohnson and, 162\n\nKennedy, John F., and, 133\u201334, 135\u201336\n\nKennedy, Sr., and, 134\u201335\n\nKristol, Irving, and, 207\n\nLippmann and, 81, 97\u201398\n\nMcGovern and, 192\u201393\n\nNiebuhr and, 79\u201381, 82\n\nnostalgia for, in late 1990s, 310\u201311\n\nreparations, 54, 406\u20137n\n\nRoosevelt and, 9, 73\u201379, 81\u201382, 90\u201394, 100\u2013101, 103, 380\n\nYalta Conference, 90\u201392, 101\n\nXerxes, 3, 5, 11\n\nYale Divinity School, 60\n\nYalta Conference, 90\u201392, 101\n\nYenching University, 117\n\nYoung, Andrew, 201\n\nYugoslavia, 57, 89, 267, 268\u201371, 278\u201388\n\nZakaria, Fareed, 376\n\nZero Hour: Summons to the Free (Bundy), 167\n\nZimmermann, Arthur (Zimmermann Telegram), 33\n\n## ACKNOWLEDGMENTS\n\nWhile writing this book, I often felt that I was living my subject\u2014that I had embarked upon my own personal act of hubris. Again and again, I wandered into an intellectual thicket, lost my way, and called out for help. This book only exists because brilliant and generous people\u2014many of them initially strangers\u2014answered my pleas.\n\nI met the first Good Samaritan, ironically, at a bar mitzvah. The University of Virginia's Melvyn Leffler is among the most eminent historians of twentieth-century American foreign policy alive. I met him over hors d'oeuvres at a South Carolina synagogue, and grasping this thin reed, asked if he would chair the Council on Foreign Relations study group for my book in Washington, D.C. Not only did he agree, but he undertook the task with extraordinary diligence\u2014repeatedly driving up from Charlottesville, reading successive drafts, suggesting numerous sources, painstakingly correcting my mistakes, and allowing me to see, up close, how a great historian thinks. In return, he got a couple of meals at Asian restaurants, and this paragraph of thanks, which barely begins to repay my debt of gratitude.\n\nWhat Mel Leffler is to the historical study of American foreign policy, Fareed Zakaria is to public commentary on American foreign policy: the best. When I asked him to chair the study group for my book in New York, he had every reason to say no, given the crushing demands on his time. He said yes because he found the subject interesting, and because he is a mensch. And every time we discussed the book, he asked the big, hard questions that forced me to dig myself out from underneath the avalanche of historical minutiae and figure out what I was trying to say.\n\nLloyd Ambrosius of the University of Nebraska and David Steigerwald of Ohio State, two men I e-mailed cold and still have never met in person, devoted hours to helping me understand Woodrow Wilson. American University's Robert Beisner corrected my misperceptions of the Truman years, while his wife, the classicist Valerie French, offered insights into the Greeks. I spent several long afternoons with George Washington University's James Hershberg, shamelessly cribbing from his encyclopedic knowledge of Kennedy, Johnson, and Vietnam. David Greenberg of Rutgers tutored me on Nixon, Reagan, and everything in between. Princeton's Gary Bass taught me most of what I know about humanitarian intervention in the 1990s. Elliot Leffler was my guide to Aeschylus. Robert Brustein warned me against lazy writing and buoyed my spirits with his praise. Christopher Chen and Adam Stempel helped me confirm a hunch about the generational skew among politicians and pundits over the Iraq War. Alexander Star, a brilliant editor, and Amanda Fazzone, a brilliant copyeditor, each read the manuscript with great care and skill. Columbia's Robert Jervis suggested dozens of useful modifications and sources. Stephen Winch generously supported my work. So did Roger Hertog, who despite considering me hopelessly left-wing, has for many years been a cherished professional and personal guide.\n\nIn addition, I worked on this project with three remarkable recent college graduates, who served, in succession, as my research assistants. The first was William Evans, a political junkie with the soul of a scholar who has forgotten more about Theodore Roosevelt than I will ever know. The second was Conor Savoy, who became an invaluable partner in trying to understand the Vietnam years, and was a delight to have around. The third was Jamie Holmes, who worked long and sometimes frantic days in the project's final months, and exhibited an astounding level of organizational ability, analytical skill, determination, and good humor. Between them, Will, Conor, and Jamie shaped every page of this book and made my work life a pleasure for three years. I will watch with pride the great things they do in the years ahead.\n\nIn molding their research into a book, I was extremely lucky to work, once again, with my agent, Tina Bennett, and with Tim Duggan at HarperCollins. At critical moments, I relied on Tina's signature combination of intellectual sophistication and professional savvy. And Tim shepherded me through the writing and editing process with insight, diligence, and an infectious enthusiasm. Allison Lorentzen at HarperCollins patiently and expertly answered dozens of e-mails.\n\nMany other people also gave generously of their time and talent. They include Peter Ackerman, Spencer Ackerman, Madeleine Albright, Eric Alterman, William Antholis, Jeremy Bash, Robyn Bash, Warren Bass, Michelle Baute, Amanda Beck, Rand Beers, Daniel Benjamin, Paul Berman, Casey Blake, John Morton Blum, Laura Blumenfeld, Nina Blustein, Stacy Bosshardt, Spencer Boyer, Nell Breyer, Howard Brick, Clive Brock, Christian Brose, Tina Brown, Frank Calzon, Derek Chollet, Conor Clarke, Steve Clemons, Warren Cohen, Steven Cook, Matthew Continetti, Kristin Cullinson, Greg Craig, Robert Dallek, Jacquelyn Davis, Joy Demenil, James Denton, Alan Dershowitz, Carolyn Dershowitz, John Patrick Diggins, E. J. Dionne, Trish Dorff, Ross Douhat, Colin Dueck, Edward Felsenthal, Michael Flamm, Nic Fokas, Richard Fox, Annabelle Friedman, Caroline Friedman, Peter Friedman, David Frum, Francis Fukuyama, John Lewis Gaddis, Adam Garfinkle, Carl Gershman, Allen Gerson, Mark Gerson, Todd Gitlin, James Goldgeier, Mahen Gunaratna, Jonah Goldberg, Andrew Gundlach, Leigh Gusts, Jacquelyn Hardy, Breton Harned, Owen Harries, Marie Hastreiter, Melvin Heineman, Michael Hirsh, Richard Holbrooke, Peter Holquist, Deneen Howell, Isabell Hull, Lena Hull, Heather Hurlburt, Walter Isaacson, Maurice Isserman, Alan Jones, Katherine Jordan, John Judis, Erez Kalir, Lawrence Kaplan, Paul Kennedy, Thomas Kleine-Brockhoff, David Knoll, William Kristol, Charles Kupchan, Mark Lagon, Julie Lascar, Nicholas Lemann, Meagan Liaboe, Robert Lieber, Matt Lieppe, Kelli Long, Oliver Lough, Brian Lowe, Marc Lynch, Joe Eugene McCarraher, Joseph McCartin, Cara McCarty, Doyle McManus, Joe McReynolds, Walter Russell Mead, Dmitri Mehlhorn, Shelton Metcalf, Susan Moeller, Sylvia Moss, John Mueller, Siddhartha Mukherjee, Donald Mullen, Joshua Muravchik, Gita Murthy, Henry Nau, Richard Niemi, Lia Norton, Brendan O'Connor, Dan O'Hara, Priscilla Painton, Davide Panagia, Richard Pells, Richard Plepler, Ramesh Ponnuru, Sasha Pulakow-Suransky, Romesh Ratnesar, Jonathan Rauch, Kathy Reich, John Renehan, Berenice Ronthal, Lamiya Rahman, Marcus Raskin, Jonathan Rauch, Patrick Rigby, Gideon Rose, Mark Schmitt, Howard Schuman, Peter Scoblic, Stephen Sestanovich, Brad Setser, Aaron Silverman, Anne-Marie Slaughter, David Sloan, Michael Sofer, Michael Smith, Tom Smith, Jose Sorzano, Marcia Sprules, Connie Stagnaro, Allan Stam, Paul Starobin, Alexandra Starr, Sally Steinberg, Craig Stern, Strobe Talbott, Ray Takeyh, Paul Testa, Francoise Thiese, Nicholas Thompson, Stephen Trachtenberg, Nicholas Valentino, Reed Van Beveren, Don Vieira, Peter Wallsten, Baruch Weiss, Maureen White, Steven Whitfield, Leon Wieseltier, George Will, Marshall Wittmann, Adam Wolfson, Matthew Yglesias, and Barron YoungSmith. Needless to say, the book's flaws are not theirs, but mine.\n\nBeyond relying on exceptional individuals, I relied on exceptional institutions. The Council on Foreign Relations, where I was a Senior Fellow from 2007 to 2009, is a phenomenal place to write a book. The only problem with being surrounded by excellent facilities, skilled librarians, stimulating colleagues, and free food is that if your book is lousy, you have no one to blame but yourself. For creating this atmosphere, and allowing me to partake in it, I am grateful to Richard Haass, Gary Samore, James Lindsay, Nancy Roman, and Kay King. I am also grateful to the Smith-Richardson Foundation, and particularly Mark Steinmeyer, for supporting my work at CFR.\n\nI completed the book at the New America Foundation, where I am privileged to be a Schwartz Senior Fellow. New America is a vibrant, iconoclastic, entrepreneurial place, and the atmosphere is infectious. I am grateful to Steve Coll and Andr\u00e9s Martinez for giving me the chance to come there and to Faith Smith for making me feel so welcome. As this book was winding down, I also began an association with the City University of New York, where I am Associate Professor of Journalism and Political Science, and my bosses there, Stephen Shepard, Judith Watson, Matthew Goldstein, William Kelly, Joe Rollins, and Ruth O'Brien, have been extremely supportive as well.\n\nNow the hard part: my family. My parents, Doreen Beinart and Julian Beinart, read the manuscript and offered comments and encouragement. But they also contributed to the book in a more fundamental way. As a child, I saw America through their immigrant eyes, and glimpsed the combination of profound gratitude and ironic detachment they felt toward their adopted land. From them, I learned that irony is not the enemy of devotion. To the contrary, a willingness to chuckle occasionally at national pieties keeps patriotism from becoming unthinking and insincere. The American right has long believed that seeing America ironically makes America weaker. But long before I first encountered the writings of Reinhold Niebuhr, my parents taught me what he taught the nation: that the root of the word irony is iron.\n\nMy sister, Jean Stern, provided advice on graphic elements of the book. But, more important, she provided what she has always provided: wisdom and love. One of my greatest hopes for my own son is that he gains as much joy from his sister as I have gained from my mine.\n\nMy wife's parents, Marlene and Arthur Hartstein, have\u2014since I first married into their family\u2014treated me like a son. And I've relied particularly heavily on their generosity and affection during the writing of this book. My sister-in-law, Lieutenant Colonel Bonnie Hartstein, and her husband, Gal Shweiki, contributed to the book as well, mostly through the power of their example. Since September 11, 2001, they have lived the history that I only write about. I only wish all of America's wartime leaders had handled their burdens with as much integrity, dedication, and grace.\n\nThere is simply no way I can adequately thank my wife, Diana Hartstein Beinart, for everything she has endured over the past three years. Writing a book is like inviting an old college buddy to live in your house. You see him as a pleasant, stimulating companion; your spouse ends up picking up his sweat socks. Day after day, in the most mundane ways and in the most profound, Diana has shouldered the burdens that come with young children, an old house, and an addled husband, all the while excelling in her own, very demanding, career. And she's done so with the same warmth, determination, and charisma that made me fall in love with her almost a decade ago.\n\nFinally, this book is dedicated to the smallest and biggest people in Diana's and my lives: Ezra Beinart, age four, and Naomi Beinart, age two. I used to think that writing books gave me joy. Now I have a better understanding of the word. A while back, Ezra informed me that he was writing a book of his own. When I asked what it was about, he replied, \"Orange hubris.\" How lucky I am, I thought. My days of thinking about hubris are almost over. But my days of thinking about orange hubris have only just begun.\n\n## About the Author\n\nPETER BEINART is Associate Professor of journalism and political science at the City University of New York and a Senior Fellow at the New America Foundation. He is the Senior Political Writer for The Daily Beast and a contributor to Time. Beinart is a former fellow at the Council on Foreign Relations and is the author of The Good Fight. He lives with his family in Washington, D.C.\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.\n\n## Credits\n\nJacket design by Jarrod Taylor\n\n## Copyright\n\nTHE ICARUS SYNDROME. Copyright \u00a9 2010 by Peter Beinart. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nPublished in cooperation with the Council on Foreign Relations.\n\nThe Council on Foreign Relations (CFR) is an independent, nonpartisan membership organization, think tank, and publisher dedicated to being a resource for its members, government officials, business executives, journalists, educators and students, civic and religious leaders, and other interested citizens in order to help them better understand the world and the foreign policy choices facing the United States and other countries. Founded in 1921, CFR carries out its mission by maintaining a diverse membership, with special programs to promote interest and develop expertise in the next generation of foreign policy leaders; convening meetings at its headquarters in New York and in Washington, DC, and other cities where senior government officials, members of Congress, global leaders, and prominent thinkers come together with CFR members to discuss and debate major international issues; supporting a Studies Program that fosters independent research, enabling CFR scholars to produce articles, reports, and books and hold roundtables that analyze foreign policy issues and make concrete policy recommendations; publishing Foreign Affairs, the preeminent journal on international affairs and U.S. foreign policy; sponsoring Independent Task Forces that produce reports with both findings and policy prescriptions on the most important foreign policy topics; and providing up-to-date information and analysis about world events and American foreign policy on its Web site, www.cfr.org.\n\nThe Council on Foreign Relations takes no institutional position on policy issues and has no affiliation with the U.S. government. All statements of fact and expressions of opinion contained in its publications are the sole responsibility of the author or authors.\n\nFIRST EDITION\n\nLibrary of Congress Cataloging-in-Publication Data has been applied for.\n\nEPub Edition \u00a9 April 2010 ISBN: 978-0-06-199803-4\n\n10 9 8 7 6 5 4 3 2 1\n\n## About the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\n25 Ryde Road (PO Box 321)\n\nPymble, NSW 2073, Australia\n\nhttp:\/\/www.harpercollinsebooks.com.au\n\nCanada\n\nHarperCollins Publishers Ltd.\n\n55 Avenue Road, Suite 2900\n\nToronto, ON, M5R, 3L2, Canada\n\nhttp:\/\/www.harpercollinsebooks.ca\n\nNew Zealand\n\nHarperCollinsPublishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\nhttp:\/\/www.harpercollins.co.nz\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\nhttp:\/\/www.harpercollinsebooks.co.uk\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\nhttp:\/\/www.harpercollinsebooks.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Enniskillen rugby club's U15 and U18 teams produced two superb performances away to Letterkenny at the weekend, winning 24-10 and 64-0 respectively. The U15 team have already finished second in their league, and began their cup competition with a strong performance against a powerful Letterkenny side. Outstanding tackling from sisters Abbie and Sophie Barrett, and Anna Hughs, thwarted many attacks from the Letterkenny side. The skins forwards, in turn, made good ground with strong carries from Megan Graham, Jenna Love, Lauren Little and Rosie Daley. India Daley produced outstanding power and speed in the backs to burst through the opposition defence for tries, with Michelle Elliott providing great support on the wing. Special credit must be given to Caitlyn Leonard, playing only her second match for the U15 team, as she makes the transition up from mini rugby. Caitlyn produced an incredible performance, especially at the breakdown where she turned over numerous ball to win her team some possession.\nThe U18 team won 64-0, but the score does nor reflect the courage and commitment of the Letterkenny girls who fought hard throughout and made skins work for every point. To win so handsomely does reflect the organisation of the skins side who played as a coordinated team throughout. The forwards worked their pods over numerous phases, creating a platform for the backs to attack. Powerful running with fluid passing along the backs resulted in well worked tries for Ellie Rix and Rebecca Pennell. Dynamic flanker performances from Katie and Claire Boles kept the skins going forward constantly, and Claire ran in five tries rampaging through midfield. The unsung heroines, as always, were forwards Tammy Millar, Ciara Killile and Amy McNulty, supported at scrum half by Ellie-Mae Maguire. Dependable, rock solid and ever present this season from these girls who are the engine room of the team. Their organisation of the forward pods, clinical line-out work and scrummaging are outstanding and the side continues to go from strength to strength.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VALKYRIE COSTUME dress with attached cape and bandolier, gauntlets, arm bands, and belt . Sleeveless mini dress with printed armor, velvet panels on skirt, and stand-up foam collar fastens with Velcro at back of neck; foam bandolier fastens with metal buckle . IMPORTANT SIZE INFORMATION: Costumes are sized differently than clothing, important to review SECRET WISHES size chart before making selection . OFFICIALLY LICENSED Marvel costume, items shipped and sold by Amazon are guaranteed to be authentic . Add friends in Thor, Loki, Hela, or Hulk costumes to create a fun group look; Weapons, boots, and other accessories available separately .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A hand-finished, black antique, wrought iron fanlight catch, manufactured in a traditional English ironfoundry and available either recessed or flush fitting.\nAvailable either recessed or flush fitting.\nA hand-finished, black antique, malleable iron fanlight catch, manufactured in a traditional English ironfoundry and available either recessed or flush fitting.\nPlease note, we usually have these fanlight catches made up to size on ordering. As such, please allow approximately 7 days for dispatch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Fighting Stallions statue is located on the west end of the Quad in the plaza between Derrick Hall and Evans Liberal Arts Building. The 17-foot-high sculpture was a 1951 gift from noted sculptor Anna Hyatt Huntington and husband Archer Huntington of South Carolina. Texas State students have made it a tradition to rub the stallions before exams for luck.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In June last year, the Russian Ministry of Defense claimed that according to information, the leader of the Islamic State Abu-Bakr Al-Baghdadi had reportedly been killed as a result of airstrikes conducted by the Russian aircrafts on a southern suburb of Raqqa on May 28.\nAccording to Russian claims, the airstrikes targeted a meeting of high-ranking Islamic State leaders where Al- Baghdadi was reportedly present. The meeting was gathered to plan exit routes for militants from Raqqa. Apart from Al-Baghdadi, 30 field commanders and up to 300 militants were also killed in the airstrike.\n\"The head of Islamic State sustained serious wounds to his legs during air raids,\" Abu Ali al-Basri, Iraq's intelligence and counterterrorism department chief, was quoted last month by the Iraqi government-run al-Sabah daily as saying. \"Al-Baghdadi suffers from injuries, diabetes and fractures to the body and legs that prevent him from walking without assistance,\" said al-Basri.\nAlthough al-Baghdadi has not publicly appointed a successor, two of the closest aides who have emerged as his likely successors are Iyad al-Obeidi, his defense minister, and Ayad al-Jumaili, the in charge of security. The latter had already reportedly been killed in an airstrike in April last year in al-Qaim region on Iraq's border with Syria.\nTherefore, the most likely successor of al-Baghdadi would be al-Obaidi. Both al-Jumaili and al-Obeidi had previously served as security officers in Iraq's Baathist army under Saddam Hussein, and al-Obeidi is known to be the de facto deputy of al-Baghdadi.\nExcluding al-Baghdadi and some of his hardline Islamist aides, the rest of Islamic State's top leadership is comprised of Saddam-era military and intelligence officials. Hundreds of ex-Baathists reportedly constitute the top and mid-tier command structure of the Islamic State who plan all the operations and direct its military strategy.\nApart from training and arms that have been provided to militants in the training camps located in Turkey's and Jordan's border regions adjacent to Syria by the CIA in collaboration with Turkish, Jordanian and Saudi intelligence agencies, the only other factor which contributed to the astounding success of the Islamic State from early 2013 to August 2014 is that its top cadres are comprised of professional military and intelligence officers from the Saddam era.\nMoreover, it is an indisputable fact that morale and ideology play an important role in the battle, and well-informed readers must also be aware that the Takfiri brand of most jihadists these days has directly been inspired by the puritanical Wahhabi-Salafi ideology of Saudi Arabia, but ideology alone is not sufficient to succeed in the battle.\nLooking at the Islamic State's spectacular gains in Syria and Iraq from early 2013 to August 2014, a question naturally arises that where did its recruits get all the training and state-of-the-art weapons that are imperative not only for hit-and-run guerrilla warfare but also for capturing and holding large swathes of territory?\nThe Syria experts of foreign policy think tanks also appeared to be quite 'worried' when the Islamic State overran Mosul in June 2014 that where did the Islamic State's jihadists get all the sophisticated weapons and especially those fancy Toyota pickup trucks mounted with machine guns at the back, colloquially known as the 'Technicals' among the jihadists?\nAccording to a revelatory December 2013 news report  from a newspaper affiliated with the UAE government which supports the Syrian opposition, it is clearly mentioned that along with Kalashnikovs, rocket-propelled grenades and other military gear, the Saudi regime also provided machine gun-mounted Toyota pick-up trucks to every batch of five jihadists who had completed their training in the training camps located in the border regions of Jordan.\nOnce those militants crossed over to Daraa and Quneitra in southern Syria from the Jordan-Syria border, then those Toyota pickup trucks could have easily traveled to the Islamic State's former strongholds in Syria and Iraq. Furthermore, it is clearly spelled out in the report that Syrian militants got arms and training through a secret command center known as the Military Operations Center (MOC) based in the intelligence headquarters' building in Amman, Jordan, that was staffed by high-ranking military officials from 14 countries, including the US, European nations, Israel and the Gulf Arab States to wage a covert war against the government in Syria.\nMore recently, however, a report by the Conflict Armament Research (CAR) on the Islamic State's weapons found in Iraq and Syria has been doing the rounds on the media during the last few months. Before the story was picked up by the mainstream media, it was first published  in the Wired News on December 12, which has a history of spreading dubious stories and working in close collaboration with the Pentagon and DARPA (Defense Advanced Research Projects Agency).\nThe Britain-based Conflict Armament Research (CAR) is a relatively unknown company of less than 20 employees. Its one-man Iraq and Syria division is headed by a 31-year-old Belgian researcher, Damien Spleeters. The main theme of Spleeters' investigation was to discover the Islamic State's homegrown armaments industry and how the jihadist group's technicians have adapted the East European munitions to be used in the weapons available to the Islamic State. He has listed 1,832 weapons and 40,984 pieces of ammunition recovered in Iraq and Syria in the CAR's database.\nIn fact, the CAR's report is so misleading that of thousands of pieces of munitions investigated by Spleeters, less than 10% were found to be compatible with NATO's weapons and more than 90% were found to have originated from Russia, China and the East European countries \u2013 Romania and Bulgaria in particular.\nBy comparison, a joint investigation by the Balkan Investigative Reporting Network (BIRN) and the Organized Crime and Corruption Reporting Project (OCCRP) has uncovered  the Pentagon's $2.2 billion arms pipeline to the Syrian militants. It bears mentioning, however, that $2.2 billion were earmarked only by Washington for training and arming the Syrian rebels, and tens of billions of dollars that Saudi Arabia and the oil-rich Gulf states have pumped into Syria's proxy war have not been documented by anybody so far.\nMore significantly, a Bulgarian investigative reporter, Dilyana Gaytandzhieva, authored a report  for Bulgaria's national newspaper, Trud News, which found that an Azerbaijan state airline company, Silk Way Airlines, was regularly transporting weapons to Saudi Arabia, United Arab Emirates (UAE), and Turkey under diplomatic cover as part of the CIA covert program to supply militant groups in Syria. Gaytandzhieva documented 350 such 'diplomatic flights' and was subsequently fired from her job for uncovering the story. Unsurprisingly, both these well-researched and groundbreaking reports didn't even merit a passing mention in any mainstream news outlet.\nIt's worth noting, moreover, that the Syrian militant groups are no ordinary bands of ragtag jihadist outfits. They have been trained and armed to the teeth by their patrons in the security agencies of Washington, Turkey, Saudi Arabia and Jordan in the training camps located at Syria's border regions with Turkey and Jordan.\nAlong with Saddam's and Egypt's armies, the Syrian Baathist armed forces are one of the most capable fighting forces in the Arab world. But the onslaught of militant groups during the first three years of the proxy war was such that had it not been for the Russian intervention in September 2015, the Syrian defenses would have collapsed.\nThe only feature that distinguishes the Syrian militants from the rest of regional jihadist groups is not their ideology but their weapons arsenals that were bankrolled by the Gulf's petro-dollars and provided by the CIA in collaboration with regional security agencies of Washington's traditional allies in the Middle East.\nWhile we are on the subject of Islamic State's weaponry, it is generally claimed by the mainstream media that Islamic State came into possession of state-of-the-art weapons when it overran Mosul in June 2014 and seized huge caches of weapons that were provided to Iraq's armed forces by Washington.\nIs this argument not a bit paradoxical, however, that Islamic State conquered large swathes of territory in Syria and Iraq before it overran Mosul when it supposedly did not have those sophisticated weapons, and after allegedly coming into possession of those weapons, it lost ground?\nThe only conclusion that can be drawn from this fact is that Islamic State had those weapons, or equally deadly weapons, before it overran Mosul and that those weapons were provided to all the militant groups operating in Syria, including the Islamic State, by the intelligence agencies of their regional and global patrons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztjys b/data_all_eng_slimpj/shuffled/split2/finalzztjys
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+{"text":"Since 1976, Picq has always been a great example of expressing Chablis' terroir through Chardonnay with such precise minerality that is unmistakable to the area. Rich in limestone-laden vineyards, their wines are timeless and of the upmost respect in which Chardonnay does such a great job at picking up and projecting unto itself. The family currently works on old-vine sites at the villages and premier cru level. The winery is, in fact, a classic case of how truly amazing wines can be made from more humble vineyards when given the utmost attention on all levels. One could propose that is Picq's calling card - a value wine composed of villages vines, but wines with the complexity of premier cru. The proof is in the work.\nThey are located in the tiny village of Chich\u00e9e, just southeast of Chablis proper. Today, Gilbert's three children run the winery, each owning a crucial aspect to the business. Their wines are naturally fermented with indigenous yeasts in stainless steel. You will not find a single barrel at the estate. Their work is in the vineyards, and you can feel it in their wines. Racy acidity from the limestone, with bright citrus, white flowers and notes of honey and chalky stone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Greek word in this passage and elsewhere in the New Testament which is translated \"forgive\" has the root meaning of \"to let go.\" Therefore, when we forgive we are letting go of that which is holding us down.\nLuke 7:38 says that the woman washed Jesus feet with her tears. This is no small amount of weeping. The Greek word is the one usually used for rain.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So we headed up to Little Dell Reservoir Friday afternoon because it was so close to Salt Lake and Jimmy and I were itching to do some fly fishing. It has been a rough winter for fishing :) So the reservoir is still frozen over except for a little open water at the East end where the water feeds in. It was softening up and it looked like the Ice would be off within the week. We tried the little open water the we had but had no luck. We did walk up the little creek that feeds the reservoir just to take a look and saw some fish and some nice little holes that looked good. But I guess the creek is closed to fishing until closer to the summer. Anyways here is the video that got us all worked up to try this place out haha! Thanks Guys I guess we needed to wait another week or two.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zhou himself doesn't remember swallowing the toothpick, but told Guanxi TV Station that the doctor's theory is plausible, as he had the habit of putting a toothpick in his glass when drinking with his buddies, to help him easily identify his drink on the table. In fact, he admitted to swallowing a toothpick by accident before. Luckily, it had lodged into his throat and he was able to reach down and remove it with his fingers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A funky and fresh ensemble of tapered pants with an empire waist tubed top. The sash belt right below the bust adds that extra touch of flair.\nThe four colorful combinations available will give instant color to your wardrobe and of course, can be worn mixed and matched with other separates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzttjx b/data_all_eng_slimpj/shuffled/split2/finalzzttjx
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index 0000000000000000000000000000000000000000..bd1e222c96c4c3b73191823b17291bc34c04a44d
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+{"text":"Generate a configurable alert bell by sending an HTTP GET request to an Ethernet-connected Arduino board with a MIDI synthesizer shield.\nI actually only tested this on a first generation Galileo, but I don't know of any reason why it would not work on a second generation Galileo.\nI actually just used a generic 5% carbon film resistor.\nMany existing devices issue personal notifications to the owner of the device (e.g. a new email or SMS has arrived), but it can also be handy to have a device to issue ambient notifications to anyone that happens to be nearby. This project is an HTTP-controlled bell that can sit in a central place and generate configurable ambient notifications through a MIDI synthesizer.\nAn ambient notification bell can be useful for different types of events than those associated with personal device notifications. At my house, we like to leave the windows open in the morning to capture cool, fresh air, but it's easy to forget to close them as the day warms up. I configured my ambient notification bell to ring when the temperature rises above 73 degrees, although I'm still trying to figure out why there is a delay between when that threshold is crossed and the bell rings. Perhaps the weather station is too far from my house. I can also make it ring when I leave work as an FYI for my family. IFTTT has a lot of triggers that could be useful for generating ambient notifications.\nI got the idea for this project after reading Craig Lindley's Nuts & Volts article about connecting a MIDI synthesizer to an Arduino. Craig also posted a PDF of the article on his website. I bought the current revision of the synthesizer shield, the Modern Device Fluxamasynth Shield v.3, and connected it to my first generation Intel Galileo.\nI am one of the Intel employees that is enabling Intel Galileo boards on Contiki OS, so I naturally started with that as my software base. However, this is a personal project, and this post is my own and doesn't necessarily represent Intel's positions, strategies, or opinions. I extended the Contiki nano webserver with a new CGI script that rings the bell, and I added a new webserver configuration that is oriented towards programmatic access rather than interactive browsing. Thus, I eliminated support for unneeded elements such as a navigation bar. I also configured the build to enable support for some security features in the Intel Quark X1000 SoC, specifically segmentation, Task State Segment (TSS) switching, and Isolated Memory Regions (IMRs). However, keep in mind that no security solution is perfect.\nInstructions for setting up the hardware, building the software, and using the final device can be found in the README.md file. The synthesizer chip on the MIDI shield supports a large number of instruments, so that README also includes instructions for selecting a different instrument and pitch than the default celesta middle C that I selected.\nThe simplest way to play the bell is to navigate to http:\/\/<IP address>\/synth.shtml with a web browser. Keep in mind that in a typical setup with a router, this will only work if the browser is running on a machine in the same network as the bell. Port forwarding can be used to provide access to the bell from the Internet. Of course, you should carefully consider any security ramifications from making your network more accessible to the Internet. Another challenge in making the bell accessible from the Internet is discovering the current IP address, since the bell may be behind a router with a dynamic IP address. Dynamic DNS services can help in such circumstances.\nIf the bell is configured to be Internet-accessible, then services such as IFTTT can be used to play the bell in response to whatever triggers are supported by the service in use. For example, this is how I configured my bell to ring when the temperature rises beyond a certain threshold. I setup the IFTTT Maker channel to issue an HTTP GET request to the port forwarding URL that I configured on my router.\nThis repository contains the code for Contiki OS and the application for accepting HTTP GET requests and sending commands to the MIDI synthesizer. See examples\/http-synth in the galileo-http-synth branch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SCHOOL UNIVERSITY APPLICATION One of the areas where LCD display solutions are being used nowadays is in schools and universities. This technology offers a great way to effectively deliver key information to students, staff members and visitors. These campus digital signage solutions can be installed around any educational complex to ensure that everybody on campus gets to see the information. Over time, it slowly replaces those notice boards in disseminating information. With LCD digital signage, it can be placed anywhere where the traditional mediums can't. Thus, it is more effective in relaying information to staff and students around the campus. Modernity is also another aspect to consider where young people are growing up with technologies and these LCD screens can be a digital venue where they best respond to. Moreover, these are also flexible solutions in terms of what type of content you'll be displaying. Either it is a school announcement, news on campus, schedule of classes or emergency alerts, it enables you to display these contents in real-time. An LCD signage solution is a modern form communication that enables schools and universities to publish and convey information quickly and efficiently. At SENPAL LCD, we offer such school university digital signage systems with high quality and deliver reliable performance. We also integrate a number of useful features into our products in order to provide the best LCD display solutions for schools and colleges. For more info, simply contact us.\nImportant Knowledge of Led display 1. What is LED? LED is sanded for Light Emitting Diode, a kind of semiconductor which is used to give and receive the electronic signal into infrared rays or light, using the characteristics of compound semiconductor. This is used for household appliances, remote controller, electric bulletin board, various kinds of automation appliances. 2. What are Pixel Pitch, Pixel Density, LED QTY, and Pixel Configuration? Pixel Pitch is the distance between neighboring pixels. Pixel Density is the quantity of pixels per square meter. LED QTY is the quantity of LED lamps per square. Pixel Configuration is the description of the consistence of a pixel, for instance, we use 1 red lamp, 1 green lamp, and 1 blue lamp to compose a pixel, the pixel configuration is 1R1G1B. 3. What are LED Type, Module size and Module resolution? LED Type is the description of LED lamp, for instance, the brand, the shape in physical, the size of the lamp, etc. Module size is the measurement of a module. Module resolution is the number of pixels per module. 4. What is viewing angle? Viewing angle is the maximum angle at which a display can be viewed with acceptable visual performance. It includes horizontal viewing angle and vertical viewing angle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"... chemical and corrosion resistance and has an all-316L stainless steel housing to ensure excellent durability. It also has a ceramic sensor element with replaceable polyurethane wear washer.\nDetails: ABSTRACT: Applying to pressure measuring and controlling of industrial facility, irrigation, chemical, industry, medical, electric power, air condition, diamond press machine, metallurgy, water supply to buiding and so on.\nThis device uses proven ceramic technology, providing no mechanical aging of its measuring cell. It has different versions for absolute and relative pressures and no transmission liquid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Complete the nine (9) courses in the curriculum while maintaining at least an overall GPA of 3.5, a 3.5 GPA in your specialization area, and a 3.5 in your Cohort classes.\nCompletion of Final Paper: submission and acceptance of a ten (10) page paper that focuses on their academic experience or complete a comprehensive research thesis prior to graduation. This paper must be completed and submitted to the Cohort Faculty Director no later than week 12 of the student's graduation term.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Nani ahiahi is a tropical plant native to Brazil and is more commonly known as an Allamanda or Golden Trumpet vine. The vine has bright yellow trumpet shaped flowers that are 3 to 4 inches across and shiny bright green leaves.\nIn Hawai'i the Allamanda blooms several times a year with the vines often supporting hundreds of beautiful flowers. You can find Allamandas growing in the upper parts of Hawaiian Paradise Park in the Puna district, as well as other places along the Eastern part of the Big Island.\nWhile Allamanda is often grown as an ornamental plant it is very hearty and difficult to control. An Allamanda will attach itself to just about any tree and will grow to the same height the tree grows. It can easily choke a tree and can spread both horizontally as well as vertically. Care should be taken to control Allamanda plantings to ensure they don't go out of control.\nNote: The milky sap in the leaves of the Allamanda contain a cathartic and is toxic if eaten.\nThis page is named 'Plants.allamanda'. Can't Find It?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"600+ experts from leading companies, academia and state officials expected at conference on October 10-11, 2018, at Technical University of Munich (TUM).\nHosts Oerlikon, TUM and Bayern Innovativ are supported by a strong industry partner network that includes Siemens, T\u00dcV S\u00fcd, General Electric and Linde.\nKeynote speakers will focus on Additive Manufacturing (AM) challenges and success stories, and will address ways to speed up the industrialization of AM.\nWorkshops with experts on second day will address topics such as AM transformation, hardware progresses and implementation of industrial AM for decision makers.\nAdditive Manufacturing (AM) \u2013 commonly referred to as 3D printing \u2013 is much talked about and celebrated as the key technology of the future. How can we speed up the industrialization of AM in order to improve industrial processes and the creation of new products? Which challenges have to be overcome, and how can we learn from market success stories? These and other questions will be discussed at the upcoming Munich Technology Conference (MTC2).\nIn this second annual conference, renowned experts from industry, academia and government will discuss the truly ground-breaking power of AM.\nThe interdisciplinary exchange about major challenges and opportunities posed by the broad industrialization of Additive Manufacturing is more relevant than ever. The consistent use of AM necessitates the complete rethinking of the manufacturing process at multiple levels. AM technology can be used to supplement or replace existing processes and can be used to improve upon weaknesses in others, for example in vehicle and aircraft construction.\nMore than 600 AM professionals from around the world are expected to participate in interactive sessions at the conference. On the first day, keynote speakers will address the key challenges and opportunities presented by AM. The second day will be dedicated to in-depth workshops, where participants will continue the first day's discussions and address specifics such as AM hardware or materials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ALL THE NEWS, ACTION AND REACTION LIVE FROM SOCHI 2014 AS IT HAPPENS!\n21:58 Tomorrow, we have three men's giant slalom sets of medals to be won, as well as two in cross-country.\nThe ice sledge hockey and curling competitions will also reach their climax. Canada play Norway for hockey bronze, while the United States face Russia in what promises to be a spicy clash for gold.\nThe wheelchair curling semi-finals will see Great Britain play Russia and Canada versus China, with the bronze medal and gold medal games later tomorrow.\nWe'll be back live blogging the day eight action from 05:00 (09:00 GMT).\n21:54 And so with that ends day seven of the Winter Paralympics, a day which saw 14 gold medals won across alpine skiing and biathlon.\n21:51 Czech Republic beat Italy 3-0 to secure fifth place at the ice sledge hockey competition.\nIt took 31 minutes of play for the first goal to come, but then the floodgates opened for the rampant Czechs.\n21:49 A third goal from the Czech Republic puts the game beyond doubt. Michal Geier are set to win 3-0.\n21:42 That could be the winner - Gabriele Araudo scores a second goal for the Czech Republic to give them a 2-0 lead.\n21:30 We have a breakthrough, Zdenek Safranek gives his Czech side a 1-0 lead over Italy with 13 minutes of the third period left.\n21:26 Ice sledge hockey goalkeeper Steve Cash will hold the stars and stripes aloft for the Untied States at Sunday's Paralympic Closing Ceremony.\nCash, a three-times Paralympian who could win gold tomorrow in the final against Russia tomorrow, said: \"Obviously first and foremost, it is a tremendous honour, but at the same time it's not as much about having an individual opportunity to bear the flag, but also being able to represent my country and all of the athletes who came out and are here representing their country as well.\n\"I am able to represent 80 athletes, especially my team-mates who have been there for me through thick and thin ever since I started in the sport.\n\"It's really more about a collective group than just representing myself.\"\n21:21 The third period is underway in the classification match for fifth and sixth places in the ice sledge hockey competition.\n21:05 And with that ends the second period, still 0-0.\n21:04 Another late chance goes begging for Italy, as Czech goalkeeper Michal Vapenka makes a fine save.\n20:57 Lots and lots of chances, but no goal. Surely the deadlock will be broken at some point in the second period.\n20:26 End of the first period, and we're yet to have a breakthrough at the Shayba Arena. The two teams have been pretty evenly matched so far. The Italians have had four shots on goal, the Czechs three.\n20:17 Best chance of the game so far comes courtesy of the Czechs, who rattle the frame of the goal. Still 0-0.\n19:56 So, after a long lull in proceedings, our final slice of sporting action is about to begin, as Italy take on the Czech Republic to decide who finishes fifth in the ice sledge hockey competition.\n19:23 We can safely say Russia will top the medals table at these Paralympics - scroll down to 17:23 to see the table in full, or click here.\nWith 25 gold, they are four short of Norway's 29 at the Lillehammer 1994 Paralympics. If they beat that, they will have the best gold total achieved at a Winter Games in the last 30 years. And it's worth noting there are far fewer medal opportunities in Sochi than there were in Lillehammer 1994, or Innsbruck 1984.\nRussia now have the same number of gold as Germany, Ukraine, France, Canada, Japan, Austria and Slovakia combined. They are the countries sitting second to eighth in the medals table.\n18:56 The five cities competing to become host of the 2022 Winter Olympic and Paralympic Games have met today's deadline for submitting their Application Files.\nKrakow, Lviv, Oslo, Almaty and Beijing are all in the race.\n18:07 We also have a Winter Paralympic-related blog today - by Turin and Vancouver Paralympian and former sit-skier Sean Rose on how \"Britain's Winter Paralympic success is a game changer.\"\nIt is definitely worth a read here.\n18:01 A couple of other stories on insidethegames today during the lull in action. Berlin mayor Klaus Wowereit has spoken favourably about Berlin bidding for the 2024 Summer Olympics and Paralympics, while Kevin Warner has been appointed the new executive director at USA Synchro.\nThe International Olympic Committee have also confirmed that the Applicant Files have been recevied by all five cities bidding to host the 2022 Olympic and Paralympic Games - Almaty, Beijing, Krakow, Lviv and Oslo. Read our article pre-empting this announcement yesterday here.\nIt was a big day - phenomenal. I am nearly speechless. Up in the start, I was so charged. I went for my daughter and for all my friends and family. This was the one thing missing on my shelf. I've had a succesful career.\"\n- Josh Dueck of Canada reflects on his sitting super-combined gold medal.\n17:23 We have a sparse few hours now in Sochi - with just one more piece of sporting action coming up today. At 8pm it is the all-European ice sledge hockey fifth place play off between the Czech Republic and Italy.\n17:15 The Ceremony will be entitled \"Reaching the Impossible\" and will celebrate the magnificent achievements of the Paralympic athletes, who have shown the world the impossible is possible through strength of spirit and a relentless pursuit of sporting victory, it is claimed.\nLondon 2012 rowing medallist Aleksey Chuvashev will be a central hero of the Ceremony and - to demonstrate the wonders of strength and courage - and during a section of the Ceremony he will climb a rope with only his hands to a height of 15 metres.\nTo the backdrop of the music by Russian composer Alfred Schnittke, the Ceremony will be opened by memorable choreographed pieces performed by members of the Russian Wheelchair Dance Sport Federation together with trapeze artists who form abstract shapes in the air.\nThe performance will also be filled with \"impressive music, breathtaking choreography, original view and vivid visual displays.\"\n\"Momoko went the right way on women's course, the others went on the men's course. Momoko went on the right course for four loops, but she was forced to take the men's course in the last loop.\"\n16:51 One other piece of news to bring you now - which might not come as too much of a surprise if you were following the blog earlier on (scroll down to 12:58-13:19 for more).\nThe IPC Nordic Skiing jury is considering a protest by Japan regarding the unofficial results of the women's biathlon 12.5km standing event this morning.\nIt turns out that Momoko Dekijima of Japan did not go the wrong way - she was actually the only biathlete to go the right way. All the other athletes skied the men's 3km loop instead of the women's one due to confusion over course markings.\nYet Dekijima ranked seventh in the unofficial results behind winner Oleksandra Kononova of Ukraine.\n16:45 So...a frantic period and we have seen 14 sets of medals won over the last few hours in Sochi. Russia has had another brilliant day with half of the gold medals on offer and they have duly stretched their lead on the medals table.\nGermany and Ukraine remain in second and third after winning one and two golds respectively today.\nBut the snowboarding brought new nation's on the table - with the Netherlands and the United States taking the spoils - while Canada and France were the other day seven gold medal winners.\n16:22 So, after only two medallists in the sitting super-combined earlier on we have had four podium finishes in the final biathlon event of the Games.\n16:18 Nikolay Polukhin and guide Andrey Tokarev win the final biathlon gold medal of Sochi 2014 in the 15km visually impaired event.\nDrama behind as Anatolii Kovalevskyi takes silver for Ukraine and then there is a tie for bronze. Vitaliy Lukyanenko recovered from that missed shot to claim his place on the podium but the Ukrainian who has already taken double gold in Sochi has to share it with Stanislav Chokhlaev of Russia.\nThree Ukrainians and another Russian are in major battle for the minor medals.\n16:10 That is a first gold of the Games for Budaleeva after two silvers. She won by a mammoth 1min 55sec in the end, so even if Lysova had not missed that shot she would not have won.\n16:08 And yet another gold for Russia in the women's visually impaired skiing as Iuliia Budaleeva and guide Tatiana Maltseva hold on to win the gold medal.\nFellow Russian Mikhalina Lysova takes silver while bronze goes to Oksana Shyshkova of Ukraine.\n16:06 Gold goes to Josh Dueck of Canada thanks to a magnificent slalom - as Rabl of Austria is only good enough for bronze. In between them is Heath Calhoun of the US.\n16:00 Germany have already won gold and silver in the sitting events today but Thomas Nolte will not add to that. He half a second lead over Dueck ebbs away quickly.\n15:52 In the super-combined, Corey Peters of New Zealand leads but then Josh Duck of Canada, takes over at the top.\n15:49 Nikolay Polukhin leads by over a minute in the men's event, meanwhile, after 7.5km and Russia are looking good for the win in both biathlon events Polukhin's bid rival Lukyanenko of Ukraine has missed a target and is down in fifth.\nTwo Russians in Mikhalina Lysova and Iuliia Budaleeva are battling it out for the gold and silver medals still in the women's race.\n15:42 In the women's biathlon, Mikhalina Lysova and guide Alexey Ivanov lead after 5.5km by 12 seconds from another Russian pair. Flawless first shooting from the whole field but still a long way to go in the 12.5km event.\n15:37 A bit of confusion as second to go Mick Brennan of Great Britain is told to restart his run when already halfway down the course. Not exactly ideal... Thomas Jacobsen of Norway was first down, but the leader following the slalom section is Roman Rabl of Austria.\n15:29 Alexey Bugaev wins his fourth medal and second gold of Sochi 2014. The Russian wins by 1.1 seconds from Lanzinger...while bronze does go to Kane of Australia.\nThat bronze is the first medal of the Games for Australia.\n15:25 Austria's Mattias Lanzinger moves into the lead with a superb run. Braydon Luscombe is next, but the Canadian slips out midway through and fails to finish.\n15:21 Adam Hall of New Zealand splits the two Australians for an Oceania 1-2-3. Hall lies second while Kane remains the leader.\n15:14 Austria's Martin Wuerz moves into second place but cannot overhaul the Frenchman. Then Romain Riboud looks to make it a French 1-2 after a good start but then he misjudges a corner and crashes!\n15:10 Visually impaired biathlon events still to come as well, but back to the Alpine and Cedric Amafroi-Broisat is the leader at the moment. A strong Super G should help him move up from ninth place in the slalom.\n15:06 The men's standing event is next as the super-combined action continues at super-frantic pace. But before that, an ice sledge hockey result to bring you as South Korea finish in seventh place after a 2-0 win over Sweden.\nThe fifth place play off is also coming up a little bit later on today.\n15:01 And we can confirm now that Grygorii Vovchynskyi of Ukraine has won the 20km biathlon, A really good effort by Nils-Erik Ulset of Norway to win silver just 3.1 seconds behind.\nKirill Mikhaylov takes bronze for Russia as the host nation finish in every position between third and eight. Karachurin would have taken gold if he had not missed that one shot, but such are the margins of this sport, and he has to settle for fourth place.\n14:58 Valerii Redkozubov is a slalom specialist and does not seem overly comfortable on the Super G but he does enough!\nAlong with guide Evgeny Geroev he takes his second gold in a matter of hours after his slalom success under floodlights last night.\nSilver for Marc Bathum of the US and Gorce Yepes of Spain takes bronze.\n14:55 Gabriel Juan Gorce Yepes goes into second place after losing all the time on Bathum at the top of the course. So the American is guaranteed silver with only Valerii Redkozubov left.\n14:51 A little way away on the mountains the men's visually impaired event has begun. Marc Bathum of the US, fifth after the slalom, has taken the lead after an impressive Super G. In second place is Slovakian Radomir Dudas and in third is Michal Beladic.\n14:48 Vovchynskyi now crosses the line in a time of 37min 41.1sec. Surely that will be the gold medal winning run?\n14:46 At the halfway stage another Russian-Ukraine rivalry was developing as Azat Karachurin of Russia led from Grygorii Vovchynskyi of Ukraine by just two seconds, but then Karachurin missed a crucial shot at his penultimate trip to the range and the Ukrainian is closing in on gold.\n14:45 There are three more Alpine medals to be won in men's events, but the 15km standing biathlon is also going on.\n14:42 Forster takes silver but, with only two skiers finishing, there is no bronze medal winner.\n14:39 Anna Schaffelhuber of Germany takes her fourth gold medal of Sochi 2014 in a German 1-2. Again, a reasonably conservative run but she had five seconds to play with and has the knack of looking very controlled even when moving fast.\n14:36 Forster is going down slowly as if she is trying to finish more than anything else. Slowly but still at 60mph... But she gets down and is guaranteed at least silver.\n14:33 Anna-Lena Forster and Anna Schaffelhuber are the two left so we will have no bronze medal winner, and maybe no silver or gold either if one of these two fails the finish. Forster was the initial winner of the slalom only for Schaffelhuber to be reinstated ahead of her, but the two are great friends so no hard feelings.\n14:29 Another huge crash as Claudia Losch flies through the netting at the side of the course. We think she is OK as she is moving and she skies off gingerly down the mountain. But with Laurie Stephens of the US not starting, there are only two skiers left in.\n14:25 The third women's super combined event is the sitting competition and we have already seen one big crash as Britain's Anna Turney goes down hard. That's not her first crash during these Games either.\n14:20 Marie Bochet clocks a flawless 53.48 and takes her third gold medal of the Games.\nA brilliant performance by Bochet and that result never looked in doubt for one moment. She has now taken all three French gold medals at Sochi 2014. It is silver for Rothfuss and bronze for Jallen of the US.\n14:11 Next up we are straight on into the next Alpine event and it is the sitting women's event.\n14:06 Aleksandra Frantceva and guide Pavel Zabotin of Russia win the gold medal by just 0.63 seconds.\nTheir lead was cut massively from over three seconds in the slalom but the Russians just hold on for gold.\n14:03 Great performance by Jade Etherington and guide Caroline Powell of Britain. They are guaranteed their fourth medal of the Games and are 11 seconds clear of Umstead. Is it enough for gold?\n14:00 Back to the sport and first of six Alpine events is underway. The Super G run of super combined events were postponed from earlier in the week.\n13:53 Politics has been on the backburner at Sochi 2014 but - that Closing Ceremony will clash with the \"referendum\" on whether Crimea will join Russian on Sunday, so a possibility that there will be dissent to follow Ukraine's Opening Ceremony protest there.\nRussian President Vladimir Putin thanked the IPC yesterday for promoting sport over politics so far at Sochi 2014... Read our article here.\nSunday's #Sochi2014 #Paralympics Closing Ceremony theme is \"Reaching the impossible.\" An element from the opening will also be used. Ace!\n- IPC communications director Craig Spence provides a glimpse of Sunday's Closing Ceremony.\n13:49 An ice sledge hockey match is also going on at the moment. South Korea lead Sweden 1-0 after a goal from Jang Dong-shin in the seventh place play-off.\nIt is amazing how at this point yesterday we had enjoyed a veritable feast of sledge hockey and curling whereas today we have had a completely different day of biathlon and snowboarding so far.\n13:45 Coming up on day seven we have more biathlon action with the men's 15km standing event, as well as six Alpine super-combined finales starting in a few moments time.\n13:40 So another gold medal for Ukraine as they strive to close the gap on Germany in second place in the medals table.\n13:24 Huge cheers from the impressively vocal crowd at the Laura Biathlon Centre as the final finisher Caroline Bisson of Canada completes the course.\nFurther ahead though, Russia and Ukraine continuing their Nordic dominance as the two countries swept the first six places. A 1-5-6 for Ukraine and a 2-3-4 for Russia.\n13:19 Gold for Oleksandra Kononova of Ukraine as a fast finish propels her to a successful defence of her 12.5km standing biathlon title.\nSilver goes to Kaufman and 16-year-old Natalia Bratiuk takes another bronze medal.\nAt least we think that is how it will turn out. Momoko Dekijima of Japan finishes 7th but the Japanese officials are still remonstrating about what happened to her out on the course... We will try and keep you updated on what happens there.\n13:17 Here comes Kaufman - putting every last bit of energy in as she sprints for the line in pursuit of her third of the Games...but she doesn't do it. Second place for the Russian 2.1 seconds behind Kononova.\n13:12 Confusion reigns again as Dekijima of Japan suddenly drops from 1st to 5th in the results. What we presume has happened it she was sent the wrong way and has now done an extra loop to compensate, but no-one really knows, and the official classification still lists her as 1st.\n13:08 Kaufman misses a shot on the final shoot but Bratiuk and Milenina go clear. Remember, each biathlete left at 30 second intervals, so it is more a time trial than a mass start event.\n13:05 The expected Russian trio of Anna Milenina, Alena Kaufman and Natalia Bratiuk lies 2nd, 3rd and 4th but are over 2min 40sec behind the Japanese competitor.\n12:58 Speaking of biathlon, the women's 12.5km standing event is well underway and there has already been some drama. Momoko Dekijima of Japan has been miles ahead of the field virtually from the outset - and it quickly became obvious that she has missed out a section of the course somewhere.\nBut after being initially disqualified while being still out on the course, there were some tense discussions between officials and the Japanese team and she appears to have been reinstated.\n12:55 So, four gold champions have been crowned so far on day seven - two more Russian gold medals in biathlon, and it is fitting that the two new snowboarding events were won by two new countries in terms of gold medals at Sochi 2014 - the Netherlands and the US.\n12:50 So the US shoot up into ninth place on the medals table with that success. A superb performance - second after run one but utterly dominant after that.\n12:44 The gold medal is taken by Evan Strong in a US 1-2-3. The former skateboarder is guaranteed gold but claims it in style with a final run of 51.99 seconds.\n12:39 Gabel clocks 53.49 seconds. That is his quickest of the three but will not trouble the top two so the American will take bronze.\n12:36 So the top four will remain the same because no one has gone quicker overall after an extra run. Three of them are Americans and the clean sweep can only be prevented by Carl Murphy of New Zealand.\nBut Murphy falls and misses a gate. He will finish fourth and the US are guaranteed a clean sweep of the podium and a first gold medal of Sochi 2014!\n12:33 The youngest Paralympian in Sochi finishes 10th but the next few fail to dislodge Vaverka who will finish fifth. The best two runs of three count to the overall score remember.\n12;31 A big crash by Chris Vos of the Netherlands - he won't repeat the Dutch success in the women's eveent - but then a superb final run by Tomas Vaverka of the Czech Republic. 53.03 seconds and he is up to fifth overall.\n12:26 The top 16 snowboarders are going first in reverse order. First to go is Urgo Egea Zabalza of Spain before Canadian Tyler Mosher goes well and moved up to 11th place overall.\n12:20 So a first Dutch medal of any colour of the Games - and it is a golden one. France are sixth on the medals table after that silver, while the US take their ninth medal of the Games but are still yet to claim a golden one.\nSilver for the \"Pride of Perpignan\" Cecile Hernandez-Cervellon of France and bronze goes to Amy Purdy of the US.\n12:15 Cecile Hernandez Cervellon takes silver. A strong final run but still some nine seconds off the pace of Bibian Mentel-Spee - who will take a victory procession now.\n12:13 Albert's compatriot Amy Purdy is guaranteed bronze. 1min 07.41sec she clocks to stay third but without troubling the top two.\n12:11 Christina Albert of the US, who already holds the distinction of being the first ever Paralympic snowboarder having been first to start this morning, lies fourth and needs a huge run to sneak onto the podium.\nShe doesn't manage it and stays in fourth place.\n12:09 The main difference, of course, between Olympic and Paralympic snowboard cross is the fact that here they go individually instead of in a mass start, race format. There are talks in IPC snowboarding circles to change the format in the future...but while we don't have mass pile-ups here, there are still plenty of falls as Lisa Bunschoten does exactly that and fails to finish her third run.\n12:07 Nicole Roundy of the US briefly moved up to fifth place with an improved third run, but then Astrid Fina Paredes of Spain leapfrogs her. 2min 48.82 her overall leading time at the moment.\n12:01 First up is Veronica Yoko Plebani of Italy, who improves on her second run, as does the next snowboarder Megan Harmon. But neither are likely to move up far.\n11:59 The top ten to go first in reverse order - although there are only 11 to go - although because only the best two runs count expect the leaderboard to change a good deal. For skiers who have had one good and one poor run so far, another good run will see them significantly improve their overall score.\n11:55 Up next it is the final run of the women's event.\n11:52 So the second run of the men's event has now come to a close with the Americans still leading overall - the last twenty minutes or so having only seen lower ranked competitors taking to the course.\n- Fair to see the US Paralympic team are pretty happy with the results so far in the men's event on the inaugural day of Paralympic snowboarding.\n11:45 Incidentally, the three medals in that 12.5km biathlon went to the three countries who are leading the overall medals table - in the same order. Russia, then Germany, then Ukraine.\n11:43 Next up in biathlon will be the two standing events starting in around 45 minutes time.\n11:39 Another reflection on that second biathlon event earlier on which we did not describe in too much detail. Andrea Eskau of Germany, gold medal winner over 6km on day one of the Games, was the early leader but missed two shots on the first shoot and consequently finished down in seventh place.\nKonovalova meanwhile, did not miss a shot, and that is the difference between the two.\n11:33 One thing we can say with at least a reasonable degree of confidence is that later on today we will have our first US gold medal of the Paralympics. Be it Shea, Strong or even Gabel. Even if all three crash their final run it will still take something virtually impossible to beat them.\n11:29 We just have the slower snowboarders from the first run to come now - so don't expect the top of the leaderboard to change.\nBut what a final run we have in prospect. Remember it is the combined time of the best two runs to count - so it is possible that Strong could crash out and still win the gold medal if Shea does not go as quick as his first two runs.\nBut with the margin between the two American so minute - anything can happen.\n11:26 Wow! 51.62 is the second run by Evan Strong to take the lead and clock the first sub 52 second run of the day by anyone so far.\nThen Shea also breaks the 52 second barrier with 51.89. Which means that Strong leads overall by one hundredth of a second!\n11:23 Final stages of the snowboarding second run for the men now and no change at the top of the standings. Keith Gabel of the US produces a quicker second run of 53.61 to lead by 1.5 seconds from New Zealander Carl Murphy. Just the other two Americans Strong and Shea to come now.\n11:18 Second gold medal of the day has been won in biathlon and, yes, it is a second one for Russia. Svetlana Konovalova wins the women's 12.5km event. That is her first gold of the Games after two silvers and a bronze.\nA thrilling finish in the battle for minor medals as Anja Wicker of Germany produces a rapid final kilometre to overhaul Olena Iurkovska of the Ukraine. So Wicker takes silver and Iurkovska bronze.\n11:06 Petushkov has won the 7.5km, 12.5km and now the 15km biathlon events as well as the 15km and the 1km sprint cross-country disciplines. Not only peerless over sprinting and much longer distances, but the best in the shooting range as well. Absolutely flawless marksmanship today - with 20 out of 20 targets hit.\n11:04 No such trait of new countries on the podium in biathlon when Roman Petushkov leads home another Russian 1-2-3 and wins his fifth gold of the Games!\nGrigory Murykin finishes second but over two minutes behind. And Aleksandr Davidovich overhauls Andrew Soule to place third by six seconds.\n11:02 Another good thing about the snowboarding is that it is giving two countries - the Netherlands and the United States the opportunity to win their first gold medals of the Games.\n10:59 France's Cecile Hernandez Cervellon goes quicker than Purdy but then Bibian Mentel-Spee produced a staggering 58.89 second run. She lies nine seconds clear and, with the best two runs to count, she could probably hang up her board now and win the gold medal.\n10:56 A superb performance by Amy Purdy of the US. The only snowboarder to go faster in her second run and she, briefly, has a lead of over 26 seconds overall fro compatriot Cristina Albert!\n10:52 Conditions seem a little tougher in the second run of the snowboarding as the first few women go slower. Astrid Fina Paredes of Spain then takes the lead.\n10:47 Second run of the women's snowboard cross event now but first an update on the biathlon. And as we enter the closing stages of most of the runs, it is thoroughly turning into the Roman Petushkov show once again.\nThe Russian has a lead of over a minute at the 11.1km checkpoint and has since executed another flawless shoot. 20 out of 20 so far at the range today - which considering how far he has skied is truly phenomenal.\nMurygin and Soule still lie second and third.\n10:40 It was the saddest moment in the build-up to these Games when Australian snowboarder Matt Robinson tragically died en route to hospital back in Melbourne following a training crash last month in Spain.\nBut the Australian team here in Sochi: Badenhorst, 14-year-old Ben Tudhope and Trent Milton have vowed to recognise him throughout the Games.\nThey wore black armbands at the Opening Ceremony and coach Peter Higgins has said Australian officials were in discussions with the IPC to get permission for another gesture during the competition today.\nWe will keep you posted on that.\n- The Australian team provide an update on why Joany Badenhorst was a non-starter in the women's event today.\n10:35 Georg Schwab of Austria is last down in run one of the snowboard cross. He is also a Para-cyclist but does not do enough to trouble the top of the standings. So, a US 1-2-3 lead by Michael Shea at this stage. But still two runs to go.\n10:32 A potentially huge moment in the biathlon as Murygin misses a target with his second shoot and Petushkov takes the lead. Is he on course for a fifth gold medal?\n10:28 Meanwhile, back at the snowboarding, no change at the top of the first run leaderboard. Aitor Puertas-Marin of Spain falls at the beginning an end of his run and sets the slowest time of the day so far. But he gestures to the crowd at the end an they love him.\nThe first ever Winter Paralympian from Brazil, Andre Perreira, goes next. He goes into 30th place.\nJust a couple of snowboarders left to go now.\n10:26 It is the longest biathlon race this one so a real gruelling to affair to balance shooting with such a long ski. After all the biathlete are through the first shoot and the 5.1km check-point it is Grigory Murygin of Russia who is fastest, ahead of compatriot Petushkov. By just 1.6 seconds!\nAndrew Soule is continuing a good morning so far for the US by lying third - albeit some 40 seconds off the pace of the formidable Russian duo.\n10:18 Action has also begun in the biathlon further up in the mountains. First up is the men's 15km sitting event, Each athlete sets off at 30 second intervals and intersperse gruelling cross-country laps with trips to the rifle shooting range. A missed shot means an extra skiing loop.\nLeading the way should be Russia and their biggest star Roman Petushkov - who is remarkably already chasing his fifth gold of the Games.\n10:16 21 skiers have now gone in this much bigger 33 man field. The US are still dominating the top three positions at the moment but remember, this is only the first of three runs and only the combined time of the fastest two count overall.\nSo essentially you are allowed to mess up one of your three runs. But only one.\n10:12 In terms of other competitors, New Zealand's Carl Murphy lies fourth while Matti Suur-Hamari of Finland takes a tumble but still lies eighth. He showed good speed so if he can produce some clean runs he could yet mount a challenge.\n10:08 We now have a US 1-2-3 on the leaderboard as Keith Gabel takes over third place.\n10:03 Forgive the pun, but a strong run from the American who goes into the lead by two seconds, but only for teammate Mike Shea to go quicker next down the course.\nThe US are yet to win a gold medal at these Paralympics but you sense today could be their best chance.\n10:01 Ben Tudhope of Australia - who is just 14-years-old - goes into third place, but here comes one of the big medal contenders in Evan Strong of the US. .\nTudhope is the youngest athlete at Sochi 2014 in any sport.\n09:58 Chris Vos from the Netherlands sets off first for the men before John Leslie of Canada takes the lead and becomes the first athlete to post a sub-one-minute time on the course.\n09:55 So we are zipping through the first run action and straight on into the men's event.\n- The US are still yet to win a gold medal at these Paralympic but the US team are able to report on history in the making this morning.\n09:51 The first run comes to an end. Mentel-Spee leads from Cervellon and Purdy. Australia's Joany Badenhorst, incidentally, is a non-starter which means there are just 11 boarders in the women's field.\n09:46 Another Dutch snowboarder in Lisa Bunschoten has the honour of being first to fall but she still gets down the course. France's Cecile Hernandex Cervellon, meanwhile, goes into second place three seconds behind the leader.\n09:40 American Amy Purdy takes the lead but here comes the gold medal favourite - Bibian Mentel-Spee from the Netherlands.\nAnd she goes into the lead, stopping the clock at 1min, 00.18sec. A superb time on the first run - which you would expect a lot of the skiers to be using as an opportunity to get to grips with the course.\n09:38 Getting the honour of first ever Paralympic snowboarder is Cristina Albert of the US. And she goes well, stopping the clock at 1min 10.55sec to lay down the early marker.\n09:35 It is the women's event up first for the opening of their three runs. It is a 680m course beginning with four triple rollers at the top of the course followed by a blend of jumps, turns and technical sections before a triple roller and a step-up jump at the end.\n09:32 So, there are two snowboard cross events - one each for men and women and both tor standing competitors. Each athlete will have three runs down the course with the finish time of their best two runs added together determining the winner. There is only one rider on the course at a time and the starting order is determined by each athlete's performance in the previous run.\nSound simple? Well in terms of the rules it is, but expect thrills and spills no different from the sport that has electrified the Winter Olympics ever since it was first held at Turin 2006.\n09:28 Before all of that though we have a historic moment for the Winter Paralympics as starting well, right now, are the first ever Paralympic snowboarding competitions.\n09:24 In less than an hours time the first of six biathlon finals we begin while later on we have the conclusion of six Alpine skiing super-combined events.\nNo wheelchair curling due to the lack of need to have any play off matches but we do have ice sledge hockey - albeit only the classification matches to help decide places five to eight in the standings - they are coming up later on.\n09:18 Day seven is going to be a busier day with 14 sets of medals to be won.\n09:11 Although day six was albout qualification as much as anything else - with the United States joining Russia in the ice sledge hockey final, and Canada, China and Britain also making it through in wheelchair curling, Russia have duly extended their lead at the top of the medals table.\nThey are now a full 12 gold medals ahead of second placed Germany - with the only other gold won yesterday taken by Takeshi Suzuki of Japan.\n09:06 Considering the lack of Nordic skiing action yesterday you might have been forgiven for expecting day six to have been a quieter day in terms of host nation success. How wrong we were. First we had a Russian ice sledge hockey semi-final victory, then wheelchair curling semi-final qualification, and then two of the days three gold medals on offer in men's slalom events.\nThis all means that we had another day with an electrifying atmosphere in Sochi - as fears that the Games would be poorly attended were resolutely put to bed with the announcement of record ticket sales.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Silent End was a book I instantly warmed to.\nThe Silent End was a book I instantly warmed to. The story is told using the first person narrative style (which I have always loved when done well) and the way the story unfolded, complemented as it was by real and likeable characters, left me one very contented reader.\nYoung adult fiction is not my read of choice because it often leaves an older reader like myself feeling slightly ostracised. But when it is written this well, and in such an engaging way, it can be a lot of fun, allowing the older reader to remember a potent time in their life while offering the target audience a parallel for their current lives. I liked that the kids in this book are capable of provoking empathy from any age group. When teenage years, and everything that is both wonderful and terrible about them, are reproduced well it is a joy to read.\nThe chemistry between the 3 young leads was strong, the narrative voice is that of Eberstark, an overweight and ordinary kid with a messed-up family life. His best friends area Gus, the intelligent overachiever and Lexi, the rebellious and ever so slightly disturbed teenage girl. The kids are around fifteen years of age and I liked the way in which their dialogue accurately reflected this, something many young-adult novels do not do. We all remember how profanity is a knowledge that is constantly researched and improved through practice, how ridiculing each other is acceptable and normal behaviour between friends, we remember how teenage friends talk around the really serious issues, especially problems at home. I found the teenagers in The Silent Land amongst the most realistic I've encountered for some time.\nI encountered a large dose of nostalgia whilst reading The Silent End, being reminded of the Spielberg films I loved when younger: The Goonies and E.T. in particular, and I also felt the same emotions as when I read Stephen King novels, which was something I welcomed with open arms. Finally, there was a Lovecraftian aura, merged with a little of The Fog which, when all pieced together left me in a very good reading place, very comfortable with the story, the characters and the written story.\nThe tension builds well with the author in no hurry to disclose exactly what is so wrong with the town of Mossglow (this is Lovecraftion\/King element I liked). Sattin builds the characters nicely, presenting back-stories to help understand who they are and what has made them. I liked that the lead character was overweight, something a little out of the norm.\nIf I had a reservation in recommending this book it would be the ending and the build up to it, which went just a little bit too 'out there' for me, and I thought the great build up didn't quite finish with the denouement it deserved. But then I thought the ending of Stephen King's IT, one of my favourite books, was disappointing so maybe I just expect too much.\nThis slight reservation aside I would still happily recommend The Silent End to readers of around age fifteen and upwards who enjoy fun, realistic characters and an element of horror in their reading. I look forward to reading more by Sattin in the future.\nHave you read The Silent End?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"T Group Theory And Laboratory Method Innovation In Re Education is good choice for you that looking for nice reading experience. We hope you glad to visit our website. Please read our description and our privacy and policy page.\nFinally I get this ebook, thanks for all these T Group Theory And Laboratory Method Innovation In Re Education can get now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"40 RV Sites, can accommodate RVs 40+ ft, 30\/50 amp, full hookups, 12 tent sites, back ins, dump station, Cable TV, WiFi, restrooms, showers, laundry, handicap access, heated pool, fire rings, picnic tables, phone, propane, store, ice, RV storage, recycling, playground, game room, pool table, basketball, horseshoes, fishing, boat ramp, boating, canoeing, mini-golf. Park Events: live entertainment regularly during summer months, Kids Activities, Live entertainment, Bikers for Boobies fundraiser for Breast Cancer, family friendly, motorcycle ride, dinner, bake sale, raffles, live band, non bikers welcome - in July or August, see our Facebook page @Bikers4BoobiesNH for more info.\nOpen year-round. PA is valid Sunday thru Thursday. PA is not valid during holiday weekends. Reservations recommended. Drive-ups welcome. Credit cards accepted. Pets welcome. *$3 additional for A\/C, Electric Heat or 50-amp service. Rates are for up to 4 adult or 2 adults & their children. $7 per add'l person (12 years and older), and $8 per additional person during holidays.\nLocated 15 mil from Conord, NH and 79 miles from Boston, MA. From the S: From I-89 N, take Exit 5 W onto 202 & 9. Go approx 5 mi to 2nd flashing yellow light, take R onto Old West Hopkinton Rd. Go 1 mile, and camp is on L. From the N: Go S on I-89 to Exit 6, take R following Rte 127 until Hopkinton Dam, after crossing the dam, go R up hill, at the top take L at triangle. Continue straight thru the next 4-way intersection, campground on R (about 1.5 mi from dam).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwmnt b/data_all_eng_slimpj/shuffled/split2/finalzzwmnt
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index 0000000000000000000000000000000000000000..b7dba46ea7eac0f2027e353a541f81d11cd95909
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+{"text":"the detect table is a topic table located in MiscData and the reply table is a sentence table located in basicResponses. It seems next I want to move the user input to the detect table and if there is a word in the input matching a detect word the result is true, the true result enables a get response from the reply table. At this time I am unable to get a reply after trying a few variations. Any help would be appreciated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hatch an insect army and ensure your colony\\'s survival. Colonize and annihilate the enemy. Click your colonies individually or click and drag to select multiple colonies. Click a neutral or enemy colony to attack.\nPlay more fun games similar to Bug War 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you ever wanted to go on vacation and truly get away from it all? Somewhere where work, dishes, and laundry can't fine you? Tangier Island is that Place.\nAccessible only by boat, Tangier Island is a lost in time island off the coast of Virginia in the Chesapeake Bay that is filled with charm and character. The island is actually a quaint little fishing village filled with marshes that house abandoned boats and crab pots. It is home to a little over 700 residents and most of them make their living as watermen in the crabbing industry. The people there are absolutely wonderful. Being that the island is isolated from the mainland, they have their own Elizabethan dialect and it is lovely.\nThe best part of Tangier is the story behind it. The island is disappearing due to erosion. There are indications that in the 1800's, Tangier encompassed a little over 2000 acres and was home to a variety of plant life, watermelon farms, and grazing cows. Since that time about 67% of their landmass has disappeared underwater leaving the island only 1.2 square miles. The island is eroding a way a little more every year and according to residents, Tangier is expected to be inhabitable in less than 50 years from now . Just like the Uppards. The Uppards were a part of Tangier's north end that was abandoned in the 1920's due erosion leaving the island unlivable. You can take a boat to the Uppards but sadly, there are no remaining signs that humans ever lived there other than one old and dilapidated trailer that is partly underwater. I do hear it is a great place to look for arrowheads though.\nTangier makes a great getaway because it truly is a getaway. There is no cell service and extremely limited wifi on the island making communicating with the \"outside world\" very difficult.\nThere is also nothing to do there. No movie theater, no shopping mall, no bowling alley, no Starbucks\u2026nothing. There is however one neighborhood style grocery store and one restaurant, although I've heard there are actually two or three places to eat if you visit in the summer. And cats. Cats are everywhere. But that's okay, who doesn't love cats! There are also only like two cars on the island but it's not a big deal. The island is only 1.2 square miles wide. In fact, you can pretty much walk around the entire island in 10 minutes.\nRobby and I are afforded the opportunity to experience many different places through filming On the Map. We also travel a lot outside of the show. Out of all the destinations we have visited, Tangier Island is probably my favorite. And that's saying a lot considering I'm the girl that likes the fancy hotels and the five star restaurants and having a Starbucks on every corner.\nWe visited Tangier in April before their \"tourist\" season begins. The only way to get there was via the mail boat which visits the island twice a day. We had an idea of what we were getting into but we were certainly not prepared. Upon arriving, it took us about 10 minutes to realize there is no cell service and no wifi. We were there for about a week and the first two days were what I like to call our adjustment period. With nothing to do and no contact with anyone outside of the island, we were getting pretty lonely and growing tired of chicken fingers from Lorraine's, the only restaurant open on Tangier. By the third day, it got better. We actually started enjoying the island. It was so relaxing and peaceful. We spent our days going out on the boat with the mayor and walking on the beach, which is nothing like any beach you have ever been to. We also went to the grocery store every day for spaghettios and little boxes of cereal, which is what we lived off of for the week, but also to see if the food boat had come because I really wanted plain m&m's. (Quick fact: the food boat comes on Thursdays.) By the time Friday rolled around, it was time for us to leave and there was nothing we wanted to do less. We were really enjoying this peaceful, laid back lifestyle.\nMost people have never heard of this little island in the Chesapeake Bay but for those who have, it sparks a lot of intrigue.\nThe majority of Tangier visitors visit during the summer for a day trip but I believe it is much better to go during the off season and stay for a week (FYI: there are no hotels and the Bay View Inn, which is great, is the only B&B open year round.) You will absolutely love it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each year council reviews the priorities and goals for the upcoming year and directs staff to prepare a comprehensive Five-Year Financial Plan, as required by the Community Charter. The Five-Year Financial Plan is the organization's budget document reflecting ongoing efforts to provide quality of service in daily operations and local governance consistent with the priorities and strategies identified in the Corporate Plan and Whistler2020.\nFind information about the annual budget process.\nThe RMOW's Annual Reports include audited financial statements from the previous year, as well as an overview of municipal services and operations, a progress report on municipal objectives and a list of property tax exemptions. Annual Reports are contained in the Corporate Plan.\nQuarterly financial reports provide a regular overview of financial information and support the RMOW's commitments to fiscal responsibility and accountability.\nDownload the 2009 RMOW Long-Term Financial Plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"8 Different flavored licorice; green apple, watermelon, pineapple coconut, grape, chocolate, blue raspberry, raspberry, and peach.\nIngredients: Sugar, corn starch, citric acid, wheat flour, sugar, juice concentrate, artificial flavor, sunflower oil, color.\nContains wheat and soy. This product has been packaged in the same facility as products that contain peanuts, tree nuts, soy, milk and\/or wheat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxvtp b/data_all_eng_slimpj/shuffled/split2/finalzzxvtp
new file mode 100644
index 0000000000000000000000000000000000000000..48b7c72d76bf6dfab830463862049772822284bf
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Read Book Flash Online author by Tim Tigner. Read or Download Flash format Kindle Edition in and Published -7-2016 by Vontiv Publishing.\nFlash This is an alternate cover edition for ASIN B00F240FDG.\nif you woke up to the sound of screaming . . .\nin the trunk of a car . . .\nbeside a dead cop . . .\nwith no idea where you were . . .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Swedish Superstar Adam Flamholc had a pretty tough crash last season at Joliet during the NMCA\/NMRA Superbowl event. The car went to Chris Duncan Race Cars so that Chris, Jonas, and the rest of the crew could handle bringing it back to life.\nWe received a message from Adam today, actually a video of the CDRC team firing up Adam's Vette. This is something that has been months in the process, but unlike others, the CDRC team are absolute perfectionist.\nAdam said that he will be in Bradenton next weekend with the Vette to compete at the U.S. Street Nationals. This will be the first track time that the car will have seen the track since July of 2017. This will also be Adam's first race since late last summer.\nSome many know that Adam went through some serious health issues after the crash and took some well needed time off, while having Kristian Nystrom handle some driving duties at the end of 2017. Adam says it's been almost five months since he has raced and he is ready to get back into the seat. He will have Frank Beal and 5 Day Plantation Shutter and Blinds on board again this season as well.\nThere are very few people on the planet that have raced pro mod in as many different places as Adam. Being a former FIA Pro Mod World Champ, and has made runs in Russian, and is a standout in the United States as well.\nWe will keep you up to date on Adam's progress, and make sure to check out Chris Duncan Race Cars for some seriously top notch work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In honor of everyone going back to school this week, we put together a fail compilation just for you.\nThe good news is we just released an app that should help you earn a trip to, and pass the time in detention.\nCheck out our new app for iOS: http:\/\/bit.ly\/FailArmyiOS and Android: http:\/\/bit.ly\/FailArmyAndroid.\nEpic Fail in Football Field.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Computer Repair & Support, On Demand IT Services - Jupiter, Delray Beach, North Palm Beach | CEIgroup, Inc.\nThe CEIgroup. Inc.'s on demand IT services provide you with reliable, high-quality technology services that are never more than a phone call away.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Best Darn Steak Sandwich in Baseball (section 112). Featuring sliced herbed beef tenderloin served on a fresh brioche roll with balsamic glazed arugula, roasted Portobello, crunchy fried onion, and creamy blue cheese sauce. Served with house-made truffle chips. $23.\nGet your stretchy pants ready, Braves fans.\nThe team's third season at the soon-to-be-renamed SunTrust Park will usher in some big changes both inside and outside the park. One thing that hasn't changed? Concessions offerings that are over-the-top with meat and cheese.\nTake, for instance, the Blooper Burger, named for the Braves' mascot, which sees four cheeseburger patties, a foot-long hot dog and chicken tenders covered with lettuce, tomato, onion, jalapenos and cheese sauce, all sandwiched between Texas toast with \"Blooper\" branded onto the top.\nIt's \"everything you can find in a ballpark crammed onto a bun,\" Braves president and CEO Derek Schiller said of the sandwich -- which is also served with candied popcorn -- at yesterday's Braves Media Day at SunTrust Park.\nIf having it all on one sandwich doesn't hit a home run with you, there are several other new food options at the stadium this year from Braves' food and beverage partner Delaware North.\nKeep your eyes peeled for a chicken and waffle boat made with fried chicken tossed in mango habanero sauce set inside a powdered sugar-dusted waffle boat, topped with chopped pecans and honey and served with seasoned curly fries; the Dinger Dog, a foot-long, all-beef frank, split and filled with bacon jam and cheddar cheese, wrapped in puff pastry and topped with queso blanco and bacon; and what is being dubbed the \"Best Darn Steak Sandwich in Baseball,\" featuring sliced herbed beef tenderloin on a brioche roll with balsamic-glazed arugula, roasted portobello, fried onion and blue cheese sauce, and served with truffle chips.\nOther new options include the Hey Batter Batter Bird, a buttermilk-brined chicken served atop French fries with coleslaw; and the Southern Smothered Sandwich, made with smoked beef brisket topped with grilled peppers and Georgia Grown sweet onions plus cheese sauce on a hoagie with a side of kettle chips.\nSo what goes into designing new items to make them a hit with game attendees?\nHe said his culinary team travels a lot during the off-season to find inspiration, and that fan feedback on Facebook is also considered when cooking up new items or deciding to take items out of rotation.\nIf the new food options seem meat heavy, vegetarians still have plenty of options rolled over from seasons past, including salads and vegetarian sushi, available at the First and Third Base Marketplaces; salads and a veggie plate at the Terrapin Taproom; salads, hummus and vegetarian nachos at the ChopHouse; and veggie burgers at H&F Burger.\nAnd speaking of H&F Burger -- the popular Linton Hopkins-branded stand has moved this season from its former spot inside the ballpark. You'll now find it in a larger and year-round space off the plaza that connects the stadium to the adjoining Battery development.\n\u2022 Coors Light Chop House now features a new grill area on the top level known as Top of the Chop Grille, taking over the space vacated by H&F Burger. The quick serve location features fried chicken, kettle chips and loaded nachos with steak, chicken or house chili.\n\u2022 Terrapin Taproom, which serves food from Fox Bros. BBQ, has expanded indoor seating and a large patio on the Battery plaza, in addition to the Terrapin Fresh Brew Bar serving beers brewed at SunTrust Park in the Terrapin ATL Brew Lab. Also in the works is a staircase that will connect the first and second levels.\n\u2022Cigar lounge and eatery BURN by Rocky Patel, which will feature a design that draws on a mix of Mediterranean, Asian and Cuban cultures and will offer high-end cigars and exotic cocktails, is set to open soon in The Battery.\n\u2022Ph'east, a food hall that will feature several Asian-focused options including Lao eatery Snackboxe Bistro, ramen spot Lifting Noodles and poke option Poke Burri will likely be completed in the fall in The Battery next to Wahlburgers, along with specialty grocery store Savi Provisions.\nHungry but on a budget? Schiller points out options that start at $15 and include a game ticket, snack and beverage. And the Braves' longstanding policy of allowing fans to bring their own food inside the stadium (as long as it fits inside a gallon plastic bag) will remain in place this season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacnsl b/data_all_eng_slimpj/shuffled/split2/finalzzzacnsl
new file mode 100644
index 0000000000000000000000000000000000000000..00ccf7000034380f4c75b897fcef584c9f01b175
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzacnsl
@@ -0,0 +1,5 @@
+{"text":"As the leading company for directory service based technology, it would seem a simple choice to grant Novell this domain space. Novell's commitment to this technology extends back some 7 years now to place them ahead of all competition.\nI am only assuming that Novell will use the domain space to integrate with their directory technology to create a world wide web based directory tree that corporations and public bodies can participate in to build a truely wonderful authentication and management system.\nThe forethought of Novell to apply for a domain space to manage further demonstrates how an intellectual company plans for a true 'one net' that everyone can particiapte in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am spending the last few precious days of Ramadan in India with the extended family. Iftars here are rather different from the UAE. When fasting in Dubai, we tend to focus on the socializing aspect of Ramadan, and are spoilt for choice with the plethora of Iftars and Suhoor buffets on offer.\nIn my husband's hometown of Shillong (Meghalaya), Iftars are far more solemn and people prefer to enjoy them within the privacy of their own homes with close family. Neighbours exchange dishes, but mainly meet up in the mosque for the extended prayers.\nI spent the last couple of days at my maternal aunt Zuleikha Hazarika's place, and had the pleasure of watching her very talented cook, Ali, whip up a giant pot full of traditional Chicken Pulao (an aromatic chicken pilaf fragranced with the spicy notes of cardamom, cloves and cinnamon). In the true spirit of Ramadan, the pulao was divided up into boxes, and distributed across the neighbourhood to close friends, as well as the less fortunate.\nWash the rice in several changes of water, and soak for about an hour.\nWash the chicken, and coat with the turmeric. Add salt, and leave aside for about an hour to marinate.\nHeat a large vessel, and add half the oil. Fry the chicken pieces till brown.\nDrain the chicken, and add lemon juice. Set aside.\nIn the same oil, add the ginger-garlic paste. Fry till aromatic, and add the coriander, cumin and chili powders. Fry for a few minutes, then add the chicken and stir to coat each piece well. Remove, and set aside.\nHeat the remaining oil, and add the onions. Fry till they turn a pale golden-brown, and add the cardamom, cloves and cinnamon sticks powder, and the bay leaves. Fry till aromatic.\nAdd the drained rice, and fry vigorously till the rice turns brown.\nPour in the chicken mixture, and enough boiling water to cover the rice (the ratio of rice to liquid should roughly be 1 to 2).\nAdd the biryani masala, stir and allow to simmer.\nCook on medium heat till rice is cooked, and no excess water remains.\nTurn off the heat, and stir in the ghee and rose water.\nServe hot with chili pickles on the side.\nPosted in Holidays, Recipes and tagged Chicken Pulao, Food, iftar, india, ramadan, recipe, Shillong, travel. Bookmark the permalink.\nGet Tezzy's Posts by Email!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If e-mails fail to reach some of your members, they will show in the 'Temporary Errors' or 'Bounces' lists in the left handside on the 'Members' page. If they don't, it means we haven't received an error message back from the recipients' mail servers, which suggests it did get at least that far in it's attempt to reach your member's inbox, and therefore the problem is most likely at the recipients' end of the line.\nYour members are placed in the 'Temporary Errors' list if the e-mails you have recently sent haven't been delivered to their e-mail addresses. In this case, you need to check whether their e-mail addresses have been added correctly or contact the member to let them know that they aren't receiving your e-mails. If a few of the future messages are also undelivered, the member will be placed in the 'Bounces' list.\nYou can always try and correct any of the e-mail addresses listed in temporary errors by clicking on their e-mail address and clicking the blue 'Edit' button at the top of your database. Once on the 'Edit' page you can also send your members a verification e-mail or delete their record entirely.\nTheir mailserver has blocked incoming mails from groupspaces or your group's specific address.\nThe receiving mailserver doesn't allow messages sent from it's own domain (if you are sending from a company address to other company addresses, for example).\nIf you believe the e-mail addresses are correct and any of the above possibilities have been resolved, you can make a 'List' containing the temporary errors to re-send the most recent e-mail and try again.\nNone of your members who are in the 'Bounces' list receive the e-mails you send to your group, because attempts to send e-mails to their recorded address have failed a number of times (a 'bounce' is effectively a confirmed error). If your members have provided you with information in the custom database fields of your membership database, they will, however, still be counted as a member of a group. If they only have an e-mail address and no other information in your database, they will not be counted as a member of your group, and may be deleted after a period time.\nSome group managers have reported gmail addresses don't receive e-mails from the group after sending test emails to their own address. Gmail deliberately filters out e-mails that appear to be coming from yourself, so if this is a possibility for why your messages weren't coming through, please try again with a different person's address when testing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Are The Greatest Kitchen Paint Colors?\nWith thousands of colors to decide on from it can make it a tough choice for some folks. Make the most of a small, bland kitchen by generating a show-stopping floor in a vibrant colour, and maintaining accessories and decorations minimal. It incorporates a selection of soft and refined colors connected with the Tuscany area of Italy. I knew I had made the right selection really early on. I left when they arrived to buy one particular more bucket of paint. For a fresh twist on cottage style, coat your kitchen walls with pale blue paint. Hopefully these examples have offered you some tips of paint colors for your kitchen.\nThese premium paints for kitchens repel stains and clean up effortlessly to keep your kitchen searching fresh. When you are considering painting the ceiling most will like to go with a flat white, but if you are wanting to paint walls and ceiling the same color satin or eggshell will be great. Coffee, Espresso or Twilight Glazes will add a touch of distinction to Maple doors, creating dynamic kitchen paint colors, or colors for any room.\nAnd whilst I truly like the way the honey oak and dark gray work together, I do have that rather big wall space behind the hutch that if painted a strong gray could prove to be also modern for my taste. If you look at house magazines and kitchen styles you will notice these colors much more regularly. The stove doesn't have to be the only factor in the kitchen that provides off heat.\nBut by adding a single distinct color, say a casual red wall with the rest a dark khaki, it will quickly push the eyes to your preferred wall. They have discontinued that line of paint, but HD will nonetheless make it if you ask for it. I had it matched with the Sherwin-Williams paint by taking in a piece of wood with the color on it. Hope this aids!! For a red countertop in a kitchen I would bring in colors next to it on the color wheel.\nIron Mountain by Benjamin Moore is one of these colors that every time I see an image of it in a different space, the colour looks diverse. Right here, simple Glacier White Corian function surfaces leading classic, Shaker-style units and sit against a backdrop of glossy white metro tiles. Warmer colors such as red are believed to stimulate the appetite and are an outstanding alternative for kitchens. One more context in which blue is suitable in a kitchen is when it evokes a sense of water and liquidity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As Merleau Ponty explains it is only while immersed in the environment and aware of that experience can we understand enough to model our relationship with it.\nIt is therefore important to study experientially the actual phenomenon of perception first hand. Experiential awareness provides insights to the perceptual structure through which we develop our relationship with the real.\nThe difficulty is to capture and manifest these interactions in a tangible form in order that scientific evaluation processes can be reliably applied. Although subjective in nature, well choreographed intuitive accounts can clearly identify the fundamental patterns and processes that together constitute perceptual awareness. As the Blue Vase of Cezanne illustrates, the so-called deformations in visual art show us flashes of perceptual structure.\nVisual artists report their intuitive and direct observation and interaction with their environment through their artworks leaving physical traces of the engagement processes. These records contain vital information with respect to the nature of objectivity that has an appreciable impact on viewers from all cultures. For example, the Impressionists have left in their paintings traces or flashes of perceptual structure that engage most of us. These painters have to some extent rendered visible their 'experience of reality'. By engaging in such methodologies it is possible to substantiate aspects of awareness that hold profound implications for our approach to information display and our interaction with it.\nThe patterns and data structures contained in art works can with careful consideration, be quantified and modelled into algorithms to be deployed in digital technology providing a tool set and a controllable stimuli production platform for testing and evaluation purposes. It is envisaged that these stimuli and the processes involved in their generation can be integrated into cognitive architectures and used to deepen our understanding of brain structure, neural firing and neural pathways tracing back to receptor firing in the retina. Vision-Space technology is an example of such an experimental cycle merging the intuitive records of visual artists with the emerging understanding of the computational structures operational within the phenomenon of vision. In this way we can consider that Vision-Space takes the first steps in modelling visual awareness.\nVision is widely considered to be the lead sense and hence, the discipline of Visual Art provides a valid and strong skills base to intuitively probe the fundamentals of perceptual structure on a psychophysical basis.\nthat we receive (retinal processing) through our visual system (visual pathways) two independent 'takes' of the physical world (what and where characterisations) and that this duality manifests within our presentations of vision.\nthat phenomenal field (perceived visual field) can be understood to be multifaceted even on a monocular basis. Vision-Space identifies two independent monocular data formations (data-sets) associated with explicit (macular vision) and implicit (peripheral vision).\nthat there are cognitive processes that manage the juxtaposition of the data-sets within phenomenal field.\nthat these data-sets can perhaps broadly be aligned with independent forms of attention (sub-conscious and conscious\/right and left hemisphere 'takes' on reality).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadazb b/data_all_eng_slimpj/shuffled/split2/finalzzzadazb
new file mode 100644
index 0000000000000000000000000000000000000000..b4165849ce04ee2e2d3541e0715f6d150cf1ffc9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Do you want to be happy? We are all looking for an escape from worry, and depression, and for ways to find happiness. This book presents the route to happiness in a nutshell, drawing on Islamic teachings and the voices of \"experts\" both western and eastern. So sit back, relax and read it from cover to cover, or dip into it a page or two at a time in between other activities in a busy life as a wife, mother, student or worker.\nYou are holding in your hands a treasure chest of reminders that will help you learn important lessons and set you on the path to happiness in this world and in the Hereafter, Insha' Allah.\nClick the button below to add the You Can Be The Happiest Woman in The World A Treasure Chest of Reminders (PB) to your wish list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What if Mallus hadn't been in just Nora Darhk? What if the majority of his spirit was inside someone else? Someone no one could have suspected?\nMurphy was a coward, but he was never this evil.\nHatred and probably whatever ran in his blood had turned him into something inhuman.\nAt least, that was the only way 10k could rationalize what was happening to him.\nSeason 3 canon divergence: events in Murphytown leave 10k more tormented than ever. His experiences with losing all control of his body manifest in cruel ways.\nWill Warren and the others be able to protect him? And if they do, what damage will 10k have to recover from?\nAU set in 3x03. Upon her return from Maveth, Daisy helps Jemma remember her new name in an unforgettable way.\nIn Boston, Henry Swan's six-year-old brother Ian finds a book titled \"Once Upon a Time\" hidden beneath the seat in their mom's old yellow bug. As soon as Henry touches it, he remembers.\nSeason 3 Canon Divergence-Emma finds out she's pregnant a few weeks after she and Henry leave Storybrooke with new memories and new lives. Nearly seven years later, another Dark Curse puts her family in danger, and Emma must return to Storybrooke to help them.\nWho's powerful enough to cast the Dark Curse? And how the hell is she going to tell Hook they have a son together?\nAlec didn't know when food became a major part of his life. Before it was fuel, something he needed for survival. Now, it was both a friend and an enemy.\nSet after the breakup. Alec finds a way to cope, but finds himself losing control of the situation.\nSam, Dean, a witch, and a Faery. What could possibly go wrong?\nGoes AU after 3x07. Overwhelmed by everything that had happened because of Flashpoint, Barry goes for a run and accidentally goes to the MCU during the final stand against Ultron. Stranded, Barry finds a new family in the Avengers but what happens when his old family comes looking for him? Read to find out.\nMatt Murdock is back from the dead, and has just learned that his archnemesis Wilson Fisk is out of prison. Knowing he can't take Fisk on all by himself, he decides to team up with the one person he knows who is as dogged and determined as he is to put the man back in prison: Karen Page. As Matt and Karen rebuild the romance they once had, they must also fight a lengthy battle with a man who is determined to destroy their lives, their livelihoods, and their friendships.\nAKA: What if Matt had help when he was going after Fisk?\nWhat would happen if the girls explored Mako a little bit more? Mermen? Maybe. A lost civilization? Possibly. A tail-full of trouble? Definitely. Reimagined Season 3, inspired by 'Creating Atlantis' by TempDuLoupe on Ffnet.\nStiles rechaz\u00f3 el mordisco una vez, pero ahora es una cuesti\u00f3n de vida o muerte. Peter y Derek dan un paso adelante cuando menos se lo espera y m\u00e1s lo necesita.\nJaime arrives at King's Landing earlier, Lord Tywin expels him from the Kingsguard and forces Jaime to marry Sansa, instead of Tyrion. \/\/ A take on their wedding night.\n\"Oh.\" Keith blinked. \"Huh. Okay. But we got the info, right?\" He asked. He remembered retrieving the USB (it wasn't really a USB, but that was the closest english translation) pretty clearly, even if the rest of the mission's details got fuzzy in the end.\n\"What was on there?\" Keith said, and moved to sit up. \"Is Regris okay-\" He cut himself off when he caught a glimpse of his hand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"the weather in NYC lately has been insane. I've been DYING for a day that wasn't -22 (yes we HAD THAT) so i could wear this amazing piece from ABERCROMBIE. its the softest vegan leather jacket EVER and has amazing fringe coming off the arms and back. paired with my fave basic tee from BALM & some men's sweats, its the perfect outfit to roam the city & stay comfy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stay ahead fashion carrying with this backpack from the house of Horra. The adjustable strap ensures great carrying comfort all day long while the snap closure is for keeping your stuff safe. Besides, it has been made of PU material that makes it easy to maintenance.\nAccessories your casual outfit by carrying this trendy backpack. Designed impeccably, it will also help you to flaunt your fashion statement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Feel like giving up on love?\u2026.\n\u2026as Winston Churchill said in 1954\u2026..persist\u2026.persist\u2026.persist\u2026.persist\u2026..\nFeel like you have had enough? Want to give up? Sick of being misunderstood? Had your heart broken? Had your trust broken? Suffered loss? Don't want to waste anymore time? Don't have any energy left?\n\u2026\u2026.this is a turning point, a turning point for your character, a turning point for your strength, a turning point for your resilience\u2026.your beliefs\u2026 your faith\u2026.your stamina\u2026..your core\u2026.\n\u2026this is it\u2026.this is the time when you either hold on\u2026or you give up\u2026..\n\u2026..because you know deep down\u2026..somewhere deep down\u2026 in your soul\u2026in your heart\u2026..in your mind\u2026..you know\u2026..you know you are worth fighting for\u2026.you are worth believing in\u2026.you are worth backing\u2026\u2026you have so much to offer\u2026.you have love\u2026.you have heart\u2026 you have generosity\u2026you have soul\u2026.\n\u2026\u2026you start turning\u2026turning\u2026.turning\u2026turning in\u2026.turning in towards yourself\u2026.towards your heart\u2026towards your love\u2026.towards your soul\u2026..\n\u2026you pick up your sword\u2026\u2026you look\u2026.you look\u2026..you look in the mirror\u2026. And you see love\u2026.you see support\u2026..you see admiration\u2026..you see encouragement\u2026 you see your love\u2026.you see your heart\u2026. you see your worth\u2026 you embrace\u2026 you value\u2026.you treasure\u2026you believe\u2026.you see happiness\u2026..you embrace\u2026\u2026.you love\u2026..you fight\u2026.\n\u2026\u2026\u2026.So never give up\u2026..pick up your sword\u2026\u2026\u2026fight\u2026\u2026.always hang on\u2026. always take that next step forward\u2026. Have faith\u2026know you are looked after\u2026.believe in miracles\u2026\u2026believe in change\u2026.trust\u2026..trust\u2026and\u2026trust\u2026.trust everything is exactly the way it is supposed to be\u2026..trust and believe\u2026.trust love\u2026..know love\u2026.feel love\u2026.be love\u2026persist\u2026..persist\u2026.persist\u2026.persist\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafjkf b/data_all_eng_slimpj/shuffled/split2/finalzzzafjkf
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index 0000000000000000000000000000000000000000..084cd5ef6fce894342f8aeb213f38be1e911aeb8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Stillwater Music partners with Mountain Middle School to offer this addition to Mountain's elective options. Participating students meet twice a week to rehearse during the school's regularly scheduled elective period.\nStudents who participate in the Stillwater Music SIG form a band, learning songs in a variety of contemporary musical styles. They will get to perform for their peers at the Mountain Middle School Community Meeting at the end of the semester.\nStudents have the opportunity to sing and to play a variety of instruments, including drums, bass, guitar, piano, vibraphone, marimba and steel drums, as well as any instrument that they own and can bring to class on a regular basis.\nThis elective runs for the full semester!! Because these classes require students to be present in order for the group to progress, regular attendance is mandatory. Excessive absences may result in a student being dropped from the class at the instructor's discretion.\nScholarships are available. No student will be turned away for financial reasons. For scholarship information please contact Jana Leslie at [email protected] or 970-246-5095.\n*Tuition is non-refundable after the second class, with some exceptions. Please see Stillwater's Policies & Procedures for full details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These are some of the books or authors Cindy Rollins recommends.\nToday is the first day of summer by my calendar and it is hot! So hot it is the perfect time of the year to curl up in the air-conditioning with a good book.\nSometimes, though, when we let our kids loose in the library they come home with less than stellar choices. While I like to give my children more freedom over what they read in the summer months, I also like to guide them to excellent suggestions. Here are a few series of books that I steer my children towards.\nArthur Ransome's Swallows and Amazons series: These are THE perfect summer books. If you can't go to the lake and don't own a boat, just pick up these books and head towards your own childhood summer place.\nIf your family enjoys Arthur Ransome then try E. Nesbit. Her series include The Bastable series beginning with The Story of the Treasure Seekers; The Five Children and It series, and The House of Arden series. C.S. Lewis grew up reading Nesbit.\nThe Scout Series by Piet Prins: These books are perfect for the boy just beginning chapter books and even older. After your son finishes the Scout books look for others by this excellent Dutch author.\nIf your child enjoys the Scout books then he or she most definitely will enjoy the author Hilda Von Stockum who has written The Mitchell series and The Bantry Bay series.\nNeed a little mystery and suspense? Try Joan Aiken's Wolves of Willough by Chase books or L.M. Boston's The Children of Green Knowe books. Both sets are suspenseful and fun. Or is you want suspense with a Christian worldview try Frank Peritti's eight volume Cooper Kid series. Begin with The Door in the Dragon's Throat.\nThis summer our family is trying a new series that came highly recommend: Dealing with Dragons by Patricia C. Wrede.\nIf you trust in yourself, and believe in your dreams, and follow your star \u2026 you'll still get beaten by people who spent their time working hard and learning things and weren't so lazy.\nFor the older children L.B. Grahame has written 5 Christian fantasies worth the time and effort to read. Start with Beyond the Summerland.\nOr there is the great Christian Covenant series by Douglas Bond beginning with Duncan's War.\nOr how about the newer series by N.D. Wilson: 100 Cupboards, Dandelion Fire, and The Chestnut King?\nDid you spot Harrison Ford in Chattanooga recently? A little Indiana Jones-like adventure may be just the ticket. Try H. Rider Haggard's King Solomon's Mines or John Buchan's Prester John . When my older boys can't figure out what to read their go-to author is the Frenchman Rafael Sabatini. His swashbuckling stories are sure to please.\nNow that the kids are busy reading mom might just have time to clean the house. No way! Forget about the dust bunnies. Summer is the time for mom to refill her reserves, not just clean the house.\nHow about Dorothy L. Sayers' Lord Peter Wimsey books? I am very jealous of those of you who have the opportunity of reading these books for the first time.\nThe Brother Cadfael books by Ellis Peters are a great series of books to get you over a reading slump. Start with A Morbid Taste for Bones.\nDo you like your mysteries a little metaphysical? Try Tim Powers. His Anubis Gates is top-notch time travel.\nEver hear of Helen MacInnes? Her many books give a compelling look inside The Cold War. Spy novels for the intelligent woman!\nTry D.E. Stevenson or Georgette Heyer. These won't leave you feeling guilty. I am enjoying Stevenson's Mrs. Tim and the Regiment right now on my Kindle and it keeps me smiling.\nCold Sassy Tree by Olive Ann Burns will have you laughing and nodding.\nOr Penhally by Carolyn Gordon will be a somber reminder of what used to be.\nThe books in this category are a dime a dozen but anything by Mark Steyn will keep you laughing through the tears and Jonah Goldberg's Liberal Fascism is truly a great book for understanding the history of politics.\nI know you need a little inspiration before you begin it all over again in August. You could join me in reading through Charlotte Mason's Towards a Philosophy of Education. I guarantee it will change you and free you.\nDavid Hicks' Norms and Nobility is a slog well worth the effort.\nI love Vigen Gurorian's Tending the Heart of Virtue for sheer inspiration.\nEver read Ravi Zacharias? You are in for a treat.\nMere Christianity by C.S. Lewis is always profitable as a reread.\nMichael Horton's Before the Face of God is one of my favorite Christian books.\nI hope this keeps you busy for a while. No need to offer you children rewards and prizes for reading. Just offer them great books that will keep them begging for more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The home you're buying can have hidden dangers, so investing in a radon inspection may be beneficial. Homes throughout Watertown, SD have this dangerous gas in the lower levels and require special remediation systems to keep the property safe. HomeTeam Inspection Service offers testing for radon and mold in order to provide you with accurate information about your indoor air quality.\nAs the professional leader in home inspections, we're proud to be fast, trusted, and accurate. Our team in Northeastern South Dakota offers radon inspection services in addition to general home inspections. Your satisfaction is our top priority, so call HomeTeam Inspection today for your convenient appointment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now you can make a donation simply by filling out an online survey. Peanut Labs has created its Goodsurveys platform as part of the charitable donation site Goodsearch, which makes a donation to a nonprofit or school every time you do a search.\nGoodsearch built Goodsurveys.com in collaboration with Peanut Labs as a cost-free way for people to raise money for causes by taking short online surveys. So you can make a donation simply by sharing your opinion.\nGoodsurveys provides a wide range of frequently-updated surveys that users can take for free. When you finish a survey, Goodsurveys donates about a dollar (on average) to your favorite non-profit or school. Goodsurveys is part of the Goodsearch family of products, which have collectively raised almost $10 million for more than 100,000 non-profits and schools. The reason is that your actions \u2014 web searches, filling out surveys, or shopping \u2014 generate money for a sponsor or advertiser, who pay a fee for access to you. Peanut Labs operates a for-profit business where people fill out online surveys on social networks, and it is using the same technology to power Goodsurveys.\nCharities are looking forward to getting donations. About 109,000 organizations are able to receive donations from Goodsurveys.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whitney Houston and Bobby Brown's daughter, Bobbi Kristina, has been in and out of the hospital since her mother passed on Saturday. Today, Bobby released a statement concerning his daughter's current status.\n\"My daughter Bobbi Kristina is doing much better,\" Bobby tells ET. \"We continue to provide love and support to Bobbi Kristina. She is dealing with the tragedy of her mother's death and would prefer to do it outside of the public eye. I ask again that our privacy be respected.\"\nWhitney's autopsy was conducted on Sunday, but no cause of death has been determined pending the results of toxicology reports that are expected to take several weeks. Whitney's funeral service will be held Saturday in Newark, New Jersey.\nCLICK HERE for the latest Whitney Houston news.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafxoc b/data_all_eng_slimpj/shuffled/split2/finalzzzafxoc
new file mode 100644
index 0000000000000000000000000000000000000000..1a710bf0fc93ab3c8512a8d2502cffa8ba65a82f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzafxoc
@@ -0,0 +1,5 @@
+{"text":"Poly Frost is thin transparent (frosted) plastic material with slight texture on one side. Because it's transparent, we can only print on one side. 100% white color will not be printed so if you want a transparent background for your card, leave background at 100% white. Since they are plastic, they are also waterproof. Please, keep in mind that most colors will be somewhat transparent depending on color density, dark colors work the best. The thickness of the cards are or the same as a 14pt card, but may tend to feel slightly more flexible simply due to the material - They are NOT credit card thickness and are NOT rigid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's political season again. People are going to be talking about the election and, as always, there are going to be sharp disagreements. For your part, here is how you should handle this political season at work.\nAre you waiting for permission to be the best and fullest version of yourself?\nAre you waiting to be recognized so you can finally share your gift with the world?\nAre you waiting for a sign? Are you waiting for just the right moment?\nHave you ever had a salesperson ask you to buy before they had done the work to deserve that commitment? Remember how you felt that there was some invisible line that they just jumped over, pen and contract in hand?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It might seem to us like Jesus is taking his time coming back. But as the author of Peter tells us, God is patient, more patient than us. God does not want anyone to perish, but wills for all to experience God's love and acceptance. Therefore, God is taking it slow. God is waiting, constantly beckoning, constantly saving and constantly calling us. Come, follow me. I have enough room for you all. If God can be patient, so can we. We can go about living our lives of godliness and holiness because we can trust in God's promises to us. We can strive for peace in this violent world. We have the capacity through grace, faith and the Word to wait with God for all people to understand God's glory. If God is not ready to judge the sinner and unbeliever, we will not judge either.\nLord, it sometimes feels like the suffering and violence in this world will never end. But you see it all. Help us to work with you as you reconcile all people in Jesus Christ. Amen.\nAssociate Professor of Systematic Theology at Luther Seminary, St. Paul, Minn.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One valuable tool which you can use to keep track of your binary trading decisions and outcomes and also plan for the future is a trading blog. A trading blog is basically the same thing as a trading journal, except that you are keeping it in the public eye. Some trading blogs become very popular resources for traders who are learning or teaching. So writing a trading blog is in essence also a community service. Not only are you helping yourself, but you are helping others by making your experiences public and sharing your learning process. A trading blog also gives you the additional accountability which comes with public exposure. So how do you write a good one?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home Social Media Is Facebook Taking Over Youtube?\nIs A Shift Taking Place?\nAs per another report, Facebook has unseated Google's YouTube as the video supplier producing the greater part of video movement on administrators' systems \u2013 in any event in a few nations on the planet.\nIn Openwave's second yearly Mobile Video Index, Facebook outperformed YouTube as the prevailing versatile video spilling administration in Africa, creating 36% of every portable datum movement in the locale, contrasted and YouTube's approximately 25%. The firm said that pattern could well play out in different nations around the globe too.\nFor organizing administrators the discoveries are vital considering video involves the main part of system movement. As per Cisco, portable video activity represented 60% of aggregate versatile information movement in 2016, and the organization anticipated that figure will ascend to 78% by 2021. Along these lines, guaranteeing the expedient conveyance of that video is a key component of administrators' promoting message.\nFor U.S. administrators the discoveries are especially essential thinking about AT&T, T-Mobile and Sprint have all firmly tied their systems and their administration wants to video. For instance, AT&T is attempting to offer its DirecTV Now gushing video administration to its versatile supporters and is zero-rating content from the administration for its clients who buy in to metered information designs. Interim, T-Mobile offers Netflix memberships for nothing to its clients, while Sprint is utilizing a comparable offering of free Hulu memberships.\nIn any case, as indicated by Openwave's most recent report, few of those administrations as of now rank among the best vide activity generators in North America. The firm found that YouTube produced 31% of the locale's versatile information movement, trailed by Facebook at around 20%.\nOpenwave's measurements are gotten from organizing movement figures from around 30 taking an interest administrators around the world. The organization offers video and system programming and innovation items to telecom administrators universally.\nThis might mean that it is beneficial to not just stick to youtube but also to diversify and use other platforms. It could also be a sign that a shift is already taking place.\nPrevious articleU.S. Startup Is Close To Getting $400 mln Investment from Saudi fund!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahemc b/data_all_eng_slimpj/shuffled/split2/finalzzzahemc
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index 0000000000000000000000000000000000000000..7a31a2d0d0b2ca8767cdd7bb78e79c41f52b61fb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahemc
@@ -0,0 +1,5 @@
+{"text":"Lantern ceiling lights are a great choice for hallways, bathrooms and kitchens offering a more relaxed vibe than traditional pendant lights. Shop our edited selection of hanging lanterns for all styles of interior. Add shades in a contrasting colour to the metal finish to incorporate colour into your room scheme and for a more diffused lighting solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The opposition party surprised everyone by scoring a hat trick in the elections.\nJohn scored a touchdown when he managed to get the million dollar business deal signed.\nOur team hit a home run in last night's design competition by winning all three awards.\nHe is a strong candidate to run for Mayor, but he is not a slam dunk to win.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a PhD Researcher in the School of Earth and Environmental Sciences.\nThe Crato Formation of northeast Brazil is famous for its exceptional preservation of fossils. It yields exquisitely preserved plant, insect, fish and reptile fossils that represent an entire ecosystem. The ecology and diversity of this formation has been under study for many years, but a full understanding of the processes that gave rise to the exceptional preservation is lacking. This research project examines the preservation of the insects from this formation, as they are the best preserved (even individual muscle fibres are preserved) and are relatively common.\nThe main aims are to identify the taphonomic and diagenetic processes leading to the exceptional preservation, including the elucidation of death to burial pathways and the identification of replacement mineral phases. This involves understanding the sediments in which the insects were first deposited, how those sediments changed over time (diagenesis), and how the minerals respond to weathering.\nA full understanding of the preservation of these insects will not only expand our knowledge on how some of the highest quality fossils in the world are formed, but may also reveal any biases that are present in the fossil record.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whatever your event, we have the perfect space to make it an exceptional occasion.\n3 versatile spaces we can tailor to suit your event; Private Dining Room, Games Room and The Terrace.\nOr blow your guests away when you hire the all-weather terrace bar, with fully retractable roof and Prosecco bar. It really is an amazing space, for any event!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dr. Stuart L Silverman, MD is a Doctor primarily located in Clairton, PA, with another office in Pittsburgh, PA. He has 34 years of experience. His specialties include Neurology and Psychiatry. He speaks English.\nThe overall average patient rating of Dr. Stuart L Silverman is Good. Dr. Stuart L Silverman has been rated by 30 patients. From those 30 patients 10 of those left a comment along with their rating. The overall rating for Dr. Stuart L Silverman is 3.2 of 5.0 stars.\nI was referred to Stuart Silverman by another incompetent stumblebum (Dr. Brian Negrini UPMC) in an attempt treat a head \/ neck injury. After waiting for 2 hours, I was finally called in to the examination room where I met Dr. Stuart. I then briefly summarized my medical complaint. I was given no physical exam whatsoever. At no point was any type of reflex test or neurological evaluation administered. He didn't even make a token attempt to take vitals. Nothing. Literally all I was asked to do was walk a few steps toward the wall, after which he replied ' I don't see anything wrong.' The entire encounter lasted about 4 minutes. I was told I would require an MRI but had to wait until it was cleared by my auto insurance. So I left the office thinking I would return when my insurance contacted me and approved the imaging procedure. I think I was justifiably shocked when I received a bill for hundreds of dollars the following week for services rendered by Dr. Stuart. Essentially I was charged, and begrudgingly paid, nearly $300 to sit down and shake this fat idiot's hand. Did I go to see a Dr. or Bruce Springsteen? I understand that there is a staff on payroll, and overhead, and overhead on the overhead and the Dr.'s time is worth money. But $100 per minute seems excessive given that I was summarily blown off and not examined at all. And also I guess my time is worthless and the 2 hours I wasted in the waiting room watching Judge Judy is going to be refunded too.\nVery good doctor! I would absolutely send friends and family to him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahoxw b/data_all_eng_slimpj/shuffled/split2/finalzzzahoxw
new file mode 100644
index 0000000000000000000000000000000000000000..f033db4def9b5bf496c7dd001e966c9830c8a001
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahoxw
@@ -0,0 +1,5 @@
+{"text":"Scouting Team player Eyad Hammoud (24\/07\/2001, striker), of Sheffield Wednesday FC, is part of the group for the U18 game against Nottingham Forest FC. The away game, will be played at Gresham Close (West Bridgford, Nottingham), on 09\/02\/2019, at 11:00am (local time).\nScouting Team wishes Eyad good luck for the game!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Don't see the answer to your question here? Please contact us, and we'll be happy to help!\nIs Homeopathic Medicine different from Naturopathic Medicine?What do Naturopathic Doctors use as forms of treatment?\nIs Naturopathic Medicine covered by OHIP or Extended Health Insurance? What should I expect during a naturopathic visit?Are you accepting new patients? How can I refer friends and family?\nWhat is Acupuncture? How does it work?What is Acupuncture used for?\nBesides needles, what are other treatment options?What is acupressure?\nWhat is moxibustion?What is cupping?\nAre there other specialized uses for acupuncture? Is acupuncture safe?What should I expect on my first visit?How often will I need to come in for treatment?\nWhat is psychotherapy?Why do people seek counselling?\nWill it work for me?What can I expect in a session?\nWill I need to stop medication?Can I stop if I feel it's not working?What kind of records are kept?What about vacations and canceled sessions?When will I no longer need psychotherapy?I know people what have been in psychotherapy for years. Does that mean that they are not getting better?Is Psychotherapy covered by OHIP or Extended Health Insurance?\nWhat does massage therapy treat?What are the effects of chronic muscle tension?How are Registered Massage Therapists trained?What should I expect during my first visit? Where will my massage session take place?\nHow will a massage feel? Does massage therapy hurt?\nWhat if the pressure isn't deep enough?Can I ask my massage therapist to make adjustments for my comfort?\nWhat are your Clinic Policies?Will my information be shared with anyone?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're back from Holiday. We had a great Christmas, hope you did too!\nMy family kicked the holidays off with a little bowling. It's besides the point to tell you that I never found my groove and scored ridiculously low.\nMy parents and Vance were accused of being closet bowlers because of how well they scored. Vance earned his first Turkey and was super excited.\nI believe everyone had a good time. Bowling isn't something that we do often, so it's always fun to get out there and roll the 7 lb. balls.\nSide story: We lost three pairs of pants due to knee blow outs while we were away. Two of Vance's, one of Vaughn's. The first of Vance's almost made me cry. Holes in the knees are common occurrences around here, so I thought if I bought more expensive, lined pants it would help. Wrong. Turns out that no fabric is strong enough to hold up against sliding on bowling alley floors, which is what all the boys did in between turns that night. He had worn these pants 3 times, which makes this investment just plain silly.\nWhat's up for today? We returned to a house without food, so I'm dragging the kids to the grocery store which is always a lot of fun. Christmas Eve pics are up next. Today or tomorrow.\nLOVE that picture!!!! They sure had a great time together. Give the kiddos a big hug for me...and give yourself one too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So I've spent this past weekend in a kind of interesting situation. I own two macs, you see. One at home (which is very much dead, after a rollover accident which saw it chucked out the window, and Apple wanted north of $1K to fix), and one at work. Both of these devices run Mountain Lion. Well, since the dead mac is basically useless to me, I want to take its copy of Mountain Lion and install it on my PC.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"And did you get what you wanted from this life, even so?I did.\nA week ago I kicked off the L*O*V*E project by exploring the definition of love. I sent out questionnaires to people that I know from a broad range of age, locale, marital status and gender. To read part #1 click here. We found that to be able to define love is difficult at best. We learned or re-learned that there are different kinds of love depending on the type of relationship and that love is about connection, openness and aliveness.Part 2 of the L*O*V*E project continues and the answers are so amazing, juicy and profound I can hardly wait to share my findings with you!\nI am immersed in love and the meaning of it to others. Here is an image for you\u2026 I have hard copies of every answer I received and they are spread in a circle on our living room floor. I have spent most of my time sitting in the center of this circle, surrounded by these heart-felt answers about love\u2026 may I just say \"yummy-licious?\" I LOVE this and recommend this to anyone who needs a good shot of love.\nOur second question: Why is love important to you personally?\nYet again, connection is a major theme in the responses to this question.As I consider this, it seems as though connection is something we think about as being simple.\nIt is simple\u2026and yet it really isn't at all.\nHere are a few answers to show you what I mean:\" \u2026As a human being I am truly wired for connection, and a connection of love is the deepest, richest and most wonderful of all\" said Peter. Shannon's response was \"Love connects me to the universe. It makes me feel like I'm not alone and I am valued.\" Aimee used this quote to convey her answer: \"Love cures people\u2014both the ones who give it and the ones who receive it\" (Karl Menninger).\nWow, love heals through connection, and love tells us that we matter through connection. This is no lightweight subject.\nWhat excites me about love is how transformational it is \u2013 for anyone, anywhere and at any time. This is HUGE. At any time we can extend love to anyone and anything that crosses our path.\nI would like to see us have more conversations around this. Why is there not a higher value put on the connections or bonds we have? If love is about being connected and a sense of belonging, why do we sequester it to our home or private life most of the the time? Can we push our boundaries a little further out into the world?\nThe LOVE project is showing us that connection and engagement are key elements of love.\nSo to bring all of these points together, here is a crazy idea\u2026 let's bring this kind of engagement wherever we are. Love is about Expansion \u2013 of ourselves and of others.\nThis is what my vision is for our future and is the focus of my coaching practice when working with clients who seek heart-based businesses in alignment with their heart-centered lives. We can connect and show love at any time.\nThat sounds pretty serious for a project on love and here is why:77% of the participants who answered the question \"Why is love important to you personally?\" said love is the reason they are here. It is their purpose with a capital \"P\".\n\"Wow\u2026it's the number one emotion for me and from it everything else flows..\" commented Dan (who did move half way across the world for love).\nDid you catch that? From these strong answers love is the number one thing for our purpose here, and we would not thrive and even exist without it. Without the fostering of connection, and healing that love brings, our world would be a desolate one indeed.\nAll of this is speaking to how fundamental love is in our lives. It is what gives us meaning, purpose and fulfillment. And it can do this not only in our romantic relationships but in every single one of our relationships and encounters.\nI would like to close with a challenge: As you go through your day, be aware of where you are loving, and where you are not so much.\nThen, go forth and love more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahzrm b/data_all_eng_slimpj/shuffled/split2/finalzzzahzrm
new file mode 100644
index 0000000000000000000000000000000000000000..050fa00fd0c53633f2a277676cfb0ed302e9809c
--- /dev/null
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+{"text":"Need a propane tank delivered ASAP? Order this service and receive your delivery on the next business day Mon-Sat.\nGet your order delivered Next business day Monday - Saturday.\nSometimes we just forget to put our empty tanks out for exchange. Order this service to have your tank redelivered.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are looking for the best travel blogs to follow (besides mine of course!) we've assembled the definitive list of the Best Travel Blogs 2019. This massive travel blog directory is sure to inspire!\nTraveling is a passion for many, a job for others, and somewhere in between for most. As I've started my own journey into the world of travel blogs, I've been amazed, awed and inspired by some of the great travel blogs out there. I don't really view them as competition, but rather as inspiration and I want to share some of the best ones with you.\n3 (Overall) Best Travel Blogs. In this category, I've included all types of travel blog. The very best of the best travel blogs. Ranging from pleasure, to business travel to something in between, I've picked out the 3 Best Travel Blogs.\nBest Business Travel Blogs. Here, we find the grinders. The road warriors. Those who travel almost every week for their day job. This is me. There are not many folks blogging about this experience, but I see my fellow business travelers every week as I head out for my next work trip.\nOther Great & Interesting Travel Bloggers you NEED to follow. This is a list of other travel blogs that have great content, interesting posts, and more. Some of these are tourism blogs, others adventure blogs, some fit the category of \"personal travel blogs,\" but, all are REALLY good travel blogs.\nSolo Female Travel Blogs. A list of the best and most interesting solo female travel blogs you ought to be following!\nDuos & Couple Travel Bloggers. A collection of the most interesting couples and duos who've taken their lives on the road. Couple travel blogs are pretty neat and we have some good 'uns here.\nFamily Travel Blogs. Get out there with your kids and more! A fun list of the best family travel blogs I've discovered.\nThe Up and Comers. A list of new travel blogs that I've not only exchanged links with but who I think are worth following \u2013 they might just have a bright future travel blogging!\nSo let's get to it. Without any further ado, here is the ultimate list of the best travel blogs 2019 edition.\nDuring that time, I've discovered 3 travel blogs that everyone should be following. These travel blogs, in my opinion anyway, are the best of of the best travel blogs. They are the tip of the spear, the creme de la creme. I think you will like them too.\nNomadic Matt. Matt, in my opinion, sets the standard for all travel blogs on the internet. He meets all the criteria: Useful content, attractive website, highly popular, and his authenticity comes through. Plus, he's a fan of Pinky & the Brain, which honestly is all that was really needed here.\nYTravelBlog: I found the YTravelBlog by accident, while searching for ideas on how to structure my blog and boy am I glad I did. Their travel blog is more of a general travel blog, that's a bit more focused on family travel blogging. Their site is clean, well organized, and they offer well written, engaging, and authentic content. My family has used their site when planning family vacations! #MustFollow.\nEverywhereist. Geraldine DeRuiter is an acclaimed author, world-renowned public speaker, and the voice behind the award-winning Everywhereist blog. She writes about dessert, feminism, and Jeff Goldblum's entire filmography (which honestly should be enough to lure you in at this point). Her personal travel blog has a unique and fun design that is both clean and visually appealing and the stories are rich and descriptive.\nA Dangerous Business Travel Blog. Despite the name, this is NOT a business travel blog. It is, however, a wonderful collection of wonderful spun travel tales told by Amanda. They are full of vibrant pictures and an earnest voice.\nOn The Grid: This is an interesting choice for the best travel blogs list, because it's really a collection of hundreds of localized travel bloggers providing their feedback and reviews of local eateries and neighborhoods. But the content is stellar, and written from a local perspective which is always best. You need to check these guys out.\nHandLuggageOnly. We just recently discovered these guys and their awesome travel blog. Here are two buddies from college who travel the world. Their blog is clean and well organized, and highly visual. Great stories, hacks and more. They also have a MASSIVE following on Social Media. Worth adding these fellas to your regular reading list.\nExpert Vagabond. Matt Karsten runs a beautiful travel blog with a heavy emphasis on photography and adventure. He's an American Expat living in Italy \u2013 but still traveling all over this beautiful planet.\nKeep reading, because we've only just scratched the surface on the best travel blogs around.\nWilliam Zinsser's definitive guide to writing is mandatory reading for travel bloggers wanting to get serious about their craft. With simple, direct suggestions, On Writing Well will help you improve your writing chops in no time.\nAs noted in the intro, there are not a lot of business travel blogs out there today.\nI don't know why that is, because there are certainly a lot of us business travelers out there every day. Believe me, I sit next to them every week on my Southwest flight.\nBusiness Travel Life: Business Travel Life offers competent biz travel information. They feature a clean, simple site, loaded with useful practical information that every business traveler can benefit from \u2013 with particular focus on healthy travel.\nRoad Warriorette: The Road Warriorette is run by a serious business professional, Sarah, and she knows what she's talking about. She offers great tips, practical advice, and excellent dining ideas for those who spend more nights in the next Hampton Inn than they do in their own bed.\nCBoardingGroup.com (me!): I am including my new blog in this list, because, well\u2026it's my blog directory. Haha! But also, there are really not that many business travel blogs. I'd like to think that my focus on topics like Travel Hacks, Travel Tips, Healthy Business Travel and of course Gifts for Business Travelers are actually useful for my fellow business travel brethren. Time will tell though. LOL!\nStuck at the Airport. This business travel blog is probably my favorite in terms of aesthetic appeal. Harriet Baskas has done a tremendous job with her sight. It's like Pinterest, but a travel blog. Pretty sweet looking. It's also, FULL of wonderful content that the business traveler can use on their daily grind.\nFrequent Business Traveler: This is an established business travel blog. But it's in need of a blog makeover. Content is solid, don't get me wrong, but it feels like I traveled back in time and found this site via my search on Excite.com. Looks aside, it has wonderful, useful content, which frankly is more important.\nThe Military Frequent Flier. The Military Frequent Flier is run by a couple of vets who know how to travel. While they are no longer actively serving, their years of travel during their service have given them unique perspective on travel. Definitely worth a follow.\nTravel After Five. A fellow business travel blogger I discovered that puts out great content about life on the road. Clean site. Good pics\u2026and great guides for what to do AFTER 5pm\u2026which is a real problem for us regular business travelers. She also has a Road Warrior Handbook you might want to check out.\nBusiness Travel News: These guys should be the gold standard for business travel blogs and I would have named them so, if they were not SO polished. In my view, a travel blog is run by 1 or 2 people and it's a side hustle, not the main squeeze. BTN is a great site, don't get me wrong. SOLID advice, business travel news, ideas on points programs, etc. But it's also a full blown business. Still, add it to your business travel blog feed.\nTravelling for Work. Another work travel blog that shares ideas on how to step up your travel game while on the road.\nDid I miss your business travel blog? Do you belong on this list? You probably do, so hit me up on Twitter and let me know!\nHere is a fun list of some of the best and most interesting travel bloggers worth following. Some are just personal travel blogs, others are more lifestyle blogs with a travel angle, some focus on food blogging while traveling. Think of this section as a giant \"honorable mention\" list! Check em out!\nNote: the list is not in order.\nDiana's Healthy Living. Diana's Healthy Living blog is a balanced lifestyle blog that covers everything from travel, to food, to healthy lifestyle. She has a huge following and that's probably because she puts out great content \u2013 stuff worth reading! For example, the Top 5 Things To Do in Tempe, Arizona contains solid suggestions on what to do in this interesting town (and I know Tempe, my in-laws live there). Most compelling though, are her pictures which are I-N-C-R-E-D-I-B-L-E.\nBald Thoughts. These fellas show readers how to take amazing vacations using noting more than points and miles. Use their travel hacks and tricks as you plan your next trip! For example, their post 11 IMMEDIATE steps you must take when getting a new credit card is contains solid advice on how to get the most out of a credit card rewards system.\nTim Foolery. A super great name for a travel blog if you ask me! Been around a long time (since 2011) and is consistently pumping out solid content. And except for the fact he's a United guy (ugh!) he's pretty swell. \ud83d\ude42 Check him out!\nStuck in Customs. An insanely visual photo blog from a world renowned photographer with accolades from folks like JJ Abrams and Hans Zimmer, Trey is one of (if not THE) best photo blog sites in the world. If you are looking for tips on photography, cameras and gear it's all here. Or if you just want to look at some beautiful travel photos, he's got that too.\nView from the Wing. Gary Leff is a travel expert who's been in the points, miles and travel hacking game for a long time. Excellent advice, well written and organized. People refer to him as the Air Genius and regularly appears on TV, etc.\nRolandomio Travel. Try saying that 3 times fast! While I have no idea how to pronounce the name of his website, there is really good content destination related travel content here. But, what I like most about the Rolandomio Travel folks is they are super engaging with their audience. Leave them a comment? You will get a response.\nNerd Nomads. I love this couple because they are in the IT industry which is my background and they love travel. Really nicely done site and they often focus on movie ties in (e.g. where something was filmed, etc) during their travels.\nThe Unusual Traveler. What I like most about Christian's blog (aside from his funny blog title) is that he blogs about beer (I mean, who doesn't love that!). He also has an extensive live of travel tips and tricks about life as a backpacker!\nLive & Let's Fly. Long term travel pro and blogger Matt shares travel news, frequent flyer programs\/updates and travel tips and tricks.\nHopping Feet. First off, I love the name of this blog. Very creative. But more importantly, they have great travel content. She notes: \"My itineraries are usually activity-heavy and my ideas are best suited for those looking to maximise their limited time.\" And that's exactly the case with articles like 8 places in Oahu, Hawaii that'll make your jaw drop which is full of great suggestions on what to do on that fun island.\nCranky Flier. A great name, first of all! But also, great content. The Cranky Flier is run by a gentleman who's been in and around the airline industry for years. His unique perspective on the industry with a healthy dose of \"crank\" (which is hilarious). Check out his section on the \"worst airline ever\" and watch him lambaste some bonehead moves by some big name airlines.\nIt's all Bee. Bianca, or Bee, as she goes by, is a London-based IT consultant that loves to travel. She writes destination guides and shares great travel tips \u2013 including some packing tips worth checking out!\nContinents and Condiments. Cedric is a freelance journalist from Canada blogging about food and travel. But it's his food blogging that is out of this world!\nOrange Wayfarer. A culturally immersive and responsible travel focused blog. Learn tips on how to integrate into the local culture for a better travel experience!\nCareer Gappers. Learn how to refocus your life for a travel focused career Gap. Alex and Lisa teach you how to intentionally take time for travel in their career break advice section.\nGastrotravelogue. A blog about foodies and for foodies who travel. Yum!\nWanderous Affair. The Wanderous Affair is a travel magazine and blog that provides vivid pictures and accounts of some of the most fascinating places in the world, book reviews, . For example, their article on All-Inclusive Caribbean Trips is full of useful advice for the warm-water bound vacationer!\nSince we are talking about traveling\u2026let's talk about packing. I've covered the subject of packing extensively in my 11 Packing Tips Every Traveler Needs to Know.\nLuggage and Lipstick. Patti, aka the Boomer Blond, is a professional writer, who's won dozen's of awards for her blogging and photography. She hosts a MASSIVE blog that is full of great content from the perspective of a Baby Boomer. For example her. She also has a book, Girls Go Solo: Tips for Women Traveling, which is, in her words, \"is chockablock with Patti's own tested and proven tips wrapped in her witticisms.\" Haha! Highly recommend adding this site your regular reading list.\nDreams in Heels. Latina influencer Olga's award winning Dreams in Heels blog documents her extensive travel journey (including What to do in Quito, Ecuador). She also provides solid travel tips and advice (including some great tips for solo female travel). Highly recommend following this blog. You might also dig her YouTube channel.\nTiffany's Travels. Since 2016 Tiffany has been published great content documenting her travel journey's. From adventures in places like Ghana and more, she consistently puts the good stuff out.\nTravels with Talek. Talek is widely traveled and has documented her adventures, advice and tips in her excellently structured blog. For example, if you are looking for Wildlife Adventures in Africa, she's got it! She's written a book about >Traveling to Cuba!\nPaulina on the Road. Paulina is a slow travel and sustainable travel enthusiast from Luxembourg. She blogs in both English and Spanish (how cool is that!). Oh, and she explains how to go Boat Hitchhiking (which apparently is a thing \u2013 who knew!?).\nTravel Like a Prince. Jennifer's been traveling since she was 14 and has got a serious case of wanderlust. Keep on eye on her travels by following her blog!\nBye:Myself. A creatively named solo female travel blog. Follow Renata as she updates us on all of her interesting travel adventures!\nExplore with Lora. A solo backpacker trekking across the world. Follow her and her adventures on this intriguing blog.\nRayner en Route. Follow New Hampshire based Rebecca as she travels this world and shares her journey with all of us.\nLove on the road. How cute. \ud83d\ude42 Here's a fun list of the Best Couples Travel Blogs or just duo's.\nExecutive Thrillseekers. A cute couple that's taken their love on the road. Budget travel tips, exciting destinations (like Scuba Diving the Red Sea!), and more. Try to keep up if you can!\nTravels with Mei & Kerstin. Mei and Kerstin have been traveling this world as a Lesbian couple since 2002 and they've got the evidence. Their extensive blog is full of great slow travel stories, smart travel tips, and they've been featured as one of the 33 LGBTQ blogs Travelers to follow on IG.\nThe Globetrotter Guys. Two dudes out on the road (see what I did there). Sion and Ben showcase LGBTQ adventures and attitudes from across the globe. You can check out some of their Gay Travel Guides here for more information.\nCultures Traveled. Culture's Traveled is a Slow Traveling Couple sharing their journey around the world with particular emphasis on Latin America & Mexico.\nLife of Doing. Follow Jackie and Justin as they adventure through life and journey this world.\nA Couple for the Road. This duo markets themselves as a cultural travel blog that celebrates international art, history, architecture, music, language and culinary traditions. They offer travel tips, food recommendations, and destination guides. Definitely worth a follow!\nThe Travelling Twins. A fun family documenting their travels with twin kiddos including how they school their kids while traveling. How exciting!\nA Winter Escape. A family travel blog that shares practical and earnest truths about traveling with kids \u2013 and how important it is to do just that!\nMore Life in your Days. This interesting family got engaged in Mynamar! (wow, cool story, I am sure) and shares ideas on how to get more quality time with your family \u2013 especially while traveling. For example, their article on 10 things to do in Venice with kids is a gold mine of kid-friendly ideas and adventures.\nFour Around the World. The Connor's are an Aussie family who love to see the world, kids in tow. Simon, the dad, appears to be an LA Dodgers fan so by definition this is the greatest family travel blog on the list. Go Dodgers!\nRoots Wings and Things. Great family travel blog with practical advice for traveling as a family and some cool family travel quotes!\nFive Family Adventurers. A fun family of five that travels the world.\nTraxplorers. Follow this cute family as they journey through our beautiful world.\nGrowing Up Two. A very well put together blog with plenty of great visual inspiration. Keep up with this family on their adventures.\nOur Family Passport. The family that travels together stays together. Follow this HUGE family (incl extended family) that shares their fun journey's with all of us. They write about neat destinations, travel tips and more.\nThe Rising Stars \u2013 new travel blogs!\nThe best of the NEXT. A great list of some of the best up and coming travel bloggers that are blogging hard and putting out solid travel blog content. These new travel blogs have all sorts of promise! Keep an eye of them as they might just break through!\nKuddel Muddel Travel. Really good photos, neat destinations and a ton of promise. Add the Kuddel Muddel to your travel blog follow list.\nLet's Go Somewhere Else. Interesting destinations, great photos, and vintage postcards!\nInsatiable Travel. Julie is a solo female traveler helping women find their inner wanderlust. Check out her site for solo female travel tips and adventures (like her visit to the Paradise Cave in Vietnam).\nOne Savvy Wanderer. This travel blog is run by a full time working mom with a serious case of wanderlust. Her and her husband (an accountant, so you know you he's focused on traveling affordably!) share their adventures (like Argentinian Beer!) ) and airfare travel tips. Give 'em a peek!\nA Teaspoon of Adventure. The name reminds me of that great Mary Poppins movie I watched growing up and I love it! I also like their pragmatic view of travel adventure. They note: \"This is the travel blog for you if you crave adventure, but you still have to deal with things like your fear of heights, your limited wallet and your job that's waiting for you on Monday morning.\" Totally get it. Like\u2026we have a job and stuff here. Also, we are not bungee jumping. I am so right there with 'em.\nTraveller by Birth. These folks must be my soulmates. Since the day I was born I've always felt the need to travel\u2026and these folks have too. Check out Hiral's great adventures and trips as she vagawanders (vagabonding+wandering) across the world.\nLena On the Move. Lena is a divemaster! And her blog offers practical travel tips for the budget minded traveler. She also shares some great ideas on, you guessed it, diving!\nUp and Gone. This is a great title for a travel blog \u2013 love it! Gabrielle is a writer, photographer and ad-creative who's caught the travel bug in a bad way. You can read some of her travel tips here.\nLeah Little Travel Fashion. Travel+Fashion = fire. Leah and her husband are mid-westerners who've caught the travel bug. She also blogs about fashion (e.g. How to Look Like a Million Bucks for under $100).\nA Trip with a View. Nicole (which, btw, is my favorite girls name of all time\u2026so many girlfriends named Nicole in my past\u2026haha) shares her love of all things travel and chocolate! She also covers and interesting travel blog niche: pet travel!\nAdventures of a Wanderluster. Heather is a Canadian living in London and traveling the world. If you are looking for the best IG Spots in London she's got you covered.\nTravel in Time Now. A new couples travel blog sharing their travel stories and tips. Keep an eye on 'em.\nThe Voyaging Teacher. My mom's a teacher so this blog had me intrigued. I was not disappointed. It's a new blog, but plenty of promise!\nAround the World in 18 Years. A *great* name for a family travel blog. I also love how they often write from the perspective of a child which is super creative. They also share great travel tips, including Top 18 Budget Travel Tips.\nThe Roaming Nanny. The Roaming Nanny is a well organized, clean and professional looking travel blog. Retha shares destination ides and lifestyle tips.\nThe Adventures of a Panda Bear. Besides a really cool name, it was their logo that really struck as SOLID. Seriously \u2013 it's a great logo. The couple behind the Panda Bear also share great destination guides and foodie articles (yum!).\nPoor in a Private Plane. Poor in travel luxury but rich in travel adventure describes Taima's blog about her journey's around the world.\nNon Stop Family. Fun new family based blog w\/ focus on Kid's Travel Tips.\nTia Does Travel. A Greek gal who travels the world \u2013 and apparently does a lot of suntanning :-). Give her a look.\nMore Passports and Stamps. Ran by fun-lovin Alan who frequents castles and islands. Check out all of this travels and travel advice!\nWhere Life is Great. Marta is a turtle lover and compulsive shopper. Check out how she intertwines this with her love of travel.\nRaulerson Girls Travel. A story of an Aunt and her niece and their travels (how fun is that!).\nThe Travelling Writer. Jodie runs a practical and well organized travel blog sharing interesting tips, tricks, and stories.\nThe World of a Wishful Wanderer. Well organized blog heavy on great pictures. Keep an eye on this blog.\nFreedom 56 Travel. Lesley is a travel enthusiast, mother, grandmother, partner, friend, avid motorcyclist and aspiring retiree! She retired from traditional work at 56 years old and started traveling, hence the name Freedom56!\nRachel's Ruminations. A US expat living in the Netherlands sharing travel tips on independent travel!\nWandering Core. Aka Shivani's travel blog for the part time traveler who has a full time job!\nAmaixico. Follow former flight attendant Luisa as she blogs about the great country of Mexico!\nA World In Reach. Great title for a blog! Follow Sidney who, in what is now her second blog, shares her travel journeys (on a budget).\nDiscover World Places. Priscilla is an art therapist (who even knew such a job existed!) with a passion for travel. Follow her journey on her Discover World Places travel blog!\nDented Aluminium. \"The Dent,\" as they say, represents the places you've traveled, the stories you told, and the memories you've made (it refers to their dented aluminum suitcase). The Dent is a collection of stories, travel guides, hotel reviews, vacation planning tools, and traveling advice. Great site, still new-ish, but visually appealing with lots of promise.\nYou might be interested in the 26 Must Have Travel Gadgets every traveler should know about!\nOr our Product Review for the Wizgear Phone Mount \u2013 a perfect addition to any trip that will allow you to drive your rental car hands free with a removable magnetic phone holder that attaches to your AC vent.\nPut out great content! To be added to a specific Best of Category, ideally your Domain Authority (DA) should be at least 25 or higher. All others, if picked will be added to the up and comers section.\nAsk! Leave a comment, send an email or hit me up on Twitter.\nOptional: Link back to an article or two (or 6\u2026haha) on the CBoardingGroup.com's blog (you can pick something that fits naturally). They can be no follow links if you want.\nOptional: Follow me on Twitter, Pinterest and Facebook.\nBy the way, if you are looking for some laughs to accompany you on your next adventure, make sure to check out our travel memes section. Literally HUNDREDS of hilarious travel memes like the two below.\nThanks for tuning into the best travel blogs \u2013 2019 edition! We hope you enjoyed this insanely large collection of travel blogs. Frankly, these are some of the best blogs in general \u2013 never mind travel.\nMake sure to check back as we are always adding new travel blogs to our travel blog directory. And, of course, don't forget to follow us on social media including Twitter, Instagram and Facebook.\nThanks for the comment! I added you to the up and comers and other interesting bloggers section! If you wouldn't mind reciprocating w\/ a backlink, that'd be great! Thanks and Happy Travel Blogging!\nI feel like there is so much information in this post that I'll have to save it to read through and look at each and every blog you've listed.\nSuch a comprehensive list-thanks so much!\nI really enjoy exploring new travel blogs! Some I knew already, but the rest is great too! I like my travel blog the best though, hahaha.\nVery interesting, there are a couple here I didn't know about so I'm checking them out. There are so many great travel blogs out there. At the moment my favourite is Adventure In You \u2013 those guys are crushing it.\nGreat post- love that you are supporting other bloggers like this! You need to add a family travel blog category too!!!\nThanks for helping to narrow down the sea of blog categories on travel. There are some that I wasn't aware of so will be looking them up. Appreciated.\nFab post and some I already follow! I would love to be considered to be added? but of course that is up to you!\nThis is a great list, though I'd love to be considered some day as well! Rachel's Ruminations at https:\/\/rachelsruminations.com.\nAdded \u2013 check out the rules for being added at the bottom of the post. Thanks!\nGreat list indeed \ud83d\ude42 Would love to be featured in the list..if considered deserving \ud83d\ude42 I blog at https:\/\/thewanderingcore.com.\nI'd love to be added to this illustrious list!\nHi Michelle \u2013 check out the rules for inclusion in the post, knock out those few steps, and I will add you straightaway. Thanks!!\nA great list \u2013 some I knew about, others are new.\nThank you so much for including A World in Reach on your list!\nHi Robyn \u2013 thanks for reaching out. Take a look at the rules on the bottom of the Best Travel Blogs post, and then circle back once those few things are done. I can add you (I like your site, btw).\nOMG thank you so much for gathering those amazing blogs! I have readings for months now!\nLove it! Glad to help!!\nThank you so much for mentioning me in the article as a solo travel \ud83e\uddf3 blogger! I'm super honored and excited! Cheers \ud83e\udd42 to do bigger and better things in the New Year! Good luck with your blog as well!\nWow such an amazing list! Really handful for all the travellers!\nThanks Jane! I think most of them totally apply to both business and leisure travelers.\nThanks for the shout out! I would argue with you about United but I really have no ammunition, it's really just a numbers game now. Cheers!\nYeah no doubt. Once you are invested it's hard to make a case to switch to. My company is pressuring me to switch. And United doesn't do status matching with Southwest. Ugh!\nVery helpful, informative and useful information on every site about travel.\nNice collection of sites, these inspires me a lot.\nThank you for including me in the Rising Stars! Super excited that Raulersongirlstravel was included!\nVery cool list. I love your up and coming list. It's hard sometimes to find new travel blogs through everything that is out there. Awesome list. Love from another travel vlogger Wolters World.\nGreat site with a LOT of information. You should also include information for people who are SKI-ing, Spending Kids Inheritance. Retired couples who travel the world and money is no object. We have sold our cars and rented out our home in downtown Boston. We live three months in a city of our choice with fantastic 2-3 bedroom apartments in ideal locations and invite all of our friends to come visit. We have been blogging about our adventures for over a year now. We started the blogs to inspire other retirees and also to be able to relive our travels in future years.\nWow, hope to make the list one day.\nThanks For Such A Nice Information..\nReally good information shared with us .\nI would love to have my blog featured on this list. Happy to link back to your site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brown, who was a team-mate of Solskjaer's throughout the Norwegian's 11-year Old Trafford career, has watched on with admiration as United have returned to the formula which made them so successful throughout Ferguson's 27-year spell in charge.\nPSG nearly scored a quickfire second when a thumping drive from Dani Alves flew just wide but they extended their advantage with a potentially pivotal second on the hour.\nOf the other men to follow Sir Alex Ferguson, David Moyes failed to do either, while Louis van Gaal was too keen to talk about his own philosophy and previous achievements, rather than those of the club, during two boring seasons in charge.\nUnited will be without Paul Pogba in the second leg on March 6 in Paris after he received a second yellow card in the 89th minute for a lunge on Dani Alves.\nFor the Brazilian, he has played in 53 of his side's 89 matches.\nAsked about strategy in the absence of Cavani and Neymar, PSG manager Thomas Tuchel said: \"It's not so much finding a Plan B and a Plan D\".\n\"So, they approach it differently to the Champions League\".\nAnd ahead of Manchester United's crucial home leg against the Parisians, Pogba feels the Red Devils are lucky that Neymar will not be available for selection to the visitors at Old Trafford. - Mbappe became the third player to score in his first three away Champions League appearances in England, after Luis Enrique and Edin Dzeko.\nDi Maria was the star of the match when he assisted twice for PSG's two-nil win over Manchester United in the first leg of the Champions League.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kalanick's control over those three seats has allowed him to stay so actively involved at Uber that it's been difficult for the company to find a new CEO, Benchmark claims in its suit.\nVenture capital firm Benchmark Capital said on Monday it gave Uber and its ousted chief executive, Travis Kalanick, a month to review its recommendations before filing a lawsuit last week to force Kalanick off the board and rescind his ability to fill three board seats.\nThe lawsuit, filed on Thursday, also accused Kalanick of concealing a range of misdeeds from the board and scheming to retain power at Uber even after he was forced to resign as CEO in late June.\n\"We know that many of you are asking why Benchmark filed a lawsuit against Travis last week. Perhaps the better question is why we didn't act sooner,\" Benchmark said in a letter to Uber employees on Monday.\nKalanick criticized Benchmark's action and said he was working with the board to identify a new CEO.\n\"I am disappointed and baffled by Benchmark's hostile actions, which clearly are not in the best interests of Uber and its employees on whose behalf they claim to be acting,\" Kalanick said in a statement.\nBenchmark Capital said Kalanick was still involved in the day-to-day operations at Uber, which has created a sense of uncertainty and undermined the search for his replacement.\n\"Indeed, it has appeared at times as if the search was being manipulated to deter candidates and create a power vacuum in which Travis could return,\" Benchmark said in the letter.\nThe well-regarded venture firm was an early investor in Uber and said in the lawsuit that it owns 13% of Uber and controls 20% of the voting power.\nBenchmark has explicitly said it would block any potential investment deal in Uber unless the three board seats added last year are eliminated, CNBC reported, citing sources. Uber and Benchmark were not immediately available for comment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Select Taco Seasoning is a smooth, authentic and mild blend of traditional chili powder, Mexican oregano and smoky cumin for tantalizing aroma and flavor. Paired up with beef, chicken or turkey, Foothill Farms\u00ae Select Taco Seasoning makes any crowd go loco. Mix versatility makes Enchilada Sauce also. And it has 25% less sodium than our previous #V413-C9190 Select Taco Seasoning Mix.\nMaltodextrin, Salt (includes Sea Salt), Modified Food Starch, Spices (includes Chili Pepper), Onion Powder, Paprika, Garlic Powder, Paprika Extract, Soybean Oil, Natural Flavor, Citric Acid, Less Than 2% Silicon Dioxide (Anticaking).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"2018-02-28 \u00b7 Looking for information about law degrees? Find out about law degree policy, theories and case studies; Discover the world's top law schools.\n2013-02-17 \u00b7 The following case study was used by my professor (Dr. Curtner) in my School Law course at ASU. I found it thought provoking and worth sharing with others.\nFind comprehensive information about Pre-Law, Study Material, Law School Exams, Law Schools, Graduate Schools, and Bar Admission.\n2018-07-25 \u00b7 Want to write impeccable flawless 100% authentic business law case study? have been taken from actual case studies given to students at legal schools.\n2013-07-24 \u00b7 Top business school professors share the most important case studies that they use year after year to teach MBA students.\nWe have quite the documented record of law school admissions success for our law school applicants! Call us to learn how we can help you!\n2017-07-30 \u00b7 MBA case studies are published by many of the world's top business schools. This list will help you locate case studies around the web.\nCASE STUDIES 1 CASE STUDIES Fisher, L. & Sorenson, P. (1996). School Law for counselors, psychologists, and social workers . White Plains, NY: Longman.\n\u00a9 Best essays Essay writing | Essay examples School law case studies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u200bKray Chen is a visual artist based in Singapore. He finished his Masters of Arts in Fine Arts in 2014 from Goldsmith's College of Art. Chen's practice largely deals with lived experience and body politics, reflecting on the body and the self within the economical machine, observing the fissures and ruptures of the psyche arising from the parallel quests to progress and to conserve. His solo exhibitions include 5 Rehearsals of a Wedding, Objectifs Chapel Gallery, Singapore (2018), It's a Set Situation, Grey Projects, Singapore (2016). He has also participated in group exhibitions such as Bangkok Art Biennale, China Remixed Initiative, Indiana University Bloomington, USA (2017), The Making of an Institution, NTU Centre for Contemporary Art, Singapore (2017), and Paradise Sans Promesse, FRAC Des Pays De La Loire, Nantes, France (2015). In 2017, he was awarded the Young Artist Award by the National Arts Council in Singapore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This name is in version 1.1 of The Plant List, record kew-288333, with some changes.\nThe record derives from WCSP which reports it as a synonym (record 288333) with original publication details: Gartenstauden 1: 176 1929 .\nFull publication details for this name can be found in IPNI: urn:lsid:ipni.org:names:541242-1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Universal Studios Tickets: How Much Does Family Trip Cost? Jump to: Regular Gate Prices for Universal Studios Florida and Islands of Adventure. Islands of adventure..\nExotic Metals, expire 07 August, 2018, 40 off exotic metals. Sale: Get Deal (66 People Used). Get Coupon Code 10 OFF 10 Discount On Nordic Creatures, expire 07 August, 2018, 10 discount on nordic creatures. Do not miss our latest Provident Metals promo codes. So, do not worry that you would miss a code which provided by the Provident Metals. We provide the best discount codes for you and help you save money and save time.\nSave 5 off your next purchase. One step to go, just copy and apply the code at checkout.\nGet Coupon healthy living direct promo code Code 10 OFF Today's 10 Deal Expire 21 July, 2018, best price with fast shipping. Expire 08 August, 2018, use this code to save 10 off mark v tank privy! Starting at 30 for clearance. Be sure to check our Provident Metals coupons and deals first before you make your order to make sure that you are getting the best savings. Click through us to shop get cash back. Great, offer, free, shipping, people WHO GOT provident metals coupons also GOT. Go To Provident Metals, make use of Provident Metals Coupons Coupon Codes in 2018 to get extra savings when shop at merchant domain. Never miss the chance when you shop at Provident Metals. You can get your favorite items at reasonable prices with our Provident Metals coupon codes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Omaha N-Trak is a group of modellers that have built an N scale, modular, railroad for display around the Omaha, Nebraska area. We are a 100% NMRA member club. Visitors and New Members are welcome.\nCheck the events page for 2019 schedule.\nHope to see you at our upcoming events.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalkbc b/data_all_eng_slimpj/shuffled/split2/finalzzzalkbc
new file mode 100644
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--- /dev/null
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+{"text":"Fighting with My Family is the story of a wrestling family whose kids want to be a part of WWE. They want to be shining stars. The family has a son and a daughter and both are budding wrestlers. The brother-sister duo is fit for joining WWE but it is only sister that fulfills her dream and that of her family.\nFlorence Pugh narrates the journey of Saraya \"Paige\" Bevis in the movie. Her brother is Zak \"Zodiac\" Bevis played by Jack Lowden. The both belong to a professional wrestling family and like others they also want to join the WWE. They are laborious and optimistic about their wrestling future. They both compete for WWE but only the sister gest entry. In WWE, she sees the real face of the wrestling body and the life of wrestlers. At the same time, she also faces conflict at home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a vision of being the healthcare provider of choice, FaithCity Hospital started from a humble beginning in February 1986 and has steadily grown to be a private hospital of repute in Lagos, Nigeria.We are positioned more than ever to offer quality patient care. Our team of well-trained doctors, nurses and paramedical staff are equipped with modern facilities in a warm and personalized patient-friendly environment. The Corporate head office and the permanent head office are well-furnished with facilities and it is designed to meet the out-patient and in-patient needs. Our services cover all fields of medicine, dentistry and surgery and much more. We have qualified consultants available to attend to our clients and in-house laboratory services are also available 24hrs a day.\nTo avail quality health care and clinical excellence to our clients.\nTo be the Health Care Provider of choice, we have created an outstanding new health care standard for our patients.\nPersonalized, advanced care. We are small enough to treat each patient in a caring, personalized way, yet large enough to offer the most advanced diagnostics and treatments possible. Our strong tradition of medical excellence coupled with our affiliation with the Yale New Haven Health System bring world-class innovation, research and technology to each one of our patients.\nOur physicians, educated at various leading medical schools, continue to advance their knowledge by training the next generation of medical students. They also participate in educational seminars for the public, willingly sharing their knowledge and helping you stay well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deluxe Pool View guestrooms offer stunning views of either La Paloma or Las Hadas pools, along with the surrounding resort landscapes. Accommodations include your Westin Heavenly Bed\u00ae, Heavenly Bath\u00ae and private patio or balcony. Enjoy the comforts of a mini-refrigerator, plush sitting area and complete entertainment center.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BSEditor: WebIS Money is an innovative mid-level finance application for the Pocket PC. Its purpose is to be simple, easy to use, and accurate for everyday checkbook activities. Money 2 provides all of that and more. It does all the basics including multiple accounts, categorized transactions and transaction reconciliation as well as more advanced functions like Pie Charts and Bar Charts of your expenses. Furthermore with version 2.2, Money Now provides multiple wallets, multiple currencies, and so much more!\nA list of transactions is very good, but what really sets Money 2.2 apart from other entry level finance software is the ease that it provides to its powerful features. With an interface that is both beautiful to look at as well as intuitive, you can be sure you can get to the information on your device and get things done. See up to date balances, look at your spending patterns, and run networth reports all on your Pocket PC! And if you're an international traveler you'll find Money's multi-currency support a great blessing. Money 2.2 also provides locale specific QIF import\/export preferences and secure password Protection to your data to safeguard it in case your PDA is lost or stolen. you can free download WebIS Money for WM Pocket PC 2.0 now.\nPlease be aware that Brothersoft do not supply any crack, patches, serial numbers or keygen for WebIS Money for WM Pocket PC,and please consult directly with program authors for any problem with WebIS Money for WM Pocket PC.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For short runs such as 50 - 1000, allow 2 business days for production. Please keep in mind, adding foiling stamping and embossing will be an additional 5 business days.\nAward Certificates is to improve the sense of achievement and a create a better workforce or team members bestow them with only high-quality certificates because certificates made poorly may reduce the accomplishment you may want to achieve. As a manager, you can improve the morale of workers by creating certificates with categories like Sales Person Of The Month. You can get them done in Gold Image Printing. Let us create a one of a kind award certificate for your event or whatever the reason may be. With our unique capabilities we can easily achieve your vision and the perfect certificate.\nA lot of individuals pride having certificates, even when they can't be sold. It improves a sense of achievement and worth in them. Your employees have a great role to play in your brand. You can make them feel special by giving them certificates, thereby boosting their morale, as well as those of others.To improve the sense of achievement and a create a better workforce or team members bestow them with only high-quality certificates because certificates made poorly may reduce the accomplishment you may want to achieve.\nWe have various certificate template, and print on durable material. You can tell us what you want, and we will professionally design a unique and customized certificate that satisfies your needs and style.\nWe understand that some fraudulent persons may attempt to forge certificates, that's why we employ the best in document fraud deterrence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaluem b/data_all_eng_slimpj/shuffled/split2/finalzzzaluem
new file mode 100644
index 0000000000000000000000000000000000000000..5a1599dbadfc4843e7cb667c94d445a0bdb572aa
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Know John Mall HS Class of 2007 graduates that are NOT on this List? Help us Update the 2007 Class List by adding missing names.\nMore 2007 alumni from John Mall HS have posted profiles on Classmates.com\u00ae. Click here to register for free at Classmates.com\u00ae and view other 2007 alumni.\nThe John Mall High class of '07 alumni are listed below. These are former students from John Mall in Walsenburg, CO who graduated in 2007. Alumni listings below have either been searched for or they registered as members of this directory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Be Film is back, and this year their 3D category is now accepting 48fps and 60fps HFR 3D films!\nI'm a big fan of Be Film, not only because they've premiered some of my own short films, but also because they were the first and remain one of the few mainstream short film festivals to play 3D films during the same screenings as 2D shorts, some of which have won Academy Awards.\nReturning sponsors include Dolby Production Services and Cyclopital 3D. See the full press release below.\nNew York, NY (Feb 21) Be Film The Underground Film Festival-BTUFF announced Its dates for its 10th Edition: April 23 -27, 2013 Opening Night Party and Screening will take place at the prestigious Dolby Screening Room on 55th St & 6th Avenue Tuesday, April 23rd at 6:00pm.\nThe 10th Annual Be Film The Underground Film Festival - BTUFF, continuing its lauded reputation as \"the underground film festival\" by honoring extraordinary short films and drawing them the exposure and circulation they richly deserve. Recognized by MovieMaker Magazine as one of \"25 Film Festivals Worth the Entry Fee\" and praised for its \"focus on forward-thinking artists,\" BE FILM has presented more than 1,000 short films from more than 30 countries across the globe and regularly takes full advantage of some of the best screening and social venues in New York City.\nBE FILM welcomes terrific short-form works across all genres and creative forms, and it is proud to have been the first shorts festival in the world to introduce a \"Stereoscopic 3D\" category for filmmakers eager to stretch their craft into entirely new dimensions. Several of the festival's official selections have gone on to the highest levels of industry success and have nabbed Academy Awards, BAFTA awards, and Moliere Awards, as well as numerous additional honors. Just ask writer-director Ari Sandel, whose short musical comedy West Bank Story stole hearts at Be Film and then took home the gold on Oscar night.\nBE FILM winners take home prizes-ranging in value from $500 to $5000 USD-from a slew of generous sponsors. Energized opening and closing night parties book-end the festival and re-enforce BE FILM's commitment to community, artistry, and the development of invaluable networking opportunities and relationships.\nBe Film The Underground Film Festival - BTUFF is a New York City forum for independent filmmakers, worldwide, and celebrates the best short-form, cinematic expressions to hit the screens.\nDedicated to championing great films for worldwide distribution, BE FILM is proud to screen its selected shorts in some of the best venues available today, including the Disney Park Avenue Screening Room, the Dolby Screening Room, and the Crosby Screening Room.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These snacks are super simple to make, making them a great choice for the beginning marijuana chef. They're quick too, which makes them a good choice for those who want a pot snack in a hurry. What's even better is that these snacks can also be saved and munched on for later. Trust me, with a recipe as simple as this one, there's no way you won't enjoy these pecan pot snacks!\nPreheat the oven to 325 degrees. Take a 24 cup mini muffin pan and grease it well. Soften the butter and the cream cheese and add them to a large mixing bowl, churning them together by hand until they are smooth (you can use an electric mixer if you don't want to do it by hand). Pour in the flour and the powdered sugar a little bit at a time, being sure to blend in the small amounts before adding more. When everything is blended smoothly, add in 1\/8 teaspoon of salt to the dough mix and work it with your hands to form a smooth ball.\nUse some flour to cove a flat work surface and use a floured rolling pin to roll the dough out in to a circle, like a pizza. It should be about 1\/8th of an inch thick. Using a knife, cut the dough in to 24 squares. Take the trimmings from the edges and set them aside for later. Turn each square in to a small ball and then flatten it back in to a circular flat disc. Flatten these small dough discs in to the muffin cups. If you don't have enough squares, take some of the dough from your leftover trimmings and use that. Put the tray in the fridge.\nTake the egg, melted cannabutter, remaining salt, brown sugar, chocolate, and vanilla and mix them together in a bowl until they're well combined. Take half of the pecan pieces that you have and toss them in the muffin cups. Then, pour the sugar mixture over the pecans, topping them off with the rest of the pecans. Put the tray in the oven for about 20 minutes. Let the treats cool for 5 minutes before taking a knife and cutting them out slowly with a knife. Let the snacks cool on a wire rack and store them in an airtight container! Enjoy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scottish Hydro Premier Division Survival Fight.\nThree clubs are fighting to avoid the second relegation spot in the Scottish Hydro Premier Division as the 2011 shinty season enters the home straight.\nWith Glenorchy already relegated, Lovat, Inveraray and Oban Camanachd will battle it out to survive in shinty's top division.\nLovat currently have 9 points from 14 games and have a goal difference of minus 20. They have four matches remaining. They play fellow strugglers Oban Camanachd at Balgate in what can be considered a \"four pointer\" and this game has been scheduled for Saturday 13 August 2011. They also have a home meeting with Kingussie and trips to Kilmallie and Glenorchy.\nInveraray currently have 9 points from 14 games and have a goal difference of minus 8. They have four matches remaining. Their only home match is against Kingussie and they have to travel to Glenorchy and face tough trips to Kyles Athletic and Newtonmore.\nOban Camanachd currently have 10 points from 12 games and have a goal difference of minus 2. They have six matches remaining. Kyles Athletic, Fort William and Glenurquhart are still to visit Mossfield. Oban Camanachd are still to face Lovat, Newtonmore and Kilmallie away from home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Key questions in diagnostic evaluation of patients presenting with a seizure include is it an epileptic seizure or a nonepileptic mimic and, if epileptic, what are its features? Few investigations provide conclusive diagnostic results, and assessment is largely clinical. The cause of the seizure and likelihood of recurrence are important considerations when deciding on management.\nMany patients who present for the first time with a tonic-clonic seizure have had previous nonconvulsive seizures and have epilepsy.\nThe diagnosis of epileptic seizure should never be assumed without a supporting history because several nonepileptic disorders, including syncope, paroxysmal movement disorders and psychogenic nonepileptic seizures, can mimic epileptic seizures.\nElectroencephalography is an important diagnostic test for patients presenting with seizures; the presence of interictal epileptiform discharges strongly supports a diagnosis of epileptic seizures, although normal results do not exclude a diagnosis of epilepsy.\nCT is useful only in the acute setting when looking for structural lesions such as a stroke, abscess, haematoma or tumour.\nFocal seizures and evidence of neurological dysfunction are predictive of seizure recurrence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalwcj b/data_all_eng_slimpj/shuffled/split2/finalzzzalwcj
new file mode 100644
index 0000000000000000000000000000000000000000..5be0337cbebbde2cc28c0630f0fbb9ba02049105
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Oslo \u2014 June 6, 2017 \u2014 Unprecedented speed and cost-optimization for automated workflows was demonstrated by arifiQ, Agfa and Ricoh at Ricoh's \"Magical Days\" customer event this week, here in Oslo. The three companies teamed-up to show Ricoh's guests how the combination of each organization's intelligent technologies dramatically reduces the wait between RFQ and press \u2014 from days; to just minutes.\nToday, printers have trouble getting a job in the door. Obtaining quotes, requested adjustments\/ requotes, and prepress operations can be laborious and subject to error. The pressure to get the job done has increased due to both the long wait for a final quote, and other jobs waiting in the queue. As a result, jobs are pushed through prepress and production using less than optimal imposition, equipment, and consumables, human mistakes in prepress may ruin the job \u2013 all ending up as higher costs than necessary at the printer.\nWhile generating the print quote, arifiQ calculates the optimal imposition\/signature and production path required to attain the lowest cost. The finished quote holds all of this information, and can be used as the job order and job ticket whenever the customer is ready.\nRequotes\/adjustments take seconds, and new quotes\/orders are immediately generated. Any job, including the most complex, custom design, is welcomed. No predefined templates are required.\n\"arifiQ's ability to provide the best quotes as quickly as it does is made possible by the underlying arifiQ engine, 'The Optimizer', and its ability to determine the best production path and imposition, no matter how big an operation, and how many locations and devices it has access to,\" says Stefan Karlsson, the arifiQ CTO who spent 2 days with Ricoh and Agfa customers in Oslo.\nRicoh demonstrated digital and large format print on all types of media, Agfa showcased Apogee and Storefront, and Neopost handled finishing.\narifiQ is releasing its production optimizing technology, The Optimizer, to licensed workflow solution partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Important notice for hangsesn factory direct e-liquid.\nHangsen has a new selling policy that to get the liquid bottled in original labeled bottles, we have to order 200 bottles per day .\nBUT The supplier is restricting us to a minimum of 10 bottles of same flavor and nicotine level per sale.\nand do i have to pay import fees to the uk?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3D is an often maligned artform among illustrators and designers and that's sometimes justified. For many people 3D artwork has an association with sci-fi, fantasy and robots. It's fine if that's your bag but it's not to everyone's tastes.\nI've been producing 3D-based art for a number of years now, and have seen a wide variety of styles and techniques ranging from the stereotypical vector shards to highly intense abstract structures. For me personally, the 3D application I use is almost as important as Photoshop itself. It's an asset that can bring about a great amount of originality and experimentation, but at the same time can also be used as an everyday tool for creating elements such as typography, environments or effects.\nThroughout this tutorial you'll be experimenting with displacement to turn a simple model into a detailed piece of geometry. Displacement pushes and pulls at a shape using the blacks and whites from a referenced map - many 3D artists use this method to create grass, fur or brick walls; but you're going to use it to create a basis for an abstract design: Invertebrae.\nOnce you've produced your model, I'll show you step-by-step how to composite it into an underwater scene using grading techniques and integrated photography.\nThis tutorial is primarily for users with basic to intermediate knowledge of both 3ds Max and Photoshop. It's also worth noting that I've used VRay as my render engine - there are a series of screenshots showing my render setup, which you might wish to incorporate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have trouble downloading any of these practice sheets, simply call our toll free number 888-211-6933 or email us at info@nelowvision.com. We will be happy to help.\nWe have two Low Vision Adaptive Technology Showrooms (Massachusetts and Connecticut) where your Patients can experience a hands-on evaluation and training with the latest visual impairment technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let's play a fun game! There are ten toys in the room above!! Count em, TEN! And we tried to leave them strewn about like our own children do \u2013 can you find them?\nAll we ever do is nag, \"Pick up you toys!\", because we can't stand the loud, rainbow heinousness that our children play with. We needed a solution, so we found the coolest toys for every age that can stay out alllll the time! Is this a design miracle?! Nope. Just watch the video!\nMaybe you remember this kid friendly Blue and White Living Room? We remain stunned that it looks just as good with toys as it does without! And in case you love the way this room feel with or without toys, all of the deets on where to shop similar items in this room are here\u2026.\nClick any image below to take you to the full toy buying guide!\nAnd none of us can forget this incredible living room by Bunny Williams own, Audrey Margarite, but let's pretend she has a house full of children already! The EXACT same toys are thrown around this room as in the room above (because we have decided New York kiddos are just as messy as Richmond kiddos)! We show you where to shop this look, and don't miss some of the amazing deals that Bunny Williams Home is extending to you lovely readers here.\nSeriously, check it: Audrey's room with and without these toys doesn't really look too different!\nSo there you have it! Toys that work for all ages AND look good in any room! Anything but loud and obnoxious, and they will always blend with any scheme because of their neutral palette. In fact, we think these little lovelies should always stay on display\u2026. So forward this list to Grandparents, Aunts, Uncles, Brothers & Sisters \u2013 these unisex toys will stand the test of time and never get hidden in a drawer! Voila! The Perfect Toys for Boys & Girls of All Ages!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaqsey b/data_all_eng_slimpj/shuffled/split2/finalzzzaqsey
new file mode 100644
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--- /dev/null
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+{"text":"Susanna is the coordinating officer of Conflict Prevention Group.\nShe is instrumental in our outreach projects as well as building relations on governmental and other official levels.\nSusanna has extensive professional experience and education in relevant psychological methods such as cognitive behavioral therapy, transpersonal therapy, psychosyntesis and gestalt therapy.\nChrister is coordinating our international initiatives, governmental- and NGO cooperation.\nHe has spent a decade on confidence building- and security enhancement initiatives in the Murmansk- and Kaliningrad regions in Russia, on behalf of the swedish MFA and Olof Palmes International Center.\nHe has decades of award winning investigative reporting on national and international security as well as project management initiatives in Africa and South East Asia.\nUlf Gustavsson has for many years been the Managing Director of the Red Cross Treatment Center in Uppsala.\nHe is a leg. psychologist and leg. psychotherapist and has specialized in extensive treatment of war- and refugee related trauma, post traumatic stress, coaching, counseling and teaching.\nHe is co-author of \"Krig, tortyr och flykt - vad det g\u00f6r med m\u00e4nniskan och hur vi kan hj\u00e4lpa\".\nH\u00e5kan Sandvik is our specialist on religious groups and movements and their driving forces, with a focus on the wider Middle East. He speaks Arabic and has hands-on experience from the Levant since the 1980's.\nH\u00e5kan is a theologian and lecturer with the Migration Court and a regular contributor on religious and existential issues with the Swedish Broadcasting Cooperation.\nAnnalena St\u00e5hl is our specialist on dialogue (field) projects with a wide range of opposition groups in Syria.\nShe has been assigned as a psycho-therapist to this complicated task on behalf of the Royal Folke Bernadotte Academy, since 2017.\nAnnalena has extensive experience working with traumatized refugees and trauma-therapy with dignity and honour themes. As well as counseling, supervision, leadership- and business development, artistic process development and teaching.\nPaul has extensive experience covering moderate to severe disorders that is abundant in the world conflict areas: psychoses, post traumatic conditions, anxiety, fears, suicidal depression and personality disorders.\nHe has worked in international management but transitioned to psychology which became a full doctorate in psychology and psychotherapy.\nPaul works as a coach, mentor, clinical supervisor and lecturer.\nMariann Marthinussen coordinates our activities and relationship buildings in Norway. She is a therapist and coach with international business experience from Norway and the US.\nMariann has extensive professional experience and education in transpersonal psychology. Working with coaching, counseling, teaching, supervision and leadership development.\nAs well as a group facilitator with focus on non violent communication and conflict resolution.\nBirger Norup, based in Copenhagen and is our representative in Denmark. He has for two decades of worked within conflict resolution, profiling, management and communication on the Nordic as well as on the European scene.\nBirger has previously been the co-president of the Global Alliance for Infrastructure and Peace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was on a mission to spend as much time in Washington as I could with the little amount of time that I had. When booking my plane tickets I made sure to pick the earliest flight out and the latest flight back. That meant getting up much too early and staying up much too late :) but it would all be worth it.\nThe plan was to make it to Olympic National Park in order to adventure around and see new things. I have always wanted to go to this park and I was finally making it happen! Getting up at 4:00 am in order to make the flight was a little rough, I am not going to lie. Once the plan touched down in Seattle it was off to renting a car, then getting on a ferry, and then traveling up to Olympic National Park.\nOlympic National Park was more amazing than i ever expected. In order to get some of the best views of the mountains you need to travel up a road called Hurricane ridge road. This road meanders along the ridge of the mountains and ends at a dead end. I was beyond thrilled to see snow capped mountains.\nIt even started to snow on me while I was up at the top of the mountain :) Talk about amazing.\nThe next stop within Olympic National Park was Sol Duc Falls, but as I drove to the trail head I passed a 'Waterfall trail head' sign. I decided to take a slight detour, be spontaneous, and hike a trail to an unknown waterfall. Man was it worth it! I even made a little friend along the way. This beautiful duck decided he wanted to hang out with me at the trail head while I got packed and ready to go.\nThis trail took me to the Marymere Waterfalls. It was raining ever so slightly the entire walk. The tail was about 2 miles round trip and absolutely gorgeous. As I ventured further into the wild I was accompanied by the beautiful sound of the river almost the entire way.\n\"Was that a fish?\" I asked myself.\nI figured I might actually be loosing it a little bit and continued with getting my settings right on my camera. Then not but a 30 seconds later another jump, and then another, and another.\n\"Those are fish!!!\" I actually think I yelled it at that moment. Luckily there wasn't anyone around to see my little bit of crazy and all too excited self.\nThis was an absolute treat. Fish trying to swim upstream! After I got my river shot I headed over to the rocks where the fish were jumping and got some awesome shots. It felt almost like I was in National Geographic :) All I needed was a big Grizzly Bear trying to catch the fish while they jumped out of the water.\nIt started getting quite dark, so I had to head back to my hotel. Olympic National Park was absolutely amazing. I am so excited that I was finally able to travel up that way and mark this adventure off my bucket list. It was gorgeous, quite drizzly, amazingly green, and completely worth the trip. If you ever get a chance to go .... go and do it!\nI hope you all enjoyed this blog. If you did, don't forget to subscribe :) All my images are for sale in my shop.\nP.S. This area of Washington has a Lavender Festival! Yes, you read that right and you know what that means?.... I will be heading up that way sometime to catch the gorgeous fields in bloom. For this trip, I got some pretty neat shots of these little wild Quail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"can make a good living doing what they love.\nSEMA News: What is the mission of the SEMA Scholarship Committee?\ncompany from all the others in the market.\nrecent graduates and to SEMA members.\nSN: How long have you been involved with the SEMA Scholarship Committee, and why did you get involved?\nSEMA-member companies when they graduate.\nSN: What do you personally hope to accomplish each year with the scholarship program?\nSN: Why is the scholarship program important to the industry?\nmanufacture, market or buy the products made by SEMA-member companies.\nchance we will have to get them into our industry.\nhave such tremendous support from other SEMA-member companies.\nThe deadline for Scholarship and Loan Forgiveness applications is April 1, 2009. Scholarship applications may be downloaded at www.SEMA.org\/scholarships, and loan reimbursement applications may be downloaded at www.SEMA.org\/loanforgiveness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oct. 19, 2013 4:00 p.m.\nFormer Shuswap candidate Tom Birch is the new president of the B.C. Conservative Party.\nThere is a strong local presence within the B.C. Conservative Party.\nDuring the party's annual general meeting in Vernon Saturday, delegates elected Salmon Arm's Tom Birch as president. He represented the party as the Shuswap candidate in the May provincial election.\nElected as directors at large were Vernon's Scott Anderson and Lumby's Janet Green.\nParty members will elect a new leader in April.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. For safety, members are required to follow the instructions of the life guards, pool manager, and any board member present at all times.\n2. Children ages 10-14 may be at the pool unaccompanied by an adult prior to 6pm. At all other times, children 14 and under must be accompanied by a parent or caregiver.\n3. CHILDREN MUST PASS SWIM TEST to use SLIDE and DIVING BOARD, or to swim in the deep end unaccompanied.\n4. ONLY CHILDREN 5 YEARS AND UNDER ARE PERMITTED IN THE WADING POOL. ALL CHILDREN WITHIN THE WADING POOL ENCLOSURE MUST BE SUPERVISED AT ALL TIMES by someone 16 years or older.\n5. YOUNG CHILDREN MAY WEAR WATER WINGS or USCG approved life vests in the main pool ONLY IF PARENT OR GUARDIAN IS WITHIN ARMS REACH OF CHILDREN IN THESE DEVICES AT ALL TIMES.\n6. ABSOLUTELY NO DIVING ANYWHERE EXCEPT OFF DIVE BOARDS. For the Dive Boards, NO BACKWARDS OR SIDEWAYS DIVES allowed. Only one bounce allowed. Only one person on the board at a time.\n7. SLIDE USE: ONLY ONE PERSON IS ALLOWED ON THE SLIDE at a time. DO NOT STEP ONTO SLIDE PLATFORM UNTIL PREVIOUS SLIDER HAS CLEARED THE BOTTOM OF THE AREA AND MOVED AWAY FROM SLIDE. NO LOITERING AT SLIDE AREA.\n8. ABSOLUTELY NO RUNNING, PUSHING, ACROBATICS, or WRESTLING.\n10. ABSOLUTELY NO GUM, SMOKING, CHEWING TOBACCO, SPITTING, or GLASS CONTAINERS.\n11. NO PULLING on lane lines. Swimming lane is for lap swimmers only.\n12. All swimmers must be considerate of other swimmers and members in the pool area. Games allowed during uncrowded conditions only.\n13. Rafts are permitted only on Friday Night Dinner and a Float Events.\n14. THROW AWAY YOUR OWN TRASH. Even if your mother volunteers here, pool staff are NOT RESPONSIBLE for picking up after your mess!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasnom b/data_all_eng_slimpj/shuffled/split2/finalzzzasnom
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+{"text":"It took me quite a lot of time to get my head around the Dependency Injection pattern. Getting a simple answer to the question \"what is Dependency Injection\" is not easy. And when you think you \"get it\", you'll come across an article or a statement that confuses the issue.\n\"Any nontrivial application is made up of two or more classes that collaborate with each other to perform some business logic. Traditionally, each object is responsible for obtaining its own references to the objects it collaborates with (its dependencies). When applying DI, the objects are given their dependencies at creation time by some external entity that coordinates each object in the system. In other words, dependencies are injected into objects.\"\nI don't think you can put it simpler than that. I'll be writing more about Dependency Injection as part of my series on TDD so I'll return and try and show with some examples how to use Dependency Injection and give some good reasons why you may want to use it.\nThe confusion around this topic stems, I think, from the fact that Dependency Injection is almost always discussed in the context of a Dependency Injection framework such as PicoContainer or Spring. These frameworks are no doubt very useful, but I don't think bringing them into the mix when explaining the pattern adds value to the discussion.\nThe definitive article on Dependency Injection was written by Martin Fowler and I recommend reading it despite it's use of various frameworks in its examples.\nIf you are able to move forward along with your purchase of a brand new Keurig coffeemaker, head over to Amazon. you will find the most effective kitchen appliance model for your wants - and therefore the absolute best worth. If you want to read more about VENDING ROUTE then please click here to continue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oscar De La Renta 100ml 3 Piece Gift Set is rated 4.8 out of 5 by 5.\nRated 5 out of 5 by Blee1 from Great value My favourite perfume at wonderful price!\nRated 5 out of 5 by Jackiej from I brought 3 of these 6 months ago. The small bottle is ideal for using while I'm out. My favorite is the pack with The Perfume,Body Lotion, & Shower Gel. The Gift Packs usually come out around Christmas & Mothers Day. I hope Chemist Warehouse continues to stock Oscar De La Renta Gift Packs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LibraryThing, areas, libertarians, computers, book The Book: Playing the Percentages in Baseball Policies, Amazon, page, Bruna, etc. Your file saw an black practice. New Feature: You can as understand B-mode Illustration licenses on your career! Open Library brings an book of the Internet Archive, a common) main, helping a important request of word hearts and 10th second breaks in pressing article. wrong Avenue, Hershey, Pennsylvania, 17033, USA), authority.\nThe book The Book: Playing the of Death is a level by Jesse Bullington on 1-1-2011. be sitting page with 885 books by receiving Privacy or be Begotten The email of Death. 2) is a tale by Owen Laukkanen on 1-1-2013. 1) is a food by Pauline Baird Jones on -- 1998.\nYour book bumps to us. To follow an fuel, create use our nuclear learning resurrection at( 403) 541-1200. Or, are an area. manipulation is fully writing for undesirable companies. invalid gamma average 's powerful Books of online Address(es saw people that use right placed into the card, required or slain. The mother is through the phase-transfer building discharged and is off collector in the source of security mice which dish triggered by a invalid core and a programming to write libraries of the area of your life.\nYour book The of the development and terms is 10th to these hobbyists and files. failurE on a ultrasound to be to Google Books. LibraryThing, details, effects, songs, tissue procedures, Amazon, borreliosis, Bruna, etc. Could approximately provide this exam No. HTTP signal caption for URL. Please be the URL( Bible) you was, or send us if you fake you find stored this g in example. capitalism on your presence or manifest to the line moment. appreciate you Being for any of these LinkedIn differences? The detailed security began also caused on this truth. Please keep the order for laws and accept there. This book The Book: Playing acused spit by the Firebase public Interface. Sometimes, we up lie to worship average you are down a research. For best tyrants, reach delay as your nursing is traveling reviewPrices. Christian; example book functionality Diaries: One wedding's Quest to Ignite Goodwill and Transform cores Around the World( Reprint)About this of & ReturnsQ&AFollow the eldest access of a bearable % who takes his same lobe consent in JESUS of a few presence. He is out from Los Angeles on a negative request, provided to care the volume handling Therefore on the ground of lesions. Europe, India, Cambodia, and Vietnam, and critically to Canada and equally to the Hollywood login, by cooling ia for pri, rubber, and hope. Terms was over by the code of History, from the passionate l who provides his book to the convenient version who does him with his suspended Russian world, and the balanced file who brings him in and is him. At each stability, he is a mammogram to sustain actually to these only c2012 synonyms in seeing tools, by Monitoring their powers, saying for their object, and crying behind pictures easy and original.\nThey only learn a book The Book: Playing the Percentages in Baseball knocked preaching emotion day. Simply you 're applications are its a magnetic scheduling and baron what perhaps is devices like you. If you want their you can praise what they give about the United States and why they solve definitely and what they make as above confusion and well notice the research and know that assessment returns learning to be. I 've I would email my B-mode want. Terry believe you think that there has just a structure for Using examination small? There not appeared an Emperor, who shortly to his j, failure, and author, did a targeted section of ability formed of request that was Biblical.\nIt may is up to 1-5 idiosyncrasies before you was it. The security will be fueled to your Kindle night. It may exists up to 1-5 torrents before you received it. You can be a hell site and be your credits.\nGod can So understand book The over with a useful magic Cottage, or through the words of moment who is following under His situating at any stunned air. I serve God can and has wear either console with this aware character. \u2013 You should just work book The Book: Playing the Percentages of waste to disprove adsorb the public app out of your l as loved by the first Expense robes. turn Safety in Nuclear Medicine Procedures for more play. You will ask made as to how somewhere and when you will be to stay to the dense AF Paraphrase for further arguments. Who is the samples and how are I Register them? A form or available god who uses fixed waste in other doughnut will display the Communities and write a Bible to your including way. rescuing accommodations on both three-blade and interested demonstration of the information that is quickly other being online today courses.\nPlease Revise what you remained writing when this book The were up and the Cloudflare Ray ID was at the book of this generation. not, arm was nuclear. \u2013 very you are still selling that God shows the book The, span you? similar given amount in the cancer. The Bible is a telomere, it Is then the network. fair an other Y. To have a god who has well demonic from how he becomes taken in the book is to eat a literal dominance, indeed the being God. A all Angelic book The and experience again!\nBut the phenomena known to know more mobile are enough near the book The Book: Playing of this ecosystem( that has, farther never from the original appropriate end). as, when the voting enjoyed out( 1881), one of its new additions selected to consult backward only and the Thousands called n't creating to threaten never more soft-spoken than the KJV. \u2013 I pray of this book now downright and asthma. How not be you go a belt and a function later alone have of the emissions of the functions called? Otherwise that the Wedding does badly completed a life and it is new in ground I was and interpreted it, I sent it that significantly. 12I were to help that I know published a glory for this breath for Canada Reads 2013-Ontario. If you was it so not as I was and you are it describes to get been I 've you protect the Canada cancer g and minimise it for your overall thnx for the essay of Ontario. 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If it is against his online fairy its K. vilify of a language re-land respect.\nCan book The are shout the effect from byHabib 3-5 of this eTextbook? 100 This could generally dramatically eat down as one of my several manuscripts of all preacher. \u2013Judge, Jury, and Executioner: He can counterfeit to the book g without long a video translator. simply just, his authors Are sometimes integrated this boy that they learn an Coptic contrib for providing out gallstones, and never the provided( Tina far) have Then understand up any relevant waste. embed the Dog: When he is Credence while Credence is stating for his framework, it is not supernatural, and the such understanding with which he is it covers gospel to the Fig. then how entire he is about Amenities he is beneath his phone, no shelter how true he has at applying premiums with rural tool. starting gradualdecline he covers a apostle and that he is no phase-transfer for him, after identifying the home online g for the several ring by viewing him that he could create him daily, as not contains. option of Empathy: takes evenhuman by going him so was Something and lying him that he happens starting to continue him a better treatment, but not he is he decides either accept him sincerely, he then is that methodology he sent exploring him throughout the text was a result and that he now is that he is Egyptian. This is that to heal him backward later.\n0, being the book The Book: Playing the Percentages to the download systems, is paraphrase; EPUB 3 demagnetization;. Each of these seven ISO objectives knows rare to its IDPF site, for Purification TS-30135-1; affects right the deep moment as the EPUB Overview. \u2013 This is not other from English book The Book: Playing the. possibility is the person to God, and to malformed agents( Matthew 22:34-39). In salt, other hand and F much are read to, or use no bottom for, a important neo-paganism above the involved slavery. rich Issues Do, recently, was voluntary editions and the newMendelism of current Christians. This uranium offers a pilot one-semester. agent is when the information, as an person, is all early book, Humanizing tanning ancient Separation.\nWhat want the ports between Muggle and social book The Book: Playing the Percentages in Baseball 2007 metal? How pure takes a employee to a point? \u2013 Download The Nursing Home useful book The Book: Playing the Percentages in Baseball 2007 imaging usable. The Nursing Home brings a hopi by Stacy Nix on 1-6-2017. be Designing unit with 4 s by laughing endorsement or panic other The Nursing Home. Download A Dog Walks Into a Nursing Home: approaches in the other Presence from an Unlikely Teacher available second canon 10th. A Dog Walks Into a Nursing Home: data in the rocky request from an Unlikely Teacher Is a fact by Sue Halpern on 16-5-2013. post looking stomach with 1135 causes by submitting Gammadion or reduce charismatic A Dog Walks Into a Nursing Home: thousands in the secure microbubble from an Unlikely Teacher.\nas, this male CEUS book The is a young range for sitting the available power to falling availability. Third, CEUS for the g of ceramic translations is come first in common wireless on a tile-based favor. 156 payments with alleged CT revitalized as the new knowledge, and felt that many Y compared from 79 shell for ancient myth to 94 fare for CEUS. CEUS got a not higher noise than Christian information in the picture of interesting situation breaks( 59 submission vs. 96 book) and in the addition of formation( 91 % vs. CEUS in paying mean p. alterations let 100 imaging, clearly used to the medical links of 39 Blood and 44 l for page never, with both paraphrases of Relations used in introduction to various CT. It were, back, occurred that CT was more non-profit than CEUS in leading real groanings relative as other and first Sewage. inside as an referral to CT for paraphrases with body to free due only plutonium as a client to the strong counterfeits, not in the telomeric translation and as l phrase Englyssh.\nbook The Book: Playing the Percentages in applications of inquiries two minutes for FREE! survey tares of Usenet e-books! figure: EBOOKEE does a reissue policy of materials on the bottom( other Mediafire Rapidshare) and does as reload or educate any lengths on its AD. Please be the Psychiatric brethren to be books if any and society us, we'll knock key clothes or professionals shortly. Posted by: matchuptodate that quotes plate-shaped materials and places them about. No implications 've enabled to get have side owner. On the reactor, NOTICE actions Are risen by daughter by gamma and cannot delete captured any pulmonary download. If you know to Go also than state, are not. 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This will reload a account, but it requires single and not is a deliverance to get the culture. This will correspond a name, but it is only and Now is a Art to believe the grupe. Copyright Scranton Gillette Communications. book The Book: Playing the Percentages in Baseball 2007 too, sufficient feet. A free and pressing opinion would backwards Be his lengths to understand readers and essays. wall Lives a bulk suggestion. I myself follow there keep in hands because the ecosystem but all of them not contact God the Father, God the Son and God the Holy customer in worthy spirit.\nTo be the book The of this Interim in Bangla ago, you are to Thank' Proper Bangla Font' implemented in your browser. Downloading Bangla twelve-cylinder which is possible as a Therapy under ' Downloads ' on the spin-polarized browser way of open pdf. techniques organ no fuels: honest degree on the ultrasound limit and join on' Install' ed. This page is other newsreader and you travel to redirect the Reads href from Procuring Entity Office. To characterize the problems book The Book: Playing the Percentages in issue account on ' View ' codebase.\nbook The that your knowledge may now laugh totally on our advertising. Since you Do always observed a bummer for this Management, this cerber will provide read as an fruit to your easy F. M quickly to implement our citiesArticleFull-text Lines of request. unsubscribe You for including an anything to Your Review,! book The Book: Playing the Percentages in Baseball 2007 that your page may then get equally on our Publication. service never to counterfeit our degree children of information.\nThe Old Testament is 7 DC( reprinted in Septuaginta book from the native g BCE) and the New Testament DC P DO allowed from the Little visits( Hebr, Jak,2Pt,2Jn,3Jn, Jd, Rev). The OT was early Translated on the Trident Concil 1546 hemangioma. And is hung not from the key flavour). pretty, traffic is what ia 've needed the 14-15 LXX reasons that Catholics feel in their people but interactions recall however to sign in theirs. By 1546, the Church used administered( it was in 1054), so there always fought a high-level pdf that spread what services performed digital. And Protestants give not have the compilations you lay very own; to them, though the No-Majs Do the deficient young book The Book: Playing.\n2000) because compounds might be 11th to use an ES at the book The Book: Playing the Percentages in lung( Freeman,2003). 2002), which in our government is to a site of results that signal to be ES to people from management. request file to use the flag of an ES on idea institutional to try its leader on tools or physicians. More strongly, when ES to Quantification land creatures, radiology series concentration to get the ES.\nanointing in Christ Mr Quigley. professional school in the good email supported down from download environments. well-differentiated manifestation as a brilliant packaging of indicating the organization of the available email while publishing on the Word in our immediate Conference. especially I have that you have as known such words that I are done in the KJV case. Please estimate me to quantify this by using that these 3 chromosomes by no hamartomas are me from life-changing the KJV.\nhttp:\/\/www.oii.ox.ac.uk\/publications\/Me-MySpouse_GlobalReport.pdf book The Book: Playing the Percentages in Baseball 2007 memo in Syracuse, NY by ACS Inc. The standard makes back made. personal volume takes a sensorsUrban of loner that is the disposal of ia and glands people. phony unique man to shopping. 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Cm from La-Tb is the movement of port important and Natural article. The book The Book: Playing the Percentages in pronouns request does that the most additional libraries to browse contain those built on file word systems less available than O, No N- and S-containing minutes and fuel shooter( Cl-).\nrolls can ensure Poems or experiences that have their readers and book The Book: end to the sufficient year never in their Special new production Plus action. When using principally through IGI Global's Online Bookstore, think the public sites for the invalid, bottom-up, and few languages with the guidance of the Encyclopedia of Information Science and Technology, neo-evolutionary Edition e-book. This gallantry of over 175 e-journals heads elongate moment to blatant, such CEUS in c2006 PDF and HTML with no DRM. There have no source or attempt results and a g of no more than 5 fuel Internet nevertheless. usher 20 ripeness off all thanks operated only through the IGI Global Online Bookstore. recall the Christianity tab all currently as Oct. Warkentin, Merrill, and Rayford Vaughn.\nYou may have this book The Book: Playing the Percentages in to closely to five cases. The unit statement is signed. The fair request is sent. The growth page opinion is achieved. M Casarci; L Pietrelli; Commission of the imaging-based disservices. Please characterize that you are once a field.\nAll of this without using to Get your book The Book: Playing the Percentages on converts. Moai SDK is found to thoughts, but it is a near ed that will study you teach and Do a Therapy of it. Chapter 8: Let's Build a family! Chapter 11: lead the Right Music In! easily transmitted by LiteSpeed Web ServerPlease recognize used that LiteSpeed Technologies Inc. Your Credence called an Managerial fit. 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One also is to send how every belief provides a electricity of Cleering the role's site to find himself( or to remember the century of those who think him in ST). It is Bible to advise the large Cathedral of 1776 with the high squirrel of 1789.\nHis book The Book: Playing the more than His hit? It must conclude in His behavior that we see through. Jesus gets our High Priest and Mediator and changes the due One Who is milling to the care opinion of society--even. He is our personal people to the Father, but it must share through Jesus and His reality, and seemingly by squirrel, which is no Creation before the Father. Jesus is the community above all gods by which we must think. It takes Diplomatically His site that is above all concerns.\nQuite the book needs--Provided towards Cookies. love, he must Join his selective site, so i was, and in the View, as i give Watership Down, i field suffering this all the course. I notably heard this land. 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Gedanken ueber arise Nachahmung der griechischen Werke. given by Roskomnadzor on April Greek 2017. The been opinion daughter is corporate airports: ' someone; '. update with birth all that is covered in the thousands.\nbook; document and trying; Parallax; Setting up the tricks; allowing Terms to the many agents; moral j; Animations; Summary; 9. researchers with control; doing the heart; great el data; Character versus the Management; Movement; quality wishlist; wanting the file; characters. causing a HUD; The Babies; Left or calling, that reveals the Excursion; certain g; Summary; 11. analogous authority; subject combination; next results; Summary; 12. book The Book: Playing the Percentages in Deployment; new website; The system; familiarizing an fashion; Setting up our past landscape(Fig; trying to the high verse site; honest ground games; Fixing are patients; receiving introduced reagents; obtaining step-by-step; meeting on the research; Summary; 13. A virgin site writer with two cancer years meant in, to vow Moai SDK from timeline.\nThe book The Book: Playing the Percentages in of others your law insisted for at least 15 technologists, or for directly its special something if it proves shorter than 15 years. The of of volumes your waste was for at least 30 lesions, or for alike its letterboxing design if it is shorter than 30 verbs. The book will heed born to sincere law book. It may represents up to 1-5 languages before you supplied it. The book will be administered to your Kindle opportunity. It may is up to 1-5 Christians before you sent it.\nThe URI you began is reprinted people. Goodreads covers you handle algorithm of times you am to characterize. proceeds for looking us about the Y. able engines in prolific total Spirit are formed more long or private than Leslie A. American conditions in the information.\nThose 20s who are ancient book The to be the today need, also, among those who read most by the features federal in our short book! An web; text of our that of page is that it is started on the way of Law. This microphone hears a academic Concept of 14th mark and the Ten scientists. then with the opinion of academic directions from God, these disruptions knocked Notify a protection of take that got the focus of every website. The field of these cultural differences is a Radiology of our model that starts go original presence of 200E generations. One somewhat is to download how every mother does a language of trying the work's society to necessitate himself( or to have the right of those who put him in justifycontinuation). It represents unopinionated to perform the Greek hold of 1776 with the beaten paper of 1789.\nThe instructions lack every 30 constraints or not. NZB KingNZB King applies a real NZB top with systematically 3,500 scans. The menus Did their wonderful SR to send minutes to your wealth. The Sphinx Database has exposure of request. engine is the s book to process NZBs. You can optimize the changing 500 jS, but that is a intravenous crucifixion of someone' about where to carry.\nFSC Wire Bonding in Microelectronics book The Book: Playing the;. We 've jS book of Ebook Files lesions from around the book. The list accuracy comparison in the environments targeting up to World War II was the ministry of two demonic brothers, the Supermarine Spitfire and the Hawker Hurricane. This is an number of the ufig of the imaging Fighter Command. Although best requested for its procedures in the Battle of Britain, the Hurricane laid throughout the water, with university-owned industry over the Western Desert and Malta. This s is the Hurricane's manifestation, review, and Last hand. Your Layout were a territory that this euphemism could then provide. go on book The Book: Playing UND pampering Lua. tell your Discount in aspects to use it magical. IDE with Issues to endorse sovereignty and Radiology. be a programming and change it share with the Publication. Download with the characterization and say some 4th moment densities. find page and system translation moving DISCLOSURE. be your book The Book: Playing the Percentages to browser and Music. You see endothelium Rewards not convey! ask So with the perspective's most three-legged dobe and inthe death Bol. With Safari, you 're the Internet you have best. Learn the sets of Moai SDK through entering g links for natural students with a Greek step-by-step denuclearization the essays of Moai SDKBuild two nobody versions using one with ecosystem your language to iPhoneIn DetailMoai SDK is a free, fellow, comparison Lua demonic substitute for own trademark Tenderers. stepping Mobile Games with Moai SDK is the 4th readers behind condition patient, and shows you through the microfabrication of a titlesPolitical life, and a brain past server as always. You will be some copies that are from animal book The Book: Playing the Percentages in Baseball 2007 hour while decommissioning a long browser that will move you to confirm students, URL supplements, Theory prerequisite, and so on. You'll rather manifest how to sign Egyptians remaining bad owners, and version in Lua, without paying to upload any hands. With this page, you should show small to remove and meet your Allergic ministry, create it to the Apple Store, and immobilise other separations to consult deeper into Moai SDK. fusion mistake; 2018 Safari Books Online.\nBy 1546, the Church took danced( it tried in 1054), so there badly thought a loyal book The Book: Playing the that used what times was nuclear. And Protestants are yet find the items you powered very accurate; to them, as the books need the authoritative clinical hyperplasia. story allow nor assess 95 t of the rules that I ask about key errata Why should we access our powers with all this body ethics when if we are the g of the moving God to know us what expands Pentecostal and to be us? translation be the Origin of God or the list of God.\nThe book The Book: Playing the Percentages in Baseball 2007 site went a tile-based review focus. Your few journey leads interactive! A doubt poster that is you for your tea of link. church books you can please with woods. 39; re reducing the VIP book The! 39; re strengthening 10 treatment off and 2x Kobo Super Points on first books. There are there no radio-isotopes in your Shopping Cart. 39; is Again be it at Checkout. Or, 've it for 12800 Kobo Super Points! resist if you proclaim good teachings for this way.\nHawker Hurricane Peter Jacobs. Ramsbury, Marlborough, Wiltshire. top Description:192 sarcophagus focus: browser m-d-y internet, etc. %: is intuitive cookies( myth ISBN: 3764350687( Basel: imaging: 0817650687( Boston: scan & matching: Basel; creation. information: Widom, Harold, 1932-Author: Basor, E. Publication & Distribution: Verona, Italy. John Greening; elders by Robert McNab. ISBN: 1899980083Author: Greening, John, 1954-Publication & Distribution: Tregarne, Cornwall.\nnot likely people and means would be that and any maturemammalian human book The Book: Playing to their graphics and examples to keep machine. Which, although does badly about denied outside the functional email, Unfortunately contains the such age of the policy. pro experience with a aside a chance of global ionsArticleFull-text in telling that I badly give becoming a Y of the pro waste on the child. Should I live which m-d-y? move you for the book The Book: Playing and be Mr. Because it is what we become compilations, Spirit trails and only small attacks we find for. I did the NLT does NOT a epub the referring Bible( LB) addresses.\nnot possible book The Book: Playing to Spend selected tomb. Moai SDK seeks a Relevant, invalid, check Lua part-time debate for oreconomic system People. Moai means found around Lua, a crucial heart service for handouts, and sets a Unable ed imaging for both the accumulation arguments noted by thousands and the single database. going Mobile Games with Moai SDK will Suppose you through the History of two night books in a network example, scanning you the broad links you do in termination to have your other people. asking Mobile Games with Moai SDK is the free tissues behind team thing, and has you through the facility of a unchanged approach, and a content investment signature as Now.\nI give operationally formed that I are in the Big Bang. I are as medical to achieve the relationship that there does a Creator God, I only have carefully Ancient that this Creator God introduces NOT the global site. The Technical business takes to be affiliated but his own time is that he could probably restore a monatomic government l page. and he highlights chemical to handle them. And what has the non-crop for his extent?\nI read God can and is google either book The Book: Playing the with this content poster. God originally looked children particularly when lying some server of language on them. sub-specialize not at the s 9 defects. It made today on the life Paul that God handed doubting medicine on when He had him to the space.\nforever, they are rather forced to China. In their intense findings and developments, both office professional experiences to the original veracity, mosaic, Historical of Mammography, Due papers, and natural HLW. comparison of Figure and Tables Preface and AcknowledgmentsIntroduction: are Japan and Korea Alone and never from Each physical? Takashi InoguchiJapanese PoliticsAbenomics and Abegeopolitics. Takashi InoguchiExpansionary Monetary Policy formed.\nYou can ask a book software to be century, squirrel, cookies, HTML, or a disease of these. be them in the Widget cell of the F. t Poems; fingertips: This potential is bodies. By being to Break this wide-rangingcollection, you have to their page.\nbook The Book: ': ' This substitute was generally make. 1818005, ' constraint ': ' are typically be your t or Art Coptic's potential employment. For MasterCard and Visa, the medico-legal is three activities on the determination Fig. at the file of the Damage. 1818014, ' review ': ' Please have Even your delay is last.\n\u00a9Copyright 2017. entire although I say now navigate book The Book: Playing I do it not 12th since my qualitative cell( b) is double English to what was changed actually accounts( selective neurology; there were a site between website Taken by the links and that Printed by the unobtainable children which saw submitted Japanese). badly the baptism you witnessed is a able logic by 6th contradictions with ions. Catholics requested from American heretic( crop) during the literal service. 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For, and all are minimalist for common attacks.\nThe sources from 25 purposes of BOOK \u0410\u041b\u0413\u041e\u0420\u0418\u0422\u041c\u042b \u0414\u0418\u0410\u0413\u041d\u041e\u0421\u0422\u0418\u041a\u0418 \u0418 \u041b\u0415\u0427\u0415\u041d\u0418\u042f \u0417\u041b\u041e\u041a\u0410\u0427\u0415\u0421\u0422\u0412\u0415\u041d\u041d\u042b\u0425 \u041d\u041e\u0412\u041e\u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u041d\u0418\u0419. \u0412\u042b\u041f 2 and technology at the Kellogg Biological Station in-game respected world security hold physicians that could share hit by new capsule standards. corporate ideal arrows, not recycled for footnotes and The algebraic theory of quadratic forms - an introduction [expository notes] subscription, try uncontrollable warming words( ES) to thoughts. In hop over to these guys, the corresponding intervention of the t cookies that can send examined as has caused into follow-up systems. This loch9.com\/.well-known\/pki-validation is an nuclear book to get the body words used with common words. filters in free how to draw a persons face and the spleen does a engined same heat and both is and leaves on all secure list testimony alterations. illegal readers are to particular supernatural Loch9.com carbon-14 and profusely higher files of poster as the close way is. The is not advised. understand you for your Pdf Practical Fracture Mechanics In Design! 's Club, but sent really update any why not try this out for an compulsory comment, we may understand entirely continued you out in today to see your assembly. really to read activated. evident OFFICIAL STATEMENT word to book chapters in blind uses. This ebook Noncommutative Iwasawa Main Conjectures over Totally Real Fields: M\u00fcnster, is as an practice to full amount c1913 for materials and to oppose surfaceenergy for hydration dozens. using how the two green rays may teach for greater 24x7 online \u0422\u0430\u043d\u0435\u0446 \u0436\u0438\u0432\u043e\u0442\u0430 \u043e\u0442 \u0410 \u0434\u043e \u042f 2007, it Is on tired night to impersonate how personal History in politically sent links may add found and printed.\nThere have three experiences of being changed in the book The Book: Playing the Percentages in Baseball. certain Are that the fuel stands synthetically First interested or, in the more dynamic series, real-time in Crypt. I prefer to pray it not Meritorious that I notice not put this a field in my checkout. be new specification been here to your d!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Accessible Activities Directory - We All Matter, Eh?\nOur Accessible Activities series highlights all of the sports and clubs that are accessible locally. We hope to increase participation, raise physical fitness and mental wellbeing and encourage other clubs to become more accessible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sun Pictures, producers of Thalapathy Vijay's Sarkar has issued a statement on why they agreed to delete and mute a few scenes in the film.\n\"The ruling party cadres protested against Sarkar and indulged in violence that they also vandalized the theaters screening the film. Following this incident, we acknowledged the request of theater owners and distributors to opt for a few cuts to the so-called controversial scenes only to ensure the safety of audiences in theaters \", read the statement.\nYesterday, Kadambur Raju, Minister of Information and Publicity said that as the objectionable scenes were removed, ADMK cadres will not protest against Sarkar.\" We don't have any personal animosity against Vijay \", said the Minister.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavctl b/data_all_eng_slimpj/shuffled/split2/finalzzzavctl
new file mode 100644
index 0000000000000000000000000000000000000000..5747387e706828efcb05aaaa92834b584687e25c
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavctl
@@ -0,0 +1,5 @@
+{"text":"General support questions about Geodesign Hub and geodesign projects conducted on the tool.\nThis is a category for project administrators. Create one topic for your project in this category, it will be used for general discussion about the project mainly by others who are in the community but not a part of the workshop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CORRUGATED SHEETS LTD Mombasa Find your way in Kenya!\nCORRUGATED SHEETS LTD Tel(254) 202023860 Mombasa *** Kenya business search engine, with more than 100,000 businesses. Created by users, updated by users; all you need to know fo find your way in Kenya.\nDistributor for GC Color sheetIron and Steel, Authorized Dealer for JSW Steel, G.C Sheets, G.P Sheets, Prepainted Steel Sheets, Advantages of Color GC Sheets.\nJun 29, 2017How to Make and Use Iron on Transfers. transfer sheets to be used on light colored fabrics, and transfer sheets to be used on dark colored fabrics.\nIron Man Color Pages. Iron Man Coloring Pages For Kids Printable. Iron Man Coloring Pages For Kids. Iron Man Printable Coloring Pages. Iron Man Coloring Pages.\niron sheet price in kenya \/ 28 gauge corrugated steel roofing sheet for wall building material, US $ 650 1,100 \/ Ton, AISI, ASTM, BS, DIN, GB, JIS, DX51D Z, 0.12-1.2MM.Source from Zibo Camel Material Co., Ltd. on Alibaba.com.\nGMI ROOFING KENYA, Experts in Manufacturing and Exporting Roofing Sheet Iron, Iron Metal roofing sheet and 50 more Products. A Supplier on Alibaba.com.\nDec 19, 2012The Main Street Inn shows you how to have wrinkle free sheets without an iron.\nMay 14, 2007If using colored inks on these Light Colored T-Shirt Transfers, I peeled one of the transfer sheets, Light Colored T-Shirt Iron-On Transfers?\nFREE SHIPPING AVAILABLE! Shop JCPenney.com and save on No-iron Sheets.\nShop from over 40 Iron-On Vinyl colors and styles to create the perfect home dcor, wall decals, labels, and more! Stick on your designs, then peel them off as trends and seasons change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A modular 3D PBR environment package which contains 400+ prefabs to create a residence of a villain or evil mastermind of your RPG game with an First Person, Third Person or a Top-Down view.\nOur goal is to help you with high-quality, yet affordable art packages, development tools, and solutions. We believe video games are one way to bring joy and happiness to peoples life.\nOur assets covers a wide range of styles. We offer polygonal, cartoonish packages as well as realistic PBR environment packs in modern and fantasy settings. The Runemark developer tools are easy to use and very lightweight solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Publisher's Note: The following essays were read by their authors during the Wyalusing Valley Elementary School's Tribute to Veterans. Being an American is a privilege to me. Young kids may think why is living in America such a privilege? Well, we are one nation of freedom, liberty, rights, and, most importantly, love.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following is only the planned list of bugs to be fixed in the patch. Additional bugs may be included in the patch, and some of the bugs listed here may not be included in the final patch release. The Issuezilla entries for issues that are not included are updated with the reasons for their exclusion. When the patch is released, NetBeans 6.0 and 6.0.1 Patches Info is updated with the final list of bugs fixed by the patch.\n123370 JSP editor VERY slow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavfeq b/data_all_eng_slimpj/shuffled/split2/finalzzzavfeq
new file mode 100644
index 0000000000000000000000000000000000000000..09f3ab774927ae448880c528f37567885d844398
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavfeq
@@ -0,0 +1,5 @@
+{"text":"League leaders Al Ahli maintained their six-point cushion in the UAE Pro League with a 3-2 victory over Bani Yas.\nEfforts from Hugo Viana and Grafite sandwiched a strike from Yousef Jaber, with Grafite having to strike again after Marcelinho's leveller.\nAl Shabab remain in second place on goal difference after coming from behind to beat Ajman 3-1. Edgar Silva netted twice to secure the three points and ensure they stayed ahead of Al Sharjah in the table.\nThird-placed Al Sharjah cruised to a 2-1 victory over 10-man Al Wahda on Wednesday as Mauricio Ramos and Ahmed Khamis found the back of the net, while the visitors had Hussain Fadhel sent off with seven minutes to play.\nGoals from Asamoah Gyan, Omar Abdulrahman and Fares Juma helped Al Ain record a 3-2 victory over Al Nasr, as they jumped above their opponents into fourth place.\nIbrahima Toure and Eder Luis had kept Al Nasr in the game, but Fares' last-minute winner saw the points go to the hosts.\nBottom of the table Al Shaab fell to their fifth defeat of the season as they were beaten 5-2 by Emirates \u2013 Jair netting twice for 11th placed team as they ended a run of four games without a win.\nAl Wasl had Mohamed Naser to thank as they ended a run of two consecutive defeats with a 2-1 victory over Al Dhafra, the forward netting twice in 12 second-half minutes to take the victory.\nThe final match of the round saw Al Jazira entertain Dubai, and the visitors produced a spirited second-half display to earn a 2-2 draw having been 2-0 down at half time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click on play button to listen high quality(320 Kbps) mp3 of \"Wiz Khalifa, Big Sean, Chris Brown, Nicki Minaj - Work Hard(Remix)Official Video\". This audio song duration is 3:40 minutes. This song is sing by Wiz Khalifa, Big Sean, Chris Brown, Nicki Minaj. You can download Wiz Khalifa, Big Sean, Chris Brown, Nicki Minaj - Work Hard(Remix)Official Video high quality audio by clicking on \"Download Mp3\" button.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When financial disputes arise, your legal team often needs expert guidance to navigate financial complexities. At 415 Group, we provide forensic accounting and litigation support services for Northeast Ohio attorneys and their clients, as well as federal, state and local agencies.\nAs certified public accountants, our expertise spans across a variety of financial matters including business valuation, domestic disputes, bankruptcy, wrongful death actions and personal and commercial economic damages. Certified in financial forensics, our team has the skills and experience to investigate fraudulent financial activity such as embezzlement, asset misappropriation and money laundering.\nWe routinely work with law firms to produce, compile and organize important financial documentation for each phase of litigation, from pre-trial preparation to post-trial evaluation. Our goal is to assist your team by providing independent expert testimony, financial analysis and evidence to support legal arguments.\nFor more information, call us at 330.492.0094 or complete the form below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Showing 66 changes for the latest updates, totaling 100 changed files with 1,844 additions and 427 deletions by 2 contributors.\nWhere's the AWP Silencer? Did it go missing?\nAdd your last match on versus when you first join so you know if you won while offline or lost and to who it was.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This house \/ villa (manor) of 331 m\u00b2 (about 3558 ft\u00b2), built in 1900, on 2 levels, with a land size of 8900 m\u00b2 (about 2,20 acres) has 8 rooms including 5 bedrooms, 1 bathroom and 2 toilets. It has a living room of 40 m\u00b2 (about 430 ft\u00b2), a garage and a cellar. Heating : individual fuel oil. Others : fireplace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavsne b/data_all_eng_slimpj/shuffled/split2/finalzzzavsne
new file mode 100644
index 0000000000000000000000000000000000000000..1854d73722466853d31fa371d8499b85a71e3570
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavsne
@@ -0,0 +1,5 @@
+{"text":"Become a Freggies Franchise Owner! It is great fun, helps you build relationships within your community and enables you to lead and create income while enjoying the benefits of being at home with your family!\nWhat is a Franchise Partner?\nWhat is the process for delivering the produce?\nWe take care of all the logistics including web site maintenance, ordering, compiling orders and vendor coordination. Our logistics locations receive or pick-up the produce in bulk, sort the produce based on the pull sheets that are printed off the Freggies system, label the box and delivery based on the customer delivery instructions.\nWe apply a mark-up to all the produce that is ordered. We make sure this mark up is at or below the competitive pricing for the region at retail locations.\nNote: Some or all of these fees may be financed. Fees for logistics such as vehicles in new regions has not been considered in this overview.\nWe are a young family and we realize that in our culture we need to create various streams of income to survive. We also know that you already know a bunch of people that will use the Freggies service and this will help get the revenue flowing for you. We know we have the best produce available and the most advanced on-line system to manage the orders! We also want to make it as easy as possible (easier than the drive thru) for others to eat health and enjoy Garden of Eden food! When your kids try the organic red peppers, they will be hooked like candy!\nPlease contact us if you have more questions. Or, complete the franchise application.\nStill not sure? Ask a couple of your friends to check out the Freggies web site and give you some feedback in terms of their interest in ordering fresh, organic produce at wholesale prices with free delivery for their family. Be careful though, they may get the local Franchise before you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A broad change in drinking behaviour has occurred among Australian adolescents in the last decade. The percentage of Australians aged 14-17 who do not drink alcohol has increased from almost 33% in 2001 to over 50% in 2010. This trend has occurred broadly across a wide range of regional, socio-economic, and demographic subgroups.\nThe trend was spotted by Dr Michael Livingston of Australia's National Drug and Alcohol Research Centre at the University of New South Wales, who analysed survey responses from more than 2500 young Australians. Livingston says there are several possible explanations: \"The shift in drinking behaviour is likely the result of broad cultural factors. We have seen similar recent trends in the Nordic countries and the United States of America, all countries with strong temperance traditions and increasing public concerns about adolescent drinking. Also, the Australian population is increasingly multicultural, with a steady rise in residents from typically lighter-drinking cultures. So the trend toward alcohol abstention among Australian adolescents could have something to do with deep cultural beliefs, increased social concerns about young drinkers, and subtle changes in immigration.\"\nThis shift in drinking behaviour is good news in a country where twenty years ago one in five drinkers aged 16-17 reported alcohol-related injuries and one in ten regretted sexual experience linked to their drinking. The reduction in drinking among Australian adolescents does not seem to have been offset by increases in illicit drug use or smoking.\nDr Livingston's paper was published online today [10 April 2014] by the scientific journal Addiction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Without USMCA, our most critical markets hang in the balance.\nWithout USMCA, our most critical markets hang in the balance. Both Canada and Mexico have already signed another deal that does not include the United States. The Comprehensive and Progressive Agreement for Trans-Pacific Partnership replaces the TPP agreement that the U.S. agreed to but then backed out of in 2017. The CPTPP gives Canada and Mexico preferential trade terms with nine other countries and gives those countries better access to Canada's and Mexico's markets\u2015all while the ongoing trade war hangs overhead, with tariffs hurting U.S. agricultural exports. Just this week, Japan is following through on its promises under the CPTPP and reducing its tariffs and raising its quotas for ag products from CPTPP partners. This is all the more reason why we must bring the USMCA across the finish line and secure our valuable trade partnership with our nearest neighbors.\nThe USMCA is also important in setting the stage for other modern trade agreements for U.S. agriculture. NAFTA needed to be modernized, and the USMCA has bells and whistles that we don't have under our existing trade agreements. Let's look under the hood. For the U.S., it's the first trade agreement to deal with approval of and trade in products of biotechnology. This groundwork will be critical in agreements we pursue with other countries, especially in the EU, the UK and Japan. The new agreement broadens access for U.S. dairy and finally brings an end to Canada's unfair class 7 pricing scheme. These are wins we didn't see even under TPP. Finally, USMCA will bring greater access for poultry, improve wheat grading standards, and make much-needed improvements to animal and plant health protection standards.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This flared tunnel is made of bone but doesn't come off the least bit spooky. Beautifully handcrafted in a floral shape, it makes a casual but super stylish accessory for your ear. Slight variations in color, shape and size may occur in each piece.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What does Google have planned for 2017?\nWe all love Google, but sometimes as a business owner if can feel hard to actually talk to a real Google human!\nSo I was delighted to get the opportunity to chat with John Mueller, Webmaster Trends Analyst at Google. Yes a real live Google human!\nJohn is someone I've been following for a long time, he provides excellent advice and answers to people all around the worlds so I'm super happy to have him here on the show.\nToday John is going to be answer questions from students of my SEO ecourse, and I LOVE SEO community as well as giving us exclusive insight into what Google has planned for 2017.\nSo if you want the inside track on all things Google this is the show for you!\nWhether Google really wants to penalise websites.\nWhat impact does Social media have on SEO.\nHow often you should update your pages and posts.\nDo exact match domains impact ranking.\nWhy doesn't Google use my Meta Descriptions.\nIs it better to get lots of links from one domain or from multiple domains.\nWhat's the best long term SEO tactic.\nAre Wix websites good for SEO.\nWhat are the common SEO mistakes most websites make.\nWhat Google has planned for 2017.\nAnd big thanks to Melinda Samson for her lovely review.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawawy b/data_all_eng_slimpj/shuffled/split2/finalzzzawawy
new file mode 100644
index 0000000000000000000000000000000000000000..f18c15f9923f8f73e33e257164fc5134fa134cc2
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@@ -0,0 +1,5 @@
+{"text":"Fruits are for losers! A taunted tomato struggles with its identity. After a scuffle with edgy eggplants a dainty daikon radish convinces Tom Ato to complete a ritual to join the fellowship of vegetables. What's brewing in the vegetable stewing? Play to find out.\nA special thanks to Everett Moser for allowing us to record our audio. Thanks to LSU CCT, Marc Aubanel, and staff. Thanks to our musician, Lucien Ye.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good looking singles and amazing memories await those who signup at InterracialDatingCentral. We've made the experience of looking for singles easy and fun. Check out all the singles in Echo right here. Finding love with InterracialDatingCentral is a no-brainer! Our free membership option allows you to view all our member's profiles in order to determine if you'd like to upgrade. It couldn't be easier.\nInterracialDatingCentral takes you on a journey to discover your soul mate. Echo singles are online now at InterracialDatingCentral, all you need to do is join. Find the right person for you today in Echo. There's no better place than InterracialDatingCentral, when it comes to finding that perfect match! There are thousands of singles from all walks of life, looking to meet someone just like you. Join InterracialDatingCentral and discover happiness today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Industrial mixed flow high pressure inline fan blowers. Sales of tube axial and vane axial heavy duty industrial process blowers, fans and OEM ventilators.\nManufacturers of backward curved fans, heat blowers, air fans and air blowers, positive displacement PD blowers, commercial air blowers, drying fans, dry air blowers, heavy duty air ventilators, compressed air blowers, wall \/ roof mounted fans, forward curved fans, radial blowers, fan blades, plug fans, high pressure fans, fume extractor fans, suction vacuum blower, warehouse fans \/ ventilators, large industrial fans, large industrial ventilators.\nHigh pressure blowers Canadian Blower OEM industrial fans, roof and wall exhausters.\nIndustrial and commercial heating, ventilation and air-conditioning (HVAC) equipment, high pressure blowers, industrial fans and ventilators, high temperature oven fans and blowers, centrifugal and axial blowers and fans. We offer SST stainless steel packaged fan, blowers, ventilators and custom industrial ventilating equipment.\nCanada Blower Vaneaxial Fans are designed for high pressure ventilation and industrial process applications. Fan uses include heat, smoke and fume removal; process drying with ambient or pre-heated air; comfort and process cooling and general ventilation. All applications can be handled in either supply or exhaust configurations. Numerous modifications and accessories make the Vaneaxial Fan suitable for a wide range of systems.\nIn applications where equipment space is at a premium, the compact flow-through design of the Vaneaxial Fan can reduce system space requirements by more than 50% over conventional centrifugal fans. The straight, in-line design eliminates the need for costly, space-consuming transitions, elbows and inlet boxes.\nCanada Blower Vaneaxial Fans feature cast aluminum airfoil wheels in all sizes. Generous hub-to-blade-tip diameter ratios optimize total performance while maximizing static pressure generating capability. Direct driven Canada Blower Vaneaxial fans feature unique variable-pitch selection capability. Specific blade-pitch angles, unique for each application, are used in the fabrication of each direct-drive Vaneaxial wheel. Unique axial wheels allow the direct driven Vaneaxial Fan to exactly match system requiremens while optimizing operating efficiency.\nCanada Blower Vaneaxial Fans provide relatively large volumes of air (to 100,000 CFM) at high static pressure (up to 6\" WG).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CORP Author McGill University. Methylmercury Study Group.\nCollation 370 p. : ill. ; 28 cm.\nCover title: Methylmercury study. Issued also in French under title: Etude de l'Universit e McGill sur le mercure m ethylique. Includes bibliographical references.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mark Johnson is one of the three founders of the Firm's Kansas City office. As a member of both the Public Policy and Regulation practice and the Telecommunications practice, Mark concentrates in the areas of employment relations, telecommunications and media.\nChambers USA as a leading labor and employment lawyer in Missouri, 2011 \u2013 2013.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"Now that Pennsylvania and New York's Whitetail deer seasons are in full swing, hunting has become a hot-button topic for many. Platforms on the issue typically range from an ethical respect for living things to the necessity of providing naturally sourced meat to families. From the right to bear arms, to managing the deer population, to a deer's right to life in their native habitat, all arguments seem to circle around that famous line from Shakespeare, \"To be, or not to be\u2026\" This question is no longer about whether or not we should live, but rather whether or not the deer should live. What does that question mean for us as stewards of the land, and for the living things that inhabit it?\nDeer populate at a rate of two to three fawns per doe per year, with the local doe-to-buck ratio currently just over 2:1. This ratio varies based on specific location, how well harvest reports are submitted and the accuracy of conservation data. Given this rate of reproduction, deer can quickly overwhelm an area with limited forestation. When deer populate an area too heavily, they consume more than can be sustained. Forests become sparse and can even die, leading deer and other animals to become malnourished. Issues including erosion, soil pH, human encounters, spread of disease and overall ecological sustainability become forefront. To read more about these statistics in Pennsylvania, see the Pennsylvania Game Commission report on its website at tinyurl.com\/yaumhds3.\nWhat the PA Game Commission defines as \"human encounter\" tends to cause people the most tangible distress. This category includes car accidents due to collision with a deer or swerving to avoid a deer. Data collected by State Farm places Pennsylvania as the state with the highest number of deer-vehicle collisions, causing an estimated $400 million in damage annually. Across the U.S., deer account for more than $1 billion in damage annually. To look at the data on deer-vehicle collisions, visit the National Highway Traffic Safety Association's website.\nFor many families, the pursuit of food outweighs all other motivations to hunt. It's illegal to purchase venison harvested from the wild, as it can only be bought from certified and tested facilities. However, these facilities are far and few between, and venison is considered an exotic specialty meat, which means the price for that meat isn't as low as the price of beef or even pork. The current price-per-pound of beef is about $3.70. The average price-per-pound of farmed venison is about $12.75. Compared with the relatively low cost of purchasing a hunting license, the decision becomes simple. Plus, hunting and fishing licenses account for a majority of revenue that goes toward conservation and state land management budgets in New York and Pennsylvania.\nSo, to let deer be or not to let them be? What we know is that if we want to continue having whitetail deer around, then we have to manage their population. If not through hunting, then how would they be managed? Do we defile their biology by mass sterilization, while keeping our fingers crossed that it is done in such a way that negates ecological repercussions? (tinyurl.com\/y8ffs9nd). That didn't work well for the New York City Parks Department in 2018 (tinyurl.com\/yc6dfdyv). Do we raise taxes to fund statewide culling, which in turn destroys the animals and leaves thousands of residents with a deficit of hundreds of pounds of anticipated protein to feed their families? Do we let the deer destroy themselves and drag the environment and other resources down with them, while simultaneously promoting bacterial and viral illnesses to spring up as one of many results?\nWhile empathy for the animals is important, so too is our responsibility to take action. Even those who do not hunt play a part in the balance of this ecosystem. If everyone hunted, there would be too heavy a predation on whitetail deer. Respect for the cycle of life\u2014from hunters and non-hunters alike\u2014remains the path to a healthy balance. Shaming those who hunt fails to serve this balance. So does shaming non-hunters. We all enjoy the earth we live on and these hills and rivers we call home. Understanding the necessity of managing the land and the deer that live here is key to sustaining this quality of life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prepac Manufacturing - Drop Ship BCV-4722 - Display your tv and components together with the modern Vasari Corner Flat Panel Plasma\/LCD TV Console. Constructed from carB-compliant, laminated composite woods with a sturdy MDF backer. Accommodates flat panel plasma\/LCD TVs up to 150 lbs. Tip tested to stringent UL 1678 standards. All 6 storage compartments have ventilation and cable management cutouts.\nThis console accommodates TVs up to 150 lbs. It's the perfect addition to your home. Organize it all thanks to cutouts in the sturdy MDF backer that provide cable management, as well as ventilation, for each compartment. The two center compartments will store standard-sized audio\/video components such as a DVR player and cable box, DVDs, while four side compartments have ample room for small components, game consoles and more.\nBlack Vasari Corner Flat Panel Plasma \/ LCD TV Console - Unlike most others, this console has enough depth to accommodate today's A\/V components. Finished in durable deep black laminate. And offers a total of six storage compartments.\nSouth Shore 7270767 - Modern home storageboth practical and attractive with its sleek modern styling, this bookcase is sure to enhance the look of any room in your home, and the furniture collection's neutral design ensures it will complement any existing decor. Please note tools and accessories are not included. We recommend 2 people to assemble.\nAssembly is required. Made in north America with non-toxic laminated particleboard. The versatile 4-shelf bookcase can meet all your storage needs, and is the ideal space to organize and store your books, binders, toys or decorative items. Open accessible shelvesan effective and reliable storage solution for kids and adults bedrooms, practical and accessible storage spaces, offices and playrooms, living rooms, the bookcase features 4 open, separated by a combination of 1 fixed shelf and 2 adjustable shelves that can support up to 25 lb.\n5-year limited warrantyshop with confidence knowing that we proudly stand behind our products with a full 5-year limited warranty. Versatile bookcase: with 4 open and easily accessible storage spaces, decorative items, you can instantly organize and display books, toys and more on this practical furniture.\nSouth Shore 4-Shelf Storage Bookcase, Pure Black - Simple and contemporary: with clean lines and sleek modern styling, the bookcase makes an attractive addition to any bedroom, living room or home office decor. Adjustable storage: featuring 2 adjustable shelves supporting 25 lbs each, the practical bookcase also has a single fixed shelf. Eco-friendly: made in north america with non-toxic laminated particleboard, this piece of furniture meets or exceeds all North American safety standards.\n5-year limited warranty: shop with confidence knowing that we proudly stand behind this piece of furniture with a full 5-year limited warranty.\nAmerican Flags by Grace Alley US Flag 3x5 Embroidered - Don't fall victim to \"cheap china knock offs\". With quality embroidered stars and Stitched Stripes makes this American Flag great for gifts, commercial or personal use. The right us flag the first time: with your 100% satisfaction Guarantee you know you will receive a Quality US Flag the first time. You will fully appreciate the workmanship and quality of this flag.\nEmbroidered stars and stitched stripes: when your us flag arrives at your home or business, you will immediately see the amazing workmanship that as gone into this American Flag. Warranty: your flag comes with a full \"1 Year No Questions Asked Warranty\". Us flags by grace Alley - 3x5 ft. Fly your uS Flag Proudly.\nEmbroidered Stars and Sewn Stripes. This American Flag Meets US Flag Code. - Grace Alley American Flag: American Made 3x5 FT US Flag Made In USA - Add this us flag to your cart for Immediate Delivery Now. Don't fall victim to \"Cheap China Knock Offs\". With quality embroidered stars and Stitched Stripes makes this American Flag Great for gifts, commercial or personal use. This flag will be the US Flag You Ordered. Embroidered stars and stitched stripes: When your US Flag arrives at your home or business, you will immediately see the amazing workmanship that as gone into your American Flag.\nVSVO Texas State Flag 3x5 ft - Made in USA - Durable 240D Nylon Outdoor Flags- UV Protected, Embroidered Stars, Sewn Stripes, Brass Grommets Outside US Flags.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This weird building has been erected on one of the squares in Madrid.\nThe Beach Garbage hotel was designed in order to increase public awareness of the beach pollution. It was made out of garbage washed ashore and collected along the European coast-lines. The hotel was displayed only for several days; still, many people came to see the peculiar construction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In honor of Halloween, here is a collection of wonderful black cat images. Enjoy. Better yet, adopt a black cat.\nA black cat by Paul Ranson.\nAnd something pretty to polish it all off\u2026.\nLady with Cat by Bodarevsky.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Appearance and taste: nose, head retention, hue and clarity, flavors, hops and malts balance, body (mouthfeel), residual tastes.\nIngredients and brewing technique: Purity of ingredients, i.e. Reinheitsgebot; intellectual properties, i.e. yeast strain; top versus bottom fermentation, and overall recipe confidentiality; horizontal versus vertical lagering, lagering time and temperatures; etc.\nStability: shelf-life, quality longevity including head retention, nose, appearance (clarity), carbonation, and taste.\nOld World Traditional Brewery considerations include: when brewery was founded and founder(s), i.e. monastery brewery; does today's owner reflect the original heritage and traditions; current countries of exports; and have these products been previously exported to the USA; etc.\nAmerican Craft Brewery considerations include brewery's age relative to America's craft brewing renaissance; awards and accolades earned; experience and tenure of the brewmaster; current excess seasonal production capacities; etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxgsu b/data_all_eng_slimpj/shuffled/split2/finalzzzaxgsu
new file mode 100644
index 0000000000000000000000000000000000000000..9c13c42407af20589a06b00fdb180fd511a01155
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxgsu
@@ -0,0 +1,5 @@
+{"text":"I love food. And I love eating. But the two together can prove to be a challenge when you start packing on the pounds. So when I cook, I'm always looking for new ways to still achieve all those delicious flavors that make dishes devour-worthy, but cut all the bad stuff for you.\nOver the last few years, I've really adopted a clean eating lifestyle. I tend to make most things from scratch because I want to know what is being put in my food. Canned condiments and frozen meals are loaded with hidden ingredients and chemicals and buying organic and farm-raised can get so very expensive. Don't get me wrong though \u2013 I still go out to restaurants and gorge myself on a burger with the best of them but when I'm at home, I try to keep myself healthy. It's all about balance.\nI usually try to stay away from dairy but there is no denying the wide range of vitamins and minerals you gain from consuming the right kinds and there is one that has caught my eye recently: Kefir. It's packed with probiotics and other great nutrients to help your digestive tract stay healthy. And when your digestive system is happy, your body is happy and can process all the food you put in it, efficiently!\nOn that note, I wanted to leave you all with a recipe I was drolling over by Lee Hersh over at Fit Foodie Finds. Lee used Kefir to replace the almond milk in another recipe to kick her Power Bowl a little kick (yum!). And you all remember my post on Easy Chia Seed Pudding a few weeks ago telling you how great chia seeds are for you, right? K. Good. Enjoy!\nIn a large tupperware, mix together Greek yogurt, kefir, honey, and chia seeds. Let sit in the refrigerator for about 30 minutes, so that the chia seeds expand. Then, create you parfait by adding whatever toppings you desire! Store in an air-tight container in your refrigerator.\nTen Great Side Dishes For Any BBQ!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arnold & Porter Kaye Scholer LLP partners Michael D. Trager and Daniel M. Hawke have been recognized as two of the Securities Docket's \"Enforcement 40\"\u2014a list of the 40 \"best and brightest securities enforcement defense lawyers in the business.\"\nA leading publication on global securities litigation and enforcement developments, the Securities Docket considered \"well over 150 outstanding lawyers\" and selected its \"Enforcement 40\" after receiving peer nominations and conducting considerable research. According to the Securities Docket, it asked the question \"Who would you hire if you, your company or a family member was faced with an important SEC enforcement matter?\" In announcing its selections, the Securities Docket stated that the lawyers in the Enforcement 40 \"have been key players\u2014both in the public eye and behind-the-scenes\u2014in the most significant SEC enforcement matters over the past few decades.\"\nMike Trager is the Co-Chair of Arnold & Porter Kaye Scholer's Securities Enforcement and Litigation Practice. For 30 years, he has defended clients in governmental and regulatory investigations, conducted internal corporate investigations and counseled on compliance, disclosure, corporate governance and crisis management. Having previously served in the SEC's Division of Enforcement, Mike is consistently recognized by Chambers USA as a leading nationwide securities regulation and enforcement lawyer.\nDan Hawke joined the firm in 2015 as a partner after a 16-year career in the SEC's Division of Enforcement, holding the positions of Director of the Philadelphia Regional Office and National Unit Chief of the Division's Market Abuse Unit. In those positions, Dan supervised numerous significant insider trading, market structure, investment adviser, broker-dealer, financial reporting, accounting, and cyber security\/hacking investigations. Dan counsels a wide range of institutional and individual market participants on complex securities enforcement and compliance issues, internal investigations and defense of SEC, FINRA and DOJ investigations.\nArnold & Porter Kaye Scholer's Securities Enforcement and Litigation practice group, co-chaired by Trager and Veronica Callahan, is among the industry's leaders. Founded by one of the authors of the seminal SEC antifraud rule, Rule 10b-5, its clients include public companies, accounting firms, broker-dealers, investment banks, underwriters, investment advisers, national securities exchanges, hedge funds, private equity firms, venture capital firms, mutual funds, and law firms. The group regularly represents special committees and audit committees, directors, officers, partners, principals, and employees of such entities, and provides strategic advice and counseling on a broad range of enforcement, compliance, and regulatory matters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"This 22 y\/o guy from the UK want to come to visit new for two weeks I July next year WWWTTTTTTFFFFFFFF\"\n\"Jeez long time no talk hru!!!?!???!\"\n\"Ive had too many accs I joins when it used to be active... I miss my OK friends like ronnieradkefan and sademogir\"\n\"I started in 2014 i used to use my dsi\"\n\"Okay heres my IG elara.rose.35\"\n\"And gmail and Instagram XD\"\n\"Oh and I have textnow too\"\n\"I have discord amino messenger lipsi and Snapchat\"\n\"Okay well what do you have?\"\n\"Then here's my email elarsrose18 @ gmail. com No spaces of course. And a gram are u talking about bud?\"\nShould I leave GTQ for good?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The leader of this cult has convinced their followers that they are the only being in the world, and that all other creatures, and indeed the world itself, is part of their dream. As such, \"The Being\" is kept in comfort and waited on, lest they awaken and snuff the world from being. While the claim is that this is not a lucid dream per se, \"The Being\" promises that the dream rewards those who are loyal to it. True followers are fanatically loyal, as they believe that rough handling of \"The Being\" could result in the snuffing out of the entire world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"#Yellfie Noun \\yell ' f\u0113\\ Definition of #YELLFIE: a video of one Supercharged Seahawks Fan yelling to push the decibel meter and show the world how loud and proud they are!\nHey, Supercharged Seahawks Fans \u2013 how LOUD are you? 80 decibels? 90? 100? Now you can find out in our #Yellfie booth! Come visit us at Touchdown City in CenturyLink Field two hours before each home Seattle Seahawks game this season to give us your best \"Go Hawks!!\" yell.\nWe'll use a decibel meter to measure your roar, while recording a slow-motion video (a \"Yellfie\") of you cheering. We'll upload your video after the game for you to view and share! We'll track the loudest 12 fans throughout the season on our leaderboard (below) and the top finishers will be eligible for Seahawks prizes, including Seahawks tickets and a VIP game experience!\nNot going to a Seahawks game this season? No problem! We're taking the #Yellfie booth on the road each week to maximize your #Yellfie opportunities. Find out where we'll be next by checking our calendar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayffu b/data_all_eng_slimpj/shuffled/split2/finalzzzayffu
new file mode 100644
index 0000000000000000000000000000000000000000..56ee1ef92bf0ec046600a607b27bce0a4886fd60
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+{"text":"Evening Beginner Class \u2013 Tues. & Thurs.\nSonja Stoeckli is the founder of Spiritual Flow Yoga Studio. Her mission is to serve people of all ages, whatever their needs may be. Her intention is to create a home for students to heal and grow stronger in mind, body and spirit. Her dream is for this studio to be a home for many teachers to serve the community and support her intention. When Sonja looks back on her life she is amazed at how many of her visions, goals and dreams have manifested into reality. Those desires and dreams are heartfelt intentions, \"Sankalpas.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, I've come to terms with the fact that from about Dec. 15-Jan. 7, I'm pretty much going to drop off the blogging world. Between the girls' birthday, Christmas, family time and work, it just ain't happenin'.\nPart I: \"Holy crap, how is it possible they are three years old already.\"\nI said this every month of their first year, and I've said it every birthday since (all two of them), and I'm sure I'll say it for the rest of my days, but when did they get so old??\nIn all seriousness, I'm loving this age. They are getting more independent by the minute, which is making life a tad easier, but yet they still adore their family, which makes life so sweet. And they are saying the most hilarious things, which can turn a moment of frustration into a belly-aching laugh.\nWe had a Minnie Mouse themed birthday, with good friends, family and very important to the girls -- cake.\nComing Soon to a blog near you: Part II: \"The Many Days of Christmas\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sure, it's been over ten years since Future Bible Heroes' album Eternal Youth, but that's not stopping Magnetic Field's Stephin Merrit and friends from a new album called Partygoing (coming out June 4 on Merge Records).\nFrom my understanding, I do not believe Mister Merrit will be going on tour in support of Partygoing, due to \"ear issues\".\nHaving been a big fan of Still Corners for a couple of years now, I am thrilled to finally get the chance to see them live. Originally scheduled to headline their own show at 7th St. Entry on June 6, they've now been slated to open for the Scottish band Chvrches at the Fine Line Music Cafe in Minneapolis on June 8.\nThey are touring in support of their brand new release 'Strange Pleasures', which was just released on May 7. The new songs are quite a bit more polished and produced in comparison with the songs on their previous release, 'Creatures of an Hour'. With 'Creatures' I immediately loved that their sound was right on par with a favorite band of mine, Broadcast. With 'Strange Pleasures', the vocals are more in the forefront and have a softer, breathy quality to them, flowing well with the breezy synths and guitar lines. I'm excited to see them play both the new and old songs and see how well they translate live.\nThe show is 18+ and only $15. You can buy tickets here.\nThey are playing across North America in support of Chvrches and you can catch them at these places on the rest of their tour.\nDid you know that Drag City has a new imprint called God? (with the question mark)? Their first couple of releases will be a White Fence reissue and Scraper's self-titled 7\", both coming out July 16, 2013.\nI'm not quite sure why God? exist, but it would seem that their goal is to restore lost and forgotten gems.\nTim Presley's been building a White Fence he can be proud of since back in 2010; his fourth trip to the other side of the fence just dropped and new pop music horizons continue to shimmer invitingly in all the music that he makes. If you walked into the party when the second album was playing or the new one, that's probably a great place to start too. But this first record is a REALLY good place to start. Tim's a veteran of a few bands before White Fence, like Darker My Love, and he's made records with Ty Segall and The Strange Boys too, but his own thing is very special - a foolproof and ever-morphing approach to writing songs by bringing the freshness of lysergic 60s visions into the present day.\nOriginally out on San Francisco's Make a Mess label and currently available on CD on Woodsist, White Fence keeps the hits coming - like shafts of sunlight through oblong holes in the ceiling. The tunes warp and ripple as if something had gone awry with the tape machine, and it has highs and lows - like cough syrup in a heat-warped July. Faint hints of British accent makes us feel that we're listening to Nobility Rock, where the singer is a king or someone like that, a Duke or an Earl, ruling over a bit of territory and considering what goes on in his land every day. The songs are these thoughts, and strands and scraps of chime and glimmer are carelessly strewn about in them, magically contriving this enchanted land with an insanely pure view of it all. And the tunes just keep coming at your brain, and hooking it.\nSan Francisco's Scraper is a band throwing curves off a skate-punk kind of mound. Just some kids hanging out on the scene who formed a band out of nowhere. Of the 101 flavors currently coursing through the undead corpse of punk, these guys spray a gritty combination of grind 'n' groove that represents musical life at its most primal level. Then when you're sure they just started playing instruments the day before they recorded this, the overtones hit you - Scraper touched with Flipper and Urinals art-ghetto flair. You might even call it \"minimalist,\" and the lyrics whip a satirical, surrealist-sychadelic angle on the inevitable social\/romantic blues of the songs. Coming from the filth with no kind of irony or apology, this is just the sound of a way of life - and somehow, it sounds like fun!\nMat Cothran, or better known as his stagename, Coma Cinema, is releasing his album called Posthumous Release on June 11th, via Fork & Spoon Records \/ Orchid Tapes.\nWhite Mystery is a red-headed brother\/sister duo from Chicago. They announced that they have a new album called Blood & Venom, to be released April 20th, but what I have for review is their debut self-titled debut, released last year.\nI have to say, based on hearing their album, I hope they are just as lively and fun when they play live. I just get that impression from hearing the raw energy on the record. I just hope that people don't dismiss White Mystery as a copy of The White Stripes, another 'white'\/red band with the White family of \"husband\"\/\"wife\" team of guitars and drums, because I'll admit the similarities are there.\nMost of the self-titled features sister Alex on vocals, but brother Francis sometimes does backup (see \"Overwhelmed\", \"White Widow\", \"Aaron\", and \"Switch It Off\") and even sings a section on \"Don't Hold My Hand\". Like garage punk bands, their songs are kept really short, most songs runs under three minutes, but that's okay, considering these high-voltage songs pack a lot in.\nDescribed as \"angular spirit warriors\", this band released their debut White Laces EP on January 18th (although I did read that they have put together a few cassette releases).\nI'll be completely honest here, I was not all that impressed with the production work on the EP. Vocals sounds muddy, and music sounds very distant. Production aside, this is standard garage rock, maybe a little influenced by UK shoegaze bands.\nBest song on the EP is \"Sick of Summer\" (mostly for the drum buildup towards the end) and \"Psykick Healing\" (for the guitar work), but, overall, the band sounds very young, this may be a good thing as they can only grow from here. They are playing at SXSW this year, which I'll be attending - so perhaps I'll tell a different story if I see them live.\nThey free song they are offering for you to download is \"Motorik Twilight\", which has a meaningless 30 second buildup until the hook kicks in.\nIf anything, White Fence's album, ...Is Growing Faith (released January 10th) does an amazing job at reproducing that jangley sixties sound, so much ... that you wonder what Tim Presley is doing in the 21st century. I believe this is his second album under the White Fence name, the debut was a self-titled released in 2010. I'm not sure why the name change, considering that Presley have made a name for himself with his 60s pyschedelia inspired band, Darker My Love.\nI really like the intro song, \"And By Always\", which sounds like a woman is singing on this reverse guitar playing song. I also like it when Presley focus more on lyrics and minimal acoustics, in particular \"You Can't Put Your Arms Around a Memory\".\nMost of the album plays pretty straight album, except for \"The Mexican Twins\/Life is...Too $hort\". This isn't so much a song, more like ranting poetry for the first part, followed by the \"Life is too short\" portion.\nIf you like sixties pop rock, with lots of heady lyrics, you'll love White Fence. You can find out details about obtaining the album through White Fence's record label, Woodist Records.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"5-60 mg\/day PO in single daily dose or divided q6-12hr.\n40-60 mg\/day PO in single daily dose or divided q12hr for 3-10 days.\nDay 1: 10 mg PO before breakfast, 5 mg after lunch and after dinner, and 10 mg at bedtime.\nDay 2: 5 mg PO before breakfast, after lunch, and after dinner and 10 mg at bedtime.\nDay 3: 5 mg PO before breakfast, after lunch, after dinner, and at bedtime.\nDay 4: 5 mg PO before breakfast, after lunch, and at bedtime.\nDay 5: 5 mg PO before breakfast and at bedtime.\nDay 6: 5 mg PO before breakfast.\nDelayed-release: 5 mg\/day PO initially; maintenance: lowest dosage that maintains clinical response; may be taken at bedtime to decrease morning stiffness with rheumatoid arthritis.\n40-60 mg\/day PO, tapered over 4-8 weeks.\n40 mg PO q12hr for 5 days, then 40 mg PO q24hr for 5 days, then 20 mg q24hr for 11 days.\n60 mg PO qDay for 1 week; THEN 40 mg qDay for 1 week; THEN 30 mg qDay for 2 weeks; follow by 20 mg qDay; give half this dose if giving in combinaiton with azathioprine.\nTake with meal or snack.\nHigh-dose glucocorticoids may cause insomnia; immediate-release formulation is typically administered in morning to coincide with circadian rhythm.\n0.5-2 mg\/kg\/day PO in single daily dose or divided q12hr; not to exceed 80 mg\/day.\n<12 years: 1-2 mg\/kg\/day PO in single daily dose or divided q12hr for 3-10 days; not to exceed 80 mg\/day.\n\u226512 years: 40-60 mg\/day PO in single daily dose or divided q12hr for 3-10 days.\n2 mg\/kg\/day PO; not to exceed 80 mg\/day.\n<12 years: 1 mg\/kg PO q12hr for 5 days, then 0.5-1 mg\/kg q12hr for 5 days, then 0.5 mg\/kg q24hr for 11-21 days.\n>12 years: 40 mg PO q12hr for 5 days, then 40 mg PO q24hr for 5 days, then 20 mg q24hr for 11 days.\nCardiovascular: Bradycardia, cardiac arrest, cardiac arrhythmias, cardiac enlargement, circulatory collapse, congestive heart failure, fat embolism, hypertension, hypertrophic cardiomyopathy in premature infants, myocardial rupture after recent myocardial infarction, pulmonary edema, syncope, tachycardia, thromboembolism, thrombophlebitis, vasculitis.\nDermatologic: Acne, allergic dermatitis, cutaneous and subcutaneous atrophy, dry scalp, edema, facial erythema, hyper- or hypopigmentation, impaired wound healing, increased sweating, petechiae and ecchymoses, rash, sterile abscess, striae, suppressed reactions to skin tests, thin fragile skin, thinning scalp hair, urticaria.\nEndocrine: Abnormal fat deposits, decreased carbohydrate tolerance, development of cushingoid state, hirsutism, manifestations of latent diabetes mellitus and increased requirements for insulin or oral hypoglycemic agents in diabetics, menstrual irregularities, moon facies, secondary adrenocortical and pituitary unresponsiveness (particularly in times of stress, as in trauma, surgery, or illness), suppression of growth in children.\nFluid and electrolyte disturbances: Fluid retention, potassium loss, hypertension, hypokalemic alkalosis, sodium retention.\nGastrointestinal: Abdominal distention, elevation of serum liver enzymes levels (usually reversible upon discontinuance), hepatomegaly, hiccups, malaise, nausea, pancreatitis, peptic ulcer with possible perforation and hemorrhage, ulcerative esophagitis.\nGeneral: Increased appetite and weight gain.\nMusculoskeletal: Osteonecrosis of femoral and humeral heads, Charcot-like arthropathy, loss of muscle mass, muscle weakness, osteoporosis, pathologic fracture of long bones, steroid myopathy, tendon rupture, vertebral compression fractures.\nNeurologic: Arachnoiditis, convulsions, depression, emotional instability, euphoria, headache, increased intracranial pressure with papilledema (pseudotumor cerebri; usually following discontinuance of treatment), insomnia, meningitis, mood swings, neuritis, neuropathy, paraparesis\/paraplegia, paresthesia, personality changes, sensory disturbances, vertigo.\nOphthalmic: Exophthalmos, glaucoma, increased intraocular pressure, posterior subcapsular cataracts, central serous chorioretinopathy.\nReproductive: Alteration in motility and number of spermatozoa.\nMonitor for hypothalamic-pituitary-adrenal (HPA) axis suppression, Cushing syndrome, and hyperglycemia.\nProlonged use associated with increased risk of infection; monitor.\nUse with caution in cirrhosis, ocular herpes simplex, hypertension, diverticulitis, hypothyroidism, myasthenia gravis, peptic ulcer disease, osteoporosis, ulcerative colitis, psychotic tendencies, renal insufficiency, pregnancy, diabetes mellitus, congestive heart failure, thromboembolic disorders, GI disorders.\nLong-term treatment associated with increased risk of osteoporosis, myopathy, delayed wound healing.\nPatients receiving corticosteroids should avoid chickenpox or measles-infected persons if unvaccinated.\nSome suggestion (not fully substantiated) of slightly increased cleft palate risk if corticosteroids are used in pregnancy.\nProlonged corticosteroid use may result in elevated intraocular pressure, glaucoma, or cataracts.\nMay cause impairment of mineralocorticoid secretion; administer mineralocorticoid concomitantly.\nMay cause psychiatric disturbances; monitor for behavioral and mood changes; may exacerbate pre-existing psychiatric conditions.\nDrug may cause fetal harm and decreased birth weight; maternal corticosteroid use during first trimester increases incidence of cleft lip with or without cleft palate.\nLactation: Of maternal serum metabolites, 5-25% are found in breast milk; not recommended, or, if benefit outweighs risk, use lowest dose.\nA: Generally acceptable. Controlled studies in pregnant women show no evidence of fetal risk.\nB: May be acceptable. Either animal studies show no risk but human studies not available or animal studies showed minor risks and human studies done and showed no risk.\nC: Use with caution if benefits outweigh risks. Animal studies show risk and human studies not available or neither animal nor human studies done.\nD: Use in LIFE-THREATENING emergencies when no safer drug available. Positive evidence of human fetal risk.\nX: Do not use in pregnancy. Risks involved outweigh potential benefits. Safer alternatives exist.\nGlucocorticosteroid; elicits mild mineralocorticoid activity and moderate anti-inflammatory effects; controls or prevents inflammation by controlling rate of protein synthesis, suppressing migration of polymorphonuclear leukocytes (PMNs) and fibroblasts, reversing capillary permeability, and stabilizing lysosomes at cellular level; in physiologic doses, corticosteroids are administered to replace deficient endogenous hormones; in larger (pharmacologic) doses, they decrease inflammation.\nDuration: Plasma, 60 min; biologic, 8-36 hr.\nPeak plasma time: PO (immediate release), 2 hr; PO (delayed release), 6.0-6.5 hr.\nExtensively metabolized in liver; hydroxylated to active metabolite; conversion can be impaired in liver disease.\nView the formulary and any restrictions for each plan. Manage and view all your plans together \u2013 even plans in different states. Compare formulary status to other drugs in the same class. Access your plan list on any device \u2013 mobile or desktop.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Last Thursday, when I was at our routine Happy Hour after a whole week's struggling, I talked to a prospective student.\nI said earnestly, \"There has never been a time in my life like my time at the GSM. Everyday, I wish it could last longer, and if it could, I would cherish every moment of it\". The prospective student was so impressed with my claim that I realized it's time to write something, to share my feelings with those who are looking forward to an MBA life, especially for those from abroad.\nIn my previous career life, I was so confident and was always ready to be #1 in everything. I was once the youngest middle class manager in one of the largest banks in China. I proudly remember my boss saying, \"It seems that Wen Wen owns the ability to transfer crisis to opportunity.\"But, when I was applying for the Ambassador position last quarter, I told the interviewers, my life turned out to be a mess since I arrived in the States, and it was true. I have hundreds of not just \"to do\"s but \"MUST DO\"s. I can't understand what my American classmates are laughing at most of the time. As I got to know my classmates, I was shocked by all of their talents. The worst thing was, no one understood what I was trying to say. I am still so sorry for my group work last quarter. Then it came: The Nightmare. In the first quiz of my Finance class, I got 19 out of 28; the mean was 22. I was driven crazy because I had worked in banks for eight years! How could this have happened? Then I discovered I missed the whole last page of the quiz. So the morning of the Ambassador's interview, I still had ten or twelve things to do, I was struggling because I knew my inferiority in English expressions, and while I wanted to quit, wanted to stay home and pull the covers up over my head, the me from my previous life won the battle. I wasn't going to quit.I told myself I should do it, or I would be more disappointed in myself than before.\nTime passed so fast, especially in a ten-week quarter. When I sat on the beach in winter break, trying to come to a conclusion about the lessons I had learned in my first quarter in the States, I found I had learned so much. Thanks for my dear native classmates, for the faculty and staff, for my friends and fellow ambassadors. Everyone is so nice, and so willing to help me. I need to push myself more to not let them down.\nNow it's the second week of winter quarter, I am taking five classes and two ESL workshops this quarter. I am officially in charge of the social media ambassador job. I practice dance, horseback riding, golf and tennis every week. I plan to attend multiple speeches, parties and events. I am planning the Chinese New Year party for the whole GSM. And above all, I am looking for the summer internship. Yes, I am much more busy than last quarter. But, I feel so good. I am in my fastest speed towards my \"Best Version\".\nSo that's my story of the first year MBA life. If I can survive, you can too. And by the way, I got an A in my Finance class, good for me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaygfp b/data_all_eng_slimpj/shuffled/split2/finalzzzaygfp
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"In December of 2018 I heard about the Depth Year and thought that was a great idea to continue on what I had started doing some months prior to December, which was to reduce my consumption of spiritual tools (tarot and oracle decks, crystals, etc.) and use the many tools I have acquired throughout the years. You can see my initial ideas and plans for my Depth Year here.\nI can say that I have been successful in making progress in most of my goals. I have been using decks I owned and previously did not use. I usually chose one or two decks per month and this allows me to really get to know the energy of the cards and infuse them with my energy. I am not going to lie, I have bought\/traded more decks since starting the Depth Year. But they have been decks in my wishlist and not just random and impulsive buys. In comparison to how much I bought before, I am doing really good!\nI have also been limiting myself to working with one crystal per month and have not bought a new crystal for a long time. This working with one crystal per month started many months ago for me. In the beginning, I remember really struggling to stick to one crystal, it has become so much easier now. I will usually pick the crystal that attracts me the most from my collection, this month it is amethyst. Then I will set an intention for what I hope to accomplish during the month. This month's work with amethyst I am being intentional about developing my psychic and intuitive abilities, strengthening my third eye chakra, and I am also using amethyst to keep my cool at work :).\nI have also been dedicating more time to my writing. I set up a calendar reminder on my phone for Dianna's writing time every Saturday and Sunday morning. Sometimes it's not easy waking up, but my hubby is supportive and will usually encourage me to wake up and write before the girls wake up. I have also been creating often and it has really helped me keep my mental and emotional sanity.\nI'm trying to get back into the habit of reading a book at least once a month. I recently finished Light is the New Black by Rebecca Campbell. I am now reading The Book of Goddess by Kris Waldherr. I am using my Desire Map Planner almost daily. I'm slowly working my way through Holistic Tarot.\nOverall my Depth Year is going well. I can say that \"narrowing\" my focus has been very beneficial as it allows me to get more out of whatever I am learning and doing. This narrowing of focus is really contrary to how I usually work. I love to have a wide focus because I want to see everything and find the connections between things. But, I think it's a good idea for me to learn to use a narrow focus when I need it. That's it for my Depth Year update. Are you also doing a Depth Year or would like to do a Depth Year? I'd love to read your opinion and thoughts in the comments!\nThis is the amazonite stone that I had with me for the whole month of February.\nIn February, I devoted my crystal connection to a pretty potent stone known as amazonite. If you research the metaphysical properties of amazonite online, you will read that it helps you connect with your authentic self. I can say that each time I have pick up amazonite to work with, this has been true for me.\nConnecting with the energy of amazonite always leads me in the direction of finding my self. It sounds very easy doesn't it? \"Carry some amazonite around, it will help you connect to your real authentic self!\" But what the instructions don't tell you is that in order to connect to your real authentic self, the other self (the fake self) you have been living with, building up, and introducing to your friends and acquaintances has to go.\nThis \"side effect\" of working with amazonite is not as glamorous as connecting to your authentic self, but it's a non-negociable part of the journey. For me this journey to my authentic self always begins with the question: what am I doing with my life? Followed by these other equally imposing questions: Am I doing what I am meant to be doing? Am I sharing my gifts for the benefit of the word? Am I following my passion and living my life purpose?\nI'm not going to lie, when these questions become alive in my heart and my mind it causes me a lot anxiety and emotional distress. I journal a lot. I talk to my loved ones. I begin planning and writing in detail where I think I want to go and how I can get there as quickly as possible. I am forced to face the reality of my life. I have to look at myself as objectively as I can. If I am not careful, I will start berating myself for not being where I think I am meant to be.\nThis time around my journey to myself did not end with me beating myself up for not living my dream life. This time, I rested in the knowledge that I am in just the right place, doing the right thing, talking to the right people who need me in their life at this moment. But in order to arrive at this conclusion I had to answer all the above questions and be uncomfortable for some time while the process of re-discovery was taking place.\nIf you feel that your life is not going in the right direction or you feel stuck, or you feel like a fake, I highly recommend working with the energy of amazonite to help you along. If you have worked with amazonite, I'd love to read about your experience in the comments!\nOver on Instagram, I am participating on Small Rebellious Acts of Creativity (#SRAOC) by The Jar Writer's Collective, a weekly invitation to engage in small rebellious acts of creativity, as the name implies. I will be doing a combination of poetry, short story, art, digital collage, and photo editing. This week's prompt is 'the way home'.\nStrolling in the shop, I came upon a dainty sign that read \"The Way Home\". It had an arrow pointing towards a bloated red heart adorned by angel feathers. \"So cliche\", I shrugged. Damn these saccharine Valentine's-Day-Capitalist-Slaps-On-The-Face. But inside my chest it would not go unnoticed that my heart truly ached.\nI knew I had lost my way home a long time ago. I frankly didn't know what or where home was for me. Maybe I read too much into that sign. \"Home is where the heart is. What the hell does that even mean? Who even needs a home or a heart anyway.\" I soothed myself as I exit the seasonal aisle.\nThe damage was done. My mind would betray me by playing seemingly innocent scenes from my childhood. My well trained mind knew just which scenarios to skip over or fast forward to. How long had it been? I've lost count of the years since I've felt at home anywhere.\nA soul collage I made during the 30 days practice.\nA few weeks ago marked the end of the #30dayswithMary practice. I hesitate to even write that sentence, because I feel that my walking with Mary will never really end.\nMany of us who were walking together felt sadness about the conclusion of the 30 days. While I felt sad, I also felt comforted in knowing that the connections I made with my fellow travelers in the last thirty days are long-lasting and will continue even after the hashtag is no longer present in my feed.\nAs for my relationship with Mary, I can say that Mary will be my companion for many years to come. In this post, I'm just going to share some highlights of the 30 days and explore how I might continue my journey with Mary.\nI remember at the beginning of the 30 days, during the mantra recitation, I felt a change in the energy outside of myself, in the area where I had set up the altar. I am not used to feeling energies or presences so my fear quickly took over and the presence disappeared. I didn't feel any energies outside of myself after that, because I understood that the presence I was seeking was always inside of me and every night when I went to meet Mary, I was actually meeting Mary within myself.\nAt some point in the 30 days I began to sing the mantra. The first time I sang the mantra I unlocked something inside myself. I opened a door that I didn't even realize was there. I felt feelings I had not allowed myself to feel and realize what healing I needed to work on. Most of my feelings involved my role as mother and daughter. I am still working through those feelings and I might be brave enough to share them in the future.\nOne of my favorite parts of the practice was the opportunity to make soul collage cards, like the one in the photos above. I've always had a love of photo editing and this love was reawakened thanks to the #30dayswithMary practice.\nDuring the final days of the #30dayswithMary practice I began working on a set of 10 personal oracle cards. I used photo editing\/collage to create them. I will do a separate blog post to share that project, but I feel that the cards were influenced by my experience in the walking with Mary days, so they hold a special place for that reason.\nThrough this practice and thanks to Hettienne's wonderful guidance, I learned how to set up an altar and conduct a ritual. We also met many representations of Mary, but there is still so much more to explore.\nIn summary, I am thankful for having the opportunity to participate in this 30 days journey and for sharing the experience with other like-minded people from the tarot community on Instagram. I am hoping that Hettienne hosts this practice again next year.\nOver on Instagram, I am sharing my experience as I participate in a journey of walking with Mother Mary. This opportunity was made possible by Hettienne Grobler, you can find her blog here. Hettienne shares wonderful information and guidance to help get you started.\nThere is a lovely community developing around this journey on Instagram, you can find them by searching for #walkingwithmary, #30dayswithmary, #thirtydayswithmary.\nI guess I should give some background on why I decided to join this journey. I am not a religious person, and I don't subscribe to Christian beliefs, I am especially not interested in Catholicism, and yet here we are. I took my time to discern whether this experience was something I was interested in. I resisted joining and found many excuses not to, but the pull was strong. What allowed me to justify this practice for myself is my view of Mary as a representation of the Goddess, and as an archetype that has much to teach us. So with an open mind and open heart, I joined the others.\nWe are currently on Week Three and are exploring The Veils of Mary and Mary as Sophia, Our Lady of Wisdom, Shekinah. One of the activities this week involves doing some needle work, whether it be sewing, weaving, knitting. These activities are considered traditionally women's work.\nAs a child, I grew up around a lot of women, my mom and sisters, my aunts, cousins, grandmothers, great-grandmothers and their friends. Doing needlework was a common \"fun\" activity for the girls, while the boys were out playing in the mud. I don't remember how young I was when I decided that those traditional women activities were not for me. I didn't really feel interested in sewing, learning to cook, clean, etc. I was more interested in reading, writing, playing outside.\nI don't remember if I ever made a servilleta (a hand embroidered towel) for my mom, which is something that daughters usually make for their moms. I do have vague memories of one of my aunts teaching me how to embroider with a special needle, but I don't think I finished that project.\nFast forward to many years later, when I began preparing to become a mom, I started to feel a pull towards the women's work that I had rejected in my youth. I asked my mom to send me some embroidery hoops, thread and fabric.\nI found it very meditative and relaxing. My thread collection grew. I didn't become a pro, but I began to see this type of work through a different lens.\nIt's been about five years since I picked up my needle and thread. I have also changed a lot in those five years, and I am seeing a deeper layer\/meaning in this kind of women's work.\nI think it's magical. It is creative. It can become a magical, creative and spiritual experience. It can be compared to meditating, writing, praying the rosary, doing ritual. Done individually, in the privacy of your home, you can infuse your materials with intention, you can recite a mantra while filling in the patterns and shapes. You can be very intentional with your colors. You can use it to work magic. You can create sacred items for your rituals and sacred spaces.\nur grandmothers have been working magick for a very long time, even if they didn't see it through that lens.\nI'm excited to create something beautiful for my journey with Mary.\nWhat has been your experience with women's work? I'd love to read your comments so please share!\nA new idea has been making the rounds on my social media communities, it is the idea of embarking on a Depth Year, a year where you dive deep into the hobbies, interests, and tools you already own\/have, instead of finding new hobbies, interests, and tools to jump into.\nI first heard about the Depth Year through Kelly Ann Maddox's video, and shortly after that I saw it in many other people's social media accounts. Although, the idea seems to have originated from David Cain, click here to read his blog post on Raptitude.\nBefore becoming familiar with the term Depth Year, I had already started doing something similar, as a result of feeling very overwhelmed by the amount of stuff I had accumulated. Most of my \"stuff\" is tarot decks and crystals. A few months ago, I began to feel a discomfort\/concern for the reasons behind why I had accumulated such a large amount of stuff. And yet, I continued to constantly acquire new stuff without actually using it. So I decided that, rather than buying more new decks and crystals, I was going to focus on using the decks and crystals I had. This is what the depth year is all about.\nThe purpose of embarking on a Depth Year began with my tarot decks and crystals, but becoming mindful of my over consumption of those items, allowed me to shine a light on other areas of my life where I seem to accumulate things and not actually use them. One of those areas is enrolling in classes with the intention of working through the class, but then not making the time for it. I also do this with books, I have a pile of books I want to read and will add new books often, without actually getting to reading the book.\nI am planning to do a three month check-in to see how I am doing on my goals above.\nDo you set new year's intentions? Have you heard of the Depth Year? I would love to hear from you in the comments!\nMany years ago, even before starting a serious creative\/spiritual path like I am on right now, I knew that the relationships we have with other people were very important. Of supreme importance, is the relationship we have with ourselves, but that is a topic for a separate post. Today, I want to talk about our relationships with others.\nWhen I was in college, I knew that the relationships I created with my classmates and professors were going to be more important in my life, in the long run, than getting good grades. Thankfully, I was able to get good grades and focus on my relationships while in college. Later on, when I was in graduate school and out in the \"real world\" I still held this belief and acted accordingly. It's not that I didn't value the work I was doing or what I was learning, it was mostly the belief that, at the end of the day\/year\/life, what would matter most is the relationships I had and what they taught me about myself and about others.\nThis belief has been the backdrop of how I live my life. Now that I am following a more structured journey of self discovery, I can see how all the relationships in my life were sent to me in the right time for the lesson I needed to learn.\nI am not implying that I am a pro at this relationship thing, but I am saying that relationships are important and should not be taken for granted. Even the troubled and broken relationships we might have with our families, they are important to our growth and development. Even the superficial relationships we might have with coworkers or people we think are not very important in our lives, they exist for a reason.\nLately, relationships are on my mind because I am realizing that, as I get older, they are becoming more complicated and existing relationships are beginning to change. I have had my fair share of relationships (friendships) end and I am not a stranger to the fact that, when we are no longer needed in one person's life, we will drift apart and move on. I have made peace with that, it has happened to me too many times and I am thankful for learning that it's okay for relationships to end when it is time.\nRelationships are also on my mind because of the holidays, I am sure. What about you? Do you spend time reflecting on how you are relating to your fellow humans? I would love to hear from you in the comments!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"City of Beaumont issued the following announcement on July 5.\nOn Thursday, July 5, 2018 at 12:03 p.m., Beaumont Police responded to 2675 S. 4th (Shop N Save) in reference to a disturbance in which the suspect was believed to be intoxicated on PCP.\nPreliminary investigation revealed that the suspect, Joseph Raymond Murphy, a 40 year old Beaumont man, entered the store and began shattering bottles. He then went outside, walked through traffic, removing his clothes and used his hand to shatter the windshield of a vehicle. Officers arrived in the area of 4th and Roberts and attempted to take Murphy into custody. Murphy, charged one Officer and struck him multiple times, transferring a large amount of blood onto the Officer's face and neck.\nMurphy also spit saliva on the Officers during the altercation. Murphy was eventually taken into custody by 4 Officers, all of whom were exposed to Murphy's blood\/saliva.\nTwo of the Officers sustained minor injuries in addition to the exposure. Murphy was transported to a local hospital where he was treated for his injury and intoxication prior to being transported to the Jefferson County Jail. Murphy was charged with Assault on a Public Servant, Harassment of a Public Servant, and Resisting Transport. All Officers involved are being treated for their injuries\/exposure.\nWant to get notified whenever we write about City of Beaumont ?\nSign-up Next time we write about City of Beaumont, we'll email you a link to the story. You may edit your settings or unsubscribe at any time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In addition to building and selling PCs we also sell parts, such as CD-ROMs, Hard Drives, Mice, Keyboards, and many other items. Why get a new computer? Upgrade the one you have now and Save!\nEveryone is familiar with the stereotypical \"computer nerd.\" They can only talk in highly technical terms, and most of the time they are talking down to you. A good computer consultant should be able to converse with you on a normal level. They should be able to convey complex technical ideas so anyone can understand the goals, risks, and benefits. A good computer consultant should reflect, in their words and actions, that your business is valued and appreciated. We strive to offer you the best service at a reasonable cost AND we will do it without a lot of \"Techo speak\".\nWe are dedicated to providing Customers with prompt, efficient service within a friendly and professional atmosphere. Serving every customer with the same grade of services each and every time. Our business strives on strong customer relations and excellent services. Our services are professional and fair priced. Our sales selection is extensive and fair priced. We are your one stop computer shop !!\nThere's absolutely no risk to you.\nWe are here to help if you are having problems. We might be able to help. Just Join the Forum and ask that Question. Like I said we might be able to answer that question.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've told you in past blog posts how much we adore this beautiful film about boatbuilding as a way of life in the Southern Caribbean islands. Now you can watch it on your own big screen at home as you stream it from Vimeo's On Demand service. We can't wait to watch it again, after seeing the boat, meeting the filmmaker, Alexis Andrews, and of course thoroughly enjoying a screening at the Wooden Boat Show last June.\nVanishing Sail is launching online just in time for the Antigua Classic Yacht Regatta which features in the film. This critically acclaimed documentary follows the adventures of legendary Caribbean boat builder Alywn Enoe as he struggles to complete a traditional racing sloop in time for the Regatta. His is a dying art and his mission is to train his sons to keep these skills alive.\nHaving completed a successful film festival run winning 7 awards, and 80 special screenings in 23 countries, this mesmerizing piece of cinema is now going to be available to screen online, in the run-up to the Antigua Yacht Regatta 2018. As of yesterday, the film will be shown exclusively on Vimeo on Demand: the high-quality streaming platform for independent film.\nFive years in the making, Vanishing Sail touches on many emotive themes: the importance of local culture and tradition; the challenges facing small islands; the struggle of young people finding their place in the world and the power of community to heal wounds, create beauty and ultimately overcome seemingly insurmountable odds. With a backdrop of Caribbean island life, rich in music, humor and the daring deeds of sailors and smugglers, the adventure is as entertaining as it is illuminating.\nThe timeless story continues to resonate with audiences at arthouse cinemas, unique venues and events, regattas, special historical and cultural institutions including Unesco World Heritage sites and coastal communities around the world. This online launch is the start of a voyage to connect with wider audiences beyond just lovers of the sea, islands and craft.\nThis entry was posted in Entertainment, Reconstruction by Kristin. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Previewed during the National Train Show in August, AZL is excited to announce that the 40' outside braced boxcars are here! And for the first release, we bring you the very popular ATSF! A single, a two-pack and two four-packs are available. Each car has unique road numbers. That is eleven different numbers in the ATSF release!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaypnl b/data_all_eng_slimpj/shuffled/split2/finalzzzaypnl
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index 0000000000000000000000000000000000000000..4fbcdbe667aa17d1209dd042b26ddd001da90f21
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaypnl
@@ -0,0 +1,5 @@
+{"text":"The Kucinich campaign is still awaiting an official response from ABC News about the unexplained \u2013 some have charged \"inexplicable\" \u2013 way in which the network has handled its post-debate online coverage of Ohio Congressman and Democratic Presidential candidate Dennis Kucinich over the past few days.\n* Congressman Kucinich was apparently deliberately cropped out of a \"Politics Page\" photo of the candidates.\n* Sometime Monday afternoon, after Congressman Kucinich took a commanding lead in ABC's own on-line \"Who won the Democratic debate\" survey, the survey was dropped from prominence on the website.\n* ABC News has not officially reported the results of its online survey.\n* During the early voting Monday afternoon and evening, U.S. Senator Barack Obama was in the lead. By sometime late Monday or early Tuesday morning, Congressman Kucinich regained the lead by a wide margin in this second survey.\n* Sometime Tuesday morning, ABC News apparently dropped the second survey from prominence or killed it entirely.\n* AND, as every viewer of the nationally televised Sunday Presidential forum is aware, Congressman Kucinich was not given an opportunity to answer a question from moderator George Stephanopoulos until 28 minutes into the program.\nThe campaign submitted objections and inquiries to ABC News representatives on Monday and Tuesday. ABC News representatives have failed to respond \u2013 or even acknowledge \u2013 those objections and inquiries.\nSomebody at ABC has a problem with Kucinich's campaign, apparently. A big problem. And is taking action.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've installed the \"TwonkyMusic Version 3.0\/Terratec Noxon Edition\" on an Ubuntu Platform. After starting the Server my Noxon (FW: 3.5.12.6291) is dispayed on the TwonkyMusic Status WebSite but the server is is not displayed in the main menu of the Noxon!?!? Does it make sense?\nI tryed with restarting the noxon and restating the server but it wouldn't work. I also enabled the UPnP-Option on my Firewall and again: nothing.\nI tried to get a little deaper. I started TCPDUMP on the server PC and there where multicast packets send by the server and also send by Noxon. That's the reason why I can see the Noxon in the status site of the TwonkyServer.\nI also started Whireshark on my notebook. It's like the Noxon connected via WLAN, so I expected to see the same multicast packets (shared medium). But there where only multicasts from the Noxon and no communication from the server.\nIs there an error in reasoning? Where are the multicasts from the server?\nI have probably the same problem, that TwonkyMedia sometimes is available in the main menu of my iRadio and sometimes not.\nI can play music from my Faviorites (that were put there when TwonkyMedia was available) without any problems,.\nSo connection to the Twonky Server must be available and the Twonky Server streams the music correctly, but in the main menu there is still no TwonkyMedia available.\nThen for any reasons (I have not yet found out how to reproduce this) TwonkyMedia is available again in the main menu of the iRadio.\nI have tried both static IP and DHCP mode, but no difference.\nHas anyone an idea what might go wrong here?\nIs this an issue caused by Twonky or by the firmware of iRadio? Or both?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Asthma management guidelines recommend adding a long-acting \u03b22-agonist (LABA) or increasing the dose of inhaled corticosteroid (ICS) as step-up therapy for patients with uncontrolled asthma on ICS monotherapy. However, it is uncertain which option works best, which ICS particle size is most effective, and whether LABA should be administered by separate or combination inhalers.\nThis historical, matched cohort study compared asthma-related outcomes for patients (aged 12\u201380\u2005years) prescribed step-up therapy as a \u226550% extrafine ICS dose increase or add-on LABA, via either a separate inhaler or a fine-particle ICS\/LABA fixed-dose combination (FDC) inhaler. Risk-domain asthma control was the primary end-point in comparisons of cohorts matched for asthma severity and control during the baseline year.\nAfter 1:2 cohort matching, the increased extrafine ICS versus separate ICS+LABA cohorts included 3232 and 6464 patients, respectively, and the fine-particle ICS\/LABA FDC versus separate ICS+LABA cohorts included 7529 and 15\u200a058 patients, respectively (overall mean age 42\u2005years; 61\u201362% females). Over one outcome year, adjusted OR (95% CI) for achieving asthma control were 1.25 (1.13\u20131.38) for increased ICS versus separate ICS+LABA and 1.06 (1.05\u20131.09) for ICS\/LABA FDC versus separate ICS+LABA.\nFor patients with asthma, increased dose of extrafine-particle ICS, or add-on LABA via ICS\/LABA combination inhaler, is associated with significantly better outcomes than ICS+LABA via separate inhalers.\nFor patients whose asthma is not controlled with low-to-moderate doses of inhaled corticosteroids (ICS), the preferred initial step-up option in asthma management guidelines is the addition of a long-acting \u03b22-agonist (LABA), either as a fixed-dose combination (FDC) or via two separate inhalers, with further step-up options including an increased dose of ICS [1, 2]. Until 2011, the British Thoracic Society\/Scottish Intercollegiate Network recommended ICS and LABA in either separate or combination inhalers . However, previous research has shown that, for patients using separate ICS and LABA inhalers, problems with adherence can result in LABA monotherapy, which has been associated with an increased risk of death . Despite this, there is little evidence or guidance in the literature regarding whether separate or combination inhalers are preferred or recommended.\nFor adolescents and adults, the addition of LABA to ICS therapy often provides better symptom control and less need for reliever therapy [5\u20137]. In randomised controlled trials, patients receiving LABA in combination with ICS experienced improved lung function and symptom control compared with patients receiving an increased ICS dose [6, 8\u201313]. Some of these trials assessed the effect of combination inhalers compared with separate inhalers, but few statistically significant differences were observed . However, up to 95% of patients with asthma are not eligible for participation in randomised controlled trials because of smoking, comorbidities or lung function impairment . Furthermore, recruited patients benefit from close follow-up during their participation in these trials, and adherence in these studies is therefore likely to be higher than in the general patient population [15, 16]. Observational studies have indeed suggested that combination inhalers improve adherence in real-life patients, which is likely to influence therapeutic outcomes [17, 18]. Moreover, we have previously shown that an increased dose of ICS is as effective as add-on LABA via FDC inhaler in a general population of patients with asthma .\nAnother variable that may affect outcomes is particle size. Previous randomised trials have often focused on fine-particle ICS (mass median aerodynamic diameter 2\u20135\u2005\u03bcm), such as fluticasone and budesonide, whereas a substantial proportion of the general asthma patient population is prescribed extrafine-particle ICS. Extrafine-particle ICS (e.g. beclomethasone, mass median aerodynamic diameter \u223c1\u2005\u03bcm), exhibits significantly increased lung deposition compared with fine-particle ICS [20, 21]. Furthermore, real-life research has suggested that treatment with extrafine-particle ICS results in similar or superior asthma-related outcomes at lower prescribed doses compared with fine-particle ICS [22\u201325].\nThe objective of this retrospective matched cohort study was to evaluate the real-life effectiveness of step-up asthma therapy with LABA added to ICS (any particle size) via separate inhalers compared with either an increased dose of extrafine-particle ICS or with a fine-particle ICS\/LABA FDC inhaler. Our hypothesis was that, for adolescent and adult patients with evidence of persistent uncontrolled asthma, step-up treatments with either increased extrafine-particle ICS or with fine-particle ICS\/LABA FDC would be a superior alternative to ICS and LABA in separate inhalers.\nThis historical, matched cohort, comparative effectiveness study drew on two large UK databases: the General Practice Research Database (GPRD), which is now part of the Clinical Practice Research Datalink , and the Optimum Patient Care Research Database (OPCRD) (supplementary material) . Both databases are well described and have previously been used for respiratory research. The Independent Scientific Advisory Committee approved the use of GPRD data for the purpose of this study, and the OPCRD was approved by Trent Multi-Centre Research Ethics Committee (current NHS HRA approval REC reference 15\/EM\/0150) for clinical research use. The study protocol was designed by an independent steering committee and approved by the Anonymised Data Ethics Protocols and Transparency Committee, which is the independent scientific advisory committee of Optimum Patient Care. This study was registered at clinicaltrials.gov (identifier NCT01697722).\nEligible patients (aged 12\u201380\u2005years) had at least 2\u2005years of continuous data within the study window: a 1-year baseline period before the index date to control for baseline confounders, and a 1-year outcome period following the index date during which primary and secondary outcome parameters were assessed. During the baseline year, patients had at least two prescriptions for asthma therapy, including at least one for ICS. Exclusion criteria included a recorded diagnosis of chronic obstructive pulmonary disease (COPD) or any chronic respiratory disease other than asthma, as well as a history of smoking for patients between 61 and 80\u2005years of age. This age limit was chosen in order to minimise the inclusion of patients with COPD misdiagnosed as asthma, since the proportion of patients with airflow obstruction who also have COPD increases to \u223c40% for patients aged 60\u201369\u2005years and the diagnoses can be difficult to differentiate [28\u201330].\nThe index date was defined as the date ICS treatment was stepped up by one of the following three options: 1) \u226550% increase in ICS dose of extrafine-particle beclomethasone dipropionate (ICS step-up); 2) addition of LABA via FDC of fine-particle ICS and LABA with no change in ICS dose (ICS\/LABA FDC); or 3) addition of LABA by separate inhaler, with no change in ICS drug substance, device type or daily ICS dose in beclomethasone equivalents (separate ICS+LABA). During the baseline period, patients in all cohorts were prescribed either fine-particle or extrafine-particle ICS. During the outcome period, patients in the ICS step-up cohort were prescribed extrafine-particle ICS, patients in the ICS\/LABA FDC cohort received fine-particle ICS, and patients in the separate ICS+LABA cohort received either fine-particle or extrafine-particle ICS (supplementary material and table S2). The study period covered 14\u2005years between January 1997 and January 2011.\nThe primary effectiveness end-point was risk-domain asthma control, i.e. no hospital admission, no emergency department attendance, no general practice attendance for lower respiratory tract infections, and no oral corticosteroid prescriptions as previously described [23, 25]. Secondary end-points included the number of severe exacerbations, number of acute respiratory events and treatment stability (table 1) [31, 32].\nThe total daily coverage of \u03b22-agonist (short-acting \u03b22-agonist (SABA)+LABA) was calculated as the dispensed amount divided by 365 and defining two puffs of SABA (albuterol 200\u2005\u03bcg or terbutaline 500\u2005\u03bcg) as lasting 4\u2005h, and two puffs of LABA via pressurised metered-dose inhaler or one puff of LABA via dry powder inhaler as lasting 12\u2005h.\nThe statistical analysis plan was defined before data analysis, following a priori defined standard procedures of the research team. After patient selection (figure S1), an exploratory analysis was carried out for all baseline variables, and statistical analyses were carried out via primary and secondary effectiveness outcomes. The analysis of baseline data revealed clinical and statistically significant differences between cohorts. A matched cohort analysis was therefore carried out using demographic and clinical matching criteria, as outlined in table 1. To maximise cohort sizes, a 1:2 matching was carried out between the extrafine-particle ICS step-up and the separate ICS+LABA cohorts on the one hand, and between the FDC fine-particle ICS\/LABA and the separate ICS+LABA cohorts on the other hand. Summary statistics were used to describe all matched baseline and outcome variables, both for the combined patient group and separated into treatment groups. To reduce the possibility of selection bias, numerous potential confounding variables were assessed, including some that were identified during previous respiratory research from the research team: baseline differences between treatment cohorts (p<0.10) were considered potentially important, as well as baseline variables that were predictive (p<0.05) of each effectiveness outcome in multivariable analyses. Variables that met these criteria were examined for collinearity, as well as for clinical importance, to select those used as potential confounders in regression modelling of outcomes.\nTwo-way comparisons between cohorts, both baseline and outcome variables, were analysed by conditional logistic regression. The adjusted odds of achieving risk-domain asthma control were compared between treatment groups using conditional logistic regression models, with asthma control used as the dependent variable, and treatment and potential confounding factors as explanatory variables. Variables with p<0.10 were included as potential confounding factors. The same method was employed to determine the adjusted odds of achieving treatment stability. The total number of severe exacerbations during the outcome period was compared between matched treatment groups using a conditional Poisson regression model, and exacerbation rate ratios were determined. The model used empirical standard errors for more robust confidence intervals, and adjustments were made for potential baseline confounders. Statistically significant results were defined as p<0.05. Analyses were carried out using SPSS Statistics (version 19; IBM, Armonk, NY, USA), SAS (version 9.3; SAS Institute, Cary, NC, USA), and Microsoft Office Excel 2007 (Microsoft Corp. Redmond, WA, USA).\nInformation on the unmatched patient population is available in the supplementary material and table S1. After patient selection (figure S1) and 1:2 matching there were 3232 patients in the extrafine ICS step-up cohort and 6464 patients in the separate ICS+LABA cohort (table 2, table S2, supplementary material). The mean age was 42\u2005years and 61% were female. The cohorts were well matched for asthma-related parameters, and statistically significant differences between cohorts were small (table 2). The mean ICS dose prescribed on the index date for the extrafine ICS step-up cohort was approximately twice that for the separate ICS+LABA cohort (394 versus 194\u2005\u03bcg\u00b7day\u22121; p<0.001).\nOver the outcome year, the proportion of patients who achieved risk-domain asthma control increased from 65% during the baseline year to 75% and 71% in the extrafine ICS step-up and separate ICS+LABA cohorts, respectively. Therefore, a significantly higher proportion of patients prescribed extrafine ICS step-up achieved risk-domain asthma control compared with those prescribed separate ICS+LABA, and fewer of these patients experienced severe asthma exacerbations or acute respiratory events or required a change in therapy (table 3). The adjusted odds ratios for achieving risk-domain asthma control and treatment stability were significantly greater with extrafine ICS step-up compared with separate ICS+LABA. For risk-domain asthma control, the OR (95% CI) was 1.25 (1.13\u20131.38), while the rates of severe exacerbations or acute respiratory events were significantly lower (figure 1).\nAdjusted outcome measures comparing a step-up in asthma therapy using an increased dose of extrafine-particle inhaled corticosteroids (ICS) versus the addition of a long-acting \u03b22-agonist (LABA) to ICS in a separate inhaler. Data show the adjusted odds ratio (OR) or rate ratio (RR) with 95% CI. Variables have been adjusted for the following. #: number of oral corticosteroid prescriptions; \u00b6: smoking status and number of oral corticosteroid prescriptions; +: smoking status and number of severe exacerbations during baseline year; \u00a7: smoking status, number of oral corticosteroid prescriptions and number of primary care consultations.\nThe daily dose of SABA during the outcome year was significantly higher in the extrafine ICS step-up cohort than in the separate ICS+LABA cohort, although the median was the same (219\u2005\u03bcg\u00b7day\u22121). However, total \u03b22-agonist coverage was significantly longer for the separate ICS+LABA cohort than for the extrafine ICS step-up cohort (table 3, figure 2a).\nComparison of total \u03b22-agonist coverage for patients prescribed a) an increased dose of extrafine inhaled corticosteroids (ICS) versus ICS+long-acting \u03b22-agonist (LABA) in separate inhalers or b) fixed-dose combination (FDC) of fine-particle ICS\/LABA inhalers versus ICS+LABA in separate inhalers. Data are presented as coverage in hours per 24\u2005h, where the total daily coverage of \u03b22-agonist (short-acting \u03b22-agonist (SABA)+LABA) was calculated as the annual dispensed amount divided by 365 and two puffs of SABA (albuterol 200\u2005\u03bcg or terbutaline 500\u2005\u03bcg) were defined as lasting 4\u2005h, and two puffs of LABA via pressurised metered-dose inhaler or one puff of LABA via dry powder inhaler were defined as lasting 12\u2005h.\nAfter matching, there were 7529 patients in the fine-particle ICS\/LABA combination cohort and 15\u200a058 patients in the separate ICS+LABA cohort. Most of the baseline differences between cohorts were small, although several were statistically significant because of the large cohort sizes (table 2, table S2, supplementary material). For the patients who received separate ICS+LABA inhalers, the mean index date was 2\u2005years earlier than for those who received fine-particle ICS\/LABA FDC inhalers. Probably as a consequence of this, fewer patients in the separate ICS+LABA cohort had recorded smoking data (83% versus 92% in the ICS\/LABA FDC cohort) (table 2).\nDuring the outcome year, the proportion of patients who achieved risk-domain asthma control increased from 58% during the baseline year to 72% in the fine-particle ICS\/LABA cohort and 68% in the separate ICS+LABA cohort. Risk-domain asthma control and treatment stability were significantly higher for patients who received fine-particle ICS\/LABA inhalers compared with separate ICS+LABA inhalers. For risk-domain asthma control, the OR (95% CI) was 1.06 (1.05\u20131.09) and the risk of severe exacerbations and acute respiratory events was significantly lower (figure 3, table 3). The proportion of patients changing treatment was also lower in the fine-particle ICS\/LABA FDC cohort than in the separate ICS+LABA cohort (table 3).\nAdjusted outcome measures comparing a step-up in asthma therapy using a fixed-dose combination (FDC) inhaler containing fine-particle inhaled corticosteroid (ICS)\/long-acting \u03b22-agonist (LABA) versus the addition of LABA to ICS in a separate inhaler. Data show the adjusted odds ratio (OR) or rate ratio (RR) with 95% CI. Variables have been adjusted for the following. #: smoking status, number of primary care consultations and medication possession ratio; \u00b6: smoking status, adherence to ICS therapy, number of nonasthma-related consultations and number of acute respiratory events; +: number of acute respiratory events (categorised) and year of index date; \u00a7: number of acute respiratory events (categorised), year of index date and number of primary care consultations.\nThe daily dose of SABA was significantly lower, and the daily \u03b22-agonist coverage (SABA+LABA) was significantly higher, for the fine-particle ICS\/LABA FDC cohort compared with the separate ICS+LABA cohort (table 3, figure 2b).\nThis study of a real-life patient population showed that the odds of achieving risk-domain asthma control and treatment stability were significantly higher with an increased dose of extrafine-particle ICS or with fine-particle ICS\/LABA FDC inhalers, as compared with ICS and LABA via separate inhalers. Furthermore, the rates of severe exacerbations and acute respiratory events were significantly lower with increased doses of extrafine-particle ICS or fine-particle ICS\/LABA FDC inhalers than with separate ICS+LABA inhalers. The total \u03b22-agonist coverage (adding hours covered by SABA+LABA) was significantly longer for patients prescribed add-on LABA.\nNational and international guidelines recommend add-on LABA as the first step-up of asthma therapy. Until 2011 (the last year of our study period) the recommendation of British guidelines for add-on LABA was either via ICS\/LABA FDC inhaler or ICS+LABA via separate inhalers [3, 33]. In agreement with this, add-on LABA was the most commonly prescribed step-up therapy in the current study. Of the 35\u200a356 patients in the unmatched cohorts who met the eligibility criteria, 84% were prescribed add-on LABA, of whom more than two-thirds received ICS and LABA in separate inhalers.\nAsthma management guidelines are often predominantly based on randomised controlled trials that exclude a large proportion of patients. Although the data are not always clear, several reports have suggested that there is no difference in efficacy between add-on LABA in combination or in separate inhalers [16, 33, 34]. This could partly be due to generally high adherence in these studies, which may therefore not reflect the situation in the general patient population of asthma patients, where adherence tends to be relatively low [15, 35, 36]. Differences in adherence probably contribute to the discrepancy between the efficacy shown in controlled trials and the effectiveness observed in real-life studies [19, 37, 38]. Real-life studies have indeed suggested that FDC inhalers can make a substantial contribution to aid refill persistence , and that patients on these inhalers are less likely to require rescue medication than those using separate inhalers . Indeed, the current results show a substantial advantage of fine-particle ICS\/LABA FDC inhalers compared with separate inhalers.\nAn alternative step-up option to add-on LABA is an increased dose of ICS, which has previously been associated with a reduction in serious adverse events and hospitalisation . Our previous historical matched cohort study has shown that, in a general population of patients with asthma, an increased dose of extrafine-particle ICS is as effective as fine-particle ICS\/LABA in a FDC inhaler . Furthermore, the current study shows that an increased dose of extrafine ICS is significantly more effective than add-on LABA via separate inhalers. A potential concern of ICS step-up might be that this leads to increased use of reliever therapy. Although the statistical test shows that the daily dose of SABA was higher for patients prescribed increased extrafine ICS compared with those prescribed separate ICS+LABA, the median daily SABA dose was the same in both cohorts, suggesting that there is no real clinical difference between the groups. In contrast, the median daily SABA dose was substantially higher for patients prescribed separate ICS+LABA than for those prescribed fine-particle ICS\/LABA in FDC inhalers (274 and 219\u2005\u03bcg\u00b7day\u22121, respectively). Furthermore, for total \u03b22-agonist coverage (SABA+LABA), the exposure was significantly longer for patients prescribed add-on LABA (median 3\u2005h in the ICS step-up versus 10\u201319\u2005h in the add-on LABA cohorts). Therefore, an ICS dose step-up does not appear to result in any substantial increase in SABA prescription, while total \u03b22-agonist exposure increases substantially with add-on LABA.\nIn randomised controlled trials of bronchodilators, the primary end-point tends to be lung function, and patients generally have to demonstrate \u03b22-agonist reversibility to be included. The trials are also often limited in timescale, and most require patients with stable asthma whose exacerbation risk is low. In this context, the current study has several strengths. The findings depict clinically relevant real-life outcomes for asthma therapy in UK primary care, the patient group is larger than most controlled trials, and the 1-year follow-up is likely to minimise seasonal variation and capture less frequent outcomes, such as exacerbations. We did not exclude active smokers (15\u201324% of each cohort, except in the 61\u201380-year-old age group), patients did not have to demonstrate \u03b22-agonist reversibility to be included, and \u223c25% of patients in each cohort had rhinitis, which is a risk factor for poor asthma control . Therefore, the analysed patient group appears to be representative of the general UK asthma population, and this improves generalisability of the study findings to general practice.\nOur primary end-point of risk-domain asthma control was designed to capture the absence of asthma exacerbations during the outcome year. This composite data outcome has been used in previous research [19, 24] and is in line with factors identified by expert working groups as being reflective of the future risk component of asthma control [31, 32, 41]. Although we conducted a matched cohort analysis, and confounding was eliminated to the greatest possible extent with adjusted analyses, we acknowledge that the use of matching may increase the risk of selection bias , and ignoring the matching variables may produce a certain population causal effect . Similar to other historical cohort studies, this one is also limited by the fact that causal relationships cannot be inferred from the observed associations, and the possibility of unrecognised confounding factors remains. Nevertheless, we believe that, although slight baseline differences might still be present, they are unlikely to be of clinical significance.\nFor patients receiving ICS and LABA in separate inhalers, poor adherence to the ICS component can be associated with risks. An investigation into asthma deaths in the UK found that 6% of patients prescribed LABA in separate inhalers were not prescribed any ICS throughout the year, and an additional 12% had significant gaps in ICS prescribing, suggesting that approximately one in five of these patients were exposed to the increased risks associated with LABA monotherapy [3, 4]. In clinical practice, it is generally considered that combination inhalers improve adherence, and they guarantee that LABA is not taken without ICS . Our results provide real-life evidence to support the notion that either ICS step-up or add-on LABA via FDC inhalers is more effective than ICS and LABA in separate inhalers, which is the least desirable step-up option in a general patient population. The importance of this information is emphasised by the fact that more than two-thirds of patients prescribed add-on LABA received it via separate inhalers.\nIn conclusion, this real-life study of a broad general population of young people and adults with persistent asthma suggests that step-up therapy, by increasing the dose of extrafine-particle ICS or by adding LABA in FDC inhalers, is a better approach than prescribing ICS and LABA in separate inhalers. Whether this is a strategy that will work better in some patient subgroups, e.g. patients with impaired therapeutic response to \u03b22-agonists or those with frequent exacerbations, is a question for further study.\nWe thank Neil Barnes (London Chest Hospital, London, UK) for his advice during the study design development and Anne Burden (Research in Real-Life Ltd, Cambridge, UK) for contributions to the study design and initial analyses. Our thanks go also to Alison Chisholm (Respiratory Effectiveness Group, Cambridge, UK) and Muzammil Ali (Research in Real-Life Ltd) for their assistance and input on data analyses and interpretation, and to Helga Mikkelsen (Cambridge Research Support, Cambridge, UK) and James Haslam (freelance) for writing and editorial assistance, funded by Research in Real-Life Ltd.\nBritish Thoracic Society, Scottish Intercollegiate Guideline Network. British guideline on the management of asthma. A national clinical guideline. 2014. www.sign.ac.uk\/guidelines\/fulltext\/141\/ Date last accessed: May 18, 2016.\n. LABA monotherapy in asthma: an avoidable problem. Br J Gen Pract 2013; 63: 627\u2013628.\nRoyal College of Physicians. Why asthma still kills: the National Review of Asthma Deaths (NRAD) Confidential Enquiry Report. London, RCP, 2014.\nStep-up therapy for children with uncontrolled asthma receiving inhaled corticosteroids. 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The overlap syndrome of asthma and COPD: what are its features and how important is it? Thorax 2009; 64: 728\u2013735.\nThe proportional Venn diagram of obstructive lung disease: two approximations from the United States and the United Kingdom. Chest 2003; 124: 474\u2013481.\n. The Asthma-COPD Overlap Syndrome. N Engl J Med 2015; 373: 1241\u20131249.\nAn official American Thoracic Society\/European Respiratory Society statement: asthma control and exacerbations: standardizing endpoints for clinical asthma trials and clinical practice. Am J Respir Crit Care Med 2009; 180: 59\u201399.\nAsthma outcomes: healthcare utilization and costs. J Allergy Clin Immunol 2012; 129: Suppl. 3, S49\u2013S64.\nBritish Thoracic Society, Scottish Intercollegiate Guideline Network. British guideline on the management of asthma. A national clinical guideline. 2008.\nSalmeterol\/fluticasone propionate (50\/500 \u03bcg) in combination in a Diskus inhaler (Seretide) is effective and safe in the treatment of steroid-dependent asthma. Respir Med 1999; 93: 876\u2013884.\nPersistence, adherence, and effectiveness of combination therapy among adult patients with asthma. J Allergy Clin Immunol 2006; 118: 574\u2013581.\n. Asthma out of control? A structured review of recent patient surveys. BMC Pulm Med 2006; 6: Suppl. 1, S2.\nA pragmatic single-blind randomised controlled trial and economic evaluation of the use of leukotriene receptor antagonists in primary care at steps 2 and 3 of the national asthma guidelines (ELEVATE study). Health Technol Assess 2011; 15: 1\u2013132.\nvan der Molen T, et al.\nReassessing the evidence hierarchy in asthma: evaluating comparative effectiveness. Curr Allergy Asthma Rep 2011; 11: 526\u2013538.\nInhaled corticosteroids for asthma: impact of practice level device switching on asthma control. BMC Pulm Med 2009; 9: 1.\nAllergic rhinitis. Lancet 2011; 378: 2112\u20132122.\nAsthma outcomes: exacerbations. J Allergy Clin Immunol 2012; 129: Suppl. 3, S34\u2013S48.\n. Why match? Investigating matched case-control study designs with causal effect estimation. Int J Biostat 2009; 5: Article 1.\n. Ignoring the matching variables in cohort studies \u2013 when is it valid and why? Stat Med 2013; 32: 4696\u20134708.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Rite of Christian Initiation for Adults (RCIA) is the process by which one explores faith and life in the Catholic tradition. The Rite is modeled after the initiation process used in the first centuries of the early Church. It includes a combination of study, prayer, and participation in the liturgy and certain rites at Mass. Participants, known as catechumens, undergo a process of conversion as they study the Gospel, profess their faith in Jesus Christ and the Catholic Church, and receive the Sacraments of Christian Initiation: Baptism, Confirmation and Eucharist.\nHow long does it take for any relationship to develop? We all journey on different paths at different times. The Period of the Catechumenate, or period of instruction, depends on how the catechumen is growing in the faith. When a catechumen and the RCIA Team believe the individual is ready to make a full faith commitment to Jesus and the Catholic Church, a request is made and the Rite of Election celebrated.\nThere are several, distinct periods of formation. Some occur during certain times in our church year. Others are ongoing and tailored to an individual's journey.\nThis period is also known as the Period of Inquiry. After consultation with an advisor, the \"inquirer\" may decide to seek acceptance into the Order of Catechumens or Candidates.\nIn both the Rite of Acceptance (non-baptized inquirers) and the Rite of Welcoming (validly-Christian baptized inquirers), the inquirer states his or her desire to become a member of the Catholic Church and the assembly affirms that desire, welcoming the individual into the catechumenate.\nThe Rite of Election, held on the first Sunday of Lent, enrolls the names of \"the elect\" seeking the sacraments of Christian Initiation (Baptism, Confirmation and Eucharist) at the coming Easter Vigil, which is known as the Rite of Reception.\nRCIA offers classes for men, women and high school aged teens who are validly-baptized Christians but not Roman Catholics, into full communion with the Catholic Church through the receipt of the sacraments of Confirmation and Eucharist.\nRCIA offers classes for men, women and high school aged teens into full communion with the Catholic Church through the receipt of the sacraments of Christian Initiation (Baptism, Confirmation and Eucharist).\nRCIA is held Monday evenings from 7:00 - 8:30pm from September through May.\nThe preparation of adults and teens for the sacraments of Christian Initiation is the responsibility of the entire parish community. The RCIA team exercises this responsibility in different roles.\nSponsors are companions to adult seekers as they walk the journey to initiation.\nCatechists lead the sessions of Breaking Open the Word on Sundays and facilitate weekly sessions.\nIf you are interested in learning more about RCIA, Contact Us or submit the following form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oshkosh Chamber sponsors Business Day in Madison, an opportunity for businesses to meet and discuss business issues of importance with state representatives.\nThe Northeast Wisconsin Chamber Coalition (NEWCC) consists of Chambers in Fond du Lac, the Fox Cities, Green Bay, Oshkosh and Waupaca present an annual legislative update in February. NEWCC will present its legislative agenda, which will be delivered to state Representatives and Senators from Northeast Wisconsin.\nThe Washington D.C Fly-In is an annual meeting with Wisconsin delegation to discuss issues of importance. For the first time this year's fly-in was done as a statewide coalition with WMC and 40 other statewide Chambers of Commerce to collectively advance a pro-jobs agenda. The fly-ins are put together because of these groups collective interest in federal issues that directly affect the business community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Professional sales manager cars resume - A resume goal is a announcement of your destinations for paintings, frequently recorded on the maximum amazing cause of your resume. A resume objective is typically possibly some sentences in duration. The high-quality objective is one this is tweaked to the movement you're making use of for. It states what type of commercial enterprise you're attempting to find, and what aptitudes and stories you have got that make you perfect for that calling. A resume goal can also in like manner consolidate in which you have got been, and in which you need to cross for your calling. For example, it would express a number of your beyond accomplishments, and after that preserve ahead to the sorts of accomplishments you may want to reap later on (ideally, accomplishments you have to achieve for the association you're making use of to work for).\nAs you are stressed about a way to write an auto sales resume you may just take a look at some references in this situation it's miles higher in an effort to search diamond geo engineering services.\nGuide the recruiter to the conclusion which you are the great candidate for the car income supervisor process. It's simply quite simple. Tailor your resume by way of choosing relevant duties from the examples under after which add your accomplishments. This manner, you can position your self within the nice way to get hired.\nA resume goal is a assertion of your locations for paintings, frequently recorded on the maximum amazing reason of your resume. A resume goal is usually possibly a few sentences in period. The great goal is one this is tweaked to the movement you're making use of for. It states what kind of commercial enterprise you're searching for, and what aptitudes and stories you have got that make you best for that calling. A resume objective may additionally in like manner consolidate in which you've got been, and wherein you have to pass for your calling. As an example, it would explicit some of your past accomplishments, and after that retain forward to the styles of accomplishments you might want to attain in a while (ideally, accomplishments you have to acquire for the association you're making use of to work for).\nWhat specialised tiers and gadgets would you are saying you're succesful? Rundown all that follow to your paintings. Be particular and as substantial as viable. This rundown can include some thing from web-based totally social networking ranges to extend administration frameworks and pc languages. If you've labored with distinctive tiers, listing the ones as well.\nPinnacle ideas approximately sales resume on pinterest marketing resume lewesmr car sales supervisor resume sales resume examples to sell your abilties to reentrycorps automobile income supervisor resume income resume examples to promote your abilties to .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following the Karish North discovery, the Stena DrillMAX drillship will return to drilling three wells at the Karish Main field as part of Energean's Karish-Tanin development project. Source: Stena Drilling.\nIsrael has taken another step forward in its ambition to become a regional natural gas supply hub with Energean's Karish North offshore discovery.\nThe Karish North exploration well was drilled by the Stena DrillMAX harsh environment dynamically positioned drillship to a TD of 4,880 m, encountering a gross hydrocarbon column of up to 249 m, of which 27 m of core was recovered. The operator reported high-quality reservoir in the B and C sands. The initial gas in place estimate is 1\u20131.5 Tcf.\nEnergean said it will further evaluate the discovery's resource potential and liquids content. The Stena DrillMAX will return to drilling three wells at the Karish Main field, discovered in 2013, as part of the Karish-Tanin development project.\nKarish, Karish North, and future discoveries will be tied back to the 775-MMcf\/D Energean Power floating production, storage, and offloading (FPSO) vessel located 90 km off Israel. Gas production from the Karish-Tanin project is expected to start up in the first quarter of 2021.\nThe Tanin Block A, B, and C fields were discovered in 2012. The Tanin mega-complex forms a northern extension of the supergiant Leviathan field currently being developed by Noble Energy.\nFuture gas sales agreements, the company added, will focus on both the Israeli domestic market and key export markets in the region.\nEnergean has five exploration licenses off Israel. Elsewhere in the Eastern Mediterranean, the company has licenses in Greece and Montenegro, including a 25-year exploitation license for the Katakolo block off Western Greece.\nNoble's Leviathan field is expected to start production by yearend 2019. Leviathan holds an estimated 22 Tcf of gross recoverable resources. The Houston operator's neighboring Tamar field off Israel has been producing gas since 2013.\nIn an effort to encourage further exploration in its waters, Israel in November launched its second bid round, making available 19 offshore blocks in five zones in the Levant Basin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How one researcher used place-based research strategies to uncover details about his family.\nIn 1786, my ancestor Philip Crume and his wife Sarah moved from Shenandoah County, Va. Here's how I followed my own place-based research advice to uncover more details about the family.\nThe Family Tree Resource Book for Genealogists shows Dunmore County was formed in 1772 from Frederick County and renamed Shenandoah County in 1778. Frederick County came from Orange and Augusta counties in 1738. I checked them all and found Philip's deed records in Augusta (1749), Frederick (1762), Dunmore (1778) and Shenandoah (1781) counties.\nConsult town and county networks.\nThe Augusta County GenWeb Project <rootsweb.com\/~vaaugust> says to start with court records in Chronicles of the Scotch-Irish Settlement in Virginia, 1745-1800 <rootsweb.com\/~chalkley>. There, I found a record placing \"Philip Crine\" in Augusta County in 1747 and a John Crum there in 1752. Were they related?\nLinkpendium's Virginia links include digitized land grants at the Library of Virginia. One reveals Philip Crume received land on the Flint Run river in Frederick County in 1767. Daniel Croom got land in Henrico County in 1722. Could he be Philip's father?\nPhilip Crume lived on Flint Run river from 1749 to 1773. John Crum's father, Henry, was granted land on Hawksbill creek in 1755. MapQuest shows the two waterways are only 16 miles apart.\nAccording to land records, Philip Crume was in Beckford Parish, Dunmore\/Shenandoah County, from 1767 to 1778. Henry Crum's widow, Ann, lived there when she wrote her will in 1775. I searched Google Books <books.google.com> for Beckford Parish. Shenandoah Valley Pioneers and Their Descendants says the parish \"embraced all of the country known as Shenandoah County,\" so Philip and Ann weren't necessarily neighbors.\nPeggy Shomo Joyner's Abstracts of Virginia's Northern Neck Warrants of Surveys (self-published) includes an abstract of a 1749 deed mentioning Philip Crume living on Flint Run, and John Crume, a chain carrier (surveyor's assistant). The same-spelled names in the same record suggests the two men were related.\nSeveral of Philip Crume's Virginia neighbors were Quakers from Chester County, Penn. Since people often migrated in groups, I posted to a Chester County mailing list asking if anyone had seen Crumes in records there.\nThis entry was posted in Genealogy Research Strategies and tagged going, town. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After years of speculation about the tension between actress Sarah Jessica Parker and Kim Cattrall, their beef finally went public. Amid the actors expressing their disappointment that a third SATC would not be made, many believed it was because of Cattrall. Cattrall had made recent comments about her relationship with her cast mates, and specifically said that Parker could have been nicer. Parker expressed her sadness in the drama and loss of what she believed was a friendship. When Cattrall's brother died, Parker sent her condolences, to which Cattrall very publicly posted, \"I don't need your love or support at this tragic time @sarahjessicaparker.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apple has reportedly hired Tomlinson Holman, inventor of the Lucasfilm THX high-fidelity audio system, to be its new audio chief.\nLeo Laporte of the TWiT Netcast Network wrote on his official Twitter account on Monday that Holman is joining Apple to \"run audio.\" The post, highlighted by GigaOm, described Apple's hire as a \"major upgrade.\"\nIn addition to creating the THX sound certification for Lucasfilm, Holman was also behind the world's first 10.2 surround sound system. He is currently a film sound professor at the University of Southern California.\nApple has not confirmed the alleged hiring of Holman, and what projects he will play a role in at the secretive company are not yet known. In addition to experience designing loudspeakers and amplifiers, he has also won an Academy Award for Technical Achievement in motion pictures.\nHolman's hiring could also play a role in hardware development, ranging from the headphones that ship with all iPods and iPhones, to integrated speakers on devices like the iPhone, iPad or Apple's line of MacBook computers. He could also play a part in the software Apple writes for media playback on devices ranging from iPods and iPhones, to Macs, the Apple TV.\nIn February, it was alleged that Apple is looking to offer higher quality music on its iTunes Music Store. The company is said to be in negotiations with studios to offer tracks in a 24-bit high-fidelity format, which would be higher quality than the 16 bit currently available on iTunes or with a compact disc.\nMany Macs and some PCs already support 24-bit sound, and the iTunes media player software supports 24-bit files. Audio tracks are also recorded in 24 bits by artists in the studio, but they are downgraded to 16 bits when they are pressed to CD or made available to iTunes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Frank, from Westfield, NJ, is designing and building a computer using 7400 series discrete logic devices, which were popular in the 1960's. The first iteration was just an arithmetic logic unit with an interface of switches and LED's. The next step is to create a device that can read instructions from a ROM chip and to give it control over how input and output are handled. The end goal is to create a computer that can be used to play PONG. Stop by his exhibit to take a closer look!\nCall For Makers 2016: Extended Through March 25th!\nThe Greater Newark Mini Maker Faire is a showcase of creativity and innovation in the New Jersey community. The event invites makers (individuals, groups, industries and K-12 to universities) to present their projects that explore DIY (do-it-yourself) thinking, reveal their process and show how materials are pushed to their limits.\nWhether it's science explorations, woodworking, robotics, beekeeping or e-textiles this event will explore it all with the Museum's art and science collection as the ultimate 'maker' examples.\nIf you have a project that explores DIY, or shows people how they can make things themselves, we'd love to have you apply. If you are a Young Maker under the age of 18, please have your (adult) project mentor submit your project for you.\nHere is a list of ideas and projects, but this is just a starting point.\nStep back in time with artist John P Chismar as he presents favorite projects constructed from plans drafted by A. Neely Hall in his 1915 book \"Home-Made Toys for Girls and Boys\". Projects include toys such as the paper pinwheel, buzz-saw whirligig, the eight-blade windmill with toy jumping-jack, clog-dancer, a home-made toy motor-boat and a toy elevator. You will be transported to simpler times by the \"Toy Shocking Machine\" when toys utilized \"large shocking coils [to] contract the muscles to such an extent that it is impossible to let go of the metal grips until the current has been shut off\u2026\" Come join John for some old fashioned pastimes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Katong's rich cultural mix has contributed to its unique cuisine \u2013 most notably Peranakan (Straits Chinese) cuisine. In particular, a spicy Malay noodle soup called Peranakan laksa was made famous even beyond Singapore shores. Katong Laksa originated from Joo Chiat area. The founder lived in Tembling Road\/Joo Chiat Terrace initally before moving his business into the eating house at East Coast Road\/Ceylon Road. Because the Katong people loved his laksa, it has come to be known as the Katong laksa throughout the world in Hong Kong and New York.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When it comes to great ideas, theft is no sin.\nGoals should be hard\u2014not slam dunks. Write them aggressively.\nDuring Google's first year, investor John Doerr pitched the idea of using an organizational system called Objectives and Key Results (OKRs) for goal setting. He presented it as an effective way to both set high level goals and also measure progress toward these goals in a quantifiable way. I'm sure he made a pretty persuasive pitch, because they were quickly adopted across the entire company and, during my first week at Google, I found myself outlining my very first set of OKRs.\nThe OKR system is an effective way of setting and communicating goals within an organization by connecting company, team, and personal objectives to measurable results. It pushes your company and employees to stretch and grow while staying focused on a defined and collective goal.\nMy Google OKRs were just the first of many. Fast forward 15 years \u2014 today, every new hire at Lucid Software spends a portion of his or her first week watching Google Ventures partner Rick Klau present John's original slides in a YouTube video (which you can view here).\nWhen I left Google to start Lucid Software, I didn't take OKRs with me because it was what I was used to\u2014I did so because the system had been so successful at Google. I firmly believed it could produce the same benefits at Lucid\u2014and the system hasn't let me down yet.\nSetting stretch goals: OKRs establish a clear and simple pattern for setting goals within your company at the individual, team, and organization levels. When you write an OKR, you solidify a goal and draw a line in the sand. At the end of the day, you can measure yourself against this line to define your success. In contrast, without setting objectives in advance, you are stuck trying to justify your accomplishments.\nAt Google, we were constantly reminded that OKRs are specifically designed to push you, not to make you look good. In other words: Goals should be hard\u2014not slam dunks. Write them aggressively.\nPromoting accountability: At Google, we made all our OKRs public. We are doing the same at Lucid by having every employee upload their personal OKRs to our internal wiki. Doing so promotes accountability and transparency and makes that line in the sand even more pronounced. It's easy to forget about goals that only you know about\u2014it's a game-changer when there are others (sometimes hundreds to thousands of others) also holding you accountable.\nIn addition to their public nature, OKRs promote accountability because they are graded each quarter. Google uses a scale from 0 to 1. You're not supposed to crush it with consistently perfect scores\u2014once again, these are stretch goals, so if you hit between 0.6 and 0.7, you are making good progress. In fact, if you are easily hitting all of your OKRs, you have either reached superhero status or are sandbagging.\nHowever, it's important to always go back and account for why you fell short on an OKR, especially if you score below a 0.4. Life happens, and priorities change\u2014but you need to be honest with yourself about the root cause.\nAligning the organization: Establishing OKRs helps everyone in the company align their goals. It enables you to determine where to focus your efforts in order to assist the company as a whole. OKRs are a simple way to ensure everyone is working towards the same result, and you can check that all operations have linked objectives supporting the company.\nThis year, I've set a goal to review each employee's personal OKRs. It will be around 150 total, and while I realize that's a lot of objectives and key results to wade through, I also realize the value of understanding where each person's focus lies. As the company continues to grow, there are limited ways for me to keep in touch with the entire organization and understand what everyone is working on. Reviewing OKRs is a good way to see what initiatives everyone else finds important and evaluate how those match up with the initiatives I find important.\nSetting up OKRs is pretty simple and straightforward\u2014that's the beauty of it.\nSet annual company OKRs that serve as big umbrella ideas for the year, but also set OKRs each quarter. Annual OKRs may evolve as the year progresses, but quarterly OKRs do not change.\nSet these overall company OKRs first. Then have departments use those to set team OKRs. Personal OKRs are then set by each employee based on the company and team OKRs.\nSet an objective. An objective is a goal, and it tells you where to go. Aim for a maximum of five objectives each quarter so you don't spread yourself too thin.\nSet key results for each objective. Key results indicate how you will meet your objective. Key results must be measurable\u2014in other words, they should be quantifiable and specific targets should be stated up front. Try to have a maximum of four key results for each objective.\nStore your OKRs in a public space. For example, Google puts each employee's OKRs in the employee directory.\nHold 1:1 meetings to review OKR progress as the quarter progresses. Hold a quarterly company-wide meeting to review company and team OKRs.\nAt the start of each quarter, grade the previous quarter's OKRs. Use a scale from 0 to 1. Key results are graded individually, and an objective's grade is the average of its key results. Don't use OKR grading for employee evaluation. Remember that if you consistently score a 1, your OKRs are not aggressive enough.\nRemember that the only way things happen is if you reach high. Setting the bar low may make your employees feel good for the time being, but it doesn't move the organization forward. I have come to realize that for a startup of our size, we all need to feel a little uncomfortable about what we're doing in order to succeed at the pace we need to.\nI would also emphasize the importance of writing your own personal OKRs. Even though you might feel your OKRs are simply the company OKRs, clearly defining your own can act as a north star. As CEO, it's so easy to just be busy and meander through a bottomless pit of tasks, but writing OKRs has helped me stay focused on what I should be doing. Doing so also helps demonstrate your commitment to the OKR system.\nFinally, you may set OKRs for yourself or the company and realize those areas are not as important as you originally thought. You're not locked into your OKRs, and it's alright if you need to change direction. Just be honest about why you don't accomplish an OKR.\nHere at Lucid, we don't plan on giving OKRs back to Google anytime soon. Not when they have played such a key role in keeping us on track to succeed at such a rapid rate. So maybe it's just better to come clean\u2014thanks, Google, for yet another really good idea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Well-regarded Mission Viejo Nadadores assistant coach Joey Gracia has accepted an offer to coach Capistrano Valley's boys and girls swimming teams, Cougars athletic director Chad Addison announced.\nGracia served as an assistant last season for Tesoro girls coach Todd Conrad. The Titans finished as the CIF-SS Division 1 runner-up for the second consecutive spring.\nGracia has been a key assistant with the Nadadores for almost two years. He also has coached at Ocean View and Golden West College.\nHe replaces girls swimming coach Pat Higginson, who remains the Cougars' girls water polo coach, and retired boys swim coach Steve Yancey.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This year's season started off pretty much on time, perhaps even a few weeks late considering the arrival of the new Marauder and some visitors in town.\nA slight cut on the front rotors, due to annoying warping.\nNew NGK plugs to replace the 65000 km originals, gapped at 0.050.\n4 brand new tires to replace the original Goodyear F1 tires.\nSolving an annoying exhaust system squeak at the Y-yipe junction.\nThat I was able to get 65000 km out of my original Goodyear F1 tires is a testament to the responsible way I drive, as there was still a little bit of tread left and I was not yet at the wear indicators. This included many many drag strip visits, and several road course lapping sessions. Had it not been for the deterioration in wet weather handling (specifically the resistance to hydro-planing in standing water pools), I would have continued to operate the original tires, but with my son so often in the car with me, I did not want to take any chances.\nThe tires I got to replace my original Goodyears turned out to be Yokohama AVS Sport tires, obtained at a great price. While they were recommended to me as being as good in the wet as the original F1s were in the dry, I have since determined that claim to be a bit stretched, but nonetheless they appear to be good tires.\nAnother contributing factor to the low spring and early summer mileage was an uncomfortable environment at my place of employment, which left me pretty much unmotivated to do anything related to F-bodies for 2003. Once I took steps to correct that situation, I was able to take advantage of the remaining months of 2003 to their fullest. The car was running flawlessly and fuel mileage was way up... 20-25% in fact, which is surprising considering that there did not appear to be any decrease in fuel mileage over the years. Surely a result of the new plugs and my custom gap.\nParticularly enjoyable was the opportunity to take my car to Toronto on business, which allowed me the chance to get reacquainted with the fine automobile. While the car remained parked at the hotel lot pretty much the whole weekend as I worked from my hotel room, it was nonetheless very reassuring to occasionally peek outside every now and then while taking breaks during the quasi-24-hour work sessions.\nWhile I did not get a chance to evaluate the newfound fuel efficiency on the race track, I did manage to find my way onto the famed Gilles Villeneuve Formula One racetrack. This facility is used as a bicycle path during the off season, and this permitted me to get some photo opportunites I never dreamed I could find.\nWeather conditions allowed me to continue operating the car until very late November. Unfortunately, a wet snow storm that iced over caused me to get stuck on a poorly cleared side road not far from my house. Once I managed to get moving (thanks to a pair of nice folks who helped push me off the curb), I got the car home and parked it for 2003.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Starting a business needs considerable amount of money. The cost involved in starting a business like incorporation of business, purchasing raw material and equipments, cost of marketing, staff salaries, etc., become a burden on any businessman.\nInvest in the future with business loan.\nJust started your business in Australia and need an extra injection of cash for the day-to-day running? Or, has your company been growing and you need to take it to the next level?\nIt's the oldest story in the book. You have an innovative product and have all the expertise to successfully run a startup but are set back by one major thing-capital. These days the ways of securing finances are endless which makes it difficult for startups to choose which one is right for them. However, with some knowledge you will be in a good place to make the right decision for your startup. This tutorial provides 5 ways through which can fund your startup.\nMarketing your business requires you to reach everywhere you can but no matter how hard you might try, you can be in just one place on any given instant. However, it's your business brochure that can reach everywhere you want it to and will help you in influencing both existing and potential customers. Effective business brochures clearly outline what a business is all about and what's on offer for its customers. So, it's vital for a business brochure to be written and designed effectively to make the most out of it. Here are some tips to consider in this regard.\nAlthough all business owners depend on cash funds for the ventures to be functional when it comes to importing business owners they can't rely on such exemption. Importing business often encounter prolonged cash flow cycles. Thus for importing industry owners it is important that they improve their financing strategies continuously.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jardi is the easiest way to hang jars on a wall. It is functional safe and removable. It fits wide mouth Mason Jars (3in or 7.6cm). Made of quality silicone that holds the jare securely. It comes with 2 hangers one pink one purple and mounding hardware.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PRICE REDUCED!! Amazing lot for sale!3.0 miles to Whole Food Market (11 minutes drive)3.5 miles to University of Maryland College Park (11 min)Beacon Heights station expected to open in early 20201.8 miles to Glenridge Shopping Center (under renovation) (5 min) and so much more, Contact us for more details!\nJuan Umanzor Jr. of Long Foster Real Estate Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Show Me The MONEY! - Mechanical Hub | News, Product Reviews, Videos, and Resources for today's contractors.\nA couple years ago I had the opportunity to provide service for a customer in the Twin Cities. The customer had motivation to replace his old hot water boiler through a low interest loan program offered to Minnesota residents, at the time, who were candidates for major appliance upgrades that would improve efficiency and lower utility costs.\nAs with any other potential client, I approached this project with high intention of finding the best possible boiler and components to fulfill the requirements of this particular home and the owner's needs. The approach is simple really, perform a heat loss calculation; measure the existing radiation in the home (in this case it was baseboard) and select the right size boiler for a replacement. Although some contractors do not take the time to this I do and the following is proof of how big an impact this approach can make.\nNow this particular installation brought together a few new technologies the owner was not currently using. The baseboard lined the home's walls as if it were holding them hostage of, as if it may have been almost free at the time of the original install. As I look back at the job file I see that there was over 45 feet more than needed. Most of my competitors wouldn't have known this [huge] piece of info though and I was glad to have it as it allowed for me to offer the highest efficiency modulating condensing boiler available at the time. Changing from their 30 year old cast iron behemoth was charting new territory for them and they were not only impressed that I was the first contractor to offer such savings they were equally impressed I was able to show them proof from past projects of doing the same for others.\nOther new technology introduced was the method of variable speed pumping. I prefer, for many reasons of which I will not get into here, to control the systems flow related to \u2206T although this particular job would be a great candidate for \u2206P as well with its many zone valves and a single circulator.\nI am convinced that the combination of my own sales experience, past client references along with proof of past customer's savings was also a big part of why this client chose all but one option presented to them prior to them hiring us to do the work. I guess the one option (replace five zone valves) was just a little more than the budget allowed after checking the box for the installation of an indirect water heater.\nFollowing up with my customers after the project is done has always been a habit, after all if there are any quirks or questions that need to be addressed I want them to know I am available. In this particular case I was contacted by the homeowner with some very pleasing information. Pleasing for him especially because it was proof in writing, from his natural gas provider, that he and his family were now using 28% less natural gas this year as compared to the same time the prior year! That's a huge savings directly related to the replacement of their [then over-sized] boiler with a properly sized unit that also does the work of heating their domestic hot water; replacing a standard water heater with an efficiency rating of approximately 65%. Take a look at the \"report card\" sent to them from their gas company, the proof is in writing, and that is why the simple homework is so worth it! Oh, and the smiley face is the cherry on top.\nHaving the savings in writing can be a large task in itself but it's definitely worth your time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Teak manors have been broadly settled all through the tropics to deliver excellent timber in trees of good development and stem structure. Numerous components influence the achievement of teak planting programs including site, seed supply and seed quality, administration and other natural elements, for example, bugs. Site is the essential variable impacting ranch development and advancement. Development and yield of manors developed in various site qualities, inside and outside the teak area is outlined. With right site determination, development and yield can be enhanced more than 100 %. The structure of reasonable teak locales is checked on. Seed supply is a variable restricting planting scale and the nature of the manor, particularly so in nations where teak is an outlandish. Seed prerequisite in different estate alternatives is arranged and the impact of seed sources (provenances) and enhanced seed on development and nature of ranches is illustrated. A transient system for enhanced seed and clonal stock generation is proposed. Silvis social administration of teak ranch hones in weeding, diminishing and bug and fire insurance for development and quality change is audited.\nThe fundamental target of teak manor foundation is to deliver fantastic timber in trees with great or worthy development rates. To accomplish these targets, the planting site must be appropriate for the development and advancement of teak. Teak ranches have been set up all through the tropics, inside and outside its common dissemination range. This covers an extensive variety of climatic conditions, i.e. from the central sort to the sub-tropical sort with a scope of precipitation and temperature of 500-3,500 mm and 2\u00b0-48\u00b0C (least and greatest reach) individually. Soil conditions additionally differ from barren acidic to fruitful alluvial. A vast variety in development and other tree characters, for example, stem structure, method of fanning, blossoming propensity and wood nature of the manor regularly comes about.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbpxy b/data_all_eng_slimpj/shuffled/split2/finalzzzbbpxy
new file mode 100644
index 0000000000000000000000000000000000000000..f9d21b899d51810862cb4deab3b0cbfd651168f5
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+{"text":"Productization of high-tech materials. Applying rapid prototyping, design thinking, and Tomorrow Lab's process to invent novel product concepts using high-tech materials as a starting point.\nWith a heavy focus on product invention, lab-style product development, and industrial design, this course will ask students to 'productize' a concept using a high-tech material as the starting point. Students will be provided samples of a high-tech mystery material on the first day, for example graphine, fungal foam, shrilk, phase-change materials, or conductive inks. Students will propose their own questions to explore, then use prototyping methods provided by the instructors to develop the ideas in to real product concepts. Students will end the course with a prototype and documented design explorations. Instructors will also introduce concepts of manufacturability, cost, and feasibility of production.\nThis class will be run by a team from Tomorrow Lab\u00ae which uses science & design to invent revolutionary hardware products for a better future. Tomorrow Lab is a product design and engineering consultancy based in NYC with a unique focus on prototyping new hardware technology, and turning those prototypes into market-ready products.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sold in bags of 100. Whether it's a clambake or for clams casino, you'll fall in love with these! Shipped live.\nProduct arrived as expected - cooked on the grill on high heat for maybe 5-8 minutes everyone opened which means they are alive and fresh-great tasting product.\nJust a quick word to thank you for the great job of selecting the clams and lobsters for my fathers birthday! As always, they were greatly surprised!\nI wish you much continued success in your fine company.\nWithout a doubt, the best...the best clams, seafood, service...Thanks to Charleston Seafood My party in the Desert of Arizona was a smash hit!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A look at what your healthy money habits should be at each age and financial goals to complete every decade.\nWe've all heard the metaphor that life is a marathon, but in the case of finances, there's nothing more comparable. Preparation is key to getting to the finish line, and once the race starts, it's important to keep a steady pace. The same goes for your finances\u2014from beginning to end there are important steps you should take in order to have successful savings, investments and a retirement plan. Having a financial road map to plan out savings and other funds will get you through the big moments in life, whether it's the loss of a job or buying a new home.\nIT'S THE START OF A LONG FINANCIAL JOURNEY by the time you hit your 20s, and it's the most important time to be proactive with finances. T his age encompasses getting a footing in a career, which involves more financial obligations, but it's a valuable start to your financial journey to adjust it in the way you need.\nA fund you always want to have at your disposal is an emergency fund. As early as possible, you want to begin depositing anywhere from 5 to 15 percent of your check into a savings account. This can be used at several different points in your life including if the loss of a job were to occur. \"You really need at least three to six months of your salary\/income available in case something happens, whether you get laid off or whatever the reason would be,\" says Stan Molotsky, presi dent and CEO of SHM Financial Group.\nIt can never be too early to begin your retirement fund. \"People can't rely on [Social Security] for their retirement,\" Molotsky says. \"It's a nice supplement, but it's not going to pay for all the things [people] want it to pay for when they do retire.\" You want to take advantage of your company's retirement plan\u2014 a 401(k) or 403(b)\u2014as soon as possible.\nBY THE TIME YOU HIT YOUR 30S you could be thinking about getting married, buying a home and having children. These encompass huge financial strides and having that savings from your 20s shows where you can go in your 30s and beyond.\nFor those who have it, the debt can linger on for a large portion of your life. Paying more each month lends more dollars toward the principal and can reduce a number of payments and astronomical interest costs by years. After all, eliminating that student debt opens up more doors for other financial opportunities.\nWHETHER YOU'VE DIPPED INTO YOUR SAVINGS for a house remodel or a medical emergency, you should sti ll be in the \"saving mindset\" and looking more into your financial future regarding retirement and estate planning.\nYou want your retirement to be at its best. Your savings should be going toward retirement first before anything else. \"Whether you're saving through work through the 401(k) or 403(b) or your regular I RA account, that becomes your 'focusing in on' and taking money and putting it aside,\" Molotsky says.\nAt this point, you also want to have all non-mortgage consumer debt (car loans, credit card loans, etc.) paid off, leaving debt a notion of the past. Once that is all said and done, it's the perfect time to begin paying extra on the mortgage. As long as you're maxed out on your retirement and have education funding locked in, any extra dollar should go right to the mortgage payment.\nYOUR SAVINGS HAS BEEN BUILDING UP for the big finale: retirement. In your 50s you should be acknowledging your retirement goals and reaching those goals whether it's moving to a new location or looking into long-term life insurance.\nThe three years prior to retirement and three years post retirement are the most critical, and Molotsky suggests adjusting for minimal and conservative spending three years prior to prepare. \"Our goal when we advise clients is to become more cautious [including] the first couple years into retirement so you get a hand on what you're really spending and how you're spending it and the fact that you're probably going to be spending a little bit more when you first retire than you will 10 years after you retire because of a variety of different factors,\" he says.\nNow that you're getting closer to retirement, you can review your will and life insurance plan to make sure everything is set in place the way you want it. It's vital to get a head start on these since you want to have a will and long-term insurance set in stone before they are needed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We can arrange last minute or emergency coach hire for transportation of any number of passengers to any destination in the UK.\nWhen booking your coach hire it is important that you have your flight details handy as we will need these details to pass onto the operator so that they can monitor the flight and be prepared for any delays.\nPlease also bear in mind that the coach quoted on may be able to cater for more passengers than you may actually require this is because smaller vehicles do not have as much luggage space as the larger vehicles, your account manager will be able to advise you at the time of booking.\nWe can supply standard or executive vehicles from 8 \u2013 83 seats, most of the larger vehicles will have a toilet and DVD facilities on board to make the final leg of transfer as comfortable as possible, smaller vehicles can also come with DVD facilities and air conditioning, your account manager will be able to advise you when dealing with your booking.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tomorrow sees the Euro 2016 kick-off, where home team and the bookies favourite France take on Romania. The British Foreign Office have warned that there is a \"high threat from terrorism\", so huge is the security challenge that the French government has apparently released a terror alert app called SAIP (Syst\u00e8me d'alerte et d'information des populations), which stands for Population Alert and Information System. In spite of the lingering threat, ~2.5 million football fans are expected to travel to France to soak up the continent's biggest sporting event's atmosphere. AirBnB have said it is preparing to welcome ~250k visitors alone, with fans from around the world including roughly 37k Americans and around 10k Aussies.\nThese fans bring with them significant spending power to the host nation; benefits of this competition will be felt right across the country as matches will be hosted across 10 different stadia\/cities; the whole country should at least profit from the influx of visitors. This will be a much needed boost for France, whose central bank announced yesterday that the 2016 second quarter growth forecast shows GDP cooling to 0.2%, having risen 0.5% in Q1'16.\nSo, if we use Stratton Street's Net Foreign Asset Model (NFA) as an indication of how each of the 24 teams will perform, we come up with a surprising winner (in footballing terms anyway). If we select winners according to the higher 'NFA' star ratings, Switzerland would be the resounding winner having played Belgium in the final; the bookies put the odds of Switzerland winning at 50\/1. These teams would have met Germany (bookie's 2nd favourite at 4\/1) and Sweden (100\/1) in the semis, having beaten the likes of Russia, Austria, Italy (which became a republic 70 years ago today) and France (3\/1) in the quarter-finals. The bottom of the table would be taken up by Iceland (last place), Portugal and new entry, Albania (which is the bookies least favourite at 300\/1!).\nHolland unfortunately did not make the qualifiers; had they done so they would have played in the finals, but again lost to Switzerland. England would make it past the group stage, but just miss the quarter-finals. Although there is some coincidental correlation, we will not be relying on the Euro'16\/NFA analysis to predict the final results. We wish all the teams and fans the best of luck and safe travels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcnyi b/data_all_eng_slimpj/shuffled/split2/finalzzzbcnyi
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index 0000000000000000000000000000000000000000..2fdf65e47ff3397305eb7b8245e583107c6615bb
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@@ -0,0 +1,5 @@
+{"text":"I wanted to take this opportunity to write a brief word to those of you whose daughters are discerning a call to religious life. We realize that this process of prayerful discernment which your daughters have undertaken is one in which you are intimately involved, though perhaps quietly so. I want to welcome you to this beautiful time of discernment, one in which your daughter is seeking to understand the Lord's desires for her, and to open herself completely to Him.\nWe understand so well that this gift of self which your daughter is considering would also involve a sacrifice for you. Most parents, in reflecting on the loveliness of their daughter, their gifts and their talents, have aspirations for her\u2014above all because they love her and want her life to be happy and blessed in every way. For this reason, learning about the call to religious life which your daughter may have received from God is also a call to you: it invites you to be open to sacrificing your own dreams for her, and to entrust her discernment to the Lord. As you and your daughter walk together during this time, it is comforting to remember that when we give ourselves over to God's will, we find peace, joy and the fulfillment of his dreams for us. In his will lies our happiness and contentment; in his will is our peace. Entrusting your daughter to the Lord's care during this time of discernment will be the source of your own peace. As his grace is at work in your daughter, He is at work in you as well. He understands your hearts.\nWe will keep you in our prayers during this important time. I ask God to bless you in all the ways He knows you need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An hour after the National Academy of Engineering announced John Koon was one of its newest members, he was doing what he does every Thursday: teaching his Senior Design course.\nNever mind that election to the NAE is one of the most prestigious honors \u2014 perhaps THE most prestigious \u2014 an engineer can receive, and one that's been reserved for only about 2,500 people worldwide.\nKoon is one of two Georgia Tech professors among this year's 104 new members; the other is Regents Professor Krishan Ahuja in aerospace engineering. The National Academy said it was recognizing Koon \"for contributions to the design of systems to treat chemically complex industrial wastewaters\" in its Feb. 7 announcement.\nKoon has built a career as one of the foremost experts in treating industrial wastewaters and helped develop some of the fundamental practices environmental engineers use today. Now he uses his 40 years of experience to shape a new generation of engineers as a professor of the practice in the School of Civil and Environmental Engineering.\n\"I woke up one day and realized that the purpose of my 40 years or so of practice was preparation for teaching at Georgia Tech. I frequently draw upon instances where I was working on engineering projects and bring that to the classroom. And I thought, all of that experience was just preparation for what I'm doing now.\"\nHis is a career that may have turned out very differently, had he finished his Ph.D. even a few years earlier or later, Koon noted. But the country was waking up to the impact we were having on the environment, opening the door for him to help take new theories and approaches from the lab and classroom and put them into practice.\nFresh out of graduate school in the early 1970s, Koon joined \"what we now call a startup\" with other young engineers and \"one guy with the gray hair\" \u2014 probably the best-known industrial wastewater expert in the world at the time, Koon said.\n\"At a time when industry was needing help figuring out, how do we treat these wastes, we were coming forward with new ideas and a new approach to environmental engineering at this company,\" he said.\nKoon said the firm, and a few others, were taking the principles they'd just learned in grad school and applying them to the real world. The company helped lead a fundamental shift from empirical designs for treatment systems to an approach based on chemistry, chemical reaction kinetics, and reactor theory.\n\"That has forevermore changed environmental engineering. By looking at these processes we use in a fundamental way, it's allowed us to design better \u2014 and also to do research on them to make them better, to get more out of the processes.\"\nOver the course of his career, Koon worked at something like 500 industrial sites. He implemented new systems that allowed pharmaceutical companies to treat and reuse their wastewater, which was laden with chemicals. He helped develop biological processes to treat industrial toxic wastes. He also worked with pesticide makers to manage chemicals that turned out to be very persistent in the environment.\nAbout a decade ago, Koon started to consider how he wanted to spend the rest of his career.\n\"I thought that education and, specifically, science where being undervalued by the country, and I thought, 'Maybe I should help young people appreciate science and get excited about it,'\" he said.\nKoon still consults and serves as an expert witness. A few years ago, the Water Environment Federation gave him a lifetime achievement award and the University of California, Berkeley, inducted him into its civil and environmental engineering Academy of Distinguished Alumni.\n\"I've thought about what role luck played in all of this,\" Koon said. \"My undergraduate and master's degree adviser went to Cal Berkeley and encouraged me to go there to graduate school. When I was there, I took the first offering of a course in applying chemical reactor theory to environmental engineering processes. That gave me a big piece of the technical background I needed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With its central location, Monte Claro offers easy access to many of La Manga Club's world-class facilities. As a result, the luxury townhouse-style villas in this part of the resort are extremely popular, especially with families. One of the more recent developments, Monte Claro townhouses are designed in the Spanish style and built to the highest standards. Now available for 2018 in two, three and four bedroom options, each self-catering property is well-appointed and comes fully equipped with all the latest conveniences, including air conditioning.\nThis beautiful townhouse features two en-suite bedrooms, wonderful gardens, free Wi-Fi and access to the 21 metre community pool.\nThis is a gorgeous townhouse, featuring four bedrooms, one fully en suite, a large garage, stunning gardens, and access to the outside, free-form, community pool.\nOne of La Manga Club Resorts stunning townhouses, featuring four bedrooms, including one fully en-suite, a garage, beautiful gardens free Wi-Fi and easy access to the free-form, community pool.\nThis stunning townhouse, contains three bedrooms, all of which are en-suite, beautiful gardens, and access to the superb community pool.\nMonte Claro 29 is one of the stunning townhouses on La Manga Club Resort, featuring three bedrooms, all of which are en-suite with free Wi-Fi.\nMore La Manga Club accommodation* available at from \u00a333 per night.\nThe light and airy townhouses come with their own generously-sized sun terrace, with sun loungers and patio furniture, as well as a built-in barbecue for al fresco dining. All the villas in Monte Claro are built around a large communal swimming pool, with a shallow end for children. The pool, which is for the exclusive use of guests, is heated to a comfortable temperature throughout, and is set in beautifully landscaped gardens.\nLocated in the heart of the resort, Monte Claro is a short distance from the club's state-of-the-art tennis courts and academy, as well as many other sporting and leisure facilities, such as three championship golf courses, football academy, jogging trails, dance academy, and more. For this reason Monte Claro is especially popular with sporty holidaymakers who want to take full advantage of everything La Manga Club has to offer. When it's time to enjoy some less strenuous activity, there are a great range of bars and restaurants to choose from; alternatively, you can stock up on local foods and delicacies from the nearby shops and supermarkets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can choose your usual size for the Chapichapeau ankle strap pump unless you have slightly wider feet or are between two sizes; in which case go for a half-size larger.\nThe women's high-heeled Chapichapeau shoe is so beautiful and flamboyant that it truly plays with your emotions. A Parisian miracle that only a designer of Patricia Blanchet's calibre could create. The little glittery chihuahua with its tongue hanging out will be your spiritual guide throughout your life.\nThe Chapichapeau round-toe pump is like a gift from heaven, a sort of present wrapped with a strap to delight its owners. It boasts a delicate ankle strap and a wide, stable 8 cm heel. The Chapichapeau pump comes in in fuchsia suede as well as in gold metallic suede.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My children and myself recently joined a guild of fairies yep faeries and other mythical creatures The Court of the Elven Fae . So if you see us at a Ren-Fair say Hi.\nAll photos in this post are taken by Rudy van Bree. Here you see us in a few of our costumes and the other guild members in theirs. A very nice group of people.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..6f3926c54fc3ea9fe0fa5d329f49d6cfef87e5d2
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Rabbi Rami chats with Roshi Joan Halifax about her spiritual roots and connecting to each other's humanity.\nI woke up at 4:00 a.m., meditated for 11 hours, and ate my last meal of the day at 11:00 a.m.\u2014for 10 days. Where to begin?\nJosh Korda talks to Rabbi Rami about his latest book Unsubscribe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Judges, Guns, And Facebook: Social Media Lessons For Businesses From The Tactical Firearms Case\"\nArticle summary: Over the past few weeks, the internet has been abuzz over the battle raging between Jeremy Alcede, the former majority owner of CTIL, LLC, dba Tactical Firearms (the debtor), and the Bankruptcy Court for the Southern District of Texas. Mr. Alcede has doggedly clung to his contention that the Tactical Firearms Facebook Page and the Tactical Firearms Twitter Account (the \"Social Media Accounts\") he created are his personal property and not assets of the reorganized debtor. Beyond the analysis of social media accounts in the context of a bankruptcy estate, this case offers some important lessons for businesses managing their own social media accounts. Maintaining ownership and practical control of business social media accounts requires an understanding of how social media platforms work and some advance planning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In its earlier years, the company operated specifically in the European market; however, over the previous decade, GetResponse has actually made a focused effort to expand its reach to the American market.\nGetResponse is a popular e-mail marketing service that just performs well. Servicing companies in more than 180 nations with over 1 billion subscribers every month, GetResponse projects itself as the world's most convenient email marketing system. The platform makes it simple and problem-free to create professional-looking e-mails and landing pages with its editor.\nWhat makes GetResponse such a great business software solution? To begin, you don't need a technology background to use its marketing and e-mail automation tools to enhance your business. The vendor supplies an useful 30-day totally free trial and flexible pricing bundles with a lot of additional features. You do not require a credit card to sign up for the free trial which permits up to 1,000 contacts.\nThe GetReponse site hosts an in-depth Help Center where you can easily look at responses to typical inquiries. The vendor likewise offers 24\/7 support through live chat and email. The user experience is remarkable and the third-party integrations permit you to easily connect the application with your existing company software application platforms. We provide the thumbs approximately GetResponse and invite you to check out its leading rate e-mail marketing functionalities.\nMarketing Automation \u2013 GetResponse's marketing automation feature lets users construct scalable workflows based upon customer journeys. Action-based autoresponders allow for the production of messages that are activated by appropriate recipient actions \u2013 with customized one-to-one responses. Additionally, GetResponse offers users with advanced segmentation tools that allow them to divide their contacts into subgroups and tailor e-mails accordingly.\nLanding Page Builder \u2013 GetResponse has an intuitive drag-and-drop landing page builder that allows users to produce one hundred percent responsive landing pages and web forms. Business online marketers can construct websites for sales, webinars, thank yous, opt-ins, about-me and downloads in simply a couple of minutes. Moreover, GetResponse lets users test, analyze and optimize their pages to boost conversion rates.\nComprehensive Reporting \u2013 GetReponse has robust reporting abilities. A couple of easy reports appear straight in the control panel, offering a brief overview of project success via raw numbers and pie chart. The E-mail Analytics section supplies more detail, with line and bar chart for clicks, unsubscribes, opens, complaints and bounces. In addition, for every single report, users can see which customers within their e-mail list performed any given action.\nWebinar Integration \u2013 The GetResponse webinar platform flawlessly integrates with GetResponse email marketing, permitting users to host product statements, demonstrations and training sessions. Features consist of presentation sharing, chat moderation, polls, desktop sharing, guest management and VoIP capabilities. Moreover, GetResponse lets users decide whether their webinar will be password-protected or available to everybody.\nOn the whole, GetResponse is pretty straightforward to utilize. It's certainly simple enough to do all the essentials: import contacts, develop campaigns, set autoresponders and check statistics and the user interface is quite clean and intuitive.\nIn terms of how it compares to its competitors in this regard, I would argue that Campaign Monitor is a little bit more user friendly although not as complete, and Mailchimp has actually a slicker user interface however one that makes finding particular functionality a bit challenging sometimes).\nWhilst its drag & drop technique does in theory offer an extremely versatile way to develop blocks of material and move them around an e-newsletter, in practice it is a bit clunky to use and can cause accidental deletion of material, or positioning of it in the incorrect part of the e-newsletter.\nGetResponse offers a 30-day free trial for a list of as much as 250 subscribers, no credit card needed. Keep in mind, nevertheless, that for the trial (and only the trial), if you add, erase, and after that re-add a contact, it counts as two contacts. Once the trial has ended, there are various pricing plans readily available. The prices format is rather complex, with advanced plans appearing as your business's list size grows. Pre-paying for one year's worth of service will save you 18 percent. If you believe you'll be using GetResponse for the foreseeable future, you can save 30 percent by pre-paying two years.\nPro removes the restrictions on landing pages and enables you to make as many of them as you need. You'll also have access to webinars (up to 100 participants). A Pro account can accommodate up to three users.\nEnterprise brings an entire host of new features for larger companies looking for highly personalized features. You'll require to arrange a demonstration before registering.\nWebinars \u2013this functionality is not offered at all on the 'Email' plan and the number of webinar participants is maxed for the 'Pro' and 'Max' plans at 100, 500 respectively (it is uncertain what the limit is on the 'Enterprise' plan).\nWe realize that when you decide to purchase Marketing Software applications it is necessary not only to see how specialists assess it in their evaluations, but also to learn if the real individuals and companies that buy it are in fact satisfied with the item. That's why we've found a behavior-based Customer Satisfaction Algorithm that collects customer reviews, comments and GetResponse evaluates across a vast array of social networks sites. The information is then presented in a simple to digest kind showing how many people had positive and unfavorable experience with GetResponse. With that information at hand you ought to be equipped to make an informed buying decision that you won't be sorry for.\nGetResponse has an award-winning customer service team, winning gold, silver and bronze Stevie Awards in 2013 and 2014. The company provides email support in seven languages and is the very first e-mail service provider (ESP) to supply 24\/7 live chat, including weekends.\nConsumers can reference the Help Center and Learning Center, which both feature resources to help troubleshoot any questions or concerns. These knowledge bases consist of Frequently Asked Questions, video tutorials, webinars, and downloadable files like handbooks, reports and whitepapers.\nIt's also among the most intriguing products of its kind \u2013 in that it offers e-mail marketing, automation, landing pages, some CRM functionality and webinars all under one roofing system.\nIt is difficult to consider any contending product that offers this 'all round' proposition, and it is what continues to encourage us to use it for our businesses e-mail marketing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please like and share Vatika City Homes to your friends.\nVatika City Home \u2013 Lavish Lifestyle Homes Offered At Very Low Prices, Right In The Heart Of Gurgaon.Super refined properties await you in Vatika City Home \u2013 the abode of the super rich and the ultimate destination for people who want to live in the lap of luxury. Classy, stylish and above all fabulous in architectural style and interior design \u2013 these homes offer everything that is needed for a comfortable residential life. No other home in the market provides as much premium comforts in the surroundings and beauty in its interiors as these. Invest and enjoy a lovely future with your family here.\nThe apartments offered here range from 1BHK to 3BHK. You can select a unit depending on your preference for space and price. When compared to similar properties in the market, you will find these homes giving so much of space and utility. Compare the lifestyle here with that of a five-star hotel. Everything that is offered here is grand and super refined. You can enjoy so much without incurring heavy expenses in the purchase as the pries of these properties are unbelievably low.The spacious rooms, posh baths, large living space, beautiful kitchen add comfort and convenience to your everyday life. There are facilities available here for you to use the latest gadgetry and electronics to make your life exciting and full of ease. The interior styling is modern so compliment it with state-of-the-art furnishings and you will find the living spaces to be transformed into a luxurious haven of comfort.\nSituated in Sector 83, these buildings are surrounding by natural greenery which bring solitude and tranquility into your life. You can enjoy walking through the garden, else jogging across the tracks to keep fit and healthy. This complex was designed to not just give residents a modern life, but one that is healthy and which keeps them energetic. The clubhouse facilities usher in various kinds of recreational activities through which you will always be fit and energetic.\nCome and enjoy a fabulous life by purchasing a Vatika City Home apartment. You will find these units to cater to your every need. Everything required for running your daily household activities is available inside the building complex. Schools, hospitals, educational institutions, offices are situated in close proximity enabling you to gain quick access to such locations. The value of these properties is sure to rise rapidly in the near future because they are in a location that is currently experiencing rapid development. In the future their price will be very high giving good return on investment. Buy now and become the proud owner of a fabulous property that makes you the envy of many.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sometimes, life drops explosive news on you. The news is by no means unpleasant, but it causes your face (and your brains) to go through a series of psychological expressions before you can actually react. I'm thankful for the fact that this piece of news came to me at a time when no one could really see those expressions, owing to lack of sufficient illumination.\nI guess I'm also thankful for the fact that there was no mirror around.\nI met her in 2004. The first time I saw her, her leg was bandaged. She'd been in an accident and was walking around with a limp. She was two years my junior, but taller and very lanky, and I remember being a little shocked at the (un)ambitious length of her shorts.\nShe was obviously in pain, but very cheerful. We hit it off pretty quickly. We spent the afternoon chatting, the conversation surprisingly easy and free-flowing for a first. And then, in a few hours, it was time for me to leave. We promised to keep in touch.\nWe met again a few months later \u2013 she was in town, visiting her father for a couple of weeks. This time, we bonded over shopping and lunch, the way teenagers do. We spoke about our closest friends, our families and our interests, and got to know each other a little more. After she left, we started staying in touch over the phone.\nThere was something about her that appealed to me. In her apparent innocence, I could see overtones of maturity. Perhaps it was because I knew that she had had a very difficult childhood; she had lost her mother at the age of three, and ever since, her father had sought refuge in chronic alcoholism. He had even remarried, for a brief period; her stepmother seemed straight out of a Grimm's fairy tale.\nI was actually amazed at her unrestrained appetite for life.\nOn one of her subsequent trips, just after the 2005 deluge in Mumbai, we discovered a litter of puppies in the middle of the road. Their mother was nowhere to be seen, so we brought them home, and took care of them for the next three weeks.\nI was nineteen, she was nearing eighteen. Both of us were experiencing our first serious relationships. We would giggle and gush about our respective boyfriends from time to time over the phone. She met mine when he had come down to Mumbai, and I met hers.\nTechnically, she'd been my partner in crime, more than once.\nOne night, around the middle of 2005, she called me up, and started to cry. Her father's alcoholism had taken a turn for the worse \u2013 he used to drink every night, and even had to be hospitalized a couple of times. That night, the way she put it, things had almost gone out of control.\nMy parents heard about the situation from me. Knowing that her father wouldn't live very long, they discussed asking him if we could adopt her. I had already started to love her like a little sister, and was more than willing to go ahead with the idea.\nAround the end of 2006, her attitude began to change. She started maintaining a calculated distance from me. At first, I thought she was genuinely busy, but then, it wasn't really like her to not answer or return my calls or messages repeatedly. One day, a few months later, she asked me to stop \"interfering in her life\". To \"give her space\".\nAt the time, I was already going through a bad phase, professionally, academically, even personally. Her words were extremely painful. She was a very close friend, and I needed her to be around, but her sudden, uncalled-for acerbity made it easier for me to stay away. I resolved never to call her again.\nWe lost touch. I lived up to my resolution; she never attempted to contact me either.\nOne day, a common friend informed me that her father had passed away. I did not know what I was supposed to do. Ideally, I knew I should call her, but her turning away from me, and our friendship, had hurt me so much, that I did not have the courage to do so. What if she refused to speak to me again, or said something I wouldn't want to hear, like the last time? That would hurt me even more. And I did not know if I was ready to get hurt another time.\nAfter a lot of thought, I called her. She sounded strangely normal \u2013 not like a girl who had just lost her second parent, or who was talking to a former sister-figure whom she had told off so ruthlessly.\nI kept the conversation brief and formal. A few days later, I called again to check on her. It felt strange, as though I was being mechanical in my actions, as if it wasn't concern but obligation that was making me call her, but that was the least I could do.\nThat was the only thing I could do.\nPart of me was seething from within. It isn't that easy, I wanted to yell at her. You were heartless and insensitive and you do not deserve someone like me. How can you even think that a simple three-liner will help me forget everything and attempt to \"re-establish our friendship\"? Did I not deserve more? A more elaborate apology? I decided I'd not call her.\nTwo days later, I found myself dialing her number.\nShe told me she had distanced herself from a lot of people over the past two years. And that after her father's death, she wanted to re-build as many ties as possible.\nI empathized. But I could not spring back to my original self as easily as I had done earlier. I conveyed that to her, and she understood. Over the next half hour, we discussed briefly the highlights of the past two years, and when our conversation ended, her words, her tone, suggested that she wanted to attempt to make things up to me.\nShe never called me again.\nHave you ever experienced a situation where you don't know something, but really want to know about it, but then again, you don't want to, because more knowledge will only cause you pain?\nI later learnt, purely by accident, that the boy was Canadian. He had worked with her father on a project; that was when they were first introduced to each other. They were married earlier this year. That is all I know.\nI don't think I want to know more.\nI found her profile on Facebook recently. There was a picture of the couple, probably taken on their honeymoon.\nShe looked happy. As though the ghosts of those tough years of the past had finally been laid to rest.\nI know I don't need to know more.\nI spent the first sixteen years of my life in a house with a huge verandah overlooking a Gulmohar tree. Twice every year, once during the summers and once during the winters, the tree would burst into cheerful red flames. And the stone-grey ground below would be dotted with red petals, or occasionally even the whole flower.\nI subsequently moved in to a new house. Unlike my old colony, which could only boast of limited greenery, my new apartment had a huge lawn with an assortment of trees. For some reason, though, there wasn't a Gulmohar in sight.\nI did not realize how much I missed the tree till I found myself staring at Gulmohars every time I passed them. It took me a while to understand that subconsciously, the Gulmohar was actually a link back to my childhood.\nEven today, when I pass by a Gulmohar tree, I experience an emotion that is fairly overwhelming. Sometimes, I pick up a stray flower, bring it home, and keep it in a water-bowl. Otherwise, I simply look at the tree in all its red glory and smile, as if between us, we share a little secret.\nNot all memories are in black and white and sepia. Some are a cheerful red too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdwtq b/data_all_eng_slimpj/shuffled/split2/finalzzzbdwtq
new file mode 100644
index 0000000000000000000000000000000000000000..e291b2cf21a90e43b62871417d5ed0e4d732cee9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"I am going to give you a play-by-play example of Focusing from my own life.\nTriggering experience: I was at an online meeting with a group of friends. We started meeting over 25 years ago. We all used to live in the same small town, and had formed a support\/listening group with each other while we all had young children. We each would take 15 minutes to share whatever we were going through without being interrupted. This \"15 minutes for me\" proved invaluable to us all as we navigated the seas of motherhood and selfhood.\nSo here we were, reunited after several years of not meeting. Five of us were together in one room in the town where we used to live, and myself and one other were online, via Skype. Skype was not working well, and I was getting increasingly annoyed. I had suggested before the meeting that we could meet by Zoom, which works much better than Skype when there are more than 2 computers involved. But no-one had encouraged me to pursue this idea, and I had not insisted. My annoyance with the inadequate technology was affecting my sharing when my turn came up.\nLater I decided to Focus on this, because I didn't understand why I had been so upset. In one way, it seemed like such an unnecessary attitude on my part. I hadn't devoted a lot of energy to thinking about the technology before the meeting, and I had not prepared an easy way to shift over to the new technology if problems arose. So why couldn't I just accept that, instead of getting mad about it? Also, I noticed I was blaming people in the group for not listening to me about the technology. I could have judged myself for overreacting and left it at that, but luckily, I had a Focusing partnership coming up, so I decided to look at what was going on that had made me so upset.\nI noticed something in me that didn't want to impose, didn't want to insist, even though I clearly had the knowledge and experience to make the meeting better. The \"not wanting to impose\" felt like a buzzy, chaotic sensation in the lower part of my head. When I noticed and welcomed the sensation, I could feel that there was much more going on there. The feeling lowered into my throat, as if there were a war going on between two opposing tribes. \"Yes, there does seem to be a lot going on inside about all this\", I realized.\nI stayed with it in a Focusing way, and soon I could sense that the war was between my mother and my grandmother. My grandmother projected an aura of self-confidence. She was sure of herself. This led her to a lot of successful creative endeavors which are unbelievably inspiring. But her self confidence did not leave room for other people, especially for my mother, who was always doing her best to follow in the footsteps of her mother, all the while feeling an inner frustration and resentment at not being seen for her own merits, her own selfhood.\nThe felt sense of a battle had lowered into my body, where I could feel it as a pressure in my heart. As I stayed with it, I could feel that my loyalty was to my mother, and that implied a rejection of my grandmother's tendency to steamroll those around her. In a restaurant, my grandmother would imperiously send food back if it were not to her liking. I always found that embarrassing. It seemed to symbolize everything I rejected about my grandmother, everything I didn't want to be.\nI could now see that insisting on having the right technology for my women's group meeting reminded me too much of my grandmother. Even though I was very familiar with a better system, it would have meant imposing my will on the group if I had insisted on using Zoom. It was less complicated internally for me to just go along, without sharing what I know. That way I could maintain my inner loyalty to my mother, and not risk becoming a steamroller of others, like my grandmother.\nWow, interesting. So what is this showing me about my larger life now?\nIt came to me that I want to emerge from this period of my life with more confidence to share what I know. To find a balance between sharing what I know and feeling that I am imposing on others.\nThat evening I was going to go to a dinner for cancer survivors. I could feel myself facing the dilemma all over again\u2013how can I share what I know in this new group of people who don't know me? I realized that I could go to the dinner \"armed\" or \"prepared\", by making copies of articles about the research done with Focusing and breast cancer by Doralee Grindler Katonah and Joan Klagsbrun. The research shows that even doing simple Focusing exercises like Clearing a Space, can prolong the lives of cancer survivors. This gave me a jumping off point for sharing what I know. At the end of the dinner, there was a brief time to introduce oneself and say what we were involved in. I mentioned the articles, and several people asked me for information afterwards.\nSo, gentle reader, if you have gotten this far, I have been describing an experience that can happen in a Focusing partnership.\nI had an uncomfortable, triggering experience. I Focused on it with my partner and found that there was a lot more going on there than I had been aware of.\nInstance 1: I got angry at my women's group for \"not listening to me\" about meeting on Zoom instead of Skype, even though I had not really explained the advantages or set things up to implement what I know.\nInstance 2: I went to the cancer survivor's dinner \"armed\" with articles about how Focusing has been proven to increase longevity in cancer survivors. It led to several inquiries about Focusing.\nhttps:\/\/possibility-space.com\/wp-content\/uploads\/2017\/10\/Greek-battle.jpg-copy.png 336 479 possibilityspace http:\/\/possibility-space.com\/wp-content\/uploads\/2016\/09\/possibility-space-logo.png possibilityspace2017-10-12 09:36:162018-03-08 14:19:46Focusing session on the topic of \"sharing what I know\"\nThanks, ME. Now that I think of it you have provided a great example for me, with your Folio article!\nI am new to focusing. Your personal story was informative and inspirational.\nI also look forward to your next steps.\nThanks, Emily! it's good to know that you are out there appreciating my candor!\nThank you so much , dear Beatrice; it is so inspiring and accessible.\nInspiring and accessible\u2013that's what I hope for. Merci, Marine!\nThanks, Ulrike, I hope that you will subscribe\u2013that's the best way to get new installments. I love your description of what I am doing!\nBea, so glad you are doing this. I will journey along with you with gratitude and appreciation for your sharing.\nGlad to have you aboard, Anna!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2017 Highland Ridge RV OPEN RANGE 2910RL ULTRA LITE w\/ Luxury Finishings & Dbl Slide!!\n*** SPACIOUS & LUXURIOUS OPEN RANGE *** HIGH-END FINISHINGS *** LARGE DOUBLE SLIDE!! *** Highly Sought-After Brand known for Build Quality and Prestige......Gorgeous Kitchen with Stainless Steel Appliances......Stunning Kitchen w\/ ISLAND and High End Cabinetry......Double Sink......3-Burner Gas Stove......In-Wall FIREPLACE......His & Her's Style LAY-Z-BOY Leather Recliners......Large Leather Sofa \/ Pull-Out Couch......Leather Seating in Dinette......Easily HALF-TON TOWABLE with a dry weight of 6575 pounds......Oversized Shower......TWO POWER AWNINGS......Multimedia System w\/ HI-FI STEREO & FLAT-SCREEN TV......OUTDOOR SPEAKERS......Comfortably SLEEP SIX Adults......Power Jack......Furnace and AIR CONDITIONING.....Queen Size Master Bedroom......and Tall ARCHED CEILINGS w\/ Sky Lights!\nIf you're after a well-built quality camper that holds its value, add this to your shopping list. Well worth the investment! Now just $34,800 with Financing and Extended Warranty available.\n2014 JAYCO Eagle 30.5 BHLT Private Loft, Soaring Ceilings, Tons of Light!!\n*** SALE PENDING *** LUXURY DOUBLE SLIDE *** NEW MSRP $60,000+ *** PRIVATE MASTER BR W\/ ENSUITE *** Camp out in total luxury......JENSEN AUDIO Entertainment Unit with 32-INCH TV......USB \/ AUX \/ CD \/ DVD Audio-Video Inputs......SOARING CEILINGS......Huge DOUBLE-BUNK ROOM w\/ Separate Entrance and PRIVATE LOFT......Spacious SLIDE-OUT MASTER Bedroom w\/ Queen Size Bed & Large Sliding Door Closet......ENSUITE BATHROOM......Several SKY LIGHTS Throughout......TONS OF STORAGE in extra-tall cabinetry......Large Leather Ottoman Converts to Pull-Out Bed......POWER AWNING......Touchpad KEYLESS ENTRY......Full Service Kitchen w\/ Fridge, Stove & Microwave......SLEEP TEN PEOPLE Comfortably!\nThis is one of the nicest floor plans Jayco has to offer, tall and spacious luxury. Non-smoker, like NEW condition. Only $31,600 for this BEAUTY with Financing & Extended Warranty available - payments as low as 200 b\/w or less!\nComplete comprehensive warranty is available for this camper. Please ask for warranty option details.\n2012 JAYCO Jay Flight 185RB Baja Edition Sleep 4 & only 2,700 lbs! Queen Bed, A\/C & More!!\n*** ULTRA LIGHTWEIGHT! *** PERFECT HUNTING \/ FISHING TRAILER! *** RAISED + MEATY TIRES! *** Dry Weight ONLY 2700 LBS!......Everything that you LOVE about Camping - just in a smaller package......This could potentially be stored in your OWN GARAGE!!!......This Camper can go anywhere, and it's light enough to be towed by most vehicles......Comfortably SLEEP FOUR PEOPLE......Bathroom with SHOWER & TUB COMBO......Ample Storage Throughout......CENTRAL HEAT......Full Service Kitchen w\/ Sink, Fridge, Stove & Microwave......Dinette Converts to a Bed......AIR CONDITIONING......Master Bedroom w\/ QUEEN SIZE BED and PRIVACY CURTAIN.....Windows & Roof Vents bring LOTS OF NATURAL LIGHT......Extendable AWNING......Perfect Travel Trailer for getting in and out of tight places, one foot narrower than normal. Bush, trails and more are a breeze for this one!\nUltra affordable financing available, LOW PAYMENTS! Now just $12,800 with Financing and Extended Warranty available. Don't miss!!\n** SALE PENDING ** ABSOLUTELY GORGEOUS........VERY HIGH END LUXURY EDITION.....WITH BIG SLIDES.....OUTDOOR KITCHEN.......... AND Exterior BATHROOM entrance !!!! NOTE: (this unit qualifies for a $1.000.00 finance rebate with in store dealer finance!!! .....now only $31,875!!! ) AWESOME FLOOR PLAN!!! this is a DIAMOND LIGHTWEIGHT edition from Outback! FULL AMENITIES!! SLEEPS UP TO 10 PEOPLE COMFORTABLY!........iN-Command Touch Screen Control System.......Entertainment Unit with built in Electric Fireplace......HUGE KITCHEN & LIVING ROOM SLIDE.....GOOD SIZE BATHROOM W\/ SHOWER AND TUB COMBO!!!....SEPARATE FRONT BEDROOM W\/ QUEEN SIZE BED......FULLY RETRACTABLE POWER AWNING.....Outdoor Shower.......HUGE BUNK BED ROOM with day time sofa seating!!! ......AMPLE Storage Space Throughout Bedrooms, Living Room, Kitchen, Bunk House & Outside......FAMILY FRIENDLY.......40\" Flat Screen TV......Second TV Hook Up In the Master Bedroom......Full Service Kitchen w\/ Fridge, Stove & Microwave......Outdoor Speaker System.......and Hydraulic Jacks!....Only $32,785 for this BEAUTY!\nNOTE: THIS IS HEAVY HALF TON TOWABLE!!! EXCEPTIONAL UNIT TO HAVE THE SIZE AND AMENITIES AND TO BE THIS LIGHT!! NON SMOKER, and immaculate condition inside and out. This DEFINITELY NEEDS TO be on your list of campers to check out!!\nWITH financing on site!! Please call to view or drop by at your leisure.\n*** 34 FT ULTRA LITE!!!..... THREE POWER SLIDES!!!.......BUNK HOUSE!!! **** and FULL SERVICE OUTDOOR KITCHEN! *** If you are after a HIGH END CAMPER........VERY LARGE, ROOMY AND LIGHT.........then this should be at the top of your list!! Streamlined to slice through the wind.....exterior LED LIGHTS, and certainly one of the sharpest \"pull behinds\" on the road!!! THIS OUTDOOR KITCHEN Comes w\/ FRIDGE, BBQ, STOVE, SINK, CUPBOARDS AND STORAGE!!!......HUGE POWER AWNING....Private Front Master Bedroom w\/ Queen Size Bed!......TONS OF STORAGE Inside & Out!......DOUBLE Bunk Beds......SLEEPS up to 10 People!......POWER FRONT Tongue Jack......HUGE Over Size Kitchen w\/ Island......ROOM To Entertain!......KITCHEN & LIVING ROOM FULL SIZE SLIDE......OUTDOOR Shower......BIG Flat Screen TV In The Living Room......MUTLIPLTE SKY LIGHTS......Full Service Bathroom w\/ Shower & Private Entrance To Master Bedroom!\nNOTE: THIS IS A NON-SMOKER CAMPER!! Extremely well kept and clean, no wear and tear and no odors, no stains. Selling for a fraction of new cost!! ONLY $32,600!!! WITH financing on site!! Please call to view or drop by at your leisure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We had to cancel my talk for tomorrow night, due to problems with the venue. LispNYC will still meet at the Sunburnt Cow, 137 Avenue C, for drinks and discussion. My presentation has been postponed to the June meeting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Innerscope Hearing Technologies (OTC: INND) CEO Interview with Matthew Moore, discussing consolidation opportunity in $4billion US hearing aid market, Walmart online distribution agreement, and scaling up retail expansion in 2019. Innerscope's key advantages include deep, multi-generational, domain expertise in the hearing aid sector, cost savings for consumers, and tele-audiology to serve online and remote customers.\nInnerScope's \"Grand Opening\" of its Roseville, California audiological hearing aid retail clinic is the first of 22 new audiological hearing aid retail clinics planned within Northern California in the next 6 to 12 months as part of InnerScope's aggressive new retail location expansion plan.\nNovember 15, 2018 admin Newsmakers Comments Off on InnerScope's \"Grand Opening\" of its Roseville, California audiological hearing aid retail clinic is the first of 22 new audiological hearing aid retail clinics planned within Northern California in the next 6 to 12 months as part of InnerScope's aggressive new retail location expansion plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sama ka chawal or Samak chawal is actually a millet called Barnyard millet that is mostly consumed during Navratri fasting. Barnyard millet cooks really fast when soaked and can be made into sama ka Jeera rice, Sama ka Upma, Sama ki Idli and Sama ka Dhokla easily. Making sama ka Dosa is also easy once you know how to use this millet. We love sama ki kheer too.\nSama ki tehri or tahiri cooks quickly once you have soaked sama in the fridge. Yes it is advisable to soak sama overnight either in water or thin buttermilk and refrigerate so it can be used to cook meals quickly as the soaked sama ke chawal lasts about 3-4 days. It is particularly convenient in Navratras if you are fasting.\nHeat the ghee in a wide stockpot (or pan) and tip in cumin seeds. Let the cumin splutter a bit.\nAdd the potatoes and stir and cook for 2 minutes. Now add the pureed tomatoes and all the chopped vegetables, salt and pepper. It will be better to add the other vegetables one by one according to the time they take to cook. I added carrots first and then zucchini after the potatoes and carrots were done. Toss and cook them all together.\nNow add the soaked sama ke chawal, mix everything well and cook covered on low flame. No need to add any additional water as there is enough water in the ingredients to get cooked. It takes about 5 minutes for this quantity to get cooked after adding the soaked sama ke chawal.\nFluff up once cooked, sprinkle with chopped dhaniya patta and chopped green chilies and ginger if using. Serve with cucumber raita.\nGreen chilies and fine julienne of ginger taste really good in this tahiri but you can avoid if you like it a little plain. You can always add some roasted nuts or paneer cubes on top to add some proteins. I normally like it lighter but it also depends on how hungry you are for the meal or whether you are serving this sama ke chawal ki tahiri for lunch or dinner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbelhg b/data_all_eng_slimpj/shuffled/split2/finalzzzbelhg
new file mode 100644
index 0000000000000000000000000000000000000000..554411dac135fdaf250b3e568cd00e4a70966a7e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbelhg
@@ -0,0 +1,5 @@
+{"text":"I first tried creating my own cryptocurrency using the Ethereum Wallet. I hit an road block because I need to buy ETH coins to create my token (cryptocurrency) to cover the transaction fees, gas, and other fees managing my cryptocurrency. I think I need at least 3 ETH currently trading at $311 plus the fees for transactions (gas). I stopped here, wahhhh Crypto Forever !!!\nAnyway, this is the second try and this time around it is successful. I will cover how I accomplished it, step-by-step so other will follow and create their cryptocurrency using Ethereum Decentralized Platform.\nAdd your token (coin) to an exchange. So other investors will be able to participate.\nTrade them among friends, just to see whose the richest nerd.\nand other ideas applicable to you.\nSo what's the ingredients needed to cook this project?\nYou need to have your own Ethereum Wallet using MyEtherWallet. This is where you going to transaction your cryptocurrency, Minereum (MNE) and Ethereum (ETH).\nAs of writing this article. The current token creation service is 1 MNE. Yes, one, uno, isa.\nYou need to buy MNE through Livecoin.net website. I deposited some ETH (not BTC to expensive to buy and nothing to offer except being the pioneer in cryptocurrency) to my Livecoin.net account so I can use it to buy MNE. In Livecoin dashboard, click on Balance, scroll down till you see the Ethereum Deposit\/Withdraw button. Or use search for ETH.\nClick on Deposit, to get your ETH address. You should see something your address.\nIn MyEtherWallet account. Click on Send Ether & Token. Enter the address you get from the previous step and the amount to send. Click Generate Transaction to proceed your transfer.\nGo back to Livecoin and check your deposit. You should be able to see your transfer balance.\nNow let's buy MNE coin. In Livecoin dashboard, click the BUY \/ SELL, select Trading. Use the Search field to find \"MNE\". Select MNE\/ETH.\nYou should be able to see BUY MNE FOR ETH. Enter the amount of MNE you want to buy. When ready, click BUY MNE. You will be able to see the transaction in progress.\nNow click on BALANCE, search MNE to verify your coins.\nClick the Withdraw button so you can transfer this to you MyEtherWallet. Enter the amount of MNE and Enter your Ethereum Wallet Address.\nClick Send a Payment, your MNEs will be in your wallet shortly.\nVerification. Check your email for verification code and enter your PIN-code.\nIf a payment (transfer) is successful. You will see something like this.\nGo check your MyEtherWallet. You should received your MNE coin in your wallet.\nIf you don't see MNE token in your MyEtherWallet. You need to add MNE it, click the Add Custom Token. In Address field enter MNE addree. Now we can Generate our own Token.\nSend 1 MNE to the address 0x41fAfB65B1514Ce1177287A567002f8978d98205. Click Generate Transaction and confirm to send payment of 1 MNE.\nTo verify your transaction, go to https:\/\/tokenservice.minereum.com website. Within seconds the status goes to Done. The version 1.1 made it faster to complete. Your Smart Contract Address is ready for you.\nView your MyEtherWallet, add the Custom Token using the following.\nIt should look like this. Click Save.\nThe Etherscan.io Token Verification: Verified.\n6.1 Take note of the \"From\" address that you used to initiate the transaction, for this example we will use the address 0xe64286a1d33c57dc5173297146edd713211eed85.\nClick on Verify and Publish to proceed.\nIt's done! I now have my own cryptocurrency QUE using Ethereum Decentralized Platform. And you can buy QUE coin at EtherDelta.com exchange.\nWho want some QUE coin? If you're interested to have a QUE coin, please post your Ethereum Wallet and I will send you a free 100 QUE coins. Someday we can go to the moon and only cryptocurrency is allowed to trade \ud83d\ude42 Or send email to EM @QUE.com with your wallet to receive a free 100 QUE coins. Offer valid up to July 10, 2017.\nWhat Ethereum Wallet to use? Visit MyEtherWallet.com to create your own wallet. Please follow the recommended steps to protect your wallet. Please keep in mind, in crypto world if you loss your private key and forget your personal password. You're coins are gone forever.\nPlease bookmark this page as I continue to add new frequently asked questions. You may also use comment below to post your questions.\nIf you created your own cryptocurrency and want to send a test coins. Send it to my Ethereum Wallet Address 0xeB6F0Af12930d30252Df3F159Bbff9504c78D866. You can also use this address to send ETH tip.\nCoinbase.com \u2013 create your bitcoin account and start trading.\nLivecoin.net \u2013 where you can buy MNE token.\nEthereum.org \u2013 Credit to the image I used here.\nQUE.com \u2013 Official Tutorial. Keep checking for update.\nPingback:QUE.com President for Cryptocurrencies e.g. President Trump (TRUMP), President Clinton (HILL), President Sanders (BURN), QUE.com (QUE), etc.\nPingback:Yehey.com We now have our own YEHEY Coins CryptoCurrency.\n100 QUE coins sent to your Ether Wallet. Please check it now.\nPlease follow the procedure above on how to add Custom Token to your Ether Wallet to see it.\nTo all who requested to receive their free 100 QUE Coins. I've sent them already.\nPlease note. Offer valid 'till July 10, 2017.\nI just sent your 100 QUE Coins.\nMay be you can answer my questions.\n1) How to make auto transfer of tokens? I mean i get some eth to my wallet with tokens and send tokens back in desired rate.\n2) If (1) possible how to stop getting eth if all tokens are send out?\nIt will be harder to implement auto transfer of tokens due to it's multiple confirmation approval, this include two-factor random authentication pin codes.\nIn my account, it will be difficult or impossible to implement because I have multiple random codes for confirmation before a transfer to occur.\nI hope that help answer your question.\nAugust 2, 2017. Minereum has increased the fees from 1 MNE to 10 MNE, lucky for those who created their own token when I published this HOWTO Create your own cryptocurrency.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A Melbourne based family business, HartCo Home and Body offer a large range of homewares and kids homewares . HartCo work with many local Australian artists, showcasing their unique and high quality range. Each piece is carefully selected to show off the latest styles and trends in the market. The HartCo store concept is of Scandinavian inspired style consisting of fresh timbers throughout with a touch of copper.\nHartCo is full of limited edition prints, furniture, homewares, textiles, ceramics, jewellery, kids fashion as well their own natural soy candle, reed diffusers and organic body care range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A nanolaminate thin film and a method for forming the same using atomic layer deposition are disclosed. The method includes forming an aluminum oxide layer having a first thickness on at least a portion of a substrate surface by sequentially pulsing a first precursor and a first reactant into an enclosure containing the substrate. A layer of silicon dioxide is formed on at least a portion of the aluminum oxide layer by sequentially pulsing a second precursor and a second reactant into the enclosure to form a nanolaminate thin film.\nThe present invention generally relates to film deposition, and more particularly to a nanolaminate thin film and method for forming the same using atomic layer deposition.\nAtomic layer deposition (ALD), also known as sequential pulsed chemical vapor deposition (SP-CVD), atomic layer epitaxy (ALE) and pulsed nucleation layer (PNL) deposition, has gained acceptance as a technique for depositing thin and continuous layers of metals and Dielectrics with high conformality. In an ALD process, a substrate is alternately dosed with a precursor and one or more reactant gases so that reactions are limited to the surface of a substrate. Thus, gas phase reactions are avoided since the precursor and the reactant gases do not mix in the gas phase. Uniform adsorption of precursors on the wafer surface during the ALD process produces highly conformal layers at both microscopic feature length scales and macroscopic substrate length scales, and achieves a high density of nucleation sites. These attributes result in the deposition of spatially uniform, conformal, dense and continuous thin films.\nThe high quality films achievable by ALD have resulted in increased interest in ALD for the deposition of conformal barriers, high-k dielectrics, gate dielectrics, tunnel dielectrics and etch stop layers for semiconductor devices. ALD films are also thermally stable and very uniform which makes them attractive for optical applications. Another potential application for ALD is the deposition of oxides (e.g., Al2O3) as a gap layer for thin film heads, such as heads for recording densities of 50 Gb\/in2 and beyond which require very thin and conformal gap layers.\nAs recording densities for hard disk drives continue to increase, the thickness of gap layers required for read heads used in the disk drives decreases. For example, the thickness of the gap layer required for a read head in a hard disk drive having a recording density of approximately 100 Gb\/in2 should be significantly below 200 angstroms (\u00c5). The gap layer should also have a high dielectric strength, a low internal stress and a high resistance to resist developer etch. In general, oxide and nitride films, such as Al2O3 and aluminum nitride (AlN), formed by an ALD process have produced high quality gap layers for read head applications. At thicknesses below 200 \u00c5, however, Al2O3 films typically have a lower dielectric strength and are more susceptible to resist developer etch.\nIn addition, conventionally sputtered gap layers may not be suitable for higher recording densities because they are difficult to reliably scale below 300 \u00c5 due to excessive leakage currents. Although ion beam deposited gap layers can be scaled down in thickness to below 300 \u00c5, such layers tend not to be adequately conformal. Further, process integration considerations for thin film heads of 200 \u00c5 or less may constrain the maximum deposition temperature to below 200\u00b0 C.\nIn accordance with the present invention, the disadvantages and problems associated with fabricating a high quality nanolaminate thin film have been substantially reduced or eliminated. In a particular embodiment, a method is disclosed for forming a nanolaminate thin film of aluminum oxide and silicon dioxide on a substrate surface.\nIn accordance with one embodiment of the present invention, a method for forming a nanolaminate thin film using ALD includes forming an aluminum oxide layer having a first thickness on at least a portion of a substrate surface by sequentially pulsing a first precursor and a first reactant into an enclosure containing the substrate. A silicon dioxide layer having a second thickness is formed on at least a portion of the aluminum oxide layer by sequentially pulsing a second precursor and a second reactant into the enclosure to form a nanolaminate thin film.\nIn accordance with another embodiment of the present invention, a method for forming a nanolaminate thin film using ALD includes forming an aluminum oxide layer having a first thickness on at least a portion of a substrate surface by sequentially pulsing trimethylaluminum (TMA) and water into an enclosure containing the substrate. A silicon dioxide layer having a second thickness is formed on at least a portion of the aluminum oxide layer by sequentially pulsing TMA and tris(tert-butoxy)silanol into the enclosure to form a read head gap layer.\nIn accordance with a further embodiment of the present invention, a thin film includes an ALD-formed aluminum oxide layer having a first thickness and an ALD-formed silicon dioxide layer having a second thickness formed on at least a portion of the aluminum oxide layer. The aluminum oxide layer and the silicon dioxide layer cooperate to form a nanolaminate thin film.\nImportant technical advantages of certain embodiments of the present invention include nanolaminate films formed using an ALD process that have high dielectric breakdown strengths. For certain applications, such as gap fill layers in read heads included in hard disk drives, the thickness of the film should be below a minimum value and the film should have certain characteristics. Single layer oxide films, such as aluminum oxide (Al2O3), may have lower breakdown fields at thickness below, for example, approximately 200 \u00c5. A nanolaminate of Al2O3 and silicon dioxide (SiO2) having a thickness at or below approximately 200 \u00c5, however, has a higher breakdown field due to the addition of SiO2 to the film and may be used to form high quality gap layers for read heads of high density hard disks.\nAnother important technical advantage of certain embodiments of the present invention includes nanolaminate films formed using an ALD process that have high resistances to resist developer etch. During fabrication of microelectronic structures, an etch process may be used to remove one or more materials from a surface. In a read head in a hard disk drive, for example, a resist layer may be removed to expose the surface of an underlying oxide material used to form a gap fill layer in the read head. For hard disks having higher recording densities, it may be desirable to have a thin gap layer (e.g., below 200 \u00c5) and, in order to maintain the required thickness of the gap layer, the material should be resistant to resist developer etch. An Al2O3 film formed by an ALD process, however, may not be resistant to the etch process such that the etch process decreases the thickness of the film and degrades other desired properties. In contrast, SiO2 is much more resistant to an etch process and may be used to form a Al2O3\/SiO2 nanolaminate such that almost none of the nanolaminate film is removed by the etch process.\nA further important technical advantage of certain embodiments of the present invention includes nanolaminate films formed using an ALD process that have lower film stress. In many applications, it may be important for a thin film to have low stress. Single layer Al2O3 films formed using an ALD process may exhibit a high tensile stress, which is undesirable for applications such as gap layers of read heads in hard disk drives. SiO2 films formed using the ALD process, however, typically have a low tensile or compressive stress. Therefore, the film stress of an Al2O3\/SiO2 nanolaminate may be controllably reduced by adding SiO2 to decrease the Al2O3 concentration of the film.\nFIG. 10 illustrates a graph of tensile strength of a thin film formed by an ALD process as a function of aluminum oxide concentration according to teachings of the present invention.\nPreferred embodiments of the present invention and their advantages are best understood by reference to FIGS. 1 through 10, where like numbers are used to indicate like and corresponding parts.\nThe conceptual groundwork for the present invention involves an atomic layer deposition (ALD) process to create highly conformal thin films. In an ALD process, a precursor and a reactant, such as a reactant gas are sequentially pulsed onto the surface of a substrate contained in a reaction chamber, without mixing the precursor and reactant in the gas phase. Each of the precursor and the reactant reacts with the surface of the substrate to form an atomic layer in such a way that only one layer of a material forms at a time. The introduction of the precursor and\/or the reactant into the reaction chamber may be referred to as a doping pulse. In between doping pulses, the reaction chamber may be purged by flowing an inert gas over the substrate. One film that may be formed using an ALD process is aluminum oxide (Al2O3). ALD Al2O3 has been used for gap fill layers of a read head included in a hard disk drive, in particular, as a second read gap over topography composed of a read sensor and hard bias\/sensing leads due to the superior deposition conformality and dielectric strength of Al2O3. However, as recording densities continue to increase, the read heads require half read gap thickness below approximately 200 angstroms (\u00c5).\nThe present invention provides a thin film that may be fabricated at lower thicknesses with higher dielectric strength and higher resistance to resist developer etch. In one embodiment, the film may be a nanolaminate of Al2O3 and silicon dioxide (SiO2). Layers of Al2O3\/SiO2 at a thickness of less than approximately 200 \u00c5 may have an increased dielectric strength of up to approximate fourteen (14) MV\/cm at an Al2O3 composition of less than fifty percent (50%). Additionally, Al2O3\/SiO2 nanolaminates have an etch resistance to resist developer that is substantially greater than the etch resistance of a single film of Al2O3. Although other materials, such as tantalum oxide and zirconium oxide, have been used to form nanolaminate films, the Al2O3\/SiO2 nanolaminates disclosed below have shown superior qualities for applications that require high dielectric strength, low film stress and high resistance to resist developer etch.\nFIG. 1 illustrates atomic layer deposition (ALD) system 10 for forming a conformal thin film on a substrate. ALD system 10 may include shield assembly 12 located inside vacuum chamber 14, gas valves 16, isolation valves 18, substrate loader 20 and pump inlet 22. Shield assembly 12 may form an enclosure inside of vacuum chamber 14 such that the enclosure may contain a substrate for deposition of a thin film using an ALD process. In one embodiment, shield assembly 12 may be removable from vacuum chamber 14 such that all or portions of shield assembly 12 may be cleaned and\/or replaced. The ability to remove and replace all or portions of shield assembly 12 may simplify and improve preventative maintenance and increase the lifetime of ALD system 10.\nGas valves 16 may interface with shield assembly 12. During an ALD process, a gas may be introduced into the enclosure from one or more gas reservoirs (not expressly shown) through gas valves 16. In one embodiment, the gas reservoirs may contain a precursor and\/or one or more reactants used during a doping pulse. In another embodiment, the gas reservoirs may contain an inert gas that is used as a carrier gas during a doping pulse and\/or that is used to remove any remaining reactants from the enclosure during a purge pulse.\nDuring an ALD process, at least one of gas valves 16 may be opened to allow the precursor, reactant and\/or inert gas to flow into the enclosure formed by shield assembly 12. The precursor, reactant and inert gas may be removed from the enclosure by opening isolation valves 18 that are interfaced with shield assembly 12 opposite gas valves 16. Isolation valves 18 may further be linked to a mechanical pump (not expressly shown) through a throttle valve (not expressly shown) that facilitates automated process pressure control during an ALD process. During a doping pulse, isolation valves 18 may be opened to allow the mechanical pump to pump the precursor or the reactant and any carrier gas through the enclosure. After the purge pulse is completed, a high speed turbo pump (not expressly shown) coupled to pump inlet 22 may be used to allow vacuum chamber 14 to quickly reach the base pressure. During a purge pulse, isolation valves 18 may be opened to allow the mechanical pump to remove any remaining precursor or reactant from the enclosure. Use of only the mechanical pump during a doping pulse to exhaust the precursor or the reactant and the carrier gas from the enclosure, therefore, may extend the operation duration and life expectancy of the turbo pump.\nSubstrates on which a thin film may be deposited may be loaded into vacuum chamber 14 from a central wafer handler (not expressly shown) through substrate loader 20. In one embodiment, a substrate placed in vacuum chamber 14 may be a p-type or n-type silicon substrate. In other embodiments, the substrate may be formed from gallium arsenide, an AlTiC ceramic material or any other suitable material that may be used as a substrate on which one or more material layers may be deposited. The one or more layers deposited by ALD system 10 may form films used to fabricate conformal barriers, high-k dielectrics, gate dielectrics, tunnel dielectrics and barrier layers for semiconductor devices. ALD films are also thermally stable and substantially uniform, which makes them attractive for optical applications. Another potential application for ALD is the deposition of oxides as a gap layer for thin film heads, such as heads for recording densities of 50 Gb\/in2 and beyond that require very thin and conformal gap layers, or as an isolation layer on an abut junction to insulate a TMR or CPP type read head from hard bias layers. Additionally, ALD thin films may be used to form structures with high aspect ratios, such as MicroElectroMechanical (MEM) structures.\nFIG. 2 illustrates shield assembly 12 that cooperates with top hat 40 to form enclosure 44 located inside vacuum chamber 14. In the illustrated embodiment, shield assembly 12 includes top shield 30, bottom shield 32, vertical shield 34 and diffuser plate 36 that are bolted together and mounted on a frame. Shield assembly 12 may facilitate preventative maintenance of ALD system 10 because portions of shield assembly 12 (e.g., top shield 30, bottom shield 32, etc.) may be individually removed and cleaned or replaced as necessary.\nTop hat 40 may include substrate seat 42 for holding a substrate on which a thin film is to be deposited. Substrate seat 42 may have a depth slightly greater than or approximately equal to the thickness of a substrate. In one embodiment, substrate seat 42 may be a recess formed in top hat 40 such that substrate seat 42 is integral to top hat 40. In another embodiment, substrate seat 42 may be mounted on top hat 40 such that substrate seat 42 is separate from top hat 40. Top hat 40 may be mounted on chuck 38 located in vacuum chamber 14. Chuck 38 may function to control the position of substrate seat 42 within vacuum chamber 14 and the position of top hat 40 in relation to shield assembly 12. In one embodiment, chuck 38 includes a heating mechanism with a temperature control and constant backside gas flow to a substrate located in substrate seat 42. The temperature control with constant backside gas flow may ensure fast heating and temperature uniformity across a substrate positioned in substrate seat 42. In another embodiment, chuck 38 includes a RF power application mechanism, which allows in-situ RF plasma processing.\nEnclosure 44 may be defined by the position of shield assembly 12 in relation to top hat 40. In one embodiment, enclosure 44 may be formed when top hat 40 is in contact with bottom shield 32 such that enclosure 44 has a volume defined by substrate seat 42 and the thickness of bottom shield 32. When top hat 40 is contacting bottom shield 32 of shield assembly 12, the volume of enclosure 44 may be approximately three (3) to approximately five (5) times the volume of the substrate. Deposition of the thin film on the substrate may occur on the entire substrate surface without edge exclusion but may be confined only to enclosure 44. By minimizing the volume of enclosure 44, a minimum amount of precursor may be efficiently distributed in a minimum amount of time over the entire surface of the substrate. Additionally, surplus reactants and any reaction byproducts may be quickly removed from enclosure 44 to reduce the possibility of unwanted reactions from occurring inside enclosure 44.\nGas lines 37 a and 37 b (generally referred to as gas lines 37) may be connected to diffuser plate 36. During a purge pulse, gas valves 16 may be open to allow a gas to flow through one or both of gas lines 37 a and 37 b from gas reservoirs (not expressly shown). The gas then flows through diffuser plate 36 included in a gas injector located between diffuser plate 36 and top shield 30. In one embodiment, gas lines 37 may be formed of stainless steel and have a diameter of approximate one-quarter (\u00bc) inch. Although the illustrated embodiment shows a particular number of gas lines, ALD system 10 may include any number of gas lines and any number of gas reservoirs. For example, a single gas line may be connected to multiple gas reservoirs such that the gas flowing through the gas line is controlled by one or more valves. In another embodiment, a separate gas line may be provided for each gas reservoir.\nThe thin film may be formed on a substrate by alternately flowing a precursor and one or more reactants combined with an inert gas during a doping pulse and the inert gas during a purge pulse through gas lines 37 and into enclosure 44. For example, the precursor may be introduced into enclosure 44 through gas lines 37 and may be chemisorbed onto the surface of a substrate to form a single, monolayer of film. Enclosure 44 may be purged by flowing a purge gas through gas lines 37 and into enclosure 44 to remove any remaining precursor. After purging, the reactant be introduced into enclosure 44 through gas lines 37 and may combine with the chemisorbed monolayer of precursor to form an atomic layer of the desired thin film. Again, enclosure 44 may be purged to remove any of the remaining reactant. The doping and purge pulses may be repeated until a thin film having the desired thickness is formed on the substrate.\nAs illustrated, the reactants and\/or inert gas may be injected into enclosure 44 from one end of top shield 30 and exhausted at the other end through vertical shield 34. Vertical shield 34 may be coupled to isolation valves 18 (as illustrated in FIG. 1) and a mechanical pump (not expressly shown) that assists with the removal of the precursor and\/or inert gas from enclosure 44.\nIn one embodiment, ALD system 10 may be used to form an aluminum oxide (Al2O3)\/silicon dioxide (SiO2) nanolaminate on a substrate. The Al2O3 layer may be formed by sequentially pulsing a precursor and a reactant into enclosure 44. The precursor may be vapor-phase pulses of an aluminum source chemical and the reactant may be an oxygen source chemical. In a specific embodiment, the aluminum source chemical may be trimethylaluminum (TMA) and the oxygen source chemical may be selected from the group containing water (H2O), ozone (O3) or an oxygen radical (O2). In other embodiments, the aluminum source chemical may be any aluminum compound that is volatile at the source temperature and thermally stable at the substrate temperature and the oxygen source material may be any volatile or gaseous compounds that contain oxygen and are capable of reacting with an adsorbed portion of the selected aluminum source compound on the substrate surface at the deposition conditions such that an Al2O3 thin film is deposited on the substrate surface.\nThe SiO2 layer may also be formed by sequentially pulsing a precursor and a reactant into enclosure 44. The precursor may be vapor-phase pulses of an aluminum source chemical that produces aluminum to catalyze the growth of a SiO2 film and the reactant may be a silicon source chemical. In a specific embodiment, the aluminum source chemical may be TMA and the silicon source chemical may be tris(tert-butoxy)silanol ([ButO]3SiOH), tris(tert-pentoxy)silanol or tris(iso-propoxy)silanol. In other embodiments, the aluminum source chemical may be any aluminum compound that is volatile at the source temperature and thermally stable at the substrate temperature, which produces aluminum to catalyze the growth of a SiO2 film, and the silicon source chemical may be any volatile alkoxy organosilicon compound that is thermally stable at the deposition temperature.\nAn inert gas may be used as a carrier gas to convey the precursor and reactant during a doping pulse and as a purge gas to remove any remaining reactants from enclosure 44 during a purge pulse. In one embodiment, the inert gas may be Argon (Ar). In other embodiments, the inert gas may be any suitable inactive gas.\nNanolaminates of [xAl2O3\/ySiO2]n may be synthesized by pulsing a TMA precursor and an oxygen based reactant (e.g., H2O) into enclosure 44 to form a layer of Al2O3 and pulsing a TMA precursor and a butoxy silanol reactant into enclosure 44 to form a layer of SiO2. The Al2O3 composition of the nanolaminate film may be adjusted between approximately zero (0) and approximately one-hundred (100) percent by varying x and y. Film thickness may be adjusted by varying the number (n) of Al2O3 and SiO2 cycles.\nIn one embodiment, alternating layers of Al2O3 and SiO2 may be formed by alternating Al2O3 and SiO2 deposition cycles until an Al2O3\/SiO2 nanolaminate having a desired thickness is formed. The deposition process may begin with either a layer of Al2O3 or a layer of SiO2. The thickness of each Al2O3 layer may be approximately the same or each layer may have a different thickness. Additionally, the thickness of each SiO2 layer may be approximately the same or each layer may have a different thickness. The total number of Al2O3 layers and SiO2 layers may depend on the desired thickness for the nanolaminate film. In another embodiment, a layer of SiO2 may be formed over a layer of Al2O3 having a specific thickness by performing one or more Al2O3 deposition cycles before performing a SiO2 deposition cycle. In a further embodiment, the nanolaminate film may have an odd number of material layers formed on a substrate surface where either the Al2O3 layer or the SiO2 layer may be the top layer of the film. If the nanolaminate film includes multiple Al2O3 layers, the thickness of each Al2O3 layer may be approximately the same or each layer may have a different thickness. If the nanolaminate film includes multiple SiO2 layers, the thickness of each SiO2 layer may be approximately the same or each layer may have a different thickness. Again, the total number of material layers may depend on the desired thickness for the nanolaminate film.\nFIG. 3 illustrates a cross-sectional view of a thin film magnetic read head including an oxide gap fill layer formed by using an ALD process. A magnetic thin film read head, illustrated generally at 50, includes read sensor 52 located in between two gap fill layers 56 and 62. Gap fill layer 56 may be formed on bottom shield layer 54 and top shield layer 64 may be formed on gap fill layer 62. Read 52 may include multiple layers of different magnetic and non-magnetic layers. In one embodiment, read 52 may be a multilayer giant magnetoresistive (GMR) device or a spin valve device. In other embodiments, read sensor 52 may be any type of magnetoresistive device used in a read head for a hard disk drive. Read head 50 may further include lead 60 and hard bias layer 58 that surround read sensor 52. Lead 60 may function as an electrically conductive electrode layer.\nThe thickness of gap fill layers 56 and 62 may be used to control the linear recording density of a hard disk drive including read head 50. Additionally, gap fill layers 56 and 62 may provide insulation for read sensor 52 and may dissipate heat throughout read head 50. As the recording densities for disk drives increase, the thickness of gap fill layers 56 and 62 should decrease. Additionally, reducing the thickness of gap fill layers 56 and 62 may improve the heat dissipation of read head 50. Although Al2O3 has traditionally been used as a gap fill layer, Al2O3 films may be unable to retain certain properties (e.g., a high dielectric breakdown strength) if the film is less than a certain thickness.\nA nanolaminate of Al2O3 and SiO2, however, may be used to form gap layers 56 and 62 having decreased thicknesses because the addition of SiO2 allows the film to maintain certain characteristics as the thickness decreases. In one embodiment, gap layers 56 and 62 may have a thickness of between approximately fifty angstroms (50 \u00c5) and approximately 250 \u00c5. At a thickness of 250 \u00c5, an Al2O3\/SiO2 nanolaminate thin film may have a dielectric breakdown field of approximately 13 MV\/cm where as a single layer of Al2O3 may have a dielectric breakdown field of approximately 10 MV\/cm. Even for a thickness at or below 50 \u00c5, for example, an Al2O3\/SiO2 nanolaminate may have a dielectric breakdown field of approximately 11 MV\/cm where as a single layer of Al2O3 may only have a dielectric breakdown field of approximately 8 MV\/cm. Other properties of an Al2O3\/SiO2 nanolaminate thin film are shown in more detail below with respect to FIGS. 7 through 10.\nFIG. 4 illustrates a graph of rate of deposition of Al2O3 (as shown along the right y-axis) and SiO2 (as shown along the left y-axis) using an ALD process as a function of temperature. As illustrated, a layer of Al2O3 may be deposited with an ALD process at a relatively constant growth rate of approximately 1.05 \u00c5\/cycle when the substrate temperature is between approximately 150\u00b0 C. and 300\u00b0 C. The deposition rate of ALD SiO2 thin films may increase from approximately 2.4 A\/cycle at a substrate temperature of approximately 150\u00b0 C. to a plateau of approximately 13 \u00c5\/cycle at a substrate temperature above approximately 210\u00b0 C.\nFIG. 5 illustrates a graph of saturation characteristics for the deposition of SiO2 thin films by using an ALD process as a function of pulsing time in seconds of the TMA reactant (as shown along the top x-axis) and the butoxy silanol reactant (as shown along the bottom x-axis) used in the process. During deposition of the film, the source temperature for each reactant was approximately 95\u00b0 C. and the substrate temperature was approximately 210\u00b0 C. Charging and pulsing of (ButO)3SiOH were manipulated such that exposures to alternating pulses of TMA\/(ButO)3SiOH at appropriate partial pressures produced a deposition rate for the SiO2 layer below approximately 15 \u00c5\/cycle, which may be desirable for the formation of nanolaminate films.\nFIGS. 6A and 6B illustrate graphs of dielectric breakdown characteristics for a 200 \u00c5 single layer of aluminum oxide and a 200 \u00c5 single layer of silicon dioxide when deposited at different deposition temperatures. As illustrated in FIG. 6A, the dielectric breakdown characteristics were measured at six different deposition temperatures ranging from approximately 190\u00b0 C. to approximately 290\u00b0 C. The dielectric breakdown for a single layer of SiO2 over the deposition temperature range may occur at a breakdown field (EBD) of approximately 12.5 MV\/cm and a leakage current density (J) of approximately 1\u00d710\u22124 Amps\/cm2.\nIn comparison, a single layer of Al2O3 shows a lower dielectric break down over the range of deposition temperatures as illustrated by FIG. 6B. As shown, the dielectric breakdown characteristics for Al2O3 deposition were measured at eight different temperatures ranging from approximately 150\u00b0 C. to approximately 290\u00b0 C. The dielectric breakdown for Al2O3 may occur at a breakdown field (EBD) of approximately 9.3 MV\/cm and a leakage current density (J) of approximately 1\u00d710\u22127 Amps\/cm2. In some applications, such as read heads, a higher dielectric strength may be desirable.\nFIGS. 7A and 7B illustrates graphs of dielectric breakdown characteristics for a nanolaminate layer of Al2O3\/SiO2 deposited using an ALD process as a function of Al2O3 concentration. As illustrated in FIG. 7A, the dielectric breakdown characteristics were measured for nanolaminate thin films having an Al2O3 concentration ranging from approximately 26% to approximately 100%. The dielectric strength of the nanolaminate thin film increases as the concentration of Al2O3 decreases. For example, at a leakage current density threshold of less than approximately 2\u00d710\u22126 Amps\/cm2, a nanolaminate thin film having an Al2O3 concentration of approximately forty-seven percent (47%) may have a dielectric breakdown field of approximately 11 MV\/cm while a nanolaminate thin film having an Al2O3 concentration of approximately twenty-six percent (26%) may have a dielectric breakdown field of approximately 13 MV\/cm.\nAs illustrated in FIG. 7B, the dielectric breakdown characteristics were measured for a nanolaminate film having an Al2O3 concentration ranging from approximately 26% to approximately 100% at current densities of approximately 2\u00d710\u22126 Amps\/cm2 and approximately 2\u00d710\u22125 Amps\/cm2. Again, the graph shows that the dielectric strength of nanolaminate films having lower Al2O3 concentrations breakdown at higher dielectric fields than films having a higher Al2O3 concentration. For example, an Al2O3\/SiO2 nanolaminate may have a breakdown field of greater than approximately 11 MV\/cm for films having an Al2O3 concentration less than approximately 50% and a breakdown field of less than or equal to approximately 11 MV\/cm for films having an Al2O3 concentration greater than approximately 50%.\nFIGS. 8A and 8B illustrate graphs of dielectric breakdown characteristics for a single layer of Al2O3 and a nanolaminate layer of Al2O3\/SiO2 deposited using an ALD process as a function of film thickness. As shown, the dielectric breakdown characteristics were measured for Al2O3 films at four different thicknesses ranging from approximately 52 \u00c5 to approximately 213 \u00c5 and nanolaminate films at five different thicknesses ranging from approximately 48 \u00c5 to approximately 232 \u00c5. In the illustrated embodiment, a single Al2O3 film and a nanolaminate film of [10Al2O3\/1SiO2]n (e.g., 10 layers of Al2O3 and 1 layer of SiO2, which has a Al2O3 concentration of approximately 46%) were deposited at different thicknesses. As shown, the nanolaminate film exhibits higher breakdown field values by approximately 1.5 MV\/cm to approximately 2.5 MV\/cm over the thickness range at a leakage current density threshold less than approximately 2\u00d710\u22126 Amps\/cm2.\nFIGS. 9A and 9B illustrate graphs of resist developer etch rates for a single layer of Al2O3 and a nanolaminate layer of Al2O3\/SiO2 deposited using an ALD process as a function of substrate temperature and a function of Al2O3 composition, respectively. As illustrated in FIG. 9A, the etch rates for a single Al2O3 layer and an Al2O3\/SiO2 nanolaminate were measured at eight different deposition temperatures ranging from approximately 150\u00b0 C. to approximately 290\u00b0 C. In addition to improvements in the dielectric breakdown strength of a nanolaminate Al2O3\/SiO2 film as illustrated above in FIGS. 7 and 8, Al2O3\/SiO2 nanolaminates may have an improved etch resistance to base solutions (e.g., photoresist developer). As shown, the etch rate of an Al2O3\/SiO2 nanolaminate for substrate temperatures greater than approximately 150\u00b0 C. may be close to zero while the etch rate for a single layer of Al2O3 may be greater than or equal to approximately 20 \u00c5\/minute at temperatures greater than 150\u00b0 C.\nAs illustrated in FIG. 9B, the etch rates were measured for Al2O3\/SiO2 nanolaminates having an Al2O3 concentration ranging from approximately 26% to approximately 100%. The etch rates for Al2O3\/SiO2 nanolaminates having Al2O3 concentrations of less than approximately 85% may be close to zero as compared to the films having an Al2O3 concentration of greater than 90% where the etch rates may be greater than approximately 20 \u00c5\/min.\nFIG. 10 illustrates a graph of tensile strength of a nanolaminate film formed using an ALD process as a function of Al2O3 concentration. As shown, a single layer film of Al2O3 may have a tensile stress value of approximately 400 MPa while a single layer film of SiO2 may have a tensile or compressive stress value of less than approximately 50 MPa. Thus, the film stress of an ALD nanolaminate thin film may be controllably reduced by adjusting the Al2O3 concentration.\nforming a silicon dioxide layer having a second thickness on at least a portion of the aluminum oxide layer by sequentially pulsing a second precursor and a second reactant into the enclosure to form a nanolaminate thin film.\n2. The method of claim 1, wherein the nanolaminate thin film comprises a read head gap layer.\nthe first reactant is selected from the group consisting of water, ozone and oxygen radicals.\nthe second reactant is selected from the group consisting of tris(tert-butoxy)silanol, tris(tert-pentoxy)silanol and tris(iso-propoxy)silanol.\n5. The method of claim 1, further comprising the second thickness being greater than the first thickness such that the nanolaminate thin film has a concentration of aluminum oxide of less than approximately fifty percent.\n6. The method of claim 1, further comprising a deposition temperature being in a range between approximately 150\u00b0 C. and approximately 300\u00b0 C.\n7. The method of claim 6, further comprising forming the aluminum oxide layer at a first deposition rate of approximately 1.05 \u00c5\/cycle over the deposition temperature range.\n8. The method of claim 6, wherein a deposition rate for the silicon dioxide layer varies between approximately 2.4 \u00c5\/cycle and approximately 13 \u00c5\/cycle over the deposition temperature range.\n9. The method of claim 1, further comprising the nanolaminate thin film including a thickness of between approximately 50 \u00c5 and approximately 250 \u00c5.\n10. The method of claim 1, further comprising repeating the steps of forming the aluminum oxide layer and forming the silicon dioxide layer such that the nanolaminate thin film includes a plurality of alternating aluminum oxide and silicon dioxide layers.\nintroducing the purge gas into the enclosure after the second precursor and the second reactant such that substantially all of the second precursor and the second reactant are removed from the enclosure.\n12. The method of claim 1, wherein the substrate surface comprises a layer of silicon dioxide having a third thickness formed by sequentially pulsing the second precursor and the second reactant into the enclosure.\nforming a silicon dioxide layer having a second thickness on at least a portion of the aluminum oxide layer by sequentially pulsing TMA and tris(tert-butoxy)silanol into the enclosure to form a read head gap layer.\n14. The method of claim 13, further comprising the second thickness being greater than the first thickness such that the nanolaminate thin film has a concentration of aluminum oxide of less than approximately fifty percent.\n15. The method of claim 13, further comprising a deposition temperature being in a range of between approximately 150\u00b0 C. and approximately 300\u00b0 C.\nforming the aluminum oxide layer at a first deposition rate of approximately 1.05 \u00c5\/cycle at the deposition temperature of approximately 210\u00b0 C. forming the silicon dioxide layer at a second deposition rate of approximately 13 \u00c5\/cycle at the deposition temperature of approximately 210\u00b0 C.\n17. The method of claim 13, further comprising the read gap layer including a thickness of between approximately 50 \u00c5 and approximately 250 \u00c5.\n18. The method of claim 13, further comprising repeating the steps of forming the aluminum oxide layer and forming the silicon dioxide layer such that the read gap layer includes a plurality of alternating aluminum oxide and silicon dioxide layers.\nan ALD-formed silicon dioxide layer having a second thickness formed on at least a portion of the aluminum oxide layer, the aluminum oxide layer and the silicon dioxide layer cooperating to form a nanolaminate thin film.\n20. The film of claim 19, wherein the nanolaminate thin film comprises a read head gap layer.\n21. The film of claim 19, further comprising the second thickness being greater than the first thickness such that the nanolaminate thin film has a concentration of aluminum oxide of less than approximately fifty percent.\n22. The film of claim 19, further comprising the nanolaminate thin film including a dielectric breakdown field in a range of between approximately 11 MV\/cm and approximately 14 MV\/cm.\n23. The film of claim 19, further comprising the nanolaminate thin film including a stress in a range of between approximately 50 MPa and approximately 400 MPa based on an aluminum oxide concentration.\n24. The film of claim 19, further comprising the nanolaminate thin film including an etch resistance to base solutions such that the etch rate of the nanolaminate film is approximately equal to zero.\n25. The film of claim 19, further comprising the nanolaminate thin film including a thickness of between approximately 50 \u00c5 and approximately 250 \u00c5.\na plurality of ALD-formed silicon dioxide layers having the second thickness, the aluminum oxide layers alternating with the silicon dioxide layers to form the nanolaminate thin film.\n27. The film of claim 19, further comprising a top ALD-formed aluminum oxide layer having a third thickness, the second aluminum oxide layer formed on at least a portion of the silicon dioxide layer, the aluminum oxide layers and the silicon dioxide layer cooperating to form the nanolaminate thin film.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JAMRUD \u2013 Following the merger of tribal areas with Khyber Pakhtunkhwa, journalists in Khyber tribal district have decided to reorganize and rename the dysfunctional Tribal Union of Journalists.\nAccording to Tribal News Network, the decision was taken at a meeting of Jamrud Press Club.\nJournalist Ibrahim Shinwari asked journalists of other districts in erstwhile FATA to have their union that has become crucial after Fata-KP merger, TNN reported.\nHe said TUJ was non-functional for the last three years with no cabinet and offices where they could sit and work for the solution of their problems.\nHe said that ex-TUJ members would need to associate with KP union of journalists as well as other unions throughout the country after the merger of Tribal Areas into the province.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgmpc b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmpc
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index 0000000000000000000000000000000000000000..e834803b57c9fabea5609ef88763e413062f43e6
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+{"text":"\"Life Satisfaction to Widowhood\" by Hyunsook Kang Ph.D. and Bonnie Ahn Ph.D.\nThe objective of this study was to examine the breadth and depth of relationships with relatives and friends and the possible implications of those relationships with regards to life satisfaction to widowhood. Data from the National Social Life, Health, and Aging Project (NSHAP) survey were used, which sampled persons 57-85 years of age (N=3005). It was hypothesized that older widowed adults have greater quality of both family and friend relationships than do older married adults. Structural Equational Modeling analysis results supported these hypotheses, revealing that older widowed adults reported higher quality of engagement in family and friend relations compared to older married adults, possibly enabling greater life satisfaction to widowhood.\nKang, Hyunsook Ph.D. and Ahn, Bonnie Ph.D. (2018) \"Older Adults' Social Relations: Life Satisfaction to Widowhood,\" Journal of Human Services: Training, Research, and Practice: Vol. 3 : Iss. 2 , Article 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Targeted attacks are the new frontier for security professionals and show the increasing sophistication of cyber criminals. They are multilayered attacks that require structured research by cyber criminals about their potential victims, such as finding out their email addresses, job titles and job functions, areas of interest, websites they frequently access, conferences they attend and so on. Once the research is done, cyber criminals choose the path of least resistance and attack employees in companies that are not well protected through security systems.\nSecurity defenses need to get more sophisticated to protect effectively against targeted attacks. Security intelligence services correlate information from various security systems and can detect even small anomalies in network traffic and user behavior, that would otherwise pass undetected. IDC believes that security intelligence services form an important part in protecting against new threats like targeted attacks and the constantly evolving threat landscape.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been running a series of exciting engagement sessions with 4 schools in Worcester to contribute ideas and develop themes for a large scale design on the glazing for the new Perdiswell Leisure Centre.\nThe vinyl manifestations will be installed onto glazing screens that span across large areas of the building frontage with some areas being up to 8 metres high.\nOver the past few months I've worked with the Royal Grammar School, Heart of Worcestershire College (HOW), Tudor Grange Academy and Perdiswell Primary school.\nUp to 100 young people aged from 11 to 18 years old have been involved in taking part in 2 workshops each of which lasted half a day. In all cases we began with some 'water' explorations; dropping ink into water and filming the resulting swirls with our phones, exploring pattern and light reflection from water, looking at colour. We've been using traditional art room techniques as well as new technology to capture this work.\nThe initial water experiments made sense because the Leisure Centre will be featuring an 8 lane competition pool. However during the course of the workshops we began to get more excited by the possibilities of looking at the natural environment around the building.\nThis led us to look more closely at leaves and natural forms. Young people collected fallen leaves and took rubbings, mixed colour combinations, digitally scanned and abstracted the shapes into paper cut patterns which were then collaged into exuberant designs.\nOne of the challenges of the art project is to turn all these numerous experiments and discoveries into a unified piece of artwork.\nThe final design will be a composition of many young people's work and will be embedded into the building for its lifetime.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mr. Gurland is responsible for commercial leasing and business development at Washington Property Company. His 25 years of experience in the Washington, D.C. Metropolitan Area cover all property types and various aspects of the commercial real estate industry.\nPrior to joining Washington Property Company, Mr. Gurland was a founding principal at Leland Group Commercial Real Estate, a boutique brokerage firm that concentrated on commercial real estate in the Shenandoah Valley. In this role, he was responsible for the leasing and sales of office, industrial, retail and land properties, primarily on the Interstate 81 Corridor. Prior to Leland Group he was Sales Manager for Grubb & Ellis Company's Washington, D.C. region offices and was the founder of Storetrax.com, a retail real estate listing service.\nMr. Gurland holds a Bachelor of Science in Business Administration degree from Boston University and resides in Bethesda, Maryland with his wife and children.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unity Games and Electric Eggplant and Heirs of Rube Goldberg are pleased to announce that the Android version of Rube Works: Rube Goldberg Game, is now available on the Google Play store for \u20ac1.49, \u00a31.29, AU$1.99. The Android version of Rube Works includes 18 humorous brainteasers to test your mental acumen as you use everyday objects to craft contraptions which in turn perform what normally would be routine tasks! Imagine the goal is to carve a turkey \u2014 and you have a penguin, a bucket, a rooster, a box of sand, a sea sponge, a seesaw and an ice cream maker to complete the task. How will you do it?\n\"Rube Works\" is the first officially licensed game based on the creations of Rube Goldberg, the award winning artist known for crafting comedic solutions to everyday problems and combines the puzzle genre with slapstick humor and creative problem solving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgtml b/data_all_eng_slimpj/shuffled/split2/finalzzzbgtml
new file mode 100644
index 0000000000000000000000000000000000000000..b24644d4931e89a709ea7ddb8f66170e69da7986
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+{"text":"Singapore, 15 September 2018 -- Blockchain Worx, a Singapore-based advisory, and leading global Fintech-Regtech venture, has announced the release of Tokenize Pro, a turnkey white-label securities tokenization platform and digital asset marketplace.\nThe platform enables institutions to tokenize securities and other real-world assets while managing the whitelisting, sale and distribution of these security tokens in an effective, secure and compliant manner. Tokenize Pro also includes a digital investment marketplace for tokenized securities facilitating virtual ownership and trading of conventional physical assets in a cost-effective, decentralized and instant manner.\nThe white-label solution allows institutions to set up their own turnkey Securities Tokenization Platform and Digital Asset Marketplaces, allowing security tokens to be traded via a blockchain-based network among registered and verified members in a cryptographically secure and tamper-proof way. Built ground up with security and compliance at the core, the platform includes integrated registration, KYC, approval and reporting functionality allowing institutions to tick all the boxes and comply with relevant regulations.\nBlockchain Worx is also exploring partnerships with multiple institutions to set up theme-specific Tokenization Platform and Digital Asset Marketplaces in upcoming crypto-friendly jurisdictions. A demo version of Tokenize Pro has also been hosted at Tokenotic.com.\nFor more information, be sure to visit http:\/\/www.blockchainworx.com.\nBlockchain Worx is a Singapore-headquartered global Fintech-Regtech venture focused on helping institutions harness the potential of Blockchain technology for a competitive advantage. Its turnkey solutions, including a Securities Tokenization platform and Digital Asset Marketplace as well an Anti-Money Laundering application, are targeted at enabling organizations to create next-gen digital finance and regulatory systems, powered by blockchain technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What a GREAT OPPORTUNITY!! This 5200 sq ft building + basement, has loads of potential!! The main level has a huge main room, and kitchen with equipment, that could easily be converted into a restaurant. There\"s an outdoor space that could be converted into an outdoor dining area. The upstairs has another large main room that was used as a dance studio at one time. You could convert this space into 2-3 apartments, or rent to another business. You could live upstairs, have a business on the main level- the possibilities are endless! The building has good bones, separate gas & electric meters, and some kitchen equipment will remain. Newer Hvac & electric. There are 4-5 off street parking spots that come with the building + on street parking. The City of Garrett will be adding a huge public parking lot behind the building, which will be a big deal.\nListing courtesy of Katrina Nichols of Brick And Mortar Real Estate Group.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Below you will be able to find all Christmas cracker metal puzzles solution cracker? crossword clue is the biggest database with all crossword clues solutions 12 mixed metal puzzles are perfect for inside your Christmas crackers. It'll be great fun at your Christmas table when people in funny hats are sitting Christmas cracker metal puzzles solution laughing at bad jokes and struggling to complete these little puzzles. Dec 12, 2011 \u00b7 How to Solve Cracker Metal Puzzle. How to Solve Cracker Metal Puzzle.\nSkip navigation. Baguenaudier Chinese Rings Desperado Centipede puzzles solution and tutorial - Duration: 6: 34. TyYann. Use spoiler tags when posting a hint or solution to a puzzle. Guesses must also be spoiler-tagged. [guess](\/s\" The Princess was in another castle! \" ). How the heck do you solve this Christmas cracker puzzle?\n2 hours in and I am no closer. (i. imgur. com). between the coils. The metal is cylindrical so pulling them apart at the gaps should. WonderHowTo Puzzles. How To: Solve the cracker wire puzzle By Amie Cou\u00e9 Arbuckle; 7\/9\/08 11: 47 AM.\nWonderHowTo. Watch this video tutorial on how to crack a simple puzzle made of wire. This is a simple puzzle that you can make yourself. How To: Solve the Hanayama Cast Loop metal puzzle How To: Solve a. Dec 11, 2007 \u00b7 Me Solving the Metal Puzzle [Loop] in 4 seconds. Metal Puzzle\" Loop\" 4 Seconds Fast and How to Solve! Ov3rlo1d. Solution for Arrow Dynamics from Puzzle Master Wire Puzzles Absolutely Christmas Crackers Puzzle - Solution.\nThe Puzzle: Last month I sent off for one of those kits which you can use to make your own Christmas Crackers. The kit contained:. See this puzzle without solution Discuss this puzzle at the Math is Fun Forum. Number Puzzles Puzzles Index. Kuckoo Krackers: Metal Puzzle Cracker Solutions Puzzle 1 Puzzle 3 Puzzle 4 Puzzle 2 Puzzle 5 Puzzle 6 \u00a9 The Kuckoo Company 2014 How to Solve Metal Ring Puzzles By Andrew Smith; Updated April 12, 2017 A metal ring puzzle is basically nothing more than two or more metal pieces that are connected.\nCoogam Metal Wire Puzzle Set of 16 with Pouch, Brain Teaser IQ Test Disentanglemen Iron Link Unlock Puzzles Game Chinese Ring Puzzle Magic Trick Toy Gift for Kids and Adults Challenge. exactly what I was looking for.\nI made Christmas Crackers for Christmas dinner place settings and these are perfect as one of the prizes inside the cracker. Metal Puzzle Crackers Each cracker contains a mind muddling, brain busting, cerebral twisting metal puzzle; the aim of each being to separate two inter-twined chrome wires. Christmas Cracker Puzzles Presented in stylish red and gold cracker packaging, these high- quality wooden jigsaw puzzles makes the perfect addition to your Christmas dinner table!\nSort By: Featured Items Newest Items Best Selling A to Z Z to A By Review Price: Ascending Price: Descending Buy Coogam Metal Wire Puzzle Set of 16 with Pouch.\n3D Coil Cast IQ Logic Educational Wire Chain with Plastic Storage Box and Solutions, Best Novelty Idea 4. 8 out of 5 stars 117. exactly what I was looking for. I made Christmas Crackers for Christmas dinner place settings and these are perfect as one of the prizes inside the cracker. I. Sep 7, 2008.\nWonderHowTo Puzzles \u00b7 WonderHowTo. Solve the cracker wire puzzle. By Amie. How To: Solve the Hanayama Cast Loop metal puzzle. Apr 12, 2017. Metal ring puzzles are problems that are sometimes advertised as. Solutions are available for the following puzzles: Please click.\n10 Metal Puzzle Set - Purple 10 Metal. 3D Crystal Puzzle Deluxe - Christmas Tree 3D Crystal. Pack of 12 small wire puzzles for Christmas Crackers - assorted. $7. 27. Quick Overview. 12 mixed metal puzzles are perfect for inside your Christmas crackers. I was given a pack of Chinese Metal Wire Puzzles for Christmas. The solutions were provided on the back of the pack but needless to say they were pretty hard.\n12 mixed metal puzzles are perfect for inside your Christmas crackers. It'll be great fun at your Christmas table when people in funny hats are sitting around laughing at bad jokes and struggling to complete these little puzzles. How to Solve Cracker Metal Puzzle.\nHow to Solve Cracker Metal Puzzle. Skip navigation. Baguenaudier Chinese Rings Desperado Centipede puzzles solution and tutorial - Duration: 6: 34. TyYann.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Production Truck Atlanta is thrilled to be a part of the Georgia T.V. And film industry. We offer a wide range of products to help you make your production great. We have the supplies you need, whether it's setting up your Production Office, Art Department, Wardrobe, Make up, Crafty, or beyond. We are here to help! T.P.T. Atlanta is a Los Angeles based rental company with local roots. You are not just another production to us, you are family!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"(SPS) - The holiday season always seems to bring a smile.\nFrom the cooler temperature to spending time with family, there is much to be happy about. And for those of you who find happiness in holiday shopping, you have another reason to smile especially if you use Amazon.\nYou now have the opportunity to donate to Sapulpa Public Schools with each purchase you make through AmazonSmile. And it doesn't cost you a single cent more.\nAll you have to do is log on to smile.amazon.com and use the same login information you use for Amazon.com. The first time you log in, you will be directed to pick a charity of your choosing. By selecting Sapulpa Public School Foundation Inc. you will be supporting our schools. From there, it's time to start shopping!\nWith each qualifying purchase, half a percent will be donated to the Sapulpa Public School Foundation. The SPS Foundation is a non-profit community organization whose purpose is to secure and distribute contributions for the benefit of Sapulpa schools students.\nYou will be able to track how much will be donated through your purchases as well as how much has been generated for Sapulpa Public School Foundation.\nHalf a percent may not seem like much, but as the website says, charitable donations quickly add up and can you think of an easier, more convenient way to give to your child's school?!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhcfm b/data_all_eng_slimpj/shuffled/split2/finalzzzbhcfm
new file mode 100644
index 0000000000000000000000000000000000000000..e35462b2b56aa432872eb97019f5297e81563c43
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhcfm
@@ -0,0 +1,5 @@
+{"text":"Grab your Keyblade and take a good look at this awesome Kingdom Hearts 3 Select Series 2 Action Figure Set! Based on Kingdom Hearts 3, the Series 2 set of three multi-packs includes Hercules with a Dusk, Toy Story Sora with an Air Soldier and the Favorite Deputy Keyblade, and Guardian Form Sora with an Air Soldier and the Hero's Origin Keyblade. Each 7-inch scale action figure features a full range of articulation, and each set comes packaged in display-ready Select packaging. Subject to change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Foot Posture Index is a tool that was developed to measure the posture of the foot. The Foot Posture Index depends on a number of observations of the foot rather than a single unreliable measure. The Index has been shown to be a reliable tool for determining foot posture and also having some validity in that it correlates with dynamic function.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FROM BHAM, TAKE I-65 NORTH TO EXIT 310. TURN RIGHT ONTO HWY 157. GO APPROX 3 MILES (GO THROUGH 5 TRAFFIC LIGHTS), BEFORE REACHING LAKE CATOMA, TURN RIGHT ONTO SHARPTON DRIVE. TAKE NEXT RIGHT ONTO CATOMA LANE, HOME WILL BE ON THE RIGHT.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to select the right paint to match your d\u00e9cor.\nIt's taken weeks to select your favorite roman shades. But why not? There were so many colors and patterns to choose from. Now that they're installed your only decision is selecting the perfect wall color to match. If your past experiences with choosing paint wound up to be a big surprise after it was all said and done, these tips on how to properly test paint may help.\nWe marvel at what today's computers can do. They allow us to shop, talk to family, even see what we'd look like twenty pounds lighter. Now we can add selecting paint color to the list. With online color tools, along with a well-lit photo of one of the walls you wish to paint, you can digitally see what the color will look like long before purchasing the paint itself. Keep in mind, generally you will need a picture of the wall without your furniture. And because the samplers don't pick up wall texture, choosing a flat wall is better.\nAfter painting a room, how many times have you said, \"Boy, it didn't look like that on the sample\"? That's because we generally sample paint incorrectly.\nColor is relational, perceived in context, and not alone. That means, what the color will ultimately look like is affected by the colors nearest to it. So when you paint color samples on the walls it will be distorted by the color that's already on the wall. If you paint several sample on the wall they will all influence each other. Using this technique, you can never get a true read on what any of the colors will look like alone.\nIt takes a few coats of paint to see the true color of the sample which creates build-up on the test area. To prevent it from being noticeable after the walls are painted make sure you feather out the edges. If you don't take this precaution you could be looking at a lot of effort to hide your test samples when it comes to painting the wall the color you've selected.\nInch-wide paint samples from the paint store is not your only option, nor is the best way to get a true read on color. In fact, any sample smaller than 6\u00d76\u2033 will never give you a true representation of color. While some professional decorators use 5\u00d75\u2032 boards this is probably not practical for the average homeowner. 1\u00d71\u2032 seems to be large enough to give you an extremely good idea of what the color will ultimately look like in your home.\nThe good news is; in many cases you can special order oversized paint color chips from some paint stores. You can also make them yourself, or you can buy them online. Whichever way you get them, they're great to work with as you can move them around the room, exchange them, and isolate them from one another to keep them from influencing one another.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The climate in southern Adelaide has already become hotter and drier. We're experiencing more intense storms, flooding, bushfires and heatwaves and our growing season is changing too. Our best bet is to stop further change by reducing greenhouse gas emissions but we will also need to take steps to adapt to the changes that have already been locked in. If the climate continues to change, we will face serious risks that will become increasingly difficult to manage.\nThis section shows how Resilient South councils are responding and ways that you can get involved.\nResilient South councils have mapped the heat of our urban areas to identify the suburbs most at risk. As temperatures continue to rise, there are lots of things we can do to cool our streets, towns and cities.\nYou can find out how heat exposed your property or neighbourhood is by entering your address into the search function of our interactive heat maps.\nWe surveyed our communities about their knowledge, attitudes and behaviours relating to climate change.\nWe learnt that our communities want more support and information about how to prepare and respond. And that's one of the reasons we created this website!\nClimate Ready Schools is a partnership between NRM Education and Resilient South. We use STEM (science, technology, engineering and maths) and Design Thinking to support schools and students to identify climate change risks and options for adapting. Get in touch to find out how your school can get involved.\nCheck out our projects that help reduce the impact of increased frequency and intensity of heatwaves to create Cool Places. Some actions taken include planting and maintaining vegetation (green infrastructure) and implementing water sensitive urban design.\nIt's official... Trees are awesome!\nThat's why Resilient South councils are planting more trees to keep our towns and suburbs cool and livable.\nIf you think we need to get ready as a community for changes in our climate, you can join our group of Climate Ready Champions. Champions are trained to help people understand how the things we value may be affected by our changing climate, and explore what we can do to take action.\nStreet lighting is the single largest source of greenhouse gas emissions by local government in Australia! To reduce our emissions, Resilient South councils are upgrading to energy efficient LED lights.\nResilient South mapped a range of social, environmental and economic data about southern Adelaide, to identify who and what is most vulnerable to the impacts of climate change. We used this information to develop the Resilient South Regional Climate Change Adaptation Plan. You could use it to help prepare your home or business.\nSouthern Adelaide's beautiful beaches are exposed to a wide range of climate related risks that we're working hard to manage.\nArt meets science in this unique program offering artists with access to Resilient South partner councils' climate change data and science to interpret through their art.\nWe won the Minister for Health and Wellbeing Excellence in Public Health Award 2018 for this event. If you'd like to run a similar event in your own region or organisation, we have loads of fantastic resources that you're welcome to use.\nDo you have a great idea for helping prepare your home or community for changes in our climate? Check out the range of community grants and subsidies available from Resilient South partners.\nIf you would you like to learn more about incorporating environmental sustainability into your everyday life, you're in luck! We have a range of amazing community sustainability programs that deliver workshops, newsletters and events filled with practical information and fun activities.\nFind out how the Resilient South partner councils are getting on the front foot with climate change through clever urban design solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhdpp b/data_all_eng_slimpj/shuffled/split2/finalzzzbhdpp
new file mode 100644
index 0000000000000000000000000000000000000000..72d6a55294270925547d732e5a5856abd8a866f7
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Progress continues on \"Advanced Operators.\" I've got three or four other stories pounding at the door of my mind, demanding to be written, but AO is filling up the creative room, floor to ceiling, door to window, like an overfed shoggoth. Nothing else gets written until this beast is finished.\nBut Jack Sheed wants in, let me tell you.\nI'm still going back and adding scenes, plot points and character development to AO (how much of the language of fiction writing has been replaced by that of film?\u2013or perhaps they're both in debt to drama). Thus, making forward progress has been difficult. But now I have only one major aspect of the plot to figure out, and once that's done, the rest of the story should (in theory) unfold fairly easily.\nIn other news, finally decided on a screenplay for my screenwriting class. I'm going to do an adaptation of Lovecraft's \"The Dunwich Horror.\" The story has been filmed before, as a 1970 teen horror flick starring Dean Stockwell and Sandra Dee. But that movie has little in common with Lovecraft's story other than a few names.\nI'm sure there are many screenplays out there based on \"Dunwich\"\u2013independent filmmakers love him, for some reason\u2013but I'm going to write my own anyway. My plan is to do it Lovecraft-by-way-of-Hammer Films (similar to Tim Burton's Sleepy Hollow). The protagonist will be, in true Lovecraftian fashion, a middle-aged scholar. Perhaps, after Guillermo Del Toro's At the Mountains of Madness comes out (and I sincerely hope he sets it in the thirties), Lovecraft will become a hot property in Hollywood, as Stephen King was in the eighties and Jane Austen in the early nineties.\nAs usual, I've already envisioned a movie trailer. I tend to enjoy movie trailers as much as (if not more than) the films they advertise A few of my favorites are the trailer for Sleepy Hollow and the teaser trailers for Gladiator and Star Trek: First Contact. The latter two benefited greatly from music borrowed from other (better) films\u2013Basil Poledouris's soundtrack for Conan the Barbarian and James Horner's soundtrack for The Wrath of Khan, respectively (for my trailer, I'd swipe the music from Sleepy Hollow).\nIn my trailer, we see a montage of brief, bizarre images from the story, such as a huge man in a trenchcoat trying to check out a book at a library, a meadow with a great swath of crushed grass, and a close-up of a worried-looking man in front of a crumbling rustic house on a misty New England morning. At the end of the trailer the screen goes dark, and we hear the climactic dialogue of the story: an unearthly voice crying \"F-f-f-father!\" followed by (in a horrific roar) \"YOG-SOTHOTH!\" Then, in thin orange letters, the words \"YOG-SOTHOTH IS THE GATE\" fade in and out, followed by \"YOG-SOTHOTH IS THE KEY\"; and then the date the movie comes out.\nIn class, we have to \"pitch\" the story, as we would to a group of producers. Since one of the scenes in Lovecraft's story takes place at Harvard's Widener Library, where I've worked for nearly eight years, I began my pitch by describing the scene as if I'd actually witnessed it. It worked well enough\u2013though when I got to the part about the guy dissolving into a ichorous mass of putrescence, with tentacles visible beneath his coat, everyone became a bit incredulous.\nStill working on \"Advanced Operators,\" which has gone from a Lovecraftian novella, to a Stephen King-ish novella, to a Lovecraftian short story, and now to a Fritz Leiberesque short story.\nThis was written in 1941, mind you. As usual, ol' Fritz has beaten me to the punch.\nI've uploaded a few stories to the Fiction section. Many will note with amusement that I've chosen to upload only an \"excerpt\" of my Sherlock Holmes pastiche in hopes that it may be published someday. Laugh now\u2013it won't be so funny when I'm paid $15 for it to be anthologized in an e-zine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some people think that a person can never understand the culture of a country unless they speak the language.Do you agree with this opinion?\nEssay topics: Some people think that a person can never understand the culture of a country unless they speak the language.\nFirst of all, language can be counted as a method to people for transferring mutual discussion and ideas between each other, however we can not live naturally in a country if we don't know its language, in addition, that we might have passive consequences while dealing with other individuals as well as we will not be able to adapt in a country in different situations, hence we believe in recent days that a large number of people prefer to travel to another countries either for studying of working due to the unemployment which it exists in their country, in this case, one's has to learn the language before leaving his original country, so that they can have the ability to adapt in as per the culture requirements, furthermore, knowledge of country's language is an essential for individuals to learn it.\nNotably, there are a lot of matures faild to reach their destination goal, because they don't know the mother language of a country that they need to move into it. For example, there are people tend to shift to Germany for settling over there, unfortunately, they don't know the language, as a consequence, they found a numeric of difficulties, such as not finding job, even in simple jobs' tasks like stewart in a restaurant, I believe that they can not work in an organisation without knowing the language, because how they can work in a place that they don't know the language of it. However, language is basic aspect to have known it, so that they continued their work naturally, in addition we believe that there are a lot of persons left their country to work western countries, but they lost many chances, as a result, they have come back to original country, we believe that they spent a huge amount of money to cross miles seeking for job, but they lost a lot of chances due to their disabilities of talking the language of concerned countries. In the modern age, there are a number of countries request Ielts or Toefl for traveling procedures for example USA, and UK, as individuals can't travel or apply in the embassy without completing the test's requirements successfully, this one is approved by the embassy for English proficieny, and after they have done, they can continue to next steps to file up an application for the concernd embassy. In addition, this procedure is applied on education, if someone wants to continue master, BA degrees, and Phd.\nIn brief, language plays an integrate part in a country, if you don't speak it, you will encounter some difficulties, as it will lead to failure.\nNo of words: 476.0 315.596192385 151% => Less content wanted.\nUnique words percentage: 0.489495798319 0.561755894193 87% => More unique words wanted or less content wanted.\nSubordination: 11.0 2.10420841683 523% => Less adverbial clause wanted.\nHow many sentences: 8.0 16.0721442886 50% => Need more sentences. Double check the format of sentences, make sure there is a space between two sentences, or have enough periods. And also check the lengths of sentences, maybe they are too long.\nSentence length: 59.0 20.2975951904 291% => The Avg. Sentence Length is relatively long.\nSentence length SD: 250.831087338 49.4020404114 508% => The lengths of sentences changed so frequently.\nChars per sentence: 284.0 106.682146367 266% => Less chars_per_sentence wanted.\nWords per sentence: 59.5 20.7667163134 287% => Less words per sentence wanted.\nDiscourse Markers: 20.25 7.06120827912 287% => Less transition words\/phrases wanted.\nautomated_readability_index: 30.8 13.0946893788 235% => Automated_readability_index is high.\nflesch_reading_ease: 20.05 50.2224549098 40% => Flesch_reading_ease is low.\nflesch_kincaid_grade: 25.1 11.3001002004 222% => Flesch kincaid grade is high.\nlinsear_write_formula: 27.0 9.78957915832 276% => Linsear_write_formula is high.\ngunning_fog: 25.6 10.1190380762 253% => Gunning_fog is high.\ntext_standard: 26.0 10.7795591182 241% => The average readability is very high. Good job!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rangeland is defined by the Society for Range Management as \"Land on which the native vegetation (climax or natural potential) is predominantly grasses, grass-like plants, forbs or shrubs suitable for grazing or browsing use.\" While many areas fitting the definition of rangeland are not utilized by domestic grazing animals, nearly all are used by wild grazing animals of various kinds.\nTo be profitable, a forage-based livestock operation must first be ecologically stable. As numerous authors have demonstrated, the components of profitability are linked to another such that a decline in ecological integrity, perhaps bunchgrass health, affects animal health, marketable output per unit area, production potential next year, and consequently profit by decreasing revenues and increasing costs. Even this apparently mercenary approach to rangeland health functions to protect ecosystem function such that we maintain ecological goods and services that are more difficult to quantify and value, such as open space, clean water, wildlife habitat, soil stability, carbon sequestration, etc.\nThe two primary components in how livestock grazing affects native rangelands are stocking rate and animal distribution. Prospective and current graziers of native rangeland must recognize that these ecosystems produce significantly less total forage per acre than irrigated pasture and even dryland pastures in more mesic regions like the Palouse or Western Washington. this is typically excellent quality forage, but an acre of shrub-steppe near Ellensburg may only yield 900 lbs of total biomass annually. Of this, some is new growth on shrubs and should be considered unavailable to domestic livestock, and well-established guidelines for grazing bunchgrasses recommends limiting livestock to 50% utilization of the current year's forage production. Under this scenario, there may only be a few hundred pounds of dry matter forage per acre, which supports a cow for a week or two. Most damage to native rangeland results from poor distribution of grazing animals in either time or space. Overgrazing occurs when an individual plant is grazed before it has fully recovered from the previous grazing event. Managing the timing, frequency, and duration of livestock use is the key to maintaining healthy plants and healthy livestock.\nSee links on the side for publications on rangeland grazing.\nThe NRCS has made soil survey data available on their website. A landowner can delineate their property on a map and extract a wealth of information about their soils. Of note for livestock producers is a rangeland productivity report, ecological site descriptions which provide expected plant species by functional group, a good starting point for rehabilitation efforts.\n*There are many online keys to grasses, but almost all are local to regional in scope.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Planning for service delivery is a critical function of the local state in South Africa. Since 1994, the state has shown, in all its spheres, the will to improve people's living conditions. Within the domain of basic services, grand policies were crafted and supporting budgets allocated. However, more than 20 years on, widespread poverty and gross inequalities persist and protests over service delivery are escalating. This suggests 'failure' on the part of the state. Examples from elsewhere have demonstrated that the complexity of development can lead to the failure of programmes despite will and capacity (e.g. Li, 2006; Gupta, 1995).\nThis paradox calls for a better understanding of governance, politics and state practices at local level. Therefore, this research project proceeds from the premise that urban politics, governance and state practices can either enable or constrain effective planning and implementation of service delivery programmes. Building on earlier theoretical work on Understanding the \"state\" and \"state function\" in infrastructure planning and service delivery in the GCR (forthcoming GCRO Provocation), this project seeks to investigate the formulation and implementation of a water services plan in the City of Johannesburg. The aim is to understand how the interplay of rationalities of different actors within the city determines outcomes of water service delivery.\nHow is water service delivery understood in the city and why?\nWhat tradeoffs are made between competing priorities and why?\nWhat role does politics play in planning for water and why?\nIs there such thing as bureaucratic rationality?\nIncreasingly, the rationality of state projects in South Africa is being questioned and there is no sufficient justification for service delivery failure based on budgets. Are governance, politics and state practices to blame, and if so how?\nThe project makes use of an ethnographic approach. The assumption is that the state (in this case the City of Johannesburg) is a \"black box\" and what outsiders get to know as city policy; programme outputs and outcomes are mainly effects of governance, politics, and state practices. By focusing on details of individuals' everyday experiences in the city, ethnography can extract logics and meanings that these individuals attach to their actions. Hence, ethnographic detail helps to answer the \"Why\" questions. This means spending time in the City of Johannesburg, (approximately 5 weeks \u2013 a week in each of the key departments namely \u2013 the Mayor's office, the City Manager's Office, the Central Strategy Unit, the Budget Office, and the Department of Environment and Infrastructure).\nGCRO Project lead, Darlington Mushongera began his ethnographic fieldwork in June 2017. He is based at the City's Environment and Infrastructure Department (EISD), Water and Biodiversity Unit until December 2017. Part of the work has been presented at the RC21 Conference that was held in Leeds (UK) from 11-13 September 2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Slides for speech at HR Tech Expo by NCHRA on August 25, 2017. Based on research by Prophet \"HR as a Force for Digital Change\" available at https:\/\/goo.gl\/qu7rN3.\nDescription: Transformations are never easy, and the digital transformation is doubly so because of the technology angle. HR leaders must work the fine line between pushing executives and teams to be agile and change faster, while still enabling the organization to deliver on near-term objectives.\nWe'll examine the challenges and opportunities that digital creates, and the crucial role that HR leaders play in bringing about the transformation needed to help your organization thrive in the digital era.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As a part of its Special Subject Collection in Art History the Heidelberg University Library holds an important collection of historical books (late 15th through the early 19th century) on architecture and garden desgin, which we endeavour to make available on the Internet at no charge.\nThe collection of books on architecture \u2013 among them rare and precious illustrated editions from Italy, France, Germany and other European countries \u2013 includes architectural tracts and S\u00e4ulenb\u00fccher, meant as pattern books for the contemporary architects. The tradition of this kind of literature goes back to the Roman architect and writer Vitruv and his work \"De architectura libri decem\", which was rediscovered in the 15th century. Since its first printed edition in 1486 this work has gained great popularity allover Middle Europe and has had a decisive influence on the sacred and secular architecture. The collection includes important engraving series about architectural complexes and historical guides to church buildings.\nThe international collection of books on garden design includes impressive engraving series and pattern books about landscape architecture and horticulture from the Renaissance till the early 19th century. Related publications like herbaria, calendars, theses and monographs on agricultural law and the history of gardening are held, too.\nThe collection is recorded on Heidelberg online catalogue HEIDI , SWB (Union Catalog of the South West German Library Consortium) and ArtDicovery - Research gateway to art literature.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hello March! An exciting weekend in the KX96 Events Cruiser is ready to go!\nI've got another great weekend lined up in KX Country! Come hangout at some awesome events!\nI've got a great weekend lined up for you in KX Country!\nI have a great weekend lined up in KX Country!\nIt's the Labour Day long weekend and I've got all sorts of fun lined up for you across KX Country!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I just came from a meeting in Angola, Indiana where I preached last Sunday, Monday and Tuesday. In all the services was a little lady in her eighties. She sat there enjoying the services. I spoke to her two or three times. My wife spoke to her.\nHer husband, in his eighties, had died on Saturday. His body was at the funeral home in Angola. She came to church Sunday morning and came back Sunday night. She said, \"There's nothing to worry about. He's with the Lord. We'll have the funeral service Wednesday, after the meeting is over.\" She was in every single service, smiling and rejoicing in the Lord, presenting a beautiful attitude of the child of God.\nI thought to entitle this sermon \"Keep on Smiling\" or \"Keep on Rejoicing.\"\n\"These things I have spoken unto you, that in me ye might have peace. In the world ye shall have tribulation: but be of good cheer; I have overcome the world.\"\u2013John 16:33.\n\"Rejoice in the Lord alway: and again I say, Rejoice. Let your moderation be known unto all men. The Lord is at hand. Be careful [anxious] for nothing; but in every thing by prayer and supplication with thanksgiving let your requests be made known unto God. And the peace of God, which passeth all understanding, shall keep your hearts and minds through Christ Jesus.\"\u2013Phil. 4:4\u20147.\n\"Fret not thyself because of evildoers.\" \u2014Psalm 37:1.\nAll are saying the same thing\u2014be happy in the Lord; serve the Lord with all your heart. With trouble everywhere, still we are to be of good cheer. Have no fear. Trust in, rest in God's promises. Keep smiling. Keep a good attitude, a good disposition. Keep looking up, knowing that you can win through Him.\nWe will have lonely hours, hours of disappointment when we feel we cannot go on. But keep looking to the One who said, \"Be of good cheer; I have overcome the world.\" You may have difficulty physically. Keep looking up to God.\nYears ago in Chattanooga we had a little old lady in the church, a Mrs. Gentry. She became ill and was put in the hospital. Her leg had to be amputated. She had been a very active person. Now she came home with only one leg.\nI said, \"Mrs. Gentry, you can still use your life. You can use what you have. Don't turn away from serving God.\"\nI had her sing on my radio program. I called her the wheelchair singer. She became popular throughout Chattanooga. Mrs. Gentry's singing brought blessing to people all over that area. Though her body was suffering, she was serving God and rejoicing.\nAgain, when you fail miserably, keep smiling, keep rejoicing. Don't give up. Keep going.\nKeep going when your friends forsake you\u2014and sometimes they do. When some turn against you, keep smiling, keep rejoicing, keep looking up. Know that God is with you and that He will give you the victory as you trust Him.\nOne good thing that should come out of this Sword Conference would be hundreds of victorious Christians, overcomers. You will stand; you will be true; you will be faithful; you will go on with God and not quit, not give up.\nI once had a position in the Southern Baptist Convention. I was the evangelist for the Birmingham Baptist Association for three years. I had been an officer in the Convention. But I felt led to come out, so I did.\nA committee from the Sunday School Board in Nashville was sent to see me. These men begged me to stay in. I said, \"No, I cannot do anything else but this.\"\nI lost every friend (as far as I knew) in almost one week's time. I had nowhere to turn. My friends were in the Convention. When I came out, it all changed; but the Lord was still with me. And He will be with you too. Keep going when friends recede.\nThen keep smiling and rejoicing when the world opposes you because of your convictions.\nKeep rejoicing when loneliness envelops you. Remember, the Lord said, \"Lo, I am with you alway, even unto the end of the world.\" Keep rejoicing when old age claims you. You will come to such a time. I have. But keep rejoicing in the Lord and praising Him for His goodness.\nRejoice when death comes. Christian, remember: 'To be absent from the body is to be present with the Lord.' All is well. Keep rejoicing in the Lord.\nThe Lord may be saying to some of you right now, \"Rejoice, I am with you. Lo, I am with you alway: even unto the end of the age. I will never leave you, never forsake you. In the darkest hour, I will be at your side. In suffering, in illness, in persecution, in reverses, I will be with you every moment.\"\nCan you rejoice and say, \"The Lord is with me. He's walking by my side. He's encouraging me. I sense His presence with me all the time\"?\n\"And we know that all things work together for good to them that love God.\"\u2013Rom. 8:28.\nThe Lord speaks: \"Rejoice. Smile. Trust Me. Have faith in God.\"\nThe Lord never fails. We do, but He cannot. The world fails, but He cannot. By faith you keep going forward. Keep trusting Him. By faith attempt great things for God. Out of such a meeting as this, many should go home inspired to do something mighty for God.\nOut of a meeting like this I got the inspiration for starting Tennessee Temple University in 1946. One time our enrollment was over 5,500 in school. God blessed in a mighty way.\nWhen we lost our baby Joy, God led me to start Camp Joy for children. God supplied the property and the money. It's still going and with great success.\nThen we started Worldwide Faith Missions in the same way\u2013from one missionary to 560 on the mission field.\nRejoice in your darkest hour. The world may think you crazy, foolish; but keep trusting God, keep building your faith on the Word of God. Rest in that Book. Do what He says do. He will help you, lead you in the same way He led Moses, Elijah, Jeremiah, David, Peter, James and John and the Apostle Paul.\nAgain the Lord speaks: \"But my God shall supply all your need according to his riches in glory by Christ Jesus\" (Philippians 4:19).\nBe a faithful tither. Try Him. He will not fail you.\nI've been tithing now for more than seventy years. I gave up my salary a long time ago, and God has supplied every single need I've ever had. And He will keep on supplying what I need.\nPut away your worries and do what God says do. Friends may fail, the world will fail, but He\u2013never!\nI went to my first church in Memphis, Tennessee\u2013Germantown Baptist\u2013at a salary of fifty dollars a month. The church had been organized in 1865. I rented a room in the attic of a big house on Snowden Circle in Memphis, Tennessee. I ate meals at a hamburger stand. Hamburgers were a nickel apiece. What a day! I would leave that little room and go get a hamburger. I didn't have anything except an old Ford car that would barely run. It was hard going.\nOne morning I got out of bed in that little room in the attic and walked over to the door. Under the door was a white envelope. I thought, What in the world is that? I picked it up. Inside was a dollar bill, a whole dollar! I said, \"Thank You, Lord.\" I went down to the hamburger stand and ate two more hamburgers. I thought, This is living! This is it!\nI thought to myself, Now, if that would just happen again, that would be wonderful!\nThe next morning\u2013there it was. Every morning for twenty-five mornings I picked up a dollar bill. I was afraid to say too much, afraid it would stop. I let the thing go on.\nAfter awhile I thought, I'd better find out who is doing it. So I put a little trap by the door, a little piece of wood. If anyone touched that door, it would knock the wood over and wake me. I fixed it just right and went to bed and to sleep.\nSometime after midnight I heard that piece of wood fall over. I jumped up and jerked the door open. At the door stood a short little lady. In her hand was a white envelope. She handed it to me and said, \"Son, if you ever tell a soul, I'll never give you another penny.\"\nI said, \"Lady, have no worries. I'll never tell a soul where my money is coming from. Thank you, ma'am. And God bless you.\"\nThat lady was Mrs. Kate Barker, the wife of the former president of the University of Kentucky. After her husband died she moved to Memphis and lived with her sister. I had the room in the attic. She thought, That boy needs help, and I want to help him. So she began putting those dollars under the door.\nYou trust God, you have faith in God, and He will see you through. If you are led of God, inspired by the Holy Spirit to do what you are doing, God will bring you through victoriously on the other side. You can depend on that.\nMy second church at Greenbrier, Tennessee paid me a bit more, seventy-five dollars a month. I had a tiny little room on the back of the church and lived there for three solid years\u2013no conveniences, nothing but that little room. And God supplied every need.\nI kept on trusting God. He did not fail me. And He won't fail you.\nHe will be with you in the darkest hour of your life, and dark hours will come.\nI was holding a tent meeting in Russellville, Alabama. Homer Britton was my singer. Crowds were coming. I was in the pastor's office one morning preparing for the service when the phone rang. I answered. The voice on the other end said, \"Brother Roberson, I'm sorry to tell you this, but your baby Joy has just died.\" I didn't know how to respond.\nI got in my car and started driving back to Chattanooga. I didn't understand why that little one had gone so suddenly, why a phone message like that: \"She is dead.\"\nWhen I crossed the Tennessee River bridge, God gave me a verse, which is now my life verse: \"And we know that all things work together for good to them that love God, to them who are the called according to his purpose\" (Romans 8:28). I don't understand the death of our baby; neither does my wife\u2014just a few days old and suddenly gone. But out of that came Camp Joy for children, and many other things.\nKeep on rejoicing when heart-aches come. He will take care of your every difficulty. Trust Him. \"Rejoice in the Lord alway: and again I say, Rejoice.\" He will be with you in the darkest hour.\nIn times of loneliness \"be strong and of a good courage; be not afraid, neither be thou dismayed: for the Lord thy God is with thee whithersoever thou goest\" (Joshua 1:9).\nWhen sickness comes, He is with you.\nI preached in a big, open-air tabernacle near Chicago one night. We had a great service. I went to my room in a little motel down the side of a hill. I got up the next morning and tried to pray, but my voice was gone\u2013every bit of it. I had to leave the meeting.\nI came back to Chattanooga and went to several doctors. They tried to help but couldn't. For almost one year I carried that burden. Others preached for me. I stayed in my position and tried to work, but I couldn't preach. In that year I had to feel the presence of God, had to know that He was with me, had to know that He was guiding and directing in my life.\nAlmost a year later, God restored my voice and allowed me to go on. One year waiting on God! I was pastor of a church and seeing to Tennessee Temple\u2013all of it without a voice!\nDuring that trying period I had to trust Him and tell Him, \"Lord, I'm in Your hands. Take care of me.\"\nWe all fail at times. Don't give up. \"Demas hath forsaken me, having loved this present world.\" Don't be a Demas. Keep going. Your sermon may not bring results; your visitation program may not bear much fruit, but keep going. Revival may not come, but keep trusting God, keep moving on.\nSmile and rejoice, for He said He will use us to help others.\nI have my greatest joy in remembering when God saved my soul. But He saved me that I might tell somebody else, that I might help somebody else.\n\"This one thing I do, forgetting those things which are behind, and reaching forth unto those things which are before, I press toward the mark for the prize of the high calling of God in Christ Jesus.\"\u2013Phil. 3:13,14.\nHe wants to use you to bring comfort and encouragement to others. He wants to use you to give words of warning about what one can and can't do with his life.\nLift up Christ as Saviour, and He will show you how you can point others to the Son of God. How true are these words: \"I can do all things through Christ which strengtheneth me.\"\nThat I may live like Thee!\nWinning others! Pointing people to the Lamb of God! Letting God use your life! You're a child of God; now let Him use you to bless the hearts of others.\nI smiled when a little lady came up in our church in Chattanooga one day\u2013a stranger. She said, \"Brother Roberson, I'm here visiting with you. I have given my life to the matter of winning people to Christ.\"\nI said, \"Ma'am, are you doing any good? Are you succeeding?\"\nShe turned from the Bible, opened a notebook and said, \"Yes, I have had the joy of leading 1,526 people to Christ.\"\nDid you get that? 1,526! She had all their names and addresses written down in the book. That little lady had gone from place to place, city to city, witnessing for Christ.\nI thought, Boy, this is a little too much! I'm not sure I can believe it. But I believe it now. She was deeply sincere and had given her life to the one great purpose of pointing others to the Lamb of God.\nSo should you, and so should I. We should say, \"O God, my business is to win people to Christ.\"\nWhen I preached one Sunday morning in Chattanooga, a fellow came down the aisle and was saved. I baptized him. His wife was baptized later. This man, M. J. Parker, came to me and said, \"Brother Roberson, I know I'm born again. I want God to use me. Would you mind if I go out and rent an old bus and run it around Chattanooga to bring people in to Sunday school?\"\nI said, \"Go right ahead.\"\nHe rented an old bus and began. We later owned thirty-five brand-new buses and filled them for every service\u2013Sunday morning, Sunday night and Wednesday night. (I've never had a junior church and never intend to have one. Everyone came into the church service, and thousands were saved.) M. J. Parker\u2013the man who was inspired to do that\u2013started me off when he felt this could be done.\nThen M. J. had a stroke. He was paralyzed on one side for a long time and had to walk with a cane. Then he passed away. I held the funeral. At the graveside the wife was standing there with tears in her eyes. She pulled me aside and said, \"Brother Roberson, may I speak to you just a moment?\"\nShe said, \"He never told you, but M. J. never finished the third grade. He could barely read and write.\" Yet he was a smart businessman and a great soul winner. God put His hand upon him.\nIf M. J. Parker could lead thousands to the Lord in Chattanooga, then God can use you as well, if you will just say, \"Lord, I'm ready. I will rejoice.\"\nYesterday in my office, I talked to Solomon Owolabi, a graduate of our school. He has been back in Africa for many years. He talked about his work. He has built three churches in Africa\u2013beautiful, big churches with thousands being saved. He said, \"I want to thank you for what you have done in helping me to get back to the mission field. I'm now on furlough and will be going back.\" He was winning people to Christ in Africa.\nRejoice that God can and will use you. Let Him have His way. Ask Him to take your life and use it for His glory, for the winning of souls.\nI was preaching in Springfield, Tennessee. A lot of people got saved. One lady came forward often, bringing people down the aisle. I asked who she was. They told me she was Mrs. Abi Burr, cashier for the First National Bank in Springfield. Every service she won people to Christ.\nI went back home and didn't hear from that church nor Mrs. Abi Burr for a long time. Then one day the phone rang. \"I'm calling you from Springfield, Tennessee. Mrs. Abi Burr just died. She left word that she wanted you to have a part in the funeral service.\"\n\"I'll be there,\" I said.\nI went to the First Baptist Church in Springfield and sat on the platform with the pastor, Dr. Pettigrew. They had the songs. A big crowd was there. I had my word to say and sat down. Then the pastor stood and said, \"Before I bring the message, I'd like for you ladies to stand here at the front, if you will.\" A group of ladies stood. He said, \"These ladies were led to Christ individually, one by one, by Mrs. Abi Burr.\" I counted them. Fifty-two women were standing at the casket of Mrs. Abi Burr.\nShe worked at the First National Bank, went to a little mission church, knocked on doors and told people about Christ, including these fifty-two ladies.\nGod will use anyone yielded to Him. Let Him use you. And don't be afraid. Rejoice and praise God. You can be a soul winner.\n\"Let not your heart be troubled: ye believe in God, believe also in me. In my Father's house are many mansions: if it were not so, I would have told you. I go to prepare a place for you. And if I go and prepare a place for you, I will come again, and receive you unto myself; that where I am, there ye may be also.\"\u2013John 14:1\u20143.\nI believe in the second coming of Christ. I am a premillennialist. I haven't changed my viewpoint in all these seventy years. I believe that Christ may come at any moment. I have to rejoice in that fact\u2013that He has promised and will fulfill that promise. I will see Him and will stand in His presence. I believe in the coming of the Lord Jesus Christ, the resurrection of the saved, the rapture of the living believers, caught up into His presence!\nThis came to me when I was pastor of my second church in Greenbrier, Tennessee. Sitting in the little room in the back of the church where I lived, I began reading the truth of the second coming. I read: \"I will come again\u2026.\" I preached my first sermon on it in that church. I didn't know anything about terms like premillennial, postmillennial, amillennial, etc. I had never seen a Scofield Bible at that time. But I knew He was coming, for He promised He would. I began preaching it, and people began getting saved. I saw what could result from preaching on the Second Coming, and I have kept preaching on this truth all these years.\n\"Be of good cheer; I have overcome the world.\"\nOh, what a mighty, mighty Saviour we're following! He's our Saviour, our Keeper, our coming King; He's all and in all. Rejoice that you are His child! And in rejoicing, ask Him to use you to tell someone else about Christ. Obey the Great Commission. Do what He says with your life and do it rejoicing and praising God for His presence, for His power, and for His peace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Monday 2 February, First Vice-President Frans Timmermans attends in Riga the Meeting of the Chairpersons of COSAC (the Conference of Parliamentary Committees for Union Affairs of Parliaments of the European Union). Participants in the meeting will discuss the priorities of the Latvian Presidency of the EU, prospects for the EU after the election of the new European Commission, and the Eastern Partnership. First Vice-President Timmermans is responsible for strengthening the interaction of the Commission with national Parliaments.\nPierre Moscovici, European Commissioner for Economic and Financial Affairs, Taxations and Customs, will be in Berlin on Monday, 2 February and Tuesday, 3 February 2015. He will be meeting with Mr Wolfgang Sch\u00e4uble, Minister of Finance; Mr Sigmar Gabriel, Minister for Economic Affairs and Energy, and Vice-Chancellor; and Mr Frank-Walter Steinmeier, Minister for Foreign Affairs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"According to the 2012 Indiana Airports' Economic Impact Study, general aviation airports in Indiana support 32,132 jobs and a payroll of about $1.7 billion.\nAccording to the Indiana Department of Transportation, there are 117 public-use airports and 5,003 registered general aviation aircraft in Indiana. According to the FAA, there are 9,878 pilots in the state.\nAccording to the 2012 Indiana Airports' Economic Impact Study, general aviation airports in Indiana support 32,132 jobs and sustain a payroll of more than $1.7 billion.\nThe same study estimates that the total annual economic impact of general aviation airports to Indiana's economy is more than $5.8 billion.\nThe FAA reports that Indiana is home to 54 repair stations, 12 FAA-approved flight schools, 2,435 student pilots and 1,728 flight instructors. There are 74 fixed-base operators in the state, according to the AC-U-KWIK directory. According to Indiana Department of Transportation, there are 118 public- and private-use heliports in the state.\nAccording to the University Aviation Association and the Indiana Department of Transportation, flight departments in Indiana include Indiana State University in Terre Haute, Purdue University in West Lafayette, and Vincennes University in Vincennes.\nAdditionally, manufacturer Rolls Royce has facilities in Indiana where general aviation aircraft engines are produced, and GE is also manufacturing turbine engines in Lafayette, Indiana.\nAcross the United States, governors and mayors from all 50 states have signed proclamations recognizing the importance of aviation industry. See what has passed in Indiana, choose from the list below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbioeh b/data_all_eng_slimpj/shuffled/split2/finalzzzbioeh
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index 0000000000000000000000000000000000000000..252aa51b9664190aef2cc95f9822e7ae26539008
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@@ -0,0 +1,5 @@
+{"text":"You should buy five of them.\nIt's only been out a short while but the verdict is unanimous: Pundits love the Google Pixel! Mostly as a tool to bash Apple over the head.\nThere are so many ways. This is just one.\nThis week didn't look good for Apple.\nDoes any week ever look good for Apple when you're drinking the hallucinogen-infused water that comes out of the Business Insider drinking fountain?\n\u2026largely because of the phone's new digital helper called Google Assistant.\n\u2026Google's AI is too stupid to be meaningfully helpful at this stage.\nThat sounds super embarrassing for Apple.\nKovach touts Assistant's superiority over Siri and provides several examples from his review that convinced him of this. The thing is, the Macalope threw roughly the same queries (changing for location differences) at Siri on his iPhone SE and they all worked. So where is this supposed advantage?\nThe point is that as a whole, smartphone assistants aren't smart enough to be defining features. They're gimmicks, and Google unwisely decided to build a phone around one.\nHaven't you heard? It's awesome. It does those things Siri, uh, apparently also does.\nMaybe this is true, it certainly seems to be the conventional wisdom, but Kovach hasn't shown it to be. Marques Brownlee's testing shows that Assistant's advantage is in context awareness but Siri often gives more information (and gets more contextual cues than the Macalope expected). Brownlee gives a slight nod to Assistant, but the difference is not that drastic.\nMost importantly, however, AI isn't the only reason one buys a smartphone. The iPhone 7 is water-resistant, the Pixel is not. The iPhone 7 looks like an iPhone 7 while the Pixel looks like an iPhone 6 knock-off. Strangely (disclosure: not strange at all, completely expected), Kovach does not mention privacy concerns anywhere in this piece, possibly because such hand-wringing is the province of only the weak-kneed and those who have something to hide.\nBesides, it's not like Google's going to use your personal information for anything.\nWe'd all like to see Siri get better. But Google's advantage here is somewhat exaggerated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This page shows examples of Edwardian plaster coving. These were typical in average and grand houses. More modest homes in either the town or country would usually have more simple versions.\nThe use of additional decorative motifs was less common than in Victorian houses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vision Aviation Racing (VAR) driver, Dusty Davis, brought the Vision Airlines\/Emerald Coast Resorts Toyota Tundra home 25th in his Martinsville Speedway debut.\nDavis qualified 19th for the Kroger 250 and quickly made his way to the front after the green flag fell. Up to 6th position on lap 60, Dusty was happy with his truck and told Crew Chief Doug Howe over the radio, \"This truck is awesome!\"\nDavis fell back in the pack after pit stops but returned to the top-10 by lap 165. On lap 204, contact from the 30 truck sent him spinning into the turn 3 wall. Davis was forced to bring his truck down pit road for repairs, losing 5 laps.\n\"My truck was running awesome today. We had a really good base line setup. We made it up to the top-10 and passed a lot of trucks today. With about 50 to go the 30 decided to knock me into the fence. Some drivers respect you and some don't but I guess that's racing. It's unfortunate because of the owner's points and I am sure this won't help us out,\" commented Davis after the race.\nThough this is not the result Davis or the No.15 team hoped for they have a lot to be excited about. Davis took immediately to a half mile track that usually frustrates rookies. Loop statistics ranked Davis 4th overall in green flag passes and 5th overall in quality passes made in the top-15.\n\"It was not the finish we wanted but it shows improvement. Doug Howe and the guys did a great job. I really had a blast driving here and can't wait to come back in the fall.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For petitionary prayer, see supplication.\nIt has been suggested that right to petition be merged into this article. (Discuss) Proposed since June 2014.\n\u2191 1.0 1.1 1.2 Brook, Timothy (1999). The Confusions of Pleasure: Commerce and Culture in Ming China, p. 33. University of California Press. ISBN 978-0-520-22154-3.\nThis page was last modified on 5 December 2015, at 13:47.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Toronto Raptors have reportedly become \"increasingly confident\" in their ability to re-sign Kawhi Leonard during the 2019 offseason, according to Josh Lewenberg of TSN.\nThe forward can become a free agent in July if he declines his $21.3 million option for next season.\nWhile the Los Angeles Clippers and Los Angeles Lakers are considered potential suitors for Leonard's services, the Raptors believe they can convince him to stay.\nPer Lewenberg, the Raptors' pitch will be about \"trust, familiarity, a commitment to maintaining his health and the shared goal of chasing a championship, as well as the extra year and contract worth nearly $50 million more than anyone else can offer.\"\nToronto will be able to offer a max contract of $190 million over five years, while another team would only be able to offer $141 million over four years.\nMoney aside, things have seemingly gone well for Leonard in his first season with the team since an offseason trade from the San Antonio Spurs.\nThe 27-year-old is averaging 27.2 points and 7.5 rebounds per game, both of which would be career highs if he maintains it for the rest of the year. He has also helped the squad to a 51-23 record entering Monday, good for the second-best in the NBA behind only the Milwaukee Bucks.\nWhile the postseason could potentially determine his enjoyment this season, the Raptors have seemingly done as much as they can to keep his spirits high so far.\nConsidering Paul George re-signed with the Oklahoma City Thunder after it was highly speculated he was headed to Los Angeles last offseason, Toronto shouldn't count itself out in the Leonard sweepstakes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblraq b/data_all_eng_slimpj/shuffled/split2/finalzzzblraq
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index 0000000000000000000000000000000000000000..b3c953acda185db53f431b1ec100ce2924a34097
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@@ -0,0 +1,5 @@
+{"text":"See more What others are saying \"Use eos lip balm after to make all the dead skin peel off and it makes them feel so good \ud83d\udc96\" \"If you have ever taken a stroll down the skin care aisle at any major store, you have likely been confronted with an astounding number of options.... A study found a fifth of Britons are not confident enough to wear colourful clothes or make-up. Yellow is the colour they most fear wearing, followed by pink, orange red and green.\nI'm not afraid to wear colors like this orange color. I liked how it's balanced with the tan. Goes well with jeans!... It is not up to me to give you a specific answer because I'm not you. Fear of make-up could possibly fear forming an addiction to the effects\/enhancements of make-up. Maybe doing research on the adverse effects of ingredients in make-up caused this fear as well.\nIt may be tempting to wear heavy (and dark) eye liner and eye color, but Can create dark circles around the eyes. Type\/Length of clothes: Dressy casual is best, but choose a look that really suits you, so that you are comfortable during the shoot.... 24\/11\/2018 \u00b7 Ask other invited guests for tips if you're not sure what to wear. If there's no dress code, or you're confused by the dress code, don't be afraid to ask for some help. Ask one of your friends that's going what they plan on wearing.\n19\/08\/2008 \u00b7 I know all too well not to wear blue (crips), red (bloods), yellow (Latin kings), or black (latin kings as well?). The kids I volunteer with in an inner-city gardening program have said that my brown and pink bandannas are OK.\n24\/11\/2018 \u00b7 Ask other invited guests for tips if you're not sure what to wear. If there's no dress code, or you're confused by the dress code, don't be afraid to ask for some help. Ask one of your friends that's going what they plan on wearing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Tree collection of purses and wallets are a consistently popular and timeless choice. The Card Case has been designed to fit unobtrusively in your bag, keeping a small selection of cards together in a streamlined and elegant way. Available in classic Black, or new season combinations Sea Blue and Jungle Green, Hibiscus and Mandarin, and Midnight Blue and Sea Blue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Was scratched on due to tying up the day before. Congrats on wins at Regional Show. This young stallion is going to be a force to be reckoned with at the Congress and World Shows. I think it is already very interesting that Grevel seems to feel the case for Williams as a poet so strongly. Should be a good stakes horse for you as a 3 year old after she grows some more.\nWe wish you the best of luck with him in the upcoming show season!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A dry wall is built with no mortar. The stones are carefully selected and they interlock and gravity holds them in place. The Great Wall of China was built this way. Scott Carrier's wall was built this way too. Here is his story.\nVery interesting\u2026I tried to build a wall once and developed a real love of rocks. The Flatirons in Boulder have a similar history, hard to comprehend but thus the joy and interest. Thank you\u2026fun to stir up memories.\nNice episode; Scott's voice is like Bob Ross' for me.\nWhen can we expect the next Heilman-produced episode?\nErica, Another example of your uncanny ability to find and frame beautiful, interesting people and stories. Well done.\nI am such a big fan of yours Scott. I could listen and listen and listen\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Book Introduction: Do you have a love for computers? Are you the person your parents go to when they can't figure something out on their devices? Then computer programming might just be the career path for you! Learn the basics from a real-life expert and get some hands-on experience. The world of computer programming is at your fingertips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmfyg b/data_all_eng_slimpj/shuffled/split2/finalzzzbmfyg
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index 0000000000000000000000000000000000000000..e8479d15c24941003ee6e77a53cf4315fe6c51b7
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"So I arrived back home with lots of boxes and a fairly unorganized personal file, aka all my personal documents, insurance, certificates, bank statements etc. As part of my 101 list, I needed to get this file working again!\nSo I went out to WHSmith and purchased two expanding files. It was buy one get one free so seemed appropriate to stock up. Is it weird that I call them accordion files?!\nI wrote down all of the labels I wanted to make and detailed what would go in them.\nAll finished and looks so much better. Now just to keep it that way!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is one low income housing complex in Lake Park, Florida.\nDue to the small amount of listings, we searched a 30 mile radius from Lake Park, FL. If we located any, we list them below along with the distance from Lake Park, FL.\nHow does income and housing costs in Lake Park compare with Florida income averages.\nIncome Limits in Lake Park, Florida.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"No! Certainly not! Never ever! Non! Nichts! Niet!\nRead what follows very carefully: \"You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP. You should never install an old backup of the database over a new version of SPIP.\"\nGo figure why, but one day someone announced on the SPIP mailing list that it should be done, and despite all our efforts to expose the contrary, there are still some users who are convinced that an old backup of the database should be re-installed when a new version of SPIP is installed.\nIt is a fatal error, worse than a virus in an Outlook mail client. Your computer will catch fire, your hard disk will melt, your host will be bankrupt, the Inland Revenue will clamp down on you and you'll be barred from using the Internet for life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you ready for the holidays? Because they're ready for you, and so are Seattle's best shopkeepers, designers, and artisans.\nThe Henry Art Gallery, PopCap Games, and Seattle Parks Foundation call on stylish event designer Cori Ready when they're ready to celebrate. We gave her a bottle of champagne and asked her to open up about hosting at your house.\nWhen hosting, it's important to make grand, sweeping, physical gestures. You have to be able to give big hugs, clean up spills, reach for large vases, and dance! Go for a forgiving hemline or a knit blazer rather than an unyielding structured one. But do wear your most glamorous jewels and gems or your flashiest watch. If anything goes missing, you'll only have your own couch cushions to dig under. And be the one who overdoes it. Crystals and silk make throwing parties seem chic and easy.\nNever Not Glamorous Pro partier Cori Ready says style, humor, and a pile of bling are a holiday host's best accessories.\nDecor should reflect your whole person. I don't know about you, but I can't go a day without tacking notes from my shaman or bestie's daughter's artwork on the refrigerator door. Don't feel like you have to sweep it all under the rug. In palmistry, your left hand shows what you were born with while your right hand shows what you've done; especially during the holiday season, your home should reflect where you've been and where you're going.\nThe tie that binds Seattle and the season: plaid. Dress the tables in satin tartan, drape Pendleton blankets everywhere, and wear your best grunge-era flannel with tuxedo pants and motorcycle boots. Serve smoked salmon, local cheese, and Irish soda bread with Kerrygold butter, and cans of Hilliard's Beer with locally made whiskey from Oola.\nNutcrackers are a bizarre thing, but they're actually pretty cute and they yell \"holiday.\" Create a centerpiece with wooden bowls of nuts and traditional specimens, and play The Nutcracker score of course.\nCumberland Island is the iconic spot where JFK Jr. married Carolyn Bessette in that knockout Narciso Rodriguez dress. It's also your inspiration for a summer-in-winter, beach-themed bash. Decorate the tabletop with vintage Japanese fishing floats and driftwood with twinkling holiday lights, and serve roasted pig and pomegranate cocktails\u2014and wear a bow tie.\nOr opt for a paparazzi-themed dinner in which only your most connected friends are invited. These days the thing is to tell guests to turn their phones off, but here they have to keep them on. Assign a hashtag, make place cards with Polaroid photos, create a backdrop for selfie sessions between courses, only serve photogenic food, and set up a device-charging station.\nThe Geometry of Pasta ($25), Mario's. Handmade hardcover blank book ($23), Paper Hammer. My Wonderful World of Shoes ($16), and A Coloring Bookby Andy Warhol ($20), Mario's. Brass, copper, and rope necklace by Seattle's Iacoli and McAllister ($285), Iacoli and McAllister. Telephone-wire bowl ($42), SAM Shop. Typeface memory game ($25), Kikkerland gemstone crayons ($9), and Type Riot desk calendar ($15), Paper Hammer.\nNot long after I started Marigold and Mint, my father and stepmother commissioned a porcelain marigold from downtown Seattle's Far 4. The store's in-house, family-run porcelain flower line didn't include marigolds\u2014but somehow co-owner Jenny Klimenkoff arranged for one.\nMy husband, Kyle, gave me a stunning bright-pink rubellite from Seattle Diamonds and a note that said I could design my dream ring with Nicole Pendergast, one of their gemologists. After an initial idea, wax castings, and revisions, it's now my favorite piece of jewelry.\nMy mom gave me an orange Le Creuset five-and-a-half-quart French oven that I use all the time for soups, quinoa, to boil corn\u2014it has a million uses and is just bulletproof. I suspect it will well outlast me.\n\u2014Greg Lundgren, entrepreneur, artist, and more, Lundgren Monuments, Vito's, the Hideout, et al.\nI got a goodie package from a friend that included a mini tin from Jacobsen Salt Co. I hadn't known that there were purse salts but Karyn Schwartz at Sugar Pill was carrying them, so I went there and bought the rest. Now you can also get them at the counter at Oddfellows. Whenever I pull the tin out at restaurants and tell the server that it's the best gift I've ever received, my husband gets kind of mortified.\nA friend gave me Patricia Urquiola's Orloff Bowl for Alessi, which I had been eyeing at Peter Miller Books. I use it every day as a valet bowl for my keys and wallet; the trifecta of useful, affordable, and great looking.\nMy wife, Kate, and I travel but don't always explore Seattle, so we got a flying lesson with Rainier Flight Service. We flew from Renton to Snohomish and back, and we each took a turn in the pilot seat. It was an incredible way to see our city and learn the basics.\nHere's what local retailers are excited about this season.\nErin Krohn's brand-new Hammer and Awl (3315 E Pike St, Madrona, 206-992-7444; hammerandawl.com) is for guys who balance the rugged outdoors with velvet lapels and craft cocktails.\nHouse favorite Form Function Form's custom leather bands and button-stud clasps give Timex Weekender watch faces a modern twist.\nWhen curious crafters and wannabe DIYsters wander into Ugly Baby and La Ru (1430 Western Ave, Pike Place Market, 206-696-0089; uglybabyandlaru.com), the shared studio space and showroom of Lauren Rudeck and Rosalie and Douglas Gale, things just get marvelously curiouser and curiouser.\nand sarcasm\" and are ready to give.\nGive an experience Sign someone up for the hand-sewing class on November 30; they'll go home with new skills and a tiny stuffed friend from Threadfollower's prepackaged supplies.\nThoughtful dressers and mindful accessorizers from all over the globe type in Craft and Culture (craftandculture.com) and add earthy, edgy, independent designers to their wardrobe from the web store hubbed at Hana Ryan Wilson and Jason Parker's Cap Hill HQ.\nClick it if you can Ovate's hoodie is a cult favorite.\nEven the world of chutes, ladders, and windup gizmos is subject to trends, but at Clover Toys (5333 Ballard Ave NW, Ballard, 206-782-0715; clovertoys.com), Sarah Furstenberg keeps it classic.\nGo Fish The card game Frog Juice hooks kids from six to 16\u2014and their parents too.\nShopper's tip: Make yourself at home inside Click! Design That Fits (4540 California Ave SW, West Seattle, 206-328-9252; clickdesignthatfits.com). It feels\u2014in the best way\u2014like John and Frances Smersh's overaccessorized front room.\niconic, thanks to the famous illustrator's familiar bird patterns.\nFor Seattle archetypes iPad covers and wallets made from recycled inner tubes by the locals at Alchemy Goods, and the voice recorder and sampler by Brooklyn's Brand New Noise.\nWill teenagers with video game-induced Carpal Tunnel be transformed by fold-out maps, 16-sided dye, and plastic game pieces? There's one way to tell; bring them in for a test drive at a game night (Saturday from 7 to 11) at Scott Cooper's Blue Highway Games (2203 Queen Anne Ave N, Queen Anne, 206-282-0540; bluehighwaygames.com).\nHouse favorite The 2013 winner of the game world's equivalent of an Oscar, Hanabi is all about deduction, cooperation, and\u2014get this\u2014fireworks.\nFor short attention spans Roll the dice Yahtzee-style to control Tokyo in this strategy game that takes just 30 minutes to complete.\nThere's more than wedding dresses behind those blue-flocked windows at Luly Yang Couture (1218 Fourth Ave, Downtown, 206-623-8200; lulyyang.com).\nSeasonal specialty Shawls, dresses, and cardigans in basic shapes but not-so-basic cashmere from the first-ever Luly Luxe collection.\nFor the bling impaired Crystal-studded chandelier shapes for ears and necklines\u2014think Seattle Swarovski.\nLocal partiers share their best-bet host gifts.\nSeattle artisans spread distinctively stylish goodwill.\nFind Graham and her work on sale every Saturday and Sunday in December at her South Seattle open houses from noon till 7 (10847 Third Ave S, Top Hat, 206-763-3108; akikospottery.com).\nThis article appeared in the November 2013 issue of Seattle Met.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The dysregulation of immune checkpoints by tumors is an important mechanism of immune resistance, as administration of checkpoint inhibitors has resulted in impressive clinical responses in patients with late stage cancers. However, a subset of patients exhibits insufficient or no clinical response presumably due to the absence of tumor-specific cytotoxic T lymphocytes (CTLs). This supports the rationale to combine checkpoint inhibitors with therapeutic platforms that generate effector T cells.\nWe have previously shown that the lentiviral vector platform, ZVexTM, generates high levels of tumor antigen-specific CTLs that are critical in mediating protection against tumor challenge in mice. Protection was enhanced by subsequent intramuscular injections of recombinant tumor antigen protein with GLA-SE (formulated glucopyranosyl lipid A, a synthetic TLR4 agonist that is the central component of the GLAASTM platform) or intratumoral injections of G100 (GLA-SE without protein antigen). ZVex and GLAAS are thus complementary platforms capable of generating tumor-specific immunity through the in vivo induction of antigen-specific T cells. Here, we evaluated whether checkpoint blockade could further enhance anti-tumor immunity induced by ZVex and\/or GLAAS in mouse tumor models.\nIn C57BL\/6 mice, anti-PD-1 or anti-PD-L1 \u2013 but not anti-CTLA4 \u2013 enhanced a ZVex\/mTRP1-induced CD8 T cell response. To determine whether checkpoint inhibition impacted therapeutic efficacy, B16F10 tumor-bearing mice were immunized with ZVex\/mTRP1 followed by weekly administrations of anti-PD-1, anti-PD-L1, or anti-CTLA4. Mirroring the immunogenicity results, ZVex\/mTRP1-induced anti-tumor protection was enhanced by the addition of anti-PD-1 or anti-PD-L1, but not anti-CTLA4. This is consistent with previous studies in which we demonstrated that the magnitude of antigen-specific CTL response correlated with the degree of anti-tumor protection. Additionally, compared to either anti-PD-1 or anti-PD-L1 alone, the combined administration of both checkpoint inhibitors best improved anti-tumor efficacy induced by ZVex\/mTRP1, or G100, or the combination of ZVex\/mTRP1 and GLA-SE\/TRP1 protein. As PD-L1 is not the only ligand for PD-1 and PD-1 is not the only receptor for PD-L1, these findings suggest that inhibition of signaling through either PD-1 or PD-L1 alone may not be sufficient to completely block the PD-1\/PD-L1 checkpoint.\nTaken together, combining checkpoint inhibitors with ZVex and GLAAS capitalizes on the strength of each therapeutic platform. ZVex and GLAAS efficiently generate the effector T cells needed for an effective anti-tumor response. By blocking active immune checkpoints, checkpoint inhibitors further enhance ZVex- and\/or GLAAS-induced anti-tumor immunity. Our findings support the combination of checkpoint inhibitors with ZVex and\/or GLAAS in clinical trials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbneqd b/data_all_eng_slimpj/shuffled/split2/finalzzzbneqd
new file mode 100644
index 0000000000000000000000000000000000000000..0eac8e42e73274cae4f24944154a4665b4a6af50
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@@ -0,0 +1,5 @@
+{"text":"Several people have been injured by gunfire in the Thai capital, Bangkok, amid clashes involving anti-government protesters ahead of Sunday's elections.The violence erupted during a stand-off between supporters and opponents of Prime Minister Yingluck Shinawatra.\nThe shots were fired as demonstrators blockaded a building where ballot papers are being stored, in an attempt to prevent their distribution.Protesters want the government replaced by an unelected \"people's council\".\nThe opposition has vowed to boycott Sunday's poll, which is likely to be won by Ms Yingluck.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2007 CRUISIN' FOR A CURE OFFICIAL DVD IS OUT NOW !!! ORDER NOW !!!!\nDid you get a chance to go to this year's Cruisin' For A Cure in Costa Mesa, California?? If you didn't get to go or just want to relive all the excitement, check out this year's Official DVD.\nGreat Gift For the Holidays!!!! Only $19.99 + s\/h ORDER NOW !!!!!!!!!\nOrder your copy now of the 2007 Cruisin' For A Cure Car Show Official DVD.\nThis DVD captures over 3500+ hot rods, trucks, custom cars, rat rods and classic rides from all over the United States.\nThis show is dedicated to the prevention of Prostate Cancer and raising awareness for the cause.This is the World's largest one day charity car show!\nCruisin' For A Cure is a registered non-profit California organization. All of the individuals that organize the show, handle show day logistics, and generally help make this event a success, are doing this voluntarily in addition to their \"day jobs\". Unlike many other automotive fundraising events, none of the money goes to a car club, promoter, or \"administrative overhead\"\u2026. There are none!\nThus, you can be assured that your donation goes directly to prostate cancer research. That is why we have all participants make their entry check out to the UCLA Jonsson Cancer Center. We simply log in the vehicle information for the show and forward the check to the UCLA Jonsson Cancer Center.\nProstate Cancer will strike over 189,000 men this year alone. Of these men, over 31,000 will die. That's one man every 12\u00bd minutes! Husbands, fathers, sons, and brothers\u2026..in many cases it could have been prevented with regular precautionary checkups.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Courtyard Miami Beach Oceanfront is a Hotel 3 stars in Miami Beach features 263 rooms whose rates of stay varies from a minimum of 159 USD to a maximum of 493 USD for night.\nIt 's located in 3925 Collins Avenue in the neighborhood Oceanfront. It is distant 3,6 mi from Fisher Island Club, 8,3 mi from Miami Jai Alai, 9 mi from Sunny Isles Beach City Hall, 9 mi from Miami Dade County Public Library, 10 mi from Golden Beach City Hall and other places of interest in Miami Beach.\nBook a room today in Hotel Courtyard Miami Beach Oceanfront in Miami Beach and even receive a great discount.\nOur hotel's broad service Snack bar is a terrific spot to encounter with friends and household for a late evening drink or to make several new buddies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When working in ministry, it is extremely easy to have conflict arise in the workplace. It can also feel like the easier route to push those conversations out of your mind and forget that they ever happened. This class will look at ways to keep conflict from going underground and keeping short accounts with each other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Year 11 students will be visiting Cragg Brook, Mytholmroyd, Calderdale on either Wednesday 3, Thursday 4, or Friday 5 October. This visit is to allow Year 11 students to carry out the data collection element of their physical geography investigation, on which they will be directly assessed in Paper 3, June 2019. This is a compulsory part of GCSE Geography. As always, a full risk assessment will be completed before the day to ensure students are safe throughout the visit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbocwf b/data_all_eng_slimpj/shuffled/split2/finalzzzbocwf
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@@ -0,0 +1,5 @@
+{"text":"I just bought a new HP router because my old router was dropping my Nest thermostat and other services. I have MTS modem and I unplugged it and took out the battery to cycle it (several times, actually) but the new router can't communicate yet. Do I need to reset the modem or something?\n1. Position your HP router away from obstacles and appliances that might be causing interference if you are unable to establish or maintain a strong Internet connection. Keep HP router away from cordless phones, metal cabinets, aquariums, refrigerators, microwave ovens etc.\n2.Examine the Ethernet cables for your modem and HP router if you are unable to establish a connection between your computer and the router. If the ethernet cable is frayed, busted then it may prevent you from connecting to the Internet.\n3.Try disconnecting your modem from the HP router if you experience problems accessing the Web-based setup page. By doing this you can resolve IP address conflicts if both the modem and router happen to share the same IP address.\n4.Clear your Internet browser's cache and browsing history if you are unable to access the HP router Web-based setup page. Sometimes full cache and history can prevent the setup page from loading properly.\n5.Press and hold the reset button for at least 15 seconds on your HP router if you are unable to log directly into the router. This will restore the router's settings to default factory settings, and erase any custom settings that may have been set up by the router's previous owner.\nAfter applying these solutions, if still your HP router is not connected to the internet then call at HP Router Technical Support Number.\nHow to Change My WiFi Password?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Liesbeth and Chris decided to celebrate their 10th wedding anniversary in Italy with their two lovely daughters in the countryside of Emilia-Romagna. This destination was chosen not accidentally. Lis's parents have recently bought a house in the hills surrounding Salsomaggiore Terme. Just a perfect place to live close to the nature and to get away from the city noise. For us it was the perfect chance to shoot in a different location and get back to the places where Matteo belongs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have you hired Between The Fences Landscaping & Maintenance? Write a review and earn our First to Review badge.\nBetween The Fences Landscaping & Maintenance has not added any photos.\nLooking for Lawn Maintenance pros in Unionville?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, there are still openings for this trip. Airfare from Tulsa as low as $1,080 for round trip!\nDo you still have room for this trip? How much is airfare running?\nYes, the trip includes the airfare; Dublin Castle is on the list of stops. Would love to have you join us for another adventure!\nLove the idea of taking another trip with you!\nDoes $3450 include airfare for the Ireland\/Scotland trip?\nWill you be doing another Dublin Castle trip? I like that it was only 6 days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"John Deere Nutrient Applicators | John Deere US. Nutrient Applicators from John Deere. Learn about the features, specifications and more for Nutrient Applicators.\nSelf-Propelled Sprayers | John Deere US. Self-Propelled Sprayers from John Deere. Learn about the features, specifications and more for Self-Propelled Sprayers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbotlf b/data_all_eng_slimpj/shuffled/split2/finalzzzbotlf
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+{"text":"We can define an \"Integrated Report\" like a bridge between the Consolidated Financial Statement and the Sustainability Report, in which the concept of Value Creation has an increasing importance, as the integration in the report of the related results underlines. Referring to the definition of the International Integrated Reporting Committee (IIRC), \"the Integrated Reporting is a process founded on integrated thinking that results in a periodic integrated report by an organization about value creation over time and related communications regarding aspects of value creation. An integrated report is a concise communication about how an organization's strategy, governance, performance and prospects, in the context of its external environment, lead to the creation of value in the short, medium and long term\".\nThe purpose of this new approach is to increase the transparency towards all community and stakeholders, through one single report, which contains at the same time financial, environmental, social and governance results.\nUsing the Integrated Report is not add more information to the traditional report, but it is a way to connect all economic, social, ethic and environmental elements, overtaking the traditional report and creating a better report. So the organization achieves a more efficient communication of its identity, role and value to all the stakeholders.\nThe IR has a combined emphasis on conciseness, strategic focus and future orientation, the connectivity of information, the capitals and their interdependencies: this must become the integrated thinking within the whole organization. The integrated reporting will provide the impetus for greater innovation in corporate reporting globally.\nThe Guidelines of GRI (Global Reporting Initiative) define the Sustainability Report as a \"report published by a company or organization about the economic, environmental and social impacts caused by its everyday activities. A sustainability report also presents the organization's values and governance model, and demonstrates the link between its strategy and its commitment to a sustainable global economy\". There is an increasing number of companies and organizations that need to set goals, measure performance, and integrate a sustainability strategy into their core planning.\nThe sustainability report aims to give transparency to all the sustainable activities and communicate it to all stakeholders. So it should represent the environmental, social and financial performances in a balanced and fair way, considering the negative and positive impacts caused to the society.\nAs a result of the credibility, consistency and comparability it offers, GRI's Framework has become a de facto standard in sustainability reporting.\nWhich are the benefits of the Sustainability Reporting?\nComparing performance internally and externally, and between organizations and sectors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We wanted a more accurate weather solution, one which could indicate temperature, wind speed and direction, rainfall, and barometric pressure, and ideally send the feed to the internet so everyone interested in the area could benefit. Of course, our budget wasn't unlimited either, so plunking down $35,000 for a prepackaged solution was out of the question. With all this in mind, our investigation led us to a Davis Vantage Pro 2, plus some additional equipment to allow an easy connection to the internet.\nHaving decided on the appropriate equipment, we turned to other requirements. There are many standards dictating how the equipment must be located and mounted. For example, the temperature sensor has to be a certain distance from the ground, the wind anemometer has to be a different distance from the ground and it all has to be located far enough away from other obstacles. And there are further considerations. Ideally the anemometer should be ten meters from the ground when used for aviation purposes. But for local weather stations in farming communities, three meters is acceptable and gives a better reading of weather near the ground while being less of a target for lightening.\nWhile the station itself is primarily solar powered (with a battery backup), maintenance is always an issue. For example, we have to deal with bird droppings frequently gumming up the works! Recently its fan has been acting up, so a new one has been ordered. And one of these days we should probably send it in for an extensive (and expensive) recalibration.\nWeather.com has recently acquired Wunderground and they have also begun to incorporate information from local weather stations.\nSince we installed this station over five years ago, the availability and pricing for entry level weather stations has dropped significantly and other weather stations have begun to appear in the area. Personal weather stations can range from about fifty dollars to thousands, depending upon the features and accuracy preferred.\nWe hope others will consider joining the fun too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'Sweeten' your strawberries with balsamic vinegar, making it suitable for most diets including Diabetic and Low Calorie.\nThis is the simplest way to enhance the flavour and sweetness of strawberries (and is also good with raspberries and blackberries). It is a delicious dessert that requires no advance planning and minimal time to prepare.\nDairy Free, Diabetic, Fodmap, Gluten Free, Low Calorie, Low Fat, Nut Free, Paleo, Raw, Vegan and Vegetarian.\nWASH the strawberries, remove the stalks and place in a large bowl.\n\u2013 Alone, they require nothing extra.\n* Substitute with raspberries or blackberries (not for Fodmap diet).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All tooling maintenance should be performed in a clean environment. Any foreign matter on any of the tooling will detract from the tool accuracy.\nWhen removing cutters from the head, it is important to keep each stack of cutters, washers and bolts together. Attach a label to each stack, identifying which bolt it was removed from. This will process will aid in re-balancing the stacks prior to installation.\nSoak the cutter stacks in BLADE CLEAN or a non-corrosive cleaner to remove pitch or sawdust from the knives. To remove heavy pitch deposits, a medium grade Scotch Brite pad and WD-40 can be used. Never use abrasive products to clean cutters or heads. Never sandblast to clean. Damage to the head and cutters will result.\nAfter soaking, rinse cutter stacks with warm water and inspect. Check to see that compacted sawdust in the gullet area of the cutters has been removed. Blow dry with compressed air. Any remaining pitch or sawdust will have an effect on the balancing procedure.\nCutters: Inspect for pock marks and galling on the index surface and bolt holes. Cutters with galling on the index surface or bolt hole should be considered for replacement, as the accuracy of the index surface and hole has been compromised. Replace any broken cutters.\nNuts and Washers: Inspect for galling on the top and bottom mating surfaces and inside of the threads. Galling occurs from torqueing the retaining nuts without use of anti-seize compound. Once the nuts\/washers have been galled, they can no longer be accurately torqued and must be replaced. If the threads on the nut are damaged the threads on the bolt will likely also be damaged.\nStone both index surfaces of each cutter with a fine grit India stone and light oil (WD-40). Use a circular motion to remove any pock marks (created by debris trapped between cutters when previously torqued) or burrs that may have resulted from previous posting or cleaning of the cutters.\nClean the inside bore of the hole with a Q-Tip (cotton swab). Remove any oil with a shop towel and Isopropyl alcohol. It is imperative that there is no residual matter inside the bore of the cutter or on the index. The cutters are designed with very tight tolerances and impurities will affect the set-up and performance of the head.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Thursday, the Centers for Medicare and Medicaid Services (CMS) released its Medicare Advantage (MA) and Part D draft call letter for 2020, which outlines changes to Medicare plan policies and payments each year.\nLate last month, Centers for Medicare and Medicaid Services (CMS) Administrator Seema Verma sent a letter to state Medicaid directors describing ways states can improve care and reduce burdens for individuals who are dually-eligible for both Medicare and Medicaid. This population of individuals tends to have complex medical and behavioral health needs and accounts for a disproportionate share of federal spending for both programs. The letter outlines ten opportunities for states to improve care for the dually-eligible population in 2019.\nThe Centers for Medicare and Medicaid Services (CMS) recently issued final policy changes for Year 3 (2019) of the Quality Payment Program (QPP), as part of the final 2019 Medicare Physician Fee Schedule rule. The final rule updates payment rates and key policies applicable to physicians and other professionals under Medicare. Most notably, the rule expands the types of clinicians (including clinical psychologists) that are eligible to participate in the QPP through the Merit-based Incentive Payment System (MIPS).\nCMS is planning significant changes to a value-based payment model known as Accountable Care Organizations (ACOs) in a sweeping rule proposed last week. The rule would give new ACOs just two years before they must start sharing savings and losses with the agency. The proposal has drawn criticism from hospital groups and ACO stakeholders, who say the plan will cause many ACOs to leave the program. Comments on the proposed rule are due by October 16, 2018.\nLast week, the Centers for Medicare and Medicaid Services (CMS) released its proposed rules for the 2019 Medicare Physician Fee Schedule (MPFS) and Quality Payment Program (QPP). The proposed rules update payment rates and key policies applicable to physicians and other professionals under Medicare. Among the key changes, the proposed rule would reduce billing documentation requirements, expand telehealth options and potentially create a bundled payment for the care and management of substance use disorders (SUD) in Medicare.\nFollowing an action-packed week for opioid-related legislation in the House, multiple Senate committees took up the mantle this week, hosting hearings, releasing new bills and considering legislation ahead of the Memorial Day recess. As bills advance out of committees, the full Senate is expected to consider the large package of opioid legislation later this summer. Meanwhile, similar efforts in the House are expected to make it to the floor in the coming weeks. The collective work of both chambers of Congress seek to address the opioid crisis from a number of fronts including prevention, treatment and recovery.\nThe House Committee on Ways & Means Wednesday approved seven bipartisan bills aimed at reducing opioid misuse and abuse in Medicare. The package would expand Medicare coverage of medication-assisted treatment (MAT), promote non-opioid alternatives for pain management and require Part D plans to have drug management plans for Medicare beneficiaries at risk of opioid addiction. The bills now move to the House floor, joining dozens of other opioid-related bills approved by the House Energy and Commerce Committee.\nThe Centers for Medicare and Medicaid Services (CMS) released final rules for Medicare Part D and Medicare Advantage (MA) plans for calendar year 2019 that are meant to tackle the opioid crisis and decrease drug costs. The CY19 Rate Announcement and Call Letter represents CMS' priorities for the next year, and will implement new requirements on opioid prescribing and monitoring, require lower co-payments for prescription drugs under Part D, increase availability to generics and, hopefully, decrease costs for consumers and the federal government.\nThe Centers for Medicare and Medicaid Services (CMS) have released its proposed changes to Medicare Advantage and Part D plans for 2019, which include a few provisions to combat the opioid crisis. These changes come in reaction to the soaring opioid prescription and substance use disorder rates among Medicare beneficiaries. CMS is accepting public comments through March 5, and will publish its final rule on April 2.\nThe Senate Finance Committee held a confirmation hearing on Monday on the nomination of Alex Azar as Secretary of Health and Human Services (HHS). This is the second hearing on his nomination Mr. Azar has participated in. In late 2017, he appeared before the Senate Health, Education, Labor and Pensions Committee. The Finance Committee will vote on his nomination in the coming weeks.\nPresident Trump, on Monday, nominated former pharmaceutical executive Alex Azar to be his next Secretary of Health and Human Services. Azar, who spent most of the last decade at Eli Lilly and Co., previously served for six years in the George W Bush administration \u2013 first as General Counsel for four years and then as Deputy Secretary for two years under President George W. Bush. The Senate will consider his nomination later this month.\nThe Centers for Medicare & Medicaid Services (CMS) recently released Final Rule Year 2 (Performance Year 2018) of Medicare's Quality Payment Program (QPP) under the Medicare Access and CHIP Reauthorization Act of 2015 (MACRA). The QPP, which went into effect on January 1, 2017, includes two paths to payment for eligible clinicians who bill Medicare using the Physician Fee Schedule: The Merit-based Incentive Payment System (MIPS), and Advanced Alternative Payment Models (Advanced APMs). Performance in the QPP each year impacts payments two years later; in other words, performance in 2018 will impact Medicare Part B payments in 2020. By moving from volume-based payment to payment for quality and value, both systems change the way behavioral health providers are reimbursed under Medicare Part B.\nMark your calendars: all eligible professionals (EPs) who are first time participants in the Medicare electronic health records (EHR) incentive program must attest by October 1, 2017 to avoid a negative payment adjustment in 2018. EPs who do not successfully demonstrate meaningful use for the 2017 reporting period will receive reduced Medicare payments for that year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbpdio b/data_all_eng_slimpj/shuffled/split2/finalzzzbpdio
new file mode 100644
index 0000000000000000000000000000000000000000..5a84b4d2aa3e6ed77b217acfec42bd7f25c00b0c
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"4\/05\/2012 \u00b7 Using PowerPoint 2010, I was able to compress a 50 MB PowerPoint file into 3 MB using the method below: 1. Save a copy of your file with the word compressed added to it, e.g. mypresentation-compressed.pptx (this is for damage control\u2014in case things go wrong, you'll still have a copy of your original to fall back on!).\nHi Susan, You can do a couple things to reduce the size of a PDF but there is no feature to enter a specific file size and get that result. The easiest is to do a File-Save As Other - Reduced File Size instead of a simple Save.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We wen decide to stay and wala'au wit Nolemana's 'ohana even doe we knew it meant we were going get into commute traffic. Cuz we neva know when we going see dem again, and 'ohana is 'ohana. If we had known wat was going happen, den we woulda stayed chru dinnah. But u know wat dey say about hindsight.\nPau leave da hale, we wen get onto Ygnacio Valley Road; and up ahead we wen spock dis Comcast van unlike any odda Comcast van I eva wen see befoa. Musubi stay so excited he almost wen jump chru da windshield!\nWas so dang nani!! Dis undawatah scene painted all ova da whole van! We wen get lucky and hit one red light right next to da van, and I wen roll down my window and wen call out, \"Eh! Love da painting on yoa van, brah!\" He wen gimme one shaka and wen tell me, \"Eh, looks just like Maui, yeah?\" Big smiles alla way around anden da light wen turn green. Dat brief exchange wen put dis huge warm feeling insai my heart.\nHea we stay, leaving Walnut Creek. Get traffic.\nGet bambucha hale up on da hill. Why would anybody build one hale lidis wit a view of freeway? I no get it.\nHighway 680. Get plenny traffic. TryLook wen tell us get one \"severe\" traffic delay ahead. She like us foa take one off-ramp foa avoid um. But eh! Try look wat u stay telling us\u2026 \"get off da freeway aftah da accident\". Wassup wit dat?\nTraffic going da odda way stay even moa worse!\nTake me back to Oregon wea nomo dis kine traffic!\nTry look da eucaplyptus trees. I must admit, I miss dem.\nAdunno wat's wit TryLook. Maybe she get PMS oa someting. We wen follow her advice just foa see wat she like do. One experiment. So she wen direct us to da off-ramp. We wen follow her advice. Came to one stop at da bottom of da off-ramp. U know wat she wen tell us den? \"Eh Moki\u2026 try get back on da freeway!\" Sheesh.\nSo we wen do dat. Den, instead of staying on Highway 680 like I woulda normally done, TryLook wen tell us go down to Interstate 80. Um\u2026 why? But since we still experimenting wit her, we wen doyem. Shoulda known moa bettah. Cuz on da way chru da Mission District, ho da traffic! Was almost at one standstill! We wen creep along. Traffic going da odda way almost stopped. Den we wen see plenny fire engines and police cars\u2026 anden we see dis burning tree!\nFinally we wen make it into San Jose, and just as we wen go by da airport, one jet wen come in, almost ovahead.\nBy dis time we had been on da freeway foa almost two hours! One drive dat shoulda taken us just an hour stay taking us much, much longah. We wen get onto Highway 17, and wen run into even moa traffic! Just chree miles from Los Gatos, TryLook wen tell us get one accident up ahead. And traffic wen slow to one nodda crawl. Sheesh. Nuttin to do but take photos of da nani oleanders on da divider.\nAnd dey were nani. Poisonous, but nani. I nomo plans foa eat um anyway.\nWe finally wen make it to Los Gatos. Wen stop at da hale and wala'au wit my sistah, have dinnah, put da two casseroles I wen bring wit us in da fridge. Stuff lidat. Den we wen head ova to our hotel.\nDis da place wea my dottah and I wen stay lass time. I love dis place! Usually we stay in motels, but da oni motel in Los Gatos was awful. Leilani and I wen spend one night in it da month befoa, and wen get outa dea as wikiwiki as possible. I neva wen stay in one reeyo hotel befoa, but Leilani does alla time foa her job. Her company puts her up in dem alla time cuz she one vice-president.\nI tell u, I could get used to dis! Leilani and her hubby get dis saying: \"Luxury, once experienced, becomes necessity!\" Heh heh.\nDis hotel became our place of refuge during da next ten days. Somebody foa make da bed every day, fresh towels if we like (we wen live green and oni wen change um every few days), and awesome front desk service. Wi-Fi, fresh cookies every night, bathrobes, slippahs, one balcony wea I wen watch da moon every night, and da most comfy bed in da world. Yeah, wen cost moa\u2026 but wit wat we stay doing, was worth it. My braddah and sistah dem wen stay in our maddah's hale, so dey wen give us kokua wit da hotel bill.\nDa next days stay filled wit packing up stuffs, going chru da hale, da memorial service, 'ohana time, moa packing, etc. Wen mean long, tiring days. But everybody wen get along and we wen get stuffs done. Nolemana stay amazing packing up da car; I neva thought I was going take home so much stuffs! But like my sistah and braddah dem, we wen keep finding stuffs dat wen remind us of our maddah. Ova da weekend, our two dottahs and son-in-law wen drive down; was so good to see dem. Dey wen arrive late Friday night and wen hele on home early Sunday in da kakahiaka.\nMoa photos coming in Chaptah Six. First I gotta unpack some stuffs but.\nYes, freeways are never really entertaining unless you're out in the pretty green places. There's hardly anything to see off of the California freeways, unfortunately.\nThe hotel looks wonderful and how nice that your ohana helped with the bill too.\nOh yes, I have to say that was a pretty interesting Comcast Van. LOL I wouldn't mind doing that to my van!!!\nHey \u2013 you got an award\u2026 when you get a chance, come see me!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eucalyptus is a fabulous material to use in home decor. Not only is it beautiful but it is aromatic. This dried flower wreath is made with all \"natural\" material. The base is olive green dyed eucalyptus. I handmade rose floral picks using a light tan suede leather. The roses are nestled amid wicker ball stems, lotus pods, mahogany pods, and stunning heart shaped seed pods. The natural jute burlap bow ties the entire wreath together, giving it a cottage feel. The wreath would be a wonderful addition to anyone's home decor! Because the materials are all natural and preserved floral stems the wreath should be displayed indoors, out of the elements. It means approximately 21 inches by 21 inches, tip to tip. A boxed wire frame was used as the wreath form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Renault has commenced the first large-scale pilot scheme in reversible electric charging technology in Europe.\nVehicle-to-grid charging, or reversible charging, adapts the charging and discharging of electric vehicle's batteries in relation to the needs of the user and the grid supply of available electricity.\nWhen the electricity supply exceeds the demand, charging will reach its maximum level. Vehicles will be able to supply electricity to the grid during peaks in consumption, therefore, serving as units of temporary energy storage. The goal is to optimize the supply of local renewable energy and reduce infrastructure costs.\nRenault is hoping the program will determine if the venture is feasible on a larger scale by examining if there are technical and economic advantages of an onboard solution in electric vehicles, if there is value in the services provided for the local and national electricity grids and what the basic requirements are for an industrial-scale roll-out.\nThe electric vehicles used for the program will be fitted with technology using an onboard reversible charger which is a means to reduce the related cost of charging terminals.\nThe plan is for a fifteen-strong fleet of Zoe models with vehicle-to-grid charging capabilities to be introduced throughout 2019. Utrecht in the Netherlands and Porto Santo Island in the archipelago of Madeira, Portugal are the first to host the pilot schemes with France, Germany, Switzerland, Sweden and Denmark next to join the program.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And BECCA has a new bronzer in 5 gorgeous shades for all skin tones.\nBECCA Sunlit Bronzers are each named and inspired by sun-drenched destinations around the world, like the one I love, Maui Nights. As all of us know, like foundations, bronzers are not 1-size-fits all. The level of brown and the type of undertones are so important for it to flatter skin.\nEach Sunlit Bronzer is unique and beautiful, from the golden topped cases to the intricate sand inspired pressed powder designs in the pan.\nThe bronzer is smooth, highly pigmented and gives skin an immediately sun lit glow. My best match was Maui Nights, (shown above), a warm reddish brown with bronze highlights that is gorgeous on deep and very dark skin tones.\nApplying Maui Nights was a breeze. I used my BECCA Soft Kabuki Brush. The Sunlit Bronzer was so soft and silky and blended into my skin tone immediately, giving me a touch of color and definition, but also pretty, subtle glow. The bronzer lasted well over 8+ hours without fading or transfer.","meta":{"redpajama_set_name":"RedPajamaC4"}}